Wring Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣵⣶⣯⣶⣶⣤⣤⣀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢠⣄⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⣄⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⢿⢿⣶⠄⠐⠒⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠈⢀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠸⢇⠈⠁⠀⠐⠒⢲⠿⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⣀⠨⠅⠀⠈⢳
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣆
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢆
⠀⠀⠀⢠⡏⣀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢳⠄
⠀⠀⠀⠀⢹⠟⠛⠛⠶⣦⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡶⢶⠊⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⡸⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠎⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠈⠀⣐⡀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⢼⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠨⠆⠀⠀⠄⠀⠀⠀⠀⠀⠏⠀⠸⠄
⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠈⠒⠂⠤⣤⠄⠘⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣀⣀⣀⠀⠀⠀⠀⠈⠐⠂⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠹⡀⠀⠀⠀⠘⠓⠒⣂⣉⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠀⣠⡩⠏⢌⠙⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣝⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡋⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠒⠛⠿⣿⣌⣿⡓⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣁⣴⣿⣦⡤⠄⠀⠈⠙⠻⢿⣧⡀⠈⠳⡀
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠴⠒⠀⠉⠠⢾⠟⠛⠋⠛⢾⠀⣄⠀⠀⠀⠀⠀⠙⠳⢤⡈
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣅⠀⠀⠀⠀⠀⠛⢿⣷⣠⡄⣄⠀⠀⠀⠙⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠙⠿⣿⠯⣬⣀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣄⣶⡶⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠈⠓⠦⣍⣿⣦⣀⠀⠊⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣄⣤⡄⠀⠙⠟⢿⣦⣄⢀⡀⡀⠀⠀⠉⢻⣿⣷⣤⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⢐⣠⣠⡌⠙⠃⠻⣴⣼⢄⡀⠨⠏⠻⢾⣤⣤⣀⣀⡸⠁⠙⣷⣄
⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣷⣿⣧⣞⠻⣿⣧⣤⡀⠀⠹⢻⣿⣷⣿⣴⣚⣿⣿⣿⣿⣿⣿⠿⠛⠋⠙⢷⣄
⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣿⣇⣧⣀⣞⣛⣹⣻⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠈⠉⠭⠭⠭⠭⠿⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠙⠛⠛⠉⠉⠁

⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⢠⣀⣰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠰⠈⠁⠀⣐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⢠⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠇⠈⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡠⠋⠔⠀⠀⠀⠀⠀⠸⢀⣀⠀⠀⠀⠀⠀⠀⡲⣆⠀⠀⠀⠄⡄
⠀⠀⣤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⠀⠉⢀⢀
⠀⡎⠀⢡⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢻⣃⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠃⠀⠸⣸⠀⠈⡀⡄⠀⠀⠸⢀⡀⠀⠀⠦⣄⠀⠀⠛⠒⠵⣄⢀
⠀⢸⠀⠀⡏⡀⠀⠀⠹⣆⠀⠀⠹⡀⢀⠔⠠⠀⠑⠲⠠⠄⡠⣹⣬⠀⠈⠀⠐⡢⠄⡀⣂⣀⣂
⠀⠸⡀⠀⠁⠃⠀⢀⠀⠈⠂⠀⠀⡼⠏⢀⣀⠀⡀⠠⠤⠊⠀⠀⠌⠂⠀⠀⠀⠀⠠⠄⠀⢀⠨⠉⠉⠂
⠀⠀⢃⠀⢸⢸⢀⠏⠀⠁⠀⠱⢄⠃⠀⢀⣀⣀⣦⠠⠒⠂⠁⠌⠀⠀⣀⡠⠄⣀⠤⠒⠈⠀⠀⠀⢰
⠀⠀⠸⡄⠈⠸⡞⠠⠂⠀⠀⠀⠀⡏⠉⠉⠀⠀⠈⠐⠂⢀⠠⠀⠀⣑⠯⠒⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣷⣀⡀⡇⠀⠀⠒⠈⠒⠤⠁⠀⠉⠉⠓⠒⠲⠤⣌⣁⠀⠀⠀⠂⠀⢀⣀⣠⣤⠤⠴⠖⢒⡀
⠀⠀⠀⢡⡏⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠏⠉⠉⠀⠀⠀⠀⠀⠀⡌⠡
⠀⠀⠀⢸⠗⢻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⢠⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⡸⣇⠹⢸⠦⣬⣄⣒⣒⣒⣒⣀⡠⢄⡀⢀⣀⠀⣀⠔⢡⠂⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡇⠋⠇⣨⣀⣀⠀⠈⠉⠉⠉⠉⠉⠀⠸⡤⠔⠋⢀⠔⠁⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢣⠀⠆⠀⠐⠁⠀⠀⡄⡇⡂⠀⣨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠦⠤⣤⣤⠤⠴⠆⡇⠃⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢆⣐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⠟⣳⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⣷⣿⣧⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣄⠀⠙⠛⣹⣿⣿⡘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠁⠀⠀⠀⠘⣿⣿⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀⠔⠛⠛⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⢀⣀⣀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠁⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⢀⡀⠀⠀⢀⣄⡖⢤⠀⠀⠀⠀⢤⣈⢃⠀⠀⠀⢿⡿⠀⠀⠀⢾⡿⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⣷⡼⢦⠶⠒⠒⠿⠛⠋⢀⠀⣀⢸⡅⢀⠀⢠⣿⠃⠀⢾⣀⠈⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣒⡒⠐⠂⠀⠲⠤⠌⠁⠀⠀⠀⣨⠀⣯⠁⠉⠉⠡⠀⢀⣼⣿⣿⣷⣭⠈⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣀⣨⣿⣿⣤⣀⣄⠘⣀⣾⣿⡿⣿⣿⣿⣷⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⢶⠰⢶⡾⠟⠋⢽⣭⡄⣱⣾⣿⣿⠟⠁⠀⠉⠉⠉⠉⠉⠉⠐⠙⠈⠻⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠺⢼⢯⣴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠠⠂⠀⣼⣶⣾⣶⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣷⣦⣾⣶⣾⣷⠎⠙⠢⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢾⣿⢻⣿⣿⢉⡁⠀⠀⠀⠀⠈⠐⠃⠰⠤⠒⠀⠀⠀⠀⠀⢾⣷⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⢘⣿⣿⣾⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⢿⣋⣴⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⣹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⢈⣧⣾⠁⠙⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⣿⣿⣿⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠶⢷⣷⣶⣶⠀⠾⣤⣤⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⣿⣿⢲⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡿⢀⣼⣯⣿⡿⢰⠇⣿⡏⠀⠹⢄⠀⠀⠀⡃⠀⠀⠀⢀⢸⠺⣿⣧⣾⣿⣰⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣿⡿⣿⣿⠗⠋⣸⣿⣶⣤⣶⣦⣡⣦⣸⠁⠀⠀⢰⣿⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⣀⣰⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠟⠿⠿⣾⣿⣿⣿⣿⣼⣿⣿⠿⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠆⠐⠐⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⢛⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣯⡿⢿⣿⣿⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢛⢿⣿⣿⣾⣿⠟⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣸⠀⠛⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡌⠀⠀⠀⠀⠀⠐⢨⠟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⠀⠀⠛⣄⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⠀⠐⠀⠀⠘⣷⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠐⠤⣈⣋⠩⣍⡀
⠀⠀⠀⠀⠀⠀⣏⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡗⠄⡉⠄
⠀⠀⠀⠀⣠⡾⣋⣩⡄⠀⢌⠂⠀⠀⠀⠀⠀⠈⢹⣷⣄⠳⣄
⠀⠀⠀⢸⢿⡿⠟⠉⠁⠈⠒⡅⠀⡄⢠⠀⠀⠀⠈⣿⣿⡆⠙⣆⠀⠀⠀⠀⠘⣦
⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠹⣿⣿⡄⠸⡀⠀⠀⠀⠀⠈⠥⠀⡀
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⢿⣿⣷⡀⢱⠀⠀⠀⠀⠀⠈⣑⠺⡄⠀⠀⡤⢠
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇⣸⠀⠀⠀⠀⠘⣿⣿⣧⠈⡦⣄⠀⠀⠀⢠⣬⠑⠁⠀⠘⠀⣸⠂
⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠇⠀⠀⠀⠀⠀⢹⣿⠟⠋⠀⠈⢦⠠⠚⠁⠀⡁⢂⠀⠖⠀⠏
⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡈⠂⠀⣴⠇⠦⠈⣃⣴⠟⠁
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠖⢁⠴⠋⠀⠀⠀⢀⠀⢀⡜⠁⠀⠀⢿⡶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢠⡤⠞⠁⠀⠀⠀⠀⠀⠀⣴⣟⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣣⠀⠂⢰⡇⡇⠀⠀⠀⠀⠀⢀⡜⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣥⣿⣆⡀⠘⠃⠀⠀⠀⠀⢀⡴⠋⠀⠈⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠉⠛⢻⡷⣄⣀⡀⠀⢠⡞⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠀⠙⠉⠙⠛⠛⠋⠀⠀⠀⣀⣀⣤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢷⣶⣶⣶⣶⣶⡆⣾⣿⡿⠿⠟⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠛⠀⣀⣾⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣷⣶⣶⣽⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠉⠉⠉⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠱⠦⡕⠀⢘⠋⡙⠀⠀⠁⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠉⠁⠀⢀⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⡤⠀⠀⠀⣠⠞⠻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣾⣿⣧⣀⣀⣠⡴⠞⠁⠛⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⡈⠂⠀⠿⢧⠛⢿⠛⠉⠀⠀⠀⠀⠀⠘⠙⠲⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⡿⠀⠀⠀⠀⣴⠊⠀⠀⠀⢲⡄⠀⢠⠇⠀⠀⠈⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠾⠁⠀⠘⠃⠀⠀⠅⡀⠋⡀⡀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⡀⠀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠰⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣥⣭⣑⠶⣯⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠈⠀⢀⠛⠻⢷⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⠆⠀⠀⠀⠁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣤⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⢀⡁⠢⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡦⠀⠀⣼⠃⠀⠀⠁⢠⠁⠄⣉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠓⠤⠿⠀⠄⠀⢀⡁⠀⠀⢹⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠘⡄⠀⠈⣥⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠁
⠀⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣁⣄⠁⠀⠀⡀⠀⠀⣀⣀⣀⡀⢀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⣡⣖⣫⣍⣀⣤⣴⣤⣤⣼⣿⣯⣽⠷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣦⠾⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢾⣶⠷⠾⠷⠖⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⡾⣉⠺⣿⠟⠉⠑⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠋⠉⢀⡴⢋⣼⣿⣷⡖⣠⠆⡀⠀⠀⠀⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⢧⡴⢋⣴⣿⣿⣿⣿⠿⠷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡼⢋⣴⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀
⠀⠀⠀⠀⠀⡰⠁⣠⣠⠋⡴⠟⠁⠰⡰⡄⠀⠀⠀⠀⠀⡄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠂⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⢠⠃⢀⡽⠀⠻⡀⠀⠀⠀⢈⣥⣤⣶⣶⣾⣿⣿⠿⣵⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠎⣠⢾⣅⣀⣐⣡⣤⣶⣿⣿⣿⣿⣿⣿⠙⣿⣿⣷⣄⣀⠙⠓⢦⣤⡀⠀⠀⠀⣠⣆⣀⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⣿⣷⡿⠉⣿⠛⠋⠛⢿⠿⣿⣿⣿⣿⣿⣆⣿⡿⣿⣿⣿⣿⣷⣶⣾⣷⣾⣴⡾⢻⣿⡇⠀⣸⣿⣿⣿⠀⢻⡏⣿⣿⢹⢸
⠀⠀⢀⣾⠿⠋⠀⠀⠙⠻⣄⠑⠌⡿⠛⠻⣿⡛⠒⠤⣍⠻⢦⣀⠘⠿⣿⣿⢿⣿⡛⠻⠍⢿⣿⠀⠀⣿⣿⣿⣿⣷⣾⣸⣿⣿⠘⠈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⢧⡄⠀⠘⣧⣤⣤⠼⠀⠀⠉⠙⠛⠲⠖⠛⠛⠉⠛⠛⠚⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠉⠉⠁⠀⣸⠃
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠼⣿⣿⡿⠿⠋⠙⠻⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠰⣤⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⢀⣠⣶⣿⡆⠀⣀⣤⣶⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⣴⣿⣿⣿⣿⣿⣿⣅
⠀⣀⣤⣶⣿⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷
⠙⠻⠿⠿⠿⢿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣄⣀⣀⣀⡀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣶⣶⣆⡀⣠⣄⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡿⣷⣍⠙⠶⣿⣿⣿⣙⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣮⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠓⠶⠖⢶⣯⣝⣻⣟⢿⣿⣿⣿⣿⣿⣿⣿⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⢿⣿⣿⡿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠦⣙⠛⢿⡿⣿⣇⣻⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣮⢉⠲⣿⡜⣿⣿⣿⣿⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠂⠘⢿⡜⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣻⣿⣿⣆⡙⣿⡜⣿⣿⢸⣿⣿⠟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⡻⢿⡜⣿⣿⡟⠋⣼⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⣿⣿⣿⣿⣶⣿⡝⣿⣵⡶⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⢿⣷⡼⣿⡇⠸⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡻⣷⣤⣤⣬⣤⣈⣷⡹⣿⡇⢸⡟⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣛⣛⠛⢻⣿⣷⣹⣿⡿⠁⠀⣸⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡿⣿⣿⣿⣿⣿⠿⣿⣧⢹⣿⡗⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣶⣾⠟⠛⢶⣿⣧⢻⣷⣿⠏⠼⢿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣯⣅⣀⠻⢿⣧⢿⣯⡤⣶⣿⣿⡿⠃
⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⣠⡎⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣯⣙⢿⣯⣿⣷⠺⣿⡿⠛
⠀⠀⠀⣰⣿⣿⣷⣆⣄⢀⣠⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⣙⣿⣿⣾⡻⠁
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣟⣴⣦⣤⡀⠀⣸⡿⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠉⠙⢻⣏⣻⣟⣿⡅
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⡟⠁⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⡙⣿⠃
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⢰⣿⣿⣿⣿⠟⡛⢻⣿⣿⣿⣿⣿⣿⣾⣿⢇⢀⡟⠀⠀⠴⣄⡀⠀⠀⠀⠀⠀⢀⡀⠠⢄⡀
⠘⠿⢿⣿⣏⠀⢒⣠⡟⠡⣀⠙⣿⣿⠿⠉⢞⣼⠷⠀⠀⠀⠈⠙⠷⠆⠀⠀⠀⠪⡵⠐⠋⠀⠀⢀⡙⠦⢄⡀⠀⠀⠀⠁⡀
⠀⠀⠀⠠⠬⣙⣛⡻⠷⣤⣤⣼⣿⣿⣶⠞⠉⠀⠹⠓⢤⢀⠀⠀⠀⠀⠀⠀⠀⡴⠤⠄⠀⣀⠀⠀⠘⠣⣶⡀⠀⠀⢠⠀⠃
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⡝⣀⠃⠀⠀⠀⠀⠀⠀⠈⠑⠐⠢⠄⠀⠀⠈⠇⠀⠀⢀⡙⠁⠀⠀⠀⠈⠉⠶⣄⡀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠰⠀⠀⢀⣴⠀⠀⡸⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⢰⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣬⣧⣴⣧⣴⣦⣴⣿⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠦⠲⠶⠳⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠙⠃⣠⠾⡾⣋⢨⢈⣤⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣘⣡⣄⣠⡄⠉⠉⠻⡅⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⢃⠀⠈⠙⠳⠃⠦⡀⠄⣀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠰⠛⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡤⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠤⠒⠁⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠙⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠉⡀⠸⢄⣀⡴⠋⠁⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀⡴⠃⠀⠀⠐⠀⠀⠘⠁⠀⠀⠀⡠⠜⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⣀⡴⣶⠟⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢁⠃⠀⠀⠀⠀⢀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⢀⣠⠆⠑⣻⡿⠡⡀⠀⣜⠳⠒⠶⠶⠦⠤⠀⢀⣰⠇⢰⠀⠀⠀⠂⠀⠀⠈⠀⠀⠁⢠⡀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⣶⢡⡁⠀⠀⠁⠀⠀⠀⠙⢣⡤⣀⠀⠀⠁⠀⠀⠀⠉⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠟⢦⡀⠄⠀⠠⠀⢳
⣀⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢠⡉⠢⡀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠲⢤⡀⠄⠁⣠⡾⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⢀⡏⠀⠀⠑⢄
⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠊⠉⢦⡈⢂⡀⠀⠀⠀⣀⠐⡏⠀⠀⠀⢈⣵⠿⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠈⡇⠀⠀⠐
⠀⢸⡟⠿⣦⡀⠀⠀⠀⢀⠀⠀⠀⠱⡄⠀⠱⣄⣳⣄⠴⣋⡤⠖⠁⣀⣤⠾⠋⣁⣴⠋⠈⠁⠀⠈⠀⠀⠈⠀⠀⠤⡴⠀⠀⠀⠀⠀⡼⠃
⠀⢸⡇⠀⠈⠛⢷⣄⠀⠘⡄⡆⠀⠀⠘⣆⠀⣀⠤⠒⠉⠁⣠⣴⠞⠛⠛⠉⠉⠉⠈⠳⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⢀⠀⢈⡼
⠀⢸⡇⠀⠀⠀⠀⠙⠳⣦⡁⠸⠆⠀⠀⠈⠋⠁⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⣠⠶⠃⠀⠀⠙⠓⠦⣄⣀⡀⠀⠀⢀⡇⠀⡀⠠⢏⣀⠀⠀⣠⠞
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣀⣴⠾⠋⠁⠀⠀⡀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣸⠟⠳⣏⢰⠆⠀⡸⠖⠋⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠟⠋⢹⠀⠀⠀⠀⠲⣼⣶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠁⠀⢀⣿⠁⢀⡴⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⢀⣀⣤⣶⢮⣭⣤⣤⣤⣤⡤⢶⣿⠈⠉⠉
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠼⡟⠋⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠸⡄⠀⣿⡇⠀⠀⠀⠀⠀⣿⠃⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡏⠹⡄⠀⠀⠀⠀⠀⢳⠀⣿⡇⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠹⡄⠀⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⣸⡇⠀⠀⠹⡄⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢀⣿⡀⠀⠘⣿
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⣿⣀⣤⡴⠿⠛⠉⠉⢻⡏⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣀⣀⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⣠⣤⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⢀⡼⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠛⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢀⣤⡶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠾⠟⠛⠛⠻⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢀⣾⠋⠀⠀⠀⠈⣹⣿⣆⠀⠀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⢠⡿⠉⠻⣷⣄⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⠇⠀⠀⠀⠀⣼⡇⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⣿⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⣷⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠘⢿⣦⡀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣠⣴⡾⠛⠁⣴⠏⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡉⠙⠛⠛⠛⠛⠉⠁⣀⣤⠾⠃⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣶⣶⣶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣤⡀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⣿⡏⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠃⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣀⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣶⣤⣤⣤⣶⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢑⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀⡼⢁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⢀⠎⠀⠋
⢀⡴⠂⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠒⠒⠒⠒⠲⣄⡴⠒⠒⠒⠒⠂⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠐⢦⡀⢀⡼⠁⠀⡠⠃
⢸⠀⣴⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠒⠂⠀⠐⣻⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠂⠒⢲⢂⠀⣧⠞⠀⢠⡞
⢸⠀⡇⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠠⠆⢀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣼⢾⡴⠋⠀⣴⢋⡴
⢸⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⣒⡚⠋⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⡀⣸⠞⠁⢀⡾⡱⠋
⢸⡄⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠤⠤⠤⣴⠏⠀⣠⡏⠀⠋
⢸⡇⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⡾⠁⠀⡼⠋⠇
⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⡀⠀⠀⠀⠀⣠⠊⠀⢀⢼⡇⠀⡇
⢸⡇⠸⡇⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠅⠀⡠⠃⢸⠇⠀⡇
⢸⡇⢠⡁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢀⡶⠀⠀⣴⠄⠠⡇
⢸⡇⠠⣧⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠖⠋⠀⠀⠀⣻⡇⠘⠇
⢸⡇⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣿⠂⢠⡆
⢸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇
⢸⡃⠈⣦⡀⠀⠀⣀⣀⣀⣀⢀⣤⡄⠠⣤⠄⠠⠤⣼⣁⣤⠄⠠⣤⠀⠀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣿⠆⠈⠃
⠀⠳⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⡤⠎⠱⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠜




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣨⣏⡙⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠉⠉⢿⣿⣿⣿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠁⠀⣸⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⠔⠀⠀⠀⠀⢸⠋⠁⠈⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠀⠀⠀⠠⠘⠀⠀⡿⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣤⡄⠀⠀⣰⠃⢰⠀⠀⠀⣈⢓⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⣿⠃⠀⠀⠁⠀⢸⠀⣠⠊⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠚⢘⣼⡟⠀⠄⠀⡤⠔⠛⡾⠁⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⡃⠈⠀⢀⠑⠀⢠⣧⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣄⣾⡿⠁⢠⠖⠁⢀⠀⠻⣇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠂⢬⣿⣿⡿⣇⠰⣭⡀⠘⣿⣧⣤⣿⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⣀⣄⣰⣿⣿⣿⡿⠀⠀⠀⢀⠿⣷⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢹⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣧⣀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢟⠂⣀⠄⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠈⡐⣌⠃⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⣤⣀⠐⢌⡉⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠈⠒⠄⡀⠀⠀⠀⡠⠐⢉⠀⠀⡈⣁⣠⣾⣿⡟⠃
⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⣠⠄⡀⠈⠐⢄⡞⢀⠴⢪⣭⣿⣻⡝⣻⠉⠀⠈⠓⢄
⠀⡀⢀⠤⠒⠉⠀⠀⠀⢠⠄⠀⠠⠐⡈⠄⠊⠠⠔⠀⣈⣤⡾⣾⡟⢫⠅⡣⠨⠀⠀⠀⠀⠀⠀⠑⢢⢠⣤⣄
⠀⠈⠐⢤⡀⠀⠀⠐⠊⠀⠂⢁⠀⠊⢠⠄⢉⠠⠴⠟⠻⢄⣐⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿
⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⣁⠔⢁⡄⠊⠁⠀⠀⠀⠈⠉⠉⠉⠑⠒⠒⠶⠤⢤⣀⡀⣀⣀⣀⠀⢠⠃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡽⠯⠉⠉⠹⣿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠹⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⣀⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠠⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢐⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢔⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠒⠶⢤⣄⡀⣠⠏⢀⣪⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⡿⠁⡠⣵⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⢔⡟⠁⠉⠙⠛⠷⠶⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣪⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡰⡽⠃
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⠎⢀⣼⡞⠁
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⡤⠔⠊⠁⠄⠀⠀⡴⠋⣠⣿⢟
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⢀⡜⠁⣜⡵⠃⠈⠆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠠⢢⠎⢀⢮⠞⠀⠀⠀⢨
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢠⠃⠀⠂⡰⠁⡠⣳⠋⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠛⠛⠛⠛⠛⠛⠻⠷⠶
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⡄⢀⠜⠀⣔⣽⠷⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⢊⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⣶⣶⡶⠾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣿⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⡹⠶⠊⢿⡀⠀⠀⠀⣠⣂⣤⠴⠖⠿⡅⠀⠙⢷
⠀⠀⠀⠀⠀⢀⠞⠁⠀⣠⠟⢻⡖⠂⠉⠁⠀⡧⠀⠀⠀⢹⡆⡀⢸⡇
⠀⠀⠀⠀⠀⡎⠀⢂⡴⠃⠀⠀⠹⣄⠀⠀⠀⣧⡀⠀⠀⠀⣧⣤⡾⠁
⠀⠀⠀⠀⢸⣶⡴⠋⠀⠀⠀⠀⠀⠈⠳⠶⠞⠋⠙⠳⠶⠞⠃
⠀⠀⠀⠀⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠓⠚⠋⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⡀⣀⣀⡀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣠⠞⠉⠉⠉⠀⠉⢳⡄⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡏⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡇⠀⠚⠂⠀⠈⠉⠁⠀⢳⣀⣀⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⡇⠀⠐⠃⠀⠀⠘⠀⠀⠈⢩⠄⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣷⠀⠀⠐⠂⠀⠀⠀⠀⣠⣤⡴⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠁⠉⠀⢀⡴⠃⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣶⠶⠶⠚⠹⣇⣤⣼⣤⣤⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠹⢤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠓⠦⣄⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⣠⠞⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡆⠀⠀⠀⠙⢦
⠀⠀⢀⡤⠞⠁⠀⠀⢀⣤⠞⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⢦⡀⠀⠀⠈⢷
⠀⡴⠋⠀⠀⠀⣠⠞⠻⢦⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡧⠴⠚⡧⠀⠀⠀⠘⡇
⡼⠁⠀⠀⢀⠀⡏⠀⠀⠀⠑⠫⢷⣦⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⢠⡇⠀⠀⠀⠀⠹⡄
⡇⠀⠀⡆⢸⠦⠃⠀⠀⠀⠀⠀⠀⠉⠻⢕⠢⢤⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⢸⣁⣿⠀⠄⠀⠀⢻
⠙⠦⢤⡥⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣽⢰⠀⡇⣀⡼
⠀⠀⠀⠀⠈⠉⠓⠲⢤⣤⣤⣤⣄⣀⣀⣀⣤⠼⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⣿⠙⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⣀⣠⢴⣾⣭⣤⣤⣤⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⢀⣀⡤⠴⠖⠛⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⢑⣾⣿⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠠⠛⢤⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⠀⠀⠀⠂⠈⠈⠀⠀⠀⠀⢂⠀⠀⠊⠐⢀⠞⣹⠋
⠀⠀⠀⠀⠀⠰⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠖⡠⠀⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠠⡤⠀⠀⢀⡀⣀⣀⠤⠐⠐⠂⠒⠀⠉⠁⠀⠀⠀⠀⡀⠀⠄⡠⠞⡡⠊
⠀⠀⠀⠀⠀⠐⡇⠀⠀⠈⢀⣀⣀⢠⠤⠤⠐⠒⠀⠈⠁⠀⠀⠀⠈⡠⣠⠔⢁⠶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠤⠄⠀⠐⠀⠉⠙⢁⠈⣴⠃⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠉⠀⠀⠀⢀⣀⠀⠀⠀⠂⠐⢩⠎⣻⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠂⠉⠉⠀⠀⠀⡀⠀⢂⡰⢃⡔⠃⠀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠁⠁⠀⡀⠉⠃⢎⡷⠎⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢧⠀⠈⠉⠉⠉⠈⠀⠀⣠⡤⠴⠴⠶⠖⠂⠒⣘⣥⣤⢤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⢄⢠⠄⣒⣂⣩⣥⡤⢴⡖⠒⠚⠋⠙⣛⣋⣉⣭⣭⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⣷⠿⠛⠩⠕⢂⠆⣠⡤⢶⡖⠛⠛⠛⠿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣴⠶⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⡀⠀⡀⠀⢀⣀⣄⢄⠄⠠⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠉⠀⠀⠀⠀⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣠⠔⠒⠊⠡⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠚⣁⠀⠀⣘⠀⢃⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡃⠀⠀⢸⠀⠀⠀⢀⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠻⡜⣣⡟⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠈⠀⠀⠀⠇⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⣤⠀⠀⠀⠀⠶⠀⣬⠀⢷⠆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠉⠉⢸⠀⢠⠀⠀⠀⢃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⣠⣀⣼⡋⠀⠀⠀⠀⢀⠴⠶⢠⣄⠀⢸⠀⠀⠀⠀⠈⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠁⠚⠉⠑⠶⠖⠋⠀⠀⠀⠀⡈⠑⠺⣄⢀⡀⢀⠀⡸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡤⠔⠊⠀⠀⠀⠀⢰⠚⠉⠉⠲⠄⢈⣉⣿⣉⠁⣡⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢀⡀⠀⠀⠀⠈⠠⠄⠠⣤⠄⣀⡈⠀⠀⠀⢸⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠒⣀⣀⣀⠒⠉⠀⠀⠀⠀⠙⠉⢛⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠡⠉⡍⢩⠌⡗⠶⢲⡟⣼⢧⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⡆⠷⠴⠦⠣⡾⢶⠃⠶⠸⠧⠶⠰⠟⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢰⡆⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠘⠀⠀⠀⠸⠂⠀⠀⠀⠀⠑⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠎⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡰⠾⠦⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣄⢀⣀⣀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠉⠀⠀⠀⠈⢡⣼⡧⠐⠒⠒⠒⢄⢀⡜⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⡏⠀⠁⠀⠀⠉⠁⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⢀⣀⣀⡄⠀⠘⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⣀⠀⠀⠀⠉⣶⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠙⠀⠀⠀⢠⠛⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⠘⠀⢀⡈⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠄⠒⠉⠀⠀⠠⠀⠠⠔⠊


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠁⢀⠀⣿⠻⡉⠛⢹⠟⠛⡉⡟⢫⠉⠙⠋⢙⠋⡿⠛⠋⠙⢉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⣠⡔
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠦⣤⣤⣠⠞⠛⡅
⠀⠀⠀⠠⣄⣎⡾⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⠒⠛⠛⠃⠀⠀⠱⡀
⠀⠀⠀⠠⠋⠠⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠌⣤⡤⣄⡀⠀⠀⠀⠑⢄⢀
⠀⠀⠀⢤⡄⢶⢹⡙⣿⡷⠤⠔⠂⠒⠐⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠘⢙⢆⠀⠀⡀⠠⠁⠄
⠀⠀⠀⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⢧⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧

⠀⠀⠀⠀⡀⣠⠆⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠙⡁⠀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠈⠉⢉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁⠉
⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠀⠓⠚⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠴⠶⠲⠰⠖⠶⠶⠀⠀⠒⠲⠖⠒⠓⠒⠒⠖⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠑⠀⠀⠀⠈⠁⠀⠉⠉⠋⠀⠁⠘⠋⠃
⠀⠀⣠⣄⠀⡀⢠⢀⣤⣤⡄⠤⠾⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠲⠲⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⢀⣠⢀⣀⣀⣋⣀⠀⠀⢀⣀⣤⣄⢀⣀⠀⠀⠀⠠⠀⠄⢠⠀⢤⣀⡀⠀⠀⠄⠄⠤⠀⠠⠤⠄⠄⠠
⠀⠀⠀⠈⠀⠁⠈⠀⠀⠁⠈⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢶⢤⠠⠂
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠄⠀⠀⠀⠀⢀⡀⢀⠂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠐⠷⣞⣊⠉⠁⠉⠀⠈⠉⠐⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠉⠉⠉⠀⠉⣩⠆⠉⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠈⠳⣶⡟⠃⠓⠐⠐⠒⠐⠒⠘⠓⠀⠀⠀⠀⡀⠀⠁⠉⠀⠈⠁⠀⠀⠀⠈⠀⠀⠨⠁⠀⠀⠐⠀⠐⠉⠉⠉⠈⠀⠈
⠀⠀⠀⠠⠴⢤⣶⠴⠤⠤⠤⠤⠤⠢⠤⣤⠀⠶⠂⠒⠠⠴⠦⠀⠀⠀⠠⠴⠲⠐⠀⡀⠠⠖⠊⠁⠂⠂⠀⠉⠈⢰⣶⣶⠐⠂⠂
⠀⠀⠀⠀⠤⠀⠦⠤⠀⠀⠀⠀⠀⠀⢴⡿⠴⠄⠀⠀⠀⠰⣦⠦⠤⠠⠄⠀⠀⢆⠚⠃⠀⠀⠀⠀⢂⡶⠀⠐⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣂⣴⣤⣦⠄⠀⣠⢦⢤⣤⡄⣄⣄⡤⣼⣆⡀⠀⠀⠤⠶⠿⠶⠷⢶⡤⠤⠄⠈⠁⠀⠠⡒⠀⠄⡴⠧
⠀⠀⠀⢀⢤⣀⡀⡀⡉⠈⠈⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡷⠀⠀⢀⣠⣤⠤⣴⣤⠤
⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⣠⠀⢀⡀⢀⠀⠁⠀⠀⠀⢀⣀⠀⢀⡀⣀⢀⣀⣀⣀⣀⣰⣏⣈⢡⡄⢀⡀
⠀⠀⠀⠀⠀⠿⢿⡿⠇⠀⠀⠈⠁⠀⠑⠛⢛⣙⠃⠉⡉⠛⠙⠻⠿⢋⣩⠙⠉⠉⣡⣀⣀⣀⣀⡀⢀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠈⠐⠐⢈⠀⠀⠐⠂⠀⠒⢿⡇⠾⠛⠛⠟⠟⠋⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⡉⠉⠉⠁⠀⣀⡄⠀⠀⠁⡀
⠀⠀⠀⠒⠁⠀⠋⠛⠚⠚⠓⠒⠀⠀⢀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠘⠑⠈⠐⢀⣔⠀⠐⠃⠀⢀⡈⢡⠇⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⢠⠚⠓⠀⠀⠂⠀⠀⠂⠻⠷⠆⠀⠰⠟⠚⠓⠀⠂⠀⠈⠁
⠀⠀⠀⠠⠤⢴⡏⠀⠀⣠⣤⠀⠈⠀⠠⢠⠾⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⢀⡀⣤⠄⢠⠄⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠐⢂⣰⣄⡠⠄⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣴⢾⣍⣉⣉⣉⢐⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢉⣤⠴⢿⣄⠑⢌⠳⢤⡉⠙⠳⢤⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠎⡠⣚⠥⠒⠊⠙⢻⢦⠀⠙⢤⡉⠳⣄⠀⠉⠢⡈⢢⡀
⠀⠀⠀⠀⠀⢀⠔⣡⢊⠞⠁⠀⠀⠀⡴⠋⢻⡳⣄⠀⠙⠢⣄⠉⠒⠤⢝⣦⢑⡄
⠀⠀⠀⠀⢠⢫⢞⡵⠋⠀⠀⢀⡤⠊⠀⠀⠀⠙⢮⡳⣄⡀⠀⠉⠛⠒⠶⠤⢭⣍⣶
⠀⠀⠀⠀⣹⣧⣊⣀⡠⢤⡖⠋⠀⠀⠀⠀⣃⠀⡐⠉⠲⠭⣓⡒⠠⠤⠤⢤⣤⡶⣏⢦
⠀⠀⠀⣰⡏⠀⡜⠁⠀⢸⠇⠀⠀⠀⠀⠀⠩⠠⠁⠀⠀⠀⠀⠀⠉⠉⡟⠉⠈⢇⠈⣮⢧
⠀⠀⢠⢿⡆⢠⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢘⡄⢸⡜⡂
⠀⠀⣼⢸⠇⢸⠀⠀⠀⡻⠀⠀⠀⠀⠀⠀⠓⠖⠋⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⢸⡇⢣
⠀⠀⡇⡾⠀⣸⠀⠀⢐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠈⣷⢸
⠀⠀⣷⢃⡴⠹⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⣧⡀⢻⣄⡂
⠀⠀⢹⡏⠀⠀⣇⠀⡞⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⡇⠀⢰⠁⠙⣮⡏⠁
⠀⠀⠸⣇⠀⢰⠉⣦⡇⠀⠀⠈⠑⡶⠠⠤⠤⠴⠶⠤⠄⢲⠊⠉⠀⠀⠀⠇⢠⠋⡄⠀⢻
⠀⠀⣼⣸⠀⡌⢠⠁⣠⠖⠉⠉⠉⡟⠒⠒⠠⠤⠤⠐⠒⢺⠉⠈⠉⠳⣄⠘⠻⠈⡇⠀⡛⡀
⠀⠀⠀⢯⡼⢡⠇⡰⠃⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠘⢆⠀⢇⢹⡀⣷⠇
⠀⠀⠀⠀⠉⠋⡺⠁⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⡇⠙⠢⠟⠋
⠀⠀⠀⠀⠀⢰⠉⠉⠉⠒⠒⠿⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢹⡖⠈⠉⠉⠱⡀
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⢩⠟⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⣉⣳⣀⡠⠤⠔⠖⠚⣆⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣏⣽⠓⠢⢄⠀⠀⠠⢤⡽⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠋⣤⣾⡀⠀⠀⠑⢄⣹⠚⠙⣆⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⡠⠊⠁⠹⣿⠀⢾⣿⣻⡄⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⣀⣰⣃⠴⡄⣤⠐⣦⡿⠋⠐⠒⠒⠒⠒⠂⡶
⠀⣠⠊⠀⠀⠀⠀⢿⡄⠰⣿⣿⡟⣄⠀⠀⠀⠀⠀⠀⠈⣳⡉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡘⢩⡏⢩⡏⠉⠉⠙⢮⣦⣉⣤⢧⡬⢆⠀⠀⠀⠀⠀⢀⡠⠿⢖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⠒⡶⠒⢰⣿⣇
⡇⠠⠀⢸⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⢷⠂⣀⡴⠚⠉⠀⣠⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣿⣿
⢣⠐⡆⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⢀⣠⣖⣯⢿⡿⢎⡷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣿⡇
⢸⠉⡟⠉⡟⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠊⠓⣶⠛⣿⠯⠒⠉⣠⣾⠟⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠛⠒⢺⣿⣏
⢠⠰⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⡶⢶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠸⡀⣃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⢰⠉⡛⠉⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿⣿
⢸⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠘⡄⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡁⠀⠀⠀⠉⠉⢛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆⠀⣀⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⢶⡆⠀⠀⢲⣶⣿⣶⣶⣶⣶⣶⣶⣶⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡀⠀⠀⣲
⠈⠳⣦⣶⣾⣿⣿⣿⣿⡛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣿⡿⠟⠓⢲⡶⠋
⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣶⣶⣤⣄⣀⡀⠀⠀⠀⣀⣠⣤⣶⣾⣷⣦⣀⠀⠀⠀⠀⢀⣀⣤⣤⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠾⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⡷⠀⣠⣴⡆⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠰⣏⠙⡆⠀⣉⠜⠁⣼⣿⢡⠇⠀⠀⢀⣀⣀⠸⠷⢘⡆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢰⣷⠈⣿⣧⠀⣀⡈⠉⡟⣩⣵⢦⣸⣿⡷⣣⣴⠦⡄⣏⣤⡈⢧⣠⣞⢀⡴⣿⣷⡴⢻⣿⡇⢀⡀
⠀⠀⠀⠈⠛⠀⣿⣿⠀⣿⣷⢀⡇⣿⡟⠉⣿⡿⢰⣿⡏⠘⢃⣿⣿⠓⣻⢁⣿⠏⢰⣿⠋⣠⣿⡟⣼⡿⠿⠂
⠀⠀⠀⠀⠀⠀⢻⣿⣆⣿⣿⡾⠀⠙⠓⠊⣿⡇⠈⠻⠷⠖⠉⢿⣏⣠⠃⠘⠿⠀⠈⠉⢠⣿⡟⠀⢿⣇⣴⠆
⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠉⠁⠀⠀⠀⠀⠹⣷⠰⠟⡆⠀⠀⠀⠈⡴⢋⣩⡇⠀⢀⣠⠞⢿⣉⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠀⢷⣈⣫⠥⠖⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡠⠀⢠⠀⢠⠀⢠⠀⡄⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡋⠁⠈⠁⠀⠁⠀⠀⠀⠈⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⢁⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢠⠼⠞⣫⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⠿⣫⣤⡞⣀
⠀⠀⠐⠒⠶⠶⠶⠦⣾⣰⣫⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠾⢋⡥⢊⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠚⠒⠒⠒⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣡⡶⠛⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣦⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣾⣿⣶⣿⡿⣿⣿⠟⢋⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣴⣿⡟⢹⣷⣾⠃⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⡿⣿⣤⣾⣿⣿⣿⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢉⣴⣿⣿⣿⣿⣿⡇⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⠏⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣤⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣫⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣏⣴⣿⣿⣿⣿⠟⠁⢸⣿⣿⣿⣿⣿⡇⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣽⣾⣿⣿⣿⡟⠁⠀⠀⢨⣿⣿⣿⣿⣿⡗⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡷⣹⡇
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⣫⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣧⣤⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣸⣷⣿
⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣿⣽⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣯⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣿⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢹⡋⣿⣿⢾⡇⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣷⣿⣿⣻⡇⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠘⣿⣿⣿⡇⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠙⣿⠀⠀⢸⣿⣿⠁⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⢿⡀⠀⢈⣿⣿⣧⡀⣿⡛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠯⣴⣶⣶⣿⣟⣹⡿⣿⣟⣹⣷⣽⡇⠈⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣴⣶⣿⣯⣶⣾⣿⣿⣤⣾⣿⣽⣿⣿⣷⣤⣤⣬⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣉⣹⣟⣿⣿⣿⣿⣿⣶⣶⡿⠿⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⣷⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠉⠙⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣶⣶⣦⣾⣿⣦⣾⣿⣧⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⠿⠛⠻⠿⠟⠿⠿⠻⣿⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠐⠲⠶⠂⠀⠀⠀⠀⠀⢠⣤⡀⠀⠘⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⡟⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⣾⠄⠀⠀⠀⡅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡏⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠲⣦⠉⢀⠀⠀⢠⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠁⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠐⢠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⠠⠤⠤⠀⠀⠀⠙⠀⢠⣤⠃⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣄⡀⣤⠀⠀⠀⠀⠀⠀⠀⠰⠆
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠉⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠈⠉⠁⠈⠁⠉⠁⠉⠁⠈⠀⠉⠉⠉⠉⠀⠀⠀⠀⠛⠀⢀⡉⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠄⢠⡄⠉⢠⣴⣤⣀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣾⣿⣿⣽⠛⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⢻⡞⠉⠉⠀⠉⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⢀⡄⠐⠀⠀⠀⠠⠄⠀⠈⠒⠒⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠅
⣀⣀⣠⣄⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⡈⠑⢄⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⢀⢀⡀⠀⢀⡤⠀⠀⠑⠄⠂⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⡄⠀⢠⣠⠐⠒⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠃⢈⠉⠻⠿⠁⠀⢀⡀⢠⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠛⠀⠀⠀⠀⠦⡕⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠡⡀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⣀⣀⣀⡀⠈⠀⠀⢾⠀⠅⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢀⠀⠀⠄⢠⡗⠤⣠⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠢⠀⢆⠣⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠘⠠⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⠀⢜⠸⣤⣀⠀⠀⠀⠘⠸⠇
⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠑⠀⠀⠀⠀⠁⢢⠀⡀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⡀⠀⡇
⠀⢠⡜⠙⠘⢄⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⢇⣿⠤⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁
⠀⢸⡆⠀⠀⡇⢳⠀⠀⠀⡇⠀⠀⠀⠀⠀⢢⢸⠀⠀⠀⠀⠈⠷⠠⠤⣄⠈⠢⣄⠀⠀⢠
⠀⠈⡇⠂⠀⢹⡸⡆⠀⠀⡇⠰⡀⠀⠀⠀⠈⠆⠀⡀⠀⠀⠀⢸⠦⣄⠀⠀⠀⠈⠋⠑⠦⢷⡀
⠀⠀⣇⢸⠀⠘⡇⡇⠀⠀⢡⠐⢿⣄⠀⠠⠀⠈⢎⠁⡠⠂⢀⠄⢀⡠⠓⢦⣄⡂⠀⠀⠀⠀⢻⡍⠋⠐⠒⠒⡀⠀⢤⢀⠀⠀⢀⠴⠀⠀⠄
⠀⠀⢻⠘⠀⠀⣇⢸⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠈⢧⡀⣴⠡⠒⠁⠀⠀⠀⠀⠉⠉⢁⠖⠛⠚⠛⢍⠀⠐⠒⠂⠀⠔⠋⠓⠴⡒⠒⠦⠤⠦⢤⣤⣤⡀
⠀⠀⢸⡀⠄⠇⢹⠸⡄⠀⢠⡇⠀⠀⠈⢣⡀⠀⠀⣰⠯⠇⢤⣀⣀⣠⣤⠤⠀⡶⠖⠁⠀⠀⠀⠖⠌⣃⠀⠀⠀⠀⠀⢀⠤⠶⠍⡀⠀⣀⡠⠖⠀⠀⠞⠃
⠀⠀⠀⣇⠀⠀⢸⠀⡇⠀⣸⠉⡁⠒⠒⠒⠓⣄⠀⠷⠀⠀⠀⢀⠀⠀⢠⡄⠀⡴⠤⠤⠒⠃⡚⠁⠀⠀⠀⢀⡀⢄⡨⡀⠄⣂⡥⠖⠋⠀⠀⠀⠀⢠
⠀⠀⠀⠸⡀⠀⠀⡇⣿⢠⠇⠀⠀⠈⠉⠀⠀⠈⠻⣦⠤⠤⠴⠷⠒⠛⢛⣗⡈⠀⠀⠂⠠⠄⢉⡑⠄⠤⣚⠵⠚⣉⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢧⠀⠀⠇⢸⡏⢀⠔⠁⠀⠀⠀⠀⠀⠀⡟⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠚⡁⠄⠒⡀⢈⣉⠵⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣧⠀⠀⢸⡇⠄⠀⣀⠤⠔⠒⠒⠤⣀⠁⠀⠉⠉⠙⠓⠲⠶⠤⣤⣄⣀⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠄⠁
⠀⠀⠀⠀⠸⡏⣷⠼⡌⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⠑⠒⠲⠦⠄⣄⠠⠤⠶⠖⠒⠛⠋⠉⠉
⠀⠀⠀⠀⠀⡇⡇⠀⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⡇⣷⠚⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⣅⠠⣿⢸⡀⠀⠤⢤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢠⠃⠁⠀⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⢠⡇⣿⠀⠿⢸⠿⠿⠷⢶⣴⣄⣤⣤⣤⣭⣭⢭⠼⠲⢤⣀⠀⠦⣤⠀⢀⡤⠚⠁⣰⠃⠀⡀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠐⠋⠹⠀⢨⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠨⣗⣀⡤⠞⠉⠀⣠⠞⠁⠀⠸⡇⢀⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠀⡇⠰⠋⠉⠉⠓⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⣶⠒⠂⢈⠁⠀⠀⣀⠞⠁⠀⠀⠀⠀⡇⢸⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠸⠀⠀⠀⠁⠀⠀⠀⠀⡞⢠⡇⢸⢀⣀⠤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠺⣤⣀⣀⣀⣀⣀⣀⣰⠷⠘⡇⠟⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⠀⡇⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠀⠈⠉⠉⠉⠉⢍⠀⠒⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⡤⣤⠄⢀⡀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠐⠂⣠⠤⢤⣤⠤⠀⢤⠀⠀⢶⣀⠈⢆⠀⢣⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠉⠒⠂⠀⠀⠁⠀⡄⠀⠀⡀⡀⠁⠀⢀⣀⠀⠀⠀⠁⣂⠤⣵⠌⠀⠓⠮⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣵⠀⠀⠀⠀⠀⠀⠀⠰⠣⡄⡏⠃⠀⡔⠁⠀⠙⡄⠀⠈⠁⠀⠈⠂⠀⠀⠀⢀⣰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠀⠀⠱⢿⣆⠔⠀⠀⠘⠿⠆⠔⠁⠀⠀⠀⠘⢡⠀⡔⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢦⠐⠒⠀⠐⠂⠡⡈⠆⢇⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⢠⠀⠀⡂⢁⠜⠀⠀⠀⠀⠀⠀⠁⢐⣄⡀
⠀⠀⠀⢀⣀⠀⠀⠀⠤⠤⠦⡀⠀⠠⠀⠆⠀⠀⠀⠀⠀⠉⠂⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢉⠀⡰⠉⠀⠀⡀⠂⠀⡠⠂⣈⠄⠀⠀⠈⠑⢄
⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⣂⠀⠡⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠈⠀⠒⠒⠀⠉⠀⠀⠀⠁⠔⠃⠀⠀⠈⠉⠉⠉⠁⠈⠉⠈⠉⠉⠉⠉⠉⠂⡀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠁⠒⠷⣵⣀⡀⠀⠀⠀⠀⡀⠄⠔⢊⠳⣄⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⢄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠐⠂⠤⢀⣑⠈⡆⠈⠉⠈⠰⠆⠉⠀⠀⣤⡨⠶⠤⠤⠤⠒⠉⠉⠉⠁⠒⠒⠠⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠁⠒⠒⠒⢄⠀⢀⠂⠀⠀⠠⠄⠀⠀⡠⣿⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠀⢀⠠⠀⠂⢀⣉⠉⠉⠉⠉⠉⣛⡒⠲⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⡖⠁⠐⢼⠀⣀⣀⡀⠀⡤⠔⠐⠒⠒⠀⠉⠉⠀⠀⠁⠒⢤⣠⣤⣤⣤⣤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠠⠆⠀⠀⠀⠀⠀⠈⢁⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡍⣒⠀⣈⣉⡀⣀⣀⣀⠤⠤⠤⠔⠒⠒⠊⠉⠙⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠚⠷⠤⠤⠄⢀⡀⠘⣑⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠃⠀⠀⠀⠀⢀⣀⠀⣀⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠉⠈⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠀⠀⠀⠀⠀⠀⠀⠉⢃⢠⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠐⠉⠂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠀⠀⠀⠠
⠀⠀⠀⢀⠮⠀⢀⢀⢀⠄⠀⠤⢠⡄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⢄⡀
⠀⠀⠀⠈⠐⠀⠈⠉⠈⠀⠈⠀⠋⠀⠁⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⢸⡏⠀⢀⠀⠀⠀⣤⠄⠈⡉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠘⡢⠀⠀⠀⠀⠈⠡⠀⠁⠀⠀⠀⠋⡄⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢂⢋⡘⠇⠀⠀⠀⠀⠀⠀⠚⣣⣀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⣠⠅⠀⠀⠀⠀⠀⠀⠀⢀⢫⡣⢄⣀⠔⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠐⣥⠄⠀⢹⠂⠀⠠⠂⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠈⠁⢉⡀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⢀⣁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠈⠀⠈⠉⠃⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠤⢄⡠⠄⠒⢂⠀⠲⠀⠐⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠑⢄⡀⠒⢄⡀⠁⠀⠀⠀⣀⡄⠀⡴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠐⠁⠀⠀⠀⠈⠐⠀⠟⠂⠀⢄⠀⣍⠀⠀⢱⡀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⢉⣹⠃⠀⠀⠘⠒⠀⠊⠉⡨⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠰⠤⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣠⢶⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠚⠓⠛⠆⠁⠈⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⢯⡀
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⢧⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠵⠋⠉⠹⡄⠀⡠⠤⣄⡟
⠀⠀⠀⠀⠀⠀⢇⠀⢠⠀⠀⠀⠀⠀⢰⣿⠞⠉⠉⠓⢿⣾⠀⠶⣽⠄
⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⠀⠀⠀⠀⢸⠸⡄⢠⠶⣄⠼⠿⠄⠈⣯
⠀⠀⠀⠀⠀⠀⠀⡆⠸⡄⠀⡀⠀⠀⠸⡄⠹⡴⠋⠀⢀⣀⣶⠆⣏⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⠥⠞⢹⣤⠛⢦⣀⣀⢱⣎⣁⡄⠀⠈⢢⠞⣰⠋⠉⠀⠉⠉⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢴⠦⣌⡉⠉⠉⠀⣀⣀⣠⡵⠞⣹⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠉⠉⠉⠉⠉⠓⠤⠔⠊⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠔⠛⠉⢸⡇⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣠⡀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⢸⡗⢆⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⢠⠋⣇⠀⠹⣄⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⡇⠈⢣⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠛⠒⠛⠲⠦⢽⣆⡀⠀⣎⢢⣠⠴⠖⠚⠋⠁⢀⢀⣿⡀⣀⠀⠀⢸⡄⠀⠀⢀⡤⠴⡄⠀⣀⣀⣀⠀⣀⣀⣀⡀
⠀⠠⠤⠤⠴⠶⠶⠤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠛⢎⢻⡛⠓⠒⠒⠛⠉⣩⠟⣽⠛⠁⠀⠀⣟⢉⣩⠿⠀⢀⡏⠉⠉⠉⠉⠉⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠐⠺⢧⣳⡀⠀⠀⣠⠞⠁⠀⠋⠀⠀⠀⢀⡟⢸⡏⡆⠀⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⢤⣄⣀⣀⣀⣧⣀⠀⠀⠹⠻⣿⣷⠀⢸⣁⣀⡀⠀⠀⠀⠀⣰⠏⠀⢸⢻⣿⡄⠨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡼⠁⠈⠙⠓⠒⠈⠁⢸⣛⠉⠀⠀⠉⠙⠲⣤⠞⠁⠀⣤⠴⢦⣿⣧⠴⠿⡇
⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⢤⣄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⡄⠀⠀⠰⣶⡇⠀⠀⠀⠸⡖⠒⠒⠲⡆⠀⡇
⠀⢀⣠⣔⣋⠁⠀⠀⠀⠀⢠⣎⠀⠀⠀⠀⠈⠓⠛⠒⠒⠒⠒⠛⠋⠉⠉⠉⠓⠓⠾⠦⠴⠋⠀⠀⠀⠀⠀⣧⢤⣤⣤⠇⠀⡇
⠀⠀⠀⠀⢩⡏⠉⠉⠈⠉⠁⠈⠉⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣤⠤⠴⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠒⠲⠾⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠉⠉⠐⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣁⣀⣀⣀⣀⠀⢀⣀⣀⠀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣤⣤⣤⣤⠤⠽⢧⣀⠀⠙⣆⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠑⠒⠤⢍⡑⠢⢄⡉⠒⢿⣄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⡙⢷⠢⣄⡀⠀⠀⠀⠀⠀⠈⠑⠢⠭⣒⢤⡈⠑⠦⣈⢧
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡇⠈⢳⣄⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⣬⣿⣆
⠀⠀⠀⠀⠀⠀⢀⠏⠀⢀⠞⠀⣠⠞⠈⠛⠶⣤⣄⣈⡑⠒⠢⢤⣤⣤⣤⡤⢔⣶⡯⢴⡟⢦⣀⣀⡀
⠀⠀⠀⠀⣀⡴⢋⡠⠔⢉⣠⠞⠁⠀⠀⠀⠀⣀⠈⠉⡙⠓⠛⠓⠒⠒⠒⠯⠏⣌⣈⣭⠗⠀⠀⠀⡕
⠀⠀⠀⠀⠀⠙⣿⠗⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⡦⠤⠒⠋
⠀⠀⠀⠀⠀⢜⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢱
⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠳⣀⠔⣳⡤⠄
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠇⠀⠀⠀⠋⣴⠟⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⠋⠀⠀⠀⠀⠀⠀⢚⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⠟⠁⠀⠀⠀⠀⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠶⠤⢤⡤⢤⠤⠤⠴⠶⠛⣩⠴⣇⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⢻⠆⢤⣤⣤⣀⣄⣤⠤⠴⠏⠁⠀⠀⠑⢤⡀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢠⠷⠴⠤⢤⣄⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⣧
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣃⡴⠚⠉⠉⠙⠢⡄⠀⠀⢀⡴⠋⠁⠀⠈⠑⢾⡄
⠀⠀⠳⢶⢀⡄⢀⠀⣠⣀⡟⠀⠀⠀⠀⠀⠀⠘⣆⢀⡾⠀⠀⠀⠀⠀⠀⠈⣷⠒⠒⠂⠐⠒⠀⠠⠔⠲⠶⠶⠦⠶⡤
⠀⢀⠔⠉⠉⠣⠞⠲⠃⢨⠃⠀⠀⠀⠀⠀⠀⢀⣏⣈⣽⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡅
⠰⠛⡖⠒⡖⠒⠒⠒⠒⡾⠀⠀⠀⠀⠀⠀⠀⠸⠦⢤⠼⡇⠀⠀⠀⠀⠀⠀⠘⡧⠤⠤⠤⠤⠤⢤⡤⢠⡄⠀⣠⣿⡇
⡇⢠⠁⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⢷⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⢰⠀⢰⠁⠀⣼⣿⣷
⢇⠀⡄⠀⡇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⢿⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⢸⠀⠀⢿⣿⡏
⢘⠦⠷⠤⢷⠤⠤⠤⡾⠃⠀⠀⠀⠀⠀⢀⣸⠷⠤⠼⠦⠤⣧⠀⠀⠀⠀⠀⠀⢈⣳⣄⣀⣀⣀⣨⣄⣠⣧⣀⣸⣿⡇
⠸⠀⡇⠀⡇⠀⠀⡞⠉⠁⠒⠒⠒⠤⠤⠤⣭⠀⠀⠀⠀⠀⣻⠄⠒⠒⠊⠉⠉⠁⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⢰⠀⡇⠀⡇⠀⠀⡶⢢⣤⡤⢤⣄⣀⣀⡀⣤⠀⠀⠀⠀⠀⢿⣀⣠⣤⠤⣴⣒⣶⡎⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠈⡦⢵⠄⢼⡄⢼⣇⠰⠿⠟⣛⣳⣿⣿⠈⣇⣀⣀⣀⣀⣀⣨⡅⢹⣿⡻⠯⠝⠛⢋⡼⣃⣀⣀⣀⣀⣀⣀⣀⢘⣿⣇
⢸⠀⡆⠀⡌⠀⠈⠻⣷⣄⣀⣀⣀⣈⣩⡾⠃⠀⠀⠀⠀⠀⠀⠻⣦⢤⣀⣀⣤⣔⠮⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⠀⡇⠀⠇⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠈⢆⣸⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃

⠉⣤⠈⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⡉⠉⠉⠉⠈⠉⢉⣭⡉⠉⠉⣁⠁⠀⡨⠉⠀⠉⣤⠈⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠈⠉⢉⣭⡉⠉⢉⡡⠁⠀⠈⠉
⠤⠤⠤⠄⠀⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠄⠀⠤⠀⠀⠀⠀⠠⠤⠤⠤⠄⠀⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠆⠀⠤
⠀⠀⠠⠶⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠂⠀⠀⠀⠀⠀⠒
⠚⠒⠒⠒⠂⠀⠂⢋⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠘⢁⡀⣀⣄⠀⠀⠀⠀⠐⠚⠒⠒⠒⠂⠀⠂⢛⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⢀⡀⣀⣄
⣀⣀⡁⡀⢀⡀⠀⠈⠀⠀⠀⠀⣤⡀⢠⡄⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⡀⠀⠈⠀⠀⠀⠀⣠⡀⢠⡄⠀⠀⠀⠀⠀⠈⡀⠉
⢠⣽⠖⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠀⠖⠀⠀⠀⠂⠰⠂⠀⢠⣵⠖⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠀⠖⠀⠀⠀⠂⠠⠂
⠲⠶⠶⠂⠀⠖⢰⡄⠀⠀⠀⠂⠀⠐⠂⠀⢀⡀⠐⠈⠀⠀⠀⢰⣶⠆⠂⠀⠀⠀⠲⠶⠶⠂⠀⠔⢰⡄⠀⠐⠀⠀⠀⠐⠂⠀⢀⡀⠐⠈⠀⡀⠀⢰⣶⠀⠒
⠁⠉⢀⣀⣀⠀⢈⣉⣁⢀⢀⣹⡁⠀⠀⢀⣤⣤⣄⣀⠸⠃⠀⢀⣄⠀⠀⣠⡄⠀⠁⠉⢀⣀⣀⠀⢈⣉⣁⠀⣀⣹⡁⠀⠀⢀⣬⣤⣄⣀⠸⠃⠀⢀⣀⠀⠀⣠⡄
⢤⠄⠉⠉⠁⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠴⠀⠀⢤⠤⠶⠦⠀⠀⠠⠀⠀⠀⢤⠀⠉⠉⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠼⠀⢀⢤⡤⠶⠦⠀⠀⠠
⠀⢤⣶⡄⠀⠐⠠⠴⠶⠤⠠⠤⣶⠤⠴⠦⠐⠒⠒⠛⠐⠚⠠⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡄⠀⠐⠠⠶⠶⠤⠤⠤⣶⠦⠴⠦⠐⠒⠒⠛⠐⠚⠠
⠀⠀⠁⠀⠀⠺⠧⠖⠻⠶⠚⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠑⠛⢛⠒⠀⠈⠋⠀⠀⠀⠀⠁⠀⠀⠺⠦⠖⠻⠶⠚⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠑⠛⢓⠒⠀⠈⠃
⠁⠉⠛⠛⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢁⡀⠀⢀⠈⠀⠐⠉⠀⠀⠀⠀⠀⠘⠀⠉⠛⠚⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢁⡀⠀⢀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠊⠁⠀⠀⠀⠉⠓⠚⠉⠁⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠓⠘⠉⠁⠀⠁⠀⠀⠀⠀⠈
⠉⠁⠀⠘⠀⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠀⡀⠀⠉⠀⠀⠀⠁⠀⠀⠀⢀⠀⠉⠁
⣀⡀⢈⣛⡈⠁⠀⠀⠀⠒⠂⠀⠀⣀⣃⡉⠁⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⣠⡀⠉⢀⡀⢈⣛⡈⠁⠀⠀⠀⠒⠂⠀⠀⣈⣃⡉⠁⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⣠⡀
⠉⣤⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⣭⡉⠉⢉⣀⠀⠀⡨⠉⠀⠉⣤⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⣭⡉⠉⢉⡠⠀⠀⠨⠉
⠤⠤⠤⠄⠀⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠄⠀⠤⠀⠀⠀⠀⠤⠤⠤⠤⠄⠀⠤⠤⠴⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠀⠤
⠀⠐⠀⠶⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⠒
⠚⠓⠒⠒⠂⠀⠂⢋⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠘⢀⡀⢀⣄⠀⠂⠀⠀⠀⠚⠓⠒⠒⠂⠀⠂⢋⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠘⢀⡀⢀⣄⡀⡀
⣀⣀⡀⠀⢀⡀⠀⠈⠀⠀⠀⠀⣤⡀⢠⡄⢀⠀⠀⠀⠀⠈⡀⠈⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⠀⡀⠉⠁⠀⠀⠀⣤⣤⣥⣤⣄⣀⡀⠀⢀⣭⣉⠉⣩⢍⡁⣀
⠡⣽⠖⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠖⠀⠀⠀⠀⠠⠂⠀⠡⣵⠖⠄⠀⠀⢤⠠⠀⢀⠸⠀⠀⠀⠀⠀⠀⠀⡰⢂⡠⢠⠶⠤⠄⠠⠞⠰⠆
⠲⠶⠶⠂⠀⠔⢲⡆⠀⠐⠀⠂⠀⠐⠂⠀⢀⡀⠐⠀⠀⡀⠀⢰⣶⠂⠒⠀⠀⠀⠲⠶⠖⠆⠀⠔⢲⡆⠀⠈⠀⠀⠀⠐⠃⠀⢀⡀⠐⠉⢀⡀⠈⢰⣶⠀⠒
⠁⠉⢀⣀⣀⠀⢈⣉⣁⢀⣀⣹⡁⠀⠀⢀⣤⣤⣄⣀⠸⠃⠀⢀⣄⠀⠀⣠⡄⠀⠁⠉⢀⣀⣄⠀⠈⣉⣁⠀⣀⣹⡁⠀⠀⢀⣬⣤⣄⣀⠸⠃⠀⢠⣀⠀⠀⣠⡄
⢤⠀⠉⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠴⠀⢀⣤⡤⠴⠤⠀⠀⠠⠀⠀⠀⢤⠀⠉⠉⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠴⠀⢀⣤⠤⠴⠤⠀⠀⠠
⠀⢤⣶⡄⠀⠐⠠⠶⠶⠤⠠⠦⣶⠦⠴⠦⠐⠒⠒⠛⠀⠚⠠⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡄⠀⠐⠠⠶⠶⠤⠤⠤⣶⠦⠴⠦⠔⠒⠒⠛⠐⠚⠠
⠀⠂⠁⠀⠀⠚⠧⠖⠳⠶⠛⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠑⠚⢛⠐⠀⠈⠉⠀⠀⠀⠂⠁⠀⠐⠺⠇⠖⠻⠶⠓⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠑⠛⢓⠒⠀⠈⠋
⠀⠈⠛⠛⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢁⡀⠀⠀⠘⠀⠀⠉⠀⠀⠀⠀⠀⠘⠀⠉⠛⠛⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢁⡀⠀⢀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠓⠘⠉⠁⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠈⠓⠘⠉⠁⠀⠁⠀⠀⠀⠀⠈
⠉⠁⠀⠈⠀⡀⠀⠉⠀⠀⠀⠁⠀⠀⠀⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁
⣂⡀⢈⣛⡉⠁⠀⠀⠀⠒⠂⠀⢀⣈⣃⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⡄⠉⢀⡈⢈⣛⡈⠁⠀⠀⠀⠲⠂⠀⢀⣈⣃⠉⠁⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⣠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⡀⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⢀⠀⠉⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⢈⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡋⠘⣗⠲⠖⣯⡟⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⠢⣈⣷⣞⣽⣴⡞⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢈⡋⣿⢰⢹⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⣃⣀⡀⠀⠠⡇⡇⡆⢸⠀⠀⠀⠘⣟⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠤⢄⠀⠉⢻⢸⠃⡃⢸⡆⠀⠀⠀⢻⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣇⣀⡜⠓⠂⢾⣎⣠⣆⣸⣧⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣴⣧⡤⠭⠶⢶⣾⣿⠁⠀⠀⠀⠀⠀⠀⢀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠑⠒⠢⠤⢄⣀⣀⢼⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⡴⢁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠒⠒⠊⠉⠉⠁⢀⡠⠔⠚⠑⠒⠠⠤⠤⠤⢤⠤⠄⠺⢏⣉⣏⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠦⠤⠤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡴⢃⣀⣀⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⡛⢹⠀⠀⠀⠀⣠⠤⡤⠐⠒⠒⠛⠛⠒⠒⠒⠚⣿⠉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡇⢸⠀⠀⠀⠀⣟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⠀⡆⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⠃⠀⢇⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢤⢼⡀⠀⠘⣤⠄⢼⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⡠⠔⠒⠋⠉⠀⠀⢀⠇⠀⣀⠷⠀⢾⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠈⠑⠒⠀⠐⠒⢒⡶⠛⠊⠁⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠓⠈⠁⠙⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠶⠖⠒⠛⠋⠉⢙⣦⠤⠶⠶⠶⢤⣀⣤⠶⠛⣿⡤⠟
⠀⢠⡤⠶⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠀⢀⢀⣠⠶⠛⣩⣴⡖⠋⠁
⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠀⣀⣤⠶⠛⣉⣤⡶⠋⠁⠀⠹⣆
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣛⣉⣤⢶⢾⣿⡟⠀⠀⠀⠀⠀⠈⠷⣄
⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣠⣯⣀⡿⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠸⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠉⠉⠓⢦⣤⣀⣀⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⣀⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢛⣉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠈⠳⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣀⣀⣀⡠⠄⠀⠈⠛⢶⡶⢄⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡴⠊⢿⡏⠀⠀⠀⠀⢠⣴⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⡤⢾⠀⠌⠀⠀⠀⠀⠀⢤⡀⠐⡶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡞⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠛⢂⡔⠊
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠾⢯⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸
⠀⠀⠀⠀⠀⠐⠋⢀⡀⠀⠀⠸⡆⠀⠀⠹⡀⠀⢈⣦⡀⠀⠀⡾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⣧⠀⠀⠀⢳⡔⢯⠈⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⡀⠀⣼⠻⢳⣼⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢧⡸⣇⣠⠞⠿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠀⢀⡔⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡎⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⡞⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⠾⡇⠀⠐⠶⠀⢸⡇⠀⠀⠀⠀⢶⣄⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢘⣧⣀⣤⣿⣎⣀⣀⣀⣤⠶⠾⠷⣤⠤⠒⠒⠛⠽⣿⣦⡀⠀⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠐⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⢹⠀⠀⠀⠀⠀⠀⠦⣝⠢⣄⠀⠀⠙⠳⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⢀⣼⣥⣀⣀⣀⠀⢬⣽⠞⠉⠛⡦⠠⠀⣀
⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣤⡤⠤⠖⠚⠛⠛⠛⠉⠁⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⡩⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠈⢀⡒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⡷⠤⢎⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⢰⢀⡀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⡀⠀⠀⢰⢸⠀⠀⠀⣀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣇⠀⠀⡸⠖⢲⠾⠓⠯⠀⠀⠘⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠦⠤⢄⡠⡌⡹⠊⢁⣀⠤⠔⠒⠂⠉⢃⠀⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣈⣷⠊⠁⠶⣺⠀⠀⠀⢀⣀⣀⣀⡗⣿⠣⣄⠀⠀⠀⠀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡄⠈⠹⣧⣄⣀⣆⠀⠀⠀⠸⣦⠶⠺⢶⣅⠀⠞⠂⡐⠲⡀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢖⡏⠢⣈⠁⠀⠀⠉⣀⡈⠉⠈⣛⠀⠀⣈⣏⣐⣐⣂⣉⣀⣈⣢⡀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡬⠗⠈⠉⠐⠒⡖⠓⠛⠒⠒⠉⠹⣧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠃⠀⠀⠀⢸⠓⡇⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⡖⠒⠒⠒⣶⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠋⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣌⣳⣔⣦⢀⣠⣼⣄⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠉⠁⠀⠀⠈⠉⠒⠭⣒⠦⡀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠣⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠟⠂⠀⠀⠤⠤⠤⠤⠤⠭⠭⠍⠁⡖⠲⠒⠤⡄⣀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣇⣀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡜⠉⢺⡄⠀⢸⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⢄⣀⢼⡇⠄⣞⠣⢄⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⡊⠁⠀⠉⠀⠉⡷⠊⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠢⡀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢆⡀⠀⠀⠑⡄⠉⠉⠉⠁⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⡀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡚⣿⢧⠀⠀⠀⠀⠀⠀⡔⡩⣚⡟⠙⠋⣻⣔⢕⡄⠀⠀⠀⠀⠀⠀⡼⣿⢓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣉⣉⡜⠀⠀⠀⠀⠀⢠⣱⡟⢁⡀⠀⠀⢀⣈⢻⣿⡀⠀⠀⠀⠀⠀⢠⣉⣉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡆⠀⠀⠀⠀⠀⣸⠹⡇⠭⠍⠀⠀⠩⠭⢸⡏⣇⠀⠀⠀⠀⠀⢸⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠀⠀⠀⠀⡏⠒⡀⠀⠀⠀⡄⠀⠀⢀⠒⢹⠀⠀⠀⠀⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢃⡀⠀⠀⠀⠇⠀⠱⡀⠻⣉⣉⠟⢀⠎⠀⡘⠀⠀⠀⢀⡸⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠉⠉⠁⠘⠒⠒⠚⠷⡂⢒⠾⠓⠒⠒⠃⠈⠉⠉⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠒⠒⠒⠒⠒⠒⠲⠯⡉⠈⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠤⣀⢀⡀⣸⠀⠀⠀⠀⣄⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠒⢂⠀⠀⠀⢸⠀⠀⠀⢸⠒⠂⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡶⡾⠁⠀⠁⢹⠀⠀⠀⢸⢶⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡈⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠃⢸⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣀⣀⣀⣀⣀⠏⠉⢏⣀⡰⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠄⠑⠍⠂⠤⠌⠱⠄

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⢸⣿⣏⠀⣄⡀⠀⢀⣼⣿⠃⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠰⠁⠀⠘⠿⠁⢠⣿⠄⠀⠀⠀⠀⠈⠂⡈⠢⠀⠀⠀⠊⠠⡄
⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⣠
⠀⠀⠀⠀⣨⣿⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢿⣧
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣰⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⠀⣿⡘⣿⡆
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣰⡏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⢰⣿⡅⠀⠀⠀⠀⠀⣸⣿⠀⢿⣧⢹⣿
⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠿⠟⠁⢀⠀⠀⠀⢀⣿⡍⣠⡘⣿⡆⣿
⣷⠀⠀⠀⠀⠀⠀⠈⠃⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠠⠀⠀⠤⢝⠄⠀⠈⠁⣇⡉⠀⢹⣿⣿
⣿⣇⠀⡆⠀⢀⢀⠁⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠈⠸⠀⠀⠀⠀⠀⠁⠀⣬⠰⣿⠇⠀⠀⢿⣿
⠸⣿⡌⠃⠀⠀⠁⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠀⠀⠀⠘⣿
⠀⣼⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠙
⣼⣿⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣐⠛⣹⣿⡈
⣿⡟⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣴⣶⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠂⠀⣿⣿⠇⠀⠀⢀⣾
⣿⡅⠀⠐⠂⠀⠲⠟⠛⠛⠋⠁⠀⢀⣿⣿⠁⣾⣿⠃⠀⠀⠀⠀⠀⠰⡤⠀⠀⢾⡿⠁⢰⣦⣼⡏⠀⠀⢀⠞⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠘⠻⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⢿⣿⣿⠁⠀⣀⠉⠘⣿
⣿⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⡄⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢀⣴⠏⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠈⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠐⠛⠛⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡘⢯⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣦⡀⠊⣶⡀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠚⠉⠈⢿⣦⠈⠃⠀⠀⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠹⣷⣄⠁⢠⡀⠀⠀⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠻⣷⡄⠙⢦⣄⡀⡀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣝⣿⣦⡀⠻⣿⣿⣿⡀⠀⠱⡀
⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠿⢿⣦⡈⠻⣿⡽⡆⠀⠱⡄
⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢰⠆⠉⣀⡺⢿⣾⣷⠂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣽⣶⣬⣶⣤⣀⠀⢀⠀⣀⠀⢄⡀⢀⣤⣀⣀⣀⣥⣤⣭⣙⣂⠀⢇⠀⠀⣽⣿⣇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣿⣿⣿⣿⣷⣶⣶⣶⣶⠿⠿⠛⠿⠿⠟⠛⠿⢯⣿⣿⣿⣿⡛⠛⠳⣽⢿⣄⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠰⣴⣶⣬⠻⢛⣭⠷⠛⠁⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠙⠓⢚⡉⠳⣌⢠
⠀⠀⠀⠀⠀⠀⠀⢴⡿⢟⣫⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⣌⣛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉




⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡰⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⠤⠖⠉⠀⠠⠜⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⣅⠈⠁⢠⠀⠈⠁⣰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣄⠀⠠⠄⠀⣠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠀⢗⠂⠐⡿⠀⠙⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠄⣡⣌⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡴⣹⡠⠊⠉⠉⠙⠉⠉⠉⠑⠄⠏⢆⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠂⢄⠀⢀⡼⠥⠛⠂⠄⡀⠀⠀⠀⠀⢀⠠⠒⠚⠬⢧⡀⠀⠠⠐⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⢢⡠⠒⠁⠀⠀⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠒⢄⡔⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢰⡑⡐⣠⢠⠁⠈⠒⠒⠒⠒⠒⠒⠒⠂⢤⠄⠒⠒⠂⠒⠒⠒⠒⠒⠁⠈⡄⣄⢂⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠖⠂⠁⠢⢄⠀⠀⠀⠀⠘⠁⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣈⣐⣤⣀⣀⠤⠔⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣤⣾⡿⠻⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣶⣌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠓⣠⣾⣿⣿⣿⣿⣶⣽⣋⡙⣿⣿⢿⣿⡿⢿⣷⡿⣿⣿⠿⣿⣿⠛⠿⣏⣙⣿⣿⣦⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣶⣿⣷⣾⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⡋⠤⡑
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⣌⣐⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠩⠞
⠀⠀⠀⠀⣀⡐⢍⡄⠀⣰⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠉⠀⠀⠰⣿⣿⣿⡿⢓⣩⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠓⠬⠲⠅⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⣿⣿⣿⣿⣧⣤⣠⣀⣠⣶⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢛⡿⠛⠉⠁⠈⣻⣿⠏⠙⢿⣿⣷⣴⠟⠙⢿⣿⠋⠙⣿⣿⡿⠈⢻⣿⣿⣿⣶⠏⠉⠉⠉⢉⣙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣶⣾⣿⣿⢿⠀⠀⠘⣿⡟⠁⠀⠀⣿⣿⠀⢀⣿⣿⠃⠀⣾⡿⢿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⣿⣿⡄⠀⣄⠀⠉⠀⡀⠀⢘⣿⡇⠀⣸⣿⡟⠀⢰⣿⣦⣴⣿⠇⠀⢀⣁⢀⣼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⢿⣿⣿⡷⠀⢸⣿⠀⢀⣿⣤⣤⣾⠇⠀⣾⣿⠁⠀⣿⣿⠁⢠⣿⣿⡿⢿⣿⠀⢀⣿⣿⡿⠿⣿⣿⣿⣿⣿⡇⢀⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣄⣈⣀⣀⣤⣾⣿⣄⣼⣿⠛⣿⣿⣆⣸⣿⣿⣄⣸⣿⠿⣄⡀⣀⣀⣠⣼⣿⣄⣀⣀⣠⣤⠞⣿⣿⣿⣿⣿⣿⠈
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⢻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣯⣉⠈⣿⣿⣿⢿⡿⠟⢿⡿⢿⡿⠿⣿⠿⠻⣿⡿⠿⠛⡿⠻⣿⠿⣿⠿⠻⠇⣸⣿⠿⠻⣿⡿⠿⣿⠿⢿⣿⣿⡡⣉⣽⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⢀⣚⣀⣷⠀⠇⣸⠃⣐⣃⣼⡇⢰⣿⠃⣰⠃⢠⡏⣰⡶⢀⣿⠃⣴⠂⣼⠃⣼⠃⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣌⣉⣽⣿⣄⣴⣿⣆⣉⣩⣿⣄⣙⣿⣄⣠⠀⣾⣧⣡⣤⣸⣷⣌⣡⣄⣿⣄⡤⠀⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⢛⣩⣟⡛⢋⣼⣿⣿⠟⢿⡿⠿⣿⣿⣿⣯⣈⣁⣼⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢶⣶⣄⠹⣿⣿⣿⣿⣿⣿⡯⣙⠿⣟⣻⠿⣉⣭⣸⣿⣿⣿⣿⣿⣿⣿⠯⠀⠀⠀⠀⠂⠉⠉⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠉⠃⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⢀⡀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡛⠟⠙⢩⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⣈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣤⣤⡀⢠⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣄⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⡀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⠛⢿⡇⠀⢦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡸⡿⣷⠀⠘⡇⠀⢀⠂⠀⢀⣀⣿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⡷⡏⠀⠀⠑⣠⣄⣴⣾⠟⠁⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⣿⣦⡀⠀⣿⢿⣿⣿⡷⠉⢄⣸⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣭⣿⣿⣿⠙⠐⠋⠉⠉⠉⠃⠀⢸⣿⡿⢯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⣹⣿⣿⣿⣿⣦⢠⣾⠟⣿⣿⣿⡿⢿⡇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⣿⣉⡟⠉⠉⠙⢻⢸⠋⠙⠉⢻⣿⡷⢾⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠛⣙⢿⡄⠀⣠⡛⠘⣣⡀⠀⣿⣯⡁⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠘⢿⡏⣷⠀⠏⣭⣭⡌⠃⣸⣿⡷⢠⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⡌⣇⣿⣷⡀⠈⠛⠀⣰⢿⣻⠄⣘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂⢨⣭⢩⡄⡁⡅⠰⠛⠓⠒⠻⡅⠀⢠⠀⠀⠀⠶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡟⠀⢸⣿⣿⠀⢀⡀⠀⠀⠀⢀⣴⣿⡟⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣼⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠿⡿⠀⠈⢩⠃⠀⠀⢾⣿⠟⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⢆⠀⠀⠀⠀⣄⡄⢁⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠛⠀⠀⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⡁⠘⠃
⠠⠄⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠁⣰⡆
⠀⠀⠀⠀⠀⢀⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⣿⣿⡀
⠀⠀⠀⠀⠀⢺⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠄⠀⠂⠒⠀⠀⠀⠀⠀⠀⢀⣷⠸⣽⣧
⠀⠀⠀⠀⠀⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣶⡤⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠘⣿⣇⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠡⡟⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⠀⠀⡙⡿⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢿⣿⡄⢿⣿
⡆⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡼⠀⠼⠛⠟⠁⢀⡀⠀⠀⠀⠀⢰⣿⣨⢀⣤⠘⣿⡧⠘⣿
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣠⠀⠀⣠⣝⠳⣄⠀⠀⠘⠃⣿⠘⠏⠀⠹⣿⣿⣿
⣿⣤⠀⠀⡄⠀⠀⠀⠀⡂⠀⢀⣾⣿⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠃⡖⠁⠀⠀⠁⠀⠑⠌⠀⠀⠸⠀⡿⣷⠀⠀⠀⢻⣿⣿
⢿⣿⡆⢨⡇⠀⠀⢦⡰⠃⠀⢼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠈⢿⡏⠀⠀⠀⠸⣿⣿
⡜⣿⣷⠈⠂⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢻⣿
⠀⢠⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠠⠀⠀⠀⠀⠀⢻
⢀⣾⣿⠏⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⢟⣷⡀⠦
⣿⣿⡏⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢀⣴⡰⠄⢁⣾⣿⣷⡄⠀⠀⠀⠀⣠
⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⠀⠀⠀⣼⣿⣿⠆⢀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠈⣿⣿⡟⠀⠀⠀⠀⣴⣿
⣿⡏⠀⠀⠰⠞⠁⠀⠶⠿⠿⠿⠟⠋⠀⠀⠀⢀⣿⣿⡟⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠠⠆⡦⠀⠀⠰⣿⡿⠃⠀⣼⣠⣿⣿⠁⠀⠀⠀⠼⢿⣿
⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⡼⢿⡞⠁⠀⠀⠀⡀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠘⠁⠀⢸⣿⣿⣿⠇⠀⠀⠀⠚⠃⣾⣿
⣿⡇⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠀⠀⣠⡶⠀⠀⣴⡇
⠿⡇⠉⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣰⠃⡄⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⢀⣴⡟⠁⠀⠀⠹⠇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢰⣿⣿⢱⡏⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠂⠂⠀⠤⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣏⣿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢁⡀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢸⣿⣿⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠀⠀⠀⠠⡀⠀⠀⡀⠀⠂
⣠⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣁⣸⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣎⣀⣀⣘⣄⣈⣄⣀⣈⣠⣀⣀⣤⣀⣀⣀⣐⣿⣇⣀⣀⣀⣀⣤
⣿⣿⣛⣙⣟⣙⣛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣠⠀⣀⠀⠄⣈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⠀⣉⠀⠀⠀⣉⡁⢸⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⠀⠀⠀⠀⠀⠀⠀⢨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠂⠀⠀⣠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠬⡓⠒⠒⡫⠄⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⢄⡈⢆⠎⣀⠄⠀⠘⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡄⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⢠⠟⠒⠢⡀⠀⢠⡾⠴⠓⠂⠄⣀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠳⠼⣦⠀⠀⡠⠒⠚⢧
⠀⠀⠀⠀⠀⡰⠋⠐⠢⡀⡱⠐⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠑⠲⡅⡠⠒⠈⠳⡀
⠀⠀⠀⠀⣸⡱⢠⢀⡄⡹⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠸⠁⣄⢠⠰⣹⡀
⠀⠀⠀⠀⠹⡄⠇⡘⠥⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⠤⣄⡠⠤⠤⡀⠀⠀⠀⠀⠀⠡⠝⡀⠇⡼⠁
⠀⠀⠀⠀⠀⠹⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠇⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠙⣹⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠽⠦⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠖⠒⠞⠉⠓⠊⠓⠶⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣰⠴⠞⠓⠒⠋⠙⠒⠛⢦⣄⣠⣄⠀⠀⠀⠀⠈⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⢠⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⢤⡀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢾⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠓⣄⠀⡴⠶⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⠆⠈⠛⠀⠾⠁⣰⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡰⠋⠀⠀⠀⠀⠀⠘⢁⡴⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢖⡽⠋⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡶⠤⣤⣀⣀⣀⣀⣀⡠⠤⢶⠞⣉⡴⠻⣄⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠊⠑⠒⣧⠤⣄⣀⣀⣀⣀⣀⣀⡤⢼⡗⠁⠀⠀⠈⠳⣄⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠙⠛⠒⠶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡂
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⠇⣠⠴⠒⠒⠒⠢⣄⠀⠀⠀⠀⢀⡤⠚⠉⠉⠉⠑⢦⣟
⠀⠀⢾⣁⣠⠀⠀⠀⠀⠀⠀⠘⡶⠁⠀⠀⠀⠀⠀⠈⠳⡀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣨⢧⣼⡇⢰⡆⣰⠓⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡷⠢⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡿
⠀⣠⡞⠁⠀⠀⠙⠉⠉⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠒⠺⢇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⡸⠋⣿⠋⢙⠟⠉⠛⠛⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢳⡖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠲⠶⠶⠶⠶⠶⠶⡶⠶⢶⠶⠶⢶⣿⣧
⡇⠀⡏⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸⠁⠀⡟⠀⠀⣿⣿⣿
⡇⠀⣇⠀⢸⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣿⣿⣿
⠸⣄⣹⣄⣀⣇⣀⣀⣀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡀⢘⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠘⡄⠀⢻⠀⠀⢻⣿⣿
⢰⠃⢸⠁⠀⡏⠀⠀⢀⠿⠤⢤⡤⣀⣀⣀⡀⠀⠘⠿⠉⠉⠉⠉⠉⠙⣟⣀⣀⣠⣤⠤⠴⠖⠒⠚⢻⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿
⢸⠀⣿⠀⠀⡇⠀⠀⣟⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣇⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⠄⢹⡀⠀⡇⠀⠀⣽⠉⣿⣷⣖⣒⠶⢤⣤⡤⣤⡇⠀⠀⠀⠀⠀⠀⢳⡤⣲⣶⣚⣻⣭⣽⣿⣏⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⡷⢸⡧⠶⣷⠄⢼⡿⡐⠛⠛⠫⣽⣿⣿⣯⡃⢸⣦⣤⣤⣤⣤⣤⣤⣼⡄⢹⣾⡭⠿⠖⠚⠉⠉⡰⣻⣅⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿
⢸⠂⢸⠁⠀⡇⠀⠀⠻⣝⡦⣀⣀⠀⠀⠈⣉⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⢤⣀⣀⣀⣠⣴⡻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠘⣇⢸⡆⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⣀⣉⣈⣛⣚⣛⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣁

⠀⠀⠀⠀⡏⢸⢹⠋⠙⠀⡟⠉⠉⢛⣹⣿⣿⠙⡏⠀⠀⢻⢿⡇⡆⢠⠐⠀⢰⠒⠖⢐⣖⠂⢰⢶⠒⡓⡖⣾⣿⣿⡀
⠀⠀⠀⠀⡗⠴⣰⡆⠀⡆⣇⢀⣴⣿⣿⣿⣿⣾⣇⣤⣀⣠⣬⣧⣏⠸⠜⠀⠈⢁⠃⡶⡏⣾⠠⣾⣿⠉⣼⣿⣿⣿⠁
⠀⠀⠀⠀⣧⣤⣼⣧⡎⢠⣿⣿⣽⣯⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣇⠀⠀⠈⠠⢠⡇⠘⢰⣟⣯⣾⣿⣿⣿⣿
⠀⠀⠀⠀⡿⣿⣿⣿⡿⠿⣿⢿⣏⣵⢟⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣾⣠⣶⣾⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⣿⣿⡏⣻⢃⣴⣏⣾⣿⢟⣿⣿⣿⣿⣿⣿⠟⠰⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣿⢾⣿⡟⡽⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡿⢡⣿⡟⣿⠱⣻⣿⣿⣽⣿⣿⡿⣻⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡟⢠⣏⡾⣿⣿⣿⣿⠉⣿⣿⣿⣞⣽⣿⣿⣿⡿⣽⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣿⣽⡱⣿⣿⣿⣿⣇⣟⣽⣟⡿⢻⠟⣹⢋⣽⢿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠯⠿⠵⠿⠿⠿⢽⣿⡿⢛⣿⣴⡿⢸⡯⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣬⡉⠁⠉⠙⠻⠧⠿⠿⠽⠿⠟⠋⠛⠛⠁⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣄⠚⠛⠛⠛⠿⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣦⠉⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡙⠄⠀⠀⠀⠀⠀⢀⠘⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⢲⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠻⣗⢦⡀⠀⠀⠙⣷⣤⣀⣀⡈
⠀⠀⠀⠀⠀⠀⡴⠋⠹⣆⠈⠣⡈⠀⠀⢀⡏⠀⠈⠁
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠘⣆⠀⠈⠢⡀⠸⣿⠂
⠀⠀⠀⢠⣾⡥⠂⠀⠀⠀⠀⡀⠠⠶⠈⠲⢻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠻⣀⢀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠟⠁
⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠢⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⠃
⠀⠀⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢹⣯⠓
⠀⠀⠸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠍
⠀⠀⠀⢇⠘⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠿⠃
⠀⠀⠀⢸⡆⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⢀⣀⡤⠂⠀⠀⠀⠐⡄⢀⠐⣩⠉
⠀⠀⠀⠀⢿⡀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣠⡼⣿⠉⠉⠀⠀⠀⠀⠀⠀⠈⡀⠀⠁
⠀⠀⠀⠀⠘⣇⠀⠘⢿⡄⠀⠀⠀⠀⠚⢩⡇⣻⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⢻⡄⠀⢸⣿⡄⠀⠀⠀⢀⡾⠀⢻⡇⠀⠀⠀⣀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠳⣝⠀⠀⠀⡸⠁⠀⢸⢃⣠⡶⢿⠉
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢈⡧⣤⣀⡁⠀⣀⣼⠟⠋⠀⠈⡆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠉⠛⠾⠟⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⢰⡎⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⢀⠴⠊⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⢾⣟⠶⢤⡀⠀⠀⢀⠤⠂⠀⠀⠀⢀⠠⠊⠁⠀⣸⡦
⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⣀⣀⠸⣧⠀⠀⠀⠈⠉⠛⠿⣦⣾⣧⣶⣒⠒⠒⠚⠁⠀⠀⢀⠴⠋
⠀⠀⢀⣿⡭⣭⠭⣓⣒⣠⣄⣀⣤⣶⣿⣿⣥⣶⣦⣿⠀⠀⠀⠀⠀⠠⠊⠀⠉⠉⠓⠦⣍⡓⣆⠀⣠⠖⠁
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠻⠉⠉⠉⠉⠉⠉⠛⠛⠀⠀⠀⠀⠀⠀⠈⠠⠀⢀⠀⠀⠀⠉⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢧⡀⠀⠀⠀⠀⠀⠀⠀⠁⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣌⠓⢄⡀⠀⠀⠀⠀⠀⣈⠤⠗
⠀⠀⠀⢰⣦⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣚⡢⠀⠈⠛⢦⣍⠲⣤⡤⠒⣉⠤⠒⢁⠔⠒⠂⠒⠠⠤⠄⢀
⠀⠀⠀⠙⢄⠉⠢⡀⠀⠀⠙⠢⣄⠀⠀⠀⢀⣼⣿⡭⣶⣅⠀⠀⠀⠉⠻⠧⠛⡋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⢾
⠀⠀⠘⢧⠈⠡⡀⠈⠂⡀⡀⠀⠈⠻⣦⣀⣾⣿⣷⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⣀⠞⡱⠋
⠀⠀⠀⠀⠳⣄⠈⠢⣀⣠⣴⠶⣶⣿⠿⠿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣁⠒⠒⠤⠤⣀⣀⡔⡡⠊
⠀⠀⠀⠀⠀⠈⠳⣄⣴⠶⠖⠋⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠉⠉⠒⠢⠤⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⠀⠀⠀⡀⠀⢄⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⢤⡀⠀⢀⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠐⠛⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠗⠀⠀⠀⠻⠂⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⡄⠀⠀⠀⠀⠀⠂⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠉⠀⠈⠘⡀⠀⠀⠘⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⣀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠠⠄⠠
⠀⠀⠀⠘⠂⠁⠀⠐⠀⠀⠀⢉⣉⣀⣈⣀⡀⠀⠀⠀⢀⡠⢀⢀⠀⠀⠀⠀⢀⠐⣀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠠⠲⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠠⠒⠒⠄⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢤⠠⠠⢤⠀⠀⠀⠀⠠⠤⢄⠄⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠬⠤⠀⠀⠀⠀⠀⠠⠀⠀⡠⠀⠀⠀⢐⡡⢤⣤⣄⡀⠀⠀⠰⠶⣿⣤⣥⠦⡀⠀⠀⣠⠄⡀⢀⣄⡀
⠀⠀⠀⠀⠤⣤⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡈⠀⠠⠀⠀⠀⠀⠨⠄⡀⠁⠀⠀⠀⠺⢧⣠⣀⣨⠀⠀⠀⠈⠛⡿⣿⣷⡿⠇⠀⠰⣿⠷⠿⠾⠟⠋

⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠤⠄
⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠰⠀⠀⠀⠄⠀⠀⠀⠀⠎⡀⠀⠉⠀⠀⠀⠂⡰⢀⢀⠨⠈⠀⠀⢰⡃⠂⠀⠘⣲⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠠⡄⠀⠀⠀⣄⠀⠀⠂⠠⠈⠠⠄⠀⠂⠀⣠⣶⣿⣷⣷⣦⡄⠀⠀⠂⠠⠀⠰⠀⠀⠀⠀⠁⠀⡤⠄⠀⠀⠀⠀⠀⠐⠐⡀⠀⠀⠀⠀⠀⡀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠒⠒⠒⠒⠀⣱⣦⣶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⢠⠎⠀⣠⡶⢤⣤⡶⢾⣻⡽⠚⠉⠢⣀
⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠔⢋⠴⠂⠀⠈⢱⠇⢰⣎⣥⣿⣿⠽⡞⢫⠔⠂⠀⠀⠀⠈⠳⢄
⠀⢀⡀⠔⠊⠁⠀⠀⠀⢀⡠⠒⢀⠀⠖⢉⠤⠂⢀⡄⠚⠁⣀⣴⡶⡛⠻⠯⣁⣔⣁⣋⡔⠈⠀⠀⠀⠀⠀⠀⠀⠉⡆⣶⣖⣆
⠀⠀⠈⠓⢤⡀⠀⠀⠘⠉⠀⠈⠀⡠⠈⠁⠠⠂⢀⡠⠲⠛⠛⣛⣤⣌⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠒⠀⡠⠚⢁⡄⠈⠉⠀⠀⠀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠲⠦⢤⣀⡀⠀⠀⣀⡀⠀⠀⡇⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠈⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡿⡍⠉⠉⠙⣿⣧⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠋⠉⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣤⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠟⠉⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⢻⡟⠉⠁⣤⡀⠀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣀⠀⠀⢀⡀⠀⠀⠀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠋⠀⠀⠘⠃⠀⠀⠀⠈⢻⣿⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣃⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠠⠉⠉⠁⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣀⣀⣀⣀⣀⡀⠠⠶⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡏⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠓⠂⠒⠒⠂⠉⠀⠉⠁⠒⠂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠑⠆⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢱
⣤⠤⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⣠⠂⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠇⠀⠈⠒⠂⠤⢤⠜⢱⡆⠀⠀⠀⠀⢰⠒⠢⣄⠀⠀⠀⢀⡿⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⢣⡀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⠘⣤⠀⠈⠉⠀⠒⠚⠁⠾⠋⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠈⠁⠰⠶⠄⠀⢀⡀⠀⠀⠉⠁⠶⠸⣦⣄⠀⠀⠀⠀⠀⠀⠱⠾⢦⣀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠰⡆⠀⠀⠀⠀⠀⠀⣠⠊⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⢄⣀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠤⢀⡀⠀⠀⠙⠃⠤⠄⣀⡀⢀⡴⠋⠀⠀⠀⠘⣀⠀⠀⠀⣠⠞⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠄⣀⠀⠀⠀⠀⠀⣠⠜⠃⠒⠤⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⠔⠉⠲⠄⠚⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠀⠒⠁⠀⠀⠀⠀⠸⣿⠒⢠⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠉⠑⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠐⠒⠓⠛⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠶⠀⠀⠀⠀⠀⠈⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠄⢛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⢀⣀⠀⠀⠀⠀⠐⠶⠦⠤⠤⣤⣤⠤⢤⡤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⠀⠀⠀⠀⠀⠀⠀⠰⡦⢤⠤⠤⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠤⢤⠤⠤⢤⠤⠤⠤⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢠⡧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⡄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⣾⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠆⣀⣀⣀⡀⠶⠞⠁⠀⠘⣦⣀⣀⡀⠀⠀⢀⣠⠛⢀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⢀⠆⠀⣠⠞⠁⠀⠀⠀⠉⠁⠈⠁⠀⢠⣄⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠧⠀⠃
⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⣀⠔⠛⠁⡔⠋⠀⠀⠀⠀⠀⠀⣠⠄⠘⠃
⠀⠀⠀⠀⠀⠓⠤⠤⠤⠤⠄⠒⠁⠀⠀⠀⠧⠀⠀⠀⠀⠀⠤⠚⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠉⠈⠋⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠚⠋⠉⠉⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⡽⠉⣁⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣼⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠰⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢠⠏⢀⣀⠀⠀⢀⣤⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠸⣿⣷⠀⠘⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠁⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠒⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠄⠀⠀⠈⠙⠳⣿⣛⣛⠋⠉⠉⠁⢀⣀⣠⠴⠊⠁⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⠔⠁⠘⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠸⡆⠻⣄
⠀⢠⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⢳⡄
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⡎⠀⠀⠉⢻⡛⠒⠦⣤⠤⢖⣋⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠳⢤⣤⡤⠊⠀⠈⠉⠉⠉⠙⠛⠛⠻⠿⠯⠽⣷⣖⣶⡶⠤⠤⢤⣤⣈⡀⢀⣠⠜⠃
⣠⡤⠖⠒⠋⢉⠀⠀⠀⠐⠢⢀⣉⠀⠀⠄⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠒⠉⠀⢀⣀⣠⠤⠖⠋⠉⠀⠉⠙⠛⠒⠒⠒⠶⠦⠤⠤⢤⣄⣀⣀
⠀⠉⠙⠒⠦⣤⣉⡀⠐⠂⠀⠀⣀⡀⠀⠒⠂⠤⠉⠑⠀⠒⢰⣋⠁⠀⢉⣉⡿⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠤⣤⣀⡀⠑⠶⠀⠀⢀⡠⠔⠃⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣲⣶⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢳⡜⠛⠁⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠳⢦⣄⣠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⢀⣠⣾⣿⡿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢀⣠⢶⣿⣿⣿⠟⠉⠀⠀⠙⠳⢦⣄⣀⣀⠀⣤⣤⣤⣤⣄⠀⠀⢻⣿
⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⣴⢋⣵⣿⣿⠟⠙⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⡄⠀⠀⢹⡄⠀⠸⣿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⢰⣿⣷⣿⡿⠟⠁⢠⡾⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⡇⠀⠀⣿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣹⠃⣠⣿⠿⣿⡟⠛⠛⠛⠚⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⣿⡇
⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣀⡀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⡀⢠⡇⠀⢰⣿⡇
⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢠⣶⡝⠟⠉⠉⠉⠛⠛⠒⠲⠶⠖⠚⠛⠛⠓⠶⠶⠶⠶⠶⠚⠛⠃⣼⣙⣷⣟⣼⠁⠀⣼⣿
⠀⠀⠈⢿⣧⠀⠀⠀⠀⠰⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⢰⣿⡏
⠀⠀⠀⠘⣿⣧⠀⣴⠛⢀⣠⡴⠞⠛⠛⠶⣄⡀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠘⢿⣧⡉⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣤⣄⣀⣀⣀⠀⠀⢀⣀⣀⣠⣤⣴⣶⡿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣾⣩⣟⣿⣿⣾⢿⣟⣣⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣾⠿⣻⠟⠟⠻⣻⣟⣿⣷⣿⣯⣍⣟⡽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣊⣿⣿⣿⡵⢿⠃⠀⠀⠀⠀⠘⠉⠋⣼⢛⢋⣴⣻⠿⡾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣟⣾⠛⣷⡙⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠉⣿⣿⣿⣿⣵⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢿⣝⣀⠉⠃⠀⣋⢉⠃⠀⠀⠀⠈⣉⠁⢠⡚⣩⣷⣿⣗⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⡏⣃⠀⠀⠀⠰⠦⠇⠀⠀⠀⢨⡦⠅⢸⡾⣯⣷⣏⣰⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢣⣧⡙⠆⠀⠀⠀⠀⠀⣀⠒⢀⡀⠀⠀⠈⡋⠉⢿⣯⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⢷⣦⡄⠀⠀⠀⢤⣈⠀⣉⣠⠀⠀⡸⠃⠀⣈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠣⣀⠀⠀⠉⠻⠉⠁⣀⠔⠁⢀⠄⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠈⠁⠒⠒⠒⠏⠽⠀⠀⠘⢒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⡀⠀⢀⠇⠀⠑⡀⡠⠋⠡⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠁⣠⠞⠀⠀⢠⡿⠁⠀⠀⠘⠄⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢢⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠳⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠱⢤⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠖⠊⠉⠉⠁⠒⠂⠼⣀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠳⠦⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠘⠫⣵⠀⠀⠠⠤⠤⠤⠄⠠⠀⠀⠀⠘⠛⠢⡀⠀⠀⠀⠔⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠉⠉⠉⠉⠙⡄⠀⠀⠀⠀⠀⢀⣀⡀⣼⠗⢄⠀⠀⢀⡸⠳⢌⡀⠉⠉⠉⠉⠉⠈⠉⠉⢱⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠁⠀⠀⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠃⠀⠀⢀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠉⠈⠀⣀⠈⠈⠁⠀⠀⡉⢉⣉⢀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⡀⠀⠀⢀⣀⣀⣈⢀⣁⡀⠐⠁⠀⠘⣁⡁⠋⠘⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⢛⢿⢛⡻⠻⣿⢛⣻⡿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠠⠤⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣒⣒⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠲⠤⢭⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠉⠉⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⠶⠦⣤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⣈⣇⡀
⠀⠀⠀⠐⣤⣊⠉⠉⠉⠒⠒⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠃⢠⣤⣄⡀⠀⢀⣀⣤⣶⣛⡋⠥⠀⠀⠀⠂⠆
⠀⠀⠀⠀⢹⠀⠀⠉⠒⠦⣤⣀⠀⠀⠀⠉⠉⠛⠒⠶⠦⠤⣄⣀⡀⠀⠈⢸⡄⢀⣀⣽⣷⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣤⡤⠤⠔⠒⠒⠉⠉⠉⠛⠒⠿⠛⠉⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⣲⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⣧⣄
⠀⣾⠟⠷⢶⣬⣉⠓⠦⣤⣀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠄⠒⢒⣈⣩⠤⠤⠶⠒⠛⡿⠁
⢸⡏⠀⠀⠀⠀⠉⠙⠓⠒⠿⢿⣶⣤⣀⠀⡸⢡⠀⣀⣀⣠⣤⡤⠶⠶⢖⣚⣉⣩⡥⠤⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠘⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢿⣿⣭⣭⣵⠶⠶⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣠
⠀⠈⠙⠛⠷⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠞⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⣦⣟⣀⣤⣤⠤⠶⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠂⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠒⠲⣄⠉⠐⠂⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣄⠈⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⡆⠀⠠⠐⠂⠀⠀⠀⠀⠐⠀⠀⠐⠁⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⡙⣶⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⠀⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣴⠏⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡾⠃⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠢⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢲⣤⣤⡄⠀⠐⠀⠈⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⢴⠞⠁⣿⡅⠀⠙⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⡶⠶⣿⡀⠀⠸⣿⣶⣤⣤⣤⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢹⣧⠀⠀⠘⣻⠿⠟⠛⠛⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢳⡀⠀⢿⣿⣶⣾⣥⣤⣴⡶⠾⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠴⠟⠻⠛⠋⠉⠁⠀⠀⣀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣶⣶⣿⣿⣿⣦⡀⠀⣾⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣏⣈⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⠳⠶⠶⠶⠶⠞⠛⠋⠉⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣤⣤⠾⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣉⣉⡉⠉⠉⠁⢀⣠⣾⠿⣆
⠀⠀⠀⣴⣶⣶⣶⣤⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⡉⠙⠋⠉⢉⣁⣤⣤⣤⣾⡄
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⢀⣤⡇
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠋⠛⢿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⢿⣿⣿
⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻
⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠁⠀⢀⣾⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⣠⠄⢠⡞⢁⣿⣶⣾⡶⣿⣉⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⠁⣠⣿⣟⣿⣿⣿⣿⡿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⣿⣿⣿⢈⣿⢸⣏⣿⣿⣿⠃⣾⡷⣣⣔⣠⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⢻⣷⠿⠛⢸⢿⣿⠛⣿⣸⣿⣼⣿⣿⣥⣤⡼⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣼⢿⣛⣿⡆⠀⢸⣿⠟⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⣿⠁⣸⣷⣴⢿⣿⣻⣿⣿⠛⣻⡟⣫⣽⣿⣿⣿⣿⣿⣀⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠞⠛⣿⣶⡟⠿⠿⠋⠉⠉⣵⣿⡿⢿⣾⣻⣿⣿⣿⣿⢿⡟⢋⣸⡷⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣴⣿⢛⠁⠀⠀⠀⢤⣿⣿⣯⣴⣿⣿⣿⣿⠿⠟⣻⡿⠛⣻⢟⡴
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣭⡅⠁⠰⠂⠀⠀⠀⣹⣿⣯⣽⡾⠛⠙⢷⣴⠟⢡⣧⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⣐⣿⠀⠀⠀⣠⠊⢁⣰⣶⣿⣄⣠⣶⠾⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣻⡯⠇⠀⠸⠇⢂⣿⡿⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣻⠀⢠⡄⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣯⡻⠇⠹⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⠀⡵⣸⣿⣿⣷⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⣀⣀⣤⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢰⡿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣿⣶⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣷⣿⣿⣿⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣈⣅⣈⡉⠶⠖⡂
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⣿⡿⣿⢹⡰⣾⣿⣿⣿⠿⣧⣈⡻⠿⣿⣷⠾⡥⠭⠅⠀⠀⠉
⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣿⣿⢻⡜⣿⣧⠈⠁⢈⠛⠻⣿⣃⠐⠒⠐⠒
⣿⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡼⢎⡓⠫⢴⣤⡈⠑⠦⣭⠉⠁
⡄⠈⣉⡹⠛⠛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠿⣿⣿⣿⣿⢫⣉⡳⣮⣶⡠⢌⠻⠦⠀⠀⠀⠰⠶
⠏⠀⠀⠀⠀⠀⠹⢿⢿⡿⢿⢿⣿⡿⢻⣿⣿⢻⣟⡃⢸⡇⢹⣿⣿⣧⠙⣿⣛⣿⣿⠎⠁
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠿⠏⠁⣿⡏⢸⣿⡟⡉⢸⣿⢸⣿⣿⣿⣧⠜⠛⠉
⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢋⡉⠙⠻⢈⠙⠁