Orchard Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣾⠤⠆⣄⠀⠀⠂⠀⠀⠀⢔⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣙⣿⣯⣿⣿⣿⣟⡋⠁⠀⠃⡀⠀⠁⣠⠁⢏⠋⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢚⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣆⡍⣇⠺⠓⡆⠰⡦⢄⡀⢰⣲⣇⣈⠃⢠
⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣽⣿⣍⢤⣟⣿⣷⣶⣏⣯⢓⣱⣄⣀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣿⡿⠿⢿⣾⡥⣠⣄⢘⣿⣿⣿⢿⣿⣾⣶⣿⣇
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣿⣽⣆⣾⣿⣵⣶⣷⣻⣿⢻⠿⣿⣿⡿⣟⢾⠥
⠀⠀⠀⠀⠀⠰⠋⢩⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣶⣿⣿⢿⣿⠹⣻⣧⣙⠆
⠀⠀⠀⡀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⡷⢆⠈⠢⣿⣿⣿⠖⠀⠀⠀⠀⡄
⠀⠀⠀⣧⠀⠀⠀⠸⠁⡟⢻⡟⠈⣺⢟⣿⣧⡻⣿⢿⣟⣫⣿⠛⠅⠘⠿⢦⠗⢤⠀⣬⣽⣿⣿⠏⠀⠀⠀⢐⡧
⠀⠀⢀⣿⡄⡄⠀⠀⠀⠃⠈⠀⠀⠀⠀⣺⠿⣿⣿⢻⣟⡿⠁⠃⠀⡀⠀⢨⡄⠀⠀⠳⣾⣿⣿⣖⡀⠀⢀⣸⡯
⠀⠀⢸⣿⣷⣧⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⡂⠀⠀⣀⣇⠀⣀⣰⣶⣦⣭⣽⣿⣷⢤⡄⠀⢀⣿⡁
⠀⠀⣸⣿⣿⣿⣼⣿⡇⣼⣃⠆⢀⡄⠀⠀⠀⠀⣿⠀⡏⠀⣀⣷⣿⣿⣿⣿⣿⣧⠶⣫⣿⣿⣟⡛⠀⣤⣤⣭⣁
⠀⠀⠹⢿⠿⡿⢷⠿⠇⠛⢤⢠⣾⣴⣴⣾⣷⣷⣿⡆⣧⣶⣿⣿⣿⣟⠓⡑⠈⠀⠀⠁⢁⣿⣩⡥⣶⠏⣩⠀⠉⠉
⠀⠈⠉⠰⠶⢶⣶⣦⣤⠀⠀⠀⠀⠀⡀⣀⣀⣠⣿⣧⣽⣄⠀⠒⠛⠻⠶⠿⢶⢶⣴⣾⣯⣯⣥⣄⣤⣼⣿⠄⠠
⠀⠀⠀⠀⠀⠀⠈⠉⠿⠿⠿⠶⣾⣿⣿⣿⣿⣟⠛⢛⢟⣻⣿⣿⣷⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⡷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠈⠀⠀⠀⠉⠀⠉⠙⠉⠋⠉⠉⠉⠉⠁

⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈
⠀⠄⢠⡠⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠂
⣆⣦⣸⣧⣧⡼⢒⣠⡤⠂⢤⢆⡀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡓⢠⢴⣓⣾⣭⣰⡶⠂⣠⠀⡀⠁
⣿⣿⣿⣿⣾⢷⠱⠷⢠⠐⡆⡺⠛⢃⡀⠀⢰⡀⠄⢀⠀⠀⠀⠀⢼⠄⣱⣜⡿⡯⢸⣿⣩⢶⣋⣾⣷⣿⣿⡗⣾
⣫⣿⣿⣷⣊⡞⣻⣶⣻⢦⣱⣷⣴⠄⡀⠀⠀⠀⢀⢀⠀⠀⠀⠐⠵⣩⣻⣻⣿⣷⣌⢻⣿⣯⣿⣿⣿⣏⣹⣿⣿
⣿⣿⣿⣿⣿⣿⡿⣿⣴⣾⣛⣩⠏⡩⠡⢀⠄⠁⠱⡄⠐⠁⠀⠐⢚⡿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣶⣽⣾⣯⠴⣷⠿⠄⡀⠀⠀⠀⡂⠆⠟⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⢐⡀⢀⡀⠄⢘⣉⠡⢙⣉⣽⣭⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣯⢳⣇⣥⢾⡡⠀⣥⣰⣌⡉⣛⣿⣻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⢿⣿⣻⣿⣿⣿⣿⠿⣿⣿⣿⡿⠟⢻⣍⣻⣍⣿⣾⣟⣿⣿⢙⣍⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣾⡿⣿⣻
⡿⠷⣿⢿⣾⣿⣿⣴⣾⣿⣿⣿⣷⣿⣿⣿⡿⣿⣿⠿⣿⠿⣿⠿⡼⠭⢿⣿⣿⡟⢋⣹⣿⣿⡟⣿⠛⠉⢷⣿⠁
⠀⠀⠁⠀⢿⣿⠁⠙⡟⠉⠁⠁⠠⠋⠀⢹⠀⠀⠋⠀⠧⠀⠁⠀⠴⠚⠁⠰⠋⠀⢜⣹⡿⠋⣀⣿⣀⠀⢸⣿
⡀⢀⢀⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡀⡄⣠⢀⡈⢿⣤
⡷⠿⣿⣿⡿⢿⠿⣿⠷⠶⠆⠄⠤⠖⠠⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠒⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠉
⡆⠀⠐⠂⠂⣀⣀⣀⣀⣀⣀⣀⡀⢀⡀⠀⡀⢰⠂⠀⠈⣀⣀⣀⣌⣀⣈⣀⣀⣁⣀⣀⠀⠀⠀⠀⡀⢀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⢷⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠋⠀⢀⣤⠶⠾⠃⠀⠀⠀⠃⡀⠱⠘⠠⡄
⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⢠⠔⠈⡉⢸⠅⠀⠀⠆⠀⠀⣲⠄⠀⠀⠶⠂⠀⠀⣌⠉⠁⠀⠠⡾⠟⠀⠀⠀⠉⠀⠠⠈⡀⠑⡀⢒⣿⣾⣥⡔⢀
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⡢⣾⡙⣛⣷⠂⠀⠀⠀⠛⠀⠀⠀⢀⡄⠀⠀⠰⠟⡁⠀⠐⣒⣷⢤⠀⠑⠀⠀⠁⠀⠀⡄⣢⣀⢀⡿⣧⣻⡿⣚⣄
⠀⠀⠀⠀⠀⠀⡥⣆⢫⣠⣲⣧⣼⡷⢿⠆⠀⠀⠀⠀⣆⢀⠀⠀⠀⢹⠂⠀⢸⡂⠄⣠⣯⡉⣆⢁⡀⠄⠛⢿⣾⣿⣾⣿⣾⢾⣷⡿⣿⡛⠀⡾⠙
⠀⠀⠀⠀⠀⠀⠀⡙⠃⠁⡼⠂⡋⠁⠈⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⠒⠀⠀⠀⠿⡀⠠⠿⠄⠛⠟⠛⠆⠒⠗⢈⡁⠈⠹⠷⠿⠾⠚⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠈⢡⠀⡄⣧⢠⠁⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⡆⠀⠀⠀⠀⠠⠇⠀⠀⠀⠁⠀⠀⠁⠀⠆⠁⢠⠞⠀⠄
⠀⠀⠀⠀⠀⠀⢀⣯⠠⡀⡇⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⢂⣳⣦⣀⠀⣸⣀⡀⣀⠀⡀⠘⣧⡈⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⡄⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠈⠉⠉⠉⠀⠁⣀⣿⣧⣿⣴⠒⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢣⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠠⠠⣀⠁⠀⠉⢉⣉⢉⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⠀⠐⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠒

⣿⡿⢿⡿⠿⢿⣿⠿⠿⠿⠛⢿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠛⠿⢿⡿⠿⠿⢿⠻⠿⠿⠿⠿⠟⠟⠛⠿⠿⠿⠿⠿⠿⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿
⣿⡏⠂⠉⠿⡿⠿⠒⠒⠨⡙⡯⠉⣥⣀⠀⠸⡟⢀⡀⠀⡅⠀⢀⡤⠠⠁⠀⠀⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⣀⡃⠖⠀⠱⠞⣧⠀⠀⠀⠻⣿⣧⠂⡅⠀⠈⠛⠠⣀⡀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠀⠀⠀⠀⢸
⣿⣇⣎⡁⣈⡙⢰⠀⠀⠉⢻⣆⠀⠀⢿⡇⠚⠀⠑⠄⠀⠀⠀⠹⠷⣄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⣿⣷⡙⡇⣌⠀⠒⠏⠉⠻⣦⡀⢹⢀⡜⠊⠉⡓⠀⠀⠀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠆⢻⣿⠇⠅⣺⣅⠀⠀⡀⣬⣿⡷⠋⠹⡍⠀⠛⠦⢄⠀⢳⠤⣀⣈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡄⢶⡀⠀⠀⢻⡿⣈⣁⠙⠽⢿⠄⡠⠀⠀⠄⢀⡤⠄⠳⣧⣄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡄⠀⠀⠄⠀⠻⡇⠀⡹⠒⠐⠈⠂⠀⠀⠀⠀⠈⡆⠀⠀⣾⣿⠐⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡃⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣿⠇⠉⠉⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⢨⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⢡⣷⣄⣦⢴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣶⣦⡤⠘⢠⡶⢎⡉⠙⠤⠀⢌⣿⣿⣆⠄⠀⠷⣀⠀⠀⠀⠀⠀⢈
⣿⠁⣀⠀⠀⠁⢤⣀⠐⠂⠀⠀⣼⣏⣨⣿⣧⠐⢯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⣿⡅⠀⠀⢀⣾⡟⠒⠊⠁⢘⠋⣥⠁⠀⠀⠰⣿⡇⠀⠀⠀⠀⠘
⣿⡟⢁⣀⢦⣄⣀⣝⠁⢀⠀⢸⣿⠿⡿⢿⣿⢀⣀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠤⡀⢀⡙⡄⠀⠀⢺⣿⣏⠀⢀⣄⣾⡇⣽⡔⠀⠀⣸⣿⠇⠢⠀⠀⠀⢸
⣿⡇⠘⠋⢱⡟⠁⠁⠀⠀⠀⢼⡿⠀⠆⠼⠿⠛⠛⠻⠿⠷⣦⣀⠀⠀⠀⠀⠀⣌⡚⠁⠐⠀⠉⠉⠉⠉⡽⠁⠀⢩⡍⠩⢨⡍⠉⣴⡆⢀⡏⣈⡤⠀⠀⣀⣸
⣿⡇⠀⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠤⡄⣿⣷⠚⠓⠀⠀⠀⠀⠒⠠⠀⠼⠧⠤⢿⢷⣴⣿⣷⣿⠏⠁⠀⠚⠛⠛⠛
⣿⡯⣁⣒⡂⠨⠭⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⣐⠤⣐⠠⣔⠢⢅⡢⢄⡂⠽⢛⠻⠂⠄⠀⢀⠀⠀⢀⡀⠀⢀⣠⣀⣑⣫⣾⣿⠿⣿⢳⠒⠀⠀⠀⠀⢸
⣿⠯⠭⢒⣒⣉⡩⠤⠐⡒⣀⡀⠀⠄⠀⠀⠀⠀⠀⠈⢳⣌⣱⣦⠽⠶⠟⠓⠋⠉⠀⡜⠀⢀⠀⠀⠀⠀⢉⠀⠀⢸⠇⠀⣿⡇⢸⡿⠀⣿⠀⠁⠠⠀⠀⠀⠘
⣿⣟⣋⣩⠥⠶⠒⣛⣉⡭⠤⠒⢂⣁⣠⡤⠴⠖⠚⠛⣉⠉⠀⠐⠐⠀⢀⠄⡤⠐⡆⠠⠀⠈⠀⠀⠀⠀⢨⠀⠀⢸⠁⠀⣿⠀⢸⡇⢀⡏⠈⠀⢀⠀⠀⠀⠸
⣿⣯⣥⠴⠖⢛⣉⣭⡤⠶⠞⠛⠉⠉⠀⠠⠒⠀⠀⢀⣠⠄⠐⠂⠀⠀⠆⠰⠃⠀⠠⢄⡀⠐⠀⠀⠀⠀⢸⠀⠀⢸⠀⠐⣿⠀⣸⡇⢸⣇⡤⠆⡎⡀⠀⠀⢸
⣿⣧⡴⠶⠛⠋⢉⡀⠀⠀⠀⠀⠀⠀⠑⠒⣿⣦⡄⠀⠀⠀⠀⢀⠈⠁⠀⠀⠘⠀⢠⣸⢿⠈⠁⠀⠀⠀⠘⠀⠀⠚⠢⠚⠛⣲⡿⠛⢻⡿⠠⣾⣿⢆⠀⠀⢸
⣿⡇⠀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠈⠹⣿⢠⣀⡀⠀⡈⠛⠛⠁⠀⠀⠠⠄⠩⠟⠓⡄⠀⠀⠀⢀⠀⠀⢠⠄⣩⣤⣞⠄⠀⠀⢰⣷⡌⡂⠄⢁⠰⠊
⣿⡇⠀⠀⣀⠀⠀⠀⠀⠀⢲⣼⣦⡄⠀⠀⠀⣠⡀⠈⠀⠀⠀⠀⠀⠀⠔⢒⡀⠀⠀⠀⠀⠐⠀⠀⠉⠀⢀⣀⢀⠖⠁⠀⢩⢿⡀⠀⠀⣾⣿⢣⣤⣀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠔⠊⠉⠙⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣼⠔⠀⠀⠈⠉⠂⠀⠀⢀⣴⣤⡀⠀⠀⠀⠀⡀⠸⠃⠀⠑⠀⠚⢛⠁⠉⠉⠁⠀⣠⠸
⣿⡇⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⢰
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣴⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡶⢶⠖⢻⠖⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⣶⣶⣿⣿⣷⢨⣠⣤⣍⣣⣉⣲⡶⠋⠉⠻⢿⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⡟⣯⠻⣣⡶⢾⢿⡿⢻⣿⣿⣿⣿⣿⣿⣯⡿⢻⣫⣶⣆⡐⢿⣾⣯⣿⢯⡹⡷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢿⡻⣝⢷⡑⡌⢳⡿⢠⠖⠫⣾⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣹⡷⣾⢷⣿⣾⡇⢈⡹⣆⣡⡞⡹⢛⣶⣄⡀
⠀⠀⠀⠀⠀⣠⡾⣏⡳⢮⣢⢙⠮⡣⠙⢠⡿⣷⢆⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⢿⣻⠶⣾⣿⢀⠴⣋⠴⢛⣹⣦⣄
⠀⠀⠀⣠⠾⢭⣙⠲⢭⣲⢭⡓⢦⡈⢣⣿⣖⣻⣿⣬⣿⣿⣿⣿⣿⣿⣤⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⣊⡵⠞⠫⣑⣯⣿⢷⣄
⠀⢀⣾⣿⣿⣶⣮⣝⡲⠬⣙⠺⢕⡎⠙⣿⣿⣷⣿⣿⣿⣿⣿⣿⣏⢙⣿⣿⣷⣾⣿⠃⣽⣿⣿⣿⣿⣿⣿⠉⢻⣷⣼⡏⢈⣡⡴⠖⢛⣩⠗⠒⠒⠻⣦
⢠⡟⠉⠉⠙⢻⡲⠦⢭⣙⣒⢢⣤⣌⣑⠻⢟⡻⢿⣿⣿⣿⡦⡙⢿⣿⣴⣿⣉⣿⠃⠞⣽⣿⣿⣉⣿⣿⣿⡿⣿⣿⢟⣈⡭⠴⠖⣛⡏⠁⠀⠠⠀⣈⠘
⡟⠀⠀⠀⠀⠀⠫⣭⡽⠒⠉⠉⠀⠀⠈⠉⠳⢬⡜⢿⣯⣽⣿⣎⠲⣝⠻⣿⠿⠃⢠⣾⣿⡟⢻⣿⢋⣴⣯⠶⢿⠥⠞⠉⣙⣛⡯⢭⠏
⠃⠀⠀⠀⠀⠀⠀⠈⣡⣴⣦⠖⠲⢦⡀⠀⠐⠀⠳⡜⠻⣿⣧⣽⣿⣷⡄⠀⠀⡠⣵⣿⣿⣿⠿⠃⣼⠟⠉⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣫⠅⢀⠐⠺⠿⣧⡀⠀⠀⠐⠙⠲⠞⠻⠛⢿⣿⣷⣄⢄⢁⢿⣿⣛⡅⣶⠟⠁⠀⢀⡠⣄⣀
⡄⠀⠀⠀⠀⠀⢸⣯⣉⡄⠚⣶⣾⣷⣿⣿⡄⠀⠀⠠⠢⠴⠒⠤⠒⠛⠿⣿⢾⢻⣿⣿⣿⣿⡇⠀⠀⢰⣿⣟⡀⠈⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⢧⠀⠀⠀⠀⠀⠸⣿⣿⠷⣾⣿⣟⣿⣿⡟⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⢿⢸⣟⣯⣿⣿⡇⠀⠀⠘⢿⣿⡿⣾⣿⣇⣠⣤⠴⠶⠖⢒⣛⣉⣉⣩⣭
⠈⢷⣶⠦⠤⣤⣤⣹⣿⣿⣿⣿⣷⣿⠟⠭⠉⢉⣉⣉⣉⣁⣉⣉⣉⣍⣻⣿⠘⡼⠹⢉⡿⣽⡷⠶⠖⠶⠶⠾⠻⢟⠛⢻⣿⣿⣾⣿⣿⡿⡿⠿⣿⣿⣿
⠀⠈⢻⣏⣁⠀⠀⠉⠉⠛⣿⣿⣿⣿⣷⣶⣚⠲⠿⠿⠶⠶⠞⠿⢋⣩⣽⡏⡇⠁⢠⣿⢻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣭⣹⣽⣷⣾⣿⣿⠟
⠀⠀⠀⠙⢾⣏⣉⡓⡒⠦⢭⣟⣛⣬⣍⣽⣛⣳⠶⣤⣤⡴⢉⠁⣘⣯⣿⣵⣷⣄⣾⣿⢿⣿⣿⣿⣶⣼⡾⠿⠿⠶⠾⠶⠶⠶⠿⢿⣏⣛⣩⣽⠟⠁
⠀⠀⠀⠀⠀⠙⠶⣯⠭⢷⣶⣤⡈⠉⠙⠛⠛⠛⢛⣓⣻⡿⠟⠛⢋⣿⡿⣿⣿⣽⣿⢯⣻⣽⣻⣿⣿⡶⢶⣖⣒⣒⣶⣶⢦⣴⣦⡤⢄⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣉⠙⡳⠚⠓⠒⠚⠋⢁⠀⠀⠀⠚⠉⠵⠚⠋⠑⠛⠻⠘⠋⠛⠯⠙⠎⠙⠂⠀⠀⠈⠉⠉⠉⣁⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⣤⣲⣛⣠⠉⢂⡣⢀⢘⠍⡀⢢⣄⠄⢠⠔⡊⡤⢂⡣⠀⠠⠠⠀⣀⣀⣤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠷⠶⢾⣮⣧⣬⣠⣴⣿⣭⣦⣶⡿⠷⠾⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⠄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠔⠀⢤⠉⢸⣥⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⠀⠸⡆⠀⡠⠈
⠀⠀⠀⠈⠀⠐⣆⣀⣌⠹⡿⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠆⠀⠀⠀⠁⠀⢁⡀⠠⡆
⠀⠀⢀⡀⡀⠠⠆⠀⠉⠙⢿⣧⡸⠗⡀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⡄⠀⣠⢀⣧⡄
⠀⠀⠈⠃⢸⠂⣠⢰⠇⠙⠻⣿⣿⣴⣽⡏⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⠀⢠⡴⠈⠉⠁⢻⣿⣯⢀⡀
⠀⠀⠈⠟⠻⡟⠈⣥⣰⠶⢟⣿⣿⣷⣿⣿⣾⣷⣰⣷⠴⢦⡀⠛⡟⠤⣀⠀⠀⠄⠉⠀⠀⠌⡛⣛⠻⢛⡃
⠀⠀⢨⣴⣾⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠦⢍⣀⣹⣦⣰⣿⣤⣬⣄⣤⣤⣠⣽⣯⣁⣟⣽
⠀⠀⠠⠌⢿⡯⠤⠅⢹⠋⠀⢹⠈⠏⣏⣹⣹⣉⣉⣭⣍⣉⣉⡉⠋⠿⡟⠏⣉⣻⣏⣻⡿⠿⠩⢿⡿⠏
⠀⠀⠀⠀⠈⠑⠒⠲⣾⣧⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣬⣭⣥⣬⠽⠯⠍⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⠿⠿⠿⠟⠛⠛⠉⠉⠁⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠁

⠀⠁⠀⢀⣰⢶⣦⣄⡀
⠀⠀⠀⢾⣿⣟⢾⣿⣾⣷⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣾⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠙⠻⢿⣿⣟⣻⣶⣬⣑⢦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣤⣌⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⢨⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣯⡅⠀⣀⠀⠰⣶⠞⠉⢻⣾⡟⣑⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⣾⣿⣷⣀⠾⢟⣶⣌⣿⢼⣿⣿⡍⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⢉⣾⣿⠁⣩⠻⢿⣝⡌⣿⣿⣧⡀⠉⢻⡿⣿⡿⠋⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⡿⠀⠃⠀⠈⠙⠿⣮⣹⣿⣷⡄⠽⠟⠀⢀⠐⠂⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⡀⠀⠀⠀⠉⠿⢿⣿⣗⡀⠐⣶⣬⡆⡊⢡⢼⠿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠶⣖⠁⢁⣾⡊⠚⡈⢵⣅⡛⣿⣿⣿⡿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠠⢠⢦⡛⢾⢸⣿⡦⢹⡦⢡⣭⠿⡟⣶⡏⢉⡉⢙⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⡺⠙⡥⢤⣰⣦⣾⣿⣷⣯⢶⣬⣭⣤⠁⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⡐⠓⡴⠗⣾⣿⣽⡿⣿⣿⣻⣟⠠⠶⠟⠘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⡻⠀⠀⠀⠀⠠⠍⢉⡙⢛⡀⠁⠀⠵⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡇⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡀⠀⠀⠀⠀⢀⡐⠰⠦⢈⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⣀⠐⠉⢀⠀⠀⠋⣙⣛⠫⠿⣿⣷⣶⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣾⠿⢿⣶⣦⡈⢀⠈⢀⠘⢽⣥⢴⣿⡷⣉⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡙⣿⣿⣦⡉⠉⣿⣿⠄⠀⠀⠈⠰⠆⠉⣈⢹⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⢳⣿⣿⢚⣧⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠐⠿⠛⠋

⣿⢟⠛⠛⠛⢛⡛⠻⣿⡛⠛⠟⠛⠻⠛⠛⡛⠛⠛⠛⣿⠛⠻⢛⡛⠛⠛⢿⣿⡟⠟⠛⠛⢛⠿⡟⣛⠟⠛⠻⣿⣟⠿⠛⠻⢿⣿⠿⣿⠟⠛⠛⠛⠻⣿
⣿⢸⣦⡄⠀⠘⠀⠀⢸⣿⡆⢰⡠⠚⠀⠓⠊⠀⡀⠀⠀⣷⣤⣤⠷⠀⠀⠀⠟⠇⠈⠉⠀⡯⢀⡀⠈⠀⠀⢰⣿⡇⢀⡀⡀⠀⠈⢰⣾⠄⢀⡄⠀⠀⢻
⣿⠉⠠⣿⣄⡀⠀⣠⣼⣿⠷⠋⠊⠉⠙⢣⣄⠀⢐⡀⢀⣹⣿⡗⠀⠀⠀⠀⠒⠀⠀⡄⠀⠀⢸⣿⣧⣠⡄⠙⣋⠀⢸⠃⠆⠀⠀⣼⣿⡗⠀⢀⠀⢀⣽
⣿⠀⠀⢹⡟⠑⠐⡄⠈⢵⠀⠀⠀⣤⠀⠀⡀⠓⣾⣤⣠⠾⠋⠁⠀⢤⠀⠀⠀⠀⠀⢡⠀⠀⢸⣿⡯⠁⣨⠁⠀⠀⢸⡀⣨⡤⠀⢽⠛⠀⠀⠠⣈⠀⢸
⣿⠀⢀⠼⢷⠀⠀⠀⠀⠀⠁⠀⠀⣟⠈⡏⠀⠀⣿⣿⠘⠃⠁⠀⠀⠀⠓⠤⠤⠀⠀⠀⠁⠀⠈⠁⠁⠀⠙⠀⠀⢀⠀⠀⢙⣾⡦⡤⣤⢄⡤⠴⣋⣱⣸
⣿⠀⢤⢀⡀⠀⠀⠰⠀⠀⠀⠀⠀⡗⠀⠀⠀⣀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣧⢰⠠⠆⠀⠀⢹⣿⣿
⣿⢪⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠈⠉⠀⠀⣀⠠⠤⠐⠒⠀⢀⣉⣉⡀⠀⠒⠂⠤⠄⣀⠀⠀⠀⠀⠀⠈⢀⡀⢸⠿⠁⠁⠀⠃⠀⠀⢾⣿⢿
⣿⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⣀⠄⢒⣩⣠⣤⣶⣾⣿⣿⣿⠿⡿⢿⣿⣿⣷⣶⣤⣀⠍⠒⠢⣀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⢸
⣿⠀⠿⠿⠀⠀⠀⠀⠀⡤⠤⠤⠾⠴⠧⢊⢄⣰⣾⣿⣿⣏⢑⣨⣿⣿⣿⣨⣵⣾⣿⣿⡇⢼⣿⣿⣿⣿⣶⡄⡑⠴⠦⠿⠤⠤⢤⠀⠀⢀⡄⠒⢲⡯⢸
⣿⠐⠆⠀⠀⠀⠀⠀⠀⡇⢩⣿⣾⣿⣿⣾⣟⠉⠻⣿⣿⣿⠾⠟⠛⠏⠉⠉⠉⠉⠻⠛⠻⠷⣿⣿⣿⠋⣼⣻⣶⣿⣾⣷⣷⣿⢸⠀⠀⠈⠠⠆⠀⠀⢸
⣿⣹⣄⠀⢀⠀⠀⠀⠀⡇⢰⣿⣿⣿⣿⣿⣿⣼⡿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢛⢷⣽⣿⣿⣿⣿⣿⣿⣯⢸⠀⠀⠀⠀⠂⠀⠀⢸
⣿⢹⣿⡏⠀⠀⠀⠀⠀⡇⠀⣿⡇⢸⣿⢛⠿⠯⠂⠀⠀⠀⠀⠠⡀⠄⣐⡠⢴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠹⠿⠛⣿⣿⢸⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⢈⡿⠃⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⢀⣴⣶⣿⣦⣴⡽⠖⢲⡿⢛⢻⣿⡄⢃⡠⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⣀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⡏⢹⣿⠆⠀⣴⣿⣿⣿⣿⣿⣿⡿⠤⢤⣿⣿⣶⣾⣿⡿⠩⠴⠂⠀⠀⠀⠀⢸⣿⡏⢙⣿⣯⢸⠀⠀⠔⠁⠀⠈⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⣿⣿⣿⠄⠀⠈⢘⡏⠉⢹⡏⠁⠀⠠⢸⣿⡏⠙⢉⣿⡇⠛⠻⠷⠦⣄⠀⠀⢸⣿⣿⣿⣿⣷⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡏⢸⣿⣬⣤⣤⠈⠁⠀⠈⠁⠀⠀⠀⠸⠿⠃⠰⠂⠛⢃⡀⠠⢀⠀⢀⣠⣤⣬⣿⣿⢸⣿⢁⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⣷⣾⣿⣿⣿⠿⣷⣄⣈⠁⠀⠀⡄⠀⡀⠀⠈⠑⠢⣍⠒⠌⠁⢀⣤⣵⠿⢿⣿⣿⣿⣾⣿⡟⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠿⠿⠿⠿⢿⣧⡺⢂⣿⠛⡷⣶⣴⣂⣤⣤⣤⣀⣤⣤⣰⣶⣶⣿⡿⣿⡌⢛⣼⡿⠿⠿⠿⠿⢻⢸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⢀⢀⠀⠀⠀⠀⠃⠤⠄⠤⠠⠤⢌⡙⢿⢿⣯⣬⣡⣾⠋⣭⣹⣿⠻⡟⢻⣿⠉⠙⣿⣄⣥⣽⡿⡿⢝⣠⠤⠤⠤⠴⠠⠸⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡞⠀⡀⡀⠛⡦⢄⣀⣀⣀⣀⠀⠀⠀⠀⠙⠲⢤⣉⡙⠿⢿⣷⣶⣾⣿⣴⣧⣼⣷⣾⢷⠿⠟⣋⣁⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠂⠀⠀⢸
⣿⢰⡾⢽⡷⠂⠃⠈⠉⠉⠉⠉⠛⠓⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠭⠤⣤⣠⣤⣤⠬⠥⠔⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣀⡑⢋⣿⢼
⣿⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣦⡀⣿⠙⠏⠀⠿⢸
⣿⡖⠠⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣐⠠⣐⠠⣀⠠⢀⠠⢄⡀⢄⡉⠍⣒⢤⣀⡀⠀⠈⢁⣤⡄⠀⠚⠛⡛⠉⢉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸
⣿⣚⡥⠭⠔⣒⣒⠠⠄⠀⢠⠄⠂⢀⡀⠀⠀⠈⠳⣌⡑⢦⣙⣦⣭⠶⠼⠖⠚⢛⠋⠉⢩⡇⠀⠀⢰⣿⡗⠀⠠⠜⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠁⢸
⣿⠖⣒⣋⣭⠤⠶⢒⣂⣩⠠⠷⠆⠁⠘⣿⣧⠲⠒⠛⠉⠉⠁⠀⠀⠂⠀⠀⠠⠃⠀⢰⠊⠉⠄⠀⠾⠛⠁⠀⠀⠈⠀⠠⠀⠀⠀⣷⣤⠱⢒⠀⠒⠀⢸
⣿⣋⣭⡤⠶⠒⣋⣩⣤⠴⠖⠛⠀⠀⠀⠹⡿⠀⠀⣂⣴⣦⡆⠀⠀⠀⢀⠀⠄⠀⠀⠈⠀⠁⠀⠀⠀⡀⣰⣤⡀⠀⠀⠀⠀⢠⣾⣿⠇⠇⠀⠀⠉⠂⢸
⣿⣡⡴⠶⠚⠋⠉⢀⠀⠠⠀⠀⠀⠀⠠⠆⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠂⠤⠤⠄⠀⠀⠀⠀⠀⠂⠙⠛⠛⠀⠀⠀⠐⠒⠈⠛⠉⠀⠀⠀⢀⣀⠀⢸
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣼
⡿⢿⠿⠿⡿⢿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⢀⠐⠢⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠑⠂⠀⣠⣄⣀⣠⣤⣶⢦⡴⡀
⠀⠀⠀⠠⣟⣷⣿⠽⣧⣄⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣤⢶⣗⣺⣿⣿⢿⣿⣻⢛⠛⠿⠹
⠀⠀⠀⠖⡠⠋⠽⠿⣿⡿⣿⣿⣿⣿⡿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⢀⣀⣾⣿⡿⢿⡷⠿⠛⠛⠓⠋⠊⠀⠉
⠀⠀⢐⠙⢶⣂⠒⠒⣤⡂⠁⠉⠿⠛⡛⠿⠿⠷⣾⣿⣿⣦⣤⣤⣤⣄⣀⠀⠀⠀⣠⣴⣦⣴⠿⠿⠛⠙⠉⡉⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⡤⣐⣤⡄⣀⣶⣀
⠀⠀⢸⣿⣮⠿⣿⡧⡼⠷⣶⣄⠤⠄⢤⡠⠤⠀⠀⠁⠊⠉⠛⠋⢛⣿⣯⣿⣴⣤⣬⣉⣉⣀⡠⠐⠒⢀⣀⢀⣀⣤⢀⣼⢣⣾⢣⣼⣃⣽⢏⣼⣿⢉⣽⡟⢁
⠀⠀⠘⠳⣾⢴⡝⢳⣿⣷⣄⣻⣶⣶⣒⣹⣧⣴⠤⢤⡄⡒⠄⡀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠉⢀⠠⠀⠟⢼⡏⣴⣯⣾⣇⣾⡏⣼⣿⠊⣹⣿⡟⠛⣿⣿⣟⡆
⠀⠀⠀⣘⢿⣿⣿⠾⢻⠉⡿⣿⣻⣭⣈⡛⢯⣦⣶⣖⡙⣿⣦⣳⣤⡠⠰⢦⣴⣄⡄⠆⠀⡄⣚⡔⣟⠴⢃⣴⠎⣤⣶⠇⣁⣤⣟⠛⡈⣻⠿⢧⣼⣿⣿⣻
⠀⠀⠀⠘⣴⣆⠜⠛⣿⣶⣶⣈⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⢯⣟⣣⣰⡾⣂⠹⣿⣾⣾⣶⠖⢋⣶⣯⣶⡾⣁⣶⡟⢉⣈⣹⠿⠃⢾⣿⣿⠗⣠⣤⣽⡉⠁
⠀⠀⠀⠀⠐⢾⣿⡇⣿⣿⣿⣯⣿⣿⣿⣿⡟⢿⣿⣛⣷⣤⣤⣘⢿⣿⣿⣿⣻⣿⣵⣿⣧⡾⣿⣿⣿⡟⠟⣋⣩⡿⣿⣿⡏⢶⣿⣿⣏⢩⠐⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠙⣾⣿⡟⣿⡉⡻⢙⣵⣴⣧⣴⣎⢹⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⢿⠏⣧⣤⣽⣿⣿⡿⣶⠿⠿⣿⣿⣿⣿⣿⣿⣿⢤⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⢯⣎⡄⢾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⢻⣿⠟⣋⣝⡻⣿⣿⣿⡟⠾⣰⣶⣿⣿⣿⣿⢿⣿⠏⣡⢶⠌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣟⣾⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⠟⠟⠻⠪⡽⣿⣿⣿⣿⣿⣩⠏⣡⣴⣿⣿⣿⣿⣗⡷⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⡾⣿⣿⣏⣴⢟⣾⢿⣯⣿⣿⣿⣏⣽⠶⣺⣿⣿⣿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠒⠻⠿⠟⠿⠟⠛⠛⠛⠻⠋⠁⠙⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣠⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⠻⡏⢿⣿⣿⣾⡿⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠂⠁⠯⣼⣅⠀⢿⣷⣶⠟⣁⠆⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣼⣿⣿⣤⣾⣿⢿⡟⣿⣶⣶⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⣿⣯⢹⡏⢻⣿⡽⠧⢚⣋⠉⣩⢉⡡⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⠟⣧⠚⣴⡁⡾⢺⣿⣴⣬⣿⣮⡿⣷⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠸⠿⡟⠤⠙⠦⠥⣦⣼⣿⢋⣥⠿⢋⣿⢻⠏⠹⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡿⢻⡟⡷⠒⠋⡀⠀⠀⠀⠀⠀⢀⣴⣦⣦⡀⣘⢆
⠀⠀⠀⠀⠐⠠⣀⣤⠤⣲⡀⠀⠀⠉⠀⠈⡜⠁⠀⠛⢿⣿⡿⠇⠀⠻⢟⠻⣷⣿⣿⣿⣤⡆⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠐⣚⡞⠀⠀⢸⡁⠀⠀⠀⠘⠧⠀⠀⠀⠀⠀⠘⢻⠟⠉⠉⠉⣠⡶⢶⣾⢆⣀⣤⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠁⠀⣄⣾⠀⠀⠀⠠⠀⣀⠀⡀⠀⠀⠤⣤⣾⡀⠀⠀⠀⠀⠠⢿⡀⠉⢻⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠙⢣⣨⠹⡹⣦⠀⡀⠈⠁⠀⠉⠶⠦⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢙⠀⠐⠊⠳⣄⢀⠀⠀⠀⢀⠈⡁⠂⠀⣀⣤⣄⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠁⠀⠀⠀⠈⠀⠀⠀⠉⠃⠠⠿⠋⠉⠵⠳⠄




⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⣠⠤⠖⠀⠀⢀⠀⠑⢠
⠀⠀⠀⠀⠀⠂⠀⠀⢠⢥⣅⠀⠀⠀⡤⠂⠀⠐⠁⢀⣬⠁⠀⣺⠛⠀⠀⠀⠀⠀⠄⠀⢸⡟⢶⣤
⠀⠀⠀⠀⣀⡐⣀⣰⣙⡣⠦⠀⠀⠀⡀⠀⠀⠑⠄⠀⡄⠀⣴⢻⡋⠀⠀⢤⣴⣴⣷⣷⣾⣿⠟⢑⠂
⠀⠀⠀⠀⠀⠉⠀⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠀⠛⠈⠋⠁⠀⠃⠈⠀⠙⠛⠛⠉
⠀⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢨⡀⠀⣄⡀⠀⠀⠰⡀⠁⠈⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠉⠉⠁⠈⢴⡿⠤⡔⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠟⠁⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠠⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠒⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠡⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠐⠀⠀⠀⠘⡀⠢⠀⠀⡦⠀⠀⠀⠀⠘⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠘⠠⡀⠀⠀⠁⢢⡤⠀⠀⠀⠀⠀⠐⠯⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⢀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⣂⡉⢀⡀⠀⠀⠀⠀⣄⡀⠀⢠⡀
⠀⢀⡀⠀⠀⠈⠀⠀⠘⠃⠀⣀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠕⡄⠁⠀⠀⠀⠀⠖⠀⠀⠀⠀⡀⠀⠈⡃⠄⡄⢀
⠀⠀⠛⠧⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠔⠁⡠⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠰⢞⡀⡀⠈⠁⠀⠀⠀⠈⣹⢀
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠉⠙⠲⠀⠀⡆⠀⠀⠀⠀⠀⢭⡔⠀⠀⠀⠀⠀⢀⢀⠁⠂⠀⠀⠀⠀⠂⠴⣇⠘⠂
⠘⠂⠀⠀⢻⡶⠠⠀⠀⠀⠀⠀⢰⡆⠀⡀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠷⣰⡄⡀⠀⠀⠀⠀⠀⠺⠤
⠠⠀⠀⠀⠈⠁⢂⠄⠀⠀⠀⠀⠀⠀⠀⢉⠑⠀⠀⠀⠀⣠⠄⠀⠐⠀⠂⠀⠀⠁⢱⠉⠁
⣓⢀⠀⠀⠀⠀⠈⠰⢃⠠⠂⠀⠀⠀⠒⠞⠀⠒⠀⠀⠀⠀⠀⡠⠤⢀⠀⠀⠀⠉⠉⠉⠁
⠁⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠃
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈
⠀⠀⡃⠂⠈⣧⡂⡅⢈⠁⠀⠀⠐⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⠀⡔⠀⡈⢀⣇⡐⠀⠀⠀⠠⠀⠇⢠⠀⢸
⠀⠀⣇⣺⣄⣿⢿⣥⡾⠃⣏⡴⠖⠀⢦⢇⣀⠀⠖⠀⠀⠀⠀⠈⠀⣠⠀⠀⠀⠀⠀⡀⠄⠀⣰⣈⠅⣗⣾⣫⣿⣵⣥⣷⡗⡃⣰⠈⢀⠍⠐⢨
⠀⠀⡿⣷⣿⣿⣷⣾⢷⠧⣾⡆⣡⠰⡆⣺⠟⠿⡀⠀⡄⢠⣆⢰⠀⠠⠀⠀⠀⠐⠂⣷⠌⣡⣌⣿⣿⢇⢿⣿⣯⡷⡞⣋⣾⣷⣿⡿⣿⢳⣾⣿
⠀⠀⠟⣻⣿⣿⡧⠔⡾⢷⣦⣶⣷⣜⠥⣿⢌⡄⠉⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠶⣦⠓⣿⠿⣿⣾⣆⠚⠿⣿⣷⣻⣿⣿⣿⣟⠹⢽⣾⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣷⣾⡿⡗⢩⣼⡿⠿⢾⠗⡲⠂⠀⢀⠒⠠⣁⠈⠀⠄⠀⢉⣀⢴⣿⣶⣿⣷⣿⢿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣼⣻⢟⣷⣶⣼⣏⠰⣇⣭⣴⠃⣤⣈⡽⠋⠔⠀⠀⠀⠠⢚⠫⣿⣷⣿⣿⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿
⠀⠀⣇⢻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣊⣻⠛⠓⠂⠀⠀⢀⣀⣨⢖⣉⣫⣙⣱⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡛⢑⡥⠀⠤⠀⡰⢈⣉⣀⣶⢉⣤⣴⣷⣼⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣯⣿⣕⢶⣏⢠⣼⣝⡀⠈⣡⢀⣤⡋⠙⢛⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿
⠀⠀⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⡃⠱⡙⢻⣧⣽⡿⣿⣿⡟⢟⣛⣿⣿⣿⣿⣿⣿⣾⣿⣿⢍⣽⣿⣽⣿⣿⣿⢿⡿⣿⣿
⠀⠀⣷⡞⣽⣿⣿⣻⣾⣿⢟⣣⣼⣿⣿⣿⣿⣤⣶⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣣⣊⡹⣳⣯⣾⣿⣿⣿⠟⠿⣿⣿⣿⣿⣿⢿⠻⣷⣸⡿⠋⢹
⠀⠀⠇⠉⢹⠊⠻⣿⣿⠿⠿⣿⡿⠿⠟⠟⠛⢯⠟⠙⣿⠏⠉⢹⡏⠙⡟⠉⠿⠁⠙⢈⣩⠽⠛⣽⠿⠃⣴⣾⣿⣿⠟⢸⣿⠀⠀⢘⣿⡇⠀⠈
⠀⠀⡅⠀⠀⠀⠀⢿⣿⠀⠀⠸⠇⠀⠀⠀⠀⠉⠀⠀⢸⡀⠀⠀⠀⠘⠃⠀⠀⠀⠘⠋⠀⠀⠈⠁⠀⠈⣡⠞⠏⠃⠴⠾⠷⠶⠀⠀⣿⡇
⠀⠀⡅⢀⣀⣀⣀⣿⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠤⠠⠆⠠⡖⣆⣰⡠⣄⠸⣷⣦⣶
⠀⠀⡿⠿⣿⣿⣿⡿⣿⡿⣿⡿⠿⠗⡄⡎⠠⠶⠂⠄⠰⠶⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉
⠀⠀⡇⠀⠀⠦⠄⠄⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠠⠦⠄⠀⢁⢀⡀⠀⣀⡁⢀⠘⡁⠀⢈⡀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢘⣠⣀⠸⠀⠤⠀⠀⠀⠀⠀⠀⠠⠧⠄⠤⠀⠀⠀⠀⠀⢀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⡄⠀⠀⠀⠀⡀⠀⠀⠈⠃⢀⠇
⢀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢱⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠁⠀⠘⣇⡠⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣄⠀⠀⠀⠠⠩⠄⠀⠀⠀⠈⠾⠁⢠⡀⠀⠀⠀⠉⠒⠤⡆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠢⠜⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⡄⠐⠀⠀⠀⠀⠨⡤⠀⠠⠀⠀⠀⠀⠠⠦⡴⠀⠠⠇⠀⠈
⠀⠀⣄⡀⠀⠈⠁⠀⠀⠈⠁⠠⣄⠀⠀⠀⠀⠉⠖⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⡚⡆⠀⠀⠀⠀⠀⠐⠁⡈⠀⠀⣀⡄⠀⠀⠕⣒⣰⣀
⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠒⠃⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠚⢭⢠⠄⠀⠀⠀⠀⠀⣀⣸⣀
⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠈⠙⡶⠀⠐⡏⠀⠀⠀⠀⠀⠰⣧⠋⠀⠀⠀⠀⠀⢀⢀⠌⠈⠀⠀⠀⠀⠀⠢⠻⡇⠘⠂
⡘⠇⠀⠀⢘⣷⠆⢄⠀⠀⠀⠀⠀⢰⡆⠀⡀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⢰⡆⠀⠀⠀⠀⠀⢀⠀⠇⠄
⡡⠶⠂⠀⠈⠁⡰⣠⠀⠀⠀⠀⠀⠀⠀⠀⢑⠘⠀⠀⠀⠀⠀⡤⠀⠀⠠⠐⠂⠀⠀⠊⢸⠎⠈
⡶⠀⠀⠀⠀⠀⠀⠃⡼⠀⣀⠀⠀⠀⠀⠤⡼⠀⠂⠀⠀⠀⠈⠈⠉⣠⢀⠀⠀⠀⠀⠒⠘⠓
⠋⠈⢀⠀⠀⠀⠀⠀⠂⣁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡷⠤⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⢰⡆⢀⣶⠶⠒⠠⠀⣰⣶⣶⣶⣶⣶⣦⡤⠒⣠⡀⣖⣤⢖⣠⣴⠒⠀⠀⠠⢀⣀⣀⣀⡀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠘⢀⡾⣁⠀⢑⠀⣶⣿⣿⣿⣿⣿⣾⣿⡠⠛⣈⢁⣿⠀⢈⠛⡉⠝⢫⡤⠄⣀⣀⣀⠁⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⠉⠗⢶⡒⠀⣀⣀⡀⠀⠀⠐
⢠⡟⡸⠛⠡⠀⣼⣿⣿⣿⣿⣿⣯⣥⣠⣁⣠⣴⣾⣿⣧⡌⠲⡁⠀⠈⡵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡝⠉⠋⠑⠀⠀⠀⠀⠼⠿⣄⠰⠶⢤⡍⠉⠉
⢸⡇⠩⢋⡤⠁⠹⣿⣿⣿⣿⣿⠂⣠⠄⠙⣿⣿⣿⣿⢷⠄⠈⠀⠤⠤⠄⠀⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠀⢀⠴⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢲⡄⠀⣻⡂
⢠⠀⣠⣾⣡⡐⠀⠀⢛⠝⢩⣿⣾⣅⣀⡄⠠⡝⣿⢇⢫⡈⢀⣠⣄⡀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡀⣠⠄⡃⠀⠀⠀⠁⢠⠀⠀⠀⠀⣨⣴⠃⠀⠘⠥⣶⠟⠉
⠈⠀⠉⠏⠡⠀⢁⠴⠆⠁⠘⡀⢠⣾⣿⣿⣶⣴⠶⣤⣀⣀⡺⢿⣿⣿⣿⣿⣟⣀⠀⠀⠉⡁⢀⢸⠇⠀⣀⣇⣠⠴⠂⠀⢀⠛⠲⠶⠤⠒⠁⣠⠀⠀⠀⠏⠑⢦
⢠⠠⣄⡀⣠⣴⣯⡄⢁⣞⠈⠑⠻⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣧⢙⡿⢟⣻⡿⣿⣿⣶⠆⠘⢯⠀⠀⡏⠉⠟⠁⠀⠀⠀⣄⡙⠷⣀⣤⣴⢻⣟⣷⣶⠀⠀⣀⣸⡇
⠸⠓⣶⣾⡿⣦⣿⣿⣿⣯⡶⠶⠂⠈⠉⠉⠁⢸⢿⣿⣿⣿⣿⣿⠂⠒⠉⢕⡞⠹⢇⠀⠲⠀⢀⠀⣬⠑⠀⢀⡀⠀⠀⢀⣾⣯⣭⣤⣉⣉⢀⠈⠉⠀⢀⠈⡯⣟
⡇⣶⣿⡏⢰⣿⣷⡞⢁⡼⣶⣿⣦⣄⠄⠀⢨⢀⠀⠹⣿⣿⡿⠋⢀⠈⢰⠄⠁⠦⠘⠂⠒⠐⠋⠀⢘⡿⠃⠈⢳⡤⠀⠉⠉⠛⠉⠹⠚⣿⣿⠀⠀⣀⡟⠀⠀⠈⣷
⣷⣿⣿⣿⠸⣿⢡⣿⡜⠁⠈⢿⠿⠟⠊⠀⠆⣬⢢⠀⣀⣀⡀⠀⣘⡰⠃⠤⠀⢱⣴⣠⣤⣤⣷⣀⣸⣇⠀⠀⢀⡴⡀⠀⠀⠀⣠⣤⣼⣿⣍⢙⣶⣄⠀⠀⣘⣷⠏
⠹⡟⣿⡿⠀⣿⡼⠿⣧⠈⢀⣄⡀⣀⢄⡀⣠⣿⠇⠉⠉⠉⠙⠛⠉⠃⠛⠂⣻⣙⣿⠛⠉⣼⠉⠹⠟⢹⡿⣾⣷⣟⣿⣶⣷⢶⡛⠋⠉⣿⣩⣯⣼⣿⡿⢿⡿⢃⣠
⢠⠀⢹⣧⠐⠉⣧⡀⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠻⣶⡉⠉⠉⠙⠋⠛⠓⣿⣧⣠⣶⡀⢀⣿⣀⠉⠈⠁⡼⢛⡀⠀⠈⣠⠟⠛⠛⠛⠛⣸⠛⠛⢛
⠀⢰⣼⣿⠓⠖⠮⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠍⢉⣹⣿⢄⠂⠄⣠⣄⠀⠀⣸⣿⠀⠀⠀⠀⠀⠼⠦⠀⠉
⠀⠈⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠟⠈⠀⠆⣿⣯⢄⡓⠦⣿⡀⠠⠦⠀⠀⠶⠆⠀⣤
⠀⠀⢿⣿⣿⡄⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⣿⣯⡀⠙⠲⠤⢉⡟⣿⢦⡀
⡇⠀⠸⣟⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⢰⣯⣽⣿⡄⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠙⠷⠈⠀⢄⡁⢀⡃⣿⡰⣌⠑⠢⢄⡀
⠃⢀⠀⣿⠸⣿⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣫⣾⣿⠋⠁⠀⢸⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠈⠲⡇⣿⣏⠀⠳⣄⠀⠉
⠀⠸⣶⠸⣇⢿⣿⣆⠈⠀⠟⠶⠀⠈⠛⠋⠀⠁⠸⠓⢀⡴⢋⣴⣿⣿⣿⠀⢠⠀⢸⡿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣇⣿⠢⡀⠁⠘⠳⠄
⠀⡀⢿⡀⢻⣎⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⣾⣧⢞⣿⣿⡀⠀⠀⠈⠻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠫⠀⠈⠒⢄
⠀⢁⠸⣷⡀⠙⢦⣽⣿⣧⡀⠀⠀⠀⠀⣀⢀⣴⠏⠀⣂⠼⢋⣠⡟⠽⠏⠁⢀⣀⠀⢉⢺⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣈⠶⠀⠉⠢
⢀⠘⢠⡿⢿⡶⠤⣈⠉⠛⠿⠖⠀⠀⢺⣥⡌⢱⡴⠟⢁⠔⡱⢋⠎⠀⠁⠀⠘⠉⢁⣴⢈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠃⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠙⠀⠔⠉⠜⠁⠀⠀⠘⠓⠀⠀⠚⠚⠓⠘⠾⠿⠿⠿




⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣶⣴⣿⣴⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡐⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⡴⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣀
⠀⠀⠀⣀⢄⡄⠀⠠⢤⠆⠀⠇⠀⠀⠀⠀⣠⠀⢀⡠⠀⠀⠈⣦⠁⠀⠰⠄⠀⠀⠀⠘⠁⠁⠰⠀⠐⡄⢠⢰⡆
⠀⠀⠘⢆⠘⠀⠀⢀⠄⠀⠀⠀⢲⡆⠎⢰⠋⠀⢸⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⠀⡁⠀⠀⠀⠀⣀⢡⠀⠘⠁
⠀⠠⠀⢸⡗⠉⠀⠀⠀⠀⠀⠀⠀⢒⢀⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠔⠀⡀⠀⢸⡾
⠂⠀⠀⠉⠁⠈⡆⡧⠐⠀⠒⠂⠀⠼⠿⠀⠀⠢⠐⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⢸⣶⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠉⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣸⣿⡄⠀⠀⠀⠛⠁
⠀⠀⠉⠁⠀⢀⣿⣿⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⣀⣰⣦⡘⡗⠒⢰⣦⢸⠖⣦⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡶⣶⡠⣦⣀⠄⢀⣄
⠀⠀⠀⠀⠀⠀⡻⣧⣙⣽⡟⢻⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⠱⣿⣳⣿⣝⣦⣿⣿⣽⡆
⠀⠀⠀⠀⠀⠴⠱⢽⣿⢋⣽⣿⣟⣿⣿⣿⣿⣿⣧⣧⣆⡠⠄⠀⢠⠀⢄⡀⢀⣀⣴⣶⣥⣿⣷⡄⠹⢿⣿⡟⣻⠝⢱⡍⢡⣶
⠀⠀⠀⠀⣰⢈⠄⣤⣿⣿⣿⣷⣿⠿⣷⣿⣿⣿⡆⢻⡏⣿⢸⠐⠀⠀⠸⠰⣾⢹⣿⣷⢿⣿⣿⣧⡾⡞⣱⢿⣏⣷⣉⠱⣲⡧
⠀⠀⠀⠀⣼⠟⡳⠛⢻⣝⡻⠹⠓⠲⠋⠉⢱⣘⠱⣏⠉⠋⠀⠀⠀⠀⠀⠀⠈⠋⠳⠧⠘⠙⠟⠐⢦⠄⢩⠙⠷⠗⡉⢰⡷⣶
⠀⠀⠀⠀⠁⣀⣇⡀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠈⠚⣴⣿
⠀⠀⠀⠀⡀⣿⢿⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢇
⠀⠀⠀⠀⢭⣿⣛⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣬⠎
⠀⠀⠀⠀⠚⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣛⣞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛

⠆⠙⠳⣿⣆⠠⠄⠠⣔⢀⡀⠈⢿⠇⠀⠐⡀⠀⠘⠣⠀⠈⠢⠤⠀⠀⠈⠐⣧⡀
⠀⢀⣐⡀⠀⡰⠶⡄⠁⠘⣿⡦⢈⡀⠀⠓⡄⠂⠁⠀⠀⠀⠀⠀⠐⢃⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢰⣅⠀⢵⠀⠀⢈⣸⣦⠀⢹⡇⢃⠈⣐⠀⠀⠘⢳⣄⣀⠄⠀⠈⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠈⣿⡇⠾⢰⡈⠈⠀⣨⣦⡸⠦⠁⠠⣌⠀⠀⡀⠀⢻⣿⠀⠀⠀⠁⠀⠀⠐
⠠⢍⠁⠀⢸⡿⢀⠒⢿⡏⠀⠀⠁⢀⡀⠑⢤⣁⠩⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠘⠇⠈⠒⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠂⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⠀⠀⠀⢠⠜⠛⠁⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣀⠀⢀⣀⡾⠿⠛⠘⢿⣷⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⡀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢁⣷⠆⠘⠀⠟⣫⠋⠀⢀⣿⡄
⠀⢀⣤⠂⠀⠈⠲⡄⡈⠀⢠⣿⣦⣾⣧⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⢀⠉⠄⢀⣼⣿⣀⣠⣴⡿⣸⡄⠀⠰⠿⡀
⠀⠀⠿⠐⠫⡗⠘⠃⠀⠀⢸⣏⢉⢹⡷⠾⠿⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣠⣄⣀⣀⣀⣆⣀⣰⣄⣴⣷⣠⣿⣤⣧⣗⣠⣤⣴⣶
⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣤⡤⣀⣀⡀⢀⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⠀⠒
⣐⡒⠬⠥⣀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡒⢄⡢⢄⡢⢔⡠⢀⠠⣈⢑⣲⡤⠴⠞⠛⠛⠣⡴⠄⠤⠀⠄⠀⡤⠀⢻⠏⢿⡿⢿⡟⣿⢿⠇⠎⠄⠀⠤
⣒⣲⠭⠕⣒⡩⠭⢐⣂⠠⠄⡂⠀⠀⠀⠀⠀⠀⠈⠲⣍⢲⣍⡶⠽⠒⠛⠉⠁⠀⠀⠀⢀⠀⠒⠙⠀⠆⠀⠀⠀⠀⠂⠀⢸⠀⢸⡇⣼⠁⡟⠀⢀⠄⢀
⠥⢖⣋⡭⠖⣒⡭⠴⣒⡪⠥⢒⣂⠥⠀⣀⣠⡤⠶⠚⠋⠁⠀⠀⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⣶⠐⠀⠀⠀⠀⠀⠸⠄⣾⠦⣿⠴⣷⢡⣾⣶⠁
⣛⣭⠴⣚⣩⠵⢒⣫⠥⠖⣋⣩⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡧⢀⡀⠀⠐⠚⠂⠀⠀⠙⠛⠂⡄⠀⠀⠀⠀⠠⠀⣤⣴⠃⠀⠀⣶⡍⠈⠁⢠⠐
⡥⠖⣛⣭⠶⢚⣩⡴⠖⠋⠉⠤⠒⠀⠁⠀⠄⠀⠀⠀⡀⣶⡷⠀⠀⠀⠠⠠⠀⠀⠀⠀⢀⢄⠠⠄⢀⠀⠀⠀⠀⠀⠠⠄⠘⠁⢠⠜⠇⠀⣰⡿⢃⣤⡄
⡶⢛⣩⡴⠞⠋⠁⠰⠒⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠚⠁⠀⠀⠀⠀⠀⠶⠿⠇⠀⠀⠀⠖⠈⠀⠀⠀⡠⠀⠀⠀⠀⡀⠃

