Ye Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⢤⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⠀⠀⠀⠀⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡇⠙⢀⡘⠃⣧⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠛⠛⠛⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⣈⣡⠔⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢠⠀⢳⣰⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⢠⣄⠀⠀⢠⣤⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⡇⠀⢸⠀⠈⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠘⠧⣼⠀⠈⠁⠀⠀⠈⠁⠀⣿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠧⠤⠿⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠛⠛⠀⠀⣰⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠙⠲⠶⠶⠶⠛⠁⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠻⡆⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⢤⣤⠤⠆⠀⢸⠀⠀⠀⢻⣼⠃⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⡋⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠘⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⡇⠰⠀⠀⠆⢸⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⣰⣧⡀⠀⢀⣼⡀⠀⠀⢀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⡘⢃⣠⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠈⠙⠛⠋⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠉⠉⠀⢀⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢲⡆⠀⢸⠀⠈⠟⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣀⣸⣧⣀⣼⣀⣀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠚⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣽⠄⠤⠴⠂⠀⠀⠸⣦⣠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠆⣻⣶⣷⠆⠀⢤⢄⡀⡉⠑⠛⠻⠦⠘⠿⠇⠀⠀⢠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣣⢜⠿⢋⢾⣤⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⢿⣿⠟⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡏⡎⣰⠃⢸⣿⠃⡰⠂⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄⣶⣼⣿⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢿⣏⣁⡄⠞⠃⣈⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠘⠉⠭⣿⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣅⠰⠀⣈⣿⡇⠀⣾⣿⠛⡏⠉⢹⠟⠳⠶⣴⣤⣤⡀⡲⠂⡀⣋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡁⠀⠿⠟⠀⣠⣿⣇⠘⠛⢀⣠⣤⣤⣤⣤⠬⢿⣿⣷⣦⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠀⣀⣴⣿⠟⠛⠁⣴⣿⣿⣿⠟⠋⠀⣀⡾⠿⠻⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣾⡿⠟⠛⠀⢀⣷⣿⣯⣷⠶⠶⢾⣍⣉⡀⠀⠀⠀⠈⠛⢿⣿⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡶⠛⠁⠀⠀⠀⡙⢛⠛⠶⠀⠀⠀⠀⠉⢻⣿⣿⣶⣦⣤⣼⠿⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⢐⡐⠂⠂⠒⠀⠀⠀⠀⠀⠤⠖⠛⠙⠋⠉⣙⣿⣷⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠷⣶⣤⣄⣀⡉⡀⠀⠀⠀⠐⠁⠀⠀⠀⡰⣿⣿⣿⣿⡿⠋⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀⣠⠶⣦⣸⣿⡿⢟⣉⣠⣴⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣀⣀⣀⣀⣤⣶⣏⣵⣶⢟⣫⣮⣚⡋⡵⣗⠶⣿⢆⣐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡽⠷⠹⡿⠟⠛⠉⠀⠠⠶⢿⢷⣬⡢⠬⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠛⠃⠒⠒⠀⠀⠀⠀⠀⠀⠼⢏⠈⢂⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢀⣴⠀⣣⣷⡱⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠇⠈⠀⢻⡿⠇⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠀⢀⣤⣤⣄⣀⣀⡀⠀⡀
⢸⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣯⡴⠾⠿⠿⠉⠀⠉⠀⠀⢀⣮⡤⢀⣀
⢸⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⡴⠖⣛⣯⡷⢿⣿⠶⢦⢶⣄⡀⠀⠀⢀
⢸⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣭⣛⠛⠶⠶⠖⠒⠶⠾⠯⠁⠀⠀⠀⠀⢀⣠⣽⣟⣛⠛⠒
⢸⠀⠀⠀⠀⠀⠈⢿⣿⡿⠿⣿⡿⣿⣶⣷⣬⣀⣀⡀⣀⣀⣤⣤⣴⣶⠿⠿⠏⠉⠙⠻⠯
⢸⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠉⠛⢻⡏⣉⢋⣉⠛⠛⠛⠛⠋⠙⠘⠒⠒⠖⣠⣤⡀
⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣻⣿⣿⣟⣛⠛⠲⢶⣶⣶⣿⣿⡿⣿⡛⠓
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣽⣿⣿⣿⣿⣶⣄⠙⢿⣯⣝⡻⠦⣍⣙⢒⡀⠀⣀⣀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢿⣿⣿⣿⠿⣿⣿⣿⣿⣦⣍⠛⢿⣿⣶⣿⣿⣿⣿⣿⣏
⢸⠀⠀⠀⠀⠀⣤⠠⢀⣠⣤⣤⣤⣄⠀⠈⠉⠁⠀⠙⢿⣿⣿⣶⣮⣿⣽⣿⣿⣿⣷⣾⣿⣿⣿⣿⡟⠛⠙⢷⣴
⢸⠀⠀⠀⠀⠄⠹⣶⣿⣯⣥⣤⣴⡶⢤⠒⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡀⠘⣿
⢸⠀⠀⠀⣤⠟⠘⣿⢻⣿⣿⣿⡟⠁⠂⠀⠀⠀⠀⠀⢀⡌⣿⠋⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠏
⢸⠀⠀⢀⣽⠆⣈⠃⠀⠈⢟⠹⠄⠀⠀⠀⠀⠀⠀⠀⢮⢭⡜⢢⠛⡋⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⡿⠃
⢸⠀⠀⠉⠍⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⡼⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃⠚⠋
⢸⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⣠⣿⣤⣤⣼⣿⣿⣿⣿⣿⠟⠛⠃
⢸⠀⠀⠀⠀⠀⠉⠉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢁⣤⣶⡍⠻⡿⠛⣿⠏
⢸⠀⠀⠀⠀⠀⠀⠀⠛⣿⣶⠶⠄⠀⠀⠀⠀⠀⠠⠀⣿⠁⠀⠸⠏⠀⠁⠀⠉
⢸⠀⠀⠀⠀⠀⠀⠀⠘⢧⡤⠄⠀⠀⠀⠀⣠⣤⣶⠇⠿⡄⠀⠀⠀⠀⠀⢀
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣤⡶⠛⠻⣿⠟⠀⠀⠀⠀⠠⢤⡤⠴⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣶⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣆
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣧
⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢸⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣇⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠈⠀⠀⠀⠀⢠⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢰⣶⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⠴⠋⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠉⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠉⠻⣦⡀⢀⣽⣶⣶⣶⣦⣄⡀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⠿⢷⡀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣀⣠⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠀⠀⠀⣠⡴⠞⠋⠉⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⣿⣠⡴⠛⠁⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⢼⣿⣿⡿⠛⠉⠀⠀⢀⣠⣿⣿⣤⣀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠈⠙⠳⣦⣤⣶⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠙⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⣠⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⣿⣿⠿⠋⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢰⡏⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⠟⠋⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡏⠀⠀⠀⠀⢀⡴⢿⡀⣠⣴⣾⣿⣿⣿⣿⡿⠋⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠴⠞⠋⠀⠀⠹⣿⣿⣿⣿⠿⠋⠁⠀⠀⢀⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⣄⣀⠀⢀⣤⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠕⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠠⠤⠐⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠤
⣀⠤⠄
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣤⣷⣴⣶⣷⣶⣧⣴⣦⣀⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣯⠁⣠⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⡍⠉⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣷⣿⣿⣃⣀⣿⣿⣿⣿⣿⣿⣿⣏⠀⢻⣿⣧⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠿⢿⣿⠿⠟⠋⠉⠉⠑

⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠒⠀⢠⠀⠁⠀⠂
⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠀⠦⢤⡀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡐⣄⢠⣾⣷⣠⠂⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠪⡿⢸⠟⣯⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠁⢀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠇⣦⡇⢿⢠⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⢀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⡇⠸⠁⡞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠈⠋⣄⠀⠀⠀⠀⠀⡄⢻⡀⣴⠃⠂⠀⠀⢀⠀⠀⡄⢁⣀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⢢⢹⡀⠀⠀⠀⠠⠁⠀⠻⠋⢀⠁⠀⠀⠄⢀⣀⣉⣌⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠓⠈⠢⠀⠤⠖⠲⡄⠀⠀⣠⠒⠲⠤⡶⡖⡲⢖⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣀⣽⣬⣒⡲⠴⠃⠀⠀⠘⠶⢧⣾⣭⣧⣐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠔⠙⠒⢦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡠⡢⡀
⠀⠀⠀⠀⢀⠤⠊⠉⠀⠀⠉⠑⢄⠀⠀⠀⠀⠞⡠⢲⠘⠀⠀⠀⠀⡠⠊⢠⢃⠀⠀⢀⡀⠤⠐⠒⠒⡄
⠀⠀⠀⠠⠃⠀⣀⡀⠀⠀⠀⠀⠀⠑⣀⣀⣠⠊⠁⣸⣀⡀⠤⠒⠉⢀⠔⢑⡡⠔⠊⠁⢀⡀⠤⠀⠚⠃⠐
⠀⠀⢀⡃⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠆⠁⡠⠤⠒⠈⢁⣀⡀⠀⠀⣠⡒
⠠⠒⠉⢈⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠁⠀⠀⣀⣉⣁⠀⠀⠀⠀⠀⢀⣀⡨
⠓⠋⠭⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠀⠀⢀⡗⠶⠚⠁
⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢄⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⠀⠀⢀⠀⡀⣀⠠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠫⢭⢆⡑⢲⡦⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⡴⠛⠉⠉⢉⡿⠻⡏⠉⠉⠛⠶⣄⠀⠀⠀⠀⣠⠶⠛⠋⠉⣿⠿⣿⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠀⢿⡀⠀⠀⢀⣾⣁⣀⣿⡄⠀⠀⢀⣽⠀⠀⠀⠸⣇⠀⠀⠀⣰⣏⣀⣸⣧⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠉⠛⠲⣟⠛⠛⠛⠛⣻⡗⠚⠋⠁⠀⠀⠀⠀⠈⠛⠒⢾⡛⠿⠿⠿⢛⣷⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⡖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⢠⠆
⠛⠋⠙⠃⠐⠒⠠⢤⣀⡀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⢄⣀⠀⢣⡠⠐⣲⣶⡆⢠⣥⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡕⠻⢷⣜⡟⢋⢶⣿⠏⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣿⠊⣿⣿⣓⣋⠉⠓⠲⣾⡿⠃⠀⠤⠤⠄⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠓⠲⣤⡀⠀⠀⠘⠋⣑⣨⠏⣿⣿⣿⣿⣶⣶⠉⣻⡒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠐⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠠⣾⡿⠟⢻⠀⢻⣿⣿⣿⣿⠛⢻⣿⡇⠀⠀⠈⠑⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣠⣿⡟⠛⣿⣿⣫⣏⣁⣤⣴⣶⠶⠶⢦⣄⠈⠘⠂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⠶⣦⡤⣶⡿⠟⠛⡧⢤⣷⡞⠓⣺⣿⣿⣏⠉⢁⣼⡷⢄⡀⠘⠷⢄⡀⠀⠀⠀⠉⠒⠀⠤⠤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣰⡿⠋⠙⢋⣤⣿⠿⠂⣠⣿⠆⡀⠀⠙⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⡆⠀⢰⡟⢷⡶⠾⠿⠉⠋⢀⣼⡿⠟⢀⣿⢄⠀⠀⠉⠻⠿⠿⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⠈⣄⣀⣠⣤⣾⠏⠛⠁⣀⣻⣿⢘⣆⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢃⠈⢷⡄⠘⠟⠉⠉⠁⣀⣤⢶⣿⠀⢠⣴⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠀⠈⠹⡄⠈⠲⡞⠛⠛⠋⢀⣠⣴⡟⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢃⠀⠀⠹⣆⠀⠀⠑⠺⠛⠛⣛⣉⣥⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠃⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠸⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⢺⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢆⣴⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢠⣦⣼⣿⣿⣿⣉⡹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⢠⣩⣼⣿⣿⣿⣿⣿⣿⣷⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣄⠀⠀⠻⣿⣿⣿⣿⣿⣿⢟⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣄⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⢿⠟⢻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣷⣟⢢⣸⡇⢸⣟⢶⣍⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣦⠹⣆⢸⣿⣷⣟⣵⣮⣼⣿⣿⣿⣿⣿⣿⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣸⣿⣿⣿⣿⣿⡿⣯⣿⡃⢹⣾⣿⣋⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠠⠰⠄⠉⠻⠛⣹⣿⣿⣿⡿⠯⡉⠀⠀⠀⠊⠳⣼⣿⠉⢡⠊⠘⠛⣿⣿⡿⠫⠈⠙⠻⠛⠛
⠀⠀⠀⠀⠀⠀⠀⡠⠉⠈⠀⠉⡿⠿⠉⢁⠀⣴⣿⣧⠀⠀⠀⡀⢸⣿⣿⡯⠀⠀⠀⠀⣿⠡⡄⢴⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⡠⣤⣾⣿⣿⣿⣿⣿⡳⣄⣦⣉⣿⣿⣿⣧⣴⣠⠔⠛⠀⠀⠊⣦⣼⣿⢈⡁
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠈⣹⣿⣶⣿⣿⣿⣿⣿⣿⡿⠃⠀⠙⢿⣯⣿⣧⣾⡛⠁⠀⠀⠀⠀⠀⢈⣿⣻⣾⠳⡦⡄
⠀⠀⠀⠀⠀⠀⠀⢀⢤⢷⣶⣿⣿⣿⣿⠿⠟⠟⠁⠀⠀⠀⠀⢾⣿⣿⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠛⡧⠀⠁⠉⠆
⠀⠀⠀⠀⠀⠀⠉⠹⠖⠉⠘⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠙⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣴⣸⣿⣶⣤⣤⣀⠀⠀⣀⡴⠞⢁⡴⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡟
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣤⣤⣶⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⡾
⠀⠀⢀⣠⣴⣾⣿⣿⠛⢿⣿⠿⠿⠷⠿⠿⢿⡟⡾⠻⡟⣽⠏⣼⢫⡾⢫⡿⣻⢿⡿⢻⣿⢿⣷⣶⣦⣤⣀⡄⡀⠀⠀⢠⣞⣀⡼
⠰⢾⡋⠉⢠⣴⣿⠞⠀⠀⠈⣿⣶⣶⡺⠿⠟⠿⠿⠟⠛⠳⢾⣷⡦⠀⠀⠀⠁⠊⠁⠈⠁⠈⠋⠘⠛⠛⠛⢛⣛⣿⣿⣿⣿⣿⠁
⠦⣤⣛⣄⢦⠉⠁⠀⠀⢀⠀⣾⠯⣅⠀⣀⠠⠤⠤⠖⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠋⠉⠉⠙⢿⡿⡾⣆
⠀⠀⠉⠛⠳⢦⣤⣤⣄⣿⣷⣿⣿⢿⣿⣷⣶⣶⣒⣀⣈⣋⣠⣤⣤⣤⢠⣀⣠⣤⣴⣶⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⢿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣿⠿⠿⠿⠿⣿⡿⠿⢿⡁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⠉⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣷⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠛⠛⠓⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠶⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⢦⡀
⠀⠀⣀⣤⣶⣿⣭⣴⣶⣶⡶⠶⠶⠿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠈⠳⣤⡀
⣴⣾⡿⠟⠋⠉⠉⣹⠟⣉⣻⣿⠶⣄⡀⠻⣿⡄⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣀⣤⣴⣶⣶⣖⠶⢮⡙⠳⣦⡀
⠀⠙⢷⣤⡀⠀⢰⡟⣿⣿⣿⣿⡆⡀⠙⢶⣽⣧⡀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⡼⠁⣠⠴⠋⣹⠟⣿⣿⣿⠉⢷⠀⠙⠷⣌⣹
⠀⠀⠀⠈⠙⠳⢦⣻⣦⣈⣛⣫⠞⠁⠀⢀⣨⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⣧⠞⠁⠀⠀⢿⡀⢿⣿⣿⣀⣸⠂⠀⢀⡼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠛⠻⠶⢦⣤⣀⣈⣳⣤⣭⡿⠉⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⣤⣤⣤⣤⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⣸⣷⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⡼⠇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⣿⣶⢿⣿⣳⡴⣾⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣰⠃⠀⠎⠀⠀⠱⠀⢻⡃⣀
⠀⠀⠀⠀⠀⠀⢶⣦⣦⣟⢿⠻⣷⣿⣾⢹⣟⣧⠿⣿⣿⣁⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠐⠁⠉⠐⢴⠀⠄⢠⢀⡄⠈⠉⢋⣠⣴⠂
⠀⠀⠀⠀⠀⣀⣈⣿⣟⣿⣧⣽⣮⣿⣿⣼⣿⣿⣿⣧⣿⣿⣿⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠤⢀⣀⠀⠀⡈⠘⠀⣠⣈⢀⠄⣠⠋⠜⣷⡶
⠀⠀⠀⠀⣀⣬⢿⣗⢟⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⡯⣉⣿⣽⣇⣠⣄⠀⠀⠀⠀⠀⢀⣤⠜⠫⡀⠀⠨⡐⢄⣴⠶⠶⢷⣿⣫⡎⠁⠀⢀⠔⢠⣤⡄
⠀⠀⠀⠀⢉⣿⡷⣿⣷⣾⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⡷⣿⣟⠁⠀⠀⠀⠀⠀⠀⠉⠧⠀⠘⠒⠰⡶⣾⣧⣣⣝⡝⣾⣿⠠⢁⣚⠓⠀⠤⣉
⠀⠀⠐⠾⣿⣿⣷⣿⣯⣿⣿⣿⣿⣟⣟⣿⣿⣾⣿⣿⣯⣿⣿⢿⣯⣿⠶⠀⠀⠀⠘⠻⢤⡀⠀⠀⠀⠀⠍⢻⣽⢸⣝⡹⣷⣟⣢⠤⠀⠀⠀⠀⠨⠿⠖
⠀⠀⠀⠀⣠⡿⡟⠷⣾⢿⣿⣿⣿⣾⣾⣿⣾⣿⣿⣟⡻⣿⣛⣿⣧⡀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠐⠶⠋⠄⠀⢛⡿⠷⢿⢯⠉⢍⠁⠑⢄⠈⢿⡤
⠀⠀⠀⠘⠙⠛⠋⣻⣧⣫⣿⣿⣿⡟⣿⣿⣿⡻⣿⣿⣷⣲⣿⣏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢀⡏⠀⠀⠀⡴⠨⠀⡙⡜⠆⠂⠰⠠⢀⣠⣷⡄
⠀⠀⠀⠀⠀⠀⠰⠟⠓⡿⣛⣹⣿⣷⣿⡿⣿⢧⣮⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠐⠂⠁⣀⠀⠀⠠⠑⡀⠀⠀⠦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠉⣯⣾⡿⣿⡿⠛⢿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠂⠀⠀⢠⠆⣤⡌⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠏⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠃⢀⠞⠀⠀⡀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡸⠓⠂⡀⠈⠀⠀⡀⠁⠀⠵⢶⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠃⠘⠀⠃⠞⠀⠀⢠⠓⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠘⢿⠋⠁⠘⠮⡂⠈⣈⠤⠴⢳⣧⢎⠔⠋⠈⠘⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⣀⣀⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠺⢿⡀⢑⣤⣀⣀⣨⠎⣥⣄⣀⣄⠁⡿⠠⢀⣤⡈⠀⡻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣄⠀⠀⡀⠀⠠
⠀⠀⠀⠀⣤⡤⠈⠁⠀⠐⠂⠘⢿⠀⢿⣿⣿⠟⠀⣜⡒⠒⠒⠂⠈⠑⢢⣄⡀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠑⠀⠄⢉⡩⠕⡈⢣⣀⠙⠉⢀⠴⡷⠏⠨⡅⠀⠤⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠃⠀⠀⠐⠶
⠀⠀⠀⠀⠠⠾⠤⠄⢠⠌⠀⠈⠠⡃⠟⢉⠍⢏⠋⠈⡂⠀⠈⣁⡔⠛⠁⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠄⣀⠠⠘⠁⠁⠈⠸⡈⠀⠀⢡⣁⡒⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠐⠀⠀⢀⡂⠘⠄⣠⡎⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠂⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣄⠏⠙⠿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣦⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⣾⣿⡿⠀⣰⣾⠏⠀⠀⠀⠀⢀⣀⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣀⠀⠀⠸⣿⣧⡀⢀⣾⣿⣿⠃⠀⣿⣯⠀⠀⢀⣴⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⠸⡿⠿⠘⠟⠛⠛⠂⠸⠿⣿⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠉⠀⣀⣠⣤⣴⣶⣶⣶⣶⣤⡀⠘⢿⣿⣿⠃⠀⣀⣠⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢿⣦⣤⣤⣶⣦⠀⢀⣤⣾⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣦⡀⠙⢃⣤⣾⣿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠃⠀⣾⣿⣿⠟⠉⠀⣀⣀⣤⣴⣶⣤⡈⠻⣿⣿⣄⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⢸⣿⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⢿⣿⣦⠀⢀⣀⣀⣠⣴⣶⣤⣄⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣀⣀⣤⠶⠿⢿⣷⣶⣶⣶⣾⣿⣿⣿⠀⢸⣿⣿⡇⠸⣿⣿⣿⠏⠀⠈⠙⢿⣿⣿⣆⠈⢿⣿⡇⢈⣿⣿⣿⠿⠟⠛⠿⢿⣷⡶⠟⠋⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⢉⠁⠈⣿⣿⣿⡄⢻⣿⣿⣦⣤⣶⠀⢸⣿⣿⣿⠀⢸⣿⠇⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⠀⠹⣿⣿⣿⣦⡙⠻⠿⠿⠋⣠⣾⣿⣿⣿⠀⢸⠟⠀⣴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣾⡿⠟⠉⢀⣤⡀⠙⢿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⠿⠃⠀⢀⡀⠘⠛⠛⠛⠻⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡦⠀⠉⠛⠛⠛⠿⠛⠛⠛⠉⠀⠀⢀⣴⣾⣷⣤⡀⠀⠀⠀⠀⠙⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠀⣾⣷⡦⠀⢰⣤⣤⣶⠀⢴⣶⣧⠀⠙⠿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠃⠀⠀⣸⣿⠁⠀⢸⣿⣿⡟⠀⠈⠛⣿⣷⠀⠀⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⠉⠀⠀⠀⠀⣴⡿⠏⠀⠀⣿⣿⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠈⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠤⠴⡖⢿⣟⢳⡦⠴⠶⠶⣦⣤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠉⠉⡀⠰⠂⠀⠀⠈⠛⠛⠉⠀⢉⣁⣭⢥⣴⣶⣶⣆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠶⢟⣿⡿⠿⠋⡅⣠⠀⠆⠰⠀⠀⠀⠃⠈⠀⠀⢀⢴⣾⣾⣿⣿⣿⣿⣿⡿⣿⠟
⠀⠀⠀⠀⠀⠀⣶⣾⣷⣶⣾⣾⠟⢉⡔⠐⠀⣼⠃⠀⠐⠀⠂⠓⣠⠊⡀⢀⢜⣵⣿⣿⣿⣿⣿⡿⣻⣭⠞⠁
⠀⠀⠀⠀⢴⣾⣿⠶⢶⣾⠟⠱⢠⡟⠁⡀⠂⢿⡐⠀⠆⠀⠀⣼⣇⠈⢴⣣⠞⠛⠛⠢⠭⠙⢻⣿⣿⣿⣷⣶⣤⣄
⠀⠀⢀⣴⣿⣽⣶⣿⠟⠡⠀⢀⠸⣷⡀⠰⠀⠈⠳⢴⣀⣀⡀⠉⠛⠋⢉⣁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⣴⣿⣿⣿⣿⣳⣶⡶⠤⠭⠀⢠⡝⠂⠐⠦⠤⣤⣤⣤⣤⠴⢾⣿⣯⣅⣀⠀⠀⠀⠀⠈⠉⠉⠉⣻⠿⠛⠉
⠀⣠⣿⣿⣹⣿⣿⣿⣿⠿⢿⣒⣠⠝⠧⠂⠀⢄⠀⠤⠤⠴⠶⣚⣛⣭⡿⠿⣻⣿⣷⣶⣶⣶⡞⠋⠉
⠀⣹⣿⣿⣧⣿⣿⣿⣿⣿⣯⠽⠒⠁⣠⠀⠈⠈⠳⣤⣈⠉⠉⠭⠥⠴⠖⠛⠉⢁⣠⠿⣡⠎
⢰⣯⣥⡿⡀⠉⠉⠉⡍⠁⠀⣧⠘⠂⡄⠀⠊⠘⠀⠀⠙⠛⠷⠶⠶⠶⠶⠟⣻⠿⠟⠋⠁
⣾⣷⣿⠁⠙⠂⡘⠦⠀⢀⠿⠀⠀⣀⡤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⢸⣿⡏⠸⠄⠂⠑⢠⣼⢀⣤⣶⣿⣿⠕⢀⡤⠂⠀⠀⠀⠀⠀⣠⣴⢿⡇
⢸⣿⠁⣄⠂⠹⠀⠀⣴⣿⣿⠿⡫⢀⣴⠋⠀⠀⠀⠀⣠⣴⣾⡏⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠛⢷⡄
⠈⣿⠀⡀⠠⠀⡟⣸⣿⣿⣷⣮⣴⠟⠁⠀⠀⠀⠀⣼⣿⣿⣸⢳⡇⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⣿⣿⣟⢰⠃
⠀⣿⡆⠁⢲⡀⠀⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢟⣭⢷⣿⡿⢟⡏⣼
⠀⢸⣇⠸⠃⠀⠦⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠁⠿⠛⠿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢛⡥⣞⣫⣷⣿⣿⣾⣋⢱⡇
⠀⠀⢿⣆⠈⠓⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣉⠙⡁⠉⡛⠛⠿⠿⢷⠶⢶⠾⣿⣿⣶⣫⣞⣿⡿⠟⢋⠉⠀⢬⣿
⠀⠀⠈⢻⣦⠀⠘⠃⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠀⡀⠘⢀⠐⠈⠄⠐⣀⠐⢀⡈⣿⣛⣛⣷⣶⠶⠶⠆⡔⢰⣿
⠀⠀⠀⠀⠙⢷⣤⡀⢸⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢈⠀⠰⠁⠀⠂⢈⣵⡿⠿⣿⣿⣿⣾⣦⣅⡀⠊⢿⡀
⠀⠀⠀⠀⠀⠀⠉⠻⢾⣿⣿⡻⣿⣿⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣭⣻⡶⣮⣽⣲⢬⣙⠛⠳⠌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣌⢿⣝⢿⡿⣿⠿⣟⢿⣖⢶⣶⣤⣶⠶⠾⠛⠛⠉⠁⠀⠀⠀⠉⠙⠲⣽⡓⠯⣿⣿⣿⠿⣷⣤⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡻⣦⣻⣮⣷⣿⣧⣻⣷⣽⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣍⡛⠻⣾⣭⡛⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣮⣍⡛⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣽⣋⣍⣹⣝⣭⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠂⠈⢧⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣨⣷⣾⣿⣿⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⣤⣤⣀⡀
⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀⡀⠀⠘⢷⣽⣿⣿⣾⣶⣾⣿⣭⡾⠃⠀⠀⣀⣀⣠⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠈⠳⣄⡀⠀⣠⠞⠋⠉⠀⠀⠀⠈⠉⠙⠓⠶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣥⠶⠚⠋⠉⠁⠀⠀⠀⠈⠙⠷⣄⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡉
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⣿⣿⣿⣿⡿⠿⣿⣧⡝⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣍⣻⣦⣄⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⠴⠖⠋⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠙⠶⠦⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣻⣿⣿⣿⣿⣿⣶⣿⣿⣯⣟⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣻⢿⣿⣿⣿⡿⠿⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣷⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇



⡶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡶⢶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⠶⡶⢶⣶⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣚⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣴⡦⠤⠄⠐⠒⠒⠂⢙⡃⡀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠤⠀⠀⠀⠰⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣆⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⡄⠀⠀⢠⢚⠤⢁⢴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠤⠀⠈⠂⠀⠀⠀⢀⠀⠀⠈⠉⠁⣈⠉⠀⠀⠙⢎
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠈⠀⠁⠤⢈⠁⠀⣁⡞⠉⠀⠀⠀⠀⠀⢘⣆⠀⠀⠀⣒⠀⠀⠀⣜⠃⠀⠀⢀⣿⡇⠀⢰⣆⠀⠀⢀⣄⠉⠀⢨⢀⡀
⠀⠀⠀⠀⠀⣴⠦⠀⠈⠕⠂⢠⡤⡾⡭⠶⡈⠀⠈⠁⠀⢠⡄⠀⠀⠀⠈⠁⡲⢤⠤⡤⢶⠙⠀⠁⠀⠀⢘⢈⠀⠀⠀⢉⠆⠀⠀⠀⠠⠘⡃⠄
⠀⠀⠀⠀⠀⠀⣤⡇⠀⠀⠀⠆⠀⠀⠀⠀⠰⡄⠀⠀⠀⠸⢗⠀⠀⠀⠀⡐⠀⠀⠉⠀⠀⠡⠀⠀⠀⠀⠘⠛⠀⠀⠀⢼⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠛⠄⠀⠀⠀⠛⢃⠀⠀⠰⠃⠀⠀⠀⠀⠀⠈⣣⠀⠀⠀⢰⠄⠀⠀⣌⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠙⢅⡤⡞⠂⠀⠀⠀⠀⢀⠀⢈⡂⠀⠀⢠⣠⠀⡐⠉⠀⠀⠀⠀⠐⠣
⠀⠀⠀⠀⠀⠀⠐⡈⢆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢣⡀⢰⠀⠀⠁⢸⠀⢀⡁⠀⠀⠈⠄⣶⠉⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠤⡄⠰⠀⠇⢰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⠆⢸⠸⠆⠊⠀⠀⠀⠀⠈⠩⢀⣀⠀⠀⠀⠀⢀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠉⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠓⠒⠒⠒⠒⠸⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠤⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠒⠐⠀⠀⠂⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⣀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢈⣃⠀⢂⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠆⠀⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⢂⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠂⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⢠⡁⠠⠹⠀⡀⠇⠅⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⣈⠑⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⠧⠀⠀⠀⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢰⣿⣿⣿⢀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⢶⠀⢀⡉⠙⠶⠄⠀⠀⠀⠀⠊⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠻⡿⠏⠛⠀⠀⠀⠀⠀⢠⣥⠀⠀⠀⠘⣛⡀⢸⠙⠳⠄⠀⠀⠀⠀⠀⠰⠚⢻
⠀⠀⠀⠀⠀⠀⠀⡄⠘⡀⠀⠀⡄⠀⢰⠀⠀⢶⠀⠀⠀⠀⠀⢂⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡀⠀⠀⠀⢫⠧⣜⠲⠦⠀⠀⠀⠀⠀⠀⠠⢤⡞
⠀⠀⠀⠀⠀⠀⣸⣿⠀⣡⠀⡘⢰⠀⡀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠛⠅⢸⠀⣄⠀⢠⠀⠰⠐⠓⠈⢠⠀⠀⠈⠣⠨⡶⠆⠀⠀⠀⠀⠀⠀⠰⡶
⠀⠀⠀⠀⠀⠀⠹⣿⣦⣄⣓⠃⠘⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡘⠒⠃⡀⣀⣑⡁⠠⠐⣁⠌⠀⠀⠀⠀⠀⠀⠙⣤⠎⠁⠀⠹⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⢿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣇⣀⣀⣀⡙⠛⠁⠤⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣹⠃⠀⢀⡀⠀⠀⢳⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡻⠆⠀⠀⠶⠂⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠂⠀⠀⠀⠀⠀⣸⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠧⠀⠀⠀⠀⠀⠀⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⡤⢥⡴⢦⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠱⡄⠀⠈⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠘⡀⠀⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡤⠖⠛⠉⠛⢦⡀⠘⡀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢧⠀⠀⣸⠀⣠⡆⠀⢸
⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠀⠀⢰⠞⠁⢳⠀⢸
⠀⠀⠀⠀⠀⡟⢧⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢈⡇⠀⠀⠀⠀⢸⠀⢸⡀
⠀⠀⠀⠀⠀⢷⡀⠓⢤⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⣃⠀⢰⠀⠀⠘⡄⠘⡇
⠀⠀⠀⠀⠀⠀⢽⣦⡀⠉⠓⠠⠄⢳⣄⡀⠀⠀⠀⣠⡖⠋⣁⡤⢆⣉⡉⠉⢧⠀⠸⣄⠀⠀⠈⢄⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠂⠀⠀⠤⠖⠿⣀⣀⣀⣀⣀⣉⣉⡀⣠⡤⠎⠀⠀⠈⠣⣀⢀⣀⠄⠀⠈⠐⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⢀⣀⣀⣀⣤⠄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠉⠛⠋⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣠⣴⣶⣶⣶⣦⣤⠤⠴⠶⠒⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣀⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡾⠟⠛⠉⠉⠁⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠁⠀⠀⠉⠓⠦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣤⣶⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠁⠐⠒⠒⠂⠤⠤⠄⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠼⢿⣿⠟⠛⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣶⣶⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠙⠲⣤⣀⡀⠀⠀⠀⠉⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣤⡄⠀⠀⠈⠢⢄
⠀⠀⠀⠀⠀⠐⠠⢀⡀⠀⠀⠀⠀⠀⠙⢌⡒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣇⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠜⣛⣛⡳⠶⠦⢤⣤⣈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⣀⡴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠐⣾⣷⣤⡤⣤
⠀⠀⡠⠊⠉⠀⠀⠀⠀⢀⡠⠔⠒⠉⢉⡀⠄⠀⠀⠀⠀⠀⠀⠈⢻⠿⠉⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⡠⠔⠋⣡⠖⠉⠀⠀⠀⠀⠒⠒⠒⠛⢄⠀⠀⠀⠀⠀⠁⠶⡍⢒⠠⣈⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠊⠀⢀⠔⠉⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢆⢉⠳⣕⢮⡑⣆⢄⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠑⢸⡷⢽⡌⠳⡍⠢⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠹⣄⠻⡄⠘⢆⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠌⢣⡝⢆⠘⡀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⢠⡿⠛⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠉⠹⣷
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠈⢿⣿⣷⣄⣠⣴⣶⣶⣾⣿⣿⣶⣶⣶⣤⣤⣴⣿⠏⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠻⣿⣻⣷⣦⠀⠀⠀⠀⠀⠈⣿⣟⣁⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠿⢿⠿⠇⠀⠀⠀⠀⠀⣿⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠅⢀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⢀⠠⢁⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢊⡠⠊⠀⠀⠀⠀⠀⠤⠤⠞⠦⠤⠄⠀⠀⠀⠀⠙⠢⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂




⣿⣿⣿⣻⣛⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣻⣿⣟⣟⣛⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠥⠤⠠⠤⠴⡆
⠀⠀⠀⠀⠀⠀⣠⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⡏
⠀⠀⠀⢶⣒⠊⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠤⠤⠤⢶⠗
⠀⠀⠀⠀⠈⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢳⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢠⣎⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⡇⠘⠳⠞⠀⠻⠶⠃⠀⠳⠞⠀⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣀⣀⣀⣷⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣀⣀⣀⣀⣀⣷
⠀⠀⠀⠀⢸⡇⠙⠒⠃⠘⠒⠃⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠛⠚⠃⠙⠒⠃⢸
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠠⠀⠀⢸⣤⣤⡀⠀⣠⣤⡀⡇⣠⣄⡀⠀⣠⣤⡄⠀⣠⣠⣄⡇⢠⣤⣄⠀⢠⣠⣼⠀⠀⢀⠀⠠⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠒⠒⠀⠀⢸⠀⠀⢧⣤⡇⠀⣧⣧⡇⠀⢧⣤⡇⠀⢧⣤⡗⠀⢯⣧⡼⠀⢿⣤⡼⠀⢸⠀⠀⠈⠒⠚⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠈⢢⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⠢⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠐⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢻⠉⠀⠀⠀⠉⡟⠦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠂⠐⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡴⠃⠀⢸⠀⠀⠀⠀⠀⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡄⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸⠸⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠃⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣇⣀⣀⣀⣸⣀⣀⣀⣀⣀⣇⣀⣸⣀⣷⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠉⠀⠈⠈⠁⠈⠀⠁⠈⠁⠀⠁⠈⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⡔⠓⣦⢀
⠀⠀⠀⠀⠀⠀⠁⢰⡆⠀⠀⠀⡇⠀⠃⠀⣄⠂⡀
⠀⠀⣀⡠⠂⠀⠀⠈⠓⠀⠀⠀⡇⠀⢀⣼⢏⠀⠈⠢
⠀⠀⠲⣖⠀⠀⠀⠀⣀⡤⣄⠀⠹⡶⠋⠁⣾⠀⠀⠀⠘⣦⣤⣀
⠀⠀⠀⠈⠀⠔⠒⢈⠽⠶⢋⠀⠀⠀⠀⠀⣟⠀⠀⠀⣴⡟⠁⣿⣧⡈⢲⣶⣶⣶⣶⣤⣴⣶⠶⠶⡶⠶⠶⡤⢤⣄⣀
⠀⠀⠀⠀⠀⠚⠞⠃⠀⠀⠀⢠⡀⠄⠀⢲⡏⠀⠀⡼⠁⠀⣠⣿⣿⡆⠀⢹⣿⣿⠳⢦⣤⣤⣤⢶⣶⠶⠤⠄⠀⠀⠀⠈⠀⠲⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⡇⠀⠀⠀⠀⠀⡈⢻⣿⠀⠀⠸⡏⠁⠀⢰⡿⠇⠈⠂⠉⠀⠀⠀⠀⢀⡴⠀⠀⠳⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⣸⠁⢸⠏⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠓⠃⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢦⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⢻⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⡄⡑⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠱⡀⡌⢳⠀⠀⠀⢰⣏⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣾⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠶⠶⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠚⠦⡀⠀⠀⠉⣼⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢹⡆⠙⢿⣿⣦⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣼⠟⠀⠀⢸⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⢹⣿⣧⣀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⡈⢿⣿⣿⣧⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⡿⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠻⣿⡟⢳⡀⠘⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠁⠀⢱⠀⢘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣸⠀⢰⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠈⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠈⢰⠇⢀⠌⡀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠧⣤⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠁⣴⣉⣏⣃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠳⠛⠀⢰⡄⣌⣁⠂

⠒⠉⡀⣰⠗⠁⠀⠀⢀
⠊⠁⡀⠀⠀⠀⠀⠀⠀⢀
⣄⠄⠀⠀⠀⠀⠀⢀⠀⣠⣄
⢻⠀⠀⡀⠄⠀⠀⠁⠀⣿⢬⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡇
⠂⠃⠀⠀⠀⢠⠀⠀⠀⣛⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣄⣀⣴⣿⣟⢡⠃
⠀⠀⡘⠀⠐⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟
⠀⠀⠁⠐⡀⠖⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢘⠒⠂⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠙⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⠀⠤⠌⠀⣿⣿⣿⣿⠿⠿⠟⠀⠀⣿⣿⣿⣿⣿⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠰⡂⢡⠴⡀⠀⢸⣿⣿⡟⠧⠄⠀⣤⣠⠄⠉⠉⠛⣿⣿⠟⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⡀⠀⢀⣷⠋⠒⠁⠀⡀⠄⣿⣿⣤⣀⡠⠾⠆⣀⠋⢘⡶⠋⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⣷⠆⠈⠀⠀⠄⢀⣾⠀⢰⣿⣿⠃⠈⠓⠒⠚⠋⠃⠀⠀⠀⠀⠀⢠⠔⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⣡⣾⣶⣶⢾⣾⣏⠅⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⡿⠉⠀⠀⠈⣣⣤⣿⠷⠛⠁⠀⢠⡀⠀⣢⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠏⢀⠐⢋⣤⠞⠋⠁⠀⠀⠀⠀⠀⠈⢻⣶⣿⠁⠀⠀⠀⠀⠀⠀⢀⣠⠂⠈⠀⠉⠙⠹⠀⠟⠿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠂⣴⣟⠉⠀⠀⣰⣤⣶⡶⠶⠖⠾⠛⠛⢛⣏⠀⠀⠀⠀⢀⣴⣾⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠙⡻⢿⣿⣿⣿
⢦⣿⣿⣶⣾⣿⡿⠋⠁⠐⣫⠖⠂⡈⠉⠁⣟⠀⠀⠀⢀⠎⠁⠈⢁⡙⠻⠽⠟⢿⣶⣤⠀⠀⠀⠀⠀⠀⠀⢰⡷⡄⠀⢈⢹
⢀⣁⣹⣿⠟⡡⡵⢢⠄⠀⠈⠀⠉⠂⠀⢸⣿⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⡄⠀⠀⠀⠀⠀⠀⠐⡶⠄⠀⢀⡥
⣫⣤⡿⠶⠶⢆⣴⠄⣀⡐⠁⣠⠄⠐⠂⠈⣿⣄⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢀⠤⠈⠓⠀⠑⠀⠀⠀⠀⠀⠀⣷⠚⠾⠷
⠿⡿⠿⠿⣿⠿⢿⡿⠻⠿⠖⠁⠀⠀⢀⡀⢹⣿⣷⣶⡏⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚
⣧⣀⣀⣴⠇⠀⣿⠃⠀⠀⠀⣠⣀⣤⠴⠆⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⣷⣥⣤⣤⣶⠶⠿⠛⠛⠓⠒⠚⠉
⡏⣩⣁⣛⣖⣩⣏⣠⣰⣾⣿⣿⣓⣂⡒⣆⣀⡀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣿⣟⣫⣭⠟⢿⠉⣩⠟⠉⣹⠋⠩⠉⠋⠉⡩⠉⠠⢴⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣹⣿⠿⠛⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠶
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡀⢀⣹⠟⠛⠛⢛⠛⠁⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣈⣫⣦⣶⣾⣿⣀⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢆⠀⠀⠀⠀⠈⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⣠⣄⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣞⠀⠀⠀⢠⠀⠀⠈⠁⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠙⠓⠒⠒⠒⠒⠒⠚⠋⠁⠀⠉⠀⣀⡤⠋⠀⠀⢀⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠢⢤⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⢢⡀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠹⠶⠆⢀⣀⠤⣿⠀⠀⠀⠀⠀⠀⡼⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿
⠀⠀⠀⠀⠉⠁⠀⡏⠀⠀⠀⠀⣠⡞⠉⠉⣠⠆⠀⠲⢤⠠⢤⡤⠜⠀⠀⠀⢠⢧⡏
⠀⠀⠀⠀⠀⣠⠞⠀⢀⡠⠚⠉⠹⢧⡀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⢠⡏⠀⢰⠛⠁⠀⠀⠀⠀⠳⠀⠈⠢⣄⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⠈⠓⢤⡀
⠀⠀⠀⠀⠀⡇⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠢⠴⠦⠃⠀⠀⠀⠀⠈⠙⢆⠀⠀⠓⢶⠛⠁

⠀⠀⠀⠀⠀⢀⣴⡟⠻⠷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠶⠶⢶⣦⡄
⠀⠀⠀⠀⠀⣼⡇⠈⢦⡀⠀⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⣠⠞⠉⣿⡄
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠹⣄⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢸⣇
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣨⠷⠶⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠳⠶⣤⣀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠹⠆⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣴⠋⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢻⡄⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢱⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠟⠋⠉⠙⠳⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⡶⠞⠛⠶⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠙⢷
⢀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⡟⠀⠀⠀⠰⠋⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⢸
⠈⠉⠉⠛⠳⠶⢾⣧⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢧⣀⡀⣀⡼⠳⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣀⣠⣿⠶⠶⠖⣾⠛⠛⠋⠁⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡉⠓⠆⡠⠐⠀⠀⠐⠠⡀⠀⠀⣿⠉⠉⠀⠀⠈⠉⢙⡇⠀⠀⡠⠐⠒⠂⠀⢀⠀⠀⠀⠁⢰⡟⠀⠀⢰⡏⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⢻⣠⠔⠊⠉⠉⠑⢺⠁⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢸⡇⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣽⣷⡋⠀⠁⠀⠀⠀⠁⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠓⣶⣿⣁⠀⠀⠀⢸⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⣀⡴⠚⠉⠀⠈⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠙⠻⢦⣄⢸⡇⠀⠀⢠⡏
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠉⢻⣇⠀⠀⢸⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠑⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⣠⡟⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠘⠦⣀⡸⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⣠⠴⠋⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢀⡎⢀⠆⠀⠀⣸⠃⢠⠀⢀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢀⣠⡿⠥⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠾⠦⠶⠒⠛⢶⣧⣀⣇⣀⣠⡶⠿⣤⣀⣀⣀⣀⣉⣓⣚⣋⣽⠇

⠀⠀⠀⠀⣴⠟⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢶⡄
⠀⠀⠀⠀⣿⠀⠑⢄⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⢉⡤⠂⢸⣧
⠀⠀⠀⢰⣿⠀⠀⢒⡳⣄⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⢀⣴⡯⠤⠀⠈⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠩⠬⢷⡄⠀⠈⠛⢶⡶⠶⢿⣿⡟⠛⠛⣿⣿⠛⣿⠷⢶⡶⠟⠁⠀⣠⣾⠖⠚⠋⠀⠀⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠄⡒⠲⠾⠆⠀⠀⠀⠀⠀⠀⢿⣇⠀⠐⣿⠏⢠⡏⠀⠀⠀⠀⠀⠴⠳⠾⢛⣉⠀⠀⠀⣿
⠀⠀⠀⠈⣿⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡟⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⢰⡿
⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⣼⠇⠀⠀⠀⢀⣤⣶⣶⡶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⠲⣶⣤⣀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⢰⡏⠀⠀⠀⣰⠋⣾⣿⣿⣄⣀⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣄⣠⣿⣷⡙⣧⠀⠀⠀⢸⣇
⠀⠀⠀⣾⣧⣶⠖⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠐⢶⣶⣿
⠀⠀⠀⣿⡿⠁⠀⠀⢻⡀⢻⣿⣿⣿⣿⣿⡿⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿⣿⣿⡿⢁⡾⠀⠀⠈⠻⣿⠄
⠀⠀⠀⢿⣿⣶⣦⡄⠀⠙⢦⣍⣛⣛⣛⣩⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣛⣛⣛⣉⡤⠞⠀⠀⢴⣶⣿⣿
⠀⠀⠀⠘⣿⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠐⢿⣿⡿⠂⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⣽⠏
⠀⣀⣀⣤⡼⣷⣶⠒⠛⠒⠒⠒⠒⠒⠲⠄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠴⠤⠐⠀⠒⠒⠒⠚⢲⣾⠿⢠⣄⣀⡀
⠉⠀⠀⠀⠀⣈⣻⣷⣶⡂⠈⠉⠉⢀⠤⠀⠀⠀⠀⠳⠶⠞⠉⠓⠶⠞⠀⠀⠀⠀⠨⣉⠉⠉⠁⢐⣲⣶⢿⣁⡀⠀⠀⠈⠀⠁
⠀⠀⠠⠔⠉⠈⠀⠀⣹⠿⣷⣦⣮⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣤⣶⠿⣏⠀⠀⠈⠈⠑⠂⢀⣤⡶⣤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⢠⡿⠁⠉⠛⠻⠷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣶⠶⠿⠛⠋⠉⢿⡄⠈⠑⢄⠀⠀⠀⠀⢺⡇⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠊⠀⠀⢀⣿⣷⣶⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣬⣿⡄⠀⠀⠁⠀⠀⠀⠘⣷⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢀⡿⠙⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢻⡆⠀⠀⠀⢠⣾⣧⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⣿⠀⠀⠈⣿⠀⣠⣴⠋⠈⠿⣷⡾⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠐⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠃⠀⠀⠀⢸⡏⠀⠀⠀⣿⠛⢻⣿⣆⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠘⣧⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿⢀⣠⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⢻⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢠⡿⠀⠀⠀⣼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⣼⣧⢀⡀⢀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡀⠀⡀⣼⠧⣄⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠿⢾⣧⣼⣦⣤⠾⠶⠶⠶⠶⠶⠶⠾⠿⣤⣤⣯⣴⣧⠿⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡯⠙⢷⣤⡀⠀⠀⠀⠀⣠⣼⡿⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⣠⣴⣿⠻⡿⣿⢿⣿⣷⣶⣌⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠈⠉⠁⣀⡀⠀⠀⠀⢉⣩⣭⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠹⠿⠿⠀⠀⠀⢾⠿⠋⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠟⠀⢠⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠠⠀⠀⠉⠓⢶⣈⣁⣼⣛⣩⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿⠿⣿⣷⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣯⣷⣽⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣻⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡰⣟⠏⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠃⠀⠀⠀⠀⠀⠔
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣠⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠉⠉⠙⠻⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀
⠀⣴⣿⣿⣝⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠙⠻⠿⠿⢆⠀⠀⠀⢹⡄⡄
⠀⢻⡿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆⠀⠀⠀⠉⠃⢀
⢰⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣴⣧⣀⡀⠀⢀⣀⣴⡤⠤
⠈⠉⠛⠛⠛⠛⠛⠟⠛⠛⠂⠠⠤⠤⠤⠄⠰⠖⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠦⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⢤⡀
⠀⠀⠀⠠⢠⡆⢠⡆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠈⠀⠀⢠⣄⠁
⠀⠀⠀⡄⢸⡇⢸⡇⢠⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡿⣷⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣶⣄⢹⣯
⠀⠀⠀⣷⣸⣟⣸⣗⣼⡄⠀⠀⠀⠀⠀⣸⣿⠀⣶⣿⣿⣦⢠⣿⡇⠀⠀⠀⢀⣀⠀⣿⠀⣿⣿⠀⣿⠀⣀⡀⠀⠀⣰⣆⢸⣷⢿⡿⡿⡮⡿⢠⣦
⠀⠀⠀⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣠⣿⣿⢴⣶⣿⣷⣶⣾⣿⣷⡀⠀⠀⢹⣯⣼⣿⣿⣿⣿⣿⣿⣧⣿⡇⠀⠀⢻⣿⣶⣾⣾⣷⣷⡗⣶⣿⣿
⠀⠀⠀⢸⣿⡏⢹⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⠟⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠇⢀⡀
⠀⠀⢰⣶⣶⡆⠰⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⡈⣾⣦⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣾⡆⠀⠓
⠀⠀⠠⣯⣼⣤⣤⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⣏⣧⣼⠁⠀⠀⠀⠀⠀⠀⢻⣟⣀⠀⣿⣿⢀⣀⣻⡗⠀⠀⠀⠀⠀⠀⣦⡼⡿⢿⣴⠀⡀
⠀⣠⣠⣿⣿⠛⡟⣿⣿⣅⣄⠀⠀⠀⠀⣀⣺⣀⡟⠛⠛⠿⣰⣇⡀⠀⠀⠀⢀⣾⣿⣉⣉⣿⣿⢉⣉⡿⣇⡀⠀⠀⠀⢸⣧⣬⣻⣿⣟⣡⣤⣷
⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠹⣿⣿⣷⡶⣷⣾⣿⣿⠃⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠸⠿⠿⠿⠇⠈⠿⠿⠿⠿⠀⠀⠀⠀⠐⠿⠿⠿⠃⠸⠿⠿⠿⠀⠀⠀⠀⠾⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠷⠀⠀⠀⠀⠿⠿⠿⠁⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢴⠄⠀⠐⠆⠀⣶
⠀⠀⠀⠀⢰⡇⢹⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⢀⠠⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢶⣶⠀⣿⡇⠀⠀⠀⠀⠀⠠⠾⢧⠀⣼⠄⠰⠿⠆
⠀⠀⠀⣶⢸⣷⣾⣇⢰⠀⠀⠀⠀⠀⠀⠀⠀⣧⢼⡿⣿⢧⡇⠀⠀⠀⠀⠀⠀⠀⣠⣤⠁⣾⣟⢠⣿⡇⠀⠀⠀⠀⠀⠘⣟⡟⣤⣿⣥⢹⣻⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣸⣿⣾⣷⣿⣞⣿⣀⣀⡀⠀⠀⢠⡄⢸⣿⣶⣿⣿⣼⣿⡇⢠⠄⠀⠀⠀⣘⣛⣟⣛⣟⣛⣻⣟⣓⡀
⠀⠠⣤⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⠀⢘⣟⣿⣛⣟⣛⣛⣟⣻⣛⣻⡁⠀⠀⢰⣷⣾⣿⣿⣿⣿⣿⣿⣷⣾⡄⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⢀⣿⡟⣿⡿⢻⣿⢻⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣟⡿⠛⠟⠛⢙⠁⠀⠀⠈⣿⡟⠻⣟⠿⢿⢿⡟⢿⣿⠀⠀⠀⠀⢰⣿⣿⣿⡏⣿⣿⣿⣇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⣤⣤⠀⠀⠈⣿⣿⡇⠀⠀⣠⣤⠀⠀⠀⢴⣶⡦⢀⣿⣿⠀⣴⣶⡆⠀⠀⠀⠀⣤⠄⣈⣿⣿⣿⣿⡀⠠⣤⠀⠀⠀⠀⠀⠀⢰⣦⢀⣿⡦⢼⣿⠁
⢸⣿⣀⣀⣀⣿⣿⣇⣀⣀⣸⣿⠀⠀⠀⢸⣿⠷⠶⠿⠿⠶⣿⡾⡇⠀⠀⠀⠀⣿⡀⣟⣛⣛⣛⣛⣛⢀⣿⠀⠀⠀⠀⣿⡇⢸⣯⢸⣿⡇⢸⣿⠂
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠠⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣯⣤⠀⠀⠀⣿⣷⣶⣷⣶⣷⣶⣾⣿⠇
⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⡟⣻⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⢻⣿⣿⣿⣿⠟⣿⣿⣇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡏⢹⣿
⠉⠉⠀⠀⠈⠁⠀⠁⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠁⠀⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉

⠀⠀⠀⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄
⠀⠀⢸⡿⠻⣿⣷⡄⣀⣠⣄⣀⣀⣀⣴⣾⣿⡟
⠀⠀⣿⣷⡦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠇
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⢹⣿⣿⣿⣿⡋⠟⣿⣿⣿⣿⠛⢟⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⣀⣀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠻⡿⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⠲⠂⠂⠀⢀⠉⠉⠂⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣑⠀⡄⠀⠂⠈⢀⠐⢍⠻⣿⠿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣇⣄⣰⣴⣶⣄⣩⣶⣔⣜⡷⢿⢿⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣩⠙⠀⠀⠢⣜⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢥⡄⠀⠀⠀⡀⢀⠠⡀⡹⣽⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡳⠧⠄⠀⠀⡀⠀⠑⠀⡉⣳⣿⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⠈⠀⠀⠀⢀⠀⠆⠈⢈⠻⢯⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠄⠀⠀⠀⠁⠀⠀⣦⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠑⡄⠀⠠⡄⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⣾⣿⣧⣿⣯⣭⣟⣛⣿⣿⡿⠿⣿⣷⡾⢷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣄
⠀⠀⠀⠀⠀⠀⢠⣼⣯⠏⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡾⠛⠛⡁⡛⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣟⡞⢛⣿⣿⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣼⡟⠉⠀⠐⠰⡄⡇
⠀⠀⠀⠀⠀⠀⢸⢾⡯⢀⠂⠀⠀⠿⣷⡝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣿⡿⠠⠂⠀⡠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠐⠫⠀⠀⠨⠀⠑⣿⣮⢿⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⢾⠏⣼⡿⠓⡐⣁⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠈⣷⠺⠆⠀⠀⠠⠦⢬⣱⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣯⠄⠔⠀⠄⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⣻⠑⠀⠀⢠⠀⠀⠤⠈⠿⢟⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡷⣽⠓⠀⠀⠀⢔⡂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣄⣤⣶⣭⡷⢟⠿⣿⣿⠹⣿⣿⣿⣿⣿⣟⢉⣴⣾⣷⣦⡀⠉⠀⣾
⠀⠀⠀⠀⠀⠀⢀⡴⠧⠐⠀⢀⣤⡿⠿⣿⡿⠿⠿⠆⠱⠉⢻⠁⣷⢿⠋⢺⣿⣵⣿⣿⣿⣿⣿⣿⣧⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠃⢱⠙⠿⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠹⠛⠉⠙⠛⢛⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣶⣶⣾⣿⣿⣻⣿⣿⣿⣆
⠀⢀⡴⠊⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣯⣽⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⡿⠸⢻⣽⣿⣿⡟⣿
⡴⠋⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠻⠿⠟⡩⠹⠃⠀⠀⠀⠀⠀⠘⣿⣿⣛⡋⠔⠁⣠⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡆⠀⠀⠀⣿⠽⢿⣿⣿⣿⡃⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⣄⣀⣀⠀⠀⠀⠙⣿⠏⠀⠀⠄⢀⠒⡲⣾⣿⣶⣼⡯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠥⠄⠀⠀⠀⣀⣽⣀⠀⠀⡐⣄⣽⣿⣿⡿⣯⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⠀⠀⡀⢀⠀⠈⠁⠜⠷⣷⣶⠖⠋⠉⠀⠈⠉⣻⡛⠿⠍⠉⢱⡾⠁⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢋⠄⢀⡀⠀⠀⠀⢀⠈⢛⠶⣤⣤⣤⣴⢿⢏⣉⣰⣤⣶⡏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠐⠂⠐⠀⠤⠄⠀⠐⠌⡿⢿⠞⠿⠉⣧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠸⢧⣠⣤⣤⣤⡄⣠⠐⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠀⠀⠣⠖⣥⢩⠢⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠆⣀⠀⢀⣸⢳⠠⡯⡞⢣⠀⠀⠀⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡤⠶⢲⢮⠠⡶⢤⠀⣰⡖⢦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠖⡐⠀⢘⡆⠉⠀⢠⡃⠀⣊⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠃⠐⣠⠈⢹⡟⢀⠎⠁⠀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣷⣄⢀⢤⣀⣀⠹⠎⠀⢀⣠⣄⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⢿⡛⢾⣤⣠⣤⣤⣤⣽⣿⠏⢀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⠻⢦⣈⣉⣉⣡⠼⠃⢀⣤⠞⢸⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠻⢷⣄⣀⣉⣁⣤⣴⠾⠟⢁⠞⠀⣶⠎⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠳⣄⡉⠙⠋⢉⣄⣀⡤⠖⢁⡜⢰⠏⠸⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠄⠈⠛⢣⠀⠙⢉⠥⠀⠚⢋⣠⢏⢾⣆⢩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⠀⠈⠀⠀⢀⠐⣴⠟⢉⣁⣨⡼⡘⢸⣻⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣌⠛⢿⡄⠀⠄⠀⠈⣷⣡⡾⠿⠛⣋⣴⢷⠀⠉⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣦⣌⠦⠀⠀⠀⠈⡅⢠⣴⣤⣍⣥⠚⣰⡇⡈⠿⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⢙⠻⠀⠀⠀⠀⢹⠋⠀⠉⠉⣠⡄⠉⣴⢃⡤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠈⢷⣀⠀⠀⠀⠠⠀⠀⠐⠟⠋⠀⠘⠁⣊⡄⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⠛⠶⠶⢶⠁⢈⡙⠀⢹⠛⡟⠻⠛⢯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢁⣸⢁⡾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠤⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠐⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⡇⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⢸⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⣦⣸⣇⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣾⣿⣏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢻⢿⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⡘⠸⠹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠓⠁⠈⢷⡀⠀⠀⠀⡞
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢠⠁⠀⠀⠘⠙⠉⠀⠀⢠⣾⣭⣍⣉⣷⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢀⠁
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣠⠋⠻⢤⡸
⠀⠀⠀⠀⠀⠀⠸⡀⢀⠜⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡝⠒⠤⠄⣀⣀⠀⠀⠀⠁⠀⠀⠀⠤⠤⠒⠉⠱⡀⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠒⠠⠤⠐⠊⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢠⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢔⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠺⠀⠀⠀⠀⠀⠀⡄⠁⠆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠔⠁⠀⠀⠀⠀⠀⢀⡃⡠⠂⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⡀⠀⡀⠀⠀⢀⠄⢀⠁⠀⠀⠀⠀⠀⠀⢀⠇⠚⡀⠀⠀⠀⢀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠉⠀⠈⠣⠀⢄⣀⡀⠄⠠⠚⠀⠀⠈⠀⠈⠀⠈⠁

⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠘⠉⠙⠢⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠠⠤⠤⠤⠤⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠉⠉⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⣲⡶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⣶⠖⠒⠚⡹⠥⣹⣿⡾⣻⣷⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡓⣄
⠀⠀⠀⠀⠀⠀⡇⢂⠂⠀⠹⢟⣿⣫⣾⣿⣿⣿⣿⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣆
⠀⠀⠀⠀⠀⠀⠈⠒⢦⡤⠑⣺⣿⣿⣿⣿⣿⣿⣿⠿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⣿⣿⡿⣿⣷⠏⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣴⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⡾⣿⡿⠇⠀⠀⠀⠀⠀⠻⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣻⣷⣶⣶⣶⣴⣶⡶⣶⣶⣿⣿⣿⣛⡋⠉⠙⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢓⠀⠃⠀⠀⠀⠀⠀⢀⣦⣞⣿⣿⣿⣿⣿⡿⣺⣿⣿⣯⣿⣿⣿⣫⣽⣿⣿⣿⣶⣌⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣿⢫⣿⢋⣿⣿⣿⡿⠯⣄⡀⡝⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣷⢰⠀⠀⠀⠀⣘⣽⣿⣿⣾⣿⡿⣿⣿⣿⣽⣿⣽⣿⣻⣾⣿⣷⣿⠶⢐⣬⡿⠃⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡘⡧⠀⠀⠰⣟⢻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⡷⢎⠚⠁⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢧⠀⠀⠀⢻⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢛⣁⣡⣈⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢳⡀⠀⠹⣹⣿⣿⣿⣿⣿⣛⣛⣛⣋⣫⣵⠶⠖⠛⠛⠛⠛⣿⣷⣄⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠇⠀⠀⡏⠀⠉⠉⠙⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⠰⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⢶⣄⡐⢎⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⠛⢿⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠃⢀⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠀⡼⠀⠸⡎⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⣠⠾⠃⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠤⠒⠃⠀⡰⢅⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡖⣠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣶⣖⠒⣺⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠉⠃⠀⠹⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣹⡿⢆⠀⠀⠀⣼⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠟⠉⠉⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠤⣄⠀⠠⠀⠤⠀⠀⠊⣠⠉⢉⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠐⠒⡀⠀⢀⡠⠦⠄⠀⠐⠆⣠⣿⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⢻⣿⡿⠟⠛⠉⠉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣤⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣯⡁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣚⣉⣻⣿⣷⣀⠄⠀⠀⠀⠤⢾⡄⠉⠙⠓⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣯⣿⣿⡯⠛⣀⠆⠐⠁⣆⠀⠊⠀⠀⠰⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡟⡂⣻⠈⢃⠀⠀⢛⣴⣶⣦⠴⠶⢀⡈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢻⡿⢿⣆⢡⡀⠀⢸⣧⠭⠋⠚⠀⠈⠹⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠚⣃⣉⠘⡿⡂⠀⢀⠁⠀⠀⠄⠀⠰⣾⣻⡧⠋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣧⣣⡔⠃⠳⣶⠞⠀⠀⠀⢄⣾⡟⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣹⣿⣿⣿⣄⣀⡽⠆⠠⢤⣴⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣿⠿⠿⣿⣷⣤⣤⣤⣾⣿⣿⡷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⡿⣯⣐⡻⠓⢿⣛⠩⡉⡹⠛⠉⠛⠒⠂⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⢤⠤⢾⠋⠉⠀⠉⠉⠁⢆⣌⡿⠀⠀⠀⠀⠀⠀⠀⠂⠴⠀⠀⣄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⣴⣾⡣⠂⠀⠀⠀⣠⡀⠀⣈⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠠⠰⠠⠄
⠀⠀⠀⠀⠀⢀⣶⡿⠟⠀⠀⠀⠀⠤⠉⠣⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀
⠀⠀⠀⠀⠀⢈⡟⣀⣀⣠⣴⣤⣤⣀⢤⠜⠂⠘⠲⣄⠰⠆⠀⠀⠈⡄⠠⠀⣧
⠀⠀⠀⠀⠀⠼⠉⠁⠈⠀⠀⠀⠀⠹⣿⣳⠦⠂⠠⢹⣦⠀⢀⠀⠀⣷⣰⡼⣧⠇⡀
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⡡⠿⣿⣷⣄⣧⣿⢠⡀⢸⠆⣤⡿⠋⠁
⠀⠀⠀⠀⠀⠚⠟⡀⠀⠀⠀⠀⠈⠀⡀⠀⠙⠛⠻⠃⠘⠳⠼⢺⣏⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠙⢻⡟⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⠃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣀⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡆⠀⡄⠀⢀⠀⢰⡀⠀⠀⠉⠻⢷⣧⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣧⣀⣾⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠠⠂⠀⠉⠉⠉⠉⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⡉⠁⠀⠀⠀⠓⠀⠀⢀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢰⠀⠘⠛⡁⠀⠀⠈⠻⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⡆⠀⠀⠸⡀⢀⡀⠀⠀⠀⠀⢀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢷⡘⢄⠀⣈⢌⣀⣀⡤⠞⢀⣼⢦⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⣌⠾⠀⠀⠀⡥⢊⠔⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠈⡇⠀⠀⠀⠗⠁⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢇⠀⠀⢇⠀⠀⡠⠁⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢹⠀⠈⢆⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⡇⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⡾⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠱⡀⠀⠀⢡⠀⠀⠀⣇⠀⠀⠀⠀⢀⠎⠱⡀⠀⠀⠸⡄
⠀⣠⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠘⡀⠀⠀⢸⠀⠀⠀⣴⠁⠀⠀⠱⡀⠀⠀⢇
⡎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⠀⠀⢰⡇⠀⠀⠀⠀⠀⠇⠀⠀⠘⡄⠀⡜⢹⠀⠀⠀⠀⠙⡄⠀⠸
⡇⢰⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⡄⠀⠀⢧⠈⠀⡼⠀⠀⠀⠀⠀⠘⡄⠀⢇
⠳⡀⠳⢄⡀⠀⠀⣀⡇⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⣖⣋⣄⡀⠀⠀⠀⠀⠀⠱⡀⠸⡄
⠀⠙⠢⣄⣈⠉⠉⠀⣘⡦⠀⡀⠐⠺⣄⣀⣀⣀⣀⣀⣀⣀⣧⠀⠀⠹⣄⡀⢨⡣⠀⠀⠀⠀⠀⠘⡄⠈⠒⠢⣄
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠑⠲⠷⠤⠦⠃⠀⠀⠀⠀⠀⠀⠀⠹⣀⣰⠀⢈⣹⠉⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣇⢈
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⣿⡀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢶⠀⠀⠖⢒⣻⣷⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⢀⣸⡆⡀⠋⠉⢻⡏⢀⣀⣸⡆⠀⠀⠀⠀⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⢻⡇⣧⠀⠀⣸⣷⢸⠘⢻⡇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢠⣼⣧⢰⡄⣸⢿⣿⣼⣄⣼⣧⠀⠀⠀⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣤⣿⣷⡄⠀⢲⣿⣿⡾⣰⡟⣏⣿⣿⣿⢿⣿⠀⢠⣮⣤⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⢠⡀⠃⣹⣿⢉⢹⣿⣿⣿⣿⡏⢸⣿⢀⡄⣤⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⣿⡏⣹⣿⠠⣞⡟⠾⠿⡟⡿⢙⣿⢃⣿⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠙⡇⣻⡿⠀⣀⣤⣤⣀⠀⠀⢻⣿⡟⣹⣿⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⡃⠀⢘⡇⢸⡇⢸⣿⣿⣿⣿⠂⠘⠀⣿⡧⣻⣿⡏⣽⣇⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⡇⣶⣿⣿⣄⣼⡇⣹⡇⠠⣿⣿⢻⣿⠀⠒⢈⣿⣧⣾⣿⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠶⢶⣶⣿⣧⣶⣻⣿⣛⣻⡷⠞⠋⠉⠉⠉⠳⢾⣛⣛⣿⣓⣦⣼⣿⣶⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⢯⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠳⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣶⣶⣶⣜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠏⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡟⢦⣸⣦⡘⠀⠀⠀⢀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⣿⡸⡏⠛⡇⠀⣴⠺⣆⠀⠈⢀⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⠙⣯⡻⣿⣶⡙⠄⠀⠈⣟⡻⠃⠀⠀⣛⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣀⣠⡴⠾⠛⠚⠙⠛⠐⠺⢟⣛⠛⠀⠀⠀⠀⠈⠠⠔⠊⠁⣡⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⢀⠔⠁⠀⠀⠀⠈⠉⠓⠒⠠⢤⡐⢂⣤⠄⠀⠀⠉⠓⣤⢤⡀⠀⠀⠀⢀⡄⠀⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣿⠀⠀⠀⢸⣘⡷⠋⡆⠀⢠⣾⠧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣰⠁⠀⠀⠀⣾⡅⠀⠀⠀⠀⣧⠁⢀⡄⠀⠐⠀⣀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠸⠀⠀⢾⡁⠸⠄⠀⣼⠆⠀⠁⠄⠸⢁⠛⠂⠉⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠘⠀⠀⠀⠠⣸⡇⠐⡆⠀⢿⣦⠈⠳⡆⢠⣆⣀⣀
⠀⠀⠀⠀⠀⠘⡀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢰⡀⢧⢡⠀⢱⠀⢻⠊⠃⠀⢀⠀⡉⠉⠛⠇⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡀⠀⠀⠀⢀⠌⠀⠀⠀⠐⡀⠀⠀⡀⣷⢸⡄⠀⢤⡆⠀⠤⡀⠀⠀⠉⠑⠀⠢⠶⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢷⠸⣄⢱⡙⣆⢻⡄⠘⣷⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠠⡄⠈⠢⡘⢶⢇⠙⡎⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⠀⠓⠈⠈⢇⠃⠀⠀⣄⣀⣀⣀⠀⠀⠀⠀⢠⣤⣶⠶⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡭⠴⠾⠟⠀⠀⡛⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣦⣴⣿⡷⠶⠆⢰⡶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦⠼⣭⣏⣁⣁⣤⠠⠀⣤⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠤⢄⡀⠀⠀⠀⠀⠹⡉⠙⠉⠛⢠⢰⣧⣀⣴⠃⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠴⢦⣀⣷⣄⠀⣤⣼⠦⡙⠯⣁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠊⠁

⢀⣴⣶⣶⣤⣀
⠘⢿⣿⣿⡿⣿⣿⣦⡄
⠀⠀⠉⠉⠁⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢾⡿⢿⢿⣿⣿⣆⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣾⢿⣿⣶⣶⡦⣤⣤⣶⣶⣶⣶⢶⣾⣿⣿⣿⣶⣶⣶⣤⣤⣤⠤⠖⠛⠁⠏⣶⣌⣷⣾⠻⣿⣿⣾⣭⠥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⠟⠁⠉⠱⢌⣾⣿⣿⣿⣿⣿⣷⣦⣠⡄⢉⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠈⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠀⠀⠀⢀⣴⣟⢿⣿⣿⣿⣿⣿⣿⣏⣠⣾⣾⣿⣿⠉⢙⣿⣿⠁⠀⠀⠀⠀⣏⠻⣿⣿⣧⣄⡀⣠⣥⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠙⠓⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⡡⣿⡏⢰⣿⡄⡀⠀⢩⣄⠀⠀⠉⣋⣀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠉⠉⠀⢻⣿⣿⣿⣿⣿⡿⠇⠂⠀⠀⢸⣿⣿⡇⠀⠉⠐⣣⣷⠘⠻⠟⠋⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣿⣿⣟⠉⠋⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⣁⣆⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⢀⡤⣄⠀⠀⠀⢀⣤⡶⠨⠵⠀⠀⠉⠉⠉⠐⠂⠀⠀⠐⠂⠀⢈⠉⠁⠐⣲⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣯⠐⣡⠂⣀⣴⣿⣿⠟⠐⠒⠛⠙⠒⠚⠀⠊⠁⠀⠀⠀⠋⠓⠉⠀⠉⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣡⠄⡖⠋⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠻⣇⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠹⢄⡀⠀⠈⢿⣷⣄⡆⣀⣀⣀⣠⠤⠤⢤⣤⣴⣶⣿⠯⢴⢒⣲⣷⣷⠀⢻⣿⣆⣒⣲⣶⣶⡶⣶⢂⣐⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⡀⢈⣶⣶⣶⣿⣿⠽⠿⣿⣿⢷⣶⣿⣿⠾⠶⣚⡳⠦⠤⠖⠖⠚⠛⠢⠀⡡⠏⠛⠋⠉⠹⠎⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢾⣟⠻⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣮⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⠠⢦⡷⠄⣀⠀⠹⢧⡙⢦⣀⡀⡀⠀⣀⣤⣠⣀⡀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⠄⠀⠈⠒⠈⠻⠿⣿⣾⣿⣿⣿⣿⣿⣻⣫⣾⣿⣶⣦⣀⣀⡀⡀⣀⣀⣀⣤⠶⣛⣥⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠤⠤⠜⢢⣜⠀⢀⢦⣷⣽⣿⣿⣿⣿⣿⡿⣫⣻⣽⡿⣿⠿⣻⣿⣿⡿⠟⣋⠴⣾⢟⣨⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⣤⡄⠐⡊⣤⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣟⣻⣴⣴⣎⣭⡿⣋⠀⠈⠁⠀⠀⠉⠩⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡒⢥⠀⢀⢠⢿⡝⠙⠃⠙⢯⣹⢿⣏⠋⠉⣿⣿⣿⡿⣿⣯⣿⣿⡟⢡⣈⣀⣤⡖⠠⠒⠂⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⣠⠶⠟⠁⠀⠀⠘⠷⠤⣩⣌⠘⢐⣦⢸⢿⣯⣿⣿⢻⠿⠏⣶⠛⠍⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠤⢟⣋⣿⡆⣰⠶⠖⢂⣙⣁⡀⠈⠛⣆⠀⢹⠘⠸⣿⣿⠏⠀⣰⡞⠻⣟⡁⢐⣀⠀⠀⠂⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣄⣽⣍⢛⣑⠿⢴⣶⣶⢠⣀⡉⠛⢦⡀⠨⠎⣠⠠⠗⢈⠀⠒⠛⠀⠘⠋⠀⢚⡻⠅⠀⠚⢃⠜
⠀⠀⠀⠀⠀⠀⠀⠤⢾⣿⣯⣾⣟⡉⣛⣻⣷⣷⣞⡿⣅⢠⣧⠀⠀⠮⠀⠉⠀⣴⠞⠛⠓⠀⠐⠀⠶⣦⣄⠀⢰⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⡿⣿⠿⡿⢟⣙⡩⠯⠚⢻⣴⠀⠀⠀⢀⢿⡇⣀⣋⡹⣬⡽⢶⣶⢿⣧⡐⢦⠦
⠀⠀⠀⠀⠀⠀⠈⢉⣁⣤⣤⣤⣬⣭⣾⣿⡷⠿⠂⠀⠁⠀⢸⣿⣭⣄⣀⣾⢾⡷⠟⣛⡶⠿⡟⠓⠙⠓⢮⡨⠁⠄⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠠⠼⣟⣿⡿⠿⣛⣒⣻⣿⡷⠄⠀⠀⠀⠹⣿⣿⠟⠉⠀⠀⢩⣉⡐⠛⣚⣓⣶⣦⣬⣯⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣭⣀⡝⡩⣡⣴⣧⡀⠀⠀⠀⢀⣀⣠⣾⡁⠀⠀⠀⠠⢼⣲⣶⡿⠯⠻⣿⣿⣟⡛⠉⠙⠻⠧⠌
⠀⠀⠀⠀⠀⠀⠀⠰⠞⢟⠟⡩⣊⠾⠉⠉⠄⠄⠡⠤⣀⠀⠀⠉⠉⠛⠂⠠⢀⡶⠐⠻⠉⠀⠀⠀⠈⠀⠒⢍⠑⠢⢤⣀⠀⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠋⠘⠃⠀⠠⠀⠄⠂⠀⠂⠻⢷⣶⣤⣤⣤⣴⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⣽⣟⢿⣿⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠋⠘⠃⠂⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣄⡀⢠⡆⠀⡆⠀⠀⢠⡅⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⡿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⡟⠻⠛⠉⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣴⣾⣿⣶⢹⣿⣿⡿⠟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢷⣄⡘⠻⠟⢃⣼⡿⠁⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠙⣿⡛⠛⣻⠛⢁⣴⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠘⠿⡿⠟⠀⢨⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⡏⠇⠀⠀⠀⠀⢠⣿⣸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠛⠁⢀⠀⠀⠀⢀⣾⡟⠘⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢸⣷⡀⣴⣿⣿⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⠻⣿⣿⣿⡿⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⢹⡇⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠟⠉⠙⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠲⣶⣶⣶⣤⣤⣤⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢁⡀⢠⣤⢰⡏⠀⡀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⢠⣤⠄⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⠀⠀⠼⠛⠓⠒⠀⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⢠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠈⠀⢀⣠⡀⠀⠀⠀⠀⢠⢿⣿⠟⠙⠆⢸⡇⠀⠀⠀⠀⠀⣦⠸
⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⢄⢀⠞⣿⡟⠛⠆⠀⠀⠀⠘⠸⣿⣿⠗⠀⢸⡇⢀⠀⡀⠀⣠⠇⠃
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠳⣜⡀⡴⠚⠁
⠀⠀⠀⠀⠃⢆⠀⢄⠀⢄⠀⢠⢀⡇⠀⠀⠀⠀⠀⢰⣶⣮⣿⠆⠀⠀⠀⠀⠀⢠⠞⠉
⠀⠀⠀⠀⠀⠈⠓⠮⣓⣆⢀⣦⢾⣀⠀⠀⠀⣀⡀⠀⠉⢉⣁⠀⣀⠜⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠑⠦⣄⡀⠈⠐⠊⠛⠉⠹⠃⢠⡤⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡍⠙⠛⡶⢄⣀⣀⡴⠃⢀⠞⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠈⠒⠤⠴⢀⡼⠡⠴⠚⠁⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣨⠆⠀⠀⠀⠀⠀⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠑⡄⠀⢀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡇⠀⠀⠀⠈⡦⣸⠀⠀⠀⢀⡏⡇⠀⠀⠀⠀⠀⠀⡤⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⠀⢸⠇⡏⠀⠀⠀⡜⠀⢣⡀⠀⢀⠀⠀⠀⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⣼⠑⠁⠀⠀⠀⢣⢤⡔⠉⠀⠀⠀⠁⠀⢪⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣠⡀⠀⣀⡠⠾⠃⠀⠀⠀⠀⠀⠀⡖⡇⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠚⠂⠐⠤⠔⠚⠁⠈⠑⠚⠁⠈⠀⠈⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⡿⠟⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅⣿⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣾⣿⣿⣿⣿⣷⣾⣷⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⣤⠌⣋⣉⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⡿⣿⣿⣿⣟⣿⣿⣯⡇⣼⠀⠀⠀⠀⠀⠀⠘⠋⡖⠀⠀⠉⠈⠻⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⡇⣿⢹⣿⣟⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣽⣿⡇⣿⣿⢿⣿⣇⣿⠆⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡤⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⣿⣿⣿⣿⣿⣿⣿⠯⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⠈⠉⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣯⡤⠤⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠏⠀⠀⠀⡀⡄⣴⣦⣹⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡈⠐⠚⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⠀⠉⠻⢿⣿⣿
⣿⣿⣿⣿⣿⠂⢀⢠⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⠿⠂⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣧⣠⣿⣾⣧⣰⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⣠⣤⣿⣸⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠖⠂⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⢸⡿⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡘⠁⣙⡻⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣧
⠀⠀⠀⠀⠀⠀⣰⢿⣿⡷⢽⣷⣦⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠈⠙⢷⡈⠁⣀⢶⡗⠾⡶⣯⡙⠂⢠⣤⣄⣠⠄⠀⠦⠜⠋⠙⠕⠀⠈⢻
⠀⠀⠀⠀⠀⠁⠠⢿⣛⣦⠄⠀⠀⠸⠇⢠⣿⣮⣷⣴⣷⣿⢿⣶⢾⡷⠟⢁⡄⠀⠀⠀⠤⢀⣈⠀⣀
⠀⠀⠀⠀⠀⠀⣄⠀⠻⡋⠀⠀⠀⠀⠀⢹⣿⣿⡅⣹⣿⣧⢸⣷⣿⣁⡜⠁⠄⠀⠀⠀⠠⣾⡧⣴⠃
⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠘⣙⣤⠐⡇⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣷⡖⢀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠃⠰⠀⠀⠀⠀⠉⠈⣷⢤⡛⣿⣿⣷⣿⣿⣿⣿⣿⠟⣟⠛⠛⢿⣄⡀⠀⠀⠛⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠀⠙⣿⢻⣿⠟⣿⣿⡟⠿⠟⠙⠚⠦⠄⠈⠋⢀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⣠⣤⡤⣄⠈⠘⡿⠆⠻⡟⠀⠀⠀⠀⠀⠀⠠⠀⣀⠉⠀⠀⣠⣿
⠀⠀⠀⠀⠀⢀⢿⣿⡦⢽⠁⠀⠿⢻⣷⢾⡇⠘⣃⠀⣺⠁⠀⣼⣟⣻⣷⡄⠀⠻⣍⠀⠀⠀⣿⣌⠃
⠀⠀⠀⠀⠀⠹⣷⣤⠷⢻⣷⣆⡈⠻⠿⠛⢻⣴⠁⠀⢿⣀⣸⣿⣿⣿⡍⠀⠀⣀⣼⣆⣵⠀⣷⡟
⠀⠀⠀⠀⠀⠀⢻⡟⠐⢢⣴⣽⣻⣷⣶⣶⣻⡿⠁⠀⢠⣿⣿⣦⣄⣀⣄⣠⣾⣯⠁⠻⠟⢙⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣗⣀⢉⡙⠋⠉⠋⠉⠉⢿⣖⢒⡾⠛⠻⠿⢿⣿⣧⣴⣿⣧⢀⣷⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⡄⠀⠀⠀⣀⣿⣟⡀⠀⢠⣾⣿⣿⣿⡟⠋⢹⣦⣿⣿⣯⣿⣦⡀
⠀⠀⠀⠀⠀⠂⠀⢀⠷⠿⠛⠿⢯⣽⢶⣾⡋⠛⠉⠛⠛⢋⣡⣴⣿⡟⣁⣬⣷⣦⠛⣦⣽⢿⣿⢶⣿⠖⡀
⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⢀⠀⠀⣧⠈⠙⢿⣿⣷⣶⣾⣿⠿⢟⠿⡟⣯⠛⠋⠻⣏⣿⣿⣿⣿⣨⢍⣠⡔⠂
⠀⠀⠀⠀⠀⠀⠁⡠⠀⢸⣃⠆⢀⣿⢴⠂⠰⢴⠇⠀⠉⠀⠀⠈⢤⠀⡜⢢⠈⠷⡈⢿⣿⣿⣿⣿⡋⣵⣷⡄
⠀⠀⠀⠀⠀⠀⠞⠳⣆⠙⡉⠀⠉⠙⡃⠀⠈⠄⠀⢸⠀⠀⠀⠀⢸⣾⣷⠀⡗⢆⣴⢱⠏⢸⣴⣋⠀⠀⠀⠂⠶⠒
⠀⠀⠀⠀⠀⠀⠀⡪⠵⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛⠀⠁⠀⣰⠚⠀⠈⠈⠁⠀⠀⠀⠀⢲⠶⠤
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢰⡤⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣠⣐⣿⡏⠸⠀⠀⠀⠀⠈⠀⠀⠀⢀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠸⠀⠀⡏⠇⢠⠀⠀⠀⢤⣄⡲⠀⣄⢿⣿⡟⠉⠛⠢⠀⠀⠢⠀⠀⢯⡀⠈⢠⠄⠀⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠹⢿⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠋⠈⠈⠀⠀⠀⠂⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢂⠙⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⣩⡾⣷
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⢲⡓⠀⠀⠀⠀⢎⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠳⡄⢦⠰⡶⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡐⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⠠⣥⠀⠀⠀⠀⣈⣙⣧
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣉⠆⠀⣤⡀⡉⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣶⡿⣧⡀⡀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢲⡉⠁⢼⣢⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⢨⡄⢠⠈⣿⣸⣿⢳⠇⣀⠀⢰⠀⡀⠀⣄⢈⡃⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⡌⠀⢠⠀⠁⡀⣈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠻⠀⠇⠈⠀⠇⣠⣿⢻⠂⠈⢖⢸⠀⡇⡇⢾⣼⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⢀⠀⠈⣧⠓⠀⣟⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⡧⣾⡀⠀⠈⠙⣾⡇⢿⣾⣿⡇⠀⠃⠀⠀⢀⠀⠀⠀⣀⠋⣄⠐⠀⣻⠀⠀⠉⠀⠠⠟⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢣⡀⠼⠀⠿⡷⣬⢷⣼⣇⠀⠀⠀⠂⠈⣷⠀⠰⣿⣿⣿⡧⠘⣃⣶⣌⠄⠀⡇⠀⠠⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⠄⠰⠦⠻⠀⠙⣧⢰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠠⡄⢸⣿⣱⡇⢸⡟⡿⠀⠀⣷⡀⣸⡟⣥⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⣤⠀⢣⡀⡆⢸⡗⠚⠛⣾⠏⠸⡏⠀⠀⠀⠀⠀⠀⠀⠘⢠⣿⣿⣷⠊⠘⠃⠀⣳⢛⠿⠙⢋⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢠⣏⢸⣦⣕⣶⠞⣩⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣽⠆⠀⠀⠁⣼⣄⣀⣾⠇⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠂⠈⣾⠏⠸⣾⣷⣾⣿⣿⣷⣾⣿⠷⠲⠀⠀⠀⠀⠠⠀⠀⢀⠀⠸⣹⡇⠐⠉⠀⢠⠤⣘⣡⠿⠀⠁⠀⠀⠁⠀⣦
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⣀⣶⡿⠿⠛⠛⠉⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⣼⡯⠀⠻⣿⣿⡶⣲⣿⠃⠀⠀⠀⠃⢀⠀⡐⢈⡖
⠀⠀⠀⠀⠀⠀⠃⠀⢰⡆⠀⠀⠀⡆⠀⡠⠗⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣀⠀⠶⠀⣘⣧⣴⣿⣶⣿⣧⠤⠉⠀⠀⠀⠀⣀⣤⣤⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠷⠀⠛⠰⠷⠿⠿⠛⠋⠉⠁⠀⠀⠀⣤⡤⠤⠞⠛⠛⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠛⠛⠛⠓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡤⠤⣄⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢠⠞⠁⠀⠀⠀⠑⡄⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠹⡄
⠀⠀⠀⠀⠀⡸⠀⠀⠀⣰⠏⢀⠀⠀⠀⠀⠀⠘⡀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿
⠀⠀⠀⠀⠀⢵⠀⠀⣼⠃⠀⢀⣡⠴⠖⠒⠛⠛⠉⠙⠛⠒⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠳⠞⠁⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣶⠞⠛⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣀⣠⣤⠤⠤⢤⣄⣀⠀⠀⠀⢠⠃⢸⠇
⠀⠀⠀⠀⠀⠀⢀⠇⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸
⠀⠀⠀⠀⣖⠀⠀⢸⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢹⣄⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⡜⠁⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠸⣧⡾⠁⠀⠘⠀⠀⠀⣀⡤⠴⠖⠚⠛⠓⢦⡀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⢰⡇⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠣⡀⠸⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⡙⢷⣄⣀⡀⠀⠀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⣀⡀⠀⠀⢁⡔⠉⠉⠛⢛⡟⠩⠵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⣼⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⡰⠋⢀⠄⠀⢠⠏⠀⠀⠀⣴⠋⠀⠀⣀⣤⠤⠤⠤⣀⣀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠇⡰⠃⠀⠀⢸⡀⠀⣠⡞⠁⠈⠙⠋⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⢳⡀⠀⢠⠂⠉⠋⠉⠀⠀⠀⠄⠒⠒⠒⠒⠒⠦⠤⠄⡀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠑⠄⠸⡄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠥⠖⠚⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠙⠦⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠈⠻⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⡊⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢧⠀⠀⢢⠀⠀⠀⢀⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣿⣟⠀⠀⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⡀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠡⠃⡂⠀⢀⣼⢻⠟⣀⣀⠀⠀⣼⣿⣿⣿⠿⠛⠻⠿⣷⣄⡀⠻⠀⠀⠀⠀⠀⠄⠀⠀⠀⣴⠀⠀⣼⠏⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡆⣴⠀⢀⣾⠃⢸⠇⠘⠇⠀⢺⣿⣿⣿⣯⣤⣤⣄⣄⡀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⣸⡿⠀⢹⣷⡀
⠀⠀⠀⠀⠀⡀⢀⡟⣀⠀⠸⣵⡏⠀⠀⠩⡀⠚⠀⠈⡷⠰⣾⣿⠿⠟⠛⠉⠉⠉⠙⠙⠟⠀⣠⣦⡀⠀⠀⣤⠀⠀⠻⡆⠉⣶⣿⡇⠀⠀⠙⣧
⠀⠀⠀⠀⣼⠁⣾⠡⠁⠀⢸⣿⠀⠀⠀⠀⠙⣿⣿⡿⠉⠭⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠇⠀⠀⠀⢹⣷⣿⡇⠃⠀⠀⠀⠈⢀
⠀⠀⠀⠀⢻⣻⣿⣿⣿⣿⣿⡿⡟⢓⠒⠖⠶⢿⡿⠁⠀⢀⣹⣿⠀⠀⠓⠀⠀⠀⢰⠀⣷⣶⣦⣀⡀⣀⣀⣀⣀⣀⠈⢀⢙⣿⠀⢀⣀⣀⣀⣀⣀⣆
⠀⠀⠀⠀⢸⣿⣧⠬⢴⣯⣾⣿⠀⠀⣠⡄⠀⠚⠀⠀⠀⠀⢉⣷⠀⠀⠀⠀⠀⠀⠀⠀⣫⠏⠛⢻⣟⠛⠛⠛⣟⠉⠉⢹⡋⠉⠂⠸⣿⡍⠉⠉⠉⠛
⠀⠀⠀⠀⠈⣛⣽⣆⠈⠛⣿⣿⠀⠀⠹⠃⣴⡶⠖⢿⣿⣿⣿⣿⠀⣀⠀⠀⠀⠀⣄⠀⡟⠀⠀⢘⠛⢀⣀⣈⣛⣀⣀⣈⡉⢀⡆⢸⡿⠷⠆⠙⠛⠁
⠀⠀⠀⠀⠀⣬⣿⡟⢸⣿⢽⣷⠀⠀⠀⢀⡙⠆⠐⠖⠂⠙⢛⣿⡀⠙⠀⠀⠀⠀⠉⢀⠁⠀⠈⠻⠿⠿⠃⠙⠈⠉⠉⠙⠙⡻⠇⢸⡿
⠀⠀⠀⠀⠀⡹⢿⣿⠸⠿⣿⣿⠀⣿⠀⠸⣄⣀⣀⣠⣤⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠹⠿⠁⢰⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣈⢟
⠀⠀⠀⠀⠀⠿⣽⣿⢐⡻⢯⣿⠀⠀⠀⠀⢽⡟⠉⠉⣿⣿⣿⣽⡇⢸⡆⠀⠀⠀⡗⠈⠀⠀⢸⠶⣬⣤⠀⠀⠀⣤⣤⠀⠀⠯⣻⣛⠿⢀
⠀⠀⠀⠀⠀⣰⣾⣿⠀⣋⣹⣿⠀⠀⠀⠀⢘⡇⢠⠀⢸⣿⣿⣞⣇⠸⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⣥⣭⠀⠀⠀⣷⣽⠀⠀⣶⡿⣿⠅
⠀⠀⠀⠀⠀⣈⣚⣿⠀⣿⣿⣿⠀⣿⠀⠀⢨⡇⠀⠀⠀⠀⠫⣼⣿⠀⡀⠀⠀⠀⠀⠀⠹⣋⢟⡭⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⣳⣖⣂
⠀⠀⠀⠀⠀⠈⠛⣿⡏⠼⣿⣿⡄⠀⠀⣿⣿⣛⠀⠀⠐⡿⢿⣷⣿⢠⡄⠀⠀⠀⣶⠀⠀⡟⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⢠⡿⡿⣿⡃⠀⠀⠈⠉⣿⠀⠀⠰⣿⣾⣟⣿⠐⡇⠀⢀⠀⠛⠀⠀⣷⠀⠀⢰⡄⠀⠀⠀⢴⡆⠀⠀⠸⢿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠾⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠒⠘⠃⠀⠀⠀⠘⠁⠀⠀⢸⣇⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣥⣈⡅⠀⠀⠀⢀⣿⠀⠀⠀⢐⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣝⣧⢄⣀⣤⣤⣼⣤⣤⣡⣬⡷⡿⠿⠧⡤⢤⠤⠀⠀⡀⠾⠛⠛⠿⠿⠿⠷⠶⠶⠶⠶⠦⠴⠾⠛⠛⠛⠂
⠀⠀⠀⠀⣤⡶⠶⠻⠟⠛⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⡆
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⢛⡻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣍⣡⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣏⢻⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡿⡿⢿⣿⣿⣧⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣟⣨⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣝⣿⣽⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⢿⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠻⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠢⣇⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⡀⠂⠘⠀⢹⠳⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠧⠀⠸⡜⣸⠷⢼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠁⠀⠀⠀⢀⠀⡀⠐⠃⠈⠀⢈⠏⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⠁⠁⠞⠰⢀⠀⠘⢰⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠠⣄⣀⠀⠀⠙⢷⠇⠀⠈⠁⠀⣲⣿⢟⣿⣿⣿⣿⣿⣿⣭⡽⠿⠛⠙⡻⣖⣷⠾⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢿⣷⣄⠀⠈⠀⠀⠀⠀⠈⠉⢠⠾⡣⣷⠚⠋⢉⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠉⠀⠒⠂⠤⣀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⣿⣷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠫⠀⣦⣤⠀⠀⠀⠀⠉⠠⠀⠄⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣈⠑⠂⠀⡆⠐⢄⠀⠀⠀⠀⠑⠶⢴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⡄⠀⠻⡥⠄⠂⠀⠲⠒⢀⡀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠁⠒⠮⣐⢤⡱⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣤⡕⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣿⣯⣿⣷⣦⠀⠀⠀⠀⠀⠂⠀⢀⠔⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠙⣟⠂⠙⡂⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠙⠤⣛⣛⣽⠏⠙⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣄⡤⠀⣾⣿⠷⠄⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣟⠛⠾⣿⠠⠀⠀⠀⢈⠐⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⢅⢿⣦⣀⣙⣷⣅⠀⠃⠨⠳⡀⠀⣐⡒⠄⠀⠀⠀⠀⠀⠂⠃⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠌⠀⣹⡝⢛⣹⣟⠿⣭⣄⣄⣾⢆⠙⢅⠠⡈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⢋⠛⡾⠿⢿⣿⣿⣿⣮⣨⣃⣣⣤⣷⣜⢆⠀⡢⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡀⠀⢠⣁⢈⠃⠋⠭⡛⠙⠝⠎⠁⠀⠀⠄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⡠⠖⠒⠒⠒⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠲⠞⠉⠀⠀⡀⢈⣱⠤⠤⠔⠉⠀⠐⠟⠶⠴⢎⡴⠉⢑⠀⠀⠀⠈⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⡥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣸⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣴⡀⣀⡀⠀⠀⠀⠀⠀⣐⣄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡰⠋⢀⣠⡄⠈⢳⡀⠀⢰⠋⡀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡇⠀⣾⣿⣧⣀⠀⡇⠀⢸⣸⣇⣀⢸⡇⠻⣄⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠳⣄⡈⠛⠛⢁⣴⣧⣤⣬⣿⣿⣫⣼⠓⢦⡈⠳⣄⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢀⣴⣿⠁⠀⠠⠊⢉⣩⠉⠛⠉⠁⠀⢧⣤⠀⣠⠇⠀⢀⠀⣿⡀⠀⠙⠓⠒⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⣴⠟⠁⢹⡇⠀⠀⠀⠘⠑⢤⡀⠀⠀⠀⠀⠈⠉⠁⠀⡠⠋⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠹⡷⠶⠶⢶⣶⣿⠖⠊⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⢲⠦⢴⣄⠀⠈⠒⠤⠤⠙⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠑⠀⡈⠙⠒⠲⢶⠶⠖⠻⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⡜⠘⠀⢠⣽
⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⡽⡆
⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⣷⠘⢆⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠄⢸⠇
⠀⠀⠸⣆⠈⠓⠤⣤⡴⠛⠓⠋⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⡏⠀⠀⠀⠠⠛⠀⡿⣄
⠀⠀⠀⠘⠷⣤⣀⡸⠀⠀⠀⠐⢦⡀⠀⠀⣇⠀⠀⠀⠀⢸⠀⢀⣀⠔⠁⠀⢀⠇⢹⡄
⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⢳⡀⠀⡟⠀⠀⠀⠀⢸⠂⡟⠁⠀⠀⠀⣸⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⢀⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠳⣼⡇⠀⣀⣀⢀⣀⣇⡇⠀⠀⠀⢸⣇⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⢠⠈⢣⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣆⣀⣠⣇⣼⣀⠀⠀⢀⠆⠀⢠⣀⣧⣄⣀⣀⢘⣀⣸⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠁⠀⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⣸⡷⣄⠀⠀⡀⠀⠀⠀⣠⣬⡁⠀⣀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢩⣿⡿⠟⢻⣏⠹⣾⡋⢿⣼⣆⣸⠿⣯⣄⣘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣶⠶⠀⢛⠉⠢⠉⣹⣝⢻⡯⠈⣋⣇⠙⢻⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⡀⠨⠄⡇⢾⣿⣿⣽⢰⠀⡿⢠⢶⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠾⣋⢹⡼⠓⣆⠈⢉⣇⡟⣘⣤⢤⣥⣴⣖⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⠀⠀⠉⢛⣿⣯⣹⡀⠀⢽⣧⣿⣦⣾⣶⠷⣙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠉⠀⠀⠈⣹⡿⢿⡇⡀⣸⣿⣿⡟⢿⣿⣓⣻⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⡄⠀⠀⠀⠀⠈⠿⠛⠙⠻⠿⢦⣬⣭⣿⢟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢀⠀⠀⠀⠐⠓⠢⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⢶⣤⣄⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⣏⣴⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠁⢨⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⣾⢿⣼⣿⡟⣻⣿⣷⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⢒⣐⣖⣀⡀⠒⢒⠒⠒⡒⠒⠶⣤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢒⣡⣄⠊⠁⣠⠀⠀⠀⠀⠀⠉⠛⢿⣿⠁⠀⠀⠀⠈⠀⠀⠙⣆⠀⠀⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⠚⢋⠉⠀⠈⠀⠀⠀⠀⢀⠄⠀⠀⠀⠹⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠁⡽⢃⡾⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢯⠀⠀⠁⠀⠀⠀⠀⠀⠸⡎⣇⢸⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡇⠀⢠⣴⣾⡇⠀⡞⠀⠀⠀⠀⢠⣄⠀⠀⣤⣴⢂⣼⠃⠀⠀⠀⠈⠘⣆⣴⣿⣀⡲⠄⠀⠀⣿⣿⠈⠁⠀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣾⠸⣿⡟⠀⢿⣿⣿⡇⠀⡇⠀⠀⠀⣠⡿⣿⣽⣿⣿⣿⡜⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⣽⣿⠦⡄⠀⢸⣼⡆⠀⠀⠀⠸⠿⣇
⠀⠀⠀⠀⠀⠀⠀⢻⣆⢻⣇⠀⠘⣿⣿⡇⠀⣷⠀⠈⠉⠉⠁⣽⣿⠋⢻⣿⠁⠀⠀⠀⠀⣿⡀⠀⠀⢎⠻⠧⠀⠀⡠⠀⢸⡏⡇⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⠀⠀⠘⣿⣧⠀⣿⡆⠀⠀⣄⠀⠀⠀⠐⣻⣯⠀⠀⠀⠀⢀⡿⠷⠀⠀⠀⠁⢰⣶⠟⠁⠀⢸⣿⡇⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⢻⣿⠀⣿⠈⠀⠀⠘⠳⣶⡆⠀⣻⠃⠀⣠⡶⠛⠉⣉⢀⠀⣤⣤⡈⠻⣅⠀⠀⠀⢸⣿⡗⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⢸⣿⡇⣿⠀⠀⠀⠀⠀⡼⠃⠀⡙⠀⡾⠃⠀⠀⠾⢿⡛⠃⠀⣻⡷⠀⢹⡄⠀⠀⢸⣿⣷⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣸⡟⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠸⠁⠀⠀⠀⠳⠿⠇⣠⣦⣙⣿⠃⠀⠃⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣴⣿⣿⡇⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⠿⡿⠛⠁⠀⠀⢀⢺⠄⠀⡟⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⢿⣿⣷⠀⠘⠁⠘⣾⣧⠀⠀⠀⠀⢀⡁⡅⠨⢽⣿⡇⣸⣿⡒⠂⠀⢸⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⣿⢿⣆⠀⠀⠀⠻⣿⣧⡀⠀⢠⢠⡄⣴⠶⢿⠟⣿⣿⣿⣧⣉⣀⡌⡠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠸⡆⠻⣆⠀⠀⠀⠈⡿⢿⣶⣤⣀⣤⡤⢺⠛⢷⠹⠉⠈⢿⠻⠋⡴⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢹⡄⠀⢷⠀⠈⠓⢦⣄⠀⠘⠺⠿⠻⣿⡏⣱⠹⣆⠀⠀⠀⠀⠸⣳⠟⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠁⠀⠈⢧⠀⠀⠀⠹⠦⣀⣀⣀⠀⠈⠿⣿⣷⣤⣉⣁⣠⣤⣤⡟⠀⠀⠀⢀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⣀⣀⣉⣡⠀⠤⠞⠋⠀⠀⠀⠀⠈⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⢰⣰⠃⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡇⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡟⡀⠀⠀⠀⠀⡰⠟⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠆⠀⡿⠀⢸⡏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠈⢰⡇⢀⠀⠀⠀⠀⠀⠈⠻⣿⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⢿⣇⣴⡄⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⠀⠀⠀⠈⠻⡿⣧⠀⠀⠀⠀⠀⠈⠃⢻⣶⣿⡄⡄⠀⢀⣤⣄⠀⢀⠀⠀⠀⠀⠀⡈⣧⠀⠀⠀⠀⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠙⠏⠻⢿⣴⡿⠁⠙⠻⣾⠃⠙⠂⠀⠀⠹⡿⠀⠀⠀⣸⠇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠋⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⠋⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠧⠀⣤⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⡀⠀⠀⠀⠀⠈⠛⢿⣾⣷⣤⣘⣦⡀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⡷⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣿⣧⣀⣠⣼⣇⣀⣴⣄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⡀⠀⣿⣷⣶⣿⣿⣿⣿⣿⣯⡿⡿⠼⢿⠿⣿⣿⣿⣿⣇⠀⢰⣶⣶⡄
⠀⠀⣀⣠⣤⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣷⠿⠿⠿⠿⠿⣿⡟⢻⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠶⢖⢲⣶⣞⣾⣶⣲⣾⣿⣿⣷⣿⣿⣲⢒⣶⣶⣶⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠿⢿⣬⢭⣭⣍⣭⣽⣿⣛⣿⣿⣿⣿⣾⣿⣭⣿⣿⣿⣿⡿⡝⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⣿⡟⠛⠻⣿⢿⣿⡿⢟⣿⣻⣿⣿⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣾⣆⣩⣴⡺⣿⣾⡶⠶⠟⠋⠉⠉⠅⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁

⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢸⣿⣷⣦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⣠⣶⣷⠀⠀⠀⠸⣦⣄⠀⠀⠀⢀⣴⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣽⣿⡿⢿⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣯⣽⣿⣿⡀⠀⠀⠀⢰⣿⣽⣿⣯⣯⣿⣿⣷
⠀⠀⠀⢸⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠰⢿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠆⠀⠛⠛⠋⠀⠀⠀⠈⠛⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣀⡀⠀⠀⣠⡾⠀⠀⢻⣿⣿⣿⣿⣿⣵⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠙⢿⣿⣵⣿⣿⣿⡇
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⣿⣿⡿⠋⠀⠀⢠⣴⣿⣿⣷⣯⣿⣿⣽⣿⣤⣶⣤⡀⠀⠈⠛⠛⣿⣿⣿⡇
⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠟⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠻⣿⡇
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⣌⠃
⠀⠀⠀⢻⣿⣶⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣷⣶⣦⣴⣶⣿⠃⠀⠀⠀⢸⣿⣦⣤⣤⣾⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⣟⣿⣿⣿⣿⣿⣇
⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣾⣾⣿⣿⣿⣿⣄⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠿⠿⠿⠛⠿⠿⠿⠛⠛⠿⠟⠃⠀⠀⠙⠛⠿⠟⠻⠿⠿⠿⠻⢿⡿⠃⠀⠉⠉⠉⠉⠛⠉⠉⠈⠉⠉

