Arizona Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡤⢠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⡿⠚⠀⠤⠀⠀⠀⠀⠀⡀⠞⠀⠘⡇⠀⠀⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⡗⡟⣠⠠⠀⠀⠀⠀⠀⣇⣀⣀⡀⠀⠀⡄⡧⠇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠒⡹⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠿⠃⠀⠀⠤⢀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠠⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠀⣷⡄⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⡄⣿⠃⡀⠀⢨⠀⢰⠀⣶⠀⠀⠀⢰⠀⠠⠤⠄⣀⣈⡀⠀⠀⠈⢉⠁⠒⠸⠅⠀⢀⢿⠡⢯⠝⠁⠘⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⢆⠀⣿⠦⣇⠄⡈⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⠳⡜⠚⢞⢑⡗⢃⣀⠀⠉
⠀⠀⠀⢠⣿⣀⣘⠞⠀⠀⠈⠓⣾⠄⠸⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⢉⣋⣡⠑⡇⢀⠳⣤⡼⠋⠈⡷⠋
⠀⠀⠀⠀⠹⣄⡹⠤⢄⡰⡖⠀⣿⠀⡂⠀⢀⣠⠂⠀⠀⢀⠠⣄⣢⠀⠀⠀⠦⠴⣦⡀⠀⢀⠀⠀⠹⣵⢠⣤⣧⠜⠀⠀⠰⣤⡾
⠀⠀⠀⠀⠀⠀⣆⠀⠀⢹⠇⢫⣻⣿⠟⠂⠀⠉⠀⠀⠀⠈⠊⠉⠉⢲⠌⠈⢄⠀⢸⡅⡔⠙⠚⡀
⠀⠀⠀⠀⠀⠰⠿⠿⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣦⣄⣠⠼⠧⠟⠐⠇⠠⠤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠂
⠀⠀⠀⠀⠀⠀⣴⡆⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⣿⠃⣿⡿⣡⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠹⢧⣿⠀⣿⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⡀⣴⣶⣛⡀⠀⠀⣿⠛⠁⢧⣷⡇⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣷⣦⣀
⠘⢿⣧⣽⣿⢋⣿⡶⣿⣀⠀⢇⡏⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⣀⣟⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀⣀⣆
⣤⣶⣾⣿⣧⣼⣷⣶⣿⣿⣿⣶⣿⣿⣿⣷⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣙⣷⣿
⠀⠀⠀⠀⣀⡄⠘⢻⣿⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣠⣦⣤⣀
⠀⠀⠀⠀⠻⢷⣶⢸⣿⠿⠃⠀⣾⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠘⢼⠊⠀⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣠⢀⣤⣼⣿⣠⣿⣧⣀⣀⣼⣦⣤⣽⣇⣀⣤⣄⣠⣤⣶⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠛⢿⠿⠿⠿⠿⡿⠿⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣿⡿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⢠⠀⢄⣀⣀⡀⡠⣀⢀⡀⠀⢀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡇⠀⠀⠈⠁⠈⠉⠉⠀⠁⠉⠈⠉⠀⠈⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⢀⣼⠃⣄⠀⢀⢀⠔⡄⢠⢲⣠⢞⠆⡞⡽⡰⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡖⡞⠛⠁⢹⡠⢧⢞⠾⡖⡳⠊⠉⠈⠋⠉⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⣀⠜⠀⠘⠒⢀⠀⠀⡳⠚⠀⠀⠀⠁⠈⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠂⠈⠊⠃⠈⠂⠉⠙⠃⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀⣠⣶⣦⡈⠀⢀⣾⣷⣦⡀⠀⠀⢀⣸⡧⣶⣄⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣯⣀⡽⠃⠈⢻⣿⣿⣷⣪⠁⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⣟⡻⣿⣿⣇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢⠀⠀⠈⠉⠉⠀⠀⠀⠀⢹⣿⣿⡇⠈⠻⣿⣷⣿⣉⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠁⠀⠀⠀⢀⣿⠉⠛⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⢤⣇⡀⠀⠀⠀⢸⣿⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠛⠛⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⠀⠀⠀⠀⠀⠀⣉⣋⣙⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⢹⣦⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠊⠁⠉⣉⡉⠀⡀⠀⠻⠟⠀⠈⠉⠉⠀⡇⡄⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⡆⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⢨⠇⠐⠂⠉⠁⠠⡇⠀⢸⠀⡿⠀⠁⠰⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠁⣿⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠓⠀⠁⠀⠀⠃⢱⣀⣿⣲⠃⠐⠒⠒⣽⣇⡒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡿⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡿⠀⠀⠀⠆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⡶⠤⣀⣘⣧⢴⣄⡀⠛⠀⠐⠛⠂⢀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢤⡂⣍⣇⠞⣄⢴⠚⣸⡀⢀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠿⢿⣾⢿⣤⠃⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⠍⠀⡻⠇⠀⠿⠮⢾⣟⣜⣭⣳⠀⡀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠿⣣⢾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠬⣁⣀⣀⠘⠾⠃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⢉⣡⣭⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠞⣯⡙⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⣯⡌⢷⠤⣝⠳⢦⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⣀⡄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠢⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠴⠋⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⡅
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠘⠀⠀⠀⠀⠀⢈⠀⠁⠰⠂
⠀⠀⠀⢀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠒⠀⠈
⠀⠀⠤⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⢸
⠀⠀⠀⠀⠂⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠐⠀⠀⠀⠀⣀⣠⠀⠀⠀⠁⠀⠀⠘⠱⠄
⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠾⠍⠙⠂⢀
⠀⠀⠀⠤⢾⣷⢿⠄⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⢸⠀⣤⠂⠀⢀⣤⣀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⢸⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⡀⣄⡀⢀
⠀⠀⠀⠀⠀⠀⢠⠀⢀⢶⠀⡦⠀⠀⠀⢸⠀⠀⢹⠅⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⡃⠨⠋⣜⣡⠀⠀⠀⠀⣇⠀⠼⠀⠀⢸⢇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠸⠛⡃⠁⠀⠀⠀⠀⠀⠀⠙⠲⠀⢀⡜⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣠⠴⠶⠤⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⢠⣀⡠⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢐⢀⠢⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡗⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⡤⠀⠀⠀⠀⠾⠷⠀⡿⠖
⠀⠀⠀⠀⠀⠀⠤⠦⠔⠻⠟⠃⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⢠⠀⡄⠀⢀⠀⢠⠀⡀⠀⡀⢠⠀⡄⢠⡄⣄⠀⠀⢠⡄⣀⢠⡀⠀⢠⠀⡠⡄⢠⠀⢠⢀⠀⡀⣤⠀⠀⣠
⠀⠀⠀⠃⠘⠀⠚⠀⠘⠀⠃⠀⠓⠘⠐⠘⠊⠃⠛⠀⠀⠘⠀⠙⠈⠁⠀⠉⠂⠛⠃⠘⠐⠁⠘⠀⠃⠂⠹⠀⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⠈⠉⠑⠂⠀⠄⢀⡀⠠⠒⠂⠀⠠⠒⠢⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⢀⡀⠀⠀⡀⠀⢀⠄⠂⠐⠂⢄⠀⠀⡀⠄⠀⣀⡀⠕⠂⠐⠂⠄⠄⠠⢀⡀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠠⠮⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⡀⠀⠉
⠀⠀⠀⠀⠀⢀⣀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠁⠀⡢⠀⠀⠀⠃⠀⠀⠇⠀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠃⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁⠁⠒⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⠠
⠀⠀⢀⠂⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣈⠙⠂⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠄⠀⠐⠒⠀⢀⠀⠄⠀⠀⠄⡀⠀⠀⢰⠀⢰⡀⠄⠂⠁⠀⠉⠙⠢⢀⠀⠀⠀⠀⣇⠀⠤⠤⢄⡀⢠⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⠀⡀⠑⠆⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠤⠄⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⣶⡆⠑⡀⣀⠀⣁⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢈⠀⢠⣀⠀⠀⠀⣀⢠⠁⠘⠗⠈⠀⠀⠀⠀⡆⠀⢇⢀⠀⢀⠀⡀⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠒⠦⡀⠙⠀⠈⠀⠀⠀⠀⠈⠐⠀⠀⠁⠀⠀⠀⢰⠀⠃⠀⠘⢩⣄⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠆⠀⢸⠀⠀⢠⠀⠀⠀⢀⠄⠀⠀⠀⢃⠀⠰⠀⠇⠀⠀⡠⠀⠀⠂⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠢⡀⠀⠀⠀⠈⠁⢀⠀⠠⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⠠⣨⠄⠀⠀⠀⡀⠀⠀⠀⠆⠘⡀⡆⠀⠀⠀⠰⡄⠀⣄⢀⢰⡆⠂⠀⠀⢰⠒⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⢀⣸⣄⢈⠂⠉⡸⠱⣟⠤⡎⡄⡈⠠⡀⠃⠀⣀⠆⢠⣀⡤⠻⠌⠛⠓⠂⠀⠀⠸⢀⡀⢰⠀⠀⠠⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⠤⡤⢶⠉⠀⠀⠀⢺⣁⡓⡇⢐⡎⠃⠀⢉⠔⠁⢀⡧⠀⢄⠐⠒⠆⠀⠀⠀⡨⡧⠬⠄⡀⡟⠳⢀⠛
⠀⠀⠀⠀⠀⡲⢅⣠⠃⠀⠑⠛⣆⡀⠀⢸⠃⡷⠞⠣⡇⠀⠗⠒⠐⠉⠓⡤⠼⣄⡠⠆⠀⠀⢠⣀⡠⠼⡄⢀⠇⠒⠚⠄
⠀⠀⠀⠀⢠⠁⢀⠃⠀⡀⢀⠀⠘⠣⢄⡨⠟⠁⠈⠑⠇⠀⣤⠔⢳⢠⠎⠁⢀⣈⡁⡄⠀⠀⠀⠀⠀⠀⠐⢾⠂⠀⡇
⠀⠀⠀⠀⠠⠀⠰⠂⠀⠡⠎⠰⠀⠀⠀⠅⠀⠠⠄⠰⠃⡠⡇⠴⠀⠱⠀⠎⠁⠀⠀⡇⠀⠀⢠⠀⠀⠀⠀⣀⠱⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠐⠠⡀⠀⣠⣤⣤⡶⠴⠈⠀⠐⠀⠲⠤⠤⢄⠀⢀⣄⡇⠀⠀⢰⠀⠀⣠⠔⠉⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠒⠂⠀⠈⠒⠋⠙⢋⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣿⠛⠇⠀⠀⠰⠖⠂⠁⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠇⠀⠀⠀⠤⠤⠴⠾⠤⠤⠤⠼⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠤⠀⠀⠤⠤⠤⠁⠀⠶⠀⠀⠀⡀⠀⢀⠆⠈⢀⡀⠀⠀⠈⠂⠂⡀⠀⡀
⠀⠀⠀⠀⠀⣀⣀⠔⠂⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠖⠒⠊⠉⠁⠀⠑⠠⠤⠤⠤⠤⠤⠌⠄⠀⠀⠤
⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⢀⡠⠂⠄⠠⠄⠖⠉⠑⠐⢂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣐⠂⠀⠀⠀⠀⡄⠀⠀⡡⠙⠋⠩⠄⠀⠀⣀⡀⠀⢀⠤⠄⠐⢈⢁⠔⠉⠁⠢⠠⠄⡐⠊⠁⢂⠑⠒⠀⠀⣀⡀
⠀⠀⠀⠈⠢⠔⠂⠀⠀⠐⠀⢈⠑⠂⠒⠒⠂⠄⠒⠅⠀⠀⠁⢀⠀⠀⠈⠀⠀⣙⠭⠒⠲⠋⠑⠀⠠⠀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠢⠠⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠐⠤⢒⠀⠈⠉⠉⡀⠐⠂⠬⠠⠤⠴⠆⠀⠀⢀⠠⢐⠂⠀⠉⠄⠂⠀⠀⠀⠀⠀⠄⠠⠐⠁⠀⠉⠐⠒⡒⠠⣀⠀⠀⠉⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠳⠀⠒⠀⠀⢡⠀⠐⢀⡀⠂⠀⠤⠈⠉⠁⠀⠀⠉⠉⠉⠀⠀⠰⠄⠀⠀⠀⠒⠀⠀⠀⠤⣀⠑⠈⠁⠲
⠀⠀⠀⠀⠀⣀⣠⣠⣸⡧⠆⢤⡆⢀⣠⣀⡰⣀⢸⣠⡠⡆⢃⠀⢀⠀⠀⢀⠀⣖⠀⠀⡀⡀⡀⠠⣶⢠⡄⡀⢸⠀⠀⠀⠀⡄⢠⡀⢀⡰⢃
⠀⠀⠀⠀⠀⠘⠄⠙⠒⠔⡋⠵⠘⠁⢸⢓⡙⣫⣿⢁⡍⠻⡿⠉⢸⠀⢹⠈⣥⠽⢩⡉⠥⠭⠅⠀⣧⣹⢋⡟⠩⠉⠭⠈⢀⣁⠬⣅⡛⠁⠬⡅
⠀⠀⠀⠀⠀⠀⢘⠔⠪⠘⠄⠀⠈⠡⠀⠄⠑⣖⠛⢋⡈⠀⢃⠀⠉⠀⠉⠀⠷⠤⠴⠧⠤⠔⢠⠃⡈⠢⡉⠀⡐⠂⢂⠸⠀⠀⡠⠐⢠⠎⠐⠖⠁
⠀⠀⠀⠀⠀⠐⢅⡔⠀⠐⠈⠢⡀⠀⠀⠀⠂⠐⡤⢂⡀⠀⠄⡐⠄⡀⠔⢭⠢⡀⠀⠀⠀⡴⠛⠀⠠⢢⠈⠡⠀⢄⠀⢢⡰⠚⠒⢄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠁⠁⡹⠶⠄⣐⡀⠌⠀⠀⠀⠢⡀⠀⠀⢡⠀⠀⠇⠀⢡⣀⣰⠀⠀⠈⠄⠈⠀⠐⠤⠀⢣⠀⠣⡀⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡐⠁⠄⠂⠀⠀⠘⢆⠀⢀⡀⠀⠘⠀⠀⢸⠀⢠⠼⠀⣠⠃⠀⠈⠢⡀⠀⢀⠄⠀⠀⠂⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡃⠀⢠⢻⡰⠄⠋⠀⠈⠐⠤⢀⣀⠆⡰⠁⠀⠀⠈⠂⠤⢄⠀⣃⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⠠⢀⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢤⣧⡀⠀⢠⠀⠁⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠂⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠈⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡄⢀⠰⢜⠁⠀⢠⡀
⠀⢠⣠⠆⠠⡷⠆⠀⠀⠀⠀⠀⠀⠄⠀⡄⠄⠋⠖⡢⠀⠀⠀⠒⠀⠠⢘⢀⠀⠀⢰⣿⡆⠀⠀⠀⢽⡤⠄⠀⠁
⠀⠀⢁⣀⡧⠷⠀⠃⠀⠚⢢⢀⣑⡐⡀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠜⢍⣉⣾⡃⢠⠀⠀⠀⠀⢀⣤⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠿⡿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⡇
⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣿⡇⣷⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⡿⢿⣿⠸⠃
⠀⠀⠀⠀⠀⠀⣿⣇⡧⢹⣿⡇⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠿⠛⠻⠏⠀⠀⠀⠘⣿⠄
⠀⠀⠀⠀⠀⠀⣿⣿⡟⢸⣿⡇⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡤⠀⢀⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⡇⣿⢸⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⡄⢸⡿⣀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠸⣧⣙⣿⣿⡟⠃⣿⡇⠀⠀⢠⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⡏⢸⣧⡟⡀⠀⠀⠀⠀⢠⣻⣿
⠠⠄⠀⠠⠤⢀⣀⠈⠛⣿⣿⣧⣴⠟⠀⠀⠀⠀⠀⠁⠀⠀⣀⠔⠒⠒⠒⠠⢄⣀⣿⠀⠸⣧⣸⡏⢰⡇⠀⠀⠀⠀⠀⣹⣋⠀⣀⣀⣀
⠀⠀⠀⢀⣠⡀⣴⣿⠒⣿⣿⡇⠀⠤⢀⣀⠤⠤⣤⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡷⠛⠁⢀⣠⠴⠞⣛⣛⣛⡛⠿⠿⠿
⢀⡀⠀⠘⢻⣷⣿⣋⣀⣿⣿⡶⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⢸⡇⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠉⠉⢿⡿⠀⠐⣿⣿⡿⠀⠀⠒⠚⠛⣛⣛⡳⠶⠶⠶⣦⣤⣤⣤⣶⠟⠛⠁⢀⣠⣴⣾⣿⣿⡿⠿⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣬⣭⣼⣷⣒⡲⠤⣤⣶⡾⠿⠟⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣶⡿⠿⠿⠷⠦⣤⡀
⢸⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣾⣿⠀⠈⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣶⣦⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠘⠀⠀⠀⢰⣶⡆⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⢠⣶⣦⠀⢸⣿⡗⠀⠀⠀⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣾⣿⡿⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⠀⠀⢠⣦⡼⢿⠛⣧⢀⣾⣿⡇⠀⢸⣿⡇⠀⠀⠘⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⣿⡋⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠁⢸⣿⣈⠀⠀⠈⣼⡇⠸⣧⡈⢸⣿⣿⠀⠀⢸⣿⠇⠀⢀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⡀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡙⠀⠀⠘⣻⡇⠀⢹⣧⣿⣿⡇⠀⠀⢹⡿⠇⠠⠀⠉⣾⠀⠀⢠⡶⣆⠀⠀⠀⠀⠀⠀⠈⢶⣤⡼⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⠀⠀⠀⢻⡇⠀⠀⠻⠙⠛⠁⠀⠀⣿⠦⠻⠀⣤⣴⠁⠀⠀⡸⢁⡇⠀⠀⠀⠀⠀⠀⠀⢸⣤⣆⠀⠀⠀⠀⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⠀⠀⠀⠀⠳⠀⠀⢀⡀⢀⢀⣀⠀⠁⣀⣠⣾⣿⣿⠀⠀⠀⣿⣧⠀⠀⠀⢀⣀⡀⠀⠀⠈⠉⠀⠀⠀⠠⡄⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣆⠀⠀⠀⠀⠀⢁⢸⡁⣵⡈⢿⣷⣾⣿⡏⢸⢿⡟⠀⠀⢰⡟⠛⠁⠀⠀⢸⠀⡹⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣽⡄⠀⠀⠀⡀⠀⠈⢻⣽⠇⡌⢹⣶⣿⡃⣼⢈⠁⣠⣦⣾⠿⡆⠀⠀⠀⠀⢷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢻⣴⣶⠀⠁⠀⠘⣿⠋⠀⠀⣾⠿⠛⡿⢻⠟⣠⣿⣿⣧⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠀⠀⠀⠀⠱⡄⠀⣬⠓⠀⡸⠳⣄⣠⣿⣿⠟⠂⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡌⠣⣄⡀⢀⠀⣡⠀⠘⢆⠁⠀⢰⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⣿⣧⢿⣴⡿⣶⠀⠼⣀⣠⡾⠟⠋⠀⠀⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣸⢇⡀⣽⡇⠀⠉⠁⠀⠀⣶⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣻⢿⣿⡟⠀⠈⡁⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⢈⢑⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠂⠄⠀⠀⠀⡀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠠⣦⣀⡀⠒⠛⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠣⠈⠁
⠀⠀⠀⠀⠉⠚⠈⠁⠀⠙⠻⠆⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠐⠆⠀⠀⣾⣧⠀⠀⠀⣀⣀⣀⣠⣄⠀⣀⠀⠀⠀⠀⢠⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠏⠙⠀⠀⠀⠀⠀⠈⠛⠁⠉⠉⠉⠁⠘⠀⣦⡍⡀⢸⣿⠀⠘⢉⡉⣉⣽⡇⠀⠂⡈⠁⠉⠙⢿⣻⡽⢽⢩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣼⣯⠀⣴⠯⣷⠆⢸⣿⠀⢀⣸⡥⣬⣿⡇⢀⢸⠰⣷⣶⢤⣾⣿⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠺⣿⣿⣿⣤⣷⢶⡿⣶⣜⠛⠃⢸⣿⠺⢿⣿⡿⢘⣼⠎⠉⢻⣤⣤⣿⣧⣼⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠤⣭⣙⣻⡇⣀⢻⣿⣥⡤⢾⣿⠄⣀⡙⠛⢛⡥⠀⣤⣤⣭⣿⣤⣴⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠛⠉⠁⠀⠀⠈⠙⠡⠭⠤⠈⠉⠀⠛⠋⠉⠉⣻⣎⣡⣴⣺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢉⣉⣉⣹⠇

⠀⠀⠀⠀⢤⣿⡀⠀⠀⡀⠈⢀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠡⠄⠀⠠⢀⠀⠀⢀
⠀⠀⠀⠀⠀⣇⠁⠂⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⡀⡀⡀⠉⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⢀⠀⠀⢄⠃⠀⠀⡀
⠀⢀⢠⡄⠠⡷⡤⠀⠘⠀⠀⠀⠁⢠⠀⢠⢀⠐⢃⠖⡄⠀⠀⠀⠄⠀⠀⠸⠀⠀⠀⢠⣶⡆⠈⠀⠀⢨⣄⡄⠀⠉
⠀⠈⢋⠀⣆⣧⠄⠒⠀⠰⢢⠀⢐⠀⠄⠘⠒⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⠪⡸⢐⢒⣼⡇⢁⠀⠀⠀⠀⠀⣀⡔⠓⡀
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠂⠈⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠙⠛⠚⠀⠀⠀⠀⠀⠓⠓⠚⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⡀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣾⣿⣿⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠞⠁⠀⠈⠑⡄⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣟⡻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡀⢀⠤⡀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢴⠈⠢⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⡇⣺⣿⢹⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⡆⠀⢿⠀⢰⠌⠢⡀
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣴⣿⣷⣾⣿⣼⣿⣿⣮⡻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠃⠀⠉⠀⠘⠃⠀⠈⠢⡀
⣿⣿⣿⣿⣿⣧⡻⣿⣿⡏⣸⣷⣿⣿⣿⢿⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠳⣀⠄⢀⣾⡇⠉⠥⢀⣀⠈⢇⠠⢀⠜
⣿⣿⣿⣿⣿⣿⣿⣮⠻⠇⢿⣿⣿⣿⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠲⣎⣿⠘⡡⠀⠈⠁⠀⠘⡖⠁
⣿⣿⣿⣿⣿⣿⣟⣫⣷⣄⡸⢯⠿⠿⠋⣉⣛⣋⣛⣛⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠤⠞⢹⠿⣅⣡⣀⣠⡶⠦⠤⣷⠤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠜⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡤⠤⠤⣤⣤⣄⣀⡜⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠋⠉⠉⠀⠉⠀⠀⠀⠀⠀⢏⠁⠀⠈⠃⢀⠀⠐⠢⢤⡖⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡤⠀⢰⠁⡀⠀⠀⠀⢳⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠉⠀⢱⡦⢶⡉⠀⠈⢻⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢃⣾⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢄⠀⡸⠧⡀⠀⣶⣤⣾⣟⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠏⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠒⠋⠹⡄⡋⠀⡱⢟⣆
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⣡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡡⢈⠔⠀⠙⢃⡀
⠀⠀⠀⠀⠀⢀⢜⠱⠁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⣾⣷⡃⠀⠀⠀⠈⢹⡄
⠀⠀⠀⠀⢠⠃⠈⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠈⣻⣿⣧⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⠸⡄
⠀⠀⠀⢠⡎⡰⠀⠀⠀⠀⠀⣿⡇⠀⠀⠰⠿⢷⣷⣖⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠞⠙⠛⢻⣇⢀⠀⠀⢹⣿⠀⢀
⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣀⣠⣤⣽⡿⠗⠀⠀⠀⠀⠺⣯⠝⠀⠀⠀⠀⢿⡁⠀⠀⠀⣸⣿⡧⠀⠄⢸⡿⡀⠁
⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠐⠘⣏⠄⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠘⠢⡖⠉⠳⠖⠃⠀⠀⠀⠙⠓⠲⠛⡿⢛⠃⠀⢸⣿⣣⡏⠠
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠊⠀⣀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣵⣿⡟⢽
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠶⠦⣤⣀⣀⢀⢀⣤⡖⠋⠉⠉⢢⣀⣴⠉⠁⠀⠉⠻⢶⣄⠀⢀⣀⠤⠔⠋⣹⡷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⣿⡟⠀⠀⠀⠀⠀⣿⡷⢀⡀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⢠⣿⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠸
⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣿⡏⡴⢀⢀⢠⣾⡟⢷⣿⣄⣄⣄⣤⠀⣹⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⡧⠔
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⢿⣿⣽⣹⣿⣵⠟⠏⠀⠈⠛⠿⣿⣿⣾⣿⣿⣯⠀⠀⠀⡸⡿⠁⠀⠀⢰⠇⠐⠀⠀⠀⠀⠀⠀⢠
⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠻⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡀⠀⢠⡇⡇⠀⠀⠀⢸⡤⠀⢀⠀⠀⠀⠀⡀⠸
⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⣿⣿⣥⣁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⡇⠀⢸⡇⠃⠀⠀⢀⡟⣠⡞⠁⠀⢀⠀⠀⠠⡾
⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⢀⣀⣴⣾⡿⣫⠺⢻⡇⠀⢸⡿⠇⠀⢀⣼⣻⠟⠀⢠⠖⢀⠀⠀⢰⠃
⠀⠀⠀⠈⢶⢂⡀⠀⠀⠀⠘⢂⣤⠀⠀⡀⣿⠀⠀⠀⣿⣿⡻⢃⠈⠟⠁⠉⠻⣿⣿⣿⠟⠁⠀⠑⠁⢸⠀⠀⠸⣷⣀⣠⣿⣿⣳⠞⠈⠁⠐⢁⠀⢀⠇
⠀⠀⠀⠀⠀⠁⠳⣤⠄⠀⠀⠀⠸⠳⠶⢿⠎⠀⠀⠀⠘⡿⡱⡡⠁⠀⠀⠀⡀⣿⣷⡏⠀⠀⠀⡔⢠⠏⠀⠀⠀⠹⣿⣟⣿⢿⢫⡐⣀⡤⠀⠟⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠉⠾⣱⢡⠂⠎⠘⣀⠿⡗⡣⢡⠂⢄⡴⠋⠀⠀⠀⠀⠀⠈⠛⠧⢭⢏⡴⢩⠤⠜⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠛⠈⠁⠀⠈⠁⠐⠐⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣴⣶⣶⣤⣀⣀⡀⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣽⣿⣿⡿⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⡿⠋⠿⣦⢤⣆⣼⢻⣿⣿⣿⣿⣿⣿⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠡⢤⡀⠀⠀⢉⡀⠈⠿⢿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠀⠈⠁⠀⠀⠋⢿⡶⠀⠀⢻⣿⣿⣿⡛⠡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣼⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣄⡀⠀⠀⠰⠐⠢⢀⣤⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠙⠛⠃⠀⠉⠉⠀⠈⠁⠀⠀⢸⣿⠁⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣶⣶⡤⠀⡠⠀⠀⠀⠀⠀⠀⡙⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢤⡀⠁⠁⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣬⣤⠠⠰⠃⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢠⡌⠃⣠⣦⣴⡦⢷⣾⣿⡟⠛⠻⢿⣿⣿⣵⣾⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣼⣿⣂⣭⣿⣿⣿⣿⣿⡟⢀⣾⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠉⠹⣿⢿⣿⣿⠇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠁⢀⣶⡀⠀⠛⠋⠀⣤⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⡟⠁⢸⣿⣤⣴⠆⠀⠀⢨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠦⠀⠀⠘⠋⠁⢀⣤⣤⣾⣿⣿⡿⠁⠀⠉⠙⠿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠂⠀⠀⠁⠴⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢀⣩⣍⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂⠀⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⢄⣀⣀⠉⠁⠀⠀⠀⠀⠀⣈⣉⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢙⠦⣈⠛⢦⠀⠀⢀⠴⣶⠶⢤⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠁⡎⣸⣿⣿⣆⠙⣿⣞⡾⣥⣿⣍⣉⣙⣒⠲⢤⣤⣄⢂⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⡆⢫⠙⣿⣟⠛⢋⢿⣝⢿⣿⣶⣌⠙⢷⣶⣖⠚⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃⣹⢿⣿⣷⠘⢧⡀⠛⠃⣸⡼⣯⡼⡟⢻⣿⣿⣦⣮⡻⣿⣃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣱⣾⣃⣏⣻⣿⣇⢀⣹⣧⠀⢡⠄⡏⠃⣿⠇⠟⢉⣿⣿⣿⣮⡻⣤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠴⠶⠶⠦⢬⡻⠿⢿⣶⡶⠚⡈⠀⣼⣿⡿⠄⠀⠉⢹⣿⣷⣽⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣤⣒⠢⡀⢀⡀⢄⠒⠤⠄⡴⢺⣿⠟⠋⠀⢀⣴⢻⡿⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣌⡑⠛⠦⠕⠒⠊⠀⠀⠈⠓⠂⠒⢫⡍⣨⢁⣾⢻⣿⣯⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢈⡏⣼⢿⣿⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡞⢆⠰⣤⣄⣀⠀⠀⠀⠀⠀⣉⣾⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢧⡀⠳⣜⠿⣮⠙⢶⣄⠀⠀⢈⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠋⣻⣷⣦⡈⣆⣼⣿⣷⣬⡛⠿⣄⢼⣿⣷⡄⠈⣺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠿⢿⣿⣿⢿⣿⣿⣿⣾⡿⡟⠿⣹⣷⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⡀⢀⡀⠀⠀⠛⣦⣹⣿⣷⠻⣿⣿⡈⠁⠉⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢶⣷⠀⠀⣿⣿⣿⣿⣯⠈⢉⣟⣿⡶⠒⢹⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⢀⣮⣿⠀⣷⣿⣿⣿⢿⡟⢰⠿⣫⣟⣡⣤⣼⣶⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⠏⣼⣿⢇⣿⡟⣼⡠⠴⣺⢿⣻⣿⣿⣿⠻⣿⣟⠀⠙⠆
⠀⠀⠀⠐⢶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⡶⠶⠀⠀⠀⣠⣴⣶⣿⣿⡿⠋⣰⡿⢋⣾⠟⡼⠉⠀⢘⣵⢿⣿⣿⠿⠏⠙⠈⠛⣿⡄
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⢂⣼⡿⣛⣽⣾⣿⠁⡈⠉⣴⡿⠃⠈⠀⠀⣰⠟⣵⠿⠟⠉⠀⠀⠀⠀⠘⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠫⣾⣿⠿⠋⠛⠛⠁⠸⠋⠀⠀⠀⠀⠈⠁⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠉⡏⠛⠛⠛⠒⠒⠒⠒⠶⠶⠤⠤⠤⠤⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣷⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢸⣿⡄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣼⡇⣇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣰⣿⢷⢹⡀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⡠⢲⡈⠀⢀⠎⠁⠀⡇⠀⠀⠀⠀⠀⢀⠞⠀⠀⢠⠇⡟⠸⡎⣇⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢰⠇⠈⠁⢠⡞⠀⠀⠀⠱⠄⠀⢀⠀⣠⠏⠀⠀⢀⡞⢸⠃⢀⢹⠸⡄⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⣾⠀⢠⠴⠋⡀⠄⠀⠠⠀⡀⠀⠀⣶⣃⣀⣀⣀⠜⠀⣼⠉⠈⠉⣇⠱⡀⠀⠀⠈⢣⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢻⠀⠀⢰⡞⠁⠀⣠⠂⢀⠃⠀⠀⠀⠀⠀⠀⠉⠐⠒⠃⠀⠀⠀⠘⠄⠐⠒⠶⠤⢤⣝⣦⡀⣸⡤⠖⠒⠛⠉⠉⠁
⠀⣀⣀⣀⣴⣾⣇⡀⢢⣀⣴⣞⣁⡔⣿⣷⣄⣰⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⢦⣈⣺⣷⣞⣉⡫⢽⣻⢽⣟⣿⠓⢻⣿⣾⠿⠷⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢳⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠘⣟⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠘⠓⡶⠀⠀⢰⡚⠃⠀⠀⠀⠀⠀⠀⠘⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠹⡄⠀⠀⢠⠞⠁⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢤⠄⠀⣤⠷⢀⡴⠃⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⡴⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠈⢆⠺⠖⢒⡿⠁⠀⠀⠀⣶⠲⠾⠆⠀⠀⠴⠿⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠈⢣⡴⠋⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢊⠝⠁⠀⠀⠐⡟⠚⢃⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⣰⡘⢿⠋⠀⠀⠀⠀⠀⠈⢫⡑⠚⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠼⠧⣴⠆⠀⠀⢀⡼⠃⠛⢮⣧⠀⠀⠀⠀⠀⢀⣠⣽⣿⡏⣧
⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢠⠞⠁⠀⠀⠀⠘⣿⠓⠒⠒⠋⠉⣵⠏⢸⠁⢸⡄
⠀⠀⠀⠀⠀⠴⠯⣤⠄⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⡾⠁⠀⣿⠀⠀⢻⡄
⠀⠀⠀⠀⣠⣶⣏⠤⠤⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢰⡇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢀⣾⣧⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⣠⣶⣁⠀⠀⠀⠀⠳⣼⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠑⣄
⠒⣀⠤⠒⠉⠀⢿⡈⠙⠲⠤⣤⠶⠿⠠⠤⣶⠟⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠸⡏⠙⠒⠤⣀⡀⠠
⠈⠳⠀⠀⠀⠀⠀⠉⠉⠒⠒⡦⠤⠤⣤⣾⣁⣀⣿⣀⣤⣤⡤⠤⠴⠿⠿⠿⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⢩⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣻⣧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣻⣾⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣻⣾⣿⣿⡿⠿⠿⢃⣀⣀⣀⣀⣤⠴⠶⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⢟⣩⣴⣶⣶⣾⣿⡿⣻⣿⡟⠡⡜⢿⣼⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶⣤⣀⠸⣿⣿⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⢷⢹⢸⡿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣮⡉⣸⣿⡿⣿⣿⣿⣿⣿⣿⡟⠛⠀⡟⠀⡟⡾⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣇⢹⣿⡇⣿⣿⣧⣈⣉⡙⢿⣷⣶⣤⣀⡀⠀⢙⣦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣛⣛⣻⣽⡔⢮⣅⣈⣛⣛⣫⣥⣾⣷⣮⣝⠻⢿⣿⡿⠿⠛⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠿⡿⣦⣝⡛⠿⠟⠛⢋⣛⣿⣿⡿⠿⠿⠟⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠉⠉⢉⣴⠟⠉⠉⠀⠙⠃⠀⠀⠀⠐⠳⣶⣶⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣼⡿⠿⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⣿⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⣤⣤⢀⣤⣤⣤⣤⢠⣤⡄⣤⣤⡤⣤⣤⢠⣤⣄⢠⣤⡄⢀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡌⠁⢸⣿⠘⢿⣷⣬⣉⠀⣿⣦⣼⣿⠀⣿⡇⢸⣿⣿⣦⣿⠀⣿⡏⢈⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⣾⣿⠼⡿⢶⣿⠿⢰⣿⡆⢼⣿⠤⣿⣧⢼⣿⠌⠿⣿⠀⠻⣿⡶⢿⡏

⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠿⠿⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⣲⡶⠖⠒⠲⢤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠙⣦⠞⠁⠀⠀⠀⠀⠈⠑⡾⢹⣗⢤⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠊⠈⠻⡟⢆⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡇⠀⠀⣤⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⣷⠘⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⠀⠛⠀⠀⠀⡾⣤⠀⠀⠉⠀⢀⡏⠀⡇⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⢼⠉⠙⢦⣀⡤⠊⢀⡼⠁⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈⠉⣿⠛⢿⡛⠳⣄⡀⣹⢀⡤⠊⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠙⠲⠾⣭⣒⣶⣟⣆⣀⣹⣤⡤⠜⠛⠁⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠓⢄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⣀⣀⡤⠖⠛⠛⠓⠤⣀⣀⣹⡀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣄⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⣸⣆⣀⣀⡀⢀⡀⠀⣠⠴⠒⠋⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠋⠉⠀⠀⠀⠀⢹⡏⠓⢻⢤⡀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣠⠾⢷⠚⠉⠉⠁
⠀⠀⠀⠀⡼⠁⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣄⠀⢰⠃⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣘⡆
⠀⠀⠀⠀⠱⣄⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠁⠀⠀⠀⡗⠘⣆⠀⠈⠉⠀⢀⣠⡄⠀⠀⣴⡆⠀⢹⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⣄⣀⣤⣤⣄⣀⣯⡈⣧⡤⢤⣀⡤⠞⢡⠤⠼⣦⣀⠀⠀⠀⠉⠉⠀⠀⠉⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡴⠚⠉⠀⠀⠀⠘⢧⣬⡿⡷⣤⡤⠞⠀⠀⠙⢦⣀⡀⣸⠉⠓⠒⠶⠤⠤⣤⢤⣖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠠⣏⣸⠳⣌⡇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡤⢤⡀⢸⣀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠈⣇⠀⠀⠀⠀⠀⣀⡤⠽⠶⢲⡏⠀⠀⢻⠛⠛⠛⠫⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠰⢤⠔⠀⠀⢠⠀⠀⠀⠿⠀⠀⡿⠀⠀⢀⠔⠋⠁⠀⠀⠀⠈⠧⣄⣠⠞⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣰⠃⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⠤⣤⣤⡤⠤⢤⠒⢉⣸⣀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⠀⢧⠀⠙⢦⠈⣷⣋⡥⠤⣷⠀⣧⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢒⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡿⢤⡀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢼⣆⣀⠀⣀⣤⣶⣿⣟⣛⣿⣿⣿⣿⣿⡟⢿⣷⣶⡅⠀⠀⠀⠀⢀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⣼⣷⡘⢷⣽⡿⠿⠿⣛⡋⡉⠛⠛⠻⣾⣿⠏⣴⣿⣶⣄⠀⠀⠓⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣞⠉⠀⣴⣿⣿⣿⡿⠈⠁⠀⠀⠀⠀⠀⠈⣉⣉⣉⣡⣄⣜⣷⣿⣿⣿⣷⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡁⢈⣧⣾⣷⣾⣿⡯⣄⡀⠀⠀⣠⠴⡒⡋⠉⠉⠀⠈⠠⠀⠀⣉⣻⣿⠿⢿⡿⢿⡿⣿⡿⢟⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢍⣉⣿⣿⡿⠋⠀⠀⠉⣽⠟⠉⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠀⠀⠞⠐⣛⠛⠈⢀⡼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⢠⠀⣱⠟⠇⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⠀⠠⣄⡈⣿⣦⣾⣥⣇⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢫⠀⣒⠿⡈⠠⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⣛⠶⣄⠈⠁⠉⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢦⣄⡠⠀⠀⠀⣤⠀⠀⣰⡦⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡷⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⢖⣛⣿⡦⠖⢻⣿⣿⣿⠦⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡿⠉⣹⠃⠀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⡀⢰⢖⡓⢲⠶⠚⠁⠀⢰⠇⣰⠏⢇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⣿⣿⣿⣷⣀⠀⡀⡠⠚⠙⡟⡇⡿⠀⠀⠀⠀⢸⢞⣅⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⢻⣿⣿⣿⣷⣾⣄⡜⡇⢀⣿⠇⠠⣀⠀⣀⡄⠀⢻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⡱⢄⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣆⠀⢹⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡐⠀⡄⡌⠙⢻⣶⣯⣿⠿⢿⣿⡿⠿⠿⣿⣿⣿⠿⠛⡿⠲⠁⠀⡠⡀⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢳⠮⠥⡘⠃⠀⠀⠀⠙⢿⣷⣆⠈⠚⢿⣿⣿⣿⡿⠁⠀⡰⠃⠀⠀⠀⠈⠛⣀⠭⢶⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⣤⠀⠀⠀⠀⠙⢻⡄⠀⠈⠹⣿⣿⠀⠐⡏⠀⠀⠀⠀⢀⡔⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣥⠀⠀⠀⠀⠀⠀⠿⠒⠦⣤⣿⢟⠁⠀⣧⠀⠀⠀⠀⢤⡁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠻⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⠉⠈⣷⠈⠉⣷⠀⡔⠁⠈⠉⢳⣠⠊⠈⠙⣦⠀⠀⡎⠉⢹⡆⠈⢱⣄⠊⠉⠉⢧⠀⠀⡊⠉⠱⡆⠉⢹⣆⠐⠉⠉⢳⡀
⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⡟⠲⣧⣀⠐⠲⢟⡈⠀⢀⠀⠸⡆⠐⡇⠀⠈⠁⠀⢸⡝⠀⢀⠀⠘⡇⠐⡇⠀⠀⠀⠀⢸⡟⠀⠀⡀⠈⣇
⠀⠀⠀⠀⠸⡀⠀⠀⠀⢄⡿⢾⠁⠈⠀⢀⠘⠀⠀⢈⠀⠀⣹⣾⠃⠀⢸⠆⢀⠸⠁⠀⢈⠀⠀⣸⣾⠃⠀⢠⠀⠀⠸⠃⠀⠈⠁⠀⡸⣆
⠀⠀⠀⠀⠀⠹⠶⣤⡶⠟⠁⠈⠳⢦⣬⡶⠿⣦⡶⠋⠳⣤⡾⠛⢶⣤⠾⢧⣤⡾⢦⣴⠟⠳⣤⡾⠟⢷⣤⡾⢷⣤⡾⣦⣤⠟⠻⣤⣴⠏

⠀⠀⠀⠀⠀⣠⡤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢄⡀⠀⠀⠀⠀⠀⠀⡠⠂⠙⠂⢄
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠶⣤⡀
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠙⠢⠤⠔⠁⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠲⡀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠉⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢹⣆⣤⡤⠤⢤⣄⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⡴⠶⠛⠛⠛⠛⠛⣷⠞⠛⠛⠷⠀⣀⣘⠋⠀⠀⠀⠀⠈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣠⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⢰⣏⣩⡇⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠙⢧⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⡶⠀⠉⣉⠀⠙⠻⣿⡉⠁
