Charity Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒⠚⠒⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣧⡤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⡹⠀⠈⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⣪⣻⡎⠀⠀⢀⡴⠋⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣒⠭⠗⠋⠁⠀⢳⢳⡀⢠⠋⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢠⡖⡪⠕⠚⠉⡔⣩⠉⠢⠀⠀⠀⢣⢣⣿⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢣⢣⠀⠀⠀⢧⡈⢟⡰⢀⡠⠴⣚⡵⢻⠆⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢯⢧⠀⠀⣀⡬⢖⣪⡵⠚⠉⠀⠉⠻⡄⠀⠈⢧⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⣠⠞⠉⢎⣓⠭⣖⣋⣉⣉⡉⠉⠉⠉⠑⣄⢹⠀⠀⠘⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢀⣼⣁⣀⣀⣈⣀⣀⣉⣉⣉⣉⣀⣀⣀⣀⣀⣈⣿⡧⡀⠀⠀⠀⣉⠢⡀⠀⠀⠀⠀⠀⢀⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⣄⠈⢿⣦⣝⣆⣀⣀⣀⣴⣿⡷⠋⠀⠀⠀⠀⣀⡤⠖⠒⢲⠃
⠀⠀⢸⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⡇⠀⠀⠸⡄⠀⠈⠙⣻⠛⠛⠋⠉⠘⢦⠀⠀⠀⠀⡴⠁⠀⠀⣠⠃
⠀⠀⢸⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⡇⠀⠀⢠⠃⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠁⠀⢀⡴⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣴⠟⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠊⠀⢀⡴⠋
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡗⢤⣀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠤⠤⠖⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⡀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠍⠉⠁⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠈⠙⠿⢿⣿⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⠐⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠁⠈⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠻⢿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠻⢦⣄⣀⣀⣠⡴⠟⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡀⠀⠈⠉⠉⠁⠀⢀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠈⠀⠉⠀⠀⠀⠀⠉⠀⠁⠂⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠎⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⣀⣠⠴⣞⣯⣭⢧⠀⠈⠀⢄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠔⠃⢀⡤⢖⣻⣽⣶⢿⣿⣿⣿⣞⣧⠀⠀⢘⠂⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡆⠘⣆⢿⣿⣯⣿⡎⣿⡿⣟⣻⠼⠂⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣞⢿⢿⣛⣯⡖⠛⠉⠀⡀⠀⠀⢸⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⠊⠀⠹⠚⠻⠷⠶⠶⠀⠀⠀⠀⠠⡀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⢨⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠸⡀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⠎⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⠢⢀⠈⠋⠀⠀⠀⠀⠈⠋⠀⠀⡀⠔⠁⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠁⡀⠛⠶⠴⠶⠋⠀⢀⠊⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠁

⠀⠀⠀⠀⠀⠄⠀⠒⠄⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠢
⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠈⠂⠀⠀⠀⠀⠀⠀⠄⠰⠂
⠀⠀⠀⢀⠀⠁⠐⠀⠑⠄⠀⠀⠀⠀⠦⠀⠀⠐⠂⡀⠀⠀⠀⠀⠐⠀⠅⠀⠣⠀⠀⠀⠀⠀⢂⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠠
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈⠀⢠⢤⡄⣤⣈⠀⠀⠀⡀⢀⠀⠀⠀⢀⠁⠀⠀⠠⠤⣀⠀⠉⢀⣀⣀⠀⠀⠀⢀⠜⠰⠖⠀⠣
⠀⠀⡂⠀⠰⠀⢀⢴⡲⡯⣯⠛⠿⣉⡆⡿⠘⠃⢯⠿⠡⠀⠠⠀⢸⠁⠉⠀⣇⠀⠀⠀⠘⡃⠀⠈⠈⠀⠀⠀⠀⠀⠠⠂⠆⠀⠀⠀⣀⢀⠑⢀
⠀⠀⠇⠀⠀⠀⢸⠛⣿⡗⡧⣍⡒⣀⡏⠁⠀⠀⠀⠂⠔⠀⠀⠀⡼⢤⠀⢀⡀⠂⠠⠀⡀⠀⠁⠀⠀⠀⠐⠘⠁⠀⣀⡇⠀⠀⠀⠀⠀⢀⡍
⠀⠀⠀⠁⡀⠐⠉⡄⠈⠥⠝⠀⠀⢀⢀⡀⠀⠠⠤⠠⡄⠀⢀⠀⢀⠀⠠⠄⠄⠤⠠⠀⠁⢀⡀⠀⠀⠀⠀⡀⡀⠀⠁⠁⣰⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⠀⠀⠂⢐⠁⠀⠀⠀⠀⠃⠈⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⡀⠀⠀⠀⢀⡆⢠⣤⡀⠠⠄⠀⠀⠀⡤⠤⠴⠶⠆⠓⠦⠀⠀⣠⠴⣤⣠⡇⢘⠀⠀⠀⠀⢀⣀⢀⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⢀⠀⢀
⠀⠀⠰⠒⠀⠀⠢⡀⠀⠀⠀⢀⣠⡐⠒⣲⠄⠀⠀⢃⠀⠘⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠭⢛⡄
⠀⠀⠀⠀⠠⠄⢀⣄⢠⠄⠀⣬⡟⠉⠉⠈⠳⠀⠀⠈⢆⣀⣤⠊⠀⠀⡠⠒⣆⡀⢀⡀⠀⢀⣔⠈⠑⠢⠀⠸⡉⡡⢔⠵⡎⡄
⠀⠰⠒⠋⠀⠀⠀⢘⠀⠀⠀⠙⡀⠤⣤⡤⠄⠂⠀⠀⡾⣩⣻⡧⠀⡘⠒⡗⠁⠀⠀⠐⠐⠁⠀⠉⢣⠀⠃⠀⢩⡜⡼⢋⣴⢿⡀
⠀⠀⠀⠐⢀⠀⢠⠏⠃⠀⢀⡤⢿⣿⣾⡞⢟⡇⠀⠀⠁⠻⠟⡻⠀⡇⠘⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠄⢰⠀⠀⠏⠀⠀⠑⡡⠁
⠀⠀⠠⠞⠁⠀⠘⠃⠈⠀⢸⠀⢀⣸⣿⣏⠁⡀⠀⠀⠀⠀⢀⠁⠀⠰⠅⢀⣰⠀⠀⠀⠀⠀⠀⢸⠀⢠⠰⠀⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢸⠀⢸⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡟⠀⠀⠀⠀⠀⠀⠘⠙⠉⠀⠀⠀⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⡆⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⢸⡆⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠈⠀⠀⢰⠀⠀⠈⡇⢸⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠒⢰⠆⠀⠀⠀⠀⠀⠇⢀⠀⠘⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣶⠙⣦⢤⡀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇⠀⡗⢢⡤⣤⠞⠉⠀⠀⠀⠶⠶⠞⠳⠶⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣼⠀⣿⠀⡇⠀⡇⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⠀⠁⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⢸⠶⠶⠤⣀⣀⠀⠀⠀⠀⠉⠉⢉⣻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⠉⢹⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣼⣤⣄⣀⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠦⠤⠤⠄⠀⠀⠀⠀⠀⠉⠓⠒⣿⠀⠀⠀⠀⠀⠀⢠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⡄⢠⡀⢸⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠶⠰⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⡇⢈⡀⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⣶⠒⠒⠒⠀⠀⣀⡴⠋⠉⠙⠒⣇⢀⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠣⠼⠓⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡹⢿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢧⣀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⣘⣿⠿⠶⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠸⠿⠀⢄⣉⣿⣿⡃⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⠀⠻⠿⠿⠿⠃⢀⠀⠀⠈⢛⣯⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⣾⣦⣤⣤⣤⣼⣿⣿⣿⣷⣦⠔⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠿⣿⣦⣴⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠙⠛⠁⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⠙⣿⣿⡀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣠⡀⠀⢿⣿⣿⣿⣿⣿⣷⡆⠀⠰⠂⠀⠀⠸⣿⣿⣿
⠀⠀⠀⣸⣿⡇⠀⠹⣿⣧⠀⠀⠀⢠⣴⣿⡿⠟⠿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⡟⠀⢰⣿⣷⠀⠘⣿⣿⣿⣿⠛⠁⢀⣠⣄⡀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠈⠋⠀⠀⠀⠉⠁⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠈⠙⢦⡀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠈⠻⠛⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⢀⣴⠄⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡶⠦⣤⣠⠏⠙⢦⡀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁⠀⠀⠀⣀⣴⡯⠇⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣚⡭⢶⡹⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠉⠉⠉⢲⠀⢀⣠⠴⣒⡭⠵⠚⠉⠀⠀⠀⢳⡹⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⢀⣠⣾⣿⣶⣶⣶⣶⣶⣶⢟⣪⠭⠖⠋⢁⠞⡥⠉⠳⡄⠀⠀⠀⢷⢱⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣇⠀⠀⠀⠸⡀⠉⣹⢀⡇⠀⢀⣠⠼⣃⡷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣀⣿⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⡍⠉⠉⠉⠉⠹⡜⣆⠀⠀⠀⠈⠓⣚⡭⢖⣚⠭⠖⠋⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠏⠀⠀⠀⠀⣠⢽⡜⣆⣠⠴⣚⣫⣵⢾⣭⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠻⠬⠖⢻⣷⣶⣶⣶⡆⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠹⠿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠿⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛

⠀⠀⠀⠐⡂⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⣤⣤⣴⠠⠀⠀⠀⠀⠀⢀⣀⣠⡶
⠀⠀⢸⣄⠟⠀⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⠂⠃⠀⠀⠀⠀⠀⠀⣴⣀⣮⠙⠙⠛⠈⠀⠀⠀⠀⠀⠻⠛⠁
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢶⠿⠛⠛⠿⠒⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⠀⣤⠶⠶⠴⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⢚⠷⠚⠛⠀⠀⠀⠀⢸⢴⣶⣲⡶⢸⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⡶⢦⣵⣴⢴⡀
⠀⠀⣠⣤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠸⠤⢤⣤⠬⠼⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⢹⠀⠀⠀⠀⢠⣿⣬⣤⣣⣼⡄
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠓⠒⠚⠓⠚⠁

⠀⠀⠀⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⣤⣤⠼⠀⠀⠀⠀⠀⠀⠉⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠃
⠀⠀⠀⡼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢦⣴⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢀⣀⣰⠋⠭⠿⠛
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⢙⡇⠀⠀⠀⠀⠀⠀⢼⣿⢿⣭⣷⠀⠀⠀⠀⠈⠛⠛⠒⠚⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣾⣻⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡞⠛⠁⠀⠀⢠⣤⡄⠀⠀⢔⡢
⠀⠀⢸⡿⠿⠿⢿⠁⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⠿⠁⠀⠀⠀⠀⠀⠚⣿⠇⠀⠀⢸⡇
⠀⠀⠸⠷⠄⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠚⠁⠀⠀⠀⠀⠀⠀⠿⠅⠛⠛⠁⠃⠶

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⣶⡶⣶⣶⣶⡶⡶⣶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣏⣿⣿⣿⣇⣷⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠄⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⡁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⢻⠋⢀⡀⠀⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣶⣿⣷⣤⣴⣾⣷⣶⣤⣶⣿⣷⣤⣴⣾⣷⣦⡀⠀⠀⠀⠈⠀⠀⠦⠤⠈⠀⠄⠀⠀⠶⠀⠀⡉⠀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡀⠀⡀⠀⠚⠁⠀⢀⡠⠀⠙⠋⢀⠈
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠻⠛⠋⢋⣙⡛⠯⠤⠬⣄⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠐⠄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⢛⣛⣛⣉⣉⣉⣙⣓⣒⣐⣒⠲⠙⢂⠀⠀⠀⣿⢀⠀⠀⠆⠀⢀⡋⠀⡐⠂⠀⡃⠀⠀⡀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⢸⠀⠀⢠⠞⣿⠓⡄⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠐⠀⢀⠉⠉⠠⠈⠀⠀⠀⠙⠒⠂⠁⡀⠄
⠀⠀⠀⠀⠀⠢⡈⠛⠛⣼⠀⠀⠘⢄⣾⣢⠇⠀⠀⢸⠀⠀⠸⠰⠒⠒⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠓⠒⠒⠒⠚⠸⠉⠁





⡶⣶⢶⠶⡶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⢶⣶⣶⣶⣶⡶⠶⢶⡶⢶

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⢀⣠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠛⠒⠦⡀⠀⠀⡠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣠⠖⠉⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠳⣼⠁
⠀⠀⠀⠀⠀⢀⣀⠤⠔⠋⠁⠀⣀⣀⣀⣀⡤⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠙⠲⠤⠤⠤⠄⠀⠈⠑⠢⠤⣀
⠀⠀⠀⣀⣀⣨⠤⠴⠒⠂⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⠒⠶⠥⣀⣀
⠀⠀⢘⠤⠄⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⢠⠋⠀⠀⢀⠁⠀⠀⠀⠤⢤⡗
⠀⠀⠛⠤⡄⠒⠒⠀⢀⡁⠀⠀⠀⢣⠀⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⡏⣀⣀⠂⠤⠀⠉⠑⠒⢠⠤⠛
⠀⠀⠀⠀⠉⠉⠓⠤⠖⠒⠊⠉⠉⠙⡅⠀⠀⠀⠀⠀⢀⡤⠤⠤⣀⠀⣀⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠈⠉⠉⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⢸⠀⠀⠀⠈⠉⠁⠀⠀⠀⡇⠀⣀⣀⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠷⣄⠀⠀⠀⠀⠀⣀⠜⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⡀⠀⣴⣶⣴⣶⣦⠀⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣶⣶⡶⠆⠀⠀⢀⣴⣶⣶⣄⣴⣶⣶⣄
⠀⠀⠸⣿⣆⠻⢿⣿⡿⢋⣾⡿⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣶⣾⡆⠀⠀⠀⠀⢸⣿⡿⠋⠀⠈⢻⣿⣿
⠀⠀⠀⠹⣿⣷⣄⢁⣴⣿⡿⠁⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⡿⠃⠀⠀⠀⣤⣾⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⠻⣿⡀⠀⠀⣸⡿⠋
⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠹⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⠉
⠀⠀⠀⠀⠀⠉⠛⠘⠋⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣶⡦⣄⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣤⡶
⠀⠀⠀⠀⠀⢿⣿⡿⠻⣠⣠⣀⠀⠀⠀⣿⣿⣿⠃⠛⠀⠿⢿⣿⡇⠀⠀⠀⠀⠀⢸⣥⣀⣽⣿⠿⢿⣷⠀⢠⣶⣄⢀⣴⡿⠿⠿⢃⣴⡆
⠀⠀⠀⢀⣠⣴⣯⣵⣿⣿⣿⠏⠀⠀⠀⠙⣿⢨⡄⣼⠀⣇⢸⠟⠁⠀⠀⣤⣴⣦⣤⣴⣷⡆⠀⠀⠀⠀⠀⠀⢻⣿⣮⣙⠃⢀⣠⣾⡿⠃
⠀⠀⠘⣿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⠿⠛⠁⠀⠀⠀⠀⠉⠉⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡆⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡀⣶⢰⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣰⡇⣀
⠀⠀⠀⠀⠀⢱⣿⣿⡏⢀⣀⣄⡀⠀⠀⣶⣄⢴⠛⢿⡄⢠⢆⣴⡆⠀⠀⠀⣾⣿⣿⡿⠞⠛⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⢀⣠⣤⣤⣤⣾⣿⣿⡟⠀⠀⠀⢻⣿⣮⡑⠿⠟⣫⣾⣿⠃⠀⠀⠀⠻⣿⣿⠀⠀⢀⣿⡿⠃⠀⠀⠀⠘⢶⣄⡿⠛⠛⠻⡇
⠀⠀⠺⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠙⢿⣿⡆⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⠙⠷⡶⠟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿

⠀⠀⠀⢀⣤⣄⡀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⡀
⠀⠀⢰⣿⣿⣿⠛⡟⢻⣿⣿⣷⠀⠀⠀⠀⠀⢸⣇⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⡶⠀⠀⠀⠀⠀⣾⣿⣿⣟⢳⣿⣿⣿⣿⡆
⠀⠀⠸⣿⣿⡼⠃⠁⠋⣼⣿⡟⠀⠀⠀⠀⠀⠈⢿⣦⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⢇⢀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠻⢿⣿⣿⠇
⠀⠀⠀⠈⠛⢷⣤⣶⡴⠟⠋⠀⠀⠀⠀⠀⢀⣤⣾⠟⢿⣦⣄⠀⠀⠀⠀⢴⣶⣾⣿⣷⣶⣿⣿⣿⠏⠀⠀⠀⠈⠻⢿⣿⣿⣇⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠙⠃⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋

⠀⠉⠉

⠀⠀⢀⣀⠀⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧⣶⣠
⠀⠀⢘⣟⠐⣿⣿⠂⣻⡃⠀⠀⠀⠀⢀⣰⠁⢶⡶⠈⣆⡀⠀⠀⠀⠀⠈⠀⠘⣤⣀⢿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠐⠛⠛⣵⣶⣶⣮⠛⠛⠂⠀⠀⠀⠈⠹⡴⠾⠷⢦⠏⠁⠀⠀⠀⠈⠂⠄⣼⣿⡿⣸⡿⠛⠁⠀⠀⠀⠀⢷⣼⣿⠟⠛⢿⡄⠀⠀⠀⠀⠀⢷⣬⣿⠛⠛⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣗⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣾⠿


⠀⠀⠀⣠⣶⠟⢰⣶⣶⡆⠀⠀⠀⠀⠀⠀⣠⠴⠒⠦⠄⠀⠀⠀⠀⠀⢠⣴⢶⣤⣤⡶⣦⡄⠀⠀⠀⠀⠀⠀⣴⡿⢷⣦⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠻⣶⣄
⠀⠀⣼⣿⠟⠂⠀⢙⣫⣤⠀⠀⠀⢠⡖⠖⢢⠀⠀⡔⠶⢲⡄⠀⠀⠀⢿⡅⠀⠈⠁⠀⢨⡿⠀⠀⠀⠀⠀⠀⢻⢇⣼⡟⠀⠀⠀⠀⠀⠀⣼⠟⠛⠀⠀⠛⠻⣧
⠀⠀⢻⡘⢶⣤⡀⠘⢛⡟⠀⠀⠀⠀⠑⠶⠋⠀⠀⠙⠶⠊⠀⠀⠀⠀⠈⠻⣦⡀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⢟⣤⡀⠀⠀⠀⠀⠀⢻⣦⣤⠀⠀⣤⣴⡟
⠀⠀⠀⠙⠾⣿⣥⠴⠋⠀⠀⠀⠀⠀⠀⠀⠒⠦⠴⠚⠂⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠻⡿⠋⠘⢿⠟⠀⠀⠀⠀⠀⠀⠛⠿⣦⣴⠿⠋


⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠒⠤⣀⠀⠀⠀⠀⠀⣟⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢀⡤⡒⢒⢤⡀
⠀⠀⠀⣀⣀⣼⣿⣤⣤⠀⠀⠀⠀⠀⢀⣤⣤⠴⠚⠙⠛⠁⠀⠀⠀⣠⣾⣿⣿⣿⣶⡀⠀⠀⢘⠄⠀⠀⢀⡞⠁⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠲⢤⣠⣤⡀⠀⠀⠀⢿⣷⣿⣼⣷⣿⠇⣄⣴⠊⠀⠀⠀⠈⠃⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠟⠉⠛⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⣀⣄⣘⣛⣛⠁⠀⠀⠀⠀⠀⠘⠻⠿⠿⠟⠃


⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⢸⡏⢙⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⡟⡉⣽⡯⣽⣯⢉⢻⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⣿⠀⠀⠀⠀⡎⢉⣉⠉⣩⡭⡍⢻
⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣧⡁⢻⣟⣾⡟⢉⣼⠀⠀⠀⠀⣿⠠⠤⠤⠤⠈⡇⣿⠀⠀⠀⠀⣇⣐⣂⣒⣂⣒⣂⣼
⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⡆⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⢾⣷⣶⣾⡿⠏⠀⠀⠀⠀⠀⠀⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠿⠯⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⣾⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⣴⠧⠖⠛⢩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠙⠲⠼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⢀⡀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣃⣀⣀⣀⣨⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠽⠽⠿⡍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣏⣴⣒⣒⣒⣒⣒⣶⣶⣒⣒⣒⡒⠒⣦⣿⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⣀⣀⣀⣠⣾⣿⣭⠭⠭⠭⠭⠭⠭⡭⠭⠭⠭⠭⠭⣭⢹⠀⣿⣿⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⣿⣿⠀⠀⠀⢀⠞⣩⠍⠳⡄⠀⠀⠀⢿⢸⠀⣿⣿⣀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠀⠀⠀⠸⡀⢈⡷⢀⡇⠀⠀⠀⢽⢸⣤⡿⣿⠒⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣀⣀⣀⣀⣉⣒⣒⣋⣀⣀⣀⣀⣼⢸⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠲⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠙⠷⢦⣄⡀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⠀⠀⠀⠀⢀⡠⡰⠶⠚⠋⠁⠁⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠂⠈⠉⠓⠶⣤⡤⣔⠲⠋⠊⠉⠙⠋⠛⠲⠦⣼⣧⡁⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠐⠀⠒⠒⢯⣐⣛⣣⠇⠀⠀⠀⠀⢀⠤⠀⢀⠬⢙⠻⢦⣄⡠⠄⠠⢀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠰⢭⣦⣀⠀⠀⠀⢀⠀⠀⠀⡔⠠⠙⡏⢙⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⡁⠉⠢⡈⠙⠳⣤⡀⠑⣌⢳⡦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣄⡀⠑⠄⡀⣤⠀⢰⠣⠯⣤⣠⣄⠀⠀⠢⡀⠀⠋⠀⠁⠀⠀⢰⠀⠀⠐⠻⢶⣮⣆⠙⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⢳⡏⢹⠿⣹⠗⠲⠬⣤⣉⣻⠁⠀⠀⠈⠉⠛⠖⠦⣤⣑⠪⠤⠀⠀⠀⠊⠀⠀⠀⠀⢀⣅⢻⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠙⠛⢛⡏⠀⠀⠀⠀⢸⣿⣂⣀⣐⠀⠀⠑⢄⡀⢈⡉⠛⠲⣤⣀⠀⠀⠀⠀⠀⣠⡿⠡⠊⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣧⣠⣀⡐⠀⠀⠈⢏⠋⠙⠛⠛⠶⢶⢤⣌⣑⡘⠉⠐⠊⠉⠳⢶⡄⢰⣼⣟⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠉⠙⠛⠶⣦⣤⣤⣈⣁⡚⠁⠐⠒⠉⠀⠈⢙⣛⣳⣦⣄⣈⡈⠛⣿⡿⡿⠶⣄⠐⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⠒⠒⠒⠛⠛⠛⠉⠉⠉⠳⠽⠭⠽⠋⠙⠛⠛⠛⠛⠋⠉⠉⠉⠉⠛⠿⣯⣘⣟⡠⠚⢷⡠⠊
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘




⡿⠿⠿⡿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⣛⣿⣟⣟⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠠⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠔⠤⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠤⠠⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢀⠁⠀⠄⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠈⠛⠂
⠀⠀⠀⠀⡰⠀⠠⠘⢀⠄⠂⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉
⠀⠄⣠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠀⠄⡀⠀⠀⠘⠃⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣀⣠⣤⣤⣠⣤⣤⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣀⣀⣀
⣿⣿⡉⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⡏⠀⠀⠀⣿⠶⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠈⢻⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠁⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠙⢷⡀⣿⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⠿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠘⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡴⠖⠲⡄⠀⠀⠀⠀⠀⢻⡆⠀⠀⢹⣿⠀⢠⡶⠛⠛⠛⢶⣴⠞⠛⠛⠳⣦⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠳⣆⣀⡀
⠀⠀⠀⠀⣠⡴⣴⡇⠀⠀⣿⠛⢻⡆⠀⠀⠘⣷⠀⠀⠈⠟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⠀⠀⠀⠀⠀⢠⡶⢶⣿⠀⠀⣿⠋⠙⣧
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⣇⣀⡀⠀⢻⡆⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⣀⣀
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡏⠉⢻⡆⠘⣷⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⢀⣠⠾⠁⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠉⢹
⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⠙⢷⣄⢀⣴⠟⠁⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠸
⢀⣤⡀⠀⣿⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣼⠃⠀⢀⣤⣄⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⣿
⣿⠁⠹⣦⣿⠀⠈⠃⠀⠀⠁⠀⠘⠇⠀⠈⣧⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢿⡅⠙⣧⣿⠀⠀⠉⠀⠀⠀⠀⠀⠛
⢸⡇⠀⢻⣿⠀⢀⣠⣤⣀⣀⣤⣤⡀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⣇⠀⠸⣿⠀⢀⣠⣤⣄⣀⣤⣤⣄
⠈⣷⠀⠈⠿⢀⡿⠁⠀⠙⠋⠁⠈⢹⡆⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠀⠀⠻⠀⣿⠁⠀⠈⠋⠀⠀⢹⡇
⠀⢸⡄⠀⠀⠈⢻⣄⠀⠀⠀⠀⢀⡾⠃⢀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸⣆⠀⠀⠀⠻⣦⡀⠀⠀⠀⢠⡾⠃
⠀⠈⣷⡀⠀⠀⠀⠙⠷⣄⣠⡾⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠻⣦⣠⡾⠋⠀⠀⢰
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣾⣷
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠰⠶⠂⠀⣤⢦⠀⠀⠐⠒⠀⠀⢀⡤⢄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡖⠆⠀⠀⠀⠀⠀⠀⣠⢦⠀⠀⠐⠂⠀⢠⠲
⠀⢠⣦⠀⠀⠁⠈⠀⠀⡀⠀⠀⠀⠈⠁⠘⠀⠀⠈⠀⠀⠀⣠⠄⠀⣄⣤⠀⠘⠁⠀⠀⢹⠃⠀⠀⠠⠀⠀⠀⠻⡤⠀⠀⠃⠀⠀⠘⠿⠂⠀⣶⡆⠀⢰⡆
⠀⠈⠇⠀⠀⠀⠀⠀⢀⣛⣘⡀⠀⠀⠀⢀⠀⠀⠀⢠⣄⠀⠘⠂⠀⠙⠁⠀⠀⣁⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡆⠀⣼⣧
⠀⠀⠀⠀⠀⠴⡶⠄⠸⡿⢹⡇⠀⠀⠀⠄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠋⠙⠁⢸⡇⣿⠀⠀⠁⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⣐⠀⠆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠙⠀⣀⠀⠂⠐⣀⡀⠀⠀⠀⠘⠃⠙⠀⢀⠀⠠⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠸⠋⠀⠀⠀⡈⠉⠁⠀⠀⠐⠄⠀⠀⠸⠙⠂⠀⠀⠀⠀⠀⠉⠁⠀⠘⢏⡹⠀⠀⣴⠋⢹⠀⠀⠉⣷⠀⠸⠓⠀⠀⢛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠾⠛⠗
⠀⣀⠀⠀⣿⣿⣿⣷⣀⠀⠀⠀⠀⠰⠿⠂⠀⠀⣘⣁⠀⠠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠺⣿⠇⠀⢸⡇⠀⣾⣶⡂⠀⢀⣀⠀⠀⠀⠠
⠀⢀⡿⠀⢻⠟⠛⠛⠋⠀⣾⣶⣖⠀⠀⠀⠀⠀⣿⣿⠀⠈⠁⢰⠄⠀⣾⣿⠀⠀⣤⣤⣤⠀⠀⠒⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠰⢂
⠀⠀⠀⠀⠑⠋⠀⢀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠈⠀⠀⠟⠸⠀⠀⠙⠋⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⢀⣠⢴⠞⠉⡂⠀⠀⠀⠀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠴⠊⠍⣰⠁⢁⠄⠊⠀⢀⡀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢢⢀⡠⠀⠄⠀⠀⠀⢠⡐⡇⡉⠚⡁⢀⡤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢐⠡⠂⠉⢁⠄⠋⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠒⠓⠺⢤⠐⠀⠉⠁⠀⠈⢉⡹⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠠⠄⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠉⠉⠁⠀⠀⠀⠀⠀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠄⠤⢄⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠡⢄⠀⠀⠀⠀⢄⠀⠑⡄⠀⠀⣀⣀⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⠄⣀⠀⠀⠈⠒⢄⡀⠈⢑⠠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠁⠻⡿⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠉⠉⠉⠉⡉⠉⠉⠉⢉⠉⠉⠉⣉⠉⠉⠭⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠁⠒⠉⠀⠀⠀⠀⠀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠉⠐⠄⡔⠐⠄⠦⡀⡆⢐⣆⠀⠒⠉⢰⠁⠀⢨⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠱⡀⠦⠔⠁⠃⠔⠁⠀⠀⠂⠘⠈⠂⠀⠀⠘⠀⡠⠃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⠀⠀⢀⠠⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⣀⣀⣀⠀⢀⡀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡀⠀⠀⢰⣿⡧⠀⠀⠀⠀⠀⠀⢾⣿⡆⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⡇
⣠⣤⣶⣤⣠⣴⣿⣶⣄⣠⣴⣤⣄⣠⣶⣿⣦⣀⣤⣶⣤⣄⣤⣾⣷⣦⣀⣤⣶⣤⣀⣴⣾⣷⣄⣠⣴⣦⣤⣠⣴⣿⣶⣄⣤⣶⣦
⠉⣿⣿⡍⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⢹⣿⣯⠉⠉⠉⠉⠉⠉⢹⣿⡏⠉⠉⠉⠉⠉⠉⣿⣿⡍⠉⠉⠉⠉⠉⠩⣿⣿
⣠⣼⣿⣤⠀⣠⣤⣤⡀⣠⣽⣯⣄⢀⣤⣥⣄⢀⣤⣿⣥⡀⢀⣤⣤⣀⢀⣬⣿⣤⡀⣠⣤⣤⡀⣠⣼⣯⣄⢀⣠⣤⣄⡀⣠⣽⣧
⠉⠛⠛⠋⠉⢹⣿⡏⠉⠉⠛⠋⠉⠉⣹⣿⡍⠉⠉⠛⠋⠉⠉⣽⣿⠉⠉⠙⠛⠉⠉⢉⣿⣯⠉⠉⠙⠛⠉⠉⢹⣿⡏⠉⠉⠛⠛
⢀⣀⣄⡀⠀⣨⣿⣅⡀⢀⣈⣀⡀⢀⣨⣿⣅⠀⣀⣁⣀⠀⢀⣽⣯⣀⠀⣀⣈⣀⠀⣠⣽⣯⡀⢀⣈⣈⡀⢀⣨⣿⣅⡀⢀⣀⣁
⠛⣻⣿⡛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⢛⣿⣟⠓⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠛⠛⣻⣿⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⣙⣟⡁⠀⢀⣐⡀⠀⢀⣻⣟⡀⠀⢀⣂⠀⠀⣈⣻⣋⠀⠀⣀⣁⠀⠀⣘⣿⣃⠀⢀⣐⣀⠀⢀⣹⣟⡁⠀⢀⣀⡀⠀⢀⣻⣟
⠿⠿⠿⠿⠖⢻⣿⡟⠓⠿⠿⠿⠿⠚⢻⣿⡛⠺⠿⠿⠿⠷⠚⣿⣿⠓⠺⠿⠿⠿⠗⠛⣿⡟⠓⠿⠿⠿⠿⠖⢻⣿⡟⠲⠿⠿⠿
⠀⠀⠀⠀⠀⠘⣿⡃⠀⠀⠐⠀⠀⠀⢹⣿⠃⠀⠀⠀⠂⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠘⣿⡃
⠶⢿⣿⠷⠴⠿⠿⠿⠧⠾⣿⣿⠷⠾⠿⠿⠿⠦⠿⣿⡿⠦⠾⠿⠿⠷⠴⢿⣿⠿⠶⠿⠿⠿⠷⠾⢿⣿⠷⠴⠿⠿⠿⠦⠾⣿⡿
⠀⠻⡿⠃⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠘⢿⠗⠀⠀⠠⠄⠀⠀⠸⣿⠇⠀⠀⠀⠄⠀⠀⢿⡿⠂⠀⠀⠠⠀⠀⠐⢿⡿
⣴⣿⣿⣷⣤⣶⣿⣶⣤⣾⣿⣿⣶⣴⢾⣿⣶⣤⣾⣿⣿⣦⣴⣾⣿⣦⣴⣿⣿⣷⣦⣶⣿⣷⣦⣴⣿⣿⣷⣤⣶⣿⣶⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠀⠀⠀⠸⣿⡇
⣠⣶⣶⣦⣠⣶⣿⣶⣄⣴⣶⣶⣤⣰⣶⣿⣶⣄⣴⣶⣶⣄⣴⣾⣷⣦⣠⣶⣶⣦⣠⣶⣾⣷⣦⣠⣶⣶⣦⣠⣶⣿⣶⣄⣴⣶⣶
⠀⣾⣿⡆⠀⠀⣁⠀⠀⠀⣿⣿⠀⠀⠀⡈⠀⠀⢰⣿⡧⠀⠀⠈⠁⠀⠀⢼⣿⡆⠀⠀⢁⡀⠀⠀⣾⣿⠄⠀⠀⠀⠀⠀⠠⣿⣿
⣠⣾⣿⣦⣀⣤⣶⣤⣀⣤⣾⣷⣄⣠⣤⣦⣤⣀⣴⣿⣶⣄⣠⣴⣦⣄⣠⣴⣿⣦⣀⣤⣴⣤⣄⣠⣾⣷⣦⣀⣤⣴⣤⣀⣴⣾⣷
⡿⠛⢻⠿⡿⠿⠿⢿⠿⠿⢿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⡀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠀⢈⡷⣰⣶⣦⣨⣆⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠙⠀⠀⠘⢵⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠗⠐⠖⣻⣀⡶⠀⠀⣄⠻⢿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⠎⣦⣴⡷⠈⠁⠀⠈⠛⠃⠀⣿⣿⣏⣿⣿⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡄⠉⠉⢳⣄⠶⠆⠀⢓⣰⢾⣏⠁⢩⡿⡏⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢀⣸⣯⠀⠒⢿⣿⣿⡛⢽⡆⢀⣰⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣅⠀⠀⡆⠀⢀⣿⣿⣏⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣼⣷⣶⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣯⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡅
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⡀
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⢀⣠⠿⢿⡟⢛⢏⣡⡟⠎⢹⣀⢽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠐⠇⣀⡦⠉⢐⠋⢈⡿⠀⢰⠈⣅⣸⠖⢻⠞⠓⢊⡇⢙⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠑⠂⠚⠁⠁⠀⠀⠰⢰⣿⣯⠟⠃⠀⠀⢠⣗⠉⣧⠜⡟⠻⣿⣿⣿⢟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠁⠀⢀⠀⣡⠟⠋⠉⢇⣀⡵⠖⢇⡼⠙⣞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠂⠞⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠟⠳⢦⣤⠴⠞⠳⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⢁⣴⢟⡁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠏⢀⣴⠟⠁⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠻⣦⡀
⠀⠀⠀⠀⢀⣴⠏⠀⠀⠘⣧⠀⠀⠀⠉⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠻⣦⡀
⠀⠀⠀⠰⣟⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢻⣿⢆⣠⠶⠚⠛⠳⠶⣄⡀⠈⠳⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠈⠳⣄⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠈⠻⣄⠀⠈⠳⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠻⣦⡀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⠳⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠻⣦⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠈⠳⣄⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣌⣷⣄⠈⠻⣦⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠻⣦⣀⣄⣠⣄⣤⣀⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠈⠳⣄⠈⠻⣦⡀⠈⠛⠶⢦⣤⡤⠶⠛⠹⣿⣦⡀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠈⣻⠅⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⢀⣀⣀⣈⣷⣄⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⢿⡄⠀⠀⣠⣾⣁⣀⣀⣀⡀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠻⣥⣤⣭⣥⣬⣷⣦⣬⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣦⣤⣴⣿⣥⣤⣾⣭⣬⣭⣭⣭⣬⠟⠀⠀⠀⠀⠀⠈⢳⣄
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⣴⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣹
⣿
⣿
⣿⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⠇⠸⣧⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⢀⣀⠀⢀⣄⣀⡀⠉⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⢰⡟⢹⡇⢸⡏⣽⠃⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢷⣼⡇⢸⡇⢿⣀⡀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣯⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠿⣿⠓⢶⡄⢀⣄⡀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⡇⢸⡇⢸⡏⢹⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠰⢿⣅⣼⡇⢸⣧⡾⠃⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣤⣀⣀⠀⢀⣀⣠⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿
⣷⣦⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣤
⣿⣿⣿⡿⠛⠿⠟⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⣀⠀⢠⠐⡐⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠃⢀⠈⠻⢾⣏⣻⡛⢆⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣽⣿⣴⣮⣥⠀⡠⠃⠀⢻⣡⡄⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⡞⠙⠛⠃⠀⠀⢀⣀⠀⢸⠃⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⢻⡇⠘⠓⠀⠀⠀⠈⠃⠀⠀⣶⣜⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣯⢉⡡⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣦⠉⣿⣄⠀⠀⠈⠉⠀⠀⢀⡼⠿⢿⣧⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣱⣶⣈⢹⣿⢂⠘⠂⢀⣠⣼⣶⣶⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣤⣿⠛⠊⠉⢉⠁⢐⣋⠉⠀⢹⡟⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠁⠀⠀⠀⢌⡇⠀⠀⠈⢡⢿⡄⡀⠀⣸⣷⡈⠰⡄⣀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠠⠀⢀⢸⡏⠀⠀⠀⣤⡨⠂⠁⠀⢻⡌⢧⡀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⣀⣥⡞⠛⠯⣭⠭⢔⡿⠳⢤⣁⡐⠙⣓⣼⣷⡏⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠞⠸⡿⠃⠀⠙⢮⠀⢹⡞⠀⠁⠀⡀⢸⡏⢿⠙⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡿⡇⠐⢈⠀⠀⠠⠀⠀⠀⠀⡇⠀⠇⠈⣆⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠛⢊⡴⠛⠖⠛⣷⣄⣠⡇⠰⠀⠀⠁⠀⠈⠀⠀⠀⠄⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠟⠁⠈⠀⠀⢀⡟⠉⠀⡆⠀⠀⠐⢄⠀⠴⠀⣀⠊⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⠢⢄⣄⡀⠀⠈⠀⠀⢀⠁⠀⠀⠀⠀⠀⠆⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠄⠈⢉⠲⡌⡐⠀⢀⣀⣀⡀⣠⡶⠤⠤⠀⠶⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡈⠦⡀⠀⠀⠀⠀⠉⠉⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⣌⡓⢀⠀⠁⠀⠀⢀⡀⠤⠤⠄⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠒⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠐⢧⠀⡄⠈⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣾⣿⣿⣧⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢿⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⣤⠺⣭⣝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠒⠀⢀⡀⢀⠄⠠⠀⠀⠀⠈⢿⣾⢷⠀⠀⠀⢠⣴⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣴⠀⠈⡷⣬⡅⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⠙⠒⠂⢀⣠⠤⠟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣦⣴⣦⣾⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠐⠁⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠹⠆⠀⢸⣶⠎⠁⠙⣦⠀⡀⠀⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡤⠖⠋⠘⠀⠀⠙⠋⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠶⠈⠳⠿⠚⠋⠀⣠⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡤⠴⠖⠛⠁⢀⡤⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠛⠉⠀⢀⣀⣤⡴⠀⣰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⡀⠀⠀⠀⠀⠀⠠⠀⢀⢀⡀⠖⠲⠟⠗⠀⠀⠀⣀⠟⠁⢡⠀⠀⠀⠀⠀⢀⣿⣿⣷⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢯⢹⡄⠳⣁⢠⣄⠰⠂⠁⠀⠣⠀⠀⠈⠒⢤⣀⠀⠀⠰⠛⠁⠀⠀⠈⣆⠀⠀⠀⠀⣼⣟⢉⣥⣀⡍⢿⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⣎⢣⣙⢶⣛⠿⠃⠀⠀⠠⠄⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠉⠀⣀⣼⣿⣿⣿⣿⣿⣿⣦⢧⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠵⡘⢢⠘⠿⡹⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⡙⢣⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⡾⢷⠀⠀⠉⠛⠿⢿⣿⣿⣿⢶⡝⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠉⠉⠿⣿⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⢄⣀⣀⢠⡤⠤⢀⡀⠀⣀⠴⠒⠂⠉⠉⠉⠒⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣀⡀⣾⡀⣙⠁⠀⠲⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠉⠁⢄⡠⠃⠀⠀⠀⠠⡀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢠⠜⢁⠀⠀⠀⢂⠀⠀⠀⠀⠱⣄⠀⠀⠀⠈⠢⢄⣀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠃⠀⠈⠢⣀⠀⠀⠑⣄⡀⠀⢀⡼⠢⣄⣀⣀⡠⠎⠀⠑⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠉⠁⠘⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄⠀⠀⠀⠀⠀⠀⢠⠖⠚⠓⠆⠀⠀⠀⠈⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢸⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠇⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡎⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⢣⡀⠈⠀⠀⠀⠑⢤⣀⡀⢀⣀⠤⠊⠀⠀⠀⠀⣠⠊⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡴⠢⠦⢀⣀⣀⣀⣀⡠⠤⠚⢏⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠃⠀⠀⠀⣀⠠⠜⠄⡀⠀⠀⠀⣠⢧⡀⠀⢀⣠⠜⠑⠂⠤⢀⡀⠀⠈⠣⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⢰⡋⠀⠀⠀⠀⠈⠉⠉⠉⠁⡀⢉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠟⢳⡀⠀⠀⠀⠀⠀⢇⠈⠋⠀⢨⠃⠀⠀⠀⣤⡀⠀⢀⡏⠀⠁⠐⠾⠡⠄⠀⢀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⢀⣷⣄⣴⠀⠀⠀⠀⠑⢄⡴⠁⠀⠀⠀⠀⠘⡀⠑⠋⠉⠓⠰⠄⣀⣀⣀⠀⠀⠀⠀⢘⠆
⠀⠀⠀⠀⡀⠤⠐⠒⠊⠁⠀⠀⡀⠔⠊⠀⠀⡈⠀⠀⠀⠀⣀⡀⣠⠤⠠⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠀⠐⠋
⠀⠀⠀⡎⠀⠀⠀⠀⡠⠴⠊⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⢄⠀⠁⠀⠠⠁⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠙⠤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠉⠒⠔⠁⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡜⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⢀⡠⠚⢄⡀⠀⠀⠀⠀⣀⠌⠉⠑⠒⠒⠂⠈⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠈⠉⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⣀⡠⠖⠛⢄⣀⠀⠀⠀⠀⠀⣠⡎⠁⠈⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠉⠛⠰⡆⠚⠋⠀⠡⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠙⣄⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈⢢⡀⢀⡠⠗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣠⡛⣆⠁⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡘⠁⠀⠈⠳⣄⡀⢨⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⢀⡍⠻⣁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡉⠀⠀⡍⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣿⡟⠉⠈⢻⣷⡆⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⢀⣤⣤⠀⠀⠀⠀⣸⣿⠋⠁⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢠⣤⣄
⣿⡄⠀⠀⢈⣿⡏⠀⠀⠀⠀⠀⠘⠿⣿⣿⡶⣾⣿⣿⣷⣶⠶⣤⣿⣿⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠸⢿⣿⣿⢶⣾⣿⣿⣶⣶⢶⣼
⣿⣷⠀⠀⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠈⠟⠙⠀⠀⠀⠹⢿⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠸⠏⠁⠀⠀⠂⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠈⠉⢱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡛⠛⠻⣧
⣾⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠘⣳⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣿⠀⠀⠀⠀⠘
⡏⠂⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠘⠟⠁⠀⠀⠀⠘⢿⠇⠀⠐⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠚⠛⠀⠀⠀⠀⠘
⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠈

⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣾⡿⠟⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣾⡿⠟⠻⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⣿⠃⠀⠀⠸⣿⡷⠀⠀⠀⠀⠀⠠⣾⣿⣦⣤⣼⣿⣿⣦⣤⣤⣀⣾⣿⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⣼⣿⣿⣤⣤⣄⣰
⣿⣧⠀⠀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠉⡿⠛⠉⠹⡿⠛⠉⠉⣉⢻⣿⣿⣄⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⢹⡟⠋⠉⢻⡟⠏⡉⠉⡉⣿
⠙⠋⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠟⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣿⣆
⣴⣾⣿⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡆⠀⠀⠀⠠⢤⣴⣾⣷⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠁⠀⠁⣿⡀⠀⠀⠀⠤
⡟⡋⠛⢻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠠⣿⠇⠀⠀⠀⠠⣿⡟⠉⠛⢻⡿⠉⠀⠀⠀⠀⠀⠀⠀⠹⣧⠄⠀⢴⡿⠀⠀⠀⠀⢰
⠇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠈⠁


⣤⣶⣶⣶⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣶⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣟⣙⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡀⡄⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣇⠇⣠⢤⢀⠤⡄⡇⡠⢼
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⢇⢸⢀⠀⠀⡅⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠉⠀⠉⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠐⠏⢹⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⡆
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⢧⠀⠀⢠⡓⢣⢈⠀⡃⠃⢸⠙⠂⡇⢹⠀⠀⡇⢸⠀⡇⡃⢸⠘⡆⡏⢱⢸⠙⠀⠀⢸⠀⡇⡊⣱⠰⢀
⠀⠀⠀⠀⠀⠁⠀⠀⢠⡀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠈⡆⠀⠀⠦⠜⠀⠀⠧⠆⠸⠴⠂⠣⠜⠀⠀⠧⠜⠀⠧⠇⠘⠐⠃⠣⠜⠘⠀⠀⠀⠸⠴⠃⠣⠼⠀⡿
⠀⠀⠀⡀⠀⠀⠐⠀⠀⠈⠂⠀⠀⠀⠘⢄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⠔⢁⣤⠐⠂⠀⠀⠀⠀⡜⠀⠀⢀⡀⠑⡀⠀⣸⠀⢀⡤⠴⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠰⠋⠔⠁⠀⢀⡴⠀⢠⠀⠀⡇⠀⠀⠀⠈⠉⠁⡰⠃⡴⠋⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⡴⠋⠁⠀⠀⠉⠑⢦⡀
⠀⠀⡰⠁⡠⠋⡁⠀⠏⠀⠀⡇⠀⠀⠀⠀⠀⠐⠁⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⢰⠁⠈⠀⢰⠁⡘⠀⠀⠀⠇⠀⠀⠀⢀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢼⠠⠁⠀⢸⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⣀⠲⣄⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢠⠀⡏
⠀⠀⠀⠀⠀⢀⡨⠒⠤⠧⢨⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡇⢀⠇⠀⢀⠀⣰
⠀⠀⢀⠀⠐⠊⠉⠁⠀⠀⠈⠓⠒⠢⡀⠀⠀⠀⠀⠈⠑⠠⠄⣀⣀⣀⣀⣠⠴⠊⢠⠏⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡸⠀⠸⢁⡴⠁⡰⠃
⠀⠰⠁⠁⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⢀⣷⠋⢀⡴⢡⡴⣢⠆
⠀⠇⠀⠀⠉⠉⠠⠐⠊⠉⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⣼⠃⠀⠀⢀⠉⠀⣀⠀⠀⢸⡇⠀⢤⣤⣀⣁⠀⣾⠞⢋⡼⠋
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣰⠃⠀⠀⠀⠈⠁⣀⠈⠁⠀⠈⠀⠉⠈⠉⠉⢉⡁⠀⣰⠟
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠉⢇⠀⠀⠈⠉⠓⢄⠀⠘⠉⠁⠀⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠁⠀⠀⢠⠃
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⢀⠊
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠃⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠑⠠⠭⠭⠩⠀⠰⢉⡱⡶⠚⠂⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢢⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⠀⠀⠀⣀⣰⡧⠞⠁⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠄⠰⠖⠲⠆⠄⢠⣄⣀⠀⢲⣰⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣩⠖⠺⢿⠉⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠾⢓⣀⣹⢦⠀⠀⠀⠀⠀⢠⠴⣾⠁⢠⠟⠀⠲⣼⠃⣸⠗⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⣛⣲⣶⢵⣶⡛⣆⠀⠀⠀⢀⡾⠀⠀⢀⠊⠀⠁⢸⠟⣾⠁⠒⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣞⣓⣓⣚⣛⣚⣒⣛⣚⣦⠀⠀⠸⣄⡟⣟⠛⣛⣛⣛⣛⣛⣛⣛⡛⠛⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢷⡤⠀⠈⡇⣟⣉⡇⢀⣀⣀⣀⣀⡀⣏⣉⡇⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠤⠤⠤⠤⠴⠖⠋⠀⠀⠀⡇⣯⣉⣇⠠⠤⠤⠤⠤⠄⣏⣉⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠒⠒⣛⣋⡷⠖⠚⣋⣹⠆⠀⠀⣇⣧⣤⣭⣭⣭⣭⣭⣭⣭⣥⣤⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⣤⠴⠒⠋⠉⠀⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇⢺⠀⠻⡀⢁⡼⠂⠀⠀⠀⠀⠀⠀⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⣠⣤⠀⠀⢀⡴⣄⠀⠙⠋⣀⣀⠀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠸⣟⣻⠗⢀⡏⠉⠉⢻⠀⠀⣇⣨⠇⠀⠀⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⣾⠉⡙⢦⣸⣇⣀⣀⣸⣀⡴⢋⠉⢳⢠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⡭⠼⠿⠽⠧⢭⡶⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠘⠃⠇⠀⠀⠀⠀⠸⠃⠇⠀⠸


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⢼⣉⡷⣞⢹⣾⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡈⢀⠀⡉⢀⡈⣉⣈⡉⢀⣀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡼⠿⣧⡽⢺⠻⡇⡿⢷⢸⠇⡿⢯⣬⠿⡟⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⢀⣸⣿⡇⢠⠀⠀⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠙⠿⢿⣷⣶⣴⣶⣿⣿⣿⣿⣧⣀⣠⣾⣭⣿⣤⣤⣀⡀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠻⣿⣿⡏⠀⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠿⣿⡿⠿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣶⣿⡟⠁⠀⠀⠀⣾⡟⠹⣿⣿⣿⣿⣿⣿⡀⠀⢋⣴⣤⠀⣉⡉⢹⡟⣿⣿⣿⡟⣉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⠿⠿⠿⢷⡄⠀⠉⠀⠀⠻⣿⣿⣿⣿⣿⣧⠀⣌⣛⣋⡘⠿⠿⢘⠁⢿⡿⠙⣧⣉⣤⡉
⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣶⣋⣠⡄⠠⣶⣶⣷⣾⣿⣿⡿⠂⠈⠀⠀⠉⠉⠙⠀⢹⣿⣭⣿⣶⣶⣿⣿⣿⣷⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⣀⣢⣽⣿⡛⠛⢁⣤⡀⢀⠸⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢸⣿⡿⠉⠛⠿⢿⣇⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣶⣿⣿⡿⠛⢹⠀⢶⣿⣶⣿⠏⠀⠀⠀⠀⠀⠀⠤⠀⠀⣤⠀⣀⣸⣶⣤⣤⣴⣶⡅⠘⠃⠀⢲⣼⣿⠟⠛⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠾⠿⣣⣾⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⡿⣀⠀⠀⡀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣿⣣⣿⡿⠟⠉⠹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠀⠈⣿⣿⢀⣤⣤⣤⣀⣿⣷⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢀⣀⣀⠀⠀⠀⢣⡀⠀⠒⠀⢼⢿⡿⢿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠹⢿⣿⠏⠀⠀⣀⣠⣿⣿⣧⠀⠀⣴⣿⡿⠿⢷⣀⣀⠀⠀⠀⡀⠀⠈⣼⢠⡀⠀⣀⢸⡟⣧⣀⣀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⠘⢿⠀⢠⣿⣿⣿⡿⠛⠉⠶⣤⣿⠁⠀⠀⠀⢹⣿⣧⣾⣿⣿⣿⣷⣾⣀⡤⣤⢉⡼⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠀⠀⠠⢶⣿⣿⣿⠇⠐⠀⠓⠶⠆⠒⠀⠀⠒⠀⣿⡏⠉⠉⣻⣿⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠛⠛⠛⣶⣟⣷⣶⡾⣿⣇⠀⠀⠋⣿⣿⠟⠃⠀⠀⠹⠀⠀⠀⠀⠀⠶⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⢘⠛⠛⠛⣛⠛⠛⠋⣾⣿⣿⣷⣤⠄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣶⣿⣏⠉⠉⠀⠐⣶⠂⠈⠉⣉⣿⣿⣦⣤⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠸⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠿⠿⠿⠶⣶⣤⣤⣄⣀⣀⣀⣀
⠀⣀⠀⣀⣀⣀⣀⣀⡀⠀⠀⡀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣤⣤⣀
⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠉⠀⠀⡰⠋⢀⣴⣾⣿⡿⠿⠛⢀⣀⣀⣤⣴⣾⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣦⡀
⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠊⢀⣴⡿⣟⣯⣷⣾⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠹⡿⠿⢿⣿
⢸⡿⣫⣿⣿⣴⢸⣿⣿⣿⠀⢀⡠⢞⣯⣷⠿⠟⣛⣩⣥⣤⣤⣤⣤⣄⣀⡀
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣴⠾⠛⠉⠴⠶⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣀⣤⣶⣶⣦⣤⣀⡀⠀⠈⠉⠛⠻⢿⣿⠟⠁
⠀⠘⠿⠿⠛⠻⣿⣿⣿⡿⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢀⡞⠛⠉⠀⠀⢑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢸⠁⣀⣠⣴⡖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠛⢿⡟⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣤⡄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠁⠀⠀⠀⠀⠀⠀⠸⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣉⠉⠉⠉⠛⠛⠲⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣀⡈⠉⠉⠙⠒⠀⠀⠀⠀⠘⢧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡈⠉⠓⠲⠤⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣍⡉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠉⠛⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢻⡟⢹⡷⠶⣄
⠀⠀⠀⠀⢦⣀⣠⡴⠟⠁⠀⠀⠀⠀⠀⢀⣰⣟⠀⠀⣀⡠⠤⠀⠤⣤⡀⠀⠀⠀⣴⠋⢀⡞⠀⡼⠁⣸⠏⣷⡀
⠀⠀⠀⢠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠈⠉⠉⠀⠀⠀⠀⢠⡟⠛⢦⣀⡜⠁⠀⡜⠀⡸⠁⢀⠏⢀⡏⠙
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⣸⠃
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠶⣴⠇⠀⡆⠀⣸⠁⠀⠀⠀⣠⠤⣄⡀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⢸⠀⣸⠀⢠⠃⢀⠆⠀⡐⠁⠀⠈⠻⣧⣀⣤⡤⠶⠶⠒⢦⣴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠛⠳
⠀⠘⢴⡇⢀⡎⢀⡾⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠙⠛⠷⠾⠷⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⢀⡀⠀⠒⠦⣤⣍⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡆⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣄⣀⠀⠈⠒⠒⠤⢤⣀⣈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡶⠦⠤⠤⠌⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠄⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣠⠤⠄⠀⠁⠐⠂⠛⠀⠈⠀⠰⣏⢑⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠐⠎⠟⡀⠀⠀⠀⠀⠀⠀⢀⠀⢂⢈⡄⣸⣇⣿⣧
⠀⠀⠀⠀⠀⠀⢐⠼⢿⣿⠂⠐⢻⠏⣩⠶⣊⠀⢀⣀⡼⠃⣾⣼⣟⣽⠯⠘⠙⣶⡄
⠀⠀⠀⠀⠀⣶⣴⢦⣆⣦⣄⣀⠁⣺⣧⠞⠁⠀⠀⣉⣈⢸⣿⠿⡛⢥⣾⠌⣻⣿⡇
⠀⠀⠀⠀⠀⢿⣿⡾⣿⡇⢀⣉⠀⣉⢡⣄⣹⣶⣷⣿⠏⡌⠠⠤⠜⢾⣿⢀⡜⠛⡀
⠀⠀⠀⠀⠔⠚⣿⡟⠋⣱⡾⣿⣆⣹⣿⣼⣿⠟⠉⠙⣿⣿⣦⠆⢀⣬⡅⠀⢤⣄⣿
⠀⠀⠀⠀⠐⠼⢿⢣⠈⣿⠿⣯⣼⠿⠟⠛⠁⠀⠀⠀⠀⠋⣿⠆⠄⣈⠛⢠⣰⠿⢟⢢
⠀⠀⠀⠀⠀⢘⡷⠀⣤⣤⣼⣿⡿⣶⣶⠂⠀⠀⠀⠀⠀⠀⠈⢹⡎⠀⡴⠖⠃⣾⣷⠉
⠀⠀⠀⠀⢠⣾⡿⢺⣏⣿⣿⡿⠀⠘⠉⠀⠀⠀⠀⠰⣿⡿⠆⠈⠓⣾⡿⢦⣾⣿⠏
⠀⠀⠀⠀⠀⠻⠷⣾⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣧⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠀⣾⣄⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⣤⡀⠀⠈⠁⠀⠀⠀⢀⣠⣾⣿⡯⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣠⡾⢻⣿⢷⣄⠈⠙⠑⠒⠶⠶⣴⠾⠟⢁⣬⣅⡀
⠀⠀⠀⠀⠀⣼⣡⣦⣴⣾⣿⣷⣬⡓⠦⣤⣤⣀⣀⣠⣤⣴⠟⣉⣽⣷⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣽⣿⣿⣶⣦⣌⣭⣍⣥⣤⣶⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣼⣿⣿⣩⣿⣿⣿⣿⣿⣷⣭⣿⡿⠛⣿⣿⠿⣻⣽⣿⣿⣯⣥⣿
⠀⠀⠀⠀⣿⠛⠛⢛⡻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⠀⠀⠘⠁⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⣿⣷⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⠛⠳⡄
⠀⠀⠀⠂⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⣿⣛⣛⣋⣉⣹⣽⣿⣿⣿⡇⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢫⣿⣿⣿⣿⣯⣴⣶⣿⡿⠿⢛⣛⣛⣛⣻⣿⣿⣶⠀⠀⠀⠙⢦
⠀⢀⠀⠀⠀⠀⢨⢧⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⡏⠛⢿⣀⡀⠀⠀⢀⠀⠀⠼⡀
⠀⢸⠀⠀⠀⠀⠘⣿⣿⣿⣮⣟⣛⡻⠿⣿⣿⣿⣖⢺⣿⣿⣿⣿⣧⣄⠀⠉⠉⠓⣦⢃⠄⠀⠀⣘
⠀⠸⠀⣀⡀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣼⣿⣶⣷⣬⠏
⠀⡜⠃⠉⠉⠀⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⠿⠿⣿⣿⣿⡿⠿⠛⢿⣟⣿⣿⣿⣿⣹⡏
⢸⠀⠀⠀⠀⠀⠀⢻⣾⣿⠻⠿⠿⡿⠟⠛⠉⠁⠀⠀⠀⢀⣿⣦⣷⢁⣀⣠⣿⣿⣿⣿⣿⡿⠃
⢌⢰⣆⠘⣷⣻⡿⢿⣿⡿⢷⠚⠀⠀⡀⢀⣀⣴⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣇
⠀⠉⠉⣛⣽⡿⠿⣿⣿⣷⣶⣶⣶⣾⡟⠦⡍⠐⠂⣉⣩⡭⠯⢥⣾⣿⣿⣿⡶⠖⠉⠁⠻⣷
⠀⣠⠊⠙⣿⣿⣷⣮⣙⠛⠛⠿⠛⠛⠃⣀⣴⣾⠟⠉⠁⠀⠀⠈⢿⣿⣿⣿⠃⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⣤
⣼⣿⠣⣄⠉⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣍⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠀⠀⠀⠀⠀⣠⣶⡏⢀⢀⣉⣿
⣿⢂⣷⣿⣿⣦⠠⣾⣏⡉⠙⠻⠿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣌⠠⠀⠀⣀⣠⡆⢠⣤⣀⠀⠙⠲⣤⣠⣾⣿⣿⡑⣸⣯⣿⣿
⢻⣿⣿⣿⣿⣿⣷⢻⡿⣿⣯⡛⣿⠟⣿⣿⣻⣿⣦⠀⠀⢀⡆⠀⠀⠀⠈⠙⠟⠳⣶⢟⢋⡉⣴⣾⣿⣿⡗⠀⠀⣤⣿⣿⣟⣜⣿⣿⣿⣿⠋
⠀⠙⠿⣿⣿⣿⣇⠾⣧⣸⣿⣿⣿⣤⣈⣬⣽⣿⣿⣷⣄⣾⣷⢦⡂⠀⠀⠀⢀⣼⢯⠺⡝⣱⣿⣿⣿⣿⣇⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⡄⠑⢈⠀⠀⣼⣟⣷⡟⣾⣿⣿⣿⣿⣿⣙⣿⣿⣿⣿⣿⣿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠷⠀⢻⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⠁⠈⠻⣿⣿⣿⣿⣯⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⠁