⠀⠀⢀⣀⣀
⠀⠀⠘⣏⠉⠙⠛⠶⢦⣤⣀⡀
⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⢠⠟⠀⠀⢶⠶⣶⡒⠒⠦⠤⠀⠈⠛⠶⣦⣤⣤⣀
⠀⢠⡏⠀⠀⠀⠘⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣤⣤⣀⣀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⣀
⠀⢸⠇⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠈⠛⠲⣶⣦⣤⣀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠈⠁⠈⠛⣶⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡏⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⢸⣷⠟⠛⢦⣠⣿⠴⠒⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠁⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣠⠞⠙⢧⡀⠀⠉⠛⠲⠶⠶⠦⠶⠶⠶⠖⠛⠋⠉⠀⠀⣀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠶⣄⣆⢙⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠛⠛⠛⠒⠒⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⣉⠤⠊⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠿⡟⠳⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣴⠟⠁⣠⠞⠁⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠈⠑⠢⢍⠛⠷⣦⣤⣤⣶⠶⠿⠛⠛⠛⠉⠙⠙⠛⠁⠀⡰⠁⢀⡔⠚⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠶⠠⠄⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢾⡔⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠐⠒⡻⡠⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠁⠀⠀⠀⠀⠀⠀⠀⢂⣍⠑⡄⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣶⣮⢂⠀⠀⠀⠀⠀⢸⠙⠛⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠡⠜⠁⢀⠀⠄⠄⠈⠀⠁⡴⠖⠒⠂⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣀⡀⠀⠀⠤⣀⠆⠓⠀⠀⢈⠀⠀⠐⠀⠄⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⠀⠀⠐⠀⠉⠁⡠⠄⠂⠈⠀⠀⠀⠀⣸⣀⣀⠀⡀⠆⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢆⣈⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠄⠒⠁⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄
⠀⠀⠀⠀⠀⢀⣤⢶⣤⡀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⢰⡿⠇⠀⠈⢿⡀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠘⣧⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⢸⡄⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⢷⡀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠻⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠚⠁⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣦⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣗⠀⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠈⢺⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⣷⢶⡆⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣘⠀⣠⢂⣠⣎⡤⠏⠀⠻⣦⣞⠀⢀⡴⠀⣀⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣠⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⣴⠀⠀⠠⠀⠀⢠⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠦⠤⠾⣤⣤⠼⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⡒⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⡫⠍⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠀⠈⠢⡀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠠⠊⠀⠘⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⢘⡄⠀⠀⠉⠉⠁⠘⣿⠁⠉⠁⠀⠀⢠⠁⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡘⠃⠀⠀⠀⠀⠀⠀⢿⡄⢿⠀⡿⠃⠀⠀⠀⠀⠙⢆⡸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢀⣤⣦⣤⡀⠀⠀⠀⠀⠀⢀⡴⢶⣶⡦⡀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡒⠀⠀⢠⢸⣧⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣾⣿⡇⠰⠀⢐⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠉⠁⠀⠀⠢⡙⠿⡿⠋⠀⠀⢄⣀⠀⠀⠻⠟⠋⠔⠁⠀⠉⠉⠱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠉⠉⠀⠀⠀⠀⠀⡀⠔⠉⠂
⠀⠀⠀⠀⡄⣒⠰⢄⠀⠀⠀⠀⠀⠀⠀⠈⡱⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⢎
⠀⠀⠀⠀⠢⢉⣅⡊⢀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠢⣤⢈⣉⣉⡉⢀⡠⠐⠁⠀⠀⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢖⡄⢀
⠀⠀⠀⢠⠀⠄⡈⣄⢁⠃⠀⠀⠀⠀⠀⠌⠈⢢⠀⠀⠐⡃⠀⣫⣙⠀⠀⡇⠀⠀⢀⠔⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⣃⣎⠧⠡
⠀⠀⠀⠠⠁⢂⠙⡹⠂⠀⠀⠀⠀⠀⡌⠀⠀⠀⠑⠂⢄⠈⠉⠀⠀⠈⡉⠤⠂⠉⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢁⣜⡞⠰⠂
⠀⠀⠀⠀⠑⢄⠑⠞⡀⠀⠠⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠀⢡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢉⠔⢀⡬
⠀⠀⠀⠀⠀⠀⠈⠲⢡⡌⢠⠀⢀⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣯⡀⠈⠘⣢⡄⠀⡄⢂⠀⠗
⠀⠀⠀⠀⠀⠀⢣⠠⠞⡧⠁⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⠌⢢⠈⠁⠘⢆⠑⣸⠱⠒⡠⠂
⠀⠀⠀⠀⠠⡀⠠⠱⡀⢀⣀⡸⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡄⢀⠀⠀⠀⢸⠀⠃⠀⡊⢀⠀⢀
⠀⠀⠀⢀⠀⠀⠁⣢⠌⢮⡴⢳⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣁⠜⠀⠀⠀⢸⠛⠉⠙⠦⠀⠂⠁
⠀⠀⠀⠣⢀⣁⣖⠁⠄⠀⠄⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠹⠃⠀⠀⠀⢀⠖⠬⢀⠠⢨⡠⠐⠒⡄
⠀⠀⠀⠀⠰⠁⠼⡀⡐⠠⠀⢀⡼⠒⠀⠀⠒⢤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠒⠀⠘⠓⢄⠀⢀⡴⡩⠧⢆⣨⣠⠊⢰⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠘⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⢈⠁⠀⢀⠀⢀⠰⡖⠁⠀⠀⠀⠀⠀⠀⠁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⢀⣃⠀⠰⢄⡄⠀⠀⣀⠁⠀⠘⣀⠀⠀⢀⡸⠄⠀⣨⡀⣨⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠁⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠻⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢴⠊⢿⣿⣿⣿⣧⡈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⢁⠍⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢾⣅⡐⣿⣿⣿⣿⣯⡐⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡨⢉⣤⣿⠏⠙⣿⣿⣿⣿⣿⣇⠀⠀⠰⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⡾⢃⣾⣿⠤⠤⣾⣇⠉⠻⢿⣿⣿⣷⡀⠀⠐⠆⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠸⠇⠸⣿⣠⠀⠠⠋⠙⠃⠀⠀⠉⠻⣿⣷⣶⣾⣶⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⣬⣿⣄⣀⣤⡾⡿⢿⣾⠃⣀⣉⠐⢆⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⢻⣦⢉⣿⣿⠀⠀⠰⢾⡏⠀⠛⠉⠀⠀⠀⠀⣀⣄⠘⣿⣿⣿⣿⣷⣾⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣿⡷⣾⠋⠉⠶⠄⠀⡸⠰⡟⠀⠀⠀⢀⣦⣄⣤⠀⣄⠀⠟⠻⣿⣿⣿⣆⠀⣉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠘⡟⢀⣴⣤⣷⡴⣿⠈⢻⣶⣶⣄⣠⣿⣿⡶⠀⢸⡇⠀⠀⠉⢹⣿⣿⣿⡟⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠘⢷⣏⣙⣿⠋⠛⠷⣤⣤⣄⢿⣿⠛⠛⡿⢿⢷⡎⠀⠈⠰⣤⣷⣶⢿⣿⣿⣄⢀⠰⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠛⠻⠿⣤⣶⣤⠀⠈⠟⣠⣿⠇⣠⣿⠒⡿⠃⣀⣛⣿⠿⠿⠏⠀⢹⣿⡿⠿⢷⣴⠖⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠆⠀⠀⢀⡀⢀⣄⠀⣀⣤⠀⢺⡿⢿⣿⣿⣿⣧⡀⢰⣿⡟⡟⠃⠀⠐⠛⠿⣿⡇⢀⣾⡅⢀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣤⠀⣴⠞⣿⢧⣤⣶⣿⣏⡉⠻⣧⠽⣿⣿⣿⣿⣿⣿⣿⣽⢻⣤⢛⣼⣿⣷⠋⣠⣾⣿⣷⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣤⣿⡄⠉⠐⠉⢿⣷⡿⠋⣀⣹⣶⣾⣿⣿⣽⠟⣭⣿⠏⠸⣿⣿⡝⠋⣹⣤⣽⡛⠛⣯⡀⠀⢈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⣿⣿⣷⢆⣤⣥⣿⣿⣿⡀⢉⡀⢺⣿⣟⣩⣿⣿⣿⣿⣶⣿⣿⣶⠷⢎⡁⠀⠚⠉⠙⠛⢿⡿⠄⢁⣤⠺⣿⠆
⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠈⣿⣿⣿⣤⣤⣈⣀⣨⡉⢿⡿⠉⢻⡿⢿⣿⠟⠛⠿⢟⢻⡇⠻⠛⢠⣀⣿⠀⠀⠀⢺⡆⢼⣧⣄⡈⢉⣥⡀
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣯⣥⠘⣷⣀⣼⠓⢂⣠⣶⠶⡶⠋⠈⠿⠀⠀⣟⠻⢻⣤⠄⠀⠀⠛⣮⣉⣾⣧⣤⣽⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢀⠘⠙⢿⣿⣿⣿⣿⣿⢿⣿⣿⣾⣿⣿⣿⡷⢏⣹⣿⣿⣧⣼⣇⠀⠀⠾⣧⣴⣿⡁⠀⠀⠀⠀⠉⠁⢩⣙⣿⠟⠉⠀⣡
⠀⠀⠀⠀⡀⢀⡇⣀⣼⣧⣷⣤⣿⣿⡿⠛⣿⣿⠉⣿⣿⡿⠛⢿⣤⣾⣿⣿⣿⣿⣿⡻⣿⣶⣾⡏⢁⣄⠙⢓⣤⣤⠄⣤⣴⠛⠛⠻⣶⣾⣷⡿
⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⡙⣻⣿⢁⣀⠁⠙⡟⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⠟⢻⣧⣤⣶⣤⣉⣋⣛⣯⣤⣾⡇⢠⣿⣿⣿⣇
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣉⣿⣿⠖⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢙⣷⠂⣹⣿⢉⠽⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣶⣷⣄
⠈⠿⠿⢻⣿⣿⣿⠛⡜⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡿⠋⠙⣿⣶⣿⡁⢙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟
⠀⠀⠀⠀⠉⠙⠃⠰⠿⠿⢿⣿⣯⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⡿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⣾⠿⠿⣿⡟⣛⣀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠈⠛⠿⠟⠿⠿⠛⠛⠛⠻⣿⣿⠷⠿⣿⣟⣿⠖⠋⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⠏⠙⢻⡏⠉⣙⠛⢛⣏⠛⢹⣽⠟⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣴⣶⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠁⠈⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣏⣕⣿⣿⣿⠀⠀⣿⣿⣿⣍⣹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠁⣤⣤⠈⠻⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠏⢠⠀⠀⢸⡇⠀⠀⡄⢻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣄⠈⠑⠒⢥⡽⠒⠋⠀⣸⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡛⠓⠶⠤⠀⠀⠤⠶⠚⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣆⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣦⣀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣀⣀⣤⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠘⠁⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁



⠀⠀⠉⠁
⠀⠐⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠛⠿⠿⢷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣶⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣾⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⣿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⣿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⡶⢒⠀⠈⢰⢠⣷⣄⠴⠖⣲⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⠫⣅⠢⣷⡓⠀⠀⠃⠠⣼⠟⢿⣿⣿⣿⠿⣍⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢝⣥⣏⡙⠛⣿⣧⣽⡧⢤⡤⢀⣽⣧⠀⣉⠁⡈⢰⣿⡿⡟⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣥⡾⣿⣦⣩⡿⢶⠏⣻⣷⣽⣿⣓⣚⠋⠻⣷⡟⢻⡇⠀⣗⠀⣼⣉⣳⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⠜⡋⣹⣾⡟⣃⢌⣹⣽⠟⣿⠂⠿⣿⡟⢿⣿⢿⣿⣹⣬⡻⣄⣷⣤⣉⡄⠙⣿⣏⢅⠙⢿⣶⡄
⠀⠀⠀⠰⠟⠩⣍⣩⡿⢧⡡⣿⡿⢿⣼⠃⠀⣀⣨⠀⣿⡜⣿⣷⡷⣾⣿⣿⠿⣿⣿⣿⣮⣿⡟⠻⠿⣶⣎⢻⣎⡙⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠛⠀⠀⠀⠀⠙⠛⠋⢗⢈⣇⠸⣿⣷⡹⢻⣿⣶⣮⣿⡿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣩⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡉⢻⡓⠉⠹⢿⠿⣿⣿⣞⣷⣶⣌⠻⠿⠿⣿⡿⠯⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠨⠀⠀⢀⠘⠀⠛⠩⠻⢿⣏⡿⢷⣿⠓⠪⠱⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉⠉


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣦⣷⣴⣤⣶⣤⣾⣵⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⢀⢠⠔⠊⠁⠀⠉⠐⠢⣀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡀⠐⡡⠁⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠈⢀⠔⠀⠀⠀⠀⢀⠔⠀⠐⢄⠀⠈⡈⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⣿⡆⠠⠀⠰⠁⢀⡀⡀⠀⢂⠀⡇⠑⡀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠘⠿⠿⠀⢀⠁⢠⣿⣟⡈⡗⠈⡄⡇⠀⠈⠒⠄⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣾⣉⠙⠚⢦⡀⠙⠛⠃⠀⢠⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠙⢿⣿⠿⠟⠣⠀⠀⠀⠔⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠈⠉⠉⠉⠁⢀⠀⠤⡒⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢈⠡⠀⠂⡡⠔⠁⠈⠐⠠⢄⡀⠀⠀⠀⠀⠘⡀⠐⠊⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡍⠁⠁⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣞⠀⠀⠀⢆⠀⠀⠀⡠⠔⠂⠀⠀⠀⠀⠀⠀⠸⣄⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⠚⠉⠀⠀⠀⠘⡀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⡀⠈⠀⠀⠀⢰⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⢴⡀⢀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⠀⠀⠀⢀⣀⣠⣤⣶⡿⣻⣿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⠿⠛⠩⠉⠻⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⡟⣽⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡘⢋⠀⠀⣠⣤⠒⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⡿⢹⣿⣇⣁⠀⠚⠁⠀⠀⠀⠻⣿⣿⣿⣿⣍⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣽⣿⢃⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⢸⣿⠿⣏⢻⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠇⠿⢿⣼⣿⣿⡿⣿⣿⣿⣽⣶⣶⣷⣄⢩⣽⣧⡿⠿⣿⣿⣿⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣩⣿⣤⣵⠿⣿⣄⠀⠙⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠰⠛⢫⣿⣿⣭⣍⣉⣉⣀⠀⠀⠒⠹⢿⣿⣿⣿⣿⣦⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠯⠁⠀⠀⠀⠀⠀⠡⢾⡿⠿⣯⠀⠀⠈⠀⠀⠐⠉⢙⣿⡿⡿⠉⠺⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣈⠁⠀⠀⠀⠀⠀⠀⠈⠿⣖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣈⣀⣠⠴⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣗⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢀⠠⢴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣯⠀⠀⠀⠀⣀⣠⠄⠐⠟⠉⠈⠁⠀⠀⢰⡀⠀⠀⡀⠀⠀⠀⢺⣶⣞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⣴⡡⠊⠀⠀⠀⠈⢿⡄⠀⠀⠈⠻⣶⣦⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣯⣠⣶⠀⠀⠀⠀⢈⠙⣿⣧⣦⣤⣄⠀⢰⣾⢹⡄⠀⠀⣤⣾⣿⡟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠟⣠⢀⠆⠀⠀⠙⢉⣏⣾⣸⣷⣾⣿⡞⣿⣿⣮⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⣮⣵⠏⠀⢀⣶⣄⣾⣾⣾⣿⣿⣿⠻⢿⣌⣿⣿⣿⣷⡦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣉⡀⠀⠿⠿⠋⠁⡟⣽⠟⡿⢹⣿⣇⢸⣿⣟⢈⠿⡟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠠⠚⠋⠺⠋⠀⠀⠡⢾⠟⠻⣿⠟⠻⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠐⠂⠻⠐⢊⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡀⠐⠀⠀⢂⠦⠤⠄⠒⠈⠁⠀⠀⠁⠆⢄⠀⠀⠀⠂⠑⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠱⠤⣤⠄
⠀⡃⢠⠄⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡄⣃⠀⠀⠀⡀⠀⠀⠀⢀⠂⠀⠀⠠⢴⠀⠀⠀⠀⠈⠀⡀
⠀⡨⠁⢀⠞⠂⠠⠤⠤⠀⠊⠀⠢⠀⠀⠆⡁⢸⠀⠀⠀⠐⢀⠀⠐⡉⢄⠀⠀⠀⠉⠤⢀⠁⠀⡁⢀⠰
⢄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠘⠉⠐⠠⠔⠒⠁
⠀⠀⠀⠀⠀⡠⢤⡤⠄⠠⡄⠀⠀⢁⣀⠀⠃⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠈⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠂⠀⠘⢀⠰
⠀⠀⠀⠀⠀⠈⠒⠒⠈⠁⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠁⠑⠀⠤⢲⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢈⠲⢄⠀⠀⠀⠠⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⠈⠂⠀⠠⠐⠀⠀⠀⠀⠘⡂⠀⠀⡇⠀⠀⠀⠀⠠⠒⠀⠄⠐⢀
⠀⠀⠀⠀⠀⠐⠄⠄⠀⠀⡠⠠⠀⠐⢀⠔⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⡈⠀⠀⠀⠀⠀⠰⠀⠜⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⡶⠂
⠀⠀⡀⠀⡀⣀⠜⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠜⠁⠀⢠⠀⠀⠀⠀⡄⠠⡀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠐⠒⠀⢄⠀⠀⠀⠀⠀⠐⠈⠀⠀⠁⠠⠃
⠈⠀⠀⠀⢨⡀⠘⠀⠀⡠⠀⠀⠀⠀⠠⢴⡦⠖⠀⠀⠀⢀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠈⠁⠀⠈⠑⢤⡀
⠇⠀⠀⡀⠀⠁⠉⠀⠉⠁⠀⠀⠀⡀⠀⠈⠀⠀⢰⠀⠀⣞⠢⠀⠀⠀⢰⠑⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠐⠄⠀⡇⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠠⢀⡀⢀⠌⠀⠄⠂⠀⠀⠀⣀⠞⠀⠀⠀⠀⠎⢀⡀⢘⠀⢄⢀⠅
⠀⠀⠈⠅⠠⠄⢄⢀⡄⠨⠂⠀⠀⠀⠢⠘⠢⠐⠉⠃⠤⠚⢉⠉⠉⠁⠄⠊⠀⠀⠀⠀⠀⠈⠁⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣈⣷⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠉⠗⠊⠉⠀⠀⠀⠀⠀⠀⠷⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠴⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⠀⠀⠀⠀⠀⣠⠏⠀⣰⣯⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠁⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⠀⢀⣤⠞⡱⠀⠀⠸⣿⡿⠟⠀⠀⠀⠠⠄⠀⠀⠀⣴⢻⣿⡆⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⠧⠖⠋⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⡀⠀⠰⣿⣿⣿⠇⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠⠄⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆⠀⠉⠛⠁⠀⠀⣼⠀⠀⠀⠀⠀⢀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡀⠀⠈⢦⡀⠀⠀⠀⠈⢻⠛⠛⠉⠀⠀⠀⠀⠀⠀⡞⠀⢧⠀⠀⠀⣠⣎⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣷⡀⠀⠀⠉⠙⢲⡖⠻⠿⢦⣀⠀⠀⠀⣠⠒⠀⠀⢸⠀⠘⣆⡴⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⣷⡀⠀⠸⣧⡀⢀⡴⠋⠉⠉⠉⠀⠀⡀⢀⡞⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠹⣄⠀⠈⠉⠉⠀⠀⣠⡖⣀⣠⣾⣣⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠀⠀⠀⠘⡈⠳⢦⣤⠤⡶⠋⠙⠉⠈⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣋⠩⠍⠒⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⠂⠤⠄⠊⠀⠀⠀⠀⠀⠀⢹⠶⣄⠀⠀⠀⠀⢠⠋⠀⠀⣀⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣼⠀⠈⢷⡀⠀⠀⣼⠀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠹⣄⠀⠸⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠹⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠙⢄⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡜⠑⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠳⢆⣀⠀⠀⣠⢀⠔⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢧⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠈⢄⠈⠛⠑⠶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⡔⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢠⣇⡙⠳⢤⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠈⣄⣀⣀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠲⠇⠀⠀⠀⠀⠀⠀⣸⠀⠉⠀⠀⠈⡹⠀⠀⠀⠀⠀⠀⠀⢸⠊⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣀⣠⡞⡡⠒⠉⠉⠁⠂⠀⠀⢸⣄⣸⣀⣀⣆⣠⠤⠛⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠻⣇⣘⠀⠀⠀⠀⠀⠀⠀⣀⡞⠀⠀⠀⠀⢸⣸⠁⠀⠀⠀⠀⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠒⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠙⠦⢦⣤⣤⣤⠼⠤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢀⡴⠛⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⠳⣄⠀⢀⣠⠴⠖⠚⠛⠉⠉⠉⠉⠒⠢⢄⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⡹⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⢦⡀⣸⠁
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠱⡏
⠀⠀⠀⠀⠀⠀⢹⡄⢀⠞⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⠒⠠⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢳⣸⠀⠰⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣶⠆⠱⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣴⣿⣆⣠⠱⠀⠀⠀⠀⠀⠀⠣⠍⠡⠔⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢠⠻⠿⠋⡠⠀⠀⣠⣐⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠉⠀⠈⠀⠀⠀⠉⠙⣏⠀⢀⠄⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠠⠁⠀⠓⠄⠔⠁⠉⠁⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠤⠔⠚⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⣾⠁⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢻⡀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣧⠀⠀⠀⣸⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⠀⠀⢠⡟⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠸⡄⠀⠀⠘⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣀⣴⠏⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⡀⣇⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠉⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠉⡩⠛⠂⠀⢀⠇⠀⠀⠀⢠⠇⠀⠀⠀⢠⡷⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠾⣧⣤⣤⣤⣼⡀⠀⠀⡀⣸⣅⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⣶⣶⣾⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠈⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⢉⣠⣤⣤⣤⣤⣀⡀⠀⣀⣤⣶⠶⢶⣮⣝⢿⠁⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠁⠀⠻⣶⣄⡴⠋⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠈⢻⣦⠐⠛⠛⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠻⠛⠁⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⣧⠀⠀⠀⠈⢿⣿⣿⡿
⠀⠀⠀⠀⢠⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢋⣉⡉⠻⣿⠀⠀⠀⠀⠀⢀⣀⡉⠙⢶⣿⣿⡆⠀⠀⠀⠀⠻⠿⠃
⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣻⣏⣿⠆⠟⠀⠀⠀⠀⠀⣾⣏⣿⡷⡌⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣀⣀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⣾⢿⣿⣿⡿⣦⠀⠀⠀⠉⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠙⣴⣿⣿⡦⠋⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⣠⣾⣷⣄⡀⠀⠀⠀⣀⣴⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠶⠾⠻⠿⠿⠿⠿⢛⣻⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠿⠛⠉





⡶⡶⢶⠶⡶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⣶⢶⣶⡶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⡶⣦⣤⣀⠀⠀⠀⠀⣴⠿⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⣩⣿⣷⣶⣶⣾⣧⣄⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣿⡿⢿⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⢠⣿⣿⢛⣟⢻⠁⠈⡟⣿⡛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣦⣭⣾⠀⠀⣶⣭⣵⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⣀⣀⣀⣿⣿⣿⡟⠁⢦⡴⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⠆⠀⠀⠠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢠⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠀⠀⢀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⢻⣿⣿⣷⣄⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⢻⣿⣿⡷⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡇⠀⠀⠸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠛⠋⠉