⠀⠀⠀⢀⠔⠉⠈⠀⠙⠦⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢠⡿⣆⠀⠀⠈⣷
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣠⡞⠁⠙⠳⠶⣾⠃⠀⠀⠀⡀⠤⢀
⠀⠀⠀⠰⡀⠀⠀⠀⠀⢠⠆⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⡆⠀⠀⡊⠀⠀⠀⢁
⠀⠀⠀⠀⠈⠒⠤⠤⠚⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⢿⠀⠀⠣⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⢰⣿⠆⢀⡀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠛⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡠⠔⠂⠒⠒⢤⡀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁⢠⠎⠁⠀⠀⠀⠙⢆
⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⠛⠶⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⢾⡏⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢠⡟⠉⠙⠻⣧⡀⠉⠉⠉⠉⣡⠟⠁⠀⠀⣿⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠈⠻⣦⠀⠀⣴⠋⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⠢⢀⡀⢀⣀⠠⠊
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⣿⠻⣆⠀⠀⠀⠀⠈⣷⡼⠃⠀⠀⠀⢀⡾⢻⡇⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⢷⣄⠀⠀⠀⢸⣇⠀⠀⠀⣠⡞⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣆⠀⠙⠳⢦⡴⠞⠙⠶⠴⠞⠋⠀⣰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠈⠛⠶⣤⣀⣀⣀⣀⣀⣀⣠⣴⠞⠁⠙⢷⣄⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠤⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡾⠉⠉⠉⠉⠉⠹⣇⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠀⠞⠁⠀⠈⠱
⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠘⡄⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢄⠀⠀⠀⣰
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⣦⡀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠑⠐⠂
⠀⠀⠀⠀⠀⠡⢀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠈⠙⠓⠲⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⣤⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⢀⡀⠀⡀⠀⡀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⢠⡝⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠉⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿
⠀⠀⣸⣿⠈⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⡄⠀⢠⣿⣿⡏⠠⠄⠘⠣⡀⠉⠀⠀⠀⠀⠀⠀⣷⠀⠀⡀⠀⠀⠀⣺⣧⡿
⠀⣩⣿⠙⠉⡀⠠⠀⠀⢼⠇⠀⠀⣠⠾⠀⠀⠀⠺⣿⡗⠀⠀⠀⣼⣇⠀⠀⠿⢿⣿⣆⠀⠘⢀⠀⠀⣰⠀⠀⠀⠀⢸⡿⠀⠻⠅⠀⠀⢨⣿⣟⡿
⢀⣾⡿⠈⠐⠀⠀⠀⢠⡟⠈⡀⠺⢿⣦⠀⠀⠀⠀⠈⢡⣧⠀⢈⣿⠧⠀⢠⢠⣀⢀⠁⣿⠀⣄⠀⢠⣿⡆⠀⣀⠀⣿⡿⠅⠀⢠⡆⠐⢾⣿⣿⣭
⣿⣿⣯⣟⠛⠁⢀⠀⣼⡏⠀⢀⣠⡇⠉⠛⠀⠀⠀⡦⢸⣿⣤⣽⣿⠗⣂⣿⣼⣿⣿⣴⣏⣸⣷⣾⣾⣿⣫⣶⣿⣴⣿⣣⡀⣸⣌⣷⣼⣿⣿⣻⡟
⣿⠛⠁⠛⠒⠀⠀⠀⣼⣷⣴⠄⠉⠁⠀⠀⠀⠀⠀⠀⣾⢿⣾⣿⣭⣥⡴⠛⠃⠀⠉⠿⠿⣻⠈⠉⢻⣿⣶⡟⠻⠛⠛⠿⠶⢿⣿⣿⣿⣿⡿⢻⡇
⢫⢦⠀⡀⠀⠲⠎⠀⠹⢻⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠻⠛⠷⠛⠋⠀⢀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢥⣾⣧
⡡⣄⠈⡱⠓⠀⠀⡀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⣿⡾⠉⣺
⠀⠃⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠙
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⠀⠀⠐⠀⠠⠤⠠⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⣴⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠉⠉
⣀⡈⣀⡌⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣀⣀⣀⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⣀⣤⡆
⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡿⠟⠛⠋⠉⠛⠛⠛⠿⠶⢒⣶⣶⣶⣾⣿⣶⣄⣴⣦⣤⣤⣀⡀⠀⠀⠲⣤⣤⣀⣹⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣴⣤⣀⣴⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣤⡄⠉⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣧⣴⣾⣿⣿⣿⣿⣿⣿⣷
⣠⣀⣄⣤⣀⣠⣠⣴⣤⣾⣿⣿⢿⣿⠿⠿⠻⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⣦⣤⣄⣠⣤⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠙⠿⢿
⠿⠟⢻⣿⣿⣿⣿⣿⣿⠟⠳⠲⠄⠀⠀⠀⠀⠐⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋
⣾⣿⡿⠿⠿⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠔⢡⠔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠎⡐⢁⠞⡠⣊⢔⡁⢒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣰⣘⣃⣃⣊⣐⣡⣊⣊⠕⡰⠲⠕⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣴⣲⠟⢹⣿⡿⡪⠋⠛⣽⣽⡝⡟⠹⣷⣷⣯⣀⣠⡾⠪⣂⢐⢌⢂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⢀⣤⠔⠛⠿⠱⠑⡀⡢⠙⣿⡇⡄⠜⣾⣿⣟⣄⣜⣿⣿⣻⡟⠊⠻⣿⣷⡶⢵⣯⡇⠀⠀⠀⠀⠀⠀⣠⣶⣫⡂
⢀⣠⠖⣁⠣⠖⠇⠀⠀⠀⠀⠍⠉⠎⢅⢄⠌⢜⣿⣞⢀⠈⠜⣿⣽⡟⠔⠴⣽⣽⣟⣔⢜⣿⣿⢁⣦⣿⣷⣘⡮⣿⣥⠂⢀
⠫⣁⠑⢬⡁⣧⢠⠆⠀⠂⡙⠂⠃⣀⢂⡎⠊⠊⠻⠏⠔⠐⡠⣻⣿⠏⡊⡀⣾⡿⣫⠎⠨⠛⠋⠉⣉⢉⢩⣉⡡⣿⣆⠡
⢀⠈⣢⢤⠳⡈⠜⣠⠤⠈⠀⡠⢠⡾⠛⠇⠀⠀⡐⠄⡸⡈⢀⡿⠋⠈⢁⠁⠙⠈⢄⢠⣢⣥⢾⣉⠀⠀⠀⠀⠙⠯⣍⠒⡠
⠀⠀⠈⠁⠉⠐⠲⠤⢤⡄⠀⠐⠰⡋⠅⠲⡀⠄⡠⢑⠀⠀⠎⡈⣖⣱⣕⡤⠴⠚⠻⠛⠈⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢺⣷⠖⣚⣒⠛⠒⠓⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣲⡽⡟⢌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠘⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⣀⢀⡠⠞⠀⠀⠀⡤⠒⠈⠉⠁⠒⠤⡀⢀⡠⠔⠒⠀⠐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣜⣡⠖⡠⠒⠒⢤⠘⠎⡠⠄⠤⣰⡜⡗⠊
⠀⠀⠀⠀⠀⠀⠀⡎⠈⠁⠀⠉⠉⢳⠉⠉⠀⠈⠋⠙⡄⠀⠀⢀⠚⠀⠀⠀⠀⠡⠰⠁⠀⠀⢀⣈
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠉⡆⠀⠛⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⡠⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠤⣀⠔⠈⠣⣀⡠⣄⡠⠂⢉⡀⢤⣤⣶⣶⠴⠶⠶⢿⡿⠴⠶⠶⣶⣆⣤⡤⢄⠒⠒⠒⠂
⠀⠀⠀⠀⣠⠖⠛⠓⠲⢮⣀⣰⠐⠒⠒⠘⣇⣀⡏⠉⡧⣐⠥⣐⣢⣭⠓⠂⠀⠀⠁⠀⠐⠚⢭⣔⣂⡽⣂⢽
⠀⠐⠒⢸⠁⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⠀⠀⠳⠤⣈⡉⠉⠋⠉⠉⠉⠋⠉⠉⠉⣉⣩⠥⠬⠥⠤⣜
⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢠⠞⠳⡖⠋⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠈⢦⡤⠴⠒⠋⠉⠉⠉⠉⢒⡶⠤⣼⡀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠈⣡⠶⡍⠉⠉⠙⢦⠤⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⢈⡋⠰⣤⢤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡤⠋⢧⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⡇⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠉⠀⠀⠀⠐⠦⠿⠥⠞⠀⠀⠀⠉⠀⠀⠀⣸⠁⠀⢀⠒⢺⣀⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣃⠀⠀⠀⡗⠘⡚⠰⡊
⠀⠀⠀⠀⢀⠄⢄⠀⠀⠀⠀⣴⠋⠉⠙⠶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⠞⠋⠉⠈⢹⡄⠀⡬⣛⣟⣈⡀
⠀⠀⠀⢎⠉⡖⡞⢘⡆⠀⠀⠘⢦⡀⠀⠀⢈⡏⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢇⠀⠀⣀⡤⠊⠀⠀⠑⠚⠛⠒⠁
⠀⠀⠀⠸⣁⣭⡀⡏⠀⠀⠀⠀⠀⠉⠓⠦⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠋⠁
⠀⠀⠀⢔⢢⣿⠴⡢⠀⡀⣀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠉⠉⠉⡔⠒⡇⡸⠶⡄⠀⠀⠀⢀⡼⠲⢤⣀⣀⠀⠀⢀⣀⡤⠺⣄
⠀⠀⠀⠀⠀⠀⠀⢨⠂⣑⠃⢶⠁⠀⢀⡴⠋⠀⠀⠀⢠⠏⠉⠉⢹⡀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⣠⢍⣹⣃⢅⠀⠀⣾⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠓⠊⠀⠀⠙⠦⣤⣀⡤⠞⠀⠀⠀⠀⠀⠈⠳⢤⣀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠠⢄⠀⠀⠀⣴⠖⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡀
⠀⠀⢀⡴⠒⠐⠒⠜⣀⠤⣀⠀⢱⠀⠘⣇⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠛⠒⢦⠀⠀⠱⡀⢀⠎⢁⡨⢟⡤⠦⡀⠡
⠀⠀⢸⠀⢠⠖⠉⠒⠁⠀⢈⢆⡜⠀⠀⠈⠳⣄⠀⠀⠀⠘⢧⡀⠀⣀⣀⣀⣀⣀⣠⡞⠁⠀⠀⠀⢀⡾⠉⠢⡀⠗⠚⢦⣸⡒⢄⠀⢠⡷⠃
⠀⢀⠬⢦⣘⢦⡀⠀⠀⢀⢮⢎⡤⠤⠤⢄⡀⠈⠛⢦⡀⣠⡼⠛⠋⠉⠉⢿⡀⠙⡵⠿⣆⣠⢶⠺⡉⠀⠀⠀⠓⠘⠓⠢⡈⠙⠻⣷⠋
⠀⠃⡠⠤⣝⡋⣉⠉⢶⠕⡱⠁⠀⠀⠀⠀⠙⡄⠀⢈⡟⠁⠀⠀⠀⠀⠀⠀⠙⠶⣧⠀⠉⢹⡆⢇⢣⠀⠀⢀⠤⠤⢄⢀⠇⠀⢀⠇
⠀⠳⣧⠀⢈⠔⣺⣀⠞⠒⠇⡠⠒⠀⠒⡄⠀⡐⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡏⠁⠈⢧⠣⣰⣅⠠⠤⢔⣋⣠⡔⠋⠑⢆
⠀⠀⠈⠳⡾⠞⠋⢡⠖⠒⠒⠇⠀⠀⠀⢸⢠⠃⠸⡇⠀⠰⡿⠀⠀⢀⡀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠑⠒⢆⢛⠉⠁⠙⠉⠈⢱⠀⢸
⠀⠀⠀⠀⢇⠀⠀⣇⢀⣀⣀⠀⠀⠀⢀⢇⠎⠀⠀⢷⡀⢀⡀⠀⠀⠊⠓⠁⠀⢀⣀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠪⡢⡀⠀⢀⣠⡪⠔⠁
⠀⠀⢀⠔⠊⠓⢤⣈⡓⠤⠄⣙⣄⣠⢎⠎⢀⡠⠄⠒⢿⠉⠉⣧⣤⣀⣀⣀⣰⡏⠉⣻⠃⠤⢄⡀⠀⠀⠀⠀⠀⠈⠻⡿⠯⣩⠎⣁⡈⢢
⠀⠀⡇⠀⡰⠂⠒⠞⠁⠉⢻⢐⠖⠒⠁⡔⠁⠀⠀⠀⠈⠛⠛⠁⠑⣄⡠⠊⠀⠉⠛⠁⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⡎⠀⡔⠒⠊⠀⢹⡜
⠀⠀⠘⢤⡱⣀⠀⠀⠀⣠⡢⠊⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡠⠒⠉⠛⢦⣷⣾⣉⣩⠟⢢
⠀⠀⠀⠀⠈⠙⢻⠶⠞⢏⠀⠀⠀⠀⡇⠐⢧⢠⠀⠐⣷⡄⢠⡄⠀⠺⡄⡄⢠⡀⠀⠺⡄⡄⢠⡄⠀⢸⢸⠁⠀⡠⠒⠢⣥⠚⠙⠣⡀⠀⡇
⠀⠀⢀⣀⢀⣀⠾⢠⠒⢢⣓⣊⠑⢢⢣⠀⢸⡼⠈⠁⢼⡆⡟⠇⠉⠀⡟⠇⡟⠇⠉⠠⠿⠇⡟⠇⠀⡼⠨⢦⡀⣇⠀⠀⠀⠀⠀⢠⢃⠞
⠀⡰⠁⠀⠀⠀⠹⣿⡉⠀⠉⢙⠇⣸⠀⠣⡀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠜⣀⠤⣀⠻⠞⠃⠒⢦⣠⣶⠕⠁
⠀⢇⠀⡔⠁⠈⠱⠞⠻⠷⣾⢛⣩⣈⢧⡀⣈⠢⡀⠀⠀⢏⣸⠀⠐⣇⡹⠀⠸⣁⡝⠀⠀⢀⠔⢣⠰⡁⠀⠈⠒⠉⠲⣄⣀⣏⠞⣁⡈⠳
⠀⠈⠣⡱⣄⠀⠀⠀⠀⣠⢛⣾⠁⠘⠒⠲⡀⢱⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠱⣷⡀⠀⠀⢀⣴⠉⢠⠜⠾⠁⢈⡶
⠀⠀⠀⠈⠺⢷⡤⢔⡪⠒⠁⠹⣦⣤⣤⣜⠥⠊⠀⠀⠀⠉⠢⢄⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠈⠪⠶⠟⠛⠙⢤⣰⣤⣀⣀⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣀⣀⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠠⡀⠀⠑⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡟⠉⣶⣄⣻⣯⣿⠿⢟⣻⡦⠀⠀⠰⡆⠀⠄⠀⢨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡇⠀⢿⣿⣿⣏⠀⢸⣽⣽⠇⠀⠀⠀⠀⠀⠸⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡰⣿⣿⣦⣀⣬⣙⣛⣷⠿⠋⠁⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠙⣻⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠻⢿⣿⣿⠟⠛⠋⣀⣤⠞⠋⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣀⣤⠤⠴⢶⣯⣵⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⣿⣿⣿⣷⡕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⣿⣿⣿⣿⣿⡟⣻⠷⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡹⣿⣿⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢁⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⠙⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⡌⠀⠹⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠐⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣷⣆⣶⣶⣶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣧⠺⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣆⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣟⠿⣿⡿⠿⠿⢯⣭⣿⣿⣾⣿⣮⣜⡿⣆⠀⢉⣦⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣒⠚⠓⠈⢀⣀⣀⡼⢥⣿⣿⠼⠋⠛⠻⢿⣼⡄⠀⢀⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢠⣾⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠒⠒⠢⣤⣤⠰⢿⣿⡦⠌⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣤⣤⣤⡾⠙⠏⢀⣼⠅⠀⠀⠒⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣀⡔⢿⣿⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⡴⠛⠉⠉⠉⠉⠛⠒⠶⢤⣀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠒⠛⠋⠛⠛⠒⡦⠾⢤⡴⠒⢦⡀⣠⠖⠋
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠁⠀⠼⢟⠁
⠀⠈⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠁⢀⡀⠀⢀⣀⣠⠇
⠀⠀⠀⠀⠉⠙⠓⠒⠦⢤⣄⣀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣇⣀⡼⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠨⠅⠠⠀⠀⠀⠈⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢠⡴⠲⢬⠈⠀⠁⡞⠋⠳⣄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⣰⠋⠀⠀⣸⠃⠀⣈⣇⠀⠀⠈⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⢠⠏⠉⠉⠉⠙⣆⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣠⠏⠀⠀⠀⠀⠀⠘⢦⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⣄⣀⡀⡀⢀⣀⣠⡴⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⢹⠉⣹⠃⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⣿⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠛⠁⠙⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠠⣷⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣏⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⢶⣧⡗⠀⢠⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠗⠀⠀⠘⢷⣿⠷⠀⣈⣻⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣮⣤⠄⣀⣿⣷⠀⠀⠀⠀⣘⣿⣿⣿⣟⠀⠀⣸⣿⣿⣟⣛⢬⣿⣿⣿⣿⣥⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⡶⠂⢻⣿⣿⠗⠀⠀⠀⢯⣿⣿⣿⠟⠀⠀⣈⣿⣿⣿⣁⡈⢿⣿⣿⣷⡶⣀⠀⣾⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣷⣶⣤⣻⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⠄⠀⠽⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣯⣜⣻⣟⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣭⣴⣾⣿⣿⠏⢀⣶⣤⣍⡙⠿⠿⣯⠀⠰⠾⣿⣿⣿⣟⡽⣿⣿⣿⣟⣯⣉⢼⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡟⣡⢻⣿⣿⣿⣿⣿⣶⣦⣤⣤⣌⣉⣉⣭⣥⣶⣦⠙⢿⣿⣿⣷⣶⣿⣿⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠶⣾⣿⣿⣿⣿⣿⣿⠟⡴⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡕⢌⠹⣤⣴⣽⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⣿⣿⢋⠞⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠳⣄⢶⣾⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⡿⢡⢎⡄⣸⣿⣿⢹⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣆⠈⠣⡘⢿⣿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⡰⢃⡞⣰⣿⣿⡟⢸⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠠⡙⢦⡹⣿⣯⣭
⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⢏⡼⣡⠟⣴⣿⣿⣿⠇⢸⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢦⡑⣌⠫⣤
⠀⠀⠀⠀⠀⠀⠀⠛⣿⡿⢃⠞⡴⢃⣾⣿⣿⣿⡟⠀⠈⣿⣿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣯⣭⣉⣉⣉⣥⣾⣿⣿⣿⣧⡙⠎⠳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠋⢈⣴⣿⣿⣿⣿⡟⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠰⠿⠿⢿⣿⣿⠟⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢀⡾⠉⠛⢠⡦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⢿⣿⠿⢿⡟⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⡄⢿⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠘⢧⣄⢀⣾⠁⢠⡿⢸⣗⢀⣴⡆⣤⡀⢀⣀⢀⣀⣀⠀⣀⡀⠀⠀⠀⠈⠁⣀⣤⡀⠀⢰⣦⠀⠈⣿⣦⣿⢿⡘⣧⣀⣿⠆
⠀⠀⠀⠀⠀⠘⠳⠶⠏⣼⡇⠀⣼⢃⡿⢻⡾⣿⢡⣿⣧⣾⣿⢸⡏⠉⠁⣿⠙⣧⠀⠀⠀⢸⡇⠀⠀⠀⢸⡟⣷⡀⢹⡎⠛⠘⣷⠹⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠛⠛⠋⢼⠇⠈⢠⡟⣸⠇⠻⢱⡇⣼⣧⣤⠀⣿⡀⣿⠄⠀⠀⢸⡇⠀⠀⠀⣸⡷⠟⢷⡌⠿⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣦⡄⠀⠀⠀⠈⠁⠛⠀⠀⠸⠇⠿⠦⣤⠄⡿⠙⢿⡄⠀⠀⠀⠛⠶⠛⠃⠙⠁⠀⠀⠀⣤⣶⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣶⢶⣶⣶⣶⣶⣤⣤⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣽⣎⣷⣟⣿⣿⣿⡿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⣿⠛⡿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⢠⣟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠉⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿
⠀⢀⣼⣿⠈⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⢀⡆⠀⢰⣿⣿⡇⠀⠄⠈⠣⡀⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⣺⣧⡿
⠀⠽⣿⠍⠉⠀⠠⠀⠀⢼⠃⠀⠀⣠⠞⠀⠀⠀⠚⣿⡟⠀⠀⠀⣼⣇⠀⠀⠟⠿⢿⡆⠀⠘⠀⠀⠀⣰⠀⠀⠀⠀⣸⡿⠀⠻⠄⠀⠀⢬⣿⣟⡿
⢠⣿⡟⠈⠐⠀⠀⠀⢠⣍⣀⡀⠺⢻⣦⠀⠠⠀⠀⠈⢠⡧⠀⣨⡿⠧⠀⢠⢠⣀⣠⠀⣿⠀⡄⠀⢰⣿⠂⠀⣀⠀⣿⡿⠅⢀⢠⡆⢐⢻⣿⣿⣤
⣿⣿⣷⣟⠉⠁⠀⠀⢾⡿⠂⢀⣰⡇⠈⠛⠀⠀⠀⠦⢾⣿⠤⣿⣿⠏⢀⣿⡼⣿⣿⣼⣿⣼⣿⣾⣾⣿⣱⣷⣿⣼⣿⣿⣂⣼⣾⣷⣼⣿⣿⣽⡏
⣿⠛⠁⠋⡂⠀⠀⠀⣺⣿⣶⠄⠈⠉⠀⠀⠀⠀⠀⠀⣾⢿⣾⣿⣭⣥⡾⣛⠁⠀⠀⠿⠟⣽⠀⠉⢻⣿⣿⡏⠙⠉⠛⠛⠷⠿⠿⢿⣿⣿⡿⢻⡇
⣩⢶⢀⡀⠀⠙⠂⠀⠙⢻⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⠛⠷⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢤⣿⣧
⡴⠤⠈⡿⠃⠀⠀⣀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠉⣺
⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⢀⡀⠀⠀⠐⠂⠀⠦⠐⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⣾⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉
⣴⡿⣦⠎⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠘⣂⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠦⠄⠒⠀⠀⠀⠈⠀⠀⠀⠀⣠⣠⣶⡆
⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⡟⠛⠉⠉⠉⠉⠉⠉⠛⠛⠫⣿⣿⣿⣿⣿⣷⣤⣶⣶⣦⣶⣄⣀⡀⠀⠳⣦⣦⣤⣼⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣷⣴⣠⣾⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⠶⠄⠈⠉⡉⠛⠛⠛⠿⠿⠿⠿⠿⢷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣤⣦⣤⣤⣴⣴⣾⣶⣿⡿⠿⠿⠿⠿⠟⠛⠀⠀⣠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣷⣶⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻
⠛⠛⠻⡿⠿⢿⣿⣿⡿⠛⠐⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⠤⠄⡘⠉⢰
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠚⢶⠖⠐⣀⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⢀
⠀⠀⠀⠀⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣧⡀
⠀⠀⢠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡄
⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄
⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣧
⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣿⣿⣿⣿
⣿⣿⣿⡟⠀⠀⠀⢸⣀⣁⠀⠄⢀⣠⣾⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⢻⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠀⠀⣾⣿⣿⣿⣿⡿⠧⠤⠤⠤⠤⠤⢠⣶⣄⠈⠀⢻⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢸⠀⠁⠀⠀⠻⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣼⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⡿⠿⡿⠟⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⢴⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⢀⢀⡐⠊⠉⠉⠙⠛⠳⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣴⣶⡆⠱⠀⠀⠀⠀⡰⠁⢸⡵⡀⠀⡀⠀⠀⠀⠀⠈⢫⣠
⠀⠀⠀⠀⢀⣤⡼⠫⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠸⠿⠟⠀⣶⠀⠀⠀⡔⠀⠀⢸⠈⢺⡜⣷⣦⠀⠀⣦⠀⢐⡾
⠀⢀⣠⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠓⠄⠀⢀⣴⠃⢈⣂⠀⣰⠁⠀⠠⠚⢄⠉⠧⠻⣯⣵⣄⣿⣰⣿⠃
⠀⠋⣏⠀⠀⠀⠀⠰⢄⠀⠀⠀⢜⠋⠛⠻⡄⠀⠀⠀⣀⡈⢲⠟⣧⠀⠈⠛⢃⠃⠀⠈⠀⠀⠈⠉⡕⢰⣶⣘⡊⣀⠙⣏
⠀⠀⠘⡓⢤⠀⠀⢀⠤⣾⢦⡀⠘⡄⠀⠀⡀⠀⠀⠀⠀⠉⠁⠁⣌⠀⠢⡀⠘⠁⠀⢠⣤⢄⣀⠀⢇⠈⡿⠿⢿⣿⠋⠉
⠀⠀⠀⣿⣦⠤⣶⣥⣴⠟⢠⣵⡄⠵⢀⣾⢟⠯⢧⡀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠄⠈⠃⠀⠀⢱⠀⠑⡽⠾⢾⣿⠄
⠀⠀⠀⠻⠇⣠⠖⠁⠀⠀⡸⠟⠁⣠⡞⠉⡀⠀⠀⠈⣿⡶⢤⡈⠀⠀⠀⠀⡶⠉⠢⣤⠎⠉⠡⠄⠀⠀⠀⠉⠒⠚
⠀⠀⠀⠀⠋⢹⡀⢘⡰⠊⠀⠀⣰⡟⠁⠀⠈⢆⣇⠀⢙⡛⠗⠛⠶⣤⣤⣤⡿⠦⠖⠈
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣼⡟⠀⠀⠀⠀⠘⣿⣶⣬⡓⢮⣏⠒⠶⡿⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠘⠀⢀⡠⠚⠉⠀⠀⠀⠀⣀⡠⠤⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣟⣽⢠⠖⠋⠉⠀⢀⡤⠋⠀⠀⠀⡠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⠺⢎⣥⣾⢜⣓⠐⢲⡤⠾⢧⣀⣀⣴⠭⠤⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠢⠜⠃⣇⣙⣦⣿⣿⡍⠒⠛⠯⡀⠀⠀⠈⠙⠒⢢⠖⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⢻⠀⠀⠀⣰⠀⠀⠀⡎⢹⡾⢿⣏⠨⣜⢒⣀⣘⣢⣄⣀⠤⠺⠥⣤⡤⢤⢤⣀⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠁⠂⢠⣾⠀⠐⠈⠉⠀⠀⢸⢷⠋⣰⢧⡙⢦⠀⠳⣄⡀⠙⣎⠈⣆⠀⠀⠀⠈⠂⣄⠀⠀⠈⠁⠐⠢⠤⢑⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠋⠀⠀⠀⠀⠈⡶⠶⠤⠌⠇⠀⠀⠀⠀⠀⣐⠷⠓⣹⣁⣸⡿⠦⢶⡶⠿⢿⣦⡽⢢⡿⠙⠢⣀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢠⡉⡀⠀⣀⠀⢀⠔⢲⣦⡎⣡⠖⠚⠊⠈⡳⣀⠀⢀⡼⠹⢨⣷⢩⡇⠸⣿⠀⢸⣿⠀⢀⠏⢉⡾⠀⠀⠀⠀⠑⢄⠀⠀⢃
⠀⠀⠠⠀⠀⠉⠉⡾⡀⢨⢂⡔⠓⢀⠠⢤⠀⢰⡗⠃⣹⠀⣪⣟⢿⢾⢿⠈⢹⡄⠿⠀⠀⠛⠶⢷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⡂
⠀⠠⠀⠀⠀⠀⡌⠀⢸⠈⠫⣀⠀⠘⢦⣸⣂⠞⢀⢅⡝⢻⠘⡞⣋⡸⠈⣳⠾⣆⠀⠘⢋⣀⣠⣿⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢱
⠀⢂⡴⠊⠉⠛⠃⠀⠀⠙⠉⠉⠙⠒⠲⣿⡿⢦⡼⠛⢝⡞⠃⠜⣭⢅⣃⡙⢖⣄⠀⠉⣿⡿⣷⣇⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠃⠀⠀⠀⠀⠷⠦⡀⣀⡷⠈⡟⠉⣹⢓⡟⣁⡠⠾⣷⠞⠙⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠒⠤⠄⠘⣄⣀⣧⣤⡴⠷⠿⢷⡟⠋⠀⠀⡼⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣦⣀⠘⠲⢂⡴⡫⠟⢯⣀⣴⠥⠔⠚⣷⠶⠶⣶⠇⢰⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣄⠸⠶⠉⢸⠀⢱⡂⣾⠊⠀⢀⡴⢿⠀⠀⠀⠀⠸⣆⠀⡿⢀⠞⡅⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠋⠋⠉⠱⡖⠀⠘⡥⠂⠻⣿⣧⡠⠊⡀⠘⣇⠀⢀⡤⠞⣉⣿⠇⢸⣤⠾⣤⣤⣼⠃⠙⡄
⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠐⠀⠠⠃⠀⠀⠀⠀⠀⢷⣙⣿⠐⢲⡤⠙⣾⡏⠒⠛⣿⠞⠀⠀⠙⣆⠙⠚⢁⡤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠓⠦⠿⠿⠧⠜⠉⠀⠀⠀⠀⠈⠷⠒⠋⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡎⢠⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠠⠀⠀⠀⠀⢀⠀⠀⠐⠀⣀⠤⠀⠀⢀⠀⠀⠀⠘⠄⠀⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠾⠀⠉⠠⠀⠘⠓⠁⠀⠀⡐⠂⠾⡐⢄⡀⠀⢠⡘⠀⠄⢼⣷⡤⠅⢷
⠀⠀⠀⠀⠀⠠⠄⢰⡀⠀⠂⠠⠀⠀⠀⠀⣰⡿⣣⣾⣧⣁⣈⠀⣇⠉⠀⠀⠀⡀⠱⡄⠈
⠀⠀⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢀⠀⠈⠆⠠⠄
⠀⠀⢳⡀⠀⠈⠛⠿⠿⠷⠾⠲⠶⠶⠶⢶⣶⣒⣛⣛⣛⣛⣛⣉⠉⣿⠀⡸⠃⠀⠀⠀⠀⠂
⠙⠛⣉⣉⣐⣛⣛⣛⣋⣁⠀⠀⠈⠛⠿⠯⠭⢥⣤⠀⠀⠀⠀⠀⠀⢹⡗⠰⠀⠀⠀⢱⠀⠀⠁⠀⠀⣠⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣦⣬⣭⠉⠁⠀⡀⢠⠀⠀⠀⠬⣤⣴⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠘⡉⠀⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣧⠀⠀⠀⣿⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠃⢰⡀⠀⠀⠛⠁⠀⠀⢰⡀⠐⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣄⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡼⠁⠠⠄⠀⠀⠀⠀⠀⠀⠁⠐⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡓⠀⠀⠀⠄⠈⠛⠉⠓⠄⠀⠠⠀⠀⠐⠊⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠽⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣀⣀⣤⣤⣶⣦⣀⣀⣀⣓⣠⣤⣤⣤⡀⠀⠀⠶⠿⠿⠛⠛⠉⣉⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠉⠉⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀
⠀⠀⠀⠀⢀⠄⠀⢀⣀⠔⠂⠉⠀⠀⠀⠀⠉⢳⣀
⠀⠀⠀⠐⣅⣠⢴⣿⡇⠀⡀⠄⣿⣷⣶⡿⢿⡧⠤⠇⠀⣀⣴⣿⡶⠒⠂⠐⠂⢄⡀
⠀⠀⠀⠀⠙⢻⠋⠁⡠⠊⠀⠀⡻⠋⠁⠀⠀⠀⠀⠀⠀⠛⠿⢿⡟⣶⡀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠆⠀⠌⠀⢀⣴⣾⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡛⠉⣱⠆⠄⠀⢀⣱⣄⡀
⠀⠀⠀⠀⢀⠀⠘⠀⣰⡿⠋⢉⠀⣐⡉⠀⠀⠐⠤⣤⢤⠀⠀⠀⠀⠑⣮⡀⠀⠀⢸⣏⣫⣬⣾⡇
⠀⠀⠀⠀⠘⠀⡃⢰⠏⢀⠀⠘⡟⢉⠻⠀⠀⠈⠷⠛⠿⣦⢤⠀⠀⢠⠛⠻⣦⡀⠀⠲⠟⠛⠏
⠀⠀⠀⠀⠀⢂⠁⢸⠀⠸⣞⠺⡀⠈⠀⠀⠀⠀⠀⠀⠀⠹⠂⡠⠶⠋⠀⠀⠈⢿⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠑⠨⠣⠔⠈⠐⣬⣶⣄⣀⡀⠀⢀⣄⣀⠠⢀⣦⣖⡄⠀⠀⠀⠈⡇⠰⠀⠠
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠈⠉⠁⣈⠻⣿⣿⠟⠛⠛⢩⣿⠤⡜⠀⠠⠀⡴⠃⣁⠔⠁
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢃⠀⣀⠀⠀⠀⣰⠏⠡⠀⡀⠀⠀⠙⡛⠋⢲⠒⡾⠛⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡧⠤⡶⠋⠛⣶⣤⣀⣘⠀⠸⠀⠍⠂⢀⣹⣷⣦⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠉⠻⠻⡛⢿⠿⣯⣯⠟⠋⠀⠈⠉⠉⠁⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠁⠁⢂⠄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠈⠔
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠻⠿⠣⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡿⢻⣟⣻⣿⣿⣿⣟⢻⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣠⠤⠤⠠⠈⠀⠀⢤⡀⠀⠠⡆
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡿⡅⣮⡟⣿⣯⣍⣉⣿⡞⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢺⣗⠀⡰⣿⢡⢄⠐⠒⠶⠆⠀⠇⠀⢐⣗
⣿⣿⣿⣿⣿⣿⣯⣬⡿⡛⠁⣻⡼⣽⣿⣿⣿⣿⣿⣹⣿⣭⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠃⢰⣵⣿⡀⡎⠄⠀⠀⠀⠀⠘⡄⠐⡗
⣿⣿⣿⣿⣿⣿⣟⣭⣴⣇⣠⣜⣛⣙⠛⠛⣩⣬⣭⣅⣩⣝⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠤⠖⠛⠻⠛⠡⠼⢤⣤⣴⠟⠓⠒⠿⠖⠦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⡀⡀⠌⠂⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠀⠀⠀⠠⠦⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⡐⡙⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠋⠀⢀⠄⠂⠠⡀⠀⠀⢂⠣⠀⣀⠄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⢀⠄⢀⠈⠁⠄⡀⠁⠀⠑⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⠠⠶⢈⠀⠀⡀⠀⠀⠀⠈⠖⠐⠢⡀⠀⢉⡀⠰⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⠔⠁⡀⠀⢰⠑⠀⣿⠀⠀⠔⠀⠐⠄⠀⠀⠁⢄⠀⠀⠀⠀⠑⠄⠀⢰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡇⠀⣸⡆⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠠⠨⡍⠀⢭⠅⠈⠭⠰⢤⠤⢤⠀⠄⠠⠄⠤⠛⡀⠀⡀⠀⠐⡄⠠⢤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠐⠑⢀⢩⢁⠀⠁⠂⠀⠑⠀⠈⠀⠄⠠⠁⠤⠰⡮⡅⣄⠀⠰⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⢰⡆⠀⠀⠀⡀⠄⠐⠈⠀⠀⠀⡀⢀⠀⠀⠀⠀⣻⣀⣭⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠅⢸⡇⠀⡐⠁⠀⠀⠁⠀⠀⢀⠄⠀⠁⠀⢸⡆⠈⠏⠚⠟⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⡇⠀⢂⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢾⡇⠀⠀⢰⡆⠀⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠚⠂⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠉⠈⠛⠁⠉⠋⢹⠙⠋⠈⠙⠉⠉⠁⠀⠀⢸⣿⣿⣿⣷⣾⣷⣴⣿⣶⣴⣶⣦⣾⣷⣦⣾⣷⣿⣿⣿
⠀⠀⠀⠀⢠⣦⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⢿⣧⣿⢿⣦⣄⣬⡴⠶⢦⣄⣀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡉⠉⢠⣈⠉⠻⠟⣋⣉⣛⠿⢿⣿⣿⣿⣿
⠀⠀⢀⣀⣞⣀⣉⣛⣾⣭⣵⣞⣛⣛⣾⣯⡀⠀⠀⠀⢸⣿⣿⣿⠿⢱⠿⡶⣬⠑⠒⠋⣡⢶⣬⡉⠚⢿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⣀⠀⣠⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣷⣾⣷⣾⣿⣾⣿⣾⡿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢄⣤⣾⣾⣿⣿⣿⣿⣿⢿⠿⠿⠿⠿⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣤⣾⣷⣧⣿⣿⡿⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⣠⣶⣾⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣬⠾⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⣠⣴⣶⣿⣷⣾⠗⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⣴⣿⣿⡿⠛⣟⠉⠁⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⢀⣠⣾⠿⣿⣿⡥⢼⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡤⠒⠉⠉⠉⠀⣿⣿⠃⠸⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠢⠄⣀⡀⠘⢿⣦⣀⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⢸⣿⠿⠋⠉⠙⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠆⠨⢭⣑⢛⣛⣩⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡸⠃⠀⠀⢠⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⢁⡼⠁⠀⠀⠀⠰⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⡛⠈⠑⢈⠃⠀⠀⠀⠀⠀⢇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿

⢠⣴⠶⠶⠶⢦⣤⡀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠘⢏⠋⠁⠀⠺⢥⣠⠼⠆⠘⢯⠁⢙⡶⠀⢸⠀⠀⠀⠀⣀⡴⠚⠁
⠈⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣈⡁⠀⠀⠀⠛⠚⠺⠂⠀⣸⠀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠦⣀⣀⡰⢦⡀⠀⣠⠟⠉⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⡴⠾⠛⠉⠁⠀⠀⠀⠈⠙
⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉
⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡶⠟⠋⠉⠀⠀⠀⠀⠀⠈⠉⠛⣿⣶⣤⣀⡴⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠉⠀⠘⣷⣄⢀⣀⠀⠀⠀⠀⢀⣀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢠⣶⢶⣤⡽⠟⣿⠀⣠⡴⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠶⠶⢶⣿⡄⠀⠀⠀⠹⢿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠁⠀⠀⠀⣠⡴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣦⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡇
⠀⠀⠀⠀⢠⡾⣿⣀⠀⠘⣧⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⣤⠀⠀⢀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠈⠙⠷⢦⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣸⣿⣁⡴⠀⠀⠀⠙⠷⠶⠿⠁⠀⠀⠀⣾⠁
⠀⠀⠀⣼⡋⠀⠀⠀⠀⠘⢻⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⣼⡋⠀⠀⠀⢀⣠⣾⡷⣦⣿⣷⣦⣄⡀⠀⠀⠀⠀⢠⡾⠋⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⢰⠲⠤⠖⡷
⠀⠀⠀⠈⢻⣟⠛⠛⠋⠁⢸⡇⠀⠀⢠⡿⠋⠛⠳⠶⢦⣤⣼⣇⠀⠀⠀⠈⢷⡄⠀⢀⣀⣤⠾⠋⠁⠀⠀⢀⡼⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⣾⣁⣀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢻⡟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠓⠲⣄⡴⠋⠉
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠟⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣿⠀⣠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡾⠋⢰⠇⠈⠛⠚⠋⢳⣤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢸⠀⠀⣲⠲⢲⠀⠀⠀⠀⠸⡦⢤⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠈⡷⠐⠳⣤⠞⠃⠀⣠⠷⢤⡄⠀⣷
⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠶⣦⣤⣀⣀⣀⣠⣴⣿⠁⠀⠀⠀⢰⡇⠀⠀⠈⣠⡄⠈⢳⣤⠼⠆⢠⠟⡆
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠈⣿⠉⠉⠁⠀⠹⣆⠀⠀⠀⠈⠧⣄⠈⢫⢁⢨⠏⠀⠁⢀⡴⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⢰⡇⠀⠈⠉⠉⢁⣠⠔⠋⠀⠀⠀⠈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠳⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⢱⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⡀
⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠟⠁
⠀⠀⠀⠀⣿⠃⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⢀⣄⠀⢁⣤⣄⡀⠀⣤⡀⢠⣄⠀⣤⡀⢠⣄⠈⢩⡍⠁⢀⣤⣀⡀
⠀⠀⠀⢰⡟⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠋⠀⣿⠀⣿⠀⠘⠇⠀⣿⠀⢸⡇⠀⣿⠁⠸⣿⡄⢸⡅⢠⡟⠀⠘⠇
⠀⠀⠀⠸⡇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣴⠀⠀⣿⠀⢿⣦⣀⠀⠀⣿⣀⣸⡇⠀⣿⠀⢀⡻⣇⢸⠇⣾⡷⣆⣀⣀
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠛⠀⠀⣿⠀⠈⠙⢻⣷⠘⣿⠉⢹⡇⠀⣿⠀⠰⡆⢻⣼⡅⢿⡇⠀⢸⡏
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⣆⠀⠀⣿⠀⣿⠀⢸⡇⠀⣿⠀⢰⡇⠘⣿⠀⠸⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣃⡀⠀⢐⣟⣀⠛⣳⣖⠁⢀⣿⣆⢸⣷⡀⣿⣇⣸⣧⡀⢹⡀⠀⠙⣳⣞⢿
⣿⣿⡿⠿⣿⡿⠿⠿⡿⣿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣀⣀⣀⣀⡀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠉⠉⠛⠳⢦⣄⠀⠀⠀⢀⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣸⡇⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠘⠷⣦⣀⣀⠀⣀⣠⣴⠞⠛⠛⠛⠻⢦⣤⣀⣠⣤⠾⠋⠁⠀⠈⠙⠿⣦⣀⠀⠀⠀⢀⣠⣴⠾⠃
⠀⠀⠀⠀⢠⣄⡀⠉⠙⠛⠋⠉⢀⣠⣤⣴⣤⣤⣀⠈⠉⠉⠀⣠⡴⠞⠛⠻⢶⣄⡈⠙⠛⠛⠛⠋⠉⢀⣠⡄
⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⡶⠾⠛⠁⠀⣀⡀⠈⠙⠛⠶⠾⠛⠉⢀⣠⣤⣤⣀⠉⠻⢶⣤⣤⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠘⠳⣦⣄⣀⣀⣀⣠⣴⠞⠛⠛⠛⠻⢶⣤⣄⣠⣤⠾⠋⠁⠀⠈⠙⠳⣦⣀⠀⠀⠀⢀⣠⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠉⠉⠙⢦⣴⠋⠉⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠉⢳⡀⡴⠋⠁⠀⠈⠙⣆
⠀⠀⠀⠸⡇⠀⠀⠀⠘⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⢦⠖⠛⠓⣤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⢉⣶⠦⠤⢴⡧⠤⠤⢶⣊⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⢠⡏⠀⠀⠀⢠⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⢀⣭⠷⠶⠖⠒⣿⠶⠖⠶⠶⢯⣁
⠀⠀⠀⠘⣇⠀⠀⢀⡼⢣⡀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠈⠙⠚⠉⠀⠀⠉⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢀⣴⠾⠛⠛⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⣠⠖⠒⢦⣠⠖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣀⣀⣠⠞⠁⠱⢤⣀⣀⣀⡴⠋
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠉⠻⢶⣄⠀⠀⠀⠀⣇⠀⠀⠀⠁⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⢀⣀⣉⣬⣅⣀
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠈⠳⣄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠉⠉⠙⠻⣦
⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠈⠙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠻⣆⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⣘⣧⠀⠀⠀⣿⣅⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣴⠾⠟⠛⠉⠉⠉⠉⠉⠉⢹⡿⠋⠉⠛⠷⢀⣤⣝⠋⠁⠀⠀⠈⠻⣦⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⡀⠀⠀⢿⣅⣽⠃⠀⠀⠀⣠⣴⡿⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⠋⠀⣩⡅⠀⠛⢿⣿⠟⠉
⠀⠀⠀⣠⡤⢤⡀⣠⡤⢤⡀⠀⠙⣷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⡿⢷⡄⠀⠀⣿
⠀⠀⢰⡇⠀⠀⠙⠁⠀⠀⢹⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣴⠟⠁⠈⠻⠶⣾⠋⢀⡴⠒⠓⣤⠔⠚⠳⣄
⠀⠀⠈⠳⣄⠀⠀⠀⢀⡴⠋⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠸⡄⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠈⠙⠶⠖⠋⠀⠀⢸⡏⠀⠀⠀⠀⣶⡀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣰⡆⠀⠀⠘⣿⠀⠙⠦⣄⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠰⣿⡷⠀⣀⠀⠀⠀⠙⠛⠟⠛⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡿⢷⣄⠀⣠⡾⠛⠛⢶⡄⣀⣻⣶⠛⠳⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠙⠻⣿⠀⠀⠀⠈⠿⠋⠉⣉⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠀⠀⠻⣦⠀⢀⣠⣴⠶⠟⠛⠉⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠳⠶⠶⠶⠶⠶⣿⣷⣿⣭⣤⣀⣀⡀⠀⢠⡿⠀⠀⢀⣀⣀⣠⡴⠞⠋⠁⡰⠊⠉⠉⠲⣀⡖⠋⠉⠙⢦
⠀⠀⠀⢀⡤⠤⢄⣀⡤⠤⣄⡀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣻⡾⠿⠛⠛⠋⠉⠉⠻⣦⠀⠀⠀⢇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠂
⠀⠀⠀⣾⠀⠀⠀⠋⠀⠀⠈⡇⠀⠹⣦⣠⡶⠟⠛⠻⣦⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⠀⠀⠀⢀⣰⡟⠀⠀⠀⠈⣳⣤⣤⣤⣼⣤⣤⣤⣖⠋
