Connective Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢤⡄⠠⠖⠶⠖⠰⠒⠒⡖⠒⠒⠂⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢴⠀⠉⠁⠁⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⡜⠀⠈⠀⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠀⠀⠘⠖⠀⠀⠀⠀⠀⠀⠁⣲⡄⠀⠀⠀⠀⠀⣸⠃⠀⠀⠐⠀⠀⠀⠰⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣽⢹⡿⠀⠀⠀⠀⡀⣀⢇⡀⠀⠀⠀⣀⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⢳⡀⠁⢀⡀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⠬⠤⣆⠀⣀⢀⠰⠂⣽⣆⠘⢄⡠⠔⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠙⢜⢿⡌⠀⢀⠀⠀⣀⠞⠒⠒⠂⠠⠎⠁⠀⠀⠉⠐⣇⢀⡂⠈⠐⠒⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠛⠒⠒⠞⡌⠜⠀⠀⠱⡀⡁⠄⠀⠀⠀⠀⠓⠄⢃⠀⠀⠀⠈⡦
⠀⠀⠀⠀⠀⠀⡀⠡⠀⠀⠀⠀⠀⠀⠀⠀⣠⡋⠀⠀⠀⠀⠀⢱⡀⡄⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠁⠠⡒⢀⠀⡀⢴⡖⡦⠀⣜⠀⠀⠀⢰⣦⠀⠜⠁⡇⠀⠀⠀⢀⠁⠈⠠⠀⠀⠀⠀⡰⡸⢌⠤⠤⢚⡗
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠤⣈⡩⠔⠫⠬⠍⠼⠰⠆⢡⣎⠀⣰⠀⠀⠀⠀⠐⢠⠐⢹⡇⠢⡀⠐⠩⠀⠀⠉⠋⠈⠣⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠭⠥⠤⠷⠿⠶⠞⢠⠌⡁⣀⢀⡐⡡⠀⠀⠀⠘⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢖⡆⠀⠀⠀⠈⠈⠁⠀⠹⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⢄⡀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠘⠏⢁⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⣠⣯⠣
⠀⠀⡀⠀⣀⠄⠀⠀⠀⠐⡠⠐⠁⠀⠀⠍⠒⠤⣈⢰⣽⣌⣲⠤⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠈⠍⠓⠁
⠀⠀⡃⠈⠙⠟⢡⢚⠐⡎⠀⠀⠀⠀⠀⢰⣫⡦⠈⡹⠋⠉⠀⠀⠀⠀⠑⢄⠁⠀⠀⠀⠀⢀⠄⠃⣄⠀⠀⡀⠀⠀⠘⠀⠀⣀
⠀⠀⡥⠄⠀⠈⢾⡻⠂⢱⠀⠀⠀⠀⠀⠈⠉⢈⠒⠀⠀⠀⠀⠀⢐⠄⡔⣦⠐⢤⣀⡁⠖⠠⠀⠈⠃⠈⠒⠠⣀⡐⠄⠒⠉⠉
⠀⠀⠆⠁⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠠⡁⠰⠀⠀⠀⠀⠀⠈⠁⡐⠙⠄⡀⠀⠀⠐⠂⠀⠃⠀⠀⣈⡀⣐⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢘⠀⠀⠄⠐⡀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⠀⢀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⣈⠩⠀⠀⠀⠀⢀
⠀⠀⢠⣀⠂⠀⠄⠀⢂⠘⠉⠁⠉⠉⠀⠀⠀⠀⠱⠈⡀⠀⠀⠀⠀⢀⡼⠂⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠶⠂⠐⠐⠀⠈⠤⠁
⠀⠀⠀⠀⠁⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⣀⣨⠄⠐⢆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠉⠀⠀⠀⠀⠀⠈⠢⣑⠀⠀⠀⡄⢠⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒

⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⣤⠀⠀⢠⢤⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⣄⡀⣄⡀⠀⠀⠀⣼⠁⠀⢸⠁⠀⠀⣸⠆⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⣿⢀⡠⢼⣯⠠⠴⣶⠤⠴⠊⣠⣶⣶⡀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢡⡀⣸⠃⠸⣀⣀⣧⠀⣦⣯⠀⠀⠀⠛⢿⡿⠃⣇⠀⠀⠂⣤⣴⣶⡄⠀⠀⠀⠀⠀⠀⣐⣂⡀
⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⢸⡼⠛⢀⠤⡀⠀⠽⢿⣿⣿⣦⠀⠀⠀⠀⠀⠈⣯⡄⠀⠀⠈⠈⠁⢀⣀⡠⠴⠒⠊⡉⢹⣿
⠀⠀⠀⠀⠀⠀⠙⠲⢄⠘⢦⣾⠁⢀⣑⣤⠼⣼⠄⠚⢿⡏⠈⠛⠃⠀⠀⢀⣈⢉⣤⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢿⡍⠁⠀⠀⠀⣿⠀⠀⠀⠙⣷⣧⣤⣖⠉⠁⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠈⠓⣤⡀⠀⠱⡈⠿⢄⡀⢀⡠⡿⠀⢀⠔⠊⣽⠉⢷⡏⡁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿
⠀⠁⠀⠀⠀⠒⠦⣌⠻⣦⠀⠱⡄⠈⠙⠯⢘⠵⠊⠁⠀⣰⠏⠀⢸⠿⣏⢤⡀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠙⠛⢷⣄⠈⠢⣀⠖⠁⠀⢀⣴⠞⢁⡠⣶⠉⠀⠈⠳⡙⠦⡀⠀⢳⡀⠀⠀⢀⡀⡴⠊⠈
⠀⠀⠀⠀⠀⠈⠃⠤⣤⡀⠀⢹⣿⣶⠳⡀⠀⣠⠟⠁⣠⡾⠋⠀⠀⠀⠀⠀⠀⠰⢼⣦⠈⣧⠀⣠⠏⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡔⢻⡀⠓⢵⣴⣏⣠⣞⡕⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣙⣷⣏⣠⡞⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⢿⡋⠻⣉⡻⠛⠛⠛⠫⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣤⣿⣟⣿⣍⣉⣑⠒⠒⠒⢲⣶⣒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⣇⠀⠀⠀⢀⣈⣦⣀⣀⣠⠤⠶⠒⠒⣨⡟⠉⠙⣇⠈⢿⡄⠀⠉⠉⠉⠹⠟
⠠⠄⠀⠀⠀⢀⣀⣤⠄⠠⠤⠴⡾⠛⠛⠟⣿⣏⣉⠉⠁⠀⠑⣄⠀⠀⢀⣤⡾⠊⠀⠀⠀⠈⠛⠛⢿⣄⠀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠿⠿⠦⠀⠀⠈⠓⣴⣟⢻⡷⡀⠀⠀⠀⠀⠀⠀⠸⣏⠙⠦⣀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣽⠀⠀⠀⠀⠀⠀⠀⠐⠦⣴⣀⣀⠀⡧⠉⢧⢱⠌⣄⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⣤⠯⠷⣦⣤⠠⠤⠄⠀⢀⣀⣁⣡⣉⣻⠓⠦⣬⣷⣶⣽⣄⠀⢀⠔⠦⠀⠸⡇⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡀⠀⠈⠙⠷⢦⣠⣤⣠⡵⠚⠁⠈⣯⠈⠉⠉⠹⣿⠿⡍⠳⠋⡞⢓⡲⠶⢿⡖⠚⠉⣀⠤⠒
⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣞⢷⠀⠀⠀⠀⡜⢹⣿⡿⣍⠉⠉⠉⢿⣄⡀⠀⠀⠑⠂⠷⠀⠀⠉⠁⠀⠀⠈⣥⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠰⣶⣧⠀⢀⠀⠰⣸⣮⠇⠈⠛⢦⡄⠈⠳⡝⢄⠀⠀⠀⠀⠀⢀⡠⠶⠒⠒⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⡠⠩⠩⠜⠀⠓⠀⣿⡿⠀⠀⠀⢀⡨⠿⠶⣽⣦⣷⣤⡖⠒⠊⠁⠀⣀⡀⠀⠀⠀⠀⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢾⣮⠜⡇⠀⠀⡜⠁⠀⠀⠀⠀⠉⠙⠺⠿⠿⠟⠻⡿⣿⣿⣿⣶⣤⣀⠀⠈⠛⠂
⠀⣄⡀⡀⠀⠀⠀⣀⣀⣤⣶⣾⡿⠟⠛⠉⠁⠀⡇⠀⠰⡁⢀⡤⠖⣲⠶⣦⡀⠀⠀⠀⠀⠀⢨⣻⣿⡄⠀⠉⠧⢽⣦⣄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⡶⣻⡿⠛⠉⠀⢀⡴⠶⠟⡻⠟⠓⠶⣾⣿⣛⡿⠭⠬⢴⣷⠊⣢⡀⠀⠀⠀⠀⠘⢷⠙⠀⠀⠀⠀⠀⠉⠣
⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠘⠁⠀⠞⠁⠀⠀⠰⠁⠈⠁⠀⠀⠀⠀⠀⠉⠙⠮⠖⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀⢀⣀⢠⣾⠀⠀⢶⣤⠄⠀⣶⣄⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⠀⠀⠀⠀⣴
⣤⡄⠀⣠⡀⠀⠶⠶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠁⠏⠈⠁⠈⠉⠀⠁⢠⡀⠀⠁⠀⠀⠈⠉⠀⣨⡁⠀⠀⠀⠀⠀⠀⠀⠰⠁⠰⠛⠿⣄⣄⠀⣿
⠿⡇⠈⣿⡁⠻⡟⢠⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠀⠀⠀⢀⠀⠀⢀⡀⠈⠀⠉⡀⠀⠺⠀⠀⢄⣙⠁⠀⠀⠀⠀⠀⠀⠀⣿⠂⠦⠄⠀⠿⠏⠀⢾
⠀⢾⣿⠉⣤⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠙⠀⠈⠁⠀⣿⠀⠙⠆⠐⠀⠀⠈⢹⣷⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⠀⢀⠀⠀⠀⠘
⠀⠀⢻⡇⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠆⠀⠀⠃⠀⣾⠃⠀⠂⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣴⡤⠤⠀⢾⠄
⠀⠀⠉⠑⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠐⠶⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢸⡧⠀⠀⣤⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠧⠀⠐⠀⠀⠘⢁⣄⠀⠀⠀⠀⠠⡄⢈⡁⠀⠀⠀⠀⠀⠀⠀⢿⡧⠀⠠⠀⠀⠀⢰⡶
⠀⠀⠀⠈⣷⠀⠀⠈⠁⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⣤⠀⡀⠙⣿⡆⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⡀⢸⡇
⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠘⠃⠀⠀⠀⡀⠀⠟⠀⠀⠀⣀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠘⠃⠀⢷⣼⠁
⠀⠀⠀⠈⢿⡷⠀⠀⠿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠛⠀⠀⠘⣧⢀⡄⠀⠀⠀⠀⠀⠿⠀⢀⡀⠀⠀⣾⡇
⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠐⠀⠀⣠⡀⠸⠇⠀⠀⠒⠀⢠⠄⠀⠘⠃⠀⠀⠀⢀⡄⠐⠀⠘⠃⠀⢀⡙⠃
⠀⠀⠀⠀⣻⣧⠀⠀⠀⠀⢀⣼⣧⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⠀⣴⡄⠀⠀⢹⡁⣿⠀⠀⠀⠀⠄⢠⡀⠀⠀⢾⡆⠀⠀⢸⣧⠀⠀⠀⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠻⠟⠀⣀⡀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⣶⠀⠀⣨⠀⢀⡄⢘⡁⢀⣄⠀⠀⠀⠀⠈⠀⠀⢠⡾⠃⠀⠀⠀⠉⠀⣤⠈⠃⢀⡐⠻⡇
⠀⠀⠀⠀⣤⡏⠀⠙⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⢰⣄⡀⢈⡆⢸⡇⢸⣿⠀⠀⠄⢸⡇⢀⠀⠛⠁⢸⠄⠀⢰⣏⠀⠀⠀⠃⢸⠁⢺⡇
⠀⠀⠀⠀⣿⡇⠀⠀⣄⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢸⡇⢾⠀⠸⠋⠁⢈⡇⢸⠇⠈⡟⠀⠀⠀⠀⠁⡼⠀⠀⠀⠘⠀⠀⠀⢉⡀⠀⠐⠆⠈⠀⠈
⠀⠀⠀⠀⣿⡏⢰⠀⣿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⣄⠀⠀⢠⠘⠃⢸⡇⠀⠁⠀⢀⠀⡁⢰⡇⠀⠀⢸⡄⣠⠀⠀⢸⣇⠀⠀⠸⠀⣠⢀⡀
⠀⠀⠀⢘⣿⢻⣆⠀⣬⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢿⡀⠀⣼⠀⠀⣿⡇⠀⠀⣰⡏⠀⠇⠈⠁⠀⠀⠘⠃⣿⠀⢠⡌⢿⠀⡇⢸⠆⣿⢸⡇
⠀⠀⠀⢺⣏⠹⡏⠀⣿⠀⢽⡇⠀⠀⠀⠀⠀⠀⢰⣿⠛⠀⠈⡟⠀⠻⠀⠀⢻⡇⠀⠀⣸⠃⣰⠀⠀⠀⢀⠀⢠⠀⣿⠀⢸⡇⠘⠀⢇⡈⠀⠻⣄
⠀⠀⠀⢸⣿⠀⡇⠀⣿⠀⣼⡇⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⡇⠀⢠⡀⠀⠸⠀⠀⠀⣿⠀⣿⠀⢠⠀⣿⠀⠈⠀⢹⡆⠈⠃⠀⣷⠸⡇⢸⢀⡿
⠀⠀⠀⢸⣇⢰⣷⠀⢈⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿⠀⠻⠀⠸⡇⢠⡆⠀⠀⠀⣿⠀⣿⠀⠸⠀⠀⠀⡄⠀⢿⠀⠀⠠⠀⠀⠀⣷⡌⣿⠁
⠀⠀⠀⢸⣿⡀⠋⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⢸⡇⠈⡧⠀⣠⠀⠀⡄⠘⠃⠀⠀⠠⣆⠀⠿⠀⠀⠀⣿⠀⠁⠀⢸⠀⠙⠀⢠⠀⣶⠘⣷⡏
⠀⠀⠀⠈⣿⣿⠂⠀⢠⡄⠘⣧⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣘⠀⠀⢻⡞⠀⠀⠀⠀⣿⡄⠀⠀⢸⠄⣿⠀⡇⠀⣸⠀⢸⡇⢸⠀⢿⡆⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⢸⣧⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⢉⠀⠀⠈⢳⠀⠀⠀⠠⣿⡇⠘⠀⠘⠀⣾⠀⡇⢸⡇⠀⠘⠃⠘⣆⢸⣿⣿
⠀⠀⠀⠀⣿⡇⢰⡆⢸⡿⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⠛⠀⠀⠀⢸⡆⠀⠀⠀⣿⠁⣤⠀⣴⠀⣡⠀⠃⢠⡀⠀⢰⡀⢸⣟⠘⣿⣇
⠀⠀⠀⠀⣿⡁⢸⠁⠸⡇⠀⣿⠃⠀⠀⠀⠀⠀⣼⡇⠀⢹⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⣿⡦⠛⠀⢹⡄⢿⠀⠠⣉⡁⠀⢘⡇⢸⣿⣿⣿⡿
⠀⠀⠀⠀⣿⠃⠸⠀⣴⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⢧⡀⢸⣦⢀⣿⠂⠀⢸⠀⠀⠀⠀⠸⣿⠆⠀⡌⠁⠈⠀⠀⣿⡇⠀⠸⠇⠈⣿⡿⢿⡷
⠀⠀⠀⢀⣿⢰⡆⢀⡿⠀⣼⡇⠀⠀⠀⠀⠀⣸⣿⠀⢿⠀⠻⣮⠉⣦⠀⡿⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⢀⡄⢸⡇⠀⠀⠳⠀⢿⣧⣸⡆
⠀⠀⠀⣸⣿⠈⠃⢸⠃⠀⣿⠇⠀⠀⠀⠀⠀⣿⠉⠀⠘⠷⠀⢹⡀⠀⠀⡇⠀⠀⠀⠀⠀⣿⡄⠃⠸⠃⠀⢸⡇⢸⡇⢸⠀⠘⣦⠈⣿⡿⠃
⠀⠀⠀⣿⣿⠀⠀⡜⠀⠀⣰⡄⠀⠀⠀⠀⠀⣿⠀⡇⠀⢠⠀⠈⠁⠀⣾⠁⠀⠀⠀⠀⠀⢸⡇⢀⡄⠀⠀⢸⡇⢸⠆⠸⡇⠀⢿⠀⣹⡄
⠀⠀⠀⠛⠛⠻⠆⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⣿⠀⠹⠀⠸⠁⠀⠙⢬⡟⠀⠀⠀⠀⠀⠀⠘⣷⠀⠃⠀⠀⣠⠇⡀⠀⠀⠒⠀⠀⣧⣸⡇
⠀⠀⠀⠇⠀⠀⠀⠀⣤⠀⠸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢴⡄⠀⠀⠸⣷⠀⣿⣿⠇
⠀⠀⠀⣾⡆⠀⠀⢀⠉⠀⠀⣿⡄⠀⠀⠀⠀⣿⠀⣼⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣄⠀⠀⢡⠀⠉⣶⢨⣿
⠀⠀⠀⢿⠃⢇⠀⠘⠀⡀⠀⣿⣇⠀⠀⠀⢰⡿⠀⠘⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠀⠹⠃⢠⡾⠆⠀⣿⣸⣿
⠀⠀⠀⠈⣷⡈⠃⠀⢸⡁⠀⠉⣿⠀⠀⠀⣼⡇⠀⠀⢴⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⡖⠀⠀⠀⠙⡧⢀⣀⣿⣿⣿
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠃⠀⠀⣿⠀⠀⠀⣿⠁⢼⠂⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠐⢉⡟⠀⠀⢠⠞⠀⢸⣟⠙⣿⣿⡀
⠀⠀⠀⠀⢸⣿⠘⠀⠀⢰⡇⠀⣿⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣠⣽⡆⢠⡄⠐⠀⢸⡏⠀⣿⢻⡇
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠈⠃⢰⣿⠀⠀⢸⡇⠀⡀⠘⢆⠀⠀⢧⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠸⠁⠇⠸⠃⠀⠀⣼⠃⠀⣿⢸⣇
⠀⠀⠀⠀⠀⣾⠀⣸⠂⠀⣆⠈⣿⠀⢀⡿⠁⠀⠁⠀⠘⣇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣦⠀⠀⠀⠐⠿⠰⠂⢉⣸⣿
⠀⠀⠀⠀⢸⣿⠀⠁⠀⢠⡅⠀⣿⠀⢸⡇⠀⣀⠀⠆⠀⠉⣀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⠀⢸⡂⢈⠀⠀⢠⠀⣤⠀⣖⠋⣸⣧
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⣿⠄⠀⢀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⢺⠃⠘⡇⠀⠀⢀⣁⠀⣈⠀⣹⣿
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⡀⠀⣿⠀⡏⠀⠀⢀⡄⠀⣸⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⢺⡇⠸⣷⠀⠀⠈⡿⠀⠙⣿⢻⣯
⠀⠀⠀⠀⢸⣿⠀⢶⣦⠼⠇⠘⠁⢸⣷⠀⠀⠀⠁⠀⢹⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⡇⢸⠁⠀⡷⠀⠉⠠⣷⡄⠸⢿⣸⣿
⠀⠀⠀⠀⢸⣿⠀⠈⠉⢰⡄⠀⠰⢸⡇⠀⣰⠀⠃⠀⢸⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⣈⠀⠀⡇⠀⠘⠀⢸⡇⢀⣼⣿⡟
⠀⠀⠀⠀⠀⢿⡇⢰⠀⠈⠰⣦⡀⢠⣄⠀⣿⠀⢡⡄⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⣈⠀⢸⡇⠀⠀⠀⠸⠇⠸⠛⣿⡇
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣿⡇⢸⡁⠀⣿⠀⠘⡁⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⠀⢸⣿⠀⠀⠀⢠⣄⣿⣾⣿⡁
⠀⠀⠀⠀⠀⠀⢸⠀⢷⠀⠀⢸⣷⣟⠁⢠⡿⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠟⠀⠀⠀⡈⠃⠈⣿⣿
⠀⠀⠀⠀⠀⠀⣀⡴⠀⡀⠀⢽⣇⠿⠀⢸⠇⠀⠀⠀⠀⣶⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣄⠀⠂⠀⠀⠀⠁⠀⡈⠻⣿
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠼⡧⠀⢀⡎⠀⠀⡤⠀⠀⣿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⡄⠀⠀⣄⠀⠀⣹⣶⣿⡇
⠀⠀⠀⠀⠀⠈⣿⠀⢀⠁⢰⣶⡶⢀⣼⠃⢀⣄⠀⠀⠀⣿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠖⠃⠀⣤⠀⠐⣿⣀⣄⣽⣿⣿⡇
⠀⠀⠀⠀⠀⢰⡏⢀⡤⠀⢰⣿⡦⡘⠃⠀⢈⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠁⠀⠀⠠⡎⠈⠙⣋⣿⡇
⠀⠀⠀⠀⠀⣨⡁⠈⠁⠀⢃⣿⣷⠋⠀⠀⠙⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡀⠀⢶⠀⠀⠀⠀⠸⠉⣻⡇
⠀⠀⠀⠀⢀⣿⠀⢰⠀⣐⣿⣇⡿⠀⠀⢰⡇⠀⠀⠀⠀⢀⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿⠇
⠀⠀⠀⠀⣾⠏⠀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠙⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣤⠀⡄⠀⢀⣽⠆⠀⠀⢸⣿
⠀⠀⠀⢘⡟⠀⠀⠀⣿⣥⣿⠇⠀⠀⠀⢰⠆⠀⠀⢀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⣾⣿⡏
⠀⠀⢠⠿⠃⢰⣆⠀⢿⣿⡟⠀⠀⠀⠀⠈⠀⠀⠀⠸⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⣿⡇
⠀⢠⣾⣦⡀⣼⣿⠀⣼⣿⠇⠀⠀⣀⡆⣸⠃⢠⣶⠸⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⣴⠀⣾⡆⠀⠀⢿⠀⠸⠍⠋⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⣤⡄⠀⠀⡂⠀⠸⠀⠉⢀⣼⣯⠏⠀⠀⠡⣶⠂
⠀⠀⠀⠀⠢⠄⠀⠀⠁⠀⠀⠀⠀⠀⠣⠀⠀⠀⢭⠀⠀⠐⠁⠀⣤⠗⣾⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠄⠀⠀⠈⢂⠀⠀⠐⠀⠀⢈⠟⢀⣀⡿⣄⡀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠐⠀⠀⠃⡀⠀⠀⢶⢿⠊⠉⠁⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠐⠀⣥⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡀⠀⠄⠀⠠⠂⢤⠦⢰⢶⣲⣌⠀⠀⠀⠀⠀⠢⠄⠠⠀⠀⠀⡀⠀⠒⠀⠀⠀⠈⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠐⠀⠁⣀⠀⠀⡀⠀⣁⠀⠈⠈⠉⠉⠉⢀⠈⠀⠀⠀⢁⠉⠀⠀⡀⠤⠀⠀⡀⠄⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⡄⠀⡈⠀⠀⡐⠀⠀⠀⠀⠙⠄⠀⠂⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁⠒⠂⠀⡀⠠⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠀⠢⠂⠀⠀⠀⢀⠑⠄⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠓⡄⠀⠀⡀⠀⠭⠹⡚⠹⠄⠐⠂⠆⠤⠀⠘⠤⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠈⠸⣿⣾⡇⠀⠶⣴⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠈⠀⠋⠁⠁⠀⠋⠀⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠄⠉⠁⠀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⣐⣀⠀⠆⡀⠠⠤⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣛⣉⢻⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠆⠉⠀⠄⠈⠀⠀⠠⠙⣧⡠⢉⣻⠉⠛⠦⠲⠆⠡⠆⠀⠀⠀⠀⠀⠀⠀⣤⡀⠉⠉⠉
⠀⠀⠀⠒⠀⠀⠂⠀⠂⠠⠐⡏⠀⠀⠀⠀⠃⠀⠠⠈⠛⠫⠁⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠐⠲⠂⠆⠀⠀⠂⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡐⠂⠀⠠⠀⠈⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣿⣷⣭⣯⣬⣵⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣪⣤⣤⣈⣁⠀⢀⢈⣉⣤⣬⣁⠢⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡠⢂⡜⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⡳⢉⠳⡙⢦⣤⣤⣤⣤⡤⠖⠊⠁
⠀⠀⠀⠀⠀⡀⠈⠉⠙⠿⠟⢋⠟⣠⠏⣀⣼⣿⣿⠿⠿⠿⠾⢿⣿⣿⣿⣷⣦⡀⠀⠙⢯⠍⠁⠀⠀⠀⡀
⠀⠀⠀⠡⣀⠑⠀⣀⢀⢠⠔⢁⣔⣡⣾⣿⣿⣿⡏⠀⢀⣶⣦⠀⠸⢿⣿⣿⣿⣿⣶⣄⣀⠉⠐⠒⠒⢋⣡⠄
⠀⠀⠀⠀⠈⠛⢲⣀⣠⣤⣼⣿⣿⣿⣿⣿⢯⣹⡷⢀⣾⣿⣿⣧⠀⠈⠛⠗⠹⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⣯⣻⣿⣿⣿⡷⡅⠑⠻⢿⣿⣿⣿⡆⠀⠘⣿⡀⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⠀⠉⠻⣿⡿⠟⠋⢿⣿⣿⡯⢠⡞⠉⠉⠘⠉⢻⡀⠀⠹⡟⠈⠛⠿⣿⣿⠛⠻⣿⠃⠀⠀⠀⣀
⠀⠀⣺⠀⠀⠀⠸⡆⠀⠀⢻⣷⠇⠀⣿⠟⠿⣿⣻⣶⡀⠀⠀⠀⠀⣇⠀⠠⢿⠤⡄⣵⣽⠏⠠⢤⡿⠀⠀⠀⠀⠽
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠉⠉⠙⠫⣠⡏⠈⠀⢸⣿⣿⣿⣶⡀⠀⠀⢸⡀⠀⠡⣆⡰⠞⠁⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠘⠳⡄⠀⣿⣿⣿⣿⣿⣿⣶⣤⣴⡆⠀⣰⠋⠁⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡞⠅⠀⠀⠀⣮⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⠀⠑⢈⠛⠿⠿⠿⠿⠿⢛⣫⠞⠀⠀⣀⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⡀⠈⠁⠈⠀⠀⠈⠀⢀⣀⣴⠾⠋⠁⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⠵⠖⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠐⠒

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

⠀⠀⠀⠀⢴⠀⠀⠀⢀⠀⠀⠀⠆⠰⡄⠀⠐⣄⡄
⠀⠀⠀⠀⠈⠢⣠⠳⢎⡡⡀⠀⠈⢀⡘⢦⡀⠈⢆⡀⠀⠑⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣺⣿⢤⡀⠀⠣⠈⢟⣢⠨⠑⣦⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠘⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣾⣷⡄⠠⡗⠜⣧⣓⡀⠈⠣⠇⣀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⣇⠀⠁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢻⣭⣥⣿⣦⠀⠈⠀⠉⣧⡀⠀⢈⠈⢡⠀⠂⠀⠀⣦⠀⠀⠀⠀⠀⠀⠉⠄⠀⢀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣆⠀⠀⠀⢐⣵⣄⠀⢳⣄⠳⠐⢀⡁⠘⣷⣶⡨⣶⡴⡇⢀⠻⡀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⢿⣶⣿⡙⢦⡀⠀⠀⠉⠈⢡⠀⠀⠀⠀⠀⠀⠀⠈⢡⣽⣟⣿⣷⠀⠐⢇⠀⠁⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢨⢻⠻⢧⡘⣷⡀⠀⠒⠀⠀⠀⠀⠀⠠⠀⠀⢘⠄⢸⣵⠋⣠⡌⢣⢠⠘⣦⠈⠂⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣧⠛⢻⡗⣹⣧⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠸⢗⡸⣿⡄⠿⣿⠀⣧⢁⢬⣇⠘⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠹⡇⢈⣿⡼⣿⣆⠀⠀⢀⠘⠀⠀⠀⠀⠡⠤⢸⡀⠋⢻⣆⠀⠀⣿⠆⠈⠉⢠⣗⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢹⣆⠸⡇⣷⣿⡆⣀⠀⠆⢨⠆⠀⠀⠀⠁⠈⢃⠀⠀⢪⣟⣾⠛⠧⠀⠀⠀⠹⣦⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠻⡀⠈⢾⣧⣿⡄⠄⠰⣠⡷⠛⠀⠀⠀⠀⠀⠠⡀⠠⠸⣿⣿⣿⡦⠀⠀⠀⢹⣆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢷⡄⠘⣎⢯⣻⣻⠄⢹⣱⡟⢆⠀⡀⠐⠒⠀⠈⢦⡀⢠⠙⢿⣿⠄⠀⠀⠀⢿⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⠑⡔⠘⢨⣿⡇⠆⠀⢻⡌⢩⠄⠱⢀⠀⠀⠀⠀⢻⢀⠑⣆⡹⡄⠀⠀⢀⠈⢣⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠽⠀⠀⢩⠿⠲⣆⠈⠻⣌⠒⢃⣷⠊⠇⠈⠀⠀⠹⣦⡹⣵⣼⣇⠀⠈⡀⠫⣧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠤⠂⠠⡉⣿⡀⠈⠂⠀⠘⣿⣻⣿⠆⠈⠀⠀⠀⠀⠈⣧⠹⡜⢎⢡⠀⣀⣀⠙⢧⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡀⠠⠈⢧⡀⠀⠀⠀⠘⣷⡌⢾⡄⠀⠀⠀⠀⠀⢾⣇⠰⡤⢺⣧⣤⣜⢷⠘⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠈⣷⡀⠐⡈⣷⡄⠀⠀⠀⠰⡻⡎⢳⡄⠀⢀⠠⠀⠈⣿⠀⣷⠙⣏⢿⠟⢻⣄⠃⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠸⣧⡀⠘⢄⠙⠄⠀⠀⠀⠐⠀⠀⣷⠀⠀⠀⠡⠀⢸⣇⠘⣧⠘⡷⠲⡚⠃⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢘⣿⣷⣦⠀⠐⠰⠂⠀⠀⠈⠄⠁⠘⠃⠀⠀⠐⢳⡀⢻⡄⠘⣷⠀⡍⢿⣊⠢⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢻⠀⠠⠓⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⢹⣆⠘⣧⠈⠁⢻⡔⠀⠜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⣄⠐⠀⢸⠀⠀⠀⢄⠀⠀⢲⠀⠀⠀⠀⠀⢀⠀⢧⣰⣿⡆⢹⣆⠀⠈⣿⡀⠢⠨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡰⣰⣾⠀⠀⠄⠀⢃⡀⠀⠐⡀⠀⠀⠀⡄⠢⠈⡈⠋⣿⡄⢻⡀⢀⣿⣠⠐⢁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠹⠿⢦⠀⠀⠀⠁⠙⠀⠀⠈⠠⠀⠈⢴⡄⠿⡶⡈⣟⢿⡀⢛⣶⠘⡇⠘⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠋⠉⠻⠈⠀⠱⢀⠻⡀⢷⠠⠂⠄⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠛⡀⡈⢂⠀⠛⠂⠂⠀⠻⣆⢻⡀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡝⢀⠀⠀⠀⠀⠹⠀⢉⡀⢸⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠁⣄⠘⡀⢀⠄⠐⠄⠁⢻⠈⠧⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⢡⠀⣄⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⣠⠀⠀⢀⣤⣤⡄⡠⠄⠀⠀⠀⠆⠀⠀⠀⠈⠂
⠀⠀⣤⣤⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠈⠙⠉⠀⠀⠸⠿⠿⣿⣷⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢼⣿⠄⠒⠠⠀⠀⠀⡀⠀⠀⠀⠰⠉⠁⠀⠀⠀⠁⠀⠀⡀⠈⠋⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠀⠶⠤⠆⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⢀⣶⣿⣷⠀⠀⠙⣇⠀⣠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠈⠉⢹⣿⣷⠀⠛⠁⠀⠀⣠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠋⢁⣤⣄⣀⣀⠀⠹⠤⠾⣧⣜⣷⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠒⠂⠀⠀⠀⢀⣴⣾⣯⠀⠈⠂⠀⡸⣿⡿⢿⣯⣽⠀⠀⠠⠀⢈⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠦⠀⠀⠀⠀⣠⣾⣿⣿⠟⠀⠀⢠⡆⠀⠀⠀⠙⠲⣿⣶⣄⠀⠀⠙⣦⣼⠆⠀⢠⠄
⠀⠀⠀⠀⢠⣾⣿⠂⠀⣀⣄⠀⠉⠀⠀⣁⣀⠠⣾⣿⣿⠿⠛⣀⣤⣀⠀⠈⠁⠐⡀⠀⠸⠿⠟⠋⢀⣠⣾⣿⡅⠀⠀⠈⠀⠀⠀⠀⢀⣠⡤⠶⠆⠈⢻⡆
⠀⠀⠀⠠⠀⠿⡿⠂⠰⢿⣿⠗⠀⠀⠉⠁⠈⠐⠋⠁⢀⠀⠀⢿⣩⣿⡗⠀⠀⠀⠁⠀⣀⠄⠀⠀⠘⠿⢿⡿⠋⠀⠘⠂⠐⠀⠀⢲⡻⠉⣦⡀⢠⣄⠛
⠀⠀⠠⠤⣤⣄⠘⠤⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠾⢹⠃⣌⠑⠋⠬⢄⡀
⠀⠀⠀⠀⠀⠈⠱⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠓⠂⠆⠀⠀⠀⠀⠀⡀⢠⠂⠀⠀⠀⡰⠋⠚⠙⠏⠃⠀⠘⠟
⠀⠀⠀⢀⡀⠛⠀⣀⡀⠀⠉⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⡀⠀⠀⠘⠂⠀⠠⠐⠀⠀⢀⡶⠰⠆⠐⠋⠁
⠀⠀⠀⠀⠀⢀⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⣈⣀⠀⠀⠀⠰⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠐⠀⠁⣀⠀⠈⠀⢀⠀⠉
⠀⠀⢠⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠤⣈⣁⡈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠚⠃⠘⠛⠀⠈⠀⢰⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⠥⠁⣀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡴⠀⠀⢀⣲⣤⣄⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠁⠀⠀⠀⠬⠥⡤⠤⠌⠀⠘⠒⠀⠨⠀⢰⠃⢐⣥⣌⡐⠀⠀⠀⣀⠀⢈⠀⠀⠀⠸⢿⣿⣿⠇⠀⠰
⠀⠀⠀⠀⠀⠐⠆⠀⠸⠀⠀⠀⠐⠆⠀⠗⠈⣀⣠⡀⠀⠀⠀⠀⠀⢠⣶⣾⣶⠀⣿⣷⣼⣿⡿⠃⠤⣄⠀⠐⠀⠉⠁⠆⢶⣤⠀⠈⠀⠀⡀⠀⠀⠉
⠀⠀⠀⣴⣤⡄⠀⢀⡄⢀⣀⠀⠀⣠⣶⣄⣸⣧⣼⣷⡶⢄⠀⠀⢀⠈⠛⠟⢻⡄⢿⣏⡁⢀⡤⡀⠀⠈⠇⠀⠀⠰⠀⠀⠀⠀⠈⠛⢢⣤⣥⣤⣀
⠀⠀⠀⢿⣿⠇⠐⠯⠽⠦⠿⠀⢀⠻⢿⡿⠃⠈⠉⠿⠆⠙⢰⣿⣿⠻⠆⠀⠈⢿⣤⡜⠛⢷⣦⠃⠀⠀⠄⠀⠀⢠⠞⠛⠛⠓⠆⠀⠺⢿⣿⠿⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢠⡄⠀⠀⠀⢃⣴⣶⣦⠀⠀⢀⠀⠁⠈⠁⠀⠀⠀⠀⠘⠟⠀⠂⠘⠦⠀⠀⠀⠀⡀⠀⠛⠀⠀⣄⠀⠀⢠⡄⠀⠀⠶⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠘⠛⠛⡁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⣆⣹⡄⠀⠀⠙⢀⠀⠀⠀⠈⠇⠀⠂⠀⠐⢈⡀⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⡈⠑⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠓⠀⠀⠃⠀⠀⠀⠀⠇⠀⠀⢰⣿⣿⣷⣤⣄⠀⠀⠈⣀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠿⣛⣿⣿⠊⠀⠀⠀⠦⠀⠤⠀⠙⢦⡠⠀⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠋⠁⠻⣦⡳⣤⠚⠉⠑⣎⢁⠐⣤⣶⣶⡄
⠀⠀⠐⢁⣀⡦⠀⠀⡻⠟⣁⢀⣀⣀⠀⠄⠀⠀⠀⠀⢋⠀⠀⠀⠀⡠⠀⠠⣀⠀⠀⠀⠀⠇⠀⠀⣴⠋⡉⠀⡀⠂⢄⣀⠈⣏⠰⢿⡿⠟⠃
⠀⠀⠀⡈⠀⠀⠀⠀⠇⣠⠙⠛⠋⠉⠠⠀⠤⠤⠀⠀⢏⠀⠀⠀⠞⠀⠀⠀⠉⠘⠒⠀⠀⢢⠄⠚⠙⠛⣅⡈⠃⡀⢦⠁⠴⠀⠐⢂⣤⢄⡀
⠀⠀⠀⠑⠂⠀⠀⢀⣼⣏⠳⠆⠀⠐⠂⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠐⠀⠀⠀⠈⠓⠂⠀⠀⠀⠀⠃⠀⠈⠃⠀⠉
⠀⠀⠀⠀⠀⠆⠀⣿⣿⣿⡇⠂⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⡏⠀⢀⠀⠀⠀⣠⣴⣶⣤⠒⠀⠀⠀⠈⡀⠸⠷⠄⠀⠀⢰⡀⡀
⠀⠀⠀⠇⠀⠀⠀⠀⠙⠁⣠⣤⡀⠀⠀⠀⠀⣀⣴⣿⣤⠀⠀⠈⠉⠀⠀⠀⠈⠁⢀⣀⡀⠻⡟⠿⠃⢠⣶⣶⣦⠀⢰⡄⢰⣆⠀⠀⠈⠛⠉
⠀⠀⠀⡞⠀⠈⠳⠄⠀⠺⣿⣿⡋⠀⠈⠀⠀⠙⠛⠿⠛⠀⣀⣤⠀⠀⢀⠴⠂⠀⢻⣭⣷⠀⣠⠀⠀⠈⠻⣿⢿⣦⣼⣿⣞⢛⠀⠀⠘
⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⢾⣿⣿⣶⠂⠀⠀⠀⠄⠀⠀⢸⣿⣿⡇⠀⢀⣄⢀⠀⠈⠩⠙⣀⣠⣤⣤⠀⠀⠀⢸⣟⣥⡘⠟⠘⣧⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⢀⡉⣽⣿⣯⠐⠒⠢⣀⠀⠀⠈⣻⠋⠀⠀⠀⠉⠈⢁⣀⠀⠀⠛⣿⠟⢁⣠⡄⠀⣴⣿⣏⠙⣧⣴⡟⠀⠀⠀⢀
⠀⠀⠀⣠⣤⡅⠀⠀⠀⠐⠙⣿⣿⠆⠉⠒⠒⠒⠁⠈⢀⠔⠉⠀⣾⣿⣶⡄⠀⢼⣿⣷⠆⠀⠀⡀⠚⠀⠀⡸⠿⣿⣍⢀⣠⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠏⣤⠈⠁⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣿⡷⣸⡷⣦⠀⠙⡇⠙⢁⠀⠁⠀⠀⡛⠀⠀⠈⢙⡉⠰⠇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠐⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠄⢀⠉⠁⠼⡟⠋⠀⠀⠀⠤⠤⠀⢀⠀⠰⠇⣀⠤⣁⡾⠋⢰⡂⠀⠀⠘⣤⢴⣶⣾⡆
⠀⠀⠀⠆⠰⠃⠀⠀⠀⡄⠀⠐⠶⠉⠠⠈⡡⠘⠃⢤⣈⠀⠀⢀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠶⠿⣿⡟⠃⠀⣍⠀⣽⠆⣠⠚⡁⠀⠀⠀⠙⠚⠛⠉⠤
⠀⠀⠀⣶⠀⠒⢠⡀⠀⠀⠁⠀⠀⢀⣀⣌⠣⣄⠀⣄⠋⠈⢉⣥⠀⢶⠞⠋⠀⣠⣤⣀⡀⢠⠒⣮⠁⢀⡴⠛⠻⣿⠗⣙⣲⡇⠘⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠉⠀⠀⠀⠄⠀⢤⢀⡤⠀⠙⢿⣿⠀⣀⠻⠖⠂⡴⠟⠃⠰⠂⠀⠋⡈⢀⣠⠞⣀⠞⢛⡁⠂⡈⣧⡀⡶⠛⢿⣏⠉⠠⠋⠰⠀⣹⠿⠿⡇⢀
⠀⠀⠀⠀⠀⠀⢁⡀⠠⡦⡉⠡⠀⠀⢠⠈⢠⠀⠀⠶⢋⡀⠸⠿⠗⠀⠤⠄⢀⡀⠀⣀⠉⠴⠀⢠⣀⣿⡿⢇⠰⢶⣌⠣⠀⠀⢠⣿⣾⣦⠘⠀⠠⠈
⠀⠀⠀⠀⠰⠀⠈⠁⢰⣦⣴⣄⡀⠘⠀⠘⠿⠗⠠⠄⡤⠴⠚⣾⣀⠀⠀⠐⠊⠒⠀⠉⠀⠀⠀⠹⣿⣿⠞⢦⠴⠀⠉⠄⡉⠁⠈⠛⠻⠋⠄
⠀⠀⠀⠚⠀⢄⠠⠘⣿⠿⢿⡟⢠⠧⢤⠀⠀⠤⠴⠀⠀⢀⠀⣿⡿⠀⠰⡖⠀⠀⠀⢠⠤⠀⡤⠄⢠⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⢾⣷⣶⣦⣼⠆⠀⠀⠀⢀⣤⣉⠀⠀⠀⠀⠠⠋⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣁⠈⣿⣥⡄⢀⠀⠁⠿⣿⠿⠄⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢉⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠛⠁⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⡤⠴⡶⠾⠿⣥⣒
⠀⢀⡠⠀⠀⠀⠙⠂⠈⠙⠷⡜⠀⠁⢰⠺⠟⢛⡋⠉⠉⠉⠛⠻⢶⣶⣶⣆⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠁⢀
⠀⠈⠀⠀⠔⣄⠀⠠⢀⣀⡂⠨⠀⠀⡾⣄⣘⣡⣤⠴⠰⠶⠛⣠⣆⠄⢀⢤⣤⣉⣙⠿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⠀⠔⠊⡠⣄⣈⣀⠚⠉⠉⣈⣈⣉⡋⣛⣿⣷⢶⣖⣊⡡⠄⠠⠝⢛⣭⠱⠖⢛⠛⠉⡹⠙⠻⠿⢿⡲⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⡁⠐⠛⠟⣭⣽⣯⡭⡉⠅⠈⠐⣞⣛⣀⡀⢀⢀⡀⠀⠁⠈⠙⠷⣤⡀⠀⠴⠀⠀⠀⠀⠀⠉⠁
⠀⠤⠄⢤⠶⠾⣿⣯⣤⠤⠖⣋⠩⠃⠀⢀⠀⠐⠛⣋⣉⡀⠐⠀⣵⠃⡀⠀⠀⢿⣶⣦⣍⡛⠶⣤⠀⢀
⠠⠬⠴⠖⠒⠛⠛⠛⠛⠿⠿⠿⠴⠟⠛⢉⣩⣥⢾⣯⠥⠤⠄⠤⠄⢀⠀⠈⠀⠁⠊⠙⠊⠙⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠠⡤⠀⣀⣐⣸⣭⣵⣖⠋⢁⣀⣰⠶⠾⠛⠛⠋⣉⣩⡦⠑⢃⢤⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠘⠼⠷⠶⠚⠉⠘⠛⠛⠀⠐
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠶⠤⠤⢤⣀⠀⢀⣀⡤⠖⠚⠉⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⢀⣠⣴⠾⣷⣾⡀⠀⠀⠀⠀⠀⠀⢀⣀⣽⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣀⣤⣶⠿⠛⠉⠀⠀⣿⠉⠻⣶⣄⠀⣀⣤⡶⠟⠉⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⠾⠛⣿⠉⠀⠀⠀⠀⠀⠀⡿⠀⠀⠙⣿⣿⡿⠋⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢻⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢀⡴⠚⢩⡟⠉⠉⠉⠛⠲⢤⣿⠀⠀⠀⣀⣤⣶⣿⣿⣦⣀⣾⡏⠀⠀⠀⠀⢀⡠⠾⣄⣀
⠀⠀⠀⠀⠀⡞⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣈⣿⣶⡿⠟⠋⠁⠀⠀⠀⠉⠻⣄⣀⣠⡴⠊⠁⠀⠀⠀⠈⠙⡄
⠀⠀⠀⠀⡸⠀⠀⠀⠈⢧⡀⣀⣤⣴⣶⠿⠟⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢠⣧⣄⣀⣀⣤⣾⡿⠿⠿⢦⣤⣄⣴⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⡞⠈⠻⣿⣿⠛⠁⠀⠀⠀⠀⢀⣽⣿⡇⢘⣧⣀⠀⠀⠀⣀⣀⣠⣴⣿⣽⣻⣶⣦⣤⣀⣀⣀⣠⠖⠉⢰
⠀⠀⠀⡇⠀⠀⠘⣿⡇⠀⠀⠀⠀⢀⣾⠋⢠⣧⠞⠀⣹⠟⠛⠉⠉⠉⠁⠀⣠⠟⠚⠳⣌⠈⠉⠙⡿⠋⠀⠀⡞
⠀⠀⠀⠳⣄⢀⣠⠼⢿⣟⠉⢉⠟⣛⣾⣶⣟⡙⣷⠞⠁⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠈⠳⢤⣼⡥⠴⠒⠚⠣⡀
⠀⠀⠀⠀⣨⠟⠦⠤⠤⠭⠿⢿⣿⣋⣁⣤⡼⠿⡿⠛⠓⠶⢤⣤⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⡅
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠁⠀⠀⡇⠀⠀⠀⠀⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃⡿⠀⠀⠀⠀⢰⠁
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⡇⠀⠀⠀⠀⢷⢸⣇⠀⠀⠀⠀⠀⠀⣼⠃⢠⡇⠀⠀⠀⢠⠏
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣼⣿⣷⣤⣤⣤⣤⣼⢋⣴⠋⣧⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⣿⣆⣠⣾⣿⠳⠶⠶⣟⣋⣯⡁⠀⠀⣈⣩⡿⠋⠉⠉⠉⠹⣄⡴⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⣤⡴⠟⠉⠀⠀⠙⠳⣶⣄⣀⡈⣙⡿⡿⠋⠁⣿⣷⣤⣤⣤⣤⠴⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⡏⢰⠃⠀⠀⣿⡏⠉⠉⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠘⣧⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⢀⡾⢦⡀⠀⠀⣿⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠋⠀⠀⠙⠦⣄⣻⡀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⢧⠈⠁⠒⢭⡦⢠⣍⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠢⡈⠓⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡠
⠀⠪⠀⠀⡀⠀⠈⠉⠙⠔⡤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⡝⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠔⢈⢀⣠⡤⠖⠋⠃
⠀⠀⢀⠌⠁⠀⠀⠀⢀⠜⡀⠇⠘⠸⢀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠀⢀⠀⡀⠀⠤⠀⢀⠀⠀⡒⠉⠛⠉⠁
⢀⠔⠉⠀⠀⠀⠀⣠⠊⢀⡘⢂⠀⠆⠀⠀⠁⠀⠉⡐⠦⠀⡀⠀⢠⠙⡀⠀⠀⠀⠈⠉⢤⢄⡂⠉⠑⠀⠀⡃⠀⠀⠀⠀⠀⠀⣔
⠀⠀⠘⠖⠀⠂⠤⠅⡰⣉⠄⠤⢡⡁⢀⢀⠀⠉⡉⡁⢓⡂⠈⣁⡘⢣⢰⡤⠄⠀⢺⣆⠦⢶⡉⠀⠠⠤⠄⠇⠂⠐⠀⡀⠚⢩⠏⠤⠉
⠀⠀⠒⢺⢆⠊⢀⠖⠀⠒⠘⠁⠛⠓⠆⠈⢥⢀⠉⠉⠉⠩⡀⠀⠀⠀⠆⢱⠀⠀⠀⠌⠉⠁⠛⠀⠒⠈⠁⠓⠀⠠⠬⣀⡀⡾⠀⡠⢢
⠀⠀⠀⠠⠻⡀⠀⠀⢀⠃⠀⢠⠀⠀⠀⠑⣀⠄⠈⠚⢴⡒⣤⠤⠄⠀⡈⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢋⠴⠊⠅⢉⠟⠢⢤⣈
⠀⠀⠀⡀⠰⢣⠠⡶⢆⢀⢠⡒⠀⠀⠊⠀⢀⡲⠔⠋⠁⣠⠽⠀⠀⠀⠀⠈⡆⠁⠀⠄⢤⠀⠀⠀⢤⠤⣏⠲⡧⠊⠀⠀⠈⠀⠀⠀⠈
⢀⠀⢠⢀⣦⠑⢃⠈⠟⠀⡈⣄⡶⠼⠚⠉⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢊⠁⣤⣭⣴⣇⢲⡄⣢⣦⣴⣶⠵⠯
⠀⡠⠠⠑⠙⠂⠀⣴⠤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⠴⠊⠁⢠⠞⠊⠀⢂⠨⠀⠐⠐⠉⢻⡁
⢁⣀⢄⡠⠔⠊⠉⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠠⠀⢀⣼⠚⠅⢀⡤⠔⠁⠀⠀⠀⠀⡆⠀⠀⠀⢸⣷⣀
⠓⠋⠁⠀⠀⠀⠀⠁⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡄⠀⢀⣀⡤⢿⡄⠐⡖⠞⠋⠀⠀⢀⠀⠀⡀⣼⠩⢄⠀⡐⢿⠒⠈⠙⢓⢀
⠀⠀⠀⠈⠀⠈⠰⠀⠀⢵⣀⠀⠄⠀⠀⠀⢠⠟⠀⠀⠀⠀⡄⠂⠀⢀⠘⣷⠄⠡⢄⡌⠆⠐⡜⠀⠄⡶⠃⠀⠨⠆⠀⠟⠀⠀⠀⠺⠛
⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠙⡀⠀⠀⣀⣠⠾⠠⢀⠰⠂⠀⢷⡀⠆⢁⣀⠠⠗⢯⡅⣌⠈⠞⠉⠀⠀⠀⠀⠀⣶⠄⢀⡆⠀⡀⠀⠠⣀
⡀⠀⠀⠈⠉⣁⡰⠆⠵⠤⠞⠙⢃⡘⠒⣀⡀⠒⠁⠀⣂⠔⠂⠁⠀⠀⠀⠀⠀⣿⢁⢛⡀⠘⠀⠐⠁⠀⠈⠀⠀⠈⢀⠁⠀⠄⠀⠈⠠
⢴⠘⠉⠉⠉⠈⡔⢒⠒⠉⠉⠹⣩⠩⠛⠡⠤⣀⡀⠊⠁⠀⠀⠄⠒⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠤⠐⡏⠀⠀⠀⡀⠄
⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐⠠⡪⠊⠁⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠄⠠⠚⠀⣀⣽⢠⡖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡤⠀⣴⡄⠀⠃⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠐⢂⠀⠆⠖⠐⠒⠻⢤⠴⠯⢥⠾⠀⣡⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⠚⠊⠁⠐⠀⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⠋
⠂⠈⠣⠀⠐⠂⠀⠄⠈⠶⡤⠧⠤⡤⠤⠦⠄⠭⣁⡈⠀⡀⠂⠀⠀⠀⢠⠴⠧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⣸⠁⣄
⣠⠔⢀⡀⠀⠀⠐⠈⠉⠓⣚⡆⠂⠂⠀⠁⣨⠵⠞⠋⠙⠒⠾⢣⣄⡴⢀⠀⡀⠀⠠⠄⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⡋
⠀⠱⡄⠉⠄⠀⠀⠀⠀⠀⣢⠄⠂⠘⠀⠀⢉⣀⣀⠤⠀⠀⠠⠐⠀⠉⠉⠚⠗⢦⣴⣀⡀⢤⠀⠀⠀⠈⡡⡴⡤⡂⠈⡄⡰⠁⢀⠴⠠
⠁⢀⠘⣀⠀⠀⠄⠀⠈⢀⣼⠤⠄⢒⠛⠉⠉⠽⠀⣀⠀⠀⠄⠀⠀⠀⠀⠐⠃⠀⠀⠈⡉⠟⠶⡾⢌⠠⠠⣀⢘⡃⠈⡟⠂⠠⡀
⢧⠀⠀⠀⠤⠀⠒⠵⠊⠀⣰⠀⠠⠂⠰⠀⠲⠜⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠁⠀⠀⠀⠀⠀⢄⠉⠀⠲⠌⣺⢎⡀⠀⠃⠀⠈

