Sails Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡆⠀⡾⠻⢧⣒⠀⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⠀⠀⠀⠀⣸⠁⠀⠱⣆⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⠈⠢⣄⠀⢠⡏⠀⠀⠀⠘⣷⣄⠀⠀⠀⣰⠃⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⠥⡀⠀⠀⠈⠢⣼⠇⠀⠀⠀⢸⠏⠉⠢⡀⠀⡿⠀⠉⢹⠆⠂⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢄⡀⠀⣤⡞⠋⠠⣤⠑⡄⠀⠀⠀⠈⠢⡀⢀⣰⠋⠔⠀⠀⠈⠲⡏⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢼⡏⠠⡀⠀⠈⠑⠇⠀⣀⣠⣴⠿⠛⣵⣥⢂⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢨⡇⠀⠈⠳⣄⠀⠀⠐⣻⡻⣿⡿⣶⣦⣌⠀⡒⠄⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠑⢄⣰⠏⢱⣨⠛⢿⣿⣷⠀⠁⠂⢤⡀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⢀⠆⣠⢻⡇⠃⠀⠀⠀⠀⠀⠙⢦⡎⢡⡀⠈⠛⢿⠀⠀⠀⣾⠋⠑⢄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠊⠌⠳⢣⣾⠑⣷⠀⠀⠀⠀⠀⣼⣾⡇⠼⠁⠀⠀⠀⠑⢄⣠⣿⠀⠀⠀⠀⠄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡥⠤⠤⢀⠁⣾⡏⢹⠧⣀⠀⡀⠀⢠⣿⡏⡑⠀⠆⠀⠀⠀⠀⠸⠙⠿⡀⠀⠀⠀⠀⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠲⣤⡀⣰⣟⠄⠈⠃⢄⡴⣍⢰⣿⣟⢐⣥⠀⡀⢠⣄⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠋⢠⣄⣤⡤⠈⣅⡼⠾⠓⠦⠉⡡⢛⣶⠓⣄⢂⣥⠀⠀⠀⠀⠀⠈⠢⣀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⣀⡀⠋⢻⣦⣐⣿⣷⣖⠓⢶⣦⣴⣮⣉⡒⠃⠌⡼⠰⣤⡄⣠⡄⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⢉⠉⠙⠻⣶⣯⣻⣿⡁⠉⠟⠦⣍⣄⠸⢟⣿⠑⠒⠥⣀⠙⠂⠋⣾⠀⢸⡄⣷⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣻⠀⠀⠀⠀⢈⠙⣿⡟⢀⠀⠀⢉⠉⠓⠺⣧⡀⠀⢰⡟⠀⠦⢀⡻⣆⠾⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠀⢠⡟⠀⠀⠀⠀⠈⢀⣰⣿⠈⡄⡤⠈⠀⠀⠀⠈⠈⠑⠺⢇⡀⡀⠈⢳⣄⡀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡼⣅⢀⣿⣟⢾⡀⠀⠀⠀⡸⠃⣿⣰⡏⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠳⢦⣄⣹⡌⠙⠣⣀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠳⣠⣿⣻⣿⣘⡣⣞⣀⣶⠃⣾⣯⣿⠃⠀⡋⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⣿⣧⡀⠀⠈⡁⠲⢤⡀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠜⠀⠤⠟⣿⣣⣿⣧⡋⣱⢧⡿⢃⢶⣿⣿⡇⠀⠀⢻⠀⢀⠀⠀⠀⢀⡀⠀⠶⡗⢀⡝⢿⡿⣿⣐⠀⣷⡀⠀⠉⠙⠣⣀
⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⢺⣽⢯⣿⡿⣙⣿⣾⠏⢀⣾⣿⣿⣧⢹⡇⢼⡄⢼⡶⢀⡄⡈⠁⠀⢰⣧⣸⠀⣸⣿⣿⠞⠀⡇⢤⠀⣾⡄⠀⠀⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣬⡎⠉⠲⠘⣋⣋⣈⣿⣛⠻⣿⢿⡟⠛⠘⡇⠹⠇⠘⠄⠛⠀⡀⣏⣿⣋⣴⣟⣛⣛⣀⣘⡂⠠⢾⡉⠄⠀⡆⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣾⣶⣿⣷⣿⣶⣺⣿⡋⠉⠁⢈⢯⠏⡙⠻⠈⣉⣠⣤⣶⣶⡶⠶⠚⣛⣛⣛⣿⣿⣿⣿⣿⣿⣷⣿⣏⠀⠈⠉⠒⠄⠸⣄⣄⣾⠃
⠀⠀⠀⠀⠀⠀⣸⡏⠉⣩⣛⡋⠿⠿⢻⣗⣾⣷⣿⠞⢐⣠⣗⣩⡥⢟⢛⢭⡴⢶⠟⠛⠛⠛⠁⠀⠋⠩⣍⣿⣻⣿⢋⣿⣆⠀⠀⠀⠀⠀⢈⡿⠁
⠀⠀⠀⠀⠀⢀⡟⠀⣘⡷⠉⠥⠀⢀⡘⠿⢿⣟⣻⣿⣻⣏⣿⣗⣶⡄⠛⠀⠁⠀⠀⢀⣉⣉⠉⠉⠙⣛⣿⣿⣿⠏⡼⣿⢿⡆⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠈⣿⠀⠠⣄⡀⠐⠢⠤⠌⢀⡛⠉⣴⣻⡾⢿⣿⠛⠟⠋⡀⠀⠀⠀⠀⡁⠤⠤⠤⠤⣤⣭⣽⡿⢃⣾⢥⣸⠟⠁
⠀⠠⠀⠀⠀⠀⢹⣆⠑⠀⢤⡙⢉⠒⠂⠤⠂⠠⠬⣠⣌⣫⠈⢷⣠⡀⠀⠀⠀⠀⠀⠀⠲⠴⠶⠤⣬⣿⢟⣵⡟⢹⠆⠁
⠀⠀⠀⠀⠀⠀⡿⣻⡍⢑⡂⠰⠤⣉⣩⠘⠋⢐⠁⣀⢈⣥⡾⢿⡦⠤⠐⠛⣀⡀⠤⠐⠒⣒⣖⣲⣾⢟⣾⠏⡰⠃
⠀⠀⠀⠀⠀⢨⣷⡿⠛⠘⠿⢿⠶⠲⠶⠖⠂⠩⢽⣀⢄⣬⠼⠟⠓⠺⠷⠟⣢⣤⣶⣉⡥⢶⣾⣿⣧⣾⠑⣮⣥⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⠋⠉⠛⠛⠛⠻⠍⠈⠛⠋⠀⠀⠈⠓⠛⠈⠛⠻⠿⠿⠿⠥⠀⠙⠿⣿⣿⣮⣭⡭⠶⠛⠻⠛⠓

⠀⢰
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠣⡀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠑⣄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠂⠒⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⣀⠀⠰⡧⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢢⡄⠀⠈⠀⢀⡇⠀⣀⠠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢍⣭⡝⢛⣒⢦⣈⣀⡀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠛⠀⠄⢠⡄⢈⣑⣠⣴⠎
⠀⠀⠀⠀⠀⠠⠤⠶⣤⣤⣤⡀⣀⣀⣤⣀⠘⣿⣿⣾⣿⣶⣬⣭⣽⣏⣉⣾⣀⣐⣃⣀⣶⣤⣤⣽⡶⠶⠛⢻⡟⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⠀⢐⣶⣶⣶⣮⣿⣟⣻⣿⣿⣿⣿⠿⣿⣿⠫⣿⣿⣧⠖⢦⣭⣭⣭⠨⢭⣥⣶⣢⣤⣿⣟⠛⢻⣶⣶⠏⢀⣒⠂⠀⠀⠁
⠀⠀⠉⠀⠀⠚⠤⠀⠙⠛⢛⡛⠡⠐⠺⠿⠶⠦⣀⣬⠛⠛⠿⠿⣒⡾⠛⠉⠠⣾⣿⣥⢂⠀⢩⡍⠀⣉⠉⠒⠂⠉⠉⠀⠀⠘⣒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠛⠀⠲⠂⠘⠛⠛⠻⠤⠀⠀⠐⠒⠀⢲⡀⠉⠉⠩⠤⠄⠀⠀⠉⠥⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣭⣭⣽⣤⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣶⣶⣾⣷⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡆⠈⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣤⡤⠼⠗⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⠀⠀⠀⢀⠀⠀⣀⢀⣀⣼⠦⠻⠚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠴⠖⠒⠛⡟⠁⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣼⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡍⣀⡀⠀⠀⠶⢺⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣀⣠⣤⠶⠖⣶⠏⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠉⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢹⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⢷⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠘⠀⠂⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠁⠀⢀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⡠⢀⣴⠿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠤⣂⣠⡾⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡄⠀⠀⠀⠀⢀⣰⣤⡶⢞⡁⡀⢈⡟⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣃⣤⣤⠴⠆⢻⡏⠄⠋⢀⣀⣠⣄⣼⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢢⡤⠟⠞⢹⡃⠀⡠⠔⠈⣠⣴⣶⣿⣿⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⣀⠸⠗⢈⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣨⣥⣴⣾⣿⣿⣿⣿⡿⠟⠙⣿⢿⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡭⠛⠟⠛⠛⠛⢛⣛⣉⠻⠞⣻⣗⡜⢼⣟⣿⣍⠛⢿⣶⣾⣿⡛⠿⢿⣤⣴⣬⣶⣶⣋⠁
⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠀⠶⠖⠋⢲⣶⠚⣳⣶⢿⣦⡾⢿⣿⣋⣀⣴⣿⡿⠿⣿⡿⠿⢛⣡⡘⠋⢘⠛⠶⠿⠿⢿⣿⣶⣿⣷⣬⣽⣿⣿⣿⡿⠟⠁⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠥⠤⠶⠾⠛⠋⠹⣿⣿⣿⠿⠂⠒⠻⣭⣿⠿⠿⠿⢓⡠⠀⢾⣿⣿⠇⠀⢨⡛⠻⢶⣶⠀⠀⠛⢛⣉⣩⣍⡙⠛⠛⠃⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⢀⣙⠫⠥⠤⠄⠈⠉⠋⢡⣤⣤⣤⣉⡁⠀⠐⢛⣫⡵⠀⠀⠈⠻⣷⣾⣯⡭⠉⠉⠛⠿⣿⡿⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠖⠂

⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢸⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠀⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⣰⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⠀⢀⣼⣵⠟⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠃⡴⢀⡴⠟⠋⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣾⣵⠟⠁⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠟⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣼⠏⣸⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⢞⡏⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡞⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢀⡞⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢀⡞⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢀⡞⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢠⡞⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⢠⠟⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠟⢉⣼⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠘⠉⠁⣸⠁
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⡀⣰⠋⠀⠀⠀⠀⢀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⣠⣾⣫⣤⠆⠀⠀⣀⣀⣴⠿⠁⠀⠀⠀⠀⠀⠈⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⣠⣾⣿⠿⠿⠶⠚⠛⠉⣩⠿⠷⠶⠶⠶⠤⠤⠶⣶⣞⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠃
⠀⠈⠉⠁⠐⠒⠒⠒⠶⢶⣾⣷⣶⣶⠶⠗⠊⢦⠉⣽⠘⠧⠄⠉⠉⠀⠀⠀⠒⠒⣖⠒⠶⡿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠀⣭⡽⠿⢷⣶⣶⣤⣬⣍⣛⣒⣒⣒⣒⣒⣒⣒⣀⣉⣤⠴⠋⢀⡴⠀⢀⣠⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣤⠆⠀⠈⠙⢿⣶⣄⠉⠉⠽⠛⠛⠻⣿⣿⣿⣶⣶⡶⠶⠛⠁⣀⣥⡶⠞⠛⠉⠉⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠄⣀⠀⠀⠉⠻⣿⣶⠶⠖⠀⠀⣸⠏⠛⠿⠷⠶⠶⠾⠛⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠈⠀⠀⠈⠻⣿⡟⠻⠿⣿⣿⣿⡿⢿⣿⠃⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣧⠀⠁⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⠯⠿⠛⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⡿⢿⡿⠏⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠎⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠿⠛⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣷⣦⣤⣶⡚⠡⣄⡀⢀⣀⡀⠠⣤⠀⠪⢥
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⠻⣿⣿⣿⡿⠟⠓⠈⠁⠀⠀⠁⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣴⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡾⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⡿⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣿⠃⠀⠀⢠⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⡿⠀⠀⠀⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣺⡇⠀⠀⠀⠁⠀⡜⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⣿⡇⠀⠀⢨⠀⠀⠀⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣹⣷⠀⠀⡄⠀⠀⠄⢼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⢹⣿⡆⢀⠁⠀⡀⠠⢼⡄⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⢿⣿⡌⠀⠰⠀⠀⣾⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠂⠀⠀⠀⠘⣿⣷⡀⡀⠀⠀⠈⡇⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⢿⣷⡇⠀⠀⢠⣿⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢄⣦⠈⢿⣧⠀⢐⣾⣿⡄⠀⠀⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠉⠺⠾⣿⣧⠀⠁⣿⣶⡀⠀⢉⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢐⣾⣿⣿⣧⠀⢻⣿⣯⣶⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣠⣴⣾⡿⠟⠋⠁⠀⢙⠁⠉⠉⣩⣭⣍⠉⢉⣴⡿⠁⠐⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⢿⣶⣶⣾⣷⡾⠿⣦⣄⣬⡉⣿⢿⠿⠿⠿⡿⢶⣾⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣴⣮⣤⣤⣬⣤⣴⣦⣌⣙⣏⣉⡛⠻⣛⣛⡹⢟⣋⣟⣉⠉⣉⠛⣿⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⣿⠿⠿⠿⠛⠛⠛⠛⠋⠉


⡶⠶⠶⠷⠖⠲⠶⠶⠶⣶⣷⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡶⣶⣶⣶⣷⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⣠⡄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⣀⠀⠀⠀⢸⠀⠀⢠⣄⠀⠀⠀⠀⢸⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡽⡗⢄⡀⢹⠀⠀⠀⣿⣷⠢⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡇⢻⠀⠙⣿⡇⠀⠀⡟⡇⠁⠘⣷⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⢈⣿⣇⠀⢠⡇⡅⠀⠀⢀⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿⣿⠀⠸⠀⠀⠀⠀⠈⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⡟⡇⡆⠀⠀⠀⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣇⣿⠃⠀⠀⠀⠀⠀⠀⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣷⠄⠀⠀⠀⢀⠀⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⣿⣯⢲⠀⠀⠀⠀⠀⢀⠀⣀⣐⢄⢀⣱⡆
