Choco Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⣠⠏⠀⠀⠀⠀⠼⠃⠀⠀⠀⠈⠓⢶⣀⠀⠀⠇
⠀⠀⠀⢀⠐⢄⠀⠠⣤⠠⢥⣀⣀⣀⡼⣀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠀⠀⠑⠶⣶⢁⠂⠀⠀⠀⠀⠀⠓⠨⠁⠀⡠
⠀⠀⠀⠉⠑⠀⡈⠑⠀⠉⠌⠀⠧⣄⡴⠀⠉⠉⢉⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢠⠏⠀⠀⠀⢰⣣⠃⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠲⢦⣉⠂⠀⠀⠀⠀⠀⠈⠋⠀⠀⠄⢸⣄⠀⠀⠀⠀⠄⣀⣤⣄⡰⡏⠀⠀⠀⢠⢒⠏⠀⠀⠬⡁⠀⢢⠐⡄⠀⠀⠀⠀⠀⢀⣀⡠⡆
⠀⠀⠀⠀⢾⠂⠀⠉⠃⠐⠠⠀⠀⠀⠉⠀⠐⠀⡀⠀⠀⠀⠠⠂⠀⠀⠈⡙⠁⠛⠶⢶⣤⡋⠎⠀⠀⠀⣄⠈⢦⠀⠇⡇
⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⣨⡋⠀⠀⠀⠀⠉⠓⠚⠂⠃⠀⠀⠀⠀⠙⠓⠒⠂
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠃⠀⠀⢶⣤⣀⣀⣴⠏⠀⠀⠀⠀⢠⠟⠀⠀⣠⣤⢶⣤⣄⠀⠀⠀⠀⠀⠀⢠⡄⢄⢄
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠻⠿⣦⣄⣀⠀⣴⡎⠀⠀⢪⠞⡵⠚⠿⣯⣧⠀⠀⠀⠀⠰⠀⠀⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠀⠀⣴⣄⢘⢀⠀⠈⠁⣠⠦⠀⠀⠀⠸⣤⣠⣀⡀⡰
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠈⠀⠀⠀⠀⣴⠄⠀⠀⠉
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠏⠀⠀⠀⠀⠄
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠂⠀⢦⣄⣠⣔⣺⡼⣷⣄⠀⠀⠀⠀⠀⠀⣠⠤⠂⢩⣿⡍⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠇⠀⠀⣄⠀⠀⠄⢀⡿⢟⣯⠾⠛⣫⠼⢆⡀⠀⠀⡀⢺⣿⣦⡀⠈⠙⢯⣌⠙⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢗⣋⣽⡔⢛⣯⣴⣿⣭⣉⣙⡿⠆⠁⠀⠀⠙⢯⡻⡯⠈⠀⠈⠛⠒⠤⣉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠰⠄⠀⠀⠀⠀⠉⠻⠻⠟⠊⠉⠛⠂⠓⠛⠋⠹⠁⠀⠀⠀⠀⠀⠀⠙⠂⠈
⠀⠀⠀⠀⠂⠀⠀⢐⡎⠀⠀⠀⡄⢀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⡟⠒⠂⠀⣸⠎⠀⠀⠈
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⣠⠄⠉⠍⠉⣇⠀⠈⠲⠗⡉⠁⣀⣀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⢲⠁⠀⠀⠀⠠⡀⠀⠀⠠⠄⢨⠇⠀⠀⠀⠀⢠⠦⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠓⠀⠀⠀⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠐⢄⣀⣤⣤⠤⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠹⢦⡘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠢⡀⠠⠄
⠀⠀⠀⢠⣀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠈⠀⡖⠀⠀⢰⣦⡤⢄⡀⠀⣠⢟⠦⡀⠀⢄⠀⢀⠀⢀⡴⠀⠀⢀⡀⠀⠠⠀⠐⡇⠀⠀⠠⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⢈⠀⠀⠀⢀⣷⠄⠀⠈⠙⠣⠀⠈⢀⡴⠁⠀⠈⢢⣄⣧⣤⠆⠈⠀⠀⠀⢋⠀⠀⠀⠐⠔⣀⡀⠴⠤⠖⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠔⠚⠩⢤⣄⣀⡐⠂⠨⠉⠑⢒⣢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣴⡉⠧⠤⣶⣶⠮⠟⠶⠄⠉⢈⣁⡠⠔⢚⣉⡭⠴⡓⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢙⢳⡿⢟⡲⠤⣌⣙⠒⠲⠚⣋⣡⠤⢖⡛⢩⢴⠚⠋⢁⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣲⣤⣆⣂⠙⠹⡗⠪⣭⣉⣉⢽⠂⠏⠠⠀⣈⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢎⡀⠀⠐⠻⠽⣒⡦⣄⣈⢘⣀⣉⣤⣴⢶⣛⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠹⢓⠺⣝⡲⠤⣄⣀⠩⢑⠺⣭⠭⡝⠛⠋⣉⣩⣤⣾⣡⠵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠙⠛⢶⣅⣈⠉⠛⠓⡻⣿⣖⠤⠬⠤⣶⣞⣫⢷⠿⠩⠝⠛⣀⣸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠫⡧⡀⢪⣍⣛⠶⠮⣄⡉⠁⠉⡿⠉⠁⣘⣀⡤⡤⣖⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡁⣝⣄⠳⡤⠵⠒⠒⠋⠓⢶⣷⣶⡿⣭⣔⣒⣺⣸⣦⠀⠀⠀⠀⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠙⢦⡀⠀⢣⡙⢮⣒⣀⣖⣶⣬⡽⠿⠳⠒⠚⠋⠉⢉⠙⣁⣤⡼⠆⠀⠀⠠⠋⠀⣠⣽
⠀⠀⠀⠀⠀⠀⠀⣡⠘⢓⣿⠀⠀⠑⣬⣉⣁⣁⣬⣬⣤⠶⠶⠖⠻⠿⠛⠛⠋⠋⠉⠀⣀⠀⠀⠙⠻⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠦⣄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⣂⣛⣦⠀⠀⢜⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠐⠀⠐⣶⡈⠉⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣆⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠛⣻⡜⡂⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢠⣤⣤⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⢶⠤⣉⣻⡐⢠⣤⣾⡿⠿⠿⠟⠛⢿⡛⠛⠉⠉⠉⠉⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠉⠁⠚⠨⣷⡸⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠩⡒⠦⣤⣘⣷⢹⡄⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀⣀⣀⠀⡄
⠀⠀⠀⠀⢺⣿⣳⣤⡀⠀⠀⢤⣄⠺⠿⠟⣣⢷⡀⠠⠤⠶⠒⣒⣦⣼⠤⠤⠔⣒⣒⣛⡿⢃⠀⢀
⠀⠀⠀⠀⠈⠻⢿⣷⣤⠠⣂⠉⠀⠀⠀⠀⠨⠿⠿⡯⠭⠭⠽⠖⠒⠶⠛⠋⢍⠀⠀⠀⣀⠀⠀⣺⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣮⣟⠦⠀⠀⠀⠂⠈⡤⢳⠀⢰⣞⢦⣄⠄⠀⠀⡀⠙⣦⠐⠁⠀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⢠⠄⢀⠄⠀⠈⠀⠈⠻⣷⣍⠂⠀⠀⡐⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠘⠀⠀⠈⠻⣿⠂⢸⠟⢠⣿⣄⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠶⣦⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⡾⠟⠀⣠⣶⠶⣶⣄⠀⠉⠁⠀⠈⠉⠛⠻⢷⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢯⣭⡛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣫⡞⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢌⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⠟⣡⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦
⠀⠀⠀⠀⠀⣸⠋⣼⠋⠀⠀⠀⠀⢠⣶⣶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣷⣄
⠀⠀⠀⢀⡴⢁⡾⠋⠀⠀⠀⠀⠀⢺⣿⣿⣷⣌⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣷⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⢀⢹⣦
⠀⠀⣠⡞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠉⠛⠋⠀⠀⠀⣼⣿⣿⣿⠿⣦⠀⠀⠀⠘⡇⣿⡇
⠀⢰⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠛⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⡿⠀⠀⠀⢰⡇⣿⡇
⠀⣿⠃⠀⢤⣤⣶⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠋⠀⠀⠀⠀⠹⣇⢻⣇
⠀⡿⠀⠀⠈⢿⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢰⠃⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣇⡇⠀⠀⠀⣠⣾⣷⠸⡆⢸
⣿⣸⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⣿⠿⠋⠈⠀⠀⢀⣼⣻⣿⣿⢀⡇⣼
⣿⣾⡆⠀⠀⠀⠀⠀⢰⣿⣿⣿⣜⣿⠀⠀⠀⠀⠈⠻⢿⣿⠿⢟⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⣸⣷⡟
⠘⣧⢻⣦⡀⢀⣀⠀⠘⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⡟
⠀⠘⢷⣾⣿⢷⣾⣿⠿⣶⣿⣿⣍⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣰⣿⡇
⠀⠀⠈⠉⠲⣆⠙⠁⢠⡌⠉⠉⣿⠿⣷⣤⠀⢠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢶⡀⠀⠀⠀⢀⢞⣿⠇⣿⡏
⠀⠀⠀⠀⠻⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⢿⣿⣿⣿⡟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⡇⠀⠀⣠⣵⡿⢋⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⣝⡿⠿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣛⡫⠀⠀⣼⡿⣫⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠌⢻⡆⠀⠉⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣼⡟⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠢⣾⡃⠀⠀⠀⠀⢰⣿⣿⣥⣿⣧⠀⢀⣀⣠⣶⣴⠾⠷⠿⠟⣛⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⣽⣭⣤⣬⣙⢿⣿⣿⣿⠏⡰⠟⡭⣉⣭⣷⣶⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⢿⣶⣶⣶⣶⣦⣭⣭⣤⣶⣶⠿⠿⠿⠛⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣬⣿⣿⣯⣭⣤⣿⣿⣯⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣩⣍⢳⡴⢛⡛⢦⠀⠀⠀⢀⣄⣰⠏⣰⢸⣇⣰⠋⣹⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡘⢁⡏⡘⠁⢋⠍⣘⠓⠒⠚⠋⣸⡏⠐⢣⣿⣉⠁⢠⣴⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⣧⣬⡴⠁⠾⠶⢚⡻⡟⡛⠟⢫⠙⠀⡴⢋⠉⣹⢀⡟⣩⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⡿⠟⡁⢀⠀⠄⠸⢡⠀⠷⡀⠬⣰⠀⣀⣉⠀⣡⣘⣃⠱⣃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠞⠛⠚⠓⠛⢺⡟⠷⠾⠿⠛⠛⠀⠉⣤⣤⣶⠌⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⠴⠖⠛⠉⠳⠦⡭⠤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡏⠁⠀⠀⢸⡇⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡇⣀⣀⣀⣈⣇⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⡏⠉⠉⠀⢠⡏⠀⠀⠀⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠇⠀⠀⠀⢸⠁⠀⠀⠀⢿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡹⣤⠤⠤⢴⡧⣤⣶⠶⢾⡦⠶⠶⠦⣾⢿⡥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢍⠙⠷⠀⠀⠀⠀⠉⠁⠐⢬⣄⣀⣀⠔⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠰⠒⠀⠐⠂⠤⠄⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢸⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠚⠿⠛⠛⢲⣄⣄⠀⠀⠀⠀⡀⠀⠉⣹⣿⠇⣀⠀⠀⠈⢳⣶⡄
⠀⠀⠀⠀⣿⡀⢀⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡞⠀⠀⣆⠀⣿⣷⢇⠀⠀⠘⢷⣷⠶⠟⠁⠀⠙⠳⠶⠶⠾⠛⢁⣀
⠀⠀⠀⠀⣿⡄⠀⠠⠀⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣾⠃⠀⠀⢹⠀⢸⣾⡘⠀⠀⠀⠀⡀⠀⠀⠐⣿⣿⡆⠀⠀⠀⠁⠈⣸⡇
⠀⠀⠀⠀⠘⢿⣶⡀⠃⢀⠀⠁⠆⢀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠸⠀⢸⣧⡇⡇⠀⠀⠀⠷⣤⣤⡾⠿⠋⠁⠀⢠⡀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢻⣧⣆⠁⠊⣀⠀⢊⡄⠀⠀⠀⠀⠈⠘⠁⠸⠀⠀⠀⠀⢸⡇⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠡⢆⡄⠠⢆⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⡇⢰⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⣤⡄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣦⣈⠰⣦⠉⠔⠀⠀⠈⣆⣿⣤⠀⠀⢰⣇⣼⣿⡇⠀⠀⠀⠀⠀⠀⠐⢲⣦⢴⣾⣯⣁⣠⣿⡂⠁⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣬⣁⠸⠄⠀⠀⣿⡸⣿⣧⡀⢸⣿⣿⡿⠁⠀⠀⣀⡀⠐⠳⣶⡏⣀⣼⣿⣍⣤⣝⣿⣆⠉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠑⠀⠀⠘⢷⣿⣿⣇⣾⣿⡟⠀⢀⣀⠈⢉⣿⣶⣞⣫⣤⣴⣿⣿⢙⣿⣿⣻⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠖⣐⠲⠶⠶⠶⣂⠴⠶⠶⠶⣌⠙⠛⠛⠉⠀⠀⠀⠈⠙⠻⢾⣷⣿⣯⣭⣽⣿⣿⠷⠾⠿⠿⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠤⠤⣿⠤⠤⠤⠤⣽⠄⡤⠤⢌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠈⠀⢀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⠀⠀⠀⠂⣿⠇⠀⠐⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡴⣯⢤⣤⡠⢤⣿⢤⣤⡤⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⢰⣿⠀⠀⠐⢻⡯⡀⢀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠠⣇⣠⠶⠆⠛⠷⠖⠰⠖⠿⠶⠖⢲⣔⠏⢻⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡟⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⠀⢠⡴⠁⠀⠀⠀⠀⠀⠤⣴⣶⣤⣴⠒⢶⣖⡖⠚⠃⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠠⣶⣦⣾⣤⣔⣋⣤⣽⣏⠀⡀⠀⠛⠗⢦⣄⠰⣶⣤⠤⣤⣤⣀⣁⡀⠘⡃⣾⣿⣷⠄⠀⠀⣀⣠
⠀⠀⠲⣦⡀⢀⠀⠀⢠⣄⣸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣙⠶⣽⣶⠤⣌⡉⠉⠛⠂⠀⣤⠨⠿⠿⠛⠉⠉⠉⢉⣻⣶⣿⣿⣿⠆
⠀⠀⠜⠋⠻⠮⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠓⢄⠈⠳⢌⣽⠿⠶⠶⣤⣤⣀⡈⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⢀⡴⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠢⠀⠹⢦⣄⣠⣤⣴⣿⣻⣷⣶⣄⣸⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠑⠢⣾⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⠐⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠒⡒⠀⠀⠈⡉⠩⠙⣿⡄⠀⠀⠈⠀⠄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢻⣆⠀⠀⠀⠁⠌⠐⡐⡡⠁⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢊⢆⠠⠀⠈⠌⠔⠔⡀⠁⠠⠘⢿⡄⠀⠀⠀⠈⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡄⢂⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠘⡿⣄⠀⠀⢐⣒⣪⣥⣥⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠛⢿⡌⢆⢤⣤⢬⣭⣿⣶⣶⡶⠿⠽⠕⣎⡵⠛⠛⠉⠉⠉⠉⠉⠀⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢥⣄⡀⠀⠀⢱⡈⢆⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣖⠚⠼⠈⢹⠐⠪⣷⠈⢜⡄⠀⠀⠀⡀⠁⠠⠀⠂⠀⠘⣽⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢉⡗⡦⢤⣀⡘⠃⣦⣧⠈⡿⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣀⡠⠤⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠇⢨⠙⢲⣿⢧⠸⣄⠀⠀⢀⣀⣠⡄⢠⣤⢤⣄⢸⠓⠀⣀⣉⣉⠥⠤⠤⠴⠶⢲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠈⠉⢀⠀⢣⠹⣄⣀⣀⣀⣤⠭⠄⠖⠒⠒⠚⣉⣉⣉⡤⠤⠤⠖⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢻⢷⠀⢷⣯⠭⠤⠤⠶⠒⠒⠚⢋⡩⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠺⠑⠀⢿⣗⣤⡑⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠯⣅⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠦⢄⡀⠈⠁⠀⢀⠤⠒⠊⠁⠀⠀⠀⠤⢐⣂⡭⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⡒⠢⢅⡀⠀⢀⣄⡤⠄⢂⣀⡠⠒⠛⠙⠙⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⣶⣤⡈⠀⠀⢾⠀⡴⠒⠋⢉⠜⢀⣰⣦⡤⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢟⠻⣶⣤⣀⢰⣆⣀⡤⠟⠒⠉⠀⠀⠀⢀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⢲⡸⡧⡈⠂⢜⠉⠉⠁⠀⠀⣀⣀⣠⠬⠭⣽⠓⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠈⠻⡁⠈⠶⡔⠒⠋⠙⠋⠉⢹⠞⣠⠆⠰⠁⠀⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠤⠾⡗⠉⣹⠠⡑⢄⠃⡇⠠⠀⠀⣀⣤⠯⠞⠃⠒⠓⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢉⣟⡶⠀⡐⠀⠈⠱⣄⢻⠆⠉⠉⠉⠉⠀⠠⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠘⠋⠀⠀⠀⠀⠀⠚⠛⠋⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠒⠚⠓⠒⠲⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠲⠦⠤⣴⣯⣓⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢉⣑⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⣹⠋⠙⡿⣿⠙⠲⠦⡤⠖⠒⠉⠁⢀⠀⣘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣨⠞⢽⢷⣄⠀⠁⠤⣔⡢⠭⢴⠞⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠎⠀⡀⣼⣿⠀⢸⡾⠋⠓⡆⠉⠁⠀⢀⣮⡴⢎⠁
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡜⣰⠃⢸⣠⣿⣷⣦⣄⣀⡤⠶⡾⠟⠉⠀⠀⣻⡆
⠀⠀⠀⠀⢀⡾⠳⠤⢤⣀⣀⠀⠀⠀⠀⠀⣠⢎⣾⠉⢣⣿⣁⡀⠀⠀⠈⠉⠑⠆⠀⣠⠔⣡⢰⠁⢃
⠀⠀⠀⠀⣾⣕⣒⠠⠀⠀⠈⠉⠓⣆⢀⡴⢡⡏⢸⣴⡟⠛⠛⠫⠭⢭⣄⣀⡀⠀⢈⡴⢺⠡⣋⡴⠋
⠀⠀⠀⠀⠘⡄⣌⠉⡙⠦⢤⣀⡀⠚⠛⣱⠛⢇⣼⣿⣦⣤⣄⣀⣐⠉⠉⠀⠉⠉⢻⠄⣸⡟⠋
⠀⠀⠀⠀⠀⠈⠉⠒⠛⠓⢄⣙⠻⠉⢻⣧⡀⣼⡏⠀⠀⠀⠀⠀⠉⠉⠛⠒⠷⠤⢼⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠳⠿⠋⠀⡠⠶⣤⡀
⠀⠀⠀⠀⠀⠐⠒⣶⣷⠄⢈⡉⣹⢷⡀⠀⠀⡠⠊⠀⢠⣸⣿⡆
⠀⠀⠀⠀⠑⠶⠥⠾⠇⡈⠁⠀⠉⣂⡼⠁⠘⠅⣀⡀⣋⣠⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢀⣀⡀⣠⠆⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠄⣿⠁⠀⣤⣤⣶⣳⣎⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⣆⠀⠐⣤⣷⣾⣿⣆⠀⠈⢹⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠲⣀⣄⠘⠾⡁⠀⣀⠤⢉⢻⣷⣠⣾⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢆⢀⡼⢿⡄⠛⣉⣑⠛⠿⣋⢸⣿⢏⣽⠛⠋⣻⣯⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⢀⡿⠒⠋⢻⠀⣻⠿⢤⢈⡿⢋⣴⡁⣿⣷⣤⣄⢽⣎⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣥⢀⣞⣤⣤⠀⣸⠀⡍⣩⣼⣿⣇⣘⠇⣽⡌⡏⢀⣬⡞⢩⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢠⣆⠳⣤⣍⡳⣶⣾⣇⡩⠝⠿⣯⣷⡾⣿⣣⣏⡘⣛⣴⢫⣾⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠐⠶⣶⡟⠉⣁⡿⢿⡿⡇⠀⢠⡾⠋⢸⣯⢀⣴⣾⠟⣻⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣑⠺⠿⡚⢻⣿⡿⠴⠶⠂⠲⠾⣦⣲⣾⣾⣿⣷⡶⠞⢛⣩⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⠐⠒⡤⣭⣭⣭⣁⣀⣀⣈⣉⣉⣉⣭⣥⣶⣶⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠿⡄⠀⣇⠀⠀⠈⠙⠟⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢿⠁⢸⡄⢐⢰⣷⠆⠾⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠸⡏⢸⣼⣿⣧⢽⣿⣿⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡄⣿⡄⣾⣿⣿⣸⣿⣿⣿⣿⢸⣽⣿⣿⣿⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠧⠧⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠿⠿⠿⠿⠙⡂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢠⡌⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠇⠀⠈⠀⠀⠀⠀⠀⢀⠐⠁⠠⠛⠉⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⠀⠘⠶⣤⡈⠐⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⡠⠀⠀⠀⠈⠙⠳⣄⡀⠀⡠⠃⡀⠊⠀⠀⠀⠀⠀⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⢶⡦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣬⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠉⠀⣠⠟⠛⣀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠍⠓⢤⡈⠐⢄⠰⠒⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠚⠁⠀⠘⢿⡄⠁⠈⠳⣄⠙⠢⠄⠀⠀⢀⠞⠁⠖⠁⠀⠀⠀⠉⠳⣄⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⢀⠀⠀⠙⢦⡀⠀⠀⠉⠢⣀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠹⠛⢷⠀⠙⠦⣀⠀⠀⠈⢿⣥⣄⠀⠀⠀⠉⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢠⠔⠠⠀⠀⠸⠖⢳⠀⣨⣷⣄⡀⠂⠀⠈⠳⠤⠀⠀⠀⠀⢂⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠰⠁⠈⠁⠓⢂⡀⠀⠠⠾⠁⣀⣸⡙⠲⣄⡀⠀⠀⠀⢤⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠑⠊⡅⣰⠋⣼⢤⡀⠀⠻⠽⠋⢹⠂⠀⠉⠓⠦⣤⣠⠏⣹⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠉⠡⠞⡁⢖⣮⡤⠂⠀⢀⢎⣀⣠⠶⣤⠴⢤⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⡏⠺⣈⠅⠽⡏⠈⠋⠀⢀⡴⠋⠛⠒⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢁⠨⢄⠇⠠⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⣄⢠⡀⣠⣄⢀⣤⡀⣠⣄⢠⡀⠀⣠⡀⣤⣤⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⣿⣼⢿⡇⣿⣾⠈⠃⡇⣿⢸⡇⠀⡿⣇⢸⡇⢸⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠇⡿⢸⠟⣧⡿⢺⣴⠇⣧⡿⢸⣥⣴⠗⢿⢸⡇⢸⣤⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠦⠤⠤⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠄⠈⠀⡽⢇⡠⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡂⠀⠀⠀⠈⠉⠉⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠂⠀⠤⠀⠀⠀⠀⠄⠀⠀⡰⠁⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⡞⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠒⢩⠔⠒⠞⡰⠁⢠⣟⠁⠀⠠⠤⠤⣄⢀⣰⣠⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡴⠁⢠⠇⠀⠉⠁⠒⠐⠊⠀⠠⢙⠥⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⡖⠀⠀⠓⠂⠤⠼⠁⠀⡿⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⡀⠀⠀⠀⠁⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡶⠒⠦⢤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣶⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⢠⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡞⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠠⠤⠤⣄⣀⣀⣀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡦⠀⠀⠰⠦⠴⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠉⠀⡶⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⢀⡝⠉⠉⠓⠂⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠄⠀⠰⠿⠧⠤⠶⠆⠘⢆⡰⠀⠀⠀⢮⣅⣰⡀⢀⠀⢄⣀⣍⠉
⠀⠀⠀⢀⡀⠀⠠⠒⠀⠀⢀⣀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠛⠑⠒⠒⠒⠒⠚⠂⠈⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠀⠙⠉⠉
⠀⠀⠀⠰⠀⠲⢤⣤⣰⠞⢉⣁⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠈⠑⠈⠻⢶⣵⢦⣤⠆⠀⠀⠀⠀⠀⠀⠐⠙⢀⠀⠄⡤⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢩⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠋⠀⢀⡀⠀⠀⠶⢒⢴⠥⠀⠀⠀⠀⠈⢗⠀⣺⠵⠚⠁⠀⠀⠀⣠⠤⠤⠄⢀⠀⠀⡀⠀⠀⠀⠀⢣⡄⣀⣀⡥⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣄⠰⠀⢀⡂⣐⣁⣔⡡⠀⠀⠀⠀⠹⡘⡁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡼⣆⠀⢚⣡⠄
⠀⠀⢀⠀⠤⠤⠤⣀⠔⠋⠉⣌⠀⠐⠦⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣀⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⣾⠝⠋
⠀⠀⢨⠁⣋⣬⠉⠉⠀⡀⠐⠒⣷⠦⣖⠛⠃⠀⠀⠀⠠⠖⢈⡍⠁⠒⢉⡰⠂⠀⠀⠀⠀⠀⠀⠀⡦⠠⠄⢀⣀⠒⠠
⠀⠀⠈⠉⠙⠙⠦⠤⠖⠛⠚⠛⠐⠚⠀⠀⠀⠀⠀⠀⢀⡞⢠⣤⣖⠚⣥⠐⠲⠒⠦⡀⠀⠀⠀⠀⢣⢀⠀⠀⠈⠑⠒⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠡⠀⠰⢰⣰⣈⡢⠴⠃⠀⠀⠀⠀⠈⠃⠒⠒⠒⠲⠦⠘⠖
⠀⠀⠀⠀⣀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠚⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢨⢛⡗⠀⠀⠀⠀⡠⠀⠀⠀⠀⡐⠢⡀
⠀⠀⠈⣦⠉⠀⠀⠘⡄⠀⠀⠀⠀⢠⡖⠂⠄⠀⣀⠈⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⣬⠇⠀⠀⠀⠀⣇⡀⠀⠀⢠⢤⠀⠀⣀
⠀⠀⠀⢰⡀⠀⠀⠀⣰⡀⠀⠀⠀⠀⣀⡈⠀⠑⢠⠄⠀⠠⠀⠀⣠⠁⠀⠀⠀⠀⠀⠇⡀⠀⣀⢀⠏⠀⠀⠀⠀⠀⠈⠁⠀⠉⠈⠀⠀⠊⠁
⠀⠀⠀⠀⢷⠄⠀⠉⠄⠡⠀⠀⠀⠐⡏⠋⠑⠒⠀⣀⣈⣋⠠⢸⡥⠀⠀⠀⠀⠀⠈⠒⠚⠯⠶⠊
⠀⠀⠀⠀⠸⡄⠀⠈⠀⠐⣇⠀⠀⠀⢰⠀⠈⠉⠃⠁⢀⣈⡀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠙⠤⠔⠊⠉⠀⠀⠀⠀⠀⠁⠈⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡠⠤⠖⠒⠐⣮⠉⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡴⠒⠀⢎⠀⠁⠀⢀⠙⡄⢀⣠⣶⣬⣷⡖⠛⠛⠙⢆
⠀⡀⠀⠠⠤⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠘⢦⣀⣀⣠⣤⣷⡴⠶⠛⠛⠹⡍⠁⠀⠀⡀⠻⡄⣁⣠⣤⠧
⡀⢄⣀⣀⠀⠀⠱⡀⠀⠀⠀⠐⠋⠀⠁⠑⠒⠖⠒⠉⠀⠀⠀⠀⠘⣧⠴⠾⠛⠚⠋⠉⠁⠀⠀⢁⠀⠘⣀⣤⡽⠄⠒⠒⠋⠉⠉⣀⢀⣀⡠
⢩⠀⠄⣠⡴⠤⠤⠁⠀⠀⠀⠠⣤⣄⢐⡀⣠⡀⢠⡠⠆⠀⠀⠀⠸⡎⢃⠀⣀⣀⣸⣆⠤⠖⠒⠒⠃⠈⠁⠀⣠⣆⠴⠞⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢂⠈⠙⠛⠋⢀⠀⠀⠀⠀⠀⠀⠹⣴⠋⠉⠁⢀⣴⠦⠤⠟⠒⠒⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣤⣂⢘⡃⠂⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊⠉⠉
⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠖⠚⣛⣉⣉⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⢶⣶⠶⠶⠶⠶⠶⠶⣶⡄⠀⣿⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠇⠀⢹⡆⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⣼⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡀⠐⠆⣧⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡖⢸⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⠀⢀⢹⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠃⢸⡇⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⡇⠀⠈⠸⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡇⣶⣶⣶⣶⣶⣾⣿⠀⣶⣶⣶⣶⣶⣿⣿⠠⣶⣶⣶⣶⣶⣿⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣿⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠞⠗⡀⢸⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠐⠁⢸⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⠀⠈⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠙⠆⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠳⠾⣧⣤⣤⣤⣬⣥⣽⣿⠁⣤⣤⣤⣤⣤⣿⣿⢀⣤⣤⣤⣤⣬⣿⣤⢴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢉⡛⠿⢿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣴⡿⠿⠟⢛⣩⠁⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠉⠳⢦⡬⣿⡿⠋⠉⠉⣀⠀⣀⡉⠉⠠⠒⠚⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠒⠦⠴⠦⠤⠤⠤⠤⠈⠉⠛⠉⠙⠚⠛⠉⠀⠀⠀⣶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠴⠮⠤⡤⠤⠰⣄⠀⠀⠀⣜⣀⣌⠀⠀⠀⣀⡉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠀⢀⣀⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠉⠐⠒⠂⠐⠒⠂⠁⠀⠀⠀⠀⠉⠈⠉⠁⠀⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠈⡒⢦⡞⢋⡡⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠢⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠛⠛⢓⢁⠀⠀⠀⠀⠀⠀⡀⠀⡀⣘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡑⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠑⠊⠉⠀⠀⠀⠀⡀⠉⡁⠀⠄⠠⠊⣄⠀⠀⠀⠙⠔⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠘⣬⢄⠀⠔⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⣀⠐⠈⢁⡀⠐⢦⡀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣉⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡾⠛⠃
⠀⠀⠀⠀⠀⠀⢠⣈⣵⡅⠉⢠⡀⣐⠻⢓⠌⠋⠀⠀⠀⠐⢂⠡⠀⠀⡩⠔⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢠⣐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠃⡶⠊⠑⠒⠀⠉⠁⣂⠀⠀⠀⠀⢳⣀⣀⣀⣈⡉⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠔⠖⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠄⠈⠋⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⠄⠀⠀⠀⡠⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠙⡀⠀⠀⠀⠠⠒⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣇⠀⡀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢀⠴⠀⠀⠀⠀⣀⣁⠈⠁⠖⠀⡀⠂⣠⠄⠀⠀⠀⠀⠐⠀⣀⠤⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⢀⠀⠀⠈⢃⠀⠉⢁⠰⠄⠤⠒⠓⠀⠀⠀⠀⠀⠈⠉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⢀⠠⠜⠀⠀⠀⠸⠀⠤⠤⠤⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠤⠐⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠐⠂⠀⠻⡌⠀⢀⣌⣳⡠⠤⠶⠦⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠁⠀⠻⣄⢀⣀⣬⣧⡴⠶⠟⠳⡏⠉⠁⠀⠻⣄⣀⣀⣴⠄
⠀⠀⠀⠀⠀⠤⣀⠀⠀⠰⡄⠀⠀⠀⠋⠀⠉⠑⠒⠐⠈⠀⠀⠀⠀⣣⡴⠚⠓⠘⡉⠉⠀⠈⢀⠀⣂⣤⠽⠄⠒⡈⠉⢉⣀⠀⠀⠐
⠀⠀⠀⠀⠘⠀⠀⠶⠃⠐⠀⠀⠀⠀⢰⠶⠥⢤⣦⡐⠄⠀⠀⠀⠀⢳⢁⣀⠤⠴⠟⠀⠚⢉⣉⢁⣀⠀⠴⠂⠚⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢩⡀⠀⠀⠀⠀⠀⠀⠻⠀⠠⠤⠒⠛⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠚

⠀⠀⡀⠀⢀⠀⠀⢀
⣿⡏⢭⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢖⣒⠭⠥⠒⠲⠊⠉⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠉⠉⡆⠀⠀⠀⠀⢇⠀⠀⣀⣀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢹⣀⣠⠤⠴⠚⡏⠉⠁⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠴⠒⠋⠉⢇⠀⠀⠀⠀⢳⠀⢀⣀⣠⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⣸⣄⡤⠤⠖⠚⣏⣉⣀⣀⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠋⠉⠁⢣⣠⠊⠳⣀⠜⠁⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠙⠒⠛⠉⠁⠀⠀⢀⡤⠤⠴⠒⠲⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⡠⣄⡤⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣇⠀⠀⠀⠀⠀⢀⠀⡀⢠⡐⠆⠯⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⢠⠴⠆⠣⠙⠃⠉⠀⠀⠀⠀⠀⢸⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⢀⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠇⠁⢸⡷⢂⢸⠀⠀⠀⠀⢠⡷⠦⠰⠷⡧⠀⡧⠆⠀⠆
⠀⠀⠀⠀⠀⠀⡟⠛⢻⡟⡛⢻⣟⠛⢻⠀⠀⠀⠀⢸⡟⡛⠛⡟⠛⠛⡟⠛⢛
⠀⠀⠀⠀⠀⠀⣤⣤⣼⣧⣭⣼⣇⣤⠘⠀⠀⠀⠀⢸⣧⣄⡀⣃⣀⣀⣃⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡇⡠⢰⡤⢀⠀⡷⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠸⣷⡒⢪⣲⡖⢢⠓⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣀⣄⣀⡀⠴⠀⠀⠀⠀⣟⣓⡘⣚⣃⣸⣀⣀⣸
⠀⢠⠀⠀⠀⣀⣠⣴⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⡏⠀⢸⠉⡀⢸⠉⠁⢸⠀⠀⢀⡠⠐⠀⠀⢀
⢀⠜⠓⠚⠉⢹⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠖⢦⠶⠶⢦⡶⠲⢾⠀⠀⠙⠛⢦⣾⠛⠛
⠀⠀⠀⠀⠀⠘⣿⣯⣁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⡇⢀⢘⠀⠀⢸⣇⣀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣷⠤⠤⠿⣖⡛⠉⠂⡀
⠀⠀⣠⣄⣰⣶⡾⣿⣿⣿⣧⣀⡄⠀⣀⠀⣤⠄⠀⠲⢄⡤⠽⠲⣖⠀⣀⡀⠀⠈⣷⣴⣶⡶⠪⠞⠿⠟⠓
⠀⠈⠻⢿⣿⣿⠷⣶⣾⣭⣈⠭⢓⡭⠷⣤⠙⢷⣯⢁⣀⡹⠢⣶⠈⢈⡪⢠⣤⡤⠉⠁
⠀⠀⠀⠀⠉⠻⠆⠚⠙⠿⠟⠛⠁⠀⠈⠁⠀⠀⠻⣿⣿⣡⣴⣾⣶⡿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⣡⠔⠳⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⢿⣄⠈⠀⠀⠀⠈⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠔⠋⠙⣆⠙⣦⠀⢀⠀⠀⠀⢀⣽⣧⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠻⣄⠀⠀⠀⠀⠘⠀⠘⣧⡈⢁⡤⠖⣩⣄⠻⣷⡀
⠀⠀⠀⠀⢠⡞⠁⠀⠊⠉⠣⠘⣦⠀⢀⠀⠀⠀⢀⡿⢿⣇⠀⠈⠀⠈⠁⠘⢷⡀
⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠈⢷⣌⣉⡴⢚⡥⠦⡄⠻⣦⠀⠀⠀⠀⠀⠀⠳⠂⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠹⣿⡄⠠⣄⡠⠀⣠⠼⣿⣇⠀⠀⠀⠀⠀⠀⠹⢧⠀⢀⠀⠀⠁⠀⠀⢀⡀⠈⠉⠉⣉⠖
⠀⠀⠀⠀⠀⠀⠘⣿⣦⡤⠖⠫⠖⠲⣄⠹⣧⠀⢀⡀⠀⠀⣀⠤⠎⠁⠀⠀⠓⢀⠀⣀⣻⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠙⢷⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠁⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠳⠀⠀⣀⢤⡞⠁⠴⣿⣶⡄⢀⣀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠔⢋⣡⡴⠿⠶⠾⠋⠋⠀⠀⠈⠒⢦⣤⠏⠀⠀⠀⠀⠀⠀⠁⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠄⠀⢀⡠⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣂⢹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠔⠁⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⠔⠋⡠⠊⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⠀⠀⠀⠀⡠⢊⡔⢈⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠜⠀⠊⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⣠⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⢀⣠⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⢀⠂⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡎⠀⠀⠀⡰⠁⠀⠀⠰⢤⡀⠀⠸
⠀⠀⢀⠐⠄⠀⠤⡰⣄⣀⣀⡸⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡉⠀⠲⢦⠄⠀⠀⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠈⠉⠠⡈⠀⠐⠂⠒⢦⣔⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣼⠆⠀⢀⠄
⠀⠀⠀⢨⠓⠠⠄⠀⠀⠀⠀⠄⠀⠁⠘⠂⠀⠀⠀⠐⠒⢲⠊⣄⣀⣀⠌⠎⠀⠀⠑⠀⠐⡘⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠠⠀⠈⠀⠀⢀⡄⠀⠀⠈⢙⠆⠀⠀⠈⠓⠼⠄⠂⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠰⣤⣀⣠⡞⠀⠀⠀⢀⠎⠀⢠⣤⢤⣤⡀⠀⠀⠀⠀⢠⡄⡠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠙⠳⢦⣤⣠⡟⠀⠀⠞⡕⠙⢹⠿⡀⠀⠀⢰
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠛⠤⠄⠄⠀⠐⠋⠀⠀⡀⠙⠒⠒
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢦⢀⣔⣤⢾⢆⠀⠀⠀⠀⢀⡀⠀⢠⣾⣡⣀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⣠⡴⢻⡿⢊⣭⢾⣧⣄⠀⠀⠸⣿⣦⣀⠉⠲⣬⡙⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠘⢧⣾⡿⠴⠿⠛⠿⢷⢶⠏⡀⠀⠀⠈⠳⠍⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢎⠀⢀⡠⠂
⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠢⢀⡆⠀⠀⠀⠄⠀⠉⠃⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠊⠀⠀⠀⠘⣄⠀⠈⠀⠃⠀⠀⠀⠘⣧⠀⠀⠀⣀⣀⡲⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄⠠⠀⠀⢜⠳⣄⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠃⠀⠀⠈⠀⠀⠐⠁⠀⠀⠳⢾⠞⠁⠀⠀⠀⠢⠀⠀⠘⠥⠒⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⢠⣄⣠⡤⠖⠒⣒⣟⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠀⠀⠉⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡆⢀⢻⠃⠀⠇⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡄⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠇⢸⢸⡄⡆⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡼⠃⡇⢹
⠀⠀⢠⣖⣋⠀⠀⠀⠀⡀⠀⠀⠙⠉⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⣿⢀⣸⡀⡀⠇⢿⣦⣤⣤⡤⣤⠤⠤⠄⠀⠀⠠⠤⠴⠄⠀⠤⠀⡱⡌⡄
⠀⡴⢸⣟⠷⣤⣤⠊⠁⠀⢀⣀⡀⠤⠴⠶⠶⠛⣉⣿⣦⠀⠀⠀⢹⡀⣀⡑⠀⠀⠻⠥⠶⠄⠀⠀⠀⠁⠀⠀⠀⠀⣂⡀⠀⠀⠠⠀⠁⡤⣤
⣼⠁⣼⡸⠀⠈⠙⣿⣶⣤⣌⣁⣠⣤⣴⡶⠞⠋⠉⠛⢠⠀⠀⠀⠘⣷⢰⡄⠀⡄⢹⣿⡿⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⢶⣧⢸
⡏⠈⠳⣄⡀⠀⠀⢻⡿⠘⠇⣀⠀⠀⢃⠀⢀⠀⠀⠴⠻⣄⠀⠀⠀⠹⣆⣧⠀⣇⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠈
⢳⡀⠀⠀⣝⠓⢦⣀⠀⠀⠀⢿⡀⠀⠘⡀⠘⣇⠀⣴⣤⡟⠀⠀⠀⠀⢻⣼⡄⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠘⢸
⠀⠙⠶⣤⣟⠀⠀⠈⠙⠂⣠⣼⣷⡌⢄⠛⠀⢻⡶⠚⠋⠀⠀⠀⠀⠀⠀⢿⣧⠸⠀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠛⠓⠛⠛⠒⠒⠒⠒⠂⠈⢣
⠀⠀⠀⠀⠙⠻⢄⡀⠀⠀⠀⣩⣿⣿⠸⠷⢠⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢀⣾⣋⣉⣀⣀⣄⡀⢀⣠⠤⠶⠶⠶⠦⠤⠤⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⠀⠈⢩⣿⣤⡼⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠂⠢⠀⠿⢋⠔⠊⠉⠉⠉⠈⢥⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠽⠿⠶⠶⠶⠒⠒⠒⠒⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠃⠀⠈⠉⠉⠁⠀⠀⠀⢠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡪⠟⠉⠀⣀⡠⠀⠀⠀⠀⠀⢀⣠⡶⢶⠛⠁⠀⢹⡌⢉⡙⠛⠳⠶⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠷⣦⣤⣤⣭⣤⣤⣶⣶⠾⠟⢛⣉⡀⢀⡼⠂⠀⢰⣿⡇⠀⠭⠝⠓⠲⣦⡀⠤⡎⠀⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⠇⠀⠀⠀⢰⡀⠈⠁⠠⠔⣒⣫⡭⠗⠚⠉⠀⠀⢀⡼⢸⡇⠀⠀⠀⠐⣋⡭⣖⠫⠄⡈⠁⢳
⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⠏⠎⠀⠀⠀⠀⠀⠙⠶⠖⠚⠉⠁⠀⣠⣐⣈⣿⣴⣾⠿⠾⠟⠁⠀⠔⠒⠤⠃⠈⠀⠒⣫⢕⣫⠞
⠀⠀⢀⡤⠖⠋⠩⠴⠆⠘⠲⠤⠭⠿⠿⠿⠿⠿⠶⠶⠿⣿⣿⡛⠿⢿⣧⣤⣄⣠⠴⣒⠤⠶⠔⠊⠁⠆⠀⠰⣛⡭⠖⠋
⠀⠀⡞⠐⠲⢤⣄⡀⠒⠤⣄⣒⠤⣄⣀⡀⠀⠀⠀⠀⢀⠉⠀⠉⢙⣒⣈⠙⣯⠐⠉⢀⠝⠃⠀⠀⣀⡤⠖⠋⠁
⠀⠀⣇⠀⠹⣳⣌⡉⠛⠲⢤⣉⡙⠲⠯⣭⣭⡥⠴⠚⣉⣤⣶⠟⠻⠟⠃⠓⠋⠀⣉⣤⣀⡤⠖⠛⢁⡀⢠⡦⢤⣀⡀
⠀⠀⠈⠲⢤⡠⣉⠛⠆⠀⣀⡈⠛⠳⢦⣤⣀⣠⢴⣿⠿⢛⣩⡄⠀⢀⣠⠴⢒⡩⠖⠋⠁⠀⣠⠮⠉⣠⡏⠀⠀⣨⡭⠟⠃
⠀⠀⠀⠀⠀⠉⠓⠯⣕⡢⣈⠋⢍⣙⣳⡌⣿⠵⠋⡔⡊⠅⠋⠀⣀⡥⠤⠖⠋⠀⠀⠀⠀⠀⠙⠶⣼⡿⠿⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠿⣶⣤⣀⠀⢸⠀⠀⠀⣀⡤⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣲⣤⣴⣖⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠙⠒⠲⠤⢤⣀⣀⠀⠀⠀⣠⠴⠋⢧⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠷⠒⠃⠀⠀⠀⠈⠱⠤⡀⠀⠀⣀⡀
⠀⠀⠀⠀⣤⣞⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠋⠁⠀⠀⠀⠀⠀⢠⣄⣤⠖⠚⢍⠁⠔⠁
⠀⠀⠀⣸⠁⠀⠉⠙⠒⠢⠤⣀⣀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⡀⣠⣢⠏⡉⠉⠀⠅⠀⢅⡀⠄⠈⡄⠀⠂⡠⡀
⠀⠀⠀⠛⠬⢄⣀⠀⠀⠀⠀⠀⠈⠙⠻⡍⠉⠉⠁⠀⠀⢀⣒⢵⡿⠯⠀⡁⠉⠐⠉⠄⠂⣀⠤⠁⡉⠀⠆⠁⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⣀⡀⠀⠓⠐⢲⡀⠀⠄⡸⠾⠋⡁⠀⠉⠀⠤⢐⢀⡰⠌⠀⠈⠑⠊⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⢤⠬⠃⠀⠳⣄⠀⠀⠉⠐⠀⠠⠈⡀⢀⠤⠂⠄⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠑⠘⠦⣄⣀⠀⠀⠀⠀⠀⣕⣀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠙⠦⣀⣀⣀⡀⠀⠈⢏⠓⠒⠦⢬⣿⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⢀⣀⣀⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⢀⡖⣫⠄⣸⠂⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠟⠙⠲⣤⡀⠘⣧⣓⣚⣡⠖⣀⣴⠾⠛⢿⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠙⠳⢤⡀⠀⠀⠀⠙⠳⣤⡉⠉⣠⡾⠋⠁⢀⣴⠟⠋⣀⣙⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢀⣠⢖⡛⢢⠉⠳⢦⣀⠀⠀⠀⣹⣿⡏⠀⢀⣴⠟⠁⢠⠞⡭⠆⡼⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⣅⡀⠀⢼⠱⣏⡴⢋⡀⠀⠀⣉⣷⣶⠾⠛⠀⠹⢤⣾⣅⡀⠀⠸⣦⣓⣚⡤⠆⢀⣠⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠺⢯⣉⠀⠀⠀⠙⠲⣌⡛⠲⠒⠋⢀⣴⠟⠉⠀⠈⣣⣤⣤⣤⣴⠟⠈⠙⠶⣤⣀⠉⠁⢀⡴⠋⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠙⠲⣄⣰⠟⠁⠀⣠⡴⠛⠉⠀⢀⡉⠳⣤⡀⠀⠀⠀⠉⠳⣴⠋⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣠⠞⠁⢠⡴⣚⠙⡆⠀⠀⠀⠙⢳⣄⣀⠀⢀⡼⢇⠀⢀⡾⠋⠀⠀⣠⢞⡩⢉⡇⠀⠙⢦⡀⠀⠀⢰⡇⠀⠄⠀⠒⠀⢸
⠀⠀⠀⣠⣟⠁⠀⠀⣏⠸⣡⠞⣁⠀⠀⣠⠾⠋⢡⠈⠉⠉⠀⠈⠓⣾⡟⢦⣄⠀⢧⡸⠴⢊⡤⠀⢀⣤⡿⢦⣤⣼⠇⠀⠀⠀⢀⣴⠞
⠀⠀⠀⣿⠙⠳⢦⣄⡈⠓⠒⠛⢁⣴⠞⠁⠀⠀⠈⣦⡶⠚⠳⢶⣾⠋⠀⠀⠈⠳⣤⡉⠉⠉⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⢸⠐⠂⠀⠉⡛⠶⣤⣠⠞⠁⠀⠀⢀⣴⠞⠁⣠⡤⠶⡄⠉⠳⣄⡀⠀⠀⠀⠙⢦⣴⠋⠖⠀⠀⠀⡀⠀⠀⣠⡶⠋⠀⠀⠀⠀⢠⣂⣶⡀
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⢷⠀⠀⢀⡴⠋⠁⠀⣾⢡⠚⣠⠃⠀⠀⠈⠛⢦⡀⠀⠀⢠⡇⠀⠀⠀⠀⢈⠆⣠⠞⠋⠀⠀⠀⠀⠀⠀⠈⠙⠚⠃
⠀⠀⠀⠀⢷⠐⠄⠀⠀⠀⠀⠘⣇⢀⡟⠻⢶⣄⡀⠙⠦⠭⠵⠚⠁⢀⣠⠶⠛⠛⠷⠤⠞⠁⠀⣰⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠂⠀⠉⠉⠀⠀⠀⠈⠛⠷⣤⡀⠀⢀⣴⠟⠁⢠⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⢠⣶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⢀⠀⠀⠘⠀⢀⠈⠙⢷⢛⠁⠀⠀⠀⠀⠰⠃⠠⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠐⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⢸⡇⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣖⣉⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢼⣇⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠀⠂⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠺⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠲⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢤⠤⢊⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠋⠉⠉⠉⠛⠒⠶⠤⣄⣀⡀⠀⠀⠀⡤⠾⠃⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢉⡿⠷⠞⠁⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡤⠽⡒⠊⢉⠽⠁
⠀⠀⠀⠀⣼⠋⠙⠓⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡰⠖⠒⠫⠁⠂⡀⠌⢁⣀⠀⠒⢀⠠⢀⡀
⠀⠀⠀⢠⡇⠤⡀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣤⠤⠤⠤⠼⠀⠀⠀⠀⢀⠤⣪⣴⠾⠚⠥⢰⣀⡠⠦⢁⣀⠁⠃⢨⠀⠤⢃⠀⢠⠐⠙⡄
⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠈⢉⡇⠀⠀⠀⠀⠀⠀⣠⠵⣚⣿⠍⠂⣀⠤⠀⡀⣈⠙⠂⢠⠤⠔⣃⠠⢄⠀⠋⠀⠀⠀⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣄⣀⠀⠉⠊⠙⣄⡀⠀⠆⡽⠺⠋⢠⣀⠀⠀⠀⠑⠂⣁⠠⣚⠢⠤⣀⠤⠋⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⡤⠜⠀⠣⠳⣄⠀⠀⠀⠉⠉⠀⠠⠎⣀⠀⢀⠭⠒⠠⠀⠀⠀⡀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠓⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠔⠒⢦⡀⠀⠀⠀⠀⠈⠉⠛⡶⠦⢤⣀⡀⠰⡟⢋⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠙⠲⠤⠤⠤⠵⠀⠀⠀⠉⠙⠋⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⢀⣀⣀⣠⣠⣤⣲⡦⠤⠷⠒⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⡀⠤⠤⠄⠒⠂⠹⠾⠖⠒⠚⢻⣽⡏⠉⠁⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢀⡀⠀⢀⣀⠠⠤⠤⠄⠒⠒⠈⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠸⣿⡅⠀⠀⠀⠀⠀⣿⣿⠀⢀⣀⣀⣀⣼⡇
⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⣀⣀⣀⣤⣿⠷⠤⡶⠶⠒⢺⣛⣟⣋⣩⣭⡵⠶⢤⣷
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣩⣭⣭⡵⠶⠤⢾⣷⠖⠚⠋⠉⠉⢹⣿⡅⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⣻⡄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣿⡀⠀⠀⠀⠀⣸⣿⣇⣀⣠⣤⠤⠤⢿⡿⠶⠲⣛⣛⣋⣭⡇
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⠗⠚⣛⣛⣫⣭⣽⣧⠶⠤⠵⠒⠒⢳⣿⡋⠉⠉⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⣿⡟⠉⠉⠁⠀⠀⢿⣿⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⣿⣿⠄⠀⢀⡀⠀⢸⣿⡇⠀⢀⣀⣀⣠⣿⣷⡤⠤⣴⡲⠶⠿⠇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⢀⣠⠞⠇⡉⠀⠀⠀⠀⠀⡈⠐⢾⡻⠛⠛⠉⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀⣀⠀⡴⠚⢍⠀⠀⠀⠀⠀⠀⠈⠂⣄⠈⢢⠄⠂⠈⠀⣀⡤⠴⠛⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠠⠒⠒⠂⠉⠁⠀⠀⠃⠀⢀⠀⠀⠁⢠⠀⠀⠀⠀⠈⠁⣀⣳⣬⣀⡴⠚⡉⠁⠀⠀⠀⠀⢀⠠⠐⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠠⡄⠀⡠⡶⠒⠛⠉⠉⠀⠀⠈⠑⠄⡀⠑⢠⠀⠀⠈⢀⣀⡠⠂⠉⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠂⡈⠀⠈⠂⢄⠀⠀⢀⡀⠄⠀⠐⠉⣦⣰⡀⣤⠖⠉⠁⠀⠀⠀⠀⢀⠠⠈⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⢀⠀⠑⠈⠀⢀⢠⡠⠔⠂⠁⠉⠑⠧⡀⠈⠂⢄⡀⠄⠂⠀⠠⠐⠀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢑⢔⡦⢲⠞⠊⡁⠀⠀⠀⠀⠀⠀⢀⠠⠐⣄⢀⢀⠀⠀⡁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠀⡀⡀⠈⠠⢀⡀⠀⠐⠈⠀⠀⠀⠉⢺⠅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠊⠢⡀⠈⢀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠍⡀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣾⡿⠉⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣿⣿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⣾⡿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⣠⡞⠁⣠⢾⣿⣿⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⢀⣴⠏⢠⡾⠁⢸⣿⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⣄⠘⢿⣦⡀
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⡁⢠⡟⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠻⣷⡄
⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠻⣷⡟⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠙⣿⣆
⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠏⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠈⢿⣦
⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⠈⣿
⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠋⣀⣾
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠻⣷⡄⠀⠀⠈⣉⣁⣠⣤⣶⣾⡿⠛
⠘⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠙⢿⣿⡿⠿⠿⠟⠛⠋⠉
⠀⠀⠈⠙⠿⢷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠿⠿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠆⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠟⠋⢁⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣤⣤⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⣿⡿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⡟⢟⣙⠛⣻⢛⣻⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢉⣩⣭⣤⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣉⣉⣉⣉⣉⣿⡟⢉⣉⣉⣉⣉⣽⡇⠈⣿⠸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠃⢀⡸⡆⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣉⢻⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⠀⠁⣧⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠃⢸⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⠀⠐⢸⣾⣿⠿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢸⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡏⣥⣭⣭⣭⣭⣿⡏⢨⣭⣭⣬⣥⣿⣿⢉⣭⣭⣭⣭⣽⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⡟⠀⢸⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠁⢸⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⢀⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠛⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠙⠛⠷⣶⣶⣶⣶⣶⣿⡇⢰⣶⣶⣶⣶⣾⣿⢠⣶⣶⣶⣶⣾⠷⠖⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠠⢍⡙⠻⢿⣿⣷⣾⡿⠿⠿⠿⣿⡿⠾⠛⢋⣉⠥⠆⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣄⣀⠉⢛⠛⠾⠁⠀⠀⠤⣤⡤⣤⣀⣠⣭⠁⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠉⠉⠑⠉⠉⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⠤⢤⠤⠤⠤⠤⠤⠤⠤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠼⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠒⠋⠉⠉⠉⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣏⠀⠉⠒⠠⠀⠀⠀⢀⣠⡤⠖⠊⠁⠀⠀⢀⣀⠼⣃⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠨⣍⣒⠤⢀⠀⠀⢰⠁⠀⠀⣀⠀⠀⢀⣩⠤⡶⡺⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⢀⠢⠈⡏⠂⢀⣀⠚⠖⣋⣡⢤⢤⠞⠁⣈⠀⠀⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣷⣦⣀⠀⠀⡇⢸⠋⠁⠀⣤⢋⣠⠴⠿⠞⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢹⡍⠻⡷⠦⣄⣸⣤⡤⠚⠉⠁⠀⠀⠀⣀⣁⣤⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⠀⠹⣄⡿⠢⡈⠰⠧⠄⣀⢀⢀⣀⣠⣤⡤⠴⠒⢒⠖⢩⡽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⢀⠀⠙⢇⠀⠙⢶⠒⠛⠉⠟⠋⠀⢸⡟⣀⡔⢀⠊⠐⣀⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠢⠴⠯⡿⠉⣹⠂⣌⠢⡘⢸⠀⠠⠀⠀⣠⢤⠮⠚⠛⠒⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢱⣟⡆⠀⡐⠀⠀⠑⢆⡜⡇⠉⠈⠉⠉⠉⠀⠠⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠁⠈⠷⠁⠀⠀⠀⠐⠀⠒⠿⠟⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠚⢉⠵⠋⠀⠀⠀⠀⠈⠙⠯⣉⠑⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠥⠒⠒⠊⣁⣀⠤⡀⠀⠀⣀⡀⠀⠀⠀⠉⠒⠒⠒⠚⠳⠦⣀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢼⣉⡡⠤⠔⠃⠀⠀⡇⢸⠀⠀⠀⢀⡤⠊⣹⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⢠⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⠀⢰⣁⠴⠊⠁⠀⡤⢄⠀⢸⡙⣄
⠀⠀⠀⢰⠃⡸⠀⠀⠀⡠⠤⠔⠒⠒⡄⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆⠀⢇⠘⣆
⠀⠀⢠⠇⡠⠃⠀⠀⠀⠙⠒⠚⠋⠉⠁⠀⢠⠶⠒⠢⠤⠔⠺⠷⡀⠀⠀⠀⠀⠀⠀⠣⣈⠆⠈⠢⣼⡄
⠀⠀⣼⠜⠁⢰⠙⢆⠀⠀⠀⠀⢀⡠⡤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠒⢖⢦⠀⠀⠀⠀⠀⠀⡀⠀⠈⢣
⠀⠀⡏⠀⠀⠀⢣⡈⢦⠀⠀⠀⢾⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⣀⠔⢉⡽⠀⠀⢸⠄
⠀⠀⡇⠀⠀⠀⠀⠑⠚⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⣡⠔⠉⠀⠀⢀⣼
⠀⠀⡿⣄⠀⠀⠀⠀⡠⢤⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡆⠀⠀⠀⠀⠀⠀⢠⠎⢸
⠀⠀⢱⠈⢆⠀⠀⡴⢁⡜⠀⠀⢸⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠖⠁⠀⠀⠀⢠⠋⡵⢸⠀⡎
⠀⠀⠘⡄⢸⠀⠸⣠⠎⠀⠀⠀⠀⠉⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⣤⡀⠀⠀⢀⠏⣰⠁⠘⣤⠃
⠀⠀⠀⢳⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠦⠒⠒⠲⢖⣩⠏⠀⠘⠦⡈⠲⣄⠈⠒⠁⠀⠀⡟
⠀⠀⠀⠀⢻⡀⢀⡤⠤⠤⠤⢄⠀⠀⠀⢀⡴⢦⠀⠀⠀⠀⠀⢀⡖⢦⠀⠀⠈⠲⠼⠀⠀⣀⢤⡞⠁
⠀⠀⠀⠀⠀⢣⠘⠒⠒⠒⠒⠊⠀⢀⠔⢁⡴⠋⠀⠀⠀⠀⠀⢸⠇⣸⠀⠀⠀⠀⠀⢠⠊⢁⠞
⠀⠀⠀⠀⠀⠀⠙⠢⢤⡤⣄⣀⡀⠈⠒⠉⠀⢀⡤⠤⣀⠀⠀⠸⣄⡏⠀⠀⠀⠀⠀⣎⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡈⠙⠢⢄⠀⠀⠑⠢⢄⡉⣢⠀⢀⠤⠤⢄⣀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠽⠦⣄⣀⡀⢈⣡⣔⡡⠤⠔⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣄⠀⠀⠀⢀⣠⣶⠀⠀⢠⣄⣀⣀⣤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁⠀⠈⠛⠛⠛⠋⠉⡈⢛⣶⠟⠉⠉⠉⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠈⠹⢦⣄⡀⠀⠀⠰⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠈⢳⣤⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⡾⠛⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢦⣽⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⢐⠆⣠⡾⠋⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠙⠿⣦⣣⠀⠀⢤⠀⣠⣿⠋⠁⠀⠀⠀⠀⠀⠠⡈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣤⣀⣤⠞⠓⠿⠟⠛⠛⠳⣧⡀⠀⢀⣴⠛⠃⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⣤⣿⠛⠀⠐⠀⠀⠀⠀⠀⠀⣠⣾⢿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣈⣁⣀⠀⠠⠤⣤⠀⠰⡉⠙⢿⣿⠋⠁⠀⠀⠀⠀⢀⠀⠀⠀⠈⣪⣿⠿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣿⡋⣮⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠛⠋⠉⠉⢛⢶⣠⠴⠦⢀⡽⠀⠘⣿⡿⢾⣆⠀⠀⠀⠀⠀⠰⣤⣾⡿⠀⠀⠈⠙⢷⣱⡄⠀⢀⣴⣿⡱⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣰⠀⠀⠀⠀⢀⠀⢻⢳⣀⣀⣴⠾⢿⢧⣸⡀⠀⠹⣷⣶⣦⣀⣀⣶⡻⠆⠀⠀⢆⠀⠀⠀⠹⠿⣦⡾⣿⣥⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡟⠀⠀⠀⠀⠀⠀⢻⠁⠙⠉⠀⠢⡀⢳⡿⣿⣗⠀⠀⠐⢦⡀⠉⡉⠻⣤⣤⣬⣄⠀⠀⠀⢀⣾⡻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⠀⠙⠿⠷⣶⡎⠠⠶⠀⣠⡀⠀⠙⢿⣄⢆⣰⣿⣯⡷⠏
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣼⠏⢀⣀⠘⠁⠉⠀⠀⠈⢻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢸⠇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢼⣏⣰⠏⢹⣆⠀⡞⠻⠄⠀⠀⠉⠳⠛⠹⣧⡀
⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠁⠀⣀⣉⡗⣀⠀⣠⢤⡀⠠⣤⡄⠀⠈⣷
⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢛⣿⠿⣿⣿⠀⢃⢁⣀⡀⢸⣿⣶⡾⠃
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⡁⢀⣿⡏⢀⠀⢻⣏⠁⣽⡿⠟
⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡡⠀⣡⡾⠋⣿⣄⣴⠿⠿⡷⠿
⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⣴⢋⠀⠀⣀⣾⠋⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠉⢷⣄⡀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠸⡆⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⣠⠟⠡⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡹⠄⠀⠀⠀⠀⠀⠀⠈⢂⠀⢠⡾⡃⠂⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⡶⠋⠁⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣴⠟⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠴⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢰⠃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠇⢀⡟⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⡟⠀⡼⠀⢠⠏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢴⠃⣸⠁⢸⠃⢠⠏⠀⡼⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣫⠭⠂⠀⢸⠀⣿⠀⣿⠀⣼⠀⢸⠃⠀⠨⣙⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣷⣿⣥⣶⣾⣴⣿⣶⣿⣶⣿⣴⣾⣤⣄⣑⠲⡽⢊⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠦⣈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢋⣡⡴⣞⠁⣿⣟⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⣚⣋⣭⡷⢞⣫⠘⠛⠲⣷⠈⣷
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⢸⣟⡲⠄⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣉⣭⣭⡶⢞⣻⡅⢀⣢⣴⣿⢄⡟
⠀⠀⠀⠀⣤⢞⡭⢒⠭⢂⠀⠀⢻⣍⡀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠭⠤⠶⠶⣛⣫⡅⣼⣯⡷⢟⡵⠏⠘⢦⡀
⠀⠀⠀⣾⢡⡟⢰⠃⡞⢡⡆⠀⠀⠻⣦⠄⠀⠀⠀⠀⠀⠐⠒⠒⠒⣚⣛⣫⣭⢟⣼⣯⣷⣾⣯⠠⡐⣌⢆⢻⡀
⠀⠀⠀⠻⣌⢷⣜⠳⣝⡦⣝⡂⠀⠀⠙⢷⣄⣀⠀⠀⠀⠀⠬⠭⠭⣭⣷⣶⣿⣿⣿⢻⡏⣿⣾⡇⡷⠸⢀⡾
⠀⠀⠀⠀⢉⣷⣭⣳⠦⣭⣓⠒⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⡿⠿⠿⣟⣻⣽⠿⣫⡾⣣⠟⠘⢐⣡⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠹⠷⠾⣭⣥⣀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠿⣭⡭⢭⣷⣚⣛⣩⣤⣴⣶⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢛⣋⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⣚⣛⣻⣿⣿⣿⣿⣿⣿⣶⣶⡶⠶⠖⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠩⠭⠭⠭⠿⠯⠭⠭⠍⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡟⢡⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣌⡙⢿⣿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣦⣌⠛⢡⣾⣿⣿⣿⣿⣿⣿⠟⣡⣾⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⢋⣴⣤⡙⢿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠻⢿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣦⣌⠻⠟⣡⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣦⡙⠿⡿⢋⣼⣿⣿⣿⣿⣿⣿⡿⢃⣤⡙⠿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠗⣀⠛⢿⣿⣿⣿⣿⣿⠟⣠⣿⣿⣿⣶⣌⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣦⡙⠻⣿⡿⢃⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠉⠛⠻⠿⣤⣿⣿⣿⣿⣿⣿⣷⠌⠀⢿⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⢡⠞⠁
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⣰⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠁⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠁⠀⠀⠀⠀⠉⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⣾⣧⣄⠀⠀⠀⠀⢀⡴⡫⣪⠾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠉⠙⢿⣿⣶⡄⢀⡴⡫⣪⠞⠁⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠁⠀⠀⠀⠀⠙⢿⣿⡫⣪⠞⠁⠀⠀⠀⠀⠀⢉⡿⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⣾⣧⡀⠀⠀⠀⠀⢀⡴⡫⣪⢿⣿⣶⣀⠀⠀⠀⢀⡴⡫⣪⠞⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠉⠻⣿⣿⣦⡀⢀⡴⡫⣪⠞⠁⠀⠙⢿⣿⣦⣀⡴⡫⣪⠞⠁⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠞⠁⠀⠀⠀⠈⠻⣿⣿⡫⣪⠞⠁⠀⠀⠀⠀⠀⢙⡿⣯⣪⡞⠁⠀⠀⠀⠀⠀⢀⡼⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠳⢨⡀⠀⠀⠀⠀⢀⡴⡫⣪⣿⣿⣦⡀⠀⠀⠀⢀⡴⡫⣪⠾⢿⣿⣦⡀⠀⠀⢀⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⠀⠑⡙⣦⠀⢀⡴⡫⣪⠞⠁⠈⠻⣿⣿⣦⣀⡴⡫⣪⠞⠁⠀⠀⠙⠿⢿⢦⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⠀⠀⠀⠀⠐⣸⣿⡫⣪⠞⠁⠀⠀⠀⠀⠈⢻⡿⣯⢪⡞⠁⠀⠀⠀⠀⠀⢀⡴⡃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⢿⣿⣄⠀⠀⠀⠀⢀⡴⡫⣪⠾⣿⣿⣦⡀⠀⠀⢀⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢻⠛⠳⣄⣀⡴⡫⣪⠞⠁⠀⠈⠻⣿⣿⣦⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠪⣣⡿⣯⣪⡞⠁⠀⠀⠀⠀⠀⢈⡿⡻⠊
⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣪⡾⣿⣷⣄⠀⠀⠀⢀⡴⡫⠊
⠀⠀⠀⠀⠀⡠⡪⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡙⢄⠈⠻⣻⣷⣤⡴⡫⠊
⠀⠀⠀⡠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢈⡿⡫⠊
⠀⡠⡪⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣴⡫⠊
⠲⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡸⠊
⠀⠈⠮⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡨⠊
⠀⠀⠀⠉⢪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠊
⠀⠀⠀⠀⠀⠈⠪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⡢⡀⠀⠀⠀⠀⠠⡨⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⡢⡀⡠⡢⠌
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣪⣚⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⠶⠶⢶⠶⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣶⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠯⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⢻⠋⠙⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣰⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣤⣄⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠰⠃⠞⡽⣹⢻⡾⣧⣤⣤⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠁⠛⠀⠏⡼⢉⡿⣷⢦⣼⣏⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢶⠛⣦⠀⠀⠀⠀⠀⠀⠀⠘⠁⠋⠾⢸⢏⡟⢲⢦⣄⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣆⠀⠀⣠⠖⣶⠀⢀⣀⠀⠀⠀⠈⠁⠋⠸⠃⠿⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠶⠞⠁⠀⠈⠋⠉⠉⠙⠓⠲⢦⣄⣀⠀⡴⠶⢬⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡞⣴⢠⡆⡤⣰⢠⢆⡤⢠⡄⣤⢀⡄⣠⠀⣤⠀⡄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⠿⠾⠷⠿⠿⠾⠷⠿⠶⠷⠾⠶⠿⠞⠳⠚⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣶⣶⣶⡾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢸⣿⣿⣯⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣿⣿⣿⣿⣿⣿⣿⣿⢽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣽⣿⣿⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡘⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⡿⠁⠀⢹⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⢿⡆⢿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⠃⣀⠀⠀⣀⣬⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡄⠀⣿⢾⣧⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡏⠀⢈⣴⣾⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠘⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣴⠋⠝⣿⣿⠦⡜
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢹⣿⠉⢛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⡋⠀⠈⢹⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⣸⠟⠒⠀⠙⠀⢀⣈⣉⡻⣿⣿⣿⣿⣏⡴⠋⠐⠒⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠌⢹⠀⠀⣠⣾⣿⠛⠋⠛⠿⠹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⣼⠀⣴⣿⣿⡉⠀⠀⠀⠀⢧⢹⣿⣯⠹⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡀⡏⣸⠏⠙⣿⡁⠀⠘⠀⠀⠈⠂⢻⡇⠀⠀⠀⠀⢀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⠀⠀⠀⢸⡄⠀⠀⠀⠀⢀⡀⠈⢻⣤⠀⠂⠀⠈⢦⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠈⢁⡀⠀⠻⣧⠀⠀⠀⠈⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⡀⠀⠀⠀⠗⢀⠀⠈⢣⠀⠀⠐⡀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠁⡀⠀⠀⠄⠀⠁⠀⠰⡇⠀⠀⡐⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡄⠀⠀⠀⠀⠀⠀⡀⢷⡠⠦⠀⠒⠶⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠉⣀⣀⣠⣄⣀⣀⣈⣠⣷⣶⡶⠶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣿⣭⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣶⣶⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠋⣉⣩⣭⣉⣤⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣉⣉⣭⣍⣥⣴
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣈⣩⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣈⣉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⢶⣦⣤⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡿⣿⢣⣶⣾⣾⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⣵⣾⡇⣿⡸⣿⣦⠳⣴⡷⢶⣻⣮⡝⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⢿⣏⢻⣿⣿⣜⢿⣷⡙⣷⣶⡹⣮⢻⣖⣿⣄⢰⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢨⣎⢿⣷⣍⠻⣿⣦⡻⣿⣜⢿⣛⢹⠾⣿⠸⣿⣇⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡸⣿⣿⣿⣿⣿⣮⣝⠿⢿⢿⣷⣝⢿⣷⡿⢃⣼⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡻⣦⣍⡛⠿⣿⣿⣿⣝⡲⣬⣿⣿⣷⣝⢷⣿⣿⡇⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⢸⣿⣧⠹⣿⣿⣷⣦⣍⡛⠿⣿⣶⣬⣝⣛⠿⢧⠿⢟⣼⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣷⣦⣍⡻⢿⣿⣿⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗⣈⣡⣤⣴⣶⣶⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠈⢿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠟⣩⣴⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣾⣿⣶⠝⢰⣿⣿⣿⣿⣿⡿⣡⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣭⡛⠿⢿⠟⢰⣿⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡀⠀⠀⠀⠀⠀⠀⣼⡿⠟⣋⣥⣶⣾⡷⠀⣿⡆⠐⢶⣾⣦⣀⢀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣄⡀⠀⠀⠀⡏⠀⠀⠘⣿⣿⠋⠀⢀⡿⠁⠀⠀⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠁⠀⠀⠀⣿⠃⠀⠠⠚⠀⠀⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡧⠀⢰⣤⣄⡀⠀⠀⠙⢿⡟⠀⠀⠀⠀⠸⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣧⠀⠈⢿⣿⣿⣿⣦⠀⠈⣷⡤⠀⠀⢀⡴⠋⢀⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡟⠁⠀⠀⠈⣿⡄⠀⠘⣿⣿⣿⣿⡇⢠⡟⣀⣤⣾⣯⠤⠤⣾⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡉⠉⠉⠙⣧⡀⠀⠀⠀⣿⡇⠀⠀⠘⢿⣿⢿⣧⡾⠟⠉⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢻⣄⠀⠘⣿⡇⠀⠀⠀⣾⢀⡾⠋
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠛⢧⡀⢹⣇⠀⠀⢠⣿⣿⠞⠁
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢻⣼⣷⠀⢠⡿⠁
⠚⠛⠻⠿⠿⠿⠿⢿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢻⡇⣠⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠒⠀⠉⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠔⠒⠉⠁⠀⠹⣧⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⠉⠁⠹⣧⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⣠⣤⣶⠾⢿⣄
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣀⣤⣴⡶⠿⠳⠘⠉⠅⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡄⢀⣀⣤⣽⠷⠔⠛⠩⠁⠀⠀⠈⢻⡆⠈⠆⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⡾⠟⡛⠍⠐⠐⢶⡄⠀⠀⠀⠀⠀⠀⢻⣆⠈⢆⣀⣤⣴⠾⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠁⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⣀⣠⣴⡿⠧⢘⣫⣥⣖⣺⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢀⣀⣽⣆⡸⢟⣫⣭⠲⢶⢒⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⣶⡿⢟⣻⣽⠺⢿⣛⣿⠽⠗⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⡿⣿⣯⠿⠗⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠒⢲⣞⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢦⣤⠞⠉⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⢀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠁⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⢠⣶⡾⠟⢋⠩⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⣤⣴⣶⠿⠟⡋⣃⡈⠐⢂⠁⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠄⢀⣀⣤⣶⡿⠷⠜⠋⠡⠀⠈⠀⠀⠘⢷⡄⠈⢂⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣾⠿⢛⠉⠁⠒⠙⣶⡄⠀⠀⠀⠀⠀⠀⠈⢻⣄⠈⢆⣀⣠⣤⡶⠟⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠁⠈⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣀⣠⣼⣿⣦⠸⢛⣩⢥⡲⣯⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣻⣆⢀⡶⠿⣛⠫⢭⣤⠶⣾⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢀⣤⣶⡾⠿⣛⣭⣿⡶⠿⣞⣗⣥⠬⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡐⢨⣿⣷⣾⣿⣏⡦⠽⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⠾⠟⠛⠉⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢹⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⢸⡆⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢉⠙⠋⠉⠱⣦⣆⠀⠀⠀⣀⠀⠀⣰⣿⠃⣤⡀⠀⠈⢿⣶
⠀⠀⠀⢸⡷⠀⠆⠰⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡞⠀⠀⢧⠀⣿⢿⢃⠀⠀⠙⠛⠚⠋⠁⣀⣈⠙⠛⠛⢋⠁⠤⣄
⠀⠀⠀⠘⣿⣀⠐⠠⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⡀⣸⠃⠀⠀⢸⠀⣽⣿⠸⠀⠀⠀⣠⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠈⢿⣷⢀⢠⣠⠀⠁⡰⠀⠀⠀⠀⠀⠆⡏⠀⠀⠀⠘⠀⢹⣿⡇⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⢶⣤⣾⠿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠔⡀⠐⢃⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠐⢃⠈⠞⡄⠀⠀⢂⢸⡄⠀⠀⢀⡇⣼⣿⡇⠀⠀⠀⠀⠀⠀⣤⣄⣠⣽⡟⠂⢑⣦⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣑⠂⢀⠀⠀⢸⡞⣿⣆⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⢤⣤⡾⠋⣹⣿⡛⡛⢿⣧⡐⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠒⠂⠀⠈⢿⣿⣿⣇⣼⣿⡟⠁⣀⠀⠚⣻⣦⡴⢻⣁⣩⣿⣿⢻⣿⣿⣿⣵⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣀⠤⠤⠤⠤⡠⠤⠤⠤⣍⠛⠻⠟⠋⠀⠀⠈⠙⠲⢿⣿⡽⣯⣭⣻⣿⣿⣾⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⡷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⢀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠄⣿⠠⠤⠶⠤⣧⠠⠶⠄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡄⠀⠀⠀⠁⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿⠀⠀⠀⢸⡿⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠈⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⣷⠶⠦⠒⢲⡗⠶⠦⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡀⠀⠀⢻⣇⠀⣀⡘⢟⣆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠖⠋⠁⠈⠉⠁⠈⠉⠉⠉⢉⢷⡅⠘⠶⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⢀⡘⠁⠀⠀⠀⠀⠀⣀⡀⠀⣒⣒⣒⣂⣰⡇
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣄⠀⡰⠃⠠⣤⠋⠀⠀⣤⣄⡀⠀⣉⣙⡛⠛⠂⠉⠛⠃⠠⠄⣴⣾⡆
⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⣦⣿⣬⣿⣿⣿⣿⣾⣿⣿⣿⠀⠢⣄⣠⣀⣙⠓⠈⠻⢭⠉⢛⠛⣲⣶⠶⠶⠿⠻⠿⣥⣠⣶⣾⡇
⠀⠀⠼⢻⣦⣲⣤⣤⣿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠪⡉⠻⣬⡛⣶⣤⣤⣄⣈⠁⠈⠀⠀⠀⢀⣴⣾⣿⣿⣿⣯⡀
⠀⢀⡤⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠂⢀⠙⢧⣄⠀⣀⣩⣽⢷⣶⣤⡀⣸⣿⣿⣿⣿⣿⣿⡿
⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠢⢨⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⠀⡄⣠⣄⢀⣤⡀⣤⣄⢠⠀⠀⣤⠠⣤⡄⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⢸⣤⡇⡇⢸⢸⠀⠁⡇⣿⢸⠀⢰⢿⡆⢸⠀⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠟⠸⠐⠇⠷⠾⠸⠶⠇⠧⠟⠸⠦⠼⠛⠇⠸⠀⠿⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠒⠒⠂⠀⠒⢒⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠳⠒⠒⠒⠦⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠉⠀⠀⠀⠀⢂⠀⢀⠎⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠤⣀⣀⣠⢀⠎⠀⣼⠇⠀⠀⠀⠀⠀⠀⡇⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡃⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⠊⠀⡜⠓⠀⠄⡈⠁⠝⠃⠒⠷⢛⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⢀⠀⠀⢤⣀⢀⣠⠇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⢐⠇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⢤⣀⣀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⢤⡼⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠤⠤⠤⢄⣀⣀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⢀⡆⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⣾⣇⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣧⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢻⣿⣿⣶⣾⣿⡇⠈⠏⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢈⣿⣿⣿⣿⡿⠃⠀⢀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⢸⣿⣦⡀⢀⣼⣿⣿⠿⠋⠀⠀⠀⣿⡄⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⣼⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⠀⠘⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣾⠀⠀⠀⣧⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡀⢠⣿⡀⠀⣠⣿⡇⠀⢹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⣿⣿⣿⣿⠀⢿⣿⣿⡟⢿⣿⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠻⠟⠃⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣆⠀⠀⠀⠀⣼⣿⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⣰⣿⢻⣿⣧⠀⠀⢸⣿⣿⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⢿⣿⠀⠿⠟⠀⠀⠀⠙⠋⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣤⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣿⣿⣶⠿⠿⢿⣷⣶⠶⢶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⣛⡛⢿⣏⣠⣶⡄⣰⣿⣿⣆⢹⣷⣶⣶⣈⣭⣭⣙⠛⢷⣦⣀⡀
⠀⠀⠀⠀⢀⣴⡾⢛⡛⠃⣾⣿⣷⠸⣿⣿⣿⠀⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⠛⠿⣦⡀
⠀⠀⠀⢠⣿⣯⣴⣿⡿⣿⣿⣿⣿⣄⣉⠙⠻⠀⣿⣿⣿⣿⣄⠘⢿⣿⣯⣤⣶⡆⢹⣿⣇⠹⣷⡌⢻⣆
⠀⠀⣴⡟⣩⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿⣦⡀⠸⣿⣿⣿⣿⣇⠘⣿⣿⣿⡿⠀⠿⠿⢿⣿⣿⣿⢸⣿⡇
⠀⢸⣿⠇⠹⣿⣿⡟⢃⣀⣉⣉⡉⠙⢻⣿⣿⣷⠀⣿⣿⣧⠙⢿⣴⣿⣿⣿⠃⣴⣶⣶⣦⠈⠛⢁⣼⡝⣿
⣰⡿⢱⣶⣤⡈⠛⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣷⣤⣨⣿⣿⣿⠀⣿⣿⣥⣤⣴⣆⢸⣿⣇⣼
⣿⣇⢸⣿⣿⣿⡄⠸⣿⡿⠛⢉⣉⣍⠉⣭⣉⣉⣀⣈⠙⠛⠻⢿⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⠇⢸⠿⠿⢿
⣾⡿⢠⣿⣿⠏⣠⣦⣤⡄⢰⣿⣿⣿⣄⠙⣿⣿⣿⣿⣿⡿⠒⢀⣀⡉⠻⢿⣿⣷⣄⠙⣿⣿⣦⣤⣴⡶⢸
⣿⣧⢸⣿⡏⣸⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⠷⠄⠙⣿⣿⡆⢸⣿⣿⣿⣋⣴⡿
⢸⣿⣄⠙⠀⣿⡟⠛⠛⠃⠀⢿⣿⣿⣿⣿⠋⣉⣉⣉⣉⠙⠛⠿⣿⣿⣦⣶⣦⠘⣿⡷⠀⠻⠿⠿⠟⠋
⠘⣿⣿⣷⠀⣿⣿⣿⣿⣿⣧⡈⠻⣿⣿⣿⡀⢻⣿⣿⣿⣿⣷⣦⣄⡙⢿⣿⣿⡆⠙⢁⣤
⠀⠈⠻⢿⣇⠘⢿⣿⣿⣿⣿⣿⣶⣤⣈⣉⣁⣀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣧⣤⣥⣼⣿⡿
⠀⠀⠐⣶⣶⣶⣦⣬⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⠀⠿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⣌⣙⣛⡛⠻⠿⢿⣿⠿⠟⠁
⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠿⠿⢿⣟⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠙⠲⠶⠖⠛⠻⢦⡼⠛⠚⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠳⢤⡀⠀⠀⣠⡜⠋⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡙⣦⣾⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡏⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠳⣦⣀⠀⠀⡀⢀⣶⠟⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢤⣠⠴⠓⠾⠛⠛⢶⣄⡀⣠⡾⠋⠀⠀⠀⠀⠈⠻⢷⣤⣵⠟⠉⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣠⣤⡤⣤⣄⡒⠒⠀⠰⡌⢻⣿⣀⡀⠀⠀⠀⠀⠀⢀⣴⠟⠻⢷⡀⡀⠀⠈⢠⣾⣏⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢟⠁⠀⠀⠀⠀⣿⡉⠉⣸⣥⣄⣿⠉⠻⣤⣀⡀⠀⣸⡟⠃⠀⠀⠀⠙⢿⣄⣴⣿⣳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢸⠀⠀⠀⠀⠀⡟⠙⠛⠁⠀⢩⠿⣶⠂⠉⠩⡍⠙⠙⢧⣤⣄⡀⠀⠀⣠⡞⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣠⠞⢡⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡸⠀⠈⠛⠛⣿⠁⠒⣠⢤⠀⠙⢧⣀⣼⣿⡽⠋
⠀⠀⠀⠀⣠⡞⠁⠀⡼⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⣰⠏⣴⢶⡀⢠⢤⠀⠈⠻⣿⡿⣧
⠀⠀⣠⡾⠋⠀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣽⠟⠁⣈⣳⢄⠀⣠⣄⠀⣤⠀⠙⣧
⢠⡾⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⢳⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⡿⢻⣿⠃⢀⣠⡄⣸⣷⡾⠃
⢿⣇⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⣸⣿⠀⣀⣿⣄⣾⠋
⠀⠙⢦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠘⠳⠏⠀⠉⠁
⠀⠀⠀⠙⢦⣄⠀⠀⠈⡇⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠙⢷⣄⠀⢹⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⡴⠋⠀⢠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⢧⠀⠀⠀⠀⠀⠱⡀⠀⣰⠏⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣠⠞⠁⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠞⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣀⣼⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠩⠉⠉⠉⢸⣿⠉⠟⠛⠉⢸⣧⢄⠀⠙⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⢸⣿⠀⠀⢀⠀⢸⣿⣴⠯⠄⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣬⣼⣿⣤⣤⣤⣤⣼⣯⣬⣤⣬⣥⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠈⠉⠋⠉⢹⣿⠉⣉⠉⠉⢸⣧⢩⡍⠉⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⢸⣿⠔⣿⣃⠀⢹⡟⠉⠀⠊⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣼⣩⣴⣶⣦⣬⣼⣧⣤⣤⣤⠀⠙⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠩⠉⠉⠘⠛⠀⠉⠉⠉⠙⠋⠉⢁⠅⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢩⠄⠀⠀⠀⠀⠀⠑⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡋⢈⣴⠋⠀⠶⡀⢠⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡸⠁⠀⠀⠀⠀⠐⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡬⠀⠀⠀⠁⠀⠀⠀⠀⣠⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⢶⢇⣠⣴⣿⣿⣶⣶⢞⣁⣤⣴⡶⠒⢻⠁
⠀⠀⠀⠀⢑⠂⢤⣀⡀⢀⡠⠐⢂⠉⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⡸
⠀⠀⠀⠀⢸⣧⠀⠀⠈⠉⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣩⠽⠿⠶⠾⠛⠉⠁⢸⣿⣿⣿⣿⣿⣿⡿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣝⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣻⠢⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠚⠒⠲⠶⠶⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣅⡀⠤⡉⠀⠀⠀⠀⠀⠀⡀⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡝⠓⠒⠀⠈⠁⢉⣀⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣟⣙⠿⢳⣶⣽⣻⣷⣆⡀⠈⠓⠂⢀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⠀⣀⠠⠛⠦⣄⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⣦⢹⣯⠿⢥⡄⠉⢛⣿⣮⣟⣶⣤⣀⡀⠈⠑⠋⠄
⠀⠀⠀⠀⣐⡉⠀⠀⠀⠈⠙⢷⣄⠊⠁⠀⠀⠀⠈⠙⢶⣤⣶⠾⠋⠀⠀⠀⠀⠸⣿⣏⢯⣷⡋⣉⠿⢯⣀⡄⠛⢳⣽⣿⣟⠛⠆⠥⠀⠀⠐⢤
⠀⠀⠀⢾⡝⢿⣦⡀⢀⠄⠊⠀⠉⠛⢶⣄⠀⢀⣀⠐⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡻⣽⣧⣀⡞⠉⠹⢾⣁⠼⠿⣿⣷⣦⣀⣤⡤⢀⡀⠔
⠀⠀⠀⠀⠙⠳⣍⡻⣧⣄⠀⠀⠀⠀⠀⣙⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⡛⠿⣷⣾⣄⣰⠛⢲⣶⠙⣻⡿⠷⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣷⣦⣠⢶⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⣍⣙⠛⠿⣧⣬⣟⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠻⠷⠶⠶⠿⠿⠿⠇




⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢠⣀⢦⠼⠽⣒⣦⢄⣀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠫⠼⢣⣽⣾⣻⣟⣯⣿⢷⣮⣽⢇⡾⠵⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣉⠀⠄⠂⢀⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠀⣀⣠⠔⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠃⠚⠓⠛⠻⠶⠦⣤⡀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣁⣠⣶⡶⣿⢡⡀⠀⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣬⣿⣿⣿⣿⣽⣿⣿⣦⡀⠀⠀⠄⣀⣄⡣⠆
⠀⠀⠀⠀⠀⠀⢘⠀⠆⢀⣘⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠈⢯⣼⣯⣿⣿⣿⣿⣿⣿⣿⣇⣴⣯⣷⣿⠄
⠀⠀⠀⠀⠀⠀⠯⠦⠦⠶⠤⢭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠳⣿⠻⡿⠋⣯⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⠗⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡄⠀⠀⣰⣿⣿⣶⡄⠀⠀⠰⣿⣿⣿⠏⠀⣀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡀⠀⠙⠿⣿⠟⠀⠀⠀⠀⠀⠉⠋⠀⠘⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⢸⠄⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡿⢛⣋⣵⣾⣿⡇⠀⢸⣿⣿⣿⣿⡿⠛⠋⠁⠀⠁
⠀⠀⠀⠀⢀⣠⣤⣤⣴⣿⣿⣿⣿⣿⡇⠀⠀⠸⠿⣉⣩⣥⣴⣿⣿⣿⣿⣿⡇⠀⠸⠿⢋⠉⠉
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⢒⢔⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠦⠉⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠤⠒⠀⠁⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠈⠋⠐⠰⢠⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠂⠀⢸⠀⠨⠘⠁⢀⡀⢀⣀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠘
⠀⠀⠀⠀⠀⠲⠒⠩⣩⢉⠆⠀⠀⠀⠀⠘⠀⠠⠀⠙⠉⡐⢕⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠂⢀⡀⢠⠀⢠⡄⠐⠀⡇
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠡⠀⠘⠋⡐⠴⢠⠘⠘⠀⠀⠀⠸⠀⠀⠀⠃⠀⠀⠄⠋
⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⢠⢰⡆⠀⠀⠀⠀⠂⢨⠀⡄⠀⢠⠀⠐⢸⠃⠀⠀⠀⠀⠉⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠈
⠀⠀⠀⠀⠀⠀⠉⠀⢸⠀⠀⠈⠀⠀⢤⠆⢂⡖⢚⡄⡀⠀⢸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠁⡟⠀⠀⠀⠀⠀⠀⠀⠁⠂⢐⡀⠤⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠠⠀⠂⠉⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠠⠄⠊⠉⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠐⠥⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⡀⠀⡄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠹⠿⠿⣿⣿⡿⠿⠿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⠀⠨⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠄⠒⠀⠀⠀⠈⠁⠈⠉⡁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠠⢀⠀⠀⠀⠂⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠞⠀⠀⣀⠀⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠛⠙⠉⡀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⡄⢀⡀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠈⠁⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⡀⠀⠀⠀⠂⠄⠈⠉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⢀⡤⠆⠀⠒⠊⠀⠈⠀⠠⠄⠐⠂⠛⠋⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⢀
⡀⠀⠀⠈⠀⠤⠤⠀⠲⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⡀⠁⠐⠀⠀⠀⠀⠠⡄⠀⠀⠐⠀⠀⠀⠉⠀⠐⠀⡀
⠈⠀⠀⠂⠀⠀⠀⠀⠀⣀⢀⠠⡄⠐⠂⠙⠀⠀⠀⡸⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡄⠊⠀⡂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⡄⢀⠀⡄⠈⠁
⠀⢃⠀⠑⠀⠀⠒⢸⠀⠀⠀⠀⠥⠤⠄⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠂⠰⠊⡁⠀⠀⠃⠀⠀⠀⠀⠀⠈⢑⠀⠉⠀⠈⠀⡀
⠀⠀⠀⠈⠀⠤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠈⠐⠋⢀⢠⠀⠀⠀⠀⠀⠀⢠⣴⠟⠀⠄⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠠⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠆⠡⠀⠤⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠤⠄⠀⠄⠠⠴⠆⠀⠀⠀⠘⠚⠈⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠴⠐⠂⠓⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⡰⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠀⠐⠂⠐⠀⠓⠀⠛⠂
⠀⠀⡀⠀⠀⠀⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠚⠀⠠⠤⠀⠀⢀⣀⣀⡀⠀⠀⠠
⠀⢀⡆⢀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇
⢀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠰⡤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆
⢠⠀⡀⠀⠀⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈
⢨⠀⠀⣠⢀⡀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠙⣦⠄⠀⠀⠚⠀⠀⠀⠀⠀⠀⡠⠤⡀⠠⠦⠀⠀⠁⠀⠀⠀⠀⠀⡆
⢨⠐⢀⠈⠈⠈⡈⠀⠁⠀⠀⠀⠀⠀⠈⠐⠂⠤⠬⠤⠬⠤⠔⠒⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠇⠀⡠⠒⠂⠈⠉⡐⠒⠤⢀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⡀⣀⡄⢲⠂⠂⠀⠀⡇⠀⠀⠀⠀⠀⠰⢂⡆⠀⢀⠻⠁⠀⠀⠈⠀⠀⠀⠀⠠⠂⠀⣀⢈⠉⢀⠀⠁⢀⡀⣆
⢠⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⠀⠀⠀⠈⠀⠀⠠⠙⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⢈⣀⡈⠁⠉⠈⢀⠏
⠘⠀⠀⠒⠀⠐⠀⠄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠓⠤⣀⣀⠤⣀⡠⠔⠋
⠀⠠⠄⠁⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠈⠶⡄⠀⠀⠀⠀⡀⠀⠀⠔⠊

⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⡧⣻⡝⣻⣿⣿⠀⠀⠀⠙⢷⣿⣿⣿⣧⠀⠀⠀⠀⠘⡿⣿⣽
⠁⠀⠀⠀⠀⠀⠙⠿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠈⣿⣿⢧⣿⣯⢚⢿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢙⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣿⣸⡏⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣻
⡟⠃⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣿⣶⣷⣶⣾⡇⠀⠀⠀⣀⣹⣿⣿⣃⠀⠀⠀⠐⢛⣿⣿⣿
⡇⠀⠀⣨⣭⡄⠀⠀⠀⠀⢠⣿⣳⣿⣾⡿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⢻⣿⣿⠟⠁⠀⠀⠀⠸⣿⣿⣿
⠁⠀⢀⣿⣿⣿⠀⠀⠀⠀⣻⡇⣿⣿⣿⣿⣻⡾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠸⣯⢿⣿⣻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠂⢀⡄
⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣾⣷⣾⣶⡶⠆⠀⠀⢀⣴⣶⣶⡄⠀⠀⠀⢀⣴⣶⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠰⠄⣽⣷⡄
⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⢈⢐⣿⣿⡂⠒
⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⢀⣾⣿⡆⠀⢸⣿⣿⡟⠀⠀⠀⠠⠠⣿⣿⠅⡍
⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⢻⣿⣿⡇⠀⣸⣿⡿⠃⠀⠀⠀⠠⠴⣿⣿⡃⠂
⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣴⠿⠋⠁⠀⠀⠀⠀⣤⣰⣿⣿⣇⣅
⠀⠀⠀⠿⠿⠇⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣀⣻⣿⣿⣁⣀⣀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠘⠛⣿⣿⣿⣿⣿⠛⠋⠀⠀⢠⣦⣤⣬⣭⣭⣤⣤⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⣀⠀⠀⠀⠀⠀⠀⠸⣿⣇⢠⣾⣿⡆⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⡏⠁
⣿⣧⠀⠀⠀⠀⠀⠀⠘⠿⣾⣿⣿⠇⠀⣼⣿⡿⠁⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⣿⣰⣶⡀⠀⠀⠀⠀⠀⣿⣿⣿⣠⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠇⠀⠀⠀⠀⣸⣿⣮
⣶⣱⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⡷⣿⡿⣞⣷⡀⠀⠀⠀⠀⣠⣾⣿⣿⣦⠀⠀⠀⠀⣽⣿⣿
⣿⢪⡝⣿⠇⠀⠀⠀⢠⣤⣤⣭⣭⣭⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣟⣿⣯⣿⢻⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣿⣷
⠿⠿⠇⣟⣀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⣹⣟⣿⣏⣿⠗⡿⣿⣃⠀⠀⣤⣤⣿⣿⣿⣿⣯⣤⣤⠀⠀⣘⣿⠿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠁⠀⢀⡐⢠⣥⠐⣃⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣟⠛⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿
⣿⣿⣿⡿⠃⠀⠀⠀⣠⣠⣀⠀⠀⠀⠀⠀⠀⠠⠅⣿⣿⡇⣀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⠹⣿⣿
⠛⠛⠉⠀⠀⠀⠀⠈⢛⣛⡛⠀⠀⠀⠀⠀⠀⠐⠂⣿⣿⡇⠤⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠉⣿⣿⡇⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣴⣶
⣆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠁⣿⣿⡗⠀⠀⠀⠀⠀⠀⣴⣾⣷⡀⠀⠀⠀⠀⢀⡀⣾⠇⠀⣾⣿⣿⠀⠀⠀⠀⣠⣶⣾
⣿⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠿⣿⣿⣿⣿⡿⠇⠀⠀⣴⣦⣻⣿⣿⡇⣤⡄⠀⠀⢸⣿⣧⣠⣴⣽⡿⠏⠀⠀⠀⣸⣟⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⢒⡲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠁⠀⠀⠀⠀⠌⡂⠂⣉⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⢱⡄⢀⠀⠀⢀⣴⢟⣩⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠠⣀⢀⣬⣷⣀⣸⡄⠈⣀⣠⣴⣿⣿⠿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣄⠀⠈⠀⠀⠀⠮⣙⠷⣮⡙⢿⡇⠀⢻⣿⡿⠯⠔⠒⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣝⠷⢄⠀⠀⣀⠠⢄⠤⣜⣿⣷⣌⡑⠛⡁⠀⡄⠀⢉⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡂⠐⠪⠿⣿⢳⣦⠄⡥⠐⠊⠉⠉⠀⠀⠈⠛⣿⡈⡏⠀⠜⢒⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⡹⠀⠁⢀⣌⠻⣿⣷⣜⠷⣤⡀⠀⠀⠀⠀⠀⠯⠟⠧⠀⡒⠵⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⠃⠀⠀⠀⢹⠆⣈⠻⣿⣷⣜⠓⠄⠀⠠⠀⠒⠂⢈⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠰⠯⠆⠈⠻⣿⡷⣤⢠⡤⠀⢆⠀⢩⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠈⠧⠀⠘⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠃⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⠟⠀⠈⠁⢀⠀⠀⠀⠀⢀⢤⠀⡂⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣉⣉⣠⢈⣉⡉⠙⢉⣡⡈⡷⠁⠀⠀⠈⠒⠖⠦⠤⠤⠀⢴⡶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⠀⢀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠬⠙⡿⢁⢄⣀⢐⡄⢐⠘⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠉⠉⣭⢄⣀⡀⠀⠀⠀⠀⢠⡀⢢⣅⡨⠆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠓⠈⠁⠀⠀⠀⠀⢠⠈⢩⠁⠨⠀⠤⠈⠠⡀⠀⠀⠀⠻⡘⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠑⣬⡄⠀⠴⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⡰⠂⠉⠀⠀⠰⣄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉⣁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣤⡺⠟
⠀⠀⠀⠀⠀⠀⠀⡕⠒⢂⣀⠛⢃⠀⠀⠶⣶⣤⠘⠗⠁⠀⠀⠀⠤⠄⢂⠀⠠⢒⡩⠂⠀⠀⠀⠀⠀⠀⢠⡤⠀⢀⠀⠀⠤⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠧⠤⠄⠛⠒⠓⠡⠂⠀⠀⠀⠀⠀⠀⠰⠃⣤⡤⢒⣨⠄⠰⠦⢄⠀⠀⠀⠀⠀⡆⡀⠀⠈⠙⠖⢒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡀⠀⢠⣄⣀⡤⠴⠃⠀⠀⠀⠀⠈⠑⠒⠒⠲⠒⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠉⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠁⠀⡁⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⡃⠀⠀⠀⢠⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠀⠈⠁⠀⠀⠀⠀⠰⠌⡀⠀⣀⠄⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡿⠁⠀⠀⠀⢸⣀⠀⠀⠠⠤⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⠰⠄⠀⠀⠀⢀⣠⣀⠀⡀⠐⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠃⢀⣀⠤⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢉⢁⠠⠤⠬⠤⠚⠗⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⡈⠇⠀⠀⠀⠨⠀⣀⣀⣀⠠⠤⠦⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠰⡂⠉⠉⠀⠹⠆⢀⣀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢲⡀⠋⠁⠀⠱⡄⢀⣠⣤⣴⣤⠶⠾⠛⠛⢯⠈⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁⢠⠀⠀⠀⠀⢀⡶⠠⠄⣄⢀⢀⠀⠄⠀⠀⠀⠹⡆⢀⣀⣤⡡⡦⠶⠞⠛⠻⠋⠉⠄⠀⠹⡄⢀⣀⠤⠌⠳⠒⠒⠛⠁⡀
⠀⠀⠀⠀⠀⡌⠐⠒⢀⡀⠀⠡⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠠⡽⡍⠉⠀⠀⢀⡀⢀⣀⣄⣠⡀⠐⠒⠏⡉⠀⣠⠄⢀⡠⠖⠐⠒⠈
⠀⠀⠀⠀⠀⠁⠈⠈⠉⠈⠀⠀⠀⠀⠀⠀⠻⠛⠸⠴⣾⠆⠉⠀⠀⠀⠀⠀⠹⣀⡤⠖⠒⠋⢁⡈⢠⣠⡤⠠⠔⠂⠚⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⡀⠀⠰⣆⠃⠀⠀⠀⠀⠀⠀⠻⠄⠤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠙


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⡶⠶⢶⣶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠒⠈⠙⠉⠀⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⡀
⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢄
⠀⠀⠀⠀⢠⠏⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢆
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⢸⠧⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸
⠀⠀⠀⡘⢰⣾⡇⠠⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⣿⣴⢀
⠀⠀⠀⡗⢺⡾⠁⢀⣤⣶⣶⣶⣶⣶⡖⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⣶⣦⣄⡀⠀⠹⡇⠸
⠀⠀⠀⢳⣾⢱⣸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡆⣂⢼
⠀⠀⠀⢸⡏⡂⣿⣿⣷⡷⠂⠀⢩⣭⣝⡇⠀⠀⠀⠀⠀⠘⠢⣿⣿⣥⡌⠈⠙⢨⡿⣿⣯⠙⠀⣷
⠀⠀⠀⢨⠃⠀⣷⠟⠮⢧⠀⠠⡀⠘⢿⡇⠀⣠⣶⣴⣶⡄⠀⢻⣿⡿⠇⠀⠀⠀⢸⡟⣿⠀⠀⣿
⠀⠀⠀⡾⣴⠃⠙⠿⠦⠬⠷⡶⠴⠛⢉⠀⣸⣿⣿⣿⣿⣿⣆⠀⠉⠒⠶⠮⣽⢷⣾⡿⠃⠀⠀⠘⡆
⠀⠀⠀⡿⡃⠀⠀⡠⠀⠀⠀⢀⡀⠀⠉⠀⣿⣿⡿⣿⣿⣿⣇⠘⠀⠀⠠⡤⠄⠒⠦⣀⠸⡇⠀⢀⡇
⠀⠀⠀⠱⡝⢶⣄⠀⠀⠀⠀⠈⠉⠀⠀⢸⣏⣿⣶⣿⣷⣿⣿⡇⠀⠀⠀⠁⠀⠀⠀⠈⡂⢹⣤⡼⠃
⠀⠀⠀⠀⠈⠓⣷⣶⣦⣄⠛⢷⣤⣠⡶⠈⠙⠿⠿⣍⠻⠿⠛⠃⡄⢀⣴⡾⢿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⡿⢿⣿⣷⡀⡻⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣰⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢋⠈⣿⣿⣿⣽⣿⣇⡆⢸⡀⣄⡀⣆⠰⣄⣰⣀⡀⠁⣿⠿⣿⠻⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡄⢺⢿⣆⠹⢿⢍⠈⡏⢹⡏⢹⠏⢹⡏⢹⠁⡟⢿⠁⣰⡿⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠈⠻⠶⣼⣠⣄⣼⣠⣼⣄⣤⣤⣸⣄⣷⡿⠒⠿⠃⣇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠃⠀⠀⠀⣬⡸⡏⢿⢹⡏⡯⢹⢹⡏⣿⠣⠟⠀⠀⠔⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡀⠀⠀⠀⠀⠁⠙⠙⠉⠙⠉⠁⠉⠉⠋⠁⠀⠀⠀⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣎⠀⢠⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣠⣤⣤⣤⣀⠀⠀⡀⠀⣡⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣼⠣⡘⠿⠀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⡉⢔⡲⠈⢢⠘⠒⡩⠔⣉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢤⣀⠴⢚⢁⠈⢄⠑⣤⠴⢊⠡⡒⢌⠣⡘⢛⡠⢸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢣⡁⠈⠂⠉⢄⠈⠉⣀⠔⣣⡄⢊⠀⠻⠇⢈⠆⠉⣵⠊⣵⡆⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⢡⠀⠀⠄⠀⠀⠳⡭⠒⡭⠄⠈⢢⠱⢄⡪⢂⣉⠳⡈⢦⡨⣔⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⡁⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠙⢦⣈⡡⠔⠪⠦⡧⡀⠀⢊⣠⡷⠦⡘⣄⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠋⢀⠃⣀⠀⠀⠀⠈⠢⡀⠁⠢⡀⠀⠀⠱⣎⠐⠯⠂⣸⠎⠊⡟⠡⢦⢀⠽⡸⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠀⠀⠀⠀⢸⠀⣬⠾⢒⣆⠘⠄⠀⠀⢱⠀⠀⢀⠀⠀⠀⠈⠢⣔⡮⢚⡉⢦⠈⢦⡪⠔⠟⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣆⠀⠀⠀⠀⣘⢀⣠⠖⣏⡜⢠⣠⢤⠀⠀⡆⠀⠀⠉⠐⡀⠀⠀⠈⢦⡉⢊⣴⠗⢎⠣⡜⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⢧⡀⠀⠀⠈⠁⠈⢣⡄⣤⢡⠇⡸⠀⠀⠘⠦⢄⠀⠀⢁⠀⠀⠀⠀⠹⡇⠐⠯⢀⠷⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢳⡀⠀⠀⠀⠀⠀⡄⠘⠋⣠⠗⡄⠀⠀⠀⠀⠙⣆⠘⠒⠒⠲⡀⢠⠘⡦⣂⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡸⣄⠀⠀⠀⠀⠓⢤⠐⢡⣊⡡⣀⠀⠀⠀⠀⢘⠀⠀⠀⠾⠁⢀⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜⣦⠀⠀⠀⠀⠘⡄⠀⢡⢔⡡⡀⠀⠀⠀⠈⣉⣵⣀⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡳⡄⠀⠀⠀⠘⡀⠈⢉⠔⠃⢀⡠⢐⣫⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡘⢦⡀⠀⠀⠙⢊⣡⡴⣺⡥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⠙⣄⡤⢖⣫⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡾⠒⠉





⣿⠿⠿⡿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢶⣒⣻⣿⣿⣿⣖⣲⡶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⣫⡷⢟⣛⣭⣽⣿⣿⣿⣿⣿⣿⣿⣶⣬⣉⠛⠲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡷⢟⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⢷⣄⡀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡾⣫⣾⣿⣿⣿⣿⣿⣿⡽⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠳⣝⣷⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣱⢟⣵⣿⣿⣿⣿⡿⣫⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣻⣷⡀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣽⢫⣾⣿⣿⣿⣿⢏⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣻⣷⡀⠀⠀⡀⢀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣼⢣⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢻⣷⢡⣎⣴⣫⢆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⢧⣿⣿⣿⣿⣿⢱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣿⣇⢿⣿⣿⣿⣆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡟⣾⣿⣿⣿⣿⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⣿⣸⣿⣿⣿⣿⡞⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⢱⣿⣿⣿⣿⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⣿⣿⣿⣿⣧⢿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡟⣼⣿⣿⣿⣿⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣻⣿⣽⣟⣛⣛⣛⣸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣽⣇⣿⣿⣿⣿⡿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡜⡇
⠀⠀⠀⠀⠀⠀⠀⡿⣿⠾⣿⣿⣿⣿⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣿⣶⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⡿⢿⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⣸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀⠀⣿⣿⣯⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⣷⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢁⢀⣠⣤⠭⠿⠿⣿⣿⣿⣿⣿⣿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⣤⣠⣶⢞⣽⣿⡾⠟⠗⠾⠙⡦⣄⠀⠀⠀⣠⡞⠁⠠⣤⣰⡾⠋⠁⣀⣤⡞⠘⣾⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⢿⣿⢳⣿⡿⠁⢀⣤⣶⣤⣄⡀⠋⢳⡀⣴⠋⢠⣀⣲⡾⠁⢀⣴⣾⣿⠟⠀⣸⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⠏⣼⡇⣿⣿⡃⢀⣿⣿⣿⣿⣿⣿⣧⣼⣿⢃⣤⣤⣽⠏⠀⣠⣿⣿⡿⠃⢀⣾⣿⣿⢿⣿⣿⡿⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣿⠀⣿⡧⣿⣷⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⢇⣤⣿⣿⠋⠀⣴⣿⣿⠟⠀⣴⣿⡿⣫⣷⣿⣿⣿⣿⣿⣷⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⢀⣿⡇⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⣿⣿⢈⣻⣿⣿⡄⣼⣿⣿⠃⢀⣾⣿⡿⡝⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣼⣿⣸⣿⣿⡸⣿⣿⣦⠈⢿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣷⣿⣿⡇⠀⣼⣿⡟⣞⣠⣞⣿⣿⣿⣿⣿⣿⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⣯⢿⣿⣧⠙⣿⣿⣷⣤⠛⢿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣠⣿⣿⣿⣿⡏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢻⣿⣿⣯⢷⣻⣷⡙⠿⣿⣿⣿⣶⣷⣿⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣽⡟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⣿⣿⣷⣿⠟⢿⣦⣘⠿⢿⣿⣿⣿⣿⣿⣷⡙⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣴⠟⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣇⠙⠿⣷⣤⡙⠻⢿⣶⣯⣽⣭⣷⠞⠋⠹⣿⣷⡀⢻⣿⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⠿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠿⠿⠟⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠉⠙⠲⢭⣿⣿⣶⣿⣿⣿⣿⣿⣿⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⢀⡄⣠⣄⢠⣤⡀⣤⣄⢠⠀⠀⣤⠠⣤⡄⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⢿⢼⡇⡇⢸⢸⠈⠉⡇⣿⢸⠀⢰⣿⡆⣿⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠟⠹⠘⠇⠷⠾⠸⠶⠋⠷⠟⠸⠦⠼⠛⠇⠻⠀⠷⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠁⠛⠓⠒⠒⠒⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠳⠊⠑⠒⠲⠤⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠐⠀⠀⠀⠠⡤⠤⡠⢠⠂⢀⡾⠃⠀⠀⠀⠀⠀⠈⡃⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⡀⠁⠀⢠⠂⢀⡎⠑⠂⠄⠀⠨⠝⠉⠒⠗⡉⣷
⠀⠀⠀⠀⠀⠀⠀⢀⢞⠁⠀⠀⠀⣠⠄⠀⠤⢄⣀⣠⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠉⠍⠀⢀⠀⠀⠀⠐⠀⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡦⠤⣄⣀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢤⡞⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣸⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠁⠀⠀⠀⠀⠀⠀⠀⠨⡄⢠⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠤⠤⠤⢤⣀⣀⣀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⣀⡀⡀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣶⣶⡶⠶⢤⣄⣀
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠥⠀⠀⠀⠀⠈⠉⠳⣦⡀
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠲⠀⠀⠀⠀⠀⠀⠀⠀⢽⣦⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠷⣄⠀⠀⠀⠀⣀⠀⠰⣿⣿⣧
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠈⠁⠀⠶⣮⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣤⣄⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⢌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣦⣄⣀⣀⣠⣤⣶⣾⣷⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠑⢌⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢭⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⠇⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⢀⣤⡆⠀⣿⣿⡿⠛⠋⠋⠁⠉⠈⢻⣿⣿⣿⣿⣯⣽⣿⣿⣿⡿⠋⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⣿⣿⡇⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⣠⣿⡙⣛⠛⢛⣿⠋⠉⠁⠒⠃
⠀⠀⠀⠀⠀⠀⢠⣿⠿⡇⠀⣰⣿⢻⠀⣼⡏⠁⠀⠀⠀⠀⠀⠋⣤⡿⢂⡹⣄⣶⣿⣧
⠀⠀⠀⠀⠀⠀⢸⡅⣀⣰⣰⣿⡇⢆⠀⣿⡿⠇⣤⠀⠀⠘⠇⡈⠷⠀⣿⣐⡞⢿⡿⢻⡿⢂⣲⣭⣶⣶⠦
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣇⡘⢇⣿⡷⠀⣸⡆⠀⠀⠀⠉⠸⠇⠨⠉⠀⢅⢻⣄⣿⣿⡟⠹⣟⠁
⠀⠀⠀⠀⢀⣀⣀⣈⢿⣿⣿⣿⣿⣿⡸⣿⡇⢰⣾⣷⠀⠀⢤⠀⠀⠐⠃⠀⠚⢀⣼⠿⣿⣿⢾⣟⣃⣀⣴⣀⣀
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣯⡀⡈⡙⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⣤⠞⣋⣵⣿⣿⣿⠛⢉⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⢋⣛⣿⣿⣾⣿⣿⣿⣿⡟⢿⣷⣄⣠⡶⣿⣧⣶⠞⢡⣟⣿⣿⣡⣦⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⣿⣷⣿⣾⣿⣿⣿⣷⣿⣿⣿⣷⡿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠻⠻⠋⠉⠙⠛⠛

⠰⠒⠒⠒⠂⠂⠲⠀⠒⠒⠒⠒⢦⠀⡠⠠⠀⠀⢀⡀⠀⠀⢀⠤⠄⣄⠀⠐⠒⠒⠒⠀⡀⠀⡀⢀⢀⣀⠒⡀⢀⢀⠂⠀⠀⠒⠒⠒⡂⠂⢀⡀⠀⠀⠀⠀⢠
⢰⠀⠀⠠⡂⠀⠢⠀⠀⢄⠀⠀⣼⢸⠇⠂⠀⠀⠀⠈⢱⠄⢉⠀⠀⡸⠀⠀⠀⠀⠀⠘⡀⠻⠧⠇⠸⠸⠀⠟⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠀⠈⠉
⢠⡤⠀⢀⡄⠀⣠⢀⠀⣰⠀⠠⡌⠈⢀⣀⠀⠀⠀⠉⠉⠀⢀⣈⣩⣤⣤⣄⣒⡁⠀⠀⠁⢀⣀⠀⠀⠀⠀⠀⣀⡍⡁⠈⠉⠁⢀⠀⠁⠃⠀⢰⣮⣼⣿⣷⣦
⠀⠀⠀⢀⣠⣤⣴⣤⣤⣀⠀⠀⠀⢐⣿⣿⣿⣿⣷⣄⠂⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⠖⠠⡀⢜⠷⠀⢄⡓⠒⠓⠘⢀⠊⠙⣀⠺⠐⠄⠀⣿⣿⣿⣿⣿⡇
⢐⠡⢉⣿⣿⣟⣿⣟⣿⣿⡗⠀⠀⢸⣿⣿⣿⣿⣿⡋⠀⢸⠏⣿⢻⡿⢹⣟⢻⡏⣿⠘⠠⢀⠛⠉⠐⣶⠤⡫⠜⢧⠀⠈⠧⡀⣀⣀⣠⠆⠀⢻⣿⣿⣿⣿⠃
⠀⢀⠃⢇⠙⠿⣿⡿⠿⠋⠴⡀⠀⢟⣿⣿⣿⣿⣿⠁⢀⠣⡀⠉⠀⠀⠘⠋⠀⠀⡸⠂⠀⠀⠈⠑⠂⠤⣀⣃⠤⠊⠀⠀⠀⠀⠉⠉⠀⠀⠀⣘⠿⢿⡿⢟⡄
⠀⢀⠈⠉⠓⢀⠀⠀⠠⠞⠀⡇⠀⠀⠈⠈⠉⠃⠁⣄⡄⠑⠨⠠⠄⢀⡀⠀⠀⠁⠂⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡠⠄⢔⢶⡲⣤⠘⠀⠤⠰⠐⠁
⠀⠀⠑⠤⣀⠀⠀⠀⠀⣀⠔⢀⡠⠔⠒⠒⣢⡀⠜⡜⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣨⠖⢗⠂⡄⢸⡍⠁⠉⠉⣍⠀⣰⠈⢉⣀⠄⠓⠈⠁⠀⠅⢀⠤⠄⣤⡀
⠀⡤⠀⠄⠤⡄⠈⠀⠀⢠⠆⠁⠀⠐⠀⢀⠔⣄⠋⠰⠒⠢⡀⠀⣈⠁⣘⢈⠆⠀⠈⠉⠀⠀⠀⠎⠀⠠⠄⠀⠈⢀⠡⠀⣺⣀⡀⠄⠒⠁⠀⠀⣏⡘⣀⠠⡾
⠀⠁⢉⠉⠉⠃⠀⠀⢰⠁⠂⠈⣀⠤⢊⣱⠋⢸⠀⡇⠀⠂⠘⠆⠀⢀⣤⣶⣿⣿⣿⣿⣶⡀⠀⠃⠀⣀⠀⣄⡀⢸⠀⠀⢉⣀⡀⠀⢀⠔⠠⡦⠌⠒⠒⠒⠁
⠀⠀⢙⠀⠀⠀⠀⢀⣭⣈⠉⠀⠀⡔⠁⠈⢃⠼⢀⢸⠀⠈⠀⠰⠀⢸⣿⣿⣭⣭⣾⣿⡿⡃⠀⠃⡐⠦⠖⠢⠄⢸⠀⡾⠁⠄⠉⡄⠸⠈⡑⠦⠘⠦⠰⠆⣒
⠀⠉⢻⠀⠠⠀⢠⡾⠿⡟⠧⠀⡜⠀⠀⠀⢲⡶⡦⠰⢁⠀⢁⢠⠀⠈⠛⢛⣙⣋⠁⠄⠊⠀⠀⢓⠛⠱⠌⠁⢀⡼⠀⠡⢌⣐⣢⠇⠀⠱⣄⣺⡝⠂⠡⢄⣫
⠀⠀⢠⡀⠀⠀⠘⠢⠀⠄⠊⢰⠀⠀⠀⠀⠈⢆⢸⠀⠀⠑⣄⡜⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣍⣉⠉⠀⠀⠰⠈⠁⠆⠀⠀⠀⠀⠀⣀⠈⢁⠈⠉
⠀⢠⠶⠁⠀⠀⠀⣠⣶⣷⣄⠀⢣⡀⠀⣰⡀⢝⡈⣧⠰⠀⢈⡴⣛⡟⡿⠿⣛⠿⢛⣻⣛⡻⢻⣿⠻⡟⣛⠛⣶⡀⠣⢀⡀⠄⠀⠀⠀⠠⡎⠋⡂⠄⣈⠑⣄
⠀⠀⠃⠁⠀⡅⠸⡑⠟⠟⠻⠀⠀⠑⠋⠀⠐⠄⠣⢚⠆⠀⢸⣷⣭⣧⣶⣬⣯⣶⣭⣵⣝⣵⣬⣧⣶⣧⣿⣬⣿⣷⠀⠀⠀⢀⡛⠁⠀⠀⠉⠒⠨⢐⠊⠥⡹
⠀⠇⠀⠈⠸⠇⣀⠀⠤⠤⠁⢀⣀⠀⠤⢀⡀⠀⠈⠉⠀⠀⠘⢿⣿⣿⣿⣿⣟⣻⢟⣿⢛⣹⡛⢻⣿⣿⣿⣿⣿⠇⢀⡀⢀⡶⠣⡬⠃⠐⠠⡀⠀⠀⠀⡸⠁
⠀⠀⢠⣀⠐⣸⣞⣋⣋⡋⣉⣈⣈⣈⣙⡃⡌⢐⡄⣀⠴⠂⠶⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⢁⠆⠙⠈⠀⠸⠻⠀⠀⡜⠠⠈⡝⠀⡰⠁
⠀⠀⠀⠀⣻⡊⠀⠁⠀⠀⠂⠀⠀⠀⠀⠨⡰⢛⡈⠠⠈⣀⠃⣀⡀⢤⣐⣄⣠⣄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠰⠶⡀⡀⡆⠀⢴⠂⠀⣄⠀⠀⠀⢀⠅
⠀⢀⠐⠉⠀⠚⠂⠤⣤⡄⠀⠀⢠⣠⡤⢚⡃⢀⣀⠠⠖⠁⢘⠻⠏⠶⠦⠿⠽⠚⠁⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⣤⢠⣦⣴⣶⡎⠀⠀⠀⠑⠒⠐⠋
⠀⠛⠀⠒⠒⠒⠈⠁⠀⠈⠂⠤⠀⡀⢐⠀⣡⡀⡠⣄⠀⠀⠘⠒⠐⠖⠒⠂⠄⠒⠁⠀⠀⢛⣿⢹⣿⣿⠃⠀⢰⣿⣿⡿⠟⠉⣧⠀⢰⡆⠉⢋⠉⢉⡉⢹
⠀⠴⠒⠲⠖⠐⠲⠒⠲⡖⠒⠲⠀⠴⠄⠤⣈⡝⠠⢹⠀⠀⢘⠀⠀⢠⠀⠠⡄⠰⠀⠀⠰⠀⠛⠀⠸⠟⠀⠀⠀⣯⣀⣤⣶⣾⡍⠀⠀⡏⠀⠉⠀⠈⠀⠈⡀
⠀⢀⠀⠀⠀⠀⢈⠀⢀⡇⠀⣀⠃⠑⠄⣁⣠⡱⢢⣮⠠⠀⡈⠁⠀⠀⠀⠀⠀⠀⢢⠀⠈⡀⠀⠀⠀⢀⠀⠀⠀⠘⣿⡿⠿⡟⠀⠀⠸⠀⠀⠀⠀⣀⣤⣶⣧
⠀⠈⠀⠈⠁⠀⠉⠰⢎⠁⠀⠈⠀⠀⠀⠑⠮⠽⠭⠮⢉⡧⠈⠁⢐⡀⠀⢀⡈⠁⠀⠀⠀⠈⢩⠉⠉⠀⠀⠀⠀⢠⠒⣀⡘⢤⠀⠀⠀⠀⢳⣾⡿⣻⣭⣴⣿
⠀⠉⡏⢉⣉⠉⠙⣄⠩⠉⠉⠉⠋⠉⠉⠙⠀⠀⠾⠭⢱⠀⠀⠰⢥⣴⣶⣦⣤⣵⡀⠀⠀⠀⠘⠔⡄⠀⠀⠀⠀⣀⠈⠩⢉⠀⠀⠀⢠⠀⢸⣿⠿⠛⠋⠁
⠀⠀⡇⢀⣄⡀⠀⠘⢆⠘⢂⠀⠄⠀⠀⠰⠀⡐⠁⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠆⣀⣤⣶⣼⣷⣦⡀⢎⠀⢱⣐⡈⠇⠠⡀⠸⡀⡌
⠀⡀⣇⠁⠘⠑⠈⠃⠊⢣⠒⡀⣀⠀⠀⣠⠀⠴⠶⠿⠶⠶⡅⢀⠀⠂⠒⠀⠋⢹⢀⠔⢸⣿⢿⢿⡟⣿⠙⢣⠀⠑⠄⡁⠈⡖⠀⢹⠀⠈⠁⠒⠒⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠜⠁⠀⠀⠀⠀⠰⢶⣆⢰⡆⠀⠀⢣⣀⣀⣀⡤⠁⠀⢰⠈⠥⠤⠬⠥⠐⠒⢁⠠⠁⠀⠈⠑⠓⢂⡁⠀⣠⡄⠐⠴⠪⡁⠑⢄
⠀⢰⠰⠊⣭⢱⢲⠀⢠⠖⠖⠀⠑⡖⢰⠀⠀⢦⣤⣤⣤⣤⠆⠀⣤⣤⢡⣅⢀⣀⢀⣀⡁⣀⡀⣀⠀⠀⠁⠀⣀⢠⡒⡆⠀⠀⡊⠑⡤⠌⢪⡖⠀⠙⠞⠀⢀
⠀⠈⠋⠑⠐⠋⠘⠀⠀⠓⠑⠂⠐⠉⠈⠀⠀⠯⠿⠿⠿⠿⠂⠈⠮⠿⠦⠿⠦⠿⠧⠽⠧⠼⠷⠼⠃⠀⠀⠘⠒⠚⠠⠕⠀⠀⠐⠠⠧⠄⠀⠀⠀⠐⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⢉⣠⠶⠳⣄⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿⡇⠀⠒⠋⠀⠀⠀⠙⠇⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⣤⡈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⣀⣴⡞⠉⠠⠞⠉⠀⠈⢷⡀⠹⣧⡀⠀⣀⡀⠀⠀⠀⣀⡼⣿⣦
⠀⠀⠀⣀⡴⠚⠉⠻⣧⡀⠀⠀⠀⠀⠀⠈⠓⠀⠙⣷⣄⠈⠁⣀⡴⠞⣉⣄⠘⢿⣧
⢀⡴⠚⠁⣠⠴⠛⢦⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡟⠁⠠⠚⠁⠈⠳⠈⡿⣷⡀
⣿⣄⠀⠈⠀⠀⠀⠀⠻⡌⠻⣆⠀⢤⣤⠀⢀⣠⡴⠛⠉⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⣠⠶⠋⣡⠶⠛⢷⡀⠻⣷⡄⠀⡀⠀⠀⠀⠀⢀⠽⠛⠙⠒⠢⣤⣤⣄⣀⡀
⠈⢿⣿⣦⠀⢠⣀⣀⡀⠀⢀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠁⠀⣀⡤⠚⠁⠀⠀⠀⢀⠀⠀⠀⠀⠉⠉⣉⡶
⠀⠀⢻⣿⣧⡀⠉⣡⠴⢚⣩⣤⣄⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣴⠋⠁⠀⠙⠶⠀⠀⠀⠈⠻⣤⣴⠖⠛⠋⠁
⠀⠀⠀⠹⣿⣿⡏⠁⠀⠉⠀⠀⠙⠇⠀⢹⣧⠀⠑⠖⠀⣀⡤⠖⠉⠉⠛⠒⠁⠀⠀⠀⠀⠀⣤⣴⡶⠛⢧⡀
⠀⠀⠀⠀⠘⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣠⠴⢚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠏⠁⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠈⢷⣿⣧⡀⠳⡄⠀⠀⢀⣠⣶⠋⠁⠰⢿⣿⣶⣔⣀⣢⡀⢀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⢀⡤⠞⠉⢀⣸⣧⣤⣴⡾⠋⠓⠉⠀⡙⠿⢦⣤⡀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣩⣤⠶⠚⠋⠉⠁⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⡌⣿⣆
⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⣿⣿⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⠔⠊⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠙⣟⣊⣻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠁⡠⠊⠁⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡠⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⡴⢁⡠⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣷⡀⠀⠀⣠⠊⢀⠞⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⠀⠁⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⡄⠀⠀⠀⠀⠀⠀⣀⡴⣻⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⡄⠀⠀⠀⣠⣶⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣆⣠⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⣀⣀⡀⣀⢀⣤⡀⡤⣄⡴⣤⢴⠀⢰⣦⠖⣖⢲⡞⠶⣿⣶⣶⣾⡀
⠀⠀⠀⠀⠀⢸⣾⣿⣿⣅⣧⠃⣷⣹⣸⡇⣧⣷⠈⣷⢹⡼⡇⣸⣿⡆⢻⡀⢿⠒⢾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣏⣿⣀⣿⢸⡇⣧⣿⠸⣤⠾⣼⠇⠿⠿⠇⠿⠘⠇⠘⠓⠓⣿⣿⣿⣾⣇
⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣨⣉⣈⢠⣤⡀⢀⡒⣒⣦⢤⣄⢤⡶⡄⢠⢠⣼⡗⣶⣄⣾⠏⠝⠋⠙
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠹⠟⠃⠛⠚⠥⠈⠀⠀⠉⠋⠈⠙⠋⠀⠁⠀⠁⠈⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⣿⠉⢹⡏⠉⢹⠀⠀⠀⠀⠀⢀⡤⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⢤⣀⣀⡼⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⡿⠶⢾⡷⠶⢾⠀⠀⠀⠀⠀⣼⣅⣀⣿⠞⠙⢦⡀
⠀⠀⠀⠀⠀⣠⢾⡋⠉⠓⢦⣹⣁⡤⠞⠓⢦⣀⠀⠀⠀⠀⠀⢸⣤⣄⡛⠀⣸⣇⣀⣼⠀⠀⠀⠀⠀⠀⢠⡾⠥⠤⠤⠶⠽⢦⡀
⠀⠀⠀⠀⢸⣧⡀⠉⠓⢦⡴⠛⠉⠀⢀⣠⠴⠈⡇⠀⠀⠀⠀⢸⠈⠳⣍⠛⠺⢧⣄⣸⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠤⠤⠤⠤⠽⢦⡀
⠀⠀⠀⠀⠸⣄⠉⠙⠒⣾⠈⠓⢦⠖⠋⠀⠀⢀⡿⠀⠀⠀⠀⢸⠀⠀⠈⠳⣄⠀⣠⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠤⠤⠤⣤⠟
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠷⣄⡀⢸⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠛⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⢾⡏⠉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