Just jenn 100 Emojis & Symbols

⠀⠀⠀⣠⣤⣤⣤⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠿⣿⣿⣿⡿⣾⠁⠀⠀⠀⠀⠀⠛⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠠⡀⠀⠀⢸⣿⣿⢿⡿⣿⡛⢿
⠀⠀⠀⣤⣾⣿⣿⡷⢿⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸⣿⢿⣧⣾⣯⠿⢸
⠀⠀⠈⠛⠷⠿⠷⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿

⠀⠀⠀⠀⠠⠤⣄⢀⣀⡀⠀⠀⢠⣤⣴⣶⣦⣴⣤⡄⠀⠀⠀⠀⣀⣠⠄⠀⠄⣀⣀
⠀⠀⢠⡐⠐⠶⠶⠸⢿⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠓⠾⡿⠶⠈⢿⠀⠀⠀⠀⢰⣵⣶⣴⣦⡀
⠀⠀⣿⡇⠀⠀⠀⠀⢘⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⢸⣿⣿⣿⣿⠈⡇
⠀⠀⠘⠩⠶⠦⠴⣷⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣤⣀⣀⣠⣴⣾⠀⠀⠀⠀⠘⠿⣿⡿⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠁

⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠉⠛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀
⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡼⠁
⢻⣄⠀⠀⠀⠀⣠⢾⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢀⡴⢦⣀⠀⢀⡴⠷⣄⣀⣀⣠⡴⠋⠻⣄⣀⣀⣀⡴⠋
⠀⠉⠛⠒⠚⠛⡇⠈⠳⣤⣤⡴⠏⠈⠷⢶⠶⠞⠋⠠⢀⠈⠉⠉⡀⠀⠀⠉⢍⠁⠀⠀⠀⠀⠉⡟⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⠐⠀⠴⠀⡔⢁⠀⠀⡇⠀⢄⠑⣠⠊⠔⠀⠀⢰⠦⠀⠄⠀⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠸⡀⠀⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠁⢸⠀⠀⠀⢸⠀⡁⠈⢀⠇⠀⣼
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡇⠀⠀⠀⣧⠘⠀⠀⣇⠀⠈⢀⠛⡀⠂⠀⠀⣮⠐⢀⠌⢸⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢻⠠⠀⢀⡀⠙⠀⠀⡄⠀⠀⠁⠀⡏⠂⠀⢀⡄⠐⠁⠀⡞⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡄⠀⠈⢡⡀⢠⢠⠃⡄⢢⠒⠈⢃⠔⠢⡈⢁⠘⠀⢀⡇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡇⠀⢠⡀⠠⠊⡌⡰⠰⡀⠆⢀⣨⠀⣀⣈⠀⠆⠀⢸⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣁⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠄⠀⠀⠋⠀⠀⠀⡚⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⢁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉

⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠁⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠤⣠⠖⠒⠲⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⣠⠤⠐⢲⠫⠀⠀⠀⣷⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡏⠀⠈⢠⠀⣿⣒⣄⠇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣽⡆⢧⠀⠀⠀⣴⠿⡿⠋⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⢦⢸⡀⢀⣼⡇⠀⠀⢀⠂⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠘⣣⠟⠉⠉⠀⢀⠆⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⣤⣄⠀⠀⠀⠀⢠⣤⠀⠀⠈⣿⡞⣡⠄⢢⡦⡤⠂⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠈⠱⣄⠀⠀⢹⣆⠁⠀⣀⣇⣱⡹⣤⣤⠞⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢷⣄⣀⠈⢧⡀⢀⣹⣿⣿⡟⠋⠉⠛⢷⣆⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢄⡈⠙⠛⠛⠛⢟⠉⠁⠈⠻⣦⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢦⣀⠀⠀⠀⠹⠿⠿⠂⠘⣇⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠈⡀⠀⣿⠀⣠⡾⠃⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣏⠙⠻⠿⠿⠥⠴⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠂⠀⠁⠀⠀⠀⠀⣠⣤⣤⣤⣷⣶⣶⣶⣶⣶⣤⣀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣽⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠙⠛⠙⣿⡇⠀⠀⠀⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⣄⡀⢀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢹⠓⢻⡿⠛⢻⣿⡟⢿⣧
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠙⠷⢺⣇⣀⣤⣿⡀⠟⠉
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠿⣿⣿⡿⣿⠁
⠀⠀⠀⠀⢿⣿⠟⠙⠛⠛⠿⣿⠟⠀⠀⠀⠀⢻⣿⠟⠛⠋⠉⠹⠿⠟⠀⠀⠀⠀⠀⠀⠀⠐⠘⠻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⢀⠀⢀⣨⡶⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣷⣄⡀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⢠⡟⣾⣿⣿⣿⡶⡖⠀⠀⠀⠀⠀⠀⠀⠀⢘⢲⣿⣿⡟⠃⠻⡄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⠀⠉⠁⠉⣉⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠙⠋⠉⠀⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠐⢤⡔⠉⠐⠁⠰⣬⠀⠀⠀⠀⠀⢀⣩⠾⠙⠛⠛⢺⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣈⣠⠀⠠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠠⢴⣿⣿⠁⠀⠈⢹⣿⣦
⢠⣾⣿⣿⣿⡟⠛⢿⣿⣿⣿⡇⠀⠀⠀⠀⣾⡿⠟⠁⠐⠀⡈⠛⢿⡃⠀⠀⠀⠀⠀⠈⢹⣿⢻⣦⣤⣴⠛⠟⠁
⠘⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⠂⠀⠀⠀⠛⣁⣠⣿⣤⣤⣽⣦⣤⣙⠀⠀⠀⠀⠀⠀⠈⠛⠀⣿⣿⣿
⠀⠉⠉⠉⠹⢿⣿⣿⡿⠋⠀⠁⠀⠀⠀⠀⠀⠉⠙⠋⠉⠙⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⡆⠀⠀⠤⠖⢲⣾⠭⠗⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣷⠿⠖⠚⠃⠀⠀⠀⠐⠒⠒⠐⠒⠒⠒⠒⠐⠂⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠺⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣏⡴⠀⠉⠀⠀⠀⠀⣠⡤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠰⣿⣭⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢿⣿⡄⠀⠀⠀⠀⢀⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⣷⠻
⣿⣿⣿⡆⠿⠉⠻⣄⠀⠀⢤⢴⣇⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣤⡄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢰⡿
⣿⣿⣿⠈⢣⠀⠀⠁⠀⠀⠀⣄⣍⡄⠀⠀⠀⣀⢴⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣼⣿⣿⣷⢶⡤⣶⣴⡄⠄⠀⠀⠀⠀⣾⡇
⣻⣿⣿⡄⠀⣣⢸⠀⡀⡀⠓⣿⣿⡿⠂⠀⢾⡿⢻⣇⣄⢀⣤⡀⢀⡄⢠⢠⣤⣦⣠⣶⣿⣦⣼⡟⠋⣿⣿⠷⣿⡃⠀⠀⠀⠀⠀⠀⢻⡇
⣿⣿⣿⣇⠀⠙⢸⠇⣾⠁⠘⠋⠉⠛⠀⡴⢫⣇⢸⡏⣸⣾⡖⡃⠀⣇⣼⡰⢻⣇⢿⡇⢸⣼⣛⣷⣾⣣⣿⣧⣟⣻⣀⡀⠀⠀⠀⠀⠀⡇
⢿⠁⢰⣿⠀⡇⢸⠀⣧⣀⠄⠀⠀⣠⠞⠰⠂⠛⠘⢣⣟⡘⠉⢀⡀⢀⡀⣠⡇⣩⣤⠅⠈⠃⢩⠟⠙⠻⣦⡠⠘⠻⠷⠆⠀⠀⠀⠀⠀⠁
⠁⠀⢘⢻⠀⢻⠘⡆⠛⠿⠿⠉⠁⠁⢀⠚⠀⠘⢠⠏⢹⡇⣿⡗⠉⡗⠱⣏⡇⣛⡇⠀⠀⠀⠀⠀⠀⠀⢈⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢼⣤⡔⠈⠓⠀⠤⠤⠤⠤⠶⣮⣁⠀⠀⢑⣤⣶⠈⠁⠉⠃⠀⠋⠀⢉⠙⠈⢁⣤⡤⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⣧
⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⢀⣸⣫⡿⣎⡩⢿⠁⣿⠀⠀⣠⠋⢸⠉⢳⡼⠀⠀⡹⢸⣀⡀⠀⠀⠀⠄⠀⠀⢀⣶⠛⣶⣿⣁⣀⠁⣴⡄⠻
⠀⠀⠀⣿⣻⣷⡀⠀⡞⣠⣶⢿⣻⣿⡿⣿⣦⣀⠀⣹⠀⠐⣏⣠⠂⡀⢰⡇⢦⢀⡇⢸⡏⠀⠀⠀⠂⠀⣀⣴⣿⠟⢀⣼⠋⢿⣷⣾⣿⡇
⠀⠀⢸⣿⣿⣿⣧⢞⢞⡹⠃⢀⠜⣿⡁⣿⣿⠿⣰⣯⣤⡄⠟⢧⣄⡡⠋⡀⠀⡜⠀⠰⠷⠶⠖⠀⠀⠀⠘⠻⢿⣴⣿⠟⣠⡮⠈⢿⣿⣧⣀
⢠⡐⢺⣿⣿⣿⣿⣿⠈⠁⣰⠋⠀⠻⣇⣨⡅⠈⠁⠀⠈⠀⠀⠀⣠⣄⠀⢠⣗⣸⡃⢀⡴⢶⡆⠀⠀⠀⡀⠀⠚⠋⣡⣶⣿⣷⣴⣾⣿⣿⡟
⢸⣿⣾⣿⣿⣿⣿⣿⡇⠼⡡⠀⠀⢀⣉⡳⠀⠀⠀⠀⠀⠀⢠⠶⢺⡶⠀⡆⡇⢸⠀⠀⢀⡾⣧⣒⣄⠈⢙⡁⠀⠚⠻⠛⠋⢡⣿⠉⢈⡍⠁
⠀⢻⣿⣿⣿⡿⠋⣿⢧⣈⢁⡀⠀⠈⠈⠛⢀⣈⣀⣀⣀⠀⢨⣿⠸⣇⡤⡧⠇⢸⡆⣰⣿⣷⣾⣿⣶⠶⠂⠖⠒⠀⣴⣶⣷⣿⢟⣧⠛⡇
⠀⢱⣾⣿⣿⣷⣇⠈⠉⢿⡟⠛⠔⣃⠀⡽⠛⠯⠍⡽⠟⠂⠀⠤⠠⠥⠄⠀⠤⠄⠨⠭⠥⠤⣴⣆⣀⣀⣤⣤⣴⣿⣿⣿⣿⣿⠟⠃⢠⡷⢴
⠀⣿⡿⢸⣿⣿⣿⣶⣄⢸⡇⠀⠐⠛⠈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠉⠉⠉⡹⢿⣷⠶⠟⢻⣿⣿⣿⡯⢠⠀⣾⣇⣼
⠀⢻⣧⡈⠙⢿⣿⡻⠛⡆⠹⣆⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⡀⠀⠀⢀⢀⣤⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠛⠛⣉⣠⣿⣿⣿⣿⣿⣿⣿⣿
⣦⣶⣼⣿⣷⣽⣿⣿⣶⣦⣔⣼⣷⣤⣏⠀⠐⣠⠤⠴⠂⠁⠀⠀⠀⣾⣿⣿⣸⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢛⣛⣻⣿⣯⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⠻⠿⢿⣿⣿⣿⣿⣿⣯⣉⠪⠥⠈⠁⠀⠀⠀⠠⠶⠶⠶⠿⠛⠓⠚⢳⣆⠀⠀⣰⠃⠀⠀⠀⠀⣴⣾⣿⠟⣽⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⣿⣤⡀⠿⣿⣿⣿⠏⠙⠻⠖⢃⢀⡤⠤⠴⠶⠛⠋⠁⠀⣰⣄⣀⣀⣸⣿⣀⣠⣇⣀⡀⠀⢀⣜⣛⣿⣧⣿⣼⠛⢙⣿⣿⣿⣿⣿
⣿⣿⣾⣿⠊⣻⣷⣶⡾⣿⣷⣤⣤⠤⣴⣶⣦⣴⣤⠶⠤⠤⠤⠤⠤⢿⣿⣿⣯⣭⠭⢥⠿⠴⢶⣴⣶⣿⣿⡿⣿⢷⣿⣷⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣄⣴⣖⠚⠛⠛⠓⠀⣠⣯⣴⣶⣶⣿⣿⠿⠿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠺⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠴⠀⡀⠀⠈⠳⣀⠀⠐⠒⠀⠀⢀⠋⢀⣠⡴⣶⠴⠖⠒⠰⢦⣄⣀⣤⠃⠳⠀⠀⠀⠀⠀⢣⣀⡥
⠀⠀⠀⠀⠀⠀⢲⡀⠀⣠⠔⠒⠀⠀⠀⠀⠐⢄⠀⠀⠙⠛⠗⠒⢛⣡⡴⠛⠉⠀⠀⣿⡄⠀⠀⠀⣀⠾⠋⠈⢣⡀⠀⠀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡾⠁⡴⠆⠀⠀⠀⠀⠀⡄⠙⢷⣶⡤⣀⡴⢋⣁⣀⠀⠀⠀⠀⠸⣧⠀⣠⠞⠁⠀⠀⠀⠀⢣⠈⢦⣀⠀⠀⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢘⠇⠸⠀⠀⠀⠀⢠⠃⡘⠀⠀⠀⣨⡖⠁⡸⢿⣭⣇⡀⠑⡄⠀⠀⣿⡴⠋⠀⠀⠀⠀⠀⠀⠈⣇⢠⡉⠛⠿⠛⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡄⢧⠀⠀⠀⠀⣇⠀⡇⠀⠀⣼⠋⠀⠜⢀⣾⣟⢻⣿⡏⠻⠀⠀⣿⠁⢀⡀⠀⠀⠤⣀⠀⠀⢹⠀⣿⣷⣦⣤⠤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠁⠀⠀⠀⠀⠁⠉⠀⢰⡏⠀⠀⢃⠸⣿⣿⣿⠏⡠⠁⠀⢸⡇⢐⠋⣽⣷⣶⣤⣽⣧⣃⠼⢷
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⢀⠀⢀⡤⠀⠀⠘⡇⠀⠀⠈⠢⠤⠭⠀⣈⣀⠄⠐⢿⠀⠀⢸⣿⣧⣾⡟⠈⠋⠁⢸⡎⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠀⠇⠀⠸⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣀⣩⡿⣄⢠⣬⣤⡷⡀⠛⠛⠋⢀⠊⠀⢀⡾⠁⠀⠱⢄⡜⠁
⠀⠀⠀⠀⠀⠌⠀⠀⠀⢇⠀⢀⠀⠀⣏⢀⡄⠠⣶⣿⣯⡝⣟⠛⠛⠛⡩⠀⠀⠛⠀⠻⣦⡀⠙⠁⠀⠁⠀⢀⣠⠞⠁⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠁⠈⠀⠀⠈⢹⣾⠾⢻⣇⠈⣿⣯⠐⠀⠀⠐⠀⢀⠀⠀⠀⠈⠛⠶⢦⣤⠤⡶⢛⣽⠷⡄⠀⠀⢠⠈⠉⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣣⣾⡋⠉⢧⣹⣿⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆⣰⣽⡿⢷⣦⣉⡆⠀⠸⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣼⣿⢋⡴⠿⣦⢸⣯⡛⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡊⢠⣽⠋⣰⣾⣩⠙⣧⠀⠀⠙⠒⠚⣣⣤⡀
⠀⠀⠀⠀⠸⠀⠙⡆⢠⠀⠉⠁⠊⡀⢽⣿⡟⢳⣄⠸⣇⣿⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣣⡾⢃⣼⠿⣧⡹⠀⢀⣀⠀⠀⠻⡤⠏
⠀⠀⠀⠀⠀⠘⠋⠁⠈⢄⣀⠀⡔⠀⢸⡏⢹⡆⢹⡆⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠁⢸⣿⠟⣴⠟⢻⣆⢹⡇⠀⢣⡀⠳
⠀⠀⠀⠀⠘⠲⢤⣄⠀⠀⠀⠁⠀⠀⠸⠳⠀⢿⠀⣿⢸⡇⠀⠀⠀⠆⠀⠀⢠⠀⠀⠀⠄⠀⠀⢸⠏⣼⠇⣴⣟⣹⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⢸⡇⢸⡎⠁⠀⠀⠀⠈⢀⡀⠀⢀⡀⢁⠀⣀⠀⣹⢸⡏⣰⢇⡜⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢲⠁⢀⠀⢸⡇⣸⠇⢤⡜⡿⢻⠛⡆⠉⠞⢹⢳⡟⣿⢹⠆⢿⣿⢠⡟⢸⡇⠀⢋⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢤⠢⠔⠀⢀⡾⠀⣸⡷⠇⢀⣨⣅⡃⣸⣀⡇⢠⠀⣤⢸⣁⣏⣤⡀⠸⡇⣼⠁⠘⢷⡀⠈⢏⡁
⠀⠀⠀⠀⠀⠉⠛⠀⠤⠝⠂⠤⠄⣰⡿⠃⢀⠟⠁⠀⠀⠉⠋⢉⡁⠉⠀⠀⠀⠈⠉⠛⢙⠏⠀⠀⠸⣿⠀⢀⠀⠉⠀⠀⠒⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠒⠤⠤⠄⠀⠚⠃⠀⠀⠀⠀⠙⢆⠀⠂⠀⠀⠀⠀⠀⠀⢇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠒⣄⠈⡧⠀⠀⠀⠀⠀⢀⡔⠀⠆⠀⠀⠀⠈⠀⠁⢄⠀⠀⠀⢀⡄⠀⠀⢀⣀⡀⠀⠀⣠⠖⢠
⠀⠀⠀⠀⠀⢀⡠⡤⣀⠀⠀⠈⠄⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠉⠀⣰⡀⠀⠈⠀⠀⢷⣄⠀⠀⣂⡀⠞⠁⠀⡴⠁⠀⠙⡆⠀⢃⣀⢀⠇
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠾⠟⠀⢠⣾⡄⠀⠈⠛⠛⠛⠉⣠⠀⠀⠀⢣⡀⠀⡰⠅
⠀⠀⠀⠀⠘⢄⠀⠀⠀⡴⠀⠤⠐⢄⠀⠀⠀⠀⠐⠦⠄⠀⠀⠀⠀⡀⠴⠛⠉⠙⠦⣀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠚⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠊⠁⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠰⠃⢀⠀⠀⢠⠀⠆⠀⠠⠁⠰⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢴⠀⠡⡀⢀⠀⠀⠀⠂⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⠀⠀⢸⠀⠘⡀⠀⠀⠸⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠃⠀⢸⠀⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⠀⠀⡏
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠑⠀⠀⠈⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠃⠤⠐⠆⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⢀⣀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠏⣠⡗⠻⣍⡱⠀⡀⠀⠀⠀⢠⣧⣆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠
⠀⠀⠀⠀⠘⢈⡶⣦⣾⣗⠟⢉⡝⠀⠀⠎⢻⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠌⠀⢀⣿
⠀⠀⠀⠀⠀⠹⢆⣽⠉⠀⢦⣠⠀⠀⠀⠀⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠄⠁⢀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠈⠻⢶⠀⠈⠻⡄⠠⠁⠆⠀⠀⢸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠊⠀⢸⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠚⣧⠀⢀⣷⣠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠉⡇⠙⠻⠜⠉⠻⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⠀⣀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⣀⠀⢀⠀⣀⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠄⠀⠀⠀⣠⠲⠀⠀⠀⠄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣿⠀⡀
⠀⣴⣔⡱⣄⠘⣄⡹⢿⡛⠓⠓⠛⠛⡇⠀⢘⡀⢀⢤⣰⠞⠛⢧⠀⠠⠀⠀⡠⠀⢚⠀⠀⡀⠠⠻⠀⠀⠀⢘⠀⠀⠀⣸⡏⠀⡀
⠀⣿⣿⡟⢸⡆⣼⠙⠲⣝⠢⡀⠀⠀⢹⣾⢋⠐⠁⠆⢷⢠⣴⣶⣶⣦⡀⠀⣗⠀⠀⢡⠈⠀⠀⠘⣶⠁⠀⠀⢱⠠⣿⣿⠙⠑⢳
⠀⣿⣿⣧⣌⣀⡏⠀⠀⠈⡷⠇⠀⠀⠘⡿⣧⠀⠀⢸⠈⢺⣿⣿⣿⣿⡇⠀⡿⠀⠀⠀⣶⠻⡀⠀⡿⡋⠀⠀⡙⢸⣻⣿⡦⠄⠀⡇
⠀⢸⡿⠃⡀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣁⣠⣞⠀⠀⠙⠿⢿⣛⢀⡔⢻⡀⠐⢇⠀⢥⡀⠀⣴⠘⠒⠀⢠⠀⠀⠛⠛⢦⣼
⠀⠀⡧⣤⣭⠿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡭⠭⠭⣭⣽⡇⣼⡁⠀⠀⠀⠈⢧⠀⠉⠀⠈⡇⠀⠀⠈⠙⠀⡀⠀⠉⢰⡆
⠀⠐⠀⠀⢸⡄⢇⠀⠀⠀⠀⠀⠀⠀⢸⢯⣽⣿⡯⣛⣿⣿⠀⠀⠀⠘⣄⡴⠀⠀⠀⠀⣇⠀⠀⠀⠀⠄⠄⠀⠀⢞⡤⢀⠀⠀⠀⡄
⠀⠀⡀⠀⠈⠁⠰⠦⣤⣀⣀⣀⣠⠤⠚⣼⠀⢸⣿⣷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠄⡠⠠⠀⣠⣀⣀⣴⠎⣀⠀⢀⠠⣤
⠀⠸⠤⣤⣤⡤⠂⠀⢨⡉⠀⠋⡅⠀⢠⣿⣦⣼⣿⣿⡿⠉⠏⠉⠉⠁⢿⡼⠦⣤⡴⡏⠀⠀⠃⠀⠀⠉⠈⠁⠀⠈⠉
⠀⠀⠀⣻⠀⣇⢀⡴⢾⣧⢰⣤⣇⠀⠈⠙⡟⡟⢻⠋⠁⠀⢁⠀⠇⠀⢰⢀⣀⢠⡤⠃⢀⠀⠀⢀⠀⠀⣀⣀⠀⣀⠀⠀⠀⠁
⠀⠀⠈⠿⠟⠚⠓⠀⠾⢿⣿⣟⠿⠿⠦⠀⣿⣿⣽⣀⠀⠀⢠⠀⡄⠀⢤⣄⡄⠸⢤⡀⢋⣿⣷⣬⣑⣢⠜⠳⠃⠻⠛⠘⠀⠪⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⢿⢊⡛⠚⡳⠄⠛⠖⢿⡤⠚⠉⠀⠀⠉
⠀⢀⠔⠁
⠔⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣴⣿⣿⡋⠀⠀⢀⣠⣴⣶⣤⣄⠀⢀⢀⣀⣀⣀⡀⠀⠀⠐⠷⣤⡶⠀⠀⠀⢀⣤⣄⡀⠀⣠⣶⣿⣿⣷⣦⡀⠀⢀⠔⡀
⠀⠀⠀⠀⠈⠙⢿⡿⣋⣀⡀⢠⣿⣿⣿⣿⣯⣿⣆⠸⡸⠟⣋⣀⣀⠀⠸⣦⡞⢿⣇⠀⠀⠀⠸⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣷⠀⠊⠺⠿⢦⡀
⠀⢠⣾⢻⣟⡛⣮⠑⣿⣿⣿⠸⣿⠻⠿⣿⠯⢻⡿⠀⣰⣾⡟⢉⡻⣿⣦⠙⠓⠟⠛⠀⣀⣤⣤⣤⣄⡀⠘⣿⣿⣙⣉⣿⣿⡏⠀⣎⣤⢒⠀⠉
⠀⣿⣿⣼⡟⣽⣿⡇⢉⣉⣁⡀⠙⠿⣿⣿⡿⠟⠁⠀⣿⣿⣾⣿⣷⣿⣿⡆⢸⠷⢠⣾⣿⠿⣿⡿⢿⣿⣆⠈⠛⠿⠿⢛⡋⠀⠢⠯⣢⣏⠳⢀⣠
⠀⠘⢿⣿⣿⡿⢟⣴⡿⠿⠿⢿⣦⠀⠀⠐⢶⣠⡦⠀⠻⣋⣀⣉⣀⣉⡿⠀⣠⠄⣾⣿⠀⢀⠀⣀⠀⢸⣿⡆⠀⣰⡿⠿⠿⣷⡄⠚⠁⠀⠀⠸⣿
⠀⣠⣴⣦⣄⡀⢸⣷⡄⠀⠀⣰⣾⡇⠸⣦⡞⣿⡁⠀⠀⠈⠛⠛⠛⠉⠀⣿⣿⡇⢹⣿⠀⠈⠀⠈⠀⢸⣿⠁⠀⢿⣆⠀⠀⢸⡟⠀⣠⣶⠛⠓⣦
⣾⡟⠷⠶⢻⣷⠈⢿⣧⣿⣿⣿⡟⠁⠀⠙⠓⠛⠃⠀⠀⣠⣾⡿⣿⡿⣷⣮⡉⠀⠀⠙⠿⣿⣿⣿⡿⠟⣡⣴⠶⢾⣟⠿⠿⢛⣠⠀⣿⣸⠀⠀⣿
⢻⣇⣠⣄⣸⡿⠀⣀⡈⠉⠉⠁⣠⣾⣿⣿⡿⢦⡀⠀⣼⣿⡿⣷⠈⢰⣿⣿⣿⡄⠀⠀⣀⣀⣀⡀⠀⢰⣿⣤⣶⣦⣽⡇⢰⣿⣿⠇⠈⠛⠻⠛⠁
⠀⠉⠛⠛⠉⠀⣶⣿⣿⣶⡀⢰⣿⣿⡛⠉⣰⣿⣷⠀⣿⣿⣷⣌⠃⢈⣻⣿⣿⡇⢠⣾⣿⣿⣿⣿⣦⠈⢿⣁⣉⣈⡹⠃⠋⠛⠯
⠀⠀⠀⠀⠀⠀⠈⠛⢿⠟⠀⠈⢿⣧⣝⣾⣿⣿⠏⡀⠹⣿⣿⠛⠈⠛⣿⣿⡟⠀⢾⡿⣻⣿⣿⣻⣿⠀⢀⣬⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⡛⠁⢼⡿⠂⠈⠙⠻⠿⠿⢛⠉⢀⡀⠈⠷⣶⣶⣶⡶⠏⠀⣿⡋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣧⠀⣠⣶⣾⣷⣶⣄⠀⢄⢸⢞⢣⢤⡀⢠⣤⣍⠁⢀⣰⡆⠙⠳⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⣸⡟⠉⠉⠉⠉⢿⡆⠀⢍⡷⢛⠆⠁⠻⣿⡿⠀⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⣤⣄⣠⣤⣸⠃⣠⠀⡠⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⡄⠀⢸⣿⣎⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣧⠀⠈⠙⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠻⠿⣿⣿⣿⣿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢸⣿⣿⣶⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣼⣄⣈⣉⣋⣉⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡷⠶⠟⠙⠶⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⢉⡉⠀⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣉⢉⡀⠀⠀⠁⠀⠐⠀⢀⣨⣤⣤⠀⢸⠀⠄⠁⠀⠸⠿⠿⠿⠷⠠
⠀⠀⠀⠀⡀⠀⠁⠀⠐⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠦⠶⠰⡴⠀⠀⠀⠀⠂⠀⠨⠍⠩⠍⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⢦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠴⠒⢀⡄⠀⠠⡀⠀⠀⠀⠀⠐⠋⠈⠋⠀⠀⠘⠀⢸
⠀⠀⠠⠀⠄⣄⣠⣀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠄⠉⢀⡈⠁⠀⢀⡀⠉⠀⠀⠠⠄⠀⠂⠀⠀⠐⠐⠒⠒⠂⠀⠐⠀⠈
⠈⠈⠁⠉⠈⠈⠁⠁⠁⠀⠠⠀⠀⠘⠂⠀⡉⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠒⠅⠀⣤⠁⠂⡠⠂⠀⠀⠀⠂
⠀⢄⣀⡑⢦⡴⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠠⠀⠀⠀⠀⠐⠀⢠⢀⠁⠀⠀⠠⠀⠀⠀⠠⠀⠸⢸⠶⡇⡿⡄⠁⠠⡄
⠀⢀⡤⣤⣤⠶⠤⠼⠁⠈⠀⠀⡄⠀⠉⠀⠘⡇⠈⠀⠀⠀⢀⣡⣀⠈⣈⣄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠂⠁⠀⠋
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢠⠀⠀⠁⠴⠋⢳⣄⣸⠀⠉⠻⠉⠙⢁⠔⠀⠐⠲⢀⠐⠀⠀⠀⠄
⠀⢀⠀⠀⠐⠘⠂⠀⠀⢀⠀⠀⠧⠀⠔⠀⠶⢲⡤⠤⢄⠈⠁⠐⣂⡀⠢⠀⠁⠈⠀⠈⠁⠉⠀⠀⢦⠀⢠⠒
⠀⠉⠉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠃⠐⡂⠠⠅⠀⠀⢆⠀⠐⠂⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⢐
⠀⠈⠂⠘⠀⠀⠐⠅⠀⢃⠀⠀⠀⣀⠀⠀⠀⠀⠆⠈⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠖⠀⠰⠄⠀⠀⠤⠀⢀⠄⠀⠁⠘⠀⠄⠈⢀⠀⠀⠂⠀⣀⣀⡀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⣤
⠀⣀⠀⠠⣤⠄⡀⢀⠀⠈⠀⠁⠠⠀⢳⠃⠀⠀⠀⠀⠀⠀⠀⣼⠄⠨⢹⠀⡴⠚⠁⠠⢀⠀⠀⠀⢀⣈⡁⠀⠀⢀
⠀⠁⠀⢱⣶⡶⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠁⠀⠀⠡⠀⠘⠓⠒⠋⠀⠧⠀⢥⡀⣤⠁⠀⡠⣶⡒⣠⣔⢤⣤⠀⠠
⠀⢐⠀⠸⣛⡇⠀⠏⠀⣁⢈⠀⠀⠁⠀⠀⠈⠐⠈⠀⠀⠈⠀⠈⠉⠉⠀⢸⢂⡲⢒⡄⠀⠀⠀⠓⢻⡳⢚⠷⣸⢁⡀⢼⣇
⠀⠨⠄⣀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠐⠃⠠⢱⣶⡆⠄⠀⠁⢰⠒⠀⠒⢂⠈⢉⡥⠟⠉⡅⠉⠙⠀⠁⠚⠛⠉⠀⠋⠁⢀⡻
⠀⠀⠀⠀⠡⠀⠀⣀⣂⠋⢀⣠⡀⠀⠀⠀⢸⣿⣷⢀⣤⠀⠀⠠⠀⠀⢀⠀⠾⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠘⢦
⠀⠠⠤⠤⠤⠥⠀⠉⠁⠁⠈⠀⠁⠀⠀⠀⣺⡟⣿⠀⠉⠀⣉⠀⠀⢀⢘⠀⠀⠀⠀⠂⡃⠀⠀⠀⠀⠀⠀⢀⠀⡅⠀⣠⠟
⠀⢠⣤⣤⣤⣤⣄⠀⠹⣦⠎⠀⠀⠀⠀⠠⠉⠁⠁⠠⠀⠀⠈⠃⠉⠁⠁⠀⠀⠀⠀⠀⡅⠀⠀⠚⠛⠀⠴⠿⠂⠉⠀⠉
⠀⠀⠘⢛⠋⣿⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡎⠀⠂⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠂⡀⠐⠀⠰⠨⡀⠐⠳⢆⡀⠀⣀⣀⣠⣠⡀⠀⠀⠂⠁⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠛⠁⠉⠁⠀⠙⠂⠀⠁⠀⠀⠁

