Body language Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣶⡆⠀⠀⠀⢀⣶⣆⡁⠀⢀⣠⣶⡆⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣴⡦⠀⣳
⠀⠀⠀⣼⣿⣶⠀⠀⢀⣶⣾⣷⡀⠀⢀⣴⣾⣷⣀⠀⠀⠘⢿⣿⠃⠀⠈⠙⣿⣶⡄⠀⢀⡴⣿⡶⠶⠋⠀⠚⣿⣿⡄⠀⠀⢠⣿⣷⠋
⠀⠀⠰⢿⣿⡿⠇⠀⠉⢱⣿⡿⠇⠀⠉⢱⣿⡇⠉⠀⠀⠀⣿⣿⠀⠀⠀⢠⣿⣷⠇⠀⠀⢙⣿⣷⠀⠀⠀⠀⣿⣿⠇⠀⠀⣾⣿⣿⡀
⠀⠀⠀⢸⣿⡇⠀⠀⠀⢨⣿⡇⠀⠀⠀⠨⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⡟⢻⠀⠀⠀⠀⣿⣿⠀⠀⠀⠟⣿⠻⡇
⠀⠀⠀⠘⠛⠃⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠛⠃⠀⠀⠀⠘⠃⠘⠂⠀⠀⠀⠛⠛⠀⠀⠀⠀⠛⠀⠛
⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⣸⣿⣁⠀⠀⠀⣨⣿⣁⠀⠀⢠⣿⣻⣡⠀⠀⠀⠀⣺⣷⠀⠀⠀⢀⣿⣧⠀⠀⠀⢠⣿⣧⠀⠀⠀⢠⣾⣷⡀
⠀⠀⢀⣯⣿⣿⡄⠀⠘⢯⣿⣧⠃⠀⠀⢀⣿⡿⠃⠀⣠⠜⣻⣿⠣⣄⠀⢸⣿⣿⡀⠀⠀⠘⣿⣿⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⣸⡿
⠀⠀⠀⢹⣿⡏⠁⠀⠀⢸⠿⣷⠀⠀⠀⠸⣿⡇⠀⠀⠀⢀⡿⢿⡄⠀⠀⠈⢿⡿⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⣿⣿⠀⠀⢀⣰⢻⣿⢧⣀
⠀⠀⠀⢸⣿⡇⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⠁⠀⣷⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⡟⢿⠀⠀⠀⠦⣿⣿⡶⠄

⠀⠀⢀⣠⡶⢦⣀⠀⠀⠀⣤⡔⠟⠀⠀⠀⢰⣶⠄⠀⠀⠀⠀⣠⡀⢀⠀⠘⠦⣀⣴⠶⣀⡴⠃⠀⠀⡠⣶⡆⠀⣀⡄⠀⠀⠀⢠⣦⠀⣠⠔
⠀⠀⠈⠹⣿⡏⠉⠀⠀⢸⣿⣿⠀⠀⠀⠀⣾⣿⣶⠀⠀⠀⣾⣿⡷⠞⠀⠀⠀⠈⢿⣿⠁⠀⠀⠒⠚⠓⣾⡿⠉⠁⠀⠀⠀⠀⣿⣿⠏⠁
⠀⠀⠀⢸⣿⣧⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠚⢡⣿⣇⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⢀⣿⣷⡀⠀⠀⣴⣶⡞⣹⣿⣆
⠀⠀⠀⢸⣏⡽⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣿⠙⣧⠀⠀⠀⣾⠉⢻⡄⠀⠀⠀⢀⡟⢹⡇⠀⠀⠐⠓⠚⠟⠉⢷⡀⠀⠈⠋⢠⡏⠉⠙⣦⡀
⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠛⠀⠛⠀⠀⠘⠃⠀⠘⠃⠀⠀⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠘⠃⠀⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠂⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⠂⠀⠀⠀⠀⠀⠑⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⢀⠀⠀⠀⡀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⣀⡀⠀⠘⠢⠄⠤⠔⠂⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠷⠂⢉⡡⠄⠚⠀⡜⢰
⠀⠀⠀⠀⠀⠀⡇⠀⣸⠀⠀⠀⠀⠀⡀⠁⠈⠀⠀⠄⠀⠤⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠑⠢⣍⠈⠁⠒⠊⠁⢸
⠀⠀⠀⠀⠀⠀⡇⠀⡎⢆⠀⠑⠀⡀⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⣀⠈⠑⢄⣀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⢇⢠⠉⠒⠤⣈⢂⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠁⡞⠁⠀⢤⠈⠉⠆⢸
⠀⠀⠀⠀⠀⠀⢈⠀⠀⢨⠀⠀⠀⡞⣨⠁⠈⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠒⠀⠀⠌⠀⢀⠴⠂⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⠀⠀⠈⠀⠀⢀⠀⠈⡀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⠀⠀⡜⠀⠀⢸⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠐⠀⠀⠘⠀⠀⡇⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⡔⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠰⡃⡀⠤⠴⡄⠀⠈⠀⠀⠑⠚⠑⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢄⠰⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⡀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠁⠀⣠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣞⠃⠀⣀⡀
⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⡄⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠈⠭⠀⠀⠩⠅⠈⠠⠄⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠙⠃⠘⠃⠈⠛⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠸⠀⠰⠶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⡀⠀⠀⢀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠂⡿⠿⠟⠛⠄⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⡄⣿⣿⣿⢯⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⡐⠄⣀⡀⠠⠈⣰⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡉⠁⠀⢀⠐⢠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠉⠱⠞⠹⠂⠀⠈⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⠓⠀⠄⠀⠀⠚⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠄⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣤⣶⣧⣑⡒⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⡰⣡⣶⣶⣶⣮⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠛⠛⠛⠃

⠀⠀⠀⡀⠀⠀⠀⢀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡀⢠⠄
⠀⠀⠀⠹⣦⣾⣦⠏⠀⠀⢷⣾⣧⡾⠃⠰⣇⢰⡶⠀⠀⢰⡀⢶⠆⣶⠀⠀⢀⣿⡆⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠋
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⠏⠀⠀⠀⢸⣿⣷⡄⠀⠀⠙⣿⡏⠁⠀⢸⢹⣿⡏⣷⠀⢀⡤⣽⡥⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⣿⣟
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡆⠀⠀⠀⣾⢿⠛⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⢿⡇⠀⠀⠘⢀⣿⣇⠛⠀⠀⠀⣠⣼⣯⣤⠀⠀⠟⠟
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⠇⠀⠀⢰⡇⢸⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⡿⢸⡇⠀⠀⣠⠟⠁⠙⣦⡀⢀⣘⣇⣿⣿⣸⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠶⣤⣀⣾⠆⠀⢳⡄⢸⣇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣤⡀⠀⠀⡄⢠⣤⠀⠀⠀⠀⠀⡀⢶⠆⡀⠀⠀⠀⢴⡆
⠀⠀⠀⠀⣽⣿⣷⠀⠀⠙⣾⣯⣷⠀⠀⠀⠈⠳⣤⣀⣠⣏⣉⠀⠀⢠⣿⡿⠿⠂⠀⠀⠙⢲⣷⠶⡄⠀⠀⠞⠉⣿⡏⠙⠆⠀⣰⣿⣿⡄
⠀⠀⠀⢠⡟⣟⠁⠀⠀⠀⠈⡿⣯⡄⠀⠘⠛⠛⠛⠛⠛⠛⠋⠀⠀⣿⢿⡀⠀⠀⠀⠀⢀⡼⢿⠀⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠛⣿⣿⠛
⠀⠀⠀⠼⠁⢿⠀⠀⠀⠀⠀⢷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠸⡇⠀⠀⠀⠠⠞⠁⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠠⣄⢀⣄⣰⠇⠀⠀⡀⢀⣤⡄⠀⠀⠀⣀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⣤⡄⠀⠀⠀⢠⣤
⠀⠀⠀⠈⠳⠦⣼⣧⠴⠞⠁⠀⠈⠻⣿⡏⠀⠀⠀⠳⣼⣿⠇⠀⠀⠀⢻⣧⡞⠁⠀⠀⠀⠀⠀⢰⣦⢀⡆⠀⠈⠳⣿⣦⡀⠀⢀⣶⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠿⣿⡷⠤⠤⠤⠈⠛⠛⢺⣿⡏⠁⠀⠀⠀⣿⡟⠀⠀⠈⢻⣿⠋
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣀⡼⠿⢧⡀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠋⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠙⠆⠀⠀⠉⠁⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⠛⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁
⠀⠀⠀⠀⣦⠀⢠⣾⣿⡿⠾⠿⠇⠀⠀⠀⠀⠰⡆⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠰⣾⣿⣇
⠀⠀⠀⠀⠸⣀⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢻⣘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣷⣿⣇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣤
⠀⠀⠀⠀⢸⠉⠀⠈⡏⢻⣇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⢸⠁⠀⠸⣿⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠃⠘⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣶⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⣀⣸⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣯⣶⠄⠀⠀⠀⠀⠀⠀⠀⠚⠛⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣿⡟⠂
⠀⠀⠀⠀⠛⠻⠿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢽⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇

⠀⠀⠀⠀⣤⣦⡀⠀⠀⠀⣤⣦⡀⠀⠀⠀⣠⣦⡀⠀⣦⠀⣤⣦⡀⢠⠆⢿⢠⣴⣄⠀⠀⠀⢀⣴⣤⠀⠀⠀⠀⢠⣦⡄⠀⠀⠀⣠⣦⡄
⠀⠀⠀⢀⣽⣿⡇⠀⠀⠀⣽⣿⡇⠀⠀⠀⣹⣿⡇⠀⠈⠳⣽⣻⡷⠋⠀⠈⢿⣿⣷⠀⠀⠀⣨⣿⣿⡄⠀⠀⠀⣸⣿⣧⠀⠀⠀⣿⣿⡷⠒⠛⠁
⠀⠀⠀⣼⣿⣿⣿⠀⠀⣼⣿⣿⣿⡀⠀⣼⣿⣿⣿⡄⠀⢀⣿⣿⡇⠀⠀⠀⣼⣿⣿⡇⠀⠸⣽⣿⣿⡽⠀⠶⠾⣿⣿⣿⡆⠀⢸⣿⣿⣧
⠀⠀⠀⢹⣿⣿⣿⠁⠀⢹⣿⣿⣿⠁⠀⢹⣿⣿⣿⠁⠀⢸⣿⣿⣷⠀⠀⠀⣿⣿⣿⡏⠀⠀⣾⣿⣿⡇⠀⠀⢀⣿⣿⣿⠃⠀⢹⣿⣿⣿
⠀⠀⠀⠈⣿⣿⠁⠀⠀⢸⡏⠹⣯⠀⠀⠈⣿⣿⡏⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⡏⠀⠀⠀⣽⠏⢹⡇⠀⠀⠀⢹⣿⡏⠀⠀⠀⢹⣿⡏
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣤⢾⣿⣦⡄⠈⢁⣾⣿⣆⠀⠀⠉⣵⣦⡄⠀⠀⠀⣾⣿⣆⠀⠀⠀⢰⣿⣧⠀⠀⠀⢠⣶⣦⠀⠁⠀⠀⣠⣤⡀⠀⠀⠀⢾⣿⡇
⠀⠀⠀⠈⣿⣾⡏⠀⠀⢸⣿⣿⣿⠀⠀⢀⣿⣾⣷⠀⠳⣶⣿⣿⣿⡞⠀⠀⣶⣿⣿⠄⠀⠀⣼⣿⣿⡄⠀⠀⠀⣿⣿⣿⠂⠀⣠⣾⣿⣧⡀
⠀⠀⠀⢰⣿⣿⣇⠀⠀⢠⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⢠⣿⣿⣇⠀⠀⠀⣾⣿⣿⠀⠀⠀⣻⣿⣿⡃⠀⠀⢰⣿⣿⣯⠀⠈⠉⣿⣿⣿⠉
⠀⠀⠀⠘⣿⣿⠟⠀⠀⠸⣿⣿⠿⠀⠀⠸⣿⣿⠿⠀⠀⠸⣿⣿⠿⠀⠀⠀⣿⠿⣿⠇⠀⠀⠿⣿⡿⠇⠀⠀⠸⢿⣿⡿⠀⠀⠸⣿⠿⣿
⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠿⠿⠃⠀⠀⠀⠿⠿⠇⠀⠀⠀⠿⠿⠃⠀⠀⠀⠿⠀⠻⠀⠀⠀⠰⠟⠷⠀⠀⠀⠀⠺⠿⠇⠀⠀⠠⠟⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡠⠚⠂⠀⠀⠀⠐⠀⡀⠀⠀⠀⣤⠀⠀⢠⡄⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀
⠀⠀⠀⣡⣾⣿⡆⠀⢀⣤⣞⣿⡆⠀⠀⢹⣾⣿⣾⠁⠀⠸⣾⣿⡇⠀⠀⠀⢰⣿⣧⠀⠈⢀⣴⣦⣤⡀⠀⠀⠀⠰⣿⣷⠀⠸⡇⢾⣿⡇⣽
⠀⠀⠈⠉⣿⣿⣧⠀⠈⠉⣿⣿⣧⠀⠀⠀⣿⣾⡇⠀⠀⠀⣿⣿⣧⠀⠀⢀⣾⣿⣿⡆⠀⢸⣿⣿⣿⡄⠀⠀⢠⣾⣿⣿⠀⠀⠉⣿⣿⡟⠁
⠀⠀⠀⢰⣿⣿⣿⠄⠀⢰⣿⣿⣿⠆⠀⢠⣿⣿⣧⠀⠀⢰⣿⣿⡏⠀⠀⢸⣿⣿⣿⡧⠀⠀⢹⣿⣿⡆⠀⠀⢰⣿⣿⡏⠀⠀⢠⣿⣿⣷
⠀⠀⠀⠸⣿⣿⠟⠀⠀⠘⣿⣿⠟⠀⠀⠘⣿⣿⡟⠀⠀⠸⣿⣿⠿⠀⠀⠀⢿⣿⡿⠇⠀⠀⠸⣿⣿⠛⠀⠀⢺⡿⣿⠇⠀⠀⠘⣿⠻⣟
⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠿⠿⠃⠀⠀⠀⠿⠿⠇⠀⠀⠀⠿⠿⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠿⠿⠀⠀⠀⠿⠁⠿⠀⠀⠀⠰⠟⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⢀⣀⡀⠀⣀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⡆⠀⠀⠀⠀⢀⣿⠁⢺⣿⡿⠀⢻⡄⠀⠀⠹⣦⣴⣶⡆⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣆⠀⠀⠀⠈⠙⠻⣶⣶⣶⠟⠋⠁⠀⠀⠀⠙⣯⣭⣿⠾⠟⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠂
⠀⠀⠀⠀⠰⣿⣿⣦⡀⠀⠀⠐⠛⠋⢹⣿⣿⣿⠙⠛⠃⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣀⣤⣬⣯⣥⣤⡶⠶⠞⠛
⠀⠀⠀⢦⣀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠻⢿⣿⣿⡏
⠀⠀⠀⠀⣭⣥⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣤⣼⣿⣿⣧⡀
⠀⠀⠀⠀⢻⣿⣿⣿⡋⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠹⣿⣏⠁⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠸⠿⠈⠙⠿⠇⠀⠀⠀⠀⠸⠿⠻⠟⠀⠀⠀⠀⠀⠀⠀⠿⠟⠿⠃⠀⠀⠀⠀⠀⠀⠿⠻⠿⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠀⠈⠙⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠀⢀⣴⣤
⠀⠀⠀⠀⢀⣴⣿⣦⠀⠀⠀⠀⠀⢸⣿⣽⣿⠀⠀⠀⠀⣠⣿⣿⣿⡀⠀⠀⠀⢀⣼⣿⣿⣦⡄⠀⣸⣿⣿⣦⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⢀⣽⡿⠛⢿⣧⠀⠀⠀⠀⠀⣿⡇⢿⣇⠀⠀⠀⢸⣿⣽⣿⠀⠀⠀⠀⠈⢸⣿⣿⣿⠉⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⢸⣿⡀⠀⢸⣿⠃⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣿⢻⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠐⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠘⣿⠇⠀⠸⣿⠀⠀⠀⠀⠀⢿⡇⠈⣿⠆⠀⠀⠸⣿⢾⡿⠀⠀⠀⠀⠀⠸⣿⢻⡿⠀⠀⠐⣿⢿⣿⠀⠀⠰⣿⠋⢻⡄⠻⡿⠇⢸⠏⠻⣷

⠀⠀⠀⡀⠀⢀⠀⢰⣦⡀⠀⠀⣀⡀
⠀⠀⠿⠟⢸⣿⣿⣬⡉⠀⠀⠸⣿⣿⠆⠀⢀⣠⡄
⠀⠀⠀⠠⣶⣿⣿⣿⠃⠀⢠⣶⣾⣶⡶⠞⠛⠉⠀⠀⢀⣶⣷⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⣿⣻⣿⣿⡇⠀⣠⣤⣄⣄⣄⣹⣟⣡⣤⣤⣤⣄⠀⢻⡄⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣺⣿⣿⡇⠀⠈⠉⠉⠉⢹⣿⣿⡏⠉⠉⠉⠁⠀⠘⢷⣼⣿⣿⡀⠀⠀⠀⠀⠀⣸⣿⣿⣛⣿⠆⠀⠀⢀⣿⣿⣇
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣠⣤⣼⣿⣿⣧⣤⣄⠀⠀⠀⠀⠀⣿⣿⣿⡿⣧⠀⠀⢠⡿⣿⣿⣿⡏⠀⠀⠀⣼⢿⣿⣿⡿⢷⡀
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⣿⡟⠉⠉⠉⠉⢻⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣷⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⠿⠻⡿⠀⠀⠀⠀⢾⠿⠅⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⢿⠏⠛⠛⢿⠟⠁⠀⠙⢿⠟⠛⠋⢿⠇⠀⠈⠻⡿⠛⠛⠹⡿⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠈⢷⡀⠀⣰⣶⣆⠀⢀⡿⠁⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⢀⣘⣛⣋⡀⠀⠀⠀⠀⠀⠀⣘⣛⣋⠀⠀⠀⠀⣀⣀⣀⣀⣀⣙⣟⣁⣀⣀⣀⣀⡀⠀⠘⠳⣦⣙⣻⣋⣴⠞⠃⠀⠀⢷⣄⢀⣈⣻⣋⣀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡾⣿⣿⣿⢷⡀⠀⠀⠉⠉⠉⠉⢹⣿⣿⣿⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣿⡟⣷⡀
⠀⠀⠀⠀⣤⣿⣿⣿⣤⡀⠀⠀⠀⣼⣧⣿⣿⣿⣬⣷⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⡟
⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⢩⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠻⠿⠛⠻⠟⠃⠀⠀⠀⠘⠻⠿⠛⠻⠟⠃⠀⠀⠀⠀⠀⠘⠻⠟⠛⠿⠟⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠿⠟⠃⠀⠀⠀⠀⠀⠛⠿⠛⠻⠿⠋

⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠀⠀⠀⠀⣀⣴⣤⣀⠀⠀⠸⣇⠀⠀⣤⣦⡀⠀⢰⡟⠀⠀⠀⠀⢠⣴⣤
⠀⠀⠀⠀⣻⣿⣃⠀⠀⠀⠀⠀⠀⣙⣿⣋⠀⠀⠀⠀⠀⠀⣘⣿⣛⡀⠀⠀⠀⠘⢿⣍⣻⣟⣹⡿⠂⠀⠻⢶⣄⣻⣟⣡⡴⠟⠀⠀⢶⡄⠀⣘⣿⣟⡀
⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⠀⢾⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠻⠟⣿⣿⣿⡟⣦⡀
⠀⠀⠀⣿⣿⣿⣿⣟⠀⠀⠀⠀⣿⣿⣯⣿⣟⠀⠀⠀⠀⢠⣿⣿⣿⣥⠀⠀⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⣾⣿⣿⣧⡄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⠟⠁
⠀⠀⠀⢘⠛⠻⣿⡏⠀⠀⠀⠀⠈⡟⠻⣿⡏⠀⠀⠀⠀⠀⡛⠛⢿⣯⠀⠀⠀⠀⠀⢹⠛⢿⣿⠁⠀⠀⠀⠀⢸⠛⠻⣿⡅⠀⠀⠀⠀⠀⠀⠻⠛⢿⣯
⠀⠀⠀⢸⣀⡀⣿⡇⠀⠀⠀⠀⢰⣀⠀⣿⡇⠀⠀⠀⠀⠀⣇⡀⣸⣿⠀⠀⠀⠀⠀⣆⢀⢸⣿⠀⠀⠀⠀⠀⣘⣀⢀⣿⡇⠀⠀⠀⠀⠀⠀⣇⡀⢸⣿
⠀⠀⠀⠘⠀⠘⠿⠁⠀⠀⠀⠀⠸⠀⠈⠿⠁⠀⠀⠀⠀⠰⠁⠀⠻⠃⠀⠀⠀⠀⠀⠁⠀⠻⠏⠀⠀⠀⠀⠀⠏⠀⠸⠟⠀⠀⠀⠀⠀⠀⠰⠁⠀⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⠿⠃
⠀⠀⠀⣰⣿⣿⣿⣷⣄⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣷⠌⠁⠀⠀⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣠⣶⣄
⠀⠀⠸⣯⣼⣿⣿⣧⣿⡆⠀⠀⠀⣿⣼⣿⣿⣿⣼⣇⠀⠀⠀⠀⢀⣿⣿⣻⣦⣀⡀⠀⠀⠀⠀⣼⠟⣿⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⣻⣿⠟⠋
⠀⠀⠈⣿⣟⣿⣻⣟⣿⡇⠀⠀⠀⣿⣿⣿⣻⣟⣿⡟⠀⠀⣀⣀⣈⣛⣿⣟⠻⣿⣅⠀⠀⠀⠐⣏⣀⣘⣻⣿⡟⢿⣷⡄⠀⠀⣀⣀⣈⣻⣿⡟⢿⣷⡄
⠀⠀⠀⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⡿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿⠀⠈⢿⣧⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠙⢿⡆⠀⣿⣿⣿⣿⣿⡇⠀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⠀⠀⠀⠀⠀⢰⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠐⢿⣿⠇
⠀⠀⠀⠀⠀⢠⣬⣯⣤⠀⠀⠀⠀⠀⠀⢀⣬⣭⣥⡄⠀⠀⠀⢀⣤⣽⣭⣄⡀⠀⠀⠀⠀⠀⢠⣶⡍⠁⠀⠀⠀⠀⠀⢠⣴⡟⢻⡁⠀⠀⠀⣴⣶
⠀⠀⠀⠀⣰⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⢼⣿⣿⣿⡏⠻⠶⠟⠃⠀⠀⣿⣿⣧⣀⠀⠀⠀⠀⢠⣿⣿⢷⡟⠁⠀⠀⢰⣿⣿
⠀⠀⠀⢰⢏⣸⣿⣿⣿⣈⢷⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠈⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣾⣶⡆⠀⠀⣈⣿⣿⣷⣦
⠀⠀⠀⣰⣿⢛⡁⠀⢽⠻⣷⡀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠀⡜⠀⢘⣿⠀⠀⠀⠀⠀⢀⠃⠀⢣⣿⡇⠀⠀⠀⢸⠀⠈⣡⣿⠇⠀⢀⠃⠀⢋⠹⣿⡀
⠀⠀⣰⣿⠃⢸⠒⠒⢲⠀⢻⣷⡀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠗⠒⢾⡿⠀⠀⠀⠀⠀⠸⠖⠒⢾⠟⠀⠀⠀⠀⡶⠒⠒⢿⠛⠀⠀⢸⠖⠒⢺⠀⣿⡇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⡄⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⣿⡟⠛⠛⠃⠀⠀⠀⠀⠀⣀⡛⠛⠛⠛⠁⠀⠀⠀⠀⣙⢿⣿⡿⠟⠀⠀⠀⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⡆⠀⡀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⢠⣼⣿⣤⠀⠀⠀⠀⠀⠀⣹⣧⣤⡀⣴⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠺⠿⣤⡀⠀⠀⠻⢿⣈⠿⣇⡀
⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⣿⣷⠀⠀⠀⢸⣿⡀⣿⣇
⠀⠀⠀⣿⢻⡇⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⡷⠀⠀⠀⠀⣸⡿⠀⠀⠀⢈⣿⡟⣹⣿⠇
⠀⠀⢸⡟⠈⣿⠀⠀⠀⠀⠀⠀⣸⡇⣿⠀⠀⠀⠀⠀⠀⢀⣿⠙⣷⠀⠀⠀⠀⣿⣇⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⠀⠛⠀⠛⠀⠀⠀⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠀⠛⠛⠀⠀⠀⠘⠛⠃⠙⠛

⠀⠀⠀⠀⣠⣤⣶⣤⣄⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⡦⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠈⢹⠏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠈⠉⡹⠇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁
⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠰⣿⡇⠀⠀⠀⠀⠀⠙⣾⣿⣃⠀⠀⠀⠀⠀⠀⣘⣿⣣⡀
⠀⠀⣰⢿⣿⣿⣦⠀⠀⠀⠀⠀⣰⣿⢶⡟⠀⣿⣿⣿⣄⠀⠀⠀⠀⢸⣿⣿⢳⡄⠀⠀⢠⣾⣿⣿⡟⠧
⠀⠀⠙⢻⣿⣿⠁⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⢸⣿⣿⠹⠆⠀⠀⠀⢸⣿⣿⠟⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⣿⢹⡇
⠀⠀⠀⠸⡿⠃⠀⠀⠀⠀⠀⠠⡿⣿⠀⠀⠀⣿⠁⡿⠀⠀⠀⠀⠀⣿⠡⡿⠀⠀⠀⠀⠀⢸⡟⠸⡷

⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣤⡀⣀⡀
⠀⠀⠀⢰⣿⡟⠀⠀⠀⣤⣶⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⢸⣿⡇⢸⡟⣃⣛⣛⣀⡀
⠀⠀⠀⣾⣿⣇⣀⣴⣾⣿⣏⣴⣿⡿⠀⠀⠀⠀⣠⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⢸⣧⠙⠿⠿⢿⣿
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣶⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣩⣤⣍⣥⣬⣩⣴⠷⠖⣸⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣡⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⣵⣾⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁







⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⠸⣿⡧⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢰⣿⡇⢸⣿⡇⡟⣰⣿⡟
⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⢸⣿⡇⣴⣿⠟⣴
⢀⣈⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣋⣌⣛⣡⣌⣛⣤⣙⣋⣼⣿
⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣩⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣼⠇⢀⣀⠀⠄⠀⠀⠀⠀⠐⠀⣀⡀
⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠰⣿⣿⠇⠀⢀⣀⣤⠀⢀⣴⢾⣿⣿⠂⠀⠀⠀⠀⠀⠰⣿⣿⠇⠀⠀⠀⠀⢿⡄⣿⣿⡷⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆
⠀⠀⠀⢠⣶⣾⣷⣦⡀⠀⠀⢀⣴⣾⣷⣶⠞⠛⠉⠁⠀⠘⠳⣶⣾⣷⣦⡀⠀⠀⠀⢠⣴⣾⣷⣶⡄⠀⠀⠀⠈⢻⣶⣿⣶⣄⠀⠀⠀⠀⢰⣶⣾⣷⣶⡄
⠀⠀⠀⣿⢿⣿⣿⡿⣿⠀⠀⣿⢻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣷⠀⠀⠀⣿⣹⣿⣿⣟⣿⠀⠀⠀⠀⢸⣿⣿⣿⢿⡆⠀⠀⠀⠈⢿⣿⣿⣿⠁
⠀⠀⠀⣿⣾⣿⣿⡇⣿⠀⠀⣿⢸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⠀⠀⠀⠹⣿⣿⣿⣟⣿⠀⠀⠀⠀⢸⣿⣿⣿⢸⡇⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠉⣼⣿⣿⡇⠉⠀⠀⠉⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠉⠀⠀⠀⠀⢸⣿⢸⣿⡉⠀⠀⠀⠀⢸⣿⣿⣿⠈⠁⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣸⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢿⡿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠸⣿⢿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣶
⠀⠀⠀⠀⠁⣴⣶⡄⠀⠀⠀⠘⢷⣄⠀⢠⣶⣶⡀⠀⣠⡾⠃⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠈⣀⣌⠀⠐
⠀⠀⠀⣶⢀⣙⣟⣃⢰⣦⠀⠀⠀⠙⢷⣌⣛⣛⣥⠞⠋⠀⠀⣀⣴⣞⣿⣿⣿⣷⣦⡀⠀⣤⣤⣀⡀⠸⢿⡿⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣇⡀
⠀⠀⠀⠻⢿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⠋⠉⠁⠀⠀⠈⠉⠛⠛⢶⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠻⣷⣄⠀⠀⠀⠀⠀⢸⣿⣻⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⡟⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⢿⣧⡀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⠃⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠻⠗⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠈⠛⠙⠛⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠃⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠐⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⢰⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠀⠹⢿⠿⢛⣷⡦
⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡷⣦⣄⡀⠀⠀⠀⣶⣿⣿⣿⣿⢦⣄⣀⠀⠀⠀⢰⣿⣿⣿⣿⣦⠀⠀⠀⠀⣴⣿⣿⣿⠛⠉
⠀⠀⠀⣿⢸⣿⣿⣿⣽⠇⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠉⠙⠛⠀⠀⠈⢻⣿⣿⣿⠀⠉⠙⠛⠀⠀⠛⢻⣿⣿⣿⠛⠀⠀⠀⣼⠟⣿⣿⣿⡀
⠀⠀⠈⠟⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠈⡿⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢠⡟⠀⠸⣿⡿⢿⣦⡄
⠀⠀⠀⠀⣿⣿⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⣿⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠙⣿⣆
⠀⠀⠀⠀⣿⣿⠀⣤⣤⣭⣤⡄⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠸⣿⠆
⠀⠀⠀⠀⠹⠟⠀⠿⠿⠿⠿⠇⠀⠀⠀⠀⠻⠟⠿⠇⠀⠀⠀⠀⠀⠀⠀⠸⠟⠻⠟⠀⠀⠀⠀⠀⠀⠀⠸⠿⠻⠟⠀⠀⠀⠀⠀⠀⠀⠿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢀⠀⠀⣀⢀⡀⢀⠀⠀⣀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠀⠁⠁⠉⠀⠈⠀⠀⠁⠁⠁

⠀⠀⠀⠀⠘⠄⠼⠀⠆⠀⠀⠀⠀⠀⠰⢄⠸⢀⠀⠀⠀⠀⠀⠀⠆⡀⠇⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⢀⠼⢄⠨⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠀⠀⠀⠀⠀⠀⠀⢨⣘⠒⠒⠲⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⢠⠠⡤⠄⡀⠀⠀⠀⠀⠀⠀⡀⠧⠠⠠⠀⠀⠀⠀⠀⠸⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡤⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠧⡀⠀⠀⠀⠀⠀⠀⠈⡇⡘⠉⠈⠀⠑⠀⠀⠀⠐⢅⠙⣈⡀⠂⠊⠀⠀⠀⠀⠱⢀⠆⢁⠐
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⣀⠀⠀⠀⠀⠤⠎⠈⡊⠀⠀⠀⠀⠀⠀⠀⡴⠑⡤⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⢀⢄⠀⠀⠀⠀⠸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠳⠚⠲⠋⠀⠀⠀⠀⠀⠀⠈⠰⢀
⠀⠀⠀⠘⡾⡳⢀⠄⠕⠀⠀⠀⠀⠠⡄⠡⠒⢄⠀⡢⠀⠀⠀⢀⠄⡀⢠⠠⣃⠀⠀⠀⠀⠀⠀⠠⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠘⡠⠘⢄⠀⠀⠀⠠⠢⠀⠡⢑⢰⠄⠀⠀⠀⠀⠜⠁⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠉⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⢀⠖⣠⠀⠀⠂⠀⠀⠀⠀⠀⠒⡀⡶⡶⡆⠀⠀⠀⠀⠀⠀⠲⠊⠀⠀⠀⠀⠀⠀⢀⣀⠰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠼⢄
⠀⠀⠀⠀⢸⣶⠍⠁⠒⠁⠀⠀⠀⠀⢸⢠⠋⠉⠉⠁⠀⠀⠀⠀⠀⠠⣀⠁⠀⠀⠀⠀⠀⠀⠀⠉⢁⡉⠈⠀⠀⠀⠀⠀⠀⠀⠞⠖
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠈⠉⠉⠉⠁⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⣿⣿⡟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢰⣿⣿⣧⡀⠀⠀⢀⣄⣠⣬⣿⣇⠰⠂⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⢀⡀⠙⠛⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣦⣤⣾⣷⣾⣷⣶⣶⣤⡀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣀⠀⠙⢿⣿⡏⠈⠐⠞⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⢿⡿⠿⠿⣿⣿⣿⣿⡆⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣾⣿⣿⣁⡀⠀⠀⢸⣿⠏⢻⣿⣿⣿⣿⣿⣿⡇⠀⠈⣴⣶⡄⢀⣀⡀⢹⡏⣿⣿⣿⡟⣁⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⠿⠿⠿⠿⣦⠀⠈⠁⠀⠈⠻⣿⣿⣿⣿⣿⣷⡀⢢⣙⣋⣁⠹⠿⠟⢘⠀⢿⡿⠉⣿⣈⡠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⢏⣀⣀⠀⣴⣶⣿⣶⣾⣿⣿⡿⠂⠉⠀⠀⠉⠉⠙⠃⠘⣿⣯⣽⣷⣶⣾⣿⣿⣿⣷⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⢐⣨⣿⣿⡛⠛⠋⣤⣈⠁⢈⡿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢽⣿⡟⠉⠻⢿⣿⣏⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣿⣿⡿⠿⢻⠁⢶⣾⣭⣽⡿⠁⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⠀⢀⣀⣶⣤⣀⣠⣤⣬⠉⠐⠂⠀⠦⣰⣿⡿⠿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣁⣿⣷⢆⣼⣶⣾⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡶⢦⣿⣿⡿⠟⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡟⠛⠛⠋⠉⢻⣿⡏⠉⢁⣀⡀⠀⣿⣯⠁⠘⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣿⣏⣉⠀⠀⠀⠙⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⡄⠘⣿⣷⣿⣿⣿⣿⣷⣾⠿
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣟⠀⠀⠀⣠⣶⣶⣶⣤⠀⠀⠘⠃⠀⠀⠀⠀⠋⡟⠀⠙⠛⠛⢹⣿⣦⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⢻⣿⠁⠀⣤⣶⣾⣿⣿⡿⢇⡀⢰⣿⠟⠉⠉⠙⢶⣶⡄⣀⣤⣴⣤⣤⣀⡇⠒⠀⠀⠖⢸⠃⠙⠷⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⠀⢘⣿⣿⣿⣿⠁⠀⢨⣛⣛⡇⡀⠀⠀⡀⠀⣿⡿⠿⠿⣿⣿⣿⣿⣿⣶⣟⣷⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠘⠻⣿⣿⣿⡏⠈⠁⢀⣍⣉⣁⣉⣤⣀⣁⣠⣽⡇⠀⠀⢸⣿⣿⣿⣿⠟⠻⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⣌⣉⣉⣹⣿⣿⣿⣿⢟⣿⣿⣀⠀⠀⢸⣿⠟⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠛⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢀⣉⣉⣉⣉⡍⠉⠉⠁⢿⣿⣿⣿⣷⡆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡅⠀⠀⠀⠘⣿⠀⠀⢀⣠⣼⣿⣷⣶⣶⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⣿⡄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠷⣶⡄⠈⣶⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠈⠀⢀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣶⣶⣦⣤⣀⣀⠀⠀⠀⠈⠙⠻⣿⣿⣿⡷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⠟⢿⣦⡀⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠙⠻⣿⣿⣿⣿⣿⣿⣧⢹⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠘⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡘⣷⣾⣿⣿⣿⣛⠋⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣿⣿⢹⡿⠋⢀⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠿⡎⢰⣷⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡍⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡘⡿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡃⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⠤⣀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠢⡄⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡄⠀⠀⠑⠤⣔⠀⠀⠀⠀⠀⢀⡤⢤⡤⠤⢤⣼⡇⠀⠀⠀⠀⢀⡏⢠⠄⠀⢤⠈⣣⡈⠀⠀⠀⠀⠀⠀⠖⠋⠁⠈⠙⠲⣶
⠀⠀⠀⠀⢘⠰⠆⠀⠰⠆⢰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⢠⡄⠤⠤⠄⢤⠁⣿
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡸⠀⠰⠀⠀⠈⠘⢧⡀⠀⠀⠀⢀⠡⠲⠦⠤⠶⣔⠇⠀⠀⠀⠀⡰⠁⣤⡀⠀⢠⡄⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠰⡄⣁⡀⣼⠁⡔⠀⠀⠀⠀⠀⠀⠃⢤⣠⠰⣀⠓⠰⠄⠰⠶⠟⡀⢨⠙⠒⢹⡱⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠐⠎⠁⠀⠀⠈⠙⢢⠀⢤⠀⠀⠶⠄⠠⢠⠀⡔⠉⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠊⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢱⠠⠤⣴⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠇⠈⠁⠀⠀⠀⠠⣄⡁⠠⠂⠀⠀⠀⠀⠀⠀⢀⡑⠤⠤⠤⣂⠀⠀⠙⠠⠤⠤⠅⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⢄⣀⠰⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⢂⡀⠀⠀⠀⠀⠐⣉⢈⠀⠠⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠒⠤⠶⠤⠊⠀⠀⠀⡀⠀⡶⠀⠈⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠁⠄⠂⠈⠁⠑⠀⢠⠀⠀⠀⠀⢼⡀⠉⠀⠈⠁⡁⡆⠀⠀⣠⠀⣤⠀⠀⣤⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠂⠐⠲⠽⣢⡄⠀⠀⠠⠀⢀⠀⠜⢀⠦⠤⡀⠐⡄⠀⠀⠐⡞⢛⠀⠈⠀⣘⠛⡃⠀⠘⢿⣆⣀⣠⣀⣀⡴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠇⠒⠂⠀⠲⠂⠫⢀⠀⠀⠁⠀⡄⠸⠂⠀⠰⠆⢸⠀⠁⠀⠐⡀⢼⣩⡃⣤⣬⣍⣉⣉⠀⠀⠈⡁⠀⠿⠆⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⡔⣠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠊⣅⡈⠠⠀⣤⠛⠣⠀⠀⠀⠑⣄⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢦⡀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠈⠉⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠈⠙⠁⠃⠄⠀⠀⠀⢀⠀⠠⡨⠈⠒⠂⠱⢠⢤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠿⠛⠛⠛⠛⣻⣿⣛⡻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⣀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣶⣬⡉⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢰⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⣷⡄⢀⣤⣄⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣰⣿⠋⠉⢹⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠀⠀⠀⢰⣿⠁⠀⠀⢸⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠏⠁⠀⠀⠀⠀⣿⡇⠀⠀⢠⠇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⠙⠛⠃⠀⠀⠀⠀⠀⠀⢻⣧⣀⡴⠋⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣤⣴⣾⣏⠉⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣾⣿⣿⣿⣿⣿⡿⠇⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⡶⠁⢀⣀⠀⠈⢻⣿⣿⡏⠐⠠⠤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣸⣿⣿⣿⣿⣿⣿⣿⣷
⡇⠀⠀⠉⠠⣶⣿⣿⡟⠁⠄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠈⡟⢉⡈⠉⠛⠛⠛⢿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠩⠿
⠀⠀⠀⢀⣿⣿⣿⣯⣤⣤⡄⠀⠀⠀⠿⠋⠐⢿⣿⣿⣿⣿⣿⣿⠀⠀⢱⣿⣿⠇⣰⣶⣦⢸⡟⢹⣿⡿⠻⣇⠛⠀⠈⠛
⣠⣤⡀⣰⡿⠟⠛⠋⠙⠛⢷⡀⠀⠀⣀⣀⣀⣀⠹⠿⠿⣿⣿⣿⣇⠈⣦⣬⣤⣤⡈⠛⠛⣨⣴⣾⣿⣿⣿⣿⣿⣿⡽
⣿⣿⣷⣟⣠⣤⡄⠀⣾⣿⣿⣷⣾⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠹⣿⣷⣾⣿⡿⠿⣿⣿⣿⠛⠿⢿⣿⣿⣧⣶⣿⣿
⣴⣭⣿⣏⠛⠛⢁⣤⣄⠀⢀⠾⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠸⠿⠋⣤⠀⢠⡄⠀⢹⣿⠋⠰
⡿⠿⠏⡇⠰⣶⣿⣭⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣀⡀⠀⣀⣸⣦⣄⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⣿
⣷⡦⢀⣧⣴⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢁⣤⣤⣄⡀⠸⡿⠡
⡍⣼⣿⣿⡿⠿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⢠⣿⠟⠿⠟⠛⠛⣿⣿⡿⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⠢
⣾⣿⠛⠉⠀⠀⠈⠿⢿⣿⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⣧⠀⠀⠀⢀⠀⠸⣿⣧⣶⣿⣿⣷⣾⣿⡁⠀⠉⠀⢘⣿⣿⣿
⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢀⣾⣿⣿⣿⣿⡆⠀⠉⠴⠀⠀⠉⠀⠸⡿⣿⠋⠻⢿⣿⠿⣿⡇⠀⠐⢺⣿⣿⣿⣿
⢿⣿⡟⠀⠀⠂⠀⢀⣼⣿⣄⡈⠀⠀⢸⡿⠋⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠀⠀⣄⡇⣀⠀⠀⢀⠀⣿⠁⠀⠀⠈⠛⠁
⠘⢿⡇⠀⣠⣾⣿⣿⣿⡿⠿⢳⣦⣤⣼⢇⣀⠀⠀⣀⠀⢹⣿⣁⣀⣀⣀⣀⣀⣴⣇⢀⣠⣄⠈⣠⡏⠀⠀⠀⠀⢠⡀⠀⠛
⠀⠀⠁⣠⣿⣿⣿⣿⡏⠀⠀⣠⣤⣤⣤⣄⣁⣤⣄⡈⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡟⠀⠀⠀⠀⠀⠈⠆⠀⡀
⢀⣴⣶⣬⣿⣿⣿⣿⠁⠉⠀⠀⣦⣤⣤⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉
⠈⣿⠟⠛⠛⠛⠿⣿⣇⠀⢀⣾⣿⣿⣿⣿⣿⡟⠛⠉⢉⣿⣿⣿⠻⠿⠿⣿⣿⣿⣿⡄
⠀⢁⡀⠀⠀⠀⠀⢻⡿⠃⢰⡿⣫⣍⣩⣹⣿⡷⣶⠂⠀⠉⠉⠙⠀⠀⠀⣿⢿⣿⣿⠂⢀⣀⣠⣤⣤⣤⣀
⡀⢀⠁⢀⡀⠉⠀⠾⣷⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢿⣿⠟⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⣤
⣿⣞⢿⣯⣽⣦⣴⣾⣿⣿⣿⣿⣆⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣀⠀⠀⠈⠋⠀⠀⡿⠋⠉⠀⠀⠉⢻⠃⢠⣾⣿⣿⣿
⠈⣿⠈⠉⠛⠛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣴⣾⣿⣿⣿⣦⣤⣤⣶⣤⣁⠰⠂⠀⠀⠒⢀⢶⣿⣿⠃
⠖⠀⠀⠀⠀⠀⠉⠉⠙⠻⣿⣿⣿⣿⣇⠁⠀⠁⠀⠘⢡⣿⣿⠟⠉⠀⠀⣀⢹⣿⣿⣿⣿⣦⣶⣛⣶⣶⣟⣻⣿⣿⠣⡄
⠀⠀⠀⢀⣴⣿⣷⣆⠀⣼⡿⣿⣿⣿⣿⣀⣀⣀⣤⣴⣿⣿⣿⡀⡤⠀⠀⢀⣀⣿⣿⣿⣿⣿⡿⠿⠿⠋⠉⢹⣿⣆
⠀⠀⠀⠀⢿⣿⣿⣿⣦⣙⠀⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⢠⣾⣫⣿⣿⣿⣟⡛⠛⠃⢤⡤⣠⣴⣶⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠎⠙⠛⠉⠁⠠⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠒⠒⠒⠺⠇
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀
⠀⠀⠀⠀⡀⢤⠴⢧⠤⠤⢤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠐⠒⠚⠓⠀⠉⠀⠉⠐⠓⠦⣄
⠀⣠⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣦⡘⠛⠿⠿⠿⠟⠛⣡⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞
⠀⠉⠘⠓⣢⣤⣤⡄⠀⠀⠀⣀⠀⠀⠀⣠⣔⣢⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣡⡒⢄⡀⠠⣤⠤⠤⠐⠾⡇⠈⢹⡄
⠀⠀⠀⠀⣿⣤⠋⠀⠀⠈⠉⠉⠉⢉⡾⣯⠚⠁⠀⠒⠒⠂⢰⠀⠀⢀⠀⠀⠀⠀⠀⠉⠉⢎⡹⠀⠀⠀⠀⠀⠀⠈⠑⣾⡏⡄
⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⣎⠻⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢡⠛⣄⠀⠀⠀⠀⠀⠀⠈⢠⢇⡀
⠀⠀⠚⠀⡄⠀⠀⠀⠀⠀⠀⠀⣷⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡀⠀⠀⠀⠀⠀⠀⢺⠀⢣
⠀⠀⢷⢀⡇⠀⠀⠀⠀⠀⠀⣴⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡇⠀⠀⠀⠀⠀⠀⢨⣠⠆
⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠹⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡈⠃⠀⠀⠀⠀⠀⢀⡙
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠇⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢋⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⠋⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠛⢿⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠃⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠙⠻⣶⣦⣀⣀⠀⠀⠀⠀⣀⣀⣠⣶⠿⠋⠁⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠈⠉⠉⢻⡟⠛⠛⢿⠉⠉⠀⠀⠀⠀⢀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⠂⢤⡤⠄⠀⣸⠀⠀⠀⣿⠃⠠⣤⡤⠐⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⡀⠀⠀⠀⠀⢹⠁⠀⠀⢻⠁⠀⠀⠀⠀⣼⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣟⠁⠀⠈⠉⠙⠋⢻⡇⠀⠀⢹⣋⡉⠉⠉⠁⠈⢻⡄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⡿⠛⢦⡀⠀⠀⠀⠘⣃⠀⠀⢻⠉⢃⠀⠀⣠⠔⠿⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⣀⠁⠀⠀⠙⢢⡄⢀⡼⠛⠷⣤⠆⠀⠘⣷⢾⣁⠀⢐⡇⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢀⣽⣀⣀⡀⠴⠧⣤⡼⠳⠦⠀⠀⠀⠀⢰⠆⠀⠀⠈⠛⠛⠦⠄⣀⡀⠀⠀⠷⡀
⠀⠀⠀⠀⠀⢀⡮⠁⠀⠀⠀⠘⠁⠈⠉⠈⠁⠀⠀⠀⢠⣶⣤⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠻⡄
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣀⣀⣀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠁⠈⠙⠿⢲⡤⠤⣤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠐⠒⢶⡶⠶⠶⠶⠶⣶⠶⢶⣶⠶⠷⠶⢶⡶⠒⠒⢉⡴⢦⣤⣄⣈⣽⠋⢻⡷⠒⠒⠒⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢧⡸⠦⠤⠤⣾⠀⣄⣀⣀⣘⠛⠁⠀⢛⣋⣉⣥⠀⣇⢀⠀⢼⣆⢨⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⣄⢀⣶⣄⠈⠛⠁⠀⠁⠙⣷⢤⣤⣾⠉⠀⠉⠚⠃⠀⠀⠀⠈⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣼⣿⠖⠺⣿⠉⢹⡿⠳⣄⠀⠀⠀⣾⠇⣸⣇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⢙⣁⣀⠤⠴⡿⠄⢀⣭⠀⠀⢠⡏⢠⠛⠻⣧⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣗⠚⣉⡁⣤⠤⣶⠦⠤⣤⣏⠀⠀⡶⠀⣻⠀⠄⠈⣦⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣉⣉⣀⡀⢰⡏⠀⠀⣸⣧⡄⠸⣄⢰⡋⢹⡷⠀⠀⣦⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡘⣿⠛⠋⠁⠀⠛⠃⠉⠛⠋⠀⠙⣂⠀⠙⠃⠙⡇⠀⠀⢀⠛⠀⠉⠛⠋⠀⠀⠀⠀⠀⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣦
⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⣰⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⢠⡇⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠙⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⡇⠀⠀⠀⠙⣤⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠈⢁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠧⠤⠤⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠤⢚⣿⣿⣿⡿⠟⣛⣛⣯⣭⠉⣿⡟⢩⣽⣿⣛⡛⠛⢿⡿⠿⣿⡓⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⣠⣿⣷⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠃⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣡⠤⢀⣀⣀⢀⣀⠠⣦⣉⠷⠋⠱⠞⣡⡤⠄⣀⣀⣀⠀⢠⢤⣜⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡬⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡾⠀⠀⠀⠲⠦⠤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠀⠀⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠠⠛⠀⠀⠀⠀⠀⠉⠁⠐⡂⠀⠉⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⠀⠀⠀⠀⠀⠀⢿⡇⠒⠛⠃
⠀⣠⠴⠿⠄⠀⠀⠀⠀⠀⠀⠀⠿⢸⣿⠀⠀⠀⠀⠀⠀⡀⠘⠀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⡅⠁⠀⠒⠀⠀⠀⠀⠀⠀⠄⠑⢊⣁⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢠⣠
⠀⠐⡄⠠⡴⡀⠀⡀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠀⢸⣿
⠀⠁⢨⠄⠈⣝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⡄⠀⠀⠀⠘⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⡇
⠀⠇⠂⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿
⠀⠐⢦⣄⣀⣸⣧⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠉⠙⠛⠿⣿⠟⠝⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠛⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠻⣿⠀⠀⠻⠀⠀⢸⢄⠰⠈⠉⠈⣧⣾⣿⣿⣿⣿⠋⠉⠉⠙⣾⣿⣿⣿⠧⠀⠀⠀⠀⢈⡀⣡⣤⣌⡻⣿⡇⠀⠀⠀⡿⠛⠀⠀⢸⣿⣿⣿⣧⠀⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢠⣧⠀⠀⢰⡿⠁⠈⠙⢿⡟⠰⠀⠐⠀⠋⠙⠻⠋⠀⢠⣶⣦⣤⡜⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠂⠉⠑⢾⠋⠉⠋⡿⠀⠀⠙⠉⠁
⣶⠶⠶⣦⠀⠀⢰⣿⣿⣿⣿⡏⢠⣶⣾⣷⣄⠀⠀⠈⢰⣤⡎⣱⣾⡇⠀⣀⣀⠀⠟⠉⠁⠙⣿⠈⡅⠀⠀⢀⣧⡔⣶⠶⣄⢤⣄⣢⣤⠈⠃⠀⠀⠁
⠀⠀⠀⠈⠀⡀⢹⣿⣿⡿⣿⣿⣾⣿⠟⢛⡿⠇⠀⠀⠀⠙⠿⡿⠟⠀⣾⣿⣿⣷⠀⠀⠀⢠⡿⠀⠋⠀⠀⢨⣧⠈⠀⢀⣨⣿⣿⣿⣿⣥⣶⣾⣷⣄⠀⠀⣀⣤⣤
⠀⣀⣀⣀⡀⠀⢠⡈⠀⠀⠀⠉⠉⠉⠀⠛⠓⠀⢠⣤⣤⠀⣤⣤⡄⠀⠟⠛⠛⠛⠃⠜⠋⠙⣦⠀⠀⠀⠀⢸⢏⣆⠰⣾⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⠀⠀⠏⠉⠉
⣼⣿⣿⣿⣿⣶⣿⣧⡀⠀⢀⣼⣿⣷⣶⣾⣿⣷⣾⣯⣍⠁⠉⢩⣤⣀⠾⠿⢷⣦⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⣈⡉⢡⡭⡾⠀⠾⠟⠻⠋⠉⠉⠛⠃⠈
⣿⡇⠀⠀⢈⣿⣭⣯⣽⣄⣿⣿⣿⣿⣿⠏⠀⠈⢹⣿⣿⡆⠚⢿⡟⠋⠀⠀⠀⠛⣃⣿⣿⣿⡟⢈⠛⠿⠀⠿⢿⣯⣤⣴⣤⣤⡀⠳⣦⠀⠀⠀⢰⣷⠀⢀⣀⣀⡀
⣿⣧⡄⢠⣼⣿⣿⣿⣿⣿⣇⠸⠿⠿⠛⠀⠀⠀⢸⣿⠟⣁⣤⠿⣷⣤⣾⣽⣶⠶⡿⠟⠋⡙⢧⠚⠉⠀⠀⣼⣿⡿⠛⠛⠻⣿⣿⣇⠻⢡⠀⣄⣻⡃⠰⣿⣿⣿⡷
⡏⠀⠀⠈⠙⠁⣿⠛⠉⠻⡀⢀⠄⠖⠰⢄⣠⣶⣿⣿⣿⡿⠿⠶⠶⣸⣿⣿⣿⣎⠃⠀⠀⠀⠸⠅⢆⡀⠐⢿⣿⡇⠀⠀⠀⣿⡿⠃⠀⢡⣾⣿⣿⣷⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⣎⠀⠀⠀⣿⠟⠛⠿⣿⣿⢀⡀⢠⠀⡿⠻⣿⣿⠉⠀⣀⣀⣀⠰⣷⣿⣿⣿⣦⡝⠃⠀⠀⠀⠉⠀⠰⢦⣸⡿⠛⠿⣿⣧⢀⡿⣿⣳
⣀⣤⣄⣀⠀⠀⠀⠀⠁⡀⠀⠿⠀⠀⠐⡏⠀⠀⠀⠸⢃⠀⡠⢤⣠⡄⠀⠀⠀⠀⣾⠟⠛⠿⣷⠘⡟⠛⠉⠉⡃⢀⣀⣀⡄⢀⠀⣀⠀⠜⠇⠀⠀⡞⠁⣤⠘⣤⠓
⠙⠉⠉⠹⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣿⠃⣰⣶⣶⣤⠉⠀⠀⠀⣿⠀⠷⠀⠀⠰⡟⠃⠙⠉⠹⣾⣿⣶⣿⠆⠀⠀⠀⠀⠀⣉⠶⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⣿⡆⠀⠀⠈⠉⠀⠀⠀⠀⢸⡏⡁⠀⠀⠀⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⡫⠾⠛⠟
⢠⣾⡿⠖⢶⣄⢀⣳⡄⠀⣀⣤⣤⣤⣤⣤⣀⠀⣤⣾⣿⣿⣷⣦⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢽⡂⣀⣾⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢃
⣿⡏⠁⠀⠀⣿⡟⡟⠠⣶⣿⣿⡿⠋⠉⠉⠙⣿⣿⣿⡿⠿⠿⣿⣿⠆⠀⠀⠀⣠⣾⣿⣷⣦⡀⠀⠀⠀⠀⠘⠷⣿⣿⠛⠛⠹⣿⣿⣄⠀⠀⠀⣠⣄⠀⣫⣴⣤⣤
⣿⣇⠀⠀⣠⣿⡟⠁⠀⠙⠻⣿⣧⠀⠀⠀⠀⠉⠻⡿⠀⠀⠀⢸⣧⠀⠀⠀⣸⣿⣿⡿⣿⢿⣿⣄⣀⣠⣄⣀⡀⢸⣿⠀⠀⠀⢱⣾⣿⠄⠀⠀⣿⢿⣿⡉⠀⠈⢻
⣿⡿⠃⠸⣿⣿⠗⠛⠁⠲⠄⠀⠀⡴⠀⣠⡀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠚⢠⣿⣿⡇⠀⠀⠸⣿⡟⠛⠋⠉⠛⣷⡄⠈⢀⣠⡀⠘⠋⠁⢰⠏⠀⠀⢢⣿⡆⠀⠀⢠
⢻⣧⣶⣦⡝⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠠⣿⣶⣶⡤⢶⣿⢿⡇⣰⠟⠁⠙⣦⠀⠀⠀⠀⠀⠀⢸⣿⣷⡦⠰⣾
⣾⡏⠙⠻⡇⠀⠀⠀⠀⠀⠀⠀⣦⣾⣿⣿⣿⣿⣆⣠⠀⠳⣦⢤⠀⠀⠀⠀⠈⠀⣀⠀⠀⢀⣿⡍⢩⠟⢻⡀⢸⡇⣿⡀⠀⢀⣿⠀⠀⣦⠀⠀⠀⠎⠉⡿⣀⣀⣸
⡿⠓⠂⠒⠃⠀⢀⣀⠀⣤⣶⣄⣽⣿⣿⣿⣿⣿⣿⠧⠠⡤⠿⢾⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⢀⡀⠷⢾⠃⢀⣀⠙⠛⠁⠉⠉⠀⠈⢠⣦⣴⣤⡀⠀⢿⣿⠿⢻
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠘⡿⠻⠋⠉⠁⠀⠉⠛⠀⠂⠀⠂⢰⠃⢀⣀⣀⣀⣸⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⡏⠁⠾⠵⠐⠿⠖⠀⣀⣀⣈⡉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣶⣾⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⡄⢿⠋⠉⠉⠉⡇⠀⠀⠛⠙⠂⠀⣾⡿⠛⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣄⠀⠀⠀⠀⠀⡀⠀⠘⠿⠟⠛⠛⠛⢣⠃⠀⠀⠀⠈⡇⠀⠁⢀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⡈⠛⠀⠀⠀⠀⣿⠉⠀⠀⠈⡆⠀⠀⣾⢿⣿⣿⠿⢶⡀⠀⠀⠀⢼⡄⠀⠀⠀⠀⡿⠀⠙⠉⠁⢰⡟⠁⠈⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⣛⣛
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⣿⡀⠀⠀⣰⠟⠀⠀⠅⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣏⣡⣤⣼⡁⣆⣠⠀⠀⠀⣾⣧⠀⠀⠀⣿⣀⣀⣀⠀⡀⢄⣠⢠⠄⠈⠉⠉⠉
⢀⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠿⠗⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⠃⠀⠀⠀⠛⠿⠀⠀⠀⡿⠉⠉⠉⢻⡕⠉⣀⠀⠀⡄
⣾⠁⠈⠙⠁⠀⠀⠐⠂⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⢂⠀⡀⣀
⠓⠀⠀⠀⠃⠀⠖⠀⣴⠂⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣾⣷⣾⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠘⠛⢶⣶⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠿⣿⠿⢿⠿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣾⣿⠟⠻⢿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣰⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠇⠄⠀⠀⠤⠘⡇⠀⠀⠀⠀⠀⠀⢸⣿⠟⠙⠛⠏⠙⢿⡿⠀⠀⠀⠀⠀⠀⢻⡇⠠⠀⠀⠉⠅⣻⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠂⠀⠀⠂⢸⠃⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠘⣷⠐⣀⣀⣐⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠣⣀⣠⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣍⣿⣩⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣆⣀⣀⣴⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠈⠉⠉⣩⣭⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠋⠻⣿⣷⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠙⠛⡉⢿⡇⠀⠀⠀⠀⠀⠀⢸⡿⢻⠿⣿⠿⢿⣿⣇
⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⠈⠀⠀⠀⠁⢸⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⠀⢹⣿⡆
⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠄⠀⢀⣠⣿⣧
⠀⠀⠀⠀⠀⢿⣿⣿⣦⠀⢠⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⡄⠀⣴⣾⣿⣿⠛
⠀⠀⠀⠀⠀⠈⠻⠟⠋⠀⠈⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣥⣤⣬⣿⣋⠉
⠀⠀⠀⠀⠀⠐⠛⠒⠒⠒⠒⠒⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⢠⣶⣿⣿⡿⠿⢿⣿⣄⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⢻⣿⠂⠀⠀⠀⠀⠀⢸⣿⠟⠉⠛⠋⠈⠻⣿⡇⠀⠀⠀⠀⠀⣿⣿⠁⠀⠙⠀⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠹⠀⠂⠀⠀⠂⠘⠃⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠙⠏⠐⠀⠀⠐⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣦⡀⠀⠀⢀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠀⠈⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠶⠲⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⣫⣵⠶⠿⠛⠛⠛⠛⠛⠛⠷⢦⣍⡻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣥⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣩⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢟⣵⠛⣁⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠙⣆⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⣭⣶⡿⠿⠿⠿⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣦⡀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠘⣿⣿⢻⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡝⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢿⡇⠀⠀⠀⢹⣷⣼⣿⡇⠿⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⣛⡛⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠟⣭⠄⠀⠀⠀⠀⠀⠈⣿⣯⣹⡿⠚⠛⢳⣝⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡷⢸⣧⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠉⠀⢺⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⡣⡎⢠⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣿⣿⠀⣶⣿⣿⣿⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡇⠈⣾⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡅⣿⠀⠉⠋⠈⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡸⠂⠈⢠⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡆⢸⠀⠀⠀⠀⢋⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠁⠀⢀⡾⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢷⣼⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠟⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⣿⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⡇⠀⠀⠀⠀⢀⡀⣿⠤⠤⠤⣤⠶⠖⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠛⠒⠒⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⡀⢹⢸⣇⠀⠀⠀⠐⢏⠁⠀⠀⠀⠀⠀⠀⠀⠱⡌⠳⣄⠀⠀⠀⢀⠤⠤⠚⠛⠛⠛⠛⢻⣶⢽⡇
⠀⠀⠀⠀⠀⣠⠾⣫⣽⣶⣶⢮⣍⡻⣇⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⢦⡔⠚⢻⡄⠀⠤⠤⠤⠤⠄⢸⣿⢸⡇
⠀⠀⠀⠀⣴⢫⡞⢡⠞⠀⠲⢀⡉⢻⣦⢠⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠳⣄⠀⠛⠆⣼⡄⠀⠀⠀⠀⠀⠀⢸⡿⣸
⠀⠀⠀⢰⡏⣾⠀⢈⠀⠀⠀⠀⢳⠀⣿⡌⣷⣌⢻⣆⡀⠀⠀⠀⠀⠀⠀⢶⡋⠁⠀⠀⠹⢦⡀⠀⠀⠳⡄⠀⠀⠀⠀⢀⣾⣧⡝⣦
⠀⠀⠀⠸⣇⢿⠀⠘⣄⠀⠀⢀⡌⠀⣿⢣⡟⠈⣣⠝⣿⣦⣀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢷⠀⠙⠦⠤⣀⣀⣀⣀⣀⡤⠛⢀⣼⠟⣼⠁
⠀⠀⠀⠀⠹⣎⠷⣄⡈⣛⡿⠋⣡⡾⢋⡾⠁⢰⡋⣾⣿⣿⣿⣷⣦⣤⣄⣀⣀⣹⣦⠀⠸⣇⡀⠀⠀⠀⠉⠉⠉⠉⠀⢠⠏⠙⢷⡝⣧
⠀⠀⠀⠀⠀⢈⣷⠎⣙⣿⣰⣿⠻⡆⣿⠀⠀⠈⠛⠶⢮⣍⡛⢻⣿⣿⣿⣿⣿⣿⠙⠀⠸⣇⠙⠂⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢻⡌⢧
⠀⠀⠀⠀⠀⡟⣵⠟⠋⢙⡄⣟⠀⢿⡹⣄⠀⣀⣠⣤⠖⣫⣴⣿⣿⣿⣿⣿⣿⣿⣾⠄⠀⢹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢻⡜⣧
⠀⠀⠀⠀⠀⣷⣸⡧⠚⢛⠄⢛⠀⠈⢻⡌⢛⣭⣷⣶⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠻⣅⠀⠀⠀⠀⠀⠀⠘⠃⠸⠂⠀⠀⠈⢿⡸⣇
⠀⠀⠀⠀⠀⢳⡙⣷⠞⢛⡆⢠⠀⠀⢸⣇⣼⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡹⡆⠲⣀⠀⠈⠳⠶⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣧⢹⡀
⠀⠀⠀⠀⠀⢸⡄⢿⡤⠛⣹⡷⠂⠀⠀⢀⠉⠙⠻⠏⠀⠀⠉⠛⠿⠿⠟⠛⠁⢰⡆⠳⣄⣀⣀⡄⠈⡀⠀⠈⠉⠙⠓⠶⠶⡇⠀⠀⠀⣼⣿⡇⢷
⠀⠀⠀⠀⠀⠀⠹⣮⡛⠶⠿⣿⠉⠐⠦⣼⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣼⢿⣿⣿⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⣿⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣠⡞⠁⣀⣠⠞⠁⣾⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢻⡆⠀⡴⠃⠀⠀⠀⠀⠀⣠⠟⡌⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⡁⠀⠀⣿⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠻⠿⠷⠶⣶⣶⣶⡖⠚⠁⣰⠁⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢰⣿⣿⣿⣿⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢹⡇⣾⣿⣿⣿⠀⠄⠀⠈⠁⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢸⣿⣿⣿⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⣋⣾⣿⣿⣿⣿⠐⠶⠀⠀⠀⣤⠇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⣾⢿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢩⣶⣿⣿⣿⣿⣿⣿⣅⣼⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠛⠀⠀⠐⠒⢛⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⢿⣿⣿⣿⡿⢟⣭⢸⣇⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣝⣛⣭⠶⠛⠳⣮⡻⢦⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⢸⣿⣿⣿⣿⡿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣝⡓⠶⣤⣤⣄⣀⣀⡀⠀⠀⠙⠒⠀⠠⠤⠤⠤⠒⠊⠁⣀⣀⣠⣼⣿⣿⣿⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠭⢭⣍⣛⣛⣛⣛⣛⣛⣒⣒⣚⣛⣛⣛⣛⣫⡭⢭⡜⣿⣿⠟⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠻⣮⣡⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣧⣤⣼⣧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⡾⠋⠙⢷⡀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠘⣧⡀⢀⣸⠃⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠈⢹⡏⠁⠀⠀⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⢸⡇⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢀⣤⠞⠉⢸⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⠓⢲⡖⠒⢲⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣸⣇⣀⣀⣸⡇
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⠷⠶⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⣸⠃⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣉⣉⣉⣉⣁⣉⣽⡇
⠀⠀⠀⠀⣿⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢧⡀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠛⠛⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣶⣶⣦⣶⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣼⣃⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣟⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣟⣹⡿⠂⠀⠰⠿⠟⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⣫⣿⡿⠟⠉⢸⣿⣥⣶⣦⡄⠀⠀⠀⣸⣿⣿⣿⣧⠲⡌⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⢸⣿⣿⡟⠃⡀⠀⠀⢀⣀⠘⠋⠀⠀⠀⡽⢿⡉⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⡰⢋⣤⠴⠛⠛⠉⠀⠀⠀⠀⠀⣄⣸⡇⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠀⠙⢧⣻⠿⠓⣒⣈⣉⡁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡔⣡⠞⠏⠀⠀⠀⠈⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⣠⡇⠀⣀⡴⠚⢩⣼⣿⣧⡀⠀⠀⣀⠀⠀⠀⠻⣤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠓⠛⠀⢀⣙⣒⣎⡁⠈⠟⠫⡴⢋⡞⢉⡿⠁⠓⠀⠀⣿⣄⠀⠀⠀⠙⢲⠇⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⣈⡷⢀⡴⣾⠟⣽⣿⡿⠿⠀⣤⡀⠀⣟⣠⠞⠁⠀⠀⠀⢀⣿⠇⠀⠀⠀⣠⠏⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢰⡿⢚⡡⢞⣥⢠⣞⣠⣴⢂⡉⢀⡿⠀⠉⠁⠳⣄⠀⠀⣠⠞⢯⣀⡀⠀⣰⠃⡰⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢛⣙⡶⠋⢐⣎⡼⠟⠛⠻⣯⠉⠁⢾⣁⣀⣀⠤⢤⣈⠙⠛⠁⠀⣠⠟⠉⣿⠃⡰⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠾⢫⡄⢱⢋⡞⠑⢒⡤⣄⡈⠑⠒⠲⢦⡀⠀⠀⢀⠜⠀⠀⠀⠀⠷⠒⢶⡇⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠷⣾⢸⡇⡸⢏⣸⣿⣥⡤⠼⠿⠀⠀⠀⠘⣷⢠⠞⠋⢀⡶⣶⠄⠀⠀⢀⣼⣀⡀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡶⠫⣄⣿⣅⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⡓⣚⠽⠛⠁⠀⠀⠀⠀⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡤⠈⢿⣉⣙⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠶⠶⠶⣤⣤⣄⣀⣸⡃⣾⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣭⣭⢀⣀⣤⣤⣤⣤⣀⠀⢠⡟⠀⠀⠀⠀⢠⡟⠉⠉⠉⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣯⠶⠛⠉⠀⠀⠀⠀⠉⠛⣿⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣸⣇⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠃




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣙⣏⣨⣏⣹⣍⣉⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⣉⣀⣀⡀⠀⠀⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⠋⠙⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠀⠀⠙⠛⠛⠉⠹⡟⠛⠛⠋⠁⢀⣾⣿⠟⢩⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⣠⣿⡿⠁⠀⣾⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠚⢧⡀⠀⠰⣄⠀⠀⠀⠐⢲⡾⠉⠁⠀⠀⢠⣿⡿⠁⠀⣼⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⡄⠈⠳⡄⠀⢸⣿⣶⣤⣤⣾⣅⡀⠀⠀⠀⣾⣿⠁⢀⣼⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣹⣶⣾⠉⠁⣸⣿⣿⣿⣿⣷⣦⣄⣿⣿⢠⢞⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠻⣿⡟⠧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣸⣿⣇⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⣀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣿⠟⠉⠀⠀⠀⠀⠀⣄⠙⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⡀⠀⠀⠀⠀⠀⠀⠲⣌⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢷⣄⣴⣿⣿⣋⣿⣶⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣫⣟⣁⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢤⡠⠚⠁⠀⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⣭⣭⡷⠦
⠀⠀⠀⠀⠀⠀⠀⠐⠛⣓⡶⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠉⠉⣽⣣⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠬⠭⣯⡉⠛⠒⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠚⠓⢸⡀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣠⠶⠚⠒⢆⣠⠞⠋⠉⠉⠙⢧⠤⢖⣒⣋⠭⣹⠇⠀⣿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡞⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸⣈⠩⠤⠒⠒⠛⠓⠶⠋
⠀⠀⠀⢠⠞⠉⣉⠒⢤⣀⠞⠀⠀⣈⡓⢧⡀⠀⠀⡀⢰⠇⢀⠀⠀⢀⡴⢻⡀
⠀⠀⠀⢻⡀⠈⠑⠛⠆⠋⠀⣀⣼⠉⠉⢲⠙⢦⠤⠟⠋⠓⠿⠶⣶⣋⠀⠀⡇
⠀⠀⠀⠀⠉⠓⠒⠒⠒⢦⡀⠀⠉⠛⢶⡋⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠲⢤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠸⡟⠲⣦⣤⣀⣀⣀⠀⢀⣀⣰⠚⠛⠛⠓⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⣄⠘⠶⠳⣀⡝⠳⠏⡙⢯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣻⢏⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢿⠀⠀⢀⠏⠹⡄⠀⣳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⢶⠋⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣤⠴⠾⠀⠀⣸⡠⠤⢽⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠙⣲⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣧⢸⡄⠀⠀⢠⠿⡀⠉⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠻⣅⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡤⣼⡈⣧⡤⢤⡞⠀⠉⢹⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠏⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⡧⠞⠀⠈⠙⣦⣄⡈⠉⣩⠝⠁⢠⠏⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠁⠢⣀⠀⠀⣰⠛⢻⠙⠛⠁⠀⢠⠏
⠀⠀⠀⠀⢰⡋⠉⠛⣍⣓⣶⡤⢤⣄⣰⣇⠀⢸⣦⠴⠖⣚⠟⠛⠒⡦
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠳⠏⠀⠉⠙⠓⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡴⠀⠀⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠇⠀⢀⠔⠁⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠤⠂⠁⢀⠞⠀⠈⠉⢉⣥⠏⡁⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢀⢀⠀⠒⠃⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢄⠂⢄⡀⠐⠒⠒⠒⢆⡁⠀⠀⠀⠀⠀⠈⠒⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡇⠀⠀⠀⠀⠀⠀⢲⢀⠀⠀⠀⡀⠈⠐⠠⠀⠀⠀⣥⠑⠒⠒⠦⣤⢤⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⠀⠈⠐⠀⠐⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢃⠀⠀⠈⣷⠀⠈⠀⣤⣤⣴⡾⠏⠀⠀⠀⠀⠀⠙⠻⠶⠖⠂⠀⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⢢⠀⠀⢉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠨⠴⢠⣕⢄⡀⠀⠀⠀⠀⠘⠳⣦⡄⠀⠀⠀⠀⠀⣴⣿⣁⠀⠀⠀⣄⠸⠀⠀⠊⡀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡀⠙⠄⢈⠐⠀⠀⠀⠠⠖⠛⠁⠀⠀⣀⣀⡀⠀⠈⠉⠀⠀⠀⡉⠲⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⢹⠆⠈⠂⣄⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠈⠒⢀⡀⠀⠀⠀⠀⠀⢀⠔⠋⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠉⠉⠣⡀⠀⠀⡔⠀⠋⢀⡎⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠈⠒⠦⠀⠀⠀⠀⢀⠤⠖⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣠⠐⠁⡀⠤⠄⢤⣄⠀⠀⠀⠈⢀⠀⣘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣠⣤⠤⠤⠀⠐⠈⠇⠀⠀⠀⠀⠀⡈⢐⡌⠀⠀⢀⣾⢿⡗⠀⠁⠀⣸⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⠤⣀⣀⣤⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠄⠒⠊⠉

⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⡅⠱⠀⠀⠀⠀⠀⠀⠙⢟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢹⡄⠓⠂⣀⠀⠀⠀⠀⠀⢷⠨⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠤⣿⡈⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠋⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⢈⡬⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠐⠂⠀⢠⠀⠀⠰⣿⠀⠖⠀⠀⠀⢠⠀⠀⠀⣼⠀⠀⣁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⡾⠄⢀⠇⠀⠐⢰⡇⣀⡀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⢩⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢛⠛⠛⠓⠒⠚⠓⠒⠃⣼⠀⡍⠀⠀⠀⢀⡇⠀⡌⢸⡇⠀⠁⠸⠂⢻
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣯⣭⣵⣶⣅⣀⣤⣨⡥⠴⠖⠚⠋⠻⣿⣿⣿⣿⡿⢾⣷⣶⣷⣿⣯⡉⠝⠀⢡⢸⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⡟⣿⣿⣂⠀⠀⠀⠀⠀⠈⢻⣷⣯⠿⠚⠻⣾⣿⢿⣿⣿⣦⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠻⣇⠈⠁⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣌⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⣼⣿⣿⣿⣿⣿⣿⡟⠁⠉⠛⠶⣽⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠈⠉⢉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣦⠤⠶⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⣰⣿⠿⠛⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⢀⡠⠴⠒⠉⠁⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢰⡇⠀⠀⠀⢀⢀⣀⣀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⣠⢤⣿⣿⣿⣿⣿⡷⣤
⠀⠀⢀⣀⣤⣤⣤⠀⢀⣾⠏⠀⠀⠀⠀⠀⠳⢤⣴⣾⣿⡿⣿⣿⡇⡜⣈⡧⠀⠀⣀⣤⡖⠋⢠⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠡⡉⠻⡿⠴⣊⡏⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡝⡓⢸⣿⣿⣭⢤⣴⣺⡽⠛⠉⠀⣴⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢳⡀⣠⢞⠟⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠷⠃⣏⠉⠉⠉⢩⣾⠉⠁⠀⠀⣠⣾⣿⡿⡿⠿⠿⢿⠟⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⡴⢻⣿⣿⠟⠁⠀⠀⠘⡄⠀⠀⣾⡏⠀⠀⠀⠘⠋⠉⠉⠀⠀⠠⢀⠀⢠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡿⠛⠁⠀⠀⠀⠀⠀⣷⠀⠴⢿⣇⠀⠀⠔⢉⢀⣦⣔⠶⠋⠚⠂⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢻⡄⢆⠤⠊⠈⠀⢠⡄⢀⠆⡀⢐⣤⣤⡴⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠜⠀⡏⠐⠘⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣦⣴⣶⣦⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣤⢄⡄⠀⠀⠀⠀⠀⠀⡀⠀⣠⠋⠀⣠⡞⠣⡈⢢
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⣷⣄⠬⠌⠘⠘⠳⠀⠀⠀⠒⠢⡤⠄⢣⠴⣎⠀⢴⠀⠘⠺⡄
⠀⠀⠀⠀⣿⣿⠟⣋⡀⠆⠈⢉⠳⡀⠰⠀⡀⠠⠐⠒⠒⣒⡂⠀⠘⡴⢷⡇⠀⠢⢀⣶⠃
⠀⠀⠀⠀⢙⠋⣆⠬⠌⠊⢘⣸⡷⠈⡂⠉⢡⠀⠀⠀⠀⠰⠄⠀⡠⠘⠤⣨⡡⠔⡋⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⣿⣷⣵⡶⠥⢺⡟⠁⠀⠇⠀⠀⡀⠀⠠⠄⢀⠄⡪⠊⡠⠊⠀⠑⠒⠁⠀⠀⠢⡀⠈⢂
⠀⠀⠀⠀⢀⠅⠋⠁⠑⢠⠿⣷⠀⠀⠀⠀⢀⠇⠀⠀⠔⠁⠀⠀⢰⠁⠀⡰⠄⠀⠀⠀⠀⢀⡇⠀⡸
⠀⠀⠀⠀⡎⠀⠠⠔⠂⢫⠀⢸⠀⢐⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣤⠒⠁⢺⠀⠀⢠⣎⣁⢈⠕⠁
⠀⠀⠀⠀⢃⠀⠀⠀⣀⠼⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠛⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠁

⣀⢀⣠⡀⠀⠀⣀⡀⠐⠂⠀⠄⠀⠚⠀⣀⡀⠀⠀⡈⠛⠙⠉⠃⠀⠀⠀⠀⠀⠁⠀⠀⠢⡀⠁⣠⣀⠀⠁⠘⢀⡀⢀⠑⠈⠉⠀⠀⠀⠀⠁⠀⠘⡟⠄
⠈⠉⠀⢀⣾⡁⣀⠀⠱⠀⠀⠀⡴⠋⠩⣽⣿⣿⣯⡽⣆⠀⠀⠀⢨⠿⠟⠷⢤⠀⠀⠀⠀⠀⢰⠃⠬⠽⣆⠀⠀⠘⠀⠀⡀⠁⠈⠲⠖⠀⠈⠀⠀⣀⠉⠀⡈⣷
⠀⠀⠀⠸⡄⠀⠈⠀⢸⠀⠀⠐⠃⢀⠀⠆⠀⠘⠉⠄⡋⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠀⣼⠀⠀⢠⠀⠀⣧⠴⠄⠴⠤⣖⠒⠒⠒⢣⠂⠀⠀⠀⢷
⠀⠀⠀⠀⠁⠀⠥⠀⠀⠀⠀⢨⠀⠈⣀⠀⠦⠀⠀⢀⠁⠀⠀⠀⠁⠀⠤⠄⠀⠁⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⡄⣤⠄⠀⢻⡀⢠⠄⠐⣉⣉⣉⣉⣉⡀⠠⠤⠀⠊
⠀⠀⠀⠀⠀⣅⣀⣠⠀⠀⠀⢈⠀⠀⢨⠂⠀⢺⠀⠀⠀⡆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⡠⢄⠀⠙⠛⠋⠀⠀⠀⠈⠢⠇⡄⠀⠀⠈⣇⣤⠜⠁⢀⠀⠀⠒⠦⠒⠂⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⡤⣀⣀⣢⣄⠀⣀⡀⠐
⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⡙⠯⠯⡲⡄⠀⠀⠀⠀⠠⡀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠚⠝⢀⣶⠀⠀⠈⠀⠀⢰⠋⣡⠉⡆⠠⠉⠂⠋⢀⠶⠝⠧⡠⣵
⠀⠀⠐⢃⡘⠀⠀⠀⠀⣦⠠⡤⠤⢴⡅⠀⠀⠀⠸⡀⠀⠀⠀⣀⢀⠀⠀⠀⠀⡌⠀⠀⠀⠀⡅⠀⠀⠀⠀⡟⠀⠀⠀⢻⡆⠀⠀⠀⠘⡆⠀⠀⠀⠸⠀⠀⠀⠀⠐
⠀⠠⠀⠈⠀⠠⠀⠀⠠⠐⠄⠴⠄⠐⠀⠀⠀⠀⠐⠳⠀⢂⠀⠊⠀⠀⠀⠀⠀⠁⠠⠔⠤⠀⠃⠀⠀⠀⠀⡇⡀⠀⠀⣸
⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⡖⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⡀⢀⡄⠀⠀⠀⠀⠀⣇⡗⠀⠘⡯⠀⠀⠀⠀⠀⢰⠁⠀⠈⡀
⡀⠀⣀⣤⣀⣤⠀⣀⡤⠈⠂⠠⠀⠄⠀⢀⠀⠀⠀⠐⠀⠄⠀⠀⢀⠀⠀⡀⠀⠀⠒⠠⠄⠓⢀⡄⠀⢠⡀⠋⠀⢀⠠⠉⠂⠀⠀⠀⠄⠀⠠⠤⠄⠀⠠
⠉⡛⠉⠉⠇⠉⠁⠀⢰⠂⠀⠀⠀⠀⠀⢺⠁⠀⠀⡀⠀⠀⠀⠀⠨⠏⠭⠟⢦⠀⠀⠀⠀⠀⠿⠤⣀⠀⢙⡄⢀⠀⣀⠀⠠⠁⡏⠁⠀⠀⠀⠀⠀⢀⡛⠣
⠀⠁⠀⠀⠏⠉⠀⠋⠸⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠃⢀⠀⠠⠀⠀⠀⠀⢻⠀⠀⠀⠀⠐⡂⠀⡀⠀⠸⠁⠢⡐⠂⠀⠀⠄⠁⠠⡁⡞⢀⠀⠀⣾⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⡀⠋⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠘⣀⠸⡄⠀⠄⠀⡸⠈⠤⠀⠀⠈⠤⢾⠆⠠⠁
⠀⠀⠀⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰
⡤⣠⠀⠀⠀⠄⠒⠀⠀⠀⠢⢄⣀⠀⠀⠀⠀⠁⠀⣠⠀⠀⢄⠀⠁⠂⠀⠂⠁⠀⡄⡀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠘⣫⡄⠀⠀⠀⢀⡀⢾⡀⠀⠈⠀⠀⠀⠀⠰⠂⠓⠀⠀⠀⡐⠀⠀⠀⠀⢀⣀⠀⢸⡗⠀⠀⠀⠀⠀⢠⡶⠪⠳⣶⡉⠀⠀⠀⠀⢈⠃⠈⠉⣷⠀⠰⠋⠐⢴⡳
⠀⠀⢰⡆⠀⠀⠀⢠⡇⠀⠈⠀⢲⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⣺⡿⡄⢤⣿⣄⠀⠀⠀⠀⢿⢔⠀⠐⢿⡇⠀⠆⠆⠆⢰⠂⠀⠈⠀⣿⠀⠀⠀⠘⡇
⠁⠀⠘⠀⠀⠀⠀⠀⢣⠀⠤⠀⣈⠂⠀⠀⠀⠀⢲⣀⠢⠄⣰⠃⠀⠀⠀⠀⡟⡀⠰⠀⢨⠅⠀⠀⠀⠀⣏⠁⠉⠉⠉⢃⠀⠀⠀⠀⠘⠆⢀⣢⠀⠟⠀⠀⠀⠀⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠈⢗⣀⣺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⡘⠀⠀⣧⠃⠀⠀⠀⠀⢻⠆⠘⠁⠀⠻
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⠓⠛⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢀⡀⢀⠠⠤⠤⠤⠼⠠⠀⢀⡄⣀⠓⠃⠀⠀⢀⠋⠀⠀⠀⠀⠀⠚⠢⠔⠚⠀⢀⣀⣀⡤⡄
⡄⠀⠀⠀⠀⠄⢉⡁⠦⠀⠀⠀⠀⢠⠔⠇⠶⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡝⠊⠉⠚⡆⢀⡠⠀⠀⠠⠔⠋⠢⣤⡀⠀⠀⣀⡀⢎⡁⢩⡀
⠉⠀⠀⠀⢪⠀⠀⠀⠀⣤⠀⠀⢲⣎⡉⠀⠀⠁⢁⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡔⠐⠒⢂⠂⠂⡇⠀⠀⠀⡇⠈⡇⠀⠀⢰⣶⠀⠈⣧⡐⠒⠒⠺⣗⣅⠀⠈⠉⠉
⠀⠀⠀⠀⠸⡄⢀⡀⢠⡧⠀⠀⠀⠈⠆⢀⣦⡄⠠⠁⠀⠀⠀⠀⠈⠣⠼⠆⠜⠁⠀⠀⠀⠀⠀⢰⠀⠄⢰⠀⠀⠀⠀⠀⢿⠋⢃⠙⢙⠇⠀⠀⠀⠘⠻⡄⠨⠀⠸
⠀⠀⠀⠀⠠⠸⠆⠠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⢦⠀⠀⠀⠀⠀⠀⢈⠁⠀⢀⠀⠀⠀⠀⠀⡇⡄⠀⠀⡼⠐⠂⠒⠒⠒⠂⠈⠀⠈
⠀⠀⠀⠀⠁⡀⠠⡀⠁⡈⠠⠄⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠉⠒⠒⠈⠀⣠⠀⠀⣀⠐⠀⢀⣤⡀⠐⠀⣀⣀⠀⠉⠋⠀⠀⢛⠉⠀⠀⠁⠈⠉
⠀⠀⡌⠓⣄⠈⠒⣄⡐⠁⠀⠈⠑⣀⡄⠀⠀⠀⠀⠈⠀⠐⣶⣤⡀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠁⡐⠉⠄⠹⢻⠀⠘⠡⠈⠀⠀⢴⠀⠀⠀⠀⡀⣀
⠀⠀⠑⠄⢠⠈⠉⠉⠚⣆⠁⠀⠀⢀⠁⠀⠀⠀⢘⠀⠠⠀⠀⢹⡇⠀⠀⠀⠀⡆⠁⠀⠀⠀⡔⠀⠀⠀⠰⢖⠀⠀⠀⢘⡇⠀⠀⠀⠀⣠⠈⠀⠄⣠⠰⠀⠀⠀⡀
⠀⠀⠀⠰⠈⠀⠀⠀⠁⠛⠄⠲⠂⠋⠈⠈⠀⠀⠈⣤⠀⣤⠀⠸⠁⠀⠀⠀⠀⢠⡀⠂⠂⣐⠇⠀⠀⠀⠐⠘⠀⠠⠀⠠⠡⠀⠀⠀⠀⠇⠀⠄⠀⠘⠈⠈⠉⠀⠉
⠀⠚⡠⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⡍⠀⢂⠀⠀⠀⠘⡄⠉⢰⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡁⢿⠻⣿⣿⣿⣿⣿⣿⡿⡇⠸⢿⣿⣿⣿⣿⣿⣿⢿⡿⢧⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⡈⠉⠛⠛⠛⠋⠀⡇⢸⠂⠉⠛⠛⠛⠉⢁⢨⣶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡙⠇⠀⠀⠀⠀⢶⣀⠃⠈⣀⡶⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣹⣟⣟⣉⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⣿⣿⣿⣄⠀⠀⠁⠙⠛⠛⠻⠟⠛⠛⠃⠘⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢦⡀⠀⠀⠈⢹⠿⠿⠋⠁⠀⠀⣠⣾⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢮⣿⣦⣆⡀⠀⠀⠀⠀⢰⣦⢞⠕⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠛⠿⢿⣶⣶⠶⠞⠫⠐⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣼⣷⣻⣆⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡏⢷⡀⠀⠀⢀⣾⣿⣿⣿⣻⣆⠀⠸⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠾⠿⠿⠿⠿⡷⠛⠂⠰⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡛⠲⢤⣀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡒⠶⠶⠦⠤⠤⣤⢽⣄⡀⠉⠳⠏⠰⢴⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠒⠛⠀⠀⠀⠀⢀⣀⡤⠤⠴⠶⢶⡄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢄⡤⠚⠉⠓⢦⡴⠚⠉⠁⠀⠀⠀⠀⠘⣷⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠤⢤⡀⠀⠀⠀⢀⠴⠒⠲⠄⠀⠀⣟⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⣏⠀⣞⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠛⠉⢉⣕⣬⠿⠀⠀⠀⠀⢶⡋⠀⠈⠀⠀⠀⢰⠃⡼⣫⠍⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⡰⠚⡇
⠀⠀⠀⢠⣠⠶⠋⠁⠀⠰⡀⠀⠀⠀⡟⠒⣄⢲⣤⣾⠤⠤⣄⠀⠀⠀⠀⠀⠸⠶⢡⣌⡠⠟⠐⠒⠒⠤⠤⠤⣄⣀⣀⣀⡀⢀⣀⣀⣀⣀⡃⠛⠀⣰⠗⢚⡆
⠀⠀⠀⢽⠃⠀⠀⠀⠀⠉⢉⠒⠒⠲⢹⡆⠉⠀⠀⠠⠤⡴⠋⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠐⠢⠀⠀⣀⠀⠀⠉⠉⠉⢉⡍⡇⠀⠈⡇⠀⠞⢁⣠⠞
⠀⠀⠀⠈⠳⣄⡀⠰⠤⠈⠉⣀⠀⠀⡈⣇⠀⠀⠐⢒⡿⠤⠤⣤⣤⡤⣴⡞⠋⠀⠀⠈⠉⠛⠲⠤⢄⣀⡀⠉⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡇⠀⠰⢯⣀
⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⣥⣀⣀⣳⣈⠲⣉⡿⢁⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠛⠃⢸⠷⠤⠖⠓⠢⢄⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡽⠉⠙⠉⠁⣿⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣼⠁⠀⠀⠀⠀⠁⣆⡀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢠⠒⠠⠊⡟⠀⢠⠃⠀⠀⠀⢰⠒⠀⠈⢱⣀⠇
⠀⠀⠀⠀⠀⠀⢱⠄⠀⠀⠉⡇⡎⠀⠀⠀⠀⠈⠲⣀⡴⠋⡌
⠀⠀⠀⠀⠀⠀⠈⠛⠲⠚⠉⠀⣧⣤⣤⣀⣀⣀⣀⣠⡤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⠶⡤⠤⣤⡼⣼⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠀⠀⠑⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣇⠀⣠⠴⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⡄⠙⠶⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣷⣤⢤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣃⢹⡎⢣⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⣁⡤⠽⢿⣌⢇⠈⡆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⣾⠁⠀⠀⠀⠈⢾⡰⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⢤⣿⡀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣆⠉⠛⠒⠲⠒⠚⢛⣉⣡⠶⠋
⣀⣀⡀⢀⣀⢀⣀⢀⣀⣀⣀⣀⢀⣀⢀⣈⣉⣉⣛⣛⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⣀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⣣⣾⣿⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⣛⣛⡛⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠛⠛⠛⠛⠀⠀⠀⢹⣿⣿⣿⣿⣿⢸⣿⣿⡇⣿⣿⣷⢹⣿⣿⠿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣸⣿⣿⡇⣿⣿⣿⢸⣿⡇⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⣿⣿⢳⣿⣿⣿⡷⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⣿⣿⢸⣿⣿⡿⠁⠀⠀⣿⡇⣿⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢸⣿⣿⣇⣿⣿⣸⣿⣿⡇⠀⠀⡀⣿⣧⣿⣧⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⣛⣛⣛⣛⣛⡛⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣻⣿⡇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣭⣛⡛⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⣿⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⠷⢣⡖⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠏⠉⠈⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠘⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⡍⠿⣶⣅⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⠈⢢⡀⠁⠈⠉⠻⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⢢⡀⠉⢲⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⠃⠀⠀⠈⣇⠙⢦⡀⠰⠧⡀
⠀⠀⠀⠀⠀⠀⠀⡰⢱⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠰⠁⣿⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡇⢯⠸⡀
⠀⠀⠀⠀⠀⢠⠃⢸⣿⠀⠀⠀⠀⠀⣸⣠⣀⣤⠖⡀⠈⡇⡇
⠀⠀⠀⠀⠀⡜⢠⣿⣿⠀⠀⠀⠀⢠⣭⣽⡧⠔⣹⠇⣠⡇⢇
⠀⠀⠀⠀⢀⡇⠘⢻⣿⠃⠀⠀⠀⠀⡿⠛⣞⣿⣵⡿⠁⢿⣾
⠀⠀⠀⠀⠘⡇⣦⢸⣿⡀⠀⠀⠀⢠⣇⠀⠀⣼⡯⠃⠀⢸⢻
⠀⠀⠀⠀⠀⣷⣿⡄⣿⠗⣉⡑⠛⢦⣹⠀⢠⣿⣷⡀⢀⣼⠆
⠀⠀⠀⠀⠀⡘⢻⣿⣿⣾⣯⠍⠙⢦⡻⠀⢸⡈⠓⠞⠒
⠀⠀⠀⠀⢀⡇⠀⠹⣿⣿⢹⡆⠀⠈⠁⠀⠈⡇
⠀⠀⠀⠀⠈⢧⣳⣤⣹⣿⣆⣇⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠄⠀⣶⣾⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⢻⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢂⠈⠻⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡇⠁⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣦⠆⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⡻⢿⣿⣿⣧⣤⡙⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡳⣍⠀⠉⠙⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⠒⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⡭⡀⠀⠀⠀⠀⠀⠀⠀⡑⢙⠀⠀⡀
⠀⠀⠀⠀⠎⠉⣭⠆⠀⠀⠀⠀⣀⣀⠻⢡⠃⠀⠀⠀⠀⠀⢠⡶⡆⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⡅⠢⡀⠀⠀⠀⠀⢠⢜⠓⢄⡠⠟
⠀⠀⠀⠀⢀⡣⠛⡤⠀⠀⠀⠀⠀⢺⠄⡆⠀⠀⠀⠀⠀⠀⣄⠀⡵⠖⢄⠀⠀⠀⣀⠀⣪⠀⠀⡄⠐⣤⠄⠀⠀⠀⣒⡌⠀⠑
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⡀⣁⠀⠀⠀⠀⠀⠀⠹⣾⣲⢀⠀⠀⠀⠀⠈⠁⠀⡇⢈⠁⠀⠀⠀⠀⠀⠀⠀⠥⠠⠆
⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠰⢀⢆⠀⠀⠀⠀⠀⠀⠀⠸⢀⣨⠀⠀⠀⠀⠀⠀⠀⠀⡈⢢⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⡆⠃⠀⠀⠀⠀⠀⠂⡘⢀⠂⠀⠀⠀⠀⠀⢀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠤⠀⠀⠱⢁⠀⠀⠀⢀⠂⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⠲⠒⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠈⠀⠀⠀⠈⠊⠀⠰⠤⠀⠀⠀⠀⠀⠀⠲⠚⠓⠒⠆⠀⠀⠀⠀⠀⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢚⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡇⠀⠀⠀⠀⠀⢸⢠⡆⠀⠀⠀⠀⠀⢈⢹⠟⠀⠀⠀⠀⠀⠀⢘⢉⡇
⠀⠀⢠⡀⠀⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⣿⠀⠀⠀⠀⠐⠉⣏⠉⠀⠀⠀⠀⠀⢠⠟⡄⠀⠀⠀⠀⣀⠀⢠⠁⠁
⠀⠀⠀⠋⢐⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⣇⡌⠉⠀⠀⠀⠀⠈⡇⢯⡚⠁⠀⠀⠀⠀⡽⠰⠳⠀⠀⠀⠀⠶⡂⡉⠀⠓⡃
⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠈⠄⠀⠀⠀⠀⢐⡥⠄⢬⠀⠀⠀⠀⠀⢇⠖⡒⠁⠀⠀⠀⠀⠈⠅⣤⡡⠁
⠀⠀⠀⠀⢀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠁⠹⢸⠀⠀⠀⠀⠀⠀⠰⡅⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⡄⢚⡤⠊⣠⠀⠀⠐⠤⢤⠔⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠴⠐⡀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⡌⠠⠐⡀⠀⠀⣭⢉⡉⣅⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠁⠑⣰⠄⠀⠀⠀⠘⢠⠣⠈⠄
⠀⠀⠀⠐⠒⠀⠀⠑⠁⠀⠀⠙⠀⠘⠈⠀⠀⠀⠘⠀⠀⠋⠀⠀⠀⠀⠓⠈⠚⠀⠀⠀⠀⠐⠚⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠓⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠠⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠈⠳⣄⠀⠀⠀⣠⠞⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣷⣠⡾⠃⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣶⠶⠤
⠀⠀⠋⠉⠉⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡷⠖⠒⠛⠉⠉⠹⣏⠉⣹⣍⣿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣤⣼⣭⠿⠁⠀⠉⠉⠉⠉⠒⠒⣿
⠀⠀⠀⠀⠀⠸⣿⠀⢀⡠⠄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢹⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣷⠀⠁⠀⠀⢠⠀⠀⠀⠀⠐⠲⣄⡀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢽⣠⡾⠁⠘⠷⣄⡀⠀⣮⠁⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣬⠇⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠙⠛⠧⠤⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠙⠻⣿⣿⣓⠾⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⣷⣮⣝⣷⢯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠉⠛⠻⢧⣍⣷⠮⣝⣓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣯⣙⢲⣭⣛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⡒⢬⣝⡲⢤⣄⣀⣤⠶⠶⠶⢤⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⣽⣗⠮⣭⣀⣤⣶⠦⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⢿⣿⣷⢤⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⠿⠿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⠎⠉⠉⠱⢤⡄⠶⠀⠀⠰⠦⣄⠀⢠⠔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠁⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠹⠖⠈⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠙⢦⡀⣀⡀⣤⠒⢤⠀⢀⣀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠀⠀⠐⠒⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠀⠀⠀⠀⠀⠀⢠⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠤⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠄⠒⡤⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠷⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⢀⣀⠀⠀⠀⠀⠈⢶⡉⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠓⠀⠀⢸⡆⣰⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠀⠀⠀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢹⠀⠀⢀⠊⠀⠀⢓⡄⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠛⠃⠈⠀⠀⠀⠀⢃⠀⢸⠀⠀⠛⡀⠀⠰⠋⠙⠢⠤⠀⠀⠤⠤⠀⠒⠒⠛⠛⠂⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⢀⠀⠀⠀⠘⢆⠸⠀⠀⠀⠈⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠰⠿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠳⢄⠠
⠀⠠⠄⠚⠁⠀⠀⠀⠀⠀⠁⠷⠀⠀⠀⠀⠇⠀⠀⢀⡳⠆⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⢡⠔⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⢣⡀⠀⠀⠰⠀⢠⠋⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀⠈⠉⠀⠀⠰⠇⠀⠀⠀⢸⠀⠀⠀⠀⣧⣀⠤⢿
⠀⠀⠀⠀⢀⡴⠁⢀⠎⠁⠀⠀⠀⠙⢄⡴⠃⠀⠆⠀⠀⠀⠀⡴⠃⠉⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠈⠦⡀⠀⠀⠀⡞⠢⠤⠔⠃⠀⠀⠀⡸⠁⠀⢀⡌
⠀⠀⠀⠀⠸⣇⡠⠛⠀⠀⠀⠀⠀⠀⠸⠆⢤⣄⣸⣄⠀⠀⠀⢧⡀⠀⢀⣀⠀⠀⠀⠀⠑⢄⣹⡀⠀⠀⠀⠱⡄⢀⡜⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠈⠉⡿⠶⠄⢀⡀⢹⠀⠀⠀⠀⠀⠉⠑⢦⡀⠀⢱⠄⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⣠⠞⣠⠎⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠼⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⢣⣀⡀⠀⠀⠀⠀⠀⠘⠶⠶⠀⠀⠀⣼⡀⢰⠀⠀⠀⣀⠰⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠋⠘⠋⠙⢧⠀⠐⢲⠀⠀⠀⠀⠀⠀⠹⣶⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠉⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠶⢤⡀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⡄⠀⠀⠀⣠⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆⠀⠙⢤⣤⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠤⠤⠄⠒⠂⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠀⠀⠀⠀⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠸⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡌⠉⠀⠑⠒⠒⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠒⠠⠤⠤⠤⠤⡤⠋⠀⠀⠀⠀⠀⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠠⠤⠤⠀⠀⠤⠼⡄⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠘⡶⠠⠤⠄⠘⠃⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⣰⣆⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠋⠀⠈⠛⠁⠀⠈⣦⠀⠀⠀⣀⠔⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠙⠙⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠴⠦⠀⠀⠋⢹⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣀⡀⠶⠋

⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠂⠈⠉⠉⠉⠀⠒⠢⢄⡀
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⣀⣤⣤⣀⠀⡶⠤⣀⠀⠈⠲⣄
⠀⠀⡠⠊⠀⠀⠀⣠⠔⠛⠉⠉⠉⠉⠙⠳⢯⣊⣓⣄⠀⠈⠢⡀
⠀⡜⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠙⢯⣾⢧⠀⠀⠑⡀
⣸⠃⠀⠀⢀⠃⠀⠀⠀⠀⠀⡠⢊⢔⡡⠀⠀⠀⠀⠙⣯⣣⠀⠀⢷
⣯⠀⠀⠀⠈⠀⠀⠀⠀⠀⢨⠔⠁⠉⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢸⡆
⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡢⠀⠄⠀⠀⣧⡇⠀⢸⡇
⢸⣥⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢊⠄⡠⠂⠀⢠⣷⠃⠀⢸
⠀⢳⡅⠀⠀⠐⡀⠀⠀⠀⠀⡠⠊⠊⠐⡡⠊⠀⠀⠀⣾⠋⠀⢀⡟
⠀⠀⠳⡵⣄⠀⠈⠢⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠞⠁⠀⢀⣾⣷⢄⡀
⠀⠀⠀⠙⢧⡧⡄⠀⠀⠑⠂⠤⠄⠤⠤⠤⠐⠊⠁⠀⣀⣴⣾⣿⣿⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠙⠻⢥⣅⣄⡦⠀⠀⡄⢀⠀⣤⣤⣶⡿⠟⠉⣟⣫⣃⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠚⠚⠛⠉⠁⠀⠀⠀⠘⢗⢵⠤⠄⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣖⠒⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⡫⡀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡴⡀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⡣⡂⡀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡢⣂⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡷⣠⡄⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⡤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⣶⡷⠶⢶⣶⣦⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⢿⣿⣿⠓⠒⠀⠀⠈⠉⠛⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠻⠃⠀⠨⠽⢶⡆⣤⠠⠿⠿⠿⢓⣴⡮⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⠀⠀⠬⠇⠂⠀⠀⠀⠠⡫⠋⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⡞⠉⠁⢿⣶⠀⠀⠀⠠⠠⠔⠡⠀⠀⠀⣿⠃⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⢰⠁⠀⡆⠈⢿⣦⠀⠀⠀⣹⣦⣄⡀⠀⠀⢩⣘⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡀⠀⠘⣤⠀⢰⠀⢠⣿⣿⢿⣿⣿⣿⡿⠋⠉⢉⡉⠛⡙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢱⠀⠸⡁⠀⠘⣆⠏⣼⣿⣤⣿⣿⠟⠀⠀⠀⢄⡈⢢⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠙⣄⠀⢻⣼⣿⣿⣿⠟⠁⠀⠀⠀⠐⢤⣹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠆⢸⡿⠛⠋⢁⠀⠀⢀⣀⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠴⠊⠁⠀⠀⢠⠈⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣦⣶⣶⣶⣶⣶⣾⣿⡏⣴⠟⢷⡌⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣍⣭⣍⣭⣭⣽⣿⣿⣿⣧⡘⠳⠛⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠻⣿⠟⢛⣿⣿⡿⠟⢋⣼⣿⣿⣿⣿⣿⣿⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠰⣿⣭⣤⣴⣾⠟⢛⢛⢛⣛⠋⣉⣤⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣴⣿⣷⣬⣿⣿⠛⣉⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⢻⡇⢸⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣅⠸⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣤⠀⠀⠀⢠⣤⡄⠀⠉⠙⠻⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⣿⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⡁⠠⠐⠂⠤⢈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠅⣣⠖⠋⠉⠉⠉⠒⢜⣠⡈⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⣒⣦⣤⠤⠤⢖⢒⡚⡉⠠⣈⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡁⠄⣈⢙⡒⠲⠤⠤⣤⣴⣒⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡕⠉⠉⠉⠉⢹⡤⠥⠤⠶⠒⠒⠊⠁⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠑⠒⠶⠦⠥⠬⣄⡿⠉⠉⠉⠉⢪⡳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣤⡤⠤⠤⠤⠖⠚⠉⠁⠀⠀⠉⠙⠒⠦⠤⠤⠤⠤⠤⠤⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣸⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⣄⠀⠀⠀⠀⠈⡇⠀⠀⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠀⠀⣸⠀⠀⠀⠀⠀⣠⢣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡦⣄⣀⠀⠀⢳⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠚⠉⠁⠀⠀⠀⠀⡏⠀⠀⣀⣠⢴⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣆⠀⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⡼⠽⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠉⠓⠒⠦⠤⢤⣄⣀⣀⣀⣤⠤⠤⠖⠚⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠚⠋⠉⠁

⠀⠀⠀⢠⣤⣤⠀⠀⠀⣠⣶⣦⠀⠀⠀⣰⢶⡄⠀⠀⠀⡴⡶⠆⠀⠀⠀⡠⠶⠆⠀⠀⠀⠠⠶⠆
⠀⠀⠀⢸⠀⡸⠀⠀⠀⣯⠉⣸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠰⠶⠶⠂⠀⠀⠻⠤⠟⠃⠀⠀⠲⠤⠖⠀⠀⠐⠢⠤⠖⠀⠀⠀⠢⠤⠔⠀⠀⠀⠲⠤⠔⠂
⠀⠀⠀⢠⢴⣤⠀⠀⠀⢴⣤⣴⠀⠀⠀⣤⣶⣄⠀⠀⠀⣠⠶⡄⠀⠀⠀⣠⣶⣄⠀⠀⠀⢠⣴⠄
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠈⠉⠉⠀⠀⢠⣇⢀⣹⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠠⠀⠠⠀⠀⠀⠴⠤⠴⠀⠀⠀⠏⠈⠿⠀⠀⠀⠦⠭⠴⠀⠀⠀⠦⠬⠴⠀⠀⠀⠦⠌⠴
⠀⠀⠀⢀⣤⣄⠀⠀⠀⣠⣤⣄⠀⠀⠀⢠⣤⡀⠀⠀⠀⢠⣤⡄⠀⠀⠀⣠⣤⣄⠀⠀⠀⢠⣤⣄
⠀⠀⠀⣿⠉⢹⡆⠀⠀⠛⠉⢹⠀⠀⠀⠈⠀⠁⠀⠀⠀⢀⣀⡀⠀⠀⠀⢉⣀⡙⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠿⠁⡻⠁⠀⠀⠤⠀⡠⠀⠀⠀⢤⣀⡤⠀⠀⠠⢤⣉⠤⠀⠀⠀⠤⣉⠤⠀⠀⠀⠤⣉⠡⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⣣⡴⠎⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀⠀⠀⠀⢀⡀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣤⡴⣧⣤⣤⣤⣤⣤⣤⣤⣾⡧⠴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⢰⡅⣀⣀⣤⡄⠀⠀⠈⢉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠸⢄⣘⣿⡷⠃⠀⠠⢄⣀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⢸⠇⠀⠲⠖⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠸⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⢀⡇⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠷⠶⠶⠶⠶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣼⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⣨⡟⢻⣏⠉⢹⡇⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠁⢹⡏⠙⢧⣈⡇⡤⠀⠀⠀⠀⢰⡏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡻⡇⠀⢀⠸⡇⠀⠀⢹⡟⠀⠀⠀⠀⠀⠈⢣⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⡸⠁⡇⠀⠸⠀⡇⠀⠀⣾⠀⠀⠀⠀⢀⠀⠀⢸⠀⠸⡀
⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⠇⠀⡇⠀⡄⠀⡇⠀⢰⠁⠀⠀⠀⠀⠸⠀⠀⠈⣇⠀⣿
⠀⠀⢀⣀⣹⣿⠿⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣿⠀⠀⡇⠀⠇⠀⡇⠀⡜⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⠀⢸⡆
⠀⣴⣶⣿⡇⢉⣀⣀⠘⣆⢰⠛⢦⡀⠀⠀⠀⠀⣠⠖⠉⠁⠀⣠⢺⡇⠀⠀⡇⢀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠈⣧
⣷⢿⡟⠁⠀⠻⣄⡈⠓⢾⡿⠀⠠⡈⠳⡶⠖⠋⠁⠀⢀⡴⠊⠁⣼⠁⠀⠀⡇⢸⠀⠀⡇⣸⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⠀⠀⢸
⠙⢦⡀⠀⠀⠀⢠⡙⠆⣼⠃⠀⢀⡇⠀⢇⡀⣠⡴⠖⠉⠀⠀⠀⡿⠀⠀⠀⡇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠘⡄
⠀⠀⠹⣦⠀⠖⣩⠁⠀⢥⣤⡴⣏⠑⢦⣜⠉⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⡏⠀⠀⠀⣇
⠀⠀⠀⠈⢣⡄⠁⠀⠀⠀⠀⡤⠈⠙⢦⣈⠑⠦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠻⣄⠠⢄⠀⠈⠀⠀⠀⠀⢻⡟⠃⠸⠷⠀⠀⢠⡏⠀⠀⠀⠀⡟⢧⣄⠀⢸⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡜⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠈⠙⡟⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⡇⠠⢤⣀⣀⡞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⢨⡉⠛⠳⠦⢤⣼⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⠃⠀⠀⠀⠈⠓⠶⢀⣀⠀⠀⠉⠛⠛⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⠴⠖⠚⢙⡇⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠤⠤⠤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠏⠀⠉⠙⠛⠓⠶⠶⠦⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠶⠶⠶⠖⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⠶⠖⠀⠉⠉⠁⠀⠀⠀⣠⣼⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠋⠀⢘⣧⡿⠷⠶⠶⠶⢀⣤⡤⠤⠤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠒⠒⠉⣉⣉⠁⢀⣠⣤⡤⠶⠚⠉⠹⣤⣤⣤⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁
⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠙⠦⠀⠀⠘⠃⠐⠀⠒⠀⠀⠀⠀⠀⠀⠄⢈
⠀⠀⠀⠀⠀⠳⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢟⡿⠷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢔⣆⠀⢀⠐⢦⠄⣀
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢈⠠⠈⢁⠀⢧⠀⠙⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠠⠤⣤⠒⣉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠐⠒⠀⠑⢄⠀⠀⠀⢂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢢⡀⢀⡀⠀⠀⠀⢃⣼⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢣⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠅⠤⠤⠀⡀⣑⠪⢠⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠐⡄⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠈⠑⢄⡑⢄⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⣎⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⠵⢄⠱⡤⠂⠀⠁⠉⠟⠂⠀⠤⢀⠀⠀⠀⢀⠀⡈⣢⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠄⠈⢀⠀⠁⠈⠀⠀⠁⠀⠜⠁⠀⠈⠁⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡜⢻⡾⣦⣾⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣶⣶⣶⣶⣶⣶⣿⣷⣤⠀⠀⠀⠀⠀⠀⢎⠠⣄⠒⢶⠶⣶⣦⣄⡀⢀⠀⣠⣴⠆⠀⢳⣾⢧⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⢀⠀⠀⠀⠈⠀⠀⠸⠀⠑⠉⠢⣹⠈⠁⢸⣿⡁⣀⡤⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⣿⣿⣿⣿⣿⠿⢛⡛⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠿⠏⠉⠻⠋⠉⠁⠀⠀⠈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠒⠉⠀⠀⠀⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠎⡥⠀⠀⠠⠀⣀⠛⣀⠀⠒⠂⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣑⣀⡐⣠⣬⣄⣀⣀⣀⣀⣈⣵⣶⣦⣤⣤⣴⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠀⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⡀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⠶⠶⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡄
⠀⠀⠀⢿⡀⠀⠀⠀⠈⠉⠓⣿⣽⠖⠒⠒⠖⠒⠛⢛⣉⣉⣙⠶⢶⡀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⠇
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢻⢸⠛⠛⠋⠉⠉⠉⠉⠀⠀⠉⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠳⠒⠒⣒⣋⣩⡅⢬⣭⣉⣐⣒⠒⠒⠂⣲⡾⠁⢀⡞
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠀⢰⠀⠀⠀⠀⠀⠉⠈⠁⡙⢩⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣠⠞⣰⡛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠈⠉⢹⠃⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠀⡌⠀⣀⣀⣀⣀⣀⣀⣀⡇⢸⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠳⢤⡉⠉⠛⠓⠶⢤⣤⣀⣀⣠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠸⣍⠁⠀⠀⢀⠀⠀⠀⠀⢀⣀⡤⠤⢥⠘⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠓⢦⣄⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠒⠢⠈⡄⠐⠀⠉⠁⠀⣠⠔⢪⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣝⠦⣄⠀⠈⠛⠒⠒⣲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢤⡀⠰⠀⠀⢀⠠⠊⠁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⣝⣶⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡁⠀⠀⠀⠀⣠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠊⠉


⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣄⡀⠤⠤⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄
⠀⠀⠀⠀⢰⠋⢹⡄⠀⠀⣀⠤⠚⠁⢠⡇⠉⠓⢤⡀⠀⠀⠀⠀⣼⠁⡇⠀⠀⠀⠀⠀⢸⣀⣠⡶⠲⠶⠦⡔⢪⠷⢶⣶⣦⣤⣤⣤⢤⣄⣀⡴⠞⠋⢠⡟
⠀⠀⠀⠀⢸⠀⢠⡧⠖⠋⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠑⠢⠤⢤⠇⢠⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⠶⠿⢿⣷⡟⠉⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠇⠨⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣸⠀⠀⠀⠀⠀⠀⣇⣀⡀⠠⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢹⡏⡇⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠸⡄⢷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣇⡴⠁⠀⠀⠀⠀⠀⠀⢯⡁⠀⠀⠀⠀⠀⠀⠠⠤⠭⠍⠙⠳⠶⢾⣧⠇⢠⠞
⠀⠀⠀⠀⠀⠀⠙⠪⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⠄⠀⠀⠀⠀⠄⠀⠐⢒⣨⣵⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⢸⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⠶⠾⠟⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠓⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠸⠀⠐⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠠⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠋⠀⠀⠢⣦⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡌⠀⠀⠉⠢⡈⠲⣌⣦⡖⠠⢥⡒⠷⠠⠀⠀⠈⣶⠏⠀⠀⠀⠀⢁⠀⠁⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠡⠀⠀⠀⠀⣻⣋⠐⠓⢠⣐⠁⠈⠆⠀⠂⠀⡀⠻⠀⠀⠀⠀⠀⠘⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⢸⡁⠑⢌⡀⣈⠬⠘⠂⢬⠀⢀⡼⠀⠐⣇⠀⠀⠀⢸⡔⡀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠀⠌⢀⠄⡞⠀⠒⠀⠠⠤⠂⠀⠀⠊⠉⢀⠛⠀⠀⣤⠆⢀⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠈⡔⢀⡀⠛⠀⠀⠀⠀⠀⠀⢀⠒⠂⠐⠁⠀⠡⡐⠈⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢷⠊⠉⠁⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠈⠐⠒⠒⣲⠚⣯⡀⠁⠱⠒⠁⠁⠺⠀⠂⠀⠀⠠⠠⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡔⠓⠒⠭⠧⡐⠀⠀⢃⣘⡀⢬⠈⠀⠀⠀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⣀⢪⠁⠀⠔⠢⠤⠅⠀⠈⠙⡛⠁⢘⣀⠤⡐⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠒⠚⠦⠤⠄⠓⠶⠤⣲⡄⢠⠄⠀⠄⠠⡤⣀⡉⠀⣀⣡⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠄⣰⡃⠠⠀⠉⠐⠺⡁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄⡀⠀⠀⠈⠸⡁⠈⠈⠀⠤⠆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠈⢄⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠸⠛⠏⠙⠇⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠘⠁⠈⠹⠁⠀⠀⠀⠀⠰⣿⣶⣶⡆⠀⠀⠀⠀⠙⠛⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⠉⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⠈⡇⠀⠀⠀⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡦⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠐⠒⠀⢸⣿⣿⠀⠀⡇⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠃⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣘⡿⠃
⠀⠀⠀⠀⠀⠀⡸⠃⠐⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣤⣤⣤⠀⠀⢸⣿⣿⣿⣿⡁⠀⠀⠀⢀⠀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⠆⠀⢠⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⠁⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡃⠀⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣿⣿
⠀⠀⠀⠀⣀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⢀⠀⠘⢿⣿⣿⠀⠀⠀⠀⢀⣠⣴⣤⡦⠀⠀⠀⠿⠿⠰⣿⣿
⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠃⠀⠘⠢⠤⠤⠔⠀⠀⠈⠿⠥⠤⠤⠅⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠉⠉⠛⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠿⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⢆⡜⠱⠔⠶⠀⠀⠐⣿⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⠉⠀⠒⠀⣠⠞⠁⡴⡒⠢⠀⠀⠀⠇⣠⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠐⣫⣀⠀⡦
⠀⠀⠀⠀⡠⠛⠲⠎⠁⠀⠀⠣⢀⠔⠀⠀⠀⠀⠁⠀⣠⠀⠀⠀⡴⠊⠑⡤⠀⡀⢀⡀⠀⠀⠀⠀⡠⢀⣟⣊⡁
⠀⢧⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠲⠛⠀⠀⢸⠇⠈⢰⠇⠈⠀⡼⢸⢃⡎⠀⣴⣷⣯⠛⠁
⠀⠀⠁⠀⠀⢤⡄⢀⠀⢼⠀⡖⠋⠸⡆⠳⠀⠀⠀⠾⡠⠀⠀⠘⢦⠀⣾⢰⡄⢀⠧⣿⢼⡀⠀⠙⠛⠁⠀⠀⠀⠋⠐⠆
⠀⠘⢶⡄⠋⠁⡇⠈⢣⢸⡄⠒⠃⢢⣇⡐⠒⠖⠃⠀⠀⠀⠀⡀⣀⠀⠀⠉⠉⠙⠚⠁⠦⠀⢠⡖⢙⠒⠦⠀⠀⠈⠉
⠀⠀⠀⡇⠀⠀⢹⠐⣈⠔⠧⠖⠚⠁⠈⠀⣠⣤⣄⠀⠀⠀⠘⠆⠤⠃⠀⠀⣠⢦⣄⣠⠦⡄⢾⡀⠄⠁⡸⠀⠀⠀⢀⡄⡀
⠀⠀⣀⣃⠤⠞⠘⠋⠁⠀⠀⣀⣀⡀⠀⠀⠛⠟⠛⣠⢶⣶⣦⡀⠀⠀⢀⣤⠯⢄⣠⣄⠀⠑⢤⡑⠒⠚⠀⠀⠠⢊⡅⢠⠟
⠀⠀⠉⠀⠀⠀⠀⠀⢀⠔⠈⠁⠈⢻⡿⠦⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⢇⠀⢸⡀⢀⡇⠀⡼⠇⠀⠀⡰⢾⠶⣫⠖⠁
⠀⠀⣰⡆⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⠛⠛⠛⠁⣀⡀⠀⠈⣗⠀⢉⣉⠀⢠⠁⠀⠀⠀⠛⠾⠋⠁
⠀⠀⠈⠿⣷⡄⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢰⡟⠉⠛⣦⠀⠈⠓⠁⢈⡙⡁⠀⠀⠀⠆⠀⠀⠰
⠀⠀⠀⠀⠉⠀⠀⠀⠈⢣⡀⠀⡆⠀⠀⡜⠁⠀⠀⠀⠀⠀⠈⠷⣤⣄⣼⠀⠀⠀⠀⠀⠀⠰⢐⡖⠂⠰⠐⡔⣦⣼⡃
⠀⢠⢤⡔⣖⡤⢠⣤⠄⠀⢧⠀⠳⢀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⣀⣀⡀⠀⠈⠀⠉⠈⢀⠈⠀⠁
⠀⠘⣎⢱⡇⡀⠨⡥⠀⠀⣾⡘⢸⣾⠁⠀⢀⣾⡿⠿⢿⣶⡄⠀⠀⣰⠿⠛⠛⠻⠿⠆⠀⠀⠀⢸⣦⣴⣴⣶⣷⣷⣶⣿⡦
⠀⢀⢘⠈⣇⠃⢀⠈⠃⠀⢹⣯⣿⣿⠃⠀⢸⣿⠀⠀⠀⢹⣿⡀⠀⢿⣤⣀⣀⡀⣸⡟⠀⠀⣀⣈⡉⢉⣡⣀⣀
⠀⠈⠁⠀⠉⠁⠈⠉⠀⠀⠀⠉⠉⠁⠀⠀⠈⠻⣷⣄⡀⠀⣿⡇⠀⠀⠙⠛⠛⣿⡿⢀⣴⠀⢇⢈⡩⠿⠁⣁⠘⠢⡀
⠀⢀⡰⢂⣻⠃⠀⠀⢀⣀⠀⠀⠀⡖⠀⠀⠀⢀⠈⠛⠿⢷⢿⠿⠗⠂⣀⡀⠀⠙⠀⣨⠐⠠⣨⠀⠀⣤⠲⠀⠀⠐⢙⡆
⠀⠹⠟⠉⢡⠄⣤⠤⠿⢨⠀⠀⠸⡧⠈⠒⡆⠀⠀⠀⠀⠀⠀⠀⢠⡴⠃⠐⠦⠀⢰⣇⡀⢀⢸⠁⣸⢡⡈⣨⠃⣨⡜⢰⡄
⠀⠀⠠⠠⠌⠀⠀⠀⠐⠀⠀⠀⠀⢳⡀⢰⡃⠀⠀⠀⡀⣠⢤⠀⠀⡧⠖⠀⠲⠀⠀⣄⠄⠈⢺⣤⢀⢫⠇⡀⠘⠀⠀⣜⠁
⠀⠀⢸⡀⠀⡆⣆⣆⢠⡀⠀⠀⠀⣎⠀⠠⢥⠀⠀⢈⠎⠁⠐⠖⠀⠁⠰⠀⠀⠀⠀⢹⡀⠐⠉⡀⠒⣻⡞⠁⠟⠀⡄⢸
⠀⠀⠨⡿⠠⡿⠽⣿⢼⣧⠀⠀⠀⠈⠉⠁⡾⠀⠀⠈⠐⢋⡠⠤⠀⠀⠀⠀⠀⠀⠀⠈⠛⢏⡙⣍⠪⠇⠱⣇⣠⣊⠙⠃

⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢻⣿⣿⣶⣄⠀⠀⠀⠀⣸⣛⢛⣓⣀⠀⠀⠀⠀⣴⣶⠶⣆⠀⠀⠀⢀⣶⣾⣶⣦⠀⠀⠀⠀⠠⢒⣆
⠀⠀⠀⠀⠀⢻⠿⠿⠿⣿⠀⠀⠀⠛⠋⠀⣀⠈⠛⠃⠀⠀⠀⡟⠀⠀⢻⠀⠀⠀⣾⡧⠀⠠⢿⡇⠀⠀⠀⡖⠋⠋⢲
⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄⠀⠀⠀⠀⢀⣠⣄⣀⣦⡄⠀⠀⠹⣷⡄⠀⣾⠃⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠐⠄⠀⠀⠀⠄⠀⠀⠂⠀⠀⢀⠀⠀⠈⠗⠻⡛⡗⠃⠀⠀⠀⣀⠁⠀⢁
⠀⠀⠀⠀⠀⡀⠀⡀⠁⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⣷⠹⠄⠀⠀⠀⠈⠂⠔⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠁⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣶⣶⣶⡄⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⡿⡿⠟⢿⠀⠀⠀⠀⣾⠋⠈⢿⠀⠀⠀⠀⣾⠛⠛⢻⠀⠀⠀⠀⠸⡇⠀⠉⡓⠀⠀⠀⠀⡿⠛⠋⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣄⠀⠀⠀⠰⣤⣤⡼⠁⠀⠀⠀⠀⠁⡀⢠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢿⡟⠀⠀⠀⢀⠉⠉⠁⡀⠀⠀⠀⠀⠴⢠⣀⠦⠀⠀⠀⠀⣄⠈⠀⠡⠆
⠀⠀⠀⠀⢠⣾⣿⣷⣷⣿⣷⡄⠀⠈⠀⠂⠀⠂⠈⠃⠀⠀⡙⠄⠀⠼⡃⠀⠀⠀⠀⠳⠀⠠⠇
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠠⠶⠀⠀⠰⠆⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⡄

⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⡟⠛⠛⢿⠀⠀⠀⢠⣿⡿⠋⣿⡆⠀⠀⠀⢐⣋⣉⣙⣓⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⣴⣿⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠌⠉⠉⠡⠀⠀⠀⢸⣇⠁⠀⢠⡇⠀⠀⠀⠘⠁⠀⠀⠋⠀⠀⠀⠐⡟⢛⠛⣿⠀⠀⠀⢸⣿⠃⠐⢉⣧
⠀⠀⠀⠀⠀⠀⢀⠂⠐⡀⠀⠀⠀⢸⣿⠇⠀⢿⡇⠀⠀⠀⠀⠴⢀⠴⠄⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠘⢿⡦⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠁⠐⠃⠁⠀⠀⠀⠀⠋⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢐⣤⡐⠀⠐⢁⣴⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⡮⢵⣿⣿⡇⠀⠀⠀⠘⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠛⠛⠃⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠐⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢈⣠⠴⠚⠛⠛⠛⠦⣁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⠪⠋⠀⠀⣀⣀⠀⠀⢀⣈⠣⠈⠢⠀⠀⠀⡀⠠⠄⡀
⠀⠀⠀⠀⠀⡠⡒⠤⢒⠄⢠⠊⠀⠀⠀⠂⠀⠬⠑⠒⠒⢚⡛⠒⠒⠱⠄⠀⠁⢄⠀⠂⠀⠐⠈⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⣍⠀⠀⠠⡒⠒⣒⡂⠤⠭⠤⠤⠤⠀⠤⠍⠢⠈⠃⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⡇⠘⠀⢠⠐⠒⠒⢒⣒⣀⣤⣤⣤⣤⣀⣐⠢⢄⠀⠀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠈⢆⢀⠀⠀⢡⠀⠀⠀⢀⠀⠀⠠⢤⣀⣒⣶⣶⣖⣂⠠⡉⠉⠀⣆⠀⢔⡨⠊
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠣⡀⠀⠈⠉⠉⡨⠤⠄⣐⣒⣀⠤⢄⠈⠀⢀⠜⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠥⣀⠙⣄⡀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠂⠤⠄⣀⣀⣀⡀⠠⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡿⠛⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠈⠀⣠⣾⣿⣿⡿⠟⠋⠉⠉⠉⠉⠉⠛⠿⣿⣿⣿⣦⡀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⠀⢠⣄⠀⠙⢿⣿⣿⣆⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣷⠀⠀⢻⣿⣿⣆⠀⢻⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢿⣿⣿⡀⠘⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⠇⠀⣿⣿
⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣧⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡀⠀⢀⣿⣿⡿⠀⢰⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢻⣿⣿⣦⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠟⠁⢠⣾⣿⣿⠃⠀⠾⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠹⣿⣿⣷⣤⡀⠉⠉⠉⠉⠉⠉⠀⠈⠀⣠⣴⣿⣿⣿⣿⣿⣦⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣤⣤⣤⣤⣴⣶⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠂⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠓⠢⠄⣀⠀⠀⢸
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⠘⠂
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⣤⣀⠀⠉⠉⣿⣿⣶⣴⣤⣤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⢹⣿⣿⣿⣿⣿⡏⠔⠈⠙⡄⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠉⠛⢿⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣮⣴⡇⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠱⠟⠀⠀⢸⠉⠉⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⢸⣿⣿⣿⣿⣿⡄⠘⡏⠀⠀⠘⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⠙⠻⢾⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠛⠿⣟⡛⠁⠀⢹⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠈⠙⢦⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠂⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠶⠖⠒⠒⠒⠛⠛⠛⠉⠉⠛⠛⠛⠓⠒⠒⠶⠤⢄⡀⢠⠞⠙⠲⣄
⠀⠀⠀⠀⠀⠀⢠⠏⠁⠉⢷⣠⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⡶⠀⢀⣀⡤⠶⠒⠛⠉⠉⠉⠉⣑⣲⠦⠈⠉⠀⣀⣀⡸⠇
⠀⠀⠀⠀⠀⠀⠸⠤⣀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡴⠋⠁⠀⣼⠤⠚⠉⠀⠀⠀⠀⠀⢀⣤⣖⡋⠭⠄⣀⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⡀⠀⠀⠀⠀⢠⠋⠰⢃⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢀⠏⡘⠁⡇⠀⠀⡀⠀⡆⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠄⠀⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡈⠰⠿⠿⢧⠀⠀⡁⠠⠁⠀⠀⠀⠀⠀⡠⠴⠒⠋⠉⠉⠉⠉⠁⠊⠁⠀⠀⠢⡀⠀⠁⠄⠀⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣼⠀⠀⣘⡄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡠⠔⠒⠒⢮⡓⠀⠰⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⢻⡆⡞⢁⢤⠀⠀⠀⠀⠠⡓⠀⠀⠀⠀⢠⠞⠁⠀⢀⠎⠀⠀⠀⠀⠀⠹⡄⠀⠱⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡀⣿⣶⣾⣿⣿⡇⠙⠁⡏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⠂⠀⠢⡎⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠇⣅⣠⣤⣤⣼⡇⠀⠘⠷⠒⠒⣚⢳⡀⢀⡇⠀⠀⠀⠀⠀⠀⣤⠚⡖⢤⣤⣀⡀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⡇⠀⢠⠖⠉⣉⣄⣈⡉⠉⠀⠀⠀⠀⠅⠀⠀⠈⣰⡇⠉⠁⠀⠈⢉⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⣦⣤⣶⣶⣶⡇⠀⣿⠀⠈⠙⠒⠚⠉⢀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠙⠿⠶⠶⠚⣋⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⡏⠉⠉⠉⢉⡇⠀⠘⠳⠦⠤⠶⠒⠛⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠉⠀⠀⣹⠀⢀⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡁⣿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢳⣡⣤⣤⣤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠘⢀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠦⠄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠸⣸⡏⢉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠲⢄⡀⣀⣀⣤⡄⠀⠀⠒⠈⠁⠈⠒⠁⠀⠀⠀⠀⠐⠿⣿⣷⡖⠢⡄
⠀⠀⠀⠀⠀⠀⢣⢻⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠳⣄⠔⠀⢻⡠⠊⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠋⢳⣶⣿⠁⠀⠀⠀⠀⢀⡴⢢⠞⡿⢲⡆⣀⠀⠀⠘⠛⠀⡄⢣⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⣀⡤⠂⠉⠉⠈⠉⠉⢭⠉⠉⠓⠘⠢⠓⠊⠀⠁⠉⠉⠉⠉⠙⠛⠛⢆⣸⠀⡏⢳⣄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⡀⠔⠋⠁⠀⠰⣖⠒⠀⡔⠁⢸⣤⠄⠤⠐⣦⡤⠤⠤⠤⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢆
⠀⠐⠉⠀⠀⠀⠰⠚⠉⠉⠣⠀⡀⠀⠰⠊⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠑⠒⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡉⡆⠀⠀⠀⠀⠀⠚⠃⢂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢴⡥⠤⢀⣀⠉⠀⠒⠒⠚⠋⡠⠒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⡁⠒⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠉⠁⠉⢐⠉⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⣤⣀⣈⠤⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠛⢻⣟⡛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⠈⠡⠃⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠧⠤⡤⣼⣈⣷⢦⠦⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⡴⢦⣬⠶⣾⠶⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠋⠀⠀⠉⠀⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡵⠀⠀⠀⠜⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣠⠄⠀⡠⠚⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⡛⠉⢹⠉⢳⡤⠊⠀⡸⠀⠈⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣄⡀⠀⠀⠀⡜⠀⡀⢸⠠⣿⡿⠦⢠⠇⠀⠀⣇⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⣾⠇⡰⠦⠾⣄⠔⠁⢸⣷⣾⣧⠐⡎⠀⠇⠀⢸⠀⢀⠀⢠⡄⠂⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⣿⡟⣻⡜⠀⠀⠈⠛⠋⠉⡿⠀⢸⡇⠀⡀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⠀⢿⡆⠀⠀⠀⠀⠸⣋⣼⣿⠁⠀⠀⠀⠀⢀⣼⢁⣀⣾⡀⠀⢷⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⠀⢸⣷⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⢀⡞⣿⢛⣋⡟⠀⠀⢸⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⡏⢠⡞⠀⣿⠀⢸⡟⡆⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⡠⠻⡭⡤⠤⢸⡇⢀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠈⠀⠀⣿⠀⠈⠁⢹⡀⠀⠀⠀⡜⠀⠀⠀⢀⡄⠀⠀⢠⠄⠀⣼⣷⠈⢇⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣠⣤⣄⡀⣿⠀⠀⠀⠀⢳⠀⠀⡜⠀⠀⠀⣠⡞⠀⠀⢠⡟⠀⠀⣿⣿⠀⠘⢧⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣅⠀⢤⣬⡁⣿⡀⠀⠀⣤⠘⣧⡼⠁⠀⢠⣾⠋⠀⣀⠀⡞⠀⠀⢸⣿⣯⣤⡶⠦⣅⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡉⠓⠀⠀⠀⣿⡇⢸⠀⠉⠀⣹⠁⣀⢶⡿⠁⠀⠀⠛⠀⢰⠀⠀⣿⣿⣇⣀⡰⠖⠋⢡⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠛⠛⢀⣿⡇⢸⠀⠀⠀⢣⠞⠡⠋⣀⣀⣄⠀⠀⠀⢸⠆⢸⣿⡿⢋⡉⠀⢠⠾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡀⠀⠘⣶⣿⠐⠚⠛⠉⠁⣤⠀⠀⢠⠟⠀⣿⠟⠀⠉⠁⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⡏⣇⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣶⡴⠋⠀⠀⢀⣀⣰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⠐⢾⣿⣄⣇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣿⠁⠀⠀⠔⢋⣵⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⡇⠀⠀⠀⢸⣿⠉⠉⠀⢰⡧⡤⠀⠀⠀⠀⢠⣤⠀⡀⠚⠿⠿⣦⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢷⣶⠟⢿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠞⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⢸⣁⢿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣤⢸⡷⢿⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⡿⣦⣿⣤⣤⣄⣼⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠀⠆⢸⢹⡉⠉⡹⢿⠉⠉⣭⡿⠋⠉⠉⠉⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢸⠸⡇⠀⣷⠘⡆⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣘⠂⢳⠀⣿⠀⢣⠀⠀⠇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣦⣾⣧⣤⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣭⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⣇⣀⠀⢀⣴⣿⣿⣿⣷⣄⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⣤⠀⠀⠀⠀⠀⢀⣰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣴⣿⣿⣿⣿⣿⣿⣿⣷⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣷⠀⠀⢙⡟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⢩⣿⣿⢿⡿⣿⢿⣿⣮⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠰⢻⡟⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⢘⣛⣛⣚⣛⣛⣛⣛⣛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣧⣤⡄⠀⠀⠀⠀⠀⢸⣿⣿⠸⣿⣿⣼⣧⣿⣽⣿⡿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠭⣭⣭⣭⡭⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⢰⣶⣶⢳⡞⣶⢳⣶⣶⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣠⡄⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣿⣿⡇⣿⢟⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣧⡰⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⢶⣶⠀⠀⠸⠁⠀⣘⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢘⢿⣿⣿⣿⣮⡛⠃⠀⠀⠀⠀⠀⠀⣾⢫⣭⣽⣿⡇⠿⠦⠤⠤⢤⠄⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣷⣭⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢸⣿⡿⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡴⢲⡴⠋⠉⠓⠋⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⡄⠀⠸⡇⠀⢀⣤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢻⡀⠀⢘⡞⠁⠀⠉⠉⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠓⠀⢸⡇⠀⠀⠀⠀⠈⢳⡀⣠⢦⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠘⣇⠀⣤⠀⠀⠀⢸⣷⣧⡆⣷⣄⠀⠀⢨⣧⡀
⠀⠀⠀⠤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠈⠀⠀⠀⡾⠀⠉⠀⠀⠀⠈⠛⠡⠴⠃⠈⢧⣠⠏⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣨⠴⠒⠛⠛⠛⠓⠒⠦⠤⣄⣀⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣨⠟⡶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⢤⣧⣀⠀⠀⢀⡴⣣⠞⠁⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠚⠷⠶⠤⣄⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⣏⠉⡿⠉⠉⠉⠀⠀⠀⠙⢦⣀⣀⣤⠰⡄⣶⢼⢧⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡇⣷⣿⡾⢮⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠿⠋⠀⠈⠛⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⠛⠛⠓⠒⠒⠲⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠑⠦⣄⡰⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⡯⠉⠉⠉⠙⠒⠚⣒⣀⣈⡁⠉⠹⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢄⢀⡤⠒⢤⣀⡠⠔⠊⠉⠁⠀⠨⣧⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⣄⡀⠀⠀⢀⡤⠖⠦⠀⠀⣽⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠤⡀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⡯⢀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣠⠤⠖⠋⠉⠁⢸⡴⠛⠁⠀⣀⠘⠟⠀⠀⠀⠀⠀⣇⠞⠥⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⡔⢉⡇⣀
⠀⠀⢠⡶⠋⠀⠀⠠⠧⣄⣀⣸⡅⠱⠜⠛⠛⣋⡳⠀⠀⠀⠀⠀⢉⡴⠒⠋⠀⢈⡉⠉⠙⠒⠒⠶⠦⠤⠶⡖⠒⢲⠀⢀⠞⢉⡼
⠀⠀⠈⢧⡀⠀⣀⡀⠖⠀⠀⢀⢡⠀⠀⠠⢥⣞⡀⣀⣀⣀⣠⠴⠋⠑⠒⠢⢤⣀⡀⠀⠐⠃⠀⠀⠀⠀⠋⡇⠀⢸⡄⠀⣴⠋
⠀⠀⠀⠀⠈⠑⠲⠤⢤⣍⣀⠘⣆⠢⣒⡖⠉⠀⠈⠉⠉⡵⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⣄⡀⠐⠖⢠⡧⠤⠼⠢⢄⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢩⠏⠙⠋⠁⠇⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢁⡏⠀⠀⠀⢰⠸⢤⠀⡰⠁
⠀⠀⠀⠀⠀⣜⠉⠉⠾⡄⡸⠀⠀⠀⠸⣉⠀⣠⢇⠃
⠀⠀⠀⠀⠀⠪⠤⡠⠄⠃⣇⣀⠀⠀⠀⠈⢚⣁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣭⣙⣍⣭⣭⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⡆⠈⠑⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢱⠀⣴⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢇⠈⠍⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢆⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⣦⣠⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⡎⣧⠱⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣡⠔⠚⠷⡝⡀⡇⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠁⠀⠀⠀⠈⠗⠁⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⣚⠳⠤⣄⣀⣠⠤⠴⢚⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⠖⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠄⠒⠈⠉⠀⠀⠀⠉⠀⠂⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠊⠁⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠈⠁⠀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⡠⢐⠅⠃⢀⡀⠄⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠄⠀⡀⠀⠘⠃⠅⠦⣄⡀
⠀⠀⠀⠀⣠⠎⠊⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠙⠶⣄
⠀⠀⣠⠞⠁⠀⠀⠀⠀⡅⠀⠀⠀⠀⢀⣀⠀⠠⠤⠐⠒⠀⠈⠁⠀⠀⠈⠉⠀⠐⠂⠠⠄⠀⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⣦
⠀⣰⠋⠀⠀⠀⠀⠀⢰⠓⠒⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠰⠄⢧⠀⠀⠀⠀⠀⠀⣸
⠰⡇⠀⠀⠀⠀⠀⢀⣸⡤⠤⠀⠀⠀⠀⠀⠠⠤⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠠⠤⠄⠀⠀⠠⠤⣼⣀⣀⣀⣀⣠⡴⠟
⠀⠛⠦⠀⡤⠶⠞⠋⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡄⣀⠉⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠈⠉⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡉⠐⠒⠠⠤⢀⣀⣀⣀⡀⠠⠤⠤⠀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁





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

⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣶⠞⠳⢶⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⢾⡇⠀⠀⠀⡿⠶⠀⠠⠶⠶⠶⠶⠶⠶⠶⠆⠀⣴⠛⢁⣀⣀⣀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠿⢦⡴⠾⢧⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠆⢸⠇⠀⠈⠁⠀⠉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠶⠶⠆⠸⡇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⠀⠹⣦⣀⡀⠀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠉⠉⠉⠉⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡞⠉⠀⠀⠈⢳⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠟⠻⠛⠟⠿⠻⠟⠷⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⡴⠟⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣠⣤⣤⣤⣤⣤⣀⣄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⡖⣦⣰⡆⣟⢦⡄⣲⠆⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠉⠁⠁⠘⠁⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣾⡿⣷⣷⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣩⡖⠋⠁⠀⠘⣧⢳⡀
⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠒⠙⠛⠃⠒⣂⠤⠤⣄⣹⣼⡇
⠀⠀⢰⣿⡏⢹⠿⣦⠀⠀⠀⠀⠀⠀⠀⠺⠤⣀⣀⠀⡠⢤⢄⠊⠙⠳⣤⡀⠀⠉⠉⢢⣀
⠀⠀⢸⣿⣇⠛⠀⠘⢷⡀⠀⠀⠀⠀⡠⠀⠈⠁⠀⠀⢰⣶⠀⠇⠀⠀⠘⠉⠑⡶⣶⠒⠚⠃
⠀⠀⠈⣿⣷⠀⠀⣶⢈⣷⡀⠀⠀⠺⢤⣀⣤⣤⣤⢤⣸⡁⠀⠀⠀⠀⠀⣀⡠⠋⣸⣿⣶⠶
⠀⠀⠀⠹⣿⣧⢠⡇⡀⠸⣇⠀⠀⠀⠀⠀⠀⡇⠀⠂⠙⠤⣤⠖⠀⠀⠀⠀⠀⣸⣿⣿⠁⣠⠞⠃⣠⣄
⠀⠀⠀⠀⠹⣿⣿⣡⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠐⠂⠂⠚⢣⣀⠀⡠⠔⠁⣼⣿⣿⡷⠀⠘⢀⡈⠘⠉⢪⣦⣄⡀
⠀⠀⠀⠀⠀⠈⠻⣿⣧⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠦⣄⣼⣿⠿⣿⢳⠀⠐⡿⣿⠢⣤⡈⠉⠈⣙⣢
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣷⣾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡦⣤⠖⠆⠀⠀⠀⠀⠀⡰⠛⣿⣿⡿⣿⣾⣤⠤⡷⠺⠀⠐⡖⣶⠖⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣲⣡⡆⠸⣿⣿⣿⡖⢚⣣⡎⠉⣸⡆⢺⠭⠀⠀⠤⠄⠀⠀⣀⣀⣀⡎⠀⠉⢙⠛⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠉⠀⢰⣿⣿⣿⠋⠈⠋⣠⠞⠉⡇⣴⣶⠀⠀⢴⢠⠀⠀⠠⡤⠄⠀⢨⡯⠏⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⠤⠠⠊⠀⠀⠀⡇⠀⠉⠀⠀⠈⠈⠀⠀⠀⠉⠁⠀⠈⠉⠁⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⣶⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠋⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠌⢤⣄⣀⡀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⡖⠒⠓⣷⣶⣾⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣠⣾⣿⣿⣧⡀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠛⠻⣿⣧⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠻⠿⠿⠿⠟⠛⠿⠿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⡞⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⢳⡀
⠀⠀⢸⠁⠀⠀⠈⢧⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⡞⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠘⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⠃
⠀⠀⠀⢹⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡏
⠀⠀⠀⠀⠹⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠈⢳⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⡿⠁
⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣐⣄⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠛⠛⠛⠛⣿⡏
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣂⣤⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠛⠛⢻⣿⣿⣿⣿⣿⣿⠇⠀⣀⠔⠁⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢛⠋⡀⢀⣬⣦⡀⠀⠀⣟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠀⠒⠂⠈⠀⠀⣷⡿⠋⠄⠀⠀⠀⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠁⠠⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠉⠁⠀⠀⠀⠀⠐⠈⠀⠀⢈⡄⠀⣀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣷⠀⢀⠀⣶⣦⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣜⣹⣿⢻⡇⠀⠀⠀⠀⣿⡿⠛⠻⣿⠀⠀⠀⠈⠁⠀⣦⣦⣾⡇
⠀⠀⠀⠂⠐⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣤⣼⡇
⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⣿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⣁⣀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⠄⠐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⣀
⠀⠘⠿⠛⠛⠛⠃⠀⠀⠀⣿⣿⣿⣿⣿⣟⠁⠀⠠⠇⢲⣾⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⣀⣙⣛⣛⣛⢸⣿⡇⠀⠀⠀⣿⣿⣿⡇⢀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⠸⠿⠃⠀⠀⠀⣿⣿⣿⣧⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⠇⣿⣿⣿⡇⢀⣿⣿⠋⠉⠙⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣿⣿⣿⠴⣿⣿⣿⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⠤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⢹⣿⣿⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣦⠀⣿⣿⣷⣶⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠠⣦⣄⡀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⠁⠹⣇⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡄⠀⢀⡄⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣯⠉⠉⢠⡄⢠⡄⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⢠⡄⢠⡄⢠⡄⠀⠀⣀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣄⠀⠀⢠⡄⢠⡄⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣼⡷⠀⠀⠀⠀⢠⡄⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣤⣤⣤⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠹⣿⣿⣟⣛⣣⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠋⠉⠉⠉⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⠿⢶⡶⠒⣶⣶⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠇⠂⠀⠀⠀⠐⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡿⠂⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⠇⠐⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⠀⠀⠀⠀⠂⠸⢿⣧
⠀⠀⠀⠀⠀⠀⢸⣷⣤⠀⠀⠀⠀⠀⢀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡄⠀⠀⠂⠀⢀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡦⠀⠀⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠳⠄⠀⠔⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⢿⡿⠟⠃⠀⠀⠙⠻⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠿⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠛⠛⠿⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠙⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⠀⠀⠀⠈⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⠀⠀⠈⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠈⠀⠀⠀⠁⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠀⠀⠀⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠉⠉⠉⠉⠉⠘⠿⠆⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠐⠊⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠐⠋⣉⣤⣈⣉⠙⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣰⠇⣴⡾⠟⠻⠻⣿⣿⣄⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠋⠉⠉⠹⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠛⠁⠹⣿⣿⡏⠁
⠀⠀⠀⠀⠀⢠⣿⣶⡟⠁⠀⠀⠀⠈⠙⢿⣼⣧⠀⠀⠀⠀⠀⠀⠀⢸⣇⣬⣤⡄⢠⣬⣤⣽⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠈⢿⡿⠃⠀⠀⠀⠀⠈⠀⠾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠃⠈⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⠀⠀⠀⠁⠘⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠻⠦⡀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⠀⠀⠀⠀⢀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠿⠿⣿⣷⣤⢃⣔⣥⡶⠶⢄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢈⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⢦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣴⣿⣿⣿⡟⣩⡦⠀⠈⣿⣿⣿⡏⢛⣿⣿⣦⡙⠂
⠐⡀⠂⠠⠀⠂⠉⠀⠀⢸⢫⣾⣿⢃⠹⠠⢁⣀⠀⢸⡟⢸⠇⣀⠛⡜⢿⣿⡍⠒⠢⢄⡀
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠈⠀⠴⣟⠫⠥⡀⠀⠀⠘⠿⣷⠀⠈⣿⣿⡄⠀⠀⠈⠑⠢⢄
⠀⠀⠙⡦⠀⠀⠀⠀⢀⠀⣿⣿⡇⠀⠀⢰⠁⠠⠀⡀⠀⢀⡀⠘⠈⠀⢠⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⣸⠃⠀⠀⠀⢊⢊⠴⡿⢿⠃⠀⠀⠸⡀⠀⠈⠁⣀⠈⢀⡜⠀⠀⣼⣿⣏⠈⡀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⢠⡏⠀⠀⠀⠀⢠⠏⠀⢢⣄⠀⠀⠀⠀⠒⠛⠁⢤⡠⠟⠛⠈⠀⠀⢻⣭⢌⢧⢙⠀⠀⠀⠀⣰⠀⠀⣷
⠀⣸⠁⠀⠀⠀⠀⠘⣆⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡼⠋⢊⡞⠀⠀⠀⠀⢠⡏⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠈⠲⠶⠾⢧⣀⣀⠀⡴⠖⠒⠒⠚⢁⣀⡤⠖⠿⠤⠶⠋⠀⠀⠀⠀⢀⣿⠁⠀⢀⡇
⠀⣷⠀⠀⠀⠀⠀⠀⢀⣠⣐⠤⠑⣂⡥⠾⢶⣶⠂⢠⠞⠛⢿⣿⠀⠐⠢⢄⡀⠀⠀⠀⠀⢌⠆⠀⠀⢸⠃
⠀⣿⠀⠀⠀⠀⠀⠸⠶⡋⠄⠂⠁⡳⠂⣸⣦⡁⢠⣮⣶⠒⠒⢃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢀⡟
⠀⢻⡀⠀⠀⠀⠀⠀⣁⠀⠀⡠⢊⠝⣼⣿⠃⣿⡟⠸⣿⡆⠠⡀⠳⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⡾⠁
⠀⠘⣇⠀⢀⠂⢁⡎⠀⠈⠈⣀⡏⣼⣿⡏⢠⣿⣧⠀⢻⣿⡀⢱⡀⠘⣦⠀⠀⠀⠀⢠⠁⠀⢀⡾⠁
⠀⣀⠿⠶⠺⡀⠈⣇⠀⣆⠀⡏⠉⠉⠉⠓⠺⠿⢟⠒⠉⠉⠉⠉⠉⠑⠚⠢⢤⣀⠀⠘⠀⢠⡟⠁
⠈⠀⠀⠀⠀⠉⠒⠉⠐⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣴⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄
⠀⠀⢀⣴⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠉
⠀⠀⠀⠉⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⢿⣿⣶⣦⠀⠀⠀⠀⢀⡀⢰⣿⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⠉⠈⠙⠀⠀⢀⣀⣀⡽⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠂⠀⠀⠀⡼⠛⠋⢁⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠊⠷⢸⣿⣿⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠉⣿⣿⠿⠿⢿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠋⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⢀⣀⣀⣤⣤⡤⠿⣿⠀⠀⠀⠀⠀⠀⣰⣿⠋⣽⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠚⠛⢛⣿⣿⣻⣿⣿⣿⣿⣿⣶⠂
⠀⠀⠀⠈⠀⠀⢀⡈⠉⠀⠀⣀⡀⠀⠀⠀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⢛⡿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠙⠋⠇⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄
⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠚⠉⠁⠀⠀⠀⢀⣆⠀⠀⠀⠀⠉⠑⠒⠦⣤⣄⣀⣀⣽⡿⣿⡆
⠀⠀⠀⣿⡟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠚⠋⠉⠀⠀⠀⢀⡀⠄⠀⠀⠁⢀⠉⠐⠂⠄⢀⡀⠀⠀⠀⠀⠀⢹⠟⢷⡸⣿
⠀⠀⢰⣿⠃⠈⢿⣷⡀⣀⣠⢤⠖⠚⠉⠁⠈⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠭⠤⢤⡇⣿⡆
⠀⠀⢸⣿⠀⠀⠈⢿⣿⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡇
⠀⠀⣿⡿⠀⠀⠀⠈⣿⣧⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠔⠐⠋⠹⡍⠋⠑⠒⠲⠶⠤⢤⣤⣀⣀⠀⠀⠀⡸⢸⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⡇⣼⠃
⠀⠀⢹⣧⠀⠀⠀⠀⠀⢻⣷⠀⠤⠤⢔⣒⡂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠏
⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⢿⡆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⠤⠤⢤⣄⣀⣀⣀⠠⠤⠐⡾⠁
⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⢸⣧⠠⠤⠤⠤⠤⠤⠔⠒⠒⠒⠒⠒⠂⠉⠉⠉⠉⡩⠔⢋⠁⠀⠙⠓⢦⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣏⡙⣉⡒⠂⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠐⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢤⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠳⠤⣬⣦⡑⢄⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠓⠂⠐⠒⠒⠈⠁

⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⣀⣀⠀⡀⡀⠀⡀⠀⠀⠀⣀⣀⡠⠤⠤⠴⠖⠒⠒⠒⠒⠛⠛⠋⠉⠙⠛⠛⠒⠒⠒⢶⠤⣤⣀⣀⣤⠞⠉⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢳⣀⡠⠴⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣀⡤⠔⡶⠀⠀⢀⣠⠤⠶⠒⠚⠛⠉⠉⠛⢒⣢⣤⠈⠉⠀⡀⣀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡴⠚⠁⠀⣼⡡⠖⠛⠉⠀⠀⠀⠀⠀⢀⣤⣖⣛⣉⣁⣀⠀⠀⠀⢫⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⡀⠀⠀⠀⠀⠀⡴⠋⡽⢀⡴⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⣴⠚⠋⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠋⣇⠀⠀⠀⠀⢸⠃⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⡠⠽⠆⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⣿⣶⣿⠀⠀⡗⢄⣿⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠒⠒⠢⠤⠞⠉⠉⢤⣀⠁⠙⣇⡠⠜⠃⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠇⡏⣈⣉⣹⡆⠀⠹⣄⠁⠀⠀⠀⢀⠄⠊⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣤⡤⣍⡳⢄⠈⢂⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣸⣿⠿⠿⢿⡇⢰⠖⢉⡀⠀⠀⠀⠉⣐⡄⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠁⠀⠀⠀⠙⣆⠀⠈⢆⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡏⣯⣤⣤⣤⣼⡇⢸⡴⢹⠃⠀⠀⠀⠀⠙⣦⠀⠀⠀⠠⠚⣡⢤⣄⣰⠃⠀⠀⠀⠀⠀⠀⠸⡆⠀⠘⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠛⠛⠛⠛⡇⠀⠀⣿⣀⣀⣀⣤⡀⠀⢸⡇⠀⠀⠀⡞⠁⠀⠈⢻⣄⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢼⣿⣿⣿⣿⣿⡇⠀⠀⢈⣩⠤⠤⠤⠷⠤⣼⠁⠀⠀⠰⡧⠖⠚⠿⠁⡏⠹⡿⠒⠶⠤⣄⣸⠃⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣸⡁⠀⠀⣀⢀⣧⠀⢠⠏⠀⡠⣖⣒⣦⠄⠀⠀⠀⠀⠀⠳⣄⠀⣀⣴⢷⣄⣀⣀⣀⣠⡼⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⡿⠿⣿⠀⢸⣄⠀⠀⠈⠉⣀⣀⢤⡀⠀⠀⠀⠀⠘⠛⠛⠉⠀⢉⠙⠉⡉⠥⠤⠦⣄⠀⠀⡟⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢸⣼⣇⣀⣀⣄⣀⣿⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢁⣝⠐⣠⡀⠀⣀⣠⠟⠀⢰⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡏⡿⠟⠿⠟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⡇⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢷⢿⣶⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠐⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠾⣿⣿⡶⣶⣤⣤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡼⣏⣉⣠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠈⠓⢦⣀⣀⠤⢤⣴⠒⠚⠋⠉⠁⠀⠉⠉⠁⠀⠈⠈⠉⠛⠻⢿⣿⣟⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢻⡿⠿⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⠳⣤⠖⠁⢹⣤⠊⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⢳⣾⣿⡇⠀⠀⣀⣀⣀⣠⠖⣶⢻⠗⣲⣇⣀⣀⣀⠈⠛⠃⢰⢸⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⠉⠉⠉⠉⢉⠽⡍⠉⠙⠒⠓⠼⠲⠋⠀⠀⠀⠉⠉⠉⠉⠛⠛⢶⣀⣇⣸⣉⣦⣀⣀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠰⣦⠒⡒⡤⠊⠀⣧⡤⢤⠤⢲⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣞
⠀⠀⠀⠐⠉⠀⠀⣀⠤⠖⠛⠉⠣⡙⡄⠀⠈⠖⠁⠂⠀⢀⡥⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠒⠦⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠙⢃⠈⠀⠈⠀⠀⡠⠚⠁⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣀⡀⠀⠀⢀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣊⣀⠀⠉⠘⠒⠀⠤⠦⠚⠒⡠⠖⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⣈⠒⠠⢽⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠉⠉⠉⠓⠒⠒⠢⡔⠋⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⣄⣀⡀⠀⠀⠃⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠐⠂⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⢀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⣴⠋⠀⠀⠀⠉⠋⠀⠀⠀⠀⠙⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⣆
⠀⠀⠀⠀⠀⢀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠲⠤⣤⠴⠞⠙⠶⠖⠦⢤⡤⠞⠁
⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣤⣤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⡼⠓⠀⠀⠀⠀⡾⣁⡉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣤⡀⠘⡇
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠸⡅⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⢳⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣳⣬⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠁⠀⠲⠺⠯⣭⣛⠋⠉⠁⠀⣠⠏
⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠱⣜⠛⠓⠶⣄⡀⠈⠉⠛⠒⠚⡇⠀⠀⠀⢀⣀⣀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠶⠞⠁⠸⡿⢿⡦⣄⠀⠙⠲⣄⣀⣀⡼⠁⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠹⣄⣀⣀⡀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣳⠀⠀⣇⠸⣷⠈⠙⢦⡀⠀⠀⣿
⠀⠀⠀⠀⠉⠉⠙⠲⠴⠶⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢹⡄⠀⠀⠀⠀⢹⣦⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⣧⠀⠀⠀⢀⡾⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⡀⠀⠀⠀⠀⠀⢰⠛⠉⠀⠀⠀⠀⠀⠀⡴⠛⠒⠦⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠛⠲⠶⠶⠶⠿⢦⡀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⡼⠁⠀⠀⠀⠀⣸⠇⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢳⡀⠀⣸⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⡇⣰⠃⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⣿⠃⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡴⠃⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⠞⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠾⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠟⠑⠐⠉⡁⣁⣁⢩⡑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡠⠀⠔⢐⠁⢨⠀⠰⠰⠄⠠⠐⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠨⠀⢰⡅⢈⠀⢀⠀⠠⠆⠲⠀⠂⠲⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⠤⢰⣟⡀⢉⠀⠀⠄⢀⣒⡐⠂⠀⠀⠀⠉⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠒⢓⣋⣥⣶⣶⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⣁⣤⣶⣾⣿⣿⣿⣿⣿⠃⢟⣧⢻⣿⠈⠹⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⣿⣏⣸⣿⣿⠟⢇⠘⠷⠺⠿⠇⡴⣿⣁⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⢅⠿⢿⡿⠋⠁⠀⡂⠰⣴⠇⠀⠀⠀⢉⣻⠏⠀⠀⠀⠀⠀⠀⠀⢀⢠⡼⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⡀⠁⠀⡀⢰⠇⢸⡇⠀⠀⠀⠀⢻⣧⡄⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⢂⠘
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠶⠛⠃⠀⡀⠀⠀⠀⠀⠘⡄⠀⠈⢾⠀⣼⣷⣄⣀⣀⣀⣀⣿⣷⠀⠀⠀⢀⡀⠀⠀⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠁⠢⣀⠈⢢⡀⠀⠀⠀⠸⣆⠀⠈⢷⣿⣟⠁⠉⠉⠉⠉⠁⠀⢀⡟⠀⣸⠷⠂⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠑⡀⠀⠀⠀⢹⣦⠘⠘⣿⣿⣧⠀⠀⠀⠀⠀⠐⢌⣀⣴⠯⠀⠀⣀⠀⠀⢈⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣷⡀⠀⠸⡄⢻⣧⡀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣻⣿⣿⡷⠀⠀⠓⠈⣿⣷⠄⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⢀⣿⡇⢀⣀⠀⠠⢸⣇⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡿⢹⣿⠸⠟⠋⢳⣈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣼⡇⢸⣿⡇⠠⣄⠈⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⠀⠀⠐⢁⡠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢸⣿⡷⣦⢻⣧⣿⣿⣿⣿⣿⣿⣿⠀⡀⠐⠁⠀⠀⠈⠉⠁⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣿⣿⡇⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠐⠚⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣚⣒⠛⣿⣿⣿⣇⣿⣿⣿⣿⡿⠟⠻⡁⠀⠀⠀⢠⠄⠀⠀⢚⠰⠛⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢟⡽⣱⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⡘⢦⡀⡀⠀⠀⠀⠀⠈⢹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢻⣿⣿⠟⠉⠀⠀⠖⢋⠉⠁⣀⢐⡈⢳⣿⡦⠀⠀⠼⢳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⡀⠉⠹⠀⠙⣶⣿⣤⣿⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢀⡀⣄⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⢿⡻⡿⠟⢻⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣄⡖⣀⣰⣷⡆
⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡷⠲⣾⣧⣤⣤⣤⣤⣤⡤⠤⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⣿⣷⣷⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣷⣤⣿⡟⠛⠙⢿⣿⣿⣿⣿⣻⠇⢀⠸⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠈⡿⢿⣿⠈⣿⠀⠈⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⡁
⠀⠀⠀⠀⠈⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠈⢻⡷⠦⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⡀⠀⠀⠀⡈⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣤⣤⣤⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⢀⠈⠅⢼⠀⣰⠦⠈⠁⠀⢸⣿⣿⣿⣿⣿⣿⠉
⣠⣴⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⡏⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⢩⡇
⣿⣿⡟⡔⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣆⠀⠈⣷⣼⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⠀⠈⡁
⣿⣿⡀⣇⡜⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢄⣀⣀⣤⣤⣶⡶⣶⣶⣶⣶⣶⣶⣾⣿⣷⡆⢹⠀⠀⠇
⣿⣿⣿⣿⡇⡀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⠿⠏⠀⠀⠈⢹⡿⠿⠿⠿⠟⣿⡿⠳⠘⠳⠶⠂
⣿⣿⣿⣿⣧⢺⣦⣦⣇⠀⡀⢿⠀⠀⠀⠀⠀⠻⣧⣀⡿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠉⠑⣄⣸⣷⣦⣤⠤⠯⢥⡄⠀⠀⠀⠄
⠛⠛⠛⣿⣿⣾⣿⢧⣿⠀⣷⣘⢧⠄⠀⠀⠀⠀⢈⣿⣿⣿⣿⣷⣦⣀⣦⣀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣁⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢿⣿⣿⣿⣷⣿⡆⣷⣧⠘⣄⠀⠀⠀⠀⠘⠛⠉⢉⣿⣿⣿⣿⣿⣿⣶⣆⠀⣠⠂⣠⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣤⣤⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠙⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠿⠻⡿⠛⢻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠛⠛⠛⠿⠛⠛⠉⠉⠉⠉⠙⠉⠉⠁

⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣀⣤⣤⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠙⠛⣿⣿⠿⠁⠀⠀⠀⠸⡟⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⡿⠂⠀⠀⣴⣾⣿⣿⠆⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣴⣿⣿⣄⠀⠀⠀⠀⠀⡇⢨⣿⡿⠇⠀⠀⠀⠀⣴⣿⣿⣿⡀⠀⠀⠰⣿⣿⣿⣿⣧⠀⠀⠀⠀⣴⣿⣿⣯⠙⠁
⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠀⢠⣶⣿⣿⣤⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠉⣹⣿⡏⠁⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⢺⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣦⡀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣦⣾⠶⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣿⡅⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠸⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣟⠀⠀⠀⢹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠙⣿⣿⣿⣿⡄⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠈⠛⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⡟⣿⣿⡇⠀⠀⠀⠀⣿⣿⠋⢻⣿⣇⠀⠀⣸⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢀⣿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⢡⣿⣿⠀⠀⠀⠀⢰⣿⣏⣠⣾⡿⠋⠀⠀⣿⣿⣧⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⢁⣿⣿⠁
⠀⠀⠀⠀⠀⣼⣿⡇⢸⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⡏⢸⣿⣿
⠀⠀⠀⠀⠀⢻⣿⡇⢸⣿⣇⠀⠀⠀⠀⢈⣿⡟⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣷⣘⣿⣿⣤⣀⠀⢠⣿⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⡀⠀⠀⠀⢸⣿⣟⡀⠀⠀⠀⢸⣿⣿⡘⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠘⠛⠻⠿⠿⠋⠉⠉⠁⠀⠉⠻⠿⠟⠛⠀⠀⠀⠀⠀⠿⠿⣿⠿⠯⠁⠀⠀⠺⠿⠿⠿⠷⠂⠀⠈⠿⠿⠿⠿⠏⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⢖⡭⠖⠮⢝⣒⣶⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⣤⢔⣺⠭⠖⠮⣕⢦
⠀⠀⠀⠀⠀⢰⢃⠎⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠈⢆⢳
⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠘⡎⢇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠏⡞
⠀⠀⠀⠀⠀⠀⠘⢎⡣⣄⠀⠀⠀⣿⠀⠀⠀⠀⠒⠲⠤⢤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠄⠂⠀⠀⠀⠀⢸⠀⠀⠀⣀⡤⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢭⡒⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⢔⣪⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢿⡀⠀⠀⠈⠙⠒⠲⠤⢤⣤⣄⣀⣀⣠⣤⠤⠤⠒⠂⠀⠀⠀⠀⣸⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠚⠛⠉⠉⠁

⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⢀⣴⣶⣶⣦⡄
⠀⠀⢠⡿⠿⠿⢿⣿⡆⠀⠀⠿⠛⢛⠿⠀⠀⠀⣿⢁⠀⡈⣷⠀⠀⣼⡟⠋⠙⢿⣷⠀⠀⢾⡿⠿⠻⢿⣿⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢺⡇⠀⠈⠀⣿⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⢠⡏⠀⠀⣿⡀⠀⠀⢸⣿⠀⠀⠨⡅⠀⠀⢀⡅⠀⠀⠀⠈⠬⠀⠌⢁⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣴⠆⠀⠀⠀⣿⣧⠀⢼⣧⠀⠀⠛⠃⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⣧⠀⠀⠀⠿⠃

⠀⠀⢠⣴⣶⣄⡀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⣠⣤⣶⣶⣄⠀⠀⠀⡠⠴⠶⢄⠀⠀⠀⠀⣴⣾⣷⣦⡀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠹⠃⠞⠛⠛⠆⠀⠀⠀⠐⠀⠀⢠⠀⠀⢠⣿⠀⠀⡀⣿⠀⠀⠰⡅⠀⠐⠈⠃⠀⠀⢸⡟⡁⢈⡹⡷⠀⠀⠀⠋⠀⠉⠈⠁
⠀⠀⠀⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢀⣾⠿⠿⢿⣦⠀⠀⠀⣠⣿⣿⡄⠀⠀⠀⣀⣠⣼⣷⣄⠀⠀⢰⣿⠿⣿⣿⣤⠀⠀⢠⡾⠻⠿⠿⡆
⠀⠀⣸⡇⠂⠐⢂⣿⡆⠀⠀⠛⠅⠙⠛⠀⠀⠀⠋⠉⠉⠉⠛⠀⠀⣾⡕⠀⠐⢸⡟⠀⠀⠈⠃⠂⠀⠂⠁⠀⠀⠀⡈⠁⣀⠀⡄
⠀⠀⠻⠿⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢠⣿⠇⠀⠀⠀⠀⣐⣀⡂⠀⠀⠀⠀⠹⣦⣥⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⣠⡤⠤⢤⣄⠀⠀⣴⣿⠿⢿⣿⡆⠀⠀⣴⣿⣿⣷⣆⠀⠀⣴⡿⠿⠿⢿⣷⠀⠀⢸⡿⠛⠛⠿⣇⠀⠀⢀⡠⠒⠐⠠⣐⠂
⠀⠀⠰⡁⠁⠁⠁⢸⠀⠀⠸⡅⠀⠀⠇⠀⠀⠀⠋⠀⠀⠈⠋⠀⢸⣿⠀⠀⠀⢨⣿⡇⠀⢺⠃⠀⠀⠀⡿⠀⠀⠘⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠇⠀⠀⠛⠋

⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⢀⣠⣶⣦⣦⣀
⠀⠀⠀⡿⡉⠀⡈⢿⠀⠀⣾⠿⠛⠻⢿⡆⠀⠀⢾⠁⠀⠈⠇⠀⠀⢰⡟⠉⠙⢻⡆⠀⠀⢸⡿⠟⠛⢿⡗⠀⠀⢼⡟⠻⠟⠛⣿⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠿⠈⠀⠈⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠁⠈⣸⡷⠀⠀⠀⠁⠀⠀⠐⠁⠀⠀⠸⠇⠁⠀⠁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡄⢤⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠀⠀⠀⣻⡦⢱⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠠⢢⠄⠀⡀⠀⠀⠀⠀⠀⠈⢧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⢀⠀⠀⠐⣦⣤⣤⣵⠄⡒⠈⠋⠀⠀⠀⠀⠀⡅⢰⠀⢠⠘⠆⠸⠃⠀⠰⠊⣠⠸⡲⠀⡁⠔
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠒⢀⣠⣀⠑⠜⠄⠀⠀⠀⠀⠀⠀⠀⢀⣅⢠⠀⢏⠀⠀⠀⠰⢟⣁⢰⠇⠀⡃⠂
⠀⠀⠠⠂⡀⡀⣷⣿⣂⠀⠀⠆⠂⠈⠀⠀⠀⠀⠀⠀⢀⠙⠻⣭⣀⠆⠀⠀⠈⠀⠀⠔⠀⠀⣿⣿⡶⠡⠈⣀⠀⣵⣿⣾⣷⣆⣤⡁⠠
⠄⠀⠀⠀⠈⠀⡏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣦⢨⠏⢀⡤⡶⠊⠀⠀⠀⡀⠀⠈⢟⠁⠀⣤⡟⠛⢍⠉⠉⠈⠑⠉⠿⠃
⠀⢀⠀⠀⠀⠀⣗⠠⠊⢀⠀⡀⠀⡴⠂⠖⠒⢀⣐⡻⢤⡸⠛⠈⠀⠀⠀⠈⠀⠐⢛⡛⠛⠠⠎⠚⢐⣵⣿⣿⣾⠆⠙⠲⢤⣀
⠠⢁⠈⠀⠔⠀⠈⠦⠙⠁⠀⠀⠄⠀⠀⠀⢙⡎⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⣛⡓⠀⠶⠶⠶⠋⠀⠀⠀⠀⠀⠀⣠⣶⣨⡓⢤⡀
⠀⠐⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⠀⡠⠀⠻⣿⣁⠉⠐⡄⠀⠠⠽⠿⠟⠛⠛⠛⠂⠀⠀⠀⢀⠠⠐⡾⠿⠿⠛⠯⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣃⠀⠓⠥⣈⡀⠉⠍⠐⠀⠀⠀⠀⠀⢀⠀⠀⠦⠀⠒⠀⠀⠀⠀⡀⠀⠄⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠑⠛⠓⠀⠀⠀⢀⣀⠀⠀⠀⡀⠀⠀⠤⠀⠀⠀⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣒⡶⠶⠶⢖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠶⠛⠛⠛⠁⠤⠤⣤⣀⣀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⣀⣤⡤⠄⠀⠀⢒⣶⠤⣦⣄⣈⣉⠉⠓⠒⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠗⠊⠉⢾⢛⡇⠀⣠⣾⣿⣿⠟⠻⠀⠀⠉⠉⠀⠂⠠⠄⣀⡈⠉⢑⠖⠤⠀⠀⢀⡀⠀⣤⣤⣤⣶⣦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡭⠏⡛⢰⣣⡤⠭⠖⠚⠉⠉⠀⠀⠀⠀⠑⠌⢆
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣤⣤⣤⣤⠄⣬⣤⣥⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠿⠋⣸⣿⡿⠿⠿⠟⠻⠛⠻⠶⠶⣶⣤⣤⣤⣀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇
⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢻⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠏
⠀⠀⠀⠀⠀⠀⡟⠙⠒⠶⣶⣦⣟⠨⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⠜⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠏
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠙⢿⣿⢻⣿⣶⣶⣶⡿⠿⠿⠟⠛⠛⠋⠙⠉⠉⠀⠀⠀⢆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢷⣤⣀⠈⠸⡄⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣀⣠⡤⢬⠈⠀⠀⠀⠀⠀⢀⠠⣐⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠻⢿⣷⣶⣿⣿⣶⣤⣤⣴⣶⣶⡶⡞⠿⠛⠉⠉⠁⢀⠀⠀⠃⠀⣦⣒⡤⠵⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⣙⡻⣿⣿⣿⣋⢉⠁⠀⠀⠀⠀⠀⠤⠒⠈⣉⣁⠤⠴⠊⢢⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣛⡻⠿⡾⢤⡤⠤⠤⠤⠴⠖⠒⠚⠋⢉⢁⠠⠄⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⠀⠒⠒⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠐⠫⠘⠓⢣⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠤⠄⠀⠀⠠⢄⠀⠀⠀⠀⢘⡆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⢀⣀⡀⠤⠀⠀⠤⡀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⢠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠙⠁⠀⠀⠤⠀⠀⠀⡠⠀⠀⠀⠀⡠⠃⠠⡨⠐⠉⠄⠐⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⡆⠀⠈⠀⠀⠀⡠⣂⠙⢦⡐⠊⠐⠀⠁
⠀⠀⠀⠀⠀⠀⢰⡄⠜⠀⠉⠀⠀⢆⠆⠈⠉⠑⠄⠢⠊⠱⡔⠤⠠⠙⠒⠂⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⡂⠀⠀⠀⠀⠀⠠⠀⠀⠀⢈⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠒⠙⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢄⢇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠱⣁⣈⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⠦⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠴⠒⠒⠋⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡴⠀⠸
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠪⠂
⠀⠀⠀⠀⠀⡏⠑⠒⠲⠤⠤⠤⣤⣀⣀⣀⣀⣀⡀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⣀⣀⣀⣀⠤⠤⠤⠔⠒⠈⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠉⠀⠀⢀⠄⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⡀⠀⠀⣀⣀⣀⣠⡤⠤⠴⠖⠚⠉⠁⠀⠀⢀⡠⠔⠈⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⢉⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⠀⠀⠀⠀⠀⠐⠒⠈⣉⣀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠂⠋⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡶⣶⣶⣶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣯⡙⣦⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣸⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢠⣴⠾⠟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠴⣾⣿⣿⣿⠏⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡩⢓⡥⠀⠀⠰⣶⣿⡿⢫⣾⣿⣿⣿⣿⣟⣉⣿⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠛⠛⠹⣿⣿⠿⣛⣿⣿⣿⡿⠟⢹⡷⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣹⡿⠂⠉⠛⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣡⣾⣿⡿⠟⠉⠀⠀⣾⣰⣡⣾⣿⣿⠶⠆⠀⠀⠀⠀⠀⣾⠿⣿⣿⠿⡜⠎⠉⢢⡀⢹⣼⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⢸⡇⣿⠈⠐⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏⠀⣠⠔⠂⠀⢠⣴⣶⠾⠿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣧⠀⢩⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⡼⢁⣴⡾⠟⠉⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣸⠏⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠈⠳⣄⠳⡿⠿⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⠀⡨⠛⣠⣴⠒⠋⠉⠉⠉⠉⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡞⠁⠀⡱⠀⠀⠀⠀⠀⢀⣠⠤⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⢀⣴⠃⠀⠀⢀⡴⠞⠋⢀⡦⣾⣿⣿⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣧⡴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⢰⠶⠀⠀⠀⠙⠦⠤⠖⠋⡀⠙⢠⠶⣋⢀⣀⠏⣡⠞⠛⣸⠇⠘⢷⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠖⠀⠀⠀⢀⣠⢶⡿⢢⢞⣵⣾⡁⠀⠀⠀⠈⠛⠀⡼⠃⠀⣴⠏⠀⠀⠀⠀⠀⠀⡾⣼⡇⠀⠀⠀⠀⠀⠀⣼⠃⢀
⠀⠀⠀⠀⠀⠛⣠⡄⢀⣤⠛⢃⣤⠖⣋⠴⢋⡴⣣⢎⡾⠹⠃⠀⣐⣦⡀⠀⢸⡇⢠⡞⠁⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⡼⠃⢀⡞
⠀⠀⠀⠀⠀⢀⡈⠀⣾⣯⠖⠋⣠⠞⣡⠀⢀⡴⠃⢀⣠⠤⠒⠉⠀⢨⡷⠀⠈⠉⠁⠙⣦⡀⠀⠀⠀⠀⢀⡞⣿⠀⠀⠀⠀⠀⢀⡾⠁⢠⠞⣸
⠀⠀⠀⠸⢣⣬⣡⢖⡋⢀⡴⠟⢁⢺⠙⣦⡾⣲⣏⣃⠬⠴⠶⠂⢠⡾⠃⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠶⠋⠀⠈⠙⡶⠶⢤⣀⡞⠀⢠⠎⣰⠃
⠀⠀⣀⣴⠞⠋⠠⣀⡵⢋⠀⢀⣸⣆⡽⠋⠉⠉⠉⠹⣇⠀⠀⠀⠺⣧⣀⣠⡤⠤⠶⠖⠲⣤⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠟⠀⢠⠏⣰⠃
⠀⠀⠉⠀⢀⣠⠞⠋⡆⠀⢳⠟⢡⠞⠙⠒⠦⣤⣀⠀⠈⠙⠓⠒⠲⢦⣄⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⢿⠤⠀⠤⣾⠀⠀⠈⢰⠇
⠀⠀⢀⣴⡟⢁⣤⠀⡇⠀⢹⣶⣋⣐⣒⡂⠉⠁⠈⣹⣦⡀⠀⠀⠀⠀⢹⣷⠀⠀⣠⣴⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⡏
⣠⡶⠋⠛⠋⢛⡵⠀⣇⢠⠞⠑⠤⠴⠶⠶⠶⠚⠋⠉⠋⠀⠀⠀⠀⠀⠀⣿⡆⣼⠋⠀⠀⣀⡼⢋⡽⠃⠀⠀⠀⠀⠰⠾⣤⡤⣤⡘⠁
⠉⠀⠀⢀⡴⠋⣴⠄⣽⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡈⠓⠚⣉⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⣿
⠀⠀⠀⠈⠀⠀⠀⠚⢁⠈⠙⣿⡶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠾⣧⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⢸⣷⠦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠉⠉⠛⠛⠿⣷⣶⣶⣶⣤⣷⣤⣿⣥⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣤⠄⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠈⠉⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣶⠒⠾⣁⣤⡾⠟⠋⠁⠀⠀⠀⠈⠉⠙⠳⢦⡿⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣛⣋⣀⡀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣛⣻⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡂⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⠀⣿⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣶⣶⣶⡄⢸⣿⣿⣿⠀⣿⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⢰⣶⣶⡦⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⣿⡇⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠘⣿⣿⡇⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⡟⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡈⠻⡯⠀⣿⣿⣿⡇⢸⣿⣿⣿⠀⠿⠋⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⠛⠻⠿⠃⠸⠿⠛⢋⣀⣴⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⠿⠛⠉⠛⢁⣤⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠉⠁⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⢤⠈⠀⠀⠀⢸⡿⠨⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠐⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣒⣸⣟⣀⢾⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣷⣶⣶⣶⣶⠁⣈⣿⣷⠀⢴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠘⠟⠉⠛⠛⠿⠟⠛⠋⠀⣿⣿⡿⢿⣿⣿⣾⣿⣿⣿⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡟⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢸⣿⣿⠀⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⢸⣿⣿⡶⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⣿⣿⠀⠀⠀⢀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⣶⡤⢳⣶⣆⠀⠈⠙⠁⠀⠀⠀⠙⢻⡀⠂⡰⡄⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠘⠻⠿⠃⠁⠀⠀⠀⠀⠀⠂⠀⠉⠄⠀⠀⠂⢀⣀⡀