Asian Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠈⠋⣹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣶⡄⠀⠛⠛⠛⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠀⠀⠼⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢸⠇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠀⢀⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⠋⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠀⢀⣀⣀⣁⣠⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣟⣛⣉⣹⣿⣿⣿⣿⣿⣶⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⣿⣳⡿⠛⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠏⠁⣠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹⠆
⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⠟⠋⣀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢸⡿⡟⠛⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⡟⠀⡜⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠁⣠⣴⣿⡟⠁⠀⠠⣶⣿⠀⠀⣤⣀⠀⠀⠀⠀⢀⣀⣤⠤⢤⣀⣀⣀⣀⣠⣤⠞⣋⣴⡿⠀⠈⠀⠀⠀⠀⠎⢘⠀⠄
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣫⡵⠟⠀⠀⠀⣿⡟⠀⠀⠸⣯⣝⠛⡿⠛⣉⣽⣶⣣⣶⣾⣿⣭⡿⠋⣡⣴⣿⣿⡇⠀⠀⠀⠀⠰⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⢸⠟⠋⠁⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠹⢿⣧⡼⠟⠉⠀⢀⡨⠉⢻⣿⣿⣶⢿⣿⠟⡙⠀⠀⠀⠀⠜⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⡇⠀⠀⠀⠀⠀⣾⣭⣠⣤⣶⣦⠈⢀⣼⣠⡿⣟⣵⣿⡿⡟⠁⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡇⠀⠀⠀⠀⠀⣿⣿⠿⣿⡿⣿⡶⠀⢛⣿⠟⢹⣿⣿⣿⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡇⠀⠀⠀⠀⣰⡿⣿⣤⠉⡀⢿⣇⠀⣿⣶⣶⠟⠛⣿⣿⠇⠀⠀⠀⠀⠀⠀⠌⠀⠀⢀⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣽⡃⠀⠀⢀⡴⢻⣿⡾⣽⠓⠛⣿⡿⠲⣟⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⠁⠀⠀⠀⠀⠀⠁⠀⠋⠑⠐⠊⠟⢻⣿⣿⠹⣹⣿⣟⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢀⣿⣿⡏⠉⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣿⡇⡆⢰⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣻⣤⢀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣟⣧⢡⣌⢳⠀⠀⠀⠀⠀⠀⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠉⠽⠿⠿⠿⠿⠯⠼⠏⠸⠀⠀⠀⠀⠀⠤⠯⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠟⠛⠉⠛⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡗⠒⠶⢤⡄⠀⠀⣠⡴⠞⠻⠺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠛⠛⠷⠂⠀⠀⠀⠒⠺⠟⠙⠋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⡀⠀⠀⣀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⣀⡈⠉⠀⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣀⣠⠞⢽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠩⠀⠀⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰⢰⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠿⠁



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

⠀⠀⠀⡀⠤⠤⠖⠒⠒⠒⠚⠛⠉⠉⠉⠙⠛⠛⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⢀⠒⠀⠠⠤⣄⠀⠈⠢⠀⠀⠀⠀⠀⠀⠠⢄⡀⢀
⠀⠈⣮⡰⢄⡀⠨⠉⠋⠀⠀⠀⠀⠒⢀⣤⡐⠘⠃⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡍⠀⠙⠀⡤⡀
⠀⠀⢤⡉⠑⠛⠟⢒⡒⠒⠒⠒⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠔⢀⡀⠀⢀⣀⣀⠒⠡⠤⠤
⠀⠀⠀⠺⡗⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠈⢪⡢⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠀⢠⣈⠀⠀⠉⠉⠃⢀
⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⣀⣈⠉⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠂⠀⠀⠀⠀⠐⠀⠤⠔⠂⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡢⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣠⡀⠄⠐⢂⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠻⠟⠛⠛⠛⢉⣨⣟⠛⠓⠒⠖⠞⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠁⠀⠀⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠃⠄⠀⡀⠈⠂⠛⠀⠲⡍⠀⠉⠉⠘⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠐⣦⣄⣑⡦⠄⢠⣆⠀⠉⠦⠤⣌⡛⢲⠴⠀⠀⠀⠀⠀⠀⠏⠉⠁⠀⠒⠒⠒⠒⡒⢖⠂⠀⠀⠀⠁⠠⣄⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⡿⢷⣀⢿⢤⡈⠛⢳⠀⠒⠀⠘⣄⡀⢳⠀⠃⠀⠀⠀⠀⠘⢄⠉⢀⡀⠀⡀⠀⠀⠐⠀⠀⠀⢃⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣐
⠀⠐⠷⣾⣼⠶⣦⡾⠷⣄⠛⠊⠓⡷⠄⠝⠶⣟⠯⡈⠐⠀⠀⠀⠈⠢⢅⠀⠀⠂⠠⠀⠀⠔⣀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠉⠛⢧⣟⠶⢶⡜⣫⣑⠔⢊⠶⠖⢀⣀⠤⠄⠋⠁⠀⠀⠀⠀⠀⠈⠐⡂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠷⣵⣶⣶⡈⠯⠀⢀⢀⣀⢀⠐⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣗⠅⠘⠧⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠄⠀⠤⠤⢠⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⡶⠌⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⣄⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠨⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣝⣳⠤⣄⣀⣀⣀⣀⣤⠄⠀⣤⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠠⢤⡀⠓⢦⠠⣠⣄⠀⠀⠀⠀⠀⠀⣄⠸⡙⠛⡗⠛⢒⠲⠖⠉⠉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠤⠒⠄⠘⠁⠀⡁⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠄⠧⣄⡨⠴⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡀⡐⠒⠚⠛⢻⢿⣳⡿⢶⡀⠍⢂⠀⠀⠈⠈⠃⠄⡀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠷⢄⣋⣉⡉⠜⠜⠓⠶⠟⠀⡪⣢⣎⣀⠂⠀⢈⠀⠀⠀⠰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣭⣟⣐⡂⠒⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠘⠢⢬⣍⡁⢀⠀⢀⣈⠁⢉⠁⣈⡠⠄⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢞⠏⢽⡯⢀⠉⠁⠀⢉⠍⢛⣛⠶⡶⠈⡆
⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠌⠓⡈⠠⠀⠧⡴⢒⣶⠒⠢⣷⡖⠒⢀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠫⣙⠓⠢⠴⠆⠠⣿⠶⠖⠒⠐⠙⠲⠐⠂⠀⠀⠄
⠀⢀⠀⠀⠀⠤⠤⠄⠀⢀⣀⠙⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣬⣁⡂⠤⠄⠀⠈⠉⠉⠄⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠤⠿⠤⠶⠦⠤⠤⢤⣤⣤⣬⣀⣀⣀⠀⣀⣘⣒⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠛⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠯⠉⠁⠉⠋⠀⠒⠒⠒⠂⠀⠀⠐⠂⠐⠀⠁⠀⠈⠭⠭⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣾⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⡿⠋⠀⠀⠹⠛⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢋⡞⠀⠀⠀⠀⠀⠃⠀⣿⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣋⣐⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠉⠛⠛⠛⠂⠀⠀⠀⠀⣸⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⣴⢶⣦⣤⠀⠀⠀⠀⡰⠁⣀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠏⠀⠈⠀⠀⠀⢀⠜⠀⠀⠁⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⣷⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠠⠤⢀⢤⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠠⠴⠆⠀⠰⢤⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠒⠢⠒⠒⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠹⣽⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠓⠶⠤⠤⠶⢖⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠠⢂⢾⢋⣞⡝⣱⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢱⣿⢏⡜⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡼⠡⠋⠀⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣼⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⡴⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⢰⣿⢃⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠠⠔⣩⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⢾⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣀⡴⠞⠋⠁⣿⣿⣿⠟⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠎⣿⣿⢿⡟⣿⠋⠀⠀⠀⠉⠀⢀⣠⢶⣿⡿⠛⣡⡾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⢁⡟⢸⠁⠀⠀⠀⠀⠀⠔⢁⢔⡻⠋⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣔⣄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠊⠈⠁⠁⠁⢂⠄⡀⠀⡀⠠⣠⣤⣿⠿⠟⢽⠛⢛⡀⢶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠘⠙⠂⠀⢈⣀⣀⣀⠀⠀⠀⠢⠀⢶⣾⣛⠑⣀⢼⡢⠀⠀⠄⣸⠘⠏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠩⠁⠲⣣⣶⡏⣫⣽⣿⣿⣷⡀⠐⢨⣼⠡⠐⢀⣤⣤⣉⠀⠢⡄⠠⠦⡁⠲⠰⠜⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠄⠀⠀⢛⡛⠏⡻⡿⠑⠫⢿⣿⣇⠀⠐⣏⠀⠄⢠⣿⣿⠛⢓⠀⢱⣄⠀⢈⢆⠁⡤⢈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠙⠈⣠⠈⡿⣦⠄⠀⠐⠀⢩⣿⠁⠀⠌⢘⠀⢀⢸⣿⣷⠀⠳⠈⠉⢭⣑⡃⣬⣐⣿⡲⢏⡀⠔⡀
⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⣀⠀⠀⠀⢈⢁⠖⠠⠴⡕⣛⠏⠀⠀⠀⠄⣾⠃⠀⢀⡤⠺⣇⠈⡆⢿⣿⣎⠁⠄⣀⠄⠸⣷⠿⠿⡿⣽⣿⡿⣯⡍⠤
⠀⣀⠀⠀⠀⡤⣲⠇⠮⠙⣽⣹⣳⢦⣄⣦⢥⢷⡏⠡⠅⠼⠀⣦⠀⢠⣯⠀⠐⢠⡉⠆⣄⡑⢒⠘⣿⣿⣦⡀⠉⡈⢬⣥⣙⣶⣌⡫⡷⡰⠂
⠀⠔⠆⠀⡸⢪⣶⠞⠟⠀⠈⠁⠙⠆⡹⢻⣦⣻⢆⡀⠀⢶⡞⡏⣠⡀⠃⠆⣠⡈⠁⠀⣻⣿⣥⡤⠘⣿⣿⣿⣲⣉⠛⠻⠿⣿⠟⢛⠙⠁⠀⠈⠑⠀⠀⠠⠁
⠀⠠⠐⠄⡗⣿⣿⣿⣦⣀⣠⣤⣤⣠⣡⡌⡝⢷⡾⣿⡶⠏⠀⢊⣼⡂⠜⠃⢡⠞⣜⡄⠉⢡⣿⣧⠀⣽⣯⣿⣿⣿⡷⢾⣦⣄⡔⠀⠁
⠀⠀⠀⠀⠻⣜⣿⡿⣿⣿⣿⣿⣾⣿⣾⡗⠘⠮⣛⣮⣤⡴⠀⣩⡟⠀⠠⣤⠉⠀⠀⠑⣤⠉⠻⠟⢈⣿⣿⣿⣿⣻⣷⣦⢥⣸⠗⠦⢀
⠀⡀⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⡇⢦⢀⠀⠈⠺⠁⠐⠣⡀⠉⠁⠘⠢⢤⣀⡀⢀⠐⣂⣴⣿⣻⣿⣫⣾⠟⠉⠐⠀⠀⣠⠄⠀⠠⢤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠐⠐⠀⠐⠌⠠⡐⠑⠄⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠁⠁⠈⠉⠀⠀⠀⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣾⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠚⠱⠋⡼⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⠌⡑⠁⠀⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠃⠀⢀⠔⢁⡼⠁⠀⠀⣹⣿⡇⠀⢻⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⡿⠁⠀⠀⠀⣠⠊⠀⠀⠀⠀⣿⡟⠀⠀⠈⠸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠷⠿⠿⢶⠶⠇⠀⠀⠀⠀⢺⣿⠿⠿⠛⠛⠷⡏⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣠⣠⣴⣧⣄⠀⠀⠀⠀⢠⡿⢁⣤⣦⣤⣄⠀⡃⢻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠟⠿⠤⠘⠛⠂⠀⠀⣰⡟⠈⠒⠬⠛⠛⠉⠁⠇⢸⣿⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⣸⠷⢛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣽⣿⡙⡄⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣾⣿⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡾⢱⠃⢹⣧⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠴⠋⠀⠀⠀⠀⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠛⡀⠀⠀⠐⠐⠒⠈⠀⠓⠒⠂⠀⠀⠀⠀⡰⠋⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⢄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢄⡀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⠒⠒⠊⠉⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠨⠙⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⠀⠀⠀⠀⣠⡄⠀⣄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⡀⠉⢀⡀⠀⣀⠈⠁⣀⠉
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠘⠋⠀⠙⠃⠀⠛⠁⠈⠋⠀⠘
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣷⠀⠐⠟⠀⠘⠟⠀⠸⠃⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠉⣿⣿⣿⣿⣿⣿⡀⠀⠀⠠⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⢩⣄⠀⣠⡀⠀⣤⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣷⣶⣭⣀⡀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⢉⡉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠀⣀⣀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠞⠛⢉⣉⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⢀⣬⣭⡙⠛⠋⠉⠁⠉⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣬⣿⣷⡆⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⣤⣻⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣴⡶⠂⠀⠀⠈⠻⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠉⢿⣿⡿⠆
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠠⣾⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠀⠀⢀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠻⢿⣿⣿⡿⠛⠛⠁⠺⠿⠟⠀⠀⠀⠸⣶⣶⣿⣿⣿⣿⠆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠀⣰⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢻⣿⣿⣷
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⣿⣿⣿
⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⣿⣿⣿⣆⡀
⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⡿⠋⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣤⡀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢱⣦⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡾⠁⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣼⣧⢰⡆⠀⠀⠀⠀⠀⠀⢺⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⣧⣤⣶⣿⣿⣿⣾⣗⣴⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⢰⠀⠀⠀⢀⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠈⠀⠀⠀⢠⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢀⠀⠀⠠⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⢻⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⢸⣷⣄⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⣦⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠸⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠰⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠐⡧⠀⢠⡐⣚⣭⣡⣤⣭⣤⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⢀⡌⠉⠀⠀⠑⢀⡀⡀⠀⠀⢂⠀⠡⣾⣿⡿⠿⠿⠿⠿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠾⠋⢁⠀⡀⠀⠘⠀⠭⠁⠔⠀⠀⠐⢤⠀⠀⠈⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠔⠀⠀⠀⠐⠖⠛⠂⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⢘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡇⠀⠀⠈⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠄⠀⠠⢀⣀⣀⣼⣿⣿⣿⠃⠀⠀⡉⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⣬⣶⣿⣿⣿⣻⣟⣿⡷⣻⣿⢿⣿⣟⣋⣽⠈⠋⠀⢀⠘⣀⢀⠘
⠀⠀⠀⠀⠀⢀⡴⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣍⣛⠉⠹⣿⣿⣧⣤⠄⠩⠾⠇⠀⠘
⠀⠀⠀⠀⢠⡾⣷⣿⣿⣿⣿⣿⣿⠿⠛⠋⠛⠛⠛⠛⠉⠩⠍⠠⠖⠛⠻⢿⠓⡀⠀⠀⠀⠈⡆
⠀⠀⠀⢀⣿⢦⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠐⠀⠂⢠⠂⠀⠐
⠀⠀⠀⢨⣿⣤⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢄⣤⣬⣶⣦⣶⣖⣶⣶⣤⣤⣄⠠⠒⠀⠠⠐⡄
⠀⠀⠀⠘⣿⣿⣙⣿⣿⣿⣿⣦⣀⣀⣴⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⣷⡆⠨
⠀⠀⠀⠀⠹⣿⣿⣶⣿⣭⣹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠋⠋⠉⠛⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣻⣿⣿⣻⣿⣿⣿⢿⡟⡛⠉⠀⠀⠀⠁⠈⠐⠂⡄⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⠀⠀⠀⠆⠀⠃⠀⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⣶⣶⣴⣶⣦⣤⣦⣤⣲⣴⣾⣿⡇⠀⠀⣠⠀⢀⠐⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⠿⠿⢾⣿⣿⣯⣛⢿⠺⣿⠿⠁⠀⣴⠏⠠⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⡋⠑⠀⠀⠀⠀⠲⢈⠘⢹⣿⣆⣐⠀⠂⠈⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢠⡀⠀⢀⠈⠄⠐⠂⠀⠬⠈⠉⠋⠋⠠⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢷⡖⠀⢀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⢐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠂⠉⠀⢢⡀⠂⠈⠀⠀⠀⠀⠀⠣⠃⠈⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⣤⠤⣀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠈⠀⢀⣀⠟⠁⣱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠜⢩⠁⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⣫⣶⡏⠀⠀⠐⢻⣭
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠻⠟⠠⠺⠃⣴⠎⠉
⠀⠀⠀⢤⠴⢖⣿⣷⢶⠲⠤⣀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⢀⣀⣑⣿⡽⠾⣋⠀⠀⠘⠁
⠀⠀⠀⠙⠹⡶⡞⣏⣿⠀⠀⠀⠆⠈
⠀⠀⠘⠷⣾⣿⣭⡵⣖⠀⠀⢀⣀⠤⠂
⠀⠀⢤⣤⣬⣧⣷⣷⣫⠀⠀⠀⢀⢀
⠀⠀⠀⠉⢻⢻⢻⡏⡇⠀⠀⠀⠉
⠀⠀⠀⢖⡟⣯⢹⠟⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠤⡀
⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣦⣔⣦⣤⣤⣤⣀⣀⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠛⠿⣭⣽⣿⢿⡿⠻⠿⠿⠟⠻⠋⣙⣛⣿⣯⠟⠛⠉⠉⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠰⢿⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⡯⣄⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⢸⠃⠀⠈⠛⠿⠿⠅⢦⠀⠁⠀⠀⠀⠀⠀⡤⢶⣾⡿⠘⠂⠀⠀⠀⠈⢻⠐⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠆⢠⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢫⡁⠒⠂⠙⠗⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠖⠀⠀⠀⡄⠀⠐⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣇⠚⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣧⡉⢹⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣭⡉⢹⣿⣿⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡿⠛⣿⣿⣿⣿⣿⣇⡾⢿⣿⠋⠀⠉⠻⢿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣿⣿⡷⣆⢸⣿⣿⣿⢿⣿⠃⠈⣉⣦⣀⣄⡀⠀⠈⠻⢷⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠟⠀⣽⣿⣿⣿⣧⣄⣄⣀⣬⣽⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⣿⣿⣿⡋⠉⠀⠀⢉⣿⣿⣿⣿⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠈⠻⡿⢿⣿⣟⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣶⢠⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡄⠹⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣴⣏⠉⠛⠃⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣤⣤⣀⣀⣤⣤⣤⣶⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣉⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⠛⠋⠉⠀⠀⠀⠹⣿⣿⡄⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣯⣅⠀⠿⡿⣿⢿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⢿⡅⢉⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣤⣤⣴⡿⣿⣿⣿⡏⠉⠉⠉⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣿⣷⣶⣾⣿⣿⣿⣿⣿⣄⠀⢀⣾⣿⡿⠛⠉⠀⠈⢹⣿⣷⣄⡀⠀⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠁⠞⠋⠡⠟⠋⠩⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣾⣶⣶⣿⣦⣴⣤⠄⢀⣀⣨⣌⣛⠃⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠙⠉⠉⠉⠋⠙⠛⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣿⣿⡟⠛⠻⠉⠁⠈⠛⣻⣿⣿⡿⠿⠟⢻⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⡁⠀⠀⠀⠀⣠⣶⣾⣿⣿⠏⠀⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢶⣠⣿⣿⣿⠃⠀⠨⢳⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⠐⠓⠞⢹⣿⣿⣿⣿⡟⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⣤⣼⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⠀⠀⠀⣀⣀⣀⣠⣴⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣶⣿⣿⠿⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣭⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⢿⣷⣦⣤⣤⣄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⡆⠙⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣭⣶⣾⣿⣿⣿⣏⣴⣶⣾⣋⣙⠃⠀⠀⠀⣢⡴⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣶⣀⡀⣀
⠀⠀⠀⠀⣠⣄⣀⡐⣤⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⢿⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡤⢿⣠⠂
⠀⠀⠀⢸⡭⣿⣧⣿⣿⡿⣿⠛⠟⢙⣋⣙⣉⣛⡛⣻⣿⡛⠿⠿⢿⢿⣿⣽⣙⠘⢻⣴⠀⠤⢡⡄⠀⠀⠀⠀⢀⣸⣤⠀⠀⠈⠹⡇
⠀⠀⠀⢴⣺⣿⣿⣿⠙⡡⠘⣀⣩⡡⠤⠤⠾⣿⣙⢿⣿⢿⡯⠰⣦⣤⣤⣈⠙⢿⣾⣹⡆⠇⢸⠁⠀⠀⠀⢰⡏⠋⠉⢷⠀⠀⠐⠁
⠀⢠⣿⣿⣿⣿⣏⢑⠊⣰⡿⠋⣁⣤⣤⡤⠤⢤⣁⠉⢂⠈⠹⣦⡻⡏⣟⣿⣿⣌⣿⣏⡴⣀⣀⠀⢀⣷⠄⠀⠀⢈⡿⢸⡦⣴⣧⣠
⠀⢸⣷⢯⣿⢿⡀⢀⣰⠙⣧⣜⢿⣿⢿⣗⣀⠂⠈⠉⠀⠒⢀⠐⣕⣤⣿⣿⣿⣿⠋⠯⣾⡿⢿⣷⡿⠛⣁⠀⣄⢈⡥⢿⣠⣿⣿⣿
⠀⢹⣿⣿⣯⠃⡅⣾⣿⣿⣷⣽⠈⣝⡆⣿⣏⠢⠀⠄⠀⠠⡀⡜⣘⢧⡽⠛⢛⠁⣠⣴⣿⣱⡄⠂⣏⠀⣿⠀⠚⡁⠀⠘⣿⣦⣤⡀
⠀⠘⣾⢋⣸⡧⠀⢻⣮⢻⣭⡌⢭⣭⠛⢍⣹⢀⡀⠀⠀⢔⣵⣾⣶⣦⣾⠷⠵⣾⢿⣿⣿⣿⣿⣷⣿⣷⣿⣧⣾⣿⣿⣿⣿⣿⣿⣧
⠀⢀⠷⢂⣽⠧⠁⠘⣿⡿⣯⣿⣟⣛⣦⣿⢉⣽⣇⠀⠀⠚⣻⣿⣿⣽⠶⣛⣛⡛⠷⣷⣝⡉⠹⠻⣿⣿⣟⠹⣹⣿⣿⣿⣿⣯⣿⣿⠆
⠰⣿⣦⡟⢡⣾⡀⠀⠼⡿⠇⡁⠉⣿⠹⢿⣟⣥⡼⠁⣀⣼⣿⣿⣯⣶⣿⣿⣿⣿⣿⣮⠟⣋⡁⠈⠛⠹⢯⣤⣟⣃⣬⢛⣿⣿⣿⣿
⠀⠉⠉⠁⣌⣁⡀⢂⠀⠀⠀⠉⢠⣿⡆⠀⠁⢀⣴⡾⠿⣿⣿⣿⠟⠇⠨⣬⣅⠉⠛⠛⢙⣿⣰⡖⢊⣴⣿⣥⣬⣏⣳⢾⣯⣿⣜⣿⡀
⠀⠀⠀⠸⢿⡃⠀⠜⠃⠂⢤⡴⡿⠿⢇⠀⠀⠀⠀⢩⣄⡘⠷⠻⣽⠊⣙⢶⣋⣉⠀⠂⢤⠈⠙⠻⣿⣿⣯⣿⣿⡿⣿⣦⠙⣿⣿⣿⠃
⠀⠀⣀⠻⢭⠼⠽⠂⠀⠀⠀⣴⣦⣤⡈⠁⠀⠀⠀⠘⠛⣿⡿⢇⣛⡲⣛⣼⣿⣿⣷⡀⠄⠠⢀⡀⠀⢈⣿⣿⣿⡇⢸⣿⠏⢈⣿⣿⠃
⠀⠀⣼⣶⣦⣠⣶⣼⣿⣷⣴⣿⣿⣿⣿⣶⣤⠄⠀⢰⣯⣷⣿⣿⣿⡿⢿⣛⣿⣿⡟⢀⠀⠀⠠⣑⡢⣹⡿⠈⢴⡟⣸⣿⡄⢺⢿⡏
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠉⢛⣸⣧⡀⠀⡈⠉⠉⣉⣽⣿⣾⣿⣿⢻⣧⡯⠲⢒⣶⣟⣫⣥⢿⡄⠘⠟⢛⣡⣤⡆⣾⡁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡁⠀⠈⣘⣿⣿⠿⠫⠅⠈⠙⠿⠿⣾⣿⡿⣿⣿⡷⠆⠻⠿⡿⢻⣷⣶⣤⣐⣷⣿⣿⣿⣾⣿⣧
⠀⠀⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣁⣚⣿⣾⣿⣯⢵⣊⣀⣠⣶⡇⠬⢍⣩⣄⣩⣭⡦⠕⠻⠖⡿⣞⠛⠾⠿⠿⣭⡽⣻⣿⣟⣿
⠀⠀⠹⣽⣿⣿⣿⣽⣟⣭⣉⣽⣿⣿⣿⣿⢿⣿⣟⣿⣿⣷⣙⣱⡮⣖⣻⣮⣿⣦⣭⣽⣷⢦⡀⣀⢿⠀⡄⣼⣀⣻⣙⣩⣵⣿⠿⠿
⠀⠀⠁⠀⠙⠛⠛⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠉⠀⠈⠉⠉⠁⠀⠉⠉⠊⠉⠁⠈⠉⠘⠉⠛⠛⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠶⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣐⣲⣶⣶⣦⣄⢳⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣛⡟⢛⡿⢿⣿⠃⠀⠀⠀⠐⣈⠩⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⠾⠋⢁⡴⠋⡀⠀⠀⠀⠀⢀⣠⣤⣭⣻⣾⣷⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⣾⡟⢁⠀⣰⣯⣶⣯⡶⠄⠀⠀⠀⠈⠀⠉⠙⢿⣿⣿⣿⣷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣡⣾⣿⣯⠞⡞⣼⣿⣿⣿⡟⣡⢎⣶⡟⠀⠀⠀⠀⠀⠀⢻⡌⢿⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⣾⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⢞⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣎⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⣀⣤⣶⣾⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣾⠇⠀⠀⠀⠀⠀⢀⣀⡀⠀⡐⠀⠻⠛⣋⣤⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠀⠀⢀⣠⣴⡾⠿⠛⢋⠄⠀⠀⠀⢠⣿⣯⣹⠏⢿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⠟⠉⢀⣠⣀⣤⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⠁⠀⣠⣿⣿⣿⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠚⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠈⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠛⠛⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠑⠄⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡸⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠹⣆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠚⠶⠶⠶⢦⣤⣤⣤⣶⣾⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠙⢮⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠙⢿⣿⣷⣝⠿⣿⣿⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⡿⢟⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠈⠻⣿⡛⢷⣤⡙⠻⣿⣿⣦⡘⢿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⠻⣦⣌⡙⠶⡌⠫⡻⣷⡌⢻⣏⠻⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠀⠈⠉⠛⠛⠛⠲⢬⠈⠻⣆⠹⣆⠈⢿⣦⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⡄⠙⠂⠀⢻⡆⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⢀⣠⣤⣤⣤⣤⣤⡤⠶⠚⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢛⣷⠾⠛⠋⠉⣠⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⡄⠀⣴⠁⣴⡟⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣧⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣄⠀⠀⠑⢤⡀⠀⠀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣉⡛⠉⠛⠛⢋⣸⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⠃⢶⣶⣾⡟⠃⠀⣴⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣹⣯⡀⠀⠈⠀⠀⠀⠀⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⢿⣿⣿⡿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠐⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠇⠈⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣂⣶⣿⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠋⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⠿⡿⣿⣿⣳⠄⠀⢰⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠠⠀⠀⠀⢸⡿⠏⠩⠭⠉⠙⠚⠇⠀⠸⡏⠀⠁⠀⠀⠁⡿⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠠⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⣃⡙⠻⠙⠀⠀⠀⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⣲⡉⠀⣺⠀⠀⠀⠀⠠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠⠀⠀⠀⠠⠀⠀⠀⠰⠀⠄⠀⢀⢢⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⠈⠍⡤⠥⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡓⠃⠁⠘⣈⡇⠀⠀⠀⡄⠀⠀⠀⡀⠐⡄⠀⢀⠀⢂⠀⢀⠀⢢⠀⠀⠀⣶⡄⣈⠛⠒⠀⠠⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂⠀⠀⠙⠳⢿⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠉⠑⠀⠀⠂⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠈⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⢂⢲⠄
⠀⠀⠀⠀⠀⠂⠀⠚⠁⠀⢀⡇⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⣉⣉⠀⠀⠀⠀⠀⠈⠈⠀⠀⠆
⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠃⠀⠀⠈⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⣤⡄⠀⠀⠀⠀⠀⠠⡠⢤⠠⢄
⠀⠀⠀⠀⠈⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠐⠠⡤⡅⣢⣌⣅⣆⣂
⠀⠀⠀⠀⠀⡀⢷⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡂⢈⡑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡠⠈⠉⠀⣀⢀⢄⠐⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⠀⠀⠰⠉⠈⠀⠀⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠛⠂⠀⠐⠓⠒⢀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠒⠂⠀⠀⠀⠐⠀⠈⠷⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢄⠀⠾⠤⠀⠀⠀⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⢾⣦⣄⣀⣀⣤⣾⠿⡄⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡟⠁⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠁⠀⢹⡇⢲⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠿⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠀⢸⠁⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢁⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠂⠊⣾⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡞⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣷⠉⠀⢀⡟⠀⠀⠀⠀⠘⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣴⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠇⠀⢀⡞⣡⡆⢀⡄⠀⡴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣠⡾⠃⠀⡰⠈⠀⠀⠉⠉⠁⠀⠀⠀⠀⢁⠟⢀⡾⢠⡄⠀⣴⢋⡇⠘⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠿⢇⡾⢀⠀⡁⢸⠇⠄⠀⣴⠟⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢘⠿⠶⠀⠀⠀⠐⠂⠀⡠⣄⢀⡠⠊⠀⢺⠟⢁⠜⡜⢀⠌⠁⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠠⢳⡄⠀⠀⠀⠉⠁⠀⠩⠞⢑⣫⡴⢁⡤⠁⠀⠈⠀⠁⠀⠀⣠⡄⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠘⢧⠀⠀⠀⠀⠘⢤⣀⠁⠀⠀⢀⡀⠀⠈⠀⢀⣉⣁⣀⠄⠀⢠⣄⠀⠀⠀⠐⠤⠈⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⣠⡖⠀⠘⠳⠄⠀⠈⠳⢦⡀⠀⠀⠀⠉⠀⠀⠀⠉⢭⡤⠀⠳⠈⠩⠯⠁⠀⠀⠀⠈⠙⠢⡀⠐⠂⠀⠀⠀⡔
⠀⠀⠀⠀⠠⠚⠁⠀⣀⣳⣶⠤⠀⠀⠀⠀⠉⠑⠒⠀⠀⠠⠤⠤⠄⠰⠠⠤⠠⠴⠂⠀⠐⢶⡶⢂⠶⡦⠤⠀⠘⠒⠀⢀⣀
⠀⠀⠀⢀⡆⢠⠶⠦⠀⠀⠠⠤⠄⣛⣀⣤⠀⠀⡀⠀⠘⢃⡄⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢦
⠀⠀⠀⣾⠁⢚⡀⠀⠀⠀⠀⠀⠈⠉⢁⣤⡴⠛⠁⣀⠀⠀⢀⠈⢠⠃⠀⡾⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⡴⠃⠀⠐⠉⠀⠈⡇
⠀⠀⠀⢨⠓⠦⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠋⠁⠀⠀⠈⠀⡜⠀⠀⡁⠀⠀⢀⠀⣀⡀⠤⠄⠒⠂⠀⠜⠁⠀⣠⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠳⣄⠈⠃⣀⣀⣤⣤⣀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣼⡇⢀⣴⠏⠤⣝⣉⣭⣤⣤⡤⠐⠂⠠⠞⠁⠀⡠⠀⡀
⠀⠀⠀⠀⠀⠈⠁⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣶⠿⠋⠀⠉⠀⠀⠀⠀⠉⠛⠦⠀⠀⠀⠤⠶⠾⠷⠋⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠴⠶⠖⠛⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣷⣦⡀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠋⢁⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣮⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠃⡑⣄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⡿⠷⡔⣝⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠋⢳⡀⠉⠚⠁⠀⠀⠀⠀⠀⠀⢸⣳⣄⣠⣴⣾⣟⣿⣿⣿⣿⣿⡿⢋⠁⠀⠀⢘⣎⢳⣬⡻⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣼⣷⣾⠟⠙⣆⠀⠀⢀⣀⣤⡼⣿⣿⣿⣿⣿⠻⣿⣾⢻⣿⣿⡟⣾⣿⠆⣠⣼⣿⣷⡹⣿⣮⡙⢿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣘⣶⣾⣿⣿⣿⡇⠙⠶⠽⢿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢿⣿⣶⡅
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣦⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⣿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡟⣽⣾⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣧⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⣛⠿⠿⣿⡿⠿⣛⣵⣿⠟⣋⣭⣥⣭⣭⣍⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣇⢾⣿⣿⣿⣿⣿⣿⠗⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣷⣮⣭⣭⣭⣭⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠶⠶⢶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣿⣿⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⣿⣿⣿⣿⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢿⣿⣿⣿⣿⡿⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠀⣸⣿⣿⣿⣿⣇⠀⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⣃⣴⣿⣿⣿⣿⣿⣿⣦⣘⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⡠⢶⣶⡾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⢷⣶⡶⣄⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠿⠄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡄⠀⠀⠈⠉⠉⠉⠉⠁⠀⣀⣤⣴⣶⣿⣿⣶⣦⣤⣀⠀⠈⠉⠉⠉⠉⠁⠀⠀⢠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣦⣤⣠⣤⣤⣤⣶⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣶⣤⣤⣤⣠⣤⣴⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣵⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣮⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣋⣥⣾⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣷⣬⣙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⣸⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣇⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠃⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠘⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠋⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠙⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠁⢹⣿⣿⣿⡿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⡟⢉⣤⡄⠀⠉⠈⠉⣵⣬⠉⢹⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⣹⡇⠈⠛⠃⠀⠀⠀⠀⠙⠋⠀⣸⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⡟⠿⣿⣿⣿⣶⣤⠤⣤⣴⣶⣿⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠉⠻⣄⠈⠟⠉⢫⡁⠀⢤⣤⠟⠉⠙⢶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⣼⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠘⢷⣦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⠤⢄⣀⢀⣀⣀⣀⣀⣀⣠⠤⠴⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡜⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣤⣤⣦⣤⣤⣤⣤⡴⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⡒⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡳⠞⣿⠫⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣿⣿⣷⣶⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⣠⠔⠒⠚⠷⠒⣀⣹⣿⣿⣿⠧⣆⡀⠤⠀⢀⠀⣀⣀⣀⣀⡤⠐⠂⠀⢀⣴⣤⣶⣤⣀⣿⣻⡿⣠⠁
⠀⠀⠀⢀⣾⠃⠀⠀⢀⡀⠀⠉⠙⣿⣏⣴⣧⣤⠀⢤⣤⠚⠉⠁⠀⢀⣼⣿⣷⣤⣾⠿⠛⠉⠉⠉⠉⠉⠛⠛⠁
⠀⠀⠀⢸⣿⠀⠀⣠⠿⣅⠀⠀⢀⡞⣿⣿⣿⢧⠃⣼⣿⣷⠦⣶⡾⠟⠋⠉⠉⠉
⠀⠀⠀⢀⣿⠀⠀⠋⠀⢹⣿⡤⠷⢿⡿⠛⠛⣷⡿⠁
⠀⠀⠀⢸⣿⣅⠀⠀⠠⠚⠉⠀⣴⣿⣤⣀⡰⠟⠁
⠀⠀⠀⠈⣿⣿⣷⡄⠀⠀⠀⣼⠟⠈⣿⢋
⠀⠀⠀⠀⢸⣿⣿⣷⣴⣤⡴⠛⠃⠀⣿⠎
⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⣚⡋⢡⢼⠃
⠀⠀⠀⠀⣜⣿⡿⢷⣶⣾⠉⠭⣿⠏
⠀⠀⠀⠊⠈⠉⠉⠛⠓⠚⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠒⠛⠛⠓⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⣤⣶⢾⢾⡿⠿⠛⢻⣶⣶⣮⣍⠛⠷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⠋⣠⣾⡿⠉⠀⢠⠞⠀⢀⣴⠋⠁⠀⠿⠛⣷⢬⠙⢿⣆
⠀⠀⠀⠀⣠⡟⢡⣾⣿⠃⠀⠰⠖⠉⠀⢄⠊⠀⠑⠖⠀⠀⠀⠃⡀⠱⢌⢪⢷⣄
⠀⠀⠀⢠⠏⢰⣿⡅⠈⠶⢀⠀⠀⠀⢄⠀⡰⠄⡀⢀⠜⠢⡀⠀⠀⠀⡌⢂⢻⣿
⠀⠀⠀⣾⠀⣾⠉⠿⠁⠀⠀⠑⠀⠀⠀⠋⠀⠀⠘⠉⠀⣠⠌⠀⢀⠴⠀⠀⠀⢿⡄
⠀⠀⠀⣷⠀⣿⠃⠀⢀⠴⢀⠀⢀⢖⣄⡀⡴⠒⣶⣦⡐⠀⡀⢴⡃⠀⠀⢺⡇⢸⡇
⠀⠀⠀⣧⠀⣟⠀⠰⠆⠀⠀⢰⣇⠈⠛⠛⠻⣟⣉⣙⢷⣄⠈⠀⢈⡤⣀⠀⣧⢸⡇
⠀⠀⠀⣿⡀⢿⣆⠀⠀⠰⢀⡀⠉⠟⠢⣄⡀⠈⠙⢿⡾⠁⠙⢶⠄⠠⡸⠻⡏⣼⠁
⠀⠀⠀⠸⣧⡈⢧⡳⠂⠀⢀⠝⠁⠀⠀⠈⠁⠀⡠⠋⠀⠀⡀⠀⠀⢀⣄⡞⣰⡏
⠀⠀⠀⠀⠘⢷⡈⠁⢤⡜⠁⠀⠀⠰⠋⢀⣠⠞⠁⠀⣀⠞⠃⡀⣤⡨⠋⣴⠏
⠀⠀⠀⠀⠀⠈⠻⣦⣈⠃⠀⠀⠐⢄⠀⠀⠁⢄⠀⠀⠈⠑⢦⡴⠟⢠⡾⠧⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠐⠛⠦⠤⠤⠟⠐⠒⣀⣡⠴⠚⣉⣀⣀⣀⣀⠀⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠶⣶⡶⠶⠟⢛⣭⢖⡾⠛⠉⠉⠉⠛⠿⣝⣒⢦⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢙⡧⠀⠀⡴⠋⣤⣤⣞⣻⣶⡏⠀⢀⠀⠀⠈⠀⠀⠹⡟⠟⢴⣷⣄⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⣸⠇⠀⡼⠁⠀⣾⣿⣿⢹⢸⡇⠀⠀⠀⢈⢠⡀⠀⠀⣿⡆⠀⠘⡟⣷⡀⠀⢻⡄
⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⡟⠀⢸⠁⠀⣼⣿⣟⣿⢢⡘⣧⡀⠀⠀⠀⠀⠀⣀⡼⠃⢣⠀⢠⢁⡛⣷⠀⠀⣷
⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⡇⠀⣿⠀⢠⣿⡿⠹⢻⣋⣽⣤⣝⠓⢦⣤⠤⣾⢟⣤⣶⢼⣶⣶⡊⣇⠸⡇⠀⢸⡆
⠀⢠⡞⢁⡴⠖⠋⠓⠢⡀⠀⣷⠀⣿⠀⠀⣿⠁⡴⠋⠁⠀⠀⠈⡉⠻⡄⠀⣿⠋⠁⢀⠀⠀⠈⠙⢯⠀⡇⠀⢸⡇
⠀⡿⠀⡾⠀⠀⠀⠀⠀⢡⠀⢹⠀⢹⡄⠀⢿⣾⠁⠀⠐⠠⠄⠀⠀⠀⣻⣼⠁⠀⠀⠀⠀⠠⡄⠀⢸⡧⠀⠀⣼⡇
⠀⣇⠀⢣⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⣷⠀⠈⣿⠀⠀⠀⠐⠁⠀⠀⠀⢸⢿⡄⠀⠀⠀⠘⠉⠈⠀⣾⠃⠀⣰⣿
⠀⠙⣦⡀⠑⠀⠀⠀⠀⢀⡼⠁⠀⠀⠘⣧⡀⠘⢦⡀⠀⠀⠀⠀⠀⣠⠏⢸⠻⣄⡀⠀⠀⠀⢀⣼⠋⠀⣴⣿⠃
⠀⠀⠈⠙⠲⠦⠦⠴⠞⠋⠀⠀⠀⠀⠀⠈⢷⣄⠀⠙⠷⣶⠤⠴⣾⠁⠀⢸⡆⠈⢙⠛⡒⠛⠋⠀⣠⣾⣿⠃
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠈⠁⠀⠻⠄⠀⠤⠦⠀⠘⠋⠀⢀⡠⢞⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⣤⣀⣀⣀⣀⢀⣀⣠⣴⣾⡽⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⢀⠔⠀⠀⡤⠊⠁⠀⠠⠾⣖⣋⠉⠀⠀⠠⠂⠄⠸⡄
⠀⠀⠀⠀⠀⢄⡀⢠⣄⠲⢶⣶⡀⠁⠰⣶⠁⠔⢖⡋⠀⢶⣀⡀⠒⠒⠂⠒⠻⠫⣙⡃⠋⠠⡀⠒⠲⣄⠰⠒⠒⠒⠒⢲⡆⡀⡒⡆
