Tap Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⣠⡶⠛⠻⣦⣀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠈⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉⠀⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⡀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠁⠀⠀⠉⢻⡶⠶⠶⣶⠏⠁⠀⠀⠉⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣞⣋⣉⣉⣉⣉⣻⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣴⠶⠶⠶⠶⠶⠶⠾⠛⠻⣿⢦⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⣦⣄⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠿⠶⠶⠶⠶⣤⣤⣤⣤⣤⣿⠶⠛⠛⠛⠷⠶⢦⣤⣤⣤⣤⡶⠶⠾⠛⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠙⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠉⠉⠛⠒⠶⠤⣤⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣠⡤⠴⠖⠛⠉⠁⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⣀⣠⡤⠶⠒⠒⠒⠒⠲⡏⠀⠀⠀⠀⠀⠀⠀⣹⠖⠒⠒⠒⠒⠲⠦⣤⣀⣀⡀⠀⣀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣤⢤⣄⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣤⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠛⠛⠛⠛⠛⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⢻⠻⠷⡶⢶⠦⡤⢤⡤⣤⢤⡬⢿⠷⡶⡶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠛⠒⠷⠷⠾⠶⠷⠼⠦⠶⠴⠧⠾⠴⠷⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣰⠟⠁⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⢰⡏⠀⠀⠀⠀⠀⣸⠟⠶⢦⣤⠤⠤⠴⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⡀
⣿⠁⠀⠀⠀⠀⢰⡏⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⠉⠛⠛⠲⠶⠦⣤⣄⣀
⣿⠀⠀⠀⠀⠀⣸⠁⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⡀
⣿⠀⠀⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⢿⡄⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⢸⣇⠀⠀⠀⠀⠀⢷⠀⠀⢸⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⢻⡄⠀⠀⠀⠀⣸⡷⠞⠋⠉⠉⠉⠛⠛⠲⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣼⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠻⢦⣤⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠓⠒⠒⠒⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠓⠶⢤⣤⣤⣤⣤⣤⡤⠶⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⣦⣤⣤⣤⣤⡴⠶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠤⢤⣤⣤⣤⡖⠛⢶⣤⣤⣤⡤⠤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣤⣤⡤⠼⣆⣀⣼⠧⢤⣤⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠉⢳⣄⣀⣀⣀⣀⣠⡾⠋⠉⠀⠉⠙⢷⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣤⡼⠛⠛⠛⠛⠛⠻⣦⡀⠀⠀⠀⢀⡴⠛⠛⠛⠛⣆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣾⠁⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⣧⠘⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠲⣄⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠈⠉⠉⠉⢹⠀⠀⠀⢨⠉⠉⠀⠀⢀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠄⠒⠒⠚⢦⠤⢠⠼⠃⠒⠒⠂⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠈⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡕⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡐⠀⠀⢀⠈⢷
⠀⠀⠀⠀⠀⢀⠤⠤⠤⣤⠴⠶⢄⢀⣀⣀⡇⢀⡀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⠒⠂⠈⠉⠉⠉⠉⠒⠀⠠⠤⠔⠒⠋⠉⠉⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⣀⠀⠀⠀⢀⡇⢀⠀⠀⠀⠀⠀⡀⠀⠄⠀⢄⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠸⠀⠁⢀⣀⣉⣀⣀⡐⠶⠖⠀⠀⠀⣀⡤⣤⣀⠈⠓⢀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠈⠷⠲⠳⠒⠷⠶⠎⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠀⠙⣆⠀⢱⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⡂⠀⠀⠎⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠷⠦⡄⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣠⡄⢶⡭⠾⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⡔⠉⠀⠀⠐⢄⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⢙⡷⠒⣇⠲⠮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠘⠀⠀⠀⠀⠀⠘⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠘⣄⣀⡤⠤⠲⣖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠙⠂⠒⠂⠁⣸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢃⠀⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠋⠅⠂⠀⠀⢀⣀⡴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠈⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⡤⠖⠊⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠠⠀⠀⠀⠀⠀⢀⣼⡀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠤⠴⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠈⢿⠉⠁⠓⣾⡀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⡄⠠⡄⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢻⠙⠁⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠶⠶⠦⢾⣀⣀⡸⠖⠖⠲⢂⠀⠀⠀⠀⠀⠀⣠⠤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡸⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⣀⠙⠲⢤⡀⠀⠛⠭⠭⢀⣀⣒⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢐⠉⠉⠉⠒⠒⠒⠚⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⢉⠉⠀⠈⠑⠢⢄⡈⠓⠦⣄⡀⠀⠈⡇⠈⠉⡇
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢱⠀⠀⠀⠀⠀⠈⠑⠠⠀⠉⠲⢤⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠶⠶⠿⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢠⡟⠙⠛⠛⠲⠶⣾⠁⠀⠀⢻⡶⠖⠛⠛⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠙⢦⡀⠀⠀⠀⢸⣇⠀⣀⣀⣀⣤⣿⠀⠀⠀⢸⣧⣄⣀⣀⡀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠉⠉⠉⠁⠀⣿⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣰⡖⠛⠒⠒⠒⠚⠓⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠙⣇⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣘⣇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⠦⠤⠤⠾⠛⠛⠶⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠗⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠛⠛⠛⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠤⠤⠤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠐⠒⠒⠒⠀⡔⠉⠉⢲⠀⠤⠤⠤⠄⠀⠐⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡠⠀⢠⣄⣠⣠⣇⢲⣖⢯⣤⣤⠀⡄⠀⠀⢻⡍⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢙⡏⠈⠉⠉⠉⠛⠛⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⢯⠭⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⢀⡀⣸⣁⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⣉⣉⣿⣫⣩⣹⣿⣯⣽⡿⠀⠀⠀⠀⠀⠀⣠⠒⠉⠁⠁⠀⠀⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡑⠋⠉⠙⠂⠀⠀⡀⣽⡇⠀⠀⠀⠀⡠⢤⡁⢃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢀⣀⠀⠠⠤⠠⠤⠤⢤⡊⠓⠀⢠⡄⠀⠀⠀⠒⠀⢤⡹⢦⣤⣤⡴⣧⠀⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⢀⠔⠊⠁⠀⠀⠀⠈⠈⠈⠉⠈⠀⠀⠒⠁⠀⠀⠀⠀⢀⡀⠁⠀⠁⠁⠀⠁⢹⣤⣿⣺⠀⠇⠐⣒⡒⠚⠋⠀⠐⢿
⠀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡄⠀⠀⠀⠀⠀⠀⣠⠀⣈⠇⠋⠀⣤⠀⠀⣴⢸⣷⣿⣤⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⣁⢀⣯⣤⣤⣕⢆⠀⠂⠈⡄⢻⠀⠏⢀⠂⣇⣼⣆⣤⡇⣸⣿⣿⢽⠀⠥⠄⠐⠒⢀⣀⣀⣐⡛
⠀⠀⠀⠀⠀⠠⠴⣼⠟⠉⠀⠀⠀⠈⠑⢦⣄⢀⡀⣀⢀⢀⣦⠞⠉⠀⠀⠀⠙⢼⣿⡏⡟⣨⠥⠈⠁⠀⠀⠀⠐⢀⡆
⠀⢘⠀⠀⠐⠊⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠹⡿⢁⣒⠂⣈⣩⡥⠀⠀⢀⡼⡇
⢠⠸⠀⠀⠰⠶⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠭⠅⠒⠀⢤⣶⣿⠿⡇⢠
⡸⣼⠅⠀⠨⠤⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣾⣧⣾
⡇⢴⠉⠀⠀⢾⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢻⣿⡟⠋
⠈⠓⠤⠤⠬⠗⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⡀
⠀⠀⠠⠉⣦⡀
⠀⡠⡅⠀⡆⡑⢄
⠰⡷⠀⠀⡄⢞⡾⡄
⠐⠆⠀⢠⣶⣶⣟⡇
⠘⢤⡀⠈⢙⣉⠝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⡀⢀⡤⠶⢤⡀⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣠⠽⢯⡀⠀⢀⡽⠯⣄⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⣽⣴⣯⣀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠤⡤⠞⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⠊⠉⠀⠉⠓⣄⣀⣀⣀⣀⣀⡼⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⠶⠶⢦⡀⠀⠀⠀⢀⡴⠶⠶⠶⠶⠶⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠉⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠈⠳⡞⠋⠉⠉⠉⠉⠓⠢⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⡎⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣇⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠒⠓⠚⠉⠀⠀⠀⠘⠳⠤⠤⠶⠛⠂⠀⠀⠀⢀⡜⠀⠈⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠸⣄⣤⣄⣠⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⣾⠇⠀⠀⠀⡸⠁⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⣠⠞⡫⠛⠁⠀⠀⠀⣰⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠰⣇⠀⢀⣀⠴⠊⠀⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡟⠛⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠾⠛⠛⣿⠿⢿⡟⠛⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡿⠶⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠻⡝⠟⠛⠻⠛⠛⠛⢶⣅⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡶⠛⠉⠙⢷⣄⠹⡀⠀⠀⠀⠀⠀⠀⢹⡇⠹⡄⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠈⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢰⡏⢠⠇⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⣴⣾⣃⡼⠀⠀⠀⠀⠀⢀⣴⠿⡶⠾⠶⠶⠶⠶⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠀⠀⠀⠀⠀⣿⣍⠙⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡿⢦⡻⢦⣤⣤⣤⡴⠿⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣉⠓⠒⠲⠒⠒⢋⣡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣝⠛⠛⠛⠛⠛⠛⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⣧⠀⢠⠀⠀⠀⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⠀⢸⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⢸⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣿⠀⢸⠀⢀⡄⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢸⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠘⠃⢺⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⢸⡇⠀⠀⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⡇⡇⠀⠀⢹⡇⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢸⡇⢸⠇⠁⠀⠀⢸⡇⠸⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢸⡇⠘⠀⠀⡆⠀⢸⡇⠀⠐⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⢶⣤⢸⣇⠀⠀⠰⠇⠀⢸⡇⠀⡿⠿⠀⠀⠀⠠⠴⠖⠒⢀
⠀⠀⠀⠀⠀⠉⠉⠀⢀⣴⠿⠆⠀⣀⡀⠁⠀⠀⠀⠀⠀⠞⠁⠤⣄⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠀⠀⢧⠀⠀⣴⠚⠀⠙⠂⢰⣀⠀⠀⠀⠘⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢀⣠⣆⠀⠀⢀⡀⠀⣤⠀⠉⣥⣄⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠙⢶⣤⠾⠃⠀⠙⠻⠛⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢤⠟⠘⢧⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣿⣷⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠛⣦⢠⡞⠓⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⡟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⠾⠷⡄⠀⠀⣴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣦⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠛⠛⠁⠀⠀⠀⠙⠛⠛⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣠⣤⣤⡀⠀⠀⠀⣠⣤⣄⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⡇⠀⠀⠙⠓⠒⠛⠁⠀⠀⠻⠾⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠘⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣝⣫⣍⣉⣏⣩⣟⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠤⣤⣤⣤⣤⣤⣤⣴⣏⣹⣦⣀⣀⣀⣀⣠⡤⠖⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠛⠛⠛⠛⠛⠛⣿⠋⢻⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠚⠛⠋⠋⠉⠛⠛⠒⠶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠶⠶⠶⠖⠒⠛⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⢤⣄
⠀⠀⠀⠀⠀⢀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠖⠒⠒⠒⠶⠶⢶⡀
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠒⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⣇⡀
⣠⠶⢤⡀⠀⣧⡤⠤⠴⠒⠒⠛⠉⠉⡇⠀⠀⠀⠀⠘⠷⠶⣤⠤⠤⢤⡤⣤⣤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛
⢷⡀⠀⠙⣆⡟⠀⠀⠀⣀⣠⡤⠴⠶⡇⠀⠀⠀⠀⠀⠀⠀⠙⠳⣖⣲⠛⠛⠁
⠀⠙⣦⡀⡜⡧⠐⠚⠉⠉⣀⣀⡤⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧
⢠⡾⠋⠀⣳⡷⠖⠒⠉⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠆
⣿⠀⢀⡼⠃⡟⠛⠒⠒⠶⠶⠤⢤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠈⠛⠋⠀⠀⣇⢀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⡏⠀⠈⠉⠉⠉⠁⠈⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡟
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⢰⣦⡀⡇⢀⠇⢹⠀⠀⠀⣤⡄⡇⢀⡴⡆
⠀⠀⠘⣇⢻⡿⣼⠀⠈⣴⢻⡼⠁⡇⣧⡾⢠⡇
⠀⠀⠀⢹⡌⠇⠘⠀⠘⠇⢸⡇⠀⡗⢿⠁⢸⠃
⠀⠀⢠⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦
⣶⣴⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶
⣿⣿⣇⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⣶⣦⣼⣿⣿⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢹⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣠⣭⣭⣭⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠁⠀⠀⢘⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⣂⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣀⣤⣴⣤⠀⠀⡤⢸⣿⣿⣿⣿⣷⠈⠉⠁⠀⠉⠛⠛⠂⠉⠉⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢀⡴⠋⣠⣴⣿⣿⣿⣿⣿⣷⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣧⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢠⠏⢀⣾⣿⠟⠉⠉⠉⠉⠛⠛⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⠿⠻⠿⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡞⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠇⠸⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡅⢄⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠸⡞⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⣯⡞⠁⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠙⠷⣤⣴⡟⠻⠶⢶⡞⣿⣶⣶⣟⠋⠀⠁⠁⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠓⠻⣧⠀⠀⠀⠀⣿⡋⠉⠉⠛⠷⣦⣄⣠⡀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⣀⣼⡟⠁⠀⠀⠀⠀⢿⠠⠀⠀⠀⢹⡇⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⣤⣤⣾⣶⣤⣤⣴⣾⣷⣶⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣿⣿⠯⠭⠿⠿⠃⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣵⠀⠀⠀⠀⠀⠀⠐⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠻⢿⣏⣻⡯⠍⠀⠀⠀⠀⠀⢽⣿⣷⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠉⠛⠛⠳⠶⠶⠞⠛⠛⠉⠁⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠡⠤⠶⠶⠶⠶⠶⠶⠶⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠉⠛⠻⠶⠶⣤⣤⣤⣤⣤⡤⠶⠟⠛⠋⠉⠀⠀⠀⢒⠀⠒⠒⠂⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⢷⡾⠾⣿⠟⠛⠋⠁⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⢸⡇
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿
⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠛⠻⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣦⣤⣤⣿
⠀⠀⠀⢀⣸⣧⠀⠀⠀⠀⠀⠀⠐⢠⣤⣿⠃⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⠁
⠀⠀⠀⣏⠙⠿⠦⢤⣤⣤⣤⣥⣼⣼⣥⡿⣧⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠙⠿⣷⣄⣀⣀⠀⠀⠀⠀⣀⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⢹⡛⠛⡛⠛⠋⠉⠁
⠀⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣇⢀⢠
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⢹⡸⣄⣃
⠀⠀⠀⠀⠀⠀⢰⣿⠀⢠⡇⠈⣧⠈⠉
⠀⠀⠀⠀⠀⠀⣿⣷⣶⠘⠃⠀⣿
⠀⠂⠀⠀⠀⠀⢿⣟⠇⠁⠀⢠⡟
⠀⠂⠀⠀⠀⠀⠈⠻⢷⣤⠶⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣉⠉⢹⠂
⠀⢠⠖⢦⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡛⠛⡇⠀⠀⡞⠉⠰⠿⡄
⣠⠞⠛⠉⠛⡏⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠾⣇⠀⠀⠛⠛⠛⠛⠁
⢷⣄⣠⡤⣴⠥⡴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠖⣦⠀⠀⠘⠒⠒⠒⠋
⠀⣿⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⠴⢧
⠀⣿⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛
⠀⣿⠀⠀⢸⠀⡇⠀⠀⣠⡄⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⣤⣤⣀⡀
⠀⣿⠀⠀⢸⠀⣇⣀⣰⠃⠸⠋⡰⠟⢛⣧⣠⡞⠉⠙⠋⡉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠉⠓⢦⡀
⠀⣿⠀⠀⢸⢠⠋⠘⠉⠁⠀⠀⠀⠐⠋⢱⡜⠀⠀⠀⡐⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠙⣄
⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⣆
⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠈⢀⠇⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⠶⠤⢤⣀
⠀⣿⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⣿⠀⠀⢸⠈⢢⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⢠⣞⡀
⠀⢻⠀⠀⢸⢰⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠈⡅⢹
⠀⣿⠀⠀⢸⠸⠤⠤⠤⠤⠤⠤⠤⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢸⠀⠄⠀⡧⠼⢶
⠀⣿⠀⠀⢸⢤⣤⣤⣤⣤⣤⣤⣤⣬⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⣣⠀⣠
⠀⣿⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠁⠀⠀⠀⢀⣸⣀⣀⣀⣀⣀⣀⡀⠀⠀⣇⠀⠀⠀⡇⠀⠀⡇⠀⢸
⠀⣿⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇⠀⢸
⠀⠙⢶⡶⠾⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⠞
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒⢒⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠒⠒⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠤⣀⡠⡀⠀⠾⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠋⠈⣭⢀⠂⠁⢀⣗⠒⠄⣀⠀⠀⠀⠀⢀⡠⠒⠉⢁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣽⣿⣷⣀⣀⡙⠘⠀⠀⠀⠘⠋⠲⡄⠹⠶⠤⢐⠏⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡀⠀⠀⠀⡠⠊⠀⢰⣿⣿⣿⣿⣿⣿⢿⠟⠀⠀⠘⠃⠀⠦⡀⣀⠔⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣪⢻⣝⠣⠤⢶⡎⠀⠀⣀⠼⣿⣿⣿⣿⣿⣿⣶⣶⣿⡄⠀⠀⢀⢀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⣷⣻⡿⠿⢿⣏⣶⣯⡀⠀⢙⡿⠛⣻⠋⠀⠀⠀⠉⠳⡤⠊⠁⠀⠀⠀⠀⠀⠈⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠀⣀⠀⣾⣗⠋⠉⠿⡍⠁⠲⠶⣅⡀⠀⠀⡀⠀⠀⠈⢦⠀⠀⠐⠂⠤⢀⡀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⠞⣋⣭⣄⣯⢄⠀⠀⠀⠘⠲⢤⣙⢦⡙⢄⠀⠇⢀⠔⠀⠀⢣⠀⠀⠀⠀⠀⠈⠉⠒⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠴⡻⠃⢠⠄⣈⠑⠛⠒⠶⠤⡀⠀⠀⠉⡿⢋⠑⢄⣡⡊⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡻⣄⣀⠛⠐⠉⠀⠀⡀⠀⣆⡈⣑⠤⣀⢁⠃⠀⠀⠀⠙⠀⠀⢦⠀⠃⠀⠀⠀⠀⢠⠔⠒⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣛⣆⡙⢯⣳⣤⣦⠮⠤⠤⠋⠙⣌⠠⡺⠦⠤⡀⠀⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⡠⠄⠈⠙⠓⠒⠚⠻⡒⠒⢦⡀⠀⠙⢄⢀⠜⠀⠑⠒⠤⣈⡀⢀⡠⠊⢀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⡔⠁⠀⠴⠋⠀⠀⠀⢰⠀⡴⣞⣂⡟⣽⣦⡙⣄⢀⡔⠁⠀⢀⡠⢤⠆⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠳⢭⣥⠤⣿⡏⣧⠜⠁⠠⠔⠊⢁⠔⠁⠀⠀⡠⠊
⠀⠀⢀⣌⡠⠔⠒⠃⡉⠉⠐⠒⠒⠤⢄⡀⠀⠀⠀⡿⣇⡏⠀⠀⠀⠀⠀⠊⠀⠀⠀⠊
⠀⠀⠛⠒⠨⠷⠤⣀⠉⠂⠀⠀⠀⠀⠀⠈⠑⠤⡀⠈⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⣀⠀⠀⠤⢀⣀⣀⠈⢦⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠐⠒⠠⠤⠼⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⢒⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⢀⣀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠴⠒⠋⣹⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣼⣀⠤⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠉⣀⣠⠤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣁⡤⠔⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⠤⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⢢⡤⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣴⣋⣀
⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⡞
⠀⠀⠀⠙⢶⣖⠒⠒⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠈⢻⡳⡀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠾⢶⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢳⡘⢦⡀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⠋⠀⠠⡏⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢧⠘⣷⡄⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢰⢄⡔⢆⡇⢱⡄⠀⠀⠱⣄⣀⣄⠀⢲⣠⣾⣟⣤⠖⠀⠀⠀⠀⣀⠔⢹
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠙⠋⢈⣡⠬⠽⠯⠥⠧⢞⠓⠂⣸⠋⢉⣿⣿⣿⣟⠁⠀⣀⣠⣴⣊⠅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡴⠊⠁⠀⠀⠀⣀⡠⠴⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢸⡍⣛⣛⣓⣲⡶⠤⣄⣀⡤⠖⢿⡇⠀⠀⠀⣀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⢸⡯⣟⡻⣿⣽⡟⠛⠉⠁⠀⠀⠀⠁⢀⡤⠚⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡧⣔⣲⢏⢌⡝⣳⣢⣤⣀⣀⣠⣶⣯⣴⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡵⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⡿⠛⠓⢤⡀
⠀⢶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠿⣛⡛⠛⠒⠲⠶⣶⣤⣤⣤⣴⣶⠟⣛⣛⣛⣛⣛⣿⣿⣿⣛⣛⣛⣛⣛⣛⣭⣭⠤⠤⠤⠴⠶⠞⠳⣄
⠀⠠⠤⠤⠬⠭⠭⠿⠷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⠉⠉⠉⠉⠹⡏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄⡀⢳⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠁⠐⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡤⡀⠈⠀⢸⠀⠀⡇⠀⢀⠚⠠⡀⢃⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣑⠀⠀⠢⢄⠀⡀⢟⢦⡁⠡⣉⠄⠬⠓⠂⠉⠉⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢋⡀⠀⠠⣤⠴⠁⠈⠀⠙⠁⠀⢀⢀⡀⠠⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢉⠉⠐⠀⠬⠄⠒⠠⠒⡡⠚⢀⠔⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢠⡦⡄⠀⠀⢩⡉⠉⠉⠀⠀⠀⠐⠒⠠⠀⢀
⠀⠀⠀⠀⠀⢀⡐⠇⡣⠈⠁⢀⡆⠤⠜⠋⠁⠀⡀⠀⠀⠀⠀⠞⠀⠄⠀⠀⠠⠶⠦⢤⣀⡀
⠀⠀⠠⠄⠀⠊⢡⠈⠁⢀⠄⠉⡀⡀⠀⠀⣄⡴⠗⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢀⣀⡄⠤⠉⠀⠐⠀⠄
⠊⠀⠀⠀⠀⠀⠃⢀⠴⠃⢠⠀⠀⢐⠀⠈⠁⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⢉⡉⢳⡀⡀⠀⠀⠁⠢⢌⡁⠀⠠
⠀⠀⠀⠀⠀⠀⣠⠆⢀⢠⠤⠥⢠⠃⠀⢀⡼⠠⠯⢴⡞⠀⠀⠀⠀⢴⡋⠀⠀⠐⠀⢁⡌⠀⢱⡄⠀⠀⠀⠀⠙⠦⡀
⠀⡀⠀⠀⢀⠔⠀⠀⠀⠈⠑⠦⢄⠀⢠⠆⠀⠀⠂⡀⠀⠀⠀⣀⠈⠀⠁⠀⠠⠀⡀⢀⡆⢀⡴⠀⠀⠀⠀⢀⣠⣤⠕
⠸⠇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠤⣀⡠⠐⠒⠓⠻⠖⡲⠎⠀⠑⢄⣀⣤⠲⢖⣿⠁⠀⠀⠀⠀⡠⢂⠞⢁⠃
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠁⠀⠀⡠⠛⠈⠉⠙⠓⠀⠀⢀⡠⠈⠐⠁⡠⠞⠠⢄
⠀⠀⠈⠉⠉⢀⡠⠖⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠂⠁⢺⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡁⠈⠁⠀⠠⡀⠀⢠⣇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠟⠀⢼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠐⠄⠀⠀⠀⠀⠐⠀⣤⡄⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠔⠋⠑⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢂⠀⠀⠈⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠀⢸⠀⠀⠀⡇⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠁⠈⠁⠀⠀⠈⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠃⠀⠀⢂⣁⠀⠒⠢⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢀⠀⠀⢀⢀⣾⣿⣿⣷⣶⣦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠏⡰⠚⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⣴⣿⣿⡆⠸⠈⠛⠩⠙⠛⠿⠿⠿⠿⣿⣿⣿⣶⠀⡀⠀⠀⠀⠀⢿⣿⣿⡟⠁⠀⠈⠁⠀⠀⠹⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢟⣛⠘⠁⠀⠀⡂⠀⢀⣤⣶⣦⣤⣀⣠⣶⢛⣉⣀⠀⠀⠀⠀⠈⢻⡿⠀⠀⠀⠈⢿⡀⠀⡄⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⢈⣉⠛⠶⡄⠀⠻⠀⢰⠿⠛⠉⣉⠽⠿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⢰⡿⠿⠛⠛⢒⡇⠀⠀⠀⠄⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⢉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⡈⠀⠀⠀⠀⠀⠐⢠⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣸⠀⠀⠀⢀⣀⣤⣠⠔⠀⠀⠀⠙⢷⣄⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢘⡅⠙⠶⠶⠶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠹⡶⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢸⡇⠀⢀⣤⠶⠤⣔⣒⣨⣥⠾⠟⠿⠿⢶⣿⣆⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣷⡀⣸⣿⡟⠛⠉⠁⢁⣠⣤⣶⣶⣶⣿⠏⠉⡇⢀⠄⡇⠀⠀⠀⠀⠀⠀⠁⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⣎⠙⣿⣶⣶⣿⠟⠛⠛⠉⢉⠉⠃⠀⠀⡇⡸⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡜⢷⡻⣝⠛⠓⠂⣂⡩⠕⠊⠀⠀⠀⠘⢀⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢱⡉⠙⠭⠤⡴⠶⠖⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠁⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⣀⠀⠀⠀⣀⣀⣴⣾⠟⠀⠀⠀⠀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⡟⢯⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠹⣆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠘⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠛⠛⢿⣿⣶⣾⣿⠟⠉⠉⠛⣿⣷⣶⣿⡿⠛⠛⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⢀⣨⣭⣭⣅⠀⠀⠀⠀⣨⣭⣭⣥⡀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣴⣿⠟⠛⠻⣿⣦⡄⢀⣴⣿⠟⠛⠿⣿⣶⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣠⣶⡿⣿⣧⣼⣿⣿⣷⣄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣀⣀⣀⣀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⠏⠙⢿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣾⣿⠟⠁⠀⠀⠀⠀⠀⠙⣿⣷⣤⣤⣤⣤⣤⣤⣤⣾⣿⠋⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⣿⠋⠉⠻⣿⣷⣄⣀⣀⣀⣀⣠⣶⣿⠟⠉⠉⠉⠉⠉⠉⠋⢻⣿⣦⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟
⠀⠀⠀⠀⠀⠀⢰⣿⣏⣀⣀⣀⣀⣘⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠘⢿⣤⡀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⢀⣤⣤⣄⡀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⣛⣛⣛⣛⣛⣋⡉⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⢉⣿⣛⣿⡉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣟⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⣄⠈⢦⠉⠉⠉⠀⠀⠉⢳⡖⢾⡟⠛⠛⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠘⣷⠘⡆⠀⠀⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣽⠀⡇⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⣴⡟⡰⠁⠀⠀⠀⠀⢀⣼⣧⣼⣥⣤⣤⣤⣤⣤⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⣿⣙⠛⠓⠶⠶⠶⠶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣌⠷⣤⣤⣤⣤⠿⣩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣬⣙⡒⠒⠒⢒⣋⣡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡍⠉⠉⠙⠉⢉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢸⡇⠀⡄⠀⠀⣿⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⡇⠀⡇⠀⠀⢸⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠀⠇⢸⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢸⡇⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡇⢸⢸⠀⠀⢸⡇⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⢸⡇⣼⠈⠀⠀⢸⡇⠘⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠃⢸⡇⠁⠀⣰⠀⢸⡇⠀⣈⡁
⠀⠄⠀⠀⢀⣀⡀⠀⠀⣠⠞⠛⠆⠸⡇⠀⠀⠋⠀⠘⣇⠀⠛⠛⣀⠀⠀⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠃⢀⡴⠀⠀⢀⡀⢤⡀⠚⡀⠚⠲⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠓⡀⠀⠇⠀⠀⠈⠈⠣⠄⠀⠀⢈⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⠴⠾⢧⣀⣀⡾⠀⠸⣦⣤⠾⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣆⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⢻⣿⣿⡟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⣠⣾⣿⣤⣿⣿⣿⣃⣰⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⣼⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣰⣿⢉⠟⠀⠀⠈⠉⠙⠿⢿⣿⣿⣿⣏⣹⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⡀⣄⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢷⣤⣤⣄⣠⣉⣈⢳⡿⠿⠿⠟⠀⢀⣴⢿⣿⣿⣯⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⡿⣿⣿⠁⢸⣿⣿⣷⣧⣌⡓⠒⣿⣿⣿⡟⣻⡿⠟⠋
⠀⠀⠀⣀⣴⢲⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣟⣻⣿⣿⣅⠋⢻⣿⣷⣀⡀⠈⠉⠉⠃
⠀⢦⡾⣿⣿⣿⣻⣿⣿⣿⣹⣿⣿⣶⣿⣿⣿⣟⣿⣿⢻⣶⣿⣿⣿⣿⣿⣆
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠈⠻⣿⣻⣿⡿⢿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣷⣾⣿⣿⣿⣟⣹⢿⣿⣿⡿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣿⣿⣿⣷⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠛⠲⠦⠤⠀⣀⣤⡎⠉⠉⠉⠉⠉⠉⠉⢳⡄⣀⣠⡤⠶⠶⠲⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⢟⣋⣭⠤⠤⠼⢿⡇⣤⣤⣤⣀⣀⣀⣀⠸⡇⠀⣀⣰⠶⢶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠓⠚⠛⣿⣿⠿⡿⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣿⠀⠁
⠀⠀⠀⠀⠀⡴⢃⠎⠀⠀⢀⠞⠀⠐⠁⠔⠁⠐⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠈⠀⠁⠟⠀⠀⠐⠀⡀
⠀⠀⠀⢠⠊⡔⠁⠀⠀⡀⠁⠀⠂⠀⠊⠀⠀⡠⠂⠀⠚⠄⠀⠀⠀⠀⠀⠀⠀⡇⠋⠀⠀⠀⠀⠐⠈⠙⠉⢹
⠀⠀⠀⣡⠊⠀⠀⢠⠎⠀⠀⠀⠐⠁⠀⠀⠔⠀⠀⢀⠼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇⠄⡄⠀⠀⠀⠀⣿
⠀⠀⡔⠁⠀⠁⡐⠁⠠⠂⢠⠂⠀⠂⡠⠂⡀⠀⡔⠁⣰⠁⠀⠀⠀⠀⠀⠀⣴⣧⣄⣿⡇⣁⡀⠀⠀⠀⣀⡀
⠀⠜⢀⠂⢀⠈⠀⡀⠁⡰⠃⠀⠀⡔⠁⠐⢁⠊⣠⠊⣠⠀⠀⠀⠀⠀⣠⠚⠻⢭⣋⣙⡈⠭⠁⠠⠍⠛⠋⠀⣿⣀
⠀⠀⠁⠠⠂⢀⠎⢀⡜⢁⠄⢀⠎⠠⠂⡰⠁⡴⢁⠌⢘⠀⠀⠀⠀⠀⣿⣿⣯⣄⣀⡉⠛⠃⠀⠤⠴⠶⠦⠭⠖⠛⡇
⠀⠀⠀⠀⡠⠃⢀⠞⢀⠈⣠⠂⠐⠁⠌⠀⠜⠰⠁⠂⢸⠀⠀⠀⠀⠀⠙⢻⣿⣿⣟⣛⣛⣀⣀⣀⣀⣀⣀⣀⣤⡶⠃
⠀⠌⠀⡔⠁⢠⠋⡠⠃⣰⠃⢬⣠⣤⣤⣬⣤⡴⡳⠒⣻⣆⡀⠀⠀⢀⣠⠾⣗⡿⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠘⢦⡀
⠀⢀⠎⠀⡠⠃⡰⢁⣼⠋⠀⠀⠀⠀⠀⠀⠊⣴⠁⣴⠉⠈⠉⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⡀
⢀⠃⠀⡜⢁⡜⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄
⠀⢀⠎⢀⠞⠀⠊⣿⢀⡀⡀⢀⡄⡀⠀⠀⡂⡇⠸⢣⣤⣤⣠⣤⣤⣤⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⢠⠊⢀⠀⡠⠁⡰⣿⣼⣿⣿⣿⣿⣿⣿⣾⡇⢃⠀⣿⣿⣿⡿⠿⠿⠻⢿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣿⣿⣿⣞⣡⣀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠁⠀⢁⡼⠀⡐⠀⡸⣿⡿⠛⠻⠻⠹⠿⠟⢿⡘⡄⠙⠛⠈⣀⣤⣤⣤⣄⡀⠹⡆⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠿⠿⠿⣿⣿⣿⣯⡖⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠘⠀⡜⢀⡜⠀⠈⠛⣿⣶⣶⡶⣶⣶⡞⣿⣷⣀⣴⠟⠉⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠚⠛⠒⢦⣌⠘⠻⣟⣭⣴⡀⠀⠀⠀⠀⠀⢿⡀
⠀⠈⠀⡜⢀⠎⠀⡠⠁⡘⠛⣿⢟⣼⣟⣟⣼⡿⣱⠃⣠⠂⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣤⣤⣤⣤⡤⠾⠃⠀⠀⠀⠀⠈⠻⣆⠈⣿⣿⡯⠀⠀⠀⠀⠀⠘⣧
⠈⠀⠜⢀⠎⠀⡰⠁⡔⠀⡜⠁⡾⢋⣿⢞⡟⣴⣿⣟⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠨⣻⣿⡥⠀⠀⠀⠀⠀⢹
⠀⠈⠀⠎⠀⡼⠁⡼⠀⠜⠀⡔⠀⠜⢉⠞⡼⣳⣿⡟⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⣿⣿⣏⠕⠀⠀⠀⠀⠈
⠀⠀⡎⠀⡰⠁⡜⢀⡎⠀⡼⠀⠈⠀⠈⡼⣵⢟⡟⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢻⣿⠗⣫⠄
⠀⠜⠀⣰⠁⡜⠀⠈⠀⠸⠁⠘⠀⠀⡜⠁⡞⠈⣼⡟⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢾⣿⣟⡅⣾
⠠⠀⢠⠃⡸⠁⠄⠀⠰⠁⠐⠀⠀⠰⠁⠜⠀⡼⠉⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⣿⣥⣻⣏⣀⣀⣤⡾
⠀⢠⢃⡔⠀⡄⠀⢠⠁⠀⠀⠄⢠⠃⡰⠀⡸⠁⢠⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠊⡸⠁⡔⠀⢀⠇⠠⠁⡘⠀⠇⡰⠁⣰⠁⢠⠃⡰⣿
⡄⠰⠁⠀⠀⢀⡎⠀⠀⠀⠀⡞⠰⠁⢠⠇⢀⠃⣸⠃⡏
⠰⠂⡰⠀⠀⡜⠀⠄⢀⠂⠘⢠⠃⠀⡎⠀⠌⢠⠃⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠉⠉⠓⠲⠦⢤⣤⣤⣀⣰⡋⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⣤⠤⠴⠒⠋⠉⠁⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⠀⢀⣀⣤⠴⠖⠚⠋⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⢀⡷⠃⠈⠉⠙⠛⠲⠦⢤⣀⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡶⢶⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠷⠾⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠋⠉⠉⠉⠉⠉⠉⠻⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣩⣹⣉⣋⡏⢹⠛⡟⢻⠛⡟⢻⠛⡏⢻⢙⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠋⠉⠉⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⡴⠞⠛⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⢠⡿⠁⠀⠀⠀⠘⣷⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⣼⠃⠀⠀⠀⠀⢰⠏⠀⢈⡿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡖⠲⠶⠤⢤⣤⣀⣀
⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢦⣄
⣿⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⢸⡇⠀⠀⠀⠀⠸⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠈⢿⡀⠀⠀⠀⠀⢻⡤⠖⠛⠛⠛⠒⠶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⢷⣄⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠲⠶⠦⠴⠶⠖⠒⠛⠋⠉⠉⠀⠀⠀⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⠲⠤⣤⣄⣀⣀⣀⣤⡤⠶⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣤⣤⣤⣤⡴⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⠙⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠁⠀⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣉⣉⣉⣉⣉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣛⡻⣯⡉⠉⠉⠁⠀⣀⡈⣁⡀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣋⣡⣿⣯⣿⣿⣿⣿⡿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢹⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣽⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢿⣏⢉⢹⣏⠙⣿⣆⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⣿⡿⣿⢿⣾⣿⡿⣰⣞⣿⣆⠀⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣤⣤⣤⣀⣀⣀⣹⣿⠁⠀⠀⢉⣿⣧⣤⡿⠅⢘⣿⡆⠀⠸⣿⣷⣶⣚⡿⣷⢶⠾⣿⢿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⣿⡇⠀⠀⣹⣿⡟⠙⠉⠈⠈⠀⠉⠀⠘⠈⢻
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⣀⠀⡀⠀⢀⠀⠀⠀⢀⢀⠀⠄⠀⢀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣔⣽⠿⠶⠷⠶⢮⣧⣴⡶⠶⠶⠶⠶⠶⣬⣅⡐⣴⣿⠇⠀⠀⣿⣿⢿⣾⣷⣶⣥⣿⣼⡽⣶⣿⣾⠁
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣄⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⢸⣟⣀⣠⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠘⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠂⠈⢿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠈⠉⠉⠉⠉⣿⡏⠀⠀⢹⣿⠉⠉⠉⠉⠀⠀⠀⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣤⣶⡶⠶⠶⠶⢿⣷⣤⣤⣾⡿⠶⠶⠶⢶⣦⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⣿⠉⠉⣿⡇⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣤⣴⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠰⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣤⣤⣤⣤⣤⣤⣾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁⠁⠈⠉⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣷⣄⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⣀⣀⣀⣀⣸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣤⡾⠛⠛⣦⡀
⠀⠀⣠⡶⠖⠛⠶⠶⠶⠾⠛⠃⠈⠉⠉⠁⠀⠀⠀⠘⣿⣤⣤⡀
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡌⠉⠈⣿
⠀⢸⣇⠀⠀⠀⣀⣀⣀⣀⣄⡀⠀⠴⠿⢦⣀⣀⣀⠾⠃⠀⢀⣿
⠀⠀⠘⠛⠛⠛⢻⣏⠉⢻⣥⠀⠀⠀⢰⣿⠛⠛⠻⣦⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⠶⠶⠶⠖⠒⠚⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⣀⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⢀⣴⠶⠶⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⢀⣾⠁⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⢦⣤⣤⡀
⣼⠃⠀⢸⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣄⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⠻⣦⣾⡀⠀⠀⠀⠀⠙⠛⠠⠤⠰⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠙⠻⢦⣤⣤⣤⣤⣤⣤⢼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⣀⠀⠀⠀⣀⣠⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠙⠛⠛⠛⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⠟⠛⢦⡄
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⡟⠁⠀⠀⣸⡇
⠀⠀⠀⣾⠏⠉⠙⢷⡄⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⢀⠀⠀⠀⣣⣤⣄⡀⠙⠛⠛⠃⠀⠉⠁
⠀⠀⠀⣿⣄⣀⣴⠞⠛⠀⠀⠀⠀⣤⠟⠋⠙⠁⠀⠀⠀⠘⠀⠀⠀⠛⠉⠈⠛⣶
⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠙⣯⠉⠛
⠀⠀⠀⣠⠶⠶⣿⠇⠀⠀⠉⡻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⡿⠀⠀⠀⠀⢰⡇
⠀⠀⡼⠁⠀⢰⣷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠉⠀⠀⠀⠀⠀⠈⠻⠿⠟
⠀⠀⢻⣆⣠⡾⠁⠀⠀⠀⠘⢷⣀⠀⠀⣠⡄⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂⠹⣇⡀⠀⠀⠀⣠⡞⠻⣦⣤⣤⡴⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣶⣶⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣷⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠦⠐⠀⢀⣴⠄⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠁⠀⠉⢉⡉⢬
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣄⣠⠠⢄⠠⣀⠀⢠⠄⣤⡠⡀⠄⠀⠀⠀⠦⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠀⠤⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⡐⠀⢘
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠂⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣤⠀⣠⠀⢀⠀⠀⠠⠠⠤⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⣠⣄⣤⣀⣀⢄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣴⣦⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣶⣠⠔⠋⣽
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠿⣿⣿⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠉⠀⠀⢸⣿⣦⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠃⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠠⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⠀⣀⣆⣛
⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣿⣤⣼⣿⡿
⠀⠰⣿⣏⡁⢳⣀⣠⣶⣶⣦⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠋
⠀⢸⣿⣭⣴⡾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⣴⣿⡗
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠁⠂
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡴⠒⠚⢫⣥⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣴⠶⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⡀
⠀⠀⣤⣴⠶⠚⠋⠁
⠀⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢰⣂⣀⣀⣀⠀⢐⡆⠀⠀⠀⠀⠐⠀⠉⠉⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠉⠉⠉⠉⠓⠃⠀⠀⠀⠀⠀⢀⠎⢠⠎⠀⠀⠀⠀⠀⠔⠒⠒⠒⠒⠠⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡜⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠔⠀⠀⠀⠀⢀⣧⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⣿⣿⠟⠀⠀⢮⣀⣈⣉⣉⣉⣩⡇⢀⣄⠀⠀⠐⠀⠢⠤⠤⠤⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠳⢄⠁⠀⠀⠀⠀⠈⠀⠀⠀⠛⣻⡟⠀⠀⢻
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠟⠀⠀⠀⠀⠀⠀⠀⣴⣏⣀⠀⠀⣸
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⣠⣿⣿⣿⣧⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣾⣧⣄⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠹⣇⠀⠀⣴⣏⣀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣀⣀⣀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡗⠀⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⣴⣋⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣉⣉⠡⠊





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

⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠳⠦⠤⠤⠾⠋⠀⠈⠙⠦⠤⠤⠴⠟⠉⠙⢷⡀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⢀⣴⠶⠶⠶⢶⡀⠀⠀⣠⡶⠶⠶⠶⣄⡀⣀⣼⠃⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢹⡟⢻⠋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠷⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢈⡏⠉⠉⠉⠉⠛⠶⢤⡴⠶⠛⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠁⠒⠒⠒⠒⢠⠊⠉⠑⡄⠠⠤⠤⠄⠀⢐⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⢀⣄⣠⣤⣼⡰⢯⣺⣤⣠⣀⠤⠄⠀⢸⡅⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⣿⠀⠈⠉⠉⠙⠛⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠙⡔⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣅⣠⣠⣧⡬⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⢤⣼⣧⠦⠶⠿⠷⢾⡿⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡱⠉⠀⠀⠁⠀⠀⠀⠺⣧⠀⠀⠀⠀⡖⠺⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⡠⠤⠔⠒⠒⠂⠈⣿⡽⠯⠒⠁⡠⠒⠂⠀⠀⠈⠐⠩⢜⠟⠖⠖⡞⢷⠀⣧⣰⠀⠀⠀⣀⠀⡀⠤⠠⢼
⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠀⠀⠀⢀⢸⡗⣿⢹⠀⠁⠚⠛⠛⠓⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢠⡄⠐⠀⡀⠀⢀⠀⠀⠰⡂⣰⠆⠀⠀⢻⠀⠀⡟⢸⣿⣿⣾⠀⠆⠀⢀⣠⠄⠀⠤⢠
⠀⠀⠀⠀⠀⠀⢀⢀⡦⠞⠓⠛⠛⠮⣀⠀⠀⠈⠀⠁⠀⡐⣤⡶⠚⠓⠛⢧⢺⣿⡿⡇⠠⠈⠠⠄⠀⠐⠒⠒⢒
⠀⢰⠀⠀⢀⠀⢨⠏⠀⠀⠀⠀⠀⠀⠈⠛⠦⢶⣰⣶⠼⠟⠁⠀⠀⠀⠀⠀⠙⢿⣽⠃⣂⠉⠀⠀⠀⠀⠀⠀⣼
⠀⢰⠀⠀⢠⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣖⠖⠴⠒⢚⣃⣀⣤⣼⠋
⢸⢸⠀⠀⢈⣈⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹⠙⡏⣹
⡇⣙⠒⠀⠈⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠄⡅
⠱⠭⣀⣀⣐⠭⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠆⠉

⠀⠀⠠⢓⣄
⠀⠠⡁⠀⡌⠢⡀
⢠⡷⠁⠀⡀⢔⡷⡀
⠰⠆⠀⢠⣤⣼⣟⠇
⠘⢤⢀⣉⣙⣉⠍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣷⣴⣆⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉⠙⡿⣿⠉⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣷⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⢠⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠉⠙⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠛⠁⠀⠀⠀⢻⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠉⢀⣀⣤⣄⠀⠀⣀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠓⠒⠖⠀⠉⠉⠀⣿⣯⣙⣿⣿⡆
⠀⠀⠀⠀⠀⢀⡴⠊⢁⣴⣾⣿⣿⣿⣿⣧⣤⣭⣴⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣶⣤⣤⣴⣶⣶⣀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⡰⠃⢀⣾⣿⡿⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢰⠃⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉⠉⠉⠁
⠀⠀⠀⣾⠀⣾⣿⠃
⠀⠀⠀⠈⠈⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⣀⣠⣀⡀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⢻⣿⣿⠉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣛⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣭⣭⣭⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠘⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠉⠉⠓⠲⡶⠾⢟⢻⠉⠉⠀⠀⠉⢹⢛⠿⠶⠶⠒⠊⠉⠉⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠡⠀⠀⠀⠀⠀⢀⣀⡇⠀⣀⠞⡄⠀⠀⠀⠀⡞⢸⡀⠀⣀⡀⠀⠀⠀⠀⠀⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣐⣒⣒⣩⠥⠖⠛⠋⢻⡀⣇⠀⠀⠀⢠⠃⣸⠋⠙⠳⠮⢍⣒⣒⣂⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢷⣼⣤⣤⣦⣞⣴⠃⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠶⡷⠶⠶⠶⠶⢷⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⠶⣾⠒⠒⠒⠒⠒⠺⣷⠸⠧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡟⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⡷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⢀⡖⢛⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠒⠒⣚⣛⠩⠅⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⡇⢸⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⡤⠞⢋⣡⠤⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⢸⠀⠀⡆⡇
⠀⠀⠀⠀⡴⠋⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢸⢸⢸⢠⡆⡇⣿
⠀⠀⠀⡼⢁⡜⠁⠀⠀⠀⡠⠂⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⢛⣛⣿⣿⣿⡇⢸⢸⢸⢸⣿⡇⣿
⠀⠀⢸⠃⡼⠀⠀⠀⠀⢸⠀⢀⡴⠋⠉⠉⠉⠉⠉⠛⠛⠲⠶⢤⣤⣀⣀⣤⣤⣄⣒⣨⣭⠴⠖⠛⠉⠁⠀⠀⠈⣧⣸⢸⣿⣸⣿⣧⣿
⠀⠀⡿⢰⠇⠀⠀⠀⠀⣸⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⣭⣉⣉⣩⡽
⢀⣶⣇⣼⠀⠀⠀⠀⠀⣿⢀⣏
⢸⠅⡀⠈⠩⠭⠭⠭⠭⠭⠭⣾
⠘⠢⢴⣋⣩⡿⠿⢿⣯⣭⣿⡿
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⢦⣤⣿⢀⣿⣤⡴⠶⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣀⣀⣀⣤⣴⠿⣦⣤⣀⣀⣀⣀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⣿⠀⣿⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣤⣿⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⢒⠢⠤⢠⣤⠉⠉⠉⠉⠉⠉⢹⡆⢠⠤⠶⠒⠒⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠯⠵⠒⠒⠚⠻⣰⣶⣶⣦⢤⡤⠌⠃⠀⠠⠶⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠉⠁
⠀⠀⠀⠀⣠⢀⠔⠀⠀⡐⠀⠐⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⡔⡡⠁⠀⠀⠈⠀⠀⠀⠁⠀⠠⠀⠐⠃⠀⠀⠀⠀⠀⠀⡏⠃⠀⠀⠀⠒⠂⠂⢹⠆
⠀⠀⠀⠎⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠸
⠀⡰⠀⠀⢀⠀⠠⠀⡠⠁⠀⠠⠂⠀⠠⠂⠔⠇⠀⠀⠀⠀⢀⣾⢷⠾⡿⠤⠄⠀⢀⣤⠄⣠
⠐⠀⠀⠀⠀⡀⠀⡔⠁⠀⠐⠀⠀⠔⢠⠊⠐⡇⠀⠀⠀⢰⣧⣬⠙⠓⠶⡤⠀⣈⣁⣀⢀⣛⢶
⠀⠀⠀⢀⠈⢀⠎⠀⢁⠌⠀⠁⠊⠠⠡⠀⠀⡇⠀⠀⠀⠘⢿⣿⣿⡷⠶⠀⠀⠈⠉⠁⣀⣀⡼
⠀⠂⠀⠂⢠⠊⠀⢀⠀⠀⣤⠄⢤⣤⡴⠶⢒⣧⡀⠀⠀⣀⠴⢟⡟⠉⠉⠉⠉⠉⠉⠉⠁⠙⢆⡀
⠀⡐⠁⡰⠁⠌⣰⠏⠀⠀⠀⠀⠀⠈⡼⢠⠎⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⡀
⠀⠀⠐⢀⠆⠰⡟⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⢀⠎⠀⠂⠀⢸⣇⣶⣶⣴⣶⣤⣤⡇⠃⢠⣿⣿⣾⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣦⣾⣶⣤⠄⠀⠀⠀⠀⠀⠈⠳⡄
⠂⠀⢡⠄⠠⠁⣿⣿⠿⠿⠟⠿⠿⢿⡸⠀⠿⠋⢉⣁⣤⣄⡉⢲⡄⠀⠀⠀⠀⠀⠀⣾⡿⢿⡿⢿⣿⣿⣦⠄⠀⠀⠀⠀⠙⣆
⠀⢀⠁⡰⢁⠌⠀⠙⣳⣶⣶⢶⣶⡞⣷⣧⡤⠾⠉⠁⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⣩⠖⠒⠲⢤⣉⠻⢿⣥⣆⠀⠀⠀⠀⠸⡆
⠀⠁⡐⢀⠎⢀⠆⡰⠉⡿⢣⣿⣿⣼⢟⣾⣠⡆⠀⠀⠀⠀⠀⠀⠛⠲⠶⠶⠶⠶⠚⠋⠀⠀⠀⠀⠉⢧⡈⢿⣯⡃⠀⠀⠀⠀⢻
⠀⠀⢀⠊⢀⠎⡰⢁⠌⢠⠋⡼⢋⢏⣾⣿⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠨⣿⣷⢄⠀⠀⠀⠈
⠀⢀⠊⢀⠎⡰⢁⠎⢠⠃⠀⠀⢃⣎⣿⡟⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⠅⡀
⠀⠌⢀⠎⡰⠁⠊⠀⠂⠀⠀⠀⠎⠸⠙⣼⢟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣷⡫⣶
⠀⢀⡎⠠⠁⠀⠀⠂⠀⠀⠀⠈⠐⠁⡜⠉⡾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣿⣯⣿⣋⣀⣤⠴
⠀⡜⡴⠀⠄⠀⠌⠀⠀⠀⡔⠠⠀⡸⠀⠌⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⢀⠁⠀⠀⡈⠀⠀⠈⢠⢀⠁⣠⠁⡐⢠⢻⠃
⢡⠁⠄⠀⠐⠀⠀⠐⠠⢃⠂⢀⠃⠀⢠⠃⢸
⣦⣼⣤⣤⣤⣬⣤⣤⣤⣬⣤⣦⣤⣥⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣅⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⠤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡀
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣿⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠉⠀⠀⠀⠠⢤⣤⣄⡀⠀⠀⠙⢿⣿⣦⣴⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⡏⠀⡀
⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠄⠀⠀⠀⣴⣀⣀⣀⠀⣀⠀⠀⠴⠤⠀⠀⠀⠀⢀⡒⠛⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣄⣀⣿⣶⣦⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠉⠁⠈⡟⣿⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠾⣿⣆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⢿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣀⣀⡀⠠⠴⠃⢰⣦⢶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⣀⠰⠊⢀⣤⣶⣾⣿⣆⠀⠰⢂⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⠀⠈⠁⣀⣀⡀⢸⢿⣯⣿⣿⡇
⠀⠀⠀⠀⢠⠞⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⠃⢀⣾⣿⠏⠁⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠙⠛⠿⡿⠿⠿⠿⠿⠇
⠀⠀⠀⡾⠀⣾⣿⠃
⠀⠀⠀⠃⠸⠿⠇


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠛⠛⢻⣿⣿⡿⠛⠛⠛⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⢸⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠸⡿⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣭⣭⣭⣭⣭⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⢿⡿⠿⢿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣷⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⢸⣿⡀⠘⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇
⡿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⢿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠰⣙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢹⠞⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠛⠻⠻⣧⡀⠀⢀⣤⣤⣄⣀⣠⣤⣄⣀⣀⣤⡾⠋⠘⠇⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣛⣿⡅⠉⠙⠛⠛⢸⣿⣿⠿⣤⣤⣀⠀⠀⢀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⣶⡶⠛⠉⠉⠉⠉⣷⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⠿⣶⣤⣷⣄⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣿⠀⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣶⣶⣾⣶⣶⣶⣶⣶⣿⣷⡟⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⠯⠭⠽⠿⠟⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣭⣵⠀⠀⠀⠀⠀⠀⠀⠠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠶⣶⣿⠺⢿⣿⣒⠀⠀⠀⠀⠀⠀⣲⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠉⠛⠻⠿⢶⣶⣤⣤⣤⣴⡶⠿⠟⠛⠉⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⡿⠀⣀⣀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣾⠿⠏⠀⠉⠉⣉⢉⣉⣉⡉⠉⠉⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠚⠉⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⡆⡀⠀⠐⣒⣲⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠛⠻⣛⡛⠿⣉⠉⠉⠁⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢿⡇
⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⢸⡇
⠀⠀⠀⠀⠸⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡶⠟⠛⠿⣷⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣸⡇
⠀⠀⠀⠀⠀⣿⣁⠀⠀⠀⠀⠀⠀⠀⡄⣀⣠⣿⠋⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣰⠛⢿⣯⣀⣀⠀⠀⠀⡆⢐⣂⣛⣛⣿⡄⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠹⣦⣤⡀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣤⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠈⠙⠻⠷⠶⣦⣤⣤⣤⣤⣤⣾⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢻⠀⡄⢧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠘⣇⢳⣀⡁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⡇⠀⢻⡄⠉⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣷⣴⠘⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠂⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣧⣦⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣤⠤⠤⣠⣤⣤⣄⣀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⠀⠀⠀⠉⠉⢒⠚⠿⣹⠀⠀⠀⠀⢀⠏⠛⠓⠒⠉⠉⠀⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣬⠀⠀⠀⣀⡠⠤⣼⣠⣤⠃⡆⠀⠀⠀⣸⠸⣥⣀⡤⠤⢄⡀⠀⠀⠐⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠶⠒⠚⠉⠀⠀⠘⣆⢁⣀⣀⣀⡇⡼⠁⠀⠈⠙⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣏⣉⣉⣉⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢥⣴⠥⠤⠤⠤⠬⣧⠸⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⣾⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⢞⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣄⣀⣤⣤⡏⢁
⠀⠀⠀⠀⠀⠀⣠⠤⠖⢚⣉⡉⠭⠤⠄⠐⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡴⠋⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⢀⡄⢸
⠀⠀⢀⡞⢠⠊⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⣾⣿⣶⡇⢸⣿⡏⣿⡷⣿
⠀⠀⡼⢠⠇⠀⠀⠀⢸⠀⢠⠖⠋⠉⠉⠉⠙⠓⠲⠦⢤⣄⣀⣀⣀⣀⣀⣒⣬⠥⠖⠚⠉⠉⠀⠉⢇⡸⣹⣇⣿⣏⣿
⠀⢠⡇⡾⠀⠀⠀⠀⡜⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠬⠭⠭⠭⠽
⢰⠾⠡⡇⣀⡀⠀⢀⣇⡸⡇
⢸⠀⠀⣥⠀⣒⣒⡒⢦⣬⡇
⠀⠉⠒⠓⠾⠿⠿⠟⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸
⣀⢀⡀⠀⠀⠀⠀⠀⠙⡁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⠀⢀⡴⠖⠶⣄⠀⢀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⡶⠦⣤⡀⠀⠀⠀⠀⣼⠁⠀⠉⠉⠉⢻⠁⠀⠀⢸⡏⠉⠉⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⣦⠀⠀⠀⠘⢦⠤⠴⠶⠒⢻⠀⠀⠀⢸⠓⠲⠶⠦⢤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣀⣼⣤⣤⣤⣼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⣧⠀⠀⠀⠀⠀⢠⡷⠶⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠻⢤⣤⣤⡼⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠞⠁



⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣄⣶⣀⠀⣀⣤⠾⠛⠀⢀⣀⠀⣠⡀⢠⣄⡀⢀⠀⠀⠀⢀⣠⣤⢤⣠⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⠘⠉⠉⡁⠀⠀⠀⡜⣦⡼⡿⢷⢿⡿⠆⠘⡾⡶⣡⢿⢷⡇⣾⠇⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣡⡶⠶⣦⣴⡙⣦⣄⠀⠈⠈⠀⠀⠈⢀⡀⠀⣀⠀⢠⣀⡀⢨⡆⠀⣀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⠣⠂⢀⣘⠛⠈⠉⠀⠀⠀⠀⠀⠀⡛⣤⡼⡟⣷⢿⡿⠀⡞⢸⣳⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⠄⣾⠁⢳⣤⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠉⠈⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⢠⣶⢾⣾⢿⣤⠀⡄⠀⢀⣤⣴⣿⠟⠷⢦⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⢦⣿⠘⣿⣯⣙⣲⡇⢚⣫⣴⠟⠃⠀⠀⠰⠆⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⣿⣌⠫⣳⣼⣿⣭⣭⠛⠋⠉⠀⠀⠀⠀⠀⠈⠉⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣷⣿⣦⣬⣽⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠋⠛⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠃⠀⡸⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣀⣼⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠞⠛⠛⠷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠉⣿⢻⠒⠒⠒⠒⠛⠉⣿⠁⠀⠀⠀⣼⣿⡏⣿⠓⠲⠦⠤⠶⠒⣿⠉⠉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣴⣿⣿⠟⢛⠻⠷⣶⣤⣿⣀⡄⠀⣈⣱⣿⣿⣿⣷⣶⣶⣿⣶⣶⣿⣠⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠛⢽⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⡇⣸⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⣠⣰⣤⣼⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⡙⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠦⣭⣶⣿⡿⠯⣭⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⠀⣿⣿⣇⠀⠀⣸⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠓⠒⠒⠒⠂⠀⠐⠾⣿⣿⣿⣯⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⠍⠉⠉⠉⠉⠙⢋⢻⣿⣿⣿⡟⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠱⢄⡀⠀⠀⠀⠀⠀⠀⣰⠶⠆
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣄⠀⠉⠓⠶⣤⣤⡤⢾⣿⠀⢠⢴⣴⡤
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣀⣀⠤⡄⠀⡀⡄⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣿⠀⣿⢸⡇⡇
⠀⠀⢠⡾⠁⠀⠀⠀⣠⣴⣾⣿⣿⣷⣿⣿⣿⣄⣇⣧⡀⢀⢰⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⠀⣿⢸⡇⣇
⠀⢠⠏⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢾⡷⣿
⢀⡟⠀⠀⠀⢠⣿⣿⡿⣿⠯⠁⠀⠉⢝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠹⣿⠛⡏⠛⠙⠋
⣸⠇⠀⠀⢃⣿⣿⣿⣱⠃⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠃
⣿⠀⠀⠀⠸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉
⣏⡄⠀⠀⢷⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠿⢷⣶⣤⣤⣤⣶⠿⠋⠁⠀⠀⠈⠛⢿⣶⣤⣤⣴⣶⡾⠿⠛⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⣀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠾⠿⠟⠛⠛⠉⠉⠉⠉⠛⣷⣦⣤⣤⣤⣤⣴⡿⠛⠉⠉⠉⠉⠛⠛⠻⠿⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠉⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⣤⣤⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡿⠛⠉⠉⠉⠉⠉⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡛⠛⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠿⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⠻⠿⠿⠿⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⣤⣤⣤⣤⣶⣶⣶⣶⣶⡶⠶⠾⠿⢿⣿⣦⣄⣀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢶⣶⣦⣤⣀⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣶⣤⣀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠿⢷⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣸⣿⣴⡶⠿⠿⠿⠿⢶⣶⣤⣤⣄⣀⣀⣀⣤⣤⣤⣶⣶⠾⠿⠿⠟⠛⠛⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⣤⣤⣤⣴⣶⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⠀⠀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠉⠙⠻⣿⣿⣿⡿⠋⠀⠀⠉⢿⣿⣿⣿⠿⠋⠉⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠅⠀⠀⠀⢀⠰⠶⣶⣄⠀⠀⠀⠀⣠⣶⣶⣦⡄⠀⠀⠀⠈⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⢹⣷⡄⢠⣾⣿⠟⠉⠑⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣼⠟⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⣀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠋⠀⠌⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣾⣿⠛⠉⠀⠀⠀⠀⠀⠀⢾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠉⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢰⣿⡿⠻⠻⢿⣷⡄⠀⠀⠀⠀⠀⢀⣠⢎⠻⠿⠿⠿⠿⠿⠟⠿⠿⣿⣷⡀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⢸⣿⡇⠀⠀⠈⠁⠀⠄⠀⠀⠐⣴⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠛⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢓⣀⡀⠀⠀⣀⡚⠛⠁

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠰⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡏⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡏⠁⠈⠙⠻⡆⠀⠀⠀⠀⠱⡀⠀⠀⠈⢢
⠂⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢣
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⡼
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠠⠄⠀⠤⣤⠶⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⠃
⠀⠀⠀⢀⡴⠛⠁⠀⠀⢀⣀⣀⡀
⠀⠀⣴⠋⠀⠀⢀⡴⣾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠲⢦⣄
⠀⡼⠁⠀⢀⡴⢋⡾⠁⣀⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⣸⠃⠀⠀⡾⠁⠺⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⣿⠀⠀⢸⠇⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠉⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢸
⠀⠀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸
⣰⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠉⠘⣇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠴⠶⠖⠛⠋⠁⠀⠀⠀⠀⠸⣆⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⡇⠀⠀⠀⠀⢸
⣿⠀⠀⠀⣶⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⣀⣤⡤⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⣤⣄⡀⠀⠀⠹⣆⠀⢸⠇
⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡴⢾⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢳⡿⢦⡀
⣿⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⡾⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇⠀⢷
⣿⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⢰⠄⠀⠀⠀⠈⠙⠛⠚⠛⠉⠀⠀⠀⠀⠀⢸⠇⠀⢸⠇
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⡟⠀⠀⠀⢰
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠴⠋⠀⠀⠀⠀⣿
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠉⠛⠛⠛⠛⢿⠛⠛⠳⣤⣀⢰⡟⠛⠛⠛⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠉⠻⠇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠘⢷⣄⣀⣠⡴⠃⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣀⣀⣀⣀⣤⡤⠶⠛⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣭⣽⣷⣶⣶⣾⢹⣿⣿⣶⣶⣶⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠁⠀⠀⠈⣻⣛⠃⠀⠀⠀⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣛⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⣿⣿⡶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣷⣤⡄⣀⡀⣀⣠⡿⣿⣿⣿⣿⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣼⣯⠏⣿⣽⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢰⣿⢿⡇⣿⣿⣿⣿⣇⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠁⠛⠙⠺⣿⠿⠇⠋⠁⠀⠉⠉⠛⠛⠛⠛⠉⠁⠉⢳⡝⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⢀⣤⡶⠶⣶⣄⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠁⠀⠉⠻⣶⣶⣶⠟⠁⠀⠀⠀⠙⢷⣶⣶⡾⠛⠁⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢀⣤⣤⣤⡄⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢤⣤⠶⠛⠁⠀⠈⠿⣦⡀⠀⣠⣾⠏⠀⠀⠉⠳⢦⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣇⣀⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣶⡒⠒⢲⡶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠈⠉⠻⢶⣄⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⣶⣶⣶⣶⣾⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠾⠇
⠀⠀⠀⠀⠀⢠⡿⠃⠠⠤⠤⠤⠼⣧⡀
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠟⠻⣄
⠀⠀⠀⠀⠀⢰⡿⡿⠉⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠈⢧⡁⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣁⣀⣀⠀⠉⠀⠀⢀⡀⣴⠋⠉⠉⠉⣳⡄⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣽⣟⣻⣻⡻⣛⣚⣶⠟⠧⣤⣤⠄⠉⣁⣀⣀⣀⣀⣀⣀⡀⡈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠙⠛⠻⢿⣿⣆⠀⠙⠃⠀⣠⣷⣿⣶⣭⣭⣝⣛⣉⣉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⠋⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠾⠿⠿⢿⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠧⠤⠤⣤⣼⢧⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣼⣧⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣶⣒⣻⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠿⠿⢿⣟⠛⣻⣟⣛⣧⣄⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡋⢙⠙⠋⠉⠉⠉⠉⠉⠉⠀⢨
⠀⠀⢠⠠⠤⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠀⠀⣀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠠⠤⠄
⠀⠀⠐⠐⠲⠛⠚⠇⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠻⠷⣦⣀⡀⠀⠠⣀⡀⠀⠰⢀⠃⠀⠀⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣡⢡⣆⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠸⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⢠⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⠒⣦⣰⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠸⢀⣀⢠⣄⡇⡇⠀⠀⠀⢀⡀⣢⢼⣮⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣼⡿⡅⢻⡀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⢸⣿⣿⣿⣿⣇⣿⣴⣤⣴⣶⣶⣷⣬⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣖⡂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠻⣿⣷⣦⡀⠀⢠⣀⡀⠀⠀⠀⣰⡽⠛⠁⠈⠹⣿⣿⣽⣶⣆⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢧⣾⠧⠤⠤⠶⠛⢳
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡬⠿⠷⢒⣒⡓⠠⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡯⣤⣴⣶⣞⣛⣛⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⣀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⣀⡀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⡤⠤⠔⠒⠒⢢⠀⠀⠀⠀⡔⠒⠒⠢⠤⢤⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠑⠂⣶⠒⠒⠖⠒⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⠛⠤⠤⠒⢖⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠔⠁⠀⠀⠀⠀⠈⠑⠢⠄⣀⣀⣠⠄⢤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢘⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⠴⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡰⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢼⡀⣰⠤⠤⠤⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠉⠑⠒⠒⠂⠐⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠠⠤⡤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⢼⠉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢆⠀⠀⡔⠐⡄⠀⡸⠀⠀⢰⠀⠈⠉⠁⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⢦⡤⡗⠒⠁⠀⡇⠀⠀⢸⠀⠀⠀⡰⢋⠷⠴⠃
⠀⠀⠀⠀⠀⠀⡠⡀⢰⠢⣱⡀⠀⢢⠜⠉⢲⣇⠔⠁⠈⠳⡀⡰⢁⠏⡤⠒⠠⡀
⠀⠀⠀⠀⠀⠈⠢⠊⠈⢣⡈⠓⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠳⠁⠮⠊⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⢤⣀⡀⠀⠀⣀⣀⡤⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⢀⣀⡀⠀⢀⡠⠤⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢤⣀⣀⣠⢤⢤⠈⠀⣀⣾⢈⣍⠁⣀⡀⠀⠀⠀⢈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠤⠍⠑⠓⠃⠛⠑⣜⣛⡀⠀⠁⠛⠓⠻⠭⠭⠏⡛⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠐⠀⢀⣀⡛⣛⡂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⠁⠨⡉⠉⠍⠭⣛⡻⠇⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠄⠀⠉⠀⠀⠅⠀⡐⠄⠀⢱⠈⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⢀⣀⠀⠀⠀⠀⠀⠄⠀⠄⡂⠀⡾⠁⠂⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡤⢚⠉⠀⢀⢨⡀⠀⢀⠀⠀⠀⠀⠀⡇⢁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠁⠀⡐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⠎⠀⠀⢐⡨⠌⠈⠀⠀⠀⠀⡈⠐⠀⠀⣗⠪⠀⠁⠀⠀⠤⠤⠤⠀⠀⠀⠀⠰⠊⡀⠢⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡴⠁⠀⠀⠔⠁⠀⠀⢠⠀⠀⠠⠀⠢⠁⠀⠀⠧⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⢤⣤⣶⣶⡄⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠄⠀⠠⡀⠀⠀⠀⠀⢆⠁⡂⡀⠠⠀⠐⡃⣤⢤⠠⠄⠤⠤⠤⠄⡤⢠⠠⢤⠤⠴⠂⢾⠀⡇⠰⠶⠶⠶⠦⢸
⠀⠀⠀⠀⢀⠀⠀⠈⡄⣰⡀⣀⢀⡔⢀⡘⡀⠀⠀⠀⠈⠐⠨⠀⠀⠠⠈⢐⠔⠕⡡⡪⡨⡪⢋⡪⠢⢂⢂⢀⡂⠠⡀⠈⠀⡇⣘⣛⣻⣿⠗⢘⠅
⠀⠀⠀⠀⠀⡀⠀⠀⡇⢻⡇⣿⠈⡇⢸⠁⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠂⠀⠁⠀⢄⠁⠰⢀⣹⣾⣭⡭⠁⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠠⣅⢈⣁⣛⣀⣓⠘⠡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⣐⡲⠶⢒⣂⠼⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠕⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣊⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡫
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡟⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢆⡐⠠⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⡿⠿⠿⠛⢻⡟⠛⠛⠛⠛⠋⠉⠉⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠿⠟⠛⠛⠛⢛⣿⡟⠛⠛⠛⠉⠉⠉⠀⣼⡇⠀⠀⢀⣀⡀⠀⠀⣿⠇⠀⠀⣤⣴⠄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⣀⣠⣾⣿⠁⠀⠀⣴⣶⠀⠀⠀⣿⠁⠀⠀⣿⣿⠁⠀⢰⣿⠀⠀⠀⣿⣿⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⠀⠀⢸⣿⣿⣿⡿⠀⠀⢠⣿⡿⠀⠀⢰⣿⠀⠀⢸⠿⠋⠀⣠⣾⡏⠀⠀⠈⠉⠁⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⡟⠀⠀⣾⣿⣿⣿⡇⠀⠀⠈⠉⠁⠀⠀⣸⡇⠀⠀⠀⢀⣤⣾⣿⣿⠇⠀⠀⣴⣶⡆⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⢀⣿⣿⣿⣿⠀⠀⠀⣾⣿⠇⠀⠀⣿⠇⠀⠀⣴⣿⣿⣿⣿⣿⠀⠀⢀⣿⣿⣀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⡟⠀⠀⢰⣿⣿⠀⠀⢠⣿⠀⠀⣠⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⣼⣿⣿⣿⡇⣀⣀⣾⣿⣿⣶⣶⣾⣿⣾⣿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠈⣇⠹⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠉⢉⠽⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⢿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⢾⣆⠀⠀⠀⠀⠀⢀⡀⢤⣾⡀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣄⡀⠀⡄⠈⢿⣿⣇
⠀⠀⠀⢀⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠈⠂⢄⣀⠀⠀⠀⠀⠈⠁⠑⢤⣄⣀⣠⠴⠃⠀⢀⡼⠀⠀⢠⠀⠀⣸⣿⣿
⠀⠀⠀⣿⣿⣿⣫⣤⣤⣶⠂⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣦⣴⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣭⣉⣩⡤⠶⠚⠁⣹⣿⣿⣿⠟
⠀⠀⠀⢹⣿⣿⣿⡿⠟⠁⠀⠀⠴⠚⠉⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠠⣿⣿⠇
⠀⠀⢠⣿⣿⠛⠉⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠿⠿⢿⣷⣶⣶⣶⣤⣤⣄⣀⣈⣼⣿⡿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋⠉⠉⠉⠉⣠⣿⡏⠉⠉⠉⠉⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠈⠙⠛⠛⠛⠛⣻⣿⡿⠋⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⠃⠀⠀⢰⣶⣶⠀⠀⠀⣼⣿⡇⠀⠀⢠⣶⣶⠆⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⡏⠀⠀⠀⣿⣿⠃⠀⠀⢸⣿⡿⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢹⣿⡟⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠛⠋⠁⠀⠀⢰⣿⣿⠀⠀⠀⠋⠁⠀⢀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⢠⣤⣤⠀⠀⠀⢸⣿⡏⠀⠀⠀⢀⣠⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢀⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⢰⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⣠⣤⣴⣿⣿⣧⣀⡀⢀⣿⣿⣀⣤⣤⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣴⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠛⠒⠦⢤⣄⣀⣀⡀⣰⠃⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣤⠤⠖⠛⠉⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢰⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⢀⣀⣤⠴⠖⠚⠉⠉⠉⢻⡀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠙⠓⠲⠤⣤⣀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢶⠓⡦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠶⠶⠶⠶⠶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⢾⢿⠥⡤⡤⣤⣤⣤⣤⣤⣽⡶⡦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠚⠒⠶⠶⠶⠶⠶⠷⠾⠶⠶⠷⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⡶⠒⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣰⠏⠀⠀⠀⠈⢳⣤⣀⣀⣀⣀⣀⣀⣤⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⣿⠀⠀⠀⠀⢠⡏⠀⢠⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠛⠒⠲⠶⠤⢤⣄⣀⡀
⢸⡇⠀⠀⠀⠀⣼⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠸⡇⠀⠀⠀⠀⢿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⣿⠀⠀⠀⠀⢸⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⢹⡄⠀⠀⠀⠀⢧⣀⣀⣷⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢻⣄⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠶⠤⣤⣄⣀⣀⣀⣠⣤⠤⠶⠖⠚⠛⠛⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠓⠦⣤⣀⣀⣀⣀⣀⣀⣤⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣤⣤⣤⣤⡴⠶⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡄⠀⣿
⠀⠀⠀⢀⡶⠲⣦⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠛⠛⢻⡆⠀⠀⢰⡟⠉⠠⠾⢧⡄
⠀⠀⢀⣬⠷⠴⠿⢿⡴⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡶⠟⣠⣼⠃⠀⠀⠸⠧⠶⠶⠶⠾⠃
⠀⢠⡟⠁⠀⠀⠀⠀⡳⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣈⣹⡇
⠀⠈⠻⣦⠤⠤⠴⣾⠓⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣉⡉⠉⢹⡇⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠛⠁⠰⠿⡄
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠏
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⣠⠶⣆⣀⡴⢶⡄⣀⡀⠀⠀⢀⣀⣤⣀⣀⣠⣤⠴⠶⠶⠖⡶⠛⠛⠒⠒⠶⢦⣤⣀
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⣀⣀⣀⡏⠀⢹⠋⢠⠾⠋⢉⣿⣤⣤⠏⠁⠀⠋⠉⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠉⠳⣦⡀
⠀⠀⠀⣿⠀⠀⠀⣿⠀⡼⠉⡸⠋⠙⠀⠀⠀⠀⠀⠰⠛⠻⣄⡏⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⣄
⠀⠀⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠚⣼⠁⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⢀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢰⠘⠖⠛⠓⠲⠶⣤⡀
⠀⠀⠀⣿⠀⠀⠀⣿⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⣿⠀⠀⠀⣿⠀⡠⠛⠶⠶⠶⠶⠶⠶⠲⠒⢸⠇⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠶⢶⡄
⠀⠀⠀⣿⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⡄⢸⡇
⠀⠀⠀⣿⠀⠀⠀⣿⣀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡇⠀⠠⠀⢸⡦⠼⠷⢦⡀
⠀⠀⠀⣿⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⠀⢃⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⣿⠉⢹⠛⠛⠛⠛⠛⠛⠛⠛⠙⢻⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⡟⠉⠉⡟
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢀⣀⣄⣠⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣇⣀⣀⣀⣠⣅⣠⣤⣤⡇⠀⠀⡇
⠀⠀⠀⣿⠀⠀⠀⠸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⡇
⠀⠀⠀⠙⣶⣒⣶⢾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣧⣤⣤⠇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⣀⣤⣤⣀⣀⡀⠀⠀⠀⣠⠖⠊⠉⠙⠲⣄⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⣼⠁⠀⠀⠀⠉⠉⠉⠛⢻⣿⡄⠀⠀⠀⠀⠈⣏⠉⠉⠉⠉⠉⠉⠙⣷⡄
⠀⠀⠀⣿⣦⣀⣀⣀⣤⣤⣤⣤⣿⣿⣗⠈⠀⢀⡄⠀⣿⣤⣤⣤⣲⣄⣀⣴⣿⠇
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠉⠉⢿⣿⣿⣶⣿⣾⡷⠉⠉⠉⠛⠛⠻⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡦⠤⠤⠿⠷⠼⠦⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣶⣶⣦⣤⣤⣤⣼⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡽⠿⠿⠼⠤⠿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡇⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠻⠷⠦⠤⠤⠤⠴⠿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⡿⠿⠧⠤⠤⠤⠤⠤⠤⠤⠼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠻⠛⠛⠛⠛⠓⠒⠒⠒⡲⣾⠁
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⡶⠶⠶⠶⠶⠶⢶⢶⣶⠇
⣿⢛⢻⡷⣄⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⠀⠀⠀⢘⢨⣇
⣿⠀⠀⡯⠘⠛⠷⠶⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⣷⣄⡀
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣐⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣤⣄⡀
⣿⠀⢠⡇⠀⠀⠀⠀⣀⣥⣾⣶⣎⡩⠁⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣄⡀
⣿⣈⣘⣇⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣥⣄⣀⣂⡀⠤⢀⣰⣀⣀⢀⠐⠄⡀⠀⠀⠙⢿⣄
⣿⣰⢸⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⢠⠀⠀⠀⢻⡆
⣿⣿⣾⣯⠟⠁⠀⠀⠀⠀⠙⢿⣿⣻⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠉⠙⢿⣿⣷⣃⠁⠀⠀⠀⢹
⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣻⣿⡇⡆⠃⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠁⠀⠀⠀⠐⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠒⠒⠒⠒⠒⠒⠒⠻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⣿⢿⣀⠀⠤⠤⠤⠤⠤⠤⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⡓⠲⣂⠶⣲⠖⣲⠶⣖⠲⡖⠲⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⡈⠢⢈⣆⡥⢊⡡⠺⣏⢦⡈⠢⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢢⡈⢦⢘⣯⠔⣫⠔⡡⣞⢤⡙⢦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢄⠑⢄⢙⡄⠚⡡⠞⡡⡞⢄⠙⢆⠙
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣾⣶⣾⣶⣿⣶⣷⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⠶⠞⠓⠶⠶⠟⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⢀⡞⠁
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢰⡇⣀⣀⣤⣤⣄⣀⡀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠦⠶⠶⠖⠛⠉
⠀⠀⢸⣏⠉⠀⠀⠀⠀⠈⢉⣿
⠀⠀⠀⠉⠙⠛⠒⠒⠚⠛⠋⠁
⠀⠀⠀⠀⠀⣠⠞⠙⢧⡀
⠀⠀⠀⢀⡼⠃⠀⠀⠀⠻⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠘⣦
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠙⠶⢤⣀⣀⣀⣀⣠⡤⠞⠁
⣶⣶⡶⠶⠶⠾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠞⠛⢻⣿⠿⣿⠛⠛⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠾⢿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣰⣟⠉⢻⡙⠻⠿⠟⠛⠙⠻⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠟⠋⠉⠙⢷⡄⢳⡀⠀⠀⠀⠀⠀⠈⢿⡄⢻⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠘⣷⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡿⢸⠇⠀⠀⠀⠀⠀⠀⣾⠃⣼⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢲⣿⣾⣡⣟⣀⣀⣀⣀⢀⣠⡾⠿⠾⠷⠶⠶⠶⠶⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣿⠀⠀⠀⠀⢸⣿⣬⠉⠙⠉⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⢦⣙⠷⠶⠶⠶⠾⣫⠼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣌⣙⣛⣛⣛⣋⣡⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⡏⠉⠉⠉⠉⠉⡉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⡇⠀⣤⠀⠀⠈⣿⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣿⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⣿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⣟⠀⢰⡆⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡿⠀⢸⠀⣿⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⢀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢸⡇⢀⣸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢸⡇⢸⢸⡇⠀⠀⣿⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢸⡇⣸⠀⠀⠀⠀⣿⠀⠉⢀⡤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣀⠀⠁⢸⡇⠉⠀⢸⡇⠀⣿⠀⢀⣤⣀
⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⣴⠛⠛⠇⠸⠇⠀⠀⠘⠁⠀⣻⠄⠘⠋⢉⣤⡀⠀⠚⠉⠉⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠁⢠⡴⠄⠀⢀⣀⠠⣄⠘⢃⠀⠛⠦⠀⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠓⠀⠀⠟⠀⠀⠈⠁⠹⠦⠀⠀⠀⢉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⠾⣧⡀⠀⣴⠆⠀⣷⣀⣴⠿⣦⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠉

⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣥⣦⣧⣤⣦⣤⣷⣤⣷⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠻⠷⣶⣶⣴⡶⠿⠋⠀⠀⠀⠈⠙⠿⣶⣶⣶⡶⠾⠟⠛⠛⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣀⣀⣠⣤⣤⣴⣶⣦⣤⣄⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣤⣄⣀⣀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠀⠀⠀⠀⠈⠻⣿⠿⠿⠿⢿⣿⠋⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠿⠿⠿⠿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⣾⣿⣯⣤⣤⣤⣤⣤⣤⣤⣽⣿⣶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣥⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⣤⣤⣴⣶⣶⣶⡶⠶⠶⠾⠿⠿⢿⣿⣦⣄⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠶⣶⣦⣤⣀⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⣦⣄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⣿⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠷⠶⠾⡿⠛⠉⠁⠀⠈⠉⠙⠛⠛⠿⠷⠾⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣀⣀⣤⣤⣴⡶⠿