Water supply Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⡖⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠋⠻⠟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣿⣿⢿⣷⣶⣶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⢤⡤⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⣿⣤⣤⣤⣤⣴⣿⣿⣷⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⢾⠷⠶⠶⠶⠶⠶⠶⠶⠶⣦⣄⠀⠀⠀⠀⣿⡏⠉⠉⠉⠙⢷⣿⠿⠉⠉⠉⣿
⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠻⠃
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⢛⣭⣙⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⣛⣥⣾⣿⣿⣿⣷⣮⣙⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⣛⣵⣾⣿⡿⢛⣛⣛⣛⡛⢿⣿⣿⣮⣝⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⣵⣾⣿⣿⣿⣿⣿⣶⡆⣶⢰⣶⣿⣿⣿⣿⣿⣿⣮⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢟⣫⡅⣶⣶⣶⢨⣝⣛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢏⣾⣿⠿⣃⢿⣿⡿⢘⠻⣿⣷⡝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⢣⣿⣿⣶⣭⣶⣿⣿⡎⣿⣷⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡇⣿⣿⢟⣘⣛⣈⢿⣿⣿⣿⣿⣿⣿⢃⣛⣋⡘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠁⣿⣿⣷⢰⣶⢰⣿⣿⣿⣿⣿⣿⣿⣶⡶⢶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠿⣿⡿⢸⣿⠸⣿⣿⣿⣿⣿⣿⣿⠟⣼⣎⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⡀⡄⣀⣴⣿⡿⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢀⣀⣀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⠇⡇⣿⠿⠋⠀⠀⠀⠀⠀⠀⠘⠻⣿⣶⣄⠿⠿⠿⠿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇



⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠖⠚⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠶⠒⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⡶⠚⠛⠉⠉
⠀⠀⠀⢀⣠⣴⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠉⠻⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⠀⠀⠀⠀⠀⠀⠻⠦⠶⠴⠶⠴⠶⠦⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⡄⠀⠀⠀⠀⠀⠀⢀⡆⢰⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⡅⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⡃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢸⡅⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⡀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠁⢀⣠⡤⠶⠷⠶⠧⣀⠼⠶⠶⠦⢤⣄⠀⠈⠻⣎⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣀⠎⠁⠙⣦⣤⣤⣤⣤⣤⣤⡤⠤⠖⠑⢎⠛⢦⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣀⣤⣤⡤⠴⠁⠀⢠⡚⠀⠀⠀⠀⢠⠒⠋⠀⠀⠀⠀⠸⠢⡀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⣿⣤⣤⣀⣀⣈⣉⣩⣭⣿⣷⣶⣦⠌⢦⠀⠀⣀⠀⠀⠀⣠⡇⠀⠘⠓⠒⠒⠒⠒⠲⠖⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡴⣉⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠦⠤⠞⠈⠷⠤⠤⠤⠤⠤⠤⢤⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡇⠀⠈⢟⠛⣅⣈⡗⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢯⠕⢠⠝⡟⠻⠀⠀⣸⣤⣬⣥⣤⣄⣀⠤⠲⠞⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠱⣀⣠⣏⣸⣍⠇⢠⠋⠉⠀⠀⠀⠀⠘⠲⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣩⠆⠈⢿⣛⡛⠛⠛⠳⡀⠀⣠⠀⠀⠀⣠⠇⠀⠀⣀⣠⡤⠶⠛⠉⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡲⠀⠀⠳⠚⠓⠲⠶⠾⠿⠼⠦⠤⠾⠷⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠤⠤

⠀⠀⠀⡘⢉⡑⠂⠤⠤⠄⣀⣀⣀
⠀⠀⢰⢥⣤⣄⣀⡀⠀⡀⠠⠤⠤⠀⠀⠀⠈⠋⠙⣽⣷⣄⡀
⠀⢠⡃⠀⠈⠉⠉⠁⠩⠹⠗⠶⠖⠢⠤⣤⡴⣦⣤⢹⣿⣿⣿⣦⣄⡀
⢀⠎⠉⠛⠓⠶⠶⠾⡤⣤⣄⣠⣄⡉⠀⢀⡀⠀⠸⣾⣿⡫⢿⠛⠛⠿⠂
⠾⠶⣤⣤⢦⣀⡀⠀⡀⠀⠀⠀⠉⠉⠹⠙⠛⢛⢷⣿⠃⣤⣤
⠀⠀⢸⣿⢝⣿⡏⠛⠛⠓⠲⠶⠶⣾⠶⣿⣭⡿⠁⣿⢰⣿⢽
⠀⠀⣼⣿⠻⢻⣷⢷⣶⡶⣦⢤⣤⣠⣄⣀⡀⠀⠀⢸⠸⠛⢻
⠀⠀⢙⣿⣿⣿⣷⣻⣿⣿⣿⣇⣾⣏⣽⡞⣉⠉⠘⠺⣠⡲⣿⡀⠀⠀⣀
⠀⠀⢘⣿⣿⣿⡁⠀⠈⠉⠉⢹⣿⠛⠻⠿⠛⠛⠿⠆⢣⣿⣷⠾⠽⣿⣸⣣
⠀⠀⢸⡟⢹⠟⡥⠤⣤⡤⠤⢸⣿⠤⠤⣤⣀⣀⣀⡀⣷⣿⡇⠀⠀⠅⡿
⢀⠔⢺⣧⣨⣤⣷⣖⣾⣟⣿⢿⢿⣟⠉⠛⡗⠒⠲⡇⣽⣿⣷⡄⢀⢰⠇
⣼⣤⠤⠤⠽⣏⣎⣿⡿⣿⣟⣻⢸⣿⢛⠒⠧⣄⠀⣣⠘⣿⣶⣼⡌⡾
⢻⡗⢄⠀⢀⡨⠛⠓⠿⠿⢿⣿⣼⣿⣿⣯⡦⠟⠯⡛⠶⠿⢻⣟⢧⡇
⠸⡇⠀⣽⠗⠤⠤⣀⡀⣠⡾⠃⠀⠀⠀⠇⠀⠀⠀⢈⡱⣶⠮⠿⠞⠣⠄
⠀⠙⣦⣻⠀⠀⠀⠀⠈⣿⠓⠒⢒⡒⠲⡗⣒⣚⣭⡝⣬⡧⣤⠄⠊⠐⠰⢶⣉⠄
⠀⠀⡏⠛⠷⣤⣤⣀⣸⡏⠀⠀⢸⢿⢭⡀⡿⣪⣿⣿⢿⣿⠃
⠀⠀⡟⠶⣤⣸⣏⠉⠋⠻⠟⠛⣗⢛⣯⡿⡫⡋⣻⣗⣼⣿⠀⠀⠀⡀⠀⢀⠀⣀
⢠⡦⢸⣄⠀⠉⡟⠛⠲⢶⠷⠦⣵⣵⡧⣾⣾⣾⣿⡻⣻⢟⠇⢧⠻⡏⣼⡟⢔⡟⢏⢰⢄⢀⢀⡀
⠘⢐⢻⣌⡟⢶⣧⣄⣀⣸⣀⠀⢿⡹⣿⣕⣣⣪⣿⣗⣿⡧⡿⢜⣆⡇⡔⣿⣸⣔⢻⠓⡆⢀⢯⡫⢟⡆⣶⣴⠀⢀⣀
⠀⠀⠈⠙⠻⣬⡄⢩⣏⡍⣉⠙⡿⣻⢫⡛⣿⣼⡟⣻⠇⣄⠳⡸⢎⣹⣅⣄⣨⠎⣱⢢⢥⡄⢸⢩⡅⠰⠴⢀⢣⠎⠛⡥⠟⡟⣂
⠀⠀⠀⠀⠀⠈⠈⠏⠙⠗⠻⠗⠿⣼⢿⣾⢿⡾⣿⡏⠦⡜⣏⡷⠎⣯⢘⡸⡤⡒⡝⢴⡀⠦⠎⠆⠏⠌⠐⠆⠻⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⠤⠤⣤⠴⠦⠀⠀⠀⠤⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠛⠻⣅⠀⠀⣀⣀⣀⣀⣠⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⠋⠉⠉⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣉⣀⣀⣀⣤⣶⣶⣿⣿⡿⢛⣛⣫⣉⠉⣿⣿⠿⠿⢿⣿⣿⣿⣭⣿⠿⠿⠶⣶⣤⣤⣶⠋⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⣉⡉⠀⠈⠉⠙⠒⠛⠓⠻⡿⠿⠿⢥⣿⣿⣷⣶⣤⣬⣭⣿⣯⣿⣶⣶⠾⠿⠛⠛⢯⣄⣀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠛⠛⠶⠶⠶⣤⣤⣤⣎⣀⣀⣀⣀⣀⣀⣀⣀⣀⣧⠀⠀⣀⣀⣀⣀⣤⣤⣶⡶⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠈⠉⠉⠉⠉⠩⢿⡉⠉⢻⡟⠛⠛⠋⠙⠛⠛⠉⠉⠀⠀⢺⡇⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣃⡠⢄⣀⡀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⢤⣄⣁⡀⠀⠀⠀⣘⠀⠤⠀⠀⠀⢀⣀⡀⠀⠀⢸⡧⠤⣀⠤⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠉⢿⡟⠓⠶⠿⣶⣤⣬⣐⣂⣀⣀⣀⣀⣠⣼⣧⣄⣀⣠⣤⣤⣶⣶⡿⠋⠡⡴⢯⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⠀⢸⡗⠒⠀⠀⠀⠀⠉⠉⠉⠉⢿⡿⠛⠛⠻⠋⠉⠭⠭⠕⠒⠶⠘⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡧⠀⢀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢮⠙⠛⠛⠻⠿⢶⣤⣤⣄⣀⣀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠤⠀⠀⢀⣀⣤⠴⠛⢛⣭⣥⡌⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢦⠄⠀⠀⠀⠀⠀⠈⣿⠉⠉⠙⠿⣛⠛⠛⠛⠛⠛⠛⠿⣿⡟⠉⠉⠉⠠⠐⠚⠋⠁⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣀⠀⠀⠀⠀⠉⠁⣿⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣃⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣀⣿⡄⠀⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢀⣀⣤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠂⠀⠀⠀⠀⠀⠲⠒⠛⠛⠛⠉⠉⠁

⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠦⠤⢤⣄⣤⣠⡤⠤⠶⠶⢦⡀
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣠⣤⠤⢾⣤⡿⠤⣤⣄⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⣹⣏⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠤⠤⢇⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⠒⠒⠒⠚⢧⡀⢀⡴⠒⡆
⠀⠀⠀⠀⢀⠔⠈⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⡇
⠀⠀⠀⠀⠈⠀⢐⠉⠉⠉⠉⠉⠉⠙⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠯⠤⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢀⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⠀⠀⡇
⠀⠀⠀⠀⠀⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠃
⠀⠀⠀⠀⠠⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠒⢒⢒⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠭⠿⠁
⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠙⣆
⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠴⠋

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

⠀⠀⠀⢠⠋⠑⠒⠠⠤⢠⣀⣀⣀⡀
⠀⠀⢀⣇⣀⣡⡀⠀⠀⠀⠀⢀⡠⢄⡉⠉⠉⠉⠉⠛⠏⣿⣦⣀
⠀⠀⡎⠀⠙⠓⠉⠙⠉⢉⣳⠶⠤⠴⠤⢤⣀⣠⣦⣀⣸⣹⣿⣿⣗⣤⡀
⠀⡜⠛⠲⠦⢤⣤⣤⣆⣈⡀⠀⢀⠀⠂⠂⠈⠁⠈⠉⣇⣿⣿⢳⣿⠿⢿⣷⠄
⣼⢤⣀⡀⣠⠀⠀⠀⠀⠉⠉⠉⠉⠛⠒⠲⠴⠷⠤⢴⣼⡿⠃⢁⠘
⠈⠉⠙⣿⡾⢿⣶⠶⢴⣦⣤⣄⣀⣀⣠⣌⣠⠄⣢⡳⢹⡇⣈⣿⢻
⠀⠀⢀⣿⣯⣾⣿⣠⣀⣀⣀⡈⠉⠉⠉⠉⠉⠛⠛⠁⢸⣇⢫⣟⡿
⠀⠀⢿⣿⣭⣡⣿⣫⢿⢻⢯⣿⡽⢳⣾⢷⢖⡶⠤⢀⣸⢦⠈⣀⣤
⠀⠀⠀⣿⣿⣿⡿⠻⠿⠼⣿⣿⣶⡟⣧⣾⣿⣄⣤⣤⣜⢸⡎⣮⡿⠤⢄⣀⠤⡄
⠀⠀⠀⣷⣽⣿⡧⠀⠀⠀⠀⠀⢸⣿⠁⠈⠛⠉⠉⠉⢛⢈⢳⣿⡟⠙⠓⢿⢰⡿⠂
⠀⠀⢀⣿⠨⠟⢻⠄⠲⣶⣒⠒⣺⣿⠒⠒⢶⡤⠤⠤⢬⢸⣻⣿⡇⠀⠀⡂⡿
⢀⠔⠉⣿⣤⣤⣾⢶⣾⣻⣿⣿⢿⢻⣯⢍⠉⠛⠉⠑⢺⢸⣽⣽⡷⣄⢀⢡⡇
⣼⣤⠤⠤⠘⢿⣵⣳⣿⠿⣷⣏⣹⢸⣿⣛⡓⠲⢤⡀⢸⣀⠹⣷⣶⣼⡎⣸
⢻⡗⢄⠀⠀⣀⠜⠛⠺⠿⠿⣾⣿⣸⣿⣿⣿⣵⠼⠾⢝⠻⠾⠟⢻⡟⠇⡏
⠸⡇⠀⢹⡾⠤⢄⣀⡀⠀⣠⡾⠃⠀⠀⠀⠍⠀⠀⠀⠀⢉⣦⣴⢪⣿⡼⢅⣀⡀
⠀⠙⣄⢸⡇⠀⠀⠀⠈⢹⡿⠓⠦⠤⠤⠤⡶⠲⠶⢒⣎⢏⣦⣧⣏⠿⠟⠶⢦⣬⡟⣢
⠀⠀⠿⠿⣷⣄⣀⠀⠀⢸⡇⠀⠀⢸⢕⡕⡇⢸⣋⣿⣯⣾⣿⣼⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢻⣤⣀⠉⣿⠟⠿⠞⢷⡲⣦⣬⣵⣷⣳⠾⠾⠫⣿⡵⣼⣯
⠀⣀⢸⡀⠉⠛⠿⠷⣤⣄⣸⣁⣠⣊⣄⣿⣺⣖⣬⣼⡿⡿⣿⣿⡄⡆⣦⣧⢠⢶⢆⣼⣶⠔⡄
⢸⠪⣸⡟⣦⣄⣸⡀⠀⠈⢹⠉⠉⣿⡿⣯⢝⣿⢿⢿⣿⣡⣟⠎⣤⢹⡄⣿⢌⣿⠉⠚⣨⣶⣘⡈⢢⡶⡦⣤⢀⢀⡀⡀
⠀⠐⠌⣷⣿⡌⡟⣻⢷⡶⠾⠶⢤⣬⣮⣿⣿⣧⡗⣿⣿⣽⢻⢩⡏⢳⣣⢧⡎⠘⠜⢞⠌⢇⠸⠰⠐⢱⡯⡅⠣⠋⡞⢇⢄⢶⡄⢄⣄⣄
⠀⠀⠀⠀⠈⠹⠾⢤⡾⣼⣃⣶⣀⣏⠼⣽⢏⣽⡫⣿⡡⣟⠰⣇⡏⡌⠱⡼⠧⢷⡖⠋⡰⡅⢓⠳⡂⢸⡸⣦⢊⠁⡑⣎⡎⠬⠴⠎⠹⠆⠖
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠙⠉⠛⠻⣟⢿⠘⡿⠹⠿⠘⢛⡉⡦⠛⢣⠷⣂⠇⢋⢩⡵⡁⠧⠐⠓⠃⠁⠃⠈⠀⠙⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣘⣛⣛⣛⣛⣛⣛⣛⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠘⢛⣛⣛⣛⣛⣛⣛⣛⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢠⣬⣭⣭⣭⣿⣯⣭⣿⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣀⣀⣀⣀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢰⣶⣶⣶⣴⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠼⢛⣩⣭⣿⣭⣽⡟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠏⠀⠀⠀⠀⠀⢀⣴⢋⣥⣶⣶⣶⣬⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⣿⣿⡟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⢣⣿⣿⣿⣿⣿⣿⣿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢿⡸⠿⠛⣛⣛⣻⡟⠿⢇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⢸⣿⣿⡜⢧⢸⣿⡟⢿⣿⣿⣿⡎⢣⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠸⠟⠈⠉⠉⠉⠀⢸⣿⣿⣸⣿⣿⣿⡇⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢏⣿⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⢸⣿⡟⣸⡿⢸⣿⡏⣠⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣇⢿⣧⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣭⣭⣤⣭⣿⣿⣿⣿⣿⣮⣿⣿⣾⣿⣜⣿⣷⣽⣿⣭⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣭⣤⣭⣭⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣬⣭⣥⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⡿⢿⣿⣿⣿⣿⡿⢿⢿⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⢸⠀⢸⣿⡇⠀⢸⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⣼⠀⢸⣿⡧⠄⢸⢼⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⣿⠀⢸⣿⡧⠄⢼⢸⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⡿⠀⢸⣿⡇⠀⢸⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⡇⠀⢸⣿⡇⠀⢸⢸⡇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⣥⡇⠀⢸⣿⣧⡄⣸⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⣸⡇⠀⢸⣿⣇⡀⣸⢠⡇⠀⢻
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⢸⠁⠀⢸⣿⡇⠀⢸⠀⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⢸⠀⠀⢸⣿⡇⠀⢸⠀⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣸⠀⠀⣾⣼⠀⠀⢸⣿⡧⠀⢸⠀⣷⠀⠀⡇
⣤⣄⣀⣀⣀⣀⣿⣀⣀⣿⣿⣀⣀⣼⣿⣧⣤⣾⣄⣿⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⢶⡶⠶⠶⢶⡶⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⣇⣀⣀⣸⡄⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⢋⣿⣭⣭⣿⡙⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣋⣁⣀⣀⣈⣙⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢛⠛⠛⠛⠛⠲⣤⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣰⠏⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸
⠀⠀⠀⢠⡏⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⣼⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⣾⠛⠛⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⢿⣤⣤⣤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⢠⡞⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⣠⠏⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⠀⣴⠋⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⣿⠀⣶⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸
⢿⠀⠘⣧⡀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠶⠆⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⠀⣸
⠈⢧⡀⠈⠙⠛⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠁⠀⠀⠀⣰⠏
⠀⠀⠙⠳⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣒⣚⣋⣁⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⠄⠄⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄
⠀⠀⠠⠤⠤⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣜⣛⣛⠀⠀⠀⠀⠀⠀⠋⠛⠛⣿⣿
⠀⠀⢠⣤⡤⣾⣿⣦⣤⣤⠀⠀⠀⠀⠀⠄⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣦⢸⣿⣿⣰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀
⠀⠀⠸⠿⠷⢿⣿⠿⠿⠿⠀⠀⠀⠀⠀⠡⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠟⢸⣿⣿⠹⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠇

⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠧⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠐⣁⣀⣈⡂⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣄
⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠁

⠀⠀⢀⡀⠠⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣤⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡄⡀⠀⠀⠀⠀⠀⠙⣿⣟⣿⣟⣻⡟⠁
⠀⠀⠀⢸⣿⣹⣾⣹⣿⣿⠀⠀⠀⠀⠀⠀⡿⡉⠗⠀⠀⠀⠀⠀⠀⠐⣼⣿⡿⣹⣿⡄⠀⠀⠀⠀⠀⣸⡇⢸⡏⣼⡇
⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡧⠁⢤⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⣿⣿⡗⠀⠀⠀⠀⠀⣹⡇⢾⡇⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈

⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⢼⡿⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⣀⣀⠈
⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡐⢺⡗⠲⣿⠂⠀⠀⠀⠀⠀⡠⠒⠀⠒⢄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠋⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢺⣷⣶⣿⡆⠀⠀⠀⠀⣿⡏⢉⠉⡉⢉⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠶⠀⠆⠰⠆⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠇⠀⠀⠀⠀⠀⠈⠪⠅⠅⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠟⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⠶⣦⣤⣄⣀⡀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠉⠉⠛⠻⣿⣿⠿⢇⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢼⣿⣿⣷⣦⣬⣙⣿⣯⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⠉⠉⣿⠏⢩⡴⣦⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢸⣿⣏⣠⣤⣿⠀⠈⢻⣮⡹⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⣿⣿⣿⣕⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡳⠄⠀⠀⣰⡆⠈⠻⣿⡿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⢰⣿⡇⠀⠀⠈⠁⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡯⠀⢐⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠟⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠂⡀⢰⣿⣿⣿⣿⣿⣄⣤⣤⣈⣿⡇⠀⠀⠀⡀⢠⡆⠀⠀⠀⠀⠀⠀⣀⠀⣻⣷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⣿⣿⣿⣿⣿⠏⠛⠉⠋⠉⢃⣀⢀⡀⠈⢻⣷⠀⠀⠀⠀⠀⠀⣹⣷⣿⡿⠛⠛
⠀⠀⠀⠀⠠⣾⣷⡄⠀⠀⠀⠀⢀⣾⣿⡇⠀⢰⢀⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⣿⣟⣯⠙⡇⢸⠁⠀⠄⠀⠀⠀⡾⠛⣿⣷⠇
⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⢀⣴⣿⡏⣹⡃⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢫⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⡏
⠀⠀⢲⣄⣠⣤⣼⠏⢀⣤⣾⣿⡿⢷⣌⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡇
⠀⠀⠀⢹⣟⣿⠁⡀⠜⢛⠉⠁⠀⠀⢋⠀⢀⡘⢿⣿⣿⣿⣿⣿⡅⠀⠀⠀⢀⣾
⠀⠀⠀⠈⢿⣿⣿⣷⣤⣿⣦⣄⢠⣀⣌⡠⠟⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⣤⡄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣿⣿⣿⣿⡇⠀⣠⣾⣿⣴⣾⣦⣴⡶⢄⣾⣯⣴⣶⣿⣿⣿⣅⣭⣧⣤⣀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡶⠊⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿⠋⠈⡿⢿⣴⣀⠀⡄⠈⡀⡀⠈⠁⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⠇⠀⠀⢰⣦⡀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢻⣦⣝⣿⣧⢻⣦⣵⣲⣀⠀⠀⣀⣰⡶⠛⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣦⡴⠚⠋⠀⢁⠀⣿⣿⣿⣿⣿⣿⡀⠀⢺⣶⣾⣻⣿⣿⠇⠙⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⡁
⠀⠀⢀⡀⠀⠀⠀⣼⣿⣿⣿⣿⣏⠁⠀⠀⢀⡸⠀⠘⣿⣿⣿⣿⣿⣟⠀⠸⣿⣿⣿⣷⣟⡀⠀⠀⠠⠤⠠
⠀⠀⣀⠁⠀⣠⠞⢻⣿⠿⣿⣷⣉⠀⣄⠀⣼⠁⠀⠀⠸⣿⣿⣿⣿⣿⡆⠀⡿⣿⣿⣿⣿⠖⠂⠀⠀⠀⠐⠀⠠⠀⠸⠀⠀⠀⢀⢀⣤⣴⡄
⠀⠠⡿⣧⢸⣯⣄⣭⣍⡁⢹⣿⣿⣶⣿⡿⠁⠀⠀⠀⠀⠛⠋⠋⠙⠻⢿⣀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠉⠉⠻⢇
⠀⢐⢿⣯⣼⣿⣿⡂⢉⣹⡾⠿⠋⠉⠉⠀⠀⠀⡘⠀⠀⢀⣾⣿⣿⣷⣾⣿⣷⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠻⠟⠒⠒⠒⠀⢸
⠀⡬⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⢀⠂⢐⠃⠀⠀⠀⣿⣿⣦⣭⣙⣛⠛⢿⣿⣿⡿⠷⠶⠶⠶⠶⣦⣤⣄⣀⡀⣀⠀⠀⠀⠀⠀⠀⣸
⢾⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣦⣦⠜⠃⠀⢨⠀⠀⣰⣄⣼⣿⣿⣿⣿⣿⣆⠀⠹⣿⣿⣷⣶⣤⣶⣶⣦⣤⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⢻
⣽⣿⣸⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⡿⠏⠀⣯⣼⣿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⣿⡟⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⡟⠀⠀⠀⡿⠉⠀⢱⣦⡀⠀⣀⣀⠀⠀⡀⢠⠆⡨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⡿⢿⣏⡑⠀⣙⣧⣿⣿⣿⣿⣿⣿⣿⡆⣿⡟⠀⠀⠀⣀⠀⢀⡀⢀⣿⣿⣀⡍⣋⣤⠀⠸⠇
⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣉⣶⡿⠅⠀⠈⠻⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠁⣀⣤⣤⡀
⠈⣳⣿⣯⣙⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣷⢖⣟⠾⣿⣿⣽⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡟⢟⠛⠟
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣥⣤⣼⡿⢻⣿⣿⣿⣿⣿⡍⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠣⠼⠟⠞⠓
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡿⠻⠿⢿⣿⣿⣦⣬⣭⣿⣍⡻⠿⣿⣿⠿⠿⠿⠧⠄
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⣉⡉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠒⠒⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⢶⠶⠶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⢻⠇⠀⢀⣰⣀⡀⠀⢻⡛⠛⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⣤⣼⡄⠀⠉⢹⠉⠁⠀⣸⣧⣤⣄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠻⣦⣀⣀⣀⣠⡴⠃⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢸⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⠴⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣶⢂⣶⡐⡖⢰⣶⢰⣶⡆⠰⣖⣰⡆⣶⣶⣲⢰⣲⢶⡆⢲⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠘⠉⠃⠃⠈⠛⠈⠉⠃⠈⠛⠁⠙⠁⠉⠀⠘⠀⠈⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⢀⣠⣀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⢀⣠⠖⠋⠀⠈⠓⢦⣀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⢀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⢀⣠⠞⠋⠀⠀⠀⢠⠶⠶⠶⠶⠶⣦⠀⠀⠀⠈⠙⢦⣀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⣤⡞⠉⠀⠀⠀⠀⠀⠀⠈⠉⢻⠉⢻⠉⠉⠀⠀⠀⠀⠀⠀⠈⠙⣦⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣠⡤⣼⠛⠛⠛⣷⠤⣤⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡞⠉⠀⠀⣸⠀⠀⠀⣿⡀⠀⠈⠻⣆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡏⠀⢀⡴⠋⠙⢦⣄⣤⠏⠉⠳⡄⠀⠘⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⡇⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡇⠀⠠⣞⣓⣒⣛⣶⠀⠀⠀⠀⠀⠀⠀⢰⣟⣓⣒⣻⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⡖⠀⡇⠀⠀⢸⡏⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡉⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⡇⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠙⠛⠉⢸⡇⠀⢸⠀⠉⠋⠉⠙⠛⠛⠛⠁⢿⡀⠀⣽⠈⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠹⣄⣀⣀⡀⢀⡼⠁⠀⣸⠀⠀⠀⠀⠀⢀⣀⣀⠀⠈⠙⠋⠁⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⢹⠉⡟⠉⠀⠀⣰⠃⠀⠀⠀⠀⠀⣞⠁⠉⠳⢤⣠⠞⠛⠋⠉⠉⠉⠉⠀⠉⢳⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣼⣈⣧⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠙⠦⠤⠤⠶⠶⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡀⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢸⠀⠠⠀⠁⢠⢣⠀⢤⠀⠀⠠⠀⠀⠂⠀⡏⢸⢴⢠⠀⠦⠴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠘⠀⠀⠀⠐⠐⠂⠘⠀⠀⠀⠀⠂⠀⠀⠀⠠⠈⠑⠂⠀⠀⠀⠘⠀⠂⠃⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠉⠉⣿⠀⠀⠀⢀⣠⡶⠛⠉⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⢻⠋⢀⣠⠶⠛⠁⠀⢀⣠⣀⡀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⡟⠁⠈⢻⠄⠀⠀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⢀⣤⣤⣀⣀⣠⡶⣦⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣀⡀⠀⠀⠸⣇⣠⣬⣭⣽⡇⣿⣭⣭⣤⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠁⠀⣠⡼⠷⠿⣤⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠶⠶⠶⣿⠉⠁⠀⠀⠀⠀⠉⢹⡷⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠻⠶⣤⣀⠀⢀⣠⡴⠾⠃⠈⠛⢷⡀⠀⠀⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⢀⣼⣧⣤⣤⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣴⠶⣴⠾⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢹⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠾⠃⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠼⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣆⢠⠀⢄⠠⡄⢀⢄⢀⡄⠀⢴⡂⢀⠀⡀⡀⣄⢀⢠⡀⢆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⠎⠰⠽⠀⠇⠻⠌⠸⠀⠀⠠⠽⠸⠴⠆⡧⠞⢸⠴⠃⠇⢳⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⢀⣀⣀⡀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠸⠿⠿⠄⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄⠀⡄⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⣠⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⣴⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⣿⠙⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⣿⣇⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠶⠶⠶⠶⠦⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇
⠀⠘⢿⣷⣦⣽⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⠟
⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⡆⡶⢰⢶⠐⢲⠖⢰⣖⠂⣶⢲⠄⢠⣖⠂⢰⠀⣶⢰⡖⣦⢰⠒⡆⣶⠀⢶⣰⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣹⣏⣿⣃⣿⣚⣇⣸⣀⣸⣭⣄⣟⣹⣄⣀⣬⣗⣸⣤⣟⣸⣏⣁⣸⣋⣁⣿⣤⣈⣏⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡦⢠⣤⣔⣒⣲⣦⣴⣯⣭⣤⣤⣤⣤⡟⠙⣧⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢩⢤⢷⠶⢿⣿⣿⣯⡿⠟⠛⠙⣉⡉⠩⠭⠭⠭⠇⣸⠧⠤⣬⣅⣀⠀⢀⣁⡛⣷⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠐⠸⢴⣊⢙⣩⢗⣛⠛⣋⣩⠤⠒⠚⣉⡩⠤⠄⠔⢒⣒⡂⣺⡁⠀⠈⠒⠒⠤⠭⣙⡒⠤⣝⠩⢝⡻⣭⣛⠍⠒⢀
⠀⠀⠀⠀⣠⠞⠃⠐⡐⡀⠀⢀⡊⠋⣀⡴⠞⠉⠁⢒⡈⠉⣉⣉⣉⣁⠃⢉⠉⠉⠑⠒⠢⢤⣄⠈⠁⠂⠄⠀⠈⠑⠉⠑⠢⣀
⠀⣰⢃⡞⠁⠀⠄⠀⠀⢀⡀⠀⢀⡾⠉⠀⡠⠀⠀⣴⣶⣾⠿⡿⠕⠁⠀⠀⠽⣿⣷⣶⡦⢄⡀⠓⢤⣠⣀⠀⠀⠀⠀⠐⡆⠈⠂⠀⠀⠀⠠⣄
⠰⠃⣸⠐⠀⠘⠀⡀⠀⡏⠈⠋⠿⠀⡂⢸⠀⠀⣶⢸⠇⠁⠀⠀⠀⣀⣀⣴⣮⣮⣿⠈⠂⠀⠀⠀⠀⣈⣿⣷⣤⠀⠀⠀⠈⢣⠀⠀⠀⠀⠛⡌⢦
⠘⠀⢀⢰⢀⢰⢡⢡⠀⣇⢀⣀⡀⠀⢇⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⡠⠆⠀⠈⠀⠀⢨⣽⢿⣛⠄⠀⠀⢀⠈⡷⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠈⢎⢧⢻⢧⠱⡈⢮⡻⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠈⠉⠁⠀⣀⣀⡤⠤⠶⠀⢀⡾⠉⠀⣠⣾⠏⠀⠀⢀⡄⠀⠐⢸⢄⣷
⠀⠀⠀⠀⠈⢦⠳⣙⣴⣌⠂⣙⠪⣝⡷⣦⣀⠀⠀⠐⠢⠄⠀⡀⠀⢈⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠴⠋⢀⣤⠾⠋⣀⠴⢋⠤⠋⠀⠀⢀⠎⣼⠃
⠀⠀⠀⠀⠣⣀⠑⢌⠳⣗⠑⠢⣔⠠⣙⠻⢭⣟⡶⢤⣤⡀⠀⠀⠀⠀⠬⡭⠭⠭⠄⠀⣀⣀⣤⠤⠶⠚⣋⣉⠤⢖⣫⠵⢚⣡⠔⢠⠄⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠑⠂⠈⠙⠲⢭⣒⡬⢽⡓⠲⠬⣭⣽⣛⠓⠚⠛⠛⣛⣛⣩⠭⠥⠖⣒⣛⡩⠥⠖⣊⡡⠄⠚⠁⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⠀⠈⠻⢡⡀⠀⡄⠀⠀⠒⠒⢈⠉⠩⠥⠴⡒⠊⠉⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠈⠑⠈⠑⠊⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⠀⠰⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡈⣠⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢾⡠⠄⠊⣡⠞⠁⠀⠈⠀⠀⠀⠀⠀⢧⡄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣔⠀⠏⠀⢈⡶⠁⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⢳
⠛⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣿⠁⠀⡸⠂⡼⣁⡀⠀⠀⠀⣴⠂⠀⠆⠀⢄⣠⡄⠸
⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠉⠁⠀⠀⢡⣴⣿⣿⠗⢀⠙⡄⠊⠉⠀⠀⠀⠀⠀⠀⠁
⣷⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡂⠀⢫⣿⣷⠛⣣⡛⠈⠀⠀⠀⠀⠀⢀⣀⣆⠀⠈
⢼⠗⢄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡾⠟⠀⠀⠀⠈⠁⠀⣿⣿⣶⢾⢽⣷⡇⠀⠀⠀⣀⣠⡟⠉⡈⠀⢠
⣾⣷⣾⣷⣤⣤⣄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠃⠀⡰⠂⢠⠀⠀⠀⢠⣿⡿⢡⡟⣼⣽⡇⠀⠀⠀⠀⠹⣧⢽⢣
⣿⣿⣿⣋⣿⡿⠿⠿⣧⣾⣿⣿⡗⠠⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⢹⣿⣿⣆⣽⡿⣿⡁⠀⠀⠀⠀⠀⠘⠶⢧⡀⡀
⣿⡿⠛⠉⠀⠀⢲⣶⣿⣿⣿⣿⡇⣀⡨⣿⢿⣶⣬⣯⢶⣾⡿⠋⢋⣀⣀⣀⣾⢟⣟⣮⡖⠀⠀⠀⠈⢿⣿⣿⣿⣧⡈⠷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣾⣿⣇⣀⡉⠻⠿⢿⡦⠰⢯⣉⡿⠿⡿⢋⣾⣿⣿⡇⠀⠀⠀⠀⠈⢻⣏⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣿⣿⣿⣿⡿⣿⣇⡈⠁⠀⠤⣀⠒⠿⣞⠛⣾⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣧⣿⣿⣏⡳⢶⣶⠮⣯⣷⡴⠶⠏⣲⣿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠯⣭⣀⠈⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠶⠿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⢞⣛⣡⣿⡳⠻⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣷⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢿⣿⣿⠿⠛⠫⠭⢭⣿⣿⣟⠚⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⠟⠒⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⡳⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠻⠿⡿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣆⣼⣏⣝⣻⣶⣄⣶⣤⣴⣦⠀⠀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣯⣿⣯⣿⣿⣿⣿⣿⣽⣿⣧⠿⠃
⠀⠀⠀⢀⠊⠀⠀⠀⢀⣾⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⣿⠃
⣢⡬⠞⠉⠀⠀⠀⠀⠸⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣥⡴⠃⠀⠀⠀⠀⠀⢠⠀⠉⣹⣿⣿⣿⣿⣿⣿⣿⡇
⠁⠀⠀⠀⠀⠀⠀⠂⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠹⣷⣀⡀⡒⣴⣿⣻⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠃⠀⢀⢄
⠀⠔⢸⣖⣷⣻⣍⢻⣿⢹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⣀⠃⠀⠀⠀⠈⠱⠀⠄⠂⠀⠂⠨⠒
⣤⣤⣼⣿⣽⣧⣿⣟⣿⣾⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⠞⠁⠀⠀⠀⠀⡄⣀⣸⣿⣿⢻⣿⣿⣶⣴⣶⣶⣤⡆⢶⡆
⣿⣾⣿⣽⣿⠽⠗⠛⠛⠛⠉⠛⠛⠛⠿⣿⣿⣿⠟⠁⠀⠀⠀⠄⠀⢀⢀⡹⣿⣿⣿⣿⣿⣻⣿⣿⣟⡿⠟⠉⠁⠈⠀⠁⠀⠀⠈
⣿⡯⠝⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⢉⠀⠀⠀⢤⡀⢦⣀⢀⣄⢳⣻⣿⣿⣿⣿⢿⣷⣷⡟⠁
⠋⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⡘⢤⣰⣀⡀⠴⠄⢀⡀⠰⣦⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⡷⣿⡇
⠀⠀⠀⡀⠀⠒⢑⣆⣾⡟⣿⣿⣿⣿⣿⣿⣷⣀⣨⣽⣷⣿⣿⣿⣿⢿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡦⡀
⢀⣀⢀⡁⠀⠀⠒⣌⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣾⣷⠦⠄⠀⠀⠀⣠⣤
⢷⣿⣿⣿⣷⠎⣯⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⠙⠙⠛⠛⠛⠛⠻⠟⣋⡉⢉⣀⣀⣉⣩⣅⣀⡀⢶⣦⣐⠷⣶⣮⣽
⣿⣿⣻⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠀⣀⣤⣴⣦⣄⡀⣷⣶⣿⣿⣿⣷⣦⣼⣿⣿⣿⣿⣦⣭⣻⣿⣿⣿⣿⣿⣿⣮⣝⡛⢶⣭⡙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⣄⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⡉⠉⠉⠉⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠂⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⡀⣀⣀⣀⣀⣈⣁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⡂⠉⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⣄⣀⣈⣠⣤⠤⠀⠌⠥⠴⣺⠁⠀⠀⠀⠀⢀⠀⠔⠁⠀⠀⠀⢀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠀⠀⠀⠀⠐⠂⠀⠣⠁⠎⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠰⠤⣀⠀⠀⢀⡴⠁⠀⠀⠀⣀⠤⠊⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡠⠖⠉⣀⡤⠒⠈⢁⡠⢀⠔⠀⠀⠉⠉⡁⠀⠀⠁⢉⠁⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡴⠃⠀⡠⠊⠁⠀⢀⠔⠉⣠⠃⠀⠀⠀⠄⢠⠀⠸⠀⠀⠸⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡑⠀⣠⠊⢀⠂⠀⡰⠃⠀⡴⠁⠀⡜⠀⠀⠀⡎⢠⠃⡇⠀⢠⡙⡄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡔⠁⠀⠀⢀⠜⠁⡠⠊⠀⢰⢰⠁⠀⠀⡜⠀⡜⠀⣻⠀⢸⢡⠸⡄⠘⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠎⠀⢀⡠⠚⠁⠀⠀⡠⠨⠀⠸⢸⠀⠀⡜⠀⡜⠀⢈⠉⡆⢸⢸⠀⢣⠀⠸⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠐⠁⡰⠁⢀⡰⠋⡰⠁⠀⡇⡆⠀⢈⠀⡜⠀⠀⢸⠀⡇⠸⢸⠀⠸⡀⢠⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠜⣁⠔⠉⢀⠜⠁⢀⠜⢰⠁⠄⣡⠞⠀⡀⢀⣾⠀⡇⡇⢸⠀⠀⡇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⢀⠔⢃⠠⠔⠁⣠⠃⡠⠊⠁⠀⡜⠀⡜⢸⢠⢡⠁⡌⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⡇⢰⠁⠀⠀⠌⠀⡼⠀⢸⠀⠆⢠⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠐⠀⠘⠀⠀⡆⠈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⡀⠀⠀⠀⠀⠰⠁⢀⠊



⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⡇⠀⠹⣷⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠈⢿⣮⡛⢿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀⣀
⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣾⣿⣿⣷⣾⣿⣿⡷⢯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⣴⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⡿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⡦⠜⣑⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣸⢿⣮⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢛⣛⣛⣩⣭⣶⣾⣿⠿⣋⣵⣿⣿⣿⣿⣿⣿⣿⡿⢘⣿⣿⡿
⠻⠾⠟⠛⠳⠮⣍⣛⠿⠿⣿⣿⣿⣿⣿⣿⣶⣿⣭⣽⣟⣛⣛⣿⡿⠿⠿⠿⠿⢿⠿⠿⠿⣿⣿⣿⢿⣿⣿⣛⣿⣭⣵⣶⣿⣿⣿⣿⣿⣿⡿⠿⢛⣫⡴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠭⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣼⣿⡿⠿⠿⣽⡿⠿⠟⠛⠛⠻⠿⠿⣯⣥⡼⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣶⠒⠒⠲⠚⢿⡶⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣹⣶⣶⠆⠈⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⣨⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⣿⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⣻⡇⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⡇⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠂⠀⡇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⢘⠀⢀⡁⢸⡇⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠈⠀⣾⠃⠸⠃⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⠀⠨⠇⠀⠀⠀⠀⣽⠇⢠⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠐⡶⠈⡄⠀⠀⢣⣵⣾⣎⢉⣼⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⡄⠰⣤⡈⣿⣿⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠈⣶⡇⠀⣿⣌⠙⡿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠗⣻⡅⠀⣿⠊⠀⠀⠚⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⣭⠅⣀⢸⢴⡶⠄⡎⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⣤⣤⠀⠈⠁⠆⢻⡌⢡⣾⣿⡆
⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⣒⣓⣀⡀⠀⠀⠚⢁⠰⣿⣛⣛⣛⣛⣛⡆⠁⠸⠿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠲⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣷⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣋⣉⠉⠈⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠋⠁⠀⠀⠀⠀⠀⢸⣿⢻⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣼⣿⣿⣤⣤⣤⣤⡤⡤⢤⣼⣏⣘⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢙⣯⣿⣿⣿⣿⣿⣿⡯⠤⠀⠀⠀⠀⠀⠈⠉⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⣀⣀⣀⣄⠀⠀⣀⠀⠀⠉⠉⠛⠛⠛⠛⠹⠻⣟⣛⣻⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠚⠆⠄⠀⣀⠐⠂⢄⠠⠄⠀⠀⠀⠀⠌⠂⠀⠀⣢⣠⡤⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣴⣯⣶⡄⣷⣸⢿⣄⣤⣴⣷⠆⠊⣛⣍⠱⣿⣶⡘⡒⢢⢤⣄⡆⡤⠀⡀⢁⢺⣁⣴⣹⡇
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣊⣶⣷⣻⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣶⣿⣽⣿⣿⣿⣿⣯⣦⣤
⠀⠀⠀⠉⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠙⠋⠉⠁⠁
⠀⠀⠀⠀⠐⠀⠘⢁⣠⣴⣖⣀⣠⠀⣦⠀⠀⢠⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣧⣦⠈⣳⣴⡆⣯⣦⣶⢸⣯⡀⣤⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠿⣿⣟⣛⣙⣿⣿⣿⠀⢀⣰⣦⣆⢀⣾⢿⡇⢰⢼⡄⠀⠀⠸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠿⠉⢉⡛⠻⠆⠀⠀⠏⣿⣿⣿⣿⣷⡠⠼⠇⠀⠀⣆⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⠀⠀⢳⡶⢖⠪⡿⠿⠶⠶⣿⣁⣈⣛⣻⣦⣤⠤⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡜⠿⢷⠦⠴⣠⣴⣦⣶⣾⣿⣿⣦⠀⠈⠉⠉⠙⠛⠓⠒⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠄⠹⣿⣿⣿⣿⣿⠿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠥⢄⡉⠂⡇⣑⣒⣆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⠃⠳⠆⡿⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣿⣿⣿⣿⣳⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣩⣾⣿⣫⣽⣭⡽⠁⢻⣿⣿⢷⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢡⣿⣿⠓⠋⢥⣁⢠⣲⣮⣭⣟⡋⠙⠾⣯⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢰⣿⡿⢡⣬⣻⣿⣿⣿⣿⠋⠁⠀⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⣠⣾⣟⡵⢈⠻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣯⣿⠏⠀⠾⢽⣿⣿⣿⣿⠀⣠⣴⣶⡲⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣠⣿⣿⣗⠀⠉⣤⣿⣿⣿⣿⡻⡿⠛⠋⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠠⠔⠉⠤⣲⡞⠿⠛⠉⠀⣄⢀⢶⣿⣿⡿⢇⣁⠤⠤⠴⣿⣿⣿⣷⡀⠠⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⢀⠀⢶⣎⣷⣿⣿⣿⣿⡏⡓⠀⠀⠀⠀⠀⢨⣭⣻⣿⣿⣷⣶⣶⠟⢛⡿⢟⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⡧⢋⠠⠀⠀⢼⣿⣧⣶⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠈⠝⢋⣤
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣯⣭⣽⣿⣷⣶⣟⣃⡉⠀⠀⣀⣤⣼⣿⣿⣿⣿⣿⡿⠷⠉⠉⠀⠀⠀⠤⠶⣺⣿⠟
⠀⠀⠀⠀⠀⠈⢻⡿⠿⢿⣛⣿⡿⣿⠿⣥⡐⠀⠀⠀⠈⢻⣿⣿⣾⢿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁
⠀⠀⠀⠀⣴⣶⣦⣀⠀⠀⠨⠈⠃⠀⠀⢀⢠⡄⢀⢰⣶⣿⣿⣿⣟⠷⠉⠻⠿⣖⠀⠀⠀⠀⠀⢀⣤⣴⣿⠇
⠀⠀⠀⢸⣿⣿⣿⣿⣯⠄⡤⣸⣖⣶⣦⣼⣿⣿⣼⣷⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠠⠖⣛⡛⠉
⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠿⠻⠍⠛⠛⠛⠓⠒⠒⠒⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠻⠛⠿⠿⠿⠿⠿⠟⠋⠙⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠎
⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣻⠋
⣀⣤⢤⣶⣾⡿⠿⢻⡿⢋⣭⠉⠁⣀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠛⠛⠻⠿⣶⣦⣤⣤⣤⣴⣿⣿⡽⠃
⠻⣿⠀⠫⢥⣀⠒⠛⠳⣿⣿⡀⠀⠀⠀⠌⠉⠁⠃⠀⠁⠀⠁⠁⠉⠉⠁⠀⠄⠀⠠⠀⠤⠰⠨⠭⠭⠭⣼⣿⣏
⠀⠈⠓⠦⣠⠒⠒⠾⡶⠠⠒⠂⠀⠠⡄⠀⡀⠀⢂⠉⢀⣀⣀⡄⠀⣀⣀⠀⠀⠈⠀⢀⣀⡤⣶⠟⠛⠓⢾⣿⣿⣦
⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣴⣶⣶⣤⣤⣤⣤⣤⣄⣤⣤⣤⠤⠤⠤⠤⠤⠤⣶⣖⣚⣩⣵⠞⠁⠀⠀⠀⠀⠙⢮⣋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡏⠀⠀⠀⠀⠀⠈⠻⢿⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢌⣆⠤⢆⡄⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠰⠀⠀⠀⠌⠢⠈⠢⡈⠂⠂⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡵⡇⠀⠄⣀⠂⢄⠢⠀⢈⠀⣀⠠⠄⠤⢱⣄⣣⡀
⠚⠁⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⣠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠳⢄⡀
⠀⠀⠀⠀⠓⠤⢄⡀⠈⠙⠢⢤⣀⠀⠀⠀⠀⠘⣠⠴⠊⠁⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠁⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣁⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠿⠘⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⡠⠖⠀⡠⠒⠀⠀⣏⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡠⢹⠀⠀⠀⠀⠀⠨⠭⠄⡠⠾⢑⠟
⠀⠀⠀⠀⠀⠀⢸⠤⠒⡉⡀⢀⠌⠀⠀⠀⠀⢣⠄⠀⡀⠀⠄⠈⠙⢁⡘⠤⠐⠈⠁⠀⠀⠑⢌⠠⠀⠐⠄⡴⠊⣠⠔⠁
⠀⠀⠀⠀⠀⠀⣁⡠⢊⠊⢠⠊⠀⠀⠀⠀⠀⢹⡒⠁⢀⣀⣀⠴⠈⣡⢀⠄⠠⠠⠴⠄⢀⣀⢀⡉⠀⡄⠐⠒⠉
⠀⠀⠀⠀⠀⠀⡝⡰⢁⡴⠁⠀⠀⠀⠀⠀⠐⠓⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⣠⣟⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⡈⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡴⠂⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠊⠀⠀⠀⠀⡠⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠰⣟⢦⡴⣷⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣭⣍⡛⣿⡈⢰⣿⢟⣩⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⠛⠷⢬⣹⡜⠇⡾⢡⣏⡥⠾⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢹⠇⠀⠁⠸⡏⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⡆⣸⠃⠀⠙⡟⡆⠀⠀⠀⠀⢐⢻⠏⠀⠘⣧⢰⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⡏⠀⠀⠀⣇⠁⠀⠀⠀⠀⠈⢸⠀⠀⠀⢻⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠸⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣭⣽⣷⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣾⣯⣭⡙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⢸⠛⠻⣷⣶⡲⢶⢄⢀⣿⠿⢻⣄⣡⡟⠿⣿⠁⡠⡶⢶⣶⣾⠟⠛⡅⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⠈⠛⢷⣦⣧⣿⣇⣀⣀⣺⣷⣀⣀⣸⣿⣴⣴⡿⠛⠁⠀⠀⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⢸⢀⡀⠀⠀⠀⢻⣏⠉⠙⠋⠙⠙⠛⠋⠉⠛⠉⣹⡟⠀⠀⢠⠇⠀⡇⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢨⡇⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠸⠂⠀⠻⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣾⡿⠟⠛⠛⠻⠿⠿⠿⠿⠿⠟⠻⠿⢿⣷⠀⠀⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡆⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠘⠀⢸⡇
⠀⠀⠀⢀⣴⠞⢛⣛⣻⣶⣼⣷⣴⣶⣶⣶⣶⡿⢷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡾⢿⣶⣶⣶⣶⣦⣾⣧⣶⣟⣛⡛⠳⣦⡀
⠀⠀⣠⢟⡁⠐⢿⣏⢉⣁⣠⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⢶⣦⣤⣤⣤⣤⣤⡶⠤⠶⢤⢤⣤⣤⣤⣤⣤⣤⣀⣈⡉⣹⡿⠂⢈⡻⣄
⠀⢰⣟⢿⠁⠰⣦⠙⢿⣅⣰⡟⠉⠩⠝⠳⣄⡀⠀⠀⣠⠞⡩⠔⠒⠒⠢⢍⠳⣄⠀⠀⠀⣤⠞⡫⠉⠉⣹⢆⣨⡿⠋⣴⢆⠈⡿⣻⡆
⠀⣾⢻⢸⠀⠆⡿⡀⠀⠻⣿⡇⠀⠀⠀⠀⢮⢳⡄⢰⠏⡞⠀⠀⠀⠀⠀⠀⢳⠹⡆⢠⡞⠕⠃⠀⠀⠀⢰⣿⠟⠀⢀⢿⢠⠀⡇⣿⣷
⠀⢿⣼⢸⡠⠇⡇⣿⠀⠀⠙⢿⣦⣤⣤⣀⣀⣈⣻⣷⣾⣀⣀⣀⣀⣀⣀⣀⣀⣇⣻⣿⣁⣀⣀⣤⣤⣴⡿⠋⠀⠀⣿⢸⠸⠀⡇⣏⡿
⠀⠈⠛⠀⣷⢠⠱⠟⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿⠛⠛⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠻⠋⡄⣾⠀⠛⠁
⠀⠀⠀⠀⠈⡿⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡇⢠⠀⡁⢀⠀⠀⠀⠘⠀⠄⢰⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠷⠋
⠀⠀⠀⠀⠀⠀⢠⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⡇⢸⠀⡀⠀⠀⠀⠄⢠⠀⡆⢸⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡄
⠀⠀⠀⠀⠀⠀⢸⠀⡃⠀⢀⣀⣠⣤⣴⣶⣶⣿⡟⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⢻⣿⣶⣶⣦⣤⣄⣀⣀⠀⢸⠁⡇
⠀⠀⠀⠀⠀⣀⣼⣤⡿⠟⢛⣋⣭⣥⠴⠶⠶⣿⡏⠙⠓⠒⠶⠶⠶⠶⠶⠶⠖⠒⠋⢙⣿⠶⠶⠶⣭⣭⣙⣛⠿⢿⣤⣧⣀
⠀⢀⣠⣶⣿⡛⠭⠔⠒⡋⠉⠁⠀⠀⠀⠀⢀⣿⠋⠛⠒⠲⠶⠦⠤⠤⠴⠶⠖⠒⠛⠙⣿⡄⠀⠀⠀⠀⠈⠉⢙⠒⠢⠭⣛⢿⣶⣄⡀
⣠⣿⠋⠉⠛⠻⠶⢶⣤⣤⣤⣄⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣿⣇⣀⣀⣀⣀⣠⣤⣤⣤⡶⠶⠟⠛⠉⠙⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⢿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡿
⠀⠀⠉⠻⣿⡛⠛⣷⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⣶⣶⠶⣾⠛⢛⣿⡟⠉
⠀⠀⠀⠀⠙⣷⣤⣥⠔⠐⠻⣶⣄⠀⠀⠎⡭⠈⢭⡛⢤⡀⣼⠏⠒⠪⢹⡄⢀⡤⢛⡭⠑⢭⠱⠀⠀⣠⣶⠟⠀⠂⣜⣤⣾⠋
⠀⠀⠀⠀⠀⠈⠙⢿⣦⣀⠀⠈⠫⣧⣼⠸⠀⠀⠀⠉⠢⡙⢯⠀⠀⠀⢀⡾⢋⠔⠉⠀⠀⠀⠇⣧⣼⠟⠁⢀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣷⣿⣦⣤⣤⣤⣤⣤⣬⣮⣷⣤⣤⣾⣵⣥⣤⣤⣤⣤⣤⣶⣿⣶⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢋⠽⠋⣉⠟⠉⢉⠟⠉⠉⠉⠙⠻⡉⠉⠛⣉⠙⠯⣙⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢃⡜⠁⢠⠞⠁⠀⡴⠃⠀⠀⠀⠀⠀⠀⠘⣦⠀⠈⢳⡄⠈⢣⡘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠘⠀⠀⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⢡⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡄⠀⢣⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡙⠀⢠⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⣸⣇⣀⣀⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⣀⣀⣸⣇⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣟⣀⣀⣁⣉⣉⣈⣛⣃⡈⣃⣀⣒⣀⣀⣀⣀⣸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠤⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠤⠤⠤⠤⠤⠤⠀⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⢉⡁⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠉⠉⠉⠁
⠀⠀⠉⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠤⠤⠤⠤⠤⠀⠠⠤⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠔⠊⡛⢻⣷⣦⣰⣄⡀⢠⣀⠀⠀⠀⠀⢀⣀⡀⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⠡⠠⠼⠂⠀⠐⣂⡐⢀⠴⠄⠀⢠⣄⣤⣤⠧⢙⣵⢿⡿⣻⣿⣿⣿⣶⠈⠳⠀⠀⠀⠀⠉⠁⠤⣤⣶⣶⣤⣴⣿⣦⣄⣴⣦⡈⣩⣥⣆⡀⣀
⢤⣴⣶⣶⣷⡴⠈⣤⣤⣶⣴⣦⣤⣠⣼⡽⠁⡡⢖⣫⣾⣿⠟⣫⣿⣷⠀⠈⠲⣦⣄⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣛⡏⡠⣊⣝⠙
⠀⠈⠁⢀⣠⣤⣄⣀⠙⢁⠉⠙⠉⣁⠤⢚⡩⠔⣪⡷⣛⣵⣾⣿⣿⣿⣿⡄⠀⠀⢮⣧⣀⣦⣳⣰⣤⣀⣠⡀⠈⢉⣙⡛⠛⠛⠛⠛⠛⠛⠋
⠤⠤⢒⣛⠩⠥⢚⣛⣭⡅⠤⣒⣩⠔⢊⣭⠶⣛⣵⣾⣿⣿⡿⣿⣿⣿⣿⢃⠄⠀⠀⠒⠯⠹⠧⣦⡂⢥⣠⡬⠍⡉⠈⠍⠋⣠⢾⣤⡦⡤⡚
⣂⣉⠡⠴⢒⣛⣭⣴⠴⠞⣋⣥⠴⣚⡭⢖⣫⢗⣫⠟⣛⣥⢞⣫⣟⡿⣻⣷⣙⠏⠢⡀⠀⠈⠁⠪⠉⠉⠉⠚⡉⠍⠓⠈⠙⠁⠀⠀⢛⡯⠭
⠵⢖⣚⣛⠯⣉⣠⠤⢒⣫⣽⣶⣿⠵⣚⡿⢞⡋⣀⢜⣫⣕⡿⣛⣵⣿⢟⣻⣿⣿⣿⣈⣀⠈⠷⠀⢀⣀⢒⣀⠒⣓⠒⠢⠠⠤⠤⣠⣤⣤⡴
⡭⠿⣒⣛⣭⠷⠖⣛⡭⢖⣛⠭⠖⡫⠁⠾⠋⢝⣒⡽⠚⢁⣮⠟⣫⣵⠿⣫⣿⡿⢋⣽⣿⠟⣿⢿⣿⡿⢟⣿⣿⣿⣷⣾⣷⣷⣿⣿⣡⡟⣰
⠘⠋⢉⠡⢐⣂⡥⠔⣚⡭⢗⣢⡴⣖⣡⣴⠾⢋⣡⣴⠞⠋⣰⠞⣩⣵⡻⠛⣡⣶⠿⢋⡠⢊⡥⠞⣩⠞⣡⣿⣿⣯⣞⣿⣿⠟⢩⡾⠉⣡⣯
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⠀⠀⠀⠀⠀⠀⠐⠒⢶⣦⣤⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢰⣤⣤⣽⣿⣿⣿⢖⣦⣾⣿⡿⣿⣿⡟⢸⢞⢲⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠠⣤⣭⣽⣿⡿⢋⣵⣾⣿⣿⣿⢳⣿⡿⡇⠹⣙⡢⠋⠁⡬⠛⠁
⠀⠀⠀⠀⠀⠤⠶⣾⡿⢋⣴⣿⣿⣿⣿⣿⠃⠙⡛⠀⠀⠠⡀⠀⠀⠈⠉⢙⠇
⠀⠀⠀⠀⠄⠲⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⡀⠀⠀⠨⡀⠀⢁⠤⠃
⠀⠀⠀⠠⠶⢾⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⡀⠀⠂⠈⠳⠤⠝⠁
⠀⠀⠀⢀⣴⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠋
⠀⠀⠀⢠⣼⠣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⢀⡠⠄
⠀⠀⠀⢀⣼⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠀⣴⣤⣶⡿⠋
⠀⠀⠀⠁⣽⠨⣿⣿⣿⣿⣿⣿⣝⡛⣿⢕⣣⢴⠟⠛⠉
⠀⠀⠀⠀⢰⡆⣿⣿⣿⣿⣿⣿⡀⠅⠈⠉⠉⠁⠀⠀⠀⢀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠐⠀⠱⠹⣿⣿⣿⣿⣯⢢⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠻⠿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠸⣄⢣⢽⣿⣿⣿⣿⣫⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢯⠙⢿⣧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠹⡆⣬⡻⣿⣿⣿⣷⣷⣄⢀⣠⠾⠟⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠉⣿⠑⣮⠿⣿⣿⣿⣿⣀⠉⠋⠉⠙⠶⠆⠀⠀⠀⠀⠀⠀⠐⣘⣧⣤⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠜⡓⢮⡛⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠀⢤⣄⣉⡛⠿⠿⣿⣶⣶⣶⣶⣾⣷⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃⢹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠀⠀⠀⠀⠀⠙⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣤⣀⠀⠈⠁⠀⠀⢀⣠⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠭⢉⡁⠀⠀⠀⡀
⠀⠠⣂⠁⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣢⣄⣰⣤⢻⣿⣴⣶⡴⡀⠴⠀⠀⠠⠐
⠀⣲⠿⣗⠁⠀⠀⠄⠀⡠⠀⠀⡀⢯⡻⣽⡿⢏⣵⣬⠏⠙⠁⠐⠈⠄⠀⠀⠈
⣾⠁⣶⡹⣦⣀⠂⠀⠈⠀⠀⠸⡏⣮⡇⡷⠃⢈⢱⡇⠌⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⢀⣐⢀⠠⠈⠠
⠙⠞⢿⣿⡿⠟⠋⣠⠤⢤⣿⡁⢿⡟⢧⣇⢰⣾⡀⠸⡦⠀⠠⠀⠂⣤⠐⣄⠀⡀⠀⢠⣶⠆⢛⠣⠀⠀⠐
⠀⠀⠀⠈⠰⣖⠊⠁⠀⢩⠧⠍⠰⠂⠉⣅⢨⡀⢄⣲⣽⢷⡴⣷⣤⣿⣻⠿⠋⠀⠀⡟⢁⠘⢿⡷⣇⣮⣁⣀⢀⠀⠠⠑⢄⠈⠀⠠⡀
⠀⡀⢀⣻⡦⢌⣙⣒⠒⠛⠒⠁⠄⠠⣦⢽⣞⠓⠈⣮⣿⡶⠀⠈⡫⠅⢠⣖⣤⣤⠜⠀⠹⢿⣶⣯⣦⣭⣭⡅⢠⡗⣠⣦⣅⣩⣷⣴⡿⠿⠄
⠀⠈⠀⠉⢉⠉⣉⠉⠆⡀⣮⠹⣀⣶⣿⢺⣍⣙⣒⣪⢞⠁⣶⣄⠻⢷⡬⣭⣥⡄⠠⡤⠴⠶⢶⠓⠷⠶⠶⠦⠫⠶⣜⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⢠⠸⡦⣹⠳⣔⣦⢮⠙⠛⠛⣋⣉⣉⠫⢅⡐⣯⣳⣬⡿⡏⣲⡉⡓⠈⡓⣤⡤⠶⣞⢿⠻⣿⠗⠊⢰⢶⣻⡛⣻⡛⢏⠛⠛⠁
⠀⠀⠀⠀⠀⠉⠓⠦⣄⠸⣷⣦⣌⣛⠓⠲⠮⣽⡲⠿⠲⡿⣮⣻⣤⠽⡿⣿⣒⠛⠾⠿⠖⠚⠛⠉⠠⡖⢄⣈⠲⢶⡐⢄⠝⠾⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠻⠖⠉⠉⠁⠀⠀⠀⠉⠉⠁⠉⠈⠀⠀⠀⠀⠀⠁⡉⠙⠷⠖⠒⠒⠲⢶⢻⣯⣹⠯⠭⠓⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢐⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠁⢢⡀⠁⠈⠠⠊⠀⠀⠁⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠈⠀⢳⠀⠀⠀⠠⡀⠤⣂⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⢀⠀⢆⠀⠄⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠐⠀⡀⠀⠀⡠⠁⣜⣧⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣾⡷⠾⠭⣿⠇⠀⣁⣸⣿⡷⣶⡂⠐⠠⡀
⠀⠀⠀⠄⠀⠀⠁⠀⠈⠀⡉⠦⢻⡄⠀⠁⠠⠰⠄⠻⠿⠫⠐⠀⡀⠱⢃⡆⠂⡀⠁⠀⡀
⠀⠀⠀⠀⠀⢀⠠⠀⠁⠀⠐⠈⠣⡀⠀⠀⠚⠒⠲⠦⠤⠤⠤⠔⢒⡺⠏⠀⠀⠀⠀⠀⠂⠐
⠀⠁⠀⠀⠂⠠⠀⢂⣀⣤⣥⣦⣄⣀⣑⣒⢦⣤⢤⣤⣤⣤⣴⣞⣉⣀⢧⣤⣤⢤⣄⠈⠠⠌⣀
⠀⠀⠀⢀⡴⣾⡿⠿⠟⠛⠻⡉⠛⠛⠛⣋⠀⠉⡉⠉⠁⢠⠿⠟⠶⠷⠾⠿⢟⡿⡾⠿⣿⣶⣲⣔⢦⡀
⠀⠀⠀⢸⡞⠁⠁⢀⡄⣀⢀⡀⠀⠈⣏⠘⠢⣀⢈⣀⣀⣰⣞⣓⣈⣥⣀⣘⣀⣋⢁⣈⠷⠋⢹⣏⢇⡇
⠀⠀⠸⡌⡓⠄⡀⠐⠊⠁⠀⠀⠀⠠⠰⠅⠂⠠⠛⠿⠩⠰⠚⠁⠈⡀⠨⠀⠉⠁⠉⠀⠀⠠⠊⢀⡼
⠀⠀⠀⠐⢎⠀⠀⠌⠀⠂⠀⠀⠠⠤⠤⠤⠌⠄⠨⠭⠭⠭⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⢯⣒⢤⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠂⠀⠉⣁⣭⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠓⡫⠷⢢⢂⢆⡤⡴⡤⠤⠤⢤⢤⣤⡤⡤⡤⠤⠦⠤⣖⣻⢫⠟⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠋⠚⠓⠑⠓⠚⠛⠛⠛⠉⠉⠉⠁⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣦
⠀⠀⢻⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠘⣿⣿⣿⣿⣆⣀⣀⣀⣰⣶⣶⣶⣶⣾⡿⠿⢿⡟⠛⣿⠛⢛⠉⠻⠏⠁⠉⠙⠟⠛⠛⠿⠻⢿⢿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠭⠿⠀⠠⠾⠀⠀⠒⠒⠒⠂⠀⠙⠀⠀⠀⠉⠀⠀⠴⠀⠐⠇⠚⠛⠛⠿⠻⠿⠿⠩⣿⣿⣿⣶⣤⡀
⠀⠀⠀⣿⣿⣿⣷⡤⠄⠀⠀⠀⢀⣀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⠠⠄⠀⠤⣤⠀⣈⠉⠹⠯⠄
⠀⠀⣸⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⠿⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠛⠋⠉
⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠂⠈⠁⠀⠉⠁⣀⡀⠀⡁⠀⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠤⠴⠀⠀⡀⣀⠹⠿⠄⠈⢀⣠⠀⣤⠤⠁⡙⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⢆⣀⡀⠀⢀⣤⠖⠀⠙⢚⣙⠀⠀⠄⠀⠠⣤⣀⡁⠀⣀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣠⢤⣾⠁⠀⠁⠯⡿⠀⠀⠀⠀⠀⠲⢂⣾⣶⣤⠄⢠⣽⡀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠀⠀⠀⢐⠘⠃⣹⢷⡄⠀⠀⠀⣻⣧⠀⡀⠀⠂⠉⠱⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡀⡸⠮⠲⡇⠒⠼⣷⣿⠏⠀⠉⣰⣦⣺⢿⣯⢏⠛⠰⠆⠤⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⡀
⠀⠀⠠⠀⠀⠀⠀⡙⠈⢡⠄⠈⠈⡒⠄⢈⣻⠇⠀⠀⠊⠉⢼⣾⠟⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠆⡈⢹⢿⢿⡗
⠀⠀⠀⣀⠀⠚⠁⠙⠚⠣⠇⠀⡀⠷⠌⠋⢉⣇⠀⠀⠀⠀⡼⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⣤⠄⣀⡀⡀⠠
⠀⢀⣍⣻⣿⡏⢀⣓⣶⢷⣶⡏⣯⣤⣤⣶⠂⣤⢶⣾⣷⣾⣶⣀⣔⡀⠒⠀⣀⣀⣀⡀⠀⣀⣀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠈⠉⠁⠈
⠀⠸⡷⢶⠄⠻⠻⣿⣿⠏⡟⠿⠐⢚⡷⠶⣿⣟⠙⠛⠀⠛⠻⣟⣩⣐⠂⠀⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⣀
⠀⠀⠤⠤⠤⠥⠤⠭⠝⠛⠛⣁⣘⣥⠖⣖⣵⠀⠁⠀⠁⠀⢀⡘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠤⠤⠦⡄⢀⠀⠀⠀⠊⠁⠀⠀⠛⠠⠉⠁⠂⠀⠀⠀⠈⠃
⠀⠀⣀⡠⠎⠁⠀⢉⣤⡍⠀⠤⠢⣀⣀
⠀⢈⣍⠁⠚⠀⠀⠘⠓⢢⣤⡴⢠⣬⡍⢉⡂⠀⠄⠀⠀⠀⠈⠉⠁⢀⠄
⠀⠈⠁⠰⠄⠀⠢⠀⠈⠁⠀⠂⠀⠒⠘⠉⠀⠉⠁⠀⠚

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


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

⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣭⣭⣭⣭⣍⣉⡉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣫⠥⣶⣶⣶⣾⡯⢭⣙⠓⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣶⣟⡛⠉⠉⢙⣛⣻⢯⣗⠦⣙⠷⢄⡈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢢⢊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣟⡉⠉⢹⡄⢴⣟⣿⣟⢷⣮⠙⠪⣻⢦⣙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⢰⢲⠇⠈⠀⠁⠈⠀⠀⢀⣴⣶⣶⣦⣄⡀⠀⠀⠈⠛⠋⠀⢠⡇⡸⣿⣾⢡⡆⠹⣵⣄⠈⠳⣮⠑⢦⣉
⠀⠀⠀⠀⠀⠄⠀⡱⠛⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⣭⣄⡀⠉⠛⠻⣷⣦⡀⠀⠀⢀⡼⠀⢉⡩⢪⡞⣀⣉⣉⣿⣷⣤⣌⣳⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠿⡤⠈⠹⡄⠀⠀⠀⠈⠛⠶⣦⠞⣡⣔⣩⣾⠿⣻⣿⣿⣥⠤⣄⠉⠛⡝⠻⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⡒⠝⢧⡀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⣾⣷⠾⣛⠿⣫⡵⣻⠟⣫⠽⣿⣿⡌⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣞⠀⢀⡀⠙⢷⣤⣤⠏⠀⣀⣤⡾⠿⠛⣉⠤⢊⠵⢋⠔⡽⠁⢸⠁⣻⣿⣾⠷⢁⡴⢁⠜⡠
⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⡻⣝⠛⠛⠿⢿⣶⣿⡿⠷⠿⠛⢛⣛⣩⠥⠔⢒⣩⠴⢚⢁⡴⢋⡜⠁⣰⡞⢦⣙⢿⣍⣛⣛⣊⣡⣖⣥
⠀⠀⣠⣴⡿⡻⠝⣚⣉⡙⠊⣿⠢⢭⣑⣲⣶⣬⣿⣿⣿⣿⣯⣭⣴⣶⣒⣩⣩⣤⣴⣴⣯⠶⠿⢶⣿⣿⣧⣤⣭⣳⣿⣷⣶⣾⣟⡛⠫
⠰⠟⣱⢋⠊⠔⠉⣁⣍⣉⣴⣿⠴⠚⠉⠉⠥⢖⣛⣩⣙⣛⣿⣿⣯⣭⣭⣭⣄⣤⣤⠤⠶⠶⠖⠲⠤⠤⢄⣀⠀⠉⠉⠙⠛⠛⠿⠿⠷
⢀⢰⠃⠔⠈⠀⠀⣯⣿⠿⠟⡁⠄⣢⠴⠚⢋⣉⣡⣤⣤⣶⣶⠖⠛⠛⠟⠿⠟⣛⣛⣛⡛⠿⠟⠻⠿⢶⣶⣤⡭⢽⣶⣶⣶⣤⣤⣤⣤
⠈⢸⠀⠐⡐⢄⡁⠒⣂⣈⣤⠶⣚⣁⣤⡴⢞⣿⣭⣽⣿⣿⣿⢿⣷⣶⣦⣤⣄⣀⣈⣉⣉⣉⣉⣉⣭⣭⣤⣤⣭⣐⣒⠦⢄⣀⣀⣀⣀
⢈⠸⣇⠀⠉⠦⢍⡉⣉⣥⡴⠿⢛⣩⡵⠾⠛⠉⡉⠁⢀⣀⣠⣴⣶⣶⣿⣿⣿⣷⣤⣤⣤⣴⣞⠫⣅⣶⣶⠦⣌⠙⢧⡈⠍⠐⠛⠟⠟
⠐⠀⢙⣷⣤⣤⠶⠛⠉⢀⣠⡶⠛⠉⠀⢐⣨⣴⣒⠋⠩⠁⢈⠔⠑⠡⣟⡿⡯⠛⢦⣙⠻⢯⣈⠥⠚⢹⡀⣾⣼⡁⠀⢳⡀⠁
⠘⠛⠛⠉⠁⠀⢀⣠⠶⢟⠉⢐⣠⣴⣊⣉⣀⣬⣍⣁⣒⣮⣀⣤⣀⡬⠜⠋⠁⠀⠀⠙⠻⣔⡛⠶⣄⠀⠙⢮⣳⢶⣮⡤⠛⢦⡀
⠈⢀⣀⣀⣤⢖⣋⣁⡤⠴⠚⠋⡉⠩⠄⠀⠒⠐⠒⠒⠒⠬⣭⣍⣉⡛⠛⠶⠦⣤⣀⡀⠀⠀⠉⠙⠛⠻⢦⡀⢌⡙⠛⠻⠷⢶⢿⣷⣶
⠸⠿⠟⠛⠛⠛⠋⢉⣭⣭⣉⣙⠛⠒⠲⠶⠤⠤⠤⠴⠖⠒⠒⠒⠛⠛⢛⣛⠋⠭⠽⠿⠷⠶⣶⣤⣀⡀⠀⠉⠳⣽⣧⣅⡠⠄
⠠⡤⡤⠴⠶⠚⣋⣉⣉⣉⣉⣉⣉⣛⣒⡒⠶⢤⣤⣤⠴⣖⣒⣂⣉⣡⣤⣤⡤⢤⠤⠄⣒⣒⡈⢛⠿⣯⣿⡛⢲⣶⣿⣿⣿⣮⡄⢀⣀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⠟⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣾⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⡄⣻⣶⣤⣀⣀⡀⠀⠀⠀⣀⣤⠞⢉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣦⣤⡎⢀⣀⣤⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠇
⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣻⣿⣿⣟⣻⣯⣿⣿⣟⣾⣻⡿⣻⡟⣻⡿⣻⡿⣿⡿⣿⢿⣿⣿⣿⣿⣷⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⡴⠋⢠⠏
⢀⣠⡴⠾⠟⠛⣿⣶⡛⠀⠙⠻⣮⣉⣠⣤⣤⣥⣿⡬⢥⣬⣧⣭⣠⣟⣘⠛⠁⠋⠘⠋⠾⠇⠸⠟⠹⡿⢻⣿⣿⣿⡿⣶⣦⣤⣤⣾⣷⣶⡟
⠀⠙⢣⡀⣀⠻⠿⠟⠀⠀⢀⠀⣻⡟⡟⠓⠉⠉⠈⠉⠉⣁⣀⣨⠭⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠾⠿⢿⣿⣿⣻⡅
⠙⠻⢦⣥⣞⡃⠀⠀⠀⠀⣴⡀⣿⣓⣳⣤⣤⠖⠈⠉⠉⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠉⠁⠉⠀⠀⠀⠀⠻⣝⠛⢻⡀
⠀⠀⠀⠀⠉⠛⠻⠶⣶⣮⣿⣿⡟⣛⣛⣿⣿⣿⣭⣭⣭⣵⣶⣶⠶⠶⢷⣴⣤⣶⣴⣶⣶⣾⠿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⣿⣿⣯⣭⣭⣭⡭⠽⠿⠿⠿⠟⠛⠛⠛⠻⣿⡻⠏⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣷⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠿⠟⠛⠙⢿⣿⣇⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣴⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠻⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣛⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠬⢭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⠳⠄⠀⠀⣤⣤⣾⣯⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⡖⠀⢰⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⢀⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡾⠁⠀⠋⢠⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠸⣇⠀⠶⠀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⢿⡆⠀⠀⠘⠈⠃⠀⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣯⠉⠁⠀⠀⠀⣰⠃⠀⡆⠀⢀⣴⠀⠀⠀⠀⠀⢉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣌⣋⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⢀⣀⣠⣤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡀⠀⠉⣉⣉⠐⠒⠓⠒⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⢸⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣀⣈⣻⠛⠃⠀⢰⡇⠠⡄⢠⣆⠀⠀⠀⠀⠼⢫⡿⠀⢸⡇⢠⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠏⠉⠙⢢⣤⠤⡄⠹⡄⢀⡄⠉⡇⠀⣠⠶⢶⡄⠀⠀⢸⣷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⠴⠶⣄⠀⠀⠐⠃⣠⠇⠸⠁⣼⢁⡾⠀⢈⣸⠇⠀⠀⢸⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠺⣧⡶⠂⠘⣷⠀⠀⠐⠋⠀⠀⠀⣉⡴⠃⠰⠟⠁⠀⣰⠆⢸⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⣸⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⠖⠛⠁⠀⢸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠆⠀⠀⠀⠀⣠⡾⠏⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⣸⡆⢸⣿⡛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⠛⠳⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣦⣤⣴⠏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸⣿⠁⠀⠘⡄
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠤⠦⠞
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠈⠀⠒⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠉⠙⠲⠶⢤⣤⣍⣉⣙⣛⣛⣀⣀⣬⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⠋⠁

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲
⡰⠒⠢⡄⠀⠀⣀⠀⠀⠀⠀⠸⢱⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠒⡄⠧⠴
⠑⠤⠠⠎⠀⡘⠂⠃⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣩⠆⠰⠅⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢎⣠⠇
⠀⠀⠀⢠⡏⠉⠑⢦⡤⡀⠀⠀⠀⠂⠣⡀⠀⠀⠘⢍⢧⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⢀⠴⠤⢔⡺⠛⢉⡭⠝⡩⠿⠽⢦⠀⡠⢤
⠀⢀⠀⠀⠙⠛⠛⠲⣄⠙⣄⣀⣀⡼⠆⣈⡉⡒⠤⠒⠃⠉⠁⠀⠈⠉⠉⠢⠴⠊⠑⢒⣁⡴⣤⠤⡝⣯⡛⠒⢒⣉⣰⡴⠋⠀⠀⡿⢀⡾⠔⠊⠉⠁⠀⠈⠉
⠀⠈⠀⠀⠀⢀⡴⠒⠈⢳⣉⠲⢷⡖⠲⣆⣏⣁⣠⠦⣀⣀⡤⠀⠀⠒⠲⠴⠂⠀⠀⠀⠠⠀⠀⡜⢵⠋⠙⠒⠋⠀⡜⠀⠀⠀⣠⠇⢸⠁
⢀⣀⠀⢠⠒⠉⡠⢄⠤⢌⡳⣧⡀⠉⠓⠏⠙⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠢⠤⠦⠶⠤⠀⠀⣸⠁⠀⢸
⠣⠔⠁⠈⠁⠋⠀⠠⠀⠀⠳⢌⣻⣦⡀⠀⠀⠀⠀⠙⠓⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠦⣶⡦⣄⠀⠀⠀⠀⠙⠦⣤⡤⠤⣄⣀⣄⣠⠖⠶⠃⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⠀⠁⢿⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠈⠓⠶⠛⠛⠋⢩⠉⠛⠉⠅⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⠶⠤⣤⠴⠋⠳⠦⠴⠒⠐⠦⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠘⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⠀⠀⠁⠒⠂⠤⠤⢤⡀⠀⠀⠀⠀⠀⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠂⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠂⠤⠤⠤⠤⠤⠤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢒⠀⢀⠀⡇⠀⠀⠀⠀⠃⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⠉⠉⠉⢁⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠰⠐⢠⠁⠀⠘⡄⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠀⠠⠄⢤⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠃⠀⠀⠀⠀⠁⠐⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣀⠤⠤⠀⠀⠀⠈⠢⠤⣀⣀⣀⣀⣠⠔⠃⠀⠀⠐⠂⠤⣄⡀⠄
⠀⠀⠀⠀⠀⢀⠄⣠⡪⠖⠋⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⠒⠋⠉⠀⠀⠀⠉⠑⠮⡢⡀⢀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⡀
⠀⠀⠀⠀⡅⠀⢱⡀⠀⠀⠀⠘⢆⠀⠀⠀⠠⠼⠂⠈⠢⠤⢤⠤⠴⠊⠀⠸⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡀
⠀⠀⠀⠀⠘⢄⠀⠙⢤⡀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠇⠀⠀⠀⠀⠀⣠⠔⠁⠀⠔
⠀⠀⠀⠀⠀⠀⠳⠤⣀⠉⠓⠦⢤⣈⣓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣀⣠⡤⠖⠋⢁⣠⠔⠉
⠀⠀⠀⠀⠀⠠⡀⠀⠀⠉⠒⠦⢤⣀⣉⠉⠛⠛⠒⠶⠶⠶⠶⠶⠶⠶⠖⠚⠛⠋⢉⣉⣠⠤⠔⠚⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠙⠛⠒⠒⠒⢒⣒⣒⡒⠚⠛⠉⠉⠉⣀⠴⠊⠂⢞⣻⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠢⡀⠀⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⢔⢈⠎⠃⣫⠀⠀⠈⡁⠐⢶⠋⠨⢄⣀⡤⣀⠠⠑⠀⠠⡠⠊
⠀⠀⠀⠀⠀⠀⠈⠢⢌⡐⠀⠈⠉⠒⠢⠤⠤⣀⣀⣀⣀⣀⣓⣦⣖⣊⣀⣀⣡⠤⠤⠖⠂⠉⡀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢤⣌⣀⡀⠀⠂⠀⠀⠌⠉⠉⠉⠉⠁⠄⠀⠀⠀⣐⣀⣤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠖⠛⠛⠋⠉⠁
⣿⣿⣿⣿⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣿⣟⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⣾⣷⠀⣠⠆⢄⣀⣠⣾⠻⣷⠀⠀⠀⠀⠀⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⢹⣶⣼⣿⡛⣾⣿⣄⠈⢿⣿⣿⡇⣿⣶⣦⣀⠀⠀⢀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀⡄⣀⣀⣾⣿⡛⠉⠳⡌⠿⠃⢀⣻⣿⣿⣿⣼⣿⣇⣿⣧⡔⠫⠀⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠻⠆⠀⠀⠠⡘⡄⠀⠈⡏⢹⡗⠛⡿⢿⣿⣿⣿⣧⣤⣰⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠘⠛⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠃⠈⠁⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠰⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡘⣸⠀⣿⢠⣿⣿⣦⣄⠄⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⢀⣴⣒⣠⣶⡿⠟⠻⠛⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡇⣸⣧⣿⣿⣿⣿⣿⡀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠛⠋⠋⠛⠛⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡟⠠⢱⣿⣿⣿⣿⣿⣿⣿⣦⡞⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠈⢀⠀⠀⣴⠃⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠀⠀⠈⠀⠀⢿⡟⢏⣿⣿⣿⣍⣹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠠⠿⠖⠚⠛⠛⠛⠛⠻⣿⡶⠶⢿⣥⣤⠀⠀⣀⣠⣤⠀⠀⠀⢚⣿⡿⠃⠀⠀⠀⠀⢀⡀⠀⣠⠔⠀⣰⡿⠻⣿⣿⠟
⠀⠀⠀⠠⠔⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠛⣀⣀⣒⣀⡀⠀⠐⢶⣬⠁⠀⠀⠉⢹⣤⣶⠉⠉⠉⢴⣦⠖⠁⢀⠔⠛⠻⠟⢛⠶⡟⠋⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠤⢤⣀⡀⠉⠉⠉⠉⠀⠀⠀⠀⢀⣉⡿⠛⠀⠀⠉⠥⢀⣀⣠⡶⠤⢀⣾⠗⣀⡤⢀⠀⠠⣾⣿⣴⣶⡟⠁
⠀⠀⠀⠀⠀⢠⣤⡾⠋⠁⠀⠠⣄⠀⠀⠀⠀⠈⠓⠲⠦⠤⠤⠀⠀⠈⠁⢀⣬⣭⠤⣤⣤⣤⣭⣤⣤⣶⣯⣵⣿⣿⣿⣿⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢿⣿⣿⣿⣅⡀⠀⠈⠉⠙⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⡶⠶⣶⣦⣴⣛⠉⠁⠀⠈⠉⠙⠓⠦⣤⣀
⠀⠀⠀⠀⠀⠙⠦⣄⠉⠛⠛⠿⢶⣤⠀⠾⠆⠀⠀⠀⠉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣖⠲⠦⣄⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣄⠀⠀⢙⣦⡀⠀⠀⢹⡀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠈⢧⣠⣤⣽⣶⠀⠀⠸⣇⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠞⠀⠀⡼⠁⠙⠦⣄⠘⣇⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⡴⠋⠳⢿⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠟⠀⠀⠀⠀⢷⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⡴⢶⡀⠳⣄⠸⣗⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈⠳⡽⠀⠀⠉⠛⠶⢤⣀⡀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢤⣤⣀⣀⣙⡏⠉⠉⠉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢈⣩⠴⠖⠚⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠴⠋⠁⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⢦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣠⣤⣤⠤⠤⠤⣤⠤⣤⣀⡀⣀⣤⣰⣦⣴⠆
⠀⠀⠀⠀⠀⠀⠀⣰⠿⡿⣍⠻⣗⡒⠰⣟⡳⢉⣽⣟⢿⣿⡿⣶⣭⣟⠯⣫⡿⢿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⢿⢸⠐⣽⡧⠹⡿⠑⠺⡻⡿⠻⠍⡣⢛⢿⣿⡿⣿⣿⣮⣔⢩⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢘⣏⣶⣘⣿⡀⠇⢠⢀⡇⠀⠀⠐⠀⢸⣆⡠⢃⣿⣿⣿⣿⣷⣻⣽⣿⡀
⠀⠀⠀⠀⠀⠀⣴⢛⣭⣍⣉⡉⠘⠒⠋⢙⣷⠦⠤⠤⢔⣘⣠⠔⣨⠀⢛⣹⣿⣿⣿⣞⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠦⣦⡐⠀⠀⠉⠉⣛⣯⣤⣤⣄⣤⣾⣉⣁⠚⣠⣿⣿⣾⣽⣿⡿⣿⡾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⣴⡖⠮⢿⣿⡿⢿⡛⠷⠊⠁⠀⠙⢿⣿⣿⡿⢛⣭⣬⣿⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣴⣤⣄⣀⡀⠀⠈⠁⠀⠙⢛⣫⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣤⡀⠀⠙⣿⣯⡹⣿⠀⠀⠀⠀⠐⣿⢿⢿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡿⠓⠦⢼⣾⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠘⢿⣿⡄⣤⣄⡀⢰⣾⢷⡏⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢀⣀⣀⣀⣀⣀⡼⠃⣽⡿⣷⣝⣯⣿⡾⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠿⣿⣿⠀⠀⢹⣤⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡯⢿⣿⣤⡐⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⣽⢿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣡⢿⡿⠁⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⢾⢿⢎⣷⡟⢻⡅⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢱⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠋⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠷⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠙⠿⣿⡿⠋⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣼⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣤⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⣠⠔⠈⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⢂⠠⠤⠈⠀⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠒⡀⠊⠀⠁⠀⠃⢀⡴⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠀⠀⢀⠘⢤⡀⠀⠢⠀⠈⢠⠀⠀⠀⠀⠰⡃⣀⠤⣤⣤⣶⡎⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠧⣀⡅⠀⢀⣅⠐⠠⠒⠣⠤⡰⠀⠀⠀⠞⢡⠿⠆⠛⣘⢋⣅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠠⠁⠐⢀⣀⠀⡀⢺⢄⣤⡶⠃⠠⢀⠀⣀⣀⡀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢦⠀⠢⠄⠐⠒⠒⠛⠫⠛⠁⠀⠈⣉⡉⡡⠔⢁⠴⣿⣦⡥⠤⠅⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤
⠀⠴⠒⠀⠂⠂⠐⠀⠀⣀⠀⠀⠀⠴⠤⠀⣐⠒⠀⠰⠦⠶⠶⡴⠖⢚⠋⠁⠀⠀⠞⠂⠁⠙⠙⠃⠀⠀⠀⠐⠀⠐⠒⠐⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠠⠂⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠐⢂⡄⠉⠉⠉⠁⠐⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⣥⣄⣀⠀⠀⠺⣿⢶⣿⣿⠄⣠⣶⣶⠀⠀⠉⠑⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣼⣿⣿⣿⣿⣧⠀⢀⣠⣀⡠⢋⣾⣿⣿⡿⣣⣼⣿⠟⢠⣄⣤⣬⡉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡴⠀⣼⠟⡿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⠉⠉⠋⣾⣿⣿⣿⠙⠿⠿⠜⠛⠛⢂⠤⣜⡂⠤
⠀⠀⠀⠀⠀⠀⠀⣰⣫⣾⣿⢃⣼⡟⡌⠁⠋⢻⣿⣿⣿⠏⢰⠰⣴⡀⠀⠰⠀⠁⢀⠴⠒⠒⠒⠒⠀⠀⠀⢴⣿⣿⣶⡶⠖⠒⢷
⠀⠀⠀⠀⠀⠀⡼⣱⣿⣿⣷⣾⠛⢡⠀⠀⠀⠸⣻⣿⠃⢀⡟⣧⣿⣧⡀⠀⠀⠠⠃⠀⠀⡀⠊⠐⠀⣠⣶⣿⣿⣿⣿⠿⣆
⠀⠀⠀⠀⣠⣴⣷⣿⣿⣿⣿⡏⠀⢠⠀⠀⡀⡀⠈⠃⢰⣼⡇⠸⣿⣿⣷⢀⠀⠀⠀⡠⠊⠉⠀⢢⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰
⠀⢀⡴⢋⠝⣫⣿⣿⣿⣿⣿⠇⠀⢸⡀⠀⠱⡐⣄⠀⢸⣿⣿⣄⠈⠛⠿⣿⣦⠀⢸⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⡏
⣰⠋⠀⠀⠊⣡⣿⣿⣿⣿⡵⠂⢀⣾⡇⠐⢄⠘⠮⡳⢎⢿⣿⣿⣷⣦⣄⣀⠈⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⡃⡜⠀⡞
⢻⣀⠀⠀⠀⠺⣛⣭⠞⢁⣠⣶⡿⠃⠘⡄⠈⠓⢤⣘⣆⠳⢍⡻⠿⠿⠿⢿⣿⡦⠀⠀⠀⡈⠉⠒⠄⡈⠙⠻⠿⣿⠞⠀⡸
⠀⠈⠙⣶⡾⠟⠋⠀⠀⣿⡟⠉⠀⠀⠀⠈⠦⡐⢤⣈⠉⢅⣀⠈⠉⠉⠉⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠤⣉⡒⠇
⠀⠀⢸⡇⠀⢀⣴⣾⣷⠋⣽⣗⡀⡀⢀⣠⣤⣌⡒⠤⣉⠉⠩⠕⠲⣄⡀⠀⠀⠀⠉⠒⠦⠤⠤⣤⣀⣀⣀⣀⠀⠀⠀⠉⣢⠁
⠀⠀⠘⣇⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣤⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠭⠀⠀⠐⠊⠁
⠀⡴⠋⠁⠩⠾⠟⣻⣿⣎⣝⣛⣷⣆⣀⣈⣉⠉⠛⣿⣿⣿⣿⣿⠀⣶⣤⣄⣀⣤⡤⠤⠄⠒⠈⠁
⠾⠒⠒⢶⡠⡄⣍⠹⣿⠟⢿⣿⣿⣿⣫⡞⠁⠀⢸⣿⣿⣿⣿⡏⢸⣿⡟⢿⡿⠙⠀⢀⠁
⠀⠀⠀⠀⠳⡱⣌⢷⡅⠀⠸⢟⢭⣻⡿⣷⣄⠀⠀⠈⠙⠿⠟⢀⣿⣿⣿⢸⣷⡦⢔⡀⢀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⣠⣤⠄
⠀⠀⠀⠀⠀⠙⢾⡶⠇⡠⠶⣶⣦⣬⣉⡛⠻⢷⡀⣤⣀⠀⠠⣿⣿⣿⡟⣼⣿⣷⢰⣍⣾⣿⣿⣿⣿⣿⣿⠧⣀⡴⣨⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠂⢬⣛⣶⣮⠉⠛⠛⠛⠚⢻⣦⣝⢷⣤⡀⠈⠉⠰⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⡇⢰⢋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣶⣦⣽⣿⣿⣿⣿⣶⣄⣀⠈⠁⠀⠹⠟⠙⠿⢿⣿⣿⣿⢧⣓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⠙⠋⠁⣠⠞⠉⠻⢿⣿⣿⣿⣷⣦⣄⡀⠀⣠⣴⣿⣿⣿⠛⡗⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠹⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣯⣻⣿⣷⣿⡏⠒⠄⢹⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣝⢷⡝⢟⢮⡋⠓⠤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠠⣀⠙⠦⡀⢨⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢔⡀⠑⢄⣸⠿⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⠅⠈⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢂⣤⣵⠶⠊⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣈⣙⠛⠿⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⢦⣤⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡟⣉⣀⣤⡔⣀⣀⣀⣀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠿⣶⣕⢄
⠀⠀⠀⠀⠀⠀⡌⠀⠿⠿⠛⣉⠉⠻⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠙⢷⣅
⠀⠀⠀⠀⠀⠀⡇⠶⣶⡾⢃⠉⠀⣴⡖⣾⣿⣿⣷⡀⢀⣀⣀⠀⠀⣿⠿⣻⣿⣿⣿⣿⣿⣾⣿⣦⡀⠙⢿⣄
⠀⠀⠀⠀⢀⠄⢀⠔⡩⢀⣥⣴⠾⠟⣛⠛⠛⢛⣛⣳⡌⠻⣿⣦⣴⢹⠂⠐⠠⠬⠍⣙⠛⠿⢿⣿⣿⣦⡀⠻⣧⡀
⠀⠀⠀⠀⡊⠀⠀⢈⣐⠫⠍⠒⠉⠁⠀⠈⢋⢻⣿⣿⣿⡆⣉⠙⠿⣧⠣⡀⠀⠀⠀⠀⠈⠁⠂⠌⠙⠻⣿⣄⠹⣷⡀
⠀⠀⠀⠀⠙⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠟⠋⠀⠀⠑⠠⣈⣁⠈⠂⣀⠀⠀⠀⠀⠀⠀⠈⠀⢌⠻⣆⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡎⣿⣿⣖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣽⣿⣷⣿⣿⣿⣷⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⢿⡿⣿⣿⣿⣷⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠥⠄⠐⠁⠈⠒⠂⠀⠈⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠛⠒⠪⢥⣂⢄⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⠀⣿⠿⠶⢮⣑⠮⣒⢄⡀⠀⠀⠀⠀⠀⢀⣤⠶⠒⣒⣒⡒⠤⢄⣀
⢉⣉⣉⣉⡓⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⢸⡆⢿⡀⣴⣶⣬⣙⠮⣓⢭⡲⠤⣤⣤⡴⠟⢁⣴⡿⠋⢉⣉⣉⣓⠲⢭⣓⡲⠤⣤⣀⣀⣀⣀⣀
⠛⠋⠉⠉⠙⠛⢦⣑⢄⠀⠀⠀⠀⠀⣼⠃⣠⣾⠇⢸⡇⢹⣿⠟⠛⠷⣮⡓⢬⣓⠶⢤⣤⣶⠿⠟⠀⣰⡿⠋⣩⣭⣙⡓⠦⣍⡓⠶⢤⣤⣤⣤⣤
⠿⠿⠿⠿⠿⠷⢦⣌⠳⣕⢤⡀⠀⣰⠇⣸⠏⢁⣠⣾⡇⢸⣿⠀⢠⣶⣤⣙⠳⣬⣛⠶⣤⣤⣤⣤⣾⡿⠃⣸⣿⠛⠛⠛⠷⣦⣍⡓⠶⣤⣤⣤⣤
⣶⣶⣶⣶⣴⣶⣦⣬⡓⢮⡳⢬⣙⣉⣰⡟⢠⡿⠋⣁⣤⣾⡿⠀⣼⣿⡟⠛⠿⣦⣝⠻⢶⣤⣬⣥⣤⣤⣾⣿⠃⣠⣾⣶⣶⣤⣉⠻⢷⣦⣬⣭⣭
⣤⣤⣀⣉⣉⣉⣉⣉⡙⠳⣭⡳⢬⣉⡋⢀⣾⠃⣾⡿⠋⢉⣀⣴⣿⡟⢀⣴⣶⣦⣙⠻⢶⣤⣍⣉⣉⣉⣉⣀⣴⣿⠟⠋⠉⠛⠿⢷⣦⣤⣉⣉⣉
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⣭⡓⢮⣝⣛⣃⣼⡿⠀⣾⣿⠿⠛⠋⣠⣾⣿⡿⠿⠿⢿⣦⣍⣙⣛⣛⣛⠛⠛⠛⢉⣠⣶⣿⣷⣶⣤⣌⣙⠛⠛⠛
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⣦⣭⡓⢮⣝⣛⠛⢁⣼⡿⢁⣴⣾⣿⠿⠟⠋⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠛⠛⠻⠿⠿⠿⠿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣌⣙⠳⢮⣭⣛⡛⠛⣁⣾⡟⠉⣀⣤⣤⣴⣶⣾⣿⣿⣿⣷⣦⣤⣴⣶⣶⣶⣶⣤⣤⣤⣶⣶⣶⣶⣦⣤⣤⣤⣤
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠶⣦⣍⣙⠛⠛⢋⣠⣾⠟⠛⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⡀⠀⠈⠉⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠙⢿⣿⣦⡀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡦⠀⠀⠀⠀⢹⣿⣷⠀⠀⣿⡄⠀⠈⢠⣾⣷⠄⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⠿⢷⡀⠀⣠⣤⣁⣀⣿⣿⣿⣷⣄⣽⣿⡀⠀⢸⣿⠏⠀⠀⠀⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡏⠀⠀⠀⠀⣿⣿⣿⠁⣠
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣦⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿⣿⠃⠀⣠⣶⡆
⠀⠀⣀⣀⣀⠀⠀⠀⣀⡀⠙⣿⣦⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣼⣿⡏⠀⢰⠟⠉⠀⠀⠀⠀⠰⢶
⡞⣴⣿⣿⣿⣿⣶⣤⣄⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣀⣠⣤⣤⣦⣤⡀
⠙⠻⠿⠛⠛⠛⢉⣉⣩⠍⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠰⠿⠟⠀⠀⢀⣀⣀⣈⣉⣉⣙⣛⣛⡛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣭⣍⣛⠛⠿⠿⣿⣿⡀
⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤
⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂
⠀⠀⠀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠴⠀⠀⠀⠀⣀⠠⢀⣀⠀⣀⡀
⠀⠀⠠⡀⠀⠈⢄⠨⡒⢄⡠⢀⠃⢱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠲⣀⣀⠤⠒⠩⠔⠁⠀⠂
⡤⠀⠀⠀⠀⢀⠈⢂⠑⠒⠠⠤⠐⠂⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠐⠒⠄⠃⢄⡀⢀⣀⣀⠄⠀⠀⠀⣀
⠀⠠⡌⢁⣀⡡⠤⠤⠗⠒⠒⠦⣄⣀⡳⠚⠃⢄⣄⡠⡆⢧⠀⠀⡀⠠⠠⢤⣀⡠⢀⢀⣭⡤⠄⠈⠙⠀⠐
⠀⠀⠈⠠⢅⣀⣀⠀⠀⠀⠀⠈⠀⠀⡀⠤⢴⣆⣀⠔⡁⠀⠀⠰⠔⠒⠀⠐⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠐⠒⠒⠁⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣻⣿⢋⣷⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣉⠻⡀⣿⣿⡏⣼⣿⢻⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡡⠒⢧⢱⣿⣿⣿⣿⠷⡺⣙⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡞⠀⠘⣿⣿⣿⣿⠃⡆⢡⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣤⠃⠀⠆⠈⣿⣿⡏⠂⠁⠘⠀⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣀⣠⠤⡤⠜⢡⡅⠀⣤⣠⣤⣀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⡋⠉⠸⠚⠤⣴⣀⡼⠟⠀⠀⠀⠱⠀⠐⢉⣉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠍⠛⠾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⡟⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⡎⠙⢻⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⡀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇⠀⣞⡇⠀⠀⣽⣿⣿⣏⠀⠀⠀⠀⠀⠀⠏⢈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠂⠀⠀⢨⠇⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⣿⠒⠃⠀⠙⣠⠀⠉⢿⣿⣿⡿⠈⠁⠉⠘⢘⠒⠒⠐⡧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡀⠀⠀⢭⠂⢀⠙⠁⡨⣀⢀⠸⣿⣿⡗⢐⣤⠤⢄⡦⠴⡴⢄⡛⠀⠈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠻⣷⣦⣤⣤⣉⣀⣀⣀⣀⣀⣀⣀⣀⢂⣀⣀⣀⣀⣠⣤⣤⣤⣶⣾⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡎⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢛⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠀⠀⠀⠈⠉⠋⠹⣭⣯⣽⣿⡗⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⣿⣿⣿⣿⢧⠀⠀⠀⠀⠈⠀⠘⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⢷⠐⠶⠶⢶⠶⠶⠶⠶⠶⠶⠦⣻⣷⣶⣾⠐⠶⠶⠶⠶⠾⠾⠾⠾⠚⠲⠈⣄⡀
⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡽⠛⠉⠈⢀⠀⠂⠀⠠⠀⠀⡀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣀⣀⣈⠰⠀⠂⠆⠄⠠⢀⡛⠭⣝⣶⣤⣀
⠀⠀⠀⠀⢠⠿⡿⣿⣻⡶⣶⣴⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣿⣿⡿⠛⠛⡁⠀⠀⠀⠀⠀⠀⠀⣐⣨⠴⠒⣭⢥⡴⣟⣻⡄
⠀⠀⠀⠀⣟⠚⠤⢌⡈⡝⠛⠷⢯⣿⠛⠛⢻⠛⢛⡟⠛⢿⠟⠛⠻⠛⠟⠿⠻⢻⢻⠿⢟⠻⢿⠿⢻⢻⣿⣿⡟⠛⢛⡡⠷⡛⢙⡁⣇
⠀⠀⡤⣤⠧⢍⣰⠒⢭⣑⠆⠣⣀⣌⠁⣙⠉⠉⠉⣇⠉⠛⠉⣛⠉⠙⠁⣖⠛⠚⠂⡗⠓⠒⠒⢺⠚⠒⢳⠒⠺⡂⢉⠤⠔⠃⣫⡠⠺⡆
⠀⠐⢤⠿⣷⣶⣤⣹⠓⢄⡙⠒⠦⣏⣀⣈⣉⣇⣀⣀⣀⣏⣀⣀⣈⣹⣉⣀⣀⣹⣈⣀⠈⣹⠉⠁⠈⢹⠉⠉⠁⢫⣡⢤⠔⠙⣄⣴⣘⢛⣻⣋⡓
⠉⠉⠅⠛⢓⠐⠢⠧⠗⣦⠈⠂⠐⠁⠀⠸⢀⣀⠈⢀⣀⣀⣀⠁⣀⣀⣀⠯⣀⣀⣀⣸⠀⢀⣈⡉⠉⠉⠉⡃⣉⣉⣆⡤⣷⣮⡷⠶⠴⠖⠒⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⢟⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣿⣫⣍⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦⣀
⠀⠀⠀⠴⠷⠶⠾⠿⠿⠷⠟⠟⠿⠾⠷⠶⠦⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠤⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⣤⠆
⠀⠀⠀⠀⠀⠀⢀⣤⣦⣴⣿⣿⣶⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣿⢛⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠟⠛⠁
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⣿⣿⠁⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣯⣾⣷⡏⠀⠘⠁
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠿⠷⢶⣶⣾⣦⡀⠀⠀⣴⣿⣿⣿⣿⠿⢿⡿⠶⠤⠤⠄⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⡀⠀⠀⠀⠀⠀⡟⠀⠀⠈⣻⡃⠉⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢟⠛⢷⠀⠀⠀⠀⢠⣶⣄⡼⠛⠿⠀⠀⡿⠋
⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣷⡤⣤⣄⣰⣦⣀⡀⠀⠀⢀⣠⣾⣿⣿⣿⣻⣿⣿⣄⣀⡀⡀⡀⠀⠀⠻⠇⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠀⠈⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠋⣿⣿⣶⣬⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣿⣭⣭⣭⣽⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠲⠶⢾⣿⣟⠻⠿⢿⣿⣶⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⣠⣾⠿⠿⠛⠿⠿⣿⣀⠁⠀⠀⠀⢠⣶⣶⡆⠀⠀⠈⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣿⠟⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣤⣿⣿⣿⣧⠘⢆⠀⠀⠀⡄⠀⠀⠀⠀⢀⡴⠟⢛⣻⣷⣄⣀⣠⣴⣞⣀⣀⣀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⠔⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣤⣶⣷⡆⠀⠀⠀⠀⠀⠈⢀⣾⣿⠇⠀⢀⣴⡟⣛⠁⠁
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣛⠛⠂⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏⢀⣤⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠘⢿⠁⣠⣶⣶⣿⣟⣿⡶⠂
⠀⠀⠀⠀⣠⣾⣿⣽⣿⣿⢿⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣵⣾⣿⣿⣿⣿⣼⣿⢦⣴⣾⣿⡛⠀⠀⠀⠀⠐⢿⣿⣿⣮⠟⠋
⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠁⠐⠓⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠈⠉
⡾⣶⣷⢶⢶⡷⣶⠶⢶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣿⣶⣶⣶⣿⣿⣷⣶⣶⣾⣿⣾⣷⣶⣷⣾⣷⣾⣶⣶⣶⢾⣷⢶⣶⣶⣶⠶⣶⠶⣶

⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⡆
⠀⠀⠀⠀⠀⢸⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠙⣿⣿⣿⣿⣿⣿⡟⠁⠘⣿⠀⢸⡇
⠀⠀⠀⠀⠀⣸⡇⠀⣿⠏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⢻⠀⠈
⠀⠀⠀⠀⠀⠘⠁⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠙⣿⠀⠀⠀⠸⣿⡟⠁⢹⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⡟⠛⣿⡟⠀⠀⠀⣿⠀⠀⠀⠀⣿⠇⠀⠀⣿⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡿⠀⠀⢸⡇⠀⠀⠀⣿⡀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠻⠇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢢⡀⠐⡀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠢⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐



⠀⠀⠀⠀⠀⣠⠀⠀⣠⡀⠀⠀⢠⠀⠀⢀⡄⠀⠀⠈⡀⠁⠀⢀⠀⠀⢠⡀⡀⢠⠀⡀⣠⢀⢀⡔⡄
⠀⠀⠀⠀⠑⠉⠁⠄⣁⡓⣀⣀⣼⣦⠀⣢⣳⣀⣀⣬⣧⢀⣨⣼⣆⣀⣥⢵⡄⣥⢧⣼⡧⠀⠊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢨⡉⢠⡄⠀⣄⠀⢀⡈⠉⢀⠀⠀⡀⠀⣠⠀⢀⡄⢀⡄⣀⡴⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠝⠂⠹⣤⣻⣦⣈⣳⣄⣛⡧⣘⢷⣄⢻⣄⣘⣀⡨⠅⡈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⡀⠀⢔⣆⡰⣄⣴⢈⣆⠴⢡⡆⠤⠀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⢷⣰⡄⣰⢄⣶⡔⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠈⣿⡇⣿⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠆⡃⣿⡛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⢀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠘⠀⠂⣃⣀⣀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠤⠀⢤⢤⢤⢤⠤⠰⠒⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠐⠀⠤⠤⢀⣀⣀⣀⣀⠀⠠⠠⠀⠀⠀⠌⠗⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠤⢤⣤⣀⣠⣀⣀⣀⣀⣄⠀⠀⠤⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠊⠄⠀⠀⠀⠀⠀⠀⠀⠲⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠤⠒⠒⣒⠤⠀⠀⠀⡠⢴⢒⡦⡄⠀⠀⣀⠤⢶⣤⡀⠀⠀⠀⡠⢤⡤⠄⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⡄⠀⠀⠀⠰⡁⠀⠀⣠⣎⠀⢸⡋⠀⠀⣠⠞⢁⢠⠋⠀⠀⢀⠠⠊⠀⡏⠀⠀⠀⠀⡤⠊⣿⣟⠁
⠀⠀⠀⠀⠀⡀⠄⣀⡉⢙⠉⣼⡇⣜⢘⡗⢲⠊⣱⣮⣿⠨⣷⠖⡆⡹⢀⠀⡀⢉⠒⡒⠒⠉⠨⢠⡁⣏⡲⠶⣖⠃
⠀⠀⢐⡇⠀⠀⡀⠀⠄⠀⠐⢸⠏⠁⡎⡧⠃⢐⢸⡿⠋⡄⢸⡦⠃⢠⢩⠀⠇⠀⡀⢟⠄⠁⢰⠋⡡⠋⢠⢺⣇⡇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠄⠀⠀⠄⣠⠁⠀⠀⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠃⠄⠠⠐⠰⠀⠁⠁⠢⠘⠁⠀⠪⣼⠀⠀⠀⠀⠔⠃⠘⠰⠀⠁⠊⠅⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡎⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢄⣤⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⣀⣀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠠⠤⠤⠤⠀⠀⣀⠀⠈⠀⠋⠉⠉⠙⠛⠋⠛⠂⠙⠒⠒⠲⠶⠶⢶⣮⣉⣉⠉⠉
⣥⣤⣤⣄⣷⣤⣤⣶⣿⣿⣿⡿⠿⣛⣙⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠋⠍⠉
⠭⠭⣽⣯⣭⡉⠿⠿⠿⡿⠯⠽⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⢀⡀⠀⣀⣀⣸⣶⣷⣶⣶⣶⣶⣶
⣶⣶⣾⣿⠿⢭⣭⣭⣭⣥⣀⣀⠀⠤⠤⠤⠤⠶⠶⠿⠛⠛⠓⠒⠚⠛⠛⠛⢽⠽⠿⠿⠷⠖⠲⠶⠶⣾⣿⣏⣉⡉⠡⠤⠤
⠛⠉⠓⠂⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣖⣒⣉⣭⣤⣄⣠⣤⣴⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⡀⠀⠉⠭⣽⣿⣿⣟⣛⣛⣉⣉⣀⣀⣁⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⢛⡛⠿⠻
⠛⠛⠩⠭⠭⠥⣤⣤⡄⠀⠶⠶⠾⠿⠿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⣼⣤⠿⠭⠵⠶⠶
⢖⣶⣶⣿⣿⣿⡶⣶⠶⠶⠶⠒⠲⠶⠶⠶⢶⡖⢂⡀⠐⠂⠀⠀⣀⣀⣠⣤⡄⠁⠀⢀⣀⣉⣉⣉⣉⣙⡛⠓⠒⣒⣲⣶⣶
⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⠶⠶⠶⢾⣿⣿⣛⡛⠿⠿⣿⣿⣭⣍⣀⣀⠀⣀⣀⣀⣀⣈⣉⣉⣩⣤⣴⣶⣲⣶⣶⡶⠶
⣤⣤⡤⠤⠤⣴⣦⣤⣤⣤⠴⠶⠒⠒⠶⠶⠤⠀⠀⠐⠒⠒⠲⠶⠖⢺⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣶⣶
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⢿⣷⠶⠶⠶⠶⠾⠛⠟⠛⠾⠿⠗⠒⠂⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠥⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠿⢟⣛⣛⣛⣛⣛⣿⣿⣿⡿⠯⠭⠽⠟⠛⠛⠛⠛⠉⠉⣉⣛⣉⣉⣀⣈⣉⣉⣀
⠿⠿⠿⠿⠿⠿⠾⠿⠿⠶⠾⠛⠛⠛⠋⠉⠉⢉⢁⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉
⣒⣒⣂⣠⡤⣄⣠⣤⡤⠤⠤⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⠲⠒⠒⠒⠒⠂⠛⠋⠀⠒⠀⠈⠙⠛⠓⠂
⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀
⣀⣀⣀⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣭⣭⣽⣿⣿⣭⣥⣬
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀⠀⠈⠉⢉⣩⣭⣭⠭⠭⠴⠶⠚⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠉⠛⠛⢛⣛⣛⣛⣛⡛⠻⠿⠷⢶⣶⣶⡶⠶⠶⠶⠶⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠛⠛
⣀⣀⣀⣲⣀⣈⣭⣭⣭⣿⣿⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣯⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠈⠛⢿⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⢸⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠟⠻⢷⣦⣄⡀
⠀⠀⠀⠀⢻⣧⠈⠉⠻⢷⣦⡀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠈⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢀⣴⣦⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠛⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⣴⡟⠁⣀⣤⣾⠟⠋⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⢸⣿⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣴⡾⠟⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢀⣠⣴⡿⠻⠿⠿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠈⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠐⡋⠐⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢪⠖⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣁⣈⣀⡰⢁⡎⠀⣀⠀⠀⠀⠘⠂⣀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡞⠛⠙⠽⡯⠙⠙⣿⣿⠿⢷⣤⣀⡠⠀⠀⠀⠠⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⣙⣾⢄⠀⠀⠠⠁⠚⡉⠨⢢⡙⣐⠒⢹⠕⠶⣤⠴⠀⠀⠀⣠⢏⡔⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠐⠒⠀⢀⣴⡿⣙⣍⣵⠋⢠⡔⠄⢢⠀⠀⠀⠀⠝⠀⠀⠀⠱⠌⠊⠀⢤⣀⣴⣷⡿⢀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⣤⡤⠀⢀⣴⣯⣿⣟⠛⠿⠷⠤⠟⠛⠦⠀⠀⣄⢀⠤⠀⢧⢪⠔⠁⠀⠀⠀⠸⢏⢻⣿⣿⡀⠀⠀⠀⠀⡠⠖
⢀⠏⠀⢔⡕⣊⠒⣍⠿⠀⢀⡶⠀⠀⠈⠀⠠⠄⠋⠅⠰⠈⠊⠼⠇⡠⠀⠀⠠⠠⣀⣾⣿⡋⠁⠀⠀⠀⢸
⡋⢣⠀⠸⡘⠿⢀⡾⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠨⠡⠁⠡⢠⢁⣠⣀⡶⠖⠒⣶⣾⣿⣯⡁⠀⠀⠀⠀⠈⢣
⠑⠈⠓⠀⢈⢙⡉⠄⠀⠀⠀⠀⢀⠘⡠⡀⠂⢀⡤⠰⣢⣌⣋⣰⣽⠏⠱⠈⠀⠸⣿⣿⠎⠓⠄⠀⠀⠀⠀⠐⢆
⠀⠀⠲⢄⠀⢠⠀⡀⠀⠠⠠⠐⢁⣼⡇⡀⠠⠀⡰⠀⣶⣿⣿⣿⡟⠀⠀⠀⠒⠒⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠉⠒⠄⣀⣀⣤⣖⣄⢺⣿⣧⣈⢀⣀⣠⡾⣿⣿⣿⠏⠀⠀⠀⠀⠀⣸⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⠳⡿⣿⣿⣿⣷⣻⣿⣿⡿⠟⠀⠀⠀⡐⠀⢠⠇⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠙⢿⣛⠛⠛⠛⠿⠛⠦⡀⠠⠀⠈⠀⠀⢸⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠡⠀⠀⠀⠁⠙⠲⠤⡀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠡⡀⠀⠀⠀⠁⠀⠀⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠠⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠚⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠁⠀⠅⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⣀⠀⣤⠦⠐⣄⠤⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠌⠳⡄⡀⠀⠀⡕⠊⠀⠀⠀⠀⠀⠀⠀⠐⡢⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠘⠃⠀⢀⠀⠀⠈⠛⡠⠀⠀⠀⢀⠄⠤⢈⠴⠫⢀⠔⠢⠤⠁⢀⠀⠀⠀⠰⠆⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⡀⠀⠀⠀⢠⠣⡈⠢⢀⢠⠐⠁⠀⠀⠀⠤⡀⠐⠃⠀⠲⢿⠁⠂⠄⡌⠀⠠⠒⡂⠑⠃⠀⡘⠤⠃⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠉⠌⢊⠢⡀⠀⠀⠀⠑⢄⢡⡀⡀⢀⡞⢞⣳⠀⠀⠀⠐⢄⠱⡀⡆⠨⢰⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠁⠜⢀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡀⠄⠀⠀⠂⠀⠀⠺⠓⠂⠉⠀⠢⡠⠴⠀⠀⡃⢀⠃⡝⢑⠂⡰⠈⠀⠐⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⢀⠰⣀⡐⠁⠀⡠⢠⡟⡏⠅⠀⠀⠠⢈⡤⡁⠂⠄⠀⠰⠖⡐⠋⠀⡀⠔⣡⡮⠀⡑⠀⠀⠀⠠⠄⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡣⠄⠠⡄⠕⠐⣬⡂⡊⡤⠋⢈⠄⠨⡐⠪⡴⠛⣁⣤⣄⡀⢤⢲⠿⠤⠤⢤⣆⣁⣼⣀⢨⣥⠀⣉⠉⠠⠔⠄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢜⢯⡑⠞⠫⠒⠀⠐⢁⣭⡈⢀⡙⢹⡆⠀⠈⠓⢓⠈⡀⡀⠀⠠⢌⠀⢨⡼⠀⠈⣣⣰⢁⣾⡷⢎⣡⠁⡰⠀⢂
⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠑⠩⠑⢤⣀⠀⢠⡟⠄⡽⠿⣉⡉⠀⠀⣠⡤⠗⠛⠲⣤⡤⠋⠷⠔⡉⢳⠀⠉⠈⠡⠈⡋⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⣤⠀⣎⠆⠀⠆⠐⠂⠀⠰⡙⡖⠫⠉⢉⣤⠴⠃⠂⠐⠑⢀⡤⣐⡒⠖⠠⠤⠐⠒⠀⠂⠉⣀⠀⢄⠁⠉⠠⡩⠂⠀⠛⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣀⡆⢠⠒⠀⠀⠀⠀⠀⠀⠀⢰⡈⠀⠘⠚⠑⠲⠓⠉⠙⣻⣮⢔⣲⡭⠁⢠⠒⢒⡂⢁⠀⠂⠐⠀⠡⠈⠀⠀⠂
⠀⠀⢀⠀⠀⠀⡄⣡⠴⢄⠀⠀⠂⡠⢀⠆⠀⠂⠀⠀⢀⡄⠰⢆⢬⡁⠉⠀⠠⢀⠀⢗⠀⡆⢀⢀⠀⠀⠀⠀⠐⠀⠀⠔
⠀⠀⠀⠁⡄⢠⣅⡨⡗⠀⠦⡭⣤⠴⠻⠄⠀⠀⠀⣄⠴⢀⡾⠞⠛⠁⠒⠒⠂⠁⡀⠀⢇⡱⠈⠊⠀⣤
⠀⠀⠀⠂⠁⠪⡉⢫⡀⠀⠀⠀⠠⢄⡁⠀⠀⠀⣊⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠜⣆⠈⠀⠀⠑⠈⡠⠔⠂⠉⠀⠀⠀⠘⠀⣀
⠀⠀⠠⠀⠀⠀⠀⠠⣨⢗⡂⠀⠀⠀⡁⠀⠀⠀⠣⠃⠀⠀⢀⡀
⠀⠀⠀⠑⠀⠀⡠⠚⠉⠋⠐⠀⡀⠀⠐⠂⠠⠤⠎⠬⠄⢒⠄
⠀⠀⢀⠔⠰⠶⡇⠀⠒⠀⠒⠂⠀⠂⠠⠆⠒⠨⠄⠒⠠⢣
⠀⠀⠈⠉⠀⠀⠐⠈⠀⠀⠘⠄⠐⠐⠠⠄⠉⢀⣘⡄

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄
⠀⠀⠀⠀⣰⣯⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢻⣧⣼⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠰
⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣽⣿⠏⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠳⠤⠽⠯⠵⠔⠃⠓⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧
⠀⠀⠀⠀⠀⢈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦⣄
⠀⠀⠀⢀⣤⣴⣶⣦⣐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⡾⢛⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣇⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠛⠀⠀⠀⠀⠀⠀⠈⡽⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠻⢏⣉⣀⣀⣈⣹⣿⠿⠃⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠿⣟⡋⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡔⢲⡇⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣀⠀⠁⠛⠁⠀⠀⢱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠲⢶⣶⡶⠶⠚⠁⠂
⠀⠀⠀⠀⠀⠀⡀⡔⠀⠒⠒⠀⠐⠒⣀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣏⡀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣤⣿⣯⣭⣡⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣙⣻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣤⣭⣼⣂⣁⣉⣉⣙⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢈⣉⣛⣛⠛⠛⠛⢛⣛⣛⣻⠭⠾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠲⢤⣶⣾⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠀⣀⣉⣉⣉⣉⣉⣉⡁⠤⠔⠊⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠛⢻⠛⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠠⠤⠤⣤⢀⣤⣄⣢⡀
⠀⠀⠀⠀⠀⢀⠈⠀⢠⣶⣦⣤⣶⣴⣶⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠲⡿⡿⠿⣿⣿⠿⠛⠋⠄
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠈⠻⠏⠉⠁⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠐⠒⡋⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠠⠴⠶⠒⠚⢚⠃⣿⣿⣿⣶⣤⠍⡃⠘⠀⠰⢀⣠⡀⠀⠀⠀⠀⠀⣀⣤⣤⣴⢤⣦⢦⣿⡷⣴⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠤⠤⠤⣤⣤⣬⣤⣤⣤⣤⣶⣶⣶⣾⣿⡀⠀⠀⠀⢠⣉⡁⠀⠻⠿⠿⠿⢿⣷⣿⣿⣿⠿⢿⠿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠒⠀⠠⠤⠤⠭⠽⠿⠿⠿⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠉⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠚⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⢦⡀
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣿⠉⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⡇⣀⣠⡴⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⡇
⠀⠀⠀⠀⠀⠘⡇⠈⠙⠳⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣬⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