⠀⠀⠀⠀⠀⠀⣿⡵⠠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠻⣷⡔⠀⣳⣦⠀⠀⠀⠀⢀⡾⠁⠀⣦
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣤⣄⣀⣸⣧⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⡛⠿⣿⣿⣿⠁⠀⠀⣸⣿⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢩⠄⠀⣠⠜⣡⠀⠀⠀⣀⣿⣿⢗⡈⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢀⡰⠻⢠⣴⡆⢰⠊⠉⠐⢋⣀⠀⠘⢩⣿⠉⠿⠀⣟⢙⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⣠⠞⠁⢖⠄⢩⣤⠛⠃⠠⠄⠿⢿⡀⠠⣿⡸⠆⣠⠞⠀⠉⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣷⢦⣤⣀⣴⡶⠂⢀⡀⢀⡸
⠀⠀⠀⠀⡽⢦⣶⠎⢠⡏⣹⠛⢒⠐⠂⠀⠢⡝⠚⣿⣧⠰⠋⠀⠼⠁⠘⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣼⣥⠛⠻⠿⠟⣭⡸⣉⠉⡻⣿
⠀⠀⠀⠸⡇⠈⣀⠠⠟⢀⣛⠉⢉⡉⠛⠤⣀⠀⠤⠀⠀⠀⣀⡒⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣭⣄⣤⠀⢠⠀⣴⡶⠈⢢⣀⣿
⠀⠀⠀⠀⡻⢶⣿⠁⣰⣿⠻⠛⠋⣵⡖⠶⣶⠆⠀⠀⠀⠀⣭⡍⠀⣼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢻⢆⡉⢁⣿⠀⠘⠋⠀⣀⣀⠈⣿⣿⡀
⠀⠀⠀⠀⠁⠀⣿⣴⣿⠟⠀⠀⠾⢏⠀⣄⠉⠳⠂⠒⢤⣾⣟⠃⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠟⠛⠳⢖⠒⠀⠠⠈⠛⠻⢦⣽⡇
⠀⠀⠀⠀⠀⣰⢗⡄⠀⠈⠀⢶⣷⡂⠀⣀⣤⡶⢾⣧⣴⣿⢟⡐⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡄⠀⠀⠘⠀⠀⠀⠀⠀⠀⣉⣿⠁
⠀⠀⠀⣼⡿⠃⠀⠀⠉⠐⠚⠃⠈⠁⠐⠂⠀⠀⠀⠀⠰⠁⠀⢡⣾⡟⠁⢈⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠧⣕⣀⡀⠀⠀⡀⠀⠶⠟⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⣄⣴⣿⣿⣿⡿⠿⠿⠛⠹⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⢿⠐⢤⠈⣣⡞⠠⠀⠐⠖⠀⢀⠼⣷
⠀⠀⠀⠀⠀⠀⣈⠙⠓⠐⠟⠀⠀⢀⢀⠐⠿⠃⣀⣠⣶⢿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢀⣴⣿⣿⠸⣆⠠⣤⣈⣀⠀⠀⠀⠀⠈⢁⠀⢻
⠀⠀⠀⠀⠀⠀⠈⠉⡷⠖⠦⠀⠀⠈⠙⣄⣠⠆⠛⠟⠉⠀⠀⠀⣀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣇⠀⠈⠳⢬⡉⠁⠀⠀⠀⠒⠶⢤⡼⠿
⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠲⣦⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠙⠹⢧⣄⣄⡀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠧⠀⣿⡿⠷⠤⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠈⠃⠈⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠸⠃⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠈⠁⠀⢠⣾⣦⠆⠐⠀⠈⠁⠀⡀⠀⠀⠀⠀⣤⣄⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠃⠀⣀⣼⣿⡿⠁⣄⠀⣀⡀⢠⠆⠉⠋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣶⠾⠟⠛⢻⡇⠈⠀⠉⠙⠛⠫⠀⠀⢰⡃⣀⡴⠋⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠁⠐⠀⠀⠰⢿⣧⠀⢦⣤⠀⢠⡾⠁⠀⠸⡀⠉⠀⠐⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⢠⠀⢀⣤⣤⣿⡦⣀⢠⡤⢶⡆⠀⠀⠀⣿⠖⠀⢙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠙⣿⣿⣿⣿⣧⣀⠀⢀⣼⣧⡀⠀⢸⡇⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠧⠀⠀⣸⡷⣿⣿⣿⡇⢀⡤⠄⠀⣻⣉⠀⠘⣧⠄⢈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣶⡋⠙⢻⣿⣇⠀⠀⢀⣀⣉⣿⣿⣿⣄⠀⠐⢶⣿⣷⣶⠾⠄⠀⠠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⠀⠀⠈⣷⠤⠈⠙⠏⠀⠶⠟⠟⠿⣿⣿⣿⡟⠀⠁⠀⣿⣿⠟⢺⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢻⣷⠄⠀⠀⠀⠀⠀⢠⣀⡀⠀⣆⣰⣈⠻⡿⠃⠀⠀⠉⠙⠷⠀⠈⢧⣀⠀⠀⢄⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⢦⡄⠀⢀⡀⠀⠀⠀⠈⠀⠉⠉⠁⠀⠙⠶⢤⡤⢠⠄⠀⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣶⣶⡆⠀⢀⠈⢽⡿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠐⠛⠀⠀⠀⠀⣨⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⢿⠿⠻⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠿⠿⣿⠿
⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠶⣶⠶⠞⠿⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡀⣠⠶⠌⠀⠐⠤⡀⠀⠀⠀⢀⠠⠔⠒⢲⣤⣄⡀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣼⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣷⡀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢰⣿⠅⠀⠀⢀⣤⢶⣶⣤⡀⠀⠀⢠⣴⠶⣶⣦⣄⠀⠀⣹⣷⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠘⣿⣤⣀⡔⢿⣿⣾⣿⣿⠧⠤⠠⠼⠿⠿⣿⣿⡟⢳⣠⣿⠿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⡟⠉⠀⢸⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠹⢖⠀⠀⢠⠆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⢰⡇⠀⠀⠁⠀⠀⢀⣴⣶⠾⠛⣷⣦⡀⠀⠀⠀⠁⢠⡏⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠈⢻⡗⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠹⡆⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⠁⠀⠀⠀⠀⠀⠠⡇⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⣾⣿⣄⠀⠀⠀⠀⠀⠀⢀⡔⢣⡀⠀⠀⠀⠀⠀⠀⢀⣇⡀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠹⢄⣘⣦⡤⠔⠓⠁⠀⠀⠙⠑⠶⣄⣀⣜⣠⠞⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠲⠦⠦⣄⣤⠤⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠙⢦⡀⠀⠀⣀⣤⡤⠤⠤⠤⣤⣄⣀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡽⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢠⢋⣤⠬⣦⠀⠀⠀⠀⠀⢰⣥⠬⡱⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣧⠀⠀⠀⠀⠀⠈⠮⣿⡿⠃⠀⠀⠀⢀⠀⠘⠿⠿⠟⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠋⠁⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣂⣀⡧⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠁⢈⠈⠉⠀⢀⣠⡴⠋
⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠶⠤⠤⠤⠶⠖⠚⠉⠀⢳
⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⡏⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⠀⠀⠀⠀⠀⢸⠇
⡇⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠉⠉⠉⠉⠉⠙⠛⠶⢤⡀⡼⠂
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠱⡀
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⡤⠴⡶⠶⠶⢤⣀⠀⢀⠀⠠⡘⡆⣧
⢷⠀⠀⠑⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⠶⠖⠋⠁⠀⢸⡀⡆⢠⠂⣸⡀⢀⡱⢀⡽⠃⣾
⠈⢷⣄⠀⠀⠀⠉⠛⠓⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠿⠤⠶⠶⠶⠶⠶⠲⠒⠒⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠛⠻⢶⡞⠁
⠀⠀⠉⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠲⠶⠶⠦⠤⠤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠴⠶⠶⠶⠶⠖⠒⠒⠶⠶⠶⠦⣤⣤⣀⣀⣀⣼⠇
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶
⣿⣿⣴⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠬⣁⠀⠈⠁⠀⣧⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⡀⣆
⠀⢠⣤⠀⠀⠀⡀⠀⣾⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠮⢿⣿⣿⡛⠛⡄⣼⡇⠜⢳⣾⣯⣿⡁⠤⠀⠑⠆⠀⢀⠄
⠀⠀⠈⠀⠀⠈⢁⠀⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⡿⢿⣿⡇⠆⣿⣿⣸⢈⣿⠫⢿⣏⠉⠀⠲⠀⠀⡀⢰⡀
⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣇⣹⣿⡿⠅⣿⡇⠍⢹⣿⣧⢸⡿⣶⠀⡴⠄⠀⠀⠀⠁
⠀⢠⣤⣴⡤⠰⢵⡄⠀⠀⠀⠀⢰⠀⠰⠀⠀⠀⠀⣤⣿⡇⣿⣿⢟⣻⣿⡿⣿⣦⣘⡿⣿⣾⡇⣿⠀⢱⣿
⠀⡌⠛⢋⣶⡀⠘⠛⠀⠀⠂⠄⠌⠀⠀⠀⠀⠀⠀⢹⣿⠟⢿⣿⣸⣼⡿⠢⠿⣿⣣⣤⠹⡿⡁⣿⣤⡬⣷⡆
⢠⣾⣿⣿⣿⣿⣾⠋⡅⠀⠀⢠⡀⠀⠀⠀⠀⠂⠀⢸⣿⠃⣿⣯⣻⣤⣦⣭⣤⣉⣿⣿⣦⣼⣟⣿⠥⡭⡤⠤⠄⠀⠉⠁
⠀⠉⢻⣿⣿⣿⡏⠀⡇⠄⠀⠸⡇⠚⠛⠃⠀⠀⠙⣶⣟⣀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢟⡿⡿⣿
⠀⠀⠈⠉⠙⠻⠂⠀⠇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠙⠉⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠉

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠰⣏⣝⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣍⣹
⠀⠈⠻⠻⣿⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⡿⠟⠟
⠀⠀⠀⠀⠀⠻⢯⡷⣄⡀⢀⢠⠠⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡠⠤⡄⡀⢀⣴⢮⡽⠋
⠀⠀⠀⠀⠀⠀⠈⠹⢿⣏⠟⣴⠟⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⠻⣙⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣣⣼⢟⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣻⣧⡜⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠋⠈⢻⣯⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠙⣌⠂
⠀⠀⠀⠀⠀⠀⠀⡺⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠋⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠑⢄⠀⠀⡠⠊⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢎⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⣧⣀⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⢻⣧⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠙⢿⣦⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣶⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⢠⣾⢞⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡳⣷⡄
⠀⠀⠀⣰⠟⣱⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣎⠻⢆
⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣧⢰⢟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠰⠈⠀⠈⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⣀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⣀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠖⠉⢩⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣿⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⡏⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣧⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠛⠿⢶⣦⣤⣤⣀⣀⣤⠤⠖⠛⠉⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣻⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣿⣟⣛⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠁⠀⠀⠈⠉⠛⢦⡀⢠⡔⠛⠛⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⢿⣦⡀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⢶⠀⠀⠈⠳⠶⣦⡀
⠀⠀⠀⠀⠀⢠⣤⡶⠞⠋⠁⠀⡶⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣶⣄⠀⣿⡆⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⡄⠀⢠⣾⡛⢶⡀⠀⠀⠀⠀⠀⠀⡟⠙⣆⠹⡇⢹⣿⣆⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢹⡄⢳⠀⠀⠀⠀⠀⠀⣿⣤⣿⠀⣿⠀⢹⡟⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢸⠀⠀⣿⣷⣾⠇⡸⠀⠀⠀⠀⠀⠀⠹⣿⠃⣸⠇⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣠⢿⣦⡀⠘⢿⣯⡴⠃⠀⠀⠀⢀⣀⣀⣀⡀⠉⠁⠙⢾⣠⠴⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣈⡙⣷⣤⡄⠀⠀⠀⢠⣾⣿⠿⠛⠛⢿⡆⠀⠀⢈⡇⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⢹⣇⠀⠀⠀⠀⣿⣿⣷⣤⣴⠿⠁⢀⣠⡾⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⠷⣤⣄⣀⣀⣭⡴⠶⠶⠶⠶⠿⠟⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠿⢿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣿⠿⠛⠛⠛⠉⠉⠉⠻⣶⣤⣄⡤
⠀⠀⠀⢀⠄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣷⣦⣤⣴⣾⡿⠉⠉⣹⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⢸⠰⡀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠈⣻⣿⠟⠁⡇⠀⠀⣿⠀⠸⣇
⠀⠀⠀⠘⣆⠙⠦⣄⣸⣿⣧⠀⠀⠀⠀⣼⣿⠿⠶⣤⠇⠀⠀⠿⠧⡄⢉⠻⣆
⠀⠀⠀⠀⠈⠓⠦⣤⣭⣭⣭⣀⣀⣀⣀⣭⣀⣰⣷⣸⣇⣀⣠⡄⢰⣸⣼⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠉⠙⠋⠉⠉⠉⠛⠋⠙⠋⠉

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠲⠀⠀⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠄⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⢠⣤⡆⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢊⠏⠀⠀⢠⣤⡄⡈⠁⠈⠀⠀⠀⠀⠀⠀⢤⠠⠀⠀⠀⠀⠹⣦⡀⢄⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣾⣾⠀⠀⠀⣠⣶⢦⡀⢀⠀⠀⠀⠀⠀⢠⠀⠀⣠⡴⢦⣄⠂⠀⢸⣿⣄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⣽⡇⠀⡀⠐⢻⣿⡿⠃⠌⠀⠀⠀⠀⠀⠀⢠⡖⠻⣿⣷⠟⠃⢡⠸⣿⣿⠀⠀⣴⠀⠀⠈⠢
⠀⠀⠀⠀⠀⣾⠀⣠⠀⠀⠘⣿⡇⠀⠁⡀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠴⠀⢻⣿⡆⠀⢿⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⢸⣥⠷⠋⠀⠀⠀⣿⡇⠀⣐⠣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠆⢠⣿⣷⠀⠀⠀⠀⠀⠀⠠⣬⣤
⠀⠀⠀⠀⣿⣿⣦⠀⡀⠀⠀⢸⣧⢠⡈⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠐⠦⣨⡿⣿⡆⣱⣄⡀⠀⢀⣄⠀⣤⡑
⠀⠀⠀⠀⣿⢿⣿⣿⣥⣆⣦⠀⣿⢸⠡⢂⡤⠀⡀⠀⣼⣭⣿⣿⣿⣿⣿⣿⡆⠈⠂⠀⠀⣶⣵⠀⣿⣧⣿⣿⢹⡯⣷⣠⣼⡏⢹⣿⠛
⠀⠀⠀⠀⢹⣾⡿⣿⢩⣯⣼⡇⣏⡻⣄⠘⣿⢆⠄⠀⠙⠛⣿⣿⣿⣿⣿⠟⠀⠀⠈⠀⣶⠟⢉⣸⣿⣿⣿⣿⠸⣷⣿⣏⠁⣰⡟
⠀⠀⠀⠀⠈⠻⣼⣯⣾⣿⠟⠁⣿⠁⠈⠃⠘⡇⠀⠀⠀⠀⠀⠙⠟⡉⠀⠀⠀⠀⠀⣸⢃⠀⠈⠹⠋⠀⢹⣿⡀⣿⡿⡏⢸⢇⠎
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⣸⠏⣱⡀⠀⠀⢿⣦⡀⢀⣀⣀⣷⣶⣥⣀⣀⠀⢀⣰⢯⡤⠂⠐⠚⠂⠹⣿⣿⣇⢹⣷⣿⣊⡼
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣳⡯⠿⠟⠀⠀⣤⡈⠻⣿⣿⡟⠋⠉⠹⠻⠛⢻⣿⢏⡵⠋⠀⠀⠀⡀⠀⠀⠑⢜⢿⡜⠓⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠃⠀⡄⠀⠀⠀⠀⠀⠉⢻⣷⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠈⠂⠀⢠⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣶⣶⣾⠟⠡⠀⠀⠀⡄⣀⠀⢲⡀⠀⠸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⢻⣆⠀⠇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⣟⠀⠀⣵⣦⣾⣿⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡏⠀⠻⠱⠿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠁⠀⠘⠋⠛⠋⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠛⢷⣀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠢⠀⢱⣌⠀⠲⠀⠀⠀⠀⢤⣀⠀⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⡸⣥⢰⣤⣀⣶⣼⡶⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡀⠀⠀⠀⠘⠉⠈⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡆⠀⠀⢀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⢰⠄⢈
⠀⠀⠀⠀⠁⠀⠀⠠⡀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⠘⠐⡀⠀⠀⠀⠆⠀⠂⠀⠀⠁⠀⠀⠁
⠀⠤⣀⠀⠀⠀⠠⠀⠈⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠠⢠⠀⡨⠎⡀⠰⠄⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⠀⠰⠀⠐⠈⠅⠄⠀⠠⠨⠀⠑⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠿⠌⠀⠡⠤⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢰⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⢻⣧⡀⠀⢀
⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠀⠐⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢰⠀⡆⠀⠀⠂⠠⠀⠀⠀⡀⠀⠀⠠⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢷
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠮⠨⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣠⠖⠛⣿⡆
⢀⡴⣾⣿⣷⣶⣾⣿⣦⡀
⣿⡟⠁⠀⠀⠘⣧⠤⣾⣿⡄
⢹⣦⠀⠀⠀⠀⠘⣷⠟⠋⣹⡄
⠀⢻⡀⠀⠀⠀⠀⠘⢷⣽⣿⣿⡄
⠀⠀⢻⣆⠀⠀⠀⠀⠈⢻⣦⡿⠟⣛⣙⣛⡓⢦⡀
⠀⠀⠀⠹⣕⡤⣤⣤⡴⢞⣩⣶⣿⣿⣧⡀⠉⣳⠇
⠀⠀⠀⠀⠈⠙⠛⠿⠚⠋⠁⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠻⣟⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡜⢦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⠦⡀⠈⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠢⡀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠢⡀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢈⠒⢄⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡁⠢⣄⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣙⠢⣄⣙⠲⠦⣍⡒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣛⣶⣬⣟⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣿⣶⣤⣀⡀