⠀⠀⠀⠘⢦⣀⠀⠀⠀⣠⠞⠁⠀⠀⠘⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢈⣷⣄⣀⣠⣤⣤⣶⠾⠛⣿⠀⠀⠀⠀⡜⠉⠀⠀⠀⣼⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠈⠓⠶⠊⠁⠀⠀⠀⠀⠀⠙⢷⣤⣴⣾⣯⣤⣴⢶⡶⠟⢻⡏⠉⠉⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢧⠀⠀⠀⢀⠿⡄⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣨⣷⠾⠿⣶⣾⠇⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠈⠓⠒⠋⠁
⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⣀⣤⣠⣄⠀⠀⠀⠈⠛⣿⠃⠀⠀⢘⣷⣀⣀⣀⣀⣤⣶⡟⠁⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⣰⠋⠀⠀⠀⠉⢳⣀⡞⠁⠀⠀⠀⠹⣆⠀⠀⢘⣿⠶⠶⠞⠋⢹⣿⡏⠉⠉⠁⠈⣿⠀⠀⣠⠎⠁⠀⠀⠈⢳⡄⡴⠉⠀⠀⠀⠙⣦
⠀⠀⠀⣏⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⠃⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠘⣧⠀⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⡴⠃⠀⣸⠇⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⢸⣇⠘⢦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⣠⠽⠛⠛⠒⠒⣿⠒⠒⠛⠻⠭⣀⠀⠀⢿⡄⠀⠀⠀⠀⢀⣼⠿⣦⠀⠀⠀⠀⠀⢰⡟⠀⢀⡭⠿⠛⠒⠒⣿⠒⠒⠚⠻⠯⣅
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣇⠀⠈⠻⣦⣤⣤⡶⠟⠁⠀⠙⠻⠶⣤⣤⡴⠟⠁⣼⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⣧⠀⠀⠀⠀⠀⢠⢿⡀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⡿⡄⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠘⠦⣄⣀⣠⡴⠋⠀⠱⢤⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⡤⠞⠀⠙⢦⣀⣀⣀⠴⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡰⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⠤⠖⠉⠀⠠⠜⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⣅⠈⠁⢠⠀⠈⠁⣰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣄⠀⠠⠄⠀⣠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠀⢗⠂⠐⡿⠀⠙⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠄⣡⣌⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡴⣹⡠⠊⠉⠉⠙⠉⠉⠉⠑⠄⠏⢆⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠂⢄⠀⢀⡼⠥⠛⠂⠄⡀⠀⠀⠀⠀⢀⠠⠒⠚⠬⢧⡀⠀⠠⠐⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⢢⡠⠒⠁⠀⠀⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠒⢄⡔⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢰⡑⡐⣠⢠⠁⠈⠒⠒⠒⠒⠒⠒⠒⠂⢤⠄⠒⠒⠂⠒⠒⠒⠒⠒⠁⠈⡄⣄⢂⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠖⠂⠁⠢⢄⠀⠀⠀⠀⠘⠁⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣈⣐⣤⣀⣀⠤⠔⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠠⢤⡤⠄
⠀⠀⠀⠀⠀⠔⠁⠐⢩⣷⠌⠀⠘⢦⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢠⠊⠒⠂⢀⣆⣀⠰⡄⢀⠈⠵⣁
⠀⠀⠀⢰⠀⠓⢲⡏⣿⡟⣳⠷⣿⢹⢫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣂⠀⣀
⠀⠀⠀⠀⡄⠀⠈⢃⡀⣀⣀⠈⠁⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⢀⠀⣤⣶⡞⣽⢻⢿⣌⠲⢤
⠀⠀⠀⠀⡇⠀⠀⡴⠚⠋⠁⠹⣿⠿⢤⠀⠀⢀⢀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⣿⣿⠛⣿⣵⣷⣿⣷⠛⠛⣃⠈⢷⡀
⠀⠀⠀⠀⠛⠀⠸⣄⠀⣄⠀⣀⣄⣧⣎⣳⠀⠈⡻⠟⡏⣛⠒⠁⠀⢀⢀⡀⠀⢀⠓⡾⢓⢸⢟⣿⣿⡏⣿⡿⡉⠘⡛⠛⢿⠭⣵⢂⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠍⠀⢰⣊⠙⢏⡎⠉⣷⢸⣃⢾⠃⠀⠀⠀⢻⡱⢲⡀⣪⣷⠃⠉⠒⠒⠪⢿⣓⣶⣤⡾⠖⣆⠀⠓⠇⡄⠀⢀⠻⣆
⠀⠀⠀⠀⠀⠀⡀⠀⡱⠀⡀⢠⣦⣤⡥⢈⡓⢤⠿⠗⠚⠋⠀⠐⠸⡤⢮⡀⠙⢻⡟⠁⠀⢀⠄⢔⣟⠉⠉⠀⠀⠀⠠⠞⢷⡄⢇⢓⠀⠠⠒⠈⣣
⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠈⠃⠀⠀⡀⠁⠈⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠳⠷⠛⠀⠶⣟⠛⡗⠀⠑⡄⠀⠠⢆⢠⡼⣿⢧⡼⡤⢆⠠⠤⢜⣿
⠀⠀⠀⠀⠀⣀⣄⠁⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆⣀⣘⢦⠀⠀⠀⠀⠀⠀⠈⣄⠀⣇⠘⢌⠣⡁⠀⣽
⠀⢀⠀⠀⠀⠀⠈⣷⣄⣀⣀⠤⠒⢻⣦⣄⣠⣤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠟⠓⠛⠀⠀⠀⠀⠀⠀⢾⡁⠀⠉⣃⠀⠉⠲⠆⠋
⠀⠀⠉⠁⠐⡆⠀⠈⠀⢀⣱⠀⠀⢾⡁⢀⠀⠁⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠋⠁
⠀⠀⠀⠀⠀⠳⠤⠴⠂⠉⠀⠀⠀⠀⠀⠀⠈⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⢦⣀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⣿⣟⣷⣼⣿⣿⣿⡿⢟⣯⣽⡟⠟⡴⢋⣴⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣛⢷⣶⣿⣿⣿⣿⡿⠋⠑⢋⡽⢛⡅⣠⢿⠟⡉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠻⠿⣿⡘⣿⡇⡄⠀⠀⠀⢉⡞⠟⢸⣴⣷⣦⣿⡤⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⠀⣘⣶⣶⣝⠻⡧⢳⣄⣀⢀⣼⣤⡄⣨⡭⠭⣂⣀⡨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠽⣿⣿⠀⢷⣆⠛⠛⠛⢛⡭⠀⢰⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⣷⣤⠙⠿⠶⣟⣫⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠩⢭⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⣠⠴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣣⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢹⣿⣿⡿⠃⠀⠀⠀⠀⢀⣾⣿⢿⣿⣿⡏⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⢿⣿⣧⠀⠀⠀⠀⠀⠈⣿⡇⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⠈⠻⣿⣦⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠈⢻⣷⣄⠀⠀⠰⠿⣧⣀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡇⠀⠀⠀⢿⣿⣷⣦⡀⠀⠀⠉⢿⣿⡟⠋⣉⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⡇⠀⣦⡄⠈⠿⠟⠙⠻⣷⣤⣤⣾⣟⠳⣌⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣧⠀⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⢻⡝⢳⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙⢦⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⣶⣶⣾⣷⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⢋⡄⠀⠀⠀⠀⢲⡴⠋⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⣠⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠘⢃⠀⠿⢿⣿⣿⣦⡾⠁⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠻⣿⣿⡥⢾⠀⢠⣶⡇
⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⠀⠀⠈⠳⠀⠘⠟⠁⣿⠇⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡀⠈⠙⠻⢿⣧⠀⠀⠀⠀⠀⢠⠀⠀⠉⠉⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣷
⠀⠀⠀⠀⣰⠿⠁⠀⠈⠉⠀⠀⠀⠀⠈⠈⠓⢀⡀⠤⠤⠄⠻⠀⠀⠀⠸⠹⡤⠀⠀⡐⠀⠔⠈⡇⠀⠀⠀⠀⠀⠲⠈⠀⢀⣀⡀⢻⡆
⠀⠀⠀⢠⠏⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢰⡆⢛⠶⠄⠀⠀⠀⠀⢀⠑⠀⠀⣶⣄⠀⠁⡠⠂⠀⠀⠀⠀⠀⠀⠠⣍⠡⣦⡁⣀
⠀⠀⠂⠈⢢⡈⠛⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠰⠋⣠⡎⠀⠀⠀⠘⢿⡀⡠⠖⣁⡠⣴⠊⠀⠀⠀⠀⠀⠀⠙⢶⣾⣿⣶⣶⠟
⠀⠀⠐⠗⠤⣅⣀⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⠀⠀⠀⠀⠀⠻⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠉⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣽⣯⣯⣭⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿

⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠈⣙⢿⣾⣿⣿⣿⣶
⣿⣟⣴⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣿⣿⣿⣿⣿⣿⢿
⣿⣿⣿⡷⣿⡿⢶⣿⣷⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠀⠁⡀⠰⣿⣍⢛⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡿⢿⣿⣿⣿⣿⣿⣿
⣿⣯⠀⠀⠀⠈⠆⠙⠿⠿⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡠⠿⠛⠛⢻⡿⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⡄⢠⣄⠀⡴⢠⡄⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠃⠀⠀⠒⠂⠚⠉⠁⠂⢀⣿⣿⡆⠀⡏⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠠⠤⠶⠒⢂⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⣀⣠⠤⠤⠤⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⢀⠀⡠⠤⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣀⠈⣭⣿⣥⣤⠀⣀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠍⠉⠛⠂⠙⠉⠉⠀⠠⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⢤⣤⣰⣾⠃

⣄
⠫⡄⠠⡀
⠀⢹⡄⠈⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠲⠀⠀⠀⠲⢤⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠁⠀⠄⠀⡀⠁⠈⠀⣄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠒⠒⠐⠦⠴⠦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣈⠺⣿⡗⠲⠶⠦⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⣤⣤⣶⣶⣶⣾⣿⣿⡿⠿⠟⠻⠻⠷⠷⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠽⣿⣿⣧⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠊⢴⣿⣿⣧⠀⠹⣄⠉⠁⠒⠀⠰⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠇⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠲⠤⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠠⡄⠀⠀⠀⠈⠉⠒⠤⢤⣀⡀
⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢶⡤⠤⠄⠀⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠢⢤⣀
⠀⡀⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠉⠀⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⢄⣀⣀⠀⣀⣀⠤⠤⣀⣀⠀⠀⠀⢎⡴⡞⣠⠊⠉⠉⢢⡀⠏⡠⠚⠓⠢⡿⣌⡏⠉
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠈⠉⠈⣿⠁⠁⠀⠀⠉⠀⢹⡀⠀⠀⢀⡙⠁⠀⠀⠀⠀⢣⢰⠁⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠉⡆⠀⠙⠛⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠉⠁⡠⡀
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⠀⠀⠀⠀⠀⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠲⡄⠀⢀⠀⢀⡤⠴⢧⠀⠀⡀⠀⣠⠤⠞⠀⢀⣀⣠⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣈⣹⠉⠙⣏⣀⣀⣈⣱⡏⠉⢿⠥⢲⡖⢩⡗⢻⣝⠛⢫⣤⠄⠀⠘⠋⠀⠀⣤⣍⠛⢻⡟⠺⡍⠲⡄
⠀⠀⢀⣀⣰⠋⠀⠀⠀⠀⠉⠓⢶⣏⠁⠀⠀⠀⠓⠒⠚⠀⢸⡉⠒⠿⠷⠿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠿⠶⠿⠒⠋⡇
⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢆⡀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⠴⠖⠚⠛⠛⠛⠻⢤
⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⣴⠋⠙⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠶⠚⠋⠉⠉⠉⠉⠉⠉⣹⠷⠦⣽⡄⠀⣸⣀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠈⣡⠶⡏⠉⠈⠉⠙⣦⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠓⢘⡒⠣⣤⣀⣀⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⣧⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠘⠲⠴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⣶⡆⢀⡀⠀⠀⠸⣿⠇⠀⠀⠀⣸⠀⠀⠀⠀⢠⠖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢰⡉⢹⠤⡞⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣋⣀⠀⠀⠀⡟⠈⣟⠑⣏⠁
⠀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠀⢠⡞⠉⠉⠙⠷⣦⣄⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣤⣶⠞⠋⠁⠀⠈⣷⠀⢀⠭⣉⣿⣛⣁
⠀⠀⠀⢰⠉⢻⢤⡞⠒⡆⠀⠀⠀⠳⣄⠀⠀⠀⠀⣽⠁⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠘⡇⠀⠀⠀⣀⡴⠋⠀⠈⠓⠼⠿⠥⠚
⠀⠀⠀⠀⡟⠈⡞⠑⣞⠁⠀⠀⠀⠀⠈⠓⠦⣄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠤⠖⠋⠁
⠀⠀⠀⢀⢍⣚⣿⣒⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠈⠲⠽⠿⠥⢞⣀⡔⠓⡆⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣵⢲⠉⣉⠆⠀⠀⠀⣠⠜⠉⠙⠲⠦⡤⠤⠤⣤⠴⠖⠋⠈⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣨⡅⢘⡇⠀⠀⢠⠞⠁⠀⠀⠀⠀⣸⠁⠀⠀⠘⣇⠀⠀⠀⠀⠈⠳⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠚⠒⠁⠀⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠰⠟⢀⠀⠀⣀⣀⠀⠈⠃⠀⠀⠀⠠⡄⢀
⠀⠀⠀⠘⡖⠚⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡀⠀⠚⠓⠒⠀⣉⣉⣠⣛⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠠⠊⢀⠄⣀⣀⢠⣄⣀⠘⡄⠀⠀⠀⠰⢿⠚⢿⠛⠁⠀⠉⠉⣿⣹⠀⠀⠃⠀⠈⠉⠉⠈⠛
⠀⠘⡧⢄⣸⣰⣻⠀⠀⣻⡂⢂⣤⡀⠀⠀⠀⠈⠒⠻⣄⣤⣆⣲⣴⠯⠃⠀⠘⠄⠀⢀⣶⠶⢆⠀⠀⠀⠰⢦
⠀⠀⠀⠀⠹⡞⠂⠀⠀⠋⢀⠤⢹⠃⠀⠀⠀⠀⠀⢠⡭⠤⠌⣦⣬⣂⠀⠀⠰⠇⠐⣲⡄⠐⠾⠀⠀⠑⠀⠊
⠀⡄⠀⠀⠀⢁⠀⠂⠀⢀⡴⠀⡾⠀⠀⠀⠀⠀⠀⠄⡆⠄⢉⡑⢚⡛⡆⠢⠐⠶⠂⠙⠁⠀⠠⠆⣐⡎⠤⠇
⠀⠹⡶⠄⢰⠀⢻⡿⢿⣿⡧⢰⠃⠀⠀⠀⠀⠀⠌⢀⣿⡁⠉⠉⢹⣇⠁⢀⠇⠘⠻⣄⣠⢀⣠⢈⣠⠾⠂
⠀⠀⠈⠙⢸⠀⠈⠀⠘⠻⠁⢾⠀⠀⠀⠀⠀⠈⠀⢸⡟⠁⠀⠄⠈⢿⡟⠒⠀⠀⠠⣎⠉⢁⡒⠛⠁
⠀⠀⠀⠀⠈⠑⣆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡀⣘⣧⡀⠀⠀⠀⠀⠀⠀⢁⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢈⡷⠠⣀⡀⠀⠀⢲⠋⠈⠛⠉⠉⡩⠉⠀⠐⡄⠀⠀⠰⠐⠊⡁⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠰⠊⠀⡇⠀⠙⢦⡀⠞⠀⠀⠀⢀⠠⠃⠀⠀⠀⣷⠀⡀⠀⢠⠔⠀⣠⠃⠈⡦⠤
⠀⠀⠀⠀⠀⠀⠘⢳⠂⠀⠤⣄⣀⠇⠀⠀⠀⢳⠀⠀⠀⠈⢸⠀⢇⠉⠀⠀⡏⠀⠓⠒⠹⣄⣀⣠⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⢼⠔⠒⠒⠒⢺⢦⡁⠀⠀⢸⢁⠀⠀⢀⠇⠀⠈⠢⣀⡐⠃⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⡜⠀⠈⠓⠖⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⢀⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⣠⡄⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣈⠛⠻⢿⣦⡀⠀⠀⣿⡇⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⣠⣤⣤⣤⣄⣀⣉⠉⠛⠻⢿⣶⣄⠙⢿⣆⣠⣿⣶⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⠿⣶⣤⣉⢻⣷⣾⣿⠿⠛⠉⠉⠙⠛⠿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣹⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣤⡀
⠀⠀⢀⣤⡶⠾⣿⣿⣶⣄⠀⠀⢀⣠⣶⡿⠟⢛⣽⣿⠋⠉⠉⢹⣿⡿⠟⠉⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄
⠀⠀⠻⣷⣶⣾⠿⠛⠉⠉⢀⣴⠿⠋⠁⠀⣴⣿⠋⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠁⣀⣀⣠⣾⣟⠁⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⢼⣿⠇⠀⠀⣜⢳⣾⣿⣦⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣭⣭⣽⠿⠗⠀⠀⠀⠀⣠⣤⡙⣿⡄⠀⠀⠀⠈⠛⠀⠀⣠⣌⢿⣿⡻⣿⣿⣦⡀⠀⠀⠀⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⣰⡟⢉⣿⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⢿⣿⣧⠙⢷⣌⠻⣿⣿⣦⡀⠀⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢰⣿⡏⠀⢀⡀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠈⢿⣿⣧⡀⠻⣧⡙⠿⢻⣿⡄⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⢀⣠⣴⣶⣾⣿⠟⠀⣿⣿⠁⠘⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣶⣤⡈⢻⣿⡷⠀⠈⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⣿⠁
⠀⢀⣾⡟⢁⣠⣿⡿⠃⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠲⠿⢷⣦⠈⠉⢻⣷⠀⢻⣇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣸⣿⠇
⠀⠈⠿⣿⠿⠟⠋⠀⠀⠀⢸⣿⡇⢀⣀⡀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⣿⠀⠈⣿⡄⠀⠀⠀⢸⣿⣧⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠹⣿⡷⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣥⠀⠀⢹⣇⠀⠀⠀⢸⣿⣿⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠈⣿⣧⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠰⣿⣿⣦⠀⠀⠀⠀⠀⠙⣭⣤⣀⣈⣿⡄⠀⠀⢘⣿⣿⡿⠛⠁
⠀⠀⠀⣀⣤⢶⣿⣿⠃⠀⠀⢹⣿⡄⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣷⠀⠀⠀⣿⣿
⠀⠀⣼⣿⣡⣾⡿⠃⠀⠀⠀⠀⢿⣷⡀⣰⣿⠃⠀⣀⣤⣶⡿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⣿⣧⠀⠀⠸⣿⡄
⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣴⡿⠟⠋⠁⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣤⣾⣿⣷⡀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⣀⠀⠀⠻⠶⣦⣤⣀⡀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠟⠉⠻⣷⡀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠈⠙⠿⠷⠀⠙⠃⠀⠀⠀⠀⣀⣤⣶⣿⠟⠁⠀⠀⠀⠹⣷⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣄⣀⣀⣀⣀⣤⣤⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢠⠴⠒⠦⣀⡴⠋⠉⡽⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⢀⣠⠴⠗⠒⠛⠓⠒⠛⠶⣤⣤⣾⣷⡖⠋⠀⣀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⣿⣏⠻⣶⠞⢉⣝⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠙⢛⡻⣿⡁⠀⠀⢛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⢻⣭⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢚⡿⠋⣟⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠁⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠿⠇⠀⠀⠀⢀⣤⠀⠀⠀⠠⣿⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠘⠒⠻⠞⠃⠀⠀⠉⠁⠀⢀⡞⠀⠀⣀⣤⣴⣤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢰⠋⠙⣇⣀⡿⠛⣟⣧⡕
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠈⠉⢹⡟⠛⠒⠒⠶⠶⠶⠖⠒⠺⡟⠛⠲⠦⣼⠓⢺⡉⢉⡷⠴⡏⠀⠀⠀⠀⠀⣴⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⣤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⣸⠶⣾⠋⠙⣧⡤⠇⠀⠀⠀⠀⠀⠘⢄⣀⣇⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⠙⠳⣤⠜⠓⠊⠀⠀⠀⠀⢀⠔⠒⢦⣀⣴⣿⡧⢄⣀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⡀⠀⠀⠀⠀⠀⣀⡼⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠀⠀⣈⣻⣁⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠃⠀⠉⢹⠗⠛⠛⠻⣅⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⠿⠋⡏⠀⠁⣠⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡟⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣠⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠁⣠⠶⠒⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⣀⡤⠟⠚⠛⠛⠛⠛⠒⠒⠶⢤⣀⣀⣠⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡄⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢴⣷⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠛⠃⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠀⠀⠀⢀⡀⣶⠂⡀⠀⠀⠸⣿⠇⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡔⠉⢹⣦⣀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠬⢿⡷⠒⠉⠀⠈⠳⡀⠀⢙⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢋⣭⣇⠀⠀⠀⠀⠀⠀⠀⢠⢏⣖⡆⠐⠾⠟⠃⠀⠀⠀⠀⢳⠖⠋⣰⠃⠈⠉⠉⠙⠓⠒⠲⠶⠶⠶⡖⠚⠻⢧⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠕⠛⣄⣠⣄⡀⠀⠀⠀⠸⡌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢈⡿⠛⢦⣤⣾⡀⢀⡇⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣠⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢦⣄⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠘⢦⣤⣾⡀⢀⡿⠉⠙⡆⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡤⠤⠤⠋⠀⠀⢀⡽⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠉⠁

⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢁⣄⠄
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠅⠀⠀⣀⠀⠀⠀⠀⢀⡀⢀⠀⠀⢀⢇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠋⢠⠂⠁⠀⠀⠀⠀⠐⠃
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠏⠆⢀⠀⣆⡸⠥⡸⠄⠀⠈⡣⠀⠀⠀⢀⠀⠀⠀⠺⡋⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡁⠀⠀⠀⠘⠀⣀⣎⡴⠬⠗⠚⣇⠂⠀⠀⠀⠂⠀⠀⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡀⢸⡄⠠⠇
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⣿⢇⠸⡄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣖⢀⢰⠀⠀⠉⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠃⠀⠀⢠
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⣀⡀⢀⣄⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⠊⠀⠈⠉⣠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣶⣿⡧⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⣤⣆⣀⠀⣀⠦⠀⠒⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⢄⡀⠙⠁⢉⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⣀⠀⠾⣡⣤⣀⠀⠀⠰⠋⣄⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⢀⣠⡀⠰⠖⠂⣤⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⢀⠞⠁⢀⡴⠋⠀⠀⠀⣀⠐⠶⠛⡱⢦⣀⣀⡵⠆⠉⠀⠈⢳⣤⠰⣾⠷⠊⠁⠀⢀⡴⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢷⣄⠀⡛⠀⠀⡎⠀⠀⠀⡠⠊⠀⠀⠀⠀⣴⠶⣿⠋⠀⠀⠀⠀⠀⠈⡙⢟⠀⠀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⢻⡇⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⠁⠀⢰⡆⠀⠀⢰⡇⠀⠀⠁⢸⣥⠶⠋⢻
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⣴⠚⠧⣴⠋⣼⡀⢨⠀⠰⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⢻⣄⣀⠀⣆
⠀⠀⠀⠀⠀⠉⠱⢦⣄⣺⠁⠀⠀⠀⡟⠈⠻⣯⡄⠁⠀⠀⡀⠀⠀⣠⠴⠻⣤⢠⣷⡀⠀⢠⠀⠀⠀⠀⠀⣠⢹⡇⢠⡉⠹⣆
⠀⠀⠀⠀⠀⠀⢀⡀⠈⠡⠀⠀⠀⠀⢸⣆⠀⠈⠀⠀⠈⠉⠉⠛⠛⠁⠒⠒⠶⠂⠈⢧⡀⠂⠀⠀⠀⣠⠞⠁⡼⠁⠀⠹⠀⠸
⠀⠀⠀⠀⠀⠀⢸⡟⣄⢰⠀⠀⠀⠀⣮⠙⣷⣦⡄⢶⡆⢻⣶⣬⣤⣄⣀⠀⠀⠀⢀⡾⠁⠀⣀⣴⡊⢀⡤⠚⠀⠀⠀⢰⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣦⠈⢿⠀⠀⠀⠀⡿⠀⠀⣿⠀⡈⠃⣼⡇⠀⢈⡿⠉⢦⠀⠀⠸⡷⠖⠉⠙⢎⡙⣿⠀⠀⠀⠀⠀⠘⡰⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢀⣤⠀⡇⠈⢐⡟⣨⠗⠆⠉⢢⣤⡝⠁⠀⠺⠀⠀⠀⠀⣴⣶⣀⣀⠈⢻⠀⠀⠀⠀⣌⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡾⠋⠀⠀⡏⠀⣞⠁⠉⠀⠳⠆⠘⠃⠁⠲⠀⠀⠀⠀⢠⡞⢀⡙⠛⠁⠂⣽⡀⠀⣀⡖⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠘⢦⣈⣁⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠀⠀⠀⢠⣼⣇⢀⣨⣽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠻⡖⠢⣄⠀⠀⠀⠤⠃⢀⠳⠊⢉⡉⢁⣼⠋⠹⣦⡄⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⣳⡀⠀⠀⠀⠁⣀⠀⠀⠀⠈⠒⣤⠈⢳⣄⠀⠀⠈⠃⠀⠈⣹⠟⠁⠀⠀⠈⠻⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡏⠃⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠘⠗⠊⠑⠒⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⢠⣤⠐⠖⠉⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠐⢯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⠛⠷⣤⣤⣴⣶⣏⣀⣤⣤⣤⡶⠶⣤⣤⡴⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠤⢴⠟⢀⣼⠃⠈⠉⠳⢦⣴⠉⠙⢫⡽⠋⠁⠀⢸⣧⠀⠀⠀⢀⣦⡝⢻⣃⢰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢶⠏⢹⡄⠀⠀⠀⠀⣟⠀⠀⠀⢹⡆⠀⠀⣤⠀⠉⢂⠀⣸⡄⠙⢾⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠂⠉⢀⠆⠀⡰⠆⠀⠀⠀⡰⠉⠙⢏⠉⡍⠙⠢⡄⠈⠳⣀⠙⢆⠀⠹⣦⣄⡉⠓⢦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠁⠀⠀⢀⡜⠀⠀⣧⡀⠀⢀⡌⠁⠀⠀⠀⠛⠠⣄⠀⠈⢢⣀⡿⠃⣼⠷⢤⣼⣤⡉⢲⣆⠀⠈⠉⠐⠒⠂⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠠⠄⠀⠚⢡⡄⢀⠈⠉⡉⠟⠁⣀⣤⠚⢷⡄⡤⢿⠁⢦⢄⡙⠀⠀⠸⣦⠤⣤⣤⡈⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠉⠐⠠⠀⠉⠱⠖⠁⠀⠉⠹⠀⠉⠲⣤⠀⠈⡐⠃⠘⠃⠀⠀⢀⡛⣦⣤⣄⠀⢈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠈⠑⠆⠀⠀⠀⠀⠀⡇⠀⠀⠉⢶⡄⠁⠀⠀⢠⠀⠀⡀⠳⠀⠀⢀⣁⠉⣿⣥⠄⠀⠀⠀⠀⠀⢀⣼⡿⡅⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠘⠃⠀⠀⠀⢸⠀⢀⣹⠆⠀⠐⣭⣉⣁⡀⢠⣤⣦⣴⠶⠖⠚⠋⠉⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠐⠒⠂⠊⠙⢦⡀⠀⠀⠀⠀⢀⣏⣱⡀⢶⠒⠒⠀⠘⠀⠈⢿⠀⠀⠀⠀⠉⠉⠉⢀⡾⡍⢰⡷⠒⣤⣴⠒⣦⣮⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠿⡌⠓⠏⢘⠁⠀⠀⢈⠆⠀⡜⢣⠀⠀⠀⠀⠀⣠⡞⠁⠀⠈⠛⠛⠛⠛⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠡⠀⠀⠀⣴⢢⠀⠘⠀⠀⢀⡋⠀⠀⣷⠈⠆⠀⠀⠀⣄⠻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⣦⢶⡙⢯⣧⣜⠀⠀⢘⣧⡶⠂⠈⢷⠘⢂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⣿⠀⣠⢤⠀⢹⠒⠒⠸⡇⢻⣟⡄⢡⣀⠘⡄⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⡃⠀⢻⣞⡄⣸⠀⠀⠀⡀⠉⠉⠁⠀⠈⠳⣼⣧⡀⠉⠄⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠⠇⠀⠋⠉⠂⢹⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠙⡙⢦⢀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠄⡻⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣄⠢⡀⠀⠀⠀⠀⠀⠈⢦⡈⠑⠂⢤⠄⣀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⢀⡤⠛⣄⠀⠀⠀⠈⠑⢄⡂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡏⠁⠀⠀⠀⠀⠔⠋⠀⠀⠈⠓⢄⠀⠀⠀⠈⢱⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⡇⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠌⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⠀⠀⠐⡄

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⣿⣼⣿⣻⣟⣩⣿⣾⣀⢴⡿⣿⠯⢀⣩⡐⢺⠦⣿⠙⣽⣾⢻⣟⣹⡯⣿⢻⠤⠦⢀⣶⣂⠤⣤⣤⣄⠀⠃⣶⣴⡄
⠀⠀⠀⠀⠉⠉⠁⠉⠉⠁⠉⠉⠀⠀⣩⣿⣿⡿⠻⠙⢷⣄⡙⠀⣇⣼⡞⣯⣿⣷⣿⣨⣧⡏⠈⠁⠝⡘⣯⡿⢹⠋⢰⢞⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡟⢋⣠⣤⣦⣿⣿⣿⣀⠈⠭⠵⠯⠌⠸⠟⠟⠇⢤⡤⠙⣿⣿⠇⠀⣴⣗⠼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣧⠀⢸⡧⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣯⣿⣻⣿⣿⢟⣻⣿⣿⣷⣯⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣼⡇
⠀⠀⠀⠀⢀⡀⣴⢿⣿⣿⣿⣿⣿⣿⡛⠩⡽⠎⠈⠓⠊⠁⠉⠙⢼⡽⠃⢘⣿⡯⣵⠉⢻⣿⣷⣄⡀⠀⠀⠀⠀⠉⠛⣻⡷⠇
⠀⠀⢰⡾⣏⡿⢋⣿⣿⡟⣿⣿⣿⠁⠂⠀⠀⢀⣀⠀⠀⠀⠀⢠⡶⠁⠀⠈⠉⠁⠀⠀⠈⠴⠬⠻⣧⡀⠀⠀⠘⠳⠾⠙⠇
⠀⢠⢤⣷⡽⣤⣿⣿⣿⣴⣿⡿⠉⠀⠀⠀⣀⠜⢿⣦⠙⢷⣤⣸⣿⠳⠤⠀⠀⠀⣀⡔⠀⣀⣤⠴⠒⣬⣶
⠀⢹⡄⢿⣷⣿⣿⣿⣿⡿⣿⡇⠀⠀⠘⠋⣁⡈⠀⠙⠃⠘⠻⠿⠌⠳⣄⠈⠶⠬⠄⡁⠈⠙⢁⣴⢋⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠈⣿⣷⠙⣿⣻⣿⣿⣿⣿⣷⣤⠶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣤⣴⣽⣿⣦⣤⣤⣬⣷⣤⣥⣈⣙⣳⢦⡆⠀⠀⠀⣿⣿⠋
⠀⢀⣿⣿⠀⠛⢿⣿⠻⣿⣿⣾⣧⢀⣿⣿⣿⣿⣿⡿⣻⣾⣿⣿⣿⣿⠿⣻⣯⣥⣼⡿⣿⣦⢈⠉⢭⡿⠎⠀⠀⠀⠀⠈⠉
⠀⠈⣽⣿⣧⢀⣠⠙⣷⡌⠛⠚⢿⣿⣿⡇⣾⣿⣿⠷⠾⣫⣾⣿⣿⣿⣿⠿⠋⢉⣿⣿⠇⠀⠀⠀⠄
⠀⠀⠸⣻⣿⣦⡈⢁⣙⠻⣦⣀⠈⠉⢻⡇⢸⣿⣯⡿⠟⣵⣿⡿⢃⣴⣶⣶⠏⢉⠛⠉
⠀⠀⠀⠹⣿⠿⠿⢦⣝⣄⠀⠉⠳⣀⣠⣇⢸⣿⠏⢀⣠⣷⣻⠿⠋⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠸⠟⠙⠋⢂⠰⡶⠾⣿⡧⢿⡇⠀⣰⣶⣦⣤⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠵⣮⠓⠉⠁⢾⡇⠀⠀⠀⠤⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠻⢿⣿⣿⣿⣷⡀
⠀⠀⣼⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⠠⣴⠔⣶⡖⠲⣶⢶⣦⠤⣀⡀⠀⠀⣀⡤⢾⣿⣿⣿⡀
⠀⣼⣿⣿⣿⡋⠀⠀⠀⠀⠀⣀⠤⠖⠒⠊⠁⠈⣿⠀⢹⡄⢹⡇⠀⠿⠈⠋⠀⠀⠈⠉⠉⠀⢠⡞⡇⢸⣿⣷
⢰⣿⣿⣷⣤⡈⠉⠐⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠷⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⢳⣷⣾⣿⣿⣇
⣸⣿⣿⣿⣧⡉⢢⣄⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡟⠋⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⡌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⠀⠠⢤⡤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣿⣿⣿⣿⣿
⠹⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠒⠦⣄⣀⡀⠀⠀⣀⣀⡤⠚⠁⠈⠳⣤⡄⣀⣀⠤⠔⠋⠀⣰⣿⣿⣿⡿⠋
⠀⠙⠿⣿⠿⢿⣿⡿⣿⠛⠦⡀⠀⠀⠀⠀⠀⠘⢯⡉⠹⠇⠀⠀⠀⠀⠀⠻⢡⠏⠀⠀⠀⠀⣠⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠢⣄⠉⠒⠠⠶⠚⠁⢀⢴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠶⠖⠚⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢤⣤⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣎⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⡼⠁⠀⢸⡇⠀⠀⠀⠀⣠⠎⠁⠈⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔⠀⠀⠀⠀⢧⡀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣴⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠻⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣧⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠿⠃⠀⠀⠀⠉⠛⠻⠿⠛⠛⠁
⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⡀⠀⠀⠀⠀⣀⣠⣴
⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡏⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⡿⢯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣀⡙⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⡿⣗⢿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢎⢨⡀⠘⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⠷⣿⢿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠂⠀⣉⢁⣀⠈⢆
⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⡿⣿⣵⣿⣜⢽⣯⠿⡽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣄⠴⠊⠈⢣⡊⠳⣌⣧
⣿⣿⣿⣿⣿⣿⣿⣳⣫⣾⣭⣵⣿⠿⣿⣷⣦⡻⣿⣞⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡼⠔⠁⠒⠊⠀⣤⠀⠈⠙⢄⠀⢣
⣿⣿⣿⣿⣿⡿⣳⣟⢹⣿⡿⢿⣿⠭⣿⡿⢿⣿⡏⣻⣞⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⠎⠠⡆⠀⢀⡀⠀⣿⠀⢀⡀⠀⢰⠄⠱⡀
⣿⣿⣿⣿⣿⡵⣿⣗⣸⣿⣏⣹⣿⣿⣿⣏⣹⣿⣇⣺⣿⢯⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢎⠀⠸⡧⠀⠸⠆⠀⠉⠀⠰⠆⠀⢼⠅⠀⡱
⣿⣿⣿⣿⣿⣿⡝⣷⣾⣿⠟⢹⣽⣿⣟⣿⣿⣟⣷⣾⢫⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢣⠈⠃⠀⣰⡆⠒⠀⠤⠤⠄⢢⠘⠁⡜
⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⠆⣿⣿⣿⣶⣬⣾⣿⡿⣳⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢣⡀⡰⣹⠱⢢⠈⠉⠛⠃⠈⣆⠜
⣿⣿⣿⣿⣿⣿⣿⣿⠮⠍⠀⣻⡿⣿⣿⣿⡿⣿⡧⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣹⣳⣿⡆⢣⠂⠀⢀⣤⢀⣘⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧⢿⣼⣷⣭⣥⣤⡶⣴⣶⣶⣷⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⡌⠃⠚⠓⠛⠛⢩⠋⠉⠉⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡠⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⢯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡰⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢳⣄⠀⠀⠀⠀⠀⠀⢀⡤⢚⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⣾⣾⣿⣿⣿⣶⣤⣤⣾⣍⣴⠋⣀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡱
⠀⠀⢀⣠⣤⣶⡾⢿⣿⢿⣿⠿⠿⣿⡿⡿⢿⢿⡟⣻⣿⡿⣿⢿⢿⡿⣿⣿⣷⣤⣄⣀⣀⠀⠀⠀⣰⠋⣰⠁
⠰⢾⠋⠩⣽⣿⠃⠀⠉⢷⣶⣶⠶⠾⠿⠷⠚⠳⢶⣷⡌⠀⠁⠉⠚⠁⠉⠁⠋⠙⠛⠛⢛⣿⣾⣾⣿⣿⠃
⠲⢦⣧⡶⠈⠁⠀⢀⡄⣾⣸⣄⣀⠄⠒⠒⠒⣈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠟⠋⠋⠉⠉⠙⣿⠿⢇
⠀⠀⠈⠉⠛⠲⢦⣼⣿⣟⣻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣶⣶⣶⣾⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣟⠿⡿⠛⠛⠛⠛⠋⠉⠉⠉⠙⠯⣉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢬⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠔⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣏⣑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣭⣀⠀⡰⢲⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠫⣀⣤⡉⢟⠋⢁⣙⣻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠻⣿⠇⣠⣿⣿⣿⣯⠻⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠑⢄⢰⣾⣿⣿⣿⣿⣿⣿⡄⢸⣷⠲⢦⣀⠀⠀⠀⢀⡤⠴⠖⠒⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢻⢿⣿⣿⢿⠋⠉⠉⢹⢺⡟⣦⠒⠈⣷⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⣄⣠⣚⡳⢤⣼⣀⡀⠀⠐⠀⠀⠀⢰⡼⡙⢾⣅⡼⢹⠀⡼⠁⠀⠁⠀⠀⠀⠀⠀⠈⠑⢬⡂
⠀⠀⠀⠀⠀⠀⠀⡇⣶⣦⣌⠲⣍⠻⣷⡀⠀⠙⡛⠿⠶⠶⠶⡻⠶⣯⡤⣿⠀⡿⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⢧⢹⡟⠀⣇⢸⠑⠮⠳⡀⠀⠀⠁⢩⠍⣀⢀⣀⣷⣳⣏⠎⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠸⣧⠀⢽⠸⠀⠀⠀⣿⣴⠾⢙⡿⠀⣴⣿⣿⡿⠛⢧⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠓⠛⠤⣄⣀⣀⣿⣿⡽⠽⣇⣼⣿⣟⣽⠁⠀⠀⡽⠀⠠⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢀⡷⠗⣦⠛⣦⠎⠈⢹⠅⢀⣒⣚⣉⣉⣉⣠⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⡠⠤⠾⡿⠟⡿⣤⡏⠀⠀⢸⡖⠓⠒⢦⣄⣀⣠⡴⣿⠀⠀⠀⣷⠀⢰⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⠸⡄⢸⠄
⠀⡖⠊⢰⣾⠒⠒⠷⠻⣀⡇⠀⡇⠀⠀⠈⣇⠀⠀⣀⣘⣟⢿⡞⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠃⠀⣇⡟
⠀⡇⠀⢐⣿⠀⠀⢰⠀⡏⠩⢕⣧⢄⠀⠀⣸⣶⠏⠉⠉⢹⣦⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠢⡈⠑⢲⡿
⠀⠹⠤⢴⠻⣄⣀⣉⣠⢿⠒⠊⣈⠎⠀⢠⠿⠍⠆⠀⠒⠛⣿⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⢘⣇⡔⡻⠁
⠀⠰⠉⠉⣦⡝⠀⠀⠙⣬⠿⠏⠉⠀⠀⠈⣧⡀⠇⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠉⠉⠀⠒⠀⠋⣰⠃
⠀⠸⠭⠭⠏⢇⢀⠀⣀⡇⠀⠀⠀⠀⠀⠀⢸⠿⣊⣭⣭⣽⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠓⠒⢦⢄⠀⠇
⠀⠀⠀⠀⠀⠘⣼⣄⣿⡇⠀⠀⠀⠀⠀⠀⢘⠿⠋⠉⠀⠈⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⢬⣭⣈⣡⣤⡤⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⡆⣾⣿⣴⣿⢀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡀⣿⣦⣿⣷⢸⡷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣷⡹⠟⣼⣫⠟⠁⠀⠀⠈⠻⣿⣯⣽⣿⠟⠀⠀⠀⠈⢿⣝⣧⠻⢏⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⢀⣠⣤⣤⣤⣭⣭⣤⣤⣤⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⣠⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣄⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣾⣿⣿⠗⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠸⣿⣿⣶⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣦⣌⣛⠿⢿⣿⣿⡿⠿⣛⣡⣴⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣥⣚⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⢻⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠒⠉⠙⠛⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⢀⣤⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⢷⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠛⠉⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠶⢤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞⠛⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⢀⡰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡤⠴⠖⠛⠉⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡖⠛⠚⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣴⠟⠋⠉⠉⠙⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠸⣷⣀⡀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢀⣀⣿⣏⣉⣛⣛⣁⡀⠀⠀⣠⣤⣄⡀⠀⠀⢀⣤⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⡶⠟⠛⠉⠉⠉⠉⠉⠉⠙⠛⠻⣿⠃⠀⠈⢻⣦⣴⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠈⠁⢀⣀⣙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⢀⡾⠋⠉⠙⢿⡀⠀⢀⣠⣤⣤⣴⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⢾⠃⠀⠀⠀⣼⣷⡾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠲⠿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⢳⡄
⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⡾⠁⠀⠀⠀⠀⠈⢣⡀⠀⠀⣴⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⠀⠀⠀⣸⠇
