Mug Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠴⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠉⣀⣤⡤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠤⠤⣄⣀⣈⣷⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠛⠛⠛⠉⠉⠉⠁⢸⠊⠀⣀⣀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⠱⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠊⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡼⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣀⠤⠊⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⠀⠒⠂⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠳⠦⠤⠤⣀⣀
⠀⠀⠀⠀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠤⠤⠬⠍⠉⣛⡒⠒⠒⠒⠛⠛⠭⠍⠉⠤⠤⠀⠐⠂⠀⠁⠀⠀⠀⠀⢀⣠⡴⠟
⠀⠀⠀⠀⠈⠻⢍⡑⠒⠢⠤⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⠤⠤⠖⠒⢚⣩⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⣄⣀⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⣀⣠⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣶⣛⡿⠿⠷⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠶⠶⠤⣭⣉⡓⣶⣄
⠀⠀⠀⠀⠀⠀⠸⣅⠙⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢛⣹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠉⠓⠦⣬⣉⡉⠛⠓⠲⠶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠴⠶⠒⠋⢉⣠⣾⠃⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣤⡀⠀⠉⠙⠓⠒⠶⠦⢤⣤⣄⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣠⣤⡤⠶⠒⠛⠉⠀⢸⣀⣸⠇⠀⠈⣳⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠰⣿⣿⣿⠂⠀⠰⡷⠀⠀⠀⠀⢀⣀⡀⠈⠉⠉⠉⠉⠉⠉⣡⣤⣄⠀⠀⠀⠀⢸⣿⠀⣿⠀⠈⢷⠀⠈⠋⢹
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡗⠀⠀⠀⠃⠀⠀⢸⣿⣿⣿⡇⠀⠤⠀⠈⠁⠀⡟⠀⠀⢸⠀⣀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⠀⠀⠈⠀⠀⣠⣤⣄⠀⠀⠁⠀⠀⣀⣄⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⡀⢰⡇⠀⢀⡾⠘⠋⠀⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⣿⡗⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠻⠟⠀⠀⢀⣤⡀⠀⠀⢰⣿⣿⠀⠀⠀⣸⠤⠖⠋⠁⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠈⠛⠋⠀⠀⠀⡟⠰⠀⠀⣾⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⢠⣿⣿⣦⠀⠐⠂⠀⠈⠛⠉⠀⠀⠀⠀⢠⣶⡄⣸⠷⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠘⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣰⣶⠀⠀⠀⠉⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠛⠛⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡆⠀⠀⠈⠁⠀⡄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⣀⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡉⠙⠓⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠚⠋⠉⣰⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠒⠒⠶⠦⠤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣰⡤
⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠂⠀⠀⠀⠀⠀⠒⠖⠻⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡖⠒⠀⠀⠐⠒⠒⠶⢶⣷⣿⡴⠟⢻⣶⡄⠀⠀⠀⠀⣿⣤⣤⠀⠀⠀⠀⠠⠀⡤⣤⣤⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠂⠀⠀⠀⠀⠉⠈⠏⣿⡿⠀⠀⠀⢹⢹⠀⠀⠀⠀⣿⣿⢷⣦⣠⡶⠄⣴⣄⣷⣄⣰⣿⡟⠛⠿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⠁⠉⠀⠀⠃⠹⡟⠁⣿⣿⡇⠀⠐⡙
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⣿⡏⠀⠀⠀⠀⣿⣿⡍⠀⠀⠀⠀⠀⠀⡅⠀⢿⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢀⣼⠟⠀⠀⠀⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⡃⠀⣼⣿⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢾⣿⣧⣮⠏⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⢠⡇⢀⣭⣿⣇⣠⣾⠇
⠀⠀⠀⠀⠀⠀⢻⣧⢠⡆⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡀⠀⠀⠀⣀⣾⡇⢀⣿⣿⣿⠝⠋
⠀⠀⠀⠀⠀⠀⠘⣿⣷⡷⠀⠀⠀⠀⠀⢸⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⠀⠛⠁⠛⠛⠹⣷⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⡓⠐⠒⣀⣉⣙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣍⣉⡙⠛⠙⢙⣟⣟⣿⣽⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠤⠤⠤⠤⢤⣄⣈⡉⠙⠲⠤⣄⡀
⠀⠀⠀⠀⣀⡤⠖⢒⡒⠒⠦⢤⣀⠀⠀⢠⠎⠈⢇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⡉⠓⠦⣄⡀
⠀⠀⢀⡞⣡⠖⠉⠉⠉⠁⠀⠀⠈⠙⠲⢧⡄⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⡉⠓⢤⣀
⠀⠀⡾⠰⠃⢠⠴⠚⠋⠉⠉⠙⠲⡄⠀⠀⡇⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡈⠓⢦⡀
⠀⢀⡇⠀⢰⠃⠀⠀⠀⠀⠀⠀⢠⠟⠐⠚⠁⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⠉⠳⣄
⠀⠀⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⢄
⠀⠀⢧⠀⠘⣆⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠱⣄
⠀⠀⠘⣆⠀⠈⠳⣄⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⢧
⠀⠀⠀⠈⢧⡀⠀⠀⠉⠓⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⡆
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⠤⣄⣀⣀⠷⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣧⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠳⠤⠤⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⢀⢺⣤⢏⡴⠲⡄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣈⠉⠁⠈⠳⠚⠱⣅⡼⠁⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠤⠤⠤⠤⠤⠤⠶⠖⠚⠛⠉⠁⠀⠀⢀⣠⠤⢰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡏⠈⠳⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⠤⡤⠤⣄⡀⠈⠲⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⡇⠀⠀⠷⠊⠁⢀⠇⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠉⠳⣄⠘⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠹⡄⠀⠀⣀⠄⠀⢴⠟⠀⠴⢦⢠⡋⣳⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠘⡆⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠙⠛⠉⠀⠀⠀⢯⡤⠀⠳⠞⠀⠉⠁⠀⠀⠀⠀⠀⣀⣤⡎⠀⠀⠀⠀⠀⠀⣀⣰⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠉⠉⠛⠛⠛⠋⠉⡡⢋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠦⢤⣀⣀⣀⣀⣠⡤⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠐⠒⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⠛⠒⠒⠶⠶⠶⠶⠶⠤⠶⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢖⣋⣙⠲⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣃⡤⣊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃
⠀⠀⠀⢀⡠⠤⠒⠚⠉⠉⠉⣈⣉⡷⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⢯⣀⣉⠉⠉⠉⠒⠲⠤⣄
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠶⠤⠭⠭⠭⠯⠿⠷⠿⠿⠯⠭⠤⠤⠴⠒⠚⠉⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠈⠙⠒⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠴⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣉⣛⣛⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠳⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠂⠉⠀⠈⠐⠆⠤⠴⠋⠞⠀⡀⠠⠊⠀⠠⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠠⠀⢸⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠿⢏⣀⣀⠀⠀⠀⠁⠀⠉⠀⠤⠌⠁⠀⠀⠸⠄⠀⠘⠁⠀⠀⢔⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⣤⣤⣬⣀⣀⣒⣀⠢⠀⠈⠤⠭⠭⠭⠍⠉⠠⠟⢀⣆⣀⣀⣠⣈⠿⣞⡡⠇⠀⠀⣀⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣽⠙⠛⠻⠏⠹⠛⠛⠛⠛⠛⡟⠛⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡟⢻⡟⠀⡤⠖⠉⠀⠀⣀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣒⢀⣈⣽⣿⠿⠿⣟⡃⠈⣊⠀⡟⢁⡤⠞⠋⠉⠙⢳⡄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠄⠠⠤⠶⣖⣲⣿⣯⠭⠂⠈⠵⠂⣷⠋⠀⠀⠀⠀⠀⠀⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣉⠁⢠⣭⠥⠶⢖⣛⡫⠀⢚⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠄⠀⠒⢚⣛⣉⡥⠄⢈⡁⢸⡇⠀⠀⠀⠀⠀⢀⡼⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠀⢐⠛⣛⣋⣭⣭⠴⠖⠀⡭⠀⢸⡇⠀⠀⣀⡤⠖⠋⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡍⠠⠬⠥⠶⢖⣛⡋⠤⠒⠀⢸⣃⠉⠄⠀⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⣒⣛⣋⣭⡥⠤⠀⡩⠀⡟⣀⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡥⠨⠭⠵⠶⢖⣚⡋⢠⠆⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢒⣀⣲⣿⣿⡿⢃⠬⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⡤⠤⠤⠶⠶⠒⠛⢷⣄⡂⢀⠀⠀⠀⠠⡄⢰⣶⠀⣴⣶⣶⣾⣿⣏⠙⣠⣴⠿⠶⣶⣶⡶⢦⣤⣤⣤⣀⣀⡀
⠀⠀⢀⡤⠖⠛⠋⠉⠁⠀⣉⠁⠀⠀⠒⠒⠒⠛⠩⠯⡽⠒⠀⠀⠀⠀⠀⣂⣳⣤⡾⠿⠿⠟⠓⠒⠿⣿⠿⡛⠛⠛⠛⠶⠶⠦⣤⣈⠉⢉⠙⠻⣷⢦⣄
⠀⠀⠸⣧⡢⣄⣦⡀⠀⠘⠄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠙⠒⠶⠤⠤⢤⣤⣤⣤⣦⣼⣿⡿⠿⠷⠗⠚⠁⠤⠴⠂⠀⠠⠐⠀⢀⠠⠼⠃⣀⣴⣠⠼⣣⡟
⠀⠀⠀⠀⠙⠳⢯⣍⣛⠓⠒⠶⠶⠶⠤⠤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣤⣤⣤⣤⣤⣿⣿⣿⣿⠶⠶⠶⠶⠒⠛⣋⣩⣭⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠴⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢾⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣬⣉⡩⠶⠶⠾⠿⠿⠶⠶⠶⢍⣉⣵⣿⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⢛⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⢀⡼⠀⢸⣿⣿⣿⣿⡇⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⢀⠞⠁⠀⣼⣿⣿⣿⣿⡇⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣌⣀⣠⣾⣿⣿⣿⣿⣿⡇⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠶⠶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⢟⠿⠋⠁⠤⠀⠠⡈⠉⠁⠒⠦⢍⡚⢕⣶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠖⣿⡕⠁⡢⠔⠈⠀⠀⡨⣃⠀⠀⠀⠀⠐⢌⠓⣝⢷⣬⠷⠐⠠⣄
⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⣾⠏⡄⡜⠁⠠⠚⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠑⠈⠆⢿⣦⠴⠒⢘⠂
⠀⠀⠀⠀⠀⢀⡞⡔⠁⠀⢸⡸⠘⠈⠀⠀⠈⠉⠀⠀⠀⢤⠀⡈⢁⢠⠈⠀⠀⠀⠀⠘⣼⡇⣠⢾⣧
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣿⢷⢠⢀⠀⡀⡀⠀⠈⠢⠠⢀⡌⠀⢀⠀⡀⠸⠀⠀⠀⠀⣇⣿⣥⣀⡀⠀⠀⠉⠐⠆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢻⡸⡌⠆⢆⢰⣔⢄⡀⠀⠀⢀⠁⡴⣡⢘⠔⠁⠀⠀⡜⡸⣹⣿⡋⠙⠍⠟⡖⠦⠤⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣧⡱⡄⠀⠀⠙⠻⠏⣛⠲⣦⡁⣴⠿⠃⠀⠀⢀⠜⡰⢡⡿⠉⠁⠀⠀⠐⣧
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣷⣜⢦⡈⠓⠤⣀⠀⠉⠂⢑⠃⠀⠀⠀⢐⡡⢞⣴⡿⠁⠀⠀⡀⠀⠀⣸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⢮⣓⠦⢤⣉⣉⣀⣀⣀⠤⠴⣒⣩⠖⣫⠟⠁⠀⠀⠀⠁⠀⠀⡿
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠁⠹⡷⣬⣍⡓⠒⠒⠲⠒⠒⠚⢋⣡⠴⠛⠁⠐⠐⠀⢀⠌⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠡⡡⡈⠢⣙⠻⠛⠓⠚⠚⠛⠛⠉⠁⠀⠀⠀⠊⡠⢠⢊⠄⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠊⡂⠀⠑⠀⠄⡀⠀⠀⠀⠀⢀⠠⢐⠠⢊⠔⠱⠃⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠊⠐⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢽⣶⣦⣤⣀⣀⣀⡀⠀⢀⣀⣀⣀⡤⢴⣒⡭⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠿⠯⠽⠿⠿⠿⠖⠒⠋⠉⠁



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

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⣰⡁⢀⣀⣤⣴⣶⣾⣿⣿⠿⠿⠿⣿⣿⣿⢿⣿⣟⣛⣛⣛⠛⠻⣛⡿⠿⣿⣿⣶⣦⣄⡀⠀⠀⠀⡉
⠀⠀⠀⠀⣦⠻⣿⣿⣿⠿⣿⣿⣿⣿⣷⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣷⣬⣉⣙⢻⣿⣷⣆⣴⢁⡼⢀⣠⡔⠂⠉⠀⠲⣄
⠀⠀⠀⠀⢻⡷⢈⣛⡷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣥⣾⣿⣿⣣⠴⠀⠀⠈⠙⣿⡄
⠀⠀⠀⠀⢈⠁⠀⠀⠁⠀⠉⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⡛⠛⢋⢡⣴⣾⡯⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠈⠼⣿⣿⡯⢸⣿⡿⠟⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⣻⣿⣧⣾⠇⠀⠀⠀⠀⠀⠀⠊⡀⠁
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⡏⠀⠀⠀⣀⠤⠚⣡⠜
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣤⡴⠒⢊⣁⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡞⣿⣿⡿⣛⣉⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠙⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀⣀
⠀⢠⣶⣿⠿⠿⣛⣛⣛⣭⠭⠭⠭⠭⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢛⡛⢛⠿⠿⣿⣶⡄
⠀⣿⣿⠶⣯⣍⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣩⣵⠶⣿⣿
⠀⣿⣿⠀⠀⠈⠉⠩⠙⢛⣛⣛⣒⣲⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣖⣚⣛⣛⣿⠿⠭⠍⠑⠂⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣶⣾⠿⠿⠿⠿⠿⠿⣿⣶⣄
⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⢀⣤⣴⣶⣶⣶⣦⣄⠙⢿⣷⡄
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⡿⠟⠉⠀⠀⠀⠉⢻⣷⡄⢻⣿⡀
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⠈⣿⡇
⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣿⡟
⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⢰⣿⠇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢠⣿⡟
⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣠⣿⠏
⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣴⣿⠟⢁⣴⡿⠋
⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣀⣴⣾⡿⠛⢁⣴⣿⠟⠁
⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⠿⠟⠋⢁⣠⣾⡿⠋⠁
⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⡀⢀⣠⣶⡿⠟⠋
⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠋⠁
⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠉⠁
⣾⣷⣷⢿⣶⠶⣶⣶⣶⢷⣷⡶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿
⣿⣿⣦⣷⣼⣶⣧⣾⣶⣿⣯⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⠒⠶⠤⠤⢤⣤⣤⣤⣤⠤⠤⠴⠖⠚⠋⠁⠀⠻⠶⠶⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⣄⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢱⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠲⠖⠚⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⠤⣤⣤⣤⣤⣤⠤⠤⠴⠖⠚⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠄⠀⠒⠢⠤⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣶⣒⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠲⠤⠤⢄⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣑⣲⣶⣤⣤
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⠶⠶⠶⠿⠟⠛⠛⠛⠛⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠟⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⢶⣤⣶⣿⣿⠟⠟⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠉⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡴⠶⢶⣒⣶⡚⠋
⠀⠀⠀⠀⠀⠀⢀⢼⣓⠢⢄⡀⠀⠀⠀⠉⠛⠲⢤⣄⡀⠀⣿⣧⣀⣀⣀⣀⣀⣤⡤⠶⠖⣚⣋⣭⣭⣶⡾⠶⠶⠟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠙⠂⠉⠓⠦⣄⡀⠀⠀⠀⠈⠙⠓⠺⢿⣿⣿⣿⣵⠶⠶⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠐⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠒⠤⣄⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⣖⣛⣻⡟
⠀⠀⠀⠀⠀⠀⠃⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡉⠓⠤⣀⠀⢸⣿⡤⠀⠀⠀⢀⣀⣠⣤⠴⣖⣾⣿⣭⣷⠶⠿⠛⠛⠋⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠛⠿⠿⣛⣛⣯⣿⡿⠿⠾⠛⠛⠉⠉⠁⠀⠀⡀⠠⠄⠐⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣶⣿⡛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⢸⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣸⣿⣿⡅⣀⣀⣤⣤⡴⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡬⠥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⢀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡀⣀⣴⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⣀⡸⠣⠴⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠛⠻⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠞⣛⠩⠀⠀⣀⣀⣄⣀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠭⠭⠅⠐⠒⠂⠉⠁⠀⠀⠀⢸⠁⠀⠀⠀⠙⡆⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣠⣴⠟⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠾⠟⠛⢉⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠳⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⣠⠤⣔⠊⠉
⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠙⢦⡙⢅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠂⢄⠀⠉⠓⢄
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⢴⠖⠉⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢳
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⠬⠭⠭⠭⠭⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⣀⣈⠁⠒⠂⠠⠤⠄⠀⠤⠠⠤⠀⠠⠤⠀⠐⠒⠈⢁⣀⣠⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠤⢀⢐⣒⣒⣠⣤⣠⡀⠀⢀⠀⠠⡀⠒⠀⡒⠒⡂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣴⣶⠖⠃⠀⠀⠈⠚⠉⠈⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣙⣛⣯⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⡄⣀⣀⣤⠵⠂⢀⠠⠤⠄⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣭⡋⠛⠛⠻⠷⠶⠷⢶⣶⢴⡴⠶⠶⠒⠶⠶⣿⣟⠟⠛⠉⣉⡅⢰⣎⢡⢋⣤⠶⠂⢠⠄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠠⡏⠀⠐⠢⠤⣠⣴⢟⡾⣯⡖⠋⠉⠉⠉⠳⠌⢷⣉⣷⢭⠀⢄⠈⠋⢘⠀⣏⠃⠀⠀⠑⠤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠍⠉⠃⠀⡠⣤⢀⣌⢹⣠⣤⡀⢈⡹⣉⣅⣘⠃⢃⡈⡛⠄⠬⠀⠀⠀⠀⠐⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠬⠉⡐⡟⠸⣼⡖⠃⢀⣐⢊⣄⡙⠠⣭⠀⠀⠠⣿⠇⠙⡇⡟⠀⠠⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠈⢱⠀⠩⡇⢠⡎⣸⢧⡮⣻⢆⣺⣦⡀⠀⡷⠀⠀⡇⠻⠀⢈⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⢣⡝⣯⡥⠟⣠⡏⠉⣁⣄⣠⡀⠀⡇⢸⠀⠸⠀⠀⠀⠀⠀⠠⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⢘⣀⠀⣇⣀⡀⠁⠳⠊⠘⢙⣶⣦⡼⢟⠛⣿⡀⣧⣼⠀⢈⣀⠀⠀⢀⠄⠁⠁⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠐⢀⢸⡟⠳⢿⣿⢻⣦⠠⣄⣐⣿⣿⣿⡖⡸⣠⣿⣟⣹⣄⠀⢨⡀⠈⠈⠀⠀⢀⠔⠁
⠀⠀⠀⠀⢀⠀⠄⣀⣀⡐⢢⣥⡀⠀⠀⣝⢿⡋⣴⣆⠈⢻⣸⢪⠁⠛⢿⡟⠾⡟⢧⣯⢤⠄⣴⣍⠁⠀⢸⣟⣭⣥⠴⢖⣃⠤⣀⡀
⠀⢄⣬⣵⢾⠻⢏⢩⢻⡿⣝⡾⢁⠀⠠⠽⣻⡗⠻⣽⣣⣼⢻⣻⣄⠘⣿⣷⣶⣿⢈⣙⣢⣴⡏⠛⠀⢀⠌⢖⣋⡶⠶⠶⡍⣟⡶⠯⢭⡗⢄
⢄⣿⢻⣥⠗⠀⠀⢺⠉⡰⠤⠀⠈⢳⣀⠸⢶⣿⣷⣼⣈⢁⡼⣻⣯⠈⠉⢁⣹⣜⢿⣿⢿⠉⠁⠀⣠⠎⠀⠉⠀⠀⠀⠈⠵⣇⢚⠻⡑⣿⢌
⠀⣻⠼⣌⣀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠳⢄⠈⣿⣾⣿⣿⣿⡏⠙⠃⣐⣻⣇⣯⣽⣷⣃⡠⡴⠊⡠⣀⢤⠄⠀⠀⠈⠁⠐⠉⡀⠠⠚⡥⠎
⠀⠈⠙⠒⠶⢦⣤⣖⠄⣀⣤⠄⠒⠐⠀⠀⠀⡉⣶⣜⣟⠁⠙⠃⠀⠀⠹⣯⠿⠿⠿⠿⡛⠑⠁⠄⠀⠂⠥⠆⣘⣛⣤⠤⠄⣂⠥⠚⠁
⠀⠀⠀⠀⠈⠂⠈⢉⠙⠓⠒⠛⠛⠛⠓⠓⠒⠲⠶⠖⠲⣶⠭⣬⣬⡥⠤⠴⠶⠖⠶⠶⣾⣿⣿⣿⣿⣿⡿⠟⠋⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⢤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠤⢀⡶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠤⠤⠭⠭⠉⠉⠉⠀⠀⠀⠉⠉⠩⠥⠤⠤⠶⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⣠⠤⠤⠶⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠶⣶⣶⣶⠤⣄⡀
⠀⠀⠀⠀⠀⡮⣐⣤⣤⣶⣶⣶⣶⣶⣿⣷⣾⣿⣾⣷⣦⣦⣤⢤⣀⣀⡥⢿⣆
⠀⠀⠀⠀⠀⡇⡃⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠶⠒⠚⠛⠉⠉⠀⢀⢸⣿⣠⠴⢖⣒⣒⠶⣦⣄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⣒⡤⠬⣅⡀⢸⡿⣡⡔⠋⠉⠉⠙⣆⠹⣧
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⣓⡀⢸⣇⣸⠇⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣖⡂⠀⣼⠏⠁⠀⠀⠀⠀⢀⡜⢠⡏
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣀⠒⢠⡿⠀⠀⠀⣀⣠⠴⠋⣰⠟
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⡒⠀⣾⣧⢶⣚⣩⣥⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣼⠿⢫⡿⠛⠋⠉
⠀⠀⠀⠀⢀⣀⣤⡤⣧⠀⠀⠀⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⣘⡂⣼⠿⠶⠯⣤⣤⣄⣀⡀
⢀⠴⠒⠋⠉⠀⣀⡀⠸⣇⠀⠀⠀⢚⣿⢤⣤⠁⠀⣤⣈⣹⣾⣤⣿⣤⣤⣿⣿⡛⠛⠛⠛⠳⣦⠀⠠⠄
⢃⡀⠀⠀⠀⠊⠋⠁⠀⠻⣤⣀⣐⡒⠛⠒⠒⠒⣚⣛⣋⣭⣵⡾⠛⠛⠁⠀⠈⠉⠁⠀⠀⣠⡿
⠀⠉⠓⠦⣤⣄⣀⣀⣀⡀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⣀⣀⣀⣤⣤⣀⣤⣤⣴⠶⠛⠉⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠉⠙⠛⠻⠭⢭⣭⣉⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣉⣩⡭⠵⠞⠛⠉⠀⠀⠀⠒⠀⠀⠀⠒⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣀⣠⣄⣠⣄⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⡉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁⠀⠂⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠾⠿⠿⠟⠉⠩⠏⠁⠀⠀⠐⡟⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡶⢶⣶⣶⠶⢲⣶⢢⡖⠒⠒⠒⠒⠒⢶⠶⡶⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣥⣿⣤⣴⣦⣼⣄⢠⡷⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠈⠛⠛⠿⠿⢿⣿⣿⣿⢿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⢀⣠⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⠟⠛⠉⠉⠉⠉⡥⠖⠒⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠄⠠⣀⣀⣀⣀⡀⠀⢀⡤⠖⢒⠉⠀⠀⠀⠀⠀⣼⣷⡤
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠀⡏⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠑⠒⠒⠢⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠢⠤⠄⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠲⠾⢿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠴⠖⠚⣷⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⠀⣆⠀⣿⢋⠕⠛⠓⢯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡷⠼⠀⠀⠀⢈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢰⡇⠀⠀⢀⡠⢎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣾⡴⣒⣫⣥⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣩⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⠉⠉⢳⠀⠀⠀⣶⣀⠀⠀⠀⢀⣆⣸⣁⣉⣽⠿⠛⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠉⠁⠈⠣⢤⣄⣙⣒⣒⣪⣭⣭⠷⠛⠈⠁⠈⠉⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢶⣖⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⢖⣲⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠒⠒



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣧⣤⣴⣶⣶⠶⣿⣿⣶⢰⣶⣶⣶⣶⣤⣤⣤⣄⠈⣿⡆
⠀⠀⠀⢰⣾⣿⡏⣽⡿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠻⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⣤⣬⣤⠀⠏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⠀⠀⠉⣡⣝⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣷
⠀⠀⠀⠿⣦⣧⠀⠀⠀⠀⠈⠩⣗⣀⣼⣿⣿⣿⣿⣿⣿⣇⠀⠀⢦⣘⣼⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⢹⣿
⠀⠀⠀⣾⠿⣟⣄⠀⠀⠑⢀⡼⢯⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠤⡮⣸⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣽⣿⡇⢸⣿
⠀⠀⠀⣿⠀⡋⣏⠀⠁⠠⡽⡷⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣶⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⢰⠈⠀⠀⠸⡇⢂⠀⠀⠨⣽⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣴⣶⣶⣶⣶⢰⣶⣶⣶⣶⣶⣶⣤⣬⣭⣅⠸⣿
⠀⠀⢸⣇⠀⠀⠀⣿⡀⢶⡖⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢦⣿⣼⣿⣾⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⡼⠿⣦⠀⠘⣿⣷⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣤⢿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿
⠀⠀⣷⠘⠿⠠⠊⢉⣥⣤⣤⣦⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿
⠀⠀⣿⣤⣶⠤⠄⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣏⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹
⠀⢰⣟⣉⣐⣶⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠧⣼⣿⡛⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠇⡿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢰⠏⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠘⢷⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠸⠇⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡶⠛⠉⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠙⠳⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣶⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣻⣭⣭⠹⣦⠀⠙
⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠈⣿⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⣿⠿⠿⠛⠛⠛⠋⠉⠁⣸⠏⠁⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣤⠶⣿⢦⣄⠀⠀⠀⠀⠀⢠⣿⣠⡴⠟⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣆⠄⠀⠀⠀⠀⠀⣿⠀⠋⠀⣿⠀⠀⠀⠀⣠⣿⣩⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣙⣧⡀⠀⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⢀⣴⣟⡉⠁⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠉⠉⠙⠷⢦⣤⣤⣬⣥⣭⣭⣤⡤⠶⠋⠉⠉⠙⠛⣦⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡟⠛⠛⠂
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⣠⣄⠀⠀⡾⠁⠀⠀⠀⡇⢀⣤⡄⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⢸⡇⢿⠀⢠⡇⠀⠀⠀⢿⢈⡇⠰⡷⠄⠀⠀⠀⡇⠸⣷⣃⡸⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠶⠛⠀⠻⠞⠁⠸⡗⠋⠀⠀⠀⠀⠈⠋⠀⠀⣷⠀⠀⠀⠀⠃⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⣀⠀⠄⠀⠀⠀⠀⠀⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠲⢩⠄⠤⠀⠐⠂⠈⠉⠁⠁⠁⠀⠉⠀⠈⠉⠀⠂⠚⠨⠭⣙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢓⣫⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢙⠿⢥⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡫⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠉⡵⠖⠒⠂⠦⠤⠄⠠⠤⠤⠤⢤⡤⠤⠤⠠⡤⢔⡢⠦⠛⠋⠁⡤⢒⣊⣉⣙⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠓⠀⠈⠚⠉⠁⠊⠀⠉⠁⠀⠀⠀⠀⠀⢠⡴⠋⠁⠀⠙⢻⡆⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢰⢾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠙⠷⢯⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠜⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠀⠠⠶⠎⢀⣉⠘⣆⡀⡴⢶⠁⡂⣶⡄⠀⠀⠀⢢⢉⣶⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡺⠃
⠀⠀⠀⠀⠀⢀⡤⡴⡲⠿⢿⡙⣻⠭⠋⠂⠀⠀⠀⠀⠀⠀⠉⠟⣶⡗⠿⢿⣿⣣⡇⢀⣾⡿⠾⠃⠀⠀⠀⠀⠘⠻⠉⣀⡤⢊⣄⡀
⠀⠀⠀⣠⢞⠵⢋⠡⠊⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢗⣨⣿⡻⣶⣾⣿⣟⡇⠘⠏⠁⠀⠀⠀⠀⠀⠰⣫⠴⠊⣿⣾⠿⣷⣿⣷⢦⣀
⠀⢀⡞⠱⠋⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡭⠟⢣⣿⣿⠿⣯⠉⠣⡄⢺⣍⢀⣀⠀⠀⣠⠏⡁⢐⣿⠿⡛⠂⠀⠈⠛⣽⢯⣳⡄
⠀⠐⡇⠆⠀⠀⠀⠦⣐⣿⠤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠟⠋⠀⠈⠀⠀⠀⠀⠁⠈⠁⠀⣰⣿⣿⣿⣿⣿⣆⠀⣀⣤⣤⠀⠀⠉⣿⣿
⠀⠀⠙⢎⣆⡀⠀⠐⢨⠿⢾⡾⣿⣿⠥⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⣀⣀⣀⣀⣀⠠⠄⠂⢀⣬⠾⣯⢿⣿⡾⢻⣵⡾⠿⠊⠁⠀⢀⡰⣥⠞
⠀⠀⠀⠀⠙⠷⣦⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⢾⣿⣿⣶⣶⣶⣾⣿⣶⣦⣾⣿⠅⡀⠀⠀⠁⠀⠋⠴⡛⠀⠀⠀⠀⢀⣠⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⣤⣄⣀⣀⡀⠀⠀⠀⠑⠢⣄⣁⡀⠉⠈⠙⠛⠛⠻⠟⣚⣋⡴⠃⠀⠀⠀⣀⣀⣀⣠⡤⠆⠒⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠉⠙⠛⠛⠓⠒⠶⠾⠿⠿⠿⠶⠶⠶⠖⠒⠛⠒⠒⠚⠛⠉⠉⠉⠉⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠅⣒⣒⡒⠂⠤⠤⠤⠤⠤⠤⠶⣤⣀⣀⣲⣴⠶⠶⠚⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣚⡭⠵⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠲⠤⣍⣓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⢀⣀⣠⣤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠶⠤⢤⣄⡀⠀⠉⠛⢦⡙⢦
⠀⠀⠀⠀⠀⢠⡏⡏⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠉⠙⠲⣄⠀⣿⢸⡆
⠀⠀⠀⠀⠀⢸⣇⢣⡞⠉⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠒⠋⠀⡤⠤⡄⢹⡀⠀⠀⠀⠀⠀⢀⡿⣡⢾⡇⣀⣤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⢸⡈⠳⣌⡳⢤⣀⡀⠀⠀⠸⡄⠀⠀⢰⠖⠒⠚⠁⠀⢹⢸⡇⠀⢀⣠⡴⢞⣡⠞⣡⠞⠋⠁⠀⣀⣀⣀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠉⠓⠮⣍⣛⠒⠶⠿⠦⢤⣬⣷⣤⣤⣤⠤⠿⠾⠒⢋⣩⡤⠖⠉⠀⠀⣧⣀⡤⠖⠋⠁⠀⠉⠙⣦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠶⠶⠶⠶⠶⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢧⣄⣀⣀⣀⣀⡤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣨⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠈⠉⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠻⣍⠛⠋⠉⠉⠁
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⢸⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡏⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⢦⣌⡙⠲⠦⢤⣄⣀⣀⣀⣀⣀⣀⡤⠤⠖⠚⣋⡥⠞⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠤⠴⠶⠒⠋⠉⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠙⠪⣝⡲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⣚⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠺⠭⣍⣓⣒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠴⣖⣒⣻⠭⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣄⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⠼⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠛⠛⠛⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠀⡀⠀⠂⠀⠀⠠⠶⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠤⡄⠉⠻⣿⣷⣾⣀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠶⢴⠞⠁⢰⠾⡇⠀⠀⢤⡄⠀⠤⢼⠟⠛⠛⠓⠀⢀⡀⠀⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠰⠃⢻⣤⣾⣧⠆⠈⠁⠀⡀⣠⣴⡆⠈⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⡀⠐⠀⢿⣶⡦⡤⠀⠖⠀⣶⢀⡄⠈⣿⠿⠋⠷⠖⠀⠀⡙⠛⠛⠀⠀⠐⠳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠉⠁⠚⠁⢄⡀⠀⢀⣤⣼⣿⡿⡘⢀⢠⣀⣶⣤⠀⣤⠉⠀⠀⢠⠤⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄⡀⠀⠀⠀⠀⡀⠉⠩⠉⠓⢷⣿⣦⣿⣟⣛⣿⣿⣾⣆⡄⠐⠀⠐⠀⠠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠖⠀⠋⠉⠂⠠⣄⢀⣼⠟⠉⠁⠉⢻⣿⣿⣿⡿⣓⠂⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠸⣿⡇⠀⠀⠀⠀⠘⡿⢉⣿⣿⡋⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⢦⠄⠀⠞⠋⡀⠚⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⡏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⠁⠀⡾⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣰⠇⠀⣸⠃⠀⢠⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡟⠀⢰⠇⠀⢠⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠖⣾⠀⢸⠃⠀⡿⠀⢠⡟⠀⢀⡏⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣭⠴⠒⠀⠀⡏⠀⣿⠀⢸⡇⠀⢸⠁⠀⣼⠀⠀⠠⣌⣙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢾⣿⣶⣞⣋⣥⣤⣿⣤⣿⢠⣼⡇⣤⣼⢠⣤⣿⣀⡀⠒⠦⣭⣛⠆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠈⢉⣠⠟⢑⣶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣍⠓⠂⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⣉⣀⣤⡴⠾⣛⣵⡆⣚⣛⠧⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢋⣉⣉⣭⣴⡶⠿⣛⣭⠀⠉⠉⠙⣾⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠃⢸⣟⡳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⣉⣥⣴⡶⠿⣛⡅⠈⠑⣂⡴⣻⡣⢠⡏
⠀⠀⠀⠀⠀⢀⣤⠞⣋⡥⢒⡠⠄⠀⠀⢻⣟⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣉⣉⣉⣩⣭⣴⠶⢟⡋⢡⣞⣛⣽⣾⢟⣵⠟⢶⣄
⠀⠀⠀⠀⣰⠟⣡⡞⢡⡞⢉⡴⠊⡀⠀⠈⢿⣟⠒⠄⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣭⣭⣴⡶⢿⡛⢡⡿⢿⣛⣭⡶⠛⠁⣀⢢⡈⢷⡄
⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⡀⢾⡀⠀⠀⠀⠻⣯⡒⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣭⣥⣤⠴⠶⣞⡛⣱⣿⣿⣿⣿⣿⢿⡌⢳⡈⣆⢳⠈⡇
⠀⠀⠀⠀⠻⣆⠹⣧⣌⠳⢮⣛⠶⣬⣁⠀⠀⠀⠈⠻⣥⣄⣀⠀⠀⠀⠀⠀⠤⠤⠶⢶⣶⣾⣿⣿⣿⣿⣿⢸⣿⡜⣿⢎⡇⢸⠇⡏⢀⡼⠃
⠀⠀⠀⠀⠀⠈⣳⣮⣙⠷⣦⣍⣙⠓⠦⠄⠀⠀⠀⠀⠈⠙⠻⢯⣿⣛⣛⣛⣛⣛⣯⣽⡿⠿⢛⣫⣵⡿⢋⣾⠟⣰⠟⠘⠁⢋⣠⣶⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠨⠽⠿⠶⢮⣭⣙⣓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠽⠷⠶⠶⠶⣚⣛⣋⣭⡵⠾⠛⢁⣈⣡⣤⣶⣾⣿⣿⠿⠭⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣉⣹⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣛⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣟⣛⣛⣛⣛⠛⠛⠛⠛⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⣉⣩⣭⣭⣭⣭⣭⣭⣉⣉⡉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡤⠤⠴⠶⠶⠖⠒⠒⠒⠒⠚⠛⠛⠒⠒⠒⠒⠒⠒⠒⠖⡤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⣠⣶⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢩⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣀⣰⣀⢀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣶⠶⠶⠶⠶⠴⠦⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠛⡛⠛⣛⡋⢻⣯⣿⡀⢀⡴⠋⠠⠤⢌⣳⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⢍⡁⠘⠾⠽⠘⠧⠶⠒⠚⠞⠓⢒⣂⣉⣉⡉⢉⣉⠉⠁⠀⠀⠀⠁⠈⠉⠉⠈⡇⡾⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣾⣻⣿⠿⣽⣿⣨⣵⣄⠀⠀⠠⠄⠀⣦⡀⢠⣿⣿⣿⣧⣼⣿⣯⠙⣷⣷⣦⣤⡀⢀⠠⠀⢄⠀⣿⡇⠀⠀⠀⠀⠀⢸⢿⡁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣶⣿⣼⢿⣆⢹⣿⣿⣿⡽⡚⣿⣖⢿⣿⣿⣿⣿⣷⣤⡈⠁⠀⢻⡇⠀⠀⠀⠀⠀⢎⣿⠄
⠀⠀⠀⢰⣧⣿⣿⣿⣿⣿⡛⢛⠨⢿⡟⣝⠻⣿⣣⣤⡈⠉⢻⣆⠈⠈⢿⣿⣿⢿⣿⠾⣿⡛⣿⣿⣿⢿⢿⣿⣿⣦⣶⢸⡇⠀⠀⠀⢀⡜⣼⡏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡗⠐⠀⠀⢸⠀⠀⠀⠻⣿⣿⣿⣆⠀⢱⠀⢠⣾⣿⣿⣧⡉⠁⠀⠀⢁⡄⠀⠈⠠⠟⡿⢿⣿⣯⠻⣦⣀⣒⣿⣾⠋
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡷⠄⠀⠀⠀⣷⡀⠀⠒⢺⣿⣿⣿⣆⣸⠀⣻⣿⣿⣟⣿⣿⠀⠀⠃⡘⠇⠀⠀⠀⠀⠘⠀⢿⣿⠀⠀⢈⣽⠏⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⠀⠀⢠⠀⢸⣟⡀⠀⠈⠁⠻⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⡧⠀⠀⢂⣴⣂⠀⠀⢰⣆⠀⠀⠉⠀⣠⡿⠂
⠀⠀⠀⠸⣿⣿⣿⣿⣣⡿⠷⢴⣼⣿⣻⣟⠿⣗⠀⠐⣠⣮⡿⣿⣿⣿⣿⣿⣿⡿⠋⠐⠀⠀⠀⠫⡜⢹⡽⠋⠀⠀⠀⣰⣼⠟
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣇⡁⠀⠉⠛⠛⠿⠀⠀⠉⠉⠓⠁⠀⠈⠉⠯⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠈⢻⠚⢿⣿⣎⣷⣖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⢣⣿⣿⣿⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣿⣖⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡻⣿⣿⣯⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢝⣿⣿⣿⣭⣶⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⣿⣿⣟⡫⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣲⣴⣿⣿⣿⣿⢿⣯⣷⣶⣤⣤⠄⠀⠀⠀⠀⣀⣀⠀⡀⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⠻⠛⠿⠿⠿⠿⠿⣯⣿⡍⠉⠉⡉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠸⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⢀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⣀⠀⠐⠈⠉⠁⠀⠒⠒⠒⠛⠓⠲⠶⠀⠉⣉⣀⠴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠀⠀⠐⠐⠒⠒⠛⠋⠉⠉⠉⠁⠀⠀⣸⡥⠔⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⢠⡏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠉⠀⠀⢀⡾⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⡖⠁⠀⠈⠀⢹⡆⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠳⣄⠀⠀⡠⠞⠁⠀⠀⢀⣾⠡⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣄⣀⣀⣀⣀⣀⡤⠘⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣡⠶⠋⡹⠋⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⣠⠞⠂⠀⣼⠁⢠⠀⠀⠀⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠛⠀⠀⣴⢃⣠⡦⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠂⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡤⠚⠀⣠⣦⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢾⠆⠀⠀⠀⠀⠀⠀⠌⠃⠀⠁⠈⠉⢒⡆⠀⠙⣿⠀⡆⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⣤⢀⡉⠻⠀⢇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⠇⠀⢸⡿⠀⠀⢠⠗⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⣤⠦⠈⠢⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠏⣰⡿⠚⠁⠰⠟⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⢀⡤⠃⠀⠀⠉⠻⠆⠐⠶⠚⠾⠃⠀⠀⠶⠀⠀⠒⠛⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠠⡟⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⣦⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⢠⠀⠀⠀⠈⠁⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⣧⠀⠀⠶⣦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠶⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠀⠀⠀⠀⠀⠀⠐⠒⠲⠚⠀⠀⠀⠀⠀⠀⢀⣤⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢤⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠘⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣶⣶⣶⣶⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘






⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣾⣶⣶⣶⣾⣶⣾⣶⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⣻⡟⢿⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢠⡿⠁⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠘⢷⣄⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⢻⡆⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠟⠁⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⣠⣶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⠀⣸⣷⣤⡶⠶⠶⣶⣄⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⡶⠶⠿⣶⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⡄⠀⠀⠀⢻⣇⠀⠀⠈⠉⠙⠛⠛⠛⠛⠻⠿⠿⢿⡿⠿⠛⠛⠛⠛⠋⠉⠉⠀⠀⢃⣿⠃⠀⠀⢀⣾⠇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⢄⣼⡏⣀⣤⡶⠟⣁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠸⡗⠈⣿⡇⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⣉⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠄⠀⠀⠀⠙⣷⣎⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⣿⡷⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣦⡀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣿⡇⠀⠀⢀⣠⣾⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠉⠉⠉⠻⢷⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⠋⠉⠉⠙⠛⠷⣦⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣄⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⠿⠿⠷⠶⠶⠶⠾⠿⠿⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡶⠀⠀⠀⠀⢸⡇⠀⢀⣀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡄⠀⣴⠀⣾⠛⠛⢻⡇⠀⠀⠀⠀⣼⠛⢷⡀⠀⣿⠀⠀⠀⠀⠀⢸⡇⢀⡟⣹⠆⢀⡾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⣀⠈⣧⢀⣿⡀⢻⡄⢀⣼⠃⠀⠀⠀⠀⢸⣆⣼⠃⠘⣿⠛⠀⠀⠀⠀⢸⡇⠈⢿⣯⣤⠌⠷⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡶⠶⠛⠁⠀⠻⠟⠉⠁⠘⣿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢹⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⠙⠛⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⠴⠚⠋⠉⠉⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡴⠖⠚⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿⠟⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⠤⠤⠤⠤⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣴⡶⠶⣤⡀⠀⠀⠀⠉⠛⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⣻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠶⠞⠛⠉⠀⠀⠀⣿⠇⠀⠀⠒⠲⢶⣤⣀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⢹⣷⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢈⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⡴⠋⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⢀⡴⠋⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣴⠞⠛⠛⠳⢶⣤⡴⠛⠁
⠀⣰⡟⠁⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠻⠤⠤⠠⠖⠛⢿⣷
⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⢤⣤⡤⠴⠶⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⡶⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠒⠚⠉⠙⠩⠭⠉⡉⠉⠁⠤⠈⠀⠀⠈⠉⠉⠩⠍⠉⠉⠙⠒⠒⠒⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⣿⠥⠤⠒⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢵⣿⣿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣙⠛⠲⠦⠤⠤⠤⣤⣄⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣥⠤⠤⠴⠶⠞⠛⣉⣴⣶⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠛⠶⠤⠤⠶⠤⠄⠈⣉⣉⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⢈⣉⠉⠀⠤⠴⠤⠤⠴⠞⠋⢩⣿⣿⡁⠤⠀⠀⠉⢧
⢀⡄⡂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡴⢲⡆⠀⠀⣠⡄⠀⠀⠀⣉⠉⠉⠉⢉⡉⢉⡁⠀⠠⠀⠀⠀⠀⠀⠀⣴⢶⠀⠀⣿⣼⢻⢧⠀⠘⠓⠀⡼
⠘⢦⣥⠄⠲⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠷⢬⣠⡀⠀⠙⠂⢠⠟⠛⢿⣷⡄⠀⣨⠟⠄⠀⠀⢶⣦⣀⣠⣿⣿⢶⡔⠁⠀⡜⠙⠶⠿⠋⢠⠃⠀⣰⠃
⠀⠀⠀⠉⠐⣄⠈⠀⠡⡀⠀⠀⠀⢀⡀⠀⢸⡀⠀⠁⠈⣠⡴⢦⡀⠘⠄⠇⠀⣿⠁⠊⣤⣶⡄⠓⠀⢩⢿⣯⡉⠉⠃⠀⠄⠀⣼⠀⢀⣀⠠⠚⢁⣤⠖⠁
⠀⠀⠀⠀⠀⠈⠛⠻⢦⣌⠂⣠⢂⠥⠂⠁⡼⣷⡴⠀⠰⣏⣰⠀⣹⣤⡆⢀⡼⠑⠒⢠⣧⣁⣶⡤⣾⣁⣄⠛⠁⣼⡦⣄⠀⣸⡵⠟⣉⣡⢴⠺⡉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡳⣎⡙⠀⠯⠤⠆⡈⢳⠀⠀⠈⠁⠴⠛⣻⣧⢸⠆⣘⣯⡙⡆⢻⣻⠑⡋⣽⣷⠆⢰⣿⣤⠉⣰⠋⣰⣿⡉⢉⣭⡻⢮⡢⢄
⠀⠀⠀⠀⠀⠀⠀⡰⢡⢎⠔⠉⠻⣦⡑⢤⡒⠛⠏⣷⡈⢠⣄⣀⠀⠨⣿⣿⠀⢉⠛⠻⠿⣾⣿⡿⠋⣭⠦⡄⢠⣿⠁⢠⢧⣀⣉⡾⠃⠻⣄⣀⣠⡈⢦⠳
⠀⠀⠀⠀⠀⠀⠀⠆⡇⠀⣠⣶⡄⠈⢿⣆⠙⢇⠰⠼⣷⣄⠁⠋⢀⣠⡁⠈⠓⢬⣀⢀⡀⠈⣯⣀⠀⡓⢂⣀⠼⠁⢠⠏⢦⡉⠁⢀⣠⣶⡄⢹⣿⡅⠀⡇⡇
⠀⠀⠀⠀⠀⠀⢀⡃⢧⣳⣅⠈⣠⠄⣰⢻⣧⡨⣳⣦⡈⢿⣢⠀⠀⢨⠤⣦⠴⠢⣬⡍⠉⣴⠂⠹⠺⠑⠋⠁⢀⣰⠋⣾⠛⠻⣿⢻⡍⣰⣧⠼⢋⣴⣿⠇⡣
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣮⣿⠧⠤⢯⠄⢿⣿⠈⠛⠛⠓⠛⠿⠶⠦⣄⣉⣶⣂⡻⠀⠀⢾⡅⠶⠀⠤⣣⣾⣟⣁⠄⠉⠀⠀⣿⡘⠿⢃⡞⠰⣫⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⢷⣄⡰⠆⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠳⢦⣤⣤⣭⣤⣶⣶⡾⠿⠿⠛⠃⠀⣠⡤⣤⣌⣉⠋⣉⣴⠿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⡛⠳⣌⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠓⠂⠀⠀⠀⠀⠀⠐⢯⡠⠀⣀⠴⠚⣋⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢮⣝⣿⣷⡶⢤⣤⣀⣀⣀⠀⠀⠀⠀⣀⣠⡧⢀⣀⡀⠀⠀⢀⣀⣠⠤⠄⣒⣭⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠮⠭⢭⣭⣭⣿⣿⣿⣿⣞⣓⣲⣦⠭⠭⠴⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⡠⠄⠒⠀⠀⠈⠁⠀⠀⣠⣔⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠖⠉⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⠜⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡴⠃⣰⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣉⣀⢀⠀⠀⡀⡀⠐⠨⡢⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⣼⠇⢰⠃
⠀⠀⠀⠀⠀⠀⢠⡲⢿⣿⣿⣿⣟⠳⢖⣒⡚⠵⡲⣦⣄⢱⡀⠀⠀⠀⡠⠊⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣠⣿⠏⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠿⢿⣶⡾⢿⡟⠛⠛⠂⠉⠫⣷⣿⣆⣠⢊⣠⣴⠶⠛⠉⠀⠀⠀⠀⠠⠤⢀⣠⠞⠁⢀⣼⣿⠏⢠⡞
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠒⢁⡴⠊⠙⠒⠦⢄⣒⠤⣄⣈⣿⣿⣿⣿⣭⣤⣤⣤⣀⣀⠀⠀⠠⠄⠠⢤⣀⣀⢠⣾⣿⡏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⢀⡙⢿⣿⣿⡿⣿⣿⣯⣭⣭⡭⠭⠍⠉⠒⠂⠒⠈⠉⠹⣿⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠋⠉⠙⠛⠛⠿⢶⣤⣀⡀⠀⠀⠀⠀⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠤⠀⠀⢽⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⢰⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠓⢻⡆⢸⣧⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣤⡴⠟⣡⡞⠁⠙⣧
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠉⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠈⢻⣟⠶⣤⣄⣀⡀⠀⠉⠳⢦⣤⣀⣀⣀⣀⣤⣤⡴⠞⠋⠀⠀⠀⣀⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣌⡉⠙⠛⠓⠶⠶⠦⣬⣭⣭⣤⣤⣤⣤⡤⠶⠶⠚⣛⣭⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠖⠛⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠟⢠⠂
⠀⢠⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣇⠀⡀⠹⢦⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠈⠁⠉⠀⠀⠀⣦⣤⡄⡀
⠀⣏⠀⢱⣶⠀⣧⠀⢸⠁⣄⠀⠀⠀⠲⢿⣷⣶⣾⣿⣿⣛⣛⣛⣻⣛⣿⣿⣷⣶⡾⠟⣃⠸
⠀⠹⣦⡄⢹⣶⡻⠀⡖⣠⡘⢆⠀⢴⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣻⣿⠁⣆⣲⣦⣄⢲⡀
⣀⣀⣈⠳⣄⣹⡇⢰⠃⣿⡇⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢸⠋⠁⠀⡼⣸⠇
⠘⡍⢶⣄⣩⣭⣽⣿⣦⣿⣧⠟⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠇⣎⣠⠴⢊⣼⠟
⠀⠈⠢⣀⠜⠋⠹⠛⣿⠿⣁⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⣸⢋⣤⠾⠛⠁
⠀⠀⠀⠈⠐⠒⠒⠁⠀⠀⠙⠷⠆⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠏⡰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢾⣒⠁⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣫⣞⠓⢤⣤⣐⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠀⠉⠙⠋⣆⠰⢍⡛⠷⠶⠶⠶⠶⠾⡿⢿⣿⠿⠋⠻⢂⠽⠂⢉⣆⡕⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⡴⣞⡋⠠⡆⠀⢀⠀⢀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣻⣲⣿⣻⣿⣻⣻⣿⣷⣿⣿⣿⠿⣿⠛⠋⠉⠀⠀⠉⠙⠣⡀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣏⠙⠶⣤⣤⠤⠴⠒⠚⣷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣿⡀⠀⠀⠀⢀⣽⣬⠿⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠮⣿⣷⣠⠤⠚⠋⠁
⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢁⠈⠀⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠂⠀⠈⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠶⢺⡀⠀⢁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⣹⠀⠀⣏⠉⢵⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⢋⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠒⠡⠐⢧⠀⠱⣀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣀⡀⠀⡈⡗⢤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡜⢀⢊⠐⠁⠀⠀⠀⠑⢤⡈⠑⠂⠷⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠴⠊⠉⠀⠀⠀⢀⠻⠲⡐⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⡼⠉⠑⠂⠠⠀⢤⡀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠡⠁⠀⠀⠀⠀⠀⣠⠞⠀⠃⠀⢨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢒⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⢀⣁⠤⠖⣒⠼⠶⠶⠦⠤⠶⠚⠁⠀⠀⠀⠠⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⠆⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⣠⠴⠶⠒⢉⣡⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠌⡁⠂⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⠀⠔⠊⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠴⠒⢈⠠⢐⡮⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⣀⠀⠀⠀⠈⠉⠁⠒⠒⠀⠤⠤⠤⠤⠤⠤⠠⠤⠤⠶⠶⠾⠛⠛⠓⠊⠉⠉⠠⠀⠂⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠃⠐⠒⠒⠒⠒⠒⠒⠒⠂⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣡⠀⠄⠀⡄⢰⠀⠀⠢⠀⠀⣀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢠⡏⣸⡆⠀⡖⣾⣇⠀⢨⠠⣾⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⣽⠋⣿⠀⡴⠀⣿⠀⡠⠃⠰⡇⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣿⣀⠈⠉⠀⠀⢈⡁⠀⠀⠀⠉⠀⢸⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢠⠇⢸⠀⢠⡄⠐⠀⡗⠀⠀⢀⡞⠡⠀⠉⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠠⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⣾⠂⢸⡏⠀⣼⠃⢀⠀⠈⣧⢀⡀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⣤⡀⠁⣼⠁⣴⠂⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡟⠀⠾⠀⢰⡏⢀⡿⠀⠛⠀⣸⠃⠀⠀⣼⠃⣸⠃⢠⡇⠀⠊⣸⠃⢰⡇⣸⡇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⠀⣼⠁⡌⠀⠛⠀⠸⠄⠙⠧⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠈⠁⡠⠀⠒⢲⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢁⡤⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠊⠀⠀⠀⠀⣿⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⡆⠀⠀⠀⢀⡼⠃⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠂⠉⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⢤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⣠⠞⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⠤⡤⣤⢠⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⡜⡟⢯⣋⣀⠀⠈⠉⠉⠓⡲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠆⠓⡀⠈⠁⠀⠀⠀⠀⠀⣿⣽⢶⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠙⠒⢠⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⢆⠀⠈⠁⢀⣀⣀⢀⠠⢠⣝⣦⣾⢢⠁⢀⣤⢶⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿⣷⣤⣼⣥⣤⠤⠀⣀⣠⢿⣿⣿⣯⡍⢁⡁⠀⠀⢀⣤⠇⣆⡀⡾⣿⣿⠏⣿⣩⣋⣼⡿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡗⢉⣿⡇⠘⢿⣓⣲⠞⠾⡟⣸⠿⣿⣾⠿⣍⣁⣀⣉⠛⡖⠰⡟⠤⠜⠑⡇⣰⣿⠛⠋⠉⠀⣸⣻⡟
⠀⠀⠀⣀⡤⡤⠲⠲⣖⠻⡉⠊⣧⣸⠇⠁⠀⠈⠻⣷⣶⡿⢂⣁⣀⡀⠀⠀⣏⢣⣤⣉⡛⢃⠏⠸⣶⠘⠃⣣⣽⡟⠂⡀⣤⡴⢿⡿⠁
⠀⢠⣾⣿⣿⡷⡀⠀⣌⠣⣌⢠⠺⣯⣆⠀⠀⠀⠀⠈⠓⠯⠍⠉⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠉⠉⠉⡹⣸⠔⣊⣍⣭⣾⠟⢨⠙⠒⢄⡀
⠀⢿⣿⣿⣿⣿⣛⠿⠿⠯⠈⠳⠄⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠿⠷⠄⠀⠀⠀⠀⠀⢀⣰⣿⣷⠿⠿⣿⣻⡟⣩⠄⣂⣴⣖⠭⡆
⠀⠈⢿⡻⢿⣿⣯⣿⡧⣌⢄⣀⣀⢀⡈⠻⣿⣵⣄⣰⡄⢠⡄⠀⢀⢐⡀⠀⢀⠀⠀⢀⣠⠀⣴⢀⣭⡿⠛⠋⠁⠀⠛⠛⣫⣾⣁⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠉⠳⣮⣭⣙⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣾⣖⣤⣼⣧⣾⣆⣢⣷⣶⣯⣾⡷⠛⠁⠀⠀⠀⠀⣀⣈⢉⡥⠬⣴⢚⣫⠵⠋
⠀⠀⠀⠀⠀⠀⠉⠻⢿⢿⣶⣶⣮⣭⣭⣯⣛⣛⣛⣛⣿⠿⠭⠭⠭⠭⠉⠉⠩⠉⠩⣁⣒⣚⣒⣒⣤⣘⣩⠭⠭⢒⣒⢶⣻⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠯⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣭⣭⣿⢿⣛⠓⢊⣂⣤⣤⣅⣸⣤⠥⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⡀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⡴⠀⠀⠰⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡎⠀⠀⠀⢀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠠⠀⡘⢡⡷⢤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢤⡄⢖⠀⣠⣴⣶⣶⣿⡿⢷⣼⣿⣿⣿⣿⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠷⢂⢹⣧⣤⣬⠍⠉⠉⠁⠉⠉⠉⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠑⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠠⠐⠞⠻⣦⡀⠀⠀⠀⠀⠀⢀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠷⣤⣍⣓⣶⣶⡤⠴⡋⠁⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣌⠻⣿⣿⣿⣿⣦⣌⣉⣉⣉⣁⡀⠔⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⢠⡴⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠋⠉⠉⠀⠀⠀⣀⣀⣤⡔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠉⠁⠀⠈⠁


⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠒⠒⠒⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠀⡄⠁⡄⢀⠤⠐⠒⠠⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠠⠒⠉⠉⠒⠁⠀⠀⠀⠀⠀⠱⠄⠒⠬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠤⠧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⣠⠴⢶⡀⠀⠀⠀⠀⠀⠀⠀⢀⢆⠀⢀⠀⡶⠶⣄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⡰⢀⠀⠀⣸⠃⢀⠀⠀⡠⠋⠒⠂⠁⠀⠈⠉⢜⣇⠀⠀⢧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠬⢀⠆⠀⢾⡀⢀⠎⡈⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢱⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠸⠀⠀⠀⠀⠀⢀⠀⠈⠎⠀⢸⣺⠀⠀⠀⢣⡀⣀⡄⠀⠰⢄⠈⠤⡤⠼⠀⠸⠇⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢠⠀⠀⠀⠀⠀⠀⠄⠀⡇⠀⢀⠈⡆⠀⠀⠀⠈⠁⠀⢠⡀⣡⠀⠀⠀⠀⠀⠜⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠣⡘⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⢏⠔⠁⠀⡨⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⡀⠀⠈⠐⠓⠢⠤⢀⣀⣀⣀⠠⠤⠐⠚⠒⠁⠀⠀⡔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⡁
⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⢀⠀⠁⠀⠠⠀⠐⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⠒⠛⢻⠛⠛⣿⠛⠛⠛⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣄⣀⣀⣠⣤⡤⠤⠤⠤⠤⠴⠦⠤⠤⠤⢤⣤⣄⣀⡀⣙⣿⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠳⣶⠒⠒⠛⠋⠉⠁⣿⠶⢦⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡧⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣄⠀⠀⠀⠀⠀⢰⣇⣀⡤⠔⢁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣴⡉⣿⠀⠀⠀⢠⠿⠤⠶⠖⠚⠉
⠀⠀⠀⠀⠀⢀⣀⣠⣤⡼⠿⣦⡀⠀⠀⠀⠀⠀⠀⣿⢾⡓⠀⢸⡄⢀⡴⠓⠶⠤⢤⣀⡀
⠀⠀⠀⠀⣾⠋⠉⠀⠀⠀⠀⢨⣿⣶⣤⣀⠀⠀⠀⠻⠾⠶⢚⣫⣴⣿⡄⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⣶⣶⣿⣿⠿⠟⠛⠁⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠄⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠐⢄⠊⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠦⣍⣘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⣸⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⠔⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⢀⢤⣶⠿⠛⠉⠀⠀⢀⢠⠃
⠀⠀⠀⠈⢇⠀⣈⣉⠐⠠⣀⢮⠢⡐⠄⠀⠀⠀⠀⠀⠙⠓⠲⢦⡴⢒⡵⠟⠁⠀⠀⠀⠀⢄⡴⠃
⠀⠀⠀⠀⠀⠑⠄⡀⠁⠲⣈⣧⠦⠬⠱⡾⣽⡴⡦⡄⣀⣰⣯⣥⣔⣉⡀⡀⠠⢀⡂⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣉⣗⣾⡿⣿⠬⡯⢑⢔⡉⣀⠛⣧⣶⡖⡖⠒⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠤⠠⡿⠿⠹⣞⣛⢯⡉⣯⣽⣿⣘⢣⢻⣯⢏⣿⣾⣤⢀⡠⡀
⠀⠀⠀⠀⠀⠀⠜⠀⢘⢐⣝⡭⠨⣽⢛⠖⢚⣿⣓⠾⠏⠉⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠤⠂⠈⠑⠋⠉⠒⠃⠘⠯⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⢄⠀⡀⢲⣾⣻⣍⣯⣿⣦⣾⣤⣌⡓⣒⣠⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⣰⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣺⡿⠻⠩⠿⣯⡿⠛⠀⠀⠀⠠⠤⢀⣤⣦⣲⠀⠀⠉⠙⠻⢯⠿⠷⠶⠿⢿⣶⣤⡀⡞⡵⠚⠉⠉⢝⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣧⣶⠐⠗⠃⠀⠀⠀⠀⣤⣄⢸⣎⡙⣣⠜⣻⠠⣴⠒⡦⣀⠀⠀⠀⠀⠀⢠⡄⢙⣿⡿⣇⢿⡀⠀⠀⠀⠀⢹⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡯⡋⡷⢻⣿⣦⠀⠀⠐⣠⣦⡽⣿⣷⡿⢿⣧⠲⣌⠶⢏⠁⠀⠀⠀⠀⠀⢀⣴⢎⡴⢣⣿⡷⡁⢸⣷⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡻⡃⠑⡘⢛⣩⢈⠀⠀⠀⠸⠾⠃⠛⣀⠨⣅⡐⠛⠭⠈⠁⠀⠀⠀⠀⠀⠀⠙⠉⢀⡠⡾⣔⣻⠿⠶⠀⠀⢀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢶⣥⣬⣉⣊⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠆⠀⠀⠀⠀⡠⢤⣲⣖⣠⣥⠶⠿⠋⠀⠖⡦⢧⠀⠧⠄⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠟⠃⢀⠀⠈⠉⠙⠛⠛⠓⠿⠿⠛⠛⠖⠲⠟⠷⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠐⡦⠔⠀⣁⡤⠐⢫⡀
⠀⠀⠀⠀⣠⣮⠀⠀⢀⠰⡇⣀⣂⠀⠄⠀⠀⠀⠄⡀⠀⢠⣄⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣄⣦⡀⠀⠀⠀⠀⠠⢋⠴⢂⣉⣉⠀⠀⠀⢻⣚⣀
⠀⠀⠀⢠⣿⡇⠀⠀⠋⡐⢛⠉⠼⣣⠀⠀⠀⠀⣼⣇⡀⢈⣁⡜⣠⠤⣶⠚⣯⢣⢄⠀⡀⠀⠛⢾⠏⡀⠀⠀⠀⠀⢀⠊⠈⣋⢬⠏⠉⠀⠀⢀⠉⢹⣷
⠀⠀⠀⢾⡏⡀⠀⠀⠒⣤⣿⡛⣾⣀⢀⠁⠀⠘⠛⠻⠛⢀⣍⣤⢧⣉⡰⢇⡿⡞⠂⡞⣉⡶⠗⠛⠛⠀⠀⠀⠀⢾⣋⠈⠁⠉⠅⠀⠀⣂⢢⢾⣴⢎⡿
⠀⠀⠀⠘⢿⣿⠿⠦⠀⠈⠓⠉⠀⠁⢴⠆⠑⡀⠀⠀⠒⠁⠠⠀⠸⣿⠊⠁⠀⠉⣡⠟⠛⠇⠀⠀⠀⠀⠀⡀⠐⠯⣭⠔⠀⠀⠀⠀⠀⣤⣴⣿⠔⠋
⠀⠀⠀⠀⠈⠙⢶⣢⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⢠⢀⡤⠀⠀⣀⣈⣤⠼⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠾⣦⢶⣤⣀⣀⡀⠀⠀⠀⢫⡑⠂⠤⢤⣤⣠⣤⣤⣔⣶⣴⡬⠴⠖⢋⣝⣀⣤⣬⣭⠬⠖⢛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⡀⠉⠉⠓⠒⠒⠛⠒⠒⠶⠦⠴⠶⠒⠒⠒⠒⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⢶⣶⣤⣦⣤⣤⣤⣴⣰⣶⠤⠤⠆⣲⠖⠲⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⣀⠤⠔⠂⢈⣁⠨⣧⣀⢠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⢔⡠⠖⠉⠑⠠⢀
⢀⢴⡚⠉⣁⡤⣔⣲⠭⠕⠒⢉⣀⡄⠀⠀⠀⠀⢀⠄⢀⣀⣀⠤⠤⠒⠊⠁⠀⠑⢅⠀⠀⠀⠀⠀⠉⠒⢄⡀
⡇⠃⢹⡩⠾⣍⢁⡠⠴⠒⠋⠉⢚⢤⠀⠀⡀⠐⠁⡞⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀
⡇⠀⣼⠃⠀⢸⠀⠀⠀⢀⠔⣋⣑⣤⣤⠎⠀⠀⠀⠳⡀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢈⡵⠂
⡇⠀⣿⡄⠀⣾⠀⠀⢐⣤⠚⣀⢨⣯⣕⠐⠢⠁⠀⠱⡘⢆⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⣠⠴⠊⡡⢧⡄
⣇⣆⡇⡇⠀⡇⠀⢶⡉⠙⠈⠀⠀⠉⠉⠙⠲⢄⡢⢹⡀⢈⣳⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣶⠉⣀⢔⡪⠚⠉⠓⢦⣀
⠀⠈⠷⣇⣀⣧⠶⣟⢍⡓⠦⣄⡀⠀⠀⠀⠀⢀⡹⢦⠿⣉⠀⠘⢄⠀⠳⣄⣀⣠⠤⠴⠖⠒⢋⣉⢿⣔⡪⠖⠉⠀⠀⠀⣠⢴⡟
⠀⠀⠀⠁⠸⣿⣗⠦⣕⡪⡂⢄⠉⠒⢄⡠⠖⠉⣠⣜⣄⡈⠓⢤⡈⠳⣠⣏⡂⢀⠤⢄⣒⠭⠕⠚⠉⢈⣱⠆⣀⡤⠖⠋⣁⠜⣳
⠀⠀⠀⠀⠀⠈⠪⢑⢄⡉⠛⢷⣬⣀⢠⣧⠴⣫⡞⠣⣀⠉⠢⣀⠉⠲⣌⠓⠒⠒⠉⠁⠀⠀⣀⡤⠞⣉⢿⣞⠉⢀⠴⣊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⣐⠀⢈⣯⣀⠴⠊⠀⠣⣀⠈⠳⢄⡈⠑⢤⡀⠙⠢⣄⢀⡠⠖⠋⢁⠤⣪⢕⣩⡽⢚⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠁⠀⠀⠀⠀⠈⠑⢤⡀⠙⠲⣄⡈⠳⣄⣠⣏⠂⠀⣊⣵⡯⠖⢋⡨⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠙⠢⣄⠉⢓⣲⠿⠋⠁⣠⢔⡩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⣹⠫⠒⣀⠔⣪⠔⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣻⣆⣤⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡀⠄⠀⠶⠂⠖⠒⠒⠒⣶⣶⣦⣤⣄⣀⡜⠲⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡤⠀⠀⠀⡀⠠⣍⣀⣀⣤⠈⠉⠉⠂⠛⢛⡛⢿⣷⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠰⣇⠀⠀⠀⠀⣡⠀⢩⡀⠈⠹⠷⣤⣈⣁⣼⡿⢃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣒⡐⠀⠀⠀⠀⣠⣶⡀⢼⠭⢽⢟⠉⢰⣏⣹⡷⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠴⠖⠀⠉⠙⠀⠋⡿⣿⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⠶⠒⠋
⠀⠀⠀⠀⠀⢀⡔⠟⠛⠉⢛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⣶⣤⣀
⠀⠀⠀⠀⠐⠈⠀⠀⠀⣰⠻⡿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⣻⣿⠵⣺⡀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣛⣷⠋⡷⢲⣦⣧⣀⡀⠀⡀⠀⢀⢃⣠⣶⣟⢓⣏⣹⣿⣧⠆⠀⠀⣘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢋⠥⢠⠈⢉⡛⠛⠛⠿⠛⠿⠛⠻⣿⣯⣯⣯⠋⠁⠀⢀⣠⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠛⠁⠀⣀⣀⠀⠀⠀⣀⠀⠈⢀⣤⠄⢰⠿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠉⠉⠁⠀⠓⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⣼⣿⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣸⣿⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠤⠤⠴⣿⠥⠤⣾⢯⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡩⢭⣶⣶⡶⠒⠊⠉⠉⠙⢏⠉⠻⠓⠒⠒⠒⠨⠭⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣧⣜⣻⠿⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢨⣍⠙⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠉⠁⢀⣿⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⡴⠶⠶⣌⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣠⡟⠀⠀⠀⠘⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣸⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠖⣫⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⢾⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣱⠯⣅⣀⣀
⠀⠀⠀⣀⡤⣖⣺⠭⠗⠚⠉⠉⠀⠀⢸⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢧⠀⠀⠐⠚⠭⢭⣒⡢⢤⣀
⠀⢠⡞⣵⠟⠉⠀⠀⢀⣀⠤⠤⣒⣒⠚⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠦⣬⠷⠤⢤⣀⠀⠀⠈⠙⠳⣮⢳⡄
⠀⠸⡎⣿⠀⠀⢠⠖⣩⣶⡾⠋⠉⠀⠀⣿⡸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠈⠙⠲⣦⣍⠲⡄⠀⠀⠘⣰⠃
⠀⠀⠙⢮⡳⣄⠸⣄⠻⣿⣦⣀⠀⠀⠀⠈⠷⣬⣓⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠟⠀⠀⠀⠀⠀⠀⠈⠿⣣⠇⠀⣠⠞⠁
⠀⠀⠀⠀⠉⠒⠭⣂⡉⠒⠪⠭⢟⣒⣲⣤⠤⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⣀⣀⡀⠤⠴⠒⣉⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⠤⣄⣉⣉⣉⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠚⢋⣉⣉⣉⡥⠤⠤⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⢀⣀⠀⠀⣀⣀⣀⠀⠀⢰⢴⠾⢿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠁⢢⠄⠀⠀⠀⠈⡿⢉⣻⠁⠤⠀⠒⢒⣦⡤⠟⠀⠀⠀⠢⣄⣀⢀⡀⣀⣀⣂⣀⢀⣀⡀⠄⠀⠀⠂⠈⢳⡄
⠀⠀⠀⠀⠀⠀⡶⠲⠤⠜⣉⡐⠶⢶⣯⣤⢴⣶⣶⠖⠉⢉⠠⠀⠀⠀⠀⣀⠎⠈⠋⠉⠩⠥⠄⠀⣙⣟⠁⠒⢂⣠⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠉⠉⠑⠚⠳⡞⠉⠤⡧⢂⣠⠴⠓⠀⠀⠀⠙⠛⠓⠒⠒⠒⠒⠒⠚⠁⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⢄⣀⡀⠀⠀⠀⡏⣀⡤⠿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠁⠀⠀⠀⠀⠀⠀⠀⣴⠶⡂⠀⠐⠌⠉⠁⠈⠉⠏⠓⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⡾⠷⠒⠚⠛⠿⠿⡶⠦⣄⣈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠈⢣⣤⣤⣤⣤⣤⣴⣧⣄⣀⠈⠙⠲⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠭⠁⠀⠰⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠲⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠀⠀⠠⠶⠖⠶⠶⠶⠶⠶⢿⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⡀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣧⡴⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠚⠛⠛⠛⠛⠙⡿⣟⡛⠛⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⣶⣇⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠠⣄⡀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⢀⣠⠆⠀⡁⠀⢀⣤⡼⠔⡈⠀⠺⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠓⣦⣤⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠊⡍⠁⡀⠀⡿⣚⢉⡀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠑⠠⢀⠀⠀⠀⠈⠘⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢆⠀⠀⠀⡇⠀⢇⠀⠃⠀⠸⡇⠀⡁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠢⢄⣠⣶⣿⣷⡀⠐⡀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠀⠀⡇⠀⢻⠀⠁⠒⠒⠁⠀⡷⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⢸⠇⣹⠿⠋⢁⠰⢅⢀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⢀⠀⢸⠀⠀⠄⠤⣆⠀⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠠⠈⠱⣄⣴⠾⠋⠀⠀⠀⠀⠀⣦⣼⠀⢸⠀⣍⠀⠀⣿⠁⡄⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠉⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢦⠹⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⡄⣿⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢰⠃⣿⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢀⠏⣸⠇⠀⢰⢿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟⠀⠀⡞⢈⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣾⠁⠀⡼⠁⣼⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣰⠏⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠶⠖⠚⢻⡀⠈⠁⣿⠀⠈⠀⠀⠀⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣋⣥⣤⠴⠶⠶⠒⠈⠻⠦⡘⠙⢧⡘⠛⠳⠶⠶⠤⢤⣀⠉⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣥⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠾⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⣀⣼⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡇⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠖⢋⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⠶⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡁
⠀⠀⠀⢀⣀⣤⣶⠾⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀
⠀⢀⣴⡿⠋⠉⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⡶⠶⠖⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄
⠀⢸⣿⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⣄⠀⠀⠀⠀⠀⣹
⠀⠈⢿⣦⡀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⠿⣶⣄⣈⠙⠿⢷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠞⠋⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⣶⣤⣬⣉⣉⡙⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉⠉⠁⣀⣀⣠⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠉⠉⠁

⣦⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣭⣭⣿⣵⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣶⣶⣾⣷⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⡟⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⢿⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⡘⡇⡄⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣶⠛⠉⠉⠉⠉⠉⠉⠉⠛⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠩⣤⣶⣶⣶⣶⣶⣶⣶⣤⠍⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠟⠻⡆⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⣠⣌⢹⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡼⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢃⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠄


⠀⠀⠀⠠⣤⣄
⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠙⠻⠷⠢⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡠⠲⣌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢣⠀⠈⢂⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠑⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠸⡆⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠼⢷⡀⠀⢹⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⢙⣦⡀⠹⣤⠤⠀⠒⠒⠀⠒⠒⠒⠂⢀⠀⠀⠀⠀⠀⠐⠂⠒⠒⠒⠠⠤⣀⣠⣾⣠⠴⠖⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠳⠶⢯⣥⣄⣹⣆⣈⣳⣀⣀⣀⣀⣀⣀⣀⣴⣋⣉⣓⣦⣶⣶⣦⡤⠤⠤⠶⠖⠛⠉⠁⣿⣁⡤⠴⠶⣄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⡿⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣠⠞⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡿⠀⣴⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠇⣼⠁⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⡿⠛⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⣚⣉⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠚⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣀⣀⠀⠉⠉⠉⠉⠙⠛⠲⠦⣄
⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⢀⣀⣀⣤⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⠉⠉⠉⠉⠒⠒⠠⠀⠀⠈⣷
⠀⠀⠀⣴⠟⠁⢀⡠⠔⠒⠉⠉⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠲⠶⠶⠶⠦⢾⠶⠶⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⢛⣽⠋
⠀⠀⠀⠹⣗⠲⠤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⣀⣀⣀⣠⣤⣤⠴⠶⠖⠚⠛⠋⠉⠉⣀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿⠉⠉⠙⠛⢯⡉⠁⠀⢀⣀⣀⣠⣤⠤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠲⠶⠶⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣏⠀⠀⠀⠀⠀⠙⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡤⠦⡀⠀⠀⠈⠲⣄⠀⢀⣀⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⠏⠀⠠⠇⠀⠀⠀⠀⠀⠉⠁⣀⣀⣀⣀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡾⠤⠤⣤⢀⡴⠚⠓⢦⡀⣰⠖⠲⢤⠀⢸⡏⠓⣄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣧⠀⠀⣸⠸⣇⠀⠀⣸⠇⣇⠀⠀⢨⡇⢸⡇⠀⢸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠈⠓⠚⠋⠀⠙⠛⠛⠋⠀⠈⠛⠛⠋⠰⠾⠳⠒⢻⠃
⠀⠀⠀⠀⠀⠀⡴⠁⠀⢰⣦⣄⢀⣠⠾⡇⠀⠀⣠⠔⠒⠉⠓⠢⡀⠀⠀⣠⡤⢤⣄⠘⢤⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠈⠙⠟⠁⢸⡇⠀⣰⠁⠀⠀⠀⠀⠀⢹⠀⡾⢁⣀⣀⣈⣁⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠙⢄⠀⢸⠄⠀⠀⣀⡀⠇⠀⢻⠀⠀⠀⠀⠀⠀⢸⠀⢻⣄⣀⠀⠀⢈⡗⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⢰⠟⠉⠉⢳⡄⠈⠢⣀⠀⠀⠀⣠⠋⠀⢀⠈⣙⣛⠛⠉⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸⢧⣄⣀⡼⣃⣄⡀⠀⣉⣩⣍⠀⢘⠀⣸⠞⠉⢹⡇⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢈⡥⠞⠉⠉⣧⢸⡇⠀⢸⡇⢸⠀⣿⠀⠀⢸⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⣤⣤⣴⡾⠃⢸⠃⠀⢸⡇⣿⠀⣿⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢷⣀⣈⣀⠀⢸⠁⣿⠀⣿⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⡆⠀⠀⠀⣀⠤⠄⠤⠤⣀⡉⠀⠉⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠈⠉⠃⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⡟


⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⡏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠂⠀⠀⠉⠉⠉⠉⣉⡉⠉⠉⠉⠉⠉⠑⠐⠒⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢑⠈⠡⠀⠒⠒⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠒⠀⠄⠁⡒⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠉⠉⠉⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠢⡀⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⡀⠀⢸⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣹⠀⠀⣏⣉⢵⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠹⡑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡡⠔⡉⠄⢧⡀⠹⣄⠠⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⣀⠀⢀⢻⠦⡈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⡜⢁⠎⠀⠈⠀⠀⠀⠑⢤⡈⠑⠂⠵⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡡⠖⠉⠁⠀⠀⠀⠸⠗⢌⠢⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠒⠠⠄⢠⣀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠎⠀⠀⠀⠀⠀⢀⡴⠃⠐⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢱⡈⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠙⠖⠉⠐⠒⠢⠤⠤⠤⠄⠀⠐⠂⢈⡠⠴⢒⡲⠷⠶⢤⣤⠤⠖⠋⠀⠀⠀⢀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠠⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠢⢤⣤⠴⠒⠂⣉⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢑⠢⠌⡑⠂⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠈⠁⢀⠠⠒⠋⠁⠀⠀⠀⠀⣀⣀⣠⣄⣤⠤⠤⠒⡁⠤⣲⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡁⠀⠀⠠⠍⠉⠐⠒⠒⠂⠤⠤⠤⠤⠤⠠⠀⠤⠤⠤⠶⠶⠶⠛⠛⠛⠉⠉⠡⠄⠐⠀⡉⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠁



⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⣿⡿⢿⣿⣿⠿⣿⢿⠿⢿

⣿⣿⠟⠋
⣟⣀
⣿⣿⡷
⣿⣁⡀
⣿⣿⡿⠄
⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⣠⣾⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⣿⣿⣤⡀
⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣼⣿⣄⣀⣀⣤⣾⣿⣿⣁⣠⣤⣠⣤⣤⣤⣄⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⡆⡄
⠀⠀⠀⠀⠀⠀⠀⡀⣤⣻⢸⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⡇⣟⣤⢀
⠀⠀⠀⠀⠀⢀⣴⠇⡇⡟⣸⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⣇⢻⢸⠸⣧⡀
⠀⠀⠀⢠⣤⣾⣿⠀⡇⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠀⣿⣷⣤⡄
⠀⠀⠀⢸⣿⡿⢿⠀⡇⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⢸⠀⡿⢿⣿⡇
⠀⠀⠀⣿⡇⢧⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⡇⡾⢹⣿
⠀⠀⠀⣿⡇⢸⠘⢸⢡⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡌⡇⠃⡇⠸⣿
⠀⠀⢠⣿⠀⠀⡇⢸⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡇⡇⢸⠀⠀⣿⡄
⠀⠀⢸⣿⠀⠀⡇⢸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⢸⠀⠀⣿⡇
⠀⠀⢸⡏⠀⠀⠃⡌⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢡⠘⠀⠀⢹⡇
⠀⠀⣾⡇⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⢸⣷
⠀⠀⣿⡇⠀⠀⠀⠃⠀⠌⣐⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣂⠠⠀⠸⠀⠀⠀⢸⣿
⠀⠀⣿⠁⠀⠀⠀⠂⠠⠤⣀⣉⠓⠺⣵⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣮⠕⢚⣉⣀⠠⠤⠀⠀⠀⠀⠘⣿
⠀⢀⣿⠀⠀⠔⠀⠀⠒⠒⠀⠤⠭⠕⣲⣭⣛⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣛⣭⣖⠫⠭⠤⡀⠐⠒⠀⠀⠢⠀⠀⣿
⠀⢸⣿⠀⠂⢀⡀⠤⠄⣀⣒⠢⠬⣍⡒⠮⢭⣛⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣛⡭⠕⣒⣩⠥⠔⣒⣀⠤⠄⢀⡀⠐⠀⣿⡆
⠀⢸⣿⣍⣐⡒⠤⢭⣑⡒⠤⢍⣓⠒⠬⣙⡒⠬⣙⠢⢝⠻⣷⡀⠀⠀⢠⣾⠟⡩⠔⣋⠥⢒⣋⠥⠖⣚⡭⠴⢒⣊⡭⠤⢒⣒⣨⣿⡇
⠀⠈⠉⠛⠻⠿⢿⣶⣦⣭⣉⣒⠂⠍⠑⠂⠈⠁⠀⠉⠀⠁⠀⠙⢆⡴⠋⠀⠈⠀⠉⠀⠈⠉⠐⠊⠩⠐⣒⣉⣭⣴⣶⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢷⣶⣤⣤⣀⡀⠀⠀⠀⣀⣴⣾⣷⣦⣀⠀⠀⠀⢀⣀⣤⣤⣶⡾⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣶⡿⠿⠿⠟⠿⢿⣶⡶⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⢀⠀⠀⠀⡇⠀⢀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⡀⠙⢄⠀⠈⢆⠀⠀⡇⠀⡞⠀⢀⡜⠁⠀⡀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠈⠢⡈⠢⠐⡈⠄⠆⡇⠱⠱⢡⢊⠀⡠⠊⢁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠈⠂⠐⠀⠀⠀⠇⠀⡐⠁⠀⢊⠤⠂⠁⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠉⠓⠢⠄⠀⠀⠄⡠⠂⠀⠀⠀⢈⠀⠂⠁⠀⠀⠒⠉⠁⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣀⣀⣈⣉⡉⠁⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢪⠟⠛⠉⠉⠁
⠐⠋⠉⠉⠉⠀⠀⠀⠀⠈⠑⠤⢄⡀⠀⠀⣄⠠⠄⠀⠤⠐⠂⠉⠉⣡⠋⣁⠄⠀⠀⠀⠀⢀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⠠⠀⠈⠉⢦⣤⠀⠤⠤⠤⢤⡴⠒⠊⠁⠌⠁⢤⡀⣔⢊⠁⡀
⠀⠐⠂⠐⠂⠂⠀⠂⠐⠒⠀⠀⠁⠒⠒⠛⠋⠉⠀⢀⡴⢚⣤⣠⣰⣂⠀⠀⠐⠁⠓⠀⠑⠮⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠠⠤⡤⠶⢚⠚⠁⠀⠡⠄⠀⠭⠩⠝⠋⠉⠉⠛⠉⠉⠉⠛
⠀⠀⠀⠒⠋⠉⠉⠉⠉⠉⠻⠤⠀⠀⠀⠒⠒⠡⠤⠈⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢴⣄⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠈⣿⡆⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⢀⣴⠟⠁⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠸⣿⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠹⣷⡀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⠻⠿⠶⠶⢶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠶⠶⠶⠿⠟⠛⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢸⣿⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠸⠿⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⠙⠻⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣿⣶⣶⣤⣤⣤⣤⣶⣶⣿⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣍⣉⣋⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣭⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠈⠠⣉⡐⢬⣳⣄⠀⠀⣴⣫⣴⣾⡟⠍⠉⢉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠉⠛⠿⣷⣌⢻⣆⣼⣿⣟⠫⠭⠭⠻⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠄⠘⠁⠀⠀⠀⠀⠀⠉⠛⢿⠿⠿⡋⠄⠀⠀⠀⠄⠈⠉⠛⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣀⡤⠤⠄⠀⠒⠂⠁⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠧⣀⡀⠴⠂⠒⠒⠒⠒⠒⠒⠒⠚⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠲⠒⢲⠖⢉⣰⣄
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠸⠒⢇⣠⡤⠵⠶⠒⠛⠉⠁
⠀⠀⠀⣼⣇⣀⣀⣀⠀⣀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⢈⣧⠀⠀⠈⢦⡀⠀⠀⠀⠀⣀⠔⠉⣀⡴⠗⠒⠋
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠾⠿⠡⠹⠛⠛⠶⠚⠛⠿⠿⠾⠿⠿⠿⠭⠀⠀⠀⠀⠱⡄⢤⣴⡾⠵⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠁⠉⠀⠐⠂⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡔⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡂⢟⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⠳⠄⠀⠀⠀⠈⠁⠒⠀⢤⠄⢒⢍⣢⡷⠖⠛⡁⠐⡖⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⠉⠀⠀⠀⠀⠉⠙⠛⠒⠂⠠⠼⠶⠿⠋⣁⠄⠂⣁⣤⠖⣇⡀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⡜⠙⠆⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣠⡽⣾⠚⠉⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢃⡀⠀⠀⠀⠈⠉⠁⠂⠤⢈⡉⠀⠀⡀⠶⠂⠛⡋⡕⣈⡴⠄⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠠⠴⠒⠲⠤⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠈⠁⠒⠐⠠⠄⣀⠀⠀⠀⠀⠙⡁⠀⣀⣶⠷⠟⠋⠁⠀⠀⠀⠀⠀⠸⠥⠾⠤⠶⠶⠶⠶⠈⣉⣉⣑⡖⠀⠈⢢⣄⣀⣀⣠⣤⣀⣀⣈⣁⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠴⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⠀⠴⠄⡄
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠋⠀⠀⠁⣶⠂
⠿⠓⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⢠⠔⠁
⣾⣶⣖⢸⡟⣳⣶⣤⣄⣀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠌⠀⢀⠉
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠛⠿⠿⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⢰⡴⠁⠀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣴⣶⣶⣿⢯⣭⣤⡤⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣾⣁⣈⠀⢀⣤⣤⠀⢘⡿⣷⢽⣻⣿⣿⣿⣿⣿⣿⣶⣶⣾⣧⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⣤⣤⣤⣤⣤⣤⣄⣀⣀⣰⣿⣉⣀⣀⣀⣀⡙⢒⣾⣒⣺⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⣉⡁⠈⠉⠈⣿⣿⡉⠁⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⡈⠁⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠃⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠⠆⠉⠈⡈⠉⣁⠉⠀⠀⠀⠀⣼⣿⣿⠿⣽⣶⣶⣶⣮⣉⠙⠁⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠴⠀⠀⠀⠀⢀⠀⢲⣂⣀⣤⣤⣶⠂⠀⡀⠀⠁⠘⠛⣩⠉⠿⣿⣿⡆⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠄⠀⣸⠃⠀⠀⠀⠀⠀⢈⡀⠁⣙⡂⠐⠛⢿⢷⣄⡀⠈⠃⣬⣿⡿⠁⢘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢌⣙⠀⠤⣄⣀⠀⠈⠉⠀⠩⠁⣠⡀⠀⢐⠛⢛⣛⡻⠞⠋⣤⡞⢾⣏⣀⣄⣈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠻⣶⣶⠄⠀⠀⠀⡀⠐⠓⠛⠓⠂⣙⡋⠩⣷⣴⡀⢠⡿⣧⣿⡟⠁⠀⠙⣯⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠛⠁⠀⠀⠉⠄⠈⢸⠏⣿⠇⠀⠀⣠⡿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⠶⠶⠛⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠷⠟⠛⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣿⣷⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡴⣿⣗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀⣶⣿⡿⢟⣳⡄⠈⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢘⣿⣉⣻⣷⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⠀⠀⣛⡾⣿⣟⣩⣿⣥⡀⠀⠱⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢁⣯⣽⡟⠉⣻⠏⣿⣾⣧⣤⣀⣀⠀⣀⢀⠀⣀⣁⣠⣶⣿⣫⢊⡽⣒⣿⣿⣷⠞⠀⠀⠀⢘⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣥⡀⣌⠉⢙⠛⠻⠿⠿⠿⠿⠿⡿⠿⠿⣿⣟⢍⣯⣱⣝⠋⠁⠀⠀⠀⢀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠞⠁⠰⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠉⠉⢁⠀⢀⡮⢶⡋⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠤⠦⠤⠤⠤⣄⣢⣤⣄⣤⡐⠿⠯⠁⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠤⠠⠄⠀⠀⠒⠀⠀⠀⠤⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢲⣩⣤⣤⣤⣄⣀⠠⠤⣤⡴⢄⣠⣤⣀⣤⣤⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢻⢇⡛⣿⠛⠙⠿⣿⡟⣠⣞⣠⠷⡿⣟⣿⣿⣷⣾⣃⣀⢄⠼⠉⠤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⠥⣀⡀⢀⠿⠨⠥⠭⠻⠷⠿⠟⢿⡛⢛⡙⣭⡝⠒⢁⠞⠉⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⢶⣶⣶⣾⣭⣉⣀⢀⡉⣀⠁⣉⣉⣩⢤⢶⡮⠀⡖⠯⢀⣀⠀⠀⡟
⠀⠀⠀⢀⠤⣒⣻⣿⣶⣿⣾⡏⠉⠛⣻⣿⣫⣻⣾⣇⣿⣿⣿⣿⢿⣾⢿⣾⡇⠀⣀⢀⣠⣞⣕⣢⣀
⠀⢀⡞⠁⠀⠙⢿⠿⢻⡿⠻⣷⢄⠘⠉⠉⠉⠋⢿⣾⡿⣽⣿⣿⡟⠡⣼⡿⠀⣠⠟⢿⣿⣿⣿⣿⡎⢣
⠀⠘⣅⠀⠈⣙⣶⣷⣮⣻⡿⠿⠙⠠⢄⣱⣤⡀⢀⣹⣤⣿⡿⣿⣷⣿⣿⠐⢿⣷⣿⠉⣿⣽⣿⠟⡣⠏
⠀⠀⠈⠙⠻⠿⣿⣻⣦⣬⣷⣶⣤⣠⣤⠈⠻⢍⣉⣁⢙⡿⣾⣿⡿⠟⢯⣤⣴⣾⠾⣾⣻⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢚⡛⠛⠪⠭⠶⠤⠤⠤⠤⠴⠭⠿⠶⢽⣻⣿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠓⠶⠶⠶⠛⠋⠉⠉

⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⡀⠀⠈⠀⠀⠀⠂⠐⢠⠠⠄⡀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡀⠤⢄⠐⠀⠂⠀⠀⠈⠂⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⡀⠀⠀⢀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⢡
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⢊⡠⠂
⠀⠈⠀⡎⢑⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠢⠀⢁⠠⠀⠀⠀⠀⠐⠒⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠂⠀⠀⠀⡄⢰
⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⢀⠀⠀⣀⠀⠀⣀⠀⠀⡀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⢀⣀⣀
⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢀⠄⣠⠘⠒⠁⡀⠠⡀⠑⠎⢀⠄⠄⠀⠃⠀⡤⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡤⠉⡁⠘⠄⠀⠆⠀⡃⠠⠀⠠⠀⠠⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠔⠉⠢⠀⠀⠠⠊⠑⠄⢀⠀⠐⠃⠀⠀⠠
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠲⢰⠒⠐⠖⠀⠒⠂⠂⠐⡖⠒⠂⠀⠂⠒⠒⠒⠒⠂⠐⠂⠒⠒⠂⠂⠐⠒⠒⠒⠒⠒⠲⠒⠂⡆⠴⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠅⡀⠀⠀⠀⠀⡇⠀⠀⠭⠂⠀⠀⠰⠂⠀⠀⠀⠀⠠⠀⠀⠰⠀⠀⠀⠀⠀⡨⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠀⠀⠀⠀⠀⠐⠐⠀⠐⠅⢀⠔⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⡠⠐⠂⡀⠀⠀⠀⠒⠀⠀⣠⡀⠀⠀⠀⠊⠀⠈⠂⠀⠨⠈⠆⠀⠀⠀⠀⠠⠒
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠐⡆⠀⠀⠀⣆⠀⠨⠒⠒⢄⠀⡁⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠘
⠀⠀⣀⣀⠀⠑⠩⠄⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠⡆⠀⠀⠈⠀⢉⠀⠀⠀⠀⡃⠀⡂⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠂⠲⠁⠂⠀⡇
⠀⠀⠱⠀⠄⠈⡂⠠⡀⠀⠀⠀⠀⡁⠀⠀⠀⢀⠠⠇⢄⠀⠀⡖⠉⢢⠀⢀⠔⠁⠂⠀⠀⠀⠀⠤⠄⡀⠀⠀⠀⢐⠀⠀⠀⠀⢀⠒⠀⡊⢑⠀⣁⠂
⠀⠘⠀⠤⠀⡉⠀⠖⠂⠀⠀⠠⡊⠀⠠⠀⠀⠁⠀⢀⠜⠀⠀⠀⠂⠒⠀⠠⠓⠐⠀⠀⠀⠀⠢⡀⡠⠂⠀⠀⡔⠈⠑⡄⠀⠀⠀⠙⠀⢈⠁⡄⠀⠆
⠀⠀⠀⠈⠁⠈⠒⠀⠀⠀⠀⠀⠈⠆⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠈⠈⠀⠁
⠀⠀⠠⠀⠀⠀⠄⠤⠤⠀⠀⠀⠠⠤⠀⢀⠠⠄⠠⠄⡀⡠⠀⠀⠠⢀⡠⠄⢀⠀⠀⠠⠀⢠⣀⣤⠀⡀⠀⠀⡤⠐⢄⢀⠠⠀⠠⢀⠀⠀⠀⠀⢀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠘⠁⠀⡀⢀⠀⠈⠣⠀⠀⢀⠀⠀⠄⠀⠡⡀⠀⠀⢸⠀⠀⠀⠈⠢⡀⠇⠀⢘⠀⢀⠀⢄⡠⠓⠀⠀⢃⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠞⠀⣠⠀⠀⠀⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠃⠀⢸⠀⠀⠀⠀⠀⠈⠃⠀⢸⠀⠁⡐⠀⠂⠠
⠀⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠋⠀⢀⠀⠈⠂⠐⠁⠀⡠⠀⢀⠀⠐⢇⠀⢈⠓⠄⠀⠀⢸⠀⠀⠀⢸⠢⡀⠀⠀⢘⠀⠑⠀⠘
⠀⠲⣀⡐⠀⠀⠤⠤⠇⠠⠤⠄⠠⠤⠬⠁⠀⠤⠤⠄⠈⠥⠤⠌⠢⢀⠀⡤⠤⠀⠈⠢⠤⣤⠀⢠⠤⠜⠀⠈⠢⠤⠜⠐⠠⠤⠤⠐⠁⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⢿⠿⡿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣿⣿⣿⣻⣿⣟⣻⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⠰⠶⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠖⣊⣉⠭⡤⠤⢤⠤⠬⢭⣙⡒⠦⣀⠀⠻⠷⣀⣀
⠀⠀⠀⠀⠀⡰⣡⣶⣋⣭⣥⣴⣶⣶⣶⣶⣧⣔⠀⠈⠑⠪⡻⣄⠀⠀⠈⠑⣄⣀⣠⠄⠀⠤⠄⠠⢄
⠀⠀⠀⠀⢀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣺⠗⠀⠀⠀⡹⣏⣹⠦⠤⠚⡁⠀⣀⡴⠂⠈⠀⠠⠊
⡴⢊⡭⠭⣙⣿⣌⠻⢿⣿⣿⣿⣿⣿⣿⣿⢿⡇⢀⣠⠤⣚⣹⡟⠋⢀⠄⣸⣷⣚⡉⠉⠀⠀⣠⣆
⣇⢸⡀⢀⡼⣿⣿⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⣍⣴⣿⣿⣿⢧⣤⡷⢞⣉⠁⠀⠀⠀⣴⣉⣠⣄⣙⡢⢄
⠘⢦⡙⠯⣄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣿⠀⣌⢧⡀⠭⠀⢠⢶⡉⠻⠀⠉⠈⠉⠂⢘
⠀⣼⠙⡷⠦⣽⣟⠣⠀⠀⠀⠀⠀⠀⠀⠀⣌⠙⣿⣻⣿⣿⠇⢀⡿⣧⢿⣯⣭⣓⢦⣉⣀⣠⠤⠤⠶⠋⠉
⠀⢿⠠⠂⠀⠀⠙⢦⡀⣠⣄⣀⣄⣠⣀⣦⣿⣼⣿⣿⣿⣿⣶⣄⣠⡿⣻⣧⣩⣿⠟⠁
⠀⠘⢆⠠⠀⠀⠀⠀⠙⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⠤⠆⣱⣿⣍⣧⣽⣀⡀
⠀⠀⠈⠓⢤⡀⠀⢀⡀⠀⠀⠈⠙⠛⠿⠟⠿⠿⠟⠛⠉⠉⢀⣠⣾⣿⠛⢽⣿⣭⣿⢿⡆
⠀⠀⠀⠀⠀⠈⠓⠲⠤⣍⣁⣀⣀⣀⢀⣀⣀⣀⣠⣤⣶⣞⣩⣿⣿⣿⢀⡘⠿⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠛⠉⠉⠉⠈⠿⣯⠭⠶⣿⢿⣝⢧⣀⣾⣯⣽⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣎⢿⣼⠏⠳⠬⡅⠀⠊

⠀⣶⠶⢶⡶⠶⢶⣶⢶
⠀⣿⣏⢩⡅⢸⠛⢝⣿⢠⠄⠀⠀⠀⠀⡀⣤⠀⠀⢠⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡤⠀⠀⠀⢠⡄
⠐⡦⣟⡃⠎⠘⣿⠋⠻⠸⣽⠻⠟⠻⠀⠇⢯⡷⡇⢸⣧⠜⢋⠢⠶⠩⠾⠰⠅⣿⠐⠷⠀⡇⢾⠸⠿⠦⠞⢸⠸⢷⠀⡇⢾⠸⡮⡽⣗
⢰⠓⠀⡱⣆⠨⠉⡶⢽⠀⡄⡀⢀⣀⠀⣠⡀⡀⠀⠬⠁⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠈⠋⠀⠈⠛⠉⠋⠀⠋⠀⠁⠀⠋⠉⠃⠈⠋⠃⠘⠃⠘⠃⠋⠈⠘⠀⠋⠃⠘⠘⠉⠋⠁⠛⠃⠘⠉⠀⠚⠁⠘⠐⠃⠃⠃⠛⠍⠃⠛

⠀⠉⠛⣶⢌⣿⠟⡶⢬⣍⠙⢿⣯⠋⡏⠀⣻⠉⣤⡹⣿⣏⣩⣽⣿⣍⣉⣹⣿⡿⠿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠁
⠀⠀⠀⡈⠋⠉⣉⣻⣶⣿⢟⢳⣋⢂⣦⣯⡀⠙⠉⣁⠂⠉⠛⢻⣅⡉⠉⠀⠉⠁⠒⠇⠐⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠱⠀⠀⠀⠄
⠀⠢⠋⠉⠀⠀⠀⠁⠹⠿⣷⣮⣝⣿⠙⠛⢿⡀⡄⠀⡄⠀⠀⠀⠸⠀⠀⠉⠉⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠀⠀⠀⠀⠀⠐⠟⠿⣷⣽⡟⠋⠡⣤⠀⢻⣧⠁⢀⡃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⠀⣐⣶⣿⣿⣿⣿⣿⡿⢿⣯⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡿⠻⢬⠎⠃⣙⣻⢌⣛⠀⠀⠙⠒⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠘⠛⢋⣭⣍⡉⠉⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠘⢀⠸⠁⠀⠐⠀⠀⢀⣭⡿⣘⣯⠸⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⡍⢻⡄⠀⠀⠀⠀⡃
⠰⠄⢀⠀⠀⠠⡤⠄⠐⣦⠢⣤⠀⠀⣹⣧⠋⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿⠁
⠀⠠⠁⢀⠀⣀⣀⠀⢀⡁⠀⠀⠀⠀⠻⢿⡆⠀⠀⠀⣠
⢀⠠⠀⠁⡜⠋⠉⠗⢀⠁⠀⠀⣰⣿⣟⣀⠀⠐⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⠀⠄⠀⠷⠀⠀⡸⠳⠂⠀⢠⡟⢻⣿⣿⣷⠂⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡟⢧⣄⢀
⢠⠀⠀⠀⠀⠠⠾⠃⠌⠀⠀⠈⠀⠸⣿⣿⣿⣦⡀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠀⠘⠛⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠦
⢘⢲⡀⠀⠸⣦⣄⣾⠁⡆⠘⣦⡀⠀⠻⣿⣿⣿⣿⠓⢄⠀⠀⠀⠶⠞⣁⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣾⡿⣳⣦⢀
⢘⡨⠁⣀⣀⣈⣛⣛⢀⡇⠀⣿⣷⣄⠠⠙⠯⠒⠉⠃⢰⠀⠀⠈⠐⠆⣹⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠸⣇⠻⣷⠾⠃⠀⠀⡀⠀⢀
⠸⠧⡆⢈⠿⠻⠃⣿⣌⠀⠀⣿⣿⣿⡿⢦⣴⣤⣴⡄⠾⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⣿⣶⣶⣷⣽⣿⣿⣻⣿⡃
⢘⠻⠇⡈⢒⡀⠒⢿⡇⠀⢸⣧⣹⡁⠀⠀⠙⠉⠈⠁⠀⢠⡦⢀⠀⢀⠸⠀⠀⠀⠀⡇⠄⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠉⠙⠛⠉⠉⠁
⢘⣻⣛⡙⠻⣧⣗⠺⠇⠀⢸⣿⣿⣧⠀⠀⠘⠀⠀⠀⣆⠀⣇⠟⠀⣴⠃⠀⠀⠀⠈⢟⠃
⠘⠁⢿⣏⣵⣿⣾⣾⠃⠀⣸⣿⣿⣽⠀⠀⠀⠂⠀⠀⣿⡀⠘⠀⠘⠏⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠠⠞⣿⣿⣿⠀⠠⡿⢻⣿⣿⡆⠀⠀⠒⠀⠀⣿⣿⠷⠒⢴⡄⠀⠀⠀⠀⢰⣆
⠀⠀⠸⢃⣬⡿⣿⡏⠀⢸⡇⣾⣏⣽⡟⠀⠀⠀⠀⠀⣿⣿⣇⣐⣈⠉⠉⠀⠀⠀⠈⠙⠓⠓⠒⠀⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠐⠲
⠀⠀⠀⢴⠁⠡⡭⠀⣠⣺⠇⣿⡏⠉⡇⠀⠀⠀⣶⣿⣿⣇⢿⠋⠻⣟⡀⢀⣀
⡀⠀⠀⢸⣴⣶⠇⢰⡟⣿⠀⢸⣧⠀⠃⣄⠀⠀⢫⣤⣿⡟⠀⠀⠀⠈⠁
⠄⠀⠦⠿⠿⠋⣠⠟⠀⢽⠀⠈⠻⠚⢺⣿⣷⣾⢛⡿⠉⢇⠀⠀⣠⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠂⠄⠄⠀⠀⠀⡠⠄⠘⠉⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⣶⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⣀⣀⣀⣀⣀⣀
⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⣀⣤⣄⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠚⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣅⠀⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢤⡈⢓⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠁⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡶⠊⠀⠘⠀⠀⠙⢶⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣿⣋⡁⠀⠀⢣⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⠛⠿⠭⠭⣾⣒⣒⣒⣶⣶⣶⡾⠷⠿⠛⠋⣩⠷⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠞⢻⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡼⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢙⣋⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣏⠒⠤⢀⣀⣀⡀⠀⣀⡀⠤⢒⡏⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣌⣉⣓⡒⠒⠒⠶⠶⠒⠒⢊⣉⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠛⠛⠻⠿⠿⠛⠛⠋⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⢤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠛⠛⠳⠿⠭⠭⠭⢽⣭⣤⡤⠤⠤⠤⠤⠤⠖⠚⢛⡆⢀⡠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⠖⠒⠒⢄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠷⠁⠀⠀⠀⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣳⣶⣿⠀⠀⠀⠀⠀⣸⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣽⢷⣿⡇⠀⠀⠀⠀⡴⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠂⡀⠈⠀⠈⠀⠀⠀⠀⠰⠀⢀⠻⢿⣛⣻⠁⠀⢀⡠⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢠⢟⣒⣭⠵⠞⠉
⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡾⠋⠀⠀⠀⠀⠀⠀⠀⣈⣴
⠀⠀⠈⠙⠒⠶⢶⣤⣤⣤⣤⣌⣧⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣄⣀⣀⣼⣤⣤⣤⣤⣶⣶⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢿⣿⣿⣿⡯⢉⣉⣉⣉⣙⣛⣛⣿⣿⣭⣭⣽⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠶⠶⣦⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣶⣟⡿⠋⡁⠄⠀⣠⡀⠈⠁⠒⠬⢝⡺⢷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠕⣿⡷⠉⢂⠕⠊⠀⠀⠈⣃⠐⠀⠀⠀⠐⢍⠲⣝⢷⣌⠷⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡴⣫⠞⠁⣾⠏⡄⡔⠁⠠⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠈⠎⢿⣦⣤⠖⢘
⠀⠀⠀⠀⠀⢀⡞⡴⠁⠀⢸⣼⠘⠨⠀⠀⠀⠁⡀⠀⠀⣴⠄⡀⢁⠀⠀⠀⠀⠀⠀⠘⣼⡇⣠⢾⣧⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣿⢿⢀⠀⠐⡀⡀⠀⠘⠦⠄⣀⣄⠀⠈⠀⠀⠸⠀⠀⠀⠀⡇⣿⣥⣀⡀⠀⠈⠀⠒⠄
⠀⠀⠀⠀⢰⠃⠁⠀⠀⠀⢻⠸⡈⠌⢆⢳⣕⢄⡀⠀⠀⢈⠃⡴⣋⢔⠵⠁⠀⠀⡸⢸⢹⣿⠟⠻⠉⢗⡶⠦⠤⠃
⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠈⣧⠱⡀⠀⠀⠙⠻⢟⣓⠶⣤⡁⣴⡿⠃⠁⠀⢀⠜⡱⢣⣿⠛⠃⠀⠀⠈⣧
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠘⣷⣙⢦⡈⠲⢄⡀⠀⠉⠢⢱⠇⠀⠀⠀⠔⣡⢞⣵⣿⠁⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⢍⡲⠤⣌⣁⣀⣀⣀⣀⣠⠤⢖⣫⠴⣻⠞⠁⠀⠀⠀⠂⠀⠀⡟
⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢂⠈⠻⡷⣍⣙⡒⠲⠾⠭⠶⠶⠒⢛⣉⡴⠞⠁⠀⡀⠠⠀⠌⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠠⡣⡈⠮⡛⠻⠷⠶⠶⠶⠖⠛⠋⠉⠀⠀⠠⠊⠀⢀⠌⠄⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠓⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⡪⢀⠕⡡⠊⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠂⢅⡀⠀⢀⠀⠈⠀⠀⠀⠂⠉⢀⠁⠊⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠄⣢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣗⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣖⣫⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠯⠭⣿⣿⠿⠭⠭⠷⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠄⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡹⡆⠀⠀⠀⠀⢠⡀⠀⣠⠄⠶⣇⠤⡨⠂⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢀⠴⠚⠋⢙⠀⠀⠀⠈⠀⠀⢗⣆⠀⣨⠅⣀⠄⠀⠀⠀⠀⠀⢠⠏⠸⠄⠈⢊⠀⣄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⢀⠠⠴⠚⠋⠀⠀⠀⠀⠢⠀⠈⠛⠫⠀⢂⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢠⣎⣠⠄⢋⣘⠁⣄
⠀⠀⠀⠀⠀⠀⣀⣠⠄⠴⠋⠀⡀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣀⠀⠀⡤⢯⡇⠈⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠰⠛⣉⠔⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠦⠰⠚⠁⠸⣏⠲⠀⠀⠀⢘⡇
⠀⠀⠀⠀⣰⣎⣁⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⣒⠂⠀⠀⠈⠀⠀⠀⠐⠒⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡔⢢⣤⠛
⠀⠀⠀⠀⠙⢿⡧⠊⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⠭⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠳⣄⡀⢀⠠⢄⡤⠤⣄⣠⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⣀⣴⡯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠘⡆⠑⡶⠉⠑⠂⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠍⠚⡅⢰⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠙⠷⠚
⠀⠀⠀⢠⣍⣤⠞⡙⠠⡤⡌⠄⠀⠄⠀⠀⢱⠘⡧⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⢣⠏⡠⠊⠙⡆⢳⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠚⢣⡼⠲⠔⠼⠀⠀⠀⠀⠀⢃⢧⠈⠑⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⡀⠤⠤⠒⠊⠉⢀⣮⠏⠉⠁⠀⠜⢁⠌⠀⠀⠀⣠⠘⢃⠀⠙
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣟⡠⠐⣀⠄⠂⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⣡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣠⣞⠋⠉⠉⠉⠉⢉⣁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣟⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢹
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⣀⣈⣉⣢⣶⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣫⡄⢸
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠛⠋⢉⡴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⢐⡇⢸⡆
⠀⠀⠀⠀⢀⡃⣀⣀⣀⡰⠂⠀⢠⠎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⢻⣟⠁⡈⠀⠈⡇⢸
⠀⠀⠀⠀⠈⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣿⣷⠿⣡⣶⡽⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢠⠈⠁⠱⣈⠳⡴⢁⡎
⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡹⢉⣺⣷⣤⣄⠀⠀⠀⠀⠴⣒⠉⠡⣀⣆⠀⣬⣀⠀⣲⠿⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢸⠅⠒⠀⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⣵⠿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣉⣉⣀⣉⣉⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⢠⠄⠀⠄⠠⠄⠀⠄⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡴⣟⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⣥⠾⢋⡄
⠀⠀⢀⣀⣀⣀⣀⣀⡄⡠⠚⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣡⠾⣛⠴
⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⣄⣀⣤⣤⣤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡥⢚⣥⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠋⠥⠴⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠻⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢿⡘⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠰⡄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⡝⢿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣡⣿⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠀⡇⢲⠚⢓⣲⡶⠤⠤⣄⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣛⣻⡇⢾⢭⣥⠭⠿⣛⣛⣛⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡤⣼⣧⣤⣬⣁⣀⣙⣆⣒⣂⣀⣀⣠⣧⣤⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢛⠁⠙⠛⠶⠾⠿⠮⠿⠭⠿⠿⠿⠿⠙⠛⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⠉⡔⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⡾⣩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⢦⡳⣀⣿⣿⣿⢿⢿⢿⣿⡿⠿⠿⡿⢿⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠳⣽⠷⠀⠀⢀⣀⡀⡀⡀⣀⣀⣀⢀⡀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠲⠽⣿⣿⡶⣒⡶⡾⡾⣶⢾⢮⣽⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣮⣝⣂⠀⠀⠹⠌⣊⣡⢩⢉⣈⡱⠃⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠴⣒⠈⠻⢷⣿⣶⣶⣿⣷⣶⣾⠿⠋⣀⣒⢦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⣫⣽⣷⣄⡈⠀⠁⠁⠀⠈⠈⠁⣠⣾⣯⢍⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠤⠤⠬⠭⣭⡭⣭⠭⠤⠤⠴⠞⠛⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣮⣤⣚⣛⣛⣓⣒⣒⣒⣒⣚⣛⣛⣛⣤⣤⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠄⠒⠂⠈⠉⠉⠁⠒⠒⠒⠤⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⡠⠤⣖⡒⠂⠉⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠓⠒⠲⠶⠤⢤⣤⣀⣀
⠀⠀⠀⢸⢡⠀⠀⠀⠉⠉⠓⠒⠲⠤⠤⣄⣀⣀⠂⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⡟⠁
⠀⠀⠀⠸⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⢤⣄⣀⣀⡀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠸⡇
⠀⠀⠀⣀⣑⣲⣤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠷⠓⠤⢤⣄⡀
⠀⠀⠀⠙⡟⠲⠤⣄⣀⠀⠈⠉⠙⠒⠲⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠐⠚⢷⣦⣀⣀⣤⡶⠿⠿⠗⠒⠒⠊⠉⠉⠀⠸⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠒⠤⢄⣀⠀⠀⠀⠀⠈⠉⠉⠛⠓⣶⣦⣤⣄⣀⣈⣇⣠⣤⡶⢿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⡶⠶⠒⠛⠋⠉⠉⠀⠀⠀⠉⠉⠙⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣀⣈⡿⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⢶⣿⣶
⣰⡿⠛⠳⠷⣦⣭⣛⣿⣶⣤⣀⠀⠀⠀⠀⠀⣼⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠔⠒⢒⣋⣩⣭⡥⠶⠶⠛⠛⠋⢹⠃
⣿⠁⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣶⣤⣀⣠⠏⡄⢀⣀⣀⣤⣤⠶⠶⠒⣚⣋⣩⣭⡤⠶⠖⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣯⣭⣴⡶⠶⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣘⣦
⠀⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣖⣷⣀⣀⣤⣤⡶⠶⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡛⠷⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠒⠶⢤⠀⢠⡀⠙⣿⣧⡄⠀⡀⠀⢠⠊⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣼⣇⠀⣠⡀⠀⠀⠐⠋⠀⠀⠉⠁⠀⢀⡀⣀⡀⣤⣥⣄⣏⠺⠞⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⡀⠀⠙⠄⢿⡇⠀⠀⢠⡴⢤⣤⣴⣶⣶⣿⣿⠁⠀⠀⠀⡿⠛⠷⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⣿⣿⠛⠂⠘⢄⢀⠀⠈⠃⠀⠈⣿⣻⡿⢿⣿⣧⡀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠃⣠⡄⠀⠀⠉⢸⣄⠄⠁⠀⠸⢿⡻⣾⡏⠉⠉⠂⢸⣧⣼⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⠿⠘⠃⣱⢾⡿⠁⠀⢹⣄⣀⡀⠐⣻⠀⠀⠀⠀⢠⡶⠾⣿⣿⢠⠦⠀⣀⣀⣐⡀
⠀⠀⠀⠀⣠⣴⣶⡬⠚⠛⠋⠉⠉⠸⣿⣧⠀⠘⠅⠀⠀⠴⠻⠏⠀⠀⠀⠿⣯⣀⣀⣀⣤⠆⠀⢀⡛⠙⠀⢰⣾⣿⣿⠿⠿⣷⣶⣤⡀
⠀⠀⣴⣶⣼⠋⠀⠀⠀⠀⠐⠢⢴⣎⣿⣿⣷⣄⠐⠷⡀⠀⠀⠀⠀⠀⠀⠈⠁⠋⣙⣿⡦⠀⢀⠈⣠⣤⣤⣼⣿⣿⣿⡇⠀⠀⠀⠙⢳⣤⣤
⠀⠀⢻⣿⡁⠀⠀⠠⢄⣸⠗⢀⣀⠋⠉⠙⢿⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⣀⣠⣴⣾⣿⠿⣿⣏⣀⣘⣿⢁⠀⣰⠂⠀⠀⡟⠻⠆
⠀⠀⠀⠉⠳⠦⣄⠀⢸⠟⠲⠥⠀⠀⠀⠀⠀⡟⣿⣿⣿⠌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⣿⠳⠀⠀⠉⠉⠘⣡⠾⠯⢃⡵⢘⣾⠖⠛⠁
⠀⠀⠀⠀⠀⣠⣤⣉⠙⠲⠤⢐⣂⡀⠀⠀⠠⡀⠙⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠀⠀⠀⠀⠀⠀⢁⠴⠤⠅⠘⡉
⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣶⣶⣦⣤⣤⣄⣀⣈⣓⡒⠒⠀⠤⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠐⣒⣂⣀⣀⣠⣭⣤⣴⡶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⢛⣋⣭⣿⣿⠿⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⢿⡿⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⠶⣶⠶⠶⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣠⠄⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣼⡏⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢻⣧⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠻⣷⡀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢹⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⡼⠃⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠀⠒⠒⠂⠒⠒⠒⠒⠐⠒⠒⠒⠒⠐⠒⠀⠠⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡖⢉⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡍⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡰⠁⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⠤⠊⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⣀⣀⠤⠐⠁
⠀⠀⠀⠀⡠⠄⠒⠒⢈⣉⣭⣤⣬⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣧⣤⣤⣤⣉⡁⠒⠒⠠⢄
⠀⠀⠀⠀⠢⢄⣀⠀⠛⠿⠿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⢀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⢤⡀⠭⠭⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠭⠭⢀⡤⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠂⠀⠤⠤⠤⠤⠤⠤⠀⠐⠒⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣀⠀⠀⠀⢠⠶⡄⠈⠑⠦⡀⠀⢦⠀⠇⠀⡖⢀⡜⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠢⡄⠀⠀⠳⠃⠀⠈⠙⠒⢲⠟⠊⠁⠀⢀⡀⠈⠂⠀⣀⣤⣄⣀⠉⠴⠊⢀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣞⡉⢹⠓⠒⠒⠁⢀⣄⠀⣖⣆⡀⣀⣀⡜⠀⠤⣀⣀⣀⣉⠁⢠⠞⠁⠀⠀⠈⢳⡀⠊⣁⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⢀⡄⣞⢹⢿⣀⡏⢼⢁⡴⣇⣄⠀⠀⠀⠠⠤⠄⢸⠀⠀⠀⠀⠀⠀⠇⠈⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡠⠛⡇⠈⠁⠀⠉⠀⠈⠉⠀⠓⠚⠀⠀⠰⣶⣀⠀⠀⢀⣄⠀⠀⠀⢠⠖⣦⠀⣠⠲⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⣀⠀⠀⢀⣰⠉⣆⡀⢰⣶⡴⠋⡼⢀⣤⡏⢸⢀⢤⠈⡟⢠⠇⢀⡧⠾⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢩⣳⠦⣼⡀⠛⣨⠇⠀⠈⢿⠐⠃⡇⢺⡴⠃⡼⠁⡾⠈⠃⡏⡼⠸⠀⠃⢸⣠⠞⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣁⠴⠋⠉⠉⠀⠀⠀⠀⠈⠀⠸⣀⠎⠀⠀⢧⣠⠃⠀⠀⣙⣁⣀⡀⠀⣀⠤⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣠⣄⠀⠉⠳⡧⠤⠏⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣁⡼⠀⠀⠀⠙⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠀⠀⠉⠉⠉⠉⠀⠀⠐⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡨⠐⠂⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⡘⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠠⢐⠡⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡨⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠢⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠐⠁
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡴⠖⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠄⠐⠀⡀⠀⢀⡀⠀⣀⣀⣀⣀⠀⠀⡤⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛
⠀⠀⠀⠀⠀⠀⢡⣬⣭⠥⠤⠤⠤⠄⠀⠀⠐⠒⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠈⠻⡄⠀⢀⢀⡴⠛⠉⠀⠀⡀
⠀⠀⢀⠀⣈⠀⠀⠀⠀⣄⣙⣘⠀⢀⡠⠤⠀⠀⠀⢠⣐⣲⠶⣾⠀⠀⠘⢿⡵⠋⠀⠀⡠⠀⠀⠀⠀⠂
⠀⠀⡁⠀⡇⠀⠰⣆⣀⡏⢹⢨⠖⢰⣧⣤⣄⠀⠀⠜⠿⠋⠀⠸⠲⡒⠒⣦⠀⠀⠠⠪⣀⠀⠀⠀⠀⠀⠀⠈⠐⡤
⠀⠀⠃⠈⠁⠀⠀⠉⠀⠁⠈⠉⠀⠈⠀⠀⠐⡋⠀⡇⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⢈⣲⣤⡁⠂⢀⠀⠀⢀⠔⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⣸⠂⠀⠀⠀⠈⠈⠈⠀⠀⠀⡊⠳⢤⣉⡓⠦⣌⣒⣤⡞⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⢀⡀⡧⠄⡀⠀⠀⢐⠙⠳⢄⡈⠉⠓⠢⢍⣑⡴⠃⠀⠂
⠀⠐⡷⢾⡄⠠⢤⣀⣀⣾⡇⢰⣶⣶⣧⣤⡇⠀⠀⡄⡀⢹⣻⠇⠇⠉⠁⡀⠀⠀⠑⢄⠀⢝⠲⢄⡀⠰⢈⡴⠚⡠
⠀⠀⠀⠐⠃⠄⠜⠀⠐⠋⠓⠘⠂⠚⠃⠂⠓⠐⠒⠋⠀⠋⠀⠁⠁⠀⠀⠁⠀⠀⠀⠀⠑⠀⠑⠠⡉⣛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠕
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠓⠤⠀⠈⠁⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⡴⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⢸⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⢸⡇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠇⠀⠀⠿⠴⠶⠶⠾⠿⠛⠓⠶⠶⠶⠖⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣽⡃⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⣤⣴⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⠀⠀⠀⡼⠉⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⢀⠞⠀⡄⠀⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⢸⣠⣋⣀⡀⢿⠞⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⢸⡿⠿⢋⡴⠁⠀⣀⡤⠤⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⢡⣄⢸⢀⡴⡫⢄⢀⣎⠀⠀⠀⠀⠀⠀⠀⢀⣉⣩⡗
⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠛⡟⠁⢸⡄⠀⢹⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⡠⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠾⠿⠟⣇⠀⢸⠃⠀⠸⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠒⠲⠺⡦⠶⠶⠷⠴⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⢀⠴⠊⠱⢄⡈⢇⢄⠀⠀⠀⠀⠀⠠⡖⢉⣉⣉⠉⢳⡄⡇⢄⠀⠀⠀⡼⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⢡⡇⡆⠀⠀⠀⠉⠉⠻⢷⣀⣀⣀⣀⣀⡏⠉⠉⠉⠉⢹⠁⠘⢢⣴⣤⠞⠁⠀⠀⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⣌⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠹⡰⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡠⠄⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈⠛⠳⢶⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠘⠛⠛⠃⣀⣀⣀⣠⣤⣬⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠭⠭⠿⣛⣛⣛⣛⣛⣛⣛⣛⡻⠯⠭⠽⠛⠋