⠀⡠⢰⡎⢑⡆⢄⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠠⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠑⡌⢚⡗⢩⠊⠀⠀⠀⠀⠀⠀⠀⠳⡐⡀⠠⡡⠆⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⡃⠠⡇⢸⡅⠰
⠀⠀⢰⠈⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣟⡛⠀⠀⠀⠀⠀⠀⠀⡃⠀⡗⢸⡆⢰
⠀⠀⠘⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠃⠈⠓⠘⠃⠐


⠀⢠⣾⡭⢭⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠻⠉⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠰⣶⣶⠛⠳⣶⡶⠂⠀⠀⠀⠀⠀⡼⠁⠀⠄⠘⣿⠙⢦⠀⠀⠀⠀⠀⢀⣀⣶⣼⣷⣴⣀⡀
⠀⣿⣦⠐⢂⣴⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⡦⣾⣧⠀⠀⠀⠀⠀⠀⠀⢧⡀⠐⠀⢀⡿⠀⣸⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠛⠋⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠙⠓⠊⠁⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠚⠋⠁


⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢀⡀
⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡬⠮⠆⠀⠀⠀⠀⠀⠀⠀⣰⡾⡆⢰⢿⣆
⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣇⠀⣀⠀⠀⠀⠀⢠⡆⢄⠒⠀⠐⠐⡄⠀⠀⠀⠀⠀⠀⣿⣿⠇⠘⣿⣿
⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠚⠉⠛⠛⠉⠛⢻⠟⠋⠛⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢻⣿⡇⢸⣿⡟
⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠡⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⡀
⠠⠶⡴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣷⣄
⠶⣏⣁⣉⣉⣉⣹⠶⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠟

⣿⣿⣿⣿⡸⣷⡙⢷⣜⣿⣌⢳⣌⠛⠿⣶⣦⣬⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⡿⣛⣵⣿⣵⣾⣿⢟⣠⣾⣣⠟⠁⣠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠹⣷⣄⠈⢿⣿⣷⣿⣷⣦⣄⡈⠙⢻⣿⠉⣿⣿⣿⣼⣿⣿⣵⠟⣋⣭⣿⣿⣿⣿⣿⣿⡟⢁⣤⣾⡿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠹⣿⣆⠀⣿⠇⣈⡉⠛⠿⣿⣦⡈⣿⣇⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⠟⠋⢉⣁⢹⣿⣴⣿⡿⠋⣴⡿⢹⣿⣹⣿
⣿⣿⣿⣿⢿⣿⡄⠹⣿⣼⡿⢸⣫⡩⠳⢦⡀⠙⢿⣿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠋⢀⡴⢚⡩⣹⡎⣿⣿⠋⣠⣾⠟⢁⣿⢣⣿⣿
⣿⣿⢿⡿⣧⡻⣿⣆⢻⣿⡇⣿⢸⢹⣞⡎⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡪⡾⡿⠆⡇⣿⣧⣾⠟⠁⢀⣾⣧⣿⣿⣿
⣿⣿⡎⣿⡜⢿⣼⣿⣷⣿⣇⣿⢸⡶⣫⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣭⣻⣿⡆⣇⣿⡿⢋⣠⣾⣿⣿⢟⣽⣿⣿
⣿⣟⢿⣜⣿⣄⠙⢿⣿⣿⣿⣾⣆⣶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣼⣿⣿⣿⣿⣿⢟⣫⣴⣿⢟⣿⣿
⣿⡛⠿⣿⣿⣿⣷⣦⣽⣯⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿
⣿⣉⣥⣤⣭⣽⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⣄⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣦⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣭⣉⣹
⣿⣥⣶⣶⣾⣿⣭⣭⣽⡳⠆⠀⠀⠀⠀⢸⠀⠀⣿⠊⢳⡀⠀⠀⠀⢠⠏⢸⣇⠀⢈⠇⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣯⣿
⣿⡿⢛⣽⣿⣯⣭⠴⠂⠀⠀⠀⠀⠀⠀⠈⢧⡀⣿⠇⢠⠇⠀⠀⠀⢸⡀⢸⡿⢀⠞⠀⠀⠀⠀⠀⠀⠀⠲⢦⣭⣿⣿⣯⣍⡻⢿
⣿⣴⠟⣫⣿⡿⠁⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠒⠉⠀⡠⡤⠀⠀⠙⠒⠛⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠙⢿⣿⣿⣷⣍⣻
⣿⣷⣿⣿⡟⣠⠞⢋⣽⠿⠛⠉⣿⣷⢦⣄⡀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⢀⣠⡶⣾⣟⠉⠛⠿⣭⡛⠶⣌⢿⣿⠿⣿⣿
⣿⣾⣿⣿⣿⠃⣰⡟⠁⠀⢀⡼⠛⡏⠀⠈⠙⠳⠦⣤⣀⣀⢴⣿⠤⣀⣀⣤⠴⠞⠋⠁⠀⢻⠻⢦⡀⠀⠈⠻⡄⢹⣿⣿⣾⣿⢻
⣿⡿⡏⢸⣿⠀⡟⠀⠀⠀⠉⠀⠼⡏⠀⠀⠀⠀⠀⠀⠀⢉⡟⡏⢿⠁⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣧⣾
⣿⣿⣷⠀⢻⡆⠀⡀⠀⠀⠀⣀⣼⣿⡇⠀⠀⠀⠀⠀⢀⡞⢀⣷⡘⣧⠀⠀⠀⠀⠀⠀⢼⣿⣦⡀⠀⠀⠀⢀⠀⣸⢏⡟⢸⣿⣿
⣿⣿⣿⣧⡈⢻⡀⣷⣄⡐⠋⢉⣴⣿⡇⡄⠀⠀⠀⢀⡾⢠⠏⡏⢳⡈⢧⠀⠀⠀⠀⢠⣸⣿⣮⡉⠁⢀⣠⣿⢰⠟⠈⢠⣿⣿⣿
⣿⣿⣿⣿⣷⣦⣁⣙⣮⣉⣽⠟⠋⠈⣿⡇⡄⠀⠀⡼⣱⠏⠀⡇⠀⠳⡌⣇⠀⠀⢤⢸⡟⠀⠙⢿⣭⣩⣽⣃⣀⣠⣴⣿⣿⣿⣿
⣿⡻⣿⣯⡙⠛⠛⣿⣿⣿⣿⡛⠻⣦⠘⣿⠁⠀⢸⣷⡁⠀⠀⣇⠐⠀⢹⣾⡄⠀⢸⡿⢁⡴⠟⣛⣿⣿⣿⣿⠛⠛⢉⣿⡿⢛⣿
⣿⣿⣎⢻⣧⢠⣾⡿⢿⣟⢿⣿⣷⡘⣇⢸⣧⡀⠘⣿⡇⠀⠄⠉⠠⠀⣸⣿⠀⢀⢾⡇⣾⢡⣾⣿⡟⣿⡿⣿⣷⣀⣾⠏⣴⣿⣿
⣿⣿⣿⠀⢻⣿⣿⣷⣺⣿⣼⣿⢹⣿⣿⣾⠁⠙⣆⢻⣷⠈⠀⠳⠀⠀⣿⡇⡴⠃⢸⣷⣿⣿⡟⣿⣷⣿⣁⣿⣿⣿⠏⢰⣿⣿⣿
⣿⣿⣿⡇⠀⣿⣇⢿⣿⡟⢻⣷⣴⣿⣿⠏⠀⠀⠘⣾⣿⡀⠗⠀⠪⢰⣿⣿⠁⠀⠀⢿⣿⣿⣤⣿⡟⢿⣿⠓⣾⣿⠀⣸⣿⣿⢹
⣿⠹⣿⣷⡀⠹⣿⣿⣟⣁⣐⣬⣽⣿⣿⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿⣿⣯⣥⣑⣞⣻⣿⣿⠇⢠⣿⣿⠃⣼
⣿⣷⣾⣿⣿⣦⣬⣝⣛⣻⠿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣇⠀⠀⠀⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⣛⣋⣭⣤⣶⣿⣿⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠔⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣿⣇⣀⣏⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢨⠇⠀⠀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣏⣌⠥⠀⠈⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠈⠀⢸⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⣀⣠⡤⠀⠀⠀⢀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀⣤⣤⣶⣾⣿⣿⡿⢃⣗⠀⠄⢼⡄⢃⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⢀⣄⣾⠀⠈⠁⣤⡀⣀⡭⠴⣙⡻⣿⣿⡿⡟⢿⠿⣿⡟⠋⠁⢸⣿⠀⠂⣿⡇⢠⡄⠈⠑
⠀⠀⠀⠀⠀⠀⢀⣿⡀⣼⠿⠓⠁⡠⠟⠪⠴⠀⠩⠈⢁⣄⢂⡡⠈⠉⠉⢠⠀⢀⡄⠁⠊⠲⠄⢄⣠⠈⠙⠛⠻⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠸⡟⠁⠡⠀⠀⢊⠑⢠⡆⡀⠠⣀⠀⠈⠁⠀⠀⡠⠐⠈⠀⡄⢸⣇⠀⠀⠀⠀⠀⠘⠩⢂⠀⠀⡉
⠀⠀⡀⠀⡀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠸⡧⠓⠀⡀⠀⠀⠀⣸⡆⣀⠄⠀⣰⣷⣾⠿⠄⠲⢂⠀⠂⠀⠄⢸⠀⠀⠇
⠀⠀⠃⠠⠀⠀⠤⠈⠐⠒⠀⠾⢿⣿⠿⠆⠀⠈⠉⠀⠚⠃⠾⢧⠈⣉⠈⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠁⠲⠄⠀⢰⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠒⢂⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⠀⡨⠋⢵⣶⣶⣝⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⣶⣭⣵⡌⢙⣆⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠞⠛⣿⣿⣿⣿⣿⡁⢰⣮⡅⠀⠛⠁⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⠙⠂⠀⠀⠀⠀⠐⢆⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⣶⣦⣄⡀⠀⠀⠀⠀⠈⠉⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠛⠛⠛⠋⠤⠀⢒⣀⡉⠀⠈⠀⠀⠈⣀⡒⠂⠤⠄⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡩⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣁⠒⠀⠠
⠀⠀⠀⠀⠀⠀⣠⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢒⡀⠉⠀⠀⠀⠀⠀⠀⠈⡅⣒⠐⠌⠤
⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⠩⠤⠐⠂⡀⠀⠀⠀⠀⠀⡐⢒⣰⣸⣀⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⣒⣉⠭⠄⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠠⠄⠒⣒⠈⠉⠀⠀⠀⠀⠁⠉⣐⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⣀⣀⣀⠐⠈⠉⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢹⣿⣿⣿⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠀⢀⡤⠄⣠⣶⣦⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡿⠀⠀⢻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⢸⡇⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⡄⠀⠀⠀⠀⢸⠁⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣀⠃⠀⣿⠀⢠⡎⠁⣾⠄⠀⠀⣿⠀⠀⠀⡶⠀⠀⣤⠀⢠⡞⠑⣿⠈⠀⢾⡄⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰⣿⠐⢠⡏⠀⠸⣇⠀⣿⠀⠀⠀⣿⠀⠀⢀⡇⠀⢀⡏⠀⢸⣇⠀⠹⣆⠀⠘⣧⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠹⠄⠞⠁⠀⠀⠀⠀⠘⠆⠀⠀⢹⠀⠀⠈⠙⠀⢸⡇⡀⠁⠈⠀⠈⢻⡀⠂⢹⡇⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⡸⠁⠀⣠⠇⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢀⡤⢠⣀⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⣷⡀⠀⠀⠀⠀⠀⡨⠟⠓⢤⣰⣶⠀⢰⠁⠀⠀⠈⢃⠀⠀⠀⢀⠄⠠⡀⢈⠀⠁⠀⠀⠠⠀⠀⠢⡑⡀⠀⠈⢂⢀⠐⡄
⠀⠀⢿⠹⣆⠀⠀⠀⡜⢀⡀⡈⠀⢹⠉⠀⠘⣄⠀⠀⢀⡆⠀⠀⠀⣵⣶⣶⣶⠃⠀⠐⠀⡀⠀⠐⠀⠀⠑⡘⢄⠀⠨⠈⠀⠐⠄
⠀⠀⢸⡄⠘⢿⠀⠀⣹⣄⠀⠀⢀⡜⠀⠀⠀⠈⠉⠉⠁⠹⣷⣄⠀⠙⡿⢿⣯⣿⣿⣿⣾⠀⠘⠀⠀⠀⠀⠈⢄⠒⠁⡄⠀⠀⢈
⠀⠀⠸⠇⠀⠀⠰⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⡄⢠⡶⠀⠀⠈⠛⠃⠀⠀⠀⠂⠉⠉⠉⢣⣄⠀⠀⠀⠀⠀⠀⠀⠢⡀⠡⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⢀⣀⣼⣥⣤⠀⠀⢀⣴⣶⣶⣦⣤⠄⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⢀⣀⠀⠀⠀⠀⠀⣼⠋⠙⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠰⢿⣿⣷⣿⣿⡁⠀⡀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠘⢾⣿⣿⣶⣲⠀⠘⠳⢾⡷⠾⢶⣄⠀⠈⠻⠿⠿⠋⠀⠀⢰⣿⣿⣿⠿⠗⠀⠧⠰⡷⠀⠀⠰⠆⠀⠀⠠⠊⠀⠀⠀⠀⠑
⠀⠀⠀⡽⠉⠉⠀⠀⠀⠀⠸⣇⡀⣠⡟⠀⠀⢀⠀⠖⡦⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠁
⠀⢠⣴⣧⠀⠀⣤⠴⢶⡀⠀⠉⠛⠉⠀⢠⣾⣦⣤⡀⠀⠐⢳⡀⢀⠠⢀⣸⠀⣆⡀⠤⡀⠀⠀⠀⠠⢢⠀⢃⠀⠀⠀⠂⠢⢀⠆
⠀⠀⠀⠉⠀⠀⠿⣤⡼⢋⡀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢠⣴⡀⡇⠀⠑⢤⣼⣿⣧⣤⠊⠀⠀⠠⠀⠀⠀⠀⠀⠁⠂⠄⠠⠐⠁
⠀⢤⣦⡀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠈⢄⠙⢿⡄⠀⡿⡱⠁⠀⣊⠀⠼⠓⡧⣤⣑⠄⠀⠁⠀⠸⠁⠀⠀⠀⢀⣴⣤⡀
⠀⠈⠻⡟⠀⠀⠀⢲⣿⣿⡇⠀⠀⠀⢀⠀⠀⠉⠦⢦⡔⠊⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠂⠀⡠⠤⠀⠀⠀⠀⠉⠛⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠶⠷⠒⠀⠀⠶⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠂⠂⠁⠀⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣆⠀⢀⡖⣶⠆⣦⡀⠀⠀⠀⠀⠀⠀⣤⠀⣀⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⠉⠛
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠯⠽⠀⠒⠀⠀⠀⠀⠀⠰⢿⡟⢩⠧⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⡞⠛⠛⠛⠛⠛⠛⠉⠛⣷
⠀⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡀⠠⠀⢁⡀⠠⡄⠀⠀⠀⠀⠀⠦⣤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠘⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⣛⣻⣟⣛⠛⠛⠋
⠀⠀⠀⠉⠉⠙⠓⠒⠦⠤⢄⣼⠁⠀⠀⡒⠉⠙⣶⣄⠀⣤⠘⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣌⠉⠉⢉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣼⣿⡖⠁⠙⠟⢣⠀⠀⠀⣴⠟⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢀⡴⠋⣹⠆
⠀⠀⠀⠀⡠⠄⠀⢀⡀⢰⠡⠀⠀⢇⡘⠉⠑⣤⣤⡄⡘⠀⢰⠟⠀⠀⠀⠀⠀⠶⠆⠀⠀⣾⣿⡧⠀⠀⠄⠐⣏⣠⠞⡡⣪⣶⣄
⠀⠀⠀⠀⠠⣰⡀⠘⠀⠀⠀⡀⠂⠈⠻⣦⠀⠛⡿⠋⠀⠀⠀⠀⠀⠠⡶⢠⡜⠛⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⢻⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⣦⣴⣦⣰⠷⡴⠷⣴⠆⠀⠀⠀⠀⠀⠀⠀⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⠀⠀⠄⠀⠈⠀⠀⢀⡠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣰⣫⣾⠿⢷⡌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⣤⣤⠀⢸⣿⠀⠀⠁⠀⠀⠀⠀⠀⢀⣷⣿⠇⠀⠀⠹
⠀⠀⠀⠀⠀⢰⣶⣶⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⢀⡀⠀⠀⠀⣿⣿⠀⣼⣿⠀⢠⣄⠀⠀⢠⣶⡆⢸⣿⣿⡀⠀⠀⠀⠀⣤⣶⣤
⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⣿⣿⡿⠀⣾⣿⠉⠉⠍⠀⠀⣿⣿⠀⣿⣿⠀⣸⣿⡄⠀⢸⣿⡇⠀⣟⣿⣇⠀⠀⠀⢸⡏⠀⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⠀⢠⣿⣿⡇⠀⡇⣷⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⣿⡇⢻⡀⢸⣿⠃⠀⠸⡜⣿⣆⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣾⣇⢸⣿⣿⡇⠀⣟⣿⣛⣛⣛⠀⠀⣿⣿⠀⣿⡇⠀⣿⡇⠀⢧⢸⣿⠀⣾⠂⠙⣞⢿⣦⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣼⣿⣿⡇⠀⡇⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⣿⡇⠀⠘⣿⣿⠀⣟⠃⠀⠘⣮⢿⣇⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡇⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠈⠉⠀⠙⠁⠀⠀⠘⡞⣿⡆⢿⣤⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⠿⣿⣛⠀⠀⣿⣿⠀⣿⣷⠀⠀⣀⠐⠀⠐⠀⣀⡀⠀⠀⠀⠀⢳⢻⣇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⢿⣿⣿⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⣿⣤⡄⠀⠀⠀⠉⠀⠁⠀⢀⣿⣳⡆⠀⠀⠀⢸⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢀⡀⠀⠀⠈⠛⡙⣇⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢤⠀⣠⠀⢀⣄⣀⣴⣀⠼⣧⠜⠧⠜⠀⠀⠀⠉⠉⠉⠀⠀⠀⢨⠿⡄⠀⠀⠀⠙⣏⣛⣧⣀⣠⣋⣙⡛
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠉⠈⠁⠀⠁⠀⠀⣀⣀⠀⣀⣀⠀⢢⡞⠀⠀⠀⠀⠁⠀⠀⢀⣠⠀⠀⠈⠻⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠁⠀⢠⠎⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡴⠀⢠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠀⢀⡴⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢸⣯⣿⣁⠀⣀⣨⣿⠉⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣯⡤
⠀⠀⢸⣿⣿⣿⣿⡟⠛⠛⠿⢶⣶⣾⠛⠛⠛⠶⢤⣄⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⣈⠃⢀⣤⡀⠶⠀⠀⠉⢀⡍⣩⣡⣾⡀
⠀⠀⢸⣿⣿⣿⡿⠿⠶⠤⣄⣀⠈⠉⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠟⢡⡎⢀⡴⠿⠿⠶⣤⣠⢄⡿⣋
⠀⠀⢸⣏⣿⣿⣄⡀⠀⠀⠀⠈⠉⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⣿⣇⣀⣤⠀⢀⠀⠀⣀⣀⠀⠀⠀⠄⠀⠙⣤⢈⢠⣤⡀⠀⠀⠘⢘⠛⠁
⠀⠀⣿⣿⣿⠋⠉⠉⠉⠉⠙⠒⠦⢄⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣀⣿⣿⣿⣿⣾⠀⠆⢀⣀⠛⠟⠻⠧⡁⠀⢀⡠⠋⢀⠀⠙⣿⣇⠀⠀⠀⠘
⠀⠀⢻⣿⣿⠷⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢐⣒⣛⣻⣿⣿⣿⣟⣘⢷⣦⠿⠿⠷⣦⣤⣄⠀⠀⠀⠁⠀⠀⠙⠀⠀⣉⠀⠀⢀⣤⡍
⠀⠀⠈⢿⣿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⣿⣿⣿⣿⣿⢿⡿⠋⢀⣠⡀⠀⢘⣿⡂⠀⠐⡿⠷⠀⠀⠀⠀⠠⣇⠰⠟⠁
⠀⠀⠀⠀⠉⠛⠛⣿⣿⣿⣿⣿⣿⡷⠶⠶⠓⠚⠛⠛⠛⢉⣉⣭⣿⣿⣿⣿⣿⣿⣿⠂⠈⠛⠙⢌⣩⣿⣧⣤⡄⣀⢀⠀⠀⠀⠈⠛⠉⠋
⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠉⠁⠸⣄⣀⣀⣀⣠⠀⠐⠊⠉⠁⢘⣿⣿⣿⣿⡏⠁⠙⢦⣤⣴⣷⡾⠟⠹⡇⠀⠀⠘⠉⠳⣦
⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⢈⡭⠋⠉⠀⠀⠀⠀⢀⡴⠿⠿⣿⣿⣿⣷⣀⣰⠞⠛⠋⡿⠛⣷⡀⠈⠠⣴⣶⡄⠀⢿⣄⢲⣿⣿⡄
⠀⠀⠀⣼⣿⣿⠖⠒⠀⠁⠀⠀⣠⠎⠀⠀⠀⠀⠀⣠⡴⠋⠀⢀⠔⣹⣿⣿⣿⣯⠀⠀⢠⣤⡄⠀⡿⣧⣤⠀⠉⠛⡀⣀⣾⣧⠀⢁⠀⠀⡀⠛⢦⡀
⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⣸⡯⠀⠀⢀⣠⠴⠛⠁⠀⠀⠠⠂⠀⠀⣿⠟⣿⣧⠀⠄⠹⠟⢏⡁⢀⣿⣿⣻⣶⠞⠛⠋⢨⡖⠂⠉⣷⣿⣿⣧⢄⣵
⠀⠀⣿⣿⣿⣷⠂⠀⠀⠀⠠⠈⠚⠓⡿⠋⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢸⡇⠘⢿⣆⣀⣸⣄⣠⣷⣾⠿⠟⠟⢠⠀⠀⠀⠀⠂⠀⠈⠉⣿⠙⣿⣷⣾⡆
⠀⠰⣿⣿⣿⣧⡀⣀⡴⠊⠀⠀⢀⠊⠀⠀⠀⠀⣀⠄⠛⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠻⣿⡿⠯⠛⠁⠀⠘⠀⠀⠁⠀⠀⠀⠀⢀⠀⣸⡟⢠⡾⠗⠚⠃
⠀⠀⢿⣿⣿⣿⣿⣿⣆⣀⣀⣤⣾⠀⠀⣀⡠⠂⠀⢠⣤⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠷⣶⡦⢠⠄⠀⠀⠈⠁⠀⠀⠀⠀⣸⣿⣟⢲⣦⠄
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢀⣤⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⡾⠻⣥⣟⣳⠘
⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣉⣽⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⡶⢾⣟⡻⠿⢵⣦⠜⠿⠇⠉
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⡛⡳⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣫⠄⠋⠁⠙⠃⠬⣝⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⡵⠂⠁⠀⠀⠀⠀⠀⠀⠀⠙⠒⣍⡳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣍⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⣡⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⡛⢦⡀
⠀⠀⠀⠀⢀⡼⣣⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡹⣄
⠀⠀⠀⢠⠏⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡘⢧
⠀⠀⢠⠏⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⣧
⠀⢀⡟⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠘⡇
⠀⢸⠁⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⢸⡀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢰⡇
⠀⠈⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⡾
⠀⠀⠘⣦⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠎⣡⠞
⠀⠀⠀⠈⠳⣤⡙⠶⢠⣀⠀⡀⠀⢀⣀⢠⣴⣼⠟⠁⠈⠛⠾⣶⣂⣤⡄⠠⠴⢐⣛⣡⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⠭⠥⠽⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡥⢨⣙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠥⠖⠉⠀⠀⠈⠘⠲⢈⣛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢮⡙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢈⡳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢤⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢃⡾⠁
⠀⠀⠀⠐⠓⠒⠒⠂⠀⠀⠹⣆⡙⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠙⢷⡄⣀⠀⠀⠀⠀⠀⠀⣀⡄⠒⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣍⣁⠶⠆⢤⣤⠠⣤⣄⣾⠿⠋⠀⠀⠀⠙⠻⠧⢼⣿⣴⣿⣥⡭⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⠛⠛⠛⠉⠉⠑⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⢀⣀⣠⣤⣤⣤⣄⠀⠀⠀⠀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⣷⠒⠁⠀⠀⠈⠁⠀⢄⠙⠋⠢⣄⠐⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣻⣷⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⡈⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣾⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⣀⠀⠙⡄⠀⠀⠀⠈⠙⠿⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣾⣿⣿⣿⣿⣿⢿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠻⡿⢿⣿⣿⣿⡇⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣀⣤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡐⠁⠀⠉⠛⠛⠃⣤⣶⣿⣿⣿⣧⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣧⣄⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣀⣠⠴⡚⠛⠉⠉⠁⠀⠀⢠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⡿⠻⢿⣿⡿⠿⣿⣿⣿⣷⣾⣿⡿⠿⢳⡀⠀⠉⠅⠀⢠⣦⠄⡀⠀⠀⠀⡴⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⠀⠉⢰⡄⠇⠙⢛⠉⠁⠐⠀⠀⢇⠁⠀⠀⠀⠀⠀⠿⠿⠃⠀⢀⡜⣃⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡤⣷⡘⢗⣤⣆⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠂⠀⠏
⠀⠀⠀⠀⢤⠤⠶⠂⠀⠀⠈⠁⠹⠇⠸⣾⡿⡌⣿⠸⠀⠀⠀⠀⠀⠰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⢁⠀⢀⡘⣀⠅⢀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⢀⡄⠘⢣⡭⠀⡈⢡⣯⠄⠀⠀⠀⠀⠀⡀⢄⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⢀⡀⢀⠀⣰⣇⣀⠀⣴⠀⠀⠙⠁⠄⠀⠐⣆⠘⠿⠐⠟⣧⢶⠄⢠⣦⠶⣻⠀⡘⢷⣾⠁
⠀⠀⠀⠀⢀⣾⠟⠁⣼⣿⡟⢁⣶⡟⠀⠀⠀⠀⠀⠀⠄⠈⠀⣘⠛⡒⠃⠀⢴⡜⡙⠎⠏⠀⠉⠂
⠀⠀⠀⢠⣾⠏⠀⢀⣿⡟⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⣿⣟⡄⠀⣼⣿⡏⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦
⠀⠀⢰⣿⣿⣿⣐⣿⣿⡇⠈⣿⣧⠀⠀⠀⠀⠀⠀⢰⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡇
⣦⡀⣼⣿⣿⣿⠙⣿⣿⣿⠀⠙⣿⡄⠀⠀⠀⠀⠀⠈⣷⡀⠉⠑⣲⣄
⠻⣿⡘⣿⣿⣿⣦⣿⣿⣯⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠈⢻⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣂⠠⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠵⢻⠆⠁⣸⠇⠀⠀⠀⠀⠀⠀⡀⡀⢀⣀⣬⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡫⠉⢁⡭⡝⡆⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢈⣿⠟⠉⠋⠙⠻⣾⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⡀⠀⠀⠀⣨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠒⠲⢤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠰⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣩⡄⣀⠲⢠⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠭⠖⠁⠀⠀⠀⠀⠰⡞⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣦⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⣠⣾⣿⣴⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⡁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠾⠛⣻⣿⣿⡿⠿⠋⠋⠀⠀⠀⠀⢻⡉⠀⠀⠀⠀⠀⢿⣹⣇⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠱⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⡀⠀⠢⡈⠁⠀⢢⠂⠀⠀⠀⠀⠀⢿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡰⣱⡀⠀⠀⠀⠀⠀⢸⠟⢻⠷⡀⠀⠑⠀⠀⠀⠈⠄⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡅⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠤⠂⠀⠠⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠒⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠜⠃⢶⠦⢄⠀⠁
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠁⢹⣿⣿⠄⠀⠀⠀⠀⠸⣴⡢⢀⣼⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢀⡀⠀⠀⠀⢸⣿⣿⠀⠀⠘⣅⠁⠃⠘⡆⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣛⣿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠀⣿⠎⠈⡀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠘⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠀⠀⠀⠀⠀⠒⠐⠀⠀⠐⠒⠒⠒⠒⠒⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠛⠛⠛⠛⢿⣷⠀⠀⠀⠀⠀⠐⠔⣤⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠓⠞⠋⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡀⠀⢀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣅⣈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⢉⣉⣉⣉⡁⢀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠤⠤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⠟⠛⠿⠿⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⢌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳⣤⣤⣄⣀⣄⡽⠆
⠀⠀⠀⠀⠙⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⠀⠈⠈⠀⠀⣀⣤⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣀⡠⠄
⠀⠀⠀⢈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠲⣬⣉⣉⠉⠉⠉⢉⣈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠋⠀⠀⠀⠈
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠈⠙⠛⠛⠛⠛⠉⠈⡂⠀⢀⠀⠇⠐⠒⣀⣀⡤⠀⠈⠁⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢈⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠐⠒⠒⠒⠒⠊⠉⠉⠁⠀⠠⡀⠀⠀⠁⠀⠁⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠸⠄⡀⣀⠀⠀⠀⠀⢠⠀⠛⠂⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⢀⢀⣄⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣬⠠⠄⠀⠀⠀⠀⢃⠐⣄⠀⠸⠿⡋⡘⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⡷⠀⡎⠁⠠⠿⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠶⠀⠠⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠒⠃⠘⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢲⣀⠀⠀⠀⠐⠀⠂⠄⠀⠀⠀⠀⠀⠀⠤⣆⣀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠤⡀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠀⠈⠈⠂⡀⠀⡄⠠⠂⠀⠀⠢⣀⠀⠀⠃⢹⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⡜⠀⡀⠀⠀⠠⠀⠀⠀⠐⠁⠃⠀⠀⠆⠀⠀⠙⡁⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⣦⠐⠠⠀⠀⠐⠊⠀⣀⠄⠀⠀⠄⠀⠈⠀⠰⠬⠂⠀⠀⠐⠈⠀⠀⣀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠈⠁⠁⠈⠀⠀⠈⠁⠁⠉

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢀⣴⠃⠀⠤⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡄⣸⡯⠀⢰⡇⣸⡏⢰⡟⠀⠀⠀⣿⠃⣾⠁⣼⠃⢸⡟⣰⠟⠀⠞⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠘⠀⠄⠈⠁⠘⠃⠈⠁⠀⠀⠀⢿⠀⠀⠀⠙⠁⠈⠁⠘⠃⠀⠼⠋




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡀⢰⣿⣿⣿⣿⡆⠀⣾⣿⣿⣿⣿⡄⣰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠀⠘⢿⣿⣿⡿⠃⠀⠻⣿⣿⣿⡿⠁⠘⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⢀⣠⣴⣶⣤⡀⠀⣠⣴⣶⣦⣄⠀⢀⣤⣶⣶⣦⡀⠀⣠⣴⣶⣶⣄⠀⢀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣧⢰⣿⣿⣿⣿⣿⡆⣾⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠿⣿⡿⠟⠀⠘⠻⣿⣿⠿⠃⠀⠻⢿⣿⣿⠟⠀⠘⠿⣿⣿⡿⠋⠀⠻⢿⣿⣿⠟⠁⠘⠿⣿⣿⡿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠄⠀⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⢻⡋⠛⣛⣛⣒⡢⢤⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⡀
⠀⠀⠀⠀⣜⣀⠊⢗⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⡱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢈⣾⣹⣿⣶⣤⣁⢋⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠈⠑⠢⡤⠤⠀⢀
⠀⠀⣤⠒⠨⡙⠻⢾⣿⢏⠂⡀⢀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠈⠑⢂⠀⠀⣀⢀⠀⢸⡾⠀⠀⠀⠠⠀⠀⠀⣸⡆
⠀⠜⡈⠀⠴⣠⢮⠕⢀⠀⠉⣸⠩⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⣑⠈⢸⡇⠀⠀⠀⡄⠀⠀⢠⣿⠁
⠀⠑⠚⠵⣢⠄⡉⢀⠂⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠛⠉⠙⠛⠓⠳⠶⣾⡇
⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠀⠀⠴⠶⡈⢦⠀⠀⠠⠀⢀⡠⠊⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣒⣒⡠⡞⠊⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠸⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡴⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣆⣾⣀⣸⣆⣆⣀⣀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠰⡧⠀⠀⠀⠀⢀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠘⠿⠇⠀⢻⡇⣀⠤⣀⠋⣸⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⢙⢧⡀⠠⣧⠀⠀⣀⠀⢠⣶⡄⠠⠄⠔⡂
⠀⠀⠀⠀⠀⠀⠀⠖⣄⡠⡴⠊⠲⣠⠁⠀⠈⣤⣿⠋⠉⠉⠑⡄⡔⠁⠀⠉⢢⣈⡪⠇⠃⣿⡇⠀⠉⠀⠀⠉⠀⡆⣿⢞⡀⡀
⠀⠀⠀⠀⠀⣤⠀⠀⡘⣻⠼⣄⡀⢸⣀⡔⠉⠀⢸⡀⠀⠁⠀⣿⡇⠀⠀⠀⣜⠀⠈⠑⡄⣿⡿⢂⡟⡠⠀⢄⡰⣳⠏⢀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠧⣄⡠⠬⣿⠟⡄⠀⠀⢃⠀⠀⢀⠜⣿⡟⢆⠀⠀⠠⠀⠀⢀⠗⢿⣵⡟⠹⠁⠀⢸⠿⠯⡀⠁
⠀⠀⠀⠀⠀⠰⢮⣥⣤⣴⡁⢀⠠⡇⠀⠊⠀⠠⠆⠀⠀⠁⢠⣿⣧⠀⠁⠀⠀⠃⠀⠈⠄⠀⠧⢅⠀⠀⠀⠀⠀⣀⠃⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡩⢛⡽⠁⠀⢈⠀⠀⠀⠀⡢⠀⠄⡀⠸⣿⡟⠀⡰⠄⠠⡁⠀⠀⠀⢀⠀⠀⠹⣷⠀⠀⠀⠁⢑⠂⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⢠⣪⡶⠟⢣⠀⠀⢈⠖⢤⠀⠘⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠜⠀⡠⠒⢇⠀⠀⣰⠋⠀⠀⠲⣴⣮⣴⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠁⠑⠤⡄⠀⠰⠄⠀⢠⠁⠂⠀⠘⣿⡟⠀⠀⠈⢡⠀⠀⢇⠀⢀⡤⠚⠁⡣⢸⣦⡤⠶⠖⠁⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠤⡀⢸⠀⠈⠆⠀⠘⡆⠀⠀⢠⣿⣷⣀⠄⠀⡇⠀⠀⠃⠀⠈⠀⠠⠒⢳⠊⠉⢪⡉⢲⠀⠰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⠀⠀⠀⢀⡜⠁⠀⠀⢘⣿⡟⠁⠀⠀⠑⣄⠀⠀⠀⠀⠈⠀⠀⢸⣀⠛⠤⢂⣀⣀⣀⣠⠀⠠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠃⡀⠈⣄⠀⢀⡎⠀⠐⠠⠤⢾⣿⣿⠦⠄⠒⠀⠘⣄⠀⣀⡏⠀⠈⠃⠋⠱⣾⣿⠿⢛⣽⡿⣁⣀⣀⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠚⠀⠉⠉⢄⠀⠀⠀⠀⢘⣿⡇⠀⠀⠀⠀⣠⠋⠉⠀⠸⠈⢠⣭⣠⠀⠉⢈⠙⣌⠉⠉⠰⠶⠶⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⢀⡴⠋⠉⠑⢦⠀⠠⠾⣿⡟⠢⠀⡠⠊⠉⠉⠳⡄⠀⣠⠾⠟⠢⢄⣀⡾⣦⣄⡱⡀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠀⠂⠀⠀⠀⡇⠀⠀⣿⠃⠀⢠⠅⠀⠀⠐⠁⢈⢩⠁⠀⢰⠀⠀⣥⡈⢢⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⣀⠀⠈⠀⢀⠏⢇⠀⠀⠀⢀⣀⡀⠴⠊⠉⠒⢒⢚⣛⠹⣼⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠒⠉⠀⠈⠑⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⠈⠋

⠄⠀⠈⠉⢾⣿⣿⣿⣿⣿⡿⡿⠿⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣿⣿⣿⡏⠙⠻⣾⣿⣉⣽⣿⣿⢯⣥⡉⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠻⣿
⣷⣂⠂⢀⠀⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠺⢿⠟⠛⠁⠀⠀⠀⠈⠀⠀⠀⣶⣴⣉⠈⠙⣿⣿⣅⣀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠛⣿⠈⠉⠀⠀⠰⡧
⣾⣿⣶⣖⠠⠌⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡦⠀⠀⠀⠀⠀⠀⣯⣀⠈⢻⣶⣧⣀⡀⢙⣷⢿⡄⠀⠀⠀⠀⠀⠀⠀⠠⠿⠶⡦⠀⠀⣰⡶
⣿⣿⡏⠀⠠⠀⠀⠙⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠇⠀⠀⢰⠲⠶⠆⠙⢿⣿⣿⣥⣀⠈⣿⣿⣥⣤⡀⠀⠀⠀⠸⠿⢿⣶⣶⣿⣶⣶⣶⣶⣿⣿
⢻⣿⣿⣀⣀⠤⠖⠒⠋⢹⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿⣿⣿⠀⠟⠀⠢⠙⢿⣷⣯⣀⠀⠙⠇⠀⠀⠀⢀⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣤⠀⠀⠀⢀⡄⢀⣸⣿⣿⣿⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⢀⢮⣽⣷⣶⡂⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣭⣉⣉⣿⣿⣿
⠄⠘⡏⠀⠀⠀⢸⡇⣿⣟⢹⣿⡿⢹⣿⡿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⣿⡇⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉
⣶⣤⡇⢀⡀⠀⠘⠃⢸⣿⣶⣶⣶⣾⣿⣯⣁⣀⣉⣿⣿⣿⣿⣿⣿⡋⠙⠛⠃⠀⠀⠀⢀⢈⣵⣼⢿⣿⣆⡛⠃⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⣿⣷⣴⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣟⢻⣿⣿⣾⠿⠿⠿⡿⣿⣿⣷⣶⣶⣆⠀⠀⠀⠘⠻⠟⢻⠞⠻⠁⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⢹⣿⣴⡟⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡠⡶⠆⢀⠀⣤⣤⣤⣴⡶⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⠛⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⢿⣿⣿⠟⢛⠛⣻⡟⣛⣛⡋⠀⠛⠀⠈⠀⠉⠈⠁⠉⠉⠉⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛
⣅⢀⡆⡄⡠⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠇⢐⣺⣏⠻⣀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈
⣿⣿⣾⣷⣣⢇⡞⡰⢀⢀⣤⣤⣤⠸⠋⠡⠊⠁⠈⠉⠳⣮⠾⡟⣻⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀
⣿⣿⣿⣿⣿⣿⣾⣱⢃⣼⣿⣿⣿⣤⣆⡀⠀⠈⣷⣇⣸⣾⣳⣶⣿⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠃⠃⢀⣀⡀⠀⠀⠀⣰⣇
⣿⣿⣿⣿⣿⣿⣿⣷⣟⣼⣿⣿⣿⣷⣮⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣥⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⢸⣿⡇⠀⡀⠀⠘⢋⣤⣤⣤

⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠳⠦
⠀⠀⠀⡴⠉⠀⢴⠄⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠈⠀⠀⠀⠸⡄⠛⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠂⢒⠚⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣒⣒⣒⣦⣀
⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⠀⣀⠈⡇
⠀⠀⠀⠑⢤⡀⠀⠀⠀⡠⠋⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠒⠒⠒⠒⠒⠒⠂⠀⠤⠤⠤⣄⠀⠀⠀⡇⠀⠂⠀⠀⠠⠅⠀⠂
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠇⢀⣁⣈⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⢦⠀⠀⠸⠀⢀⠀⠐⠀⠀⠀⠀⠀⢠
⠀⠀⣀⡀⣀⣀⣠⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⢀⢤⡀⡰⣤⠀⣤⢶⢀⣴⠀⠏⢡⠜⡷⠋⠀⠀⢠⠀⠀⡄⠈⠀⠉⠀⠀⠀⡀
⠀⡧⠈⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠠⢈⠀⠀⠀⣦⠌⠀⡾⢁⣎⣰⠇⠰⠤⠊⠃⠘⠉⠀⠀⠀⠀⢀⠖⠁⠀⠀⢸⠀⠀⠑⠠⠤⠶⠤⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⡂⠀⠈⠀⠳⠠⠊⠑⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠴⠮⠀⠀⠀⠀⢸
⠀⠈⠀⠒⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠡⠀⠀⠀⢸⣀⡀⠤⠀⣄
⠀⠀⠀⠀⠀⠀⢰⢧⢀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠔⠀⠀⠀⢸⠋⡠⠀⡆⠡⡱⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠶⠆⠠⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠂⠄⠀⢀⠓⠏
⠀⠀⠀⠀⠀⢠⠞⠋⠉⢳⠹⠄⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣈⣀⣀⣈⣴⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⡔⠈⠂⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠰⠒⠒⠠⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢆⣀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠒⠀⠈⠙⢢⣀⠄⢀⣀⠀⠀⠘⠢⠤⠤⠤⠚⢀⣀⣀
⠀⠀⣠⠞⠉⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⣈⣉⡉⡩⠭⠛⠋⠢⠀⠀⢀⠜⠁⠀⠀⠀⢠⠊⡤⢀⠀⠀⠈⣲⡀⠀⠀⠀⢰⠁⠀⠀⠀⡿⢝
⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⡔⢘⠈⡡⠫⡃⠈⡀⠄⠀⣐⠀⡂⠀⣆⠀⠀⠀⢀⠔⠁⠮⠀⠀⠑⢀⠔⠡⠋⠀⠀⠀⢸⠀⠀⠀⠀⡧⠘
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡄⠀⡇⣂⠊⢀⠄⠀⢁⡁⠐⠠⠅⠲⠁⢄⠛⠒⠒⠒⢻⢇⡒⠤⢀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠰⠁⠁
⠀⠹⡄⠀⠀⢀⣀⠀⠀⡏⠉⠈⠇⠀⠐⠀⠁⠀⠈⠁⠀⠀⢀⠔⠁⡠⠔⢂⠑⡀⠀⠀⠀⡐⠉⠁⢀⠐⠚⢳⠀⠀⡠⢆⣠⡏⠀⠙⠒⠚⠃
⠀⠀⠈⢦⣰⠉⡀⠱⡀⠘⡈⣀⡄⠀⠀⠀⠀⢀⠤⠒⠀⠀⠙⠄⠈⠔⡨⠐⢀⠐⡄⠀⠀⡑⠠⠀⢀⠕⢀⠎⠀⠌⠀⠀⠉⠁⠊
⠀⠀⠀⠀⠙⢄⠈⢄⢠⠀⠑⠉⠈⠂⠢⡀⠀⢃⠀⠀⠀⠀⠀⠈⢀⠈⠔⠈⢀⡤⠊⠀⠎⠁⠀⠀⠁⡔⠁⠀⠀⢆⡀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢢⠀⠀⠀⠀⠀⠈⢆⡀⠂⠁⠀⠀⡜⣫⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠡⠀⢀⠠⠐⠀⠉⠀⠀⠀⠀⢀⣡⠏⠐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⢠⣀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢀⢈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠥⠂⠈⠁⠐⠰⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⣀⡀⠀⠀⢀⣤⡔⠂⢸⠞
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢠⠀⠈⣿⣷⠆⠀⠈⠻⠟⠁⠘⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣦⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣄⠀⠄⣤⠄⠠⣀⠀⠀⠀⠀⢀⠀⡼⡄⠀⠀⠀⠀⠀⢰⠒⣲⡄⠀⠀⠀⠀⠀⡟⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⡀⠑⢄⡀⠨⠤⠀⠀⠀⠈⢀⠀⠘⠀⣰⣆⠀⠀⢹⠛⢛⡆⠀⠀⠀⠀⠀⡇⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠫⣁⣄⣠⡀⠀⠀⡀⠀⠀⠀⡰⠧⠼⡄⠀⠀⠉⠉⠀⠀⣀⣀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⣷⣬⣦⣝⠩⠒⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⣴⠋⠁⠀⠀⢰⡇⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠥⣀⣨⠉⠐⠂⢈⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠤⠄⢸⡇⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠴⠐⠂⢀⠀⠐⠂⠀⣁⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣖⡀⣀⠀⠒⠀⣈⢀⣐⣢⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠃
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⢈⣉⠋⠋⠙⠙⠉⢉⠁⠀⢀⢈⡉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠁⣧
⠀⠀⠂⠀⠁⠀⠃⠀⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣡⠀⠀⠀⠻⠀⠁⠖⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠖
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⡄⢀⠂
⠀⠀⠀⠀⠀⠀⠑⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡐⡠⠁⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠡⡀⠆⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠠⠊⢀⠔⠉
⡔⠀⠑⢄⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⠔⡠⠊
⠀⠐⢄⠀⠢⡀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣐⠁⠈
⠀⠀⠀⠈⠢⢈⠢⡀⠀⠀⠀⠀⠀⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⢤⡀⠀⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠑⠰⣄⠀⠀⠀⠀⣀⠔⠁⠀⣀⣤⣤⡄⠀⠀⠀⢠⣾⣿⣷⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢀⡔⠁⠀⢠⡾⠛⠋⠉⠙⠀⠀⠀⠀⠀⠀⠉⠻⣇⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠟⢀⠄⠀⠀⠀⢀⠀⡠⠐⠂⠐⠠⡀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣀⣠⡁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸⠀⠀⢰⣳⡆⠀⢸⠀⠀⣶⣷⠀⠀⡁⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠋⠀⠀⠙⠦⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⠘⡀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⢀⠁⢀⡀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡀⠀⡸⠀⠀⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠋⠉⡀⠀⠀⠢⣀⢀⡀⠄⠀⠑⠄⣀⣀⠤⠁⠀⠈⠙⠓⠀⠀⢸⠁⠀⠀⠈⢆⠀⠈⠂⠑⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠃⣄⠀⠀⠀⡈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⡏⠀⠀⢀⠇⠀⠀⢀⡴⢃⠌⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⠀⠘⣿⣿⣷⣄⣀⣀⣄⡀⢀⣀⣄⣠⣤⣾⣿⣿⡿⠀⠀⢀⣾⡒⠒⠊⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣿⢦⡀⠹⣿⡿⠟⠛⠛⠻⠟⠛⠛⠛⠿⣿⣿⣿⡟⠁⠀⡠⠋⠛⠓⠒⠈
⠀⠀⠀⠀⣀⠀⢀⠎⢀⠔⠋⠀⠀⠹⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠀⣠⠊
⠀⠠⠊⠁⠀⠈⠉⢰⠃⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⢀⠖⠉⠉⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠎⠀⠀⢀⣴⡂⢠⠀⠀⢅⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠘⠢⠔⠿⠠⠁⠈⠦⡀⣠⠀⠀⠀⠀⠀⡇⠀⠈⠉⠀⠒⠒⠒⠀⠒⠀⣀⠀⠠⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠠⠀⠂⠀⠀⠀⣀⣀⡀⠤⠤⠄⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠐⠒⠂⠈⣉⢁⣀⠀⠤⠤⠄⠒⠒⠚⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠉⢉⣁⠀⠠⠤⠀⠒⠂⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠦⣤⣄⣀⣀⣀⣤⣴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⡿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠢⠀⠀⠀⠀⠠⠛⠁⠘⡧⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠰⡄⠀⢀⡤⠀⠀⠀⠀⢰⠋⠡⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⢦⢄⠀⠀⠀⠈⠳⠄⠀⠂⠀⠀⠀⠀⠈⠐⠉⠀⠀⠀⠀⠠⠃⠀⢆⡟⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢐⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⠄⠊⠀⠀⢠⣾⠻⠏⠀⡀⠄⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⢀⠎
⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠁⡐⠀⠀⠀⠽⢦⣄⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠇⡀⠀⠀⠂⠀⠀⠀⠀⠀⢠⣿⣏⠀⢐⠁⡔⠀⠀⠐⠀⠸⠄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⡄
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣬⣶⡁⠂⠀⠀⠀⢀⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠄⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡛⠙⣿⣲⣆⣤⣴⣤⣼⣧⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠑⣤⣭⣦⣤⣵⣤⣤⣤⣤⣄⣀⣀⣿⣿⣿⣧⡀⠈⣿⣿⣿⡿⢼⣿⡿⠀⣚⣀⣀⣀⣀⠀⠀⠀⠀⠀⠻⣄⣠⣤⣤⣤⣤⣤⡍⢤⣴⠡⣄⠂
⠀⠀⠀⠀⠉⠀⠊⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⠸⣿⣿⣷⡾⠛⠀⣴⠋⠀⠀⠀⣀⡄⠒⠀⠀⣸⣶⣿⠁⠀⠀⠀⠀⠀⠑⠀⠈⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡄⠀⠀⠀⠀⣹⠉⠙⠛⠛⠻⣃⣀⣀⠀⠈⢃⡀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡿⠋⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣⢀⠀⠀⡰⠉⠉⢼⣷⣶⣏⣼⣷⠂⠤⠀⢾⣿⣿⡿⠛⠁⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠤⠄⠈⠙⠛⠿⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠒⠲⠦⣤⣀⣠⠖⠛⠉⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⡇⢀⡼⣟⣋⡤⠖⢿⣛⡤⠔⣿⣠⣾⣇⣠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠖⠒⠋⠁

⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⡿⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣏⠀⣸⢿⡀
⠀⠀⠠⢙⣉⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⢶⣦⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⢸⢸⢰⣅⣀⣵
⠀⠀⠀⢀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣳⡄⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⠞⢸⡿⠿⢿
⠀⠀⠾⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢇⡀⠉⠁⠀⠀⠀⠀⠀⠸⠿⠻⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣀
⠀⣀⣤⣄⡀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢾⣶⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⢰⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣿⣿⡟⣀
⠈⢯⣿⣯⡴⢩⣭⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⡄
⠀⠀⠉⠻⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣶⣶⣶


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡰⡦
⠀⠀⢀⣶⡀⣐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣾⡿⣧⡤⠄
⢀⣀⠞⣿⠳⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣂⣀⣀⣀⣀
⢾⡇⠀⣿⠀⢹⡏⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠁⠀⠉⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁


⠀⠐⡉⡉⢉⣉⠉⡇⡀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣩⣉⣉⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⢰⠀⠄⠀⠀⢀⣇⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠶⠶⠶⢠⢳⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡶⣺⠛⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠞⢠⠀⠀⠀⢸⡸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⣭⣭⠁⡌⡜⠀⠀⠀⠀⠀⠀⠀⠄⢻⣿⣷⣌⣣⡾⠿⠿⠂⠀⠀⠀⠀⠀⠀⠈⣟⠛⠛⠛⠛⠋
⠀⠓⣒⣒⣒⣒⣋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠓⣒⣒⣒⣒⣩⠇⠀⠀⠀⠀⠀⠀⠀⠈⣨⡭⠉⠉⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠩⡿⠉⠉⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢦⡀⠀⠀⢀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡄⠀⠀⠀⢙⣦⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡣⡋⡄⠀⠀⠊⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠢⠰⠆⠀⠀⠀⠀⠐⠠⠿⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠒⠀⠀⢀⡀⢀⣀⣒⠀⠀⠀⠤⠀⠐⣀⢤⣤⠷⠟⠍⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠉⠓⠓⠶⢤⣠⠀⢀⠀⠀⠤⣝⡈⠛⠛⠠⣀⣀⣤⡶⠿⠛⠉⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠀⠀⠈⠑⠓⠒⠲⡘⠀⡼⠟⠛⠋⠁⠀⢂⠨⠄⠁⠢⠀⡠⠞⠰
⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠄⠒⣀⠄⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⢈⠁⠁⠶⢤⡄⠜⢸⣏⡀
⠀⠀⠀⠀⠀⡬⠐⢧⡀⢠⢊⠰⣢⠀⠀⢠⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣆⡀⢀⢄⠀⠸⡧⠆
⠀⠀⠀⠀⠀⠡⡈⣰⣿⠀⠀⠀⢂⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁⠓⠂⠀⠀⠳⡆
⠀⠀⠀⠀⠀⠀⠈⠚⠛⠀⠀⡤⠀⠀⠈⠫⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠛⣙⢰⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢿⠂⠀⠀⠀⠀⠀⠀⠀⠘⠉⢉⣠⠖⠦⠉⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠈⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠠⠘⠀⠁⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⡎⢰⠄⡆⠀⠀⠀⠀⡠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠢⠤⠄⡹⢍⠄⠠⠖⠀⠉⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⢧⠀⠘⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠻⠆⠀⠈⠉⠁⠀⠀⠃⢰⣆⠂⠀⠈⠙⠉⠛⠛⠃
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠃⠀⠀⠀⠓⢶⡶⠒⠘⠀⠀⠈⣿⣄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣉⣉⣉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠃⠀⢐⡖⢀⣹⣿⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⡃⠀⢸⣇⠀⣻⣿⡟⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠇
⠀⠀⠀⣼⡇⠀⡸⡿⠛⣿⣟⢷⡀⠀⣿⠀⠀⠀⢀⣠⣴⣶⣶⣶⠶⣶⣶⣶⣶⡤
⠀⠀⣶⣿⡇⠀⣇⠃⢰⣿⢻⣾⠁⣿⣿⡀⣠⣾⣿⣿⢿⣿⣿⣿⣤⣄⢀⡀⠀⠀⣜
⣶⣾⣿⣿⣧⢰⣿⣷⡺⠛⠈⢻⣿⣿⣿⣷⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣄⣸⣼
⣿⣿⣿⣿⣧⣼⣿⠻⣿⣤⣤⣤⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡿⠉⠉⠉⠉⡟⠉
⣿⣿⣿⣿⡿⣿⣟⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣶⣶⡂⠀⡑
⠀⠀⠀⠘⠳⠛⠓⠉⠉⠙⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠛⠃⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢴⡆⠚⢳⣸⢀⡐⣔⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡁⢀⠘⢕⠋⠅⠰⠜⠟⢡⠟⣹⣫⣭⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠿⢭⣿⡄⠘⢷⢠⠸⡆⢲⠀⡐⡆⢐⣚⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⢂⠀⡄⢊⢀⢈⠀⠀⡀⠠⠈⡇⠀⠙⠤⠄⡒⠀⢄⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠞⠀⠀⠉⠀⠀⠢⠀⠀⠀⠀⠈⠀⠀⠀⠠⠌⠄⢠⠌⡼⡄
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠰⣮⠄⠀⠀⠀⠀⢀⠀⢀⢠⠠⠠⠔⠢⢀⠈⡱⣧⠤⠀⢨⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⠠⠀⠀⠀⠊⠀⠀⣀⢀⣀⣂⡼⣉⡇⠐⢧⡿⣿⠨⣹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⣁⢄⠐⢶⢳⠏⡻⣧⡸⡳⣝⣧⡉⠠⠤⣛⣈⠙⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡹⣆⢮⡩⢼⣲⣿⣿⢿⣚⡿⡤⢳⡆⢸⡀⣸⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠁⣴⠩⡭⡓⢈⡛⠍⣁⣤⢵⣬⡗⠲⠛⠙⠀⠀⠐⢔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣤⡾⠓⠬⡜⠘⢋⡡⠶⣛⣨⣴⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡷⢞⣒⣶⠺⠻⠹⣥⣫⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡟⣪⣋⡀⣀⣬⠵⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠾⠵⢷⣞⠩⠙⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⢉⣉⣤⠴⠚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⣉⣉⡠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⣀⠀⠀⠀⠀⠀⣤⣦⣤⣶⡄⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⣼⣿⣿⣿⣇⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡘⠿⡿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡏


⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣋⣴⡄⣿⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⣼⣿⣿⣿⣿⣷⡄⠀⠀⠀⠿⠿⢿⡿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠙⠁⠙⠛⠃⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁


⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢰⣶⡆⢶⣶⡆⠀⠀⠀⠀⠀⠀⠐⠗
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⣶⣶⣶⡄⢦⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠽⠿⠿⠇⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠁⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢴⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠝⢻⡍⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠄⠀⠠⠀⣠⡿⠤⠀⡀⠀⠀⠀⠀⠀⠈⢆⠀⠋⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠉⣁⣀⡀⠀⠀⢠⡋⠀⠀⠀⠉⡀⢤⣤⡄⠀⠀⠈⠁⠀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⠉⣦⣾⠁⠀⢹⡆⠀⣼⣿⣿⣿⣷⣶⡗⠘⠛⠁⠀⠒⠒⠂⠀⠀⠀⠙⢷⠄⢷⠇⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠤⠶⠁⠘⠦⠴⠾⠄⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠓⢦⣄⠀⠀⣆⢠⠄⢦⢠⠄⠂⠀⠀⣠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣲⡒⣰⠆⠀⠀⠀⠀⠈⠉⣿⣿⣿⣧⣰⡋⢉⠶⡉⢉⡆⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠙⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⣤⣏⠀⠀⠀⠀⠀⠀⢀⣨⣭⠉⠉⠉⠁⠀⠀⠀⠈⢁⣀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣤⣤⠹⡆⠀⡈⠛⠁⠀⠀⢿⣻⡆⠀⠀⠀⠀⠀⢀⣶⣾⣅⠀⠀⢹⡆⢠⣴⠀⢀⣴⠟⠉⣠⣾⡁⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⢦⣀⣀⠀⠀⢺⣿⣿⣿⣿⣿⠹⣿⣷⠀⠀⠀⠀⠀⠸⣿⣿⡏⠀⠀⠘⣿⠀⠉⣴⡟⠁⠀⠀⠻⠟⠡⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠘⠋⠀⢠⡞⣀⣉⢳⡀⠠⣾⣿⣿⣿⣿⡇⢻⣿⡇⠀⢿⠇⠀⠀⠈⡄⠀⡤⠀⠠⣿⠀⣸⡟⠀⣤⣤⡄⠀⠀⠀⡰⠟⠁⠀⠙⠷
⠀⠀⠀⡀⠀⢠⣤⡄⠀⠈⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⢉⠁⠈⢉⣁⣀⡀⠀⠀⠰⠆⠀⠀⠀⠶⢸⣿⢀⣿⠃⠀⣿⣿⡇⠀⢰⡖⠀⠀⠀⠀⠀⠀⢰
⠀⢀⡞⠀⠀⠈⣿⠅⡀⢠⡄⢀⠀⠀⠀⠙⢶⡀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⡆⠀⠆⠀⣼⡏⣸⡿⠀⠀⠻⠿⠇⢰⣦⠈⢆⠀⠀⠀⠀⠠⠊
⠀⠙⠛⢻⡶⠀⠠⢄⠙⠀⠁⠈⠠⠆⠀⠀⢸⡇⠀⣘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢰⣿⢃⣿⡇⠀⠀⠺⣷⣿⣞⣿⡆⠀⠿⠂⠀⠺⠇
⠀⠀⢀⡞⠁⠀⠒⠂⠀⠀⠀⠀⠀⠚⠁⠀⢰⡇⣸⣿⣿⣿⣆⠀⠈⠐⠆⡀⠀⠀⢻⣷⠀⣠⣿⡏⣼⣿⢀⣦⠀⠈⠛⠛⠛⠙⠃⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠈⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣙⣀⠀⢸⡇⠈⠉⠉⠉⣁⡀⠀⠂⠀⠀⠀⠀⢸⣿⣾⣿⣿⣼⣿⠇⣼⠿⢶⣁⠄⠀⠀⠚⠻⠿⠿⠿⠿⠛⢉⡅
⠀⢠⣶⠿⠀⣬⣐⡀⠀⣿⣾⠇⠀⢡⡞⠁⣾⡇⠰⣶⢿⣿⣿⣿⣿⣶⣤⠀⠀⠀⣸⣿⣿⣿⣿⡿⠋⠀⣀⣀⡼⢣⣆⠀⠀⠐⠒⠂⠀⠤⠐⠊⠁
⠀⡘⠁⠀⠀⠻⣿⡇⠀⠈⠉⠀⣠⡈⠀⠀⢻⣇⠀⠀⠸⢿⣿⡿⠏⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⢨⣿⣦⠈⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠾⣿⣦⠀⠛⠁⠀⠀⠀⢠⣿⣶⡖⡆⠘⣿⣦⡀⠀⠀⠀⣴⣾⡀⠀⠀⣰⣿⣿⠟⠉⣠⠔⠀⠄⠀⠀⠀⡤⠀⠐⠀⠀⠤⠀⣉⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⠀⢠⡟⠀⠀⠈⠻⣿⣷⣤⣄⡙⠛⠀⢀⣼⣿⡿⠁⡀⡀⢀⠀⣀⡀⠀⢰⣿⡇⠀⠀⠀⡀⠐⡀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠉⣿⣯⡀⠀⠀⠀⠂⠀⡙⢿⣿⣿⣷⣤⣾⣿⡟⢀⡄⠀⡰⠀⢀⡔⠁⢠⠀⠉⠀⢤⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠉⠉⠁⡴⠄⠀⠀⢠⠁⠀⠈⢿⣿⣿⣿⣿⡇⠢⠤⢴⡤⠤⠋⠀⠀⣼⡆⠀⠀⣰⣄⠀⠀⢰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⢠⣿⣿⣷⠀⠀⠀⢸⡄⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠚⠓⠀⠀⠀⣠⡿⠀⠶⠁⠀⠀⠑⠖⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠏⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⠶⠶⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⣰⣿⣿⣿⣿⣿⣿⣿⡏⠀⣀⣤⡤⠶⠶⠒⠒⠛⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⢠⡞⢁⣀⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠹⣆
⠀⠀⠀⠀⣠⣿⣞⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠈⠛⠳⢦⣼⢀⣠⣴⢖⣛⣋⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣉⣛⣶⣿⣆
⠀⠀⠀⢠⣿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠲⠿⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡄
⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢽⣿⣿⣿⣧⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⢿⣿⣿⣿⡿⠟⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⢷⢶⣶⣤⣤⣀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⣟⣻⣯⡽⠿⡗⠻⡿⣦⡠⠐⠁⠀⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⡟⢻⡏⣹⣿⡄⣅⢹⡄⠘⡜⣷⡴⢶⡄⠀⠄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠄⢴⣇⣽⣿⣷⣄⣱⣿⣾⠷⠾⣧⣾⢷⢚⣦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣸⣧⣶⠼⣿⣛⣿⣯⣿⣿⣶⣿⣿⣿⡋⠌⠌⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣷⡟⢿⣿⢿⣿⣿⣻⣽⣿⡿⢹⢹⣇⢃⢁⢊⡇⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣇⠛⣿⣦⣿⡾⣿⣿⡟⠛⠙⡄⡈⣎⣿⣾⣦⣬⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢹⠈⣁⣇⣬⣿⣷⣾⡿⢿⣿⣿⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⢷⣾⣻⣟⣹⣭⠿⠶⢿⠛⠛⠛⠁⢸⣯⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡏⠙⠛⠋⠁⢳⣶⣾⣿⣄⣀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣆⠀⠀⣠⣿⣿⣿⡿⡿⠛⠃⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠉⠉⠛⠿⠟⠁⠀⠀⢀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⣠⣿⣿⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣷⣾⣶⣿⣿⣿⣿⡿⠀⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣤⣶⣿⣿⣿⣿⢟⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠠⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠻⠉⠉⠉⠉⠉⠉⠙⢉⣉⣉⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡊⢽⠭⣿⣿⡄
⢶⠄⠀⠤⠦⠤⠤⠤⠀⠠⠶⠶⠶⠶⠦⢰⠠⠶⠶⠶⠶⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣤⣤⣤⣤⣄⣄⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⡛⣿⣿⣿⣿⣶⣆⣀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⣵⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣷⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⠀⠀⡀⣀⣀⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢿⣿⣿⣿⣮⣵⣿⣿⣿⣿⣯⣿⣿⣿⠟⠉⠉⠉⠉⠀⠀⠁⠈⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣦⢸⠀⠠⠤⣤⠄⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠈⡦⠀⠀⠌⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⡀⢠⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⢀⡀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⢂⠀⠀⠁⠀⠂⠀⠀⣀⠌⠀⢀
⡡⠊⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⡈⢀⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⡈⠀⠀⠁⠤⠈⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡤⢊⠽⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄
⠀⠠⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠐⠃⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠓⢄⡀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣴⠆⠀⠠
⠀⠀⠀⣀⠤⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠠⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⡤⠴⠋⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢄⠘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡐⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡂⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠊
⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠈⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠈⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡴⠻⣄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⣶⠀⡀⠈⠐⠋⠀⠀⠈⢷⡄⠀⠠⠀⡀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠘⢶⣤⣄⡀⠀⡜⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⣀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⡀⠀⠀⠀⠉⠙⠛⠾⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠙⠒⠒⠋⠙⠃
⡀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠂⠀⠀⡄⠀⠃⠀⠀⢀⠀⠀⠀⢀⠀⡄⠀⠠⠐⠂⠄
⠀⠀⠀⠀⠐⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠠⠀⠈⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⢀⠀⠄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠠⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣈⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣔⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠠⠀⠀⠀⠀⡆⠀⠀⢸⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⣌⣉⣉⣉⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢸⡸⠀⢸⡁⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠟⠛⠛⠻⠛⠇⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⢱⡇
⠀⠀⠀⠀⠠⠂⢠⡖⠒⠲⡄⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠃⠐⡠⢃⡄⠀⠀⠀⠀⠰⠪⢿⣄⣀⣴⠟
⠀⠀⠀⠀⠠⡀⢀⠻⠻⢟⠃⣀⠄⠀⠀⠀⣧⣀⣀⣀⣠⣄⣀⣀⣀⣠⡇⠀⠀⠀⠛⠛⢏⠔⠩⠬⠤⡄⠀⠀⢀⣤⣤⣤⣭⣥⣤⡀
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣶⣶⣶⣀⣀⣨⣶⣶⡆⠀⠀⠀⠀⠁⠀⣤⣶⣶⣦⡀⠂⠀⠀⠀⢀⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⠉⠉⢻⣄
⠀⠀⠀⠀⠩⡭⡭⡭⡭⠭⠭⠭⠁⠀⠀⠀⠀⠀⢸⣿⠿⠿⣿⡷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠘⠟⢂⣠⣤⣶⣦⣐⠚⠟
⠀⠀⠀⠀⢀⣃⣓⡓⣓⣒⣒⣒⠀⠀⠀⠀⠀⢀⠈⢻⢸⡟⡼⠃⠀⠀⠀⠀⠀⢣⡀⠀⡐⠀⢀⡠⠀⠀⠀⠀⡏⢌⡟⠛⠛⠛⣿⠩⡃
⠀⠀⠀⠀⠠⠆⠦⠆⠁⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠷⠷⠇⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠔⠃⠀⠀⠀⠀⣛⣚⠛⣛⣛⣛⢛⣓⡳
⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠒⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠊⠛⠗⠀⠠⠀⠀⠀⠀⠀⠀⡠⠀⡴⢁⠀⠲⠄⢤⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢰⣾⠖⠚⠛⠓⢾⣶⡆
⠀⠀⠀⠀⠀⢁⠀⠠⠠⠄⢀⠀⠀⠀⠀⠀⠘⣇⠀⡏⠀⢢⡀⠀⢀⠏⠀⠀⠀⠀⠀⠲⣶⣶⠆⠀⠀⠀⠀⠀⢨⠃⠀⠀⣀⡤⠂⢹⡀
⠀⠀⠀⠀⠀⣀⣤⣀⣀⣠⣆⠀⠀⠀⠀⠀⢠⡜⠓⣧⡙⠛⡡⠔⠋⣆⠀⠀⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠸⡄⠀⠀⠁⠀⠀⣸⠁
⠀⠀⠀⠀⢠⠒⠛⠛⠛⠛⠓⢐⡄⠀⠀⠀⠀⠙⠒⢻⡉⠉⠓⠃⠜⠁⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⢘⡦⢤⣤⠤⢾⠁
⠀⠀⠀⠀⠀⠃⠙⠃⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⣶⠿⠛⠛⠿⣷⣄⠀⠀⠀⢰⠿⠟⠃⠀⠙⠿⢷⡆
⠀⠀⠀⠠⠤⣶⣶⣿⣿⣿⣿⣶⣶⠤⠄⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡙⢿⣵⢆⡀⠀⠀⠘⣿⡆⠀⠀⢸⠀⣤⣤⣤⡄⠀⢸⡇
⠀⠀⠀⢀⠀⣿⣶⣮⣭⣭⣵⣶⣿⠀⠀⠀⠀⣿⡇⠐⠂⣃⡀⣠⡄⠀⠀⠀⠡⠐⢝⢿⣵⣆⡀⢀⣿⠇⠀⠀⢸⠀⠶⠶⠶⠂⠶⢸⡇
⠀⠀⠀⣸⠀⠻⠿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠿⠧⠿⢧⠀⠀⠀⠀⠈⠢⡁⢝⢿⣯⡾⠋⠀⠀⠀⢸⠀⣏⣉⣉⡁⣋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠥⠙⠟⠁⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠖⠒⠚⠛⠉⠉⢉⣨⠷⠖⠒⠒⠒⣦⡤⠶⢛⣹⡷⠞⠁
⠀⠰⡖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠴⢛⣩⠴⢾⡉⠁
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣀⣤⠖⠛⣉⣤⣶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣺⠵⣦⣶⣻⣻⡿⠃⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢠⣾⣭⠶⠻⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⢻⡙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢄⠸⣉⠆⣤
⠀⠀⠀⠀⠀⠀⡔⠒⢄⠀⠀⠀⠀⠀⠇⢂⠑⢄⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⢶⠀⠀⠀⠀⠀⠀⡠⠒⢢
⠀⠀⠀⠀⠀⠀⢣⠀⠀⢇⠀⠀⠀⠀⡇⠀⢢⠀⠑⢤⡀⠀⠀⠀⠀⢀⡴⢫⠃⠀⠀⠀⠀⠀⡘⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠘⡄⠀⠀⠀⢡⠀⠀⠱⡀⠀⠈⠢⡀⠀⡰⠋⢠⠃⢠⠀⠀⠀⠀⢠⠁⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⠀⠀⠀⠈⠄⠀⠀⠈⢆⠀⠀⠘⡌⠀⡔⠁⢀⠎⠀⠀⠀⠀⠃⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡔⠒⠢⢄⠈⠢⡀⠀⠀⠱⡀⠀⠸⠈⢀⡠⠊⠀⡠⠔⠒⢢⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠱⡄⠀⠀⠉⠒⢌⠢⣀⠀⠰⡀⢸⠀⠁⡠⠒⠁⠀⠀⢠⠎⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠈⠢⠀⠀⠀⠀⢣⠀⠉⠢⡡⡆⠀⡜⠀⠀⠀⠀⠔⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠇⠀⠀⠘⠧⠸⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⢤⠶⠆⣤⠀⢠⠀⢠⣄⢠⢠⠶⠆⡴⠦⢠⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢿⡞⢸⣋⡀⣟⣀⢸⣀⢸⡟⣾⢸⣃⡀⣈⣳⢈⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠉⠀⠀⠀⠀⠉⠁⠈⠁⠀⠉

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡔⢦⠀⢄⠀⠀⢠⠎⠉⠀⣉⣈⠤⠏⠀⠀⣠⠠⡰⡄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠠⠀⠀⠀⢀⢤⠌⡭⠐⠋⣧⠈⠳⣄⡏⠀⡴⢻⢟⡇⠀⢀⡴⠋⠞⠇⠐⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣧⠀⠀⢸⠀⣤⠉⢡⠸⡇⠀⢈⠀⠀⢸⠓⠐⢀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠜⡇⠈⠓⠀⠞⢀⡏⠀⠀⠁⠉⠰⠋⠀⠀⠈⠳⠖⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊⡐⣿⠍⢻⣄⡴⠒⢦⠀⠠⠀⣀⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤⠀⢿⡀⠸⡶⠋⣼⠁⠀⣾⣰⢾⢧⠏⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⢹⠈⡟⠀⠹⠤⠜⢡⠏⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⢁⣄⠠⡀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⣍⡃⣀⡤⠖⠒⢋⡑⡆⠀⠈⡅⢷⣃⡀⠀⡔⣋⡑⠒⠶⢄⣀⢺⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠁⠁⠁⠒⠑⠋⠀⠈⠛⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⡟⢻⡽⠛⠻⠀⢹⣿⠁⢿⣿⡆⠀⣾⣿⠈⣿⡉⠉⡇
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢼⣷⠀⠀⠀⢼⡯⠀⣹⣻⣿⣠⣿⣿⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣯⠀⠀⠀⢸⡇⠀⢾⠀⢿⣿⠑⡟⠀⢻⣅⣀⣤
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠈⠉⠀⠊⠃⠀⠀⠈⠉⠀⠉⠉
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠈

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠤⢠⠀⠀⡒⡆⠀⠠
⠐⡆⠀⡄⠀⢀⡠⠄⠀⠀⡀⢠⣾⣿⣛⣛⣛⣛⡛⣿⣷⡄⢀⢁⠠⣀⠔⠀⠁⠐⠀⠊⠀⠀⠀⡁⠀⠀⢀⠃
⠀⠹⡀⠀⠀⠠⠄⢀⡀⠠⠄⢸⣛⡛⡿⠿⣻⠻⠟⣻⢻⡇⠘⡀⠀⢀⠀⢀⠀⠀⠀⠒⠚⠀⠆⠂⠀⠃⠱⠰⠒
⠀⠨⢷⠀⠀⢀⣈⠂⢒⡁⠀⠘⣿⣷⣾⣷⣶⣿⣷⣾⣾⠃⠀⢈⡀⠒⠀⠁⡀⠂⠀⠀⠈⠃
⠀⣄⠂⠁⡀⠀⠸⠒⠐⠣⠀⠀⠈⠙⠛⠛⠿⠛⠛⠋⠁⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠆⠆⠀⠀⠀⢀⢀
⠀⢃⠀⠀⠈⠰⠸⠀⠰⠀⠂⠁⠀⠘⠃⠀⣀⠀⠒⠬⠁⠀⠀⠀⠀⠀⠀⠀⠉⠤⠂⠢⠶⠠⠥⠅⠠⠄⠤⠠⠤
⠀⡤⠁⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⠀⠀⡠⡀
⠀⠀⠀⢡⡀⠠⠤⠀⠀⠀⠔⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠉⠀⠀⠀⢀⣤⠀⠀⠀⠀⣋⠀⠴⠰⠤⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠄⠬⠄⠀⠀⢀⠀⠀⠀⠀⠀⢠⣠⠃⢘
⢀⡀⠀⠀⣰⠀⠁⢀⠀⠀⠘⠐⡀⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠸⠀⠀⠀⠤⠔⢨⣸⢰⣈
⠈⠁⠀⠀⠈⠀⠀⢉⢼⠄⠀⠀⠁⠀⠐⠅⠀⠁⢀⡤⠤⠤⠀⠀⢀⣀⡀⠀⠠⠐⠒⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠐⠉⠉⠉⠉⠁⠐⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⠀⡀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠂⠈⠋⠉⠁⢰⡿⠙⠛⢻⡏
⠀⠀⢠⡤⠤⠤⠤⣤⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⢀⠀⠀⠰⡀⠀⢣⠀⠀⠂⠀⡀⠀⢆⣼⣅⣸⠐
⠀⠀⡂⠀⠀⠀⠁⠀⠀⠘⠒⠒⠀⠀⠒⠂⠀⠐⠲⠂⠀⠀⠘⠄⠀⠀⢃⠀⠸⢷⡀⠠⡄⠁⢀⡈⠛⠛⡁
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠠⠀⠪⠂⠀⠀⠀⢠⠐⠂⡄⢀⠀⠠⠐⠈⠁⠛⠑⡄⠲⠓⠄⠀⠄⠀⠐⠀⠀⠈⠁⠄
⠀⢀⡁⠐⠀⠀⠀⠀⠁⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠐⢀⡆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠂⣀⡖⠤⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠃⣨⡄⠀⠀⠁⠦⠐⠠⠰⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠿⢖⠋⠀⠀⠁⠁⠠⠀⠀⠀⠀⠀⠀⢀⠱⠴
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣶⣤⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠂⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠈⠙⢿⣧⡀⠀⠀⠹⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⢀⡾⠁⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⣀⣴⠾⠟⠛⠻⠷⣦⣄⠀⠀⠀⠘⢷⡀⠀⠀⣸⠁⠀⢀⣴⠟⠋⠁⠀⠀⠀⠈⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣇⠀⠁⠇⠀⢠⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀⠀⠸⡀⠀⠀⠀⠊⠀⢀⣴⡾⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠷⠶⣤⡀⠀⠘⠄⠀⠃⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠐⠂⠀⠀⠉⠲⠄⠀⠀⠀⣼⣾⣀⠞⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⣰⣶⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠴⠚⠉⠙⠛⢿⣦⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠃⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⣿⠟⠉⠀⠀⠉⠀⠀⠀⠸⠧⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠸⠁⢠⣤⠀⠀⢰⡿⠁⠀⠀⠀⢠⣶⠶⠆⠀⠀⠀⠀⠈⠁⠈⠙⣷⡀⠀⠀⠀⠀⢹⡆⠀⢠⣿⣿⡄⠀⠀⠆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠁⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠼⡿⠆⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠀⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⣤⠀⠀⢠⡀⠀⢠⡄⠀⠀⣄⠀⠀⠀⠀⠀⠠⢤⡀⠀⣤⠀⠀⣤⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⣿⠄⠀⠀⣿⠀⢸⡇⠀⠀⢸⡇⢠⡎⠀⠀⠀⠀⠹⣦⠹⣇⠀⠘⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠈⠣⡄⢸⡇⠀⠀⢭⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢹⣆⠀⠸⣇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡿⢸⡇⠀⠀⠸⣧⠘⢷⡀⠀⠀⠀⢀⠟⠀⠀⢻⡄⠀⢹⣆⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⠀⠀⠀⣈⠁⠘⣃⠀⠀⠀⠈⣃⠀⠉⠒⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠁⠀⢀⠘⠃⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠹⠀⣿⠀⠀⠀⠀⠹⠀⣰⠂⠀⠀⠈⠱⠀⠀⠀⠘⣷⠀⠀⠀⣿⡆⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⢀⡀⠀⠀⡀⠀⠀⠀⠘⣧⠀⠀⣿⠁⠀⠰⡋
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠉⢻⡄⣿⠀⠀⠀⠀⠀⠈⣷⡀⠈⠀⠀⢸⡇⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠙⠢⠄⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠈⠓⠦⠀⠀⠈⠁⠀⠀⠀⠀⠀⠻⠆⠻⠀⠀⠀⠈⠻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣆⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢀⣴⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⣷⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⠋⢀⣠⣾⣶⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣾⣁⣠⣾⣿⣿⣿⣦⡀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠈⣻⡟⢉⣵⣿⣿⠟⠁⠀⠀⣼⡟⠀⠀⠀⣀⣠⣤⣶⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢻⣿⡿⠛⠉⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠻⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡷⠜⣷⠀⠀⠀⢰⣟⣀⣀⣴⣦⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⠿⠟⠋⠉⠀⠀⠀⢹⣧⠀⠀⠀⠛⠛⠋⠉⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⣴⣶⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⣋⣥⣶⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣿⣿⣿⡿⠟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣿⠿⠛⢉⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠹⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣠⣶⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣴⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣤⡈⠻⣿⣿⣿⣿⣿⣿⣇⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⣉⡉⢉⣹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠏⣠⣶⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢰⣿⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⣰⡿⢻⣧⠈⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⢻⣧⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣰⡿⢁⡀⢻⣆⣈⣉⣉⠙⣿⠋⣠⣤⣤⣾⠇⠀⢿⣦⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣀⣾⣧⣈⣉⣉⠉⢿⣆⠀⣰⡟⢁⣤⣤⣴⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢻⣾⡟⢀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣉⣠⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⣀⡀⢀⠀⢀⠀⠀⣀⢀⡀⣀⣀⢀⣀⠀⣀⡀⠀⢀⠀⣀⣀⢀⣀⡀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠻⡾⠨⣗⡂⢸⣀⢸⣅⡀⡟⢾⡇⣟⣒⢘⣳⡄⣓⣦⠀⢸⠸⣇⣀⢺⣁⡽⢸⠛⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⢛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠔⠉⠉⠑⣄⣀⡀⢀⠔⠊⠑⠢⠤⠤⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠈⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⠗⠚⠉⠉⠀⠀⠉⠉⠐⠢⢒⠄⠀⠀⠘
⠀⠀⠀⠀⠀⡰⠃⣀⣀⣠⡔⠉⠹⡖⣈⣈⣉⣐⣓⣄⠀⠀⠀⠀⠀⢠⠽⠞⠁⠀⠀⠀⠀⠀⠀⡔⠐⢄⠀⠑⠐⡄⠀⠘⢢
⠀⠀⠀⠀⢰⠁⠸⡀⠀⢠⠔⢤⣠⠏⠀⠀⠀⠀⠈⠫⣣⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣌⠑⠂⠠⡘⡄⠀⢀
⠀⠀⠀⠀⡼⠀⠀⠑⠤⠃⡶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡆⠀⠀⠘⢤⣀⣀⣀⠀⠀⢀⣀⣀⠤⠒⡏⠃⠀⠴⠁⢱⠒⠉
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡇⠈⠁⠋⠒⠤⣀⣠⣀⣀⣀⡶⠁⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠒⠀⠀⠀⠃⠈⠁⠀⠀⡌
⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⠃⠀⠈⠁⠀⠀⠀⢠⠀⡌⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠐⠀⠀⠀⢆⣀⡠⣄⡴⠁
⠀⠀⢀⡼⠀⠀⠀⠀⢀⠎⠀⠀⠈⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠲⡤⠒⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠈⠑⡄⢀⠀⢠⠪⣶⡄⢀⠀⠉⠂⢂⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⣤⠀⢒⠦⠅⡀
⠀⠀⠀⠀⠈⠁⣷⠃⠀⠁⠉⠈⢻⠋⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠇⡀⠀⢀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡰⠀⢀⠀⠘⣄⠀⢰⡀⠁⠦⢄⡀⠀⠀⡀⠠⠂⠀⠀⠀⠀⠀⠄⢀⠈⢰⠀⠀⠀⠀⢀
⠀⠀⠠⠀⠀⠉⢠⠈⠢⣀⠀⠉⠚⠭⠶⣄⢀⠈⠐⡦⢄⡀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠝⣀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⢰⠀⠀⠀⠘⠀⠀⠀⠉⠒⠒⢄⣀⣨⠖⠑⠒⠁⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠠⢄
⠀⠀⢸⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⠀⠀⠀⠀⡀⠴⠆⠀⠀⠀⠘⠀⠁⠉⠄⠂⠉
⠀⠀⠸⠄⠀⠀⡇⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠇⠤⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⢐⡊⠍⠒⠀⠀⠀⠀⠐⢀⡀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠥⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣀⣬⣿
⠀⠀⢰⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿⡿⠉⡄
⠀⠀⢀⠱⣀⡀⠀⣤⠔⣂⠥⠄⢀⠀⠀⠀⠀⠘⢛⡛⢛⡛⠓⠀⠀⠀⠀⠸⠄⠀⠀⢀⠈⠁⠀⠠⠃⠀⠀⠀⣶⣤⣄⣀⣠⡄⡀
⠀⠈⢄⠣⣒⣒⣐⣒⣫⠔⠊⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢾⣳⢀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠐⠒⠓⠒⠈⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡻⠴⢻⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡍⣟⣛⣯⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡄⠠⠄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠸⠛⠓⠈⠛⠁⣿⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡇⣀⣤⣶⣤⡀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠧⢸⣽⣿⢽⠥⠿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣋⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠾⠾⠾⠿⠆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣷⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⡷⢶
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣟⣛⣯⣛⣛⣿⣛⣻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⣏⣹⣉⣉⣏⣹⣉⣉⠇⠀⠀⠀⠀⠀⠀⠀⢸⣾⢷⣷⣶⣦⣶⡾⠿⡇
⢀⠀⠀⣀⣾⣿⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣥⣨⣅⣨⣅⣼⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣷⣾⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠵⣷⣾⠮⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠟⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠰⢄⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⠂⠀⠀⠀⠀⠀⣠⠏⡐⠶⠃⠀⠀⠀⢀⡐⠂⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⢀⣞⠉⠳⠞⠀⠀⠀⠀⠀⠀⠀⠡⡌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢦⡵⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠸⠃⠀⠀⠀⠀⠀⠀⠀⣾⠤⣌⠆⠀⠀⠀⢰⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢥⠊⠀⠀⢔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣤⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢠⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⣀⡀⠀⠀⠀⢤⡐⠀⠁⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡀⠀⠈⠢⡤⠀⢄⠀⠀⠀⠀⠀⠀⠠⣎⢂⢈⡰⠂⠁⠀⠀⠀⠀⢈⡤⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡐⠉⢆⠀⠀⢄⡀⣠⠀⠀⠀⠀⡖⢢⠀⣩⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡐⠀⠀⠀⠑⡄⠀⠀⠀⠑⢤⡈⢆⢙⡸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⠒⠣⣰⣁⠁⠑⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡐⠁⠀⠀⡠⠊⠈⠢⡀⠀⠀⡠⣅⡜⠉⠀⠀⠀⢀⠊⠈⠲⠀⠀⠀⠀⢀⣠⢉⠀⠐⠗⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠀⠀⢀⠔⠁⠀⠀⢀⠜⠓⢄⠑⠉⠈⢢⠀⠀⠀⠀⠢⠄⠘⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⡰⠃⠀⠀⡠⠖⠁⠀⠀⠀⣁⠦⢀⡀⠡
⠀⠀⠀⠀⠀⠀⡌⠌⠀⡠⠊⠀⢀⠤⠊⠀⠀⢀⠠⠒⠉⠀⣀⠠⠘⠋
⠀⠀⠀⠀⠀⡐⠌⢀⠔⠁⡠⠐⠁⢀⡠⠐⠈⣀⠠⠐⠊⠉
⠀⠀⠀⠀⢰⠏⡠⢃⠴⠊⡀⠄⢂⠡⠔⠒⠉
⠀⠀⠀⢠⣏⣨⢔⣡⠔⠍⠒⠉
⠀⠀⢠⡿⠟⠛⠉


⣿⢿⠿⡿⡿⠛⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⠟⠉⢀⠉⣀⢸⠸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣉⣿⣿⣿⣿⢁⠀⢀⡏⢀⡇⢨⣶⡰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⢎⡽⣿⣿⣿⣧⡟⢀⡞⣠⠞⣠⢎⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣰⢃⣾⣿⣿⣿⣿⣫⣴⣫⣴⣯⣾⡋⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⣾⢠⡏⣼⣿⢧⣿⣿⣿⡟⠉⠁⠀⣿⡦⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⡏⣼⢡⡟⣿⢸⣿⣿⠿⠗⠀⠀⠀⠁⠀⠀⠘⠦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⠃⣿⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⢸⠁⡇⢸⠀⣿⠸⡇⣿⣦⡄⠀⠀⠀⠀⠀⠀⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⡐⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢻⠀⡇⠸⡀⢡⠀⠸⡈⢿⣤⡀⠀⣾⣶⣶⡶⠆⢰⠇⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⢸⡆⠱⢆⢣⠈⣷⡀⢣⠀⠈⣿⡇⢹⣿⣿⠁⢠⡞⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢳⣄⡃⠀⢆⡘⣷⣤⣷⡾⠛⠀⠀⠈⢀⡴⠋⠀⠀⠀⠀⠀⠀⡜⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢻⡿⠛⠉⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠋⠊⣞⢁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⠈⠙⠾⠀⢠⠆⢀⠀⠀⠀⠀⠀⢠⡟⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠈⠳⣄⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⠽⢷⢲⣄⢄⠈⠀⠃⠷⢢⡦⠀⢠⡟⣴⣿⠋
⠀⠀⠀⠀⠀⣀⣤⡾⢫⣽⣷⣄⠀⠉⠒⠤⠤⠔⠁⢸⡛⢶⣦⣄⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠜⠾⡖⠀⠀⠈⠂⢠⠟⣴⣿⠃
⠀⠀⢀⣴⠟⢿⣷⣮⣳⣿⣾⣿⢿⡖⠄⠀⠀⢠⠤⠻⣇⢿⣾⣿⣧⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⣿⠏
⠀⢀⡿⠁⣤⠈⠠⠙⣿⣿⣿⣿⢆⢻⣆⠀⠀⠀⠀⠀⢿⣧⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡼⣻⠋
⠀⣸⠇⣮⠁⡟⣷⣽⣹⣿⣿⣿⣜⡆⢻⡄⠀⠀⠀⠀⠸⣏⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢞⣿⠃
⠀⣿⢰⡟⢼⡿⣿⠟⢹⣿⣿⣿⣿⣷⠈⣷⠀⠀⠀⠀⠀⣿⠀⣿⡶⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣄⠀⢠⡿⠱⣱⠇
⠀⣿⣼⣇⡾⢿⣿⣧⡏⢹⣿⣿⣿⣏⢀⢻⡄⠀⠀⠀⠀⣿⣠⡼⠷⠚⣻⡝⠳⢦⣄⡀⠀⠀⣤⡞⢺⡄⢻⡆⢳⡿⠡⢠⠏
⠀⢸⣿⣿⣀⠾⡦⢤⡄⢹⣿⢛⣿⣿⣹⣾⣧⠀⠀⠀⢀⡞⠁⠀⠠⠾⠿⠷⠛⣉⣽⠟⢷⣶⡆⢣⠀⢇⠀⠃⠈⣷⠁⠏
⠀⢸⣿⣹⣿⣿⡛⠈⢁⣿⣿⣿⣿⣿⣻⠏⢿⡀⡠⣄⡜⠀⠀⠀⠀⠠⠤⠒⢊⡡⢾⢶⣬⣙⣿⠘⠃⠈⠀⠀⠀⢸⠇
⠀⣸⠇⣿⣿⢻⡧⠘⠛⣿⣿⣿⣯⣬⣵⡾⠿⠻⣄⠈⢆⠀⠀⠀⠀⠠⠔⠒⣉⣴⣻⣿⡿⠿⡟⠀⠀⠀⠀⠀⠀⡼
⠀⢹⡆⢿⠛⠛⢗⣴⣿⣿⣿⠟⠋⠁⠉⠀⠀⠀⠘⡇⠈⠆⠀⠀⠀⣐⣂⣩⣴⠟⠿⡿⡂⢰⡛⢄⣀⣀⢀⣠⠞⠁
⠀⠀⢻⣾⣶⡰⣾⢿⣿⡿⠋⠀⠀⠀⠀⠀⣀⣀⡀⢹⡀⢸⠞⡛⢋⢹⣿⣷⣿⡇⠖⡓⣬⢦⠳⣤⣬⡭⠋⠁
⠀⠀⢸⡯⣿⣷⣴⣿⣿⣁⣄⣠⣤⣶⢿⣿⣿⣿⣿⢸⣿⣿⡀⠀⠸⠌⣿⡿⣶⣥⣤⣤⣬⡶⠟⠉
⠀⠀⠀⢻⡘⢟⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣯⣭⣴⣿⠋⣿⠃⠂⠂⠀⣿⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠉⠀⠈⠙⠛⠟⠻⠶⠟⠛⠛⠛⠉⠉⠉⠗⠀⠿⠀⠀⠀⠀⢺⡇


⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⡶⠛⠀⠀⠨⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠞⠛⠁⠀⢀⣴⡾⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠐⢿⣯⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣷⣦⣄⠀⠀⠀⠀⠀⠙⣷⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠿⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢠⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⣿⠁⠀⠀⠀⠀⠀⣼⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⢀⣼⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢸⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⢴⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⣾⣿⣷⣦⢸⣷⣴⣦⣤⣤⣄⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣟⡉⠉⣉⠉⠀⠀⠉⣛⡻⣿⣿⣿⠉⠉⠙⠛⠛⢿⡿⠿⢿⣿⠿⣿⣶⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣿⣿⣿⣿⣿⣿⣿⣷⣬⣭⣀⣀⣀⣀⣩⣿⣿⣟⣁⣀⡀⣀⣀⣀⣉⣉⣹⣦⣥⣶⣾⣿⣿⠛⣻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡛⣿⣿⣿⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣝⢋⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⡟⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⡇⣼⡟⠉⠙⠻⣿⣿⠛⠋⠩⠝⢿⣿⠿⠋⠉⠉⠙⢻⣿⡄⠀⡀⢘⣷⣯⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⣿⣿⣿⣧⣹⣆⣴⣿⡇⢽⣧⣠⡄⣿⣿⡄⢺⣤⡄⠀⢸⣿⣷⣮⣤⣿⡿⢿⡏⠉⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣾⣏⠉⠙⢻⣿⣷⣾⣿⣿⣿⣼⣧⣿⣴⣿⣿⣷⣦⣽⣤⣴⣿⣿⣿⣿⣿⣏⠁⠀⢹⣄⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣿⣿⠆⣀⣾⣿⠙⠛⠉⠙⣿⠿⣿⣿⠿⠟⠋⠙⢻⡿⠛⠋⠙⣿⠋⠁⠀⠸⣧⣤⣼⡟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣀⣿⣧⣾⣧⡀⠀⠀⠀⣸⣷⣀⣀⣴⣿⣦⣠⣤⣺⣿⣧⣿⣧⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣿⣿⠿⡿⠿⠟⠿⠛⠛⠹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⢁⠘⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢐⡒⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠇⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠖⠄
⠀⠀⠀⢰⠀⠀⢈⡀⡇⢸⠀⠀⠀⠀⠀⡴⠂⡱⠃⠀⠀⠈⠠⢠⢸⠀⠀⠀⠀⠀⡀⠌⡤⡀⠀⡀
⠀⠀⠀⢸⠀⠀⡏⢸⡇⢸⠀⠀⠀⠀⠀⠳⠤⡁⢀⡐⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠖⠄
⠀⠀⠀⠀⠀⠀⠧⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠿⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙


⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠢⠄⠀⠩⠄⠀⠀⠀⠀⠀⠀⠁⠀⢢⠰⠒⢲⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠆⠒⠢⠂⠠⡀
⠀⠀⠀⠀⠠⠄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠸⠂⠀⠈⠀⠁⠀⢂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢄⠁⠉⠁⠀⠉⠁⠡⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⢢⠰⢐⣵⠂⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁
⠀⠀⠀⠀⠀⠈⢧⣼⠇⠀⠀⠀⠀⠀⠀⠀⠠⠃⠄⣄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢜⣛⢄⠀⠀⠀⠀⠀⠀⠀⠃⡈⠀⠀⠀⠙⢉⠀⡀⠀⠀⠀⠀⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⢀⡞⢿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠛⠎⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⡛⠍⠀⢠⣬⡷⣿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⢿⣿⣶⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠉⠛⠛⠉⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣋⠓⢆⠀⠀⣤⡶⣤⠀⠙⠛⠁⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡇⠉⠛⠋⠀⠐⢤⣤⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣄⢀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢴⣶⡦⠄⠀⠀⠀⣠⠎⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣆⠀⠀⢀⣀⡴⠞⠁⠻⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠺⠭⠀⢾⣧⡻⢿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⡟⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⡆⠀⠀⢸⠉⠁⢀⡹⠀⠀⠀⢀⣀⠀⠀⠀⢠⠄⠀⠈⠙⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠈⠀⣴⠏⠀⠀⣀⠀⠈⠿⢆⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠃⣀⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡄⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠞⢤⣤⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⠣⠀⠂⢤⡶⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠶⣬⣿⡟⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣇⡠⠒⠒⠂⠠⢍⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣿⣿⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⣀⣄⠀⠀⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠐⡄⠂⠐⠂⢄⠀⠘⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣄⠀⠀⠀⢠⡎⠠⢀⠒⢣⠀⠇⠀⠚⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣇⡀⠰⣿⠀⠀⠀⠀⠰⡘⠀⠁⠶⡦⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⠘⠛⠙⢄⠀⠀⡀⣴⣄⠀⠈⠈⣷⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡄⠀⠁⢨⣤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⢡⡀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⣠⠶⣋⡉⠁⢠⣤⣬⣉⣉⣹⣿⣿⣋⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⢿⡷⠶⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢢⠄⢠⣀⣀⣀⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⠛⠛⠋
⠀⠘⠀⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠉⠉⠉⠁
⠀⢠
⠀⠘

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡏⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⠀⡴⠶⠶⠶⠶⠶⢦⡈⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⠀⡇⠀⡴⠒⢦⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠂⠒⠒⠒⠀⠀⠀⢸⠀⡇⢀⣷⣤⣼⡀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠉⠀⠉⠉⢉⠉⠉⠉⠁⢸⠀⣇⣯⣀⣀⣀⣹⣼⠇⡇
⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⠀⣤⣀⠀⠀⠀⠀⢀⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠓⠒⠒⠒⠛⠀⠋⠁⠀⠐⠛⢻⡭⢥⣀⡬⢭⠟⢳
⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⢯⠉⠉⠉⠉⠉⠉⠁⢸⡧⠤⡤⠤⢼⠀⠯⠇
⠀⠀⠀⠀⠀⠀⠓⠶⠶⠶⠶⠶⠶⠶⢶⠶⠾⠿⠶⢶⠶⠶⠶⠶⠆⠘⢇⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⣉⣉⣉⣉⠉⠉⢱⡦⠤⠾⠦⠄⠀⠀⢀⢤⡾⣟⠛⣹⢶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠤⠤⠤⠤⠤⠤⢸⡟⠒⠤⣄⡀⢀⠜⠁⠀⠑⣌⠉⡡⠋⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠶⠶⢦⡀⠀⢸⡧⣄⠀⠀⠉⠁⠀⢰⡆⠀⢸⠙⡇⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⣀⠰⡃⣎⢙⡆⡷⢀⡸⠃⠀⠀⠲⢄⣀⡤⢺⡇⠀⠀⠁⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⢮⣭⠖⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⣾⢸⡶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⢿⣿⣿⣿⣍⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠿⠿⣿⣿⡅
⣿⣿⡏
⣿⣿
⣿⣿
⣿⣿
⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣧⠤⠄⠀⠀⠀⠀⠐⢁⡐⠀⢌⠀⠀⠉⠀⠁⠠
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠃⠀⠀⠈⠀⠘⠈⠀⠇⠈⠇⠂⠀⠠⠔⠠
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠾⠇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⢤⡀⠀⠀⠀⠀⠶⠶⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠋⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⠂⠀⠈⠁⠀⠀⠸⣟⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡶⢶⣿⣿⣇⡀⠀⠀⠀⠀⢰⣯⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠆⢹⡿⢿⡿⢿⣤⣀⣀⣠⣶⣿⡛⠛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣧⣎⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣀⣀⣈⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⠋⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠖⠲⠦⠴⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠿⠿⠷⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿
⣿⣿⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⢤⠀⠀⠀⠀⠀⡞⠈⠛⠒⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠀⠈⢹⠆
⣿⣿⠀⠀⠀⠠⠖⠆⠀⡀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠶⠄⠀⠀⠀⠀⠀⢘⣿⣷⣤⣶⣶⣶⣶⣿⡇
⣿⣿⠀⠀⠀⠀⠐⠀⠀⠀⠛⠀⠀⠀⠀⠀⡀⠀⠀⠰⠀⠀⠀⣿⣾⣶⣶⣶⣦⣤⣞⠀⢽⣿⣻⣿⣿⣿⣷⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣿⣿⣿⣿⣿⣇⡘⠿⠟⠛⠻⣿⣯⣽⠇
⣿⣿
⣿⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠙⡋⠪⠙⠅⢙⠃
⣿⣿⠀⠀⠀⠀⢿⣿⡗⣠⣤⠀⠀⠨⣵⠶⠚⠶⢥⠆⠀⠀⠀⠀⢰⣶⣶⡄
⣿⣿⠀⠀⠀⠠⣶⣶⣶⣬⣯⡄⠀⠰⡶⢸⡆⣥⣼⡇⠀⠀⠀⠀⢾⣿⣿⣇⠀⠀⠀⠄⠰⠀⡆⠀⡷⠀⠀⡄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉
⣿⣿⠀⣀⣀⣀⣀⣀⣀
⣿⣿⠶⠿⣿⣿⣿⣿⣿⣛⣛⣛⣛⡛⠛⠛⠛⠛⠒⠒⠒
⣿⡏⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⢿⣿⣦⣤⣤⣶⣶⣶⡖⠂
⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡖⠒⠒⣒⣶⣶⣶⠖⠒⠒⠒⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠠⠤⢼⡿⠭⠤

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣴⣾⣿⣿⣶⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⣿⠀⢸⡟⢛⣛⣻⡟⠛⣿⡟⢻⣟⡛⢛⣻⠛⢻⡟⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⣴⠀⢸⡇⢈⣉⣿⠁⡀⢹⡇⢸⣿⡇⢸⣿⠀⢨⡅⢸⣿⣿⣿⣿⣷⡆
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⣤⣿⣤⣼⣧⣬⣭⣯⣴⣶⣤⣧⣬⣭⣧⣼⣿⣤⣼⣧⣼⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣜⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠹⣿⣿⣿⣿⡿⢿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠈⢿⣿⣿⣇⠘⠀⠈⠃⣸⡇⠰⢶⡇⢸⣿⠀⣿⣿⠐⡆⢸⡇⠰⢶⠀⠶⠤⡇⠰⠦⣼⣿⣿⠏
⠀⠀⠀⠀⠀⢹⣿⣿⡀⢰⡆⢀⣿⡇⠸⢿⡇⠸⢿⠀⠿⢿⢀⣇⢸⡇⠸⢿⡿⠿⠀⣿⠿⠇⢸⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⣭⡇
⠀⠀⢠⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢾⡿⠋⢡
⠀⠀⢀⠳⣀⠤⠤⠤⣂⠬⠄⣂⡄⠀⠀⠀⠀⣛⣛⢛⣛⡛⠀⠀⠀⠀⠘⠠⠀⠀⣀⠀⠀⠠⠘⠀⠀⠀⢰⣦⣤⣤⣀⡄⡀
⠀⠈⢄⣒⡪⠭⠭⢙⠵⠊⠁⠀⠀⠀⠀⠀⣾⡿⠿⡿⠿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢺⣷⢀⡄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣙⣺⣹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⡏⣿⡭⢿⠺
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠀⠀⠰⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠈⠋⠉⠁⠉⠀⣿⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣴⣿⣷⡦⢸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣷⡆⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣯⣿⣿⡇⠛⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣨⣵⣒⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠉⠉⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢩⡄
⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠿⡶⠿⠷⠶⡿⢶⠾⢾⠄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣙⣛⢛⣉⣤⣤
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⢳⠒⡗⠒⠗⢺⠒⡺⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣫⣭⣿
⠐⠀⠀⠐⣿⣿⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⣫⣛⣹⡹⣋⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⠿⠟
⣤⣤⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⡀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠸⢷⣄
⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢞⠛⢶⠹⡊⡴⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⣶⢏⢹⣿⡐⠢⣜⠐⠀⣣⠔⢒⣿⡏⡹⣖⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⠁
⠀⠀⠀⠀⠘⣿⣏⣾⣺⠁⠙⢽⠻⡟⠑⡼⠂⣰⢿⡟⣧⠙⢿⣦⡑⠂⠘⢊⣴⡾⠋⣾⢛⡿⣆⠐⢤⠊⢹⠿⡯⠋⠉⣗⣷⣻⣿⠃
⠀⠀⠀⠀⠀⠙⣿⣡⣿⣭⣒⡬⢄⡁⠠⣕⣻⠃⢾⠧⠟⣤⣦⡍⠛⢦⡴⠛⢩⣴⣤⠷⢾⣷⠘⢟⡫⠄⢈⡠⣥⣖⣽⣿⣬⣿⠋
⠀⠀⠀⠀⠀⠀⠘⣿⣉⡽⣿⡿⣷⣌⠄⠩⢙⡲⠛⠉⢰⣿⣿⡏⠀⠘⠋⠀⢹⣿⣿⡆⠹⠛⠖⢏⠌⠐⣡⣾⢿⣿⢧⣘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣞⢟⡄⢙⣿⡌⢠⡆⢀⡀⠰⠄⢸⣿⡇⣤⣠⣆⣤⢸⣿⡃⠰⠄⢀⡀⢰⠄⢁⣿⡋⢠⡻⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⢰⣷⡇⠀⠁⡿⡏⢠⢒⡟⢢⢹⣾⠳⢞⣷⡏⡔⢳⡒⡆⢹⣿⠈⠀⢸⢿⡄⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣗⡇⠀⠀⡇⢠⠌⠉⢸⡟⣸⠛⠃⠘⠛⣇⢿⡇⠉⠡⡄⠺⠀⠀⢸⣸⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⢿⣳⣶⣾⣧⣿⡀⠀⠀⢀⣿⣼⣷⣦⣞⡿⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⣷⣽⣆⣸⢿⢧⡀⢀⡼⡿⣅⢰⣯⣾⠷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠉⠁⠀⠀⠉⠉⠙⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠁⠀⠀⠉⠉⠙⠻⢶⣤⣀
⠀⠀⠀⠀⢠⠴⠟⠉⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⡠⠖⠒⠊⢹⠉⡇⠀⠀⣀⠈⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡆⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⢸⠀⠀⢠⡄⢸⠀⡷⢳⢸⡰⠃⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⣾⠈⠣⢄⠼⠁⠘⡄⡇⠸⠜⠦⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢻
⢸⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣤⣴⡶⠶⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⣀⡀⠀⠀⣿⣴⡟⢹⣿⠀⠀⠀⠀⠀⠀⠘
⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⣤⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⢰⣶⡿⠛⣿⡄⠀⣿⣿⠃⣼⠏⠀⢰⡄
⢸⡇⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠈⢻⣦⠀⠀⣿⣇⣀⣀⡀⠀⢸⡇⠀⢸⣿⡇⠀⢸⡇⠀⢸⣿⣿⠋⠀⠀⢸⡇
⢸⣇⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣿⡆⠀⠀⢹⡇⠀⠀⠀⢻⡆⠀⠘⣿⠛⢻⡇⠀⠸⣿⠀⠀⣿⡇⠀⢸⣷⠀⢸⣿⠙⣷⡀⢀⣿⠃⠀⠀⠀⢀
⠈⣿⡀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿⠀⠀⠀⠈⣿⠀⠀⣿⠀⠘⣿⠀⠀⣿⠀⠀⣿⡇⠀⠀⣿⡀⣸⣿⠀⠈⠛⠛⠁⠀⠀⠀⠀⣸
⠀⠘⣷⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⢸⣧⠀⠀⣼⡇⠀⠀⠀⠀⣿⠀⠀⠻⠃⠀⣿⠀⠀⣿⡇⢀⣿⡇⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠘⣷⡀⠀⠀⠈⢿⣦⣀⠀⠀⢈⣿⣤⣾⠏⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢻⣇⣰⡟⠿⠿⠋⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠘⢷⡄⠀⠀⠀⠈⠙⠛⠿⠿⣿⠋⠀⠀⣀⣠⣤⣤⣾⠇⠀⢾⡆⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⣴⡟⠻⣦⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⣰⡾⠛⢉⣩⡿⠻⣷⡀⢸⣇⠀⠀⠀⠀⠀⠀⢀⣴⢶⣦⡀⠀⢻⣿⠀⠀⢻⣇⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣠⣴⣿⣴⣿⣶⠾⠛⣭⡀⢀⣿⠇⠘⣿⢀⣠⣤⡀⠀⠀⣾⡇⠀⠹⣷⠀⢸⣿⠀⠀⠈⣿⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠛⠟⠛⠁⠀⢿⣆⠀⠀⠉⠛⠛⠉⠀⠀⣿⣿⠏⠙⣿⠀⠀⣿⣷⡀⠀⣿⡆⢨⣿⠀⠀⢀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠛⢷⣦⣄⡀⠀⠀⠀⠀⢹⣿⠀⠀⣿⡆⠀⢻⡏⠻⢶⣿⡿⠟⣿⣧⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢀⠈⠙⠛⠷⣦⣄⠀⠸⣿⠀⠀⢸⡇⠀⢸⣿⣀⣀⣿⠁⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⢠⣾⠟⠁⠀⠀⠀⠈⢻⡆⠀⣿⠀⠀⠘⣷⣤⣿⠋⠛⠛⣡⣴⠟⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢸⡟⠀⢹⡇⠀⠀⠈⠉⠀⠀⣠⡾⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⡉⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣴⠟⠉