⠀⠀⢀⡴⠋⠀⠀⠀⠉⠻⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡟⠛⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣸⣟⠃⣠⠀⠀⠀⠀⢾⣿⡿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠁⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣰⠚⠁⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠤⣤⡟⠉⠉⠛⠛⠿⢿⣷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢀⡴⠖⠒⠶⣼⠇⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡀⠀⠀⠀⠀⠀⠀⣽⠟⠛⠻⠶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠾⠋⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣀⡀⣰⡟⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⢻⣿⠿⠿⣿⣥⣀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠈⠙⢷⣦⠀⠈⠳⠦⣤⣀⣀⡀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠉⠙⢦⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠈⠉⠉⠑⠛⠁
⠀⠀⠀⣠⠶⠖⠶⢴⡏⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⠛⠻⠷⠶⠾⠟⠀⠀⣀⣀⣀⡀
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣴⠿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠙⢦⡀
⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⣴⠾⠛⠁⠀⠀⠉⠛⣿⠶⣶⣶⣶⣶⣶⠾⠛⠁⠈⠻⣦⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣷⣀⡤⣤⣀
⠀⠀⠀⠛⠦⣀⣀⠀⠀⠀⠀⢀⡾⠁⣰⡟⠁⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠙⢿⣤⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠋⠀⠀⣿⡁⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⣤⣀⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⣀⣼⠇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠿⠟⠛⠁⠀⠀⠀⠀⠀⠱⠤⠤⠤⠤⠤⠤⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡈⠙⠻⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠉⠻⢿⣶⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⢿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡿⠿⢿⣶⣦⣄
⠀⠀⠀⠀⠈⢿⣧⠉⠛⠻⢿⣶⣄⡀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠈⠙⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠉⠻⢿⣦⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⢀⣠⣴⡿⠟⠛⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⢀⣿⣟⣠⣴⣾⡿⠟⠋⠀⠀⠀⠀⠈⠙⠿⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⢾⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⡿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⣀⣤⣾⡿⠛⠛⠛⠿⠿⠿⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰⡀⠀⠀⠁⠈⠁⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣄⣠⣶⣶⣷⣶⣶⣦⡀⠀⠀⠙⠒⠀⠂⠁⠤⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣴⡿⠃⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠋⠀⠀⢿⠆⣤⢸⣿⣿⡿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣰⣿⣤⡄⠀⠀⠀⠀⠘⣿⣿⣷⣶⡀
⢠⡶⠂⠀⠀⠀⠀⠀⠻⠃⡠⠀⢀⢀⣤⣤⣀⠀⢠⣿⣿⣵⣶⣾⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣣⠀⠀⠀⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡁⠉⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⠉⠉⢰⣶⣶⣴⣶⣤⠀⣉⠋⠀⠀⠀⠀⠀⣤⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣴⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⢿⣿⣿⣿⡦⡤⢤⣀⡀⠀⠀⠀⢀⣸⣿⣶⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠒⠒⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠏⠉⠉⠉⠉⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠻⣿⣿⣇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣸⠿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂⠀⠀⢸⠧⠆⠠⠀⡀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢀⣀⣡⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠐⠒⠂⠉⠋⠁⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⣩⠎⠀⠀⠀⠀⠀⢀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⠞⠁⠀⠀⠀⠀⢀⣀⠆⠠⡀⣆⠀⠀⠡⠀⠀⠀⠀⠀⢣⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡐⢰⠃⠀⠀⠀⠀⠈⠀⡈⠀⠀⠀⢻⣆⠀⠀⠑⠄⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⠀⠀⠁⠑⠄⠀⠀⠀⠀⡀⠀⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⡀⠀⠑⣀⠴⠒⠚⠛⠛⠂⠀⠀⠖⠚⠋⠛⠒⢼⠇⣾⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡽⢿⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠁⠂⠤⣀⣀⣀⣀⠄⠀⠀⠀⠀⠀⠘⠀⠀⠑⠦⡁⠠
⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⡧⠉⠉⠀⢤⠄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠄
⠀⠀⠀⠀⢀⠌⠀⠀⡐⠁⠀⠀⠀⠀⠘⢣⡤⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠙⢮⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠁⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡌
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠴⠚⢦⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠡⠀⠀⠀⠀⠁⠀⠁⢀⠀⢠⡞⠀⠀⠁⠂⠤⢐⠡⠤
⠀⠀⠀⠀⠀⡀⠈⠢⡀⠀⠀⠐⠀⠀⠀⢂⠀⠀⡀⢀⠀⢄⠰⡀⠀⡠⠁⠌⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠁⠀⣤⢔⠖⢣⡀⠀⠀⠀⠁⠂⡄⠂⠘⠁⠀⠈⡁⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⢆⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠐⠃⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠁⠀⢀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢀⡀⠀⠀⣀⠘⠻⢛⠛⠋⠉⠉⠉⠉⠁⠐⣒⡄⠀⢄⣀⡀
⠀⠀⠀⠀⠸⣷⡴⠾⠳⠦⠾⠛⠉⠉⠀⠀⠀⠀⠈⠀⠀⠄⡀⠀⠀⠑⠿⢿⡯⠝⣚⣩⢠⠄
⠀⠀⠀⢀⡀⢽⡇⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⠀⠃⣀⣉⡴⢀⣤
⠀⠀⠀⠸⠟⠛⠋⠀⠀⣠⣌⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠠⢐⠁⠀⠈
⠀⠀⠀⠀⠀⠐⢠⠤⢿⠀⢦⠀⠀⠀⠠⣲⡦⠤⠄⠀⠀⠀⠀⣰⡠⠟⠓⠁⣰⠄⡿⠤⡄⠂
⠀⠀⠀⠀⡠⠒⠁⠀⠸⠖⠉⠁⠀⠀⠲⠀⢀⡀⠀⠀⠀⢀⡀⠀⠔⠂⠀⠈⠉⠲⠇⠀⠈⠐⠄⡀
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢒⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄
⠀⠀⡶⣪⣭⣭⣭⡲⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⡇⢸⡇⠀⠀⠀⠀⠀⠀⠄⢸⠃⠀⠀⠇⡠⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⣿
⠐⢾⣸⣿⣿⣿⣿⣿⣸⡖⠀⠀⠀⠀⠀⠀⣿⠀⡇⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣠⣿
⠀⠿⡸⣿⣿⣿⣿⡿⣸⠷⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⢸⡇⠀⠀⠀⠀⠀⡠⠐⣎⠀⠉⠀⢰⡖⢤⠀⠀⠀⠀⢀⠠⠟⠁⠀⠈⠻⠄⡀
⠀⠀⠟⢮⣽⣿⣭⠞⠇⠀⠀⠀⠀⠀⠀⠀⡏⢸⣁⡱⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤⣴⣿⡇⠀⡇⠀⠀⠀⡇⣠⣷⣤⣀⣤⣶⣆⢨
⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠒⠚⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠓⠀⠃⠀⠀⠀⠃⠘⠛⠛⠉⠛⠛⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⡀⡐⡞⣎⢤⡲⣤⡀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠈⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⣀⣞⣗⡀
⢀⣾⣹⡇⣿⢹⣏⠿⠜⠿⠀⠀⠀⠀⡞⠀⠀⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀
⠘⠛⠉⠁⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠀⣠⠀⣠⠀⣠⠀⡇⠀⠀⠀⠀⠀⢨⢠⠴⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⡇
⠀⠀⠒⠒⠖⠒⠳⠒⠒⠒⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣀⠨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⡤⢴⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣬⣤⡅⣤⣬⡄⠀⠀⠀⠀⠀⢀⣠⣤⣭⣬⣭⣥⣤⡀⠀⠀⠀⠀⠀⢰⣤⣥⣦⣬⡆⣼⠇
⢸⣿⣿⡟⠙⠋⢻⣿⠉⠉⢣⠀⠀⠀⠀⠀⠸⣿⣿⣸⡿⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢿⢿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣇⣀⠼⢿⡇
⢸⣿⣿⣧⣤⣤⣼⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⢻⣿⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⢢⣼⡇
⠾⢿⣛⡿⠿⠿⠿⠿⠿⣟⣻⠻⠇⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠀⠉⢃⣀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⢀⡠⣤⠤⢀⣀⣀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⣠⣖⡽⣾⣿⡿⣟⠔⠋⠀⠀⠀⢠⣿⣿⡿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣤⣀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⠿⠿
⠀⡜⠇⠀⠉⠉⠉⠛⠋⠁⠀⠀⠀⠀⢠⣿⣿⢿⡇⡿⣻⣿⣧⠀⠀⠀⠀⠀⠺⣿⣿⠆⡙⢛⣋⠀⠀⠀⠀⠀⠀⢠⣟⣻⡄⠈⠙⠻
⠉⠫⣤⡀⢠⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⣾⣿⡿⠃⠀⠀⠀⠀⠀⢰⣬⣉⠶⡏⠻⠏⠀⠀⠀⠀⠀⠀⣤⣘⢻⣀⣤
⠀⠀⠁⠀⠋⠉⠉⠉⠩⠛⢄⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⠖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠒⠒⠚⠒⠓⠒⠂⠀⠀⠀⠀⠀⠐⠒⠒⠚⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠋⠀⠈⠳⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⢀⣀⠤⠤⠤⠄⣀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣤⣄⣠⠞⠁⠀⠀⠀⠀⢸⣁⡄⠀⠀⠀⠀⠈⠙⠢⡀⢠⠖⠋⠁⠀⠀⢠⣀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⢰⢉⣴⡟⣠⠖⠊⠙⢢⡀⢸⠃⣠⠔⠒⠤⣰⣆⢹⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠁⠀⠀⠛⠉⠑⡖⠉⠉⠀⠀⠘⠚⠉⢦⠀⠀⠈⠁⠘⠶⠁⠀⠀⠀⠀⠹⡀⢰⠁⠀⠀⠀⠀⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⢰⣖⡄⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠘⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠰⣚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⢠⡀⠀⣠⠒⠚⢧⡀⠀⣀⠀⢀⠶⠐⠋⠀⢀⣀⣀⠤⠴⠶⣤⣤⣤⣴⡿⣧⣤⣤⣤⠶⠶⠤⣀⣠⣤⣄⡠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⣈⢹⠁⠈⢯⢀⣀⣀⣀⠉⡿⠀⠹⠯⠐⢲⠊⠩⡔⠂⣭⠙⠛⢣⣤⡄⠀⠀⠛⠃⠀⠀⣤⣤⠛⠛⣭⠑⠒⡍⠑⡆
⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠈⠙⠳⢶⣞⠁⠀⠀⠀⠀⠓⠒⠲⠃⠀⢸⠉⠒⠬⠥⠴⣞⣛⣣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣛⣛⡶⠤⠭⠔⠊⡃
⠀⠀⠘⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⢤⣄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠖⠶⢶⣅
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⢁⡶⠛⢷⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣀⣠⡤⠶⠶⠶⠶⠶⠶⠤⣤⣄⡀⢸⡅⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠻⠷⠀⠀⣯⠴⠶⠤⣤⡀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⡀⢠⡞⢙⡆⠀⠀⠀⠀⢹⡶⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠀⣉⠋⠰⣦⣤⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣴⢻⡆⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠻⢤⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⢠⣄⣻⣿⣀⡶⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⣾⠀⠀⠀⣠⠤⣾⠁⢹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠣⣄⢴⣛⡏⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣥⣤⡀⠀⠀⢰⡅⢀⣭⠀⢹
⠀⠀⠀⠀⠀⠀⢀⡔⢢⡀⠀⠀⠀⠀⠀⣾⠉⠀⠉⠙⠻⢶⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣤⣴⡾⠛⠉⠀⠀⠀⢹⡄⠀⡠⠭⣉⣿⢉⠡⡀
⠀⠀⠀⠀⢰⠋⠙⣤⣴⠗⠲⡄⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⣯⠀⠀⠀⠀⢀⣠⠟⠀⠀⠘⠲⠼⠿⣥⠔⠁
⠀⠀⠀⠀⠈⡳⠈⠦⠛⠰⡞⠁⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣠⠴⠞⠉
⠀⠀⠀⠀⠀⠣⠤⣿⡦⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⣉⢲⣿⡶⢊⡶⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠑⢉⡴⠴⠇⠀⣿⠠⣄⠀⠀⠀⠀⠀⠀⣹⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠮⣹⢇⣠⠜⠀⠀⠀⢀⣠⠞⠁⠀⠈⠉⠓⡶⠒⠒⠒⣶⠚⠉⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣰⣆⣀⡽⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠤⣸⣯⡠⢤⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠛⠐⠊⠀⠀⠀⠙⢦⣀⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣆⡀
⠀⠀⠀⢀⡶⣄⡀⠀⣀⣴⣾⣿⡿⠛⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⠿⣿⣿⣦⣄⠀⠀⣀⡴⡆
⠀⠀⢙⣺⣷⣿⢿⣿⣿⣟⣉⣁⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⣉⣙⣿⣿⣿⢿⣷⣷⡊
⠀⠀⠀⠀⠈⠁⢾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠆⠉
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢛⣿⣿⣿⣿⣿⣟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠁⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠁⠀⠀⠀⠀⠻⠿⠿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠠⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣿⣃⠈⠲⡄⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠁⠀⠀⠀⠀⢀⠀⠈⠻⢿⣿⡅⠀⠈
⠄⠀⠀⠀⠀⠀⠈⢱⠶⠲⠜⠛⠃⠠⣀⠀⠀⠀⠀⣄⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠆⠀⠀⠈
⠀⠀⠐⠂⠀⢠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠶⠶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡿⠛⠛⠉⣀⣠⣤⣄⣄⣈⠉⠙⠛⠿⢷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣿⣆⣴⠾⣶⠜⠛⠙⠛⠛⢿⣿⣷⣶⣶⣤⣄⡙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠋⠙⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣧⣼⠛⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣷⣀⣠⣬⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣷⣾⣠⣿⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⣿⠻⠟⠛⠛⠋⠉⠉⠉⠙⠛⠱⠄⠈⠉⠀⠸⠿⣧⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡀⢀⣀⣀⣾⠇⢀⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣾⣿⣷⠶⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⣿⣿⣿⠁⣤⠘⢻⣿⣿⠟⣷
⠀⠀⠀⠀⠀⠀⢠⡄⢀⣿⣿⣇⣤⡌⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⠀⢻⣿⣦⠈
⠀⠀⠀⠀⠀⠀⣿⠇⢸⣿⠏⠘⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠻⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠹⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠀⠠⠄⠀⠀⠀⣀⡀⠀⣀⢀⣀⠀⠀⠤⠤⠤⠤⠶⠆⠴⠶⠄⠀⢻⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠶⠶⠀⠀⠉⠛⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⢸⣿⠀⢠⣤⣶⠖⠀⢷⣄⠀⠠⠆⠀⣴⠆⠠⠤⠐⠀⠀⠀⠠⢘⣿⣦⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⡇⠸⣿⠟⠁⠈⠁⠀⣀⡘⢿⣆⢀⣤⠈⠋⣀⣤⣴⣶⣶⣶⠄⠀⣿⣿⣿⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢃⡼⠿⣧⣾⣿⣿⣿⠿⠿⣿⣿⣷⡄⠻⣿⣽⣿⣷⣿⣿⣿
⠀⠀⠀⣤⣤⣤⡄⠰⡿⣿⡆⢸⣿⣶⣤⡾⢶⡤⣾⠿⠀⢸⣿⣿⣿⣄⣀⠀⣴⣿⣿⣿⡷⠀⠘⣿⣿⣿⣿⣿⣿⣦⠄
⠀⠀⠈⠛⠉⠀⠀⠀⠀⠈⢀⡀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⢀⡈⢩⠍⠉⠀⠀⠀⣿⡟⠻⠿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣶⣾⣿⣶⠀⠀⣀⠀⢸⣟⠉⢙⠋⢻⣀⣠⣄⣿⡇⠀⢿⠳⣶⣄⠈⢿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣠⡿⢃⣸⠟⢹⣉⣹⣿⣿⣿⣇⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠘⠀⠈⢿⡿⠀⠈⠙⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡟⢀⣾⡟⢰⣾⢉⣉⣉⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠋⠰⡿⠟⠀⣼⣿⠿⢿⣿⣿⡟⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠃⢰⣀⣀⣀⣠⡆⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⢠⣶⠄⠀⠠⣀⣰⠟⣿⡄⠘⠛⠛⠁⠀⣰⣧⠀⠀⢤⠞⢲⡄⠀⠀⠈⠉⠁⠙⣷⡞⠛⠟⠉⠉⠃
⠀⠀⠀⠀⠀⣼⢃⣰⣿⢃⡈⠻⠊⢻⣷⣶⣿⣿⡖⣿⣿⣿⣧⣀⣀⠀⣺⣿⣤⣤⣴⣶⣶⣿⣿⣿⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⣴⡇⢸⡟⠀⠿⠁⠀⠀⠀⣥⣻⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠿⢀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠙
⠀⠀⠀⠀⣿⠀⠀⢀⡾⠀⠰⠇⣰⡄⣿⣧⣿⣿⣟⣿⣿⣿⣿⠿⠿⠿⠿⠟⠻⠟⣻⣿⣿⣤⣿⡟⠀⠀⠀⣼⠀⠀⠀⠀⢀⣿⡇⠀⢸⡇
⠀⠀⠀⠀⢉⣹⠃⠘⣷⣌⠀⠈⣻⣿⡿⠛⠻⡿⠟⠛⠻⠟⠛⠿⠛⠛⠻⠓⠀⠻⠋⠛⠃⠀⠀⠀⠀⠠⠖⠀⠀⠀⢀⣰⡿⠋⠀⠀⢸⡇⢠⣾
⠀⠀⠀⠀⠘⢿⣷⣶⣞⣛⡷⣾⣿⠋⠀⠀⠀⠀⠀⣴⣶⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⢠⣤⡿⠇⣾⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣾⣿⢰⣄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠤⠄⠊⠉⢁⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣹⡄⠈⠛⠷⠂⠲⠂⠀⣀⢀⣀⡀⠀⠀⠀⠀⠤⠤⠤⠖⠂⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⡇⣨⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⢀⡀⣿⡇⢠⡴⠶⠂⠲⠖⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠀⠈⠉⠉⠁⠀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⡗⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣤⣤⣤⣄⠴⠶⠿⠿⠃⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣤⣿⣿⡄⠀⠀⠚⠛⠛⠓⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠰⠞⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⠀⠀⠀⠚⠃⠀⢸⣿⠟⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡾⠿⠿⠿⣧⡄⠀⣤⠀⢠⣶⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡀⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⣩⠙⠛⢾⣵⠛⠳⣍⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⣡⣿⣿⡟⣤⡀⠳⡘⡷⣤⣟⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⢶⣦⣶⡟⠉⢾⡿⣹⡟⠠⡉⠱⡀⠉⠈⠪⣽⣧⠶⢦⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⢿⣿⠚⢽⣷⡂⢰⠟⢀⡷⣄⡖⠉⠀⢠⡧⣆⣈⣥⡾⢚⡥⠲⠂⠳⠬⣿⢦⣝⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣭⢿⣿⠦⠭⣿⡷⣿⣟⠛⡁⣤⠙⡓⣛⣛⢶⣯⣑⣊⠉⠙⡇⠀⠨⣯⢖⣿⣿⣷⣿⢿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠡⣿⣖⣾⣿⠍⠻⠿⡿⢺⡗⢋⣤⣈⣉⣁⣀⣀⣍⣀⣉⣍⣛⡓⣿⡄⠸⡟⠾⡿⣿⢽⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠐⣻⣧⣥⡿⠾⠿⣛⠛⠛⣛⡛⢋⣭⡉⣭⣅⢨⣭⣭⡉⣭⠉⣭⠉⣉⠛⣛⣛⠳⣿⣾⣿⣟⡂
