Epistle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⣺⣍⣑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⠒⠒⠒⠒⠒⠲⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠛⢺⢸⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡴⢁⢄⠔⡀⠑⣄⣰⠋⡠⢀⢆⡌⢦⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢸⠁⡣⢊⢞⡔⡕⡈⠃⡴⣡⢋⢎⢞⢸⠆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠸⡄⠴⣡⢊⢞⠜⣜⡟⡰⣡⣣⢣⠎⡜⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠻⣤⢀⠀⠀⠀⠀⠹⡔⢡⣣⢊⢎⢎⡜⡰⡱⡡⢣⠞⠀⠀⠀⠀⠀⠀⠀⢸⡷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⢮⡠⢀⠀⠀⠈⠲⣅⠣⢋⢎⣜⠜⣰⠗⠁⠀⠀⠀⠀⠀⢀⣠⠖⠋⢔⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠣⣎⡄⣀⠀⠀⠙⠢⢎⡼⠋⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⢸⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢥⣣⢆⡀⢀⣠⣄⣄⡄⠀⢀⣤⠖⠉⠀⠀⠀⠀⠀⠀⢐⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠟⠋⠁⠀⠀⠈⠙⠓⢯⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⢦⣤⠀⠀⠀⠀⠀⠸⠽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡨⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢶⡤⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠔⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⣌⣊⡊⠐⠀⠂⠄⠄⠄⠄⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠊⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣠⠔⣠⢄⡠⢂⡤⢀⠔⡠⢂⡤⢀⠔⣠⠔⡠⢀⡠⣀⠄⡠⢀⡠⢀⠄⡠⢄⡤⣠⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠈⠀⠈⠀⠁⠀⠁⠈⠁⠈⠈⠁⠈⠀⠉⠈⠁⠈⠀⠁⠀⠁⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⠒⢲⢙⡽⣿⣭⠻⢖⣖⣤⢤⣄⡀⠀⠀⠀⣠⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣧⣤⠀⢿⣨⠭⠚⠛⠉⠉⠙⠓⠪⢿⣴⡋⡕⠢⡞⢿⣅⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠿⣊⡶⠙⠃⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡾⢟⣁⣏⣉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠁⣷⢒⣰⡇⠈⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⠹⣏⢁⢿
⠀⠀⠀⠀⠀⠀⠀⠠⢳⢻⡷⣿⣍⢱⣲⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⡇⣸
⠀⠀⠀⠀⠀⠀⢠⢛⣦⠀⠤⠤⡈⢻⢷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢜⠇
⠀⠀⠀⠀⠀⢀⡿⢽⠃⢻⣕⣐⣃⡰⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⢸⣇⠠⠛⢲⣬⣑⣐⣮⡇
⠀⠀⠀⠀⠀⡞⠱⣄⠀⠀⣀⡈⣉⢈⡇
⠀⠀⠀⠀⠀⠳⢒⣺⣵⣤⣶⣨⠿⠻⡇
⠀⠀⠀⠀⠀⢣⣯⡎⡀⠀⠙⠻⣦⠈⢻
⠀⠀⠀⠀⠀⠸⣱⣵⣧⣾⢙⠛⣿⣆⠈⡀
⠀⠀⠀⠀⠀⠀⢦⡎⠀⠀⠨⢶⡾⣝⠇⢣
⠀⠀⠀⠀⠀⠀⠘⣿⠄⢠⡾⣅⣹⢸⣇⣊⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠙⠛⠃⠉⢈⣿⡖⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠧⣭⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡶⢶⣀⣀⣴⢨⡉⠓⢯⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⣌⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢥⡁⡀⠘⠻⢇⡵⣉⠼⣳⠦⢄⢀⣀⣀⣀⣠⣤⡞⠑⢂⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣜⠛⡽⢟⣯⣞⣴⢶⠟⢋⣁⣏⡤⠏⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠁⠞⠉⠉⠀⣈⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⢈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠁⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠉⠀⠄⡀⡁⠒⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠊⠀⠂⠀⠀⠀⠀⠀⠁⠂⣄⠐⠄⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡹⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⡆⠀⠄⠀⠀⢰⣃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠰⣄⠀⢂⡔⠀⠀⢀⡞⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⡄⠀⠀⠀⠀⠄⠀⢲⠀⠀⠀⢀⠄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁⠀⡄⠀⠈⠀⠀⠀⠠⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠂⠀⠀⢀⠠⠜⠀⣀⠀⠀⠠⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣁⠀⠀⡀⠠⠀⠀⠁⠀⢸⡇⠀⠈⠂⠀⠀⠂⠀⠀⠀⢌⠀⠈⠠⠖⠀⠀⠀⢤⡀⡐⠀⠪⠹⢁⡄⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠋⠁⡀⠀⠀⠈⠀⠀⠀⠐⠄⠀⠀⠀⠀⡀⠀⠙⠁⠀⠠⠀⠸⣶⠃⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠰⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠃⣈⠁⠀⠀⣧⡀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠀⢀⠀⠀⠁⡈⢀⡀⠀⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⣀⢀⡀⢂⡀⠖⠉⠈⠀⠀⠀⠀⡀⠓⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣉⢉⣂⣙⣀⣈⣙⠓⣁⣈⡇

⢀⣴⡟⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⣼⠃⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠏⠉⠉⢻⡆
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⣿⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣸⣿
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠃
⢻⡇⠀⠀⢰⡟⠀⠀⢀⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⠛⠛⠻⣦⠀⠀⠀⣿
⢸⡇⠀⠀⢸⡇⠀⠀⢸⡏⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⣿
⢸⣇⢀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⣿
⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⡇⠀⠀⠀⣿⡇⠀⣠⡿
⠘⣿⠉⠉⣿⠃⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⡿⠿⠿⠿⠿⠿⠟⠛⠁
⠀⣿⠀⠀⣿⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⡇
⢰⣿⣶⣶⣿⡇⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⡇⠀⠀⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤
⠈⣿⠙⠋⢿⠁⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⡇⠀⠀⣿⡷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿
⠀⢿⣄⣠⡿⠀⠀⠀⢸⡇⠀⠈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⡇⠀⠀⣿⡇⠈⠙⢿⣦⣄⠀⢀⣠⣶⠟⠋⠀⢸
⠀⠈⢿⡿⠁⠀⠀⠀⢸⡇⠀⠀⣉⣉⣉⣉⢉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠀⢸⡇⠀⠀⣿⡇⣠⣴⠾⠋⠙⠿⠟⠉⠛⠷⣤⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁⠀⢸⡇⠀⠀⢿⣿⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣸⣧⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠁⠀⢸⡇⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣦⡶⠟⠁

⠀⠴⠾⣦⡀⠀⠀⠀⠀⠀⣤⣂⠑⡆⠀⠀⢠⡖⠉⠉⢐⣴⠂⠀⠠⡖⠀⠙⠲⣄⠀⠀⢲⡆⠀⠀⠀⠀⠀⣤⠒⠐⠦⠄
⠸⣄⣀⠸⣷⡀⠀⠀⠀⠀⢸⢹⡷⡁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣻⠄⠀⢸⡧⠀⡶⠀⠀⠀⣻⢀⡀
⠀⠳⠃⢰⢿⣷⠀⠀⠀⠀⢸⠋⠀⡿⠀⠀⢻⣄⠀⠀⠀⡀⠀⠀⠀⣿⠀⠀⣠⠟⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠁⠉⠁
⠀⠀⠀⡞⠀⢿⣇⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠉⠁
⠀⠀⢸⣃⣀⣘⣿⡄⠀⠀⠀⣠⠄⠀⠰⠀⠀⣤⠀⠀⢰⡖⠀⠀⢰⣶⠀⠀⠀⠀⣰⠆⠀⣶⠀⠀⡰⠂⠀⠰⣶
⠀⢀⡏⠀⠉⠀⢹⣷⡀⠀⠘⡇⠀⠀⢤⠀⠀⣿⠀⡠⢨⡇⠀⠀⢸⡟⠀⠀⠀⠀⢸⠀⠀⢙⠂⢦⡀⠀⠀⠀⣟⠀⠠
⢀⣾⠀⠀⠀⠀⠀⠿⢧⠀⠀⠈⠀⠀⠈⠀⠀⣿⡀⠁⢀⣷⠀⠀⢀⡇⠀⠀⢠⠀⠂⠀⠀⢸⡀⠀⠱⠀⡄⠀⡿⢠⡬⠂
⠘⠛⠃⠀⠀⠀⠀⠘⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢠⠤⠀⠀⠤⠄⠀⠀⠀⠀⠀⢠⡶⠀⠄⣶⠀⠀⠀⠀⡠⠔⠂⠒⢤⣔⠀⠀⠀⣶⡆⠒⢦⡀⠀⢀⡔⠊⠉⠙⠲⣄
⠐⠛⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢸⣿⡄⠀⣿⠀⠀⠀⣼⠁⠀⠀⠀⠀⢹⡆⠀⠀⣿⣤⣄⡼⠃⠀⣿⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡇⠹⣦⣿⠀⠀⠀⠹⡀⠀⠀⠀⠀⣼⠃⠀⠀⣿⠁⠀⣀⡀⠀⠻⣄⠀⢀⠀⢀⡾⠁
⠀⠤⠂⠀⠈⠁⠀⠸⠓⠀⠀⠀⠘⠓⠀⠈⠛⠂⠀⠀⠀⠉⠒⠀⠐⠊⠁⠀⠀⠀⠛⠃⠀⠈⠀⠀⠀⠈⠉⠀⠈⠉⠃
⠀⡤⡤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⢀⡀⠀⠀⠀⡀⢀⣀⣀⡀⠀⠀⠀⠀⣤⣤⣄⠀⠀⢤⣤⠀⠀⢤⡄⠀⠀⡀⢠⡄
⠀⣿⡇⠀⢘⣟⠀⠀⠀⠀⠀⠀⣾⡀⠀⠁⠀⠀⠈⠉⡀⠈⠁⠀⠀⠀⠀⠀⢡⡞⠁⠀⠀⢸⡏⠀⠀⠈⣷⠀⠀⠀⡎
⠀⣿⡇⢤⡚⠁⠀⠀⠀⠀⠀⠀⠈⠚⢔⡀⠀⠀⠀⣜⡃⠀⣔⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢨⡇⠀⠀⠀⠸⣧⠀⡜
⠀⣿⡇⠀⠙⣷⣄⠀⡄⠀⠀⢄⠀⠀⢈⠇⠀⠀⠀⠻⣧⡀⠈⠀⢀⠀⠀⠀⢿⣆⠀⠀⠀⢸⣇⠀⠀⠀⠀⢻⡷⠁
⠀⠛⠋⠀⠀⠀⠉⠛⠃⠀⠀⠋⠑⠒⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠁
⠀⣤⣤⣤⣤⡤⣤⣤⡤⠀⢰⣤⡄⠀⠰⣶⣶⠀⠀⠠⢶⠶⣦⠀⢀⣤⠢⣤⡀⠀⠀⠰⣦⡄⠀⠠⡤⠐⠦⠤⠤⢤⣰⠆
⠀⠁⢠⡞⠀⠀⢀⡞⠀⠀⠈⣿⡇⠀⠀⠈⢻⣧⡀⢠⠎⠀⠁⠀⢈⡿⠀⢿⡇⠀⠀⠀⣿⡇⠀⠀⠑⠃⠀⠀⣠⡿⠃
⠀⢀⣾⠃⠀⠀⣾⠁⠀⠀⠀⣿⡆⠀⠀⠀⠀⠙⣽⣎⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢐⡫⠃
⠀⠘⣿⡇⠀⠀⢿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⠋⢾⣧⡀⠀⠀⠀⠀⠀⠈⠿⣦⡤⠜⣿⡇⠀⠀⠀⠀⣰⡗⠁
⠀⠀⠘⠻⠦⠤⠼⠻⠦⠤⠶⠿⠧⠀⢘⢿⣴⢧⠀⠀⢻⣟⣄⠀⠀⣶⡆⠀⠀⠀⠀⠀⣟⡇⠀⠀⠀⣰⣟⣠⣄⣀⣀⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⣾⡇⣿⣿⣿⣿⡟⠿⠿⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠻⣤⣾⣷⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡗⣀⣿⡿⠋⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠁⠉⣿⠋⠉⠀⠀⣀⡤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠈⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡀⠀⠀⠀⢀⠔⠒⠤⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⠞⠛⠛⠛⢦⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⢀⠀⡎⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣿⣿⣷⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠁⠀⠀⠀⢸⡄⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠻⡿⠛⣿⠛⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⡇⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠓⢦⡀⠀⠀⣀⣀⣠⡼⠥⣨⣧⣀⡳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⡞⠀⠀⠀⠈⡇⠘⣿⣿⡀⡤⠤⠤⠤⠤⠤⣤⣴⣟⣻⣭⣁⠀⠒⠦⣄⡀⠀⠈⠈⠙⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⠁⠀⠀⠀⠀⢷⠀⣽⣿⣷⣦⠴⠶⠀⢀⠢⢶⠿⡹⠉⠈⣻⣷⡄⠀⠀⠉⠓⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣈⣃⡹⣿⣷⠀⠀⠀⢀⣀⠤⠺⠟⢂⣠⠞⣛⣋⣬⠤⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠉⠉⣠⠿⠛⢐⠂⠉⠉⠀⢀⡠⠖⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣤⣀⡀⠀⠀⠘⠋⠀⠀⢸⡆⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣿⣿⣷⣄⠀⠀⣀⢀⣦⣿⣿⣅⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣾⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠒⠈⠉⠻⢿⣿⣷⡾⠿⠿⠛⠋⠉⢹⡶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠑⠒⠦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠠⠤⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⠆
⡆⣴⣾⣀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣷⣦
⢧⠄⢀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢊⡀⠄
⠈⠑⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣖⡄
⠀⠘⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⣴⣦⣼⣶⣾⣶⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣧⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⡤⠀⠀⠤⠤⠄⠤⡤⡤⠤⡤⠄⠤⠤⣤⠤⠤⠤⠤⠤⣤⡤⠤⠤⠤⠄⡤⢠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠐⠀⢀⠀⠈⠃⠀⠉⠀⠔⠄⠀⠀⠁⠶⠄⠇⢠⣌⡀⢀⠈⠀⠁⠪⡀⠁⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠐⠀⡰⠁⡆⠀⠀⠀⠀⡐⢉⢋⠀⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⠀⠐⡆⠀⢐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡏⡀⠠⡄⠀⢀⠜⠀⡀⠀⠀⠀⠠⠅⠀⠀⠀⠌⠐⠰⠏⠤⢀⠀⢠⠀⠀⠀⠀⠀⠇⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠈⠆⢊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢂⣈⠥⡹⢢⠀⢈⠀⠀⢈⠄⣈⠀⢀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠄⢠⠄⡉⠀⠊⡁⢂⠀⠀⠀⠀⠐⢀⡀⠚⢆⢣⡄⢁⣶⣒⣲⠶⠇⠀⠘⠿⡃⢸⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⣠⠚⠀⠀⠀⠀⢊⡄⠈⣀⠴⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⣀⠀⠀⠀⣀⠄⣁⡈⣠⣋⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠄⠀⠁⡈⠥⠀⠀⠀⠑⠘⠓⠀⠀⠐⠓⠒⠂⠐⢠⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⢄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⢄⠀⠂⠁⢆⡇
⠀⠀⢀⣤⡀⠀⠀⡴⠞⠓⠻⣿⡶⠶⣤⡀⠀⠀⠀⠀⠀⠈⡥⠤⠂⠀⠀⠀⠀⠀⠀⠆⠒⡀⠀⠴⡀⠀⠀⠀⠀⠀⠀⢄⡀⣘⠂⡶⡇
⠀⠀⠘⢯⣁⠀⠈⠳⠿⠀⠀⠣⡿⠳⢄⠙⢦⡀⠀⠀⠀⠀⡓⠒⠄⠀⠀⠀⠀⠀⠀⠄⠤⠉⠀⠲⠀⠀⠀⠀⠀⠀⠀⠈⠁⢫⠁⠧⡇
⠀⠀⠀⠀⠈⠉⠙⠒⠲⠤⣄⣠⣶⠲⢄⠑⢆⠑⢄⠀⠀⠀⡍⠊⠀⠀⠀⠀⠄⠀⠠⠀⠁⠀⠀⡈⠒⠀⠀⠀⠀⠀⠀⠂⠀⠄⡠⠈⡇
⠀⠀⠀⠀⢠⡞⡋⠙⠛⠶⣤⡙⢯⡀⠀⠑⢄⠱⣄⡑⣾⣶⠃⠀⠀⠀⠐⠂⠀⡀⢁⡒⠀⠀⠠⡠⢤⠀⠀⠤⠤⠤⠤⠤⠠⠴⠤⠚
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠙⠷⣝⢦⡀⠀⠑⢌⠻⢭⣿⡀⠠⠀⠉⠀⠀⠀⠉⠢⡉⢁⠘⡉⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣍⠢⣀⠀⠙⢯⣿⠀⠀⠠⠀⡙⢤⠈⠀⠀⠀⣌⠃⢀⠁⢲⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠻⣗⢦⡙⠻⣟⣿⠀⠀⠀⠃⠙⠀⣁⠀⢄⠁⠈⠠⡀⠀⠀⠄⠈⠁⢥⠄⡜⠁⡈⠯⡭⠽⠶⡤⠀⢠⣼⡁⢸⠯⡽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡉⠉⠓⢦⡐⢟⠃⠙⠳⣞⣻⡀⠀⠀⢀⠰⠆⢌⠁⢀⠀⠠⠀⠈⠀⠀⠀⠀⠐⠀⠌⢉⡞⣱⠔⠀⠀⠀⠈⠀⠀⣉⠀⠈⣀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠙⢦⠉⠒⢶⡞⣿⠌⡄⠁⠰⠁⢀⠈⢢⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢂⠠⠰⣆⠒⠈⡁⠜⠀⠀⠈⠀⠀⠆⠀⠐⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⢈⣻⠆⠈⡀⠀⠀⠀⠀⠀⠣⠀⠄⠀⡀⠀⠀⠁⣘⡌⠀⠀⠀⢸⣄⠀⠀⠀⠀⡀⠀⠀⠠⠅⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⢿⡄⠀⠇⢀⠀⠠⡀⠀⠂⢠⡆⠐⡀⠠⠠⠂⠈⠠⡀⠶⠀⠆⠙⠉⠀⠀⢁⢀⡀⠔⠁⠀⠈⠠⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠚⠓⠒⠐⠛⠒⠂⠒⠓⠓⠒⠓⠐⠓⠒⠛⠒⠓⠒⠒⠀⠛⠓⠀⠒⠒⠂⠒⠐⠒⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⢆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠄
⠀⠀⠀⠀⣄⠀⠀⡦⢦⢰⠂⡴⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡦⠗⣉⡴⠋⡇⢀⡇
⠀⠀⠀⠀⡧⣷⠀⡇⢼⡿⣾⢃⡅⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⠥⠖⠋⠁⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⠄
⠀⠀⠀⠀⢠⣿⢟⡇⣿⠿⢹⡄⣐⡟⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⣀⣴⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⠻⣿⡯⢀⢠⠈⢷⡘⠁⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⣀⣠⣴⣚⣫⣶⡇⠀⠀⠀⠀⢠⣴⣿⣭⣽⣻⣷⣿⢿⡆
⠀⠀⠀⠀⠈⡓⠿⠀⡸⢨⡀⡿⠁⠀⠀⠀⠀⠓⠒⠒⠚⠿⣿⠟⠛⠒⠒⠃⠀⠀⠀⠀⠀⠦⠭⠭⣭⣭⣭⣭⡵⡖⠀⡄⠀⠀⢈⠻⢿⣿⡽⣛⣟⣿⡿⠟
⠀⠀⠀⠀⠀⠸⣤⡄⠁⣿⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠂⠀⠀⠄⠠⠀⠠⠄
⠀⠀⠀⠀⠀⢀⠉⠉⠈⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⡼⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠌⠀⡇⠀⠇⠀⠀⠀⠀⠀⢀⡤⢤⣤⠤⣤⠤⠤⡤⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢘⣀⠁⡄⢀⠀⢠⠀⠀⠀⢸⡇⡜⠁⠀⠀⠀⠈⠿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⠀⠄⠠⡀⠀⠄⠠⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠄⠠⠠⠠⠤⠀⠐⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢱⠐⡐⠂⠈⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢣⣀⠀⢤⡤⠀⡀⢨⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⡄⡀⣀⠀⠀⠀⠀⠀⠀⠠⢤⣾⣿⣴⡾⠶⠶⠿⢿⣿⠇
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡶⢶⡶⢷⡄⠀⠀⠀⠀⠀⣠⢤⣤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣿⣿⢟⢡⣀⢀⠀⠀⠀⠀⠠⣾⠟⠋⠠⠀⠀⡒⠐⠢⣄⠑⠄
⠀⠀⠀⠸⣇⣀⣀⣹⠆⠀⠳⣄⣃⡾⠁⢀⠾⡄⠀⢰⡥⠄⠁⢾⣿⡀⠀⠀⠀⠀⠆⣠⡴⠪⢹⣗⢛⣲⣷⣿⡇⠀⠀⠰⣧⠋⠀⢀⠀⠁⠀⣀⣠⠀⠈⣳⡌
⠀⠀⠀⠀⠈⣿⡏⠁⠀⠀⠀⢘⣿⠀⠀⠘⣶⡟⠀⠈⠻⢾⣷⠺⠋⠀⠀⠀⠀⠁⠎⢀⣷⣶⠿⣷⡀⣿⣿⣿⡇⠀⠀⠃⡏⠀⢀⠀⠀⡜⠉⢉⠙⡆⠀⠀⣷
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢉⡿⠀⠀⠀⠀⠀⠀⢈⠓⢤⣈⣾⣿⣿⡇⣿⣻⣿⠁⠀⠀⠠⣧⠀⠀⠀⠘⣏⠈⠛⢁⡟⠀⠈⡿
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⣿⠀⠀⣀⣿⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢬⣉⠒⢈⡿⣿⣽⡇⣿⣿⣿⡁⠀⠀⠀⢿⡆⠐⠓⠀⠈⠉⠋⠉⢀⠀⠰⢁
⠀⠀⠐⢶⣲⣿⡇⠀⠀⢸⡿⠉⣯⠀⠀⠉⠿⠀⢀⣀⣤⣴⣿⠀⠀⠀⠀⠀⠀⢺⢄⡑⠠⣿⣟⣿⣯⢹⡿⠛⠁⠀⠀⠀⠈⢹⣦⣀⠐⠈⠀⠔⠀⠀⢀⣤
⠀⠀⠈⠛⠛⢻⡇⠀⠀⠈⠉⠉⠿⠀⠀⠀⠀⠀⢠⣼⣯⣿⣿⠀⠀⠀⠀⠀⠀⠘⠀⠈⠐⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠈⠉⠒⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠤⠀⠀⠶⣤⣤⠖⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠉⠉⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⢀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠁⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠏⠒⠢⠤⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢀⠑⢄⠀⠀⣀⠠⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠊⣀⠀⠀⠀⠀⠀⢀⠐⠂
⠀⠀⢀⠑⠄⠈⠘⡧⠴⠶⠚⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠢⠀⠐⡧⠤⠖⠚⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠂
⠀⠀⠀⠀⠀⠀⠐⠧⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠜⠂⠀⠀⠻⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠻⠀⠒⠊⠉