⠀⠀⢜⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠀⠀⠤⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠿⢀
⠘⠁⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⣈⠁⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡐⠠⠀⠀⠹⡀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⢷⡀⠀⠀⠀⠐⡄⠀⠀⠀⠢⡄
⠂⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⣦⠀⠸⠀⠀⡧⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢅
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣠⢞⡡⠂⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⠙⠷⣄⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠛⠀⠀⠀⠁⠉⠀⠀⠀⠉⠒⢤⠤⠠⠀⠀⠀⠈⠉⠀⠠⠀⠀⠘⠃⠖
⢀⣤⠒⠀⠠⣤⣀⠀⠀⠀⠈⠋⢀⣀⡤⢾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⠀⠀⠠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠉⠀⡀
⡛⠀⠀⠀⠀⢈⣿⡇⠀⢠⡴⠛⠉⠀⠀⠀⡽⢡⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠱⠀⠀⠀⠈⠁
⣧⣀⣀⣀⣀⡿⠋⠀⣰⠋⠀⠀⠀⠀⠠⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠂⠀⠀⠀⠀⠄⢈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠰
⠈⠙⠛⠋⠁⢈⣠⡴⠗⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠈⠉⠀⢀⣠⣤⣤⢤⣄⡀⢀⠀⡀⠀⠀⠀⠀⠀⠑⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠤
⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⢠⠀⢰⠋⠀⠀⠀⡀⠀⣄⠀⠈⠋⠀⠀⠀⠀⠈⢀⣄⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡄⠳
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⡀⠀⠀⠀⠀⠀⡹⠀⠀⢀⠀⠀⠈⠀⠐⣄⡀⠀⠁⠀⠀⠀⢀⠀⠀⠐
⠀⠀⠀⠳⠤⠖⠂⠁⠂⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⣫⠂⠀⠀⠀⠈⠐⠠⠤⠤⠊⠀⠀⠀⠀⠀⠁⠀⢀⠐⠀⠀⠀⠀⠀⡄⠀⠀⠁
⠤⠤⠀⠀⠰⠀⢈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣐⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠄⠀⠀⠀⠀⠈⠊⠁⠢⢀⠀⠀⢄⣵⠀⠀⢰⠏⠀⠀⠀⣠⢀⠆⠀⠀⠀⠠⠀⠀⢀⡤⠂⠀⠀⠂⠀⠀⠀⢀⡀⠀⠂⠀⢰⠋⠉⠓⠀⠠⡀⠸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⠠⠀⡿⣡⠄⠀⠀⠠⠤⠶⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⢻⣄⠀⠀⠀⠐
⠄⠀⠀⠠⢀⠤⠀⠀⠀⠀⠀⠠⠀⠐⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⢀⠀⠈⠹⡆⠀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣇
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡟⠀⠀⠀⠈⠙⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠘⢦⡀
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⢻⡄⠀⠀⠀⠐⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠙⠦⣄
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⣠⠀⢀⠓⠂⠀⠤⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠰⡄⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡠⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀
⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠍
⠀⠀⢱⡀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⡆⠀⠐⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠇⠀⡀⢹⠀⠀⠀⠀⡼⢷⣶⣧⣾⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣈⣾⣯⠻⠀⠠⣽⣸⠼⣿⣿⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⢻⢿⣿⣿⣿⢇⡄⣼⣿⣧⣿⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣤⠀⣀⣰⣤⣿⣿⣿⣿⣿⣿⣦⣼⣿⠛⣿⣿⡷⠧⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣶⣿⣷⣿⣧⣽⣿⣿⣿⣍⣿⣿⣿⣽⣿⣅⣿⡏⠀⠀⠀⢀⠀⠀⢅⣠
⠀⠀⠀⠀⠀⠀⡄⡂⠀⠀⣴⠯⠻⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣦⣴⣤⡜⣤⣿⠟
⠀⠀⠀⠀⠀⠀⢠⣿⠶⢏⣁⡄⢻⣾⣿⣿⣿⣿⣿⣿⣯⠛⢿⣿⡿⣿⣟⢻⣏⣼⣎⣿⣿⣿⣤⠾⠀⣀⠀⢀
⠀⠀⠀⠚⣄⢂⢿⣿⣮⣉⡛⣿⣿⣿⣿⣶⣿⣿⣿⣝⢻⡗⠸⣏⢳⣸⠋⠙⢛⢿⣿⡿⡛⣹⡋⣀⢀⠡
⠀⠀⠀⠺⣽⣧⣸⣿⣿⣽⣯⣽⡿⢿⣿⣿⣿⣿⣿⣮⣿⣏⣠⢋⣼⣇⠀⠖⣹⣯⣎⡠⣞⣙⣓⠍⣍⠁
⠀⠀⠀⠠⠍⠙⠿⢿⣿⣿⣿⣿⡾⠯⠶⣿⢿⣿⢿⣿⣿⣯⡟⠁⢿⡿⠳⢤⠁⢸⣣⡞⠛⠈⠛⠛⠂⡤
⠀⠀⠀⠀⠀⠈⠀⢦⢼⣎⣙⣳⢬⣷⣺⣶⡘⠙⣦⡽⣿⣿⠀⠀⠀⣇⠀⢸⠷⣾⣹⠛⣇⠀⣜⡀⠀⠀⠀⠀⠀⠀⡠⣤⣴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣎⡛⠙⢹⣛⣿⣿⣟⠻⣿⣿⠀⠀⠀⢿⡰⠋⠀⢸⣏⠠⠀⠀⠘⢹⠛⡁⠀⠀⠀⣼⢿⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠈⣿⣦⡤⠞⠛⠳⣿⡛⢿⣿⡆⠀⠀⣼⡁⠄⠀⢸⢧⠀⠖⠙⣳⡯⢾⣇⣻⠶⢛⣿⣏⣉⣮⣻⢧⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⣧⡼⠀⠈⠳⡄⣿⡇⠀⠀⣿⠁⠄⠀⡞⢸⣧⣴⣿⣿⡟⠻⣯⣉⣴⣿⣿⡟⠛⠚⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠿⣦⠉⢔⠿⣿⣿⣂⣼⣿⡀⣹⣿⣷⣾⣿⠵⠀⢉⡁⣠⣾⣿⠿⣷⡦⠶⣤⣶⡛⠑⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢻⣖⢨⣿⣿⠴⣿⣿⣟⣽⢋⣘⣿⡒⢀⣶⠋⢠⣂⣀⣴⢻⠞⣪⣽⡇⠉⠊⠕⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣇⣼⣿⣧⣀⡽⣿⣿⣆⣿⣿⡿⠟⠋⢙⣯⣵⠏⣁⠠⡶⠛⠉⠚⢯⢉⠉⠃⠪⢯⠐⠆⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣛⠻⠟⠛⠿⣿⣀⠁⣦⣹⣿⣿⣿⣿⣷⣴⠂⣲⣿⣷⣞⣡⣾⣡⣴⣾⢦⣾⣬⠭⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠊⠁⣀⡀⠸⢢⡌⢿⣏⣹⠿⣻⣿⣿⣿⣅⣤⣹⣿⣿⣿⣿⣿⣿⣿⣾⣷⡿⣋⣨⠀⣀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠁⠀⠀⠘⡩⠄⣿⡇⠀⢸⣿⠎⠛⣯⣠⣿⣻⡿⠿⠻⠟⠻⠟⠻⠻⠛⠿⡿⠿⢶⣍⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠤⣶⠁⠂⠸⣧⠀⠺⣿⡄⣴⣿⣿⠟⠋⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠱⡑⠤⠽⡬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⣿⣿⣿⡟⠀⠀⡠⢶⠚⠛⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⡀⠰⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣿⣿⣿⣁⣠⣀⡤⠼⣀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣴⣿⠤⠿⠿⠷⠶⡿⠿⣿⣿⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠿⠾⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠠⠤⠤⠠
⠀⠀⠀⠘⢷⣛⠓⣂⠔⠀⠀⢀⡼⣿⠀⠀⢐⠀⠀⠔⠉⠛⠁⠀⠁⠀⡠⠀⠀⠀⢾⡀⠐⢀⢂⣀⡔⢰⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼⠟⠐⠁⠀⠀⠘⠦⠤⠂⠐⠢⠀⠀⠐⠊⠀⠀⠀⠀⠈⠉⠓⠊⠉
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢢⡑⠂⠀⠀⠧⠥⠤⢤⠂
⠀⠀⠀⠹⠶⠄⠀⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣨⠇⠀⡄⠀⠀⠀⠀⢠⣴⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠧⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⣨⣿⣀⠀⠀⠀⠈⠂⠄⢀⣈⠭⠋⠀⠀⠠⠤⣤⣤⣾⠟⠁⠀⠀⠀⠀⠀⠈⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠈⠑⡀⡀⣦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠠⣄⣀
⠀⠀⠀⢦⣤⠤⠴⠆⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠁⠀⠈⠀⠀⢸⠀⠀⠂⠀⠐⡊⠀⠔⠀⠈⠻⢦⡀⣀⣠⠾⠃⠀⠘⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠰⠀⠀⠀⢀⡠⠄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣴⠖⠁
⠀⠀⢠⠂⢠⣷⠤⣀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⢀⡤⠀⢤⠀⠁⡐⠒⠲⢦⠀⠀⠀⠀⠀⠙⠛⠀⠁
⠀⠀⠘⢶⣿⣁⣠⣻⠇⣄⢰⣿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⡀⠘⡀⠀⣀⡾⠀⠀⠤⣤⣴⣶⣶⣶⣶⣶⢀⣀
⠀⠀⠀⠀⠀⠙⠛⠋⠀⣈⡛⠋⠀⠀⠀⠈⠓⢄⣀⣀⠄⠀⠀⠀⠀⢙⠚⠋⠀⢀⠈⣈⠉⠁⠀⢶⣾⣿⣿⣼⣿⣿⣿⣿⣿⡿⠄
⠀⠀⠀⣀⣀⣴⡇⠀⠒⠛⠃⠀⢧⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣾⣅⠀⢐⠀⠀⠀⡌⡇⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠊⠀⠐⠒⠛⠁⠀⠀⠋⠉⠉⠉⠉⠉⠉⠙⠃

⠀⠀⠀⣀⡠⠄⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠤⢍⣩⠩⣽⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣣⡄⠀⠀⠀⠀⣾⣷⣾⣂⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠨⠿⠞⠛⣟⣻⣿⣿⣧⡤⣀⡀⠀⠀⠒⣻⣿⡿⠅⠀⠀⣰⣾⣿⣿⣿⣿⣶⣦⠀⠀⠼⣿⣿⣟⣻⡿⣿⣿⡉⠃
⠀⠀⠀⢀⣀⣤⣴⣿⣿⣟⢿⣿⣿⣟⡿⢷⣤⣀⠈⠇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⢸⡏⠁⠉
⠀⠀⠐⠋⠉⠉⡬⣭⣿⠿⣟⡯⢿⣿⣷⣾⠛⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⢃⣀⡀⣐⣦⠤
⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⢠⡄⢀⣀⠀⠉⠛⠉⠉⢹⠏⠉⠙⠋⠁⠀⠀⠀⠀⢠⣿⣿⣿⣧⣿⣯⣵⡆⡻⢶⡀
⠀⠰⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⣠⣤⣶⣿⢐⣬⠶⢉⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⣀⣴⣶⣦⡄⠀⠀⠋⠛⠛⢿⡻⡿⠿⠿⠃
⠀⠘⡿⣽⣿⣿⡿⠄⠀⠀⠀⢠⣿⣾⣿⣿⣏⣿⣿⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣜⢻⣿⣿⡟⠛⡃⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠘⠉⣉⣉⠁⠀⠀⠀⠀⠾⠘⠋⠿⢿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢯⣈⣻⣿⣿⣭⣤⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠠⣽⠦⠀⠀⠀⣠⣤⣤⣤⣤⣠⣀⡀⠈⠉⠱⡿⠉⠀⠑⠀⠠⣤⣶⣦⣤
⠀⣠⣴⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣏⣻⣇⠤⠄⠀⠀⠀⠀⠀⣾⡿⢿⣿⣿⣿⣷⣦⡄
⠀⣿⣯⠽⠿⠿⠾⣿⣿⣿⣿⡟⠿⣿⣿⣿⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣄⣀⠀⣴⢶⡾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠈⠀⠀⠀⠀⠿⠿⠯⠟⣿⠧⢴⣿⣿⣿⣿⣦⠀⠀⠉⠈⠉⠻⣻⣧⡿⠋⠉⠈⠉⠉⠁⠀⠀⠀⠈⠁⠈⢻⡿⢩⠋⠻⠟⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⢻⣿⣿⣿⣿⠿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣷⣿⣿⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⣴⣼⣿⣿⣿⣿⣿⣿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣷⣤⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣟⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⣻⣿⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⢰⣿⡿⣿⡿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡏⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣿⡇⣿⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡾⠛⣷⡙⠀⢠⢸⡟⣰⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣽⡧⠆⢼⣇⣿⣿⣾⣧⣿⠋⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⣿⣿⣣⣀⣾⣿⣿⣿⣿⣿⣿⡿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡇⣸⣿⢟⣼⣿⠻⣿⣿⣿⣿⡇⠀⢻⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣶⣿⠏⣼⡿⠃⢘⣿⣿⣿⣿⣿⡀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣽⡟⢁⣼⡟⠑⣀⣿⣿⣿⣿⣿⣿⣧⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣡⣵⡿⠿⠷⠾⠿⠛⠛⠁⠀⠈⠻⠿⣄⣀⡂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠼⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⠀⠀⣠⠔⠚⠛⠛⠒⠦⣤⣀⣀⣀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⢻⡇⢰⠁⡤⠶⠶⠦⢤⣀⣀⣀⣤⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣘⣇⠘⢛⣥⣦⣄⣠⣀⠀⠀⠀⠉⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⢻⢠⣟⠉⠀⠀⠙⠳⢮⡵⠶⠛⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣿⠖⠀⠂⠀⠀⠀⠀⠀⠀⠬⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⣠⣷
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⡆
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣧
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡃⡏
⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡴⢠⠇
⠀⠀⠀⠀⠀⠀⠈⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⣿⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢳⡽⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣾⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠆⠀⢀⡡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⡤⠤⢤⠤⠤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⠴⠂⠀⠀⠀⠀⠉⠋⠁⡄⠀⡀⠀⠀⠀⠀⢠⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⣧⠞⠛⢫⠄⠀⣄⠀⡠⠀⠓⠀⠙⠖⠦⠀⡀⠀⡷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢨⠀⠀⢀⠀⠀⠹⣧⣣⡀⠀⠀⠠⡄⢀⠀⠻⡦⢀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠇⢀⣨⣻⡀⠘⡞⢠⡀⣰⠋⠉⠉⠀⠙⠶⡿⠋⠀⡰⠋⠀⣀⠀⠀⢸⡇⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠺⡄⣠⠀⠀⠀⣑⣄⠀⠳⣄⡀⠈⢻⡇⠀⠀⠠⠀⠀⠀⡇⣠⠞⢠⠆⠀⣹⣀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⢴⠶⢼⡟⠀⠀⠀⠀⠈⠑⣤⣙⣷⣄⢸⠀⠠⣄⠀⢷⡀⣠⣿⡧⠶⠚⠋⡙⠉⠀⠈⠓⠀⠀⠈⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠱⣄⠀⢀⡄⠀⠛⠁⠀⠀⡉⢻⣇⢠⠁⠀⢨⣿⠟⡁⠀⠀⠘⢦⡷⢚⡀⠀⢰⡇⠔⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⣈⣳⣼⣟⢠⣄⡀⣹⠙⠁⢀⠹⣷⡀⣰⣿⢁⠠⠇⠒⠀⢀⡜⠀⢈⡇⠔⠊⠀⢤⡌⣒⣀
⠀⠀⠀⠀⠀⠀⠴⠀⠀⠐⠋⠀⠠⠾⠛⠦⣄⡈⢏⠉⠀⠘⠢⣹⣷⣿⠋⠀⠀⣿⣀⣤⠾⠖⠚⠉⠛⠒⢰⣄⠀⢘⠋⠁
⠀⠀⠀⠀⠀⠚⠲⠛⠀⣰⠄⠀⠀⠘⠆⣀⠀⠭⠿⠷⣦⡀⠀⠘⣿⣿⠀⣠⡴⠟⠡⠤⢀⠠⠋⠀⡄⢀⠜⠁⢲⠌⢀⡀
⠀⠀⠀⠀⠀⠀⡘⠣⡐⠚⠣⡀⠀⠀⠐⠃⢀⠀⠀⢠⡬⠿⣦⡀⣿⣿⡾⠋⠀⠀⡤⠀⠈⠃⠀⣀⣿⣫⡤⠶⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣩⡷⠶⢬⣶⣤⣤⣤⣬⣧⣤⣄⣀⡀⠹⣿⣿⣿⢁⣤⣴⡾⠿⠿⠿⣛⠋⠉⠉⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠊⠀⠀⢀⠴⣎⣉⣠⠖⠛⠛⢿⣿⣿⣿⠟⠉⠉⠉⠻⡶⠞⠋⠑⢄⣀⠀⠀⢡⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡦⢠⡀⠀⠒⡿⠁⠀⠀⢱⠁⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠘⡄⠀⣤⠀⠹⣟⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡞⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣷⣀⠈⠀⠀⠚⠁⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠂⠀⠉⠀⠀⡀⡰⠋⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠈⠣⡴⠂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠹⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣧⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠄⠠⠁⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠤⡤⠥⠤⣤⣀⡀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠁⢪⡐⠠⠐⢀⠌⢉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⠬⠤⠴⠁⠈⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠛⠓⡲⠤⠧⠐⠒⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⢀⡀⣠⠏⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠀⠀⡇⣧⠴⠒⠛⠉⠉⠉⠉⠓⢦⡀⠀⠀⡘⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠲⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠀⠀⢀⡴⠒⠋⠉⠉⠛⠒⠒⠼⠤⠀⡲⠒⠒⠲⢤⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠙⣦⠰⠊⠀⠀⠀⠀⠀⠘⢆⠀⠀⢀⠏
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡜
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⠁
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠀⠂⠀⠀⠀⠀⠀⣰⠗⠁
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣀⠈⠳⢄⡀⠀⠀⠀⠄⠉⠐⠄⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠛⠉⠁⠀⠀⠀⠀⠈⠑⠛⠒⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣟⣻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣩⡽⠿⢾⣍⣿⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡶⠿⢿⣟⣻⣿⣽⢻⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡿⢟⣻⣿⣿⣿⣿⡇⠀⠀⠀⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⢿⡀⢀⡞⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⡼⠒⢤⣤⣻⣾⣶⡿⠍⣀⢀⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣤⣄⢀⠈⠉⠉⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣤⣿⣽⡿⠆⠀⠀⠀⠀⠀⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢰⡄⣤⣶⣶⣾⣧⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠙⡇⣌⣃⠈⠀⣿⢿⣋⣽⣽⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣸⢰⡇⡟⠟⠀⠀⠃⠎⠿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡌⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡻⣿⣿⣿⣿⣿⣿⣿⣏⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣹⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣄⠿⠀⠀⠂⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⢿⣿⣿⣿⣿⣿⣷⣸⠀⣀⣴⣄⣑⣴⣧⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣽⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⣛⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠛⠻⠿⣿⡿⢿⠿⣶⣿⠿⣿⣿⠯⠅





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⣀⣀⣤⣤⣾⣿⣷⣤⣴⣶⣶⣿⣯⣀⡀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⣷⣦⣤⣾⠟⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⡀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⡿⠋
⠀⠀⠈⠙⠷⣦⣤⣀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⣀⡀
⠀⠈⠉⠉⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠉⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠈⠙⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣶⣦⣤⣤⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⡁⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠉⠙⠛⠛⠛⢻⣿⡿⠛⠉⠉⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠈⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⣤⠔⢠⠀⢀⡄⣀
⠀⠀⠀⣦⠅⠀⡀⡖⣞⡘⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⣊⠠⢆⠐⢶⠹⣠⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠄
⠀⠀⠘⢄⡢⣦⢠⡾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢆⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠓
⠀⠀⠀⠀⠑⣤⣤⣻⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠉⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠠⡄⠀⢀⣤⢄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣌⠘⣈⠳
⠀⠀⢲⡉⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠸⠦⠶⢢⡝⠘⢐⡀⠀⠀⠀⠀⠀⡄⡀
⠀⣅⡙⠡⡿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠦⢢⣱⡁⣢⣌⣛⢎⡀⠀⠀⠀⠀⢼⡏⢦
⠀⠼⢛⡏⣓⣍⠀⠀⠀⠀⠀⠀⢠⢾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢅⠨⠃⣓⠍⡸⡄⡀⠀⠀⠀⠀⣿⡇⡿⠆
⠀⢃⣻⡇⡇⢸⠀⠀⠀⠀⠀⠀⢘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡕⠀⠐⠂⠈⠀⠀⠀⠀⢈⡿⠏⣽⣱
⠀⠈⢾⡇⣟⠛⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢌⢾⠣⣉⢵
⠀⠁⣸⡶⡇⠚⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⢞⠺⠡⠘⢏
⠀⠀⠙⡆⠉⠀⠀⠀⠀⠀⠀⣽⣯⣿⣷⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢲
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣾⣟⣷⡯⢿⣷⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣚⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⠋⠉⠉⠓⢲⠏⠀⠀⠀⠀⢀⡴⠷⠿⠦⣤⠴⠃⠀⠙⣆⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣏⠀⠀⠀⠀⠉⠀⠈⡇⠀⢀⡴⠀⠀⢀⡴⠓⠒⠒⠠⠤⠄
⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⢀⠗⠂⠄⠀⣿⣆⢀⡆⠀⠀⠀⢀⣷⠞⠛⠉⢳⣖⡭⠠⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣼⣧⡀⠀⢀⣠⡏⠀⠀⠀⢠⡼⠛⠛⢧⡀⠀⠀⠈⢷⣀⣀⣀⣼⣧⣤⣀⢀⠀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠊⠉⠉⢢⣀⣘⣿⡷⠞⡽⠁⠀⠀⠀⣾⠀⠀⠀⠀⠙⣆⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⠷⠶⠶⠛⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⣄⣀⡠⠤⠒⠒⠒⠒⠒⠲⢾⡀⠀⠀⡴⠋⠉⠑⠄
⠀⠤⠤⠤⠤⣄⡀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⢠⠁⠀⠀⠀⡄
⢣⡄⠀⠀⠀⠈⠙⢦⡀⠸⡄⠀⠀⠀⢰⠃⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢀⣤⠤⢤⣄⠀⠀⠀⠀⠀⣸⡇⡆⠀⠀⠸⢧⣄⡀⠠⠄⣄
⠀⠉⠐⠢⣄⠀⠀⠀⠹⡄⠈⠢⣄⣀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢰⠉⠀⠀⠀⠈⠙⠶⢤⣤⡴⠛⠀⠙⣄⡀
⠀⠀⠀⣠⠏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠄⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⣇⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⣀⣤⡴
⠀⢀⡎⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠁⠀⠘⢦⣀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠈⠣⣄⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠄⠀⠐⢀⠀⠀⡀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⡏⢹⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡷⢹⡀⠀⢰⠁⠈⡆⠀⢀⡠⡎⡟⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢓⢤⡀⠀⠀⡇⡄⢧⡀⡸⠀⠀⢇⡰⢉⠊⡜⠁⠀⢀⡤⡚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢌⠓⢤⣧⣘⠤⢽⣇⣠⣄⣸⡧⠼⣀⣇⡤⠚⡡⢊⠞
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠙⣄⠑⢄⣈⠗⠊⠉⠀⠀⠀⠀⠉⠑⠺⣁⡠⠊⣠⠋⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⢍⠒⠒⢺⢦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡴⡟⣉⠭⠉⣙⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢌⣉⣇⡎⠀⠐⣋⣭⡛⢦⡀⢀⡴⢛⣭⣙⠂⠀⢱⣸⣠⠞⠉⠁
⠀⠀⠀⢀⣀⣀⡠⠤⠔⠒⠒⠊⠉⣹⠀⠀⠸⢯⣿⠟⠀⡇⢸⠀⠿⣿⡿⠇⠀⠀⣏⠉⠑⠒⠒⠢⠤⢤⣀⣀⡀
⠀⠀⠀⠈⠉⠉⠓⠒⠢⠤⠤⢄⣀⣹⠀⠀⠀⠀⠀⠀⡰⠁⠈⢆⠀⠀⠀⠀⠀⠀⣏⣀⡠⠤⠤⠔⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⡏⢇⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⡸⢹⣉⡑⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣫⣍⣀⣒⣉⣼⠞⢣⡀⠀⠀⠈⠩⠭⠭⠍⠁⠀⠀⢀⡔⠳⡧⠤⠤⣑⠵⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣠⠊⡠⠊⢉⡶⢄⣀⠀⠀⠀⠀⣀⡠⢴⡉⠑⢄⠙⣄⠀⠈⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⢊⡤⠚⢹⠉⡖⢺⡏⠙⠋⢹⣗⠒⡍⢻⠓⢤⡑⢌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡬⠚⠁⠀⢀⡜⡠⣁⠎⢱⠀⠀⡎⠈⢳⠘⢸⠀⠀⠈⠓⢥⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣼⡸⠊⠁⠀⠸⡀⢀⠇⠀⠈⣇⢾⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣇⡸⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢋⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⣽⣽⡇⠂⠀⠀⠀⠀⠀⢠⡿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠤⠀⠀⠀⠠⣄⣀⠀⠀⡁⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢤⡀⠀⠀⠀⠄⠠⠀⠀⢠⠀⠀⠀⠀⠀⣀⠠⡶⠆⠀⢆⠿⡁⡅⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠠⠇⠀⠀⠀⠀⠀⡠⠆⠀⠈⠁⣀⠀⠀⠐⠒⠹⠁
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⡀⠀⠀⠀⣄⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡄⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢀⠶⣭⠙⡁⠀⠀⡀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢿⠀⠀⠴







⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠶⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈
⠀⠀⠀⡠⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣖⠀⠐⣷⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠
⠐⠀⠐⣴⠔⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣮⣯⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠤⠼⠂⠀⠀⠀⢤⣤⠂⠀⠀⣰⡿⠀⠠⠼
⠀⠀⠀⠈⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⣀⣄⠀⠀⣀⠀⡀⠀⠀⣀⣀⣄⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣴⡓⢀⣲⣀⣄⠀⠀⠀⠀⡀⠀⢀⣀
⣿⡯⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠃⠀⣀⣴⣿⢿⣀⠈⡤⢋⣤⣥⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠔⣿⣷⣿⣿⣿⣿⣾⣿⠿⢿⣾⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⠿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⣀⠀⠀⠀⢠⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠛⠁⣹⣿⣟⣷⣄⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠀⠀⢀⣼⣿⣿⣿⣿⣼⣿⣦⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡏⣘⡟⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⠃⣸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣠⣿⣿⣿⣿⣿⠟⠉⠻⣿⣿⣿⠁⢀⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⣰⢀⣘⣿⣿⣿⣿⠏⢡⣤⡖⣾⣿⡅⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⣿⣿⣛⣿⣶⣿⣿⣏⣸⣿⣴⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠻⣿⣿⣷⣿⣦⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣭⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡷⠶⠤⣴⣿⣿⡿⢿⣿⣿⣿⠿⢿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣀⣿⡁⣿⠚⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣿⣿⡇⠀⣾⡿⢫⣤⣿⣿⣿⣿⣿⡟⣿⣿⣟⣣⣾⣿⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣝⣻⠉⣀⢀⣏⣀⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⠿⣟⣿⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣾⣿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠰⠛⠙⣿⠃⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠉⠁⠀⠀⠀⠀⠀⠀⢹⠛⢿⣿⠁⠀⣸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⠀⣻⣿⣲⣿⠻⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡷⣯⢹⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣛⣲⣶⣦⣴⣶⣿⣷⣽⣿⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠁⠈⣀⣀⣤⣼⣿⣿⣿⣿⣻⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠴⠿⠛⠛⠛⠛⠛⠂








⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡛⠏⠙⢻⡋⠉⡟⠛⢛⡛⠏⣻⡿⢛⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⣉⣿⣷⡧⢢⠂⡀⠀⠀⠀⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣣⣌⣼⣻⣷⣿⣷⢋⣸⣳⣤⣦⣶⣿⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠾⣷⣿⣿⣞⡱⡥⠛⠝⢯⠻⢏⣠⣽⡀⡀⡴⣖⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣰⢵⣿⣷⣶⠉⡛⢾⣿⣫⠞⠀⡈⣣⣶⣿⣿⣽⣿⣼⢹⠹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠸⠐⢺⡍⣹⣮⣿⣿⣿⠳⡟⣯⣕⣐⠒⣴⣭⣿⢛⣾⣛⣧⡌⢤⠴⢶⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢱⣬⡟⠿⣏⡿⡫⡤⠽⡷⠿⡿⣅⣰⣛⣻⣿⣿⡿⠿⢽⣿⡞⠡⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣧⢠⢿⣆⡰⡬⠇⣫⡑⡗⣹⣿⡿⣼⣿⣟⣿⣿⡧⣾⢧⣤⠠⠀⠴
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⣀⠸⢤⣽⣼⢷⣾⢾⢿⣿⡿⣿⡿⠦⢾⣽⣿⢞⡿⣻⡶⢂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣰⣾⣯⣡⣜⡘⣱⢼⡼⣟⠟⣻⡍⢯⡡⣧⣼⣷⣾⣿⣁⣿⣯
⠀⠀⠀⠄⠀⠀⣠⣞⣋⢻⣯⣿⣷⣫⢾⡷⣈⣛⡱⢿⡛⢉⡱⠚⣥⣶⣿⠿⢿⠻⣿⣿⡟⠈
⠀⠀⠀⠀⠀⠉⠉⣿⣿⡼⡷⣾⣿⠿⣉⣿⣷⣾⣯⣸⣷⣾⡔⠀⢀⢻⣿⣜⣇⣫⢽⡮⣽⣾⣆
⠀⠀⠀⠀⠀⣤⣬⣿⢟⣯⢟⡻⣣⢾⣭⣋⣹⣿⣿⣿⣿⠷⠀⠟⢋⣛⣿⣿⡏⣵⡶⢁⢿⠻⢿⡿
⠀⠀⠀⠀⠈⠻⢿⣽⣠⡛⣟⣿⣿⣟⣿⡿⣿⡿⣿⣿⣿⡷⢀⣄⣿⣿⣿⡿⣻⢟⢯⣸⣧⣃⡿⣷⢦⡄
⠀⠀⠀⠀⠘⠾⠡⣿⣿⣙⠿⠙⣿⣞⣯⢡⡞⠉⣿⣿⣏⣿⣈⣹⣇⣽⣧⣿⢗⣾⣿⣍⡉⣷⠛⣻⠘
⠀⠀⠀⠀⠀⠀⠁⠟⠋⠀⠘⠙⣿⡿⠁⠙⠛⠂⢩⣿⠏⠀⠏⠉⠁⠑⠀⠻⠿⠇⠂⢿⣿⣷⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⣽⣤⠀⠀⠀⠈⠀⠀⠀⠠⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⣸⣿⡀⣀⣀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣧⣸⣥⣤⠔⢩⣞⣷⣿⠘⣇⣻⣟⣤⣄⣤⣤⠤⠄⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠅⢁⢠⠤⡑⠞⢳⢀⣿⣛⡁⠉⠛⠛⠻⠛⠖⠋⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠋⠘⠒⠛⠃⠙⠓⠚⠐⠀⠑

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠠⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠤⠒⢆⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠋⠁⠀⠊⠱⠠⠄⠃⠀⠀⠀⠈⠀⠑⠄⠠⣀⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠠⠤⠂⠐⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⣠⠊⠀⠀⠒⣎⠀⠀⠀⠀⠀⠀⠀⢀⡖⠊⠉⠚⠉⠑⡄⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠐⠒⠆⠑⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠂⠀⠂⠀⢙⠀⠁⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⢉⠀⠀⡀⠀⠀⠙⠢⠀⠀⠀⠂
⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡐⢉⠲⢤⣤⡤⠚⠀⡀⠀⠂⣀⠀⠀⠁⢀⠴⠤⡤⠤⢤⠁⠀⡀⠀⠠⡁⠀⠀⠀⠐
⠀⠀⡀⠀⠂⠀⠁⠀⠀⠶⠠⢐⠀⠀⠄⠀⡚⠙⠉⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠬⡒⠀⠀⡇⠀⠀⠀⠀⠀⣇⠈⠀⣀⠀⡁⠀⠀⠀⠀⠠
⠀⠀⠀⡜⠀⠀⠀⠀⢀⠀⢢⠀⠀⠀⠀⠀⠇⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢂⡀⠀⠀⠱⣄⠀⠀⢀⡴⠃⠀⣁⠀⠘⠁
⠠⠀⠀⡡⠀⠀⢀⡀⣈⣀⣤⡀⠂⠀⠀⠀⢆⣀⣀⣑⡾⠁⠀⢀⠀⢀⡴⢤⠄⠀⠀⠀⠀⠀⠀⠀⠄⠈⠝⠋⠁⠀⠀⣠⣉⣀⠀⠀⡀⠄
⠂⠀⠰⠀⠀⢀⠁⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠉⠁⠀⠀⣀⡜⠛⠻⡄⠀⣇⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠰⠂⠄
⠐⠀⠁⠌⠁⠀⢣⡀⠀⠀⠀⣰⠃⢀⡀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⢳⠀⢹⡄⠀⠀⠀⡑⠦⣀⡀⠄⠀⠀⢀⣀⡠⠁⢀⠀⠀⠀⠀⠂⠄⠀⠂
⠀⠀⠀⡇⠀⠠⠀⠉⠒⠒⠉⠀⠀⠈⠀⢀⣠⠤⡠⠜⢹⠀⠀⠀⠀⠀⠈⡇⠈⣷⠀⠀⣰⠁⠀⡀⠀⡀⣠⠀⠀⠈⠙⢤⡞⠀⠀⠒⠀⠀⡎⠀⠐
⠀⠀⠀⠇⢠⠄⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣀⣠⠴⠇⠀⠘⢷⡴⠇⠈⠀⡄⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⡀⠈⠀⢠⠇
⠀⠀⠀⠀⠀⢄⠀⠀⠀⡀⠀⠀⠤⠀⠦⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢶⠓⠤⡀⠀⠀⠤⢠⡆
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠊⢁⠔⠙⠒⠒⠒⠢⡄⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⢰⠀⠀⠀⠀⠰⣇⣀⣀⣀⡼⠀⣸⠀⡄⠀⡬⠀⠔⠀⡟
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⠠⡄⠀⠀⠀⠀⢀⡗⠀⠃⠀⠀⠀⠀⢻⠀⠀⠀⠸⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⠟⡠⠛⠃⠶⠒⠢⣀⠃
⠀⠀⠀⠀⠀⠀⠈⠒⢤⠀⠖⠀⠙⢦⣀⣀⡤⢞⣀⠄⠀⠤⣀⠰⠤⡟⠀⠀⠀⠀⠁⠀⠈⠀⡟⠙⢻⡛⠛⠉⠁⢰⡇⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠲⣀⢀⡀⡀⠀⢀⠅⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡄⢀⠀⢀⣼⠇⠀⠀⠉⠉⠉⠛⠉⠙⢤⣀⣀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⡘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠌⠀⠀⠀⠀⠀⠀⢠⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠐⠀⠀⠀⠀⠀⠀⠀⢸⢇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢸⡄⠀⠀⠀⠀⠀⠀⠀⠊⠀⢦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡇⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢰⠀⠈⠀⠀⡐⢀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠁⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠸⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣿⣾⣿⡿⣿⣦⠀⣠⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣛⣿⣿⣟⣿⣶⣿⡟⣯⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⣻⣿⣿⣽⣿⣿⣿⣧⣴⡟⠈⠛⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠿⠍⡻⢿⣿⣿⡿⣿⣿⡆⣵⢿⣿⣏⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣟⣩⣾⣿⣿⣿⣿⣿⣿⣧⣜⣟⠿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣥⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣿⣿⣿⡿⢻⣟⣛⣻⣟⢻⣿⢿⣿⣾⣿⣿⣽⣿⣯⣅⣤⢌⣋⣠⣄
⠀⠀⠀⠀⠀⠀⢠⣤⣖⣛⣉⣭⣽⠟⢻⡽⠿⣿⣿⣷⣿⣿⣯⣿⣿⣿⣿⣯⣿⣿⣝⡒⠛⠛⠁
⠀⠀⠀⠀⠀⠠⠚⠿⠿⠿⢲⣾⡭⢩⣿⣷⣿⣿⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣉⢙⡁⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣶⢿⣾⡿⠿⠿⠷⣿⣙⣻⣿⣿⡏⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢶⡾⠿⠛⣫⣭⣴⠞⣿⣿⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠒⠓⠚⠛⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⣿⣿⣷⣶⣬⣑⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠉⠋⠛⢻⣿⢿⡉⠛⠙⠻⠿⠿⠟⠂⠛⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠟⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣚⠛⠛⠻⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠉⠄⠉⡍⠙⠛⠛⠉⠉⠋⠉⠁


⣿⣿⠻⠿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣀⡀⢄⣀
⠀⠀⠀⠀⠈⣿⣿⡁⠀⠈⢁⡟⢦⡀
⠀⠀⠀⠀⠀⠉⢿⣿⣄⠤⠀⠁⠠⡘⣇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣄⠀⠈⠁⢜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⡀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠸⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⣄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣗⠐⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡌⣙⠘⣿⣿⣳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡧⡀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣷⡆⢀⣾⣿⣽⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣱⠅⠀⠀⠈⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣿⠆⠀⣈⡍⢿⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣠⠀⠐⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡊⠘⠻⣿⣷⣿⣿⣿⡽⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⡀⣺⣿⣿⣿⣿⣹⠷⡋⢿⣿⣿⣿⣿⣿⢿⢿⡇⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠀⠀⠸⢻⣿⣿⡿⡿⢣⡌⠠⠇⠀⠉⢻⣿⣿⣿⡖⠫⠣⠀⠼⡁⢠⣤⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠄⠀⢀⣿⣿⡿⢣⠑⠈⠁⠁⠄⠀⠀⣈⣿⣿⣿⣇⣀⠀⠈⢓⣯⣽⣿⡆⢬⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠂⠀⣸⣿⣿⡗⠋⠀⢀⣀⣀⡊⠀⠀⢀⣾⣿⡿⠛⠛⠛⠛⠛⠛⠻⢿⡏⠼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⡀⠀⣼⣿⣿⣶⣴⣾⣿⣿⣿⣿⡆⠀⢠⣿⣿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠐⣿⣿⣿⣿⣿⣿⡽⠿⠁⣿⡇⢀⣾⡛⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⣿⣿⣿⣿⣿⡿⠉⠈⡰⡿⢁⣾⢿⠃⠡⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣿⣿⣿⣿⣟⢃⠀⣠⣿⢃⣯⣿⣆⡩⣶⡿⠁⠀⠒⠒⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠐⠀⠀⣀⣀⣀⠘⠻⣷⣿⣿⣿⣿⣿⣧⣴⠟⣃⣰⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⡄
⠀⠀⠀⠀⣀⣀⡀⠀⠈⢉⡙⠛⠿⢿⣿⣿⠿⠿⠿⣿⣼⣽⣾⣿⣏⣙⡿⠰⠶⠶⠤⢦⣤⡀⠀⠀⠀⠀⠀⢀⡀⢈
⠀⠀⠀⠀⢀⣠⠽⠃⠀⡼⣿⣶⣶⢄⣀⠁⠢⠄⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠼⢐⠃⠈⠲⣆⡀⠀⣸⡿⠉
⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠹⠌⠋⠁⠘⠀⠀⠛⠳⢿⣄⠰⠿⢷⠶⠦⡄⠠⠠⢀⣤⣄⣀⠀⣬⣿⣶⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠠⢀⠀⠀⠀⢀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢉⣠⢤⣧⣌⠉⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠘⣧⡁⠈⢢⡀⠀⠀⢠⠞⠛⠂⢈⠿⠗⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠤⠘⢧⣼⣤⡄⢠⡿⣿⡿⠖⠛⠒⠶⠬⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠷⠟⣷⣿⡇⠀⠀⠀⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⠈⠓⠀⠀⠚⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠠⠀⠠⡀⠀⠠⠀⠂⠘⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠀⠐⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠰⠈⠀⢀⠢⣀⠀⠀⠀⠀⠀⡖⠉⠉⠁⠈⠂⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠁⠊⢀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠵⣄⠀⢀⡠⠀⠀⠀⠀⠀⠠⠀⣈⢀⣂⡐⠄⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠤⠀⡄⠀⠀⠰⠒⠒⠑⠊⠀⠀⠈⠁⠀⠀⠂⠀⠈⠄⠀⠈⠀⠈⠀⠈⡆⠀⢀⠈
⠀⠈⠄⠁⠀⠀⢀⠀⠂⠀⠀⠀⢸⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⠴⠃⢠⠀⠘
⠀⠀⠀⠀⢠⠄⠀⠔⠂⠌⠀⠀⠀⠂⠴⠾⠁⠀⢀⡠⡔⠓⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡔⠤⠀⢀⠐
⠀⠀⠁⠀⠀⡀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠹⡀⢧⠀⠀⠀⢢⡀⠀⡀⠀⠀⠀⠀⠃⠁
⠀⠀⢄⠀⠀⠈⠒⠒⠊⠀⠈⠀⢀⠤⢠⠜⡏⠀⠀⠀⠀⣇⠘⣆⠀⢠⠁⠈⠁⣀⡀⠀⠉⠳⢤⠂⠀⠀⠀⡈
⠀⠀⠘⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠤⠔⠋⠀⠙⠶⠃⠀⢠⠀⠀⠱⣄⠀⡀⠀⠀⠀⠈⢠⠂
⠀⠀⠠⠀⠖⠀⠀⠢⠤⢀⠠⢑⡀⣀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⡇⢹⠉⠑⠀⠀⠀⢺
⠀⠀⠀⠀⢲⠀⠀⠀⠀⢈⠀⠀⠀⠀⡧⠰⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⢀⡼⠀⠧⠬⠤⢄⠘
⠀⠀⠀⠀⠀⠑⠀⠀⠆⠘⠢⢄⠤⠜⡁⠤⠀⠀⠤⢰⠃⠀⠀⠀⠀⠀⢨⠛⠳⣖⠛⡉⢠⡇⠀⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡟⠀⠀⠈⠁⠈⠁⠈⠢⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠄⠀⢈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠸⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢸⠀⠀⠀⠀⠀⠀⠀⠘⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠄⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠱⠀⠀⠀⠠⠀⠀⠀⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣣⣤⣠⣄⣀⣮⣠⣀⣀⣀⣀⣀⣀⣠⣤⣴⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢷⡀⢣⠘⢷⡄⠀⠐⣄⠀⠀⠸⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⣤⢀⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⢀⣤
⠀⠀⠀⠀⢀⠙⣆⠳⡌⢿⣦⡀⠈⡆⠀⠀⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣰⣿⣷⠈⠂⠀⠀⢰⠏⠀⠀⠀⠀⠀⢠⠤⠀⡀⢀⣴⠏⢠⡿⠁⠀⠀⢀
⠀⡄⠀⠀⠈⢣⠘⠀⣿⠸⣿⣅⠆⠀⠀⠀⠹⣄⠀⠀⠀⢸⡟⠀⠀⠀⢹⢈⡟⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣀⡰⠁⣾⠁⣠⠎⡄⠀⢀⣴⣿
⠀⠀⡀⠠⠀⠀⠑⣸⠋⢤⣟⠉⠺⣦⠀⠀⠀⠈⠃⠀⠀⠈⣷⠄⠀⠀⢨⢘⣿⡆⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠋⠀⢰⡏⠰⠈⠆⠀⣴⡿⠛⢡
⠀⠀⢀⣤⠀⠀⠀⠀⢇⠈⢹⣆⠐⠟⢷⣀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⢸⠸⣇⠴⢀⡾⢃⠀⠀⠀⠀⠀⠴⠁⡄⢠⡟⠀⣧⡘⠀⣸⠏⣀⣴⣿
⠀⠠⡀⠀⠈⠳⣌⣠⣿⡀⠀⠻⡄⣇⠈⢳⡄⠀⠀⠀⠀⠀⣸⡇⠀⠀⢠⡾⣰⠏⢸⣼⡧⠀⢠⠜⠀⡀⠐⠀⠀⣰⣿⠇⢀⣿⠁⢠⡏⣰⢋⣼⠇
⠀⠀⠈⠂⠐⠢⠤⢁⡍⢻⡄⠠⡘⡀⡆⠀⠗⠀⠀⠀⠀⠀⢹⣧⠠⠀⠀⡆⢹⠀⣧⣿⠁⣄⠄⠀⢰⣰⠃⠀⢠⣿⠏⠀⣼⠋⠠⢋⡴⣧⠟⠁
⠀⠀⠀⠠⡄⠀⡀⠀⠘⡄⣿⣄⡐⢰⡷⢄⠐⠀⠀⠀⠀⠀⣿⡁⠀⡄⢠⠃⣼⠀⢻⡅⠀⣇⠀⢀⡟⠀⣠⠀⢫⠇⣰⣟⣡⠎⢠⠋⠘⠁⠀⠔
⠢⣄⠀⠀⠘⣂⠈⢦⠀⠘⢬⡛⢿⡄⠳⣈⡇⠀⠀⢀⠀⠀⢈⣿⣶⠑⠈⢾⢃⣴⠿⢃⣼⠟⢀⡾⠃⢾⠁⡠⠋⣠⡷⠋⡸⢀⡄⠀⠀⠐⠂⣄⡴
⠀⠘⡃⠈⠀⠹⢆⡀⠂⣀⠀⠉⣿⠃⠀⠠⡉⢦⡀⠀⠀⠠⣿⣿⣇⡀⠐⢹⣿⡇⠀⣼⠁⢠⡿⢀⡶⠃⠈⢠⡾⠋⠀⠀⢠⡟⠁⡰⣂⣤⠞⡝⢣
⠀⠀⠈⠲⣐⠀⠀⠙⠂⠙⢦⡄⣿⣇⠀⠀⠈⠜⠽⣢⠐⠀⠈⠙⣿⣿⣦⠈⠹⣿⡀⣿⣴⡿⠁⡀⠁⢲⣶⠟⠃⠀⠀⣠⢞⣠⠀⣴⢋⡤⣞⡶⠆
⠀⠈⢢⣁⢨⣇⡀⠀⠐⢦⠘⣯⠈⠻⣷⣦⣄⣀⠠⣽⣀⠁⠀⠀⣼⣿⣿⡇⠀⣿⡇⢹⣿⠁⠰⣁⣠⣾⠃⠀⠀⢀⠞⣁⣾⡥⢾⣿⣩⡶⠋⡠⠖
⠀⠀⠈⣏⠉⠙⢿⣅⠀⠈⢣⠙⠲⣤⣀⠈⠻⢿⣷⡌⢻⣷⢇⣴⣿⣿⡿⢠⣾⣯⣎⣾⡿⠀⠀⣿⠉⡠⠀⠀⡀⢀⣴⣾⣏⣴⠾⠛⢁⡀⢈⣤⣾
⠀⠶⢤⣈⡛⠛⢆⠨⠳⣄⠘⠳⣄⠀⢸⡇⠀⠘⣿⡿⢸⡟⢻⣿⡟⣿⢡⣿⣿⣿⣿⣯⣤⣾⣤⡿⠀⠃⡠⢂⣤⡾⣻⡿⠏⠁⣤⠔⢃⣼⣾⠷⠉
⣄⡀⠀⠀⠙⠶⠾⣟⢿⣉⢓⣦⠸⣄⣀⣙⣶⣴⣿⠁⣾⣷⠙⣿⣇⢿⣿⣿⣿⣿⣿⣿⡟⣛⣉⣡⡾⠋⣰⣿⠍⣰⡟⠀⠴⣉⣤⣿⡿⢋⡁
⠀⡉⠑⠄⠀⠀⠀⠘⢦⣙⣿⣿⣷⣤⣄⠀⢹⣿⣿⣧⡘⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⢋⣭⣭⣥⣾⣯⣥⠾⠫⢤⣾⠿⠿⢛⡿⠛⠋⠁⠀⡠
⢦⣄⡀⣈⠁⢀⡀⣄⠀⠀⠀⠙⠛⠛⣿⣇⢿⣿⣿⣿⣿⣿⠟⠋⠩⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣟⣁⢠⣶⣶⣟⣡⡶⠾⠋⣀⣠⡔⣂⣻⣷
⣀⡀⡀⠀⠈⠁⠉⠛⠷⠶⠶⠾⢶⣄⡹⢿⣬⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣤⣼⡿⢿⣛⣿⣭⣭⠿
⣋⡉⠁⠂⠉⠛⠶⠶⠶⣦⣤⣤⠶⣿⣿⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣟⣿⣭⡿⣧⠿⠶⠟⣉⣉⡄⠐⠂
⠙⠾⠶⣿⣁⠴⠶⣤⣤⠴⢦⣄⣀⣬⣿⣿⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⢛⣉⣹⣧⣤⣴⡚⢛⣷⣾⣧⣄⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⣏⠉⠛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⠟⣿⠟⣛⣛⠋⣻⣿⣟⢉⣉⠙⠿⠿⢛⣁
⠠⠀⠄⠤⣀⣀⣀⡀⣀⣽⠷⣦⣌⣛⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠛⢿⣛⠛⡿⠿⢿⠿⠿⣟⣛⣛⣋⣩⣍
⢀⡤⠌⠀⠉⠛⠑⣉⠉⢡⣶⣄⠉⢉⣼⣿⣿⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⢻⣬⡟⠲⢦⣉⠛⠛⠳⠾⠓⠚⠛⠛⠓⠢⠀⠄
⣀⣀⣤⢶⠾⠟⠾⠛⠳⠿⡥⠈⠟⠿⣿⣼⣟⣿⣿⣿⣿⣿⣿⣶⣤⣶⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣮⡛⠿⣦⣉⡉⠛⡢⣤⢀⣀⡀
⠋⠉⠀⠀⠀⠀⠀⠀⠐⠀⣁⢀⣶⠶⢿⢠⣼⡿⣿⡏⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣌⠉⠻⣿⣿⣷⣌⡙⢻⣶⣤⣠⡀⠀⠈⠛⠿⠛⢣⣶
⠀⠀⠀⠐⠉⠁⠀⢀⡄⣴⠏⠉⣁⠔⣤⢶⣿⣴⣟⣸⡏⣿⣏⣿⠏⢘⣯⢿⣿⡟⢧⡻⣿⣿⣧⣀⠈⠙⠳⠦⡷⢶⡍⢻⣯⡉⠤⡀⠠⢀⡈⠀⠈
⠀⠀⠀⠀⣠⡤⠴⠏⠀⠀⠔⠉⠰⠞⣁⣼⡏⢁⣼⠏⢀⣾⣏⣿⠀⢻⣇⢨⣿⢃⠸⡋⣿⣯⡛⢻⣷⣄⡀⠀⠀⠁⠙⠦⠹⣿⣤⡈⠉⠠⠀⣀
⠀⣄⡲⠿⠋⠀⠀⠀⠀⣠⢴⠿⣋⣾⣿⠛⠐⡏⠁⠀⣿⢹⣿⠃⠀⠸⣯⠹⣷⡘⢆⠙⣋⢿⠿⣿⣿⣿⣿⣌⡲⣦⡀⠑⠢⣝⠻⡿⢶⣶⣄⡀
⠚⠁⠀⠀⠀⠀⠀⠀⢈⣡⣁⣼⣿⠏⠉⠂⡔⢁⠀⢀⣾⣦⣿⠀⠀⢰⣿⡇⠈⢻⡎⡆⠹⡈⠀⠈⣿⣿⡏⢿⣷⣜⢷⡄⠀⠻⣄⠘⠊⢻⣏⠻⣆
⠀⠀⠀⠀⠀⠀⣰⠟⠋⢉⡟⢻⡉⠠⠀⡠⠃⠀⠀⢸⣿⡟⠁⠀⠀⠀⣿⠇⣀⣾⡃⢨⡀⠁⠀⠀⢻⣎⡳⣤⣙⠿⢷⡍⠻⣦⠀⠙⣦⡈⠉⢷⡌
⠀⠀⠀⠀⡀⢨⣿⢀⣸⡏⢀⣭⠋⠀⣼⠀⠀⠀⠠⢾⣿⡅⠀⠀⠀⠈⢻⣤⡙⡯⣿⠘⣧⡀⠀⠘⠙⢈⠙⢦⠈⢇⠀⠰⡆⣹⢷⣄⡘⠷⣄⡀⠈
⠀⠀⢠⣤⣤⣿⠁⢾⣩⣾⠟⠁⢀⢾⡏⠀⢀⣦⣴⢾⣿⠟⠀⠀⠀⠀⠀⣿⠟⠁⣼⠃⠈⣿⣦⡀⠀⠈⠀⠈⠂⡀⠐⠄⠀⠘⠷⣄⡙⠳⡀⠹⣧
⠀⢀⣿⠁⠀⠀⢈⣾⠛⠋⠀⢀⡟⢸⡇⢠⢻⠉⠀⢀⣿⠀⠀⠀⠀⠀⠀⣹⣷⡄⡈⠃⠄⠈⠚⣿⡆⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢌⠻⣆⡡⠀⠻
⡰⠟⠋⠀⠀⢠⣞⠅⠀⡀⢠⠊⢠⡇⠀⠸⠀⣧⠀⢸⠇⠀⠀⢀⡠⠖⠈⢿⡟⢧⣁⠀⣴⡀⠐⠿⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠌⣻⢷⣄
⠁⠀⠀⠁⠀⠼⠃⠀⢠⡸⢀⡤⢿⠀⣠⣴⠾⠛⠀⡞⠀⠀⢀⡏⠀⠀⠀⣟⢧⡘⠏⠐⠨⣿⣄⠀⢦⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡙
⠀⠈⠀⠀⠃⠀⠀⢤⠋⢠⢸⡴⠃⢸⡏⠀⠀⠀⠸⠇⠀⠠⠊⠁⣄⠀⠀⢹⠆⠣⠈⢠⡄⠀⢻⡆⡘⠌⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠈⠛
⢠⠄⠈⠀⡀⠀⠰⠃⠀⢸⣼⠃⠀⢼⡇⠀⠀⠠⠀⠀⠠⠀⠀⢠⡁⠀⠀⠹⢦⡀⢣⠈⠉⢦⢼⡇⣶⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠁⠀⠀⠀⠀⠀⠀⣠⠞⣱⠏⢀⠔⡘⠀⠀⢠⠀⠀⠀⣰⠀⠀⣸⡇⠀⠀⠀⠀⡅⠈⢄⠀⠸⡈⢿⠻⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⡞⠁⡼⠃⠐⠁⠖⠁⠀⠀⠃⠀⠀⡀⢙⠂⢀⢳⠈⠇⠀⠀⠸⡃⠀⠀⢠⡀⢸⡇⢣⠀⡄⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣜⠷⠈⢰⠀⢿⠀⠀⠀⠑⠀⠀⠀⠇⠸⢷⡠⡋⠡⡀⠁⠀⠀⠀⠀⠈⠋⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿⠀⠙⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣾⡄⠻⡀⠀⠀⠀⠀⠀⡃⠰⠒⠂⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣀⠠⡍⣧⣼⣿⢟⣶⢲⡞⣾⣿⣥⣄⣤⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣟⠯⣿⢿⣿⣿⣿⠻⣬⠃⡼⢋⣾⡳⠻⠎⣽⣮⣵⠇⠺⣣
⠀⠀⠀⠀⠀⠀⣀⣜⠆⡟⡞⢻⡷⢺⡋⢫⣜⣿⢿⣳⣿⣤⣾⢿⣟⣤⢬⡏⢟⣚⡿⣩⣻⣆
⠀⠀⠀⠀⡠⣴⣿⣫⣶⠶⢏⣿⣥⣵⠹⢽⣿⣷⣾⢯⣿⣿⣿⢿⣿⡛⣷⣷⢞⢿⢿⡯⡅⢫⢌⣦⠠⠄
⠀⠀⠀⢀⢵⠭⡋⣹⣟⠿⣾⣿⡿⣿⣳⣿⠿⡽⢽⡮⣗⣹⣿⣆⣋⣘⢱⣻⡿⣿⣿⣣⢼⡗⢺⣯⣶⣦⡄
⠀⠀⢠⣞⠛⢿⣙⢽⣁⡟⣿⣿⡽⠹⠼⡿⢿⣤⠿⣤⣟⣆⢿⡟⣽⡿⠻⣉⣻⠿⣙⡝⠻⣏⣿⢻⡶⣳⣀
⠀⠀⢈⢷⣾⡛⢹⣿⣿⣿⠿⠗⣱⠺⢬⣶⣶⢿⡜⣿⣿⣿⡧⡞⣺⣷⡾⣾⡟⠛⠾⡸⣽⠿⣷⣻⡤⢻⣺⠆
⠀⠀⠚⣶⣿⣿⣺⣏⣣⣤⣮⣾⠟⣻⣿⢉⠟⣛⡏⢹⣝⣟⣿⣿⣽⢿⡿⣻⡯⣾⣇⣘⣿⣷⠿⡛⢣⠞⣧⡀
⠀⠀⠀⢩⣯⣻⣿⡿⡿⢿⣷⡾⣾⣽⣷⣽⡻⣿⣿⣶⣶⡿⢟⣫⣽⣭⣷⠻⣽⣯⢿⣫⢿⡽⣿⣧⣮⣶⠻
⠀⠀⠀⠀⢼⣜⡁⣖⣿⣿⣬⣿⣅⣟⣻⣿⣹⡮⡻⠇⠉⠳⠉⠙⢟⣿⣿⡁⠹⣿⠟⣿⣿⡻⢍⡳⣹⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠋⠛⠊⠒⠫⠃⠀⢂⠘⠐⢐⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠙⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠡⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣀⣤⣥⠤⣀⣀⣀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣲⣠⠴⠒⠋⠙⠧⣄⣀⡀⠀⠀⠀⠀⠉⠓⠒⠦⢀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣽⣦⠈⣧⠤⠶⠶⠶⠤⣤⣍⣉⣉⣛⣛⡛⠛⠒⠶⠦⣄⡑⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠶⠶⠤⠬⠧⠷⠄⠒⠒⠒⠒⠉⠉⠀⠀⠀⠈⠉⠝⠢⣄⠀⠀⠉⠳⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠀⠈⠉⠓⠲⠤⣤⣤⣤⠤⠶⠤⠴⠶⠞⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⣈⣟⣟⣛⣟⣯⣛⣏⣻⣋⣛⣛⣋⣛⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠰⠀⡀⡀⠀⢆⡀⠉⠐⠈⡄⠓⠂⠄⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠔⠀⠀⢧⠀⠊⠀⠀⠤⠀⠀⠠⠠⡀⠀⢘⡄⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠀⢰⠀⠀⠀⠀⡀⠀⠀⠀⢁⠗⠐⠃⠀⠙⠦⡤⢤⡈⠀⠀⠀⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠉⠀⢎⠀⠀⠈⠃⠑⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢫⠉⠢⠤⣄⡠⣄⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠀⠀⠀⠀⠠⢀⡀⠌⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⠢⠈⡀⢈⡕⡆⠏⢑⣃⡄⠀⠀⠀⡀⣀⡇⠘⠊⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠟⠂⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠈⡇⠀⠀⠀⠁⢘⢠⠇⠳⠙⠊⠉⢙⣒⣲⠁⠧⠀⡀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠈⠂⡐⠀⠀⠀⠀⠀⠤⢄⠀⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠲⣀⢀⠤⡇⠉⠀⠁⠀⠀⠀⠉⢇⠀⣹
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠓⠃⠀⠈⠨⠀⠀⢠⣀⢌⡠⠃⠀⠂⠠⡀
⠀⠀⠀⠀⠀⠈⠂⠐⠆⣨⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠊⠁⠀⡀⠀⠀⠀⠀⠀⠠⢀⠀⠁⠀⠀⠀⠀⠀⠂⠆⠀⠀⢀⡀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠄⠠⠀⠄⠀⠉⡑⠀⠀⠀⠀⠠⠀⠀⠠⠀⠝⠐⡃⡀⠀⠁⠀⢢⠈
⠀⠀⠀⠀⠀⠀⠤⠚⠀⠁⢄⠀⢀⡁⠠⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⠀⠉⠉⠶⠚⢛⠄⡔⠠⡆⠀⠀⠀⠀⠀⠉⠁⠀⠈⡦⢀
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠸⠀⠀⠀⡈⠀⠀⠀⣤⡰⠄⠎⠉⠉⠇⠀⠈⠉⠙⠐⠘⡔⠉⠀⢧⡀⠰⠄⡖⠦⠤⠄⠀⠠⠦⠈⠔⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⢠⡀⢀⠀⠀⠀⠈⠙⠆⠶⡑⡄⠴⠒⠂⠀⠀⠀⠀⠀⠀⠈⠂⠑⠚⠤⢰⠇⠄⠐⠣⠄⡀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⡀⢀⠄⠠⠙⠁⠘⣀⣤⠀⠄⠦⠐⠁⠀⠈⠢⣀⠰⡄⣄⣀⣄⡞⣄⢠⣰⣂⣂⠔⠉⠋⠈⠀⠠⢌⠑⠒⠒⠘⢆
⠀⠀⠀⠀⠀⠈⠀⢂⡀⢀⠤⠀⢄⡨⠅⠀⠀⠀⠀⠀⠀⠀⡄⣁⣀⡀⠀⢁⡊⢀⡅⠬⠂⣉⠅⢠⣠⢄⡠⡠⢀⠘⡢⠀⠐⠀⢀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠐⠓⠊⠒⡈⢒⠆⠠⡀⠎⠈⠁⠋⣠⠒⠩⡀⣄⣠⠀⢄⢠⡄⠛⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠁⠀⠁⠀⠀⠀⠢⠈⠀⡘⠀⠑⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠘⠂⠇⠐⠐⠲⠦⠋⢂⠧⠘⠛⠃⠿⠤⢴⣠⡄⣰⢶⡏⠦⠸⡔⢴⡶⡰⡧⠠⠼⠶⡼⠆⢲⠶⠶⢤⠤⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠁
⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⢿⠿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⣾⣿⣿⣿⠟⠋⠁
⠀⠀⠀⢸⣿⣿⣿⠃⠀⣀⣤⣴⣶⣦⣤⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢸⣿⡿⣡⣴⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣶⣶⠀⣠⣿⣶⣦⣄
⠀⠀⠀⠀⢩⣾⣿⣿⣿⣆⡀⠀⠉⠙⢿⣿⣿⣿⠛⠿⠟⠛⠛⠃⠠⢏⢻⣷⡆⠉⠿⠦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠅⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠈⢠⠀⠀⠈⠙⠐⢹⠀⠀⠐⠞⠀⠀⣀⣴⡴⠿⠶⠶⢤⠠
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⣻⠁⠀⠀⠀⠀⢀⣀⡐⠀⠀⠀⠀⠠⠤⠁⠈⠁⠀⠀⢸⡿⠩⣐⠒⠂⠁
⠀⠀⠀⠀⠘⢿⣿⣿⡿⠿⠊⠀⢀⣜⡁⠠⣔⢶⣶⣿⣟⣅⣄⢸⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠡⠖⠂
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⣶⣛⣫⣤⣿⣿⣷⣿⡟⢿⣷⢊⠰⠿⠇
⠀⠀⠀⠀⠀⠀⠸⠀⢀⣈⣴⠾⠋⠛⠉⠉⠉⠉⠉⠉⠛⠻⠿⣦⣄⠀⠀⢀⣀⣀⣀⣴⠿⢶⣦⣤⣤⠤
⠀⣤⣤⣰⣦⣠⣴⡶⠟⠋⠁⢀⣴⣦⣉⣿⣅⣀⠀⠀⠀⠀⠀⠀⠙⠻⣏⣲⣿⠿⠟⠙
⠐⠋⠉⠛⠓⠋⠉⠓⢦⣁⠀⠠⠁⢼⣿⣿⣿⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⡀⠠⠀⠙⠳⠉⠟⡿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣾⣛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠹⢷⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣄⠀⠠⠔⠒
⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠙⠶⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⡴⠤⣀⣰⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡤⠿⠷⡖⠛⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠚⠀⡀⠀⢸⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠁⠀⠊
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣨
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡀⠠⢂⣴⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⣀⠄⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠀⣸⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣡⡀⠀⠀⠀⠀⡄⠛⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣁⠉⠉⠋⠐⠒⠒⠴⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⠉⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠢⣀⠀⠀⠀⠀⠀⠉⠲⢌⡈⠸⣶⡀⠀⠄⠠⠞⢧⣀⣠
⠀⠀⠀⠀⣀⡼⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⣀⠀⠀⠀⠈⡙⠿⣧⡀⠀⠀⠀⠀⠈⣿⠆
⠀⠀⣤⡾⠋⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠤⣀⡏⠱⢦⡀⠀⠀⢸⣿
⣴⠿⡃⠀⠀⠀⠀⠘⣷⣴⣖⣼⣛⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⠤⢤⠤⠶⠛⠳⠶⢿⣷⣤⣾⣏⠆
⡏⠈⠀⠀⠀⠀⠀⠀⠘⢦⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢉⠟⢿⣿⣿
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠹⣿
⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠹⣄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⢳⣄⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡘⠀⠀⢀⣄⠀⣠⡴
⣖⠀⠀⠀⠀⠀⠀⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠈⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣼⡿⠛⠁
⣿⣆⢆⠀⠀⢠⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢤⣤⣧⡀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⢀⣀⡤⠼⠒⠛⠉⠉⠉
⠙⣿⣦⣄⠀⣼⠏⠀⠀⠀⢀⡀⠀⠀⠑⢀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠊⢿⣶⣄⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠈⠉⠁⠀⠀⠀⣀⣠⣤⣴⣶⣷⣶⣤⣄
⠀⠘⣿⣿⣿⡿⠦⢶⠏⠩⢩⣷⣈⠤⠒⠈⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣤⠀⠀⠀⠀⠀⠀⣀⣤⣠⣤⣴⣤⣶⣶⣾⣿⣿⠿⠿⠟⠛⠛⠛⠿⣿⣿
⠀⠀⢸⣿⠏⠙⠒⢶⣤⣤⣶⣿⡷⠤⣔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠿⠶⣾⠿⠿⠛⣿⠛⠛⠛⡟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣾⡟⠀⣀⣤⡾⠟⠛⡏⠀⠀⠀⠀⠹⢶⠤⠀⠠⠤⠄⣀⢀⣀⣠⡴⠞⠋⠁⠀⠀⢰⠇⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣼⣿⣷⣾⠿⠋⠀⠀⡼⠁⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⣤⠞⠉⠉⡀⠀⠐⠀⠀⠀⠀⠈⡆⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄
⣾⣿⡿⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠈⢇
⡿⠋⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⠀⢀⠔⠁⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⢦
⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⢦⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠚⠁⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠇

⠀⠀⠀⠀⡄⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⣀⠀⠀⠀⠴⠓⠂⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⠂⠀⠀⠀⠀⠀⠴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠐⣀⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⢄⡀⠀⠀⡄⠀⠀⠀⠈⠁⠠⠧⠀⠀⡀
⢀⠀⠀⠀⡀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠠⠠⠀⠀⠀⠊⠀⠁⠀⠈⢣⣠⠆⠀⠀⠀⠀⠀⠀⠀⠻
⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣠⣄⠀⠀⠀⠀⠌⣡⡄⠀⠀⠀⠈⠱⠍⠀⣀⠀⠀⠀⠀⠈⠓⠢⢰
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣄⠽⢠⠄⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠈⢄⡀⠰⡂⠀⠀⠀⢀⣡⡤⠀⢠⠀⠀⠀⠀⠀⣡⣄⡆⠀⠠⡆
⠀⠀⢀⠀⠀⠀⠙⠁⠀⠀⠘⠃⠀⢀⠀⠀⠀⠀⠀⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠥⡄⠀⠀⠀⠀⠀⠀⠵⠂⠀⠀⠀⠀⠀⠀⠀⠚⠢⠄⣠⠄
⠀⠀⠹⢳⡄⠀⠀⠀⠀⠀⠀⠀⣀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⣴⠒⠀⡀⠐⠉⠀⠀⠀⠀⠁⢻⠋⠁
⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠲⠷⣦⣀⠀⢀⡤⠀⠀⠀⠀⠀⠀⢻⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠳⠠⠀⠀⠀⠀⠈⢆⠙⣼⡇⠠⣄⠀⠀⠀⠀⠒⠛⠒⠃
⢠⡄⠀⠀⠴⣿⣁⠀⠀⠀⠀⠀⠀⠐⠁⢀⠈⠃⠀⠀⠀⠀⠀⠀⠙⠿⠀⡀⠀⠀⠀⠀⠀⢒⢔⢁⡀⠀⠀⠀⠀⠀⠀⠉⠘⡳
⡆⠁⠀⠀⠀⢬⡟⠀⠓⠀⠀⠀⠀⠀⠀⠹⠀⠄⡆⣀⡄⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⣪⢧⠠⡀⡤⠀⠀⠀⠂⠀⠉⠛⠁
⠴⠛⠃⠀⠀⠀⠠⠞⡴⠀⡀⠀⠀⠀⠀⠀⠁⠀⢠⡈⠀⠀⠀⠀⠀⢀⠶⢀⠀⠈⠂⠉⠀⠀⠀⠁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣲⠀⠀⠀⠀⠀⠀⠀⠡⡾⠀⢠⠐⠀⠀⠀⠠⠶⠾⠀⠃⠂⠀⠀⠀⠈⠈⠈⣠⡄⡀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠛⠈⠁⡀⠀⠀⠀⠀⠀⠖⣁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠶⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣽⡄⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⣠⠴⡒⠈⢛⡋⢐⠒⠦⢄⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⠀⠀⠘⠁⠶⠶⠤⢤⡉⠁⠢⠈⣢⠍⠢⡀⠀⠀⠀⠠⠤⠄⢄⣀
⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠠⠑⠦⣀⠒⠈⠲⣄⠠⠘⠀⠀⠁⠀⡠⠐⠈⢉⢓⡲⣤⣍⡢⡀
⠀⠀⠀⡼⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠸⡄⢤⢦⡀⠳⢤⡈⠀⠑⠂⠀⠀⠀⠀⣷⠒⠋⠉⠉⠉⠉⠉⠙⠛⠦⣄⡀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠋⠉⠀⠀⠀⠀⠈⢄⠣⢜⠃⢤⠚⠒⢤⣀⠀⠀⠀⠂⢀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠒⠊⠉⠀⠀⢀⡀⠤⠤⠤⣄⡀⠀⠀⠀⠈⠣⡀⠳⢤⡉⠁⣠⠐⠀⠀⠀⣦⢌⠓⠶⠤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠊⠉⠁⠀⠀⠁⠀⠐⠙⠆⠀⠀⠀⠀⠈⠢⣄⣉⠒⠦⠤⠌⣍⠢⣉⠒⢭⠒⢤⠬⠉⠳⣄⣀⣀⣀⡀⠤⠖⡫⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠶⠛⠋⠉⠓⢤⡈⠐⠤⠀⣀⡀⠀⠠⠀⠀⢀⠀⢒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀⠀⠙⠁⠀⣩⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⣤⡤⠤⠒⠋⣁⣤⠖⠒⠒⠒⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠄⡠⠤⠴⢋⢁⡠⠤⠐⠊⠉⠙⠁⠀⠆⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠖⠉⢁⣀⣀⠠⠀⠀⠀⠂⣀⣬⠵⠖⢒⠛⠿⠅⡈⠳⠤⠾⠀⠀⢳⡄⠀⠀⠀⢀⡔⡉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⢠⠋⠔⠚⢋⣁⡀⠤⠀⣤⠀⠀⢀⣀⡀⠠⣄⠀⣀⠆⠀⠥⠤⠀⢀⠀⠀⡜⠀⠀⠀⠀⡞⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡏⠠⠖⢰⣊⣁⠩⠤⠴⠚⠊⣉⠃⠁⠐⠚⠁⠤⠠⠄⠀⠠⡐⠂⠣⢀⠞⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⢻⡉⠋⠈⣈⣀⣠⠴⠔⠊⠉⠀⠀⠀⠠⠄⠀⠀⠀⠃⠀⠐⠀⣣⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⡀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠑⢄⠀⣀⣀⣀⠤⢀⠈⠀⣀⡤⠚⠁⠀⠁⠒⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠼⠤⠖⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠄⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠃⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠀⠘⠁⠄⠀⠀⠆⠀⠂⠁⠀⠤⢀⠁⢆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⢘⡄⠀⠀⡙⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠀⠐⠐⠄⠀⠀⠀⠀⢄⠀⠇⠀⠀⠈⠀⠀⠀⠀⠀⠈⠘⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠊⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠀⢠⣧⡀⣹⢿⡇⠠
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡳⠀⠀⣆⡀⢀⠀⠀⠆⠀⠀⠀⠀⠀⠙⢏⠙⠌⣷⠀⣦⠂⠀⠀⠀⢀⣀⠠⠄⠀⠤
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠶⠀⢂⠀⢸⣷⠇⢠⡴⠚⠁⠀⠠⢦
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠠⡤⠀⢀⡀⣀⡈⠁⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⣨⠷⠦⢥⣄⠀⠀⠀⢠⡆⠀⠉⠿⠀⠀⣀⣤⣴⡦⣤⣤⣄⡤⠄⠈⠀⠀⠈
⠀⠀⠀⠀⠳⠀⠀⠀⠀⡤⠀⠄⢀⠀⠀⠈⠀⠀⠐⢊⡀⠀⠀⠀⠉⠑⠦⣀⠉⠁⠀⠀⠀⠀⠀⠟⠉⠉⠁⠣⠀⡀⠈⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⢄⣈⠀⠀⠘⠸⣇⠀⠀⠀⠀⠀⣿⡆⠀⣴⢦⣀⡀⠀⠀⠀⠀⠀⠂⠀⡁⠠
⠀⣤⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠑⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠑⠀⠏⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⠀⠐⠂⠀⠀⠀⠀⡠⢿⡋⠙⠋⠉⠉⠉⣒⣶⡶⣞⡓⠂⠀⠀⠀⠒⠿⣤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⣤⠁⠈⠃⠀⠘⣁⠔⠁⠀⠀⠉⠑⠦⣤⠀⠀⠀⠉⢲⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⢰⡀⠀⠀⠀⠄⠀⠁⠀⠀⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠈⢿⡜⠀⠀⠉⡉⠁⢈⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⣅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠃⠈⠁⢀⣈⣉⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢈⣄⡀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⡉⠁⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠘⠅⠢⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⣽⢇⣆⡀⠀⡀⠀⠐⠲⠛⠛⠛⠗⠤⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠠⠀⠀⠀⡄⢠⢿⣷⡀⠁⠀⠁⠀⠐⡀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣤⡀⠀⠂⠀⠀⠀⠿⠸⢿⣇⠀⠀⠀⠀⢀⡴⡫⣷⠈⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⠀⠹⡌⢎⢳⣤⠐⠀⢀⢸⠁⠀⣆⢹⡀⠀⠀⣰⠞⢱⢣⡟⠀⠀⠀⠈⡐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⠀⠐⠀⠀⠀⠸⡇⠈⢄⠟⣷⡆⠀⠇⠀⠀⡇⣼⣷⠀⡴⠃⢀⣏⢎⡟⠀⠀⠀⠈⠀⣀⠈⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⢿⡳⣴⣄⣀⠀⠐⣇⠀⠘⣞⣰⣽⣿⠤⠤⠶⠻⠿⠿⡿⣅⢀⡞⡿⣜⡟⠀⢁⣠⠤⠚⣡⢾⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢮⠪⣝⣗⢶⣾⣀⠴⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⠔⠋⠁⣠⢞⣵⡏⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠄⠐⠠⡀⠀⠘⢧⠀⠙⠪⣪⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣠⣞⢕⣿⡏⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠘⢧⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣀⡄⣠⡤⠤⠶⢦⣴⡶⠂
⠀⠀⠙⠻⣿⠯⣟⣻⢏⣓⠿⣖⣻⠿⠀⠀⢀⡤⢴⠶⠶⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⣀⣀⠀⠀⠀⠹⡅⠀⠀⣀⠤⢒⣭⠾⠛
⠀⠀⠀⠠⠉⠓⢦⡉⠙⠛⠿⢶⡿⡀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣟⢷⡀⠀⠀⢀⡴⠟⣥⣀⣀⣴⣉⣀⠀⠀⠀⠀⠸⣦⠛⣠⣴⠞⠋⠀⠄
⠀⠀⠤⠄⠀⡀⠀⠉⠳⢤⡀⣼⠁⠀⠀⠀⠀⠁⠀⣻⣿⣿⣇⣽⣿⣷⠀⠀⣼⠻⠓⠛⠹⣿⣿⡟⣨⡗⠠⠀⠀⠀⢻⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⣿⡄⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠠⠀⠸⡅⠉⠙⠓⠶⠆⣠⣀⠀⠁
⠀⣀⣠⣶⠶⠞⠋⠭⣭⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣧⣤⣤⣶⣶⣶⣶⣴⠿⠖
⠀⠉⠿⠏⠩⢉⣀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⣶⠖⠚⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠉⠑⣲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠢⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠀⠀⢐⣰⢿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⢀⠈⠲⢦⡀⠀⠀⠐
⠀⠀⠀⠀⠀⣠⡴⣏⡬⠗⠋⠉⠙⡄⠈⠀⠀⠀⠀⠀⠀⠀⠲⠔⠓⠒⠦⠤⠚⠛⠛⠒⠒⠀⠂⠀⠀⠀⠀⠀⢠⣿⣷⣿⣟⣾⣥⣂⣽⣷⣄⠂
⠀⠀⠀⣠⣾⣗⣋⡥⢤⠤⠴⠖⠶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣄⣀⠋⣤⢊⠂⢀⠀⠀⠀⠀⢀⡴⠉⢻⡀⠁⠀⠉⡁⠉⠙⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣼⠫⡫⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠠⠐⠀⠀⠀⢀⣴⢿⡢⡀⠀⠩⡄⠈⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⢀⠄⠊⠀⢀⣼⡋⣠⠞⠁⢀⣹⣷⣦⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⣀⡤⠞⢿⠷⣴⣝⢮⡲⢄⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⠞⢁⣠⢖⠋⠘⣯⢋⣻⠋⠛⢳⣶⣦⣤⣤⠤⠴⢶⣶⣿⠋⠄⠀⢸⡄⠌⠉⠻⠿⢦⣕⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠵⠞⡁⠀⠀⠀⢸⢇⢈⠇⠀⣰⠋⢸⡏⢿⠀⠀⠀⡜⠘⠿⣆⣌⠄⠈⣏⠂⠀⠀⠀⠀⠁⡉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⢀⠀⠀⢸⡏⡌⠀⡞⠀⠀⠈⣿⢸⠈⠀⢰⠁⠀⢠⠘⢿⡌⠄⢧⠀⠀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣾⡝⡰⠃⠀⠀⠀⠀⠘⣷⢠⢀⡇⠀⠀⠀⠀⠀⠹⣤⣬⠀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡀⠀⠐⠀⠀⠀⠀⢻⣼⡘⠃⠀⠂⠀⠠⠈⡄⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠂⠈⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠤⢱⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣺⢥⣿⣂⣃⠣⡥⣁⣀⣃⣰⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢇⣡⠒⡀⣴⡏⢯⣉⠁⡷⣟⣤⣸⡗⣬⡝⠧⣀⣲⠊⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⡀⠀⠀⢄⡀⡄⢀⡤⢄⡠⣀⣰⢉⡀⠤⡇⢈⢷⣍⡿⠣⣤⣆⠼⣽⢁⣹⣵⠔⢽⣶⣿⠚⢆⡁
⠀⠀⠀⠀⠀⠀⠠⣤⣀⠠⣗⣇⣁⣷⣶⣖⠫⢭⣾⢿⠌⣡⣉⢹⡾⣟⠋⢡⣾⢰⡿⠾⣧⣼⣜⣿⣷⣽⣩⣿⡶⠾⣿⣿⠾⡨⢱⣔⣤⢄⢘⢓⣠⡖⠏
⠀⠀⠀⠀⠀⠀⡸⣿⡩⢿⡖⠌⢱⣞⡏⢻⡼⢶⣷⣶⣦⣼⣿⡛⣶⣿⣷⣞⣻⣻⣜⢹⣾⣿⡿⠾⣿⢟⣟⢫⣤⣿⣿⣶⡶⣿⣧⣴⣏⠷⣻⣷⡂⣥⡦⡆
⠀⠀⠀⠀⠀⡀⠈⠱⡾⣉⢑⣼⣾⣿⣧⡞⣿⣿⣻⣿⣿⣿⣿⣷⣿⣭⣿⣿⣿⣿⣿⣿⣿⢯⡟⣛⣵⢿⣿⣷⣿⣿⣿⣿⣿⣿⣷⢋⣟⣴⡿⢿⣍⠃⠁
⠀⠀⠀⢀⢹⣿⢀⣰⣾⣮⣙⣿⣾⢏⣟⣿⣿⣿⣻⣯⣽⡿⣿⣿⣿⣿⣿⣟⣿⣿⣷⣿⣿⣶⣾⣿⡿⢲⣿⡟⣿⣬⣻⢿⣾⣿⡟⠋⢿⣷⣬⢫⡾⡿⢧⠀⢠⣀⢲
⠠⣠⡷⢻⣾⡻⣫⣄⣰⢨⣻⣿⠿⠿⣿⣿⢿⣿⡿⢻⣧⢐⢟⢩⡹⣿⣿⢿⣿⡿⠿⢿⣿⣮⣿⣿⣧⡟⢛⡿⠟⢾⡹⣮⣯⢛⣫⣿⣿⢾⣿⣿⠃⡈⢓⡐⣟⡞⠜
⠀⠀⢈⣙⣏⣁⣿⣟⠿⣾⣽⣽⢲⣦⣽⣽⣿⣻⢿⣿⣇⠘⠷⣧⡄⣾⣹⣧⣱⡏⢸⣿⣟⣿⣟⠛⠻⡟⣈⡤⢠⣿⣷⣟⣧⣿⣿⢟⣿⣟⣿⣵⣻⣿⣿⡧⣾⣷⠁
⠀⡀⠤⣦⣿⡟⢷⣿⢳⣿⠻⡿⣄⠡⡿⣇⣸⣭⣬⣿⢟⡿⢦⣼⢻⣿⡀⣻⣧⡽⢟⢛⡭⣵⠘⢳⣿⣣⣿⢿⣻⡴⣯⡷⠿⣯⠿⠛⠿⣿⡛⠻⠿⠿⣿⢃⠈⠉
⠁⠙⡿⣏⣛⣊⣽⢿⡁⣹⡟⣿⡷⣿⠷⣾⣿⣏⢩⡝⠻⣷⣺⢿⣵⣟⡁⣶⣙⣦⣜⣿⡾⣿⣾⠫⡮⡏⠉⠀⠛⠒⢮⡷⣚⠁⣰⣾⣺⣿⡏⣔⣶⣐⣿⢘⠂
⠀⠀⠀⣠⡸⣿⣿⡾⢿⢅⣹⠟⠃⠀⠀⠈⣿⣿⣿⣿⠒⢿⠭⠷⢶⢏⣝⣿⡟⢛⣯⣟⣯⢛⢶⠛⣥⣅⢒⣀⣘⣴⣀⣀⣉⣿⣯⣬⣽⡗⣿⠻⠿⢽⣳⢡⣄
⠀⢠⣽⣿⢻⣿⣋⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⢻⣳⠞⣆⠀⠙⢰⣶⣷⣟⢯⢽⡿⠃⠛⣻⠅⣵⠍⢂⡉⠉⡷⡚⠿⢥⣄⣀⣈⣱⡿⣼⣽⣓⡾⠷⣿⢉⡀
⠀⠈⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡝⣦⠾⣮⣿⣿⣿⠏⣨⣋⣈⢟⠩⠿⡇⠫⢓⡈⠂⠈⠃⠀⠀⠈⠙⠖⢢⡸⠿⣇⡻⣗⡾⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣐⠫⠈⡿⣿⡿⠀⠀⠉⠀⠀⠤⠀⣃⠳⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡏⠀⠐⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⣠⣿⡟⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡽⠐⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⢘⢳⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣼⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢧⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⠖⣸⣿⣿⣄⢠⣴⣼⣿⣷⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⢠⠁⠁⣰⣿⣿⣿⡧⠺⠉⠉⠀⠛⢠⡤⢼⠄⠀⣤⠀⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⣣⣦⣀⣄⣧⣴⣏⣸⣾⣿⣟⢿⣿⣿⣤⢁⢠⣴⣧⣧⣆⣀⣈⡩⢽⡟⠉⠉⠉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠛⠈⠉⠉⠙⡉⠘⢿⠈⣻⣭⡉⣙⠲⠢⠴⠿⠻⠆⠀⠒⢿⡉⠋⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⠀⠀⠀⢀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⣿⠀⠀⣴⡿⣻⢿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⣀⣿⣀⣾⠋⡼⠹⣼⡆⠀⠀⠀⠀⡸⡇⡴⢿⣾⡥⣶⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠿⠛⠛⠛⢹⠁⠐⢷⣸⣿⠀⠀⠀⢠⣿⠙⠏⠀⠿⠁⠀⠈⠹⡼⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⡠⠀⠀⠀⠀⡼⢋⠏⠀⠀⠀⢿⠯⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠁⢸⡀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠜⡽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠐⠉⠉⠁⠀⠀⠀⠀⢧⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡂⢻⠶⠚⠀⠰⠁⠀⢀⠀⠀⢀⠀⠀⠀⠈⢷⣄⠹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠿⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡈⠳⠦⣤⣔⣀⣐⣄⣨⠶⢏⠀⠀⠀⠀⠀⠙⠷⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢹⡄⢀⡴⠃⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡴⠒⠛⠉⠀⠀⠀⠀⠀⣀⣴⣋⣤⣤⣄⡀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣾⣏⣀⣠⡴⠒⠒⠒⢋⡽⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡟⠀⠈⠆⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡀⠀⠀⠀⠀⠀⢠⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢠⣤⠽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣙⣦⣐⣀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⣠⣄⡲⡶⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢀⣍⣩⣤⣬⣭⠿⢾⣦⣀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⣰⠖⠛⠛⣉⣀⣀⣀⡤⠞⠛⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣩⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⣶⣠⡆⡆⢀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠰⢶⣶⡆⣲⣀⣿⣿⣿⠿⠿⠳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣥⡀⡓⢻⣭⠀⠘⣷⣿⣿⠓⠹⡿⠿⠋⠀⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⣻⣿⣷⣏⣿⣿⣏⠹⣉⡝⣿⣶⠿⠷⢶⡞⠉⠻⠿⣮⣶⡀⢍⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⡿⣿⣿⣿⣶⣚⢻⣿⣧⣦⣴⣿⠇⢀⡀⠀⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠋⠀⢏⣉⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⣀⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣾⣷⣾⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢻⣿⣿⣍⠻⠻⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⢉⣨⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣺⢦⣶⣿⣧⣼⣿⣍⡃⣹⣦⣉⠙⣽⣿⣿⣿⣳⣿⡟⡋⠡⠌⠉⠈⠿⠙⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⡗⠺⡿⣿⣿⡟⣿⣿⣿⣿⣿⢟⣽⣿⣿⣧⣶⡞⠄⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠊⠋⢀⣹⣮⣿⣿⡇⣿⣿⢿⡿⢟⡠⠀⣤⣸⣷⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢾⣿⣿⡿⢷⡇⢻⣿⣿⣿⣿⣟⠒⢻⣿⠛⠛⢿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣦⣿⣶⣾⣐⢲⣟⡏⠃⠈⣿⣘⣍⠹⢛⣻⡟⠒⠈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡾⢺⣯⡏⢛⣿⠛⢻⡶⠀⣿⣽⠿⡵⣏⣿⣿⣿⡧⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡚⠉⠀⠈⠁⠀⣿⡇⠀⠀⠈⠫⠥⡍⠈⠛⣿⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠁⠀⠀⠀⠀⢀⣟⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣚⣛⣛⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢻⡿⣻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠉⠀⠒⠒⠦⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢸⠁⠀⢀⠤⠐⠒⠊⠁⠀⠀⠀⠀⡔⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠒⠁⠀⡰⠁⠀⠀⠀⢀⠀⠀⠀⠀⡴⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⢀⠎⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡄⠀⠀⠜⠀⠀⠀⠀⠈⠀⢀⠖⠉⢁⠀⠈⢀⠎⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠂⠒⠁⢀⠞⠁⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠛⠤⠀⠀⠁⣠⠊⠐⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⣠⠤⠐⠒⠊⠀⠀⠀⡀⠀⠀⠀⠘⠁⠢⠀⠀⠀⠀⡠⠒⠈⠁⡀⠀⢀⡼⠀⠀⠀⠀⠐⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔⠁⠀⠀⠀⠀⡀⠀⠀⢈⡤⠀⠠⡄⠀⠀⡠⠄⠀⠈⠀⠀⠠⠀⠈⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡀
⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⣀⠤⠒⠁⢀⠀⠀⠀⣀⠜⡀⠀⠀⠦⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⣀⠀⠀⠀⠈⠢⡀
⠀⠀⢰⠁⠀⢀⡰⠃⠀⡠⠀⣠⠁⠀⠀⡔⠁⢀⡀⠀⠈⠁⠐⢊⠑⠄⠈⢀⣀⡠⠴⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⠀⠱⡀
⠀⠀⡆⠀⠰⠁⠀⡠⠎⡰⠋⠀⡠⠶⡊⠀⠀⠀⡠⠄⠀⠀⠀⢀⠔⠂⠉⠁⠀⠀⠀⢃⣀⣀⠤⠤⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠃
⠀⢠⠁⢀⠆⠀⠘⠀⢀⡀⠤⠊⠀⠀⠀⡠⠔⠊⠀⠂⡰⠀⢡⠃⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡸⠁
⠀⠸⡀⡜⠀⠀⠁⠀⡏⡀⠀⠐⠀⢀⢎⡀⠈⠂⡠⠊⠉⢠⠂⣠⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⢤⣀⡀⠑⣄⠀⡀⠠⠄⢒⡡⠊
⠀⠀⢣⡱⡀⠀⡔⠋⠀⠈⠂⠀⣀⠜⠀⢀⣠⠎⠑⢀⡴⠁⡜⠁⠀⠀⠀⢀⣀⡤⠴⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢮⡆⠀⠐⠈⠁
⠀⠀⠀⠱⣁⠘⠀⢀⣀⡀⣠⠎⠀⣀⠴⠁⢀⡠⠖⠁⢀⡞⠀⢀⡀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸
⠀⠀⠀⠀⠈⠓⠦⠥⢀⣈⣀⣒⡊⠤⠤⠒⠉⠀⠀⠀⣎⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⡀⠀⠈⠛⠋⣉⣁⡤⠤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣠⠖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣷⠀⡑⠀⠙⢴⡕⠁⢀⠶⢁⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⢌⢀⠠⠜⢺⣤⣄⣄⡗⠟⠁⡠⠄⢈⠤⣲⣀⡀
⠀⠀⠀⠀⠀⢦⡀⡀⠀⢠⠀⠶⡀⠀⠛⠀⠤⢤⡱⣿⣮⢳⣇⡞⠀⠠⣨⠒⡋⠉⠠⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠐⠱⣿⡄⡀⠂⠀⢸⡦⣠⡀⠀⠀⠀⢽⣿⣿⡯⠀⣠⡞⠛⠍⡄⠁⠀⠀⠀⣾⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣓⣳⡄⠠⠥⣽⣶⡀⠀⣸⣹⣿⣧⡽⢯⠗⡀⠌⣁⢀⣠⠶⠊⣀⡠⢞⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⢀⣻⣦⡱⢿⣝⣷⣤⣈⣹⣿⠏⠀⠈⣶⠿⠶⠄⣠⣖⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣤⠶⠛⠿⢿⣿⣿⢿⣿⣽⣿⣤⣶⣶⡟⢀⣴⣾⡅⠈⠉⡉
⠀⠀⠀⠀⠀⠀⠈⠹⠟⠊⠉⠈⡉⠰⠲⣶⠶⠿⣧⠻⣿⣿⣻⣿⣿⣿⠛⣽⡿⠷⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⡶⠖⠾⠷⣾⣿⡏⣿⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣲⠶⠶⠥⠼⠾⣝⣂⡀⢿⣷⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠳⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⢤⣤⠠⢤⠼⠷⠦⠐⠀⠀⠈⠀⠉⢙⠩⠍⠙⠙⠛⠛⠛⠓⠲⠲⠦⢀⡀⠠⠌
⠀⠂⠡⡄⡐⠀⠀⠘⠃⠀⠀⠠⢁⠀⣀⣀⢀⠦⠤⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⡀⢀⡀⠀⠀⠀⣈⡻⠂⣠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠿⣃⡀⢀⣠⣉⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡜⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⡄⣠⡆⠀⠀⠀⠀⠀⠰⠿⠗⠀⠉⡤⢠⡉⠉⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠴⠮⠀⠐⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠉⠴⠂⠀⠀⠀⠀⠀⣴⣶⣶⡤⠀⣬⢃⢂⡀⢈⠉⠁⠀⠀⠀⠀⢠⣤⣰⡇⡠⣤⣶⣦⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡛⠀⣠⡀⠀⠀⠙⠿⠟⠁⢇⡰⠃⠘⠇⠿⢀⣀⠀⠀⢀⠀⣤⡛⣧⡜⠁⠻⣿⡿⠁⠀⣷⡄⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠐⢄⠟⣡⡤⠀⠀⠀⠻⠗⣼⠘⠛⠛⢀⡄⠘⠁⢰⣄⠻⣃⡴⠚⢁⣛⡸⠄⠀⣀⠶⠦⠈⠀⢶⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠴⢆⡑⢭⡀⣴⠀⠀⠚⠃⠹⣌⡻⠆⠉⠀⠀⡼⠀⣱⡾⠛⠋⣉⣉⠀⣼⠇⠈⠛⠇⢀⢎⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⣀⡙⠦⣝⠀⣾⠀⡶⢪⡿⣦⠈⠛⢛⡱⣄⡟⠰⡆⢾⣿⣿⡇⢀⣤⠀⣾⢀⡞⠸⣿⣿⡟
⠀⠀⠀⢀⣠⣤⡀⢠⡄⢀⡄⠀⠀⠀⠀⠀⣿⣿⣿⠆⠈⣙⠦⣄⡰⡆⢠⡿⣤⡬⢉⣡⣼⡗⢀⡀⠈⠙⢉⣀⣛⣣⣤⠶⢛⡰⡆⢀⡀⢰⡇⣴⣾⠿⠂
⠀⠀⠀⠈⠙⠛⣋⡈⠁⠿⠃⣠⠀⠀⠀⠀⠈⠙⢉⣀⠘⡋⠰⠈⠻⣶⣿⢀⣠⡶⢛⡉⢵⣄⠈⠓⣴⡿⠟⠛⠛⠻⠷⢦⣌⣀⣀⣺⣣⠌⢡⣦⡀
⠀⠀⠀⠀⠀⠚⠛⢱⡆⠈⠲⢥⡀⢹⣦⠀⣠⡜⠃⠙⠃⠛⢀⣈⣳⣘⣿⣿⡏⠀⠘⠃⠀⣉⡀⢸⣿⢀⣛⠇⢠⣶⣶⡄⢰⣎⠉⣉⠀⠸⡇⣉⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣆⠙⠳⣥⣀⠉⠀⠞⠀⢶⡄⠀⠖⢀⣩⠻⣿⣧⠛⠁⠀⠀⠘⠛⣼⣿⣬⣭⣀⠘⠿⠿⠃⠀⢉⣀⣟⣘⣃⣀⠟⠰⠟⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃⣰⠿⠀⣍⠛⠛⣛⡓⠲⠶⣶⡄⠘⠋⢼⠊⢿⣧⠀⠀⣠⣴⣿⣿⠟⠁⣨⠙⡛⠛⠛⣋⡍⠉⢉⣁⡀⠠⣄⠀⠰⣦⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢰⡄⢠⡆⠉⠤⣶⠍⣡⡾⠛⠻⠿⣶⣶⣦⣤⣈⣿⣧⣼⣿⠟⠉⠀⠀⠈⠉⠸⡿⠀⠐⠛⠁⠀⣿⣿⣿⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⠛⠈⠵⠒⠓⠒⠶⠶⠿⢫⣄⠀⠀⠀⣶⢠⡍⠛⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡾⠿⠋⠀⠻⠆⢠⣶⣿⣶⠀⠺⣦⠉⠁⠀⠀⠀⠈⠃⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢴⣦⠈⡄⠀⠀⠀⣶⣶⠏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⣀⡠⣶⡀⠀⠀⠀⠀⠀⠀⠛⣷⡈⢀⢀⠠⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣉⠝⠻⢸⣷⠀⠀⠀⠀⢡⣠⣿⣨⡿⣦⣹⣾⣔⣾⠇⣿⠆⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢻⠟⣻⡛⢤⠀⠉⠀⠈⢳⡄⠉⠛⢿⣿⣷⣬⣿⣿⣿⣷⣾⣯⢠⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣸⠼⠿⠃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⠛⠛⢿⣿⣿⣿⣿⣾⣿⡿⢂⡔⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠐⠈⠉⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠿⣿⣿⣿⣿⡿⠃⡷⠷⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⣀⡀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣷⣧⢰⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠒⠀⠉⠁⠀⠀⠀⠀⠈⠀⠁⢉⣤⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠅⢼⣿⣿⣿⣿⣿⣿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢄⣉⠐⢶⣄⠀⣠⠀⠀⠀⣀⣾⣿⠀⠐⠀⠀⠀⠁⠀⠄⠉⠀⠢⠹⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠣⡀⢄⠈⠑⢬⣷⠆⢴⣻⡟⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⢡⡄⡀⠀⠉⠩⠀⡄⡣⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠃⢀⠈⠜⣧⢀⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠖⢹⣟⠈⢀⠀⠐⠀⠁⢀⢰⣿⣿⣿⣿⣿⣿⡿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠣⣼⣿⣇⣤⣌⣦⣤⣠⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⣿⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢆⠀⠶⠾⠁⠊⠀⡻⡿⠿⣿⣿⣿⣟⣿⣿⡿⣖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠈⠙⠈⠓⠋⠀⠈⢸⠁⢙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⠴⠚⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣠⣴⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡞⠁⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠦⠼⠷⠾⠿⠶⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⠤⠖⠲⢤⣤⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⠛⠿⢿⣿⠿⠟⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢸⠀⠀⠀⢠⣤⠠⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠀⠀⠀⡏⠁⢠⡀⠀⠀⡘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢾⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡈⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⠉⠂⠀⠀⠀⠀⠀⡡⠀⠀⢼⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠐⠲⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠⣼⠟⠁⠰⣒⡒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠦⡄⠀⠣⣇⢀⡀⣤⣀⡼⠧⢶⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣂⣈⣀⠰⠋⠰⡆⠐⡠⠠⠌⠟⠀⠹⠈⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠉⠀⠀⠔⠂⣀⢙⣆⠀⣴⢶⠾⠷⠶⠒⢀⣤⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠿⠟⠒⠉⠁⠀⠀⠐⠂⠑⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷⠼⠿⣿⣿⣷⣬⡿⣿⣿⣯⣥⣿⣿⣧⢧⡾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡫⠉⠉⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⡀⢀⡏⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠋⠀⠀⠀⠀⠀⡴⠚⠉⠓⠲⠶⠶⠛⠉⠀⠀⠉⢢⠀⣠⠞⠉⠙⣿⡇⠀⠀⠀⠈⠒⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠛⣿⠁⠀⠀⠀⢠⠖⠛⣇⣀⣀⣀⣀⠀⣠⠔⠒⠲⠦⣤⣴⠾⠋⠀⠀⠀⣿⠣⡀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⣿⡀⠀⠀⠀⢸⣴⢞⠋⠉⠈⠉⣩⠿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠙⢗⠢⣠⠖⠋⢳⡀⠞⠉⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠋⠁⠀⠈⠈⠙⠒⣤⠟⠑⠁⠀⠀⢀⡞⠑⠀⢻⣀⠀⠀⠀⠀⠀⠀⣀⠔⢋⠐⡲⢤⣄⣠⠜⠁⠀⠀⢸⠃⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣯⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡾⠰⠀⠀⠀⠈⡙⠛⠒⠒⠚⠻⣤⣰⡇⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠈⠙⠻⠿⣿⣧⡀⠀⠀⢿⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⢀⣠⣯⡴⠞⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠈⠉⠛⠒⠾⠷⣤⣄⣤⣾⣷⡀⠀⠀⠀⢈⣇⣀⣤⡀⠀⠀⠀⢈⡃⣉⣓⣶⡶⠶⠞⠛⠉⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠒⠶⠾⠿⠷⠶⠚⠓⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣟⣛⡻⠿⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠖⠚⠛






⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣶⣤⡀⠀⠀⣄⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣦⠀⢹⣷⣄⠀⣿⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⡈⢻⣿⣿⣿⡆⣾⣿⣿⣴⣿⣿⡏⠀⢀⣾⡇⠀⠀⣠⣾⣿⠋⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⡿⠀⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢿⣶⣶⣾⣿⣶⣤⣀⠀⠹⣿⣿⣿⢿⣫⣽⣶⣶⣶⣾⣭⣟⠿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⣿⣷⣾⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢿⣿⡿⠟⠉⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣭⣿⣿⣏⣾⣿⣭⣥⣭⣝⣿⣿⡿⠿⠻⠿⣿⣧⢻⣿⣿⣷⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠿⠿⣿⣿⣿⢹⣿⣿⣿⣟⣧⣿⣿⣿⣾⣿⣿⣻⣾⣿⡟⣿⣿⣛⣉⣀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣯⡉⠉⠙⠛⠛⠋
⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠟⠉⢁⣴⣾⣿⣿⣯⡻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣾⡿⠟⠛⢋⣿⣿⣶⣽⣛⣿⠿⠿⣿⣛⣭⣾⣿⣿⣿⣿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣠⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃⢸⣿⣿⠏⣿⣿⣿⠏⢿⣿⡆⠀⠀⠻⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠸⣿⠃⠀⣿⣿⣿⡄⠀⠉⠛⠶⠄⠀⠀⠀⠀⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠹⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⣀⣀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠘⣿⣦⡀⣴⣿⠀⠀⠀⠀⣀⣠⣄⣹⠋⢀⣛⡛⠋⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⠀⠘⢉⣤⣶⡶⠂⠈⠛⠻⠟⠋⠀⠘⢦⣾⡁⠀⠀⠙⢦⠈⠻⠿⠷⡀⠀⢸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠓⢖⡛⠛⣉⣀⣀⣀⡀⠠⡄⠀⠀⢠⣾⣿⣿⡆⠀⠀⠘⡆⢻⣴⣋⠉⢳⡌⠁⣄⠀⣠⣾⡇⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠀⠀⠀⠀⣹⠿⠛⠉⠉⢉⡟⠻⠿⣦⣤⣀⠻⠿⠿⠃⠀⠀⢰⡿⣿⣿⣿⠂⢰⡇⠀⠙⢧⡿⠋⠀⠀⢸⣿⠃⣀
⠀⠀⠸⣷⣄⢀⣾⣿⣿⠇⠀⠀⠀⢸⡏⠀⣴⣶⣤⣄⠃⡴⠒⠛⣯⠙⢿⣦⡀⠀⠀⢀⡾⠁⢙⠛⠋⣰⣿⣤⣤⣄⡔⢠⣤⣤⣤⣸⠏⠀⢿⡆
⠀⠀⠀⠈⠻⠼⠿⠛⡁⠀⠀⠀⠀⢘⣇⠀⠀⠉⠉⢉⢸⡃⢠⣶⣿⣷⣶⠙⣿⣦⣴⠏⠀⠐⢋⣧⡾⠋⣠⡶⠀⢈⣳⣦⠼⠉⠁⠈⡂⢀⡬⠓⢾⣿⣿⠿⠃
⠀⠀⠀⠀⡄⢰⣶⣾⣷⠀⠀⢠⣾⣿⠋⠳⠤⣤⠖⠃⡈⢷⡈⢿⣿⣿⡟⠀⠘⣿⣿⠀⣀⣴⡿⢋⡸⣷⡿⠁⠀⢿⣿⣿⡷⢀⣤⢴⣋⢀⣴⡏
⠘⠿⣿⠿⢧⡘⢿⣿⡟⠀⠀⠟⠛⠁⢠⡾⠋⠉⣩⣶⣷⡈⣿⣈⣉⣭⣄⣀⠀⢹⣿⣿⣿⠋⠀⡸⢧⠉⣠⣴⣄⠘⠛⠛⡱⠃⠀⣀⣽⣿⣫⣀
⠀⠀⠀⠀⠀⠙⠶⢦⣤⣤⠶⠶⠶⢤⣿⡇⠀⠀⣿⣿⣿⢧⡿⠋⠁⢀⣦⠉⢳⣸⣿⣿⡇⡜⣽⣭⠹⡄⠹⢿⠟⠀⠀⣰⠃⢠⡀⢿⣿⣿⣿⣿⢀
⠀⠀⠀⠀⠀⢀⣀⣷⣾⣅⡀⠀⠀⠀⠙⣧⡀⠀⠈⠉⠉⣾⠁⠀⠀⠸⣿⡇⢀⢿⣿⣿⡇⢱⡟⠋⢀⡗⢀⣀⣠⡴⠞⠁⠀⣿⣷⠘⢿⣿⣿⠏⢸⡇⠀⣀⣤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣤⣀⢰⡇⠀⠙⠛⠶⠶⢶⣿⡀⠀⣤⣶⡾⠓⠀⣼⣿⣿⡇⠘⣃⣴⣿⠛⣛⠛⠲⠦⣄⣀⠀⠙⠻⣀⡀⢦⣤⣀⡸⢇⣼⣿⠏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠁⠈⠉⠋⣠⣶⠖⠈⣷⠀⣠⡌⠻⣦⣄⣀⣀⣀⣴⣿⣿⣿⣷⣾⠋⢠⣶⣿⣧⠰⠃⢀⣠⣭⣭⣉⠀⠙⠻⠦⠉⠉⠉⡈⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣄⠉⠁⠀⠀⠨⡚⠋⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⡃⠀⠸⣿⣿⣿⣇⣴⠟⠁⠀⠀⡈⠙⣦⣄⣀⣀⣠⣴⣁⡀⣴⡿
⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠾⠶⣤⣤⣤⠤⠷⠦⠤⣤⣤⣤⣀⠀⠈⣿⣿⣿⣿⣿⠿⠶⣤⡤⠴⢋⣾⠃⠀⣶⡄⠀⢻⣷⣧⡄⠉⠉⠀⠀⠀⠹⡄
⠀⠀⠉⠉⢡⣀⣀⣤⣴⢾⠏⣀⠀⠀⠀⣠⠖⠋⠉⢿⣦⢀⠀⠀⠉⠻⣿⣦⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⡾⠁⢦⣤⣹⠇⣼⣿⣿⣿⣿⣧⠀⠰⣄⠀⠼⠃
⠀⠀⠀⢠⣿⣿⠀⠀⠀⣼⣾⡋⠀⠀⢠⡇⠀⠀⠀⣠⣽⣧⣄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⣀⣤⣴⣾⣏⣀⠀⢀⣀⡕⠀⠸⣿⣿⣿⣿⠃⠀⣴⣿⣷⣦
⠀⠀⠀⢸⣿⡟⠀⠀⠀⠿⠿⠃⠀⠀⠘⣧⠀⠀⠀⢻⣿⣿⡿⠀⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣟⠻⠿⢯⣅⠀⢶⣄⠈⢉⣭⣥⣤⡀⠹⣿⡿⠃
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢳⣤⠞⠛⠻⢷⣦⣄⣀⣩⣭⣴⡿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠋⠀⢿⣿⣿⡷⠀⠀⣬⠛⢦⡬⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠘⠛⠛⣁⣀⠀⠛⠀⠀⢹⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠛⢛⣦⣀⣠⠜⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⢺⣦⠙⣴⡾⠀⠀⢠⣆⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠰⣶⡜⠟⣁⣀⠀⠀⠉⠁⠉⠀⣰⡄⠘⢏⣯⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⠶⣋⣋⠀⢀⠉⠉⠘⠛⠋⠀⠀⢀⡇⠀⠀⣟⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠈⠒⢦⡀⠀⠀⠀⣼⡥⠚⠉⠀⠀⠀⠀⠀⠀⣄⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⣀⢀⣙⣆⠀⢰⡟⠀⢰⣷⢀⣤⠀⠀⠀⠀⠻⠫⠷⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢫⡾⠀⠀⠀⣴⡿⠈⠹⣧⣿⠁⢶⣦⡟⠿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⣤⡀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⢻⣿⠀⠀⠉⠀⠀⢀⣀⠤⠤⠖⠀⠀⠀⣉⢽⣦⣄
⠀⠀⠀⠀⠘⠿⠿⣎⣙⠀⠀⠙⠇⠀⠀⣤⣤⡻⢇⣶⠆⠀⠀⢸⣿⠀⠀⢀⡶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠻⠇⠉⠁
⠀⠀⠀⣤⡀⠀⠚⠛⠁⣀⣠⣤⣬⣴⣤⣈⣛⡛⠈⠁⠀⠀⠀⢸⣿⡄⣰⠏⠀⠀⠀⣀⢼⡇⣀⠀⠀⠀⠀⣴⣠⣶⠇⠀⠀⣀⣤
⠲⣶⣤⠛⠃⢀⣠⣀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣄⡀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠻⠇⠟⠋⠀⠀⠀⠀⠙⠱⡶⠆⠀⠀⢛⠣⣶⣶
⠀⢀⣾⡟⠀⢸⠏⠻⠀⠀⠀⢦⣤⡀⢀⡠⠒⠚⠛⠛⠿⣿⣦⡀⣿⣿⠀⠀⠀⠀⣀⣴⣶⡶⠶⠶⣶⠒⠲⠒⠒⣒⠈⠀⠀⠘⢿
⠀⠀⠁⠀⠀⠈⠀⠀⠘⠻⣳⡄⠉⠉⣴⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡆⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣄⣻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⣠⠴⠖⠚⢉⡟⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢀⡜⠁⠀⢀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠤⠒⠉⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⡀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣯⡀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣹⣦
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢄⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠈⡆
⠀⠀⠀⠀⠈⢢⡁⠀⠀⠀⠀⠈⠳⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢱
⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⣀⣠⣤⠽⠶⠟⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠀⠉⠙⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠈⠻⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢠⠞⠋⠀⠀⠀⠔⠂⠉⠉⠉⠉⠉⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⢀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠔⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠒⠒⠀⣷⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠠⠊⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⣄⣀⣀⣀⠀⠐⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⣀⣤⡴⠞⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣄⡉⠉⠙⠛⠶⠶⠶⠶⠶⠶⠶⠶⠞⠉⠀⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⢤⣀⡀⠢⠤⢤⡄⠠⠄⠒⠂⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡒⠚⠙⠶⡚⠛⠦⣄⠀⠀⠀⠀⠉⠓⠶⠤⠴⠶⠚⠋⠛⠋⠉⠀⠀⠐⠤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠠⣇⣴⠶⣄⠙⢦⡄⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⠀⠀⣀⣀⣀⠀⠀⠄⢠⠂
⠀⠀⠀⠀⠀⠀⢀⠴⢦⠙⢄⡤⢬⠁⠀⠉⡷⠀⠧⢀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⡄⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⠂⡎⠉⠀⠠⠎⢱⡟⠓⠚⢦⠣⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣻⡀
⠀⠀⠀⠀⠀⠀⢀⢤⡂⢠⠞⠁⠀⣈⠒⠦⣌⠳⡄⠀⠀⠀⢁⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡴⡆⣸⣶⠇⡀⡀⢀⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠈⣦⠁⠀⠣⣠⠜⢀⣉⣀⡈⠲⢬⡁⠂⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⣿⢻⠃⠀⠘⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠲⠤⠆⠰⡉⠀⠈⠙⣇⡀⠙⢲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠇⠈⠀⡠⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⠀⡌⠀⠒⠒⠉⢻⡆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠐⣊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠒⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠛⠋⠁