⠀⠀⠀⠀⠀⠀⠀⡾⢛⢿⣯⢗⠂⣾⠛⠷⠀⣿⣧⢸⡟⣧⡿⣿⢸⣧⣼⠃⣿⠀⣿⣧⣿⢸⡏⡻⠂⠒⠛⢽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣧⣶⢷⢻⣁⡄⢻⣇⣶⢸⡿⢻⡌⡧⠹⠇⠿⠸⡇⠀⠀⠿⠀⡿⠹⡏⢹⣯⣿⠃⠀⠾⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣦⣛⣼⠯⣿⣂⣭⣥⣤⠴⣒⣤⠴⠶⠒⠒⠚⠚⠛⠛⠛⠒⠒⠒⠒⠶⠶⠶⠤⣭⣭⡭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠀⠀⠻⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠂⠀⠰⠆⢠⣩⣅⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠄⣀⣀⠀⠁⠀⠀⠀⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⣿⠋⢀⣴⡄⠠⣀⡀⣀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣿⠏⠒⠛⠉⠀⠀⠀⠀⣉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣠⣾⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢿⡄⠀⠀⠐⣦⣿⣧⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠈⠻⣦⡀⠀⠀⠈⠛⠋⠓⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠹⣿⣿⠀⠀⠀⢀⣠⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣦⡀⠀⠉⠁⠀⠀⠀⠈⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠁⠙⠷⣦⣄⣀⡀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡶⢄⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡤⣀
⠀⠀⠀⠀⡠⠇⠀⠀⢸⠀⠀⢠⡄⠀⠀⠀⠀⢠⡤⠤⠋⠹⣄⡀⠀⠀⢸⠀⠈⠳⣄⠀⠀⠀⠀⠀⣰⣲⡤⣀⣀⠴⠊⢹
⠀⠀⠀⠈⠑⢢⠀⣠⠤⣷⠤⠇⠘⠒⢲⠄⠀⠀⢹⠆⠀⠀⡴⠋⠀⠀⣸⠀⠀⠀⠀⠑⠖⠒⠊⠉⠁⠀⡎⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠉⢲⠀⠀⠀⣏⠀⠀⠀⠛⠒⠙⠢⠆⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠈⠳⣄
⠛⠳⠶⣤⣄⡀⠀⠀⣤⠤⠼⠶⠋⠉⠻⣀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡀⠀⠀⠀⢀⡠⠤⠔⠃
⠀⠀⠀⠀⠉⠙⢷⣄⡘⢦⠀⠀⠀⠀⠀⠈⠉⣱⠆⠀⠀⠾⣁⠀⠀⠀⢠⠟⢦⡀⣄⠀⠀⠀⠀⠀⠙⣌⠉⢧⠀⢀⡎⢀⣀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡃⠀⠀⠀⠀⢰⠚⠁⠀⠀⠀⠀⠈⠙⠳⢞⡋⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠘⢆⣠⡷⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡶⢤⡀⢨⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣙⣀⠴⠤⠷⠀⣠⠶⠤⢤⣤⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⢋⣠⣴⠶⠞⠛⠛⠉⠙⠉⠉⠙⠛⠻⠶⣶⣾⡁⢀⣴⠿⣟
⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍⠛⠛⠁⠀⢿⣄⡀⠀⡀
⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⢉⣿⠟⣿⠀⢀⣠⣤⣴⠶⠶⠛
⠀⠀⠀⠀⠀⠙⠷⣄⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣀⣄⠻⣏⠉⠛⠁⠀⢻⣾⡛⢯⣀⣀⣀⡀
⠀⠀⠀⣀⠀⢀⠜⠉⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠻⣿⠆⠀⠀⠀⠀⢈⣹⡷⠀⢀⡴⠁
⠀⠀⠹⣄⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣠⣤⡀⠀⣾⠋⠁⠀⠀⢮
⠀⠀⠀⢈⠆⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠈⠉⠁⠀⠙⢶⡿⢀⡤⠤⣀⡈⡆
⠀⠀⠀⡜⠀⣀⣼⡇⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⢸⡗⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠉⠉⡀⢸⡇⠀⠀⠀⠀⠀⠈⢿⣿⠇⠀⠀⠀⠀⠀⠀⣾⣷⠄⢀⡀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⢘⣷⠤⣠⠔⡆
⠀⠀⠀⣀⡜⠙⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠾⠷⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠆⠀⠀⠓⣄
⠀⠀⠘⢦⡀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠓⠒⢄⢰⠊⠉
⠀⠀⠀⠀⡥⠖⠒⠼⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠷⣤⣀⣀⣠⡶⠛⣿⢀⣰⠟⢡⠤⢄⣈⠵⢢
⠀⠀⠀⢀⠤⣀⠀⠀⠀⠀⣙⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠉⠉⠁⠀⢀⡿⠟⠁⠀⠀⡇⠀⠀⠀⢦⡀
⠀⠀⠀⠘⡄⠈⠑⠒⢒⣉⣤⣷⠤⠿⠿⢶⡶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠠⣎⣀⡀⠀⢀⡤⠽⠂
⠀⠀⠀⣠⠇⠀⢠⡾⠋⠉⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⢀⣤⡴⢦⣌⠻⣦⣀⡜⠀⠙⣄⡾⠄⢤
⠀⣴⠊⠁⠀⠀⢺⡁⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⢦⣤⡄⢠⡟⠁⠀⠀⠉⠻⣾⠋⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠉⠛⠒⡄⠀⠛⣷⠶⠶⠶⠶⠾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣇⠀⠀⠀⠀⠀⣿⢦⣠⣄⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠸⡀⡰⠃⠀⠀⢠⣤⣤⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡷⣦⣤⣤⠾⡏⠉⠁⢸⡖⠲⢦⣬⠆
⠀⠀⠀⠀⠀⠀⠙⢁⣤⠶⠟⠛⠋⠁⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸⠀⠙⣶⣁⡀⠀⡤⡾⠷⠎⠘⢤⣀
⠀⢰⡛⠒⠂⠶⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣤⣀⣀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁⠀⢸⠀⠀⠀⠉⠻⠶⠷⠺⠆⠀⠀⡰⠋
⠀⠀⠳⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠈⠉⠙⠛⣿⠛⠛⠛⠋⠙⣧⠀⠀⢀⡹⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⡶⠖
⠀⠀⠀⠱⡀⠀⠙⢧⣄⠀⠀⠀⠀⣀⡴⠟⠙⠲⢤⡀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢻⣦⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⢀⠇⠀⠀⠀⠉⠛⠳⠷⠛⠋⠀⡤⢄⡠⠖⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⢀⠎⢠⣀⠀⠀⡀⠀⠀⠀⠀⢀⡤⣻⠀⠀⠀⠓⣤⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢀⡎⠀⡼⠈⠉⢉⣇⠀⠀⠀⠀⢸⠐⠧⠰⣄⣰⠋⠁⠀⠈⢿⣄⠀⠀⠀⠀⠀⢠⡟⠀⠘⡇⠀⠀⠀⠀⣠⠤⣀⣀⣀⣀⣘⠆
⠀⠀⠙⠻⢤⡀⢀⣀⣻⠷⣄⠀⠀⡘⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠙⠳⠶⣤⣤⠶⠟⠁⠀⠀⢱⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠳⠊⠀⠀⠀⠈⠳⣄⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠚⠛⠛⢀⠀⠘⠛⠛⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠲⠶⠖⠈⡇⠀⠳⠶⠃⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠘⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⠓⠒⠋⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣥⣀⣀⣀⣬⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡿⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠳⣇⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠋⢷⡀⠀⠀⠈⠓⠦⣖⣒⠒⠶⣤⠶⠋⠀⠀⢀⡴⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣀⣀⣀⠀⠙⠳⢤⣀⣠⡶⠤⠭⠇⠀⠀⠀⢀⣠⡴⠋⠀⠀⢀⣀⡈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣈⡉⠓⠲⠶⠶⠶⠒⢛⣉⣥⠴⠖⠚⠋⠉⠉⠉⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠓⠶⠶⠒⠒⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⠀⠀⢀⢀⣀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⢠⡾⣿⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠹⣟⠀⠙⣧⣠⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢰⡏⠀⠀⢻⡄⠀⠘⣿⠀⢻⣤⣤⣤⣤⣤⣴⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡍⠉⠛⢾⣇⣠⡴⠾⠓⠛⠲⠿⣤⡼⠿⢤⡀⠀⠀⠈⣿⢀
⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣈⣷⡀⠀⣠⡟⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡍⠉⠛⠛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⣤⣀⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⢾⡇⠀⠀⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠉⠙⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠚⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠞⠋⠻⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠛⠛⣿⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⣠⣤⠿⣳⣦⣄⣀⣀⣀⣤⣾⠀⠀⢸⡇⠀⠀⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠶⢻⡟⠁⠀⣰⡟⠀⢨⡏⠉⠉⢻⣾⠀⠀⢸⡇⠀⣀⣉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⢸⡇⠀⢠⡟⣷⠀⠀⣷⠀⠀⢸⣿⠀⠀⢸⡇⠀⠉⣉⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣾⠁⢸⡇⣰⠟⢷⣄⣿⢹⠀⠀⢸⡇⠀⠀⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠈⣿⠏⠀⠀⠈⢉⣿⠀⠀⠸⣇⠀⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣠⡾⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣅⣀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⠈⠻⣿⣷⡄
⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠁⠘⢞⢿⣿⢶⡀⠀⠀⠀⠀⠀⠀⠀⠻⢷⡦⠖⠀⠀⠀⢻⣿⣧
⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠈⠳⡝⠂⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆
⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⢤⠞⡄⠀⠀⠀⠸⣆⠁⢀⣤⣌⢳⡄⠀⣠⣶⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣸⣿⡇⠀⠀⠀⠀⣰⣦⡾⢳⣄⣠⡾⠻⣷⣧⣮⠆⠀⠀⠀⠀⢀⣴⠟⠹⡿⣿⣿⣿⡟⠈⢻⡿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠁⠀⠀⢀⣾⠏⠈⠁⣸⡿⠋⠀⠀⠀⢹⡷⣶⣄⡀⠀⡴⠟⠁⠀⠀⠙⠈⠻⣿⣿⣤⡀⠀⢈⢿⣦⡀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⢀⣴⣿⣷⢃⣴⠞⠋⠀⠀⠀⣠⠔⠁⠀⠈⠙⢿⣿⣄⠀⢀⣠⠶⣄⠀⠀⠈⠁⠙⠻⣶⣌⠳⣟⣿⣦⡀⠀⠀⠀⣿
⣿⣿⣴⣾⠗⠋⠴⠋⠁⠀⠀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠼⠛⠉⠀⠘⠿⣷⣤⡀⠀⠀⠀⠀⠉⠳⠈⠙⠮⣻⣦⣀⠀⣿
⢿⣿⡏⢀⡆⠀⣸⡀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠘⠻⢦⣀⠀⠀⠀⠀⣿⡀⠀⣬⡝⠻⢿⣿
⠸⣿⣧⣸⣧⢠⣿⣇⠀⣿⡀⢠⣷⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⣤⣤⣀⠀⢀⡀⠀⠀⠀⣀⣰⡄⣨⣿⡅⢴⣿⣧⠀⣾⣿
⠀⢻⣿⣿⣿⣷⣿⣷⢺⣿⣯⢰⣿⠆⡆⡄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⠀⣿⣀⣿⣿⣧⣽⣿⣯⣾⣿⣿⣴⣿⡟
⠀⠈⢿⣿⣿⣿⣿⣷⣾⣿⣿⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠻⣿⣿⣍⣉⣉⣉⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣿⣉⣉⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠈⢿⣿⣷⣤⢤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣷⣽⠷⠦⠤⣤⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⠿⠟⠛⠉

⠀⡄⠠⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀⣀⡀
⠀⡇⠀⠄⢘⠀⡆⢠⠀⠀⡇⠀⠀⠀⢀⣤⢾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢋⡉⠘⢉⠁⠀⠀⠤⠤⠄⠤⠤⠤⠤⠤⠠⠄⠠⠠⠤⠠⠤⠤⠤⠤⠤⠤
⠀⠇⢠⡁⠠⠀⡇⢸⠀⠀⡆⠀⠀⠀⠰⣧⣼⢻⣿⣿⣿⡟⠀⠀⠀⠀⠰⠶⠒⣏⣆⢩⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⡀⢘⠀⢰⠀⡇⢈⠀⠀⡇⠀⠀⠀⢐⣿⡿⠞⠋⠁⠈⣥⠀⠀⠀⠀⠀⢓⡵⣏⡽⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉
⠀⡄⠸⠀⢸⠀⡇⢨⠀⠀⣇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⢰⠤⠈⠢⢜⠋⠙⠟⠗⠈⠦⠒
⠀⡇⢐⠂⠠⠀⠁⠉⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⠃⠀⠊⠀⠑⠂⠀⡁⠀⠀⠀⠀⠀⡉⠀⠁⡀⢀⢀⠀⢀⠀⠀⠀⣀⣤⡤⠤⢄⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⢨⡀⠀⡀⠀⠀⠀⠀⠅⠈⠂⠀⠀⠀⠃⠛⠈⠃⠂⠀⠈⠀⠀⠀⣾⠟⢃⠀⠀⠀⠙⢢⡆
⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠼⠀⠀⠀⢁⠠⣄⣸⣷⣿⣿⣿⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣧⣤⠋⠀⠠⠄⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣹⣈⠀⢀⠀⣼⣠⣿⣿⣿⣿⣿⣿⠀⠀⠠⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⡄⢀⠊⠀⠀⠀⠀⡇⢀⡅⠎⣰⣾⣿⣿⣿⣿⣿⣿⣿⠀⢀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣳⠀⡀⠀⠀⠀⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠉⠀⠱⠄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⠀⡀⢐⡀⠀⠂⠐⠀⠀⠀⠀⠀⠈⠳⠖
⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠄⠤⠤⠄⠀⠄⠄
⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠒⠀⠂⠀⠒
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠘⠃⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣽⣺⠀⠀⠐⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠁⡀⠤⠠⠤⠄⡀⠉⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⢀⠔⠁⠀⠀⣀⠀⠀⠀⠑⢂⠀⠀⠀⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠎⠀⢀⠔⡩⠁⠑⢄⠀⠀⠀⢡⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⠔⡙⠁⠢⡀⢰⠄⠒⠁⡠⠁⠀⠀⠀⠈⠁⠀⣸⠀⠀⠀⣀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⠤⢃⠀⠀⠈⠉⠢⡤⠊⠀⣀⠀⠀⠀⢀⢀⠀⠈⢂⠠⠊⠡⣕⠠⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡈⣠⠃⠀⣿⠀⢀⢔⡂⠈⠀⠀⢀⣿⠤⡀⠔⢁⡆⠈⠒⣿⠄⠒⠊⣿⠀⠀⣭⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡶⣿⠂⠒⣿⡃⠁⢸⣇⡀⣰⠔⠊⣿⠀⢑⣶⣞⠓⠂⠀⠛⠒⠶⣶⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⢀⠐⢻⡇⠀⢽⡅⠀⠿⢀⠼⢿⣯⠳⣄⠀⠀⠀⠀⠉⠯⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠿⠄⠀⠈⣦⠀⠼⠅⠀⣻⣆⠀⡠⢂⢢⢁⠘⡆⠈⠫⠐⠀⠀⠀⠈⠢⡀⠓⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⠄⠠⡞⣛⣧⠆⠀⠐⢚⡲⢊⡠⠁⡀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠒⢨⡶⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠰⣾⡈⠩⢢⡿⠀⣀⣤⠀⠡⠄⠰⠃⣉⣃⠀⢘⡃⠀⢀⣀⠁⠌⠬⠃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠚⠛⠛⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠤⠀⠀⠠⠄⡀⠀⠀⠀⠀⡀⠔⢈⡀⣁⡉⠒⠤⢀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠒⠀⢄⡁⠒⠒⢉⠄⠊⠀⠀⠀⠀⠉⡲⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⡀⠀⠉⠉⠀⠀⠀⢀⠀⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣘⡻⢝⠻⡙⣌⠦⠰⠒⠋⠋⡇⠉⠉⢉⣀⣂⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠒⠦⢈⠓⢓⡙⠁⢠⣇⠀⠀⠀⠁⠀⠀⡟⢟⢉⡇⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠼⠷⠭⢍⣢⣄⣀⠴⠋⣢⠔⠃⢹⡄⠀⢀⣄⠀⠀⠙⠒⠛⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠠⣞⣭⣍⣑⡶⠋⢀⣤⣶⠛⠁⠀⠀⢰⡇⢠⠞⠉⠳⣄⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡯⠖⠒⣲⠏⠀⠀⠐⠋⠛⢷⡀⠀⢀⡾⡿⡧⡀⠀⠀⠈⣇⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠁⢠⠟⠒⠒⠶⣤⣄⡀⠀⠿⡤⠋⠈⠀⠁⢳⣠⠖⠀⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⣏⠢⡄⢻⠀⠀⣀⡤⢺⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡧⢴⠶⣾⠲⠒⠢⢜⡄⠘⣼⣠⠚⠁⠀⢸⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠱⡀⠀⠀⢡⢀⡟⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⠦⣧⣄⡀⣼⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⣾⣿⣦⣑⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⢻⢿⢿⡺⣷⣤⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠟⡇⠃⠈⢫⡳⡑⠢⡉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇⠘⠀⣀⣳⡈⠢⣈⠂⠌⣢⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡜⠀⠀⡸⠀⠈⠑⠤⠤⠌⠭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⠜⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⣏⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠀⠉⠉⠙⠲⠤⡄⢸⡤⠖⠒⠈⠉⠉⠉⠉⠓⠦⡀
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢇⡜⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣉⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⠖⠉⢫⡤⠚⠉⠀⠀⠈⠓⡽⢲⢄⠀⠀⠀⠀⠀⠀⢓
⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⠊⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠒⠋⠑⢟⢆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⠁⠀⢴⡄⠀⠀⡀⠀⠰⡆⠀⢸⠈⡄⠀⠀⠀⣸
⠀⠀⠀⠀⠘⡀⠀⠀⠀⢇⠀⠀⠀⠀⢧⡀⠀⠀⠀⠘⡵⢤⡀⠀⣠⠎⣰⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⠢⡀⠀⠀⠀⠉⠒⣦⠤⡄⠳⣄⠘⣯⢀⠔⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠈⠑⠲⠾⠶⢴⣟⣄⣈⡦⠤⠝⠛⠁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⢘⠀⠀⠱⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠉⠁⠀⠀⠈⠉⠀⠚⢦⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⣰⣀⣀⡀⢀⠀⢀⠴⠖⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠈⡋⠋⠧⡄⢀⣀⢠⠜
⠀⠀⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢀⠀⠀⠀⠀⠀⠑⡖⠊⢹⡁
⠀⠀⠀⠸⡀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠆⢸⠀⠀⠛⠀⠀⣀⠀⠀⣀⠈⠰⡆⠁
⠀⠀⠀⠀⠙⠢⡀⠀⠀⠳⣀⣀⠀⢀⣐⡏⢣⡠⢄⣀⠔⢁⠤⢵⣄⠀⠀⠀⠉⠀⠀⠋⠀⡰
⠀⠀⠀⠀⠀⠀⠈⢢⠔⠋⠁⠀⠀⠘⢤⡼⡿⢤⠴⠁⠀⠘⠢⠤⣠⠋⠁⠒⠒⠀⢴⠶⢎
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠸⠴⠉⢲⡁⠀⠀⠀⠀⠀⢸⡀⢀⡴⠒⢦⣸⠤⣄⡕
⠀⠀⠀⠀⠀⠀⢈⠀⢀⣀⡀⠀⠀⠀⠀⢰⡆⠀⡇⠀⠀⣀⠔⠊⠉⠁⠸⡃⠀⣸⠀⠀⠈⠉⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠚⠓⠀⠀⠀⣰⠃⠀⣼⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⠤⢤⠤⡤⠤⡖⢊⣼⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠀⠘⣄⠈⢦⢸⠯⠐⠒⢇⣻⣤⠄⢢⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠤⠤⠈⡑⠬⠮⠄⣀⡠⠎⢻⡚⢍⡒⠳⠭⠄⠱⣤⣔⣉⠉⠶⠶⠓⣂⠵⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⣉⠐⠒⠒⠂⠀⠐⠒⠒⢈⡁⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠉