Foul Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶⠖⣏
⠀⠀⠀⢰⠶⠶⡶⣄⣀⣀⣠⣀⣠⣤⣤⣴⣶⣶⣶⣿⢿⣦⣤⡾⢿⣶⣦⣄
⠀⠀⠀⠀⠑⠒⠂⠀⠀⢀⠀⢀⣘⡷⠶⠶⠖⢻⢸⡇⢸⡇⢸⡇⢸⡇⣷⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢻⣿⣷⢸⡇⢸⡇⣼⣿⣿⣝⣦⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⡇⢸⡇⣿⣿⡏⠻⡾⠋⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⢸⡇⣿⣿⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣸⣇⣸⣇⣿⣿⠀⠀⢹⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣻⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠀⠀⠿⠿⣿⡷

⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆⣴⣶⣶⣶⣶⣶⣶⣶⣶
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⠿⠿⠿⠿⢟⡛⠛⠿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣤⣤⣴⠶⠛⠛⠻⠁⠀⠀⢤⣽⣿⣿⣿⡟⣰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠾⣿⣿⣁⡀⠉⢿⣿⡇⡀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣬⣭⣤⣀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⣠⠿⠛⠉⠉⠉⠉⠉⠛⠿⣿⣿
⠀⠋⠙⠛⠛⠳⠶⠄⠁⢠⣬⣭⣄⣀⢀⡙⢛⣙⡿⣿⣿⣿⣿⠏⡈⠙⠛⠛⠻⠿⠿⢿⢿⣿⣷⣶⣶⣶⣦⡤⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠛⠀⠀⡄⠈⠀⠀⠀⣤⣀⢀⡌⠛⠋⠀⣮⠻⠿⠬⢿⢿⠏⠈⠻⠀⠒⠀⠰⠀⠐⠀⠃⠀⠀⠉⠍⠉⠙⠡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⡷⠤⢠⠤⣄⣀⠈⠹⠟⢃⣁⡄⠀⢀⠀⠀⠈⢀⡀⠀⠀⢰⣀⣠⡄⣀⡀⠀⠀⢀⢀⡀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣼⣧⣞⣗⣀⠀⠉⣤⡴⠄⠋⠁⠘⠿⠿⠷⠂⠀⠀⠁⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠈⠈⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿
⠀⣿⣿⣿⡿⠟⠀⠀⢀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣘⣿⣷⢿⣿⣿⣿⣷⣶⣶⣶⣶⡆⠀⠀⣺⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⠀⠛⠉⠀⠀⠀⣠⢔⡡⠶⠞⠃⠀⠀⠀⠀⢀⣠⡞⠀⠿⠿⠿⣿⣯⡆⠉⠉⠉⢿⣿⣿⡿⠟⠁⠀⣼⢻⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⢿⣽
⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢀⣦⣾⣿⣿⣷⣿⣿⡆⠀⠀⢺⡿⠯⠁⠀⠀⣠⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠞⢀⣿⣶⣤
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⣿⣿⣶⠦⠀⣰⣿⣿⣿⣿⣿⣿⣷⣦⣄⣠⡄⢠⣿⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠾⠟⣴⣶⠂⠀⠀⠀⠀⠀⠠⠂⠀⣾⣿⣿⣿⣿⡿⠿⠟⠁⠞⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣿⠿⣿⣿⣿⣿
⠀⣄⠀⠀⠀⢸⣿⠃⠀⣠⠀⣠⡆⠀⠀⠀⠾⠟⠛⠋⠉⠁⠀⠀⣰⡇⠀⠇⢶⡈⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⣿⣧⡀⠀⠸⠛⡀⠐⠋⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢀⡆⠈⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⣀⣀⣴⠿⠋⠀⢀⣾⣶⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿
⠀⠏⣱⣾⡷⠀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠛⠉⠁⠀⠀⠀⢀⣾⣿⣿⡆⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⡴⢋⣾⣿⡆⢀⣶⣾⠟⣹⡟⠀⠀⠀⣀⣀⠀⠠⢆⢄⣀⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿
⠀⠵⠿⠿⠾⠶⠿⠷⠷⠶⠟⠀⠀⠀⠉⠀⠠⡴⠾⠿⠟⠋⠀⠂⠀⠀⠀⠀⣼⢿⡿⢿⠿⣿⡷⠄⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡧⠸⠿⠿
⠀⣶⡲⢶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣀⡀
⠀⠛⢿⣶⡌⠙⣿⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠱⡌⢻⣷⡀⢠⡀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢀⡀⠙⠁⢠⣽⣿⣿⣛⣛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣌⣿⢛⣧⡇⣶⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂
⠀⣿⠿⠀⢰⣾⠏⠙⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢡⠛⢸⡟⠃⢟⡼⣣⡦⠁⣀
⠀⠀⠚⠳⢾⣿⣧⠀⠸⣿⣿⣿⡿⠓⠀⠀⠀⠀⣸⠛⢻⣟⠓⢆⠀⠀⠘⠁⠀⠈⠰⢋⣴⠇⠈⠀⠀⠀⠀⣠⣤⣴⣶⣶⣦⡴⠦⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⠿⣦⣄⡀⠀⠀⣰⣿⣄⠀⠀⢀⣈⡓⠀⠀⠐⠄⠀⠸⠏⠀⢀⡴⠀⠀⠀⠀⠻⣿⡿⠫⠟⠿⠿⠧⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣆⢺⣿⣿⣿⣷⣦⣄⡀⠀⠀⣀⠱⡸⣇⠀⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣄⣀⠐
⠀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠿⠿⠿⠿⠿⠿⠇⠀⠙⠃⠁⠙⡼⡷⠀⠀⠀⠀⠁⠀⠀⠀⠀⣨⣿⡏⢰⣶⣿⣶
⠀⠠⠿⠿⠛⠓⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠈⠙⠛⠿
⠀⠀⠀⠈⠉⠁⠈⠀⢈⡉⢉⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⣀⣠⣤⣀⣀⣀⣄⣀⣀⣀⡀⣠⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣀⣾⢾⣷⣾⡿⠶⠆⠠⣦⣤⠀⢲⣦⡄⠀⣴⣶⣶⣿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠾⠿⠿⠖⠚⠙⠉⠛⢛⠛⠿⠿⢷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⢠⡏⡄⠎⡜⠁⠁⠀⢠⡄⢹⣿⠀⠈⠻⠇⠀⠈⢡⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣶⣤⣤⡍⠉⠙
⠀⣿⣶⣶⠆⠸⠀⠃⠀⠂⠀⠀⠀⠀⠳⣈⠋⠀⠀⠀⠀⠀⣺⣇⣿⡀⠀⠷⠧⣤⠀⢀⡀⢠⡤⠘⠿⠉⠀⠰⠄⠀⢀⠀⠀⠐⠆⠀⠀⡀⠂⠈
⠀⣿⣿⣿⡆⠀⠁⠀⢠⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢲⡆⠀⢶⡶⣿⠂⠀⠀⠄⠈⠀⠀⠁⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣉⠀⠀⠄
⠀⠿⣿⣿⣷⡄⠐⠂⠠⠶⠆⠀⢀⠀⠈⠂⠀⠀⠀⢸⠇⠀⣸⢉⣿⠀⠀⠀⠂⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠏⠀⡀⠄
⠀⠀⠛⠛⠛⠉⢀⡐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣿⣶⣿⠀⠀⠀⠁⠀⢠⠀⠀⢀⡀⠄⣠⣿⠄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⣤⣄⣈⠙⠛⠷⠂⠀⠀⠀⠀⠀⠀⠀⠆⢰⣄⠀⣿⢿⣿⠀⠀⠀⠠⠇⠀⠀⠀⢀⣷⣾⣿⡿⠶⠶⠦⠀⠀⢴⣶⡶⣶⣴⡟⠀⠋
⠀⢀⣤⣼⣯⣽⣟⣿⣷⣶⣄⣀⠁⠀⠀⢀⡇⠐⢀⣼⡟⠀⣿⣆⣀⣤⠄⢠⠁⠀⠀⠀⠀⠙⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⠐⠒⠀⠠
⠀⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠷⠀⠀⠿⠷⠀⠘⠏⠀⠀⠿⠿⠿⠿⠞⠃⠀⠸⠂⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠆⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡄⠀⠀⠀⠀⠀⠱⣇⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⡾⣦⣄⡀⠀⠀⢀⣿⡛⠛⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⢷⣬⣙⢻⣿⣿⠻⣷⣷⣤⣾⢟⣿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣭⣛⢿⣧⣿⡇⢹⡇⢸⡷⢸⣿⢸⣯⡛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡇⢸⡇⢸⡇⢸⣿⢸⣿⣷⡼⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⢸⡇⢸⡇⢸⣿⢸⠿⣿⡀⠀⠀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⢸⡇⢸⣿⣾⠀⠈⠙⠓⢶⣶⣀⢈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣸⡇⢸⣇⢸⣿⣿⠀⠀⠀⠀⠀⠹⢝⠗⠦⣌⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣽⣧⣼⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⢹⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠿⠿⠟⠀⠀⠻⠿⠿⣿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠄⠰⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣤⣀⡴⠋⢳⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣼⣇⠴⠋⠛⠉⠀⠀⠀⠑⠾⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⢻⠀⠀⡖⠉⠈⠉⠑⠂⠀⠀⠀⠂⠉⠁⠀⠈⢳⠀⡏
⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢸⡆⠀⠀⠀⠘⡆⠀⡇⠀⢀⣠⣤⡀⠀⠀⠀⢤⣴⣀⠀⠀⢸⣠⠇
⠀⠀⠀⡠⠚⠁⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠘⡦⠇⠀⠀⠘⠉⢀⡴⢀⣀⡠⠥⠤⠤⠤⠼⢯⡴⠲⠤⠤⣄⣀⡀
⠀⠀⠊⠀⠀⢀⡴⠊⠉⢐⡄⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⣴⡞⢉⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠉⠉⠲⣄
⠀⠀⠀⢀⠴⠉⠀⠀⠒⠊⢀⣀⠀⡀⠀⠀⠀⠀⠀⠑⠤⣾⠟⠋⣩⠤⠀⢀⣀⠀⢀⡤⠖⠷⠤⢄⣀⣀⠀⠀⠀⠀⠀⢘⡆
⠀⠐⠚⠁⠀⠀⠀⣀⠤⠚⠁⢀⢀⡸⠀⠀⠀⠀⠀⢀⡠⠞⠰⣏⣠⠖⠋⠁⢏⣠⠎⠀⠀⢀⡤⠚⠁⠈⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢈⣁⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠛⠙⠀⠀⠀⠀⠀⢲⡠⠶⠞
⠀⠀⠀⠀⠀⠐⠻⣥⡴⠶⠟⠛⠛⠀⢀⡐⠲⢮⣁⠀⠀⣀⠀⢀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠋⠂⠾⣋⠉⠁⡀⠘⠛⠂⠀⠈⢳⣆⣜⢲⣾⣿⠇
⠀⠀⠀⠀⠀⠀⡀⣒⠀⠀⠅⢸⠷⠆⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⢩⠹⡤⠀⢾⠀⠀⣸⠁⠀⠀⠀⠀⠀⠙⣿⠟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⡘⠃⠘⢟⠙⣄⡤⠒⠻⡅⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠍⠀⠀⠀⠈⢣⡀⠀⠀⠄⠀⢰⣆⠀⠀⠀⠀⠀⠀⢠⡤⣘⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡀⠓⣀⣌⢠⡼⢷⣸⡀⠀⠀⣘⠳⠁⠀⠠⣀⡀⠀⠀⠀⡿⠁⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠆⠉⠁⠠⠓⠈⠁⠰⠀⡾⢻⡆⠀⠀⠀⠘⢿⣷⡶⠶⠶⠤⣇⣠⠤⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠒⠀⠓⠂⢰⠂⡀⠀⠀⠀⡀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠶⢤⣤⠙⠢⣘⣧⡀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣇⠀⠀⠀⣧⣿⠀⠀⠀⠀⢸⡟⠀⠠⣠⠀⠀⠀⠀⠘⠻⣿⣾⣿⣧⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠐⠀⣠⠈⣯⠀⠁⠀⢀⣿⠏⠀⠀⢀⠀⢸⡇⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠘⢧⣛⣿⣿⣿⣧⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠹⠂⠈⢂⣄⡀⠊⠀⣤⠀⠘⠘⠂⠈⡇⠀⠀⢰⢀⡴⠀⠐⠢⠀⠀⠀⠀⡈⠙⠻⣿⣿⠇⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⡄⠀⠀⢠⡏⠀⢸⣾⠀⢿⣄⣾⣷⣆⠀⠃⠀⠀⠀⢿⣿⣶⡄⢀⡷⠀⠀⠀⠀⠀⠀⠉⢻⣿⡀⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⢀⠀⠰⠿⣇⠰⠋⡹⠀⠈⣿⣿⢿⡀⠁⠀⠀⠀⠀⡻⣿⣿⡧⠏⢁⠀⠤⢂⠀⠀⠀⠀⠀⢿⠆⣸⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠒⢿⣄⠟⣷⡄⢸⣯⠐⠂⠀⠀⠀⣻⣿⣿⣇⠀⠀⠀⠀⠀⢢⡌⠉⠁⣠⡀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠂⡎⠉⠹⡷
⠀⠀⠀⠀⠀⢰⣖⣿⡆⠰⡀⢿⣿⣆⠈⠓⠀⢻⣧⠀⠀⠀⣴⣿⣿⠀⠉⠀⠀⠀⠀⠀⠈⢿⠀⢴⣷⠀⠹⣿⣿⡿⠇⠀⠀⠀⠑⠀⢡
⠀⠀⠀⠀⠀⠀⠙⠿⠇⠂⠁⠟⣿⣿⣧⠀⠀⠀⡿⡆⢀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠈⠻⣶⣌⠒⠛⠁⠀⠀⠀⠐⢶⣶⣶⣤⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠸⣿⣿⣷⡀⠀⠀⠈⠀⠐⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣰⣶⣾⣿⣧⠈⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠸⠀⢀⣹⣝⡿⣿⣄⠀⢀⠀⠀⢹⡧⠀⠐⠄⠀⠀⠀⠀⡀⠀⠀⠠⣬⣽⣿⣿⣿⣿⣇⣠⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠡⡄⠀⠀⠀⠙⢿⣾⣿⢷⡄⢣⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡞⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⡨⣿⣿⣿⣧⠀⢷⣄⠀⠈⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠰⢀⡀⠀⠰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠀⠹⣝⡿⠋⢹⣇⢸⣿⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠙⠦⠀⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠙⠋⠀⠂⠈⠀⣿⡀⠀⠀⠀⣷⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄⠀⢰⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⠆⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢻⠳⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⠙⠄⠰⠀⠀⠀⢠⣿⣿⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉

⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡟⣽⢧
⠀⠀⠀⣿⣿⣿⣿⣿⠉⠻⣿⡇⠀⢸⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡾⢰⡏⣠⡀
⠀⠀⠀⢻⣿⣿⣿⣿⡸⣆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠃⡟⢰⢻⡇
⠀⠀⠀⠈⣿⣿⣿⡿⠁⠲⠀⠀⠀⠀⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠉⣼⠁
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⡇⠀⠀⠀⠀⠀⠀⠸⡀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣰⠀⡇⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⢀⠆⠀⠒⠛⠲⢤⣙⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣴⠃⣰⠇⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡴⠋⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⠂⠤⠤⣉⡒⠒⠶⣤⣀⠀⠀⢰⣧⠞⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡀
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣔⡋⠀⠉⢲⡿⠁⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⢠⠇
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠢⣠⡞⠀⠀⠀⠀⢀⣿⠃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠄⠀⠒⠒
⠀⠀⡇⠀⠀⠀⣀⣠⣤⣤⡴⢤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠤⣤⣀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠙⠒⠚⠁⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠁⠈⠀⠉⠀⠁⠸⠿⠛⠛
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠃


⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠫⣄⡁⢄⡰⠲⡄⡖⢦⠠⢒⡄⡇⠀⢸⢢⡔⢦⢠⠀⡄⡇⠀⠀⢠⢠⡖⡦⢲⢠⠰⡇⠔⢲⠰⠆⣔⣢⠀⠀⡖⢢⢰⠀⡆⣔⠢⢸⠖
⠣⠤⠇⠂⠣⢴⡇⠇⠸⠸⠩⠇⠇⠀⠸⠐⠦⠜⠸⠤⠇⠇⠀⠀⠘⠀⠇⠀⠸⠨⠀⠇⠏⠹⠀⠆⠯⠭⠀⠀⡧⠼⠸⠤⠇⠬⠽⠸
⣤⣤⣤⣤⣼⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣭⣬⣯⣤⣭⣽⣥⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⠁⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣏⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠟⠋⢧⣄⡀
⠀⠀⠀⠐⣟⠛⠛⠟⠢⠤⠴⠤⠶⠤⣶⠶⠾⠿⠿⢿⣿⣿⡟⣿⡶⣾⡟⢻⡟⣿⠿⠿⠿⠶⢶⡦⡤⠤⠶⠦⠴⠞⠿⠛⢛⡁
⠀⠀⠀⠀⠀⠉⠑⠒⠂⠀⠄⢀⣀⣀⣹⣛⣛⣛⣛⣻⣿⣿⡇⣿⠇⢿⡇⢸⡇⣿⣛⣛⣛⣛⣻⣅⣀⣀⠠⠄⠀⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠛⢻⣿⣿⡇⣿⠀⢸⡇⣸⡇⣿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⣿⠀⢸⡇⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⢿⡇⢻⡇⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⢿⠀⢸⡇⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⠀⢸⣿⣿⣿⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣛⣋⣀⣈⣛⣻⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⡠⠚⣯⣤⡔⠊⠛⢦⡀⠀⠀⠀⢀⢴⠊⠁⠀⠀⠈⠳⢄⠀⠀⠀⠀⡠⠊⠉⠒⢄⡈⠳⡦⡀
⠀⠀⣰⡡⠚⢿⠿⠓⠤⣀⣸⠘⡄⠀⢀⠎⠸⡀⠀⠀⠀⠀⢠⠀⢣⠀⠀⣜⠌⠉⡱⢄⠀⠑⡀⢐⡵⡀
⠀⠀⣿⡀⠀⢸⠀⠀⠀⣸⣿⡗⡇⠀⢸⠀⠀⠑⡀⠀⠀⠰⠁⠀⢸⠀⠀⡏⠀⠎⠀⠠⠒⠐⠓⠉⢀⡇
⠀⠀⠸⡈⢲⣿⣷⡖⠉⠀⢹⡰⠁⠀⠀⢇⠀⢀⠃⠀⠀⢠⠀⢀⠌⠀⠀⠱⡸⠀⠀⠃⠤⠤⠄⠂⡹
⠀⠀⠀⠈⠲⢏⣉⣑⣤⡶⠛⠁⠀⠀⠀⠀⠑⠧⣀⣀⣀⡠⠕⠋⠀⠀⠀⠀⠙⠣⢄⣶⣠⣤⠴⠊

⠀⠀⠀⢀⠴⠚⢭⣙⣍⡱⠦⡀⠀⠀⠀⠀⠀⠀⡠⡔⢺⣿⣿⣍⢱⠀⠀⠀⢠⡔⠈⠉⠉⠉⠒⣄
⠀⠀⢰⡥⠊⠉⢀⠇⢩⠹⡑⠳⡄⠀⠀⠀⡠⣫⣼⠟⠉⠀⠙⣿⣿⠀⠀⡰⠁⠙⡆⠀⠀⠀⡞⠁⠱
⠀⠀⡏⠀⠀⢠⠎⠀⢸⠀⠱⡀⡇⠀⠀⣜⢱⠟⠁⠀⠀⠀⠀⢈⠇⠀⠀⡇⠀⠀⣷⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠱⡀⢠⠃⠀⠀⢸⠀⠀⣧⠃⠀⢰⣿⣯⡀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠱⡀⣰⠇⠀⠀⠀⢧⡀⡰
⠀⠀⠀⠑⠾⣀⠀⠀⣜⣠⠔⠁⠀⠀⠸⡈⠻⣿⣦⡤⠔⠊⠀⠀⠀⠀⠀⠀⠘⠣⣀⣀⣀⣀⡨⠛
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⣤⡔⠢⠀⠀⠀⠒⠴⡤⢀⠀⠀⠄⠀⠀⠄⠀⠀⡄⠂⠐⠀⠀⠀⠀⢀⡲⢶⡀⠀⠠⡀⠀⢠
⠀⠰⠀⣉⠀⢾⠆⠀⠁⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡅⢠⡼⠋⠀⠀⠃⠀⠀⡁⠀⡇
⠀⠀⢠⣟⣣⡴⠀⠀⠐⣤⣠⣤⠈⠀⠀⣤⣀⣠⠄⠀⠀⢢⣤⣤⡄⠀⠈⢻⣦⣤⡀⠀⠀⠐⣥⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣣⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠁⠀⠀⠀⠀⠀⢠⣿⠉⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⠀⠀⠨⣽⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⣣⣀⠀⠀⠀⠀⢻⣤⡤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣯⣙⠻⣶⣶⡶⣿⣇⣀⣼⢶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣟⡷⣿⣿⡇⢸⡏⢹⡇⢸⡇⣿⡻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡇⢸⡇⢸⡇⢸⡇⣿⣿⣎⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⢸⡇⢸⡇⣸⣧⡟⢿⣿⡷⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⡇⢸⡇⣿⡿⡇⠀⢹⠄⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢸⡇⢸⡇⢼⣥⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣷⣾⣷⣿⣿⠀⠀⢨⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠀⠀⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠿⠿⠀⠀⠿⢿⣿⡿⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣭⣛⠦⢋⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⡴⠟⠃⠀⠀⠀⢰⣿⡿⠿⣛⣫⣅⢭⣥
⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣥⣾⣿⣿⢏⣵⢨⡻
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣤⣤⣄⠙⠿⠿⣣⣿⡏⠞⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣴⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣤
⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠀⣠⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠈⠙⠛⠛⠋⢁⣤⣾⣿⣿⣿⡿⢟⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣫⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⠁⣀⣤⣄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢰⣿⣿⣿⣦⣄⠀⠀⠀⠀⠈⠉⠉⢀⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣶⣄⡀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡿⠛⠁⠈⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣝⠻⠿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠉⠙⠻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄⠀⠀⠀⠘⢿⣿⣿⣿⣷⡄⠀⠀⠀⡰⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠀⢀⡴⠁⠀⠀⠀⣴⣿⣿⣿⣿⠟⠁⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⠴⠋⠀⠀⠀⠀⠸⣿⣿⣿⣿⡋⠀⠀⠀⠸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡠⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⠀⠀⠀⢹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠒⠲⠶⠴⠶⠚⠉⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡄⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠘⢿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢀⠇⠀⠀⠀⠀⠀⠀⢠⢖⠟⠉⠲⡄⠀⠀⠀⠀⠀⠀⠸⡎⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠈⡀⠀⠀⠀⠀⠀⢀⣾⡃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⡏⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⢡⠀⠀⠀⠀⠀⠀⢽⡇⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣼⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣤⡀⠀⠀⠀⠀⠈⣿⣤⣶⢼⠁⠀⠀⠀⠀⢀⣤⣟⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣴⣿⣍⡻⣶⣤⣠⣤⣾⣇⠂⠀⣸⣷⣤⣄⣤⣶⢿⣫⣿⣄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣤⡙⠻⣦⣿⢹⣿⠈⣿⡿⢿⣿⠁⣿⡏⣿⣷⠟⢋⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣌⣻⣿⣿⠀⣿⡇⢸⣿⠀⣿⠇⣟⣡⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⠀⣿⡇⢸⣿⠀⣿⢠⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡄⣿⡇⢸⣿⢠⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⡇⢸⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡇⣿⡇⢸⣿⢸⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣧⣼⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠟⠀⠀⠻⠿⠿⢿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠀⢈⠀⡆⠀⠀⠐⠀⠘⠀⠀⢠⠤⡄⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠚⠀⡤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠃⠈⡄⠀⠀⠂⠀⠀⠀⠀⠀⢁⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠼⠁⠲⣤⡦⢄⣤⠀⠀⡖⠸⠿⠃⠀⠀⠀⠀⠀⠀⢸⡟⠛⢷⡄
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⢀⡤⡒⢮⡹⡶⢒⣿⣤⡀⡀⡠⡦⡄⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⢷
⠀⠀⠀⠀⠀⠠⠃⠀⣀⡠⠀⢀⠀⠤⠀⠡⠴⠏⠈⣰⡾⠻⡿⠸⢻⡾⣘⣥⠚⠏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⡀⠃⠀⠤⠤⠾⠡⠾⡐⢃⠠⡼⠷⠀⠵⠦⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣿⣟⣻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡞⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠇⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⣨⠋⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠂⢸⡟⠉⠀⠀⢰⠷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣾⣇⢀⣀⣤⣾⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠻⢿⣧⣬⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢺⣿⣿⣀⠀⡇⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠸⠇⠈⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢤⣄⠀⠀⢀⣀⡐⢢⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠟⢻⡷⡐⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇⠀⠀⢧⢿⣄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⢄⡀⠀⠉⠠⠀⠀⠀⠓⡛⢋⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠑⢄⠄⠀⢀⠀⠀⠀⠒⡽⡿⣷⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣻⠆⠀⠀⠀⣀⣳⣤⣄⣄⠀⣀⣀⡈⣥⠆⣷⣶⣻⣫⣀⣄⣀⣠⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣷⣧⣴⠊⢻⣿⣿⣿⣿⠟⠿⠛⠋⠉⠃⢈⣤⣽⣷⠿⠿⠛⠿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣛⣿⣿⠟⠛⠋⠋⠀⢸⣶⡿⠋⠉⠀⠀⠐⠂⠈⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠓⠸⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠟⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⡤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣁⠀⢀⣸⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣿⣿⠈⣿⠟⢿⡏⢸⣿⢉⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⡾⣿⣼⣿⠀⣿⠀⢸⡇⢸⣿⢸⡷⣦⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⢋⣴⣿⣿⣿⠀⣿⠀⢸⡇⢸⣿⢸⣷⣌⠻⣮⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉⢻⣴⡿⠋⢻⣿⣿⠀⣿⠀⢸⡇⢸⣿⢸⠉⠻⣷⣾⣋⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⢧⡀⠀⢸⣿⣿⠀⣿⠀⢸⡇⢸⣿⣿⠀⠀⢠⠟⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠱⡄⢸⣿⣿⡄⣿⠀⢸⡇⢸⣿⣿⠀⡴⠋⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡈⠛⣿⣿⣧⣿⣤⣼⣧⣼⣿⣿⠛⢃⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠈⢻⣿⣿⣿⣿⣿⣿⠟⠁⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣿⣿⣿⣿⣿⣿⣶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡇⠀⠀⣿⣿⣿⣦⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣟⣛⣃⣀⣀⣛⣛⣿⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⠤⢒⣰⠚⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠎⣠⡴⠞⠛⠛⠛⠲⠏⠀⠀⠈⠉⠛⠋⠃⠀⠘⠋⠷⠚⠛⢳⣎⣓⣒⣒⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠒⠒⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠁⠀⠉⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣦⢰⡆⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠖⠂⠀⢀⡾⢋⣁⡈⢹⡄⠀⢿⠸⡇⠀⠤⢾⠶⣷⠒⠀⠀⠀⠀⠀⠀⠀⣀⡞⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⠀⢸⡇⣿⣀⣿⣸⠃⠀⠸⡄⡇⠀⠒⢻⡛⣿⠉⠁⠀⠀⠀⠀⠐⠺⡅⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣀⡀⠀⠀⠀⠀⠀⠀⢹⡏⠁⠀⠀⠘⢷⣌⠉⠈⣀⠄⠀⠀⡀⠄⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⣀⡴⢦⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣰⡶⠶⠶⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡴⣫⣓⠶⠶⠶⠒⠋⠉⢻⡆⣰⠶⠟⠦⣤⣤⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣞⣁⣠⠴⠋⠘⠁⠈⠉⠉⠁⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠻⣶⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣷⣴⠿⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠘⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⠃⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣆⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠙⣷⡀⠈⠻⣦⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⢀⣴⠟⠁⢀⣾⠋⠀⠠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢰⡷⠀⠈⢻⣆⠀⠙⣷⡄⣰⡟⠁⠀⠈⠻⣧⣠⡾⠋⠀⣰⡿⠁⢴⡄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣼⡇⠀⠀⠀⠹⣧⡀⠈⠻⣿⡀⠀⠀⠀⢀⣿⠟⠁⢀⣼⠏⠀⠀⢸⡇⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢹⡇⠀⠀⠀⠀⠈⢿⣄⠀⠙⣷⡀⠀⢠⡾⠃⠀⣠⡿⠁⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠿⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⠁⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣄⡀⠀⣠⡿⠷⢶⣶⡶⠟⠙⢷⣤⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢸⡟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⢀⣠⡿⠁⠀⣶⠶⢠⡶⣦⢰⡆⢰⢠⡆⠀⢰⡆⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣿⡀⠺⢿⣍⠀⠀⠀⣿⠤⢸⡇⢸⢸⡇⢸⢸⡇⠀⢸⠃⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⣶⡄⠀⠀⢹⣇⠀⠈⢻⡆⠀⠀⡿⠀⠸⢧⡾⠘⢧⡾⠸⣧⡤⢨⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣾⠛⣷⡀⠀⠈⣿⡀⠀⢸⣯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣸⡏⠀⠹⣧⠀⠀⢸⣇⠀⠘⠋⠉⠛⢿⣆⣀⣤⣤⣤⣄⡀⠀⣠⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢰⡟⠀⠀⠀⢻⡆⠀⠈⣿⡀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠉⠛⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢠⡿⠁⠀⠀⠀⠈⢿⡄⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢀⣾⠃⠀⠀⠀⠀⠀⠘⣷⡀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢹⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣤⣤⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣀⣠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⢀⡀⣀⣀⣀⠀⡀⢀⣀⢀⣀⢀⡀⣀
⠀⠈⠋⠉⠉⠉⠋⠃⠈⠉⠋⠉⠉⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⢋⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠀⠁⢸⠛⡌⠤⠰⣖⡒⠒⠲⢀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡐⠈⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠆⣜⠇⠀⠀⠀⠀⢣⡻⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⡀⠄⠘⣧⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠌⠀⠀⠈⠻⢆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠶⠤⢄⠢⠄
⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠹⠐⠀⠁⠀⠀⠙⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠊⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠑⠿⠷⠔⠀⠀⠀⠀⠀⠀⠀⠐⡸⠟⢛⣁⠀⢀⠀⢀⡀⠂⠀⠀⠀⢰⣒⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠻⠛⠛⠿⠛⠛⠛⢛⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠙⢷⠶⠦⠰⠶⠾⠿⠿⢿⣏⠉⠉
⠀⠀⠀⠀⠀⠀⠁⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⣶⣴⢶⣾⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⣠⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠿⢋⣴⣷⠈⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⢀⣾⣧⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣳⠖⠿⠵⠿⠀⣿⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣽⠉⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⡃⢀⠀⠀⠀⠀⠉⣠⣴⣶⣤⣄⣀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⠙⠿⠉⡀⣀⠴⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣾⠇⠀⢀⣾⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⡟⠀⣠⣿⣿⣿⣿⠿⠛⠋⠉⠀⠰⣿⣷⣶⣿⣿⣿⣿⣦⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣝⣾⣿⡟⠋⢠⣾⣿⡿⣻⣽⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠋⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⡠⠒⠺⠿⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣒⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠙⠛⠿⠿⠿⠇⠀⣧⣄⠈⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢹⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠙⢷⡀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⡄⠀⠀⠀⠀⣷⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⡇⠀⠀⠀⢠⡏⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⠁⠀⠀⣠⠟⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⠀⠀⣰⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⠀⢠⡇⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⢧⣦⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣏⠀⠀⠚⠉⠙⠛⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⢀⣠⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⡀⠀⣴⡆⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⢿⣿⣿⠇⠀⠸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⢤⣿⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣸⠀⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⠓⠀⠈⠀⠀⠙⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣘⡰⠋⢀⡤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠛⠶⣤⡀⠀⠀⠾⢀⠞⠉⣠⣖⣩⡤⠔⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡀⠠⠤⠤⠀⠤⠄⣀⡀⠀⠀⣼⠗⠚⠁⠀⠀⠀⠀⠀⠁⠈⠻⣦
⠀⠀⠀⠀⢀⡼⠃⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠁⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢀⡞⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠉⠉⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⣾⠔⠁⠀⠀⠀⠠⠶⠚⠋⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⡤⢤⣀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢤⡀⠙⢦⡀⠀⠀⠀⠀⢠⡞⠀⣠⢞⡀⠀⠈⠳⡄⠀⠀⠀⠀⠀⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⣙⠦⣀⠙⠦⠀⠀⠀⠛⢀⠞⣵⣿⣿⣠⠀⠀⢹⠀⠀⠀⠀⠀⢸⡛⣦
⠀⠀⡏⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⣿⣿⣤⡌⠳⡄⠀⠀⠀⠐⣏⠀⠹⢿⣿⠟⠀⠀⣼⢠⡤⠀⠀⠀⢸⡇⣹
⣰⠒⣇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⠿⠾⠟⠀⢠⠇⠀⠀⠀⠀⠈⠠⣄⡤⠴⠒⠛⠁⠀⠀⠀⠀⠀⠀⢸⣧⠟
⣯⠀⢿⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠘⢦⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠈⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠚⠉⠉⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣀⣀⡀⠀⠀⠀⠶⠶⠿⢷⡀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠁⠀⠀⢀⣀⣀⣀⣀⣀⣳⡀⠀⠀⠀⣰⠆⠈⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢰⣥⠤⠴⠒⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⠠⠼⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠉⠻⣦⠀⠀⠀⠀⣀⣨⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠉⠛⠲⠦⣤⣄⣀⣀⣀⣀⣀⣠⣤⡴⡖⠚⢉⠔⢀⡏⠀⢀⣘⣧⠀⠀⠛⠉⢀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠙⠻⣬⣉⠉⠩⠙⡇⠀⠈⠲⠋⠀⣸⠁⡶⠋⠁⠙⠶⠶⠖⠒⠲⡄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⣤⣤⣤⣤⠤⠿⠚⠀⠀⠀⠀⠠⡴⠶⠶⢶⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⢈⣉⣡⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⠀⣭⣻⣿⣿⣿⡟⠚⠛⢛⣻⣧⣤⡿⠋⠉⠉⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠉⠚⠙⠟⠿⠧⢄⣴⠟⡉⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢻⡾⠃⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⣤⣠⡴⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣀⣀⣙⣶⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⣀⠀⠀⢀⠤⢤⠀⠀⡤⠤⡀⠀⠀⡀⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣹⡇⣿⢎⠀⠀⣱⣿⢸⣏⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣄⠀⣑⣦⣀⣼⡏⣬⠿⣂⣒⠿⣥⢻⣧⣀⣴⣊⠀⣠⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣾⢙⢓⣻⠛⠳⣿⣿⣿⣿⣿⣿⠞⠛⣟⡚⡋⣷⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣴⡿⠘⣝⡧⠬⢼⡐⣿⣿⣿⣿⣋⡧⠧⢼⣫⠃⢿⣦⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣨⣿⣿⢸⡿⠿⣶⠀⣉⢷⣿⣿⡿⣉⠀⣶⠿⢿⡇⣿⣿⣍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⢿⣯⡴⢉⡝⢣⡼⢾⣤⡌⢡⣤⡷⠧⡘⢫⡉⢦⣽⡿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠇⢾⢲⡓⠞⠲⣌⠛⠛⣿⡅⢈⣿⠛⠛⣡⠖⠲⠚⡖⡷⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⠷⠆⠀⣑⠉⢿⣾⣿⠁⠈⣿⣷⡿⠉⣋⠀⠸⠾⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣿⣵⣬⣉⠛⠀⠀⠛⣉⣥⣮⣿⣿⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠉⣿⢋⣽⣥⣮⣯⠙⣿⠉⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠿⠾⠗⠓⠘⠺⠇⠿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⢢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠀⠀⠙⠃⠀⠀⠘⠋⠀⠀⠸⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠙⠳⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡫⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡠⠂⠀⠀⡠⠀⣠⢾⣀⣞⡴⠋⠙⣆⡄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏⢀⣠⣴⠯⠖⠋⠀⠛⠋⠁⠀⠀⠀⠘⣷⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠁⠰⠦⢄⡀⠀⠀⠀⠀⢀⡴⠒⠃⢻⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣶⣶⡍⢠⡇⠀⠈⢡⣶⡆⠀⢸⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠬⠭⠀⡼⠀⠀⠀⠘⣛⠥⠀⢸⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡀⣀⡤⠤⠤⢤⡄⡄⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣤⣤⣤⣤⡤⢤⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢴⠞⢻⠁⠸⠀⠀⠀⠈⡏⢳⡦⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠁⠀⠀⡜⠀⠘⢆⠀⠀⠀⠀⣸⠁⠈⡇⠀⠀⢉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠱⡀⠀⠀⢹⠀⣀⡈⠳⡄⣠⠞⢁⣀⣸⠃⠀⠀⡎⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠑⠀⠀⠈⠉⠘⠧⣄⣸⣁⣀⡼⠀⠁⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢩⠩⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠈⣀⠠⠤⠤⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⡟⠠⣄⡀⠀⢀⠀⡇⠀⠀⠀⢀⡃⠀⠈⡇⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡤⠞⣷
⠀⠀⠀⠀⠀⠀⢹⠶⠦⣭⣙⣻⠀⠑⠦⠤⠴⠊⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢰⣏⣩⣥⠴⣾⠋
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⣀⣸⣏⡄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠛⠳⠶⣤⣶⣶⠶⠶⢤⣤⣧⣧⣤⣴⣶⣚⣋⡉⠁⠀⠀⠀⠀⢸⣀⣀⣀⣀
⠛⠛⠛⠛⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⡿⢿⠉⠉⠉⠉⠉⠉⠛⢿⡂⠀⠀⠀⣀⡼⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿⣿⣤⣀⣀⠀⠀⣐⡮⣷⣿⣶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠙⣆⡀⠀⠀⠀⣀⣤⡤⠶⠶⠶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠲⠶⣯⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⠀⢀⣠⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠛⠛⠓⠛⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⢀⡟⠻⢷⣤⣄⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡀⢸⠇⠀⣀⣬⣍⣉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠸⡌⢧⣿⢠⡞⠉⠀⠀⠈⠁⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠛⠷⠶⣶⠶⢿⡏⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣹⣦⣿⠞⣋⡁⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⠴⠞⠛⠉⠁⠐⠒⠲⢦⡀⠀⢸⡇⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠸⢁⡟⠁⠉⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⣧⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⣾⢁⡼⠃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠓⠲⢤⣀⠀⠀⠀⠈⣧⠀⠀⢘⡇⠀⠀⠀⣀⣨⣿⣷⣤⣾⣶⡆⣠⡏⣿⠀⣀⢀⠀⠀⠀⠀⣀⣀⣈⠺⣵⡟⣋⡴⠃⣟
⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠛⠁⡀⠀⠀⠀⠀⠤⣍⡉⠒⢤⡀⠘⢷⣴⠋⠀⠀⠀⠸⠣⣄⣾⣻⡿⢿⣿⠟⠋⠀⠹⣾⣿⣿⣧⣤⣶⠿⠛⠉⠉⠑⢹⡟⣯⠀⠀⢿
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⣿⠃⠀⣀⣀⣀⠀⠀⠈⣷⠀⠀⠱⡀⠨⡇⢀⡠⠄⠀⠀⠀⠈⠉⠛⠒⠋⢹⡆⠀⠀⢀⡾⣿⠛⣿⣻⠍⠓⣄⠀⠀⠀⢸⠇⠸⠀⠀⣾
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣰⢏⠅⠒⠈⠁⠀⢢⣸⠀⠀⠀⠈⢢⣿⡏⠀⠀⠀⠀⠀⢀⣀⣀⡙⢶⣼⠃⠀⠀⣼⢁⣀⡙⠛⠛⠒⠋⠋⠀⠀⠀⢯⡀⠀⣠⡼⠃
⠀⢀⠔⠋⠣⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠱⣹⣦⠀⠀⠀⠚⡉⢰⠋⠉⠀⣿⠀⠀⢠⡿⠋⢥⣄⡀⠀⠀⠀⠠⢄⡀⠀⠈⣷⠟⠉
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣰⠇⠀⡠⠀⠀⠀⠀⢻⣇⣀⣠⣤⣾⣷⡦⢼⡳⣼⡇⠀⠀⢸⠃⣀⠀⡏⡙⢦⠀⠀⠀⠀⠈⣦⡼⠃
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣤⣶⣏⣠⠜⠁⠀⠀⠀⠀⠀⠹⡟⠛⢻⡻⣿⣝⢿⣿⣿⠀⠀⠀⣿⣯⣽⣖⠤⣽⡀⠁⠀⢀⣤⡶⣻⡇
⣸⠉⠑⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠙⢷⣀⢠⠆⠀⠀⠀⠀⠀⡇⠀⠀⢡⠹⣏⠿⣿⣿⣀⠀⣰⣿⣿⣋⣭⣿⢷⣿⣦⣤⣹⣟⣴⡿
⡇⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⣿⡏⣀⡤⠄⠀⠀⠀⣇⠀⢀⠀⣇⢻⡂⠄⠁⠙⠋⡛⢙⢹⣽⠟⠁⠃⠀⠉⢹⡿⠛⠉
⠰⡄⠀⠀⠠⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠛⠒⠀⠀⠀⠀⢻⡄⢸⠀⠘⣧⠻⣤⣀⡈⢀⣁⣤⢞⣡⠊⠀⠀⠀⢠⡟⠁
⠀⠙⢆⠀⠀⠹⡄⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⢿⡌⠂⠀⠈⠛⠦⣭⣉⣉⣭⡴⠛⠁⢠⠇⢀⡴⠋
⠀⠀⠈⠳⣄⠀⠹⣄⠀⠀⠑⠦⣤⣤⣂⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢷⡀⠀⠀⠀⠐⠒⠤⠀⠀⠀⠀⠰⢋⡴⠋
⠀⠀⠀⠀⠈⢧⡀⠘⢦⡀⠀⠀⠀⣀⡟⠀⡎⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠚⠁⠀⠘⣧⠀⣴⣋⣽⣿⡝⣦⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⠦⣄⣽⢛⣳⠞⠁⠀⠀⡇⠀⠀⠀⠀⠴⠓⠉⠀⠀⠀⠀⠀⠀⠀⠘⢷⣏⣿⣿⣿⣿⣿⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⡍⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠠⠤⠀⠀⠶⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠑⡀⠀⠠⠫⠹⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡎⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⡙⢿⣿⠀⠀⠀⢀⣀⠰⠄⠚⠻⡃⠀⠈⠁⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⠀⠀⡀⢀⠄⠀⢾⡿⠢⡟⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠚⠀⠀⠈⠀⢰⡇⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠣⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣀⠀⢀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠁⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⢨⠓⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠄⠁⠀⠀⠱⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⡠⢤⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠔⠀⠀⠀⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡧

⠀⣼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠄⠀⢀⡀⢀⠀⡀⠀⠐⠂⢀⢀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠃⠐⠀⠀⠀⠀⠀⠙⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡀⠀⠀⠂⠘⠂⠒⠒⠊⠓⠘⠀⠃⠚⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣦⠀⠠⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡃⠀⠀⠀⠀⠀⠿⠇⠠⠭⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⢶⢤⣤⣿⣼⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⢀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠠⣦⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⠆⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢶⠖⠒⠒⠂⠒⠆⠀⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠒⠠⠋⢻⠀⠀⠀⠀⠀⡒⠺⠋⠂⠰⣶⢤⣀⣶⣶⢚⣶⣛⣦⣤⣄⡄⠀⠙⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡾⠿⠻⠿⠷⢶⡟⡛⠻⠏⣉⡛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡾⠛⢛⣙⣃⣘⣿⣿⣿⣷⣦⣽⣷⣦⣼⣿⣦⡘⢿⠾⠟⠛⠻⠿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣶⠾⠟⣛⣫⣴⣾⡿⢟⣛⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣶⣿⣿⣥⣶⡶⣈⣽⠇
⠀⠀⠀⠀⠀⠀⠸⣧⣄⡲⠿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣛⠲⢦⣬⣽⣶⣿⣿⣿⠟⠽⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣷⣌⠻⠶⢶⣄⡀
⠀⠀⠀⠀⠀⢰⡟⢡⣾⣿⣿⣿⣿⣿⡟⠋⢠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⡦⠀⢹⣿
⠀⠀⠀⠀⠀⣿⢡⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢉⠍⣩⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣰⡟
⠀⠀⠀⠀⢸⣏⠈⣸⣿⣿⡿⣱⣿⣿⣿⣿⣿⣯⣾⣫⣾⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠙⣷
⠀⠀⠀⠀⠀⢻⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣍⠱⣾⡏
⠀⠀⠀⠀⠀⢸⣇⢻⡟⣹⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⣿⣿⣿⣿⣿⣿⡏⢩⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠻⣦⡄⣿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡹⡾⠿⢿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠛⠻⡯⠙⢿⡿⢉⠀⢸⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠹⡏⠈⢿⠉⠛⠻⠿⠿⠯⠭⠽⠟⠛⠛⠛⠭⠙⠛⠀⢀⠤⠀⠘⠧⠈⡠⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢣⡀⠀⠀⠁⠐⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⠀⠀⢄⢰⢃⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠱⡐⢄⡃⠀⠀⠀⠾⠛⠿⠿⠟⠋⠀⠀⠀⠘⠛⠛⠿⠏⠹⠂⠀⠀⠀⠈⢸⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣭⡚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢂⣈⠠⢤⣄⠀⠀⠀⠀⠀⠀⡸⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢈⣆⠀⠀⠀⠀⠀⠀⣾⣷⣦⣤⣤⣶⣿⣿⡄⠀⠀⠀⠀⠴⣁⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢋⠖⠁⠈⢆⠀⠀⠀⠀⢀⣿⣿⠛⠻⠟⠙⣿⠿⠇⠀⠀⡠⠊⠀⠀⠙⢎⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢡⠃⠀⠀⠀⠀⠀⢄⡀⠀⠸⠏⣿⣦⣤⣤⣴⣿⠇⣀⣠⠚⠀⠀⠀⠀⠀⠈⢃⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠃⠀⠀⠀⠀⠀⠀⠀⡍⠓⠶⠠⣂⠉⠉⠩⠍⢐⡜⠻⠀⠀⡆⠀⠀⠀⠀⠀⠈⡀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢠⠀⠀⠀⠀⠀⡄⠀⠀⠇⠀⠀⠠⡀⠉⠀⠐⠂⠁⡠⠃⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡰⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⡞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠲⣤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠴⢶⢋⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⡼⢿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⢸⡇⠳⠭⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡀⠠⠄⠀⠀⠀⠄⢀⡀⣠⣤⣤⣤⣀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣀⣀⡀⠀⠀⠠⠄⠀⠈⣻⣿⣿⣿⣿⣷⡀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠤⠀⠀⠁⠐⠀⠄⡀⠈⠉⠛⠿⣿⣧⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⠁⠂⢀⠀⠀⠙⠢⢀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠈⢫⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠄⡈⠂⢄⠀⠀⠁⠢⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠆⠀⠳⡀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠐⡠⣀⠀⠑⠌⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⡘⠀⠀⠀⡘⠲⣄⠀⠀⠈⠀⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠿⠒⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⢀⠌⠀⠀⢀⠔⠀⠀⠀⠙⠣⠀⠠⠀⠀⠀⠈⠁⠂⠀⠀⠤⠀⣠⠤⠤⠀⢒⣫⠶⠆⠚⡀⠘⢿⣿⣿⣿
⣿⣿⠟⠁⠀⢀⠔⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠌⠡⠄⠀⠀⠬⠀⠠⠔⡒⠛⠀⠀⠐⠂⠈⡰⠀⠀⠈
⣿⣥⣤⡐⣈⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡉⢃⠀⠀⠀⠀⢀⣴
⣿⣿⡿⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣿⣶⣶⣾⣿⣿
⣿⣿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠈⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠤⠈⠂⠀⢠⠀⠀⠀⠀⠑⢢⡀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡓⠂⢬⣉⠀⠀⣉⡚⠢⢄⣀⢗⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠈⠀⠀⠀⠀⠀⣀⡀⠀⠀⢤⣈⠙⠀⠀⠀⠀⠀⠀⠀⠀⢑⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢐⡦⠲⠤⣤⠀⣿⢁⠀⠀⠀⠒⠁⡀⡰⠀⠀⠀⠂⠀⢠⠊⣡⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡌⠀⢀⡴⠁⠀⣿⡀⠧⠀⠀⢀⣠⠤⠶⠶⠶⠶⠶⠶⢶⣦⣼⠂⠀⠛⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠙⡄⠀⢼⠁⠀⢀⣯⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠉⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⣇⠈⡆⣧⠀⢀⠀⢽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠠⢤⡋⢦⡀
⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⡀⠀⠀⠀⠨⢆⡘⢻⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠒⢤⡇⠀⠳⡄
⠀⠀⠀⠀⠈⠀⣀⠴⠀⠀⡏⠀⠀⡀⠀⠈⠷⠤⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠠⠔⠁⢈⠐⠈⠹⢆⠀⢄⠀⠀⡂⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⢴⣧
⠀⠀⠀⠀⠀⠀⠐⠂⠒⠁⠀⠀⠑⠀⠀⠚⠑⠀⠀⣷⠀⠀⠀⠠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠌⠃⢸⡀⠀⠀⢰⡿⣿⠿⡖⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣡⡞⠀⡆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡥⠖⠂⢹⠃⠀⠀⠀⠱⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢡⣿⣿⡇⢠⣧⠃
⠀⠀⠀⠀⠀⠀⠤⠀⠈⢀⣠⠤⣧⣀⡔⠋⠀⠀⠀⡟⠀⠀⠀⠀⢀⠈⢀⠀⠀⣀⣠⣤⣄⡀⠀⢀⠀⣰⠰⠁⣿⣿⡟⢁⣾⡁
⠀⠀⠀⠀⠀⠀⠀⢰⣒⣁⣠⠟⠀⡽⠀⠀⠀⡠⢞⣻⣾⣷⣶⡶⠣⠀⠀⠉⠉⢿⣛⡽⠄⠉⠓⠁⠠⠃⠀⠈⡋⠁⢰⡿⠯⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣷⡠⢠⡞⡴⠉⢀⡞⠛⠛⠋⠀⠃⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⠀⠘⡇⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣿⠘⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⣾⠳⣄⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢄⠀⠀⠀⠀⣀⡀⠀⢀⣤⣤⡼⠁⠀⠀⠀⠀⠀⠀⢸⠆⠀⠈⢣⠑⠌⢷⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⡆⠀⡤⠤⡈⠙⠲⠛⠁⠀⠀⠀⠀⠴⠒⣦⣤⡔⠃⠀⠀⡀⢀⡇⠀⠈⣷⢸⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣦⡐⢄⠀⠀⣧⡼⠓⣦⣀⣀⣤⣴⠒⣶⣿⠀⢸⣿⠃⠀⠀⡆⠀⣷⣾⣅⢠⡄⣿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠙⣶⢵⣦⣼⣁⣴⣿⣿⣿⣿⣿⣼⣿⡿⠀⠿⠃⢀⠀⢰⢀⢀⡿⢳⡙⣿⠁⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠀⠀⠀⣨⡛⠛⠻⠍⡻⡏⢹⣿⣿⡿⠁⠀⠀⠀⡀⢸⣸⠿⠟⠁⢰⠃⢸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⡔⠀⠀⠈⠙⢛⠓⢦⡈⠣⠈⠿⠏⣄⣀⡀⠀⠀⣷⠿⠃⢀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⠰⡇⠀⠀⠀⠀⡀⠀⠁⠀⠰⠒⠛⠉⢁⠄⣿⣿⣁⡤⠖⠁⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢧⡆⠀⠀⠈⠳⣀⠀⠀⠀⢀⠔⠚⠙⢲⣷⡶⣶⣶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⡄⠀⠈⠀⠀⣠⠀⠀⠀⠀⠀⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠘⠛⠁⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠶⠶⠖⢀⠀⠀⢸⡆⠀⠀⠰⠀⠀⠄⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⢰⡇⠀⠚⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⢀⠀⢐⡀⠀⢱⠼⣄⠗⠀⡌⢀⡀⣁⠀⢆⣨⣔⠁⡀⠀⠀⣊⣀⡔⢉⣩⠝⠛⢳⢶⡄
⠀⠀⠀⠀⠴⢛⠉⣉⢉⡶⣤⣾⡿⠃⠀⠒⠀⠙⣦⠈⠄⢨⡷⠀⠀⣸⠁⠀⠠⠃⠀⢠⠛⡆⡙⢩⣄⠀⣬⠾⠏⠉
⠀⠀⠀⠀⠀⠻⠲⢪⠄⢀⠄⠀⠊⢳⢀⡇⠀⠛⠻⢷⠐⣿⣟⠀⢸⠟⠓⠐⠀⠛⠷⢊⢀⡇⠘⠂⡁⠀⠀⢠
⠀⠀⠀⠀⠀⠞⢆⠁⢸⠏⠲⠇⢀⣴⡟⡤⠀⠀⠀⠈⠣⣘⣍⡲⠿⡀⠂⠀⣄⡀⢆⡇⠈⠃⡀⠾⠃⡆⠈⠀⠀⠀⣠⡆
⠀⠀⠀⣧⣠⠀⡺⢀⠌⠀⠙⠆⠈⢀⣦⢭⡀⡆⠀⠆⠀⢸⠎⠉⢳⠂⠀⠀⠠⣢⣯⣀⠘⠟⣡⠄⠳⡁⠀⢱⠀⣾⠟⠁
⠀⠀⠀⠛⠃⡜⣷⣻⡆⠀⠘⢾⣟⠉⠈⢀⣘⣥⣤⠞⠋⢻⠖⠦⠼⢦⡶⣤⠂⡹⠎⠉⠀⣤⣤⣾⠧⡀⢀⡈⢆⢁⣴⠇
⠀⠀⠀⠀⣴⠀⢀⠿⠃⣄⠴⠿⢭⣝⢛⣿⠏⣽⡏⢀⣼⣫⣷⣤⣤⣾⣧⠈⢷⣾⣷⠻⠟⢫⣦⣀⠀⠟⠛⠁⠪⡘⠋
⠀⠀⠀⠀⠘⠃⣠⠞⢻⠛⠻⢶⣶⣌⣿⡏⠠⣿⡇⠀⠁⣰⡇⢀⠀⠉⣿⡇⠘⣧⢁⡴⠟⠛⠛⠛⠛⢣⣄⣠⠀⠇⠀⢀
⠀⠀⠀⠀⢇⣴⣤⡾⠛⠻⠛⣶⣦⣽⡿⡇⢀⣿⣿⠀⢤⣼⢧⡀⢀⣠⣿⡇⠸⣷⣋⣤⡶⠿⠋⠛⠚⠻⢿⣏⣤⠸⣾⠏
⠀⠀⠀⣦⡾⠛⣥⣴⣿⣿⠗⢶⣽⣿⡿⢷⠀⣹⣿⣷⣾⠷⣞⣿⡀⢹⣟⠃⣼⢻⣷⢀⣤⣶⣾⣿⣶⣄⡀⠙⣆⠁⡏
⠀⠀⠀⣾⢁⣾⣿⣿⣿⠁⢘⣾⢩⣷⡄⠸⢧⠸⠟⠁⠻⣶⢛⣿⠣⠈⢻⡝⢩⣟⣿⣿⠸⠃⣸⣷⣿⣿⣷⠄⢸⡄⡏
⠀⠀⠀⡇⢸⣿⣻⣿⣿⠀⠙⢹⣿⣿⡇⡀⠀⠀⢀⣤⣴⣟⣸⠁⠄⠤⠆⣰⡾⣿⣿⠟⣩⠿⢻⣿⣿⣿⣿⠇⣼⡇⣜⢠
⠀⠀⠀⣧⠈⣿⣿⣿⣿⣤⡆⠀⣿⠟⣡⣽⣾⡿⣿⣿⡿⠛⠁⢠⣤⣶⣶⣶⣦⡝⠿⣄⣩⡞⢁⣿⣾⣿⡏⠠⣿⠁⡈⢱
⠀⠀⠀⢻⣂⣹⣿⣿⣿⣏⠀⠀⣌⠠⣿⣿⡿⢻⣿⣿⡗⢀⡄⣿⣿⣿⣾⣿⣿⡏⠈⢮⣟⠡⢼⣿⣿⣿⣶⣾⣯⣀⡼⠆
⠀⠀⠀⢿⣿⣀⡀⠙⠛⢻⣄⠰⡙⠀⠙⠓⠞⠛⢋⣍⢰⣿⣷⠈⠉⣛⠿⠟⠛⠁⠁⣾⡝⢃⣾⡿⠟⠛⠛⡹⢿⠁⣴⣆
⠀⠀⠀⣿⠘⣿⣿⣶⣄⠀⠙⢿⡵⢄⣉⣠⡘⠶⠀⢱⣾⣿⣿⡄⠀⠆⠐⢉⣀⣀⣴⡟⣠⠾⠏⠀⡄⣰⣾⢓⣾⢿⢿⡌⠁
⠀⠀⠀⢹⢧⢛⣽⣿⣿⣶⣀⠈⠹⡄⢭⠟⣧⠀⠉⠸⣿⡟⠻⠃⠀⣀⢆⣻⡷⢡⢎⡾⠻⠆⠀⢴⡞⢛⠁⠈⠉⠿⡁⠁
⠀⠀⠀⢠⡈⠛⢛⣿⢻⣿⣆⠀⠂⣷⠀⣺⠸⣷⣶⣤⣦⣶⡄⣄⢢⣽⣾⣧⠁⠰⡿⠓⠀⠀⣵⠏⡹⠋⠀⢰⡀⣦⠄
⠀⠀⠀⢈⣄⠀⠘⢛⣷⣿⣿⡴⣇⢸⡀⢸⠇⣿⣍⣼⢩⣿⣿⡿⠛⣯⡿⢱⠀⡘⣟⠊⠁⣰⡏⣰⡇⠀⠀⣼⡟⢁
⠀⠀⠀⢸⡟⢦⡴⠛⣛⠟⢻⣿⣷⠾⠁⣻⡇⠃⠻⢿⣿⣈⠛⣷⠷⣿⠡⠸⡙⠻⠻⠶⣤⣹⡿⠋⢣⣔⠂⣝⣠⠙⠳⠄
⠀⠀⠀⠉⠀⠸⠀⣸⡉⢆⢠⠹⠿⠒⢡⣿⠋⠲⠀⢄⣈⡉⢁⡀⠀⠀⢠⠔⣱⡶⠛⠛⣯⡁⢀⡁⠻⠞⠓⢧⠄
⠀⠀⠀⠀⠀⡽⢳⢿⠃⢈⣿⣄⠟⠛⠛⢿⣏⠋⠑⠀⠤⠴⠂⠁⠠⠘⢱⣿⠟⠒⢳⡿⡤⠟⡁⠐⠀⠂⠀⠐
⠀⠀⠀⠀⠀⢦⠛⣠⠆⡘⢿⡎⠐⢀⠂⡀⠬⠅⠠⠆⡀⢠⡄⠠⠄⢀⠉⣬⠂⣱⣿⡷⠙⡤⠁⠤⡀⠔⢷⠶⠂
⠀⠀⠀⠀⠀⢴⣗⠣⠀⠀⠈⢷⠀⠀⠲⣂⣀⣠⡁⠀⠀⣾⣽⠂⠂⢈⣡⣀⡚⠙⠋⡀⠀⠀⠈⠰⣿⣿⠞
⠀⠀⠀⢀⠀⢰⠋⣅⣿⡄⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⢻⡆⢳⣶⣿⣿⣿⣿⣿⣏⡉⠀⠀⢰⣶⠻⣷⣤
⠀⠀⠀⠀⠀⠈⢀⠉⠙⣧⣠⠀⠣⠘⠋⠙⣿⣿⣿⣿⣿⣿⡶⣾⣿⣿⣿⣿⣿⠟⠛⠔⠁⣠⣾⢟⠴⠐⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠸⣀⠁⢈⢭⠀⡁⡘⠡⠀⠩⡩⢉⠛⣹⣟⣃⣿⢋⠈⢅⠀⠉⢉⠛⠤⠀⠩⢁⠄⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠰⢀⠀⠠⢤⡄⠀⠀⠀⠀⠀⠳⡙⢿⣿⢿⡍⠀⢀⡀⠀⢠⠄⠒⠂⢠⠀⠒⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠃⠀⢠⡄⠘⠀⠂⠀⠀⢀⡀⢀⠐⣈⠁⠠⠄⣤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⣿⣿⣿⣷⡄⠀⢀⣀⠤⠤⠔⡾
⠀⠀⠀⠀⠀⢀⡔⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦⠋⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⡞⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡶⣦⣤⣤⣀⣀⠀⢷⠐⠒⠙⢦⡀
⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⠿⠟⢯⠿⠿⠛⢹⣿⣿⠟⢹⣿⣿⣿⡿⠋⠀⠀⠉⠛⠿⢿⣿⣞⡲⠤⣀⣀⡨⠗
⠀⠀⠀⡞⠁⣿⣿⡛⠉⠁⠀⠀⣀⣀⠀⠀⢠⠿⠛⠁⠠⠿⠛⠋⠁⠀⣀⡀⠀⠀⠀⠀⢸⣿⡇⠈⢣
⠀⠀⢸⡄⠀⢸⣿⠃⠀⠀⠀⠨⢭⣿⡿⠶⡄⠀⠀⢠⠄⠀⠀⣠⠾⣿⣿⡭⠅⠀⠀⠀⠸⣿⠇⠀⢸
⠀⠀⠀⠣⣀⣀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⣀⣀⠞
⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛
⣤⠤⠦⠤⢤⡀⠸⡧⣾⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⢸⡇⠀⠸⡳⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢹⡄
⣿⠀⠀⠀⠀⠙⣦⡟⢳⣠⠞⠁⣤⡏⢀⡔⠋⠁⢸⣷⠶⣦⣰⠲⣆⣞⡉⢩⡿⢾⠈⣛⣷⣰⠋⢻⣤⠖⢶⠖⠒⢦⠀⣠⠴⠟⠉⢻⠀⡇⢰
⣿⠀⠸⠟⠀⢠⠙⠃⢸⠛⠦⠤⡄⠈⣾⡀⠻⠃⠀⣿⡀⠉⠁⠀⡏⡴⠚⠀⢙⠸⡄⠳⠶⣤⡀⠈⡇⡄⢰⢲⡀⠸⡏⢧⠀⠛⠀⢸⣠⣟⣿
⢿⣀⣀⣀⡤⢋⠀⠧⠼⢃⡏⠉⠁⣀⢇⠙⠚⡆⠀⡇⠙⠦⡤⠞⠀⠓⠒⠒⢋⡀⠙⠒⠚⠃⠙⠉⣀⠉⢉⣈⠙⢛⣁⣌⠉⡇⠀⠘⡏⢣⡼
⡼⠮⠭⠤⠔⠋⠲⠶⠞⠦⠭⠭⠭⠕⠋⠙⠒⣇⠀⢻⠓⠦⠤⠴⠋⠑⠒⠒⠋⠙⠒⠖⠊⠉⠙⠋⠉⠋⠋⠈⠉⠉⠀⢸⠀⡇⠀⠀⡟⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠤⣼⣁⡇⠀⢸⠃
⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⠶⠋⠀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡾⣏⠀⠈⠉⠀⢀⣾⡀
⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⢠⠟⣗⠦⠤⠤⢞⣿⡀⣠⢄⠀⠈⢠⣄⠀⢀⠀⠀⢀⢦⣀⠀⡄⠉⢀⣀⣈⣃⣈⣉⣃⣒⣾⠿⢹⡇
⢸⡀⡠⣄⠀⢸⠁⡇⠀⢰⠻⡄⢸⠀⠀⠉⡟⠋⢉⡇⠓⠃⠸⡆⠄⡏⠈⠓⠋⠹⡆⢸⠀⠀⠀⣿⠀⢸⠉⠈⠉⢸⡇⠀⣿⢹⡀⢨⠇
⠀⠉⠀⢸⡀⣸⠀⢹⣀⡟⠀⠳⠎⠀⠀⠀⢳⠀⢰⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⡀⡸⠀⠀⠀⠘⣇⠀⡏⠀⠉⠉
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠎⠀⠀⠀⠀⠳⡴⠁⠀⠀⠀⠀⠧⠏⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡆⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢻⡄⢜⡵⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⣄⣈⠳⣯⣬⣿⠀⢧⠀⣺⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢧⣾⣿⣿⣿⣿⣿⡏⣿⡿⠃⢠⣼⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⢇⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢸⢹⣿⣿⡿⣼⣿⠃⠀⢰⣽⣿⣿⣿⡿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⡀⠀⠀⠙⠄⠀⠀⠀⡜⣼⡿⢫⣾⣿⣿⠀⠀⣿⣿⣾⣿⣿⣿⣟⢿⣿⣿⣯⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠛⠀⢀⡰⣾⠀⢀⡼⠰⠯⠴⠛⠉⠀⠈⢀⣠⣿⣿⣿⡟⣿⣿⣿⣯⣻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠦⡄⠀⠘⢣⡌⠉⠙⠁⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣷⣻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡦⢄⢀⣏⠀⠀⠀⠐⠂⠀⠈⠉⠀⠀⠙⠿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⠠⠤⠤⠤⠄⠀⢀⣠⠔⠺⣿⣷⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⣉⠛⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢀⣀⠀⢀⣚⢁⡠⠋⠐⠒⠠⠤⣤⣀⣀⡀⠈⠁⢀⣀⠼⠟⠉⠀⠈⢫⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣬⡿⣿⠛⠋⣿⠿⣿⣿⣿⣿⣿⡇
⠩⠟⠓⠉⣠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⡜⠡⠾⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣶⣦⣤⣴⣶⣾⣿⣿⣿⣷⣤⡎⠀⢀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠚⠿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠠⠤⠴⠚⠛⣯⣿⣿⣿⣿⣿⠿⢿⡗⠙⢿⣿⣿⣷⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠠⣴⠶⠒⠀⠀⠀⡀⠤⠚⠉⠁⢈⣿⠉⡿⣅⠀⠀⠸⡀⠀⢻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠲⠒⠒⠚⠛⠓⠂⠂⠀⠒⠛⠁⠀⠐⠓⢺⡀⠀⢀⢱⡀⠘⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠘⠓⡄⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⣀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⡤⠤⠴⠃

⣿⣿⣿⣿⣿⣿⠿⠟⠑⠿⢻⣟⣿⠿⠿⠻⣿⣿⣷⢹⠻⠷⣋⣛⣦⣍⠂⣦⣽⢽⣽⣿⣹⣷⠾⡿⠈⣿⣞⣿⣿
⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⣿⣿⣾⣿⣷⣯⠻⠿⠿⡃⣊⠙⣁⢙⠿⢷⣻⢸⣿⢷⡍⢾⣿⣿
⡿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢿⣋⣿⣴⣶⣿⣷⣧⡫⠛⣿⣧⣟⣥⡘⠻⢆⣿⣿⣷⣤⢿⣟
⠃⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣹⣻⣿⣽⣿⣯⢯⡄⣏⢥⣽⣿⣽⣷⣄⡂⠼⣯⣾⣆⠸
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣸⢻⣿⣿⣿⠿⢞⣧⣿⣿⣾⣿⢿⣷⡄⡑⢶⣌⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣼⣿⣯⢛⣷⣾⣯⢻⠿⣿⣿⣷⣿⡁⢿⣿⣿⣻⡏⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢸⣿⣟⣿⣣⣾⣿⣯⣟⡻⢿⣷⣿⣿⣻⣿⢸⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣏⣿⠀⢠⣾⣿⢍⣹⣿⡯⣿⣏⡛⣿⣷⣿⣝⡧⣿⣿⣾⣿⣟⢾⢩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡿⢏⣸⣯⣿⣶⣽⣝⡿⣦⠉⣿⣿⢹⣿⡏⣹⢿⡯⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠚⠟⠂⠀⠀⠈⢉⣿⣼⣿⣿⣿⣿⣿⣿⣾⠂⣼⣙⡷⠿⡅⣼⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⣿⣿⣿⣿⣼⣯⣿⠷⡟⠿⣧⠘⠁⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠟⢿⡂⣷⣿⣂⠐⣡⡷⣿⡷⣰⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠐⣻⣶⣿⣾⢧⡭⣿⠃⣴⣿⣷⢌⡹⣿⣿⣷⣉⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣩⣾⣿⣿⣛⣦⣹⣷⢻⢸⣷⣿⡟⢸⣷⣼⣿⣿⣿⣷⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢟⣩⡿⣿⣿⣁⠉⣌⣼⣿⢑⣿⣾⠉⠉⣿⣽⣽⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣿⣿⣿⣯⣼⣿⢿⠟⣀⡉⠀⡶⢛⣫⣴⠺⣷⣙⣿⠸⠿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠂⣸⠾⠷⠚⠋⠈⡱⠈⡇⡀⠁⠠⢤⠠⡎⣿⣛⠁⣛⣙⡇⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⢟⡓⢀⡴⣾⢿⢋⣠⣤⣬⣟⠹⠇⠐⣒⢠⢀⡀⠿⣿⡇⢿⢟⣽⢢⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠛⣽⡤⢺⡟⣶⢿⣿⣿⣿⣼⠷⣷⠈⠋⣿⣋⣿⣯⣸⡷⠀⢰⣾⣿⢨⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡕⣌⠳⡞⠾⢟⣽⡛⣿⣿⣏⣶⣻⣴⢳⣿⣽⠟⣿⣶⠰⠃⣼⣷⡿⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣣⣽⣦⠘⠙⣼⣿⣿⣏⠙⠣⣿⣿⡹⣅⣿⣿⠇⠘⢱⡏⠉⣧⣷⣷⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠆⠈⣙⣥⣾⡧⣔⢿⣿⡇⠀⡁⣶⣶⣧⡌⣤⣤⣿⣽⡇⣻⣟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣈⡠⣤⣤⡝⠶⢿⢿⡟⢋⣵⣤⣿⣿⣿⣫⡆⣿⣿⣿⠷⢻⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣿⣿⡈⣥⡄⠒⠨⠸⢾⣿⣟⢷⣻⣽⣿⡏⣣⠘⣟⣭⣆⣾⡷⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⠿⣿⣿⣶⣼⣷⣀⡙⠿⠟⠿⢳⡏⠓⠛⣃⠉⠠⠹⣿⡃⣿⣽⣿
⣾⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⣖⠿⡿⢿⣿⡻⠟⠇⠄⠈⣉⠀⣁⢈⣭⣋⡄⢰⣷⢹⣇⣿⣾⣿
⣯⣿⣿⣇⣿⣳⣦⣀⠀⠀⣀⣄⣶⣿⣮⣍⡹⢶⣖⠦⣀⡀⠄⠒⢻⣿⣷⣦⣿⣿⡷⢯⣹⢷⣿⣿⣿⣽⣿⣿⣿
⠉⢩⠉⠁⣭⢥⣿⣿⣿⣿⡿⣛⣿⣿⣿⠿⠟⣋⣵⣮⣿⣿⣿⣟⣲⣼⣽⢻⣿⣷⣅⣿⣿⡆⢽⢿⠛⡇⣹⣿⣿
⠸⠠⠃⣶⡿⣸⣻⣿⣿⣿⠿⠿⣛⣭⣶⢾⣿⣿⣿⣿⡿⣿⣿⣿⣟⣿⣏⣹⣿⣯⣼⣷⣿⡷⣻⣿⣇⢩⣿⣿⣿
⢢⣾⠴⡦⠀⢿⠛⠛⠙⢑⢢⣿⣿⣽⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣷⣮⡹⣋⡇⣿⣿⣿
⣶⣒⣸⣷⡘⣯⡇⣿⣵⣿⣿⣿⣿⣷⢺⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⠻⢿⠿⠿⣿⡿⢿⣿⣿⡿⣷⣞⢟⡳⢿⣻
⣿⣶⣶⣿⣷⢿⣿⢸⣿⣿⣿⣿⣿⣿⣇⣛⠻⣿⣿⣿⣿⣿⣷⣿⣿⡗⣿⣿⣿⣿⣿⠷⠞⠿⢶⣯⣭⠭⣷⠻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⢀⣴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣤⣿⡇⢠⣾⡿⣧⣶⠰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⣺⣿⣿⣿⣷⣿⣷⣿⣿⣶⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⣭⣽⣿⣤⣤⣤⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⣿⡏⣿⡟⣿⡟⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡇⣿⡇⣿⠁⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢸⡇⣿⡇⣿⢰⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣇⣿⡇⣿⢸⡏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣷⣿⣼⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⢻
⣀⣠⣤⣤⣀⡀⠀⢸⣇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⠀⠀⢸⠧⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡆⠀⢸
⢿⠀⠀⠀⠀⠉⢳⣈⣷⣾⡀⣠⠞⠋⠙⡗⠀⢀⠴⠞⠛⢳⣀⣤⣄⢠⣤⣤⢀⡖⠛⠛⣶⣸⠇⠻⢤⡄⢸⠷⢿⡀⣠⣤⣄⡤⢤⣀⠀⠀⢀⣠⡖⠚⣷⠈⡇⠀⣿
⢸⠀⠀⣖⡄⠀⠈⠿⡇⠈⡷⣇⠀⠘⠛⠳⣤⡏⢠⣤⠀⢸⣿⠈⠻⠼⠀⢸⠞⣓⣢⠈⠳⡸⡇⢴⣒⡓⠛⠀⠈⡏⡇⠈⢁⡀⠀⢹⡶⢾⠉⠡⡆⠀⢻⠀⣧⣤⡏
⢸⠀⠀⠉⠀⠀⡼⠀⡇⠀⡇⢈⣍⣩⠆⠀⡟⢧⣀⣁⠀⢸⡹⣆⠀⠀⠀⡜⢸⣁⡀⣀⡠⠃⢣⡀⠉⣹⠿⣄⣀⠇⠧⢤⠇⢹⣀⣈⡇⠘⠦⣤⠀⠀⢸⣶⡧⢾⡅
⢸⡤⣤⠤⠴⠋⣠⠀⠙⠛⡁⢾⣀⣀⣀⡴⢣⣄⠀⢹⠀⠈⣇⡈⠛⠛⠋⢀⣀⡈⠉⠉⣠⣄⠀⠉⠉⣩⣀⣀⣀⣤⣀⣀⡰⢦⣀⣠⠤⠶⡄⢸⠀⠀⢸⣇⣨⠟⠁
⡞⠓⠲⠶⠒⠋⠈⠓⠚⠛⠓⠶⠶⠶⠖⠚⠁⠈⠙⢻⡆⠀⣿⠙⠲⠶⠖⠋⠀⠉⠉⠉⠁⠈⠙⠓⠋⠁⠈⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡏⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⡇⡼⠀⠀⢸⠁
⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠶⠾⠁⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢠⢞⡏⠀⠀⠉⠀⠀⢀⣿
⢻⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⢠⠿⡷⢤⣀⣀⣠⢞⣏⠀⢀⣄⠀⠀⠃⢀⡀⠀⠀⠁⠀⠀⣠⡄⢠⢤⡀⠐⠀⡀⢀⡘⢦⣙⠒⠦⠤⠤⣒⡿⢹⡄
⢸⠀⠀⡀⠀⠀⢰⠛⡆⠀⠀⡴⣄⠀⢸⠀⠉⠒⣦⠥⠶⢻⠉⢧⡞⠈⡆⡀⢰⠏⠙⣤⡼⠶⡆⠀⡟⠈⠁⠀⢷⠀⠀⡟⠚⠙⠚⢹⡏⠛⢻⠻⡇⠀⢸⡇
⠘⠧⠞⠉⡆⠀⣟⠀⢧⠀⢸⠀⢸⣀⡿⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇⠀⢸⠃⠙⠦⠼⠁
⠀⠀⠀⠀⢳⣀⡇⠀⠘⠳⠞⠀⠀⠉⠁⠀⠀⠀⠘⣇⠀⡾⠀⠀⠀⠀⢻⠀⢸⠀⠀⠀⠀⠀⣟⢀⡇⠀⠀⠀⠘⣦⡼⠁⠀⠀⠀⠀⠳⣤⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⣁⣢⡀
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠈⠉⠲⢄
⠀⠀⠀⠰⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⣨⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⡤⠰⠇⠐⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈⠁⠀⠀⠈⡇⠀⠀⠀⠃⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠙⡋⠈⣆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⢄⡰⠤⠤⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠉⣦⠈⠢⡀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢸⠀⠀⠘⢄⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠔⠁⠀⠀⠀⠀⡠⠂
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠀⠀⠀⠈⢂⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠔⠈⠁⠒⠒⠒⠒⠈⠁⠘⡀
⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⢃⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠙⠲⠤⣀⣀⣀⡀⠠⠐⠊⠑⡀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠢⡀⠀⢤⡀⠀⢠⠄⠈⠁⠈⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠐⠒⠒⠒⠀⠀⠠⠤⠤⠤⠄⠀⢀⣉⣉⣀⣀⣀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠀⠀⠸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⡆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢘⢇⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⠀⠀⠸⠈⡀⢨⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⢡⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠸⡀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⡇⠀⠀⠀⠨⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢫⣠⣔⠂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒
⣿⣿⡿⠻⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡷⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⡗⠀⠀⠀⠀⣠⠞⠋⠉⠉⠙⠳⣄⠀⠀⠀⠀⢻⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢹⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣗⠀⠀⠀⠀⠙⢦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⣺⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡧⠀⠀⠀⠀⠀⠀⠈⣉⣉⡁⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣠⡞⠋⠉⠉⠙⠳⣄⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀⠀⡷⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣸⠀⢀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣏⠀⠀⠀⠀⠙⢦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⣽⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣗⠀⠀⠀⠀⠀⠀⠈⢉⣉⠁⠀⠀⠀⠀⠀⠀⢻⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⡶⠛⠉⠉⠛⢶⣄⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀⠀⠀⠀⠀⡧⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⣇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⠀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣇⠀⠀⠀⠀⠹⢦⣀⠀⠀⣀⡴⠏⠀⠀⠀⠀⢼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠟⠷⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⠶⠶⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣤⣤⣶⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠗⠲⠶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣁⣤⣤⣤⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠓⠶⢶⣬⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠁⢀⣀⣀⣿⣿⣦⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠛⠳⣦⣌⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⠇
⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⢻⣿⣷⡾⠋⢉⣁⣤⣿⣿⣧⣀⣴⣾⣿⣿⠿⠿⠿⠿⠿⣿⣶⣤⡀⠀⠀⣠⣿⣿⡛⠛⠶⣄⠘⣿⣿⡇⠀⠀⢀⣴⣿⠟⢹⣿
⠀⠀⠀⠀⣸⡿⣿⣧⠀⠀⠀⠈⣿⣿⣷⡿⠛⣉⣠⣤⣿⣿⡿⠟⠉⠀⠀⠀⠀⣀⣤⣤⣾⣿⣿⣷⣴⣾⣿⣿⣅⠙⠻⣦⣈⣿⣿⡿⠀⢀⣴⣿⡿⠋⠀⢸⣿
⠀⠀⠀⢀⣿⡇⠸⣿⣧⠀⠀⠀⠘⣿⣿⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⣹⣿⡿⠁⠉⠙⢿⣦⣈⣿⣿⣿⣷⣾⣿⡿⠋⠀⠀⠀⢸⣿
⠀⠀⠀⣼⣿⠁⠀⢹⣿⣦⠀⣠⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⣄⡀⠀⠉⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⣸⣿⠇⠀⣤⡀⢻⣿⣿⣿⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠉⠉⠉⠉⠙⣛⣿⣦⡀⠀⠉⠈⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⣰⣿⡏⠀⠀⢹⣷⠀⢻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠛⠛⠛⠛⠀⠀⠀⠀⠙⢿⣿⡄⠀⠀⠀⠀⠀⢸⣿⡆
⢠⣿⡿⠀⠀⠀⣼⣿⣇⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⠀⠀⠘⣿⣷
⣼⣿⠃⠀⠀⢠⣿⣿⣿⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⠿⢿⣿⣿⣦⣄⠀⠀⢠⣿⣿
⣿⣿⡀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠈⠻⣿⣿⣆⣠⣾⣿⡿
⢿⣿⡇⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠛⠛⠛⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣀⣠⣀⡀⠀⠀⠀⢹⣿⣿⣿⣿⠟
⢸⣿⣧⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠻⣿⣧⠀⠀⠸⣿⣿⡿⠋
⠈⣿⣿⡄⠸⣿⡟⠉⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢹⣿⡄⠀⠀⣿⣿⠁
⠀⠘⣿⣿⡀⠉⠁⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⣀⣴⠇⠀⠀⠀⠀⢸⣧⣴⣶⣶⡄⠈⣿⡇⠀⢀⣿⣿
⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠈⠛⠗⠀⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣷⣶⣦⣄⠉⠛⠛⠛⠉⠀⢠⡆⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⠃⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⣠⣾⣿⡿⠿⠛⠛⠛⠛⠻⣿⣿⣦⡀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⣼⣿⠀⠀⣼⣿⠁
⠀⠀⠀⠀⣿⣿⡟⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠛⣶⣾⡿⠟⠋⢁⣀⣀⣤⣤⣶⡾⠿⠿⠿⠿⠷⠀⠀⠀⢻⣧⠀⢀⣼⡿⠿⠿⠿⣷⣿⡏⠀⣰⣿⠃
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠁⠰⠶⠛⢻⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠈⠀⠀⠀⠀⠀⠀⠉⠀⣴⣿⣧⡀
⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣴⣶⡄⠀⣠⣤⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠃⠀⠀⠀⠀⠀⠸⣿⡄⠀⠿⠿⠰⣿⠇⠀⣼⡿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠛⠳⢶⣤⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣾⣿⠿⣆⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⣿⣿⠃⠀⠿⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡻⣿⠿⣿⡿⢿⡿⠟⣿⠉⣿⣴⣿⠏⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⢻⣿⣿⣶⣿⣦⣼⣷⣤⣿⣶⣿⡿⠃⠀⢠⡀⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣴⣿⣧⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠋⠀⠀⢠⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⢰⡆⠀⠀⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⢸⣷⡀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄⣿⣷⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣷⣼⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣇⠿⠀⠀⡀⠀⠀⠀⠀⠀⢿⣸⣿⣷⣮⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢋⡽⠃⣻⣾⣷⣰⢄⣾⣆⡆⠘⢧⡜⢿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⣛⣿⡹⣯⡀⠀⣿⣿⣿⣿⣿⣿⣿⡷⠀⢀⣼⡏⣵⣟⣭⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣩⣥⣴⡿⠃⠀⢿⣿⡿⣿⣟⢻⣿⡿⠀⠋⢼⣷⣤⣭⣽⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠘⣧⠀⠀⢠⣞⠀⠁⠻⣿⠈⠋⠱⡁⠄⠀⢰⡟⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣰⣯⣤⣁⣀⣿⡄⠀⣿⡏⠀⣰⣏⣀⡀⣤⣼⣷⡜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣟⡛⠟⠙⣏⠹⣷⣷⣿⣷⣾⡟⢡⠏⠀⠀⢺⣛⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⠃⠐⢤⡽⠶⠿⠛⠛⣿⡿⡛⢿⡒⠴⠀⠈⠏⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠃⠁⠁⠈⠋⠉⠉⠀⠻⣦⠀⠀⠈⠂⡀⠈⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢄⠀⠀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡜⣀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣯⠀⠴⠳⢦⣠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡤⢤⡆⣠⡸⣆⠁⠀⠀⠀⠀⠀⠹⣶⡀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁⠠⠄⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⣼⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠐⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠀⠀⠀⠠⣿⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⠈⠀⠀⠀⠀⠀⠀⣀⡄⣲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡌⠨⠃⠀⠀⠀⠀⠀⠀⠠⠌⣱⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠏⠁⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡶⠖⠤⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⡁⣐⠒⠂⡙⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⢬⠉⠙⠉⠩⡀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣧⣄⠁⠀⠈⠀⠀⠈⠛⠉⠀⠀⠀⠀⠒⠒⠂⠀⠀⠁⠀⠀⠀⠀⠰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠀⣀⣄⠀⠀⠀⠀⡈⢈⡀⠆⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠀⠛⠁⠐⡀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠃⠀⠀⠋⡈⠁⠀⠀⠒⠠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠦

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⢠⣤⣤⣄⣠⣤⣤⡀⠀⣠⡶⣦
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠛⠀⠀⠀⠀⠙⢶⣦⠄⠉⠀⣀⢠⡀⠀⠀⡀⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡈⠃⠀⠀⠀⠀⠀⠀⢙⡣⡄⢲⣅⡀⠀⠀⠀⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⠀⠀⠉⠁⡉⠙⠂
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠄⠀⠀⢰⣿⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠘⢿⡿⠀⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢸⡇⠀⠐⠆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⢰⠀⠆⠀⢰⠀⠀⢀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡀⠀⠀⠘⢥⠀⣘⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠈⠀⣿⣿⣷⣦⠔⠀⠀⠄
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡀⠾⠇⢠⠀⠈⢿⣿⣿⣶⣌⡁
⠀⠚⠛⠙⠻⣿⡿⣿⡿⡟⢿⣿⢟⣻⣿⡟⠛⢻⡟⠛⠛⠟⠛⠛⢦⡄⠀⠀⠈⠀⠀⠈⠛⠻⠿⠿⠿⣶
⠀⠀⠀⠀⠀⢻⠚⣿⣿⠃⢺⣿⠋⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⡆⢻⡇⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡍⠁
⠀⠀⠀⠀⠀⣼⠁⢸⡇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠃⠐⢿⣧⣀
⠀⠀⠀⠀⠀⢻⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠁⠀⠀⠀⠀⠛⢯⣻⣦
⠀⠀⠀⠀⠀⡄⠀⠂⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡆⠀⡈⢿⣿⡆
⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠃⣀⡀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡏⢉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⢀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣐⣆⣸⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠂⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠈⠣⡒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠮⣍⡙⠒⠦⢬⣦⡀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠒⠚⠛⠦⣄⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣚⡩⠭⠤⠤⡀⠀⠀⠀⠀⠳⡺⠷⠄⠀⠀⠀⠈⢧⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⡴⠊⠁⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡞⠀⠀⠀⠀⣠⣀⡴⠃⠀⣀⣉⣉⠉⠉⠂⠈⠑⠦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡟⢸⠁⠙⢆⠙⠦⣀⣀⣠⠽⠋⢴⣊⣁⡀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⡲⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠓⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘⡱⠃⠀⠀⠀⠀⢀⠃⠀⠀⠀⡠⡔⢳⠀⠀⠀⠀⠾⠅
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣠⢟⡼⠀⠀⠀⠀⠣⡵⠚⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣱⣦⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢄⣀⣠⠖⠁⢸⠼⠄⠀⠀⠤⠴⢡⢊⣹⡏⢒⠆
⠀⢠⡖⢄⠀⢀⡀⠀⢺⠀⠀⠀⠀⠉⠦⣄⠀⠈⠒⠊⠱⠀⠀⠀⠀⠀⠀⠀⠤⣄⣤⡉⠲⡄⠸⢲⠒⠒⠒⠐⠐⢄⢉⣴⠣⣹⠂
⢮⡀⠓⠈⠂⢡⢔⡒⠚⡆⠀⠀⢀⡸⠀⠀⠉⠒⠢⠤⢄⣀⣀⡠⠤⠔⠛⠦⣄⣀⣀⣁⣠⠗⠲⠞⠀⠀⠀⠀⠀⠈⠍⡾⢃⠀⢼⡀
⠐⢏⡉⠴⠒⠢⠤⠜⠀⠘⠒⠉⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡨⢟⢏⣉⠟⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢦⠁⣄⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡏⠹⠉⠉⠉⠉⠉⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠆⠀⠀⠀⠲⠗⠰⢧⠤⡀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠷⡹⣽⠒⠤⢀⣀⡇⠀⢸⡀⠀⠀⢀⣀⢼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠭⠥⢒⣒⣒⣒⣒⣸⡄⠀⠀⣏⡉⠉⠉⠀⠀⠀⠉⡠⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⡿⢿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⣰⣿⠇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢿⣷⡀
⢠⣿⡏⠀⠀⠀⠀⠀⠀⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠃⠀⠀⠀⠀⠀⠈⣿⣧
⣸⣿⠁⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⢹⣿
⣿⣿⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠙⠛⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⣄⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⣶⣿⠟⢿⣷⣾⡿⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁
⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⡿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢀⡴⠚⠉⠁⠀⠈⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢚⠉⠁⠀⠀⠈⠉⠉⠓⢦⡀⠀⠀⠰⠋
⠀⣰⠋⠀⠀⠀⢤⡀⣤⡠⣄⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⣸⠃⠀⠀⣈⢳⡈⢳⣌⢿⢿⡗⠄⠀⢧⠀⠀⠀⠀⠀⢀⡞⣀⠀⠀⠀⡀⢀⣤⠀⡴⣠⠞⢧⠀⠀⢸⡇
⡏⠀⠐⣦⡘⢷⣽⣦⣻⡻⣧⠉⠀⠀⢸⠀⠀⠀⠀⠀⢸⡏⠁⣀⣴⡯⠔⠋⠿⠟⠋⠁⠀⠈⢷⢛⠚⠧⣀
⣇⠀⠻⣮⣿⢮⣿⡻⣝⢷⡌⠃⠀⠈⠳⡀⠀⠀⠀⠀⠈⣗⠋⢉⣥⣀⡀⠀⠀⠀⠀⣀⣤⡤⢸⢸⠀⠀⢸⠀⠀⠠⠖⠶⠴
⠹⡄⠐⢽⡝⢷⡙⠷⠈⠣⠀⠀⠠⣄⡀⠙⢦⡀⠀⠀⠀⠻⡆⠀⢠⣤⡅⢀⡇⠀⠈⢉⣀⠀⡆⢸⡠⢄⣸⠀⠀⠀⡀⢀⣀
⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠁⠉⠓⠢⠳⠀⠀⠀⠀⡇⠀⠈⠉⠁⣸⠀⠀⠀⠙⠛⢠⠋⠉⠒⠀⠻⣄⠀⠀⠈⠉
⠀⠀⠀⠈⠙⠓⠒⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⡎⠀⠘⢦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢠⠴⢛⡍⠛⠦⣠⠘⣄⣠⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡑⠂⠀⠀⠀⠀⠀⣁⠜⠁⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡓⢲⠒⠒⡖⠋⠁⠀⠀⠈⡆⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡇⠀⠀⠀⢸⡤⢄⡀⠀⠀⡋⣉⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠉⢇⠸⣄⠀⠀⠀⣀⠇⢰⠋⠑⠲⡧⠔⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⢦⣀⢽⡒⣛⢥⡴⠃⠀⠀⠀⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠇⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⠀⠀⠀⠀⢸⣘⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⣀⠤⠤⠤⣾⠀⠀⢧⠤⠄⠀⠀⠈⡏⠓⠤⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⢇⠀⠀⢨⠆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⣿⠀⠀⠀⠀⠀⠀⠀⠳⡴⠁⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣄⣀⣀⣀⣀⣆⣀⣠⠤⠤⠔⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⣿⣿⣿⡟⠉⠀⠀⠙⢿⣿⣿⣿⣿⠷⠶⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠛⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⣸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⣤⣿⠏⠉⢿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣦⣤⣤⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣿⣿⣿⣿⣄⡀⢀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠴⣶⣾⣿⣿⣿⡿⠋⠉⠉⠻⣿⣿⣿⣷⣶⡦⠄⠀⠀⠀⠀⠀⠀⢈⣷⡴⣿⣿⣿⣿⣿⠟⠋⠙⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡁⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠟⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣄⣀⣀⣤⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣿⣿⣿⣿⣦⣄⣠⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣿⣿⣿⠟⠛⠛⠿⣿⣿⣷⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⣀⣨⡿⠛⣿⣿⣿⣿⠟⠉⠙⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠃⠀⠀⠀⠀⢹⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⢸⣿⣿⣇⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀⠀⠀⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣼⣿⣿⣿⣦⣄⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣿⣿⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣶⣿⣦⣼⣷⣤⣄⣀⣠⣶⣶⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⡀⠀⠀⠀⠸⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⣠⠤⠶⢤⡤⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⣨⠿⠿⣿⣿⣿⣿⣷⣄⠀⠀⣴⣿⣿⣿⣿⣿⡿⠿⣧⡀⠀⠀⠀⢀⣿⣷⡀⠀⠀⠘⣇⠄⠀⠀⠀⡠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⡾⠁⢀⠴⠞⠛⢿⣿⣿⠏⠀⠀⠘⣿⠿⠿⡿⠛⡦⡄⠀⢹⡄⠀⠀⠊⠁⣴⣩⡷⢖⣋⣁⣀⡀⠀⠀⢀⣸
⠀⠀⣀⣠⠶⠶⢦⣄⠀⠀⠀⡴⠚⣛⠛⢧⡀⠀⠀⠀⡇⠰⡇⢠⣄⣿⢂⠇⡇⠀⠀⠀⠀⢹⠀⠘⣿⣶⣿⠇⠀⢸⡇⠀⠀⠀⠸⠟⠉⣹⠋⠉⠈⠷⣤⣤⣼⠿⠁
⣰⠟⠉⠉⠀⠀⠀⠻⠖⠖⠚⣧⣴⠟⢳⠈⠁⠀⠀⠀⠹⣄⣙⣲⣶⣚⣋⣸⠃⠀⠀⠀⠀⠘⣷⡤⠀⠉⢀⡠⢔⡿⠁⠀⠀⠀⢠⡤⠞⠃
⠘⢶⠖⠀⠀⠀⠀⣠⡴⠚⠙⣇⠘⠆⠋⣠⠄⠀⠀⠀⠀⠈⠛⠒⠒⠛⠛⠁⠀⢀⣠⣤⣤⣀⠈⠛⠲⠶⠛⠉⠈⠀⠀⠀⠀⠀⢸⡆
⠀⠻⣦⣀⡤⠀⠀⢀⡿⠀⠀⠈⠛⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠒⢭⣁⣀⣈⠉⠛⠻⠿⣶⣦⣀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠖⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⠴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠉⠉⠉⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢰⣶⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⣼⣿⣀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠋⠛⠃⠀⠀⠀⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠀⠐⠁⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⢀⣠⣴⠿⠛⠛⠛⠛⠛⠛⣿⠟⠀⠀⠀⡶⠻⣿⣶⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠛⠛⠛⠻⠻⢶⣤⡀
⣤⣶⠟⠉⠀⠀⠀⠀⠀⣤⣤⣤⡏⠀⠀⠀⠀⠖⠀⠈⣿⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⡄⠀⠀⠀⠈⠙⢷⣦
⠈⠛⠷⣦⣀⠀⠀⠀⠀⠈⠉⢹⠇⠀⠀⠀⠀⠈⣄⠀⠀⠀⠉⠉⠉⠀⠉⠙⢶⡄⠀⠀⠀⠀⢫⠉⠉⠁⢀⠀⢀⣠⡶⠟⠋
⠀⠀⠀⠀⠙⠻⠿⠶⠶⠶⠶⣿⠀⠀⠀⠀⢠⠰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢈⢷⠾⠶⠾⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠠⠄⠀⡠⠂⠀⣀⣬⡆⠀⠀⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠞⢹⣷⣄⣀⠀⠀⠀⠀⠀⣠⡾⡏⡸⣿⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣄⠀⢊⢠⣿⠻⣿⠆⠀⣠⣿⣻⠀⢠⠇⠙⠀⠀⠀⠀⢆⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠘⣆⠃⠀⠈⠒⠐⠊⠏⢀⠀⡙⠁⠉⠁⠞⠀⠀⠀⠀⠀⠀⠸⡆⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡇⠀⠀⠀⠘⣆⠁⠀⠀⠀⢀⠀⠀⠀⢣⢀⡀⠀⢃⡇⠀⠀⠀⠀⠀⠀⣯⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠇⠀⠀⠀⠀⣿⡄⡀⣀⣠⣼⣦⡀⠀⢸⣎⣁⣠⢸⡇⠀⠀⠀⠀⠀⠀⣶⠀⡄⢸⡏⠢⡀
⠀⠀⠀⢀⡤⠂⠁⡌⠙⠀⠀⠀⠀⠀⠀⣰⠟⣹⠇⠈⢷⣮⣭⡿⣶⣾⣫⡟⠉⡃⠃⠀⠀⠀⠀⠀⠀⡏⠀⡀⢸⡇⠀⠘⡆
⠀⠀⠀⡎⢸⠀⠀⢣⠀⠀⠀⠀⠀⠀⣨⣴⣾⣿⠀⠀⠀⠙⠿⢿⣿⠿⠛⡀⣰⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⣷⣿⡁⠀⠇⡇
⠀⠀⠀⠘⢆⠀⠀⠀⢣⡀⠐⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠲⠤⠤⠴⠊⠀⡿⠀⢠⠁⠀⠀⠀⢠⠏⠀⢀⣿⡿⠁⠐⣰⠃
⠀⠀⠀⠀⠀⢧⠀⠀⠀⢷⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣿⣿⣦⣾⠁⠀⡎⠀⠀⠀⢠⠏⠀⠀⢸⠟⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⣇⠀⢠⣻⡛⠿⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⡰⠋⠀⠀⡰⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠘⠳⣦⣄⣽⠀⠀⠹⣿⣦⣀⡀⠀⠉⠉⠁⠀⠘⢿⣿⣿⣷⠀⢸⣷⣤⡞⠁⠀⡠⠊⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⣤⣀⣼⣿⣅⠀⠀⠢⢤⣀⠀⠀⠀⠈⠉⠹⠀⢸⣿⡟⣠⠖⢊⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⠭⠑⠊⠛⠉⢻⠶⠔⢺⣦⣄⣤⣠⡶⠀⣾⣿⡧⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⣿⠈⠻⣭⡥⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡠⠀⠀⠀⠀⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠒⠂⠀⠀⠐⠒⠒⠒⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⣾
⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⢠⡟
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠀⣸⠃
⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠶⢶⣤⣤⣀⣀⣀⣀⣀⣤⡴⠶⠶⠖⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣧⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⠇⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⣠⣴⠶⠶⠚⠛⠛⠛⠛⠛⠛⠶⢶⣤⡀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣾⣋⣁⣀⣀⣤⣤⣴⠶⠶⠶⣤⣤⣀⣀⡀⠙⣷⣄⣤⡾⠋⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣓⣒⣦⣔⣶⣤⣴⣶⣞⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡼⠶⡄
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣰⠏⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⣠⡴⠶⡶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠉⠀⣰⠏⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡾⠁⠀⢹⣄⣀⡀⠀⠀⠀⠀⠀⢸⠇⠀⠘⠁⢀⡼⠧⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡼⠁⠀⡴⠋⠁⢀⣿⣀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠘⠁⠀⣀⡴⠟⠀⢈⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣁⣀⣀⠀⢀⠞⠉⢠⢼⣷⣤⠾⣿⠛⣿⣇⠀⠀⠀⠀⠀⠉⢀⣤⠶⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⣉⣀⣯⡤⠿⣟⠉⠉⠀⠀⢀⣤⠤⠤⠤⠶⠖⠛⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡴⠟⠳⠀⠀⢰⠏
⠀⠀⠀⠀⣰⠏⠀⠀⣤⣄⣠⣶⣦⠀⠀⠈⢷⣄⠀⠀⣄⠀⠀⠘⠋⠻⠂⠀⠀⠀⠀⡿
⠀⠀⠀⢠⡏⠀⠀⠀⢙⣿⣿⣿⣅⠀⠀⠀⠈⠻⣷⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⢸⠁⠀⠀⠀⠻⠟⠛⠛⠛⠀⠀⠀⠀⠀⢹⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠏⠀⠀⠉⠉⠛⠉⠉⠹⡆
⠀⠀⠀⢸⣧⠀⣠⡴⠞⠛⠓⢦⡀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣀⠀⠈⠛⠓⠚⠛⡇
⠀⠀⠀⠀⠹⠿⣯⣤⣤⣤⡀⠈⣷⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⢈⡷⢦⣐⡛⠓⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠟⠀⠀⠀⠀⠀⣀⣠⡤⠶⢾⡉⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠚⠛⣏⠹⡄⠀⢦⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣽⡶⠶⠟⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣾⣿⠂⠀⠀⠀⢨⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣷⣦⡄⠀⢀⣀⡀
⠀⠘⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠘⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⣿⢿⠇⠀⢸⣿⠇⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⡀⠉⣄⠉⠈
⠀⢀⡀⢺⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠈⣿⣶⡆⠀⢰⣶⡆
⠀⠀⠀⣻⡄⣀⣤⢀⡃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠐⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠿⠿⠇⠀⠨⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠸⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣀⣀⡿⢿⣿⠇⠀⠀⠀⠰⣦⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢹⡿⠳⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⢰⡄⠀⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢸⣄⠰⢶⣶⠖⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⠿⠾⠗⠛⠛⠉⠉
⠀⠀⠀⢸⣇⣀⡼⠦⠷⠶⠟⠛⠃⠀⠀⠄⢸⡟⠶⢀⡁⠒⠿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⣀⢀⠀⢀⠀⠀⢀
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠴⠾⠷⠦⠌⠀⠀⠀⠀⠀⡟⠻⣄⠀⡠⠘⠃
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣆⣠⠼⠓⣤⣀⠀⠀⠀⠀⠀⢳⣤⣤⡤⣤⡄
⠀⠀⠀⠸⠻⡟⠀⠀⠀⣤⣠⣀⣀⠀⠀⠀⠸⠛⢣⣠⠛⢃⣠⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠛⠃⠀⠀⠀⠀⢸⣿⣿⡗⣿⡇
⠀⠀⠀⢰⡀⣤⠄⠀⠀⠛⠉⠈⠉⠀⠀⠂⢸⠷⣏⣉⠶⢆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣿⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠛⠀⠘⠃⠀⠀⠀⠀⠀⣷⠶⠖⠶⠶⢶⡆⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⡄⠀⡀
⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣀⢀⣀⣀⠀⠀⠀⠰⠀⣿⠀⠿⡿⠇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⠀⠀⠈⠁⠐⡀⢀⣀⠄⠀⠀⠀⠀⠀⠉⠿⢘⣿⣷⠀⠀⠀⠀⠀⠿⠛⠿⠿⠿⠿⠇⠀⠀⠀⠀⠒⠀⠀⢠⣤⣤⣤
⠀⠀⠀⠘⠀⠀⠐⠋⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣸⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⣤⣤⠀⠀⠀⠠⠀⣧⣤⣴⣿⣤⣤⡄⠀⠀⠀⠄⠀⠄⠤⠀⠀⠠⠠⠀⠄⠄
⠀⠐⠀⠀⠀⠀⠀⠀⣬⠛⠛⠛⠓⠀⠀⠀⢸⣿⡋⠉⠉⢹⣿⠀⠀⠀⠀⠀⡆⠀⢈⣯⠀⠀⠀⠀⠀⠐⠀⠀⠆⠀⠀⠀⠐⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣠
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠉⠉⠉⢹⣿
⠀⠀⠀⠇⠀⠀⢸⠀⠹⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⡠⠊⢀⠾⠃⠀⠀⠀⠀⣿⣿⣶⣶⣶⣾⣿
⠀⠘⠀⡆⠀⠀⢸⣆⣴⡇⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⠀⠀⢁⠴⠀⠀⠀⢀⡀⢉⠀⠀⢀⠀⡀⠉⠀⠀⣀
⠀⠀⠀⠃⠀⠀⠘⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠁⠐⠋⠘⠃⠀⠀⠀⠀⠀⠀⠤⠀⣄⢁⠀⠀⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⡇⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢾⠆⢾⠆⢾⠀⠀⠀⠀⠀⣿⣄⠀⠀⢀⣼⠆⠀⠀⠀⠀⡇⠀⣤⣤⣤⡄
⠀⠀⠀⡇⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠃⢸⡄⣠⠀⣴⠀⣤⠀⠀⠀⠀⠀⣿⣿⡿⢶⣿⣿⡇⠀⠀⠀⠀⡇⠘⠛⠛⠛⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠛⠀⠛⠒⠛⠀⠀⠀⠀⠀⠛⠛⠚⠛⠚⠛⠃⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⢀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠀⠠⠠⠤⠄⠤⠄⠤⠀⠀⠄⠤⠤⠀⠤
⠀⠀⠀⠀⠀⠒⠀⠀⠸⠶⠲⠀⠀⠒⠀⠀⠰⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢾⣿⡷⠀⠰⠂⠀⠰
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠹⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣶⣦⣶⣶⣿⣿⠀⠀⠀⠀⠀⢠⡀⠀⡠⠀⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣷⡄⣀⠀⠀⢿⣿⣷⣿⣾⣿⣿⣿⠀⠀⠠⡄⠀⠀⠙⠋⠁⠀⠀⠠⠀⢾⢴⡿⠃
⠀⠀⡀⠀⠀⣠⡀⠀⠀⠀⣄⠀⠀⡀⠀⠈⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠤⣶⣶⣦⣦⣤⡀⠀⠀⢈⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠐⢧⡓⠁⠀⠀⠀⢛⣼⠂⠀⠀⠀⠈⠛⠁⠋⢀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⡸⠻⢿⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣰⣧⣄⠀⠁
⠀⢀⡀⠀⢠⣤⣤⣤⣤⣤⣴⣶⠀⢈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠺⣞⠙⢿⣿⣷
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠈⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿
⠀⠠⢄⠄⠀⠀⣠⣶⣦⣤⣦⡄⠀⠀⠀⠀⠀⠀⢡⠈⢀⣴⣦⡀⠀⢀⣀⠀⠀⢀⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠘⠋
⠀⠀⠀⢀⣼⣿⡿⣟⠍⢿⣿⣧⡀⠀⠀⠀⠀⠀⠘⠀⣿⣯⣿⡇⠀⣿⣿⡇⠀⠙⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⡀
⠀⠀⢠⣾⢟⣻⣜⢘⢧⣶⣿⡿⠟⠒⠄⠀⠀⠀⠀⠀⠉⠛⠋⢰⣶⠈⠉⠀⠀⠘⠄⠀⠠⢰⣿⠋⠀⠀⠀⡁⠀⠀⠀⠀⢀⣦⡽⠤
⠀⠀⠘⣿⣮⣛⣼⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣤⣄⢠⣴⣄⣿⡆⠀⠁⠀⠁⣼⡟⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠘
⠀⠀⠀⠙⠛⠉⠛⠋⠀⢀⠀⠀⢸⣷⣤⡀⠀⠀⠀⠀⠹⠿⠉⠟⠿⠍⠿⠟⠁⠀⡆⠀⢡⡿⠁⢀⡀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡷⠀⠀⠘⠀⠀⠐⠄⠐⠀⠀⠂⠀⠐⠀⢀⠞⠀⠀⠘⡗⠀⠀⠀⠀⣰⠋⠀⣠⠤⠄⠀⡠
⠀⠐⢫⠦⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠉⠓⠀⢧⡤⢂⡴⠁
⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠰⣦⣼⢂⣠⠀⣿⡄⢶⢸⡆⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⡔⠁⠀⠀⢸⣧⠀⠀⠀⠱⡀⠀⠀⠀⠀⢈⡏⢸⣤⠰⡏⠣⢸⡏⣧⠈⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡴⠆
⠀⠀⠀⠀⠁⠀⠀⠀⠹⡿⠀⠀⠀⠀⠁⠀⠀⢠⡄⠘⠁⠈⡧⠤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢀⠂⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢈⡲⠀⠀⠀⠀⠁⠀⠀⠀⠒⣊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠈
⠀⠐⠀⠘⠃⢠⣴⣦⣤⣤⣤⣴⣦⡀⠛⠀⠐⠀⠀⠀⠉⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⢸⡿⠟⠀⠀⠈⠿⣿⠇⠀⠀⠀⠀⠀⣰⣿⢻⣟⣯⢿⣿⣿⢻⣟⣿⣆⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢠⠀⠤
⠀⢠⡄⠀⠀⠀⢠⡀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⢿⣿⣮⣷⣹⣾⣿⣿⣾⣿⡿⠃⠀⠀⠀⠀⠀⠉⠒⠂⠀⠒⠂⠁⠀⠂⠠⢈⠐
⠀⠀⠈⠀⠀⠀⠘⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢹⡆⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣗⠀⠀⠀⠀⠀⠀⢸⡇⠀⡤⣰⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⢸⠃⠀⠈⢽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣀⣿
⣠⣶⣤⣄⠀⠀⠀⠀⠀⠀⢸⠀⢀⣶⣿⣧⠀⠀⠀⠀⠀⣾⠀⠀⠀⣂⣿⠆⠀⠀⠀⠀⠀⠀⢀⡾⠀⢀⣀⣳⣿
⢟⠁⠈⣿⡆⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⢐⣿⠇⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⣾⣿⠇
⢸⡀⠀⢼⣿⡀⠀⠀⠀⠀⢸⡀⠀⠀⠰⢿⡆⠀⠀⠀⠀⡇⠀⠀⠀⠁⣿⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢠⣼⡟
⠈⣷⠀⠾⣿⣷⠀⠀⠀⠀⢸⡇⠀⠀⢴⣿⣇⠀⠀⠀⠀⣇⠀⣤⡴⣋⣿⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡂⣽⡇
⠀⣿⠀⠀⠠⠽⣧⠀⠀⠀⠈⣷⠀⠾⠿⠿⣿⠀⠀⠀⢰⠇⠀⠀⠈⠙⣿⡄⠀⠀⠀⠀⣸⠃⠀⠀⢒⢼⡿⠁
⠀⢹⡇⠀⠈⡻⣿⡇⠀⠀⠀⡇⠀⠀⠀⢈⢸⡆⠀⠀⣼⠀⠀⠀⠀⢍⣺⡇⠀⠀⠀⢠⠃⠀⠀⣈⢡⣼⡇
⠀⠀⢿⠀⣀⣾⡿⢿⡄⠀⠀⡇⠀⠀⠀⠠⠸⣧⠀⠀⡿⠀⠀⠀⠀⢀⣿⡇⠀⠀⢀⠏⠀⠀⠀⠀⠿⣿
⠀⠀⠘⣇⠉⠀⠰⢄⢿⡀⠀⡇⠀⠀⠀⠐⠺⢿⡀⠀⡇⠀⠀⠀⠀⠘⣸⡇⠀⠀⡞⠀⠀⠀⠀⡀⢐⡇
⠀⠀⠀⢻⡀⠀⠀⠈⢺⣿⣤⣇⠀⢀⡀⠰⠟⣾⣿⡾⠅⠀⣤⣤⣤⣋⢹⣇⣤⡜⠀⠀⠀⠀⠀⠌⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣷⡀⢀⣶⣿⠟⠛⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠯⠻⠵⠶⠀⠐⠂⠄⣱⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠓⠈⠉⠉
⠀⠀⠀⠀⢸⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⣀⡤⠶⠿⠤⠤⠀⠐⠒⠠⠤
⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠⣿⠀⣠⡴⠖⠋⢁⣀⠀⠐
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠒⢀⣿⣻⣅⣀⣀⡀⠀⠀⠈⠀⠀⢀⣀⣤⠤⠴⣤⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣀⣤⣤⣤⣤⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⣀⡠⠀⠀⢀⣀⠀⠀⡤⠠⢸⣏⣩⣿⣿⡍⠉⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⡈⣿
⠀⠀⠀⠀⠀⢹⣇⠀⠀⢨⠡⠒⠋⠉⠀⠀⠀⠀⠀⢀⣀⡠⠤⠚⠉⠁⠀⠀⣠⠈⠉⠀⡀⠀⠀⣽⡇⢉⡿⣿⡿⢦⡀⣲⠟⠁⠀⠀⠀⠀⡀⢠⣶⡏
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⡠⠄⣒⡾⠟⠉⠀⠀⠀⠀⠀⣠⠄⠈⢠⠒⠀⠉⣩⣤⠬⣿⣧⠰⢟⠛⣙⣠⡾⠃⠀⠀⠀⠀⠀⠀⣡⡼⠋
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠠⠖⠈⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠠⠀⠀⠀⣠⠞⠋⠀⠀⠀⠈⠉⢿⡶⠋⠁⠀⠀⠻⣄⠲⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⢰⠎⠀⣠⡾⠓⠉⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⢀⠀⢀⣿⠰⡏⢀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⡒⠟⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣴⡟⣯⢢⠘⡀⠀⠰⠸⡄⠰⠀⢧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠮⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⣴⣟⢧⢴⣶⣆⠆⠁⠘⠀⠀⢻⠀⠠⡌⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣘⣻⠟⣇⢻⢸⡆⣿⣿⢰⢸⢀⠃⢸⠘⡂⠂⢺⠙
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣤⡄⣮⣿⣿⠀⡏⢹⠱⡿⣿⣹⢠⠘⢨⢸⠀⡀⠷⠰⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣯⣷⣿⡟⠹⣿⠈⡇⣼⢸⡇⠟⠘⣿⢠⢸⢸⠀⡇⠀⡀⣸⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⠄⠀⠀⢀⢀⠰⢈⠰⠸⣶⣿⡿⠋⠀⢠⡟⢸⢡⡇⣸⠁⠀⢀⡏⠀⠸⡸⢠⡇⠀⢁⡛⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠘⠂⠆⣠⡷⠖⣾⣿⣿⣤⣤⠖⢋⣠⣤⣶⣾⣿⠟⢁⡀⠀⢀⡾⠁⢆⡞⡰⢁⢀⠄⣼⠁⢀⠆⠀⠘⢀⣀⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣋⣘⠛⠻⠿⠿⣿⣟⢻⣿⣿⣿⣿⣿⡿⠟⣛⠥⠒⠀⠀⠀⢀⡾⠁⢀⠎⠄⡰⢣⠎⡼⢁⢀⢎⣠⡶⠗⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⢶⣦⣤⣀⣌⣿⣿⣍⡓⠀⠉⠀⠀⠀⠈⣀⣠⡴⢋⡴⠀⡡⠂⠈⣰⢋⣞⣡⡦⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿⠟⠋⠁⠔⠋⠴⣋⣤⣴⠾⠕⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⣍⣉⠛⠻⠿⠿⠛⠛⣋⣉⣠⣤⣤⠶⠾⠛⠉⠋⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣾⣿⣿⣿⣿⣿⣵⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⠉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠈⠯⡑⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢌⡉⠙⠓⠦⢬⣦⡀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⣄⡀⠀⠉⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠿⠳⢄⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠛⣋⡁⠀⠀⠀⠀⠀⠀⠤⠉⣦⣀⠀⠀⠀⠀⠘⣧⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⣉⠴⠚⠉⠉⠉⠉⠙⡄⠀⠀⠀⠀⠀⠀⢄⠉⠀⠀⠀⠀⠀⠈⠿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠄⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡀⣀⠀⢸⡀⠀⠀⠀⠀⣴⣦⠼⢁⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣤⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢺⠀⠈⢳⠀⠙⠦⠤⠤⠴⠚⠁⠰⠯⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡝⠁⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⣾⣡⠇⠀⠀⠀⠀⠣⣠⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠋⠈⣅⣆⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄
⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠌⠀⠀⠈⠉⠙⠋⡉⠁⠈⠁⠀⠄⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃⢄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⢀⢺⠄
⢀⡤⠘⣦⠙⣄⠔⢂⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣀⠀⠀⠀⠙⠀⠀⢃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⢂⣵⡖⠓⠢
⠀⢑⡢⠤⠀⣀⡐⠋⠐⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠛⠋⠉⠁⠀⠈⢻⠖⠒⠒⠒⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⡌⣅⢤⠤⣭⡦
⠀⠈⠂⠐⠊⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠛⡌⢢⡔⠃⠈⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠐⠤⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡴⠄⠤⣤⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣮⢹⡅⣤⣄⠀⠀⠀⡄⠀⠀⡇⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠴⠇⠉⠉⠁⠀⠀⠉⢲⡇⠀⠀⢳⠦⠤⠶⠒⠋⠉⠑⠂⠐⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠒⠒⠂⠀⠀⠁⠀⠀⠈⠒⠢⠤⠤⠄⣀⣀⣠⠔⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠅⠃⠈⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡄
⠀⠀⠀⠀⠀⡆⠀⣧⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣴⠛⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡇⢀
⠀⠀⠀⠀⠀⠙⣦⠘⣷⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⣷⠀⠢⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡾⠀⣸
⠀⠀⠀⠀⠀⠀⢸⠀⣼⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠙⣾⣇⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⣰⠃
⠀⠀⠀⠀⠀⠀⠋⢰⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⣏⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢧⡀⠙⣦⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⣦⡢⠔⣣⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠳⠄⢰⡄
⠀⠀⠀⠀⠀⠙⢶⡄⢹⡆⢳⡄⠀⠀⠀⠀⠀⠀⠀⢼⠉⠉⢻⡻⣼⡧⣂⡽⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢀⠀⣟
⠀⠀⠀⠀⠀⠀⡼⢱⠞⡵⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠀⠃⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⠃⠈
⠀⠀⠀⠀⠀⠀⠉⠈⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄


⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠘⢦⡐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡀
⠀⠀⠀⠀⠀⢠⡀⢻⡆⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡅⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⠇
⠀⠀⠀⠀⠀⠀⢹⡄⡿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⡜⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠸⠃⢃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠁⠀⠀⠀⠀⠎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠹⣄⠡⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡈⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠔⠁
⠠⡄⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣶⠞⠁
⡆⢰⠀⠀⠀⠀⠀⠀⠀⣸⠉⠛⠿⠷⠶⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠑⠈⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠘⣜⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡁
⠀⠀⠘⣮⠢⡀⠀⠀⠀⢿⡀⠀⠀⠀⢠⡥⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠎
⠀⠀⠀⠈⢷⡄⠀⠀⠀⢸⡇⡄⠀⠀⡟⠀⠀⢀⣠⣴⠶⠶⠶⣶⣤⡀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣠⠞⢡⡾
⠀⠀⠀⠀⠀⠘⢧⡀⠀⢘⣇⢳⡄⢸⠃⠀⠔⠋⠁⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⡇⠀⠠⠈⣡⠶⠩
⠀⠀⠀⠀⠈⠀⠘⡇⠀⠀⢻⡄⠹⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢻⣿⣶⡆⠁⠀⠀⠀⠀⠀⡎⠀⠀⠀⣀⣠⡄⠀⠷⠀⢀⣼⠃
⠀⠀⠀⠀⠀⡀⠀⠁⠀⢹⣮⣷⠀⣿⠀⠀⠀⢀⣠⠔⢋⡀⣀⣀⣀⠉⢢⡀⠉⢿⠁⠀⠀⣀⣀⣀⣀⠀⣴⡶⠟⡉⢀⠀⠀⢰⡄⡎⠀⠀⡰
⠀⠀⠀⠀⠀⠱⡀⠀⠀⢸⣿⠏⢻⡏⠀⠀⠀⠀⠈⠃⠠⠄⢻⣶⣋⣸⡁⠲⠀⠀⠁⠒⠉⠀⠀⠈⢻⠞⢛⢊⡥⣤⠀⠙⠀⣼⢣⠀⠀⣎
⠀⠀⠀⠀⠀⠀⠙⢄⠀⠈⣿⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠔⠀⠀⠀⠀⠀⠀⠀⢀⡞⠐⠀⠤⠙⠋⠤⠀⢀⡉⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⠓⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠄⠀⠓⠤⠤⠤⠴⠂⠀⣷⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⢀⣸⣇⣠⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⢦⠀⠀⠀⠀⠀⠾⠶⣤⡀⠀⠀⠀⠀⠀⠀⢸⠆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠚⠁⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⢀⡆⠐⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢹⣯⣐⠂⠀⠀⠤⠴⠒⠒⠉⠉⠉⠓⢤⣤⡀⠀⠀⠀⠀⢰⣨⠘⠁⣀⣵⣤⣄⣴⡟⠀⠀⡁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣿⠉⠛⠟⠛⠛⠛⢻⣿⡛⠻⣶⣤⣤⣿⣷⣄⣀⣀⣠⡾⢳⣶⣾⡟⠋⠠⠏⠙⠛⠳⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠀⣄⠉⠐⠎⢛⡋⠈⣿⣿⣿⣿⠋⠀⡀⠙⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄⡈⠳⣄⠀⠀⠀⠙⣿⣿⣿⠏⠈⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠂⠀⠀⠙⠲⣬⡳⢄⡀⠀⠙⠿⠋⠀⢀⠔⠁⠀⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠀⠀⠀⢦⡀⠀⠙⠲⡍⠛⠀⠀⠃⠀⢀⠌⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠰⢄⡈⠒⠦⠴⠆⠘⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⣀⣤⣴⣾⣿⣿⣷⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣁⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢲⡀⢰⡆⠀⠀⠀⢰⣰⡇⣴⠀⠀⠀⠀⢠⡀⢰⢀⡄⠀⠀⠀⠀⣤⢀⡆⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⢠⣴⣼⡇⠀⠀⠀⠀⣘⣧⣼⠁⠀⠀⠀⠈⣿⣧⡟⠀⠀⠀⠘⣆⣧⣼⣾⠃⠀⠀⠐⣆⢹⣸⣷⠇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠈⣿⣿⣿⣄⣴⠖⠀⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠘⣶⣿⣷⣠⡾⠋⠀⠀⠀⠰⠞⣛⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⣀⣈⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⡾⠋⠛⣿⣿⣷⡀⠀⠀⠲⣄⣴⣿⣿⣦⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⣦⣴⣿⣿⣿⣷⠀⠀⠀⠈⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⢿⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣾⠿⠛⠛⠛⠛⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⡶⠶⠶⠆⠀⣀⣨⣿⣿⣦⣤⣤⣀⣀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⣿⣆
⠀⠀⠀⢀⣄⡚⠋⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡿⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀
⠀⠀⠐⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⣾⣿⣿⣿⠿⠿⠛⠛⠃⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋⠉⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡄
⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠈⠙⠻⠿⠷⢦⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣿⣿⣧⡀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⢻⣿⣿⣶⣦⡀⠀⠀⣠⣶⣾⣿⣿⠋⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⣿⣿⣿⡿⠁⠀⠀⠀⠉⠙⠻⣿⣿⠀⣸⣿⡿⠛⠋⠁⠀⠀⢶⣾⡿⣿⣿⣿⣶⣶⣿
⠀⠀⠀⠀⠀⣼⣿⣿⡿⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⢹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⣿⠿⠿⠿⠟⠛
⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠈⣿⣿⣷⣄⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⣿⣇
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⣀⣤⡶⠶⠟⠛⠛⠻⠶⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⣰⠟⠁⠀⠀⠀⠀⠀⢀⡀⢠⣄⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣿⣷⣶⣦⡀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢀⣠⡤⠤⣤⣄⣀⠀⣀⣤⠴⡖⠒⠶⢤⣀
⣿⠁⠀⠀⠀⠀⠀⡀⠘⢷⣦⡙⢿⠻⣦⡙⠃⠀⠀⠀⢸⡇⠀⠀⠀⣰⠋⠀⠀⠀⠀⡏⠉⠉⠉⠀⠀⡇⠀⠀⠀⠉⢳⣤⡀
⣿⠀⠀⠀⢠⣤⡈⢻⣷⣬⡻⣝⠻⣷⣌⠃⠀⠀⠀⠀⢸⡗⠀⠀⠀⣏⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢸⠀⠻⠛⠓⢦⡀
⢻⡄⠀⠀⢤⡙⢿⠳⣽⣧⡙⢿⣷⡈⠛⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣸⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⣸⢻⡄
⠈⢷⡀⠀⠀⠉⠻⢷⠈⠙⠛⠀⠈⠁⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⣇⠀⠠⣄⣤⠶⢦⣄⣠⠴⠚⠉⠉⠛⢦⣄⣀⣠⣤⣀⡀⠀⠃⣼⠁⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢦⡰⠋⠀⠀⠀⠀⠀⠹⣆⠀⢈⡇⢀⣀⠀⠀⠘⠛⢋⡙⠛⠀⠀⠀⣀⡀⠈⢻⠶⡦⣼⣧⠀⠀⠀⠀⠀⠠⠞⠋
⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⣀⣀⣈⣹⠷⢦⣄⡸⣧⠀⠀⠀⠀⠀⠀⠀⡟⠢⣼⠀⠈⠙⠛⠶⠀⠀⠀⠀⠀⠰⠾⠛⠛⠁⠀⡿⢸⡇⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠻⠿⠆⠀⠀⠀⠀⠀⠸⡇⠀⢸⣀⣀⣴⣶⣶⣀⣀⣆⡀⠀⣠⣶⣶⡆⠀⠀⡇⢸⠁⠀⠀⢸⠇⠀⠀⠠⣄⣀⣴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣸⠉⢏⡀⠙⢉⡏⢹⠈⠉⢻⠁⠙⠋⣹⠖⢸⡇⢸⣀⣤⡀⢸⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⠀⠀⠉⠉⠉⠀⠣⣄⡀⠈⠑⠒⠚⠁⠀⣸⠗⠛⠧⠄⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⣠⡖⢺⠉⡏⠙⡋⠙⠒⣄⠀⠀⡿⠀⠀⠀⠀⠀⠀⢹⠆⠀⠈⠓⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠾⠴⠦⢚⣶⣷⣶⣷⣾⠶⢾⠧⠘⣇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠰⠄⠋⠁⠤⠶⠀⠉⠁⠛⠀⢀⣧⣀⣠⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠈⢻⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⠶⣶⠶⠖⢲⡞⠉⠀⠀⠀⠀⢸⣆⣀⣠⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⡆⠀⠘⡇⠀⠀⠀⠀⠀⢸⣈⡁⢣⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢾⡀⠀⠀⠀⢀⡷⢤⡀⠀⠀⠀⠀⡏⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⢺⠧⡿⣄⠀⢙⡶⠤⣶⡋⠀⣼⢿⠋⠒⠦⢤⣇⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⢸⡋⣧⠈⠙⠉⡇⣠⠀⠈⢹⣏⣿⠀⠀⠀⠀⣿⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠹⡆⠀⠀⠀⠀⢯⡽⣆⠀⠀⣇⠈⠀⣠⠿⣼⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⣉⢷⣄⠈⣡⠾⣩⠿⠃⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠾⣭⣯⠽⠛⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⣧⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠉⠳⢤⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠤⠤⢤⣤⠤⠄⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠳⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⡟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣀⣤⣾⣿⢻⡙⡄⢣⠸⡌⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢠⣾⢿⢻⡱⡈⢆⢳⣸⣬⡶⠿⠛⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⣿⡌⣎⣧⣵⠾⠛⠋⠁⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠰⣚⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣷⡀⠀⠀⠀⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣍⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡑⢌⣳⡿⣿⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⠋⠀⠘⣿⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢠⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣸⣷⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠸⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⢾⠍⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⠿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣽⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢘⣿⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢀⣾⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⣼⡟⣿⡆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣿⡇⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠊⠀⠀⣼⡟⡀⠀⣹⡇⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⡇⠀⢀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣿⡇
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡷⡛⠩⠉⠝⢛⡷⢦⣄⠀⡠⠀⠀⠀⣸⡟⠀⢠⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢻⣧⡀⠀⠀⠀⠀⠀⠁⡙⣧⠀⠀⢀⣼⡿⡷⠶⠦⣄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠠⢼⣇⣤⡾⠋⠀⠀⠀⠘⠨⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣶⣶⣶⣿⠛⠉⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣤⣤⣶⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢀⣀⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⡀⣸⣿⣇⣤⣶⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠛⠉⠉⠛⠛⠉⠀⢿⣿⣿⡿⠋⠀⠀⠀⠀⠙⢉⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣠⣤⣶⣦⣄⠀⣼⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⠁⢀⣴⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢿⣿⡆⠙⢿⣿⣿⣿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⣿⣿⣿⣿⠆⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠻⠁⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢸⣿⣿⠏⢰⡟⠋⠉⠛⣿⣧⠀⣰⠂⠀⠀⢀⢠⣿⡿⠃⢀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⢀⡴⠛⠙⠻⡿⠿⠋⠀⠀⠀⠘⠛⠿⠛⣍⢉⠉⢻⣦⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠎⠐⢶⠗⠀⡄⠀⠀⠀⠉⠁⠀⠀⠢⠀⠙⠋⠀⣰⠋⠀⠀⢸⣿⣿⣤⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⠀⠀⠀⠂⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠀⣾⣿⠇⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠙⠿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡿⣠⡤⠶⢦⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡔⠋⠙⢢⡙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣼⠷⢦⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠁⢠⠖⠚⠷⣇⠀⠀⠀⠀⠀⠀⢧⣀⢠⡀⠀⠀⢠⡄⠠⠜⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠸⣆⡀⣀⣿⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠄⠀⠀⠐⡟⠈⠓⠒⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡉⠉⢸⡆⠀⠀⠀⠂⠀⣴⠖⢻⠛⠛⢻⠓⠓⠶⠶⢶⣦⠀⠀⠀⠰⡿⠓⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣧⠔⠀⠀⠀⢰⡿⡖⠺⡗⠛⢻⣋⣙⣟⣓⣿⣿⠀⠀⠀⣸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣦⠀⠀⠀⠘⢧⡵⠖⠛⢛⡩⢭⣭⡉⠉⠉⠁⠀⠀⢢⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⡀⠀⠀⡆⢩⣀⠀⠀⠀⠀⠁⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⠀⠀⣼⣿⣿⣿⣄⣀⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⣿⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠛⠛⢳⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⡩⠀⠀⠀⠀⠀⠁⠀⠃⠀⠘⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣤⡄⠀⡀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡠⠊⠁⠀⠀⣠⠔⢀⣴⣿⢀⣴⣫⡴⠛⠁⢻⡄⡇⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⣀⣴⣊⣥⠶⠋⠁⡿⠟⠋⠁⠀⠀⠀⠈⢷⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣿⠧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡆⠀⠀⠉⠙⠳⢤⡠⠀⡄⠀⠀⣀⡴⠛⠉⠀⠀⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⢠⣿⣷⡶⠀⢸⠃⠀⠀⠀⣴⣶⣶⠀⠀⢸⠀⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠠⣙⣛⡅⢀⡟⠀⠀⠀⠀⢙⠿⣋⡄⠀⢸⣀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠧⣤⡀⠀⠀⠀⠉⠁⠀⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⡄⠤⠤⠶⠶⠶⠶⠶⠀⠆⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡿⠉⡟⠛⠛⠉⠉⠻⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠛⡿⠁⢸⡄⠀⠙⠀⠀⠀⠀⢠⡇⠈⣿⠓⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠀⠀⢰⡇⠀⠈⢷⡀⠀⠀⠀⠀⢀⡾⠁⠀⢹⠀⠀⠀⠀⣀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠐⡄⠀⠀⠀⠸⡇⠀⠀⠀⠻⣄⠀⠀⣠⠞⠁⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⡄⠀⠀⠀⠷⠖⠛⡇⠀⠈⢳⠾⠋⠀⢸⠛⠺⠃⠀⠀⠀⠌⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠓⠦⣼⣤⡴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠌⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡇⠀⠀⠸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣈⠙⠲⢤⣄⣰⡆⠘⣇⠀⠀⠀⠀⣠⠇⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡤⠴⠚⠉⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠛⠶⠶⣤⣽⡇⠀⠈⠙⠲⠖⠋⠁⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣤⠶⠾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣅⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠒⠶⠾⢿⣤⣄⣀⣀⣤⣤⣤⣤⣀⣀⣀⣇⣿⢀⣀⣀⣤⣤⠶⠞⠛⠋⠉⠉⠀⠀⠀⠀⢻⡄
⣤⣤⣤⣤⣤⣤⣤⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⣬⣭⣭⣿⠿⠿⠿⠟⠛⠻⠶⢶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡋⣻⠄⠀⠀⠀⠀⠀⠀⠀⣬⡻⣧⡀⠀⠀⢀⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣄⣀⠀⠀⢀⣒⠶⣝⢮⣻⡟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠶⣯⣿⣾⡷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢢⠤⠒⠒⠒⠂⢒⠖⠁⣇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠞⠀⠀⠀⠀⠀⢇⣀⡔⠀⠑⢄⠀⠀⠐⢁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⠦⡀⠀⠀⠀⠀⠀⠀⢃⠀⠐⢺⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣉⠁⢀⠤⡒⠄⠀⠀⠀⠀⠄⡰⠂⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠟⠀⠎⠀⢻⡇⠀⠀⠀⠀⠀⠁⠀⠀⡤⠑⢁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡩⠦⠴⢀⠰⢠⡄⠤⠁⠀⠀⠀⠀⢀⠃⠀⡠⡅⠀⡐⠀⢸⠂⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠚⣏⠀⠀⠀⣠⡔⠁⠀⠀⠀⠀⠀⠄⠔⠁⠀⡰⠯⠧⢏⢀⡼⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡑⡀⢌⠒⠀⠚⠻⣇⠀⠀⠀⠀⠀⢠⠖⡦⢠⡌⠀⠀⠀⠀⠁⠀⢀⢂⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⠄⠳⠄⠀⠤⠎⠁⠂⠀⠀⢀⠃⠎⡠⠢⠊⠀⠀⠀⠀⢀⡰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠐⢳⣶⣶⣦⠀⢀⣠⡤⠐⠁⠜⢰⠚⡄⠀⠀⣠⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠰⣿⡿⣋⠀⠀⠘⡞⠀⠀⠀⢀⡜⠀⠙⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠚⠁⠀⠈⠓⠒⠤⣘⣶⡒⠈⠒⠀⠀⠀⣛⣁⢀⠀⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⡀⠠⣀⢀⡀⠀⠀⠀⠀⠒⠒⣒⣛⣆⠀⠀⠀⠀⡵⠐⠒⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣀⠀⣀⣁⠸⡤⠒⠒⠂⠈⠉⠉⠁⠀⠀⠀⠉⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠠⢘⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡤⢤⠀⢂⠀⠀⣤⠤⠤⢶⡆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀
⠀⠀⠀⠈⠉⠷⠊⠛⠁⠀⠀⠀⠀⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡏
⠀⠀⠈⠠⠋⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠤⠀⠒⠂⠀⠀⢰⡟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠡⣥⣄⡀⠀⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠙⠻⠷⣤⣄⣠⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠲⣦⣄⡀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣷⣤⣀⡀⠀⠀⣼⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⠛⢶⣦⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠁⠱⣾⣝⣻⠓⣾⠃⠈⣰⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠁⣠⣟⣹⣿⡿⠷⣦⣄⡁⠀⠀⠀⣠⣿⣿⣿⣿⡇⠀⡈⠀⢀⣹⣿⣿⣦⣀⠉⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠈⣷⣶⣿⣿⣿⣿⣿⡁⠀⠀⢉⠛⠛⠦⠀⣿⣿⣿⣿⣿⠿⠈⠁⢰⣿⣿⣿⣿⡏⠙⠿⢷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⣼⣟⢡⣿⣿⣿⡿⠃⠉⠀⣀⠔⠀⠂⣰⣿⣿⣿⡏⠁⠄⠀⠀⢈⣿⣿⡟⣿⡇⠀⢀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣿⡟⢧⣤⣁⠀⠊⠀⢀⣼⣿⣿⣿⣿⣇⣠⣶⣿⣿⣿⣿⡏⢸⣿⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⢀⣼⣷⣾⣿⣿⡟⠛⠳⣬⣝⠿⣷⣤⣸⣿⣿⣿⣿⣿⣿⣿⡏⣠⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⣤⣿⢿⣿⣿⣿⣿⡇⠀⠀⠂⠉⣠⣾⣿⣿⢿⣿⣿⣿⡟⠁⠻⣷⣿⢯⣿⡿⣿⡇
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⣿⣏⣾⣿⣿⣿⡟⠡⠂⠀⠁⢸⣿⣿⣿⢏⣾⣿⣿⡟⣛⠳⢶⣿⣿⣿⡧⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⡰⠃⠀⠀⢸⣿⣿⠋⣾⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⡿⣿⣿⣾⣿⣿⡿⣿⣰⣾⣿⠿⣿⣿⡻⠷⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠻⢿⣿⣿⣧⣤⣤⣠⣾⣾⣍⠉⣿⣿⣿⣽⣼⡿⣿⡿⢡⣾⠡⠄⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⣀⣤⡦⠈⠀⠸⡭⠂⢰⣯⠙⣿⣟⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣇⣿⣷⣞⣤⣛⣠⣤⣶⣶⣶⣷⣦⣄⡀
⠀⠀⠀⠀⣼⢟⣥⣴⣴⣶⣾⣇⡀⢸⡟⠙⣿⣿⣿⡸⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠠⠀⠀⢀⣹⣿⣦⣄
⠀⠀⠀⢸⣿⠈⣽⢭⡣⣞⠂⢌⠻⢿⣯⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣾⡿⢿⣼⣷⣾⣿⠟⣿⣿⠋⠉
⠀⠀⠀⠘⣿⣄⣀⣀⢀⣠⣷⣮⣕⣤⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣩⣉⣹⣿⣿⣿⣿⣷⡿⠋
⠀⠀⠀⢠⠉⠛⠿⣿⢿⣿⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣟⣿⠋⠑⢄
⠀⠀⠀⠘⠃⠀⠀⠋⠛⠿⠿⠿⣿⣿⣿⠿⠟⠛⢉⣹⣭⣥⣿⣿⣿⢿⣿⣶⡖⠂⢻⣿⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠤⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠄⠀⠀⠀⠀⠐⠻⢶⣿⣿⣿⣿⣿⠟⠁⢈⠋⡁⠁⠀⠀⢙⢻⣿⣿⣿⣿⣧⣤⡤⢬⣿⠟⠒⠒⠲⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣶⣶⢿⠟⠋⠉⣛⠋⠉⠱⠎⠁⠀⠀⡀⠀⠀⢌⣿⡿⠟⠻⠿⣿⣿⡿⠋⠉⠑⠲⠦⢤⣄
⠀⠀⢀⣀⣀⢀⣀⡀⣤⠚⠂⠰⣀⣀⣀⣀⣠⣤⡴⠖⢫⢹⡁⣀⣐⣢⠀⠀⠰⠶⣶⠔⠉⠀⠀⠀⠀⠀⠀⠛⠁⠀⣀⡀⠀⠀⠀⠘⠃
⠤⠴⠛⠛⠟⠛⠻⠛⠛⠶⣶⢶⣿⣿⠿⠟⠿⠛⢡⣄⢘⠓⠃⠁⠉⠁⠀⠀⠉⠛⠃⣀⡠⣶⣿⣿⣿⣿⣶⣤⣀⣾⣿⣿⣶⣆⣀
⠒⣀⠀⢀⣀⣀⣀⣀⣀⣀⣘⣠⣀⣩⣀⠀⣀⣂⣀⣿⣿⡄⠀⠀⠀⠀⢦⣙⡶⢤⡀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠆⣀⡀⢉⣉⢉⣉⣀⣀⣀⡀⡀⣁⣈⣁⡀⠻⠿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠛⠉⠉⠙⠿⢷⣤⣄⠉⠋⠋⠙⠛⠟⠻⢿⣽⣿⣿⣿⡄
⠈⠁⠀⠙⠛⠛⢛⣻⡛⠉⠁⡙⠛⠛⠛⣿⣛⠉⣁⡀⠈⣀⣀⠀⣀⣀⣤⠄⣐⣂⣤⣀⠀⠈⢀⣀⣤⣦⣤⣤⡄⣀⠀⠉⠉⠙⠉
⠀⠀⠀⠀⠚⠋⠉⠋⠀⠀⠘⠋⠉⠛⠳⠿⠟⠳⣷⣔⣿⣿⣿⠛⠛⠁⠀⠀⠀⠀⠉⠙⠛⠶⣤⣈⡉⠙⠳⠟⠻⠿⠿⠶⠶⣶⠾⠗
⠀⠀⠀⣀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣰⣴⡄⢤⣀⠀⠀⠈⠙⠛⠒⠂
⠀⠀⠀⠇⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⣼⣷⣎⣿⣿⣿⣿⣿⣿⣶⢦⣤⣤⣤⣤⣦⣶⣶⣶⣶⣲⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⢛⣿⠶⠞⠛⠛⠛⠛⠻⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⢸⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⣿⣧⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠹⣆⠀⣿⠻⣆⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠙⢧⣹⡆⠹⣦⡀⠀⢿⡘⢷⣄⠀⠀⠀⠀⣴⠀⠀⣿
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢀⣄⣤⣤⡴⠶⠛⠉⢀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠈⠙⠷⣼⣧⠀⠙⠳⢦⣤⠾⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⢹⡇⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠶⢾⡇⠀⠀⠀⠀⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢸⡇⠀⠀⠀⠀⣤⠽⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⠛⠁⠀⠀⠀⣸⠷⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠀⠀⡟⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣧⠀⠀⠀⠀⠀⠀⣰⠋⢀⡇⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣠⡏⠀⢾⡞⢁⡤⠟⢦⠀⠀⠀⠀⠀⠀⠀⢀⡿⠤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡜⢸⠃⠀⠉⠀⠋⣠⠶⠛⣧⠀⣂⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠰⠀⢸⣆⠀⠀⠀⠀⠠⠴⠞⢻⡆⢙⠀⠀⣰⠟⠀⠀⠀⣠⠶⠛⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠐⡶⠞⠁⢈⣠⠞⠁⠀⠀⠀⠸⠛⠒⠶⠦⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣰⣧⡤⠶⠋⠁⠀⠀⠀⠀⠀⠰⣶⡶⠦⣤⣀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠻⣄⣀⣀⣀⡀⠀⡀⠀⠀⠀⠀⢹⡆⠀⠙⣧
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠶⠶⠶⠒⠛⢫⣿⣿⣿⣿⣿⠙⢶⣄⠀⠀⠀⢻⣄⣀⣸⠇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣴⣿⣿⣟⣼⣧⣉⠳⢦⠈⠳⣄⠀⠀⠈⠉⠁
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣠⣾⣿⣿⣿⣿⣧⣄⣉⠓⠀⠀⠀⣼⡁
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣧⣤⠀⠉⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢀⠄
⠀⠀⣠⠖⠋⠁⠀⠀⠀⠉⠳⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⡀
⠀⢰⠃⠀⠀⠀⠶⣷⣶⠄⠀⠸⡄⠀⠀⠀⠀⢀⡴⠊⠉⠀⠈⠉⠉⠢⡀
⠀⣿⠀⠀⡀⣬⣻⣟⢯⠁⠀⠀⡇⠀⠀⠀⣰⠋⠀⠀⠀⣀⠀⢀⣤⠀⠈⡆
⠀⠸⡄⠀⠹⣟⠳⠉⠃⠀⠀⡸⠁⠀⠀⠀⡗⣀⣤⠦⠒⠳⠛⠁⠀⠳⠒⠳⠀⠀⠀⠠⠤
⠀⠀⠙⠢⢄⣀⣀⣀⡐⠖⠦⣷⠀⠀⠀⠀⢻⠁⠐⡒⠀⡀⠀⠤⠤⠀⠀⠀⠈⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠛⢠⡃⠀⠸⠇⠔⠛⠈⠣⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⣈⣀⡀⠀⡀⠀⠀⢐⠀⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡐⠈⠀⠁⠁⡡⠊⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠙⠉⠀⠀⢨⡍⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢖⢸⠀⠀⠀⡋⡰⠂⠘⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠢⠜⠂⠃⠔⠁⠀⢈⡀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⣀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠘⣿⣷⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣶⠾⠛⢻⣿⡄⠀⢾⣷⠀⠀⠀⠉⠉⣛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢹⠀⠀⠀⢸⣿⠃⠀⢸⣿⠀⠀⠀⣰⠟⢿⣷⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⢸⡇⠀⠀⣸⣿⠀⠀⡼⠃⠀⢠⣶⠏⠀⢸⣿⠋⢹⣷⠀⠈⠻⣶⡀
⠀⠀⠀⢠⡾⠋⠀⡆⠀⠀⠈⣷⠀⢀⣿⠇⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⠟⠀⢸⣿⠀⠀⠀⠈⠻⣦
⠀⠀⣰⡟⠀⠀⠀⡇⠀⠀⠀⢻⣧⣼⠏⠀⠀⠀⠀⠀⠀⠈⡿⠇⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠙⣷⡀
⠀⣰⡟⠀⠀⠀⠀⠹⣦⣄⣀⣠⠿⣿⣦⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠘⣷
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣠⢄⠀⠀⠀⢸⣿⡇⠀⠀⢀⠀⠀⠀⠀⢸⣇
⣸⡇⢀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⡿⠧⠉⠀⠀⠀⠸⣿⣧⣀⣰⠏⠀⠀⠀⠀⠀⣿
⣿⣰⣿⠁⠀⠉⠁⠀⠀⠀⢀⣤⠶⡷⡶⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⢹
⣿⢿⣿⣄⠀⠀⠀⠀⠀⠀⣾⣏⢸⣄⠀⢹⣦⣿⣿⠀⣰⡖⡆⠀⠀⣶⡖⠀⢀⡠⣤⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡈⠻⣿⣷⣄⡀⠀⠀⠀⢿⣿⡄⠙⠛⡿⠉⢿⣿⣠⣿⡗⠁⠀⠰⣿⣧⣀⣾⣧⠟⠀⠀⠀⠀⠀⠀⠀⣼
⢸⡇⠀⠈⠙⠻⢿⣷⣦⣄⡘⢿⣿⣤⠼⠁⠀⠀⠉⢹⣿⡇⠀⠀⢀⡟⠛⢻⣿⠇⠀⢰⣿⡆⠀⣴⠛⣆⣿
⠀⢿⡀⠀⠀⠀⠀⠈⠙⠻⣿⣦⠉⠁⠀⠀⠀⠀⠀⠘⣿⣷⣀⣀⠞⠀⠀⢸⣿⡀⠀⢸⣿⡇⢰⣿⠀⣿⠃
⠀⠘⣷⣴⠋⠉⠛⢷⣦⡀⢹⣿⠇⠀⠀⣀⣤⣤⣤⣤⣬⣉⠉⠉⠀⠀⠀⢺⣿⣇⠀⢸⣿⣇⢸⣿⣶⠟
⠀⠀⢸⣿⣆⠀⠀⠀⠙⠿⠞⠋⠀⡄⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⡀⠀⠀⠈⠿⡿⠶⠋⠻⠿⢿⣿⡏
⠀⠀⠀⠻⢿⣷⣦⣤⣤⠤⠄⠀⢸⡃⠲⣦⡀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿
⠀⠀⠀⠀⠀⠘⠳⣦⡀⠀⠀⠀⠘⢷⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⢀⣤⣞⣿⡏⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣭⠾⠃⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⡶⠞⠛⢹⣿⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⣸⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢭⡙⠲⣾⠀⠀⢸⡀⣧⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡘⠓⠶⣧⣦⠙⠆⢦⠈⢷⣌⠳⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣁⠀⠀⠀⠀⠛⠂⠈⠳⢤⣈⡉⠀⠀⠙⠷⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⢀⡴⠋⠁⠠⠒⠉⠉⠉⠁⠀⣀⡤⠖⢒⣠⡤⠤⠄⠀⠀⠉⢙⡛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⣰⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠙⠛⠓⠂⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⡰⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠲⣄
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣆
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡀
⠀⠀⠀⠀⠀⠀⡖⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇
⠀⠀⠀⠀⠀⣸⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⡇
⠀⠀⠀⠀⠀⣿⡄⠀⢸⠁⠀⠀⠀⠂⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⠀⢀⡼⣡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢢⡀⣀⢸⡇
⠀⠀⠀⠀⠀⣿⠃⠀⢸⡆⠀⠀⠀⢀⣤⣤⣤⣤⣤⠴⠖⠒⢾⡆⠉⠛⠋⠛⠁⠀⠀⣄⣀⣠⡴⢻⣿⣷⣦⣄⣀⠀⢹⡏⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠠⠟⠀⠀⠠⠤⠤⠔⡂⠔⠈⠁⠀⠀⡀⢠⡀⠀⠀⠈⣹⠟⠓⠾⠿⣿⣧⣄⣉⣷⠘⡇⢰⠇
⠀⣠⡴⠶⠶⣿⣤⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠀⢠⠊⠀⠲⢤⣤⣤⣤⡄⠀⠀⠀⢸⣇⡾
⡾⠁⣀⣀⣀⠀⠈⣿⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠂⠀⠀⠠⠋⠀⠀⠀⠸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠛⠛⠳⢦
⠇⠚⠉⠉⠙⢷⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⡶⠶⣤
⠀⡀⠀⠀⠘⠻⣿⢀⠀⠀⢀⠄⠀⠀⠀⠀⣰⠟⠁⠸⣆⠘⠿⢷⡄⠀⠀⠀⠀⠀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⠂⠀⠀⢀
⠀⠹⣆⢀⠀⠀⢿⢸⡀⢠⡏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⣀⠚⠋⠉⠉⠘⣿⠀⠀⠀⠀⠀⢸⠀⢀⡼⠀⠀⣀⠔⠁
⠀⠀⠈⠻⣆⠀⠼⢸⡇⣾⠀⠀⠀⠐⠁⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⡇⠀⠋⠀⠀⠉
⠀⠀⠀⠀⢹⡄⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⡴⠟⠛⠹⣄⣴⠉⠳⣄⡙⠒⠤⠤⡀⠀⠀⠀⠀⡟⠀⠀⠀⢸⣾⠃⢀⡤
⠀⠀⠀⠀⠀⠙⠶⠶⢻⡇⠀⠀⠀⣠⠞⠁⠀⠀⣀⣤⡴⠿⣄⠀⠸⡟⠒⠤⣤⣤⣀⠀⠀⢸⠇⠀⠀⠀⣾⡟⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⡞⠁⠀⠀⡴⠛⠉⠀⠀⠀⠀⠉⠀⠃⠀⠞⠙⢿⣿⡆⢀⡟⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡿⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢴⣻⠖⡼⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠈⠁⢀⡼⠁⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠞⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣤⠶⠛⠉
⣿⣿⣿⣿⣷⡾⢿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣷⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⡿⡿⡿⠿⠿⣿⣷⠿⠿⣿⠿⢿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣘⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣭⡀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣥⣦⣼⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣶⣇⠀⠁⣸⣶⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠻⣉⠉⠙⠋⠓⠒⠒⠒⠒⠒⠲⣞⠛⠟⠛⠛⠛⣿⣻⣿⠋⣿⡿⢿⣿⠉⣿⡏⣿⠛⠛⠛⠛⠻⣷⠖⠒⠒⠒⠒⠒⠞⠑⠋⠉⣀⠌
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠀⠠⠄⣀⣀⣀⣽⣛⣛⣛⣛⣛⣻⣿⣿⡆⣿⡇⢸⣿⠀⣿⠇⣟⣛⣛⣛⣛⣛⣯⣀⣀⣀⠠⠄⠀⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⣿⣿⣿⡇⣿⡇⢸⣿⠀⣿⢠⣿⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⡇⢸⣿⢠⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⡇⢸⣿⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⡇⢸⣿⢸⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣧⣼⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠟⠀⠀⠻⠿⠿⢿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠒⠒⠒⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⠠⠄⢀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣰⠃⡖⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢱⢀⠞⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠱⠒⠉⢱⡨⠀⠀⠀⠀⣠⡀⠀⠘⡆⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠰⠄⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠄⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠠⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⠀⢀⠀⠂⠀⠤⢀⣀⡄⢄⣸⠛⣯⡲⢂
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠔⡁⠀⠉⠱⠇⠀⢀⣀⠈⠀⠀⠈⠓⠈⠁⠁⡀⠈⣀⠀⠈⡀⠠⠠⠀⡄
⠀⠀⠀⠀⠀⢸⠀⢠⡰⠀⠀⠀⠂⠄⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠙⢠⡄⠁⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠱⡌⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠔⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠂⢀⠒⠁⠀⠀⠀⠀⢠⠀⠀⢀⠀⢤⢀⢲⠀⠠⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠰⢸⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⣀⡀⢙⢚⢀⣀⢘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢙⢛⡟⠛⠛⠿⣛⠛⠻⢿⣉⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠉⠻⣿⠻⣿⠟⠛⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⣀⣄⡀⠀⠻⠿⠶⠠⢼⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠶⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢴⠀⠀⠀⢀⡴⠋⢠⡄⠘⠋⠀⠀⠀⠀⠀⠓⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠆⠀⣾⣿⡇⠀⠀⢰⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣴⣤⣄⣀⣤⡄⠀⠀⠀⠀⢿⠟⠁⠀⠀⢼⣿⠟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧⣠⣿⣶⣄⠀⠻⠁⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠀⢀⡝⠛⠀⠈⠉⠙⢳⠀⠀⢀⣀⣰⠷
⠀⠀⠀⠀⢠⡄⠀⠀⣀⠈⠁⠀⢠⣿⡿⠶⠶⠶⢀⣠⡌⠀⠀⠀⠈⢀⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⣇⠀⠐⣿⣀⣀⡀⠀⣹⣷⣄⠀⠀⠀⣿⡀⢀⡰⠶⠀⠙⢻⣶⠀⢶⡆⣀⣀⠀⠀⠀⠀⠠⠴⠶⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⡷⠀⠀⠀⠈⠙⠁⠀⠀⠙⠛⠛⠻⠛⢿⣿⢷⣶⣾⣿⡿⠋⢀⣤⠄⠀⠄⠉⠀⣠⠀⠉⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⠟⠛⠛⠁⠘⠋⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⠖⠀⠀⠀⠐⢿⡄⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠀⢿⣷⡄⠀⠰⠊⠁⠀⠀⠈⠙⠿⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷⠀⠠⣦⠀⠀⠀⠠⢤⣄⠙⠛⠷⣶⣶⣦⣀⣀⣀⣀⣀⣀⠤⣶⣶⠎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⠀⢰⡀⢸⣿⠀⠁⣤⡄⠙⠀⠀⠀⠀⠀⠀⠘⠿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠸⣷⡄⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠃⠀⠈⠻⣆⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣧⡀⣀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⣿⣇⠛⠁⠙⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠙⢿⣷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣿
⠀⠀⠀⠀⢀⡀⠀⢠⡶⠀⠘⠛⢂⣀⡀⣾⡷⠀⣾⡏⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⡇⠀⣿
⠀⠀⠀⠀⠸⠇⠀⠀⠙⠷⠶⠶⠟⢻⣿⣿⣇⠀⢸⣿⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⢠⠈⢳⡆⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⠏⠹⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⠛⠀⢸⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠜⠃⠀⠀⠉⢸⣿⡷⠈⢡⡄⠀⠀⠀⠀⢀⠛⠀⢠⡌⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡏⠀⠀⣼⡇⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠹⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠀⠀⠀⠘⢇⠀⠀⣤⠁⠀⠀⣰⡆⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢿⡆⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡾⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⢀⣀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⣀⣀⣀⣀⣀⣀⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⣁⣀⣀⣀⣀⣀⣼⡗
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⢿⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣦⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣶⣶⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣺⠭⣒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠋⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢇⡄⡀⡠⠀⣀⣤⣀⣠⣀⡀⠈⠙⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⢹⠃⡧⢨⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣇⠇⠀⠘⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷⠈⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣩⠿⡿⠀⣰⢟⣿⠻⢿⡟⠻⠿⣿⣿⣿⣿⣷⣾⣿⣿⣄⡀⣹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⢸⣿⡇⢨⣵⣿⣿⢦⡶⠚⠲⣶⡍⢿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣈⠀⠀⢈⣿⡇⣸⣿⣿⠇⠈⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣽⣿⣿⢣⡄⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣹⣏⣿⣤⣴⣬⣿⣷⠿⠿⢿⣿⠟⠀⢀⣀⡄⢰⣿⣿⣿⣿⣿⣽⡿⣿⣤⣤⣼⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣺⣻⣿⣿⣷⣾⣿⣿⣞⢦⡀⣿⠁⠀⠛⠂⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣯⡟⠀⠀⠄⢠⣿⣿⣿⠛⠿⣿⣽⣧⣾⢋⣾⣿⡿⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣟⡆⢹⣿⣿⣿⣿⣿⣬⠻⣿⣿⡏⠀⠀⣰⣿⣿⣿⣿⣠⣄⣤⣿⣿⣿⣾⣿⡡⣠⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⣵⣺⣿⣿⣟⣿⠿⢿⠷⣼⣿⡄⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣺⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⡿⣿⣟⠽⢛⠿⣷⢂⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣷⣿⣿⣿⣦⣼⣿⣻⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡘⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠛⠛⢿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠒⠒⠒⠒⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⣀⡤⠴⠦⣄⣀⣀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠁
⠀⠀⠀⡞⠁⠀⢠⡀⢢⡘⣦⣲⣄⠀⠙⡄⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⡄
⠀⠀⢸⠁⢀⡀⢦⡹⣆⢻⣮⣏⠷⠀⠀⢱⠀⠀⠀⠀⠀⣼⠐⠀⠀⡠⣂⡴⣟⣴⠵⠋⠘⣧⣄⣸⡇
⠀⠀⣿⠀⢦⡹⣦⡻⣿⣗⢷⡙⢦⠀⠀⣌⠀⠀⠀⠀⠀⢻⠤⠶⠛⠋⠁⠀⠉⠀⠀⠀⠀⢈⠁⡆⠈⠳⠀⠀⢀⣤⣀⡀
⠀⠀⢸⡀⠈⣿⠳⣝⢷⡝⢦⠁⠀⢀⠀⠈⠳⡄⠀⠀⠀⠸⡇⠀⢛⣛⠃⢀⡆⠀⠐⠛⠉⠸⠀⠁⡀⢸⠀⠀⠀⠀⠉
⠀⠀⠀⠹⢄⡈⠁⠈⠀⠀⠀⠀⢀⡤⠉⠓⠦⠙⣆⠀⠀⠀⠃⠀⠈⠛⠃⣸⠁⠀⠀⠿⠃⡴⠚⠧⠈⠻⡀⠀⠀⠒⠋⠁
⠀⠀⠀⠀⠀⠉⠒⠶⠤⠴⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⡃⠀⠀⠀⠀⡇⠀⠰⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⠴⢚⡛⠲⢄⡀⢁⣀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡐⠀⠀⠀⠀⠀⢈⡴⠋⠈⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠒⠒⡏⠁⠀⠀⠀⣃⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡇⠀⠀⠀⢳⠂⢤⡀⠀⠉⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠀⠀⠰⡈⠢⣀⣀⣀⠔⢀⠎⠀⠉⠛⠈⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠑⠖⠛⠤⡜⠒⠁⠀⠀⠀⠘⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢰⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⡼⠀⠀⣆⡀⠀⠀⠀⠗⠦⡀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⡀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⢀⣀⣀⣀⣄⠠⠤⠤⠐⠂⠉

⣿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣿⣿⣿⣿⣛⣻⣿

⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠂⠐⣚⠃⢀⡄⠀⠘⠂⠐⠒⠒⠒⠀⠀⠉⠉⠀⠀⠸⠀⠀⠙⠂⠄⠀⡀⠀⣀⠀⠤⠐⠊⠁⠂⢠⠁⡀⠀⡀⢀⢁
⠀⠀⢀⣐⠒⢀⣒⣀⣈⣀⠁⠁⠀⠀⠀⠒⠒⠂⣀⡀⠀⠀⣀⣀⣠⣤⠠⠀⠀⢀⠀⢀⠁⡀⠒⠀⠀⠤⠀⠀⠀⠂⢀⡀⠠⠀⡆⢀⠁⢠⠀⠈⡈
⠁⢸⠀⠃⠰⠀⠀⠀⠀⠀⠐⠒⠒⢒⣂⣈⠉⠥⠤⠐⣚⣡⠆⠀⠰⠐⡆⠀⠃⠤⠔⣁⠋⣈⠑⠒⠀⠤⠤⠐⠒⠉⠠⠀⠄⢰⠀⡈⠀⡆⢠⠁⣅
⢠⠀⠀⠰⢘⠀⡋⣴⠤⠤⠤⠥⠤⠤⠤⠶⠖⠒⢚⣩⠥⠶⠆⠀⡇⢀⠇⠘⠒⠚⠛⡉⠿⠉⠙⠛⠒⠒⠒⠛⣻⠀⠄⠰⠀⡄⢀⠀⢰⠀⡌⠈⢁
⠸⠀⠀⠀⢸⠀⡇⣿⢠⠆⠠⠀⠀⠒⠚⠛⠉⠉⠀⠀⠀⠀⠀⠈⠁⠉⢰⠒⠒⠀⠉⢹⢐⠟⠛⠛⠛⢻⠀⠀⣿⠀⠀⡄⢠⠀⡀⠀⡄⠀⡅⠐⠁
⠀⠀⡄⢀⢸⠀⠇⣿⢸⡇⠀⠀⠀⣀⡀⠀⠀⡤⠒⠐⠚⠛⠃⠀⠀⠀⢸⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⡟⠀⢸⡏⠈⠀⠁⡌⠀⠁⠀⠁⠀⠠⠒
⠀⠀⠁⠘⠀⢀⠀⡿⣸⡥⢶⣚⡋⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠃⡄⢸⠀⠃⢸⠀⠀⠂⠀⠀⡀
⠆⢰⠀⠄⡇⡘⢸⡧⠖⠋⠯⢸⡇⠸⠀⠀⢠⠀⠀⢀⣀⣤⠀⠀⠀⢠⠁⠀⠀⣿⡄⢀⡴⢆⠀⠀⠀⡇⠀⣿⠀⠄⡴⢰⠀⠀⠀⡀⠔⠀⠀⠀⡀
⠀⡘⠰⢀⢀⡇⠐⡀⢡⣤⠶⠜⢃⡼⠀⠀⠘⠛⠋⠉⠁⠘⠀⠀⠀⠎⠀⣀⣰⡇⢳⣞⠁⢼⠀⠀⠀⣿⠀⡇⠀⠂⠇⠘⠀⠓⠈⢀⠠⠄⠂⠁⢀
⢠⠁⠆⡮⠀⡶⠈⡏⢡⠴⠒⠋⠁⠀⠀⠀⠀⣀⣠⠤⠶⠂⣀⣀⣘⣩⡭⣭⣭⣽⢢⠛⣆⣈⡛⠲⢦⣽⠀⡆⠀⢠⠆⣆⠀⠀⠈⠀⢀⠠⠐⢊⡅
⠃⠼⠀⣲⠚⡀⣸⠁⡜⠀⢀⣀⠤⠄⠒⣊⡉⠤⠐⢒⣉⠉⠤⠤⠤⠄⠀⠠⠤⠼⠐⠓⠒⠀⠠⠭⣐⠊⢤⣅⠀⢸⠀⢉⡠⠤⠒⢉⠠⠔⠊⢠⠂
⢐⠆⡰⡉⡘⣰⠃⢘⣃⡭⠤⢔⣒⠫⠍⠒⠂⠀⣁⠠⠤⠐⢒⣒⣠⡎⣁⣉⣀⣒⡿⢉⣒⠠⢄⣀⠀⠈⠐⠰⠆⢈⣉⠠⠄⠒⠉⣠⣴⡆⢀⠆⠰
⠀⡐⡝⡐⠰⢛⡋⠭⠐⠀⠉⠀⠀⢀⣀⠀⢀⣸⠁⡴⠚⢉⣡⠤⠤⣬⣉⢠⠤⠤⢬⣙⡋⣉⡒⠦⠤⣍⣉⡑⢈⣠⣤⡤⠶⠞⢋⣡⡿⠀⡜⢠⠃
⠜⡘⣰⠕⢂⡡⠄⢒⣂⣈⠐⠒⣨⠥⠒⢊⣉⠉⣍⣁⡞⠁⠀⠀⠀⠀⠈⠻⡀⠀⠀⠀⠉⠻⣏⠙⠶⠤⣀⢳⡞⣠⠤⠶⠒⢻⡟⣾⠃⡜⢠⠃⠔
⣰⡀⢀⠔⢁⠴⣛⠭⠤⠤⣭⣙⣡⠖⠉⠀⠀⠀⠀⠈⠙⡄⠀⠐⠒⢄⠀⠀⢀⠀⠰⢦⠀⠀⠘⣇⠀⠀⠹⡄⣰⠃⠀⠀⢠⠟⣼⠏⡜⢠⢃⠊⢀
⢠⠁⠆⡰⢁⡞⠀⠀⠀⠀⠀⠙⢧⠀⠀⢀⡤⢄⠀⠀⠀⠸⠀⣾⠀⣸⠀⠀⡘⠀⡄⢨⠀⠀⠀⣿⡄⠀⠀⠙⠁⠀⠀⢠⠏⣴⢋⠜⡰⠁⠂⡰⠁
⠘⢸⢠⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡇⠀⠀⠀⠀⡋⡄⠁⠀⠀⠁⠀⡀⠀⠀⠀⠠⠁⠹⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡔
⡇⢨⢸⠀⡇⠀⠀⡴⠚⢠⣴⠃⠀⠀⠐⣆⣀⡾⠀⠀⠀⢰⠀⠁⠀⠀⢠⠇⠀⠀⠁⠀⡠⠞⠀⠀⠀⢷⡀⠀⠀⣼⠁⡖⣰⠀⡖⡐⠀⡄⠀⡜
⡇⠘⢸⡀⡇⠀⠸⣁⣀⣤⡴⠶⠶⣤⠀⠀⠁⠀⠀⠀⠀⡞⠀⠀⠀⣠⢻⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⢸⡇⠀⠀⡟⢠⡇⢿⠀⠀⠃⠀⡇⢰
⠠⠀⡆⢧⠘⢦⣀⡈⠥⣀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠹⣾⠀⠀⣀⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⡇⢸⡇⢸⠀⠀⢰⠀⠇⢸⠀⢰
⠔⠊⣠⠞⣁⠀⠀⠀⠀⠈⣧⠀⠀⠀⡗⠀⢀⣀⠴⠛⠀⠀⢠⡀⠀⠀⢹⠀⠀⡏⢣⠀⠀⢱⠀⠀⠀⣾⠀⠀⠀⡇⠘⡇⢸⡄⡇⠘⠀⢠⠈⠀⢸
⡴⠟⣁⠔⢹⠀⠀⠀⠀⢀⡿⠀⠀⢀⠛⣉⣉⣤⠀⡇⠀⠀⣸⢱⡀⠀⠀⡆⠀⡇⠈⡆⠀⠈⠀⠀⠀⡿⠀⠀⠀⣧⠀⣿⢸⡇⠃⠀⡄⢸⠀⡇⠘
⠴⠛⠁⠀⠘⢦⡀⢀⣠⠞⠁⠀⢀⠎⡾⢁⢀⡞⣼⠀⠀⠀⡿⠀⢧⠀⠀⢡⠀⡇⠀⢸⠀⠀⡆⠀⣸⠇⠀⠀⠀⣿⠀⣿⢸⡇⢰⠀⡇⠈⠀⡇⢸
⠘⡄⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⢎⡼⢁⡏⡾⢰⠧⠤⠤⠤⠇⠀⣼⠀⠀⠸⡀⣇⠀⠀⡇⠀⢁⠠⠿⠒⢒⣉⣭⠭⠴⠛⣀⡇⠈⠀⠃⠀⡄⠠⢸
⡄⠐⢄⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⣡⠏⣼⡧⠴⠖⠒⠛⠛⠓⢶⠉⡇⠀⠀⠏⠁⠀⠀⢸⠀⠈⡄⢰⡋⢭⠐⣒⠨⠍⠁⠀⠀⠐⠀⠠⠀⡇⢸⠈
⠐⡄⠀⠑⠒⠒⠒⠂⢉⣠⠞⣫⠞⡰⠀⣉⠠⠄⠒⠂⡈⠉⠁⠸⢠⡇⠀⠀⠰⠀⠀⠀⠀⡆⠀⠹⡄⠳⣌⠓⢌⠂⠀⠈⠋⠅⡀⠒⠺⠀⢠⠀⠆
⢀⠀⠐⠂⠀⠀⠒⠛⣉⠤⢚⢁⡼⠓⣉⠠⠄⠐⠊⠁⠃⡘⠀⡇⢸⡇⠀⠀⠀⢣⠀⠀⠀⡇⠀⠀⠙⢦⡈⠛⠶⣬⣒⠤⡁⠂⠬⣑⠠⣈⠛⢄⠙
⠠⣌⡐⠂⠀⠀⣂⠩⠤⢒⡏⠤⠒⠉⠀⡀⠤⠂⡇⡘⢠⠁⣼⠀⣾⠁⠀⠀⠀⠈⠣⣀⢠⡇⠀⠀⠀⠀⢙⣷⠶⢀⣉⣽⠶⠌⠒⠠⣑⡢⣑⢦⡑
⠀⠀⠄⢤⠠⠀⠀⠒⠓⢀⠀⠀⠂⠀⣡⡄⠀⠰⠀⢀⠂⣰⠃⣼⠃⠀⢀⣠⡤⠶⠚⠉⡸⠀⢀⡠⠔⢊⣩⠤⠖⣋⡥⠴⠒⠊⠉⢁⣉⢁⡈⢱⡌
⠀⠄⠀⠐⠐⢋⠋⠈⠃⣀⠀⠐⠀⠉⠔⠀⠠⡳⠁⠌⢰⠃⣼⠷⠒⠋⣉⠤⠐⢚⡏⣰⡧⠚⣉⡤⠖⢋⠡⠖⠉⣁⠠⠤⠒⠘⠉⣈⡁⠀⠤⠤
⠐⠒⠂⠙⠁⠖⠒⠈⠀⢀⡤⠀⢀⠎⠀⠰⠡⠁⠌⢠⠃⢀⣠⠴⠚⣋⠤⠒⠃⡜⠀⢁⡴⠞⢉⠴⠊⣁⠤⠒⠉⢀⠠⠄⠒⠉⠀⠀⣀⠠⠤⠤
⠀⡙⠀⡀⠠⠀⠀⠉⠀⠀⠀⠀⠁⠀⠄⠀⠀⠄⢀⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⠀⠁⠐⠈⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠒⠒
⣐⡀⠠⠄⠐⡆⢀⡴⠀⢀⠄⠀⡠⢂⠂⢀⣎⡠⠄⢊⡠⠔⢂⠀⡰⠀⡌⢠⡷⢃⠔⢀⠤⠂⢀⠠⠂⠀⡀⠀⠂⢀⡠⠄⠂⣀⠠⠀⠒⠂⠈⢉⣉
⠀⠀⢀⠀⢊⠔⠉⣀⠔⠁⡠⠊⡰⠁⠀⢉⡠⠔⠊⣁⠤⢲⠋⡐⠀⠰⠀⢈⠔⢁⠔⠁⡠⠊⠀⡠⠐⠉⢀⠄⠊⢁⠄⠂⢉⠠⠄⠒⢈⡉⠁⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣛⠻⣷⣦⣄⣀⠀⢀⣿⣋⣩⣽⠶⢦⣤⣴⣿⣿⠿⠶⣿⣿⣿⣷⣿⣿⣿⣉⣭⣍⣀⣀⣀
⠀⠀⠀⠀⠀⣀⣠⣶⡶⢶⣦⣤⣽⣿⣿⣿⣿⣿⣿⣟⣫⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⠧⠽⠷⢦⠤⠤⠄
⠀⠀⠀⠀⠀⢉⣭⣿⣿⣿⣿⣻⣿⠟⢿⣿⣯⣭⣍⣭⣍⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡿⢿⣿⣿⣿⣷⣶⣶⣤⣠⣤⠔⠒
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⣛⠿⢿⣿⣿⣿⣿⣻⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⢿⣛⣛⣋⣛⡣⡿⢛⣽⣿⣿⣿⣿⣟⣻⠏⣦⣡⠘⢶⠀⡀
⠀⠀⠀⠀⠀⣿⠘⠏⢩⣿⠛⠿⢃⠠⢶⣿⣭⣭⣽⣿⣿⣭⣽⣶⣶⣿⣿⣭⣿⣪⠟⠋⠁⠀⠀⠉⠻⣦⢛⣵⣿⣿⣿⣻⣯⡛⢆⠋⠈⠻⣼⢋
⠀⠀⠀⠀⢠⡟⠐⠁⣾⡿⢘⣻⣿⣶⣦⣾⠟⠉⠙⠿⣿⣿⣷⣿⣿⣿⣿⠿⢫⡏⠂⠀⠀⠀⠀⠀⠀⠈⣏⣵⣶⣿⣿⣦⡌⢷⡈⡄⡔⢷⡟⢘⠆
⠀⠀⠀⠀⢸⡟⠀⢠⣿⡇⠁⣿⣏⢩⡇⠀⠁⣤⣄⣀⠈⣉⣢⣝⢿⣿⣿⣇⢺⡃⠁⠀⠀⠀⠀⠀⠀⠀⡏⣶⣶⣿⣿⢋⣀⣾⣿⣾⣇⢾⡿⢻
⠀⠀⠀⠀⣿⢋⠀⢸⣿⣇⠠⣿⣧⢸⡧⣥⣶⠦⢼⣥⡿⣛⣯⣩⣹⣿⣿⣾⢾⡿⠓⡶⣤⣀⣠⠖⢢⡜⣴⣿⣿⣿⣿⣻⣭⣿⣿⡿⠃⢸⣷⢩⡣
⠀⠀⠀⠀⣿⡎⠙⡘⢿⣿⡝⢿⣿⣼⣷⣿⣷⣖⣛⣫⣶⣿⣿⣿⡿⠭⢭⢽⣶⣟⣻⠷⣿⣆⣜⣿⣿⣿⣿⣯⣿⣯⡽⣛⣅⣈⡏⢀⢰⠈⢻⣾⠃
⠀⠀⠀⠸⣿⣿⡟⠃⠸⣿⣿⣾⡿⠿⡛⢿⣿⢿⣿⣭⣟⣿⣿⣿⣿⣯⡿⣿⣟⠳⢟⣿⣿⡿⣻⣶⣬⣿⣿⣿⣿⣿⣧⣙⠢⡌⢃⠘⠻⣴⠓⣿⣆⡀
⠀⠀⠀⠀⣿⣯⣿⢸⣤⣿⡛⣩⣶⣾⣞⣿⡏⣋⣭⣿⣿⣿⣿⣛⣛⣫⣾⢤⣿⣿⡿⣿⢋⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⡝⢫⣍⣇⣿⣯
⠀⠀⠀⢠⣿⣿⣽⣿⡿⡮⣿⣿⣿⣿⣷⡿⣱⣼⣿⣿⣯⠾⣿⣛⠏⠟⢁⣼⣷⣤⣶⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣝⠿⣯⡻⠍⠿⢹⣾⠧⣿⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣵⣿⢏⣵⣢⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⢷⣝⢿⣎⢻⡄⠀⠀⡇⢀⣿⠏⡀
⠀⠀⠀⠘⣿⣿⡿⡱⣯⣿⣿⣫⡿⠿⢋⠻⠛⠿⣿⣟⡿⣱⣿⣿⣿⡋⢙⣿⣿⣿⠟⢻⣿⣽⣿⣿⣿⣿⣿⣿⣧⢻⡇⢿⠈⡇⢤⠐⣹⣿⣏⣸⡥
⠀⠀⠀⠀⢿⣿⣽⡣⣿⣻⣿⠿⣒⣒⣉⣁⢠⢞⣿⣿⣷⣿⣿⣿⣿⣶⣿⡟⢻⣧⣿⣘⣿⣿⣿⡽⣯⢻⣿⣿⣿⣿⡗⣼⡽⣽⡻⢿⣿⣿⢉⢁
⠀⠀⠀⠀⣿⣿⣿⢫⣿⣿⣾⣿⠿⣿⣿⡿⠃⣼⣷⣾⢹⡿⣿⣿⣿⡷⢞⣧⣄⣋⣽⣿⣿⣿⣿⣿⣿⣼⣿⣿⡿⠛⣾⣿⡾⡟⡧⣾⣿⣧⡼⡟⠈
⠀⠀⠀⠈⣿⣿⣿⣧⣙⣛⣿⣾⡿⣸⣯⡾⣃⣸⡿⠟⢨⠿⢿⡄⢺⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠈⢶⡟⣇⠻⣿⡁⣿⣿⣵⣻⠖⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡾⣽⣿⣿⣿⣿⢫⣴⣿⣿⡇⡀⠁⢼⣿⣿⢿⣟⣺⢿⢃⣿⣿⠿⡟⠉⠹⣿⣧⠀⢬⣛⣆⠄⠘⣿⣿⣿⡿⡧⠄⠄
⠀⠀⠀⠐⣿⣿⣿⢸⣿⣿⣿⣼⣿⣿⣿⢿⣇⢿⣿⡿⣫⣼⣳⢇⢀⣽⣿⣾⣿⠉⣎⣼⣿⣿⡇⠠⣤⣀⠘⠛⢧⡈⢻⡅⠃⣠⣿⣿⡿⠋⢁⡁
⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⡧⣛⡟⣿⣟⣿⡿⡂⠛⡱⣿⣿⣿⢷⢛⣾⡿⠉⣾⢹⣏⣾⣿⡿⠳⣄⠀⠙⠆⣀⡾⢡⣾⣷⣾⣿⣿⡿⠃⡴⢛⠃
⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣯⢻⣿⣯⢯⣾⣧⢠⣙⡻⠶⠍⠛⠀⠙⠃⠀⠛⠸⠟⠀⠋⣠⠀⠏⡴⠊⠟⢉⣴⣿⣽⣿⣏⣺⣿⠶⢛⠛⡌
⠀⠀⠀⠀⣿⣿⣿⣯⣿⠻⡿⠸⠎⣿⡇⠺⡟⢋⡾⢤⣷⡖⣶⢶⣷⣾⠿⣖⣾⡗⣾⣿⣢⣟⣛⡷⠲⣶⣲⡶⢶⠙⢻⣿⣿⠛⢣⣤⣾⠃⠠
⠀⠀⠀⠀⣿⣧⣿⣿⣻⣄⣽⢠⣧⠽⣷⡄⡡⠚⣃⠈⡟⢒⡟⠙⢻⠿⠷⠿⠟⠛⠹⠷⣾⡧⠤⠷⠲⠟⠏⠓⠾⢀⡿⢿⣿⣤⣴⣾⡟⠿⡄
⠀⠀⠀⠀⣿⡿⣿⣿⣧⣿⡉⣥⠀⡂⠰⠀⣁⢘⣿⣧⠳⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣀⣠⣤⣾⡿⢡⣿⣷⣻⣿⣿⣿⣚⢵⡄
⠀⠀⠀⠀⣿⣿⣿⡿⣦⣹⣇⢳⣰⣯⡆⢿⣿⣿⡿⢟⣤⡖⢛⠛⠓⠒⠒⠂⠀⠒⠐⠠⠾⠚⡛⠻⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣻⣿⡿⣿⡆⢹
⠀⠀⠀⠀⣿⣿⠿⢟⣮⠉⠋⢫⢻⣯⢿⡴⣿⣿⠾⣿⣿⣿⡜⣀⠒⠃⠀⠣⠀⠙⠁⠃⠀⠃⠙⣸⣿⣿⣿⣿⣇⢂⣿⣿⣿⣿⣿⣿⣷⣿⣷⠈⡇
⠀⠀⠀⠀⢹⣾⣷⣶⣦⣴⣤⣤⣀⡈⠉⠁⠹⣿⣿⣿⣿⣿⠟⠍⣧⡈⢸⣧⣽⠍⠀⠀⠊⣡⣤⣾⣿⣿⣏⣸⣽⠿⠿⠟⢿⡿⣿⣿⣿⠿⢿⣦⡅
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀
⠀⠀⣀⣤⣶⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⠟⠛⠲⠦
⠀⠈⠁⠀⠀⣾⠉⠙⠛⠻⠿⢿⣿⣿⣿⣶⣦⣤⣀⣀⣶⣇⣀⣤⣶⣾⣿⣿⡿⠿⠛⠛⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣏⢙⣿⡟⠛⠿⠿⣿⡿⣿⠿⠿⣿⠛⣿⡷⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣼⠃⠹⣆⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠤⠴⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⠤⠦⠤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⢲⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣀⣾⣀⣀⣀⣤⣤⣤⣯⣤⡤⠤⠤⣤⠴⠷⠶⠶⠶⠒⣶⠚⠛
⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⣿⠻⣄⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣀⣠⣿⣀⣀⣀⣀⡀⢸⡇
⠀⠀⠀⢸⡀⠀⣿⡀⠘⣧⣤⡤⠶⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠒⠢
⠀⠀⠀⠈⢧⡀⠈⠙⠋⠉
⠀⠀⠀⠀⠀⠉⠛⠒⠂⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⣅⠀⠀⢠⣶⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⢸⣿⣀⠀⢸⡉⡁⠀⣀⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣁
⠀⠀⠀⠀⠀⠛⢮⣿⣿⠛⣾⢹⡭⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⠐⣿⢹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⢻⢸⡀⣿⣾⢳⢱
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠾⣽⣇⣿⡿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣾⠯⡷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢟⣯⢼⣛⠿⢳⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⣶⡄⠀⠀⠀⠀⠀⠀⠘⠓⢻⣿⠀⡏⡟⠚⠀⠀⠀⠀⠀⠀⠀⠀⣷⠦⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡟⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠇
⠀⠀⠀⠀⠀⢠⡴⡿⢾⡶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣴⣾⣖⡶⣤⠃
⠀⠀⠀⠀⢀⣿⡇⣧⢸⣇⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⢸⡇⡟⠂
⠀⠀⠀⠀⠸⣝⡳⣿⢸⡿⣋⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢨⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⡇
⠀⠀⠀⠀⠀⠈⡿⠮⠾⣚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠚⠛⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣄⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠟⠛⠛⠋⠙⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣇⣀⣀⡀⠀⠀⣀⣤⠤⠄⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠏⢀⣤⣍⠉⠀⠀⠈⣠⣤⣄⠸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠈⠁⠁⠀⠀⠈⠈⠁⠀⠀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⠄⢀⠠⠀⠀⠀⢰⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⢀⡤⠀⢀⠔⠀⢀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⠀⠀⠁⠀⠀⠁⠀⠀⢨⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢋⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣯⡐⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠁⠀⠀⠀⢀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠃⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⡦⠉⠉⠉⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⢀⣶⠀⠀⠀⠀⡀⠀⠀⠀⣄⡀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣶⣿⡏⠀⠀⢀⣴⣷⠀⠀⣰⣿⣷⡄⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣴⣾⣿⣿⣿⣀⣴⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠟⠛⠻⠿⠷⣶⣤⡀⠀⠀⠀⣿⣿⣿⡿⠛⠉⣁⣀⣤⣤⣬⣉⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣤⣀⠉⠋⠀⠀⠀⣿⣿⣿⣶⣾⠿⠟⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⡟⣉⠙⣿⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⣴⢶⡄⠀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣬⡴⠛⠋⠁⠀⠀⠁⠀⣿⣿⣿⣿⣷⣦⣤⣙⢚⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠢⣤⣠⠟⠛⣻⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣴⠛⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠒⠫⠭⠶⠶⡶⣾⣉⣙⣛⣛⣓⣦⣽⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠉⠉⣶⣾⣿⣿⣿⣿⣿⣿⠿⢛⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣄⡀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠠⣿⣿⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠄⠀⠝

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢀⡞⣰⢂⡀⠀⣀⢲⡘⣆⠀⠀⠀⠀⠀⠀⠀⠰⠶⢄⡉⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⣉⣉⣉⣙⠳⢦⡀⠀⠀⠀⠀⠀⣀⡴⠶⠖⠲⠶⣦⣀
⠀⠀⢸⠁⡇⢾⠰⡶⢸⡆⡇⢸⠀⠀⠀⠀⠀⠀⠀⠘⠲⢦⠙⣦⠹⡆⠀⠀⠀⠀⠀⠀⠀⠚⢋⣡⣤⣤⣉⠓⠀⠀⠀⠀⠀⠀⠀⠉⢠⡴⠶⠶⢦⡄⠉
⠀⠀⠈⢧⠳⠌⠁⠀⠈⠴⢃⠏⠀⠀⠀⠀⠀⠀⠀⠾⠇⠘⠇⠸⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⣠⢞⣩⠥⠬⣍⡓⢄⠀⠀⠀⠀⠀⠀⡴⢁⠄⠀⠀⠀⠀⠀⠠⡈⢦⠀⠀⠀⠀⢀⡞⢀⡀⠀⠀⢀⡀⢳⡀
⠀⠀⢠⠇⡞⢡⢀⣀⢠⡙⡆⢧⠀⠀⠀⠀⢸⢡⠎⡴⢚⡓⢦⠹⡌⣇⠀⠀⠀⠀⢸⠁⡇⠀⠀⢀⣀⠀⠀⠂⢸⠈⡇⠀⠀⠀⢸⠀⡏⠀⣶⣶⠀⣻⠀⡇
⠀⠀⠸⠀⣇⢳⡘⡟⢸⠃⡇⣸⠀⠀⠀⠀⢻⠨⡀⣇⢻⡟⣸⢁⡇⡼⠀⠀⠀⠀⢸⠀⡇⢀⠀⠘⣿⠁⠀⡀⢸⠀⡇⠀⠀⠀⠸⣆⠙⠂⢸⡇⠐⠋⣰⠃
⠀⠀⠀⠳⡜⠂⢰⣷⠀⠚⣴⠃⠀⠀⠀⠀⠈⢧⠙⠀⣸⣇⠀⠊⡴⠁⠀⠀⠀⠀⠈⢧⠑⠄⠀⢠⣿⡀⠀⠠⠊⡼⠁⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠚⣡⡴⠖⠒⠶⢤⡙⠃⠀⠀⠀⠀⠐⠋⣡⠴⠖⠲⠶⣌⠙⠂⠀⠀⠀⠀⠀⠀⠚⢉⡴⠖⠒⠶⢤⡉⠂⠀⠀⠀⠀⠀⢸⠃⡼⠁⣀⣀⠘⢧⠘⡇
⠀⠀⠀⠀⠀⠴⠛⠛⠳⢄⣁⡀⠀⠀⠀⠀⠀⠀⠁⠶⠛⠛⠲⠈⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠛⠳⠄⠠⠀⠀⠀⠀⠀⠀⢸⡀⢷⡀⠙⠋⢀⡾⢠⡇
⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠟

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠿⢛⠻⠿⣿⢟⣿⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣀⣄⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⢤⠈⠑⠀⠀⡄⠐⣌⠋⠁⠀⠀⡂⠀⠀⡄⠠⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠄⠀⠀⣄⠉⠢⣄⠘⠂⠀⠀⠀⣸⠀⠀⠀⠴⠃⠀⠀⠙⠂⡏⠠⠋⠜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢁⡔⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠁⠀⠘⠀⢀⣀⡈⠒⠂⠀⠀⢘⠀⠨⢁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⣤⠀⢀⡀⣤⠀⢀⡀⠀⠀⠀⠩⢥⣄⠀⠀⡔⠁⠀⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠦⠀⠙⢄⡄⢠⣤⣄⠀⣼⡟⣶⢠⣿⡰⣻⢿⣿⡰⣻⣏⡀⠀⢀⡀⠀⠀⠀⠀⠣⠀⠊⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠜⠀⣿⣆⣿⠀⣿⣼⣿⠃⠉⢸⣿⠃⠉⣼⡏⢀⣾⣿⣀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠸⣩⠿⠋⠙⠚⠁⠈⠋⠀⠀⠘⠃⠀⠀⠘⠧⢪⣿⢉⠁⠀⠀⠀⢀⡠⠴⠦⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼⡏⢸⠀⠀⠀⠀⠀⠻⠆⢰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡤⣠⡄⢴⣦⠤⠤⠄⠀⢸⣿⣷⡆⠀⠀⠀⠀⢿⣇⡞⠀⠀⠀⠀⠤⡀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠫⠾⠋⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⢀⡀⠀⠀⠁⠀⠀⠀⡀⠀⠁⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⡈⠆⠀⠀⠐⢄⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠃⠀⠌⢆⠐⠲⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⢠⠂⠘⠁⠘⠀⢠⣄⠁⠀⠀⠀⠀⣤⡤⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⣠⣤⠶⠶⠶⠶⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⢀⡞⠁⠀⠀⠀⠀⠀⠀⣤⢠⡄⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣾⠁⠀⠀⡀⢀⣄⠰⣆⢸⣿⣿⣤⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠉⠉⡏⠓⠦⠤⢶⠒⠛⠓⠦⡀
⣿⠀⠀⡌⢿⣄⢿⢦⢻⢳⣯⢻⡌⠃⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⠤⠤⠞⢹⠀⠀⠀⢠⠃⠀⠀⠀⢸⠀⠀⠀⠀⢹
⢻⡄⠀⠈⢾⡌⢿⡎⢻⣇⠹⡆⠁⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⢎
⠀⠻⣄⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⠄⠁⣀⠤⠤⠤⠤⠞⠉⠉⠙⠲⠤⠴⠒⣦⠤⠀⣸
⠀⠀⠈⠳⣤⣀⠀⠀⢤⡀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⡿⡀⠀⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⣰⠃⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠉⠉⠛⠊⠻⣦⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠁⢰⠇⠀⠉⠉⠛⠀⢀⠀⠀⠘⠉⠉⠉⢸⡏⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣼⣄⣀⣰⣿⡿⣤⡾⠤⣤⠴⣶⣶⣀⣨⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡇⠘⢧⣀⠤⢾⣇⣀⡘⢦⣍⡡⠇⠀⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢀⡖⡋⢉⣡⣄⣈⣉⢙⢲⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⣤⣄⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡆⠸⢚⢋⠭⢿⣿⠭⣍⢙⠓⠗⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠐⠒⠂⠼⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠀⠀⠈⠁⠀⠀⠈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⢸⢸⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢲⡖⠒⠒⢺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣤⣄⣀⡀⠀⠈⡄⡇⠀⠀⠀⣠⠤⠶⠒⠳⢦⠲⢶⡶⣞⠁⠀⠀⣨⠗⢲⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠐⠒⠚⠛⢦⠀⠇⢷⢀⡴⠛⢁⣠⣤⠀⠀⠈⠻⣧⢣⣈⣳⣴⣊⠁⣠⠞⣼⠀⠈⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⢉⣉⣉⣽⠂⢸⢸⠘⠓⠛⢉⡾⢧⡀⠀⠀⠀⠈⢻⣧⣼⡙⠈⠙⠁⣰⠃⠀⠁⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⡟⠛⠋⠁⠀⢸⢸⡇⠀⣠⠏⠀⠈⠷⢤⠾⠷⣤⢾⣹⠇⠉⠲⢤⡞⠁⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⣧⣤⣄⣀⣀⣈⣄⡇⣴⠋⠀⠀⠀⠀⣸⡆⠀⠘⢶⠋⠀⠀⠀⠀⠙⢦⣀⠐⢼⡄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠚⠋⠉⢻⡀⠀⡼⠁⠀⠀⠀⢀⡾⢹⠇⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠉⠳⢾⣇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢳⡾⠁⠀⠀⠀⣠⠟⠀⢸⠀⠀⠀⠀⠀⠀⢨⠳⣄⡀⠀⠀⠀⠀⠉⠳⠄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⠃⠀⠀⢀⡴⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠳⣦⣤⣤⣤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠋⠀⠀⠀⠀⠀⠀⣿⠤⠭⣭⠶⠀⠀⢸⡄⠀⠀⢀⡴⣖⣒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⠀⠀⠀⠸⡇⠀⠀⠀⠉⣀⣠⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠉⠻⠿⣿⣿⡿⠿⠿⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠑⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣔⡀⠀⠀⠀⠀⠘⠁⠀⣟⡀⠀⢡⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⣈⠁⠀⠐⢂⠓⠀⡀⢀⡨⠷⠥⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠷⣤⣾⣿⢿⡷⠚⠁⣽⣿⢷⣶⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⢿⡅⠁⠀⠀⠹⠂⠑⣷⠀⣯⣿⠀⠻⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⣷⡔⠀⠀⠀⠀⢀⡟⠀⢷⣧⣀⢼⡺⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⣴⣺⠘⣷⣾⠏⠈⠛⠩⠹⠋⢼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⢷⣽⣇⠘⢉⣼⣤⣀⣶⣴⠚⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡈⣿⣿⣿⣿⣯⡍⠉⠉⠛⠁⣴⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⡙⠊⣭⣉⡠⠀⠀⢀⣴⣾⣿⣿⢳⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⠮⣽⣵⣯⣶⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠉⠉⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⣲⣀⡀⠀⠀⠀⠀⠀⠀⣀⣌⣹⡟⢛⣲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⢸⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠹⣿⡇⠻⣿⣿⣶⣍⡓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⣿⣁⡾⢻⠀⠀⢀⠴⠒⠊⠙⠒⢄⠀⠀⢹⡎⡻⣮⡻⣿⣿⣿⣷⣮⣓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣳⣿⣿⣿⣟⠘⠀⢠⣿⠄⠀⠀⠀⠀⠀⣷⠀⢸⠏⣿⣿⣿⣌⢿⣿⣿⣿⣿⣻⣿⡢⡀
⠀⠀⠀⠀⠀⠀⢨⣿⣿⢣⣿⣿⣿⠀⣦⣼⣿⠀⠀⠀⠀⠀⠀⢸⠀⡘⢀⣿⣧⢿⣿⣦⢻⣿⣿⣿⣿⣿⣿⣮⡢⡀
⠀⠀⠀⠀⠀⢠⣿⣿⡟⣾⣿⣿⣿⣇⠸⣦⡉⠀⠀⣠⣀⣄⣀⣰⡴⢁⣾⣿⣿⣾⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣮⣦⡀
⠀⠀⠀⠀⣌⣿⣿⣿⢣⣿⣿⠟⠛⠛⠀⢿⣿⠆⠈⠈⢋⠈⠁⠟⠀⠛⠛⠻⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠰⣼⣿⣿⣿⣿⣿⠃⠀⠈⠻⠄⣿⣿⣗⠲⡌⢻⣤⠛⢀⣠⠀⠀⠀⠘⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣆⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡷⡀
⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⡄⠢⣀⠀⠀⠀⠈⠿⠿⠿⠀⠀⠀⠀⢸⡀⠀⠀⢠⠋⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢄
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡀
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡀⠀⠈⢻⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡔⡄
⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⣿⣿⡞⣿⣿⢷⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⡀
⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠛⠀⠀⠀⠆⠀⠤⠤⠀⣿⣿⣿⣿⣇⣿⣿⣾⢣⡟⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣷⠁⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣽⣷⡀⠀⠀⠀⡄⠀⠀⠀⣒⡛⠛⣶⣶⣿⢸⣿⣿⣿⣿⣿⣿⣼⢳⡏⣿⣼⢿⡏⣽⡿⣿⣿⣿⣿⣇
⡟⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⣟⠦⣄⡀⡇⠀⠀⢀⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⢻⣿⡟⣿⣼⣇⣿⣿⣾⣻⣏⣿⠏⣿⣿⣿⣿⡄⢇
⡇⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣟⣷⣿⣿⡇⠀⠀⠸⠛⠿⣿⣿⣿⠿⣿⣿⣿⣿⢸⣿⣇⣿⣿⣼⣿⣿⣿⣿⣼⣯⣼⢻⣿⢻⣿⣷⢸
⡇⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣷⠀⠀⠸⣿⠙⠉⠀⠀⢿⡇⠀⠀⠃⠀⠀⡠⠈⠁⠆⠘⣦⢹⡿⢸⡿⠋⠉⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⣿⣾⣿⣿⣾
⡇⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣷⠀⠀⠿⣀⠀⠀⠀⣼⠁⠀⢀⡤⠐⠁⠀⢀⣣⢼⠄⢆⣠⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡯
⠇⣿⣿⣿⣿⣿⣿⣿⣿⣌⣷⠿⠟⠉⢻⣿⣴⡆⠀⠀⠐⠀⢠⡝⡀⠀⡉⠁⢠⣀⠔⠋⢙⠀⠀⠈⣧⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢀⣿⣿⣿⣿⣿⠟⠋⠁⠉⠋⠾⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠑⠄⠁⠘⣥⠃⠀⠀⠈⠒⠒⠋⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢰⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⡆⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡟
⠘⢹⣿⣹⡏⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢨⡟⢻
⠀⡼⣿⣿⡃⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡎⠀⣿⣤
⠀⠠⢿⣿⡇⠀⠀⠀⠀⠀⠀⡻⠀⠠⢆⢳⠀⠈⠛
⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢷
⠀⠀⠈⢫⣿⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄

⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠶⠶⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⡄
⠀⣰⠟⠁⠀⠀⠀⠀⠀⡀⢠⣄⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠐⠋
⢰⡏⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⣿⠾⣷⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠉⠀⠀⠈⠙⠛⠛⠳⢦⡀
⣿⠀⠀⠀⠀⠀⢀⠀⠻⣶⣌⡻⣝⠷⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⡯⠴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣿⠀⠀⠀⢠⣄⡙⢿⡶⣌⣳⣝⠻⢷⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠹⣆
⢻⡆⠀⠀⢄⡙⢿⠲⣽⣦⡙⠻⢷⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⡟⣠⠔⠀⠀⠀⡠⠂⣠⣾⠀⣠⣊⡴⠋⠹⡆⠂⠀⢸⣿
⠈⢿⣄⠀⠀⠉⠛⠷⠀⠉⠋⠀⠀⠀⠀⠀⠠⡀⣼⠃⠀⠀⠀⠀⠀⢸⡟⠁⠀⣀⣴⣫⠴⠞⠁⠾⠟⠛⠁⠀⠀⠀⠹⣦⢀⣤⣿⣀
⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠘⣷⠶⠛⠋⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⢻⡿⠀⡇⠈⠑⢦⠀⠀⠀⢠⠾⠻⠞
⠀⠀⠀⠀⠈⠙⠳⠶⠦⣤⠤⠶⠖⠛⠁⠉⠛⠾⣿⠄⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⢉⣉⡙⠃⠀⡤⠀⠀⠀⠋⠉⠀⠀⢸⡇⢸⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠸⣿⡿⠃⢸⠇⠀⠀⠀⣾⣿⡇⠀⢸⡇⢸⣀⣤⡀⢸⠀⠀⠀⢤⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠐⠆⠀⠀⣾⠀⠀⠀⠀⠨⠍⠀⠀⣸⠗⠛⠷⠄⠙⢯⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⣠⡴⠖⢒⡛⠛⠿⠿⢷⣶⡶⢤⣄⡀⣽⠀⠀⠀⠀⠀⠀⣹⠀⠰⣄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠛⠡⠃⣴⠁⣠⡴⠶⠶⠦⣄⢙⡆⠙⠙⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⠻⣶⠁⠐⠛⠋⠀⢀⡽⠇⠀⠀⡻⢤⣴⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢀⠀⠈⠙⠛⠒⠚⢛⠉⢀⠀⠰⣀⣹⠏⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠤⣼⣀⣀⣇⣀⣸⣠⣼⣶⠞⠋⠁⠀⢠⠷⡶⡖⢚⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢨⠉⡏⠈⡄⠀⢸⠀⠀⠀⠀⠘⡶⠧⠵⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣇⡎⢰⠀⢸⠀⣧⢘⡴⠳⠤⣀⡀⠀⡇⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠈⠑⠻⠦⠾⠴⠞⠋⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠇⢹⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣄⣷⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠦⣄⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⣀⣀⣄⣀⡀⠀⠀⠀⢸⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠉⠉⠉⣿⡇⠀⠀⣴⡿⠋⠉⠉⠙⠻⣦⡀⠀⢸⣿⠉⠉⠉⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⣿⡇⠀⣸⡏⠀⠀⠀⠀⠀⠀⢹⣷⠀⢸⣿⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⣿⡇⠀⢻⣇⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣿⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠈⠻⣦⣀⡀⢀⣀⣴⡿⠁⠀⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⣿⡇⠀⠀⢀⣴⡶⠿⠿⢶⣦⡀⠀⠀⢸⣿⠿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⣿⡇⠀⢰⡿⠁⠀⠀⠀⠀⠈⢿⡆⠀⢸⣿⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣿⡇⠀⠸⣷⡀⠀⠀⠀⠀⢀⣾⠇⠀⢸⣿⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠈⠻⠷⣶⣶⠾⠟⠁⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⡇⠀⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⢸⣿⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠉⠉⠉⣿⡇⠀⢀⣾⠟⠉⠁⠈⠉⠻⣦⡀⠀⢸⣿⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⣿⡇⠀⣾⠇⠀⠀⠀⠀⠀⠀⢹⣧⠀⢸⣿⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣿⡇⠀⢿⣇⠀⠀⠀⠀⠀⠀⣸⡏⠀⢸⣿⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠈⠻⣦⣄⣀⣀⣠⣾⠟⠀⠀⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠉⠙⠋⠉⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛
⣀⣤⡤⢤⣄⡀⠀⢻⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⠧⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻
⣿⠀⠀⠀⠀⠉⢧⣨⣶⣇⠀⡴⠋⠉⣹⠂⢀⡤⠞⠋⠹⣆⣤⣤⠀⣤⣤⡄⡾⠛⠛⣷⣼⡇⠻⠶⡄⢸⠟⢿⣀⣤⣤⣤⠤⢤⡀⠀⢀⣀⡴⠚⢻⡇⢸⠀⢸
⣿⠀⢠⣶⠀⠀⢸⢿⠀⢸⠾⣁⣀⡉⠙⢦⣼⠀⣴⡄⠀⣿⡇⠘⠶⠃⢨⡟⣙⣲⠀⠳⡄⡇⢰⣚⣛⢻⠀⠘⡏⡇⠈⣡⡀⠀⢹⠖⡟⠁⠴⠄⠀⡇⢸⠦⣼
⣿⠀⠀⠀⠀⢀⠎⢸⠀⢸⡄⡨⠤⠽⠀⢸⠛⢤⣠⡀⠀⡏⢷⣀⠀⠀⡸⠀⣇⣀⣀⡴⠃⠹⣄⣈⣹⠟⢤⡤⠇⠷⠤⠇⠹⣤⡼⠀⠙⠤⡆⠀⠀⣷⢾⡖⣧
⢹⠦⠤⠔⠚⢁⣤⠈⠉⢉⠐⠧⣄⣀⡤⢊⡤⣄⣀⡇⠀⢳⣀⠉⠛⠋⢀⣤⣀⣀⣀⣠⢦⣀⠀⢉⡠⣤⣀⣀⢶⣀⣀⡖⢦⣠⠴⠶⢲⠀⣧⠀⠀⢹⣀⡽⠃
⠀⠙⠓⠚⠋⠁⠈⠉⠉⠉⠙⠒⠒⠒⠊⠁⠀⠈⠉⣷⠀⢸⡇⠙⠒⠒⠉⠀⠀⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⢾⣣⠇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠒⠃⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⠤⢤⡤⢞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣌⣙⡒⢒⣒⡽⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠙⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠⣤⣶⠀⠀⠀⠀⠈⠛⠛⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣦⣀⠀⠈⠉⠙⠢⣤⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠉⢱⣶⣾⡿⠿⠿⣷⣤⡌⠁⣠⠀⠈⠛⠛⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠉⠙⠻⣧⣴⡚⠃⠀⠀⠛⠓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢷⣤⣤⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⣈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣦⡀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣴⠿⣀⠀⠀⢠⣤⣤⣄⣀⣀⠀⠀⠀⣴⣿⣄⠀⠻⣦⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⢰⣿⡀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠛⢿⣦⡈⠻⣄⠀⠘⠓⠚⠃⣸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠙⢿⣦⣈⣢⡀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣶⣾⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⣿⡟⠛⠛⠿⠿⣿⣿⡿⠿⢻⣿⠿⣿⠛⠛⠛⠻⢿⣿⣿⡟⠿⣿⠗⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠈⢿⣦⠀⠀⠀⠙⠛⢛⣴⡿⠃⠀⠻⣦⡄⠀⠀⠈⠛⠛⢓⣴⠟⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⠛⠛⠛⢻⣿⠟⠀⠀⠀⠀⠀⠙⠛⠛⠳⠚⠛⠛⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⣠⠛⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠻⣤
⠀⠀⠀⠀⠀⢿⣀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠛⠟⠻⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⣤⣤⣤⣤⡼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣤⡞⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠻⠿⣶⣶⠦⠶⠤⢤⣶⣶⠤⢶⣶⡾⠿⠿⠻⣿⡟⠛⠛⠛⠛⠉⠉⠀⠀⠀⣿⠟⣤
⠀⠀⠀⠀⠀⠀⠀⣶⠉⣷⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⠁⠀⢀⣿⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿
⠀⠀⠀⠀⠀⠀⣾⠁⠀⡟⠀⠀⠀⣴⠉⠀⠀⠀⠀⠀⢸⠀⠀⣼⡏⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⡆
⠀⠀⠀⠀⠀⣤⠋⠀⣼⠀⠀⠀⠀⣿⠀⢠⣤⠀⠀⠀⠙⠃⠀⣿⠀⠀⠀⠀⣤⣤⣠⣿⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⣿
⠀⠀⠀⠀⠀⠛⠀⠀⠿⠀⠀⠀⠀⣿⠿⣿⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⣬⠛⠋⠋⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⣶⡀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠠⡤⠤⣼⣿⠀⠀⠀⣿
⠀⣠⡄⠛⣿⠟⠃⢀⣴⣾⣷⣀⠀⠀⠀⠀⠙⣶⠀⠀⠸⣿⠇⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⠀⠀⠀⢿⣄
⠀⢻⣶⠟⠁⠀⣤⡟⠉⢀⣴⣿⠀⠀⠀⠀⠀⠈⣷⣀⢀⣿⡄⣀⣠⣴⣿⣤⣤⣤⣤⣤⣤⣤⡟⠋⠁⠈⢹⣿⠀⠀⠀⢸⣿⠃⠘⠁⣀
⠀⣾⠁⠀⢀⣾⠋⠀⣠⠿⠉⠻⣷⡄⣤⣤⡤⠛⠛⠛⠛⠋⠉⠉⠀⠉⠀⠀⠀⢀⣠⣤⡿⠟⠛⠛⠛⠛⢻⣿⠀⠀⠀⢸⣿⠛⠛⠛⠻⣤
⣿⠀⠀⣤⠟⠁⠀⣤⠋⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⣿⠃⠛⢦⣤⣭
⠈⠻⣾⠟⠀⠀⣴⠅⠀⢀⣿⡁⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⠿⠿⠿⢿⣥⣤⣴⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣠⣿⠃⢠⠀⠈⠛⢧
⠀⢠⣿⠀⠠⣿⡁⢀⣴⡿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⣿⡇⠀⠙⣷⣤⡎
⠀⠀⠻⣷⣤⣼⠟⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠷⠉⠁
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠲⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣶⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡄
⠀⠀⠀⠀⠰⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⣶⣾⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣾⣿⡿⠿⣿⡿⠿⠛⠻⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⢈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠧⠴⢿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠀⠈⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⣀⡀⠀⠀⠈⠛⠛⢷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣤⡤⠛⠋⠀⠀⠀⢀⣴⣿⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢈⡿⠻⣦⣤⡀⠀⠀⠀⠀⠈⠛⠛⠻⣤⣤⣤
⠀⢠⣤⣤⡈⠛⠃⠀⠀⣠⣤⣤⡾⠿⠛⠁⠈⠿⠧⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠉⠛⠛⠛⠶⠴⣶⣤⣤⡤⠌⠛⠛
⣤⣤⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠉⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣴⣦⣤⣤⣶⣶⣶⣄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣤⣜⠀⠐⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⡇⡿⢳⡞⡏⣸⢸⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⡇⡇⢸⠇⡇⣿⢀⣿⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠇⠧⠿⣼⡆⠧⠯⠸⠇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⣶⢶⢻⡏⡍⣉⣙⠛⡉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣀⣸⣇⣇⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢸⣿⣷⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠖⠒⠞⠹⠟⠷⣾⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠦⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣳⣦⠄⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢸⣀⣀⣀⣀⣀⡀⠠⠤⠤⠤⠤⠔⠒⢻⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡷⢄⣴⣾⣿⣄⠰⠦⠀⠀⠻⢤⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠧⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢧⡀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⢇⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠳⢤⡀⣀⣀⣠⡤⠖⠋⠁⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣁⡔⠁⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⢄⠀⠀⠙⢆
⠀⠀⠀⢀⡾⠳⠦⣤⡤⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡬⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⣇⠀⠀⠈⢳⡄⠀⣠⣤⣾
⢠⡤⠶⠋⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⢩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⡀⠀⠀⠀⠹⠞⠙⠁⣾
⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢦⡾⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢠⡤⠤⠤⠖⠊⠁⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠅⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⣀⠀⠀⠀⠘⡆⠀⡼⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣧⣀⣀⣠⣤⣤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣠⡞⠋⠉⠛⣦⠀⠀⠙⠞⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣠⠞⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣠⠴⠒⠒⠋⠉⠉⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠞⠁⠀⠀⠀⣀⡤⠴⠋⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣠⠶⠿⠥⠴⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣦⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣽⣷⣤⣄⣀⣤⣴⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⠀⠀⢾⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⡆⣭⣿⣿⣤⣶⣄
⠀⠀⣀⣠⣤⣴⣿⣧⣿⣞⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⠿⡄⣸⠃⣌⣽⣿⣿⣿⣿⣿
⠀⣸⣿⣿⣿⣿⣧⢿⠺⣹⣦⡀⠀⠀⢰⣿⢟⣥⣤⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢴⣷⢄⣟⣌⣿⣼⢫⣿⣿⣿⡟
⢰⣿⢻⣿⣿⣿⣿⣿⡇⣻⣿⢻⣄⠀⢸⣿⣿⣿⣧⡟⠆⣀⠈⠉⢿⣿⣿⠋⣁⢠⣿⣾⣿⠘⣿⣼⡟⢛⣵⡿⣿⡟⠁
⠈⠻⣿⣿⣿⣿⣿⣿⡟⣿⣿⣵⣽⣷⢸⣿⣿⣿⣯⣷⣥⣿⣧⣰⣿⣿⡇⣸⣬⣟⣿⢿⣿⣧⠋⣿⣿⣭⣭⣿⣿⣤⣄⡀
⠀⠀⢈⣿⢿⠻⣿⠟⣷⣾⣿⣿⡿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⣿⣟⣽⡿⣿⠏⣚⠿⣿⣆
⢠⡾⢿⣿⣿⣿⣿⣿⣷⡜⢿⡿⣣⣦⣹⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣇⣿⠹⣿⣿⣿⣿⢃⣾⣿⣻⣟⣻⣿⣿⣿⣷⣦⣿
⠀⠀⣨⣿⡿⠿⣿⣿⣿⣟⣈⢄⣿⣿⣧⣽⣿⡿⢿⡿⣟⡻⠮⠝⢿⣿⣿⠰⠞⠿⣟⣿⡇⣸⣯⣽⢹⡟⠛⣭⣻⣿⠛⠛⠉
⠀⢸⢛⣩⣴⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠫⣝⡲⠦⠍⢁⠤⠎⣴⣿⣿⣻⣿⣿⣿⣻⣷⣾⣿⣿⣿⡆
⠀⠻⢿⡿⣿⣿⣿⣿⢿⣿⣭⣍⣿⠻⣿⢿⣿⣿⣿⣿⣿⣧⣀⠄⢭⢤⠵⣂⣼⣿⡟⣱⣿⣿⣿⣿⣿⣌⣿⣷⡋⣿⣧
⠀⠀⠀⠀⣿⣿⣻⡗⠲⣽⢷⣮⣿⣿⣦⣄⣿⡿⣹⣿⣿⣿⣟⠿⠿⠿⠿⢛⣽⣿⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⠷⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⢟⣼⣿⢀⣼⣿⣿⣿⡿⣿⡇⣿⣿⣻⣿⣿⣷⣾⠛⠛⢿⣿⠛⣿⣿⣿⣾⣿⣧⡽⠿⠷⢶⣾⡛⠉
⠀⠀⠈⢿⣿⢿⣷⣄⣹⣡⡾⣿⡿⢛⣿⣿⣿⣳⣿⣿⣟⣷⡄⠈⠁⠀⠀⠀⠀⢰⣯⠿⣿⣿⣿⠿⢷⣶⣿⠿⠋
⠀⠀⠀⠀⢹⣿⣮⡙⠛⠛⣛⣽⣿⠿⠛⣿⣿⣿⣿⠫⣾⣿⣿⡄⠀⠀⠀⠀⠀⣿⡿⢣⣞⢷⣶⣦⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⠟⠻⣿⣷⣘⣿⣿⡇⠀⠀⠀⠀⠠⣿⡴⠿⠋⠀⠈⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠻⡿⡿⠿⠛⠁⠀⠀⠈⠉⠛⠿⠋
⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠋⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢀⠄⠀⠙⠻⣿⣷⣤⠀⢠⣶⣿⡿⠛⠁⠀⠀⠄⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡛⠁⠀⢀⠀⠀⠀⠈⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⡔⠁⠀⠛⠛⠀⠠⠀⠀⠀⠈⠁⠀⢠⠀⠀⢃⠀⠀⠈⠀⠀⠀⢠⠀⢸⣿⣿⠿⠧⢄
⠀⠀⠀⠀⠀⠀⠒⢰⡆⠀⠀⠀⠁⠂⠀⠀⠀⠐⢃⡀⠀⡀⠂⠄⠀⠀⠠⠔⠁⠀⠘⠛⢋⡀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⡤⠂⠀⠀⠀⢀⡈⠀⠠⠂⠀⠂⠀⡀⠀⠀⠀⠀⠀⢸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢠⠎⠆⠐⠀⠂⠀⠀⠀⠀⠠⠀⠠⠄⠀⠈⠠⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠐⠀⢆⠀⠈⠀⠀⠠⠀⡀⠀⠀⠃⢀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠈⠀⠤⠠⠊⠀⡈
⠀⠀⠀⢀⠞⠁⠀⠉⠀⠀⡂⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⡠⠤⠎⠁⠉⠳⡄
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠑⠂⠤⣄⣀⣀⣀⣀⡠⠤⠒⠀⠉⠁⠀⠀⠖⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠑⠤⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠒⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠚⠋⢩⠉⡭⠉⣳⡄⠀⠀⠈⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⣠⠒⠨⠅⠀⠤⣒⣒⣉⠩⠭⠔⠒⣉⡴⠃⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠸⣭⣉⣉⣉⣁⣁⣠⡤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⢀⣤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⢸⠀⣠⡞⠁⠀⢀⡀⠈⠳⡄⣰⢺⠋⠀⠀⠀⠙⢦⡟⢰⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⣴⠃⠀⢀⣴⠮⠴⠋⡴⡛⠀⢠⡾⣅⠉⡆⠀⢹⠁⡇⠀⣰⡋⠳⡄⠀⣧⢸⠀⢠⡙⣆⠀⢹⡀
⠀⠀⠀⣸⠉⠀⢀⡏⣀⣤⣤⣴⡇⡇⠀⢸⠀⠘⡇⡇⠀⢸⢸⠀⠀⣏⢹⠀⡇⠀⣹⢸⠀⢠⣷⢸⠀⠘⡇
⠀⠀⠀⣿⢃⠀⠈⣿⢻⣀⠀⠀⡇⡇⠀⠘⣆⢸⣧⠃⠀⡜⢸⠀⠀⢿⢸⢠⠇⠀⣿⢸⠀⠘⣿⡞⠀⢰⡇
⠀⠀⠀⢿⠘⣄⠀⠈⠑⠚⠀⠀⡇⠸⡄⠀⠈⠛⠃⠀⣰⣧⠀⣇⠀⠈⠛⠊⠀⢠⣿⢸⠀⠀⠉⠀⣠⡟
⠀⠀⠀⠈⢧⡈⠓⠤⢄⣀⣀⣠⡿⣄⠈⠲⢤⣄⣤⣴⠏⠘⢧⡈⠳⢤⣀⣀⣤⠟⣿⢸⡤⠤⢒⣿⠏
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠋⠀⠈⠓⠶⠶⠶⠛⠁⠀⠀⠀⠉⠓⠒⠒⠛⠁⠀⠻⠶⠖⠚⠋⢀
⠀⠀⠀⢠⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡳⣄
⠀⠀⠀⡏⠀⠳⣄⡀⠀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣄⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⣀⡼⠀⢹
⠀⠀⠀⢷⡀⠀⠈⠉⠙⡅⠀⠀⣀⠀⠈⠳⡄⠀⡴⢻⠁⠀⠀⠘⡇⢠⡎⡇⠀⠀⠀⠉⠓⢏⠁⠀⠀⡼
⠀⠀⠀⠀⠙⠦⣄⣠⠂⡇⠀⢰⣦⠕⠀⠀⡇⢰⢇⡟⠀⢰⡆⠀⢿⢸⡇⡇⠀⢰⠒⢤⡀⠀⢳⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠠⣾⢃⡟⢸⠁⠀⠺⠇⠀⢸⣾⡇⡇⠀⠘⡷⡄⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠠⡇⠀⢐⠒⢄⠀⠈⣿⢁⡇⠀⠀⣀⡀⠀⠀⣿⡇⡇⠀⠀⣧⡧⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠇⠀⠿⠷⠈⠀⠀⡸⡼⠀⠀⢸⡇⢳⠀⠀⢹⡇⡇⠀⠀⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣅⣀⣀⣀⣀⣤⣾⢱⣁⡀⢀⡟⢷⠸⣄⣀⣈⡇⠧⡠⠤⠤⢴⣲⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⠴⠖⠋⠙⠶⠶⠤⠟⠁⠘⠒⠒⠲⠞⠓⠒⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣿⡿⠻⣿⡿⢻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠁⠘⠋⠁⠀⠙⠁⠀⠈⠀⠙⠉⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⢿⣿⠇
⠀⠀⣀⠄⠀⠀⠁⠠⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠛⠃⠀⠀⠙
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⠀⣠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⡔⠀⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⣇
⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⠋⠀⠀⠀⡀⠀⢀⠐⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷
⠀⠴⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠠⠄⠐⠺⢿⣿⡿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠤⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠛⣿⡟⠉⢿⠏⠻⠿⢿⡿⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀⠆⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠁⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠉⠙⢿⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⢿⠃
⣾⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡀
⣿⣿⡇⠀⠀⠀⠀⣀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣧
⣿⣿⡇⣴⣶⡄⢸⣿⡧⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣿⣿⠀⢰⣿⠈⢿⣿⡇
⣿⣿⡇⣿⣿⡇⢸⣏⢰⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⣿⡇⣴⣶⣶⣶⣶⣤⡀
⣿⣿⡇⢿⣿⡇⢸⣿⡆⢙⠛⠛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣿⣿⠀⣿⣷⣌⡛⠛⠛⢿⣿⣷
⣿⣿⣷⣤⣭⣴⣦⣭⣴⡿⠟⠛⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣥⣶⣬⣥⣾⣿⣿⣿⠿⠟⠃⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣡⣶⣿⣿⣿⣿⡿
⢻⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⡇
⠈⢿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢠⣴⣄⠀⣠⡴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠈⠙⣛⣦⡼⠿⠷⣾⢋⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⢱⣶⣶⠻⣯⣉⠉⣹
⠀⠀⠀⢀⣀⠀⢀⣤⡄⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⠀⠀⠙⠛⠋
⢀⣀⠀⣾⣿⡇⢸⣿⠟⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⣦⡀⣾⣿⡇⢸⣿⣿⠀⣤⣶
⣿⣿⡇⣿⣿⡇⢸⣯⠰⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⣿⣿⡇⢸⣿⣿⢸⣿⣿
⢿⣿⠇⢿⣿⠇⠸⣿⡆⢉⣛⣛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣀⠻⠟⢃⠙⠿⢃⣼⣿⡿⢸⣿⣿
⣦⣤⣾⣦⣤⣶⣶⣤⣶⠿⢛⣋⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⣉⣤⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⢡⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⣿⣿⣿⡏
⠈⢻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡠
⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢰⠋⠀⠀⠀⢠⣶⣤⡀⠀⠹⡄⠀⠀⣀⢀⣀⠀⣀⡤⠤⣀⠀⠀⠀⠀⠁
⠀⠀⡟⠀⢀⡠⣌⣳⣿⣟⠅⠀⠀⡇⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠓⠤⢄⡀
⠀⠀⢹⡀⠀⢿⡻⠯⠳⠈⠀⠀⣰⠃⠀⡔⠀⣀⣀⣀⡠⠤⢄⣀⠀⠀⠀⡁⠀⠀⠀⢀⡄
⠀⠀⠀⠙⠦⣀⠀⠀⠀⣤⡀⢶⠁⠀⠀⢱⢄⡅⠤⣄⠀⠀⢀⡤⠄⠘⠒⠺⠄⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠛⠀⠀⠀⠸⣀⠦⠰⠶⠠⠠⡰⠶⢄⡀⠀⠀⠈⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⣀⠠⠤⠀⡀⠀⠔⠛⠈⠣⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠚⡃⠶⠶⠶⢺⠂⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⣀⣀⡠⠔⠁⠃⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠃⠈⡀⠀⠀⢨⡍⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣖⠏⣱⠤⠆⣹⡖⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⠀⢳⢄⠀⡈⣠⡿⠃⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣷⠾⠋⠁⠀⠀⠀⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠐⠀⠉⠉⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣧⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣿⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠢⣿⠀⠀⠀⠀⢸⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠀⠸⣿⣦⣤⣼⡟⠀⠀⠀⢀⠌⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠘⣶⣄⠀⣰⣿⡏⠈⣧⠀⠀⢉⣻⣿⡟⠁⠤⠤⡄⠁⠀⠈⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢿⠈⢿⣿⣿⡀⠀⠘⢿⣾⣿⡿⠋⠀⣀⠀⠀⠃⢀⣆⡼⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⢸⠀⠈⢿⠈⢳⠀⢀⠀⠈⠁⠀⠀⠀⣸⣿⣶⡶⠟⠋⠀⠀⢘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠙⠳⢦⡀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠈⡓⠤⡀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⢸⠀⠱⣄
⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣾⠀⠀⠀⠈⣇⡾⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠲⠛⠃⠘⣲⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠿⣄⠀⢀⣿⠀⠀⠀⢀⣈⣤⠎⠀⠀⠀⢰⣾⠀⠀⠳⠦⠴⠀⠀⢦⣼⠁⡀⠀⠀⠀⡀⠀⠀⢘⡇⣹⣷⡀⠳
⠀⠀⠀⠀⠀⢸⠀⠈⠙⠛⢁⡤⠶⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⡄⠀⠈⠠⣤⣄⣶⡿⡁⢰⡇⠀⠀⣴⡇⠀⠀⠀⠟⢉⡿⡇
⠀⠀⠀⠀⢠⠆⠀⠀⢠⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠂⠤⣼⣿⣿⢂⠀⠀⠠⣴⠏⣴⣇⠞⠃⢀⠂⣹⠁⠀⠀⠀⣠⠟⠁⡇
⠀⠀⠀⣰⠃⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠛⠀⢷⣦⣾⠿⣆⠈⢁⣀⣀⠂⢠⠃⠀⠀⢠⠞⠋⠀⠐
⠀⠀⢰⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠘⠉⠇⠀⢀⣾⠟⠛⠃⠀⣮⣄⠀⠀⠀⣠
⠀⠀⢸⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠸⠋⣿⣷⣄⠜⠿
⠀⠀⠀⠑⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⡀⠀⠀⠀⠀⢰⡄⠸⣇⣀⠀⠀⠀⠀⡏⣽⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⠀⠀⠀⠀⠀⣴⡿⣿⣿⣿⣿⡄⠀⢀⣀⣸⣿⣤⠟⠁⠀⠀⠐⠊⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⠀⠀⠘⠁⠀⠀⠉⣻⣿⣿⡿⠿⠟⠛⠋⠁⠀⠀⠀⢀⣴⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⢠⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢺⣻⣿⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠁⠀⠀⠹⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⣿⠟⣿⣦⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡗⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠇⠀⠸⣿⠻⣿⣧⡀⠀⠀⠀⠀⣀⣴⣿⡿⠟⠋⣽⣿⡇⠀⠀⠀⠀⠀⠀⢀⡞⠁⣼⡇⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⢻⡆⠈⠛⢿⣦⣤⣴⣾⡿⠿⣷⣦⣤⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⠈⠉⣼⠏⠀⠀⠀⢨⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠣⠀⢰⣏⣠⣄⠂⠈⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡟⠉⠉⠉⠉⠁⠀⠸⣿⣁⡀⢀⠄⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡜⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣭⣥⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⣠⣶⠉⠠⠖⠛⠋