Buns Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢠⡖⠉⠀⢠⡟⠀⠀⠀⢀⡧⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠑⠀⠀⢻⡀⠀⠀⠙⠢⠄⠒⠋⠀⠀⠀⢸⡏⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠀⠘⢧⡀⠀⠀⠈⠒⠦⣤⢄⣀⣀⣀⣀⡤⠀⠊⠀⠀⠀⡝⠈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⢆⠀⠀⠀⠘⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⢀⡿⠈⠢
⠀⠀⠀⠀⠀⠀⠀⡰⢻⡀⠀⠈⠳⣀⠀⠀⠀⠀⠉⠋⠀⠠⠦⠤⠤⠴⠤⠀⠒⠂⠀⠀⠀⠀⠈⠀⠈⣴⢣
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠣⡀⠀⠀⠀⠉⠒⠦⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠁⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⡞⣦⡀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠀⡠⠲⡄⡀
⠀⠀⠀⠀⠀⠀⢰⡇⠈⠉⠳⣄⠀⠉⠑⠒⠄⠀⠀⢀⢀⣀⣀⡀⣀⡀⠤⢀⡀⠀⠒⠘⠋⠃⠀⢠⠃⠀⢃⡃
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠘⠷⠖⠂⠉⠀⠲⣄⠈⠉⠁⠀⡀⠀⡠⠔⣄⠀⢀⣀⠀⠀⠀⠀⣼⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⠉⠋⠀⠀⠸⠛⠋⠀⠻⣦⣴⢶⡏⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠐⠐⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢂⣀⣩⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣴⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣶⣾⢖⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣞⣼⣿⣿⣿⣿⣿⣿⢿⣷⣽⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⣿⠁⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣧
⠀⠀⠀⠀⠀⠀⢠⢏⣿⣿⣿⣿⣿⣿⣿⣿⣞⣯⣿⣽⣾⣿⣿⣷⣤⣄⣀⠀⢀⣠⣴⣬⣽⣿⣿⣷⣿⣿⣴⣿⣿⣿⡿⢉⣿⠛⣿⣿⣿⣧⣾⡆
⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡿⣿⣿⡿⣿⢸⣿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣟⣭⣷⣿⣟⣯⣭⣭⣭⣿⣿⣿⣿⣷⣶⣿⡿⣡⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⡝⡝⣿⣿⣷⣶⠷⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⢿⣿⣿⢻⣿⣿⣿⡵⢻⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⣻⣶⣿⠟⠉⣶⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣾⣿⣿⣧⠻⣿⣧⣔⣛⣵⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣅⠀⣿⣿⣿⣿⣿⣿⣿⢟⡯⣙⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⡇⡙⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣿⣿⣿⣿⣿⣯⡿⣫⣾⣿⣿⣿⠿⠛⠻⢿⣿⣿⣿⣿⣿⡻⣷⣝⢿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢫⣾⣿⡿⣻⠟⠁⠀⠀⠀⠀⠙⢿⡻⣿⣿⣿⣌⠻⣎⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⢟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡝⣿⣿⣿⣧⣹⡎⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣻⣿⣿⣿⣿⣯⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣿⣿⣿⣿⣿⢸⣿⣿⣿⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣿⣿⣿⣿⣾⣿⣿⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⣿⣿⡏⡽⠙⠛⠿⢿⣿⣷⡀⠀⠀⠀⠀⣰⣾⣿⡿⠟⠻⣟⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⣿⣿⣇⡇⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠹⣿⣧⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢻⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣿⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣤⣴⣾⣿⠓⠀⠀⠀⠀⠀⠐⢻⣿⣶⣦⣤⣴⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠻⠟⠻⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠟⠟⠉⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣤⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡤⡤⠤⣀
⠀⠀⠀⠀⠀⡀⢤⠒⠋⢹⡖⠂⠈⢳⡌⢂⠙⣿⠣⡀⠀⠀⠐⡟⠡⠔⣛⣫⣶⣦⠀⠀⢀⠎⠀⣀⠤⠜⠀⠧⣀⠀⠡⡀
⠀⠀⡠⠒⣭⢄⠸⡍⠳⡈⢳⡀⠀⠀⢻⡌⡄⠘⣄⡇⠀⢸⠀⠃⢠⠊⠉⠉⢉⣈⣁⠀⣸⡀⠚⠀⠀⡄⢠⠀⠀⠱⠀⡧
⠀⠰⠁⠀⢿⠀⠂⠹⣆⢱⡈⢷⡀⠀⠘⡇⣧⣶⠟⠀⠀⠘⢧⡀⠐⠔⠒⠉⠉⢉⡟⠀⢿⣧⡀⠀⠀⠸⠂⠀⠀⣀⣴⠇
⠀⢠⡀⠀⠈⠓⠄⣦⣘⢦⣧⣀⣷⣶⡶⠟⠋⠁⠀⠀⠀⠀⠀⠙⠲⠤⣤⣤⠴⠚⠁⠀⠈⠻⣿⣷⣶⣶⣶⣾⣿⡿⠋
⠀⠀⠉⠛⠓⠒⠛⠛⠛⠛⠛⠋⠉⠀⡠⠒⠀⠁⠀⠠⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢉⡩⠭⠟⠫⢤⣀⣀
⠀⠀⣠⠒⠫⣅⠉⠉⠱⣶⢤⠀⠀⡌⠀⠀⠀⠀⠀⠀⠳⠀⢀⠄⠀⠀⠙⡿⠉⠱⡀⠀⠀⢀⣴⡙⠀⠀⠀⠀⢨⠀⢨⣉⠵
⢀⣞⠣⠄⢈⢮⡀⠀⣠⡇⢠⠳⡄⡇⠁⠀⠀⠀⠀⠀⣸⠀⣸⠶⢦⣄⡰⠁⠀⠀⣷⠀⣰⡋⠀⠘⢎⢂⠀⠀⣸⠀⠊⠁
⠠⠁⠀⠉⣷⣿⣿⡾⠛⠶⠾⢤⠁⠘⢄⣀⣀⣀⣠⠴⠃⠀⡇⠀⠀⠘⡇⠀⠤⠞⡿⠠⠀⠬⢷⣤⠼⠿⠶⠛⠁
⠘⠦⠒⠛⣉⣁⠀⠄⡠⠠⣀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠰⣤⡀⠀⠰⣄⠀⠚⠀⠀⠀⠀⠤⠶⠾⠥⣄⡀⠈⠙⠲⣄
⠀⠀⣴⠍⠁⠀⠀⠀⠀⠀⠛⣯⠀⠀⠀⠀⠀⠀⠀⠤⡤⢤⣤⣤⣭⣉⠉⠁⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠙⢲⡀⠀⠈⡆
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣤⠈⠢⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢲⣧⡤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡴⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⡱⠀⠀⠠⠁⠀⢀⠀⠀⠀⠀⠀⢹⠁⠀⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⠦⠄⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⡾⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣸
⠀⠀⠀⠙⠢⠤⠴⠄⠂⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠒⠒⠒⠋⠀⠀⠀⠀⠘⠓⠢⠤⠤⣄⣀⡀⠀⡼⠖⠉
⡶⠒⢶⢶⡶⠶⠶⣶⢶⠶⠶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⢶⣶⠶⠶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⡿⠀⠀⠀⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⡿⠁⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠘⣧⡀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⣷⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⣠⠟⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠰⠏⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠋⢉⣽⠟⠋⠉⠛⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⣠⡾⠋⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣾⠁⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⠋⣩⡿⠛⠉⠛⠛⠶⢦⣤⡀⠸⢧⣤⠶⠶⠛⠉⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣽⣆⡀
⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣴⠟⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⣀⡴⠟⠉⠀⠈⠉⠙⠛⣶⣤⣀
⠀⠀⣠⡾⠋⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠈⠻⣦⠀⠀⢀⣴⠟⠉⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠙⠷⣄
⠀⣰⠏⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣴⠟⠁⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠘⢷⡄
⢠⠏⠀⠀⠀⠀⠀⠀⠷⣤⡴⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠈⠉⠛⠛⠛⠓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠖⠒⠚⠛⠛⠛⠛⠋⠉⠈⠛⠶⣤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠟⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠤⠔⡈⠉⢭⠅⠀⠀⠉⠉⠙⠒⠒⠂⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠁⠀⢀⠀⠠⠀⣈⣉⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠉⢀⠀⠤⠀⠀⠀⠉⠙⠓⠒⠒⠚⠒⠒⠒⠒⠈⠉⠀⠀⣀⡠⠔⠁⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⢀⠀⠈⠓⠲⠤⢤⣤⣀⣒⣒⠤⠤⠤⠀⠀⠖⠒⠒⠂⠀⠈⠁⠀⠀⠀⣀⡤⠂⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢉⠑⠒⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠠⠤⠐⠚⠉⠀⠀⠀⠀⠀⠀⣀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠲⣀⠀⠀⠉⠓⠒⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠛⢷⡀⠠⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠗⠶⠶⣶⣤⡼⠁⠀⠀⠈⢷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⡆⢀⣄⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⢀⡴⠚⠋⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰⠃⠘⢻⠁⠀⡞⠉⠉⠀⠁⠙⠓⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⢠⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠸⡆⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠦⠀⠀⢀⣤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠑⠂⠚⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣤⣬⣅⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣵⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣷⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢙⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⠾⣋⢿⣽⣟⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣡⡾⣵⣿⣿⣿⣿⣿⣿⡿⣿⣿⡻⣿⢿⣿⡝⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣵⣿⣽⣿⣿⣿⣿⣿⣿⣿⣸⢸⣿⣷⢹⡇⠃⢹⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⢟⣛⣿⣿⣿⣿⣾⣿⣿⣼⡇⣀⢸⢼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣛⣻⣷⣾⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢏⣾⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠉⠻⢶⣭⣙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣯⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣾⣿⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣾⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣼⠷⠾⢿⣶⣶⣤⣀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡿⠿⠶⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⠀⠀⠀⠀⠉⠛⠿⡧⠀⠀⠀⢸⠿⠛⠉⠀⠀⠀⠀⢘⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣹⣧⢶⣦⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣶⢸⡏⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⠏⠈⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠛⠁⡼⡇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠉⠉⠀⠀⠀⠈⠉⠙⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠄⠀⠀⠀⠀⠐⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀⠀⠐⠄⠀⠀⠀⠀⠐⢀⠀⠀⠐⠀⠀⠀⠀⠀⡦⠀⠈⢆
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢂⠀⠀⠀⠐⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠸⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠃⠀⠈⠐⠀⠀⠀⢀⠀⠀⠀⠂⠀⠠⠐⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠂⠀⠴⠀⠈⠀⠀⠀⡀⠀⠁⠂⠀⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠠⠐⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠘⣗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢊⠏
⠀⠀⠀⠀⠀⠀⠈⠢⡈⠑⠒⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠔⠒⠉⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠤⠄⣀⣀⣀⣀⡀⠤⠤⠤⠔⠒⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠒⠊⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⡇⢀⠃⠀⠀⣀⡀⣰⡍⢉⣩⠀⠹⡄
⠀⠀⢀⠠⠔⠊⠉⠀⠀⠀⠀⠀⠀⢀⠠⠁⠀⠀⠑⠀⠀⠀⡀⠀⢀⠀⣡⡴⢦⡴⣶⠃⡘⠀⢠⡞⢫⣿⢹⣟⣻⡿⠁⢠⣧
⠀⠈⠂⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠔⠀⠀⠠⠋⢠⡞⣹⠁⢹⡻⢿⠎⡟⠀⠇⠀⢸⣍⣀⣐⠋⠀⢀⣀⡤⢻⠇
⠀⠀⠀⠈⠙⠒⠄⠀⠀⠠⢀⠠⠀⡠⠀⠄⠁⠀⠀⢰⠁⢰⡟⠦⣟⣴⣭⣿⠏⠟⠁⡐⠀⠀⠈⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⣽⠍⠗⣠⡠⠜⠁⢀⡠⠊
⠀⠀⠀⠀⣀⡤⠖⠒⠚⠛⢓⠶⠤⡀⠀⠀⠀⠀⡜⠀⡼⠟⠋⠁⣀⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠄⠀⠀⢀
⠀⢀⡴⠚⠁⠀⠀⠀⠐⠀⡾⠀⠀⠀⠡⠀⠀⢨⠃⢠⠄⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⣖⠃⠀⠈⠆⢀⠀⠂
⢠⠊⠀⠀⠀⠀⠀⠀⡀⠂⣷⣤⠀⠀⠀⠀⠉⠙⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠙⠍⠀⠀⠀⠈⠀⠀⡀⠤
⠼⠊⠉⠑⢢⡀⣀⡠⠔⠊⢠⠃⠀⠠⠀⠀⠀⠀⠀⠀⢀⢀⣀⡔⠀⠀⠀⠱⠂⠀⠀⠀⠀⣀⠀⠀⠐⠈⠁
⢠⢀⠀⠀⠺⠫⠥⢄⡀⠀⠀⠁⠀⠀⠀⢀⡀⠐⢦⢞⠀⠀⠈⠂⠀⠀⠀⠀⠀⠐⠒⠈
⠈⠳⣄⡀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠐⠐⠪⠄⠀⠀⠉⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⣀⢀⣀⠔⠒⢶⡤⡤⠔⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡠⠂⢀⣿⡟⠁⠀⣰⡼⠊⠀⠀⢀⣴⠱⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀⠀⠀⠀⠀⢰⠁⠀⠘⠃⠀⠀⠀⠋⠀⠀⠀⢀⡾⠖⠉⠀⠈
⠀⠀⠀⠀⡀⠔⠂⣭⣯⣀⣒⠢⡀⠀⡀⠐⢀⣀⡤⠦⠤⣦⣀⣈⡁⠢⡀⠸⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⣀⠔⠉⠐⠲⠶⢤⣤⠈⣙⡵⠃⠰⠁⠀⠿⠰⡦⠬⠦⠁⣤⣼⡻⠀⠐⠀⠀⠕⠒⠤⠤⠄⣀⣀⣠⡤⠤⠤⠒⠒⠋⢩⠃
⡼⢅⣬⠭⢛⡓⠶⢀⡬⠊⠁⠀⠀⢀⠀⠐⠀⠈⠑⠒⠓⠚⢚⢁⠢⠁⡀⠀⠈⣆⠀⠁⡀⠂⠀⠐⠀⠐⠀⠐⠀⠀⢡⠏
⠣⠟⠛⠛⠗⢰⠖⠁⠀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠁⠀⠀⠀⠀⣀⡤⠊⠀⠀⠀⠈⠛⠦⠤⣤⣁⣈⣈⣈⣈⡠⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣞⢩⣭⣿⣣⢠⣀⣿⣷⣾⢶⣶⣿⠏⠇⢤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣡⣴⣬⡙⠻⢿⢿⡙⠉⠉⠐⡙⠿⣿⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠉⠉⠀⠈⠿⠿⡷⣀⠻⣿⣾⣴⣼⣿⣄⠈⣷⡖⠋⠉⠹⠺⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠄⠀⠀⠀⠀⠀⢀⡤⠺⠿⣿⣦⡙⢿⣿⡟⠛⠉⠛⠂⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠧⢹⢸⢸⢸




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⠞⢋⣩⡴⠶⢖⠶⠶⣭⡙⠻⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⣡⣴⠖⢛⣩⣭⣍⣙⠓⢦⣌⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣰⣿⣏⠀⢾⣿⡀⠺⣿⣥⣴⠿⠋⣀⣿⣿⣷⡄⠀⠀⠀⢸⣿⣿⣿⠏⠀⢸⣿⡁⠐⣿⣅⡐⠛⢛⣀⣼⡿⠀⡀⠉⢻⣿⣿⣿
⠀⠀⠀⣿⡏⠻⣦⣄⡙⠛⠷⠶⠶⠶⠶⠟⢛⣡⡾⢻⣿⠀⠀⠀⢸⣿⣿⣿⠀⣶⣄⠙⠿⣶⣤⣍⣉⣉⣉⣉⣥⡴⠞⢁⡄⠈⣿⣿⣿
⠀⠀⠀⢻⣧⠀⠈⠙⠛⠿⠶⠶⠶⠶⠶⠿⠛⠋⠀⣼⡏⠀⠀⠀⢸⣿⣿⣿⣆⠸⣿⣷⣦⣤⣀⣉⣉⣉⣉⣉⣁⣤⣶⣿⠃⣸⣿⣿⣿
⠀⠀⠀⠀⠙⢷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣁⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠷⠶⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣤⣤⣬⣉⣉⣉⣩⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⠋⣿⢁⢀⣤⡶⢶⣖⣄⢦⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡇⡀⠠⣿⣷⣿⡿⠀⣸⣿⣿⣿⣿⡹⡆⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣾⡏⡏⣼⣿⢿⢧⡅⠀⣿⣿⣿⣁⢠⡟⣿⣿⣿⡟⣷⡃⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⡿⣇⢷⣿⣻⣏⣧⣷⠀⢻⣿⣿⢣⣿⣿⣿⣿⠀⡇⣿⣿⣼⣿⣼⣾⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠘⡇⢻⣿⣿⣿⣿⣍⠹⣧⠈⣿⣯⢞⣿⡿⣿⠿⠗⢻⠿⠟⠙⠋⣙⣉⣥⣿⠟⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⣆⠉⠸⡙⡿⡍⠻⠿⣧⣿⣶⣿⣿⢋⣮⢞⣴⢀⠀⠁⠀⡀⢚⡸⢿⣽⡟⠁⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⣤⣻⣧⠀⢻⣤⣜⣝⣦⣹⣿⣿⣿⡷⣏⡻⠃⡀⠀⠀⣴⣦⣴⣿⣥⣿⣯⡶⠛⠀⡐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣬⣤⠿⠛⠿⢧⡿⣉⣭⣽⣿⣿⡿⠿⣿⣤⡌⠣⣬⡅⢀⣰⣿⣍⡙⣿⣿⠏⠡⠢⢠⢾⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣤⡀⢶⡿⠃⠀⠀⠀⠀⠘⠟⢛⣿⣿⣿⡄⠀⠉⠁⣠⣤⡿⠶⢿⣛⣿⣿⠁⠴⣿⣦⣴⣶⣿⣾⠋
⠀⠀⠀⠀⢀⣴⡾⠋⠁⠒⡶⠚⠉⠀⡀⠀⡠⠴⣾⣿⣜⣿⡍⠀⣰⣠⣿⡷⢶⣔⠿⣥⢤⣅⠰⠟⢻⡿⠿⠿⢟⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠘⠉⣙⢬⣵⢏⠀⡄⠀⠀⣨⣍⡋⠑⠉⠫⣓⣋⠚⠿⣿⣿⠦⣴⣂⣨⣽⠖⠀⠻⣿⣿⣿⣊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡿⠃⠂⣠⠆⢀⡜⠯⠻⣿⣯⣤⣼⣿⣿⣷⣦⣄⠻⣿⡶⠿⣿⣧⡀⢈⠢⣘⣿⣿⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠫⡏⣧⠂⠰⡏⢀⣾⣠⣀⠀⣹⣦⣤⡉⠉⠙⠿⠿⢷⣤⡠⣽⣦⡈⠋⢳⣻⣷⣾⠏⣿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠳⡤⡄⠀⢱⡉⣫⡻⢿⠻⠿⢛⠻⠿⠧⠤⣀⡈⠒⣤⡠⣻⣧⣻⣄⠀⣿⣿⢿⣥⣿⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⡁⣧⠸⣮⠓⡈⢻⣶⣴⠂⠌⠻⢿⡦⠄⣀⣘⣿⣿⣿⣿⣿⣿⣿⢿⣻⣿⢘⡿⠁⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣾⠷⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠈⠙⢿⣿⣵⣿⡇⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢋⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠉⣑⠰⣤⣶⡿⣷⣈⣤⠀⠀⠀⠀⠀⢀⣀⡈⠳⠀⠀⠀⠀⠀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣿⣓⣾⣿⣤⠉⠉⠌⠁⠀⠀⠀⠀⠀⠉⠯⣠⡰⣾⣿⣧⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠻⠏⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⡉⠤⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢿⠛⠂⠀⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⡆⢀⣀⠀⠀⠀⠀⠀⡠⠞⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡆⠀⠀⠆
⠀⠀⠀⠀⠀⣼⣷⣥⠄⠀⢨⠐⣢⡔⠊⠀⠀⠀⠀⠀⠁⠀⠤⠄⠀⠀⠀⠀⠀⠀⢰⡖⠧⠀⡦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⢠
⠀⠠⠂⠀⢀⣒⣤⡍⠀⠀⠚⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠤⠀⠀⢀⡀⠐⢠⡇⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠠⠃⠀⠆
⠀⠈⠙⠃⠈⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠄⠀⠀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠐⠠⣤⡀
⠀⠀⠀⠀⠀⠀⢠⣆⣠⣆⡀⢄⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⣀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠖⠬⠉⣉⠁⠤⠐⠂⢄
⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠑⠀⢃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠐⠀⠉⠠⠄⠐⠂⠈⠃⡀
⠀⠀⠀⣸⠀⠀⠀⠀⠀⡀⠀⢠⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢃⠀⠈⠠⠄⠐⠓⠄⠈⢀⠏⠀⠀⠀⠀⠀⠑⠂⠀⠀⠠⢤⣄⡀⠀⢐⣠⠀⢀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠲⠶⠦⠤⠶⠒⠉⠀⢀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⢀⡒⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢠⣄
⠀⠀⠀⣀⣉⡀⠀⠀⠀⠀⠠⠂⠄⠀⠀⠀⠀⠀⠀⠀⠈⢠⠍⠀⢀⠴⠓⠐⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠉⠋⠂⠀⠈⠃⠀⢹⠀⣦
⠀⠀⢘⣉⡀⠀⠁⠀⠀⠠⠀⠄⠉⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠃⠀⢀⠄⠀⠐⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠉⠉⠀⠀⠈⠀⠀⠀⠀⣀⠴⠋
⠀⠀⢀⡀⠀⡐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢠⠆⠀⢠⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠐⠒⠀⠀⠐⠒⠃
⠀⠀⠀⢀⠤⠀⠀⠀⠀⢤⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢡⠀⠈⠀⠀⠘⠁⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⣠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠣⠀⠀⠋⠨⠗⠀⠀⠀⠀⠀⠀⠑⠚⢢⣀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⣴⠂⡴⠃⠸⠋⠀⠉⢀⠀⠤⠚
⠀⠀⠀⠀⠀⠠⢈⠀⠀⠆⠀⣀⡀⢠⠀⡀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠠⠋⠠⠋⠀⠀⠁⣀⡀⠤⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡤⠤⠤⠐⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣠⠞⢉⡞⢁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢿⡞⠡⠁⠘⠛⠉⢉⠉⠉⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠲⣤⡀⠀⠀⢰⠏⠀⠆⠈⠀⠃⡠⠐⠈⠀⠀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡙⢦⡀⣿⠀⠀⡄⢀⠔⠉⠂⠀⠀⠀⠘⠛⢒⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣬⣿⠍⠛⠳⢦⣄⠙⢦⡙⣿⠀⠀⡴⠡⠄⠀⠀⠀⣠⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢾⣯⠁⠀⠠⠀⡀⠀⠉⠂⠈⠳⣄⠹⡌⢧⣈⡤⠶⣤⡴⠶⣄⣈⣩⠿⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠐⠂⢈⠀⠁⠀⠀⡨⠓⢢⡼⢧⡟⠉⠛⣆⠀⡏⠀⠀⢸⡏
⠀⠀⠀⠀⢠⣏⣤⡤⢤⡄⠀⠁⡀⠀⠀⠁⡴⠶⠿⣤⣸⡄⠀⠀⢸⣦⣷⣀⣀⣼⣁
⠀⠀⠀⠀⠈⠀⠀⢰⣯⠤⠶⣺⠏⣀⣠⡼⡇⠀⠀⠈⢻⣷⣦⣤⣿⠁⠀⠙⡿⠁⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠁⠀⣽⢦⣄⣠⣾⠀⠈⢹⣿⣄⣀⣠⣧⡀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⣿⣿⣦⣤⣼⠋⠉⠹⣿⠟⠙⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡶⠃⡴⠛⠛⢿⡀⠀⢀⣟⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣇⠀⠀⠀⣿⣶⣾⡿⠦⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⡶⠖⠛⢻⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣸⣷⣄⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⡟⠉⠩⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣤⠶⠒⠞⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣏⣩⡍⣭⢉⣒⣖⣯⠭⠆⡛⠂⠛⠘⠃⢼⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣨⠅⢶⠆⠠⠀⠮⠄⠰⠒⠈⠉⠀⠀⠁⠂⠰⢉⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⠇⠘⠁⠀⠀⠀⠀⠦⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⡠⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢆⡀⠠⠀⠀⣀⠔⠋⠁⠀⠈⠁⠀⠐⠃⠈⢁⠤⠒⠒⠒⠀⠒⠂⠤⠀⢀⠤⠔⠒⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠀⠀⠀⠀⠀⠀⢄⠀⣂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⣈⡉⢠⡀⠀⠀⠀⣀⠀⣀⠁⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡦⡤⡀⢀⡀⠀⣀⢠⣤⣦⢤⠁⣤⠀⣠⡄⣠⡌⠤⢼⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣥⣩⣥⣬⣭⣤⠶⢿⣿⣾⡞⣲⡄⢴⣦⣶⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠒⠒⠞⠛⠓⠒⢦⣀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⡠⡀⢠⣀⣦⣴⣀⣀⣟⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡻⠎⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠉⠛⠛⠳⣦⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣮⣄⠀⢀⣀⣀⣠⣴⣾⣿⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣤⣶⣽⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⣽⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠚⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣤⣁⡀⠀⢀⢀⣠⣶⠟⠁⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠓⠤⣤⣤⣤⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⠄⠀⠾⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⣿⢿⠿⡿⠿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣛⣟⣿⣿⣻⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠆⠤⠶⣄⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠦⠈⢄⠠⠀⣤⡶⠿⠀⣪⣍⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣱⠄⠩⣠⡀⠀⠈⠉⠓⠾⠛⠂⠀⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠉⠛⠧⡀⠀⠀⠀⠀⠀⢤⣌⣡⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⣀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⢰⢤⣴⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠜⠉⠀⠀⠺⢷⣌⠻⢷⣦⣬⣀⠀⠀⠀⢁⢀⠸⠀⠀⠀⠙⠛⣿⣿⢿⢻⣳⠂⣾
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠓⢹⣿⠛⠛⠛⠛⠒⠂⠉⠹⠀⠀⠀⠀⡀⡙⣿⣅⣾⢱⣼⠇
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠐⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢐⢰⠰⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠶⠛⠁⠀⠀⠀⢶⣶⡄⡄⣄⢠⡠⡀⣈⢑⣂⣛⠥⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣆⣦⠤⢮⢹⡜⣿⣺⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠯⠽⠞⠓⣿⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣪⠟⠉⠀⠀⠀⠀⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣀⠒⠤⢤⣀⣀⢉⠉⠛⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠄⢀⠒⡢⠤⡀⣀⣀⢀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⡀⠁⠀⡀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠈⣀⡞⠉⠉⠉⠉⠀⠈⠙⠢⣐⠀⡀⠂⠀⠁⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⡴⢃⣀⠀⡤⠿⠿⠖⠂⠀⠠⣄⠀⠀⠀⢀⡬⠷⢶⡤⠤⠤⠴⠶⠚⠁
⠀⠀⠀⠀⠀⠘⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠛⣁⠤⠤⢒⠭⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠸⡘⢄⣀⣀⠀⠀⠀⠀⠀⢀⡤⢚⠔⠁⠀⠀⣔⡁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠑⠄⡀⠈⠑⠒⠶⠂⠉⣩⠔⠁⠀⠤⠤⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠉⠀⠰⠇⠀⠀⠀⠀⠈⠑⠨⡑⠂⠤⠤⠤⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠤⠤⠄⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⣰⢀⠀⠀⠀⡀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠈⠃⠇⠃⠃⠃⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⢵⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢠⣧⠤⣤⣦⡽⠃⠀⠀⢶⣀⠀⠀⠀⠐⢠⡐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠐⠒⠻⡾⠀⠀⠀⠀⠈⣋⠀⠐⢦⣾⣯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠇⠀⠀⠀⠀⣻⡀⠀⠀⠀⢠⠇⠀⠀⢠⡆⠀⠀⠱⡶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⢨⡖⠐⢢⡀⠀⠀⢀⠜⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠞⠋⢿⠁⠀⡌⠀⠀⠀⣹⣿⣿⣷⣄⣀⣀⣤⡾⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢆⡀⢀⡀⠀⠰⠒⣉⠛⠉⠈⠉⠛⠉⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠑⠶⠶⠾⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⣿⠀⠀⠀⠀⠀⢀⠿⢶⡞⢃⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠢⣄⣀⣠⠄⠋⠃⠀⠹⡿⠋⠛⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡄⢸⣦⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡆⠀⠀⠀⢀⡞⠟⠛⠶⡤⣤⡖⠛⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣄⣴⠛⠁⠀⠀⡀⢠⡌⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠛⠁⠙⢄⠠⢀⠰⠀⡼⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣭⠴⣧⠀⠀⠑⢲⡶⡞⠀⠘⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢛⣧⣤⣠⣬⡌⠑⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠋⠀⠘⠇⢹⣶⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣼⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠐⠫⠹⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠈⠀⠀⠀⡀⠀⠈⠁⠈⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠖⠒⠉⢉⣩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠙⠻⡏⠀⠀⠀⠀⢀⣀⣀⠀⠀⠓⢢⣄⠀⠈⠑⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⡀⠀⠀⠉⠲⢤⡀⠈⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⢀⡀⠁⠉⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠉⠁⠀⠀⠀⠈⠙⠲⠶⠶⠶⠶⠶⠶⠛⠛⠉⠉⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠸⠇
⠀⠀⠀⠀⠘⠿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡐⠲⠃⡠⠀⠀⠀⢹⠇
⠀⠀⣦⣄⠀⠀⠀⠉⠙⠛⠫⠧⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠐⠀⠀⠀⣀⠤⠚⠀⣀⢸⠀⠀⠸⠗⠀⠀⠀⣾
⠀⢄⡉⠛⠋⠶⠴⣦⠀⣿⠒⠒⠞⠋⠀⠀⠀⣀⣀⡀⠉⠂⠀⠀⠀⡀⠤⠒⠉⠀⠀⠀⡜⠉⠀⠳⣴⠁⠀⠀⠀⡔⠁
⠀⠈⠛⠷⠶⢠⣤⣄⠠⠏⠀⠀⠀⠀⠀⠒⠂⠀⠀⢁⣀⠠⠔⠂⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⢀⡤⠈⠀⠀⠐⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢹⣆⠻⠀⠀⠀⣀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠉⠀⠀⠀⢀⠀⠀⠀⠐⠒⠤⠄⡀⠀⠀⠀⠀⢡⣾⡏⠁⠀⠀⡹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠘⠓⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⡀⠀⠀⢻⣿⣿⣿⠷⢤⣀⠀⠀⢀⠀⠀⢚⡄⠀⠀⠀⠀⠀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠉⠛⠿⣶⣬⡟⠃⣀⡠⠤⠟⠀⠀⠀⡠⠀⢌⡁⠀⠀⠀⠘⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠠⠛⢛⡶⢶⡊⠀⢹⡄
⠀⠀⠘⢦⣀⣀⣀⡀⠀⢀⣀⣀⣠⡴⠃⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣠⠀⠀⠀⠘⡏⠀⠀⣸⠁
⠀⠀⠀⠀⠉⠙⠛⠓⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠤⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠹⢄⣄⣀⣤⡑⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠤⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⣀⣀⣀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠁⠀⢀⣀⣀⣀⠠⠤⠤⢅⣠⣄⣈⡓⢦⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠢⣄
⠀⠀⠀⠀⣷⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠈⠉⠉⠉⢀⡀⠀⠀⠀⠀⣉⣁⡒⠠⢀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⣀⠠⠄⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠀⠀⠀⠒⠒⢲⣤⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠀⠀⠀⠀⠀⠐⣂⡈⠄⠂⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠒⠠⠤⠤⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠂⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠁⠀⠀⠆⢀⠀⠛⠉⠙⠲⢤⣔⡊⡉⠁⠠⠀⠀⠀⠀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⣠⠞⠁⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⣆
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⠤⠐⠒⠒⠒⢢⣤⡀⠀⠀⠀⠀⠀⠈⢧⠀⢀⣀⣀⡀⠀⠀⠀⢀⠀⠄⠘⣧
⠀⢀⡾⠁⠀⠀⠉⢀⡤⠂⠀⠀⠀⠀⠀⣴⠏⠈⢹⣦⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⢸⣧
⠀⣼⠁⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⣼⠁⢀⠀⠀⠘⢷⠀⠀⠀⢀⠀⠸⡇⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢿⡆
⢰⠃⠀⠀⠁⠀⣾⠀⠀⠀⠀⠀⠀⢰⣧⠀⠈⠀⠀⠰⡿⣇⠀⡀⠈⠀⠐⢻⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠸⣷
⢸⡇⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠀⠀⢸⠀⢸⠌⠁⠀⠂⡀⢼⡄⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣿
⢸⡀⠀⠀⠐⠀⢣⠀⠀⠀⠀⠀⠀⢸⣯⠀⡀⠐⢀⠨⠀⢸⠀⠀⠠⠀⠉⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠒⠠⣿
⠘⡇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣿⡀⠁⠀⠀⠀⣦⣿⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣨⡟
⠀⣷⡀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢀⡴⠁⠀⠀⠀⢐⣿⠃
⠀⠸⣧⠀⠀⠀⠀⠀⣳⣄⠀⠀⠀⠀⠀⠸⣧⡀⠈⠀⠀⠘⢧⡀⠀⠀⢠⠇⠀⢀⣠⠔⠋⠀⠀⠀⠀⢠⡶⠃
⠀⠀⠹⣆⡄⠀⠀⠀⠹⢟⠳⣦⣀⠀⠀⠀⢻⣿⠀⠀⠀⠆⠈⢷⡄⢺⡿⠒⠛⠉⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠲⣿⣆⠀⠂⠀⠀⠀⠛⣧⡀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠙⠻⣧⣄⠀⠀⠀⠀⢀⣠⠴⠊⠀⠙⣦⠀⠀⢀⠀⠀⠈⠳⢦⣀⡀⢰⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡩⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⠀⠀⠉⠛⠓⠲⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣭⠿⠿⠽⢿⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣦⣤⣴⡾⠟⠙⠛⠉⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣤⣤⣶⣶⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⢀⣀⠀⠀⠀⠀⣿⣶⣄⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⠓⢦⣀⠀⠀⢀⡀
⠛⠿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢭⡉⠛⠒⠚⠀⠀⣌⠓⠒⠋⠙⡆
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠉⠓⢤⣸⡄⠀⠀⡀⢿⠀⠀⠀⣠⣤⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣴⣿⣿⣮⡷⠀⠀⠀⠀⠀⠀⠀⢀⡟⠠⠤⠤⢤⣌⣳⡄⠀⡇⢸⢀⡤⠞⠙⢦⣹⠆
⠀⠀⠘⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⣒⣦⡀⠀⠀⠈⠙⣦⠇⣸⠟⠀⢀⣴⣞⡁
⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⡴⠋⠉⠉⢉⡷⠚⠛⠛⠻⣴⠃⢀⡴⢻⡶⣈⡉⢩⣝⢲
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣧⣤⣿⣏⣈⣿⡷⢹⣿⡇⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⡷⠻⢦⣾⢸⣅⣹⢸⠏⣧
⠀⣰⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠿⠿⠟⢁⣼⣿⠃⠀⢀⡞⢧⡀⠀⠀⢻⣄⡤⠤⣤⡀⣠⠇⠀⠀⠙⡷⠮⠶⠋⣠⢏
⠀⢿⣿⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⣿⣧⣷⣴⣿⠀⠀⠀⠘⠛⠁⠀⠀⠸⣇⣠⠭⠿⣶⡟⠁⠀⠀⠀⠙⣧⠀⠀⠀⢀⡇⠀⠀⠀⠛⠋
⠀⣼⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⣸⣤⣤⠴⢿⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢳⣤⣀⣀⣀⡴⣇⠀⠀⠀⠀⣿
⠀⠙⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣀⣠⡟⠀⠈⠉⠀⠀⢸⡆⠀⠀⣠⠏
⠀⢰⣿⣿⣿⣿⣿⣿⣶⣻⣿⣿⣿⡟⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠉⠀⢷⡀⠀⠀⠀⠀⣸⠓⠒⠛⠁
⠀⠸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⣹⠶⠶⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠋⠙⠛⠛⠛⠛⠻⠿⢿⣿⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡭⠶⠶⠶⠶⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠉⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠁⠉⠁⠀⠀⠀⠛⠛⠛⠲⠶⠶⣤⣤⣀⣀⠢⣄⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⣁⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠉⠁⠀⠐⠒⠛⠲⠶⢦⣤⣈⣉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢼⣶⣄⡀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠚⠉⣭⣉⣙⠛⠲⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⠶⣬⣍⣛⡒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠉
⠀⠀⠀⠀⠰⠋⠠⠖⠚⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⣶⡄⠀⠀⠀⠀⣀⡴⠋⠁
⠀⠀⠀⢀⠀⠀⠀⠘⠓⠶⣤⣀⠈⠙⠳⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣄⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠷⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⣀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⣀⣀⣀
⠀⠀⠀⠀⠺⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠜⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠈⠙⠲⢦⣀⠀⠰⠦⢀⡤⠒⠉⠀⠀⠀⠀⠀⠠⠁⠀⠧⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠴⠒⠚⠋⠉⠉⠀⠐⠉⠀⠉⠉⠒⢄⠀⠀⠀⠂⠀⠀⡀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡀⠀⠂
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⠁
⠀⠀⠀⢸⠀⠀⠈⠉⠛⠛⠒⠒⠂⠀⠈⠹⠀⠀⠈⡄⠀⠀⠀⠀⠀⠉⠙⠲⢄⣀⠀⠀⠀⢀⠀⠀⠀⠚⠁⠀⢀⡀⠄⣀⠴⠁⠀⠀⠰
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⡁⠄⠀⠀⠐⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠺⠅⠢⠰⠄⠿⠥⡤⠖⠛⠉⠁⢀⣀⣀⠠⠂
⠀⠀⠀⢻⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⢤⡀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⡤⠴⠖⡂⢲⣆⣲⣖⠲⠄⠀⠀⡇⠀⠀⠀⠀⠀⠐⠀⢰⠈⠁⠈⡁⡂⠐⠂⠀⢖⠐⠂⠀⠆⠀⠀⠂
⠀⠀⠀⢸⠀⠀⠀⢀⣘⣛⣁⣉⣓⣀⣸⣤⣀⣀⡀⡇⠀⠀⠀⠀⢁⠈⠰⠀⠠⠀⡀⠄⡀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⢸⠀⠀⢀⠸⠄⠤⠤⠄⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠠⠈⠀⠤⣩⠡⠀⠤⣀⠁⢬⠤⠈⡄⠤⠄⢀⠀⠀⢀⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢐⠀⠀⠁⠀⠀⠀⠅⠐⠀⠁⠀⠀⢰
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠛⠀⠛⠌⠛⠂⠒⠐⠃⠀⠋⠀⠑⠤⠤
⠀⠀⠀⢼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠂⠐⠂⠒⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡤⡀⣤⡄⡄⣽⢳⣀⡖⣆⢰⠏⣤⡄⢸⡇⠰⡀⣎⢢⣿⡇⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⢺⣦⠿⡽⠿⠷⠟⠛⠟⠛⠉⠙⠋⠃⠙⠋⠀⠈⠙⠷⣧⣨⠞⣼⣾⣜⡁⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡵⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠛⠶⠾⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣾⣷⣦⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠙⣛⣫⣽⣭⣭⣩⣽⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣯⣍⣽⣿⣴⣾⡿⠷⣶⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⣿⣿⣿⣿⡥⢀⣩⣭⣁⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⣿⣿⡿⠄⣉⣽⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⠿⠤⠖⣿⣿⣿⣿⣿⣿⡟⠓⠃⠠⣐⠲⣶⣤⣬⣽⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⢿⣛⣩⣉⣈⣶⣼⣿⣿⣷⣮⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢭⣴⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢟⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⡛⢟⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⣠⡿⣿⡿⣻⣿⣿⣿⣏⣿⣿⡛⢿⣿⡿⡿⣿⢿⣿⣿⣾⣿⡆⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⡾⠋⠐⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣏⣿⣿⢿⣿⣷⠀⠈⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣵⣾⠋⢀⠄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⢿⢧⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠂⣠⣟⢿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣧⠈⢣⢰⣎⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡎⢄⡄⣴⣿⠏⣠⢿⡟⢡⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡀⢀⢿⣖⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢏⣞⣾⣿⢏⣴⠋⢈⡴⢿⠟⢝⡿⣹⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠙⠰⡄⠜⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣾⢇⣴⢫⡵⠋⡠⢫⠞⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣥⢡⢿⡜⣻⣿⡆
⠀⠀⠀⠀⠀⣸⣹⣿⣿⣿⣿⣿⣿⢿⣣⣾⣿⣋⡠⢈⠔⡡⠀⢈⠼⣿⢿⡿⣻⣿⣿⣿⣿⠟⠣⠀⠈⣻⣇⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡿⣼⡿⣹⣿⢟⣵⣿⣿⣿⣟⣥⠞⣫⡤⠀⠀⠠⢊⡵⣫⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⢀⠀⣀⡴⢋⠔⠋⠊⣸⠿⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠁⣿⣾⣟⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡚⠍⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⢠⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⢸⠀⣿⣿⣿⡿⠋⠉⠁⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠛⠇⢻⡇
⠀⠀⠀⠀⠀⢸⣮⡇⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣠⡸
⠀⠀⠀⠀⠀⠀⠀⠛⣏⢻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⣀⠀⠤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡆⠀⠰⠃⢀⠠⠀⠀⠈⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢠⠀⣸⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣜⣰⠟⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⣀⠄⡤⠤⣀⣀⢀⣇⡼⠋⠁⢀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠊⠙⢻⡷⣿⠛⣛⠿⡟⢿⣛⠛⠉⠉⠁⠀⠀⠀⠴⠶⣭⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀
⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠂⠀⠀⢘⣘⣇⢿⠟⠁⣰⡇⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⢰⠁⠀⢀⠄⠠⠂⠘⣋⠟⢹⡈⠀⣠⠾⡄⠀⠜⢸⡟⣿⣙⠃⠁⠀⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⠀⠂⠀⠀⠸
⠀⠀⠀⠀⢰⡇⠀⠈⢁⠀⠀⠒⠈⠉⠐⠈⡇⣾⡻⠀⢴⠀⠀⠐⣇⠸⡟⣷⣤⣀⣀⡀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠊⡘⠃⠐⠚⠦
⠀⠀⠀⠀⢠⢁⠀⠀⠀⢄⣱⠆⠀⠀⠀⢸⠇⠹⡎⢴⠘⢧⠀⠈⠏⢈⡾⣻⠍⠷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠂
⠀⠀⠀⢀⡤⠚⠲⣦⣀⣴⠿⠴⠮⣄⡡⢞⣴⢠⡳⣿⡀⡠⣡⠀⣀⠀⣐⣿⡧⣤⣽⣦⠤⣈⡚⠋⠃⠀⠀⠀⠀⠀⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠲⠶⠶⢼⣿⢟⠌⡟⢟⣶⠎⠀⠀⠀⠙⢋⡿⠁⠀⠀⠈⠑⡄⠀⠘⣦⡲⠒⠖⠀⠀⠀⠈⢀
⠀⠀⠀⢇⠀⠀⠀⢀⣜⠃⠀⠀⠀⢸⡏⠛⢶⡇⣎⠸⢂⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡄⠀⢠⣯⠙⢶⣤⡥⠀⠀⠀⠹⡀
⠠⠂⠉⠙⠳⣦⠎⠁⠀⠈⠳⡀⠠⠜⠀⠀⠀⣿⣁⡽⠗⠒⠠⣠⣾⠭⠦⣀⢀⡀⣠⠴⠶⢾⠉⠸⡇⠀⡛⡳⢤⣐⠀⠒⡟
⣧⠀⠀⠀⢰⣥⠀⠀⠀⠀⠀⡿⢑⣲⢄⣀⣀⣿⣽⣃⠀⠀⠀⠀⡀⠀⠀⠈⢯⢍⡵⠒⠐⠚⠢⡀⢹⡀⠸⣇⠈⠛⣦⢀⠘⢆
⢸⠷⠀⠐⣪⣵⣄⡀⣀⢀⡠⠃⠀⠈⠻⣷⠏⠀⠀⠈⠱⠀⣠⣘⣃⠀⠀⢀⣞⠾⠂⠀⠀⠀⠀⠙⣄⠀⡄⠌⢤⣀⣟⣑⠢⢄⠣
⢸⠀⢀⡞⠁⠀⠈⢻⣿⡛⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⡆⠁⠉⠀⠙⢶⠎⠉⠙⣧⡀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢳⡞⠁⠀⠀⠀⢸⡍⠣⢄⣤⣤⣶⣿⡿⣦⣴⣠⣴⣿⠀⠀⠀⣠⠤⠘⠦⡤⠤⣷⠓⠙⡋⠉
⠀⠀⠛⢄⠀⠀⡠⡎⠀⠐⡞⠁⠀⠀⠉⣿⣿⣿⡿⠿⣿⣷⣧⣼⠁⠀⠀⠀⢸⠀⠈⣆⠔⠁
⠀⠀⠀⠀⠙⣄⠀⣹⣦⣤⡇⠀⠀⠀⢀⣿⣿⣿⡗⠀⣀⠴⠛⠛⠦⡀⠀⢀⣼⠀⣠⡷⢄
⠀⠀⠀⠀⠀⠀⠉⡔⠀⠀⢑⠦⠠⠴⠋⠀⠀⠘⣇⡰⠉⠀⠀⠀⠀⠘⡷⠃⠀⠀⠀⠈⡆⡇
⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠑⠠⣆⠀⠀⠀⢀⣼⣧⠁⠀⠀⠀⠀⢀⠗⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⡀⣤⠚⠉⣱⠖⠒⠉⠛⢿⣦⣄⣀⣀⣀⣾⣷⠖⠛⠛⢶⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⡀⢰⡃⠀⠀⠀⢠⠊⠀⠈⠉⢻⣿⣿⡏⠀⠀⠀⢈⠁⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠚⢷⣄⣀⣀⣼⡀⠀⠀⠀⣸⠟⠛⠻⢦⣤⡴⠎⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣈⣽⠉⠀⠉⠒⠲⣾⡇⠀⠀⠀⢈⣿⡷⠶⠶⠦⣤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣰⡿⠷⣄⣀⢀⣼⡿⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢻⠃⠀⠀⠈⢹⡟⠋⢹⣄⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⢀⡞⠀⠀⠀⠉⡛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⣀⣀⢀⠴⠁
⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢤⠆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣋⣭⡤⣍⣍⠐⠀⠤⠤⠤⡤⠶⢖⣛⡭⠷⠒⡛⢛⠛⠦
⠀⠀⠀⠀⠀⣾⠀⠴⢃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠈⢁⣠⢤⢦⡬⢀⠇⠀⠐⣂⣟⣻⡩⢥⠶⠖⠺⡏⠉⠙⠁
⠀⠀⠀⠀⣸⠃⠀⢰⡇⢊⡜⢡⡆⣼⠇⢹⢀⡞⣰⠃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢣⠤⡤⣤⢤⣤⣤⠤⠤⠼⠤⢷⠓⢒⢒⠂⢨⠉⣦⠀⡀⠄⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠘⣛⡗⣿⠶⣶⣶⡶⠶⣖⣶⣳⣛⣋⠏⠉⡁⡉⢈⠀⣙⣀⣥⡤⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣁⠙⣿⡂⠀⠀⠀⠀⠉⠛⢿⣯⠓⠓⠓⠚⠤⠧⣍⣀⣉⣉⡉⣩⣀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⣐⡒⠒⠂⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⡠⠛⠋⠉⠀⠀⠀⠀⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢷⣦⣄⡉⡑⠒⠢⣄⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠉⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠫⠴⠶⢦⣍⡳⣤⣀⠀⠀⠀⠀⣀⣀⣠⡴⢟⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣌⡀⠀⠀⠀⠀⠀⢀⡀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣯⣍⣓⣛⣛⣋⣩⣅⡶⠋
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⠑⠢⢄⠀⠴⠃⠀⠀⠈⠙⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠐⠲⢼⡄
⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠉⠙⠲⣤⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⠶⢶⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣠⡾⢉⣔⡉⠀⠂⠀⠄⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠂⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠄⠼⠒⡾⠀⠀⡄⠀⠀⡠⢁⢸⠽
⠀⠀⠀⠀⠀⠀⠁⠒⠢⠤⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠂⠐⠀⠀⠘⠀⠀⠰⠣⣀⡏⠰⡀⢀⡴⠚⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠐⠀⠀⠀⠀⠀⠀⠀⡀⠃⢘⡞⢠⠋⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⣣⣄⠈⠀⡀⠀⠀⢠⠀⢐⣠⠴⠚⠉⣠⡇⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣨⠋⢀⡜⠁⠀⣠⠋⠀⠀⢂⡤⡨⡿⢁⡴⠂⠁⠉⢠⡀⠀⠀⠀⢀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⡃⠀⣜⠀⠀⣼⠃⡀⠀⢀⡼⢠⣻⢁⡾⠀⠀⠀⠀⠀⢶⡀⠀⢀⡜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣽⠀⠀⡇⠀⣰⠇⡀⢈⡤⢎⣱⢿⠇⡼⠁⠀⠀⠀⠀⠀⢸⣧⡴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⣆⠀⢧⡀⡝⠠⠴⣩⡶⠟⠋⢰⠀⣇⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣢⡄⠣⢈⡴⠋⠁⠀⠀⠀⠀⠂⠻⣤⣤⣤⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⡿⣷⣦⣤⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⢠⣞⣿⢿⣿⣝⣿⣎⢻⣿⣿⣄⠀⠀⠀⠀⢸⣾⣿⢻⣻⣿⣿⣿⣿⣯⠛⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣾⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠸
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⣠⣾⣿⣫⡏⠀⣾⣟⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣰⣿⣿⣽⡏⢀⣿⣹⣿⣿⢻⣿⣿⣦
⠀⠀⢠⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣡⣿⣿⣿⡆⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣧
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠸⣿⣿⣿⠋⠉⠁⠀⠀⠙⠯⠛⠛⣿⣿⡇⠀⠀⠰⣿⣿⡿⠟⠉⠀⠀⠀⠀⠹⢭⠉⠻⣿⣿⡟
⠀⠀⠀⢻⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢹⠇
⠀⠀⠀⢸⠃⠈⠙⠲⢤⠀⠀⠀⡴⠖⠋⠀⡜⠀⠀⠀⠀⠀⡜⠁⠀⠉⠓⠦⠀⠀⠠⠖⠋⠁⠀⡱
⠀⠀⠀⠇⠀⠀⠦⠤⠆⠀⠀⠀⠶⠤⠶⠀⠈⠀⠀⠀⠀⠀⢨⠂⠘⠳⠞⠂⠀⠀⠀⠘⠲⠞⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠁⠀⢀⠤⠀⠀⢀⡠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠈⠀⠀⠀⠈⠁⠀⠀⠂⠁⠀⠀⠀⠀⠀⠁⠐⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠯⡈⠑⡤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢀⣀
⠀⠀⠀⠀⠀⠀⠱⡀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⠴⠋⠀⠀⠑⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠹⡀⠑⡄⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠡⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⡷⠚⠉⠙⠲⢤⡀⣠⡤⠴⢧⡈⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⡴⠛⠉⠓⠦⣶⡁⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠙⡆⠀⠄⠀⠀⠀⠢⠀⠀⠀⠀⠀⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⡤⠒⠻⣄⠀⠀⠀⣀⣀⣈⠓⣤⠤⠤⣄⠀⠀⡼⠀⠀⢀⡿⠋⠁⠈⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠠⠖⠊⠁⠀⠈
⠀⢸⡃⠀⠀⠈⠑⢲⠞⠁⠀⠀⡟⠀⠀⠀⠀⠙⢏⡔⠋⠉⠉⠓⢄⠀⠀⠀⠀⣷⠤⣄⠀⠀⠀⠠⠀⠄⠋⠁⠀⠁
⢀⡤⠛⠉⠉⠉⠉⢿⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣧⣀⣠⠔⠃⠀⠈⢳⠀⣀
⢸⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠈⠓⢤⣀⣀⡤⠞⣄⠀⣀⣀⠀⢀⣿⠤⠤⣄⣀⠀⢀⡼⠀⠀⠆
⠸⣄⠀⠀⢀⡴⠖⠋⠉⠉⠁⠀⣠⠊⠀⠀⠀⠀⠀⡼⠋⠉⠈⠙⢾⡁⠀⠀⠀⠈⠑⢯⡄⠠⠀⠄⠰
⢠⠞⠓⠒⣾⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⢻
⡟⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⡸
⢷⡀⠀⠀⠀⣨⠿⠒⠒⠒⠒⠀⠈⠓⠢⠄⠒⠀⠀⠀⠙⠢⠤⢶⡋⠀⠈⠙⠒⠲⡖⠋⠁
⠈⠳⠦⢤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⡇
⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⢸⠶⠤⠴⠚
⠀⠀⠀⠀⢹⣦⣀⣀⣠⡴⠋⠀⠻⣄⡀⢀⣠⠋⣨⠷⢦⣄⣀⣀⣠⡞
⠀⠀⠀⢰⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠇⠀⠀⠀⠈⠁⠀⠘⡆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠈⢷⣄⠀⠀⠀⢸⣄⠀⠀⠀⠀⡠⢆⠠⠊⠉⠁⠲⢤⣤⠴⠟
⠀⠀⠀⠀⠀⠈⢩⡟⠋⠉⠈⣱⠒⠊⠛⢦⢰⠃⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣸⠁⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠘⠦⣄⡀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⣹⡀⠀⠀⠀⢀⣎⠀⠀⠀⠙⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠙⣦⠤⠖⠋⠀⠙⡄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣰⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣤⣤⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠠⠖⠿⢗⠀⠀⡐⠀⠂⠀⠀⠀⠀⠈⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡴⠄⠀⠀⠀⠤⠀⢀⠀⢀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠠⠀⡀⠀⠀⠕⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁⠈⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⢀⠀⠀⠈⢀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡸⠄
⠀⠀⠀⠣⡀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡀⠤⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠟⠃
⠀⠀⠀⢆⠈⠂⠤⣀⠀⠀⠀⠉⠐⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠬⠌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠆⠊⠁
⠀⠀⠀⠀⠑⠢⣄⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠈⠐⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⢤⠖⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⣀⠀⠈⠁⠐⠂⠀⠀⠀⠐⠂⠀⠀⠀⢀⣠⡤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠦⠤⣤⣤⠤⢤⣄⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

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





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⠤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀⠀⢀⣠⠴⠖⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠛⠛⠓⠶⣤⡁⠀⠀⠀⠀⠀⠀⠈⢻⣆⢀⣠⠴⠶⠒⠲⠶⢦⣄⠁⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣸⢿⣄⠀⠀⠀⠀⠀⠀⠀⠹⣏⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠂⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠘⢧⡀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀⢀⡼⠁⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠛⢦⣔⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢯⣀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⢦⣤⣀⣀⣀⣀⣀⣀⣀⣠⡬⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢻⣄⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠻⣮⠐⡄⠀⠀⠀⠀⠀⠀⠀⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⡀⠀⠀⠀⠀⢀⡾⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣼⠷⣄⠀⣀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣜⠀⠰⠐⣠⠞⠁⠀⠉⠙⠛⠓⠶⠶⠦⠤⠤⠤⠤⠶⠶⠖⠒⠛⠉⠉⠁⠀⠀⠀⠈⠳⣬⣁⠂⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠂⠑⢄⠀⠀⠉⠭⢭⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠳⢤⣄⣀⣤⡴⠋⠀⠀⠀⢀⡴⠁⠀⢠⠀⠀⣹⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⠤⠖⠚⠁⠀⢀⡴⠁⠀⢀⣿⠓⢮⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡶⠾⠿⠟⠛⠛⠛⠉⠉⠉⠀⠀⠀⣀⣠⠤⠚⠁⠀⠀⢀⣾⡏⠀⠈⢷⡘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⡴⠶⠶⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣠⣴⡿⠋⣿⠀⠀⢸⣧⢸⡷⢦
⠀⠀⠀⠀⠀⠀⠀⠞⣋⣭⣥⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠿⠛⠁⣠⣾⠃⠀⠀⣸⡏⢸⡇⠈⢧
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⠿⠛⢁⡰⠁⣰⡿⠁⣼⠃⠀⢸
⠀⠀⠀⠀⠀⣿⡏⠀⣀⣤⣶⣶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⢀⣠⠔⠋⣠⣾⠟⠀⣴⠏⠀⢀⡿
⠀⠀⠀⠀⠀⢿⣧⣼⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠖⠚⢉⣠⣴⡿⠛⠁⢀⡾⠃⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣶⣶⣶⣶⠿⠿⠟⠛⠛⠉⠉⣀⣀⣤⣴⣾⣿⠋⣁⣀⣤⣶⣯⣤⣶⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⣿⠉⠛⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠟⠋⣹⡇⠀⢸⠇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠸⡇⠀⢸⡇⠀⠸⣷⠀⠀⣿⠀⠀⢸⡇⠀⠀⡿⠀⠀⣿⠀⠀⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢳⠀⠀⣷⠀⠀⣿⠀⠀⢿⠀⠀⢸⡇⠀⢸⡇⠀⢠⡇⠀⢸⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⡆⠀⢸⠀⠀⢹⠀⠀⢸⠀⠀⢸⠁⠀⢸⠁⠀⠸⠀⢀⣦⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣦⣤⣤⣸⣀⣀⣸⣀⣀⣸⣀⣠⣼⣤⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⢤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂⠉⡀⠤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⢈⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⡀⠀⠀⢀⠀⠀⠀⢀⠔⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⢁⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⠀⡈⠛⠋⠲⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢩⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠐⠀⢨⠀⠀⠀⠘⢧⣁⣤⣔⠂⠀⠀⠀⢴⡀⠀⢰⡆⠀⠐⠀⠀⠙⠳⠿⠿⠶
⠀⠀⠀⠀⠀⣠⠒⡁⠒⢄⠀⠀⠈⠓⠤⣁⡀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣴⣠⣴⣾⠁⠀⠈⢧⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠳⠤⠤⠄⠀⠀⠀⢀⣴⣄⠄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡍⠁⠀⠈⣽⡆⠀⠀⠀⠀⡠⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢹⠹⡄
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⡄⠀⡀⠀⠀⣴⠚⠉⠁⠀⢀⡄⣶⡼⠃⢀⡀⣠⠎⠀⠀⠀⠀⣻⠀⠀⠀⠀⣠⡾⣏⠀⣐⣿⠑⡇
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠡⡀⠀⠀⢸⠇⠀⢀⣤⠒⠋⠛⠻⠒⠃⠀⠀⠀⠐⢡⣴⣖⣯⠏⠀⠀⣠⡞⠁⠀⣿⣿⣿⣯⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢁⠀⠀⢸⣾⣴⣿⡧⣂⠀⠀⢀⠀⡄⠄⠀⡐⢞⣣⠿⠟⠃⠀⠀⠀⣿⠀⠀⡠⠏⣫⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⠀⠀⠀⠉⠉⠉⠹⣿⣤⣫⣾⠾⠿⠶⠷⠿⠟⠃⠀⠀⠀⠀⠀⠀⡙⠛⢋⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⣾⣿⣿⣶⣤⣯⣶⣮⣷⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⡿⣿⣿⣿⣷⣮⣴⣭⡻⣿⣮⡻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣷⣌⢟⠻⣿⣿⣝⡛⢝⠻⣿⣷⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⣿⢿⣿⣿⣿⣿⣿⣷⡹⣿⣷⣕⣌⡙⢿⣿⣶⣵⣬⡻⣿⣿⣯⢳⣶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣼⣿⣿⣿⣿⣿⣿⣷⣌⠿⢯⣻⣿⣦⣍⠻⣿⣿⣿⣾⣿⣿⣎⣿⣷⡱⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⡿⣿⣷⣿⣿⣿⣾⣽⣿⣿⣿⣿⣿⣿⣿⣷⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣧⢻⣿⣿⣿⡤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣶⣤⡀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⣿⣿⣿⣿⣿⣿⣿⡟⣿⡹⣿⣿⣿⣿⣿⣏⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣿⣿⣿⣿⣿⣦⡀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⢰⣿⣿⣿⣿⣿⡏⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣆⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣿⣿⣿⣿⣧⢿⣎⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡳⣿⣿⣾⣿⣿⣿⣿⡿⠉⠉⠙⠋⠉⠈⠁⠀⠀⠀⠀⠀⠈⠩⠾⠛⠁⣸⣿⣿⢧⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣵⣿⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢙⠷⣬⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣍⣻⣿⣾⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠀⠀⠀⠉⠉⠓⠦⣖⢺⠍⠁⠀⠀⠈⠉⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠡⠤⠀⣠⣠⡦⠤⢤⠀⠀⢤⠄⠈⠳⣄⣀⣀⣀⣒⠀⠛⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⣤⡶⢫⣵⡾⠟⢻⣷⣦⣄⠀⣀⠀⠘⣿⣿⣿⣿⣟⣦⣄⠀⢀⣀⠙⣦
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠛⣽⠟⣴⡟⠁⠀⢀⡏⠀⢩⣽⣧⠉⠀⠀⠸⣯⠉⠈⠉⠻⣿⠇⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⡏⢠⡸⢢⡿⢸⡿⠀⠀⠀⢸⠁⡄⠀⣛⣿⠀⠀⠀⣇⣿⠀⠀⠀⠀⠈⣷⠀⢀⡀⡀⢸⡆
⠀⠀⠀⠀⠀⢰⡇⠉⠀⠸⡇⣾⡇⠀⠀⠀⢸⡀⠶⠀⢻⣿⣆⠀⠰⣿⣿⠀⠀⠀⠀⠀⢸⡄⠈⠼⠃⠘⡇
⠀⠀⠀⠀⠀⠘⡇⠀⢛⣀⠁⢻⡇⠀⠀⠀⠘⣇⠀⠀⢀⠙⠙⣦⣼⣿⠇⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⢷⠀⠈⠁⠀⠈⢿⡀⠀⠀⣠⠿⣦⠀⠈⠀⠀⠈⠙⢻⣄⡀⠀⠀⠀⢠⡟⠃⠀⠀⣸⢿⡇
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣤⠈⢷⣠⢞⣿⠀⢘⣷⣤⡀⠀⢹⡴⠂⠉⢉⡓⢦⣤⣟⠀⠀⣠⣴⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠁⠀⢸⠷⠦⠁⢀⣼⡿⠛⠙⠲⢤⣀⠀⠀⠠⠆⠀⠈⠙⠳⣶⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⣿⠀⠀⣠⣾⣏⡀⠀⠀⠀⠀⠈⣙⣶⣦⣤⣄⣀⠀⢀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠘⣦⣼⠏⠁⠀⢉⡍⠛⠛⠋⢉⣉⠰⠀⠀⠈⢉⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⣀⠈⠁⠀⠀⣤⢀⣀⣀⣀⣤⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣷⣾⣿⣿⣿⠿⠟⠋⠁



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

⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⢶⣿⣿⡿⠛⣿⣿⠛⠛⢷⠶⠶⠶⢦⣤⣀⡀
⠀⠀⣠⡶⠻⠉⠈⠀⠀⠠⠆⠁⠀⡀⣀⠉⠀⠀⡀⣿⠿⠓⠒⠶⠤⠉⠛⠷⣦⣀
⠀⣸⠟⠀⠪⠀⠀⡀⠀⠀⠀⠀⠈⠀⠉⠀⠠⠨⠀⠀⢀⣤⡴⠶⠛⢛⡛⠛⠛⠛⠿⢶⣄
⠀⣿⡌⠀⠀⡁⡜⡍⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠞⡩⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠠⠙⣧
⠀⢻⣷⣾⡆⣠⢰⢸⢰⢾⡜⡎⢀⢀⢬⠡⠠⣸⡏⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢻⠟⠿⢿⣿⣾⣿⣿⣿⣿⣧⣿⣾⣀⠸⣿⡇⠀⠀⠀⠀⠅⠀⠁⠀⣀⣀⣀⣀⡀⠀⣾⠃
⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⢿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣠⣶⣿⠿⢛⡉⠉⠉⠙⠻⣄
⠀⠀⣸⠇⠂⠀⠀⠀⠀⠐⠀⠄⠐⠂⠦⠄⡈⠉⠙⢷⡰⡠⢠⣿⣿⠟⠁⠀⠀⠁⠀⠀⢀⣠⣽⡷⠟⢛⡉⠉⠩⠛⠳⣄
⠀⢀⣿⢀⡀⠰⠀⠃⠰⠀⠐⠀⠄⠀⠀⠀⠀⡄⠀⢨⣷⠀⣼⠿⡇⡄⠠⠀⠀⠀⢉⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⢂⠀⢸⡇
⠀⢼⣯⣼⡗⣆⢀⢀⠈⢡⠀⠀⠀⠀⠀⠀⢰⢀⡖⢾⢹⠀⢸⡆⢡⡱⠀⠀⠀⠀⣾⢻⡇⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠙⠿⢿⣿⣿⣿⣷⣼⣰⡄⣴⡇⣀⠀⢸⢨⢻⣿⣿⠀⠀⠻⣿⣿⣶⣤⡐⠀⢿⡸⡄⠄⠠⠀⢀⠀⠀⠀⠀⠈⠐⠘⣇
⠀⠀⠀⠈⠉⠉⠛⢿⣟⣿⣿⣧⣿⣧⣿⣷⣧⣾⣿⣿⣯⠀⠀⠀⠈⠻⡏⠙⡇⠀⡈⠻⢿⣦⣄⡀⠀⠀⣀⠀⠁⠒⠂⢀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠩⠭⠭⠭⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⡹⠀⠀⠀⣀⣠⣿⣦⣵⢣⡀⠀⠀⢻⠛⢻⠀⠀⠒⠃⠀⠀⠀⠀⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠒⠒⠒⠲⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣷⣝⣂⣀⣸⣷⣼⣇⢀⠀⠀⠀⠀⠀⠀⠀⠐⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⣛⣻⡟⠛⠛⠛⠛⠛⢻⣿⣿⣦⣓⡀⠀⢀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠐⠒⠀⠀⠀⠶⠰⠒⠆⠐⠐⠂⠐⠐⠰⠀⠀⠂⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⢀⣤⣀⣤⣾⣿⣤⣤⣤⣤⣤⣆⣀⡀⣀⡠⠤⣤⣀⡄⠀⠀⠀⢀⣤⣄⣄⣀⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⠉⠇⠈⠒⢼⠷⠌⠐⠀⠀⠀⠀⠀⠈⠁⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠁⠀⠁⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠯⣷⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⡿⠋⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠘⣏⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⡿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠩⠘⠏⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⢢⡀⠆⣨⡏⣾⣦⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⢿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⢛⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣾⣿⣾⣿⣦⣤⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⢀⢻⣿⠃
⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠉⠿⠛⠟⠛⠛⠋⠻⣿⡀⢀⣠⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠄
⠀⠀⠀⠀⣿⣿⣅⡄⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠘⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⢿⣿⣯⠁⠀⠀⠀⠀⠀⢈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⡅
⠀⠀⠀⠀⢸⣿⣂⠀⠀⠀⠀⠀⠀⠀⢀⠗⠂⠀⡄⠀⠀⠛⡿⠉⠲⠀⠀⠠⠐⠀⣴⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡇
⠀⠀⠀⠀⢸⣟⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⠀⠀⠀⠐⠶⡆⠀⠀⠀⠈⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠆⠀⢀⣶⡆⠸⣷⡾⠁⠀⠀⠀⠀⠀⠞⠋⠉⠀⠈⠉⠛⠋
⠀⠀⠀⠀⠀⣿⣿⡧⠄⠀⠀⠀⠀⠀⠀⢶⡀⠘⢿⣿⣷⠖⠀⠀⠀⠐⠋⠉⠀⣠⣿⣷⣄⣀⠄
⠀⠀⠀⠀⠘⣿⣿⣇⡄⠀⠀⠀⠀⠀⠄⠀⠳⠘⢿⣯⠤⠀⠀⠀⠀⠀⠀⠐⠃⠠⠈⠀⠈⠈⠀⢤⡁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠈⣀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢈⣶⣄⢀⢀⠀⢀⢀⣽
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡾⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣽⣿⣟⣻⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣞⢵⡘⣥⡀⣤⡄⠈⠀⠀⠀⠀⠈⢓⣀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣸⠋⠴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⣿⢿⣿⣿⣿⣿⣿⣾⣯⣽⢶⠤⠦⣄⡀⣀⣢⣀⣀⣿⣤⣀⠀⣀⣠⣦⣰⢆⣴⣏⡶⢚⣵⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡤⠖⠋⠉⠉⠁⠀⠀⠈⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠶⢤⣄⣀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⣠⠞⠁
⠀⠀⠘⠦⠤⣄⡀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢀⡴⠉⢀⠀⠂⠈⠁⠂⢄⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡞⠀⡠⠁⠀⠀⠀⠀⠀⠀⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢸⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⡏⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⢀⠌⢀⡞⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡠⠞⠁
⠀⠀⠀⠀⣰⠃⠀⠰⡀⠀⠀⠀⢀⡠⢀⡴⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⣰⠃⠀⢀⣀⣈⠂⠁⠈⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢠⠃⠠⠊⠀⠀⠀⠈⠑⢄⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣀⠀⠀⠀⠀⠀⣀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠦⣄⠀⠹⣄
⠀⠀⢻⡀⠠⠀⠀⠀⠀⠀⠀⡰⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠈⣇⠀⠈⠳⢤⣀⠀⠀⠀⣀⡴
⠀⠀⠀⠱⣄⠑⢄⠀⣀⠤⢊⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠓⠦⣅⣀⡴⠋⠀⠀⠀⣠⠖⠛⠛⠛⠲⠦⢤⣤⣀⣀⣀⣀⣠⣤⠤⢤⣤⣄⣀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠸⡄⠀⠀⠀⣀⡤⠖⠉⢀⡤⠖⠋⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠑⠚⠉⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⠤⣤⣀⣀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⣠⣞⣿⣭⣿⣿⣿⣿⠛⠋⠹⠟⢿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣿⢿⢿⣿⡛⠿⠛⠛⠛⢻⣿⢿⣿⣦⡀⠀⠀⣼⢿⣿⣿⣿⣿⡟⠀⠀⣀⠀⠀⠃⠀⠘⠛⣻⣟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⢟⣷⣼⣿⣿⡁⠂⠀⠒⢄⠐⠁⠻⣞⡟⣷⣄⣼⡿⣾⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢩⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⠘⠻⢻⢯⡿⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠐⠛⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣀⣛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢢⠀⠿⡇⢹⢿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⢰⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⢙⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢠⣄⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠉⢽⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣾⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⣄⠘⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⢰⡃⣸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⢰⠀⢸⣿⣿⣿⣾⣦⡀⠇⠀⠀⠀⠀⠀⠀⢀⣿⣻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢹⡋⠩⢿⣿⣱⢄⡀⡀⠶⣾⣧⡀⠉⠺⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢠⣾⣿⢟⣿⣿⡝⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠈⣤⢼⡏⠐⠈⢡⣴⠒⣿⠿⢵⣤⠀⠒⢿⣽⣿⣿⣦⣤⣀⣀⣴⣿⡿⠛⣿⣷⡧⠼⢷⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠔⠈⠀⠠⣴⡆⢿⢻⣷⡜⠁⠀⠀⠀⠀⠽⠛⠿⣿⣿⡿⢿⡿⣷⣶⣮⣿⠟⠑⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠖⠺⠖⡩⠾⠛⣿⣿⣷⣶⣤⣤⣤⣤⣴⢆⡀⠉⠳⠈⣻⣧⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄⣀⡼⢊⡴⡛⠀⠀⢷⣿⠿⣿⣿⣿⣿⣿⣿⣉⠋⠀⠠⢀⠀⢠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠖⠁⠈⠓⠷⡐⠋⠁⠀⠀⠀⠀⠉⢶⠿⡏⠀⠙⠈⠢⠘⠊⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⠤⣄⣀⣀⠀⣀⣈⣤⡤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⢛⢦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠅⡚⠟⢃⠇⠀⡆⡄⢣⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢊⡔⠈⢀⠤⠊⢀⠔⢠⠁⣰⠛⢍⢣⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠭⢀⣒⠚⠙⠳⠴⠖⠷⠒⠓⢉⠔⡀⠊⠰⢀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠤⣒⣾⣮⠐⢈⠁⠠⠤⠀⠀⠀⠀⠐⠒⡈⠕⢊⣀⡔⣡⣪⠃⡜⢢
⠀⠀⠀⠀⠀⠀⢸⠭⠬⠉⢉⣙⠋⠥⠈⢛⣉⠨⠌⢚⡋⠭⠑⠋⣉⠠⠐⢉⠤⠊⡠⢊⡇
⠀⠀⠀⠀⠀⠀⠘⣈⢈⠡⠄⢀⣒⠀⠭⠀⠒⠈⣉⡀⠤⠐⢂⡉⠤⠒⣈⠤⠐⣈⣔⠅⠣⡀
⠀⠀⠀⠀⠀⢠⠊⠈⠂⠢⠭⠤⡄⠐⣒⣈⣩⡅⠀⣒⣈⢭⡀⠒⠬⠵⠒⠚⠉⠉⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠸⡖⢢⠄⢠⡀⢀⣀⠀⡀⡀⡀⠀⠀⠀⠀⢀⢀⣀⣀⡀⢀⡠⠠⡤⠀⠶⠈⣿
⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⡇⠀⠉⠀⢸⠀⠸⠁⠘⠁⠈⠁⠈⠁⠈⠀⠀⠀⠀⠁⠀⡆⢠⡇
⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠁⠀⢰⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢠⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠘⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡄⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⡃⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢀⣇⠀⡄⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⡀⠤⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠁⠈⠁⠈⠁⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣶⠛⢿⣷⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⣸⡟⠀⠀⢀⡀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢸⡇⠘⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⠻⣄⣈⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⢦⠉⠛⢿⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣰⡿⠛⠛⠛⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠾⠛⠁⠀⠀⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢿⣄⣀⡀⠀⠀⢀⣀⣀⣀⣀⠠⠄⠀⢀⣀⣠⣤⡴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣤⣭⣭⣭⣭⣭⣤⣤⡶⠶⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⢉⣠⠾⠋⠉⠉⡙⠻⣷⣦
⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢒⣉⣤⡴⠞⠋⠁⠀⠀⠀⠀⠙⣦⠈⣿⣇
⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⣿
⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⠶⠶⠶⠞⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⡿
⣸⡏⣼⠁⠀⠀⠀⠀⠀⣠⠖⢉⣠⡴⠶⠞⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡿⣧⣤
⣿⡇⢿⠀⠀⠀⠀⠀⣸⢃⣼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣤⠞⠋⠀⣨⢻
⢻⣷⡸⣆⠀⠀⠀⠀⠟⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠞⠛⠋⠉⠀⠀⠀⢀⠞⣸
⠈⢿⣷⣝⠦⠤⠄⠀⣀⣾⣿⣄⠀⠀⠀⠀⢀⣠⣤⣍⡉⠓⠦⣤⣀⣀⣀⣀⣀⣀⣀⠤⠔⠊⢁⣀⣤⡴⠶⠛⠉⠉⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⠴⠞⠉
⠀⠀⠉⠛⢻⣿⡟⠛⠋⠀⠙⠻⠿⠶⠶⠛⠋⠁⠉⠙⠻⠷⣶⣤⣤⣭⣭⣥⣤⣤⣴⠶⠾⠛⠋⠙⠿⠷⠶⠶⠶⠶⠛⠛⠉⠉⠉⠉⠉⠀⢀⣠⡄
⠀⠀⠀⠀⠘⣿⣿⣷⣤⠶⠻⢿⣶⠾⠿⣷⣦⡴⠿⢿⣦⣤⡴⠿⣿⣬⣉⣥⣤⣄⣀⣠⠶⢶⣶⣄⣀⣤⠶⣶⣄⣀⡴⢾⣦⣴⠟⠿⠶⠾⣻⣿⠃
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣤⠀⠀⠀⣷⠀⠀⠀⣷⠀⠉⠁⢸⡆⠈⠉⠉⣴⠈⠛⠛⠁⢰⡇⠉⠛⠋⠁⡖⠈⠙⠉⣴⠈⠉⠁⢸⠇⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⢹⡆⠀⠀⢿⡀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⡿⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠈⣷⠀⠀⢸⡇⠀⠀⢹⡄⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢹⡆⠀⠘⣧⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢀⡟⠀⠀⠀⣼⠁⠀⢀⡿⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠈⣷⠀⠀⣿⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⡟⠀⠀⢸⡇⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢻⡆⠀⢸⡇⠀⠈⣧⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⢀⡿⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠘⣧⠀⠘⣧⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⢀⡟⠀⠀⠀⣼⠀⠀⢸⠇⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢻⡄⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢀⡟⠀⠀⡿⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠘⣧⠀⢸⡇⠀⢹⡄⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣼⠀⠀⠀⢸⡇⠀⢸⠇⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢻⡄⠘⣧⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⠇⠀⢀⡟⠀⠀⠀⣾⠀⠀⡿⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠘⣧⠀⣿⠀⢸⡇⠀⠀⢿⡄⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢀⡟⠀⢸⠇⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠛⠀⠸⠇⠀⡷⠀⠀⠸⠇⠀⠘⠃⠀⠀⠀⡿⠀⠀⠺⠀⠀⠀⠸⠇⠀⠿⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠾⠿⠃

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣶⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣯⣿⣿⣿⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⢜⠀⠀⠀⠠⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠳⠐⠊⠁⠀⢸⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣈⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣟⣿⣿⣿⣿⣿⣿⠻⢻⣾⣿⣟⣡⣤⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠻⠟⠉⠉⠉⠉⠉⠸⣿⣿⣿⣿⣿⣿⣿⣿⣤⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⡀⣾⣿⣿⠈⣻⣿⣋⡙⠿⠟⠁
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⡄⠋⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠟⠛⢛⣿⣿⣿⣉⣿⣷⣤⠹⣿⣿⡗
⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠙⢿⣿⣿⢆⢛⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠁⠀⠀⣷⡄⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⡄⢰⣿⣿⣦⢾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠀⠀⣦⠀⠘⠿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡿⣿⣶⣾⢿⣿⣟⢠⣭⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⣻⣤⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠻⣿⠟⢠⣿⣿⣧⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠉⣿⡟⠀⠈⠙⡃⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢡⣾⣷⡆⣘⣿⣿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡔⠉⢀⣀⡀⠀⠀⠀⠀⠄⠀⠐⠛⠃⡀⠀⢠⡟⠒⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣼⣿⣿⡄⣾⡧⣽⣧
⠀⠀⠀⠀⠀⠀⠘⢀⡞⣁⠠⠃⠀⠀⠈⠀⠠⠘⡇⠀⣠⡟⠀⠀⠸⡄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⢟⣀⣿⢾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉⠈⢳⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠆⠀⠀⠀⠀⠀⠈⠄⠠⠋⠀⠀⢰⠁⢰⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡍⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⠙⡃⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣶⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⡛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⡏⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠋⢳⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⢿⠿⡿⠿⢿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣿⣿⣿⣿⣟⣿⣿⣿⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⣠⣾⣿⣿⣿⣿⣿⡈⠻⠿⠿⠟⣛⣛⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⣾⣿⣿⣿⣿⡇⢻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⢸⣿⣿⣿⣿⣿⣿⣦⡉⠻⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⢿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣭⣭⣛⡿⢿⣿⣿⣷
⠀⢻⣿⣿⣿⣷⣌⣛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣻⠄
⠀⢀⡿⠉⠉⠙⣻⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣉⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⣸⡇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣌⢻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⡇⠀⢀⣀⣀⣀⠀⠉⠉⠉⣭⣭⣥⣴⣶⣶⣿⣿⣿⣿⣿⣷⣶⣤⣉⠻⣿⣿⣿⡆⢿⣿⣿⣟⣩⣭⣭⣭⣄⡀
⠀⢸⣧⡾⠛⠉⠉⠉⠉⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣷⣌⢻⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠘⣿⠠⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⣭⣭⣭⣿⣿⡿⠁⠀⠀⠀⠈⢻⣿⣿⣿⡆⢿⣿⣿⢫⣿⠟⣩⣤⣶⣬⡙⣿⣿⣿⡀
⠀⠀⣿⠀⠘⣿⣩⠟⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⢦⠰⣆⠀⢸⣿⣿⣿⠇⣿⣿⡇⣿⣿⢸⣿⣿⣿⣿⡟⣸⣿⣿⡇
⠀⢀⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⢀⣼⣧⠾⢀⣾⣿⣿⣯⣾⣿⣿⣧⠹⣿⣦⣙⣛⣛⣩⣴⣿⣿⣿⠇
⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣾⠋⠙⢻⣿⣿⣿⣿⣿⣷⣌⠛⠿⠿⠿⠿⣛⣿⣿⠏
⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣿⡋⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡇
⠻⣦⣔⡛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠈⢻⡆⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠈⠉⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⣸⠇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠙⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠋⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠙⢻⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡙⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⣀⣤⣶⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠿⠶⠿⠟⠛⠉⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⢛⣫⣽⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⣠⣴⣶⠿⠛⣉⣴⣾⠟⠋⠁⠀⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣤⣶⠾⠟⢋⣩⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣋⣭⣴⣶⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⣖⠋⠉⠀⢹⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⢶⣿⣿⣿⡷⣶⣶⣄⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠘⡄⠀⠀⠈⢦⢀⡀⠈⢳⠀⠀⠀⠀⣠⣾⣿⣿⣿⣎⢿⣿⣿⣷⡹⠿⠿⣿⡆
⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠘⢆⠀⠀⡸⠉⠉⠑⠞⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣮⢿⣿⡿⠃⠀⠀⠈⠁
⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡎⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡍
⠀⠀⠀⢹⣄⡀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⢛⢿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⢸⠀⠈⠑⠲⠤⢤⣤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣮⣭⣛⣻⠿⠿⠃
⠀⠀⠀⠸⣄⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠿⠟⠁
⠀⠀⠀⠀⢧⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠓⢄⣀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠐⠚⠒⠯⢭⣉⠁⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡪⠤⠤⠤⠤⠤⢤⣄⣀⡀⠀⠀⠀⠬⠙⠲⢆⡀⠀⠀⡈⠢
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣐⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠢⣄⠀⠀⠀⠀⡙⠲⠈⡀⠀⠘⢄
⠀⠀⠀⠀⠀⢀⡴⠝⠋⠉⠀⡀⠉⠉⠉⠉⠛⠛⠚⠲⢄⡀⡀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠙⠤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢨⣄
⠀⠀⠀⠀⢀⣹⣦⣀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠙⢷⡄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⡾⠂⠉⠀⡀
⠀⢠⡴⠚⠙⠁⠘⠋⡙⠛⠓⠐⠶⢤⣤⣀⠈⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⢿⠀⠀⠁⠀⠀⠀⠀⠀⢀⡰⠎⠁⠀⠀⠀⠈⠁
⢰⣍⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⢀⠙⠳⠦⣄⠀⠀⡄⠀⠈⠀⣽⠁⠄⠈⠀⢄⣀⡤⠖⠋⠡⠀⡀
⠸⠤⣈⡙⠲⢤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⣱⣤⣄⣠⢤⡾⣃⣄⣤⠤⠒⠉⠀⠀⢀⣀⣀
⠀⠀⠀⠈⠓⠢⢤⡉⠓⠢⠄⣀⠀⠀⠀⠀⠀⠄⠂⠀⢀⣞⡇⠉⠈⠛⠋⠥⠤⠪⠥⠤⠉⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠴⢄⣙⠓⠲⠖⠀⠶⠢⣞⣛⡽⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠓⠒⠷⠿⡶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣤⣤⣴⡤⠄⠐⠒⠚⠋⠭⠭⠿⠿⠿⠿⠶⠶⣦⣤⣄⡂
⠀⠀⠀⠀⢠⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠙⠷⠦⣤⣄⡉⠉⠙⠛⠛⠃⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⣀⠠⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒⠒⠠⠤⠄⠀⠀⠒⠐⠒⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠒⠒⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡾⠷⠶⢶⣶⣄⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠐⠢⠀⠈⠸⡄
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠐⢀⠙⠙⠻⣦⠀⠰⠃⠀⠀⠀⠀⠀⠓⠁⠀⠙⠄⣠⠐⠀⠀⠢⠀⠘⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣁⠄⠀⠀⠀⠀⣰⠀⠀⠣⣀⡀⠀⢀⢄⠀⠀⠀⠁⠀⢠
⠀⠀⠠⠁⣀⣤⣤⣴⣦⣤⣀⣀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢄⠀⠀⠁⠀⠀⠀⠀⠁⠀⡡
⠀⠀⡀⠙⠻⠿⢍⣉⠉⠉⠈⢙⡛⠷⠰⡾⠟⠀⠀⠀⠀⠀⠀⠰⡀⣸⠻⣀⠀⠈⠀⠄⠂⠀⠐⡄⠀⢀⠀⠁
⠀⠀⠇⠀⠀⠀⠀⠈⢉⣈⣩⣽⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠃⠀⡀⠀⠀⠀⠀⠀⠆⢀⠌
⠀⠈⠛⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠼⠁⠉⠗⠆⠒⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡶⠀⠀⠀⠀⠈⠉⠉⠻⣿⠄⠀⠀⠀⠀⠀⣧⠄
⠀⠀⠀⠀⠀⠀⠈⠉⠳⣦⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢠⣣⠈⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠢⠄⢀⠀⠀⠸⡇⢀⡹⠀⠀⠀⠀⠀⠀⠀⠸⣹⡀⠀⠀⠀⠀⠀⢰⣗⣙⡆⢀⣀⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠗⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⠅⠈⢦⣀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⡀⠀⠈⠑⠲⣄⣑⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠸⠂⢀⠠⠀⢸⣿⠋⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⠀⣁⡤⢚⣇⡔⢂⣤⠤⠴⣂⣼⠻⣄⠀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⣸⠀⣠⠞⠉⠀⠀⠈⣦⣀⣴⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⣰⠏⠀⣏⠀⠀⠀⠀⠀⣿⠻⡿⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢀⣠⠗⠀⠀⢀⣠⠴⠁⠀⢿⡁⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢀⣀⣀⣀⣀⣀⣀⠀⠠⠔⠊⢀⣩⠤⠶⠶⠚⠉⠀⠀⠀⣰⣿⣵⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠻⢄⠀⠉⠀⠀⠀⠀⣠⡤⠞⠉⠀⠀⠀⠀⣠⣴⣶⡾⡿⠿⡿⡿⢊⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠉⠳⠶⠶⠶⣾⣿⣦⣤⣤⣴⣾⣿⠟⠿⠋⠁⠊⣩⢄⣴⣼⣶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠲⣤⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠠⡠⠀⣀⣀⣧⣶⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⠸⡏⠘⡷⠒⢿⠶⣿⠾⢿⠿⣿⠿⣿⠿⢻⠛⢻⠁⣿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⡆⢿⠀⣿⠀⢸⠀⣿⠀⣾⠀⣿⠀⣿⠀⣽⠀⣿⣿⣿⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠐⣸⡇⢸⡆⠸⡇⢸⢄⣻⡀⣹⠀⣿⢰⡿⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣿⢸⣧⠀⣷⣼⣆⣿⣄⣿⢤⣿⣺⣧⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣟⣾⣿⡐⣿⣿⣇⣿⡽⣿⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣿⣿⣇⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣽⣿⣧⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣵⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢗⡚⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠛⠻⠛⠛⠟⠛⠋⠉⠉⠉⠁



⣿⠿⡿⠿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻⣿⣛⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠶⠶⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠟⠛⠛⠳⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠶⠶⣤⣤⣀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⣀⣤⣴⠶⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠀⠀⠙⣷⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣿⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠙⢷⣄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢹⡆
⠀⢰⡏⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣧⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠸⣇
⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠸⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠹⣦⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢰⡟
⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣸⡇
⠀⠀⠸⣇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣧⡀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⡿
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠘⢷⣄⠀⠀⠀⠀⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠹⣧⣀⣀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⢀⣽⢷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣠⣾⡃⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠈⢿⣄⠀⠀⢰⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡆⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠻⣦⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⢶⣤⣤⡶⠶⠶⣿⣏⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣬⣽⠿⠷⠶⢶⣤⣤⣴⠾⢋⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠳⠶⠶⠞⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡆⠀⠉⠙⠒⠦⣄
⠀⠀⠀⠀⠀⠀⣀⣬⣭⣿⣿⣿⣿⣧⣤⣄⣀⠈⠥⢜⢇⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣯⣼⣭⣻⣿⣿⣿⣿⠀⠐⣶⠮⠁⠈⠉⠒⠀⠀⠒⠈
⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⡟⣿⣿⠟⡟⢻⡓⢠⡄⠈⢠⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠙⢡⣾⣿⣿⢹⢟⢣⣶⠈⠈⠇⢸⣟⠳⣾⡆⠀⣿⡀⣰
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣾⠀⣀⣿⣆⠀⣀⣼⣿⢠⣾⠟⠛⢻⡞⠛⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢃⣤⣿⣯⣌⣾⣿⣿⣄⠀⢘⣄⣀⣄⣿⣆⣴
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⣿⣿⣛⣿⣿⣿⡿⢿⣿⡌⠁⢻⣿⡿⠛⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣏⢹⡏⠀⠀⢨⣿⣦⣎⠈⢿⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⠿⠾⢷⠋⠀⢸⣿⠷⣿⣋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⣤⣼⠷⠒⠻⠃⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⢿⣷⣠⣼⡟⢻⣷⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⣾⠋⠉⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣻⡿⣦⣠⠜⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⢦⣀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⣛⣛⣭⣭⣛⣛⣒⠶⢤⣄⡀⠀⠀⢀⣠⡤⠶⣒⣛⣛⣭⣭⣛⣓⡲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠵⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢭⣓⣚⡭⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⠮⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡄⠀⠀⠀⠈⢦⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣉⠉⠀⠀⠀⠺⠿⠆⠀⠀⠀⠀⠈⠈⠀⠘⠀⠀⠀⠘⡇⣷
⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠙⢆⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣟⢹
⠀⠀⠀⠀⠀⠀⠀⣷⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠛⠀⣇⠀⠃⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⡗⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⣀⡔⠹⣄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠉⠛⠋⠉⠀⠀⠈⠙⠛⠉⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢙⠇⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸
⠀⠀⠀⠀⠀⣀⡤⢶⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡶⢤⣀
⠀⠀⠀⣠⠞⠁⣠⢼⠇⡇⠀⠀⠠⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⣄⠈⠳⣄
⠀⠀⣼⠁⢀⡞⠁⢸⢰⡇⠀⠀⠈⠒⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⡇⠈⢳⡀⠈⣧
⠀⢠⡇⠀⢸⠀⠀⢸⢸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⡇⠀⢸⡄
⠀⠘⡁⠀⠸⢦⡀⢸⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⡇⢀⡴⠃⠀⢈⠃
⢀⡞⠁⠀⠀⠀⢧⣀⣳⡹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣞⣠⡼⠀⠀⠀⠈⢳
⣼⠀⢀⣴⡦⣄⣀⣠⠇⠙⢮⣑⡢⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⢔⣫⡵⠋⠸⣄⣀⣠⢴⣦⡀
⡇⠀⢸⠇⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⣶⠶⡶⢶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⣾⠸⡇
⠙⠲⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⣿⠀⡇⢸⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠈⠉⠙⠓⠿⠀⡇⢸⠀⠿⠚⠋⠉⠁⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡗⢺⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠃
⣿⣿
⡟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⣦⠀⠀⢀⠀⠀⠀⠰⠀⠈⠄⠉⠁⡦⠀⠀⠀⠀⠰⠧⠤
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠠⠄⠤⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⣴⡀⠀⠀⠐⠀⣠⡿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⠃⠀⠀⠀⠀⠀⠠⠄⠀⠤⠊⠀⠀⠙⠤⠤⠄⠠⢞⠋
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⣦
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⢠⣤⠏⠀⠀⠀⠀⠄⠀⢰⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⣀⣀⡼⠋⠁⠀⠀⠀⠀⠀⢀⠴⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠀⠄⠀⠀⠀⠀⠀⢀⡜⢠⠀⣀⠀⠒⠒⠀⠀⠔⠀⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢒⣠⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠒⠀⠀⠀⠀⠀⠀⢀⢠⡷⣾⣋⠁
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠤⠑⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⣼⣻⣛⣶⣴⠶⡤⠆⠀⠀⠀⠀⠀⠀⠐⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠰⠲⡄⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⣸⠟⠋⠁⡈⠉⠁⠀⠀⠀⠀⠀⠀⢴⠖⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠦⡀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁⠀⡹⠂⠀⠀⠀⢠⢸⣆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠇⠤⠔⠃⠀⠀⠀⠀⠘⠻⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠄⠀⡀⢀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠠⠀⠀⢠⡀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠉⠋⠀⢀⣀⡔⠀⠀⣠⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠓⠤⠄⠃⠀⢀⣀⣂⣀⣈⣥⣴⠶⠞⣫⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣉⢀⡀⣼⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⢀⣀⠄⠈⠉⣉⠉⣁⣤⣤⣄⣾⡃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠛⠟⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣀⠀⠀⠀⠲⠞⢉⡛⠫⠄⠀⠉⠉⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠇⣀⡀⠀⠀⠀⠀⡀⢠⣔⣤⣿⠿⠋⠀⠀⠀⠀⠀⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠷⠿⠿⡾⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⢠⣾⢻⡏⠙⣿⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣁⣸⣿⠾⣿⡶⠋⠀⠀⠀⠀⣈⠀⢀⣀⣠⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡏⢉⣿⡁⠘⣿⡇⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢷⣦⡾⠃⠀⠀⠀⠀⠀⠦⠤⠠⠀⠀⠀⣶⡴⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣤⡀⢠⣤⣤⣤⡄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠉⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠙⠳⢦⣤⣀⣀⡀⠀⠀⢀⣀⣀⣠⣤⣴⠶⠞⠛⠛⠛⠛⠛⠉⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠳⢦⣄
⠀⠀⣀⣴⢿⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠉⠻⣦
⢀⣼⠋⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⣹⣷⡄⠀⠀⠀⠀⣴⣶⣄⠀⢀⣴⣦⡀⠀⢀⣴⣦⠀⠀⣠⣶⣄⠀⠀⣴⣶⡄⠀⢀⣴⣦⡀⠀⣠⣶⣦⠀⠀⣠⣶⣄⠀⠀⣠⣿⡅
⢸⡏⠀⠻⣦⡀⣠⡾⠃⠀⠹⣶⠟⠁⠈⠻⣶⠟⠁⠈⢷⡾⠋⠀⠙⢷⡾⠃⠈⠻⣶⠟⠁⠈⢻⡾⠋⠀⠙⢷⡾⠋⠀⠙⣷⠾⠋⠈⣿
⠀⢿⡄⠀⠈⠻⡏⠀⠀⠀⠀⠸⡆⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⢸⠁⠀⠀⠀⣼⠁⠀⠀⢠⡏⠀⠀⣼⠏
⠀⠈⣿⡀⠀⠀⢹⡄⠀⠀⠀⠀⢷⠀⠀⠀⠘⡇⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢀⡟⠀⠀⠀⢠⡇⠀⠀⠀⣼⠁⠀⣰⡏
⠀⠀⠘⣷⡀⠀⠀⢳⡀⠀⠀⠀⠸⡇⠀⠀⠀⢷⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⣼⠀⠀⠀⣸⠃⠀⠀⠀⡼⠀⠀⠀⢠⡇⠀⣰⡟
⠀⠀⠀⠘⣧⠀⠀⠈⣧⠀⠀⠀⠀⢻⠀⠀⠀⢸⡀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⡏⠀⠀⢀⡟⠀⠀⠀⢰⠇⠀⠀⠀⡾⠀⢰⡟
⠀⠀⠀⠀⠹⣇⠀⠀⠘⣇⠀⠀⠀⠈⣇⠀⠀⠈⡇⠀⠀⡇⠀⠀⠀⢸⡇⠀⢸⠃⠀⠀⢸⠃⠀⠀⠀⡾⠀⠀⠀⢰⠃⢠⡟
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠹⡄⠀⠀⠀⢹⡀⠀⠀⢿⠀⠀⡇⠀⠀⠀⢸⡇⠀⣾⠀⠀⠀⡟⠀⠀⠀⢸⠇⠀⠀⢀⡟⢠⡿⠁
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢹⡀⠀⠀⠀⣧⠀⠀⢸⡄⠀⡇⠀⠀⠀⢸⠇⢀⡇⠀⠀⢸⠃⠀⠀⠀⡟⠀⠀⠀⣸⢃⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⢷⠀⠀⠀⢸⡆⠀⠘⣇⠀⡇⠀⠀⠀⢸⠇⢸⠁⠀⠀⡟⠀⠀⠀⢸⠃⠀⠀⢠⣏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣬⣧⠀⠀⠀⢷⠀⠀⢿⠀⡇⠀⠀⠀⢸⠃⡿⠀⠀⢸⠃⠀⠀⢀⡟⢀⣠⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣼⣆⣀⣸⡆⡇⠀⠀⠀⢸⢠⡇⠀⠀⣟⣀⣤⣤⠾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠉⠉⠉⠉⠉⢛⣓⣲⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠉⠀⣀⣀⡤⠤⠼⠓⠒⠒⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠖⠋⠁⣀⣠⠤⠖⠒⠋⠉⠀⠀⢀⣀⣠⠤⠴⠒⠒⠒⠒⠒⠛⢓⣢⣤⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠠⠾⠕⠒⠚⠉⠁⠀⠀⠀⠀⣀⡴⠖⠋⠉⠁⠀⠀⣀⣠⠴⠖⠚⠉⠉⠉⠉⠉⠉⠉⠙⠒⢤⣀
⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⢀⣀⡠⠴⠒⠊⠉⠁⠀⠀⠀⣀⣤⠤⠖⠒⠒⠛⠛⠛⠒⠒⠮⠷⣄
⠀⠀⠀⠀⠀⢻⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠽⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠙⢷⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⣀⣀⣠⠤⠤⠖⠒⠒⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠶⠶⠶⠦⠤⠤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠯⣤⣀⠤⠀⣠⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠀⠀⣀⣀⣠⡤⠤⠤⠤⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢶⣇⣔⣁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠶⠶⠶⠒⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣥⣔⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠧⣄⣤⣀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠺⠷⠶⠶⠶⠾⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠒⣛⣛⣛⣛⡒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣋⠴⠚⠉⠉⠀⠈⠉⠁⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣡⠞⠁⠀⠀⠶⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠳⣄⠀⣀⣠⠴⣖⣻⠍⠉⠁⠀⠉⠉⠙⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⡀⠁⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡼⢛⡭⠖⠋⠁⠀⠀⠀⠀⡄⠀⠀⠰⠂⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣸⢱⠇⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣤⣄⡀⠀⠀⠀⢀⡴⣫⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠙⢦⠀⣰⠋⠞⢡⡄⠀⠀⠀⢀⣠⠤⠤⠤⣤⣀⠀⠀⠀⠀⠰⠆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣁⠀⠀⠈⠀⢀⡤⠞⣯⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⡏⠀⢀⡤⠞⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠈⠁⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣻⠖⠋⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⢴⠆⠀⠀⢠⠿⣆⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢰⡆⠀⠀⠘⣧⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⢀⡀⠀⠀⠀⣰⠏⠀⠈⢧⡀⠀⠀⠀⢠⡏⠘⠁⠀⠀⠖⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⠞⠁⠀⠀⢠⡶⠀⠀⠘⠋⠀⣠⠞⠁⢀⠀⠀⠀⠳⡀⠀⢀⡟⠀⠀⣀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢠⢞⡅⠀⠀⠀⠀⠀⠀⣀⠀⣀⡴⠚⣍⢧⡀⠈⠃⠀⠀⠀⢱⣀⠞⠀⠀⠀⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠳⡄⠀⠀⢀⡏⡞⠀⠀⠀⠀⣀⡄⣈⡵⠚⠁⠀⠀⠈⢧⡳⡄⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡙⣆⠀⠈⢧⡑⠦⠤⠴⢚⡥⠚⠁⠀⠀⠀⠀⠀⠀⠀⠑⣍⠀⣄⠀⠀⠀⢸⠀⠀⡄⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠳⡀⠀⠉⠛⠒⠋⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣨⠷⢬⣉⣉⡴⠋⠀⠀⠁⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⡀⠀⠀⠀⠀⢀⡄⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠈⠀⠀⠀⠀⠀⡤⢀⣀⣠⠤⠶⢚⣉⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⠤⠴⠶⠒⠒⠚⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠘⠖⠒⠦⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠐⡀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠛⠲⢤⣀⠀⠀⠀⢳⡀⠀⠀⠠⠄⠀⠸⡀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢳⡀⠀⠀⠁⠀⠀⢃⠀⠀⠄⠀⠀⠀⠘⠒⠦⡀
⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⠚⠒⠦⣀⠀⠈⢦⠀⠀⢳⠀⢲⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢢
⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⠈⠳⡀⠈⢧⠀⠈⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⢆
⠀⠀⠀⡸⢄⠀⠀⠀⠢⡄⠀⠈⠈⠀⠀⢳⠀⠈⡇⠀⠘⠀⠀⠄⠂⠀⣧⣠⠤⠦⣄⡀⠄⠂⠀⠀⠀⢸
⠀⠀⢰⠁⠀⠙⠒⠢⣄⠈⠂⠀⠀⠀⠀⠘⡆⠀⠉⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⣠⠗⠂⠉⠉⠀⠀⡼
⠀⠀⡇⠀⠀⠀⠀⠠⠀⠑⠢⡤⠴⠒⠉⠉⠑⠒⠒⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠱⢄⣀⠀⣀⠠⠊
⠀⠀⠈⡆⠀⠀⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⡘⠤⠤⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠳⠤⠴⢄⠀⠀⠈⠀⠀⠀⡘⠒⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⠤⠤⠓⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⣄⣀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⢪⡭⠅⡀⠙⢯⡐⢆⠉⢿⣳⢦⠀⠀⠀⠀⠠⣶⠏⢉⠥⠶⠖⢦⣄⡀⠀⠀⠀⣠⠊⠁⠀⠀⢀⠇⠰⡀⠀⠈⠢⡀
⠀⠀⠀⠀⡀⠔⠊⠁⣦⠠⡀⠈⢷⡀⠀⠀⠈⢷⠀⠆⠀⢳⠀⣧⠀⠀⢰⠀⣇⠔⢁⠰⠻⠾⠟⠛⠷⠀⠀⢰⠃⢀⡄⠊⠁⠀⠀⠀⠉⠑⢄⠀⢱⡀
⠀⠀⡠⠊⠀⡿⠙⠄⠹⡄⠘⢆⠈⣷⡀⠀⠀⠸⡇⢸⠀⠈⣶⡏⠀⠀⢷⠀⠀⢠⠂⠀⢀⣀⡠⠤⣤⡆⠀⢸⣆⠀⠀⠀⠀⢹⢀⠇⠀⠀⠈⠃⢸⠇
⠀⢰⠁⠀⠀⢧⡀⠐⡀⠙⣆⠘⡄⠘⢿⡀⠀⠀⢹⣸⣶⡿⠋⠀⠀⠀⠈⢷⣄⠀⠑⠉⠁⠀⠀⠀⣴⠇⠀⢸⣿⣦⡀⠀⠀⠈⠎⠀⠀⢀⣠⣤⡿
⠀⠰⣄⠀⠀⠀⠙⠢⢐⣄⠈⠳⣷⣀⣨⣷⣿⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢤⣤⣤⠤⠞⠉⠀⠀⠀⠻⣿⣿⡷⣶⣶⣶⣶⣿⣿⣿⠟⠁
⠀⠀⠈⠛⠷⠶⠶⠶⠶⠿⠿⠟⠛⠛⠋⠉⠁⢀⠤⠂⠒⠒⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⠭⠭⢍⣁⡀
⠀⠀⠀⢀⣤⣔⠂⠐⠒⠒⢦⣤⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡀⠀⠀⠈⢻⣿⠛⠳⡄⠀⠀⠀⠀⠀⡴⢎⠁⠀⠀⠀⠀⢸⠉⠙⡒⠒⣄
⠀⢀⣴⠋⠀⠠⡙⠄⠀⠀⠈⣿⠉⢣⡀⠀⡇⠴⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⣁⡀⠀⠀⢀⡟⠀⠀⠸⡀⠀⠀⣤⠚⠙⢦⠠⡀⠀⠀⠀⢸⠀⢠⠷⠊⠁
⠀⡭⠭⠭⣀⡢⢹⣤⡄⢀⣶⠏⢀⡆⢹⠀⣇⠀⠀⠀⠀⠐⠀⠀⢰⠇⢀⡼⠛⠛⠻⢦⡎⠀⠀⠀⣠⣿⠀⡞⠓⣀⠀⠀⠱⡔⡀⢀⣀⠾⠂
⠀⠆⠀⠀⢠⣿⣼⣿⡷⠟⠛⠛⠛⠣⠶⠀⠈⠢⢤⣀⣀⣀⡤⠞⠋⠀⠀⡇⠀⠀⠀⠉⡄⠁⠐⠊⢹⠇⠸⠄⠶⠚⠾⠦⠞⢛⣛⣛⣉⡀
⠀⠦⠴⠚⠋⢉⣁⡀⠠⠀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠰⣄⡇⠀⠀⠱⣄⡀⠐⠃⠀⠀⠀⠀⠠⠴⠶⠿⠥⣄⡀⠀⠈⠉⠓⢤⡀
⠀⠀⢀⠔⡈⠁⠀⠀⠀⠀⠀⠀⠸⣹⡂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣙⡒⠒⠚⠉⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠉⠳⣤⠀⠀⠀⠹⡄
⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠐⠙⠻⣿⡏⠁⠢⣄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡇⠀⠀⣸⠃
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄⠈⢣⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⡟⠋⠉⢿
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⣞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣧⢄⣼⠀⠀⠀⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⠀⠉⠓⠦⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠃⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⡟
⠀⠀⠀⠀⠓⠤⠤⠤⠤⠤⠤⠖⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠢⠤⠴⠚⠋⠀⠀⠀⠀⠀⠘⠲⠦⠤⣄⣠⣀⣀⡀⡀⠀⣼⠖⠋⠁
⣤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣀⣤⣯⣼⣭⣥⣥⣬⣷⣯⣥⣥⣯⣭⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣿⣿⣿⣿⣭⣭⣽⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣽⣿⣿⣿⣿⣿⣿⣷⣞⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⣿⣿⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣅⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣭⣴⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣷⣿⣿⣿⣿⣿⡍⢿⣿⣿⣿⣿⣿⣿⣷⣎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢼⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠞⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠣⠋⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠟⠋⠙⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣶⡖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⣀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⢀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢃⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠃⣴⢃⡀
⠀⠀⠀⠀⠀⠁⡀⠘⣇⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠵⣄⠃⣪⢁
⠀⠀⠀⠀⠀⠀⣰⣦⡈⠃⠁⠠⠀⠂⠀⠀⠀⠀⠀⠀⢩⣛⣷⡄⣳⢂⠘⠀⠀⠀⢀⣠⡤⠀⠶⢤
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⢀⣀⢰⡄⠀⠈⠉⠻⠆⠛⠙⠳⠶⢷⣾⢶⡷⠠⠄⢈⠁⢁⠐⢚⣻⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⡀⣹⠗⣠⣤⡐⠛⠙⠟⠰⠛⠓⣦⠀⠤⢬⡀⠐⠒⠶⠉⡤⠶⡶⠉⣴⢿⣄⡷⠀⠀⠀⠀⡀⠀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⢻⡏⣠⡥⠌⠒⠛⢻⡖⠚⠛⣷⡦⢠⣶⣶⣶⠰⠾⠛⠿⣂⣠⣶⢟⣀⢿⣵⣿⣶⣈⠨⢤⡍⣤⣄⠁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣧⣵⡶⢦⣍⠙⠻⠄⢢⣄⠻⠋⠭⠀⠲⠎⢽⣶⣿⣿⣿⣏⣵⣿⣫⣤⣿⣿⣃⣬⣤⣀⢤⣀⣩⣤⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⡛⠶⠦⠴⢿⠷⣿⣿⠀⣤⠶⠶⠈⣙⠛⢰⡝⢿⣿⣿⣟⣯⣶⣿⣿⣯⣉⣉⣬⣭⠰⢾⡍⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠋⠉⠉⠉⠉⠛⠶⣶⣶⣿⣿⡏⢰⣷⣾⣷⣿⣽⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣷⡶⠿⠿⠧⣿⣷⣮⡯⠟⠋
⠀⠀⠀⠀⠰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢟⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣨⡭⠽⠿⠛⠛⠫⠤⣀
⠀⠀⠀⠈⢧⣝⠢⢄⣀⣀⠀⠀⠀⢀⣀⣄⣀⣤⣶⣿⢿⣿⣿⣿⣽⣯⣽⣿⣿⣿⣿⣿⣅⣀⠀⠀⠀⠀⠀⠀⠈⣷⣤
⠀⠀⠀⠀⠘⡼⣿⣧⣭⣭⣷⣶⣾⣿⣿⣟⣛⣿⣿⣧⣿⣿⣭⣿⣿⡿⠟⠋⠉⠉⠉⠉⠙⠛⢿⣦⣄⢀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠐⢮⣿⡿⢿⣽⠯⣭⠔⠛⣛⡥⠛⠉⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠃⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⡸⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣆⣀⣠⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠿⠿⢿⣿⣿⣷⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢸⣿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⣿⣷⣿⢶⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⣼⡇⠀⣼⢧
⠀⠀⠀⠀⠀⠀⢀⡿⠉⠀⠀⠉⠙⠿⣯⡻⣦⡀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣀⣀⣠⣾⣿⣥⣼⡏⣸
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣧⣀⣀⣀⣠⣤⣶⣽⡿⠿⠿⠿⠟⠛⠉⠛⠛⠻⠟⠃
⠀⠀⠀⠀⠀⢀⡇⠀⠀⣀⣀⣀⠀⠀⠀⠈⣿⣿⣦⣀⣤⣴⣾⡿⣷⣾⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⢿⣧⣄⢀⣴⣿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⣻⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣲⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠉⠉⠀⠀⠀⠀⠈⠉⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠓⠶⢤⣄⠉⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⢦⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⣯⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡟⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣤⣤⣤⣀⡙⠲⣄⡀⠀⠀⠀⠀⠀⠀⢻⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢰⠃⠀⢸⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⠙⠲⣄⠙⣆⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⣰⣋⣀⡴⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣠⠾⠷⠶⠶⠒⠋⠉⠉⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⣼⠷⠦⠤⠶⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡴⠚⠛⠲⢦⡀⠀⠀⣠⡿⠓⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣹⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣠⡾⠃


⣿⣿⣿⣿⣷⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠭⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣸⠙⣏⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠈⠀⣠⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣄⠈⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢂⡤⢈⣟⣿⡃⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠰⠀⠠⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢁⡀⠑⠀⠲⢀⣀⡀⠀⣀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠛⣕⠈⠘⠃⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠈⣠⠀⠾⠂⠀⢴⠗⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠘⠃⣠⡀⠂⠀⣴⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠛⠃⣨⡀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⠙⠁⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣍⠕
⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠻⠉⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠰⡿⠀⠀⠀⠈⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⣀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⢀⠀⠀⣶⣶⠄⠀⠀⠻⠿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠙⠋⠀⠈⠛⠀⠀⠙⠁
⠀⠀⠀⠈⠛⠛⠛⠶⠞⠳⡶⠟⠶⠖⠲⠞⠁⠀⠀⠀⠀⠔⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠄⠊⡁⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠁⢀⣀⠤⠔⢋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠃⢉⡠⠄⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⣿⢿⠿⠿⢿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠞⡀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠀⡀⠀⢀⣀⡈⠂⠀⠂⢠⡃⠀⠀⠜⠁
⠀⠀⠀⠚⠀⠀⠀⢠⡾⠋⠉⠉⠙⢻⡁⠀⠀⠉⢲⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⢀⠀⠀⠀⠀⡈⣳⡀⠀⡄⣅⣀⡠⠖⠀⠐⠪⣠⡞⠁
⠀⠀⠀⠈⢁⠀⠀⠙⠛⠒⠠⠤⠤⣚⢙⣒⢒⣶⠞⠀⠀⠈⢄⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣰⡏⠉⠉⠉⠙⣿⣁⡀⠁⠀⠀⠀⠀⡅⠈⠂
⠀⠀⠀⠤⠄⠀⠐⠁⠀⠑⠠⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠒⠃⠀⠀⠀⠈⠃⠉⠉⠀⠒⣉⠍⠉⠀⠀⠀⠠⢤⡚⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠐⠤⣀⠀⣦⠀⠀⠀⢀⠖
⠀⠀⠀⠀⣀⣠⠤⠤⣤⡞⠉⠀⡀⠄⢹⠃⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢀⠗⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠁⠉⠐⠄⢠⠈
⠀⠀⠀⣮⠉⠀⠀⡠⠒⠑⠒⠛⠃⠀⠋⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⠀⠀⡬⡶⠁⠀⠀⠀⠀⠀⠀⢀⠄⠀⠚⠁⠀⡀⠀⢀⠀⠀⡼⡯⠀⢀
⠀⠀⠸⠈⠉⢉⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⡀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠤⠂⠀⠀⠃⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠲
⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠊⠙⠀⠐⠂⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠠⠴⠁⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠂⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠀⠀⢨⠅⠀⢐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠴⢄
⠀⠀⢀⠔⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⡆⠀⡠⠁⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠐⠈⠁⠀⠀⠙⢤
⠀⠀⢈⠀⢀⣤⣤⣠⣴⠂⠀⣀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠑⠦⣄⣻⡀⡧⠴⠋⠀⢤⡄⠀⠀⠢⣄⠀⠀⠸⡄⠀⣀⣀⣄⣀⣀⣀⣤⡀⠐⠂⡄
⠀⠀⢰⡋⠁⠀⢀⡨⠷⠶⢾⠁⠸⠂⠐⠀⠀⠀⠀⢢⠀⠀⣠⠖⠚⠉⠙⢻⣁⣠⡔⠀⡨⠁⠀⠀⠀⡀⠑⠶⣤⣿⠋⠀⠈⠉⠛⡅⠀⠈⠉⢲⡄
⠀⠀⠧⠉⠉⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⠈⠀⢠⡻⠤⢤⠔⠒⠋⠤⠘⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠛⠂⠲⠶⣚⠉⣓⠒⣦⡾⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠠⠄⠊⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠢⠄⠠⠐⠁⠀⠉⠒⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⢛⣛⡙⢿⡿⣿⢿⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣤⣙⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⣿⣿⣿⣿⣿⡻⢿⣟⢿⣷⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣭⣭⣍⣙⣡⣭⣵⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣽⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⡏⣼⠋⣭⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⢷⡽⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⢻⣦⣿⣷⣿⣿⣻⣭⣭⣟⣿⣿⣿⣿⣿⣷⢣⣷⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠛⢿⣿⣿⣿⣿⣿⣆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢿⣿⣿⣿⣿⢟⣿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⡿⣣⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⢱⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣿⣷⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⡟⠈⠙⠻⢿⡆⠀⠀⢰⡿⠟⠋⠁⠹⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣯⣀⡀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠻⠟⠛⠋⠁⠀⠀⠀⠀⠈⠙⠛⠻⠟⠋⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠠⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡏
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇
⠀⠀⢀⡔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⣇
⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢧⡘⣆
⠀⠀⠀⠀⠀⡠⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⡀⠀⠀⠀⠀⡀⠀⠀⠈⢳⡘⣧
⠀⠀⠀⡀⠊⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⡦⠶⠶⠦⣤⡀⠀⣠⡶⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⢦⣀⡾⠋⠀⠀⠀⠀⠈⢻⣤⣿⣤⣀⠀⠀⠀⣨⣧⣀⡀
⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠉⠻⣦⠟⠉⠀⠈⠙⠳⣄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⣿⠀⠀⠀⣠⡶⠋⠉⠉⠛⢦⠀⣠⣤⠤⠼⣧⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠹
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠘⢧⣀⣠⡏⠀⠀⠀⠀⠀⠈⡟⠉⠀⠀⠀⠀⠙⢧⡀⠀⢀⡇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣧⠀⢀⡴⠋⠀⠘⣇⠀⠀⠀⣀⣠⣄⡁⠀⠀⠀⠀⠀⠀⠈⣧⠴⠾⠷⠦⣄⠀⠀⠀⢠
⠀⠀⠁⠀⢀⠀⠀⠠⠠⠄⠂⠂⠒⠂⠉⠀⣀⠀⠀⠀⠀⠙⢦⣼⠁⠀⠀⠀⠈⠳⣴⠛⠁⠀⠀⠉⠳⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣤⠶⠋
⠴⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠄⢻⡄⠀⠀⣠⡤⣼⡇⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⣄⣼⡃⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠂⠀⠀⠀⠀⠙⢳⠞⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠈⠙⢷⡄⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⡋⠙⠳⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡴⠖⠂⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⠀⠀⣀⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⣴⠞⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢛⣡⣤⣤⣀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠉⠀⠀⠀⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⢻⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⣀⠀⣀⡀⠀⠚⠉⠛⠦⠀⠀⠀⠀⣤⣾⣁⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣠⠞⠉⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⣤⠴⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⣤⠶⠛⢉⡿⠒⠛⠁⠀⠀⠀⢤⠀⠰⠞⠳⢤⡀⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⣹⠷⣄⣀⣀⣀⣴⢿⡄⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠛⡟⠀⠀⠉⠉⠉⠀⠀⢹⡶⣴⡶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⣀⣀⣤⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣄⠀⢹⡆⠀⠀⠀⠀⠀⢀⠀⠀⢠⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⠇
⠀⠀⠀⠀⠀⠈⠋⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⢹⣆⣀⣠⠄⠀⠀⡀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⠈
⢀⡤⠈⠀⢀⠠⠀⠈⠛⠷⠴⢤⡀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠻⣿⣧⣴⣴⣤⡽⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠁
⠋⠀⠀⡔⠉⠀⠀⠀⠀⠀⠐⣆⠹⣆⡀⠀⠀⠀⠀⠀⠸⣦⣀⠘⢟⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣶⣶⣤
⣄⡰⠋⣶⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⣷⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣿⣄⡰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⢀⣾⠃⠀⠀⠀⠀⠀⠀⢠⠂⢠⣿⠟⠋⠀⠀⠈⠉
⣾⠁⠂⣄⣰⠖⠂⠀⢀⣲⣶⣾⣟⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣯⣿⠶⠖⠂⠀⠀⠀⠒⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⡇
⠙⠃⣼⡿⠛⠛⠛⠛⠋⠉⠀⠀⣿⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡿⢿⣇
⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣆⠀⠹⡄
⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠹⣶⣶⣋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⢠⣶⣿⣶⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⢿⣿⡿⠟
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣐⠟⠀⠀⠀⠀⠀⠀⠀⢀⢞⡾⢀⣴⡿⠿⠛⠛⠛⠻⢿⣷⣄⣀⣢⠤⠖⠈⠉⠁⢀⠀⠉⢙⣧⡀⠀⠀⠀⠀⠀⠀⢻⣶⣄⣽⣏
⣿⣷⡶⠶⠂⠀⠀⠀⠂⠁⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⣌⣸⣿⣼⠏⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠁⠀⠀⣀⣤⠴⠀⠀⠀⠾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿
⠿⠿⠷⢶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣣⠁⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⣃⠴⠒⣺⠉⠀⠀⠀⠀⠀⠀⠘⠷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃⠄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⡆⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠀⠀⣿⡀⠀⠀⠀⠀⠀⡀⠀⠐⠃⣠⣴⣷⣴⠷⢦⣀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⣀⠀⠀⠀⠀⠀⢹⡀⠀⢿⣿⣏⠉⠉⠈⠀⠀⣀⣤⣷⠿⠋⠙⠛⠷⣤⣬⣉⣀⣲⣦⣤⣿⡇⢀⣿
⠤⠄⣀⠀⡀⠀⣠⣾⡿⠿⠿⣿⣿⣆⠈⣵⣦⠀⠀⠀⠸⣷⡀⠘⣿⣿⣿⣿⣤⣽⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣿⣹⣧⣼⠇⠀⠀⠀⠀⠀⣤⠤
⡀⠀⠀⠈⠳⢾⡿⠁⠀⠀⠀⠀⠈⠹⣧⣼⣆⢂⠀⠀⠀⣿⣷⣄⣿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⠀⣼⣷⡀
⠉⢓⡆⠦⣀⡦⣦⠀⠀⠀⠀⠀⠀⠀⢹⠟⠙⣍⡄⠀⠀⠘⢿⣿⣿⣇⡰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡍⠠⠋⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠘⠋⠀⢌⢳⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⡾⢠⡄
⠴⢛⣦⣿⣦⡈⠃⠀⠀⠀⠀⣀⣀⣀⣾⡅⠀⠻⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠟⠷⠶⠀⠀⠀⠀⠀⠤⠴⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡁
⠶⠛⠋⠉⠙⠻⣶⣤⣄⡀⠀⣀⣈⣽⡿⠁⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⣄⠀⠀⠀⠀⠀⠀⠐⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣷⣤⣶⣤⣬
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠿⢿⣿⠇⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣭⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢿⣦⡀
⣤⣀⡀⠀⠀⠀⠀⢠⣤⠾⣷⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠂⠀⠘⠿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿
⡀⠀⠠⠤⠀⠀⠀⠀⢀⣴⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣷⣦⣤⣀⡽⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠡⣌⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠉⠑⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣼⣿⠟⠋⠁⠀⠉⠛⠟⠠⢀⠔⢪⡄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡐⣿⠁⠀⠀⠀⠀⠀⠀⣿⣱⡏⡠⢈⠁⣀⣀⠀⠀⠀⣀⣀⣸⣿⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠁⡾⣴⣯⣄⣠⣥⣴⠾⠟⠿⣭⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠃⠀⠂⠀⠀⠈⠀⠐⠀⠃⠀⠀⠀⠂⠘⠀⠀⠀⠘⠀⠂⠐⠈⠠⠀⠄⠐⠁⠁

⢠⠤⠤⠤⢤⠤⠤⠤⠤⠴⠶⠄⠐⠀⠂⠀⠀⠀⠀⠰⠶⠶⠤⠤⢤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢀⠀⡀
⠀⢀⠔⠀⠀⢠⠦⡤⠶⡆⠀⣇⠤⣤⣴⡆⣶⠀⠠⡤⣤⣄⢀⣤⠀⠁⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⡉⠉⡉⠛⡋⡉⡉⡍⠉⠁⣄⣉⡈⠁
⠀⠀⢀⣤⡄⣈⢀⣀⣀⠀⠠⣧⣤⣄⠀⣀⣀⣰⠆⠀⠀⠁⠀⠀⠀⠂⠀⠀⢸⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠉⠃⠈⠀⠉
⠀⠠⡀⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⣀⣀⠉⠉⡁⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⢸⢼⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⢀⣀⠀⠰⠶⠇⠀⠐⠆⠈⠙⠛⠙⠋⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠈⢻⣄⣠⣤⣤⣄⠀⠘⢿⠃⠢⠀⠂⠖⠶⠀⠸⠐⠳⠇⠛⠀⠃⢀
⠀⢀⡤⠠⠀⠀⠒⢤⠠⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⣼⣿⣿⢿⣿⣿⣿⣀⣴⠞⠁⣀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠉⣦⠀⡄⢠⣠⣌⣀⠀⣀⡀⢀⡀⣀⠀⣀⣀⣀⣠⠤⠤⠤⠤⠤⠠⠤⢤⡈⠘⠀⢠⡿⣿⣿⣾⣿⣿⣿⠇⠹⣆
⠀⠀⠩⢄⣵⠴⡤⠆⠬⠀⠨⢤⠊⢡⠨⢼⠋⠍⣀⣀⣀⣠⡤⢤⣤⣴⢟⣷⡇⡆⠀⢿⡀⠛⠛⣿⣿⣿⡗⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀
⠀⠀⠀⢀⡄⠀⠁⠀⠈⠀⠈⠙⠀⢸⠈⣹⡆⠀⣬⡡⠀⣤⣶⣿⠟⢋⣨⣽⡇⡆⠀⠈⠛⠶⠟⢹⣿⣿⣷⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⢸⡄⠄
⣉⡂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢹⣳⢅⣿⣴⠚⠛⣫⣵⣾⡿⢿⠽⡇⢃⡀⠀⢚⠀⠀⣺⡟⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢄
⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⢟⠀⠈⠈⠉⠉⠉⠉⣀⣀⣠⣤⡇⢸⠄⠀⠑⠀⢠⡿⠀⠀⢸⡇⠀⠀⠀⢰⣀⠀⢀⢠⣍⠀⠀⠀⠀⠁⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠑⢓⣶⠀⠉⠉⠁⠀⠀⠹⣈⡇⢸⠁⠀⠀⠀⣾⠁⠀⠀⢸⡇⠀⠀⠀⠸⡃⢥⠑⠈⣉⠤⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⢥⡔⢧⣤⠤⠄⢲⡉⣼⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⠀⠀⠀⠀⢹⡆⠀⠀⠘⣷⠀⠀⠀⠀⠀⠈⠉⣠⢤⡀⠆⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⡴⠋⢸⠀⠀⠀⠀⠀⠀⠀⢸⠆⢸⡀⠀⠀⠀⣼⠇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠑
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡘⠒⢒⣒⣈⣉⣁⣀⣀⣇⣀⣀⡀⠀⠤⠤⠤⠼⠀⠀⠇⣀⣀⣀⣀
⠤⠤⠤⠤⠀⠀⠤⠤⠤⠤⠭⠭⠥⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⡆⠐⠒⠂⠤⠤⠤⠭⠭⠭⠍⠉⠉⣉⠉⡉⣉⡉⣉⢉⣉⣉⣉⣉⣉⣉⡉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⢀⠀⢠⣀⣀⠀⢠⡤⣀⡀⠀⣶⠄⡄⠴⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣄⣄⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠘⠂⠀⠀⢀⣀⣠⣤⡴⣖⣿⣿⡟⡋⠀⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠐⠒⠒⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⢛⣽⣾⣯⣿⣿⣲⣶⠂⣿⣿⣿⡄
⠀⠀⠀⣀⡠⠤⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠁⠀⢿⣾⣿⣿⣿⣇
⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⡏⠉⠀⢀⠀⠀⠀⠀⣒⣤⡲⠶⡀⠈⠁⠀⠀⠀⠘
⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⢶⣶⡄⣇⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⢱⣠⣶⡓⠠⡀⠀⢰⣭⣿⣿⡆⣘⠀⠀⠀⠀⠀⠀⡇
⠈⢆⠀⠀⠀⠀⠀⣠⣤⡀⠀⠒⢄⠀⠀⣏⠀⠹⣾⣿⠇⢸⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠈⣷⣾⣿⣠⠇⠀⠀⠙⠛⠋⠉⡀⠑⠀⠀⠀⠀⢠⠇
⠀⠘⡄⠀⠀⢠⣾⣈⢿⣷⠀⠀⡸⠀⠀⠀⡉⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⢳⠀⡀⡀⠄⠐⠊⠉⠉⠀⠁⢸⡆⢀⣀⣤⠤⠊
⠀⠀⠹⡀⠀⠈⠻⠿⠿⠿⠔⠊⠁⢀⠠⠀⠸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢿⣭⣄⣴⣶⣶⠶⠤⡴⠚⡻⠉⢧⡀
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢷⠒⠒⠁⠀⡇⠀⢸⠳⡄
⠐⠈⠁⠀⢣⡀⠀⠀⠀⠀⢀⣀⡀⢤⠔⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⠀⠀⠀⡗⠐⢸⠀⢹
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⢸⡀⢸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠾⠛⠓⠶⠶⠶⣤⣄⣀⠀⠀⠀⠀⣀⣤⡴⠶⠟⠛⠛⠛⠛⠓⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣴⠞⠋⠁⠀⠀⠀⠐⠂⠀⠃⠀⠀⠀⠀⠈⠙⠷⣄⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⢠⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⢀⣀⣀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣠⣤⠶⠶⠶⣤⣄⡀⠀⠀⠖⠀⠀⠀⠀⠀⠙⣷⠟⠋⠉⠉⠉⠛⢶⣄⡀⠀⠀⠀⢀⠀⠀⠘⣷
⠀⣾⠁⠀⠀⠒⠀⠀⠀⠀⠀⣠⡾⠋⠉⠀⠀⠀⣼⠋⠈⠙⠷⣄⠀⠀⠀⠤⠤⣀⡀⠘⣷⠀⠀⠀⠀⠀⠀⠈⢳⣄⠘⠀⠉⠀⠀⠀⢸
⣸⠇⠀⠀⠀⠀⠀⠒⠀⠀⣼⠏⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠘⣷⡀⠀⠈⠭⠭⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠠⠆⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠋⠀⠀⢰⡟⣷⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠰⠆⠀⣸
⣿⠀⠀⠀⠀⠀⠒⠀⠀⠀⣷⠀⠀⠀⠀⠀⢰⡏⠀⠀⠄⠀⠀⢸⡇⣿⠀⠀⠀⢀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⡿
⢻⡄⠀⣠⢆⠀⠀⢀⡀⠀⠙⣧⡀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠈⣷⡿⠀⠀⠀⠈⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⣰⠟⠀⠀⠐⠀⠀⣼⠃
⠘⣧⠈⢡⠏⣠⠀⠀⠀⢀⠀⠈⠳⣤⡀⠀⠀⢿⡀⠀⠉⠀⠀⠀⠈⣧⡀⠀⠚⠀⠀⣀⢀⡿⠀⠀⠀⢀⣠⡾⡋⠀⠀⠀⠀⠀⣼⠃
⠀⠹⣇⠉⠰⠃⠀⠀⠀⠈⠀⠀⠀⠈⠙⠷⣦⣼⣧⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⢠⡀⠀⣼⠃⣀⣤⡶⠛⠁⠀⠁⠀⡄⠀⢀⡾⠃
⠀⠀⠙⣦⡀⠀⠀⢀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠘⠀⠀⠙⠳⣤⣀⣼⠟⠛⠉⠀⠀⠀⢀⡀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠈⢳⣄⠀⠈⠀⠀⠀⣀⠀⠀⠀⡤⠀⠀⠀⠘⢷⡀⠀⠀⠀⢀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣀⠀⠀⠈⠁⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⢦⡀⠀⠈⠀⢀⠀⠀⠀⠒⠀⢀⠉⠉⠛⠛⠛⠳⠿⠷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠙⠷⣄⡀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢤⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣶⣴⠟⠉⠀⠀⠈⠀⠀⠀⠀⠀⠈⣻⡷⣤⣀⠀⠀⠀⠀⠀⠠⡄⢳⡘⣆⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠆⠀⠀⠀⢠⠀⠀⠀⠀⣴⠿⠶⠶⠿⠛⠶⣦⣄⣀⡀⠙⠀⠁⠈⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠤⠀⠀⠀⡀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠈⠁⠀⣀⣴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣶⣶⣶⣶⣞⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠠⠠⣀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠉⠉⠛⠛⢛⠒⠢⠤⡀⠀⠀⠀⠀⠀⣠⡤⠶⠲⢤⣄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⣀⣠⣤⣨⡾⠳⠦⣄⠀⣏⢮⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣐⣈⣆⠀⢀⡤⠻⢿⡄⠀⠀⢸⡿⠓⠦⡀
⠀⠀⠀⠀⠀⠀⢿⡦⡊⠁⠀⠉⠟⠛⠀⡀⢀⠸⡆⠸⣷⠇⠀⢠⣴⣟⣗⡆⠀⠀⢨⣽⣿⠋⢀⡾⠆⠀⠀⠃⠀⠀⢸⠀⠀⣠⡿⡄
⠀⠀⠀⠀⠀⠀⠈⣷⡇⠀⠀⡸⢀⠀⠀⠈⠳⣿⠂⠀⢨⠀⠀⠀⠈⠏⠁⠀⠀⠀⠀⣹⠏⠀⣎⠀⣈⡶⠤⠾⠶⠶⠛⠒⠾⢧⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⣠⣜⣇⣑⣐⣀⣤⠟⠀⠀⠘⠳⢤⣤⣴⠀⢀⣀⣠⣤⡬⠏⠀⠀⠰⠜⠁⠀⠀⠀⢀⣀⣀⣀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣴⣾⠛⢻⣿⣮⡵⣄
⠀⠀⠀⠀⠀⢀⡤⠤⡤⠀⡶⢤⣤⠖⠒⣶⢶⣶⠶⠋⢉⣩⣟⡲⠚⠉⣹⢒⣳⣶⠾⠉⠓⠒⢆⠀⢠⠓⠒⠉⠉⢻⣿⡟⠉⠙⠇⠀⢷
⠀⠀⠀⠀⠀⣏⡀⠠⠴⠋⠛⠁⠤⠒⠛⠛⠉⠀⠐⠚⠙⠉⡉⢀⣒⢉⡋⢉⡉⠀⢀⠀⣀⢄⡼⠀⠸⡀⠀⠀⠀⠟⠀⠙⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠙⠛⠛⠒⠓⠒⠓⠻⠿⠶⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠈⠒⠶⣤⣤⣀⣤⣤⣴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⢒⡲⠶⠶⠖⢒⣢⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣠⠤⣤⣀⣀⣀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣎⢀⣷⡦⣰⣯⣄⣀⡴⣮⢬⣉⣀⣀⣀⣱⡄⠀⠀⢀⠤⠂⠉⢡⣫⠟⠋⢁⣴⡫⠿⡋⢀⡴⡿⠟⠋⠁⠢⡀
⠀⠀⠀⠀⠀⠀⣾⢸⡟⠻⣆⠙⠽⢧⡌⠛⠷⣌⣀⠈⠳⢤⡈⠉⠙⢦⠀⢺⣠⠀⠀⠟⠁⠀⠀⣸⡿⠁⠀⠀⠚⠉⠂⠀⠀⡀⠩⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣄⣘⣤⣄⠈⠻⡀⠀⠈⠛⢅⡂⠀⣙⣦⠴⠋⠀⠀⠙⠷⣆⣀⠀⢀⣤⡅⠀⠀⢀⣀⣀⠀⣀⣤⣯⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠓⠒⠚⠑⠒⠛⠚⠉⠉⠉⠁⣀⣀⣀⣀⣠⣤⣀⡈⠉⠉⠙⠛⠒⠚⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⢤⡀⠀⠀⠀⠀⠀⠀⡠⠒⢗⠖⢛⠍⡭⠊⢡⠔⠂⠈⠛⡆⠀⠀⠀⠀⠀⣀⣤⡤⠀⡒⢢⣤⢄⡀
⠀⠀⠀⠀⠀⠀⣰⢇⡴⠊⠛⢄⠙⣆⠀⠀⠀⢀⠎⢰⠫⠊⡠⣣⣿⣴⠞⠁⠀⠀⠁⢠⠃⠀⠀⢀⢔⠟⢛⣽⠛⡄⡦⢠⢘⡦⠙⣆
⠀⠀⠀⠀⠀⠀⡏⣼⢇⣀⡤⠤⠶⢿⡀⠀⠀⣾⢾⡇⣾⠸⡃⡇⠘⣷⠀⠂⠀⠀⠂⠘⠀⠀⣴⠃⡆⢠⡋⣂⡬⢔⡥⠂⢼⡗⠀⢼
⠀⠀⠀⠀⠀⠀⣇⠸⡃⣌⣉⣓⣶⠶⡈⢦⠀⠹⣌⠣⡱⠄⠳⡘⢷⡝⡆⠀⠠⠀⠀⢰⠀⢸⠁⡀⠑⢀⣀⣀⣠⠵⠂⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠘⢦⢷⣅⠙⠈⠀⠀⢸⠋⢧⠀⠈⠱⡌⣎⢆⠘⡀⠃⡇⠀⠀⠀⢁⡎⠀⠘⣷⡈⠲⣦⣤⣤⣤⣤⡤⢴⣾⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡁⠗⠰⠖⢁⡰⠃⠀⠀⠀⠸⣌⣘⡄⡇⡄⢿⡄⠀⡠⠊⠀⠀⠀⠈⠑⠲⠼⠿⠾⠷⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠓⠋

⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⢰⣟⡺⢭⣍⡙⠿⣍⣉⡛⠶⣍⣉⡛⠣⣝⢛⠓⢬⣍⠛⠯⣽⣷⠀⠀⣀⠴⡦⠞⠋⡷⣄⠽⢷⣶⣆⣤⡀
⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠘⢿⣶⣶⣤⣶⣦⣤⣤⣶⣾⣿⣿⣿
⠀⢀⣴⡒⡂⠀⠈⠴⠀⢐⣲⡄⢀⡄⢀⠀⠀⢠⣴⡄⠀⠀⣤⢶⣄⠀⠀⠀⠀⠈⢁⠉⠉⠭⠩⣉⡉⠁
⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⢨⠀⢸⣧⣼⡇⠀⡗⠨⣷⠀⠀⠉⢀⣹⣆⠀⠀⡠⠊⠀⠀⡀⠀⠐⠀⢉⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡇⢸⡥⠀⣿⠀⢀⡔⠁⠈⣿⠀⢀⠁⠀⣀⢀⠜⠁⢄⡴⡈⠀⠉⠄
⠀⠀⠈⠒⠒⢀⣁⣀⠀⠚⠃⠀⠀⢹⢻⠃⢸⠀⠀⣿⠀⠈⢀⠴⡊⣿⠀⠏⠀⠀⣨⠁⢦⡠⠃⠈⠣⡀⠀⠠
⠀⠰⠈⠠⠴⠿⣿⢁⣭⠟⢳⣶⣄⠀⢠⠄⢸⡞⠀⣻⠀⠰⠃⢀⡠⢿⠀⢠⡄⠸⠃⠀⡴⡀⠢⣄⠀⠈⢸⠏
⠀⠀⠀⠀⠐⠠⡇⡌⠺⢋⠟⣟⡿⠟⠀⠀⠀⣇⠀⣿⠀⠀⡔⠁⢐⡿⠀⠀⠻⣷⣦⠞⠀⠀⠀⠈⣗⣵⠟
⠀⠀⠠⢀⣤⣤⣬⣧⣧⡟⠛⣛⣇⣒⠐⠶⠀⠸⣄⠏⠀⠀⠦⢐⠽⠃⠀⠀⠀⠈⠛⠿⠶⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⣻⠟⢿⢿⣿⣿⣿⣿⡏⠿⠇⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣺⣿⢀⣠⢸⣿⡧⣹⣿⣿⢸⣄⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠟⢛⣋⣜⣛⣛⣋⡈⣥⣼⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣠⢢⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠈⠁⠈⠉⠙⠛⠿⣷⡄⠀⠀⢠⣾⡿⠟⠋⠉⠁⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠃⠀⠀⠀⠀⠘⠻⠿⢿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣿⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⡠⠊⢸
⠀⠀⠀⠀⠀⠀⠀⣎⡁⣅⣀⡸⠛⠢⡤⠻⡇⠀⢢⠀⠠⠀⠀⠀⠀⠀⢀⠄⢘⠤⠠⢁⠀⠦⣮⠆⠑⢼⡄
⠀⠀⠀⠀⠀⠀⡆⠀⡌⠀⠀⢱⠀⢀⡿⠀⠁⢤⠐⢢⠀⠁⠀⠀⢀⠎⢘⠄⣎⠀⠀⡎⠀⠀⢘⡆⠀⢲⠋⠀⠀⢆
⠀⠀⠀⠀⠀⢄⢸⠁⠀⡤⠤⣞⠀⠈⠻⠄⣠⠞⠀⡜⢢⠀⠀⠀⠀⠁⠺⡀⢸⠁⠀⠈⢠⠤⠎⣇⠀⢀⡇⢀⢀⠎⠀⢱
⠀⠀⠀⠀⠀⡆⢠⠃⠺⣀⠀⣸⠆⠐⢄⣠⠿⣄⠠⣇⠰⠀⠀⠀⠀⠀⠀⠀⢈⠄⡀⠠⡆⠀⠀⢹⠁⢹⡄⢨⡇⠀⣱⠁
⠀⠀⠀⠀⠘⡀⡨⠒⠞⠁⠙⣧⡀⢀⣸⣁⣀⢼⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢣⢰⡁⠀⠑⡄⠄⠃⢠⠈⠀⠣⢱⠐⠁
⠀⠀⠀⠀⠀⠀⡱⢤⠊⠀⣾⠁⠈⡟⠁⠈⡧⠰⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡐⠁⠀⢢⠠⠎⣀⠠⠋⠁
⠀⠀⠀⠀⠀⠀⢇⣀⡦⢴⠁⠀⣏⡩⠄⡈⢹⠖⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠒⢧⠀⠀⠜⠀⠀⡸⠘
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⡓⠂⠑⣄⢀⡏⢹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣰⠋⠈⢹⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠱⠄⠰⠁⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⡢⠀⢻⠄⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡝⠂⠺⣀⢠⡏⠀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠐⡁⠀⡸⢀⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⣸⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡸⠀⢤⠎⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⡟⠉⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⢦⣤⡠⢤⡤⣄⢠⢴⠀⠀⠀⠀⠀⢀⡤⠔⠛⠈⠢⢀⢀⣀⡀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠉⠀⠀⠀⠀⢀⡠⠾⠀⠀⣀⣤⠞⡩⠉⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢄⣀⡴⠖⠁⠀⠀⠉⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⡼⠋⠀⠀⠀⠀⢀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⡤⠚⠉⠀⠀⠀⠀⣀⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⢀⣀⠀⠀⠀⠀⠀⠈⠢⢀⡀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠙⠉⠉⠙⠂⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠉⠒⠲⢤⣀⣤⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⢰⣀⡀⠀⠀⠀⠐⠀⠀⠀⢈⢣
⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠉⠓⠆⢭⡠⠆⠒⠐⠰⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠄⠀⠐⠀⠈⠉⠐⡉⠀⢀⡀⣈⠔⠁⠀⠀⠀⠀⠀⠙⠲⡅⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⣤⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⡤⠀⠀⠀⠀⠀⣀⠀⣄⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠈
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠋⠉
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠟⠛⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡤⠒⠂⠘⡿⠋⠉⠉⡩⠋⠀⠈⠉⠭⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣦⣴⣶⣶⢖⣶⣲⣿⣿⣿⣶⣤⡀
⠀⠀⠀⣠⠚⠉⠈⠀⠀⠘⠀⠀⢀⠎⠀⠀⠀⠀⢀⠀⣤⣄⣨⣧⠀⠀⠀⠀⢀⣤⡞⠛⢿⡿⢻⣟⣿⡟⣵⣿⣿⣿⡿⠿⢟⣛⣛⣿⣦
⠀⠀⠀⡿⠁⠀⠀⡀⡀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⣿⣿⣿⢷⡀⣆⣿⢫⡇⣼⠿⣿⠟⠉⢐⣫⣭⣿⣿⣿⣿⡀
⠀⠀⠀⣇⠀⠠⡀⡱⠳⠇⢀⣠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⣿⣬⣍⡀⢻⣿⣿⢸⢰⡇⣼⣧⡮⢪⣾⣿⠿⢿⣿⣿⣟⡳⡄
⠀⠀⠀⠘⡆⠀⣹⠀⠀⠀⠀⠉⢼⣠⡇⠀⢀⠄⠀⠀⠀⠀⠀⠒⢦⠰⠀⠀⠘⣿⠄⣿⠎⠀⠀⠈⠙⢿⡿⣿⣡⣾⣿⡤⢷⣶⣶⣶⡯⣝⢻⡀
⠀⠀⠀⠀⠈⠐⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⡏⠀⠀⠀⠀⠀⣐⢤⣸⡏⠀⠀⠀⠘⠣⠟⠀⠀⠀⠀⠀⠈⠙⠙⠿⣿⡵⠾⠿⣿⣿⣿⣿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠭⡉⠁⠀⠀⡀⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣞⡿⢷⡒⢭⡙⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠀⠀⠀⠀⠀⣦⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠚⠉⣹⣿⣶⣿⣦⠙⣦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠐⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠷⠿⣿⠿⠿⠛⠉

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠓⣦⠀⠀⢀⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡏⢀⠖⠉⢰⠁⠀⢀⣠⠖⠛⠉⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⠀⡇⠀⢀⣷⠋⠀⠀⢸⢀⡴⠋⠀⠀⠀⣠⠏
⠀⠀⢀⡶⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢳⡀⠀⠀⠀⢰⠃⠀⢸⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⡰⠃
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⡜⠀⠀⠀⢧⠀⠀⠀⣼⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⡠⠤⢄⣀⡀
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢀⠇⠀⠀⠀⢸⠀⢀⡞⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⢸⣤⠋⠀⡇⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠉⠉⠓⠦⣼⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⡤⠤⠤⠤⢄⣀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁⠀⠀⠀⠀⠉⠓⢴⠋⠀⠀⠀⠀⠀⠀⠈⠑⢦⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡄⠀⢀⣀⣀⡀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢳⣄⣤⠤⢤⣄⣀⠀⠀⠀⢳⡄
⡴⠖⠒⠚⠛⠋⠉⣡⠖⠋⠉⠉⠉⠉⠙⡇⠀⠀⣠⡤⠴⠶⠦⢤⣀⠀⠀⠀⢸⣆⣀⣀⣀⠀⢠⠖⠋⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⣷
⠘⢦⣀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢧⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⢦⡔⠋⠉⠉⠉⠉⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⣿
⠀⠀⠀⠉⠑⡞⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢷⠃
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⣸⠓⠦⡀
⠀⠀⣠⠞⠉⢧⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⢦
⠀⡼⠁⠀⠀⠈⢦⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣠⠎
⢰⠃⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠤⠤⢤⡤⠖⠋⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⠒⠊⠁⠀⠑⠦⣄⣀⣀⣀⣀⠴⠚⠁⠀⠀⠀⠈⡇
⠀⢧⠀⠀⠀⠀⠀⠀⢀⡼⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠈⠣⡀⠀⠀⠀⢀⡞⠀⠀⠙⠲⠤⣤⣀⣤⠤⠖⠋⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢀⡷⢤⣀⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠈⠓⠢⠤⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⢀⣀⠴⠋⠲⠤⣄⣀⣀⣠⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠈⠉⢙⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⠤⣤⠒⠊⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣄⣀⣀⣀⡠⠴⠊⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⡀⠀⠀⠀⣀⡤⠷⣄⡀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠉⠁⠀⠀⠀⠉⠓⠒⠒⠒⠚⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠑⠦⣄⣀⣀⣀⣀⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠤⠖⠊⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⣀⠤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⡶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⣿⣿⢿⣿⡜⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⢿⢽⣿⣇⣿⣿⣿⡿⢷⣝⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡼⣧⣻⣿⣷⣿⣿⣤⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⣿⣸⣾⣿⣿⣾⣿⣿⣿⡄⣿⣼⣿⣿⣿⠟⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⣾⣁⢀⣿⣇⣿⣿⡿⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣥⣏⠀⣸⣿⣸⣿⣿⡟⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠋⠉⠁⠀⠀⠙⢯⣛⠛⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠟⠉⠉⠁⠀⠉⢻⡛⠻⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠈⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠠⢤⣄⡀⠀⠀⠀⢀⣠⡤⠄⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠇⠀⠤⢤⣀⠀⠀⠀⠀⣀⣠⠤⠄⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⣀⠀⢈⡉⠀⠀⠀⢉⡁⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⢀⣀⣀⣉⠁⠀⠀⠈⢉⣀⣀⡀⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⠙⠋⠉⠀⠀⠀⠀⠈⠙⠋⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣉⣽⡟⠻⠷⠄⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣽⡟⠻⠷⠄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡉⠉⠠⠷⢶⡄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡉⠉⠀⠷⢶⣄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢉⣉⡒⠒⢲⣤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⣙⡒⠒⢒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠿⠟⠛⣠⣉⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠰⠿⠿⠻⣠⣉⠙⠷⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢾⠿⣿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢶⠿⣿⠻⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠶⢤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢽⡳⣤⡴⢶⣾⠿⠛⠉⠉⠉⠛⠲⠦⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⠀⢀⣤⣶⣾⣿⣿⣷⣷⣄⠀⠉⠺⣦⣭⣄⣄⣀⣀⣀⣀⡀⠀⠈⠀⠙⢢⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣰⣿⢿⡿⠟⠛⠛⠛⢿⣿⣷⣄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⢳⡵⡄
⠀⠀⠀⠀⢠⠇⠂⠀⠀⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡄⠀⢙⡄⠀⠀⠀⠉⠻⣿⣿⣦⡀⠀⢳⣽⡀
⠀⠀⠀⠀⣸⠰⡂⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⡟⠉⢻⣿⣷⠀⢸⣿⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠈⠃⣧
⠀⠀⠀⠀⣿⢸⡇⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠘⣿⣯⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⣿⡗⠀⠀⠘⣿
⠀⠀⠀⠀⢿⢸⣧⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠹⢻⣤⣿⡟⠀⠀⠀⠀⠀⠀⢀⠿⠇⠀⢰⣦⣿
⠀⠀⠀⠀⢸⡌⣿⡆⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⢷⣹⣿⡄⠀⠀⢳⡄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠛⢤⡀⠀⠀⠀⠀⡰⠁⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠈⢧⣿⣿⣧⡀⠘⣿⣆⣀⠴⠊⠙⠉⢀⣾⣿⣿⣿⣦⡀⠘⠛⢶⣤⣀⡜⠁⠀⢀⣾⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣤⡬⠛⠁⠀⠀⢀⣶⣿⣿⠿⠋⠙⠿⣿⣷⡄⠀⠀⠙⣦⣀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣻⣿⣍⡀⣀⣤⣶⣿⣿⣋⡁⠀⠀⢀⣀⣈⣽⣷⣅⢀⣀⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡟⠛⠋⠉⠉⠙⠻⠟⣛⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣶⣶⣶⣴⣶⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠠⠀⢄⠀⠉⠣⡀⠀⠈⡑⠖⠍⠌⠒⠉⠐⠀⠀⢘⡀⠀⠤⡀⠀⠀⠀⠠⠂⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡦⠀⠐⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠈⠑⠂⠒⠄⡀⠀⠈⠢⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢆⡀⠀⠀⡰⡄⠀⠀⠀⠀⠀⠀⠀⠤⣤⡤⡔⠈⠦⡀⠀⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⢀⠴⠋⡄⠀⠹⡄⡔⠁⠈⠒⡠⣤⡠⠐⠃⠀⠀⠀⠀⠀⠉⠢⡠⠟⠉⠑⠂⢤⠀⠊⠁⠙⢆
⠀⠀⠀⣠⠖⠁⠀⠐⠀⠀⠀⠹⢀⠀⠐⠁⠀⠀⠈⠢⠀⡀⠀⠀⠀⠀⠀⠀⠳⣀⡀⠀⠀⠀⠱⡯⠀⡀⠀⢇
⠀⠀⡜⠁⠀⠀⠀⠇⠀⠀⡠⠎⠀⠀⠀⠀⠀⢀⠸⡇⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁⠪⡂⢸
⠀⡘⠀⠀⠀⠀⢰⣀⡠⠄⢀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⡜
⠀⢇⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠁⠀⠀⢔⠉⣄⠀⠀⠀⠀⠀⣇⢀⣇⠠⢀⣀⢀⣠⠴⢊⠗⠢⢄⡀⢀⡠⠁
⠀⠸⡀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠄⠂⠠⠤⠒⠁⠀⠀⠀⠈⠐⠠⣠⠉⠉⠀⠀⠀⠀⢠⠃
⠀⠀⠑⢄⠀⠀⠈⠢⡀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠉⠀⠒⠠⠬⢄⡀⡰⠋⢘⣄⣄⠀⠀⠀⠀⠀⡠⠂⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠊⠁⠘⠃⠀⠄⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠒⠊⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢤⣤⡤⠐⠒⣒⣶⣤⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⡠⣺⠉⣀⠔⡪⢁⡠⢤⣤⣤⣄⣀⣀⡒⠋⢉⣉⠤⠬⢭⣶⠤⣀
⠀⠀⠀⠀⡀⣼⢃⠞⡱⠊⡔⠁⠼⡳⠄⢒⣒⣒⣚⣛⡳⠦⣄⣐⠀⠀⣀⣀⣀⠱⡀
⠀⠀⠀⠀⢇⠀⠁⠊⠀⢸⠀⠀⠤⠒⠉⠁⠀⠀⢀⡠⠤⠤⠤⠬⠍⣒⣤⣀⠀⠀⡇
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠸⠀⠀⠀⠀⠤⠶⠤⠤⠒⠒⠉⢉⣉⣉⠀⠠⢀⣈⡑⠦⡁
⠀⠀⠀⠀⢀⠤⢀⠔⠀⠐⠢⣀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠠⠤⠒⠒⠉⠉⠉⠛⠋⠺⠶⣄
⠀⠀⠀⠀⠇⢠⡁⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠈⠁⠀⠀⠀⢀⡠⠤⠀⠀⢈⣉⣁⣒⠢⢬⡳⡄
⠀⠀⠀⠀⢗⡸⣈⠑⠢⣄⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠘⠋⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠙⠚⢆
⠀⠀⠀⠀⠀⠑⢨⡛⠓⠺⣗⠤⠤⠤⠤⣔⡇⠠⢄⡀⠀⠀⠀⠀⠀⠀⠶⠶⢀⣀⣈⣉⡁⠒⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠈⣟⠿⠿⢭⣴⠞⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⠔⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠶⠟⠛⠋⠉⠉⠉⠉⠉⣉⡭⠟⠉⠉⠉⠉⠉⠛⠛⠛⠶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⠁⣀⠔⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⡁⠀⠀⠀⠀⠠⠤⠤⠤⣀⡴⠞⠉⠀⠀⠉⠙⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⢀⡴⠊⣁⣀⡠⠤⠐⠂⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⣀⣠⡴⠟⠊⠉⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠾⠷⣄
⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⢀⣠⠴⠒⢋⡽⠋⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢀⡴⠋⠀⠀⡴⠞⠉⠀⠀⢠⠞⠀⠀⠀⣀⣀⣠⣤⣤⠴⠥⠤⠴⠶⠒⠒⠒⠒⠒⠒⡶⠚⠀⠤⠤⠤⠤⠤⠀⠀⠀⣠⡾⠋⠀⢀⢤⡀⠀⠀⠲⠀⠀⠀⣿
⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠒⠛⠉⠉⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⢀⣾⠃
⠀⣿⠀⠀⠀⢀⣠⣤⣤⣤⣤⡿⠤⠤⠤⠤⠤⠤⣤⣬⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠐⠊⡗⠀⠀⠀⠀⣠⡞⠁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢾⠀⠀⠀⠀⠀⠠⠄⠀⣠⠟⠁
⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠤⣤⣄⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠷⣤⡀⠀⢸⡇⠀⠀⠀⠓⠂⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠙⠦⠀⠀⠀⠀⠰⣒⠀⠀⠀⠈⣿⣦⠘⣧⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡟⣧⣙⣷⣤⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠚⠀⣸⠃⢸⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⡤⠀⠀⠀⠀⣠⡾⠁⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⡾⠡⠤⠤⠤⠤⠶⠶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠲⠖⠒⠒⠒⢲⠋⠁⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠈⢷⡤⠤⢤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠙⠛⠛⠶⠶⠶⠦⠤⢤⣄⣀⣀⣿⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠛⢛⠛⠶⠶⣚⡛⠛⢿⣿⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣶⠀⠀⠉⠀⠀⠉⠉⠃⠁⠀⢀⣉⣿⣿⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡉⠉⠉⢻⣿⣿⢻⡆⠀⠀⠀⠀⠠⠶⠶⣶⠤⢤⣈⠛⣿⡿⠋⠉⠉⠙⡟⠳⡄
⠀⠀⠀⠀⠀⣀⢀⣈⣷⣿⡆⠙⠿⠿⢿⣿⡏⣿⣦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢘⣻⣿⠟⠛⠻⣶⡄⢠⣿⣤⣤⣀⣀⡀
⠀⠀⠀⢠⠞⡩⠙⠏⢻⣯⣅⠀⠀⠀⠐⢻⣷⣨⣿⠀⢸⡆⠀⠀⠀⠀⠀⠀⣄⣼⣿⡯⠀⠀⠀⢠⡇⣾⣿⡶⠶⢆⣭⣽⡄
⠀⠀⠰⣿⣾⡇⠀⠀⢘⣿⣿⣆⠀⠀⠀⢼⣿⡿⣿⠀⠈⣷⡄⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠎⣴⣿⠏⠀⠀⣘⣿⣿⣿⣦⣤⣄⡀
⠀⣠⣤⣿⣿⣧⠀⢤⡨⣿⣿⣿⣦⠀⢰⣼⣿⣏⣿⡧⠀⢿⠁⠀⠀⢀⡆⢀⣿⣿⣿⣷⡄⣦⢀⣼⣿⣿⠀⠀⢠⣿⣿⠿⠦⣈⠲⣿⣿
⣸⣷⢚⣋⣻⣿⣷⣦⣿⣿⣿⣿⣿⣿⣼⣷⣿⣿⣿⣷⣏⣤⣄⣀⣀⣸⣸⣿⣿⣾⣿⣿⣷⣷⣿⣿⣿⣿⣇⣀⣿⣿⣿⣟⣀⢈⣙⣎⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠈⠉⠉⠉⠛⠿⠿⠿⠿⠟⠛⠿⠿⠿⠿⠻⣾⣛⣛⣿⣻⣻⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠉⠛⠛⠛⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢊⢉⣭⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢸⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⢻⠉⠩⢿⣿⣿⣧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⡇⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠛⠛⠉⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠡⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡄⠀⠁⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣬⠠⠆⠤⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸⠦⠤⠄⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠍⠁⠒⠲⠂⠠⠤⠤⠀⠀⠀⠀⠀⠤⠄⠄⢐⣂⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣾⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣷⣿⡏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣖⣿⣟⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢷⣿⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣜⣿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⡷⡑⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢿⣝⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣿⣿⣿⣿⡓⠁⣴⢠⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⣤⡄⠀⠤⠀⠀⠄⠀⢀⠀⠀⠀⠀⠘⡵⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣦⠘⡿⡟⣿⣿⡄⠀⡷⢸⡟
⠀⠀⠀⠀⠀⢦⢁⣠⣥⣤⣿⣿⣿⣿⡿⠿⠶⣶⣬⣀⠈⠢⡀⠀⠀⢱⢬⣷⡀⠀⠀⠀⠀⠀⠀⠴⣬⢣⡘⣴⣇⡏⠐⠀⠀⣷⣿⠇
⠀⢀⢔⠁⢠⣶⣿⠿⠛⠋⠉⠈⠙⣿⣧⠀⠀⠀⠂⠀⠀⠀⠈⠢⠀⠘⡇⢿⣥⣤⡀⣄⡀⣤⣰⢿⣿⣿⡝⣦⣻⣷⡀⢀⣀⡿⠟⣠⣶⣽⡿⠛⢁
⠀⠀⠀⢀⡟⠋⠀⡀⠀⠀⠀⠰⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣗⠺⡷⠿⣿⣿⣝⣿⣿⢾⣿⣿⣿⡿⢹⠟⠋⠉⢈⡩⣿⠛⠛⠺⠯⡉
⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠉⠙⠒⠶⠥⠬⠭⣽⣮⣭⣿⣟⡂⣿⣶⣶⣾⣿⣿⣿⣿⣷⣤⣤⣬⡆
⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣟⡇⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡋⠈⠛⠻⠿⠿⠝⠷
⠸⣏⣀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡠⢴⣺⡿⠁
⠀⠘⢿⣿⡍⠑⠤⢀⣀⠄⠀⠀⠀⠀⠘⠛⠛⠀⠀⢐⣾⣿⡯⠟⠁⠀⠁
⠀⠀⠀⠈⠙⠒⠲⠤⠤⠤⣤⠤⠤⠔⠒⠒⠒⠂⣙⣉⣉⡓
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⢉⠉⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢀⠶⠚⠋⠀⠀⢀⣀⣤⣄⣀⡈⢳⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠅⠂⡰⠁⠀⠔⠉⢀⠞⠃⠀⠒⠂⠉⠙⠣⣄⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⣠⣇⡀⠀⠀⢠⡏⠀⡠⠂⠁⠀⢀⠀⠀⠀⠑⢄⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⢀⡄⢿⠁⠀⠙⢦⠁⣸⠀⡰⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⢧⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠀⣆⠈⢧⡀⠀⠘⣅⠟⢠⠁⠀⠀⠀⠀⠀⠊⢀⠀⠀⠀⠈⡗⢄⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠤⡇⡔⠀⠔⢹⡄⠀⠙⢦⣄⣹⡀⠀⠤⠤⠤⠄⠐⠂⠉⠀⢀⣀⣀⠘⠃⠈⡄⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣀⠀⠀⠀⠀⠀⢄⡇⠀⡄⠀⠰⡀⢂⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⢀⡄⠀⠀⡟⠢⣀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⣠⠷⠐⠈⠁⠀⣀⣀⠠⠤⠚⡰⠀⠡⠀⢆⠹⡀⠢⠀⠀⠀⠈⠙⠲⠦⣤⣤⠐⠒⠑⢉⡠⠊⠠⡄⠀⡇⠀⠈⠷⣸
⠀⠀⠀⠀⢀⠔⠊⠀⢀⡠⠔⠊⢉⠄⠀⠀⠀⠀⠱⡡⡀⠁⠀⠀⠘⢄⠈⠂⡀⠂⠤⢀⡀⠀⠈⠉⠉⠛⢥⠀⢀⠀⠘⢦⡁⠀⠀⠀⡟
⠀⠀⢀⡔⠁⠀⠔⠊⠁⠀⠀⠔⠁⢀⡀⠤⠔⠒⠈⠱⡐⢄⠈⠀⠀⠀⠑⢄⠀⠐⠀⠀⠀⠉⠒⢤⡀⢀⠈⠇⠸⠀⠄⠈⢻⣀⣠⠜⡃
⠀⠀⡮⠀⠀⢀⡀⠀⠀⠊⣀⠤⠘⠈⠀⠀⠀⢀⡤⠚⠉⢦⡑⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠙⡀⠃⢠⠇⠀⢈⠀⠈⡍⠀⡸⠁
⠀⠀⢆⣠⠞⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⢦⣑⠀⠀⠀⠀⠀⠄⠀⢀⡠⠆⠈⠀⢁⣴⠏⠀⠀⡜⢀⠀⡷⠊
⠀⠠⠊⢀⠤⠂⠀⠀⠀⠀⢀⠠⢂⡤⢲⠏⠀⠀⢀⠄⠀⠀⠀⠀⠈⠉⠒⠶⡖⠲⠦⠍⢥⡤⠴⠖⠉⢁⡜⣀⡤⠊⢀⠎⢠⠁
⡌⢀⠔⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⡰⠃⠀⠀⠠⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣤⣀⣀⣠⣤⠖⠋⠀⠠⠴⠖⢁⡴⢻⡀
⢣⠃⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⡴⠁⠀⠀⡠⠁⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⡄⠃⠉⢲⠒⠒⠖⠚⢋⢰⠀⡇
⠀⠓⠀⣤⠖⠉⠣⣀⣀⣀⡤⠊⠀⠀⢀⡜⠁⠀⡰⠁⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠇⠀⠀⢸⠇⢸⠀⡘⠀⠀⠀⠀⢸⢠⠀⡇
⠀⠀⠀⢀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡴⠁⠀⠀⠀⡀⠐⠉⠀⠀⠀⢀⡜⠀⠀⠀⡾⠀⠀⡰⠁⠀⡠⠂⠀⡘⠀⡸⠇
⠀⠀⠀⠈⡗⡀⠢⢀⣀⣀⣠⡄⠊⣀⡀⠴⣊⣀⣀⡤⠶⢉⡀⠀⠀⠀⢀⡠⠊⠀⠀⢀⡜⠀⠀⡟⠀⡠⠊⠁⠀⡰⠁⡰⠃⢰
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠱⡀⠀⠀⠉⠉⠉⠀⠀⠀⣀⠴⢋⡠⠃⢠⠀⠁⠀⠀⢀⠜⡡⠚⠁⠀⠸
⠀⠀⠀⠀⠸⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠱⡀⠀⠀⠤⠤⠴⠒⢉⠡⠖⡁⠄⠠⠀⠀⠀⠀⠂⡡⠊⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠰⠀⢱⡀⠀⠀⣀⣀⠉⠠⠐⠈⡠⠂⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡇⠀⢳⡀⠀⠀⠀⣠⠑⠢⡌⠱⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇⠀⠀⠱⡀⠀⠀⠉⠒⠀⢸⢀⠀⠀⠀⠀⠈⠐⠲⠒⢊⡩⠃
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠁⠀⠀⠀⢡⠀⠰⠖⠤⢀⢜⠎⠀⠀⠀⠐⠒⢲⠐⠂⠉
⠀⠀⠀⠀⠀⠁⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢇⢆⡀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⢂⠀⠐⢄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⣣⡀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠛⠉⠉⠁⠀⠙⢤⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠋⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣷⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠴⣖⠦⠬⣍⣉⠓⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠌⢉⣉⡰⠖⢦⣤⣀⠁⠒⠨⠍⡒⣄⠈⠛⠄⠀⠀⠀⠀⠀⠰⣿⣿⣿
⡛⠛⣉⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠚⠒⠒⠒⠦⠤⣀⠀⠉⠐⠢⠄⣁⠂⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠉⠛
⠁⢣⣿⡏⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠁⠀⠉⠀⠁⠂⠠⠄⣉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢸⢿⡄
⢠⣿⣿⠏⣾⡀⠀⠀⠀⠀⢠⠓⠀⠀⠀⠒⠒⠢⠀⢀⠀⠀⠀⠀⠉⠁⠀⠀⢀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣷
⢸⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⢀⡀⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣻
⠸⣿⡇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⠀⠤⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠛⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢠⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠂⠀⠀⠒⠀⠀⠀⠀⠀⠤⠐⠀⢠⠄⠀⠐⠂⠀⠠⠒⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠀⠊⠀⠀⡄⠀⠄⠀⠀⠐⠢⠈⠀⠤⠂⠀⠀⠐⠂⠀⠀⠤⠀⢠⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⢀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣀⠀⠠⠤⠄⢀⠀⠀⠚⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢀⠄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠐⢀⠀⠐⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⡆⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢
⠀⠀⠀⠀⠠⠀⠀⠀⡠⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⢀⠀⠀⠠⠄⠀⠀⠈⠐⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⡂⠀⠀⠠
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⢅⠀⠀⠀⠀⠐⠄⠀⠀⢠⠀⠀⠀⢤⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠀⠁⠀⠦⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠰⠀⠀⠀⠀⠔⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⡁⠀⠀⢠⠀⠀⠀⠀⣀⠀⠀⠀⠰⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