⠀⠀⠀⠀⠠⣐⡆⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢽⠽⠳⣠⣄⠁⣆⠀⠰⠀⠀⠀⠀⠀⡄⢸⢸⢿⠀⠀⣀⠄⢠⠀⠀⢠⠀⠀⠀⠠
⠀⠀⠈⠉⠈⢾⢿⣅⠀⠲⣤⣿⠰⡘⠀⠁⢦⡾⢄⢇⠀⠂⠇⠆⠀⣿⠁⠀⡀⠀⠳⢻⡈⠀⠀⢀⡄⠀⢄
⠀⠀⠀⠀⠀⠈⠋⠉⡳⣄⣿⣿⡌⠂⢠⢡⠇⡧⡏⡀⢭⢠⡼⠀⠀⡏⣴⠀⠃⠀⠰⡿⠀⠀⠰⡟⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠠⣴⣷⠈⠙⣧⠨⠀⡌⡻⠈⠤⡽⢘⠘⡬⢀⡇⠸⢱⣞⠉⠠⠂⢂⠄⠀⠀⢰⢅⡆
⠀⠀⠀⠀⠀⠀⠐⠺⢿⣶⡆⡆⠀⠀⡼⡓⠃⠐⠑⠈⠃⠣⡀⠱⡤⢻⣑⠘⠀⠇⡇⠀⢨⠀⠸⣜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⡑⣿⠔⣣⢿⠀⠀⡀⠀⠀⠰⢸⠑⠀⠀⡌⠓⢼⠗⠀⣤⠀⠈⠀⠘⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠎⣾⣿⡟⡟⢿⡘⠐⣣⡇⠰⡷⢾⣦⡀⠀⠈⠁⠠⡎⢤⠊⠀⠀⠀⠄⠃⢴⠖⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⣇⠒⠀⠦⠁⠀⢼⣅⠀⢰⠀⠉⠢⠷⠀⠀⠀⠆⠐⣀⠀⡀⠂⢸⢀⠚⢀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⡈⠕⠱⡘⡄⠀⠀⠀⠈⠀⠘⠃⠀⠀⢀⢃⠀⠘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⢣⠠⠀⠐⡀⣄⢘⡓⠀⢀⡐⢁⠀⠀⠀⠀⢀⠀⠀⢀⠆⠀⠈⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣃⣀⣀⣀⣀⣈⣁⣀⣄⣀⣀⣑⣂⣀⣐⣀⣈⣀⣀⣐⣂⣀⣀⣀⣁⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⢸⣿⣷⣄⡀⣀⣤⣤⣼⣿⠀⠂⢠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣤⠄⢈⣹⣿⣿⣿⣿⣿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣶⡄⢀⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣼⣿⠇⠀⠈⠈⣿⡟⣿⣿⣿⣿⣿⡇⠀⠙⣩⣈⠙⠛⠻⣿⣾⣿⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⠿⠿⢷⡄⠀⠉⠀⠘⢿⣿⣿⣿⣿⡀⢌⡛⣋⠸⣿⠆⡃⢻⡿⢻⣌⣡⡉
⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣷⣋⣤⡀⢴⡶⢿⣾⣿⣿⠓⠈⠀⠀⠉⠉⠃⠸⢿⣯⣷⣶⣾⣿⣿⣷⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢿⣋⣠⣤⣾⣿⠉⢉⡴⢦⣠⠜⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠟⣀⠉⡛⢿⣴⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠈⠿⢟⣿⡍⢸⣠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⢂⣴⣶⣿⣷⣾⣿⣷⠀⠀⠀⢹⣿⡏⠁⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣻⡭⣵⣿⡿⠿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⠙⣿⡿⠙⢀⡀⠀⣿⡏⠑⠂⠀⠒⠁
⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣍⠀⠀⠈⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠠⠄⣹⣷⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠘⠟⠉⠙⢿⣿⡟⠀⢀⣀⣼⣿⣧⠀⠀⣾⣿⠿⢷⣀⡀⠉⠀⠀⠀⠈⡟⣀⠈⢉⢸⡿⣦⣀⣀⡤
⠀⠀⠀⠀⠀⠀⢠⠀⠉⠀⠻⠀⢰⣿⣿⣿⠋⠉⡲⠼⡏⠀⠀⠀⢹⣿⣶⣿⣿⣿⣷⣧⣠⢦⣈⡼⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠐⢾⣿⣿⡟⠐⠀⣙⣛⣉⣁⣀⠉⣈⣿⠀⠀⢹⣿⣿⣿⠟⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⣿⡏⠀⢠⣭⣭⣽⣿⣿⣿⣻⣿⣧⡀⠀⢻⡟⠉⠀⠀⠈⠁⠀⠀⠀⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣤⣴⡶⠆⠒⠂⣤⠄⠛⠛⠿⣿⣇⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⣽⡀⢠⣶⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠐⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠤⠤⠄⠀⠤⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠀⠁⠁⣀⣀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋
⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠑⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠱⠊⠀⠀⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠐
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠤⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠒⠤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠀⠀⠀⢀⡀⠀⠀⣠⠖⠀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠤
⠀⢀⠀⢀⠰⠤⠢⠤⠄⠂⡉⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡞⠉⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⢠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠆⠀⢂⡀⠂⠀⠀⠀⠘⡤⡏⠀⠁⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠙⢀⠀⠀⡀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠈⡅⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢩⠀⢀⠀⠀⠀⠀⡈⠀⠀⢀⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢠⠊⠈⠉⠀⠀⠁⠀⠀⠃⠉⠁⠠⠁⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⡀⠤⠄⠀⠀⠀⠀⠉⠐⠄⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡠⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⡀⠤⠒⠋⠉⠉⠉⠉⠙⠒⠀⠠⠀⠀⠀⠀⠈⠀⠀⠁⠁⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠂⠀⢀⡀⠀⠠⠤⠤⠀⠁⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠆⠈⠁⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠄⠀⠤⠤⠄⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢠⠆
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣄⣰⣿⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠞⠋⠉⠀⠀⠉⠙⠳⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠘⣧⣴⠄⠀⠀⠀⢀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡇⠀⠀⢀⣤⣼⣧⣤⡀⠀⠀⢸⣇⡀⢀⣤⣀⣿⠷⠶⠿⠶⣾⣟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣸⠋⣡⣀⡽⠋⠀⠀⣠⣤⡀⠀⠈⠻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣆⠀⠸⠿⠿⠿⠿⠇⠀⣰⠟⠿⠉⡿⠁⠀⠀⠀⠻⡿⠃⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⢻⣦⣤⣤⣠⣤⣴⠛⠿⠆⠘⠻⡇⠀⠀⢠⣾⣿⣿⣿⡆⠀⠀⢈⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠙⢳⣦⠿⣣⡈⠛⠀⠀⠀⠀⣶⣿⡀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡴⠟⠛⠛⠛⠛⠿⣇⣠⣄⠀⠀⠀⠀⣠⣿⣤⡈⠋⠛⠛⠛⠃⣠⣾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⠋⠀⠀⢠⣶⣦⠀⠀⠈⢻⣅⣀⠀⠀⠀⠈⠀⢸⡿⠛⣶⡶⠺⣿⡁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⣈⣛⣋⡀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⣸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢷⡀⠀⢸⣿⣿⣿⣿⡇⠀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢦⣈⡈⠉⠉⣈⣠⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠙⣿⠋⠹⠷⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠿⠿⠿⠷⠖⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠰⠶⠂⠀⣤⢦⠀⠀⠐⠒⠀⠀⢀⡤⢄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡖⠆⠀⠀⠀⠀⠀⠀⣠⢦⠀⠀⠐⠂⠀⢠⠲
⠀⢠⣦⠀⠀⠁⠈⠀⠀⡀⠀⠀⠀⠈⠁⠘⠀⠀⠈⠀⠀⠀⣠⠄⠀⣄⣤⠀⠘⠁⠀⠀⢹⠃⠀⠀⠠⠀⠀⠀⠻⡤⠀⠀⠃⠀⠀⠘⠿⠂⠀⣶⡆⠀⢰⡆
⠀⠈⠇⠀⠀⠀⠀⠀⢀⣛⣘⡀⠀⠀⠀⢀⠀⠀⠀⢠⣄⠀⠘⠂⠀⠙⠁⠀⠀⣁⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡆⠀⣼⣧
⠀⠀⠀⠀⠀⠴⡶⠄⠸⡿⢹⡇⠀⠀⠀⠄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠋⠙⠁⢸⡇⣿⠀⠀⠁⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⣐⠀⠆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠙⠀⣀⠀⠂⠐⣀⡀⠀⠀⠀⠘⠃⠙⠀⢀⠀⠠⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠸⠋⠀⠀⠀⡈⠉⠁⠀⠀⠐⠄⠀⠀⠸⠙⠂⠀⠀⠀⠀⠀⠉⠁⠀⠘⢏⡹⠀⠀⣴⠋⢹⠀⠀⠉⣷⠀⠸⠓⠀⠀⢛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠛⠗
⠀⣀⠀⠀⣿⣿⣿⣷⣀⠀⠀⠀⠀⠰⠿⠂⠀⠀⣘⣁⠀⠠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠺⣿⠇⠀⢸⡇⠀⣾⣶⡂⠀⢀⣀⠀⠀⠀⠠
⠀⢀⡿⠀⢻⠟⠛⠛⠋⠀⣾⣶⣖⠀⠀⠀⠀⠀⣿⣿⠀⠈⠁⢰⠄⠀⣾⣿⠀⠀⣤⣤⣤⠀⠀⠒⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠰⢂
⠀⠀⠀⠀⠑⠋⠀⢀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠈⠀⠀⠟⠸⠀⠀⠙⠋⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣺⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣿⠿⣿⢿⣷⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠀⠀⠀⢀⣤⣤⣶⣿⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣿⠟⠁⣠⣧⡀⠙⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣭⠶⠷⢮⣷⣄⡀⢨⣿⣿⠟⠉⢉⡟⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣷⣶⣶⠸⢿⣿⡟⠛⣿⣿⡟⠛⣻⣿⠿⠁⣤⣀⣤⡄⠀⠘⣏⣟⠙⢦⣴⠋⢹⣼⢡⣾⣿⡏⠙⢲⣿⣄⣀⣿⣿⡿⠇
⠀⠀⠀⠀⠀⢀⣰⣿⠯⣤⣞⠹⣿⣦⢀⣿⣷⣦⣀⣇⣠⣶⣿⣇⢠⣾⡿⡛⣻⣿⣶⢄⣦⢷⣴⠋⠉⢢⣼⣹⡏⠹⣿⣿⣄⣼⠁⢀⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣦⣜⣉⣽⣿⠃⠈⣿⠋⠛⣿⣿⡟⠋⢻⡏⢸⣿⣤⠛⢣⣼⣿⠀⣤⠙⣭⡛⠟⣭⡟⣡⡄⠀⢾⡿⣿⣿⣿⣿⡿⠟⠟
⠀⠀⠀⠀⠀⠀⢶⡟⠛⢿⡿⠛⢹⡷⠀⠸⣇⠀⣨⣭⡀⢀⡾⠀⢶⡉⢿⠿⠿⠟⢀⡖⠹⣇⠈⣤⣤⡀⢀⡿⠀⠀⠐⣆⠀⠈⠛⠋⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠈⢧⢰⣿⣿⡄⡼⠁⠀⠀⠹⣆⠿⠿⢇⡾⠁⠀⠈⣧⢰⣿⣿⠄⡾⠁⠀⢹⡌⠿⠿⢃⡾⠁⠀⠀⠀⠹⣆⢼⣿⣷⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⠀⢻⣷⣶⣿⠃⠀⠀⠀⠘⣧⣭⣭⣼⠃⠀⠀⠀⢻⣷⣶⣿⠁⠀⠀⠀⠀⠀⢹⣬⣭⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠀⠸⣿⢿⡏⠀⠀⠀⠀⠀⢸⡿⣿⡇⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢁⡴⠖⠲⣄⠀⠖⠒⢦⡀⠐⣶⠒⢶⡖⠲⣆⠰⣖⠀⢲⡖⠀⣲⠆⣠⠖⠲⢦⡀⢲⡶⠖⠀⡇⣰⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣟⠚⠒⠛⢰⡏⠉⣻⡇⠀⡇⠀⢸⠇⠀⡿⠀⠹⣤⡟⢹⣰⠏⠀⣧⠀⠀⣸⠇⢸⡇⠀⠀⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠉⠛⠛⠁⠈⠙⠛⠛⠃⠘⠛⠀⠛⠃⠘⠛⠀⠀⠙⠀⠀⠋⠀⠀⠈⠛⠛⠉⠀⠛⠃⠀⠘⠛⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡛⠛⠉⠁⣀⠀⠀⠀⢀⣴⣷⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡶⢻⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠛⠛⡟⣿⣿⠟⠛⢿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⣶⣶⡀⠀⠙⠘⠿⢿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣶⣷⠉⠈⠛⢿⣿⣿⠋⠉⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣘⣿⠿⠛⠁⠀⠀⠀⠈⢿⡇⠀⠀⠀⢠⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡾⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⢺⡇⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁


⠀⠀⠀⢀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢀⡀
⠀⣀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⠇⠀⠑⣄⠀⠀⠀⠸⣟⡋⠀⠀⠀⠀⢦⡀⠀⠀⠀⠋⠚⢃⣀⣾⣿⣶⡀
⢠⣿⡿⠻⣿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠈⢿⡛⠃⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣿⣿⣦⡄⠈⠁⠀⠀⠀⢠⠶⣿⣿⣿⣿⣿⣷
⠈⣟⠀⠀⠁⠀⠍⠤⠀⡄⠀⠀⠀⠀⠀⠀⠈⣰⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠁⠀⠀⠀⠀⢸⣿⣷⣶⣿⡿⠛⠟⠩
⠀⠉⠀⠀⠀⠀⠀⠴⠾⠆⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠚⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⡀⠀⠀⢰⣿⡧⠀⠀⠀⠀⠀⠀⢾⣿⡆⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⡇
⣠⣤⣶⣤⣠⣴⣿⣶⣄⣠⣴⣤⣄⣠⣶⣿⣦⣀⣤⣶⣤⣄⣤⣾⣷⣦⣀⣤⣶⣤⣀⣴⣾⣷⣄⣠⣴⣦⣤⣠⣴⣿⣶⣄⣤⣶⣦
⠉⣿⣿⡍⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⢹⣿⣯⠉⠉⠉⠉⠉⠉⢹⣿⡏⠉⠉⠉⠉⠉⠉⣿⣿⡍⠉⠉⠉⠉⠉⠩⣿⣿
⣠⣼⣿⣤⠀⣠⣤⣤⡀⣠⣽⣯⣄⢀⣤⣥⣄⢀⣤⣿⣥⡀⢀⣤⣤⣀⢀⣬⣿⣤⡀⣠⣤⣤⡀⣠⣼⣯⣄⢀⣠⣤⣄⡀⣠⣽⣧
⠉⠛⠛⠋⠉⢹⣿⡏⠉⠉⠛⠋⠉⠉⣹⣿⡍⠉⠉⠛⠋⠉⠉⣽⣿⠉⠉⠙⠛⠉⠉⢉⣿⣯⠉⠉⠙⠛⠉⠉⢹⣿⡏⠉⠉⠛⠛
⢀⣀⣄⡀⠀⣨⣿⣅⡀⢀⣈⣀⡀⢀⣨⣿⣅⠀⣀⣁⣀⠀⢀⣽⣯⣀⠀⣀⣈⣀⠀⣠⣽⣯⡀⢀⣈⣈⡀⢀⣨⣿⣅⡀⢀⣀⣁
⠛⣻⣿⡛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⢛⣿⣟⠓⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⣻⣿⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⣙⣟⡁⠀⢀⣐⡀⠀⢀⣻⣟⡀⠀⢀⣂⠀⠀⣈⣻⣋⠀⠀⣀⣁⠀⠀⣘⣿⣃⠀⢀⣐⣀⠀⢀⣹⣟⡁⠀⢀⣀⡀⠀⢀⣻⣟
⠿⠿⠿⠿⠖⢻⣿⡟⠓⠿⠿⠿⠿⠚⢻⣿⡛⠺⠿⠿⠿⠷⠚⣿⣿⠓⠺⠿⠿⠿⠗⠛⣿⡟⠓⠿⠿⠿⠿⠖⢻⣿⡟⠲⠿⠿⠿
⠀⠀⠀⠀⠀⠘⣿⡃⠀⠀⠐⠀⠀⠀⢹⣿⠃⠀⠀⠀⠂⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⡃
⠶⢿⣿⠷⠴⠿⠿⠿⠧⠾⣿⣿⠷⠾⠿⠿⠿⠦⠿⣿⡿⠦⠾⠿⠿⠷⠴⢿⣿⠿⠶⠿⠿⠿⠷⠾⢿⣿⠷⠴⠿⠿⠿⠦⠾⣿⡿
⠀⠻⡿⠃⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠘⢿⠗⠀⠀⠠⠄⠀⠀⠸⣿⠇⠀⠀⠀⠄⠀⠀⢿⡿⠂⠀⠀⠠⠀⠀⠐⢿⡿
⣴⣿⣿⣷⣤⣶⣿⣶⣤⣾⣿⣿⣶⣴⢾⣿⣶⣤⣾⣿⣿⣦⣴⣾⣿⣦⣴⣿⣿⣷⣦⣶⣿⣷⣦⣴⣿⣿⣷⣤⣶⣿⣶⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠀⠀⠀⠸⣿⡇
⣠⣶⣶⣦⣠⣶⣿⣶⣄⣴⣶⣶⣤⣰⣶⣿⣶⣄⣴⣶⣶⣄⣴⣾⣷⣦⣠⣶⣶⣦⣠⣶⣾⣷⣦⣠⣶⣶⣦⣠⣶⣿⣶⣄⣴⣶⣶
⠀⣾⣿⡆⠀⠀⣁⠀⠀⠀⣿⣿⠀⠀⠀⡈⠀⠀⢰⣿⡧⠀⠀⠈⠁⠀⠀⢼⣿⡆⠀⠀⢁⡀⠀⠀⣾⣿⠄⠀⠀⠀⠀⠀⠠⣿⣿
⣠⣾⣿⣦⣀⣤⣶⣤⣀⣤⣾⣷⣄⣠⣤⣦⣤⣀⣴⣿⣶⣄⣠⣴⣦⣄⣠⣴⣿⣦⣀⣤⣴⣤⣄⣠⣾⣷⣦⣀⣤⣴⣤⣀⣴⣾⣷
⡿⠛⢻⠿⡿⠿⠿⢿⠿⠿⢿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣤⣶⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⡀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣿⣿⣿⣯⡀⠀⠀⠀⢻⡗⠀⠀⠀⡀⠈⣻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣟⠀⠀⢾⡷⢾⣿⣶⣿⠚⠿⠚⢿⣿⣿⠟⠀⠀⠀⠀⣤⣶⣶⣄
⠀⠀⠀⣁⠀⠀⠀⣠⣴⣶⣤⣿⣿⡿⣿⠟⠻⠿⠃⠀⠀⠀⢴⣿⠿⣿⠆⠀⠐⢾⡟⠡⣿⣦⡀⠀⢸⠏⠉⠛⠻⠄
⠀⠀⠀⠙⣷⡀⢸⡿⠋⠉⠉⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠈⢷⣦⡻⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣴⣦⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣧⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠻⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣤⡄
⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⣤⡉⠁
⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣼⣿⣧⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣯⣬⣿⣿⣧
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠈⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠓⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠲⠤⢄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠄⠀⠀⠀⠀⠀⣔⣠⡀⣣⣠⠤⠤⠠⠀⠀⠐⠂⢀⣤⠔⠂⢑⣝⠗⡂⢤⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣎⠡⣼⢶⣵⠟⠈⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⢀⠎⣠⠞⡁⢈⡌⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⣆⣀⣴⠯⡓⡾⠋⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠻⠂⢡⠎⠐⣏⢡⣣⢄⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣈⠹⢫⡱⠊⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠰⠰⡜⣘⢇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣠⠄⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢄⠀⠀⠀⢼⠀⣜⢬⣃⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣀⠴⣈⣰⣥⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠊⣴⢭⣶⣄⠀⢘⠀⡅⠻⠄⠀⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⠁⠀⠀⠀⢈⠵⢎⠡⣾⡆⣣⡞⠀⠀⠀⠀⠀⠀⠀⠠⣷⣭⣾⣴⠙⣮⠙⠈⢣⣠⢶⣁⠀⠀⠀⠀⢠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢤⡇⠀⠀⠀⠀⡎⠀⠣⣀⡭⠓⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠯⣍⣀⣠⠇⠀⠈⡝⢇⠍⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⠈⣾⠀⠈⢸⠁⠀⠀⠀⠀⠈⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣈⣃⠀⠀⠀⠈⠁⣐⠆
⠀⠀⠀⠀⠀⣨⠛⢻⠀⠀⢤⠀⠀⠀⢀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⡔⡋⠰
⠀⠀⠀⠀⠀⠘⠶⣦⠠⡃⣾⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⢸⠮⠘⢤
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⢸⡄⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟⢈
⠀⠀⠀⣀⡀⠤⢠⠀⣤⠌⠈⢧⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠖⡥⠊
⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠴⠛⠛⠛⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⡔⣭⣚⠧⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠙⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⠥⠀⠀⠠⠀⠔⠀⠹⡅⠀⠀⠀⢂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⡛⣛⠇⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠘⣤⡠⠀⠛⠃⠀⠀⠀⠠⢄⣀⡘⢓⠛⠉⠉⠁⢀⡵⡄⠴⠁⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠐⠉⠀⣤⢚⢹⠀⠀⠀⠰⠠⢀⡄⠀⢀⣀⣉⣙⣻⣧⣴⡿⠁⣰⢷⠈⠉⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠂⠀⠉⠥⣀⣮⣄⣀⢠⠖⠀⣁⣠⡴⠛⢋⠀⠐⠋⣗⣄⡀⠰⢇⣠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠁⡂⡉⠀⠾⠞⠚⠃⠈⠉⠉⠉⠁⠈⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⠶⠒⠲⠦⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⣠⠏⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣸⠆⠀⠀⠀⣠⠴⠶⢦⣄⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠿⠛⠓⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠁⠀⠀⠀⣼⠁⠀⠀⠀⠈⢧⣀⣀⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣼⠇⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⢀⡾⠉⠉⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⡏⠀⣹⡂⠀⣠⠖⠋⠉⠉⠉⠙⠲⣄⣀⣤⡾⠟⠋⠛⠶⠦⠶⠋⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⡴⠋⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠙⠊⠉⠛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢰⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⠤⠖⠋⠉⠁⠀⠀⠀⠀⠉⠙⠛⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣾⡁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣦⣀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⡴⠋⠀⠉⠻⣗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⣶⠛⠉⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⡴⠋⠁⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⣤⣤⡾⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠹⣄⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⠛⢦⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⠤⠞⠁⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠓⠦⠤⠤⠤⠤⠤⠤⠤⠖⠚⠉⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⢱⣤⡶⠶⢷⡀⠀⠀⠀⠀⢀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠋⠀⠀⠀⠀⠙⠲⠦⠤⠖⠋⠀⠙⠿⣦⡤⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡀⢀⣶⣶⣶⣶⣶⣶⠒⠒⠒⠒⠒⠋⠀⠀⠙⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⣠⣀⡀⢸⣿⣿⣿⣿⣿⣧⣀⣀⣼⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣷⡀⠀⠀⢠⡞⠉⠙⠓⠚⠁
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⡿⠁⠀⠀⠘⢧⣀⣠⠴⢦⡄
⠀⠀⠀⠀⠈⠉⠉⠁⢸⣿⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠛⣤⣤⣤⣤⣤⡄⠀⠀⢠⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⣿⣿⡇⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠖⠋⠉⢳⡄⠀⠀⢀⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣄⣀⡼⠃⠀⠀⠈⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⡟⠉⠉⢻⣿⣿⣿⣿⣿⡇⠈⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⣄⠀⠀⣠⠤⠤⠤⠤⠤⠿⠿⠿⠿⠿⠿⠁⠈⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠿⣯⠉⢉⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⣨⢷⡋⠁⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣷⣦⡴⢿⣿⣁⣀⣻⣾⣿⢡⣤⣼⣷⣴⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣡⡜⢻⣿⡎⠉⠛⠻⠋⠋⢠⣶⡿⣫⣽⣭⡹⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⡄⣾⣿⣿⡠⢻⠃⣴⣿⠇⠀⠀⠀⠀⠀⢠⣿⡇⢿⣯⣽⡿⣹⣿⡁⣠⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⣱⡿⢋⣿⣿⣿⡿⡟⣯⣴⣦⣴⣀⡀⠀⠀⢹⣿⣦⣭⣯⣷⣿⣿⣇⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⡿⠋⠀⠀⣀⣿⢻⣧⡶⣾⣿⣷⡆⠀⠀⠀⠛⠉⠻⠻⣿⣬⣿⣷⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢼⡇⢾⣏⠶⣩⣷⣻⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣆⠀⠀⠀⠀⠀⠐⢿⣾⡛⠻⣛⣷⣿⠉⢀⡀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠈⣿⠃⠀⣀⣀⣶⣤⣼⡍⠛⠻⠏⣛⣸⣷⣾⠿⢿⣿⣶⡆⠀⠀⢰⡿⠋⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⢰⡿⢠⣤⣿⣿⣿⣿⣿⣿⡟⠀⣀⣿⡿⠻⣷⡀⠀⣨⣿⣿⣿⠀⢸⣿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⣶⣿⣶⣿⣤⡄⢠⣿⣿⣿⠁⠀⣿⣿⣿⠟⣹⣿⠁⠀⣈⣷⡾⠋⠁⢹⣿⠗⠈⠛⣠⣀⣌⢛
⠀⠀⠀⠀⠀⠀⠀⠠⣴⡿⠋⠀⠀⠀⠈⢻⣷⡆⣹⣿⣿⣷⣾⣿⣿⠿⠃⢙⣿⣶⡿⠋⠁⢻⣦⣀⣾⡿⠦⣤⡿⠛⠛⠛⢿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⡀⠀⠀⠀⠀⠀⣿⡷⠀⠈⠿⠛⣿⠛⠛⠀⠀⠈⠙⣿⣷⣦⣤⣤⣿⣿⠟⠇⢺⣿⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⣴⣿⣿⣦⢿⣿⣇⠀⠀⠀⣀⣼⡿⠇⢻⣧⢀⣴⣶⣄⠀⣿⠇⠀⠀⠁⠈⠛⠙⣿⡏⢩⣴⣦⣝⣿⣄⢀⣄⠀⣠⣿⡋
⠀⠀⠀⠀⠻⢿⡿⢋⣼⣿⢻⣿⣿⡿⣿⠛⠋⠀⢸⣿⠸⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⡿⠈⢿⣿⠿⡿⠛⠃
⠀⠀⠀⠀⢠⣶⣶⣾⣿⣥⣾⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⣬⣭⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣭⣭⣤⣶⡿⠃
⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢀⣿⠇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿
⠀⠀⠀⣼⡿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠙⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠘⠟

⣿⡟⠉⠈⢻⣷⡆⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢀⣤⣤⠀⠀⠀⠀⣸⣿⠋⠁⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢠⣤⣄
⣿⡄⠀⠀⢈⣿⡏⠀⠀⠀⠀⠀⠘⠿⣿⣿⡶⣾⣿⣿⣷⣶⠶⣤⣿⣿⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠸⢿⣿⣿⢶⣾⣿⣿⣶⣶⢶⣼
⣿⣷⠀⠀⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠈⠟⠙⠀⠀⠀⠹⢿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠸⠏⠁⠀⠀⠂⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠈⠉⢱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡛⠛⠻⣧
⣾⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠘⣳⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣿⠀⠀⠀⠀⠘
⡏⠂⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠘⠟⠁⠀⠀⠀⠘⢿⠇⠀⠐⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠚⠛⠀⠀⠀⠀⠘
⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠈

⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣾⡿⠟⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣾⡿⠟⠻⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⣿⠃⠀⠀⠸⣿⡷⠀⠀⠀⠀⠀⠠⣾⣿⣦⣤⣼⣿⣿⣦⣤⣤⣀⣾⣿⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⣼⣿⣿⣤⣤⣄⣰
⣿⣧⠀⠀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠉⡿⠛⠉⠹⡿⠛⠉⠉⣉⢻⣿⣿⣄⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢹⡟⠋⠉⢻⡟⠏⡉⠉⡉⣿
⠙⠋⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠟⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣿⣆
⣴⣾⣿⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡆⠀⠀⠀⠠⢤⣴⣾⣷⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠁⠀⠁⣿⡀⠀⠀⠀⠤
⡟⡋⠛⢻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠠⣿⠇⠀⠀⠀⠠⣿⡟⠉⠛⢻⡿⠉⠀⠀⠀⠀⠀⠀⠀⠹⣧⠄⠀⢴⡿⠀⠀⠀⠀⢰
⠇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠈⠁


⣤⣶⣶⣶⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣟⣙⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠝⠁
⠀⠀⠀⢰⣷⠄⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠟⠀⢀⡀⠀⠀⠠⠆⠀⠈⠛⠋⠀⠀⠿⠇⠀⠀⠀⠀⠀⠈⠛⠁⠀⢰⣶⠄
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⣠⣴⣄
⠀⠀⠀⠀⠀⠸⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠒⠂⠈⠋⡀⠀⢀⠖⠒⠂⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡠⠖⠉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠊⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢨⠠⠄⠀⠀⠀⠀⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠠⠀⠀⠀⠈⠁⠀⠆⠀⠀⠀⠀⡔⠒⡀⠀⠀⠀⠀⡈⠈⡃⠀⠀⠀⠠⠢⠀⠀⡄
⠠⠀⠀⢀⡆⠀⠀⠀⢀⠀⠀⠤⠦⠤⠤⠤⠄⣀⡀⠀⠀⡁⠀⠀⠐⠀⠿⠛⠁⠀⠀⠀⠀⠟⠧⠃⠀⠀⠈⠁⠀⠀⠀⠰
⠀⠀⢀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡔⠘⠀⠀⠀⠀⠀⠩⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡦⣄
⠀⠀⠀⠀⢀⠎⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⠄⠁⠈⢦⠀⢠
⠀⠀⠀⠀⠀⠑⠢⣄⠀⠀⡠⢒⡆⠄⠤⡠⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠤⠄⠀⠠⠀⠀⠈⢢⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⠘⠒⠚⠋⡠⢳⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⢇
⠀⠀⠀⡐⡇⠀⠀⠀⠀⠒⡡⠀⠀⢀⠄⡰⠊⠀⠈⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢠⠀⢈⠐⠀⠀⠈⠄
⠀⠀⠀⠈⠀⡐⠀⢀⡴⠊⠀⠀⢠⠃⠀⢱⠀⠀⠀⠀⠀⠈⠂⣄⠀⠀⠀⠀⠀⠘⠂⠀⠐⠃⠀⠀⠠⠠⡀⠀⡀⠀⠀⡮⠖⠁⠀⠀⡠⠒
⠀⠀⠀⠀⠀⠃⠀⠊⠀⠀⠀⠠⠋⣆⠀⠅⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡠⠄⠒⠉⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠔⠁⠀⠈⠢⠀⢇⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠌⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠒⠀⠒⠈⠀⠀⠀⠀⠀⠀⠑⠦⡉⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠴⠀⠀⠐⠤⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠈⠐⠪⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠋⠁⠀⠠
⣀⣀⣀⣡⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣒⣒⣂⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣴⠀⠀⠀⠰⢀⠀⠀⠀⢈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠐⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⣒⠀⠀⢰⣘⢗⣀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⡀⠀⣾⣤⣼⡏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠞⠛⢠⣿⣻⠆⠰⠄⠘⣻⣶⣶⠠⠂⠀⠴⡂⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣧⣤⣾⣧⣾⣭⣯⣥⣦⣬⣽⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠿⣿⣶⣷⠄⠀⣠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣡⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠆⠀⠀⠊
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⢻⣿⣿⣧⠀⢀⠘⢋⣿⣿⣿⣿⣿⠃⠀⢻⣧⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠈⡃
⠀⠀⠀⠀⠀⣸⣿⠿⣿⣿⣿⢸⣿⣿⣿⣾⡿⡷⠟⠛⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠸⠄⠀⢠⡃⠀⠠
⠀⠀⠀⠀⠀⣿⣿⣤⣿⣯⣤⠈⣿⣿⠙⠋⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠙⣋⣿⣿⣿⠀⡿⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡯⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⡗⠿⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠸⡷⠾⠿⠿⢿⡇⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠛⠛⣿⣷⣦⣤⣀⣈⠄⠇⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣤⣴⣶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⢿⣿⡿⠁⠈⠈⣗⣀⣤⣴⣾⣧⣤⣤⠤⠤⠀⠐⢲⠒⠂⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⠟⠁⣼⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡥⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⠙⠿⣿⣻⡿⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡃⠀⠀⣿⡉⠀⠀⠀⠀⠿⠿⠿⠿⢿⡟⠋⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⡆⠀⡸⣿⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣿⣴⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠇⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⢀⣸⣿⡇⢠⠀⠀⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠙⠿⢿⣷⣶⣴⣶⣿⣿⣿⣿⣧⣀⣠⣾⣭⣿⣤⣤⣀⡀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠻⣿⣿⡏⠀⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠿⣿⡿⠿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣶⣿⡟⠁⠀⠀⠀⣾⡟⠹⣿⣿⣿⣿⣿⣿⡀⠀⢋⣴⣤⠀⣉⡉⢹⡟⣿⣿⣿⡟⣉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⠿⠿⠿⢷⡄⠀⠉⠀⠀⠻⣿⣿⣿⣿⣿⣧⠀⣌⣛⣋⡘⠿⠿⢘⠁⢿⡿⠙⣧⣉⣤⡉
⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣶⣋⣠⡄⠠⣶⣶⣷⣾⣿⣿⡿⠂⠈⠀⠀⠉⠉⠙⠀⢹⣿⣭⣿⣶⣶⣿⣿⣿⣷⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⣀⣢⣽⣿⡛⠛⢁⣤⡀⢀⠸⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢸⣿⡿⠉⠛⠿⢿⣇⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣶⣿⣿⡿⠛⢹⠀⢶⣿⣶⣿⠏⠀⠀⠀⠀⠀⠀⠤⠀⠀⣤⠀⣀⣸⣶⣤⣤⣴⣶⡅⠘⠃⠀⢲⣼⣿⠟⠛⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠾⠿⣣⣾⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⡿⣀⠀⠀⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣿⣣⣿⡿⠟⠉⠹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠀⠈⣿⣿⢀⣤⣤⣤⣀⣿⣷⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢀⣀⣀⠀⠀⠀⢣⡀⠀⠒⠀⢼⢿⡿⢿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠹⢿⣿⠏⠀⠀⣀⣠⣿⣿⣧⠀⠀⣴⣿⡿⠿⢷⣀⣀⠀⠀⠀⡀⠀⠈⣼⢠⡀⠀⣀⢸⡟⣧⣀⣀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⠘⢿⠀⢠⣿⣿⣿⡿⠛⠉⠶⣤⣿⠁⠀⠀⠀⢹⣿⣧⣾⣿⣿⣿⣷⣾⣀⡤⣤⢉⡼⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠀⠀⠠⢶⣿⣿⣿⠇⠐⠀⠓⠶⠆⠒⠀⠀⠒⠀⣿⡏⠉⠉⣻⣿⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠛⠛⠛⣶⣟⣷⣶⡾⣿⣇⠀⠀⠋⣿⣿⠟⠃⠀⠀⠹⠀⠀⠀⠀⠀⠶⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⢘⠛⠛⠛⣛⠛⠛⠋⣾⣿⣿⣷⣤⠄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣶⣿⣏⠉⠉⠀⠐⣶⠂⠈⠉⣉⣿⣿⣦⣤⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠸⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⠛⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢸⡃⠈⡷⠴⡏⠀⢹⠀⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⠀⠙⠛⠁⠀⠉⠛⠉⢀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⡀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⡟⠒⠶⠶⠶⠚⡏⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣻⠇⠀⢻⠉⣻⠿⡏⢹⠃⠀⢿⡳⠤⣄⣀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢴⡃⠀⠀⠈⣇⢳⣶⢋⡏⠀⠀⠀⣹⠄⣰⠋⠳⡄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦⣀⠀⢹⡞⠘⣿⠀⠀⣠⠞⠁⢠⡇⣾⠁⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢷⢰⢇⡴⠋⠁⠀⠀⠘⡇⠉⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠇⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣤⠴⠞⠙⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣄⣽⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣇⣀⣷⣶⣿⣿⠛⠛⣻⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⣿⣿⡙⠀⠀⠈⠁⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠉⠽⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠅⠸⠿⠿⣿⣇⠀⣤⡄⠀⢠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣦⣾⣿⡇⠀⠀⠀⠀⠀⢻⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣿⣀⣾⣶⣿⣿⠿⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⠛⠻⠿⠿⠉⠿⠿⣿⡏⠁⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣇⣠⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⠷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⡷⠀⣀⡀⠀⠀⣤⣿⣿⣿⣿⣿⡏⠉⠉⣿⠏⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠇⠈⢱⣶⣿⣷⣶⣶⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠐⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣶⡆⠀⣸⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠂
⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⡀⠂⠀⠁⠖⠒⠛⠛⠛⠟⠁
⠀⠀⠠⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠐⠻⢦⠀⠀⠀⠀⠀⠀⠑⠂⠄
⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡤⠐⢤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⡻⠀⢀⡞⠀⡠⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢾⡄⡔⠀⢻⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡿⠃⠀⠀⠈⣧⢁⠀⠀⢻⡞⡆⠀⠀⣠⣤⣶⣦⣤⡤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢦⠀⠀⢀⡤⡞⠋⠀⠀⠀⠀⠀⠀⢸⡆⡆⠀⠀⣷⣷⣶⣾⠏⠙⢩⣿⠟⠟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠸⣶⡺⠟⠉⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣶⡶⣿⣿⡟⠿⠻⣷⠀⠈⠳⣌⢂⠀⠉⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⢸⣁⣀⢀⣀⠀⠀⠀⠀⣰⡿⠿⠊⠋⢻⣿⣿⢸⣿⢂⢀⢡⡀⠀⠀⠈⠈⢢⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠺⠛⠛⠉⠁⠀⢀⣤⣼⣯⡀⠀⠐⠀⡀⢻⣟⣾⠻⣧⠠⠀⠡⠡⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠈⠀⣆⠀⠀⠀⠀⢀⣼⣿⡿⣿⣷⣄⠀⠀⠀⠈⠉⢉⠣⡘⣧⠠⠠⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⠤⢤⣌⢳⠀⠀⠀⣾⣿⠉⠀⠉⢿⣝⣆⠀⠀⠀⠀⠀⠀⠁⠘⢧
⠀⠀⠀⠀⠀⠀⠀⢠⢟⡆⠀⢰⣏⡏⠀⢀⣴⣿⡉⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⢀⡟⣸⠀⢠⡟⣸⢀⣶⡿⡟⡩⢷⡱
⠀⠀⠀⠀⠀⠀⣸⢣⠃⠀⠈⣹⣷⣾⣿⡄⠘⡓⡀⠧
⠀⠀⠀⠀⢀⣤⣿⣶⣶⣶⣿⣿⡿⠛⣿⠙⡄
⠀⠀⠀⢀⡾⣿⣿⣿⢂⠙⢝⣿⠣⠨⡿⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠚⠀⠀⣠⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡰⠞⠛⠋⠛⠛⢶⣄⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣰⠎⠳⣆⠀⠀⠀⣠⠞⠙⣧⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡟⠀⠀⠈⠳⣤⠞⠁⠀⠀⢸⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢻⣤⣤⣤⣤⣤⣾⠁⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣗⠲⠶⠤⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡁⠸⣏⣉⣛⣛⣺⡇⠀⢛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⡜⠁⠀⠻⣍⣉⣉⡽⠁⠀⢯⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⠀⠀⢠⡄⠈⠉⠉⠀⠀⠀⠈⣇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⢤⣜⡧⠤⣤⣤⣀⡀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠉⢧⡀⠈⠓⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠻⢤⡀⠀⠀⠀⢠⠇⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡇⠀⢀⡴⠋⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣀⡀⠀⠙⠚⢉⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⢄⠀⠀⠀⠀⠀⠀⡔⠉⢣⠀⢠⠇⠀⡤⠚⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡎⠀⠀⠀⠀⠀⠀⠂⠄⢤⠙⠁⠀⠘⢆⠀⠀⡨⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠒⠋⠀⠀⠑⠒⠒⠊⢳⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠉⠉⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⣀⡀⠀⠀⠀⠀⢀⠤⢄⠈⡆⠀⡠⠊⠀⢀⠤⢤⡀⠀⠀⠀⠀⠘⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⡆⠀⠀⠀⡇⠀⠀⠉⠀⠀⠳⣄⠀⡊⠀⠀⢘⠆⠀⣀⠤⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⣀⡰⠀⠀⠀⠀⠑⠤⠒⠉⡆⠀⠀⢀⣉⡁⢠⡒⠉⠀⡰⠁⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣅⠀⡸⠀⠈⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠉⠉⡇⠀⠀⢏⠉⠉⢉⠧⣀⣀⣀⡸⠐⣅⣀⣀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡀⠀⠀⣀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠖⠢⠤⠞⠀⠀⠈⠉⠉⠀⠀⠀⠸⢄⠠⠲⡄⠀⠀⠀⠰⠁⠉⠒⠎
⠀⠀⠀⠀⠀⠀⠸⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⢀⡇⠀⠀⠀⠰⡀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢱⠀⠀⡴⠊⠉⠐⠄⠀⢈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⢱⠄⠀⢠⠃⠀⢸⡤⠤⠒⠦⠀⠀⡤⠒⠒⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⢠⠃⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠔⠁



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

⠀⠀⠀⠀⠀⣥⢊⢸⡀⢀⠀⡗⢛⡷⠋⡀⢀⠀⠀⢨⢤⡴⠀⢢⠚⢛⡆⠊⢑⡖⣉⡩⢠⠀⠠⠈⣤⠔⢿⡇⢠⠴⠋⠛⣤⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠤⠤

⠀⠀⠀⠸⠞⠛⣫⣭⣭⣭⠍⠉⠉⣙⣀⣒⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢀⠀⠀⠀⠀⢍⠉⠘⠛⠙
⠀⠀⠀⠰⠭⠰⠶⠒⠀⠄⠩⠉⠋⠙⢛⣋⢹⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣿⠿⠋⠀⠀⠀⣀⡀⣀
⠀⠀⠀⠸⠿⠛⣛⢛⣀⣀⣉⣥⣭⣤⡭⠷⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢊⠁⠀⠀⠐⠁⠀⠉⠁⠛⠓⠶
⠀⠀⠀⢸⣭⣭⠾⣛⠭⠉⠙⠛⠋⠀⠤⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣵⣾⡿⠃⠰⣄⠂⠠⠠⠠⠤⠤
⠀⠀⠀⠸⠻⠛⠛⠉⠀⠩⠍⠀⠁⠈⠈⠀⠺⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠀⠀⠀⠀⠀⠀⠀⠺⠋⠁⠈⠍⠏⠀⠡⠐⠶⠶⠦⠶
⠀⠀⠀⠀⠀⠖⠖⠒⠶⠀⠒⠢⠆⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠲⠶⠀⠒⠲⠰⠔⠲

⠀⠀⠀⠀⠈⠀⠀⡀⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣄⣠⡀⣀⡀⠀⣈⣀⣁⣀⣀⡈⢀⠀⠀⣀⢀⢀⣠⡀⢀⢀⠀⢈⣀⣈⠁⡀
⠀⠀⠀⠀⢶⡂⠘⠂⠅⡄⠀⣰⣦⡆⡀⠀⠀⠀⠰⣋⠶⣻⢛⣡⣷⢛⣽⡬⢛⣋⣴⠋⠀⠀⢘⣩⡜⣞⣇⡇⢤⢠⣾⢟⣵⢺⢆
⠀⠀⠀⠀⠸⡃⠀⠀⡆⠀⠡⠸⠻⠋⠀⠀⠀⠀⢸⠿⣶⣗⣻⣧⣟⡋⠴⣟⠩⣿⡿⢋⠀⠀⠘⠓⣏⣶⠟⠇⡖⣿⣿⠂⠲⡱⠁
⠀⠀⠀⢠⣢⠀⠀⠐⠇⠂⠈⠋⠀⣤⠀⠀⠀⠀⢸⣊⣵⢛⣙⣺⣍⣴⣶⠛⠳⢞⣊⣭⠀⠀⣴⣞⡯⣏⣾⡙⡥⠴⢺⣡⣁⡟⢸
⠀⠀⠀⠀⠍⠁⠀⠲⠂⠀⠠⡄⠐⠹⠅⠀⠀⠀⠸⢼⣷⣽⣿⡿⡓⣫⠽⣶⠷⡾⣿⣟⠀⠀⢹⠃⠛⠃⠉⢳⣴⣎⢻⡧⠱⠀⢺
⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠚⠛⠛⠋⠀⠚⠛⠛⠛⠀⠀⠈⠒⠀⠀⠐⠛⠁⠒⠉⠚⠁
⠀⠀⠀⠈⠑⠉⠉⠉⠁⠁⠉⠉⠉⠙⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠉⠉⠙⠋⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠀⠋⠉⠘⠛⠁
⠀⠀⠀⢀⣀⣀⡀⢀⡀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⢣⣜⣻⢻⣧⡒⣾⣷⣦⣿⣿⣯⣿⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠂⠀⡀⠒⡖⢦⢄⡀
⠀⠀⢸⣆⣯⠛⣬⢡⣿⢗⡾⣿⠿⣟⠿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢏⣨⣾⣢⡋⢕
⠀⠀⢸⣪⣷⣇⣴⣾⣿⣻⢿⢶⣗⣸⢑⣛⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠉⠁
⠀⠀⢘⡣⣹⠯⢭⣟⣿⣥⠚⣈⣻⣿⣿⣿
⠀⠀⠘⠂⠉⠛⠛⠊⠛⠑⠛⠊⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠁⠉⠉⠋⠉⠉⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠙⠋⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⣀⣀⢀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠰⡄⠇⡰⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠁⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠊⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠠⡀⡆⢠⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠀⠘⡶⠁⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠇⡠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡀⡇⢠⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢰⠂⠐⠂⠁⠬⠐⠀⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠂⠀⠱⣶⠃⠈⠂⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠂⠀⠀⡨⠠⠞⠀⠆⠇⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠦⠀⠅⠀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠈⡌⡔⠀⡤⠤⠤⢤⣀⠀⠀⠈⠀⢳⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠭⠭⣙⣒⠀⠸⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⠭⢝⣒⣂⠀⠰⠿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⣩⣍⣙⣓⣒⠦⠥⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣒⣒⣒⠭⠭⣓⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣒⣒⣒⣒⣒⡦⠭⢭⣒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠉⠙⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣉⣀⣉⣉⣉⣉⣓⣒⣤⣼⣤⣤⣐⣀⣀⣀⣀⣀⣀⣀⣈⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠂⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠈⠀⠀⠁⠀⠐⠃⠈⠀⠠⠀⠀⠌
⠀⠄⡀⠀⠀⠀⠩⡀⠄⠀⢀⠌⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⢀⢠⠀⠀⠀⠀⡀⠀⠐⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠄⠂⠃⢀⡀⠘⠒⠠⠀⠀⠀⠠
⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠄⠐⠁⢀⠀⠀⠀⠀⠀⠀⡃⠈⢢⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠐⠈⠀⠀⠀⠀⠁⠄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⠈⠀⠀⠀⠐⠠⠐⠂⠄⠀⠀⠀⠀⠀⡀
⠀⢡⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠀⠀⠀⢀⡐⠀⠀⢊⡀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⡀⢌⠀⠀⡱⢀⠀⠀⠀⠁⠀⠁⠉⠀⠂⠀⠀⠀⠀⡈
⠀⢁⠌⡄⠃⠀⠐⡀⠀⠀⠀⠄⠀⠐⠀⠀⠐⠠⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠌⡑⠊⠁⠀⢀⠀⠀⠀⠀⡀⠀⠀⠘⠠⠁⡈
⠀⠈⠀⠀⠀⠐⠀⠐⠂⠐⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⢊⠀⠁⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠑⠀⠀⡀⠀⠀⠃⠀⠀⠘⠁⠀⠀⠀⠀⠠⠒⠐⠂⠄⠀⠀⠀⠃⠀⡰⢆⡀⠀⠐⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠢⠀⢀⠐
⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠖⠂⡀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠑⠄⡀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡐
⠀⠀⡁⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢈
⢀⡀⠁⠀⠀⠀⠊⠀⢄⠀⠀⠐⠀⠑⠄⡀⢀⠀⢀⡀⢀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠢⠡⠀⠀⡄⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠡⣀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠠⢀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⣀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠤⠀⠀⠀⠀⠂⠈⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠄⠑⡀⠀⠀⠀⠀⠠⠀⠠⡀⠀⠈⠁⠈⢀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠰
⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠁⠈⢉⡁
⠈⢠⠀⠀⠔⠀⠀⠀⠀⠀⠁⠀⡀⠠⠀⠠⠈⠉⡀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠊⠈⠁⠀⠂⠀⠄⠨⡀⠂⠀⠀⠰⢀⠁⠠⠀⠀⡄
⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠈⠢⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⡈⠀⠀⠀⠈⢀
⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⡀⠐⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠃⠁
⠀⡠⠁⠀⠀⠀⠀⠠⠀⠀⠄⣀⠀⠀⠄⠀⠡⠀⠠⠄⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠡⠀⠀⠈⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⠀⠀⠈⢄
⠀⠐⠂⠀⡀⠀⠁⠀⠀⠀⠀⠨⠀⠀⠀⠤⠊⠀⠀⠑⠤⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠠⠀⠆⠀⠀⠀⠀⢀⡀⠰⢀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠆
⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢄⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣰⠉⣄⠀⠀⡠⠉⢆
⠀⡀⠀⠀⠀⠀⢤⠆⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢆⠀⠀⠰⢀⠀⡀⠀⠀⠀⢀⡀⡀⠀⠀⠀⡀⠤⠃⠀⠀⠘⠄⢈⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠜⠣⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠰⠨⡄⠀⠁⢠⠄⠂⠀⠀⠀⢂⠀⠀⣁⢐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⡄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠡
⠀⢊⠀⠀⠀⠀⣄⠀⢀⡀⢀⣠⠀⠀⠀⢀⡕⠀⠀⢊⠀⠀⠀⢠⠀⠔⠑⠀⠀⠀⡀⠀⠀⠈⡅⠀⠀⣨⠀⠀⠀⠀⡂⠈⠀⠠⠂⢤⠀⠀⠀⠀⡑
⠀⠐⠁⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠐⠣⠜⠀⡐⠣⠜⠂⠀⠄⠠⠀⠀⠀⠀⡀⠀⠠⠐⠈⡁⢉⡅⠈⠁⠒⠄⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⠀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠉⠉⢙⡯⠛⠂⠀⠂⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣀⠀⠀⠀⠀⠀⢠⠊⠄⠐⠆⠘⡅⣄⠀⠀⠀⠠⠀⡼⠀⠄⠀⢀⡀⠀⠀⠀⠈⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⡄⠀⠈⡶⠉⠀⠐⠒⠲⢸⡀⢀⣤⠀⢀⣼⠁⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠈⠉⠐⠂⢲⠀⠀⢙⡁
⠀⠀⠀⠀⠀⠀⠀⢀⠁⡼⣤⠄⠀⢐⠀⠀⠀⢀⠀⠀⢗⣠⣦⣔⡺⠃⠀⠀⠀⢀⣀⠤⠤⠤⢄⡀⠀⠀⠀⢻⡾⠛⠆⡀⠈
⠀⠀⠀⢀⠀⠀⠁⠜⠶⣶⠃⠀⠀⠈⢃⡀⠀⠀⠀⠀⠜⡹⠷⠀⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠘⢆⢀⡀⡀⠁⠀⢠⡜
⠀⠀⠀⠁⠀⡐⠁⠀⠀⠈⠀⠀⠀⠀⢀⠔⠊⠉⠁⠀⠢⡀⢀⠀⠀⠀⠂⠀⠃⠀⠞⠀⠙⠒⠀⠀⠠⢸⠋⠀⢰⣚⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⡆⠀⠄⠀⠁⡤⠀⠀⠀⢆⣀⠀⡄⠀⡆⠀⠘⣤⣤⡇⠀⠀⠀⣸⠛⠁⠀⠈⣠⠠⢤⡀
⠀⠸⠁⠀⠀⠀⠀⠀⠀⠊⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⢄⡘⠤⡲⠃⣈⡙⠢⠔⠃⠾⠁⠀⠀⠀⢆⣀⢀⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢇⣀⠀⠲⠆⠀⠀⠀⢠⠾⠅⠀⠀⠘⠉⠑⠠⠥⠥⠤⠴⠒⠁⣀⣤⣀⡀⢄⠙⠛⠛⠇
⠀⠀⠀⠠⢃⡀⠀⠀⢀⠔⠊⠉⠉⠳⣤⡑⠒⠶⠦⠴⠂⡠⣞⠀⠀⠸⠶⠢⠤⢀⡀⠀⠀⠀⢀⠀⡜⠠⠘⠲⠈⢢⣣⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠁⠠⠀⡎⠀⠀⠈⢺⡆⠈⣏⠋⠒⠒⢿⠁⠀⠀⠠⠀⡠⠂⠅⠈⣐⠋⠛⠀⠐⠚⠘⠂⠐⣶⣶⠀⢰⣿⡶⠴⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣸⠤⡟⠂⠀⠡⠀⠀⠀⡀⠀⠀⠀⠀⢰⠀⠚⠓⠦⢺⠇⠈⢯⡅⠈⢌⠧⡔⣿⣯⣦⢾⡿⠀⠜⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣑⡐⠒⠒⣠⠔⠓⠒⡄⠀⢹⣄⡀⠀⡀⢸⠀⠀⢀⣀⠀⠀⠀⠀⣆⠀⠀⠑⠒⢡⣭⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠈⠁⠉⠋⠀⠀⠀⠚⠏⠛⠀⠀⠀⢠⣀⠢⡄⡈⠏⣀⣀⠼⡵⠓⠁⠀⢀⠐⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠎⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⠥⠤⠖⠺⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⢀⣀⣀⠀⠀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⣀⡻⣿⣿⠟⣁⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣤⣬⣛⣛⣥⣤⠀⠀⠀⣠⣬⣛⣟⣫⣤⠰⣿⣿⣿⡆⣠⣬⣻⣟⣫⣄
⠀⠀⠀⠀⣾⣿⣿⣶⣶⣿⣿⣷⠀⠀⠀⣰⣶⣭⣿⣵⣶⡽⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⢿⣵⣽⣛⣯⣶⣝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⠿⠛⠁⠀⠀⠻⠿⣿⣿⠿⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠉⠀⠀⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠋


⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⠀⠀⠀⣴⣿⣶⠀⠀⠀⢰⣾⣷
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣴⣽⣿⣵⡄⠀⢠⣮⣿⣯⣦⠀⢠⣬⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⢠⣼⣿⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣺⣿⡛⠛⣻⣿⡛⠛⣻⣿⡀⠀⠀⠀⠀⠉⠉⣵⣾⣽⣷⣮⠉⢹⣾⣽⣷⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡗⠸⣿⣿⡿⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠀⠸⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠂⠈⠋⡀⠀⢀⠖⠒⠂⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡠⠖⠉⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠎⠀⠀⠀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢨⠠⠄⠀⠀⠀⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠈⠁⠀⡆⠀⠀⠀⠀⡔⡐⡄⠀⠀⠀⠀⡈⠉⡃⠀⠀⠀⠠⠂⠀⠀⡄
⠀⠀⠀⢀⡆⠀⠀⠂⢀⡀⠤⠤⠶⠤⠤⠤⠤⣀⡀⠀⠀⡇⠀⠀⠐⠀⠿⠙⠁⠀⠀⠀⠀⠿⠷⠃⠀⠀⠈⠁⠀⠀⠀⠰
⠀⠀⢀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡔⠘⠀⠀⠀⠀⠀⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡧⣄
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⣠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠈⢧⠀⠠
⠀⠀⠀⠀⠀⠙⠂⢄⡀⠀⡠⢒⠆⠀⠤⡠⠊⡇⠀⠼⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠠⠢⠄⠀⠠⠀⠀⠈⢢⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⠘⠒⠚⠉⠐⢳⠌⢲⠒⠒⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇
⠀⠀⠀⣔⡇⠀⡀⠀⠀⠒⡡⠀⠀⢀⠄⡴⠉⠀⠈⠓⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢠⠀⢈⠑⠀⠀⠈⠄
⠀⠀⠀⠈⠀⡜⠀⢀⠔⠋⠀⠀⢠⠁⠀⣸⠀⠀⠀⠀⠀⠈⠒⣄⠀⠀⠀⠀⠀⠘⠒⠀⠀⠃⠀⠀⠀⠤⡀⠀⡀⠀⠀⡦⠚⠁⠀⢀⡠⠊
⠀⠀⠀⠀⢠⠅⠀⠋⠀⠀⠀⡠⠋⢆⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡜⠀⡠⠔⠂⠁⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠈⢢⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠎⠀⠀⠔⡆
⠀⠀⠀⠀⠀⠑⠒⠒⠈⠀⠀⠀⠀⠀⠀⠙⠢⡈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠔⠁⠀⠑⠒⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑⠢⠷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠔⠋⠁⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠙⠁⢀⡀⠀⠀⠀⠀⢀⡀⢠⠴⠊⠡⠄⡀⠲⠀⠀⠀⠀⠀⣶⣿⣿⣿⣆
⠀⠀⠀⢠⡀⠒⠀⠐⠚⠀⠀⠀⠀⡈⠛⢇⣀⠄⠀⡠⡦⠒⡏⠀⠀⠀⣀⠘⡞⡻⠋⠀⠀⠐⠚⠓⢀⣠⣤⠀⠀⠀⢀⣿⠋⣹⢿⣆⠀⠀⠀⠀⠘⠃⢐
⠀⠀⢤⣈⣤⡤⠤⠀⠀⠀⠀⠀⣀⣉⠁⢀⣴⠊⣽⢦⣤⡶⠁⠀⠀⠰⠟⠻⠟⠁⣠⠀⠎⣂⠀⠀⡀⠲⠍⠃⠀⠀⢾⡿⠀⠉⠀⠀⠂⠀⣰⠂⢀⣸⣷⠀⣀⡄
⠀⠀⠈⡽⡹⠗⠂⠉⠀⠀⢀⠀⠙⠛⠋⢠⠞⠈⢙⣿⣤⠇⠀⡀⠀⠠⡤⠐⢚⣽⠡⠎⠎⠙⠂⠚⣿⡧⣄⠟⡿⠀⠀⠐⠠⠖⠆⣠⣤⣾⡿⠿⠛⣿⡁⢰⣿⡇
⠀⠀⢀⡟⢡⣀⣯⣄⣀⡤⠞⠁⠀⠀⠐⠛⠉⠳⠋⠁⠀⢀⣴⢃⠀⠚⣿⣿⡏⠀⠀⠀⢠⠀⣤⠀⡄⠀⠈⠡⠟⢃⡀⡈⢀⣤⣶⣿⣶⣟⣾⣿⣿⣿⣇⡧⣁⡇
⠀⠀⢸⡇⠀⠉⠛⠉⠉⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⣠⣯⡷⢛⣷⠞⣛⡽⠆⠀⠀⠀⣼⠃⠶⢶⠛⠀⠀⠀⠐⠋⠘⢋⠡⢾⣿⣽⣿⡿⠟⠛⢁⣿⡁⣠⠤⠂
⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⢰⠛⣿⠟⢠⡿⠳⣾⡿⠀⠀⠑⠒⣻⣯⡤⠾⠟⣀⠶⠂⠋⢠⢄⣤⣾⡿⠟⢛⢋⣟⡶⡀⠀⣼⡿⠋⣱⣶⡇
⠀⠀⢤⣸⠇⠑⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⡁⠀⠠⠈⠀⠀⠀⢸⡴⣿⣿⣩⣵⣶⠀⣴⣿⡈⠈⠔⠛⠉⠉⠉⠀⠀⢹⣮⣿⣦⢃⣼⠏⠀⢠⣿⣿⡇
⠀⠀⢈⡷⣾⣿⠘⣄⡄⠀⠀⠀⠀⠀⠀⠀⠈⣁⢠⠞⠀⠀⠀⣀⢤⣶⡾⢿⣿⣿⣿⣿⣿⣷⣿⡿⡟⢀⣀⣤⣢⠤⣤⣀⣰⣤⣀⣌⣙⡥⣼⠏⠀⠀⠀⠻⣿⡇
⠀⠀⠀⠙⢿⣏⢰⡏⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⡏⠀⣀⡔⠈⠠⠞⠁⣰⣟⣉⣿⣿⣿⣿⠋⠁⠀⠀⢀⣿⣿⡿⢷⣿⡇⠙⠻⢶⣿⣿⣤⠤⠿⠓⠶⠿⢊⠒⠄
⠀⠀⠀⠀⠀⠈⢿⡅⠀⠀⠀⣾⣿⣦⣶⣹⣷⣖⣁⣽⢟⠑⠂⠀⠰⠻⣿⣿⣿⡟⠛⠙⣿⡿⠀⢀⣠⣾⣿⠿⣷⣶⣦⡄⠀⠀⣀⠀⠀⢀⠀⠀⠀⢈⣵⣶⣔⠂
⠀⠀⠀⢠⡄⠀⠀⠹⢷⠀⠀⢈⣽⣶⠾⢿⣿⣿⢋⣴⣧⣶⣶⣶⣤⣿⣿⠟⠃⠂⠐⢄⠄⣀⠀⠘⠁⠙⠃⢀⡀⢉⡵⠶⢿⣿⡿⣧⡰⣌⠙⠲⢦⣬⣿⣿⠟
⠀⠀⠀⠉⢀⣴⠀⠀⠀⡀⠀⠀⠀⣸⡿⣿⣿⠁⣿⣿⠏⣤⠘⣛⡾⣭⣠⣰⡞⠀⠰⣃⣾⡷⡟⠒⠀⠉⠿⢛⣿⢧⣾⣧⡾⠛⣛⣛⡛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⣦⣠⣼⣏⡶⠋⣿⣆⠉⢿⣧⣉⣠⣾⠟⠀⠀⠹⠁⠀⠀⠿⢿⠷⢃⣀⣤⣤⣶⠭⠥⠞⡋⠙⠿⢿⣟⣿⣷⣒⣿⠴⠶⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣺⣿⡟⢩⡀⠀⢸⡿⣀⣀⣉⠙⠉⠁⠀⠀⠀⢀⡤⠤⣴⣶⠋⢹⣋⣡⡼⠃⠀⠀⠀⢿⣿⡧⠂⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⡙⠛⢦⣄⣸⣿⣿⠍⠁⠴⠾⡶⠓⠻⢿⡿⣿⡿⠿⠻⠿⢭⣠⣤⠤⣦⣀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡥⢢⠐⠛⠿⣿⣹⠛⠛⠻⢢⣼⣄⠚⢾⡷⠿⠤⢤⠖⠆⣯⡬⠉⠁⠈⢁⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⢸⡇⠀⠀⠀⠁⠉⠁⠐⢀⠀⠀⠀⠈⢙⢢⣾⣶⢓⠀⠀⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠃⠀⢠⡀⠳⣄⠀⠈⠹⢱⡯⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠃⠀⠀⠀⠀⢲⣴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠋⠉⠉⠉⠉⣩⡛⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⣷⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣉⡙⠿⣿⣭⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⣦⣬⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠣⣴⣤⣄⡀⠀⠀⠀⠀⠀⠈⢻⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠰⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢻⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⠶⠶⠟⠛⠋⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣤⠀⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⣤⣄⡀
⠀⠀⠀⢀⡘⣿⡙⠷⠟⢸⣧⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡾⠋⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⣿⣙⠛⢡⡶⠶⣦⠉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⣨⡾⠛⠛⠛⢷⣄⠀⠀⢹⣆⡀
⠀⠀⠀⣩⡽⠇⠸⣧⣤⠿⢀⣙⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠻⣆
⠀⠀⠀⠹⡷⠞⣷⢀⣴⣄⢻⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠁⠀⠀⠸⣧⠀⢤⣴⡄⠀⣼⠃⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠘⠻⠾⠃⠹⠛⠁⣶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⣿⢰⡞⠁⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⣦⣀⣿⠀⢀⣿⣠⡾⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢰⡾⠷⠿⠾⣷⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠉⠁⠀⠀⠉⠋⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣭⡛⢿⡄⠀⠀⠀⠘⠿⣾⠿⡷⠟⠀⣼⠛⠛⠉
⠀⠀⠀⠀⠀⢠⣤⣤⣴⠟⠀⢠⡾⠛⠛⢶⡄⠈⢿⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⣨⡗⠈⠻⢶⣤⡴⣦⣀⡀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⢸⣇⣀⣀⡀⠀⢿⡄⠀⠀⢸⡇⠀⢀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠉⣩⡷⠀⠈⠛⠶⠾⠋⠀⠀⣿⡉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠻⠶⠟
⠀⠀⠀⠀⠀⠀⠀⢾⣏⠀⣠⡶⣤⠀⠀⣴⢦⣄⠈⣻⡦⣠⡶⠛⠉⠉⠉⠉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⣿⠀⢀⣿⠀⠙⠿⠋⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠰⠟⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣄⣠⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢠⡞⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣀⣤⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠃⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣄
⠀⠀⠀⠀⠀⢠⣄⢀⡤⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢸⡇⠀⣀⣀⡀⣠⣤⡀⠀⠀⠀⣴⠶⠾⠋
⠀⠀⠀⠀⠀⢀⡭⢨⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⢸⡏⠉⠿⠋⢀⣿⠆⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⣨⡿⠃⠀⠀⠘⣧⣀⡼⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠻⣦⡀⠀⠀⠀⠀⠉⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣷⠄⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠈⠻⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣧⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣄⣼⡇⠀⠀⠀⠈⠻⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⠀⠀⠉⠉⠉⠉⠀⠀⠀⠐⡄⠀⠀⠈⠉⠀⠐⠀⠐⠒⠒⠒⠒⠁⠀⠀⠐⠒⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢲⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠈⠑
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠣⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠸⠀⠐⠒⠒⠒⠒⠄⠀⠀⢀⠤⠤⠤⠤⠄⣀⢐⠀⠀⠠⠤⠤⠤⢄⠀⠀⠀⠠⠄⠄⠤⠤⠄⢘⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠒⠂⢄⠀⠘⣄⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠉⠒
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠒⠄⠔⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠒⠤⠀⠀⢡
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠢⠀⠀⠒⠀⠀⠀⠓⠠⠤⠀⣀⡀⠀⠇⠀⠀⠰⠀⣀⣀⣀⣀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣤⣤⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⣨⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢚⡀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢈⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠈⠉⠁⠀⠀⠰⠾⠭⠭⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢃⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠨⣁⠠⠀⠉⠀⠀⠐⠡⠀⠀⢤⣶⣚⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠒⠀⠐⠌⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠂⠋⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠼⠇
⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢤⠟⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠖⠓⠒⠒⠠⡀
⠀⠀⠀⠀⢐⡂⠉⠀⠐⠀⠰⡃⠀⠀⣴⠒⠉⠀⠀⠀⠀⢀⠔⠁⠀⠈⠢⡀⠀⢊⡠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠪⡇⠈⠉⠀⠐⠂⠰⠅⣀⡂⠠⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⢄⠀⠀⠀⠀⠀⢈⡄⡽⠍⠍⠉⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠅⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠁⠒⠂⠤⡀⠔⠀⠀⠐⠀⠀⢀⠀⣠⠎⠀⣹⠄⠉⠁⠀⠋⢋⢀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠁⠀⠀⠀⠀⢸⠉⠐⠂⠀⣞⠇⠐⠀⣸⠁⠀⠀⠀⠐⠁⠑⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠀⣾⠀⠀⠀⠀⡇⡈⠀⠀⣼⠕⠀⠀⠀⠨⠀⠀⠨⡏⠁
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠠⠊⠀⣠⠞⢽⠀⠀⠀⢠⠷⠤⠴⠤⣼⢔⢥⠀⠀⡠⡢⢤⣼⣇⡀⠠⠤⠄
⠀⠀⠀⠀⠈⠐⠠⠀⠐⠀⠰⢸⡶⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠒⠒⡛⠫⠀⠠⢺⢀⠀⠀⠠⠤⠠⠄⠤⢾⢤⣤⡀⠀⢝⣵⢟⣾⡷⡾⠿⣛⣒⣂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠈⠣⢀⡀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠄⠄⠐⡀⣸⠅⠄⠀⠸⡓⠚⠐⢶⠟⣿⣛⠗⠐⠑⢑⣿⠟⠉⠉⠑⠤⠀⠀⠒⢓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠃⠥⠄⠀⢈⠠⠀⡠⠀⢀⠀⣼⢄⠀⡀⠸⠀⠀⡼⠃⠀⠀⠉⠁⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⠔⠁⣡⠅⣜⠁⡈⠄⠘⣠⠜⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⡀⠊⠨⠁⣠⠏⠙⠛⠳⠼⠯⠀⠀⠀⢀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢂⢈⠀⢄⢄⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠓⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⣠⣴⣶⡿⠿⠿⢿⣶⣦⣄⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⠉⠀⣀⡀⢀⠀⠀⠉⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢰⡶⣛⣛⢿⣥⡀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢰⣿⠁⠀⠀⢿⡗⣿⣿⢇⣿⡅⠀⠀⠀⠈⣿⡆⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⠀⠀⢀⣜⣟⣶⡶⢿⢽⠞⣓⢷⡄⠀⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠠⡏⣶⢝⡇⣠⣤⣹⣮⣋⡼⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠙⠖⠛⠙⢿⣻⠇⠈⠁⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡷⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⢾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠙⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣿⣶⣶⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⣿⣿⣿⣿⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣌⠰⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⢄⠛⡠⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⡇⢲⣲⣷⡀⠀⠈⢀⠜⠉⡱⢫⠍⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡝⠀⠈⣿⣿⡇⠀⡠⠁⠀⠀⠑⠔⢊⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⡗⢁⡠⠐⠀⢢⣤⣾⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠮⠡⡀⣻⣿⣿⠁⠀⢀⣴⣿⣿⠛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣐⡫⠞⠀⣿⣿⡿⢠⣶⣿⡿⠋⡕⣥⢈⠆

⣴⠈⠀⠀⠀⠀⠈⠙⣿⣿⣿⡏⠐⠀⢤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⡇⠀⠀⠘⠃⢀⣠⣴⣿⣿⡿⠃⣀⠀⠀⠀⠈⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⠉⢻⡿⠛⠉⠛⠿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⣳⣶
⠁⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⡀⠀⠀⠀⠀⣾⣿⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠸⢠⣶⣶⣄⠀⢀⣀⡀⠈⣿⡟⣿⣿⣿⣿⣿⡅⣤⠀⠀⢤⡎⠁
⠀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣧⠀⠀⠀⠀⠙⠁⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⠘⠿⠿⠛⠐⣿⣿⣿⠀⡟⠀⣻⣿⣿⣉⣹⣇⣀⣀⣄⡀
⣴⣴⣦⠀⣾⡿⠋⠉⠁⠀⠈⢉⢻⣧⠀⠀⢀⣤⣤⣤⣤⣄⠈⠛⠛⠿⠿⠿⠿⣿⠀⠸⣷⣶⡶⢶⣄⡈⠙⢉⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠞⠁⢀⣀
⣿⣿⣿⣷⣏⣠⣤⣄⠀⢰⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⣿⣷⣶⣿⣿⡿⠿⢿⣿⣿⣿⠛⠻⢿⣿⣿⣿⣧⣶⣿⣿⣿
⢐⣍⣻⣿⡟⠿⠟⠛⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠸⣿⡿⢃⣀⠀⢀⣀⠈⠁⢻⣿⣿⠫⢄
⣿⣿⣷⠿⡇⢀⣀⣴⣟⣛⣷⣶⡞⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠸⡏⣢⣥⣥
⣟⡙⠑⢺⠇⠈⣻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠐⠶⠀⣠⣤⣶⣿⣿⣿⣶⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠤⠀⠀⠀⠀⠀⢻⣿⣿⠟
⣿⠏⣂⣴⣿⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⠷⠀⠀⣀⣼⣿⣷⣦⣄⣀⣣⡃⠞
⡤⣵⣿⣿⣿⡿⠿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⢸⡿⠛⠛⠛⠋⠉⠉⢿⣿⣿⡏⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠲
⣾⣿⡿⠛⠁⠀⠀⠀⠹⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⢡⣶⣿⣿⣿⣶⣾⣿⠁⠀⠈⠉⠀⠈⣿⣿⣿⣿
⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣆⠀⠀⠓⣠⡀⠀⠐⠛⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⡿⠀⠀⣀⠀⠀⠀⢻⣿⠟⠃⠆⠀⠀⢰⣿⣿⠟⠛⠛⠛⠻⣶⣶⣄⠉⠀⠀⠂⠀⠀⠀⠛⢸⠇⠀⠀⠉⠉⠉⠀⣿⡇⠀⠀⠀⠸⣿⠟⠋⠉⠙
⢹⣿⣿⠀⠀⠀⣁⣀⣤⣴⣿⣿⣿⣶⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⢺⠐⠖⠀⠀⠠⠦⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠙⣿⠀⠀⣴⣿⣿⣿⣿⣿⠿⠛⠋⠿⠶⠶⠶⠇⣤⡄⠀⠀⣤⡄⠀⣿⣿⣤⣤⣤⣤⣄⣀⣤⣴⣿⣆⣠⡴⢶⣄⢀⣼⠂⠀⠀⠀⠀⠀⢸⡔⠀⠈⠁
⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡇⠀⠀⢀⣴⣦⣤⣤⣆⣀⣀⣤⣄⡀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠧⠀⢠
⠀⣠⣴⣤⣭⣿⣿⣿⣿⡟⠃⠙⠁⠀⢠⣤⣤⣤⣼⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠛⠛⠇
⠐⣿⣿⣿⠿⠿⠿⣿⣿⣿⠀⠀⠀⢢⣾⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠐
⠀⠻⠋⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⣿⣿⠿⠿⠿⢿⣿⣿⣄⣀⣀⠐⠚⠿⠿⢿⡀⠀⠀⠉⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠴⠀⠀⠀⣀⠀⠀⣿⡏⠀⠀⣿⣥⣥⣲⣴⣶⣽⣿⣩⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⣿⣿⡿⠀⣠⣴⣶⣶⣶⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⡀
⣀⠀⣀⢀⣀⡀⠈⠀⠐⢻⣿⣿⣿⣿⡿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣤⣤
⣿⣧⢻⣿⣯⣽⣶⣤⣶⣿⣿⣿⣿⣿⣷⡰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠛⢁⠀⠀⣿⠟⠁⠀⠀⠀⠈⠹⡟⠀⣠⣾⣿⣿⣿⣿
⠈⢿⡇⠙⠛⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠉⠉⠉⠉⡟⣠⣶⣿⣿⣿⣷⣄⣀⣀⣴⣄⡀⠃⢠⡤⠀⠀⠠⠦⠀⢰⣴⣿⣿⡟⠁⠉⠉
⣠⠈⠇⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠐⠆⠀⠠⡤⢠⣿⣿⣿⠿⠛⠉⠉⠉⢿⣿⣿⣿⣿⣷⣄⣀⣠⣤⣤⣀⣴⣦⣿⣿⣿⡇
⠉⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⡀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠀⠖⠈⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⢶⣿⣿⠟⠁⠛
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆⠀⣾⡟⢿⣿⣿⣿⣿⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⡄⠶⠂⠀⢀⣠⣤⣼⣿⣿⣿⣿⣿⣿⠙⠛⠋⠁⠀⠀⢻⣿⣷
⠀⢀⠄⠀⠈⢿⣿⣿⣿⣿⣦⣙⡀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣴⣟⣡⣿⣿⣿⣿⣟⡋⠉⠉⠠⣤⡆⣠⣴⣶⣾⣿⣿⣇⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⢄⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢴⡁⠀⠀⠀⠈⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠤⠄⠠⠄⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠁⡠⠐⠢⡀⠙⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⢠⡀⣀⠄⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠢⢀⣈⣁⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠐⠠⣤⠀⠀⠤⠔⠚⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠳⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⠤⠄⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣭⠆⠀⠈⢩⠋⠝⣋⡛⢍⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢁⢠⠸⡘⢅⠸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢀⠔⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⣤⡄⠊⠀⡈⢆⠢⢍⣒⣒⡡⠊⠳⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⢀⠀⠘⡌⢆⠀⠀⠉⠢⡀⠀⠈⠢⡀⠀⢀⠊⠀⠀⠀⢠⠊⠉⠑⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡼⠠⠁⠀⠀⠀⠑⠃⠀⠘⡌⢆⠀⠀⠀⠈⠦⡀⠀⠈⢦⡌⠀⠀⠀⠀⠐⢄⣀⡠⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⢡⠃⠀⢠⡆⠀⠀⠀⠀⠀⠘⡜⡆⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣦⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⣀⡀⠀⢤⣄⣀⡠⠤⠿⠐⠒⠺⠄
⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⠱⡘⣄⠀⣀⣀⡀⠤⢤⣗⠒⠦⠭⠙⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢰⢃⠂⠀⢠⢢⠀⠀⠀⠀⠈⠛⠁⠀⡠⠠⡀⠱⡘⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢠⢃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠡⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣣⣤⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⢳⠀⠀⡠⠄⢿⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠸⡆⠉⣹⠀⡜⠀⠀⠈⡦⠊⠈⠱⡀⠀⢠⠊⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠻⠖⠁⠀⠈⢢⠀⢰⠀⠀⡀⠀⡇⠀⣊⠑⣠⣎
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢢⠀⡀⠀⠀⠀⣄⠀⠳⡏⢀⠀⡇⢠⠃⡔⠞⠉⠀⣁⠀⢀⠐⠂⠲⡄
⠀⠀⠀⠀⡠⠤⢤⣸⠤⡴⠁⠇⠁⢄⠀⠀⡏⡧⡀⠱⡟⢠⠇⡼⠀⡧⠂⢀⠀⡏⠀⢘⢄⣀⡀⡇
⠀⠀⠀⢸⠀⠀⢠⢻⢢⢨⠏⠀⣠⣀⠑⠲⢇⡇⠱⣀⡅⡘⢀⡧⠊⣣⢔⣋⣰⠥⠆⠀⣉⠈⣨⠁
⠀⠀⡔⠉⢂⣤⠎⣤⣀⡉⠀⠠⣇⠀⠑⠢⠴⣑⣤⣶⡿⠟⢍⠉⠭⠤⠳⠤⢤⠤⠄⣼⡀⠀⠈⡆
⠀⢸⠁⠀⠀⠀⠀⢀⡨⠐⠒⠛⠊⠀⠀⠈⢉⠽⠫⡀⠕⢶⡰⣉⠒⠧⢀⣀⡎⠦⣀⣡⡭⠦⢼⠃⠈⢆
⠀⠈⢇⠀⠀⠀⢀⡎⠀⠀⣼⡤⠤⣔⡲⠚⣣⠊⢠⢃⠘⡆⠉⠊⠑⢦⡀⠈⠉⠉⣵⠋⠐⢄⠀⠀⠀⡸
⠀⠀⠈⢆⠀⢰⢡⢣⠖⢉⠭⠔⠈⠁⣠⡞⡡⢤⠃⠈⡄⠘⡄⠀⠘⣄⣙⣯⣐⠒⣿⡄⢀⣸⡄⡀⣰⠁
⠀⠀⠀⠘⡗⢶⢸⡇⠀⠀⠀⠀⣠⢊⠎⡜⠀⠀⡇⠀⢱⠀⠘⢄⢰⠋⠀⠈⡆⠀⠀⠀⠀⣇⡇⣧⠇
⠀⠀⠀⠀⡇⠈⠄⢷⣄⠀⠀⠀⣧⡇⠀⢧⣀⣠⠇⠀⠈⡆⠀⠈⠛⣦⣀⡴⡃⠀⠀⠀⠀⠉⢀⡏
⠀⠀⠀⠀⡇⠀⠰⡆⢹⠓⠂⠉⢸⡇⠀⠀⠉⠒⠈⠐⢄⢸⡀⠀⠀⠈⠉⠀⠑⢦⠀⠀⠀⢹⠋⠇
⠀⠀⠀⠀⠃⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⡀⠀⠁⠀⠀⠌⠀⠀⠠
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠠⠀⢀⡀⠀⠀⠀⠋⢁⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢶⣖⣯⠍
⠀⠀⠀⠀⡄⠀⠀⡀⠀⠚⠀⡀⠈⠀⢰⣛⠀⠀⠀⠐⠀⠀⢩⠠⢤⡤⠈⠀⠐⠀⠀⠀⠀⠀⣰⠀⠒⠋
⠀⠀⠁⠀⠀⠀⠀⠉⠀⠁⠀⠁⠄⠀⠀⠈⠀⠀⠠⡤⠲⠈⠀⠀⠄⢤⠐⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠒⢂⠀⠀⢠⠀⠠⠄⠀⠈⣃⠀⠁⠠⠀⡄⠀⠀⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⡀⢉⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠐⠀⠈
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⣀⢈
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴

⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣀⠀⠀⢀⠀⠀⢀⠀⣠⣤⣠⠀⠀⢠⠤⣀⠀⢀⣀⢤⠀⠀⣀⣀⠀⣀⣀⣀⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⢹⠀⠉⢢⠀⠀⠀⠀⠀⢸⠀⢠⠋⠀⣇⣀⡀⠀⠀⣧⠤⠚⠀⠺⣄⡀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠘⡦⠊
⠀⠀⠀⠀⢸⠀⠀⡜⠀⠀⠀⠀⠀⠈⣤⠃⠀⢠⠃⠀⠀⠀⢰⠏⠢⡀⠀⠀⢀⡽⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠸⠔⠊⠀⠀⠒⠒⠂⠀⠀⠉⠀⠀⠈⠀⠉⠁⠀⠈⠀⠀⠈⠀⠈⠁⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣄
⠀⠀⢸⣶⣄⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶
⠀⠰⣾⣿⡿⠀⠀⣶⡿⣷⡄⠀⢹⣿⣿⡏⠀⠀⣶⣾⣶⡄⠀⢹⣅⣤⣇⠀⠀⠀⡀⠀⠀⠀⠀⢹⡉⢙⡖⠀⠀⣄⣦⣶⡆
⠀⠀⡝⡩⠛⡄⠀⠈⣷⣞⣳⠀⡰⠿⠛⢲⠀⠀⢻⣿⡏⠀⠀⠸⠒⠒⠚⡄⢻⣿⣿⣇⡀⠀⠀⡼⠿⠯⠽⢄⠀⢹⣿⣿⣧⡆
⠀⢸⡜⢁⠴⡇⠀⠀⡇⠀⠀⠃⠧⠀⠀⠸⡄⠀⢺⢿⡇⠀⠀⢠⠀⠀⠀⡇⠀⢻⣿⠋⠁⠀⢸⠀⠀⠀⠀⠸⠀⠀⣿⣿⣿⣤
⠀⢸⡴⠁⢠⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⠀⡇⠀⢸⣿⡇⠀⠀⢸⠀⠀⠀⡇⠀⢸⣇⠀⠀⠀⢀⠶⠤⠄⠲⡁⠀⠀⣿⠉⠀⢸
⠀⠸⢀⠔⢹⠀⠀⠀⡇⠀⠘⠀⢸⠀⠀⠀⡵⠀⣼⣿⡇⠀⠀⢸⠀⠀⠀⡇⠀⢸⡿⠀⠀⠀⢸⠀⠀⠀⠀⢱⠀⠀⣿⢰⢸⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⠀⠀⠀⠀⢠⡄⠛
⠀⠂⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠈⢀⠂⠀⠐⠀⠀⠀⠦
⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁
⠠⠠⠄⠋⠀⠲⠹⣦⣄⢀⣀
⠀⠀⠤⠀⠀⠀⠀⣹⣿⣿⣧⣤⣄⣴⡶⢲⠆⢀⡄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠄⠀⠀⣽⡝⠈⢦⡌⣻⣧⣤⠀⠀⣬⣟⡞⠉⢳⣄⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠔⠀⠀⠀⠀⠉⠁⢀⣸⣤⣉⡛⠿⡄⠀⡡⠟⠉⠀⠀⡈⠀⠀⠀⠀⠀⠄⠀⠀⢀⣴⣿⣿⡏
⠈⠁⡀⠁⠰⠆⠀⠀⡄⠀⣻⠉⢘⠢⢄⣷⣻⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⡰⠏⠀⠀⠀⠀⠰⣤⡀
⣤⣄⣈⣠⣄⣤⣤⣤⣤⣤⣿⣶⣦⣤⣄⣨⣿⣦⣄⣀⣀⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⣠⣤⣴⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⠶⣶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣻⣿⣿⣿⣿⣯⣦⣤⣬⣭⣭⣙⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣯⣽⣯⣤⣿⣿⡟⠛⠋⠙⠋⢁⣽⣿⡇⢹⣿⣏⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡤⢿⣹⣿⣿⣿⣿⣿⣤⡀⣠⣶⣮⡛⠛⠁⠀⠈⠻⣿⡿⠉⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⣿⢿⣿⣿⣿⠟⠛⠛⢿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⢿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠘⢫⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢿⡟⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣴⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣿⣶⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠯⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣤⣀⣠⣀⡀⢀⠀⣤⡶⠀⠀⠀⣀⣀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠖⠛⠛⠛⠉⠁

⠀⠣⠤⠴⡆⠀⠀⣀⢀⡀⠀⠀⠶⠤⠤⠃⠀⠘⠤⠤⢶⠀⠀⢀⡀⣀⠀⠀⣰⠦⠤⠜⠀⠀⠣⠤⠴⡆⠀⠀⣀⢀⡀⠀⠀⡶⠤⠤⠃
⠀⠀⠀⠀⠇⠀⠘⡇⢀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⣸⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⢀⠇⠀⠀⠇
⢸⡉⠉⢙⠋⠐⠒⠋⠈⠓⠒⠈⢋⠉⠁⡹⠀⢏⠈⠉⠻⠁⠒⠚⠁⠙⠒⠂⠹⠉⠉⢉⠇⠸⡁⠉⢩⠉⠐⠒⠋⠈⠓⠒⠈⢋⠉⠁⡹
⠔⠃⠀⢸⣀⠤⢄⠀⠀⢀⠤⣀⣸⠀⠀⠓⠠⠚⠀⠀⣇⡠⠤⡀⠀⠀⡠⢄⣀⡆⠀⠘⠢⠔⠁⠀⢸⣀⡤⢄⠀⠀⢀⠤⣀⣸⠀⠀⠓
⡀⠀⠀⢠⠤⣄⡰⠀⠀⠸⣀⡤⢤⠀⠀⢀⣀⠀⠀⠀⡤⢤⣀⠅⠀⠀⢄⣠⠤⡀⠀⠀⢀⡀⠀⠀⢠⠤⣄⡸⠀⠀⠠⣀⡤⢤
⢨⠆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢏⠀⡹⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠸⡁⢈⠆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢏
⠘⠂⠒⠒⡓⠈⠉⡗⠰⡋⠉⠐⡗⠒⠀⠚⠀⠓⠐⠒⢺⠀⠉⢹⠆⢾⠉⠁⢺⠒⠒⠐⠃⠘⠂⠒⠒⡗⠈⠉⡷⠰⡏⠉⠐⡗⠒⠀⠚
⠀⢀⣀⡀⡇⠀⠘⠧⠠⠇⠀⠀⣁⣀⣀⠀⠀⠀⣀⣀⢸⠀⠀⠸⠄⠼⠀⠀⢼⢀⣀⡀⠀⠀⢀⣀⣀⡇⠀⠀⠧⠠⠃⠀⠀⡇⣀⣀
⠀⢇⣀⣨⡅⠀⠀⠀⡀⠀⠀⠀⣭⣀⣀⠇⠀⠸⣀⣀⣭⠀⠀⠀⢀⠀⠀⠀⢨⣅⣀⡸⠀⠀⢇⣀⣨⡅⠀⠀⠀⡀⠀⠀⠀⣭⣀⣀⠇
⠀⠀⠁⠀⡇⠀⠠⡏⠉⡆⠀⠀⡄⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⢸⠉⢱⠀⠀⢙⠀⠈⠀⠀⠀⠀⠁⠀⡇⠀⠀⡏⠉⡇⠀⠀⡇⠀⠁
⢰⠒⠒⠲⠆⠦⠤⠟⠘⠦⠤⠆⠗⠒⠒⢲⠀⡖⠒⠒⠾⠰⠠⠼⠃⠻⠤⠴⢸⠒⠒⠒⡆⢰⠒⠒⠒⠧⠦⠤⠟⠘⠧⠤⠖⠗⠒⠒⢲
⡨⠇⠀⢸⠀⣀⡀⠀⠀⠀⣀⠀⢸⠀⠀⢧⣀⠽⠀⠀⡆⢀⣀⠀⠀⠀⢀⡀⠈⡄⠀⠸⣄⡠⠇⠀⢸⠀⣀⡀⠀⠀⠀⣀⠀⢸⠀⠀⢧
⠀⠀⠀⢈⣉⡁⢘⠀⠀⢸⠀⣉⣉⠀⠀⠀⠀⠀⠀⠀⣉⣉⠈⡇⠀⠀⡁⢈⣉⠁⠀⠀⠀⠀⠀⠀⢈⣉⡁⢸⠀⠀⢌⠀⣉⣉
⢑⡆⠀⢸⠀⠉⠁⠀⠀⠀⠉⠁⢸⠀⠀⡖⠉⢲⠀⠀⠇⠈⠉⠀⠀⠀⠈⠉⠀⠃⠀⢰⠊⠑⡆⠀⢸⠀⠉⠁⠀⠀⠀⠉⠁⢹⠀⠀⡖
⠸⠤⠤⠤⠆⠖⠒⡦⢠⠖⠒⠆⡦⠤⠤⠼⠀⠧⠠⠤⢴⠰⠒⢲⡄⣴⠒⠲⢸⠤⠤⠤⠇⠸⠤⠤⠬⡆⠆⠐⣦⢠⠖⠒⠆⡦⠤⠤⠼
⠀⠀⠀⠀⡇⠀⠰⣇⣀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢸⣀⣸⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣇⣀⡇⠀⠀⡇
⠀⡎⠉⢑⠃⠀⠀⠀⠁⠀⠀⠀⣓⠉⠉⡄⠀⢰⠉⠉⣚⠀⠀⠀⠈⠀⠀⠀⢙⡊⠉⢱⠀⠀⡎⠉⢑⡃⠀⠀⠀⠁⠀⠀⠀⣓⠉⠉⡆
⠀⠈⠊⠁⡇⠀⢀⡔⠒⡄⠀⠀⡋⠉⠋⠀⠀⠀⠙⠉⢹⠀⠀⢰⠒⢢⠀⠀⠺⠉⠙⠁⠀⠀⠈⠋⠉⡇⠀⠀⡖⠒⡄⠀⠀⡏⠉⠋
⢠⠤⠤⢤⡅⣀⣀⡧⠰⣅⣀⡀⡧⠤⠤⢤⠀⡤⠤⠤⢬⢀⣀⣸⠆⢾⣀⣀⢸⠤⠤⠤⡄⢠⠤⠤⠤⡇⣀⣀⡷⠰⣇⣀⡀⡧⠤⠤⢤
⢘⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣎⠀⣳⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰⡁⢘⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣎
⠉⠀⠀⠘⠒⠊⢑⠀⠀⢰⠉⠒⠚⠀⠀⠈⠉⠀⠀⠀⠓⠒⠉⠆⠀⠀⠊⠑⠒⠃⠀⠀⠈⠉⠀⠀⠘⠒⠊⢱⠀⠀⠐⠉⠒⠚⠀⠀⠈
⠢⡀⠀⢸⠉⠒⠊⠀⠀⠈⠒⠋⢹⠀⠀⡤⠒⢤⠀⠀⡏⠑⠒⠁⠀⠀⠑⠚⠉⡆⠀⢠⠔⠢⡀⠀⢸⠉⠒⠊⠀⠀⠈⠒⠊⢻⠀⠀⡠
⢰⣁⣀⣨⣄⡀⠤⣄⢀⡤⠄⣀⣌⣀⣀⣱⠀⣎⣀⣀⣠⡀⠠⢤⡀⣠⠤⢀⣰⣀⣀⣈⡆⢰⣁⣀⣘⣄⡀⠤⣄⢀⡤⠄⣀⣌⣀⣀⣱
⠀⠁⠁⠀⡆⠀⢠⡇⠈⡆⠀⠀⡇⠀⠀⠁⠀⠈⠀⠀⢸⠀⠀⢸⠁⢹⠀⠀⢸⠀⠈⠈⠀⠀⠁⠁⠀⡇⠀⠀⡏⠈⡆⠀⠀⡇⠀⠈⠁
⠀⡔⠒⠶⠇⠀⠀⠉⠊⠁⠀⠀⠶⠒⠒⠄⠀⠠⠒⠒⠾⠀⠀⠈⠑⠉⠀⠀⠹⠔⠒⠢⠀⠀⠔⠒⠶⠇⠀⠀⠉⠊⠁⠀⠀⠷⠒⠒⠄
⠀⡀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⠂⠀⠀⣰⣿⣿⡿⠋⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏⡀⠀⠀⢹⣿⣀⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢛⡁⣨⣽⠃⠀⠀⠀⡀⠀⠀⢀⣠⣶⣷⣿⣾⣶⣼⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠋⠉⠉⣠⣴⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠈⣿⣿⣿⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠂⠈⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠆⠀⠀⣠⣿⣾⡷⠶⡎⠉⠁⢹⣿⣿⣿⡟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣿⣿⠃⠀⠀⠈⠄⠀⠀⠀⠀⠹⠁⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡟⠁⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣼⣿⣿⡀⢀⠀⠈⠀⠀⠀⡀⠀⣤⢸⣿⡿⠿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠤⠀⢠⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠁⠀⠀⣼⣿⢷⣶⠄⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⢸⣿⡟⠛⠿⠿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⣤⡌⠀⠀⠀⣀⣀⣀⡀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⡿⠀⠀⠀⢀⣠⣴⣶⣦⣿⣿⣿⣾⡿⠀⠀⢠⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠘⢿⣦⡄⠀⠀⠙⠉⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠋⠻⣿⡉⠉⠀⠀⠀⣿⣿⣿⣿⠛⠻⠿⡟⠋⠉⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠠⣿⠇⠀⠀⠀⠀⢿⣿⠉⠁⡄⠀⣀⠀⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⢀⡀⠀⠆⠀⠀⠀⣿⣿⡏⠙⠀⠐⠂⡿⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⣠⣿⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠈⠁⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠲⠀⠀⠖⠀⠀⠀⠀⠀⣿⣿⡿⠀⢠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⡀⠀⡀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠆⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠘⠁⢀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⠈⠁⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠤⠀⠈⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⡠⠄⠁⠀⠀⠈⠀⠀⠀⢀⠀⠠⣀⠀⠈⠉⠀⡀⠂⠀⠠⠄⠀⠀⢀⠰⠀⡁⠐⠊⠀⠀⠀⠀⠐⠂⠀⢀⠈
⠀⠀⠀⠁⠀⠀⠖⠆⠀⠠⠀⠀⠈⠀⠀⠀⠀⠈⢀⠀⠀⡀⠀⠀⠀⠃⠐⠂⠀⠈⠁⠒⠀⠀⣀⠀⠀⠀⠀⠠⠀⢠⠀⠁
⠀⢀⠀⠀⠆⠀⠀⡉⠀⡀⠀⠀⠀⠈⢠⠀⣀⠓⠈⠁⣀⠁⠈⠀⠄⠈⠀⠀⠀⠀⠘⠀⢀⡀⠈⠁⠀⠈⠁⠀⠀⡀⣀
⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢘⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠰⠀⠀⠀⡈⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠆
⠀⠨⠀⠀⠆⠀⠰⠀⠉⠁⣌⠀⠈⠀⠀⣀⠀⠀⢀⣀⠀⠈⠀⠰⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢠⡄⢠⡄⠀⠀⠉
⠀⠐⠀⢀⠀⢀⠰⠀⠀⠀⠀⢠⠀⠠⢤⠀⠐⡀⠀⢀⠀⠀⠀⠀⠀⠀⠤⠠⢀⠀⠀⠁⠀⠀⡐⠂⠀⠀⠀⠉⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠠⠄⠒⠀⠰⠈⠀⠰⠀⠉⠀⠀⠀⢠⠀⣘⠀⠁⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⡘⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠄⠄⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⡀⠀⠠⠄⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⢀⡀⠀⠐⢠⠠⠄⠀⠀⠀⠀⠀⠀⠀⡁⠔⠂⠀⣄⠀⠁⠐⠂⠀⠀⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡄⠀⠐⠂⠤⠀⠄⠀⠨⠀⠀⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠒⠀⠀⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃
⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠣⠤⠔⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠒⠁⠀⢀⡀⠀⠈⠢⠤⣄⣀⠖⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⡔⠁⠀⠱⠀⠀⡴⢪⡇⢸⡱⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⡀⠀⠘⠤⠤⠊⠀⠀⠳⣄⣘⠖⢤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⢈⡆⠀⣀⡀⠀⡴⢊⡶⠝⠀⠁⠉⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠕⠈⠉⠁⠘⠵⢮⠀⠈⠆⠂⢀⠅⠀⡎⢉⠆⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⡐⠉⠑⠀⠀⡎⠀⠀⠀⠀⠈⠑⡄⣈⡁⡠⠬⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠤⠀⢠⠃⠀⠐⠤⠔⠁⢀⢉⠆⠀⠀⠀⢀⡀⣍⠀⠑⠁⠀⠀⠀⢠⠤⠋⠑⠊⡆
⠀⠀⠀⠀⠀⡌⠜⡠⡾⡍⢑⠀⡤⠤⣀⡅⢔⢄⠀⠀⠀⠘⣃⡬⠀⠀⠀⠀⠀⠀⡴⠀⢎⢉⠆⢨⠃
⠀⠀⠀⠀⠀⢣⠑⢴⠁⠟⠀⠉⠀⠀⠀⠀⠈⠪⡒⠀⠐⠂⢸⡆⠒⠤⣀⣀⢔⡲⠀⢇⠤⣀⡔⠚⢠⢖⣄
⠀⠀⠀⠀⠀⠀⠱⡀⢸⢈⡑⠠⢐⡩⠃⠀⠀⠀⠀⠀⠘⠀⢸⠁⠘⠐⠒⠒⠁⠀⠀⠀⢔⠤⡀⠀⠳⣰⡃
⠀⠀⠀⠀⠀⠀⠀⢃⡼⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠆⡄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⠭⠥⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡳⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢐⠠⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⣉⡡⠂
⠀⠀⠀⠀⠀⠀⠀⡠⢃⠄⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄⢈⢹⠰⡄
⠀⠀⠀⠀⠀⠀⣎⠔⠁⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠘⡈⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠃⠀⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠤⡀⠤⡀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡤⣒⣈⠉⡩⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡜⠀⣠⢢⠁⠀⠀⠀⠀⢀⣤⠶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣾⡿⠁⢘⣅⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠆⡇⢠⢏⠇⠀⠀⠀⠀⠀⣾⠃⠀⠀⣼⠀⠀⠀⡀
⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⠈⠁⡰⢆⢃⡀⠀⠀⠀⢀⡤⠜⠁⣀⣌⠒⡒⢆⡄⢱⢊⠆⠀⠀⢀⠀⠀⠀⣿⡀⠀⣰⡟⠀⢀⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡼⠁⠀⠣⡁⢄⠀⣞⠏⠈⢁⣰⣿⠾⠷⠔⢛⡖⣡⣓⣒⡀⢀⣖⣒⠀⣄⣈⣛⣾⠯⠤⢤⣎⡼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠛⠋⢳⡌⢢⡽⢚⣫⣭⣥⣲⡾⠿⠝⡫⠫⠘⡃⡇⠀⣀⠤⠐⣊⠫⠭⠥⢖⡚⢍⠉⠍⡌⠉⠁
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢸⡇⠀⠀⢘⡧⠐⠋⠉⠢⢭⣑⡦⢤⣔⣫⠚⢀⡠⠗⣫⡥⠒⡨⢟⡒⢤⠀⠀⠀⠈⠂⡜⡄
⠀⠀⠀⢠⠎⡶⣶⡝⢦⡀⠈⢿⣀⠀⣸⠇⠀⠀⠀⠀⠀⢀⠽⡷⢢⡝⣊⠥⡊⠉⢎⢇⠀⡇⣯⣻⠆⡇⠀⠀⠀⡰⡰⠁
⠀⠀⠀⠀⢣⠻⠿⠁⠀⠉⢶⠠⣉⡉⠁⠀⠀⠀⠀⢀⢔⡵⡫⢚⡥⠊⠀⠑⠮⡑⠬⣌⢆⡇⠈⠁⠀⠘⡄⠀⠀⠦⠁
⠀⠀⠀⠀⠈⠱⢀⡀⠀⠀⢠⠋⠒⠨⠭⡜⣐⣒⣐⢳⡮⢊⠴⠡⢤⢀⣀⠀⠀⠈⠐⢍⠪⡗⢦⠀⠀⠀⠇⠀⠘⡘
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠋⠀⠀⠀⠀⠀⠀⠌⣹⢏⡔⠉⠉⣈⡚⠒⠦⢍⡐⠢⣀⠀⢳⡌⢆⠑⠒⠊⠀⠀⡅⠃⢠⠊⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠙⢦⡀⠀⠀⠀⡌⡼⢁⠎⠀⠀⠀⠈⠻⠿⣑⠒⠺⠶⣤⡑⢺⡸⠦⡱⡀⠀⠀⢈⢳⠀⡇⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢻⡄⠀⢘⡸⢡⠂⠀⠀⠀⠀⠀⠀⠀⢈⡆⠘⢿⣮⣻⡢⡙⢆⠱⡑⡄⠀⡈⡈⢰⢃⡤⣤⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢀⠛⢷⣄⣠⡼⠃⢠⠟⡠⠁⢀⠤⠤⠤⢄⡀⠠⠀⠈⠑⠢⣄⠙⠋⠇⡍⣎⠣⡑⡜⣤⢢⠁⠰⡘⠯⢟⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⢁⡞⠀⣴⢷⠖⠀⠀⠀⠑⡄⠀⠁⠀⠀⠈⢆⡤⠆⠅⠄⠳⠃⣸⢎⢣⠀⠀⠈⠉⠉⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⡴⠱⠁⠰⠙⠁⠀⠀⠀⠀⢀⡖⠫⢵⣯⡷⡄⢸⡇⢰⢰⠀⠀⡱⢁⡌⠱⣕⠤⡀⢀⣠⠤⣲⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢊⠔⢪⢡⠃⠀⢰⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠉⢸⠀⠀⡄⡀⡠⢊⠔⠑⢍⠦⡀⢰⢞⣭⠀⡜⠁
⠀⠀⠀⠀⠀⠀⢒⡀⠤⠂⠁⠀⢣⠃⠤⠁⠀⠣⣄⠀⣠⡤⠼⣇⠀⠀⠀⠀⠈⡸⠀⠀⣡⢋⡔⠁⠀⠀⠀⠁⢨⣧⣿⠏⢀⠼⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⢃⢦⣾⠟⠲⢤⣀⠀⠀⠀⢀⠀⠻⣀⣀⣀⣠⠜⠁⠀⡐⢡⡎⠀⠀⠀⠀⠀⠀⠈⠉⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠌⠈⠦⣌⢠⠒⢪⠙⢲⡴⢥⣀⠀⠀⠉⠉⠁⠄⠀⣰⣡⠣⢠⢀⣀⣠⣤⣄⡀⠀⠓⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠘⠙⠻⢮⣼⡏⠺⠉⢊⠹⣶⠤⣄⡀⠀⠠⠃⠃⠘⢸⡏⠀⠀⠀⠉⢹⡆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠬⣖⣋⢐⠉⡻⠓⠦⡀⠀⠀⠈⢿⣤⣤⣤⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣥⡈⣠⡧⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠻⢷⣶⣤⣴⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣤⣦⣠⣤⣤⣤⣬⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠑⠂⠀⠀⣸
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠒⠒⠲⠶⣤⣤⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⢻⣿⣿⣿⣿⡟⠩⠐⠆⢻⣿⣿⠇⠀⠀⠀⠐⠛⠻⢷⣤⣄⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⣆⣴⠀⠸⣿⣿⠀⠀⠀⠀⠀⠲⠆⠀⢸⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢸⣿⣿⣿⣿⡆⠸⠇⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠙⠻⢿⣿⣿⣿⡿⠠⡄⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠆⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡟⠲⠄⠀⠀⠀⠀⠀⠀⣈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⠀⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢸⠀⠀⠰⢿⠿⠿⠿⣿⣧
⠀⠀⠀⠀⠠⠔⠂⢠⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍
⠀⠀⢀⠀⠀⠄⡀⠀⠆⠀⠀⢳⠀⠀⠀⢀⣤⠄⣽⣿⡒⠒⢲⡄⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⣀⡀⠀⢄
⠀⠀⠀⠀⠀⠂⠀⠀⠇⠀⠀⢸⠀⠀⢀⡼⠋⠀⠛⠓⠶⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣙⢦⠈⠆
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⣻⡴⠆⠀⠘⠃⠙⢻⣦⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⣿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢤⣿⡀⠀⠤⠄⠀⢈⡼⣿⣿⡀⠈⠈⠀⠀⠀⠀⠀⢄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠛⠟⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢄⠀⠀⠀⠂⣳⣹⣆⠀⠀⠈⠀⠑⢿⣮⡿⠀⠀⠀⠀⢀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠦⠈⠃⠀⠀⠀⡿⣿⠏⠀⠀⠀⠀⢠⣿⣯⠁⠀⠱⠄⠀⠋⠁⠀⠀⠀⢱⣆⠀⠀⠀⠀⠀⣀⣀⠀⠈⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠊⠀⠂⠂⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠙⠧⠀⠀⢀⣄⢀⠀⠀⠀⣀⣠⠎⠟⠀⢀⣀⣔⣛⣛⡛⡳
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⣿⣶⣤⣾⡿⠿⠛⠀⡠⠀⡿⠟⠿⠿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠴⠀⠐⠓⠂⠈⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢀⠀⠀⠀⣠⣤⣄⣶⣶⣄⠑⠀⠀⠄⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡄⠀⠈⠀⠀⠀⠀⠱⠀⠀⠀⠈⡆⠀⣼⣿⡿⠟⠻⣿⣿⣄⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⣻⣿⣟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⢠⡟⠁⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠀⢱⠒⠀⠠⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣷⣼⣇⠀⠀⠀⡬⠄⠆⠀⠀⠀⡀⠀⢸⡇⠐⠀⠀⠀⠁⢀⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠋⠉⠉⠉⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠠⠄⠀⣼⣟⢿⣿⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠩⣿⣇⠀⠀⠘⣯⣿⡂⡿⠟⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢸⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠘⡇⠀⠀⠘⠀⠀⠀⡟⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⣀⡴⠖⠛⠛⠛⠓⠶⣄⡀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⡀⠀⣠⠞⠁⣀⡀⣰⡖⣶⡀⣀⡀⠙⢦⡀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⠏⠀⠸⣏⠙⣩⠤⢬⡙⢉⡿⠀⠈⢷⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⡀⣿⠀⢰⠞⠃⡞⠁⠀⠀⠙⡆⠓⢦⠀⢸⡇⣠⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⠳⣆⠳⡀⠀⠀⡰⢃⡶⠋⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠘⣧⠀⠸⣧⡴⢬⠉⢩⡴⣤⡿⠀⢠⡟⠰⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣈⢳⡀⠀⠀⠘⠓⠛⠀⠀⠀⣴⢋⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢹⡄⠀⠀⠀⠀⠀⠀⣼⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣛⣛⣛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣛⣙⣛⣋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⣤⣤⣤⣤⠀⠤⠤⠤⠤⠤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⠀⣭⣭⣭⣭⣍⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣛⣛⣛⣛⣀⣒⣒⣒⣒⣒⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⣛⣿⣿⣛⡛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣆⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⠀⠀⣀⣀⣀⣈⣃⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⡏⣭⣭⡍⠩⠭⠭⠭⠭⠭⠭⢹⠀⠀⡏⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠩⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⡇⢘⣛⣛⣛⣛⣛⣃⢸⠀⠀⡇⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢠⡇
⠀⠀⠀⠀⠀⠀⢸⡃⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢸⠀⠀⡇⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢘⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⣛⣿⣿⣛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⣛⣿⣿⣛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠂⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢲⠀⠀⡰⢳⡹⣦⡀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⡴⠊⠁⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠊⠁⠀⠀⠀⠀⡜⠀⡐⠻⣶⣱⡑⡙⣧⠀⠀⠀⠀⠀⠀⠘⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⠞⣆⠀⠀⠀⠀⠀⠀⠀⢇⠈⠩⠭⠭⢻⣷⣠⡟⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⢈⠎⢇⠀⠀⠀⠀⠀⠀⠈⠢⢤⠀⠭⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⢀⣀⡼⠁
⠀⠀⠀⠀⠀⢠⡃⠙⣦⡄⣈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡑⡰⠅⠑⠩⢉⢉⡿
⠀⠀⠀⠀⠀⣦⠴⠒⠚⠛⠶⠮⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⢓⣧⣷⣷⣾⡇
⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠈⠳⣴⢶⣚⡤⠃
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⡝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠉⠉⠙⢻⣦⠀⠀⠀⠀⠀⠀⠈⠫⡀
⠀⠀⣤⢧⢹⢦⡀⠀⠀⠀⣀⡴⣏⡎⠘⡜⢳⡀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣼⡠⣄
⠠⠄⢹⣆⢊⢊⠋⠛⠹⢉⠁⢈⡿⠿⡾⡀⢨⢳⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢀⣠⣴⠾⠛⠁⠀⠀⠑⢄
⢹⡃⠀⠛⣮⣇⢊⠁⢡⣛⣱⣟⣛⣓⣿⡰⢃⠚⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⡀⠀⠘⣿⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠒⠠⠤⠈⠙⠻⠿⠿⠿⠷⠶⡴⠒⢻⡇⠀⠌⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠈⠀⠐⠀⠭⠤⠛⠉⠀⣀⠀⢸⣧⠈⢈⣾⠷⢶⣶⣶⡶⠾⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢋⠙⡷⠶⠤⡴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣐⣿⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢾⣮⣎⡣⠀⡑⣼⠃⡴⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠐⣻⡿⣹⡇⠀⠀⠀⠀⠀⢀⡤⢴⠶⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢭⣔⣟⣻⣿⣾⡯⣼⠃⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⣷⠆⣻⡀⠀⠀⠀⢰⠟⡅⢄⠂⠀⠀⠘⢳⡀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠓⠒⠊⣰⠋⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⢀⠈⣷⡂⢀⠿⣄⠀⠀⣾⠿⠛⣻⣿⣷⣟⢐⢎⡗⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢂⣘⣷⢃⢈⠊⢳⡾⠉⠉⠛⠉⠍⠭⢿⣎⢄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠍⠠⠬⠳⣦⠽⠟⠀⡠⠤⠤⠀⣰⣶⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠥⣄⣀⠁⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⢒⠚⠛⠙⡉⠈⠌⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⠄⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠉⠓⠳⠦⣦⣴⣤⠴⠾⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⢻⡆⡔⠱⠳⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠏⢙⡿⠀⠀⠀⠀⠀⠀⠀⠉⢳⣅⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣁⠀⠹⣾⡕⡐⠙⢦⣄⠀⠀⣠⠶⠛⡀⡀⣤⣾⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠬⢿⣄⠄⢌⢋⠛⠛⠅⢀⣰⣴⠛⣛⣻⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⢙⣮⣧⣢⣀⣤⣶⠛⣉⢭⠭⠕⣿⠀⡟⣦⣄⣀⣀⣀⣠⡴⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⡀⢀⠛⠋⠉⠉⠋⠉⠀⠈⠙⢋⡻⣾⣜⣆⢌⡍⡌⢁⣣⡾⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠲⠚⠉⠁⠀⠀⠀⠀⠀⣐⣚⣙⣏⣿⡿⠾⠋⠭⠭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⢒⣒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠐⢀⣠⡄⠀⢀⠤⣤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣼⣿⣦⣄⣀⣼⢠⠁⣠⠛⠒⠪⣕⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠸⣿⣿⣇⣿⣿⣿⣿⣿⣿⠀⠀⡆⠀⠀⢸⠹⡌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⠐⣾⣿⣿⡯⠿⣿⣿⣿⣿⡟⠀⢤⠃⠀⠀⠚⢴⠃⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢰⣿⣿⣿⠀⠀⣿⣿⣿⣿⣧⠤⠼⠧⠀⠀⠀⢸⠀⠈⠹⡾⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠳⠧⣀⣸⣿⣿⣿⠀⠀⣯⣿⡿⠛⡇⠀⠀⠀⠔⠚⠋⠉⠓⢄⠀⢷⢸⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⠄⠠⠬⣍⣛⡟⠀⠀⣿⣿⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⢹⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠁⠙⠊⡌⠣⡀⢀⣿⣿⡇⢸⡷⠖⠒⠒⠮⠀⠀⡀⢀⡦⠴⠃⣰⠞⢳
⠀⠀⠀⠀⠀⠀⢸⣯⢁⠎⠀⠐⠄⠀⠀⠀⢘⡞⠛⢻⡇⢸⡇⠀⠀⠀⠀⠀⢀⡍⠉⠁⠀⠀⠻⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠌⠙⢀⠔⠈⠀⠀⠀⠘⣷⠀⠈⡇⢸⡇⠀⠀⠀⢰⡴⠊⠀⠀⠴⠒⠢⡄⠀⢼⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢄⠤⠐⠀⠀⠀⠹⡄⠀⠃⢸⣷⠶⠶⠶⣏⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⣽⠀⠀⠀⠀⣤⣤⡇⢀⠄⠀⠁⠀⠀⢸⡇⠀⠀⢀⠈⠙⠒⠒⠒⠋⠀⣶⠋⣸⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⣷⣷⡇⠈⠒⠀⢈⠀⠀⢸⡇⠀⠀⣸⠷⠤⠤⠾⠦⠤⠤⠻⣦⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠧⣍⡇⠀⠄⠀⢸⣴⣷⣼⡇⠀⣰⠏⠀⢰⣄⣠⠖⠒⢦⡀⢘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢝⣦⢬⡑⣴⠁⣠⠂⢀⡏⢻⣿⣿⣧⠾⠯⡤⠔⠋⠈⣷⠀⠀⢀⣷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡙⣧⡰⢋⣴⣾⣷⣿⣿⣿⡇⠀⠀⢤⣄⣴⠾⠋⢀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡿⣿⣿⣿⣿⠏⡷⠶⠛⡋⠉⢀⣆⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠋⠀⠘⠦⠶⠿⠖⠛⠉

⡇⣿⣿⣿⡄⠀⠀⣈⠀⢰⡘⣿⣿⣿⠇⠀⠀⠀⠰⠄⠀⠀⠀⣀⠀⣀⣠⣤⡤⠰⡄⠷⠀⠀⠀⢠⡾⠋⠁⠶⣰⣿⡆
⣓⠙⣛⣛⣃⠀⠀⠙⠀⠘⣇⢻⡛⢃⣤⠍⠀⠀⢀⣴⣆⠀⠻⠏⢼⣻⣯⣷⣷⠀⢻⡧⠀⠀⠌⠀⣠⣴⣾⣿⣿⣿⡇
⣿⠇⢻⣿⣿⣇⠀⠀⠀⠻⠙⠀⠀⢈⠀⡴⠞⢀⠈⠛⠉⠀⠀⠐⣿⣿⣿⣿⣿⡇⠸⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣝⠻⣆⠶⣯⣝⣆⠀⠀⠀⠀⢠⡀⣡⣾⣿⠃⠈⠀⣠⣔⠂⠀⠀⠸⣿⣿⠿⠋⠁⠀⠀⠀⢐⡀⣆⣿⣿⣿⣿⣿⣿⢰
⣄⡰⠦⡀⠰⣶⡿⠇⠀⠀⠀⣀⡧⠈⠛⠃⢀⣐⢘⣛⣻⣗⠀⠀⠀⠛⠁⠀⠀⠀⡀⠰⡶⠿⠷⠄⠻⢿⣿⣛⣛⡏⣨
⣿⣿⣿⣦⡄⠀⠀⠠⠤⠀⠀⢻⡍⢺⠆⢐⢮⣭⣘⣻⠿⠿⠇⠁⠀⠀⠀⢀⣴⣿⣿⡄⢿⣿⢻⡇⣶⣌⠻⣿⣿⡇⠻
⣿⣿⣿⣝⠁⠀⡤⠀⢦⡀⠀⠈⠃⢴⣾⡘⢦⡻⢿⡿⠋⠀⢀⡄⠀⠀⠀⠛⠿⠿⢿⣇⢸⣟⣾⢰⣿⣿⣷⡌⢻⡇
⣿⣿⠟⠈⠀⠀⣴⣿⣮⡓⠀⠀⠈⠘⢿⣇⠈⠿⠈⠀⠀⣰⣿⠏⠀⠀⠀⠸⣆⠻⢷⣶⡌⣭⠁⠺⠿⠿⠿⣟⡀
⠋⠀⠀⠀⣀⣾⣿⣿⣿⣿⠄⠀⢀⠀⠐⠁⠀⠀⠀⣠⡘⣿⡟⠁⣰⣷⠀⠀⠙⠓⠆⠈⠃⠀⢀⣠⣤⣤⣤⡄
⠀⢀⣴⣶⣿⣞⣿⡿⢋⣴⣦⡀⠈⠀⠀⠀⠂⣠⢈⡛⠿⠎⠀⠈⣭⣴⣦⠀⠀⢀⣶⣶⣷⠈⣸⣿⡟⠋⠀⠀⠀⠀⡀
⠐⠸⢿⣿⡿⢋⡀⢴⣿⣿⠟⠃⠀⢠⡀⠀⠀⠹⡎⠟⠁⠀⣠⣘⠻⠿⠿⣇⠀⠀⠻⣯⣙⠇⠙⣈⠀⠀⠀⣤⠦⢾⣿
⡀⢱⡶⠍⣁⣈⣛⠂⠻⠃⠀⣠⡀⠸⠃⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⡹⣶⣦⣤⡀⠀⠻⣉⡆⣀⠈⠀⠀⣾⣿⣍⣳⠳
⠦⡀⢙⣛⣛⠻⠿⠋⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢤⣙⠿⣿⣿⣿⣌⠻⢟⣻⡄⠀⠈⠃⠸⠀⠀⢰⣿⣿⣿⣿⣿
⡾⠿⠀⠀⠉⠀⠀⢀⣀⣤⠄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⣀⢭⣁⡛⠓⠙⠿⠛⠀⠀⠀⠀⣠⣤⡈⠛⠛⠛⠉
⠀⠀⣀⠀⣀⡀⢨⣉⣭⣅⠙⠛⠁⣀⡀⠀⠂⠀⠧⠀⠄⠈⠻⠿⠷⣝⠻⣿⠶⠀⠐⠀⠀⠀⢀⣤⣤⣤⠄⠀⢠⣤⣈
⣰⣿⡟⣿⠸⠟⠃⡉⢀⣤⣼⣻⣿⠿⠇⠀⠀⠀⠿⠆⠘⠂⠘⢛⣋⣠⣀⠐⢠⠀⢠⣘⣷⠀⠀⢉⡉⠴⢶⠀⠈⢿⣿
⢛⣉⣁⣀⣀⢐⣂⣐⣂⢉⣩⡉⠉⠀⣀⠀⠀⠀⣤⡄⣄⠡⣤⠀⠁⣀⣠⣤⠀⡀⠠⢤⣬⣠⠀⠠⣤⣤⣤⣄⠠⠄⠉
⣶⣤⠸⢿⣯⣀⠻⢿⣿⠷⠉⠀⡀⣾⡏⠀⠀⠀⢹⣧⢙⠃⠈⠱⣄⠈⠁⠂⠀⠒⠂⢳⣬⡳⠀⠀⢹⣿⠿⢋⣄⠐⢷
⠿⣛⣱⣤⣬⠉⠐⠓⢠⡄⡀⠈⢸⣿⣧⢠⡄⠀⠀⣡⢰⣶⣿⣧⠈⠀⠀⣄⠘⢿⣿⣯⡻⣿⡤⠀⠀⠠⠶⠛⢋⣉⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣤⡶⠾⠛⠛⠛⠛⠛⠲⠶⣤⣀⣠⡴⠶⠚⠛⠛⠛⠛⢿⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⢀⣀⣀⣀⡀⠀⣠⠾⠋⠁⠀⠀⠀⣀⣀⣀⣀⣸⡇⠀⠀⣿⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣤⠶⢻⡏⠉⠉⠉⣹⠟⠁⠀⠀⣠⡴⠞⢿⡉⠉⠉⠉⣿⠀⠀⢰⡏⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⣴⢿⡇⠀⠀⢻⣄⠀⣼⠏⠀⠀⣠⠟⢧⡀⠀⠈⢻⡄⢀⣼⠃⠀⠀⣾⢷⣄⠀⠀⠻⣆
⠀⠀⠀⠀⠀⣾⠃⠀⢀⡾⠁⠀⢻⣆⠀⠀⠙⢾⣏⠀⠀⣼⠃⠀⠀⢳⡄⠀⠈⢿⡞⠁⠀⢀⡾⠃⠀⠹⣆⠀⠀⢻⡆
⠀⠀⠀⠀⢰⡏⠀⠀⣼⠃⠀⠀⠀⠙⢷⣀⠀⠀⠉⠻⠶⣯⣤⣤⣤⣬⣿⠀⠀⢸⡇⢀⣰⠟⠁⠀⠀⠀⢻⡄⠀⠈⣷
⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⢘⣿⠟⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⢳⡶⠶⠶⠶⠶⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢿⡄⠀⠀⣸⠇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⠻⣤⡼⠋⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠙⠳⣦⣄⣀⠀⠀⠀⢻⣦⠀⠀⠈⠻⢦⣤⣾⠃⠀⠀⢀⣀⣤⡶⠛⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠉⠉⠙⠛⠛⠉⠉⠳⣦⡀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣄⣀⣀⣀⣠⣤⡴⠾⠛⠳⠶⣤⣄⣀⣀⣀⣀⣤⣤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⣬⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⢐⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠠⠐⠈⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠠⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠰⠁⠀⠐⠊⠀⠈⠀⠈⠁⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠠⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⡠⠊
⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣽⣽⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣷⣾⣿⣷⣶⣶⣶⣷⣾⣿⣿

⣿⣿⣿⡿⠛⠋⠀⠙⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⡟⡠⠄⠀⠀⠤⡌⢿⣿⣿⣿⡀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣄⡀⠀⠀⠀⣰⣶⣾⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⣠
⣿⣿⡇⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣷⣶⣤⣶⣼⣿⣿⣿⣿⣷⣴⣶⣶⣴⣄⡀⢠⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⠿⢿⣿⣿⡟⣻⣿⢿⢿⡿⣛⠿⠛⣿⣿⣿
⣿⣿⣿⣦⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠋⠀⠀⠀⠀⠹⣿⡏⢹⢁⣤⡄⠀⠀⣤⣄⠹⣿⣿
⢿⣿⣿⡿⠀⠀⠀⠀⢹⣿⣿⡿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠐⠀⠀⠀⠐⠀⠀⠙⠿
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠙⠛⠋⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⣀⣀⠀⠀⡠⠤⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣠⣶⢗⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠐⠂⠀⠀⠒⠊⢹⣿⡄
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠁⠀⠀⠈⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶
⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣿⢋⣀⡀⠈⣩⣍⢻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿
⡟⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⠀⠀⡀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐



⠀⠀⠀⣠⣤⣤⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⣤⣤⣤⡤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠉⠐⠛⠛⢻⡟⠛⠋⠀⠰⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣴⣾⢿⣶⣄⣬⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⣴⣿⣏⣼⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣛⣋⣀⠰⠈⢯⣿⣿⣿⣛⠋⢿⣿⢻⣿⣿⣿⣧
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣈⠀⠈⠟⠹⠿⢿⣿⣿⡏⣿⣿⣿⣿⡇
⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡻⣿⣿⣿⣿
⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣷⠀⢻⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⡿⠟⠁⠀⢸⣿
⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠛⢁⡶⠋⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠿⣿⡿⠿⠟⠋⠀⢀⣉⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠙⢿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠳⣄⠀⠀⡘⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠢⡀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⣎
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠰⣀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⡀⣀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⢣⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠘⢄⣀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⢰⠋⠀⠀⢀⠔⠢⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⣠⣄⠀⠀⠀⠀⠀⠀⣱
⠀⠀⠀⠀⠀⠹⣄⠀⣀⠆⠀⠀⠈⠢⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣼⣿⣿⣷⣄⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠠⣤⠐⠒⠒⠒⠐⠂⠒⢲⠒⠒⠒⠂⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢲⡀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠎⠀⠀⢈⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⢘⣢⣧⠀⠀⠀⠀⠀⠼⢇⠀⠀⠈⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢆⠀⢠⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠠⡟⠿⠀⠀⠀⠀⠀⡠⠓⢄⡀⠀⠀⢲⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢢⡀⣰⡦⡀⠀⠀⠀⠀⣳⣀⡀⠀⠀⡠⠒⠺⢆⡐⡐⠘⠁⢾⠷⡈⠂⢐⠉⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠱⣙⣇⣠⣠⣀⣀⠶⢉⠀⠀⡀⢹⠀⠀⠀⠀⠀⠱⠀⢇⠀⠀⠁⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠀⢀⢍⢢⠃⠀⠀⠘⠌⠐⠀⠀⠀⠀⠀⠥⣨⡄⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠠⢄⠈⡀⠀⠀⠀⣀⣀⠀⠐⡇⠀⠀⠀⠀⠀⢸⠆⡆⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠡⡀⡀⠀⣀⡄
⠀⠀⠀⠀⢀⠀⢀⠀⠲⠄⠄⡀⠿⡝⡡⠜⠤⢤⣄⣘⠟⡠⠃⢀⠃⠀⠀⠀⠄⠂⢠⣥⢀⠀⠀⡰⠜⠑⠩⡤⠰⡇
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⢲⠉⠀⠈⠉⠉⠀⠁⠈⠳⣀⠺⠀⢀⡤⣤⣀⣇⠀⠛⠀⠑⢀⠴⡁⠠⠀⠤⠀⠈⠢⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⣁⠉⠩⠁⠀⠋⠘⠘⠲⠊⠈⠀⠀⠀⠀⠀⠀⠀⢑
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⢰⡟⠗⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⣔⡆⠈
⠀⠀⠠⠀⠀⡀⠀⠀⠀⠄⣈⠔⠁⠈⠑⠢⢀⡠⢀⣲⡐⢤⣉⠷⡀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠐⡛⠽⠂
⠀⠀⠈⠘⠘⡖⠂⠴⠤⡞⠀⠀⠀⠀⠀⡔⣢⠉⢺⠟⠛⠋⠀⠀⠈⠢⡀⠆⠀⠀⠀⠀⢀⠎⠐⠀⠀⠀⠀⠀⠰
⠀⠀⡀⡀⠀⠐⣼⠾⠀⡧⠀⠀⠀⠀⠀⠛⠑⡡⠂⠄⠈⠁⠀⠀⠀⡤⡌⠂⡀⢒⢠⠄⠄⠂⠛⠀⠠⠈⡀⢀⡀⠄⠒⠋
⠀⠀⠄⠃⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠘⠁⠨⠙⠀⠉⠀⠀⠂⠀⠐⠀⠀⢀⠀⠈⠁
⠀⠀⠂⠂⠀⠀⠀⠀⡀⢊⠀⠠⠀⠐⠐⠁⠐⡄⢂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⣇⠉
⠀⠀⢠⣀⠐⠠⠄⠀⢁⠚⠉⠀⠀⠀⠀⠀⠀⠘⡈⠀⠀⠀⠀⠀⢀⠜⠀⠇⠀⠀⠀⠀⠀⠀⠀⠠⠄⠶⢶⠒⠒⠒⠀⠘⠄⠁
⠀⠀⠈⠀⠉⠉⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⣀⣨⠀⠼⢄⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠠⠀⣄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