⣀⢀⣠⡀⠀⠀⣀⡀⠐⠂⠀⠄⠀⠚⠀⣀⡀⠀⠀⡈⠛⠙⠉⠃⠀⠀⠀⠀⠀⠁⠀⠀⠢⡀⠁⣠⣀⠀⠁⠘⢀⡀⢀⠑⠈⠉⠀⠀⠀⠀⠁⠀⠘⡟⠄
⠈⠉⠀⢀⣾⡁⣀⠀⠱⠀⠀⠀⡴⠋⠩⣽⣿⣿⣯⡽⣆⠀⠀⠀⢨⠿⠟⠷⢤⠀⠀⠀⠀⠀⢰⠃⠬⠽⣆⠀⠀⠘⠀⠀⡀⠁⠈⠲⠖⠀⠈⠀⠀⣀⠉⠀⡈⣷
⠀⠀⠀⠸⡄⠀⠈⠀⢸⠀⠀⠐⠃⢀⠀⠆⠀⠘⠉⠄⡋⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠀⣼⠀⠀⢠⠀⠀⣧⠴⠄⠴⠤⣖⠒⠒⠒⢣⠂⠀⠀⠀⢷
⠀⠀⠀⠀⠁⠀⠥⠀⠀⠀⠀⢨⠀⠈⣀⠀⠦⠀⠀⢀⠁⠀⠀⠀⠁⠀⠤⠄⠀⠁⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⡄⣤⠄⠀⢻⡀⢠⠄⠐⣉⣉⣉⣉⣉⡀⠠⠤⠀⠊
⠀⠀⠀⠀⠀⣅⣀⣠⠀⠀⠀⢈⠀⠀⢨⠂⠀⢺⠀⠀⠀⡆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⡠⢄⠀⠙⠛⠋⠀⠀⠀⠈⠢⠇⡄⠀⠀⠈⣇⣤⠜⠁⢀⠀⠀⠒⠦⠒⠂⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⡤⣀⣀⣢⣄⠀⣀⡀⠐
⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⡙⠯⠯⡲⡄⠀⠀⠀⠀⠠⡀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠚⠝⢀⣶⠀⠀⠈⠀⠀⢰⠋⣡⠉⡆⠠⠉⠂⠋⢀⠶⠝⠧⡠⣵
⠀⠀⠐⢃⡘⠀⠀⠀⠀⣦⠠⡤⠤⢴⡅⠀⠀⠀⠸⡀⠀⠀⠀⣀⢀⠀⠀⠀⠀⡌⠀⠀⠀⠀⡅⠀⠀⠀⠀⡟⠀⠀⠀⢻⡆⠀⠀⠀⠘⡆⠀⠀⠀⠸⠀⠀⠀⠀⠐
⠀⠠⠀⠈⠀⠠⠀⠀⠠⠐⠄⠴⠄⠐⠀⠀⠀⠀⠐⠳⠀⢂⠀⠊⠀⠀⠀⠀⠀⠁⠠⠔⠤⠀⠃⠀⠀⠀⠀⡇⡀⠀⠀⣸
⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⡖⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⡀⢀⡄⠀⠀⠀⠀⠀⣇⡗⠀⠘⡯⠀⠀⠀⠀⠀⢰⠁⠀⠈⡀
⡀⠀⣀⣤⣀⣤⠀⣀⡤⠈⠂⠠⠀⠄⠀⢀⠀⠀⠀⠐⠀⠄⠀⠀⢀⠀⠀⡀⠀⠀⠒⠠⠄⠓⢀⡄⠀⢠⡀⠋⠀⢀⠠⠉⠂⠀⠀⠀⠄⠀⠠⠤⠄⠀⠠
⠉⡛⠉⠉⠇⠉⠁⠀⢰⠂⠀⠀⠀⠀⠀⢺⠁⠀⠀⡀⠀⠀⠀⠀⠨⠏⠭⠟⢦⠀⠀⠀⠀⠀⠿⠤⣀⠀⢙⡄⢀⠀⣀⠀⠠⠁⡏⠁⠀⠀⠀⠀⠀⢀⡛⠣
⠀⠁⠀⠀⠏⠉⠀⠋⠸⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠃⢀⠀⠠⠀⠀⠀⠀⢻⠀⠀⠀⠀⠐⡂⠀⡀⠀⠸⠁⠢⡐⠂⠀⠀⠄⠁⠠⡁⡞⢀⠀⠀⣾⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⡀⠋⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠘⣀⠸⡄⠀⠄⠀⡸⠈⠤⠀⠀⠈⠤⢾⠆⠠⠁
⠀⠀⠀⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰
⡤⣠⠀⠀⠀⠄⠒⠀⠀⠀⠢⢄⣀⠀⠀⠀⠀⠁⠀⣠⠀⠀⢄⠀⠁⠂⠀⠂⠁⠀⡄⡀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠘⣫⡄⠀⠀⠀⢀⡀⢾⡀⠀⠈⠀⠀⠀⠀⠰⠂⠓⠀⠀⠀⡐⠀⠀⠀⠀⢀⣀⠀⢸⡗⠀⠀⠀⠀⠀⢠⡶⠪⠳⣶⡉⠀⠀⠀⠀⢈⠃⠈⠉⣷⠀⠰⠋⠐⢴⡳
⠀⠀⢰⡆⠀⠀⠀⢠⡇⠀⠈⠀⢲⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⣺⡿⡄⢤⣿⣄⠀⠀⠀⠀⢿⢔⠀⠐⢿⡇⠀⠆⠆⠆⢰⠂⠀⠈⠀⣿⠀⠀⠀⠘⡇
⠁⠀⠘⠀⠀⠀⠀⠀⢣⠀⠤⠀⣈⠂⠀⠀⠀⠀⢲⣀⠢⠄⣰⠃⠀⠀⠀⠀⡟⡀⠰⠀⢨⠅⠀⠀⠀⠀⣏⠁⠉⠉⠉⢃⠀⠀⠀⠀⠘⠆⢀⣢⠀⠟⠀⠀⠀⠀⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠈⢗⣀⣺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⡘⠀⠀⣧⠃⠀⠀⠀⠀⢻⠆⠘⠁⠀⠻
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⠓⠛⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢀⡀⢀⠠⠤⠤⠤⠼⠠⠀⢀⡄⣀⠓⠃⠀⠀⢀⠋⠀⠀⠀⠀⠀⠚⠢⠔⠚⠀⢀⣀⣀⡤⡄
⡄⠀⠀⠀⠀⠄⢉⡁⠦⠀⠀⠀⠀⢠⠔⠇⠶⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡝⠊⠉⠚⡆⢀⡠⠀⠀⠠⠔⠋⠢⣤⡀⠀⠀⣀⡀⢎⡁⢩⡀
⠉⠀⠀⠀⢪⠀⠀⠀⠀⣤⠀⠀⢲⣎⡉⠀⠀⠁⢁⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡔⠐⠒⢂⠂⠂⡇⠀⠀⠀⡇⠈⡇⠀⠀⢰⣶⠀⠈⣧⡐⠒⠒⠺⣗⣅⠀⠈⠉⠉
⠀⠀⠀⠀⠸⡄⢀⡀⢠⡧⠀⠀⠀⠈⠆⢀⣦⡄⠠⠁⠀⠀⠀⠀⠈⠣⠼⠆⠜⠁⠀⠀⠀⠀⠀⢰⠀⠄⢰⠀⠀⠀⠀⠀⢿⠋⢃⠙⢙⠇⠀⠀⠀⠘⠻⡄⠨⠀⠸
⠀⠀⠀⠀⠠⠸⠆⠠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⢦⠀⠀⠀⠀⠀⠀⢈⠁⠀⢀⠀⠀⠀⠀⠀⡇⡄⠀⠀⡼⠐⠂⠒⠒⠒⠂⠈⠀⠈
⠀⠀⠀⠀⠁⡀⠠⡀⠁⡈⠠⠄⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠉⠒⠒⠈⠀⣠⠀⠀⣀⠐⠀⢀⣤⡀⠐⠀⣀⣀⠀⠉⠋⠀⠀⢛⠉⠀⠀⠁⠈⠉
⠀⠀⡌⠓⣄⠈⠒⣄⡐⠁⠀⠈⠑⣀⡄⠀⠀⠀⠀⠈⠀⠐⣶⣤⡀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠁⡐⠉⠄⠹⢻⠀⠘⠡⠈⠀⠀⢴⠀⠀⠀⠀⡀⣀
⠀⠀⠑⠄⢠⠈⠉⠉⠚⣆⠁⠀⠀⢀⠁⠀⠀⠀⢘⠀⠠⠀⠀⢹⡇⠀⠀⠀⠀⡆⠁⠀⠀⠀⡔⠀⠀⠀⠰⢖⠀⠀⠀⢘⡇⠀⠀⠀⠀⣠⠈⠀⠄⣠⠰⠀⠀⠀⡀
⠀⠀⠀⠰⠈⠀⠀⠀⠁⠛⠄⠲⠂⠋⠈⠈⠀⠀⠈⣤⠀⣤⠀⠸⠁⠀⠀⠀⠀⢠⡀⠂⠂⣐⠇⠀⠀⠀⠐⠘⠀⠠⠀⠠⠡⠀⠀⠀⠀⠇⠀⠄⠀⠘⠈⠈⠉⠀⠉
⠀⠚⡠⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⡍⠀⢂⠀⠀⠀⠘⡄⠉⢰⡎

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⣰⣧⡈⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⣰⣿⣿⣷⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠙⠋⣁⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢠⣤⣤⣤⣴⡀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡐
⡀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠢⠀⣶⣫⠭⠾⡧⠀⠠⠋⠁⠒⠠⣀⠀⠔
⠀⠁⠒⡀⡀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠁⠻⠀⠀⠀⠐⡀⢈⡒⠒⠄⡀⠀⡋
⠀⠀⠀⠇⠀⢈⠐⡤⣀⠀⠀⠑⢄⠀⠀⠀⠐⣆⢀⣄⡀⣱⠋⠀⠀⢀⠞⠊
⠀⠀⢀⠂⠈⠀⠀⠟⠾⢉⠖⠤⠄⠈⠢⢀⣄⠭⢟⡾⠞⠉⠀⠀⢠⡃
⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣨⠡⡴⠋⠀⠀⠀⠀⢠⣗⣅
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡿⢋⣤⠂⢀⠀⠀⢀⢷⡥⣏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⣶⡿⢃⣴⠏⣰⢃⠎⣜⢽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠱⢵⣯⣋⣼⣇⠎⡜⠉⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠎⠀⠙⣆⠀⠘⡿⠯⣧⠉⠀⣠⣾⠟⠙⢆
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⢄⠀⠀⠸⠒⠈⠀⢠⣽⣶⢶⢿⣧⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢈⡀⠙⠉⠀⠀⠀⠈⠘⠉⠁⠸⠢⡳⣦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢌⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠁⠀⠀⠑⢍⠣⣠⡀
⠀⠀⠀⠀⢀⠄⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡀⠀⠀⠀⠑⠈⠻⡆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠂⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠈⠀⠀⠁⠀⠐⠃⠈⠀⠠⠀⠀⠌
⠀⠄⡀⠀⠀⠀⠩⡀⠄⠀⢀⠌⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⢀⢠⠀⠀⠀⠀⡀⠀⠐⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠄⠂⠃⢀⡀⠘⠒⠠⠀⠀⠀⠠
⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠄⠐⠁⢀⠀⠀⠀⠀⠀⠀⡃⠈⢢⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠐⠈⠀⠀⠀⠀⠁⠄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢀⠈⠀⠀⠀⠐⠠⠐⠂⠄⠀⠀⠀⠀⠀⡀
⠀⢡⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠀⠀⠀⢀⡐⠀⠀⢊⡀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⡀⢌⠀⠀⡱⢀⠀⠀⠀⠁⠀⠁⠉⠀⠂⠀⠀⠀⠀⡈
⠀⢁⠌⡄⠃⠀⠐⡀⠀⠀⠀⠄⠀⠐⠀⠀⠐⠠⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠌⡑⠊⠁⠀⢀⠀⠀⠀⠀⡀⠀⠀⠘⠠⠁⡈
⠀⠈⠀⠀⠀⠐⠀⠐⠂⠐⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⢊⠀⠁⠀⠀⠀⠀⠁⠊⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠑⠀⠀⡀⠀⠀⠃⠀⠀⠘⠁⠀⠀⠀⠀⠠⠒⠐⠂⠄⠀⠀⠀⠃⠀⡰⢆⡀⠀⠐⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠢⠀⢀⠐
⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠖⠂⡀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠑⠄⡀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡐
⠀⠀⡁⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢈
⢀⡀⠁⠀⠀⠀⠊⠀⢄⠀⠀⠐⠀⠑⠄⡀⢀⠀⢀⡀⢀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠢⠡⠀⠀⡄⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠡⣀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠠⢀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⣀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠤⠀⠀⠀⠀⠂⠈⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠄⠑⡀⠀⠀⠀⠀⠠⠀⠠⡀⠀⠈⠁⠈⢀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠰
⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠁⠈⢉⡁
⠈⢠⠀⠀⠔⠀⠀⠀⠀⠀⠁⠀⡀⠠⠀⠠⠈⠉⡀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠊⠈⠁⠀⠂⠀⠄⠨⡀⠂⠀⠀⠰⢀⠁⠠⠀⠀⡄
⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠈⠢⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠀⡈⠀⠀⠀⠈⢀
⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⡀⠐⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠃⠁
⠀⡠⠁⠀⠀⠀⠀⠠⠀⠀⠄⣀⠀⠀⠄⠀⠡⠀⠠⠄⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠡⠀⠀⠈⠀⠀⠀⠀⢁⠀⠀⠀⠀⠈⠀⠀⠀⠈⢄
⠀⠐⠂⠀⡀⠀⠁⠀⠀⠀⠀⠨⠀⠀⠀⠤⠊⠀⠀⠑⠤⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠠⠀⠆⠀⠀⠀⠀⢀⡀⠰⢀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠆
⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢄⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣰⠉⣄⠀⠀⡠⠉⢆
⠀⡀⠀⠀⠀⠀⢤⠆⠀⠀⠰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢆⠀⠀⠰⢀⠀⡀⠀⠀⠀⢀⡀⡀⠀⠀⠀⡀⠤⠃⠀⠀⠘⠄⢈⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠜⠣⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠰⠨⡄⠀⠁⢠⠄⠂⠀⠀⠀⢂⠀⠀⣁⢐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⡄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠡
⠀⢊⠀⠀⠀⠀⣄⠀⢀⡀⢀⣠⠀⠀⠀⢀⡕⠀⠀⢊⠀⠀⠀⢠⠀⠔⠑⠀⠀⠀⡀⠀⠀⠈⡅⠀⠀⣨⠀⠀⠀⠀⡂⠈⠀⠠⠂⢤⠀⠀⠀⠀⡑
⠀⠐⠁⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠐⠣⠜⠀⡐⠣⠜⠂⠀⠄⠠⠀⠀⠀⠀⡀⠀⠠⠐⠈⡁⢉⡅⠈⠁⠒⠄⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂

⠻⠛⠋⠉⠉⠉⠉⠛⠷⣄⡀⠈⠣⡀⠉⢿⣿⡟⠋⠙⣿⠺⣿⡯⣷⡗⠀⠀⣏⣿⣿⡻⠀⣿⣵⣍⣄⣔⡀⢂⣙⢻⢿⣾⣥⣿⣸
⠀⠀⠀⠀⣀⣀⣀⣀⡀⡈⢷⡄⣹⠀⣀⠀⠀⠀⣠⠴⣋⣣⢻⣷⣵⡇⠀⠀⢸⣀⣻⣷⢊⣢⣾⠿⠟⠛⠛⠛⠛⠿⢷⣬⣙⡛⢿
⣴⣿⣟⠟⠛⠋⣛⡛⠿⠿⠿⡧⢥⣤⠞⠀⠀⢰⠏⢲⣏⠹⠞⣿⣿⣿⣦⠀⢺⡜⠬⢏⡿⠛⠋⣀⠀⣀⠀⠀⠀⠀⠀⠈⠻⣶⣿
⣿⢏⡠⡄⠙⣡⣯⣽⣿⡃⠀⢧⡀⣼⣚⣶⠆⣼⠶⣞⣋⠉⣳⣿⡿⣿⠖⠀⣈⣷⠀⡿⠀⠀⢰⣿⣿⣿⡿⠷⢿⢶⣦⣤⢀⢸⣿
⣿⠠⠛⡒⣸⡽⠟⠿⠿⠃⠀⡀⠑⢪⣟⠀⢠⢇⣾⣯⣵⣿⣿⠟⣺⣿⣠⣤⣽⣿⣎⡇⠀⠀⢸⡿⠡⡐⠤⠴⣓⡲⢂⣛⡿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣥⣀⣠⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⢯⣄⠀⠀⠈⠓⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠘⡆⠀⠙⢷⣀⡠⠤⣄⠑⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡸⣅⠀⠀⢰⠃⠀⠀⠀⢱⠀⠑⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣮⡳⣄⠘⣄⠀⠀⠀⡸⠀⠀⠀⢱⠈⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⣙⣮⡳⣌⠑⠒⠉⠙⢷⣄⠀⢰⠀⣠⣞⣀⡀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈⠉⠁⠙⠈⠓⠀⠀⠐⠈⠋⠃⠋⠘⠉⠉⠉⠁⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠳⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣳⣾⡖⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠁⠙⠛⢮⡁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⡠⠛⢖⢦⣠⡀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠣⡀⠪⠟⠳⠊⣠⠎⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣷⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⣿⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠈⠑⠲⠶⣾⠞⠋⠉⢷⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡆⠀⡀⠀⠀⠀⠀⠀⡀⢸⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢣⠀⠛⠗⠂⢰⠚⠛⠂⡘⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠄⠀⠠⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠠⠀⠁⠀⡀⠀⠰⠊⠉⠃⠀⠀⠀⠀⠀⠀⠀⡏⠑⠂⠈⠀⡠⠈⢁⠄⠓⠐⠲⢌⠂⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⠇⠒⣛⣓⠤⣄⢅⠀⠈⠢⢄⠘⢦⢳⡄⠀⠐⠎⠀⢀⡜⡰⠂⢀⠔⠁⠀⣀⣤⠤⠒⠛⠓⠰⠀⠁⠰⠂⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢰⢁⡔⠋⠉⢁⠝⠉⠛⠲⠤⢀⡒⢄⣢⣵⣧⡂⠤⠤⠄⣀⣼⣅⡪⣖⣡⣤⠾⠿⠒⠋⠉⠑⣂⣄⣀⣀⠀⠈⠛⠂⠈⢳⡀
⠀⠀⠀⠀⠀⢸⡏⠀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠈⠙⠛⠛⠿⢧⡤⠤⡴⠿⠟⠛⠋⠁⠀⠀⠙⠢⡄⣤⠒⠉⠀⠀⠀⠈⢹⢣⠃⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢸⡇⠘⠀⠐⠀⡠⠋⠉⠉⠉⠉⠉⠲⣦⡀⢄⠀⢱⡞⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⡀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣰⠟⠷⣴⠋⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠙⣄⠈⢺⡗⠁⠀⠀⠐⠀⠀⠀⠀⠀⢘⣿⢥⡤⠙⠢⢀⣀⡠⢧⠀⢠⡀⠀⠀⠀⡇
⠀⠀⠀⠀⡰⢃⠀⢐⠃⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆⢰⡇⠀⠀⡠⠀⠀⠀⠀⠀⢤⡾⢿⢦⡀⠀⠀⠀⠀⡀⠀⠑⠀⠁⠀⠀⠺⠁
⠀⠀⠀⢀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⡿⠦⣄⣀⣀⣀⣀⡤⠔⡿⠁⡼⢸⠙⠲⡖⠂⠒⢺⡓⠢⠤⠤⠤⠤⠃
⠀⠀⠠⠀⠑⡤⠐⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡇⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠃⡰⠃⢸⠀⢠⣿⠀⠀⠈⡇
⠀⠀⢰⠀⠉⠀⠀⠀⢤⡈⢀⠔⠚⠑⡀⠀⠀⠀⠄⠀⢀⡞⠒⠚⡟⠒⠒⠀⠀⠀⠨⠶⣾⠞⠁⠀⠀⠀⠈⡟⡇⠀⠀⢿
⠀⠀⠀⠱⣄⠀⠀⠀⠄⠉⠁⠀⢀⠝⠀⠀⡄⠀⣀⠴⣋⡀⣀⣨⣇⣀⠀⣀⠐⠀⠀⠈⡇⠀⠀⠀⣇⠀⠀⠀⣷⠀⠀⢸⡀
⠀⠀⠀⠀⡏⠳⣄⣀⣴⡤⣤⠚⠁⢠⡄⢻⠛⠉⠙⠄⢁⠀⠀⢀⡇⠀⠀⣸⠀⠎⠀⠀⡇⠀⠄⠀⣼⠀⠀⠀⢹⠀⠀⢸⡇
⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⣿⠀⠀⢧⢸⡄⠀⠀⠀⠘⠉⠉⠹⠏⠉⠉⢱⠀⠀⠀⢀⡇⣸⠀⠀⣿⠀⠀⠀⠸⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠾⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠘⠂⠁⠠⡌⠛⠃⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⣖⠉⠁⠀⠀⠀⠀⠀⡆⠀⠉⠀⢀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡞⡿⠹⠗⢀⣇⠀⠀⠀⢀⣀⠀⠀⠀⢸⡆⠐⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢁⣴⠋⡘⠁⣀⡀⠸⡟⠳⠶⣚⡋⠉⣙⠓⠖⢿⡟⢦⣀⠀⠃⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠂⠁⣀⣴⡟⠁⠐⢧⠀⠈⠃⠀⠙⠀⠐⠁⠤⢦⡌⠁⠀⠘⠀⠀⠈⣠⠆⠈⠁
⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⢀⣤⣾⣿⠏⠀⠀⠀⠈⠳⣄⡀⠀⠀⠄⠀⠀⢤⠢⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡼⠁⠀⠀⢀⣤⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣶⣤⣼⣿⣤⣶⣶⡀
⠀⠀⠘⠁⠀⠀⠀⠀⠈⠙⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⠿⠛⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢿⣷⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢣⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣎⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠜⠁⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣼⠀⠀⠀⠀⠀⠀⠀⣏⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣯⣟⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⡠⡈⡍⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢑⢾⣤⠃⣰⠥⠚⢣⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢢⣴⣍⣻⣥⣤⣤⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣼⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠋⠋⠋⠉⠐⣀⣈⢹⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠲⡶⠀⠀⠘⠛⠁⣼⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢀⣂⣀⡴⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⣄⣑⠚⣀⣤⣾⡟⢻⡿⣿⡿⠿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠛⣫⡿⠘⣯⡅⠈⢋⣭⠀⢈⠁⠿⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⣇⣦⢀⡉⠁⣀⣠⢉⣀⠛⢃⡴⠖⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠒⣄⢀⣀⣠⣟⣿⠘⠛⣀⠛⠁⡛⠿⢤⣏⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣿⢢⠀⠀⠀⠐⡷⡋⠀⣀⣈⣩⠤⣤⠞⠻⡷⠀⠛⠀⠘⠛⠀⠾⢂⠗⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢹⢸⢧⡀⠀⣀⡽⠞⠋⠉⠉⠉⣿⡤⡦⠑⢶⣤⠾⣛⡁⠺⣬⠙⠁⠀⠀⢀⡼⠉⣿⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⣾⡄⡟⠉⢻⣄⠀⠀⠀⠀⢠⣿⣷⠶⠦⣌⠉⢀⡙⣡⢒⠁⠀⠀⢀⣾⢏⣤⡄⢄⠀⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡇⣿⣷⣰⠁⠈⢳⣤⠤⢒⣩⠔⠉⠀⠀⢸⣦⡛⠃⠈⠲⠪⡖⠔⠉⣀⣀⠉⠀⠀⠉⠲⢞⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠛⠛⠉⢹⡄⠀⠈⣷⣖⠻⣄⠀⠀⠀⠀⡰⠋⠑⢤⣀⠾⠷⠀⠀⠀⠛⠛⠀⠀⣤⣄⣀⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⢸⣿⣹⡆⢻⠀⢀⡠⠊⠀⠀⠀⠀⠈⠙⠛⠲⠶⠾⠶⠶⠶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⡧⠿⠓⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢠⣶⡦⠀⠀⠀⠀⠀⢰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣤⠚⠁⠀⠀⠘⠿⣿⣷⠀⠀⣀⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣀⠈⠙⣆⠀⠙⣏⠀⠀⠀⠀⠀⠈⠋⢉⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⢯⡀⠀⢸⠀⠀⠈⢦⡀⠀⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠉⢀⣠⠏⠀⠀⠀⠀⠹⣿⠖⠋
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢩⣭⡙⢿⡟⣩⣭⡙⣿⠉⢻⣿⢹⣿⡿⡉⣿⣭⡍⣭⣽⢩⣭⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢸⡰⣿⣿⡗⢸⢸⣧⠹⢼⣿⢃⣟⠸⣿⡇⣿⣿⢨⣭⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣵⣿⣷⣭⣫⣵⣿⣼⣿⣷⣼⣯⣾⣿⣧⣿⣧⣿⣿⣬⣭⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠐⢨⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠁⢀⡴⠛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣰⠋⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠹⣄⣠⣾⣿⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢁⠀⠀⢨⣿⣿⣿⣿⣿⡃⠈⠀⠘⠓⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⠁⠀⠀⣀⣴⣿⣟⠛⣿⣿⣿⠟⠁⠈⠰⠞⠀⢒⣛⡛⠋⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣈⠄⢘⠛⢛⣿⠿⠃⠉⢩⢩⠈⣖⠀⢂⠆⠀⢠⣿⠈⢿⣏⠓⢄⠻⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣁⡾⢃⣠⠾⠢⢍⠀⠀⠀⠀⡇⣶⣠⣿⣶⠊⠛⢀⣼⡇⠀⠈⠻⣧⡄⠑⡄⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠑⢿⣿⣾⣦⣄⣢⣀⡀⠐⡖⡾⢻⣿⡟⡅⡆⣾⠏⠀⠀⠀⠀⣲⡇⣤⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠘⣿⣿⣿⠉⠛⢻⠳⠆⡇⣇⢸⣿⢁⣇⠃⡿⠀⠀⠀⠀⠀⠈⢛⡟⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢇⣇⠀⣿⣿⣿⣤⣀⣬⣤⠖⠛⠛⠻⡿⠿⢾⡿⠀⠀⠀⠀⠀⠐⠻⠎⣠⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢿⠛⡿⠉⢻⣿⣿⣿⣷⣦⣤⣤⣾⣧⣴⣾⠇⠀⠀⠀⠀⠀⠉⢀⡄⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣷⠈⣿⣿⣿⣿⣦⣤⡶⣾⠤⣶⠏⠠⠀⠀⢀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢺⡟⠉⣿⢂⣭⣭⣽⣦⣤⡶⣿⣶⣾⣴⣾⣿⡟⡣⢈⢁⢙⠷⢄
⠀⠀⠀⠀⠀⣠⠀⠰⠂⠘⣧⠀⠀⠈⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣯⣙⣿⣿⢿⡳⠶⡮⣬⣄⡀⢀
⠀⠀⠀⠀⣾⠏⠀⠀⢀⣼⣿⣧⡀⠀⠀⢿⣿⣿⠿⠛⠛⠛⠛⠛⣷⣯⣍⣻⢿⣥⣏⠀⣼⠃⣸⣩⣿⣿⣿⣶
⠀⠀⠀⢰⣿⣴⣴⣾⡿⣿⣿⣿⣷⣄⠀⠸⢋⠡⠂⠀⠀⠀⠉⠀⠀⠙⣿⣽⣧⡭⢿⣿⣿⣛⣛⣛⣵⡿⠟⠁
⠀⢠⣾⠟⢷⠿⠛⠁⠀⣘⣿⡿⣿⡟⣀⣤⢪⣴⡦⡴⠀⠀⠀⠀⠀⠀⠈⠙⣿⣧⡀⠀⠉⠉⠹⠛⠁
⠀⠘⣁⣀⣀⢠⣶⣾⣿⣿⡿⠟⢍⡇⢹⡟⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠹⠛⠉
⠀⠀⣿⣿⡿⢢⠟⠛⠉⠀⠀⣠⡯⠸⣺⠋⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈
⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢘⠁⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡠⡥⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣏⡄⣘⣇⣚⣃⠀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⣟⣻⡻⡟⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⣿⣿⣿⣿⠘⣻⠛⠂
⠀⠀⠀⠀⠀⠀⠳⣄⠈⢹⣷⣝⣟⣽⡇⠉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⠉⠉⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢣⣬⣿⣿⣽⣵⣶⣘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠉⠉⠁⢀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣿⡿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣛⣛⣛⣣⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠛⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣮⡟⣯⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣬⣭⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⣴⣦⣾⣦⣶⡄⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⣧⣼⣻⣥⡟⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠉⠁⠈⠀⠉⠁⠰⠋
⠀⠀⠀⠀⠀⠀⢸⣿⢾⣧⢿⡏⡊⣿⢧⣿⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢩⣭⣭⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣷⣽⣺⣵⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣮⣛⣫⣾⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠰⡿⠯⠭⢽⣿⣆⣴⣒⡒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⢸⢸⣿⠆⢺⡝⠀⢈⣆⡤⢄⢠⡤⣄⠠⠤⡀⡦⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⣀⣸⣸⡿⢯⣸⣇⡤⠞⠘⠤⠼⠸⠀⠸⠺⠭⠇⠧⠘⠯⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉



⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣴⠈⠀⠀⠀⠀⠈⠙⣿⣿⣿⡏⠐⠀⢤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⡇⠀⠀⠘⠃⢀⣠⣴⣿⣿⡿⠃⣀⠀⠀⠀⠈⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⠉⢻⡿⠛⠉⠛⠿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⣳⣶
⠁⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⡀⠀⠀⠀⠀⣾⣿⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠸⢠⣶⣶⣄⠀⢀⣀⡀⠈⣿⡟⣿⣿⣿⣿⣿⡅⣤⠀⠀⢤⡎⠁
⠀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣧⠀⠀⠀⠀⠙⠁⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⠘⠿⠿⠛⠐⣿⣿⣿⠀⡟⠀⣻⣿⣿⣉⣹⣇⣀⣀⣄⡀
⣴⣴⣦⠀⣾⡿⠋⠉⠁⠀⠈⢉⢻⣧⠀⠀⢀⣤⣤⣤⣤⣄⠈⠛⠛⠿⠿⠿⠿⣿⠀⠸⣷⣶⡶⢶⣄⡈⠙⢉⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠞⠁⢀⣀
⣿⣿⣿⣷⣏⣠⣤⣄⠀⢰⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⣿⣷⣶⣿⣿⡿⠿⢿⣿⣿⣿⠛⠻⢿⣿⣿⣿⣧⣶⣿⣿⣿
⢐⣍⣻⣿⡟⠿⠟⠛⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠸⣿⡿⢃⣀⠀⢀⣀⠈⠁⢻⣿⣿⠫⢄
⣿⣿⣷⠿⡇⢀⣀⣴⣟⣛⣷⣶⡞⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠸⡏⣢⣥⣥
⣟⡙⠑⢺⠇⠈⣻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠐⠶⠀⣠⣤⣶⣿⣿⣿⣶⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠤⠀⠀⠀⠀⠀⢻⣿⣿⠟
⣿⠏⣂⣴⣿⣿⣿⣿⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠿⠷⠀⠀⣀⣼⣿⣷⣦⣄⣀⣣⡃⠞
⡤⣵⣿⣿⣿⡿⠿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠀⢸⡿⠛⠛⠛⠋⠉⠉⢿⣿⣿⡏⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠲
⣾⣿⡿⠛⠁⠀⠀⠀⠹⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⢡⣶⣿⣿⣿⣶⣾⣿⠁⠀⠈⠉⠀⠈⣿⣿⣿⣿
⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣆⠀⠀⠓⣠⡀⠀⠐⠛⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠤⣴⣾⣿⣿⣿⣿
⣿⣿⣿⡿⠀⠀⣀⠀⠀⠀⢻⣿⠟⠃⠆⠀⠀⢰⣿⣿⠟⠛⠛⠛⠻⣶⣶⣄⠉⠀⠀⠂⠀⠀⠀⠛⢸⠇⠀⠀⠉⠉⠉⠀⣿⡇⠀⠀⠀⠸⣿⠟⠋⠉⠙
⢹⣿⣿⠀⠀⠀⣁⣀⣤⣴⣿⣿⣿⣶⠀⠀⠀⠸⡿⠁⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⢺⠐⠖⠀⠀⠠⠦⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠙⣿⠀⠀⣴⣿⣿⣿⣿⣿⠿⠛⠋⠿⠶⠶⠶⠇⣤⡄⠀⠀⣤⡄⠀⣿⣿⣤⣤⣤⣤⣄⣀⣤⣴⣿⣆⣠⡴⢶⣄⢀⣼⠂⠀⠀⠀⠀⠀⢸⡔⠀⠈⠁
⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⡇⠀⠀⢀⣴⣦⣤⣤⣆⣀⣀⣤⣄⡀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠧⠀⢠
⠀⣠⣴⣤⣭⣿⣿⣿⣿⡟⠃⠙⠁⠀⢠⣤⣤⣤⣼⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠛⠛⠇
⠐⣿⣿⣿⠿⠿⠿⣿⣿⣿⠀⠀⠀⢢⣾⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠐
⠀⠻⠋⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⣿⣿⠿⠿⠿⢿⣿⣿⣄⣀⣀⠐⠚⠿⠿⢿⡀⠀⠀⠉⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠴⠀⠀⠀⣀⠀⠀⣿⡏⠀⠀⣿⣥⣥⣲⣴⣶⣽⣿⣩⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⣿⣿⡿⠀⣠⣴⣶⣶⣶⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⡀
⣀⠀⣀⢀⣀⡀⠈⠀⠐⢻⣿⣿⣿⣿⡿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣤⣤⣤
⣿⣧⢻⣿⣯⣽⣶⣤⣶⣿⣿⣿⣿⣿⣷⡰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠛⢁⠀⠀⣿⠟⠁⠀⠀⠀⠈⠹⡟⠀⣠⣾⣿⣿⣿⣿
⠈⢿⡇⠙⠛⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠉⠉⠉⠉⡟⣠⣶⣿⣿⣿⣷⣄⣀⣀⣴⣄⡀⠃⢠⡤⠀⠀⠠⠦⠀⢰⣴⣿⣿⡟⠁⠉⠉
⣠⠈⠇⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠐⠆⠀⠠⡤⢠⣿⣿⣿⠿⠛⠉⠉⠉⢿⣿⣿⣿⣿⣷⣄⣀⣠⣤⣤⣀⣴⣦⣿⣿⣿⡇
⠉⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⡀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠀⠖⠈⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⢶⣿⣿⠟⠁⠛
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆⠀⣾⡟⢿⣿⣿⣿⣿⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⡄⠶⠂⠀⢀⣠⣤⣼⣿⣿⣿⣿⣿⣿⠙⠛⠋⠁⠀⠀⢻⣿⣷
⠀⢀⠄⠀⠈⢿⣿⣿⣿⣿⣦⣙⡀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣴⣟⣡⣿⣿⣿⣿⣟⡋⠉⠉⠠⣤⡆⣠⣴⣶⣾⣿⣿⣇⡀

⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⠖⠒⢲⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠂⠀⠀⠀⠀⣀⢤⣀⢖⣤⢄⠀⠀⠠⠊⠀⠈⢢⠀⠀⣠⠾⠭⣱⡀
⠀⠀⠀⠀⠘⡇⠀⡼⠀⠀⠉⢠⠇⠀⠔⠂⣲⠖⡙⠓⣚⣑⢦⢀⠎⠏⠀⠀⣼⡻⢝⠴⠨⢌⠽⣵⡀⢨⠀⠀⠀⢸⠀⣸⡀⠉⠉⡠⢅
⠀⠀⠀⠀⠀⡇⡸⠁⠀⠀⠀⡎⠀⠀⠢⣀⢸⠀⡝⡤⡽⠏⠀⡇⣠⠀⠀⢸⣽⠂⣏⠀⠀⠈⡇⢽⠃⠀⠓⠤⡔⠱⡚⠓⠒⠚⠊⡓⣹⡝
⠀⠀⠀⠀⠀⢳⠇⠀⠀⠀⠀⢳⡀⠀⠀⠀⠹⢄⠀⠩⠉⢀⡴⠛⢡⠤⢤⠀⢣⡫⠎⡓⡒⢎⡢⡿⠁⠀⠀⠀⡇⠀⠱⠀⠀⠀⢀⢡⠿
⠀⠀⠀⠀⠀⣏⣀⣀⣀⣀⣀⣀⣣⠀⠀⠀⠀⠈⠆⠀⠀⠘⠀⠀⠣⣀⡠⠃⠀⠉⠛⠽⠭⠓⠊⠀⣦⢐⠀⢀⡁⠀⠀⠑⠒⠒⠚⠈⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⡤⠔⣂⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⡄⢡⠭⡀⢸⠇⢨⠑⠒⠐⠐⠒⠒⠘⠁⡅
⠀⠀⠀⠀⠀⡀⠀⣦⣀⡀⠘⣀⢀⡜⠀⢧⡾⢵⢶⡍⡆⠀⢰⠒⢠⣒⡂⢰⣒⠀⠀⠀⠀⠈⠛⠁⠈⠛⠀⢸⡃⢸⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⡇⠐⣧⠀⣽⠀⠀⠀⠀⠀⠈⠳⢍⣭⠰⠃⠀⠘⠀⠘⠂⠂⠘⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡸⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠙⠁⠛⠀⢠⠀⠀⣶⡦⣤⣤⣤⢴⠀⠆⡀⠀⠀⠀⡀⠀⠀⢠⠄⢠⠀⡄⢀⡆⢠⠀⠂⠀⡀⠀⠀⠑⠊⠉⠉⠉⠉⠉⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠢⢨⣿⣤⠘⠿⢽⣬⡾⠞⠀⠀⠀⠀⢀⣀⡀⠀⠀⠐⠀⠸⠐⠈⠁⠇⠸⠤⠀⠀⠀⢀⡀⠀⠀⣀⠴⠷⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠤⠤⠔⠊⠣⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢫⢃⡘⠑⠒⠀⠒⠊⢢⠖⠄
⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠀⠉⠓⡄⠀⠀⡖⣞⢖⠆⠀⢤⡤⠤⠤⠤⠤⠤⠤⠄⣶⣀⣀⡀⠀⡜⡎⡆⡠⠄⢸⢸⡇⠀⢀⣀⡀⠀⠈⢀⡄
⠀⠀⠀⠀⠀⠣⠄⣀⣠⣀⣀⣬⡤⠞⠀⢰⠁⠀⡸⠀⠀⠀⢡⠀⡲⠲⠶⠲⠀⢠⡧⣴⠾⠇⣦⣵⣷⣯⡦⠂⢸⠀⡇⠀⣿⡿⣵⡆⠀⠸⡁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠤⠛⠃⠀⠀⠀⠑⠐⢆⡀⠰⠚⠁⠀⠀⣀⢀⠙⠻⠿⠀⠀⠀⠘⣄⠃⠀⠉⠹⠋⠀⣠⠠⠇
⠀⠀⠀⠀⢠⠋⠲⠦⠴⠦⠤⠤⢤⠀⠀⡰⠊⣉⡑⢆⠀⠀⢐⠐⡆⠀⠀⠀⠀⠀⢀⣠⠁⣈⣁⠀⠀⠀⠖⠉⠢⠈⠙⣤⠀⠀⠀⡰⠃
⠀⠀⠀⠀⢸⡫⠀⠈⣉⣀⣀⡉⢛⠄⠀⣇⠘⠄⠜⣸⠀⡔⣄⠉⢰⠩⣷⠀⠀⠀⠘⠈⡏⠉⡙⠀⠀⠀⠓⠀⠊⠀⠀⠙⠛⠛⠃⠁⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁⠈⠉⠀⠀⠈⠓⠶⠚⠁⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠟⠛⠛⠻⠶⠶⠶⣶⣤⣤⣤⣄⣀⡀
⣀⣀⣀⣀⣀⡀⠀⣀⣠⣤⡶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣦⣀
⣿⠉⠉⠁⢸⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⡀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⢀⣠⣤⡶⠶⠶⢶⣤⣍⡛⢷⡄
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠋⠉⠀⠀⠀⠈⠙⢻⣶⡾⠛⣩⣤⡶⠶⠶⢶⣤⣌⡙⠿⣿⡀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠶⣦⣤⣤⣀⣀⠀⠀⣠⡿⢁⣴⠟⠋⠀⠀⣠⣄⠀⠀⠉⠻⣦⡈⢿⣄
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⣦⡾⠁⠀⠀⣴⠟⢻⡟⠻⣦⠀⠀⠈⢳⡄⢻⡆
⣿⣤⣤⣤⣼⡷⠶⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢿⣄⣸⡇⠀⠙⠀⠀⠀⠘⣷⠈⣿
⠀⠉⠉⠈⠁⠀⠀⠀⠀⠉⠛⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠙⢻⡟⠷⣦⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠓⠛⠿⠿⠿⢿⡟⢻⡇⠀⠀⠰⣦⠀⢸⡇⠀⣽⠇⠀⠀⢰⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠻⣦⡀⠀⠙⠳⢾⡷⠞⠋⠀⠀⣰⠟⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣈⣿⣦⡙⠻⣦⣄⣀⣈⣁⣀⣠⣴⠟⢋⣴⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠉⠁⠀⠀⠈⠙⢶⣤⣌⣉⣉⣉⣉⣡⣤⡶⠟⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⠻⢶⠾⠋⠉⠉⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣾
⣿⣿⡟⠛⠛⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⢀⠀⡀⠟⠧⣀⢀⠀⢠⣤⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣾⣩⠉⢩⣉⡇
⠀⠀⠀⠀⠀⠀⠌⠛⠃⠻⠛⠀⠀⠘⣧⠟⢼⣻⡏⠀⠀⠃⠙⠂⠀⠛⠐⠀⠀⠈⠉⠠⢈⣽⠆
⠀⠀⠀⠀⠀⡤⠀⠟⢨⠔⢓⡀⠀⠀⠶⠂⠼⠻⠀⠀⠢⠞⡗⢠⡄⢺⢦⠀⠀⠸⣷⡐⣱⢽⣷⡄
⠀⠀⠀⠀⠀⣿⠀⡛⢁⠁⡌⠋⠀⠀⢒⠺⠷⣒⠀⠀⠀⠀⠞⡀⠀⢸⠀⠀⠀⠀⠋⠑⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠃⡊⢂⢂⠀⠀⠀⢴⠆⠀⠢⠀⠀⠀⠀⠠⠁⠀⠈⠀⠀⠀⠀⠜⠀⠘⠇
⠀⠀⠀⠀⠀⡄⠀⠀⣀⠀⠀⠀⠀⠀⣀⣤⠤⣤⡀⠀⠀⢀⡀⠀⠀⣄⠀⠀⠀⡀⠠⢤⣀
⠀⠀⠀⠀⠀⡁⠂⠀⠀⠀⠂⠀⠀⣼⠿⣿⠿⠻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠳⠒⢻
⠀⠀⠀⠀⣠⣾⢿⣷⢰⣿⡟⠀⠀⠂⠲⠂⠀⠖⠸⠀⠀⠠⣤⠀⣤⡤⠀⠀⠁⠻⠀⠘⠃⠃
⠀⠀⠀⠀⠈⠋⢢⣦⣴⢄⢁⠀⣀⠀⠘⠀⠐⠀⢀⡀⠀⢀⣤⣀⣠⣄⠀⠀⠀⠡⠀⠀⠉
⠀⠀⠀⠀⣀⡤⡀⠙⠝⠉⠻⠀⠂⠑⠦⣤⡤⠔⠳⠂⠈⣁⡾⠙⢹⣎⡃⠀⠀⠓⠈
⠀⠀⠀⠀⠁⠈⠻⡄⠀⠀⠀⠀⠀⡀⠀⠚⠀⠐⣄⠀⠀⠛⠇⠲⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⡇⠀⢸⡗⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⡇⠀⢸⡇⠀⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⡇⠀⢸⡇⠀⣿⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⢸⡇⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢤⡀⢻⠀⠀⡇⠀⠸⠇⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢳⣼⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠲⣄⣀⠀⠀⢻⠀⠀⣿⢀⡶⠋⠙⠲⠞⠋⠛⢦⠀⣿⠀⠀⠀⠀⡀⡶⠛⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢹⡇⠀⡇⢸⣇⠀⠸⡇⠀⠈⠸⡄⠀⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⣾⠉⡇⠀⣯⠀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡇⢸⡏⢹⡄⢷⠀⠀⠀⠙⢦⣀⠀⠀⣠⠞⠁⠀⡏⠀⠀⠀⣿⠀⡇⠀⣿⠀⡏⠙⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⡇⠀⡇⢸⡇⢸⡇⠈⢧⠀⠀⠀⠀⠈⠳⠞⠁⠀⠀⢰⠇⠀⣀⠀⣿⠀⡇⠀⣿⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢺⠉⢷⡇⠈⠁⠀⠁⠘⠃⠀⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢸⡍⢳⣿⠀⠁⠀⠀⠀⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠘⡇⠘⡇⣴⠚⠳⠖⠛⢦⠀⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠿⢰⠚⠛⠞⠛⢳⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠹⣄⠀⠀⣠⠟⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⠘⢦⡀⠀⣠⠞⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠓⠞⠁⠀⢸⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢳⡀⠀⠀⠙⠚⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⠔⢀⡠⠊⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣶⠁⠔⠁⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣈⠝⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠾⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠞⠁⢀⣼⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡦⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⠔⠀⠉⠉⠀⠀⠰⢿⠾⠁⠀⠄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⣸⠗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡁⠀⠄⠀⢀⠀⠀⢀⢀⡠⣊⣴⡾⢛⡭⠂⠀⢀⡤⠊⠁⠀⠀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⢰⠃⠀⣰⣷⣧⣾⣿⡭⢚⡡⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣤⠟⢀⣼⣿⣿⡿⣻⣭⠉⠀⠀⠀⠀⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣾⣿⣷⣶⣶⣚⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣩⡴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠷⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣝⣛⢦⣄⠈⠓⢤
⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⡻⣦⡏⠀⠉⠉⠉
⠀⠀⠀⠀⢀⠔⣿⠋⢠⣿⣿⣿⣿⣿⣿⢹⣿⣿⡁⠹⣇
⠀⠀⠀⡰⢃⡞⠁⢀⣿⣿⣿⣿⣟⣿⣿⣾⢿⡏⠁⢀⠛
⠀⠀⠈⠁⡜⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠈⣇⠛⠃⢻⣽⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢹⡆⢀⠀⠉⢣⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⢂⣬⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠞⠛⠛⠋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠴⣤⡀⣠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠉⠁⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⡀⣈⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⢤⣽⢿⣿⣿⣿⣿⣿⣿⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣧⣻⡇⡆⣄⠈⠘⣿⡿⠟⠋⢉⣍⢻⠿⠿⠿⠷⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡦⣬⣥⡬⠍⡀⠁⠨⣥⡉⠀⠀⠀⣀⣤⣶⠶⠞⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⡼⠁⠀⠑⢄⢈⠀⠀⣸⣿⢀⣠⠾⠛⠉⢠⠀⠀⠀⠀⠙⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣰⣿⡇⠀⠀⠀⠀⠙⠳⠾⠟⠛⠉⠀⠀⠀⠀⣿⣿⣦⣀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢻⣿⣷⣀⢀⣀⣴⠿⠓⢦⣤⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣯⡉⠉⠀⣰⠀⠀⠀⠉⣙⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢁⣤⠾⣿⡶⠦⣤⣄⡉⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢠⣿⣇⠀⠀⠉⣻⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣠⠴⠖⠋⠉⠉⠉⠉⠛⠓⠒⠲⢦⣄
⠀⠀⠀⠀⠀⠀⡏⠉⢩⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⢀⣠⠤⠤⠬⣗⣦⡀
⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢶⣏⣁⡀⠀⠀⣨⠟⣩⠖⠚⡛⠓⠲⣍⢳⡀
⠀⠀⠀⠀⠀⠀⡇⠀⢨⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⣞⠀⢠⠞⣿⢳⠀⠈⢧⢹⡄
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⠓⣿⢦⡀⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠹⡏⢧⠀⠸⣤⣿⡼⠃⢀⡞⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⢤⣤⠤⠽⢦⣙⠦⣤⣥⡤⠴⣋⡼⠥⠤⣤⡤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⣀⣀⣀⣉⣙⣒⣒⣚⣋⣁⣀⣀⣀⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⣀⣀⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠉⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⠤⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠙⢹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⢴⠻⠻⠃⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⠀⠀⠀⠈⠻⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢀⠀⠙⠆⠀⠐⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⢳⡀⠀⠀⠀⣀⣤⡴⠆⠒⠒⠻⡆⠈⡆⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠔⠛⠋⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠁⣠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠱⣄⠀⠈⠂⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠊⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠒⠂⠀⢀⡠⠄⠀⠉⠀⠀⠐⢮⡀⠀⠀⠀⠀⠀⠙⣤⡀⠀⠀⣀⠀⠀⠒⠁⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⡐⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠃⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠱⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⣀⣤⠔⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣆⠀⢷⣠⠤⠤⠤⠞⠛⠉⠀⠀⠀⠈⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢦⠀⠀⢀⠀⠘⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣣⡀⠀⠂⠀⢈⡷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢠⢠⣠⠦⠗⠋⠁⠈⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⠿⠿⠿⢿⡛⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣻⣛⣿⣟⣻⣛⣿⣛⣻⣟⣛⣿

⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣀⠀⠀⢀⠀⠀⢀⠀⣠⣤⣠⠀⠀⢠⠤⣀⠀⢀⣀⢤⠀⠀⣀⣀⠀⣀⣀⣀⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⢹⠀⠉⢢⠀⠀⠀⠀⠀⢸⠀⢠⠋⠀⣇⣀⡀⠀⠀⣧⠤⠚⠀⠺⣄⡀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠘⡦⠊
⠀⠀⠀⠀⢸⠀⠀⡜⠀⠀⠀⠀⠀⠈⣤⠃⠀⢠⠃⠀⠀⠀⢰⠏⠢⡀⠀⠀⢀⡽⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠸⠔⠊⠀⠀⠒⠒⠂⠀⠀⠉⠀⠀⠈⠀⠉⠁⠀⠈⠀⠀⠈⠀⠈⠁⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣄
⠀⠀⢸⣶⣄⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶
⠀⠰⣾⣿⡿⠀⠀⣶⡿⣷⡄⠀⢹⣿⣿⡏⠀⠀⣶⣾⣶⡄⠀⢹⣅⣤⣇⠀⠀⠀⡀⠀⠀⠀⠀⢹⡉⢙⡖⠀⠀⣄⣦⣶⡆
⠀⠀⡝⡩⠛⡄⠀⠈⣷⣞⣳⠀⡰⠿⠛⢲⠀⠀⢻⣿⡏⠀⠀⠸⠒⠒⠚⡄⢻⣿⣿⣇⡀⠀⠀⡼⠿⠯⠽⢄⠀⢹⣿⣿⣧⡆
⠀⢸⡜⢁⠴⡇⠀⠀⡇⠀⠀⠃⠧⠀⠀⠸⡄⠀⢺⢿⡇⠀⠀⢠⠀⠀⠀⡇⠀⢻⣿⠋⠁⠀⢸⠀⠀⠀⠀⠸⠀⠀⣿⣿⣿⣤
⠀⢸⡴⠁⢠⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⠀⡇⠀⢸⣿⡇⠀⠀⢸⠀⠀⠀⡇⠀⢸⣇⠀⠀⠀⢀⠶⠤⠄⠲⡁⠀⠀⣿⠉⠀⢸
⠀⠸⢀⠔⢹⠀⠀⠀⡇⠀⠘⠀⢸⠀⠀⠀⡵⠀⣼⣿⡇⠀⠀⢸⠀⠀⠀⡇⠀⢸⡿⠀⠀⠀⢸⠀⠀⠀⠀⢱⠀⠀⣿⢰⢸⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣶⣿⣿⣧⠀⠀⠀⠀⢰⠀⠀⣠⢴⣶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣝⣿⡌⢻⣿⣿⣧⠀⠀⠀⣿⠇⣰⢃⡞⠀⠉⣯⠈⢲⠀⠊⠉⡅⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⢿⣎⣻⣿⣿⣇⠀⢀⠃⢁⣿⡟⠀⢀⠄⠹⣧⠞⠁⠀⡀⣀⠤⠀⠒⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣄⠀⢸⡇⠁⠀⡜⠀⢰⠇⠀⡄⠂⢽⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣅⠤⠖⣩⢽⣴⡁⢠⠀⠀⠊⠿⡀⠀⠀⠀⢀⣠⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠉⠉⠛⠛⠿⠥⣤⣜⡓⠛⠶⢟⣿⠀⠳⣄⣀⣤⠴⠟⠛⠉⠁⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢲⣾⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⡦⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡏⠱⡀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠁⠒⠀⠠⢀⣠⠾⠀⠀⡅⠀⠘⣆⣀⡠⠔⠂⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠙⠐⠀⠀⠀⠈⠀⡆⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠞⠀⠀⠀⠸⠈⠉⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠀⠀⣀⣀⡀⠃⠀⠀⠀⠀⣀⡤⠔⠒⠉⠁⠀⠀⣠⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⢿⠴⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣻⢿⠀⠀⠰⢞⡥⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠈⢻⡄⠀⠰⠔⠛⠘⠃⢀⣸⠤⡘⣧⠀⠀⠀⠀⠀⠘⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣴⡄⢠⣶⢛⠀⠁⢠⠀⠀⠀⣳⠀⢸⠁⠸⠋⠀⠀⠠⠶⡞⠱⠛⠋⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣥⣟⡙⠤⠀⠸⠅⢸⢆⠀⠀⠀⠀⠀⢉⣤⣄⠠⣀⣰⣀⣀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣾⠉⠀⠈⢁⣈⣴⠿⣽⣮⠁⠀⠀⠀⠀⠾⣟⠁⠀⠸⡀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢀⣿⣶⣿⡏⡿⣦⡈⠿⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣯⢴⠣⠀⠑⠄⠳⠖⠶⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⣿⣧⣈⣀⠀⠀⠀⠐⠲⠞⠐⠃⠈⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⢽⡿⠀⠠⠀⠀⠀⠀⠀⠀⢀⣴⠂⠉⠑⢀⠀⢀⠀⠐⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⣿⣿⠟⢢⣼⠂⠀⠀⠀⠀⣀⢹⣷⠀⠀⠶⠙⠂⠻⣦⠤⡤⠐⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⣸⠿⠛⢺⣷⠄⡀⠀⠀⠐⠁⠀⠉⠀⠀⠤⠀⠀⠀⡟⢀⣠⠀⢸⠂⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠘⠉⡵⢧⣄⠄⠀⠀⠈⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢉⠻⠞⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠑⣀⡀⠀⢹⣦⣤⣀⣀⣰⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⠁⠀⠀⠩⠩⡉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⢰⠀⢰⣦⠆⡀⠀⠀⢈⠁⣤⡈⠐⣋⠄⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣜⠃⢀⡷⠄⠐⣦⠁⠐⠶⠟⢳⣾⢧⠀⢩⠓⠠⣶⣆⠀⠀⢀⣐⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢋⡙⢻⡿⡷⠀⢀⠘⡗⠞⣷⣶⣿⠙⢛⠀⠀⠀⣸⣿⣾⣷⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢘⠛⠿⠿⢯⣴⣷⣴⣎⣴⣇⠀⠙⣿⣿⠠⠀⣶⣤⣴⣿⣿⡇⢙⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⡀⠀⠉⠉⠉⠉⠙⣿⣻⣶⣿⢛⠂⠀⠿⠛⣋⣿⠻⠁⢈⣿⠟⠀⠀⠀⡀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣸⣤⡀⢁⡠⠁⠀⡀⠀⡿⣧⣄⠀⠀⠄⠀⢀⣠⣾⣦⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢢⠀⢠⡄⠀⢤⠄⢙⣿⡿⠿⡿⠋⠀⠀⢻⣅⣀⢀⠈⠛⠻⣶⣾⣷⣟⣯⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⢈⡀⠈⠀⠙⠻⣿⡞⣷⣠⣄⡀⢀⡉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠛⠿⢭⠀⢈⢀⠀⢻⣟⣿⣿⠿⠟⠀⠛⠉⠀⠀⠀⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠛⣷⣜⣿⣿⣿⣟⢂⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⣤⠀⢘⡃⠀⡀⠘⠁⠀⠀⠀⣸⣿⣮⣭⣉⣽⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠃⠀⠀⠀⢀⣰⣿⢿⣯⣉⣽⣟⠀⠀⠀⢴⣶⠦⠠⠤⠀⠤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣤⢀⣴⣿⠟⠀⠀⠀⠙⣿⣿⣧⡀⠀⠘⠻⡿⠶⠦⠴⢶⠞⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠘⢿⣿⢿⣦⢴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢁⣟⣠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢾⣽⣿⡀⠀⠀⠀⠀⠭⢁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣦⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⢟⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠉⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠓⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡋⢸⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠁⣡⡆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣅⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠾⠿⡶⠆⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⢿⠋⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⠂⠀⠀⣰⣿⣿⡿⠋⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏⡀⠀⠀⢹⣿⣀⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢛⡁⣨⣽⠃⠀⠀⠀⡀⠀⠀⢀⣠⣶⣷⣿⣾⣶⣼⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠋⠉⠉⣠⣴⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠈⣿⣿⣿⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠂⠈⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠆⠀⠀⣠⣿⣾⡷⠶⡎⠉⠁⢹⣿⣿⣿⡟⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣿⣿⠃⠀⠀⠈⠄⠀⠀⠀⠀⠹⠁⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡟⠁⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣼⣿⣿⡀⢀⠀⠈⠀⠀⠀⡀⠀⣤⢸⣿⡿⠿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠤⠀⢠⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠁⠀⠀⣼⣿⢷⣶⠄⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⢸⣿⡟⠛⠿⠿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⣤⡌⠀⠀⠀⣀⣀⣀⡀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⡿⠀⠀⠀⢀⣠⣴⣶⣦⣿⣿⣿⣾⡿⠀⠀⢠⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠘⢿⣦⡄⠀⠀⠙⠉⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠋⠻⣿⡉⠉⠀⠀⠀⣿⣿⣿⣿⠛⠻⠿⡟⠋⠉⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠠⣿⠇⠀⠀⠀⠀⢿⣿⠉⠁⡄⠀⣀⠀⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⢀⡀⠀⠆⠀⠀⠀⣿⣿⡏⠙⠀⠐⠂⡿⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⣠⣿⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠈⠁⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠲⠀⠀⠖⠀⠀⠀⠀⠀⣿⣿⡿⠀⢠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠉⠉⠉⠉⠉⡩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠛⠦⡀⠀⠀⠀⠀⠀⡠⠊
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠈⠲⣄⢀⡤⠊
⡏⠙⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⣠⠃⠀⠀⠀⠀⢠⠊⠉
⡇⠀⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢼⠃⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⣠⠎⠀⠀⠀⣠⣾⡎⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁
⠃⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠉⠑⠢⢄⣀⠀⠀⠀⠀⠀⡘⠀⠀⣠⡖⠁⠀⠀⡠⠊⣁⡴⠁⠀⠀⠀⠀⢀⡤⠒⠉
⠀⠀⠀⠉⠙⠒⠦⣤⣀⢀⡞⠁⠀⠀⠀⠀⠲⣄⠀⠈⠑⠢⡀⠀⠀⠣⣘⣿⢻⣀⣀⡴⣾⡖⠋⡹⠀⠀⠀⣀⡤⠔⠚⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢢⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠘⢆⠀⠀⠈⢿⣄⠉⠘⣣⣿⡟⠉⣠⠴⠚⠉⢀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⢣⡀⠀⠈⠛⠛⠛⠛⠁⡠⠋⠀⠀⠀⡴⠁⠀⠀⠀⠀⢀⡬⠃⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣦⣀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣰⠁⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡉⠓⠦⣀⠀⠀⠈⠻⣷⣶⠆⡇⠀⠀⠀⢀⠎⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠐⢺⣿⠒⠤⣴⡁⣿⡏⠀⠀⠀⠀⠈⠀⣀⣤⠔⠋⠀⠀⠀⣀⡴⢯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡾⢿⣧⣤⣯⣴⠿⠤⠤⠤⠤⠤⢦⣽⠟⣇⠀⢀⣠⣔⣋⣥⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣄⠉⠹⢏⣼⣿⡠⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠴⠯⠭⠭⠭⠭⠭⠭⠭⠭⠿⠷⠖⠟⠛⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⢰⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠙⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣾⣞⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡉⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠆⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠂⠙⢃⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠳⣦⠀⠀⢀⡴⠟⠛⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣉⣶⠛⢶⡄⠸⣧⣠⡾⢡⡶⠶⠆⠀⠸⠛⠛⠶⢦⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣤⣤⣤⣴⠾⠛⠉⠉⠀⠀⠀⢠⣿⣿⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⣆⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢨⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠠⣦⡶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⣀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠈⠁⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⢀⡀⠀⠿⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠘⠟⠀⠀⠻⣇⣀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢠⣼⡋⠀⠀⢶⣦⡦⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⣤⣾⡆⠀⠀⠀⠘⠋⠉⠀⠀⠀⢨⡅⢀⣀⣀⣀⡀⠈⠉⠀⠀⠈⠛⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠙⠻⠃⠀⠀⠀⠀⢀⣤⠶⠞⢛⣿⣻⣯⣉⡉⠉⠙⢷⣄⣀⣀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣠⣤⣤⡴⣿⠁⣠⠾⣻⣿⣿⢿⣿⣙⠻⢦⣿⡄⠉⠉⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁⠀⢀⡄⢸⣧⡾⢁⣴⠏⣿⠹⣏⢻⣝⠳⠦⡿⠃⠀⠲⣄⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣴⠋⠀⠸⠿⠾⢻⡟⢠⡟⠀⢻⡄⢙⣦⡀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣸⠇⠀⠀⠀⠀⢠⣿⣤⣼⠃⠀⠀⢻⣿⡟⠿⠆⠀⠀⠀⠀⣷⡀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣀⣤⣾⡦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⢿⣶⣤⣄⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡾⠋⠉⠙⢳⣄⠀⠉⠻⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⢀⡾⠋⠀⠀⠉⢳⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡃⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⣸⠇⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⣠⣄⠀⠀⢸⡇⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣿⣤⠶⠞⠛⠁⠉⠙⠛⠛⠛⠛⠉⠉⠙⠳⠶⢿⡦⠴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢸⠀⠀⠰⢿⠿⠿⠿⣿⣧
⠀⠀⠀⠀⠠⠔⠂⢠⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍
⠀⠀⢀⠀⠀⠄⡀⠀⠆⠀⠀⢳⠀⠀⠀⢀⣤⠄⣽⣿⡒⠒⢲⡄⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⣀⡀⠀⢄
⠀⠀⠀⠀⠀⠂⠀⠀⠇⠀⠀⢸⠀⠀⢀⡼⠋⠀⠛⠓⠶⣿⣾⣇⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣙⢦⠈⠆
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⣻⡴⠆⠀⠘⠃⠙⢻⣦⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⣿⣿⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢤⣿⡀⠀⠤⠄⠀⢈⡼⣿⣿⡀⠈⠈⠀⠀⠀⠀⠀⢄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠛⠟⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢄⠀⠀⠀⠂⣳⣹⣆⠀⠀⠈⠀⠑⢿⣮⡿⠀⠀⠀⠀⢀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠦⠈⠃⠀⠀⠀⡿⣿⠏⠀⠀⠀⠀⢠⣿⣯⠁⠀⠱⠄⠀⠋⠁⠀⠀⠀⢱⣆⠀⠀⠀⠀⠀⣀⣀⠀⠈⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠊⠀⠂⠂⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠙⠧⠀⠀⢀⣄⢀⠀⠀⠀⣀⣠⠎⠟⠀⢀⣀⣔⣛⣛⡛⡳
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⣿⣶⣤⣾⡿⠿⠛⠀⡠⠀⡿⠟⠿⠿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠴⠀⠐⠓⠂⠈⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢀⠀⠀⠀⣠⣤⣄⣶⣶⣄⠑⠀⠀⠄⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡄⠀⠈⠀⠀⠀⠀⠱⠀⠀⠀⠈⡆⠀⣼⣿⡿⠟⠻⣿⣿⣄⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⣻⣿⣟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⢠⡟⠁⠀⠀⠀⠈⠻⣿⣧⠀⠀⠀⠀⠀⠀⢱⠒⠀⠠⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣷⣼⣇⠀⠀⠀⡬⠄⠆⠀⠀⠀⡀⠀⢸⡇⠐⠀⠀⠀⠁⢀⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠉⠋⠉⠉⠉⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠠⠄⠀⣼⣟⢿⣿⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠩⣿⣇⠀⠀⠘⣯⣿⡂⡿⠟⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣤⠶⣿⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⢷⡶⠀⠁⠀⣀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠭⣤⡀⠀⣈⣿⣧⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠉⠁⠐⣿⣿⡭⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠰⣴⣿⠏⠙⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠐⠀⠀⢀⣤⠒⢰⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⣀⣠⣄⣼⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡿⢿⣇⡰⠆⣠⣠⠞⠃⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡀⠀⣠⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣄⠀⠋⣠⣾⣿⡅⠚⠏⠋⠀⠀⠀⠀⠐⠒⢃
⠀⠀⠀⢸⠏⣀⡄⠁⠀⠀⠀⠀⠀⠀⠀⡴⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉⠋⢉⡉⠁⠀⠀⠀⠀⠐
⠀⠀⡠⠘⢰⡝⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⡀⠀⠠⣤⣼⣧⡀
⡴⢋⣤⡆⠺⡇⠀⠀⠀⠰⡇⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⣿⣿⣷
⣼⡿⠛⠁⠀⣿⠀⠀⠀⢀⣧⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⣿⣷⠀⠀⠀⠙⠶⠶⢶⡿⠛⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣽⣿⣿⣿⣿⣿⣿⣿⣦⠀⣤⣾⣿⣿⣿⣿⣿
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⢿⣿⡿⣣⣾⣿⣿⣿⣿⢻⣇⠸
⣀⠀⡀⢀⣴⡆⢀⣤⣶⣀⣠⣴⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣄⣀⣀⣀⣠⣴⠞⣉⣾⣿⣿⣿⣿⣿⡁⣾⣿⣆
⠀⠀⢱⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠤⠭⣭⠍⠉⠀⠀⣨⣿⣿⠛⠋⠙⢻⡇⣿⣿⣿
⣄⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣷⠄⠀⠀⡸⢁⣿⣿⣿
⡙⠳⠀⠐⣿⢿⣿⣿⣿⣿⣿⣿⡌⢿⣿⡟⢋⠁⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣟⡁⠀⢀⣼⣿⣿⠏⢙⣿
⣿⡄⠀⠀⢿⣌⣿⢻⡏⣿⣿⣿⣷⣄⠹⣯⡸⣄⠀⠀⣶⣿⡿⠏⠁⠀⠀⠀⣰⠀⢀⣼⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⢃⣾⣿
⠀⠀⠀⠀⠈⢿⣿⡿⣷⡿⣯⢹⣿⣿⣦⡙⣿⣿⡟⢻⣿⣿⠃⢀⡀⠀⢀⣼⣿⣶⣿⣿⣿⣿⣧⣾⣿⢟⣩⣾⠟⣡⣿⣿⣿
⣿⣶⣄⠀⠀⠀⠙⢧⢻⡇⠸⣇⠻⣿⣿⣷⡜⢿⣷⣾⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣦⡀⠀⠀⢨⣏⣄⢻⡄⠹⢿⡿⠷⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⡿⣿⠇⡉⠻⠿⠿⠿
⣿⣿⣿⣟⡻⢿⣦⣀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠉⠛⠿⠿⣿⣿⡿⠛⠛⠛⣉⠀⣿⣿⣿⣿
⠀⠉⠙⠛⢿⣶⣮⡝⠛⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⠿⢿⣷⣦⣤⣼⣿⣷⣾⣷⣶⣿⣿⣾⠿⠛⠉⠀⢠⡆⠀⠀⠀⢤
⠀⠀⠀⠀⠈⠉⠛⢁⡀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠛⠛⠿⠟⠋⠉⠀⢀⠀⠀⠀⢸⣇⠄⠀⢀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣄⣀⣀⠀⠠⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⢸⣿⣶⣶⣿⣤
⠀⠀⠀⠀⠀⠀⠠⠀⠈⣿⣿⣿⣿⣿⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢲⣿⣻⡿⡾⠛⠂⠀⠀⣿⣿⣿⣿⣿