⣿⣿⡿⠿⣿⠿⠿⠿⡿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣿⣶⣷⣷⣶⣿⣷⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⢖⣫⠄⠲⢪⢟⣝⠿⠲⡄⢬⣛⢶⡶⢄⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡴⣢⠴⣛⡾⣭⣍⡒⢵⢷⢡⢊⢯⢣⢱⢷⠅⣂⣭⡝⣞⡳⢦⡱⢆⡀
⠀⠀⠀⠀⢀⡴⢋⡾⣡⠊⢇⡇⡏⡗⠬⡶⢛⡛⢎⣀⡟⣛⠺⣎⠔⠏⡆⣇⡏⢱⡙⣦⠡⡀⡀
⠀⠀⠀⠀⡎⠰⢪⠷⢃⢂⠻⣹⠷⠳⣵⣷⣩⢩⣽⠿⣯⢻⣳⣷⡧⠲⠷⡩⠛⡐⠺⢼⡑⠄⡏
⠀⠀⠀⡼⢡⠟⡽⠙⡿⡖⠚⢇⡘⡗⣿⠥⠙⠛⠁⠁⠉⠙⠃⠻⠗⡻⣂⠟⠒⡶⡻⢱⢝⢷⡐⡈⡀
⠀⠀⢰⠗⡟⡌⠱⣵⢝⡾⢖⢿⣟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢗⠺⣔⠥⣧⠊⡌⣗⢿⠃
⠀⠀⣸⠰⡇⣫⠔⡋⠺⣆⢟⢌⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡜⡇⣼⠊⣑⢬⡇⣿⢸
⠀⠀⣁⢺⢸⡱⢿⠶⢉⣚⣿⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣂⠱⢾⠷⣹⢊⠂⡷
⠀⠀⢸⠰⡇⣏⣢⣭⢾⡇⣶⢊⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢱⡇⢻⢮⡥⣊⡇⣿⢘
⠀⠀⠸⡦⣧⢣⡔⡝⡪⢻⠆⣿⣯⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣗⠲⡫⡘⡝⢤⢃⡯⣮⡆⠀⠀⠀⢴⡖⠄
⠀⠀⠀⠳⡘⠧⣵⣰⣷⠃⢐⡎⢱⣅⣺⣑⣦⣤⡄⡄⣤⣤⣄⣲⣃⣵⠏⣶⡀⠳⣶⡸⡦⡞⡰⠁⠁⠀⠀⠀⠀⢀⡆⢠⣤
⠀⠀⠀⠀⢇⡐⢜⡲⡂⢤⣶⡹⡶⡖⠻⡟⢭⣜⣿⣶⣟⣼⡽⡿⠗⠶⡶⣙⡦⠤⡲⣺⡵⢂⡧⠀⠀⠀⠀⠀⣠⠎⠀⠠⠊
⠀⠀⠀⠀⠀⠹⣌⠳⣜⢄⡎⡇⣇⡧⢂⢿⢦⡴⢎⢉⢷⣬⢴⢏⡢⣇⡇⡏⣧⠜⣱⢏⡔⠁⠀⠀⠀⢀⣤⡮⢵⠞⣣
⠀⠀⠀⠀⠀⠀⠈⠛⠉⠳⢮⠷⣛⣭⠔⡺⡾⡈⢎⣎⠜⡸⡾⣗⢬⣝⣣⢯⠴⢛⡽⣿⠷⡀⢦⠰⡖⡿⢋⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠶⠍⠂⠞⠚⣦⣯⣞⠖⢃⣊⣥⠞⢒⢋⡔⡡⢮⡑⢊⣠⠓⡲⢸⢟⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢖⣪⡅⢦⠴⡳⢏⣸⣉⠦⢚⣁⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠿⠖⠤⠔⠂⠿⠞⠛⠋⠁

⠀⠀⠀⠀⡠⢤⡤⣀⣀⣀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣀⡀⣀
⠀⠀⠀⠀⠹⠾⢷⣿⠻⠟⠿⠻⣿⣥⠙⡟⣿⢩⣉⡯⢭⠻⣁⠡⣩⡱⠙⢤⠤⣴⡿⢋⡽⣿⣿⠖⢶⣤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠊⣉⡛⢿⣀⠀⣠⡟⠷⠓⢶⠁⠉⠉⠉⠉⠉⠉⠛⠲⠥⣌⣄⣠⠣⢄⣼⢱⡆⣰⢓⣲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣭⠁⠀⠈⠙⠰⢛⠄⠈⢣⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⡘⡷⢺⣏⠿⣻⣭⠟⠋⢱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⣴⠄⠊⠂⠠⠒⢤⣾⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡞⠁⢀⡿⠲⢤⣠⡼⠃⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣈⣟⣤⣶⣖⠢⠄⣀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⢭⣗⣺⣆⢈⢰⢛⡿⠊⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢞⡉⡀⠠⡀⠤⠶⠛⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠉⣽⣻⣦⣀⣴⠐⣻⢖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⠁⠞⡟⣳⢴⡦⠲⢯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⠏⠵⠲⣽⣿⣻⢯⡮⣬⡋⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠹⣷⠁⠟⠿⠿⣴⣶⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠰⢺⣉⡿⠫⠒⠄⣀⣠⣍⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡾⠋⠕⣢⠖⠲⢽⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⡢⡝⠀⢡⡶⣤⠻⠼⣛⠎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢃⢧⡷⢦⣾⠄⢀⣽⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠽⣇⢈⡁⠀⢻⡧⠛⣫⣎⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⢻⣿⡎⡷⡯⠾⢹⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⣻⣾⡄⢀⡤⣃⣦⣾⣷⡏⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣜⣩⠜⣧⠟⢽⡃⠘⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢝⣿⣷⢋⣀⡉⠷⣨⣻⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢤⡈⢆⠀⠲⠈⢩⠒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⠏⢣⡛⢀⡀⠀⢘⠒⣧⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢌⣁⣠⣾⡀⣀⣄⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠂⣸⣿⠊⠍⢀⣅⠀⠚⢥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⢴⠶⡟⠻⢾⣅⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠰⡞⠁⢛⠛⠿⠛⡏⠀⠈⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⣜⣔⠂⢹⠇⢙⠠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⢻⠁⢤⣚⠁⡀⢀⠙⣺⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢆⡠⢿⣿⠀⠀⡠⡚⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣿⡟⢛⣶⢻⣇⡯⣩⠶⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣴⢖⣌⢃⡜⢛⣿⡭⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣧⢀⠻⡇⢸⡏⣷⠗⣂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⡀⣥⣼⣾⢀⣾⣻⡿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣾⢴⣙⡈⠓⣿⣾⣾⣿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣘⢧⡦⢹⡀⢾⠃⡻⢧⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⡾⠿⢿⢽⣷⣿⡿⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⣇⠠⢀⠿⢤⢨⠾⣇⣠⠇⡷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣉⣼⠻⢆⣶⢶⡆⢸⡼⠞⠉
⠀⠀⢠⠤⡤⢴⣶⠊⢷⡿⢧⡀⢂⣸⣯⡯⢞⡋⣁⣜⠤⡆⢉⣕⣒⣒⣶⡲⢴⠒⣿⣭⡿⣦⡴⠻⣶⡀⢮⣿⡧⠞⠁
⠀⠀⠘⠪⠭⠬⠅⠶⠦⠷⠤⠷⠾⠿⠃⠘⠽⠟⠼⠽⠥⠶⠿⠬⠷⠤⠷⠥⠮⠶⠤⠬⠽⠿⠛⠛⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⡤⠤⠤⠀⡀⣠⡀
⠀⠀⠀⠀⢀⣴⣿⡆⠀⠀⠀⢸⣀⡃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠈⠀⠀⠈⣿⢷⡀
⠀⠀⠀⢠⣾⡟⠉⣀⠀⠀⠰⠋⠁⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⣁⣇⠀⠀⠀⠐⠉⢻⣿⡄
⠀⠀⠀⣼⠋⢤⠄⡍⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⠉⠂⠀⠀⣀⠀⣤⠙⣷
⠀⠀⠀⡇⡇⠀⠀⡃⠀⠈⠆⠀⠀⣀⢴⣿⡀⢀⠀⡀⢀⠀⡀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⢀⡾
⠀⠀⠀⡇⡳⣄⣦⡃⠀⠀⢀⡀⠀⠀⠀⠀⠗⠀⠊⠋⠘⠡⠄⠲⠹⡦⣀⠀⠀⠐⠀⠀⢈⣤⣄⢞⢰
⠀⠀⠀⠀⡀⠀⠀⠛⠀⠀⠀⠄⠀⠀⠀⠰⠂⢈⠀⠰⡱⠀⠉⠐⠆⠀⠀⠀⢀⡀⠀⠀⢈⠀⠀⢘⠈
⠀⠀⢠⢀⡿⠀⠀⠇⠀⠀⡀⠀⠀⠤⢤⡤⠦⠀⠄⡄⢠⠐⠂⠴⠀⠀⠀⠀⠀⠂⠀⠀⠛⠀⠀⢸⡄⣄
⠀⠀⠘⡿⠁⠀⠀⠅⠀⠐⠀⠀⠀⠀⠀⢱⠈⠉⠉⠉⠉⠉⠉⠁⢦⡤⠴⠀⠀⢀⡀⠀⠘⠂⠀⠈⢻⠋
⢠⡦⣠⠃⠀⠀⠀⠒⠀⠀⢀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠂⠀⠨⠀⠀⠀⠘⣆⠴
⠀⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⢳⠖⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢰⠲⡟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢷⢂⡂⣠⣄⣤⣀⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣘⣀⣑⡀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠋⠛⠛⠛⠛⠃⠉⣹⣶⠶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⢀⡔⠉⠀⣠⠤⣤⣤⣦⣴⢾⣻⡽⠞⠋⠣⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢁⡤⠀⠀⠙⢢⡎⠀⡤⠚⣉⣶⣾⣻⡵⠾⣋⠽⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⢀⡠⠔⠚⠁⠀⠀⠀⠀⢀⡤⠀⠀⢀⡤⠂⢈⡄⠘⠁⢀⠔⠂⠀⣞⣠⣼⣿⣿⠿⠛⣭⠔⢻⢀⡫⠂⠀⠀⠀⠀⠀⠀⠀⠙⠲⡄⢤⣤⣀⡀
⠀⠈⠒⢌⡀⠀⠀⠀⠀⠀⡴⠔⠈⣠⠰⠂⠀⠤⠚⠁⡀⠰⠊⢀⣀⣴⣾⣿⠋⠲⡄⠈⠉⠚⠓⠒⠓⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⣿⣾⣧
⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣠⠀⠘⢁⣠⠖⠉⠉⠀⠀⠬⠽⢶⣦⣤⣇⣁⣀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠉⢀⡤⠊⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠦⣤⣀⣀⣀⣠⣄⣀⣀⠀⡿⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⠆⠀⠀⠀⠙⣿⠃⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠋

⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠆⢤⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢛⡿⠋⠉⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣤⠀⠀⠀⠀⠀⠀⠀⣠⡎
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡉⠁⠈⠇⢠⣤⣤⠀⣰⠾⠁⢹⡀⠀⠀⠀⢀⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣣⣴⡶⢶⣦⣩⣶⠞⢉⡀⠄⢈⣩⡶⠶⣦⡘⣿⠀⠀⠀⠀⢀⣤⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡇⢀⡼⢧⣽⠁⠐⠂⠁⡼⠛⠁⠀⠀⠀⠁⠿⣤⣤⣴⠿⢻⣿⣿⣿⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠋⠀⢸⣿⣿⡹⢿⡟⠛⠋⠛⠳⠶⣤⣤⡀⠈⠻⣦⠘⣦⣭⡁⢛⣵⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⠟⣤⡾⠁⠀⠈⠀⠀⠀⠀⠀⠉⠓⠂⠈⢿⣾⣾⣁⣿⢿⣿⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣿⠇⠀⠸⠆⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣯⣿⣿⣿⣿⠗
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⠆⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⢄⠀⠀⢻⣭⣟⠛⠿⠿⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣼⣿⣏⢩⡟⠀⠀⠀⠀⠀⡴⠗⠀⠀⠀⠀⠀⠈⠻⡆⠀⠻⣟⣻⣷⣦⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⡵⣿⣿⡿⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠙⠻⢟⡛⠋
⡇⠀⠀⠀⡀⠤⠀⠀⠠⢄⣀⣀⢠⣴⡿⢿⣷⣿⣿⣙⣯⣶⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠲⡄⢠⣤⡀⢡⠀⠀⠀⠉⠛⢦⡀
⡇⠀⠀⠜⠛⠂⡄⠀⠀⠀⣿⣯⣾⣿⢿⣿⣿⣯⡀⠉⠉⣿⣀⣠⣤⣶⠀⠀⠀⠀⣀⣤⠔⡌⢠⣇⣀⡤⠀⣹⣿⡈⠀⠀⠀⠀⠀⠀⢳⡄
⡇⠀⠀⠳⢤⡔⠁⣀⣨⣴⣿⡣⣻⣿⣣⣿⣿⢻⣷⡀⠰⣿⠍⢹⣿⣟⢻⣿⣿⣿⣿⣥⣾⠀⠘⠉⠉⡄⣴⣿⣿⣿⣶⠾⢷⣿⣷⣶⣦⣴⣄
⡇⠀⠀⠀⠈⠑⠛⣿⠉⠙⢷⣿⣿⣿⣭⣿⣿⠻⣿⠀⠀⠉⠀⢸⣿⡿⠻⣿⣿⣻⣿⡟⠁⠀⠀⠀⠀⠸⣿⣿⣿⡎⣿⣦⠈⠛⣿⡙⣌⠻⡞⠃
⡇⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠈⠉⠁⠘⠛⢳⣿⡀⠀⠀⠀⢨⣷⣾⣷⠼⣿⣽⣿⠀⠀⠠⠀⠀⢠⣷⣙⣶⢉⣼⢏⣿⣇⠀⠈⠁⠘⠆⠃
⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢾⣏⡿⣸⣿⣷⣾⡟⠀⠀⠀⢀⣤⣿⢯⣿⣴⡟⣡⡟⣿⣿⠀⠀⠀⠀⢱⢐⡇
⡇⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣇⣿⣿⢷⣿⣿⠁⠀⠀⠐⣩⣶⠀⣿⢠⣿⣿⣿⣿⣿⠏⠀⣄⠀⠀⢸⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡿⠿⠻⢿⣋⣴⠟⢾⣧⠊⠻⣡⡀⡇⠸⡸⣏⢽⣿⠋⠀⡹⠁⠀⠀⠆⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠁⠀⠀⠴⠋⠀⠀⠀⢸⢷⣾⡏⡀⢲⡄⣠⣿⠞⠁⠀⣤⠀⠀⢀⠂⡰⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠐⠀⠰⠷⠖⠳⠀⠀⠀⣹⠄⠛⠁⠴⠞⡹⠛⠶⠊⠀⠀⢀⡀⢊⡰⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⠀⠀⠀⠀⣴⡤⠀⠀⠀⠀⢠⡤⢶⡶⠷⠿⢿⡷⠾⠿⡶⠶⠶⣶⣤⣤⣄⡀⠀⠒⠋⡀⠛⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⡇⠠⠆⠀⣿⡍⢻⣴⣾⣵⣾⣿⣶⣌⣉⠛⢿⣷⣦⣀⠀⠑⢤⡈⠑⢤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣟⣰⠇⣈⠀⡈⠉⠻⣯⡹⠿⠿⢿⣿⣿⡿⣶⣌⠻⣿⣷⣤⡀⠀⠀⢀⠉⠢⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⣨⣿⡟⠀⠀⠀⠀⢸⣿⣼⣶⣡⡾⠀⠀⠀⠀⢸⣦⣰⣾⡿⣿⣿⣿⣿⣟⣨⢿⣿⣷⠀⠀⠈⠳⡀⠀⢢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⡸⠷⣿⡇⠀⠀⠀⠀⢸⣿⣿⡿⣏⡶⠀⠀⠀⠀⢠⡟⢛⠛⣦⣛⡿⢿⣷⣿⣷⡈⢿⣿⣧⠀⠀⠀⠹⡄⠀⣧
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⠓⠼⣿⡇⠀⠀⠀⠀⢸⣿⣿⡷⠘⠁⠀⠀⠈⠀⠰⣦⠸⣇⠘⣿⣥⠘⣿⣽⡿⣷⣾⣿⣿⠀⠀⠀⠀⢱⡄⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠀⣎⣿⡇⠀⠀⠀⠀⢸⣷⣿⡗⠂⠀⠀⣠⠁⠠⠀⣿⢀⡿⣧⡙⣿⡇⠀⢿⣿⡯⣽⣿⣿⠀⠀⠀⠀⢸⡇⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠉⢽⣟⣿⡇⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠧⠀⣷⠀⣿⣈⣿⣿⣧⣿⣿⠃⣬⣿⣷⣿⣭⡿⠀⠀⠀⠀⢸⡇⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⡈⢲⡾⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⡇⠀⡴⠀⣀⡇⢹⠄⣹⡧⣿⣷⡟⢻⣤⠂⣸⣿⣿⣿⣿⠇⠀⠀⠀⠀⣼⠀⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢤⣼⠿⣿⡇⠀⠀⠀⠀⣿⣿⣟⠁⣴⣿⠀⣨⠀⢸⣦⣿⣼⣿⣿⣿⣿⡟⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⢠⠏⣰⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠶⠋⠀⠀⠀⠀⠀⢿⠿⠿⠧⢤⢿⠶⠿⣐⣿⣉⣝⣣⣿⣏⣻⣿⣴⣷⡿⠟⠛⠁⠀⠀⠀⢀⡴⢃⡴⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣂⣀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⡠⠴⣈⡴⠃
⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠀⣿⣿⣿⣿⣿⡟⠛⠛⠋⢻⣿⣿⣿⣿⣯⣭⠤⠶⠶⠶⠶⠶⠶⠶⠦⠤⠍⠛⠛⠉
⡇⠀⠛⢋⠈⠱⣆⠀⠤⠐⠤⠤⠤⣄⣀⡤⠠⠄⠈⣤⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⢿⣷⣿⣽⣻⣿
⡇⠰⣶⣿⣿⣷⣿⣦⣤⣤⣤⣤⣤⣤⠠⠤⣤⣤⣤⣿⣿⣭⣿⣷⠯⣆⡀⢀⡄⢀⠀⢿⣿⠉⣿⣿⡀⣀
⡇⠀⠀⠌⠙⢛⣻⣿⣿⣿⣧⣀⠛⢧⣾⣆⡠⢼⡿⣟⠉⣧⣬⠨⣤⣿⡅⠀⠁⢈⡀⠼⠛⠀⢙⣿⢧⣭⣇⣀⡤⠄⣤⡤⠄⠀⢀⡀⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⣿⣶⣤⣬⣭⣋⣛⣋⣻⡦⢬⣴⣷⣼⣇⠶⣛⡶⠿⢀⣤⣄⡈⢸⣿⣿⢿⡿⠛⡳⣿⣿⣶⠿⣥⡜⢶⣦⡼⢿⣦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⣷⣿⣿⣿⠏⣻⣝⠛⠋⠀⡀⠀⠀⣉⠁⣘⣿⢷⡛⠛⢿⣿⠷⠾⠿⣷⣾⡿⣿⣶⢶⡾⣶⡷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠛⠿⠶⠿⠟⠛⠛⠛⠛⠛⠉⠛⠛⠿⠟⠿⠿⠓⢶⣶⠶⠆⠙⢿⣿⣟⣯⣦⣶⣌⡻⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡏⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⣀⣀⠀⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⣿⡇⠀⠀⣿⡟⠛⢿⣦⢸⣿⠀⠀⣿⡇⠀⠀⣸⣿⣆⠀⠀⣿⣏⣻⣷⠀⢰⣿⠟⠛⠃⠛⢻⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣿⣆⠀⣿⡇⠀⠀⣿⣧⣀⣼⡿⠸⣿⡿⠿⣿⡇⠀⢰⣿⠛⣿⡄⠀⣿⡟⠛⣿⣆⢸⣿⡿⠿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣿⠿⠿⠿⣿⡄⣿⣧⣤⠀⢸⣿⠛⠉⠀⠀⣿⡇⠀⢸⣿⢠⣿⠿⠿⠿⣿⡀⣿⣷⣴⣾⠏⠸⣿⣤⣤⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⢸⣴⣦⢴⠰⣆⣴⡤⡴⠠⡆⠀⠀⡴⢦⡄⠀⡴⢦⢦⢸⠶⣤⣶⣤⡆⢰⡆⠀⠀⡴⡆⢰⢦⣠⠶⡇⠀⠠⡆⢤⡠⣴⢦⢦⣸⠶⣤⠶⣬⠴⡆
⠀⠀⠀⠉⠃⠈⠀⠁⠙⠁⠁⠈⠉⠂⠀⠁⠈⠙⠁⠁⠈⠈⠉⠉⠈⠙⠀⠁⠘⠁⠀⠀⠑⠋⠈⠈⠈⠉⠁⠀⠀⠋⠘⠁⠉⠈⠈⠉⠉⠈⠉⠉⠐⠁
⠀⠀⠟⠀⠛⠛⠘⠛⠃⠘⠛⠃⠛⠛⠀⠛⠛⠘⠛⠃⠚⠛⠂⠛⠛⠘⠛⠃⠘⠛⠃⠛⠛⠘⠛⠓⠘⠛⠃⠛⠛⠂⠛⠛⠐⠛⠃⠘⠛⠃⠛⠛⠀⠻
⠀⠀⡇⠀⠀⣰⡄⠀⣀⣀⠀⠀⢰⣶⡄⢠⣆⡀⠀⠀⢀⡴⠶⡄⢀⣀⡀⠀⠀⣶⢦⡄⣀⣰⠀⠀⠀⣴⡶⠀⣠⣀⠀⠀⠀⠀⣶⠆⢠⡆⠀⠀⠀⢸
⠀⠀⡇⠀⢠⠿⢿⠼⣅⡿⠀⠀⢸⣏⡿⢸⣇⡽⠀⠀⠘⢧⣤⠄⠺⣤⠄⠀⠀⢿⡼⠛⣇⣽⠀⠀⠀⢾⣭⠘⢷⡻⠀⠀⠀⠸⡏⠁⢸⠇⠀⠀⠀⢸
⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠇⠀⢀⡴⠶⠄⠀⠀⠀⠀⠠⡄⢠⡀⣄⡀⠀⠀⠴⣦⠄⠀⠀⠀⢰⡆⠀⠀⠀⢰⣠⠆⣤⢀⠀⠀⣤⠀⢠⠀⠀⠀⣤⢀⣤⠀⣀⣀⣀⠀⠀⠸
⠀⠀⡇⠀⠸⣦⣲⡶⣞⢙⡆⠀⠀⡟⢻⡇⡇⢹⠀⠀⣀⣧⣸⠇⠀⢠⣤⠇⢰⠀⠀⢸⢿⡄⠿⢯⠀⠀⢿⡀⢸⠀⠀⢠⡿⣾⢻⡆⡏⢹⠈⡇⠀⢸
⠀⠀⡆⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡃⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⢘
⠀⠀⠇⠀⢸⡿⣾⡇⡟⢳⠀⠀⠰⣏⢀⣿⢰⡏⣳⠀⠀⢻⣬⠇⣶⢲⡄⠀⠀⣿⣠⣽⣦⡞⣷⠀⠀⢸⣷⡟⢸⠃⠀⠀⠀⢀⠻⡄⢾⡂⠀⠀⠀⠸
⠀⠀⡇⠀⠀⠁⠈⠃⠁⠈⠀⠀⠀⠉⠉⠁⠀⠉⠁⠀⠀⠈⠀⠀⠟⠋⠀⠀⠀⠈⠉⠙⠃⠉⠹⠀⠀⠈⠁⠁⠈⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⢸
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡃⠀⠀⢻⡇⣴⠀⠀⢸⡀⣷⡄⢠⡀⠘⣇⣼⢠⢀⡄⠈⢧⣼⣴⢇⡀⡀⡄⠀⠈⢳⡞⠀⢤⡠⠀⠐⢧⡾⢀⠀⡀⠀⠀⢚⡿⠀⣤⡀⠀⠀⢘
⠀⠀⠇⠀⠀⠸⠇⠸⠀⠀⠘⠷⠟⠳⠾⠃⠀⠘⠏⠈⠟⠀⠀⠘⠋⠿⠀⠿⠻⠃⠀⠀⠏⠹⠆⠞⠓⠀⠀⠸⠃⢈⡟⠁⠀⠀⠿⠶⠀⠾⠄⠀⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⣦⠀⠾⡆⢠⡆⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠰⠃⢸⡇⠀⢀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠚⠟⠀⠈⠉⠁⠀⠈⠛⠀⠠⠛⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⢘⡁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠀⠈⣷⠀⠀⠐⢋⡷⠀⠀⠩⣿⠀⢀⣾⣧⠀⠀⠛⢧⠀⢰⡯⣏⠀⠈⢉⡏⠀⢰⣯⡇⠀⠸⣏⣽⠀⠀⢸⠉⣧⠀⠀⠀⠀⠀⢸
⠀⠠⡆⠀⠀⠀⠀⠀⠀⠙⠀⠀⠐⠛⠒⠀⠈⠓⠋⠀⠀⠀⠛⠀⠈⠓⠋⠀⠈⠛⠃⠀⠀⠘⠁⠀⠘⠓⠃⠀⠀⠛⠋⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⢰
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠉⠁⠉⠉⠈⠉⠁⠉⠉⠁⠉⠉⠈⠉⠉⠈⠉⠁⠉⠉⠁⠉⠉⠈⠉⠉⠉⠉⠁⠉⠉⠈⠉⠉⠈⠉⠁⠉⠉⠁⠉⠉⠈⠉⠉⠈⠉⠁⠉⠉⠈⠉

⠀⢀⡄⣶⠀⠀⢰⡆⠰⠀⠀⢠⠆⢠⠀⠀⢴⠰⠆⠀⠀⣀⡆⠀⢀⡄⠀⠦⠀⠀⠦⠀⡆⠠⠆⠀⠀⣴⠒⠀⠀⠀⠠⠀⠐⣦⠀⠀⡆⠀⣶⠀⠀⣶⠀⠀⠀⣶
⠀⣴⠃⣿⠀⠀⢸⡇⢠⡀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣠⠉⡇⠀⠸⡇⠈⢳⠀⠀⢀⠼⡿⢄⠀⠀⠀⠈⠑⢤⡀⠀⠀⠴⠦⣏⠀⠀⡷⠀⣿⠀⠀⣿⠀⠀⡤⠛⢢
⠀⢹⠀⣿⠀⠀⢸⡇⢸⡇⠀⢸⡇⢽⠀⠀⣿⠀⠀⢰⡇⠀⣇⠀⠀⢃⠠⠖⠀⢠⡟⢀⣇⠈⣧⠀⠠⠆⠀⠠⠟⠀⡤⠀⢀⡼⠃⠀⠧⠀⠿⠀⠀⠿⠀⠰⠇⠀⠸
⠀⣀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⢀⡀⠀⠀⣈⠀⣀⠈⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⣀⢀⡀⠀⠀⠀⣀⠀⠀⢀⡀⣀⢀⡀⠀⢀⡀⠀⢀⡀⠀⠀⣀
⠀⣿⢀⡿⠀⠀⠀⠀⢼⠀⠀⠀⠀⣸⡇⢸⡇⠀⠀⣿⠀⣾⠀⠀⣼⠀⢷⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠿⠀⣰⠃⠘⠃⠀⠈⠁⣿⠀⠁⠀⠈⣧⠀⠈⠀⢀⡄⣿⢠
⠀⣿⠀⢹⠄⠀⣰⠃⢸⠀⠀⢠⡞⢸⡇⢸⡇⠀⠀⣿⠀⢽⠀⠀⠻⠀⣸⠁⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠘⠇⢰⡆⠀⠀⠀⣿⠀⠀⠀⠀⠉⡃⠀⠀⠿⡀⣿⢀
⠀⠉⠀⠈⠁⠀⠉⠀⠈⠁⠀⠈⠁⠈⠁⠈⠁⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⣀⠀⢀⡀⠀⢀⡀⢀⡄⢀⡀⠀⢠⡀⢀⡀⠀⢀⡀⢀⡀⠀⠀⣀⠀⣀⠀⣀⠀⠀⣀⠀⣀⠀⣀⠀⠀⣀⠀⣀⠀⠀⢀⡄⢀⡄⢀⡀⠀⢠⡄⠀⠀⠀⠀⣠
⠀⠹⣦⠀⠁⠀⣾⠁⢸⡇⠀⣷⠀⢸⡇⢸⡇⠀⢸⡇⢸⡇⠀⠀⣿⠀⣿⠀⣿⠀⠀⣿⠀⣿⠀⣿⠀⠀⠀⣼⠃⠀⠀⠀⢀⡟⠀⢸⡇⠀⢸⡇⠀⠀⠸⠇⣿⠀⠃
⠀⡤⠹⣆⠀⠀⠹⡄⠸⣇⢠⠇⠀⢸⡇⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿⠀⣿⠀⣿⠀⠀⣿⠀⣿⠀⣿⠀⠀⢰⡏⢿⠀⠀⠀⣼⠙⣇⢸⡇⠀⢸⡇⢸⡇⠀⠀⣿⠀⣿
⠀⠀⠀⠉⠀⠀⠀⠉⠀⠈⠁⠀⠀⠈⢥⡈⠷⠀⠀⠀⠈⠁⠀⠀⠉⠀⠉⠀⠉⠀⠀⠉⠀⣿⡀⠉⠀⠀⠉⠀⠈⠁⠀⠀⠁⠀⠉⠈⠁⠀⠈⠁⠈⠁⠀⠀⠉⠀⠉
⢀⡀⠀⠀⢀⡀⠀⢀⡀⠀⠀⢀⡀⠀⠀⣁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⡀⠀⢀⡀⠉⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠐⣦⢒
⢸⡇⠘⢃⣸⡇⠀⠈⡇⣸⡁⠬⣅⠀⠀⣿⢀⡎⠠⣆⠀⠀⠈⣧⣰⠃⠀⠀⠀⣯⠀⠈⣣⣰⠋⠀⠀⠀⠀⣀⣙⣆⠀⠀⠀⣿⠀⢀⢈⡳⣄⠀⠀⠚⣁⣠⡗
⢸⡇⠀⣿⢸⡇⠀⠀⡇⠘⡇⢀⡉⠀⠀⣿⠀⢷⠀⡿⠀⢀⡞⢸⡇⢳⡄⠀⠀⡯⢀⡾⠁⡇⠹⡄⠀⠀⡞⠀⡇⠘⡇⠀⠀⣿⢠⡏⢸⡇⢸⡆⠀⢾⣁⢀⡷
⠈⠁⠀⠁⠈⠁⠀⠈⠃⠀⠃⠈⠁⠀⠀⠛⠀⠀⠀⠀⠀⠘⠃⠈⠃⠈⠃⠀⠀⠋⠘⠃⠈⠃⠀⠋⠀⠀⠁⠀⠃⠀⠁⠀⠀⠉⠈⠁⠈⠁⠈⠁⠀⠯⠈⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣿⠀⣤⡄⠀⢠⢠⡀⠀⠀⢤⠀⢀⢴⠀⣠⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿⠀⠀⠀⢠⡏⠈⣧⠀⠀⢸⡇⠀⠀⠀⠿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⠀⠀⠈⠋⠻⠀⠋⠿⠀⠸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⢀⠀⠀⢰⠉⠀⡀⠑⠆⢰⠁⢰⠖⠎⠀⠀⡞⠁⠀⠁⠢⡀⢸⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠡⡀⡘⠀⢀⡀⠐⢇⠸⡀⠈⠒⠐⢒⠀⠇⠀⡌⢳⠀⢸⢸⠀⠀⠀⠼⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠞⠉⠃⠐⠁⣇⠀⠈⠁⢠⠎⠀⠙⠲⠦⠴⠋⢸⡀⠀⠉⠁⡠⠊⠀⡀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠤⢤⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠁⣀⡤⠄⠀⠀⢀⠇⠀⠈⠀⠀⠀⠀⢠⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⠆⠀⡇⠀⠛⠀⠆⠈⡄⡸⠀⠀⣀⠀⠀⠐⠀⠀⢸⠀⠈⣄⠀⢀⡼⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡼⠀⠀⠀⠙⢤⣀⣀⡤⠴⠃⠳⠤⠘⢀⡀⢀⠄⠀⠀⢯⠀⣀⡸⠀⣇⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⡔⠀⠆⠀⠀⣠⠄⢠⡀⠀⠀⠀⠀⠁⡠⠤⢤⠈⢉⡥⠀⢄⣉⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠨⠀⠀⠀⠅⠀⠸⠀⠠⠀⠀⠀⠊⠀⡀⢸⠀⡸⠀⠀⠈⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠑⡄⠸⠀⠀⠇⠀⠰⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠐⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠞⠒⠒⠁⠷⠤⣀⡀⠄⠰⠄⠠⢄⡠⢄⢀⡜⠰⠦⠴⢳⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⡠⠂⠒⠀⠀⠀⠀⠠⠀⠀⠒⠀⠀⠀⠀⠀⠒⠠⡀⠉⡩⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠄⠀⡅⠀⠀⢰⠲⡄⠀⠀⠀⠀⠀⠆⠀⠀⠠⡁⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⢠⠠⠊⠀⢳⡀⠈⠒⠃⢠⠂⡀⠀⠀⠀⠀⠀⠀⠘⠑⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠶⠃⠀⠦⠴⠃⠀⠀⠀⠀⠙⠒⢤⣀⡼⠀⠛⠒⠉⠦⢤⠖⠀⠳⠤⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠄⢠⠄⠂⣔⠁⠱⠀⡤⠀⢄⡤⠒⢲⢀⠒⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⢠⠤⠂⢃⠀⠈⢸⠀⠀⠈⢆⠀⠘⠃⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠘⣆⠀⠈⠁⠠⠀⠈⠄⠀⠀⠰⠀⠀⣇⠀⠀⡀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠈⠀⠀⠀⠈⠁⠒⠋⠀⠀⠀⠈⠒⠒⠀⠀⣀⣈⡀⠈⢑⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠙⠉⢠⠏⢠⠂⠈⢖⠉⢩⠃⣎⠁⠀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠀⢇⠀⠀⠳⡠⠀⠀⠇⠀⠈⡹⠃⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣄⠀⠈⣆⠀⠀⡇⠀⠀⡆⠀⡜⠁⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠊⠈⠓⠒⠋⠀⠀⠱⠴⠞⠁⠀⠙⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣟⣿⢻⣿⣿⣿⡏⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢀⠠⡉⢀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢿⠟⠛⡄⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠈⠀⠈⠙⣿⡏⠀⠄⠀⠀⠉⢿
⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠲⠀⠀⣀⣀⡠⠀⠀⣿⠉⠉⡄⠀⠰⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠠⠇⠀⠸⠄
⠀⠀⠀⠀⢀⠉⠉⠀⠀⠉⠉⡀⢐⠀⠀⢠⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠿⢦⣴⡇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠠⠀⠸
⠀⠀⠀⡠⠚⣁⣀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄
⠀⠀⡀⠀⣈⠀⠈⠊⡵⠒⠉⠉⠢⡀⠀⠀⠀⠀⠀⡠⠀⠀⠄⠤⠤⡀⠀⠀⠀⠀⠠⠤⠆⢄⠀⠀⠀⠀⠀⠀⢀⠀⠆
⠀⠀⢉⡁⠀⠀⢀⡚⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠱⠀⠀⠀⠀⠀⠀⢀⢠⢀⠀⠀⠀⠀⢸⢀⣰⠒⠀⠀⠀⢸⣀⣠⠀⠀⠀⠀⢠⣄
⠀⠀⠈⠍⠛⠢⠀⡇⢀⡀⡀⣀⠀⡇⠀⡀⢀⣠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠈⡀⠉⠀⠀⠃⠀⠈⠀⠈
⠀⠀⠀⠀⠈⠀⢰⠇⠈⠉⠉⣀⠀⡇⠠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣇⡄⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⡄
⠀⠀⠀⠄⠀⠀⠘⡂⠀⠀⣀⡠⠀⡛⠀⠀⠐⡁⠀⠀⠸⠀⠂⠂⠄⠂⠀⠀⢌⠀⠀⢀⠆⠀⠈
⠀⠀⠀⠀⠀⠀⢰⠘⠉⠁⠀⠀⠀⡇⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⢀⠴⢂⣈⣐⠲⡆⠀⠀⠀⠀⢀⡀⠐⠀⠄⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⢸⡠⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢃⠆⠀⠀⠀⠙⠘⡆⠀⠀⠀⠀⠙⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠦⠠⠀⠀⠒⠆⠀⠀⠀⠢⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⣧⠠⡄⠀⠀⠀⠀⠀⡄⢠⠃⠀⠀⠀⠀⠈⠀⠀⠀⠘⠦⣀⢀⠀⠔⠓⠀⠀⠀⠀⠋⠉⠉⠉
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠁⠀⠀⠀⠀⠀⣀⣀⢀⡀⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡏⠀⡇⠀⠀⠀⡔⠂⠀⠀⠴⠪⠒⠁⠀⠠⠀⠁
⠀⠀⠀⡇⢸⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⠀⣀⡀⠀⠛⣧⡄
⠀⠀⠀⠧⠼⠀⠀⠀⠈⠀⠀⢹⠀⠀⠀⠳⠤⠇⠀⠀⠀⡧⠔⠋⠁⠀⠘⣶⣤⠀⠠
⠀⠀⠀⠀⠀⠾⠐⠀⠀⠀⠀⠺⠀⠂⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠐⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⢦⣼⢳⡀⠀⠀⠀⢀⡴⠋⠁⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡙⣸⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡴⠲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡇⣈⠵⠃
⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠈⢧⣡⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠉⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣯⠤⠤⠤⠤⠤⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⣿
⠀⠀⠀⠀⠀⠀⢸⠛⢦⣄⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⢀⡀⢀⡀⢒⡒⣒⠂⠀⣀⡴⠚⠁⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⢦⡀⠀⠀⣤⣀⡄⠀⠀⠈⠯⠟⠀⠘⢭⢟⡤⠞⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠶⣄⠈⠳⠁⠀⠢⠄⠀⠀⢄⡠⠖⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⣠⠴⠶⢤⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡟⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣙⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣟⣛⣟⣛⣻⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⡀
⠀⢠⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣾⣿⣿⣿⠿⠛⠛⠛⠙⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠁

⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⡆
⠀⠀⢘⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠟⠛⠛⢿⣿⣿⣇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣴⠞⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⡍⠛⠖⠀⠁⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣤⡤⠀⠀⠁⠛⠲⢤⣠⣄⡀
⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣴⠋⠀⠇⠀⡌⠈⠀⠈⠲⣄
⢀⠔⢖⠴⠷⠛⠼⠶⠶⠶⠴⣖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣻⣗⢹⠀⢴⠁⢀⠀⠀⠀⠼⣄
⢸⠀⠈⠛⢏⠉⠀⢈⠈⢉⢉⠉⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠁⠙⢻⣿⠊⠀⢠⡠⠀⠀⠀⠀⠀⠹⣦⡀
⢸⠀⠀⠀⠈⣧⣀⣈⣁⣀⣉⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⠁⠀⠀⠋⢀⢀⠀⣤⣤⣀⠂⠀⢹⣄
⢸⠀⠀⠀⠀⠁⠀⢀⣀⣀⣀⣀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣧⢀⠸⠏⠁⠠⠏⠀⣘⡎⠀⠟⣿⣂⠤⣂⣸⣿⡆
⢸⠀⠀⠀⠀⠠⠀⣠⡏⠥⠬⠙⠃⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡼⣆⢤⡖⠀⢀⢘⠀⠀⠁⠈⡿⠃⠈⠹⢝⣿⠇⢄
⢸⠀⠀⠀⠀⢰⠀⠠⠥⠤⠦⠴⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣧⠉⡆⣀⠸⡆⠈⠉⠙⢷⠖⠲⢤⣀⣻⡿⠿⢴⠐⠢⣤⡀
⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⣠⡔⠊⠀⢀⣉⡿⠆⣥⡀⢀⣧⠰⠀⢠⠤⣀⠀⣤⣾⠟⠀⠀⠀⠀⢀⣼⣅
⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡠⠼⢅⣀⡠⠠⢴⡉⠀⠙⢆⣨⣭⣿⢇⢠⠏⡉⠈⠉⠁⠀⠄⠀⢸⡧⠟⠁⠀⠀⢀⣤⠶⠏⠁⠈⢣
⠀⠓⢲⣶⣶⣾⣦⣶⣶⡖⠒⢲⣍⠀⠐⠠⠌⠒⠢⣄⢣⢀⣤⣾⠟⠉⠐⠀⣸⣿⣦⣄⣈⠀⠂⠈⣠⡿⠉⠛⠷⠶⠾⠋⠁⠀⠀⠀⠀⠀⣧
⠀⠀⠈⠉⠀⡇⠉⠙⣿⡇⠀⠈⠉⠉⠒⢤⣀⡴⠚⠉⠈⠙⠋⠀⠀⠐⠀⠀⣿⠃⠈⠹⣿⣶⣴⠊⠻⣷⣄⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⡇⠀⢰⠈⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢸⡿⠀⠀⠀⢈⠿⣿⡄⠀⠙⡟⠷⣶⡀⠈⢆⠀⠀⠀⠀⢀⡠⠋
⠀⠀⢸⡀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡿⠁⠀⠀⠀⠈⡂⠛⢿⣄⣀⣹⣶⣄⣀⠀⠈⠡⠀⢀⡤⠎
⠀⠀⠐⠂⠀⠇⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠁⠀⠀⠀⠀⣰⠳⡀⠀⢛⠛⠿⣿⣿⡿⣏⣠⠖⠋⠁
⠀⠀⠀⡄⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣾⠏⠀⢱⠀⠈⡀⠀⢿⣿⡇
⠀⠀⠀⡇⠀⡀⠀⠀⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠆⠀⠀⣰⢻⡇⠀⠀⠀⡇⠀⢹⠀⠀⠟⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⡀
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⣰⡶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⢶⣆
⠀⠀⠀⢻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡟
⠀⠀⠀⠀⢀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀
⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⢠⡶⢦⡄⠀⣤⢶⡄⠀⣴⠶⠄⠠⢤⡤⠄⠀⠀⣤⠶⣄⠀⣤⠤⠤⢠⡤⠤⠄⣤⠀⣠⡶⡤⢀⡤⠤⠄⠀⠀⠀⠈⣿⢹⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⣇⣸⡇⢸⡇⠀⣿⠀⢷⣄⠀⠀⢸⡇⠀⠀⢠⡏⠀⢻⡀⣿⣀⣀⢸⣇⣀⡀⣿⠀⣿⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡏⠉⠀⢸⡇⠀⣿⠀⠀⠙⣷⠀⢸⡇⠀⠀⠸⡇⠀⣼⠁⣿⠉⠁⢸⡏⠉⠀⣿⠀⣿⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠸⠃⠀⠀⠈⠻⠶⠋⠐⠶⠶⠋⠀⠸⠇⠀⠀⠀⠻⠶⠛⠀⠻⠀⠀⠘⠇⠀⠀⠻⠀⠙⠷⠶⠘⠿⠶⠆⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⢰⡶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⢶⡆⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠂⠀⣿⠀⢰⡆⠀⠀⢰⣆⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢶⡀⠀⠀⢰⡆⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⠈⣿⠀⠀⠀⣿⡀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⣇⠀⠀⠘⣧⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⠀⢹⡆⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⠀⢿⡄⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⠀⠘⣧⠀⠀⠈⣿⠀⢸⡇⠀⢸⡇⠀⢠⣤⡤⠀⢸⡇⠀⢤⣤⡄⠀⢸⡇⠀⢸⡇⠀⠀⢹⡇⠀⠀⠸⡇⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⣠⣼⣧⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣼⣧⣄
⠀⠀⠀⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢶⣄⡀⢀⣤⠾⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡴⠟⠉⠛⠋⠉⠳⣤⡀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⢿⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣼⡷⠶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⡤⠠⡄⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡷⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⣶⠀⢰⡶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⢠⡤⠤⠤⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⠸⣧⠤⠤⠤⢤⣾⠇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⠀⣿⠠⠤⠤⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⣇⣀⣀⣀⣀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⠈⠉⠉⠉⠉⠉⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢀⣿⣒⣒⣒⣺⣇
⠀⡀⠀⣀⣀⡀⣸⣟⡛⡛⠛⡛⢛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣛⣀⢛⢂⠛⡛⣻⡏⣉⣉⣉⣉⣹⡇⣀
⠀⠉⠀⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠚⠉⠁⢀⣨⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠓⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠴⠚⠁⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⡟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⣼
⠀⠀⠀⠀⠀⠀⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⣤⣤⣤⣀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⢯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣼⣿⠛⣩⣤⣤⣤⡙⢿⣷⡀⠀⠀⣠⣾⡿⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⠇
⠀⠀⠀⠀⠀⢰⣿⡇⢸⣿⠉⠀⠉⡇⢸⣿⣧⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠈⣿⣷⡈⢿⣦⣤⣤⣴⣿⣿⠿⠛⠁⢠⣤⡄⣤⢠⣤⣤⣤⣤⣤⠄⣠⣴⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⠄⠉⣍⣉⡁⠀⠀⠀⠀⢸⣿⡇⣿⢸⣿⣿⡿⠋⠁⠸⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣈⣿⢸⣿⣿⡀⠀⠀⠀⠀⣿⡇⡿⠘⠛⣁⠀⠀⠀⠀⠈⠻⠿⢿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡿⠋⢾⣿⣿⠀⠀⠀⠀⠀⣿⠇⣁⢰⡿⠿⢷⣄⠀⠀⠀⠀⠀⢀⣤⢿⣿⡄
⠀⠀⠀⠀⠀⠀⣴⣿⠿⠋⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⣵⡆⢿⠀⣇⣸⢇⣿⣷⡤⠀⣠⠶⢋⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⣠⣴⣿⡇⠀⠀⠀⠀⣿⡇⣿⠀⢟⣥⣾⠿⢋⡴⢞⣥⢶⡍⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡿⠀⢿⣿⣿⡇⠀⠀⠀⠀⣿⡇⣿⠰⡿⠋⠁⠀⠀⠙⠻⣷⣬⣥⣾⡿
⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⣿⡇⢉⠀⣤⣤⣀⠀⠀⠀⠀⠀⠈⣩⣿⡿
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⣠⣴⣿⡇⠀⠀⠀⠀⢙⡇⣿⠀⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⣾⠿⢿⣿⠃⠀⣀⣴⣾⣿⡇⣿⠀⠋⢧⡝⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠿⠆⡼⠋⢤⣾⣭⣙⠻⢿⡇⣿⠀⣷⡾⢃⣾⠀⠀⠀⠀⢸⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠉⠀⠀⠀⠀⠉⠛⠿⣶⡄⢿⠀⣋⣴⣿⠇⠀⠀⠀⠀⠸⡿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⢃⣤⠤⢤⣄⡀⠀⠀⠀⠀⠀⢀⣠⠶⢟⡻⢿⣶⣤⡀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡘⠇⠀⣠⣿⣿⣷⣦⡀⣠⣾⣩⣴⣿⣿⠉⢃⣿⣿⣿⣄⢀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁⠙⢿⣿⣿⠿⠿⠛⠹⢿⣿⣿⠿⠋⠻⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⢠⣷⣰⡿⠆⣿⠈⡟⢠⣷⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁⠁⠀⠉⠀⠁⠈⠀⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡔⢩⡌⠱⡈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣱⣄⣋⡔⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢰⣖⣻⣳⣷⢿⣖⣖⣲⡆⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⢸⡿⢽⢼⣿⣾⡯⡿⣿⡇⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠻⠯⠯⠿⠏⠿⠭⠯⠿⠯⠯⠿⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⣉⠒⢠⠒⠳⡄⠶⠒⢆⡀⠀⠀⠲⣄⠤⠤⣀⡴⠚⢞⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣆⡓⢡⠌⠳⡚⠁⡎⢱⠄⢺⡀⠀⢠⠇⣤⣴⠪⠏⢪⣠⢗⢝⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠽⡥⢂⡼⠼⣇⠪⡁⢠⡞⠀⠀⠊⠙⣅⠜⢦⡤⡞⢔⡵⢋⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣠⠎⡎⠠⣖⠈⡆⢣⡁⠀⠀⠀⣠⠖⣉⠗⠋⠀⣈⡡⠔⡁⡅⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡢⠧⢈⣉⡠⠧⡞⠉⢲⠀⠀⡠⠚⡡⠢⠔⢊⢈⠀⠀⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢢⣠⠖⠒⠧⣤⠞⠑⠆⣠⣪⣼⣸⣸⣸⣨⣤⣤⣇⣇⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠑⠲⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠚⠙⡆
⠀⢀⡤⢄⣾⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣤⠴⠊⠁
⠀⣏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠖⠒⠒⠒⠛⠳⢤⡀
⠀⠘⠦⠤⠤⣴⠇⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠖⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⢀⣠⡤⠤⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⡷
⠀⠀⠀⠀⠀⣀⣉⣳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠃
⠻⣏⠉⠉⠉⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢸
⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢦⣀⣀⡀
⠀⠀⠀⢳⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡠⠚⠉⢧
⠀⠀⠀⢸⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⠊⠀⠀⠀⢸⡀
⠀⢀⣠⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠸⡇
⢰⠋⠁⠈⠁⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠸⣆⠁⠀⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠶⠛⠓⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠶⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣰⡥⠤⠖⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣰⠿⣧⡀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡟⠁⢀⡼⠋⠀⠈⢷⣄⠀⠹⣿⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠛⠳⣦⡀
⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣠⡟⠁⠀⠀⠀⠀⠹⣦⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠙⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠁⢀⡾⠃⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠈⢻⣄⠀⠹⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠃
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠁⠀⢸⡟⠳⣤⡀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣿⠀⠀⠙⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⢸⡇⠀⠈⠛⣦⣄⠀⢀⣴⡞⠋⠀⠀⣿⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣠⡾⠋⠙⠻⠛⠉⠻⣦⡀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⡞⢉⡀⣀⣀⣀⣀⣀⡀⣈⣛⣦⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣹⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⣧
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⣶⡶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢸⡇⠀⠀⠉⠻⢦⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣀⣀⣈⡀
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⡇⠀⠀⢠⡶⠛⠉⠉⠉⠉⠛⢶⣄
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡏⠀⠀⠀⠀⠀⠀⣼⠃⠹⣦⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣴⠾⠷⠶⠶⠾⢷⡄⠀⠀⠀⠀⣿⠀⠀⢸⡇⠰⣾⡷⠶⠶⠶⠶⠶⠶⠶⠶⢶⣷⠶
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⠀⢀⣤⣤⣤⣤⣤⣤⣀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣠⡿⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⠀⢸⣇⣀⣀⣀⣀⣀⣿⠀⣿
⠀⠀⢸⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠁⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⣿
⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣇⣀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢾⣀

⠀⠀⠀⠀⠀⠀⠀⠀⢐⡋⠀⢸⠈⠳⠀⡆⠘⠋⢀⠺⠀⢰⣋⡶⢰⠀⠀⣏⠁⠀⠰⣎⠀⠈⡛⠀⣊⣑⠄⠞⠈⢢⢰⣠⠂⢸⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣅⣀⣸⣀⡔⢁⡆⢀⡀⣜⠃⣳⣘⣅⡷⣸⣀⣀⣏⣀⡀⢀⣈⡗⠀⣃⢀⣋⢙⡄⣇⣀⡜⢸⡉⢆⣸⣅⣀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣶⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⠀⣀⡀⠤⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⢀⡀
⠀⠀⠀⠀⠀⠘⠀⠰⡀⠰⠀⢀⡇⠀⠄⠃⢄⠘⠋⠁⠒⠂⠄
⠀⠀⠀⠀⠀⢰⡀⠀⠑⠲⠒⢋⠀⠀⠸⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠐⣦⠀⠠⠉⡐
⠀⠀⠀⠀⠀⢸⠑⠀⠀⠀⠳⣌⠀⡠⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠶⠞⠇⠀⠀⠀⠳⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⡀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠂⠄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠰⣔⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠰⠀⠰⡄⠈⠀⠀⠀⠁⡇⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣀⠀⠀⠀⠀⠗⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡆⠀⠈⢀⣄⠜⠁⢀⠰⠻
⠀⠀⠀⠀⠀⠀⠈⡀⠀⡜⠁⠀⠀⡠⠂⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠂⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢧⠎⢀⣀⠤⠈⠀⡄⡀⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠂⠸⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⡈⢄⣀⠀⠴⠀⠀⡈⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠑⡄⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠄⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⢀⣿⠀⢑⡘⣠⡅⢤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠤⠀⠀⠀⠤⠘⡛⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠙⠲⢄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⠀⠀⠀⠀⠀⠙⠲⣄⡴⠋⢿⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⡠⠂⠁⠀⠀⠀⠀⠀⢂⡴⠋⠀⠀⠀⠙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡸⠂
⠀⠀⠀⠀⣠⠊⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠞⠣⡀
⠀⠀⠀⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡴⠋⠀⠀⣸⡆
⠀⠀⠀⡏⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⠋⠀⣀⠴⠊⠁⣇
⠀⠀⠀⡇⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⠞⣡⠔⠋⠁⠀⠀⠀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⡀⠉⠳⢤⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣴⠗⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣡⠖⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠲⢄⡀⠀⢀⡠⢤⣀⠀⣀⡤⠚⠁⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠽⠛⠉⡀⠄⡈⠙⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⠀⠐⠀⠀⠀⠀⠐⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀⠀⠀⠀⠀⡏
⠀⠀⠀⢸⡄⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⡇
⠀⠀⠀⠈⠷⣮⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣽⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶
⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⢀⣀⡀⠙⠁⣀⣀⠀⠀⠀⠀⠀⣾⣿⡶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣠⡀⠉⠉⠀⠀⠀⠀⣼⡏⠹⣷⡀⠉⠙⠛⠿⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⢰⡿⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣦⣤⣀⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⣦⣤⣀
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣤⣄⣀
⠛⠛⠀⠀⠘⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣶⣤⣀⡀
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠉⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠙⠷⣦⣤⣀⣀⣀⣀⣤⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣷⣶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⢸⣿⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣹⣿⣿⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⢸⡇⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣿⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⣼⠏
⠀⢸⡇⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⣿⣄⡀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢰⡟
⠀⢸⡇⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⢿⣿⣷⣾⣿⣿⠂⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⣿⠉⠛⠻⠶⣦⣤⣀⡈⢻⣆⢀⣿⠁⠀⠀⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠛⢿⡿⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⠃⠀⠀⠘⠛
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠻⣦⣤⣀⣀⣀⣀⣠⣴⡾⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢸⡇⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⢸⡇⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠻⠗
⠀⢸⣧⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣠⣶⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧⣤⣤⣤⣤⣤⣶⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠉⠀⠀⠀⠙⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠉⠉⠉⠛⠛⠛⠛⠿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣿⣿⣿⣿⣧⣤⣀⣠⣤⣤⣤⣤⣶⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⢰⣿⣿⠛⣿⣷⣄⡀⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠙⠻⠿⠿⠿⠿⠛⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣷⣶⣤⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠁⡟⠡⠏⠀⠀⡀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠬⡾⠛⣺⠛⢾⣔⡈⠀⣄⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠤⠀⡆⢀⡀⠈⠀⠐⡉⠩⠖⣥⢤⠾⡍⢈⢠⣈⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠘⠁⠶⠁⣀⠈⠠⡄⠀⡄⡄⢸⡆⠀⡠⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢦⡹⠖⢫⠻⢛⣥⠞⣁⠔⡁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⣉⣶⡀⠄⡹⡄⡐⠨⣯⠛⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠐⠂⠀⣰⣼⣇⠐⢨⠆⡄⠌⡄⣺⡛⠩
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠠⠔⢻⣐⣉⠄⠄⢡⡼⢧⠉⠓⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠠⠟⢩⠓⣶⠗⡿⠙⢆⠀⠡⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠰⠀⢸⡀⢸⠀⠈⠁⣀⣋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠐⠀⠀⠠⢀⣤⡚⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠲⡄⠀⢀⢠⠟⡶⠹⢧⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢨⠍⠆⠀⠀⠃⡈⠀⡇⠀⠀⢱⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣷⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⠿⠛⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⢀⣴⣿⡗
⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⢠⣾⢂⣤⣶⣾⣿⣶⣦⡤⠤⠤⢽⣿⣿⣾⣿⡯⠧⠤⠤⠤⠤⠤⠤⢿⣶⣶⡦
⠀⠀⠀⣠⠞⠋⣿⡇⠀⠀⠀⣾⡇⠀⠈⠉⠛⠛⠉⣹⡶⠀⠀⠈⢿⣿⣿⠋⠀⠀⣀⡀⠀⠀⠀⠀⠈⠉
⠀⢀⣾⠃⠀⢠⣿⠁⠀⠀⢠⣿⠀⠀⠲⣶⡄⣤⠀⣿⠁⣴⢶⡆⠀⠛⠁⢰⡗⢠⣿⠁⠒⣶⡆⢠⡤⢀⣶
⠀⠈⠛⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⣾⡇⣿⡇⣿⠀⣿⢸⣟⡾⠁⡄⠀⠀⣿⡇⣸⣿⢸⣿⢸⡇⣾⡇⢸⡿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⢠⣿⡇⣿⠁⢿⡄⡟⢹⣏⠀⣸⠀⠀⠀⣿⣇⣿⣿⢸⡇⢸⡇⣿⠃⣿⡇⡸
⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠛⠚⣹⡓⠋⠀⠈⠛⠃⠈⠛⠛⠁⠀⠀⣀⣉⣫⣼⡟⠈⠛⠛⠀⠙⠟⠈⠛⠁
⠀⠀⢠⣀⣀⣤⠶⠶⠶⠶⠶⠤⢄⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣟⣿⣿⠇⠐⠒⠚⠛⠛⠛⠶⠤⠔
⠀⠀⢠⣤⣭⠀⠀⠀⠀⢠⣤⣤⠀⣠⣄⠀⣠⣴⡄⠀⢠⣴⣤⠀⠀⣠⣤⣬⣭⡁⠀⠀⣶⣦⠀⣀⠀⣴⣶⡄
⠀⠀⢠⣿⡅⠀⠀⠀⠀⢸⣿⣿⣾⠟⠁⠀⢹⣿⣿⡀⢸⣿⣿⠀⣾⣿⠇⠈⣿⣿⡄⢸⣿⣿⢠⣿⡆⣿⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠻⣿⣦⠀⢸⣿⣿⣧⢸⣿⡿⢰⣿⣿⠀⠀⣿⣿⡇⢸⣿⣿⣾⠟⣿⣿⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⢸⣿⣿⢿⣿⣿⡇⠈⣿⣿⠀⠀⣿⣿⡇⢸⣿⣿⡟⠀⢹⣿⣿⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⢸⣿⣿⠈⣿⣿⡇⠀⠹⣿⣧⣠⣿⡟⠀⢸⣿⣿⠁⠀⠀⣿⣿⡇
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠀⠀⠉⠉⠀⠈⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡋⠀⠀⠀⣀⡤⠤⠤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⠑⢦⡀⠀⠀⢀⡤⠚⠁⠀⠙⢦⠔⠋⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠳⠔⠋⠀⠀⠀⠀⣰⠃⢠⠴⠒⠲⢄⡤⠖⠒⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠳⡄⠀⠀⠀⠀⠀⠀⣠⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣴⢋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⠀⠀⠈⠢⣀⠀⠀⣠⠞⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄⠀⠀⠈⠳⠚⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⠤⠴⠖⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠐⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣰⠖⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠛⠒⠛⠛⠛⠛⠛⢱⡆
⠀⠀⠛⠀⠂⠒⠀⠂⢐⡀⠘⠀⡂⠀⢂⠀⠂⠛⠀⠃⠐⠀⢻⠁⠀⠐⠂⠃⠐⠃⠀⢂⡀⠃⠀⡀⢀⢀⠈⠀⠐⠂⠐⠃⠚⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡉⠀⠘⠀⠈⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡀⢀⠀⠈⠀⠓⠀⠃⡀⠈⣁⢒⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣁⣑⣀⣘⣀⣀⣀⣀⣀⣀⣀⠀⢸⠀⠀⠀⠀⢐⣁⣉⣘⣂⣊⣙⣋⣂⣅⣀⣋⣊⣃⣙⣋⣁⠀⠀⡇
⠀⠀⠀⠀⣏⣡⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢸⠀⠀⠀⠀⢸⣿⣵⣧⡆⡨⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇
⠀⠀⡇⠀⣿⢾⠓⠥⠄⠆⠠⠠⠴⠐⠠⠀⢰⠦⠴⠆⠾⠀⢸⠀⠀⠀⠀⢸⡿⡟⠧⠿⠖⢲⠶⠰⡶⠶⡖⠀⡖⠲⡶⠶⠀⠀⡇
⠀⠀⡇⠀⡻⢾⠖⠲⠒⠖⠖⠒⠶⠒⢺⠒⢶⠒⢲⠒⠶⠀⢸⠀⠀⠀⠀⢸⠶⠬⠦⠶⠤⢤⠤⠤⠦⠤⡥⠤⡤⠤⠤⠶⠀⠀⡇
⠀⠀⠁⠀⡇⠺⣿⣾⣷⡆⠁⠀⡽⠀⢸⠀⢸⠀⢸⠀⠸⠀⢸⠀⠀⠀⠀⢸⠀⢼⠀⢵⠀⢸⠀⠀⡃⠨⡇⠀⡇⠀⡇⢰⠀⠀⡇
⠀⠀⠀⠀⣅⣸⣯⣩⣼⡷⣷⠀⣼⣿⢽⢢⢸⠀⢸⠀⢠⠀⢸⠀⠀⠀⠀⢸⠀⠡⠀⠨⢰⢸⠞⢒⡁⠀⡇⠀⡇⠀⡇⠘⠀⠀⡇
⠀⠀⠀⢸⣿⣿⣿⣥⣶⠶⡇⠀⢹⡟⢻⠈⢸⠀⠘⠀⠸⠀⢸⠀⠀⠀⠀⢸⠀⢘⠀⠀⠘⢿⠋⡸⠀⠀⡇⠀⡇⠀⡇⢠⣀⠀⡇
⠀⠀⠀⠀⣀⣽⣟⣟⣁⠀⠇⢞⡋⠰⣹⢠⢰⡮⢠⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⠀⢀⡰⠒⢹⢄⠒⡅⠀⡇⠀⣧⡀⢹⣧⣌⣀⡇
⠀⠀⡤⠈⠁⠀⠀⠀⠁⠱⣕⠀⣓⠀⢹⠠⢸⡗⢱⠀⠸⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⠈⢐⠆⠀⠃⠀⡇⠙⣆⢩⠈⠛⠀⡀
⠀⢠⠗⠀⠀⠀⠀⠀⢘⠀⠀⠸⡿⣾⡼⢤⡬⠰⠎⠀⣨⠀⢸⠀⠀⠀⠀⢸⠀⠰⠀⢘⠀⢸⡗⠀⠄⠀⡃⠀⡇⠀⠜⠻⠒⣶⣾⠃⠀⠐
⠀⢸⠰⡀⠀⠀⠀⢘⣿⡗⣮⡀⢹⠖⠛⣉⡈⡟⢛⡛⠻⠀⢸⠀⠀⠀⠀⢸⠛⠛⠚⡛⠛⠺⡗⠚⠓⣒⣓⡲⠟⠀⠀⠀⠘⣿⠂⡌
⠀⠀⢻⠛⠖⠂⠋⣠⣾⡇⠀⢉⣥⣶⣴⡴⠶⠿⠓⠀⠈⠀⢸⠀⠀⣀⠀⠘⠊⠁⠀⠁⠈⠉⠙⠛⠐⠬⢈⡀⣠⡞⠛⠀⠀⢿⡱⣧
⠀⠄⣼⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠦⠷⠧⠴⣧⠞⠀⠈⠣⠥⠥⠭⠍⠁⠬⠭⠉⠌⠉⠩⠽⠿⣿⠃⡀⠀⠀⢹⠿⡯
⠀⢰⡇⠀⠀⠀⠀⠀⢿⡧⣤⠤⣬⣤⡤⡤⡤⠤⠤⠤⠤⠤⠏⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠼⢦⡵⠶⠾⣶⣷⠂
⠀⢸⠁⠀⠀⠀⢰⠀⢸⠆⠀⢠⢟⣿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢸⣻⣧
⠀⡘⢰⠀⠀⡇⠀⠀⢸⡆⠀⡀⣾⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠅⣿⡇
⠀⠇⠈⠀⠀⡇⠀⠀⢸⠃⢰⣧⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⢹⣧
⠀⡴⠀⠀⠅⠇⠀⠀⣰⣀⣭⣿⣿⢟⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠃⠘⠸⡆
⠀⡇⠀⢠⡀⠀⠀⢰⣿⣿⢃⣿⢯⣿⣣⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣄⣈⠀⠁
⠀⡷⠶⢰⣤⣴⣶⣿⣿⣿⢸⡷⢱⢻⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⢿⣿⣿⣿⣿⠟⠃⡄
⠠⡇⠀⠀⣻⠘⣿⠹⡟⣿⠃⢿⣷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠃⣽⣿⣿⠶⡋
⠰⠇⠀⠀⠅⢀⢿⡞⣦⣿⠉⢻⠋⣽⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡿⣿⣏⢠⠖⠁
⠀⠀⢤⣀⣀⣀⣞⣣⣿⣾⣿⣿⠿⠛⠛⣛⣛⣛⡛⠛⠒⣛⣿⡟⢛⣛⣛⠛⣛⣓⣛⣛⣛⡛⠛⠓⠛⠛⠛⠛⠛⢛⢽⣿⣿⣿⢿⣽⡿⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⡀⢀⣀⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⣿⡏⢻⣿⣿⠛⣿⣿⡟⠙⠻⣷⣾⡟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢹⣷⣿⣇⣿⣿⡟⠀⠈⣿⡇⠀⠀⢹⣿⣯⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣤⣿⣿⡏⣿⡟⣿⡄⢠⣿⡇⠀⠀⣼⣿⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠘⣿⣧⣼⣿⣿⣥⡿⣿⣧⣤⣴⡿⢻⣧⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⣳⡟⠛⠻⣿⡟⠀⠀⢹⡏⢸⣗⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣿⠃⠀⡀⣹⣯⣤⣤⣼⡇⢸⡏⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠙⣿⡄⠈⠛⣿⣿⠉⠉⢹⡇⢸⣏⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠻⣷⣤⣤⣿⡷⠀⠀⢸⡇⢸⣿⣷⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣡⡾⢿⣿⡇⣿⣿⡅⠈⢠⣿⣿⠈⣿⣿⡀⠈⣿⣵⡞⠛⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣾⠋⠀⢸⣿⡇⣿⡏⣿⣀⣿⢹⣿⠀⣿⠈⢷⣄⣿⣿⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⢿⣷⡀⢸⣿⡇⢿⠇⠸⡿⠇⠸⡿⢀⣿⠀⠈⢿⡿⠛⠷⣤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⣿⣾⣿⣇⣀⣀⣀⣀⠀⢠⡾⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣴⠾⢿⣍⡽⠿⣯⣿⠿⢦⡼⣷⣄⣀⠀⠛⢻⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⣿⡇⠀⣿⣿⣤⣾⣁⡀⠉⠛⣿⡆⢸⡇⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⣴⣿⣿⠀⢹⡿⣷⣤⣤⡿⠃⢸⡇⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⢰⣿⣿⠀⠀⣼⣿⡁⢨⣿⡆⠈⣿⣧⡉⢡⣤⠀⢨⡄⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⢻⡇⢠⡿⢹⡇⣸⠟⣿⢰⡟⠈⢷⡟⠙⣧⡞⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⢻⣇⡀⣸⡟⠈⣿⣾⠃⠈⣿⣿⠀⢿⣿⠇⢠⡟⢷⡀⢸⡁⢀⣾⣏⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠀⠀⠉⠁⠀⠈⠉⠀⠉⠀⠈⠁⠈⠀⠈⠉⠉⠉⠉⠉⠉




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣿⣽⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿

⠀⢀⣀⢀⡀⣀⡀⣀⡀⣀⠀⣀⠀⣀⠀⣀⢀⣀⢀⡀⢀⡀⢀⡀⢀⡀⣀⠀⣀⠀⣀⠀⣀⢀⣀⢀⣀⢀⡀⣀⡀
⠀⢘⠁⡠⠀⠀⠀⠸⠀⠨⠀⠅⠀⠀⠀⠄⠀⠁⠀⠄⠀⠀⠀⠀⠈⠀⠨⠀⠀⠀⠨⠀⠇⠀⠇⠄⠁⠀⢄⠈⡃
⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⢨⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡅
⠀⢨⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⢉⠛⠧⠭⠸⠒⣄⢀⣀⡠⢧⠀⠩⠉⠒⡄⠀⠀⠀⠀⠀⢀⢀⡅
⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠁⠀⠀⡅⠥⠁⢀⡼⠛⠉⠈⠁⠈⢇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠠⡄
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠫⠑⠁⠀⠀⠳⡃⣶⠋⠀⠀⠀⠀⠀⠀⢸⠒⡰⠵⠈⡇⠀⠀⠀⠀⠀⠀⠀⡄
⠀⢠⠄⠄⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡀⡄⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⣗⣈⣀⣀⡇⠀⠀⠀⠀⠀⠀⠠⡄
⠀⢰⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⠀⡅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⠍⣏⠍⠂⠀⢸⡶⡤⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠆
⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠐⠤⢤⡿⣟⠙⠽⣇⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠰⠂⠂⠀⠀⠀⠀⠀⢸⡂⡄⣀⠀⠀⢙⠷⠧⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠸⠂⠀⠀⠀⠀⠀⠀⠸⡒⠒⠚⠁⠀⠏⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠘⠂⠂⠀⠀⠀⠀⠀⠀⢣⠄⡀⠀⠀⠐⠂⠙⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠉⡏⠀⠀⠀⠀⠀⠐⠐⠃
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡎⠄⠀⠀⠀⠀⡰⡽⣄⠀⠀⠀⠀⠀⠀⢀⢇⡂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠘⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡗⡒⡈⠷⢤⣀⠀⠀⣀⡜⠘⢋⠀⠀⡇⠀⠀⠀⠀⠀⠐⠀⠃
⠀⢘⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠘⠤⡆⣒⢰⡤⠋⠈⠳⢄⠄⢀⣡⠬⠃⠀⠀⠀⠀⠀⠈⠉⡃
⠀⢘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠓⠒⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡃
⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃
⠀⢨⡀⠑⠀⢀⠀⢰⠀⢰⠀⠂⠀⠀⠀⡂⠀⡀⠀⠀⠀⠀⢐⠀⠀⠀⢐⠀⠀⠀⢰⠀⡂⠀⡆⠀⠀⠀⠈⢀⡅
⠀⠈⠉⠈⠁⠉⠁⠉⠁⠉⠀⠉⠀⠉⠀⠉⠈⠁⠈⠁⠈⠁⠈⠁⠈⠁⠉⠀⠉⠀⠉⠀⠉⠈⠉⠈⠉⠈⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⢺⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⠏⢜⢆⢇⠇⢞⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠺⡫⡑⡜⡸⡘⡌⡌⡜⡌⡌⡷⣄
⠀⠀⠀⠀⢰⠒⠀⠤⠤⠤⠤⠾⢯⣦⣧⣣⣳⣰⣱⣰⡸⢰⡑⡘⡜⡜⡜⣷⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠓⠓⠻⠾⠼⠼⢼⣦⣀
⠀⠀⠀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⡆
⠀⣴⣮⣧⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠙⠀⡇
⠀⡇⠈⠳⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⡇⠀⠀⠀⠙⢯⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡫⠳⡀
⠀⠇⠀⠀⠀⠀⠀⠉⠲⣭⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⡿⢷
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⡪⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠯⠓⠉⠀⡆⡄
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣒⠥⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠉⠀⠀⠀⠀⠀⡃⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣕⣢⣄⣠⣀⠄⣠⣶⠽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇
⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠈⠉⠻⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇
⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡇⠅⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠅⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠄⠀⠀⠀⠀⠀⠀⠀⡈
⢷⡠⠒⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡔⡀⠀⠀⠀⠀⠀⡇
⠀⠉⠙⠒⠯⠬⢔⣐⣢⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢖⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠥⢰⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣥⠀⠀⢰
⠀⠰⠤⢀⢤⡢⡶⢔⠦⡲⠔⡢⣔⠤⡂⢄⡢⡈⢉⣐⡲⢢⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠈⠀⠁⠉⠘⠑⠃⠊⠚⠱⠫⠟⠵⠲⢶⠦⡤⣤⣽⣯⢔⢔⣴⣖⢔⡠⣠⢄⣤⣀⢀

⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⣾⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣧
⠀⠀⠈⢉⣽⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣯⡉⠁
⠀⠀⠀⢹⢹⠀⠀⠀⡖⢲⡄⡴⢲⡀⡖⠂⠐⣶⠂⠀⣰⠲⡄⣴⠒⢠⡖⠂⣦⢠⠖⠂⡶⠒⠀⠀⠀⡏⡟
⠀⠀⠀⣼⢸⠀⠀⠀⡗⠚⠁⡇⢸⡇⠙⢦⠀⣿⠀⠀⣿⠀⡇⣿⠒⢸⡗⠂⣿⢸⠀⠀⡿⠒⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⠀⠀⠃⠀⠀⠓⠚⠀⠒⠚⠀⠛⠀⠀⠙⠚⠃⠛⠀⠘⠃⠀⠛⠘⠓⠂⠓⠒⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢀⣤⣤⣤⣤⣤⣤⣤⣄⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⣦⠀⠀⣆⠀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⢰⡄⠀⢰⡀⠀⣿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⢹⡀⠀⢹⡀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⠈⣇⠀⠘⡇⠀⣿⠀⡇⣿
⠀⠀⠀⣼⢸⠀⣿⠀⠘⡇⠀⠘⡇⢸⡇⠀⡇⠀⣤⡄⢸⡇⢠⣤⠀⢸⠀⢸⡇⠀⢻⠀⠀⢻⠀⣿⠀⡇⣏
⠀⠀⠀⣿⢸⠀⠻⠶⠶⠶⠶⠶⠶⠾⠇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠸⠷⠶⠶⠶⠶⠶⠶⠿⠀⡇⣯
⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⣤⣿⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣼⣧⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣿⣤
⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⣤⠤⢤⣄⣀⣀⣀⣀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢸⣿⣿⡿⠛⠛⣛⣻⡿⠿⠿⠿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿
⠀⠀⠘⠿⣿⡄⠀⠀⠀⠀⠀⠉⠹⡄⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣧⣀⡀⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⢸⠃⢀⣤⣤⣠⣤⡄⠀⡇⠀⢹⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⣿⡿⠛⠻⠟⠛⢿⣿⢸⣿⡆⠀⠀⣿⣿⣿
⠀⠀⠀⠀⡼⠀⠈⢿⣿⣿⡿⠃⢰⠇⠀⢸⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⢣⣿⣷⡀⠀⠀⢀⣼⡏⣿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠋⠀⠀⢸⠀⠀⢸⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⣸⣿⣿⣿⣷⣴⣿⣿⡇⣿⣿⡇⠀⢠⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣸⣄⡀⢸⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⠻⣟⣿⣿⣿⣿⣿⡿⠇⠿⣿⡇⢠⣿⣿⣿⣿
⠀⠀⠀⠀⢳⡀⠀⣿⣿⣿⣿⡿⠿⠟⠃⠌⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡌⢿⣿⠍⠀⠀⠀⣀⣀⣤⣾⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣵⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣶⢒⣪⡟⠙⠟⢹⢺⡿⠖⣒⢶⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⠛⠉⣔⣇⣊⠶⠗⠀⠛⠛⠒⠛⠲⠬⣤⣾⣭⣓⣴⡒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⣭⠍⠓⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣙⠸⣍⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣤⡷⠹⣵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢙⣰⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠃⣤⣼⣨⠑⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⠹⡇
⠀⠀⠀⠀⠀⠀⣼⠁⢰⣟⣏⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠁
⠀⠀⠀⠀⠀⣸⠯⡤⣨⡬⣆⠏⡏
⠀⠀⠀⠀⠀⠁⢾⡎⢹⣷⡧⣸⠃
⠀⠀⠀⠀⢸⡇⣀⣝⠺⠻⠛⣽
⠀⠀⠀⠀⢸⣯⠙⡏⢀⢧⡄⢻
⠀⠀⠀⠀⢸⡶⣾⠀⢹⢤⡁⢻⡀
⠀⠀⠀⠀⠘⣯⡇⠐⣋⡃⠁⡘⡃
⠀⠀⠀⠀⠀⢻⠓⣀⢠⣩⢟⣧⢥
⠀⠀⠀⠀⠀⠈⢷⡸⢂⣤⠰⠛⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠈⢣⠇⢲⢦⠿⢿⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⣀⣧⡌⣏⡉⡓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣁⣴⢼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⣪⣂⡿⠧⠑⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣎⣰⢩⣼⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⣖⣈⣁⠫⣠⠙⣗⢒⣶⠤⠰⢶⠒⠫⠙⣻⣩⠧⢹⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠳⠶⠾⠯⠥⠴⠧⠼⠼⠿⠒⠒⠋⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠒⠒⠒⠚⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⢠⠈⢶⠒⠒
⠀⠀⠀⡟⢍⠉⠉⠉⠉⢉⠝⡇⠀⠀⠀⠀⠀⢸⠣⡀⠀⠀⠀⠀⡠⢻⠀⠀⠀⠀⠀⠀⡇⠋⠙⠘⠉⠋⡏⡙⡃⠀⠀⠀⠀⡾⠍⣀⠀⠀⢀⠦⣀⡈⣇⠁
⠀⠀⠀⡇⠀⡱⢄⣀⠴⡁⠀⡇⠀⠀⠀⠀⠀⢸⠀⢈⠖⠈⠐⢎⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠒⠃⡇⠀⠀⠀⠀⠘⡄⠀⡝⠘⠉⣀⡠⠔⠛⠐
⠀⠀⠀⣧⣊⣀⣀⣀⣀⣈⣢⡇⠀⠀⠀⠀⠀⢸⣔⣁⣀⣀⣀⣀⣑⣼⠀⠀⠀⠀⠀⠀⣥⣀⣄⣄⣠⣠⣀⣄⡇⠀⠀⠀⠀⠀⠸⣴⣤⣖⣋⣉⣀⣀⣀⣀


⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢘⡆⠟⠒⠓⠚⠇⣞⠀⠀⠀⠀⡴⠋⣀⠤⢄⡈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠲⠖⠒⠲⣶⠶⡄⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⢨⠆⠄⠀⠀⠀⡅⢮⠀⠀⠀⢸⠀⡌⠀⠀⠀⠘⡀⢸⠆⠬⢉⣈⣀⡩⠀⠀⠀⣇⣀⣸⣃⣆⣀⣈⣣⡇⠀⠀⠀⠀⢰⠋⠉⢉⠉⠉⠉⡹⢛⡆⢸
⠀⠀⠀⢨⡃⠀⠀⠀⠀⠄⣫⠀⠀⠀⠸⡀⠡⡀⠀⠀⡰⠀⣸⣓⠒⠒⠀⠀⠒⠀⠀⠀⡇⠈⠚⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠉⠈⡇⢸
⠀⠀⠀⢰⣃⣷⣒⣖⣢⣃⣳⠀⠀⠀⠀⠙⢦⣀⣉⣁⣠⠾⠁⠤⠄⣈⣐⣂⡉⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠸⣀⣀⣈⣀⣀⣀⣀⣀⡗⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢈⠍⢹
⠀⠀⠀⡞⠥⢴⡷⠶⠶⠠⠝⡆⠀⠀⠀⠀⠀⠀⠊⢸⠘⠀⠚⢹⠈⠂⠀⠀⠀⠀⠀⠀⡏⣀⡀⢀⣀⠀⢀⡈⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠁⠀⡸
⠀⠀⠀⡇⠀⠘⠂⠤⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⡏⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠋⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀
⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⡇⣀⣀⣀⡀⠈⠀⠀⡇⠀⠀⠀⠀⠀⡰⠋⢉⠉⠉⠉⠉⠉⡉⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⡇⣅⣀⣀⡸⠀⠀⠀⡇⠀⠀⠀⠀⣰⢡⠁⢠⠀⢀⡄⠂⢉⠙⡆⠘⡇⠀⠀⠀⡞⠉⠹⠉⣉⣉⣉⡙⡆⠀⠀⠀⠀⢀⣤⠾⣭⣽⠉⡏⠉⠉⠉⢢
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⢣⣀⠤⠖⠉⠀⡇⠀⠀⠀⡇⠀⢀⠀⠀⠀⠈⠃⡇⠀⠀⠀⠀⢸⠉⣢⣀⣰⡄⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⡀⡀⣀⡀⢀⡇⠀⠀⠸⣄⡴⣿⢦⡤⠤⠤⠤⡤⣤⣮⢷⣄⡽⠀⠀⠀⠧⠤⠤⣤⢤⡤⠤⠤⠇⠀⠀⠀⠀⠘⠾⠤⠤⠴⣤⠤⡤⠤⠤⠼
⠀⠀⠀⠉⠈⠁⡿⠨⡇⠉⠁⠁⠀⠀⠀⠀⠙⠓⠋⠀⠀⠀⠀⠀⠈⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠇⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇

⠀⢀⠤⣦⡀⠀⠀⠀⠀⠀⣠⠐⢤⠀⠀⢀⡤⠒⠒⢠⡤⠀⠀⣠⠄⠐⢦⡄⠀⠀⢴⠀⠀⠀⠀⠀⣀⠤⠲⣤
⠀⢧⣀⡘⣿⡄⠀⠀⠀⠀⣿⣳⣞⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⡄⠀⢸⠤⠀⡄⠀⠀⢸⡄
⠀⠘⠞⢁⡿⣷⡀⠀⠀⠀⣿⠁⢸⠇⠀⢹⣆⠀⠀⠀⠀⠀⠀⢸⡆⠀⣠⡟⠀⠀⢸⠀⠀⠀⠀⠀⢸⡉⠙
⠀⠀⠀⣸⠁⢻⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠈⠀⠈⠁⠀⠀⠈⠁
⠀⠀⢠⣧⡤⠀⣿⡄⠀⠀⢀⠄⠀⠀⠀⠀⢠⠀⠀⢰⡆⠀⠀⠰⠀⠀⠀⠀⢰⠀⠀⡒⠀⠀⠀⠀⠀⣿
⠀⠀⡼⠀⠀⠀⠘⣷⡀⠀⠸⡄⠀⢰⠆⠀⢸⠂⠔⠸⡃⠀⠀⠸⠀⠀⠀⠀⢸⠀⠀⣿⠀⢢⠀⠀⠀⣨⠀⢀
⠀⣼⡇⠀⠀⠀⠀⣛⣳⣀⠀⠠⠓⠀⠀⠀⠼⠆⠀⠀⠃⠀⠀⠰⠄⠀⠸⠂⠁⠀⠀⠿⠀⠈⠃⠀⠀⠈⠀⠂⠁
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⡀⠀⠄⡀
⠀⣀⠄⠀⠀⠈⢹⡄⠀⠀⠀⠀⣿⡀⠈⣹⠁⠀⠀⢠⠊⠀⠀⠈⢳⡄⠀⠀⣿⡁⠀⣻⠀⢠⡟⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⣿⠻⣄⢹⠀⠀⠀⢅⠀⠀⠀⠀⢀⡇⠀⠀⣿⠋⠉⠁⠀⢸⡇⠀⠀⠀⠀⣼⠃
⠀⢀⣸⡀⠀⠂⠀⠈⠁⠀⠀⠀⠽⠄⠈⠿⠆⠀⠀⠈⠂⠄⠀⠀⠘⠀⠀⠀⠿⠄⠀⠂⠀⠀⠈⠂⠀⠀⠚⠇
⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⢸⣾⠀⠈⣿⠆⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢉⣩⠛⢉⠁⠀⠀⠀⠀⡜⠁⠀⠀⢸⠁⠀⠀⢿⡄⠀⠀⡌
⠀⢸⣿⠤⣞⠋⠀⠀⠀⠀⠀⠀⠓⢷⣄⠀⠀⠀⣾⡃⠀⣺⠧⠀⠀⠀⣾⡇⠀⠀⠀⣸⡇⠀⠀⠈⣷⡀⡸
⠀⢸⣿⠀⠈⢣⣄⠠⡆⠀⠀⣄⠀⢀⡿⠀⠀⠀⠹⣧⡀⠈⢀⡔⠀⠀⠻⣧⣀⣀⠀⢹⣅⠀⠀⠀⠸⡷
⠀⠈⠉⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠈⠁
⠀⢰⠶⠆⠒⠖⠲⢠⠆⠀⢲⢰⠀⠀⠻⣿⡀⠀⠀⠶⠓⠀⠀⠀⡖⢲⡄⠀⠀⠘⣶⠂⠀⠂⠀⠐⠒⠒⣲⡜⠁
⠀⠀⢀⡾⠀⠀⢠⡏⠀⠀⢸⣿⠀⠀⠀⠘⢿⣆⡰⠂⠀⠀⠀⠘⠁⢸⢿⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⡰⡟⠁
⠀⠀⢼⣇⠀⠀⣿⡄⠀⠀⠨⣿⠀⠀⠀⠀⠈⡾⣣⡀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢠⣿⠀⠀⠀⠀⠀⣐⠍
⠀⠀⠸⣿⣄⠀⢹⣧⣀⣀⣸⣿⠀⠀⢠⠀⣜⠀⠙⣿⣄⠀⠀⢀⡀⠀⠈⠙⠛⠁⣿⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠈⠉⠁⠀⠀⠛⠋⠀⠀⠘⠚⠃⠀⠈⠋⠀⠀⠀⠀⠈⠛⠁⠀⠀⠒⠛⠛⠛⠓⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀
⠀⠀⠀⠀⠀⣠⣴⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣤⡀
⠀⠀⠀⠀⠀⣿⠉⠉⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⣀⣀⣀⡀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠋⠉⠉⠻⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣶⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣷⣾⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣿⠿⡿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣛⣻⣿⣛⣿⣿⣻⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣀⣀⣄⡀⢠⣤⣤⣤⣤⣶⡄⢰⣶⣶⣶⠶⣿⢠⡾⠿⠿⠿⢷
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣶⠀⠀⢴⣶⡾⠿⢷⡆⠀⡾⠟⠛⠛⢿⠀⠻⡿⠛⠉⠙⡇⠈⢹⡋⠉⣡⠙⠃⠀⣿⣀⣠⣶⠈⢸⡇⠀⠠⠶⣶
⠀⠀⠀⠀⠀⠀⠀⢹⣏⣈⣏⠀⠀⠀⣯⣤⣤⣼⡇⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢸⡋⠉⠙⢠⡄⢀⣿⡀⠀⠉⠀⠸⣧⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⣹⣄⠀⢀⣯⣀⣀⣀⡿⠀⣧⣤⣤⣤⣾⠀⣴⣿⣶⣶⡾⠇⠰⠿⠿⠿⠿⠿⠃⠘⠛⠋⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠿⠛⠃⠛⠛⠛⠛⠋⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣄⠀⠀⠀⢠⣤⣤⡄⠀⠀⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⣤⣴⡄⢰⣶⣶⠀⠀⠀⠀⢿⡿⠂⠻⣿⠇⠀⠘⣿⠃⠈⢻⡏⠀⠀⠀⠈⢻⢻⡀⠀⢀⡞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⢈⡏⠀⠀⢹⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠶⠶⠶⣏⠀⠀⢸⡇⠀⠀⡀⠀⣸⡀⢹⣤⡞⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⢉⣇⠀⠀⣼⡀⢠⣄⣀⣀⣸⡇⠀⣠⣿⡄⠀⢰⣿⡆⢰⣿⣷⣶⠿⠇⠸⠿⠿⠸⠿⠛⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠻⠿⠇⠘⠛⠛⠀⠛⠛⠛⠋⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⣀⣠⣤⣤⡀⢠⣤⣴⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⠀⣀⣤⣤⣤⣄⠀⣤⣶⣶⣶⣶⣄⢀⣶⠿⠿⠿⣷⠀⠿⡿⠛⠛⠛⡇⠀⣾⠋⠉⠉⠉⠃⠘⠋⠉⡏⠀⠈
⠀⠀⠀⠀⠀⠀⢿⡷⡄⠘⢿⠋⠀⡟⠉⠉⠉⢹⠀⠈⣟⠀⠀⢀⡟⢸⡇⠀⠀⠀⣿⠀⠀⡷⠶⢶⡞⠃⠀⠈⠛⠛⠛⠛⡆⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠹⡄⢸⠀⠀⡇⠀⠀⠀⢸⠀⢀⣿⠉⠉⠉⠀⢸⣇⣀⣀⣠⣿⠀⣴⣷⡄⢰⣷⣦⠀⠻⣶⣶⡶⠿⠃⠀⠀⠻⠿⠃
⠀⠀⠀⠀⠀⠀⣼⣷⠀⠙⣿⣶⠀⠿⡶⠶⠿⠟⠀⠻⠿⠇⠀⠀⠀⠀⠙⠛⠻⠯⠥⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣀⠀⢀⣀⡀⠀⣠⣤⡀⢠⣤⣄⢠⣴⡆⢰⣶⡆⠰⣶⣶⠀⢸⡿⠇⠘⠿⣿⠀⢘⡿⠋⠀⠟⠉⢉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃⠀⢿⡿⠁⠘⢿⠃⠀⢹⡟⠁⠈⢻⡁⢸⠿⡇⢨⡏⠀⠀⠀⠙⣦⡊⠀⠀⠀⠀⠘⢷⠎⠀⠀⠀⢀⡴⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠸⡆⠀⣼⠀⠀⠀⠈⣧⣾⠀⣷⣼⠀⠀⢀⣠⣾⠁⢹⣦⡄⠀⠀⢠⣾⣦⠀⠀⠀⣿⣷⣶⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣼⠇⠀⠀⢴⣷⣾⣿⠆⠀⠀⠠⠿⠿⠀⠿⠟⠁⠀⠘⠛⠋⠀⠈⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠃⣰⠏⢳⡀⠻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⡼⠁⠀⠀⠹⣆⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠏⠀⠀⠀⠀⠀⠈⢧⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⣀⣀⣀⣀⣀⣀⣀⣀⡞⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⣆⣀⣀⣀⣀⣀⣀⣀⣀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠻⣤⠞⠁⣾⡻⢿⣿⣿⣿⣿⡿⣛⡆⠙⢦⣼⠏⠉⠉⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡖⠒⠒⠒⠒⠒⡆⠀⠀⣿⣿⣦⠙⠿⠟⠡⣾⣿⡇⠀⠀⢰⠖⠒⠒⠒⠒⠲⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⣿⢋⣴⣿⣷⣿⣷⣌⡻⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡟⠛⠛⠛⠛⠛⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⠛⠛⠛⠛⠛⠛⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠞⠋⢹⠋⠛⠲⣤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⣰⠏⠀⠀⠀⠘⠀⠀⠀⠈⢳⡄⠀⢸⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⢀⡞⢧⡀⠀⠀⠀⣷⠀⢸⠀⢠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢰⠟⠒⠚⢳⡆⠀⠀⣿⠀⢸⠀⣿⠛⠛⠛⠛⠛⠛⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣇⣀⣀⣀⣀⣀⡇⠀⡇⠀⠀⠘⠓⠒⠒⠚⠃⠀⠀⣿⠀⢸⠀⣿⢰⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠁⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠛⠒⠒⠂⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⣤⣤⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠒⠒⠒⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠑⠂⠈⠠⡆⠀⠀⠀⠀⠀⠀⢀⡠⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⡀⢳⡀⣀⡤⠤⠇⠀⠀⠀⠀⡠⠖⠉⠀⠀⠸⡄⠀⠀⡰⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠐⠚⣀⡬⠇⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢀⡠⠔⠊⠁⠀⢰⡀⠐⠄⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠊⠁⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠈⠓⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠙⢆⠀⠀⠀⠀⠀⠀⣀⡤⢴⡄
⠀⠀⠀⠀⠀⠀⣠⠔⢛⢡⡄⡜⠳⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⡀⠀⠀⣀⠴⠚⠁⠀⡼⣇⣀⠤⠤⠤⢤
⠀⠀⠀⠀⠀⡴⠁⠀⠘⠒⠒⠃⠀⢹⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⢸⠀⡇⠀⠀⢳⠀⠤⢤⠜⠁⢻⡀⠀⠀⠀⡞
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡗⢤⢾⠀⡞⢇⠀⡄⡇⠀⠀⠀⢸⡀⢱⠀⠀⠀⢣⠀⣠⠴⢲⠉⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⡇⠀⠈⣄⠗⠚⡄⠇⠧⠖⠛⠀⢸⠄⢸⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢸⠀⠉⠉⠀⠒⠒⠐⢢⠀⡟⠀⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⠰⠖⠁⠀⠀⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠘⣯⠉⠉⠁⣀⠒⠀⠌⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⢰⠊⠉⠁⠀⣠⠎⣷
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠻⠧⠴⠇⡄⠐⠈⠀⠁⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⢀⠀⠝⠒⠛⠃⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠊⠁⡀⠸⡄⠀⠀⠸⠧⠔⠒⠒⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⢉⠁⠀⠀⢀⠔⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠙⠢⠊⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠛⠓⠒⠒⠲⠦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠁⠀⡀⠠⠖⢀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣲⣒⣀⠀⠀⠀⠀⠁⠓⡠⠆⠀⢈⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⣀⡶⠤⠤⠄⠐⠓⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⡛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠁⠂⠄⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠄⡀⠀⠀⢀⡀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠈⢂⡐⠁⠀⠀⠀⠀⠈⢂
⠀⣀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⡗⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄
⡇⢰⠀⠉⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠁
⡇⢀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⠀⠀⠀⡇
⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠠⠀⠁⠀⠀⠀⠀⠀⠀⢀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⡀⠀⠀⢀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⡇⠀⠠⠤⠄⠀⠤⠄⠀⠠⠤⠀⠠⠤⠀⠀⠤⠄⠀⠤⠤⠀⠠⠤⠀⠀⠤⠄⠀⠤⠤⠀⠠⠤⠀⠠⠤⠄⠀⠤⠄⠀⠠⠤⠀⠐⠁
⣬⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴
⣿⣿⣡⣭⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠐⠀⡸⢸⠀⠘⡉⢁⡀⢸⠀⢠⢄⠈⠆⠣⠀⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠳⣄⣀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡠⠂⣀⣄⠠⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⡼⠀⡇⠀⠋⠀⠀⢁⢰⠀⠀⠀⠀⠀⠀⠀⢙⡀⠰⡄⠀⣠⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠧⡤⠃⠀⠀⠙⠦⠤⠤⠔⠋⠘⠀⠀⠀⠀⠰⠀⠀⢸⣀⣀⠖⢰⣅⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠒⠀⠃⠀⠀⠔⠀⡄⠀⠀⠀⠀⠠⠒⠒⡀⠀⠔⠒⢠⠤⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇⡘⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢠⣀⣠⠂⡃⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇⢀⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⠀⢀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠉⠁⠘⠒⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠐⠰⠃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠴⠂⠀⠦⠴⠃⠀⠀⠀⠙⠒⢤⡠⠇⠘⠒⠊⠢⠤⠄⠀⠦⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⠀⠄⡴⠀⠢⠃⠈⢀⠔⠀⢀⠖⠀⠂⡔⠀⠆⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⢀⠖⠁⠃⠀⠠⠇⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣨⠀⠀⠘⠆⠀⣀⠴⠀⠀⠀⢀⡀⠀⠀⠰⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠠⠀⢀⠔⠂⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢰⠁⠈⢦⠀⠀⢀⠂⠀⠦⠤⠈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠂⠀⠀⠇⠀⠸⠀⠀⠎⠀⠀⠁⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣦⣄⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣿⡟⠛⠛⠛⠟⠛⢛⡛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠙⠛⠛⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏
⠀⠀⠀⠀⢀⣤⣾⠋⢻⣇⠀⠀⢸⣿⠋⠀⠀⣼⣿⠿⣶⣦⠀⠀⢠⣦⠀⠀⣰⡷⠀⢠⣶⡦⠀⠀⠀⢰⣿⠟⢿⣶⣤⡀
⠀⠀⠀⠀⢸⣿⠿⣷⣼⣿⠀⠀⢸⡇⠀⢀⣼⣿⡇⠀⠀⣿⡇⠀⠈⣿⡆⢠⡿⠃⢀⣿⣿⠃⠀⠀⠀⣸⣿⠀⠀⣽⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠈⠻⣿⣄⡀⠸⣿⡶⠟⠋⠙⢿⣶⠾⠋⠀⠀⠀⠘⢿⡿⠁⠀⠘⠻⠿⠆⠀⠀⠀⣿⣇⣠⣾⣿⣿
⠀⠀⠀⠀⣿⣧⡞⠁⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣼⣿⡇
⠀⠀⠀⠀⣿⣏⣾⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⣶⣦⡀⣠⣶⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠘⣻⣿⡇
⠀⠀⠀⠀⣿⡿⢃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⡀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠀⠀⠸⠛⣿⡇
⠀⠀⠀⠀⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⢻⣿⡟⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⣰⠞⣿⡇
⠀⠀⠀⠀⣿⡿⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄⣀⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣿⡇
⠀⠀⠀⠀⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢃⣿⡇
⠀⠀⠀⠀⣿⡏⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠐⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣿⡇
⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣄⠀⠀⠀⠀⠀⠈⣠⣾⣿⡇
⠀⠀⠀⠀⢿⣯⣼⠇⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⢋⣴⣿⠁
⠀⠀⠀⠀⢸⣿⠏⠀⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢿⣽⣿
⠀⠀⠀⠀⠸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡠⠆⣰⣎⠕⠒⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢸⠃⣩⢯⠥⡟⢤⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠈⢉⡣⠚⣤⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠔⢪⣽⣥⡚⠻⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢰⣡⠞⢲⠺⣌⡹⢘⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢈⡒⠋⢲⠸⡇⡇⢀⠜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⡏⢆⠈⠔⡗⡠⢊⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡄⢸⠀⣠⡿⠛⢩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠀⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠇⣼⢠⡇⠸⡀⡘⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠐⢼⡸⠀⠀⡝⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠈⡇⢠⢊⡤⠚⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢰⡵⠋⠀⠀⠀⠀⠉⠒⠢⠤⠤⠤⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡜⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣮⣿⣿⣿⣇⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢿⣿⡜⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠉⢿⣷⡜⠁⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣻⡏⣿⡇⠀⠈⣿⣷⢻⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣗⡈⠀⠀⠀⠸⣿⠟⡇⢻⢴⣮⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣿⡿⣿⣿⠀⠀⠀⠀⠀⢐⡆⠦⢈⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⠼⡄⣿⡿⢿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⡦⠁⣿⡟⢩⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣟⣻⣿⣿⣟⢳⡒⠶⣶⣐⣒⢶⡄⣶⣾⡆⠀⢿⣟⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣥⣼⡻⣿⡻⣿⣦⡾⢯⣅⣠⡈⢯⣿⡀⠮⡁⢽⣿⠿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⢿⣿⣿⣿⣿⣿⣿⡄⠲⣄⢸⣆⢂⣘⠂⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⠉⠉⠀⠉⠀⠁⠈⠉⠀⠁⠉⠉⠀⠉⠀⢹⠀⠋⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣿⣿⣯⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠿⣂⠒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⡧⢿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣆⠀⠀⠀⠀⠀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠏⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠈⠻⣦⣀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢀⣽⠛⠉⠀⠀⠀⠉⠳⣦⡀⠀⢀⡴⠚⠋⠉⠛⠻⣍⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢷⣴⠋⠀⠀⠀⠀⠀⠀⠘⣧⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢸⡏⠀⢀⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⢸⣇⠀⠘⠋⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠹⣦⡀⠀⠀⠀⠀⢀⣴⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⣀⠀⠈⠳⣤⣀⡀⠀⣀⣠⡴⠋⠀⠀⠈⠛⠷⠶⠶⠾⠋⠁⠀⠀⠈⠙⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠁⠀⠀⠀⠈⠻⣄⠀⢀⡤⢍⡉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣲⡄⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢧⣄⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠁⢉⣧⠞⠋⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠦⣤⣤⣤⣤⣤⡤⠶⠷⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠟⠛⠋⠉⠙⠛⠻⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣿
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣴⣶⣾⣿⡿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡀
⣿⡇⠀⠀⠀⢠⠞⠉⠑⣄⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣇⠀⠀⠀⠰⡄⠀⢀⡟⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⢸⣿⠀⠀⠀⠀⠈⠉⠁⢀⣀⡀⠀⠀⠈⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠘⣿⡆⠀⠀⠀⠀⠀⠀⣯⠀⣹⠀⠀⠀⠀⠉⠛⠷⠶⠶⠶⠶⠶⠶⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁
⠀⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣤⠤⠄
⠀⠀⠀⠀⢰⠋⢹⡄⠀⢰⣏⣽⡀⠀⠀⣠⠒⠒⠀⠀⢹⡏⠉⣷⠀⠀⢈⣧⣤⠄⠀⠀⢯⠀⣀⡀
⠀⠀⠀⠀⢻⠉⠹⠇⠀⠈⣇⣠⠽⠀⠀⠧⠤⠖⠂⠀⠨⡷⠚⠁⠀⠀⠀⣧⡤⠄⠀⠀⠘⣏⠁
⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⣄⠀⠀⠀⠠⠶⢦⡄⠀⠀⠀⡀⢀⡀⠀⠀⠠⡄
⠀⠀⠀⠀⡇⢀⣠⣤⠀⠀⢻⡴⢺⡄⠀⠀⢹⡀⠀⠀⢀⠀⢸⡇⠀⠀⠀⢹⣎⡀⠀⠀⠀⢳⠀⣠
⠀⠀⠀⠀⠳⣤⠴⠃⠀⠀⠘⠇⠀⠃⠀⠀⠈⠇⠀⠀⠹⣤⡼⠀⠀⠀⠀⠈⠇⠉⠀⠀⠀⠈⠟⠁

⠀⠀⠀⠀⢀⡀⠀⣤⠀⠀⠀⠀⠀⢀⠀⠀⡠⢤⡀⠀⠀⡠⠒⠲⡄⠀⠀⢀⠶⢦⡀⠀⠀⢠⠶⠒⢦
⠀⠀⠀⠀⢸⡝⠲⠛⡇⠀⠀⢹⡦⣼⠀⠀⢇⠀⢹⠀⠀⢡⡀⠜⠁⠀⠀⠰⣠⣤⠇⠀⠀⠘⣇⣤⣟⣀
⠀⠀⠀⠀⠀⠇⠀⠀⠻⠀⠀⠈⠃⠀⠀⠀⠈⠒⠃⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠘⠆⠀⠁

⠀⠀⠀⠀⠀⠀⢀⠤⢤⠀⠀⠀⢀⣠⠴⠀⠀⠀⡄⠀⣦⠀⠀⠀⣀⠀⠀⣴⠀⠀⠀⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠉⠘⣆⠀⠀⠀⢿⠀⠘⡇⠀⠀⠘⢦⠀⡿⠀⠀⠀⢻⣀⣶⣸
⠀⠀⠀⠀⠀⠀⠀⢤⣬⠇⠀⠀⠀⠀⠙⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠻⠃⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⠀⠀⠀⠀⠐⢦⣀⡟⠀⠀⠀⠀⠠⠚⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣅⠀⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⣾⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⠙⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠉⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣖⢶⣶⣶⣶⣶⣶⡶⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⡄⠀⠀⣠⣶⣶⣶⣄⠀⠀⢀⠤⠞⢸⣿⣷⣭⠻⣿⠟⣭⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠺⠿⠾⠿⠷⠿⠗⢀⠎⠀⠀⢸⡿⣫⣴⣿⣶⣿⣶⣝⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⣠⡔⠋⠀⠀⠀⢀⡙⠛⠛⠛⠛⠛⠛⠙⣫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢲⣿⣇⡈⠉⠀⣠⣿⡖⠲⢄⡀⢸⡇⠀⠀⡤⠤⠴⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠱⣼⡿⠿⣏⠉⠉⠻⣿⠇⠀⠀⠙⢾⣇⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠘⢦⠀⠈⢦⣀⡞⠀⠀⠀⠀⠀⠘⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠳⡀⠀⠹⡄⠈⠉⠉⠁⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠚⠓⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⠑⣦
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢳⡀⠀⠀⠀⠀⣂⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠋⠁⡛⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡧⠀⣀⡴⠋⠁⠀⠀⠀⢠⠐⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠈⠁⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⣿⠀⣁⠔⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠘⠈⠁⠀⠀⠀⠐⠨⠷⢄⠀⣠⣤⣠⢤⡞
⠀⠀⠀⡴⠊⠁⠀⠙⠲⣦⡿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠡⡀⡀
⠀⠀⠀⠀⢀⣠⡶⠴⠛⠁⠀⠀⠀⠀⠀⠀⠠⠖⠂⠀⠀⠀⠀⠀⠠⠀⠀⡀⠙⠀⠀⢐⠀⠀⠀⠀⠀⠀⠃⠇⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣔⢀⠂⢠⣾⠞⠀⠀⠀⠀⠀⠀⠉⠀⡀⣠⠏
⠀⠀⠀⠀⠀⡀⠐⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠷⣄⠀⠈⠋
⠀⠀⠀⣰⠀⠀⠀⠰⣄⢠⠂
⠀⠀⠀⢀⠙⠓⠒⠂⠐⠛⡋⠀⠀⠀⠀⠀⠲⡀⠀⢀⣀⠀⠀⠄⠀⠀⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⣤⠀⠀⠀⠀⢉⠀⠀⠘⠟⠁⠀⢸⠀⠀⢁⠀⢀⠄⠀⠀⢀⡄
⠀⠀⠀⠁⠀⣀⠖⠋⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠓⠁⠤⠠⠄⠈⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣶⣶⣤⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠉⠉⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣤⣤⣤⣤⣶⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠘⠛⠻⠛⠛⠋⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⠞⢁⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠊⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⡀⠀⠀⠀⠀⠀⡼⣿⢠⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡬⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠿⢃⠀⠀⠀⠀⡔⠁⢹⡿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠃⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢣⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠦⢄⣠⠄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠸⠀⢀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⠀⠀⠂⣠⠊⠀⠀⡸⠁⠈
⠀⠀⠀⠀⠀⠀⢰⠇⠀⡀⠈⠆⠀⠈⠓⢄⡀⡆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠐⣤⠃⠀⠀⡰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⠀⠈⠢⡀⠀⠀⠌⠣⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠈⢈⠆⠈⡇⢀⠞
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡀⠙⡆⠀⠀⠀⠀⢱⣄⠀⠀⣀⠀⠀⢀⣀⣀⣀⡀⢀⠄⠀⠀⠀⠌⠀⠀⢛⡡⢤⠌
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⠜⠚⠀⠀⠀⢠⠀⠀⠢⠍⠁⠀⠀⠀⠀⠀⠀⠀⡱⠋⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⢂⠀⣠⠅⠀⠀⠈⠢⠐⠀⠀⢀⡀⠀⠀⠂⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⡀⠀⢀⡀⠀⠀⠀⠀⡐⠁⠐⠀⠐⠒⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠇⠀⠀⡠⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠘⣉⢁⠀⠀⡀⡠⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠖⠉⠃⠄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠠⡠⠁⠀⢁⡤⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⠆⠀⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⠜⠁⠀⠴⠋⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡑⢢⣵⣐⡄⠈⢢⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⢀⡞⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡄⠉⡁⠀⠀⡟⠀⠀⡆⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠂⠀⠀⠂⠀⠀⠁⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠐⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠌⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢘⠃⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢣⡠⠆⠀⠀⢀⠇⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂⡀⠀⠈⢀⠀⠀⠡⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⡄⠀⠊⠉⠀⠒⠃⠀⠀⣀⠜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠖⠈⠉⠁⠀⠳⡄⠀⢘
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠂⠈⠄⣀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠌⠀⠀⠀⠀⣠⠤⠦⠤⣤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠒⠊⠛⠁⠠⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡇⠀⠀⠞⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠀⣠⠴⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠙⠢⣄⠀⠈⠉⠉⠉⠉⠈⠉⠈⠉⠀⠉⠉⣁⠴⠊⢹
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⡏⠀⠙⢦⢠⠞⠁⠀⢠⠃⠤⠖⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠦⡀⠀⢀⣀⠀⢀⣀⡀⣀⠴⠊⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⡏⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠃⠀⠳⠁⠀⡙⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠣⡀⠀⠁⠀⡜⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⡀⠀⢀⡰⢅⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠈⢦⡀⡼⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠈⠒⠋⠀⠀⠈⠓⠤⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠱⠇⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢇⡠⠖⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢉⣲⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠆

⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠄⡔⠄⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢠⠄⠐⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠘⢿⠟⠀⠠⠁⠂⠄⠀⢴⣶⡆⠀⢂⠐⠰⠀⠤⠘⢿⡗⠀⠀⠈⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠛⠁⠀⠀⠒⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠠⢀
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠠⠌⠦
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⣀⠀⠀⠀⠀⠀⠀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⢠⠀⠀⠀⠰⠂⠀⠀⡀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢢⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⢘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠜⠀⠀⠀⠀⠀⢈⡈⠆⠈⠀⠹⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠂⠀⠀⠀⡠⠊⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠞⡄⠀⠀⠀⠔⢀⠻⠇⠀⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢄⢀⠀⠀⢀⠄⠊⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢉⠰⢉⣠⡂⢀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠎⢔⠪⠤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣾⡆⠘⠀⢀⠙⢅⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠋

⠀⠀⠀⣼⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡄
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢀⣀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣆⣀
⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠙⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛
⠀⠀⣶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⡆
⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⠛⢷⡄⢠⡾⠻⣦⠀⣾⠛⠓⠀⠛⣿⠓⠀⠀⢠⡾⠻⣦⠀⣾⠛⠛⠀⣿⠛⠓⢰⡆⢀⡾⠛⠂⢰⡟⠛⠂⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⣤⣼⠇⢸⡇⠀⣿⠀⠻⣦⡀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⣿⣤⣤⠀⣿⣤⡄⢸⡇⢸⡇⠀⠀⢸⣧⣤⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⠀⠈⣻⡆⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠟⠀⠀⠀⠈⠻⠾⠃⠀⠳⠾⠛⠀⠀⠛⠀⠀⠀⠈⠻⠾⠋⠀⠻⠀⠀⠀⠛⠀⠀⠘⠃⠀⠛⠿⠃⠘⠷⠶⠆⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡏
⠀⠀⣿⢸⡇⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣦⠀⠀⣶⠶⠶⠶⠶⠶⠶⣶⡶⠶⠶⠶⠶⠶⢶⡆⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡂⠀⣶⠀⠀⠀⢰⡆⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠰⣦⠀⠀⠀⣶⡄⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡇⠀⢹⡇⠀⠀⠘⣿⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⡀⠀⠀⢸⣇⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡇⠀⠘⣿⠀⠀⠀⢻⡆⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡃⠀⠀⢿⡆⠀⠀⠸⣷⠀⢸⣿⠀⠀⣿⠀⠀⣀⣀⡀⠀⣿⡇⠀⣀⣀⡀⠀⢸⡇⠀⠀⣿⠀⠀⠈⣿⠀⠀⠀⢹⡇⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⣁⣀⣀⣈⣁⣀⣀⣀⣉⣀⣸⣿⠀⠀⣿⠀⠀⠉⠉⠁⠀⢻⠃⠀⠉⠉⠁⠀⢸⡇⠀⠀⣿⠀⠀⠀⠉⣀⣀⣀⣈⣁⣀⣸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠁⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⣶⠾⠿⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠾⠿⢶
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⢢⣀⡤⠖⠉⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠠⠤⠀⠀⣀⠱⡄
⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠇⠀⣀⠑⡄⠠⠠⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠠⡎⠀⠀⠀⠀⢠⠀⠀⠚⢀⡀⢀⠜⠁⠘⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠄⠊⢁⡆⠀⠈⠡⠉⠀⠐⠦⢤⣖⠁
⠀⠀⢀⡀⠤⠒⠉⠀⠀⠀⠠⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⢶
⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠐⢄⠀⠠⠀⠁⠀⡇⠀⠀⠀⠀⣀⣀
⠙⠆⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⢨⠄⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⢀⡴⠊⠸⠈⠑⢄
⠀⠈⢦⠀⠀⡘⠀⠀⠁⠀⣀⠤⠒⠋⠓⠒⠒⠤⠤⠄⣀⣀⣰⠇⢀⡴⠋⠀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠱⣀⣁⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠒⠂⠀⠉⣩⣅⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠀⠀⠀⠤⠀⠈⠀⠀⣠⡞⣁⣀⣀⣀⣀⣀⠀⠀⣀⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢄⠀⠀⠀⠀⠀⠀⣠⠆⠉⠀⠀⠀⠀⢀⠔⢁⠔⠉⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠊⠑⢄⣀⡀⣠⠎⠁⠀⠀⠀⠀⠀⢠⠃⠠⠁⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣤⣤⣄⣐⣂⣉⣉⡀⠀⠀⠀⠁⠀⠊⠀⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣉⣉⣀⣈⣁⣀⣀⣈⣉⣉⣛⣛⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢲⠞⡄⠀⠒⡐⠏⡵⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡀⠈⠚⠀⡀⠇⠀⠠⣀⠀⡄⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠒⠀⠙⠧⠝⢼⡧⠤⠗⠤⠤⠀⡃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠠⠤⠤⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⡁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣤⡤⠶⠒⠛⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡆⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠘⢧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡀⣸
⣴⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣥⣤⣤⣤⣦⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣁⣠⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠉⠉⣉⣉⢉⣉⡉⣉⡉⣉⣉⠉⢉⣉⠉⣉⡉⣉⡉⡉⢉⣉⢉⣉⠙⠉⢹⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⣿⠞⢹⣀⡇⠙⣦⠀⡇⠀⢾⣈⡇⡷⠂⡷⠂⡇⢯⡀⣽⡂⠀⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠀⠻⠿⠿⠿⡿⠧⠤⠤⠤⠤⠤⠭⠤⠬⠥⠤⠤⠤⠤⠭⠤⠤⠤⠤⠤⠤⠤⠭⠬⠥⠤⠤⠼⢿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢲⠀⠀⣶⠒⠒⢲⠀⠀⡖⠒⠒⣶⠀⠀⡖⠒⠒⡆⠀⢰⡖⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⡇⠀⠀⣿⠀⠀⡇⠀⠀⡇⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣸⠀⠀⣿⣀⠀⢸⠀⠀⡇⠀⠀⣿⠀⠀⣇⡀⠀⡇⠀⢸⡆⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⢹⠉⠉⠙⡏⠉⠉⣷⢸⣿⣿⣿⣿⡇⣿⠉⠉⢹⡏⠉⠉⠏⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⡇⠀⠀⣿⢸⣿⣿⣿⣿⡇⡟⠀⠀⢸⠁⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢠⡇⠀⠀⣿⢸⣿⣿⣿⣿⡇⡇⠀⠀⢸⡆⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⣏⢸⣿⣿⣿⣿⡇⡇⠀⠀⢸⡇⠀⠀⡇⠀⠀⣸
⠀⠀⠀⠀⢀⡀⠀⢀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣙⣉⣁⣸⣿⣿⣿⣿⣇⣉⣉⣉⣉⣉⣙⣉⣉⣉⣉⣉⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠚⠒⢤⡀⢀⡤⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢱⡎⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡇⠀⠀⠀⠘⠃⢸⡇⠀⠀⠈⠃⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⡀⠙⢦⣀⣀⣀⡴⠃⠘⠲⠤⠤⠞⠁⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠑⢤⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣀⠴⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠶⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠟⠛⠛⠻⢶⣤⡀⠀⠀⢀⣀⣀⣤⣤⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⣴⣶⣷⡶⠚⠋⠉⠉⠉⠙⢻⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠉⠉⠉⠉⠉⠉⢉⡿⠀⠀⠀⠀⢀⣤⣤
⠻⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⣀⣀⡤⠞⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⣀⣠⣤⣶⡿⠿⡇⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠛⠁
⠠⠶⠿⠿⠟⠋⠉⠀⠀⣷⠀⠀⠀⢀⣠⠞⠛⠻⠶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⠾⠿⠿⢥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣶⣶⣶⣶⣤⣤⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⠒⠐⢲⣤⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⢀⠔⠁⢀⡤⣤⣠⣦⡶⣞⡯⠟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⡀⠴⠊⠁⠀⠀⠀⠀⠀⠀⢀⠴⠚⡠⠄⠀⠈⢲⠏⢰⣖⣩⣵⣾⡭⡖⢋⡑⠂⠀⠀⠀⠈⠢⣀
⠀⢀⠀⠤⠒⠉⠀⠀⠀⠀⡠⠖⠀⠀⠔⢈⡠⠂⠈⡀⠐⠉⢀⣮⡴⠿⠿⢟⣉⡵⢈⢗⡜⡊⠀⠀⠀⠀⠀⠀⠈⠙⡆⣶⣦⣄
⠀⠈⠉⠢⢄⡀⠀⠀⠘⠋⠀⠐⠈⡠⠐⠁⢠⠄⠈⡠⠤⠞⠛⠻⢄⣙⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠄⠈⣠⠖⢁⣀⠐⠊⠀⠀⠀⠀⠈⠉⠉⠋⠛⠒⠒⠲⠤⠬⣄⣀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠈⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣯⡉⠉⠉⠙⣶⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠹⣷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣤⠶⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠐⣺⣿
⠀⠀⠀⠀⠀⣿⠈⢳⣄⡾⠁⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⡶⠛⠳⣄⠀⠉⠈⠁⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⡴⠋⠉⡉⢻⡁
⠀⠀⠀⠀⠀⢹⡄⠀⣿⠇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢠⡟⠀⠉⠳⣾⠃⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⣦⣸⠁⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢳⡀⠙⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠉⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠈⠃⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⣙⠳⣤⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠷⣄⡀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠉⠓⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢦⡀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⡾⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢻⣄⡟⠁⠈⣷
⠀⠀⠀⠀⠀⣾⠀⠲⢙⢷⣸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⢠⡇⠀⣠⠄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⣿⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠻⠇⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠐⡇⢠⡌⢻⡇⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠶⠀⠀⠰⠟
⠀⠀⠀⠀⠀⡴⠒⠲⣄⠀⠀⡴⠋⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠈⠳⡄⢠⠞⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⠀⢀⣴⠿⢷⡆⣠⣤
⠀⠀⠀⠀⠀⢧⠀⠀⠈⢧⣼⠁⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢠⣤⡹⢏⣴⡄⣰⠃⠀⠀⠀⠀⠀⠀⠀⠛⠁⣾⡟⠻⣷⣾⡏⠀⣼⡇⠉⠁
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡻⣿⣿⢟⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⣷⣄⠈⠛⠁⣰⡿⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣜⢫⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠻⢷⣤⣾⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⣤⠀⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣟⣛⡿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⠁
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡟
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣏⠹⠾⣿⣿⡿⣟⣫⡒⢿⣿⣿⣿⠃⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠾⣇⣿⡿⣻⡭⡍⣿⣽⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⠦⣿⣯⣾⣿⣿⣢⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣟⣷⣢⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠷⠆⢀⣾⣿⡄⠀⣿⠛⢻⡆⣿⡇⠛⣿⡟⠃⢠⣿⣷⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣠⡀⣼⣧⣼⣷⠀⣿⠛⠛⠁⣿⡇⠀⣿⡇⢀⣿⣧⣿⣧⢸⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠈⠉⠀⠀⠉⠁⠉⠀⠀⠀⠉⠁⠀⠉⠁⠈⠉⠀⠈⠉⠈⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠿⠻⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣰⣆⣀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⣾⡋⠉⠉⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⠛⣷⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠻⣧⠀⠀⠛⢿⡟⠋⠀⠀⣰⡟⠀⠀⠀⠀⠚⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣼⡿⠛⠁⠀⠙⢷⣦⣄⣈⣁⣠⣴⡾⠋⠀⠀⢀⣀⣠⣾⠟⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣶⣤⣼⡿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣡⣄⠀⣀⣠⣿⣋⣿⡇⣀⣴⣆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⡶⠿⠛⠛⠉⠉⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢀⣾⣏⣹⡟⠋⠉⠉⠛⠋⠀⠈⠉⠛⠛⠿⢷⣤⣤⡄
⠀⠀⠀⠀⠀⣸⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣥⣿⣷⣤⣤⣶⣦⣶⠟⠉⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣤
⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⠟⠉⣉⠁⠀⠈⢻⣧⣾⠇⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⡀
⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⡿⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧
⠀⠀⠀⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢿⠀⠀⠀⠀⢿⠇⠀⠀⠀⢸⠇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⢠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣼⠟⠁
⣀⣀⣀⣀⣀⣉⣀⣀⠀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠟⠛⢷⣄⠀⠀⢀⣼⠟⠻⠷⣶⣄
⣿⠉⠉⡉⢉⡉⣉⠉⡀⢉⣀⢈⣉⣉⣉⣉⡉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⣸⡏
⣿⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⣿⡇⠰⢶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⣤⠀⠴⣶⣦⣤⣀⠀⠘⠻⣦⡀⣀⣀⣄
⣿⠀⠸⡷⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⣶⡿⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠶⢶⠟⠁⣠⣶⠟⢀⣀⠀⣀⡀⠀⣈⡙⠻⣦⡀⠈⠛⠛⠉⢹⣧
⣿⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣿⡇⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⣴⡟⠁⠀⢸⣿⠛⠋⠛⠛⣿⠃⠀⠈⢿⡆⠀⠀⠀⣠⣿
⣿⠀⣻⡇⠀⠀⠀⠀⠀⣿⡃⢠⣄⢀⣄⠀⣿⢀⣿⡷⠟⠛⠻⠷⣦⣀⠀⠀⠀⠀⠘⠷⣦⡄⠀⢰⡿⠀⠀⠀⣠⣿⠶⠶⠶⢾⣿⣀⠀⠀⠈⣿⡀⠀⣿⠋⠁
⣿⠀⣸⡇⠰⠶⠶⠷⠀⣿⡁⢈⣿⣿⣅⠀⣿⡿⠁⣠⡶⣶⡶⣦⡈⢻⣦⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⣠⡾⠋⠀⢀⣴⣦⣄⠈⠻⣷⡀⠀⢹⡇⠀⣿
⣿⠀⣼⣇⣀⣀⣀⣀⣀⣿⣇⣈⣁⣈⣉⣀⣿⠁⣼⣯⣤⣿⠆⠘⣷⠀⣿⡆⠀⠀⠀⠀⣸⣷⠀⢸⣧⢠⣿⠁⠀⠀⣿⣍⣉⠁⠀⠀⠘⣿⠀⣿⠃⠀⠛⢷⣦
⣿⠀⢼⡟⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⣿⡄⠹⣯⡉⠉⢀⣰⡟⢀⣿⠃⠀⠀⢰⡿⠋⠁⠀⠀⢻⣾⣿⠀⠀⠀⣈⣉⣹⣷⠀⠀⠀⣿⣾⠏⠀⠀⠀⢠⣿
⣿⠀⢸⡇⠘⠻⠛⠟⠀⣿⡅⢠⣼⣧⣄⠀⣿⢿⣦⡈⠛⠻⠛⢋⣠⡾⠋⠀⠀⠀⠈⢿⣄⣀⣀⣄⠀⠙⢿⣦⡀⠀⠛⠻⠟⠁⠀⣀⣼⡿⠋⢀⣴⣶⣦⣾⠃
⣿⠀⣸⡇⠰⠾⠶⠷⠀⣿⡇⠈⠹⠏⠁⠀⣿⠀⣿⡟⠿⠷⠿⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠙⢷⣄⡀⠉⠛⠿⢶⣶⣶⠶⠿⠛⠁⠀⣴⡿⠋
⣿⠀⢹⣷⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⡿⠀⣿⡇⠀⠀⠈⣿⣶⣤⣀⣀⣀⣠⣤⣾⡟⠀⠀⠀⠀⣽⡇⠀⠀⠀⣀⣀⣀⣠⣄⠀⠀⠀⢻⡆
⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇⠀⠀⠀⠈⠀⠉⠿⠉⠙⠟⠁⠈⠁⠀⠀⠀⠀⠿⢷⣤⣄⣼⡟⠉⠉⠉⠻⣧⣀⣤⡾⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣯⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⣉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠿⠻⢿⣿⣿⠿⠛⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⠟⠀⠀⠀⠀⠉⠁⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠠⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡀⠀⠂⠀⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡡⠁⢀⡀⠶⠀⠁⠐⣄⡀⠝⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠘⠁⢠⠁⢀⣿⠵⠤⣀⣈⡻⡐⢏⣀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠁⣧⡀⠀⢀⡤⠒⠀⠉⠻⢷⡄⠐⡄⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠒⢹⠀⠀⠈⠀⠀⠀⠂⢉⡈⠙⣶⠁⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⡀⠌⠀⠂⠱⣤⠄⢀⡀⠀⡜⢩⠀⣠⠃⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⠄⠁⢀⡠⠀⠀⠇⠀⠠⠘⢦⠀⠀⠀⡍⠀⠀⠀⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⠃⢀⠀⢸⠀⠀⠀⠀⢀⠀⡀⠈⠣⣄⡜⠀⠀⠀⠀⡂⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢻⠁⠀⡀⠀⠀⠂⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠀⡠⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠰⠀⠀⠀⢀⠁⠂⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠆⠂⢀⠏⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⡔⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠃⠀⠘⠁⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⡇⠀⠃⠀⠀⡴⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠐⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⡐⠀⠘⠀⡇⠀⢠⢃⠔⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠻⣰⠇⠀⠀⠀⠀⡄⠀⠀⠀⠬⠦⢙⣌⡢⡡⡁⠀⣸⡎⠀⠀⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢜⠂⣁⠃⠀⠀⠀⢀⠀⠀⠀⢀⡰⠖⠂⠀⣀⠁⠙⢿⣄⠋⡅⡜⢁⡀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠈⠈⠉⠉⠀⠀⠀⡠⠀⠀⠀⢀⡤⠖⠚⠉⠉⣑⣻⠿⣧⡰⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠴⠤⠀⠀⠠⠤⠀⠀⠤⠴⢆⡈⣁⠀⠠⠐⠊⠀⡠⢀⠴⣍⣿⣦⡀⠈⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⡀⠃⠀⢸⠊⠀⢙⡒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠈⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⠁⡌⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢳⡦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⢀⠠⡀⡁⠀⠀⠘⠒⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠊⠀⠀⠈⠒

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣶⠿⠶⣦⡀⠀⣠⡶⠿⢶⣄⠀⢀⣴⠾⠷⣦⠀⠾⢿⣿⠿⠆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣀⣀⣼⠇⢀⣿⠀⠀⠈⣿⠀⠸⢿⣤⣀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⠛⠉⠁⠀⠀⣿⡀⠀⢀⣿⠀⢀⡀⠈⠙⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣇
⠀⠀⢠⣤⣶⣾⠿⣿⣿⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠙⠿⠶⠿⠋⠀⠘⠻⠶⠾⠋⠀⠀⠸⠟⠀⠀⠀⠀⠀⣿⡿⠿⣷⣶⣤⡄
⠀⠀⢸⣿⠁⠀⠀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠈⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⣿⡇
⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣶⣶⣶⣶⣶⣶⠀⠀⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⠀⠀⣶⣶⣶⣶⣶⣶⣦⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⣸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠻⠿⠿⢿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠻⠿⠿⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣤⣤⣤⣼⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣧⣤⣤⣤⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠦⣀
⠀⠀⠀⠀⠀⠀⢠⣼⠷⣼⠀⠀⠀⠀⠀⠀⠀⠀⡽⠣⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⠉⡷
⠀⠀⠀⠀⠀⠀⣤⠙⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢠⡽⠀⠠⣯⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣴⡦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠱⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣤⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢸⡱⠷⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣇⠰⠶⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠂⠀⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⠳⠶⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣤⣤⣌⣹
⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⠉⢉⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣄⠛
⠀⠀⠀⠀⠀⠲⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠸⣍⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⢻⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡕⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⣼⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠒⣤⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠈⠲⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⡴⠦⠀⠀⣠⠶⡆⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢪⡴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠋⡱⠂
⠀⠀⠀⠀⡇⢠⡣⡔⡡⠂⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⢣⡄⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣀⢣⠌
⠀⠀⠀⠀⡇⠈⠑⠊⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠈⢣⡼⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⡄⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠁
⠀⠀⠀⠀⣷⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃


⠀⠀⠀⠀⡠⢎⠉⠋⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠘⢁⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢤⠀⣤⣤⡆
⠀⠀⠀⠀⣷⠈⠀⠀⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡃
⠀⠀⠀⠀⠡⡖⡴⠶⣶⠍⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⢡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢋⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇
⠀⠀⠀⠀⠀⠉⠰⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠡⡬⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⡔⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠰⠇⡄⠀⠀⠰⡖⠀⠀⠀⠀⠀⠀⠀⠈⠋⡆⠀⠀⠀⢀⡈⡇⠀⠀⠀⠀⠀⠀⠀⠈⠱⡑⢄⣤⠬⠊⠁
⠀⠀⠀⠀⠂⣠⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠁⡀⠌⢡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣀⠴⢤⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢂⡈
⠀⠀⠀⠀⠀⢋⠠⠤⠂⡙⠀⠀⠀⠀⠀⠀⠀⠈⡤⠛⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣩⠞⠢⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠄⠀⠑⢌⡆
⠀⠀⠀⠀⠀⠈⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⢆⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⠛⣛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⢳⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠻⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣷
⠀⠀⠅⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠋⠉⠉⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⡶⢿⡉⠻⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⣀⡀⠀⠐⢳⢄⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣁⡴⠂⠀⠀⡜⡉⠉⠢⣄
⢀⠴⡂⢶⠚⡼⠴⠶⣤⣶⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣿⡚⠃⠴⠃⠀⠀⠀⠸⣆
⠈⠀⠉⠻⡍⠁⢠⡌⢍⡍⠁⠙⢅⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠁⠋⢻⠏⠀⢠⡄⠀⠀⠀⠀⠘⢧⡀
⢸⠀⠀⠀⢹⠤⠤⠤⠤⠤⠤⠦⢾⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⢸⠠⠀⣠⠁⢠⠂⢰⣶⡦⠄⠀⣹⣆
⢸⠀⠀⠀⠸⠀⢰⠶⠒⠶⡶⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣠⡘⢀⡀⠉⠀⠘⠣⠈⢽⡷⠨⢷⢾⣿
⢸⠀⠀⠀⢸⠀⢙⣈⣩⣀⣐⠁⢨⠀⠀⠀⠀⠀⠀⠀⢠⣀⣸⡏⢳⡉⠀⡎⠑⠤⢴⣄⠬⣁⠀⢤⣾⣏⡈⢤⣀
⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠔⠊⠉⢙⡿⢌⣈⠁⣽⡠⠀⣠⢁⡀⢠⣽⠟⠁⠀⠁⠀⣩⡃
⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡤⠸⢄⣠⠤⠴⡈⠀⠱⣤⣭⣽⢇⣸⢉⠁⢉⠁⠠⢄⠀⣷⠞⠁⠀⠀⣠⡴⠞⠁⠈⢂
⠈⠉⢳⡶⣾⣷⣶⣿⠃⠨⣷⣀⠂⠀⠉⢒⣦⢱⣠⡾⠟⠉⠐⠀⣾⠿⣶⣤⡀⠀⢀⣴⠋⠉⠛⠛⠛⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠉⠲⠖⠋⠀⠀⠀⠀⠀⢠⠀⢠⡿⠀⠀⠻⣿⣿⠁⠻⣷⣄⣀⠀⢂⠀⠀⠀⠀⠀⢀
⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣼⠃⠀⠀⠀⡏⠿⣦⠀⠹⣌⡛⠀⠀⠢⡀⠀⢀⡴⠋
⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠃⠀⠀⠀⢀⡦⠀⠙⠿⢿⣿⣿⣶⣆⣠⠴⠒⠉
⠀⠀⢠⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⠀⣤⡞⠀⢣⠀⢠⠀⢹⣿⡇⠉
⠀⠀⢸⠀⢀⠀⠸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡂⠀⢀⡜⣿⠀⠀⠀⡄⠈⡇⠀⠛⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⢦⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠁⠀⢂⠆⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⢰⠀⠱⢄⠀⠀⡀⠀⠀⠀⠒⠀⠀⠉⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠄⠀⠁⠀⠀⠀⠀⠇⠌
⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⢠⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢂⠀⠄⢠⢸
⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠑⠀⠣⡼
⠀⠀⠀⠀⠀⠀⢀⡄⢠⢠⠆⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠠⠂⠀⠀⠀⠦⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⡄⢠⠊⠀⠤⠀⠀⠀⠁⠀⠒⠒⠊⠏⠀⠀⠀⠀⠀⠤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠃⠠⣤⠀⠀⠀⠀⢥⠀⠈⡇⠀⢀⠀⠁⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠀⠀⢠⠆⠀⠁⠀⠀⠀⠀⠀⠈⠓⠰⡀⠄⢀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠑⠂⠄⠀⠋
⠀⠀⠀⠀⠀⠀⡀⠀⢠⠖⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠔⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠂⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠤⠂⠈⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⡆⠈⠴⣠⡀⡀⠀⠀⣀⣀⣈⡁⠤

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠉⠛⠿⣦⣴⠿⠛⠉⠉⠛⠿⣶⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⣀⣤⣶⠿⠟⠋⣿⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⣿⠉⠻⠿⣶⣤⣀
⣀⣤⣶⣿⣛⣉⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠏⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣉⣛⣿⣶⣤
⣿⡿⢿⣿⣝⡛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⢛⣹⣿⡿⢿
⣿⡇⠀⠈⠙⠻⢷⣦⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⣴⡾⠟⠋⠁⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⢿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠈⠙⠻⢷⣦⣄⡀⢀⣠⣴⡾⠟⠋⠁⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢸
⣿⡇⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢸
⣿⣇⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣸
⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣸⣿⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣛⣡⡾⠋⠀⠀⠀⠀⠀⠀⢸⡿⣶⡀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿⠋⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣼⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣧⣠⣤⣤⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠸⠟⠃⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⣤⡴⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⣾⡏⠈⣿⠀⠀⠀⣠⣶⣄⣴⠞⢻⣧⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠻⠇⣰⡏⠀⣀⣠⣿⡏⠻⠇⠀⢰⡟⠀⣤⣤⣤⠀⣠⣴⣤⠀⠀⠀⠀⣼⠃⣿⡷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣠⣾⣫⣴⡿⠿⠟⠛⠷⣦⣤⣴⠟⠁⠀⣿⡏⢿⣿⡟⣹⡿⠀⠀⢀⣼⣇⣀⣩⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡿⠟⠉⠀⢀⣀⠀⣰⣿⠟⢷⣆⠀⠀⠘⣷⣈⣹⣾⠟⠀⠀⠀⠈⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⠀⠀⢠⣾⡿⢿⣿⡇⠀⣸⡏⠀⠀⠀⠼⠟⠋
⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠸⣿⡇⠈⢿⣧⣠⣿⠇⠀⢠⣶⣶⡄⠀⢀⣤⣄
⠀⠀⠀⢀⣴⡿⠏⠀⠀⠀⢀⣴⣶⣄⠀⠀⠘⢿⣄⡀⠉⣿⡏⠀⠀⣿⡏⠘⣿⣶⡿⠋⠙⣿
⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⢸⣿⠉⣿⣄⣤⣤⡀⠈⠿⣶⣿⠁⠀⠀⠘⣿⡀⠘⠿⠀⢀⡾⠋⢠⣄
⢰⣿⣏⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⣿⠟⠁⣨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠟⠁⠀⠈⠙⢷⣤
⢸⡟⠻⣶⣀⠀⠀⠀⠀⠀⢸⣧⢈⣡⣴⠾⠋⠀⠀⣠⡿⠿⣶⣄⠀⠀⢀⣠⣬⣍⡀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣄
⢸⡿⠀⠈⠙⢷⣤⡀⠀⠀⠼⠿⠟⠋⠁⠀⠀⠀⢸⣿⠃⠀⠘⣿⣤⣴⠟⠉⠀⢹⣧⠀⠀⠀⠀⣠⣶⣿⢿⣿⡿
⢸⣿⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠃⠀⠀⠀⣼⡏⠀⣠⣴⡿⠟⠛⠁⠰⣿⣷
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣴⣿⠟⠋⠀⠀⠀⠀⠀⣿⣷
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣠⣴⣶⣶⣿⣆⠀⠀⠀⠀⠀⣼⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣿⡇
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡿⠋⠁⠀⠉⠛⠻⣶⣤⣴⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⣿⡿⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠰⣿⡇
⣿⡷⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠰⣿⡇
⣿⣇⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⢰⣿⠇
⣿⣿⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⣸⣿
⣿⣿⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣾⣿
⢿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠈⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⢶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣿⣾⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠉⠙⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠦⣄⡀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠓⢦⣍⡓⠲⢤⣄⣀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠳⢭⣙⠒⣺⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠛⠀⠀⠨⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣄⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠽⠒⠻⣝⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹
⠀⠀⠀⠀⠀⢀⡴⣻⠓⢤⣀⡀⠙⠒⠆⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠘⡎⡆
⠀⠀⠀⠀⢠⣾⣾⠇⣿⠀⠈⢯⠟⠛⢿⡟⡿⢦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⣷⣷
⠀⠀⠀⠀⡞⣿⣟⡀⢻⡄⠀⠀⢀⡠⢊⠞⠁⠀⠉⠛⠓⠶⠶⢦⡶⠖⠱⣀⡀⠄⠐⢊⠝⠀⢸⣿⠀⠀⠀⠀⠀⣿⡟
⠀⠀⢀⣰⠛⠉⠉⠉⠉⢷⣤⢖⣫⢶⡏⢶⠂⠀⠤⠤⢀⣀⡀⠈⠓⠒⠒⠂⠤⠤⠤⠊⠀⠀⢨⠟⠶⠾⠿⣭⣭⣿⣧⠤⢤⣤⡤⠤⢤⠴⠶⠶⠶⠶⠶⠶⠶
⠀⢀⣾⡇⠀⠀⠀⢀⢄⡾⠿⣯⣤⣊⣀⡈⠣⠴⠒⠈⢁⠼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣼⡇⠀⠘⣿⡇⠀⢸
⠀⢸⠉⢷⣄⡄⣀⣨⠏⠀⢰⡇⠉⣖⠒⠤⠥⢍⣀⣒⢊⣜⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣄⠀⠀⠀⣠⣾⠟⠀⠀⠀⣿⡇⠀⣸
⠀⠘⣆⠘⠿⠟⠉⠀⠀⠀⡾⠀⢠⠏⠀⠉⠀⠀⠐⠂⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣤⣶⠿⠛⠁⠀⠀⠀⠀⣿⡇⠀⡿
⠀⠀⠈⠓⠦⢤⣤⠤⠶⠾⢥⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣇⣄⣧⣶⠏
⠀⠀⠀⣠⣞⣛⣙⣳⡆⢀⣾⣛⣋⣛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⡟⠋⠛⢱⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠋⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⣤⠾⠻⣶⡀⢀⣿⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⠁⠀⣠⣾⠃⣸⡇⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢸⡇⠀⠀⠉⠀⢀⣿⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢿⣄⠀⠀⠀⣸⡇⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠶⢶⣾⣧⣀⠀⠀⠀⠙⠻⠶⢶⣿⠟⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣼⡏⠛⢷⣄⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢀⣴⣟⠛⢷⡄⠀⣠⣶⠀⠀⢀⣶⣶⠀⢠⡶⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣾⢻⡿⠶⣴⣿⡾⠛⣿⠀⢀⣾⠋⣿⣠⡿⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠃⢸⣇⠀⢀⡿⠀⠀⣿⡄⣼⠏⠀⠈⣿⠟⠛⠁⢀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣸⠏⠀⠀⠻⣶⠿⠁⠀⠀⢸⣿⡏⠀⠀⠀⢿⡄⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⣀⣴⠟⠁⠀⣶⠀⠀⢀⡄⠀⠀⠀⠀⢿⡄⢰⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠸⣧⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⢿⡀⠀⣸⡇⠀⠀⠀⠀⠈⢷⡿⠁⠀⢺⡟⠛⢻⡆⠀⠀⣰⡶⣦⡀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⠻⣶⠟⠀⣶⢷⣆⠀⢶⡄⢀⡶⠀⢹⡷⢿⣟⠁⠀⣰⠏⠀⢸⡇⠀⢀⣠⡶⠟⠋⠉⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⣶⠟⠀⠈⠻⠟⠃⠀⠘⠃⠀⠛⠀⢰⡟⠀⠀⣾⣡⣴⠟⠉⠀⠀⠀⢰⡟⠃⠀⣿
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣠⡶⠶⠶⣦⠀⠀⢸⡇⠀⠀⣿⠇⠀⣼⡿⠋⠀⠀⠀⣀⣤⡀⠘⢷⣤⣼⠏
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣰⡟⠀⠀⠀⣀⣤⡀⠀⢀⣠⣄⢠⡾⠋⠀⠀⢰⡟⠀⢀⣀⠀⠀⢠⣿⠀⠘⠋⠀⠀⠀⣰⡟⠋⣹⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠚⠋⠀⢀⣴⡾⠛⣿⡄⣠⡿⠉⣿⣿⠁⠀⠀⢠⡿⠁⢀⣾⣿⠀⠀⣸⣿⠀⠀⠀⠀⠀⣸⡿⠿⠿⠋⠀⠀⢀⣤⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⣀⣴⠟⠁⠀⠀⢹⣧⡿⠀⠀⣿⠇⠀⠀⢠⡿⠁⠀⣼⠿⣿⠀⢠⣿⣿⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡁⠀⠀⠀⠀⠸⣿⡇⠀⠀⠟⠀⠀⠀⣾⠃⠀⣰⡟⠀⢿⣤⡾⢹⣿⠀⠀⠀⢠⡿⠙⣿⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠉⢿⣄⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢰⡿⠀⢰⡟⠀⠀⠀⠉⠀⠈⣿⠀⠀⢠⣿⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⡿⠁⠀⠀⠀⠀⠀⠀⢻⣇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⡶⠟⠁

⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⠀⠀⠀⣀⣤⣤⣶⣶⣶⣾⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣠⣶⣿⠿⠛⠋⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠚⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣠⣴⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢒⡒⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠯⢙⡆⢣⠀⠀⠀⠀⠀⠀⢠⠏⢼⠉⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⡠⠚⠓⠒⢤⠀⡠⠔⠚⠤⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢀⠀⠻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⡀⠀⠀⠈⠀⠶⡀⠁⠀⢀⠴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣀⡴⠋⠀⢤⡀⠑⠢⠤⠔⠊⠀⠈⠓⠚⠉⢀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⣹⠏⠀⠀⠀⠀⠉⠂⠄⠀⣀⣀⣀⣀⣀⡤⠔⠋⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⣀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠙⠲⢤⣀⣠⡾⠋⠀⠀⠀⠀⠀⠙⠳⣴⠶⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣦⣼⠃⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠸⡿⢩⡏⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⣀⣠⣤⡤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⢀⣠⣏⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣿⡟⢿⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣀⣀⣀⣠⠤⠚⢦⡀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠉⠁⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣤⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⠛⠿⢶⣦⣤⣼⣧⣤⣤⣤⣶⣶⣿⣻⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢶⡶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣹
⠀⠀⠀⠀⠀⣼⡅⢠⡶⣶⡆⢰⣆⣶⡄⠀⠀⠀⠀⡴⠿⣿⣿⡶⢂⣏⣹⢻⡶⣤⣶⡏⣀⡄⠀⠀⠀⣯⣄⢀⣤⢀⡤⣄⢀⣄⡀⢀⣿⠃
⠀⠀⠀⠀⠈⠸⣧⠿⣭⡟⠹⠞⠻⣯⡤⠖⠀⠀⠀⠀⣴⣿⠶⠚⠛⠛⠿⣬⣄⣫⣿⡿⠏⠀⠀⠀⠸⠃⣏⣾⣫⣏⣀⣟⡼⣼⣁⣾⢉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣠⣀⠀⢴⣢⠵⡀⠀⠀⠀⠀⡀⢀⣤⣶⠀⢿⡿
⠀⠀⠀⠀⠀⢰⠋⠀⠉⠲⠋⠀⠈⢳⠋⠑⠀⠀⠀⠀⠀⠻⣿⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠄⢠⣶⣶⣄⣴⣶⣦⡀⠤⠞
⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢀⣄⡀⠀⡀⠀⠀⠀⠶⠄⠈⢿⣿⣿⣿⣿⡿⠁⠤⠶
⠀⠀⠀⠀⢀⢀⡤⠿⣇⠀⢀⡴⠃⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⣐⠆⠘⣲⣄⢿⣀⡟⠜⡡⠄⠀⠀⠠⠤⠤⠀⠙⢿⡿⠋⠠⠤⡤
⠀⠀⠀⠀⠛⠛⠀⠀⠈⠦⠞⠀⠀⠀⠀⡞⢩⡉⠶⢋⣍⠹⡄⠀⠀⠊⠁⠈⠉⠁⠀⡅⠀⣠⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣟⢿⡶⣿⢋⡼⠁⠀⠀⡾⠉⣰⡇⣶⢺⡏⣸⣗⣡⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣰⣒⣶⣄⣴⣳⣦⠀⠀⠀⠀⠈⢦⣘⡵⠋⠀⠀⠀⠀⠙⠛⠉⠋⠛⠈⠋⠁⠉⠀⠀⠀⠀⠀⠝⠒⠀⠀⠀⠀⢀⡴⠓⠦⣀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⡿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠲⢄⣠⠖⠒⢢⡀⠀⠐⣶⡀⠀⣴⣋⣀⣀⣀⣀⣑⡦
⠀⠀⠀⠀⠀⠀⠈⠻⣾⡿⠋⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⣿⠤⣤⠈⠁⠀⠀⣰⠃⠀⠀⠁⠀⠀⡇⠀⣴⣤⣶⡄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⣤⢼⡄⠀⣀⠀⣤⡀⠀⠀⠀⠀⠸⣇⠀⠁⠀⡼⠀⠀⢀⡼⠁⠀⠀⣴⡀⠀⠀⡇⠀⠈⠻⠋⠀⠀⣿
⠀⠀⠀⠀⠀⢰⠀⠀⣀⡀⣤⠦⣄⢰⡎⡇⢨⡟⢦⡿⠼⠀⠓⠁⠀⠀⠀⠀⠈⢣⣀⡼⠳⣄⡴⠋⠀⠀⠀⠀⠋⠁⠀⠀⠓⠒⠒⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠐⢺⠉⢀⡟⣿⠻⣤⣾⠟⠋⠙⠋⠁⠀⠀⠀⠀⠀⣠⢀⣤⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠾⠆⢀⡀
⠀⠀⠀⠀⠀⠘⠒⠋⠉⠁⣠⠞⣻⠀⣿⠀⢀⣀⢀⣤⣰⡆⣻⠻⡼⠙⠚⠃⠀⠀⠀⠀⠀⢠⣄⣤⢰⠛⡦⠚⣧⡆⠀⠀⠀⠀⠠⠶⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠟⠐⢿⣱⢿⠿⣼⠿⠋⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣧⠘⠞⣡⡀⠉⠃⠀⣴⠚⠛⠲⣄⡠⠖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡤⡄⠀⠀⠀⠰⣏⠀⣿⢀⡀⠙⢁⣀⠀⠀⢻⡀⢀⠁⣈⠁⠘⢃⠀⡇
⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠻⡉⣹⠘⢦⣸⠃⡖⢺⠆⠀⠈⠙⠃⠿⠿⠷⢾⣯⡄⠀⠀⠙⣌⠀⢉⠀⠖⢠⠞⠁
⠀⠀⠀⠀⢀⡷⠄⢰⣿⣿⣦⣴⣿⣿⡆⠠⠾⣁⠀⠀⠀⠀⠀⠈⢁⠀⠀⡀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣠⠔⠁
⠀⠀⠀⠀⠹⢤⣤⡄⢻⣿⣿⣿⣿⡿⠣⣤⡤⠟⠀⠀⠀⠀⠀⡟⢭⣉⢉⣉⡽⢻⡆⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⢀⣀⢠⡄⠛⠁
⠀⠀⠀⠀⠀⠈⠷⠤⠤⠙⢿⡿⠋⠤⠴⠾⠃⠀⠀⠀⠀⠀⠀⡇⡠⠿⣿⠿⣄⢸⡇⠀⠀⠀⠀⠀⠀⣿⢁⣾⣻⣶⡀⠈⣿⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠚⠛⠃⠀⠀⠀⠀⠀⠚⢧⡼⢧⡽⠃⢧⡼⠙⠯⠴


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⠿⠿⠿⠿⠿⠛⠛⣛⡛⠛⡛⢛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣿⠿⠿⠿⠟⠛⢛⣛⡉⠉⢩⡭⠁⠠⠴⠆⠀⠐⠒⠀⠀⠙⠉⠁⠈⠉⣁⢀⣵⠟⡅⣿⡇
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⠿⠿⢛⡛⢛⡭⠭⠀⠐⠓⠀⠛⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢌⣵⠟⠁⠀⣁⢿⣿
⠀⠀⠀⠀⢺⣿⣟⡛⢭⡉⠐⠒⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣴⠟⠁⠀⠀⠀⢀⢸⣿
⠀⠀⠀⠀⢸⣿⡍⠛⠷⣦⣜⡃⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣴⠟⠁⠀⠀⠀⠀⠀⠘⠸⣿⡇
⠀⠀⠀⠀⠈⣿⡇⠓⠀⠀⠉⠛⢷⣬⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⡇
⠀⠀⠀⠀⠀⣿⣧⠺⠀⠀⠀⠀⠀⠈⠙⠻⣦⣔⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣍⡀⢤⡀⢀⣴⠶⠾⠷⣦⣤⡾⠛⠛⠻⣦⣘⣵⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣧
⠀⠀⠀⠀⠀⢸⣿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⣦⣾⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⣿⠋⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢿⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢸⣿
⠀⠀⠀⠀⠀⠀⢿⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⡀⠁⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠠⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⢤⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠸⠁⠀⠀⠀⠀⠈⠻⣦⣄⡀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣝⢷⣄⠀⠀⠀⠀⠀⢈⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⡇⢧⠀⠀⠀⠀⠀⠀⠀⣴⠏⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡝⢿⣄⡀⠀⠀⠸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⢀⠀⠀⠀⠀⠀⢀⣼⢏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠝⠻⣦⡀⠐⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⠃⠀⠀⠀⢀⣾⢃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⡻⣦⣄⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠻⠀⠀⢀⣾⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣤⡄⠀⠠⠤⠀⠶⣒⣙⣀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⡄⢠⡾⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢠⠤⠀⠀⠶⠆⢀⣘⣛⣁⣀⣩⣤⣤⣤⣴⣶⣶⣶⣿⠿⠿⠿⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⢡⡿⠱⠇⠀⠴⠀⠐⢒⣀⣀⣩⣭⣤⣤⣴⣶⣶⣶⣾⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣶⣶⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⢆⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⢠⡏
⠀⠀⠀⠀⠀⠀⣀⡨⣷⠄⠀⠀⠀⢐⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⠴⡄⠐⠋⣿
⠀⠀⠀⢀⠴⠋⠁⠰⡇⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠠⠀⠀⠀⠻⡀⠈⠀⡃⠁⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⠴⣧
⠀⠀⠠⠀⠀⠀⠀⠀⠈⠉⢀⠀⠀⠀⣀⠀⠀⠀⠀⢠⠀⠀⠇⠀⠈⣢⠀⠀⠀⠀⢺⡙⠂⢨⠄⠀⠀⣠⡶⠶⢦⣄⠀⠀⢀⣤⠶⠶⠷⣤⡀⠀⠁⠀⡠⢃⡆
⠀⢠⡇⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠱⡀⠀⠀⠸⣀⠆⠀⠐⠒⠁⠀⠀⠀⠊⠦⠑⠄⠀⠀⠀⣾⠋⠀⠀⠀⠈⢳⡼⠋⠀⠀⠒⣦⠈⣷⠀⠀⠈
⠀⢸⡗⠤⣴⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣸⠀⣿
⠀⠈⢯⡐⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁⠒⠒⢤⣀⠀⠀⢀⣴⣾⠟⠃⠀⠀⠀⠀⠀⠀⣷⡰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⠏
⠀⠀⠀⠙⠂⠈⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣝⢆⠀⠀⠀⠀⠀⠀⠐⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣥⡀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⡠⠀⣿⡀⠤⠄⠒⠙⠀⠀⠀⠀⠀⠀⢄⣴⠟⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠹⣆⣠⠞⠁⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠒⠛⡯⠉⣹⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠋⠁⠀⢀⠔⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⢀⣖⡟⢀⠁⠀⠘⠋⠀⠀⢱⠀⠀⠀⠀⠈⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠚⠛⠉⠉⠉⢩⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠒⠙⢲⡀⠀⠀⢊⠜⢁⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⡇⠀⠀⠀⠀⠀⠀⠰⠊⠉⠙⢦⠀⠀⠀⠀⣧⠀⠀⠉⣰⠷⠶⠚⠛⠋⠉⠉⠉⠉⠛⣶⡀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠘⠄⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⣑⢀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⢄⠀⠀⠦⢤⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢹⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠢⢄⡇⠀⠀⠀⠀⠀⠀⡏⠀⠙⠀⠙⠀⠀⢀⡤⣤⠀⠀⠀⠀⠀⢈⣇⠀⠀⠀⠇⠀⠈⠲⣔⡀⠀⠀⠀⠀⣠⡞⠀⣾⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠑⠒⠊⠀⢳⠀⠀⠀⠀⠀⠈⠲⣄⠐⠁⣀⠔⠉⠀⠈⡗⠄⠀⠀⠀⠀⠘⡄⠀⠀⠆⠀⠀⢠⠊⠉⠓⠢⠴⠊⠀⠈⢄⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠋⡠⠚⠁⠀⠀⠀⠀⡇⠈⢦⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠳⠤⣤⣤⣤⡤⠤⠶⠖⠋⡴⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⣰⠚⠝⢦⣀⡇⠀⠀⢣⠀⠀⠀⠀⠀⣠⠤⠤⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⣀⣀⣧⠀⠀⠀⡿⠉⠀⠀⢀⣣⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠉⠈⠁⠀⠒⢲⡦⣄
⠀⡠⠀⠀⠀⢀⡴⠋⠉⠁⠁⠀⣰⠋⠀⠀⠀⢸⠁⠀⠀⠀⠀⢠⡇⠀⠀⡠⣢⠃⠀⠀⠀⡇⠀⢂⠀⢠⠖⠲⣄⣤⣄⠀⠀⢀⢏⣼
⢰⠁⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠻⡤⢄⣀⣀⣈⣢⣤⣤⠤⠤⠼⠀⠠⣪⠞⠁⠀⠀⠀⢠⠃⢀⡠⠓⢴⡄⠀⠉⠀⡸⠃⣀⡴⢋⡟⠀⠀⢀⡀
⠸⣆⠀⠀⠀⠀⠙⢢⠄⠀⠀⠀⠀⠑⢄⠀⠀⠘⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⢸⠔⠁⠀⠀⠀⢹⡶⠞⠛⠒⠉⠃⢀⡞⠀⠀⢀⠞⡇⠀⠀⢀
⠀⠈⠲⠤⣄⡄⠀⢸⡀⠀⠀⠀⠀⠀⠀⠳⣄⠀⡄⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠁⢐⠒⠤⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⡞⢀⡧⢒⡸⠃
⠀⠀⠀⠀⣟⠀⠀⠀⠙⠒⢺⡀⠀⠀⠐⠀⠈⠦⣧⣤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠁⠀⠀⠀⠈⢱⠾⠡⠎
⠀⠀⠀⠀⠘⠦⣀⣀⣠⡀⠀⠉⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠀⠀⠀⠀⣀⠴⠒⣒⢦⠀⠀⠀⠀⠀⢀⡴⠓⠂⠁⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⣀⡀⠤⢂⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠔⠋⠁⠀⠀⠀⠀⠀⢠⡴⠚⠁⠀⠀⣨⣞⡀⠀⠠⠀⠈⠁⠀⡠⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢒⠁⠀⠀⠀⠐⠲⠤⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠈⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠋
⣶⠒⠒⣶⠶⠶⣶⠶⢶⣶⠶⣶⠶⠶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⠶⠶⠶⢶⣶⠶⠶⠶⠶⢶