⠀⠀⠀⠀⠀⠀⠑⠤⠀⠢⢀⣙⢣⠀⠀⠉⢧⠀⠈⠱⡄⠀⠀⠈⠁⠀⠀⠀⠀⠁⠏⡆⠀⠀⠀⠀⠉⠀⠉⠁⢠⠀⠀⠀⠐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⣀⡀⠈⠈⠂⠀⠀⠀⠈⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⢀⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠀⢀⠱⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠀⠑⠄⠘⠡⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡶⠀⡂⠀⠙⠒⣠⡀⠈⠢⢈⡆⠀⠀⢀⣀⠀⠀⠀⠄⠀⢀⣀⡀⡀⠀⢠⡀⠀⠀⠀⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠙⠒⠒⠒⠒⠚⠛⠚⠛⠛⠀⠀⣿⣿⣧⠀⠀⠐⠚⠃⠤⠉⠛⣠⠝⡳⠆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⡀⠀⠠⠁⠀⠀⠀⠐⠊⠀⠔⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣷⣟⣿⣷⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠛⠛⠋⠉⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡿⠿⠁⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠁⠀⠰⣶⣶⠦⢴⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠃⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡠⠞⠻⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠁⢀⣾⣄⠀⠀⠀⢀⠰⣶⢾⣶⣿⣿⣶⣦⡀⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢺⣿⣿⣷⠀⣄⡈⡈⠁⣾⣿⣿⣿⣿⣿⣷⣄⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⢀⠀⠒⠰⠒⠔⣼⡿⠋⣤⣦⣿⣿⣦⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⢟⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣠⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣭⡶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢀⣤⣿⣿⣿⢿⣿⣿⣿⣗⠀⠀⠀⠉⠉⠁⠸⣿⠹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣄⢀⣿⣾⣿⣿⣿⣋⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣿⠀⠻⠧⠕⠊⠑⡄⠠
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⢁⣙⣿⣿⣿⣿⡿⠻⠛⠋⠀⠄⠀⠀⠀⣸⣿⣧⠀⠐⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢁⣾⣿⣿⡟⣼⡟⠁⠀⠀⠀⠀⠀⣀⣠⠔⠁⠙⣿⡷⣴⠀⠀⠀⠀⢈⣦⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠀⠀⠀⠀⠀⠀⠀⠙⢁⡾⣿⣿⣿⣇⣿⡇⣀⣤⡶⠚⣿⣿⠿⠋⠀⠀⠀⠈⠑⢿⡆⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢀⠠⠖⠁⠀⠀⠀⠁⢛⡛⢻⣿⣿⣿⠿⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠠⣄⣤⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠮⠀⠀⠀⠀⠀⠀⣠⡟⢁⡿⠉⠉⠀⠀⠠⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣀⠀⠀⠀⠀⠤⡍⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠺⠿⠀⣀⣴⣾⠟⢀⣾⠏⠀⠀⠀⢀⠀⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠐⡳⢿⠛⢠⣾⠟⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣴⣷⣮⣥⣶⣒⣖⣶⣶⣮⣿⣤⡦⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢹⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣥⣴⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⢡⣈⡁⠘⣿⠿⡀⢻⣿⣿⣿⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⢿⡀⡀⣿⣿⣷⡸⡆⢰⣷⣿⣿⣿⣿⣿⡝⣿⡷⠈⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣝⢷⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⢿⣿⡈⠁⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡻⡛⠃⠙⢷⣌⣻⢿⣿⣿⠿⠋⠛⠛⠁⠉

⠀⠀⠀⠀⠀⠀⡠⠐⠊⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠤⡀
⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡀⠂
⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⢠⠔⡡⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡠⠂⠂⠾⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠒⠈⢉⠁⡄⠠⠂⠀⠀⠀⠀⢀⢁⠈⠀⠀⠠⠖⠈⢡⠞⣓⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⢂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⡞⠀⠀⠀⠀⠀⢀⡈⠅⠂⠀⠀⡆⠀⢀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠐⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⣴⠟⠣⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠐⠀⠀⠀⢀⡠
⠀⠀⢀⣴⠚⢣⣅⠀⠘⠋⠀⠀⠀⠙⠉⠠⠈⢀⠀⠀⠀⢀⣄⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠸⠀⣀⣧
⠀⠀⠟⠁⠀⣴⣿⡠⠀⠀⠔⠂⠀⠓⠚⠑⠈⠉⢴⣄⡄⠈⠁⠀⠀⠀⠐⠒⠂⠉⢀⣀⡀⢀⡠⠊⠁⠀⠀⠈⡉⠀⠀⣀⠵⠤⠊⠁
⠀⠀⠀⠀⠀⠉⠙⠛⢆⣀⠈⠀⡀⠒⢀⣔⢁⣠⢾⠎⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠈⢡⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠒⠒⠲⡾⠋⠀⣴⡷⠁⠀⠀⠀⠀⠀⠀⠂⠛⠋⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⢐⡀⠀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⣠⠤⠖⠊⠉⠉⠁⠀⠀⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠦⠐⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⡒⠒⠠⠭⠖⠁
⠀⠀⠀⡴⠋⠁⠀⡀⢤⣶⣤⣖⡒⠠⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠀⣀
⠀⠀⡞⠀⢀⡴⢕⣚⣩⡝⠳⠦⣠⣴⠾⣛⢽⣿⣥⡤⠤⡀⠀⢀⠀⠀⢀⢧⡇⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⣀⢀
⠀⠀⠃⠀⣊⣠⠚⢃⡯⣵⢤⡀⠽⠛⡛⠓⠏⣉⠭⠅⠀⢀⠀⡜⠀⠀⣸⢸⠃⠀⠀⠀⡼⠃⠀⡐⡀⠁⠀⠀⢁⡊⠈⠁
⠀⠀⠘⠸⡁⠀⢀⣾⡇⠃⠇⡱⠃⠈⠀⣀⠀⠀⠀⠀⠀⣳⡾⠁⠀⠀⡇⢸⠀⠀⠀⢸⠇⠖⢁⡤⠄⠀⠀⢀⠞⠀⠀⠀⢠⠀⠀⢰
⠀⠀⠀⠹⡦⣐⠺⢟⡀⠀⠒⠀⠀⠀⠘⠁⠀⣀⡠⠴⣺⣵⠃⠀⠀⢰⠁⡎⠀⠀⠀⢨⢲⣀⠍⠀⠀⠀⠀⠄⠀⠀⠀⣄⠮⡀⢄⣼⡆
⠀⠀⠀⠀⠙⠀⢑⡒⠠⠭⢉⣁⣒⣛⣈⣩⠭⠴⣒⡫⡶⠃⠀⠀⠀⡌⠀⡀⠀⠀⠀⠈⡎⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠠⠂⣀⣞⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠤⠶⠦⠒⠒⠉⣩⠔⠁⠀⠀⠀⢠⠁⠄⠇⠀⠀⠀⠀⠃⠀⠀⠐⠩⠠⢠⣄⣀⣠⣤⠤⣾⠟⣡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⢤⣀⡴⠋⠀⠀⠀⠀⠀⠀⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡯⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣳⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣷⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⠀⠀⠀⣴⣶⣦⣄⣀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⠀⢳⣄⠀⠈⣉⠉⠁⠉⡇
⠀⠀⠀⠀⠀⠀⣀⣤⠖⠢⢤⡴⠒⠒⠒⢊⣡⡾⢯⣿⣧⠀⠙⠀⠊⠙⠢⢤⣼⣁
⠀⠀⠀⠀⢠⣾⣿⡟⠀⠀⠀⠙⣶⣾⣿⢿⡿⠃⢸⣷⣿⣷⣄⠀⠲⣯⣽⣿⡀⣈⣱⡦⢤⣀
⠀⠀⠀⢀⣿⡇⣿⡇⠀⠀⠀⠀⡿⠤⠤⢀⠀⠢⠈⡏⠙⠿⣿⣦⡙⢲⠶⣿⣿⠉⠀⠉⠂⠈⠙⢦⡀
⠀⠀⠀⢸⣿⡇⣿⡇⢠⡄⠀⣸⣁⠀⠀⠀⠈⠐⢤⡁⢀⡤⢞⣻⣽⣟⢤⣿⣿⣦⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢏⣿⣿⣷⣾⣿⣿⣷⡀⠁⠀⠀⠀⠀⢰⡽⢫⣾⣿⣿⣿⣿⣿⣌⢻⣿⣷⣄⣀⠀⡄⠀⢸⡆
⠀⠀⠀⢀⡾⣿⣿⡟⠉⠉⢻⣿⣿⣶⣄⡀⠀⡹⢋⣴⣿⣷⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⣧⣴⣿⣇⣀⣀⡀
⠀⠀⠀⣾⠀⣿⢸⣿⡆⠀⠈⣷⣝⠻⢿⣿⠞⣱⣿⣿⣿⡻⣻⣿⣿⣿⣿⣿⠀⣿⡿⣆⠉⡿⣻⣿⣿⣿⣿⣦⡈⠳⡄
⠀⠀⠀⣿⣤⣿⢸⣿⣧⣀⣤⣿⡻⣿⡷⠋⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⠀⣿⣷⡽⣿⣿⡿⣿⣿⣿⣿⣿⣿⡄⠈⢆
⠀⠀⠀⢿⣿⣿⠀⢻⣿⣿⣿⡿⠋⡁⣠⣦⣻⣿⣿⣿⣿⣿⡿⢉⣽⣿⣛⣋⣠⡿⢋⡟⢿⡟⣾⣿⣿⣿⣿⣿⣿⣿⡄⠈⡆
⠀⠀⠀⢸⣿⠁⠀⠀⠀⠉⠉⣧⣿⣿⣿⣿⣷⡻⢿⡿⢟⣩⣾⡿⠿⠿⠿⠛⠋⠀⢺⠀⠈⢿⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⢱
⠀⠀⠀⢸⣿⣿⣶⣤⣤⣴⠿⣻⣿⣿⣿⣿⣿⣿⣷⣖⣻⣿⣿⣿⣦⣤⣤⡄⠀⠀⠸⡆⠀⠈⢷⢿⣿⣿⣿⣿⣿⣿⣿⡇⡸
⠀⠀⠀⠈⢻⡿⠿⢟⣋⣵⠾⣻⣽⣿⣿⣿⣿⣿⣟⠛⠋⢻⣿⡛⠛⠉⣿⣠⠆⠀⠀⣧⠀⠀⠀⠳⢭⣛⣻⣯⣭⢭⠖⠊⠁
⠀⠀⠀⠀⠘⣷⣯⣭⣽⠾⠟⠛⣿⣿⢿⣝⡏⢿⣿⣦⣤⣼⣿⣷⠶⣶⣟⣡⠄⠀⢀⡟⠀⠀⠀⠀⠈⠛⠻⠿⠗⠋
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢻⡹⣆⠹⣿⡆⣿⡿⠛⢻⣾⡞⠋⠉⣿⠏⠀⠀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣌⢳⣼⣇⣿⣷⠀⡸⢟⣁⣀⣤⣿⡤⠖⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣝⡻⢷⣦⣴⣿⣿⣿⣿⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠳⠄⠀⠀⣠⠄⢾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢊⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣀⡀⠀⠀⠈⠉⠙⠆⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿⠂
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠆⠀⠀⠀⢨⡏⠁⠀⠀⢀⣀⡀⠀⠀⠐⠀⢹⡷⠶⢦⡾⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⣿⣿⠇⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢀⡶⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⡀⠄⠒⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⣞⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⢠⡀⠀⠀⠀⠘⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠋⠀⠀⠀⠀⠼⠁⠈⠙⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢀⡱⠀⠀⠀⡄⠀⠀⠛⠋⠛⠻⢿⣿⣦⠀⠀⠀⢰⣿⠟⠋⠉⠉⠙⠁⠀⠀⠀⠀⠀⢀⠻⢤⣀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡶⠶⠶⠶⣤⠈⠉⠀⠀⠀⠈⠁⣤⠶⠶⠶⣦⡄⠀⡆⠀⢠⡄⢲⣶⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⣠⡔⢢⣦⠀⠀⠁⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⣤⠀⢣⡄⢘⠇⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢐⡦⠤⠀⠁⡤⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢨⣶⣄⡀⠀⠀⠀⢀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠒⠀⠈⠀⠐⠢⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⣺⠁⠀⠹⣶⣾⠟⠉⠀⠁⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡊⠀⠀⠀⠀⠀⢀⣤⡄⢀⡌⠁⣦⣔⡆⠀⠀⠀⠀⠀⠀⠀⠲⢤⡴⠂⠀⠀⠀⠀⠀⠀⡘⢯⠏⠰⠏⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣢⣄⣀⣠⠾⠟⢻⣏⠙⠦⢩⡍⠀⣀⠀⠀⠀⠀⠀⢀⣠⣀⣤⣄⡀⠀⠀⠀⠀⣰⡟⠀⠀⠖⠂⠀⠀⠀⠀⢹⣆⣤⣠⣤
⠀⠀⠀⠀⠀⠀⢠⠜⠉⠉⠁⠈⠉⠀⠀⠀⠈⡼⠆⢠⠜⠁⠀⠙⣷⡄⠀⠀⠘⠿⣿⣿⣿⣿⠟⠀⠀⣀⢰⣦⠀⠀⢹⠄⠀⠀⠀⠀⠀⠠⡟⠉⠙⠉⠀⢀⡀⠂
⠀⠀⠀⠀⡠⠀⠀⣀⠰⠆⠀⠀⠀⢀⡄⠀⠀⠀⠉⣸⠀⠠⠲⠤⠛⣿⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢺⣟⡀⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⢸⣦⡀⣀⢀⠀⠈
⠀⠀⠀⠈⣠⡜⠋⠀⠀⠀⠀⠀⠠⣿⣧⡀⠀⠀⠀⠘⢦⣤⡶⣧⢰⡟⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠛⠻⠃⠂⣠⣟⡀⠀⠀⠀⠀⠀⠀⣸⠇⢈⠋
⠀⠀⠀⠉⡴⣙⠦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣀⣠⡾⠇⢠⠜⠓⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢤⠆⣝⠋⠀⠀⠀⠀⠀⠀⢰⣯⡔⠁
⠀⠀⠀⠀⠀⠛⢦⣤⣤⠀⠐⠒⠋⠁⠀⠀⠀⠀⠈⠻⠋⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹⡆⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠉⠉⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠦⠶⠆⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠋
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⣼⣿⣿⠟⠁
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠁
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣶⣶⣶⣶⣿⣿⠿⠛⠉
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⢷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⣿⣿⠃⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠻⠛⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠋⠀⠈⣧⠀⠀⠀⠠⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⡤⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡟⠁⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠿⠋⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠁⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠉⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢈⣧⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡄⡤⠤⣤⣤⣤⣤⠴⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡍⠻⣿⣷⣿⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⣀⣀⢿⣿⡘⣟⠻⣦⣄⣀⣀⡤⣀⣀⠀⣀⡴⠿⣿⡄⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠾⢏⣼⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣷⣠⣾⣿⣾⣿⡿⣿⣯⣿⠳⠀⠀⠀⠈⠉⠉⠀⠘⠛⠋⠉⠀⠀⠙⣿⣦⡀⠀⠈⠣⣀⠀⠀⠀⠀⣠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠇⣿⣿⣿⠁⠀⠀⠈⠉⠒⠊⠀⠒⠂⠀⠀⠀⠀⠀⠈⠻⣿⣶⠀⣠⡟⠀⠀⠀⣘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡏⣠⣿⣿⣿⡀⠀⠀⠒⠒⠒⠃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣧⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣿⣿⣿⣿⣿⣿⣷⣶⣤⣴⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⠰⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠆⠘⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⢿⣿⣿⣿⣿⣿⣷⡦⠱⣌⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣿⣿⣿⣿⣿⣿⣿⣃⣨⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⢻⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡷⠈⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⡴⠃⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣴⣟⣼⠃⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣳⡄⠻⣿⣧⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢀⡀⠈⠻⣿⣦⣀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⢷⣤⣀⠉⠁⢀⣿⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠉⣿⣬⣉⠷⣸⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠄⠁⠀⠈⠻⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣶⣦⣤⣄⣂⣀⣀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣿⣥⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣗⢾⣴⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⣽⡽⣷⣾⡿⠮⢶⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡶⢶⣦⡿⠿⣄⣨⡽⢦⣤⣼⡍⠑⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⡷⠶⢻⣈⣅⣼⣄⣀⣿⣨⢟⣹⡛⠛⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⢿⣟⡷⠖⣿⣿⣷⣀⣟⣟⢇⣉⣯⣿⡟⠿⣟⣿⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⢿⢷⢿⢹⣷⣄⡤⣯⣵⣯⣉⣉⣹⣿⢾⡷⣤⣴⣿⢹⢷⢿⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⡿⣻⢾⣻⢻⣿⣷⣿⣭⣬⣿⡋⣹⣏⢁⣿⣏⣌⣽⣶⣿⣟⠻⣻⢾⡛⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣿⣭⠷⣾⠋⣻⣇⣀⣀⣿⣁⣈⣽⠋⠉⢻⣁⡀⣽⣄⡀⣀⣿⡋⢫⡾⢋⣽⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⡛⢪⡶⠓⣾⢋⠆⣿⡩⢑⣯⣩⢟⣿⠓⡏⣝⢹⣖⠀⣹⡁⢌⢻⡖⠳⣮⠚⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣄
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣷⣾⠕⠉⠉⠀⠉⠉⠉⠓⠊⠓⠒⠛⠒⠂⠉⠉⠁⠈⠁⠙⢽⣴⢾⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣋⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣏⡿⡿⠻⠿⠿⠿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠘⠿⡿⠿⠿⠟⠻⣿⣿⣿⣿⣿⣿⢻⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡯⣿⢻⣿⣷⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣽⣿⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣙⣿⣿⣿⣿⣿⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣻⡥⠬⣏⣵⠀⠔⠊⠉⠉⠙⠓⠦⣄⠀⠈⠈⠀⢀⡴⠓⠚⠉⠉⠁⠲⠄⢰⣉⡿⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠰⢢⡘⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣰⠖⢸⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠈⠁⢹⠀⠐⠀⠦⠀⣤⠄⠤⠤⠀⠀⠀⠀⠠⠤⠄⢤⠄⠠⠆⠐⠀⢸⡀⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣰⣿⠿⠿⠿⠿⠿⠿⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡆⠀⠀⣄⠀⠀⠀⠀⠈⠀⠉⠙⠋⠉⠁⠉⠀⠀⠀⠀⢀⡄⠀⠀⠛⠻⠿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢿⣿⡇⠰⠀⣿⣦⡀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣠⣾⡇⠰⠀⣄⣀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⣠⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣠⣤⣶⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠛⠛⠉⠉⢙⡟⠉⢉⣭⣭⣤⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡴⢶⡽⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠄⢀⡀⠀⠀⠀⢀⣠⣠⣀⡀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⢇⡄
⠀⠀⠀⠀⠨⠵⢞⣳⡟⠻⠂⠀⠀⠀⣠⣀⣤⡀⠀⠀⠀⠈⢢⣄⣠⡄⠃⠀⠀⠀⣿⣿⡟⠁⠀⣀⣾⣿⢋⠈⠀⠀⠀⠀⠀⠀⠐⣯⠁⡄
⠀⠀⠀⠀⠀⠀⠁⢻⠀⠀⠀⠀⠀⠰⣿⠿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡇⠠⠔
⠀⠀⠀⠀⠀⠀⠀⡎⠰⠃⠀⠀⠀⠈⢾⣿⣽⡞⡗⠀⠀⠀⠀⢲⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⢀⠀⠀⠀⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠉⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⣤⠀⢘⣦⠾⠁⠀⠠⠐⣦⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⢠⣤⢞⠀⠀⢀⠈⡄⢠⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠔⠋⠀⠊⠀⠀⠀⠀⠀⠀⠅⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡟⠋⠉⠁⠀⠀⠈⠀⢑⡺⡍⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡇⠀⠀⠀⠀⠀⠈⠰⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢀⡄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣨⠁⣉⠀⠀⠀⣶⡶⠀⠀⠀⡇⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⡀⢉⠀⠀⠀⠀⠆
⠀⠄⠀⠀⠀⠀⣀⠀⡦⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡃⡿⣃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⠀⠀⠀⠾⡷⠄
⠀⠂⠀⠀⠀⠀⠈⢰⡖⠚⠀⠀⠸⠿⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣾⠆⠄
⠀⠂⠀⠀⠀⠀⠀⠘⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉
⠀⡂
⠀⠁
⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠒⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠁⢀⣀⣤⣤⣤⣤⣤⡤⢤⣀⠀⠀⠉⠛⢦⣀
⠀⠀⠀⠀⠀⣠⠞⠋⠀⢀⣤⠞⣡⠞⠋⠀⠀⠀⠈⢹⣤⣬⣝⣲⢤⡀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⡴⣤
⠀⠀⠀⠀⣴⠋⠰⣴⣶⣯⣿⠾⣿⠀⠀⠀⠀⠀⠀⠈⢹⠉⢙⠏⠷⣿⣦⡀⠀⠹⣦⠀⠀⠀⠀⠀⢀⡇⢸⡇⠀⠀⠀⡼⢀⡟
⠀⠀⠀⣼⠃⠀⣰⣿⣾⡇⡏⠀⢻⣤⠀⠀⠀⠀⠀⠀⣞⠀⣴⠋⠙⠋⠙⢿⡄⠀⠘⣇⠀⠀⠀⠀⣸⢀⡿⠀⠀⠀⢰⢁⡼⠁
⠀⠀⢸⠇⠀⢰⣿⣿⡿⢷⡤⣠⣞⡿⣶⡶⠒⠚⠋⣷⠻⡞⠉⠀⠀⠀⠀⠀⢿⡀⠀⢹⡄⠀⠀⢀⣏⣸⠃⠀⠀⢀⣿⡿⠁
⠀⠀⣿⠀⠀⣿⡏⠀⠀⠀⠀⠈⣹⣷⠃⠀⠀⠀⠀⠙⣦⣿⣄⠀⠀⠀⠀⠀⢸⡇⠀⠸⣧⠀⠀⣸⢧⡟⠀⠀⠀⣾⣿⠃
⠀⠀⣿⠀⠀⣿⠁⠀⠀⠀⠀⠀⣧⣾⠀⠀⠀⠀⠀⠀⠘⣧⡙⣧⠀⠀⠀⢰⡞⠧⠀⠀⣿⠀⢠⡏⣼⠁⠀⠀⣼⢻⠇
⠀⠀⢻⡄⠀⢸⡆⠀⠀⠀⠀⠀⠘⡏⢷⠀⠀⠀⠀⣀⡴⢻⡁⠳⠶⡦⠞⠋⢁⠁⠀⢸⣿⠀⣼⢳⡏⠀⠀⣰⣯⡟
⠀⠀⠘⣧⠀⠈⢿⣶⣀⣀⣀⣴⡞⣠⣼⣿⣓⣲⣟⣯⠴⢺⣵⣤⢄⣕⢰⡟⡘⠀⢀⣾⡏⢠⣏⡾⠀⠀⢠⣏⡾
⠀⠀⠀⠹⣇⠀⠘⢿⢭⠛⢭⢫⡖⢤⣿⠈⠉⠁⠀⠀⠀⠀⢸⡟⠛⡇⢚⡴⠁⢀⣞⡿⠁⣼⣿⠃⠀⢀⡞⣾⠁
⠀⠀⠀⠀⠹⣧⡀⠀⠻⡀⠀⡏⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⠧⠛⠀⣠⣾⡿⠁⢠⣯⡟⠀⠀⣜⣼⠃
⠀⠀⠀⠀⠀⠈⠳⣤⡀⠈⠛⢧⡄⠀⢘⠳⢤⣤⣀⣀⣀⣴⠻⠒⠀⠀⣠⣖⣿⡟⠁⠀⡞⣾⠁⠀⡸⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠉⠉⠀⠈⠉⠉⠁⠀⢀⣠⣴⣞⣵⠟⠋⠀⠀⢰⢳⡏⠀⣰⢃⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣶⣶⣤⣤⣤⣤⣤⣶⣾⣽⠿⠛⠉⠁⠀⠀⠀⢀⡏⡾⠀⢰⢇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣩⣭⣤⡴⠤⠤⣤⣤⣀⡀⠀⣸⣹⠃⢠⡏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉⢉⣩⣤⣴⣤⣤⣤⣤⣈⡙⠛⢳⣯⣏⠀⡞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢁⣠⣶⣿⠿⠛⢻⠀⠘⣿⡄⠰⠛⠛⠳⣾⣼⠛⡿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢁⣶⠟⠉⠀⠀⡄⠀⠀⡃⠀⠀⠀⠀⢀⣀⣰⣿⠟⣿⣻⠟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⣿⡯⠥⠴⢖⠙⠑⠀⠀⠠⠀⠀⠒⠋⠀⠀⣿⣽⢠⢿⡿⢦⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⡿⠁⠀⠀⠀⠨⡀⠀⢈⠀⠀⠀⢀⢀⣠⡀⣰⣿⣏⣿⡟⠀⣠⣆⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣷⡶⠖⢏⠁⠀⢀⡤⠘⠂⠀⠈⣹⣆⠀⣻⣿⣿⣽⣿⠋⠉⠀⠙⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡷⢦⣿⣁⠀⠀⠼⢄⠀⠈⡅⠀⢀⣠⡴⣼⡟⠀⢸⣿⣽⡿⠁⠀⠀⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⠟⠃⠀⠀⠠⠀⠒⠋⠉⢠⣣⡾⢻⣇⠀⠻⠇⠛⠓⠚⠁⠀⢱⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⡄⢹⠊⠛⠠⠀⠀⡀⢀⠤⠠⠘⢿⡄⣰⣿⠀⠀⠀⣀⣤⠴⠆⠀⠀⣿⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⡄⢷⠀⢃⠀⠀⠀⠀⢀⡀⡀⠈⡿⠼⠟⠓⠉⠉⠁⠀⠀⠰⡀⢠⠇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⡄⠣⠈⠤⠰⡖⠉⠙⢃⠀⠀⢷⠀⠀⢀⣀⡤⢸⡄⠀⠀⡏⠉⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡑⠀⣀⣀⠀⠀⠠⡞⠂⠚⠁⠀⠀⠹⠁⠀⣇⢀⠐⢡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣆⠀⠙⠷⣤⣀⣷⡐⠒⠀⣤⢄⣾⡴⠶⢛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣍⣉⡁⠈⣉⣉⣉⣤⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠸⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠧⣄
⠀⠀⠀⠀⠴⠁⢰⣤⣶⣀⣀⣀⠀⣀⠄⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⣀⠀⠀⢠⣤⣄
⠀⠀⠀⢀⠠⡀⣨⣙⡌⠹⢷⠾⠿⠿⠟⠓⠉⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢻⣷⣦⠈⠈⣿
⠀⠀⡀⢻⣆⠰⠿⣯⡚⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐⢦⡀⠀⢙⣥⣖⣛
⠀⠀⢧⣴⡪⠿⣻⠽⢢⣦⡶⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠉⠫⠀⠾
⠀⠀⠀⠱⣛⢣⣠⣴⣾⣿⢛⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠸⢇⣸⣿⣶⣦⣶⡆⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠊⣉⠀⠀⠰⠶⢚⠆
⠀⠀⠀⠀⠀⠚⠛⠉⠉⠛⢿⣻⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⢀⣐⣤⡰⠾⠮⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠉⢳⡶⠆⢀⠀⠀⠀⠤⢀⣀⣄⣤⣤⢰⡿⠦⠦⠿⠷⡶⠊⠀⢨⠉
⠀⠀⠀⠀⠀⠀⠈⠓⠻⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⢊⣁⣠⡤⠚⠛⠫⡀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢛⣿⢏⣿⡏⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠋⠀⠘⢁⣾⡟⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⠛⠠⠏⣾⢻⣿⣇
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠸⣧⣴⣶⣿⡶⠀⠀⡀⠀⠉⢈⡾⣿⡄
⠀⠀⢀⣴⠞⠋⠉⠉⢻⣆⠀⢨⣿⣿⡋⠀⣠⠎⠀⠀⢸⡏⠀⠸⡟⣦⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⢠⡟⠁⠀⠀⠀⢰⡀⣿⠀⠼⢽⡛⢿⣿⠿⠀⠀⠀⢈⣷⠀⠀⠃⠈⣿⠿⠛⠋⠉⠁⠉⠛⠋⠉⠉⠓⢦⡀
⠀⢸⠁⢀⣠⣤⣠⣾⡇⣿⠀⠀⠀⠻⢾⡷⠆⠀⣠⣶⣿⣿⣷⢤⣀⡀⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠙⣆
⠀⢸⡀⣿⣿⡿⠿⣿⣿⠃⠀⠀⠀⠀⠈⠓⠶⢞⣻⠻⣿⣿⣿⡄⠉⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⢸⡆
⠀⠘⣇⢻⡟⠀⠀⠈⡇⠀⣀⣤⣶⠶⠶⢿⡿⠟⠁⠀⠉⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣈⢿⣷⣄⡀⠀⢾⣧⡀
⠀⠀⠸⡌⢻⡀⣀⣼⠃⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣿⠛⠛⠋⠸⣷⣿⠟⠋⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣷⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⢸⡄⠀⠀⣀⣿⡏⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣴⡿⠁⢸⣿⣧⡀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⣿⠿⣿⣿⣿⣇⠀⠀⠀⢰⡿⢻⣧⡀⠀⠀⠀⠀⢀⡼⠿⣿⡛⠶⠶⣶⡿⠛⠉⠀⢀⣾⠛⣿⡟⣷⡶⠶⠞⠛⠛⢻⡆
⠀⠀⠀⠀⣿⠀⠀⠀⢹⣿⣆⠀⣠⣿⠃⠀⠙⢿⣶⣤⣤⠖⠁⠀⠀⠈⠙⠀⠀⠀⠉⠳⣦⡴⠾⠟⠛⣿⡉⠁⠀⠀⠀⣀⣴⣿⣿
⠀⠀⠀⠀⢸⡄⠀⠀⢸⣿⣿⣾⣿⠏⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠘⣷⡀⠀⠒⠋⠉⠁⣰⡿
⠀⠀⠀⠀⠀⠻⣆⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⣿⡄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⠀⠀⠀⢠⣿⣷⣶⣶⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠻⣦⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡙⠿⢷⣶⣶⣶⠿⠿⠛⢛⣉⣡⣿⣶⣶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣤⣤⣤⣴⣶⣶⣿⣿⣿⡿⠟⠋⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣧⣬⣙⣿⣿⣟⣛⡛⠉⠁⠀⢀⣀⣠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣷⣿⣿⡿⠋⣩⣿⡟⠻⢿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣷⡾⠿⣿⣿⣿⣿⣿⣿⣟⣻⣷⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⡿⣯⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣧⡙⣿⡟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠘⠻⣿⣿⣿⣷⣼⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⠶⠆⠀⠀⠚⠛⠉⠁⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡇⠀⡆⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⠿⣿⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⡿⣿⣿⣷⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣨⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣷⣾⣿⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⣿⣼⡛⢿⣿⣟⣻⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣏⣿⠆⢉⣻⡿⠿⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⡷⠆⠀⠒⠀⠀⠀⠀⠀⢰⣄⣿⣿⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠩⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⣧⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠁⠄⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣷⣶⣦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣷⣶⣶⣿⣶⣦⣤⣀⣀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠤⢖⣚⣿⠭⣝⣛⡒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣄
⠀⠀⠀⠀⢀⡴⣫⣵⡎⠡⣄⢂⡅⠘⠉⠈⠙⠂⢝⠦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣙⠦⣑⢄⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⢠⢎⡜⠀⣰⣽⠶⢦⠰⡌⠀⠀⠀⣀⡀⠀⣷⡝⣆⠀⠀⠀⠀⠀⠀⠒⠒⠒⠺⠷⣮⣿⣿⠟⠁⠀⠀⠈⢻⡿⡿⢡⠆⠙⠉⠑
⠀⠀⢀⡏⣾⡗⠀⠃⠀⠀⣼⡏⢿⣦⣄⣀⣀⡠⢺⡿⢻⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣰⣿⣿⡏⠀⠀⠀⢀⡀⠀⣿⠦⠼⡀⠠⠀⣔⢰⣄
⠀⠀⢸⠃⡏⠁⠀⠀⣴⣋⣋⡏⢻⡤⣮⣿⡏⠀⠀⠀⠀⠇⣧⠀⠀⠀⠀⠀⠀⠀⠀⣧⡻⣿⣿⣿⣾⣧⣾⣿⣶⠟⠀⠀⠀⢹⣤⣼⡶⢏⡜
⠀⠀⠸⡄⣿⣿⣀⣴⡿⣿⡯⢜⡾⣇⣾⣿⠃⠀⠀⠀⠀⡆⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣮⣭⣿⣛⣛⣛⣛⣓⣂⣀⣀⣮⣭⣵⣾⣿⠁
⠀⠀⠀⢧⠹⣿⣿⣿⣯⣟⡶⣶⣿⣿⣿⣇⢀⠀⠀⢀⡸⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠳⡝⢿⣿⣧⣿⣶⣿⣿⣿⣿⣿⣼⣀⣤⢟⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠲⣍⡳⠽⣿⣿⣿⣿⣿⡿⢟⣫⠵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣭⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⠶⠶⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣲⣦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⢶⠓⠐⠀⠉⡡⢛⠀⠢⣄
⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠥⡀⣰⣄⣀⢄⡀⠀⠀⠀⠀⡠⣐⡈⠠⠖⠛⠉⠀⠙⣶⣶⡞⠃⠀⠀⢀⣤⣀
⠀⠀⢀⢰⣿⡿⣿⠿⢿⣿⣻⣿⡿⢿⣿⡿⣿⠿⢧⡀⠊⠉⠉⣿⣿⣦⠂⠀⠀⠐⠋⠉⠇⠀⠠⠤⣤⣴⠞⠉⠀⢀⠀⣀⣠⣼⡿⢻⠄
⠀⠀⠀⠑⢽⣻⠏⢀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⣆⠀⢀⡠⢿⣻⠕⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠣⠤⠄⠀⠶⢭⣭⣽⣯⠭⠞⠃⠀⡠⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠯⠭⢭⣿⣶⣶⣶⣮⡤⠥⠭⠗⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠤⠀⠠⠤⠤⠤⠈⠁⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠨⠅⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⣀⠀⠀⠀⠀⠃⣐⣂⢱⠌⢀⣀⠘⠄⠀⠀⠀⣀⠀⠐⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠃⠀⢄⠊⣁⠁⠉⠉⢀⡄⣤⢰⡿⠍⠭⢿⡆⣤⢬⡀⠈⠉⠈⣈⠱⡠⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠎⢠⠤⢅⡸⣷⣦⡣⠠⠔⠙⣿⡗⢸⣿⠃⠣⠄⢜⣶⣾⢇⡨⠠⠄⠱⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢶⣮⢦⢓⣻⡇⠄⠲⠀⠀⡉⠁⠀⠀⠖⠠⢺⣿⡊⡡⣵⡶⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠼⡻⠿⠾⠐⠀⠀⠀⢠⡀⠀⠠⠄⡄⠀⠀⠈⠢⢿⡿⢟⠧⡀⠘
⠀⠀⠀⠀⠀⢀⠤⠂⠀⣯⣀⠆⢠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡄⠰⡀⣽⠀⠰⠤⡀
⠀⠀⢀⠂⢐⠂⠰⢸⣴⡶⣶⣤⠂⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠂⠀⣤⣶⢶⣦⡇⠆⠐⡀
⠀⠀⡁⠀⠆⠆⡑⠶⡃⡅⢩⣉⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠰⣉⣅⢸⢸⠰⢉⡰⡰⠀⢈
⠀⠀⠈⠀⠸⠄⠐⢸⠻⠷⡿⠛⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠛⠿⠾⠟⡇⠂⠠⠁⠀⠁
⠀⠀⠀⠀⠀⠈⠒⠄⡀⣿⠉⠆⠘⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠃⠰⠉⣻⢀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢲⣵⣶⡶⠢⠀⠘⠘⠀⠀⠀⠀⠀⠀⠃⠁⢠⠔⣶⣶⣮⡖⠁⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠾⢟⠎⡠⣾⡧⠂⠴⡀⠀⢂⡐⠀⠀⠦⠐⢸⣿⢅⡐⡻⠷⠀⢈⠄
⠀⠀⠀⠀⠀⠀⢀⠀⢆⠐⠒⡊⢱⡿⠿⡕⠐⢢⢠⣿⡇⢸⣿⡄⡔⠂⢪⠿⢿⡎⢱⠐⠂⡰⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠊⢆⡉⡁⣁⡀⠈⠃⠛⠸⣷⣒⣐⣼⠇⠛⢚⠁⣀⡀⢀⢉⡰⠑⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄⠀⠉⠀⠀⠀⠐⡄⠩⠍⡈⢆⠩⠉⢰⠀⠀⠀⠀⠉⠀⠠⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢔⡂⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⢀⠄⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡀⣀⡉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⢀⡆⠀⣠⡴⠆⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡀⠀⣤⠀⠀⠀⠀⠀⡰⠋⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢸⣿⣿⣿⣿⡇⠀⠑⠂⠀⠀⠀⠘⠁⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⠋⠀⠀⠈⣄⠀⠈⣵⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⡉⠉⠉⠉⠁⠀⣤⣴⣿⠇⣠⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣠⠞⢁⡀⠀⠀⠀⢈⣿⣟⣡⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⡏⠀⠈⠁⠀⢚⣤⠾⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠴⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠁⢠⡴⠛⠉⠋⠙⠲⡀⠠⠀⠀⠀⠀⠀⠀⢀⡀⠘⠋⠉⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡴⠋⠁⢀⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⡟⠠⣀⡴⠋⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⠈⠛⢲⣦⡀⠀⠀⠀⠀⠏⠀⣀⠤⢄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡆⠀⠀⠚⠻⣿⣿⣶⣶⣀⣀⠴⠂⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠸⣿⡤⣄⡀⣴⣿⣿⠛⢿⡿⠋⠀⣠⣾⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⢈⣉⡁⠠⣿⣿⡏⣠⠞⠀⣠⣼⣿⣿⠋⢀⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢃⢀⣠⡴⠾⠛⠛⢻⣿⠋⠀⣸⣿⣿⣿⣿⣶⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠾⢋⣡⣴⠶⣿⣿⣿⠃⢀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣼⣿⣯⣶⣾⣿⣿⠃⠀⣸⣿⣿⣿⣿⣿⣿⠿⠟⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⣽⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⡿⢙⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢀⣠⣤⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣿⣟⣸⣷⣿⣷⣶⣿⣿⣿⠏⢸⣶⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣃⣠⠞⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⣷⣍⠍⡛⠛⣿⣿⣿⣿⣿⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⡄⠈⠀⠀⢩⣿⣿⡇⠸⣿⡆⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣿⠀⠸⠀⠀⠒⣾⣿⣿⠀⠀⠈⣿⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⠀⡄⠀⠀⣼⢿⣿⡇⠀⠀⠀⠸⣿⣶⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢰⠀⠀⠸⢧⣿⠁⠀⠀⠀⠀⠈⠻⢿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⡀⢸⡄⠀⠇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠿⠇⢘⠀⠘⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆⠐⠉⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣸⠏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡿⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠃⠀⢙⣿⣷⡄
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⠤⠀⠒⠈⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⣤⡤⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠙⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠈⢿⡆
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡀⠀⠈⠻⣄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣦⣤⣄⣹⣆
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⣨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠟⠀⠈⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆⠀⠀⠀⠲⠀⣇⣀⣤⠤⠶⢶⡶⠚⠉⠉⣇⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠏⢡⠞⠃⢠⡀⠀⢤⡶⠋⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠎⠉⠣⠀⠀⠘⢦⠀⠀⠙⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢀⠇⠀⠀⠀⠀⢠⠂⠀⡄⠀⢀⣄⣀⣸⠇
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⢠⡏⠀⢸⣃⡴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡾⠁⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⡙⠉⠉⠙⠛⠓⠶⢒⡶⢄⡀⠀⠠⠏⠀⠀⠀⢀⡾⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣤⣤⣤⣤⡄⠀⠀⠢⢍⡂⣤⠀⠀⠀⣠⡾⠁⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠔⠁⠐⡁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣮⣅⡀⠀⠀⠀⠀⢀⣴⠊⣠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⠀⠈⠛⠿⣿⣶⣶⣶⣿⣿⣿⣿⣯⣉⡙⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⠻⣷⠀⠀⠀⠀⠀⢸⣿⠛⠿⠛⠛⠛⠛⠿⢿⣿⣿⣮⣙⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠁⠀⠀⠘⣆⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠙⠛⠳⠎⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠶⣤⡀⣾⣧⠀⠀⠀⠀⣠⣤⣤⣴⣧⣤⣤⣀⣀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋⠙⢻⣿⣤⣴⠶⠛⠉⠉⢿⣛⠒⠶⠶⠶⠦⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⢠⡾⠛⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢀⡿⠀⠀⠀⠀⢠⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣸⣇⡔⠀⢰⡆⢸⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣿⡟⠀⠀⢸⣇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠓⠀⠀⠀⡆⠀⡇
⠀⠀⣼⠏⠀⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢠⡇⢀⡇
⠀⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⣄⣸⡇⢸⡧⢄
⠀⣼⣁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠒⠚⠃
⢼⣋⣉⣉⣓⣦⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣶⣾⣟⠛⠲⠶⠶⠤⣤⣄⣀⡤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠉⠓⠆⠀⠀⠀⠀⠀⣠⠤⠶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⡶⢲⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣤⡾⠃⠀⠉⠻⠆⢠⡀⠀⠀⠀⠉⠉⠉⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⡜⣿⠀⠀⠀⡇⠀⠀⠀⢠⢶⣆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠛⠛⠀⢀⣼⠇⠀⠀⠀⢻⣾⠏⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠙⠳⣄⠀⠈⡆⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⠤⣤⣀⠀⠀⠀⠀⠀⢰⣿⣟⣷⠀⠘⢷⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⢷⣄⢀⣴⠏⠀⠀⠀⠀⡠⠘⣿⣌⡁⣀⣠⡾⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⠛⠛⠁⠀⢀⣠⡶⠋⠀⢸⡈⠙⠛⠛⠉⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣤⣤⡶⠟⠉⠀⠀⠀⢸⣇⠀⠀⠀⣠⠃⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⠀⠀⠀⠀⠀⠈⢿⠀⣠⠞⣡⡾⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠞⢁⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠇⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣧⡀⠙⢷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠉⠙⣷⠄⠀⠀⠀⠀⠀⠀⠀⠷⠶⠋
⠀⠀⠀⠀⠀⠀⢀⣀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⡴⠞⠋
⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⣼⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠘⢷⣄
⢀⣠⣀⠀⠙⢷⣄
⣿⡹⣿⠇⠀⢈⣿
⠘⠿⢦⣤⡶⠾⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢾⣿⡗⢰⣿⡶⢿⡿⠀⢸⣿⡷⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣰⣿⣷⣄⣠⣬⡄⢠⣄⡀⢠⣤⣶⠠⣭⣄⢄⣾⣿⡇⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠺⠿⠋⠉⠁⡆⠙⢛⣡⡘⠿⠻⣌⠻⠃⣄⡙⠟⢸⠈⠙⠙⠿⠿⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⣿⠿⢃⡠⠐⠒⠀⢵⡰⣿⣿⠃⢴⣿⣯⠳⡀⢿⣿⠇⡼⠀⠂⠒⢠⣀⠿⣿⢇⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡙⠛⣁⡄⠀⢻⡿⠃⣴⣿⣷⠻⣌⣡⣴⠾⠿⠿⠶⣷⣄⣡⠞⣱⣿⣦⠐⢿⡟⠀⢠⣌⠛⢋⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⢎⠻⢇⣶⣆⢀⣄⠀⣍⠁⢠⡾⠋⠀⠀⣠⣄⠀⠀⠙⢿⣆⠀⣩⡀⢀⡄⢠⣶⡜⠿⣣⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⠄⠈⠳⣜⣛⠋⠾⠿⠚⠿⢇⣿⠁⠀⠀⣴⣿⣿⣧⠀⠀⠀⢿⡞⠿⠇⢿⠿⠘⢟⣫⠔⠁⠀⣿⣿⠆
⠀⠀⠀⠀⠀⠀⢀⣄⡐⣶⣦⢀⣔⣉⠈⣭⣽⠉⣉⣛⠻⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣸⡟⢛⣉⡉⣫⣭⡉⣉⡳⡀⣴⣶⠆⣠⣀
⠀⠀⠀⠀⠀⠀⠈⠟⣡⠈⣥⡋⠻⣿⠀⠿⠛⠈⣿⠟⠀⢿⣆⠀⠀⠛⣿⣿⡟⠀⠀⣠⡿⠁⠹⣿⠃⠙⠿⠇⢿⠟⢙⣮⠉⣀⡻⠃
⠀⠀⠀⠀⠀⢀⣠⣬⡛⠀⠛⠋⠀⠀⠸⣿⣶⡄⠀⣠⡶⠾⢿⣷⣶⣾⣿⣿⣷⣶⣶⡿⠷⢶⣄⡀⠀⣶⣿⡟⠀⠀⠈⠛⠂⢛⣡⣄⡀
⠀⠀⠀⠀⠀⠀⠹⣿⠃⡀⣰⣷⢰⣷⣦⠉⠉⣠⢞⣩⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣠⣬⡻⣦⡉⠋⣴⣾⡇⣼⣧⢀⠜⣿⠟⠁
⠀⠀⠀⠀⠀⠠⣤⣄⢠⣬⡻⢭⣀⣛⣫⠶⢚⣁⠘⣿⡿⠀⣼⣿⢻⣿⣿⣿⣿⣿⣿⣿⡄⠻⣿⠇⣈⣛⠶⣝⣛⣁⣩⢟⣥⡤⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠛⣃⠈⠛⠓⠉⢤⣭⣶⠀⢿⣿⠃⠀⠀⣠⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠃⣴⣮⡥⡈⠚⠙⠁⣈⠛
⠀⠀⠀⠀⠀⠀⠘⠿⠃⣼⣿⣞⢀⡙⠟⠀⠘⠁⠀⠀⣰⣿⣯⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠁⠀⠙⢏⣁⢱⣿⣷⠀⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠘⠿⢿⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⡿⠿⠆⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠂⠀⠀⣀⣠⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣀⡀⠀⠀⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠈⠑⠦⢶⡋⠹⠲⣤⣠⣴⠿⢋⣿⣿⡝⠻⢦⣄⣤⠖⠻⠙⡶⠴⠂⠁⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢦⣄⢈⣀⣀⣴⠏⡡⡇⢀⣼⢹⡏⢷⡀⢸⢆⠙⢦⣀⣀⡀⢠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⢸⠷⡀⠀⠠⢏⠘⢘⡇⠂⣸⠆⠀⢀⡴⡇⢸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠙⠀⠀⠘⡀⠘⠇⢀⠃⠐⠀⠊⠡⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⣀⠀⠀⡠⠴⠂⠀⠀⠀⠘⢳⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣠⠀⠠⠊⢃⣠⠀⣀⣀⢤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⡄⠉⠁⠀⠀⡄