⠛⠀⠀⠀⠀⠻⣿⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⠄
⣄⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⣠⠂⠀⢄⠀⡀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⣿⣷⣤⣀⠀⠀⠳⡜⠿⡆⠀⠀⠀⣼⠇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣤⣶⣾
⣿⣿⣿⣿⣿⣶⣄⡘⠋⢻⠀⠀⠼⠉⠄⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⣀⣴⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣭⣿⣿⡛⢸⡆⣆⡀⠈⠀⠀⣀⣴⠸⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣠⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⡋⠁⠀⠁⢸⣿⣆⠚⠀⣿⠁⠤⠀⡀⠋⠀⠀⠀⠀⠀⠀⠐⢠⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⣿⣿⡿⡿⣯⠀⣘⠃⠀⠀⠉⠻⠀⡀⠘⡆⠀⢀⣼⣷⢀⣠⡄⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿
⣿⢟⣫⣿⣷⣿⡎⣀⠏⣀⢀⣀⢴⣠⣸⠋⢳⡟⣗⣛⠿⠿⢛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣧⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠗⣿⣿⣿⡿⣿⣼⠃⣾⣷⠉⠉⠀⢧⣼⡄⢀⠀⢀⠔⠉⣬⣻⣯⠟⠛⠻⠿⣿⣿⣿⣾⠟
⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣏⡟⣾⣿⣿⠀⠀⠸⣌⠿⣿⣟⣀⣠⣤⢾⣿⣿⣧⣦⣶⣶⣤⠀⠀⠈⠁⠀⢠⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢸⣿⣿⣿⣦⣾⡦⢹⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣁⡈⣳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠛⠉⠉⠉⡙⢛⡻⠿⢾⣿⣿⣿⣿⣿⣟⣫⣶⣿⡟⣃⡘⠁⠄⠀⣀⠐⢢⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⠿⠓⠲⠚⠋⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⡉⠀⠀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠤⡐⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢰⠃⠙⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⡗⣀⣀⣀⣹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢄⣿⡄⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡡⠀⣿⠇⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⠟⢰⠇⢸⣿⠀⠀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⣿⣿⣤⠀⡞⠀⠘⡏⠀⠀⠀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡏⠀⠀⠀⠉⠛⣿⠃⠀⠠⠀⠀⠀⠰⣸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⢻⣿⣿⠏⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠐⢀⠀⢠⣦⣀⠀⠀⠀⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⢻⣟⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠁⠀⠘⠻⠛⠂⠀⠀⠻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⠀⣰⣿⣿⣿⡇⠈⠻⢿⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⣦⣿⣿⣿⣿⡇⠀⠀⠀⠙⠳⠽
⠀⠈⢙⣿⣿⣿⣿⠿⠏⠁⣠⣆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣴⢻⠀⠀⠀⠀⠀⠀⠀⣂⣿
⠂⠂⠈⠉⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠻⠟⢹⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣾⠂⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⣶⡿⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠀⠀⠀⠷⣶⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⠒⠲⠦⠤⢤⣀
⠀⠀⠀⠀⠀⡏⠉⣿⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⣀⡤⢖⣒⡮⢷⣆
⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠛⠲⠤⢤⣄⣞⡵⠋⣩⣤⡉⠓⣝⢦
⠀⠀⠀⠀⠀⠧⠤⠟⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⢧⣧⡉⠀⠘⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠒⠲⡺⡄⠐⣄⣇⡼⠀⣸⢳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⣤⠤⠽⢌⡲⠤⠥⠤⣚⡵⠧⠤⣤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠤⠤⠤⠬⠭⠭⠭⠥⠤⠤⠤⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠓⠋⠉⠳⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡆⠀⡀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣿⣇⡀⢾⣷⡿
⠀⠀⠀⠀⠀⣠⣦⣀⢀⣻⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢦⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣤⣶⣿⣿⣷⣿⣿⡟⣚⣒⡶⣦⡀
⠀⠀⠀⢀⣴⠋⣾⡿⠿⠋⢻⣿⣿⣿⣿⣿⣦⣉⠻⢦⣄⠀⠸⣇⣰⢶⣾⣄⠀⠀⢀⣠⠞⣫⣿⡿⠁⢹⣿⣿⣿⠙⢿⣿⣻⡀⢻⡄
⠀⢠⣿⣿⠃⢰⣿⡏⠀⠀⠀⣿⣿⣏⠙⢿⣿⣿⣿⣿⣿⣷⣶⡿⣿⢺⣿⣿⣷⣶⣿⣷⣾⡿⠋⠀⣰⣿⣿⣿⣿⡀⠀⠙⢻⡇⠈⠻⣶⣤⣤
⢠⡟⣹⠃⠀⢸⣿⠃⠀⠀⢰⡿⣿⣿⣷⠢⢤⣉⣹⣿⣿⣿⣿⡆⠀⠈⠁⢠⠃⣽⣿⣿⣿⡿⠄⣶⣿⣿⠿⣿⣿⡇⠀⠀⢸⣷⠀⠀⠑
⠏⠀⠁⠀⠀⢸⣿⠀⠀⠀⠸⡇⠹⣿⣿⡆⠀⠀⠉⠙⠉⠉⠉⠀⠀⠀⠀⠀⠘⠟⠛⠋⠀⠀⣼⣿⡿⠋⢸⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢼⣿⣄⠘⣿⣷⡀⢠⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⢠⣿⣿⣿⡇⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣾⣿⣿⣾⣿⣿⣇⢸⡆⠀⠀⠀⠃⠀⡀⠀⢀⡄⠰⠀⠀⠀⡾⠃⣿⡇⣰⣿⣿⢇⣿⠀⠀⣠⣿⣿⠀⠀⠀⠀⢀
⡆⠀⠀⠀⠀⠀⣷⠀⠀⣶⢹⣿⣯⢻⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠈⡿⠛⠀⠀⠀⠀⠀⠀⢀⣿⣴⣿⣿⡏⣸⣿⣀⣴⣿⣿⡏⠀⠀⠀⠀⡺
⢿⡀⢀⠀⠀⢰⣸⠀⣾⣿⡎⣿⣿⣧⢹⣿⣿⠻⣇⠀⠀⠀⣐⣁⣘⣁⣀⣀⣀⣠⡀⣀⠀⢸⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣿
⣸⣷⣸⡆⠀⠀⣿⡇⣿⣿⣷⣿⣿⣿⣧⣿⣿⣷⣿⣦⣶⣿⡿⠛⠛⠛⠛⠋⠛⠻⠿⠏⠀⣼⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⡿⠀⢀⠀⠀⣽⣧
⣿⣿⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣠⣿⢿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠇⣰⣿⠀⣼⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣶⠟⢁⣿⣿⡇⣾⣿⣿⣿⣿⣿⣟⣴⣿⠃⣰⣿⣿⣼
⢹⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠏⠁⠀⠀⣸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠇⣰⣿⣟⣿⣛
⣬⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⡙⣿⣿⣿⣿⣷⣾⣿⡿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⡿⠟
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣷⠘⣿⣿⣯⢿⣿⣿⠃⡤⣘⠂⣿⣿⡿⣾⠇⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⡟⣹⣿⣿⣇⠘⣿⣿⣿⣿⣏⣼⡑⠀⢀⣿⣿⣠⣿⣀⣿⣿⣿⣿⠿⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠿⠉⠀⣀⣀⣠⣤⣤⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣾⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣄⠀⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣼
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠟⠿⠻⠟⠿⠟⠿⠿⠿⠿⠿⠿⠿⠛⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠐⠄⠰⠀⠀⠀⠰⠀⠂⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⢙⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢋⡙⠃⠈⠀⠀⠰⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡷⠇⠀⠀⠀⠀⠖⠁⡀⠈⠛⣣⠦⡗⠀⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⣿⣟⠹⠻⠀⠀⠀⠀⠈⠀⠄⠀⠉⠉⠀⡆⡇⠀⠀⠀⠀⣼⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣽⠿⣿⠐⠅⠀⠀⠀⠀⠈⢀⣾⣀⣀⣠⣾⠀⡁⠀⠀⠀⠀⠘⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⡠⣿⣲⠀⠀⠀⠀⠈⠀⡞⠉⠛⠋⢹⠏⠀⡟⠀⠀⠀⢀⣠⠘⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠇⠛⠃⠀⠀⠀⣰⠃⢰⣃⡀⠀⢠⠊⠀⠘⠁⠀⠀⠀⠙⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⡿⠉⢷⢀⣥⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢾⣿⡟⣿⣊⣿⡏⢸⡇⠀⠀⠀⠀⠀⡀⠀⢀⣤⠆⠀⡾⠂⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢙⣿⡇⣿⣿⣿⠇⢸⣇⠆⠀⠀⢼⡿⠿⠧⠉⠀⠀⠙⠛⠲⠶⠶⠋⠄
⠀⠀⣐⠀⠀⠀⠀⠀⢿⠀⣷⡶⣶⣿⢾⠶⠾⠿⠿⠗⠀⠈⢿⣤⣾⡿⠷⠂⡀⠴⠤⠄⠁⡤⣤⣄
⠀⠀⠁⠠⢀⡀⠀⠀⠙⠀⢻⡃⣿⡟⢢⣆⠀⠠⠶⠀⠀⢀⣰⠒⠀⠉⠀⢻⡀⢄⡀
⠀⠢⢲⣆⠀⠀⡀⡀⠀⠀⠈⠉⢿⣗⣠⣙⠓⣀⣀⠀⠂⣾⣟⣶⠦⢦⣤⣼⣟⣂⡱⠒
⠀⠀⠀⠙⠛⠛⠻⠿⠶⢄⣄⡀⠀⠀⠈⠁⠚⠛⠛⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣿⠿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣦⣤⣄⣀⡀⠀⠀⠀⣀⡀⢀⠀⠀⠀⣼⡟⠁⡸⠀⢸⠈⠹⠛⣼⣿⣿⣿⣿⣿⣿⡀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⢿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠃⠀⠌⠀⡤⠀⢨⣿⣿⣷⣤⣤⠈⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⣿⣿⣿⡄⢰⣆⠀⠀⠀⠃⢠⣾⣿⣿⣿⣿⣿⠄⠀⠙⠚⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡿⣴⣇⡀⠀⢸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠐⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⢿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠇⠀⡻⣿⣿⣿⣿⣿⣏⠀⠀⠀⠰⠾⠛⠋⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡇⠋⠉⠀⠀⠁⠈⢻⣿⣿⣿⡿⠿⠶⠤⠴⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣄⣀⣤⠀⠀⠀⣀⡀⠀⠀⢸⣿⢹⡴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⡇⠀⠀⠀⠀⠉⢻⣦⣼⣏⡼⣣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣽⡟⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣻⣆⠀⢀⠀⠠⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣉⣳⠀⠀⠀⠀⠀⣼⣿⡿⢸⣿⢹⣿⠀⣾⠂⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠳⣶⣶⣤⣾⣿⣿⣷⣾⡿⢸⣿⢰⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠿⣿⣿⣿⡇⠀⠀⠐⠿⠿⢿⡿⠿⠿⠿⠫⠛⠛⠧⠺⠷⠀⠉⠉⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⡿⠛⠋⠀⠙⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⡟⡠⠄⠀⠀⠤⡌⢿⣿⣿⣿⡀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣄⡀⠀⠀⠀⣰⣶⣾⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⣠
⣿⣿⡇⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣷⣶⣤⣶⣼⣿⣿⣿⣿⣷⣴⣶⣶⣴⣄⡀⢠⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⠿⢿⣿⣿⡟⣻⣿⢿⢿⡿⣛⠿⠛⣿⣿⣿
⣿⣿⣿⣦⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠋⠀⠀⠀⠀⠹⣿⡏⢹⢁⣤⡄⠀⠀⣤⣄⠹⣿⣿
⢿⣿⣿⡿⠀⠀⠀⠀⢹⣿⣿⡿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠐⠀⠀⠀⠐⠀⠀⠙⠿
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠉⠙⠛⠋⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⣀⣀⠀⠀⡠⠤⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣠⣶⢗⣀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠐⠂⠀⠀⠒⠊⢹⣿⡄
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠁⠀⠀⠈⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶
⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣿⢋⣀⡀⠈⣩⣍⢻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿
⡟⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⠀⠀⡀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐



⠀⠀⠀⣠⣤⣤⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⣶⡶⣶⣶⣶⣶⣶⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣿⣇⣿⣿⣻⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠐⠀⡀⠀⠀⢀⠄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⢨⡁⠀⢠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠁⠁⡫⠉⢀⡠⠂⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⣴⣿⣿⣶⣴⣾⣿⣷⣦⣶⣿⣿⣶⡄⠀⠀⠀⠈⠂⠀⠲⠴⠊⠀⠄⠀⠀⠶⠀⢀⡍⠀⠐
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠄⠐⠚⠁⠀⠀⡔⠀⠉⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⠉⠉⠉⢉⣉⣉⠩⠥⠌⡄⡂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣿⢛⢛⣋⢉⡉⣉⡉⢁⡂⠀⣀⠲⠛⢆⠃⠀⠀⡇⣇⠀⠀⠆⠀⢀⡉⠀⠘⠂⠀⡋⠀⠀⡀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⢸⠀⠀⢠⠞⣿⠕⡆⠀⠀⢸⠀⠀⢸⠁⠀⠀⡇⠂⠁⠠⠀⠉⠉⠠⠂⠀⠀⠀⠉⠒⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠢⠈⠛⠻⢸⠀⠀⠘⢤⣾⡣⠏⠀⠀⢸⠀⠀⣸⠨⠩⠭⠥⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠒⠒⠒⠒⠛⠓⠒⠒⠒⠛⠜⠉⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⢀⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣄⢀⣴⣿⣿⣿⣴⠶⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢻⣿⣿⣿⣿⡇⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠐⠶⡾⣿⡛⢾⣿⣿⣾⣿⡿⠟⣿⡛⠁⠀⠤⠒⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣿⡿⣄⠀⠀⠁⢿⡇⠈⠻⡟⢀⠉⠀⢸⣿⠁⠀⠀⠀⢀⡴⠛⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠿⢷⣦⡀⠈⢿⣮⣧⠀⠀⢾⢷⣠⠆⠈⠁⠰⡄⣾⢻⣆⢠⡆⢀⠋⠀⣴⣿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⣀⣜⣿⣹⡆⠀⢸⡎⢃⠁⢀⡀⠈⣸⢃⡞⣿⢸⠁⠄⣀⣼⠋⠀⠼⠓⠈⠁⠒
⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠛⠓⠼⢿⣷⣀⣳⡒⣳⡞⢁⣎⣷⡿⠾⠛⠛⠉⠉⠻⠆⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣠⣎⣾⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣀⡀⠀⠹⣦⣀⣸⡇
⠀⠀⠀⠀⠀⠀⣽⢾⣦⣤⠾⠛⠛⠛⢳⣮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣿⣿⠟⠿⠷⣶⡿⠿⣿⠁
⠀⠀⠀⠀⠀⡾⠁⠀⠉⡁⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢰⡇⠀⢄⣀⣷⠀⠀⠀⡠⠄⣿⣾⣿⣿⣶⣶⣦⣤⣤⣤⣤⠤⠾⣷⣄⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⢀⠛⠀⠀⠀⠈⢳⠀⠀⡾⠀⠀⠙⢻⣿⠀⠈⠀⠀⠉⠀⠀⢀⣀⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉⢀⣿⣿⣿⣄⠀⠀⠘⠀⠀⢀⡃⠈⠶⢶⡄
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠁⡾⠋⠀⠠⣤⠖⠹⠿⢷⣶⣤⣴⣾⣿⡿⠿⠿⠿⠿⠿⣿⣶⣶⣤⡀⢀⣴⣿⣿⣿⣿⡿⣿⠦⠦⣠⡔⠉⠈⠳⡄⠀⠙⠆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡠⠊⢹⡞⢷⣶⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⢿⡟⠁⢁⡿⠁⡠⠊⠉⠣⠀⠀⠀⠙
⠀⠀⢀⡋⠀⠒⠂⠀⠀⠀⢣⠀⠀⠈⠀⠠⠊⠀⡰⠋⣿⣿⣦⣀⣀⣀⣠⣤⠴⠤⣤⣤⣀⡀⢀⣾⣿⡇⣿⡇⢘⡤⠗⠊⠑⠆⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⠦⢤⡀⠀⠀⠊⣀⣴⣿⣿⣿⠉⠁⠀⠀⠀⣀⠀⠀⠀⠈⣻⡟⣿⡿⢹⡇⣇⠘⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠁
⠀⠀⠈⠣⣀⣀⣠⣴⠖⠒⠀⠀⣰⡶⠛⠉⠂⢻⣿⠋⠘⢯⣿⣿⣦⣤⣤⡴⢧⣤⣤⣤⣼⢋⡼⠋⠀⢸⡇⡏⣰⠠⠓⠄⡀⣠⣄⣤⣀⡬⠶⠂⠐⠋
⠀⠀⠀⠀⠀⡏⠈⠉⠙⠒⢲⡟⠩⠀⠀⠀⠀⡎⠙⠁⣴⣮⣻⡁⠀⠀⠀⢰⡆⠀⠀⠀⢠⡞⣰⣿⣄⠞⡇⠁⣿⠇⠀⠀⠀⠁⢸⠆⠀⠀⢠
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢸⠃⡇⠀⠀⢀⠀⡿⢦⡀⠀⠉⠛⠳⠶⠶⠶⣾⣧⣴⡶⠶⠿⠿⠛⠋⠁⣠⡇⢰⡟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢸⢀⠁⠀⠀⠀⢸⠇⠀⠉⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⡤⠖⠉⠀⣷⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠲⠶⠖⠚⠊⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠶⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢊⠁⠀⠀⠀⠀⡀⡀⠀⠑⣄⣀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⡎⡜⠀⡀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡏⠐⠐⠊⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠻⠖⠠⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠈⠐⠃⠶⠢⢀⡀⠀⠀⡠⠒⢼⠢⣀⠀⠀⠀⠀⣠⢆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠓⠒⠦⠤⣀⡀⠀⠀⠀⠡⠊⠤⠀⠀⠈⠢⡉⠐⣖⢉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠶⣤⠤⠤⣤⡌⣩⠝⣃⠖⠁⠄⠀⠀⠠⣀⠤⠾⠬⣉⡠⠀⠀⠛⠢⢄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣄⣀⠞⠁⣠⠞⠁⠀⠀⢨⢄⡿⠚⠳⢊⡦⠦⢤⢠⠎⠁⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⢢
⠀⠀⠀⠀⢀⠴⠋⠀⠀⣠⠞⠁⡠⠊⠁⠀⠀⢀⢔⡵⠋⠀⠀⣠⠊⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣦⠴⠒⠙⠢⣄
⠀⠀⠀⣞⠁⠀⠀⠀⠞⠃⠀⢺⠁⠀⠀⢀⡔⢱⠋⠀⠀⠀⠀⢇⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡚⠀⠒⠀⠅⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠈⠓⢄⠀⠀⠰⠒⠒⠺⠀⠀⠀⠈⢀⡀⢳⣒⡪⣦⠀⣈⣦⡠⣦⢤⠣⣄⠀⢀⢀⣀⡀⢀⡼⠀⠈⠓⢄⠠⠄⠋⠺⠍⢱⠀⠀⠀⢈⣹⠆
⠀⠀⠀⠀⠀⠀⠙⢦⣸⡀⠀⢀⣀⠀⢀⡀⠀⣓⡞⢉⣴⠞⣁⣼⠋⣐⢃⣎⢀⠈⠑⣞⣸⣴⣚⡉⢑⣀⡀⢀⠀⢀⣞⣀⣀⣄⣈⡤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡅
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢾⣠⣠⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⢶⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠤⡰⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢀⣀⠀⢀⣀⡀⡀⠀⠀⢀⡤⡀⢠⠤⡀⡄⡀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠉⢢⠋⠉⣌⠧⠀⢀⣡⠀⡇⢸⠎⠀⠡⠙⢄⠀⠰⠖⢦⠇⠀⡇⡸⡅⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠸⠤⠂⠣⠔⠙⠀⠛⠈⠈⠃⠃⠘⠈⠒⠁⠁⠈⠁⠀⠑⠁⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⢀⡀⣀⢀⡄⠀⠀⣀⣀⠀⢀⣀⢠⡀⣀⣀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣗⡇⢿⣼⢹⣏⡁⣿⡿⠃⠻⢯⣸⡎⢹⡏⠉⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠘⠇⠘⠋⠸⠷⠆⠛⠛⠀⠷⠾⠛⠃⠘⠃⠘⠻⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⡄⠀⠀⠀⣤⣤⣦⣤
⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠈⢿⣿⣿⠿⠁⠀⠁⣀
⠀⠘⡙⠛⢻⠇⢀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠚⠂⠀⠀⢀⣼⣿⣦⡀
⠀⠀⠀⠀⠘⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡀⠀⠀⠀⠀⢹⣿⠛⠛⠀⠀⠀⠀⢠⡾⠛⢷⡀⠀⠀⠀⢠⣿⠁⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣶⣻⠇⠀⣠⣴⣶⣄⠀⠀⠙⠒⠝⠚⠃⠀⣠⣴⣦⡌⢳⣄⣢⠀⠀⠀⠀⢸⡀⠀⢸⡇⠀⠀⠀⡈⠉⠁⠀⠈⠉
⠀⠀⠀⣴⣶⣦⡄⠀⠀⠙⠋⠀⠀⠀⠿⠛⠻⠿⠀⠀⠀⠀⠀⠀⠀⢸⡿⠅⠈⡿⠀⠈⣁⣀⡀⠀⠀⠀⠑⠒⠊⠀⠀⠀⢀⣿⡿⠿⠗⠀⠀⠀⣠⣤⡀
⠀⠀⠸⠏⣮⣹⠟⠀⠀⣠⣤⣦⣄⠀⠘⢶⠶⠂⢠⣴⣶⣶⣤⠀⠀⢸⣿⡄⠀⠀⢠⣾⡿⠿⢿⣷⡀⠈⢀⣴⠶⢦⡄⠀⣼⣿⡀⠀⠀⠀⠚⠛⣿⡿⠃
⠀⠀⣰⡟⠙⠉⠀⠀⠀⢫⡬⢡⡍⠀⠀⠀⠀⠀⢿⡟⠛⠻⣿⠃⠀⠸⢿⠗⠀⠀⠸⣿⠀⠀⢀⣿⠇⠀⠈⠳⣂⠶⠁⠀⢿⣿⠇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠶⠴⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠙⠿⠞⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⢿⣯⣅⡈⣆⣠⡄⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣟⡭⠄⣲⣿⣿⣿⣷⣿⣟⣁⣼⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠂⠀⠛⠻⣿⣿⣿⣿⣿⡀⠘⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⣠⣴⣾⣿⣿⣶⣾⣿⣿⣿⣿⣿⣦⣜⡿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡿⢫⣿⣟⣔⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣮⢻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⠟⣡⣿⣿⣿⠇⢻⣿⠿⠋⠀⠉⠛⠿⠿⠛⠉⠀⠀⠀⠀⠈⠻⣷⡹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⣰⣿⠋⣿⣿⢀⣨⣥⣄⡀⠀⠀⢔⣲⣶⣤⡀⠀⢀⣴⣾⣷⣤⣻⣧⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡴⣾⠟⠁⣼⣿⡇⣿⣿⣿⣿⣏⠺⣏⡥⢐⡲⠬⣹⢺⣿⡿⠛⠻⣿⡿⣿⡞⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣾⣿⡆⠀⣠⣾⣿⣿⢡⢿⣿⠁⠈⠻⣷⣜⢞⡵⢜⡳⣲⣿⠟⠁⠀⣠⣈⠛⣿⣷⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣋⣵⣾⠟⣻⣿⡟⠸⠾⠛⠷⣤⡀⠈⠙⢧⣾⣧⡽⠟⠁⢀⣴⠾⢋⣙⣷⢿⣿⣧⡹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣶⣿⡿⢛⣥⣾⣿⡿⢁⠧⣶⣿⣶⣮⣽⣶⣄⠔⠙⠋⠳⣤⣴⣿⣷⡾⢿⡿⠟⣼⣿⣿⣷⣜⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⠿⠋⠜⠛⣠⣿⣦⠉⠻⢧⣉⠉⠙⠿⢆⣤⣤⣰⣿⠟⠉⢠⣴⠞⠁⣼⡿⣿⣿⣿⣿⣷⣍⡻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢏⣼⣿⡿⠋⠀⠀⠀⣀⣴⣿⣏⢿⣷⣄⣀⣀⣀⣤⣶⣿⡿⠿⣿⣷⣦⣄⣀⣀⣠⣾⣿⣿⣿⡿⢫⢿⣿⣿⣿⣮⣻⡄
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣦⣄⣀⣠⣾⣿⣿⣿⣿⡆⣻⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⡏⣿⣿⣷⠘⠈⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢳⣿⣿⣿⣇⠀⠰⠛⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣿⣇⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⢟⣿⣿⣿⣿⣿⣷⣀⣰⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣮⣻⢄⠀⠀⠙⣯⡙⠛⢋⣽⡟⠁⠀⢘⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⡟⠁⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠙⠛⠛⠉⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣹⣇⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣽⣿⣿⣿⣿⣯⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⢻⣯⣉⣉⣉⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣼⠃⠈⠙⠿⣿⣿⣿⣿⣿⠿⠛⠁⠈⠈⣿⣝⠿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠙⣿⣿⣿⡿⠟⣋⣥⣤⢭⣿⣀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⣽⣿⣷⣮⣽⡻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⣈⣭⠭⠓⠛⠛⠉⠤⠤⠤⠀⣭⠁⠒⠀⠤⠀⠀⠤⠄⢐⠊⢩⣤⡄⠠⠤⠬⠙⠛⠳⠭⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣴⣶⣶⣿⠿⠿⠿⠟⠛⠛⠛⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠓⡄⠀⠀⠀⡜⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣰⣁⠀⠀⢰⠁⡀⠀⠘⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⣀⣀⣈⣇⢀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠈⠀⣠⠶⣖⠂⠓⠒⠒⠒⠂⠂⡶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢣⠆⠀⠀⠀⠈⡿⢀⠃⠀⠀⠀⠀⠀⠀⢀⡔⠊⠳⢄⡈⠙⠯⣉⠉⣩⠟⠉⢀⡤⠛⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢀⡇⠈⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠉⠢⣄⡼⡻⣄⡠⠚⠁⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠐⠲⠙⡄⠀⠀⠀⠀⣞⣒⣒⣒⣒⣒⣒⣒⣒⣒⣺⢸⠀⠀⠀⠀⠀⠀⢺⠲⣧⠖⢉⠇⠀⠀⢀⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣧⠤⢤⠀⠐⠒⡼⢖⣋⠑⠦⣠⣳⣜⠀⢸⢀⠀⠀⠀⠀⢀⠤⠚⠷⠧⠴⠗⢤⡤⠞⠁⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⡇⢠⠞⠀⠐⢲⠁⠉⠀⠀⠀⢵⠀⢸⠀⢸⠈⠀⠀⠀⢰⠁⠎⠁⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣄⣀⣀⣀⣿⣡⣠⣬⣉⣿⣄⣀⣀⣀⣀⣽⣀⣬⣀⣼⣄⣠⣀⣀⣼⣠⣤⣄⣠⠖⠁⠀⠀⠀⠀⠀⠀⡤⢀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢆⠀⠀⠀⠀⣀⠔⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⠘⡆⠀⠀⠘⡡⠚⢉
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢨⠀⠹⣀⠔⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⡤⠖⠒⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠒⠒⠲⠤⣀
⠀⠀⠀⢠⠞⠁⢀⣀⣤⣤⣀⣀⡀⠀⣀⡩⠷⠶⠶⠖⠲⠶⠦⠤⣄⡀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠖⠲⢶⡤⠞⠉⠀⠀⢀⣀⣀⣤⣤⣀⡀⠈⠳⡄
⠀⠀⢠⠏⠀⢰⠁⠀⠀⠀⣀⠴⠛⠉⠀⣀⣀⠤⠤⢤⡤⠤⣄⣀⠀⠉⠳⡄⣠⠞⠁⠀⣀⣠⠤⢤⡴⠋⠀⢀⣠⠴⠚⠛⠦⣀⠀⠀⠀⠈⡆⠀⠸⡄
⠀⠀⢸⠀⠀⣿⠀⠀⣠⠞⠁⢀⡤⠒⠉⠁⠙⢦⡀⠀⢳⡀⠀⠀⢙⡆⠀⣸⣇⠀⢰⡋⠀⠀⢀⡞⠀⢀⡴⠋⠀⠉⠒⢦⡀⠈⠳⣄⠀⠀⣿⠀⠀⡇
⠀⠀⢸⠀⠀⢸⣀⡜⠁⢀⠞⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⢇⠀⡰⠋⠀⡰⠃⠘⢦⠀⠙⢄⠀⡼⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⢣⣀⡇⠀⠀⡇
⠀⠀⠘⣇⠀⠈⢿⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⠎⠀⣠⠊⠀⠀⠀⠀⠑⢄⠀⠳⣇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢻⠁⠀⣸⠃
⠀⠀⠀⠹⡄⠀⠘⢧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣇⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣇⢠⠏
⠀⠀⠀⠀⠙⣆⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠘⡀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢆⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢣⠀⢸⡏
⠀⠀⠀⠀⢸⠏⠳⣄⠀⠈⠓⠦⣄⣀⠀⢀⡠⠞⠁⢀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠣⡀⠈⠳⢄⠀⠀⣀⣠⠴⠚⠁⠀⣸⠀⠸⡇
⠀⠀⠀⠀⢸⡆⠀⡏⠙⠲⠤⣄⣀⣠⠝⠋⢀⡠⠶⠉⡀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡈⠶⠐⠒⠉⠉⢀⣀⣠⠤⠖⠋⢹⠀⢰⡇
⠀⠀⠀⠀⠈⡇⠀⢳⠀⠀⢀⡠⠊⠁⣠⠔⠉⠒⠀⠀⠀⠁⠊⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠈⠀⠀⠀⠒⠘⠫⣍⠈⠑⢄⠀⠀⠀⡞⠀⢸⠁
⠀⠀⠀⠀⠀⠹⡄⠀⠳⠴⠋⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠙⠦⠜⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠲⣄
⠀⠀⠀⠀⠀⣰⠃⠀⡴⠲⣄⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⣠⠖⢦⠀⠘⣆
⠀⠀⠀⠀⢀⡇⠀⡸⠀⠀⠀⠑⢄⡀⣙⣦⡀⠤⠀⠀⠀⡀⢤⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⡠⢀⠀⠀⠀⠤⣀⠔⠋⢀⡤⠊⠀⠀⠀⢧⠀⢸⡀
⠀⠀⠀⠀⢸⠇⠀⣇⣠⠴⠒⠋⠉⠁⣀⣀⠤⠐⠶⡈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⣁⠶⠊⠁⣠⣔⠋⠉⠙⠒⠦⣄⣸⠀⠸⡇
⠀⠀⠀⠀⢸⡆⠀⡏⠀⢀⡤⠖⠋⠉⠀⠀⠑⢦⡀⠈⢣⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡜⠁⢀⡴⠊⠁⠀⠉⠙⠲⢤⡀⠀⠙⢦⣠⡇
⠀⠀⠀⠀⣸⡇⠀⢣⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠱⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⡄⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠹⣄
⠀⠀⠀⣰⠃⢱⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⡀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⡄⠀⠘⣆
⠀⠀⢠⡏⠀⠀⣧⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢹⢆⠀⠙⢄⠀⠀⠀⠀⡠⠋⠀⡴⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⣷⡀⠀⢹⡄
⠀⠀⢸⠀⠀⢸⠉⢣⡀⠈⢦⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡞⠀⠑⣄⠀⠱⡄⢠⠞⠀⣠⠊⠀⢳⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⡜⠁⡇⠀⠀⡇
⠀⠀⢸⠀⠀⣿⠀⠀⠙⢦⡀⠈⠓⠤⣀⠀⣠⠞⠁⠀⡼⠁⠀⠀⣈⠇⠀⢹⡇⠀⠸⣅⠀⠀⠈⢧⠀⠈⠳⣄⢀⣀⠤⠚⠁⢀⡴⠋⠀⠀⣿⠀⠀⡇
⠀⠀⠘⣆⠀⠸⡀⠀⠀⠀⠉⠲⣤⡤⠖⠋⠁⠀⣠⠞⠓⠒⠚⠉⠀⣀⡴⠃⠙⢦⣀⠀⠉⠛⠒⠚⠓⠒⠒⠉⠉⠀⣀⣤⡖⠉⠀⠀⠀⢀⠇⠀⣰⠃
⠀⠄⠀⠘⢦⡀⠈⠉⠛⠛⠉⠉⠁⠀⠀⣀⡴⠞⠷⠤⠤⠶⠖⠒⠋⠁⠀⠀⠀⠀⠈⠙⠒⠲⠶⠦⠴⠶⠶⢶⣛⠉⠀⠈⠉⠉⠛⠛⠉⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠉⠓⠦⠤⠤⠤⠤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⠤⠤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣄⣤⣤⣤⣤⣀⣠⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣰⣯⡤⠤⢤⡴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⡼⠃⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠓⠦⠤⠤⠤⠤⠤⢤⡤⠤⠴⣶⠖⠒⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣳⣄⣀⣠⣤⠤⠤⠤⠤⡤⠤⠤⢤⡀⠀⠀⠀⢀⡴⠋⡠⠚⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢀⡴⠋⠀⣴⠃⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣴⡷⠋⠀⢀⡼⠁⠀⠀⣰⠏
⠀⢀⣠⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⠉⠙⢯⡉⠉⠉⠉⢉⡽⠋⠀⠀⢀⡞⠀⠀⠀⡴⠃
⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠓⠛⠒⠛⠋⠀⠀⢀⡼⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠋
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠉⠉⠉⠛⠛⠓⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠋
⠀⢸⡄⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⣇⣠⡴⠞⠋
⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣁⣀⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