Studying Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠂⠀⠀⠀⠀⠀⣠⠖⠲⣄⠠⣖⠒⠒⠒⢒⡆⠀⠀⠀⡦⠴⠤⣄⢀⠤⣄⣀⢶⡄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⡇⠀⠺⠥⠤⡄⠹⠀⠏⢩⡄⠀⡤⠄⡇⢠⢤⠈⠇⠑⡌⢡⠊⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠀⠀⠀⠀⠙⠛⠒⢲⡀⢈⡀⡇⡆⢣⠱⠴⠃⡄⡆⢸⡤⢀⡆⠀⡇⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣰⡒⢒⣊⡠⠞⠀⠧⠇⠈⠓⠶⠚⠁⠷⠤⠔⠋⠀⠀⠙⠚⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣇⢸⠩⠀⢰⠋⣉⠆⢰⠋⡉⠳⡠⡖⠒⠲⢤⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢅⠀⡆⢸⠀⣉⠘⡌⠀⠋⣴⠃⠀⢘⢢⠀⢣⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢸⠀⡇⢸⠀⡟⡄⢻⠀⡷⡈⢢⡇⠘⠋⢀⡎⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠈⠳⠃⠘⠚⠁⠘⠋⠓⠃⠙⠊⠱⠶⠖⠋⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⠴⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣄⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣑⠲⠦⠤⠤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠚⠒⠒⠒⠒⠲⠶⠄⠀⠤⠤⠄⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠶⠤⣤⣤⣤⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠶⠶⠾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⠀⠀⠀⣀⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠖⢉⣀⠀⠀⠀⠀⠀⠀⠑⢤⣄⡀⠀⢴⣂⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡐⠆⠀⢠⠃⡴⠉⠈⢦⠀⢶⡀⠀⠀⠀⠈⣿⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⠃⠀⠀⠀⠑⢬⣯⠲⣄⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⣀⠀⠀⠀⠈⡁⠀⢙⠒⣼⠀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠉⠀⠠⠀⠀⠁⠀⢸⡞⠓⢤⡈⢃
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠢⣄⠀⠀⠀⢀⣠⠔⠣⣀⠀⢀⡧⠃
⠀⠀⣖⣩⡄⢨⣑⣩⣭⣍⡒⡄⠀⡸⠁⠀⠀⠈⠛⠝⠁⠀⠀⠀⠈⢆⠀⠀⢸⣉⣉⣉⣉⣉⣹
⠀⠀⠈⢎⣀⣀⣀⣀⣀⣱⣈⠁⠀⡇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡧⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⡇⠀⣟⢀⣪⣆⣀⡀⠀⠀⣀⣀⠤⠤⢄⠇⢀⠋⠉⠉⠉⠉⠉⠹
⠀⢀⣠⠚⠒⠒⠒⠒⠒⠒⠒⣇⣀⣻⣲⠫⠝⠀⠈⡉⡉⠤⠹⢯⣂⣸⣀⣀⠗⠀⠀⠀⠀⠀⠰⣄⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠊⠉⠀⠉⠒⠊⠉⠉⠉⠋⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠠⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠁⠀⠀⠀⠀⠈⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢀⡀⠘⡄⠀⠀⠀⠀⠀⢀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠄⠂⠀⠁⠀⠃⠨⠐⠀⠀⠒⠚⡑⠈⢤⠊⠡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣈⠀⠀⠀⠐⣄⡄⠀⠀⠢⡀⠀⢀⠀⠀⠀⠔⠀⡀⠁⠐⣰⠃⣀⣀⣡⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠉⢁⠃⠠⣌⠉⡀⠠⠀⠈⠂⠤⢠⣤⠤⠔⠊⠀⠐⠊⠉⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠤⡂⢀⠲⣌⠓⣷⡦⠴⠦⠤⠀⠠⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠙⠊⠡⠄⠀⠀⠀⠈⠁⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡒⣒⢒⣒⣒⣀⣀⣈⡁⠀⠐⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⡀⠶⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⠲⡖⠀⡀
⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠊⠁⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠖⠓⠒⠒⠒⠿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠐⠆⠀⠖⠈⣦⠠⡄⠶⠴⠲⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢄⣀⠜⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⢠⠀⠀⡤⠀⢰⡇⠐⠟⢠⠦⡀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⡆⠀⡄⡛⠉⠉⠙⠛⠛⡟⠒⠒⠒⠒⠒⠀⠐⠚⡇
⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠂⠀⠘⠀⣈⣤⣤⣶⡄⠘⢧⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠾⣄⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠿⠍⡉⠛⠛⢳⠲⠶⠤⠤⠤⠄⠀⠀⡏
⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⡔⠉⠙⣿⣿⣿⣿⣿⣿⣇⠐⠂⠈⠁⣀⡠⠂⠑⡀⡀⠀⠀⠀⡀⠃⠀⠘⠀⢘⠁⠀⠀⠀⠻⣦⣄⡘⣄⢂⠀⠀⠀⠀⠀⠀⢃
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⢿⡀⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⡄⠐⠡⠀⠀⠀⠀⠀⢁⠀⢰⡆⠠⢘⡇⠀⠀⠀⢠⡏⠀⠈⠹⠃⠠⠠⢤⡜⣉⠀⠉
⠀⠀⠀⠀⠀⢶⠀⠀⠀⣀⣘⣧⠀⣴⣿⣿⣿⣿⣿⡟⢂⣠⡀⠀⠀⠀⠀⠄⠂⠀⢠⣴⠀⠈⡀⢸⣇⡔⠋⠀⠒⠒⠂⠈⠀⠀⡀⡀⠀⠀⠀⠤⠀⠀⣀
⠀⠀⠐⡀⠀⠈⠇⠀⠀⠀⠉⠙⡆⣿⣿⣿⣿⣿⣿⢀⡀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢀⢧⡽⠂⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠉⠒⠂⠒⠀⠀⠉⣤
⠀⣴⣿⣿⣦⠀⠈⠀⠛⢲⠀⠀⠡⢈⣿⣿⣿⣿⡇⠈⡀⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠆⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠈⠁⠒⠀⣤⣀⡀⠀⢹
⠰⣿⣿⣿⣿⣷⠀⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡿⠽
⠀⣿⣿⣿⣿⣿⣿⠀⡸⢃⣠⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢣⠃
⠀⠟⠛⠛⠛⢻⣉⢱⣧⣬⠛⠋⠉⠉⠉⠉⠛⠟⣿⣷⣶⣾⣿⣾⣿⣿⣻⣿⣶⣶⣸⣿⠀⡄⠚⢼⠀⠀⠰⠃⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣼
⠀⠠⡄⠡⠀⢀⡛⠉⠘⠙⠀⣀⠀⠀⠀⠀⢀⣀⠉⠋⠛⢻⣿⡶⠒⠛⠻⡿⠿⠶⢿⣿⠀⠘⠀⢸⠷⠠⠃⠘⡎⡐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡏⣞⠁
⠀⠀⠚⠓⢶⣤⣄⣤⣦⡤⠞⠛⢪⡵⡟⠻⢿⣿⣿⣶⡤⠞⠙⡁⠤⠀⠀⠀⠀⠀⠈⠉⠐⡂⠐⠿⠃⠀⠀⠄⠘⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣼⠿⠁
⠀⠀⠀⠰⠹⢿⣟⣿⡿⠓⠈⠑⠀⣠⣄⠀⠤⠦⠀⠈⠚⠓⢀⠀⠀⠀⢠⣢⠄⠀⠶⠖⠂⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠄⠀⠀⠀⠀⢠⣿⡦⠤⠶
⠀⠀⠀⢢⡀⠏⢡⡏⠵⠀⠠⡄⢀⣉⠀⠀⠀⠀⠀⠀⠀⠉⠏⠀⡀⠀⠀⠐⠂⠀⣼⡟⠀⣤⣄⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⠶⠆⣁
⠀⠺⠊⠉⣭⡄⠁⠐⠀⠤⡀⠁⠀⠀⠉⠀⠀⠀⠀⢠⣀⢠⣄⠀⠀⢀⣀⠀⠀⠀⠈⠉⠀⠻⢿⣯⡛⠦⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⢋⣀⣀⡤⢤
⠀⠀⠀⠀⠛⠘⠛⠲⠦⠄⠀⢹⣶⣶⣤⣀⣀⢀⣠⣤⡞⠁⠉⠍⠉⠉⠉⠀⠤⠠⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⣶⢤⣄⣠⣶⠟⠿⠶⠶⠖⠋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠛⣿⠛⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⢿⣥⢀⠀⠀⠀⠀⣀⣨⠥⠆⠒⠛
⠀⠀⠀⠀⠀⢤⠤⢄⣀⠩⠁⠒⠀⠀⠀⢀⠏⠐⠀⠀⠀⠀⠓⠒⠒⠒⠒⠀⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠀⠈⠙⠖⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠤⠄⠠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠂⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣹⣦⣄
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣾⣿⣿⡿⣆
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣄⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣈⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣟⢻⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠸⣿⣿⡿⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣌⣛⣦⡤⢤⣀⣀⣀⣀⠀⣀⣀⡀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣳⣿⠻⢷⣽⠭⣿⠼⣯⣉⠉⠳⣦⠀⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣿⠉⠻⢦⣉⠲⣄⡀⠀⠉⠓⠒⠛⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣹⠆⠀⠀⠀⠉⠳⣬⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⡄⠀⠀⠀⠀⠀⠀⠙⠶⣬⡑⢦⡀⠀⠀⠀⣤⡄⠒⢲⣬⣭⡵⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢾⠛⡛⢻⣿⢦⡀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣌⣩⣭⣵⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠀⣽⠀⢃⠸⡇⠙⢧⣙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡏
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠙⢦⣈⣹⡄⠀⠀⠙⠳⢦⣝⣛⣡⠤⠤⣤⣤⣛⡿⠛⢳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠙⠉⠉⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠰⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢢⠀⣿
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣅⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣸⣿⠀⢸⡇
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠻⣆⠀⠀⠀⠀⠀⠈⢿⣿⠏⠀⢸⡇
⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠉⠉⢻⡿⠛⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢧⡉⠛⠛⣷⠀⠀⠀⠀⠀⢻⡋⠉⣠⡶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⡎⠙⠷⣾⣧⣦⣀⣤⣀⣀⣌⣷⡞⠋⡆⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⠁⠀⠀⠈⠉⠈⠛⠈⠉⠉⠉⠉⠀⠀⠁⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠻⣷⡶⢤⣀⣀⠀⠀⠀⠀⣀⣠⣷⣶⡿⣋⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠲⠶⠶⣶⠛⠉⠁⠀⠀⠀⠈⠉⠑⢲⡖⠋⠉⠁⠀⠀⠀⠉⠉⠓⣶⠶⠶⠶⢦⣄
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⠀⠘⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⣷
⠀⠀⢀⡼⠃⠀⠀⠀⠀⠐⢷⡄⠈⢉⣉⣉⣉⡀⠶⢀⡀⠘⠃⢀⡀⠤⣄⣉⣉⣉⡉⠁⢠⣾⠀⠀⠀⠀⠀⠘⢇
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠛⠛⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠟⠛⠿⠿⠿⢾⡇⠀⠀⠀⠀⠀⠀⠘⣦
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠙⠶⢤⣤⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⠶
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢉⡏⠁⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁⠈⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠿⠓⠉⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠓⠚⠿⣦
⣤⣤⣤⣤⣤⣤⣤⣽⣷⣶⣶⣮⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣵⣶⣶⣾⣭⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠟⠻⠿⠿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣀⣤⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠋⠠⠄⠛⠋⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠺⠃⠀⠀⠀⠉⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠠⡄⠀⢠⡄⠈⠇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⡼⢣⡀⠀⠀⠃⠀⠀⠀⣠⢶⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠰⡎⠀⠀⠙⠢⡤⠒⠂⠉⠉⡤⣣⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢴⣆⠀⠀⠀⠘⠢⠤⣀⢸⠀⠀⢠⠔⠺⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⣣⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⢄⣀⣀⡇⠹⣷⡄⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡇⠀⠘⢾⣆⠀⠀⠀⢸⠀⠀⠀⠀⢱⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠚⠧⢤⠴⠊⣉⡳⢦⡀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⡀⠀⢻⠀⠀⠀⠀⠀⢠⠀⡒⠤⢽⣾⡃⠘⣄⠀⠀⠀⢀⠇
⠀⠀⠀⢸⠉⡩⠭⠛⠉⠉⠉⠁⠀⠉⠉⠉⠚⢿⡒⠠⠤⠰⠚⠢⠭⠿⠞⠙⠿⠉⠉⠙⠒⠶⣛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⣄
⠀⠀⠀⠇⠛⠒⠤⢤⡤⠤⠴⠒⠒⠒⠒⠒⠤⠤⠬⣦⣀⣤⠤⠤⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠬⠷⠄⠀⠀⠀⠀⣀⣠⢤⡀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣒⡩⠕⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠕⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⢸⠂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⡒⣒⣒⣒⣒⣒⣛⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠒⠒⠒⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠲⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣦⣶⣶⣾⠿⠿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⣤⣀⣀⣠⣤⣤⣴⣶⣶⡶⠶⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣖⠼⠄
⠀⠀⠀⠀⠀⠀⠀⢐⣤⡀⠀⠀⠀⠈⠛⠲⢦⣄⡀⠀⠀⠀⢸⣿⠆⠀⠀⠈⠁⠀⠀⠀⣀⣀⣠⣤⠤⠴⢖⣚⣟⣛⣻⣭⣭⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠰⠋⠒⠬⣙⠲⢤⣀⠀⠀⠀⠀⠉⠙⠲⢤⣀⣿⣿⣶⣶⣶⢶⣒⣛⣯⣭⣭⣴⠶⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠉⠙⢻⣿⡟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⣷⡤
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣈⠑⠢⢄⡀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⢖⣚⣯⣭⣭⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠈⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡈⠑⠠⢄⣻⣿⣯⣀⣀⣤⣤⠤⣶⣞⣿⣯⣿⡶⠾⠟⠛⠋⠉⠁⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀⢈⣉⣩⣭⣿⠷⠾⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⢸⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⢸⣿⢋⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠼⣿⣿⣄⣠⣤⣶⠶⠖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⡿⠻⠟⠋⠀⠙⠿⠻⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⠋⠈⠐⠀⠀⠀⠀⠀⠂⠀⠹⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠉⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠋⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⢀⠈⠉⠁⡀⠀⠀⠀⠀⠹⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣶⣿⣄⠀⣠⣷⣦⣤⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣞⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡛⠛⠛⣛⣛⣛⣛⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣥⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⢀⠠⠀⠴⠀⢸⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣄⣹⣿⣿⣳⡄⠀⠀⠀⠀⠉⠳⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣭⣌⠙⠿⢷⣮⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣾⣶⣿⡿⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠛⡗⡮⠙⠋⢀⣂⠖⠀⠀⠀⠀⠰⣆⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡚⠀⠀⠀⠑⠀⠀⠀⠀⢀⡀⠀⠀⠀⠂⢀⣀⡵⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣎⠀⠙⢿⢢⡀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠤⢟⣭⣥⣶⣟
⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢙⣦⣀⣳⡙⠢⠌⠁⠀⠤⠀⠀⣘⠯⠈⠙⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠖⠁⠄⠀⠀⠤⣙⢦⡝⠹⣤⣻
⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠈⠑⣝⣆⠹⣟⡤⠀⠒⠒⢲⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢧⢹⡅⠀⠀⠀⢸⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⢄⣀
⠀⠀⡼⠃⠀⠀⢄⠀⠀⠀⠄⠀⣀⡀⢤⣹⠃⡈⡃⠀⠀⠀⠘⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠑⠤⡀
⠀⡰⠃⢀⠈⠢⢈⢢⡀⠀⠀⣀⣀⣭⣓⠚⢳⣁⣷⣄⠀⠀⠀⡇⠚⣡⣵⡶⢠⠀⠒⠘⢄⠀⠀⠀⠀⠀⠀⢀⠖⠁
⠞⠁⠀⠀⠙⠢⢤⣁⠻⣯⡉⠀⠀⠠⠤⠐⠚⠻⣷⠬⠀⠀⠀⡞⠂⠀⢁⠠⠄⠀⠀⠀⢀⠀⠀⠀⢀⢄⢴⣉⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢳⣄⠀⠀⠀⠀⠀⠀⢘⡇⣀⣀⣠⣁⣀⠡⣌⡃⠀⠀⠐⠀⠀⠀⠀⣀⠴⢧⣎⡀⢀⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢞⣰⡿⠀⢠⠀⠀⠀⠀⢫⠢⡄⠳⣄⠀⢀⣢⣛⠆⡀⠀⢠⣄⣤⢀
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠴⠚⠁⠎⣠⠔⠁⠀⠀⠀⠀⠈⡇⠁⠀⠈⠐⠋⠠⡙⠴⢀⠄⠔⣭⣭⠢
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠈⠳⢦⣀⠐⠊⣁⣀⣀⣀⣀⣤⡤⠤⠁⠀⠀⠀⠀⠐⠄⠈⠱⡄⠐⠂⠊⠁⠐
⠀⢾⡒⢒⠀⠀⠀⠀⠠⠠⠄⠀⢁⣀⣀⣀⡀⠙⢍⠣⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠢⡀
⢰⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠒⢯⡳⠄⠣⡈⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⠊⠉⠙⢢⡀⠀⠀⢠⠖⠈⠑⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠉⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⡾⠘⠊⠉⠉⠀⠈⠉⠉⠒⠔⢢⠀⠀⠀⠱
⠀⠀⠀⠀⠀⢠⠎⠀⣀⣀⣠⠖⠉⢷⠒⢒⣐⣐⡚⠳⣄⠀⠀⠀⠀⠀⢀⣘⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀⠈⠃⠰⡀⠀⠘
⠀⠀⠀⠀⢀⠇⠰⡏⠀⠉⡤⠦⣀⣸⠉⠀⠀⠁⠀⠙⢍⢆⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⠈⡒⠠⢄⠱⡀
⠀⠀⠀⠀⢸⠀⠀⠙⢄⠜⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠆⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⢀⡀⢈⡡⠘⠛⠀⠀⡰⠁⡧⠔
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠑⠚⠒⢤⣀⠀⣀⠀⠀⠀⣔⠃⠀⠀⠀⠀⢸⠙⠉⠉⠉⠉⠁⠍⠀⠀⢸⠀⠑⠈⠀⠀⡇
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡞⠀⠀⠒⠀⠀⠀⠉⣉⠈⡙⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⣀⠀⡰⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⠃⠀⠀⠂⠀⠀⠀⠠⠌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠠⣀⡠⠄⠀⠀⠀⠀⠉⡉⠜⠉
⠀⠀⠘⠣⡄⠀⠀⠀⣴⢧⣀⠀⠀⠐⠢⠠⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠌⣀⣀⡀⠀⢤⣔⡊
⠀⠀⠀⠀⠙⠚⢢⡼⠀⢀⠑⠭⠶⣦⣒⣂⡬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠳⠴⠋⡀⠀⠂
⠀⠀⠀⠀⠀⠀⡘⡄⠀⠘⠀⠀⠀⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⡆⠀⠀⠁⠰⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠀⠘⠄⠀⢆⠀⠉⠂⠄⣗⠄⣀⠀⠑⠢⢄⣀⠂⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⢐⠄⠀⠀⠀⠄⠈⠀⢀⠄
⠀⠀⢨⠀⠀⠀⠈⠁⠀⠑⠢⠤⢀⡀⠉⠙⣢⡰⢄⣠⠏⠑⠦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⡀⠀⠐⠉
⠀⠀⢸⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠈⠑⠲⠋⠁⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠲⠤⠄⢀⠠⠆
⠀⠀⠈⡄⠀⠀⢸⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠
⠀⠀⠀⠣⠀⠀⠜⠀⠀⠀⠐⠄⠀⣀⡠⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠋⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⢂⠉⠄⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢀⡀⠁⣀⡠⠭⠒⠂⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠶⠖⠀⠀⠀⠀⠀⠀⠲⠶⠆⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⣠⣄⠂⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠛⠟⡀⠀⠤⠀⠈⠻⢃⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠠
⠀⣠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠎⠉⡦⡀⠀⠀⠄⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠠⢄⡀⠀⠀⢀⠀⠀⠀⠀⠒
⠀⠁⠰⠐⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠠⠀⠀⠄⠀⠀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢀⠈⠀⠀⠈⠀⠐⠃⠀⡄⠀⠀⠈⢀⠐⠁⠀⠀⡀
⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⠀⡇
⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡦⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⢀⡀⠱⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⡀⠊⠈⠉⠁⠈⠁⠐⠚⠆⠀⠀⠀⠀⠲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠰⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠄⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⢀⡤⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠆⠀⠀⠀⠃
⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘

⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠐⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠖
⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀
⠀⠀⠀⣷⣤⣶⣤⣤⣄⠀⠛⠛⠛⠋⠉⣥⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠍⠩⠭⠉⠭⠤⠥⠤⠤⠆⠀⠀⡀⠈⠁
⠀⠀⠀⠟⠉⠉⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠃
⠀⠀⠀⠶⠲⠶⠶⠶⠶⠖⠒⠒⠂⠀⠄⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣀⡀⣀⣀⡀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⡦⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⢩⣉⣉⣉⣉⡁
⠀⠀⠀⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⣤⣄⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠛⠛⠛⠛⠛⢷⣶⣆⣉⠩⠤⠤⠄
⠀⠀⠀⣦⡤⠠⠄⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⡃⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣉⣙⣛⣛⣻⣿⣿⣿⣯⣭⣽⣯⠙⠛⠛⠛⠛⠛⠛⠛⢓⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣈⣈⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢀⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⠴⣯⠉⡠⠀⠀⠀⢄⠉⢿⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠊⠀⠀⠀⠀⠈⠑⠠⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢰⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⠃⠀⢼⣷⠀⠀⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠨⣀⣀⣰⣿⠀⠀⠀⠙⠏⠍⠀⠀⠘⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠸⣯⣽⠀⠀⠀⢠⣶⡄⠀⣠⣼⠃⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⠿⠛⠉⠉⠁
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠉⢀⣀⣴⠏⣠⣷⣬⡿⡟⠦⠐⠊⠀⠈⣿⣿⠿⠟⠉
⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣏⣤⠜⠃⠤⢀⡀⠀⠀⠀⠀⠀⠘⠂
⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣏⡀⠀⣀⣼⠧⣄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢙⣿⣿⣿⣿⡟⠃⠄⡼⣷⣴⠆⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⣶⣿⣿⣧⣃⠤⣤⡾⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡀⠀⠹⣿⣿⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠓⢿⣶⡿⠀⠀⢸⣶⡞⢰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⢏⣾⠗⠀⠀⢸⣿⡷⠋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⣿⣿⣿⡏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠛⠛⢉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⢫⣙⣁⠀⣀⠤⠎⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣦⣵⠀⠛
⠀⠀⢶⠀⢈⠂⠀⡄⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡦⢠⡄⣭⡰⠀⢸⣶⡶⠴
⠀⠀⠐⠃⠀⠸⡆⠉⠒⠘⢉⠄⠀⠀⡆⢁⠀⠴⠄⢶⡀⢀⠀⠀⠰⢲⠀⠀⠀⢸⢓⡇⠈⠻⠿⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠑⠄⠀⠑⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣾⣿⡛⠋⠉⠀⠀⠀⠒⠶⠬⣝⣓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⢖⣩⣾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠮⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢏⣿⣧⣤⠤⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣛⣛⣋⣼⣿⢻⣿⠀⡄⠀⠀⠀⢠⡇⢸⣷⠀⠀⠀⠀⣤⠙⣧⠀⣶⣿⣿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣽⣿⣿⣿⡿⢷⣾⣿⣤⣧⣤⣄⣀⣸⣅⣼⣿⠀⠀⠀⢰⣿⡇⣿⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣬⣤⣔⣾⡿⠷⢯⣍⣙⢿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣀⣀⡀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣬⣛⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡙⣯⡉⢉⣀⠀⠀⠀⠀⢻⣿⣿⣿⠟⣩⠲⢦⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠘⠧⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠙⢧⠀⢿⣏⠙⣿⣿⡛⢿⣿⠉⠉⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠘⠋⠩⣨⣾⣿⣿⣿⡿⠁⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣶⣤⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⢘⣦⣤⣾⠟⠋⠹⣿⣿⣿⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⢻⣿⣿⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣋⣵⠟⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣠⣴⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⣴⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠙⣟⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⡿⠟⣫⡵⣾⡷⣿⣿⣯⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⢀⠔⣵⣿⡿⠊⠀⠉⣡⠞⣱⣿⣷⣿⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⣦⠀⢀⠔⢡⣾⡿⠋⣠⣄⢠⠞⣱⣿⣿⣿⠿⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⣾⣿⣿⣿⡷⠋⠀⣿⡿⠁⡼⠻⢣⠏⠾⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠴⠒⠚⠉⣉⡭⡻⠻⢳⢲⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⡿⠛⠁⡴⠋⠀⠀⠀⣠⢾⢡⣱⠷⠟⠋⠛⠙⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣾⣿⠯⠁⢀⠞⠀⠀⠀⠀⣼⢃⣷⡛⠀⠀⠀⠀⢀⡤⣖⡶⣿⣿⣶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢱⣿⡏⠀⢠⠎⠀⠀⠀⢀⣾⢣⣟⠒⠀⠀⠀⢠⢞⣵⡹⠞⠟⠏⠋⢋⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠟⠀⢠⠋⠀⠀⠀⠀⡼⢣⣟⠂⠀⠀⠀⣰⣷⣷⠻⠑⠈⠀⠀⢀⢢⠱⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠎⠀⠀⠀⠀⣼⡳⡿⠀⠀⠀⠀⠐⠿⠋⠃⠀⠀⠀⠀⠀⡌⠆⠇⠡⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡼⠀⠀⠀⠀⢰⡛⡽⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢰⢨⢳
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠃⠀⠀⠀⠀⡶⣙⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⡄⡄⠀⣠⢸⢸⣸⢳⣄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⠀⠀⢸⣗⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢠⠀⠃⠇⡄⡆⣞⣼⡟⣁⣟⠚⢛⣂⣲⣶⣶⣦⣄⡀⣀
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢀⠇⠀⠀⠀⠀⣮⢺⡯⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠎⠆⠌⡘⡸⢰⣰⣷⠟⢡⠸⣿⣯⠤⠀⣀⣀⣒⣒⣒⣒⠒⣒⠂⠰
⠀⠀⠀⠀⠀⠀⢻⡀⡠⡀⢸⠀⠀⠀⠀⢰⡧⣽⡢⠀⠀⢀⢀⠀⡀⢀⢀⠂⡐⡜⡼⣴⣵⣷⠯⠛⡁⢪⢨⣼⣯⡿⠩⡭⢴⠒⠒⠒⠒⠀⣈⣉⠭⠬⠩
⠀⠀⠀⠀⢀⡤⠊⣷⡕⡑⣅⣀⡀⢀⢠⣸⡑⡯⡢⠣⢃⢊⠎⡐⠡⢂⠎⡔⠠⣨⠀⠃⢀⠄⠦⠷⠰⢾⣿⡿⡿⢧⣬⣥⣤⣤⣴⣿⣿⣖⣒⣒⠒⠭⠉
⠀⠀⢀⠔⠁⠀⠀⠀⠈⠛⢻⣻⣿⢕⣟⣿⢪⣟⡔⡱⡡⢂⡔⡠⠡⢂⠎⡴⣱⣻⢀⣀⣸⣼⡖⠶⢶⣶⣳⣦⣼⣾⣶⣶⣒⣛⡭⠭⠿⠟⠂
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠚⠛⠻⠿⠿⠟⠛⠋⠭⠹⠿⠿⠟⠛⠛⠋⠑⠚⠿⣽⣻⣛⣛⣋⠭⠭⠭⠛⠓⠊⠉
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠕
⠻⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⠿⠓⠁
⠀⠈⠓⠭⣒⠢⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⣞⠍
⠀⠀⠀⠀⠀⠉⠑⠒⠨⠭⠵⣶⣶⣤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣾⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠀⠀⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣄⣰⠷⣆⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣌⣻⣿⡟⠹⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣼⣿⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣶⣻⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢹⣶⣿⡟⢿⣿⣿⣿⣿⢱⣯⣤⠶⢄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⡾⠿⠛⣨⡶⢾⡇⣿⣿⣿⣿⣷⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⡾⠏⠁⣴⣿⣿⣿⣷⣿⢹⣿⣿⠉⣡⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣽⣿⠠⠈⣄⠈⠉⢛⣿⣿⣷⣦⣿⣿⣷⣿⣿⣿⡂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢤⣤⣾⣿⣿⡿⡄⠀⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣠⡀⣰⣾⣖⣤⣤⡴⠾⠿⠿⠛⠁⠈⠀⢈⣹⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⢻⣿⣿⣇
⠀⠸⣻⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢨⠿⠿⣿⣿⡿⣻⠁⠁⠀⠀⠀⠙⣿⣿⣶⣦
⠀⠀⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠛⢛⠿⠄⠀⠀⠀⠀⠰⠶⣿⡛⠉⢉⠙⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡦⠀⠀⠀⠀⠀⠀⠀⡒⠯⣅⠘⠃⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⠆⣤⠾⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠙⢭⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢸⡇⠀⠈⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠓⠀⣉⣁⣀⣀⣀⣁⡀
⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠮⢴⡶⣿⠋⠩⠭⡍⡭⡟⠋⠁
⠀⠀⠀⠀⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⣎⠈⠀⠀⡠⠀⠁⣼⣶⡶
⠀⠀⠀⠀⣶⣦⣼⣿⣌⠢⠀⠀⠀⠀⠀⢀⣽⠱⡀⠀⠀⠛⢀⣿⠿⢧⡀⣀
⠀⠀⢠⣤⣿⣿⣿⣉⠻⢟⣦⡑⢤⡤⠴⡏⠸⡄⠈⠑⠒⢲⡞⢿⣷⣤⢀⠄⡠⠈⠑⡄
⠀⠀⣸⠀⠀⢀⣈⡽⠿⢦⣉⠛⢦⠼⠒⠉⠳⡹⣄⣀⣀⡟⠀⢸⣿⣿⣼⣄⡐⠛⢆⡱
⠀⠀⣇⡠⠖⣋⣀⣀⣠⣤⠼⠋⠀⠀⠀⠀⠁⠙⣾⣯⢋⣴⣿⣾⣿⣿⣿⣿⣿⣦⠀⠈⢢⣽⡓⠲⢤⣀
⢀⡼⠛⠋⠉⠉⠁⠀⠀⡇⠀⢀⣀⣠⣴⡛⠒⠓⠚⠇⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣷⣤⣿⣿⣇⡈⠁⠈⠓
⣿⠀⠀⠀⠀⠀⠀⡀⠰⢳⣾⠋⠉⠁⠈⢻⣦⣀⢰⣾⣿⣿⣿⣿⣿⣿⠏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⢻⣄⠀⢤⣤⣤⣤⣾⣶⣾⣿⠀⠀⠀⠀⠀⡟⣟⢿⠿⠟⠢⡙⢿⣿⣏⣀⣿⠙⠛⠛⠿⠿⠿⣿⣿⣿⣿⡿
⠀⠉⠛⠶⠶⠶⠛⠋⠁⢸⣿⠀⠀⠀⡠⠔⠁⠘⣷⣕⢄⠀⠈⠛⠀⠙⢦⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠰⢃⢿⢷⡉⠀⠀⢀⣀⠈⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⡄⠀⡄⠀⠀⠀⡞⠈⣿⢀⡆⡎⢸⣷⠷⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢠⡇⣸⠃⣼⠀⠀⡀⠀⣿⡜⡇⠱⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣵⣿⣰⡟⠀⢀⣿⣸⣿⣷⣄⠀⠈⠉⢉⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠿⠷⠶⠿⠿⢿⣿⣿⣿⣿⣶⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣼⣾⣶⡆⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣀⣀⣀⣀⣀⣀⣀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢫⠹⣿⣿⣿⠫⠍⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⢿⣿⣿⣿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉

⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣦⣾⣿⣏⠉⢹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄⡀
⣾⢃⣆⠀⠀⠀⣠⣤⣤⣴⡾⠻⣿⣿⠟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠃⢸⣿⠀⠀⠀⠈⠉⠙⠋⢻⣶⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠙⢻⣧⣤⣀⣀⡀
⠀⣸⡿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠋⠁⠈⠙⢿⣿⣶⣤⣶⣶⣴⣶⣾⣿⣿⠋⠉
⣴⣿⡃⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⢀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀
⡿⠙⠻⠃⠀⠀⠀⠀⠀⣧⣿⣯⣀⠹⢧⠀⢹⣿⣿⣷⣴⢛⣿⣳⡿⣿⣿⣿⡿⠋⠙⣻⣏⢿⡿⠀⠀⠀⠀⢘⣟
⠣⠀⢰⠀⠀⠀⠀⠀⠀⠇⠈⣛⣿⣿⣿⣿⡷⣾⣵⣟⡍⠙⣿⣿⣵⣿⣿⢷⡻⣷⣿⢿⣾⣹⡏⠀⠀⠀⠀⠈⢿⣧
⡗⠀⢺⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡿⣬⣀⡀⢻⣿⣿⣶⣿⣿⣿⣏⣤⣾⡟⢫⣁⣾⣧⠼⣇⠀⠀⠀⠀⠀⠀⢻⣷
⠁⢰⣾⠀⠀⠀⠀⠀⠀⠀⢿⡓⣱⣸⣥⣬⣽⡿⠿⢻⣿⣿⣏⣒⡻⠿⣿⣿⣖⣺⣟⣺⣗⡓⣿⠆⠀⠀⠀⠀⠀⠈⢿
⢀⣿⣿⠀⣤⠀⠀⠀⠺⣷⣿⠟⠛⣽⡿⠉⠴⡶⠃⠘⢛⣿⣿⣿⣥⣤⢀⡉⠙⣿⣿⣿⣿⣷⣿⣶
⢸⣿⡇⣾⣿⠀⠀⠀⠀⠙⠻⢿⡻⣿⣷⣶⡮⢱⣾⣿⣿⣿⣿⡿⠿⠟⠃⢹⡛⢉⣁⣙⣾⠿⠛⠉⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⡀⠀⠉⢻⣿⣯⣿⣿⣿⣿⣿⣟⠃⠀⣀⣶⢶⣶⣾⣻⣿⠏⢡⣶⡄⠀⠀⠀⠀⠀⠀⢸⡄
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠛⠃⡤⠀⢸⡏⠻⣿⣿⣿⡿⠛⣿⣿⢛⣿⣩⣽⠏⣾⣿⡏⠠⠤⢉⠀⠀⠀⠀⠀⠀⠘⣿⣧
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠰⠄⠀⢠⢸⡇⢀⣼⣷⣿⣷⣾⢸⠉⢨⣿⣿⣿⡿⠋⣿⡃⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡠⠖⣈⡣⠿⢿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠃⠀⣿⠿⣿⠇⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡇⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⣿⡏⢿⣿⣄⣿⡟⠁⠀⠀⠀⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡻
⣿⣿⣿⣧⣾⣿⡷⠀⠀⠀⠀⠀⠀⠈⣰⠀⢸⣿⣿⡇⢸⣿⡿⢛⣶⣶⡖⠀⠀⠟⠠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⠀⠸⢿⣿⡇⠸⡇⠀⠈⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⣀⡆⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠈⠊⠹⠀⠚⠆⠈⠒⠲⢾⠤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠸⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠧⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣀⡀⠄⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⡠⠲⢄⢀⡽⡖⢲⣒⠎⠉⠳⠴⢚⣇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⡖⢣⣟⢋⣉⠉⣉⡉⡩⣑⢤⠎⠀⠀⢣⣌⣀⣀⡀⠀⠀⣀⡉⠢⠥⢿⢀⡄⠀⣀⠀⠀⢰⡎⠀⠀⠀⡀⠀⢰⠒⣒⢺⠁⠀⡷
⠀⠀⠀⡇⠀⠉⠘⠤⠃⢳⣛⣓⡚⢸⠀⠀⠀⠑⢯⣛⣩⠃⠀⠀⢯⣛⣩⠇⠈⠩⡤⣺⠍⠃⠀⡜⠁⠀⡴⠒⠿⢳⠃⠠⠄⠀⣹⠋
⠀⠀⠀⡇⠀⠈⢉⣉⣉⣀⣀⡀⠀⢸⠀⠀⠀⢻⡀⠘⠡⣖⠀⠀⠀⠙⠃⠀⠀⠀⠀⡋⣠⠃⠀⠑⡄⠀⡇⢰⣲⣻⣶⡶⠶⡤⠏
⠀⠀⠀⠠⠡⠇⠐⠒⠀⠒⡉⠁⠀⠸⠀⠀⠀⠸⡁⠈⠀⠈⠀⠀⠀⠁⠈⠁⠀⠀⡜⢉⡄⠀⠰⣮⡀⠀⡇⢸⣾⣺⣿⡇⠀⡇⢀⣴⢳⠤⢴⢢⣤
⠀⠀⠀⢰⠨⠅⠨⠗⠒⠒⠀⠀⠁⠀⣇⣀⠀⠀⠱⡀⠀⠀⢠⣤⡄⠀⠀⠀⣠⠔⠉⣏⠙⠒⠚⠁⠁⠀⡇⠘⡟⠚⠛⣃⠀⡇⠸⠘⠻⡤⠤⠍⢸
⠀⠀⠀⢸⠸⡽⠀⠒⠒⠐⠛⠀⠀⠀⢳⡜⡉⢱⠀⠈⠒⠤⠤⠤⠤⠴⠒⣏⠀⠀⢀⣼⠑⢆⡀⠀⠀⠀⠙⠒⠛⠒⠚⣿⠚⠁⢰⡀⠨⠤⣀⡀⢸
⠀⠀⠀⢸⠀⣽⠀⠘⠛⠀⠐⠂⡀⢀⣨⢋⣀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡞⠩⠶⠮⠗⡀⠀⠀⠱⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠨⠁⠐⠂⢸
⠀⠀⠀⠈⠒⠒⠒⠶⠒⠒⠚⠛⠉⠹⠤⠖⠒⠊⠁⠀⠀⠀⠀⠀⠀⢸⢘⠀⢠⠒⢺⣷⠼⡆⠀⠱⡖⠁⣘⢆⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⠀⠀⠀⠀⢀⡰⠇⢸⠀⢸⢰⢉⣍⠓⣳⠀⠀⣱⣾⣩⣂⢻⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣣⢴⡖⣤⡲⠉⠀⠀⣸⠀⢸⡆⠠⣭⠀⢻⠀⠀⠙⡇⠀⠈⠇⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡎⠀⠘⠳⣇⣼⣠⠤⢊⠏⡀⢸⠃⣓⠆⠰⢿⡆⠀⠀⡇⠀⢸⠛⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠢⣤⡦⠊⠀⠀⠠⡋⣀⠈⠙⠤⣍⡦⢵⣪⣇⠠⠀⡇⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡝⢩⠉⠙⡷⡶⠳⢶⡃⠀⣹⣛⠷⣴⠏⠀⠻⣓⡒⠛⢛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠂⢀⠸⢃⣧⠀⠀⢋⠉⠅⡌⠳⡌⠑⢤⡀⠈⠓⠚⠁⠀⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠰⡤⢄⠀⠀⠀⢙⠂⠃⠀⠈⡀⠀⠑⣄⠀⠀⠀⠀⢧⣀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠄⠀⠀⣰⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠙⢄⢸⡀⠉⠀⠂⠀⢠⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠦⣄⠀⠈⢆⡧⠤⢄⡸⡀⠘⠆⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠑⣄⠁⢀⠜⠛⠛⠓⢄⠑⣄⠀⠀⢀⣀⡀⠀⠀⠀⠈⡏⣝⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠶⠶⠖⠚⠒⠚⠦⠤⠤⠴⠚⠉⠀⠑⠲⠤⠤⠤⠤⠤⠴⠊⠓⠲⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠐⠒⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠠⠤⠤⠄⢒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣈⢀⡴⠛⣦⢀⡇⣳⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠈⠻⡀⠸⣿⠉⢆⣛⣡⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣈⡀⠀⠈⠉⠁⠃⠁⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠤⠤⠤⠒⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠄⣀⡀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠲⡙⢄⣀⣈⡠⠞⡤⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠀⠀⠀⠈⠀⠐⠒⠒⠉⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⣀⣀⡀⢀⠀⠀⢀⠀⢀⡌⠑⣄⣀⡀⠀⠀⢰⠔⡀⠝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣁⣒⠂⢭⠛⣷⡄⠈⠣⡄⠀⠀⢠⠞⠀⠀⠀⢸⣥⠂⠉⠑⡀⠀⠀⣀⠀⠀⠀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⢫⠀⠹⡓⡄⢠⣀⠔⠒⠠⣥⠀⠀⠀⠘⠀⠣⡀⠀⢋⣉⣁⡠⢤⡀⠀⠂⢤⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠇⠊⠉⠁⠚⠊⠉⠉⡙⢮⡀⣀⣀⢀⣀⣀⣀⣀⡇⣀⡤⠚⠉⠠⣶⡒⠉⠐⠪⢅⣙⠋⡭⠓⠛⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠘⠤⢀⣠⣤⠦⣀⣀⠐⢦⡁⣵⠈⠉⠒⠠⢄⣀⠀⣠⣅⡒⠀⠥⣉⠉⠀⠔⠐⡫⠔⢂⡞⣤⡤⠤⠤⢀⣘⡟⢲⠂
⠀⠀⠀⠀⠀⢀⡔⠁⠀⢀⣶⣴⣾⣷⡞⡆⠑⠢⣀⠀⠉⠉⠙⠻⣧⠴⠊⣁⠤⠊⠀⠈⠉⠑⠓⢶⢦⣌⣁⡐⢉⡔⣈⣀⡀⣹⣄⠠⠤⠄⠀⠒⢎⠁
⠀⠀⠀⠀⡠⠊⠀⠀⠀⣻⠿⠿⠿⠻⡀⡇⠀⠀⠸⠑⠤⡀⠀⠂⠤⠒⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠬⠉⠁⠠⠤⠤⠤⠟⠂⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠊⠀⠀⠀⠰⡅⠃⠀⠂⠀⠈⣰⠁⠀⠀⠀⣄⠀⠈⠉⠄⠠⠤⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⢰⠁⠀⠀⠀⠀⢀⣈⠀⣀⣉⡉⠁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠙⢯⡉⠉⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣧⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠹⠐⢊⣉⣀⣀⣀⣘⣀⠀⠀⠀⠀⣀⠄⠐⠒⠒⢄⠀⠀⢀⣀⣀⣠⡤⠤⠴⡦⠤
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡶⣾⡟⠉⡭⣍⢩⢹⠛⠉⠀⡀⣀⣠⡄⠊⠘⠰⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠛⠻⣿⣄⠀⠀⠀⠀⠀⠀⢀⣀⣇⠋⠁⠀⢀⠀⣀⢸⠉⠉⠀⠀⠀⠀⠘⠼⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣶⣦⣸⣿⣆⡢⡀⠰⠁⠁⠀⠀⢸⠛⣄⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣤⣼⣿⣿⣉⠻⣟⡦⡙⠦⣤⠤⡾⠉⣇⠈⠑⠒⢶⡶⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⠃⠀⠀⣉⣹⠿⢤⣉⠛⢷⡬⠖⠛⠳⣜⢆⡀⠀⣸⠃⠀⣿⣿⣿⠿⡁⠐⠲⢤⡤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡞⣠⠴⠚⢉⣀⣀⣀⡬⠗⠀⠀⠀⠀⠁⠘⣿⣯⡽⣡⣶⣶⣿⣿⣿⠀⡇⠀⠀⠀⠙⣿⠀⢣⠉⠉⠙⡖⠒⠶⠤⣤⣼⣇⣀
⠀⣴⠿⠓⠛⠉⠉⠉⠉⣿⠀⠀⠀⢀⣠⡖⠒⠒⠚⢿⣸⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇
⣾⠁⠀⠀⠀⠀⠀⢀⠠⢽⣴⡞⠛⠉⠉⠻⣦⣀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠀⣇⡀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡇
⢿⡀⠀⢀⣀⣀⣤⣼⣶⣶⣿⠀⠀⠀⠀⠀⡿⢾⡻⡿⡿⠧⡙⢿⣿⣿⣀⣀⡏⠉⠙⠒⠲⠤⣤⣀⣀⡠⠞
⠈⠛⠦⣤⣬⡭⠿⠛⠋⠉⣿⠀⠀⠀⢀⡠⠃⠈⣷⣕⢆⠀⠈⠲⠋⠉⢯⡉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠁⠀⠀⠀⡟⣽⣷⡝⠂⠀⠀⣀⠀⢳⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⠀⢀⠀⠀⠀⢸⠃⢻⡆⡔⢰⠉⣷⡼⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡾⢠⡏⢠⡆⠀⠁⠀⣸⣇⣇⠘⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣇⣾⢇⣾⠀⠀⣸⡄⣿⣿⣄⠀⠈⠛⠛⢛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠿⠿⠾⣯⣤⣴⣿⣿⣿⣿⣿⣷⣧⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣧⣦⣆⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡠⢀⡀⠀⠀⠀⣬⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠃⢀⣤⠀⠠⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠑⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡏⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣴⣶⣷⣶⣄⣀⣀⣴⣶⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠉⠝⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠸⣿⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠉⠉⠐⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠐⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠢⡀
⠀⠀⠀⠀⢀⡠⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢰⠀⠀⠀⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠤⣀
⠀⠀⠀⢰⠋⠀⠀⣀⠀⠀⠀⡆⠀⣀⣀⠀⢸⠀⣤⠤⡀⡆⠀⠀⡆⢠⠤⡀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠸⠀⠀⠀⡇⠀⣦⠀⢡⠀⣯⣽⠃⢈⡠⡇⠀⡇⠆⠀⠀⠇⡿⠀⠉⣈⢀⡞⠓⣦⢱⠁⣀⡀⠀⡇⠀⠀⢸
⠀⠀⠀⢠⠰⡄⣳⡄⢸⠟⢧⢸⡀⡿⠈⠧⠀⡇⠓⠴⢃⣷⠀⢰⣆⠧⠴⠂⡇⢸⠁⢠⡇⣸⠀⣿⣹⠇⠀⣄⡀⢨
⠀⠀⠀⠀⢂⢫⠈⢷⠸⠀⠈⢹⣗⡁⢀⣀⡼⠷⠶⠖⠛⠘⡇⡟⠘⠓⠶⠶⠿⣎⣙⢋⣐⣿⠼⠋⡏⡘⣸⣅⢉⠂
⠀⠀⠀⠀⠀⠁⠀⢀⣰⠤⠤⠞⠋⠉⠉⠏⠀⠀⠀⠀⠀⢠⠇⢧⠀⠀⠀⠀⠀⠈⠏⠉⠉⠙⠦⠤⢵⣑⢷⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠴⠤⠖⠋⠀⠈⠓⠦⠤⠤⠤⠤⠃⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠄⠠⠔⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠲⡖⠙⢉⠓⢴⠁⠀⠀⠀⠀⠐⠠⠄⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⠂⠀⠀⠆⠀⢀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠈⢙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠂⠀⠀⣀⠀⠀⠁⠀⠐⠀⠶⡰⠀⠘⠀⠁⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠉⠈⠒⠋⢺⠁⠀⠀⠀⣡⠸⡧⣠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠦⣀⣈⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⠀⠓⢆⠠⢄⠀⠀⣉⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢀⠀⡀⠤⠵⡀⢠⠘⠦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢰⠃⠀⠀⠀⠈⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⢃⢴⡆⠀⣠⣤⡂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⣞⡇⠀⠀⠀⡀⡀⠀⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⢉⡀⠀⢀⠩⡅⢠⠁⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢻⠐⣇⡈⠀⠑⠢⣀⣉⣠⠂⡼⠀⠀⠀⠈⢲
⠀⠀⠀⠀⠀⠀⠀⡤⠊⠁⠀⢸⠀⠐⡝⢢⠀⠀⢸⠟⡡⠀⠁⡀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⢘⡜⠀⠀⠇⣠⠃⠀⢠⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⡸⠁⠀⣼⠜⡟⠀⠀⠸⣧⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢘⡆⡰⠁⠀⢀⡏⠀⣷⠀⠀⠀⣷⣳⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡾⠁⠀⠀⢸⠁⠀⢻⠀⠀⠀⡗⢸⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⠜⠀⠀⠀⢀⡏⠀⠀⢸⠀⠀⢠⠇⠀⡆⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢀⠤⢵⠀⢸⠀⠀⠀⠀⡜⠙⠒⠒⠒⠒⠒⠲⡖⠒⠁⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢀⠆⠀⠮⢇⠘⡆⠀⢀⡞⠦⠤⢀⡀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⢦⡘⠒⣺⠁⠀⠀⠀⠀⠉⠁⠒⠠⠄⢧⣄⣀⣀⢀⠃⢡
⠀⠀⠀⠀⠰⠗⠒⠒⠒⠲⠒⠟⠛⠳⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠖⠒⠟⠷⠖⠲⠲⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢖⣶⣞⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠉⠉⠉⠉⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠉⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣄⣀⣀⣀⣀⣤⠴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠊⠉⠑⠄⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⠓⢦⡀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣓⡲⣄⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠙⣄⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠀⠀⠀⠀⠀⠿⠋⠉⠓⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣠⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠁⠈⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠛⠉⢷⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣈⣉⠉⠙⠓⠲⢤⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣴⠂⠈⣧⠀⠀⠀⠀⠀⠀⠸⡍⠛⠻⣿⠋⠉⠛⠛⡿⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣀⣠⠼⢧⡀⠀⠀⠀⠀⠀⠙⢦⣀⠘⡆⠀⣀⡼⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣄⣤⡤⠤⠖⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⠶⠶⠖⠲⠶⠦⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣀⣤⣴⠶⠾⠛⠛⠛⠛⠛⠚⠷⠾⣭⣟⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡷⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢀⣸⣿⣀⠀⠀⡊⠉⠁⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣄⠀⣿⡀
⠀⠀⢀⣠⡶⠟⠁⠀⠈⠉⠻⣅⣀⡀⠀⣸⣄⣀⣀⣀⣀⣰⠚⠒⠒⠒⣆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⣿⠇
⠀⠱⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠉⠙⠶⠤⠤⠤⠛⠙⠉⠙⠛⠋⠉⠶⠶⠄⠐⠋⠉⠈⣻⣦⡀
⠀⠀⠈⠙⠛⠛⠛⠛⠛⠒⠛⠓⠚⠓⠒⠛⠓⠚⠛⠒⠛⠓⠚⠛⠒⠺⠛⠻⠒⠻⠿⠒⠺⠿⠿⠿⠓⠛⠟⠙⠋⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠛⠲⢤⣀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠏⠀⢀⣤⠴⠦⣄⡀⣸⠋⠀⣨⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⡟⠀⢀⣠⠼⢿⠉⠙⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣇⠀⠈⣧⠀⠈⠁⢀⡼⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠦⣄⠀⠹⣄⠀⠈⠛⠲⠖⠋⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢹⠀⠀⠈⠳⢦⣤⠖⢶⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⣨⡤⠴⠖⠛⠉⠁⠀⠈⠉⠛⠓⠶⠤⣌⡀⠀⠀⠀⠀⢸⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠙⠒⢶⢤⣄⣀⡀⠀⠀⣀⣠⡤⣴⠖⠛⠁⠀⠀⠀⠀⢸⠀⢀⡷⠶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⣶⠀⢸⠀⠀⠈⠉⠛⠉⠉⠀⠀⣿⠀⢰⠀⢠⠖⠲⡄⢸⠀⠐⡇⠀⣿⠀⢸⠋⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠈⣧⠀⠙⣾⠀⢘⡇⠀⣿⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣸⠀⠀⠘⣆⠀⠈⠀⠀⠁⠀⠉⠀⠈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠃⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⢀⣨⣧⣠⣤⣤⣤⣤⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⡀⡀⣤⠀⡄⢀⠄⠀⠀⠀⠀⠀⠠⠀⡤⢀⠄⠀⠠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠸⠀⠃⠄⠀⠘⠘⠈⠈⠠⠀⠀⠀⠀⠑⠀⠀⠸⠀⠀⠀⠀⠛⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠲⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠤⠤⢤⣀⣀⣀⠀⠀⢀⣠⠴⠒⠋⠉⠙⡆⠀⢹⡀⠀⠀⠉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢳⡏⠀⠀⠀⠀⠀⠀⢸⡀⠀⣇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⢀⡤⠴⠒⠒⠒⠒⠒⠲⠦⡀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡄⢲⣿⡂⠒⠻⣦⡀⠀⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢹⠀⠈⡇⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⠈⠁⠀⠀⠀⠈⠁⠀⡇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢸⡄⣀⠱⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⠤⠤⠴⠶⠒⠒⠒⠲⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣧⠋⠻⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠄⠀⠠⠤⠤⠤⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠤⠤⠶⠶⠶⢖⣒⣦⠤⠤⣤⣤⣄⣀⣀⣇⣠⢔⠚⠙⠉⠁⣂⠀⢠⣤⣤⣀⡐⠒⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⢀⣀⣀⣈⣉⣉⣉⣉⣋⣐⣾⠟⠉⠙⢷⣤⠴⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠉⠉⠉






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⠗⠀⢀⣀⣦⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠜⠇⢰⣿⠿⣟⣛⣻⣿⣿⣿⣿⡶⢀⣴⣶⣶⣤⣀⣀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣿⣿⡇⠐⠿⠦⠶⠿⣉⣉⣉⣛⢛⠻⠏⠀⠈⢿⣿⣿⣿⣿⣿⣷⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠭⣍⣙⣿⣻⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠸⣶⢾⣶⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣷⠀⠀⢠⠄⠀⠀⣄⠱⠘⢛⠋⠉⠁⠈⠀⠀⡀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠴⠶⣷⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡄⠀⠀⠘⠷⣦⣀⠉⠙⢛⣿⠻⠷⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠙⠛⠛⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄⠀⠀⠄⠀⣀⣄⡴⢺⡆
⠀⠀⠀⠀⢀⢀⣀⣤⣤⣤⣴⣾⣿⣿⣯⣿⡀⠀⠈⠻⣿⣿⢿⣿⣿⣿⡟⠛⣢⣤⣙⠛⠛⠟⠋⠀⣾⣿⠀⢰⣦⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠘⠫⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠻⣿⡟⢿⣿⣷⣾⣿⣿⡻⣷⡄⡀⠀⠀⠘⠏⠀⠀⢻⣿⣿⣿⣿⣻⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠄⠹⠚⠉⠛⢟⠫⠃⠉⠉⠉⠁⠒⠀⠈⣠⣶⡒⠀⢻⣿⣿⣏⡝⡟⠻⣺⠏
⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⣿⣿⣿⣿⣿⠿⣿⠀⠀⢀⠎⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠾⠿⠿⠷⠀⠀⣿⣽⣯⡾⣴⣶⣿⣄
⠀⠀⠀⠀⠀⠀⠰⡌⡻⣛⢧⣻⣿⣿⣿⡶⢺⡆⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠯⠋⣑⡠⡎
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠿⠷⠨⠺⢿⣿⣿⣼⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⢢⣧⠆⠋⢞
⠀⠀⠀⠀⠀⠀⠀⢼⡛⠓⣝⣋⡻⣦⣿⣏⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣬⣇⣌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡩⢏⡛⠪⢷⣯⢔⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⡉⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠅⡢⠁⠉⢹⠊⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣹⠇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⡤⡤⠐⡄⢄⡠⢀⠂⢂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠁⣴⣛⡯⠛⠘⣁⣹⣾⠻⡌⠆⣙⣷⣎⠸⠇⢡⡔⢨⠂⢒
⠀⠀⠀⠀⠀⠀⠀⣰⣴⣾⡶⣾⣿⣏⡱⣤⡛⢭⣿⣿⣿⣿⣿⣿⣿⣟⡮⡷⣯⢴⣞⣦⣤⠣⣀
⠀⠀⠀⠀⠀⢠⣮⣀⡿⣯⡽⣿⣿⣷⣿⣿⣿⣿⣿⣿⠋⠉⠉⠙⣿⣿⣿⣷⣿⣼⣿⣿⣿⣿⠧
⠀⠀⠀⠀⠀⣭⣿⣯⠷⣣⣿⣿⣿⣿⢿⣿⣿⣿⣿⣇⠀⠀⣀⡈⡉⣿⣮⡟⣷⣿⣿⣿⣿⣯⡇⠁
⠀⠀⠀⠀⠀⢸⣿⣻⡽⢛⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣤⣽⣱⠇⣸⢿⣿⣽⠟⢻⣿⣿⣿⣧⡌
⠀⠀⠀⠐⠒⢤⣧⣟⣵⢿⣿⣿⣿⣿⣿⣿⣤⣏⣠⣵⡝⣿⣿⣿⢹⣿⣿⡿⠏⣀⣼⣿⣿⣿⣿
⠀⠀⠀⢂⣤⢸⣿⣿⣧⣭⣽⣿⣿⣟⡿⣻⣿⣾⣿⢿⣿⣞⢿⡏⢼⣿⣿⢏⣼⣿⢺⡿⣿⣿⡟⠓
⠀⠀⠀⠠⠈⡼⣡⢟⡵⢿⣿⢿⣿⡟⣽⣿⣿⣿⣿⣿⣾⣾⣼⡳⢿⡛⢃⢾⣶⣿⢻⣿⣷⣻⣯⢁
⠀⠀⠀⠀⢸⡗⠡⣫⣴⣯⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⢿⣷⡾⣿⣿⣯⣿⣿⣿⣿⡇⠄
⠀⠀⠀⠀⢸⡇⣼⠯⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢜⡌⢻⡼⠿⢿⡟⢻⣿⣿⢿⢃
⠀⠀⠀⠀⢘⣫⣧⣾⠷⢛⡠⠭⠷⣌⠿⡙⣿⣿⣿⣿⣿⣿⣯⣿⢝⢆⠀⣇⣀⣷⣤⣿⣿⠿⡘
⠀⠀⠀⠀⠺⠿⠿⠷⠟⠓⠚⠿⠻⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠀⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⣀⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⢿⡇⠀⠒⠀⠀⠀⠀⢈⣽⣿⣿⣧⡝⠿⢻⣿⡀⣤⣤⣤⣦⠀⢀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢶⣴⡅⠀⠉⢰⣤⣽⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⠺⢿⣟⠉⡆⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢿⣿⣷⣤⠙⠃⠀⠠⢤⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⡿⣿⣿⡕⠂⠹⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠹⣷⡾⡿⢿⣭⣾⠄⣵⣿⣣⠟⠈⠁⠀⠉⠙⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣤⣶⠖⠚⢉⡤⠞⠁⢀⡀⠉⠀⠀⣿⣿⣀⠀⠀⠀⠀⠀⠀⠠⣭⣽⣹⡆⢀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣰⣯⣴⡟⢡⣘⠋⠀⠘⢻⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⡇⠹⡊⢱⠿⣿⡀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡿⢀⠀⢲⡏⠀⠀⠀⡼⠁⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠻⣼⡷⠀⠀⠈⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣯⣿⡿⢿⣟⣟⣿⣧⢰⠀⠠⠂⠀⠀⠀⠀⠀⣨⣤⣤⣄⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣿⣸⣿⠉⠢⣍⠉⢻⡟⣏⣤⡾⠀⠀⠀⡀⠀⣰⣿⣋⣽⠿⣻⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⣡⣾⣣⡾⠉⠀⠀⢀⡼⠁⣴⡟⢡⡴⠶⠛⠁⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⣧⠀⣼⣿⢯⣿⠠⢤⣀⡤⣀⣠⠀⢹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⢨⣾⣿⣿⣿⣿⣿⣿⣿⣦⣬⣍⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⣿⣿⣿⡿⠛⠻⠿⠛⠻⠿⠿⢷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣷⣶⣴⣶⣾⣶⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡭⠽⠷⢤⡈⠻⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣗⣂⠛⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣦⠽⠶⠶⠛⣿⣷⣦⣤⠀⣀⣀⣀⣠⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠛⠿⠟
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⡀⠀⢄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠃⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠪⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⡿⣿⡿⠃⠀⢍⠀⢾⣷⡄⠀⠄⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⣶⣿⣿⡏⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠇⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⢠⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢛⡧⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⠀⠀⠀⠈⢿⠉⠳⠦⠀⠀⣀⡠⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⡷⠚⠳⠄⠀⠀⠈⢁⣀⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣿⢷⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⡛⣛⡀⠀⠀⠀⠹⠌⠛⠈⢁⣠⡄⠀⠀⣄
⠀⠀⠀⠀⠀⠈⣉⣉⣽⠂⠀⢀⣀⣤⡴⠞⠋⠉⠀⠀⠀⠹⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢉⣠⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠛⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠊⠉⠉⠉⠉⠉⠑⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠐⠂⠤⣀⡰⠁⠀⣠⠔⠊⠉⠉⠉⠉⠁⠐⠂⠈⠓⢄
⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⣀⠤⠒⠒⠂⠈⢇⡀⠀⠁⠔⠀⠀⠀⠀⠀⠀⠀⠤⠬⠥⣀⠑⠢⡀
⠀⠀⠀⠀⣠⠔⠁⠀⣀⠔⠋⠀⠀⠀⠈⠉⠀⠀⣉⠢⣄⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠒⠢⠤⠈⠑⢤⡀
⠀⠀⢰⢎⣁⠀⠀⠈⠀⢀⠄⠀⠀⡀⠀⠀⠀⣀⡠⠤⠄⠑⠢⢄⠀⠀⠐⠁⠀⣀⠄⠒⠈⠉⠉⠒⠒⠦⠬⡦
⣀⠤⠺⣀⠀⠉⠢⢄⡀⠀⠀⠀⠈⠀⢀⠔⠊⠀⣀⠤⠔⠒⠒⣒⣛⡦⣀⡠⢪⡆⡠⢔⣊⡉⠉⠉⠑⠒⠶⢷⣄
⠷⣦⣀⠀⠉⠢⣄⠄⡉⢲⣄⠀⠀⠐⠁⠀⡠⠊⠁⠀⡀⠄⢈⣁⢾⣽⡛⠉⢹⣎⣭⣥⣤⣤⣤⣤⣤⠤⠤⠶⠿
⠀⠈⠙⠷⣦⢄⠀⠑⠮⡀⠈⠛⢦⡀⢀⠎⡠⠀⣀⠤⠒⣊⣥⠶⠛⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠑⠫⢶⢄⡈⠑⠤⡀⠉⡹⢀⠔⢋⣠⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⡈⢒⡷⣕⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⢉⡗⡤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣿⡷⡄
⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠁⠂⠀⠀⣼⠟⠛⠛⣿⣿⣾⣿⠁⠈⠢⡀⠀⢀⣀⣴⣿⣿⣿⡳⣭⡉⠔⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⠀⠀⠀⠸⣼⡷⠀⢀⣿⣿⠟⠛⢻⡴⣀⣤⣟⠓⣻⡿⣿⡟⣟⠻⣷⣿⣿⣆⠀⣾⠈⡇⠀⠀⠀⢀⣀⡤⠔⠒⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⢸⣿⣿⣷⣆⣸⣾⡯⠚⡟⠉⢉⠓⢼⣿⣾⣆⡈⠻⣿⣿⣦⣬⣍⣀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡿⣷⡂⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⠚⣋⣀⣈⡑⠦⣿⣿⣿⣵⣷⣼⣿⣿⣿⣟⣷⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡄⠠⣤⣄⣽⣶⣾⣿⣿⣿⣿⢃⣿⡟⣴⡊⢘⣛⣛⢿⣶⣝⣿⣿⣿⣿⣿⣿⣯⣿⡿⠃⡄⠀⣻⠇
⠀⠀⠀⠀⠀⠀⢀⡀⣚⣷⠶⣾⡗⢙⢻⣿⣿⣿⣿⡇⣠⣾⣿⣹⣿⢡⡾⢿⠏⣧⣿⣿⣿⣿⣿⣿⣿⣟⢿⡟⠁⠈⣧⣴⣿⣀
⠀⠀⠀⠀⢠⣴⣿⣛⢫⣜⣤⠃⣠⠚⣉⠿⣻⣽⣶⣣⣿⣿⣿⠘⣿⣜⣧⣾⣿⣯⣼⣿⣿⣿⣻⣿⣿⣿⠿⣴⣾⣿⣽⣿⣿⣯⣤⣤⣄
⠀⠀⠀⠀⢴⣉⣻⣳⠿⣼⣍⡇⠘⣉⣮⣧⣿⣿⣿⣯⣽⣾⣿⣷⣙⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣓⣿⣯⣿⣆⣿⣾⣃
⠀⠀⠀⣾⢢⣅⣗⠻⠀⠻⢦⣙⠢⠻⣷⣿⣯⣿⣿⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⣯⣽⡿⠃⣿⣟⢻⡦
⠀⠀⢀⣻⣆⣤⠄⡛⢶⠶⣤⡀⠘⠷⢼⣿⣾⣿⣿⣦⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣭⣿⡿⣿⡿⣽⢷⣧
⠀⠠⠗⠛⠀⣓⣀⣀⠀⠀⠰⠝⠟⢦⡘⢿⣿⣿⢿⣾⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣟⣷⣿⣭⣿⣿⣿⣆
⠀⠀⠈⡉⣌⢻⣭⣿⣿⡓⢶⣬⣠⢊⡹⣾⣿⣧⣼⣟⡀⠀⠾⡏⠙⠛⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠟⠛⠉⠉⠈⠙⠒⠋
⠀⠀⠀⠙⠄⠈⠻⣿⣿⠿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣳⠦⣤⣤⣀⣘⣛⣛⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠠⠀⠩⠋⠀⠈⢹⣿⣿⣿⣿⣿⣦⣼⣿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢹⣿⣿⡿⢿⣿⣿⢿⣿⣾⣿⡴⠻⣿⣅⠀⠀⠀⣈⣿⠟⢇⣨⣶⣿⣿⣿⣿⣿⣿⣿⣽⣟⣿⣿⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠠⠇⠀⠀⠁⣠⣿⡿⠃⣼⣶⣿⣿⣿⣿⡿⣿⣅⣀⡈⢻⣷⣤⣾⡟⠁⣠⣼⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡿⠫⡞⣠⣿⣿⣿⣿⣿⡏⢀⠻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡆
⠀⠀⠀⠀⠀⠈⠻⢯⡙⠿⣁⣈⣣⣿⣿⣿⡌⣿⠏⣰⣿⣷⠹⣏⣱⣾⣋⣿⣿⣿⣿⣿⢏⣾⣿⣤⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⠀⣈⡄⠱⠈⠀⢿⣿⣿⣗⠰⣿⣿⣷⣿⡦⣻⣈⣧⣽⣿⣭⣽⡻⠟⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⣆⠀⣴⡄⢀⣀⣀⠀⣨⠈⢀⠀⢄⢈⣿⣿⡉⠖⠀⣾⡿⢻⣷⣿⣿⡿⠛⠉⠉⣻⣿⣦⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗
⣿⣿⣿⣿⣿⣿⢻⡷⣜⠢⡷⠄⢸⡈⠌⢿⡆⠤⠀⠙⣿⣽⣿⣿⣭⡿⠻⠿⡊⢏⣤⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣩⣧⣴⣧⣶⣼⣛⣶⣤
⣿⣿⣿⣿⣿⣟⣿⣿⠈⣷⣤⠀⡌⢻⢈⣿⣿⣿⣷⣤⠉⢻⣿⣿⣬⡉⡅⠀⠑⣼⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠏⠁
⢿⣿⣿⣿⣿⣿⣿⡀⠀⠈⢿⣾⣿⡄⠒⠩⢟⣿⣿⡻⠛⢻⡿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⠟
⠸⣿⣿⣿⣿⣿⣏⠑⢄⠀⠀⠃⠙⣿⣟⢦⡈⣎⡉⠈⠖⢟⡉⠲⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠇
⠀⢿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠈⢷⣮⡁⢾⣿⡆⠀⠚⢷⣠⡌⣹⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣁⣤⣈⠉⣿⣿⡿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢌⠻⢷⣆⣻⣇⠀⢴⠆⠙⢳⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⢠⣿⣿⣿⡇⠀⠀⢀
⠀⠘⣿⣿⣿⣿⣿⣿⣍⠃⠀⢀⠀⠠⠀⠀⢫⡸⣿⣿⣦⣄⣀⢀⠄⠘⡀⠋⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⠇⣠⠆⣠⡇
⠀⠀⢿⣿⣿⣿⣿⣿⣟⠁⠂⠀⠀⠀⠀⠡⠠⣶⢀⣄⣀⣈⠉⣀⢀⣶⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠏⢸⣿⣿⣿⣷⣧⣤⣿⠇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠐⠒⣺⣿⣍⣯⣿⢿⡤⠾⣟⣮⢽⣿⣿⣿⡟⢿⠛⠋⠉⠀⠀⠀⠀⠀⣾⣿⣿⠛⣛⣛⡟⠁
⠀⠀⠀⣿⣿⡏⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠀⠈⢉⠀⢳⣿⣼⣷⣤⣮⣿⣻⣿⣿⣿⠧⣼⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣷⣿⣿⣿⣿⡧⡀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠿⣿⣶⣿⣿⠿⣧⢻⣤⣤⡀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⠿⠁
⠀⠀⠀⠘⣿⣿⣿⢿⣿⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢫⢛⣾⣿⣿⣿⡿⢾⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⢴⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠰⠆⠀⠐⠆⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠀⠀⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠘⠊⡁⠀⠀⠠⠂⠀⠚⠁⣒⠆⠀⠀⠀⣀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⣷⠀⠀⣱⡀⣠⠇⠀⠀⠓⠀⠀⠀⠀⠀⠀⢹⡇⢈⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢻⣚⡽⢻⡃⠂⠉⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠉⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢻⣥⡘⣯⡀⠀⠈⠁⠀⢸⡄⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠏⠀⠀⡀⠀⠀⢸⡇⠠⠚⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡗⣶⡍⠁⠀⠐⠱⠖⠃⠀⠀⢠⡀⠀⠀⠰⠄⠀⢀⣀⣀⣨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠇⠀⠀⠤⣄⣠⣴⣾⣶⡄⣆⠀⠀⠀⢀⡘⠁⣀⠈⠉⠲⡀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⣴⠖⠀⣿⣿⣭⣿⡙⢿⡄⠀⢰⡿⣶⣿⡟⣻⣶⡀⢵⠀⠁⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠀⠛⠁⠈⠻⠿⠿⠿⠃⣸⠇⠀⣿⠅⢿⣿⣿⣿⡇⠙⣾⠄⠴⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣠⣤⡤⠀⠀⠀⡀⢀⠂⠉⠀⠀⠉⠙⠛⠀⠘⢿⡇⢴⠄⠀⠀⠀⠀⠀⢀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠛⠛⡀⠀⠀⣀⣀⡈⠀⠀⠀⠀⢠⣴⣶⣆⣀⡀⠘⠀⠀⠀⠀⢀⡄⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠚⠉⠉⠙⠛⠿⠿⠿⢶⣤⣤⡀⢸⣿⡋⠉⠁⠀⠀⠀⠀⠀⠲⠏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠓⠁⠀⠀⠀⠀⠀⠀⠊⠀⢨⠀⠘⠛⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠻⠆⠀⠈⠃⠀⠀⣀⣀⣀⠀⠀⠀⢁⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢂⣀⣀⡀⢀⣨⣭⠭⠉⠙⠛⠊⠉⠿⣦⡤⠀⠀⠀⠀⡤⠀⠀⠀⠌⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢠⣿⣿⣿⣿⣿⣿⡃⢀⣸⠀⢀⣄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣄⠀⡘⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⣀⠀⠀⠈⠻⣿⣿⠛⠋⡤⠀⠀⠀⠁⡄⠀⠀⠀⠀⢀⠸⠀⠀⠀⠀⠀⠀⢻⣿⣿⣮⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⠄⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠔⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⢀⡤⡌⠘
⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠤⡆⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⡀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠈⡉⠁⠀⠈
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠢⡀⠀⠀⢸⡤⣐⠀⣁⣤⣴⣿⣤⣤

⠀⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⢀⠸⠿⢀⡀⣼⡿⠀⣀⠿⠿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠆⠀⡀⠀⡁
⠀⢀⣬⣴⣤⣴⣥⠟⠀⠀⠉⠀⠀⠈⠰⠆⠀⠀⠶⠴⠠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⣀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢾⣿⡷
⠀⠀⠀⠀⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠈⠁⣉⡈⣨⠀⠀⢨⠀⠀⠀⠀⠀⠀⠒⠒⠂⠉⠁⠀⣀⢉⣄⣀⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠛⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠊⠑⠙⠋
⠀⠀⠰⣳⡶⣆⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⠀⠀⢠⡶⣶⣶⣲⡆⢴⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢩⣬⢀⣄⠀⠈⠉⠉⠛⠀⠈⠉⠉⠉⠋⠀⢀⡠⢐⠂⢀⡀
⠀⠀⠰⠴⢀⠴⠃⠀⠀⠀⠘⠄⠠⠄⠀⠀⠘⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠈⠈
⠀⠀⢀⣀⣀⣀⠀⠀⠀⣴⣄⣠⣤⣤⣤⠀⠀⠀⠠⠀⣤⠀⠀⢴⡄⡄⣤⣤⣤⠀⠠⠄⡤⠀⠀⠀⠤⢠⣀⠀⠄
⠀⠀⢸⣿⡿⡞⢃⠀⠀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠈⠉
⠀⠀⠘⠂⠆⠄⠞⠓⠀⠻⠇⠉⠉⠉⠉⠀⠀⠀⠉⠀⠈⠉⠁⠀⠰⠆⠁⠀⠁⠀⢀⠁⠀⡀⠀⠁⠀⡀⣈
⠀⠀⢠⡄⢤⢰⡆⠀⠀⢠⣤⣤⢰⠆⠀⠀⢠⠀⠀⠀⠰⠶⠦⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠰⠶⠂⠊⠀⠀⠀⠠⠿⠀⠊⠀⠀⠀⢠⠴⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠇⠀⠀⢰⣶⠀⠀⠀⠀⣶⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⣀⠿⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢈⣇⠈⠀⠀⠀⣾⣿⠢⣌⠂⠁⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢹⣏⠀⠀⠳⣤⡙⣇⠀⠉⠳⣤⡾⠂⢸⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⣳⠄⠀⠀⠐⡿⠏⠢⠀⠀⠀⠘⠈⠓⢄⣔⠀⠀⠀⠘⠳⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⠰⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣧⠀⠀⠀⠀⠀⢠⠆⢀⠀⣀⠀⠀⠀⠀⠈⠁⠀⠀⠹⢦⡀⠀⠀⠀⠘⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⣠⠎⠀⠈⠓⠙⠑⠈⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣶⣾⣿⣿⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣷⡀⠀⢀⣴⣾⣿⣿⣶⠀⠠⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⠀⢠⣿⠄⣀⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡷⠄⠀⢸⣿⣿⣿⣿⣿⡆⢀⣿⠄⠀⢠⣤⣴⣶⣤⠀⠀⣿⡆⢐⣿⣧⣹⣾⣿⣿⣿⣿⣿⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣷⠆⣀⠘⣿⣿⣿⣿⡿⠃⢈⣿⡗⠀⢈⣿⣿⣿⣿⡇⢴⣿⣥⣻⣿⡍⠹⣽⣿⢿⡿⢿⣿⣶⣶⣾⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣘⣻⣿⣿⣿⣿⣯⢙⡟⠋⡁⢸⣯⣿⣷⡀⠈⣿⣿⣿⣿⡿⣼⣿⣿⣼⣿⣿⣀⣠⣤⣼⣷⠾⡿⠿⢿⣇
⠀⠀⠀⠀⠀⣘⣿⣛⣛⣛⣛⣹⣿⡻⣟⣛⢷⠦⢿⣿⣿⣿⣿⣗⠀⠋⠙⢻⠋⣉⣿⣿⣿⣿⡿⠯⠕⣒⣒⣈⣉⣉⣉⣉⣫⣄
⠀⠀⠀⠀⠀⣴⣯⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣛⣲⣮⣭⣟⡿⣧⡀⢰⣀⣼⣿⣯⣼⣟⣋⣳⣬⣭⣥⣀⣀⣀⣀⣒⣂⣂⣸⣷
⠀⠀⠀⠀⠀⣧⠄⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣉⡉⠙⠛⠻⠿⢷⣯⣿⣭⣿⠿⠟⠛⠋⢉⣉⣁⣀⣀⣀⣀⡀⠀⠀⠀⠀⠠⢤⡇
⠀⠀⠀⠀⠀⣷⣶⢶⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⢶⣞⣛⣛⡶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠿⢭⣥⣒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣶⣦⣤⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠈⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠐⠆⠈⠀⠙⠋⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⠁⠀⠰⡆⣰⠀⠶⠀⠀⠈⠉⢉⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠈⠓⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠦⠴⠂⠀⣀⣴⡙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠑⡖⠤⠤⠔⡞⠁⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠘⠦⠤⠞⠁⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡼⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠼⢷⠃⠠⡀⢳⣦⠤⠤⠤⠤⠤⠤⣤⣤⣾⣥⣀⠀⠀⠀⢠⣟⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⣕⣰⡆⢘⠦⣐⣻⣹⠟⠙⠒⠲⠴⣶⡾⡉⠀⠉⠉⢻⣇⣤⠞⠁⠀⠀⠀⢀⡜⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠐⠤⣫⠔⠡⠠⠄⠀⣙⡒⠤⢀⠎⠀⠸⣴⣵⣵⣣⠴⢊⣵⡟⠀⠀⠀⢀⡴⣫⠞⠁
⠀⠀⠀⠀⠀⢀⣞⣁⣀⣀⣀⣀⣀⣯⠭⠭⠿⠿⠷⢶⣾⣧⣤⣤⣶⠶⠶⠦⠤⣅⡴⡫⠊⠀⠀⢀⡴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⢤⣤⣬⣭⣍⣉⣉⣛⣓⣒⣒⣚⠛⠒⠒⠲⠿⠿⠶⠾⠯⠤⠤⠤⢤⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠓⠒⠒⠒⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⠉⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠰⣗⣦⢀⣶⠦⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠠⡄⡀⠉⢀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⢔⠂⡡⡨⡺⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣾⡏⢿⣦⣤⢤⣿⣼⣼⠆⡷
⠀⠀⠀⠀⠀⠀⠀⢀⡔⣾⠿⠿⠟⠋⡝⢹⡇⡇⠈⢢⡀⢾⣄⢠⠀⢰⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡁⢸⡀⠀⠀⠰⠁⠀⠀⢷⠰⣻⣟⠈⣿⣄⠀⡨⣦⠀⠈⢙⡗⡆
⠀⠀⠀⠀⠀⢀⡾⠁⢟⡈⡇⢀⠒⠐⠒⠆⠀⢸⣴⣿⣿⣷⣿⣿⠈⣰⠁⠑⡀⠜⣠⡇
⠀⠀⠀⠀⣰⡛⠡⢄⢀⣹⣧⠸⣧⠀⠀⠀⠀⠈⣿⣿⣿⡿⣿⣿⡀⠘⣧⠀⠈⢺⡹⢇
⠀⠀⠀⢠⣿⠳⢄⣈⣻⢟⣿⠀⢸⣆⠀⠀⠀⠀⣿⣿⡿⠀⠹⣿⣷⡀⠈⠀⠀⠈⣧⡸⡄
⠀⠀⣠⣿⣷⣾⡓⠶⢾⣿⣿⠀⣾⠛⢆⠀⠀⠀⢸⣿⠇⠀⠀⠹⣿⣿⣦⠀⠀⠀⢸⡹⢾⡄
⠀⠰⣿⡛⠀⠀⠉⠳⣿⣿⣧⣤⡟⠀⠘⣆⠀⠀⠘⣿⠀⠀⠀⢀⣾⣿⢷⡁⠀⠀⠀⢇⣿⡽
⠶⠴⣟⣁⡴⠉⡀⢔⣴⠋⢤⠹⢽⢳⣶⣾⣶⣶⣶⣿⣤⣤⣶⣾⣿⠿⢮⣿⣶⣤⣀⣠⡿⠷⠶⠶⠴⠶⠶
⠀⠀⠘⠻⠷⢶⣧⡫⣿⣬⡲⢌⣒⣹⡟⠛⠋⠉⠉⠉⠁⠀⠀⠀⠉⠓⠦⣄⡉⠉⠉⠁
⠀⠀⠀⠀⣤⡒⠋⠁⠀⠑⠛⠛⠉⠈⠁⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⢀⣠⣄⣠⠤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⢀⣀⣀⠤⠴⠚⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⣛⡛⣛⣛⣛⣛⣛⣛⣛⣛⡛⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡇⢸⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠇⠸⠿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠙⠛⠻⠿⣿⡿⠿⠛⠋⠉⠁
⠀⠀⢀⣾⣿⣿⣿⡿⢋⣥⣴⣶⡆⢰⣶⣶⣬⡙⠿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⠀⢀⣾⣿⣿⡿⢋⣶⣿⡍⢩⣉⣡⣌⣭⡍⢉⣿⣷⡌⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣼⡇
⠀⣾⣿⣿⡿⢡⣿⣿⣿⡿⢂⠻⣿⣿⡟⠁⢿⣿⣿⣿⡎⢿⣿⣿⣿
⢀⣿⣿⣿⡇⣾⣿⣿⣿⢡⣿⣧⡙⠏⣴⣿⡆⣿⠈⠻⠿⠸⠿⠿⠟⠃
⠘⣿⣿⣿⡇⣿⣿⣿⣿⠸⣿⣿⣷⣾⣿⣿⠇⣿⢸⣿⣷⣶⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⣤⣾⣿⣿⣿⣿⣿⣷⣶⣾⣿
⠀⣿⣿⣿⣷⠸⣿⣿⣿⣧⣙⠻⠿⠿⠿⢋⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⣿⣿⣿⣧⡙⢿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⢿⣿⣿⣿⣦⣙⠻⢿⣿⣿⣿⣿⡿⠿⢋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣶⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⡄⠀⣤⠤⠀⣤⡄⢠⠤⡄⣤⡀⡄⣤⢠⡄⣤⢀⡤⠤⠀⢀⡤⠤⢀⡤⣄⠀⣤⡀⢠
⠀⠀⠀⠀⠀⠀⣧⣀⣿⣋⢰⠷⣧⢸⠛⡇⣿⢻⡇⣿⢸⠹⣿⠸⣄⣷⠀⠻⣄⣷⠻⣄⡼⢰⠷⣧⢸⣄
⣿⣿⠟⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⡤⠀⣴⣾⠿⠛⠛⣇
⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠀⠀⠀⠀⢀⣤⢾⡿⠟⠉⠁⠀⠀⡼⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠄⢀
⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⣠⢶⡷⠖⠉⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄
⠀⠈⠙⢿⣶⣄⡀⠀⣀⠀⢴⣆⠔⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⡿⠋
⠀⠀⠀⠀⠈⠛⢿⣦⣈⠁⠀⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣀⠁⡄⣠⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣉⣉⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣼⣿⣉⢰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣽⣒⢦⣄⡀⠀⠁⠂⠠⣀⣠⣴⠀⣠⡾⠋⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢵⣶⣤⣄⡀⠀⣧⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠛⣻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⠀⣀⡤⠤⣤⣤⣤⠶⠒⠛⠛⠋⠛⠛⠓⠲⢦⣄⡀⠀⠀⠀⠀⢠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣥⣶⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠽⠍⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⢰⣿⣷⢠⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠈⣛⠋⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠾⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣵⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢛⡛⣷⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢫⡅⠐⠊⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣇⠘⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠀⠿⠛⠋⠓⠀⢈⢃⣼⡿⣛⣭⣭⣭⡝⣿⣿⣿⣶⣆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⣶⣦⠀⠈⣾⣿⣿⣻⣯⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⣿⣿⢧⡛⠛⠛⠛⠛⠛⠛⠛⠛⢛⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠂⠀⠀⠀⠀⠘⢿⣿⣿⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠻⢇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⢠⣶⣶⣶⣶⣶⣶⣦⡀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⢿⣿⣷⣅⠀⣠⡀⠀⠀⠰⢀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⠟⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠼⠿⠿⠿⠿⠾⠶⠶⠶⠾⠿⠿⠇⠈⠉⠉⠉⠉⠉⠉⠉⠀⠼⠌⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣟⣻⣽⣭⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠉⠙⠟⠋⠋⠉⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⣀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇⠰⠌⢷⠀⢘⡉⠓⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡇⠀⠠⠆⠠⠀⠀⢠⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠐⣒⠂⠈⠔⡩⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣟⠻⢧⣀⠀⠀⣨⡌⣜⣴⡌⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⠿⢟⢿⡿⢸⠀⠪⡉⠉⢱⡏⠓⢉⠋⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⢋⠁⠈⠁⠀⢀⠆⠘⠃⢸⡀⢀⣼⣦⡉⢿⡆⠃⠀⠘⡉⠛⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢦⡀⠸⡀⠀⠀⠀⡎⠀⠀⠀⠈⡇⢊⣽⣯⠁⢸⣿⡆⢀⠄⡘⢂⠀⠀⢀⠈⣀
⠀⠀⠀⠀⠀⠀⢀⣾⢇⠨⣅⡀⡇⢀⡜⠒⠚⠓⠂⠀⠀⢧⣼⣿⣿⣇⣿⣿⣓⠀⣰⠁⠀⠣⡀⠂⣠⡞
⠀⠀⠀⠀⠀⣴⠛⠣⣄⡀⠀⢻⣶⡀⢿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣇⠀⠈⣆⠀⠀⠙⢞⢫⢇
⠀⠀⠀⠀⢠⡏⠠⡀⠀⠛⢯⠻⣾⠃⠈⢷⡀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠈⢿⣿⣆⠀⠈⠀⠂⠀⠘⣇⠀⡀
⠀⠀⠀⢠⠾⣃⣤⣬⣵⣒⣦⣾⣿⠀⠀⣿⢳⡀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠈⢿⣿⣦⡄⠀⠀⠀⠀⢨⠳⡌⡄
⠀⠀⣰⣵⣯⠚⠲⢦⣌⢉⣼⣿⣿⠀⢸⡇⠀⢳⡀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠈⢿⡿⣟⢦⠀⠀⠀⠈⡇⢀⡟⡄
⠀⠀⠛⢋⠊⠀⣀⠄⠈⠛⢿⣿⠷⣢⣿⡀⠀⠀⢳⠀⠀⠀⠘⣿⠀⠀⠀⠀⢠⣿⡷⠛⣧⡀⠀⠀⠀⠘⠼⣱⠃
⠶⠶⢷⣀⢠⠞⠁⡠⠔⣪⣾⠁⢦⠈⠲⡎⢶⣶⣶⣷⣶⣶⣶⣿⣶⣶⣶⣶⣿⡿⠿⠷⣼⣟⣳⣤⠀⣀⡼⠷⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠈⠛⠿⢶⣦⣵⡼⣙⣏⣘⡲⠀⠐⠤⣻⡿⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠒⠦⣄⡀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⡀⠤⠝⠛⠀⠉⠮⠥⠄⠐⠉⠀⠁⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣽⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⣄⣠⠔⠰⠪⠛⠒⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⠀⠀⣀⣠⠤⠤⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⠉

⠀⠀⡟⣧⣷⣿⣯⢇⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠐⣒⣒⣉⡩⠄⠚⠉⠉⠀⠉⢻⠙⠓⠶
⠀⢰⡇⢸⡏⣿⣿⠘⣶⣿⡆⠀⢸⢋⠤⣴⡖⢾⣉⠉⠉⠉⠀⠀⠉⠛⠭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡇⠍⣷⢿⣿⣴⣿⣿⣷⢀⣼⣄⣨⣿⠧⠀⠈⠁⠀⠀⠀⠀⠀⠶⠖⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰
⠀⠘⠃⠀⣿⣿⡿⠻⣿⣿⡿⠁⡟⢻⠆⣈⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⡀⢀⠀⠀⡶⠄⠀⠀⢸⠀⠂⠀⠀⠀⣰⠃
⠀⢰⠀⠀⢸⣿⡇⢀⣿⣿⠇⢀⠃⠘⣻⡝⢯⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣾⡷⢰⣿⠿⡇⠀⠨⢠⡆⠀⠀⠰⠃
⠀⢸⠀⠀⢿⣿⣷⣥⡈⠉⡀⢸⠀⠀⣹⡧⠻⠇⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣯⡀⠛⠀⠁⠁⠀⢰⠘⢷⠶⣤⣆⢀⡼
⠀⣼⣶⣶⠜⢻⣿⣿⣿⣯⣦⣬⡀⠀⣾⠇⣤⣠⣿⣼⣭⣷⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⢤⣆⣀⡈⢈⡿⠛
⠀⢻⣿⣿⠄⡌⢹⣿⣿⣉⢈⡉⠁⠀⡿⣺⡁⢸⣿⣿⣏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡟⣿⣿⣿⡇⠈⡘⣼⢫⢛⡿⠃
⠀⠀⣠⡁⠀⠀⢹⡿⠛⠻⣦⢹⣶⡀⣲⣿⣿⣾⣿⣿⣿⣀⣿⣿⢿⣿⣿⣿⣿⣿⠿⣿⡷⠿⠉⠀⠀⠠⣼⢃⣯⣾⠧
⠀⠀⣿⣿⡦⠀⠀⣧⠀⠀⢿⣿⡿⠉⠀⠒⠚⠛⣻⣿⣿⣥⣿⠀⠰⠛⠛⠉⠉⠀⠈⠁⠀⠀⠀⢀⣰⣾⢃⣿⣽⡏⠀⠀⢀
⠀⢸⣿⣿⡇⠀⠸⣿⠇⠀⠀⠛⠃⠀⠀⠀⠈⠉⣉⡙⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢿⣿⣿⣿⣿⣦⡼⣿⡿⠁⠀⠀⡏
⠀⠈⠉⣿⣷⣶⣾⣧⣤⠀⠀⠀⠟⠻⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠾⣿⣿⣯⣿⣿⣿⣿⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠻⣿⣿⣿⣿⣿⡝⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⡄⢠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⡶⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠛⠛⠛⠛⠉⢉⣉⣙⠻⢿⣿⣿⡿⠟⣋⣉⡉⠉⠛⠛⠛⠛⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣶⣶⡶⠶⠟⠛⠉⠉⠉⠉⠀⠙⠋⠀⠈⠉⠉⠉⠛⠻⠶⢶⣶⣶⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠁⣿⡹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁⣸⣿⡇⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡯⢻⠿⠀⠉⠙⠉⢡⣤⣽⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⢉⡶⠄⣀⣀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠞⢡⠃⠀⠀⠀⠀⠀⢠⠋⠋⠛⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠋⠁⠀⠀⡄⠀⠘⠀⢀⡀⠀⢀⣤⠾⠐⠋⠀⢸⣿⣷⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⣇⠀⠙⣶⣤⣽⣿⡀⠀⠀⠘⣿⠁⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⢲⠄⠀⢸⠆⢘⣿⣢⡿⠛⠛⢆⢀⠄⠘⣇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠀⠀⠀⠰⠋⠀⠀⢸⠈⢿⣿⠏⠁⠀⠀⠨⠎⠀⣰⠈⢧⠀⢸
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢸⡇⠘⠏⠀⠀⠀⠀⣸⣎⡴⠁⠀⠀⠃⠘⡄
⠀⠀⠀⠀⠀⢠⡟⠂⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠀⠀⠀⢀⣿⠑⠡⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠉⢨⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⣠⡄⢸⣿⣷⡀⠄⠄⠀⠀⠀⠘⡷⡀
⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⣾⣇⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⣼⣿⣧⢸⠙⣿⡟⠀⠀⠀⠀⠀⠀⠷⠸⡀
⠀⠀⢀⡎⠀⠒⠒⠲⠶⠜⢹⣿⣿⠀⠀⠀⠀⠘⣧⣼⣿⠃⢸⣿⣿⣿⣆⣀⣽⠿⣄⠀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⢤⠤⠖⠈⠻⢿⠐⠛⠛⠉⠉⠉⠁⠀⠀⠈⠳⣀⣀⣀⣀⣠⠏
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡼⠁⠀⠀⠀⠁⢀⡑⡀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⢀⣀⣀⣀⢀⢤⡤⢤
⠀⠀⠀⠀⠘⠒⠒⠶⠤⣴⣤⣺⡷⠿⠶⠄⣠⣈⡂⠄⣐⣅⡀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠢⢤⣀⠈⠉⠉⠉⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡤⢄⠀⠀⠀⢀⣠⠤⠖⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢙⠓⠲⠤⣀⡀⠁⠐⠀⠠⠤⢀⣀⠀⣩⡴⠛⠐⠒⠉⠁
⠀⠀⠀⠀⠀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢐⠤⠈⢉⠃⠠⢀⡀⢀⡀⠀⣀⣬
⠀⠀⠀⠀⠀⠛⠭⢤⣀⣀⡁⠂⠤⢄⣀⠀⠀⠀⠀⠀⠄⠂⣀⣞⣃⠀⠈⠑⠆⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠃⠂⠬⢙⠳⠦⣂⣦⡼⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠘⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣏⡀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠘⠻⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠁⠀⠀⠀⠠⣤⡖⢀⡀⠀⢀⡄⣴⣦⡀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⣀⠀⠁⠛⠘⠃⠀⠨⠄⠻⠛⠁⣀⣤⣶⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⡶⠦⠤⠤⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⡿⠟⢛⡉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⢹⣿⡟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣬⣿⡇⠀⠈⢡⡀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⣾⣿⣷⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⢀⠄⠀⠀⠀⠘⠀⠀⠀⠀⠤⢀⣼⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠿⠿⠻⣿⣿⣿⡶⣄⣀⠀⠀⣤⠀⠀⢀⣀⡴⢻⣿⣿⠿⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠉⠛⣇⣀⣉⣙⣗⠒⣲⣏⣉⣀⣀⡼⠋⠀⠀⠀⢀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠠⠤⠤⢄⣀⠀⢴⣏⠉⠀⠀⠉⢻⠉⠀⠀⠀⠉⠀⢀⡠⠖⠋⠉⠀⠀⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠀⠀⠀⠀⢀⡼⠓⢾⣿⣄⣀⣀⣀⣼⣄⣤⣤⡤⠶⣏⠉⠳⡄⠀⠀⠀⠀⠀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⢠⠏⢀⡴⠁⠀⢉⣧⠠⠴⠶⢀⠰⣏⠁⠀⠈⣧⠀⢹⡄⠀⣀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠤⡿⡷⠶⠶⢾⡇⣀⣉⣉⣉⣁⣈⣉⡛⠋⢹⡛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠤⠤⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠇⢀⡄⠀⣸⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡛⠛⣿⣶⣤⣌⠀⠀⠀⠀⠀⠙⠺⢟⡉⣿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠿⠻⣿⣿⣦⣛⢿⣷⣾⣷⡀⠀⠀⠀⠀⠀⠀⠙⢧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣾⣦⢀⡛⣹⡿⣷⣿⣾⣿⡗⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣿⡟⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⠙⡗⣶⠉⠙⠁⢀⣂⡲⠁⠀⠀⠀⠀⣰⣆⣀⣀⣐⠃
⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠱⠭⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄⠠⠀⣘⣴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠙⢗⢦⡀⠀⠀⠀⠀⠀⠀⠤⠬⠁⠀⢀⠀⢠⡶⠒⠋⢥⡴⡧⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠓⢦⡀⠈⢳⡙⢦⡀⠑⠤⢀⠀⠀⠀⠀⢀⡤⠤⠱⢾⠛⠻⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠒⢽⡒⠦⡽⣄⠈⣰⠀⠀⠐⠉⠀⠁⠰⢈⠙
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠐⠀⠉⠁⠒⠤⣙⢶⡇⠘⣶⣬⠄⠄
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡹⣆⠘⣏⣧⠆⠒⠒⠙⢳⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠂⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⢇⠸⣏⠀⠀⠀⠀⠸⡆⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀
⠀⠀⢀⡾⠁⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⢻⠏⡀⢿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠐⠀⠀⠉⠳⣄
⠀⠀⡾⠁⠀⠠⣀⠘⢆⠀⠀⠀⠈⠉⠐⠲⠦⣭⣿⣤⠱⣸⡀⠀⠀⠀⠀⢧⢀⡤⠐⠠⠤⡀⠠⢊⡄⢤⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠴
⢀⡞⠁⠀⠲⣄⡈⠑⠬⣣⣤⠤⠤⠖⠒⠚⠳⠶⢀⣈⣷⣿⡻⠦⠀⠀⠀⢸⣀⣒⣓⡛⠋⠐⠄⠀⠀⠀⡡⠊⠀⠀⠀⠀⠀⢀⣴⠁⠀⠀⠠
⠋⠀⠀⠀⠀⠀⠙⠲⠦⣬⣙⣷⡄⠀⠀⠈⠉⠉⠁⠀⠉⢿⣟⠒⠀⠀⠀⡸⠀⢀⡀⢤⡐⠉⠀⠀⠠⠊⠠⠀⠀⠀⠀⠰⡠⢻⡤⠤⠄⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣻⢀⣀⡠⠴⠧⠤⠬⠤⢤⣅⠀⠀⠀⠆⠀⠀⠀⠀⡠⡪⠚⢧⣧⡄⠀⣀⡀⡀
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠼⣃⢶⢷⠀⢀⡆⠀⠀⠀⠀⢳⡑⣄⠜⢶⣄⠀⡠⣊⡞⣱⠆⡈⠂⢠⣄⣀⡤⣀
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⢦⡀⠀⣠⠶⠋⠡⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⣇⠸⠀⠀⠈⠢⠚⠁⡙⢇⣄⢀⡄⣢⢍⣍⡑⢄
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠙⢦⣀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⣀⠇⠘⠀⢀⠀⠀⠠⡀⠈⠲⣄⠘⠠⠄⣈⡛⠃⣸
⠀⣹⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠙⢷⡶⢦⠒⠒⠒⠒⠚⠋⢉⠉⠉⠉⠀⠁⠀⠀⠀⠀⢀⠀⠑⠄⠈⠃
⢠⠙⠦⣂⡀⠈⠁⠀⡈⡉⢉⣉⠁⣀⣐⡒⠒⠢⠤⢤⣄⠙⢦⠑⢄⠀⠐⢆⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠈⠣⣄
⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠳⠌⠃⠈⢣⡈⠢⡀
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆

⠀⠀⠀⠀⠀⢠⡴⣶⠆
⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⣤⣤⠄⠠⠀⠥
⠀⠀⠀⢀⣤⣠⡇⠀⠀⢸⠩⠉⠀⠀⣀⠤
⠀⠀⠠⣼⠒⢺⡇⠀⠀⢸⠀⣀⠀⠸⡇⠅⠀⠀⡀⢀⠀⠀⠄
⠀⠀⠐⣿⠂⢸⡇⠀⠀⢸⡒⠢⠀⠀⡇⠀⠀⠀⠁⠀⠀⡀⠸
⠀⠀⠀⣯⠀⢸⠇⠀⠀⢸⡀⠀⠀⠀⣿⠀⣄⠀⠈⠀⠀⠁⠀⠃
⠀⠀⠀⣿⠀⣾⠃⠀⠀⢸⡇⠀⡀⠁⠉⠑⠃⠐⠀⠀⠀⠀⣀⠜⠐⠒⠠⢄⣀⡀
⠀⠀⠀⡇⠀⢸⡇⢀⠀⢸⡗⠒⠀⠀⣥⣤⡄⠀⠤⠤⠀⡠⠁⡀⠀⠀⠀⠀⠀⠀⠉⠐⢒⣀⣀
⠀⠀⠀⣇⠀⣸⠂⢸⠀⢸⠋⠀⠆⠘⠛⠋⠀⠀⠀⠀⡔⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣺⢻⠇⠀⠀⣀⣀⣬⡥⡴⠂
⠀⠀⠲⡇⠀⣿⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡿⡞⠖⠚⠛⠛⠉⠉⠀⣇
⠀⢄⢠⡇⠀⣿⣊⠟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠎⠸⠻⠉⠀⠀⠀⠀⠀⠀⠀⡐⢝⢘⣶⣌⣠⡄⢀⣀⣠⣴⠤⠌⠁
⢬⣶⣺⣇⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢤⠀⠀⠆⠀⠀⠀⠀⠀⣈⠈⢀⠼⢿⠒⠛⠻⠋⠉⠉⣹⣀⡀
⠀⣬⣷⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⣌⡀⠬⢤⣤⣶⣲⣾⣯⣽⣶⡶
⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠒⡀⣀⠀⣠⢀⠄⣀⠀⠀⠀⠀⠀⠀⠀⡠⣡⡹⢃⣸⣯⣽⠟⠉⠉⠁⠀⢀⣀⣠⣤⠖
⠀⠩⠿⠾⠿⠾⠯⣭⡭⣭⣴⣾⣯⣻⣿⣣⣑⡬⢤⣲⣲⡦⢤⡤⠐⣰⢡⢁⣸⠛⢛⣿⣷⣔⠠⠞⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠟⠒⠶⠮⢷⣞⣣⠍⡀⣼⠻⢿⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠳⢶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠉⠉⠉⠉⠙⠛⠓⠾⠭⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠂⠠⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡈⠙⠛⠿⠿⠿⣿⣿⠿⠿⠿⠛⠉⢺⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⡀⠀⠘⠣⢤⠀⢐⢸⡷⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⠤⠤⠤⠴⠶⢾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⣟⠉⠩⠀⠀⠀⠀⠀⢀⠀⠀⣿⣏⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡀⠈⠁⠀⠀⠀⠀⠈⠄⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠛⠛⠿⣿⡖⠤⣀⣰⣀⡠⠴⠊⡿⠛⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⣀⣀⠀⢠⡓⠒⠚⢲⠚⠂⠐⠊⢀⡠⠔⠂⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⡠⠛⡺⠧⡤⠤⠼⠤⢤⡔⠪⡉⢣⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢴⣅⣮⣄⣠⡏⠈⠛⠒⢑⣤⠤⢷⠤⠧⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠂⠐⠒⠒⠒⠚⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠒⠉⠉⠀⠀⠀⠀⠀⠉⠉⠓⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠤⢄⣀⡀⠀⠀⠀⠀⠀⢀⡠⠴⠓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⢤⢞⣉⡀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠒⠂⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⠀⢸⠀⠀⠈⠉⠒⠢⠒⠊⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⣉⣉⡙⠲⣤⣤⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠴⠊⠉⠉⠉⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣒⢒⡂⠀⠀⣶⠋⢲⡒⠒⠀⠀⢹⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠧⠤⠠⠤⠀⢾⠨⠛⡇⢱⠤⣤⠰⠃⠤⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠂⠀⠐⠚⣆⠀⠇⡜⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠭⠭⠭⠅⠀⠀⠨⠽⠤⠞⠁⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠾⢤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠮⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣓⣒⣒⡂⠀⠒⣒⣒⣒⣂⣀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠗⠀⠒⠂⠀⠀⠒⠒⠒⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣁⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣻⣍⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⡏⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⡔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢠⠧⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠭⠭⠤⠬⠥⠤⣽⣶⣶⣿⣷⡦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠷⠤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢟⣫⣥⣴⣶⣶⣦⣭⣙⠻⢋⡄⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢋⣴⣿⣿⠿⢛⣛⣛⡻⠿⣿⠐⢿⠟⣡⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢡⣿⣿⢏⣴⣿⣿⠿⢿⣿⡷⣀⠰⣶⣶⡘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⣿⣿⠇⣾⣿⢋⣴⣾⡷⣀⠸⣿⣧⢹⣿⣧⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⠀⣿⣿⢸⣿⣿⣾⣿⢃⣿⣿⢸⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢿⣿⣧⠻⣿⣧⣝⣛⣛⣡⣾⣿⢇⣾⣿⡏⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣌⢿⣿⣷⣍⡻⠿⢿⠿⠿⢛⣡⣾⣿⠟⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣙⠿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣍⣉⣉⣩⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠒⡒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⣡⠎⠀⠀⠀⠈⠂⢠⠹⡝⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢦⠇⠀⠀⠀⠀⠀⠀⠸⠀⠈⣦⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣥⣤⣾⣼⣧⣀⠀⠀⠀⡀⠀⣀⠀⠀⠹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⣄⣤⣴⣾⣿⣴⣶⣶⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣃⣈⣉⣀⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣯⣿⣿⣿⣿⣿⢿⡿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⡿⣿⣿⣼⠇⠀⣠⡴⠶⣶⣶⣦⠀⣠⣶⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⡟⢿⡎⠻⡇⠀⠀⠀⠒⠛⢿⡝⠁⠀⣯⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⢌⠧⢀⠀⠠⣄⡀⠀⠀⠀⠀⡀⠂⠸⡆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣴⣿⡄⠀⠀⠀⠀⠀⠠⠚⠁⢒⣴⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡈⢷⡀⠀⠀⠀⠀⡴⠤⠤⠤⢵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣏⠃⠈⠳⣦⣀⠀⠀⠀⠀⠈⠻⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠈⠙⠿⢷⣦⣤⣤⣿⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣏⡿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡇⢸⣿⣿⡝⢷⣄⠀⠀⠀⢀⡟⠁⣼⣿⣿⣷⠙⠘⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡕⢈⡏⠹⣿⣿⣧⣀⣻⣟⣥⣮⣿⣷⡀⠀⠘⣇⢠⣿⡿⠟⡱⣣⣿⣷⣵⡈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠀⠀⠈⢿⣮⣿⣿⠿⢿⣿⣿⣷⣿⣦⠀⠈⠁⣿⠇⢾⣾⣿⣿⡟⠛⠗⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡏⢻⠻⢿⣿⣯⡢⣠⡟⢰⣿⠋⠹⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠈⠻⠛⠷⠜⠃⠿⠏⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣾⡿⠍⠀⠀⠀⠀⠑⠒⠯⣗⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⣹⣿⣶⠀⠀⠀⠀⡀⢰⡀⠀⠀⢀⢠⡀⢠⣤⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣾⡿⢵⣿⣤⣄⣀⣀⣇⣼⡇⠀⠀⣾⡆⣧⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⢿⡟⠛⠛⠛⠛⠿⣿⣿⣷⣶⣶⣶⣦⣼⣿⣗⣣⠭⢿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⣶⠶⠀⠀⠀⠈⢿⣿⣿⡿⡻⣝⢿⣿⣿⣿⣿⣿⣾⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠇⠉⠀⠀⠀⢸⣿⠋⢀⢳⠈⣿⡙⣿⣟⢿⡟⠉⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡈⢁⣴⠿⣿⣿⣇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠁⠀⠀⠀⠀⠀⢸⣿⣶⣿⠋⠁⠀⠙⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡈⠀⠀⠀⠀⢀⣼⣩⠟⠁⠀⠀⠀⠀⠸⣿⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢀⣠⣤⣶⣿⣿⠗⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣟⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢿⠁⠀⠀⠀⠀⠀⡠⣾⣿⠿⣛⢽⣿⣿⣾⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣄⠀⠀⡠⢪⣾⠏⠀⢀⠔⣩⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⣿⣦⠄⢰⡿⢃⡾⢷⢫⡞⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠦⣤⣀⣀⣤⠤⠦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡏⠉⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠈⠉⢹⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣀⣀⣤⣤⣀⢸⣇⣀⣤⣤⣀⣀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠛⠉⠁⠀⠀⠉⠛⠛⠉⠀⠀⠈⠉⠛⠛⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠄⠀⠀⠀⠀⢠⠤⠤⡄⢠⡖⠒⠒⠒⡆⠀⠀⢠⡤⠤⣄⢀⡠⣀⣀⢤⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠚⠑⠤⡈⡇⠈⢩⣅⠀⡤⠀⠃⡤⡄⢡⠑⡌⢱⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⢱⠀⡁⢠⢰⠸⡘⠔⢡⠇⡄⠧⢂⡼⠀⡇⢸⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠭⡭⠥⠚⠁⠘⠊⠀⠉⠒⠉⠈⠓⠒⠉⠀⠀⠈⠉⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡤⡄⠀⣠⢤⡀⣀⡤⣄⡀⢀⣀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠶⠃⠇⠀⠃⠶⢣⠃⢰⠄⡅⠋⢠⡌⠑⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⢠⢀⠀⡶⠈⣦⢠⡈⢧⣠⠸⠜⠀⡇⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠣⠼⠸⠴⠃⠳⠿⠼⠘⠴⠻⣤⡤⠞⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠒⠒⠒⠒⠒⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣑⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⢄⣀⣀⣀⡀⠀⠀⠀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠭⣭⣍⣉⣁⣐⡒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⢤⣭⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠓⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣤
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⡿⠿⠿⠿⠟⠻⣿⠿⠻⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠻⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣶⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⣶⣶⣶⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠞⢄⡀⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠿⠿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢁⣘⣀⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⠶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡠⠦
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠶⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤

⠀⠀⠀⢸⣀⣀⣀⣀⡀⠀⠒⠶⠶⠶⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣂⡐⠒⣀⡀⣀⣀⣀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠹⠿⠿⠿⠿⠿⢶⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠈⠉⠁⠀⠀⠀⠰⠠⡆
⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⢘⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠰⠤⠤⠤⠤⠦
⠀⠀⠀⠸⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠂⠀⠴⠶⠤⠤⠴⠶⠄⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠈⠉⠉⠉⠉⠉⠻⠿⠇⠄⠀⠐⠒⠃
⠀⠀⠀⢴⠶⠀⠐⠂⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡟⠉⡀⠀⠀⢀⡉⢹⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣈⠤⠊⠀⠀⠀⠀⠑⠀⣀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠻⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⢐⣉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠂⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠬⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠰⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢈⠉⠀⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠂⠆⠄⠆⠅⡄⠀⠄⠠⠀⠄⠠⠀⠀⠀⠨⠀⠒⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠾⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⡌⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⣻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠙⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢰⡆⠸⣧⠀⠀⠀⠀⠀⠀⠃⠀⡐⠀⠀⠀⢿⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠼⠇⠀⠈⠳⠦⠤⠄⠀⠁⠀⠠⠁⠀⠀⠀⠈⠷⠀⠼⠧⠀⠀⠴⠀⠦⠤⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡧⠀⡄⠠⠠⡀⠀⠄⠀⠀⢀⠀⢤⡀⠄⠠⠠⠀⢠⠠⠄⣶
⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠁⠀⠁⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠂⠂⠀⠀⠀⠀⠀⠀⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠐⠐⠀⠆⠀⠂⠀⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⡀⡀⠀⡀⠀⡀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠆⠈⠀⠁⠁⠑⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠



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