⠘⣭⠁⠀⠀⢀⠤⠀⠀⠀⠀⡠⠂⠀⣀⠵⠛⢿⡁⢀⠄⠁⠸⠋⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠠⠁⢈⠁⠀⠤⡇
⠘⠫⠀⠀⠀⠀⠀⠀⠤⠤⢀⢰⠗⠂⠀⠀⠀⣀⠄⠁⠀⢀⠄⠐⢡⠀⠀⠀⢀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⢌⠀⠀⠠⠈⠀⠀⠄⠄⠀⠀⠀⠴⠖⠋⠀⠀⠀⣀⠉⡔⠃⠹⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠁⡐⠀⠄⢀⠀⠠⠉
⠀⠀⠈⠐⠒⠒⠓⠒⣚⠉⡇⠀⠐⢠⡈⠁⠀⠀⠀⠀⡤⠖⢀⡀⢠⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠠⢀⣉⡥⣦⠠⠔⢛⠀⡆⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣠⣾⣿⣽⣽⣯⣽⣿⣿⣿⡿⠋⠉⠉⠉⠉⡉⠉
⠀⠀⠀⠀⠀⠀⢉⣍⠀⠀⡞⡇⣠⠎⢀⡶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⡄⠀⠄⠠⢀⡤⠀⡁⠀⠓⠈⢀⡜⢸⠀⠄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⢀⢲⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠤
⠀⠙⠀⣠⠀⠈⠀⠀⠛⠂⠀⡄⠊⠀⡏⠀⠀⠐⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⢋⠘⡀⠀⠀⢀⠄⠠⠀⠀⠀⠀⠀⡤⠀⠂
⠀⠀⠀⠘⠁⠀⠀⠀⠀⡀⠀⡆⠀⢸⠁⡆⠀⠀⠀⠀⠄⠀⢠⠀⠀⠀⠀⠀⠐⠃⠀⠀⠐⢋⠠⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠂⣠⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠈⠀⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⡰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣤⣮⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣝⣩⣭⣹⣍⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⠦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠻⢿⣿⣿⠿⠟⠋⠀⠀⠀⠿⣿⠏⡐⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣀⠀⠀⠀⢀⣤⣴⠖⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠙⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠙⠲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣙⣄⢐⣀⣂⡁⠀⠀⠀⡴⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠉⠙⢧⠉⠉⠁⠀⢀⡜⠀⠀⣠⠞⠀⠀⠀⢀⣴⠶⠟⠛⠲⠾⣿⣿⣄
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣶⠾⢃⣤⡶⠋⠁⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⠋⠒⠋⠉⠚⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⡟⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡄
⢰⣏⠀⠀⠀⠀⠀⠈⢻⣦⣤⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⣆
⣾⣿⣷⣤⣤⣤⡄⢸⣹⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠈⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⣿⠻⢿⣿⣿⣿⢇⡼⠁⣿⣿⣿⣿⡄⠀⠀⠀⠀⠻⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢹⣀⠀⠀⢸⡀⠀⠀⠀⠈⣆
⣿⡛⠢⠤⠤⠴⠋⠹⣠⣿⣿⣿⠋⢻⣦⣀⠀⣀⣠⣾⣿⣿⣿⣄⡀⠀⠀⠀⢀⣠⣿⠏⠀⠀⠀⠀⡜⠉⠉⠛⠛⠿⢯⣭⡉⠛⢾⣆
⠜⠁⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⠟⠁⠀⠀⢀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⢸⡄⠀⠀⠀⠀⠀⠀⢸⡟⠁⣠⠋⠹⣿⡿⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢹⡅⠀⠀⢠⠀⠙⠛⠛⠻⠛⠿⢿⣿⣿⣿⣦⣤⣾⣿⣦⡀
⠀⣷⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠻⡀⠀⠘⣧⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⢦⣀⣀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠄⠸⡷⠶⠤⠀⠀⢦⡄⠈⠙⠋⠀⠀⠈⠻⣿⣿⣿⠈⠛⠢⠤⢤⣀⣀⡀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡦⠀⠀⠇⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⡀⠀⣀⣈⠻⡿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠷⠤⠤⠤⢤⠤⠤⠒⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣧⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢿⣇⠻⠀⠀⠀⢀⣷⠀⣠⣤⡤⠤⠄⠒⠒⢲⡶⠒⠒⠶⣶⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⡄⢀⡈⡿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢀⠙⡄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢀⣾⠃⠋⠀⠀⠀⠀⠀⢀⣠⡶⠃⠀⢠⡾⠁⠀⠀⠀⠀⠸⠀⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣡⣾⣿⣾⠀⠀⠀⠀⢀⣴⣿⠏⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣿⠟⠁⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣀⣴⣿⡿⠃⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠃
⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠿⠿⠿⢿⢿⣿⣿⠿⠿⡿⠿⣿⣿
⣿⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿

⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿
⠀⠀⣠⣴⣿⣷⣶⣦⣤⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣙⠛⠛⠋⠉⠁⠀⠛⢿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣦⣭⣥⣤⣤⣤⣴⣤⣤⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠓⠚⠋⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠈⠁⢈⣿⣿⣿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣄⠀⠀⠀⣾⣶⣿⣿⣿
⠀⠀⠀⢠⣾⣿⣿⣥⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠓⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠉⠉⠁⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣷⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⡀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣴⡿⠿⠋⠁⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⢰⣿⣿⣿⣿⣷⣤⣄⡈⠛⢿⣿⣿⣿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⢤⣴⡟⠁⢀⣀⣤⣤⣶⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠙⠿⠿⠿⣿⣷⠈⠛⢛⣻⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣶⣾⣶⣶⣿⠏⠛⢿⣷⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠆⠀⠀⠀⠉⠀⣀⣴⣆⠈⠉⠉⠉⠉⣠⣿⣿
⠀⠀⠘⠟⠛⠻⣿⡏⠀⠀⠀⠹⣿⡏⠁⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣾⣷⡆⢀⣾⣿⣿⣿⠀⠀⣤⣤⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⠿⠃⠀⠀⠀⠀⠉⠧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣱⣾⣿⣿⣿⣿⣧⣘⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢍⡱⠋⠉⠀⠀⠉⠒⢰⠆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⠻⠿⠓⢟⣀⠬⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⣳⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢿⣠⣤⡀⢼⣇⠀⢀⣞⠀⢲⣾⣿⣶⠛⠃⠀⣤⡙⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣧⣶⡿⠊⣡⣤⣤⡐⢀⣀⡤⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣏⢹⠉⠁⣞⣵⣷⣤⡎⣿⣿⣿⣿⣿⠯⠃⠉
⠀⠀⠀⠠⠤⠤⡄⢸⠀⠀⣆⡖⠀⠁⠃⣸⠘⣿⢻⡋⣰⡏⠻⠛⡌⣽⣿⣿⣿⡅⠎
⠀⠀⠀⠀⠈⠂⠀⢸⣇⠀⣡⠁⠀⠀⠂⠘⠆⢻⣆⣿⣿⣧⢀⣰⢧⣻⣿⠯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠛⠗⠀⠀⠀⠀⠐⠂⠹⣦⣽⣿⣮⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠓⠀⡤⠊⠁⠀⠀⠀⠀⣄⣾⣿⣿⣿⠒⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢿⡧⠄⠀⠤⠀⠀⠀⠀⠀⠘⢛⣿⠛⣿⣽⣿⡿⠘⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣤⣠⣼⠷⠶⠓⠒⠀⠀⠀⠀⠀⢀⡟⠙⢯⣜⣥⣿⢥⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠂⠈⠉⠙⠙⠆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠐⢀⠸⡇⠈⠑⢄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠳⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡈⠃⠀⠀⠤⠀⠀⠈⠀⢀⠀⠱⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄
⠀⠀⠀⢀⠐⠂⠐⠂⡃⢸⠀⠀⠈⠀⠀⢀⢀⡀⡀⠀⠀⠙⠢⣽⣀⡌⠀⣀⠶⡤⢤⣀⠀⠀⠀⣶⢰⠟
⠀⠀⠀⠀⡠⠐⠀⠀⠁⢨⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣣⠏⠀⡄⠀⠀⠀⠀⠤⣴⣯⣀⢀⡀⠀⢀⠶⡃
⠀⠀⠀⠀⠀⣠⠖⣃⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣯⣭⣭⠛⠀⠁
⠀⠀⠀⠀⠀⠨⠁⠙⡂⠀⠈⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠛⢿⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠂⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠢⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠛⠛⠻⠛⠸⠚⠀⠸⠜⠗⠪⠟⠛⠟⠃
⠀⠀⠀⠀⠀⠀⢀⣀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⠂⠑⠜⠣⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⣿⣿⡷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⣀⡀
⠀⠀⠀⠀⠀⣤⣮⡿⡿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢠⢼⡎⠳⢴⣤⡄⢳⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⣋⣻⣾⠏⠕⢀
⠀⠀⠀⠀⠈⠚⠓⠟⠛⠋⠿⠛⠡⠒⠂⠀⠀⠒⠛⠻⠳⠿⠿⠷⠲⠟⠛⠓⠂⠀⠀⠀⠀⠈⠙⠊⠀⠛⠛⠓⠐⠛⠋

⠀⠀⠀⠀⠀⢠⣶⣖⢶⡗⡖⡶⣄⠀⠀⠀⠀⠀⠀⣄⣤⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠯⣤
⠀⠀⠀⠈⠁⠞⠸⣼⡜⠟⠧⠃⠛⠟⠁⠀⠀⠀⠀⠄⠌⠀⠨⠃⠀⠉⠉⠀⠀⠤⠀⠀⠄⠘⢄⣤⣤⣤⣄⣁⣀⣌⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣴⡆⠀⣰⠒⣦⣟⡛⠓⠛⣷⣴⠁⠀⣤⣄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡗⢦⠄⠘⠿⠞⠁⠀⠘⠶⢿⣿⣷⣤⣶⡿⠋⠀⠀⠹⣿⠞⠀⣰⡀⠒⣒⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢺⠟⢟⠀⠤⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⡀⠒⠒⠄
⠀⠀⠀⠀⠀⠀⢀⠈⠂⠈⠆⠀⠀⠀⢀⠅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡟⡉⠉⠂⢰⠤⠐⠓
⠀⠀⠀⠀⠀⠀⠈⠁⠸⢄⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⠃⠀⠠⠈⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⢀⠀⠀⠀⠠⡀
⠀⠉⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀
⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡄⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡠⠀⠤⠶⠈⠊⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⡆⢀⡀⠑⠠⠐⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢦⠧⢄⠉⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⣠⡤⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⡀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠑⢂⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠋⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠚⠛⠛⠉⠙⠀⠀⢼⠹⣿⣷⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣪⣶⡿⣿⣾⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡀⡀⢸⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣿⣿⣿⠹⢭⡈⡛⠷⠜⡿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⢿⣷⣿⡆⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠏⢹⣿⣿⣧⠀⠀⠈⠿⠒⠋⠀⠉⠀⠀⠀⠀⠀⠀⠰⠁⢷⣿⣟⣁⣼⠟⢹⠁⢸⣿⠟⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠃⠀⣼⣿⡏⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⠀⢸⣿⠇⣿⣿⣿⣆
⠀⠀⠀⠀⢠⠞⠁⣴⡿⠀⠀⢀⣿⣿⢣⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡔⢰⣇⠀⣾⡟⡄⣿⣿⣿⣿⡄
⠀⠀⠀⠀⡈⠀⣴⠸⠁⠀⠀⣼⠋⣿⣿⠀⢱⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⣸⠂⣿⠁⠃⣿⡟⢿⡏⢷
⠀⠀⠀⠀⠃⣀⠏⠀⠀⠀⠀⣿⠀⣿⣿⡄⠈⠐⣦⡀⠀⠀⠀⠀⢤⣤⣴⡀⠦⣀⠀⠀⠀⠀⠀⡌⢠⢡⣿⣶⡟⠀⢀⣿⡇⠈⣿⠸⡇
⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⢰⣿⠀⢹⣿⡇⠀⠀⢷⣿⣦⡀⠀⠀⠀⠈⠏⠉⠈⠁⠀⠀⢀⣠⣾⢁⢖⣿⣿⡿⠁⠀⠘⣿⠁⠀⣿⠀⡧
⠀⠀⠀⠐⣤⠘⠀⠀⠀⠀⢻⣿⠀⠘⣿⡁⠀⠀⠈⣿⣿⣿⡶⢄⡀⠀⠀⠀⠀⢀⣠⠖⠋⣼⡇⣸⣿⣿⡿⠁⢸⠀⡰⠀⠀⠀⣸⡾⠃
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⣿⡄⠀⢻⡇⠀⠀⠀⣿⣿⣿⠃⠀⠙⠶⠤⠴⠖⠋⠀⠀⠀⣿⡇⣏⣿⣿⢣⠀⡆⢠⠇⠀⡴⢀⣿⠁
⠀⠀⠀⠀⡄⢀⣾⠀⠀⡀⣿⣿⡇⠀⣧⠁⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣿⣿⡇⠀⢃⡿⠀⣴⡇⣼⣿
⠀⠀⠀⠀⢀⡄⠀⡀⠀⡇⣿⣿⡇⠀⡏⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡞⠀⣾⠇⣸⠟⣱⣿⡏
⠀⠀⣠⠖⠛⠉⢿⣿⠀⢇⣿⣿⠇⢸⠁⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣷⣿⣿⡿⣾⡏⢰⣏⣼⣿⡿⠁
⠀⡞⠁⠀⠀⠀⡏⢻⣧⣘⣿⡿⠀⣸⠀⡆⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⣿⣿⣷⣿⠁⡟⣼⣿⡿⠷⢤⡀
⠐⠀⠀⠀⣠⠈⠀⠀⢿⣿⣿⠃⢀⡇⡄⢠⣿⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠸⠻⡷⣼⣿⣿⣿⡇⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣾⠞⣡⡿⠋⠐⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⠠⠂⠀⠀⢾⡇⣽⣿⣿⣿⡟⠀⡠⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣼⣿⠺⠋⠀⠀⠀⠀⠀⠉⠛⠒⠤⠤⠤⠤⠴⠖⠚⢉⠀⠂⠁⠀⠀⠀⠀⣸⣷⣿⣿⣿⠟⡁⣏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡆⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⢀⠀⣿⡟⣿⣿⣿⣿⡷⣰⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⣿⡿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡽⣧⣹⣿⣿⡿⠛⠁⠀⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⡄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡶⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡯⣄⣼⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣾⣗⣲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠉⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⢠⡄⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⡀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢨⣿⣾⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠆⣶⣶⣾⣷⣄⠀⠓⢯⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠓⠿⠞⠘⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⠀⠀⠀⠀⢸⡇⢿⣿⣽⣿⣿⣗⣦⡀⠙⠂⠀⠲⣶⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⡄
⠀⠀⠀⠀⠀⡀⢤⣿⡿⡇⢠⡄⠀⠀⠈⣿⣅⣽⣯⡟⢉⣯⣿⣿⣢⣀⠈⠲⣿⣫⡴⠀⠀⠩⠀⠀⠀⠀⠀⠀⠠⣿⠀⡇⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣸⣿⣦⣾⣵⣿⣿⣗⠋⠘⣛⡽⣿⣿⣱⣋⣵⣄⡈⠙⢿⡷⠁⣀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠰⠇⢠⣤
⠀⠀⣠⣾⠿⢃⠀⠀⣾⠸⠲⠿⣿⣿⢿⣿⣿⣿⡟⠿⠛⣴⣿⣿⣿⡿⣻⡏⣹⡶⣞⣷⣾⣿⠃⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⢾⠟⠁⠀⠀⢷⠞⣹⡗⠀⢀⣈⡿⢨⣱⣷⣽⡿⢟⣚⣵⣿⣿⣿⣷⣿⣵⣿⣿⠾⠿⣿⡏⠀⠐⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠘⣯⣟⣳⣢⡟⠻⣶⣿⣿⣯⣟⣿⢺⣯⡟⢿⣁⣿⣿⣿⣿⣿⣿⣦⡀⢈⠃⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⢿⠿⣤⣿⡿⠿⠟⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢢⡀⠀⠀⠋⡀⠀⣠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣀⠀⠘⠿⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣯⣿⣯⣽⣿⠇⠀⠐⠂⠀⠊⠛⢿⣿⣿⣷⣄⡀
⠀⠀⠀⢶⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣾⠻⣿⣿⣿⣿⣿⡷⣦⡀⠀⠀⠀⠀⠀⠈⠛⢿⣏⣽⡆
⠀⠀⠐⣲⣆⡚⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡏⢿⠛⣻⡗⢾⣟⠳⣄⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠉⠀⠁⠀⠀⠀⣠⠞⠋⠟⡿⠉⠉⠻⠛⠛⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⡳⠷⡘⢌⣽⣦⡙⠇
⠀⠀⠀⢠⣄⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢿⣚⣿⣿⣿⣿⣿⣿⣿⣿⣾⠉⠿⣮⣻⣽⣷⣄⣀
⠀⠀⠀⠺⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣽⣿⣿⣿⠀⠁
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢀⡀⠀⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣿⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡼⣻⢿⣤⣾⣿⠀⢹⣿⣿⣿⣿⣿⡻⢱⣾⡿⣿⣿⣟⢻⣿⣿⣿⣷⣤
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⠾⣻⣿⣟⢀⣶⣿⠿⣿⣿⣯⣿⣟⠾⠛⢌⡻⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠐⠟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡿⣿⣦⣼⠿⣻⣿⠈⠀⣀⠀⠑⢤⣙⣿⣻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠀⠈⠀⠉⠒⠿⠃⠀⠴⠿⠶⠀⠈⠻⣿⡿⣏⠙⠿⣿⣿⣇⡀
⠀⠀⢀⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢗⠄⠙⠏⠙⢆⢀
⠀⠀⠀⠙⠛⠉⠁⠀⣾⢹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣹⠷⠾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣼⠷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢿⣿⣯⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢻⣬⠿⣏⢈⣛⡻⠻⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣃⣟⠋⢹⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠉⠛⠣⠻⣿⣏⣄⣠⢾⣄⣸⠿⢽⡦⠀⠀⣀⡀⢈⣱⠾⣍⠰⠗⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⣄⢀⢸⣄⠀⠀⠀⠀⠀⠐⠀⢈⣙⣛⠾⠶⢤⣄⣬⣇⡀⠻⡬⢵⡆⣄⠀⡀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡀⠉⠀⠻⣇⣰⣀⣀⣠⣿⠤⣯⣽⣟⣹⡄⠀⠈⠉⠙⢿⣴⣿⣧⣁⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠉⠉⠉⣥⣰⠐⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣧⣤⣿⣯⡙⠀⣤⡇⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⣮⠻⣶⣤⣤⣶⠶⠛⠁⠀⠻⢷⣇⡐⣦⣤⣀⠀⠀⠀⠀⠉⠙⠿⢁⠀⠈⠙⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠒⢻⠿⠿⣷⣶⠶⠆⠀⠀⠀⠀⠈⠹⣾⣿⣿⣶⠀⠀⠀⠀⠀⠀⠨⢳⣼⣿⣶⣶⣗⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡃⠉⠁⠠⠾⢿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⡥⠤⠄⠉⠛⡿⠧⢦⠤
⠀⠀⠀⠀⠀⠀⠀⢐⣈⡁⠀⠀⠀⠀⠀⠀⠀⢽⡁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢂⣤⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣼⠁⠉⠀⣀⠴⠋⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠉⠀⣀⠞⠁
⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡀⠀⠀⠀⠁⢠
⠀⠀⠀⠀⠈⠀⠀⠀⠃⣠⢾⡇⠀⠀⠁⢆⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠈⣷⡦⠀⢀⣾

⠀⠀⠀⢠⡄⠀⢤⡀⢤⣤⠶⣦⢤⡴⢤⡤⣤⢤⡖⣶⣠⢤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⡎⡌⡆
⠀⠀⠀⠀⣦⡭⠼⠀⠚⠀⠒⠠⠼⠷⠞⠧⠽⠧⠽⠁⠸⠄⠃⢀⡤⠒⢒⣶⣦⣄⠀⠀⠀⡀⠀⠀⠀⠛⠁⠀⠀⣤⢄⢀⡀⣀⡀⠀⠀⠰⡷⠁
⠀⠀⠀⠀⡛⡷⠀⠀⠀⠀⠀⢀⣠⢔⣶⣲⠠⢤⡀⠀⠀⣀⢀⢋⣤⠀⣾⣿⣿⣿⣷⡀⠘⠿⠿⠯⠭⠭⠭⠭⠭⣽⡟⣿⣿⣿⣷⣐⡐⣦⢸⣠⠊⡆
⠀⠀⠀⠀⡇⣵⠀⣏⡭⠥⠭⣝⣚⡅⠀⣀⢨⠟⠁⠀⠀⠋⠸⡀⠁⠀⠘⠿⣧⣤⣿⡇⠀⢶⣶⣗⣀⣈⣀⠀⠠⣿⣧⣿⣿⡿⡷⠶⠶⠆⠀⢻⡧⠃
⠀⠀⠀⠀⡿⣏⠀⠈⢆⣀⡀⠀⡱⠗⠘⠋⠁⠐⠂⠀⠀⠀⠀⠑⢤⡀⠀⢀⣽⣿⠟⠁⠀⠀⢀⠀⠀⠀⠀⢀⠀⢸⣸⢸⣿⡙⡖⠀⠀⢠⠒⢬⡄
⠀⠀⠀⠀⣧⡾⠀⠀⠀⠑⠒⠊⢁⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡈⠉⠉⠉⠀⠀⠀⠀⠤⣼⠆⣔⠰⠦⡾⠀⣣⡁⢸⡄⣿⠃⠀⠀⠈⠂⠽⢧⡖⢲
⠀⠀⠀⠀⡏⣻⠀⠀⠈⠁⠀⡜⠁⠀⠈⠑⠀⡴⡟⠀⠀⢈⡀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⡠⠤⡀⠀⠈⣿⠀⣿⣾⠀⠀⠀⠀⠀⠀⣿⠧⠊
⠀⠀⠀⠀⡇⠷⡆⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⣱⠟⣇⣠⢿⠶⠖⢧⣀⣾⢁⡴⣇⣀⠾⣅⣠⠖⠀⠀⢇⠀⠹⡄⠀⠃⠀⠁⢙⣆⠀⠀⠀⡵⠀⠣
⠀⠀⠀⠀⡇⣪⡀⠤⢤⣶⠄⠳⠤⠴⠚⠁⠀⠁⠀⢈⡁⠀⠀⠀⠀⠁⠀⠁⠀⠉⠁⢀⠀⠀⠀⠀⠀⠀⠑⢄⢣⡀⠀⠀⠀⢸⡇⠀⢀⣰⣦⣄
⠀⠀⠀⠀⢣⣶⡁⠱⣤⠟⠀⠀⠄⠀⠀⣀⡄⠀⠀⠘⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⡰⡏⡛⡖⢤⣠⣀⣀⡀⠀⠱⡑⣄⠀⠀⠘⢣⢊⣿⢁⣿⢸⣿⠢⡀
⠀⠀⠀⠀⠨⡚⠁⠀⠀⢀⡀⠀⠀⢀⣼⠋⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣠⡴⢾⣥⣤⣀⠈⡴⣿⣟⠻⡟⠳⣦⡈⢮⣷⡄⠀⢈⢸⣿⢸⣿⢸⣿⠀⡔
⠀⠀⠀⠀⠘⠚⠀⠀⠀⡄⣿⡀⠀⣾⠗⠊⣀⣀⡩⠉⠙⠛⠓⠀⠀⢠⢾⠊⢰⢱⠻⢸⠑⡙⡇⠂⠀⠁⢸⠀⠀⣽⠀⠑⠚⠀⠈⠙⠻⣼⣧⠾⢛⡋
⠀⠀⠀⠐⠀⠰⣶⡀⣸⠇⣸⣧⠞⢶⣌⡉⠀⠉⠻⢶⣤⣄⠀⠀⠀⠸⣮⡀⠸⣄⠀⠘⠀⣡⠗⠤⠤⠔⢃⣀⣸⠏⠀⡀⠠⢀⠀⠀⠀⢸⠁⠀⠈⠂
⠀⠀⠀⠀⡀⠀⠈⢿⣞⠋⠈⣿⠀⠀⠉⠻⢦⣀⠀⠀⠀⠉⠁⠀⣠⣄⠙⠷⢶⣄⣩⠭⣉⠠⣄⣀⣴⣤⠟⡻⠁⠀⠀⢰⢸⠈⠆⠀⠀⠰⠃
⠀⠀⠀⠘⣧⡀⠀⠀⠙⣧⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠞⢏⣿⠂⠀⠀⠉⠙⠙⠛⠛⠛⠛⠉⠉⠁⡤⣄⢀⠴⣄⣀⢄⡀⠀⠀⠀⠀⢮⣷
⠀⠀⠀⠀⠙⣷⡄⠀⠀⠈⢦⡀⣅⠀⠀⠀⣠⣴⡶⡶⢻⣧⠀⠈⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠛⠀⢀⣤⣧⣾⢡⡞⢡⠙⣦⢹⣊⣦⡀⠀⣾⡽
⠀⠀⠀⠀⣠⡞⠛⠒⣄⡀⠈⢷⣿⠀⠐⠋⠁⢀⠜⠀⠸⣿⠀⠀⠀⢠⣿⠀⠀⠀⣼⠆⢰⡆⠀⠀⢰⢻⢧⣟⡿⢿⢇⣊⡴⢫⣾⣙⠁⠙⡄⣎⠘
⠀⠀⠀⢠⡿⠁⠀⣴⡟⠉⠒⢌⣿⡇⠀⠀⢠⢾⡆⠀⠀⠻⠀⠀⠀⠘⠷⠀⠀⠀⠻⠀⠸⡇⠀⠀⠘⢎⡈⠻⣿⣿⠾⡏⢙⢻⠧⠴⢃⡴⢁⡜⢑
⠀⠀⠀⠸⠁⠀⢰⢏⡀⠀⠀⠀⠹⣧⠀⣠⠃⢾⣧⠀⠀⠀⠀⣠⣴⣶⣞⡟⡟⣹⢿⡻⡻⡲⢶⣄⡀⠀⠻⣷⣾⣯⠤⠬⢲⣷⣶⡾⠏⠀⠈⣩⢂⡀
⠀⠀⠀⠀⠀⠀⠙⠘⡄⠀⠀⠀⠀⠻⣶⡇⠀⢸⣿⠀⠀⣀⠀⣷⣷⡿⡻⢾⣰⡛⣧⠹⣵⣵⠙⣼⣷⠀⠀⠀⠀⠉⢉⡉⣭⣥⠤⢄⣀⠀⠀⢷⣾⡇
⠀⠀⠀⠀⠀⣀⣤⢾⡷⣤⣀⠀⠀⣀⢹⣷⠀⢸⡏⠀⠈⠃⠀⣿⣻⣿⣟⣛⣳⣶⣾⣷⡿⠷⢿⣯⡿⠀⠶⠀⢀⡔⢁⣈⣁⡀⠀⢀⡝⠀⠀⢟⡩⡇
⠀⠀⠀⢀⠞⣽⡇⢸⡇⢹⣏⢣⠀⢸⡆⢿⡇⠈⣇⠀⠀⠀⠀⢹⣷⣤⣓⠚⠛⣄⣠⠞⠲⢤⣧⣼⠃⠀⠀⣀⠔⠉⠁⠀⠀⠀⠀⢙⣦⠀⠀⣴⡿⡃
⠀⠀⠀⠘⢆⣿⣧⢸⣿⢸⣟⡸⠀⢸⣷⠘⣧⡀⠈⠀⠀⠀⢠⢆⠹⣌⡙⣛⣛⣛⣒⡛⠛⣋⡽⠃⢀⠀⠀⠛⠢⠬⠥⠤⠭⠤⠔⠊⠀⢠⡀⢱⡄
⠀⠀⠀⠀⢀⡍⠉⠾⡿⠏⠁⠀⠀⠀⣿⡄⢸⣇⠀⠀⠀⠀⠀⠑⠀⠀⠙⠻⠿⣿⣿⡿⠏⠁⠀⠀⠈⠀⢀⡀⢀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣱⣗⠁
⠀⠀⠀⠀⠀⠁⠀⠀⡼⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡡⢤⠟⣜⣧⣸⣯⢹⣇⢹⠐⣂⣖⠠⢍⠰⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠦⡀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠉⢠⣄⣀⠀⠀⠀⠀⢱⣼⣷⣾⠄⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡀⢤⣚⣿⢿⡟⠀⠀⠀⠈⠀⠀⠀⠑⠄⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠑⠤⣄⡀⠀⠀⠀⠀⠀⢈⡄⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣿⣿⣻⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⡠⡀⠀⠀⠂⠄⠐⠰⢄⣀⢀⡠⠀⠠⠉⠀⠐⠀⠀⢹⢿⣿⡿⠄
⠀⠀⠀⠀⠀⠸⠅⠀⠀⢀⠄⠉⠀⠀⢰⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠏⠃⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠁⠀⠀⠀⠸⡺⣄⠀⡀⠀⠀⢃⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠤⠄⣀⠈⠻⢿⣾⣷⡤⠂⠁⠀⠀⠸⠁⠀⣤⣀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠀⡠⢮⡿⠁⡀⠀⠠⠬⠿⢻⣿⣿⢷⡦⠤⠖⠚⠁⠙⢷⡏⠿⠛⠀⠀⡁⠀⠐
⠀⠀⠀⠾⠁⠉⠀⠀⠈⠁⠀⠀⡄⢀⣨⠶⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠜⠃⢨⠀⣠⠀⠀⠂⠀⠀⠀⠀⠀⠀⣰⠌⠀⢀⡏⢓⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠠⣿⡃⠀⠀⠑⠒⠀⠒⠂⠁⠀⠀⠀⠩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⡶⠟⠳⢦⡀⠀⠋
⠀⠀⠀⠀⣠⡞⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠉⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠂⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⣴⣾⡿⣭⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠘⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠆⠀⠈⢛⣿⣯⠾⠃⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⢺⣦⡀⠀⢸⣿⣧⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠿⠟⠀⣤⡀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣼⣿⣅⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⡾⠁⠀⠀⠻⠿⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠈⠁⠀⠀⠀⠘⣿⡯⣿⣿⡿⠂⠀⣀⡴
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠀⠀⠀⠑⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣸
⣿⣤⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣶⣿
⠻⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣤⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠰⣾⣿⣿⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢈⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⡄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠁⢠⠀⡀⢠⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠠⠃⠈⠑⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡶⠀⠀⣀⣤⠆⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡻⠻⠆⠀⠀⠀⠀⠀⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⠀⠀⢀⣠⣆⣠⠀⠀⠃⠀⢀⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⡿⠁⠀⢠⡋⠉⠹⢃⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠶⠆⠂⠀⠺⢿⢇⡀⡀⠀⡸⠿⠛⢀⡿⢶⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⢻⡃⢠⠇⠀⢠⠂⠀⠠⠀⠀⠀⣴⣿⠟⠲⣄⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⠏⠀⠀⡇⠀⢠⣿⠟⠻⠀⠀⢹⠆⠀⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢠⣄⠀⠘⢀⡄⠀⠃⠀⠉⠉⠀⡴⠞⠀⠀⠀⠀⢠⡆⠈⢢⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⠀⠀⠀⠞⢷⡛⠱⠄⠀⠀⡴⠀⠀⠀⠀⠀⠀⠐⠿⠃⣀⣸⡟⠦
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡄⢲⡆⠀⢠⣷⢀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⢸⡋⠁
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡎⠑⠇⠀⢘⠉⠘⡇⢢⣿⡶⠊⠀⠀⠀⠀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠃⠀⠠⡘⢾⡄⠀⣏⡼⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⣷⡄⡟⢀⣶⣇⠀⠀⠀⠲⢀⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⡀⢠⡄⢱⠙⡄⢸⣿⢋⡀⣘⣀⠀⠘⠛⠛⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠱⣾⡇⢸⡄⠁⠀⡇⢈⠷⠛⡟⢃⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⣤⣾⣶⡷⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠀⠈⡇⣸⣷⣷⡿⠟⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⢳⡏⣿⠥⣶⣶⣽⣿⡴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠠⣄⡀⠀⠀⠀⣸⠇⣿⣠⣨⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠙⢦⡀⠀⣿⡔⡿⠃⠛⠛⢿⣼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢿⡆⢿⡿⡇⠀⢀⣀⡀⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣿⠾⠃⠀⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣛⡻⣿⣭⣯⠻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡏⠁⠀⠀⠀⠈⠉⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣗⣿⠁⠀⠀⠀⠀⢀⡀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣿⣿⣿⢿⣟⣒⣂⠀⠀⣫⣭⣥⣄⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⢻⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⣿⡿⣹⣿⣿⠟⠀⠀⠙⠿⠿⠋⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣿⣰⣿⣟⣿⣧⡀⠀⠲⠶⠂⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⣾⣵⣟⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣤⠤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢳⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣾⢿⢻⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠸⡿⠓⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠐⠃⠀⠀⠈⠉⠉⠉⠉⠋⠉⠙⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⠿⢶⢤⣦⣤⣄⣄⣀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠉⠁

⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢦⣽⣿⣿⣄⡸
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⡿⠛⠁
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣦⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠁
⣿⣿⣿⡿⠛⠛⠛⠿⣿⣿⣿⣷⣄⠘⠿⠗⠀⠀⢠⣴⣶⣶⣿⣿⣶⡿⣶⣶⣤⡀⠀⣾⣿⣷⡶
⣿⣿⣿⣧⡀⠀⠀⠀⠘⣿⣿⣿⣿⣧⠀⠀⢀⣼⣿⣿⣿⡏⠁⠁⠀⠀⠀⠉⠻⣧⠀⣿⠟⠉
⣿⣿⣿⣿⣿⣂⣀⣀⣺⣿⣿⣿⣿⣿⠀⠀⠘⣵⣿⣿⣿⣷⣶⣀⠀⠀⠀⠀⠀⠘⠀⣿⣿⡿
⣿⣿⣯⣍⣙⣋⣉⣉⣉⣬⠻⣿⣿⣿⡄⠀⠀⠙⣋⣻⣿⣿⣿⣿⡷⠄⠀⠀⠀⠀⢰⣟⣁
⣿⣿⠛⠋⠉⠉⠉⠉⠁⠀⠀⣽⣿⣿⡇⠀⠀⠀⠈⠙⠻⠿⣷⣶⡶⠶⠂⠀⠀⠀⣼⠿⠛
⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀
⣿⣿⡇⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠋
⣿⣿⣇⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⣿⣿⣿⠀⠀⠀⣀⠀⣴⣿⣿⣿⣿⣿⡇⠀⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⣿⣿⣿⠆⠀⢀⡿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁⠀⠉⢧⣄⣀⠀⠀⠀⠀⢠⣿
⣿⣿⣿⠚⣿⣿⣿⣷⣼⣿⣿⣿⠁⠈⣿⠀⠀⠀⠀⣺⣿⣿⣿⣿⡷⠀⣰⣿⡇
⣿⣿⣿⠀⠈⢹⣿⣿⣿⣟⣿⣿⣷⣦⣴⠿⠶⠆⠸⣿⣿⣿⣿⣿⣟⢴⣿⣿⠀⠀⡀
⣿⣿⣿⣇⠀⠈⢻⣿⣿⣿⣧⣄⡈⠉⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠱
⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡷⠶⠇⠀⠀⠈⣿⣿⣿⣿⣿⡟⣿⣷⣤⣴⣾
⣿⣿⣿⣿⣿⣶⣾⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠋⠙⠻⠿⠿⠿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣦⣤⣴⣶⣿⣿⣿⡿⣯⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣾⠉⢿⡶⢤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣿⠟⠁⠸⠈⢿⡀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⣷
⠛⠿⣿⣿⣧⣬⣍⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠐⡆
⠰⠆⠀⠉⠉⠙⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⡆⠀⠀⢰
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣾⡏⢠
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢰⣿⣡⢯⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠘⠃⠀⠤⠂⠀⢀⠀⢀⡀⠚⡀⠀⠁⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠀⠠⡀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠀⠂⠀⠀⠀⠒⠀⠸⡀⠄⠁⠀⠀⠀⠈⠀⠀⡀⢀⠀⠀⠀⠠⡆⠀⠀⡀⠊
⠀⠀⠀⠠⠄⠂⠀⠀⣂⠂⠀⠁⠁⢀⠤⠄⠀⠂⣀⠄⠀⠢⢄⠀⡴⡄⠀⠀⠀⠉⡉⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠨⠃⠀⠀⠂⠀⠒⠁⠀⠀⣂⠀⠀⠀⠀⠔⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄⢀⣠⣴⣶⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⡿⠍⠙⠛⠛⠿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠻⣿⡿⢿⡟⠛⠋⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⢀⣀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⠀⠀⢀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⢸⣿⠿⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣟⠁⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢦⣀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣄⠀⠀⠀⠰⣦⡄⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⣧⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠛⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢿⣿⣦⡀⠀⠀⠀⢀⠀⠀⠀⠒⣀⡀⠀⠀⠀⠀⢸⡇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢿⣷⡀⠀⠀⠈⢶⡄⠤⠽⠋⠀⠐⠓⢶⠂⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣙⡛⢀⠀⠀⠘⣧⣶⣤⣤⣶⣾⢟⣿⣷⣿⠋⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣰⠖⢙⣻⠿⣿⣟⡟⠂⠒⠀⠀⢹⣿⣿⣤⠈⣀⣼⣿⣿⣿⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⣍⣹⣿⣦⣻⣿⣿⣿⡟⠃⠀⠀⢠⣾⣿⣿⣯⡠⢾⣿⣿⣿⡿⠉⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠙⠉⠁⣸⣿⣿⡷⠀⢠⡀⣀⣬⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠈⢻⣯⣸⣿⣛⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⢴⣿⣏⢸⠃⠙⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⢠⣼⢿⣿⣿⣦⡀⠀⠙⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠀⠀⠁⠈⣿⣿⣿⣷⣄⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣆⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠙⠿⣿⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢠⠄⠀⠀⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧⠀⠀⢀⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣾⠁⢙⠴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣀⢋⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⣀⣠⣤⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⣄⣤⡾⣻⠀⣶⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣾⣷⣿⣿⣾⡿⢿⣿⣿⣷⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢛⢻⣿⠿⠻⠏⠀⠀⠘⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠾⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢄⡀⠀⠀⠀⠀⠰⢿⡶⣤⢤⣶⠏⠀⠀⣀⣄⣀⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠎⠀⢸⣇⠀⠀⠀⠀⠀⠀⡠⠀⠘⡿⠁⠀⣾⣿⣿⣿⣿⣦⢌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠈⢿⡄⠀⠀⠀⡠⠎⠑⠲⣶⣇⢀⢀⣭⣭⣛⠿⣿⣿⣿⣷⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠘⣦⡈⠻⢤⡀⠀⠀⠀⢬⠭⢹⣿⣷⠎⠙⢿⣿⣿⣮⡻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡠⠒⠊⠁⠀⠀⠀⠀⠀⠠⣄⣙⣻⣆⠀⣿⣷⣶⣤⣄⣠⣿⣷⣿⣾⣷⣤⠀⠙⢿⣷⣼⣿⣷⡆
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠉⠋⠳⡆⣼⡟⣷⠀⣿⣿⣿⣿⣿⣹⣧⣿⣿⣷⡏⢼⣿⣿
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡈⢳⣻⡀⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠃⡀⢀⣹⣿⣻⣿⣿⡿⠛⣿⣿⣏⣿⣿⠟
⠀⠀⠀⠠⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠈⠋⣿⣮⣟⣿⣿⣿⡋⠀⣠⠀⠉⠉⠉⢙
⠀⠀⠀⠠⣜⢿⣷⡀⠀⠀⠀⠀⠀⢀⣴⣿⣇⣀⡀⠀⠀⠀⠀⠠⣜⠿⣿⣿⢏⣾⣿⢣⡔⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢳⠀⡿⣿⡦⠀⠀⠀⠀⠁⠊⠀⢻⣿⣿⣶⣦⣤⡴⠓⠿⣿⣶⣯⣾⡿⠃⠸⣇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⣇⣷⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠋⠁⠀⠀⠀⠀⠉⢙⣟⠁⠀⠀⠙⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠘⣿⣷⡹⣧⡀⠀⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣦⡀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣜⣷⣤⡲⢤⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢹⠏⠛⠻⠿⢿⣤⣄⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣮⣥⠀⠀⠀⠀⢀⣀⣤⠄⢀⡾⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠏⠳⣍⠛⠁⠀⢴⣶⣿⡿⠛⣁⡴⠋⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⢤⣀⠀⠛⠁⣠⠞⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠉⠛⠒⠚⠁⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠖⠊⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠐⠂⣠⣄⠀⠀⠀⢀⡀⠀⠀⠀⣠⠤⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⢫⡀⠀⠀⡘⠀⠀⠀⢸⣦⣄⠠⠂⠄⢶⣆⠛⢶⢒⡀⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⡴⠃⠀⠀⠀⢃⠸⣹⠗⠚⠿⡭⢿⡟⡳⠵⠏⠅⠃
⠀⠀⠀⠀⠀⠀⠀⣤⡾⣦⣼⠿⠶⠂⠀⠀⠀⠀⠀⠀⠀⣨⠉⠀⠙⣧⡀⠀⠐⢄⣀⠀⠀⠀⠀⠀⢠⣾⠍⠁⠀⠀⠀⠀⠀⠑⢄⠈⠉⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠘⠃⠀⢸⠛⣤⠀⠀⠐⠀⠀⠀⠀⠀⠀⠼⠁⠁⠀⠘⠋⠹⠀⠀⢀⠙⠿⠒⢺⡿⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠠⠄⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠴⠶⠲⠖⠒⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢀⠀⢠⣁⡤⢤⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⣉⣲⡎⢡⡈⣦⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠂⠐⠚⠻⡷⠂
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⡀⠀⠈⠣⡀⠀⠀⠁⠈⠓⠂⠐⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠰⠶⠎⠀⢠⠄⠼⠦
⠀⠀⠀⠀⠀⠃⢚⣦⠀⠀⠀⡀⠀⠀⠑⣄⠀⡀⣠⡞⢂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⡙⢋⡿⢥⣾⣶⢶⠀⣄⡀⠘⠃⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⡈⢷⣄⣴⣼⣦⣶⠶⠛⠛⠋⢉⠁⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠫⠃⣤⢶⣾⣿⠤⠀⣨⠂⠀⠀⠀⠀⢁⠅
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠘⠉⠉⠤⠤⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣴⡆⠀⠀⠀⠀⠀⡀⠴⢦⡞⠀⣀⣠⣿⣇⡏⠃⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⡷⡀⠀⠀⠀⢈⡀⠀⠀⠀⠈⠑⠚⠟⠏⠁⠀⠆⠀⠀⠔
⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⡘⠁⠀⠀⠀⠀⠈⠑⠦⣄⡀⠌⡠⠀⠀⠀⠘⡠⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠉⠒⠟⠁⠀⠀⠀⠀⠂⠀⠀⠠⢠⠆⠀⠈⣡⡶⠦⠴⠀⠀⡈⠀⡀⢉⣉⣀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⡀⠐⢈
⠀⠀⠀⠀⠀⠒⠶⠀⠴⠒⠃⠀⠀⠤⠀⠀⠙⠛⠍⠀⠀⠀⠀⠀⠀⠳⣀⠂⠐⠄⠀⣰⣬⠉⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡠⠀⠀⠀⡄
⠀⠀⠀⠀⢀⡤⠖⠊⠉⠈⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠐⠒⠂⢈⣁⠤⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠲⠇⠀⠀⣠⡇
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢀⡀⡀⢀⢀⣀⣀⣀⢀⡀⠙⢷⡄⠀⠀⠀⢨⣤⠤⠶⠖⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣿⡦⠤⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⢘⢡⠁⠀⣀⣠⣔⢾⣟⡉⠛⣼⡶⡬⢉⡀⢈⣁⣤⠿⡿⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠈
⠀⠀⠘⣧⡀⠸⣟⡶⣴⡦⣮⣙⡋⣿⠟⢋⣈⣉⠉⣭⣷⣴⠃⠀⠀⠀⠀⠀⠉⠉⢀⡄⠀⠀⠀⣦⡀⢀⡠⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⢉⠓⠿⣬⣅⠶⡄⡻⠟⢛⣚⠉⣩⣽⠶⠟⡡⠀⠀⠀⢠⠀⠁⠀⢀⣠⠀⠀⠀⠈⠀⣹⠀⠀⠀⠀⣀⣀⠀⠒⠀⠀⠀⠀⠀⡀⠀⠀⠈⠱⣤⡄⠠
