Lay down Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⣀⣀⠠⠶⠿⠛⠠⠀⠈⠈⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠉⣷⠀⠀⠀⠀⠐⣺⣿⠿⠟⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⡀⣀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠏⠉⠉⠙⠀⠀⠀⠀⠀⠘⠡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠁⠉⠀⠀⠈⢢
⠀⠀⠀⠠⢜⠦⠀⠀⣀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠄⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⢧⣀⣀⣀⡀⠙⠲⠶⣿⡏⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⣔⢤⠀⠀⠀⣠⣤⣀⣀⣤⠤⠴⠶⠤⠴⠶⠖⠒⠒⠈
⠀⠉⠁⠀⠉⠉⠁⠒⠉⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡈⢄⠀⢀⣴⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆⠀⠀⢀⡠⠞⡀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣤⣬⡠⠋
⠀⠀⠀⠀⠀⠀⠀⡰⠶⠀⠀⠀⠀⢀⡴⢷⣶⣮⣷⣄⣀⣀⣀⣤⣤⣤⣶⠶⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢸⣊⣁⡤⠤⢤⣴⠟
⠀⠀⠀⠀⢀⠀⠁⠀⠀⢀⡠⠞⠁
⣤⣤⣤⣤⣬⣿⣶⣶⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⡶⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⣹⡆⠀⠀⠀⣀⡤⠖⠋⠍⠈⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⣀⠠⠢⠈⠃⠙⠈⠒⠆⠚⣇⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣵⠋⠀⣠⠖⠛⠩⠀⠠⠀⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠿⠛⢳⣀⣀⣀⡀⣯⠀⢀⠔⢸⠇⠀⠀⢻⡀⠀⠀⠀⠀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡠⢠⠀⣸⠀⠀⠀⠉⠛⠏⠡⠘⣾⠀⠀⠀⠀⢳⡀⠐⠄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠠⣄⠀⠀⠊⠀⠀⢈⠃⠠⠁⠀⠀⠀⠀⠀⠂⠘⡇⠀⠀⠀⠘⡇⠀⠀⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢄⠠⡑⠆⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⠀⠀⠀⠀⢹⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢂⠀⢁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡀⠁⠀⢸⡀⠀⠀⠀⠘⡇⢰⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠌⢢⡀⠫⠃⠀⠈⠙⠲⣄⡀⠀⢷⠸⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄⠀⠀⠐⠀⠐⠀⠀⠀⠈⢂⠀⠀⠀⡐⠀⠈⢀⠀⠑⠀⠀⠀⠉⠛⠺⠄⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⣀⡤⠤⡤⠦⣄⣀⣀⣀⣀⣀⣿⠀⠀⠀⠈⡿⣄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣧
⠀⠀⢠⡞⠉⠀⢤⡆⠠⣌⣉⠉⠉⠁⢀⠉⠂⠀⠀⢠⣇⣘⣦⡀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⡧⠀⠰⢄⣱⣀⣀⣀⣀⡀⠀⠀⠀⢀⡤⠠⠊⠉⠉⠀⠀⠀⠀⠀⠄⠈⣠⣀⣈⣀⣀⣀⣔⣀⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⠞⠁
⠀⠀⠀⠙⠛⠛⠉⠉⠀⠀⠉⠉⠙⠛⠲⣤⣰⣀⡤⠾⠖⠚⠛⠒⠒⠶⠦⣤⣀⣾⠋⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡨⠟⠒⠐⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠡⠆⠀⠀⠀⠀⠀⠀⢰⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠈⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢠⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠂⠀⠀⠀⠀⠤⠄⢠⡤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠈⢓⠆⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠁⢀⣠⠄⠂⠁⠐⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠪⣀⡠⠀⠁
⠀⠀⠀⠀⠀⡀⠠⠐⣾⡁⠀⠀⣀⡀⠤⠘⠀⠀⢀⠤⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠄⠒⠐⠂⠈⢁⡠⠔⠃
⠠⠄⠂⠉⠄⠀⠀⠀⠀⠙⠒⠈⢀⡠⠴⠀⠉⠁⠀⠀⠀⠀⠀⠀⠤⠀⠦⠄⠤⠐⠒⠉⠀⠀⠀⢀⠀⠄⠒⠙⠢⠀⠘⠈⡄⠀⠀⠀⠀⡄⠂⢀
⢸⢢⡀⠀⠀⠀⠀⠀⢀⠄⠀⠉⠀⠀⠀⢀⣀⠠⠄⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠐⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠠
⡄⠃⠈⠂⢄⠀⢀⡔⠉⠀⢠⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠐⠕⠀⡀⠀⠑⠢⡙⠶⠴⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠆
⠀⠀⠈⠪⠢⢀⠀⠈⠑⢄⠀⠀⠀⠀⣀⠠⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠉⠁⡍⠉
⠀⠀⠀⠀⠀⠀⢑⢄⡀⠀⠈⠐⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢅⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠉⡢⠴⡔

⠀⠀⠀⠀⠀⠀⠀⡀⣨⣹⣿⣿⣾⣿⣿⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣯⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠛⠁⠀⠈⠻⣿⣿⣿⣿⣾⣾⣿⠻⣭⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣟⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠘⠏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠁⢀⣩⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠉⠉⠉⠉⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⠛⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣤⣤⣤⣜⡿⢿⣿⣶⣦⣤⣄⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠇⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⣿⡿⠋⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠟⠋⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣀⣀⣀⣈⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣄⣀⣀⣀⣴⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠈⠉⠉⠉⠉⢹⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⣶⣶⠶⠶⠶⢾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⢉⠻⣿⣿⣿⡻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⡔⠁⠀⢀⣤⣴⣿⣿⡅⠋⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⣻⣿⢋⣤⡜⠀⠀⢠⠋⠉⠉⠉⠉⠙⠢⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣤⡀⢠⠟⠓⠻⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⡟⠱⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⠓⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⠿⢧⢀⠎⠀⢁⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⡥⠬⠃⠽⢿⣻⠀⠀⠀⠣⡀⠀⠈⡄⠀⠀⡘⠀⠄⠀⠂⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠅⢠⠂⠻⡙⢻⡀⠀⠀⠀⠈⠲⢄⠈⠢⣸⠀⠐⠀⠀⢘⠆⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠉⠣⣅⠸⠥⢌⣒⡟⠈⠢⡀⠀⡀⣄⠤⣫⣦⣿⣶⣶⣶⣶⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠈⠙⠒⠬⠤⠆⢀⣠⣈⠑⠌⠦⠗⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣚⡱⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠤⢀⣀⡀⠀⠀⠴⢖⣆⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⡿⠿⠿⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣭⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠙⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⡟⠋⠁⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⢸⣿⣿⠀⠀⠀⢘⡍⢉⠀⠀⠀⠀⠀⠀⠶⠄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⡴⠁⠀⠀⢐⠀⠀⠀⠀⣴⡒⢢⠘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢳⠂⠀⠉⡠⠀⠀⠀⠀⠑⠬⠋⠀⣇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠿⠀⠀⠀⠐⠈⠀⠀⠀⠀⢳⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠴⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⠇⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⠙⠚⠉⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡴⣿⣧⡄⠀⠀⠀⡴⠋⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠓⠙⠿⠁⠀⡠⠊⠀⠀⠐⠃⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⢶⣉⣳⣄⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠂⠀⠁⠢⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠔⠂⠐⠁⠠⢄⣼⢩⣿⡟⠹⣖⡾⠋⣿⣷⣀⣀⣠⣔⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢨⠃⠄⡠⠚⠳⡤⢿⣾⡷⠁⡜⠉⡇⠀⢻⡇⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡰⠁⠀⠀⠘⢿⡃⠀⠘⠁⠀⡥⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣦⣊⠉⠉⠓⠚⠣⠤⣄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡕⢠⡦⣄⠀⠘⣫⠈⣢⣄⡀⠀⠀⠢⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠫⠊⣤⠞⠁⠈⠦⠟⠁⠉⠀⠀⠉⠙⠓⠛⠻⣽⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣉⡛⠛⠋⢉⠗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠾⠦⠯⣭⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢶⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⢶⠃⠀⠸⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠳⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡇⠀⠀⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣄⠀⢶⣄⣀⡸⠛⢢⣤⣠⣴⣛⡦⣤⣤⣼⢳⣦⣤⠟⠲⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⢀⠀⠀⠀⠀⢸⠙⠋⢠⡤⠔⠊⢁⣀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠉⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢸⡉⠭⢦⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠂⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠉⠉⢱⠖⠚⠁⠀⠐⠮⡠⢄⡀⢀⡀⠔⠀⠀⠢⠤⠀⠀⠀⣠⠖⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⠀⠀⠈⠁⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠰⣥⣈⣵⡤⠇⠀⠀⠀⠀⠀⣀⡼⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠁⣄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⣀⡴⠛⣇⠀⠀⠀⣰⣧⠞⠀⠙⠒⠦⢬⣭⣉⣉⣻⠉⠉⠉⠉⣉⣀⣀⣀⣠⡼⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠈⠛⠛⠛⢡⡟⠀⠀⠀⠀⠀⢀⣀⡀⠀⢸⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠐⢰⡖⠋⠉⠉⠇⢚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢧⠀⠀⠀⢸⡼⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⢇⠀⠀⠈⣇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⠀⠸⣤⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣨⣦⡀⠘⠓⢦⠤⠼⠷⢤⡾⢻⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡤⠚⠉⠀⠀⠀⠙⠛⠛⠋⠙⠛⠒⠚⠁⡞⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⠟⠋

⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣚⣓⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣷⣦⡀
⠀⡠⠞⣋⣩⣤⣤⣬⣉⡙⠲⢄⠀⠀⠀⠀⠀⢰⣿⣿⠿⠿⠿⠛⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠿⡅⠙⣧⢿⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⠿⠿⠟⠻⣿⡆
⢀⣴⠾⠛⣿⡿⡛⢿⣿⠛⠷⣄⠀⠀⠀⠀⠀⢸⡟⠈⠳⢄⣀⡀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⡟⣿⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢻⡇
⠈⠳⣄⡀⢿⣿⣿⣾⡿⢀⣠⠞⠁⠀⠀⠀⠀⠸⣇⠘⠛⠈⢿⣿⣆⣹⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⡼⣱⠏⠀⠀⠀⠀⠀⠀⠀⢻⡀⠛⠃⠀⠛⠋⢨⡇⠀⡀
⠀⠀⠈⠙⠛⠛⠻⠛⠛⠩⣄⣠⠀⠀⠀⠀⠀⠀⠹⡄⠀⣸⡅⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠠⣄⡴⠀⠀⠀⠀⠀⠀⢷⠀⢀⣭⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠳⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠟⠁⠀⠰⠋⠳⠀⠀⠀⠀⠀⠀⠈⠑⠦⠤⠴⠋

⠀⠀⣀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⣏⣹⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣙⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⡋
⠀⠀⢻⣿⣿⠷⠶⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⢷⣄
⠀⠀⠸⣟⣿⣷⣶⡎⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⣟⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⠸⠇⣿⣿⡄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠛⢸⣿⣷⠀⠉⠓⠤⣀⠀⣴
⠀⠀⣰⣾⣿⣿⣽⠇⣼⣿⣶⠄⠀⠀⠀⠀⠀⢸⡇⠀⠙⠛⠛⢳⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣷⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⣸⡟⢿⡇⣠⣤⣶⣶⣾⣿
⠀⢾⣹⡟⠛⠛⠛⠛⠛⢿⣹⡇⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⡤⠚⠏⠉⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢻⡇⠀⠈⢧⠀⠀⠀⠀⠀⢠⡿⠁⢸⡇⢡⣞⠉⠉⢹⡇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢰⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡀⠀⠀⠀⠀⠀⠀⠠⣾⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⡀
⠀⠀⠀⠀⣠⣶⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠂⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡄
⠀⠀⠀⠀⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏
⠀⠀⠀⠐⠟⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⣿⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⡰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡃
⠀⠀⠀⠀⠰⠿⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠦⡄


⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢲⣒⣲⡆
⠀⣴⣿⣿⡿⢿⠿⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⢸⣿⣤⡞⠉⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⣾⣿⣿⣷⡄⢸⣿⠀⠀⠀⠀⠀⠀⢈⡿⢾⣍⠀⠀⡀
⠀⢿⣿⣟⣻⣿⣟⣧⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡯⣾⣎⡿⠶⢦⠆⠀⠀⠀⠀⠐⣿⣿⡿⠀⢿⣿⣿⣿⠿⠇⠸⠇⠀⠀⠀⠀⠀⠠⣿⠩⠭⢿⡇⠈⠥
⠀⠀⢩⣭⣭⣽⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⣻⡿⢿⠓⢦⠀⠀⠀⠀⠀⠀⢀⣘⣛⣁⣿⡜⢃⣶⣶⠖⠒⠐⠃⠀⠀⠀⠀⠀⠀⠻⣦⣹⠿⠃⠈⠂
⠀⠀⠀⠈⠉⡞⠁⠀⠀⣹⣾⡁⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣟⣳⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⢀⡀⠀⠀⠒⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣴⣿⣿⠋⠀⣠⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣾⣿⣶⠒⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢘⠯⠿⠿⠿⡏⠠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠇
⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣛⡂⠀⠀⠀⠀⠀⣿⣿⣿⣆⡂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠿⠉⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⣤⣤⣤⠀⠀⢠⣄⣠⡘⠓⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢉⣠⣦⣠⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠉⠁
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢁⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠃⠀⠀⠐⠛⠃⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠘⠋⢩⠖⠘⠛⡿⡟⠿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣾⣦⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠰⡫⢹⠄⢴⠀⠈⣟⣛⢃⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠋⢸⣿⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠈⠓⠀⠀⠵⢔⣉⣅⡱⠃⠘⠧⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠺⠿⠏⢸⣿⠀⠀⠀⠐⢿⣷⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⡀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⣄⠙⢋⡼⠂⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⢐⣀
⠀⠀⠀⠀⠉⠀⠀⠀⠰⣿⣿⡿⠃⠀⠀⠀⠀⢀⣼⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠠⠟⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣍⣫⣍⣉⣯⣻⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠲⡄⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠒⠀⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠄⡔⠒⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠒⠒⠒⠒⡖⠒⠒⠒⡖⠲⠒⠒⠂⠐⠂⠒⠀⠐⢺⠁⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⡀⠀⢰⠀⠃⠀⠀⠀⠘⠀⡇⠀⠀⠀⡇⠀⡇⠀⠀⢰⠀⠃⠀⠀⠀⢰⠀⡄⠀⠀⡈⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⠀⠈⡀⢰⠀⠀⠀⠀⠀⢁⠀⠀⠀⡇⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⡄⠀⠃⠀⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡇⠀⠀⡇⢸⠀⠀⠀⠀⡇⠸⠀⠀⠀⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠧⠤⠔⠃⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠒⠒⠚⠒⠒⠚⠚⠒⠒⠒⠒⠓⠚⠒⠒⠒⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠐⠒⠂⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠒⠒⠠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢀⠎⣀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⢤⡈⢇
⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠐⠒⠒⠒⠖⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⡒⠒⠒⠒⢲⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠀⠀⠀⠀⠠⣤⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠈⠳⣄⠀⠀⠀⣠⠞⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠙⣷⣠⡾⠃⠀⠀⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⢀⣠⣤⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣶⠶⠤
⠀⠀⠋⠉⠉⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡷⠖⠒⠛⠉⠉⠹⣏⠉⣹⣍⣿⠟⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣤⣼⣭⠿⠁⠀⠉⠉⠉⠉⠒⠒⣿
⠀⠀⠀⠀⠀⠸⣿⠀⢀⡠⠄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⢹⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣷⠀⠁⠀⠀⢠⠀⠀⠀⠀⠐⠲⣄⡀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢽⣠⡾⠁⠘⠷⣄⡀⠀⣮⠁⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣬⠇⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠈⠙⠛⠧⠤⠤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠙⠻⣿⣿⣓⠾⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣈⣷⣮⣝⣷⢯⣝⡳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠉⠛⠻⢧⣍⣷⠮⣝⣓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣯⣙⢲⣭⣛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣮⣿⡒⢬⣝⡲⢤⣄⣀⣤⠶⠶⠶⢤⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢧⣽⣗⠮⣭⣀⣤⣶⠦⡄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⢿⣿⣷⢤⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠈⠻⠿⠿⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠁

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉
⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠉⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡰
⠀⠀⢀⠁⠀⠀⠲⠤⣄⣀⠀⠀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠁⠂⡄
⠀⠂⠈⠀⠀⢀⠀⠀⠀⠚⠉⠉⠀⠀⠙⢦⣄⠀⠀⠀⢨⠠⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠄⠠⠌⠐⠠⡳⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠋⠁⢠⣀⣄⠀⠀⠠⠤⠀⠀⠰⠖⠒⠂⠀⠂⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠐⡆⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⢠⠄⢀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢀⠀⠀⠀⡄
⠀⡤⣿⣒⡀⠀⠀⢰⣧⠀⠷⢄⠀⠀⢺⡆⣸⣆⡀⠀⢀⣧⠀⠀⣿⡇⠀⣸⣇⠀⢸⣿⠀⠈⣢⣤⣤⣾⠃
⠀⢧⡀⢀⡽⠄⠀⢸⢸⣨⣶⣮⡇⠀⣏⣣⣣⣤⣓⠀⠡⣉⡹⠆⡏⡇⠀⠇⣉⠽⠜⡟⠀⠀⠛⠀⠐⠙
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠄⠒⢔⠃⠀⢀⣼⣶⡦⢄⠀⠀⠀⠐⠐⢰⠋⠀⠀⠀⢠⣏⠀⠀⠀⢧⠀⢀⠀⠀⠀⠀⢠⠀⠀⠂⠂
⠀⢀⡴⢀⠍⠀⠀⠛⢿⣿⣟⡠⠀⠀⢀⡴⠀⠌⡀⠀⠀⠀⡞⡇⠁⠀⠀⠀⠉⡼⠧⣄⠀⠀⠾⢦⡀
⠀⠋⠀⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠘⠀⠛⠀⠀⠀⠀⠀⠑⠀⠊⠀⠀⠂⠔⠁
⠀⢀⠀⠆⢀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣖⢲⡄⠀⡀⠀⠀⢄⡀⠀⣀⠄⠀⠀⠀⣲⠀⠀⠀⠀⢠⠄
⠀⡁⠰⢄⠀⡅⠀⠀⠀⠌⠙⠆⠀⠀⠏⠉⠀⢸⡇⠀⠀⠈⠡⠀⠉⠀⠀⠀⠘⣿⠓⠀⠀⠀⢻⡧
⠀⠐⠀⡄⠐⠃⠀⣆⣧⣧⣧⣯⡇⠀⠀⠠⢎⠉⣹⠀⢀⠜⠃⠀⠓⠄⠀⠀⡼⠉⣇⠀⠀⠠⠎⢧⠰⠒⠇
⠀⠀⠀⡅⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣼⣿⡀⠀⠀⠀⢀⣾⣧⠀⠀⠀⠀⣨⣿⣄⠀⠀⠀⣸⣇⠀⠀⠀⠀⢀⣴⢿⡉⠀⠀⠀⠀⢀⣆⣯
⠀⠀⢹⢿⠁⠀⠀⠀⠈⣿⡯⠀⠀⠀⠈⣩⠉⣍⠃⠀⢀⡿⡆⠁⣄⠀⠀⠈⡇⢐⣦⡄⠀⠀⠀⡴⠾⢏
⠀⠀⠚⠘⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠘⠀⠓⠀⠉⠁⠀⠀⠃⠀⠀⠀⠀⠀⠘⠁
⠀⢀⣴⡂⠀⠀⠀⠀⠀⢀⣴⡂⠀⠀⢀⣤⣤⡆⠀⠀⠀⠀⢀⣠⣦⡄⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠂
⠀⢺⣿⠈⠀⠀⠀⠀⣾⣧⢸⣿⠀⠀⠀⠀⢤⣄⡀⠀⢀⣀⣸⡇⣁⣀⠀⣀⠀⠀⢿⣿⡄⠀⠀⢰⣷
⠀⢸⢹⣀⣀⡀⠀⠀⣿⡏⠘⠹⠀⠀⢴⣿⣶⡿⠟⠀⠈⠻⣿⣯⠟⠁⠀⠙⠿⣾⠟⠁⠀⠀⠀⠈⣟
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⢠⣿⠤⣀⡀⠀⠀⣰⠀⠀⠀⢐⡆⠀⠡⠤⠃⣹⣃⠀⠀⣾⡀⠀⣽⡄⠀⣤⣦⣶⣿⣿⠆⠀⠀⣬⣳⡄
⠘⢿⡶⡆⠀⠈⠀⠛⠻⢀⠰⠻⠃⠀⠀⢠⣶⣿⡟⠀⢀⠿⡄⣾⣿⡇⠀⣿⣿⣿⡿⠟⠀⢠⠲⠊⣿⠳⠂
⣀⢀⢀⢁⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡘⣁⣀⣀⣀⣈⣀⣁⣉⡉⡁⣀⣉⣁⣀⣀⣀⣀⣈⣀⣁⣁⣁⣈
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠂⡄⠰⢄⣴⣯⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢻⣿⣿⣿⣯⣿⣿⣿⣿⣦⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣿⣿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣮⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠀⠛⠙⠯⠿⠽⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁

⠀⠀⠀⣠⣤⣾⠃⠀⠀⢀⣰⠟⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠂⠀⢀⣶⣖⣶⣆⢠⡀⠀⠀⠀⠀⢀⡀
⠀⠀⣼⣿⡿⠁⢀⣤⠶⠛⡁⠄⠒⠛⣿⡿⠃⣀⡤⡶⠦⣀⣺⠶⠚⢉⠁⠀⠀⣉⣯⣟⡻⠶⣄⡰⡄
⢀⣼⡿⠋⢀⡶⠋⠀⠀⠁⠁⠀⠀⠈⠙⢀⣢⣭⣿⡐⢛⣋⡄⢐⢎⡴⢒⡽⣚⡵⢋⡥⣒⠾⢛⣻⡄
⣾⡟⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣫⡵⢋⣴⢾⡫⠅⡰⠃⡾⣣⡟⢱⡷⡐⢉⡞⣡⣾⡿⡋⣧⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠏⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠣⣢⠽⣾⣁⡐⣚⠁⡸⢥⣏⢰⡏⣰⠣⢸⢰⠟⢋⠄⢠⡼⣿⣂
⠀⢀⣿⡇⠀⣤⠀⠀⠀⠀⠀⠀⢀⠟⠀⠐⣱⢞⡟⡱⣱⡇⢰⠃⠀⣡⠸⢰⢧⣤⢰⢀⠄⡀⠀⢀⡴⣿⢽⡰
⠀⣼⣿⠁⠀⣿⣆⡀⠀⠀⠀⡄⡛⣀⢂⣰⠷⠊⣴⣣⠟⣡⢇⡀⢰⢇⠀⣎⣟⡿⡿⠈⣴⠃⠂⢠⣆⣿⣷⠹
⢠⣿⠃⠀⢠⣿⣿⠛⣦⡀⠀⠀⢚⣦⠌⡋⠠⡠⠓⣫⣰⢫⠎⠜⣠⡎⢸⣹⠻⣱⠇⠀⣿⣷⠁⢊⢎⣯⣤⠐⠆
⣼⠃⠀⣠⣿⡿⢁⣾⡿⠋⢓⣶⠈⠀⢨⠁⢨⣰⢰⢗⢣⡍⡀⠀⠟⡄⠠⢧⠄⡃⡌⠀⢻⠋⣤⠆⠈⠛
⡏⠀⣴⣿⡟⢀⣾⠏⢀⣴⣿⠏⢠⡰⢸⡇⢎⡏⣼⢊⡴⢳⢡⡄⠾⢁⢖⠄⡐⠸⣵⡴⡼⢚⡄
⢇⣼⣿⣿⠠⠾⡋⠀⣼⡟⠁⠀⠀⠀⠸⢏⡜⠰⢃⣼⢡⢋⢫⠔⠰⠁⢀⡾⢿⢹⢏⠍⠀⠀⡀
⠀⣿⣿⣿⡄⠀⠀⣾⡏⡠⠀⠀⠀⠀⠀⠰⡆⠀⠘⠇⢣⡏⠀⠀⡰⢠⣼⡇⠾⠀⡌⠀⠀⠀⠂
⡀⠈⢻⣿⣷⣀⠀⡿⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢫⡀⠇⠁⣿⠰⡁⢸⠀⠀⠁⡀
⣷⡀⠀⠙⠿⣿⣷⣞⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠋⠇⠘⢰⡜⢾⠁⠀⠀⠀⠀⠉
⣿⣿⣦⡀⠀⠀⠉⠏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠀⠃
⡘⣿⣿⣿⣦⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣌⡛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠟⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠞⠀⠉⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠀⠀⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⢀⣾⣿⣿⡿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠘⠟⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⡈
⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡷⠆⠀⠤⠀⠠⠂⠀⠀⠀⠀⡰⠒⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣿⡇⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠠⠃⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠰⢻⠃⠀⠘
⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠈⠙⠧⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢸⠀⠀⢀⠄
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡠⠃
⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠄⢠⠑⠤⢀⠌
⣿⣿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠁⠂⠀⠀⠄⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠁
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡟
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠉⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣤⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣾⠃⠀⠈⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠒⠒⠲⠒⢶⣄⡀⠀⠀⠀⠀⠹⣾⠋⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⠶⠶⠶⠾⠋⢛⣦⣄⡀⠀⢀⡿⠀⢶⣤⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣠⡼⠷⣤⣴⠿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠋⠀⣠⡶⢶⣄⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣿⠀⢸⡇⠀⠀⠸⣇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡆⠈⣷⠀⠀⠀⢿⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⡇⠀⠘⣧⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣧⣼⠛⠛⠛⠛⠛⢻⣦⣈⠙⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣧⣀⣿⣈⠛⠛⠛⠛⠛⠛⠁⠉⠻⣦⣀⣼⠋⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⢉⣉⣉⣈⣉⠉⠛⠳⣦⣀⠀⠀⠀⠀⠀⠸⡟⠻⠶⠾⠃⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣠⡴⠞⠛⠉⢻⡉⢹⡟⠳⣦⣀⠙⢷⣄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠻⣤⡀⠀⠀⢸⡇⠈⠛⠛⠛⠛⢷⡀⠹⣿⠛⠛⠛⠛⠛⢻⣦⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⡾⠁⠀⠈⠳⣆⠀⢸⡷⠶⣶⠶⠶⠶⠾⢿⡀⢹⡶⠶⠶⠶⣶⠞⠙⣧⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⣇⣀⣀⣀⣀⡈⣻⡾⢷⣞⣁⣀⣀⣀⣀⣸⡇⠈⣷⠀⠀⠀⣿⠀⠀⠘⣧⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡏⠉⠉⠉⠉⠉⣹⢷⡾⣏⡉⠉⠉⠉⠉⢹⡇⠀⣿⣀⣀⣀⣿⠀⠀⠀⠘⣧⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⣧⠀⠀⠀⣠⠞⠁⢸⡇⠈⠻⣦⡀⠀⠀⣾⠃⣸⠏⠉⠉⠉⣿⠀⠀⠀⠀⠹⣆⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠘⢷⣤⠞⠁⠀⠀⢸⡇⠀⠀⠈⠻⣦⡾⠃⣠⡟⠀⢀⣴⠞⠛⠳⣦⡀⠀⠀⠹⣆⣠⡴⠟⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠷⣤⣀⣀⢸⡇⢀⣀⣤⠾⠋⢀⡼⠋⠀⠀⣾⠃⣾⠛⢷⡈⣷⠀⠀⠀⠹⣏⡀⣀⣠⡴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⡉⠉⠉⠛⠉⢉⣀⣤⠾⠋⠀⠀⠀⠀⠹⣦⡙⠛⢛⣰⠟⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁

⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀
⠀⠀⣀⡸⠿⣿⣧⣀⠀⠀⠀⢀⣀⣀⡸⠿⠿⠿⠿⢇⣀
⢠⣤⠛⢣⣤⡟⠛⠛⠀⠀⠀⠘⠛⠛⠃⠀⣠⣤⠀⠘⠛⣤⡄
⢸⣿⣶⣾⣿⡇⢠⣶⠄⠀⠀⢰⣶⠀⠀⠀⠉⠉⣶⡆⠀⣿⡇
⢸⣿⣿⡏⠉⠀⠀⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠿⢇⣀⠀⠸⠿⣀⣨⠿⠇⠀⠀⠀⠀⣀⣀⣿⡿⣿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⣀⣀⣀⡀
⠀⠀⣤⡼⠿⣧⡄⠀⣿⣿⠀⠀⠀⠀⢀⣤⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣧⣄
⢰⣶⠉⠁⠀⣿⣷⣶⣿⣿⣶⣶⣶⣶⣾⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣶⡆⠀⠀⠀⠀⠀⠈
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⠿⢿⣀⡀
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣼⠛⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉
⢸⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣿
⠀⠀⠿⠿⠿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡿⠟
⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⠛⠃
⠀⠀⠀⠀⠀⠉⢹⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⠿⣿⣿⣀⣸⣿⡿⣿⣿⡿⠿⠿⢿⢿⣿⣿⠿⣿⣿⠿⠿⢿⣿⣿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⠻⠿⠿⢿⣿⠀⢸⣿⡇⠀⠀⠀⠀⣿⡷⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣿⡇⠀⠀⠀⠀⢸⣿⣤⣼⣿⡇⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡏⠉⠁⠀⠀⠀⠀⠉⢱⣶⣿⡇⠀⠀⠀⣿⣷⣶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣿⣿⣿⡄⠀⠀⠀⠆⠀⠈⡀⠀⠀⢼⣷⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠀⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣠⠿⠃⠿⠿⠿⠿⣆⠀⠀⠰⡀⠀⠁⠀⠀⣾⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⠈⠀⢀⣼⣿⣿⡇⣿⣿⣿⣿⣿⣧⡀⠀⠱⡀⠠⠂⢀⣭⣭⣏⣉⣉⣙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠄⠁⠀⣰⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣦⡤⠊⠀⣴⣿⡉⠀⠙⢿⣿⣿⡿⠀⠀⠀⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⢀⣠⠚⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⡟⠠⠂⢸⡿⠃⢤⡄⠀⠀⠹⢿⣷⡄⠀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⢰⣏⣉⠀⣩⣭⣭⣭⣭⡁⣩⣭⣭⣭⣭⣭⡸⠀⠀⡈⠀⠀⠠⠄⠀⠀⢰⡀⡘⠉⠀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⡆⠀⣀⣀⣀⣀⣿⣿⡇⣿⣿⠿⠿⠿⠃⠛⣻⠿⠿⣿⣿⡇⠀⢀⠀⠀⠤⠀⠀⠀⠐⠀⢐⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠃⠉⠁⠀⠀⠀⠉⠉⠙⠓⠢⢄⡀⠀⠀⠀⠉⢬⠀⠘⢁⡿⠙⠁⢆⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠉⠳⡄⠀⠀⠀⠚⠀⠘⢇⣠⣤⣈⣦⣤⠀⠊⢀⠞⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢻⠵⠶⠷⣖⣛⠃⠚⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⢃⠀⣀⣉⡙⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠉⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣋⠄⠀⠀⠀⠀⠀⠑⡅⠐⠴⠀⠀⠉⠙⠝⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠛⢥⢀⣀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢀⣀⡀⠠⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠰⠊⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠆⠀⠀⠀⠙⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⣀⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠲⣤⣀⣤⠴⠂⠐⠦⣤⣠⣤⠖⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠹⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣧⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⣼⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣩⡄⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠛⢻⠀⠉⠀⠀⠉⠀⡟⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢚⣛⡳⣦⣴⠖⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠲⣦⡀
⠀⠀⠀⢀⣀⣤⠤⠤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⡀⠀⠀⠀⣴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠉⢻⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣿
⠀⠀⠖⣉⣥⣶⣶⣶⣶⣦⣌⡙⠢⠀⠀⠀⠀⢀⣚⣃⣈⡀⠀⠀⢀⣁⣈⣻⡂⠀⠀⠀⠀⠀⠀⠀⣠⡴⠏⣼⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⢠⡞⠉⢰⣿⣏⡽⢿⣿⠀⠙⣷⡄⠀⠀⢰⡟⢉⣭⡉⢻⣶⣶⡟⢉⣭⡉⢹⡆⠀⠀⠀⠀⣠⡾⠋⠀⠼⣣⠏⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⢠⠤⣄⣿⣿⣿
⠀⠈⠛⢦⣼⣿⣿⣶⣿⣿⣤⠾⠋⠀⠀⠀⠘⢷⣈⣛⣁⣼⠏⠉⢷⣈⣛⣁⣼⠇⠀⠀⠀⠘⠋⢰⡄⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠅⠃⢸⣀⡠⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠋

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠋⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡘⣧⢀⣤⡀⢀⣀
⠀⠀⢀⣾⣵⣚⣂⠀⠀⠀⠀⢠⠦⡀⠀⠀⠀⠀⢠⡏⠀⠀⠄⠴⡀⠈⣳⠀⠀⠀⠀⠀⠀⠀⠒⢶⣼⣧⣄⣀⡄⠀⠆⠀⠀⠀⠀⠀⠈⣹⣵⣿⣯⣟⣵⠟⠁
⠀⠀⣼⣿⣿⣭⣭⠀⣠⣶⣶⣿⣦⣈⡦⠀⠀⠀⠘⣧⡈⡀⣉⢅⣸⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣟⡛⠻⡟⢡⠃⠀⠀⠀⠘⠷⣾⢻⣿⣿⣿⣿⠁⡤⠤⠤
⢀⣤⣿⣿⣿⣀⣼⠻⣧⣀⣽⣿⣿⠁⠀⠀⠀⠀⠀⠘⣷⣤⣄⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀⡾⠋⣶⣿⡝⣦⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⡿⣧⡀⠖⠿⠒⠖
⠙⢄⠙⡟⠋⠉⠁⠀⠦⠽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⠉⢉⣿⢹⣧⠀⠀⠀⠀⠀⠀⠀⢿⣴⡟⠹⣿⡇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠛⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠞⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠛⠁

⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣬⡛⠋⠀⠤⠤⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡇⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⢠⣿⣿⣿⡀⠀⠦⠴⣤⠞⠁⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣁⣸⣿
⠀⠀⣼⣿⣇⠘⠷⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣣⣄⣪⣶⣍⣙⣹⡇⠀⠀⠀⠀⢸⡏⡏⠛⡏⠛⣯⢹⣿⡟⢇⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣷⣦
⠀⠀⢸⡟⣿⠀⢀⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⢹⢩⢽⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⢸⡿⠻⣿⠛⢻⣿⠛⣿
⠀⠀⢸⡇⣿⠀⢠⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣶⣿⡿⠃⠀⠀⠀⠀⠀⠘⠻⣏⣹⠛⠛⠛⠛⠛⠻⣏⣻⠛⠃⠀⠀⣸⣿⣾⣿⠀⢸⣿⣶⣿⡀
⠀⠀⠈⠁⠉⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⢀
⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢤⡄
⠀⠀⣴⣿⢿⡿⢛⠟⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢾⣿⣿⡷⢿⣦⣀⠀⠀⠀⠀⢠⣶⣶⣶⠶⠆⢸⢰⢿⠈⡇⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⣠⣤⡀
⠀⠀⢈⠃⠘⠀⠀⠀⢺⣯⡉⠁⠀⠀⠀⠀⠀⠐⠿⣿⣶⣤⣭⣭⣤⣶⣿⡿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣶⡜⡓⣒⣚⢳⠀⠀⠀⠀⢀⣤⣿⣿⣷⢠⢤⣿⣿⣷⣠
⠀⠀⠘⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⢹⠿⢿⡿⠿⣏⢁⡀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡿⢓⣂⣻⣟⣸⠀⠀⠀⠀⣾⢸⣿⣿⠙⢺⠚⢹⣿⣿⡼⡇
⠀⠀⠀⠀⠠⠖⢿⣿⣿⣿⣿⣯⡃⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⢩⠿⠉⡅⠀⠀⠀⢰⡇⠀⠙⣉⡀⢸⠀⣈⡙⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠷⠙⠷⠉⠁⠀⠀⠀⠀⠸⠿⠿⠷⠶⠾⠷⠶⠶⠿⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠬⠿⠴⠃⠀⠀⠀⠾⠷⠿⠵⠦⠦⠿⠧⠤⠤⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢺⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⣇⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠌⢀⠞⠉⢧⡀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⡎⠉⠑⢦⡀⠀⡿⠀⠘⠀⠈⠀⠀⣴⠛⠶⣄⡀⠀⠁⠒⠠⠄⣀⣀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠈⡇⠀⣠⠒⠉⠙⡆
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⠀⢨⣿⠿⠿⠶⢤⣤⣄⣀⣁⣀⡀⠀⠀⠀⠀⠈⢢⠀⣿⠈⠉⠉⠉⠉
⠀⠀⠀⠀⢴⠉⣉⡵⠂⠀⢸⡇⠀⠀⠀⠀⣼⠃⠀⠦⣀⣈⣥⣤⣀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠿⣦⣀⠀⠂⡏⠀⠀⠶⣉⠉⢳
⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠘⣇⠀⠀⠀⢸⠇⠀⠤⣤⠟⠁⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⣠⠤⢤⣠⠝⣧⣼⡇⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡿⠀⠀⢠⡏⠀⠀⢀⣀⣀⣤⡷⠀⠀⠀⠀⠀⠰⡧⢤⣀⣹⠆⢹⡿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢰⡇⠀⠀⠀⢿⠛⠉⠉⠈⠋⣠⠇⠀⠀⠀⣶⠀⠀⠳⢤⠴⠋⠀⢸⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣼⡇⠀⠀⠀⠈⠳⢤⣤⡤⠞⠃⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢸⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢫⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣤⠟⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣉⣧⠐⡄⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⢀⠀⢿⡀⠀⠀⠀⠀⢠⣾⣿⣷⣭⣿⠀⠀⠀⠀⠀⢀⡾⠃⣿⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣤⣅⣄⣀⣘⣧⠀⠀⠀⠀⣿⣘⣻⣿⣿⠏⠀⠀⠀⢀⡴⠋⠀⠀⢿⣐⠏⠉⡷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠉⠉⠉⢻⡄⠀⠀⠀⠀⠉⠉⠛⠿⠀⠀⠀⣰⣟⣡⣅⣀⣠⡟⠉⠛⢺⠀⢠⠛⣦
⠀⠀⠀⠀⠀⣀⣀⣀⡤⠖⢋⣩⠽⠚⠛⠻⣶⠒⠺⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠉⠉⣷⠀⠚⡟⠲⣇⡠⠏
⠀⠀⣠⠖⠋⠉⠉⠁⠀⣴⠋⠀⠀⠀⠀⠀⠛⠶⢶⡞⠻⢦⣄⣀⣀⣀⣠⡴⠛⢧⡀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠄⠐⠉
⠀⠀⣯⣠⠄⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⠉⠉⠉⠀⠀⣷⣼⣿⣄⣀⣀⣠⡴⣋⡴⠞⠁
⠀⠀⠀⢿⣠⡔⠀⣴⠤⠟⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠘⠷⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣴⠏⡔⠙⣇⠀⠀⠀⠀⢸⡿⠛⠛⠲⠶⠤⣤⣄⣀⡀⠀⠀⣸⠃
⠀⠀⠀⠀⣠⡟⣡⠊⢠⡴⢻⡆⠀⠀⣠⡼⢷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⠏
⠀⠀⢀⣾⢋⡴⠁⣼⣿⢿⡅⢻⣄⣸⣿⡀⠈⢻⣦⣄⡀⠀⠀⠀⠑⠲⢶⣯
⠀⣰⠟⣡⠛⡄⢠⡞⠃⠀⢻⣄⣉⡽⠋⠙⠞⠋⠀⠈⠙⠳⠶⣤⡴⠾⣿⣻⠶⢦⣄⡀
⣾⣋⣞⣁⣠⠿⠋⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠆⣸⡇
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠶⠾⣿⣾⡷⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⢤⠤⠤⠍⣁⣀⣀⣰⠋⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠶⠶⢦⣤⣯⣭⣉⣗⣚⣧
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⢈⣉⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠿⣿⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠠⣶⣄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢀⣄⠀⠀⠘⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠈⢻⣷⣄⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣶⣶⣾⣿⣏
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠙⢿⣦⡀⠀⠈⠻⠿⣷⣶⣶⣶⣤⡀⠀⣼⡿⠋⠁⠀⠈⠙⢿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠉⢻⣿⣾⣿⠁⠀⠀⠀⠀⠀⠈⣿⣇⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣷⣶⣶⣶⣤⣴⣿⠏⢹⣿⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⣿⣶⣶⣾⣿⠁⠀⠀⠀⠀⠈⠉⣿⡇⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣄⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⣿⣇⣠⣴⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⡏⠀⠙⢿⣦⡀⠀⠈⠻⣿⣆⠀⠀⠀⢀⣿⡿⠟⠉⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⣾⡿⣿⣦⡀⠀⠈⣿⣧⠀⢀⣾⡟⠁⠀⣠⣶⣿⠿⣷⣦⡄⠀⠈⢿⣧⠀⠀⠀⠀⠀⣿⣇⡀
⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⣼⡿⠁⠈⣿⡇⠀⠀⣿⡏⠀⢸⣿⠀⠀⣼⣿⠁⠀⠀⠀⠙⣿⡆⠀⠘⣿⡆⠀⠀⠀⠀⠉⠛⢿⣷⣦⣀
⠀⠀⠀⢀⣾⡟⠀⠀⠀⣼⡿⠁⠀⠀⣿⡇⠀⠀⣿⡇⠀⢹⣿⠀⠀⢿⣧⠀⠀⠀⠀⢀⣿⡇⠀⢀⣿⡇⠀⠀⣠⣤⣶⣶⣤⡈⠙⠻⣿⡆
⠀⠀⠀⣼⡿⠁⠀⠀⣸⣿⠁⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠘⣿⣆⠀⠈⠿⣷⣤⣤⣴⣿⠟⠀⠀⣼⣿⣀⣀⣼⡿⠋⠁⠉⠹⣿⣆⣀⣿⡇
⠀⠀⢸⣿⠁⠀⠀⣰⣿⠃⠀⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⠀⠘⢿⣦⡀⠀⠈⠉⠉⠉⠀⠀⣠⣾⡿⠛⠛⠛⣿⣇⠀⠀⠀⠀⣿⡟⠛⠛⠃
⠀⠀⠈⢿⣷⣤⣶⡿⠃⠀⠀⠀⠀⠀⢿⣷⣤⣾⡿⠁⠀⠀⠀⠀⠙⠿⣷⣶⣤⣤⣴⣶⡿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣤⣄⠤⠀⠁⠀⠛⠻⢛⠉⣫⣤⣾⣿⠟⠋⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣟⣛⣟⣿⣿⣟⠂⣄⡀⢀⣨⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣀⣀⠀⠀⠀⠀⠀⢴⣦⣤⣶⣾⣿⣿⣿⡿⣿⣿⠛⢻⣿⣿⣿⣿⣯⡿⣿⣿⢿⡿⠁
⠀⠀⠀⠀⢂⣼⣤⣶⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣾⣿⣿⣿⣿⣿⠟⡋⢉⢁⡎
⠀⠀⠀⠀⡜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣻⣿⣿⣿⣿⣿⡿⣿⣟⣿⣷⣿⢛⣩⠏
⠀⠀⠀⡀⠳⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠁⠾⡿⠛⠿⣿⠟⠃⠉
⠀⠀⠀⠁⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣋⣝⢀⡄⠀⢔⠡⣦⣁⢄⠔⡀
⠀⠀⠀⢠⠈⡄⠸⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⠀⠈⠉⣻⣵⣾⢷⠊⠉⠉⠘⠛⠻⠂
⠀⠀⠀⠀⠀⢿⡀⡻⠻⢾⣿⣿⣿⣿⣿⡟⠉⠉⠉⠓⠀⠀⠠⠺⣿⢿⠟
⠀⠀⠀⠀⠀⠀⢻⠅⠀⠸⣿⢿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⢠⡆⠓⠀⡿⠘⠉
⠀⠀⠀⠀⠀⠀⠈⠡⣀⢠⡇⠀⠀⠀⠀⠀⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠐⣿⢆⣿⣄
⠀⠀⠀⠀⠀⠀⣀⡠⡿⣹⣧⡸⠗⠆
⠀⠀⠀⠀⠀⢶⡯⢞⣿⣿⡧⢻⠇
⠀⠀⠐⢴⢃⡏⠛⣿⣿⣿⠗
⠀⠀⢀⣀⣼⡓⠀⠉⠉
⠀⠀⠈⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠀⠀⠀⠀⠀⢻⣟⠃
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⣤⣶⣶⣶⣶⣶⣶⣿⣧
⢸⡟⣷⣦⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⣀⣀⣤⣤⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣉⠉⠁
⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠁⠀⠀⠀⠀⠰⣿⠿⣿⣿⠟⠛⠛⠻⠛⠿⣿⣿⣶⣄⡀
⢸⣧⣼⡿⠛⠉⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠷
⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡴⠶⠀⠀⢀⣠⣴⠶⠛⠃
⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣿⣿⣥⣴⣶⣶⣶⣿⡿⠏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣶⣾⣧⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀
⠀⠛⠛⠛⠛⠛⠋⠉⠛⠀⠉⠛⠟⠋⠉⠀⠈⠃⠀⠀⠀⠛⠛⠿⣿⣿⣛⣛⣿⣿⡿⢿⣿⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠁⠀⠀⠉⠙⠛⠛⠶⠶
⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠛
⠀⠀⠀⣹⣧⣄⣤⣤⣤⣤⣶⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢶⣄⠀⠀⣠⣴⣶⡄
⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⡀
⠀⠉⠉⣿⣿⡿⠟⠻⠿⠿⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡿⠛⠶
⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣦⡀⠀⠀⠀⠸⣧⣿⠟⠁⠀⠀⠉
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠃⠀⠀⠀⠈⠁
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⡿⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⠀⠀⠀⠀⣘⣿⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⢀⣾⣿⣇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣶⣦⡀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢠⣾⣿⣿⣿⣷⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣟⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⡏⠃⠀⠀⡟⣿⣿⣿⣿⢹⠀⠀⣼⢿⣿⣿⣿⣿⡆⠀⠀⠀⠋⢸⣿⣿⣿⢻⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣄⠀⢸⣿⢹⣿⡇⠀⠀⠀⡇⢹⣿⣿⣿⠀⡆⠐⢛⣾⣿⣿⣿⣿⡟⠒⠀⠀⠀⠀⠀⣿⡏⠈⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣟⢻⣿⣿⡇⢸⣿⠘⣿⡇⠀⠀⠀⡇⠈⣿⣿⡇⠀⠃⠀⢸⣯⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢹⣿⣿⣿⠇⠘⣿⠰⣿⠀⠀⠀⢰⠁⠀⢹⡿⠗⠀⠈⠀⣼⢸⣿⣿⣿⣿⢸⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠛⠋⠋⠀⠙⠛⠛⠋⠀⠐⠛⠀⠛⠀⠀⠀⠘⠀⠀⠘⠃⠀⠀⠘⠂⠁⠘⠃⠀⠀⠘⠃⠁⠃⠀⠀⠀⠀⠛⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣶⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⢤⡯⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⠟⠛⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢿⣿⣿⣷⢟⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢋⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⣼⠛⢻⡏⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠈⠳⢤⣿⡹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣀⣀⣀⣠⣿⣿⣆⣀⣀⣀⣰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢐⣟⣿⣻⣛⣛⣿⣟⣿⣻⣿⣻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠟⢿⠟⠁⠀⠀⠀⠀⠀⠙⢿⠇⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠈⠉⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡤⠒⣛⣓⣦⣤⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⠞⠋⢉⡟⢉⣩⡽⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓⠒⣶⠟⠋⠉⢁⡴⠁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣰⠃⣀⣠⠴⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣉⣀⡀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡆⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⡤⠔⠒⠒⠒⠒⠁⠐⠚⠁⠀⠻⠹⠇⢤⠤⠤⠤⠖⠒⢂⡽⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⣀⠿⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠶⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡖⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⣀⠴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⢀⡤⠶⠒⠉⠑⢶⠃⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢬⣦⡄
⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠚⠁⠀⠀⠀⠳⣀⠇⣄
⠀⡴⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡴⡏⠀⠀⠀⠀⣠⠞⢁⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⠻⠘⣆⠀⢀⠀⠀⡀⠀⠀⢀
⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣠⠶⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢸⠄⠀⢹⡄⠀⠘⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⣠⠔⠒⠊⢹⠀⣠⠞⠁⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠀⠀⠀⢰⣘⣇⠀⠈⢧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡰⠃⠀⠀⢠⠇⠊⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠘⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡼⠃⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⢇⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾⠁⠀⠀⠀⠀⢀⡜⠁⡀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠀⠃⠀⠀⢸⣤⠁
⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⢧⡄⠀⠁⠀⠀⢸⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⣼⡤⠆⠀⠇⢰⠏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞
⠀⠘⢆⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠠⡿⠁⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠈⢷⠈
⠀⠀⠈⢣⡀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢦⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠋⠁⠉⠙⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠙⢦⠀⠻⣄⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⣤⣤⣤⣤⡤⠤⠤⠘⠇⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣦⡀⠀⡼⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡜⠀⠙⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠚⠻⠧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠛⠻⣍⠉⠑⠶⣴⠲⠶⠖⠒⢫⣉⠁⠉⠑⢦⡀⠀⠀⠉⢲⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠳⡄⠀⠈⢳⣤⠤⠤⠒⠉⠳⣄⠀⠀⠀⠀⠀⠀⡟⠀⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠙⠆⠀⠀⠙⢦⡀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠙⠚⠁⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠘⡆⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⡆⠀⠀⠀⠀⠙⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⠞⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⢀⣠⣤⣤⣤⣶⣶⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⣀⣴⣿⣿⣿⣿⣿⣿⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣾⡿⣿
⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢠⣶⣶⣶⣤⣀⡀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠁⠐⠖⠶⠿⠟⠿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣏⣠⣿⣤
⠻⣿⣿⣿⣻⡏⠀⠀⢰⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡋⠉⠉⠁⠀⠀⠁⠀⠠⠤⠶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⠟
⠀⠀⠹⣿⣿⣇⠀⠀⠀⠈⠉⠀⢿⣿⣿⣿⡿⠿⠿⠿⠛⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈⠙⠻⣷⣴⡟
⠀⠀⠀⠈⢙⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠀⢠⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⣴⣿⣿⡟⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣅
⣿⣿⣿⣿⣿⣿⣿⣿⣦⠄⠀⣤⣤⣤⣤⣶⣤⣤⣤⣤⣶⣦⣤⣤⣤⣴⣾⣿⣿⣆⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⣀⡀
⠙⢿⣿⣿⣏⣉⣿⡟⠉⠁⠐⠿⠷⠶⢾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠇⠀⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢉⣽⡿⠃⠀⠀⠀⠀⢨⣿⣍⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⠟⠛⠛⠿⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠧
⠀⠀⢀⣠⣤⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀
⣴⣿⣿⣿⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠘⠷⠶⣶⣦⣀⡀⠀⠀⣀⣤⣶⡾⠿⠋⠁
⢿⣿⣿⣿⣿⣿⡻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣶⣾⣿⣿⣯⣿⣤⣄⠀⠀⢀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣽⣿⣿⠇
⠀⠈⠙⠻⠿⢿⣿⣦⡀⠀⠀⠀⠀⠠⠤⠴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠋⠀⠀⠈⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⠀⠀⠀⠀⢀⣩⣷⡍⠉⠉⠁⠀⠀⠈⠉⣿⣿⠏⠀⠀⠀⠀⠀⠐⠛⠛⠛⠉⠉⠀⠀⠀⠉⠙⠛⠛⢻⣿⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢀⡀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠉⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⣀⠀⠄⠐⠀⠀⠀⡀⠀⠀⢀⠛⠁⢂⠀⠀⠀⠤⠜⠉⠀⠀⠀⠈⠀⠀⠂⠀⠉⠄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠲⠞
⡔⠈⠁⢱⠀⠀⢲⠂⠀⠀⠁⡄⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡀⠀⠀⣌⡀⠀⠀⣄⠀⠠
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠐⠄⣀⣀⣠⢴⠢⢄⠀⠀⠃⠀⠀⠀⠀⠀⢨⠁⠈⠉⠃⠀⠆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠘⠁⠐⠀⠀⠈⠛⠁⠠⠖⠀⢀⠀⠀⠠⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠃⠈⠀⠆⠀⠀⠀⠀⠀⠁⠀⠒⠢⡀⠀⢄⠀⢀⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀⠈⠁
⠀⠀⠀⡤⠯⠤⠤⠤⠄⣀⣀⣀⣀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡓⠀⠂⠀⠂⠀⠀⠆⠀⠀⠃⠐⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠘
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠰⠀⠀⢀⡀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⣀⣠⡴⠶⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⣠⣴⠛⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣀⣠⡴⠾⠟⠛⠛⠛⠛⠳⠶⢿⣋⠉⠛⠛⠛⠳⠶⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⣼⠋⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⢀⡾⠃⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢸⡇⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⣿⠁⠀⢠⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⡴⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⣀⣀
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠉⠉⠉⠉⠻⠶⣦⣀
⠀⠈⢷⡄⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠟⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣤⣶⣶⢾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠋⢹⡟⠋⠉⢻⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣆⠀⠀⠀⣾⠃⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣍⠻⢦⣴⡟⠀⠀⠀⢀⣼⠟⣷⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠹⢦⣀⣿⠀⠀⠀⠀⣾⣿⠀⠘⢷⣄⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⡏⠀⠀⠀⢰⡏⠹⣆⡀⠀⠙⠛⠛⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠐⢷⡀⠉⠳⠶⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⣀⣀⣀⣤⣀⣀⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⣀⣠⣤⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⢉⣤⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣶⣤⣤⣤⣶⣶⠶⠾⠛⠛⠋⠉⠀⠠⣿⡁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠿⠿⠿⠛⠛⠛⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⡿⠿⠋⠉⠉⠉⠉⠛⠈⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⣀⣀⣠⣔⣤⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⡍⠙⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢲⣀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠿⢿⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⠃⠀⠀⡀⠀⢼⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠷⠈⠙⠠⠤⠤⢀⡀⠀⠀⠁⠀⠀⣾⣿⣿⣿⣿⣿⣟⡉⠁⠀⠀⠀⠀⠀⠀⠀⠛⠒⢿⣿⣧⣤⠬⢭⣉⣺⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣾⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢀⡤⠀⠀⠠⣻⠦⠖⠲⣿⣿⣿⣦⠀⠈⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠶⢶⣿⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠘⠃⠀⠀⠶⠛⢲⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠂⠈⠁⠶⠶⠾⠦⠤⠀⢠⣤⡄⠤⣄⠀⣀⠀⠈⢹⣿⣿⣿⡇⢀⣤⣆⠀⣶⠀⣲⣴⣾⣿⠿⣷⣦⣝⣿⡏⠙⠷⢘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣇⠀⢹⡛⠷⠿⢳⣿⡭⣵⣾⣯⣶⡿⢿⠧⣦⡼⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣿⠁⠉⢳⡄⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡏⠀⠀⠀⢻⡄⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣇⠀⠀⠀⠈⣷⢀⣤⡤⣤⣿⠀⠀⠀⠀⠀⣼⠋⢁⣀⣤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣤⣤⣤⣿⡴⠛⣟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠈⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧⣀⣀⣀⣀⣀⣀⣀⣀⣠⣟⣀⣸⣃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠳⣤⣀⣠⡶⠛⠉⠛⢦⣄⣀⣤⠞⠋⠙⠳⣤⣀⣠⡴⠛⠉⠛⢶⣄⣀⣤⠞⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠤⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣤⣤⣤⡤⠶⠞⠋⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⡀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣟⠛⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣯⣙⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⣠⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⢶⣄⠀⣾⣷⡄⣠⡶⠃⠀⠀⠀⠀⠀⠀⠀⣰⠆⠿⡿⠃⠀⠀⠀⠀⠀⠿⡿⢋⡷⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⢛⣿⠋
⠀⠀⠀⣿⡇⠀⠀⠀⠙⢷⣭⣭⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢾⣿⣿⡷⣦⠀⠀⠀⠀⢰⣿⡏⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠛⠁
⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⡩⡿⠋⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣯⢻⣿
⠀⠀⠀⣿⣇⣶⣶⣆⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⠟⠘⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⢿⡿

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⣠⣶⣳⣶⣤⠀⠀⠀⠀⠀⣴⠶⠶⠾⠻⠷⣶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠸⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃
⠀⠘⠿⣿⠉⣿⢸⡇⠀⠀⠀⣸⡏⠀⠀⠀⠀⠸⣿⠇⢹⣆⠀⠀⠀⠀⠀⣴⣿⡏⣿⣦⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣶⣾⣿⡄⣠⣶
⠀⠀⠀⣿⣿⣿⢘⣇⠀⠀⢰⡿⣶⣿⣿⣿⣿⣿⣿⣿⣶⢿⡆⠀⠀⢀⣤⣬⣿⣧⣿⣤⣽⣿⣿⣄⠀⠀⠐⠒⢶⣶⡿⠛⠛⢉⣹⣿⡇⣿⡏
⠀⠀⠀⣿⣿⣿⠸⠿⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⢹⢹⣿⢿⢿⣿⠉⠉⢹⡇⠀⠀⠀⠀⠀⣿⠀⢠⣾⠟⣛⣿⣥⣿⡇
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢘⣿⡟⠉⠉⠉⠉⠉⠉⠉⢻⣿⠃⠀⠀⠀⣾⠈⣿⣾⣼⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣼⡇⣾⣿⣿⣿⣿
⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠈⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣐⠯
⠀⢰⣾⣿⡄⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣀⠠⠊⠃
⠀⠀⢻⣿⣿⡄⣾⣶⡶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡿⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢰⣶⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣶⡄⢀⠀⠀⠀⢸⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⠿⠿⠛⠛⠻⠿⠟⠸⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠿⠃⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢹⣿⣿⣿⣿⣿⠿⣿⠿⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡏⠚⠀⠀⠀⠙⣿⣿⣿⣿⣆⠀⠀⡻⠿⠿⠿⣿⣿⣿⣿⣦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⢿⣿⣿⢿⣿⡟⠁⠀⠀⠀⠀⠈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⢀⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠚⠂⠑⠛⠘⠀⠃⠀⠃⠂⠃⠋⠞⠁⠐⠃⠀⠋⠘⠁⠐⠑⠃⠀⠂⠘

⠀⠀⠀⠀⠀⠀⠈⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠢⡀
⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢄
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠢⡀
⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠐⠤⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠀⠒⠒⠤⡀
⠀⠁⠀⠀⢠⠒⠀⠀⠀⠀⠐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠒⠛⠒⠀⠀⠀⠐⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠉⠀⠁⠀⡀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢚⠀⠂⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠔⠁⠂⡄⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠨⠀⠀⠄⢠⡄⠀⠀⠀⠤⠈⠩⠐⠂⠅⣀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢠⠋⠒⠤⡀⠀⠀⡠⠒⠁⠀⢀⠊⠀⢰⠀⡇⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠀⢀⣸⣀⡁⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⣀⠠⠼⠤⠤⠀⠂⠐⠶⡤⠠⠄⠀⣀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠄⣀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢄⡀⠀⠀⠁⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜

⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠛⠛⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠳⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⣤⡤⣤⠤⠖⠂⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⡀⠀⠀⠀⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠙⣦⣀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠛⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣾⠛⠳⢦⣤⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⠀⠀⠀⣠⡾⠋⢀⣀⣀⠀⠀⠀⠀⠀⣉⠳⢦⡀⠀⠀⠀⠀⣼⠀⠀⠀⢠⡿⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣿⠀⢀⡾⠋⠀⣰⠟⠉⠉⠃⠀⠀⠀⡞⠉⠳⣄⠙⠳⣄⠀⢠⡇⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⢻⣴⠏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⠳⣾⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⣤⣴⡖⢶⠀⠀⠀⢀⣶⣶⠒⣆⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡿⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢠⡴⠖⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣷⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠘⣧
⢻⡷⠿⠿⣷⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⡀⠈⠳⣄⡀⠀⠉⣷⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⢻⡄⠠⣿⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢴⣞⣛⠁⠀⠀⠈⠛⢶⣤⣇⠀⠀⠀⠀⠀⣿⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠻⣆⠀⠀⠒⠒⠒⠊⠀⠸⣿⡀⠀⠀⠉⣹⠃⠀⠀⠀⠀⠀⠈⠙⠛⠶⢶⣤⣤⣾⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⢀⣠⣽⡆⠀⡤⠶⠒⢉⣀⣤⣿⣿⣦⡀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⢰⡟⣿⣿⠟⠀⠀⠠⠊⣡⡟⣿⡏⠀⠙⠻⢷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠛⣿⣤⣄
⠘⠛⠛⠁⠀⠀⠀⠀⣰⠏⣸⡿⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠷⣶⣶⣶⣶⣶⣤⣴⣶⣶⣶⠶⠾⠟⠛⠉⠁⠀⠀⠀⠸⣷⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣰⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈⣿⡀
⠀⠀⠀⠀⠀⢰⡟⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠸⣧
⠀⠀⠀⠀⠀⣸⡇⠚⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⣇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠻⣆⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣀⢀⣤⣴⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⡟⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⠻⢿⣿⣿⡿⠟⢵⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠉⠻⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⠈⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠶⠈⠉⠉⠉⠉⠉⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠞⠋⠀⠀⠀⠙⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠶⠚⠛⠁⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⡠⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠀⢀⣴⠛⠁⡈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠢⡄⠀⠀⠀⠘⠀⠀⣁⡤⠴⣆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣤⣤⡤⡴⠞⠋⠀⠂⣀⡄⠲⡀⠀⠀⠀⠠⠀⠂⠤⠀⠀⠀⠀⠀⢶⣄⡼⠃⠀⢄⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡀⢿⣿⣆⠃⠀⠀⢸⠀⣾⣟⡀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠒⠒⣨⠔⠂⠀⠈⠢⡜⢋⡵⠞⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⡑⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⡁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⢁⠈⣧⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣤⠖⠉⢀⣼⠇⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⢄⣁⠖⠉⠁⣠⡾⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠲⡈⠉⠉⠀⢀⣤⡴⣿⡿⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠈⠒⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢻⣷⣿⡿⠋⠁⣀⣴⠃⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠙⢿⣄⠉⣁⡴⠃⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⡀⠀⠁⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢶⠤⣀⣄⣀⣤⣤⣴⡖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠈⠁⠦⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⠴⠋⠀⠁⠀⠙⢦⡀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⠀⠀⠀⠀⢀⣀⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⡇⣀⣀⣀⣀⣠⠶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢻⡀⠀⠀⢠⠙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢸⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣀⠀⣸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠉⠉⠙⠉⠀⠀⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⡤⠊⠀⢀⡆⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⣉⣥⣴⠂⠀⣀⡤⠒⠊⠉⠉⣀⣀⣀⣤⣠⡌⠀⠀⠀⡜⠓⠢⣤⣀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣡⠖⠋⠁⢀⣾⡴⠊⠁⣠⡤⠖⠚⠉⠉⠁⠀⢀⠛⠀⠀⢀⠜⠓⠦⣤⣀⠈⠳⢦⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠂⠀⠀⠶⠏⠴⠋⠀⡀⣀⣤⠟⠁⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠁⠀⠀⠀⠀⠉⠳⢤⡀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⠒⠒⠶⠶⠒⠀⠀⢠⡞⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⠋⢀⡾⠁⠀⠀⠀⠀⠁⡌⠁⠐⢦⡶⠶⠾⠓⠖⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠙⠀⠀⠀⠀⠀⣸⠃⢠⡞⠀⠀⠀⠀⣤⣄⡴⢀⡴⢊⡠⠄⠈⠉⠉⠈⠁⠀⢄⡑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢰⠀⠀⠀⠀⢠⠋⢀⡟⠀⠀⠀⠀⠀⠀⠀⢠⢋⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡙⣄⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠈⠁⠀⠀⢀⡎⠀⡼⠀⠀⠀⠀⠀⠀⠀⠰⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠆⠀⠀⠀⠀⠀⠀⠀⢹⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡄⠀⢠⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⣇⠀⣸⠀⠀⣼⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⡿⠀⣻⠀⠀⢙⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠄⠀⠀⠶⠤⠀⠀⠀⠙⠛⠒⠒⢻⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠰⡄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣀⠃⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⢀⡛⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠅⠀⢼⡇⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠱⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰
⣤⡼⠤⠤⠄⠀⠀⠈⣀⣐⠒⠒⠒⠀⠁⠀⢾⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣐⠂⠠⠤⠤⠤⠄⠒⣨⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡾
⠷⢤⣤⣤⣤⣤⣤⣄⣉⣉⣉⣉⣉⢁⡀⠒⠚⣷⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡉⠻⣍⠉⠉⠉⠙⢧⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠳⣄⠀⠀⠀⠳⣄⠈⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⡈⠳⢦⣄⠀⠘⠷⡄⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣌⡉⠑⠶⠻⠷⣄⡀⠙⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠉⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶⠶⠛⠶⢦⣄⡀⠉⠉⠑⠒⠲⠶⠖⠒⠊⠉⠁⢀⣠⡴⠞⠉
⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣥⣤⣶⣶⣶⣶⣶⣤⣌⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⠟⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣀⣀⣤⣤⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣾⠿⠿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣶⠿⠿⠟⣛⠛⢉⣍⠉⣤⠄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣶⡿⠿⠿⠛⣛⣋⠉⠉⢡⣴⡄⠀⣾⡏⠻⣷⠸⣿⣰⡏⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⠾⠿⠟⠛⢛⣋⢈⣭⣥⣤⣄⠀⠀⠘⣿⡉⢻⣷⠀⣿⡟⣷⡀⢻⣇⠀⣀⡀⣿⡟⢿⣆⠀⠀⠀⠀⠀⢻⣧
⠀⢺⣿⠛⠛⠛⠉⢩⣥⠀⠀⠀⣿⣿⡀⠈⣿⡄⣿⡆⢸⣿⠀⠀⠀⢻⡇⠻⣷⡄⣿⡷⢿⣷⠸⣿⣄⣽⠇⠸⠧⠈⠛⠃⣀⣀⣤⣤⣼⣿⡀
⠀⠸⣿⡀⠀⠀⠀⠈⣿⡄⠀⠀⣿⠙⣿⡀⢿⣇⢹⣧⠀⣿⡇⠀⠀⢸⣷⣴⡿⠃⠿⠃⢀⣛⣁⣬⣥⣤⣴⣴⡶⠾⠟⠛⠛⠛⠋⠉⠉
⠀⠀⣿⡇⠀⠀⠀⠀⢿⡇⠀⠀⣿⠛⠻⣷⠸⡿⠸⠿⠛⢛⣁⣀⣤⣤⣤⣶⣶⣾⠿⠿⠟⠛⠛⠋⠉
⠀⠀⢹⣿⠀⠀⠀⠀⠸⠟⢛⣋⣉⣠⣤⣤⣶⣶⣶⠿⠿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠘⣿⣦⣶⣶⣶⠿⠿⠿⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠉⠉⠁
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠛⠛⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠿⠋⠀⠀⠀⣀⣀⣊⣄⣀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠿⠛⠉⠀⣀⣴⣿⣿⣿⣿⡿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢾⣿⠻⠦⠄⠀⠀⠀⠀⢠⣤⣤⣴⣶⣤
⢀⣠⣴⠿⣻⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣁⠹⣿⣿⠇
⠿⠛⠛⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⡤⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢀⣤⣄⠀⡈⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣵⣿⣿⣿⣥⣀
⠀⠀⠀⠀⣾⣿⣿⡆⠉⠀⠉⠃⠀⠀⠠⠂⠀⠀⡈⠀⠀⠀⠀⢰⠇⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣻⣿⣿⣿⠏
⠀⠠⠄⢠⣿⣿⣿⠁⠀⠀⠀⠀⠂⠀⠈⠈⠑⠔⠁⠀⠀⠀⣰⠟⠀⠀⠰⠞⠀⣦⠀⡄⣀⠀⠀⠄⠀⠀⠀⠀⠀⣠⣾⣯⣿⣿⣿⡿⠀⠀⢀
⣤⣄⣤⠟⣿⣿⢻⡇⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⡀⢠⣤⣿⡃⠀⠀⠀⠀⠈⡁⢨⠹⡗⠀⠰⣖⣠⣎⠰⠚⠛⣾⡷⣿⣿⣿⣿⠁⡀⢠⠌
⠀⠉⠀⢀⣹⣿⡀⠉⠛⠛⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠠⠿⡋⠀⠀⠀⢇⣠⣴⣿⠟⠛⠀⡀⠀⠘⣿⣿⣿⣿⣷⡴⣃⣒
⠀⠀⠀⠉⠛⠛⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⢦⡤⠴⠞⠛⠛⠛⠋⠉⠁⣿⠀⠼⣉⣷⠀⢠⣿⣿⣿⡯⣸⡛⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠳⣤⣄⡐⠂⠀⢃⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣿⡀⠐⠈⠁⠀⣼⣿⣿⡿⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠷⣰⣿⠿⢿⣿⣭⣯⣉⣉⠁⠀⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠹⠿⠿⢷⣤⣀⣠⣴⣿⣿⣿⣷⡾⠥
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠉⣻⣿⣿⣟⡿⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⣀⠀⠀⠀⠀⠉⠉









⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⠛⢳⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠶⠃⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡆⠀⠀⣇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠐⣦⡀⣳⢄⣀⠼⠉⣲⣤⣤⣴⣋⡷⠤⠤⠾⢽⣶⣴⠋⠒⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣀⣀⠀⠀⠀⢸⠉⠉⢰⡤⠒⠉⢠⡄⠀⠀⠀⡂⠀⠀⣤⠀⠀⠈⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠸⣁⠺⢧⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠠⠃⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠉⠉⢹⠒⠋⠀⠀⠀⠢⠐⠤⠀⠀⠤⠒⠀⠀⠒⠠⠤⠄⢂⡠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠸⠥⣤⠽⠶⠃⠀⠀⠀⠀⠀⣠⠞⠁⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠁⣦⣀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⡾⠋⣇⠀⠀⢀⣴⣧⣀⡤⠖⢉⣀⡀⢈⣉⣩⢭⣭⣍⡙⢅⣠⣀⣀⣠⣤⠞⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠛⠉⠁⠀⡾⢡⡞⣉⠔⠋⠉⠀⠀⣀⡬⠗⠛⠂⢹⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠉⠀⣀⣠⠴⠒⠉⠙⠢⣤⣀⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠃⠀⠀⠀⠀⠀⠀⢿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣶⠶⢶⡶⠶⠶⣶⠶⠶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⣾⠁⠀⠀⠘⣶⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇⠀⠀⠀⠀⠘⣇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣡⡼⠀⠀⠀⠀⠀⠀⠸⣄⣹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠂⠈⠉⠉⠉⠉⠉⠉⠉⠓⠒⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠀⠉⠓⣄⠀⠀⠀⣿⠀⠀⠀⡇⠉⠙⠲⡄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣆⣀⡠⠏⠀⠀⠀⣿⠀⠀⠀⣇⣀⡤⠔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠿⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠈⠳⣤⠀⡄⠀⡏⠠⠄⠰⠇⠀⠤⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢆⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠁⠈⠄⣀⣀⣀⣁⣀⣀⣀⣀⣒⣀⠀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠤⣄⠀⠀⠀⠊
⠀⠀⠐⠲⡄⠀⢀⠟⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠒⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠂⢀⡤
⠀⠈⢷⡄⢠⡆⡸⠀⡀⠀⢠⡀⠀⠀⢀⣀⠢⢀⠐⠀⢇⢈⡀⠀⠀⠀⣄⠀⢀⠄⠀⠀⡠⠒⡀⢠⠀⠀⠀⢇⢔⠜⠁
⠁⠀⠘⡘⠎⠂⠀⠨⠴⠀⠠⠱⠰⠄⡈⠀⠙⠀⠹⠇⠂⠀⠛⠒⠀⠀⠈⠀⠑⠈⠠⠊⠄⠀⠣⠀⠃⠀⢀⠀⠀⠒⠃⠀⢀
⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠐⠂⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢂⠁⢄
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⢃
⠀⠀⠀⠀⠠⠀⠀⢀⣀⣀⡀⠀⠀⠀⣈⢀⡀⠀⢀⡠⣄⣀⠠⢁⢀
⠀⠀⠀⠀⠁⠀⠀⠀⡇⠁⡉⠄⠠⠀⢸⠀⠀⠀⢨⠂⠀⡀⠢⠀⠠⠁
⠀⠀⠀⠀⠃⠆⠀⠀⡇⠟⢕⠀⠀⠀⢸⠀⠀⠀⢸⠠⠛⠓⠀⠀⠰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠆⠸⠠⠠⠄⠼⠰⠄⠄⠼⠰⠄⠤⠈⠀⡠⠪
⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡂
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⢀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡩
⠀⠀⠀⠀⠇⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠠⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⡄⠚⠚⠐⠐⠉⠉⠉⠀⣏⠓⠒⣐⣤⡤⣠⣀⠀⠀⢀⡇
⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠓⠀⠀⠀⠂⠀⠀⠧⢀⣋⡀⠲⢙⠽⣆⣀⡆
⠐⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠒⡠⣰⠀⠴⡈⠉⢏⣿⡟⢶⣤⡀
⠀⠀⠹⡀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⡀⠈⠂⠀⠁⠈⠀⠿⢓⣘⠂⠟⠲⠖⠙⠛⣦⡀⠄⠊⠡⠐⠀⢀
⠀⠀⠀⠈⠓⠤⢄⢀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠋⠲⣖⢐⣠⠔⢺⠞⠁⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠖⠲⠖⠲⠒⠲⠶⠦⠦⠖⠛⠒⠖⠓⠉⠀⠈⠁⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⣼⣏
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠘⠛
⠀⠀⡀⡀⠀⠀⠀⢀⠀⣤⠄⢀⠀⠀⡄⠠⠠⠀⠄⠀⠀⠈⠃⠘⠁⠀⠀⠀⠀⣠⠛⠇
⠀⠸⣀⠝⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⢇⣤⣀⠄⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠀⠀⠄⠀⠄
⠀⠀⠐⡔⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠰⣉⡗⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢬⠷⢇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢑⠒⡄⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠠⡄
⠀⠀⠄⠀⠀⠀⠀⠀⠂⠠⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠙⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⠄⠁

⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡘⣀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⢜⢢⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⢠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠀⠀⠀⠀⠀⡠⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠐⠀⠀⠀⠀⠓⠒⠚⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠐⠢⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠖⠀⡀⢠⢀⠅⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠀⠀⢀⣁⠐⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠈⠉⠐⢒⠢⢄⡀
⠀⠀⠀⠠⣖⠐⠆⠘⠀⠁⠀⠐⠀⠀⡔⠁⢹⠀⠸⡏⠍⠢⢀⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠉⣆⠈⢳
⠀⠀⠚⠄⠀⠀⠀⠀⠀⠰⡁⣀⠀⢸⡀⠀⡼⠀⠀⠈⠀⠑⠒⠠⠈⠧⢄⣀⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠘⠒⠂⠀⠀⠀⠀⢀⣀⠤⠒⠉⠘⠦⣔⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠉⠒⠢⢄⣀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰
⠀⠙⠙⠀⢀⡤⠒⠉⠁⠀⠀⠐⢪⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⠀⡀⠀⠉⠒⢤⡀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⢀⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢃⠀⠀⠀⠀⠀⡯⠄⠤⢀⣀⠈⡆⠀⠀⠀⢠⠃
⠣⠤⢠⠤⠬⠵⠶⠤⠶⠾⠭⠬⠭⠉⠁⠒⢶⠤⠤⠄⣀⡁⠀⠀⣇⡤⠈⣿⠀⠀⠀⢀⡧⠤⠤⣁⠀⠈⠉⠴⠒⠴⢇⡀
⠀⠀⢸⠀⠀⠀⠠⢆⣰⠀⠀⠀⠀⠷⡀⠀⠆⠀⠰⣆⡲⠘⡡⠷⡚⠆⠀⠀⢏⠁⠀⢸⠤⠔⠈⠀⢀⠎⠀⡱⠒⠀⠒⠁
⠀⠀⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠟⠀⡬⠤⠤⠤⠤⠤⠤⠥⢦⠼⡟⢦⠤⠘⣆⢠⡋⠀⠀⣀⠴⠋⠀⢸⠁
⠀⠀⢸⠀⠀⠀⢀⡤⢄⠀⠀⠀⠹⣤⣠⠁⠀⠀⠀⠐⡚⢶⡄⠚⢀⠃⠈⠢⠁⡇⢈⣪⢀⠜⢱⠦⠖⠂⠉
⠀⠀⢸⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⡈⢷⠀⠪⡀⠀⢀⡕⠈⠉⠀⢀⡱⠻⠠⠄⠀⢄
⠀⢀⣸⠦⠤⠄⠀⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⠀⠀⠀⠱⡞⢇⠀⢾⣶⡋⠀⠁⠀⢸⢋⠰⠳⠋⠉⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠑⠋⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎⠀⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠠⠋⣷⢐⠃⡀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢈⣭⣍⡛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠐⡄⠀⡀⠀⠀⡠⠀⠀⠠⢛⣽⡷⣿⢛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣹⣾⠁⠀⠀⣽⠀⠠⢲⣿⣿⣿⣏⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⣻⣮⡄⠀⠼⣧⣠⠃⠨⣿⣿⣿⣿⣏⠸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⡗⡟⢫⣅⠚⠋⠉⠀⢀⣈⠙⠳⢿⣿⣯⠄⠀⠰⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣤⡾⠐⠀⠀⠀⠀⢀⣽⣤⣦⣾⣿⣷⣶⠋⠁⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣏⠉⠉⠹⠗⠊⠿⣦⠄⠀⠘⠿⣇⢺⢿⣿⣿⣿⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣧⠀⠀⠀⠀⠀⣹⣃⣀⢀⣀⣤⡌⢧⡽⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⡴⠖⠁⠈⠀⠉⠹⢿⣉⠀⠀⠹⣇⣿⣿⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠘⣿⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⠆⠀⠀⢹⣾⡅⠀⠀⠀⢘⣠⣤⡠
⠀⠀⠀⠀⠀⠀⢀⣱⣾⣦⣤⠀⠀⠀⠀⠀⠈⣇⡀⣅⣀⡀⢀⣀⣀⡀⠀⠀⡐⠾⠷⠦⣄⣼⠟⠘⠒⠒⠒⠋⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣠⡀⡀⠀⢀⡀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠹⡉⣀⣀⠀⣀⠀⢀⠀⠀⠀⢀⡠⣶⣶⣷⣿⣛⣥⣭⣴⣶⣾⣿⣿⣿⣷⣿⣿⣿⡛⣋⠉
⠀⠀⠀⠀⠀⠐⢶⠶⠿⠾⠷⠷⢿⡿⠾⠟⠿⠿⠟⠛⠛⠛⠒⠂⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠈⠀⠈⠀⠉⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠁⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠒⠉⠁⠀⢀⠎⠀⠀⢀⡄⠀⠀⢀⣆⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠐⠋⠑⠒⠠⣀⠀⠀⠀⣌⢀⠀⢩⢭⣭⠑⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠁⠀⢰⡍⡀⠰⠓⣚⠲⠄⠒⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠞⣳⠃⠈⠚⠁⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧⣀⠐⠂⡂⠀⠀⠀⢤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⡀⠀⠀⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡆⠱⠀⠀⡼⡝⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢷⠀⢆⠀⠑⠃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⡆⠘⢓⣂⠚⣽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⣇⠀⠇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡖⠻⠀⠇⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡰⣤⣄⣐⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠆⣗⠾⣿⠋⠙⡀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⡄⢼⠁⣻⠱⢛⠀⠀⢡⠀⠀⠀⠀⠀⠀⠈⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢚⡀⣠⡕⡏⠑⢉⣑⠢⢤⡑⢄⠀⠀⠀⠀⠀⢀⣀⠽⠦⡀⠀⠀⠀⠀⠀⠀⡠⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠋⡰⠣⠜⠨⢻⠂⠀⠐⠀⠘⠣⡠⡀⠀⣠⣶⠉⠀⣤⠂⣈⣲⠀⠀⠀⠀⠀⠇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠂⡎⠀⠀⠀⠀⠀⠀⠟⡾⣾⣽⣿⡫⢶⡏⠁⠈⢇⠁⠀⠀⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠻⢿⠓⢌⡰⠀⠀⠀⠙⢌⠢⣀⠈⡓⠢⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠑⢌⠣⣇⠀⠀⠀⠀⠑⠬⡑⠇⠀⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣄⠀⠤⠤⡤⠱⣌⠢⡀⠀⠀⠀⠀⠈⢄⢠⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⡴⠚⠛⠀⠀⠒⠒⠒⢰⡶⠒⠒⠿⡆⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠀⠒⠒⠘⠇⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⢁⡾⠁⣀⣒⣲⠆⠀⡱⠯⠀⠀⠐⠀⠈⠀⣈⡀
⠀⠀⠀⠀⠀⡄⠒⢤⠀⠀⠀⣀⣀⡤⠴⠒⣲⠟⠉⠉⢡⣿⣥⠄⠀⠈⠉⠉⠙⠒⠒⠠⠎⠀⢨⠆
⠀⠀⠀⠀⠀⠣⡀⠀⠓⠒⠉⠁⠀⠀⣠⣞⣡⣤⠤⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠌⢢⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⡜⠀⠀⢈⠔⢀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢧⠀⠀⢀⠠⠀⢀⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡀⠀⠈⠄⠀⠁⠠⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢀⠔⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠢⣀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡠⠋⠈⠉⠓⠒⠒⠶⠶⠖⠒⠒⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⣴⣾⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⢈⣿⣿⡿⠛⠁⣠⡄⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⡾⠀⣸⣿⣿⣦⣤⣾⣿⡇⠀⢻⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠟⠁⣠⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠻⣿⡿⠀⠘⠛⢉⣠⣼⣿⣿⣿⣧⠀⠈⠁⣠⣾⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⡟⢀⣼⣿⣷⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⣀⣉⣉⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠟⠀⠉⠛⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣆⠀⠀⠉⠛⠛⠋⠁⠀⠈⠻⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣦⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⣾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠄⠀⠀⠀⠀⠀⣀⠀⠘⣻⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠰⣿⡧⠀⠀⠀⢁⡽⠯⡖⠀⠀⠀⠀⠐⢿⠗⢈⣿⡿⠀⠀⠀⠀⠀⠀⠀⣶⣦
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⣤⢶⡆⣶⣦⡾⠓⠏⠀⠀⢹⠀⠀⠀⢀⢰⣿⡞⠛⠉⠃⠀⠀⠀⠀⠀⣴⣿⠋⠁
⠀⠀⢀⡀⠳⣶⣶⡆⠀⠀⠀⠀⡀⠹⢿⣿⣶⣦⡄⠀⠀⠰⣧⣿⣿⣿⣇⠀⠀⠀⠀⠀⠘⠀⠀⢀⡀⠻⢿⣷⣦⣤⡀⠀⠀⢀⠘⠾⣿⣿⣶⣷⣶⠄
⠀⠀⢈⠀⠀⠐⠋⠑⡀⠀⠀⢰⠁⠠⠀⠟⠉⢻⣿⠀⠀⠀⣼⢿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠄⠐⡋⠙⣿⣇⠀⠀⡏⠀⠀⠘⠉⠙⣿⡇⠑⢄
⠀⠀⠙⠲⠄⠊⠀⠃⠐⠐⠀⠀⠒⠤⠐⠁⠘⠈⠛⠂⠀⠀⠻⠘⠋⠛⠸⠀⠀⠀⠤⠤⠀⠀⠀⠈⠐⠄⠊⠀⠃⠘⠛⠀⠀⠑⠢⠀⠋⠐⠃⠙⠓⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠈⠉⠉⠁
⠀⠀⠀⢀⣠⠏⠀⠀⠉⠁⠀⠀⢣⣀⡀⠀⠀⠘⠿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠴⠶⢠⠶⣿⡿⢂⣾⣿⢿⡇⠀⠀⠀⠀⠀⢀⣘⠿⠃
⠀⠀⠀⢹⠧⣤⣤⣤⣤⣤⣤⣵⡤⢿⡁⠶⠶⣤⡶⢿⣿⢀⠔⠀⠀⠀⠀⠒⣀⣿⣏⣠⣆⡀⢠⣦⣤⣠⣀⣀⣰⣤⡀⠀⠀⠀⢢⡀⣾⡿⢷⣶⣋⠁⣴⣀
⠀⠀⠀⣿⣶⣶⣿⣿⣿⣿⣿⣷⣶⣾⡇⠀⢠⣶⣾⣿⠿⠋⠠⡄⠀⠀⢀⠄⠙⠻⣿⣿⣿⡇⠀⠉⠀⠉⠉⠉⠈⠉⠀⠀⠀⢠⠄⠛⢻⣿⣿⣿⡀⢰⣿⣿⡆
⠀⠀⠀⢻⡿⠿⠿⠤⠤⠤⠼⠿⠿⣿⠃⠀⣼⡟⢀⠘⡁⠐⠀⡤⠀⠀⢠⠀⠂⢀⠇⡀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠂⣘⠀⡈⢿⡇⢸⣿⣿⡇
⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠉⠁⠀⠀⠉⠒⠋⠀⠀⠀⠀⠑⠒⠉⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠁⠀⠁⠈⠉⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠛⠓⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⣦⣠⣬⣿⣥⡀
⠀⠀⠀⠀⠀⣴⣶⣍⣿⣿⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣤⣤⠻⠟⠀⣸⣿⣇⢀⣀⠀⠀⠀⠀⢀⣉⠉⠉⠉⠉⠉⠁⠘⠇⠀⠀⠀⠀⡜⢻⣯⡿⠂
⠀⠀⠀⠉⣄⣿⣿⣿⢿⢿⣿⣀⣀⣀⣿⡇⠀⠀⠀⠉⣦⣾⣿⣛⣦⡄⠀⣿⣿⣿⣬⣭⠀⠀⠀⠀⢸⣿⣄⣀⣀⡀⢰⠶⠶⠶⠆⠀⣾⣿⣧⡼⠛⠑⠦
⠀⠀⠐⠊⠈⠹⡿⠿⣿⡏⠉⠉⠉⠙⠛⠏⠀⠀⢠⠊⠈⠉⡿⠿⢿⡇⠀⣿⣵⣿⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⡏⠁⢸⠀⠀⠀⡇⢸⣿⠀⡀⢃⡀⠀⠠⠃
⠀⠀⠰⣀⣀⡤⠁⠀⠸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⡠⠂⡤⠸⠿⠀⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡒⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⢢⡀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣉⠓⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣳⡄⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢔⣂⠂⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢉⡤⠊⠁⠀⠀⠐⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡯⠤⡀⡼⠀⠘⢄⠀⠀⣀⠀⠀⡘⢦⡈⢳⡀⠀⠀⠀⠀⠘⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠈⠄⠀⠁⠸⢄⣀⣀⠄⠀⠀⠀⠀⢈⠆⠀⠀⠹⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⠦⡤⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠐⠣⠈⡏⠂⣰⠁⣨⠄⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⢐⠿⠀⠀⠉⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠢⢇⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡀⢑⠀⠀⢀⡠⠃⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⡎⠈⠙⠓⠒⠲⠦⠤⠤⠒⠒⠚⡒⠤⠤⠥⠊⠁⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠁⠀⠀⠀⠻⠑⣗⠚⣎⣻⠒⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡤⢰⠁⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⢸⠁
⠀⠀⠀⠀⠀⠀⠘⢅⣈⢸⣉⡉⡑⠒⠢⠤⠤⢤⣄⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠀⠀⠀⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣍⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⢀⣾⣋⢿⠿⠅⠀⠠⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⢀⡎⠀⠀⠀⠲⠦⠤⣌⣀⣉⣢⣄⣀⡠⠼⠦⠤⠴⠦⠤⠤⠤⠤⠀⠉⢁⢄⠢⠎⢀⡤⠚
⠀⠀⠀⠀⠀⠀⣟⣃⣚⣦⠶⡟⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠒⠋⠁
⢆⠐⠐⡀⠢⣀⠀⠀⠀⠈⠉⠘⠒⠤⢄⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁
⠀⠀⠐⠥⠂⠈⠁⠂⠤⣀⢀⡀⠀⠀⠀⠀⠉⠣⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠐⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⡠⠔⠁⠀⠀⢀⣀⠀⠤⠤⠤⠤⠄⠀⠘⠂⠒⠒⠒⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⡆⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣠⣤⡶⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⢸⣿⠁⢀⣿⣿⠇⣀⣀⣀⡀⣀⣴⣿⣇⠀⠻⡟⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠞⣏⢰⡿⠋⠀⡀⢱⣤⣿⣄⡁⠈⠻⣿⣦⢺⣿⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣶⡾⠃⠀⢀⠀⢎⠘⢻⣿⠉⢫⠄⢀⠜⢿⣯⣿⠋⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠯⠀⢀⠠⣤⠤⣼⡆⢸⡇⢀⣶⢴⣇⣠⠄⢹⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⢉⣘⣀⣸⣷⣸⡇⣿⠁⣸⠟⠛⣆⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⠾⠚⢻⠿⣋⣿⠉⠋⢀⢻⣟⣿⣿⢧⣽⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣨⠞⠁⠀⡺⠂⠀⣿⡟⠚⣡⠿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠠⠀⢘⣉⡄⠀⠈⠉⠁⠀⠘⠙⣧⣿⡏⢠⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡀⠟⠀⠀⢠⠞⢷⣶⣄⠀⢿⣿⣡⣯⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢠⣿⡧⠀⠀⠀⠘⣿⣿⣿⡟⠀⣸⢯⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢾⣿⣿⠁⢪⣿⣷⣄⠀⢀⣰⣿⣿⡟⠆⠀⢏⣸⣿⣷⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⣿⢏⡟⠀⢾⠋⢿⣿⣽⠶⣦⣄⣤⣤⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣾⠏⣾⠁⠀⠀⠠⢀⣻⣿⡷⣄⣩⣻⣻⣿⠙⢴⣿⣿⣿⠟⣸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⢸⣿⢀⡟⠀⠀⠀⠉⠉⠑⣻⣿⣿⣻⠟⣿⣏⣼⣼⣿⡿⠍⠁⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⠸⠇⠀⢐⠀⢀⠀⠀⠁⠈⠿⠿⠴⠟⢿⣿⣿⠛⠀⠀⠠⣐⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⢻⣞⠀⢀⣬⣁⣙⡃⠤⡶⢠⡀⠀⠀⠘⠀⠁⠀⢀⡀⠴⣶⣺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⣸⠃⠀⡀⠀⠛⠛⠻⢿⣿⣿⣿⣦⡀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⠋⢩⠀⣠⠂⠀⠀⠀⠀⠀⠙⠿⣿⣿⣄⣰⣼⣿⣿⣿⣿⠟⠉⠀⠈⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣵⡄⠀⠀⠐⠀⢀⣣⠀⢈⠉⠉⠉⠉⠉⠹⡟⠁⢠⠀⠀⠄⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⣶⣆⣀⣈⣋⣀⣴⣶⠦⡤⠀⠀⠠⠀⢀⡏⠀⠀⣴⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠾⠿⠷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡤⠀⠀⠀⠀⠀⢀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠈⠛⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡆⠀⠀⠀⢠⣄⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡇⠀⠨⡄⠂⠀⠀⢸⣷⣀⠀⠀⠀⠙⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣶⣽⣛⣿⠇⠀⠁⢤⠀⠠⠀⢸⣿⣟⠁⠀⠀⠀⠠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣦⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⠉⠉⠀⠀⠀⠉⠉⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣠⣶⣶⣤⡀⠀⠀⣤⡀⠀⢠⣶⣶⣦⡀⠀⠀⠀⣧⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⣿⣬⣿⠃⠀⠀⣿⡇⠀⢸⣿⣀⣽⡿⠀⠀⠀⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠻⣷⡄⠀⠀⣿⡇⠀⢸⣿⠛⠛⠁⠀⠀⠀⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠸⠟⠀⠻⠇⠀⠀⠻⠃⠀⠘⠿⠀⠀⠀⠀⠀⠀⣿⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠀⠀⣟⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣞⡿⠞⠃⠀⢰⣯⣙⡏
⠀⠀⠀⠀⠀⠀⣄⠀⣿⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⢸⣧⣹⡇
⠀⠀⠀⠀⠀⢤⣿⢷⣿⢀⣾⣇⣠⡶⢀⡤⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⢠⣄⠀⢸⠷⣽⡇⣀
⠀⠀⠀⠀⠀⠈⢿⡄⠙⠚⠙⠛⢻⣷⣯⣤⣤⣤⣤⣿⣦⣸⣧⣤⣤⣿⡿⢿⠳⣾⠞⠿⣿⡿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⢢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢬⣄⣀⣀⡀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠊⠋⠁⠀⠀⠉⠀⠈⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠖⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠎⠀⠀⠰⡀⠀⠀⢠⠀⠀⠀⢦⢸⢠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠄⢂⣠⣜⢀⣨⣥⠐⠊⣵⣤⣌⣤⣃⡌⠠⠀⠙⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠦⡀⢆⠌⠒⠀⠙⠎⠉⠉⠀⠀⠉⠉⠱⠋⠀⠂⠭⡰⣁⠴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠠⠄⠀⠈⠂⠄⠈⠀⠠⠐⠁⠀⠠⠄⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢂⠀⠀⠀⠀⠀⠀⢀⠀⠡⣣⡀⢀⣜⠌⠀⡀⠀⠀⠀⠀⠀⠀⡐⠄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠈⠄⠀⠀⠀⠀⠀⢈⠶⠀⠀⠑⠛⠀⠀⠶⡉⠀⠀⠀⠀⠠⠠⠁⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠘⠄⠀⠀⡠⠆⠀⠀⠔⠃⠀⠀⠈⢦⣀⠀⡰⢄⠀⠀⠠⠁⠀⠄⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⡠⠇⣒⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠐⠌⠒⠼⢄⡀⠎
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⡀⢀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠂⠀⠀⠈⢄⠀⠀⠀⠀⠀⢀⠈⠐⡄
⠀⠀⠀⠀⠀⢀⠠⣋⠀⠖⠀⢰⠀⠀⠀⢀⠃⢠⠤⠒⠊⠁⠂⠑⣄⠀⠀⠈⠄⠀⠠⢈⣁⠀⠘⠓⠛⠦⠄
⠀⠀⠀⠀⠢⣕⠊⠀⠀⠠⠊⠉⢢⣀⢀⡍⠀⠈⠁⡘⠂⢀⠠⢺⠏⠀⢀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠈⠡⡄
⠀⠀⠀⠀⢀⡖⠀⠀⢰⠃⠈⢁⣠⠜⡸⠁⠀⠀⠈⠗⠊⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠁⠀⠐⡀
⠀⠀⠀⢀⠎⠀⠀⠐⠉⠀⠈⠉⠀⠁⣨⡠⠀⠲⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⢀⠂⠀⠀⠀⠀⢄⠡⡀⠀⠀⠈⠈⢁⠴⠁⠐⠁⢩⠀⠊⠉⠢⣀⡀⠀⠀⠀⠀⠀⠳⠤⠀⡂⢭⠀⠁⠀⠰⡀
⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠈⢁⠄⢂⡆⡀⣠⠗⠠⡀⢀⠔⣀⠑⠢⢀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠠
⠀⠀⠢⠀⣁⣀⣀⣠⣤⣀⣀⣂⣀⠀⠌⠐⠈⠀⠑⠂⠁⠀⠀⠈⠁⠉⠀⠉⠂⠠⠌⣀⣀⣀⣀⣤⣄⣀⣀⡀⠊⠔

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢉⠋⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣧⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⣿⣿⠟⠀⢀⣤⣤⠁⠀⠀⠠⠀⠤⣤⣶⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡖⢢⠁⠛⡩⠀⠉⠱⠀⠈⣯⡞⠀⢀⠖⠉⠢⡸⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠓⠖⠋⠀⠀⠀⣰⡏⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢠⡏⢩⠘⠣⠄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣾⡟⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⢀⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠫⠀⡇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⠊⠀⠀⢲⣿⡘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢇⠅⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⠦⣤⣶⣯⢨⠡⠤⠤⠤⠤⠤⠤⠤⠄⠀⠤⠀⠸⠆⠄⣀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠠⣄⠴⡿⢃⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢹⠁⠒⠦⣝⠢⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠄⠉⠛⠓⠉⢹⣿⠀⠀⠀⠀⡀⠀⡴⣏⠀⠀⠀⢨⠀⠀⠀⠀⠱⡈⢢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⡀⠀⠙⢻⣿⣻⣾⣿⣒⣒⡶⣴⢃⣞⣰⠚⢷⡖⢶⡞⠀⡰⠀⠀⠀⠹⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⢀⠀⠀⠀⠀⠊⠴⣿⠁⠉⠉⠀⠅⢸⠀⠀⠀⠀⠙⢆⠀⡀⠇⣀⡤⠤⠶⣤⠸⡆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠘⡤⠀⠀⡀⠈⠀⠀⠀⡐⠁⠄⢸⠀⠀⠀⢰⠆⠈⠙⠑⠒⠒⠤⠤⣿⢓⠉⠀⠀⠀⠀⠸⠀⡇⠀⢣
⠀⠀⢀⠤⠄⠀⠀⠊⠀⠀⡤⡤⠐⠀⡄⠀⠀⢰⠁⢠⠀⢸⠀⠀⠀⡸⡄⠀⠀⠀⠀⢀⣂⠴⠋⣆⠙⠄⠀⠀⠀⡈⢈⡇⠀⠈⢆
⠀⠀⠬⠿⠿⠷⠶⠾⣄⡸⡠⠠⠤⢤⠆⠠⢀⢃⡴⠀⠀⡌⢁⠒⡶⣄⣷⠀⢆⠀⣠⠞⠡⢾⠤⡅⠀⠀⡴⠦⡠⠃⡼⠵⠀⠀⠈⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠔⠀⠠⢀⡤⠉⠁⢢⡟⠠⠀⢢⠔⡾⡟⣩⠙⡈⢣⡈⠞⠁⠤⣤⡼⠞⡇⠀⠀⣇⡰⠃⣰⠡⠚⠀⠐⠲⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣤⢴⡂⠥⡀⢠⠹⠃⠀⠀⠀⠰⠳⣿⡧⡇⠥⠀⠳⢄⡀⠀⢇⠠⠤⡯⡤⠔⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠍⠶⠃⠀⠀⠀⠀⠀⠀⠲⡤⠷⡤⠃⠀⠀⠀⠉⠒⠠⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣍⠙⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠳⣄⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣕⠲⠼⣦⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡬⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠐⠒⠒⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣉⡥⠶⠒⠒⠢⣄⢠⡀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⣠⠾⣄⠀⠀⠀⠀⠀⣹⠀⠉⠀⡀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠸⣆⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⠤⠤⣄⠀⡇⠀⠈⢳⡀⠀⠀⢠⣏⣀⣀⣈⠻⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢣⠀⠀⢹⠈⣇⠀⠀⠀⢳⡴⠚⠉⠀⠀⠀⠈⢳⠀⠉⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢰⡀⠈⠃⠀⠀⠀⠈⠉⠒⢲⠏⠙⣄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡇⢀⠀⠙⢳⠒⢦⠀⠀⠀⠀⠀⢸⡀⠀⠘⣆⠀⢀⡴⠋⠀⠀⠀⢀⣀⣀⣀⣹⡄⠀⠀⠀⠀⣼⠀⣠
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠛⠳⡄⢹⠈⡇⢠⠇⠀⣠⡇⠀⠀⠀⠀⠀⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⣀⡽⠗⠀⠀⠀⠀⠀⠀⠀⠀⡿⠚⣽
⠀⠀⠀⠀⠀⠀⠀⡇⢰⣂⣀⠇⠚⠠⣇⣞⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡼⡄⠉⠙⠦⠀⣠⠏⠀⠛⢛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⣏⠙⠒⠒⠒⢺⡳⠤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⢤⣄⠀⠀⠀⠀⠀⠀⡼⢣⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠉⠐⠶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣲⡄⠈⣇⠀⠀⠀⣠⠞⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⡼⠀⠉⠛⠓⠒⠶⠦⠤⠤⠤⠤⠤⠴⠖⠺⡶⢤⣤⣀⣥⠴⠋⠑⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣯⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣀⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⠉⠐⠒⠂⠤⠤⠘⣷⠶⣶⠶⢶⠴⠶⣧⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⠂⠀⠀⠀⠀⠀⠀⠁⠙⠦⠴⠓⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣚⡐⣇⣀⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠰⣍⡉⡉⣉⠓⠒⠲⠶⠤⢤⣤⣄⣠⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠓⠚⠂⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⠓⠒⠂⠠⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⣠⣶⣿⣿⣧⣽⣿⢿⣦⣠⣄⡀⠀⠀⠀⠀⠀⠀⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢠⣶⢤⣤⣾⣿⣿⡯⠛⠁⠀⠀⠈⢉⣙⡛⣿⡌⠉⠤⠤⠤⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡔⠁⠀⣼⣿⣿⣿⡿⡏⠀⠀⠀⠢⣄⠀⠀⠀⠻⣿⢿⣇⣀⠤⠔⠂
⠀⠀⢀⠤⠖⠲⣶⣦⣶⣉⠀⠀⠈⠳⣼⣿⣿⣿⣿⢿⠛⠃⠀⠀⠙⠄⠈⠀⠀⠀⢀⡿⠛⠉
⠀⠀⠀⣴⣂⡤⠈⠙⠿⣿⡷⣤⡀⠀⢸⡙⠿⠟⠛⢫⠀⠀⠀⠀⠀⠀⠐⣅⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠙⢿⣷⣶⡀⠉⠻⣿⠿⣦⡌⡇⠀⠄⠀⠘⢆⠀⠀⢿⡦⠘⢀⣈⠐⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠠⣤⣀⡀⠉⠻⣿⣮⡀⠈⠛⢾⣿⣅⠀⠇⠀⠀⣈⣧⡄⠠⠴⠊⠵⠁⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠈⠻⢷⣧⢀⠀⠉⠻⣾⡤⠀⠉⠙⠳⢄⡐⠿⠟⠉⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠻⠷⣄⣀⠀⠀⠉⠑⠲⣤⣤⡀⠉⠒⠦⣴⠚⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠤⣂⡀⠀⠀⠹⣿⣾⣆⠤⠀⠻⣦⣀⠉⠓⠦⣄⠀⠀⠀⢀⣴⡷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⠤⠀⠈⠻⢿⣦⠐⠂⠘⢿⣦⡀⠀⠀⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣰⣃⠀⡀⢀⣀⣈⣀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡋⣁⠀⢠⣶⣤⣤⠘⠛⠻⣿⣿⣿⡞⠉⡸⣯⣿⣟⣀⡤⠰⠃
⠀⠰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠈⢯⡙⠔⠠⡍⠾⠿⢷⣀⣤⣻⡟⣿⣾⣿⢻⣿⣿⣟⣿⣾⣅⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣳⣶⣤⣇⡀⣦⠃⠸⠂⣷⡿⠋⣿⣷⣿⠛⢷⣴⣿⣸⣿⣿⡿⣿⡻⠒
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⢿⣿⠛⠿⠯⣑⢤⡦⣘⣗⣤⣸⣿⣿⣤⡖⢻⣶⣾⢛⡁⠀⠁⣀⢀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠊⠻⢄⡈⢹⣂⡂⠖⠛⢿⣷⣾⡉⣹⣿⣿⡏⣹⣿⣿⣿⣾⣏⠡⣤⣖⡣⠜⠟⠰⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠦⣱⠶⠠⠆⠈⠐⢾⡉⠭⠁⡀⠄⡀⠉⠹⣟⣿⣿⣿⡿⠁⣀⣿⣋⣽⡛⡛⠟⠋⠁⢤⡖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠙⢠⢶⡄⢄⠀⠀⠈⣧⠰⢈⠂⠀⡀⠉⢁⢻⣯⣿⣿⣿⡿⣻⣿⡿⣿⣟⣋⣤⡄⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠠⠆⡀⠀⠀⠂⢀⠀⡰⢺⣦⣴⣄⡃⠘⠳⣎⠈⠗⠉⠻⣤⣇⣹⡟⣿⡟⢶⠉⣷⢐⣑⣮⣏⠟⣃⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠆⢀⡄⠤⠊⠁⢠⠖⠉⠙⠻⢶⡛⣻⠾⠶⣷⣿⣯⣿⣧⣽⣿⣦⣴⠏⢈⠉⡤⢃⣼⠇⠏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠾⠂⠀⠋⠀⠀⡰⠃⠀⠘⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣹⢿⣾⠏⢀⣴⡛⡠⢷⣾⠁⠀⠖⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠐⢇⠀⠠⡄⠀⠀⠀⢽⣾⣿⣤⣿⡀⣀⣵⢋⠉⠏⠁⠀⣤⣀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣤⣿⣦⡄⠀⠘⣿⣿⣤⡾⠟⠋⠉⠫⡝⠑⠢⠤⡝⠍⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⢀⣤⣾⣿⣿⣯⣳⣴⠔⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠛⠙⠛⠙⠉⠻⠏⠻⠛⠷⠶⠤⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