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

⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣤⣄
⠀⠀⠀⠀⠠⠿⠛⢻⣿⣿⡗⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⣿⣿⣿⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣷⣶⣶⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠷⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠒
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠙⠐⢺⡀⢈⠒⠰⠶⢄⣀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠘⠁⠠⠄⠙⣙⠷⠓⠐⠒⠄⠓⡢⢴⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣠⠐⢦⢴⢈⠇⠸⣴⣖⠚⠥⣔⠀⢰⠦⠵⠞⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠶⣇⣈⠚⠖⠀⠀⠾⡀⠛⠊⣀⠫⣤⡈⠀⠀⢠⡄⠀⠲⢣⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣛⣂⡈⡀⣸⢦⢻⡳⢆⠩⢴⡂⠛⠷⢂⣝⠀⢀⣸⣀⠀⡄⠈⠡⠆⠴⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠛⠰⢩⡇⠶⢀⠠⠄⣖⠘⣆⣀⠀⠀⣸⠡⠉⢦⠠⠥⠐⠥⡚⠁⠀⠀⢹⢳⠄
⠀⠀⠀⠀⠀⠀⠀⢰⢬⡏⠀⠓⠀⠘⡇⠈⠒⠓⣋⠳⢼⡧⠁⠀⡎⠀⠂⠀⣾⢂⠓⡛⠑⠂
⠀⠀⠀⠀⠀⠀⠀⡄⡴⢇⡙⠏⠀⠑⢇⠀⠀⠀⢹⡗⣦⠀⠀⠀⠃⠀⢀⡼⠁⠀⠋⠘⡅⣆
⠀⠀⠀⠀⠀⠀⠠⠋⠒⠈⠉⠀⠀⠀⠈⠂⠀⠀⢸⡇⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡀⡄⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⡽⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡀⠀⠤⡹⢿⣿⣦⠄⣂⡐⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠁⠒⠀⠀⠁⠀⠀⠈⠁⠠⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⡤⠤⢤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡞⢁⡇⠀⡼⠁⣠⡇⠀⠀⢈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡅⢸⡃⣸⣧⠼⣿⣃⡉⠙⠒⣲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⠀⠀⠀⠀⠀⣰⡇⢸⣧⢿⡗⠙⠿⣦⣀⡉⠙⣛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠳⡀⠀⠀⢰⣹⣿⡛⠷⢤⣙⠳⠶⣄⣨⠽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⢀⣴⣟⣲⣬⣿⣷⣶⣜⣋⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠛⠉⠉⠉⠙⡝⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⣉⣑⡒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢍⣤⣶⣶⣷⣄⡠⠤⣴⣿⣼⣷⣿⣷⣶⣶⠷⠖⠂⣩⣿⣿⣿⣯⣽⣄⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢩⡿⣫⣿⠟⡩⢟⠔⢆⠔⣠⣤⣤⡄⣦⡄⢠⠀⡀⢨⠫⡙⠝⠕⠂⠙⢧⡀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠐⢁⠞⠉⠀⠄⡴⠣⠂⠈⠄⣾⡟⣿⡇⠻⠈⠄⢷⡘⡔⡑⡘⢎⢂⠀⡀⠀⠙⣆⠀⢢⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⠂⠂⠂⠀⠀⠀⠀⠀⠀⠐⠘⠇⣿⡟⡇⠀⠀⠈⠘⣧⣸⡄⠐⡈⠆⠀⠀⠀⠀⠘⣧⠀⠂⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡇⠇⠀⠀⠀⠀⠸⠀⢇⢀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠇⠀⠁⠀⠀⠀⠀⠀⢀⠀⢸⠇⠀⠀⠀⠸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠰⠀⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠀⠀⠀⠀⣸⢀⢸⠀⠀⠇⠀⡇⠀⢰⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡇⢇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⡆⠀⠀⠀⢰⡸⡘⠈⠀⠘⡀⢠⠁⠀⢸⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣸⠀⠀⠀⢀⡆⠀⠀⠀⢀⢠⠀⡄⠀⠀⠀⠀⢰⢸⠀⡇⡇⢰⠀⠀⡇⡇⡇⠀⠀⢠⠇⡌⡘⠀⡏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣧⠀⠀⠈⣿⡀⠠⠀⢸⢸⠀⡇⢸⠀⠀⠄⢸⢸⢶⠃⢣⡞⠠⣼⣼⢸⠁⠀⢀⡾⠐⢰⠁⡸⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⢹⣥⢠⢀⣜⣯⣤⡇⣸⡀⢰⡆⣧⡇⢸⢰⣼⣇⢷⣗⡇⣾⢠⣧⣯⣷⢣⡃⢰⢷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⢷⣰⣿⡎⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇⡟⡎⣿⣾⣿⣿⢠⣧⣿⣻⣿⢣⠏⣱⢫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣯⣻⣾⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⡇⣾⣿⣷⣿⣷⢣⡞⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⠿⠛⠁⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⣀⠀⠀⣸⣿⠀⠀⡀⠀⠀⠀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠸⣿⣇⠀⢿⣤⡄⣿⣿⠀⢰⣇⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠈⢳⣔⣦⣿⣿⣦⣼⣿⣿⣿⣿⣦⣿⣿⣧⣿⡿⠁⢀⡄⠀⠀⠀⢀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⣻⣿⣿⡿⠿⠛⠛⠛⠛⠛⠻⠿⣿⣿⣿⣇⣴⢟⡆⢀⣤⣶⡿⠟⠁
⠀⠀⠀⠠⣤⣤⣀⣀⡀⠀⠀⠐⢦⣔⡶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣿⠟⠋
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣷⣦⣤⣀⡈⣻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇⣤⡶⠚⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣟⣀⣤⣤⣤⣶⣶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣶⡶⠦
⠀⠠⠴⠶⠛⠛⠛⠻⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣺⣽⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠿⢿⣿⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⢶⣤⣀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠿⢛⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣏⡉
⠀⠀⠀⠀⠀⢀⣴⠿⠛⠋⠉⠀⠀⣨⡿⠛⢹⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣯⡉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⡿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡟⠮⠻⢦⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢻⡟⠀⣿⣿⢿⣿⡟⠻⣿⣿⢿⣻⣆⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠚⠀⢠⣿⡟⠈⢻⡇⠀⢻⣿⠀⠁⠹⠄⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠃⠀⢸⣿⡆⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣠⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠻⢿⣄⡀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠉⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠞⠉⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⠀⠀⠀⠀⠀⡤⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⣷
⠀⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⢿⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢿⠲⢤⣀⣀⣀⣀⣀⣤⡶⠀⠀⢀⡇⠀⠀⠀⠀⠐⠛⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠓⠤⣀⣉⣉⣩⠵⠋⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠈⠀⠀⢀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠁⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⢀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠆⠠⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠛⠁⠉⠄⠂⠀⠄⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠚⣸⢹⣾⠀⠀⠀⠀⠀⢀⡄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢻⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⠄⠀⠀⠈⠣⠀⠀⠀⠙⠂⠀⠀⡀⠀⠀⠀⢠⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡀⣀⠀⠇⣠⢀⡆⢼⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⡐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣿⣿⣟⣷⠈⢠⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠄⠀⠛⠰⠀⠼⠤⢄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⡻⢇⠼⠁⡾⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠾⣶⠤⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡅⢉⡰⠋⠾⠘⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡞⣷⠒⠃⠀⠠⢀⠴⠀⠀⠀⠀⠀⣿⡇⠸⠗⣁⣤⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣇⣶⣿⣿⣿⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣃⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠰⢫⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡈⠳⣦⣠⡄⣆⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⢵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣟⡷⡂⠉⣉⣿⡿⠷⠞⠋⠉⠛⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠐⣦⡩⡁⠀⠀⠀⡀⠠⠐⢣⡔⠸⣄⣠⣧⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣄⣀⣠⣆⡀
⠀⠀⠀⠀⢿⣧⣤⠄⠀⣄⡫⣜⡉⡇⢌⢒⠂⢘⢁⠘⡗⠥⡾⣤⡄⢠⡀⠀⠀⠀⠈⠈⠟⠹⠉⠉⠁
⠀⠀⠀⠐⠉⠉⠀⢀⡀⢯⣟⢿⡎⠯⠣⠄⠨⢇⣨⠂⢠⣀⡔⠈⢢⡖⣟⢰
⠀⠀⠀⠀⠀⣀⣰⠎⣱⠟⢿⣷⠊⡳⠞⢎⣀⢏⣷⣯⡧⠀⠚⠒⡮⢄⣈⣲⣮⡀
⠀⠀⠀⠀⢀⣩⣡⡿⢿⣷⣟⣏⣸⣠⢶⠓⢲⡛⢻⣿⣿⣷⠟⢾⣳⢞⣾⣦⣾⡗⠀⠀⠀⠀⠀⠘⠓⠢⡀
⠀⠀⠀⠀⣐⠽⣿⣧⣾⣯⣩⣿⣿⠟⢾⣷⣿⣿⣾⣿⣍⡿⢷⣼⣿⢿⣿⢫⡍⠉⠀⠀⠀⠀⠠⠀⠀⠀⣷
⠀⠀⠀⠀⠘⣻⢿⣿⣤⣿⣿⢿⣿⢿⡛⠿⢏⢹⡼⠇⢨⡯⠿⣙⣟⠻⣥⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠘⠸⢧⠛⠙⠥⣈⣳⡈⢱⠁⠉⢀⢷⠖⠁⠀⠀⠈⠓⠀⠀⢀⣤⠋⠀⠀⠀⠈⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠱⣤⠀⠀⡧⠀⠀⠀⠀⢀⣠⣤⡔⣈⡉⣰⣦⢤⡀⠀⠀⢀⡶⠤
⠀⠀⠀⠀⠀⠀⡧⠂⠀⠀⠀⠀⠀⠀⠻⣄⠘⠐⠀⠀⠀⣼⣿⠟⠁⡸⠉⠉⠉⠻⣯⣿⠀⠀⠘
⠀⠀⠀⠐⠐⣾⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡆⢰⠀⠀⢸⣿⣇⠴⣾⡅⠀⠀⠀⠀⣿⡇
⠀⠀⢀⣠⠾⣿⣄⠀⠀⠀⣠⠾⠁⠀⠀⣜⡇⢀⠀⠀⠈⠉⢡⣾⣿⣷⣦⣤⡴⠂⠉⠛⠓
⠀⠀⠸⠿⠚⠋⠀⠁⠐⠛⠁⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠸⢿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣢⢤⣄⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢤⡀⠀⠀⢠⡐⠄⡀⠀⠀⠀⠸⣁
⠀⠀⠀⠀⠀⠀⢀⠴⠤⠀⠘⢒⠒⢢⣬⣯⡕⠀⡄⠀⠠⣀⡐⠀⠬⠒⠀⠀⠀⠀⠀⠈⠳⢮⡆⠇⠳⣆⠄⡀⢄⢀⠙⣆⠁
⠀⠀⠀⢀⢀⡀⣠⠀⢦⡴⠼⡾⠳⠋⢉⠃⠀⠀⠆⠀⠀⠀⠉⠙⠂⠀⠸⡄⠀⠀⠈⠂⡀⠀⢙⣀⠀⠘⣄⣮⣺⡆⢀⣻⣼⡄
⠀⠀⠈⠁⠒⠀⢄⣀⠀⠀⠀⠀⡰⠂⡡⠀⠰⠶⠿⠶⠄⠤⢶⣦⠈⠁⠀⠐⠀⠀⠀⠀⠉⢒⠋⢝⣓⣤⡬⢿⡙⣿⠂⣋⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⠀⠠⣀⡒⠆⠉⣼⣱⠂⣰⢋⠆⠉⠉
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⢉⡠⠖⠫⠰⣀⡴⠋⠀⢠⠃⢼
⠀⠀⠀⠀⠀⠀⠀⠒⠦⢤⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠈⡠⠀⠴⠂⠀⠀⠀⢁⠃⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣄⣀⣀⣀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠛⠋⠉⠉⠀⠀⠀⠠⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢠⠔⠀⠂⠠⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⣤⠒⠇⠋⡡⠔⠉⠀⠀⠄⠀⠀⠶⠶⠿⠦⠄⢠⣦⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠒⠒⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⢰⣿⡄⠀⠀⠰⢿⠇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠈⠙⠛⠛⠛⠋⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠘⣿⣷⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠈⠻⣷⣦⣤⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠣⣄⡀⠀⠀⠀⠀⢀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠈⠛⠛⠙⠛⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡐⠀⠄⠀⠀⠀⠘⢦⣤⣀⡀⠀⠀⠀⣀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠃⠐⠈⠂⠀⠀⠀⠙⠿⠛⠛⠻⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀⠀⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠉⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠰⣄⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠀⠙⠛⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠋⠀⠀⠀⢀⠀⠀⠀⠀⢀⣦⣶⠂⠀⢰⣿⠆⠀⠀⠀⣼⠃
⠀⠈⣆⠀⠀⠀⠀⠀⠠⣴⣾⡖⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠈⠀⠀⢀⣾⡟⣠⣀⣠⣼⠏
⠀⠀⠈⠢⣀⠀⠀⠀⠀⡸⢿⡼⠁⠀⠀⠀⠀⠀⠀⠲⢴⣇⠀⠀⣀⣠⣾⡿⠂⠀⠀⠐⢠⣼⣄⣀⣴⣿⡟⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠉⠑⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠲⠟⣛⡲⢖⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠄⠒⠒⠂⠤⠈⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣁⣤⡶⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡞⣾⣾⢛⡵⠊⣀⡀⠄⠀⠀⠀⠀⠀⠄⢀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⠟⡠⠊⢁⣀⣀⣀⠀⠀⠀⠀⢸⣼⡰⠀⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⢫⡿⢈⣴⣿⣿⣿⣿⣿⣷⣄⠀⠈⡟⡏⡇⡆⣿⡇
⠀⠀⠀⠀⠀⢸⢹⡇⣾⢁⣿⣿⠟⠋⠙⠪⣿⣿⣿⡆⠀⠁⠀⠇⠃⣿⣷⠀⠀⠀⠀⠀⠀⠀⢰⠳⣄⠀⣼⠆⡀
⠀⠀⠀⠀⠀⢹⢺⢀⡏⣼⣿⠃⠀⠀⠀⠱⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⠰⣦⡙⢇⣀⣹⣆⣀
⠀⠀⠀⠀⠀⠸⣼⠈⡇⣿⠃⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣄⠘⠷⠈⠁⠨⠉⢙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠱⠇⠀⣿⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⡇⡇⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠐⣨⡙⣦⠹⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠠⣧⢿⠀⠘⡇⡇⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢼⣿⣿⠏⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠀⠀⢰⠀⡄⣼⡿⠁⠀⠀⠀⠀⠀⠀⣰⢏⠁⠀⠀⠐⡄⣠⡁⡀⢀⣀⣠⠀⠀⠀⠘⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠈⡇⠀⠀⠸⢠⣼⣿⡟⠀⠀⠀⠀⢀⣴⠞⢣⣄⣀⣀⠀⠁⠀⣿⣿⣿⣿⣿⠏⠀⠀⠀⢻⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢻⣷⠃⠀⠀⢀⣾⣿⡿⠀⠀⢀⡤⠺⣍⣭⣵⣿⣿⣿⣟⠉⠒⠀⠙⣛⣛⣋⣥⣴⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⣿⠀⠀⢀⢸⣿⣿⣇⣠⠾⡩⢂⠳⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡿⠀⠀⣼⣿⣿⣿⠟⡁⣈⠐⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠀⢣⠈⢇⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢠⣇⣿⣿⣿⠋⠠⠀⠐⣠⣾⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣶⠀⠈⣧⠈⠣⣀⠀⢀⢞⡽⠋⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⠃⠀⠀⢀⣾⣿⣿⣿⡿⠏⠁⠸⣿⣿⣿⣿⣿⣿⣧⡀⠈⠳⣤⣈⣹⣟⣿⠁⠄⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⡏⠀⠀⠀⢊⣿⣿⣿⡿⠟⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀⠈⢉⡉⠁⠀⠈⠭⠁⣦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⡿⠀⠀⠀⢾⣿⣿⣿⡿⠁⠀⣀⡴⠞⠉⠉⠀⠈⠉⠻⡿⠿⣿⣿⣿⣿⡭⢥⣴⣶⣶⣮⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣰⣿⣿⡿⠛⠡⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠾⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠇⠀⠀⣨⣿⣿⣷⣶⣶⣦⣤⣶⣞⡀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⠄⣀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⣷⠀⠀⠀⢙⣛⣿⣿⣿⣿⣿⣿⣿⣿⢿⡋⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣇⠀⠀⠀⢤⣺⣿⣿⣿⣿⣿⣿⣿⣿⣦⠔⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿⣿⣿⣄⠀⠀⠈⣭⣿⣿⣿⣿⣿⣿⠿⣋⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣦⣀⣉⠉⠉⣙⣻⣭⣷⣿⣿⠯⠭⠭⣛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣯⣥⣬⣭⣭⣭⣥⣤⣤⣄⣐⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⢰⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠥⡀⠀⠀⡀⣤⡂⠀⢀⣀⠐⣴⠤⣂⣤⣀⢀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠄⠀⠀⠀⡀⠁⠀⠄⠙⣹⣿⣤⣄⠬⢿⣯⠙⣷⣴⣧⠠⢀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⢈⣸⣦⣄⡀⡀⠀⣀⡔⢠⡃⠀⠀⠀⠩⢹⡶⣿⣿⣿⣿⢿⡷⣶⣏⢡⣤⣠⣏⣰⠰
⠀⠀⠀⠀⣀⠀⠀⠃⡴⠉⠉⠉⢸⣿⣷⢆⠃⢼⣦⣶⠶⠀⠀⣩⣛⣻⣽⣿⣞⠟⣿⣿⣿⣽⣿⣍⣈⣶⣶⡘⠄
⠀⠀⠀⠀⠞⢷⠋⠩⠟⣃⣘⣷⣾⡿⢿⣵⡖⢻⢏⠀⠀⣀⣤⣿⣿⡾⢿⣿⣶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡵⠄
⠀⠀⠀⠀⢠⣄⠘⠀⠀⣤⠉⠙⡹⠀⠰⣠⡝⢏⣏⢿⠟⣋⣟⡟⢻⣿⣶⡀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣵⡋⡇⢀⠤
⠀⠀⠀⠀⠞⠁⠤⡾⠇⢰⣇⣤⢉⣄⡠⠘⣧⣾⣿⠞⠃⠀⠿⠙⢳⣿⣿⣿⣦⣻⠟⢿⣿⣿⣿⣿⣿⣣⠞⣻⠇⡀
⠀⠀⠀⠀⣴⢦⡐⠚⣿⣿⢿⣇⢸⣿⢀⣸⣿⣿⣍⣠⠄⠈⠀⣾⣧⣿⣏⡻⠿⣿⣷⣿⣿⣿⣿⣿⣿⣯⣠⡝⠬⠇⡂
⠀⠀⠀⠀⡤⠚⣥⠛⣛⣛⣺⡷⢉⡡⠇⣙⣶⣿⣿⣿⣾⣦⣶⣿⣾⣿⠿⣧⣶⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⡖⠷⡛⠂
⠀⠀⠀⠈⠀⢀⡚⢥⠟⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⣆⣻⣽⣿⣿⣿⣿⣿⣿⣿⡌⠻⣿⣟⢉⠂
⠀⠀⠀⠀⢤⠼⠁⠋⠞⢡⣿⣿⣻⢫⣿⢿⣻⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣽⣿⣿⣿⣿⣿⣿⠶⡯⠡⣾⣉⡀
⠀⠀⠀⠀⠀⠀⠀⣉⣠⢟⡁⣍⠰⠟⠉⠘⠉⠈⡷⠯⢀⣿⠟⠉⠛⢋⢿⠛⠹⣍⠻⣿⣿⣿⣷⣿⣿⣿⣶⣤⣴⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠤⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢠⢿⠋⠀⠀⠀⠀⠂⠂⠀⠬⢅⠀⠗⡬⡈⣿⠛⢛⡿⢿⢏⠳
⠀⠀⠀⠀⠄⠠⠠⠀⠀⠀⠀⠀⠀⠀⡀⣤⢤⠠⣀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⢐⠚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠸⠀⠐⠒⠒⠂⠒⠊⠒⠒⠚⠚⠛⠿⠿⡿⠿⠿⠿⠛⠽⠿⠮⠽


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⢀⡴⠚⠋⠉⢀⡼⠉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⠴⠂⠀⠀⠀⠀⢳
⠀⢠⠏⠀⠀⢠⡴⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡤⠖⠋⠁⠀⠐⢒⠀⠀⢠⡟
⠀⠁⣀⡰⠋⠁⠀⠀⠀⠀⠀⠀⢀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣧⣀⠈⠀⠠⠄⠂⣁⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⣈⣱⣄
⠀⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⢹
⠀⠘⢌⠢⡀⠀⠀⠀⠀⠑⢄⡀⠀⠱⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠓⢤⣀⡀⠀⠀⠄⣀⣉⣶⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠀⠀⠉⠉⠉⠙⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⢰⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⢀⡀⠀⠀⠉⠉⠁⠀⠀⠀⠐⠄⠙⠷⣦⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡌⠰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⠶⠋⠁⠀⠀⡠⠤⠶⠄⠒⠒⠒⠦⠄⠀⠀⠘⣧⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠀⢹⡀⠀⠀⠀⠀⠀⠀⠑⠦⣄⡀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣁⣀⠀⠒⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠖⠂⣀⡴⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⠒⠶⢦⣤⣤⣌⣁⡀⠀⠀⠀⠀⠐⠛⢀⣁⡤⠴⠖⠛⠉⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠀⠠⠄⣠⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠉⠛⠲⠤⢤⣑⠒⠒⠶⠤⠔⢂⣉⣉⣠⣤⠾⠋
⠀⠀⠀⠀⠀⠀⣠⡔⠢⣍⠛⠉⠙⢛⡉⠉⠗⢦⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠛⠉⠉⠉⠉⠀⠀⠀⢀⡓⢦⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠰⣁⠁⠀⢿⣤⠟⢦⣄⠙⢶⠄⠀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡆⢀⣀⣀⠠⡄⠀⠀⠀⢠⠂⠁
⠀⠀⠀⢀⠖⢦⡈⢦⡤⠦⡄⠀⠀⢹⡇⠘⠆⢀⠀⠸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⡤⠘⠲⣄
⠀⠀⠀⠀⠀⢀⡽⠇⢴⠉⠁⠀⠴⠋⣼⠟⠓⠞⢤⠳⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⣼⠃⠀⠀⢸⣆
⠀⠀⠀⢀⣤⠆⠀⡴⠋⠀⢈⠓⠦⢄⡈⠳⡀⠀⠀⠀⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣦⡆⣸⣴⠏⡀⠀⠀⣾⠻⣆
⠀⠀⠀⠙⣎⠁⠀⠣⣀⣠⠋⢙⡂⠀⠱⢤⣑⠂⠠⡠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⣷⢿⠋⠀⠑⣿⢸⣿⠀⠈⠁
⠀⠀⠀⠀⠫⡀⠤⣄⡀⢠⡒⠋⠈⠙⠳⡄⠈⠳⢤⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⡇⠉⠀⢀⠀⠘⢿
⠀⠀⠀⠀⠀⠈⠃⢀⢀⠀⢈⠂⠀⣤⡴⠻⣶⡄⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⡀⠔⣉⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⣈⣀⣠⣄⣀⣤⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠦⢤⣤⣤⠤⠶⠚⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⠦⠀⠐⠀⠛⠻⠇⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠉⠁⠀⠀⠀⡀⠰⣾⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣆⠀⠀⠀⢰⣶⣷⣦⢳⣼⣾⣿⣷⣆⠀⡀⠰⣴⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣟⠻⠃⠀⠀⠈⠋⢿⣽⣿⣷⣿⣏⣿⣋⣿⣷⡇⠺⢻⣦⡄⠀⡷⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⣰⣶⣶⣶⣿⣿⣧⣴⣿⣿⣿⣿⣿⣷⣀⣿⣿⡅⠀⢠⠙⢀⣀⡀⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣰⣧⣀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠰⠂⠀⠀⣠⡈⠹⢿⠛⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣼⡿⠋⣉⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠙⠛⠀⠀⠁⢊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠸⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣹⣷⣿⣖⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⣿⣿⣿⢦⠀⠀⠀⠘⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠁⠈⠻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣳⣦⡤⠶⠞⢳⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣶⣶⣤⢺⡿⠷⣿⡿⠋⠐⠚⠿⣿⣿⣿⣿⠿⠉⣠⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠙⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠙⠛⣿⠃⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠇⢰⡟⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢨⠍⠀⣼⣿⣿⣿⣿⠙⠀⣾⡂⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠇⢸⣿⣿⣿⣿⣆⠀⠉⢠⡆⣀⠀⠀⡰⠃⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠻⣿⣿⠿⣿⠂⣴⡿⠧⠾⣆⣸⣿⣿⣿⣇⣀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⠶⠶⠿⠆⢭⣴⣴⣶⡭⠿⠿⠿⠿⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡒⠒⢦⣄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠢⡈⢷⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠠⣦⠀⠀⠀⠀⠀⠀⠙⢦⣘⠂⢻⣴⠁
⠀⠀⢀⡠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⣋⡭⠭⠉⠉⠙⢷⣤⡏⠀⠀⠀⣀⠴⠒⠛⠛⠛⠳⣾⠷⠦⣄⡀
⠀⠀⠈⠙⢆⠱⡀⠀⠀⠀⠀⠀⢀⡴⠋⠤⣤⠞⠁⠀⠀⠀⠀⠀⠀⢙⣗⠶⣴⠊⣀⢤⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠑⢱⢀⣠⠤⠤⠶⠾⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⡄⣹⠋⠁⡸⠀⠀⠀⣠⡤⣈⠁⠉⠒⠀⣈⣧⣀⣀
⠀⠀⠀⣀⡴⠶⢾⡿⠃⠀⠀⢄⠀⠀⠀⠙⢦⡀⡀⠀⢀⡠⠀⢀⣀⣀⣸⣿⠁⣀⡴⠃⠀⠀⠀⠈⢙⣮⣳⡄⠂⠉⠁⠀⢀⡤⠋⠙⠲⣄
⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠈⢳⡙⢿⡯⠖⠋⠉⠁⠈⠉⠻⡏⠁⠈⠉⠓⢦⣀⠞⣩⠖⠛⠉⠒⠢⠀⠀⠞⠐⠢⢀⠓⠌⢳⡀
⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⡄
⢰⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣷
⠈⣿⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⣄⡀
⠀⣽⠻⠿⣥⣢⣠⣀⡀⠀⠀⠀⠀⠀⠀⢀⡼⣱⣟⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠐⡄⣦⡆⡄⠆⣀⠀⠀⣀⣀⣀⣀⣰⡯⠟⠁⠉⢷
⠀⠹⣦⣄⠀⠈⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⢿⣥⣽⣯⣿⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⠿⠴⠶⠧⠶⠶⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⣀⢠⣀⣄⣼
⠀⠀⠀⠹⣿⣻⣶⣶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠼⢷⡟⠛⠋⠉
⠀⠀⠀⠀⠈⠻⢷⣮⣉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠾⠶⣷⣾⣼⣶⣤⣄⣀⣴⡆⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢀⠀⡀⡀⢀⣀⡀⠀⡀⠀⢀⣠⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠿⠿⠾⠷⠷⠾⠾⠶⠾⠾⠿⠾⠿⠿⠷⠾⠿⠿⠟⠛⠛⠋⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒
⠀⠀⠀⢀⠀⠀⠀⢀⠀⣴⡤⠴⠯⣿⣦⣚⡿⣽⣶⠘⢷⡶⢶⣶⣄⣤⡄⠰⣽⣀
⠀⠀⣀⣈⣷⢤⣠⣿⡀⣦⣴⣧⣼⣻⣟⠿⣶⣧⣟⣺⣿⣅⡈⢛⣿⡿⡑⣲⡿⠿⣛⡄⠰⣶⠆
⠐⡰⣶⣿⣷⣈⣽⣿⣥⣿⣿⣟⣿⣿⡏⠀⣼⣿⣟⠛⣝⡛⠻⣿⣿⣷⠩⢿⠹⣷⡽⢿⣾⣿⣦⡖⣿⡁⠀⠠
⠀⣸⣷⣯⠟⢋⣿⣟⠋⣙⣾⣿⣄⡙⣣⣞⢀⣴⣴⢾⡿⠷⣽⣟⠋⣡⣄⣰⣫⡿⠻⢈⣳⣿⠿⣷⣶⢿⡇
⣏⠘⠿⣿⣴⣿⣊⣟⣾⣿⠟⠛⣿⣛⣶⣷⣄⣾⣴⠿⢳⣶⣿⣿⣾⣿⢧⣼⣿⣼⣶⣿⣿⣿⣾⣟⣾⡋⣦⣢⣠
⠘⠆⣀⣹⡄⠝⢰⡿⠋⡸⠀⠙⣷⣋⣼⣿⡿⠷⠛⢺⣿⣿⡿⣞⠟⢙⣁⢉⣬⣼⣻⡿⠈⠙⠈⠻⣿⣽⡋⠉⠁
⠀⠀⠀⠉⠙⢦⣾⠉⠁⢀⠀⣠⡟⠉⢹⣷⣲⣦⠖⢙⣛⡿⠣⣤⣆⢡⣩⡢⠿⠟⠿⠳⣧⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢸⣏⠀⢀⡞⠋⢁⣠⠴⠞⠛⠋⠀⢀⣦⡅⣰⠬⠑⠁⠈⠛⠋⠃
⠀⠀⠀⠀⠀⢸⣷⡾⠛⠉⠉⠉⠙⠒⠲⠦⠤⠤⠤⠴⠊⠹⠓
⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠠⣿⣇
⣀⣀⣀⣀⣀⣹⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣙⣻⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⣠⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣤⣶⣾⣷⡎⠁⠀⢁⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣥⣶⣹⣍⣿⡦⠀⠀⠀⠀⠜⠉⠀⡔⣮⠒⠀⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⣼⠁⠘⢣⠿⣿⣦⡉⡉⠀⠰⢇⠀⣀⣿⣮⣀⠁⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣧⡟⢲⢶⣱⣷⡆⣛⠘⢿⣄⣌⡀⣀⣆⣞⢋⡷⠈⠲⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⢻⡂⣡⡋⠈⣻⣿⣿⣿⣥⣈⣋⣈⠃⣘⠏⠉⠸⣷⡀⠀⠀⠂⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣓⡶⣿⣿⡟⢿⣿⣯⠡⡾⠻⢿⣎⣽⣿⡿⢗⢿⣿⣿⣦⠁⣐⠀⠈⡎⠓⠀⠃⡀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢻⣭⣶⣿⢻⡦⣾⣫⢠⠼⠟⠀⣉⣯⡉⢆⠥⠀⠀⣄⣿⡇⢀⠀⠀⠈⠀⢀⠀⠠⡜⢧⠄⠀⠀⢀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣽⣿⣿⣿⠗⣿⣷⣄⠟⣀⠀⠛⡋⠙⠎⠟⣥⣌⡡⣾⡆⠀⡄⠀⣷⠀⠂⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠰⠿⡿⢷⣾⡿⠛⣿⢿⣿⡏⠘⠡⢿⣿⡀⢀⡀⠓⢠⣼⠹⣷⠛⢛⣦⢀⣀⣳⠟⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣶⣠⣿⣿⣿⠟⠀⠞⢿⣿⣴⣓⣾⣿⣿⡟⠉⠤⢀⣛⠂⠈⠟⠊⢭⠿⠿⠿⣿⢿⣄⡀⡀⠄⠙⢀⠀⠀⣤⡆⢢⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣤⣞⡩⣿⣿⠱⣶⣤⡤⢧⣽⡟⠻⢁⠈⠹⣿⠱⢂⠌⣭⣈⠃⠀⡀⣤⣶⠃⠘⠂⢙⡁⣉⠁⠈⠀⠣⠂⠀⠀⠀⠀⠀⠈⢃⡀
⠀⠀⠀⠀⠀⠀⠘⠣⣿⣟⣿⣿⣰⡿⠟⣛⢋⣫⣤⢁⡈⠹⣇⡹⠛⠿⢻⡮⣭⢌⡴⣅⣻⣿⣿⡤⣧⡌⠲⢏⠀⠀⠀⠀⠀⣾⢐⢀⠠⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⢠⡶⢻⣿⣯⣿⠹⠳⣦⣿⣿⣼⣷⡊⣅⢀⠀⠀⠰⡆⢀⠀⣻⣾⣷⢽⡽⢿⡿⠱⣿⣛⣠⡆⣀⠀⠀⠀⠀⢻⡒⠜⠒⠂
⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⡿⣿⣿⣿⡿⡌⠻⣟⣿⢥⣰⣮⡽⢻⣿⣼⣿⠉⢹⣿⠂⠀⣼⠃⠀⠛⠙⠿⠷⣯⣦⡇⢠⠖⢯⢙⣥⡀⢄⠀⡶⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣶⣿⣦⣽⣷⣷⣾⣽⣯⠞⣤⢽⣷⠿⠿⡋⢻⣄⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠈⠃⢙⣻⠄⠀⠈⠛⠧⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠂⡸⠛⠛⠋⠋⢻⣍⠁⠘⠁⠮⠳⡶⠆⠀⠀⠙⠟⣿⣷⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⠄⠁⠀⠀⠀⠀⢘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⡴⡩⠂⠀⠀⡀⣰⠀⠀⢠⠞⢻⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⢧⡰⢿⡽⣯⡽⠏⠾⠡⠀⠁⢸⠘⠱⠏⠾⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠲⣼⣷⣶⣾⡷⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠒⠐⠂⠂⠉⠁⠂⠉⠐⠀⠐⠂⠒⠉⠐⠒⠒⠒⠄⠤⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢑
⠀⢀⡀⠄⢀⡀⠀⠀⠈⠀⠀⠈⠁⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠔⠀⠒⠒⠀⠉⠓⠢⠜⠀⠐⠲⣄⡱⠀⢀⡀⢈⢉⣀⠀⠀⠠⠂⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠹⠯⠉⠳⡀⠀⠀⠀⡠⠀⣎⡷⠠⢄⠀⠀⠀⠀⠀⠀⠀⠚⢄⠀⠀⠀⠀⠈⠂⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⠄⢳⡿⢈⠉⠆⠀⠀⠀⢣⠘⠀⠀⠀⢀⠃⠀⠀⠰⠀⠀⠉⠁⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠈⢢
⠀⠐⠂⠈⠐⠃⠠⠊⠉⠀⠀⠀⠀⢸⠀⠀⠀⠈⠂⡘⠀⠀⠀⠀⠑⠤⠀⠐⠁⠀⠀⠀⠀⠣⡀⠀⠸⣤⠌⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠱⡀⠀⠀⠀⠀⡀⡄⠒
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠑⠄⠤⠀⠐⠀⠀⠂⠐⠊⠁⡀⠄⣴⡀⡤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠖⠒⣾⠗⠒⢄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠁⠈
⠀⠀⠀⠀⠀⡱⠐⠒⠄⠠⠠⣄⠀⠀⠀⠀⢀⡀⢠⠂⠁⠀⠀⠀⡌⠀⠀⠛⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠉⠲⣤⠂
⠀⠀⠠⠊⠉⣀⡄⢢⠤⢄⠀⠈⠋⠑⡤⠒⢺⣘⡊⠑⢢⡀⢀⠀⠱⡀⠃⠀⠀⠀⡜⠀⠀⠀⠀⡠⠒⠒⢤⠊⠉⠣⣅⠀⢀⠰⠁⠀⠑⠄⠠⡀⠀⠀⢸⠐⡁⠢
⠀⠀⢀⠀⡎⠀⠘⠋⠐⠄⢡⠀⠀⠘⠀⣄⠈⠀⠀⠀⠀⠀⠈⠤⡀⠈⠒⠂⠀⠉⠀⠀⠀⠸⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⣔⡧⢤⡇⠀⠀⠈⢢⠀⠁
⠀⢠⠊⠀⠱⡀⠀⠀⠘⢀⠆⠀⠀⠘⢄⠀⢦⠀⠀⠀⡰⠀⠀⠀⢁⡀⠒⠐⠂⠂⠀⡀⠀⣀⢲⠀⡀⠀⠀⡄⠒⣾⡗⠒⠄⢈⠀⠀⠉⢡⠄⢱⠒⢤⢀⡘
⠀⠘⢄⠀⠀⠈⠑⠁⠀⠁⢀⠄⢀⡀⠀⠑⠒⠂⠈⠉⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⡌⠀⠀⠚⠃⠀⠈⢂⠈⡀⠀⠀⠀⠤⢸⠈⢦⣀⡀⠀⡠⠂⠀⠀⠈⢱
⠀⠀⡊⠀⠀⣀⣀⠄⠀⠀⠀⠭⠀⠀⠱⠀⠀⠀⠀⠀⠀⣀⢫⣖⠂⠒⢄⠀⠀⢀⢇⠀⠀⠀⢀⠌⠁⡜⠀⠑⢄⣀⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡄
⠀⠀⢸⠆⠈⠀⠀⠀⢸⠐⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⡚⠀⠘⠉⠀⠀⢘⢻⠙⠁⢠⠓⢤⠔⢬⠥⢮⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣤⡀⢀⠀⠀⠀⡸
⠀⠀⢈⠖⠁⠀⠀⠀⠀⠑⠤⢀⠄⠒⠆⠠⠔⠈⠁⢂⠠⡘⠺⠀⠀⢀⡜⢸⠀⠦⠃⡰⠁⠀⣀⠔⢁⡘⢢⡀⠀⠀⠀⠀⠀⡠⠊⠀⠠⢋⠀⠛⠂⠀⠃⠀⠀⢣
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠒⠈⠑⠒⡏⠘⡀⠈⠀⠀⠀⠁⠠⠊⢀⡴⠃⣀⠾⣀⠀⠀⠀⠀⣀⠘⠀⠀⠐⢄⠤⠀⠀⢀⠆⠀⠀⡈
⠀⠀⠜⠁⠀⠀⠀⠀⢀⣀⡰⣂⣀⠀⠀⠀⠀⠀⠀⠤⠴⡁⠀⠀⢰⠀⢡⣼⠀⠀⠀⠀⠀⡠⠊⡠⠞⠁⡠⠊⠑⠄⠠⠔⠁⠀⠀⠀⠀⠀⠉⠂⠉⠁⠀⠀⡃
⠀⠐⣄⠀⠀⠀⠀⡐⠁⠀⠾⠇⡀⠙⡄⠀⢀⣀⣴⠀⠀⠘⢄⣀⡠⡇⠀⠛⠀⠀⠀⠀⣴⡤⠊⠀⡤⠊⢀⡀⠀⠀⢀⢀⡀⠀⠀⡠⠒⠤⠀⡠⠞⠑⠒⠊
⠀⠀⠈⠀⣱⣄⣀⣡⡀⠀⠀⠀⠊⢠⣁⡠⠏⠀⠀⠑⢄⠀⠀⠑⢄⢠⠀⠀⠀⠀⠀⠈⠁⠀⡠⠊⠀⠀⠁⣨⠉⠉⡠⣶⡄⠈⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄
⠀⣀⠀⠀⣀⣔⡄⣠⠙⠢⠴⡦⠔⣁⠀⠀⢀⡀⠀⠀⡠⠓⢄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⣴⣇⡃⠀⢀⣿⣛⠠⠤⠀⠀⠒⠒⠂⠀⠑⢬⣷⡎⢈⠉
⠀⠀⠀⠀⠐⣺⢷⢅⡦⢠⡄⠀⠀⠀⢡⣀⣉⠀⠈⠀⠂⠤⠈⣣⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⢤⠀⡴⡂⠿⠏⠉⣧⡆⣉⡁⠀⢀⣀⣀⠀⠀⠀⡄⢤⣰⣿⣉
⠀⠀⠀⡀⠀⠀⣋⣿⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⢆⠀⣀⡀⠈⢧⠀⠀⠀⠀⠀⠀⡜⠀⡰⢰⣹⣇⠉⠀⣤⣑⡜⠘⢫⠁⠀⠈⠁⠀⠀⠀⠀⢐⠠⣿⠁
⠐⢈⣿⡷⠂⢌⠁⠉⢀⣀⠀⠠⠄⠀⠀⠀⠀⠛⡼⢿⡐⡀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠁⢰⠠⠀⠁⠀⠜⠈⠿⠉⠙⡆⠀⠘⣰⢀⣠⡔⠦⡀⠀⠈⣍⠛⠉⠁
⡀⠤⠾⣸⡟⠛⠺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠸⠑⠒⠤⠤⠦⠥⠤⠤⠐⠓⣉⠥⠛⠃⣘⢈⠰⡈⡢⢠⣏⣆⡀
⠀⠀⣤⣿⣡⠈⣰⣲⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠙⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠇⠠⣀⠀⢀⣀⠀⢀⣠⡎⣁⠤⠸⠀⢩⠤⢰⡿⢁⠀⠻⠛⡄⠀⠠⠄
⠀⠀⠀⠀⠀⡜⠀⠘⠎⠁⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠘⡀⠀⠰⠂⢤⡀⠀⠁⠀⠀⠀⣇⠠⠤⠐⡇⠘⡦⢈⣁⣦⣕⣢⣄⢖⠢⣬⣼
⠀⢠⠤⣠⣤⠰⠤⡀⠀⢂⠜⠀⠀⠀⠀⢆⢤⡒⡤⠤⢄⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡇⠀⣨⠄⣼⡽⠀⣔⠀⠀⡀⢱⠀⢀⡀⢸⡄⢻⠁⠈⠛⡌⢻⠁⡌⠐⠟⢺
⠀⠀⠙⠉⠛⠃⠒⢸⠃⠁⠀⠀⠀⠀⠀⠘⠉⠓⠃⠠⣀⠁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠸⣰⠁⠀⠘⠁⠰⠈⡃⠈⠁⠈⡁⢿⣦⠀⢼⡄⣱⣄⡘⣢⠊⠢⡀⠀⢀⡰
⠀⠀⠣⡀⠀⠀⠓⡼⠀⠀⠀⡀⠀⠀⠀⠰⣄⠀⠀⠀⣡⡄⢰⢃⠀⢀⡀⠀⢀⠀⡀⣀⣹⣆⠀⠀⠀⠇⡰⠁⠀⠀⢈⠠⠈⠉⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⣤⢀⡈⠁⠊⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠁⠘⠉⠉⠀⠉⠉⠋⠁⠉⠉⠉⠉⠀⠁⠈⠛⠉⠉⠁⠉⣀⡄⠀⠀⠈⠢⣙⣀⣀⠴⠁⠀⠀⠀⠀⠛⠚⠖⠲
⣴⣤⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣯⣧⣿⣦⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣁⣠⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣦⠀⢀⠀⠀⢀⢤⣶⠤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠟⡀⢐⡛⠀⠀⡘⠃⠈⣠⣴⣦⡦⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠷⡟⠀⣳⢠⡌⠳⠟⠘⠿⣦⢈⡹⣃⠀⣴⠘⠃⢻⣿⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⡇⠲⡏⠸⡟⠰⣿⣤⡄⢠⡶⣠⢠⣍⣬⣄⠻⠂⠀⠷⣠⠊⠀⠘⠃⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠈⠁⣠⡄⣦⠙⣤⣾⠁⠐⠛⠳⢇⡀⣽⠻⠧⢀⡶⢾⡤⠛⠻⣿⣤⢘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢯⣤⣯⡀⢠⣤⣿⠋⠙⢷⡆⣶⢸⢿⠇⣤⠷⠙⣷⢾⡄⣤⣤⡄⠋⢻⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠰⠬⣷⣟⡿⠿⣼⠇⠈⡀⠰⠀⢸⡄⠁⠀⣠⡏⣀⣀⠉⡹⢇⡏⠘⠋⠾⠛⠁
⠀⠀⠀⠀⠀⠀⡀⢀⢠⡤⠟⢁⣄⣀⠀⣄⢰⣶⠇⣨⠻⢾⣿⣧⡆⣾⣇⣠⡀⠻⣧⠹⠿⢎⣴⣯⢰⣧⠀⣴⣶⠆
⠀⠀⠀⠀⠀⢠⣿⣿⠇⠁⠸⠏⣛⠃⢶⣿⡧⠉⡘⢻⣶⣦⡙⣿⡜⢿⣿⡿⣀⠻⢿⣧⢷⣾⠉⠉⢀⡓⠀⠉⡉⢸⣦⠄⢠⣦⠄
⠀⠀⠀⠀⢀⡰⣿⡁⠀⢶⡆⠁⠙⠀⠸⠋⠀⢴⠠⣦⠙⢋⣠⡌⢿⣿⢃⣀⠙⠀⣤⣿⣿⠃⠿⠸⣿⡆⢀⡶⠻⠈⠁⢀⠈⢹⣶
⠀⠀⠀⠴⠾⠇⠀⠸⣷⠀⡷⣾⠆⢁⠀⠸⠃⠾⣧⢉⣦⠘⠋⣤⡘⣿⡆⢙⣧⡤⢠⣿⣷⣶⠦⣠⣿⣯⣼⡆⢸⣀⡴⠾⠿⣌⢉
⠀⠀⠀⠐⠿⠂⡒⢺⡿⠲⡇⢠⣄⢸⣆⠀⢹⡿⢻⠈⠉⣸⢠⣯⠀⢹⣿⠊⣠⣴⣿⠛⠛⣳⣤⡟⣠⡈⢋⣄⢸⠛⢛⢸⠇⠻⠏
⠀⠀⣠⣿⣦⠸⣇⡠⣤⢀⢹⡄⢀⡿⠿⡁⠀⣴⣿⣧⠞⣅⠙⠛⢠⣿⣿⣿⡿⠟⣉⠀⢄⢈⢸⣿⠿⠃⢈⣡⠟⢿⡟⠄⢀⣻⡏
⢤⣿⠆⠉⣭⡀⢛⣤⡄⢴⡌⢻⡞⠀⢼⠿⠀⠢⡅⢸⣿⣧⣴⡆⠈⢻⣿⣿⣶⡿⣫⡄⠸⡟⣹⣿⣾⠛⢿⣵⠀⣨⡧⠀⠘⠟⠃
⠀⣠⣄⣦⠬⠧⣤⡷⠞⠛⢻⡿⣿⣦⣰⣷⣴⠆⢹⠀⡀⢾⣝⣿⣷⣾⣿⣇⢠⣔⠉⠰⢿⣿⣿⠃⣴⡆⢨⣠⠎⠀⠀⠿⠆⢰⣧
⢰⡿⠉⠫⣤⣠⡛⢃⣜⠷⠙⠃⠙⠿⠛⠻⠿⣶⣼⣿⠃⠈⣹⡏⠈⢻⣿⣿⣴⣿⣶⣿⣿⣿⣷⣶⣿⣿⣋⠁⠀⠀⣆⡄⠂⢠⣤
⠀⠀⢴⣶⡄⢙⡃⢀⡋⠀⢠⣶⣦⣰⣦⣴⣶⣶⣿⣿⣷⡀⠙⢻⡄⣿⣿⣿⡿⠿⢿⣿⣍⣽⡇⠀⠀⠾⢫⡟⢟⡋⢡⣤⡐⢿⡿
⠀⠀⢠⣍⠀⢈⣿⣿⣈⣦⣈⣿⣷⣿⠛⠉⠹⣿⣿⠿⣿⣷⣄⢸⣿⣿⣿⡟⠀⠀⠀⠙⢻⣷⠲⣥⠀⡠⠀⢙⡘⠿⢰⣿⡇
⠀⠀⠿⣷⣶⡆⠘⠋⠸⠙⣿⢿⡆⠻⠃⠀⠈⠋⠉⠀⠈⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠘⣿⣿⡷⣶⡟⣷⠲⠛⠁⠀⣤⡌⠁
⠀⠀⠀⠀⠋⠴⣶⡄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢿⠇⠈⢀⣐⣶⢰⣦
⠀⠀⠀⠀⠀⠀⠉⠀⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⡿⠃⠑⢆⣤⣀⠛⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠰⡞⠷⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠊⠙⠛⠛⢻⣿⣿⠿⠿⠿⠿⠿⠛⠛⠓⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣿⣽⠙⢴⢳⢤⣱⠄⡄⠄⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⢂⣀⣦⡀⢳⠀⢂⢿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠆⡡⣌⠻⡛⡿⡀⠀⠀⠀⠔⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⢈⠀⠀⠀⠘⠙⡨⡧⣹⡇⠀⠀⠀⠀⠀⠀⣼⣷⣴⣿⣷⣿⣶⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢶⣷⣟⠉⠀⠀⢄⠦⠀⠁⣀⠀⠀⠂⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⠉⠀⠀⠠⣧⠆⠀⠀⠙⣣⣄⢀⣀⣤⣤⣤⣿⣿⡻⣿⣿⣿⣿⣿⡿⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⠃⠀⢀⢀⣿⣷⡀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢛⣿⣿⣿⣛⡷⢌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠄⠀⣄⣿⣽⣿⣿⣿⣿⣿⣿⣟⣿⡿⢿⢻⣽⣿⣷⣿⣟⡛⡏⠉⢷⠀⢺⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢹⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⡀⠘⢿⣿⣯⠀⠙⠖⠀⠀⠘⠉⢘⢋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⡿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷⣑⣠⠚⠛⠉⠀⠈⠉⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣂⣀⣂⠙⠛⠻⠲⠄⠀⠀⠀⣀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⣹⣿⣯⣝⡿⠷⠿⣿⣿⣿⡿⠍⠀⠀⠀⠲⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠙⣿⣾⣿⣿⣿⣿⣿⣿⢿⣿⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⢆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣳⠾⠃⠀⣀⠀⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠒⢄⠘⣷⢶⡞⠓⢳⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⡿⠟⠛⠟⠛⠋⠛⠉⠙⡯⠉⠷⢾⠁⠋⠀⠀⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⢳⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠂⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣽⡟⠛⠳⣦⢹⣀⡧⢀⣀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⢀⠀⠘⢄⠀⠚⠓⠦⠄⡠⢀⠀⠀⠀⠘⠷⠟⢸⡛⠢⢟⡀⠉⡠⠋⢉⡇⠀⠀⠁⢠⢂⠀⠀⠀⠀⠛⠀⠨⠀⠄⢀⠦⡄
⠀⠀⠀⠀⠀⠄⣀⠀⠀⠉⢳⡀⢀⣀⡀⠀⠑⠀⠀⠀⢠⠂⢈⣿⠶⠶⠃⠰⠥⠴⠟⠁⠀⠀⢠⡎⠀⠀⠀⠀⠘⠀⠋⡄⠀⡀⠀⠄⠦
⠀⠀⢣⡀⠀⠀⠈⢰⠳⠀⠀⡿⠉⠀⠀⠀⡰⠀⠀⠀⠈⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢤⣤⡘⢷⣦⡀⠀⠀⣀⣤⣢⠿⡀
⠀⠀⠀⠙⠲⠖⠚⢁⠀⢀⡼⠀⠁⠀⠐⠋⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡬⣍⣙⢉⣉⣈⣵⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⡄⠈⠐⠢⠀⠀⠠⠀⠀⠀⠀⠭⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠃⠀⠀⠄⠁⠀⠀⠀⡁⠀⠀⠀⠀⠀⣠⠐⠄⠀⢀
⠀⠀⠀⠀⠀⢀⠴⠞⠋⠓⢦⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠶⠞⠁⠀⠀⠀⣀⣀⣤⠤⠚⠀⠀⠀⠀⠈⢆⠀⢀⣀⣰⣄
⠀⠀⢀⡤⠚⠋⠁⣀⣤⡆⡈⠉⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⠖⠛⠈⢀⠈⠁
⠀⠀⡃⣄⠀⢀⡞⠁⠉⠉⠛⢦⣤⣤⢄⠀⠀⠘⠀⠀⠰⢻⡇⡾⡀⣀⢰⠆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠂⢄⢰⡾⠿⠃
⠀⠀⠑⠦⠠⠶⢆⠰⡤⢤⣄⡀⠀⠈⢉⡷⠒⠁⠀⠀⠀⠿⠸⠗⠇⠏⠾⠠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠛⠶⢞⡉⠉⠀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⣠⢄⡤⣰⢆⣆⡤⣠⠄⡖⡌⣠⢄⡤⡄⠀⠀⠈⠓⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠓⠙⠚⠛⠃⠛⠚⠓⠓⠓⠙⠃⠓⠀⠀⠀⢤⠖⠀⡀
⠀⠀⠀⠀⣀⠄⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠦⡄⠀⢠⡇⡠⠊⢡⡀
⠀⠀⠀⢰⢇⠀⠀⠀⠀⢈⡶⠟⠛⠳⢤⡀⢀⡔⠃⠠⠞⠋⠐⠂⢸⠀⠀⠀⠀⠀⠀⢠⡾⠋⠈⠀⠘⠀⠀⡴⠋⠀⢠⠇
⠀⠀⠀⠘⠈⠓⠤⠶⢶⡏⠀⠀⠀⠀⢀⡏⠀⠳⠦⠀⠀⠀⠤⢖⠋⠀⠀⠀⠀⠀⠀⠈⠙⢀⣀⣀⣴⡀⡔⠀⢀⣤⠋⠀⠲⠖⢂⡁
⠀⠀⠀⠀⠐⢄⡀⠀⣠⡷⠄⠤⠐⠆⠻⢷⠀⠀⠀⠒⠒⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠳⠶⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡴⣾⡿⠇⢋⠁⡒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣩⣥⣬⣍⡒⢴⡿⠞⠀⠀⠈⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⡟⠃⠉⠀⠉⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⢆⠘⠀⠀⠀⠀⠀⢠⠀⣿⣶⠆⠀⠀⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠿⠉⠉⠉⠙⣄⡀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠘⡆⠑⢤⡀⠀⠀⠒⢁⡰⠁⠉⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⣩⣖⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢲⠛⢶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⢟⠟⢡⠂⡰⢹⢻⠟⣦⡀⠀⣠⢞⠏⠍⠀⠀⠀⠈⠀⠁⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⣀⠀⢀⣾⠏⡌⡎⢀⠏⢰⠃⡎⣼⢀⢇⢻⣶⠣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠉⠉⢉⡟⡏⡼⢸⢀⠞⠀⠀⡸⡐⡏⠸⡌⢠⢻⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠰⡄⣼⢡⡇⠇⣜⠂⠀⡆⢀⡟⢰⠃⣴⠁⡈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢨⡇⠘⢰⢠⢿⠀⡸⠃⡘⢀⠏⡸⢃⣴⢃⡿⡀⠂⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣼⠁⠀⡆⠌⡞⢠⢡⡶⠁⡞⠀⢀⠏⡈⣼⡇⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠴⠒⠚⠋⣿⡸⠀⣤⠰⢡⠇⡞⠀⠸⢃⡴⢁⡜⣼⠓⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢸⡇
⠀⣀⣠⠤⠖⠚⠋⠉⠀⠀⠀⠀⠤⢀⡀⣿⡇⠀⠇⠀⡏⢰⠇⠠⢟⡽⡠⢊⡼⠃⠠⠀⠹⣆⠠⡐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠙⠳⢦⣀⠀⠀⠀⠀⠈⠦⠀⢀⡀⠙⣿⠇⡄⠀⡼⣡⠞⠀⡠⠂⣶⣡⢾⡀⠀⠀⠈⠦⢌⢧⡈⢦⡠⠀⠀⠀⠀⠆⠀⠀⠀⠁⣾
⠀⠀⠀⠀⠈⠙⠲⢦⣄⡀⠀⠐⠉⠀⠀⣿⠀⠇⠘⡰⠁⡠⠚⢁⣴⠿⠓⠒⠛⠶⣤⠤⠤⠤⠷⠽⣦⣙⠢⡀⠀⠀⠀⠀⠁⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣀⡀⢹⣄⠀⡰⢁⠜⢁⣤⠞⠁⠀⠀⣤⠔⠊⠁⠀⠀⠀⠐⠚⠃⠈⠳⢤⡀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣶⡵⢾⣞⣋⣀⣀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⣀⣠⡤⠴⠒⠋⠉⠛⠲⠶⠤⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢔⣬⣍⢋⣽⢫⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢾⠟⠣⣞⣛⣋⡄⠉⠀⠑⠉⠓
⠀⠀⠀⠀⠀⠰⣆⠀⠓⠤⡘⠛⠉⡀⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⢴⣦⣴⣶⣶⠇⠀⢀⠄⠀⠀⡀⢴⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠈⠀⠈⠁⠀⠀⠠⣿⣿⣶⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡠⠷⣭⣽⣿⣿⡇⡀⠀⠀⠰⡄⠀⠐⠆⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⢸⣿⣴⣿⣿⣿⣿⠇⡃⠀⠀⣄⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⣖⣿⣿⣿⣿⣿⢯⣾⠃⠀⠑⠎⣿⣿⣧⡂⣀⣀⣤⣴⣆⣠⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣄⣤⣾⣿⣿⣿⣿⣿⣷⠟⠁⢠⡆⠚⢢⣾⣿⣿⣿⢏⣛⣶⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣖⡏⠛⠛⠛⠛⡿⡿⠟⠋⢉⣀⣤⣶⠟⠁⣡⣾⣷⣿⣿⣿⣟⢯⣟⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠋⠀⠀⠀⠀⢸⣸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠁⠚⣨⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠐⠀⠐⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡰⢆⣤⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣴⣾⣽⣯⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣠⣠⢹⣷⠜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⡩⢁⡠⢀⣠⣴⠶⢶⣾⣿⣿⡿⢿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠈⠛⢨⠬⣡⠄⠀⣷⡨⠋⠠⡿⠀⠀⠀⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⣄⠀⠶⠐⠒⠸⣠⠾⣃⡀⣝⡦⠀⠀⠰⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣷⣬⡷⣈⣵⣇⡜⠳⡏⣾⣿⣷⢠⠟⠂⠐⢶⣟⡅⠀⠘⢃⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢜⠿⠿⣁⠚⠉⠽⠧⣄⡷⠈⠛⠁⠼⣟⠱⣦⡸⢡⣵⣯⡤⠦⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⡼⠆⠉⢠⢦⣿⣇⣀⣶⠰⠾⣇⡙⠈⠀⠲⢼⣗⣨⣧⠘⠿⠿⣅⢒⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⠀⠸⠃⢀⡁⢐⠋⢣⣀⣶⣿⡷⠟⢴⠘⢾⡏⣁⠀⠀⣦⢲⣽⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠲⣄⣲⡀⠌⣈⠱⠞⢿⣴⠎⣷⠀⢀⣑⠿⢿⠛⢠⡄⣽⠋⣌⠁⠀⡶⠂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⣀⠘⠋⢼⠿⠐⡾⠹⡍⠽⠛⣿⣿⡆⣾⣿⣶⠈⣷⣬⠷⠷⣶⠉⠻⠍⠳⢄⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⢄⣿⠃⠀⠀⢰⠄⠛⢡⡌⠛⠻⣿⣿⡇⠘⠛⣃⡾⠋⠀⠸⠃⣶⠀⠾⠃⢼⣿⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢝⣰⠇⠘⠃⣀⠐⣿⣿⡇⠀⢸⣿⣧⣴⡾⠏⣘⠂⠀⢀⡔⠁⢠⡀⠀⢠⡍⠁⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡆⢧⡃⣤⠛⠀⠱⡌⠉⠀⠻⢼⣿⣿⣯⡤⢈⡿⠴⠶⢞⡓⢿⣚⠤⡐⠚⢇⢚⠆
⠀⠀⠀⠀⠀⠀⢠⣶⣄⢛⠋⠀⢀⠹⣏⣀⣸⣆⣿⡰⠆⠙⢿⣿⣿⣏⣡⣟⢱⡆⣶⣿⣷⠘⠦⡼⠛⠠⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣈⡀⠲⠀⢀⢶⠟⣻⡋⠉⣩⣿⢿⣿⣶⣿⣿⣿⡿⢁⣭⢁⠴⡘⠛⣃⡄⠀⣥⢀⣤⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠁⣠⣔⡁⠘⠀⠟⠁⠀⠛⠋⢀⣽⢿⣿⣿⣿⣇⣨⣿⣋⣸⣃⣠⣟⣁⡤⠴⠞⣥⡀⠠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢿⣿⣿⣿⠿⠛⢻⣿⡄⠹⢷⠀⠀⢁⣴⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠛⠛⠛⠛⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠘⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠰⡄⠀⠀⠀⡟⠀⠀⣾⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠀⠀⣄⠀⠙⣄⠀⠀⠹⡄⢀⠀⡇⡀⢰⠇⡀⠀⡴⠁⠀⢀⡤⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠈⠳⢄⠈⢢⠐⢄⠱⠸⠄⡇⡇⡞⡜⢠⠎⡠⢀⡴⠋⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠠⡑⠄⠑⠌⠀⠁⠀⣇⠀⠐⡰⠃⢊⠔⠉⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣄⡀⠨⣑⠈⡀⠀⠠⢀⠠⠤⠤⠄⡈⠢⠀⡀⠔⣋⠤⠂⣀⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠲⠶⠶⠤⢤⣄⣀⣀⡨⠁⠂⠁⢀⠋⡐⠁⠀⠀⠀⠀⠈⢢⠁⠀⠠⠀⠂⠉⣀⣀⣤⢄⣠⣤⣤⠖⠒⠶⠤⠶⠒⠲⠦
⠀⠀⠀⢀⣀⣀⣠⠤⠤⠤⠤⣀⡀⠀⠈⠉⠈⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⡖⠉⠉
⠐⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠓⠤⢄⡀⠀⠀⠀⢣⣀⣀⣀⣀⣀⠤⠚⠂⠉⠉⠉⣡⠏⢁⣠⠀⠀⠠⠤⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⡀⠄⠂⢀⡀⠀⠀⠀⠀⢀⠀⠉⠑⠒⣍⣁⣀⣀⣀⣀⣀⣀⣀⣠⠤⠴⠚⢁⣐⠊⢀⡀⠀⣠⠤⠒⠉⠉
⠀⠀⣀⡀⣀⡀⢀⠀⠀⡁⢈⣁⣐⢒⣤⣍⣀⣀⣈⣨⣽⣦⠄⠀⠀⠀⣀⡼⠁⠄⠐⣀⠀⠀⠀⠀⡩⠄⣲⠉⠲⡒⡢⣀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⡼⣃⡤⠷⣬⠶⠷⠬⣁⣀⣀⣐⣠⣅⣀⣄⣈⣉⣀⣠⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⢖⡒⠒⢦⠶⠚⢛⡛⠉⠠⠤⠤⠤⠤⠤⠴⠀⠨⠉⠁⠀⠀⠈⠉⠉⠀⠉⠉⠉
⠀⠀⠀⠀⠴⠚⠉⠉⠉⠉⠉⠉⠙⠯⢤⠀⠀⠀⠀⠤⠶⠮⠍⠤⠤⠥⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣶⣄⠀⠀⠀⢰⡄⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⣄⣿⠃⠡⠞⢁⣈⡛⢀⠀⠀⢈⡱⢭⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⡀⢰⣄⣿⠇⠀⠀⠐⠎⢀⢀⠀⠀⠉⣡⣾⠀⠙⣖⠉⠱⢀⡬⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠀⠋⠀⠀⠀⠀⢠⠇⠀⠀⠸⠟⠃⠀⠀⠸⡄⠀⣬⡙⠂⠀⠘⢇⠀⠞⣱⡿⠁
⠀⠀⠀⠀⠀⠐⠛⠷⢀⡍⠁⠒⠢⢄⡀⠀⢸⠘⠛⠀⠀⠐⠦⠔⠂⠒⣧⠀⠘⠿⢄⡞⠁⣈⣑⡀⠰⣦
⠀⠀⠀⠀⠀⠀⠲⠦⠸⠁⢰⣿⣷⣷⠈⠳⢼⡀⠀⠀⠀⣾⣶⣿⡆⠀⢿⠀⠀⠀⢸⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢀⣴⣾⡿⠂⡀⠀⠻⠿⠟⠀⠀⠀⠙⣆⠀⠀⠻⠿⠟⠁⠀⢸⠀⠀⣤⢸⡇⠀⠉⠙⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠦⣀⣀⣀⡤⠤⣤⣀⠘⣧⠀⠀⠀⠀⠀⢦⣸⠀⠀⠙⣸⠁⠀⠀⢠⠖⣀⠘⠵⠿⠋
⠀⠀⠀⠀⣤⠀⣿⠀⢉⡄⢂⣉⠀⢡⣀⠀⠀⠈⠑⢾⡆⣾⠟⠀⠀⢀⣿⡴⠖⣟⡉⠉⠉⠉⣭⡁⠀⢿⣷⣄
⠀⢰⣦⠀⣀⣈⣑⢄⠉⠀⠘⠁⠆⠘⢿⣷⠀⠀⠀⠈⣧⠀⠀⠀⣴⣿⣉⣀⣀⣈⠁⠀⠀⠀⠈⠃⠀⠀⠉⠛
⠀⣈⡻⠸⣿⣿⣿⠊⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡀⠀⣼⠏⠀⢀⣀⡀⠉⠑⠔⣛⠁⠀⠀⢀⣤
⠀⠉⠉⢒⠩⠭⡩⠤⠴⠛⠛⠛⠛⡻⠓⠶⣦⣌⡇⠀⣿⡇⢰⡟⠀⠀⠙⠋⠵⣴⣶⣶⡆⠀⠛⠊⢸⡿⣴⠟
⠀⢀⣾⡟⠁⠆⠀⠀⠀⠀⢀⣭⠀⣤⡀⠀⠀⠙⠻⣦⣸⣷⣼⠇⠀⠀⠀⣀⠀⠙⠿⠿⠃⠀⠀⢠⣠⠂⢲⣂
⠀⠀⣱⡶⢉⣉⣑⣒⢤⡀⠉⠁⠀⠻⠇⠀⠀⣀⣀⣹⣿⣿⣿⣠⣤⣤⣤⣭⣄⣀⣀⣀⡀⢀⣴⠋⢀⡀⢾⠁
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠉⠒⢦⣤⣀⣠⠶⠛⠉⠙⣿⣿⣿⣿⠟⠉⠁⠀⠉⠉⠹⡉⠉⠉⠉⣡⣄⣭⡂⠈
⠀⠀⠀⠀⠀⠉⠉⠱⠞⠉⣩⡏⠉⣴⣶⣶⡄⠀⠀⢹⣿⣿⣿⡄⠀⠀⠀⠀⣴⣿⢁⠶⡄⠐⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠻⣿⣿⠃⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠰⠟⠁⠘⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣠⣤⣤⣤⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣄⣐⣿⡿⢿⠿⣿⣿⡿⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠛⠛⠻⠿⠿⣴⣶⣯⣝⣿⣿⠟⠛⠛⠀⠀⠀⠀⣤⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⣶⣶⣦⣤⡶⣿⣿⣿⡛⣻⣝⣳⣀⣶⣶⣦⣤⣿⣿⢿⡿⠿⠖⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠹⠦⠰⢾⡟⢯⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡆⠀⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⣦⠛⠻⢷⣶⣶⣾⣿⣿⣿⣿⣿⣽⣿⣿⣾⣎⣹⠻⡿⢷⣿⣿⣿⣿⣶⡾⠂⢠⣤⠦
⠀⠀⠀⠀⠀⠀⠺⠿⢷⣶⣾⣶⣞⠓⣾⣴⣧⣬⣛⠛⠛⣿⣿⣟⣿⣿⣿⣾⣿⣿⣧⣶⣾⠛⢲⠦⢴⣷⣾⣦⣾⣿⣶⣿⣋
⠀⠀⠀⠀⣠⣤⣤⣤⡀⠘⢋⠉⡝⠛⠓⣾⣉⣻⣿⣤⣷⡮⢿⣿⣿⣯⣽⣯⡙⠟⣿⣿⢿⣿⣷⣶⣤⡼⠩⠽⣿⠇⠉⠁⠉
⠀⠀⠀⠀⠻⢿⡿⠀⠀⠠⠛⠒⠻⠿⠿⣤⣽⣿⣿⣿⡿⠛⠰⢶⣿⣛⣻⣿⣿⣶⣿⣩⣾⠌⣿⠟⣁⣠⣠⣴⣠⡀⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠐⢲⣾⣯⡴⣆⡘⢙⣟⣻⣧⣿⡟⠻⣛⣳⣟⢻⣿⣿⣿⣿⣿⣿⣿⠿⢏⣁⡘⠻⠻⢿⣿⣻⣏⣉⣩⣿⣶⡆
⠀⠀⠀⠀⠀⠀⣾⡶⠉⠀⠐⣶⡿⣯⣼⣿⡤⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⣯⣽⣿⣿⣿⣴⡶⠊⣫⣑⣀⣀⣼⢅⡈⠻⠿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠶⠾⢷⠾⢷⣶⣩⠟⡻⣟⠻⢿⣶⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⣷⠶⠛⢿⣿⣿⣷⣿⡗⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣹⣟⣻⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣩⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣠⣄⣤⠌⢻⡟⠒⣉⣛⣩⣉⡿⣿⣤⣁⣬⣽⣿⣿⣿⡟⢯⣝⣿⣿⣿⣗⢁⣀⠀⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣄⡥⢾⣿⣿⠿⣿⡛⣴⣶⣶⢿⣿⣿⣿⣧⢀⣤⡿⢿⣿⡿⠿⣿⣿⣿⣿⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⡡⠶⠁⡄⢖⠈⠀⠀⢈⣉⣻⣶⣿⣿⠉⣻⣿⣿⠾⠓⠛⠃⠀⠰⢿⣿⣿⣿⣿⠟⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣤⣾⣿⠛⠂⠀⠀⠀⠛⢻⡿⠃⠁⠙⢶⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠈⠟⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣧