Fighting Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠠⣀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁⠀⠀⠈⠓⠄⣀⠀⠈
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⢀⣀⠤⠀⠀⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡐⠱⠒⠀⢠⠀⠂⠤⠀⠐⠒⠂⢠⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⡐⠒⠐⠈⠲⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠔⢻⢴⡤⢄⠀⠀⢀⠴⠄⠀⠁⠀⠀⠘⡁
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠐⠉⢲⣄⠞⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡜⠦⢴⡏⢰⠀⠈⠀⠀⠀⠰⠄
⠀⠀⠠⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⢁⢀⠈⠢⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡀⢀⠀⠀⠒⠀⡄⡙⠛
⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⡀⠁⠀⢠⡤⣄⠉⠀⠇⠀⠀⠀⠀⢀⠀⡀⠀⢀⣀⠀⡄⣄⠂⠀⠀⠀⡘⣄⠀⠀⠀⠀⢀⣂⣀⣠⣆⠈
⠀⠀⠀⠀⡇⠀⠀⠀⡘⣠⢀⡀⠀⠸⠀⡀⢰⠞⢐⢀⠀⠠⠐⡢⣄⡔⠈⠀⠤⠅⠄⠈⢣⡀⢀⠌⠵⢶⣥⠦⠤⠅⠊⠥⠄⢄⠈⡇
⠀⠀⠀⠠⠁⡀⠀⠀⠀⣴⠈⣀⡤⠒⠂⠡⢭⠴⠿⠬⡌⠀⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠒⠁⠀⠑⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠑⠚⠉⢀⠊⠀⢀⠊⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⣠⢀⢀⡀⠀⢀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⣤⠔⠀⠞
⠀⠠⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⡀⠢⡤⢤⡤⢀⡀⠦⠤⠔⠒⠚⠁⠀⠀⠀⠀⠈⠚⠉⢲⠦⠞⠂⠀⠈⠉⠁⠀⠀⠀⠰⠀⡧
⠀⠀⠀⠀⠀⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠣⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠄⣀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠈⠆⠀⠐⠤⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡆⠐⠂⠉⠁⠀⠐⢶⡶⠶⠾⡆⢀⠄⠀⠀⠀⠀⠄⡀⢘⣄⠀⠀⠀⠀⠀⠀⢠⡄⠒⠛⠤⠀⠠⡄⢨⠟⠓⠒⠃
⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠘⡄⠀⢰⣰⠋⠘⢰⠆⢀⠃⠊⠈⢁⠀⠳⢄⡀⠀⡠⡲⠋⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⢀⢀⠴⠒⠁⠀⠀⠀⠀⠀⢀⠇⠀⠀⠉⠒⠀⠈⠢⣀⠁⠀⠀⠀⠁⢠⠊⠉⠉⠘⣤⡀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠘⠁⠈⢄⠀⠀⢀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⣀⠊⢀⡠⠴⠘⠁⠈⠑⢒⡶⠤⣄⣀⡇
⠀⠄⠀⠀⠀⠀⠨⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⠊⠀⠈⠀⠀⠋⠀⠐⠟
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠛⠉⠃⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠂⠀⠀⠒⠤⡀⢀⡀⢀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠐⠠⠂⠀⠠⠒⠂⠀⠀⠀⠚⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠄⠀⠀⠀⠀⢠⠖⠉⠁⠀⠀⠀⠀⠀⠀⢀⠴⠰⢄⠀⢢⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠷⠚⠀⠀⠀⠀⠁⠀⠀⠀⣀⠜⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠺⣄⠀⡴⠒⠲⠄⠀⠀⠀⢀⠆⡆⠀⡀⡀⠉⠉⠈⠑⢤⠇⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⠂⠡⡠⣀⠀⠄⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢚⠁⠀⠑⢄⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⡰⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢄⠀⠈⢠⣄⠀⠀⠣⠀⠀⠀⢨⠀⠬⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠦⠀⠁⠈⠀⢀⡞⠀⠀⠀⠀⠀⠀⠓⠤⠤⠤⢶⠇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠒⣤⡤⠐⢏⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⢢⡀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠄⡀⠀⠀⢠⠇⠱⡄⠀⠀⠀⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠈⠉⠩⠖⠲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠈⠙⠒⠋⠀⠀⠙⢦⡀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠀⠀⠔⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠒⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⠁⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡠⠤⠴⢁⠀⠀⠈⢰⠦⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢆⠈⢴⠒⠒⠒⡄⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⡄⠐⠁⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⠊⠉⠀⠀⠚⡀⢇
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⠀⠀⠀⣀⣀⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⡇⠀⠀⠀⠀⢀⣰⠎
⠀⠀⠀⠀⠀⠈⠉⠐⠂⠀⠀⠀⠀⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠀⠀⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⢀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢯⠄⣬⣁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠽⠀⠀⠠⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣴⠀⠀⠀⠀⠀⠒⠻⡷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⣧⠘⠀⠀⠀⠀⡓⢉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠞⢻⣶⣤⣤⣀⠀⢶⣶⣃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣧⡀⠁⢀⠀⡘⡗⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⢴⢟⠀⢀⣴⣿⣿⣿⣷⣿⣿⣿⣋⠛⠇
⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⡻⣿⣿⣷⣏⣿⣯⢿⠀⠀⠀⠂⠠⣄⠀⠀⡀⠀⣀⡾⠻⠛⠁⠀⠈⣻⠁⠀⢠⠁⠉⠹⢿⣧⡇
⠀⠀⠀⠀⠀⠀⣜⣸⣧⣤⡄⣼⣿⢻⠿⠿⠙⠉⠉⡀⠀⠀⠀⠀⠳⠾⠿⢳⠶⣅⣀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⡏⢻⢦⡔⠀⡀
⠀⠀⠀⠀⠀⠈⠋⣍⠻⠏⣻⡛⣿⣟⠰⡄⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠙⠲⠤⣀⠸⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇⠁⠈⣁⡀
⠀⠀⠀⠀⠀⠺⣇⣿⣿⣄⣮⡿⢿⡛⣹⣧⠀⡄⠀⠀⠆⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⢦⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠟⠃⠀⠉⣹⡟⠸⡇⢾⠟⢃⡄⠀⠀⠸⡀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⢡⠀⠁⠀⠀⠠⠠⠠⠀⠰⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸
⠀⠀⠀⠀⢘⠲⠄⠀⠀⣰⢿⠐⡀⠀⠀⠀⠀⠃⠀⠀⠀⠣⠀⠐⠀⠁⠄⠀⠀⠀⣰⣧⡀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣿⡦⠀⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣰⠦⡄
⠀⠀⠀⠀⠀⣹⣿⡀⣸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣈⠐⠲⣦⣤⣀⣀⣀⣀⣀⣀⣀⡤⠆⠀⠤⠋⠙⠙⢱⡂
⠀⠀⠀⠀⢰⠝⡍⠻⢆⠦⠶⠖⠂⠀⠀⠢⠤⠴⠶⠿⢿⡅⠄⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠀⠐⠀⠀⠴⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠜⠠⠧⠴⠮⠤⠐⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠁⡀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠔⠂⠄⠀⠀⠀⢈⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢀⠁⣠⡀
⠀⡠⠄⡀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠋⠀⠀⠀⠀⠠⠤⠠⠆⠀⠄⢀⡀⠐⠦⡠⢈⠀⢉
⠈⠡⠐⠀⠀⡠⠢⠄⠀⠀⢀⠔⠲⠀⠀⠠⠁⡐⠂⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⢀⡐⢀⡠⠀⠁⠀⠘⠂⠀⠈⠂
⠀⠀⠀⠀⠀⠐⠒⠀⣀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠀⢀⡄⠀⡀⢀⠄⠐
⠀⠀⠀⠀⠀⠀⠔⠀⠋⠀⠀⠀⠀⠀⠀⠀⡐⠠⣀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⡃⠀⠁⠀⠒⠁⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠢⢄⠀⠀⠀⠀⠀⠦⠀⢤⠀⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠘⠢⡀
⠘⡡⠆⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠙⡲⠄⠐⠂⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠒⠉⢢⡀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠘
⣘⡀⠀⠀⠀⠐⢆⠀⠃⠀⠀⡀⢀⡠⠁⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⣀⡀⠀⠀⠀⠀⠀⢀⠔
⠉⠀⠀⠀⠀⠀⠛⠀⠀⠊⠀⠈⡃⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡉⠩⣀⠉⠉⠁⠉
⠰⢇⢀⡠⠀⠀⠴⠀⠈⠄⣀⠔⠁⠐⠒⡬⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠢⢄⠁⠀⠀⡀
⠀⠀⠈⠀⠈⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢙⡤⠴⢬⡦
⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠐⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠐⠒⣀⡀⠂⢠⠤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣶⣦⠤⢶⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢱⠰⠄⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠸⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠁⣄⡈⠀⠰⣤⠀⠀⠀⠀⣠⢊⡑⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⢀⠠⢄⠀⠀⠀⡀⡻⠳⡏⠀⢤⣀⠀⠀⠘⠀⠀⠐⡀⢠⠘⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⣄⠠⠤⠆
⠀⣀⠀⡀⢀⣄⡀⠀⠀⡐⠁⠀⠀⠐⢄⣰⣑⠀⠀⡐⡀⠀⢄⡙⠅⠘⠀⠀⢰⠁⠀⡄⠀⠀⠀⠀⣀⠀⣠⠎⠀⠀⠀⠀⠀⢤⠦⠆
⠺⢥⣐⣀⡀⠀⠀⠀⠍⠉⠄⠀⠀⠀⠀⠀⠈⠂⢄⢡⡈⢀⡀⠀⠀⢠⠀⠀⠀⠃⠈⠀⠀⠀⠀⢰⠉⠈⡉⠛⠒⠦⢤⣤⣤⣶⣤⡤⠴⠗⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠋⠀⠀⠊⠀⠐⠐⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⢸⣷⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠋⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢟⣁⠀⡀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⢀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠰⡟⢲⠃⡎⠈⠁⡇⠈⢡⠀⠀⠀⠀⠀⠀⢀⣀⢠⡏⠁⠀⠀⠀⠀⠹⣉⠂
⠀⠀⠀⠀⠀⠀⢳⡎⠀⠈⠒⠚⠀⠀⠘⣄⡠⠤⠒⠂⣉⠥⠓⢪⠃⠀⠀⠀⠀⠀⠀⢫
⠀⠀⡠⠀⡤⠀⠸⠀⠀⠀⠀⠀⠀⣀⠀⢫⢀⡠⠺⣽⠦⡀⠀⣧⠤⡄⠀⢀⣀⡀⠀⠈⠆
⠀⡜⠀⡜⠀⢠⠃⢀⣤⠤⠤⠤⢤⠏⠙⠺⠋⠀⠀⠀⠑⠪⣑⠢⢄⣹⣻⢹⣨⡽⠀⠀⢰⠀⠀⠀⠀⠀⡄
⢸⠀⣸⠀⠀⢺⢤⣀⠀⠉⠒⠢⣼⠀⠀⠀⠀⠀⢰⣀⣀⣀⣀⠉⠒⠬⡏⠁⣀⠤⠒⢹⢣⠀⠀⠀⠈⢣⠘
⠉⠀⠉⠀⠀⡎⠀⠉⡻⣶⢤⣀⡇⠀⠀⠀⠀⠀⡠⢠⠀⠉⠙⢿⣦⣄⡜⠉⠀⠀⣠⠃⠀⢣⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢸⠁⠀⢀⣈⠉⠓⠢⠤⠤⠤⠤⠒⣋⠝⠈⠀⠀⠀⠀⠹⡿⠀⠀⣠⠔⢁⡴⠃⠸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⡏⠀⠒⠂⠉⠉⠀⠀⠠⠄⠀⠀⠀⠐⣥⠤⠊⣡⠔⠋⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠒⠒⠒⠒⠁⡄⣖⡩⠔⠒⠛⠤⡀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⡌⠉⠑⠒⠒⠤⠤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⣸
⠀⠀⠀⠸⣤⡤⠄⣠⢦⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠆
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣤⣀⣀⣀⣀⣀⡀⠀⣸⠀⠀⠀⠀⠀⠠⣄⣀⡀⢠⣤⣤⣤⣴⣶⣶⣾⡏⡿⢁⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⠹⡇⣿⡿⠟⠿⠿⠿⠿⠷⠙⠉⢀⣀⣤⣀⣀⣀⠈⠉⠁⠀⠋⠉⠉⠉⠉⣿⣿⣸⡇⣾⣿⠉⠉⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠻⠿⠿⣿⣿⣿⣶⣦⣤⣤⣄⣀⢀⣿⣿⣿⢠⣿⣿⣶⡞⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣿⣿⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⢇⣼⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠛⢳⣼⣿⢻⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣶⣤⣀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠛⠻⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠙⠻⣿⣿⡿⠃
⠀⠀⠀⠀⣠⣴⣿⡿⠟⠁⠀⠀⠀⠀⠉⠙⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⢀⣤⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄
⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⢨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠛⡁⡀⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠈⠉⠒⠒⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡄⣘⠀⡀⡀⠂⠆⠛⢤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣡⣄⠀⠀⣰⠊⠁⠈⢦⡀⠀⠀⠈
⠀⠀⠀⠀⠀⡸⠇⣴⣠⠀⠀⢂⡞⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⢆⣰⣈⣀⣀⠀⢘⠀⢰⣷⣾⣿⣷⠀⣠⣄⢀⣀⠀⡀⠀⢀⡀⢀⣄
⠀⠀⠀⠀⠈⠔⣴⢻⣿⠀⣠⠞⠀⠀⢖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠟⠋⠀⠀⠀⠀⠛⠘⠛⡛⠁⠀⠀⢦⠙⠉⢁⡄⠀⠀⠁⠈⠋
⠀⠀⠀⠐⠀⠰⠁⣠⠗⠐⡿⠀⠈⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠞⠀⠀⠀⠀⠀⠀⠅⠴⠰⠒⠒⠠⢶⣴⣛⠆⠀⠂
⠀⠀⠀⠀⠈⠀⠀⠁⠙⢘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀⠒⠂⠈⠀⠀⢀⣤⣤⣤⣤⣶⣬⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠙⢷⣄⡀⣰⠿⠋⢀⣛⣛⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⡎⡉⠀⠀⡼⠃⣴⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⢿⣏⠀⠀⢠⣶⣿⡿⠋⠙⣟⣿⣿⣿⡇
⠀⠀⠀⠀⠃⠁⠀⠀⠃⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣦⣤⣄⡀⠀⠉⠳⢦⣶⠟⠋⠀⠀⠀⠈⠙⠌⣿⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠁⢀⡀⠀⠀⠀⠀⠀⣠⠴⠀⠀⠀⠀⠈⠩⣿⣿⠛⠙⢖⡢⠄⠀⠙⢄⠀⠀⠀⠀⠀⠀⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⣀⠈⢹⡏⠀⠀⠀⠙⣷⣦⣤⠀⠑⡄⠀⠀⠀⠺⢵⡻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⢠⡞⠀⠀⠀⠀⠰⣿⡿⠛⠀⣶⠃⠀⠀⠀⠘⠜⣭⡖⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠖⠈⠀⠀⠀⠀⠀⠀⢿⠀⠀⡺⠃⠀⠀⠀⠀⠀⠀⠘⢿⡶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢰⠃⠀⠀⠀⣀⠀⠀⠀⢀⡀⠙⢾⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⠶⠦⢥⣤⡶⠖⣦⣄⣀⠀⠀⠢⠦⠿⣡⡟⠿⠿⠳⠿⠷⣂⣤⣀⣭⣦⡠⠉⠀⠀⠑⡖⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠺⡻⣆⣠⣀⣠⣤⠿⠿⠷⠂⠀⠒⠂⠄⠀⠀⣁⣭⣥⠤⠤⠦
⠀⠀⠀⣠⣤⡭⠶⠆⠈⠁⠀⠀⠀⠀⠐⢆⡤⢠⣔⣄⡀
⠀⠀⠀⠤⠄⠖⢀⠐⢡⢭⣿⣶⣀⣰⣿⣦⣬⣻⡿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⢈⠉⣛⠛⠿⠿⣿⡿⠟⠛⡛⢿⣿⣯⣭⣯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠛⠳⠶⠿⠷⠂⠈⠹⠟⠛⠛⠣⢍⣀⣿⣷⣧⣾⠾⠟⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣴⠶⡴⢦⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢉⡄⠀⠀⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⢰⣿⣿⡿⢻⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠋⡴⠆⣀⣀⡇⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⣿⣿⣿⠶⢿⡿⢒⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⣿⡿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣙⣯⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣦
⠀⠀⠀⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⣿⠉⠉⠠⣄⡀⠸⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣥⣾⣘⣿⣿⣿⣇
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⢿⣿⠏⠀⠀⢀⣬⠁⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣧⣼⣿⣿⣿⣿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠈⠛⠀⠀⠀⢾⣿⠀⠀⣧⠀⠀⢀⣀⡤⠤⠀⠀⠀⠀⠀⢰⣿⣿⠀⠈⠙⠂⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣼⣿⠄⠀⠀⠀⣴⣦⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠃⠀⠀⠀⠀⢀⡴⠶⠦⣴⠏⠀⠀⠘⠛⠓⠒⠀⠀⠀⠀⠀⠀⢻⣿⠀⠙⢋⣤⠀⠀⠀⠿⠿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⡀⠈⠓⠒⢦⡟⠀⠀⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣬⣇⣀⣀⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠧⣤⣀⢀⡞⠉⠓⠶⠶⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠤⠤⠶⠀⠈⠧⣽⣤⣈⡉⢀⣹⠶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠈⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⠙⠿⠽⣯⣥⠶⠚⢿⣛⡛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠰⣄⠸⡆⠀⠀⠀⠀⠀⢀⣠⠴⢋⡿⠁⠀⠀⠀⠀⠀⠋⠁⠀⢠⠖⠀⠀⢀⡴⠛⢧⣀⣠⡴⠟⠙⣧
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⣀⡤⣾⠁⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⣼⣃⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢦⡀⠀⠈⠛⠚⠋⠁⢀⣨⠟⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣆⠀⢹⡄
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⣠⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡇⠀⠀⠀⠀⠀⠀⢿⡀⠘⣧
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢨⡗⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣾⡇⠀⠀⠀⠀⠀⢠⡟⠀⣼⠃
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⣳⣤⠞⠁⣾⠁⠀⠀⠀⣀⣰⣟⢀⣼⠃
⠀⠀⠀⢸⣇⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠁⠀⠀⠸⣧⣄⣀⣀⣼⡋⠉⣽⣿⣽⡇
⠀⠀⠀⠀⣿⠉⠉⠙⠛⠛⠒⠲⠶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠉⠙⠋⠁⠀⠀⡇
⠀⠀⠀⠀⣿⠀⠀⢀⡀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠈⢷⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⣧⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠈⣷⡀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⣾⣧⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡇⠀⠀⠀⡿⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⣷⠀⠀⢸
⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⣸⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣏⢸⠀⠀⢸
⠀⠀⠀⠀⢸⠃⠀⠀⣼⠁⣿⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⡯⠸⣦⣤⣼⠆
⠀⠀⠀⠀⣿⠀⠀⢀⡟⢸⡇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⢹⠀⢸⡄
⠀⠀⠀⠀⣿⣤⣤⣼⠇⢼⣦⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢀⣸⡇⠀⡼⠀⢸⣇
⠀⠀⠀⣰⠛⠛⠛⠻⢶⡾⠷⢦⡾⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠿⣚⣫⣥⡿⡿⠵⠚⣉⡼⠂
⠀⠀⠀⠘⠓⠚⠛⠛⠛⠳⠶⠶⢤⣤⣤⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠀⠿⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⣠⡖⡦⠶⡄⣀⠀⠀⠀⠀⣀⡤⠶⠛⠛⠛⠛⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡤⡤⣾⣼⣇⠀⠀⠀⠀⠀⣗⠿⣴⡖⢟⡼⣛⣦⡀⣾⡏⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠐⣆⠀⠈⠳⢦⠀⠀⠀⠻⢭⣿⡇⠋⠙⠷⣺⣼⠏⢻⣸⡔⣆⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣄⠀⣿⢀⡀⠀⠈⢧⠀⠀⠀⣴⡏⠙⢶⣖⠉⠁⡾⠀⠀⠙⠗⠈⢳⠀⡴⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⠀⠛⢀⣩⡷⠀⠈⢹⣤⠞⠁⠈⢳⡞⠉⠓⢦⡟⣻⠁⠀⠀⠀⠼⢿⣁⣤⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣟⠀⠀⠀⠈⠀⣠⠖⢚⡇⠀⠀⣰⠏⠀⠀⣀⠠⣟⠀⠀⠀⠀⠀⠀⠸⠇⠀⣽⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠀⠃⠀⡼⠁⣠⣞⣁⡀⣾⠛⠛⠳⡜⣿⠓⠦⠀⠀⠀⠀⠀⢺⡅⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⢀⣈⣿⠖⠀⠀⠀⠀⢀⣠⠞⣡⠞⡿⠋⠛⠙⣯⣤⣤⣄⣽⣈⣳⣦⣤⣤⠴⠶⠛⠛⠙⠓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣏⣉⣽⠷⠒⠒⠓⠲⠿⠶⠾⠷⠶⣧⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣹⡶⠖⠳⠦⠤⢤⡤⢤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠶⣶⠒⠚⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⠀⠀⠀⠙⣆⠀⠀⣀⣠⡤⠖⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⣤⠴⠚⠳⠶⠦⠴⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⣀⣠⡤⢿⡿⠋⠉⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⠞⠋⠉⠁⠀⡰⠊⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣽⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢰⡇⠙⠳⣄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⢀⡼⠁⠀⠀⠈⠳⡄⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣾⣓⣶⠤⣼⠀⠀⠀⢀⡇⠀⠀⠀⣧⠀⠀⠀⠀⢧
⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠹⣿⣍⠓⠮⠷⢤⣤⡬⠇⠀⠀⠀⠸⡆⠀⠀⠀⠈⢧
⢀⡤⠶⣯⡀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠈⠺⢷⣆⢄⡀⢸⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠘⢦
⣾⣇⠀⢈⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣧⠴⢯⣉⣿⠀⠀⠀⠀⠀⠈⠓⠶⠟⠀⠀⠀⠀⠀⠀⠸⣇⣀⣠⣤⠖⠻⣄
⠈⢿⣦⠸⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⡤⠾⠷⠖⢋⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⣀⠴⠃⠀⣠⠼⡇
⠀⠈⠻⣷⣿⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠀⣠⢴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣉⣽⣋⣥⠶⣚⡩⠖⠋
⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠓⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⣀⡄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠈⠁⠈⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌⢀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠈⠀⠀⡀⠀⠈⠢⣀⠂⠐⠄⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠀⠀⡀⠄⠘⠀⢀⠊⠀⠀⠀⢀⣀⠀⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠰⠁⠀⠴⣤⡤⠀⠀⠀⠐⠉⠀⠑⡀⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⢌⠀⢀⡤⠤⣄⠀⠀⠀⢀⠀⠄⠀⠀⠐⠐⠒⠂⢄⡏⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠑⠘⣀⡀⠀⠄⠔⠀⠊⠀⠁⠈⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠐⡋⠀⠑⢄⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣠⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠤⢄⠀⠰⣤⡀⠀⠱⠀⠠⢀⡥⠠⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠘⡤⠊⠀⠁⣀⡞⠀⠀⠀⠑⠊⠚⠒⠶⠒⢲⠋⠈⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡀⠀⠀⠀⠀⠈⢳⢶⠀⠻⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠻⢄⡱⢄⣀⡠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠤⣀⣀⠟⠀⢣⡀⠀⠀⠛⣆⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⢘⡍⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠀⠀⠀⠂⠀⠀⠀⠀⠙⢦⣤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢸
⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⢠⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠙⣠⠤⠤⢤⢸
⠀⠀⠀⠀⠊⠀⠀⠀⠑⡄⠀⠈⠀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⢎⢸⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⡇⠀⠀⠀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠀⠀⠚⡂⠘⠥⠘⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣘⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⢠⡀⢀⡐⠀⠘⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⠀⢀⠠⢰⠀⠠⢐⡀⠘⠻⡈⠑⠱⠀⠀⠀⢀⠀⡁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢄⡰⠒⠤⢀⠀⡀⠀⠄⠀⠀⠁⠀⠈⠀⠀⠈⡂⢄⠀⠀⠀⢀⢁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⠒⠂⠀⠀⠀⠀⠠⠔⠂⡀⣀⡂⠂⡁⠀⢰⠄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⠀⠀⠈⠀⠤⠀⠀⠤⠐⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⣀⡀⠆
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠠⠀⢠⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣔⡤⣤⣤⡀
⢰⠇⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠈⠐⠀⠀⠀⠤⠚⠀⠈⠁⣀⠀⠀⢀⠀⠀⠀⢻⡵⠞⠡⣄⣀⣉⣺⣿⣿⠶⡄
⢀⠀⠀⠀⠀⠤⠂⠀⣱⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠒⡔⠀⠀⣤⠀⣀⡧⠀⠀⠀⠘⢦⠀⢠⠂⠒⠒⡍⠉⡯⡸⠃
⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠈⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣶⣖⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣜⠛⠣⠾⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢷⡆⠺⣴⣭⠍⣥⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠀⠀⣐⠐⢠⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠉⢚⠉⠚⢹⡉⠀⣾⠛⠓⠒⣄⠀⠀⠀⠀⣏⠈⡆⠀⣠⢀⡼⢓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⠾⢤⠀⣄⠀⢠⡇⠀⠀⣤⠈⠓⢲⣀⣀⣈⣦⠷⠶⠶⠙⠛⢾⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣼⠞⠂⢚⣱⣀⣠⣼⠋⠀⠬⣿⣤⣸⣏⠀⠈⠉⢙⣆⠀⢀⠀⠀⠈⠀⠈⢿⡶⢄⡀
⠀⠀⠀⠀⠀⠀⣴⠋⠙⠖⠒⣾⣼⣏⣁⠈⠳⢾⡋⠈⠳⣌⠉⠛⠛⠃⠴⢶⡶⠊⠀⠀⠀⣹⠆⠀⢻⡄⠘⢂
⠀⠀⠀⠀⠀⢠⣯⠀⢠⠞⠉⠁⠉⠉⠀⠙⠲⠶⠟⠠⠤⠈⢦⠀⠀⠀⠀⣰⠍⡀⣀⣠⠖⠁⠀⠀⠀⠙⢦⡘⢧⡀
⠀⠀⠀⠀⠀⢸⡇⢰⡟⠀⠀⠀⠀⠀⢀⡤⠆⠀⠀⠀⠀⠀⣈⡇⠀⠀⠀⢹⡈⠉⠁⠀⠀⠀⠀⠀⠈⠀⢠⡇⠳⡙⢦
⠀⠀⠀⠀⠀⢸⡇⠆⢷⡀⢀⠔⠈⠙⣿⣖⡖⠛⠛⠛⠛⠈⠉⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢳⡄⠀⠀⢷
⠀⠀⠀⠀⠀⣾⣹⣇⠀⠙⢦⠤⠴⠟⠉⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⡈⠑⣁⣠⣶⡾⠿⠻⣆⣀⡤⠜
⠀⠀⠀⠀⠀⠀⢻⡟⠄⡀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠔⠛⠉⠂⠀⠀⢀⣠⣼⡧⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠠⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠀⢠⣤⡴⠚⠉⠀⠈⠀⢺
⠀⠀⠀⠀⠀⠀⠀⣽⢯⢠⡀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠁⠀⠀⠂⠀⠀⠀⠀⠎⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡀⡄⠉⠛⠒⢋⠉⣿⠏⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⢷⡇
⠀⠀⠀⠀⠀⠀⠘⣿⠉⠓⠒⠶⠖⠓⠲⠉⢃⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠰⣯⠃⠀⠈⠂⠀⠀⠀⠀⠂⠈⢙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠜⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⡠⡀⠠⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠀⠢⣴⡋
⠀⠠⡀⠀⡠⠀⠐⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠁⠀⠈⠁
⠠⡈⠀⠀⢠⠀⠒⠈⡁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄
⠈⡄⠀⠀⠀⡀⢀⣠⣔⡉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢉⣻⠇⡠⠃⠈⠢⠀⠠⠒
⠀⠀⠁⢀⠣⡄⠻⢹⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡠⠁⢨
⠀⠀⠀⠀⠀⠀⣅⢀⡨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⠓⠂⠚⢄⠄⠀⠀⠀⠀⠄⡈⠖⣀⠀⡆
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠤⠁⣐⡀⢤⡤⡈⣂⢢⡇⠀⠴⢮⣸⣠⡄⠲⠒⠂⠁⡀⠘⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⢨⠀⠅⣂⠀⡈⠑⠛⠋⢡⠤⢀⠔⠀⠀⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠈⠃⠐⠀⠊⠀⠀⠁⠀⢇⠀⠀⠀⡸⠀⠀⠀⠀⠘⠀⠒⠐⠒⢺
⠀⠀⠀⠀⠀⠀⠈⠲⠀⠐⠒⠃⠀⠀⠀⠀⠀⠀⠀⡊⠑⡆⢈⡇⠀⠀⠀⠠⡀⠀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠀⠐⠄⠄⠀⠀⠀⠀⠀⠉⠈⠈⠉⠀⠀⠀⠀⠀⢈⡓⠒⠲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠀⠈⢈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡱⠛⠀⠅⢂⠀⠀⠀⠀⠀⠀⢀⠄⠺⡗⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣅⣠⣤⣤⡼⠛⠀⠀⠀⠀⠀⠀⠙⠄⠀⠤⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⣴⣮⣿⣥⣦⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⠟⡿⢿⣿⣿⣿⣇⡀⡠⣴⢵⡦⣠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣀⣴⣤⠚⠻⣷⠂⠸⣿⡿⠸⠣⠴⠿⣿⡛⢷⣿⣿⡷⣇⣾⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠚⢫⣤⠹⠿⣿⡿⠂⠒⡋⠀⠀⣠⡔⣲⣌⢛⣯⣿⣋⠿⢎⢿⡋⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢠⣾⠯⠄⠀⣠⣴⠿⠛⠷⣦⣔⣛⢷⣿⡇⣼⣿⡿⣌⣀⣼⠃⢹⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⡄⠀⠀⢠⠞⠀⠀⠀⠀⠈⠉⠛⠷⠊⠘⠻⢋⡰⠉⣹⣻⡀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⡙⣻⣭⡳⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠿⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢊⣯⡿⣿⣥⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢀⣽⠛⢻⣂⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⢠⡿⢸⣷⡄⣥⣮⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁⠊⣋⡴⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡄⣝⢷⣾⣿⣷⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⡀⣡⣪⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇⣯⣶⣿⣿⣿⣿⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⣴⠔⣘⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣵⣿⡇⠘⠋⠉⠛⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠽⡀⠸⣋⣍⣴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⢻⡿⣿⠃⠀⠀⠀⠀⠀⠹⣿⣝⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠉⠻⢿⠿⢲⣶⣶⣶⡀⣀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⣴⣴⡿⠃⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⡋⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠗⠄⠀⠀⠀⠀⠀⠀⠤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡹⣇
⠀⠀⠀⠀⠀⠀⣸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣁⠀⠐⢀⠀⠀⠀⠖⣨⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⡘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁⣀⣄⣠⣶⣿⣿⠆

⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣼⣷⣮⣿⣶⣬⣾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⡀⢄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣂⡊⠀⠀⠑⡍⠉⠇⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠉⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠈⠁⠀⠉⠒⢤⠁⠀⢰⠀⠀⠀⠀⢠⣇⣀⢀⡔⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢪⢇⡠⡤⠜⢯⢀⡀⢘⠀⠀⠀⠀⠘⠃⠛⠠⢊⠤⢀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠒⢉⣁⡍⢃⢰⡬⠀⠠⠀⠀⠤⠤⠀⣦⣥⠀⡼⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⣎⣽⠟⠉⣖⣊⡀⢀⣀⡀⣀⠀⣀⠴⠈⠐⠂⠈⠐⠈⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣲⠁⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⡌⠀⢰⣰⠋⠉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⡐⠁⡀⣤⠄⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡅⠖⠉⠈⡦⠴⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠑⠀⠀⠀⠀⠈⠉⢀⡠⠔⠃⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠤⠤⠊⠀⠀⠊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠁⠒⠒⠒⠉⣨⡀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢹⠁⠀⠀⢀⡠⠒⠀⠐⠐⠪⠁⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢀⡠⠤⠒⠒⠂⠀⠒⠥⣀⠀⡆⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡗
⠀⠀⠀⡀⠠⠔⠃⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⢊⠡⠔⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⣔
⠀⠰⢁⣀⣀⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣅⠀⢀⡠⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣀⣀⣀⣉⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣴⢦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⣿⣷⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣷⡿⣇⣨⡿⠛⠀⠀⣤⣄⠀⠀⠀⠀⢡⠝⣸⠿⡃
⠀⠀⠀⠀⠀⠀⠀⢬⠁⠀⡀⠘⠙⠿⠶⠖⠛⠶⣯⣻⣃⠀⠰⡆⠰⡞⠷⡘⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠈⣾⠃⠂⢻⡆⣀⠀⠀⣴⣠⣖⣶⢦⠈⠙⠛⠻⣦⣨⣿⡃⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠈⠉⠉⢩⡁⢸⣿⣿⣿⠀⠀⠀⠀⢸⣭⣿⣟⡣⢀⣴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⣷⣦⡶⠶⡚⠁⠈⠛⢦⣐⢦⣤⡀⠀⠀⠀⠉⠛⢀⠀⣉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠬⠋⣩⣿⣯⣉⠀⠀⠀⠀⠘⣿⣿⣿⠆⠀⠀⠀⠀⠈⠙⠛⠿⣿⣧⢁
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠉⠉⢙⡏⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠈⠉⠀⣀⣨⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣹⠃⠀⢀⠠⠂⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠏⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠠⠟⠃⢠⠀⣠⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠠⣀⡀⠀⣰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⡋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡶⢦⣆⠰⢾⢻⡿⠄⠀⠀⠈⡵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢺⡏⠀⠀⠁⠀⠀⠀⠀⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⣠⠔⠀⠀⠀⠀⠀⠀⡇⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣔⣒⠒⠀⠀⠀⠀⢀⡼⠁⠀⠘⠃⡄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⣰⠀⢁⠔⠁⠀⠠⢄
⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀⣾⣿⣷⣶⣷⡶⠟⠁⣤⠀⣀⠔⣡⠊⠁⣠⠞⠀⠀⢀⣥⠀⡀
⠀⠀⠀⠜⠠⠄⣀⢨⡆⠀⠀⣿⠘⡇⠘⠛⠃⠀⠸⠸⠧⠊⢁⠴⢿⠴⠋⠁⠀⡤⠀⠈⢉⠀⠙⡄
⠀⠀⢣⠀⠤⢀⣀⣹⠋⢠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣆⠀⢀⡤⠊⠀⠀⠠⢾⠆⠀⠘⣆
⠀⠀⢡⠀⠐⠀⠀⢽⣠⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣞⠉⠀⣠⠔⠀⠀⠈⡄⠀⠀⠘⡀
⠀⠀⠘⡄⠠⢔⣂⣼⡿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠚⠁⠀⢀⣤⡾⠂⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠛⣿⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣶⡿⠟⣿⣿⠯⠂⠀⢹⡀
⠀⠀⠀⠀⠠⠀⠀⠡⡹⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠙⡶⠤⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠸⠇⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⠀⠈⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁⠀⡐
⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠁⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢀⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⣤⠴⠋⠁⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⢀⠀⣀⣀⠀⢀⣐⠒⣒⣒⡒⡒⠒⣒⡒⠒⢒⠒⢒⠒⣒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣷⢸⡇⢸⡇⣿⣉⠁⣿⣉⠈⢹⡟⠉⣿⢸⠏⢻⡆⣿⣇⣿⢸⣏⡙
⠀⠢⢄⠀⠀⠀⠀⠀⠀⣿⣀⡿⢸⣇⣸⠇⣿⣉⡀⣈⣻⡇⢸⡇⠀⣿⢹⣆⣸⠇⣿⢹⣿⢀⣉⣿
⠀⠀⠀⠑⠢⢀⣀⣀⣀⡈⣹⠧⠀⠉⠉⠀⠉⠉⠁⣉⣉⣀⣈⠁⠀⠉⠀⠉⠉⠀⠉⢀⣉⡀⠉⠁
⠀⠀⠀⠀⠀⡆⠀⣚⠛⢒⡒⢒⡂⢒⣀⡂⣀⠀⡐⢀⡐⣀⣀⠀⣀⣐⠒⢂⣀⡒⠒⠚⠛⠓⠂⠀⠐⢄⡀
⠀⠀⠀⠀⠀⠁⢠⡿⡇⢸⣿⣸⡇⢿⣉⠁⢿⢸⣿⢸⠇⣿⣉⠀⣟⣹⠆⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⣼⠷⣿⢸⡏⢿⡇⣈⣹⡇⢸⣿⢸⣿⠀⣿⣩⡀⡟⢻⡄⣌⣹⡇
⠀⠀⠀⠀⠐⠤⠩⢤⣬⠌⠁⠈⠁⠈⠉⠀⠀⠁⠀⠁⠀⠉⠉⠀⠁⠀⠁⠈⠩⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠙⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠒⣻⣿⣿⣿⡿⢧⣾⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇⢻⠉⠙⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠈⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠀⢀⣀⣠⣤⠤⠶⠖⠒⡿⡙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢁⣾⢿⠋⣹⣿⣿⣿⣿⣿⣿⣟⣥⡾⠖⠛⠉⠉⠀⠀⠀⠀⢀⠜⠀⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡼⠀⣿⣼⣿⣿⣿⣿⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠉⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠀⣾⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣄⠀⠀⠐⠀⢀⣤⠞⠁⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣤⡖⠋⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣤⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⠐⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣠⣾⡿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⠁⠚⠒⠘⢿⣿⣿⣿⣿⣿⠿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠉⠹⠟⠛⠉⠉⠙⣿⣿⣿⣿⣿⠛⠛⠋⠉⠁⠀⠀⠈⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⢿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⡿⡿⠿⠏⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⣀⠀⠀⠠
⠀⠀⠀⠀⡠⠲⠄⠖⠑⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠀⠀⣀⡈⠑⢾⡅⠁⠀⠀⠈
⠀⠀⢂⠁⢀⠔⠒⠀⠁⠉⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠰⠀⠀⠀⠀⠀⠀⠇⡀
⠠⡌⠀⠀⠀⡅⠀⠋⠉⢁⠠⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠔⠀⠀⠀⠀⠠⡤⠀⠀⡇⠀⠀⠐⢀
⠐⡡⠀⠀⠀⢀⠀⠀⢀⡠⠂⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠒⢻⣟⠆⢈⠇⠀⠣⣀⠀⣀⠖
⠀⠈⡙⠀⠠⣚⠁⣾⠻⠿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⡊⠁
⠀⠀⠈⠀⠑⠘⠂⡄⠉⠹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣄⡠⣠⡂⠑⡄⠀⠀⠀⠀⠀⡠⢊⡅⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠐⠷⠤⠔⠁⠑⠠⠄⡀⠀⠀⠀⡠⢄⠀⠀⠋⠠⠀⠠⠘⠎⡀⠀⢀⣀⠀⠨⠒⠉⡠⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠉⠒⠂⠀⠄⢽⠂⢠⠴⠂⠐⢄⡀⣌⣙⡤⠆⠺⠁⢹⠀⢀⣠⠐⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠢⢄⣀⠀⠀⠀⢸⡀⠑⢬⠤⠤⡄⠈⠉⠉⠠⡖⠧⠴⠓⠈⢀⠂⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⡍⠈⠁⠈⠈⠉⠁⠀⠀⡄⠀⠀⠀⢀⠂⠀⠀⠀⠀⠌⠁⠒⠒⠒⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠄⠒⠚⢃⠀⠀⠀⠀⠀⠀⠀⢠⠋⠓⡆⠈⣳⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡌⠀⠀⢆⠡⡀⠀⠀⠀⠀⠀⠈⠒⠚⠉⠚⠁⠀⠀⠀⠀⠀⢹⠳⠢⠴⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠡⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠑⢌⣦⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣴⣽⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠴⠚⠁⠀⠙⢄⡇⠀⠀⠀⠀⠀⠀⢨⡔⠁⠙⠋⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢸⣷⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢲⡄⠀⠈⢿⣷⡀⢻⣇⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣤⡴⠶⠿⣿
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠈⢻⣿⣄⢻⣆⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠀⠀⡀⢂⡀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⣀⣤⣴⣿⡀
⠀⠀⠀⠀⢤⡀⠘⢿⣷⣄⠀⠹⣿⣦⣻⣿⣿⣧⡀⠀⠀⢀⠰⠖⠁⢀⡄⠀⠀⢠⡶⣻⣆⠀⠀⠀⠀⠀⠀⣠⡴⠟⣋⠵⣾⣿⡇
⠀⠀⠀⠀⠈⢻⣦⣈⠻⢿⣷⣦⣈⠛⢿⣿⣿⣿⣿⣆⠠⠎⠀⣴⠀⠀⣠⣀⠀⠀⠈⠉⠉⠐⠀⠠⣄⣴⠞⢉⡴⠚⣡⣾⣿⣿
⠀⠀⠀⢀⠀⠀⠙⠿⣷⣤⣙⠻⢿⣷⣦⣝⣿⣿⣿⣿⠀⠀⣈⣿⠦⠶⣿⠟⠂⠀⠀⠀⠀⠙⠀⠸⠏⢀⡴⠋⣠⣾⣻⣿⣿⣿
⠀⠀⠀⠈⠻⠷⣶⣤⣌⣙⠻⠿⣶⣬⣭⣿⣿⣿⣿⡿⠀⠀⣴⣷⠀⠀⠰⣆⠀⠀⠁⠀⠀⢠⣤⣀⣴⡟⣡⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⡀⠀⠉⠛⠛⠿⠿⠷⠶⠿⠿⣿⣿⣿⣿⡆⠀⣿⣮⠞⠡⣼⠏⠀⠐⠀⠀⠀⢀⣩⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠙⠛⠿⠶⠶⢶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣾⣿⣧⣴⣾⣿⣶⣦⠀⠀⠀⠀⢡⠿⢳⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶⠶⠾⢟⣛⣭⠷⣿⣫⣽⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢰⡍⣶⣿⣽⣿⣿⡟⣹⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠠⢴⠶⠖⠻⠛⣩⡴⢾⣿⣿⣷⡾⣿⣿⣿⡏⣿⣿⣿⢿⣿⣿⣀⣄⢸⣿⣿⣿⣿⣿⡿⣱⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⣉⡥⠾⣫⢴⣿⢿⡔⣿⣿⡏⡁⣼⢿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣊⡵⢊⠵⢮⡾⠋⡈⠀⠀⠘⢻⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⢿⣿⣿⣧⡿⢠⣿⡿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣋⣿⣿⣿⡿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⡙⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣢⣶⣶⣾⠆⣽⡿⢫⡆⣠⣨⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠟⢛⣻⣿⣿⣿⣿⣿⡿⣫⣾⣿⣷⣿⣱⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢟⣉⠁⠒⠚⢛⣩⣽⣟⣻⡛⠟⠉⣶⣫⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⣀⣤⡾⢛⣹⣿⣿⣿⠟⣴⡇⣿⣿⡿⣿⢻⡟
⠀⠀⠀⠀⠀⠸⠁⢀⣤⠾⠛⣡⣴⢟⣿⣿⣿⣿⣿⣿⣧⣿⣿⢰⣏⡼⠁
⠀⠀⠀⠀⠀⠀⡰⠋⠀⢠⡾⢋⣴⣿⢿⣿⢿⣿⡿⣿⢿⣿⡇⣿⣿⠁
⠀⠀⠀⠀⠀⣸⠇⠀⠰⠋⣰⡿⠉⢡⣿⠏⠸⡿⣡⣿⢿⣿⡇⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⢰⡟⠀⠀⣾⣿⡄⠀⢰⣿⢣⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⡏⣿⠀⠀⢸⣿⣿⣿⠈⣿⣿⣯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⢸⠏⣿⠐⢸⡄⢿⣿⣿⠂⣿⣿⣿⣿⣷
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠘⠀⣿⠀⠙⠿⠟⣿⢹⡀⣿⣿⣿⣿⢿⣦⣀⣠⠏
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣀⠀⠐⣿⡈⠑⣿⡌⢿⣿⡀⠙⠟⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣆⠀⠹⣧⡀⢿⣇⠀⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢿⣆⠀⠈⠛⠘⣿⡄⠀⠁⠊⠛⠂⠤⣠⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠘⢿⣦⣀⠀⠈⠉⠉⠉⠉⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠠
⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣤⠀⠀⣜⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣄⠑⠈⠃⠀⡑⠀⢀⠔⠈⠁⠀⠀⣉⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣉⣀⣀⣠⠄⠀⠀⠀⠐⠁⠀⠀⠀⠠⣄⣈⣁⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠞⠋⣡⡄⠀⠘⠛⣋⢲⠂⣙⠛⠃⠀⢠⣌⠙⠻⢄⠁
⠀⠀⠀⢀⠀⢀⣀⣀⣀⣀⣤⣼⣿⣇⠘⢻⣷⣁⠀⡿⠓⠫⠔⠚⢿⠀⣨⣾⡟⠃⣸⣿⣿⣇⣀⣀⣀⣀⡀⠀⡀
⠀⠀⠸⠿⢟⠛⠛⠟⣋⣭⣿⣿⣾⡃⠀⣿⣬⣭⣯⢴⣭⡜⢧⣬⣦⣽⣯⣥⣿⠀⠙⣷⣿⣿⣭⣙⠻⠛⠛⡻⠿⠇
⠀⠀⠟⠋⠉⢀⣉⣠⣤⣤⣌⡛⢟⣁⣾⣿⣧⣤⣠⣠⣦⡜⢧⣴⣄⣀⣤⣼⣿⣷⣈⡻⢛⡡⣤⣤⣤⣉⡀⠉⠙⠻
⠀⠀⠰⠶⠞⣛⣛⣿⡷⠾⠿⢿⣮⣈⠉⠙⠷⣤⣅⡈⣉⡐⠈⢉⣁⣨⣤⠾⠋⠋⢁⣱⡿⠿⠷⢾⣿⣛⣛⠳⠶⠆
⠀⠀⠀⠈⠛⠋⠉⠀⠀⡭⠄⣠⠔⠉⠙⠉⡀⠘⠈⠂⣰⠄⠁⠰⡌⠀⠁⠀⡉⠋⠉⠀⠠⠀⠀⠀⠀⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢛⡝⠁⠀⢠⠢⡏⠀⠲⢀⠸⡜⠂⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⣦⣽⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⠋⠁⠉⣿⣿⣿⣟⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢶⣤⣤⣴⠆⢹⣿⣿⣿⣦⣙⢦⡀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣅⣸⣄⢰⡿⣿⣿⣿⣿⣿⣷⣿⣆⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡠⣄⣀⠤⠤⢤⣀⣾⣷⣯⣟⣿⡾⣧⣿⢭⠿⠿⢿⣿⣿⣿⣷⠊⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣦⠙⢷⡀⠙⣇⠀⢈⣿⣧⡘⢿⡿⠋⠀⣼⣷⣀⣀⣠⡿⠛⢻⡟⠁⠴⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠳⢄⠙⢦⣨⡶⢿⣩⡇⠁⠘⠇⠒⠋⣨⣿⣉⡛⢷⣄⣠⠟⣠⠾⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠺⢷⡾⢻⠀⢸⡇⠀⠹⠒⡆⠀⠉⠀⠀⠈⠁⢸⡿⠃⣰⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠷⢶⡾⢡⣾⣆⣸⡥⣀⣀⣴⣇⣀⡀⣀⣤⣤⣤⠞⢀⡾⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢚⣻⣿⣀⣼⣿⡏⢻⣆⣀⡦⠈⡇⠀⠀⠀⢀⣾⠏⣠⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠛⢻⣿⣿⡿⢿⣷⣼⣧⣀⡈⠳⡗⠀⠒⣻⣿⣧⣴⣿⡟⠎⠛⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣧⠀⢠⡾⠿⠤⠤⠤⢤⣤⣌⣹⣿⣛⠻⠏⣽⡓⣿⠿⢏⣷⠀⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣿⣿⣬⣝⣶⣟⣻⡃⠄⠋⢿⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠒⢻⣿⣿⡟⠛⢿⣿⣿⣿⣿⠟⠉⣀⣤⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣶⣤⣿⡟⠏⢙⣿⣷⣷⡾⠿⠟⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠓⠶⣿⠻⣟⡊⣀⡾⡏⠉⢹⠄⠀⠀⠈⣡⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠛⠲⢶⣿⢾⣹⡟⢻⠀⠀⠀⢸⡆⠀⠀⠉⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠷⣦⣤⣀⡿⠀⠀⠀⢸⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠹⠷⠀⠀⠀⠘⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠘




⣿⣿⡿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠇⢠⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢃⠈⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢷⡀⠈⡀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⣇⠀⠠⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢷⡔⢿⣆⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠻⣌⠻⡆⠣⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⢦⡘⢦⠹⡆⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⡌⠂⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⡊⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡜⠡⠞⢀⠔⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠳⣄⠈⠓⠆⠀⠀⠀⠀⠀⣠⣆⡴⠛⠉⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣾⡟⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠲⢌⣁⠀⠀⠀⠀⠀⠀⢰⠟⠋⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⢀⡄⣼⣿⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣌⠁⣈⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠦⡀⢤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠟⠳⠶⠖⠈⠑⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣞⣁⣀⣤⠤⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠁⣠⣤⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⢀⣠⣤⠾⠛⠉⠉⠉⠙⠙⠛⠻⢿⣿⢿⣦⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⡀⣀⣤⣠⣾⠿⢋⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠂⣀⣴⣿⣿⣿⣿⣿⣿⣿⠟⣿⣟⣯⡥⢶⣟⠛⠻⣿⣿⣿⡓⠀⠀⠀⠀⠀⠀⢀⠹⣦⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡤⠖⠋⢡⢿⣿⣿⣿⡿⠁⠿⢷⣦⣭⣭⣤⣤⣤⣬⣥⡀⠈⠿⠀⠃⠀⠀⠀⠀⠀⠀⢸⣦⣿⣷⡀⠀⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣤⠔⢃⣾⡿⢻⣿⠃⠀⠀⠀⠀⠀⢠⠾⠛⠋⠹⣿⢿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣀⠀⢻⣿⣿⣿⣿⣿⠃
⠀⠀⢲⣄⡀⠀⠀⠀⠰⣤⣀⡈⠀⠉⠙⠃⠈⠛⠁⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡈⢳⠌⠙⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣛⣯⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠹⢿⣷⣶⣤⣀⣈⠙⠿⣿⣶⣶⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣮⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⢀⣼⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣦⣄⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣈⠉⠉⠛⠛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠙⠛⣛⣛⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠟⢿⡿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀
⠀⠀⠀⡜⠙⢻⣿⣿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢦⠀⠀⠘⠸
⠀⠀⢠⡃⠀⡿⢿⠋⠁⠀⠐⠆⠀⠀⠀⠀⠁⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⢃⠀⠀⠀⠉⠀⠀⠀⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠡⡈⠀⠀⠀⠀⠌⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠐⢀⡼⠋
⠀⠀⠀⠀⠀⠘⠀⠀⡀⠤⠤⡤⠆⠀⠀⣄⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠋⠀⢐
⠀⠀⠀⠀⠀⢸⣿⣶⣿⢦⣤⣤⣤⣤⣤⣷⢸⠏⠭⠭⠄⠨⠇⠀⠀⠀⠐⠀⢉⣾⣓⡆⠄⠤⠴⢶⡶⡆⠀⠈⠉⣉⠀⡘⢀⣰⣀⠀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⣿⡿⠿⢿⡸⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢯⢀⡠⠆⠉⠀⠀⠀⠀⡉⠉
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢸⡇⡅⠘⠷⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⣭⣙⠂⢸⠸⣿⠀⠀⠀⠀⣡⠴⠞⠁
⠀⢀⣠⣾⣿⣿⣿⣿⣿⡿⠿⣿⣿⣷⣄⡈⠃⠁⠈⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡉⠝⠁⠀⠁⢁⠑⠀⠀⠂
⠰⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠈⠙⠻⠿⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡀⠔⠁
⠀⠈⠹⡿⢿⣿⣿⠟⠃⠀⢠⡀⠀⠀⠀⠀⢀⠉⠀⠤⠆⠚⠒⠓⠒⠒⠒⠚⠓⠓⠒⠂⠆⡀⢀⡠⠤⢖⡭⠂⠀⠀⣰⡀
⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠔⠊⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢠⢉⣑⣀⠄⡄⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⠦⣄⠀⠀⢀⢀⠳⣄
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠜⠑⢤⡀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠈⡷⠒⠁⣴⠁⠠⠁
⠀⠀⠀⠀⠀⠙⢤⣄⣬⣄⠀⠀⠙⠢⢴⡥⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠴⢋⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠄⠲⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣷⡠⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⢻⡆⠀⠛⢿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣴⣄⡄⠀⠀⠀⢻⣤⣤⣴⣇⡹⣻⠁
⠀⠀⠀⠀⠀⠀⢀⣦⡴⠾⣄⣻⣿⡇⠡⢄⠠⣄⣼⣿⣿⣿⠋⠁⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⣻⣿⡇⠀⣀⣴⣿⣿⣿⡿⠷⠄⠴⠆⢠⣴⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⢲⣿⣟⣿⣿⡿⠁⠀⠠⠀⠀⣾⣿⣿⣿⣷⡀⠀⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⢰⣌⢛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⢀⣼⣿⣿⣿⣿⣿⣇⣰⣿⡇⠀⠀⠐
⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⠇⠀⠄⠚
⠀⠀⠀⢀⣴⣿⣿⣛⡉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠸⣿⣿⣿⣿⠂
⠀⠀⢀⣿⣽⣿⣟⣻⡿⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⢀⣀⠉⠁⠉⠀⠀⠀⡀
⠀⠀⠀⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣭⣤⡁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠿⠿⠟⠛⠿⠿⠿⢿⣿⣿⣿⡇⢀⣀⣠⡖⠈⠙⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠙⢿⣿⣿⡀⠀⠀⠀⠀⠉⠻⠿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⢀⠀⠙⠿⣷⣄⣀⠀⠀⠀⠀⣦⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄⠀⠈⣋⠟⠚⢣⠀⠈⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠐⠀⠹⣧⣤⡼⠗⠂⠀⢠⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠉⠁⠀⠐⠿⠛⠋⠁⠀⠀⠀⢂⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⡾⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿
⠀⢀⣀⣀⠀⠀⠀⡀⣀⣴⣶⣿⣿⡿⠿⠿⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⣸⣿⣿⣶⣶⣾⣷⣿⣄⣛⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣠⠴⠦⣀⢲⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⡁⠚⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣛⣓⠒⠀⠱⡀⠰⠆⠀⠀⢀⡀⠀⠀⠠⠄⠀⠀⠈⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡴⠎⢩⡽⠿⠇⢷⣀⠈⠹⣯⡀⣀⣀⡀⠀⠀⢀⠈⠀⣡⠖⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡐⢤⣴⡏⠀⢠⠎⠀⠀⠁⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠀⠀⣀⣴⡋⣁⠀⠈⠉⠑⠢⡀
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡁⠀⠛⠀⠀⠀⠀⠀⢀⡛⠂⠀⠀⠀⠠⢀⣤⣾⢤⠿⠛⠛⠉⠉⠁⠉⠘⠦⣄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢼⣧⠀⠘⢧⣄⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠈⠁⠐⢈⣂⠴⠆⠐⠲⠀⠀⠀⢸⠁⠀⠀⢹
⠀⠀⠀⠀⠀⢸⣿⣦⡀⠀⠙⠴⣄⠐⠀⠠⠀⠀⠀⣒⠂⠈⢀⠠⠂⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠱⣈⠙⢦⣀⠀⠈⠑⠲⠶⠤⠐⠚⠁⠀⠀⠀⠠⠆⠀⠀⠀⡠⠖⠊⠲⠶⢆⡀⡤⠀⠟⠀⠀⠀⣼
⠀⠀⠀⠈⠀⠁⠀⠸⡌⠚⠺⣶⠤⣀⡀⠀⠀⠀⠀⠀⢀⠐⠛⠃⠨⠀⠀⡞⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠠⠤⠂⠀⠀⠙⠄⠀⠏⠀⢀⠀⠁⢈⡉⢐⠄⠀⠀⠤⠐⠶⠀⠸⡇⡀⠀⠀⠀⠠⠀⠀⠀⢀⣠⠀⠈⠅
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢚⠃⠀⠀⠀⠐⣤⣿⣦⡄⠀⠀⠈⠀⠀⠀⠸⠇⢁⡀⠀⢠⠴⠆⠀⠈⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⡬⠥⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠂

⠀⠀⠀⠀⠀⢀⣠⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣔⣒⣓⠂
⠀⠀⠀⢤⣾⠿⠿⣿⣿⢿⣦⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢲⣿⠿⠿⠛⠛⢏⡉⠉⠉⠉⠙⠉⠓⠣⢐⠀⠀⠀⠂⠀⡠⠠
⠀⠀⠀⠀⠁⠀⠀⠈⠉⠛⠿⣟⢾⣷⣾⣷⣾⡿⠛⠉⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣫⣿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⣰⡶⢾⣯⡉⣧⣝⠉⣛⣿⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡄⢸⡿⢁⠻⣽⢋⠉⡙⢟⡿⣯⣽⣶⣦⡀⠀⠀⠀⠀⠀⢠⠏⠀⣿⣠⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢳⠀⢰⣿⣿⢸⣄⣐⣢⡎⡐⠠⠈⡟⣴⠋⠀⠀⠀⠀⠀⠀⠀⠺⣿⢿⣿⣟⡿⠁
⠀⠀⠀⠀⠰⣄⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣽⡇⢸⣛⡫⢚⣉⡉⣉⣉⡪⣭⣴⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡟⠁
⠀⠀⠀⢆⠀⣸⠟⠀⠘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⠛⠋⠽⠟⢹⣯⣽⠟⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠤⡶⠒⢶⣂⢶⣾⣶⣿⡷⢶⣒⣂⣶
⠀⠀⣀⠈⣹⣿⡆⠐⣄⡇⡰⠀⠀⢀⣴⣦⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠈⠀⠀⣼⠟⠙⣿⣷
⠀⠀⠘⣄⠀⠙⣿⣴⣯⠞⠁⠀⢀⣾⣿⣻⣿⣿⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣎⢀⣀⠀⠀⠀⠀⠀⠀⢸⠟⣠⣶⣿⢿⣇
⠀⡀⠀⠈⣷⣠⣿⣿⠏⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡷⠀⠀⠀⠀⠀⢀⢄⠙⢻⣟⢻⣿⣿⢿⣿⣦⡀⠀⠀⠀⡼⠖⠋⡇⠻⡌⣿⡀
⠀⠀⠀⠐⠘⣏⣿⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠁⠀⠤⡴⠂⠀⠀⠀⣌⣞⣀⠘⢧⢀⣸⡿⠚⢿⣽⣷⠀⠀⠀⠁⠀⠀⡇⠀⠀⠁⠁
⠀⠀⠀⠀⠠⢻⠉⠹⠇⠀⠀⠘⠿⣿⣿⣿⣿⣿⡿⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⣷⣤⣿⣿⣧⣶⣾⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠛⠷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢬⢉⠙⢋⣭⣵⣿⣿⠃⠀⠀⠀⠀⣀⣠⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠀⠉⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠀⠁⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⠃⠀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠄⠀⠀⢀⣠⠆⡃⣠⣀⢰⠀⠀⢈⠴⠁⢁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠃⠀⢰⣚⣩⠅⠀⠀⠈⠀⠄⠐⠂⠐⠁⠀⠀⠐⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠠⡤⢀⡉⠀⠒⠁⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰⢀⠀⠀⠒⣢⠊⠁⠀⠈⠑⠠⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠰⠀⠐⠀⠀⠀⠀⠀⠘⠃⡖⠠⢄⡀⢀⡀⢀⣄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢩⡀⠈⢯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣃⠀⢦⣄⠀⠀⡈⠀⣄⠀⠀⠀⠀⠀⠀⠸⠂⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⠛⠷⣄⡀⠀⠀⠧⡄⠤⣽⡿⠂⢧⣀⠘⠀⠀⠀⠀⠰⠍⠀⠤⠖⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣀⣀⣐⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣫⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣽⣽⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⣀⢄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣂⡎⠀⠀⠑⠍⠁⠇⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠈⠁⠀⠉⠐⢤⠀⠀⠐⠀⠀⠀⠀⢀⣆⣀⣀⡔⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢪⢇⡠⡤⠔⢧⢀⡀⠀⠀⠀⠀⠀⠘⠃⠛⠠⢊⠤⢠⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠄⠒⢉⣉⡍⢃⠐⡬⠀⠀⠠⠤⠤⠤⠀⣶⣥⠀⡸⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣎⣵⠟⠉⣖⣊⣀⢀⣀⡀⣀⠀⣀⠴⠈⠑⠒⠒⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠌⠀⢰⣠⠋⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⡀⠀⡐⠉⠉⣴⠄⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⡆⠔⠉⠈⡦⠤⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠑⢄⠀⠀⠀⠈⠉⢀⡀⠔⠃⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠠⠤⠊⠀⠀⠊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠁⠐⠒⠂⠈⣠⡀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⠃⠀⠀⢀⡠⠒⠒⠒⠒⠪⠁⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⡀⠤⠒⠒⠂⠀⠒⠥⣀⠀⡇⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡕
⠀⠀⠀⣀⠠⠴⠃⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⢔⡡⠔⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⣔⠁
⠀⠰⢅⣀⡀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣅⣀⣀⡠⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣉⡲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣿⣿⣷⣶⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣄⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠻⠿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⣠⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠶⢶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⢹⣿⣿⣷⡀⢀⣴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣤⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠙⠛⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⢿⣿⣿⢿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉














⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⠙⣀⡉⣿⡁⢀⣿⠉⣉⣯⡙⢃⣯⡿⢋⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⣦⢀⣀
⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠛⠀⠪⣡⡦
⠀⠀⠴⠚⣹⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣷⣩⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⠦⢤⠄⠈⠀⣦⣀⠈⠲⣌⣠⠖⠂⠀⢀⣀⣀⠀⠀⠤⠐⡀⠝⠶⠦
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠀⠆⠀⠢⠀⠀⠀⠀⠀⠀⢸⡿⣷⢖⣶⣷⣆⠀⠙⠉⢀⡀⢹⡅⠀⢀⠈⠀⠀⠈⠓⢀⡔⠁
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⡅⡀⡾⢟⣿⠃⠀⠀⠀⠳⣄⡁⣆⠀⠀⠀⠀⠀⠀⠈⢣
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠓⠳⠏⠠⣄⠀⠀⣀⠀⠈⠈⣿⠛⠂⠀⠀⠀⠀⠡⠀⠀⡇
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠖⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⡠
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡠⢐⡶⢄⣀⠀⣀⡀⠀⢠⣄⣀⡤⠔⠋⣸⡎
⠀⠀⠺⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣀⣀⢊⡸⠗⠁⠀⠀⠀⠡⠋⠀⣴⠟⡹⠁⠀⠀⣠⡡⠃
⠀⠀⠀⠻⣿⣿⡀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⣿⣿⣇⠀⠀⠀⠀⢯⣌⡰⠁⠀⠀⠀⠀⠀⡀⠀⠔⠉⠉⠙⠃⢀⡤⠊⠋
⠀⠀⠀⠀⠙⢿⣇⢀⠙⢿⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡛⡡⠊⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠰⠿⠿⠗⠀⢙⣿⣿⣶⠆⠀⠀⠀⠀⠀⠀⠘⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣠⣤⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠼⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠉⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠁⠀⠀⠀⠀⠀⠠⢬⣻⣦⣝⢶⣶⣶⡖⠚⠉⠁⠈⠉⣉⡓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠊⢋⠘⠛⣉⡄⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⡋⠀⣀⣠⢂⡤⠂⠀⠀⠀⠨⠙⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠠⣻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠕⠪⠙⠎⠙⠁⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣒⡠⠼⣯⣳⣄
⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠁⠀⠃⣌⢆⠄⠀⠀⠀⢝⣿⡟⡖⢀⣀⠀⠀⠀⠢⠈⠀⠀⠀⠀⣰⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣦⣽⣿⣆
⠀⠀⢀⡴⠚⠉⡀⣀⣠⣢⢠⣮⠀⠏⠀⠀⠀⠠⣒⠌⠻⢷⣷⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⠄⠀⠀⠀⠀⠒⢒⣂⣀⣉⣲⣿⣿⡇⢻
⠀⢠⠟⠀⢰⣿⣾⢟⡿⣡⣿⣿⡤⠀⠀⠀⠀⢀⡂⢍⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⣐⣺⣿⣿⣿⣿⡵⠒⠂⠀⠀⠀⠀⠒⠬⣍⣛⡿⣿⣿⠈
⢠⡏⠀⠀⠀⠈⣱⢟⣼⣿⣿⣿⡇⠀⠀⠀⠐⠂⠈⡓⢽⣦⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣿⡿⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣧
⣼⠀⠀⠀⠀⠈⠀⣸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⢧⡹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⣉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷
⣏⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠠⣑⠝⢮⣻⣿⡆⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟
⣿⠀⠀⠀⠀⠀⠀⡞⣽⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⡁⣲⣄⡠⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣯⠆⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⡇⠀⠀⠀⠀⠀⢀⠜⣽⣿⣿⣿⡏⡄⠀⠀⠀⠀⠀⣈⡙⠻⢝⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⢋⠴⠒⠛⠋⠑⠒⠀⠀⠀⠀⠀⢞⣿⡿
⡇⠀⠀⠀⠀⠀⠀⠘⡱⣿⣿⣿⡏⡀⠀⠀⠀⠀⠀⠀⠍⣓⢤⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣯⣴⣶⠄⠀⠀⠀⠈⠉⠉⠀⠀⠀⣸⣿⠇
⣷⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡅⠁⠀⠀⠀⠀⠐⠢⡉⠊⠙⢿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠋⣨⠿⣻⣿⣿⣿⣥⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⣿
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠆⠀⠀⠀⠀⠀⠀⠠⠅⠈⢺⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣯⠭⠥⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠠⠀⠀⠀⠀⠀⠨⠥⢤⡀⢲⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣉⣹⣿⣿⣿⣟⠻⠦⠀⠀⠀⠀⠀⢀⣴⡟⣿
⠀⠸⣆⠀⠀⠀⠀⠀⠀⠈⢋⣿⣿⡅⡀⠀⠀⠀⠀⠈⠁⠀⢴⣶⣿⣿⣿⣿⣿⣶⣤⣖⣤⣶⠟⠋⣛⣿⣿⣿⣿⠭⠥⠤⠀⠀⢀⢠⣴⡿⠟⢹⡄
⠀⠀⠻⣷⣆⡒⠀⠀⠀⠐⣛⣫⣿⣷⡁⠀⠀⠀⠀⢐⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣽⣿⣯⣭⣾⣿⣿⣿⣤⣤⣤⣤⣶⣿⡟⢁⠙⣆⣧
⠀⠀⠀⠈⠙⠿⣷⡒⠀⠉⢭⣿⣿⣿⣿⣶⣤⣀⣀⣠⣼⡿⢟⣻⠿⠛⠛⠉⠉⠉⠀⠀⠈⠉⠛⠅⠀⠀⠉⠉⠙⣿⣿⣿⡻⠛⠋⠉⠀⠀⠱⡜⣿
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢀⣈⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢬⣝⢯⡉⠢⡀⠀⠈⢆⠀⢹⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢮⣝⡛⠿⣿⣿⣿⠋⠀⢠⠋⠐⠀⠐⠀⠙⠏⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢻⡄⠱⣦⣵⢸⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠾⢿⣏⡄⢀⡏⠀⠀⠐⠂⠄⠒⡀⠀⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢹⣿⣾⢇⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⣸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡟⠀⠀⢒⠦⣴⣤⡀⠀⠀⢰⡏⡰⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⣿⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠀⣀⣀⠉⣻⣿⣦⣄⢀⡾⣰⣷⢢⢴⢠⢀⡄⡀⢀⣼⠇⠀⠀⠀⠀⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣬⣍⣙⣛⡛⠛⠛⠋⠚⠟⠱⠫⠯⠏⠞⣴⡟⣼⣿⡇⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠶⠶⠦⣤⣥⣽⣾⡷⠤⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⢀⢈⠓⠀⠓⠛⣤⠀⠀⠀⠈⠁⣶⣶⣲⣤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⡈⠀⠈⠀⠀⠀⣦⠀⠉⠈⠉⠀⠈⠁⠀⠐⠀⢠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣅⠠⠀⠀⠀⠀⣿⢠⢠⣴⡄⠀⠀⠀⠀⠀⠀⠘⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣿⠈⠛⠛⠀⠀⠀⠀⠀⠡⣤⢖⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡆⠀⠀⠷⠀⠄⣀⣠⣦⢿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠈⣷⡋⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⣼⡇⠃⠀⠀⠀⠾⠿⠿⠿⣼⠛⠠⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⡀⠸⠂⢙⢶⣿⣿⣷⣶⣶⣦⣤⣤⡄⠀⣿⣏⠀⢠⣤⢀⠀⠀⠀⠀⠈⠋⠳⣶⠀⠀⢠⠿
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣾⣿⠠⠀⠷⣄⣀⣿⣿⣿⣿⠋⣈⣳⠀⠀⠀⠀⢸⠋⠀⠸⡟⠉⠉⣿⣷⡤⠀⣀⠀⠻⠸⠇⠈⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠐⡤⣶⣿⣾⣿⡀⠄⣠⣬⣿⣿⣿⣿⣿⣦⡌⠁⠀⢀⡶⠆⣑⢤⠖⠀⠀⠀⢀⢿⣿⣷⡀⣹⣆
⠀⠀⠀⠀⠠⠄⠬⢥⡼⠅⣤⣭⣭⣿⣿⣿⣿⣿⣿⣿⣣⡀⠀⠒⠀⢨⢦⣰⡜⠪⣅⠀⡀⢀⣆⠘⠉⢿⣿⣇⡀⠛⠱⡄
⠀⠀⠀⢠⣆⣒⣾⣲⣗⣣⣘⠻⣿⣿⣿⡟⣻⣿⣿⣿⡏⠈⠒⠚⠋⠀⠀⠉⢻⣀⠈⠙⠑⢶⣇⠀⢨⣄⢹⣷⣿⣤⣈⠏
⠀⠀⠀⠸⣿⡬⡿⢿⣭⣯⣭⣷⡜⠛⡉⣾⡀⠁⠈⠩⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢦⣾⣿⣿⣶⡿⠻⣆⠉⠁
⠀⠀⠀⠀⣿⣿⣭⣭⣭⣽⣿⣯⣦⠁⠀⢿⣿⣶⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⣿⣿⣷⢤⡈⠣⡀
⠀⠀⠀⠀⣿⣿⣷⡖⠒⢒⠒⣲⣦⣶⣶⣿⣿⢻⠛⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣟⣃⠀⣿⣿⣿⠀⢳⠀⠠
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⣿⣿⣷⣿⣭⡍⢹⣷⣄⣿⢸⠇⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⢘⡟⢁⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠄⠐⠒⢷⢶⣿⣸⣗⡢⣤⣀⡀⠀⢸⡟⣿⠀⠀⠀⢹⣿⣿⣿⠀⠀⠈⠉⠛⠋
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣭⣭⣭⣭⣍⣭⣍⣩⣉⣴⣄⡏⣿⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣷⣿⡽⣤⣤⢀⠸⡇⣯
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠴⠤⣷⣿⠶⠦⣄
⠀⠀⠀⠀⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⠖⠲⠶⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠒⠒⠛⠛⠛⠒⠒⢲⠀⠀⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠛⠛⠉⠙⠋⠛⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠑⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣙⣛⣛⣛⣛⣛⣛⣛⣛⣉⣛⣛⣛⣛⣙⣋⣉⣉⣉⣘⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢈⠈⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣧⠀⢸⣦⡀⢀⣾⢂⣼⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣰⣿⣿⣯⣿⣿⣿⣿⣿⣦⠸⢿⣷⣿⡋⢺⣿⣿⣿⣿⣿⣿⣿⠿⠷
⠀⠀⠀⠈⠉⠉⠙⠻⡿⠿⢿⣿⣿⣿⢇⠉⠛⠋⢡⣿⡿⣿⣿⡿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⣿⣷⣶⣶⠟⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣟⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⠇
⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⣼⢧⡀⢀⣠⣾⣿⣿⣿⡿⠛⠋⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢘⣿⡀⠈⢃
⠀⠀⠀⠛⠛⠿⢿⣿⣿⣯⣀⡄⡸⠃⠘⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣧⠀⣄⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠀⠀⠰⣩⡿⠛⢀⠀⠀⠀⠀⠀⠀⢸⠿⠿⡟⠋⢻⣿⡿⣿⣷⣹⣿⣾⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠛⠿⡏⠰⠍⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠾⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⣰⣲⡴⢦⣀⡀⠀⠀⠀⢀⡤⠖⠛⠛⠋⠉⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠾⠥⣿⣀⠀⠀⠀⠀⣗⣻⣶⠛⣣⢛⡵⡄⣰⣏⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⡄⠀⠉⠳⡄⠀⠀⠙⠶⣿⡀⠀⢉⡺⢷⡟⠸⣾⡇⢦⡀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡆⢸⡇⠀⠀⠀⢹⣀⠀⣠⠾⢅⠉⣷⢅⡀⢸⣁⡀⠈⠁⠀⢹⣠⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡇⠈⠀⠐⠞⠀⠀⣸⡶⠃⠀⢀⡽⠁⠀⠙⢺⠸⠀⠀⠀⠀⠙⢿⡴⠲⡄⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡴⠾⠇⠀⠀⠀⠀⡴⠋⢨⠇⠀⣠⠞⠀⣠⣠⣤⡙⣄⣀⡀⠀⠀⠀⠘⠀⣠⠃⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠣⣄⡀⠀⠀⠀⠀⠀⣠⠎⣠⣾⠷⣶⢮⡇⠀⠀⢱⠙⢆⡉⠀⠀⢀⣀⣤⠽⣴⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠶⠲⣯⣤⣤⣤⣄⣤⣞⣥⣞⣁⡇⠀⠀⢨⡏⠙⠛⠛⠛⠚⠛⠛⠋⠉⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠓⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣼⣇⣀⣀⣀⣀⣀⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢀⢤⡤⠤⠤⠶⠖⠒⠛⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⠉⠉⠙⠂⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠹⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠳⡄⠀⠀⠀⠀⠀⠀⢀⡴⠛
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠋⠁⠀⠀⠀⠹⡄⢀⣀⣠⠴⠚⠉⠀⢰
⠀⠀⠀⠀⠀⠀⠸⣦⠤⠤⣴⠚⠋⠙⠒⠒⠒⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣀⣀⡤⠴⠒⡻⠋⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢰⠋⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⣰⠃⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⢀⠼⠁⠀⠀⢠⡇⠀⠙⠦⡀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡆⠀⠀⠀⡿⠀⠀⠀⡇⠀⠀⠀⢰⠋
⠀⠀⠀⠀⣠⠖⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⡄⠀⠀⠘⡇⠀⠀⠀⠀⠀⢿⢞⠯⣝⢧⡀⠀⠀⡇⠀⠀⠀⢳⠀⠀⠀⠈⢧
⠀⢀⣤⡞⠁⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⠫⣗⣤⡈⠉⠉⡟⠁⠀⠀⠀⠘⡇⠀⠀⠀⠈⢧
⣴⡁⠀⢻⣦⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣀⣴⠒⢦⠀⠀⠀⠀⠉⠲⢵⣴⠃⠀⠀⠀⠀⠀⢻⠀⠀⢀⣀⠼⣇
⠙⣷⡄⢸⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣠⣾⡥⠞⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⢉⡽⠁⠀⣨⣧
⠀⠘⠿⣮⣿⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⢀⣠⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣊⣷⣚⣡⠴⣒⡭⠖⠃
⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠛⠛⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣠⣼⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⣶⣶⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢸⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠇⠀⢸⣿⠉⠉⠉⠙⠛⠛⢿⣦⣄⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣦⡾⠁⠀⣿⡿⠟⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣿⠁⠀⠀⠈⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣥⣤⣤⣤⣤⣤⡀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠈⡂⢠⣤⠶⠂⠀⠀⠀⠀⠀⠀⠈⠁⠒⠛⠶⣨⡷⠀⠀⠀⣾
⠀⠀⢠⡆⠀⠀⠀⠀⢀⣤⡾⢟⣃⢀⣠⠍⠛⠀⠀⠀⠀⠀⠀⠀⠐⠶⣦⢴⣄⠾⠚⠋⣅⣀⣠⠼⠋
⠀⠀⠀⠃⠀⠀⢠⡴⠟⡽⢪⠟⢊⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⢄⡘⠛⠙⢦⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣾⢃⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡈⠙⠻⣿⣗⠹⣆
⠀⠀⠀⢀⣾⠛⠄⠀⠀⠀⠀⠹⣇⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⠀⠀⢸⠶⣄⠀⠙⠷⣘⢦⣀
⠀⠀⠀⣼⣷⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣼⠟⡟⠉⠙⠻⣤⠀⡄⠈⢣⡀⠀⠈⡀⠉⠛⠒⠂⠀⡠⠔⠁
⠀⠀⠀⣿⠿⢿⣀⣷⣄⠀⠀⠑⠈⠀⠀⣄⡀⠀⠀⠀⠀⢐⠀⣿⣡⣧⡤⠤⠔⢿⣶⠟⠀⠘⣿⠉⣸⠻⠦⡀⠁⠀⠁
⠀⠀⠀⣿⠀⣸⣿⢻⣌⠳⠀⠀⠀⠆⠀⠸⣧⠀⠁⠀⠀⢸⠀⣿⠉⠱⡄⠀⠀⠘⠈⢷⡀⠀⢼⣰⠏⠀⠀⠀⠀⠀⠀⣠⣤⢤⣀
⠀⠀⠀⣿⠀⣿⣿⡄⠹⣧⠀⠀⡜⠀⠀⠀⠘⣧⠀⠠⠀⢸⠀⠸⣿⠀⠀⠀⠀⠀⠈⠳⢓⠄⠜⠁⠀⠀⠈⠧⠀⠉⠉⠀⠀⠀⠉⠓
⠀⠀⠀⢹⣿⣿⣿⣷⠀⠙⠷⢀⠠⠀⠀⠀⠀⠙⢷⡀⠀⠸⠀⠀⠈⢦⣿⠀⠶⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⣠⡀⠰⠂⣀⡶⢤⡀
⠀⠀⠀⢸⣿⣿⣿⣛⣠⡴⠞⠿⢄⡀⠀⠀⠀⠀⠘⣧⠀⠀⡇⢸⣤⡀⠹⢦⣀⠀⠀⠀⠨⠦⠤⠀⠀⠀⠀⠀⣠⠀⠉⠀⠀⠀⠈⠀⣠
⠀⠀⠀⢸⣯⡤⠿⡟⠁⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⢹⡁⠀⣧⠀⠛⢻⣦⣄⠀⠙⠛⠃⠤⢤⣀⠀⠀⠰⠃⣞⠁⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⡌⠁⠀⠈⠁⣀⡐⠦⠙⠃⠀⠀⠉⢢⣤⠀⢨⡇⠀⢹⡄⢸⡄⠙⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠀⠈⠉⠐⠒⠊⠁⠀⡞⠁
⠀⠀⠀⢸⠉⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣾⣇⠀⢰⣷⣄⠹⡆⠈⠀⢤⣀⣀⠀⠀⠀⠀⠋⢰⠞⠃⠠⡤⢤⣤⠤⠊
⠀⠀⠀⠘⡀⠀⠈⠁⠀⠀⠀⢀⣀⠤⠐⠂⠈⣉⣀⣿⠈⠀⣸⣿⠙⢦⡈⠀⡀⠀⠀⠈⠁⠂⡄⠀⠀⠀⠂
⠀⠀⠀⠀⢣⣀⣤⣷⡶⠚⠛⠁⡄⠀⠀⠀⢸⣧⢻⣿⢀⡴⠟⠛⢉⠉⠙⠛⠉⠓⢦⡁⠀⠐⢦⣀⣠⡴⠃⠐⢤⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢹⣷⡿⠁⠀⠀⢸⡇⡀⠀⢀⣾⠏⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⣻⡀⠀⠀⠽⠷⠤⣤⣄⣀⠙⢦
⠀⠀⠀⠀⠀⢸⣟⢡⠀⠀⠀⠉⢰⡇⢀⠞⢁⣸⠿⣿⡀⡤⠶⠐⠒⢦⣀⠀⠀⠀⠁⣀⣇⠀⠀⠀⠀⠀⠀⠉⢮⣅⠀⠻⠆⠤
⠀⠀⠀⠀⠀⢸⡇⠸⠀⠄⠀⢠⡿⠀⠈⠀⢨⠿⣀⣿⣷⣄⠀⠀⠀⠀⠙⢆⠀⠀⠀⠘⡏⠁⠀⠀⠀⠒⠂⠀⠀⠈⠱⡆
⠀⠀⠀⠀⠀⠸⠇⡆⠀⠀⢀⣿⡿⠀⠀⠰⣿⣤⣼⢃⣿⠿⣦⡀⠀⠀⠀⠘⠆⠀⠀⢀⣧⣄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⡟⠃⠀⣰⣿⣟⡄⢀⡄⣴⢫⣿⣿⣿⠫⠖⠻⠇⣀⣀⣀⠂⠀⠀⠀⢈⣙⣿⣿⢦⣀⠀⠀⠀⠀⠀⠀⠀⡀⢹
⠀⠀⠀⠀⠀⢀⣿⣤⡶⢳⣿⡾⢠⡟⣼⢃⡾⠟⠉⠀⢀⣠⠤⠤⠒⠒⠒⠂⠀⠐⠒⠀⠻⠿⣿⣯⡿⠷⣤⡀⠀⠀⠀⠀⠹⡌⡄
⠀⠀⠀⠀⠀⢸⠏⠋⢀⣿⣿⣣⣯⣾⠟⠉⠀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠛⣦⠈⢿⡦⠀⠀⢀⠀⢻⠧
⠀⠀⠀⠀⠀⣾⠀⠀⢸⣿⣷⢿⡿⠁⠀⣠⠖⢶⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠤⠀⠐⠚⠋⠁⠀⠀⢻⠀⠈⠻⣷⣄⠈⠀⠈⢷⣧⡀
⠀⠀⠀⠀⣼⡇⠀⢠⣿⠃⠉⡹⠀⢀⡴⢏⠠⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⢀⠉⢤⣀⠀⠀⠈⠻
⠀⠀⠀⣼⡿⠀⠀⠂⣻⠀⡄⠀⢠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠘⣤⡀⠻⢦⡀
⠀⠀⣰⣿⠇⠀⡄⢀⡿⡜⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠀⠀⣄⠀⢸⠉⠣⢼⣿⡀
⠀⢠⣿⠃⢀⠒⢀⣾⠃⠁⢠⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⡇⠀⠀⠹⡀⡌⠀⠀⢀⠈⢻⣄
⢠⣿⠃⣠⡎⠀⢸⡏⠀⢆⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠇⡇⠀⠀⠀⠣⣀⠙
⡿⠃⢰⡟⠀⣠⡜⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠉⢠⣾⠆⡿⣿⠳⡄⠀⠀⡶⡇⠀⠀⠀⠀⠀⠙
⠁⢠⡞⠀⠀⣿⠁⢀⣦⣜⡛⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⢨⡏⡸⢠⡄⠐⢷⡄⢠⠃
⠀⠎⠡⠀⣼⡏⢀⣿⢻⣿⠋⣿⠃⠱⣄⠀⠀⠀⠀⠀⠀⣀⢠⡖⠉⠀⠀⠀⠀⠀⠀⠀⣸⠀⠿⠘⡇⠀⠀⢻⣆⠀⠐⡀
⢰⠄⠀⠰⣿⢃⣾⠇⣶⡿⠣⡘⢧⡀⠈⢣⣀⣤⠶⠚⠛⠁⠀⠰⣦⣄⡀⠀⠀⠀⠀⢠⡏⠀⢻⡀⢇⠀⠀⠀⠙⢷⡀⠈⠣
⠃⠀⠀⢹⡟⣸⡇⢸⠠⠇⠀⠈⠙⠻⠯⠼⠿⠶⠒⠒⠒⠈⠁⠀⠈⠉⠉⠀⠐⠒⠛⢻⡇⠀⠈⣷⣜⠆⠀⠀⠀⠈⢿⠀⠀⠀⠀⢀
⠀⠀⠀⣿⢣⡿⠀⡏⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢏⣷⠀⠀⠈⢧⠑⠡⡀⠀⠀⠸⡆⠀⠀⠀⠀⠐
⠀⠀⢸⡏⣸⠃⣰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠄⣾⠀⢻⣆⠀⠀⠀⠀⠠⠈⠀⠀⠀⢿
⠀⠀⣼⠡⡟⠀⠉⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⡷⠀⠀⠉⢦⡀⠀⠀⠀⠀⢤⡀⠀⢸⡄
⠀⢠⡏⠀⠀⣴⣤⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠈⣷⠌⠂⠄⠀⠀⢳⣤⣌⠁
⠀⣾⠀⠀⢸⣏⡗⠀⡖⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠛⡤⠀⠀⠀⢀⣀⣽⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⣤⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣆⠀⠀⠀⠀⢰⣇⡴⠃⠸⣷⠞⢳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡞⠀⠀⠀⠘⢧⠀⠀⠀⣸⠏⠀⠀⠀⠈⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣰⡌⣧⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⠤⠤⠤⣤⡀⠀⠀⠀⠀⢀⠔⠁⠀⢸⣧⡟⠀⠀⠀⠀⠀⣠⠤⠤⠖⠚⠋⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠑⢤⠐⠈⠀⠀⠋⢠⠀⢹⣇⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣰⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢣⠀⠀⠘⠀⢀⡰⠀⠀⠀⠹⣆⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣉⠓⢾⣇⠀⠀⠀⠊⠑⠀⠈⠰⠆⠻⣦⡿⠀⠀⠀⠀⠀⠀⠣⡄⠀⢀⡴⠚⠁⠀⢀⣴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣾⣤⣄⠀⠙⢷⣗⠀⠀⠄⠈⢀⠀⠀⢀⣽⠇⠀⠀⢸⠀⠀⠀⠀⣧⠀⡎⠀⢀⣴⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠴⠷⠴⣿⠀⠀⠀⠉⢷⡄⠄⠀⠈⠀⣠⡾⠁⠀⠀⠀⠘⠳⢦⠀⠀⢸⡞⣠⣴⡿⢿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠖⠚⠋⠀⠀⠀⠀⠈⢿⠀⠀⢀⡾⠋⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠉⠀⣀⣀⡷⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣾⠃⠀⠈⠻⣷⡀⢀⣤⣤⡄⠀⣸⣧⠴⠚⠉⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⠀⠀⠀⠀⠸⠟⠁⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠛⢤⣀⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⣴⡿⠿⠛⠿⠛⢛⣷⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠦⢽⣦⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣉⣠⣂⡀⢀⡰⠉⠀⠀⣀⣨⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⢸⣿⣏⠉⠘⠁⠉⠀⠰⠋⠀⣠⣶⣤⣿⣽⣇⠘⣟⠶⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠙⢷⣄⠀⢿⣟⣟⡁⠂⠀⠈⠀⠀⢈⡻⣋⣽⣿⣭⣿⡇⠹⠆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣬⣧⣠⣤⠀⠈⠻⣧⡈⢿⣿⣧⠈⡋⠀⠀⠀⠈⢿⣿⣭⣋⣭⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣹⣿⣋⣽⣧⠀⠀⠘⢿⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣛⠚⣫⣿⡿⠀⠀⠀⠀⠹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣷⣦⣤⣤⣤⣴⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣶⣶⣶⣶⣦⣤⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠄⠽⠀⠀⠀⠀⠀⢀⠒⠩⠠⢂⡌⠑⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡐⢦⢁⣀⠈⣑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢄⢁⡀⠀⠀⠀⠀⠀⢀⣼⢘⡷⢡⠮⡠⠀⡀⠀⠀⠀⠀⠀⠀⠄⡠⠊⠈⠁⣀⣀⢙⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⣙⠀⠀⣀⣤⣿⠿⢾⠟⣠⣉⣈⣠⠔⠁⠀⠀⠀⠀⠠⢂⠔⠀⠀⠀⠘⣉⣯⢩⣧⠀⢁
⠀⠀⠀⠀⠀⠀⠀⡰⢨⢶⣄⠀⠉⢜⡉⣁⠠⠦⠈⠦⣄⠁⠄⠀⠀⠀⣠⡴⢂⣉⠖⠠⠀⠀⠀⠀⠀⠢⠀⢨⣿⡇⠀⢼⡀
⠀⠀⠀⠀⠀⠀⢀⣃⡀⢹⣻⠷⢄⣎⡀⢬⠔⠻⣅⠶⠋⠀⠀⠀⠀⠀⠃⠀⠀⠀⠄⠀⠠⠤⢤⡤⠤⠀⠛⠛⠙⠂⠀⠀⠹⡀
⠀⠀⠀⠀⠀⢰⢫⠛⠈⠐⠛⠁⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⡀⠀⠀⢠⠔⠂⠀⠤⠤⠤⠠⣤⣤⣤⣀⠀⠀⣙⣦⡀
⠀⠀⠀⠀⠀⢸⡶⣾⠀⢸⠀⢈⣱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⠶⣾⣡⠄⠍⢦
⠀⠀⠀⠀⠀⢸⢃⣿⠐⡄⠀⠈⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠞⠋⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢸⠨⠟⠄⠈⠢⣄⡀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⡋⠑⠂⠤⠤⣴⢂
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠚⠀⠀⠀⢹⡿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢻⣷⣤⣴⣿⠞⢋⡇
⠀⠀⠀⠀⠀⠀⢰⠿⠻⡽⢻⠛⠀⠀⡇⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣀⠜⠋⡛⢯⣙⡀⢉⡭⠑⠦⣄
⠀⠀⠀⠀⠀⣠⠋⡀⠂⠀⠀⠑⠤⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢀⠞⢳⡰⠞⠀⠀⠀⠈⠉⠑⠒⠲⠬⣵⠠⢤
⠀⠀⣀⡀⠰⡁⠀⠀⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢚⣐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠈⠉⠒⠲⢮⠤⠀⠑⠲⠶⠤⣤⣄⣀⣇⣈⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⠥⣤⠶⡶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣁⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⠰⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⢀⣠⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠉⠉⠁⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣇⢘⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠏⠈⠻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⠀⠀⠀⠹⣿⣿⣿⣿⣿⣆⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄⠀⣰⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠘⢿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠈⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆
⠀⠀⠀⣴⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷
⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁

⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⢂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⠀⠀⠀⢀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⣽⣶⣿⣯⣵⣬⣥⡴⣬⣴⣟⣉⣀⣤⣥⣤⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⠿⣿⣿⣿⣿⣥⣴⢾⣿⣿⣛⣻⠟⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⣉⣿⡷⣌⡹⢿⣿⣿⣧⣿⣍⠀⣤⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⠾⠿⠿⡿⠛⢻⣿⣿⣍⠁⣈⣄⣀⣤⡴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⠀⢰⠆⢠⣿⣿⣿⠿⠻⣶⡄⠉⠑⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⢹⣿⣿⣿⣶⣴⣿⠉⣿⣿⣄⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⣶⣿⣿⣿⠀⠀⡛⠁⣸⣧⣤⣙⣃⡀⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣻⣿⣷⣤⣧⣾⣿⣿⣿⡿⢻⣿⠋⠁⠀⠀⠀⢀⡠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣋⠩⠻⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⠀⠀⢀⣤⢞⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⠇⣠⣾⣟⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣴⣾⣿⣾⣿⣿⣿⣿⣿⣤⣴⡿⠞⣣⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠙⢿⣿⠻⣿⣿⣿⠿⠟⢙⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⠀⠀⠀⠹⣄⠈⢻⣿⣶⣠⠾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠠⠀⢀⣶⣿⣿⠿⣿⣽⡏⢐⢳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣯⣴⣀⣠⣿⣿⠛⠁⠀⠘⢿⣇⣷⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢉⡿⠉⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣉⣀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠁⢈⣉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢻⣿⠇⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠊⣼⣷⣿⣏⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢰⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣹⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀
⠀⠀⠀⠀⠀⣠⠴⢺⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿
⠀⠀⠀⠀⡎⠡⣘⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣽⡿⠃
⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⢘⣿⣷⡄⠀⠀⠀⠈⠉⠉⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠙⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⠙⢿⣿⣿⡀⠀⠀⢀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠛
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⡴⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⣰⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠦⣀⡾⠋⠀⠀⠀⣿⠃⠀⠀⠀⢀
⢿⣦⣤⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠁
⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠃⠀⠀⣀⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠃⠀⠀⠙⠛⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠿⢿⣷⣦⡀⠀⣸⣿⠟⠉
⠀⠀⠀⠀⣠⣤⣤⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⠀⠀⠘⢷⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣿⡇
⠀⠀⠀⠀⠁⠈⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⡶⢦⠀⠀⠀⠀⠀⠀⠙⠋
⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡿⣿⣿⣷⣤⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴
⠁⠀⠈⠉⢿⣷⣄⠀⠀⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠈⠻⣀⣠⣶⣶⡄⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠿⠁⠈⠀⠀⠀⠀⠀⠀⢠⣾⣿⡏⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⢀⣤⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁
⠀⠀⠀⣼⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠗⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣴⣾⣿⠋
⠀⠀⢸⡿⠉⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠻⣿⠿⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣄
⣦⣤⣟⠀⠀⠈⠉⠙⠉⠻⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⣿⣿⢿⣿
⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⠀⠀⠀⠘⣿⣷
⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣠⣿⠿⠁⠀⠀⠀⠀⠀⠀⣻⣄⣠⣤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠛⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠰⣤⣄⠀⠀⠀⠀⠀⢀⣤⣄⡀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⠿⢿⣯⣴⣿⣯⢱⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣻⣯⣿⡟⣿⡿⠯⡭⡏⣯⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⣏⡍⣿⠋⡇⡿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⡷⣗⣿⣿⣖⡧⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⣿⣿⣿⣷⣿⣿⣿⣿⢷⣿⣿⠃⠀⢀⣀⣤⡆
⠀⠀⠀⢿⣟⢻⣿⣿⣿⣭⣭⣷⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞
⠀⠀⠀⠀⠙⢷⣬⡛⠿⢿⣿⣿⡿⠟⣡⣾⣿⣷⣿⣷⣻⣯⣻⣿⡽⠋
⠀⠀⠀⠀⣔⣡⡟⣿⣷⣶⠾⠗⢺⣿⡅⠤⠿⠿⣿⣿⣟⣻⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠘⠧⡽⣿⢇⣀⣈⣼⣹⣇⣤⣿⠿⠿⣿⣿⡿⣿⡏⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢾⣿⠉⠀⢠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠆⠨⠜⠛⢛⣫⣄⠀⢰⣿⣿⣿⣿⢸⣧⡊⣷⠀⠀⠀⠀⣀⣠⡴⣶⠚⠋⠩⠅⢀⣈⣡⡽⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢻⣤⠀⢒⣠⡾⣛⡛⠿⣦⣿⣿⣿⣿⣿⣿⣿⣦⣽⡧⠖⠚⠋⠉⣒⣧⣿⣤⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣿⣻⣿⣿⣷⣎⣽⣫⡿⠋⢵⣛⠫⢍⣂⣤⣴⠶⠞⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⣸⣿⣿⣧⢧⠘⠶⢾⣿⣭⣥⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣟⣿⠿⠿⠛⢿⣿⣿⣿⣷⣴⣿⣟⠻⣶⣯⣝⡻⢿⣶
⠀⠀⠀⠀⢠⣶⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣆⠾⣿⣿⣿⣵⣿⣿⣿⣷⡀⢻⡇
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣑⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⡿⠿⣿⣿⣿⢟⣡⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣥⣴⡟⠉⠻⠿⣽⣛⡷⢂⣼⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣏⠋⢙⢻⡿⣿⣦⣴⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠷⡾⠷⡿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣽⣿⣽⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢹⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣽⣀⣭⣼⣯⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⢿⣟⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⢿⣿⣿⣿⣿⣻⣿⣿⣿⣶⣦⣴⠆
⠀⠀⠀⠀⠀⠀⢀⣴⣾⢠⣼⣿⣿⢯⣿⡹⢿⣿⢿⣿⢿⣇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡞⠿⢟⡁⣴⣾⣭⣿⣷⣶⢱⣿⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠞⠋⠘⢛⣛⣛⣛⡛⠛⠘⢟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⢲⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠇⢿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣿⡆⠀⠀⠀⢀⣤⡤⠴⠒⠋⠉⣙⣧⣤⣶⠿⠃
⠀⠀⠀⠀⠀⢻⣿⣯⣾⡿⣻⣶⣯⣿⣿⣿⣿⠿⢿⣇⣴⣍⢻⡷⠔⠒⠉⣉⣯⣧⠶⠶⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣋⣰⣿⣿⡟⣴⣿⣿⡀⣮⣡⣤⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣻⣇⣿⣿⣻⠿⣿⣿⣧⠻⣿⡿⢿⣄⡼⠁
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⢀⣤⢞⣿⢿⣿⢟⣽⣿⣿⣿⣯⡻⣿⣯⡽⣿⣿⠟⣠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠈⠉⠛⢻⣽⣽⢫⣿⣿⣿⣿⣿⣿⡞⣿⣧⠨⢿⡜⣿⣿⣿⣿⣿⣿⣿⢹
⠀⠀⠀⠀⢠⣾⣿⣿⣆⢿⣿⣿⣿⣿⣿⢟⣿⣛⣯⣭⣷⣾⣿⣿⢟⡽⣿⡟⣸
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠿⠛⣱⡿⢻⣿⣿⣷⣝⣿⣋⣥⣴⣾⠽⠚⠁
⠀⠀⢰⣶⣶⣶⣿⣤⣿⣭⣭⡿⠛⣱⣿⡆⠀⢙⣾⣿⣿⣿⣛⣉⣁⣀⣀⣀
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣝⣛⣿⣯⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⣠⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⣿⡿⡿⠿⠿⣿⣿⠿⠿⢿⣿⡿⢿⣿⣿⠿⣿⢿⣿⣿⣿⣿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⡴⢶⠀⣠⠶⣤⠀⣦⠤⠀⠴⣶⠦⠠⡦⢦⡄⠀⣤⡄⠀⢠⡤⠀⠠⣴
⠀⠀⠀⠀⠀⠀⠀⣝⢳⡆⣧⣀⣸⠐⣿⠂⠀⢀⣿⠀⠀⣟⣻⡆⣰⡿⢿⡀⢐⣇⣠⠀⣿⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠀⠀⠉⠀⠀⠈⠉⠀⠈⠉⠉⠀⠉⠁⠈⠁⠈⠉⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠸⡠⡀⠀⠀⠘⢦⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣡⠞⠀⠀⠀⡠⡸
⠀⠀⠀⠀⠀⠀⠈⠪⡢⡀⠀⠀⠙⢦⡀⢀⠠⠀⠒⠀⠀⠐⠂⠠⢀⠀⠀⣠⠞⠁⠀⠀⡠⡊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢀⠀⠀⢀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾⠁⠀⠀⢀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡔⠁⠊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠑⡄⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠩⠄⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠴⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠄⠑⠢⠄⣀⣀⣀⣀⡀⠤⠒⠉⠀⠄⡡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡑
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁
⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣛⣿⣻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⣿⣷⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡟⣯⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡻⣷⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣷⣿⣿⣿⡿⠿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣾⡟⠻⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⠉⠛⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣆⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⡟⠹⣽⣞⣇⠀⠀⠀⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⡿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⢿⣾⣧⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠈⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⠻⣯⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⣿⣽⣻⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⣿⣿⠉⠹⣿⣿⣿⣿⣿⣿⣧⣀⠼⠿⡏⣻⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢻⣿⣿⡿⠟⠛⢉⣠⡀⠚⠷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⠠⠘⢋⣡⣤⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠟⠀⠉⠀⠀⠈⠈⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢉⣠⠔⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⣾⡏⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡻⣫⣯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡿⣿⡆
⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣷
⠀⢀⣠⣴⣾⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛

⠀⠀⠀⠀⠒⠛⢿⣿⣿⣿⣿⣶⣴⡆⠀⠀⠀⠀⠀⠀⣠⣤⣀⣠⣤⣶⣶⡶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣷⡀⢠⣄⣠⡆⢰⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⣿⢿⣶⣿⣿⣤⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢻⣿⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣠⣴⣶⡶⠒⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢠⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠈⠛⠻⠿⢿⣦⣤⣴⣿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠐⠛⠛⢿⣿⣿⣿⣿⣆⠲⣴⣧⡿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠻⣿⡛⠆⠀⠀⠀⠀⠀⠀⠀⣴⣶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣄⣤⠀⠀⠀⠀⣀⣀⣀⠀⢀⣤⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣧⠀⠀⠉⠙⠛⠛⠛⠉⠙⠃⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣤⣀⢀⣴⣶⣦⣤⣄⠀⠀⠀⠀⠀⠙⠛⠿⣦⣀⡀⢀⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠟⠛⠻⠟⠛⠟⠛⠟⠳⠀⠀⢀⣠⣤⡶⠂⠀⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣴⣄⢀⣀⣼⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠟⠀⠀⠀⠀⢀⣴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠷⣶⣶⡀⠀⠴⠶⠤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣄⣠⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⡀⠀⠈⠉⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡙⠳⣶⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠠⣄⡀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠷⡄⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⣼⢣⣶⣶⣄⠀⠀⠀⠀⢀⣬⣷⡤⠄⠘⠛⠛⣲⣶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠛⠋⣿⣿⠟⣩⣛⢷⡄⣰⡟⠁⠀⢰⣶⣿⣿⣿⣉⣡⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠻⣶⡾⢻⡏⣾⣿⣿⠟⢹⣯⣾⣿⣦⠈⠛⠛⢋⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠉⣠⡟⠀⣼⡏⠁⠀⠀⠉⠛⠛⢿⣄⣀⣠⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣯⣤⣴⣿⡀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⣠⡾⠛⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⢻⣿⡿⣿⣿⠁⠈⣿⡄⣼⠏⠀⣠⣼⣿⠟⣿⣏⡀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠏⢀⣴⠟⠉⠀⠈⠙⣷⡄⠸⣿⡏⢠⡿⠋⠁⠀⠀⠀⠉⢿⣤⡈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⣼⡏⢀⣾⠇⠀⠀⠀⠀⠀⠈⣿⠀⠹⣧⡾⠃⠀⠀⠀⠀⠀⠀⠀⠘⢶⣄⠘⢿⡄
⠀⠀⠀⠀⠀⢀⣿⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣹⣷⣄⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣷⡀
⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄
⠀⠀⠀⠀⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉
⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠺⠿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣠⡶⠞⠛⠛⠶⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠋⠀⢀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠏⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡷⠞⠛⠛⢶⣄⡾⠃⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢷⣄⠀⣀⣤⠀⢀⡿⠀⢠⡿⠁⠀⠀⠀⠀⠹⣧⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣦⠙⠟⣫⡁⠀⠸⢷⣴⠟⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠶⠆⠉⠁⠀⣀⣼⠷⢶⣤⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠟⠛⠛⠳⢶⣤⡟⠉⢻⡄⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠙⠷⣤⣤⣤⣴⠟⠀⠀⠀⢿⡀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢶⣤⡀⠀⠀⠀⠸⣟⠻⣦⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣰⡏
⠀⠀⣠⣄⡀⠀⠀⠀⢀⡀⠀⠀⠀⢠⡾⠁⠀⠀⠈⠙⢷⣄⠀⠀⣿⠀⠘⢷⡄⠀⣿⠀⠀⣸⡷⣤⣀⣀⣴⠟
⠀⠸⣯⡈⠻⣦⣀⣴⠟⠻⣦⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢹⡄⠀⠈⢿⡄⣿⠀⢠⡟⠀⢀⣭⠿⠛⠛⠛⣶⡀
⠀⠀⠈⠻⣦⣬⠟⠁⣠⡾⠋⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⡇⠀⠀⠘⣿⡇⣠⡿⣧⣴⠛⠁⠀⢛⣷⠀⣼⠃
⠀⠀⠀⣠⡞⠃⣠⣾⠛⢷⣄⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣸⠇⠀⠀⠀⣿⣵⠟⠀⢹⣇⣠⣤⠶⠛⢁⣴⠿⠶⣤⣀
⠀⠀⠀⠻⣦⡼⠋⠙⢷⣄⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⣰⣿⠋⠀⠀⣸⣏⣁⣠⣤⠶⠛⠙⢳⡆⠀⢹⡆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢷⣽⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⣀⣠⣿⣡⣤⣴⠾⣏⡙⠛⠛⠳⠶⠶⠶⠛⠁⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣍⠉⢹⡟⠛⢿⡛⠛⠛⠛⣾⡇⠀⠉⠉⠉⠁⠀⠀⠀⢹⣿⢻⣶⣶⣤⣤⣤⠴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣾⠁⠀⠈⢷⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⣿⢳⣌⠻⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠻⣦⣄⣀⣈⢿⡄⠀⠀⢿⡀⣀⣀⣀⣀⣤⣤⡴⠾⡇⠀⢻⡶⣽⣷⣦⡈⠙⠛⠳⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠈⠉⠉⠛⢿⡄⠀⠸⣟⠉⠉⠉⢿⣿⡄⠀⠀⣷⠀⠘⣷⡀⠉⠛⢿⣷⣤⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠙⢷⣄⠀⠘⣯⢻⡄⠀⣿⣤⣤⣼⡷⠀⠀⠀⠈⠻⣿⡁
⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣠⣤⣝⣷⡦⢿⡄⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⡟⠀⠉⠉⠀⢸⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⢸⡇⠀⠀⠀⠀⣼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠟⠀⠀⠀⠀⢠⡟⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⣀⠀⠀⣀⣀⣼⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣿⣿⠟⠛⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⣠⡶⠄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⢦⣶⠞⠋⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠦⣄⠀⠀⠤⣄⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠘⢷⠀⠀⠀⠙⠶⠋⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠁⠀⢠⡀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠂⠀⠈⠉⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠉⠉⢿⣿⣿⣿⡟⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⠀⠀⠈⠻⣿⣿⡇⠀⠙⠿⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣿⣸⠇⠀⠀⠀⠀⢸⡟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠻⣧⣄⣠⣴⠦⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡰⢄⣈⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠕⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣰⡀⠀⠀⠀⠀⣿⣿⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦
⠀⢂⣄⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⡖⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠁⠉⠀⠀⠂⠄⠐⠁⠀⠀⠀⣀⡀⣰⡜⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠓⠂⠐⣠⠀⠀⠀⠀⢀⠀⣰⠋⠉⠹⠿⡀⠛⠉⢿⣿⡿⠿⢿⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄
⠶⢨⡕⢋⠸⣮⡁⠀⠈⢻⣿⣶⣷⣶⢤⣶⣄⡀⠙⠛⠀⠀⣼⣷⠂⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣀
⢀⡄⠀⡋⠘⠯⠁⣠⣀⢸⡅⣀⠀⠀⣀⡘⠧⢈⠉⠐⢲⣤⣌⣁⢶⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⠀⣹⣧⡀
⢈⠁⠈⠁⠠⣶⠂⠀⠉⠺⣿⠟⣀⣰⣿⡶⣀⠈⢸⣇⢉⠙⠁⣈⣵⣤⣿⡏⠹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣟⣶⣦⣤⣴⣿⢷⣦⣴⡆⣷⠀⠣⣿⣿⣧⡬⢿⡿⢿⣿⣜⣻⢉⣉⠈⣿⣧⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⢹⣿⡿⣿⡟⣾⣿⣿⣿⣩⣧⣠⣾⣿⣿⡇⢺⡷⠀⠙⢿⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠉⢻⠯⢴⣿⢧⣿⣿⣿⣿⣿⡿⠛⠟⢡⠀⢷⣤⡴⢒⣮⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠂⠀⠆⠀⠀⠀⠛⢙⡟⣻⠷⢦⣍⡉⢰⡟⡀⠘⣿⠞⠋⣹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣤⣤⢬⡿⠾⣷⣷⠀⠹⢀⣴⣿⣿⠛⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠋⠁⣹⡷
⠀⢀⠀⢀⣤⣤⠆⠔⠐⠈⠙⢿⣷⣿⣧⣠⡕⣩⠄⠀⠈⡙⢿⡿⠿⣿⠟⠙⢿⣧⣿⣷⠋⢛⡁⠀⡈⢿⡶⣀⣸⣏⡅⠀⠀⢲⣄⡀
⢟⡞⢀⣾⣿⢏⠀⠿⠃⠀⠀⠀⠙⢿⣿⣧⣾⣿⣅⠀⠀⠹⣦⣀⣿⠏⣤⣴⣿⣏⣻⣿⡐⠾⢤⡶⣈⡀⢀⠀⠈⡩⢄⢠⡄⢻⠏
⡟⣠⠾⡿⢗⠾⠃⣴⡶⣤⣴⣶⠏⢀⣿⣿⣿⣿⣟⠃⠀⠀⣿⠻⣿⣧⡙⢿⣿⣿⣿⠻⢋⣑⢛⡳⠿⢬⣽⣷⣾⣡⠚⡼⣷⠁⠀⠀⢀
⠀⠀⣽⠋⠘⠃⣶⠉⠚⡟⣻⠿⣷⣾⣿⣿⣿⡗⠙⠀⠀⠀⠈⠃⠈⠛⢷⣄⣈⣿⣿⣷⣿⠁⠛⠁⠟⠰⠞⣸⡿⠛⠘⠘⠁⠛⠀⠀⠈
⢀⡸⠃⠂⠀⠀⡐⠀⠘⠀⠁⠀⠉⠉⠉⠁⠀⠀⠀⠀⠾⠄⠀⠀⠀⠀⠀⠛⣿⣿⡩⠁⡈⠀⠀⠀⠐⠄⠀⠀⢠⠤⠐⠀⠄⢀⠀⠄⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠠⠀⢤⡤⢶⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⢭⠀⠀⡀⠴⣽⠀⠐⠈⡔⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⢀⠠⡘⡀⢰⠏⡀⠘⠃⠀⢀⢤⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣉⢠⠸⡇⡤⡈⡀⠀⠀⠤⠀⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠐⠦⠴⢀⡄⠀⠀⠀⠀⠀⠈⠃⢀⠊⢀⣦⠀⣀⡤⠀⡁⠀⢀⡉⢁⠂⠴⠂⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⡦⠀⠁⠀⠀⠀⠀⠁⠺⠂⠀⠀⠀⠀⠀⢸⣃⣩⡿⠠⠀⠐⠻⠁⢆⠈⠉⠁⠔⣊⠓⠀⡀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢀⠀⠤⢐⣁⣀⡈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠂⢀⠀⠀⠀⠀⠠⣤⡀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠚⠕⢊⣕⢨⡀⢀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢅⣄⡀⣀⠈⡦⠀⠅⠀⠀⠀⠀⠈⠙⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣠⣤⣌⠑⢦⠐⠨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠨⠦⢁⠂⠓⠪⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠑⣂⠠⢄⠀⠨⢭⠱⡬
⠀⠀⠀⠀⠀⠀⠀⡘⠀⣠⠴⠒⠀⡠⠼⣥⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠤⠧⠀⠁⠂⡉⠒⢦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠉⠅⠒⣡⠂⠻⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡒⠄⢤⠁⠡⣄⠲⠈⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⡀⠀⠐⠒⠉⢘⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⠥⢄⠉⣆⠀⡀⠒⠀⠀⠀⠠⢡
⠀⠀⠀⠀⠀⠀⠀⢠⢂⠀⠀⠏⠐⢊⠅⢂⣉⢀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠭⣉⡷⠗⠛⢱⠄⣀⠀⡄⡘⢦⠂
⠀⠀⠀⠀⠀⠀⠀⢺⡸⠀⠀⢀⢴⠃⠄⠡⠖⣀⢞⡛⠋⠠⡄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣻⡿⡁⠀⠚⣿⠃⠀⠁⢁⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠜⠁⡢⠊⠀⠀⠀⠀⠉⠀⠀⠐⠠⢠⠀⢠⡀⠀⣀⡤⠆⢀⣤⣶⠀⠀⠀⠀⡤⢦⣝⠳⠈⠄⢊⡀⠀⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠞⣋⣁⡀⠀⠀⡀⠀⠀⣄⠀⠀⠈⠀⡀⠈⢰⡿⠏⠀⠀⠴⠆⠀⠀⠀⠀⣴⣦⣼⣈⠷⠋⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠀⠀⠀⠀⠀⠑⠳⠌⠀⠂⠀⠦⡵⢆⠘⢿⣦⡆⠀⠀⠀⣤⠀⢤⣲⡀⠋⣼⠁⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⡀⡀⢠⠂⠀⠃⠀⠀⠂⠀⠃⠀⣨⠁⠀⠈⠁⠀⠀⠀⠛⠃⠘⢿⡕⠦⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠜⠂⠁⠈⠃⠐⠈⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠐⠛⢄⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢀⠀⠀⠀⠀⠀⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣙⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠻⣎⠪⢓⡀⠀⠀⠀⠙⡆⢎⢇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⠀⠀⠘⠀⠋⠀⠀⠀⡐⠐⠜⠁⠀⠀⠀⣰⡗⡵⣵⡿⠃⣠
⢄⠀⠀⠀⠈⠣⠀⠐⠀⠀⠀⠀⠘⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢯⢮⣾⠋⠀⠜⢁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠵⣵⠟⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⠈⠁⠀⠀⠀⢀⡤⡪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡨⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠪⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢥⣤⣄⣀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣽⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠉⠛⠁⠀⠀⠂⠀⠈⠙⠳⢦⣄⣀⣤⡶⠿⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⠴⠶⠶⢤⣀⠀⠀⠀⣨⠛⠻⠄⠀⠀⠉⠈⠉⢻⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⠃⠀⠀⠀⠀⠀⠁⡆⠀⠉⠀⠈⠟⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣄⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠙⢷⣤⡄⠀⠀⠀⠀⢀⠔⠺⢧⣄⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢻⣷⡶⠖⠀⠰⠉⠀⠀⠀⠈⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⣿⠶⣤⣀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠤⡀⠀⠀⠀⠀⣀⡀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠆⡄⠀⠀⢠⣿
⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢰⠃⠀⠀⢸⡇
⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣄⢝⢦⣄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠖⠋⠗⠊⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⣀⠠⠄⠀⠠⢄⡀⠀⠀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠠⡀⣀⠁⠀⢀⡀⠀⠀⠈⠢⣀⠏⡁⠢⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠀⠀⠀⠊⠁⠀⠀⢀⠠⠌⠇⠀⢠⠋⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⡀⠂⠀⠀⠀⠀⡰⠁⠀⣀⣆⣀⡆⠀⠀⠀⠀⠔⠋⠀⠓⡀⠈⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⡘⢏⠁⠀⠀⠀⣈⠀⠌⣀⠀⣵⠊⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠢⡀⢰⠋⠉⢛⠂⠀⠀⠀⣀⠖⠸⡀⡠⠤⠄⠀⠀⠀⠀⢱⡏⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠘⠌⢣⠤⣄⠐⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡇⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⢻⠀⠀⠑⢦⠀⠀⠀⠀⡀⢀⡀⣀⣀⣀⡠⢄⡰⠄⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠠⠤⡀⠀⠈⣤⣄⠀⠀⢱⠀⠀⠀⠀⣅⣠⣜⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠸⡀⠔⠁⠈⠁⢀⡼⠀⠀⠀⠀⠀⠐⠰⠦⠤⠤⠤⢴⡞⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠃⢤⣀⡀⠴⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠰⡀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⠄⡀⠀⠀⠀⠀⢸⠋⢦⠀⠈⠉⠉⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡞⠉⠒⠒⣏⡤⢤⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⠀⠁⠒⠤⠤⠏⠀⠈⢳⣄⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⡠⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢤⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠙⠷⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡄⠀⠆⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⡰⠀⠐⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠚⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠇⡅⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⡤⠒⠒⠓⠤⡀⠀⡀⡰⠓⠛⠛⠳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁⠀⣐⡏⠉⡉⢹⠀⡇
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⣆⠐⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠋⠀⠀⠀⠈⠙⡄⢧
⠀⠀⠀⠀⠀⣤⠀⣀⣀⣀⠀⠸⠀⠀⢀⣀⣀⠤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣇⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠐⠓⠀⠀⠀⠒⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠘⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠠⣤⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢼⣢⣭⣜⡂⠙⡎⢿⡟⢦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠰⢆⠛⠋⠻⠎⡟⡄⠧⢤⠤⡎⣷⢹⡟⣟⢷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠨⣿⡟⢣⡄⠁⠈⠁⠈⠁⠁⡍⣰⢠⢏⠞⣿⣧⣹⣇⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⢀⣤⡾⠁⠒⢌⡀⠀⠀⠀⠃⢩⠀⣿⡟⡿⣩⣽⣿⣏⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⢺⣣⠼⠉⠂⠀⠀⠀⢹⢻⠃⠁⠶⢬⣫⢿⣸⡆
⠀⠀⠀⠀⣠⡤⢠⣀⠖⠒⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡴⠶⢦⠀⠀⠀⠀⠈⠀⠀⣛⣦⢣⢟⣣⣝⣼⡆
⠀⠀⠠⠴⠉⢂⡁⠈⠀⠀⡤⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠽⢙⡛⠁⠀⠀⠀⠀⠀⠀⠈⠹⣦⡘⢡⠶⣛⣻
⠀⠀⣱⡞⢰⣿⡏⠀⣾⡓⣗⡼⠆⠀⠀⠀⠠⢈⡀⠀⠀⠀⠀⠀⠂⠩⠀⠰⢦⡄⠁⠀⠀⠀⢚⣵⡿⠦⣜⠋⡬⢱⣄
⠀⣸⡏⣧⠈⡙⠳⣇⠀⠑⠠⠀⠀⠀⠀⠒⢸⠿⠁⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠜⠁⡀⣸⠟⣦⠐⠆⠀⠀⠀⠀⠄
⠀⣿⢮⡙⠂⡈⠑⠂⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠂⠈⠁⢀⣠⡞⠋⠉⠀⠀⠀⠀⠀⠁⣤⡾⣻⠴⠀⠀⣀⠐⠐
⠀⢹⢦⡽⠾⠘⠀⠀⠀⢼⣩⠶⣦⣄⡀⠀⡀⠀⠀⢠⠍⠙⠀⠁⢀⣤⣯⡙⠄⠂⠀⠀⠀⠀⡠⢚⡯⢽⣾⠃
⠀⢸⣧⠓⠀⠀⠀⠐⠁⡿⢡⠞⣣⢻⢟⠛⠃⠀⠀⠀⠀⠀⠀⢠⠟⡛⢧⡙⠀⠀⠀⣸⠀⠀⠀⢉⣴⣩⣭
⠀⣼⠼⣇⣂⠀⠀⠀⠀⠁⠸⣧⣼⡌⠀⠁⠀⠀⠁⠀⠀⠀⠀⣸⣯⡙⡄⢀⡄⡄⢀⠁⠀⠀⣀⣴⡏⣹⣮⡄
⠀⠸⣿⣿⣅⡀⠀⠀⠀⠀⠀⠐⠌⠐⢦⣀⡐⣷⣤⡀⠀⠀⠀⣯⠬⣧⠞⣸⠗⠋⣯⣶⡄⠀⠀⠠⣏⣭⣝⠇
⠀⠀⢻⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣹⢁⣿⣷⢹⣦⡀⠀⣟⡳⠎⠘⠁⠀⠀⠛⠉⠀⠀⠀⠉⢡⡰⡽⠃
⠀⠀⠈⢳⣆⣀⡀⠀⠀⠂⠀⠀⠀⠀⠐⠁⣾⠻⢋⡞⣸⣷⠀⠿⠍⠀⠀⠀⠀⠀⠀⢤⠤⢦⡇⣼⢠⡇⠁
⠀⠀⠀⠈⠻⣏⡈⢀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠃⢸⢫⡵⢂⣄⡀⠀⠀⣀⣀⣒⠀⢲⣶⡷⣄⡀
⠀⠀⠀⠀⠀⠈⠳⢎⠂⠀⠘⢗⡀⠀⠀⠀⠀⠀⠀⠀⣬⡴⠿⢋⣩⡇⠀⠀⠋⠉⠙⡀⠀⠙⢶⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣳⣤⣀⣈⢶⡀⠀⠀⠀⠀⠀⠠⢡⡏⡼⣩⠿⣏⡀⣈⡁⠀⢀⠁⠀⠀⠀⠙⣧⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢴⣅⠈⠀⠉⠀⠀⢢⠀⠀⠄⠀⠙⠲⡥⣈⠙⣿⡉⠀⠁⠋⠀⠀⠀⠀⠀⠈⡉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠸⢷⢯⡆⣘⠿⣳⡆⡀⠀⠀⠀⠀⠀⠀⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢿⣏⠙⡛⢤⡿⠆⠀⠀⠀⠀⢠⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢀⣀⡀⠀⠀⠘⣿⠱⣆⣶⣾⠀⠀⠀⢠⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠘⠿⠃⠰⠿⠇⣸⣿⢘⣿⣂⠈⠤⠀⠀⠀⡰⣿⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠀⠀⠀⣸⣄⡌⠁⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢣⣤⠐⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⣤⠉⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⠠⠄⠠⠀⠀⠈⠲⠿⠟⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠉⡓⠲⠤⣄⡀
⡆⠀⠀⠀⠀⠀⠀⠀⣴⢾⣶⣄⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠘⠢⢜⢹⣶⣶⠞⠃⠀⠀⠉⠓⠦⣤⣄
⢹⡄⠀⠀⠀⠀⠀⠀⠸⠏⠹⢋⡴⢧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠏⢳⠆⢰⠂⠤⣀⡀⣀⠈⣽⢷⡀
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢴⡟⣳⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢂⠈⠻⠀⠀⠘⣦⣬⣼⢻⣿⣴⠃⠀⠈⠢⡄
⠀⠀⠹⣆⣄⠀⠀⠀⡀⠀⠈⠒⢿⣿⣅⠀⠀⠀⠀⡀⠀⠀⠀⠐⣧⣤⣿⠀⢰⢤⡀⣀⣿⠀⠀⠚⠛⠁
⠀⠀⠀⠈⣿⣦⣶⣮⣤⣾⠂⠀⠀⠃⣿⣇⡴⢢⣄⣙⣿⠃⣌⣼⡏⠁⠸⡅⡈⠈⣿⣿⠃
⠀⠀⠀⠀⠘⣿⣿⡿⠀⠸⣇⠀⠀⣾⡟⠙⢥⣼⠻⠀⠙⡆⠛⠉⣣⢀⣠⠿⠿⠻⣿⣇⣀⠀⠠⠄⠀⠀⠀⣀⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣜⣶⣿⣿⣷⡤⠊⠁⠀⠀⠀⠀⠀⠀⢁⡴⠋⢠⡆⠉⠻⣏⢟⠲⣤⡠⣄⣿⣆⣽⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠠⣲⢾⣥⠏⢻⢻⡿⣷⣄⠀⠀⠀⠀⠀⠀⠐⣉⠂⢀⣿⠇⣀⣡⣤⣾⡷⠄⣏⡝⠟⢸⣿⣿⢿⣦
⠀⠀⠀⠀⠀⢀⡠⠚⠙⠋⠀⠰⠃⣾⣷⣥⣽⠟⠀⠀⠀⠀⠠⢆⣴⠟⣿⠃⠀⠿⢣⠀⣿⠀⢸⠩⣸⠖⢰⣿⣿⣾⠁
⠀⠀⠀⣠⠔⠉⠀⢐⠶⢤⣤⣶⣶⡏⢻⣿⣿⢷⣦⡾⠀⠀⠡⣼⣇⠀⠀⢠⡦⠀⠈⡇⠼⣇⣬⠉⣻⡀⣿⣿⣿⡀
⠀⠠⠊⠁⠀⠀⠰⢷⣤⠊⠀⠉⠉⠀⠀⠀⠀⠲⢿⡆⠀⠀⠀⡙⣿⡇⠀⠀⡶⠁⠀⠙⠻⣿⣿⣢⣸⣿⣼⣇⣾⢷⡀
⠀⠀⠀⠀⠀⡀⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⣿⠁⠀⠀⠀⠤⠁⠸⢹⠋⠻⣿⡟⠛⠺⠟⠛⠃⠳
⠀⠀⡀⠀⠮⣵⣀⣀⣀⣀⣀⣤⣄⠀⠀⠀⠀⡴⢷⣀⠀⢀⣀⠠⣾⡄⣁⣀⣀⣀⣀⣀⣈⣀⣀⡿⠷⣄⣀⣠⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠿⢻⡯⢿⣿⣿⣿⣷⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣏⠉⠀⠉⠀⣸⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⢻⣇⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠎⠀⠀⠀⢡⣶⣆⣤⠴⢛⣥⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⣰⠟⣱⡾⢷⣶⣿⠿⠻⢇⣼⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡔⠋⠀⢠⣧⣿⣃⣠⣦⣄⠀⢸
⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⡰⠀⣄⠀⠀⠀⡿⡿⢻⣿⠇⣀⠀⣀⠀⣷
⠀⠀⠐⠖⠀⢰⡅⣛⢇⠉⢲⡄⠡⠀⠀⠀⢸⣷⠚⠀⠃⠀⢰⡷⡿⣛⣿⡠⠎⠙⠉⠁⢹⣆⠀⠀⠀⠀⣀⣤⠆⠀⠙⠛⠀⡀
⠀⠀⠀⠀⠀⠋⠙⣿⣦⡱⣌⠻⠄⠀⠀⣠⠏⠘⠀⠀⠀⠀⣿⢷⣠⡭⢿⣷⠋⠈⡙⠂⠀⣿⣷⡖⢺⠇⠀⠢⠄⡄⠠⠖⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣫⡝⢣⡒⠉⠀⠀⢛⡀⠀⠀⠀⠀⠀⣱⡾⡄⠀⣼⣿⣆⣀⠗⠀⠀⢀⠹⣿⣿⡀⠀⡶⠀⠐⠆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡕⠑⢶⡀⠀⠀⠀⣀⠀⠀⠀⠀⡿⣏⣁⡐⠊⢻⣿⣧⠀⠀⣲⠄⠈⣿⣇⣠⣾⣤⣀⣼⠇⠐⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠈⠱⠆⢠⣬⠀⠀⠀⠀⠀⢠⣿⣍⡀⠐⠚⣿⣇⢀⠶⡀⠀⠿⠋⢿⠿⢿⡩⠋⣿⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⢠⣤⢖⣶⣶⡦⢀⡠⠰⠍⠛⠘⢁⣖⠴⣿⣿⣿⣆⢀⣴⠆⠀⣞⡀⠈⡷⠡⣯⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠸⠉⣿⡙⠛⠛⠋⠁⢀⡤⠘⠇⠈⠛⠀⣿⡋⠁⢼⣮⠰⠀⠘⠘⠁⣠⢤⠄⣿⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡧⡀⠀⠀⠠⠉⠑⣶⢾⣿⣦⡁⠀⠈⠀⠀⠙⢻⢿⠆⢸⣅⠀⠚⠀⠸⠀⠉⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⣙⢦⠀⠀⠀⠀⠍⠀⠙⠻⣶⡂⡄⠀⠀⠠⠏⠀⠈⢹⡦⠴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠢⠀⠀⠀⠀⢀⠀⠀⠀⢈⠻⢶⡆⠀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⡀⠂⠀⠀⠀⣠⣾⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⢀⠀⠀⣠⣤⣤⣄⠀⠀⠀⣄⠀⠀⢠⣶⣶⡄⠀⣠⠄⠀⠀⢀⣤⣤⣦⣄⡄⠀⠀⠀⠀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⣙⣿⣿⣤⣤⠴⠋⣠⠐⣿⣿⣿⣿⠀⠀⠀⠙⢦⣄⣘⣿⣿⣃⡴⠃⠀⠀⠀⢸⣿⡿⠋⠋⠀⢠⠀⠀⠀⣿⣿⡿⠃
⠀⢠⡤⠴⠾⢿⣿⣿⡏⠁⠀⠀⠸⣄⣬⣿⣿⣶⣤⣄⣤⠀⠀⠙⢿⣿⣿⡟⠁⢀⣀⣠⠴⣿⣿⣿⣦⡀⠀⠈⠷⠶⢿⣿⣿⣿⣷⣄⣠⠖
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡧⠀⠈⠁⠀⢠⣿⣿⣯⠀⠉⠛⠂⠀⠀⢸⣿⣿⡇⠀⠉
⠀⠀⠴⣦⣠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⠆⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⢾⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⢿⣿⡄
⠀⠀⠀⠈⠙⠛⠁⢿⡇⠀⠀⠀⠀⠀⢀⡟⣠⠾⠃⠀⠀⠀⠀⠀⢸⠋⠙⡏⠀⠀⠀⠀⠸⣇⠹⣧⠀⠀⠀⠀⠀⠀⣿⠇⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⢸⠁⠃⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢿⡀⠀⠀⠀⠀⢸⠆⠈⠱⠄⠀⠀⠀⠠⠏⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