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠉⠙⠛⠉⠉⠉⠀⢀⡀⠐⠁⠀⢀⡀⠈⠓⠄⠀⠀⠉⠒⠀⢀⣤⠴⠛⠀⠀⠄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⠀⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠄⠀⠀⠤⠤⠶⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣠⣤⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⠿⠿⠿⡿⠛⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠷⠆⠦⠘⠃⠠⢳⣿⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢆⠀⢀⣴⡿⠦⠘⣿⣇⠀⠀⠀⠀⠀⣴⠀⠉⠀⣸⣷⣦⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠙⢁⡞⢁⣾⡿⠁⠀⠀⢸⣿⣧⣀⡀⣀⡔⢡⠀⢠⣜⣿⣿⡿⠉⠙⠫⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⣼⣿⣅⠀⣀⣀⣨⣿⣿⣿⣿⢋⣶⠀⠀⢁⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⠻⠛⠋⠉⠁⠁⣼⣿⣿⡷⠙⠿⠛⠟⠀⠀⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣰⣄⠀⠀⠀⠀⠀⠀⠀⠛⠉⡅⠀⠈⠻⣶⢤⣴⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠏⠀⠀⠀⣆⠀⠀⠤⣄⣀⣤⣤⠄⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠋⠀⠀⠀⠀⠘⢷⣦⣄⣠⣬⡿⠋⣠⡾⠛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣼⣦⣄⠀⠀⢸⣦⣀⣀⣀⡉⣹⣿⡽⠷⣿⡷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⠀⢠⣾⡿⠻⠛⠿⢷⣍⠀⠀⠀⣈⣤⣤⣤⣤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣷⣦⣤⣀⣀⠀⠀⠁⠀⢻⣷⣿⣿⣦⡄⢻⠀⠀⠀⠀⢀⣹⠀⣶⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣦⠀⣀⠀⢐⣻⣵⠶⠺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠌⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣠⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠿⢟⣉⣁⣀⣀⡀⣀⣀⣶⣶⠾⣿⠿⠿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡽⠟⠛⣉⠁⢽⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠁⠀⠀⠀⠀⠀⠈⢻⣯⡁⠀⠀⠀⠀⠀⠀⠀⠟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡙⠂⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠟⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⣷⡀⠀⠀⠀⠀⠀⠀⢀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⣀⣙⢁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣄⠀⠀⠀⠀⠀⡌⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠋⠘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠻⣿⣿⣶⣤⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣤⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢿⡄⠙⢿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣄⢰⣿⣧⠀⣀⣀⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠈⣧⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⡿⣦⣍⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⡙⠻⢷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣄⠈⠛⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠹⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣷⣄⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣆⣤⣤⣼⣄⣴⣄⣤⣶⣌⣴⣶⣡⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠛⠛⣶⠛⠛⣦⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡏⠉⢻⠶⠶⠛⠶⠶⢧⣀⣼⠟⠓⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣰⡏⠉⢻⠞⠋⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⡟⠓⣦
⠀⠀⠀⠀⠀⢀⣰⡏⠉⢻⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡿⠓⢦
⠀⠀⠀⠀⢰⡏⠉⣻⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡾⠓⢦
⠀⠀⠀⣴⠛⠻⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡾⢦⡀
⠀⢀⡴⠿⣦⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣠⣷⡀
⢀⣼⣧⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠀⢹⡆
⣿⡀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⠛⠙⣦
⢈⡿⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⡶⠿⢦⡀
⠘⢧⣤⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣼⡃
⠀⢰⡏⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⢹
⠀⠀⠙⣻⠋⠙⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠾⣏
⠀⠀⠀⠘⠳⢾⠋⠉⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣠⡿
⠀⠀⠀⠀⠀⠈⠓⢾⡋⠉⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠋⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢺⡏⠈⣻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⢷⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡏⠀⣹⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣀⣠⡿⠒⢦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢧⣤⢾⡏⠀⣹⡶⠒⢦⣀⣄⣀⡴⠲⢶⡏⠙⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⢷⣤⠾⡏⠀⣹⣧⣤⡾⠙⢦⣌⡿⠶⢤⣤⣀⣠⡤⠶⠚⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣹⣇⡀⠀⠀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠛⠒⠶⢽⣷
⣿⣿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠈⠡⠀⢷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣤⣦⣤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠢⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢏⡜⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠂⠈⠙⡆⠈⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡼⣄⣀⡂⠀⠀⠀⣇⠀⠀⠠⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣡⠤⠶⠚⣟⠁⠀⠀⢠⠝⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠾⠵⠒⠶⣅⣀⢸⡆⡀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠁⠈⢣⡀⠀⢰⠀⠈⠉⠑⢢⡤⠂⠀⠀⠀⡤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢦⡤⠀⠀⢀⠀⡀⠈⠳⡀⠀⠀⢸⡆⠀⠀⢁⢀⠀⠀⠇⠀⠀⠀⡔⡠⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣠⠀⠀⠀⠀⣀⣀⡀⠐⠄⠀⠀⠙⠀⠐⢌⡴⢂⠀⠀⠀⠀⣐⣎⠋⠀⠉⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠈⠢⠀⠀⠸⡄⠀⠐⠀⠀⠈⢢⠹⢦⣾⠠⠂⠀⠀⠠⠀⠐⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠁⢀⡊⠀⠀⡄⠤⠆⠀⣤⡀⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠀⢠⡀⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁⠠⠶⠂⠒⡻⢆⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠂⣷⡋⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠐⠚⠞⠁⠀⣈⣀⠤⠤⠬⢶⡀⠀⠀⠀⠀⣤⢤⡀⠀⠀⠀⠀⠆⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⣀⡤⠂⠉⠁⠀⡀⠠⠀⠐⠳⠆⣀⠀⠀⡇⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⢀⠄⡜⠱⠶⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢄⡄⠤⣠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠖⠉⢶⠞⠁⠀⠀⠀⠀⠀⠀⠡⡀⠀⠙⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⠦⣴⡟⠠⠄
⠀⠀⠀⠀⢀⠀⠂⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠆⠀⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⡈⣤⠀⣤⣄⡂⠀⠀⠀⠀⠀⢀⢤⠔⠳⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠒⠉⢡⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣶⢖⣿⣷⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃⢾⡿⠿⠋⠉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣌⠳⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⣿⣭⣁⠀⠀⠀⠀⠀⡀⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣶⣟⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣷⣝⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⢻⣿⠟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠁⠀⠀⠈⠢⣄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⢴⣾⣿⣿⠀⠘⠈⠙⠓⠀⠀⠒⠀⠀⠀⠐⠁⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⡿⠀⡀
⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⠀⠀⠀⠄⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠋⡟⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾
⢀⠔⠈⠁⠈⠁⣀⠀⠀⠀⠀⢠⣖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢀⣾⣏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠤⠀⡴⠊⠉⠉⠐⢄⠀⠀⠀⠁⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣀⡀⢀⣤⣿⣿⣟⡡
⠁⠉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠦⣄⡠⠲⠒⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⡀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠑⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡻⣿⣿⣷
⠙⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠴⠊⠁⠀⠀⢀⣿⡟⠃⠀⠃⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀
⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠈⢆⠀⠉⠁⠀⢄⠵⠶⣦⠀⠀⠀⠀⠀⢀⠀⠀⢽⣿⣿⡇⠀⣾⠇⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⣀⢀⡀⠉⠉⢻⣿⣿⣧⢠⣿⣷⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠻⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠳⡀⠀⣰⠁⠀⠠⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠃⠀⠁⠈
⠀⠐⢄⢿⡄⡀⠀⠀⠀⠀⠕⠀⠄⠀⠀⠀⠀⠀⠀⠒⢂⣩⡴⠋⡉⠀⠙⢷⡀⠈⠀⠀⠐⠒⠒⠄⠠⠄⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠆
⠀⠀⠀⠀⠙⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠋⠅⡔⠁⠀⠋⠀⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣴⣶⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠖⠀⠀⠀⠀⠀⠀⠁⠰⣀⠀⢀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠈⠙⠙⠓⠚⠠⢄⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠉⠉⠒⠆⠒⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣦⢠⠀⢀⣴⣿
⠀⠀⠀⠀⠇⠑⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⣿⣿⣿⠏
⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠸⠁⠀⠀⢀⣤⣞⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣟⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⠀⠀⠀⠈⢻⣿⣷⣀⢠⠀⠀⣤⣿⣿⠋⢀⠀⠠⠠⠀⠠⠄⠀⡸⠉⠐⠙⠧⣐⠀⠤⠤
⠲⢀⠈⠁⠀⠁⠀⠀⠠⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠈⠻⣿⣿⠛⠃⢸⣿⣿⡡⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠒⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣷⣶⣦⣉⡀⠃⢌⣿⣟⡡⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠐⠚⠉⠉⠉⠻⠛⠛⠛⢻⡿⢿⠁⣠⡆⢫⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠉⠉⠐⠋⠉⠙⠲⡀
⠀⠀⠀⠀⢀⠔⢚⠛⠉⡉⠩⠙⠢⡀⠀⠀⠋⠀⠀⠒⠀⡀⠀⢀⠟⠀⠡⠀⢀⣸⡚⠠⠄
⠀⠀⠀⠀⠋⢀⣼⠏⢠⠁⣠⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⡄⡀⡠⠈⢻⣷⣵⣔⡀⠒⠀⠀⠙
⠀⠀⠀⠀⠀⠘⠻⣄⠀⢠⠏⠁⠀⠀⡼⣄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣠⠇⠀⠀⠙⣿⣧⣁⢀⠀⠤⠀⠒
⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠙⢷⣄⡀⠀⠀⠀⠀⠀⡀⠖⠋⠀⠀⠀⠀⠈⠙⠛⠈
⠀⠀⠀⠈⠳⠦⠀⠀⠀⠀⢀⡤⠋⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⡠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣄⠐⠲
⠀⠀⠀⠀⠀⠀⠐⠒⠀⠛⠺⡆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠒⢯⠤⠄⠂⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠲⠄⠐⠋⠀⠀⢀⣀⣀⡈⢡⠴⠂⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠀⣀⠔⠚⠙⠀⠀⠐⠀⠄
⠀⠀⠂⠐⠒⠋⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠉⠙⠒⡤⠒⠉⠈⠉⢀⣤⡀⠀⠀⠀⢴⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠠⠤⠤⠖⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠛⠉⢀⣴⣦⢄⣠⣄⣀⣐⡒⠒⠒⠿⠗⠉⠔⠈⢠⠀⢄
⠀⠀⢀⢔⣪⡶⠶⣶⠶⣶⣤⣄⠀⠀⠀⠀⣠⣤⠿⢯⣽⣿⡻⣿⡟⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⢡⣾⠛⢷⣤⣼⣦⣼⠉⠹⣿⡄⠀⠈⠇⠀⠀⠀⠀⠹⡿⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣾⣿⡦⠐⠻⣷⣶⡀⠤⢆⣀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠒⡀⠀⣠⣩⣤⣄⠀⠄
⠀⠀⢻⣿⣷⣠⣆⣿⠋⠙⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣰⡾⠁⠀⠁⣤⠭⡤⢉⡐⠓⠂
⠀⠀⠀⠙⠿⣿⣏⠉⠓⣤⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠙⠅⠀⠴⠟⣿⣖⠹⢃⢶⡶⠂
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠋⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠾⠿⠳⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣿⠿⢶⠿⢿⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⣾⠛⣶⠛⣶⠟⢳⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣉⣿⠛⣿⠛⣿⡟⢻⡏⣻⣏⣹⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣉⣿⣏⣹⣿⣉⣿⣏⣹⣿⣉⣿⣏⣹⣏⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣨⣿⣉⣿⣏⣼⣿⣨⣿⣉⣿⣅⣽⣍⣽⣍⣽⣅⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣽⣇⡼⢧⣼⣯⣬⣿⣤⣿⣧⣼⣧⣤⣿⣉⣿⣅⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣩⡿⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠻⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣙⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣿⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡽⣿⠀⠀⢀⡀⢀⡀⠠⠀⠀⠀⡀⠀⠀⠀⠀⢸⣶⣅⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢼⡆⠀⠀⠉⠉⠁⠀⠀⠄⠀⠉⠙⠋⠀⠀⢸⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠟⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⠆⠀⠀⠀⠀⢑⠀⢀⠁⠀⠀⠀⠀⡌⠢⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣆⠀⠀⠈⢩⠉⠉⠛⠁⠀⠀⣼⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡼⠋⠱⣄⡀⠀⠀⠀⠀⠀⣠⠞⢹⠑⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣤⣆⠀⠀⠀⠀⠙⠶⠶⠶⠖⠛⠁⠀⠀⠳⣀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡎⠀⠀⠀⠈⠉⠛⠻⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣵⣾⠿⢉⣿⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣹⣿⣶⣤⣠⣤⡤⣤⣤⣴⣾⠿⢛⣡⡶⢛⣴⣿⡇⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣋⣉⣑⣛⣶⣶⣿⣿⣿⠿⢟⣻⣭⣿⣿⠿⠛⢋⣡⡴⠞⢋⣡⣶⣿⣿⡟⢠⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢉⣉⣤⣶⣿⡿⠟⠋⠉⣀⣴⢞⣋⣥⣶⣿⣿⣿⣿⣿⠟⢠⣿⠟⢁⣾⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⢦⣄⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⠶⣾⣿⠿⠛⣉⣁⣤⣴⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠃⣰⠟⠁⣠⠟⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠟⠛⣛⣩⡥⠶⠛⠙⢛⣛⣛⣭⡽⠾⣛⣯⣷⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⡼⠃⠀⣴⠏⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣒⣀⣩⣥⡤⠶⠶⠞⠛⠋⢉⣁⣤⣶⣿⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠁⠀⠾⠁⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠛⠉⠉⠉⠁⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⡿⠿⠿⠿⠶⠶⠶⠶⠶⠿⠿⠿⠿⠷⠶⣶⣶⣶⣶⣶⣶⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⢀⡼⠋⠁
⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣶⣄⡀⠀⢠⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣶⣌⣻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠉⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⣻⡿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡻⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠙⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⣾⣿⣿⣾⣦⠀⠀⢀⣀⡸⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢛⣯⣏⠙⢿⣿⡗⣶⢒⣿⣿⣧⣾⣿⣿⡣⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡶⠚⢻⣧⣬⣿⠠⠼⠋⠀⠀⣉⣉⡻⢦⣿⡿⢿⡝⢿⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣿⡃⠀⢉⣡⣴⢾⣧⣀⠼⢧⣾⡏⣸⣿⣟⠂⢨⡎⢳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣯⠁⠀⢀⡽⠁⠀⠀⠈⠙⠛⠶⠝⠘⠿⣋⡌⢛⣻⡄⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣈⡻⣯⣳⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⠿⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢸⣿⣻⣧⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣼⠛⣛⣂⢼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣼⢃⣿⠷⣼⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⡡⢖⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣹⢘⣟⣻⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄⣴⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣽⣿⠻⣿⠿⢿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⡇⠰⡾⠿⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡿⠀⠀⠀⠀⠉⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠀⠓⣷⣿⣿⣤⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣪⣡⡾⠙⠃⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠉⠀⠉⠙⠛⠷⠟⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠘⠁⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠄⠀⠀⠀⠀⠀⢠⣠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣻⡇
⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⠀⠀⠰⢂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠾⠿⠇

⣿⣟⣻⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢐⡉⠀⠈⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡔⠒⣼⡀⠔⠺⡘⢍⣉⡁⢀⣽⠗⠀⠀⠀⠀⠀⢠⠿
⠀⠀⠀⠀⠉⠑⡤⠀⠀⠁⠂⠠⢼⡿⣗⠂⠀⠀⠒⠐⠒⠚⢦⠣⡼⠁⠆⠀⠈⢅⢙⡆⠈⣽⠀⠀⠀⠀⠀⢀⣍⡠
⠀⠀⠀⠀⠀⠀⠡⡄⠀⠀⣀⠀⠀⠈⠐⠒⠠⢄⡀⣀⡀⠀⠘⡏⢀⣄⠈⡄⠀⠆⠈⠷⡹⠹⠒⠂⠂⠤⡴⠈⠑⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⡷⠸⡆⠀⠀⠛⢻⠻⢦⠘⡄⠘⠆⢘⢃⠃⠀⠀⠀⠀⠃⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢡⠀⣷⡀⢀⣀⠘⢀⠌⠀⠐⠀⢀⢂⠂⠀⠐⢀⠜⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠢⡀⠀⡄⠁⠁⠀⠀⠈⠁⠀⠀⠀⠀⢌⠆⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠢⠈⠀⠙⢀⡆⠀⠀⠀⠀⠀⠀⢀⢟⡦⣤⢤⡄⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⡠⢁⠆⢲⠉⠉⠺⡣⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡐⠁⠚⠄⠬⡤⠄⢀⠜⠅⠋⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⡠⠖⠀⠀⠀⠲⡐⠠⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡄⠀⠀⠀⠀⠀⠸⠀⠙⠘⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠒⢠⡄⠀⠀⠀⢀⡆⡤⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠅⢀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠠⢤⡔⠒⠶⠋⢩⠖⠒⢦⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⡐⠃⠙⡇⠀⠘⢇⣀⠼⠂⠞⠢⠴⣇⠀⣀⠗⠚⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠘⠁⢈⠖⠒⠙⠮⣄⡌⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⠖⠢⡎⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠐⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⢳⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠢⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠓⢤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⣄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠊⠛⣤⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢧⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢹⡤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡤⠳⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⠉⠳⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠁⠝⠶⠉⠙⣦⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢱⡶⠚⠀⠀⠀⠀⠀⠀⣀⠀⣀⢄⡀⠂⠆⠈⠙⣇⠀⢹⡤⠔⢺⠦⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠈⢻⣋⣀⠴⢦⡖⠓⢫⠙⡏⠁⢙⣅⢀⡼⣀⠼⠦⠴⠚⠛⠁⠁⣈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⢑⡶⢯⡀⢸⣄⡠⠷⠔⠛⠒⠉⠊⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⣤⠶⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡐⠒⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣱⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠠⠄⠐⣀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠞⠤⠄⣠⠒⢸⣿⣿⣿⣿⣿⣧⣐⣒⣒⠲⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠾⠛⠛⠛⠛⠭⠁⠲⠌⠀⠈⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠙⠛⢿⣿⡆⠀⠀⠫⠭⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢀⣀⠀⣸⣿⣧⠀⠀⠀⠀⢠⣴⣤⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠈⠉⠀⠉⢹⣿⠀⠀⢠⣤⡄⠽⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣸⣧⠀⢺⡟⠛⠁⠀⢨⣻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣾⣿⣿⠀⠀⠁⠀⣼⣿⡿⡏
⠀⠀⠀⠀⢰⣿⣷⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠀⠀⠀⣇⣾⣿⡟⠁⡇
⠀⠀⠀⠐⠶⡿⠋⢁⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡷⠀⢙⣷⣤⣶⣦⡄⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠿⠿⠂⠀⠓⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⡴⢤⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⣀⡠⠐⣻⡖⠀⠀⠀⠀⠀⠀⠞⠁⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠀⣤⡴⣀⣤⢊⡤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠉⠛⢶⣤⣿⣭⣾⣯⣤⣴⣿⣷⠆⣲⣿⣿⣿⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⡄⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠹⣿⡾⢺⣿⣿⣿⣿⣿⠟⣉⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣀⡀⢺⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣸⣿⣿⣿⡿⢧⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠏⠿⠋⠀⣀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠙⠛⠻⠿⠿⠿⠿⠛⠛⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣦⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⠀⣸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣯⣿⣿⡿⡽⣿⢿⢯⡿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣽⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡟⠉⠉⠉⠉⢉⡉⠉⠉⠉⠉⢻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⠁⠚⡉⠙⠳⡌⣱⠛⠉⣉⠓⠘⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⠀⠀⠉⠋⠁⢣⡿⠈⠉⠁⠀⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⡇⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢸⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠿⣄⠀⠀⠘⠿⠽⠃⠀⠀⣠⠿⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⡛⠶⠤⢤⡤⠤⠶⢛⣇⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⢉⠉⡉⠓⢦⡘⣇⠹⣝⣳⢲⠶⠶⡖⢞⣫⠏⣸⢁⡴⠚⢋⠉⡉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡇⢸⠀⡇⢸⠀⠹⡜⣄⠈⠓⠾⢬⡤⠷⠚⠁⣰⢣⠏⠀⡆⢸⠀⡇⢸⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢀⡇⢸⠀⡇⢸⢰⡀⣷⠘⢧⣀⠀⠀⠀⠀⣀⡼⠃⣾⢀⡆⡇⢸⠀⡇⢸⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠸⡇⢸⠀⡇⢸⢸⡇⣿⠀⠀⠙⡷⣶⣶⣾⠋⠀⠀⡟⢸⡇⡇⢸⠀⡇⢸⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⢄⠀⠀⡇⢸⠀⡇⢸⢸⡇⢹⠀⠀⢀⣷⡇⢸⣾⡀⠀⠀⡇⢸⡇⡇⢸⠀⡇⢸⠀⠀⡴⠂⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⣄⠳⡀⡇⢸⠀⡇⢸⠈⡇⢸⠀⠀⢸⣿⠇⢸⣿⡇⠀⠀⡇⢸⠀⡇⢸⠀⡇⢸⢠⠞⣠⠄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣸⠛⠳⣌⠱⣿⡇⢸⡀⡇⠸⡀⡇⢸⠀⠀⡾⡽⠀⠀⣯⢷⠀⠀⡇⢸⢠⠇⢸⢠⡇⢸⣿⠊⣠⠞⠹⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣇⠸⡇⢸⡇⣧⠰⡇⡇⢸⠀⣼⣹⠇⠀⠀⠸⣞⣧⠀⡇⢸⢸⠂⣼⢸⡇⢸⠃⣸⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⡇⢸⡇⢸⠀⡇⢱⠘⣟⡽⠃⣀⣀⣀⣀⠘⣮⣿⠃⡏⢸⠀⡇⢸⡇⢸⠀⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢹⠀⢷⠈⣧⠘⡄⢳⡈⢧⡘⠿⣍⡁⠀⠀⢈⣩⠿⢃⡼⢁⡟⢰⠃⣼⠀⡾⠀⡇⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⣼⣄⠸⡆⠸⡆⠹⡄⠳⣄⠙⠦⣌⣉⣛⣛⣉⣡⠴⢋⣠⠞⢠⠏⣰⠇⢰⠃⣠⣧⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠘⠁⠀⠀⠉⠙⠻⢦⣹⣆⠙⠲⢬⣙⠓⠶⠤⠬⠭⠤⠶⠚⣋⡥⠖⠋⣰⣏⡴⠟⠋⠉⠀⠀⠈⠃⢸⡇
⠀⠀⠀⣀⣤⠶⠾⠷⠦⣤⣄⣀⠀⠀⠀⠀⠈⠉⠳⢦⣀⠀⣉⣉⣑⣒⣒⣒⣉⣉⠀⣀⡴⠚⠉⠁⠀⠀⠀⠀⣀⣠⣤⠴⠾⠷⠶⣤⣀
⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⡀⠀⠀⠈⠛⢩⣵⣶⣿⣿⣶⣮⡍⠛⠁⠀⠀⣀⣀⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⣸⠃⠰⣄⠀⠀⢧⡀⠀⠀⢦⡀⠀⠀⠀⢸⡸⣻⡿⡖⢾⣦⣄⡙⣻⡿⢿⡟⢋⣠⣴⡷⢺⢿⢿⢇⡇⠀⠀⠀⢀⡴⠀⠀⢀⡼⠀⠀⣠⠆⠘
⣿⠀⠀⠙⢷⣄⠀⠙⢦⣄⠀⠙⠦⣄⡀⠈⣧⢧⢳⡙⢦⡙⠿⣭⣍⣀⣀⣩⣭⠿⢋⡴⢫⡎⡾⣼⠁⢀⣠⡴⠋⠀⣠⡶⠋⠀⣠⡾⠃
⢻⡀⠀⠀⠀⠈⠛⢦⣄⡈⠛⠶⢤⣈⡙⠳⢿⡎⢧⡙⢦⡙⠳⠶⣤⣭⣭⣤⠶⠞⢋⣴⢋⡼⢱⡿⠖⢋⣁⡤⠶⠛⢁⣠⡴⠛⠁⠀⠀⠀⢀
⠈⢳⣄⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣬⣉⣛⣒⢿⣌⠳⣄⠙⠲⢤⣤⣀⣠⣤⡤⠖⠋⣠⠞⣱⡿⣖⣛⣩⣥⡤⠶⠚⠋⠁⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠙⠳⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣻⣦⠈⠳⢦⣄⣀⠈⠁⣀⣀⣴⠞⠋⣴⣟⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⢦⣄⣈⠉⠙⠋⠉⣉⣠⡴⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡴⢦⡖⢲⡏⣹⣁⣽⣉⣽⣁⣽⣁⣻⣁⣽⠉⣗⠙⡶⠒⣤⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠲⡋⢹⣅⡼⠶⡛⣮⣉⣇⣸⣇⣹⠤⠽⢤⡼⢧⡼⢧⡼⣅⣽⣉⡿⢛⡶⢧⠤⣏⢨⠗⣲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠒⢻⡤⣼⢶⡉⣣⠠⠷⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠦⡞⢙⠮⡉⠛⠧⣜⣛⢦⡀
⠀⠀⠀⠀⠀⢠⡞⠳⣴⠓⢏⣣⠼⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢤⠛⢢⠀⠈⠣⡖⢳⡄
⠀⠀⠀⠀⢀⣎⢉⡿⢭⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣉⣆⡀⠀⢹⣩⣽
⠀⠀⠀⢀⣇⣈⣏⣙⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⡔⢲⡤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⢧⠀⢸⡖⢼
⠀⠀⠀⣸⢄⣽⠶⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠽⣤⡓⠘⠤⢼⢁⡟⢙⡖⢢⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⢾⣠⢏⣉⠎
⠀⠀⠀⣧⣤⣿⡖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣠⣏⢹⡋⣹⠉⣏⣈⡏⣹⢾⣴⢋⣳⡀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡉⢣⡠⠋
⠀⠀⠀⢳⠖⢫⣽⢉⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠈⠁⢉⣷⡚⣯⢫⠤⣗⠀⠀⠀⠀⠀⠀⠀⢠⣏⠜⠋
⠀⠀⠀⠈⠱⣏⣹⢽⢤⠟⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⠻⣤⣿⠿⣍⣳⠋⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⢧⡴⠳⣝⠧⣼⠛⣢⠤⣄⣀⣀⡀⠀⠀⣀⣀⡠⣤⠤⡞⢻⣄⣟⡾⠿⣉⡷⠋⠀⠀⠀⠀⠀⠠⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠺⣉⡟⢚⠯⢷⣶⡻⠤⣏⣨⣏⣹⣇⣸⣦⣾⣴⡯⠻⡉⣢⠽⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠢⠿⣰⡋⣹⢁⣝⠉⣹⡉⣹⣁⡽⠤⠷⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠈⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠞⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⡀⠀⠀⢀⡄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠘⡀⠁⠀⠀⠀⠀⠀⢉⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠻⠟⠀⠀⠨⠀⠄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡠⢒⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⢄⢀⡠⠰⠚⠓⢺⡀
⠀⠀⠀⠀⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠰⡆⠀⠀⠐⠀⠀⠁⠀⠰⠄
⠀⠀⠀⠰⠃⠐⠣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠉⠀⠢⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠨⠀⠀⠀⠀⡀⠀⠈⠛⣉⠁⠀⠀⠀⠤⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢂⡤⠠⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⡀⠆⠈⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⠈⠀⠰⠀⠠⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢂⡀⠑⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴
⠀⠛⠍⠀⠀⠀⠐⠐⠀⠊
⣷⣿⣿⢶⣶⠾⣿⣷⣾⣾⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣶⣶⣦⣶⣦⣾⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⣹⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣿⣿⣿⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣆⠽⠟⠋⠛⠀⣨⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⣿⣿⣿⣿⠗⠶⠂⠀⠀⠀⠀⠈⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⠈⢻⡟⠁⠀⠀⠀⠀⠀⢲⣄⠀⢀⣿⣿⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢲⣾⣿⡿⠃⠀⠸⣦⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣹⣿⣿⣿⣦⡤⠀⠻⢷⣶⣾⡿⣿⣿⣷⡈⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣟⢻⣿⣤⣦⡀⠀⠀⠀⠀⠨⣺⣿⢿⣷⣤⡈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢼⣧⣤⣤⣀⠀⠀⠀⠀⢠⣿⣿⠟⠉⠁⠠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠏⣾⡟⠁⠈⢉⣠⣴⣶⣶⣾⣿⢿⣦⣀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⣸⣇⣀⢠⠏⠁⠀⠀⠀⠀⠘⣶⡬⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⢾⣄⠀⠀⢀⣠⣤⢞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠛⠛⠓⠹⣿⣾⣿⣫⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠠⣿⠿⠷⣛⣣⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠈⠀⠐⠐
⠀⠀⠉⠂⣄⣀⢀⣀⣀⣄⠀⠀⠉⠓⠚⠋⠉⠁⠀⠀⠀⢀⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠉⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠉⠀⠀⣀⡰⠤⠐⠀⠀⢀⢀⣤⣦⠄⠀⠀⠀⠀⠀⠀⢀⣠⡤⠄⣀⠀⠀⠄⠀⠀⡠
⠀⠀⠀⠤⢀⠀⠀⢀⡤⠀⣆⣀⣠⡤⣀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠡⠀⠀⠀⠀⠀⠀⠿⠞⠀⠀⠀⠀⠀⠀⠀⠙⢼⣬⠞⣧⣧⠀⢀⠀⠊
⠀⠀⠠⠄⠀⠀⠐⠿⠴⣶⣤⣤⣽⠷⠦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠘⠂⠼⠆⢰⡤⢀⠄⠠⠀⠀⠘⠿⠟⠀⠀⠀⠀⠴
⠀⢠⠘⠉⠀⠀⠉⠓⢄⠀⢀⡀⠀⠠⡀⠀⠰⣷⡛⠛⡻⣿⠁⠀⠄⠀⠀⠀⠀⠳⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂
⠀⠲⠢⠤⣄⡤⠴⠆⠖⠀⠨⡀⠀⢀⡈⠀⠀⠙⠿⣽⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣴⣶⣤⣴⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⡆⠠⠈⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣤⣠⣤⣄⣄⣀⠀⠁⠀⠀⣀⣠⣤⣤⣤⣀⢀⠀⠀⠀⢀⠴⠀⠂⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠺⠿⠿⠟⠯⠿⠟⠟⠀⠀⠀⣈⠉⠉⠛⠛⠋⠛⡋⠀⠀⠀⠀⠀⠀⡀⣀⣤⡤⠀⠀⠀⠇⠁⠤⣀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⡀⠀⠀⠀⠺⠉⠀⠀⠀⠀⠀⠋⠉⠉⢰⠋⡓⠀⠀⠀⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠈⠰⢧⡤⢣⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣋⠝⢃⠂⡀⠀⣀⣀⢴⡠⠀⠠
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠤⠤⠀⠁⠀⠈⣴⢤⡀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⡠⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠱⡛⠣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣉⣤⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠊⠁
⠀⠀⠀⠀⢀⠈⠉⠀⣬⠧⠀⠀⢤⡄⠀⠀⣠⡄⠀⢀⡀⠀⡀⠀⢤⡄⡄
⠀⠀⠀⠀⠀⠀⠔⠂⠉⢷⡀⠢⠾⠷⠫⠞⠿⠅⣤⢛⠉⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣄⣠⠴⠈⡈⠀⠄⠲⠄

⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢰⡿⡿⢉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⣸⠃⡀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣿⣿⣧
⠀⠀⠀⢠⠊⠈⡑⠚⢯⠀⢗⣄⣀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢉⣼⣇
⠀⠀⠀⠘⣄⠀⠙⢦⢨⡖⢀⣍⢷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢎⠈⣟⣭⠟⡟⠓⢀
⠀⠀⠀⠀⠘⣆⠐⡵⠢⢜⡇⣿⡿⢻⡷⣰⠢⢄⡤⠤⠤⢠⣄⡔⢠⡆⣱⠃⠉⡏⢉⠁⡜
⠀⠀⠀⠀⠀⠈⠓⢾⣿⣿⣾⡏⢀⣵⣿⣃⣀⣤⣴⣣⡴⠟⢋⡩⡖⡠⡻⣶⣦⣤⣥⡼
⠀⠀⠀⠀⠀⠀⠀⢀⢰⣀⣿⠖⡿⠛⠛⠉⠉⠀⠀⠘⢼⣴⠀⣡⣿⣤⣖⣉⡿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠙⢋⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡼⣁⠀⣼⢻⡟⢛⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠬⣹⡿⢺⠟⠒⠚⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡇⠀⠀⠀⠀⠀⢠⠊⢀⠀⠀⢀⣨⠷⠶⣖⡒⠒⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡆⡇⠀⠀⠀⠀⢀⠇⠦⢈⢉⡞⠉⠀⠀⠀⠈⢻⡯⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣡⠇⠀⠀⠀⢀⠎⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⣽⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⡇⠀⠀⠀⠀⢘⣦⣍⣐⡋⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢀⡇
⠀⠀⠀⠀⠀⠴⠿⠿⢶⡞⠀⡷⠶⣶⠖⠊⣀⣈⡷⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⡏⠙⢯⣴⣶⣶⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣿⣿⠟⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠘⠛⠁⠀⠀⠀⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠦⠀⠀⠀⠀⣀⡀⠈⠛⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢦⡄⠰⠿⠿⠇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠘⡇⠀⡀⠀⠀⠼⠛⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠖⢣⡖⠂⠀⠀⠀⠀⢀⡟⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣪⣭⡍⠉⠀⠀⣀⠜⠀⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⣤⣤⣠⣾⣿⣄⣀⣀⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠶⢄⡀
⠀⠀⠀⠀⠀⢀⣤⡄⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⣴⣲⣄⡀⡄⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣼⣿⣧⣹⣿⣿⣿⣿⢿⡇⠉⠉⠉⠀⢀⠀⠉⠻⢿⣿⣷⣮⡅⢄⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢏⡾⠁⠀⠀⠀⣶⣾⣷⣤⢀⡀⠀⠛⠿⣿⣦⠡⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⡭⠊⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⢄⣷⣿⣇⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⢘⣿⡟⣿⣿⡿⣾⡇⠀⠀⠀⠀⠀⢸⣿⣽⡿⣿⣿⣿⣿⢏⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⢠⣶⣯⢀⣻⣿⣷⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣶⣿⣿⣿⣿⣿⡀⠀⢀⡀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠸⣿⡿⢰⣿⣿⣿⢿⣿⣄⠀⠀⠀⠀⠀⠹⣿⣿⡿⣿⣿⣿⣼⣿⣿⣿⡿⢇⣴⣀⠘⣦⣀⠀⠀⠀⢷
⠀⠀⠀⠈⠈⠁⠈⠋⠉⠁⠞⣟⢿⣷⣤⣤⡄⠐⠘⠀⠑⢄⡈⠻⣿⣿⣷⡰⣶⣶⣿⣿⡿⠃⠀⠙⠳⢦⡀⢸
⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣫⠞⢻⡟⠿⠁⠀⠀⠰⠀⠀⠀⠉⠈⠻⢿⣿⡟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠁⢸
⠀⢸⠀⣾⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠳⣐⠆⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠘⢷⣥⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠹⣦⣁⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠙⠷⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣶⣤⣀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⡇⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡗⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⣻⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣛⡛⢟⣿⣿⣿⣿⣶⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢹⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⠿⣿⣿⠻⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⣴⣆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣼⢻⣿⣿⣿⠿⣿⣻⣏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣶⣿⣿⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠛⠟⠿⠿⠿⠿⣿⣿⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣄⣀⣀⡀⠀⠀⣀⣤⡀⠰⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣿⣿⡯⠉⣿⣿⣿⣿⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣾⣿⣿⣟⠿⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠶⢼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠿⠿⢿⠛⠛⠛⠛⠻⠿⠇⣰⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⠀⠀⠀⠠⣦⠐⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣮⣥⣒
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⢰⣿⣦⣀⠀⢠⣟⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠰⡒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣽⣾⣷⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡄⠛⠍⢙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⡀⣰⢀⠁⠁⠘⢧⠴⢦⣀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⠟⣄⠠⠁⢠⠀⣠⠃⠀⠙⣴⣯⣾⣷⣡⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣽⠷⠾⠓⠠⠼⣋⣁⣁⠠⠔⠉⠹⢿⠛⠭⡛⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⢀⠄⣀⡀⠀⣻⣗⣀⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠱⡐⠋⠉⠠⠆⠉⠿⢿⣃⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⢿⣀⠀⠈⣁⣠⡶⣶⣶⣏⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⡽⠿⠋⠻⣽⣟⣽⣠⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠲⣤⣶⠘⠟⣧⣷⣚⠁⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡿⣭⣀⡀⠀⠀⠈⢻⡄⠆⢹⡟⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠄⣈⡁⠒⢦⣆⣻⠈⠀⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠛⣷⣰⣿⣷⠶⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣈⣽⣿⡽⠐⡏⠉⠃⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡻⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠁

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


⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⠿⠿⢿⣿⡿⠿⢿⣿
⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