⠀⠀⠀⠀⠀⠀⠀⡼⣃⠀⠀⠀⠀⠰⢴⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⣸⣿⠘⡀⠀⠀⠀⠀⣨⢄⠁⠈⠁⠁
⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠠⠀⠀⠲⢶⣿⠿⡋⣽⣶⢦⡆⠀⠀⠀⢹⣿⡆⡇⠀⠤⠂⢀⣤⡿⠃
⠀⠀⠀⠀⠀⣰⣿⣧⠴⣋⠀⢀⣤⣤⣬⣿⣿⣷⣿⠒⠒⠚⠓⠓⠂⢾⣿⠿⢧⣠⣴⣾⣿⡿
⠀⠀⠀⠀⠀⠉⠁⠉⠀⠐⠒⠒⠀⠀⠀⢽⣿⠿⢿⣇⡀⠀⠀⢀⣶⣿⣷⣾⣿⡿⠿⢿⣿⠃⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣤⣾⣿⣤⣶⣾⣿⣿⣿⡛⣟⣥⡤⠀⣤⠁⢰⣧⠀⢨
⣿⣿⣿⣶⡬⠑⠤⣄⡑⣠⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⢫⣽⣷⣶⣿⠀⣾⣿⠆⣿⣦⣀⠀⠀⠀⣀⠄
⠛⠛⠛⠦⠉⠀⠀⢰⣥⣉⡙⠓⠶⣿⣯⣯⣽⣿⣿⣯⣴⣿⣿⣿⣷⣦⣿⠿⣿⣿⣿⣿⠂⢿⠀⠀⣿⣿⣿⠻⣿⣿⣶⣏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢶⣦⣈⠙⠛⠛⠻⢯⣭⡍⠉⠁⠀⠀⠈⠙⠓⠀⠈⠀⠀⠀⠀⠀⠂⢻⣿⣿⠃⠐⢯⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠹⣆⠀⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢻⡄⠈⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡜⣷⠀⠈⣿⡟⡿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠍⣹⡇⠀⠘⡄⠁⡈⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⠈⢴⣿⣏⠀⠀⠘⠂⠀⠀⠈⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡇⠀⠀⠀⠀⢸⣇⢀⣸⣿⣿⡆⠀⠀⢰⡀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠐⠢⠤⠶⠚⠋⠁⠀⠀⠁⠀⠀⠀⠀⣸⣗⣤⣾⡿⠻⣧⡀⠀⠀⢳⠀⠀⠀⠀⡑⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⢸⣿⣄⠀⠈⢃⠀⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⢸⣿⠛⣄⠀⠀⣦⣿⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠎⢿⣰⣿⠀⠀⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⡟⠘⡄⠀⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢿⠃⠀⣴⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠈⠀⢠⣿⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠌⠀⠀⣠⡿⠁
⠀⠀⠀⠀⢀⣀⠤⠶⢾⣿⣿⣿⣿⣿⣿⣶⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠒⠒⠲⠧⣬⡭⢭⣿⣽⣓⣂⣀⠀⠠⠀⣤⡎⠀⠀⠀⣴⣏⣀⣄⣼⠛⠃⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠻⣿⡇⠀⡠⢾⣿⣿⣿⣿⣿⡷⠦⣄⠀⠀⠀⠀⠂⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡥⠊⠀⠀⠈⠙⠛⠿⠿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⡇⢰⣿⣧⢲⣶⡄⣤⣹⣄⣀⣀⣀⡔⣁⣀⡀⠠⠤⠄⠀⠀⣀⣀⣠⣤⡾⠁
⠀⠀⣠⠰⢶⡤⣤⣼⣿⣿⡿⠿⠿⠿⠍⠉⠉⠃⠀⠉⠉⠉⣩⣥⡈⢩⣭⡭⢩⣿⢿⣉⣴⣶⣶⠶⠾⠟⢛⡫⢵⣶⠟
⠀⠀⠹⣿⣿⢟⣿⣿⣿⣿⣧⠀⢀⠀⠀⠀⢀⠀⠤⠄⠠⠶⠶⠶⠖⠒⠚⢿⠟⠛⠛⠩⠏⠁⠀⠂⠁⢀⣤⣤⡿⠁
⠀⠀⠀⠙⢿⣿⣿⡿⠿⠿⣿⣧⢿⣶⣦⣤⣤⣤⡤⠤⣤⣀⣀⣀⢀⣀⠀⣀⠀⠀⠀⠄⢀⣴⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⣀⣙⣿⣷⣶⣶⣿⣿⡔⣀⣠⡤⣤⣤⠀⠀⠉⢉⣭⣉⣉⡩⠌⣉⣁⣀⣀⣠⣉⣽⡿⠿⠿⠧⢤⣀⣀⣀⣀⣀⡀
⠠⠤⢤⣤⣬⣭⣶⣿⣧⠤⠟⠛⠉⠁⠀⠀⠀⠤⠭⡿⠶⠟⠋⠉⠀⠀⠀⠀⠉⣉⡻⠿⠋⠁⠀⢤⣤⣤⠴⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣒⣒⡢⠤⠤⣤⣤⣤⣤⣤⣴⣦⡤⠄⠒⠈⠀⠀⢀⣠⠴⡲⠻⠛⠻⢭⣴⣀⣄⡀
⠀⠀⠀⠠⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠤⢤⣩⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠸⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡇⠀⢧⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡇⠀⢸⡀⢣⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇⠀⠀⡇⠈⡇⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠀⡇⠀⢹⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡇⠀⢰⠃⠀⢸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⡇⠀⡼⠀⠀⠈⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠⡇⢠⠃⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⣧⡏⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⣦⣥⣶⣶⠿⡿⠭⠿⠟⠛⠒⢲⣼⣿⣧⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⣒⣋⣉⣁⣠⣤⣤⣤⣶⣶⣶⡿⠿⠟⠛⠉⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠛⠛⠛⠛⠛⠋⣉⣉⣉⣁⣠⡤⠤⠤⠖⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠤⠜⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⣿⡓⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⣿⣇⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢸⣿⢻⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢸⣿⢸⡀⠀⠀⠀⣠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢸⣿⠸⡇⠀⢀⣼⠟⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣿⠀⣇⣴⡿⠁⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠊⠉⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠋⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⠀⠀⡠⠋⡰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⢠⠞⢡⠞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⢀⡴⢃⡴⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⣠⠊⣠⠎⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡟⢁⡜⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡶⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣢⣤⣼⣻⣤⡤⠤⠬⠭⠭⠭⠭⠭⠭⠭⠭⠤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠙⠛⠙⠛⠓⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⡱⣄⢀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⠀⢀⣠⠃⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠉⠑⠊⡉⠒⠋⠙⠒⢉⠑⠊⠉⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⠁⠀⠀⠈⠉⠋⠉⠉⠀⠀⠀⠀⠀⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠉⠓⠋⠀⠀⠈⠛⠉⠙⠋⠙⠚⠁⠀⠀⠀⠀⠈⠓⠋⠙⠋⠉⠋⠁⠀⠀⠙⠚⠉⠛⠋⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⢀⣀⣀⣤⣤⣄⡀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⢸⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣀⣠⣴⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣾
⠘⠛⠛⠛⠛⠛⠛⠛⠁⠐⠒⠛⠛⠛⠟⣀⣀⣸⠤⡤⠤⠤⢴⠶⠶⠖⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⠛⠛⠲⠾⠿⠿⢿
⢠⣿⣤⣶⣶⣦⣤⣿⢓⣶⣶⣶⣶⣶⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣧⣬⣿⣿⣯⣤⣤⣤⡌⣢⠃⣾
⢀⣟⢻⣿⣿⣿⣿⣿⠧⢀⣈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣁⠀⢀⣤⣽
⠐⣷⣾⣿⣿⣿⣿⣿⣜⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣟⣻⡿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⣿⣿⠿⠂⠀⠀⠰
⠰⣿⣿⣿⣿⣿⣿⣿⣯⡁⠁⠉⠀⠀⠒⠂⠀⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⠀⠐⠀⠀⣀⢰
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣍⣁⣧⣴⣶⣷⣾
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⡀⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠦⠄⢠⣤⣼⣿⡋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻⣍⣁⠀⠠⣶⣦⣽⣭⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿⣛⠛⠛⠿⠷⠶⠶⢶⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣍⣻⣟⣿⣶⣶⡶⠶⠦⢀⣠⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠛⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⣀⣨⠧⠔⠶⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠛⠉⠉⠁⠀⠀⠀⠐⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⡀⠀⣀⣣⡤⠖⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡦⠤⠖⠚⡟⠉⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢀⠀⢤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⣟⣤⣄⠠⠶⠒⣻⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣠⡤⠶⠶⠛⢻⠏⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣧⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢻⡀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠁⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠘⣠⡿⠋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠴⠚⠓⣾⠟⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⣀⡠⣤⠖⠛⠟⠐⠃⠘⢰⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣀⣠⢥⢷⡎⠉⠁⡠⠘⢉⣠⣴⣶⣿⣯⣿⣦⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠉⠀⠐⢈⡧⠐⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣒⣋⣡⣴⣾⣿⣿⣿⣿⠟⠻⡿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢈⠛⠿⠿⠿⠛⠛⣛⣛⠻⠞⣷⣦⠓⣿⣿⣿⡛⢿⣿⣿⣿⡿⠶⣶⣤⣼⣶⣶⡖⠒
⠀⠀⠀⠀⠐⠾⠃⠀⠀⠀⠲⠖⠂⣰⣶⢚⣳⣾⣷⣠⣿⣿⣚⣀⣴⣿⡿⠿⣿⡿⠿⢋⣤⠘⠉⡛⠶⠾⠿⢿⣿⣷⣶⣶⣭⣿⣟⣿⢿⠿⠋⣀
⠀⠀⠀⠀⠀⠠⣤⣤⠄⠿⠶⠼⠟⠛⠉⢹⣿⡿⠿⠷⠒⠚⣿⠿⠿⠿⠟⣛⡄⠀⣺⣿⡿⠀⠀⣝⡛⢷⣖⠀⠈⣙⣋⣩⣍⣉⠛⠛⠛⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣀⡙⠛⠒⠒⠀⠈⠉⢁⣠⣴⣤⣤⡁⠀⠐⠻⡭⠆⠀⠀⠙⠿⢷⡿⠔⠋⠉⡙⠻⠿⠿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⢤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⣤⡀⠀⠀⠀⣿⠒⠂⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣿⣅⠁⠒⠒⢻⠀⠀⠀⠀⠀⣀⣤⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣷⣄⣠⣼⣦⣴⣶⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⢀⣀
⠀⠀⠤⠍⠀⠀⠛⣒⣒⣦⣥⣤⣼⣟⣛⣻⣿⣯⣿⣿⢿⣿⣿⣿⣿⡿⠿⠿⣛⣋⣉⣀⣴⣾⣿⣿⣿⣿⠦⣤⠀⠁⠀⠭⠤⠤⠭⠍
⠀⠀⠀⣒⡚⠛⠛⠛⢿⣿⣋⣭⣭⣹⣿⣿⣿⣿⣿⣇⣼⣿⡶⠓⢙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣋⣁⣀⡀⠠⠤⠤
⠰⠶⠽⠿⠿⣿⣶⣶⣶⣿⡟⠛⠛⠛⠽⠿⠿⢷⣶⣶⡷⠟⢛⣿⣿⣿⣿⣽⣭⡽⠛⠛⠛⢳⣖⣻⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠤
⠀⠀⠒⠶⠦⠤⠤⠀⢀⣈⣙⠒⠶⠶⠶⠂⠀⠉⠿⠿⠷⠦⠸⠿⣿⣿⠉⠀⠠⣤⣤⣄⣤⣭⠭⠭⠭⠼⠿⠿⠆⢀⣠⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠶⠶⠖⠀⣀⣀⣀⣀⡉⠉⠉⣻⣿⠁⠀⠀⣀⣀⣀⡀⠤⠤⠶⠛⠋⣀⣀⣀⣀⣀⣀⣤⣄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡤⠀⠠⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠈⠀⠀⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠤⠄⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢡⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣠⠄⠀⠀⣀⣠⣠⣤⡶⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣴⣤⣤⣄⣤⣶⣶⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠉⠉⠭⠭⠩⠭⠍⠍⣉⠉⣉⠐⣯⣮⣉⣙⣿⣾⠆⢉⠻⣎⠓⡻⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠊⠉⣉⣉⣉⣩⣥⣶⣤⠌⠙⠿⠛⠛⠿⠒⠈⠀⠈⠉⠉
⠀⠀⠀⠑⠒⠒⠒⠀⠀⠀⠸⠍⢀⣀⣈⡉⠉⠉⠁⠉⠙⠉⠀⠉⠀⠀⠐⠒⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠈⠀⠀⠈⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢀⡞⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⣠⣯⠾⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⢃⡞⢀⠾⠋⠁⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣾⡶⠉⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣻⡟⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⡟⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠈⡜⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⡜⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⡞⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢀⡞⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⢀⡞⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠛⣡⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣠⠞⢁⠀⠀⠀⠀⢰⣤⠏⠀⠀⠀⠀⠐⣄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣠⣾⣿⣟⣭⡤⠴⢖⣾⣯⣁⣤⣀⣀⣀⣀⡤⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟
⠀⠐⠛⠛⠉⠭⠤⢤⣤⣤⣮⣤⣤⣤⠄⢠⡐⢲⠽⣟⠒⠂⠀⠀⠀⠀⠤⠤⠤⣤⣴⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠝⠉⣙⣿⣶⣦⣤⣅⣈⣓⣒⠢⠤⠤⠤⠤⠤⠤⣙⣀⡤⠇⢀⡄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⠄⠀⠙⠻⣿⣍⠉⠉⠿⠛⠻⢿⣿⣿⣭⣤⡴⠶⠋⢀⣨⣥⠶⠚⠋⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠄⠀⠀⠈⠻⢿⣶⠶⠂⠀⣀⡿⠙⠻⠷⠶⠶⠞⠋⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠤⠀⠠⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣓⣒⣒⠢⣀⣀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡇⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠛⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⣀⣀⢀⡦⡄⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠿⢿⣿⣿⣿⡿⠟⣿⣗⡶⣿⠽⠿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡶⠶⠶⡇⠄⡀⠀⠈⠀⠀⠀⣧⣤⣽⡴⣶⠿⢿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⡿⢻⠷⢾⣿⣦⣴⣗⡪⢟⣫⠤⠒⠉⡀⠄⢚⣯⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⣉⠿⡾⣬⣠⢸⣂⣩⡴⠞⠉⣁⠀⠐⣀⣤⠶⡫⠚⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡧⠖⢋⣧⡗⠊⠧⢤⣉⣻⡟⠀⡄⠀⢀⡥⣖⠾⢋⠵⠋⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⣁⡴⠚⠡⠈⡉⠀⡀⠄⠊⢹⣿⠒⢧⡪⠕⢊⡥⠚⠁⢀⡴⢛
⠀⠀⠀⣄⢀⡤⠚⠉⣀⠤⠚⠁⠂⠀⠀⠀⠀⠀⣀⣤⡾⠴⠛⠋⢉⡤⠒⠁⠀⢀⡴⠛⠚
⠀⠀⠀⠸⣗⠒⠒⠚⠧⠖⠒⠒⠒⠒⠓⠚⠉⠉⠁⠀⣀⡤⠔⠪⠁⠀⠠⣠⢾⠍
⠀⠀⠀⠀⠈⠳⣤⡤⣀⣀⣀⣀⣀⣠⣤⠤⠖⠒⠊⠉⠀⠀⠀⠀⠀⡠⠞⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⣎⣈⡁⠀⠀⠀⠀⠀⠀⢀⡴⡞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠬⣿⡧⠤⢀⣀⣠⣤⡔⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠛⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠒⠒⠒⣿⡂⠤⠤⣯⠀⠀⠀⠀⣀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣦⣤⣼⣴⣶⣾⣿⣿⣿⠏
⠀⠀⠈⠁⢀⣀⣀⡀⠙⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣋⣯⣽⣻⣇⠀⠀⠀⠀⠀⠲⠄⠐⠒
⠀⠈⠡⠴⠶⠶⣾⣟⣛⣛⣿⣿⣿⣿⣟⣽⣿⠟⠿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠈⠍⣉⠁
⠒⠚⠛⠿⠷⠶⠾⣟⡛⠛⠫⠛⠻⣷⣾⠿⢛⣿⣿⣿⠿⠯⠟⠛⢛⣷⣛⡿⣿⣿⠉⠉⠁
⠀⠀⠉⠛⠛⠀⠐⠦⠭⣍⣉⣁⠀⠈⠛⠋⠥⠽⣿⡧⠀⠐⠒⠒⠒⣒⣀⡭⠉⠉⠀⠐⠒⠒⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠛⠃⠀⠀⠉⠑⠒⠒⠀⠀⠈⠉⠙⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢷⠀⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣦⠀⢿⠏⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣽⡄⠈⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⢨⣿⡄⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠄⠐⠒⠀⠈⠉⠀⠀⠀⢸⣃⣾⠟⢳⡄⠀⠐⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⣿⠄⠀⠀⢀⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢿⢸⡆⠀⢫⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣶⠾⠿⠯⣿⢿⣦⣀⣀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠃⠒⠒⠶⠤⠤⣤⣴⡇⠀⣠⣾⣦⣴⣯⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠉⠙⠿⠿⢯⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⡿⠰⢿⢰⣶⠤⣤⣤⣤⣀⣀⡄⠀⠀⢀⣀⣀⣤⠤⡶⠃
⠀⠀⠀⠀⠀⠰⣠⣰⣶⣿⣿⡏⠉⠉⠀⠀⠀⢀⡀⣀⣒⣒⡐⠺⣥⡿⣷⠶⠞⠛⠫⠉⠁⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⠿⣿⡤⣦⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠨⠽⠷⠶⣾⣧⠠⠔⠒⠤⠄⣀⣀⣤⠤⠀⠀⠤⣤⣤⣤⡴⠛⠛⠁⢠⣤⠤⠤⠄
⠀⠀⠀⠀⠈⠙⠛⠚⠛⠋⢉⠉⠀⣀⣀⣀⣀⣉⣉⣉⣀⠀⠀⠀⠀⠀⣉⣡⣤⣄⡓⠛⠉
⠀⠀⠀⠀⠀⠀⠠⠔⠛⠋⠉⠀⠈⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢸⡇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠃⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⢀⡼⣃⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇⢀⠀⠀⣠⣾⠿⠋⠁⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⢠⡟⣠⡾⠋⠁⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡟⣹⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣼⡟⢀⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⠜⡿⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⡼⠁⠈⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⡾⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⡾⠁⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⡾⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⢉⣴⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠉⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢠⡀⣰⠏⠀⠀⠀⠀⠀⢀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⣀⣴⣋⣡⡤⠀⠀⠀⢀⣀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠘⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⣠⣾⣿⣿⣿⡷⠶⠶⠚⠛⣹⠿⣷⣶⣶⣦⣤⣤⣤⣤⣴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠁
⠀⠀⠚⠛⠋⠡⠤⠤⢤⣤⣤⣤⣾⣥⣤⣤⡤⠤⠤⢶⠒⠒⡆⢿⣋⠛⠒⠒⠒⠀⠀⠠⠤⠤⠤⢤⣤⣴⣶⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡹⠯⠛⠻⣷⣦⣤⣅⣈⣁⠘⠳⠦⠬⢥⣤⣀⣀⣀⣀⣀⣐⠺⠗⠀⣀⡵⠀⠀⡈⠉⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡟⠋⡉⠙⠛⠿⣿⣟⡛⠛⠲⠶⢶⣶⣶⣶⣶⣶⣶⣖⣚⣋⣉⣠⣤⠞⠀⠤⢒⣋⣭⣤⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠈⠻⣿⣦⣄⡀⠋⢀⡀⠈⠹⣿⢿⣿⣿⣛⣛⣉⣁⣤⣴⠾⠛⠉⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠀⠀⠀⠀⠈⠙⠿⣯⣍⣉⣀⣠⣴⠟⠀⠉⠙⠛⠛⠛⠋⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠒⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⢄⢠⠄⠀⠀⢸⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠀⠤⢠⠂⠀⠁⡬⠁⠀⢨⡀⢣⠀⠀⠈⠀⢎⠳⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣷⣶⣧⢰⣿⡄⠀⢀⠇⠘⡇⠀⠀⠀⠘⡆⠈⢊⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣂⢀⣼⡆⠀⠀⠀⠀⠀⠀⠈⣢⣀⡇⠀⠀⠀⠀⢻⡀⠘⢷⡀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢸⣿⠍⠐⠶⣖⣂⣂⠧⠀⠀⢳⡄⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⣻⠀⠀⠀⠀⠀⠀⠁⡆⠀⠈⢿⡄⠀⠀⠹⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣰⡟⠀⠀⠀⠀⢀⠄⢠⡾⣿⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣍⠿⠿⣯⡀⠒⠛⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢘⣀⡴⣿⠇⣀⣤⡬⢤⠀⠤⠽⢷⡟⠐⣌⠢⠀⣀⣀⡀⢨⡇⡰⠃⠈⠉⢉⣉⠿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢘⣍⣀⣀⣀⣿⠉⠀⠸⠇⢸⠀⠀⠀⢠⠁⠤⢶⡞⣯⢿⣇⣀⣼⣼⣤⣴⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⡧⠘⢛⢙⠛⠛⠟⠛⠲⠾⠶⢶⠖⣛⣒⣚⣻⡿⠛⠭⠿⠻⠛⣛⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⡬⠤⣤⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢰⣶⣿⣿⡿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢠⠃⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣀⡀⠠⡯⠣⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⠀⠀⠀⢸⠀⠀⠠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣼⣏⣩⣖⢦⣤⣄⣀⣀⠀⢈⡇⠀⠀⠁⡀⢀⣆⣩⣤⡶⠃
⠀⠀⠀⢀⣀⣊⣛⣿⣷⣴⣶⣶⣦⣼⣿⣿⣿⣿⣓⡻⠶⠾⠶⠾⢒⡚⢳⣿⣿⣭⣉⣙⠝⢁
⠀⠒⠀⠀⠿⡉⠹⠿⠶⠟⡉⠽⢯⣭⠋⡐⠿⠿⠧⠽⠿⠟⢀⣽⣿⡋⠙⣛⡂⠘⠛⠯⠐⠛⠀⠀⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠀⠖⠐⠛⠛⠧⠄⠀⠛⠂⠰⠦⠉⠉⠡⠄⠀⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢾⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢸⡏⢹⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⢸⡇⣰⠌⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⢸⡏⠁⠨⠶⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣾⠇⠀⠀⢸⡇⠀⠀⢐⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢠⣿⠀⠀⠀⢸⣿⢶⣞⣯⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⣿⠀⠀⠀⢸⡇⢻⡅⠞⠃⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⢸⣿⡀⠀⠀⢸⡇⠀⢂⣠⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢸⣿⠁⠀⠀⢸⣗⡀⠈⣿⡯⠖⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⠀⠈⡶⣒⣈⠄⠀⠈⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠸⣟⢿⡄⠀⢸⣯⣀⣤⠀⠺⡿⠂⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⠄⠀⠀⠀⠀⠀⠀⣿⢘⣷⡀⢸⣿⣫⣥⡀⠀⠸⠀⠀⠀⠀⠀⢀⣱⡀⢀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠡⠀⠀⠀⠀⠀⠀⠠⠈⢿⣿⣷⣸⣿⣿⣿⡁⠀⠀⣡⣤⣶⠾⠛⠋⠙⠉⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⣀⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⡷⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠐⢒⣖⠠⣤⢠⣿⣿⣤⣶⠾⠿⠟⠛⠛⠛⠛⠉⠉⡉⠉⢻⣿⠀⠀⠀⠀⢀⡀⣀⣀⣀⡀⠀⠀⢀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⢿⠾⠶⠶⠶⠶⠶⠶⠷⠾⣿⠿⠿⠿⠿⠿⠶⠶⠷⠟⠿⠿⡟⠿⠷⠶⢶⡶⠶⣿⠷⣦⠔⠃
⠀⠀⠀⠀⠀⠠⠤⢤⣤⣵⣤⠈⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠛⠒⠒⠀⠀⠙⠛⠋⠀⠀⠛⠛⠓⠈⠛⠛⠛⠾⠟⠛⢋⣷⣤⣤⣤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣷⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡤⠤⠤⠴⠶⡾⠿⠯⠥⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠬⠉⠙⠛⠖⢛⣻⣿⣟⣙⠛⢻⣯⣭⣄⣉⡙⢛⣯⣭⣥⣭⣭⣥⠶⠾⠟⠛⠛⠛⠛⠒⠶⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢉⡙⣛⣉⣉⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣄⣘⡛⠻⠿⢿⣿⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⢉⣉⠁⠀⠤⠖⠒⠒⠛⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣤⣤⣶⣶⣶⣾⣻⣂⣀⣀⣀⠀⠄⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠉⠉⠉⠉⠉⠉⠥⠤⣤⡀⠉⠑⠓⠲⣶⢶⣶⣶⡶⣶⣶⣶⣦⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⠖⢛⣩⠤⠀⠀⠐⠀⠐⠛⠛⠛⠛⠛⠛⠉⠁⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠉⣀⡀⢀⣀⠈⠃⣀⡉⠀⠀⠀⠀⠒⠒⠂⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⢄⠖⠉⠳⠄⠀⠀⠙⠀⠀⣦⠴⢶⣦⠶⣑⠟⢙⣇⣴⣤⣄⣠⣺⣷⣂⢄
⠀⠀⠀⠀⠐⠛⠚⠛⠭⠯⢍⣴⣶⣾⣯⣵⣿⣷⣷⢾⣖⣲⣶⣾⣿⡷⣶⣤⣴⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⡃⢰⡗⣰⣦⣄⣀⣴⡄
⠀⠀⠀⢀⢀⣄⣤⣴⡆⠄⠴⣿⣍⣿⣿⣿⣿⠾⡶⢾⣿⣭⣿⢿⣿⣿⣞⣩⣾⡿⠴⠧⢿⣿⣿⣿⣿⣷⣦⣽⣿⣿⡷⣿⣿⣿⠿⠿⠃
⠀⠂⠰⠗⠶⣾⠛⣛⠿⡿⢷⣶⣿⣿⡟⢻⡟⢺⣿⣾⣿⣩⠇⠀⠀⣠⣿⣿⣷⠁⠀⠀⠀⢀⣼⣿⣟⣿⣻⣿⣟⡿⡟⢹⡂⠀⠀⠄⠀⡀
⠐⠂⠲⣦⣨⣨⣥⣼⡚⠻⢿⣿⣟⣻⣿⣽⣿⣿⣿⣽⣿⠟⣲⡿⠿⠿⢿⣿⣿⡶⠶⡶⢶⠽⢿⣿⠉⢉⠉⠉⣻⣿⣿⣿⣿⣿⣯⠷⠀⠻
⠀⠀⣠⣟⣻⣿⣏⣩⣹⣿⣟⣻⣏⢻⢿⣾⣿⣷⣿⠿⠋⣵⣿⣀⢀⡀⣼⢟⡉⠓⡠⠀⠀⠀⠠⣿⡤⢤⣤⣶⣿⣿⣿⢻⣍⠉⠃⢿⡈
⠀⠀⠀⠀⠀⣉⣩⣟⣛⣻⣿⣿⣯⣶⣭⣿⠟⠉⠀⢀⣼⣿⠏⠉⠙⠉⣳⣯⠀⠀⠃⠀⠀⠐⣶⣿⠇⠀⠀⠀⣹⣿⣿⣶⣾⣿⣶⣴⣄⡀
⠀⠀⢐⡿⠻⣿⣾⣿⣿⣿⣿⣾⣿⠿⢋⡤⢀⡴⢁⣾⣿⣿⠶⠦⣤⣶⣿⣿⡠⠰⢶⡶⣶⣱⣿⣿⣧⣀⣄⣶⣿⡏⢹⣿⣿⣿⣿⣿⣷⣦
⠈⠀⠀⣯⠖⠷⢶⣿⣿⣿⠿⠋⢡⣶⠟⣠⡟⣠⣿⣿⡿⠁⠀⠀⠀⠀⢈⣿⡏⠀⠁⠀⠀⠀⠈⢹⣿⡙⡛⠛⠻⡇⠈⣿⣿⣿⣿⣿⣿⡗⠂
⠀⢈⢉⣿⣭⣿⣶⣿⣯⣥⣔⡚⡋⠽⠾⢿⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡯⡆⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⣷⠀⠘⢿⣿⣿⣻⣿⠷⡦⠁
⠀⠈⠀⠉⣟⣿⣣⣿⣵⣿⣿⣽⣷⣲⣶⣿⣿⣿⣿⣿⣳⣶⣤⣤⡀⢀⣿⠿⣷⣿⣷⣶⣤⣶⣶⣿⢻⣶⠒⢦⣕⣿⣇⣀⢌⣿⡡⠘⢧⣀
⢤⣼⣤⣥⣭⣭⣭⣿⣿⣿⣿⣿⣿⣭⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣦⣤⣤⣤⣤⣦
⠀⠈⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣟⣿⣿⣿⣟⣿⡏⠉⠁
⠀⠀⠀⠘⠛⠛⠛⠻⢻⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣭⣿⣿⣿⣏⣾⣿⣿⢛⣤⣴⣋⣹⠾⠛⠿⠮⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⠺⠛⠳⠛⠛⠻⠟⠻⠟⠿⠿⠿⠎⠹⠤⠌⠽⠉⠛⠉⠛⠉⠁⠈⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣣⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡾⢋⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⣀⡴⣫⡾⣫⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢃⠀⠀⠀⠀⣠⠞⣫⢾⣫⡾⣻⠿⣋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠀⣠⡴⢛⡵⢚⡵⢟⡥⢞⡡⢞⡥⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⣟⣡⠖⣫⠞⣩⢖⡫⢖⡩⢖⣡⠞⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢻⡴⢚⢵⡪⢒⣩⠖⣋⡴⢛⡵⣫⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠀⠀⠀⠀⠀⠀⠀⢺⡕⣫⠖⢋⣴⠞⡩⢖⣭⢞⣩⢞
⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣻⣵⡟⠀⠀⠀⠀⠀⠀⠀⠀⢊⣠⠾⣋⠔⣫⠖⣫⠶⣫⣴⣿
⡶⠮⣍⡛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠟⣃⡭⠾⣋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⢞⡵⣊⡵⣻⡵⠞⣉⣤⣴
⣉⡛⢦⣍⡳⢬⡙⠷⣦⣀⠤⠤⠤⠤⠶⣶⡿⠿⠛⣉⡡⠔⢚⣉⣥⣶⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣾⡿⣺⣵⠾⠛⢋⣩⣴
⡯⣝⡓⠮⣝⡶⣌⡳⢬⡙⢷⣦⡤⠭⠵⠖⣚⣋⣭⣴⣶⡿⠿⣛⣩⠵⣒⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣈⣠⣤⠶⠟⣛⣽⣿
⣭⡓⠭⡓⢮⣙⠦⣝⠶⣝⠦⣈⠻⣿⡛⠉⠝⣓⡫⠭⠔⣒⣩⠥⠒⠋⣉⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣡⣴⡶⢟⣿⣟⡵
⢌⡙⠷⣦⡑⠈⢑⠈⠳⢾⣿⡮⣓⠌⠳⣌⠡⠶⠒⠓⢉⣁⣤⠴⠞⣛⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⣠⣴⡿⣫⠞⣡
⠶⢦⣅⡈⠙⠷⣦⣈⠢⣄⠙⢿⣮⣷⢤⡈⢣⡐⢒⠈⠡⣁⣢⡼⢟⣋⡁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢛⡿⢋⡼⢡⡞⢁
⣿⣷⣮⣝⡳⢦⣄⠉⠻⣮⣓⢷⡿⣛⣷⣹⣤⡴⠶⢾⣛⣉⠙⠲⠦⣬⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⠔⢉⡴⣫⡶⠋
⠭⢍⣛⣻⣿⣷⣬⣽⡤⠶⠿⠛⠛⠯⠭⣉⡓⠲⠶⣤⣍⡙⠛⠶⣤⣀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣘⣥⠞⣫⡾⢋⣤⠾
⢷⣶⣤⣐⠲⢤⣤⣀⣀⡀⠐⠛⠛⠛⠒⠶⠦⣭⣝⣒⠾⠿⣷⣦⣤⣉⡛⣿⣀⣀⠠⠤⠄⠀⠀⢴⣢⡼⠛⣁⡴⣛⣡⠆⠙⠴⣋⡵⢛
⢦⣌⠓⠮⣽⣒⠶⠬⣭⣝⣛⣛⡛⠛⠛⠒⠲⠶⠾⠽⠿⠷⣶⣾⣿⣿⣿⣿⡋⡏⠉⠉⠉⠀⠀⠒⠒⠒⣈⠡⠬⠭⠭⠥⠵⡯⠵⠚⢋
⡳⢭⣓⠦⣬⣉⡛⠶⢤⣮⣭⣭⡭⠿⢿⣿⣶⣖⣒⣒⣲⣶⣶⣶⣯⣽⠟⠉⣇⢱⠀⠀⠀⠀⠀⠀⠭⠀⠀⣒⣠⣤⣶⡶⠶⠷⠖⣛⣋
⣍⣓⠮⣭⣛⣿⡿⣿⣦⣶⣤⣍⣩⣭⣭⣧⣤⠤⠶⢿⣿⡿⣟⣫⣭⠿⢷⣾⣿⡘⣤⣶⣤⣀⣠⣴⣶⡾⠟⠛⢉⣁⣤⣴⡶⠿⢛⣛⣉
⣚⢿⣿⣿⣿⣿⣿⣿⣛⣛⣿⣿⣿⣶⣶⣶⡶⠿⢛⡩⠥⠴⠶⣖⣒⣛⣉⣽⠿⣱⣭⣿⡿⠟⠋⢉⣠⣤⣶⣿⣿⡿⠿⠿⠛⠛⢉⣉⣉
⠿⣿⣶⣾⠭⢭⣭⣭⣽⣿⠿⠿⣛⣋⣩⣴⣶⣬⣥⣬⠄⠩⠭⠭⠭⢖⣪⣶⠿⠛⢉⣁⣤⣶⡿⠿⠛⣋⣉⣡⠤⠴⠒⠚⠛⠛⠉⠉⠉
⣽⣶⣶⣿⣿⣿⣯⣭⠶⢂⣉⣡⠤⠤⠒⣒⣒⣒⣒⣒⣒⢒⣿⡿⠿⠋⣁⣤⡴⠾⢛⣉⣥⠴⠖⠚⠉⣉⣀⣀⣤⣤⣤⡶⠶⠶⠶⠟⠛
⣭⡭⠭⣷⢒⣪⡭⠖⣛⣩⠥⢖⣛⣋⣉⣭⣭⣭⣉⡭⠟⢋⣡⡴⠾⠛⣉⡥⢖⣚⣩⣥⡶⠶⠟⠛⣛⣉⣭⣭⣴⠶⠶⠶⠶⢞⣛⣛⣛
⢖⡶⠟⣋⡭⢄⣒⣭⣶⡶⠿⠿⠿⢯⣤⡤⢞⣫⣭⠶⣛⣩⣥⣶⣾⡿⠶⢛⣋⣩⣥⣴⠶⠾⢛⣛⣉⣭⠤⠴⠖⢒⣛⣉⣉⣉⣉⣉⣉
⠛⠚⠛⠑⠚⠛⠻⠟⠛⠻⠿⠷⣶⣿⣿⣛⣫⠵⣖⡿⠿⢛⣋⣭⡴⠶⠛⣛⣭⡭⠶⠖⢚⣛⣭⡭⠶⠶⠚⣛⣛⣉⣭⣭⣭⣥⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⣿⠈⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢒⣛⡓⣒⠒⠒⠛⠒⠒⠶⠶⠾⠶⠶⠶⠶⠶⡀⠔⠊⠀⣷⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣙⣿⣿⣿⣿⣿⣿⣿⡟⠿⡷⠶⣶⣾⣷⣶⣶⣶⣶⠿⠶⠶⠶⠼⠶⠶⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠿⠿⠿⡿⠿⠿⣿⡿⢋⣴⣿⣿⣿⡿⠿⠿⠛⢻⡿⠿⠿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠒⣒⣚⣛⣛⣒⣒⣒⣒⣒⣻⣟⠒⠒⢲⣶⠿⠛⠛⠛⠒⠒⠒⠒⠀⣼⡇⠀⠀⠀⠈⠙⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⢛⣛⣓⣒⣒⣒⣒⣒⣒⣲⠈⠓⢲⡶⡶⠶⠶⠶⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠛⢛⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⠶⣛⢅⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡏⠀⠀⠀⣠⠾⣫⣵⡿⢣⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣻⡇⠀⠀⠈⠛⢛⣻⣿⣿⣿⣿⣿⡇⣿⣿⣿⡟⠀⢀⣴⢟⣵⣾⣿⡟⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠯⢿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⠟⢀⡴⢋⣵⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⢋⡴⣋⣴⠿⠿⠿⠿⠋⢰⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢳⣶⠀⠀⢀⣛⣻⣿⣿⣿⣿⣿⣿⡿⢡⣟⡵⢋⣡⠥⢶⣒⡚⣛⠃⠀⠀⣹⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⣽⡇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⠿⢛⣫⣵⣶⡿⠟⣋⣱⠃⢠⣆⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣛⣿⣥⠾⠟⠋⠉⠁⢸⡟⢀⡤⢖⣫⣵⣾⡿⣟⣫⣵⣾⣿⢋⡟⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣾⡟⠀⠀⠀⠀⠀⢀⡿⣛⣥⣾⣿⣿⣿⣷⣿⣿⣿⣿⣶⡟⣾⠁⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣭⠿⠧⠤⠤⠤⠤⠤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⠇⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣭⣴⣶⣿⠛⣛⡛⢛⣟⣳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠏⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⣿⣧⣼⢟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⠏⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣳⠏⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡽⠁⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓⠋⠠⠴⠿⠿⠟⠛⠛⠛⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠽⠿⠿⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣾⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢸⠂⢐⢧⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⢸⡌⢀⣤⡞⠐⠃⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠐⡂⢀⠔⣸⣿⡽⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⡉⣰⡿⣿⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣤⠾⢂⣾⣿⠀⡟⣧⠀⠀⠀⠈⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⢈⠘⠁⠀⣾⣏⡏⠀⡇⢿⡀⠀⠀⠐⠒⠀⠀⠀⠸⠆⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡤⠁⠀⠀⣾⣿⣿⡇⠀⡇⢸⡇⠀⠀⠀⠀⠑⡀⠀⢠⠅⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⢎⡀⠀⢀⡿⣿⣿⣿⠄⠠⡇⠀⣿⡂⠀⠀⠀⠀⠈⠆⢀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣤⣉⣀⠀⡀⠀⡇⠀⢻⣏⠀⠀⠀⠀⠀⠀⠸⡷⡶⢦⣤⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣦⣤⣼⣶⣤⣤⣤⣃⣀⣀⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⢠⣴⠶⠾⠿⡭⣤⣒⡚⠛⣛⠛⠛⠻⠿⠟⠛⠻⠿⠿⠿⣿⣿⠟⠿⠿⠿⠿⣿⣿⣿⠁⠤⠤
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠃⠀⠤⠖⠊⠉⠩⠵⠖⠛⢳⡖⢀⣤⡴⠶⠶⢟⣩⠴⠤⣤⣴⣶⡆⠨⢤⡤⠉⠀⢀⢀⡤⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠴⠤⠈⠭⢉⣉⡥⠴⠶⠒⠒⠒⠲⠄⠈⠉⠉⠀⠀⠀⠉⠛⠃

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠉⡟⢉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠢⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡒⠒⠚⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠄⠀⠀⡏⠀⠉⠉⢉⣭⣭⡉⠉⢩⣭⣭⠉⠉⢩⣭⣭⠉⠉⢩⣭⣭⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⡐⢠⠁⠀⣷⠀⠀⠀⠸⠿⠿⠇⠀⠸⠿⠿⠀⠀⠸⠿⠿⠀⠀⠸⠿⠿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⣶
⠀⠀⠀⠠⠀⡘⢤⣤⣷⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⠁⠀⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⣿⣿⣿⡇⠀⠀⣾⣿⣿⡷⠀⠀⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⣀⣀⣈⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣊⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡠⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠱⠄⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⢦⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠂⢾⡷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣿⡇⠙⠆⢆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠸⢻⡏⠀⡇⠸⣀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠁⡌⡇⠀⢧⠀⢩⣄
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⢸⠇⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢘⡀⠀⡈⢿⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⣇⠀⠦⠈⠟
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢻⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⢀⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⡑⢀
⠀⠀⠀⠀⣸⣿⢦⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠷⢀⠀⠀⠀⠀⢀⡹⢦⣶⡀
⠀⠀⠀⢀⣿⣿⣦⠂⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠲⠶⣽⣾⣿⣿⣦⡀
⠀⠀⠀⠈⠛⠛⠷⠾⠷⣤⣤⣤⣀⣀⣸⣻⣰⣀⣀⣠⠀⡀⠀⠀⠈⢂⠀⠀⠀⠀⠈⠉⠙⠛⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣄⣼⣭⣭⣿⣿⣡⣶⠰⣧⣧⣤⠧⣀⣸⣦⣤⣴⠶⠒⠒⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣴⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠙⠹⠟⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣸⣿⣿⣟⣛⣉⣭⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠾⠟⠛⠛⣛⠒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠿⠿⠿⠿⠟⠛⠉⠀⠒⠒⠒⠉




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣍⣤⣬⣽⣏⣴⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⣴⡖⢤⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣦⡀⠀⠀⠀⢸⡆⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡷⠘⣏⠑⢄⡀⢸⡇⠀⠀⠀⣿⡿⠁⠢⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠘⠀⠘⡀⠀⠙⢾⣧⠀⠀⠀⡏⠁⠀⠀⠈⢂⣄⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡏⠀⠀⢀⠄⠀⠀⠀⠀⠀⢉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡆⠀⢸⠀⠀⠀⠀⠀⠀⠋⢹⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢼⡇⢀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢹⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣾⢷⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⡿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠃⠀⠀⠀⠀⠀⠁⢈⣷⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢻⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⡛⠠⠁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⢿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣼⠀⢣⠀⠀⠀⠀⠀⠀⠈⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⠀⠀⠀⠒⠻⣿⠟⣋⠴⣶⠶⠦⠄⠀⠀⠀⠀⠀⠾⢶⡁⠀⠀⠀⠐⠂⢀⣠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⣟⣀⠠⠘⣁⡀⣀⣀⣀⠤⣈⣿⣿⡿⢿⣿⠀⠒⠒⠛⠀⠀⠀⠀⠿⠿⠿⠶⢀⣠⣴⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠐⠛⠐⠊⠤⠤⠤⠄⠀⠀⠀⠀⠉⣻⣻⡿⣾⣿⡆⠀⠀⠀⠀⠀⢠⡤⣟⣭⣴⣾⣿⣿⣿⣿⣿⣿⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡃⠀⣸⣿⠏⢀⣀⣤⣶⣾⣿⡿⢿⣿⠿⠛⠃⠀⠈⠙⠃⡠⠎
⣀⣀⣀⣀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣬⣭⣶⡿⠛⠀⣴⡆⠀⣺⣿⠂⢀⣄
⣿⣿⣿⣿⣿⣖⠀⠑⠪⠄⡀⢀⣤⣀⠀⠛⢿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⣟⣻⣿⣭⠉⠁⠉⠉⣴⣶⣾⣿⣿⡇⢸⣿⡿⠗⢸⣿⡷⣄⣀⡀⠀⢀⡠⠀⡀
⠛⠛⠛⠳⠆⡀⠀⠀⠀⢰⣬⣀⠉⠙⠛⢶⣶⣭⣭⣭⣭⣭⣿⣿⣭⣤⣼⣿⣿⣿⣿⣷⣦⣿⢿⣿⣿⣿⣿⣿⡟⠘⢿⠀⠀⢾⣿⣿⣏⠻⢿⣿⣷⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣄⠈⠉⠻⠿⠟⠻⣯⣝⣛⡛⠛⠋⠉⠀⠀⠈⠉⠙⠷⠀⠉⠉⠉⠀⠀⠀⠀⠀⠆⠸⣿⣿⣛⠂⠀⢨⣝⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⣄⡀⠀⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⢾⠗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠿⣿⢤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡖⢻⡆⠠⠈
⠀⠀⠀⠀⠀⠀⠙⢗⠒⠒⠶⠼⣧⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠁⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠉⠁⣿⡿
⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣿⣴⣿⣛⣻⣿⡿⣿⡾⣦⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡵⢌⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠲⠤⠶⣼⠇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡀
⠀⠀⠀⠀⠀⠰⠧⠐⠒⠲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⢐⣃⣹⣿⣠⡤⠬⣽⡿⠶⠖⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⢀⣿⣄⣠⣴⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣻⡷⣧⣙⠤⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠾⢿⣛⣿⣿⣿⣿⣍⡉⣴⣜⣯⡾⠞⠓⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⠀⢻⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⠛⠻⣿⣍⢙⡿⢿⣿⡇⢠⣿⡿⣼⢀
⠀⠀⠀⠀⠀⠀⣨⠿⣿⣿⣿⣯⣮⡀⠛⢛⣤⢀⠈⠛⠲⢿⡟⠓⢿⣇⠳⣄⡀
⠀⠀⠀⠀⢠⣶⣿⠆⠠⠀⠘⠉⠉⠉⠒⠀⠀⠈⠠⠂⠀⠠⠄⠀⢘⠻⠿⢮⣉⠁⢤⣴⣶⣤⡤
⠀⠀⠀⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣤⣽⣄⣈⠙⠳⠦⠀⠀⢀⣠⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠋⠉⠐⠀⠈⠙⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣧⣸⣯⣝⣿⣭⣉⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠓⠒⣺⣟⡐⠒⠤⢸⠀⠀⠀⠀⠀⣀⣠⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣳⣄⣀⣼⣤⣴⣶⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠲⠤⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢟⣛⠿⣷⡄⠀⠀⠀⠀⠀⠠⢄⡀⠠⠤
⠀⠀⠀⠁⣀⣈⣉⣽⣶⣶⣶⣿⣭⣽⣯⣭⣿⡿⠿⣿⢿⠿⣿⣛⣛⣫⣥⣤⣴⣶⣿⣿⣿⣿⣖⣦⠀⠀⠠⠤⠤⠤⠤⠄
⠀⢀⣀⣭⣍⣉⣉⡻⣯⣥⣶⣾⣿⢟⣿⣿⡿⠧⠾⡿⣏⣠⣭⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣭⣥⣤⠤⠐⠒⠒
⠈⠉⣉⡉⠛⠛⠛⠛⠻⢌⣉⣉⣐⣋⠉⠻⣿⣿⡛⠉⣴⣿⣿⡯⠛⠓⠊⠀⠉⠩⠭⠭⠭⠽⣿⣿
⠀⠀⠈⠉⠉⠉⠁⠀⠲⠦⢬⣍⣉⣁⡀⠀⠉⠉⠉⠥⠬⢽⣿⠧⠀⠀⠛⠓⠒⠚⢊⣉⣩⡭⠉⡉⠀⠐⠚⠛⠓⠒⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠒⠒⠂⠀⠘⠿⠃⠀⠀⠀⠑⠒⠂⠨⠁⠀⠀⠉⠙⠒⠲⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡂⠀⠀⠀⢸⣦⣄⣹⠉⠻⡇⠀⠀⣸⣶⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣧⣄⡀⠀⢀⣿⣿⣿⣷⣤⡀⠀⡰⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⣸⣿⣿⣿⣿⣿⣿⡆⢳⣦⣤⡃⠙⢿⢠⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀⠀⠟⠛⡛⠿⡿⢜⡛⠻⠿⢿⣿⣿⣿⣷⢨⣿⣿⣿⣶⣦⡀⢷⡀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣴⡗⠀⠁⠈⣿⣶⣤⣿⠂⠉⠻⣿⠚⠛⢿⡿⣿⣿⣷⣜⣿⡄⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣷⣦⣄⡹⣷⣦⣼⡇⠈⠹⣿⣿⣿⣿⡄⢹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣤⡙⣿⣿⣿⣿⣆⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⠎⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠿⠿⠿⠇⠀⠀⣸⣿⡿⠿⢿⠛⠛⢿⣿⣿⣿⣿⡿⠿⠿⣿⠁⠀⠈⠀⠀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣧⣴⣤⣭⣤⣄⣀⣸⣀⣀⣈⣛⣉⣉⣠⣦⣤⣤⣥⣤⣤⣤⠴⠒⣽⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠴⠿⢛⡛⠻⠿⠟⠛⠛⠿⠛⢛⡛⠻⠿⠿⠛⡛⠿⠿⠿⠛⣛⠋⠀⠀⠒⠒⠒⠒⠒⠒⢒⡀
⠀⠀⠀⠒⠀⠀⠀⠁⠀⣀⠤⠤⠀⠀⠀⠀⠒⠋⠃⡀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠐⠂⠀⢀⡤⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⠶⠤⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠛⠒⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⢀⢠⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⡠⠖⠲⣿⠒⠐⠂⠀⠀⠀⠡⡰⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⣹⡏⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣄⣀⡀⠀⠀⠂⠒⠛⣿⠒⠂⠀⠂⠀⠀⠀⠀⠠⠄⠴⣾⡦⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠐⣿⢀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠙⡏⢀⠀⠀⠀⠀⠀⠀⠀⣻⠈⠙⠉⠉⢹⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠚⠁
⠀⠀⠀⣴⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⣶⣶⣶⣶⣿⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⡿⠉⠀⠄
⠠⣤⠄⠈⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡠⠤⠤⠤⠴⠶⠤⠤⠤
⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣯⣿⣿⣿⣿⣵⣦⣤⣤⣤⣄⣀⣈⣈⣀⣠⣤⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠫⣿⠏⢩⠝⣋⡥
⣿⣯⣛⡛⠿⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣮⣷⣦⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡟⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠉⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠆⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠚⠈⠀⠈⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣀⣀⣄⠀⠀⢄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡴⣿⢻⣏⢙⣁⠀⠈⠈⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠵⠓⠛⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡠⠀⣸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠈⣤⡀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣀⣴⣺⡥⢿⣤⣉⣰⣄⣠⣨⣦⣼⣅⣀⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⡴⠞⣛⣩⡤⠷⠞⢻⠚⠛⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⢿⠚⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣃⣢⠘⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡄⣘⡿⣿⣿⣿⣿⣯⣝⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⠞⠁⣹⣁⡟⣻⣔⣻⣿⣿⣿⣷⣮⣟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⣦⣤⢿⣻⣮⣷⣿⣿⣿⣿⣿⣿⣿⡏⠈
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠀⠛⠉⠉⠙⢿⡿⠋⠄
⠀⡂⠀⠀⠀⠀⢤⠤⣤⣤⠤⠶⠦⢤⣤⣬⣿⢿⠿⢁⣤⡀⠀⢀⢢⣼⡇⠀⡄⠖⠠⣤⠄⠀⠀⠀⠠⠶⠦⠠
⠀⠂⠀⠀⠀⠀⠀⠀⠡⢶⣾⢂⣶⢶⣦⣤⡤⠀⠐⢺⣽⡿⠈⣽⣾⣿⣇⣀⠻⢿⣿⣃⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢶⣶⡬⢙⣠⣴⠚⠿⠃⠀⠈⣛⣯⡌⠻⣿⣦⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠋⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⢠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠆⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢃⢠⠜⢀⡀⠠⡄⠀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢠⠀⣸⣼⡟⢛⣻⣿⣿⣾⣿⡿⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠛⠉⣰⣿⣿⣿⠛⣦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣠⣦⡀⠀⢸⣿⠟⢯⢃⡴⠞⢋⣴⣿⣏⣴⡿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣛⣻⣿⣥⠶⠋⣷⣶⠖⣩⣴⣶⣿⠿⢿⣿⣿⡿
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣯⣹⣿⢽⣿⣿⣾⣯⣥⣤⠟⠋⠩⣥⡾⣿⣿⣿⣧⣄⠐⠂
⠀⠀⠀⠛⠾⠷⣶⣶⣿⣿⣦⣭⣿⣿⣾⣧⠈⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠈⣻⣷⠶⠄
⠀⠀⠀⠀⠀⠀⠩⠷⠋⠉⠉⠉⠀⠉⠉⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠢⠤


⣿⡏⠉⢻⠟⡿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠛⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⠷⠐⠀⠀⠀⠀⠀⢀⣀⢼⠷⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢀⡀⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣙⠶⠟⠊⠉⠀⠀⠀⠀⣀⣧⡄⣰⡼⢤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⢀⣶⣤⠤⠔⠚⠉⠁⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠛⠀⠀⢠⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠐⠈⠉⠀⠀⠀⠠⣶⠀⢸⠀⢀⣀⣠⣤⠤⢤⠘⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⡆⠶⠒⠂⠀⠀⠀⠀⢀⠀⠀⣰⣿⠀⠾⠛⠉⠉⠀⠀⠀⠀⣤⠀⠹⢦⡀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⢀⣆⡠⢾⡶⠛⠛⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣧⡀⠈⠛⠻⠛⠃⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣰⠭⠀⠒⢈⣽⠷⠶⠛⠀⠀⠀⠀⠀⢀⣀⣄⣤⣷⠿⠷⠆⢀⣀⡀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠦⠶⢒⡃⠈⠀⠀⠀⢿⠀⢀⣠⠂⣀⠠⠖⠚⢻⡉⢻⣧⣤⣶⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠚⠇⠀⠀⢠⠀⠘⠀⠀⠀⠀⠀⠸⡆⠀⢧⡾⠁⠀⠀⠀⣨⣷⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣧⠀⣈⣥⣴⣶⣷⣿⣿⣿⣛⡹⠿⢿⣿⣿⣷⣾⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠯⠼⢿⣶⣶⣿⣿⠭⠍⣉⣨⣽⣿⣿⣿⣿⣿⣿⣿⠟⣯⣷⣶⣶⣤⡀⢹⣿⣿⣿⣷⠆
⠀⠀⠐⢿⣿⣦⣤⣀⠀⠈⢿⣷⠇⣰⣶⠶⣾⣿⡿⢿⡿⣭⣭⣭⣛⠛⠛⠛⣙⣧⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣯
⠀⠀⠀⠐⠋⠉⠙⠛⠿⠷⣶⣾⠟⡋⠙⠋⠻⣍⡁⣤⣠⠤⠀⣤⣴⣶⣤⣾⠿⣿⣿⣿⣿⣿⣿⠏⠘⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠘⠋

⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣄⡀⢈⣛⡛⠿⣿⡿⠛⠛⠻⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣽⣿⢿⣿⣿⣏⢻⣷⣶⣿⠐⠂⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⡗⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠘⠿⠲⠛⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣠⣿⢷⣿⣿⣿⣿⡏
⢸⣿⣿⣿⣧⢀⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡟⠿⠃⣸⡿⠿⠟⠹⠗
⢸⣿⣿⣿⣿⣾⡿⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣬⣿⣿⣟⠿⡿⠛⢻⣿⣿⠀⠀⠁
⢸⣿⣿⣿⣯⣽⣀⡈⠏⠉⠀⣤⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⡦⠤⢴⣿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠤⢤⣿⠇⢀⠀⠀⠀⠄⣀⣀⣀⣴⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⢻⡿⠽⡿⠿⠛⠛⠉⠉⠉⠉⢸⡿⠀⡈⠀⠀⢀⣺⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣾⣇⣶⡇⠠⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣀⣀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⢰⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢀⣉⣀⣀⣸⣃⣼⣿⣿⣿⣹⣽⣯⣽⣿⣿⣿⣿⣯⣽⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⣉⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡉⠉⠉⠉⢩⣉⣉⣭⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⡀⡀⢠⣐⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣧⡀⠀⠀⠲⣟⣻⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣂⣿⣿⣽⣟⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠚⠷⠈⠛⠀⢲⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣧⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠙⡷⢄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠷⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡓⠲⠦⣤⣇⣀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠊⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠴⢦⢤⠀⠀⠀⠀⠈⠉⠙⠛⢆⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⠀⠉⣟⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⣼⣦⡠⠀⠀⣀⠀⠀⢴⡀⠀⢀⠀⠀⢸⠈⠉⠒⠻⢦⣈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢷⠦⠤⣤⣀⣀⣿⣿⡇⠈⠀⠀⣼⠀⠀⠀⠀⠀⠀⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢳⡀⠀⠀⠀⠉⠉⠙⠒⠒⢦⣼⠂⠐⣄⣀⣠⠀⢀⡆⣠⢸
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠐⠒⠚⠿⠿⣤⣀⣁⣛⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢶⣟⣳⡶⠤⣤⣿⣿⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⠀⠀⠈⢉⠏⠀⠀⠀⠀⠀⠀⠀⠀⣄⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣧⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠃⠀⠀⠛⠀⣴⡖⣤⢠⣼⡏⠀⠒⠂⣶⡶⣶⣶⣿⠒⠚⠿⠀⣼⣰⢿⡆⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⠀⣰⢨⣽⣏⠙⠒⠒⠒⠿⠧⠿⢬⣿⡙⠛⠒⠒⠻⠷⠾⠿⠿⣤⣤⣄⣀⣿⣃⡈⡇⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⡏⠀⠀⠀⠀⢱⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣷⣿⣹⡇⠀⠀⠀⠀⠀⠀⢰⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣿⣸⠇⠀⠀⢀⣠⣼⡗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣀⡾⠵⠚⢻⣿⠉⢿⡏⠃⠀⣰⢀⣀⠆⣀⢼⡇⠀⢀⠄⢀⣠⡄⡀⣠⠂⣀⡐⠀⣠⣾⣹⣿⣿⣷⣶⣿⣯⣰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⢶⣾⢿⡦⣾⣟⣴⣄⣩⣿⡇⠰⣿⠆⣠⡼⠋⣾⣿⠁⠀⢹⣏⣼⣿⣶⣾⣿⡿⢶⣞⠉⠀⠀⢹⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⣼⣿⣺⣗⣿⣟⡹⣿⣿⣿⣇⡀⣿⠞⣿⡇⢠⣿⣿⣶⣶⣿⣿⣿⠣⡿⡿⡿⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣤⣼⣷⣼⣿⣿⣿⣿⣿⠿⠿⠻⣴⣾⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣾⣿⣿⣿⣛⣿⡟⢻⣿⠋⢻⣿⠛⢻⣿⡟⠻⣿⣏⡉⣿⣿⣼⠿⠿⢶⣾⣿⣿⣏⣼⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠉⠻⠛⠿⠿⡿⠿⢿⣿⡿⣿⣿⠿⣿⣯⠿⣭⣭⠿⣶⣤⡤⢤⢶⢾⡟⠻⣿⣿⠿⠿⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠂⠁⠐⠂⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠑⠢⣼⣆⣀⣀⣀⠀⠀⠀⠀⠀⣄⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠁⠲⠢⠬⣦⡀⠘⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠐⠀⠑⠐⠒⠓⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣱⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠉⣉⣉⣉⣉⣙⣛⡲⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣈⣦⠖⠋
⠀⠀⠀⠀⠀⠉⠀⠀⠔⠚⠉⣉⡠⠤⠤⠤⠬⠭⣽⣓⡻⠽⣟⣓⡶⠶⠶⠶⠶⠶⠛⣋⣩⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠪⠭⢭⣙⣛⣛⣋⣉⣡⠤⠔⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⣴⠰⣾⣾⡇⠆⢀⠤⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠈⠇⠠⡻⣺⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⠀⠐⣠
⠀⠀⠀⠀⠀⠈⠑⠿⠷⡇⠀⢰⠀⠀⠀⠀⠀⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⢴⣤⣼⠄⢸⠖⢿⣶⣠⣤⠆
⠀⠀⠀⠀⢸⣦⣀⡀⠐⠀⠀⠀⠠⣀⣠⡄⠈⠙⣿⣿⣶⡄⠀⠀⠀⠀⠀⣼⣿⣿⣶⣭⣟⣂⣉⣸⣬⣿⣿⠏
⠀⠀⠀⠀⢾⣿⣿⣿⢷⣤⣄⠀⠘⣻⡿⣷⣠⣾⣿⣟⣻⣿⣦⡀⠀⠀⠀⣿⣿⣿⣿⢿⣛⣿⠻⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣵⢿⣿⠛⣶⢶⡀⣿⣼⣿⣿⠛⢻⣷⣖⠠⣄⡀⠀⣿⢠⠏⣾⠼⢻⣭⣶⣾⣿⠿⠇
⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣷⡾⠿⣭⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣾⣢⣴⣜⣿⡆⢿⣷⣿⣿⣏⣠⣦
⠀⠀⠀⠀⠛⠻⢿⣿⣿⠿⣷⣿⣄⣸⣿⣿⣿⡿⠛⠈⢉⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣶⡶⠄
⠀⠀⠀⠀⢲⣷⣶⡿⣅⠀⠰⣶⣯⣿⣿⡫⣭⣶⣿⣿⣿⣿⣯⣿⣟⣙⣛⡻⣟⣩⣭⣭⠭⠀⠀⠲⠿⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠠⠜⠻⠿⠿⢟⣛⠛⠛⠻⠿⠿⠿⠿⣿⠿⣿⠉⣉⡉⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠂⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠛⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠨⡗⠀⠀⠀⠀⠀⠀⣀⡀⠼⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⢀⡀⠋⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣹⠶⠛⠊⠉⠀⠀⠀⠀⢀⣧⣄⣲⡬⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⢀⣶⣬⠄⠒⠘⠉⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⠀⣠⡲⠟⠛⠛⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠔⠂⠈⠁⠀⠀⠀⠠⣷⠀⢨⢀⣀⣀⠤⠤⠤⢦⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠦⠖⠒⠂⠀⠀⠀⠀⢀⠀⠀⣸⡿⠀⠚⠋⠉⠁⠀⠀⠀⠈⢆⡀⠸⢷⡀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠆⢠⣇⠤⢶⡞⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⡀⠀⠉⠛⠛⠃⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠍⢂⠂⢉⡴⠷⠖⠋⠄⢀⠀⠀⠀⣀⣠⣄⡤⠷⠿⠿⣇⣀⣀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠶⠖⢂⠍⠁⠀⠀⠀⢻⠀⠠⢰⢂⡠⠐⠒⠋⢹⡄⢛⣷⣴⣶⣿⣿⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢰⠀⠈⠀⠀⠀⠀⠀⠸⡆⠀⢷⡛⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣆⠀⡀⢀⡀⠀⣀⣀⣷⢀⣨⣵⣾⣿⣿⣿⣿⣿⣛⣹⠛⠿⣿⠿⣷⣷⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣳⠾⠿⠶⠶⠿⠿⢏⣉⣡⣤⣽⣿⣿⣿⣿⣿⣿⣿⠛⣿⣾⣶⣷⣦⡀⢹⣿⣿⣿⣷⠆
⠀⠀⠘⠿⣿⣶⣦⣄⣀⠈⠻⠿⢏⣠⣞⡻⣿⣿⡿⠿⠿⠯⠮⣭⡭⠛⠛⠉⣩⣷⣾⣿⣿⣿⣿⣿⢸⡿⠿⢿⡷
⠀⠀⠀⠐⠉⠉⠉⠉⠻⠿⠿⠿⢟⠍⠉⠁⠹⣄⠀⠤⠤⠆⠀⠤⠶⡷⢶⠿⠿⣿⣿⣻⣿⢿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠙⠁

⠀⡀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⢀⡖⠈⠁⠠⠔⢋⠺⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠻⣕⠢⢄⡀⢸⠖⠘⡟⡃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠈⠑⢦⡈⣿⠀⣀⠁⠀⠀⠀⠀⠀⣤⠐⠖⠀⠀⢰⠏⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠉⠣⣀⣽⠛⣄⠠⢀⠀⢸⣇⠠⣴⠁⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣄⠀⠀⢀⠞⠁⠀⢀⠀⠀⠀⠀⠀⢈⡟⠀⠈⠑⢦⡀⣿⠉⠄⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣶⣗⠒⠒⢰⡅⢀⡀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠈⠻⣌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⢸⡏⠙⢦⣾⠀⠀⠉⢳⣴⣋⣀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢸⡷⠀⠀⡏⠓⢦⣀⣿⡟⢦⣈⠉⠒⢤⣀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣸⡗⡀⢠⠀⠀⠈⡟⣿⠇⠀⠘⠷⢤⡀⠈⢳⣤⡰⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠻⣿⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢦⣼⡃⠙⢦⡰⠄⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢏⡛⠲⢦⣄⣼⠃⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⠀⠙⠦⣀⠹⣄⢸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠙⠲⣄⡼⢣⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢳⢾⣞⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣠⠞⠠⠀⠀⡐⠁⠀⠀⣠⣏⣀⣀⡀⡠⠃⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢹⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣖⠲⠦⢼⣅⣀⠀⠔⢹⡏⠫⣍⠉⠛⠶⢤⣀⣴⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⠈⠙⠲⣤⣀⠉⠻⢶⣿⣾⠀⠈⠻⠲⣄⠀⡝⢻⣦⡀⢀⢣⠀⠀⠀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⢸⠀⠀⠀⠈⣉⠛⢶⣿⣿⡟⠀⠀⠀⠀⠈⠻⠷⣾⡆⠈⠻⡀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⠇⠀⠀⠀⢻⢀⣾⡿⠟⣷⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣄⠈⢠⡎⠀⢸⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⡘⠀⠀⣸⠃⠀⠀⠀⢀⣾⣿⡟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣻⢦⣵⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠐⠌⠀⠀⡟⠈⠀⠀⠀⣾⣛⡺⠃⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠳⠾⣿⡀⠀⠀⠀⠀⠀⢀⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠠⠀⢸⡇⠂⠀⠀⡼⣽⢯⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠀⠀⠀⠈⠑⠦⣀⠀⠀⣠⠟⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠛⠲⢤⡿⠀⠀⢀⣼⣱⣏⠏⠀⣸⢃⠁⠀⠀⠀⠀⢀⠂⠀⠀⡀⢿⡾⠀⠀⠀⠀⠀⠀⠀⠙⠿⢇⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠄⠃⣼⢣⠟⡞⠀⣰⢏⠳⠀⠰⠂⠀⢀⣎⣀⣀⣐⣐⣿⠳⠃⠀⠆⠀⠀⡀⠀⠀⠀⠀⠙⠒⢄⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⡞⣁⠀⣀⣼⣯⣯⣼⣥⡴⠏⣀⣠⠴⡟⠚⠙⠍⣿⣿⠉⣉⣽⣟⣛⣷⣟⣒⢲⣶⣤⣄⡀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠾⢽⣿⡯⢍⣉⣠⢇⡾⢰⢁⣾⡷⠊⠉⢙⣳⣦⣤⣞⣛⣯⣽⡟⠛⠛⡛⠓⠛⠾⢯⣿⢿⡄⠈⠙⢦⣼⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠐⠮⠽⣛⣾⢿⠿⡷⡶⢿⣿⣗⣛⠿⢽⡛⣯⣁⠀⠈⠙⠞⠋⣩⣭⣭⣭⣉⣓⢞⣵⡿⢿⡆⠀⢈⠆⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⢑⠂⠬⢉⣉⣉⣭⢉⣭⢁⡷⠠⠟⠚⠋⣡⡤⠶⠐⢒⡻⢭⣁⣀⣀⣈⣉⣋⣾⡯⠱⡎⢁⣠⠾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⡀⡉⠐⠒⠶⠦⠄⢠⣀⢴⠆⡐⠛⠋⣉⠀⠄⠀⠀⠨⢛⣋⠠⠤⣤⣭⣿⣾⠷⣤⠋
⠀⠀⠀⠀⠀⠀⣠⣤⠤⠚⠚⠛⢷⣄⣉⣱⣶⣶⣶⠦⠤⣲⣶⣂⣙⣁⣀⣠⣤⣖⡚⢛⣋⣛⣛⠟⠿⠿⠿⣿⠶⠗⢒⣋⣀⣀⠄
⠀⠀⠀⠀⠀⠉⠓⠈⠒⠂⠀⠀⠡⠟⠛⠋⠉⠓⢂⣠⣀⣬⡟⠛⠉⠁⠀⠉⣭⠽⠿⠯⠤⠀⠀⢈⡉⠤⠤⠤⠤⢉⣉⠀⠀⠠⠌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡖⠶⠔⠂⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠉⠉⠉⠑⣀⣘⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣛⣓⣼⣔⣶⣒⣤⡇⡃⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢣⠀⠀⠀⠀⣼⡧⠤⢶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠷⠶⣐⣾⠤⢤⡤⠤⣄⡇⠀⠀⠈⠯⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡲⠤⣼⠀⠀⠈⠫⠉⠉⠉⢭⣽⠷⠶⠶⣶⣤⣙⣦⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠞⠒⠋⠑⠒⡇⠀⠀⠀⠈⢆⠀⠀⠈⠻⡲⢄⣣⠀⠀⢈⣑⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢱⠠⢼⣇⣾⣻⣟⣥⡲⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡼⠀⣀⣀⣀⠀⡇⣠⢏⣘⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢸⢃⣠⡴⢒⠾⡟⢻⣿⠿⣽⣽⠗⣋⢭⣿⣌⡝⣻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡾⣛⡛⡓⡚⣶⣷⣾⣇⣀⣾⣿⣿⣿⣿⣾⣷⣷⣿⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣵⣓⣐⣐⣸⡿⢿⠿⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠄⠠⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣂⣐⡿⢿⣿⣿⣿⣷⣾⣿⣿⡿⠿⠿⠟⠋⠉⣭⣩⢉⣁⢻⡑⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠭⠭⠥⠷⠒⣦⡀⠙⠛⠓⠠⠤⠍⠀⠐⠂⠀⠴⠚⠃⠼⠃⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠈⠀⠀⠉⠁⠀⠒⠒⠒⠂⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⡀⠐⠢⢄⡀
⠀⠀⠀⠀⡀⠀⠁⠁⠢⠄⡈⠑⠢⠄⡀
⠀⠀⠀⠀⠀⠈⠠⢄⡀⠀⠀⠈⠐⠠⢄⠉⠐⠠⡀
⠀⠀⠀⠀⢆⠀⢀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠒⠠⣀⠙⠂
⠀⠀⠀⠀⠨⡀⠸⠧⡌⠃⠒⠂⠀⠓⠀⠀⢘⠀⠀⣘⡖⠠⡀⠐⠠
⠀⠀⠀⠀⠀⢿⠀⢁⠀⠂⠀⠢⡈⠉⠢⣀⠀⠁⠠⢀⠀⠀⠀⠁⠢⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⡝⠂⠄⢂⡀⠀⠈⠢⡀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢀⠙⠠⢀⠁⠢⣀⠉
⠀⠀⠀⠀⠀⠀⠈⢌⠓⣀⡥⢀⡰⠄⠘⠂⠀⠀⠉⠁⠀⠀⠀⠋⠀⠀⠀⣀⡄⠑⠢⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠙⠻⣥⣠⠀⠄⠐⠂⠀⠀⠀⠂⢁⠀⡀⠀⠀⠉⠢⣀⠀⠀⡉⠢⠄⠀⠰⡤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠉⠛⠲⢦⣀⡀⠠⡀⠒⠀⠑⠄⠑⠀⡀⠀⠘⠁⠀⡇⠀⠀⠀⢠⠁⠰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠙⠓⠾⢦⣀⣀⠀⠀⠀⠀⠀⠐⠀⢀⠁⠀⠀⠀⡀⢀⡇⢀⣤⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠈⠉⠛⠳⠦⣄⣀⠀⠀⠀⠀⣂⣴⠶⠿⠟⠛⠛⠓⠛⠋⠉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⡀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢀⠀⠑⢄⡀⠀⠀⠀⠀⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢈⡑⢄⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠳⠊



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⣴⠆⠰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⣾⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢲⡯⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠋⠉⠀⠂⡀⠉⠀⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠠⠄⢀⣔⡏⠀⢰⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢿⣇⣀⢸⡇⢰⡶⣄⣀⠐⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠈⠙⡛⣸⠃⠀⠉⠙⠻⡾⠧⣤⣛⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⣤⠄⠀⠀⢨⣵⡏⠀⠀⠀⠀⠀⢹⡄⠀⠈⢿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠚⠂⠲⠀⡜⣨⡿⠀⣀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠛⠛⠿⠿⣿⣿⣾⣶⣿⣥⣶⣿⣿⡛⠛⠻⠿⣿⣆⡉⢰⡦⢿⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢩⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠯⠿⣿⠃⠀⠀⠀⠀⠀⠉⠙⠳⢾⡧⣤⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠚⣇⠀⠀⠀⠀⠀⠀⡠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣠⣾⠟⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠈⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣰⣃⢋⢌⣼⣦⣗⣴⣷⣧⣾⣿⠋⠀⠐⠇⠴⠃⠀⠀⠀⠀⠀⢰⣽⠃⠀⣀⣀⠐⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣤⣬⣅⣟⣻⣟⣿⡿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣶⣤⡀⠀⡀⣀⢀⣼⡿⢀⣴⠟⠉⠓⢢⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣉⡉⠛⢻⡿⣿⣿⣷⣮⣿⠋⠰⢟⡿⠿⡿⣿⡿⣿⣷⣦⣤⣤⣽⣿⣿⣷⣿⣧⣦⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⠷⢤⢄⣉⠛⠃⠀⠀⢈⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⠆
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣠⣞⣛⢻⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠏⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⣀⣸⠃⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⡅⡟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣭⡤⣄⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⢂⠀⠀⠀⢴⢞⣛⣩⣥⣶⣤⣿⣿⣿⣿⣿⡿⣿⢟⣩⣾⣏⠀⠀⠉⠿⢿⢿⡿⠿⠿⢿⡅⠾⢿⠀⠀⠂⠀⢀⣤⡞
⠀⠀⠀⠀⠀⠈⠳⣤⣀⣐⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣟⣛⠛⠿⣿⡅⠀⠀⠀⠀⢠⠷⣍⠢⣶⣿⣧⣀⢤⣀⣀⣰⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠐⠓⠶⢚⣋⣩⣭⣭⣭⣏⣉⡛⠿⣦⣝⠂⠀⢐⣕⣾⣶⣮⣷⣮⣽⣿⣿⣿⣿⣿⣟⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⣀⣁⣐⣨⣽⣿⣿⣿⣿⣿⣿⣿⣄⠈⠛⢷⣄⠀⠉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣭⣭⣤⣶⣾⣿⣿⣿⣿⢿⡏⠁⢾⣆⠙⠶⣤⣈⣛⠿⠿⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢈⠙⢿⣿⣧⡍⠙⠀⠶⢿⣷⣴⣤⣉⣉⣛⡛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠻⠶⠦⠉⠀⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠚⠉⠀⠀⠆⠀⢈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡤⠀⣀⡀⠤⣿⣒⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⡿⢤⣤⠴⠟⠀⠉⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⡶⢦⣀⡿⢋⣉⣽⣯⣗⣤⣀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⠀⠀⢿⣋⣠⠾⠷⢦⣿⠏⠁⠁⠀⠀⠀⠛⠋⢀⡀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈⢹⣟⣠⡴⠚⠋⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⡥⠔⢀⣰⠋⢀⡀⢀⣀⣀⣀⣀⠸⣷⠶⡾⢷⣦⣄⡀⠀⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠢⣀⠀⠀⠀⢸⣿⡶⣠⣴⣻⠃⢀⡼⠟⠉⢛⣽⣿⣿⣳⣶⣾⣷⣿⣿⣿⣿⣶⣽⣿⡅⠀⢾⡇
⠀⠀⠀⠀⠈⠑⢦⡀⠘⣿⣿⡿⠟⠛⣶⠞⠒⠛⠉⠉⠉⠉⠙⢻⣿⠟⠉⠉⠉⠉⠉⠛⢻⣿⡇⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢌⡻⠅⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⢀⠀⣼⣿⣧⠀⣸⣿⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⡈⢷⣴⡷⠿⠷⠶⣤⣄⠀⢿⡿⣿⣿⣶⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣷⢿⣄⣠⣤⣀⣻⡟⠿⣮⠀⠀⠀⢹⡟⢻⣦⣀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡼⣟⣿⣿⣿⡿⣷⣤⣌⡛⢦⠀⢸⣇⠀⢫⣿⣷⣿⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣺⣿⢾⣿⢿⡑⠉⣿⣿⣾⣽⣹⣟⡟⠾⢻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⣷⣍⣹⣶⣿⣿⣾⣇⣿⠏⣹⣿⠛⣶⣶⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢅⠒⢼⣿⡏⢙⠛⠿⡿⠛⣿⣿⢿⣿⣿⣿⠟⢿⣿⠿⠟⣻⢿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠁⠈⠉⠉⠀⠈⠀⠐⠃⠀⠉⠓

⣿⣿⠛⠿⢻⠿⡿⢻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀⣟⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⢸⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⠃⠀⠀⠎⠀⠀⠳⣾⣷⠷⠟⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢤⣴⠄⠾⢿⢿⠤⡤⣼⣿⡦⠤⡄⡀⢀⣀⣀⣀⠓⡲⠶⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡇⠀⠀⢶⠶⠀⠀⣹⢿⡇⠀⠀⠀⠩⠉⠀⠉⢹⢿⠑⠂⠰⠔⠦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢧⣸⡀⠀⠀⠉⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡄⠀⣿⡀⠀⠀⠂⢼
⠀⠀⠀⠀⠀⠐⠀⠀⣀⣀⣠⣧⣤⠀⠀⠈⢀⢸⢧⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣽⠀⢠⠀⠐⢸⠈⠳⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣹⣿⡆⠀⠀⠀⢸⠀⠀⠑⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⣈⡵⡀⠂⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣯⣤⣄⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢾⢿⠏⣤⢂⠀⣎⠜
⠀⠀⠀⠀⠀⠀⠀⣰⠋⣸⠻⡛⠟⡿⣧⡈⣇⠀⠀⠁⠀⠀⠁⣄⠀⡀⣀⢀⠂⠀⡆⠀⠀⢸⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣿⣿⣿⣷⣟⣓⠳⠶⠤⢄⣔⠙⡺⠦⠤⠤⠄⠀⠬⠑⠀⠞⠐⠛⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⣿⣿⢻⣯⢙⠟⠛⠛⠿⠷⣷⣞⠛⠂⠂⠂⠀⠠⠤⠄⠰⠄⠐⠂⠠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⡛⠛⠛⣻⣦⠸⠆⠰⠀⢤⠙⠿⠿⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣿⣷⣿⣾⢿⣦⣀⣀⡀⠀⠀⠰⣶⠆⠀⠀⢰⣦⠀⠀⠀⠻⠀⠀⠀⠀⡠⠇
⠀⠀⠀⠀⠀⠀⠠⠀⣨⣟⣿⣹⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⠞⣓⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠠⠍⣭⢙⡿⠛⣿⣿⡛⣿⣿⣿⣿⢛⣿⣻⣟⡉⢉⠉⠩⢁⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠚⠑⢟⣿⠋⢿⣟⠊⠙⠩⣶⣟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠻⠿⣿⣆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣤⣤⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣣⣿⣧⣤⣤⣤⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⡿⠟⠃⢰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣤⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡛⠿⠚⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣿⠋⢹⣷⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⠶⠦⣾⣤⣬⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠐⠶⢿⡀⠀⠀⠀⠀⠉⠁⠁⠉⠈⠙⠉⠛
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠈⠆⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣀⣤⣴⣾⣷⣿⣿⣿⣯⣭⣧⡀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⠿⠯⣥⣬⣿⣿⣛⣿⣟⣛⣿⣀⣀⡀⠀⠀⠂⠄⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⢆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠒⠶⠶⢶⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢀⡚⢀⡀⢀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⡀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⡠⢔⡲⠚⣿⣿⠁⣉⣀⣈⣁⣿⣧⣤⠤⠤⠄⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠶⠿⠶⠒⠛⠛⢻⣿⠋⠉⠉⢀⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣾⣿⣷⣲⣿⣿⣿⢦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⡦⣉⠲⢆⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡌⣿⡿⠿⢟⣛⣛⣻⣿⣌⠛⠋⢑⣤⠻⣯⣴⣾⠷⠶⠐⠖⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⣋⣤⣶⣾⣿⣿⣿⣿⣿⣿⠉⠻⣦⣻⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣽⣿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⣿⣿⡏⠙⠿⣿⣧⡄⢛⢹⢿⣿⣿⡏⠀⢰⣿⡿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣖⣋⠀⠛⠻⢇⡈⠀⠀⠙⠿⢦⣶⣿⡿⠵⣾⣿⣆⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣛⣣⠀⠉⠉⠙⣿⠿⠿⠿⢶⣄⡀⠉⠛⠐⠃⣀⠀⠀⠀⠉⠀⠀⠈⣿⣿⣦⣍⡛⠶⠦⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠉⠈⠀⠋⠓⠀⠀⢤⣉⠛⠛⢿⣦⣀⠀⠳⠿⠿⠿⠿⠗⠂
⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⣶⣦⣬⣤⣀⡀⠉⠛⠻⠶⠀⠀⠀⢠⣠⣶⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡿⠿⠟⠛⠛⠓⠢⠀⠉⠛⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⠐⠒⠀⠀⠀⠀⠀⠚⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣀⠀⠀⠛⠀⠀⠀⣀⣠⣀⣀⣀⠀⠂⠀⢀⡰⡆⠀⠨⢀⠠⠤⠌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣶⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠉⠁⠀⠁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠐⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠲⠊⠁
⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠂⣸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠩⡧⢑⡏⠀⠀⠀⠀⠈⠀⠐⣂⡀⠠⠀⠹⣀
⠀⠀⠀⢸⣛⣻⠶⡶⠦⠶⠆⣆⣰⣴⣶⣤⣤⣼⣄⣜⣁⣀⣠⣦⣤⣤⣤⣷⣿⣴⣶⣷⠾⢷⣦⣴⣶⣾⣿⢷⣿⡿⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠉⠛⠛⠲⣶⣦⣤⣤⣤⣎⣃⣇⣁⣀⣉⣉⣉⣉⣉⣉⣁⣖⣐⣜⣐⣐⣒⣉⣭⣭⣭⣭⠤⠖⣂⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⡛⠿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⢿⡿⠿⠿⠿⠿⠿⠿⠷⠴⠀⠴⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠀⠈⠉⠐⠈⠉⠓⠾⠟⠋⠘⠷⠖⠀⠉⠉⠙⠓⠒⠛⠂⠂⠉⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⠠⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠠⠒⠈⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣀⢴⡤⠴⠓⠀⠀⠀⠀⢸⠀⠀⣀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⠐⠛⠉⠀⠀⠀⢀⡀⡀⠀⣈⣺⠼⠓⠚⠉⠙⣦⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠀⠀⠀⠀⠀⠀⢀⡰⠍⠋⠁⠀⢈⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⠉⠀⠀⠀⠀⠀⠀⢺⡇⠀⢈⣤⣤⣶⠶⠓⠒⠚⡂⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠿⠑⠛⠉⠁⠀⠀⠀⠀⠠⡄⣀⣠⣿⠇⠀⠋⠁⠀⠀⠀⠀⠀⠀⢰⣄⠀⠛⢧⣀⣀⣀⡀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣇⡯⠴⠾⣟⡋⠉⠩⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠈⠉⠋⠉⢁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡬⠉⢒⠈⠁⣥⠟⠷⠛⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⠾⡾⠿⢿⣀⣀⣀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠲⠶⠖⢒⡫⠈⠀⠀⠀⢸⡆⠀⡀⡴⢀⡄⠴⠒⠋⢹⡅⠹⣿⣤⣶⣾⣿⣿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠻⠈⠀⠀⢤⠀⠘⠀⠀⠀⠀⠀⠀⣷⠀⠀⡴⠛⠀⠀⠀⠀⣠⣷⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣫⣠⢴⣶⣶⣿⣿⣿⡿⢻⠿⠿⣿⣿⣿⣵⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣯⣼⣯⣤⣥⣾⣷⣖⡚⠻⢉⣑⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣭⣤⣤⣬⡁⠈⣿⣿⣿⣿⣦⡄
⠀⠀⠀⣶⣶⣤⣀⡀⠀⠀⠙⣶⣶⠒⢀⡤⣤⣶⣶⣶⣿⣿⣟⣛⣛⣛⠻⠿⠿⠟⠻⣂⣽⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣏
⠀⠀⠀⠈⠽⠟⠛⠿⢷⣶⣤⣬⣭⣶⠿⠿⠖⢸⡋⠋⠀⡉⣈⡉⠉⢉⣩⣄⣀⣴⣿⣿⣿⡿⣿⣿⣿⣿⠃⠟⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠁⠀⠀⠀⠓⠂⠈⠉⠁⠀⠈⠉⠉⠉⠉⠉⠈⠻⠛⠿⠛⢫⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠈⠉⠁⠐⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠞⠛⣻⣷⠖⠚⠻⣶⠤⣄⣀⢸⠛⠛⠛⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠉⠁⠀⠐⠒⠀⠀⡸⠀⠉⠉⠒⠛⠤⠤⢙⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠴⠶⠶⠐⠶⠶⠒⠒⠚⢀⣇⣄⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄
⠀⠀⠀⠀⣠⡴⠛⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⢤⡬⠁⣼⣿⠿⢿⣿⣾⣷⣧⣶⣄⡀⢁⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠐⣲⠞⢿⣿⣆⡀⠉⢻⣷⣆⠀⢡⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼⠃⠀⠈⠙⣧⠈⠉⠉⠓⢥⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢀⠔⠀⠀⠀⣿⠀⠀⠀⠀⠈⡀⢀⣼⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢿⣷⣶⣀⡄⠀⠸⠋⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡴⠋⠀⠀⣠⠄⢀⣠⣠⡶⢰⣿⣿⣿⡿⣷⣼⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸
⣿⡑⠶⠶⠲⣄⠀⠀⠀⠀⠀⠂⠀⣀⣴⡿⠴⢶⣿⣾⣿⣿⣿⣿⣿⠁⠂⠀⠀⠈⠉⠐⠻⠀⠀⢸⠀⠀⠀⠀⠀⢀⡤⠴⠒⠢⢄⣀⠀⡰⠉⠀⢹
⢸⣯⢖⡂⠀⠨⡇⣀⠤⢤⣤⠴⠲⣽⣷⢦⣖⠛⠺⣿⣿⣿⣿⣿⡧⠼⠀⠈⠀⠀⠀⠀⢘⠀⠀⠀⡇⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠈⠋⠁⠀⢀⣿
⠀⢻⣅⠀⠀⠀⠉⠉⠉⠀⠈⠆⠀⠀⠛⢾⣭⢷⣴⠿⠿⣿⡿⠟⢿⡷⢾⣷⣴⣤⡄⢀⠇⡇⢀⣄⣷⢶⡶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣿⣷⣴⣟⣀⣀⣸⣷⣾⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣅⣨⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠁
⠀⠀⠀⠈⢿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣾⣿⣿⣿⣿⣯⣿⣯⣽⣿⢿⣿⣿⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣻⣿⡾⠋
⠀⠀⠀⠀⠀⠈⠻⣷⣖⠀⣤⣶⣄⣀⣁⣾⣶⣀⣿⣿⣿⣿⣿⡿⣿⠿⠛⠛⠿⠿⣿⣿⣿⣿⡆⡸⠶⣤⣤⣴⡿⠿⢦⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣮⣭⣭⣿⣷⣆⢤⣟⠿⠿⡿⢻⣷⣶⠦⠴⠶⣾⣿⣿⣿⣿⣯⣽⣿⣟⣉⣽⣿⣽⣷⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢿⣿⣿⣿⣿⣶⣦⡴⢶⣶⣾⠟⠻⢿⣯⣝⣿⣶⣶⣾⣭⣤⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠲⠿⠿⠿⠶⠾⠿⠛⠛⠛⠛⠉⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢛⣛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣖⣯⠭⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⢲⡄⣿⠁⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣖⣾⣿⣧⡿⠀⠀⠀⢀⣀⣸⣧⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⢀⣀⣿⡷⠶⢚⣛⣉⣉⣀⣠⣤⣤⣤⣤⣤⣤⣤⣽⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣰⡏⠉⣟⣋⣉⣭⣭⣥⢤⢤⢶⢶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣬⣭⣉⠙⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⠶⠖⠚⡛⠉⠩⢹⡇⠀⡿⠷⠷⠓⣛⣛⣛⣉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣉⣛⣛⣤⠇
⠀⠀⠀⠀⠀⠀⢸⣐⣀⣁⣩⣬⠤⠴⢖⣒⣿⢀⠉⡟⢛⣛⣉⣉⣩⣭⣤⢤⡤⣤⢤⡤⡤⡤⣤⢤⢤⣤⣤⣬⣽⡇
⠀⠀⠀⠀⠀⣯⣭⣵⣶⠶⠶⠶⠿⢟⣛⣛⡿⠀⠤⣿⢹⡇⡇⢸⣀⣇⣸⣸⣇⣿⣼⣥⣧⣧⣿⣸⣼⣿⣷⠿⠿⠗⠚⠲⣦⡀
⠀⢀⣤⣴⠚⡓⢒⠒⡉⣉⢉⡉⣉⠁⠄⠠⠀⠆⢀⡟⠛⠛⠋⠉⢉⣁⣀⣤⣤⠤⠶⠶⠚⠛⠋⠉⠉⠀⠀⠀⢀⣀⣀⣀⣟⣿⠆
⠀⠘⠦⣤⠤⠥⠬⠤⠤⠤⠶⠶⠶⠖⠒⠒⠛⠉⠉⠙⠒⠚⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠒⢚⠋⠉⠉⣁⣸⡿⠃
⠀⠀⠀⣇⣐⣀⠀⠀⠀⠀⠰⠐⠐⠂⠂⠂⢀⣀⣁⣈⣀⣠⡤⠤⠤⠶⠖⠒⢛⣉⣉⣡⣤⠤⠴⠶⠒⢾⣿⡟⠉⠉⠉⡿⠁
⠀⣀⣀⢸⡟⠛⠛⠿⠿⠿⠿⠭⠭⠭⠭⠭⠭⠭⠴⠶⠖⠒⠒⠒⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠳⣌⣯⣤⡄⠀⢸⠇
⠈⠉⠉⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡿
⠚⣉⣱⣢⣿⣽⠷⠲⢶⣦⠤⢤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡷⢤⣤⠖⠲
⠚⠁⠀⠈⠑⠦⣀⣀⡞⠙⠻⢷⡲⠶⠋⢈⡽⠿⠭⣭⣽⠛⠻⢷⣒⠒⢒⡖⠶⣖⠒⠲⢲⡶⢶⡒⠒⢒⣿⡿⠭⣿⠛⠳⣤⣀⣀
⠀⠀⣠⢤⣀⡀⠀⠀⠀⠀⠀⠀⣈⡛⠒⠋⠀⠀⠀⠀⢈⣉⡓⠒⠚⠛⠋⢀⣀⠈⠙⠲⠶⠃⠀⠉⠳⢮⣍⡽⠂⠈⡷⠦⢤⡉⠁
⠉⠉⠁⠀⠀⢈⣉⣉⣉⡒⠒⠚⠉⠉⠙⠒⠦⠤⣄⡤⠞⠉⠉⠛⠲⠦⠤⠜⠉⠙⠒⠦⣄⣀⣠⡟⠙⠲⠤⣄⣠⠴⠃⠀⠀⠙⢦
⠀⣀⣠⡴⠞⠋⠁⠀⠀⠉⠙⠒⠤⣄⣀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠉⠒⠒⠤⢤⣀⡀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠉⠓⠒⠤⢄⣀

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣸⣀⠀⠀⠀⠀⠀⠠⠦⢤⣿⣀⣀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢀⠀⢹⡇⠀⠰⠤⢤⣿⣧⠀⠀⠀⠀⠀⠐⠲⠾⢿⣷⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠘⠓⢺⡗⠐⠂⠀⡜⢹⠀⠀⠒⠀⠀⡂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠸⣥⠆⠀⠴⠀⠸⢠⠀⢀⣠⣤⣦⣤⣤⣤⣼⣾⣶⣶⣶⣾⣿⡟⠁
⠀⠀⠰⢶⣶⣿⣿⣿⣶⣾⣟⣻⠻⢟⣿⣿⣿⣭⣿⣿⣿⣭⡯⣿⢽⡿⣿⣿⣶⣤⣤⡀
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠸⠛⣭⠍⠉⡉⠙⡁⢰⠶⠧⢶⣾⠿⣿⣿⡷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄⠀⠀⠒⠂⠀⠀⠀⠠⠀⠀⠐⠒⠈⠑⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠶⢢⣴⠄⠀⠀⠀⠀⠀⠀⠘⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⢸⡀⠀⢀⡀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢿⠢⣀⢸⡇⠀⢸⣿⡆⢄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⠈⠆⠈⣻⡇⠀⢰⠟⠀⠀⠱⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⢸⠐⠀⠀⠀⠾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⡈⠀⠀⠀⠀⠀⡿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠥⢧⠁⠀⠀⠀⠀⠀⢿⢏
⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⠀⠀⣸⣯⡝⠀⠀⠀⠀⠀⠀⣸⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢅⣿⣿⠀⠀⠀⠀⠀⠀⠀⣹⣧⢰⠀⠀⠀⠀⢀⡄⠋⠁⠀⠁⠁
⠀⠀⠀⠀⠀⢠⡶⠃⠀⠀⠀⠐⠶⣿⠿⡚⣥⡤⠄⠀⠀⠀⢸⣿⡈⠀⠠⠄⢀⣤⡿⠁
⠀⠀⠀⠀⢠⡿⢧⠖⠡⠤⠤⠤⠦⣿⣿⣿⣿⠀⠐⠃⠀⠐⢚⡻⢛⣤⣴⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⢻⠛⢻⣦⠀⢀⣠⣼⣿⣿⣿⠿⠛⠛⠿⠁⡸
⣤⣤⣄⡀⢀⠀⠀⢀⠀⠀⢀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡵⠾⣂⣰⡆⣸⣿⠀⣤
⣿⣿⣿⠛⠁⠈⢑⡀⠿⠦⣬⣛⣿⣿⣿⣿⣿⣟⣻⣿⣿⣧⣀⣤⣾⣿⣿⣿⡅⣿⠋⢹⣿⣷⣦⣤⣤⣖⠅
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣄⡙⠻⠿⠿⢿⣟⣿⠿⠛⠉⠉⠙⠻⠍⠙⠛⠋⠁⠈⠀⠘⣿⣿⡇⠉⣿⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠄⠈⠋⠁

⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉
⣏⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠻⠟⠿⢻⣿⡿⠛⠛⠛⠛⠛⠛⢛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⢿⣿⣿⣤⣶⣤⣦⣤⣤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠁⠀⠰⣷⠀⠀⠀⠀⠀⢐⡈⠙⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢰⣶⣿⣦⣀⠀⠀⠉⣉⣋⣀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⠛⢚⣳⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⣀⠴⠒⠉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢾⠘⣿⠳⣄⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠠⠀⠃⠹⣧⠈⢃⠄⠀⠀⠁⠀⠀⠀⠀⠀⢠⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣼⡏⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠁⠐⠀⠀⢻⣆⠀⠀⣱⣇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣶⣶⣦⣰⣤⡠⢤⣤⣌⣿⣿⣷⡄
⠀⠀⠀⠀⡘⠀⠘⡄⢀⣀⣻⣄⣴⠟⣨⠓⢄⠀⠀⠀⠿⠿⢻⣟⣿⣿⢹⣿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣠⣇⠀⠰⢃⣼⣿⣿⣿⣿⣧⢹⣧⣌⠳⣄⠈⠼⡀⣾⣿⣿⣿⣸⣿⡇⠀⠈⠙⠻⠝⡻⢿⣿⣿⣿⣿⣧⠀⠀⠄
⠀⢀⣴⡿⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⣳⣆⣥⣿⣿⣿⣿⣿⣿⣷⣶⣴⣦⢤⣀⣉⡳⠿⠿⣿⣿⣿⣧⠀⠰⡄
⠀⣸⡟⠀⠀⠀⣾⠟⢩⣿⠟⠉⠹⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⢸⣿⠀⠈⠉⠙⠛⠿⢾⣿⣿⣿⣧⢰⣿⣦
⠀⣿⠁⢴⠀⣠⣸⣇⣛⣀⣀⣀⣀⣹⣆⣏⣛⣤⣿⣿⣿⣿⣯⣥⣼⣧⣤⣤⣄⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣈⣙⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠙⢉⣿⣿⣿⡿⠿⠻⠏⢿⣿⡏⠉⠉⠉⠉⡻⠏⠉⡟⠻⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⢷⡀
⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⡴⠁⠀⠺⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⡏⠋⠉⢉⣉
⣧⣴⣦⣤⣤⣤⣼⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⢦⡤⠤⠴⡦⠶⠤⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣌⣾⣇⠀⠀⠀⠁
⢥⠀⠀⠀⠀⠀⢰⡇⢶⡀⠀⠀⠀⠀⠀⠀⢿⣿⡄⢀⠶⣤⡄⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠈⣿⣿⡏
⠀⠃⠀⠀⠀⠀⠈⣿⣶⣾⢶⣤⠀⣠⣦⣤⣾⣻⣿⣟⣴⣯⣤⣤⣴⣤⣶⣶⣶⣶⣷⣶⣦⣤⣤⣀⣴⣷⣶⣤⡜⣧⣿⣿⣿⠁
⠀⠀⠀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠹⣿⣿⠿⠟⠉⣿⡟⠻⠿⣿⠏⢙⣿⣿⣿⠿⠻⢿⣿⣿⠿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⢹⠗⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣀⣿⣧⡄⠀⢀⡀⢸⣿⠘⡀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢀⣶⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣧⣾⣿⣤⣦⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡖⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣄⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠶⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡉⠉⠛⠛⠟⠛⠉⠛⠢⠦⣤⣀⡀
⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣤⣤⣄⣤⣄
⠀⠀⠀⠀⠈⠉⣛⠿⠛⠛⣉⣉⣀⣀⡈⠋⠙⠛⠛⠛⠛⠛⣻⣿⣿⠛⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠻⣿⣋⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠟⠋⠉⠛⠻⠷⢶⣦⣴⡶⠾⠿⠛⠿⠟⠛⠛⠿⠿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢋⡛⠋⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠅⠒⠁⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿⣷⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠀⢴⠒⠚⠓⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⠀⠀⢸⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡈⢸⡀
⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⢻⠘⠀⠀⠈⠁⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⢸⡇
⠀⣠⠶⠴⣧⡸⣒⣒⠒⠒⠒⠚⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣿⣀⠀⠀⠙⢹⣯⡠⠂⠀⢀⡴⠋⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡟⠀⠀⠀⢸⡇
⠀⡏⡂⠀⠀⠁⠄⠈⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠞⣫⠏⠀⠀⣠⠔⡿⢁⡀⠀⢸⠁⠀⣹⠀⠀⠀⣀⣠⠤⠚⢻⠂⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⡇⡇⠀⠀⡁⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠠⠊⣴⣣⢴⡾⢳⠃⣀⣽⡈⠀⠀⠈⠞⣽⣻⠶⠒⠉⠁⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⡄⠀⠀⡇
⠀⣷⠁⣀⣀⢀⡇⠠⠶⠃⠀⠀⡀⠀⠀⠀⠐⢦⣀⠙⠒⠲⣤⣤⢡⠞⢉⡴⠋⠀⠘⠛⢙⣾⡅⠀⠀⠀⠀⣸⣿⠄⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⣇⠀⠀⠀⢀⡇
⠀⠈⠉⠙⣏⣸⠛⠉⠉⠙⠛⠓⣌⠀⠀⠀⠀⠀⠈⠑⠦⣛⠧⢄⡀⣰⠎⠀⠀⠀⢰⠋⠁⢸⠃⠀⠀⢠⠞⠁⢸⢠⠀⠀⠀⢀⠠⠀⢸⡄⠀⡀⠀⣿⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠃⠀⠈⠱⣦⣈⢱⣶⠶⢤⣤⣸⣤⣤⣼⣇⣠⣀⣸⣄⣠⣾⣛⣛⣛⣿⡛⣛⡛⢟⣛⠙⡟⠀⢻⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⢽⡞⠃⠀⠀⠛⠉⠓⠀⠇⠰⠦⠔⠆⠰⠯⠝⠆⠐⠩⠷⠍⠈⠄⣯⣥⠇⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣲⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⡀⠀⠉⣠⣼⡃⠀⠀⠀⡈⡇⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡂⠀⠀⠀⠀⠀⠀⠀⠻⣧⣐⣀⣱⣂⣈⣀⣀⣀⣀⣀⣤⣈⣁⣀⣠⣭⣴⠞⣏⡡⡌⣧⠀⠀⠀⠀⢻⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣄⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⠀⢠⣈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