Microbiologist Emojis & Symbols

⠀⠀⠀⣠⣶⣤⡄⠀⠀⠀⢀⣶⣾⢺⡷⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⣾⣿⣷⡄⠀⠀⠀⠀⠰⠀⢰⠀⠀⠀⠀⢀⣴⣲⡶⣤
⠀⠀⠐⣇⠙⠃⡧⠀⠀⠀⠀⠴⣷⠚⣿⠅⠀⠀⠀⠰⣯⢯⡧⠀⠀⠀⠀⠀⠩⣍⣉⠀⠀⠀⠀⣴⠚⠛⠛⣿⠀⠀⠀⢈⣭⣭⣭⣭
⠀⠀⠀⠉⠋⠋⠀⠀⠀⠀⠀⠈⠹⠛⠛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣸⣿⡧⠀⠀⠀⢀⣴⣿⣷⣦⠀⠀⠀⢸⣿⣶⡿⠿⠶⠀⠀⠀⣠⣴⣶⡶⢦⠀⠀⠀⢻⣶⠶⣶⠶⠉⠀⠀⢀⡦⣵⠟⡂⠄
⠀⠀⠠⣾⣿⡿⠃⠀⠀⠀⠸⣇⠛⠛⣺⠁⠀⠀⠈⣽⣿⠷⠟⠃⠀⠀⠐⣇⣩⣁⣰⡞⠁⠀⠀⢸⣧⣻⣼⡀⠀⠀⠀⣰⣿⠽⠻⠿⠆
⠀⠀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⠟⠁⠀⠈⠱⠆⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⢀⣶⡿⢻⣆⠀⠀⠀⢀⡴⣴⠚⠃⠀⠀⠀⠀⣸⡇⠀⠀⠀⢾⣿⠷⠂⠀⠀⠀⢸⡏⠉⠀⠀⠀⢦⣴⣿⣄⡆⠀⠀⠶⢳⠾⣅
⠀⠀⠘⣏⠀⢀⡿⠀⠀⣀⡤⠛⠋⠀⠀⢠⣤⣤⣴⡿⠀⠀⢐⣲⣾⣿⣷⣾⠀⠀⢠⡟⠁⠀⠀⠀⠐⢲⣿⣿⡷⠂⠀⠀⠀⠈⢧⠁
⠀⠀⠀⠈⠩⠉⠀⠀⠀⠻⠿⠀⠀⠀⠀⠘⠻⠛⠋⠀⠀⠀⠀⠀⠙⠻⠏⠉⠁⠀⠈⠻⠦⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⠟⠿⠿⢿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠉⠛⢿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣍⡻⣿⡿⢸⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠀⠈⠙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣭⣴⣿⡿⢛⣛⣿⠿⠿⣿⣿⣿⣿⣶⣄⣀⠀⠀⢨⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣶⣿⣿⡏⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⣫⣥⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣆⠀⠀⢸⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣷⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠠⠀⠈⢿⣿⣿⣷⠀⠀⠙⣿⣿⣿⣿⡙⣿⣿⣿⠿⠿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠛⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⣻⣿⣿⣦⣀⣠⣿⣿⣿⡿⢃⢿⣯⣴⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠈⠀⠸⣿⣿⣿⣿⣿⣿⣧⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠈⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⣯⠀⠀⢀⣿⣿⠟⠉⡉⠙⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠿⢿⣿⡀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣾⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠂⠀⢿⣿⣷⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀⢀⣹⣿⣿⣷⣾⡀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣷⣤⣤⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣄⣀⣾⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠖⠀⠀⠰⠂⠄⠀⠀⠠⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⡀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⣶⣶⡄
⠀⠀⠀⠀⠀⠠⠆⠀⠀⣿⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠐
⠂⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣤⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠠⠔⠒⢒⡖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⡥⠀⠀⠻⠃⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠈⠁⠀⠀⠂
⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⢠⠂⠐⠄⠀⠀⠀⠀⠠
⠖⠀⠑⠀⠀⠀⢩⠂⠀⠀⡼⠃⠀⠀⠀⠀⠀⠁⠀⣀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⣿⡷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠙⠆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠀⠀⠀⢀⠀⠀⠺⠂
⠀⠀⠀⠀⠀⠀⠈⠅⢠⠄⠀⢀⠠⠀⢄⠀⠀⠀⠢⣀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠠⠀⢂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠂⡠⢀⡠⠀⠕⡠⠂⠁⠀⠀⢄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠈⡀⠀⠀⠀⠀⢀⠰⠀⠀⠨⠀⣀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⢀⠰⠀⠠⠀⠀⠀⠀⠀⠀⣿
⡤⢤⣀⠀⠀⠀⠢⠀⠀⢡⠀⢀⡀⠌⠁⠐⠁⠀⡘⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠀⠪⠀⠀⠀⠛⠀⠀⠀⠀⣀
⠀⠠⠈⢷⠀⢀⠜⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠇⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⣐⠋
⠀⠀⠀⢸⠀⠘⠀⠐⠀⠀⢀⡀⠐⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⢰⣶⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠓⣀⠀⣋
⣀⡠⠴⠉⠀⠀⠀⠀⠀⠀⠀⠂⡀⡰⠀⠀⠒⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠀⠀⠀⠀⢀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠈⣿⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠚⠀⡀⠀⠀⠀⠀⠢⢀⠀⠀⢿⣶⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠹⣷⡀⠉⠀⠀⠀⠀⢠⠄⠂⠀⠃⢀⡀⠀⡠⠊⠀⠀⠂
⠀⠀⠐⠀⠀⠱⡄⠀⠀⠀⠀⠀⠡⠀⠀⠙⢿⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⡉⠙⡆⠀⠀⠸⠇⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠈⡀⢸
⠀⠀⢠⠀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠁⡀⠀⠀⠄
⠀⣀⠶⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠁⠀⠀⢀⠀⢀⠀⠀⠐
⠀⠐⠀⠀⠇⣤⣉⠀⠀⠁⠀⠀⠠⠆⣸⠁⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡄⠀⠃⢂⠁
⠀⠀⠀⠀⢀⠠⠄⢀⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠴⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠶

⢐⡸⡀⠀⠀⠉⠁⠀⠀⠰⠐⢜⣛⢅⠆⢠⡎⠈⠐⠃⠀⠀⠀⠠⠔⢣⣑⠊⠐⡰⡀⠀⠀⠉⠁⠀⠀⢐⠀⢝⣛⢅⠇⢠⡎⠈⠐⠃⠀⠀⠀⠠⠔⢣⣑
⢀⠀⡉⢖⢀⠀⠀⠀⠀⣸⣦⠤⠔⠋⠀⠀⣴⠁⠀⠀⠀⡀⠀⠀⠀⠁⠀⡃⢈⡈⡈⢖⢀⠀⠀⠀⠀⢸⣤⠤⠔⠃⠀⠀⣴⠁⠀⠀⠀⡀⠀⠀⠀⠁
⢨⣾⣮⢈⠇⠀⠀⠀⡴⠋⠀⠀⣀⡀⣀⠀⠀⠀⠀⠔⡡⣀⢭⠲⠀⠀⡉⡍⢨⣾⣎⢈⠇⠄⠀⠀⡴⠋⠀⠀⢀⡀⣀⠀⠀⠀⠀⠔⡡⣀⠭⢲⠀⠀⡉
⢦⠬⡤⢎⠁⠂⠀⢸⠁⠀⠠⣊⠤⠒⠤⠵⠀⠀⠀⢆⡡⠛⠜⣡⠁⠀⠀⡜⢦⠬⡤⢎⠁⠂⠀⢸⠁⠀⠀⣊⠤⠒⠦⠵⠀⠀⠀⢦⠡⠋⠔⣸⠁
⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠊⠁⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠒⠊⠁
⠀⠀⠀⠢⣠⣠⠂⠀⠀⠀⠠⣰⢞⠺⠶⢤⣀⠄⠀⠀⠀⡠⣄⡀⠀⠀⡄⢄⠀⠀⠀⠢⣠⣠⠂⠀⠀⠀⠠⣰⢞⠚⠶⢤⣀⠄⠀⠀⠀⡠⣄⠀⠀⠀⡄
⠡⠀⠀⠵⣅⣹⢗⠀⠀⠀⠒⠹⣷⡥⠆⢀⣽⡂⠀⠀⠸⣀⣬⠇⠀⠀⠘⢆⠱⠀⠀⢵⣅⣹⢗⠀⠀⠀⠒⠹⣿⡵⠆⢀⣽⡒⠀⠀⠸⣀⣬⠇⠀⠀⠙
⠒⠀⠀⠀⠈⠈⠀⠀⢠⠀⠀⠈⠐⢹⢺⠿⡋⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠘⠈⠀⠀⢠⠀⠀⠈⠀⢹⢻⠿⡙⢿⡄⠀⠀⠀⠁
⠀⢀⠀⠀⠀⠀⠰⡀⢸⠀⡀⠀⢀⠀⠀⠀⠀⠈⢟⢄⠀⠀⢀⠖⠁⣱⠀⠀⠀⢀⠀⠀⠀⠀⠰⡀⠸⠀⡀⠀⢀⠀⠀⠀⠀⠈⢗⢄⠀⠀⢀⠔⠁⣱
⠠⢁⠃⢀⣀⣀⢣⢔⡚⠲⣤⢒⡁⠀⢎⠲⡀⠀⠀⠉⠀⢀⠂⠀⢀⠏⠀⠀⠰⢁⠃⢀⣀⣀⣣⢕⡚⠺⣤⢊⡁⠀⢎⠲⠀⠀⠀⠉⠀⢀⠂⠀⢀⠏
⡇⡎⠀⠀⠠⠄⢧⠻⠟⢀⡸⠤⠀⠀⠈⡆⢡⠀⠀⠀⢀⡎⠀⠸⠃⠀⠀⠀⡇⡎⠀⠀⠠⠄⢯⠺⠟⢀⡫⠡⠀⠀⠘⡆⢡⠀⠀⠀⢀⡎⠀⠸⠃
⠉⠀⠀⢠⠄⠊⡡⠓⠶⠚⡁⠢⢄⠀⠀⠳⠜⢀⡠⢂⢞⢧⡾⠁⠀⢀⠀⢀⠙⠀⠀⠠⠔⠊⡡⢓⠶⠚⡉⠢⢄⠀⠀⠣⠜⢀⡠⢐⢺⢧⡾⠁⠀⢀
⣠⠆⡀⠀⠀⠀⠁⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⡱⡡⠣⠁⡄⠠⢄⡨⠦⠮⣀⠆⡀⠀⠀⠀⠁⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡱⡡⠡⠃⡄⠠⢄⡨⠦
⡎⢯⠥⠀⠰⡉⣱⠀⠀⠀⡠⢴⣖⡒⡄⠀⢠⠊⡴⢡⠃⣾⠀⠤⠰⢡⢿⡍⡎⣯⠥⠀⠐⣉⣱⠀⠀⠀⡠⣴⣖⡲⡀⠀⢀⢊⠔⢡⠃⣼⠀⠤⠰⢱⣵
⠐⡰⡀⠀⠀⠉⠁⠀⠀⢰⠐⢝⣛⢄⠅⢠⡎⠈⠐⠃⠀⠀⠀⠠⠔⢣⣑⠊⠐⡸⡀⠀⠀⠉⠁⠀⠀⢐⠀⢜⣛⢅⠇⢠⡎⠈⠐⠃⠀⠀⠀⠠⠜⢣⣑
⢀⡀⡉⢖⢀⠀⠀⠀⠀⣸⣦⠤⠔⠃⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡃⢈⠈⡉⢖⢀⠀⠀⠀⠀⣸⣦⠤⠔⠃⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈
⢨⡾⣎⢈⠇⠄⠀⠀⡴⠋⠀⠀⣀⡀⣀⠀⠀⠀⠀⠔⠡⣀⢬⠲⠀⠀⡩⡅⢨⣾⣎⢈⠇⠄⠀⠀⡴⠋⠀⠀⣀⣀⣀⠀⠀⠀⠀⠔⡡⣀⢬⢲⠀⠀⡉
⢢⠬⡤⢎⠁⠂⠀⢸⠁⠀⠠⣊⠤⠐⠦⠵⠀⠀⠀⢆⡡⠋⠜⣰⠁⠀⠀⡜⢦⠬⡤⢎⠁⠂⠀⢸⠁⠀⠀⣊⠤⠐⠦⠵⠀⠀⠀⢆⠡⠛⠔⣠⠁
⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠊⠁⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠉⠒⠂⠁
⠀⠀⠀⠢⣠⣠⠂⠀⠀⠀⠠⣰⢾⠚⠶⢤⣀⠄⠀⠀⠀⡠⢄⠀⠀⠀⡄⢄⠀⠀⠀⠢⣠⡠⠂⠀⠀⠀⠠⣴⢞⠺⠶⢤⣀⠄⠀⠀⠀⡠⣄⡀⠀⠀⡤
⠣⠀⠀⠵⣅⣹⢓⠀⠀⠀⠒⠹⣶⡵⢆⢀⣼⡒⠀⠀⠸⣀⣤⠇⠀⠀⠘⢆⠱⠀⠀⢵⣅⣹⢗⠀⠀⠀⠒⠹⣿⡕⢆⣀⣽⡒⠀⠀⠸⣀⣤⠇⠀⠀⠘
⠒⠀⠀⠀⠘⠈⠀⠀⢠⠀⠀⠈⠐⢱⢻⠿⡉⢿⡄⠀⠀⠀⠁⠀⠀⠀⠀⠈⠒⠀⠀⠀⠘⠈⠀⠀⢠⠀⠀⠈⠀⢱⢻⠿⡙⢿⡄⠀⠀⠀⠁
⠀⢀⠀⠀⠀⠀⠰⡀⢸⠀⡀⠀⢀⠀⠀⠀⠀⠈⢞⢄⠀⠀⢀⠖⠁⣱⠀⠀⠀⢀⠀⠀⠀⠀⠐⡀⠸⠀⡀⠀⢀⠀⠀⠀⠀⠈⢾⢄⠀⠀⢀⠖⠁⣱
⠰⢁⠆⢀⣀⣀⢣⢕⣚⠺⣤⢂⡁⠀⢎⠲⡀⠀⠀⠉⠀⢀⠂⠀⢀⠏⠀⠀⠰⢁⠂⢀⣀⡀⢣⢕⡛⠺⣤⢚⡁⠀⢎⠢⡀⠀⠀⠉⠀⢀⠂⠀⢠⠏
⡇⡎⠀⠀⠀⠄⢮⠻⠟⢀⡹⠥⠀⠀⠈⡆⢡⠀⠀⠀⢀⡆⠀⠸⠃⠀⠀⠀⡇⡎⠀⠀⠀⠄⣮⠹⠟⢀⡻⠥⠀⠀⠘⡆⢡⠀⠀⠀⢀⡎⠀⠸⠃
⠙⠀⠀⢠⠔⠊⡠⢓⠶⠚⡅⠢⢄⠀⠀⠣⠜⢀⡠⢐⢚⢧⡾⠁⠀⢀⠀⢀⠙⠀⠀⠠⠔⠊⡠⢓⠶⠚⡉⠢⢄⠀⠀⠣⠔⢀⡠⢂⢾⢧⡾⠁⠀⢀
⣀⠆⡀⠀⠀⠀⠁⠈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠙⡱⠡⠣⠃⡄⠠⢄⣨⠆⠮⣀⠆⠀⠀⠀⠀⠁⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡱⠡⠣⠃⡄⠀⢄⣨⠆
⡎⢯⠥⠀⠠⢉⣱⠀⠀⠀⡠⣴⡒⡒⡄⠀⢠⠊⡰⢡⠂⣼⠀⠤⢰⢱⣷⡍⡆⣯⠅⠀⠀⠉⣱⠀⠀⠀⡠⣴⣒⡲⡀⠀⢠⠊⡔⢡⠃⣼⠀⠄⢰⢡⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⢶
⠀⠀⠀⠀⠀⠀⢀⡤⠊⠉⠀⠀⠀⠀⣦⣯⣥⣯⣥⣯⣥⣶⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣾⢀⣀⡘⠃⣹⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⡏⣧⡌⣧⡆⢽⡏⣿⠀⠀⠀⠀⠀⢿⣿⣿⠆⠀⠀⠀⠀⠙⢮⣙⣁⡴⣋⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤
⠀⠀⠀⢴⡧⣤⢤⡤⣤⠀⠀⠀⠀⠀⣇⣙⣃⣙⣃⣘⣃⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠁⠉⠈⠁⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣢⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠋⢽⣿⣯⠢⠄⠀⠀⠀⠀⠀⠄⢸⣿⣿⣿⠀⠀⠀⠀⠀⢀⣶⢿⣏⣿⣯⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠸⡀
⠀⠀⠀⠀⣴⠿⠿⢇⠀⠑⠀⠀⠀⠀⠀⠈⠌⢿⣿⠃⠀⠀⠀⠀⠀⠀⢿⣟⣾⣻⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠛⠁


⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡰⠊⠉⢀⣭⠑⢆⠀⠀⠀⠀⢀⠔⣉⣁⡄⠉⠲⡄⠀⠀⠀⠀⡴⢫⠉⠀⠉⠓⣄⠀⠀⠀⠀⠀⢳⣝⢫⡆⠀⠀⠀⠀⠀⠀⢀⣾⣏⣀⣤⡀
⠀⠀⢸⢡⣶⡄⠘⠛⠁⠈⡆⠀⠀⠀⡞⠰⣿⣿⣷⡖⠀⢹⠀⠀⠀⢸⠀⠈⢦⢢⡴⠉⠸⡄⠀⠀⠀⠀⡀⠬⠍⢳⠀⠀⠀⠀⠀⠀⠘⠛⠉⢉⡟
⠀⠀⠘⣆⠉⠁⢴⣿⡷⣰⠃⠀⠀⠀⠹⡀⠛⠙⣿⡿⢂⡞⠀⠀⠀⠘⣄⠀⢸⡋⠀⠀⡸⠀⠀⠀⠀⠀⢑⡬⢅⡀⠀⠀⠀⠀⠀⠀⠴⣦⣴⠟⠁
⠀⠀⠀⠈⠓⠦⠤⠵⠚⠁⠀⠀⠀⠀⠀⠉⠲⠤⠤⠖⠋⠀⠀⠀⠀⠀⠈⠓⠤⠭⠔⠊⠀⠀⠀⠀⠀⠀⠚⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠶⠿⠷⠄

⠀⡁⠈⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⠀⣀⠂⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠙⢿⢟⡉⠀⠀⠐⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠈⡀⠀⠀⠀⠀⠉⠉⣦⠈⢜⣠⢴⠾⠃⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠒⠀⠀⠀⠀⠐⣌⣩⣾⣿⠶⡗⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠘⠟⣻⡔⡈⠉⡀⢐⠅⣟⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠀⣸⡂⣿⣽⠒⠀⠀⢀⡄⠈⢉⡀⠀⠀⠀⠀⡀⠀⠀⠸⡿⢧⠀⠂⣪⠴⢶⣿⠋⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡂⢻⡇⠀⠀⠁⡤⠀⠀⠀⢤⠀⠁⠀⠀⠰⡤⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠃⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠈⠊⠀⢸
⠀⠀⠀⣶⡄⠀⠈⠁⠀⠄⠀⣴⣭⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡟⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⢀⣀⣀⣔⣦
⠀⠀⠀⣿⣷⠀⠀⠀⠀⢡⣴⠟⠋⠏⠀⠀⠀⠟⢀⡀⡀⠀⠀⠀⣎⠈⠛⠀⠀⠀⠀⠘⠳⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣿⣿⡿
⠀⠀⠀⣿⣿⠀⠀⠀⢠⣿⠉⠀⠀⠀⠀⠀⠐⠀⠹⠗⠁⠀⠀⠐⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡄⠀⠀⡆⠀⠀⠐⠀⣰⠁⠉⣹⣄
⠀⠀⠀⣿⣿⡄⠀⠀⠈⠳⠦⠞⠁⠀⠈⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣾⣿⣆⠀⠀⢉⣱⣄⡾⣱⠋
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⡏⠀⠀⠰⣛⡻⢰⡇⢀
⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠰⠆⢙⠋⠀⣀⣤⣈⣧⣤⣤⣄⣠⠇⠀⣾⣶⡣⠀⣼
⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⡆⢀⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠹⢄⣸⣛⡀⠛⠻⣿⡿⢿⠇⠀⠨⣿⣿⣧⣰⠃
⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⢣⡞⠀⠶⠔⢓⡀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠀⠀⡀⠈⠋⠃⠨⣤⣁⣤⣬⣷⣷⣾⣿⣿⣿⡿⢼⣦
⠀⠀⠀⠚⣿⣿⣿⣿⣷⣰⣄⠀⣠⣦⣾⡄⠀⣸⣿⡿⡇⢀⡀⣀⣿⠇⠀⠀⠀⢀⣰⣾⣶⡄⢀⡀⠀⡴⠀⠀⠀⠀⢻⡟⠛⠛⢿⠿⡿⠟⠛⢋⡡⠲⠋
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⣡⠍⠄⠀⣿⣿⠁⣁⠘⠛⢛⣿⣯⢀⠀⡔⡍⠁⠀⠙⠇⠀⣣⡖⠃⠀⠀⠀⠀⣾⡇⠀⠀⢴⣦⡄⠙⠒⠁⠠⠂⠀⣀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠟⢻⣆⣸⡟⣿⡇⠀⣸⣿⠿⠋⠉⠈⠈⢻⣿⠀⠀⠀⠀⡀⢸⣿⣷⣿⣶⣦⠀⠀⣿⣧⡰⣶⡼⠷⠀⠀⠀⠀⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠛⣭⣿⣿⣿⠻⠟⣾⣿⣯⣤⡿⠀⠉⠙⣿⠀⠀⠀⠀⠀⢸⣿⣀⠀⠀⣄⡇⠀⡀⣿⡽⡿⠧⠘⣶⣿⣿⣿⠿⣿⣧⡀⠀⠀⢰⠋⠀⣀⠊
⠀⠀⠀⠀⠀⠀⠀⣻⣿⡟⠉⠀⠀⠀⠙⣿⣿⡇⢠⡀⠀⡿⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠋⡇⠀⣿⠿⠀⠱⣤⡤⢋⣉⣛⣀⠀⠻⠙⠧⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⢦⡀⠀⠰⢐⣿⣿⡇⢸⣿⣾⠁⠀⠀⠀⠀⠀⠀⢸⠉⠐⠋⠁⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠷⠀⠋⠉⠓⠲⠶⢶⣶⠾⠟⠋⠀⠂⠀⠀⠀⠚⠛⠛⠛⠋⠉⠉⠉⠉

⡀⠀⠀⣣⡀⠌⠛⠄⣀⠀⠃⠀⠀⠇⣛⠜⠂⢀⣔⠶⠾⠇⠀⠳⠂⠀⠀⠛⠠⠀⠀⠉⠣⡆⠉⠳⡞⠁⠐⠀⠵⠠⠅⠀⠈⠁⣠⡥⠄
⠦⡉⠀⠀⢛⡄⠲⠈⠁⠀⠀⡂⠀⠀⠐⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠘⠲⠉⠀⠠⠀⠈⠁⠀⢠⠎⠙⢈⡇
⢍⡟⡏⠐⠄⠘⠳⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⠛⠀⡝⠊⠀⡀
⣀⡙⠢⠄⠄⠀⠀⠀⠀⠀⠀⣀⠀⡀⡀⣀⡀⣀⠀⣀⠀⣀⡀⢀⡀⢀⡀⢀⡀⣀⢀⢀⢀⡀⡀⢀⢀⡀⠀⠀⠀⠀⠀⠐⠃⣀⠠⣄
⠣⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠏⠞⠅⠟⠥⠘⠻⠽⠤⠟⠧⠌⠔⠆⠧⠠⠞⠣⠍⠺⠸⠹⠶⠗⠪⠈⠖⠀⠀⠀⠀⠀⠀⠸⢧⣦⠾⠉
⠀⢀⡠⠈⠀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡀⡀
⠐⢽⡤⠔⡨⠃⠀⠰⣄⣧⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢊⡵⠌⢃⡈⠁⠈
⢀⡄⠀⠊⡁⡀⠀⠴⠁⣀⠤⢧⣀⠀⡀⠑⠂⠀⠀⢀⣄⠀⠀⠀⢠⠀⢀⣔⠀⠀⠀⢀⡔⢒⠑⠄⣤⠀⡈⡡⠁⢀⡀⠑⢐⡄⣤⠙⠄
⣯⣤⣤⣼⣤⣼⣤⣤⣯⣤⣤⣮⣤⣤⣬⣤⣬⣴⣤⣼⣾⣤⣤⣤⣼⣥⣬⣤⣾⣭⣤⣶⣬⣤⣤⣴⣧⣴⣭⣤⣼⣿⣤⣤⣤⣥⣦⣮⣤
⣿⣿⣅⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠐⢀⠘⠀⠀⠀⢃⠰⠀⠀⣎⠁⠀⠀⠉⠑⠢⠄⠠⠄⠊⠈⠋⠑⢤⠀⠀⢀⡦⢓⠃⠛⢢⠀⠀⠀⠀⡌⠒⠂⠐⠤⢀⠀⠀⠠⢠⡀⣠⢃⠀⠄
⠀⠒⠂⠀⠀⠀⠀⠀⠂⠂⠀⠈⠑⠉⠐⠦⣀⠀⢀⣀⡤⠤⠤⠀⡠⠜⠀⠈⢴⡄⠀⠈⠀⣘⠂⠀⠀⣉⠀⠀⠀⠀⠀⠅⠀⠀⠀⡜⠃⠈⠰
⠀⠰⠐⡄⠀⠀⢀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⢀⠀⠀⡀⠀⠀⠀⠈⠒⠦⠴⠆⠋⠀⠀⠀⠈⡀⠀⠀⠀⢠⠲⠀⠀⠀⠀⠐⣞⠀⠀⠂
⠀⠀⠈⠀⠘⠀⠈⠁⠀⠀⠐⠀⠈⠡⠔⠠⠀⠀⠀⠀⠀⡈⠀⢈⠁⢑⠀⠀⠀⠀⠀⠀⣀⣠⠄⡀⠀⠀⠀⠁⠚⠉⠀⠂⠀⠀⠂⠀⠀⣹⡄
⠀⠀⠀⠠⢄⠀⠀⠁⠀⡀⠀⡂⢀⠄⠀⠁⠀⠂⢀⢀⡀⠉⠅⢀⠀⠀⡀⠀⠀⠖⠀⠐⡂⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠈⠀⡀⠁⠀⢠⠀⠀⠀⠄⠀⠒⠀⠀⠀⠐⠀⠀⠀⠀⠈⣀⠀⠐⠀⠁⠀⠀⠀⠈⠁⠞⠘⠀⠀⢳⣄⡈⠛⠲⢦⣄
⠀⠀⠄⠀⠀⠆⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⡀⣠⠠⠔⠒⠒⠄⠀⠀⠀⢀⠈⣿⠀⠀⠀⠹⡢⠀⠀⠀⠄⠀⢀⠄
⠀⢘⠀⠀⡈⠀⢀⠊⠉⠂⠀⠀⠀⢀⠰⠂⠀⡄⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⠀⣠⠾⠀⠸⡄⠀⡿⠀⠀⠀⠀⠀⠈⠀⠈⡀
⠀⠈⡦⠂⠀⠀⡄⠀⠀⠎⠀⠀⠀⠀⠀⠀⢂⠈⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⣀⠀⠤⠐⠉⠀⠬⠋⠀⠀⠀⠀⣰⠍⠀⠀⠀⠐⠀⠀⠨⡴
⠀⠀⡁⠀⠀⠀⡀⠀⠀⠁⠀⠀⡀⠀⠀⠄⠈⠁⠀⠀⠂⠀⠄⠀⠉⠑⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋⠉⡀⠘⠀⠀⡫⡈⠀⠀⠁
⠀⠀⠅⠀⠀⠀⠣⡀⠜⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣴⣶⣿⡿⢿⣷⣦⡄⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠘⠀⠤⠀⡀⠀⠀⣰⡿⡟⡉⠉⠩⠉⡉⢏⠁⣀⣛⡛⢷⣄⠌⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⠀⡴⠚⢢⠄⠈⠁⠀⢰⣿⠀⠃⣧⢠⡀⣌⣙⣌⣃⣸⣍⣁⣴⣿⣿⢿⠁⠹⣷⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⢀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠤⠀⠀⠱⠦⠴⠁⠈⠂⠀⢸⣿⣦⣼⡿⠻⣿⠋⣙⣯⡉⣽⠉⠤⠧⠐⢀⡄⠇⣆⣿⡀⠀⠀⡁⠈⠀⠌⠀⠈⠀⠈⠚
⠀⠀⡠⠤⠤⣀⠀⠀⠀⠀⠀⠐⠁⢀⡄⠀⠃⠀⠀⠈⢿⡔⣀⠧⢠⡌⠃⠻⠟⢿⡘⣇⠰⠾⢇⢀⢓⢀⣿⣿⠀⠀⠀⠐⠀⠈⠈⠃⡀⠊⣑⣂⠕
⠀⡜⠀⠀⠀⠈⡆⠀⢀⠀⡠⠀⠀⡄⠀⡀⢀⡀⠀⠀⠀⠹⠮⣤⣦⣿⡶⠷⡶⠿⡷⡿⡶⣶⣾⣿⣿⠿⠛⠁⢀⡀⡠⠤⠀⠀⠀⠒⠂
⠀⠙⣄⣀⣀⡰⠃⠀⠀⠀⠀⠐⠂⡤⠤⡀⢈⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⢍⣰⠡⣄⢀⠀⠀⠀⠀⢀
⠀⢀⣀⡀⠀⣀⠤⠠⠤⡀⠋⠀⠈⠧⡄⠞⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠐⣶⣄⡽⠞⠋⠉⠁⠀⠀⠀⠀⠂
⠀⢏⠀⡹⢰⠃⠀⠀⠀⠘⡄⠀⠁⢀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⡜⠁⢹⠀⠀⠀⠈⠣⠚⠉⠀⢀⣀⣀⡀⠀⣀⠐⠀⠁
⠀⠀⠁⠀⠘⣄⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠀⠀⣸⠀⠀⠀⠀⠈⠒⠁⠀⠀⣠⠒⣶⠶⣞⢹⣧⢎⡇⠙⠀⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠘⠀⣀⠀⠈⠑⠒⠊⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠉⠊⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠾⠃⠿⠉⠁⠀⠁⠀⠄⠀⠀⠀⢀⢀⡀
⠀⠀⢀⠀⡄⠒⢀⠆⠤⠤⠋⡄⠐⡀⠀⠀⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⡠⡪⡐⠀⠂⡕⡄
⠀⡐⠂⠔⠒⠒⠓⠐⠊⠈⠀⠂⠲⢅⡆⠀⠀⠀⣠⠤⠲⠐⣆⠀⠀⠀⠀⡠⠄⡀⠀⠀⠀⠀⠆⠀⠀⢐⠺⠀⠀⠀⠑⠐⠀⢐⠉⠀⠀⢀⠈⠀⠈
⠀⡐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠁⠀⡰⠊⢁⣀⣠⠴⠏⠀⠀⡠⠊⠀⠀⡉⠀⠀⠠⠀⠀⠀⠀⢘⣁⠀⠀⠀⡚⠀⠀⠈⠐⢄⠀⠈⢀⢄⠌
⠀⠑⡊⢂⢆⢀⠀⠤⠀⡐⣈⠄⠡⠌⠀⢸⣠⠞⠉⠀⠀⠀⡔⠊⠁⣀⡀⠴⠊⠀⠀⠀⠤⠀⠀⠀⠀⠀⡹⠐⠒⠘⠀⠀⠀⢀⠀⠀⠀⠐⠂⠁
⠀⠈⠁⠈⠠⠰⠈⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠐⣶⢠⢦⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣠⣤⣴⣿⠂⠀⠰⠿⠿⠷⠐
⠀⠀⠀⠀⠀⠀⠈⠉⠓⠚⠓⠊⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⠿⠿⠟⠁⠀⠀⠀⠀⠒⠦⠠
⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡄⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⢰⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⡿⠀⠀⠀⠻⣿⣭⣽⣿⡆⠀⠀⠳⢦⡝⠛⣋⡀⠀⠀⢼⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠠⠥⠾⠄⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠚⠛⠉⠁⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⣴⣾⣷⣦⡀⠀⠀⠀⢀⣿⣶⣴⠆⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⠷⠀⠀⠻⠿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠛⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⣠⣤⣾⣷⡄⠀⠀⠀⠀⣠⣤⣶⣶⣄⠀⠀⠀⠀⢰⣿⣿⣷⠀⠈⠂⢠⣶⣦⡜
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣠⡾⠻⠿⠋⠀⠀⠃⠹⣿⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁⠀⠀⠀⠀⠙⠛⠋⠉⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠄


⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠲⡋⠒⠠⠢⡔⠂
⣷⠀⠀⠐⢘⡶⠤
⠉⠙⡤⠱⡏⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣦⡀⣤⢰⡖⠂⢰⠀⢠⡆⠀⣦⢰⠖⠂⣶⡄⢠⡆⠒⣶⠆⢠⣦
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠱⣾⢸⡏⠁⢸⣀⡀⣧⣠⡟⢸⣋⡀⣿⠙⢾⢃⣼⣁⢀⡿⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠝⢦⡗⣴⢀⡔⠀⠀⠀⠀⠀⠈⠛⠁
⠠⠈⣈⠁⣼⡦⢀⠤
⠐⠀⢀⡄⠁⢠⣥⠄
⠅⠐⠀⢀⠐⠓⢦⣀⣀
⠀⠐⠀⠀⠙⣤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠
⠘⠀⡶⣰⡷⠟⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠃⠈⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠻⠗⢿⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠚⠈⢉⠷⠄⠀⠀⠀⠀⢤⣄⠀⣤⠓
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⡞⡟⠃⠀⠀⠀⠀⠀⠀⠀⠉⠁⣷
⠀⠀⠀⠀⠀⠀⠠⣀⣶⣶⡾⣴⡥⠂⠀⠀⠀⠀⠀⢠⠀⠻⠀⢰⠆⢠⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠖⠂⠉⢢⡄
⠐⠼⠀⠀⠀⠀⠀⠘⣹⣧⣴⣿⡓⠦⠀⠀⠀⢀⣤⢀⣱⠴⠟⠚⣿⠿⠂⠴⠃⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢈
⠀⠀⠀⠀⠀⠀⠈⠁⠈⢉⠿⠛⡅⠀⠀⠀⠠⣄⡽⠻⠀⢻⡀⠀⢉⠀⠀⠲⠛⢳⡞⠉⢀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⣄⣀⡴⠏⣉⠳⡀⢠⠀⠟⠀⠰⠛⠘⢛⡹⡆⢉⡁
⠀⠀⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢬⡥⠀⠤⠀⠈⠁⣀⠈⠁⠴⠆⠉⠛⠛⡏⠉⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⣴⢢
⣤⠰⠖⠶⢳⠄⣐⠃⠀⢀⡤⠀⠀⠀⠀⠀⢀⣤⢐⡤⠆⠂⠀⠈⠀⠄⢀⡰⢤⣀⣛⣇⠐⠀⠀⠀⠀⠀⠀⠰⢜⢓⡼⠀⠀⠀⠀⠈⠁
⠘⢇⠀⠀⠈⠡⣟⠰⡔⠃⠀⠀⠀⠀⠘⠛⠁⠹⡏⠀⣠⠀⠸⠂⣀⠰⣄⠙⢀⡙⣿⡉⠉⠺⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀
⠁⡄⠀⠛⠀⠂⠈⣀⣶⡴⠂⠀⠀⠀⠀⠀⠠⠚⠉⢳⣿⠀⡰⡆⠘⠀⡙⢀⣬⡟⠂⠙⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠿
⢀⡄⠐⠀⠘⠒⠢⠬⢀⣧⠀⢀⣀⠀⠀⠀⠀⠀⠐⠋⠀⢹⠛⢻⠖⠞⠻⡅⠀⠱⠦⠀⠀⠀⠀⡆⢀⢀⣀⢸⡂⠀⠀⠀⠀⠀⠶⢁⠤
⠋⠠⠀⢹⠀⢢⡐⠶⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠘⠀⠓⠀⠀⠀⠀⠀⠀⠀⢴⡞⠁⠀⠸⠄⠀⠁⢶⣆⠀⠀⠀⠀⣻⢤
⡔⠆⢠⡤⠔⢌⠀⠴⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⢀⠁⠠⠀⠀⠀⢠⠀⠀⢈⣴⠊⠀⠀⠁
⡙⠒⢪⠃⠐⡉⠱⠂⠈⠐⠆⠀⠀⠶⣀⡸⢦⡄⠺⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡁⠠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠐⠋⠃
⠃⠀⠈⠀⠀⢡⠀⠀⠀⠀⠀⢀⣀⣀⡼⠤⠠⠄⢋⡈⣋⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠄⠐⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⣿⣃⠰⢀⠉⣉⡟⠒⠂⠀⠀⠛⠁⠀⠀⠀⠀⠓⠀⠀⠋⠀⠀⠀⠉⠀⠀⢀⠄⢐⠃
⠀⠀⠆⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠚⠁⠙⢢⠬⠴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠱⣤⣄⠀⢀⡀⠀⠀⠛⣥⣴⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢉⡐⠊⠅⠜⢢⡈⠉⠁
⣠⣠⡆⢀⣯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠈⣦⠸
⠉⠘⠑⠉⡥⠆⠄⣠⠆⠀⠀⠀⠀⢴⣲⠄⠀⠀⠀⠰⢀⠀⢀⢘⡄⠘⠈⠋⠈⢻⡧⢄⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠘⡆
⠀⠠⡈⠀⢀⠀⠙⣋⣠⠀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠠⡀⠀⠀⢐⡀⠀⠈⠀⠈⢱⣖⠁⠀⠀⠀⠀⠀⢀⡱⠀⠀⠈⢙⣏⢁⡄
⠀⠀⠀⠀⠉⠁⠀⡡⠙⡆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠈⠃⠄⠀⣠⠄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⢴
⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣷⣿⣿⣷⣶⣿⣷⣷⣾⣶⣷⣾⣿⣾⣿⣿⣿⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣾
⣿⣿⣯⣁⣬⣉⣄⣤⣁⣽⣎⣠⣿⣫⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠂⢠⠀⠀⠀⠙⢆⡈⠃⠀⠀⠀⠀⣀⠄⠈
⠀⠀⠀⡀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠀⠀⠀⠀⠀⠀⡀⠉⡀⠀⠀⠀⠀⡀
⠀⠀⠈⠃⠀⠀⠀⠈⠁⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠂⠀⢄⠀⠀⠀⠀⠈⠠⣄⠀⢠⠀⠀⠀⠀⠘⢀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣤⣤⣤⣬⣄⠠⠀⠀⠀⠀⠀⡈⠀⠀⢼⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⢀⠀⠀⠀⠘⠀⠀⣿⣿⡆⠠⡈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠂⠀⠀⠀⠈⠉
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⢀⣀⠘⠁⠀⠀⠀⠀⠀⢠⡄⠀⠀⠋⠀⠈⠐⠤⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠀⠀⠒⠀⠀⠀⡀
⠈⠋⢠⡀⠀⠀⠀⠀⠀⠀⡖⠆⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠀⠀⠠⠀⢰⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⡀⠂⠀⠀⠀⠀⠀⠆⠀⡇⠀⢀⣀⡀⠀⠀⣿⡗⠀⠀⠀⠈⠂⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠁⠀⠀⣐⠉⢰⡀⠂⠀⣠⠀⠈⠙⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠒⠀⠘⠒⠊⣀⠈⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣶⠛⠉⠋⠲⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⢌⡉⣳⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠁⠀⠀⢸⡅⠀⠀⠀⠀⣷⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡤⠈⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡄⠀⢣⡀⠀⢀⡴⠋⢀⡀⠀⠀⠈⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡀⠀⠀⠈⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⢙⣣⡾⠁⠀⠀⠀⠀⠀⠀⠉⠀⠔⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠄⠀⠁⠀⣠⠘⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠀⠀⠀⡀⠀⠀⠀⢀⠤⠔⠂⢀⡠⠃⠀⠀⠀⡤
⠀⠀⠀⠀⠘⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠋⡀⠈⠀⠀⢤⣄⣀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⢠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠂
⠀⠀⠀⠄⠀⠀⢀⡀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠟⡟⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠤⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⡀⠀⠈⢡⣶⡆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠠⠊⠀⠃⠀⢀⡴⢄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠇⠠⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⡠
⠀⣥⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠫⠤⠤⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠴⠄⠀⠀⣀⡀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠈⢣⠀⠂⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠃⠀⠀⠀⠁⠀⠐⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⣿⣿⣿⣿⡿⣿⣏⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠈⠉⠈⢉⣉⢹⣾⠁⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⣿⣿⣿⣿⡿⣿⡏⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡶⢸⡟⢸⡟⠀⠀⠀⠀⣿⠉⠉⣿⠀⣀⡀⠀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠘⠇⠘⠇⠸⠇⠸⠇⡀⠀⠀⠀⠿⠿⠿⠿⠼⠿⠿⠦⠿⠿
⠀⠀⠀⠿⠿⠟⠛⠃⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⣼⡫⠋⠁⣼⣿⣿⠋⢉⣿⣿⠀⠀⠀⠀⠀⠀⢀⠀⢀⣹⠟⠛⠋⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⣿
⠀⠀⠀⠀⣸⣷⣶⣶⣶⣿⣿⣇⣀⣾⣿⡏⠀⠀⠀⠀⢠⣮⣴⣶⣶⣬⣳⠀⠀⠀⠀⠀⠀⢠⣶⣿⠋
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠋⠉
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢯⣈⣸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣟⣀⣀⣗⣀
⠀⠀⠀⠘⠀⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠠⣤⡄⠀⠀⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⣀⣀⣀⣀⣠⢀⢀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿
⠀⠀⠀⣐⢒⢒⢂⢂⢂⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⣶⣿⣶⡆
⠀⠀⣰⢂⢂⢆⢆⢆⠒⠀⠀⠀⠀⠀⠀⠀⠀⠎⠂⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠟⠛⠻⠃
⠀⠀⠉⠈⠈⠈⠈⠈⠉⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠴⠆⠶⠠⠆⠐⠠⠀⠂⠰⠂⠀⠄⠀⠠⠆⠀⠤⠒⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡀⢀⠀⡄⢰⠀⣄⣘⣄⣡⣤⡤⠧⢮⣴⡔⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢠⠀⡄⠘⡘⣀⣆⣸⣤⣥⠴⠾⠖⠛⠛⠋⠉⠉⠩⠀⠆⠀⠀⠀⣀⠙⢶⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⣘⣤⡷⠴⠷⠛⠛⢩⠉⠉⠁⠀⠤⠀⠆⠎⠑⠃⢠⠀⠉⠀⠐⠀⠀⠀⠀⠆⢀⠈⣿⣛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⣀⠐⢤⠟⠭⡄⠀⠘⠃⠀⢂⣙⡙⢉⡴⢆⠢⠀⠠⠀⠈⠴⠀⠀⠉⠠⠎⠀⠀⠘⠀⢀⡉⢠⣟⡒⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢀⣩⡏⠠⠆⠴⠀⠀⠀⠀⢰⠀⣷⣀⡄⠔⠰⠀⠈⠀⡀⡀⠀⠀⠀⠄⢠⠁⠤⠙⠀⢀⣠⠿⢕⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢘⣇⢀⣤⠑⢀⡠⠆⠀⠈⠛⠁⠁⠔⠒⠃⠤⠁⠒⠐⠀⣀⣀⣠⣤⢤⠴⡶⠞⢻⠫⠑⠁⢀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢠⠜⣮⣀⠃⠀⠐⠆⠀⢄⣈⣀⣠⣤⣤⡴⡶⠲⠚⢻⠙⠉⠉⠁⠁⠈⠀⠁⢈⣤⡶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠹⢛⠛⠟⡟⠛⠹⠉⠇⠡⠈⠀⠃⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣤⣀⣀⣀⣼⠏⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⣉⣉⣤⣤⣤⣤⣤⣤⣌⣉⠛⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣡⣴⡾⠟⠛⠉⠉⠉⠁⠈⠉⠉⠉⠛⠻⢷⣦⣌⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⡿⢛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⠀⠀⠈⠛⢿⣦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⣰⡿⠋⣴⡿⠋⠀⢸⣇⡾⠃⣀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠙⢿⣦⠙⢿⣆
⠀⠀⠀⠀⣰⡿⢁⣾⠟⠀⠀⢀⣾⣿⠿⠛⠉⢰⡞⠳⢶⡀⠀⣹⡷⣿⡀⠀⣰⠶⠛⣶⠀⠻⣷⡈⢿⣆
⠀⠀⠀⢰⣿⠁⣾⠏⠀⠀⠀⣾⣡⡟⠀⠀⠀⠘⢷⣤⣬⣟⠟⢫⠀⣈⠛⠿⣯⣤⣴⠟⠀⠀⠹⣷⠈⣿⡆
⠀⠀⠀⣿⠇⢸⡟⠀⠀⠀⣸⢏⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡤⠦⡍⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡇⠸⣿
⠀⠀⢠⣿⠀⣿⡇⠀⠀⠀⣿⣾⠃⠀⠀⠀⠀⢀⣴⠶⢾⣯⣬⡳⠥⢟⣠⣬⣻⠶⢶⣄⠀⠀⠀⢸⣿⠀⣿⡄
⠀⠀⢸⣿⠀⣿⡀⠀⠀⢸⣇⣿⠀⠀⠀⠀⠀⠸⣧⣤⡾⠃⠈⢹⡇⣿⠉⠀⠻⣦⣤⡿⠀⠀⠀⢈⣿⠀⣿⡇
⠀⠀⠘⣿⠀⣿⡇⠀⠀⠰⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠃
⠀⠀⠀⣿⡆⢹⣧⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢰⡿
⠀⠀⠀⠸⣿⡀⢿⣆⠀⠀⢹⣎⣷⠀⠀⠀⠀⣤⣰⣦⣀⣀⣀⣠⣤⠶⠶⢦⣄⣠⡦⠀⠀⠀⣰⡿⢀⣿⠇
⠀⠀⠀⠀⠹⣷⡈⢿⣦⠀⠈⣿⢻⡄⠀⠰⢶⡾⠋⢉⣍⠉⠉⠉⠀⠀⠀⠀⣹⠿⢶⠀⠀⣴⡿⢁⣾⠏
⠀⠀⠀⠀⠀⠹⣷⣄⠻⣷⣄⠻⠾⠇⠀⠘⢛⣷⣘⠁⠠⠀⠀⠀⠀⣀⣠⡴⠋⠀⠀⣠⣾⠟⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠻⣷⣤⡀⠀⠀⠙⠉⡿⢻⡟⠛⠚⠛⠛⠉⠁⠀⢀⣤⣾⠟⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡙⠻⢷⣦⣤⣀⣀⣀⡀⢀⣀⣀⣀⣤⣴⡾⠟⢋⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣉⡉⠛⠛⠛⠛⠛⠛⢋⣉⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠛⠛⠛⠉⠁

⠀⠀⢠⣶⣟⣻⢷⣄⠀⠀⠀⠀⠀⠀⢠⣠⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣆⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⢀⣤⡄⠀⠀⠀⠀⠀⢰⣾⡦⢨⣾⣷⡤
⠀⠀⣿⡱⢽⣷⡥⣿⡆⠀⠀⠀⠀⠈⠋⠀⠉⠫⣿⡤⠀⠀⠀⠀⠀⠀⢨⣿⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⣡⣄⡘⠻⠃⠀⠀⠀⠀⠀⠀⠉⣁⣀⠹⠚⠉
⠀⠀⠙⢿⣟⣹⣷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠐⢿⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⣯⣽⠗⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠆
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣤⡄⠄⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠶⢀⡀
⠀⠀⣼⣿⣿⡱⢎⣷⠀⠀⠀⠀⠀⠀⠌⣀⡀⢈⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣔⡿⣿⣮⠁⠀⠀⠀⠀⠀⣠⡾⢟⣛⡛⢷⣄
⠀⠀⢽⣿⡹⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣽⠟⠶⠀⠀⠀⠀⠀⠻⣧⣘⣛⣃⣴⡟
⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖⠐⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁


⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⣼⠛⣉⢿⡀⠀⠀⠀⠀⠀⠀⢠⠖⠊⢉⠲⡄⠀⠀⠀⠀⠀⠀⢀⣤⣾⣟⣿⣿⠀⠀⠀⠀⠀⢠⡏⠉⠙⠆⠴⠏⠀⠀⠀⠀⠀⠀⠀⣿⠁⠘⣦
⠀⠀⠀⣾⠠⠠⢸⠇⠀⠀⠀⠀⠀⠀⢿⠈⠐⡀⡒⡯⠀⠀⠀⠀⠀⢠⡿⣯⣟⣏⣿⠇⠀⠀⠀⠀⠀⠈⠳⢦⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⡿
⠀⠀⠀⠙⠷⠶⠟⠁⠀⠀⠀⠀⠀⠀⠉⠳⠥⠤⠚⠁⠀⠀⠀⠀⠀⠈⠻⣷⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠈⠻⡤



⠀⠀⠀⠀⣶⠐⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣏⣶⣦⡀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡸⠂
⠀⠀⢀⣴⡟⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣞⢙⣏⣸⣷⠀⠀⠀⠀⠀⢸⡀⠂
⠀⠀⣾⡏⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣻⣏⣹⣿⠇⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⡀
⠀⠀⠘⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣄⠀⠀⠀⢠⣴⣶⣦⡄⠀⢸⡏⠀⣠⣶⣶⣦⡀⠀⠀⠀⢸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣦⡀⠀⠸⣿⣿⣿⠃⠀⣸⣇⠀⠙⣿⣿⡿⠃⠀⠀⢀⣾⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣠⣽⣿⣿⣿⣿⠿⠿⣿⣿⣿⣄⡀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣄⣀⣼⣿⣿⣿⣿⣶⡟⠁⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⣿⣦⣤⣿⣿⠛⠉⠙⠻⣿⣿⣿⣿⣿⣿⡟⠉⠙⣿⣿⣿⣆⠀⣰⣿⣿⣿⡆
⠀⣴⣿⣿⣿⣷⣄⣀⡀⠀⠀⠀⠀⢨⣿⣿⣿⣄⣀⣀⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿⠿⠛⠿⠟⠃
⠀⠻⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⣏⠀⠀⣹⣿⣿⣿⣿⣿⣏⠀⢈⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣿⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣶⣴⣦⣶⣶⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣿⣿⣿⠃⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⣶⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠉⠛⠿⠿⣿⣿⣿⣿⠿⠿⢻⣯⡀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⢻⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠻⢿⣿⠟


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⣶⡀
⠀⠀⠀⠀⢀⠤⢲⠉⠽⠍⢵⡒⡄⠀⠀⠀⠐⣼⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢦⣵⣶⣾⡴⢛⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣿⡿⠁
⠀⠀⠀⡰⢩⠊⠁⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠘⣆⠘⢆⡀⠀⠀⠀⠀⠀⠀⣤⠭⣿⣿⡿⣟⣿⠭⠴⠀⠀⠀⠀⠀⠀⡠⣿⣟⡿⠁
⠀⠀⠐⠥⠃⠀⠀⢀⠔⡥⠃⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠉⣷⣀⠀⠀⠀⠀⠀⢒⡹⢿⣷⣿⢿⣑⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉
⠀⠀⠀⠀⠀⠀⣆⡱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠕⠆⠀⠀⠀⠀⠀⠙⠁⠃⣼⠘⠀⠉⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢠⠔⣉⠉⢉⡙⠢⡀⠀⠀⠀⠀⠀⠀⠀⣱⠟⣻⡇⠉⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢽⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢄⡀
⠀⠀⢠⠃⣠⠘⠂⠋⠀⢦⢹⠀⠀⠀⠀⠀⠀⣴⢋⡶⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢊⣥⡶⢂⠇
⠀⠀⠸⡄⢁⣠⠈⠉⢁⡀⢸⠀⠀⠀⠀⠀⠀⣇⠎⢰⢀⣘⠂⡄⠀⠀⠀⠀⠀⠀⢿⠀⢸⡇⠁⠀⠀⠀⠀⠀⡰⢊⡵⣞⠽⠒⠉⠁
⠀⠀⠀⠑⢤⡀⠰⠃⣚⡴⠋⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡗⠠⠀⠀⠀⠀⠀⠀⠈⠒⠿⠁⠀⠀⠀⠀⠀⠀⠳⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠹⠇⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆
⠀⠀⠀⠰⣤⠞⠋⠉⠙⠺⡅⠀⠀⠀⠀⢠⣿⣿⣧⠀⡠⠔⠊⠙⠆⠀⠀⠀⠀⣾⡿⠘⢛⡛⣦⡀⠀⠀⠀⠀⣠⡀⣰⣶⣾⣶⣦⠄⠒
⠀⠀⠀⢸⠃⠀⠀⢀⡀⠆⣿⠉⠀⠀⠀⠀⣿⣿⣿⠎⠀⢀⡀⢤⠇⠀⠀⠀⢰⣿⠋⣾⣿⣿⢈⣏⠀⠀⠀⠀⠉⢹⣿⣿⢛⣿⢿⡆⣀⡀
⠀⠀⠀⠘⣆⠀⠀⠀⠰⢢⡏⠀⠀⠀⠀⠀⠙⠛⡇⠀⠀⠀⡠⠋⠀⠀⠀⠀⠈⠛⣾⣮⣭⣽⣿⠿⠀⠀⠀⠀⣤⠐⠲⣿⣿⣿⠞⠁⠈⠁
⠀⠀⠀⠀⠋⠙⢲⠖⠺⠁⠉⠀⠀⠀⠀⠀⠀⠀⠣⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⢩⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⠦⢄⡀
⠀⠀⠀⠀⢀⣵⣦⣀⣤⣤⣠⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⢼⢋⣡⢤⠞⠀⡤⠷⣦⠀⠀⠀⠀⢠⢎⣀⠃⠨⡁⠠⠝⣆
⠀⠀⠀⢀⣿⣶⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣼⠾⣿⣿⣧⣤⠀⠀⠀⠀⠈⠉⠀⠀⣠⠎⠀⡠⠏⠀⠀⠀⠀⢸⠀⠠⡀⠖⠀⠀⠄⢸
⠀⠀⠀⠐⠻⣽⣿⡿⣿⣿⠿⡀⠀⠀⠀⠀⠀⠼⣷⠼⠿⣛⡿⠂⠀⠀⠀⠀⠀⠀⢴⠁⢀⠾⠁⠀⠀⠀⠀⠀⠘⣄⠃⠀⠃⣀⠰⢀⠏
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⠛⠏⠁⠀⠀⠀⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠴⠚⠁

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

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⢷⡀
⠀⠀⠀⠀⣠⡴⠋⠙⣷⠀⠀⠀⠀⠀⠀⠈⠳⣶⠀⢠⡶⠟⠀⠀⠀⠀⠀⠀⠀⢀⣤⢦⣄
⠀⠀⠀⣾⠋⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣀⣿⣀⣨⣇⡀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠉⢣⣀
⠀⠀⠀⠙⠳⠛⢷⣄⠈⠻⣦⣀⣤⠶⠞⠛⠉⠉⠉⠉⠉⠙⠛⠶⢦⣄⠀⢀⣴⠟⠁⢀⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠟⠁⢀⡴⠋⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⠆⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢀⡴⣯
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⡾⠋⢉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣧
⢰⠟⠛⣦⣀⣀⡟⠀⠀⢠⡟⠁⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⣦⠀⠀⠸⣇
⣿⠀⠀⠉⠉⢹⠇⠀⠀⠈⠳⠶⠛⠷⣤⡀⠈⠀⠀⠀⠀⠰⠶⠚⠛⠁⠀⢹⡆⠀⠀⢿⣀⣰⡟⠛⣷
⢿⣀⣰⠟⠓⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣤⣤⠶⢶⣄⣨⡗⠀⠀⢸⠉⠉⠀⠀⣿
⠀⠉⠁⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣿⠛⠻⣇⣀⡿
⠀⠀⠀⠀⠀⠈⣷⣠⡴⠟⠁⠀⠀⠀⠀⠀⣶⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠈⠉
⠀⢀⡴⢶⣤⡴⠛⠁⢀⣠⠶⠋⠀⠀⠀⠀⣿⠀⢸⣇⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⡟
⠀⠸⣧⠀⠀⢀⣤⠾⢿⣅⠀⠀⠀⠀⢰⡟⠉⠀⠀⠉⣷⠀⠀⠀⣤⡀⠈⠳⣴⠏
⠀⠀⠘⢧⡀⢸⡇⠀⠀⠙⠷⣄⡀⠀⠈⠛⠒⠓⠒⠚⠋⠀⠀⠀⠈⣻⣦⡀⠘⢷⣤⣤⣄
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠙⢳⣶⣤⣤⣀⣀⣀⣠⣤⡴⠶⠛⠉⠈⢻⣄⠀⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠃⠀⣿⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢺⣅⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⠀⢉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡿⢛⣯⢸⡇⣿⠇⣤⠁⡀⠀⠀⠈⠀⢀⡰⠨
⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⣄⡄⠀⠀⣤⢠⣤⠀⠀⠀⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠔⠎⡛⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠃⠇⠘⠂⠉⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⡀⢸⣳⡠⠔⠁⠀⠀⢀⡜⠓⢤⡀⢰⠋⠀⠀⠠⠀⠠⠀⡇⠀⣰⣦⡁
⠀⠀⠸⢇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⣼⣠⠏⠀⠀⠀⠐⠀⠈⠀⠀⠈⠀⠸⡆⢄⠀⢅⠀⣐⠼⠁⠀⡇⡜⠃⠀⠀⠀⣀
⠀⠀⠀⠘⡆⡜⠀⠀⠀⠀⣤⠀⡰⠂⠀⠀⠼⢾⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠒⠒⠟⣶⠛⠛⠉⠁⠀⠀⠘⡇⣇⠀⢀⡰⢶⣝
⠀⠀⠛⢄⢻⡇⠀⠀⣠⡞⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⢉⠦⠤⠤⠴⢠⡤⣧⡄⢼⠆⠀⠀⢰⠉⠂⠌⣐⠀⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢳⣈⠉⢉⣀⠜⠋
⠀⠀⠀⠈⢢⡇⠀⠀⢈⣧⡀⠀⣸⡄⠀⠀⡠⡄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠘⣆⠌⠐⠀⢁⠂⠈⠂⡼⠀⣠⣤⣄⠀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢳⠀⠀⠁⠘⠛⠙⡍⠀⠀⠀⠉⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⡽⠀⣀⡠⢤⠤⡬⠷⣤⠤⠤⠤⠖⠋⢀⢮⠞⠁⠀⡴⢹⢀⣤
⠀⠀⠀⠀⠀⢹⢠⠶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡿⠋⠀⢸⠁⠠⠀⠀⠀⠄⠈⢦⠀⠀⠀⠀⢸⠘⠒⣦⡸⠁⠓⠊⡼⠀⣾⣿⡆
⠀⠀⠀⠰⢆⢸⡏⠀⠀⡠⠤⠤⣀⣀⠴⠊⠳⡀⠀⠀⠀⠀⣀⡠⣤⠄⣀⣀⣀⠘⣦⠈⡀⠀⠀⠄⠀⣘⠇⠀⠀⠀⠈⠓⠊⠡⡷⠒⡆⣸⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠈⢾⡇⠀⠘⣅⠀⠀⡼⠉⠓⢦⣸⠁⠰⠀⠖⠉⠃⠀⠀⢀⠠⠐⠉⠉⠉⢷⠚⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⠖⢲
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⢦⡞⠁⠺⠇⢠⡏⢢⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⣤⣤⠀⢠⣤⡄⠀⣠⣤⡀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠠⠂⢼⠙⠲⢄⣠⠋⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣠⡇⠀⢸⢹⠀⠀⠀⡿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡴⠞⠁⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣾⠀⢸⣿⡇⠀⢸⢸⠀⠀⠀⣙⠇⢸
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀⠠⠍⡇⠀⠀⢀⢏⣌⢧⠀⠀⢸⣿⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣧⠄⢾
⠀⠀⠀⠀⠠⠀⡷⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠛⠀⡇⠀⠀⢸⢸⣸⢸⠀⠀⠘⠋⠀⠈⠛⠁⠀⠘⠛⠀⠀⠀⠿⠤⠼⠰⠆
⠀⠀⠀⠀⠀⠀⡧⡇⠀⠁⠀⠀⠀⠀⠀⠀⠐⡿⠷⢶⠂⠀⠀⢸⢰⡏⠙⢆⡇⠀⡆⠘⣆⣡⠞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⠀⠀⠀⠀⠀⠀⢀⠇⠂⠘⡆⠀⠀⠀⠶⠁⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⢠⣾⡁⢀⣀⣠⠔⠛⠓⢤⣠⠎⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⢹⡀⠀⠀⠀⢀⠀⢀⠀⠀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠈⠛⠿⡀⠀⠀⠀⠈⠋⠀⠤⠆⠀⠈⠁
⠀⠀⠀⠀⢀⡞⠤⣈⣆⠀⠀⠀⠀⠀⠀⣸⠃⠠⣕⠀⠀⢧⠀⠀⠀⠉⠀⣿⠀⠀⠀⢰⡆⠀⢇⣤⣤⣀⣀⣀⣀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡞⠈⠘⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⠀⣤⡄⠀⢶⠀⠀⠀⠣⠤⠄⣀⣀⣀⣀⣤⠀⠀⢠⣤
⠀⠀⠀⠀⢣⣀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⠐⣿⠀⠀⠀⢸⢿⠀⣿⡇⠀⣿⠀⠀⡤⡄⠀⠀⠀⣶⠀⠀⡷⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣠⡀⠙⠿⢿⡿⢿⡿⠿⠿⠿⠿⠛⠀⠀⠀⢿⠀⠀⠀⢸⢸⠀⡇⡇⠀⣿⠀⠀⡇⡇⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢖⣢⣀⣀⣹⣷⣤⡤⠼⢷⣦⠤⠤⠄⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⢸⢸⠀⡇⡇⠀⡿⡆⠀⡇⠇⠀⠀⢸⢸⠀⠀⣷⡇⠀⠐⢶
⠀⠀⠀⢀⣤⣤⠔⠊⠉⡿⠃⠀⠐⠖⠁⠀⠺⠊⠀⠀⠀⠀⠀⢤⡄⠤⠤⢾⠤⠶⠶⠾⠾⠶⠷⠷⠶⠻⠗⠦⠗⡖⠒⠒⣼⠛⠒⠲⠟⠓⠿⠆⠙
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠘⠈⠀⠈⠉⠁⠀⠁⠀⠀⠈

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⡤⡄⠀⠀⢠⣶⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⡀⢀⣤⠀⡀⠀⠀⠄⢐⠠⠄
⠀⠀⠀⡄⠈⢁⠀⠀⠀⢸⡎⠀⠀⣶⠀⠀⠀⢀⠀⢁⣄⠀⠀⢸⡏⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡙⠀⠁⠉⢁⡀⠀⠆⢿⠗⠂⠀⠀⠠⠋⡀⡓
⠀⠀⠀⠠⢈⡀⠐⠀⠀⠸⣿⣄⠀⠏⠀⠀⢠⡂⢰⣿⠏⠀⠀⠈⠛⠶⢶⣤⡀⠠⠺⣆⠀⠀⠀⣠⢠⠄⡠⡌⠁⠀⠀⢿⡿⠒⠀⠀⠀⢀⣃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠁⠉⠁⠐⠀⠀⠃
⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣼⡷⠶⠿⠟⡀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠄⠰⡐⠁⠀⢠⠴⠂⡄⠀⠀⠀⣘⣿⢿⠀⠀⠀⣠⠄⠀⠀⠠⠄⠀⠧⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢂⠠⢀⠀⡀⠀⢈⡀⠀⠀⡠⠠⠡⣼⡇⣒⠀⠀⣀⠻⠛⠁⠀⠀⠀⠀⠂⠸⠂⠀⠐⠀⠀⢠⣄⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⣞⠇⠀⠀⠂⠈⠁⠢⠊⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢁⠅⢠⠋⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⡄⢴⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⢀⣠⠤⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠆⣠⠀⠀⠠⣠⣄⠂
⠀⠀⠀⢼⢡⠁⠄⠛⡀⠀⠀⣱⣿⣦⢝⣤⡀⠀⢸⣧⣀⣾⠀⠀⡞⠹⡅⢦⠀⠀⠀⠀⢀⠀⠀⠤⠁⠀⣀⠐⣊⠀⠀⠔⣶⡀⠀⠀⣀⠀⠠
⠀⠀⠀⠖⢞⠀⣈⠅⠉⠀⠠⠹⣿⠁⠿⠾⠁⠀⢼⣷⠿⠋⠀⠀⠁⠭⠖⠂⠀⠀⠀⠀⠀⠀⠀⠉⡥⢁⡏⡔⠂⠀⠀⣖⣼⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠋⠚⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⢀⡀⠠⡀⠴⢀⠄⠀⠀⠀⡤⠒⠒⠢⢀⢀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⡄⢤⡄⠀⠀⠀⠀⠀⡀⡀⢁⠀⢀⠀⠀⠀⠐⠒⢀
⢀⠁⠂⠀⠀⠀⠐⣉⡀⠈⠧⢀⢄⣀⠀⠈⠀⠈⠀⠀⠀⠀⠀⢱⠀⠀⣐⢁⠀⠀⠁⡆⠀⠀⠠⠆⠀⠀⠈⠁⠄⠀⠀⠈⢰⠐⣆⡀⠁
⢀⠠⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠑⠢⠀⠒⠉⠈⠘⠁⠈⠁⠀⠀⠛⣠⠀⣀⢰⠂⠀⠀⢀⠀⠀⠀⠀⢠⡁⠐⠀⠈⠁⢠⡄
⠈⠀⠂⢐⡀⠰⠄⠀⠀⠀⠐⠠⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠈⠉⠁⡀⠀⠐⠀⠐⠦⠠⡠⠈⡀⠀⢀⡀⠀⠈⡇
⠐⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠈⠀⠁⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⡀⠀⡤⠓⠁⠒⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠜⠋
⠀⢀⠂⠈⠀⠀⠀⠀⠁⢀⡀⡀⠄⠀⠀⠀⠀⠈⠡⠘⠀⠈⠀⠀⠀⠁⠀⠀⠀⠐⢢⠤⠌⠁⢶⡀⠀⠶⡤⡀⠀⠀⠀⠂⠈⠀⠁
⠀⡈⠀⠁⠀⠀⠉⠀⠀⠀⠂⠂⠀⠀⠀⠂⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⣀⠄⠠⢄⠀⠀⠀⠀⠉⠆⠀⠀⠉⢳⡀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠔⠒⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠁⠉⠉⠁⠀⠀⠀⠀⡆⠀⢀⢔⠂⢩⠄⠀⣲⠀⠁⠀⠀⠀⠘
⠀⢢⠀⠁⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠀⠀⠀⠀⠀⣀⢀⠤⠐⠀⢠⠗⠁⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⡀⡠
⠀⠈⠀⠀⠀⡄⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠰⠀⡄⠀⠑⠂⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠊⠛⠀⠐⠀⠀⠓⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠁⡀⠄⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣀⣀⣄⣠⣤⣤⣄⣤⣤⣶⣶⣿⢿⣶⣦⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠄⠀⠁⠠⠀⢀⠀⠀⣠⡾⡿⠉⠉⠨⢉⡙⣍⢠⣘⣛⠻⣦⠕⢻⣿⣿⡆⠀⠀⠀⠉⠁
⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⢠⠒⠑⠀⠀⠈⠀⠀⣿⡁⠃⣧⢘⣀⣌⣡⣈⣀⣯⣬⣤⡿⢿⠿⡇⠈⣷⠀⠀⠀⠠⠠⠀⠀⠀⡀⠀⡁
⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠈⠐⠒⠃⠀⠉⠀⠘⣿⠷⢾⡟⠛⢿⢩⣭⣭⡉⣇⠰⠾⡄⠂⣄⠁⢶⣿⡅⠀⠀⠁⠀⠀⠀⠀⢂⡀⠈
⠀⡤⠒⠒⢤⠀⠀⠀⠀⠀⠊⠀⠤⠀⠈⠀⠀⠀⠻⣆⢠⠓⠰⣄⢀⡻⠛⣷⢹⡄⠛⠛⡀⣌⣀⣼⡿⠀⠀⠀⠀⠀⠀⡀⢂⠐⠠⠀⠄
⢸⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠰⠀⠠⠀⡄⠀⠀⠀⠈⠓⠶⠷⠿⠛⠛⠛⠛⠛⠻⠿⠿⠿⠟⠛⠋⠀⠀⣀⠐⢠⠀⠀⠈
⠈⠳⠤⠠⠜⠀⠀⠀⠀⣀⠉⣐⠒⢲⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣑⠂⡦⢨⠮⠤
⢀⠀⢀⠀⡠⠂⠉⠒⢄⠀⢀⠄⠳⠊⡀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀⠀⠀⠀⣄⢁⡀⠀⠁⠀⠀⠀⠀⠀⠁
⠘⠤⠜⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⢀⠖⠈⢢⠀⠀⠀⠘⢄⣀⠇⠀⠀⢀⡀⠀⣀⢤⡄⣴⡄⢄⠀⠀⠁
⠀⡄⠀⠀⠳⢄⣀⠠⠒⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠐⠤⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠙⡯⡛⡼⡇⠙⠃⠈
⠀⠀⠐⠀⣀⠀⠀⠀⠀⠄⠀⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⡀
⠀⣘⠂⣁⠀⢂⠁⠩⠅⠂⠣⡰⢀⠀⠀⠀⠀⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣠⠀⠒⠐⢄⠠⠀⠐⠁⠑⠀⠀⠒⠐
⠣⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡨⢰⠀⠀⡠⠚⠉⠀⠀⡵⠀⠀⢀⠔⠉⠁⡆⠁⠀⡀⠀⠀⠀⠸⠂⠀⠀⠀⠴⠀⠀⢀⢂⠐⠀⠂⠀⢄
⠂⠌⢠⡄⠀⠀⠀⠄⠀⢀⠐⣁⠂⠀⡎⣡⠔⠉⠉⠉⡠⠄⠒⠁⠀⡠⠘⠀⠀⠀⡀⠀⠀⠈⠀⠑⠀⠤⠰⠂⠀⠀⠀⠑⠀⠐⠀⠐⠁
⠀⠊⠐⢠⠠⠐⠀⠎⠀⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠑⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐
⣿⣿⠟⠟⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⣠⠴⠶⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢠⠞⠁⠠⣾⠇⣀⡈⢦⠀⠀⠀⠀⠀⠀⠀⣠⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⢝⡧⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀⣿⣿⠁
⠀⠀⢸⠘⠛⡓⠀⠀⠙⠗⢸⠄⠀⠀⠀⠀⠀⠲⣿⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⠈⠉⠉⣿⣿
⠀⠀⠘⣆⠸⠿⠀⠺⠿⢆⡞⠀⠀⠀⠀⠀⠒⠷⣶⣦⣼⠇⠀⠀⠀⠀⠀⠀⢀⠜⠛⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⠶⠂⠀⠿⠿⠁
⠀⠀⠀⠈⠑⠶⠤⠴⠖⠉⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⢀⣼⣷⣏⢹⣶⣧⣀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠙⠋
⠀⠀⠀⠻⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠁⠀⢇⢡⡌⣱⠇⠈⠀⠀⠀⠀⠀⢸⣿⢿⡿
⠀⠀⠀⠀⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⠀⠀⢀⣴⣾⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣶⣣⡤⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⢀
⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣯⢞⡁⠀⠀⠀⠀⣄⣼⣿⣿⡤⣜⣿⣿⣇⡄
⠀⠀⠘⢿⣿⣿⣯⠛⠋⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⢹⣿⣿⡟⠁⠈⠉⠙⣿⠃
⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣴⣧⡀⠀⠀⠀⠀⠀⠀⠀⣄⣿⡟⣻⣥⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⡀⠀⠀⠀⠀⠀⣴⡶⠶⡶⠶⠶⢶⣦
⠀⠀⠀⠀⠈⠙⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣽⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠟⠁⠀⠀⠀⠀⠀⣿⣀⣂⣐⡒⠂⠸⠛
⠀⠀⠀⠿⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠍⠁⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦⡶⠂⠀⠀⠀⠀⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠝⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡿⢿⣷⣶⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡫⠖⣏⡟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣵⣆⢳⠟⣹⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠡⢅⠞⠁⢸⡿⢳⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣷⢦⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣬⠏⠀⠀⠘⢷⡠⡎⣿⠀⠀⠀⠀⠀⣠⣶⢴⣄⠀⠀⠘⠛⠏⠁
⠀⠀⠀⠀⠀⣀⠤⣤⡀⠀⠀⠀⠀⠀⠀⢹⣳⠆⢿⡗⠀⠀⠘⡟⠊⣁⢩⣿⠆⠀⠀⠀⠀⠀⠀⣠⣀⣰⣆⣀⡀
⠀⠀⠀⠀⣰⢋⠛⡚⣛⢳⣄⠀⠀⠀⠀⠘⠛⢼⠞⠃⠀⠀⠐⣿⣦⢹⡊⣿⡀⠀⠀⣠⣴⡖⠿⢩⣥⢉⡹⢻⡀
⠀⠀⠀⠈⠻⢜⡞⠀⠸⠈⠉⠳⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠇⡹⣯⠀⠀⢴⠿⢢⠰⠎⣃⣨⣍⣵⣾⡇
⠀⠀⠀⠀⠀⠀⠙⣧⡀⠴⠖⠒⢰⡎⢹⡶⣤⣀⠀⠀⠀⠀⠀⡼⣁⠟⣶⣿⡀⠀⠘⢻⠞⠛⠋⠙⠋⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⣶⠀⡘⠁⠸⡇⠙⣻⡄⠀⠀⠀⣾⠇⠹⠆⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣽⠆⢰⠀⣈⡽⠁⠀⠀⣰⡦⠞⠠⣖⣸⡟⠀⣀⣶⣾⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠻⠒⠺⠃⠀⠀⠀⠸⣧⣄⣴⡾⠉⠀⣴⢿⣷⣬⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡴⣶⢾⡗⢲⢖⣶⢤⣤⡀⠀⠀⠀⠀⠈⢀⣀⡶⠝⠓⣬⣽⠉⠀⠀⢀⣴⠷⢶
⠀⠀⠀⠀⠀⠀⠀⠰⣟⢣⠖⣠⡄⠃⣤⠰⢚⠊⠰⣷⠶⠀⠀⣠⡴⣿⠟⠐⢟⣉⣿⠍⠀⠀⢀⡼⣫⢖⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠠⢻⣯⣬⣩⣤⢿⡬⠵⠾⠽⣧⠟⠀⠀⠐⢿⣰⢋⠀⣿⣨⠟⠃⠀⠀⠀⣸⡧⠡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⣴⣦⣦⣀⣀⢀⠀⠀⠀⠈⠙⠛⠉⠉⠈⠀⠀⠀⠀⠀⠀⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢨⣥⡹⠟⢿⠧⣤⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⣥⡐⠯⡄⠇⠞⠢⠏⠽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠻⠗⢾⡧⡾⢤⣿⡟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⡄⠀⠀⣀
⠀⡀⠀⠂⠀⠰⠄⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠉⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠙⠄⠀⠀⠀⠀⠀⠀⠐⠁⠤⠀⠀⠀⠀⠀⢰⢯⠃⠀⠀⠀⠀⠀⢣⣍⣷⣌⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠊
⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠰⢈⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⡾⠀⠀⠀⠀⠀⠈⠲⠖⠂⣥
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢀⠇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠃
⠀⢀⡤⠆⠀⠀⢀⠀⡀⠀⠀⠙⠂⠀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣰⡟⠶⣄
⠀⠀⠀⠀⠃⠀⡏⠀⠁⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣦⣴⣴⡦⠀⠀⠀⢠⡿⠀⠤⠿⠀⠀⠀⠀⢀⠠⠴⣄
⠀⠀⢤⣠⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⡇⠀⣽⢱⠀⠀⠀⡔⣂⠠⣺⡿
⠀⠀⠀⠈⠁⠚⠙⠙⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠛⠿⣿⣿⡿⠛⠁⠀⠀⠀⠳⠘⠃⠁⠀⠀⢧⡴⣂⣵⠿⠃
⠀⠀⠀⠀⠀⣀⣤⣀⣀⡀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⢠⢤⠀⠉⠈⠉⠙⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢦⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⣐⡤
⠀⠀⢨⠸⡀⠀⣀⠀⠀⠀⠀⠈⠁⠀⢐⣒⠀⠀⠀⠀⠀⣀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⢸⣕⣼⡧⣻⠇⢡⡄
⢀⡠⢨⣄⣧⡜⡠⠁⠀⠀⠀⠀⠀⢰⢻⠉⡗⠀⠀⠀⠀⠈⠛⠀⠙⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⣏⡻⢿⡿⠿⡏⠓⣬⢬⣷
⠈⠧⢤⠿⡟⢯⣷⡑⢦⡀⠀⠀⠀⠋⣇⣼⠁⠀⠀⠀⠀⠀⠒⣔⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⢒⡥⢨⠿⣿⠗⠱⡤⢺⠍
⠀⠀⠀⠀⠹⣤⢇⡽⢦⠏⠀⠀⠀⠰⣽⢿⡄⠀⠀⠀⠀⠀⠀⡜⠂⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣀⠘⣂⣘⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠀⠀⠀⠀⠙⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⡖⠚⠈⠄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠐⠀⡈⢠⠀⠁⠠⠠⣤⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢀⡠⠰⠁
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡃⠠⢀⠀⠉⠀⠀⠠⡇⣴⣇⡍⢀⠀⠀⠀⠀⠀⠐⢦⠞⢈⣑⠁
⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⠀⠀⠀⠐⠀⠈⠀⠍⠉⠐⠋⠰⠁⠀⡁⠀⠀⠀⠀⠀⠀⠜⠛⠂⡿⣿⠉⠑⠤
⠀⠐⢠⢤⡀⠀⠠⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠐⠀⠂⢰⡂⠀⠂⢀⣠⢈⡀⠄⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⡈
⠀⠀⠀⠀⠰⠒⢠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⡿⠛⠓⠀⠀⠀⠀⠈⠉⠙⠛⠛⠂⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠛⠿⠛⢩⠤⠆⠀⠉⠁⣤⣶⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⢄⠀⠀⢷
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠠⠰⠆⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠄⢀⣿⣷⣶⠀⠄⢀⡀⠀⠀⠀⠀⠀⡂⠀⠘⣇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣴⣾⣿⣿⣿⡄⠀⠀⡐⠁⠀⠀⠀⠔⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⡠⠊⠀⠀⠀⢀⠊⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⣿⣧⡀⠀⠰⠿⠟⠃⠀⢀⣾⣿⣿⣿⣿⣿⢯⡄⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡆⠀⠀⠀⠀⡶⠿⠿⠛⠉⢙⠟⠋⠀⢳⡄⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇⠠⢠⡔⠈⠁⠀⠀⠀⠈⢷⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⢶⡀⠈⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠙⠦⣄⣀⣀⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢁⢐⡄⠁⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠰⠉⠛⠿⠷⠂⠀⠀⠀⠰⠂⠀⠠⠀⠀⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠴⢤⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣠⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠀⡖⠒⠂⢲⡄⢰⡆⣶⠒⠒⢠⡖⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⣟⣉⡁⠈⣧⡾⠀⣿⣉⣉⢸⡟⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠁⠀⠈⠁⠀⠈⠉⠉⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⢀⠄⠀⠀⡀
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠆⠀⠀⠈
⠀⠀⢰⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗
⠀⠀⠞⢀⡖⠁⠈⠉⢀⠀⠀⣀⠠⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀
⠀⠀⠂⠈⠁⠀⠀⢀⠌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠄⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠
⠀⠀⠀⠀⠄⢁⠀⢀⠰⠁⠀⠀⠀⠀⠀⠀⠀⣠⠆⠲⠒⢿⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠐⠦⠠⢤⣀⠀⠀⠀⠈⠀⠀⣤⡀⠀⠀⠀⠀⠠⠆⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⣹⠁⠀⠀⠀⠀⠠⠊⠁⠐⠀⠀⠀⠀⠀⠐⠠⠓⣄⠀⠀⠀⠠⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⡐⠇⠀⠀⠀⡀⠀⠈⡄⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⠖⠁⠀⠀⠀⠀⠀⢠⠇⠀⠀⠐⠀⢈⠀⠠⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢻⣄⣀⡀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠄⠃⠀⢀⠂⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠅⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⢀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢀⠀⠀⠀⠀⠀⠃⠀⠄⠀⡱⠊⠀⠀⠀⠀⠐⠀⠀⢀⠀⡀⠀⡀⠤⠄
⠀⠀⠀⠀⠑⠀⠠⠙⠂⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠠⠂⠈⢂⠤⠊⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⣄⡀⠀⠀⠀⠐⠀⣀⢀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠘⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⣀⠀⠀⠘⠀⠀⠀⠐⠊⠈⡁⡠⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠀⠀⠀⠀⢀⠀⠀⠀⡂
⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠉⠀⠀⠓⠉⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠛⠙⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠐⢒⠒⠶⠶⠂⠀⠈⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠐⠦⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣀⡀⢀⣀⡀⣠⡤⢀⡤⣀⣀⡀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣀⠁⠠⠖⣋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉
⣿⣿⡿⠿⣿⡿⢿⣿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣶⣿⣷⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⠂⠐⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣀⠀⠀⡥⠀⠀⠀⠀⠤⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢠⡎⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢃⠀⠀⠊
⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠐⠂⠄
⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⡃⠀⠀⠀⡤⣖⡧⠀⣂⣀⠔⢒⠄⢀⣠⠀⠀⠀⢠⠀⢀⢢⣄⣐⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠁
⠀⠀⡀⠀⠠⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠨⢰⡌⢇⠀⢀⣼⠀⢠⠔⠀⢠⡜⢸⢀⠇⢀⡠⠮⠂⠀⠀⡇⠀⠀⠙⠦⠀⠙⠁
⠀⠀⠃⠃⠉⠁⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠋⠁⠚⠁⠐⠚⠀⠙⠀⠋⠀⠀⠊⠁⠛⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠆⠀⠀⠁⠶⠦⠠⠶⠦⠼⠿⠧⠷⠄⠴⠶⠴⠄⠠⠶⠶⠦⠿⠇⠀⠘⠀⠀⣖⠀⠤⠀⢐⡀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣀⠀⠀⠈⢠⡄⠀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠉⢀⣀⣀⡀⠊⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣶⣶⣶⣧⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣛⣿⣿⣿⣿⣏⣿⣷⣯⣽⡍⣿⣿⣿⠿⠟⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣺⠭⣛⠍⠀⠀⠀⢉⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠀⢀⡜⣤⣀⣠⠃
⠀⠀⠀⠀⢀⣴⠎⢅⠪⠓⡿⠁⠾⠀⣰⠀⠈⡎⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⡛⠉⠉⡚⠕⠀⢀⢤
⠀⠀⠀⢠⡾⢹⢪⠏⠀⢰⠃⢀⣀⣴⣿⠀⠹⣽⡟⠀⠀⠀⠀⠀⠀⠀⠐⡢⡞⠃⠀⠀⠂⢀⣤⠗⠓⠁
⠀⠀⢀⣓⣷⣃⣚⡢⠔⠁⠀⠈⠁⠉⢁⣴⣣⣿⡇⠀⠀⠀⠠⢢⠀⠀⢠⠟⡿⠂⠀⢄⣼⣹⠟
⠀⠀⣸⣳⣞⢪⠋⠰⠟⠀⣸⠀⢀⡞⠉⠠⣸⡟⠀⠀⠀⠀⠀⡐⠣⣤⣙⠀⠀⠀⠐⠚⣳⠭⠄⠄
⠀⠀⣼⣾⡌⠀⠀⠀⠀⠰⣣⠄⡸⠁⣠⣤⠟⠁⠀⠀⠀⠄⠀⣱⡞⠉⠀⡠⡠⢧⢺⣷⠁
⠀⠀⢱⡯⢥⡀⠀⠀⠀⠺⠋⠰⣛⡵⠟⠁⠀⠀⠀⠀⠈⠉⠰⡶⡎⠀⠐⣵⣗⡶⠋⠈⠛
⠀⠀⠀⠻⣮⠽⣦⢤⣀⣀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣶⣿⡯⠾⢧⡀
⠀⠀⠀⠀⠀⠉⠙⠻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠹⡇⠀⠀⠉⠁

⠀⠀⠀⡀⢠⠀⡀⠀⠀⠀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠁⠀⠀⠀⠀⠀⢂⠀⠨⣀⠀⠀⠀⠁⠂⠀⠀⠀⠈⠀⠁⠀⠀⠰⠊⠁⠉⢡⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢠⠀⠌⠀⠂
⠀⡁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠠⠤⠀⠂⠀⠀⠐⠈⠀⠀⠘⣀⠀⠀⠔⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠂⢀⡀⠁
⠀⠀⠘⠀⡄⠐⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠒⠀⠐⢄⠀⠂⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⡀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠄⠁⠈⠂⠀⡃
⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠈⠈⠀⠀⠉⠀⢀⢀⠤⠞⠀⠀⠀⠠⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⣀⣀⣤⣤⣴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠘⠀⠄⠀⠀⠀⣠⣾⠟⠛⠫⠛⡛⡛⢉⣙⡻⢯⡀⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⠐⠢⠀⠈⠀⢠⣿⠘⢸⡄⡄⣌⣳⣜⣈⣯⣩⣼⠿⡿⡏⠉⣿
⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠁⠒⠂⢀⠁⠀⠸⡿⠶⢾⠛⠻⡏⣭⣭⡩⡇⠒⢓⡐⢠⡀⢰⣽⠄
⠀⡠⠂⠀⠢⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠻⣌⠌⠂⢦⣀⣝⣉⣷⣹⣌⢉⣣⣠⣥⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠐⠀⠄⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠙⠉⠋⠛⠛⠛⠉⠁⠀⠀⢀
⠀⠀⠁⠀⠁⢀⠀⠀⠀⠀⠀⢍⢈⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⠱⠃⠚⠁⠀⠀⠀⠄
⠀⡊⠀⠀⠌⠀⠀⠀⢀⠀⠂⢀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠐⠠⠛⠁⢀⡀⡀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠂⠀⠀⠀⠀⠈⠀⠀⠐⠰⠖⢲⢸⡣⢈⠄⠂
⠀⠈⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠘⠁⠁
⠀⢀⠂⢐⠈⠀⢃⠂⠀⠐⢈⠄⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡠⠐⠁⠀⢀⠃⠀⠀⠀⠊⠐⠆⠀⠀⠀⠀⠀⠀⠄
⠀⠠⠢⠀⠀⠀⠀⢄⠀⣀⢂⠐⠀⡜⡠⠐⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⠀⠄⠐
⠀⠈⠀⠡⠠⠐⠀⠊⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⢘⣿⣿⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠉⠉⠀⠈⠉⠉⠛⢿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠀⣾⣿⣷⠀⠀⠀⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠘⣿⣧⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠸⣿⡟⠋⢁⣀
⠀⠀⠀⠀⣠⣶⣾⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⠀⠀⢹⣿⣶⣿⡿⠃
⠀⠀⠀⠀⠈⠉⠉⠈⣿⡇⠀⠀⠀⠈⠛⠛⠛⠛⢿⣿⠇⠀⠀⠀⠻⣿⣅
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣦⣄⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠘⠛⠋⠉⢿⣧⠀⠀⠀⢰⣿⣿⣶⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠙⠻⢿⣶⣾⣿⠋⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣧⠀⠀⠘⠿⠿⠟⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠉⢿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣤⣶⣾⣶⣄⠀⠀⠈⠻⣿⣦⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣦⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⣀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠟⠉⢻⣿⣶⣄⠀⠉⠛⠿⠿⠿⠟⠛⠁⠀⠰⣿⣿⡇⠀⠀⣸⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣰⣿⣿⠿⠿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⣠⣿⡿⠻⢿⣿⣷⣦⣄⣀⣀⡀⢀⣀⣠⣴⣾⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⢀⣾⣿⠛⢻⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⢂⣔⣒⣄⣀⠒⢒⠒⠒⡒⠒⠶⣤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢒⣡⣄⠊⠁⡀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠁⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠛⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⠛⢋⠉⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠹⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠁⡽⢁⡾⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢧⠀⠀⠁⠀⠀⠀⠀⠀⠸⡎⣧⢸⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡇⠀⢠⣴⣾⡇⠀⡞⠀⠀⠀⠀⢀⣄⡀⠀⣤⣴⢂⣼⠃⠀⠀⠀⠈⠘⣆⣴⣿⣀⡰⠄⠀⠀⣿⣿⠈⠁⠀⠉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣾⠸⣿⡏⠀⢿⣿⣿⡇⠀⡇⠀⠀⠀⣠⡿⣽⣹⣿⣿⣿⡜⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⣽⣿⠦⠄⠀⢸⣼⡆⠀⠀⠀⠸⠿⣇
⠀⠀⠀⠀⠀⠀⠀⢻⣆⢻⣇⠀⠘⣿⣿⡇⠀⣿⠀⠈⠈⠉⠁⣽⣿⠋⢿⣿⠁⠀⠀⠀⠀⣿⠀⠀⠀⢎⠻⠆⠀⠀⡀⠀⢸⡏⡇⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⠀⠀⠘⣿⣧⠀⣿⡆⠀⠀⣄⠀⠀⠀⠀⣻⣯⠀⠀⠀⠀⢀⢿⠷⠀⠀⠀⠁⢰⣶⠟⠁⠀⢸⣿⠇⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⢻⣿⠀⣿⠘⠀⠀⠘⠳⣶⡆⠀⢻⠃⠀⣠⠶⠛⠉⣉⠀⠀⣤⣤⡈⠳⣅⠀⠀⠀⢸⣿⡇⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⢸⣿⡆⣿⠀⠀⠀⠀⠀⡼⠃⠀⡙⠀⡾⠃⠀⠀⠾⣿⡛⠃⠀⣻⡷⠀⢹⡄⠀⠀⢸⣿⣷⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣸⡟⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠳⠿⠇⢠⣦⣙⣿⠃⠀⠃⢀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣴⣿⣿⡇⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⠿⡿⠛⠁⠀⠀⢀⢺⠄⠀⡟⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡍⢿⣿⣧⠀⠘⠁⠙⣾⣧⠀⠀⠀⠀⢀⡁⡅⠩⢿⣿⡇⣸⣿⡒⠂⠀⢸⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⢿⣆⠀⠀⠀⠻⣿⣧⡀⠀⢠⢠⡄⣴⠶⢿⠟⣿⣿⣝⣧⣉⣀⡌⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠸⡆⠹⣆⠀⠀⠀⠈⡿⢿⣶⣤⣠⣤⡤⢲⠛⢷⠽⠉⠈⢿⠻⠋⡴⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢸⡄⠀⢷⠀⠈⠓⢦⣄⠀⠙⠚⠿⠻⣿⡏⣳⠸⣆⠀⠀⠀⠀⠸⢳⠏⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠁⠀⠈⢧⠀⠀⠀⠙⠦⣀⣀⣀⠀⠈⠿⣿⣷⣤⣉⣁⣠⣤⣠⡟⠀⠀⠀⢀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⣀⣀⣉⣡⠀⠤⠖⠉⠀⠀⠀⠀⠈⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⢰⣠⠃⠀⠀⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⠇⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⡟⡀⠀⠀⠀⠀⡰⠟⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠆⠀⡿⠀⢸⡏⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠈⢰⡇⢀⠀⠀⠀⠀⠀⠈⠻⣿⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⢿⣇⣴⡄⠀⠀⠀⠀⠀⢸⣿⣿⠀⢠⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⠀⠀⠀⠈⠻⡿⣧⠀⠀⠀⠀⠀⠈⠃⢻⣶⣿⡀⡄⠀⢀⣤⣄⠀⢀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⠀⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⣀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠙⠎⠻⢿⣴⡿⠁⠙⠻⣾⠉⠙⠂⠀⠀⠹⡿⠀⠀⠀⣸⠇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠋⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠘⠋⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘⠧⠀⣤⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⡀⠀⠀⠀⠀⠈⠻⢿⣾⣷⣤⣘⣦⡀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⡷⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠓⣤⠤⠤⠤⠤⠶⢍⡀⠀⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠓⢄⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⡤⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢚⠦⣀⢀⠓⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢸⠀⠀⠁⠀⢱⠀⠑⠂⠔⠂⠤⠤⠤⠴⢂⠀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣂⣀⡀⢀⡤⠐⠒⢷⡦
⠀⠀⠀⠀⠀⠀⠘⠴⡀⢠⢧⡄⢆⠰⠤⣀⠀⡷⠶⡏⠁⢠⠀⣹⠹⣄⠀⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠳⡳⠈⠉⠀⠀⠘⠦⠤⠔⠣⢀⠌⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠽⣝⠉⣝⢉⡎⡃⠀⢤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠛⢆⠈⣷⡱⣤⠜⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢄⠀⠀⠀⠀⠉⠀⠀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⡦⠤⣀⠀⠀⣀⣰⠖⠉⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠤⠇⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⡑
⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠜⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢸⠀⠀⣀⣀⠤⠄⠃⠀⠰⡤⠤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⡩⢷⠈⠉⠉⣹⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠀⠀⢀⠔⠊⢆⡀⠑⠒⢶⡁⠉⣉⠽⣄⠀⠀⠀⢀⠄⠀⢇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⠈⠀⠀⠀⢋⠉⠉⣷⠀⠙⠤⠤⠋⠳⡀⠐⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠑⠊⠁⢇⠀⠀⠀⠀⡰⠁⠃⠀⠀⠀⠀⢀⠇⠐
⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠊⠀⠑⢄⡀⠀⠀⠀⡠⠊⠒⠒⠒⠉⠀⠀⠈⠢⠄⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡆⠀⠀⣤⣤⢀⣿⠋⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢷⣀⠛⢿⣽⣿⣶⣴⣾⣧⣾⣇⣠⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣋⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠾⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠯⣉⣭⡙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡾⠹⣶⣯⣽⣿⠿⢿⣿⣿⣟⠛⠻⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⠁⠀⢠⣿⣿⡿⠇⠀⠙⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⣿⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⣭⠙⠒⠒⠒⠒⢉⠙⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⠉⠉⠙⢢⣄⣀⣤⠬⣁⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢃⡼⠁⡏⠀⠀⡀⠈⢻⢶⣧⠀⠀⢀⣠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠋⠩⡀⠀⣸⣤⣀⣉⡰⠃⠸⣄⡤⠖⣉⡽⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠀⠀⠀⠑⠦⣝⡋⠀⠀⠀⠀⠀⠀⢀⣀⣤⢚⡡⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣞⠣⣄⠀⠀⠀⠀⢀⡎⠉⣿⣒⢒⣺⠋⡹⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠗⠀⠏⠑⠠⡔⠉⡇⢀⠂⠀⢀⠃⣞⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠳⠞⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⡘⢠⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠒⠤⢀⡀⢴⢰⠃⠀⠀⠀⡇⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡠⠴⠋⠀⠀⠀⠀⡆⡃⠀⠀⠀⠀⢠⠔⠋⠂⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⠀⠀⠀⣀⠴⠥⡀⠀⠀⠀⡠⠃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡶⢄⣈⡷⠀⠀⠈⣢⠤⠒⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠈⢳⠧⠶⢊⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠃⠀⠀⠀⣴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⡟⠀⢦⣀⣀⠀⠀⠀⣀⣀⠠⠜⢄⠀⠐⡠⠀⠀⠀⢢⠉⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢧⣀⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠈⠉⠒⠒⢲⠁⠀⠀⠀⠠⠀⢀⠀⠀⡇⠀⠀⢸⠀⠈⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡞⠒⠦⢄⣀⠀⠀⠀⠀⠀⢱⠀⡠⠊⠀⠀⠑⢄⠀⠀⠌⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⢀⠂⠀⠐⠒⠩⡉⠉⠉⠉⢣⡀⠀⠀⠀⠈⠉⠉⠙⣳⠃⠰⡀⠀⠀⠀⠀⠀⡆⢠⠂⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠸⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠊⠉⠁⠀⠈⢆⠀⠀⠀⠀⡰⢁⠆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⢨⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠀⠐⠂⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡏⠻⠽⠿⢷⡄⠀⠀⠀⠊⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠸⠆⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢿⣗⣜⢃⡚⣜⣿⡇⠀⠀⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠁⠠⠈⠘⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣾⣷⣼⡿⠃⠀⠀⠂⠀⠀⠀⠀⢄⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠄
⠀⢀⣞⣷⢾⣿⣿⣷⣟⡦⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⢄
⠀⠈⡼⣿⣽⣿⣿⣿⣯⡶⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡄⠀⠀⠀⠘
⠀⠀⠘⢻⢿⣿⣻⢿⡑⠃⠀⠀⠀⠀⠤⠤⠖⠒⠒⠒⠒⠲⠤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠒
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⢀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠆⠠⢄⠀⢀
⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠸⣗⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⢰⡆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⡷⡂⠀⠀⠀⢀⣻⡭⠀⠀⠀⠰⣤⣀⡀⡀⠀⠀⠀⠀⢺⣿
⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢵⣤⢀⢐⢎⠟⠃⠀⠀⠀⠀⠈⠉⠛⠛⠶⠾⠾⠿⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣄⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⢀⣤⣼⣿⡇⣻⠁⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠉⠛⢛⣳⣿⣧⣈⡛⠃⠀⠀⢀⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣿⣟⣻⣿⣿⣿⣿⣦⣤⣴⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⣾⣦⣤⣿⣿⣻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣤⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⢿⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠿⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢧⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⣿⣿⠏⣽⣿⢿⣿⢿⣧⡀⠈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠉⠁⠈⣧⣸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠐⠿⠿⠉

⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⣼⣿⡟⠀⠀⠀⣀⠀⢀⣠⣤⣴⣄⡀⣀⡀⠀⠀⠘⣿⣷⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣸⣿⡿⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⣄⠀⢹⣿⣧⠀⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣿⣿⣧⣠⣿⣶⣿⢿⣿⣿⣿⠟⢿⣿⣿⣿⣟⣷⣾⣧⣸⣿⣿⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⣿⣿⣿⠙⣿⣮⣽⡿⢋⣿⠋⠀⠀⠻⣿⡿⣿⡯⣾⡛⢿⡿⣿⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣴⣿⣿⣿⡄⢸⣿⣿⠀⡼⠁⠀⠀⠀⠀⠘⠧⣼⣿⣿⢀⣼⡇⣿⢠⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⣟⣿⣿⣼⣿⡧⠚⠀⣀⣤⣾⣿⣶⣤⠀⠹⢿⣟⣼⣿⡇⣿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡏⢻⣿⣟⣿⡁⡀⢰⡟⣉⣙⣛⣩⣹⣧⠀⠸⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣜⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⡂⢸⣿⣿⡿⠻⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢨⡿⣿⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⢿⣿⣇⠑⣽⣿⣿⣿⣿⣾⣽⣿⡿⣿⠿⣡⣿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣿⣿⣶⣤⣾⣿⣿⣛⣛⣿⣿⣿⣷⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣄⢸⣿⣛⣛⣛⣛⡙⢳⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣧⡾⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣮⣿⣿⣿⣿⣹⣿⣿⠻⠇⣿⣷⣿⣿⡏⣱⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠰⣄⢹⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣷⣙⣾⣿⣿⣿⣿⣿⣣⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣤⣴⣾⠏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠰⢶⣤⡀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⢺⡟⣠⣾⠟⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢻⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢠⣦⣭⣼⣷⡬⠄⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣶⡶⢸⡿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠾⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⡿⠁⠀⠀⠀⣠⡶⠂⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠿⠛⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⠀⠀⠀⠸⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⡾⢛⣃⣀⣴⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠠⣾⣿⠷⢬⣝⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣰⡶⢸⣯⡸⣷⡌⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣿⡇⠈⠉⠁⠘⣿⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠙⠻⢶⠦⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠺⡶⠿⠿⡷⠀⢠⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢠⠄⠤⣀⡤⠔⠢⡀⠀⠀⠘⡷⠤⠒⠉⠀⡀
⠀⠀⠀⠈⠁⡀⠀⡿⠀⠀⡇⢀⡀⠀⠀⠀⢀⡎⠉⠑⡍⠀⠀⠀⢀⣧⢠⡼⠵⢤⣀⣱⠀⠄⠊⠀⠀⢘⠉⢵⠈
⠀⠀⠀⠀⠀⠁⡰⠥⠀⠀⢣⠀⠁⡠⢤⣴⠟⠻⠀⢀⣼⠀⠀⠐⣅⠈⢲⡀⠀⣤⠑⣾⠉⠛⢆⠀⢠⢬⠉⢁⣤⡀
⠀⠀⠀⠀⠀⢠⠃⣠⣀⣠⣌⣆⠀⠣⡴⡅⠀⠀⠙⡿⠿⠀⠀⠀⢈⡉⢶⠬⠦⣄⣠⡏⣦⠤⠚⠀⡈⢊⡴⣋⠜⠁
⠀⠀⠀⠘⢁⣿⣿⣿⣿⣿⣿⣿⣆⠀⣷⠹⡄⠀⣐⣒⣒⡄⠀⠀⢀⣸⠈⢆⣀⡤⠛⠒⠊⠀⠀⠀⡤⣋⠜⢁⣀⡀
⠀⠀⠀⠀⢎⢻⣿⣿⣿⣿⣿⡿⢈⠀⢸⠖⠚⠚⠒⠒⠒⠚⣆⠀⠀⠙⠀⠀⠀⠀⣠⣴⠶⢒⡚⠿⣿⠁⠀⠀⢸⣏
⠀⠀⠀⠀⣈⠑⠒⠂⡒⠒⠒⠈⠁⠉⠘⠒⠦⠤⠤⠦⠆⠒⠃⠀⠀⣤⡆⠀⠀⡾⢋⠖⡒⠒⡌⢢⠙⡄⠀⠀⢸⣍⠁
⠀⠀⠀⠈⠚⠁⠀⠀⠻⠯⠀⠀⠀⠀⠀⠄⠀⠀⠀⠔⠒⠄⠴⠀⠀⠈⠁⠀⠀⡇⠸⣀⡛⠘⡜⡸⠀⡇⠀⡆⢸⠮
⠀⠀⠀⠀⢴⢀⡶⠚⢲⡀⠀⠓⠺⠰⠀⠀⠀⠀⠀⠊⠒⠤⠴⠂⠀⢠⣤⠀⣄⠹⣤⠈⢉⠉⠐⣡⠞⠀⠀⠧⠿⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠀⠀⢠⠴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⣠⠞⠛⠳⡄⠀⢰⡆⠉⠂⣴⠶⠶⠄⠈⢀⡀⠀⢀⡄⠀⣠⠤⢤⡒⠀⢸⣂⣀⣀
⠀⠀⠀⠀⠐⣆⠀⠀⢀⡇⢰⡇⠀⠀⠀⠉⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⣾⢧⠀⢸⡇⢰⡇⠀⠈⠁⠀⢸⠉
⠀⠀⠀⠀⠀⡌⠛⠒⠋⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣟⠛⠛⠂⠀⡟⠈⣇⢹⠃⢺⢁⠀⠀⢀⡀⠸⠶⠖⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢧⣀⣀⣠⠄⠀⣸⡇⠈⠁⣿⣤⣤⡄⠸⠃⠀⠘⠟⠀⠈⠦⠤⠖⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⢒⣤⣤⡄⢀⡴⣊⠥⢄⠉⢆⠲
⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠠⡆⠀⠀⠘⠀⠈⢫⡂⢠⠛⠒⠠⢼⡀⠛⠑⠙⠀⠀⣀⠁⡀⡄⢠⣾⡔⠁⠀⢀⠃⣼⢠⣤
⠀⠀⠀⠀⠀⠁⢰⠭⡆⢰⠯⡆⠀⡟⡇⠀⠀⠀⡸⠡⠀⠀⠠⠀⠈⠟⢠⣄⣤⠀⠳⣄⢀⡗⠋⠀⠀⠀⢴⣧⡴⠃⢸⠏
⠀⠀⠀⠀⢶⣾⣯⡿⡽⠽⠯⡿⠿⠿⣿⠟⠀⠘⢇⠀⠀⠀⠀⠀⠀⡆⠈⠻⠃⠀⢂⡬⠛⠬⡀⠀⠀⠀⠔⠈⠀⢻⣮⡄
⠀⠀⠀⠀⢸⣿⠀⣶⡇⠀⣿⡇⢠⣿⣿⠀⠀⢀⡠⠣⡀⠀⠐⣂⣀⡇⠀⠀⠀⡰⠉⠀⠀⠀⠉⢦⠀⠀⠀⠀⣦⣠⠟⠁
⠀⠀⠀⠀⢸⣿⠀⣿⡇⠘⠿⠇⠀⠿⣿⡆⠀⡉⣀⢀⠙⠉⠀⠀⠀⠀⠐⠀⠙⠁⠀⠀⠀⠀⠀⠸⠟⠄⠀⠀⠸⠛⠋
⠀⠀⠀⠰⠿⠿⠿⠯⠿⠿⠯⠿⠽⠽⠽⠿⠃⠈⠁⠈

⠀⠀⠀⠀⡖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡇⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠒⠺⠟
⠀⠀⠀⠀⢹⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃
⠀⠀⠀⠀⡉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⡅⠀⠀⠙⣴⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠀⢿⠃⠀⠁⠴⢀⡀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣖⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠐⠀⠀⠀⠔⠄⠀⠀⠀⠒⠀⢻⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠁
⠀⠀⠀⣜⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠋⠐⠁⠀⠀⠀⠀⠀⠀⠀⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢂⠤⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠠⠠⠀⠀⠀⠀⠀⠀⠰⡭⠂⠀⠾⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⣐⠲⣮⣈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⡀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢴⠀⠀⠁⣷
⠀⠀⠀⠐⢋⠾⠿⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠂⣀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢲⣀⡆⡲⠟
⠀⠀⠀⡞⠉⣿⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⣿
⠀⠀⠀⠙⡿⠛⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢘⡓⣿
⠀⠀⠀⠘⣦⣠⣄⡅⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⣏⣏⡹
⠀⠀⠀⠀⠈⠀⠈⣇⢀⣴⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⠀⠀⠀⠻⣿⠇
⠀⠀⠀⠀⠐⠶⠿⢻⣿⠋⡄⠱⠀⠀⠀⠀⢁⣠⡴⢤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎
⠀⠀⠀⠀⠀⠤⣀⣤⠏⠄⠈⠀⠀⠀⠀⠀⠀⢃⣀⡼⠗⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢁⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠋⠰⠚⠉⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⣁⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠉⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠰⡄⠒⠒⠠⢤⠀⠀⠀⠀⠀⠀⠀⠐⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣀⠀⢀
⠀⠀⠀⢀⠃⠀⠀⠰⠀⠰⠄⢰⠀⠀⠀⠀⠐⢀⠆⠈⠀⠀⠈⢡⠄⠀⠀⠀⠀⢠⢟⡶⡟⢷⡐⠽⡷⡀⠀⠀⠀⠀⠀⠐⠀⠄⠁⠈⠣⡘
⠀⠀⠀⢘⠀⡉⠀⠀⣀⠠⠄⠀⠇⠀⠀⠀⠀⠋⡐⠀⠄⠤⠀⡠⠀⠀⠀⠀⠀⢾⣳⣂⡝⠀⢠⡲⣠⣧⠀⠀⠀⠀⠀⠀⢰⣄⡬⠄⠒⢳
⠀⠀⠀⠘⣄⣁⠐⢀⡐⠀⠄⡛⠀⠀⠀⠀⠀⠁⠙⣀⣀⡄⠴⢄⠀⠀⠀⠀⠀⠈⢉⣾⣿⣯⢤⣇⣰⠏⠀⠀⠀⠀⠀⠀⢸⡧⣄⠈⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠰⠀⠅⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢟⣣⣤⣶⣶⣶⣶⣶⣶⣤⣄⡸⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣦⣤⣀⠀⠀⢀⣤⣾⣿⡿⠟⠛⠋⠉⠉⠙⠛⠻⢿⣿⣶⣄⠀⠀⢀⣴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⢇⣴⣿⡿⠟⠁⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠙⢿⣿⣆⢴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⡀⢀⣠⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣋⣵⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣼⣿⡏⣶⣦⣴⡶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⢿⡿⠀⠀⢀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠀⠀⠀⢀⣾⣿⡿⣡⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⢲⣿⣿⠀⠀⠀⣠⣶⣿⣷⣦⠀⠀⠀⣴⣶⡄⠀⠀⠈⠉⠀⠀⠀⣰⣿⣿⠋⠀⠈⠉⠙⠋⠉⠛⠻⣦
⠀⠀⠀⠐⠶⢤⣤⣴⣾⠿⠛⠋⠉⢸⣿⡇⠀⠀⠰⣿⣿⣿⣿⣿⡇⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⣰⣿⡿⣱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠙⢿⣿⡿⠟⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠱⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⢀⣼⣿⡿⠁⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢋⣄⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣝⠿⣿⣷⣶⣤⣄⣀⣀⣀⣀⣀⣤⣴⣾⣿⡿⠛⠁⠙⢿⣆⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠈⠙⠛⠿⠿⢿⣿⣿⣿⠿⠿⡛⠋⠁⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠿⠿⠿⠛⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⠉⠁⠉⠹⣿⣿⣿⣿⣦⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠄⠀⠸⣿⣿⠏⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⡄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⣠⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠒⠂⠀⠀⠀⠀⢀⣀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠈⠂⠀⠀⠤⠊⠀⠀⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢊⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠒⠙⡒⠠⠀⠀⠀⠀⠀⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠚⡇⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣀⡀⠀⠀⠀⠀⠀⠀⠀⢃⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠂⣧⣤⡇⣿⣭⠐⢯⣃⠉⡟⠁⠀⣾⣹⢆⡾⡄⢸⢸⢧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃⠃⠀⠃⠓⠒⠈⠒⠋⠀⠃⠀⠀⠛⠀⠘⠉⠙⠘⠘⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⣉⡁⣤⢀⡀⠉⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⣠⡍⡿⠀⣾⠋⣄⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠈⠐⠁⠆⠘⠄⠉⣨⢷⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⢠⡿⣷⠀⠰⠻⠻⠳⠀⢰⣿⢽⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⢃⣿⡀⡶⢠⣇⢠⠀⢸⣿⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⠷⡴⢫⣝⢦⡤⣼⢯⣿⡆⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⠁⢻⣿⠟⠃⠈⣱⣿⣿⠇⠘⠆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⢈⡇⠀⠀⢠⣿⣿⡿⠀⠀⠀⢻⡇⠀⠀⢀⠀⠀⠀⠀⠀⠰⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣶⣾⣿⣆⣴⣿⣿⣿⠁⠀⠀⠀⣾⠀⢠⠀⢸⣤⣦⣴⠀⠀⠀⠛⠇⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠙⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⣸⣿⠀⣾⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠀⠀⢀⠈⠙⢿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⢀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⡸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣀⣠⠤⠸⢧⣤⣄⣀⣀⣤⣤⣤⣴⡾⠃⠀⢸⣿⣿⣿⠿⠋⠙⠛⢛⡁⠈⠻⡷⣷⡀⠐⡀⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣤⠾⠃⢀⣠⠞⠀⠀⠀⠀⠙⠻⠿⢿⠿⠟⠛⠉⠀⣀⠀⠀⠋⠁⠀⠀⢀⠀⢀⣼⣿⠦⠚⣡⠾⠃⠀⠀⠀⠙⢿⣿⣿⣷
⠀⠀⠀⠀⢀⡴⠉⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⣰⡤⠞⠉⠀⠀⠀⠀⠀⠀⠈⠤⠉⠛⠁⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿
⠀⠀⠀⠈⠉⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠈
⡴⢠⢀⣄⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁
⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⣠⠄⠈⡁⠀⠘⠀⠾⠂⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣦⡠
⠀⠀⠀⠀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣷⡀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⠷⠾⢓⣆⠀⠂
⠀⠀⠀⢠⡤⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣦⠖⣾⣯⡉⠀⠀⡀⠀⣴⣷⣿⣼⣿⠗⠀⠀⡀⠀⠀⠀⠀⠀⠟⣷⢿⡏⠉⠂
⠀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⠐⣴⣷⣏⣩⡅⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⣄⠀⠀⠀⠀⠀⢤⠿⠙⠍⠋⠈⠀⠀⠀⠀⣠⣤⢰⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⡀⢰⡤⠀⡀⠀⠀⠀⠀⢺⡃⢰⡆⠀⠈⡀⠀⠀⠀⠻⠀⠀⡅⢻⠋
⠀⠀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠰⠆⠀⠀⠈⠁⠀⠙⣦⡀⣿⠀⠀⠀⠘⢶⡰⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢠
⠀⠀⠀⠰⠂⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢶⣴⣇⣀⠀⠀⠀⡀
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⣾⣭⣻⢻⡷⡔⠚⠁
⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠈⠀⠀⠐⣟⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⣼⣩⡄⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢿⡿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠁⣼
⠀⣀⠀⠀⠀⠰⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠐⢀⣠⣞⠁⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠘⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠿⠀⠀⠀⠀⠄⠉⠙⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢈⠀⠀⠀⠀⠰⠦⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠈⢘⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠁⡄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣲⣾⣿⣞⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡚⢲⡄⠀⠀⠘⠟⢻⣿⠛⠃⠀⠀⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠎⣰⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠉⠳⡖⠀⠀⢠⠆⠀⠈⠀⠂⠀⠀⠀⡀⠠⢥⣮⣿⣿⣷⡀⠀⠠⠄⠀⣇⠀⠉⢸⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⢠⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠻⠻⠟⠑⠦⠀⠀⠀⠉⠀⠀⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣴⠆⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠒⠀⠀⠠⣤⠉⠀⠀⠄⠀⢨⠀⠀⡀⠀⢼⡴⠁⠀⣀⣀
⠀⠀⠀⠀⠁⠀⠀⡼⠀⠀⠀⠀⠀⣀⡴⠒⣫⡟⠁⠀⠀⠀⠀⢰⡀⣠⡔⠀⠀⠀⠛⠀⠀⠀⠀⠀⠗⠀⠞⠀⠀⠈⠁⠐⣴⠛⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠞⣁⡴⠚⠁⠀⠀⠀⠀⠀⠐⠘⢿⣿⣆⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢄⣸⣿⣯⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣸⣧⠖⠚⠙⠋⠉⠁⠀⠀⠀⠀⠀⠙⠲⠦⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢴⣶⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⣀⡀
⠀⡀⠀⠀⠀⠊⢸⣆⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢰⣇⠀⠀⠀⣼⡟⣿⠄⠀⠀⠀⠿⡔⠀⠀⠀⠀⢀⢐⣶⣿⡇
⠈⣿⣦⣴⣴⣄⣀⡀⠀⠀⠀⠀⢋⢻⣠⣀⣀⣠⡎⡈⣻⡾
⠀⠈⠈⠚⠟⠋⠉⠀⠀⠀⠀⠀⠀⠙⠽⡿⣧⣩⣿⢷⣛⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠂

⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠡
⠀⠀⠐⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣻⣿⣿⣿⢟⣫⣷⣿⣿⣿⣿⣦⣤⣄⡀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣟⣭⣿⣿⣿⣿⣿⠿⠋⣾⠏⠛⠙⠛⠷⣤⣀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣩⣾⣿⣿⣿⣿⣯⣿⣿⣶⣼⣿⣴⣷⣦⠀⠀⢀⢙⣇⣀⢀⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣋⣾⣿⣿⣋⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⡀⢸⣿⣿⣿⣾⣿⣷⣤⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⡟⣿⣿⣿⣿⣿⣿⣡⣾⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠻⣽⣿⢻⣿⣿⣿⡿⢏⣼⣣⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⣴⣾⣿⣿⣉⠀⣠⣿⢹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣿⣿⠛⣿⣿⣿⣿⠿⠿⠿⠟⢻⣿⣻⣿⠿⣿⣿⢳⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⠏⢠⡿⠋⠉⠀⠀⠀⠀⣺⣿⣿⣿⣿⣷⣿⣿⣶⣾⣿⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣶⣴⣦⡀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿
⠀⠀⠀⠀⠀⢰⡏⢸⣿⣿⣦⣌⡙⢿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿
⠀⠀⠀⠀⠀⣼⠇⢸⣿⣿⣿⣿⣿⣾⡇⢿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣶⣶⡄⠻⣿⣿⣿⣿⡿⠃⢨⣇⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢿⢿⣿⣧⣽⣉⣭⣷⣶⣖⣛⣛⣣⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⢠⢗⡜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠘⣷⣝⣓⢤⣍⠛⢱⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠨⣿⣛⣿⣿⣷⢾⣛⣿⣿⣿⣿⣛⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢈⡿⠛⠻⠛⠄⠺⠟⠛⠛⠉⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⡇⣾⣯⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⢀⣼⣷⣿⣿⣿⣿⣯⣸⣶⣀⡰⡶⠈⢻⣿⣿⣿⣿⡿⠿⠟⢛⢛⡡⠀⠀⣀⣀⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣷⣿⠟⠿⢿⣯⣿⣿⣿⡷⣾⡟⣿⡛⠛⠁⣚⣀⣩⣤⣶⣾⣿⣿⣿⣿⣧⡀
⠀⠀⢀⣾⣿⣿⣿⣿⣿⡏⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽
⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠤⠀⠠

⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠹⢿⣿⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠂⢱⠁⣸⠤⡀⠀⠀⠀⠀⠀⢠⣶⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣠⣤⡀
⠀⠀⠀⠀⠈⠸⣯⣽⣦⠀⠀⠀⠀⠀⠀⡀⠀⢈⠁⠁⠂⢁⡀⠁⠀⠀⠀⠀⠀⠀⢍⠻⣟⣎⠂⠀⠀⠀⠀⠀⠀⣙⣿⠟⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠈⠈⠻⠑⠄⠀⠀⠀⠀⠀⠚⠏⢠⡌⡄⢄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠁⢹⣻⠀⠀⠀⠀⠀⠀⠀⠻⣿⢾⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡄⠒⡘⣐⠠⠄⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠄⡄⠀⢄⣀⠨⠀⠀⠀⠀⠀⠀⢀⣴⣴⣤⡀
⠀⠀⠀⠀⡘⡈⢁⠐⢀⠐⠘⡀⠀⠀⠈⠷⢦⡄⠻⠀⠘⢋⠀⠀⠀⠀⠀⠀⠠⠀⠑⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠤⠐⠀⡐⠀⠄⠠⠀⠀⠀⠀⠀⣀⣴⠂⢸⡆⠈⠻⣤⠀⠀⠀⠨⠙⡏⠀⢀⠀⡠⢒⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠆⠡⠰⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠊⠃⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠘⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⣭⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⡾⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠰⡤⢄⡀
⠀⠀⠀⠀⢸⣻⣿⠉⢹⡦⠀⠀⠀⠀⠀⠐⢄⠀⠘⠀⠐⠃⠀⠀⠀⠀⠀⢠⡴⣿⡟⠁⢰⡿⠀⣰⠆⠀⠀⠀⢐⡌⠀⡨⢄⠀⠳
⠀⠀⠀⠀⠀⠻⣿⣶⣿⡻⣆⠀⠀⠀⢀⠄⠀⡁⠀⠀⠀⠠⣊⡥⠀⠀⢰⠟⠁⠈⠁⢀⣬⣿⡶⠇⠀⠀⠀⠀⢨⣄⢁⠪⠐⠄⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠼⣺⡯⠌⠀⠀⠀⠀⠋⠩⠙⢂⢠⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢦⠅⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⠸⠁⠦⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⢠⣀⣠⠀⠀⠀⠀⠀⠀⢲⡶⣿⣻⡷⣴⠀⠀⠀⠀⠀⠀⠐⠉⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢚⠀⢀⠀⠀⡔
⠀⠀⠀⠀⢸⣶⣿⠿⣿⣦⡄⠀⠀⠀⠒⣿⣻⡽⡾⣿⣻⡷⠀⠀⠀⠀⠀⠀⢄⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢲⠸⠢⠬⡄⢄⠈⠁
⠀⠀⠀⠀⠘⢿⣿⣀⣼⣿⠇⠀⠀⠀⠀⢹⣯⣿⣻⣯⣿⡁⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣤⡀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠱⣉⣠⠂⡆
⠀⠀⠀⠈⠂⠘⠛⠻⠇⡀⠀⠀⠀⠀⠀⠈⠀⠉⠯⠀⠈⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠨⠕⠀⠀⢤⢙⣮⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠈⠒⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠜⠁⠀⣠⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠘⡿⠉⣤⠉⠙⢺⣣⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀⠀⢀⣺⣿⣴⣤⡸⠂⢙⣋⠀⠀⢰⢁⡞⠀⠀⠀⢀⣰⣦⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠲⠀⠈⠉⡿⢿⣿⣵⡺⠾⠙⠁⠀⣆⣾⠀⠀⢀⠴⢛⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠘⠃⠉⠁⠀⠀⠀⠀⠻⠋⠀⠀⠑⠒⠁⠀⠀⢶⡴⡦⣴⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⢰⡷⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢶⢧⢠⡉⣽⣿⣏⡀
⠀⠀⠀⠀⠄⠒⡫⠟⠀⠀⠀⠉⠀⠀⠀⠀⠈⠹⢿⠄⠀⢸⣥⠄⠃⠀⠀⠀⠀⠀⠀⠘⠻⡆⢨⣤⠑⣻⣿⣥⠀⠀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠁⠒⠈⠀⠀⠈⠙⠉⣀⠀⠀⠀⠀⢰⢻⣧⠘⠛⢶⡤⣶⠛⠃⠀⠀⡜⣺⡏
⠀⠀⠀⠀⠀⢀⡿⠏⡈⣛⣿⡴⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⣦⣿⣤⣶⢀⡀⠈⠫⠸⠇⠀⠀⠀⣀⡀⢀⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣽⠠⡥⢻⣿⡗⠀⠀⠀⠀⣿⢻⠀⠀⣤⣟⠓⠋⠿⣿⢟⣿⠅⠀⠀⠀⠀⠀⠐⠿⠷⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠛⠶⠎⠋⠉⠀⣄⣶⠄⡽⠠⠀⠀⠀⣷⡦⠙⠡⡶⠈⣾⡄⠀⡰⣶⠂⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣤⣄⢀⣄⠀⠈⠚⠃⠀⠣⠜⠀⠀⠀⠉⠿⠓⣶⡖⠿⠀⠀⣰⣱⠃⠀⠀⠀⠀⠀⢉⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⢀⠠⣶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣈⠛⠉⠀⠀⢠⡄⣤⣠⡄⠈⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⡤⢤⡄⠀⠀⠀⠀⠀⠛⢻⡇⠀⠀⠀⣤⣽⣍⣋⣿⣄⠀⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⠀⣴⣿⣔⢃⢘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣟⠁⠆⠹⢠⡯⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⣻⢿⡦⡴⠇⠀⠀⡰⠀⠀⠀⠺⡷⠄⠈⠀⠿⢶⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠀⠙⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢸⡿⠀⢀⣾⠋⠀⢀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⣾⡷⢶⣾⣧⣄⣠⣿⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢀⣴⠟⣋⣤⣤⣶⣶⣦⣤⣉⠛⢷⣤⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⢈⣿⠟⣡⡾⣋⠁⠀⠀⠀⠀⠈⡙⠻⣦⡙⢿⣄⢀⣴⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⣿⠟⣡⡾⠋⠘⢛⣡⣤⣤⣤⡀⠈⢿⡄⠈⢿⡄⢻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢻⣶⠟⣡⡾⠋⠀⠀⢠⡿⠋⠀⠀⠉⢿⡄⠀⠀⠀⠈⣿⡀⣿⣦⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣴⠟⣡⡾⠋⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣿⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢀⣼⠟⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⢙⣷⣦⣤⡾⠟⠀⠀⠀⠀⢀⣿⠁⣿⠛⠷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡀⢀⣿⠟⣡⡾⠋⠀⢠⡤⠀⠀⠀⠀⠀⣴⠟⠛⠛⣷⡀⠀⢠⣤⣤⣠⡾⢃⣼⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣈⢉⣿⠟⣡⡾⠋⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢿⣄⠀⢀⣾⠃⠀⠀⣠⡾⢋⣴⠟⠁⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡿⢡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⣠⡾⢋⣴⠟⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢶⣤⣿⢀⣿⠁⢀⣤⡶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠟⠻⣧⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣿⠀⣼⡏⠀⠀⠈⣿⠀⠀⠀⠻⠶⠀⠀⠀⠀⣠⡾⢋⣴⠿⣧⡀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠿⣷⠘⣿⠀⠘⢷⣤⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⣟⠁⠀⠈⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠹⣧⡀⠀⢈⡉⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⡟⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠟⠉⠻⣧⡈⠿⣦⣘⡿⠀⠀⠀⠀⢀⣠⡾⢋⣴⠟⠻⣧⣀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠻⣶⣌⡙⠛⠿⠷⠾⠟⠛⣉⣴⠟⠿⣦⡄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣿⠋⠛⢿⡶⠶⢶⡾⠟⢿⣇⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠁⠀⣠⡿⠁⠀⣾⡇⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⣶⡄⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠄⠄⡀
⠀⠀⠀⠀⠀⣴⡄⠉⠁⢀⡃⠂⠈⡋⢁⣶⡄⠀⠀⠀⠐⢀⢶⣦⣶⣶⡄⢂⠀⠀⠀⠀⠄⡢⠆⠔⠔⡠⡀⡀
⠀⠀⠀⠀⣀⠉⡁⠰⠷⣴⠇⣻⠃⣤⡤⡉⣡⡄⠀⠀⠠⠸⠿⣿⣽⣿⠇⡘⠀⠀⣐⠜⠁⠀⠘⠀⢃⠀⠑⢒⢀⠀⠀⠀⠀⠀⠀⣰⠎
⠀⠀⠀⠘⠟⠱⠸⠋⣴⣦⠘⠻⠃⣽⡄⢜⢙⡃⠀⠀⠀⠡⡈⠛⢛⠅⠒⠀⠀⢀⠾⠀⡃⠀⠃⡀⠈⠀⠂⠠⠦⠀⠀⠀⠀⢀⠰⠏
⠀⠀⠀⢰⣷⠁⠘⣛⢠⣦⠐⠿⢃⣬⢁⠀⠿⠇⠀⠀⠀⠀⠀⠁⠁⠀⡀⠀⠀⠀⡢⠀⠄⠼⠀⠀⢀⠀⡄⠠⢑⠀⠀⠀⠠⣮⠃
⠀⠀⠀⠀⢡⣴⠀⢅⠰⠷⠐⢿⠆⠅⠘⣿⠆⠀⠀⠀⠀⠀⠠⠐⠋⠉⠁⢢⠀⠀⠁⢣⢀⠂⠀⢁⠀⠰⠠⠁⠄⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠈⠉⠀⣴⣴⠁⣥⡁⠈⣷⡆⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠈⡁⢂⠠⡠⡘⠃⠉⠀⠀⠀⢱⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠃⠀⠀⠀⢀⡠⠒⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡳⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠋⠠⠖⠀⠂⠊⠀⠀⠀⠀⠀⠀⠀⣠⣖⡭⣛⡤⠀⠀⠀⢠⢀⡀⠀⠀⠀⠀⠀⠑⠅⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⡴⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡡⠟⣙⠋⣵⠃⠀⢀⡞⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠴⠃⣀⡀⠠⢼⠜⠊⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⡀⣀⣈⠫⠾⣋⢾⡛⠃⠁⠀⠀⡨⠇⣁⡘
⠀⠀⠀⠀⠀⠈⠀⠀⡠⠃⠀⢤⡑⠺⡑⡯⠀⠀⠀⠀⠀⡾⢗⣩⢉⣶⠾⠟⠉⠀⠀⠀⠀⡂⠀⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠
⠀⠀⠀⠀⠀⠀⠔⠊⠀⠠⣹⡄⢾⠧⡈⠂⠀⠀⠀⠀⠐⢋⡘⢡⣤⠟⠀⠀⠀⠀⠀⠖⢀⠉⠈⡓⣁⢀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠛⡦⣍⠂⠀⠀⠀⠀⠀⠈⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠦⠈⠳⠈⢈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠈⠀⠈⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⡑⠲⣄
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡾
⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀⠀⠀⠀⠀⠀⣤⠀⣀⢠⢘⠁⠀⠀⡀⠀⠀⢀⠤⠦⣄⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣰⢱⡞⡧⣹⠀⠀⠀⠀⠀⢀⢔⠱⢇⡂⠓⢆⠈⠁⠀⠀⡌⢘⠐⠂⢱⠀⠀⠀⢰⣾⣦⣀⠀⠀⠀⠀⠀⠐⠲⡒⢦⡁
⠀⠀⠀⠀⠀⠀⠿⣈⢡⡦⣹⠀⠀⠀⠘⠃⢸⡀⠀⠋⢂⠕⡞⡀⠀⠀⠀⢷⠈⡀⠂⠀⡅⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⢠⢃⠞
⠀⠀⠀⠀⠀⠀⡷⠿⣤⢹⠏⠀⠀⠀⠀⠀⠀⠙⠢⠬⠥⠈⠀⠁⠀⠀⠀⠘⣆⠁⠐⠠⢇⠀⠀⠀⠁⠙⣿⣷⠀⠀⠀⠀⠀⢸⣻⣄⣀
⠀⠀⠀⠀⢀⡼⢰⡗⢫⠁⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡈⡄⠂⢸⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⡆
⠀⠀⠀⠀⢸⣵⠆⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡁⠖⢉⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡾
⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢰⡖⠤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡶⠋⠀⢀⢸⠀⡠⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡴⠿⠛⠛⢛⠛⠛⠋⠈⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠖⠉⢰⡴⠚⠁⠀⠀⠀⠘⠾⠊⠀⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠐⡀⣰⠁⠀⠀⡠⡀
⠀⠀⠀⠀⡀⠄⢠⣤⠔⠉⢀⡤⠞⠁⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠳⠿⠥⠀⠀⠶⠃⠸⠀⠀⠈⠣⣀⠀⠀⢀⣨⠔⠁⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⠋⠀⠀⠀⠀⠶⠂⠀⢠⠏⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⡤⡸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠀⠸⢦⣾⣥⢤⣄⠀⠀⢆⠠⢴⢀⢀⠎⠀⠉⠒⠢⢄⣀⠀⢀⣀⠀⠀⠀⠀⠙⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠋⠀⢰⣖⡒⢤⡀⠀⣄⢠⠀⠀⠀⣾⢁⣀⣠⣟⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠟⠢⠀⣠⠀⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠉⢹⠗⠋⡀⠐⠂⠂⠀⠀⠉⠛⢡⠏⢀⣽⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⢀⠤⠐⠋⠁⢀⠏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠋⠉⢳⡄⠀⠀⠈⠢⡐⠀⠀⠠⢀⡸⠀⠀⠀⠀⢀⠀⣹⠧⢒⠆
⠀⠀⠀⢰⣿⣶⣤⣄⡀⠘⢄⠀⠀⡆⢀⠀⠀⢠⡄⠀⠀⠀⠸⣅⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠒⠊⠉⠀⠀⠀⠀⠀⢺⡷⠛⠊⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⡼⠀⠀⡏⠙⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡞⠙⡆⠀⠼⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢄⣀⠀⢠⣤⡌⢀⡄⠀⠀⠀⠸⡿⠀⠀⠀⢀⣼⠙⡣
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠛⠃⡀⣀⠀⠀⠀⣞⣽⣿⣿⣿⢶⠚⠁⠘⠀⠉⠀⠁⠀⠀⠀⠀⢿⠀⢀⣴⠟⠊⢹⡅
⠀⠀⠀⠀⠄⠸⣔⡽⠀⠀⠠⢀⣘⣧⣤⠤⠴⠞⠁⠀⠀⠀⠀⢹⢹⠁⣀⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠭⡟⠁⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢳⣏⠉⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠉⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠹⣧⠀⠀⠂⢉⣻⣦⣤⡼⠀⠀⣀⡀⠀⠀⠀⠀⢃⢿⣿⠀⣿⠀⠀⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠸⣍⣀⠈⡧⠀⠀⠘⠉⠀⠃⠀⠀⠄⠸⡸⣿⡇⢻⡇⠘⠋⠉⠻⡁⢘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠧⡡⠃
⠀⠀⠀⠀⣀⡀⠀⠀⢈⡴⠒⠛⠻⣏⠃⠀⠀⠀⣀⡠⢠⡀⠀⠠⠀⡇⢿⣿⣘⣿⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠑⠁⠈
⠀⠀⠀⠀⠛⠁⠀⠀⠾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢼⣟⡩⠋⠀⠀⠀⠰⠃⠀⠳⣄⠀⠀⣠⣐⣒⣒⢒⡄
⠀⠀⠀⠀⢞⣹⠀⢴⡦⠀⠀⠐⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⠆⠀⢀⡠⠖⠉⠀⠀⠈⡟⠛⠛⡟
⠀⠀⠀⠀⠀⣀⣶⠶⠦⢤⣤⡤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠙⠙⠈⠂⠾⠃⠀⠀⢰⣹⠆⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⣀⣤⣤⣤⣤⣖⠚⠉⠉⠉⠉⠉⣷⠴⠶⠶⠶⠶⠚⠃⢀⠀⠀⢠⢊⣁⣀⣨⣌⣲
⠀⠀⠀⠀⡀⠸⣄⣀⣀⣠⠇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠸⢿⣿⣿⣿⣿⠟⠰⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠘⠒⠒⠒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢦⡀⠀⠀⠀⣸⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⢳⣠⡽⠶⢶⡶⠿⣼⡀⠀⠀⣀⡴⣾⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡓⠦⣄⡴⠋⢡⠞⠋⠉⠉⠙⢦⡙⢦⡞⢉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡞⠁⠀⠻⠴⠖⠛⠓⠲⠼⠃⠈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⣄⣀⣀⡞⢀⣀⡀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠈⡇⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣽⠃⠻⡼⠇⠀⠀⢸⠁⠀⢹⡆⠀⠀⠀⣿⠚⣩⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⡟⠀⠀⠀⠀⡀⠀⠈⠳⠶⠚⢀⣤⣄⠀⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⡒⢾⠃⠀⠀⠀⣏⣉⡇⠀⠀⠀⠀⠈⠳⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡏⠀⠀⠀⠀⠈⠉⠀⠀⠀⢰⣏⣻⠀⠀⠀⡷⠒⣒⣒⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠏⠀⢠⣶⡆⠀⠀⢠⠖⠲⣆⠀⠉⠁⢀⣀⠀⣷⠚⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⠚⣉⡼⠃⠀⢀⠀⠉⠁⠀⠀⠸⢦⣤⠟⠀⠀⢰⠋⠉⡧⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣽⠁⠀⠸⣍⡗⠀⠀⣠⣤⡀⠀⠀⠀⡤⣤⠈⠛⠚⠁⠀⢿⠛⠓⠒⣲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠐⣇⣀⡿⠀⠀⠈⠳⠚⠀⢀⣀⠀⠀⠸⡷⠚⠋⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠻⣝⡓⠒⠚⢻⡇⠠⡄⠀⠀⢾⣉⡇⠀⠉⠀⢀⡤⠤⣄⠀⠀⠻⠽⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠺⣇⠀⢻⡄⠀⠀⠉⠀⠀⠀⠀⣟⠀⠀⢸⡇⠀⠀⠀⢀⠀⢰⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⠲⠶⠛⠀⠀⣀⡴⠋⠀⣾⣈⣉⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠙⠳⠦⣤⣀⣀⣀⣀⣠⡤⠶⠚⠁⠀⢀⡼⠁⠀⠈⠉⠻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⣁⡤⠞⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⢉⡟⢻⠖⠒⠒⠒⠚⠛⠉⢳⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠉⠳⣄⠀⠀⡴⠋⠀⠀⠀⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣰⠖⠓⢦⡀⢰⡇⣶⠛⡇⢸⠀⢸⠃⢸⠙⠳⠟⢹⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢋⣉⠳⡄⢰⠃⡟⢙⡆⣧⢸⡇⣨⣉⠁⢸⠀⢸⠀⢸⡀⠀⡴⠾⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⣦⢸⡇⠯⠼⠃⡇⢸⠀⠉⠉⠀⣿⢸⡇⢧⡘⡆⢸⠀⢸⡄⠀⣙⠒⠛⢦⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⡞⣳⢸⠘⡇⠀⠀⢀⡇⢸⡆⠀⠀⢠⡇⠀⡇⠀⠉⠀⢼⠀⢈⡇⠸⣍⡹⠂⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠈⠁⢸⠀⣧⠀⠀⠈⣄⣠⠇⠀⠀⠀⠙⠚⠁⠀⠀⠀⠈⠓⠋⠀⠀⠉⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸



⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⡖⠒⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⠒
⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⢠⠤⢄⠀⡀⢤⠤⠀⣠⠀⢠⠤⢄⢠⠀⠀⡠⠄⠀⢀⠠⠄⠠⡤⠄⡤⢄⡀⡠⠤⡀⢠⠀⡀⢠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢸⠀⢈⠀⡇⠀⡂⢰⠄⡆⢸⠒⢣⢸⠀⠀⡗⠀⠀⠈⠑⠄⠀⡇⠀⡇⢺⠠⡇⠀⡸⢸⠻⡀⢸
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣷⣶⣶⣶⣶⣶⣾⣷⣷⣾⣾⣶⣿⣶⣶⣶⣷⣶⣶⣷⣶⣷⣶⣶⣾⣶⣷⣾⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠄⠤⠤⠤⠄⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣈⣥⠄⠖⠚⡘⣡⠷⢯⠋⡘⠰⢲⢂⠬⡍⣁⡒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣡⠄⢞⢀⠂⠊⠈⠀⢠⠃⠀⢈⡆⠀⠁⠀⠈⠈⠀⠀⠘⠙⠲⢍⡒⢄⡀
⠀⠀⠀⠀⠀⠀⡠⠊⣠⠎⢀⠢⠀⠀⠀⠀⠀⠀⠙⠖⠋⠁⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠁⠙⠢⣌⠢⡀
⠀⠀⠀⠀⠀⠔⠁⠰⠣⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⡀⣀⡀⠈⠈⠀⠘⠀⠀⣎⠈⡇⠀⠈⠳⡜⢦⡀
⠀⠀⠀⢀⠎⠠⡀⡑⡐⠀⠑⠊⠀⠉⠀⠀⡠⠒⢉⣠⣀⣀⣀⣀⡠⠓⠀⠀⢄⠈⡂⠀⢸⣀⡄⠀⢠⠀⠙⢆⠳⡀
⠀⠀⠀⡎⢠⠙⢌⠌⠀⠀⠀⠀⢀⡠⠐⠊⢀⡰⠊⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢄⠀⠈⢂⢳⡀
⠀⠀⠘⠠⠏⠢⠂⢀⠘⠃⠀⠀⠎⠀⡰⠋⠁⠀⠀⠘⠃⠀⠀⠘⠀⠀⡖⠄⠀⠀⠀⠰⠀⡀⠀⠸⡄⠈⣆⠀⠈⡄⣧⠂
⠀⠀⠆⡴⠑⠁⠀⠣⠤⠀⠀⢨⠁⠀⡃⠀⠆⠐⠂⢀⣄⣀⠐⠝⠀⢀⡇⠸⠀⠀⢀⣀⣜⠊⠀⠀⢸⠀⢹⠀⠀⢐⢹⢂
⠀⠀⠀⢎⠌⢀⢤⠀⠀⠀⠀⡾⠀⢀⠃⠀⠀⠀⡴⣡⠖⠦⡑⢤⡤⠊⡠⠃⢀⡞⠁⡤⠞⠀⢀⣀⡸⠀⢐⠁⠀⠀⢸⢆⠂
⠀⠀⠀⡃⠀⠀⠁⠘⠃⢠⠜⠀⢀⠚⠀⠀⠀⢰⠁⡇⢎⡑⠭⡒⠒⠊⠁⠀⠈⠦⠎⠀⠀⠀⡇⢠⡤⣤⠎⠠⡄⠐⢸⢃⢂
⠀⠀⡇⢇⠀⠀⢠⡀⠀⠹⡤⠔⠁⠀⠴⠀⠀⠘⣆⠧⣄⡨⠃⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⡆⠀⠠⠄⢠⠀⡴⠡⠃
⠀⠀⢡⠹⡄⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠈⠓⠒⠒⠉⠀⠀⠀⠖⠐⡄⠀⠀⢤⠀⠀⡇⢰⠇⢀⠀⠠⠋⢠⡇⠡⠁
⠀⠀⠀⢣⠱⡀⠀⢪⣀⣠⠍⠀⣠⠞⢁⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢰⠃⠀⠐⠒⠀⠸⣄⢼⣤⢈⡥⣾⢁⡾⡴⣡⠃
⠀⠀⠀⠀⠣⡙⢄⠀⠀⠀⠀⢸⠁⣼⠁⠀⠀⠈⠀⡀⠀⡌⠁⠀⠀⠈⠉⠁⠀⠰⠀⠀⠀⠀⠛⠺⠛⢘⣶⢃⣾⡝⡽⡥
⠀⠀⠀⠀⠀⠙⢄⠑⢄⠒⠀⠀⢧⡈⠓⠤⢤⣀⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠙⢁⢎⠌⠌⡘⠁
⠀⠀⠀⠀⠀⠀⠈⠑⣄⠑⠤⡂⠀⠙⠖⠤⠤⣄⡈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢂⡠⠁⡡⠊⣴⢋⢂⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣌⠓⠤⣀⠀⠀⠀⠉⠲⢄⣈⣉⡹⠀⠀⠠⢄⠀⠀⠔⠤⠃⡠⠊⡠⡺⡡⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⢒⠤⣉⠒⠢⢄⣀⡀⠀⠀⠀⠀⠀⠈⢁⡄⠐⠤⠒⡁⢰⢊⠜⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠒⢢⠤⣀⣉⢉⡉⣉⣉⢉⡁⣀⠤⠐⠞⡐⠁⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠐⠁⠊⠌⠐⠠⠁⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⠟⠿⠿⢿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣍⡻⣿⡿⢸⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠀⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣭⣴⣿⡿⢛⣛⣿⠿⠿⣿⣿⣿⣿⣶⣄⣀⠀⠀⢨⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣶⣿⣿⡏⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⣫⣥⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⢸⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣷⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠠⠀⠈⢿⣿⣿⣷⠀⠀⠙⣿⣿⣿⣿⡙⣿⣿⣿⠿⠿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠛⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⣻⣿⣿⣦⣀⣠⣿⣿⣿⡿⢃⢿⣯⣴⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠈⠀⠸⣿⣿⣿⣿⣿⣿⣧⣬⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠈⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣯⠀⠀⢀⣿⣿⠟⠉⡉⠙⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠿⢿⣿⡀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣾⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠂⠀⣿⣿⣷⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀⢀⣹⣿⣿⣷⣶⡀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣷⣤⣤⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣄⣀⣼⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⣀⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢸⡿⡘⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣀⣛⣢⣤⠀⠀⡇⢧⣠⠿⠭⢭⡱⡄⠀⠀⠀⠀⣿⣿⣿⡿⠿⠥⠝⠟⣿⣿⢸⣜⠿⣀⣛⣓⢮⢻⣿⣿⣿
⠀⠀⠀⠉⠓⡼⠁⠀⠀⠀⣇⣀⡇⠀⠀⠀⡇⡇⠀⠀⠀⠀⣿⣿⣿⣶⣼⢛⣿⣿⣿⣿⠸⠿⢸⣿⣿⣿⢸⢸⣿⣿⣿
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠚⠛⠂⠀⠀⢀⡇⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣥⣤⣽⣿⣿⡿⢸⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢲⡶⠶⢶⢞⡩⠞⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣍⢉⣉⡉⡭⢖⣡⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠛⠘⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣤⣧⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠷⠶⠶⠶⠷⠶⠶⠆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣬⣭⣭⣭⣬⣭⣭⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠈⠹⡄⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠂⠀⠀⠴⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠘⠲⠐
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⡜⢲⠊⢱⠊
⠀⠀⠀⠀⠀⠰⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠘⣿⡿⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣼⠉⢹⠋⠙⠓
⠀⠀⠀⠀⠀⠀⠀⣷⣤⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠂⠘⠋⠀⠀⠀⠀⠀⣶⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠘⠃⠀⠀⠀⠀⠐⣿⠗⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠛⠃⠀⠀⠀⣿⣿⣿⣿⣿⠉⠛⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢠⠀⠀⠀⠀⠻⠂⠀⠠⣿⡷⠀⠀⠙⠛⠃⠀⠀⢼⣿⠀⠀⠀⢀⠀⣿⣿⡿⠿⠿⠀⠀⠀⠘⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⢀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⢿⠆⠀⠀⠀⠛⠀⠀⣾⣿⣿⠀⠐⠉⠀⠁⠀⢹⡋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣆⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⠀⠀⠈⠉⠃⠅⠀⠤⠤⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠒⢸⣿⡇⢸⣿⡇⠐⣿⡗⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⣄⣤⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⣿⡇⢸⠀⠀⠐⠀⠀⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⢠⣬⣭⣤⣤⣭⣥⣤⣭⣥⣼⠀⠀⠀⢠⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣤⣤⣬⣭⣭⣭⣥⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠂⠠⢄⣀⣀⠠⠤⠶⢀⠀⠤⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠒⠻⢤⣀⠀⣀⣐⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⢠⠏⢿⠁⠀⠉⠙⣶⣦⡀⠀⠀⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠊⢅⠀⢘⢦⣀⣁⡼⠁⢇⣠⠔⣉⢽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠀⠀⠀⠈⠑⠬⣙⡁⠀⠀⠀⢀⣀⢠⠤⠎⠡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡾⣝⠢⣀⠀⠀⢀⡼⠀⡝⠭⠭⡟⣁⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⢈⢀⠃⠈⠐⠁⢰⠀⠃⠀⠰⢀⠇⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢱⡀⠀⠀⠀⡘⡠⠀⠀⢠⡆⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡜⠀⠈⠁⢉⠁⠁⠀⠀⠀⠃⡠⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢈⠄⠀⠀⢀⣠⡊⠀⠀⠀⢀⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⢄⡀⡯⠀⠀⠑⣄⠠⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠏⠀⠈⢳⠤⠖⡊⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠣⠄⠀⠀⠀⢠⠄⠒⠁⠤⠀⠤⠀⠀⠀⠳⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠤⢀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⡀⠀⠀⡈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠠⠀⠀⠉⠒⠒⠠⠐⢦⠀⠈⠉⠒⠒⠂⠤⢤⡄⠐⠁⠀⠀⠀⠈⠀⢀⠄⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠒⠉⠀⠨⡀⠀⠀⠀⢀⠂⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠁⢠⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣟⣟⣿⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣻

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠺⣿⡿⠃⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢿⣷⡄⠀⠀⠀⠘⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⢻⣿⣧⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⢀⣀⣤⣿⣶⣶⣶⣾⣿⣤⣀⠀⣰⡟⠁⠀⠀⠀⠀⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣤⣾⡿⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠻⢶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⡶⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠾⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠷⣦⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⡿⠛⠁⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠈⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⠀⠀⠀⠀⢀⣼⠏⠀⠉⠛⣿⣿⠿⠿⠿⣿⠛⠉⠀⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⢻⣿⣧⠀⠀⠀⢰⣿⡇⠀⠀⠀⢿⡄⠀⠀⠀⠙⣿⣷⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣼⣿⠁⠀⠀⢠⣾⣿⡆⠀⠀⠀⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢰⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠠⠁⠠⠬⠐⢂⠀⠀⠀⠀⡦⠀⡐⠉⢂⢀⡀⠀⠀⠀⣀⣧⡵⠖
⠀⠀⠀⠀⠀⣥⠜⠁⢢⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠸⡀⠁⠐⠀⠂⢉⡜⠀⠀⠀⠈⢢⡠⣯⡟⣍⣈⠏⠀⠀⢼⠁⢈⡇
⠀⠀⢀⠀⡀⠀⠀⠰⢀⠀⠀⠘⡁⣤⠁⠠⡀⢀⠀⠀⠑⠤⡄⠀⡠⠎⠀⠀⠀⠀⠐⣁⣸⠨⢵⡋⠑⡀⢀⡴⠚⢯⠉
⠀⠀⠈⡉⠅⠀⠠⢂⠌⠀⠀⠀⠐⢄⠀⠀⠐⣄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠈⠉⠀⠀⠈⠉⠚
⠀⠀⠀⢰⣶⣿⣷⣷⣀⠀⠀⠀⠀⠈⠰⢤⡾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠤⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⠑⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⡈⢀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠠⠘⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⡠⢊⠆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠉⡇⠠⠀⠀⠀⠀⠀⢻⣷⠄⠀⠠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡦⢄⠤⠖⠁⠀⠀⠀⠀⠀⠠⣠⣶⣶⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠢⠄⠄⠊
⠀⠀⠀⠀⠀⠠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡡⠀⠀⠀⠀⠈⢤⡐⢀⠆
⠀⠀⠀⠀⢠⠁⠏⠀⠀⠀⠄⢀⠴⠀⠀⠀⢠⠿⢿⠿⣿⣿⡿⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠂⠐⠔⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠒⢀⠸⠈⠂⠀⠀⠀⠀⠀⠀⢂⡀⠀⡰⢀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣈⠁⢈⣤⣤⡗⠀⠀⠀⠀⠀⢠⠊⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡍⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⡆⠀⠄⣀⠀⠀⠀⠀⠀⠀⡠⢐⣤⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠐⠁⠀⠑⢩⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇⣀⣀⠀⡄⠀⠀⠀⢰⡸⠏⠟⢀⢰⠀⠀⠀⠈⡀⣬⢢⠀⠀⡠⡂⠥⠀⠠⠁
⠀⠀⠀⡁⡄⠀⠐⠒⠄⠅⠀⠀⠀⠀⠀⣠⣴⡾⠛⠻⢿⠃⠀⠀⠀⠀⠀⠃⠈⠄⢉⡀⠂⠀⠀⠀⠐⠠⢙⢠⣌⠜⢁⠠
⠀⠀⠀⠑⠈⠀⠒⠨⠈⠀⠀⠀⠀⠀⠠⡈⠛⠃⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⣀⣰⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠓⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢢⡀⠀⠀⠀⡐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠠⠀⠐⠢⢄⠀⠀⠀⠀⠀⠀⠀⠱⠀⠃⠀⠀⠀⡇⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢤⠔⠃
⠀⠀⠀⠁⢰⠓⢢⠀⠓⠀⠀⠀⠀⠀⠀⠰⠀⡂⠀⠀⠈⠳⠆⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣠⠀⢸
⠀⠀⠘⡶⠋⠀⠀⠑⢄⡀⠀⠀⠀⠀⠔⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠀⠘⠂⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⠀⠠⠀⠊⠀⠀⠀⠀⡰⢄⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⣌⠀⠸⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠊⠀⠀⠀⠔⠀⠂⠀⠄⠀⠀⠄⢀⠀⠀⠀⠀⡃⢸⠀⠀⠀⠀⡁⠇⠀⠀⠀⠀⠀⢢⡼⢖⠣⠀⠀⠀⠠⠂⠂⠀⠄⠂
⠀⠰⠁⠀⠀⠀⠜⢀⠀⠀⠀⠀⠁⢸⠀⠸⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠇⢸⠃⠀⠀⠀⠀⠀⠀⠈
⠠⠤⠄⠀⠀⠀⢀⠈⠂⠴⠀⠀⠀⠘⠀⡀⠀⠀⠀⠤⠇⢰⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠤
⠈⠁⠐⠀⠀⠤⠠⠂⠀⠀⠁⠊⠀⠸⠀⠁⠀⠀⠀⢲⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠘⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠠⠒⠀⠉⠉⠉⠉⠉⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⡀⠀⠘⠉⠀⠀⠀⠐⠦⠀⠀⢀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢠⠀⠰⡿⠃⠀⠀⠀⢐⢖⠀⠀⠈⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡌⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡁⣇⠠⡄⠀⠙⠤⠒⠀⠀⢀⡴⠀⢸⡄⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡹⢦⠙⣀⡖⠋⠒⢲⢚⠝⢁⠔⢻⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠱⣼⠀⠀⠀⢸⢁⠔⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⡀⠀⠀⡇⠀⠀⠀⢀⠁⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢣⠀⠀⠱⣄⣀⡠⠊⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡈⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢠⠇⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⠟⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢄⠀⠀⠀⠀⢠⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠱⡀⠀⠀⠀⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡜⢇⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⡄⠱⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⣠⠞⠀⠘⣆⠀⠀⠀⢱
⢠⢶⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⡇⠀⠀⡰⡇⠀⠀⠀⠈⢧⠀⠀⠸⡆
⡇⢘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢧⠀⡰⠁⣷⠀⠀⠀⠀⠀⢣⠀⠀⢇
⡇⠸⡄⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠸⡄⠀⣠⠇⠀⠀⠀⠀⠀⠀⢧⠀⠘⡄
⠹⣄⠙⠦⣄⣀⣀⣀⡼⡆⠀⠁⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢻⣫⠥⠤⡀⠀⠀⠀⠀⠀⠈⢦⠀⢳⡀
⠀⠈⠳⢤⣀⡈⠉⠀⢀⣹⡆⠀⡀⠈⠘⢤⣄⣀⣀⣀⣀⣀⣀⣀⣽⠀⠀⠀⢧⣀⣀⣪⡇⠀⠀⠀⠀⠀⠀⢳⠀⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠓⠦⠷⠤⠷⠼⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⡆⠀⣌⣫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣈⣆⣈
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⢘⣫⣵⡶⠶⠶⣾⣥⡀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⣴⣿⡿⠁⠀⠀⠀⠀⠹⣿⣦⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢈⣾⣿⣿⣿⣶⣿⣿⣿⣷⣾⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡄⣼⡿⠛⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⠀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣿⣿⣶⣿⣿⣿⣧⠀⠀⢠⣿⣿⣿⣿⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⠋⣻⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢧⣿⣿⣿⣿⣧⣤⣿⣟⣿⣿⠿⢿⣿⣿⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⢸⠟⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⡴⣰⣿⣿⣿⣧⣽⣿⢿⣟⠀⠀⢹⣿⣿⡿⠛⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⣵⣿⣿⠟⢿⣿⣿⣿⣾⣿⣷⣶⣿⣿⣿⣧⣀⣠⣿⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣾⣿⣿⠏⠉⢻⣿⣿⣿⣁⣹⣿⣿⣿⣿⣿⣏⣿⠿⠛⠛⠷
⠀⠀⠀⠀⠀⠀⠠⣦⣀⣀⣀⣄⣾⣿⣿⣿⣿⡿⠛⢿⣷⣴⣾⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢹⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡟⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣋⣹⣿⣧⣀⣀⣰⣿⣿⣿⣿⣿⣿⣇⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠮⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠁⠀⠀⠀⠈⠉⣛⣛⠿⠿⠿⠿⠟⠛⠋⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋

⠀⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⡀⠀⡠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠦⡀⠀⣠⠴⠶⠶⠶⢶⣶⣶⡶⠶⠶⠶⠶⠶⠦⡄
⠀⡇⠀⢻⣽⠓⠒⠒⢳⡀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣀⡀⠤⢤⣖⡲⠭⠭⡽⠅⡇⠀⡇⠀⠀⡠⠖⡩⠖⠒⠚⠀⠀⠀⠀⠀⠀⡇
⠀⡇⣀⠯⠼⢄⡀⠀⠀⠀⠁⠄⡀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣯⠒⠉⠉⣿⠀⠀⠀⠈⡜⠀⡇⠀⣇⠔⢋⡴⠊⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡇
⠐⣿⣁⣀⣀⣀⡧⠀⠀⠀⣀⣀⣀⣀⣀⣀⡇⠀⡇⠀⠀⠀⣫⠔⠚⠃⣿⡀⠀⠀⠀⣿⡆⡇⠀⣧⠔⠉⠀⠀⠀⠁⠈⠀⣁⣀⣀⣀⡀⣀⡇
⠀⣿⠀⠀⠀⠀⣷⠀⠀⢸⣒⠒⠒⠂⠤⠒⡇⠀⡇⠀⠒⠉⣾⠀⠀⠀⣿⠇⠀⠀⢀⣿⠆⡇⠀⡇⠀⢀⡀⠀⢀⠠⠔⠩⣗⠒⠂⠤⠤⠖⡇
⠀⣿⠀⠀⠀⠀⡟⢠⣒⠥⡚⠁⠀⠀⠀⢀⡇⠀⡇⠀⠀⢀⡥⠒⠁⠀⣿⠀⠀⠀⠀⣿⡆⡇⠀⡇⠀⠈⠀⠀⡨⢠⣖⡭⠖⠋⠀⠀⠀⠀⡇
⠀⣿⠒⠒⠒⠒⢃⡾⣒⡪⠕⡀⠀⣀⠔⠋⡇⠀⡇⠛⠉⠁⣞⠤⠤⠤⣿⣀⣀⣀⣀⡿⠁⡇⠀⡇⠀⣆⢰⠀⠣⡸⢕⣊⠥⣂⠀⢀⡴⠊⡇
⠀⣏⠓⠒⠒⠒⠋⠉⠓⠾⠭⠒⠋⠀⠀⢀⡇⠀⣇⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠂⠤⠼⢁⡇⠀⣇⠀⠀⠀⠀⠀⠉⠓⢶⡭⠕⠊⠁⠀⢀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⡞⠉⠉⢉⡩⠭⠭⠟⠛⠻⢭⡉⠉⠉⠙⡆⠀⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆
⠀⡇⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡇⠀⡇⢀⡴⠚⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡜⠉⠙⣆⠀⠀⠀⠀⠀⠀⡞⠉⠙⡆⡇
⠀⡇⢀⡇⠀⡴⢄⣼⠢⣼⠳⣄⠀⠈⡆⠀⡇⠀⡇⡞⠒⠒⠚⠤⣀⡀⠀⠀⠀⢀⣠⣄⡀⡇⠀⡇⠑⠤⠴⠃⠐⣲⢦⢤⢤⠀⠳⠤⠴⠃⡇
⠀⡇⠈⣇⡞⠀⣤⠀⠀⠀⢠⠈⠙⣆⡇⠀⡇⠀⣷⠁⠂⠀⠀⠀⠀⠈⠉⠉⠛⡇⠀⠀⢱⡇⠀⡇⡴⠋⠙⣆⢈⡉⠉⠈⢈⡀⡔⠋⠑⡄⡇
⠀⡇⠸⡉⠙⠚⠋⠉⠉⠉⠉⠛⡎⢉⡕⠀⡇⠀⡏⡆⠀⠄⠀⠀⠀⠀⠀⠀⢐⡇⠀⢠⠇⡇⠀⡇⡇⠀⠀⡏⠙⠉⠉⠉⠙⠁⡇⠀⠀⡇⡇
⠀⡇⠀⠈⢲⡀⠀⠀⠀⠀⠀⠀⡷⠋⠀⠀⡇⠀⡇⢱⡀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠔⠁⠀⡇⠀⡇⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡇
⠀⡇⠀⠀⠀⠈⢳⠶⠤⠔⡖⠉⠀⠀⠀⠀⡇⠀⡇⠀⠉⠒⠤⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⡇⠀⡇⢸⣤⣤⠇⠀⠀⠀⠀⠀⠀⢿⣤⣤⠇⡇
⠀⠧⠀⠠⣶⣭⣽⣤⣤⣤⣼⣭⣵⣦⣤⡴⠃⠀⠣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠃⠀⠳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠇