Sip Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⢔⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣹⣾⣿⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⣿⣿⣿⣅⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣬⣿⣿⣿⣟⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣽⣿⣿⣿⣿⣷⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⠟⢻⡋⠙⣿⣿⣿⣿⣷⣝⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠹⣶⣾⣿⣿⣿⣿⣿⣿⣦⣾⣋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡟⢿⠻⣿⣿⡛⠿⣄⢹⣇⢈⣿⣿⡟⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⡛⣿⡿⠶⣿⣶⣿⣿⣿⣿⣿⣿⢿⠷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡿⣿⣷⣤⣼⣿⣿⠛⣿⣺⣷⣄⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣇⠙⢙⣿⣇⣸⣿⣿⣾⣿⢿⣹⣿⣻⣿⡜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⢺⣹⣿⢻⣿⣿⣿⣿⣿⣟⣿⣽⣿⢫⢾⣟⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠻⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣶⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⠟⠛⠛⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠻⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢷⡄⠘⢷⡀⠀⣻⠀⡼⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣱⠀⠀⣟⣠⠏⠈⣧⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠃⠀⠸⠛⠁⠀⠀⢹⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢋⣩⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣬⡙⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠙⣿⠉⠉⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠙⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⢀⣰⠏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠋⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠿⢷⣶⠶⠶⠶⠶⠶⣶⣾⠿⠧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠛⠶⣦⡒⠒⢶⣤⠶⠞⠃⠀⠀⢈⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣯⣭⣭⣭⣭⣉⣉⣩⣭⣩⣭⣭⣽⡿⠟⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢸⡇⣤⠀⣠⢀⣤⣄⠀⠀⠀⠀⠀⣴⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⢀⣴⣿⣿⡿⠟⠋⠉⠛⣿⣷⠀⠀⠀⣾⡗⠺⢻⡿⢸⡇⢿⣤⡟⢹⣷⠟⠀⠀⠀⢀⣴⣿⠾⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣀⠀⣰⣿⣿⣿⠋⠀⠀⠀⠀⣀⣾⣿⠀⠀⠀⠘⣷⣠⡾⠃⠘⠃⠘⠿⠁⠈⠻⠟⠁⠀⠀⢹⡿⠀⠀⠻⣧
⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⠿⣶⢶⡾⠿⠿⠟⠋⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠈⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠙⠻⣦⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⣀⣤⣴⣶⣶⣤⡀⠀⠀⠀⢻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⢻⣿⣿⣿⡄⠀⢠⣤⣥⣄⠀⠀⠀⣠⣤⣤⣄⠀⠀⣀⣤⣤⣤⣄
⢤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣤⣿⠟⠉⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⢸⣿⣿⣿⠀⠀⢀⣄⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⢸⣿⣿⣿⠁⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⡟⠀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠈⣿⣿⣿⠄⠀⠀⣰⣿⣿⣿⣿⢠⣶⠿⣿⣿⣿⣿⠿⠿⠟⠋⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣴⣿⡿⠋⠀⠀⠀⠀⢻⣿⣿⣧⣤⣾⡿⢻⣿⣿⣿⣿⡇⣰⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⢉⣴⣿⣿⣿⣿⠙⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣼⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡇⠀⠙⢶⣄⠀⣀⣀⣀⣠⣤⣤⣤⣤⣀⣀⣀⣴⠟⠁⠀⠘⣧⣷
⠀⠀⠀⠀⠀⢀⣠⣤⠤⣴⣿⡟⠀⣀⣤⠴⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠶⣤⣀⠙⣿⣦⣤⢤⣄⡀
⠀⠀⠀⣠⡴⠋⠁⣀⣾⠛⢻⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⣀⣿⣦⣌⣙⣦⡀
⠀⠀⢰⣿⠶⣟⣻⠿⠛⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣍⠛⠷⢬⣽⣿⡄
⠀⠀⡟⠹⣏⠉⠀⠀⣸⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⣼⠋⢿
⠀⠀⡇⠀⠙⢷⣄⢀⡿⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣴⠟⠁⠀⣸
⠀⠀⣿⠀⠀⠀⢉⡿⠀⠀⠀⠀⠀⠀⢀⡏⠙⠳⣤⣀⣀⣙⣷⣤⣀⡀⠀⠀⠀⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣀⠀⡿
⠀⠀⢸⡆⢀⣴⠟⠁⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠉⣩⣉⠉⠀⠉⣽⣛⠛⠛⠋⠁⠙⠳⢦⣀⠀⠀⠀⠀⢀⣤⠟⣸⠇
⠀⠀⠀⣿⠘⢷⣤⣤⣤⣤⡴⠶⠛⠁⠀⠀⠀⠀⠀⠈⢃⣄⠀⠀⠀⣠⠙⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢻⡟⢷⠀⡿
⠀⠀⠀⢸⣧⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣀⣀⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣿⠇
⠀⠀⠀⠀⣿⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⣿
⠀⠀⠀⢀⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡀
⢶⣤⣴⠞⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⢶⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠈⠳⢦⣤
⠈⠻⣦⡀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣴⠟
⠀⠀⠈⠛⢦⣤⣀⣀⣈⣻⣦⡀⠀⠀⠀⠀⠀⠀⣠⠶⢦⣴⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠙⠷⣤⣀⠀⢀⣴⠿⠀⠀⠀⠀⠀⠀⠹⢦⡄⠀⠀⠀⢀⣠⡴⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣾⣷⣶⣤⣤⣤⣤⣤⣤⣴⢾⣷⡴⢶⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⣹⠿⡉⠙⠋⠉⠉⠉⠉⠙⠛⠁⣽⠶⠛⢿⡟⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢠⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣷⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣾⠁⠸⣇⣀⣤⣄⠀⠀⠀⠀⠀⣠⣟⡀⠀⠀⢿⡆⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠻⣶⠞⠋⠉⠀⠹⣆⣀⣀⣀⣴⠏⠈⠙⠛⢦⡼⠟⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⣡⣤⣿⡁⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡷⠦⣼⣧⣀⡀⠀⠀⠀⠀⣈⣉⣹⡆⠀⠀⠀⠀⣀⣾⣥⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠙⠛⠦⣤⣀⢉⣿⠉⣀⣠⡴⠞⠛⠉⠉⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⡬⠿⠛⠿⢟⣉⠁⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠋⠉⠛⠛⠛⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠓⠶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠶⠞⠛⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢻⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣀⣀⣀⣠⣤⣤⣤⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠾⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣜⠛⠛⠛⠛⠛⠛⠛⠛⣡⣿⣦⡙⠛⠛⠛⠛⠛⠛⠛⢛⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣛⠶⠶⢤⡤⠶⠶⣛⣥⠟⢦⣝⡲⠶⣤⣤⣤⡴⢖⣫⡽⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⣶⣦⣤⣄⡀⠀⠀⠀⣀⣀⠀⠀⣠⡀⣀⣤⣄⠀⠀⢀⣰⣿⣀⡀
⠀⠀⢀⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠙⢿⡄⣴⡟⠋⠙⢿⡆⣿⣿⠟⠉⢻⡇⠀⠈⢹⡿⠉
⠀⠀⠻⠿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⢀⣠⡿⠃⣿⠀⠀⠀⢸⡿⣿⡟⠀⠀⢸⡇⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠿⠿⠛⠉⠀⠀⠙⠷⠶⠶⠟⠁⠻⠃⠀⠀⠸⠃⠀⠀⠈⠻⠾⠛⠀⠀⠀⣠⣾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠛⢶⡄⠀⢀⣶⠓⠆⣤⣠⡄⠀⢀⠆⣀⣀⠀⠀⠀⣰⡞⢆⠀⠀⠀⠀⠹⡟
⠀⠀⠀⠀⠀⠶⠶⢦⣄⠀⠀⢠⣾⣿⡟⠀⠀⣸⠇⠀⢸⣿⠀⣤⣿⡏⢿⣄⠎⣾⣇⣨⠃⠀⠀⣿⡧⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⣾⣿⣿⡇⠀⣴⠏⠀⠀⠈⠻⠴⠃⠿⠃⠈⠛⠀⠙⠯⠖⠃⠀⠀⠿⠀⠘⠇
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠳⣿⣿⣿⣷⠞⠁⠀⠀⠀⣴⣶
⠀⠀⢀⣼⣿⣦⡀⠀⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠚⠿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⣀⣤⣤⡀⠀⠀⠀⣿⣿⣿⠀⠀⣠⢶⣶⣦⡀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠘⡿⠁⠀⠀⠀⢀⡀⠀⠀⢹⣿⣿⣷⠀⠀⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⢀⡾⠁⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⣿⣿⡄⠀⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣼⠃⠀⠀⣿⣿⣿⠃⢀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⣿⠀⠀⣿⣿⣿⡄⠀⢀⣾⣿⣿⣿⣏⣴⠶⣻⣿⣿⡟⠛⠋⠁
⠀⠀⠀⠀⢀⣤⡀⠀⠀⢹⣄⠀⢀⣴⣿⡿⠃⠀⠀⠸⣿⣿⣧⣤⠟⢹⣿⣿⣿⢿⣁⣴⣿⣿⠟
⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⣿⣿⣿⣿⠈⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⣦⠀⠀⠀⡴⠋⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣿⡥⠋⢀⣴⡆⢀⣾⡞⣠⢴⣿⡷⢦⣤⣤⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠉⠁⠀⢹⣷⡴⠛⣿⢁⣼⣿⡷⣧⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⡞⢿⣷⠀⠉⠋⠈⢿⡇⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡼⠟⠀⠀⠀⣠⡄⠉⢀⣀⠀⠘⣯⣏⣀⣀⠀⠀⠀⣠⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣆⠀⠀⢀⡀⣀⡀⢠⣦⠻⣏⣀⣿⡭⡧⠖⠁⣸⣿⠃⠀⣠⡾⠛⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠖⠒⠚⣿⠁⣠⣴⠟⣹⣿⢁⣾⡇⣸⠁⢸⡟⢠⠃⠀⠀⠻⣇⡠⠞⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⣠⢾⣿⠀⠻⠚⠘⠷⠎⠸⣷⠃⠀⠀⠉⠁⠀⠀⠀⢀⡤⠴⠤⣤⣿⣯⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⢀⣤⣄⣤⡄⢠⣿⡟⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⡆⠉⠉⠁⠀⢀⡀⠀⢀⣀⡴⢻⣆⢠⣿⣯⠞⣰⣿⢋⣼⡿⢡⣾⡿⠀⢀⣴⠾⠋⠉⠉⠃
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣼⣯⠴⢃⣴⡤⢠⣾⣏⡽⢠⣾⡟⣀⣾⡟⢻⣿⠀⡰⠻⠷⠋⠸⠷⠃⢻⣇⡴⠛⠁
⠀⠀⠀⠀⠀⢸⡟⠉⠁⠀⠀⠙⠛⣿⡿⠁⡾⣿⣉⠴⣿⡿⢞⡽⠋⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡜⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⣀⠁⢀⡆⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠘⢧⡀⠀⠀⠀⣠⡄⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡔⢺⡏⢀⣼⡷⣫⣴⠏⠁⢀⡀⢀⣀⣠⣠⠇⠀⠀⠀⠀⠹⣦⡤⣺⠟⢁⣠⡿⠋⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢻⡸⠤⢊⡿⠀⠈⠁⠀⠀⠛⠇⠛⡿⠧⢻⠀⠀⠀⢀⡴⠊⠉⣷⠘⠒⠋⠈⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⡾⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⡴⠃⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⡜⠀⢀⣠⡴⠛⠁⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠉⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣮⣶⣦⣼⣾⣾⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠄⠠⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠲⠴⠤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⠔⠀⠀⣠⠞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⣀⠔⠁⣀⠔⡩⠊⠀⠘⣄⠱⡄⠣⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⢀⡠⠔⠊⢀⡠⠊⡡⠊⠀⠀⠀⠀⠈⢦⠈⠢⣈⠢⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠠⠤⠒⠋⠁⣠⠴⠚⠁⠴⠊⠀⠀⠀⠠⣶⣦⡀⠀⠑⠦⡈⠁
⠀⠀⠀⠀⠀⢀⠔⠉⠈⠑⢄⠀⠀⢀⣀⠤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⠑⠦⠒⠀⠐⢢⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠂⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠹
⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠈⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⠴⠤⠐⣤⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢀⣣⣀⣀⠀⠀⠀⠀⠀⢣⡀⣀⠤⠤⠤⢄⣠⠎⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠑⢄⢀⠴⠊⠉⠀⠀⠀⠈⠳⡀⠀⢀⡀⠙⠁⠀⠀⠀⠐⢣⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠺⢄⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⢀⡜⣁⣀⣀⠀⠀⠀⠀⠀⠀⣹⢀⣀⣀⡀⠀⠀⠀⢀⡠⠊⣀⣀⠤⠔⠋⠀⠀⠀⠀⢀⡇⠀⠉
⠀⠀⠀⠀⠀⣖⣉⣁⠀⠠⠄⢣⡠⠄⠴⠴⠒⠋⠀⢤⣬⣍⣉⣹⠉⠀⠸⡁⢀⣀⣀⣀⠠⠤⠄⠒⠈⠁⠀⠀⡤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢸⣀⡠⠄⠒⠀⠈⠉⠀⠀⠠⣄⠘⠢⠤⠐⢃⡄⠉⠉⠀⠒⠦⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠐⠂⠁⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⢰⡃⠀⠀⠀⠀⡰⠆⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠑⢦⡀⢹⠀⠀⠀⠀⣜⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡜⠁⠺⠀⠀⠀⠀⠘⡤⠁
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠢⢀⣀⣀⡰
⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⢀⡴⠆⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠛⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢁⡀⠀⠀⠀⠀⠘⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⡰⢡⠃⠄⠀⠀⠀⠀⠀⡀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠰⢀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠂⢁⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⢡⠐⠢⢄⠠⠀⠰⢀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠃⠠⠨⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠤⠛⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠲⡤⣄⣠⣄⠀⠀⠀⠈⢡⡇⠈⢳⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⡇⠀⠉⠉⠲⠒⠒⠖⠁⢀⠀⠀⣄⠀⠠⠀⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣬⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠠⣿⣿⣦⠀⢻⠇⣠⣶⣿⣿⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⢀⣀⡈⠀⠀⠀⠈⠛⠻⡇⠀⣰⠿⠛⠋⠁⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⠻⡇⢀⣴⣾⠛⠉⠉⢳⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡴⠒⢶⣄⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠛⠃⠀⣾⣿⡇⠀⠀⠀⣸⠃⠀⠀⣿⡿⠀⠀⠀⠀⢠⡟⠀⠀⣸⣿⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⣤⣿⣿⣧⣤⡤⠶⠋⠀⠀⣠⣤⡀⠀⠀⣾⣷⡞⠀⠀⠀⣿⣿⠁⠀⠟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠋⠁⠀⠀⠘⢿⣿⣦⠀⠀⠀⣠⠾⣿⣿⠁⠀⣸⣿⣿⢁⣀⣀⣸⣿⡟⠀⠀⠀⢀⣼⣿⠆⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣠⣤⠀⠀⠈⢻⣿⣷⣄⠜⠁⠀⣿⡟⠀⢠⣿⣿⣿⡟⠁⣰⣿⡿⠉⠙⠓⠻⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠰⠆⠀⢙⣿⣿⣆⠀⠀⠻⣷⡠⠏⢸⣿⣧⠙⠾⠟⠋⠀⠀⠀⣴⡷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠰⠆⠀⠀⢠⡞⠙⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⣠⡤⣤⡀⠈⠀⣀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⡀⠀⠀⢸⡀⢀⣿⠟⠀⠀⢠⣤⣶⣶⣄⠘⢿⣧⠀⢰⡏⠀⣾⣿⠀⠀⣿⡿⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⣾⣿⡿⠀⠀⠈⠙⠋⠁⠀⠀⠀⣿⣿⠉⠉⢸⠀⠀⠀⠀⠘⣧⣾⣿⠏⠀⠀⡿⠃⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⠀⠹⣿⠁⠀⠀⣀⣤⣤⣤⣀⠀⠀⠹⣿⡀⣠⠏⠀⠀⠀⢀⣾⣿⡟⢡⣤⠀⢀⠀⠙⣛⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⠁⠀⠀⠀⢠⣾⣿⠋⠀⠀⠙⡆⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿⣿⠏⠙⠢⣿⣀⠟⠀⠈⠙⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠉⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣰⠇⠀⠀⣿⣉⣁⠕⠀⠀⠀⠹⢿⣄⣠⠔⠁⠀⠀⠀⠀⠀⡀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠁⣠⣶⣿⡿⠿⠷⠺⣿⣷⡶⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⢾⣷⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠋⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⣴⣾⠟⢳⡆⢠⣾⣿⡟⠉⢩⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠈⢿⣿⣆⢀⣸⣿⣿⠤⠞⠁⣼⣿⡿⠤⠶⠛⠁⠀⣠⣶⣾⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠟⠀⣠⡴⠾⢿⣿⡏⠈⢿⣧⣀⣠⠞⢻⣿⡇⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⠀⢰⠃⠀⠀⣼⣿⠇⠀⠀⠀⢀⠀⠀⠀⠛⠷⢤⠴⠞⠉⠀⠀⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠾⠛⠉⠀⠀⠀⠰⣿⠗⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠋⠉⢻⡟⠉⠻⣿⡿⠋⠙⢿⠋⠉⠛⣿⡟⠙⠛⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠀⠘⠃⠀⠀⡿⠁⡀⠀⠀⢂⠀⠀⠙⠀⣸⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⢸⡇⠀⠀⠃⢀⣅⡀⠀⠈⣆⠀⠀⣰⣿⡀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣿⣿⣶⣾⣷⣿⣿⣿⣶⣾⣿⣷⣾⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⡿⠛⠛⡿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠁⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠐⢶⡆⠀⠀⡇⠀⠀⠀⠶⠀⣹⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠁⢘⡃⠀⠀⠸⣿⣿⣿⣿⠓⠆⠀⠀⠀⡁⠀⠀⡇⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣶⣶⣿⣿⣿⣿⣷⣶⣤⣶⣾⣷⣶⣶⣿⣶⣶⣾⣿⣿⣿⡿⠟⠋
⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣔⠻⢿⣿⣿⣿⣶⣦⡀⣀⣀⣀⣠⣴⠶⠂⠀⠀⠐⠶⣦⣔⢶⣴⣦⣶⣶⣶⣶⣠⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢺⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣿⣛⢿⣿⣿⣿⠻⠿⢛⣻⣿⣯⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⣌⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠒⠋⠻⠿⣟⣿⡮⠀⠀⠀⠀⣰⣿⡿⣷⣶⢶⣤⠀⠀⣸⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠉⠀⠀⠁⠈⠉⠀⠠⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⣀⠀⣠⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠰⠋⠁⣀⠀⠈⠻⡿⠿⠃⠈⡂⢀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⢷⠖⠒⠒⣒⡈⣉⣉⣉⣘⣛⠛⠛⠒⠤⠤⣤⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠉⠉⠉⠉⣃⡙⠉⠁⠉⠁⠁⠀⠈⠉⠉⠙⠙⠋⠓⠒⠤⢌⣑⡆
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠤⠐⠒⠈⠉⠉⠉⠙⠛⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠃
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠠⢴⠒⠚⢉⡉⠖⢂⡦⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢙⡿⠋⠀⣠⠴⣶⣴⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠈⠻⣿⡿⠦⠀⠠⢤⣤⣤⣀⣀⡀⠀⢀⡴⠃
⠀⠀⠀⠀⡠⡞⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠛⠶⠤⢤⣤⣀⣴⣿⡿⠿⠟⠁
⠀⠀⠀⣰⢹⡇⠀⠀⠀⠀⠀⠀⡞
⠀⠀⣠⡟⠀⣧⠀⠀⠀⠀⢀⡜
⣤⣾⣿⣷⠀⠊⠳⣴⢤⣶⠏
⠻⣿⣿⠟⠦⠠⠀⢈⠙⡝
⠀⠈⠙⠁⠀⢀⡴⠷⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⢀⣴⠿⣿⣿⣿⣿⣿⠏⠀⣠⠶⠶⣌⠛⠙⠻⠟⠛⢩⠤⢭⠀⠈⣿⣿⣿⣀
⠀⠀⠀⠀⡟⣴⠶⠘⣿⣿⠟⠁⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⢻⣿⢸⠇
⠀⠀⠀⠀⣧⡉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡤⠀⠘⠛⠀⠀⠀⠀⣾⣡⠞
⠀⠀⠀⠀⠈⠛⠻⣯⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠉⠁⢀⡴⠀⠀⠀⢀⡼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⡴⠋⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣤⣤⣄⣤⣼⣿⣧⠴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣀⠀⢈⣿⣯⣧⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⣿⣦⡙⠻⠿⠿⡿⢟⡁⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⣄⠈⠛⢿⣷⠶⠒⠻⣝⣯⣠⢄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡉⠙⢻⣿⣷⣤⣏⡀⢀⣀⡿⢿⣀⣠⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠿⣿⣿⣄⡉⠛⢿⡏⠉⠛⢟⡻⠉⢹⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠘⢿⣻⣿⠿⢺⣏⣛⡒⠒⠒⠚⢹⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣀⡀⠀⢐⣺⡅⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣯⣭⣧⡲⡝⠃⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⣇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣄⣠⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⡴⠛⠋⠉⠙⠛⠻⠿⣶⣶⣿⣧⣀⣀⣀⣀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⣼⣿⡟⠛⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⣸⠃⠀⢸⣿⣇⠀⠀⠀⠀⠀⢀⡀⠀⣰⣶⡆⠀⢰⣿⣇⡼⢩⣿⣧⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⡟⠀⠀⢸⣿⣿⠤⢶⣾⡟⢋⣿⡇⠀⣿⣿⠃⠀⣿⣿⡿⠀⣼⣿⡇⠀⢰⣿⣿⠇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡇⠀⣠⣾⣿⡇⢰⣿⡟⠀⣼⣿⡇⢀⣿⣿⠀⣰⢻⣿⠇⠀⣿⣿⠃⢠⢿⣿⡟⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣼⢃⣿⣿⠁⢸⣿⠁⡰⠁⢿⣇⡜⠹⣿⡴⠃⠈⠙⠀⠀⢿⣿⣠⠋⢸⣿⠃⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⣇⣼⡿⠃⠀⠈⠛⠚⠁⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠻⠤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠈⠉⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⣰⣿⠏⠀⠈⣿⡄⠀⢸⣿⡇⠀⢠⣦⡄⢀⡴⠚⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⣾⠀⠀⣿⣷⠀⠀⠀⠀⣿⣿⡀⠀⣼⣿⡇⢠⣶⣤⠀⢀⣿⣿⢣⠟⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⢻⠀⣼⣿⡟⠀⠀⠀⠀⠹⣿⣇⠀⠉⢉⡴⣿⣿⠃⢀⣾⣿⣧⠏⣀⣠⣼⣿⡏⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠀⠀⢈⣿⣿⠏⢀⣄⠀⢀⠀⠀⢻⣿⣆⡴⠋⢸⣿⡟⢀⡞⣿⣿⣿⡞⢉⣼⣿⡟⠙⠳⢦⣤⣤⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⠦⣌⣿⢀⡿⠃⠀⣠⣿⣿⡆⠀⠀⠛⠓⠊⢸⣿⣿⡏⠷⠾⠟⠋⠀⠀⠀⠀⠈⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⣠⠋⠁⠀⠀⡴⠃⢸⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠙⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠉⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠂⠀⠤⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛⠓⠒⣾⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⢉⣉⣛⣻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⡛⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣋⣙⣛⡛⠇
⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⠀⡠⠉⠿⠃
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠿⠄⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢺⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣤⣄⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢠⠀⢰⢀⠀⢀⠄⠀⡀⠀⡄⢨⠰⣭⣰⢾⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠙⢛⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠄⡠⢿⠈⠤⠼⠰⣦⠀⠀⠀⢠⣀⣤⠀⠀⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠤⠶⠒⠖⠲⠶⠒⡶⠾⠶⢶⡶⠘⠛⠛⠛⠛⠃⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠐⠂⠆⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⡀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣠⣤⠀⠾⠁⠶⠴⣿⣿⣿⡇⠙⣿⣯⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠿⠛⠛⠛⠉⠋⠂⠀⠀⠰⠒⠒⠒⠒⠲⢂⠈⢙⣙⡒⠒⠂⠀⠐⠚⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠙⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠭⠉⠁⠅⠀⣿⣯⣽⢿⢿⡿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠓⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⣈⠀⠒⠶⢦⣬⣤⣴⣬⣟⣃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⣧
⠀⠀⠀⢀⣘⠛⣛⢿⣷⣶⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣻⣻⣿⢿⣟⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇
⠀⠀⢀⣨⣿⡏⢹⣿⣿⣿⣽⡇⡟⣿⡏⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠉⠉⠙⡆
⠀⠀⠈⣿⣿⣷⣾⣷⣾⣿⣥⣧⣿⣽⣿⣿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠈⢓⡾
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡺⠤⠤⣶⣾⣧⡀
⠀⠀⣘⣿⣿⣿⣿⣿⡟⣛⣫⢭⢾⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠇⢀⣀⣉⣉⣛⣿⣷⣦
⠀⠀⢺⣿⣿⡇⠙⢸⡇⣿⣿⢸⢸⣷⡇⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠩⢽⣍
⠀⠀⢼⣿⣿⡇⣦⣼⡇⡾⣿⢈⢸⡟⡇⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⢠⣿
⠀⠀⠀⣿⣿⣷⣿⣶⣷⣶⣿⣾⣾⣷⣶⣿⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠰⠰⡍⢴⢸⣿
⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠁⠀⠀⠀⠲⠀⠀⠲⠶⢶⠶⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇⣰⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⣠⣤⣤⢴⣾⣯⣄⡀⠀⠀⠀⠀⠀⣨⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢀⠍⠭⠩⡯⠀⠿⠛⠛⠫⡤⠠⢤⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠌⣽⠧⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠘⢛⡁⡃⠀⠐⠈⠉⠁⠒⠤⠀⡀
⠀⠀⢀⣠⣴⠇⠀⠐⢄⢘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⢶⠷⠟⢁⣈⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡄⢰⣿⣦⠀⣰⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠚⠛⣿⣛⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠶⠤⠤⠂⠀⠲⣟⠻⣿⡿⠿⣉⠐⠒⢦⣝⢿⣿⣿⣿⢸⣿⠇⠳⡄
⠀⠀⠀⠀⠀⠀⠒⢲⠠⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⡁⠉⠀⢨⣉⠉⠉⠙⠟⠿⠏⢛⢣⣤⣠⣿
⠀⠀⠀⠀⠀⣿⠀⠸⣄⡙⠟⠛⢿⣧⣄⢠⣄⣠⣼⠇⠈⠁⠀⠈⠳⢦⣤⢽⣔⡄⠈⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠶⠾⠻⠛⢋⠲⣌⠻⠿⢞⡉⠉⠀⠀⠀⠈⣿⠶⠖⠀⢸⣿⣷⠈⡽⣹⣿⠇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠑⠌⠛⠛⢋⣠⣤⡀⠀⠀⠀⠈⢑⣠⡄⣼⣿⣿⡆⢸⣿⡟⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠘⣤⠲⣦⠀⠀⠀⠀⠀⠘⠛⠁⠈⠉⠀⠀⠀⠉⢉⠉⡰⠃⣽⣿⠇⠸⣿⣧⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢯⣀⣤⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠻⠞⠧⡴⢛⣿⠈⠳⣌⣹⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢱⣿⠀⠀⠀⠀⠀⠁⣿⡿⠆⠀⠀⠀⠀⠀⡟⠁⣿⣾⠇⠙⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡽⠃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠿⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠙⠆⠐⠆⠑⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⢾⣋
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠃⣀⣤⣤⣄⣀⣀⡤⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠈⣧⡀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠘⠿⠿⠋⠀⠀⠀⢠⡔⠒⠉⠉⠉⠉⢉⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣹⠉⠙⠛⠛⠛⠋⠉⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠠⠶⠶⣏⠘⠀⠀⠀⠀⠀⣀⠀⠀⠗⠚⣶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠈⢻⡅⣀⣙⠀⠏⣩⡟⣁⢀⡴⠋⠁⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡆⠀⣖⡚⠇⢹⣿⣇⠀⠁⣛⡟⢃⡴⢊⡴⢧⠀⢀⣠⠞⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡾⠁⠀⠀⠙⣿⣾⣿⣿⡆⠘⣻⣶⣫⡴⢋⣤⡾⠶⣫⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣦⡄⢀⡾⠁⣿⠀⠀⠀⠀⠀⢨⣿⣿⣭⠀⣀⠀⣻⣟⠶⠋⣡⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⡇⢹⡾⠁⠾⣧⣀⡀⠀⠀⠀⠈⣟⣛⣋⠀⡈⢙⡛⡋⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⠀⠀⠠⣬⣭⣿⡄⠀⠀⠈⣿⣿⠾⠿⠓⠾⢷⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠰⢤⣈⣹⠀⠀⠀⠙⠿⠶⠶⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣀⣀⣀⣈⠉⣳
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣰⡏⠀⢈⣥⠞⠋
⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⣴⠟⠀⠀⠈
⠀⠀⣴⢶⣯⠀⠀⠀⢶⡟⠁
⠀⠀⢰⡞⣻⣷⣀⠀⠘⠃⣶⡄
⠀⠀⠀⠉⣅⣻⡿⣿⢋⣼⣏⠁
⠀⠀⠀⠀⠙⠉⢠⣿⡆⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠈⠉⠉⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠛⠀⠀⠭⠷⠀⣾⠯⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⢠⣄⣠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠈⠃⠐⡂⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠈⠂⢈⡂⠠⢄⡀⣠⣴⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠈⠁⠀⠀⠀⠀⠀⠀⠙⠂⢀⣩⡛⣿⣿⣿⠋⠁⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⣿⠙⠘⣇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⡠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⡀⠀⣿⣧⡀⠸⡄⢻⣦⣄⢹⡄
⠀⠀⠀⠀⠀⠀⢻⣸⠃⠀⠀⠀⠀⠀⠀⢧⢡⡀⠀⢣⡀⣿⣿⣷⣴⡇⢸⣿⣿⣿⣷⠠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣟⢃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⢸⡟⣿⣿⣿⣿⡃⢸⡿⠿⣟⡉⠀⢸⠂⠖⠘⠆⢘
⠀⠀⠀⠀⠀⠀⢀⣿⣌⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⠿⠿⠿⠟⠛⠛⠳⠀⠀⠀⣍⠉⣴⢷⣶⣶⣴⡶⡚
⠀⠀⠀⠀⠀⠀⣼⠋⢻⣦⠀⣠⢂⣴⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠸⠀⣎⠘⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⡿⠀⠀⢹⣷⣕⠟⡿⣿⣿⢂⠀⠀⠀⡀⠀⢀⣀⠀⣦⣰⣶⣶⣶⣆⠉⠛⠹⢿⡿⣠⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠿⣷⣭⣸⣉⣬⣈⣥⣤⣶⣶⣶⣾⣦⡬⢿⡿⢋⠀⠈⠁⢠
⠀⠀⠀⠀⠀⠀⣷⡰⠃⠀⠀⠒⠺⠿⠿⢿⣿⣿⢿⣻⣿⠟⠀⠀⠠⠄⠀⠑⢋⡀⣂⢆⣄⡅
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠁⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⠴⠶⠛⠻⠷⣶⢦⣰⣄⣨⣻⣷⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠛⢛⣿⣿⡿⢿⠟⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠋⠉⠉⠛⠿⠿⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⢸⣿⡄⣸⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠃⠞⠁⠟⠹⡆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠀⠀⠂⠀⠀⠀⠀⠃⠉⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣶⢦⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠶⡡⢿⡞⡏⠀⠀⠀⠀⢰⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⢀⠟⡉⠀⠁⣘⢳⢿⡄⢸⠀⠀⢨⠻⠻⣤⣇⢀⣖⠦
⠀⠀⠀⠀⠀⠀⣴⣥⠄⡀⠀⣹⡗⣡⣾⣿⠃⣰⡇⡸⠀⣿⠿⣄⣠⣿⢣⡀⠈⠹⣾⠁⠀⠈
⠀⠀⠀⠀⠀⠠⠋⠉⠛⠳⢾⣯⣾⣻⢳⠃⢠⣿⠇⡇⠀⢿⣾⡘⡈⢻⣇⢻⣤⠬⢶⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠉⠭⠙⠒⠋⠛⢻⣿⡄⠚⠛⠓⠓⠘⠻⡏⢁⡤⠔⠈⣷⣀⠄
⠀⠀⠀⠀⠀⠢⡀⠀⠀⡀⠘⠇⠀⠀⠀⠀⠀⠀⢸⢸⡇⠖⠂⠀⠀⠀⠀⣹⣀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠈⠑⠤⠷⠞⢀⣀⠀⠀⠀⠀⠀⡄⢸⠃⠀⠀⠀⠀⢀⠌⠀⢠⠋⠱⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⠉⠀⠉⠉⢉⡉⠉⢩⡀⠴⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⣤⠇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡸⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢰⠃⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡏⠀⢀⡠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠓⠒⠤⠤⣤⣤⣄⣤⡮⠤⡼⠖⠚⠉⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠮⢟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣷⡶⣯⣭⣿⣿⣿⣯⣭⣽⣿⢿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣏⠀⣈⣿⠾⢯⣿⣉⣿⣿⡀⢀⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡽⢿⣿⢻⣤⣼⡟⠉⢹⣿⡟⢻⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⣦⣼⣿⣿⠁⠀⢹⡿⣿⣿⣹⣿⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠭⣛⣛⣛⣛⣛⣛⡸⠽⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠀⠀⠀⢀⠀⢐⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⣝⡀⠀⢀⠁⡊⠀⠀⠈⢻⢦⠀⠴⢂⡿⡄⠀⠀⠹⡦⡐⠐
⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠈⠓⢍⠓⢤⣁⠀⠀⠀⠀⡟⢀⠀⢀⠞⠀⡟⠀⠀⢠⠀⠹⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⢧⠀⠈⠳⡀⠐⣦⡙⢮⠢⢸⠁⢠⠃⠀⡠⠋⠀⠈⡇⠀⢀
⠀⠀⠀⠀⠀⠀⢖⠈⣀⣀⡀⠀⠀⢸⡤⠀⠀⢧⢠⣇⠹⡌⢳⠘⣆⡿⢀⡞⠁⠀⠀⢀⠆⢀⠎⢣⠀⡀
⠀⠀⠀⠀⠀⠈⠀⣈⠉⢳⡎⢢⡀⢸⠀⠶⡀⠈⠻⠂⠀⢱⠀⠀⡿⢽⣼⠀⢠⠀⢀⡌⢀⠠⡀⡏⠐⠁
⠀⠀⠀⠀⠀⠀⠈⢹⠙⠸⡇⢀⠱⣾⡄⢁⠈⠢⡀⢠⡄⡞⠀⢀⠇⡄⠋⡠⢁⢣⠏⠀⢸⢣⢰⠇⢀⢦
⠀⠀⠀⠀⠀⢀⣆⢺⡊⠀⠹⣇⠰⠠⢄⠀⠂⠀⠰⠂⢠⠃⢀⡞⡐⢸⠐⠀⢠⠃⠀⡰⠃⣸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠘⡏⢦⣀⠀⠀⠈⠳⣝⢤⡈⠲⠀⠀⡄⢸⢠⠟⠐⠀⢾⠀⠂⡇⣠⠏⠀⣀⡎⠀⣠⢀⡸
⠀⠀⠀⠀⠀⠀⠱⣄⠈⠑⠢⢤⡀⠘⡄⠱⠀⠀⠀⠆⠈⠿⠀⠀⠀⠀⠀⡆⢹⠇⡄⢠⠎⢀⡰⣻
⠀⠀⠀⠀⠈⢻⠆⠈⠲⢄⠀⣄⡙⠶⣗⠀⠁⠀⠀⠘⢄⠈⠂⠰⠀⠀⠀⠐⡤⠊⢄⣮⠖⢋⡜⠁⣦⡀
⠀⠀⠀⠀⠀⠘⠒⠠⠱⠀⠱⡀⠂⠑⠄⠀⡄⠀⠀⠀⠀⠉⠂⣤⠀⠀⠀⠀⠄⠐⠎⠥⢈⠏⡀⠀⠁⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠢⡀⡇⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢐⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠈⠨⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠓⢤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠂⢀⠁⠀⠀⡀⠁⠀⠔⠀⡀⠀⡀⠀⠀⢐⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠐⠀⠀⠈⠈⠀⠀⠁⠀⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠉⢀⣴⣿⣿⣟⡛⠋⠉⢉⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⢎⠁⠈⠃⠀⠀⢀⡤⠶⢤⠀⡰⠟⠛⠛⠛⠿⣿⣿⣿⠿⠛⠉⠁⠉⠉⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⣇⣀⣀⠇⠀⠀⢀⠀⠀⠈⢡⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣀⣀⣀⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣏⠓⢤⣠⠏⠀⢀⣀⡩⠋⠀⠀⠐⢮⣄⣀⠀⢸⡇⢀⣀⣀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠘⡄⠀⡹⣄⠀⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠯⡀⡠⠖⢤⠀⠀⠘⠂⡇⠘⣆⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣠⠶⠋⢊⠴⣼⠱⠁⢠⢃⣀⠤⠤⢖⡶⠄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⡤⠾⠿⡇⠐⠓⠁⢀⡈⠑⠒⠋⢉⠞⡟⣹⢁⣠⣴⢮⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢯⡄⠢⡚⢉⣇⡠⠔⠉⠈⠒⠂⠉⢹⠀⠈⡟⠉⣀⣼⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⣿⣿⠃
⠀⠀⠈⢎⠉⠙⣍⡠⠧⡄⠀⠀⣁⣤⡴⠋⠁⢀⡗⠈⠻⢿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⡟⠁
⠀⠀⠀⠈⠣⡀⠘⡄⢚⣷⠴⠚⠁⠈⠀⢀⡴⠋⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠛⠉
⠀⠀⠀⠀⠀⠈⠲⠚⠋⠙⢤⣀⠀⠀⠀⢸⠗⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⢟⡀⠀⠀⠀⠉⢹⢯⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦⠶⡉⠀⠀⢳⢤⡀⠀⠀⠀⠀⠀⢀⣀⣀⣳⡈⢹⠉⡉⠉⠉⠉⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡢⡌⢢⡴⠃⠀⠹⡄⠀⠀⢀⡴⠋⠀⠘⠷⣇⣼⣀⣇⢀⣀⣀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠁⠀⠀⠀⠀⢹⡀⣠⠞⠀⠀⢠⠄⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠑⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠈⡏⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢸⣿⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡇⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠼⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⠂⠀⢠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢀⣴⣿⣿⣿⣿⣿⡀⠀⣹⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣆⠂⣼⣿⣿⣿⣿⣿⡯⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⡟⣡⣼⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠋⣙⠻⣿⣿⣿⣿⠟⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠁⠉⠛⠋⠀⠀⠀⠀⠀⠈⡙⢂⠆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧⣈⡀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠈⠉⠉⠉⠀⠀⠀⣿⠿⠿⠋⠉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠂⢐⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⣐⡞⢤⡤⠶⠀⠀⠀⠀⢸⢇⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⢄⡀⠀⣀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢱⣦⣄⣀⣠⡴⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠋⠉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠧⠀⠀⠒⠂⠀⠒⠀⠄⡀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡄⠀⢀⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⣠⣿⡿⠀⠀⠸⣿⣿⡇⠀⠀⠀⢤⢾⣧⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡅⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠿⣿⡿⠀⢀⣴⣦⡄⠀⣴⣶⡄⢠⣄⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡙⢣⣤⣄⡀⠀⠀⠀⠈⠻⠿⠟⠀⠻⡿⠇⠘⢿⡇⠀⠐⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣼⠿⠿⠃⠒⠈⠀⠈⠋⢠⡄⠀⠀⢠⣶⣄⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠋⣆⣰⡎⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⠸⣿⠟⠀⣾⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⢠⣴⣤⠀⠙⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠻⠿⠟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢀⣤⡶⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⣰⡄⠉⠁⠀⠈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠿⠇⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣦⣶⣴⣶⣦⣴⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄⠀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢟⣫⣄⣴⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠕⠿⠿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢻⣿⠁⢴⣿⣿⣿⣿⣷⠀⠐⣾⣿⣿⣟⡒⢘⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣶⠀⠉⠋⠀⠀⠉⡇⠀⠙⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣹⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⠧⢤⣤⠶⠂⠀⠀⠜⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠒⠳⠶⠤⢤⡆⠠⡾⠀⠀⠀⠀⢀⠄⡤⢛⡻⢥⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣠⠀⠀⠀⠀⠀⣇⠀⣿⠀⠀⠀⠀⠈⢀⣃⣤⣼⠈⠀⠀⠈⠙⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠻⣿⣷⣄⠀⠀⠀⠘⣦⠙⣄⠀⠀⠀⢀⡄⣡⣿⡉⠙⠛⠛⠁⠰⣦⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⣯⠘⢷⣙⣿⣿⠿⡿⠿⠟⠈⢷⢠⠀⠀⠀⠰⠇⠹⢿⣧⡤⠆⠐⠀⢰⡇⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠀⠻⣿⡀⠀⠙⢮⣗⠀⠀⠀⢀⠄⠺⠤⠧⠤⠤⠖⠀⠀⠈⡿⣷⠖⠒⠀⠀⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠉⣳⣦⣶⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠘⠛⠳⣶⡶⠀⠁⠈⠀⠀⢀⠷⡀
⠀⠀⢀⣤⠶⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣦⡀⢰⣿⣿⣿⣿⣆⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢲⡖⠂⡠⠊⠀⠈
⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠻⣿⣿⣿⡅⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠛
⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠉⠁⠀⠀⠈⠁⠒⠤⡀
⠀⠀⠀⢀⣀⡀⠀⡠⠊⣁⣀⣀⠀⠐⠒⠤⠤⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⡥⠄⡶⠉⠀⠢⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢉⠄⠀⠀⠀⠈⠑⢍⠁⠒⠠⣀
⠀⠀⠀⠀⠰⢡⠀⠀⠀⠙⡆⠀⡗⠒⠠⠴⠤⠤⠤⠤⠽⠉⠐⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⠖⠠⠃⠀⣴⠂⠀⠀⠀⠀⠑⡀⠀⠈⢳⣄⣀
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠙⠀⢰⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣂⢀⡁⢀⡌⡇⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⢯⠉⢆
⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠁⠀⢀⠖⠒⠒⠀⠓⠀⠉⡏⢑⡄⠀⠀⠀⠀⡍⠉⠀⠀⠀⠀⠈⠙⠃⠒⠀⢤⢀⣸⠀⠀⠀⠀⣼⠀⢨
⠀⠀⠀⠀⠈⠆⠀⠀⡜⠉⠱⠉⠀⠀⠀⠀⠀⠀⡀⢼⠀⠐⣆⠀⡤⠴⠾⣏⡁⠁⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢈⠰⢄⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠈⢆⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠣⣼⣆⠬⠁⣰⡇⠙⠢⠴⡖⠁⠀⠀⠀⠘⠈⠀⡄⠀⠀⠀⠓⠀⠀⠀⠀⠘⠤⠀⢄⠀⠀⠀⡌⢰
⠀⠀⠀⠀⠀⠀⠀⠑⠄⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢚⡩⠋⠉⡆⠀⠸⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⠜⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⡄⠀⠀⠲⡒⠉⡇⠀⢀⠜⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⢄⡠⠐⠈⠉⢘⠅⠀⠀⠀⠀⠀⠀⠀⢈⠦⠻⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠒⠀⢤⣿⡦⣤⠴⠋⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠎⠀⡠⠶⣶⢻⠀⠀⠀⣀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠒⠢⠼⠛⠉⠱⡀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠈⡎⠉⣿⠁⠀⠀⣸⠦⡀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢰⠁⠀⠀⠀⠀⠀⡸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⣇⠀⠘⠳⠶⠚⠋⠀⠈⢢⠀⠀⠀⠑⢄⡀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⠄⠀⠀⠒⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣼⠀⠀⠀⠀⣀⣀⠀⠀⢡⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⣂⡀⢰⡀⣀⠤⠚⡅⠀⠀⡠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣟⠀⠁⠀⠀⠀⠀⣀⠬⠦⠞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡨⠋⠉⠘⠃⠉⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠲⠶⣐⣂⣙⢲⣖⣖⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠈⠀⠔⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠒⠀⢠⢠⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⣉⣉⡉⠈⠉⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠸⠖⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⢀⣀⢀⣀⣀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣈⢉⣉⣀⡀⠬⢥⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠸⠀⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⢆⡀⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠤⠴⠃⠀⠀⡗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠄⠤⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣅⣀⣀⣰⣿⣿⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⠛⠛⠛⠙⠛⠋⠉⠉⠁⠹⣿⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢁⠤⢤⣄⠀⠀⣴⠶⠒⠢⠀⠀⠀⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠶⠿⢿⡇⠀⠀⠞⠛⠂⠀⠀⠀⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠺⢇⣀⠤⠂⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠰⢃⣀⣀⣀⣀⡤⠄⠁⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠠⠴⠀⠀⠀⠀⠀⢠⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠃⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠸⣿⡷⠦⣤⣤⠴⠒⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀
⠀⠀⠀⢠⣆⣀⣀⠀⠀⠀⠀⠀⠚⠉⠀⡇⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠀⠀⠀⠀⠀⠃⠀⡄⠀⠙⢦⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠠⠏⠀⠀⠶⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢣⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⡔⠓⠒⠠⠤⠤⣀⣹⣿⣏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢱⠠⠦⠤⠤⠤⠤⣤⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⣦⣤⣤⣤⣤⣄⣉⣠⠄
⠀⠀⠏⠻⠼⠁⠈⣇
⠀⢰⠀⠀⠀⠀⠀⢰
⠀⠘
⠀⠀⠁

⣿⣿⠟⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢄⡐⠁⠈⠄⢠⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡈⠀⠸⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣧⡤⣄⠀⠔⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠲⣄⠀⠀⡄⠀⠀⣠⣤⣶⡵⢚⣉⣠⡴⠋⠀⢠⢊⡤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⢦⠀⠈⣶⡄⣿⣿⣽⣿⣹⡿⣀⠀⠐⣰⠷⢋⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠣⠈⠹⡇⡨⣟⣉⣻⣯⣶⣟⣀⡼⠁⢀⣎⠤⢶⡆
⠀⠀⠀⠀⠀⢎⣉⡉⠉⠉⠉⠁⠐⠀⠱⢤⣰⣿⣯⠾⢿⣷⡦⠿⣿⣿⠋⣀⡴⠋⢀⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠭⠿⢛⡓⠖⠒⡿⣾⡿⠇⣷⢋⡙⢧⣝⢟⣾⣋⣤⡾⠋⢉⠎
⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⢀⣀⣠⣤⠶⣾⣿⡏⣼⣯⣅⡼⣗⣹⣫⣿⠟⠉⢩⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠁⣀⣤⠚⠋⠻⠧⠭⠾⠥⡞⠉⢁⡜⢁⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⠀⠈⠋⠉⠀⠀⠠⠔⠊⢤⣴⣟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠁⠀⢠⣤⣤⣤⣤⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠘⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠻⣿⡄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⡀⣰⣿⡿⠃⠀⢠⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡼⣿⠁⠀⠀⠸⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠘⣿⣿⣷⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣦⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⢤⡀⠀⢀⣿⣷⡀⠀⠀⣸⣿⣿⡟⠀⢤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⡇⠀⠀⠀⢱
⠀⠀⠀⢤⡀⠀⣴⡿⠀⡀⠙⣿⣿⣿⣿⡆⠀⠀⢸⣷⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⠁⠀⠈⣿⣷⡄⠀⠀⠙⠻⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣷⣤⣀⣤⣿
⠀⠀⠀⠈⣿⣶⠟⠀⠀⣿⣦⣾⣿⣿⣿⠇⢀⣴⣿⠟⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⣿⡀⠀⠀⠀⠀⠉⣰⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⣇⠀⢀⣿⣿⠀⡀⠀⠘⢿⣿⣿⣿⢟⣴⣿⣿⠋⠀⢠⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⢿⣿⣿⣿⣿⣇⣀⣽⣷⡀⠈⠙⠿⢋⢸⣿⣿⣿⣷⣤⣾⡇⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⣿⣌⣿⣿⣿⣿⣿⡿⢡⣾⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⢠⠀⠙⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⡟⠀⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣷⣶⣿⣿⣿⣿⡟⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣘⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⡟⠛⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠘⣇⠈⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣸⠀⠀⠀⠀⠀⠀⠀⠰⠋⡇⣦⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠰⡆⣨⢛⢥⡶⠀⠀⠀⠀⠀⠀⢸⢸⣄⣧⣠⠀⠀⠀⠀⠀⢀⠀⣲⠛⡏⠀⠀⠀⠀⠀⠀⡾⡿⠜⡄
⠀⠀⠀⣷⣼⢀⡾⣤⠦⡼⠀⠀⠀⠀⢸⣾⠈⡟⠙⢻⠀⠀⠀⠀⠀⠘⢟⢫⣿⣀⡀⠀⠀⠀⢀⠠⢇⠠⣤⢷⡄
⠀⠀⠀⠘⠦⠾⠇⠘⠃⠁⠀⠀⠀⠀⠀⠁⠰⠛⠖⠁⠀⠀⠀⠀⠀⠀⠀⠱⠐⢯⠃⠀⠀⠀⠘⣷⠛⠘⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀
⠀⠀⠀⠀⠘⡿⡏⡅⠀⠀⠀⠀⠀⢀⢀⠀⣿⢣⣀⡀⠀⠀⠀⠀⠀⠀⠎⣼⣿⢰⡀⠀⠀⠀⠀⠀⡄⣾⡼⣃⡋
⠀⠀⠀⠀⠸⣧⠛⣇⠀⠀⠀⠀⠀⠈⠉⢷⠾⡌⠛⢛⡁⠀⠀⠀⠀⢰⣜⣇⢹⡾⣧⡀⠀⠀⠀⠀⡁⢏⢁⢹⠠⠇
⠀⠀⠀⠸⣆⣿⢶⣼⠀⠀⠀⠀⠀⠀⣻⡛⠀⡿⣆⠛⠇⠀⠀⠀⠀⠸⣿⠹⢰⡀⠻⠀⠀⠀⠀⠠⣼⡾⢿⡘⢾
⠀⠀⠀⠀⠀⠰⠏⠋⠀⠀⠀⠀⠀⠀⠈⠃⠻⠅⠙⠋⠀⠀⠀⠀⠀⠀⠙⠘⠋⠃⠀⠀⠀⠀⠀⠀⠈⠃⡎⠻⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣗⠂⠀⠀⠀⠀⠀⠀⠀⢨⡷⢘
⠀⠀⠀⠀⢻⣍⣏⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠇⠀⠀⠀⠀⠀⠀⡔⠊⠛⠋⠳⡄
⠀⠀⢠⣶⠞⠀⠀⠙⠂⠀⠀⠀⢰⣿⡿⠒⠁⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⢤⡄⠟⠁
⠀⠀⢠⠀⠠⠖⠀⠀⡄⣤⠀⠀⢸⣿⠁⠀⠀⡂⣀⠀⠀⢀⠀⠀⡆⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠒
⠀⠀⠘⡀⠀⠀⠀⢠⠇⣟⠈⡁⠀⣿⣀⣀⠀⡇⠈⠛⠝⠋⠀⠠⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣤⡤⠀⢠⡄⡀⣿⡒⡦⠇
⠀⠀⠀⠛⠛⠒⠂⠛⠀⠛⠉⠁⠀⠛⠳⠷⠒⠃⠀⠰⠓⠀⠀⠀⠚⠂⠐⠀⠀⠁⠘⠀⠀⠀⠓⠛⠛⠛⠓⠂⠚⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⢺⣿⢋⣍⣉⣍⣉⣉⣩⣩⣍⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡝⠻⢿⣿⣿⣿⣿⣿⣿⠿⣋⣵⣿⢸⣿⡌⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⢋⢱⣿⣶⣦⣭⣭⣍⠱⣶⣿⣿⣿⡟⣸⣿⡇⣛⣭⣙⡻⢿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣍⣡⢠⣿⡇⢿⣿⣿⣿⣿⣿⣷⣶⡍⣭⣭⡘⣿⣿⡇⣿⣿⣿⣿⣶⣬⣥⡄⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣦⣍⣥⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡗⢸⣿⣇⢻⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠉⢰⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⠸⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⠿⠟⠁⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣦⡺⣿⣿⣿⣿⣿⣿⡿⢃⣴⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣮⡛⠿⠿⠿⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⢀⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠉⠹⠿⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠒⠀⠒⠶⠾⠿⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⣠⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠄⢀⣤⣠⣠⣤⣄⡉⠀⠀⠈⠈⠈⠉⠉⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠀⠒⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠿⠿⠟⠃⠀⡄⣀⢀⣤⣤⣶⢰⣾⡗⠀⠀⠀⠀⠀⠀⢲⣿⣇
⠀⠀⠀⠀⢠⣴⣶⡶⠶⠛⠓⠀⠀⠀⠈⠁⣈⣉⣛⣋⣏⠀⠿⢿⣗⣀⣀⢀⣀⣴⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠤⠶⠶⠶⠶⠚⠛⠛⠩⠿⣿⣿⣽⣿⣿⠾⠎⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⣀⠀⠀⠆⠀⠀⣼⣤⣶⣶⣶⣶⣶⣾⣿⡖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡁⠶⠂⠉⡑⡖⠒⠠⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡰⠋⠈⢄⢣⡀⠇⡇⢀⡠⢊⠦⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢰⣀⣀⣀⣈⣧⠙⠠⠖⣩⠖⠁⠀⢱⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆⠐⠒⠀⢠⡄⢀⡀⢈⡉⣀⣐⣒⣚⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⢤⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⣰⡀⣀⣴⢾⠶⠋⣽⡌⠣⡀⠉⠉⠹⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠍⠣⡉⠔⠀⠀⠀⣿⠈⠣⡹⢄⡰⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠈⠐⠀⠀⠄⣹⡀⡠⠜⢋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠶⢤⣤⣠⣤⠤⠶⠒⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠦⣄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢸⠀⠀⠀⠀⠀⢀⡠⠔⠈⠉⠑⡉⠒⠢⠤⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠈⣳⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠐⠁⠀⣠⠴⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠃⠀⠈⠀⠀⠘⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠐⣬⠒⠒⠒⠒⠒⠒⢦⡀⠀⠀⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠀⠀⠀⠈⢆⢄⡀⠀⢀⡠⠔⢸⠀⠀⢠⠀⠀⠀⠀⠘⢆⠀⠀⠀⣼⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠀⠈⠢⠤⠚⠉⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡄⠀⠀⠀⠀⠀⠀⣠⠒⠢⢄⡘⢠⣀⠈⠀⠀⠀⠀⢀⢸⠀⠀⢠⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣷⠀⠀⠀⠀⠀⡰⠁⠀⢀⠀⠈⠙⠷⠄⠀⠀⠀⠀⠊⠜⠀⠀⢸⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢱⠀⠀⠀⠀⡐⠁⠀⡠⠉⠐⠢⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⢃⠆⠀⠁⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠈⢄⡀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢻⠀⠀⠀⠀⠀⠉⠂⠄⡠⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣈⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠋⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠦⣈⠉⠙⠛⠛⠛⠛⠛⠋⢉⣁⠴⠖⠋⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⠤⠤⠤⠤⠤⠶⠖⠋⠁


⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣭⣭⣬⣭⣭⣉⣍⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣴⣦⣤⣿⣿⣿⣭⣿⣿⣽⣿⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠒⡶⠒⠀⠠⡀
⠀⠀⠀⠀⢀⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠑⣄⠀⠀⠸⠔⠊⠙⢄
⠀⠀⠀⠀⠼⠏⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠋⠉⠁⠈⠀⠈⠀⠉⠁⠐⠦⢬⣦⣀⡀⠀⠀⣀⢈⡵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢈⠉⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡠⠔⠂⠉⠁⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠊⠁⠀⠀⠀⣀⡀⠀⠤⠤⠤⢀⡀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠣⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣀⡤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠹⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣉⡀⠴⠄⠚⠃⠈⠀⠁⠈⠀⠓⢤⡀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⢀⡠⠔⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠑⣄⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⠔⠉⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⢢⠀⠀⠀⠀⠀⠱⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⢣⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠇⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠙⠄⠀⠀⠀⢀⣠⠠⢄⣀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠑⢤⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠲⠶⠖⠂⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠙⠂⢡⠀⠀⠀⠀⠀⠀⠀⢁⠀⡘⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠂⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣶⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⡿⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡟⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣠⣭⠴⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣖⠀⠀⠠⠤⠤⠤⠀⢀⣀⣀⡀⠠⢾⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣯⣭⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣿⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢹⠏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠘⠃⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣤⡤⠄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡀
⠀⠀⢀⡠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠽⠽⠿⠍⠉⠀⠀⠀⠀⣀⣀⠀⢀⡀⠈⣠⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢧⣿⣿⣿⣌⣷⡦⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⢿⣿⣿⣿⣿⣷⣌⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣿⣧⣊⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣿⣿⠻⣿⣿⣿⣿⣿⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣴⠋⠉⠉⠁⠉⠉⠉⠁⠈⠙⠛⠛⠿⣿⡼⣸⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢛⣛⡿⢿⣿⣷⣶⣾⣷⣰⣶⣦⣤⣤⣯⣠⣀⣀⠀⠘⠛⢶⣭⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠚⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡍⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⢀⣤⣰⡄⠬⠭⠍⠉⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠟⠉⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣉⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣷⣾⣿⡿⠿⢿⠿⠿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡛⢻⠋⢀⣤⣤⢄⣀⣦⣤⣤⡀⠍⠛⢿⣿⣿⣿⣿⣿⣅⣙⠃⠘⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⣮⣙⠿⣿⣿⣿⣿⣶⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠟⠛⠉⠙⠛⠛⠛⢯⣭⣿⣿⣿⣿⣿⣿⣿⣦⣀⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⡀⠀⣠⣴⣲⢤⣼⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣠⣾⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⣶⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡾⢽⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣭⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢻⢹⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⡀⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣠⡇⠀⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠋⠀⢀⣾⠟⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⢀⣴⣿⡏⠀⠀⣠⣴⡏⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⢳⣿⢀⠠⠾⢻⣿⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⣿⡿⣱⣿⡿⠃⠀⠀⣼⣿⣼⠠⠄⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣽⣿⣿⣿⣿⣆⣠⣾⣿⢿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢟⣿⡿⣿⣿⣿⣿⣿⢣⣼⣿⡎⢷⣇⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⢏⣾⡟⣸⣿⣿⣿⣿⡇⡾⣿⣿⢻⡌⠻⣆⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⢠⠁⣿⠟⣿⡿⣰⣿⣿⣿⣿⣿⡘⣇⢸⣿⣿⣿⠀⠙⣷⣄⠠⡱⣄
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣿⣾⣿⡇⣿⢸⣿⣿⣿⡿⠧⠀⣼⡈⣿⣿⡇⢢⠘⣿⣧⠙⣿⣦
⠀⠀⠀⠀⠀⠀⢸⡀⣼⢿⣿⣿⠀⣿⣼⠳⠏⢠⣷⢀⣼⣿⠃⣸⢹⡇⣸⡄⠸⣿⡇⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣧⢹⡜⣿⣿⡄⢹⣿⠀⠀⢸⣾⡈⠻⣿⢀⡟⣹⠇⣿⣥⠀⣿⣷⠀⢻⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠹⣿⣿⣿⣦⡿⣷⣄⡈⠻⠏⠀⡹⣾⣱⠏⣼⣿⣧⠀⣿⣿⠀⢸⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⣻⣯⣛⣿⣿⡝⢷⣂⠠⣼⣷⣿⢏⠼⠁⢸⡇⢰⣿⡟⠀⣸⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠀⠈⠋⣉⣿⣿⣿⣷⡙⠿⠿⠯⠁⠀⢀⠴⠋⣴⣿⠟⠁⣰⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⣿⣿⣿⣿⣷⣦⣤⣤⣤⣷⣼⣿⣿⠗⢀⡴⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣹⣿⣿⣿⣿⣿⣿⣿⠿⢛⣡⠴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣧⣴⣿⡭⠤⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠿⠟⢻⠟⢿⠿⢻⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢇⣤⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⢠⠟⠙⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⡀⢠⡆⠀⠀⠀⠈⣧⠄⠐⠸⠀⡤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠀⡰⠉⠷⠀⠐⠑⠀⠃⠀⣤⡠⠞⡷⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠸⢤⡇⠀⠄⠀⠀⠀⠀⠀⠀⠉⠀⠐⡁⠘⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠈⠁⡐⠂⡰⠁⠀⠀⠀⠤⠂⡳⠀⠇⠆⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣦⡱⡄⠲⠇⠘⠀⠀⠀⠀⠀⠀⢀⠀⠞⠀⠔⢁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠌⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠊⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠈⡤⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢥⡀⠀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠮⠴⠖⠖⢤⣼⢶⡿⣷⣾⣷⠋⢻⣿⣷⡖⠦⢤⡄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣤⣤⣴⠞⠋⠀⣸⠁⢀⣿⢻⣆⠀⠙⢯⣟⠉⣷⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢹⣁⣀⣠⣶⡇⠀⠸⣿⠄⣻⡆⠀⠈⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠈⣿⠷⠏⠛⠉⠘⠶⠶⠞

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠿⡻⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⠉⣨⣴⣿⣿⣿⣿⣿⡟⢿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣷⣶⣼⣾⣷⡟⢩⣿⣿⣿⣛⢛⣭⣿⠎⣉⡠⢧⡃⠉⢻⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⣟⣟⣭⣭⣽⢻⣬⣷⣾⣿⣿⣿⣿⢿⣿⣿⣿⣭⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠞⠁⢁⣬⣶⣿⣿⣿⣿⣿⢿⣯⢿⣿⢿⣿⣿⣦⡥⢿⣿⢿⣿⣿⣿⡿⠒⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⡟⣼⣿⣿⢿⣹⣿⣟⣻⣿⢹⢿⣿⣿⣿⣿⢸⢻⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠷⣼⣿⣿⣿⣆⣻⣿⣿⣖⢿⡜⠘⠏⢹⣿⣿⣌⡀⠀⠹⠛
⠀⠀⠀⠀⠀⣼⠿⠟⠏⠈⢠⣿⣿⡝⠯⡧⢹⣿⣯⡓⠸⣷⠀⠀⠏⢿⢿⡓⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡿⡝⠆⠀⢸⣿⣦⡉⠀⣿⢀⣤⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠑⠈⠀⠀⠀⠻⠇⣩⣶⣿⡝⢿⣿⡿⣶⣤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⡿⣫⠵⢿⣹⡇⠀⠀⠹⡄⢹⠷⠄⠀⣀⣙⠿⣔⣲⣤⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠛⠛⣿⣷⣾⣖⣲⣤⣄⣿⡗⠛⠛⠓⠂⢲⣶⡼⠶⢦⣸⣷⣶⡿⠿⠿⠿⠛⠛⠛⠻
⠀⠀⠀⠀⠛⠉⠛⠒⠒⠒⠚⣁⣤⣤⠤⠤⠌⠈⠉⠒⠒⣿⡇⠉⢁⣷⡆⢠⠈⣡⠈⣋⣉⣣⠤⠴⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡾⢸⢻⢃⣮⢞⢥⠀⠠⠤⠭⣭⣿⣿⣯⣈⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣬⡏⢀⡟⣯⣈⣬⡃⢻⣾⣯⣀⠀⠐⣾⠿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠒⠊⠉⢛⡽⠋⠋⠀⠸⠘⠹⠉⠳⠈⠙⠛⠿⠿⢽⣿⠿⠿⠓⠒⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⡆⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣤⣿⣿⣤⣶⣶⣶⣯⣴⣷⣭⣿⡆⠀⠛⢲⡀⢛⣒⣶
⠀⠀⢀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⡏⢿⣿⣿⠀⢀⢀⡐⢾⢈⣦⣠⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⣰⣶⠟⣤⢿⡏⢈⡷
⠀⢀⣾⣿⣿⣧⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠋⠀⢀⡌⣦⣠⢷⣏⣿⣯⣯⣧⡐⠀⠀⠉⢈⣍⣉⣩⣍⣭⣥⣤⡄⠀⣜⣿⣶⣼⠟⢹⣏⣸⡅
⠀⣼⣿⠏⠀⢿⣿⣇⠀⠀⠸⣿⣿⢿⢿⣿⣿⠃⠀⠀⢀⣜⣾⣷⣿⣿⣿⣛⣿⣿⣼⣿⣿⣷⣦⡰⣼⣿⣍⠓⢺⣟⡋⠉⠉⠀⠘⠿⠞⢿⣶⣿⠿⠛⠁
⠀⠻⢧⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⣁⣸⣿⣿⡟⠛⠈⠑⠀⠀⠀⠀⡀⠀⠘⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⡀⣴⢠⡀⠀⠀⠀⠀⠀⠀⠆⡐⠂⡆⠐⠀⠀⠀⠀⢰⠀⠀⠀⣿⣿⣿⣿⠇⢀⠀⠀⠀⠀⣠⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⣸⢼⣠⢧⣟⣳⢼⡰⣅⣷⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠉⠁⡄⣑⣘⣄⠀⣀⣙⣦⡴⣃⠀⠀⠀⢠⡀
⠀⠀⠀⢸⡆⠀⠀⣯⣻⣾⣾⣿⣿⣿⣾⣷⣿⣷⣿⡄⠀⠀⠀⠀⠀⢀⣼⣧⣶⡤⣄⠀⠀⠀⠀⢀⠙⠐⠀⠑⠿⠟⠀⢠⣞⣿⡻⣞⡽⠁⠀⢠⣤⣽⠆
⠀⣰⢠⡿⢁⡀⠀⠈⠛⠛⢉⠉⠉⠉⠙⠛⠛⠛⠋⠀⠀⠀⠀⢠⣼⣽⣿⣿⣿⣿⣿⣤⠀⠀⣀⡴⠛⢦⡞⠀⠀⠀⠀⠀⣹⣟⣿⣏⠀⠀⢰⣛⣦⣹⣦
⠀⢿⡈⢷⣼⣄⠀⠀⠀⣠⢿⣆⣤⠀⠀⠀⠀⣦⣷⡄⢣⣀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣞⠀⡄⡟⢷⡜⣮⣷⣴⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⢈⣽⣿⠁
⠀⢸⡿⢸⣿⣿⠇⠀⣰⣯⢾⣿⣏⣦⠀⠀⠲⣬⡟⣷⣦⡙⣷⠀⠁⢸⣿⣿⣿⣿⡟⠃⠸⣧⣿⣿⣵⣿⣟⣿⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⣿⡟⠋
⠀⣾⣷⣾⣿⣿⠀⠀⣿⡇⠀⠋⢹⡿⠀⠈⣧⣿⣷⣿⣼⣷⣡⣧⠀⠻⣿⡟⢽⣿⡅⠀⠀⣽⣯⣿⣿⣿⣿⣻⡇⠀⠀⣀⣤⣴⣆⡀⠀⠀⠀⠉⢻⣿⣦
⠀⠈⠛⠛⠛⠋⠀⠀⠈⠛⠀⠐⠋⠁⠀⠀⠙⠛⠉⠋⠋⠙⠛⠋⠀⠀⠘⠀⠘⠋⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠃⠀⠾⠿⠿⠿⠿⠟⠛⠗⠀⠚⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠉⠉⠉⠉⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠴⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⢤⡐⠄⠉⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢹⠀⠀⠀⢀⡴⠋⠒⢤⡀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⡠⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠼⣄⠀⠰⣷⠜⠂⠉⠉⠉⠉⢳⡄⠄⠈⠑⠤⣀⠀⠀⠀⠀⠘⡞⠁⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣫⡁⢰⠉⠓⠌⠙⢿⠧⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠘⡄⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣴⣫⡭⢭⣭⣽⠀⠀⠉⠒⠀⠀⠈⡦⠴⠒⠒⠒⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠲⠄⠀⠀⠀⠀⠀⠀⠸⡄
⣤⣄⣀⣀⡀⠀⢰⠋⡏⠀⠈⢢⡑⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠧⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⢧
⡇⠈⠙⠒⠯⠭⣯⣸⣤⣀⠀⠀⠉⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠒⣀⠟⠀⠀⠀⢠⡄⠀⢸⡈
⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⣀⣀⠠⠼⢧⠈⠉⢒⠶⠦⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠴⠒⠋⠉⠓⠚⠛⠃⠤⠶⠄⡠⠊⠧⠤⠼⠉
⡇⠀⠀⠀⠀⣸⠷⠈⡹⠧⠐⠒⠒⠈⣻⣤⣞⡥⠾⡏⠀⠀⢉⢀⡔⠁
⡇⠀⠀⠀⠀⠮⣁⢐⣋⣉⣉⣩⠭⡤⠤⠖⠒⠒⢾⣁⠴⠚⢡⠎
⠑⠦⢄⣀⣀⣰⠛⠋⠀⠀⠀⠀⠺⡤⣀⣀⡀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠠⠹⡍⠉⠉⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣓⣤⣤⣵⣀⣀⣀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠓⠛⠉⠀⠈⠑⠿⢷⣒⣉⠆

⠀⠀⠀⠀⠀⢀⡀⡠⠐⣲⣷⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠭⠁
⠀⠀⠀⠀⡴⠛⣻⣟⣿⣡⢾⠏⠚⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣷⠏⣯⡉⠀⠀⣠⣶⡒⣂⣤⣤⡀
⠀⠀⠀⠀⠀⢸⠟⠹⠓⡹⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⡙⢀⠀⠀⠋⠘⢷⠈⠋⠗
⠀⠀⠀⠀⢀⠎⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⡀⠀⠑⠠⡀
⠀⠀⠀⡠⠃⠀⡠⠋⠀⠤⠀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠑⠀⡀
⠀⠰⠞⢁⣴⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠗
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⢿⠿⡿⡿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡄⢀⢄⢀⠀⠀⠀⠀⠀⠀⡖⠈⣀⢀⢰⡆⠀⠀⠀⠈⠁⠈⠉⠙⡇
⠀⠀⠀⠀⠀⠀⣟⣛⡟⠚⣿⡇⠀⠀⠀⠀⠀⡀⠰⠀⠌⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⠀⠀⡃⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢾⠃⠀⠀⠀⠀⠀⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠓⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠑⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠰⠆⠈⣴⠟⠀⠀⠀⠀⠀⠀⠳⠄⣀⣠⠾⠀⠀⠀⠀⠐⠦⠀⠐⠶⠃⠀⠀⠀⠀⠀⠠⠐⣠⣴⠇⠀⠀⠀⠀⠀⠸⠦⠀⣀⣀⣀⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⢀⠀⣀⡀⠀⠀⠀⠀⢀⢀⣀⣀
⠀⠀⠀⠀⠰⣾⠀⠀⠨⢷⣩⡗⠀⠀⠀⠀⡋⠀⠀⠀⠁⠀⢹⠀⠀⠘⢆⠀⠀⠀⠀⣼⠃⠀⠀⢸⡛⠀⠒⠒⠒⠛⠻⠂
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠠⠋⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠀⠈⢢⠀⠀⡜⠁⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠰⠀⠀⢠⣷⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⢃⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⣿⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣏⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢃⣀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠐⠄⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠘⠤⠄⠀⠐⠀⠀⠀⠀⠀⠀⠢⠀⠠⠀⠂⠀⠀⠀⠀⠛⠀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⡶⠊⠉⠉⣀⣨⣷⠀⠀⠀⠀⢠⣇⣤⣾⣧⣤⡄⠀⠀⠀⢠⠀⠀⠄⠀⣄⠀⠀⠀⠀⢰⠏⠉⠉⢭⣿⠀⠀⠀⠀⢠⣄⣒⣒⢒⢲⣦⡄
⠀⠀⠀⠀⠈⡇⠠⠆⠤⠖⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣁⠀⠀⠈⣿⠀⠀⠀⠀⢰⠂⣠⢿⡴⣣⢰⡇
⠀⠀⠀⠀⠀⢸⡀⠀⠤⠀⠀⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣗⠀⠀⠀⡟⠀⠀⠀⠀⢸⠢⠔⠪⠃⡼⠀⡇
⠀⠀⠀⠀⠀⠀⣇⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡟⠀⠀⠀⠀⠀⡀⠁⠀⢀⡅⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⢘⡀⠈⠉⠉⠁⣐⡀
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠚⠃⠀⠀⠀⠀⠀⠀⠘⠂⠐⠚⠃⠀⠀⠀⠀⠀⠂⠀⠐⠛⠀⠀⠀⠀⠀⠀⠘⠀⠀⠚⠁⠀⠀⠀⠀⠈⠓⠂⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⡄⠠⡀⣀⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢐⠃⠇⠁⡇⠀⡇⠀⣼⠏⠀⠁⠀⣦⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⢉⣥⡁⠀⠒⠀⢰⠻⠀⠀⠀⡐⡝⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⡟⠉⠙⢡⠣⢀⠁⠀⠐⣸⠁⠀⠀⠀⡌⠀⠀⠀⠀⠠⠀⠀⠀⠰⡀⠀⠀⠆⠀⠀⢀⠄⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠁⠄⠄⣈⢐⠍⠀⠀⠀⢰⠀⠀⠀⠀⠀⠰⢦⠀⠀⠀⠁⠀⠘⠀⠀⠀⣰⢆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⢧⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⠙⠋⠀⠀⢸⠀⠀⠀⠀⠔⠁⣠⠑⠄⠀⢠⠀⡇⠀⢀⠄⣠⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠈⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠓⠚⠆⠀⠀⡀⠀⢀⠌⢀⠜⠀⠑⠌⠢⠈⠀⡇⠠⢂⠔⠀⠑⢄⠈⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⣧⡤⢤⣤⣀⠀⠁⠀⠌⡠⠂⠀⠀⠀⠈⠢⡘⡀⡱⠡⠂⠀⠀⠀⠈⠢⡈⠆⢀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠨⡀⡜⠐⠁⠀⠀⠀⠀⠀⠀⢱⡜⣰⠁⠀⠀⠀⠀⠀⠀⠐⠌⡌
⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⠃⠀⠀⠀⠀⠀⠀⢀⠏⡸⡉⢇⠀⠀⠀⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⠀⢸⡇⢰⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠈⠄⠀⠀⠀⠀⠔⢡⠎⠀⠈⢢⠑⠄⠀⠀⠀⠀⡐⠠
⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⣀⣀⣠⣤⡶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠠⠓⠐⠒⠒⠀⠤⠀⠀⡇⠀⠀⠀⠘⠊⠀⠀⠀⠀⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠍⣐⣂⣒⣛⣛⣛⣛⣛⣛⣛⣓⣒⣀⠨⡁⠢⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠣⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡘⣐⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘⠸⡀⠀⠀⠀⠀⠀⡀⠠⢀⠀⠀⠀⠀⠀⠀⡇⡟⡈⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠑⠀⠠⠤⠤⢌⢧⣀⠀⠀⣀⢎⣔⣀⣢⡡⣀⠀⡀⣀⡼⣰⠇⠧⢀⡀⠀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⠉⠩⠐⠀⠀⠀⠀⠈⠂⠽⡟⢋⡊⡐⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡋⡸⢶⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⣡⣈
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠰⠄⠀⣵⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠘⢀⠀⡤⣤⣤⣤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢼⣿⣇⠀⠀⣰⠻⠟⠻⡇⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⢠⣿⡿⣻⢟⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠻⠆⠒⠉⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠐⣸⡿⢟⠁⠙⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠠⣀⢿⣠⡀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢿⣿⣷⣭⣬⣵⣿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣁⣻⣇⠀⠀⠀⠀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⢀⡟
⠀⠀⠀⠀⠀⢾⡂⠈⠉⠉⠉⠩⠅⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣧⣀⡀⢀⣀⣧⣴⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⣿
⠀⠀⠀⠀⠀⠸⣦⣤⣤⣤⣤⣤⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣉⠀⠈⠉⠉⠁⠀⢒⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠑⡠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡏⠀⠹⣿⣿⣿⠏⠉⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⡐⠿⠻⣄⣀⠐⠒⠚⢀⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠛⠛⠁⠐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠓⠒⠀⣘⡃⠀⠀⠀⠀⠀⠀⣨⠀⠀⢀⣀⣤⣤⣀⣀⣀⣀⣀⣡
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀⣚⠀⠀⠀⠀⠀⠀⢰⣿⣄⣀⣀⣠⣀⣀⣀⣀⣤⣤⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠿⣿⣣⠄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⣿⠃⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣷⠰⠀⠱⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣿⣁⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠰⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠰⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠴⠾⣖⠛⠰⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠰⢿⢼⣧⣤⣠⡈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠂⠈⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠻⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡴⠀⠠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠀⠀⢤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡚⠁⠀⠀⢀⣀⣠⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣴⢶⣠⠄⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠁⠉⠉⠘⠡⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠒⠀⠤⠄⠀⢄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⡆⠉⠉⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢘⡶⠿⣶⣶⡶⠃⠀⣧⣤⣄⣀⣀⣀⣀⣀⣸⣿⡇⠀⢇⣀⣠⣀⣄⡠⠀⠠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣆⠀⢈⣽⡟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⡿⠋⠙⢷⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⠿⠿⠿⣿⣿⣿⠃⠀⠀⠑⠪⣭⣛⣛⣛⣛⣛⣛⠽⠀⠀⠀⣿⣷⣤⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣦⣤⣴⡿⠿⠟⠀⠀⠀⠀⣾⣿⡿⢻⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠉⣹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠉⢩⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢻⣿⡆⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⡟⢻⣿⣆
⠀⠀⠀⠀⢀⣀⣀⠿⠟⠁⠀⢻⣿⠄⠀⠀⠀⠀⠀⣻⡿⠁⠀⠀⠀⠹⢿⣟⠀⠀⠀⠀⠀⣾⣿⠁⠀⠙⠿⣣⡀
⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⢀⣰⣷⠄⠀⠘⠳⠿⢿⡷⠀⠀⠀⠀⣀⣼⣿⡆⠀⠀⠶⣴⡶⠁⠀⠀⠀⠰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣉⣀⠀⠀⠈⠳⡄⠀⠀⠠⠂⠀⠠⠀⠀⠐⢁⡀⠁⢀⠀⠀⠂
⠀⡐⠀⢀⠀⠀⠰⠀⠡⠀⠀⠀⠀⠋⠀⠀⠀⠑⡆⠀⠀⢹⠀⠀⢀⠀⠀⠀⠀⠁⠄⠈⠀⠀⠀⠀⠀⠀⠘⠀⣶⠾⠛⠛⢛⠒⠦⣄
⠰⠀⠀⠀⠢⡔⠁⠀⠀⡅⠀⠀⠀⡄⠀⠀⠀⠀⣸⠀⠀⢸⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰⠀⠀⠹⣄⣀⠔⠛⡆⠀⠀⣷
⠀⠑⠒⠒⠉⠈⠑⠒⠚⠀⠀⠀⠀⠐⣄⠀⢀⡴⠃⠀⢀⠜⠀⠀⠀⢰⠄⠀⠰⠀⢠⠀⠀⣠⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠙⠶⠚⠉
⠀⣏⠐⠒⠀⠀⠒⠂⢈⠇⠀⠀⠀⠀⠈⠉⠃⠒⠐⠊⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⢢⠴⠁
⠀⠀⠠⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⡙⠀⠀⠀⠠
⠀⠀⠀⠐⠄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠈⠑⠢⢀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠓⢦⠀⠀⠀⠀⠀⠈⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠰⠒⠀⠀⠠⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⣄⠐⡦⡌⢀⢴⠀⡀
⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠢⡀⠀⠀⠂⠀⠀⠃⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠤⡴⡼⠋⠁⠉⠙⢤⢠⠄
⠀⠙⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⠏⠀⠀⠀⠈⠢⡀⠀⠀⡘⠀⠀⠀⠀⠀⠈⠙⡖⠒⢊⠒⣺⠉⠁⠀⠀⠠⠆⣹⠀⠀⠀⠀⠀⢸⡉⠤
⠀⠀⠙⠿⠿⠟⠁⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠘⢦⣀⠇⠀⠀⠀⠀⠀⠀⠀⢻⢀⠠⢐⡇⠀⠀⠀⠀⠀⠚⠰⠣⣀⠀⢀⣠⠺⠔⠂
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⠀⠀⠀⠈⣇⠀⡽⠀⠀⠀⠀⠀⠀⠀⠋⠸⡞⢃⠝⠮⠀⠃
⠀⠀⠀⠈⡄⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠘⠃⢀⡤⠀⠀⠀⠀⠀⠘⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⢶⠿⠀⠁⠰⡀⠅⠀⠀⠀⢴⠿⠿⣿⣿⣿⠿⢗⡆⠀⠈⠀⠀⠏⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠉
⠀⠀⠀⠈⡄⠀⠀⣴⠁⠀⠀⠀⠀⠈⣖⠒⠂⠒⠒⠂⡉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣄⢸⣷⣄⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠠⠁⠀⠀⢻⠄⠀⠀⠀⠀⠀⡷⢌⢄⠀⠀⠠⠂⠀⠀⠀⠠⢀⠀⢰⣿⠿⠛⠉⠉⠀⣶⠀⠈⠂⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⢸⠀⠁⠐⠒⢰⠀⠀⠰⢐⠪⠜⠀⠘⠁⠀⠀⠀⠀⣼⣿⣧⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠈⠂⠩⡭⠐⠁⠀⠀⠀⠀⠀⠈⠂⠤⠠⠄⣊⣀⠀⡐⠁⠀⠀⠀⠀⠂⠀⠀⢀⣼⣿⣿⣿⡔⠀⠀⣀⣂⣴⣿⣀⣀⣀⠴⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠒⠺⠿⠟⠛⠁⠀⠀⠀⠈⢙⡙⠛⢋⡋⠛⠋⣙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠇⠀⢀⣤⣄⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠟⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⢸⣇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⣤⣄⡀⠀⠀⠀⠀⣀⣈⣉⣀⣀⡴⠋⠉⣷⠀⠀⠀⠀⡿⢩⠭⣍⠓⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣻⠀⠤⠤⠤⡀⢻⠋⠉⠉⣉⣉⠉⠉⠀⠀⠘⠂⠒⠉⠉⠉⠉⠙⠃⠸⣀⣠⠀⣏
⠀⠀⠀⠀⠀⢠⡞⠩⢬⠀⣁⠤⠔⠇⢸⠉⠉⠋⠀⠀⠱⠊⠉⠙⠲⠒⠛⠲⠞⠛⠒⠉⠉⠳⢢⣀⣠⠌⢉⡧
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠉⠙⠒⠚⠘⠦⠤⠤⢄⣀⣀⣀⣀⣀⣀⠤⠤⠤⠴⠖⠒⠒⠚⠉⠉⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡒⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠁⠒⠒⠤⢌⣉⣑⡒⠒⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⢒⣒⣚⣉⠭⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠒⠒⠒⠒⠒⠂⠉⠉⠉⠉⠉⠁⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠉⠙⠒⠲⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠋⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⢰⣷⣿⠇⠀⢻⣷⡆⠀⠀⢰⣷⣿⠿⣿⣿⣆⡄⠀⠀⠀⠀⢻⣿⣿⣿⣶⣇⠀⠀⠀⠀⢸⣿⣿⠻⣿⣶⠀⠀⠀⠀⡄⣿⣿⠏⢿⣿⡆
⠀⠀⠸⣿⠁⣰⣦⠈⣿⡇⠀⠀⢸⣿⠿⠀⠘⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠐⣶⡿⠁⡀⠛⣿⠀⠀⠀⠀⣿⡿⠁⠀⢸⢿⡿
⠀⠀⠀⠙⠦⡙⢛⠴⠋⠀⠀⠀⠀⠻⠄⠀⠀⠠⠟⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠁⠀⠀⠀⠀⠹⠄⢾⡿⠀⠋⠀⠀⠀⠀⠙⠧⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⣆⢸⣿⡄⡄⠀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⠀⢠⡄⣾⣿⢿⣿⣀⡄⠀⠀⠀⠀⣶⣻⣿⣷⢀⡄
⠀⠀⠀⢸⣿⣿⣿⣇⣷⠀⠀⠀⢈⣿⣿⠏⠻⣿⣿⣶⠀⠀⠀⣼⣿⣿⠋⢻⣿⣧⠀⠀⠀⣿⣿⣿⠟⠈⣿⣿⣿⠀⠀⢰⣦⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣷⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⡇⢀⣦⠀⣿⠇⠀⠀⠀⢻⣿⠁⠀⣼⢿⡟⠀⠀⠀⠸⣿⠉⢀⣆⠉⣿⠏⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⠀⠀⠀⠈⠁⠘⠟⠁⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠛⠟⠀⠉⠀⠀⠀⠀⠈⠙⠚⠛⠛⠋

⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⢀⢀⣼⣿⣷⡄⠀⠀⠀⠀⡀⠈⣿⣧⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⠀⡀
⠀⠀⢀⣿⣿⣿⢿⣿⣿⡄⠀⠀⢀⣿⣄⣽⣿⡟⢀⡀⠀⠀⠀⡀⢻⣿⠛⢿⣿⣆⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⡀⣿⣿⣿⣾⣿
⠀⠀⠸⣿⡟⠉⠀⠹⣿⡇⠀⠀⢸⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⣿⣾⠋⢠⡀⢻⣿⠀⠀⠀⢠⣸⣿⡏⠿⣷⣾⡆⠀⠀⢸⣿⣿⣿⠛⢿⣿⣿
⠀⠀⠀⠻⣧⡀⠀⣰⠟⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⢿⣄⣿⣧⡼⠃⠀⠀⠀⠀⠻⡏⠁⠐⢻⡟⠀⠀⠀⠈⢿⡿⠷⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠖⢹⡆⠀⠀⢀⡴⠀⠀⣠⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⢻⣿⠏⠀⣾⡇⣠⢄⠘⣃⣀⣰⣟⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣀⣼⡟⢸⡯⠚⣸⡏⣰⣿⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠸⣿⠈⠸⣷⡀⠉⡌⠙⠃⠘⣿⣀⠟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⣴⢦⠀⣈⠛⢊⠁⢠⠖⣦⢀⣤⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡜⢀⣙⠁⣿⡷⢁⣴⣿⢺⣿⠁⢰⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⡟⠀⣼⡟⢰⣿⡗⠷⠟⡡⢾⣿⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠻⣇⠜⠀⢙⠁⣿⡇⢀⡘⠥⠾⢋⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣤⣤⣦⠀⢠⠤⡌⢀⣬⠁⠈⠃⠀⠀⣾⢡⣆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⢀⣿⠇⠀⣈⠣⡧⣬⡍⣰⡦⠊⣿⢠⣶⢲⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⢀⢃⣿⢡⣿⡇⢸⣿⢻⡟⢁⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⠋⠙⠊⠈⠛⠃⠘⠃⠈⠿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡆⢃⣾⠉⣹⡆⠸⠿⢠⣄⡴⢲⡆⢰⣏⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⢉⢵⣿⢣⣿⣏⠔⣾⡟⣸⣷⠲⣦⠐⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣧⠘⠷⠃⣿⡟⠚⠋⡜⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⣨⠟⠀⠀⠀⠹⠇⠀⣀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣀⠤⠔⠒⠉⠉⠉⠉⠉⠉⠉⠑⠲⢤⡀⠀⠀⢀⣤⣾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⡠⠊⠁⠀⠀⢀⡠⠤⠤⠤⠤⠤⡀⠀⠀⠀⠈⢢⠀⠚⢩⣿⡟⠁⠰⠟⢉⣿⠟⢿⣿⣿⣿⣿⣿⣦
⠀⠀⢀⠇⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡆⠀⢸⡟⠰⣿⡶⠃⠘⠁⢸⣬⣿⣿⣿⣿⣿⣿⣀
⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠄⠀⠀⠀⢄⠇⠀⠀⠀⠀⠸⡄⠀⠈⡇⠀⠘⠷⠀⠠⣶⣼⠿⠿⠻⣿⣿⣿⣿⡟
⠀⠀⢇⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠇⢀⠔⠁⠀⠔⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⠉
⠀⠀⠈⠳⡀⠀⠀⠀⠀⠘⠦⠄⠠⠤⠆⠀⠀⠀⠀⢀⠞⠀⡎⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠻⣿⠃
⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⠤⡖⢅⠀⠀⢣⡘⢿⣷⣶⣿⣶⡄⠀⠀⠀⠀⢀⣒⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⢮⠤⢊⠐⡲⢀⡙⣾⣿⠏⠛⠛⠁⢀⣀⡤⣲⠇⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠉⠑⣶⢿⠻⣿⠹⡖⣁⡔⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⢷⣕⠈⠁⠀⠉⠸⣀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠏⡵⠗⡆⠀⢠⠄⠈⠒⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣁⣇⣀⣠⠏⠀⠀⠀⠁⠀⠀⠀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⢠⡤⢴⡤⢦⡄⠀⢰⡷⠄⣿⠴⣄⢨⡅⣤⠴⣄⣴⠆⢰⡷⠀⣄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢸⡇⢸⡇⠀⠸⣇⡀⣿⠀⣿⢸⡇⣿⠀⠀⣙⣷⠸⣇⡀⢹⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣘⣷⣤⣧⣾⡻⠋⠉⠙⠛⠲⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣣⠅⠼⠘⠞⠒⠝⠸⠞⢀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣬⣭⣻⣿⣿⠵⠾⠿⠿⠒⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⢾⡿⢛⡿⣯⣛⠟⠻⣶⣤⡈⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠞⣵⣿⡆⣇⠙⣦⡀⠈⠙⠆⠙⠅⠀⠀⠀⠀⣠⣼⣷⡛⣚⠷⠦⠀⢀⣳⣠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⡄⣤⢻⣿⡄⠹⢶⣽⣿⣄⠀⠀⠀⠐⡀⠀⠐⣣⢞⢿⣷⣬⠈⢳⢤⣴⠞⣻⣿⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢡⠻⠘⣿⡇⠘⠀⢹⡷⠙⠐⠆⠀⠀⠀⠀⣸⠧⠈⠈⣿⣟⢷⡀⢸⡏⢺⡏⢌⠓⣆⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⠀⠀⢠⡀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠸⣿⡁⠀⠀⠀⠁⠂⠰⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠦⣄⡀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠠⡀⠀⠈⠙⢦⣀⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠤⠬⠯⠿⠿⠤⠴⠖⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣆⣤⡀⠀⠀⠹⣿⡄⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢈⣉⡀⣠⣤⠴⠚⠋⠀⠀⢠⢴⠖⠈⠀⠀⠤⠐⠂⠙⢿⣯⢦⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⡤⠶⠶⠶⣶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⠐⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣯⣷⡀⠀⠈⠳
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠤⢤⠤⣤⡤⠂⢀⡤⠖⠲⠿⠟⠛⠋⢀⡾⠀⠀⠀⠀⠀⢀⣤⠖⠒⠈⠁⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⢞⣋⣩⣥⣤⡤⣶⢶⡶⠶⢶⣦⣤⣄⣀⣀⣀⣠⣤⡤⡶⠚⠋⠀⠀⣀⣀⠤⠚⢉⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠠⠤⠤⠤⠤⠴⠶⠖⠒⠒⠂⠉⠉⠉⠉⠀⣀⣀⡤⠨⠀⠀⠴⠒⠊⠁⢀⡀⠄⠚⠁⠈⠀⠀⢀⣜⠛⣋⣲⣆⣤⡤⠟⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢉⣉⠥⠀⠀⠀⠀⣀⠠⠄⠀⠲⠉⠀⠀⠀⠀⠀⣠⡖⣥⣦⡾⢸⡞⠈⢙⡓⠢⠄⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒⠒⠒⠒⠚⠉⠀⣠⠴⠒⠋⠉⠀⠐⠂⠀⠠⠂⠀⢀⠤⠐⠛⠱⠋⠽⠇⠀⠙⠋⠙⠛⠑⠃⠀⠀⠀⠀⠈⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢫⣭⣙⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣀⢁⣂⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠶⠶⣴⡏⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡶⠛⠿⠻⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠟⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⢣⡃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠆⡆⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡌⡈⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠊⠘⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠰⢠⠅⠀⠀⠀⠐⣷⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢠⢢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⡄⡘⠀⠀⠀⠀⠀⠫⠆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢡⠁⠀⠀⠀⠀⠀⠸⢁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠘
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢰⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢦⠦⠤⠐⠒⠀⠐⠂⠤⠤⠒⢒⡠⠃⠀⠀⡇⠈⠢⠤⠤⠤⠤⠔⠂⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠚⠚⠀⠀⠁⠉⠀⠀⠀⠈⠀⠀⠋⠀⠈⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢿⣿⣿⡏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠤⢦⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⡄⠀⠀⠠⠦⠴⡄⠀⠀⠀⠀⠀⢀⡤⠒⠀⡈⠳⢮⣧⣶⣾⢾⠇⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠠⠒⠃⠀⠀⠀⠀⠀⢀⢈⣴⣾⣿⣿⣿⠂⠘⠟⠻⠿⠒⠶⡌⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣭⡏⠻⠀⠀⠀⠀⠀⠀⣆⢘⣁⣠⠤⠟⠛⠫⠿⣹⠿⠟⠋⠑⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⣑⠷⠄⠀⠀⠀⠀⠀⠀⠈⠃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⠛⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⠶⣦⡀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠈⠻⠒⠒⠶⢲⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠙⣄⠀⠀⠀⠀⠀⠰⡇⠀⠐⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢄⠀⠀⠑⠢⠤⠤⣀⣀⠁⣿⣿⠁⠀⠀⠀⠙⡉⠂⠄⠠⠤⠤⠀⠀⠀⢀⡔⢸⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠑⢄⡀⠀⠀⠀⢠⣞⢷⣽⣿⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⡈⠣⡀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠼⠙⢷⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⠲⢤⣠⡤⡇
⠀⠀⢀⠊⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⠢⣑⠐⠢⠤⠔⠁
⠀⠠⠃⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣧⡀⠀⠀⠀⣀⣴⣭⠴⠄⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⡿⠛⠛⠒⠒⠒⠒⠊⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣷⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⠀⠈⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⢀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠰⠳⣄⡎⠀⠀⠁⠠⠀⡜⢆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠈⠆⠀⠀⢀⠀⠰⠁⢸⠈⠰⠃⢰⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡏⠲⢦⠀⠀⠀⠀⠀⠈⢄⠁⠀⢸⠀⠀⠀⡌⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⡘⠘⡐⢚⡇⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⡀⠀⠳⠉⠀⠌⡰⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⣄⡀⠀⠀⠀⠀⠂⠄⠐⠂⠀⠀⠁⠄⠂⠀⠀⠤⠊⠉⠈
⠀⠀⠀⠀⠀⣼⠹⣭⢍⣠⢴⠚⢠⣀⣤⡤⡖⠢⣤⢼⣐⠒⣦⠀⢲⢢⡀⠘⠛⣻⣻⡄⢠
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠻⠼⠛⠛⠊⠳⣾⡴⠃⠰⣌⣠⡼⠷⣤⣸⡖⠲⠮⠯⠟⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣔⠻⢿⣿⣿⣿⣶⣦⡀⣀⣀⣀⣠⣴⠶⠂⠀⠀⠐⠶⣦⣔⢶⣴⣦⣶⣶⣶⣶⣠⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢺⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣿⣛⢿⣿⣿⣿⠻⠿⢛⣻⣿⣯⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⣌⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠒⠋⠻⢿⣟⣿⡮⠀⠀⠀⠀⣰⣿⡿⣷⣶⢶⣤⠀⠀⣸⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠉⠀⠀⠁⠈⠉⠀⠠⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⣀⠀⣠⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠰⠋⠁⣀⠀⠈⠻⡿⠿⠃⠈⡂⢀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⢷⠖⠒⠒⣒⡈⣉⣉⣙⣘⣛⠛⠛⠒⠤⠤⣤⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠉⠉⠉⠉⣃⡙⠉⠁⠉⠉⠁⠀⠈⠉⠉⠙⠛⠋⠓⠒⠤⢌⣑⡆
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠤⠐⠒⠈⠉⠉⠉⠙⠛⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠃
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠠⢴⠒⠚⢉⡉⠖⢂⡦⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠐⢙⡿⠋⠀⣠⠴⣶⣴⢞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠈⠻⣿⡿⠦⠀⠀⢤⣤⣤⣀⣀⣀⠀⢀⡴⠃
⠀⠀⠀⠀⡠⡞⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠛⠶⠤⢤⣤⣀⣴⣿⡿⠿⠟⠁
⠀⠀⠀⣰⢹⡇⠀⠀⠀⠀⠀⠀⡞
⠀⠀⣰⡟⠀⣧⠀⠀⠀⠀⢀⡜
⣤⣾⣿⣷⠄⠊⠳⣴⢤⣶⠏
⠻⣿⣿⠟⠦⠠⠀⢈⠙⡝
⠀⠈⠙⠁⠀⢀⡵⠵⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⠀⠈⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⡰⠋⠀⠀⠈⠉⢢⠀⠀⠀⠀⠀⢀⠞⡰⠊⠉⠀⢼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢐⣁⣀⣀⣻
⠀⢰⠁⠀⠀⠀⡀⠀⢈⣳⣾⣿⣿⣷⠏⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠩⠆⠀⠀⠈⠑⢄
⠀⠈⡄⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠛⠚⠛⠉⡜⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠁⠈⠘⠁⠀⠀⠀⠀⠈⢒⠆
⠀⠀⠘⠢⠄⠠⣠⡀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢳⠄⠀⠀⠀⠀⠀⠀⠀⢱⠤⠠⠀⠀⠀⢀⣴⡏⠀⡴⠖⠁
⠀⠀⠀⠀⠀⡔⠁⠈⡆⠀⠀⠀⠀⠀⠀⡇⠀⢘⠀⠀⠀⠀⠀⠀⡠⠄⠀⠠⣾⣿⠿⠿⡿⠿⡇⠘⠃⢲⠀⠀⢀
⠀⠀⠀⠀⢈⠀⠠⡤⢀⠀⠀⠀⠀⠀⠀⠳⡀⢨⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠙⠒⠳⠄
⠀⠀⠀⠀⠀⢉⠎⠀⣠⠀⠀⠀⠀⠀⠀⠀⡴⢥⣶⠒⢄⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⣀⠤
⠀⠀⠀⠀⠀⠂⠀⣰⠋⢉⡂⠀⠀⠀⢀⣀⣇⡀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⣰⡁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠅⠀⣈⠭⠄⠉⡁⠠⠋⠀⠈⢿⠀⠴⠃⠈⠂⠀⠀⠀⠀⠀⠀⠀⠁⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠒⢆⠀⢠⠊⠲⣄⣀⢠⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢀⣴⢇⠇⠀⠀⠉⠈⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠐⠴⢧⠀⠘⠀⢆⠀⠀⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⠀⠀⠀⠀⢨⠆⠀⠀⢳⠀⠀⠈⠑⠊⡹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠸⠑⢄⠀⠀⠘⡆⠀⠊⠁⢀⠄⠤⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡆⠀⠑⣄⠀⠘⡶⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢡⠀⠀⠈⠳⡀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⢣⡀⠀⠀⠙⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣒⠊⠉⠀⠀⠀⠤⠵⠀⠀⠈⠒⠤⣀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠀⠀⢿⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠙⢦⣌⢷⣄⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⣆⠙⠷⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢛⡓⠦⡀⠈⢷⡀⠀⠀⠀⣿⠀⠀⠘⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⡄⠀⠻⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡆⠀⣠⠟⠀⠀⠀⠹⣄⡀⢻⠀⠸⣇⠀⠀⠀⠀⠀⢰⡇⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⠟⢰⡏⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠘⢷⣄⠀⠀⠀⠀⠻⣄⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⡉⠀⠀⠻⢦⣄⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠉⠳⢦⣄⠀⠀⠈⠳⡄⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⠀⠈⠉⢻⠓⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠃⠀⠀⠀⠀⢠⣸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢠⡞⠀⠀⠀⢀⣤⠟⣿⠃⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠀⠀⢹⡆⢀⡤⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢁⡖⠉⠀⣀⡴⠞⠋⠀⣰⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠘⡇⢸⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠘⢧⣠⠞⠁⠀⠀⠀⠘⠁⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢰⠇⢸⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠁⠀⠀⠀⠀⠀⢠⢀⠞⠁⠀⣰⠋⣠⡾⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠃⢀⣠⠄⣀⣴⠆⢀⠀⠀⠀⠞⠀⠀⠀⠀⠟⠉⠁⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠰⣶⣤⣛⠀⠈⢁⣾⣿⣷⣿⣿⠃⠀⠀⣀⠀⠖⢀⣀⡀⠀⠀⠀⠀⠀⣰⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣍⣻⣦⣄⡀⠀⣀⠐⣀⣈⠛⠻⣟⣻⣿⣿⡿⠟⣋⠀⠀⣠⣤⣤⣤⡶⠿⠛⠁⠀⠀⠀⣀⡼⠯⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠴⠶⠛⠻⢶⣤⣄⣈⣁⣰⣿⡿⣯⣥⣤⣀⡛⣿⣿⣶⣬⣍⠀⠻⠷⠶⢶⣆⣤⣴⣾⣿⣶⣶⣶⠤⠴⠶⠖⣶
⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣤⡤⠤⠶⠶⠚⠋⠉⠉⠉⠉⠁⣀⡤⣿⣿⣿⣿⣿⣿⣿⡿⠿⢶⠲⠶⣿⣿⣭⢤⣭⣍⣛⡛⠛⠶⠦⢤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠋⢑⣻⡿⠟⠋⢁⣀⡈⠉⠉⠉⠋⠉⠛⠒⠾⠭⣅⣀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⣯⣉⣀⠤⠖⠚⠉⠉⠀⠠⠖⠛⢉⣀⡙⠚⠙⠛⠶⠶⢦⣄⣠⣤⡴⠶⠯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁⢀⠤⠜⠓⠐⠀⠀⠀⠰⡇⣦⡔⡉⠤⠠⢾⠀⠀⠀⣀⠃⠀⠀⠈⠈⣡⠃⠀⠈⠀⢀⠤⠚⠃⠒⢀⠀⠀⠸⡑⣤⢐⠩⠄⠠⡎⠀⠀⠀⡐⠁⠀⠀⠉⢈⡔⠁
⢿⡃⠀⠀⠀⣠⣾⣄⠀⢀⠜⠁⢀⠤⠄⠀⣨⢀⡠⢴⡉⠡⠖⠊⠉⠈⠉⠉⡹⠶⣾⠁⠀⠀⠀⣠⣿⡀⠀⣠⠋⠁⢠⠤⠀⢀⡌⢀⠠⢞⠉⠰⠒⠊⠉⠉⠉⢉⡵
⠀⣱⠖⠦⠐⠋⠈⠉⠙⡿⠓⠒⠛⠓⠢⣤⠼⣏⠀⠀⠀⠀⠀⠴⠶⠦⠐⠾⠀⠀⠀⡶⠖⠆⠘⠉⠉⠉⢙⡟⠒⠒⠛⠒⢦⡤⢿⡅⠀⠀⠀⠀⡠⠶⠶⠆⠰⠮
⠨⢂⠀⣀⡴⠤⢤⣤⠜⠁⢀⣠⠄⣄⣀⣀⣠⠌⠉⠟⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠹⣀⠀⣀⡴⠤⢤⡤⠞⠀⣀⣠⠤⣄⣀⣀⣠⠈⠉⠟⠉⠀⠀⠀⣀⡀
⠛⠋⢉⣼⣄⠀⣀⣠⠀⠉⠈⢂⠀⠀⠀⠀⠴⠞⠀⢀⣀⡤⠔⠋⠉⠉⠀⠀⣨⠚⠛⠋⣁⣼⡀⠀⣀⣠⠂⠁⠙⢄⠀⠀⠀⠀⠴⠇⠀⢀⣀⡤⠒⠉⠈⠉⠂⠀⡥
⢴⡊⠀⠀⠈⠉⠁⠘⣦⠤⠐⠈⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⢀⡠⠐⠒⠚⠳⠦⣴⠊⠀⠀⠉⠉⠀⢹⣄⠄⠒⠋⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⣀⠤⠒⠒⠛⠲
⣀⣧⣀⡀⠀⢀⠠⢘⣻⠶⠟⠋⠀⠀⡠⠴⠦⣾⡤⠾⠿⠛⠞⠉⢿⠀⠀⠀⢀⣀⣈⣧⣀⡀⠀⢀⠤⣚⡷⠶⠚⠁⠀⢀⡠⠴⢴⣦⡴⠾⠟⠳⠞⠙⢗⠀⠀⠀⣀
⠀⠀⠀⠈⡹⡡⠞⠛⢦⡤⠖⠙⠒⠋⠀⠀⢀⡏⠀⠀⢀⡴⡀⠀⠀⠈⣻⠋⠁⠀⠀⠀⠀⠈⡟⡴⠚⠛⢤⡤⠊⠙⠒⠉⠀⠀⢰⠃⠀⠀⢀⠜⡀⠀⠀⠙⡿⠉
⢄⠀⢀⡴⠗⠁⠀⠀⣠⠗⠊⠀⢀⣀⠀⣠⠞⠀⢀⣄⢀⡈⡤⠖⢉⠚⠧⠤⣠⠄⠄⠀⢀⠼⠊⠀⠀⠀⣸⠗⠁⠀⣀⣀⢀⣠⠋⠀⢀⡆⢀⠀⡤⠚⢁⠛⠧⢄⣠
⠔⠛⡋⠀⠀⣀⡤⠚⠁⣀⠔⠈⠀⡘⠉⠛⠦⠒⠋⠉⠠⠀⣸⣤⣾⡀⠀⠀⠀⡠⠚⠛⡃⠀⠀⣀⠄⠛⢁⣠⠀⠁⢀⠘⠉⠓⠄⠚⠉⠉⡤⠀⣧⣤⣆⡀
⣠⣒⠁⠐⠈⢿⠆⠉⠀⠉⢧⠀⡸⠁⢀⠔⠀⠀⣀⡀⠤⡦⠋⠁⠀⠈⡣⠐⠔⢁⣴⣒⠥⠐⠉⡿⠊⠁⠈⠙⣆⠀⡞⠀⣠⠄⠀⠀⣀⡠⢤⠜⠉⠁⠀⠉⡦⠂⠊
⠀⠀⠀⢀⡤⠻⣀⣀⣀⣤⣊⠴⠁⣲⣁⠄⠚⠉⣡⣴⠊⠀⠀⠐⠥⠺⠤⣤⠔⠉⠀⠀⠀⢀⠔⢧⡀⣀⣠⡴⣣⠾⢐⣞⣁⠔⠚⠉⣠⣴⠃⠀⠀⢀⠕⠺⠤⣤⠞
⠀⠀⡠⠊⣀⡔⢹⠁⠀⠀⣠⠖⠋⠉⣡⠤⠒⢈⡡⠋⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⡐⠁⣀⡌⢹⠁⠀⠀⡤⠚⠉⠉⣠⠔⠂⢉⡵⠋⠀⠀⠀⠀⣀⣤⡾⠃
⢠⠞⠀⡈⠘⠁⣼⡀⠀⣠⡧⠀⠲⢔⣡⢶⣯⠥⠤⠤⢤⣀⠠⠊⠀⣠⢧⡠⢤⡄⡠⠎⠀⡁⠙⢀⣾⠀⢀⣼⡤⠒⠦⢎⡠⣶⣯⠤⠤⠤⢤⣀⠔⠈⢀⡼⢇⡠⢤
⣴⣀⣘⣀⡴⣚⣀⣉⣉⣙⣦⣐⣚⣋⣠⣮⣌⣿⣀⣀⣀⣠⢂⣐⣉⣀⣤⣠⣞⣁⣄⣀⣚⢠⡖⣋⣀⣉⣉⣻⣤⣒⣛⣉⣴⣧⣬⣟⣀⣀⣀⣸⣂⣚⣉⣀⣄⣰⣊
⣿⢿⠿⠿⢿⠿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣻⣛⣿⣟⣿⣿⣻⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡁⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣤⣯⣼⡴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⠿⢿⣿⣟⣋⣽⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣄⡀⠉⠋⣤⣿⢷⣄⡀⠀⢦⡀⠀⢎⠃⠀⡀⣀⣘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢰⡾⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⣠⣄⣬⡄⠘⣒⣹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠈⢩⡟⠛⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠚⠉⠉⠀⠀⠒⠲⠤⣀⣀⠤⠀⠀⠀⢀⠀⠀⠠⠋⠀⢠⢿⡏
⠀⠀⠀⠀⠀⠰⣡⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⢀⠀⠙⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠚⠳⢤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡁⠐⠚⠉⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⢤⣥⣸⣧⠐⠀⡂⠀⠠⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣴⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⠉⠁⠀⠈⡽⠳⢄⡀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⡀⡀⡞⠁⠀⠀⠈⠓⠦⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠆⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠂⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠢⠴⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⡀⠀⠀⠀⠀⠀⠀⣠⣠⠴⠶⠖⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣭⣥⣶⣖⣶⣶⣬⣵⠀⠀⢀⣀⣴⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⢿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⣠⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠉⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠋⠉⠉⠉⠉⠉⠑⠒⠲⠤⣄⡀
⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣠⠔⠒⠂⢄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⡇⠰⢻⡟⢲⣷⡖⢘⣃⠒⣄
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⢂⣉⣭⠭⠿⠄⡀⢀⠩⢾⣇⣻⡉⠃⢸⡇
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⢀⣤⣴⡟⠋⣩⣶⣾⠟⠿⠋⠉⠉⠀⣈⣙⣛⣤⣾⠃
⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢲⣾⠟⠉⠀⢤⣾⣿⡌⠁⠀⠀⠠⣤⣉⣛⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠘⠛⠃⠐⠛⠻⢶⣝⠋⠙⠂⠀⠀⠀⠀⢲⣶⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠉⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠈⠙⢩⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠖⢦⠀⠀⡤⠤⠤⠤⠴⠒⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢲⠾⠀⠀⠀⠐⠻⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠘⡇⠀⠀⠀⠀⢀⡷⣄⠀⠀⠀⠀⠀⣀⣤⠶⠶⠀⠀⠀⠀⠸⠆⠀⠀⠀⣀⣴⣛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⣦⡀⠀⠀⠀⢅⢈⡃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡉⢭⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢻⡏⠙⠷⠀⠀⠀⠀⠀⠙⠛⠿⠂⠀⠀⠀⠈⢿⡟⠋⠉⠁⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⢓⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⢻⡙⠛⠉⢉⣱⠖⠀⠀⠀⠀⠀⢀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠻⣿⣿⠿⠋⠀⠀⠀⠀⠀⣠⡞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⣁⡤⠤⠤⠠⠄⠼⠄⠀⢠⢤⡄⠀⠀⢀⣴⣾⠟⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣴⣾⡿⠿⠋⠀⠀⠀⠀⠀⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⢀⡀⠀⠳⣦⣠⡀⠀⠀⠀⠀⠀⠀⢠⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡁⣄⠀⠀⠀⠀⣀⠀⠁⠀⠠⠬⠉⠓⠢⣄⠀⠀⢠⣴⠛⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢸⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⣴⡟⠉⠀⠀⠀⠀⠀⣀⣠⣷⣶⣶⡖⢘⣧⡄⢰⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⢀⣾⣿⣿⡀⠀⠀⠀⡀⣀⣤⣶⣾⣿⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⣿⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⣿⣷⣄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⣀⣀⠀⠀⢠⠀⠀⢄⠀⠀⠐⡄⠀⠀⠀⢀⡠⠤⠄⠠⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣄
⠊⠉⡀⠀⠀⠀⠀⠈⠱⠒⠦⡀⠀⠆⢀⠀⠚⠀⠀⣀⠨⠤⠀⢀⠠⠀⠛⠁⢄⡀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠂⠀⠈⠤⠶⢤⣖⡀⠀⠉⠈⠀⠀⠲⡀⠔⠓⢤⠚⠃
⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠈⠢⠚⠀⠀⠀⠄⠀⠀⠐⡖⠂⣠⣤⠀⠀⠀⠀⠙⣤⠤⠀⠒⠂⠀⠉⠁⠀⠀⠀⡀⠐⠀⠀⠀⠉⠢⡀⠀⠀⠴⠦⡁⠀⠀⠀⠡⠶
⠀⠀⠀⠀⢌⠀⠀⠀⢁⢀⣀⠀⠀⡡⠀⣛⡀⠀⠀⠐⠃⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⢄⠀⠀⠀⠀⠀⠂⠁⠀⡠⠠⠀⠀⠄⠀⢈⠳⢄⠀⣀⡕⠄⡠⠔⢥⡀
⠀⠐⠒⠂⣀⠔⠂⠉⠈⠀⢉⠈⡙⣤⢤⠇⠙⢤⠠⠄⢀⡀⠋⠀⠐⢄⠒⠒⠀⢀⠠⠼⠀⠀⠧⠄⣀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡀⠀⠀⠡⡀⠑⢌⠀⠐⠗
⠀⠀⡠⠀⠓⠀⠀⡤⠀⠒⠉⠀⢀⡈⣎⠀⡠⠊⠉⠉⠀⡇⠀⠒⢀⠀⠑⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠡⠁⠀⠀⠀⠧⠀⠀⠀⠐⠄⠀⠑⢄⠀⠀⣀⠄
⢀⠎⠀⡰⠉⠈⠣⣀⠀⠀⠀⣩⣄⡄⠈⣱⣷⡄⠀⢴⠏⢀⣄⣀⠀⠈⠀⠈⠆⠈⠂⢀⣤⠀⠀⠀⠀⠀⠀⠀⠝⠳⠄⠀⢤⣀⠀⠈⡀⠀⠀⠀⠄⠀⠠⡑⢎
⢃⠔⢨⠅⠠⠀⠊⠀⢁⡦⠁⡸⠂⠘⡏⠀⠀⢰⠀⣈⣶⠂⠈⡄⠀⠉⠁⠒⠬⡄⠒⠀⠀⠀⠤⠤⠤⠤⢤⠎⠰⠇⠀⠀⠀⠀⠓⠢⣄⠠⠤⠤⢈⣂⠀⠀⠄⠳⡄
⠠⠄⠚⠀⠀⠀⠀⠈⢣⡀⠀⢠⣴⡈⠉⠑⠲⡗⠻⣄⢈⣶⣍⠈⢑⠶⠀⠀⠀⠀⠒⠢⠀⠐⠒⠒⠒⠒⠚⠒⠓⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠠⠀⠈⢩⠉⠉⠁⠚
⠀⠀⡆⠀⠀⠀⠀⠀⠀⢹⠂⠉⠛⠉⢲⠖⠋⠀⠀⠈⠁⣉⡇⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠆
⠀⢠⠄⠀⠀⠐⠠⠄⠀⠨⠤⢤⠚⠤⣈⡆⠀⣴⣦⠉⠉⣶⡎⠉⠁⣰⠲⡀⠀⢰⠋⠉⠉⠙⢲⣼⠋⠉⠉⠑⢦⢰⡖⣄⠀⢠⢤⡄⠀⠀⠀⠀⡁⠪⠥⠤⠤⠤⠄
⠀⢸⡠⠀⠀⠢⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⣏⣿⢀⣠⡏⡇⠀⢠⠏⣀⢳⠀⢸⡇⠀⠛⠀⣸⢻⠀⠈⠛⠀⣸⠃⠹⣼⢶⣫⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠖⠆
⠀⢻⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣉⢹⠛⠒⡗⡇⢠⣏⣀⣉⡀⢧⢸⢰⡖⠒⠋⠁⢸⢰⠖⠒⠋⠁⠀⠀⠘⣲⡇⠀⠀⠀⠀⠀⠀⣄
⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⢨⣿⠀⠀⣷⠇⣾⠼⠁⠀⠱⠼⠦⠶⠇⠀⠀⠀⠸⠾⠀⠀⠀⠀⠀⠀⠘⣯⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣀⢠⣤⡀⣠⢤⠀⠀⡴⢦⣴⢲⡀⠀⠰⠒⠆⠖⣦⠀⢀⡖⢶⢠⣤⣤⣤⣤⡀⣤⣤⣤⣀⡀⠀⠀⠀⡗
⠀⠀⡆⠀⢀⠠⠀⡀⠀⠀⠾⠀⡇⢠⠏⡴⠒⢦⣿⣜⢺⠀⠀⡏⣿⡿⣀⣇⢠⢷⡀⣆⡆⣸⡄⡸⣧⢸⢾⣰⠒⠚⠚⠃⡆⠀⣤⠀⠙⡆⠀⠀⡇
⠀⠈⣵⠚⠀⠀⠀⠈⠁⠀⢸⠀⡇⠈⠳⢯⣷⠲⣄⢸⣼⠀⠀⡇⣼⣯⣿⣸⡼⣸⠞⢃⣤⣧⣷⡇⣿⢺⢸⢈⣯⣉⣹⠀⡇⡀⢀⣠⠜⠁⠀⠀⡃
⢀⠞⢰⡀⠆⠁⠀⠀⠀⠀⢺⠀⡇⠸⡟⢧⣬⡷⣹⢳⡜⠦⠴⢃⡽⡇⣺⣏⢣⡟⣤⣾⠀⡿⡈⣼⢋⣸⢸⣹⣀⣀⣀⡀⠇⡟⢦⡑⢦⡀⠀⠀⡅⠀⠈
⠊⠀⠀⣇⠰⠀⠀⠄⠀⠀⢸⠀⡇⠀⠉⠓⠒⠛⠁⠀⠙⠓⠒⠚⠁⠻⠏⠙⠚⠀⠛⠛⠿⠇⠓⠋⠘⠛⠸⠾⠶⠚⠺⠋⠛⠃⠀⠉⠉⠁⠀⢠⠄⣀⠤⢄
⠠⡀⠀⢹⠀⢃⠘⠀⠀⠀⠘⠤⠇⡀⣀⠀⢀⢀⡀⠀⠀⡀⡄⠀⠀⣠⢠⡄⠀⠀⠀⠄⠀⠄⠒⠀⠀⠀⠤⠤⠤⠄⠤⠰⠀⠠⠖⠀⠐⠐⠒⠘⠁⠁⠂⠀⠀⠐
⠀⠈⠚⠀⢃⠈⢄⠠⠂⠀⠀⠀⠄⠠⠀⢀⠸⣀⣠⣶⡄⡅⡇⠐⠴⢁⠎⠄⡀⠀⠀⡀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠠
⢀⠖⠀⠀⠘⣄⠚⠄⠀⠀⠀⠀⠤⠠⢰⡁⡀⠈⣧⡉⠀⢱⣇⣠⢁⡆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⢠⣔⡂⠁⠀⠀⡤⠓⢌⠑⢤⡀⠀⠀⠀⠈
⠁⣀⠀⠀⠀⠸⡀⠚⠀⠀⠀⠀⠀⠠⡉⠀⠀⠠⠿⢶⠀⠉⠐⡄⠉⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡗⠉⠀⠀⠀⠀⠀⠦⢤⡼⠁⠐⠙⢆
⠙⠛⠛⠁⠀⠀⢱⠀⠀⠀⠀⣶⡀⠐⠉⢢⣰⠁⠀⠀⣣⣠⡾⢔⡊⣡⣷⡆⠀⠀⠀⠀⠀⢀⡠⠒⠒⠆⠀⠀⠀⡎⠀⠀⠀⢀⠂⡀⠀⠀⠀⠀⠀⠀⠱⡈⠆
⠀⠀⠀⠠⠄⠤⠀⢧⠀⠀⠀⠽⡓⠉⠞⠙⠴⢇⢲⡖⣏⠀⠀⠀⠵⠈⠁⠐⢄⠀⠀⠀⢠⠋⢀⠎⣁⠛⡄⠀⢸⡇⠀⠀⠔⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠈⠐
⠀⠀⠀⠀⢀⠠⠀⠀⢣⡀⠈⠓⠿⣶⣀⠀⠀⠘⠷⠎⠁⠀⠈⢳⣤⡆⠀⠰⠦⡀⠀⠀⠘⢆⠈⠎⠾⣰⠃⠀⠘⡇⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠃⠒⠊⠁⠀⢣⡀⠀⡰⠋⠛⠄⠀⠀⢴⡰⣀⠀⡠⠊⠙⢃⠐⠀⠀⡘⠴⠆⠱⠀⠙⠒⠊⠁⠀⠀⢀⡐⣆⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡰⠁
⠉⠀⠀⠀⠀⠀⠄⠂⠀⡐⠁⢲⡀⡀⠀⠘⠒⠖⡁⢠⡌⠉⢀⡀⠀⠀⣦⠴⠚⠁⠀⠀⠀⠀⠀⠐⠒⠂⠁⠂⠁⠀⠈⠢⣀⠀⠀⠀⠀⢀⠌⠀⠀⡠⠞⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠤⠘⠉⠑⠠⠤⠤⡼⢀⠄⠈⠉⠣⢈⠁⢀⠁⠀⠀⠀⠀⢀⣠⠄⠀⡤⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠤⠾⠐⠒⠉⢀⡀⠤⠂
⠀⠀⠀⠁⠉⠐⠈⠀⠀⠁⠀⠀⠀⠐⠒⠀⠐⠒⠂⠈⠁⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠘⠓⠒⠒⠒⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣁⠀⠀⡀⣀
⣿⣿⡟⠻⠟⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠠⠀⠀⠀⠤⠀⠀⠠⠀⠀⠠⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠐⠄⠁⠀⠀⠂⠀⠀⠠⠀⠀⠀⠠⠖⠒⠓⠒⢄
⠀⠀⠀⣀⠀⠀⠂⠀⠀⠈⠂⡀⠂⠀⠀⠐⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢀⡀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣠⡦⠶⣂⡠⣀⣰⠦⢴⡄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠀⠀⡀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⣗⣾⠀⠀⠀⠘⠋⠈⠀⠀⠠⠀⠁⠀⠀⠂⠀⠀⠬⣘⠒⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣀⠑⠂⠴⠊⠁⠈⠱⢦⠀⢄⡤⠀⢆⠁⠀⠀⠀⠀⡌⠈⠀⣀⡠⠋⠁
⠀⠀⠀⠀⠀⠅⡀⠘⠀⡀⠊⠁⢠⠀⠃⣀⣾⣒⣛⣋⠉⠉⠉⣙⢂⣋⣛⣺⢄
⠀⠀⠀⠀⠀⣀⠁⠠⠂⠀⠰⠀⠀⠐⠀⠀⠀⠀⠁⠉⠈⠉⠉⠉⠉⠈⠀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣈⠂⡀⠈⢠⠀⠁⡀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂
⠀⠀⠀⠀⢀⣐⣄⠐⠂⠤⠌⠠⠤⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣠⢦⣐⠂
⠀⠀⠀⠈⣞⠐⡼⠆⠐⠆⠀⠀⠠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⣂⢸⠡
⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠓
⣦⠤⠦⠤⠤⠤⢤⡤⢤⡀⠖⠒⠶⠤⠦⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠛⠒
⠀⠶⠶⠶⠦⠠⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠦⠤⠤⠄⠀⠀⡀⠠⠄⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⡀⢀⠀⣤⣤⣄⣀⣀⣀⣀⣀⡀
⣦⣿⡿⠛⠿⠯⠿⠿⣿⣿⣿⡟⠛⠒⠒⠛⠙⠛⠉⠉⠉⠉⠉⠛⠛⠉⠁⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠍⠩⠛⠛⠓⠀⠛⠉⠛⠛⠛⠛⠛
⣿⣿⣿⡿⠶⠶⠶⢶⡶⣖⣀⣠⣀⣀⠀⠀⠀⠀⠤⠄⠀⣠⣤⣀⣀⡀⢀⡀⠀⠄⠀⠠⡄⠀⠠⠄⠀⠤⠤⠄⠀⢀⡀⠤⠠⠤⠤⣤⠄⢀⣀⡀⣀⣀⣀⣀⣀
⢠⣤⣤⣄⣀⣀⣴⠔⠀⠀⠀⠤⠤⣤⣤⣤⣠⣤⣤⣄⣠⣄⣠⣄⣀⣀⡀⣀⣀⠀⣀⣀⡚⠛⡛⠚⠛⠒⠒⠂⠲⠤⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤⠬⠷
⠀⠀⠀⣀⣈⣱⣶⠶⠶⠶⠶⠶⠆⠀⠐⠶⠄⠀⠠⠩⣭⣴⣶⡦⢤⣭⣯⣍⣭⣉⣉⣉⣙⣛⣛⣛⠛⠓⠒⠀⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠘
⠀⠀⠶⠿⠿⠟⠉⠉⢉⡙⠙⠋⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⢀⠀⠀⠈⠀⠈⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉
⣿⡟⠒⠶⠦⢤⡄⠀⠈⣄⡁⠀⠀⠀⠀⠀⠄⠀⠀⣤⠀⠀⠀⠛⠓⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠀⠀⠀⠀⠠⠀⠀⠐⠂⠈
⠐⠒⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⠄⠄⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠠⠤⠒⠀⠠⠤⠤⠤⠤⢤⠀⠒⠒⠒⠀⠠⠤⠄⢠⣄⠀⠀⠲⠴⠀⠠
⠤⠄⠀⠀⠀⠈⠛⠛⠛⠋⠁⠁⠀⠀⠀⢀⣀⠀⣀⣀⣀⣐⣒⣶⣶⣦⣤⣄⣹⡟⠛⠙⠀⣐⣒⣲⣶⣶⣶⣶⣦⡴⠀⣀⣠⣄⣘⣓⠀⠠⠬⢤⣭⠙
⠀⣀⣀⣀⣀⣀⣀⣲⣤⣤⣤⣤⣤⣤⣤⡶⢷⣾⣿⣻⡿⢛⠛⠛⠶⠶⠶⠶⢤⣤⣄⣀⡛⠛⠛⠛⠿⠃⠠⢤⣤⣤⣤⣍⣉⣛⣉⠉⠙⠚⠛⠻⠗⠒⠒⠂
⣭⣭⣭⣍⣉⣙⣛⡛⠿⠿⠶⠶⣶⠿⣿⣿⣶⣾⣯⣭⣭⣭⠽⠽⠿⠐⠒⠶⠶⠶⠾⢯⣭⠉⠉⠙⠿⠶⠶⠶⠶⠶⠾⠍⠉⠉⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⢀
⠆⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠠⣄⣤⣤⠤⠶⠶⠷⠿⣿⣿⣟⣛⣛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣾⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣏⡟⠉⠑⢄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠟⠛⠛⠛⠋⠙⣿⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣃⣀⣀⣀⣠⣤⣀⣰⡄⢸⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⠁⠀⢠⣴⣶⣖⡤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠉⠚⠻⣧⣀⠀⢀⠀⡠⠊⡼⠀⠳⡤⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡇⢿⠳⣌⠻⣿⣿⡁⢸⢷⡀⠀⠀⠀⠀⢀⢀⡴⠒⡜⢁⣼⡷⡠⠀⠀⠘⡷⣤⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣮⡑⣿⡁⠹⢿⠏⠀⠈⠀⠀⠀⠀⠀⠀⢁⠴⣶⠴⠛⠛⠛⠓⠀⢄⠀⠘⠎⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠉⠛⠢⣄⡘⠃⠉⢰⣾⠂⠁⣤⣄⡈⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⢃⠀⠡⢽⣼⣶⣬⡛⣻⣷⣶⢟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣿⢻⣿⠟⢷⣤⣄⣀⣀⠀⢀⣠⣤⠤⠐⠃⠀⠀⠀⠀⠙⢶⣄⠀⠉⠙⠻⡈⠿⣷⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠉⠈⣿⡆⠀⠀⠉⠛⠛⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⠀⠀⠘⠶⠋⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠘⢷⣠⠀⠸⣿⠀⠀⠀⠀⠀⡼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣵⡆⠀⠀⢇
⠀⠀⠀⠀⠀⢀⣠⠟⢆⠀⠀⠙⢷⣴⣿⣦⠀⠀⠀⣼⡁⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣷⣟⣠⠴⠛⡆
⠀⠀⠀⠠⠚⠉⠀⠀⠈⢣⡀⠀⠀⠈⠉⠛⠧⣄⣰⠃⠉⠆⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢰⠋⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢰⣶⣆⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠠⣄⠀⠀⠀⢸⣻⢶⡠⡀⢹⠄⠀⠘⠇⠀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠐⠀⠠⡟⠀⣤⠂⠈⡇⠀⠀⠘⣿⡗⠋⠡⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣵⣄⣀⡀⠀⠀⡀⣀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢨⣍⣀⣀⡀⠀⠀⠀⢈⣻⣍⣏⣐⠞⠁⠀⠀⠀⣿⣇⠶⢦⠠⠈⠳⠈⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⠖⠂⣤⠄⠀⢠⠋
⠀⠀⠀⠀⠀⠀⢸⣡⣍⠁⠿⢀⠀⠀⠈⠠⣿⠋⠁⠀⠀⠀⠀⠀⢺⣇⠳⣀⠶⠦⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣌⠁⠖⡔⠁
⠀⠀⠀⠀⠀⠀⠸⣿⢿⡷⠾⠽⠶⢶⠖⠈⣻⠀⠀⠀⠀⠀⠀⠀⣾⣄⢴⣟⡀⢠⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠈⠏
⠀⠀⠀⠀⠀⠀⠀⢿⡟⠓⠂⠀⢶⠀⢠⠋⡏⠀⠀⠀⠀⠀⠀⠀⡿⣟⣟⠋⠀⠻⠄⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⢸⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠟⠳⢴⡀⡁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠸⡏⠀⠀⠈⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⣄⠀⠣⡖⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣋⠠⠃⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠄⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡛⠛⠓⠓⠛⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⢀⠠⠐⠀⠀⠡⠄⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡄⠠⠠⠄⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡟⠛⠉⠀⠀⠈⢦⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣤⠀⠀⠠⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡆⠒⢀⡬⠷⣄⣻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿⣀⠀⢀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⡛⠳⠖⠉⠙⣣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠿⠛⠋⠉⠁⠿⠀⠀⠀⠀⠀⠀⣄⢦⢀⣠⠀⠀⢠⣠⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠯⢅⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠎⣤⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⣧⣄⡰⠂⠁⠈⠣⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⡀⡀⠂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠹⡀⠶⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⡀⠀⠈⢀⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠾⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠶⠄⠸⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠀⠐⠁⠀⠀⠀⠀⠀⠀⠙⠚⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⢀⣀⠀⠀⣀⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣿⠽⣷⠀⠀⠀⠀⠀⢸⡏⢹⡇⣸⢿⡀⢸⡏⣿⠈⢹⡏⠉⣷⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣾⣿⠀⠀⠀⠀⠀⢸⡷⠾⠃⣿⢸⡇⢸⡷⣿⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣾⡄⣿⠀⠀⠀⠀⠀⠸⡇⠀⠰⡟⠙⡧⢸⠇⢿⠀⢸⡇⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠑⠛⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⣤⡄⣤⣤⠀⢠⣤⡄⢠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⡇⣿⣿⡆⢸⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡹⡀⠀⢀⣀⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⣿⡽⣇⣿⢻⡇⢸⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⠂⠀⠀⡈⣆⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⣿⡇⣿⡏⢸⡇⢸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠘⠉⠐⠒⠃⠀⠈⠑⣄⠀⠀⠀⠀⠛⠀⠀⠙⠁⠙⠁⠙⠁⠘⠃⠘⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢀⠜⠃⠀⠠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠈⠗⠀⢀⠤⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⠀⠀⠀⠀⠀⣀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉⠁

⠀⠀⢀⡀⠀⡀⠀⠀⠠⢀
⠀⠀⠀⠘⣠⠆⠀⢰⠔⠃⠀⣀⠤
⠀⢷⡀⠐⢿⣆⢀⠜⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⢳⡿⠀⠈⣿⣿⢠⡴⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠘⢷⡀⠀⣿⣿⠟⠀⠀⠠⠄⠀⠠⠏⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡀⠀⠀⠀⠄⠀⠐
⠀⠀⠈⠻⣿⣿⢃⣠⡀⠀⠀⠀⠀⠒⠛⠃⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠈⣉⣀⠀⠤⠄⠀⠀⠉⠉⠁⠀⠀⠘⠛⠛⠂
⠀⠉⠀⠒⣿⣿⠀⠀⣀⣀⡤⠠⠤⠄⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⣤⣤⣤⣠⣄⣀⣀⣀⣀⣀⡠
⠀⢀⣤⣾⣿⡿⠞⠂⠀⠀⠀⣀⣀⣀⣀⣀⡀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣶⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⢹⣿⠏⠛⠲⣤⣤⣤⣠⣤⣀⣤⣄⣤⣴⣆⣀
⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢀⡀⠀⠀⠘⢛⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤
⢿⣿⡿⠓⠀⠀⣀⡀⠀⠁⠀⠀⠀⠀⠉⠒⢤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠶⣤⣄
⣿⣿⣟⣦⣴⠿⠋⠀⠀⠀⠀⠀⠰⠦⠀⣰⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⢄⠐⠀⢀⠀⠀⣀⣠⣼⣿⣿⣶⣿⣿⣦
⣿⣿⣿⣿⣧⣤⣤⣄⠀⠀⣀⣀⣀⣤⣾⣿⣃⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣰⣾⠷⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢
⠀⠀⠀⠀⢘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠪⣷⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⣄⠀⠀⠀⢀⡀
⠀⠘⠷⣄⠂⠈⢽⣇⡕⠏⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠳⠲⠄
⠀⢠⣄⠙⢻⡀⠱⢃⣴⣤⣧⣦⣶⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⣀⡀
⠀⠀⠨⢧⠈⢥⣠⢀⣌⡀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠑⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠈⣣⠜⣉⡁⠙⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡜⢲⠟⠛⠷⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⡆⠃⢀⢠⠇⡄⢠⠃⡄⢀⠄⠀⠀⠀⠀⠂⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡰⣌⢆⠣⢱⣠⣘⣘⡰⠡⢃⠎⡰⢋⠀⠀⣀⠀⠀⠀⢀⠀⡀⠀⢀
⠀⠀⠀⠠⠔⠀⢀⠀⠈⣇⠀⠀⠀⠐⠦⣄⡐⠦⣀⠦⡑⢌⠾⠋⠉⠀⠀⠀⠉⠓⢮⢊⠴⣋⠴⠊⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⢈⠳⠀⡿⠒⠒⠒⠠⠤⣀⣉⠓⠬⡑⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠵⢚⣁
⠀⠀⠀⠀⠁⠀⠈⠂⢧⢈⡉⠓⠶⠆⠀⠠⠬⢭⡑⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠭⠤⠒⠒⠉⠁
⠀⠀⠀⠛⠀⠀⠀⠨⠶⢦⡉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈
⠀⠀⠐⠂⠀⠈

⠀⠀⡀⡀
⣤⣤⣉⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⠴⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢁⣴⠋⠀⡎⠀⠇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠁⢀⣼⣠⡴⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡰⠀⠀⠀⠀⠐⠄⠐⠀⠒⢭⡀⠐⣿⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⢈⠛⠄⢄⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡿⠀⠀⡶⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠈⠒⢠⣀⠀⠀⠀⢀⡞⠀⠘⢱⠿⠌⢀⠚⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⢀⡖⠋⠀⠀⠉⠃⠘⠊⠙⠁⠀⠈⠂⠀⠉⣡⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⢟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠄⠀⠀⠘⠀⢹⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣜⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠻⡄⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢰⣴⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⢈⣄⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⣶⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⢀⣾⣿⣿⣦⣀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣏⢸⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⣜⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡄⢨⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣇⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣼⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⢀⡴⠀⢀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡏⠀⠐⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠀⠀⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣄⣀⣀⣀⣤⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠉⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢿⣿⢟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣛⣛⢫⣶⣿⣶⡝⢛⣛⣿⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢸⣿⣿⣿⡿⢿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣮⣛⣿⣛⣵⣿⣿⣷⠀⢛⣛⣟⣛⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⠀⣿⣿⣿⣿⣿⣶⣭⣿⣭⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠉⠉⠙⢿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⡚⣛⠛⡳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣯⠐⠉⠁⠰⠙⠀⠀⠈⠰⠬⡛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⡞⠠⠀⠴⠠⠔⢀⠁⡹⣶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢟⠲⣿⠶⠶⢶⠶⣶⣴⢶⢷⣼⣧⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠒⠒⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠣⣸⡮⠤⠒⠉⠐⠠⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠙⣟⠗⠒⢤⠭⣄⣈⡐⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣟⡉⠉⠉⢣⡠⠐⠀⠈⢹⢇⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⢲⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠛⢛⡷⠾⠻⣾⣧⣴⣮⣾⣧⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠴⠶⠖⠒⠋⠉⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⡞⠋⠀⠀⠀⠀⢻⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⡠⠀⠀⠀⠀⡀⣠⠃⠆⠐⠂⠀⠀⠉⠡⣵⢀⡾⠫⠙⠢⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣷⣤⠠⠒⣉⣤⡏⠻⡄⢠⣤⣀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⠄⠒⠉⠀⠀⠹⣄⡽⠃⣽⠦⢍⡒⠶⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡅⠀⠀⠀⠀⠀⢀⡟⢁⠞⠁⠀⠀⠉⠓⠮⣕⣆⠀⠀⠀⠀⠀⠀⣠⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⣷⣀⠀⠀⠀⢀⠏⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠀⠀⠀⢀⣼⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣷⣦⢀⣠⣈⣶⢻⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⡲⣭⣀⣱⣟⣿⣿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣾⣿⣿⣿⣮⣳⢄⡀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣮⣽⣾⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠷⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠶⠶⠖⠛⠋⠁