⣿⣭⡿⣀⣾⢿⣿⣹⣞⣷⣴⣭⣯⢟⡻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠙⠋⠩⠉⠀⣘⡿⠿⠿⣿⣿
⣿⣿⣵⣻⢾⣽⡿⣻⣽⣿⡿⣿⡧⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀
⣿⣿⣿⣿⣏⣻⣝⣯⣧⣽⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶
⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⡽⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠐⠒⠂⠀⠀⡀⢄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡔⠮⠛⠁⠂⠁⠋⠁⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣻⣿⣿⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠀⡀⠀
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁⠀⠀
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⢁⠀⠀⡀⠠⠀⢀⠈⠠⠀⡠⠀⠀⢀⠀⠀⠀⢀⠀⢂⣓⠊⠁⡁⠊⠙⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠉⡁⠈⠀⡅⠘⠀⠈⠂⠀⠘⢁⠀⠀⠁⠀⠀⠎⠀⣛⠍⢠⠠⠔⠂⠀⠄⠀⠉⠉⠉⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⢀⡀⠀⠀⠀⣤⠁⠀⡔⠀⠀⠱⠀⠀⠀⡠⠵⠂⠀⠉⠀⠀⠀⢀⠔⠎⢢⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⠀⣀⠀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠸⠁⢀⢘⡁⢁⠀⢸⣷⠕⠀⡀⢀⡀⢂⠀⠘⠐⣷⠀⢀⠄⠀⠀⠀⣐⣯⠄⡂⠄⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⠉
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠁⡀⠁⠸⠀⠀⠀⠀⠀⠠⠀⣋⡔⢠⠐⠁⠀⠁⠀⢳⡴⠀⢁⡤⠃⠒⠠⠀⠀⢃⠀⡅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀⠀⠀⡄⠀⠰⠉⠚⠡⠂⠇⡆⡣⡀⠈⠋⠖⡉⢀⠀⢰⠐⠀⢢⠃⠰⠛⢙⠢⣄⠐⠀⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣠⠀⠌⠀⠀⠇⠀⠐⠀⠀⠀⡀⠀⠀⠀⠈⠉⠀⠀⠠⠘⡝⠋⠀⠀⠠⠀⠸⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠖⡘⠀⠀⠀⠀⠀⠀⢀⠀⢰⠅⠀⠀⢀⠀⠀⠊⠀⡀⠀⡀⠀⡌⠀⢀⠀⡠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠐⠀⠆⢠⢁⢰⠀⠀⠰⠁⢠⠀⠐⢸⡦⢇⡀⠾⡇⠠⡀⠐⠀⠠⡙⠰⠀⡀⡹⠊⠁⠁⠀⠀⠀⠐⠚⠁⠈⠊⠑⢦⢀⢀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠰⠈⠌⠈⢸⠀⡐⢠⣆⢠⠰⠟⣡⡾⠂⠀⠞⠁⡇⠁⠀⠀⡀⠀⠀⠞⠈⠀⠀⠀⠀⠄⠀⣀⣤⣤⡀⡀⠀⠠⠀⠉⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⣀⠃⢸⠫⠁⣐⠜⡩⡕⡀⠁⡀⠀⡐⡀⠐⠁⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠠⠛⠛⠛⠛⠲⠥⡒⢀⠀⠁⠂⠀⠀⠀
⡄⠀⠀⠀⠀⠀⠡⠀⠀⢀⡊⠀⠀⠀⠀⠀⢀⠀⠟⠀⠀⠀⠀⣏⠀⠀⠁⢰⠋⠀⠌⠈⢠⠎⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠔⠂⠀⡀⠀⠀⠀⠈⠐⠳⣄⠀⠀⠀⠀
⢣⠀⠀⠀⡃⢀⠘⡌⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠄⠐⠀⠏⠀⡀⣠⠞⠁⠀⠂⡡⠂⡀⠔⠀⠀⠀⠐⠈⣐⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢢⢠⠀⠂⠈⠢⡀⠀⠀
⠸⠀⠀⠀⠀⠈⠀⠹⡇⠀⠀⠀⠀⠀⡠⠀⢀⢠⠟⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠘⠁⠀⢀⣄⢱⣴⠉⡐⢀⢀⠀⠀⠀⢊⣀⢀⠀⠀⠐⠢⠀⠀⠀⢠⠀⠀⠀⠑⠀⡀⠀⠄⠈⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠰⣠⠀⠁⠰⡼⠀⠀⡀⠀⡰⠀⢀⢤⠀⢛⣽⡆⣣⣽⣷⣷⠗⡀⠠⠐⠓⢒⠁⢀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢄⢀⠀⠀⠀⠌
⢅⠀⠀⠀⠀⠀⠀⠀⣫⠀⠀⠀⠀⠀⠀⠈⠀⠀⣟⠀⠈⠳⣇⠀⠀⢄⠘⣠⠀⢂⣤⢰⢿⣿⠃⠯⣨⣿⣿⣋⡀⠀⠂⠀⡠⠔⣻⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠈⠸⡀⢠⠘⠏⠛⠋⠁⠐⣉⣥⣄⠀⠅⣱⣿⣿⣾⠂⢀⢜⠑⣂⣱⠀⠀⠀⠀⠀⡀⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⡫⠈⠀⠀⡌⠑⣂⡄⠀⠀⠐⡿⠿⠿⠛⣁⣶⣿⣿⣿⣿⢗⣀⡇⠈⠽⠃⠀⠀⠀⠀⠐⠈⠠⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢅⣶⣟⡡⠀⠐⣄⣥⣤⣤⣶⣿⣿⣿⣿⣿⠟⠨⠉⠁⠀⠀⠀⢀⣀⣀⠤⢄⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡀⠀⠀⠀⠀⠀⠀⢹⡈⠀⢀⣬⣿⣿⣵⣿⢀⠀⢭⣿⣿⣿⣿⢿⠿⠛⠁⠀⠀⠀⢀⡠⣖⣿⡦⠗⠒⠊⡙⠋⠁⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠳⠀⠁⠀⠠⢤⠀⠀⠈⠀⠀⣘⣿⣿⣿⣵⣄⠄⢷⣿⠿⠋⠉⠁⠀⠀⢀⣀⢐⣪⣷⠖⠋⠉⠁⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠈⠠⠡⠒⠀⠀⠀⠀⠀
⠀⠁⠀⠀⡠⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠾⠿⠟⠋⠁⠁⠀⠁⠀⠐⠀⢀⣀⠄⠎⢮⠿⠋⠑⠀⠀⠄⠄⠀⠉⠈⠠⠄⣀⣀⣼⣷⣶⣶⣦⣄⠀⠀⠒⠚⠀⠐⠂
⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡯⠀⠐⠢⢥⣀⡀⠀⠀⢀⣐⣊⣵⣶⠶⢒⠛⠉⠀⠀⠀⠀⠀⠀