Gelatine Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⠭⣽⢯⠥⠤⣹⢋⠥⠬⠝⣏⡥⠭⠿⣯⠭⡗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡄⠀⡟⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠁⢀⡇⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡖⠀⢸⠃⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠘⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⡇⠀⣸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠸⠁⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡇⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡇⠀⠀⠸⡆
⠀⠀⠀⠀⠀⢀⣀⣼⣀⣀⣀⣼⣀⣀⣀⣀⣸⣅⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣧⣀⣀⣀⣧⣀⡀
⠀⠀⠀⠀⠀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠙⡏⠉⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠆⣠⠶⠛⠻⢷⡆⣠⡶⠛⠛⠶⣄⠰⣾⠟⠛⠲⡄⠰⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣴⣶⡶⢠⣷⣾⣿⣿⣶⢠⣿⣶⣿⣿⣶⣿⡆⣶⣾⣿⣷⣾⡄⢶⣶⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣿⣿⡇⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⠇⣾⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣾⣿⣿⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⣿⢰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⠇⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⣤⡴⠞⠛⠛⠳⢦⣴⠞⠛⠛⠳⢦⡴⠞⠛⠛⠳⢦⣤⠶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣵⠶⠶⣶⠏⠴⠛⠛⠻⢦⡟⠁⠾⠛⠛⠳⠈⢻⡤⠞⠛⠛⠷⠹⣶⠶⠶⣮⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢠⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡏⣰⠄⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣸⠇⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⣿⢠⡏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⡏⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣼⠃⠛⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢿
⣠⠶⠶⠾⠿⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠾⠷⠶⠶
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠿⠷⠲⠶⠶⠶⠶⠿⠳⠶⠶⠖⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠿⠿⠿⠛⠛⠿⠿⠿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣠⣾⡿⠋⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⢠⣴⡾⠿⠛⠛⠛⠛⠛⠛⠿⢷⣦⣾⡿⠋⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⢠⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣾⡟⢿⣶⣤⣤⣀⣀⣀⣠⣾⣿⣿⡿⢻⣷⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣸⡿⠀⢠⡄⠉⠉⠛⣛⣿⡿⠋⠉⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢠⣿⠃⢠⣿⠇⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡏⠀⣾⡟⣠⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠘⣿⣄⡀⢩⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠃⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠈⣹⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡿⠿⠋⠁⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣠⣾⡿⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢷⣶⣶⣤⣤⣤⣤⣶⣶⡾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⢀⡈⠉⠉⢉⠉⠁⣀⣀⡀⡀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢩⣅⢸⣯⣭⢸⡇⠀⢀⡞⣧⠀⠉⡏⠁⣿⢸⡿⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠛⠘⠷⠶⠘⠷⠖⠚⠉⠙⠃⠐⠃⠀⠛⠘⠃⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠆⢰⠶⣦⢰⡶⠶⢠⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠃⣸⠻⣏⢸⣟⣛⢰⣟⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠈⠀⠉⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⣶⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠈⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⣶⣶⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⣴⡾⠿⠿⣶⡿⠋⠀⠀⠀⠈⠻⣷⡾⠿⠿⣶⣤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⢿⡿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠹⣿⠟⠛⠻⢿⣦⡀
⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠹⣷
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢀⣿⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢹⣇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⡿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠸⣿
⠀⠀⠀⣾⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⣿⡆
⣴⣶⣶⣿⣷⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣿⣷⣶⣶
⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⣀⣙⣻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣛⣁
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠉⠉⠂⠊⠀⠀⠀⠔⠀⠈⠐⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠐⢀⠐⠂⠁⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠠⠀⠀⢀⡇⠀⣗⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⡄⠀⠀⢐⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠈⠉⠓⡀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣐⠆
⠀⠀⠀⠂⠀⡇⢀⡆⠞⠉⡐⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠀⠊⠁⠀⠀⣀⡄
⠀⠀⠀⠡⣄⠈⠛⠚⠷⠠⠤⠦⡈⠙⠛⠳⢄⣀⣀⣀⣠⣄⡀⠀⠀⠀⠀⣀⣠⣤⠀⡘⠻⠀⠂⠀⢀⣴⣄⡀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠈⠙⠢⢤⣀⡀⠀⠀⠈⠐⠒⠒⠢⠤⠄⠀⠤⠤⠭⠭⠩⣙⣉⣩⡑⠀⣠⡤⠶⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠤⠤⠤⢤⣀⣀⣀⣀⣀⣀⠀⠀⢈⠉⢀⣁⣀⣀⣀⡠⠤⠤⠤⠒⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣲⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⣠⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣷⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡷⣿⣿⣇⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⡜⢿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⣴⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⣄
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠙⠋⠛⠙⠃⠘⠃⠈⠉⠉⠉⠛⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⡇⠀⣶⣶⡀⡖⢀⢀⣶⡆⢰⡖⠃⢰⢶⡀⠀⠀⣴⣆⠀⢷⣄⢰⡏⣶⠀⠀⣷⢷⡆⣾⢴⡇⣾⠀⢿⠗⢰⡶⠂⠀⣿⣿
⠀⠀⣿⡇⠀⠉⠉⠈⠉⠉⠈⠉⠉⠈⠉⠁⠈⠈⠁⠀⠈⠁⠉⠀⠁⠁⠈⠉⠁⠀⠀⠀⠈⠀⠉⠈⠁⠉⠀⠉⠁⠈⠉⠉⠀⢹⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠘⠒⠀⠀⠐⠀⠂⠐⠃⠀⠀⠀⠀⠂⠀⠀⠒⠂⠐⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠁⠈⠀⠀⠁⠀⠈⠀⠁⠀⠈⠉⠉⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⣿⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⠀⡀⠀⠀⠀⣴⠋⠙⡃⠀⡀⠀⣿⠀⣀⠀⣰⡀⠖⢀⢀⠀⠀⢀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⣿⠻⣆⠈⡟⢹⣦⡿⢹⣧⠹⣾⠋⠀⠺⣯⡀⠀⠀⣿⠀⠘⣿⣾⡽⠃⣿⣨⠽⡇⣿⠁⣿⢼⠛⣿⣼⡽⠗⠀⢸⣿
⠀⠀⣿⡇⠀⠛⠀⠻⠓⠗⠘⠃⠛⠞⠃⠛⠘⠗⠀⠒⠚⠃⠀⠀⠘⠓⠚⠋⠙⠓⠊⠛⠙⠚⠓⠛⠒⠛⠚⠃⠻⠘⠓⠚⠀⢸⣿
⠀⠀⣿⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⢀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⡀⠈⠁⡈⠉⡀⠀⠀⠀⢀⠀⢁⠀⢀⠈⡁⠉⠀⠉⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠰⠄⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠄⠠⠀⠠⠀⠀⠠⠀⠠⠄⠄⠀⢸⣿
⠀⠀⣿⡇⠀⠒⠂⠐⠀⠆⠀⠐⠀⠒⠐⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠂⠂⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠑⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠃⠀⠐⠈⠛⠀⢸⣿
⠀⠀⣿⡇⢀⡉⡁⢈⠁⠁⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⡀⠀⠀⡀⠀⠀⡀⠀⠀⢀⢀⣉⠉⣁⡀⢀⠁⠉⠀⢸⣿
⠀⠀⣿⡇⠈⠁⠠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠄⠠⠀⠄⠀⠀⠀⠀⠠⠠⠀⢸⣿
⠀⠀⣿⡇⠀⠐⠒⠀⠐⠀⠀⠀⠐⠀⠈⠂⠀⠀⠀⠀⠀⠀⠂⠀⠂⠐⠒⠀⠐⠀⠀⠐⠀⠀⠂⠐⠐⠀⠛⠘⠛⠛⠃⠚⠀⢸⣿
⠀⠀⣿⡇⠀⠉⠀⠈⠀⠁⠁⠀⠀⠀⠁⠀⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⡸⣆⠀⢸⠥⢿⢰⢲⢸⡂⠀⡺⣇⣶⣴⢶⢶⢴⣶⣷⣤⠀⠀⡇⠁⣷⡗⡿⢉⣿⢉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⡀⠀⢁⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠈⠀⠀⠀⢀⠁⢈⡀⠁⠀⡉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠋⠀⢸⣿
⠀⠀⣿⡇⠀⠀⣀⠀⢀⠀⢀⡀⠀⢀⢀⣀⠀⡀⠀⡀⢀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⣸⣿
⠀⠀⢿⣷⣄⣃⡙⠘⡈⡀⣈⠃⠀⠘⠈⠉⠀⠁⡀⠁⠘⢛⡛⢛⢁⡀⠘⠉⠙⠐⠀⠛⠛⠛⠛⠛⠋⠛⠛⠂⠈⠁⠈⢃⣠⣿⡟
⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⡽⠞⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢼⣀⣠⣼⣥⣀⣠⠼⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣇⠀⢀⡿⠁⠀⠀⣹⣅⠀⠀⠀⢸⠏⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠉⠛⠉⠷⣤⡤⠞⠁⠘⠦⠤⠶⠋⠀⠀⠀⣸⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠓⠶⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠛⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢀⡤⢶⡟⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠹⡄⠀⠀⠸⣧⣤⡀
⠀⠀⠀⠀⣴⡟⠀⢾⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡿⠈⠻⣆
⠀⠀⠀⢸⣿⠁⠀⠘⠦⣤⡴⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇⣠⡾⠁⠀⠀⣿⡄
⠀⠀⠀⢸⠘⣇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣼⣄⣀⠀⠀⢀⣠⠾⠛⠁⠀⠀⠀⣼⢻⡇
⠀⠀⠀⠘⣇⠈⠳⣄⡀⠀⠀⠀⠀⠈⠙⠦⠤⠤⠴⠞⠉⠳⠶⠤⠤⠶⠚⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⣠⠞⢁⡾
⠀⠀⠀⠀⠘⢧⡀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⡀⠉⠙⠳⠶⠤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠞⢛⣉⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠦⠤⣤⣤⣀⣈⣉⣉⣉⣉⣉⣉⣩⣭⣭⠭⠴⠶⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣭⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⣀⣤⣤⣾⣶⡶⠾⠷⠶⣶⣶⣤⣤⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠁⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⢠⣿⣿⣿⠏⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣛⣛⣛⠛⣛⡓⡒⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⡟⠟⢻⠛⠛⠛⠛⠛⠛⠛⢻⣿⠟⠛⠛⠛⠛⠛⣿⡿⠻⣿⡟⠛⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⡧⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⣧⡴⠻⢿⣶⡄⠀⠀⠀⣿⠀⢀⣴⠟⢿⣷⡄⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡌⢀⣿⣷⣤⣾⣿⡿⠀⠀⠀⣿⠀⢸⣿⣤⣾⣿⣿⠀⠀⢇⣿⡀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠐⠂⣀⣠⢾⣿⠇⢸⠏⠻⠿⠶⠟⠁⢤⣀⣀⣀⣴⠀⠛⠿⠷⠞⠁⢇⠀⢸⣹⣇
⠀⠀⠀⠀⠀⠀⣴⡟⠉⠑⠋⠊⡠⣸⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⡀⡘⣆⠀⠻⣿⣦⡀⠀⠶
⠀⠀⠀⠀⠀⠰⣿⢂⠄⠀⢀⡴⡣⠃⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢳⡈⠳⠄⠈⠺⢽⣦
⠀⠀⠀⠀⠀⠀⢻⣞⢆⢰⡋⠀⠀⢀⠀⠀⠁⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⡀⠀⢰⠈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠿⠒⢶⣏⠀⠰⣄⠀⢀⣤⠫⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⢀⣠⣶⣇⡠⠞⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠾⠛⠋⠉⠳⣤⣤⣶⠟⠳⠶⠶⠾⠟⠻⠿⣿⣿⣯⠶⠞⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣦⠴⣛⣛⠓⣦⠞⣛⣛⠲⣴⠚⢛⣛⢦⡴⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠒⠋⢹⡏⠚⠉⠉⢹⡇⠘⠉⠉⠃⢸⡞⠋⠉⠉⢹⡏⠙⠃⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⠆⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⠀⠀⣾⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣷⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⡟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢻⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠇⠀⢸⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⠀⠀⢸⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣇⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠁⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⠾⠧⠶⠶⠶⠾⠷⠶⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠃


⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠘⠯⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⡀⢀⢀⣀⢀⡀⠀⣀⡀⡀⠀⣀⠀⠀⣀⡀⢀⣀⢀⣀⡀⣀⣀⠀⡀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢐⠀⡹⢠⠠⡇⡼⢱⡀⡓⡇⣇⠀⣗⠂⠀⠙⣦⠀⡇⢸⠲⡙⡅⢀⠇⡿⣄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠈⠁⠀⠈⠁⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠘⠀⠀⠀⠐⠀⠂⠘⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠘⠂⠘⠁⠃⠈⠂⠁⠊⠐⠁

⠀⠀⠀⠀⠀⠀⢀⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⣤⠀⢢⡴⠂⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠇⠀⠀⢀⡄⠑⠀⠀⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⠐⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠁
⠀⠀⠀⠠⠀⠀⠀⠀⢀⢀⢀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⣀⣉⣿⠾⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡗⠀⠀⣴⣁⢀⣀
⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡶⠴⣮⠤⠀⠀⠹⣥⣎⣡⠀⠀⠴
⠀⠀⠀⠀⠀⢀⢸⠀⡖⣣⡀⠀⢲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⢀⣰⣀⠀⠀⠀⠉⠁⠀⠀⡄
⠀⠀⠀⠀⠀⣄⢸⣷⡏⠻⠹⠶⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠾⡻⣦⠀⠀⢰⣆⠠⢤
⠀⠀⠒⣶⣶⡿⠀⠋⠐⠿⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠆⠀⢹⡅⠸⠷⠀⢀⠉
⠀⠻⠿⢿⣿⠷⠃⠀⢌⠀⡀⠈⢃⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⢸⡇⠀⠀⠈⠋⠀⡀⠠⣏⢁⡀⠉
⠀⢠⣄⣼⠏⠀⠀⠀⢶⣬⣿⣧⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠺⡅⠀⠀⣬⠠⠤⠅⠀⠙⢍⡀
⠀⠀⠿⠃⠀⢦⠀⠐⠄⢹⣯⡿⠋⢫⠀⣀⡀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⢁⠀⠀⣠⡷⢖
⠀⠀⠀⠀⠀⠀⠂⠀⢴⡿⠀⠀⠀⠀⢄⠈⠛⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⡀⠀⣋⠀⠀⣿⣲⣴⡷⠦
⠀⠀⠀⠀⠀⠀⠀⢀⡌⢙⡿⡗⢉⠶⡄⠑⠆⠀⢀⣤⣴⡏⢀⣄⡀⢄⠀⠀⠀⠀⢀⣠⡴⠖⠒⢶⣀⠠⢀⠙⢿⣝⢂⣰⠃⡼⡏⢻⢿⣯⠴⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⢀⣭⣩⡆⢠⣤⣞⢶⠿⣿⣿⠃⠠⢤⣬⣙⣿⠀⣰⣾⣿⡿⠓⢀⡀⣸⣿⣄⡀⠈⢹⣟⣄⡈⢠⡅⢠⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣦⣸⣿⣮⠀⣿⣿⣟⣿⣷⣾⣿⢿⠟⣻⣿⣟⠿⣿⡿⣿⣟⣼⠏⡀⠀⠘⣄⡀⠀⠀⠙⠂⣱⠄
⠀⠀⠀⠀⠀⢀⢀⠂⠀⠁⠀⠀⠀⠘⠉⠉⠻⠛⠇⠈⠉⠛⠋⢉⡿⠛⣾⠀⠉⣯⠙⠀⠈⠉⠉⠉⠹⠟⠉⠁⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠘⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠁⠙⠁⠀⠀⠀⠠⣰⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⠶⣿⣍⣉⡛⠲⢤⡴⠚⠋⠉⠛⢲⣤⠖⠚⠛⢛⣲⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⣀⣤⡤⠤⢤⣬⣉⠓⢦⣙⢦⠀⠀⢠⠞⠁⢀⡴⢚⣉⣡⣤⣤⣄⣈⣉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣠⠶⠋⠁⠀⢀⣠⠶⠚⠛⠛⠛⠛⢾⡶⠛⠉⠛⠲⣿⡛⠉⠉⠙⠓⢦⣄⠈⠉⠙⠲⢤⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠁⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⡈⢳⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡞⠁⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠳⡄⠙⣦
⠀⠀⠀⠀⠀⡼⠁⠀⣠⠏⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠹⣆⠈⢷⡀
⠀⠀⠀⠀⡼⠁⠀⣰⠃⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠘⣆⠀⢳⡄
⠀⠀⠀⣸⠃⠀⡼⠃⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⢧⠀⠹⡄
⠀⠀⣰⠇⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⢧⠀⠹⡄
⠀⢠⠏⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⢧⠀⠹⡄
⢀⡟⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠘⣧⠀⢻
⢸⡁⣰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠘⣇⢨
⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠸⣟
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠹
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣀
⠀⠉⠙⠛⠒⠒⠒⠚⠛⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠧⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⠛⠶⢤⣄⣀⣀⣀⣀⣤⡤⠶⠛⠁⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣤⢀⡀⠀⠀⣀⣀⣤⣀⡀⠀⢸⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⠉⠻⣿⣤⣴⠿⣿⣿⣿⣿⢿⣧⣨⣿⡅⠀⣰⣶⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢄⠲⣦⠩⢾⣿⣿⣿⣿⣿⠳⡄⠹⣿⡿⠿⣋⣿⣽⠻⡶⢘⣿⠛⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡽⢆⣤⣷⣮⣿⡣⠎⡵⣹⣿⣿⠟⢡⠻⣠⢹⢁⡶⠏⣼⡽⡓⣁⠼⡷⠪⣙⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⡝⠻⡷⣼⡌⣾⣿⣷⣶⣾⣿⣷⣿⣭⣷⣯⠾⢿⣿⡧⢠⣟⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠹⣆⠈⣾⣳⣿⣿⣿⡿⠿⠿⢿⣟⣿⣿⠋⠖⠧⢈⢀⣨⢿⣿⡷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⠀⠻⣮⣾⣿⣽⡿⢉⡶⢛⣛⠳⡜⣿⣿⣧⠀⠀⢱⣁⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣸⣬⣿⣷⣸⣿⣻⣿⢿⡇⣷⣇⢿⣧⣴⢇⣿⣿⣿⠿⢧⠉⣹⣗⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠛⢿⡿⣿⣹⣛⠿⡿⣮⣇⢛⢟⣶⣤⣴⣿⣿⣿⠣⠹⡰⠞⠻⣿⡇⣠⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠙⢿⢎⡻⣶⣿⠃⡬⣹⣞⡯⠇⠣⢽⣿⡋⢗⠂⣤⡧⣄⣤⣿⣿⣿⠟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⣦⢿⢫⠀⠈⢿⡜⣽⣯⡏⢠⡒⠖⡄⢫⢷⣤⡴⣻⣵⣿⣿⣿⣿⣿⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⢎⠉⠈⣈⣎⢯⠠⡒⣻⣟⣿⣧⣘⣾⣿⣂⢼⡿⣑⣾⡽⣿⡿⣿⣿⣿⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⡂⢁⠵⠋⣁⣤⡾⣿⣿⢿⣿⣿⣿⣾⣿⣿⡷⡿⣟⠟⡵⣚⠝⡿⢿⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣾⡓⠅⣠⠋⣰⡾⣯⠷⣛⣭⣭⣍⡻⣿⣿⣷⣿⣽⣿⢳⣎⡞⡛⣫⣼⡇⠈⠱⠆⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⣾⢅⡀⢱⠂⣼⣿⡿⣡⣾⢿⣿⢡⣽⢇⣿⣟⢾⣿⣷⡉⣻⣼⠞⢱⣯⣀⠑⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢨⣯⠅⢐⠇⢐⣿⣾⢡⣿⢛⡿⠛⣳⠶⣏⠷⣭⣾⣿⡁⣹⣿⡿⠟⠚⠉⠛⢳⣄
⠀⠀⠀⠀⠀⠸⣿⠶⢴⠀⢸⣿⣿⢸⣿⠿⣿⢿⣍⣓⣤⣿⡟⢫⣀⢝⣵⠁⣴⠄⢠⠀⠡⠀⣿⠄
⠀⠀⠀⠀⠀⠐⣻⡯⢀⡄⠀⣿⣿⡌⡿⣿⣛⠗⣶⣽⣏⠛⠛⠉⠐⢿⣿⣷⣿⡀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠒⣿⠥⡺⡄⢹⣿⣷⣹⡱⢏⡾⠿⢿⣮⢿⣄⠀⠀⣼⣿⢿⡟⠻⠶⠶⠶⠟
⠀⠀⠀⠀⠀⣀⣀⠰⣿⣴⡟⠶⠙⠁⠛⠍⡝⠀⢂⡆⢸⣥⣿⣤⣾⣿⣳⠯
⠀⠀⠀⢠⣾⣿⣹⣿⣝⠃⠁⠀⠀⢠⠻⣼⠷⣤⣍⣀⣼⣿⣿⣿⡫⠾⠁
⠀⠀⠀⠐⣿⣿⣽⣿⢇⡀⠀⠀⠀⢧⠪⠴⠔⣿⡽⣿⠞⠛⠃⠉
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠷⠿⡴⠦⢿⡆⠀⣰⡏⠀⢠⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣷⣶⣿⣿⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡵⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⣀⠀⠐⣾⡀⠀⣩⠆⢀⣀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⠤⠖⠤⣬⠓⠚⠉⠉⠉⠉⢓⢻⡤⠶⠶⣌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⠁⠀⣠⠖⠉⠉⢱⠖⠀⠐⣎⠉⠉⠑⢄⠀⠀⠳⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡼⠀⠀⣰⠃⠀⠀⠀⠏⠀⠀⠀⠘⡆⠀⠀⠈⢧⠀⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢰⠁⠀⢠⡏⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡆⠀⠀⠂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣸⠁⠀⠀⠇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⢱⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡿⠀⠀⢸⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠘⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠇⠀⠀⣼⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢿⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⠆⠀⠊
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⠢⢀⣀⢀⣀⣀⠤⠚⠀⠀⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠀⠠⠤⠄⠀⠠⠤⠄⠀⠀⠒⠈⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠇⠀⠀⠠⡄⡀⠀⢠⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠁⠀⠀⠀⡄⠀⣀⠀⠀⠀⠀⢀⢤⣌⡛⢏⡤⢏⣠⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣵⡔⠀⠀⠘⢁⢾⢠⣤⡀⠐⠀⢸⣷⠈⢿⣾⣧⣬⣇⣘⡷⠀⡖⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢮⣿⣷⠀⠠⠈⢿⣿⣿⣿⡷⠀⠀⠀⢋⡑⣚⠷⢾⠋⢋⡉⠠⠛⢩⡏⢹⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⢿⣧⠀⠻⡞⢷⣻⣿⣥⠀⠀⠀⣼⣿⣍⣤⣀⠁⠻⠲⠀⠀⢠⡍⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠒⠰⠁⠈⠉⣼⢿⣷⣤⡈⠀⠈⢉⣉⡀⠴⠊⠉⠉⠈⠲⣍⣻⡇⢤⡀⡤⠠⢕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣵⡋⣙⡠⢸⠛⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⢦⣳⠐⢉⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠀⢀⠈⠀⠆⠀⠆⠀⠀⠀⢀⣀⡀⢀⡆⠀⠸⢧⠀⠀⢐⣀⣌⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠺⠁⠀⠀⠀⢟⠛⠮⠼⢷⠁⠀⠀⠀⠀⠀⢻⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠓⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠈⢠⠄⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠋⠀⡀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣀⣄⣀⣀⣠⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠰⠀⠀⠀⢀⡼⠇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢇⣽⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠻⠀⣼⠀⢦⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠽⣷⡞⢦⡄⠰⠋⠑⠀⣀⠀⠀⠀⠀⠘⢿⣶⡶⠶⠶⠶⠾⢿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡝⠈⣋⢿⡛⠧⠀⣤⡀⠀⠙⢻⡆⢠⣦⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠶⢷⡈⠿⣆⣠⡼⢋⡄⠀⠈⠁⠀⠋⠛⠛⠿⢷⣄⣀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡸⠁⢀⠋⡰⠢⣽⠁⠬⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠓⡤⢀⢀⠾⡀⢠⡄⠘⢲⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣶⠈⠁⣬⢱⢾⣤⣤⢤⠤⠀⣤⠀⠀⠒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⢈⡧⠀⡤⠈⠓⠄⢀⣈⠷⣀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣨⠇⢠⡄⠰⠚⢠⠽⣭⡴⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⣤⣶⣶⡶⣦⣤⣀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣀⡀⠀⡆
⠀⣀⡑⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡔⠀⠀⠛⠇⠀⢀⣿⣧⠿⠿⠛⠳⠿⢾⣿⣃⠀⠀⠀⠀⠀⠀⣤⡿⠯⢻⠆⠃
⠀⠀⠈⠉⠁⠀⢹⣦⣤⣀⣀⠀⠀⣿⡟⠃⠀⠀⠀⠀⡠⠔⠊⢁⠴⠚⡆⠀⠀⠀⠀⢸⠿⠿⠛⢀⡄⠛⠃⣀⠘⠿⠿⠏⠀⠀⠠⡶⣼⣥⣴⣶⣍⣀⡀⡀
⠀⢀⠀⠀⠐⣄⠀⠈⢿⣿⣿⣧⠀⢻⡇⠘⠀⠀⠀⠀⣠⠤⠚⠉⣠⠋⠀⠀⠀⠀⠀⣀⣀⡤⣠⣾⣿⣿⣿⠿⠧⠐⢐⣶⠂⠀⠀⠀⠀⠈⣳⣴⣋⠉⠋⠛
⠀⠈⠀⠀⠀⣻⣿⣦⣄⣘⣿⣿⡆⢸⣿⢘⣤⣶⣦⠀⠉⠑⠒⠛⠁⣤⣶⣶⣧⡄⠀⠈⠉⢻⣭⣥⣿⢿⣿⣷⣿⠃⡡⠎⠀⠠⢶⡇⠀⠀⣿⣿⣸⠆⢀⡀
⠀⠀⠙⠲⠿⣿⣿⣿⣾⣿⣿⣿⣿⣦⣙⣿⡿⠋⢉⡧⢀⣀⠰⠀⢐⣿⣣⣿⣻⣿⡃⢀⣶⣾⣿⣿⣻⣋⣿⣯⡻⣧⣤⡀⠀⣰⡇⢠⡖⢲⣿⡏⠁⠀⠼⡛
⠀⠀⠀⠀⠀⢀⠀⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⡶⠂⢰⠈⠉⠤⠀⠈⠿⣿⣶⡿⠿⠠⠞⢛⢋⣘⡛⣻⠟⠿⠛⠙⠏⠛⠛⠂⠹⣿⣾⣶⣿⣿⣷⣜⣳⣴⡇
⠀⠀⠀⣀⢴⣿⣦⢀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠒⢤⣄⣀⣀⡀⠀⠈⠀⠀⠀⠀⠀⠸⠻⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⡎⠙⠛⠛⠻⡇
⠀⠘⢿⣿⣿⣿⣷⣿⡷⠂⠀⠤⠖⠂⠠⠄⠀⠀⠀⠀⠀⠉⠟⣿⣿⣿⣿⣤⡐⢢⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⢠⣤⣄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠈⣿⣿⣿⣿⣿⠁⠀⢠⠀⢰⠐⡄⠀⡆⠀⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣀⠸⣿⣿⣴⣶⡄
⠀⠀⠀⠛⠛⠉⠛⠻⠀⠀⠑⢀⡌⠉⢀⠐⠁⠀⢽⣦⡀⠀⠀⠀⣸⣿⣿⣿⡾⠶⠶⡼⢿⡿⠀⠀⠀⢀⣠⣀⣀⡤⠄⡠⠤⠀⠀⠘⣿⡿⠿⠛⠁⠙⠛⠃
⠀⠀⠀⣀⣀⣤⠶⣶⣄⠀⠀⠀⠁⠀⠀⡀⢠⣴⣿⣿⣝⢦⣤⡄⢁⣤⣝⣿⣇⠀⠀⠃⠀⠀⠀⣎⢝⣽⣭⢽⣳⡖⡷⢁⠀⠁⣼⣦⣀⣀⡀
⠀⢠⡾⣽⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠯⣿⡟⣠⣿⣿⣟⣹⣿⣿⠟⠀⠀⠀⠀⠙⢾⣽⣿⣿⣿⠾⠃⠀⠠⣶⡿⣿⣿⡟⠀⢰⣿⣦⠀⠐
⠀⠈⢿⣿⣿⣿⣿⣿⡿⠀⠀⢰⣿⣷⣤⣴⣦⠀⠀⠀⠀⣴⣵⣿⢿⣷⣿⠿⠟⣁⠀⠀⠀⠀⢀⠀⠀⠙⠻⢿⣋⡀⠀⠀⠀⠈⢿⣿⠻⠿⠀⣿⣿⣿
⠀⠀⠀⠙⠻⠿⠿⠟⠀⠀⠀⠈⢿⣿⣿⣿⠟⠀⠀⢠⠞⠛⠉⠀⠀⠀⢠⣶⠿⢻⣷⡀⠀⡀⠀⠀⠀⠐⠿⠿⠿⠛⣀⣀⣤⣤⣌⠀⠀⠀⠀⣿⡿⣿⡄
⠀⠀⢀⣤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⣠⡶⢶⣀⣤⡀⠀⠀⠻⣿⠿⠛⠑⡿⠙⠿⠆⠀⣀⣤⣶⣾⢻⣿⣹⣧⣿⣿⡆⠸⠧⣤⣿⢷⣿⣤⡤⠿
⠀⠀⣦⠘⡇⠈⡆⠀⠀⠀⠀⢰⣄⣀⡄⠀⠀⠀⠀⢻⣷⣾⣿⣬⣿⠀⠀⠀⢀⡴⠃⠀⣿⣿⢿⠿⣿⣯⣫⣻⡿⠟⠋⢹⣟⣂⣿⠆⠀⠀⠈⣿⣏⣽⡏
⠀⠀⠹⡄⢸⠀⣁⠀⠀⠀⠰⢿⣭⣿⣄⠀⠀⠰⠀⠀⠛⡿⠿⠋⠀⢀⠠⢒⣽⡎⠀⠀⠙⠻⠿⠿⠟⠛⡛⠍⡄⠐⠀⠈⠉⠉⠀⠀⠀⠀⣀⠙⣛⠻⠃
⠀⠀⠀⠗⢠⡄⢰⠀⠀⣰⠄⠀⠟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠎⢡⣴⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡥⢀⣶⡄⠓⠀⠀⠉⠀⠀⠂⠀⠀⢈⠘⠸⢡⠆
⠀⠀⠀⠈⢾⡤⠂⢀⣼⠏⢀⠀⠀⢸⡅⠀⠀⢚⣿⠷⠂⣀⣀⠀⠀⣿⣿⣿⠁⠠⡄⠀⠀⠀⡀⠀⠀⠐⠀⢠⠉⠁⠄⢈⡀⢰⣦⠀⣀⠀⠀⢸⣿⣿⡖⠂
⠀⠀⠀⠀⠋⠀⠀⢸⣿⠁⠿⠀⣿⢼⣷⡄⠀⠈⠉⢠⠼⠛⢿⣿⣾⣿⣿⠃⠀⠀⣀⣤⢞⣭⣤⣤⠄⠀⢀⠀⠐⠂⠀⣘⣿⣾⣿⣾⠟⡀⠀⠈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣠⣀⣠⣾⡏⠀⠀⠀⠀⠀⢠⣤⡾⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⠛⠉⠁⣀⠤⠄⠀⠀⠀⠙⠛⣻⣿⣾⣿⡛⠛⠀⠀⢻⣿
⠀⠀⣿⣿⣶⣄⡀⠀⠈⠙⠿⠟⠛⠋⢀⣤⣾⣿⡟⠀⠀⠙⠇⣿⣿⣿⡿⠟⠁⠀⠄⠀⠀⣠⠔⠊⡀⠴⠆⡟⠉⠉⠀⠺⠟⢹⣿⠙⠿⠀⠐⠀⠈⡿⠀⠀⡄
⠀⠀⠸⣿⣿⣿⣿⣦⡀⢀⣀⡀⣠⣶⣿⣿⣿⣿⠁⠀⢠⠊⣰⣿⡏⠀⠀⠀⠀⡀⠀⠀⠈⠁⡠⠊⢁⠔⢂⠄⢠⣄⠀⠀⠀⢈⡁⠄⠐⡇⠄⣀⡀
⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⠇⢸⣿⣿⡇⠀⢀⣀⣀⡀⠀⠀⠠⣟⠠⠄⠂⠀⠏⢀⡎⠁⡁⠀⠀⠈⠑⢆⡠⠇⡴⠋
⠀⠀⠘⣿⣿⠿⣿⡿⢟⣩⢿⣛⡿⣿⣿⢿⣿⡿⠀⠀⠈⢾⣿⣿⠀⠀⣿⣿⣿⣿⣆⣀⣀⠀⠀⠀⠀⢘⢁⡜⢀⠜⠀⠀⠐⡃⠚⢒⡸⢗⡓⠓⡚⠂
⠀⠀⠀⠈⢋⣴⣿⣶⣾⣛⣟⡋⣶⣶⣿⣮⠉⠁⠀⠀⠀⠈⢿⣿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠈⡾⠤⠊⠀⠀⠀⠀⢠⡀⠃⠠⠄⠙⢠⡀⠀⢀
⠀⠠⠄⠀⢸⣿⣿⣿⠏⠈⠉⠀⢻⣿⣿⣿⠀⠀⠐⡿⠃⠀⠘⡟⠀⠀⠀⠈⠁⠸⣿⣿⣿⣿⠃⠀⠀⠈⠀⠀⠀⠀⠒⠀⠀⠈⠈⠀⠢⡖⠀⠈⠁⠀⢜⣸⡁
⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠙⠻⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣠⠀⢀⠀⠀⠻⣿⣷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠀⠙⠳⢦⣥⣀⠀⠈⠻⢧⣍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠠⢤⣀⡀⠀⠀⠀⠉⠐⢶⣄⠀⠁⠰⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣖⡐⠀⠀⠀⠀⠉⠙⠓⠆⢢⣄⠀⠀⠀⢴⣤⢠⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠿⠿⠿⠷⠶⣶⣤⣤⣀⡀⠈⠁⡲⢦⣜⡲⣿⣶⣷⣶⣬⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⣤⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢙⣳⣮⣿⣿⣿⣾⣿⣿⠟⠋⢡⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠈⠙⠉⠃⠚⠛⠂⠶⠒⠠⠀⣈⣛⣿⣿⣿⣿⡿⠛⠁⣘⣠⡴⠾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣁⣠⣤⡤⣤⣤⣤⠄⣠⣀⠄⣀⢀⣀⣸⣿⣿⣿⣿⣿⢫⡄⢈⡽⠛⢁⣠⣀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠈⠀⠀⠀⠀⠀⣀⣀⣈⣻⣿⣿⠿⠿⢯⡽⢤⣶⠋⠀⠀⠼⢿⣿⠙⢆
⠀⠀⠀⠀⠀⠀⢀⡠⠀⡀⡀⠠⠀⠀⠀⠀⡀⡉⠤⠮⠉⠽⢿⣿⡿⠡⠶⣶⣤⡹⡟⣷⡄⠀⠀⠀⠀⠁⠀⠈⢆
⠀⠀⠀⠀⠀⠀⢠⠒⠛⠁⠐⠀⠐⠀⠀⠀⠀⠀⡀⢀⣴⣴⣾⣿⣧⡴⠿⠿⢉⡼⠃⠘⠷⠀⠀⠀⠀⠀⠀⠄⠈⢇
⠀⠀⠀⠀⠀⠀⠋⠀⠀⡄⢂⣀⡀⠀⠂⣀⠤⠞⠀⠁⠀⢉⣩⣽⣿⣿⡗⢲⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠋⠉⠈⠁⠀⠀⡀⢠⡄⠀⠚⠜⠛⣛⣫⣿⠇⠈⠟⣿⣇⠀⠀⠀⠀⠀⠀⠰⠖⠶⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣶⡯⠓⠋⠀⠀⣠⣴⣾⣿⣿⣿⣦⣄⠀⡁⢹⢦⣄⡀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠐⠊⠀⠀⣤⠚⠉⠁⠁⠀⠀⣠⣖⢾⣽⣿⣿⣿⠟⢛⡿⣿⡿⠋⠀⠀⠀⠚⠉⠓⠶⠦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⠛⠉⠐⠚⣻⣿⣿⡇⢀⠀⠀⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⠿⠀⠀⠀⠀⠀⠈⠓⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⢦⣄⡀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣤⢠⢀⡄⣀⢠⣤⡀⣤⡄⢠⡤⠀⣤⣄⠀⠀⢠⣤⢠⢤⡀⣤⠤⢀⡤⣤⢠⡄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⡏⢸⢼⡇⣿⢸⣤⠇⣧⡿⢸⠶⠀⡇⢸⠀⠀⣿⠈⢸⣼⠃⣿⠆⢸⣧⣼⢸⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠃⢸⠘⠇⠿⠸⡇⠀⠯⠀⠸⠤⠄⣧⠞⠀⠀⢻⡴⠸⠘⠆⠿⠤⠸⠇⠸⠸⠇⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠆⠀⡆⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⠀⠇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠘⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⡻⠀⠀⠀⠁⠀⢰⠀⠀⢰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠥⢀⡔⠁⠀⢀⠌⠀⠀⠀⠀⠀⢸⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠚⠁⠀⢀⠠⠋⠀⠀⠀⠠⠀⢀⠌⢠⠃⠌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢀⠔⠁⠀⠀⠄⡀⠔⠁⡠⠂⡴⠃⢠⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠑⠀⠀⠀⢀⠠⠔⠊⣀⠄⠊⣠⠀⠀⠀⢠⠀⢰⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⢀⠂⠀⠈⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠊⠀⢀⠠⠐⣈⣉⣀⡠⠤⠔⠊⠁⠀⠀⠀⡀⠔⠁⠀⡠⠀⢀⣼⡅⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡥⠚⠉⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⠀⢀⡠⠊⡠⠞⠁⢸⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠉⠀⠀⠠⠒⠀⠒⠒⠒⠀⠉⠀⠀⠀⠠⠔⠊⡁⠀⠈⠀⠀⠀⡸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠄⠐⢒⡀⠀⡈⢉⣉⠥⠄⠒⠒⠒⠀⠁⠀⢀⠔⠀⢀⡴⠁⠀⠀⠀⣾⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠘⠃⢀⣠⠔⠚⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣠⠴⠋⠀⠀⠀⢀⡜⠁⢇⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠈⠀⠀⢀⠤⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠖⢹⠀⠀⡸⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠒⠀⡀⠀⢠⠞⠉⠉⠓⠒⠢⠤⠤⠤⠔⠂⠁⠒⠁⠀⠀⢀⠀⠀⠁⠀⠀⠀⣠⠃⠀⣠⠃⢀⠂⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠐⠈⢙⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠁⠀⠀⢀⠤⠊⠀⠀⠀⠀⣀⡤⠚⠁⢀⠔⠁⡐⠁⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠁⢀⣠⠴⠒⠋⠉⠀⠀⣠⠆⠀⠀⠈⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⠀⡠⠒⠉⠉⠉⠉⠁⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣔⣋⡠⠴⢾⡉⠙⠂⠀⠛⠉⠀⠀⠀⠀⣠⣴⣒⣋⣉⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣄⣀⣀⣀⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠶⢶⣶⡟⠛⠛⠷⢶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠷⠦⢤⡀⢀⣨⣷⣄⡀⠀⠸⢉⣠⡤⠴⠾⠿⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣡⡴⠶⠚⠲⣶⣃⠀⠀⠀⠀⠀⠀⠀⣀⣹⣶⠚⠙⠛⠶⢌⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠞⠁⠀⠀⢀⡴⠛⠉⠉⢳⡶⠛⠁⠰⡟⠁⠈⠙⢧⡀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢠⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⣠⣶⢿⡿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢶⣄⡀
⠀⠀⢀⣴⠟⠋⢀⣼⠇⣼⠁⠀⠀⢀⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡆⠀⠀⠘⡆⢹⣆⠈⠙⠶⣄
⠀⣰⠟⠁⠀⢀⡾⣿⠀⡿⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢷⢸⣿⢦⠀⠀⠈⠳⡄
⣰⠋⠀⠀⠀⡾⠁⢹⣾⣇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣾⠏⠈⢧⠀⠀⠀⠘
⡏⠀⠀⠀⠀⡇⠀⠀⠹⣿⡀⠀⠀⣇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⡼⠋⠀⠀⣸
⣿⡀⠀⠀⠀⠻⣄⠀⠀⠈⠻⠶⢶⣿⡀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⣿⣦⠤⠊⠀⠀⠀⣠⠏⠀⠀⠀⢀
⠘⣷⣄⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠿⢿⣦⣄⣀⣀⣀⣀⣀⣤⡴⠿⠶⠶⠶⠚⠛⠉⠀⠀⠀⢀⣠⠞⠃⠀⠀⠀⣠⡾
⠀⠈⠻⣷⣤⣀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠋⠁⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠈⠛⢿⣿⣶⣤⣀⡀⠀⠉⠙⠛⠷⠶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠖⠛⠉⠁⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣷⣶⣦⣤⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣠⣤⣴⣶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣿⣛⠲⣄
⠀⠀⠀⠀⡴⢋⠩⢉⠍⡉⠍⠙⢦⠀⠀⠀⠀⠀⣠⣖⣛⣛⣛⣳⡀⠀⠀⠀⠀⠀⢀⡇⠘⡇⠀⠀⠀⠀⠀⢠⠖⣫⡥⢾⢳⣗⣯⣙⣮⡇
⠀⠀⠀⢸⡧⠬⠤⠤⠤⠥⠤⠤⢼⡇⠀⠀⠀⠀⢧⡤⠤⠤⠤⢤⡟⠀⠀⠀⠀⠀⣼⣉⠉⢻⡀⠀⠀⠀⢠⠏⡾⡥⣌⢸⢸⠿⣚⣭⢶⡀
⠀⠀⠀⢸⡗⠚⠒⣖⣛⣲⠒⠓⢺⠅⠀⠀⠀⠀⠸⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣏⢈⡇⢸⡇⠀⠀⠀⢸⠸⡇⢓⡛⠾⠖⣫⣅⢸⢸⡇
⠀⠀⠀⠀⢿⣙⣯⡉⠉⢉⣽⣋⡿⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢹⠉⠀⢸⠀⠀⠀⠀⠘⣇⠳⣜⠿⢊⣤⣙⣡⢏⡼
⠀⠀⠀⠀⠀⠉⠉⠙⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣰⡇⠀⠀⠀⠀⠀⠀⣟⠀⠀⢸⡄⠀⠀⠀⠀⠈⠳⢬⣛⣛⣚⣋⡵⠚
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⣠⣖⢳⡏⣿⢓⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⣟⢦⡀
⠀⠀⠀⢸⠷⠾⢦⣷⣯⠼⠶⢻⠂⠀⠀⠀⠀⠀⣠⣤⣭⣼⣤⣤⠀⠀⠀⠀⠀⢀⡴⠋⡁⠂⠐⢈⠙⠦⡄⠀⠀⠀⠀⢀⡞⡁⢈⠀⡉⢺⡇
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⠋⣳⠀⠀⠀⠘⢶⠒⢶⠖⠒⣶⠒⡶⠃⠀⠀⠀⠀⣸⡀⠄⠠⢀⡠⠏
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣀⣻⣀⣀⣀⣀⣿⣛⣁⠀⠀⠀⠀⠘⡇⢸⡆⢀⡏⢰⠇⠀⠀⢀⣠⡴⣋⡽⠶⠚⠋
⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠼⠁⠀⠀⠀⠀⠻⣄⣀⣀⣀⣀⣀⣀⣠⠽⠁⠀⠀⠀⠀⠹⢤⣧⣼⡧⠟⠀⠀⠀⠈⠳⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣙⣦⡀
⠀⠀⠀⠀⠀⢀⣀⡤⠞⢧⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⢦⠀⠀⠀⢠⣯⣤⣬⣷⣬⣷⡀
⠀⠀⠀⠀⣰⠯⠤⠤⠤⠬⢿⡀⠀⠀⠀⢰⠏⠁⠀⠟⠁⠀⠿⠁⠀⠉⢧⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠓⠺⡆⠀⠀⠈⢷⠛⢓⣺⠷⡾⠁
⠀⠀⠀⠀⣷⠶⣤⠴⢤⡴⢶⡏⠀⠀⠀⣞⣶⡒⣲⣒⣒⣖⢒⣖⣒⣲⣺⡆⠀⠀⠀⠀⢀⣴⠋⠉⠳⢤⣠⠇⠀⠀⠀⠸⡖⢋⣥⢶⠇
⠀⠀⠀⠀⢹⡀⢸⠀⢸⠁⣸⠃⠀⠀⠀⢿⠲⠭⣽⠮⠥⣬⠭⢾⡭⠵⣻⠇⠀⠀⣀⣴⡏⠉⠳⡄⠀⠀⡽⠀⠀⠀⠀⠀⢻⣉⡤⡞
⠀⠀⠀⠀⠈⣧⣸⣆⣸⣀⡏⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠚⠛⠉⠉⠀⠀⠈⠳⢤⣽⣄⣀⡽⠶⠚⠁⠀⠀⠀⠀⠀⠈⣇⣸⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢶⣤⣄⠀⠀⠀⠀⢀⣤⡀⠀⢀⣤⡀⠀⠀⠀⠀⢠⡤⠤⣤⠀⣰⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢺⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⢧⡼⣷⣦⢷⡄⠀⠀⣠⠏⢠⠙⢶⠏⢠⠙⢦⠀⠀⠀⠈⣿⠉⡏⠘⣷⠶⡿⠀⠀⠀⠀⠀⠀⢀⡴⣳⢛⡞⠁⠙⣦
⠀⠀⠀⠀⢀⡴⠋⠀⠀⣸⠦⢾⣹⠏⠀⠀⠀⡇⠀⡼⠃⢸⡇⡼⠃⢸⡇⠀⠀⠀⡿⠀⣧⡞⠃⠀⠛⡆⠀⠀⢀⣠⢖⣫⣞⡵⠋⠀⠀⢠⡟
⠀⠀⠀⣰⠋⠀⠀⢀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠻⣄⠓⢀⡼⡄⠓⢀⡾⠀⠀⠀⣾⠁⠀⠈⡇⠀⠀⠀⡇⠀⠀⢸⡖⣻⠖⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠘⢦⣀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠶⠋⠀⠘⠶⠋⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠙⢧⡀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠙⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠳⠶⠒⠶⠃⠀⠀⠀⠀⠀⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠴⢾⣯⣉⡛⠓⠦⣤⡶⠛⠋⠙⠛⢦⣤⠖⠚⠛⠛⣲⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⢀⣀⣤⣤⣤⣤⣉⡙⠲⢬⡳⡄⠀⠀⣠⠏⠀⢀⡴⠚⣉⣉⣀⣀⣀⣉⡉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡤⠞⠋⠁⠀⢀⣤⠶⠚⠛⠛⠚⠳⣿⠶⠚⠛⠓⠶⣿⠚⠋⠉⠛⠳⢦⣀⠈⠉⠛⠶⣤⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⢀⡴⠋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣴⠋⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠳⡄⠙⢦
⠀⠀⠀⠀⠀⣰⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⣰⠇⠀⢀⠞⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠘⢇⠀⠹⡄
⠀⠀⠀⢠⠏⠀⢠⠏⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⢦⠀⠹⣆
⠀⠀⢠⡟⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⢧⠀⠹⣆
⠀⢀⡞⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⢧⠀⠙⡆
⠀⡼⠁⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠈⢧⠀⠹⡄
⢸⠃⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⣧⠀⣻
⠈⢷⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠘⣧⠏
⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠸⡆
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢹
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢈
⠘⠶⢤⣤⣀⣀⣀⣤⣤⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⣤⣤⣤⣤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣄⣠⣤⣤⠤⠶⠒⠋⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠉⠓⠶⠤⢤⣤⣤⣀⣤⣤⠤⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣶⣶⣶⣶⣶⣞⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⡶⠶⠶⠶⠶⠞⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠲⠶⠶⠶⠶⣦⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠙⠻⣦⡀⢀⡀⠀⠘⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠗⠀⣀⢀⣠⡾⠛⠉⠉⠙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠻⢶⣴⠟⠛⠷⢶⡄⠀⠀⠀⠀⠶⠟⠛⠛⢛⣴⠾⠛⠛⠛⠻⢶⣄⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣀⣴⠿⠟⠛⠿⣶⡶⠿⠛⠻⠶⣶⡶⠛⠛⠿⢷⣄⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⣴⠟⠁⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢰⡟⢠⣿⠀⠀⢸⡟⠀⣤⠀⠀⢸⡏⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⣿⠀⠀⠀⠀⠈⣿⠀⠀⢸⡆⠘⣷
⠀⠀⠀⠀⠀⠀⣸⠃⣸⡇⠀⠀⢸⡇⢰⣿⠀⠀⣿⠃⣠⡀⠀⠐⡟⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⡗⠀⢀⣄⠀⢻⡆⠀⠘⣷⠀⣿⠀⠀⠘⣿⠀⢿⡀
⠀⠀⠀⠀⠀⠀⡿⠀⣿⠁⠀⠀⣼⠀⢸⡇⠀⠀⣿⠀⣿⠁⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⣿⠀⢸⡇⠀⠀⣿⡄⢸⡄⠀⠀⢻⡇⠸⡇
⠀⠀⠀⠀⠀⢸⡇⢸⡿⠀⠀⠀⣿⠀⣾⡇⠀⠀⡿⠀⣿⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⢻⡇⢸⡇⠀⠀⠸⣷⠀⣿
⠀⠀⠀⠀⠀⣾⠀⣼⡇⠀⠀⢠⡇⠀⣿⠀⠀⢀⡇⢀⣿⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡀⠀⣧⠀⠀⢸⣇⠀⣧⠀⠀⠀⢿⡆⢸⡆
⠀⠀⠀⠀⢠⡏⢀⣿⠀⠀⠀⢸⡇⢸⡿⠀⠀⢸⡇⢸⣿⠀⠀⢸⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⣿⠀⠀⠈⣿⠀⣿⠀⠀⠀⠸⣧⠈⣧
⠀⠀⠀⠀⣸⠃⠈⠋⠀⠀⠀⣾⠀⢸⡇⠀⠀⢸⡇⢸⡟⠀⠀⢸⡇⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡇⠀⣿⠀⠀⠀⣿⡆⢸⡄⠀⠀⠀⠻⠀⢻⡀
⠀⢀⣠⡾⢿⡄⠀⠀⠀⠀⠀⣿⠀⠻⠃⠀⠀⣾⠁⢸⡇⠀⠀⣾⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇⠀⢻⡆⠀⠀⢹⡇⢸⡇⠀⠀⠀⠀⠀⣼⠟⢷⣤⡀
⢠⡾⠋⠀⠈⠻⣦⣀⠀⠀⢸⡟⠀⠀⠀⠀⠀⣿⠀⠘⠃⠀⠀⣿⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡧⠀⢸⡇⠀⠀⠸⠗⠸⣷⠀⠀⢀⣠⡾⠋⠀⠀⠙⢷
⣿⠁⠀⠀⠀⠀⠈⠙⠳⢦⣼⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿⣤⠶⠛⠉⠀⠀⠀⠀⠀⠈
⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢦⣴⣧⣀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⣨⣷⣴⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠶⠶⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⠶⠶⠶⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⡆⢀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠑⠒⠰⠀⠄⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠁⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠠⠤⠀⠀⠐⠀⠀⠀⠀⠀⠄⠐⠢⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠄⢀⠀⠆
⠀⢀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⢂⠘⠢⠊⠠⠀⠀⠁⠲⢆⡶⠜⠛⠁⠀⠀⠀⠀⠀⠀⠒⠀⢨⠀⡀⠂⠀⠀⢀⡀
⠀⠨⣀⠀⣀⠀⠠⠄⠒⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠃⠀⢒⡤⠈⠁⠀⡀⡀⠀⠀⠀⠀⠀⢀⡘⠡⣏⠟⠉⣢⠀⠃⠁⠈⠊
⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠁⠀⢀⠈⠁⢂⠘⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠀⡄
⠀⠀⠈⠁⠀⠀⠀⠀⡘⠡⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠊⠃⠄⠀⠁⠀⠀⠀⠀⠐⠀⠁⠀⠀⠄⠀⠀⠀⠀⠅
⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠈⢀⣀⣀⣤⣤⣤⣀⣀⣀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⡠⠚⠁⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢒⠀⠀⡊⢀⠀⠀⠀⣰⣾⣿⣿⣏⣉⣉⣙⣿⣿⣿⣿⣦⠀⠀⠀⠁⠀⢄⠀⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠅⠀⠀⠀⠠⠀⢰⣿⢩⠛⡏⣯⢩⣿⡩⠍⠛⠙⠉⢿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⢠⠀⢠⣠⡤⠠⣄⠀⠠⠤
⠀⠀⢂⢰⠀⠀⠀⠀⠈⣿⢰⣀⠷⢹⢸⣿⢰⡀⠀⠄⠘⣸⠇⠀⠀⠀⠀⠁⠀⠀⡀⠘⠀⠀⠠⠀⠀⠀⠀⠁⠀⢂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⡀⠀⠀⠔⢀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣷⣿⠾⠟⠋⠀⡄⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠄⠀⢀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⢠⠠⠰⠄⠘⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⠄⡢⠂⠉⠁⠂⠀⠀⠐⠐⠄⠀⠀⠁⠉⠉⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠄⡀⠀⠀⡐⠀⠀⠀⠐⠢⠀⠀⠄⠀⠀⡀⠀⠀⠑⠞⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠁⠀⠀⠀⠈⠄
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠠⣀⢀⠼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠐⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⢀⠀⠠⠤⢀⢀⠀⠀⠀⠀⠐⢀⡀⢀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠅⠀⠀⠀⢠
⠀⠉⠀⠔⠀⠂⠈⠈⠠⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⡄⠀⠀⠀⠁⠀⠈⠑⢧⠋⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠂⠀⠀⠰⡀⠀⠀⢀⠄⠀⡀⠀⠀⠀⠀⢀⠀⡈⠁⢀⢁⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⡄⠐⠈⠀⠀⠀⠁⠠⠀⠑⠢⠄⠐⠀⠀⢠⢰⠚⢣⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⣀⡀⠀⠐⠃⠀⠊⠡
⠀⣀⠀⢂⠠⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠐⠀⢀⠀⠉⠑⠰⠼⠓⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠢⢀⠐⠀⠀⠁⠄⠀⠁⠀⠀⠖
⠀⠀⠀⠈⠀⠀⠤⠄⠐⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠀⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡏⢻⡆⣀⢠⣄⠠⣤⡀⠠⣀⢀⣀⡀⠀⣰⡴⣶⠛⢂⣒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⢒⡾⣟⣻⣯⠻⣿⣾⣿⡇⣘⣷⣾⢿⣓⣭⣴⣶⣾⣿⣽⣺⠃⣨⡿⠏⢠⠖⠀⢀⠈⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⠿⠿⠿⢿⡜⢿⠛⠉⣱⢿⠟⢻⣿⣝⠿⣏⣛⣶⣾⣯⣿⠿⢿⠃⡖⠃⡀⠐⠧⠄⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠠⣤⣀⣀⡴⠃⢘⠶⣞⠓⢣⡒⣋⣻⣿⣷⣭⣯⣿⠿⠋⣉⡷⣆⡘⠀⠤⢈⣈⡊⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠉⠁⠈⠉⠉⠉⠓⠋⠀⠸⠿⠿⢿⣶⣶⡶⣶⢫⣄⣸⣛⠿⢷⣷⠤⢄⣤⠐⢰⡾⠃⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠁⠀⠻⢿⣿⣿⣿⠷⠊⠑⠤⠔⠛⣢⡾⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶⠤⠤⢤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠠⠤⠤⠶⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠋⠉⠉⠉⢙⡀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠐⠊⠈⠊⠑⠂⠢⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡟⠋⠁⣀⠠⠤⠐⣀⠈⣀⣀⣤⠆⣑⠢⢀⠀⣀⣰⣿⣯⣿⣷⣄⢀⡀⢄⠠⠀⠀⠈⠱⡈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢸⠖⠉⣤⣔⣢⡜⢛⡁⣙⠛⠛⣠⠹⣶⣶⣳⠾⣾⡿⣿⣿⣿⣿⣵⡟⠀⠀⢀⡎⠀⢠⠄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢤⣀⢀⡈⠉⠵⠙⠚⡾⠀⠿⡭⠿⠋⡠⠼⠈⠚⡮⠭⢾⣄⡟⠛⣏⠱⠆⣸⠂⡄⠴⠀⢈⠃⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡨⠻⠞⡀⠴⠒⢂⡒⢀⠠⣜⠋⠀⣭⠹⠂⢶⣮⠛⣛⠿⡧⢳⡞⠋⠛⠻⢧⠀⠰⣠⠇⢀⢄⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀⠀⠀⠀⠈⠒⠋⠙⠚⠷⠯⢭⣛⣳⠀⢠⣤⣄⣀⡉⠁⠰⡊⠀⠀⠀⠍⢩⣶⣤⣶⣶⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣀⣀⠀⠀⠀⠀⠀⠘⠤⠤⠨⠇⠁⠀⠙⠳⠦⢬⠇⢀⣉⣁⠬⠟⠛⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡤⠖⠒⠲⣤⠴⢒⠲⢤⡴⠒⠒⠦⣤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠚⠛⡿⠘⠋⠉⢳⡇⠚⠉⠙⠀⣷⠚⠉⠙⠸⡟⠛⠊⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣶⠀⢠⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⢻⠀⠀⠀⠀⣧⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠇⠀⢸⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠀⠀⡿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⡇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡏⠀⢠⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣧⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢻⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢀⡤⠤⠾⠤⠤⠤⠴⠧⠤⠤⠤⠤⠤⠧⠤⠤⠤⠤⠤⠤⠷⠤⠤⠤⠤⠤⠷⠤⠤⠤⠼⠦⠤⣄
⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠾⠤⠤⠤⠤⠷⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠟



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⣤⣤⣤⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣭⠉⠉⢉⣉⠉⠉⠉⠉⠉⢉⣍⢹⡇⠀⠀⠀⢀⣴⠶⣤⣤⣠⠶⣆⣠⣭⡄⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⡏⡓⠦⣄⠀⠀⠀⠀⠈⡁⠀⠀⠀⡏⢿⢸⡇⣿⢸⢸⡇⣿⢸⠃⣿⢹⡞⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠐⣠⡼⣻⠆⠀⠀⠀⡄⠀⠀⣴⡷⠾⠾⠷⠿⠾⣾⡷⠿⣿⡶⠿⠿⠾⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣟⣋⡥⠞⠉⠀⠀⠀⠀⣧⠀⠀⢹⡙⠷⣄⣠⠴⠛⠛⠶⠞⣩⡶⠾⣦⡀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⣀⣈⣉⣀⣀⣀⣀⣀⠀⠀⠀⣿⠀⠀⠈⢷⡀⠈⠁⠀⠀⠀⢠⡾⠋⠉⠀⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⢶⡶⢶⡶⠶⢶⡶⠶⣶⡶⢶⠶⢾⡁⠀⠀⠀⠀⠉⠓⠖⠒⠒⠒⢿⠉⠉⠉⠉⠉⠉⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠙⢦⣙⠓⠺⠓⢛⣩⡴⠋⢀⣼⠃⠀⠀⠀⢀⡤⠆⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⠋⠛⠙⠉⠛⠋⠉⠉⠛⠛⠋⠁⠀⣀⣀⣀⣸⣇⣀⣀⠀⠀⠀⠀⠛⠶⠀⠂⠰⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠶⠶⠶⠶⠶⠾⣧⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣋⣋⣙⣋⣛⣙⡟⠀⠀⠀⠀⠀⠀⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠄⠀⠀⣰⠞⠻⠻⠛⠟⠳⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⣿⣛⣛⣛⣛⣛⣛⡇⠀⠀⠀⠀⢸⢡⡴⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠶⠶⠶⠶⠶⢶⡟⠠⠶⠶⠶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⣷⣼⢸⡦⣼⢸⣿⠇⣼⣷⢸⡇⠘⣿⠑⣧⣼⡏⢷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠘⠁⠙⠘⠁⠘⠘⠛⠛⠋⠙⠋⠛⠃⠋⠈⠃⠘⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⣾⠛⣦⣾⠛⣧⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠛⠁⠘⠛⠃⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠁⠀⠀⠀⠀⠀⠀⡠⠆⠤⡀⠀⠀⠀⠀⠀⣀⢤⡤⢠⣆⣠⠤⡄
⠀⠀⠀⠀⠀⡔⠋⠀⠄⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⣠⣖⣛⣒⣲⣀⡀⠀⠀⠀⣔⡉⢉⣎⣑⡄⠀⠀⠀⠀⣨⠞⢀⡎⠲⣁⡞⢻⣉⠆
⠀⠀⠀⠀⢸⡤⠤⠤⠤⢤⠤⢤⠤⢤⡇⠀⠀⠀⠀⠨⣅⣀⣀⣀⡀⣠⠇⠀⠀⠀⢳⣾⣛⡳⠶⡇⠀⠀⠀⠀⠹⡓⠾⠽⠤⠿⠚⠉⢩⠃
⠀⠀⠀⠀⠀⣿⣉⣩⣭⣭⠽⢭⣙⣲⠎⠀⠀⠀⠀⠀⢡⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢟⣉⠷⡾⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠘⢤⣀⣀⣙⣇⢛⡧⠇⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣍⣴⠃⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠤⠤⠞⠀⠀⠀⠀⠀⠀⠀⠸⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⡠⠊⡡⢤⠤⠤⣌⣱⣠⠖⢆⠀⠀⠀⠀⣠⠊⠉⠈⠙⢦⠀⠀⠀⡔⠱⠆⢠⡍⠰⠗⠄⠀⠀⠀⠀⣀⠤⢄⡀⠀⢀⣀⢀
⠀⠀⠀⠀⡜⢰⠛⣲⣁⢀⠴⣿⠿⠃⢱⠸⠀⠀⠀⠀⡇⠀⡀⠁⡁⢈⠆⠀⠀⢳⠲⡖⠦⡲⠖⠲⠇⠀⠀⢐⢉⡿⠿⣶⠾⡽⢷⣶⢾⡛⡂
⠀⠀⠀⠀⢇⢸⠈⠔⠁⠉⢁⡞⢻⠾⠍⠉⠀⠀⠀⠀⠃⡀⠀⠀⠄⡜⠀⠀⠀⠀⡇⢱⠀⡃⡸⢸⠀⠀⠀⠀⢻⠀⠂⠀⠉⠀⠉⠀⡁⢹⠁
⠀⠀⠀⠀⠘⣌⠻⣅⣠⣱⣀⡰⠃⡼⠀⠀⠀⠀⠀⠀⠀⠑⢴⠒⡼⠁⠀⠀⠀⠀⠑⠠⢄⠵⠒⠃⠀⠀⠀⠀⠘⠦⣈⣀⣀⣀⣀⡠⠔⠃
⠀⠀⠀⠀⠀⠈⠑⠢⠤⠭⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠺⡭⣭⠎⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣞⣋⣹⠀⠀⠀⠀⠀⢀⡀⠀⢀⡰⢄
⠀⠀⠀⠀⠀⢠⣾⠥⣄⡴⢮⡱⢄⠀⠀⠀⠀⠀⠀⡴⠶⠶⠷⠾⠷⠖⣦⢤⠀⠀⠀⢀⡼⢀⠟⠁⠈⠦⢤⠀⠀⢠⠧⡌⠛⢧⡀⡆⠙⢦
⠀⠀⠀⠀⢀⡏⡰⠋⠀⠉⠑⠂⠺⡓⡆⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣰⡿⠵⠃⠀⠀⢩⠀⠘⡄⠀⠀⠀⢸⠀⠀⡇⠀⡟⠀⠀⢃⢱⠀⠀⡇
⠀⠀⠀⠀⢸⡴⠃⠀⠀⡀⠀⡀⠂⠙⡗⠀⠀⠀⠰⣚⠓⠒⠈⢉⣉⣉⣉⠏⠀⠀⠀⢺⠀⠀⡇⠀⠀⠀⠈⡇⠀⠑⠤⢝⡂⣠⢧⣈⡴⠊
⠀⠀⠀⠀⠸⡤⠤⠤⠄⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠈⠉⠉⢀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠒⠒⢧⠠⠤⠄⠀⠁⠀⠀⠀⠀⠙⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠉
⠀⠀⠀⠀⠀⠀⢀⡠⠤⡔⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⣿⡿⡇⠀⠀⠀⠀⠀⠀⢰⠒⠟⣻⣓⡆
⠀⠀⠀⠀⠀⡔⠉⠀⠀⠘⣄⠘⡄⠈⢢⠀⠀⠀⠀⠀⡟⡇⣣⠀⠀⠀⠀⠀⠀⢘⢁⠿⡵⠉⢹⠀⠀⠀⠀⡴⠒⢤⠔⠒⢲⠔⠲⡤⢲⢦
⠀⠀⠀⠀⡰⠻⣄⠀⠀⠀⢈⡏⠉⠓⠋⠀⠀⠀⠀⢜⣉⣉⠈⠑⡆⠀⠀⠀⠀⢸⠀⡜⠀⣀⢸⠀⠀⠀⢀⣇⣀⣁⠤⠤⣁⡠⠤⢤⠤⣴⡄
⠀⠀⠀⠀⡟⠦⣈⡱⠦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢈⠆⡜⠀⠀⠀⠀⠀⠘⡞⠠⠀⠀⠀⡇⠀⠀⢮⣵⣒⡬⣉⡡⢔⣂⠭⠭⡭⢧⠜
⠀⠀⠀⠀⠸⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⠀⢇⠀⠀⠀⠀⠀⠀⡃⠀⠰⠀⠄⢱⠀⠀⠀⠲⠤⠒⠒⠒⠒⠒⠊⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⣠⡨⠇⠀⠀⠀⠀⠀⠧⠤⠒⠒⠒⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⢠⣿⣧⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣌⣿⣿⡿⣀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢰⣶⠿⣸⣿⣿⣯⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢌⣱⡞⢿⠛⠟⢻⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠐⡚⠛⣋⠩⣴⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠛⢷⣶⣴⣾⠖⢠⣼⢹⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠐⢴⠃⢌⣭⡾⠷⢺⣛⣱⡿⠈⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⡕⠊⠉⠀⠀⢘⣠⡞⢁⣀⣤⡾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠉⠉⠙⠲⠖⠒⠉⠙⠛⠿⠛⠉⢶⣶⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠆⠐⠐⠄⢠⠀⢀⡀⠀⡀⠀⡠⣦⡔⢾⢙⣧⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣠⠀⣆⢔⣀⣀⡆⠀⡇⣰⡆⠹⠳⡿⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢇⢻⡀⣄⠙⣿⣀⡿⢁⡗⣾⢦⣷⣼⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢼⡾⡏⣭⠓⢿⣀⣧⣾⡧⣼⣾⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣧⣷⢻⣧⣼⡍⣿⣸⣏⣿⣹⣿⣿⡗⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢼⣧⣹⣟⣿⣻⣿⣿⣿⣿⡿⠽⣋⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⡗⠻⢭⠄⠊⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣸⣿⡃⣼⠃⠀⠰⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣻⠏⠀⡀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠘⠋⠰⠏⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠠⣶⠆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢫⡿⣁⣤⣴⣶⣶⣦⣀⠀⠀⠀⣰⣿⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢸⠃⣿⠉⠈⠻⠟⠈⠛⠇⠀⢰⣿⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢈⠀⠈⠁⠀⠀⠀⢠⠔⠃⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠿⠋⢽⣷⡀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣇⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⢿⠇⢹⣿⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣀⠀⠀⠀⠀⠙⠿⣷⣤⣀⣀⣠⠴⠋⠀⢸⣿⡌⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣼⡟⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠠⠞⠋
⠀⠀⠀⠀⣠⣶⣾⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠻⣿⣟⡻⠿⠖⠂
⠀⠀⠀⣾⣿⣿⣿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣝⠿⢿⣶⣶⣤⣄⡀
⠀⠀⠸⣿⣿⣟⡁⠄⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠉⠙⠛⢿⣦⡀
⠀⠀⠀⢻⣿⣿⣷⣶⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠈⢦⡀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠈⠛⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⣶⡿⠛⠻⢿⣦⣄
⣾⡟⢠⡖⠂⢀⠉⢿⣷⣄
⣿⣿⢸⠀⠀⠈⠂⠀⠻⣿⣧⡀
⠸⣿⡌⣧⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠻⣷⡌⢷⡄⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠙⣿⣦⡙⠦⣄⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠈⠻⣿⣦⣌⠓⢤⡀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⡙⢦⣄⠀⢿⣷⡄⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣶⣾⣿⣾⣿⣷⡆⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣌⡓⠀⢻⣿⣦⣶⣿⣿⠟⠛⠛⠉⠉⠁⠀⠀⢾⣷⠀⠘⣿⣿⡿⢿⣿⣛⣛⣛⣛⣉⣉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣶⣿⣿⣿⣯⣻⣧⣤⣶⣶⣦⠀⠀⠀⢸⣷⡶⠿⣿⠿⠿⠿⠿⠟⠿⠛⠻⢿⣿⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠛⠉⣉⣽⣿⣦⣼⣿⣦⣴⣶⣿⣿⡿⣿⣿⣇⠀⠀⠀⠀⢺⣛⢦⡈⢿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⢸⣿⠿⠛⠛⠛⠉⠉⠉⠁⠀⠻⣍⣻⡞⣿⣿⠀⠀⠀⠀⠀⠉⢿⡇⢸⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠄⠀⠀⢸⣿⡛⠣⠈⠀⠀⠀⠀⠀⠀⠀⠈⢿⠆⣿⣿⢀⠂⠀⠀⠀⠀⣿⠁⢸⣿⣿⣿⣦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⣿⠆⠁⠀⣸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣿⣿⣾⣤⣄⣤⣤⣤⣶⣶⣿⣿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠛⠻⠷⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣿⠛⠛⠿⠛⠟⠛⢻⣏⣉⣉⣼⠏⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠷⠂⠀⢀⣤⡤⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⣿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣯⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠳⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠀⠀⠀⠀⠘⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⢀⣿⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠁⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⣴⠛⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢻⣆⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣵⠆⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢈⣿⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠳⣶⣶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⠾⠋⠀⠀⣠⣶⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣤⣦⣤⣦⣴⣶⣶⣾⠿⣿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠉⢿⡿⠋⠉⠉⢹⣏⡽⠉⣉⣉⣿⣧⣿⣷⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡶⣾⣷⣶⣶⣶⠿⠿⠛⠟⠛⠛⠛⠋⠉⠉⠉⠙⢿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣯⣶⣴⣶⣶⡾⣿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡴⢶⣦⣶⡶⢶

⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠚⠛⠛⠛⠛⠻⠶⠶⣦⣤⣤⣄⣀⣀⣀
⠀⠀⠀⣠⡶⠛⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠶⠶⠶⣤⣤⣄⣀
⠀⠀⡾⠁⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⢸⡇⠀⠀⠀⠈⠓⠲⠶⠶⠦⠤⢤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⢹⡄
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⠲⠶⠶⠶⠒⠒⠛⠋⠀⠀⠀⠀⠀⣼
⢀⣼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⢸⡇⠀⠙⠒⠒⠒⠒⠶⠶⠶⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⢶⣏
⠀⣻⡦⠤⠤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣿
⢠⡏⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⢹⠆⠀⡏⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⣤⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⡟⠀⠀⢧⣀⣀⡤⠞⣻⠁⠀⠀⠀⠀⠀⠀⢹⢦⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⡇⠀⠀⠀⠈⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⣴⠛⣦⠀⢀⣤⣀⠀⠀⢀⡶⣦⡀⠀⠀⠀⢿⡀
⣿⠀⠀⠀⠀⠀⠀⣴⠶⠶⠞⠛⠛⣻⠛⠛⠛⣿⡟⢻⡇⠀⢸⡟⢻⡏⠐⣿⡁⠙⠳⡶⠋⣠⡾⠁⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠛⠛⢻⡄⣿⠛⣿⢰⠋⠛⠉⣧⠸⡇⠀⢸⡇⠘⣇⠀⠈⠻⣦⠀⠀⣴⣿⢦⠀⠀⠀⠀⠘⣇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⢻⠘⠖⣲⠀⢻⠀⣷⠀⢸⡇⠀⣿⠀⠀⠀⣿⠀⠀⡇⠳⠼⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⢴⠶⣆⣀⣼⠃⢸⡇⢸⡄⣯⣉⣀⣸⡄⠻⠶⠶⣧⠀⠛⠛⠛⣧⣿⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢻
⢻⠀⠀⠀⠀⠘⢧⣌⣉⣁⣤⠟⠀⠸⣧⣬⣭⣤⣼⣧⣤⣤⡴⠾⣤⣤⣴⠶⠾⢹⣦⣤⡿⢹⡀⠀⠀⠀⠀⠀⢸
⢸⡆⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠙⠃⠀⠀⠉⠑⢆⠀⠀⠀⢸
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢸
⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⣀⡀⠀⠀⠀⠀⢠⡇⠀⠀⢸
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⢀⡼⠀⠀⠀⢸
⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠉⠀⠀⠀⢠⡿
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠟⠁
⠀⠀⠀⠀⠉⠙⠛⠶⠶⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⠶⠶⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢂⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⢀⡜⠹⠋⠟⡆⠀⠀⢠⣉⣀⣀⠀⠀⠀⢀⣀⣀⣀⢁⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠄⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣢⡈⢂⠀⠀⢀⠃⠀⠠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠅⠀⠀⠀⡴⠟⠁⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⣀⡴⠊⠀⢁⠐⢎⡠⠚⠀⢀⠈⡇⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⡀⠐⠀⡸
⠀⡠⢔⣂⠍⠁⢀⡤⠆⠈⡔⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧⠀⠁⠀⠄⠀⠀⣀⡀⠀⡄⠀⠀⠀⠀⢼⠿⣶⣶⠤⣀⣤⢀⣀⣀⢠⡤⣔⠚⡄
⠘⠐⣈⡄⠖⡀⠩⡀⢀⠀⡃⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢻⡤⠤⠤⠤⠤⠒⠁⠀⢀⠁⠀⠀⠀⠀⣾⢦⠥⢌⣾⠛⠛⠀⠀⣢⡀⠀⠨⠼⣧⠄
⡸⠋⠀⠰⠈⠁⠀⠀⠀⠊⠀⠀⠀⠈⠀⡰⠁⠀⠀⠀⠀⠀⢸⠀⢀⣀⡤⠄⠀⠤⠖⠩⠀⠀⠀⠀⣀⠧⢈⠉⠣⠜⠓⠨⠭⣽⣚⡤⠎⠔⡨⠟
⢫⣄⠀⢀⠠⣀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⡁⠈⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠘⠠⣄⠄⡀⢈⣁⡀⡠⠞⠡⠤⠄⠂⠀⠀⢠⠐
⠀⠁⠒⠒⠑⠢⠭⣉⣉⡀⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣂⣀⣀⣀⣀⣀⣰⠁⠀⠀⠀⠀⠀⠈⠒⠳⠭⠤⣄⡀⣀⡀⣀⠀⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀
⠀⠀⠀⣀⠜⠁⠀⠀⡀⠢⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣗⡈⠉⠀⠀⣞⣽⣷⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠗⠀⡄
⢀⠔⢪⡀⠀⠀⠆⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡑⠦⠌⠭⠉⠉⠩⠍⠄⠀⠉⡀⠀⠀⠀⠀⠀⠀⢠⠆⠁⢀⠄⠀⠀⢀⠠⠗
⠘⠀⠗⡹⢄⠀⠀⠀⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠠⠄⡌
⠀⠀⠀⠀⠈⡆⠀⣴⠃⠉⠀⠁⠠⣀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠢⠀⠂⠀⢀⠀⠀⢀⠄
⠸⡦⡀⠀⢠⣧⠞⠁⠈⠲⢄⡀⠀⠠⡑⠄⡀⠀⠀⠀⠀⡟⠉⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢀⡈⠇⣀⠂⠆
⠀⠘⡊⢗⢍⠡⠀⠀⠀⠀⠀⠈⠢⣄⠔⢀⣼⠀⠀⠀⠀⣻⡀⠀⠀⠠⠐⠊⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⢔⠉⡰
⠀⠀⠑⢀⠙⠫⡂⠀⠀⠀⠀⠠⠀⡠⣔⡝⠁⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠤⢉⠃
⠀⠀⠀⠀⠁⠠⡀⠀⠀⡂⠤⢐⠬⠊⠜⠀⠀⠀⠀⠀⠀⠸⡓⠀⠀⠀⡀⠄⠂⠁⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣷⡿⢋⣤⣴⣶⣶⣶⠖⣠⣀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⢀⣀⡈⠁⠟⠁⠁⠀⣠⣿⡟⠉⢿⣿⣿⣇⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣾⠡⠉⠀⠀⠀⠀⣰⣿⡏⠈⢀⣾⡏⠉⣠⢤⡙⢉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢠⣿⠋⣽⣿⣿⡿⠣⠊⢠⣾⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢸⡟⢹⣿⣿⣿⣤⣀⠀⠘⠛⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⢀⣤⣿⡿⣿⣿⣟⠀⠘⠻⢷⣦⡀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⡟⢡⡖⣍⢻⣿⣶⣤⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⢰⣿⡷⢶⣉⢶⢸⣦⣌⡈⣩⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾⣿⣧⣘⢙⣂⣾⢿⣯⣭⣁⣰⣿⠀⠀⢀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⢿⣿⣿⣿⣁⣾⠀⠀⣿⠟⠁⠀⠀⠀⠀⣿⠃⠀⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡃⡾⠻⢻⣿⢿⡀⣠⠀⠀⢠⡾⡆⠀⠀⢀⠀⠀⠀⣠⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⡟⠟⠀⠀⠀⡏⢘⣧⣝⣵⣿⡟⠀⠀⡀⢀⣾⣷⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣬⣿⣟⡀⠀⠀⠀⢰⣿⣟⡿⣿⣿⣷⣾⣿⣿⣏⠉⣉⣶⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢉⣩⢴⡿⣿⢛⣿⣷⣄⠀⠀⣻⣯⣿⣾⣿⣿⡙⣿⣿⡿⠓⣿⣿⣿⠟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣟⣼⣿⣿⣿⠦⠿⣧⡀⢿⣿⣿⠟⣿⣿⣿⠏⠉⢀⡀⠈⠙⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠉⠀⢨⣷⡄⠀⠼⣷⣾⣿⢟⠀⣉⣏⠡⠀⢴⡋⠉⣧⠀⡙⢿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⢸⣿⣿⣄⣶⣟⡿⠁⠀⣺⣿⣿⣿⠀⠈⠓⠚⠃⢀⣴⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣿⣿⣿⠃⣠⡼⠛⣿⣋⣽⣿⣷⡐⣠⣶⣾⡻⣿⡿⠛⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠈⠻⠈⠻⠋⢀⣠⢬⡟⠻⢿⠿⠩⠿⣿⣿⡷⠟⠁
⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⣇⢀⣽⠀⠀⠀⠀⠀⣸⣷⣤⣄⡀
⠀⠀⠀⣾⣿⣿⣿⡿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⣤⣄⠀⠉⠀⢠⣿⠃⢀⣾⣿⣿⣷⠈⢿⡄
⠀⠀⠀⣿⣿⣿⠃⠀⠉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠋⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡏⠀⠈⠁
⠀⠀⠀⢻⣿⣿⣷⣶⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣹⡿⢿⠁
⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠛⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠉⠁⠀⠀⠹⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡤⣤⣄⣄⠀⢠⠀⣀⡼⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⣄⣀⡀⡀⢀⢠⢀⣼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠉⠉⠛⠋⠉⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠉⠙⠛⠚⠛⠋⠀⠚⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠛⠓⠶⠶⠴⠤⣤⣦⣦⡤⠼⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⠷⢶⣴⣴⣴⣠⣀⣆⣄⣶⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢸⣸⣠⢀⢀⡃⣀⡰⠈⠉⠉⠩⠅⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠯⠍⠉⠉⠛⠛⠛⠳⠶⡖⣴⣴⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⢸⣯⡝⠛⠛⠛⠿⠿⠿⢿⣷⣶⣶⣶⣶⠏
⠀⠀⠀⠀⠀⠀⠀⢼⣁⠀⢀⡀⠀⠀⠀⠀⠸⣅⣸⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣾⠍⠀⢀⣀⠀⠀⠀⠀⢸⣇⣿⠀⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣉⡽⢿⡚⠳⢿⡵⣦⣤⣄⣒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠛⠛⠻⠿⠿⠿⣶⣶⣯⣁⣀⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢙⣄⡤⠷⡖⠛⠻⣅⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⢀⠤⠄⠀⠢⠈⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢉⣧⡤⠞⢾⠉⣙⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠉⢀⠀⠐⠔⠀⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⡉⢀⣳⣤⠴⠷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⢀⡀⠀⢤⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠛⢯⠁⣘⣦⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠒⠀⠁⠀⣐⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡖⠚⠻⣅⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠢⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⣦⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠐⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣤⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⡀⠈⠀⢀⡀⠠⠀⠀⠀⠀⠀⣠⣤⣤⠶⢦⠂⠐⠀⠀⡀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠄⠀⠠⡀⠠⠊⢀⠤⠒⣢⡀⠰⠿⠿⠋⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠁⠄⠀⠀⢀⠀⣀⢀⠐⢄⠀⣠
⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⢮⣴⠋⢙⡿⢤⠀⠀⠈⠀⠀⡀⠀⠀⢀⠀⠀⠁⠀⢲⠶⠟⠛⠀⢀⠚⠁⠆⠀⠀⠖⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠹⣷⠄⢀⡙⠀⠀⠀⠐⠁⠁⠀⠀⣠⣴⠂⢀⡖⠁⡠⠀⣀⣀⠀⠀⠈⢀⣀⠊⡈
⠀⠀⠂⠀⠀⠀⢠⣐⡄⠳⠀⠀⠀⢻⡄⡿⢋⠘⠀⢉⠘⢢⢄⠈⠀⡿⠃⠀⠉⠠⠃⠀⢀⣿⣿⣿⣦⡴⠻⢄⡈⠳
⠀⠀⠀⠀⢘⣠⣴⠘⠃⠠⡞⢱⠠⠊⡀⡁⠄⢐⠂⢀⡁⡔⠲⠀⠰⠀⠀⠀⠀⠀⣀⠔⢛⢛⠉⠋⠉⡀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⢿⣍⠹⠗⠀⢀⠀⢀⠐⠀⣀⠄⣀⠵⠉⠀⢣⠘⡀⠠⠀⠀⢀⠄⠀⠂⠁⠀⠀⠀⠐⠀⡔⠑⠂⠁⢰⣿⡇
⠀⠀⠌⠀⠉⠉⠀⢀⣠⣀⠘⠂⠐⠂⡭⠪⡈⢀⠑⡀⠀⢣⠐⠀⠁⠠⠃⠀⠀⠀⠠⢀⣥⡶⠉⣀⣉⢲⠂⢀⣿⣏⣒⡀
⠀⠀⢀⡠⠴⠂⢀⠀⡀⠉⠘⢆⣠⠃⠀⠀⠀⠀⠀⠈⠄⠀⢡⠐⢈⡀⠂⠤⢀⣴⣾⠋⠀⡀⢘⠭⠐⠀⠀⠀⠉⠙⢿⡇⢀⠄
⠀⠀⠋⢀⠀⠠⠂⠈⡼⠀⣷⣼⡟⢳⣄⠀⠐⢀⠠⠀⠒⠀⠃⠐⠠⡈⠲⠀⠄⠄⠀⠉⠻⡎⠁⠀⢀⠀⠀⣆⠀⠠⠁⠑⣌
⠀⠀⠀⠀⡁⠈⡿⡶⠉⠈⠉⠛⠓⢼⣿⠃⣔⠀⠀⠀⠆⠀⠀⠃⠀⠈⢰⠈⠃⠀⠀⠀⠀⠀⠀⣤⠀⠋⠲⢇⡀⠀⠄⠀⠘
⠀⠄⢠⠀⠈⠉⢀⡃⠀⠀⠀⠀⠀⠂⠀⠸⡟⠁⢤⠆⠀⢂⠐⠄⣀⠠⠰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡖⢨⡙⠂⠁⠐⠄
⠀⠀⠀⠀⠈⠉⡛⣿⣦⣀⡀⠀⠀⠀⠀⣸⣧⣀⣚⠀⠄⠀⢙⠦⠀⣀⡠⠐⠡⠀⠀⠂⠁⠁⠀⠀⠰⠉⠈⠊⠁⠈⠐⠀⠰⠐
⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⠀⠀⢀⠼⠛⢋⠉⠉⠈⢁⠒⢠⡀⠐⠀⠀⢻⡶⠾⠶⣶⣦⣄⢤⡄⠘⠀⠊⠀⠀⠀⡀⡆⠀⠁
⠀⠀⠀⠀⠀⠠⠿⠁⠀⠀⠠⠀⢀⠠⢒⠉⠀⠻⣻⡍⢲⠀⠀⠐⠀⡀⠀⠀⢻⣆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣠⠏⣠⠁⠀⡀
⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⡀⠄⠐⠈⡀⠐⣄⠐⠌⣙⢊⠠⠠⢀⠇⠀⠀⠀⠀⠹⣦⠀⡁⠀⠀⠀⠀⠀⢀⠍⡠⠔⢁⡎⡀
⠀⠀⠀⠘⢿⡿⠄⢀⣀⠀⠤⠐⠀⠋⠄⣐⠀⠃⠂⠀⠐⠀⠀⡀⠓⢠⡀⣀⠀⠀⠙⣷⠀⠀⠀⠂⠐⠀⠈⠁⣠⠆⠀⠀⡀⢀
⠀⠀⠐⠠⠀⠀⡀⠘⢂⠄⠀⠦⡀⠀⢘⠥⠔⠀⢠⡀⠀⠀⠀⠀⡀⢠⠄⢤⣤⡄⠀⠊⢠⡄⠀⠀⠠⠖⢶⡎⠀⠑⠈⠁⠄
⠀⠀⠀⠀⠁⠀⠀⢈⡁⠠⠄⠂⠀⠈⠁⢀⠀⠀⠀⠀⠀⠄⠀⠄⠀⢸⣡⡟⢉⠣⠀⠀⠀⢱⣶⡘⣳⠄⠀⠑⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠶⢄⠀⠈⡠⠒⠀⠀⠀⢀⡀⠀⠀⢀⠎⠙⠛⣈⢼⣆⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡤⠀⠹⠤⢤⣤⣤⣄⡂⠃⠀⠀⠀⠴⠿⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠈⠓⠒⠁⠀⠀⠀⠀⠀⠙⠋⠉⠀⡀⢉⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠈⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠢⠤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢰⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣿⡏⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠛⠉⢲⠇⣼⢸⠿⡂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡩⢷⡰⠀⠂⢐⣿⢸⡆⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣼⠁⠀⡹⠁⠀⢾⣿⠈⡆⠀⠠⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢐⢺⡇⠀⠀⡇⠀⠀⠘⣿⠀⢿⣄⠀⠊⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠄⠐⣄⠀⠳⣄⠀⠙⠹⣧⡈⢿⣼⣄⣀⡼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⢠⣶⠶⢈⡉⠒⠺⣟⣶⣶⠶⢷⣆⡈⠃⠒⠒⠺⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠘⠿⣼⠁⠀⠀⢀⢀⡀⠈⠉⠛⠁⠠⠤⢀⡀⠀⠀⠒⠊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢘⢧⣄⠈⠢⠤⡔⠋⠀⠙⠲⢄⡀⠀⠀⠀⠀⠉⠲⣏⣿⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠉⠻⢦⣄⠀⠀⠀⠀⠀⠀⢀⠉⠙⠲⠤⢤⣤⣀⣈⢙⡛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⣀⣀⠙⠓⠾⣦⣠⡆⡀⠈⢉⠑⡒⠂⠢⠤⠈⠉⢉⣡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠛⣿⡦⠤⣤⣭⣭⣿⣷⣦⣤⣤⣤⡴⠶⠞⣮⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⡇⠀⡇⠀⣿⠀⣾⠉⢹⠇⠀⡿⠀⢸⡇⣼⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣿⠀⢁⠀⣿⠀⣿⠀⢸⠀⢸⡇⢀⣿⢠⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹⡇⠘⠀⢹⠀⡿⠀⡼⠀⣼⡅⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⢧⠀⠀⢸⠀⣷⢠⣧⣰⣿⢡⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣻⡆⠓⢺⣶⡿⢸⡟⣿⣿⣿⡿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⠋⠓⠚⠚⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢶⣶⣶⣶⣶⣴⣶⡶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡿⢱⡯⣿⣼⢿⡟⣿⢽⣷⢘⢿⡽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢹⡧⣿⢽⡏⣿⣿⡇⣿⣿⣿⣦⣿⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⣿⡇⣿⢽⡇⣿⣿⡇⡇⣿⡟⡇⣿⣿⢹⡄
⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⢧⢻⣿⢷⣿⣿⣷⣿⣾⡇⡇⣿⣷⣇⣿⣻⡜⣧⢿⣿⣿⣶⣶⣦⣤⣀⡀
⠀⠀⣴⣾⣿⣿⣿⠿⣛⣽⣿⣷⡆⡜⣾⣿⣸⣿⣿⢿⠟⣻⡇⡇⣿⣿⢹⢻⣯⣧⢸⡞⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠻⣿⣿⣿⣯⣺⣿⣿⣿⣿⡳⣷⣿⡏⣿⣿⣿⢸⣷⣿⡇⡇⣿⣿⡼⣸⣿⣿⡞⠿⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠉⠛⠻⠿⢷⣾⣿⣯⣛⣺⣭⣭⣟⣛⣛⣻⠿⠿⠷⣛⣻⣿⣿⣋⣻⣯⣵⣿⣭⣽⣶⡿⠿⣟⣻⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠯⠭⣭⣭⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣯⣭⡭⠽⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠭⠭⠭⠭⠭⠭⠍⠉⠉⠁

⠀⢀⠔⠂⠈⠉⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⠶⣋⢛⢦⡀
⠀⡃⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠃⠘⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠸⠪⠝⣃⡥⢚⡡⠖⠛⢇
⠀⡉⠀⡀⠉⠉⠀⠀⣀⣉⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠘⢈⡀⢈⣉⡩⠤⠒⠉⠀⠀⠀⡜
⠘⠒⠢⠂⠄⠐⠐⠤⠠⢶⠂⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠍⣉⣀⣐⠒⢒⣒⣒⡾⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠂⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠐⠀⠐⠀⠀⢨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⡠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡔⢪⠔⠒⡶⡄
⠀⠀⠀⠀⡔⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠐⢉⡉⠀⠁⢑⣻⡇
⠀⠀⠀⢀⠗⢐⠢⡀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⠃⠀⠁⡥⠜⠉⠤⠌⢣
⠀⠀⢀⠞⠐⠀⠇⠈⠆⠈⢇⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠒⠒⠒⠒⠒⠒⢺⢁⡤
⠀⢀⠞⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠀⠀⠀⠀⠀⠄⠐⠐
⢀⠚⠁⢪⣹⡠⠮⠒⠛⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠆⠀⠀⠀⣸
⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠈⠒⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⣠⠴⠊⠽⢣⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⢄⡀
⢀⣴⣋⣀⣀⠀⠠⢥⠤⠤⠤⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢐⡀⠥⠭⠤⢐⠙⢦⡀
⠈⡤⠤⠄⠤⣀⡠⠔⠢⠤⠴⡇⠀⠀⠀⠀⠀⠀⡠⠔⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠖⠁⠀⠀⠀⠀⠀⠈⢺⡇
⠀⠧⠀⢀⣀⣀⣀⣀⣀⣀⠠⠇⠀⠀⠀⠀⠐⣊⣀⣀⠀⠠⠀⠀⠀⠀⠤⢝⠢⠀⠀⠀⠀⠀⠈⠁⠀⠠⠤⠤⠤⠤⠤⠄⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠉⠀⠐
⠀⠀⠀⠀⢀⠆⡐⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣯⡀
⠀⣠⠀⢀⣀⡀⠐⢒⡒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⡄
⠀⠘⡉⠁⠀⠒⠒⠂⠀⣻⠯⣱⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⢀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⢀⡀⠱⣤⢀⣀⣀⡀⣴⢉⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠒⠒⠁⠀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠉⠀⠀⠀⢁⠆⢀
⠘⠠⣤⣤⡤⠤⠤⠤⠤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠂⡀⠈⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠠⠈⠰⠋
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⣶⠿⠿⠿⣶⣄⡀
⢀⣿⡇⣰⠖⠂⠈⠻⣿⣦⡀
⠈⣿⡇⣇⠀⠀⠁⠂⠈⢿⣷⡄
⠀⢹⣿⠹⣄⠀⠀⠀⠈⠀⠻⣿⣆
⠀⠀⢻⣧⡘⢦⡀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠻⣿⣦⡙⠦⡀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠈⠻⣿⣦⡈⠲⣄⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣌⡙⢦⡀⠘⣿⣦⠀⠀⠀⣀⣀⣤⣤⣶⣶⡶⠿⠿⢿⠿⠿⣿⣿⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣬⣀⠘⣿⣷⣾⣿⢟⡛⠉⠉⠀⠀⠀⠀⠀⢻⣷⠀⣸⣿⣿⣟⣻⣿⣭⣭⣭⣭⣭⡍⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣷⣼⣷⣶⠾⢿⣶⠀⠀⠀⣸⣿⣟⣛⣟⠛⠛⠛⠛⠛⣉⡉⠙⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⠟⠋⠉⣀⣤⣼⣿⣷⣾⣿⣶⠾⢿⣿⣿⡟⢿⣿⡇⠀⠀⠀⠀⠻⣝⣦⠘⣿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⣿⣛⡉⠅⠀⠀⠀⠀⠀⠀⠑⢦⣿⠘⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣗⠠⠀⠀⠀⠀⠸⠃⣠⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣏⣀⣀⣸⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣠⣬⣥⣤⣿⣿⣿⣶⡶⠾⠿⢿⠿⠟⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠯⠈⠍⠉⠉⠉⠀⠻⣶⣶⠶⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠁⠀⢠⡾⠋⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠹⣦⣤⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢹⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡾⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢀⡤⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⣤⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⣤⠀⠀⡟⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣀⠀⢷⣄⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣠⡾⠃⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠟⠛⣻⣿⠟⠛⠛⢻⡟⡿⠉⠉⠙⣿⣡⣦⣴⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣷⣶⣦⣼⣷⣤⣤⣶⡿⠿⠷⠾⠿⠿⠛⠛⠋⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⡡⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣿⣤⣤⣶⣶⢶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⢶⣶⣴⣶⢶⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⣤⣴⣶⡾⠿⠟⠛⢛⣛⣿⣷⣶⣦⣤⣀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠁⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠁⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⢾⡷⠀⠀⢀⣿⣿⣿⡏⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣇⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⣶⣤⣤⣬⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠈⢠⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠈⠿⠃⢹⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠀⢸⢀⣤⣤⣄⠀⠀⠀⠀⡏⠀⠀⣀⣤⣤⣀⠀⠀⢸⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⣿⡀⣹⣿⣷⠀⠀⠀⡇⠀⢸⣏⣀⣽⣿⡆⠀⠸⢸⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡤⠀⠀⢀⣾⣷⠀⡟⢿⣿⣏⣹⠏⢀⠀⠀⠛⢀⠘⢿⣿⣏⡿⠃⠀⠀⡜⣷⠀⠈⠋
⠀⠀⠀⠀⠀⠀⣠⡶⠚⡩⠯⠃⢀⡇⠀⠈⠉⠀⠀⠘⠿⠶⠾⠋⠀⠀⠈⠁⢀⠀⢧⠀⠹⣿⣦⠀⠀⢠⡀
⠀⠀⠀⠀⠀⢰⣏⢀⠀⠀⢀⡼⡸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⡌⠳⣄⠈⠫⣻⣦
⠀⠀⠀⠀⠀⠘⣿⢰⡀⣴⠋⠀⠀⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢦⡀⠀⠀⡌⢻⡇
⠀⠀⠀⠀⠀⠀⠈⠳⢭⣭⡶⢤⣶⠁⠀⡀⠀⠀⣠⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⢀⡴⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠬⠷⠞⠙⢷⣄⣀⣴⡿⠶⣤⣤⡾⠿⢶⣿⣿⣯⣭⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠚⠻⠭⢍⣓⠶⡎⠉⠉⢙⡶⠚⢉⣉⠽⠶⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠴⠒⠛⢙⣳⡶⠬⢵⣼⣄⡤⠮⣄⣴⠯⠴⠶⣖⡒⠒⠦⢬⣙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⡠⠊⠁⠀⢀⡔⠋⠀⠀⠀⠀⡼⠁⠀⠀⠀⠙⣆⠀⠀⠀⠉⠲⡄⠀⠈⠳⣌⠳⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠞⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⢦⠀⠀⠈⢦⠘⢦
⠀⠀⠀⠀⠀⢀⡞⠀⣰⠋⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢳⡈⢳⡀
⠀⠀⠀⠀⠀⡞⠀⡰⠁⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠱⡀⠳⡀
⠀⠀⠀⠀⡼⠀⡸⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠹⡄⠹⡀
⠀⠀⠀⣸⠁⡸⠁⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠹⡄⢳⡀
⠀⠀⢰⠃⣰⠃⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢱⡀⢳
⠀⠀⠈⢳⠃⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢳⠏
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢧
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠘⠲⠤⠤⠤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⣀⣀⠼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢧⣀⣀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠈⠑⠲⠤⠤⣤⠤⠤⠖⠊⠁⠀⠀⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠈⣿⣿⣦⠀⠀⣠⡾⣿⡇⠀⠀⠀⢠⣦⣤⣄⡀⠀⠀⢀⣤⣤⣶⠀⠀⠀⣿⢫⠤⡀⠀⠀⠀⠀⡰⣬⢻⠀⠀⠀⣿⣷⣤⡀⠀⠀⠀⣠⣴⣿⡗
⠀⠀⠀⠀⠀⠛⡇⠹⣷⣷⡿⢉⡿⠁⠀⠀⠀⠀⠿⢅⣀⡙⣶⣶⣋⣀⠨⠟⠀⠀⠀⠸⣾⣦⣿⣢⣤⣴⣺⣴⣿⠞⠀⠀⠀⢻⣿⣿⣿⣦⣀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣾⠇⣤⠚⠟⢢⡄⣿⡆⠀⠀⠀⠀⢠⣿⠾⢻⡟⢿⣿⠾⣷⠄⠀⠀⠀⠀⣠⣞⣯⢋⣿⢉⣽⣷⣦⠀⠀⠀⠀⠀⣾⣿⡌⠿⠿⠟⣸⣿⣦
⠀⠀⠀⠀⠀⣻⣷⠄⠀⠀⠀⢴⣿⡁⠀⠀⠀⠀⠘⠳⠾⠋⠀⠀⠛⠶⠞⠁⠀⠀⠀⠀⠸⢿⡞⠃⠀⠈⢳⣿⠟⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠸⢿⡿⠃


⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢄⡀⠀⠀⠀⢀⡠⢤⣦
⠀⠀⠀⠀⢹⣷⢿⣱⣄⣄⣴⣹⢷⣿⠁⠀⠀⠀⠇⡀⡀⠀⢀⡀⠀⡀⠀⠁⠀⠀⠀⢸⣇⠈⠁⢂⣠⣠⠊⠉⢸⡿⠀⠀⠀⢀⠄⠲⢶⢄⣤⣠⡤⠄⢀
⠀⠀⠀⠀⠀⣹⣿⣏⣿⣿⣏⣿⣻⡁⠀⠀⠀⠀⠀⢀⡀⠀⢿⡧⠀⢀⠀⠀⠀⠀⠀⠀⣩⡶⠂⣾⣿⣿⠂⠲⣌⠁⠀⠀⠀⠷⣤⣤⣠⣾⣿⣦⣀⣀⣤⠗
⠀⠀⠀⠀⠀⢿⣿⣿⠀⠁⢨⣿⣿⠇⠀⠀⠀⠀⠀⠋⠀⡀⠸⠃⠀⠀⠇⠀⠀⠀⠀⠀⠻⣦⣸⡿⠹⠻⣗⣠⡟⠁⠀⠀⠀⠀⠈⠏⣉⠝⠻⣍⡉⠝
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠉⠙⠁


⠀⠀⠀⠀⣴⡶⢤⣀⠀⠀⣀⡤⢠⢶⠀⠀⠀⣠⠤⠖⠲⣤⣀⣀⣤⠖⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⣢⣄⠀⡀⢀⣤⣆⠐⡆
⠀⠀⠀⠀⠘⣦⠈⠙⣿⣿⡛⠉⣠⡟⠀⠀⠀⠈⠑⢤⡀⠈⢹⡏⠁⣀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠘⢶⠊⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣟⣻⣿⣟⣿⣿⠆
⠀⠀⠀⠀⠀⢸⡙⡇⠛⠛⢲⢋⡿⠀⠀⠀⠀⠀⠀⠀⢻⡾⠃⠹⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⠘⣿⣬⣭⢿⣭⣴⡿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠈⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠥⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⢴⠀⡦⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠂⡀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⠰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠨⠉⠉⠁⠒⣒⣒⣒⠒⠀⠉⠉⠁⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⢀⡼⢷⡀⠀⠀⠀⢀⡼⢷⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢁⣴⣶⣦⡈⠁⠈⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠙⠀⠐⠈⠉⠁⠀⠤⠀⠀⠀⠀⠀⠘⣿⡟⢻⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠁⣀⠀⠀⠡⠀⠀⠀⢀⡌⠀⠀⡈⠀⠁⠀⠄⠈⠓⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠂⠀⠀⢢⠀⠀⠘⠁⠀⢠⠒⢠⠀⢀⠠⠐⠀⠠⠀⠂⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠘⠀⠀⢸⡆⠀⠀⠀⠀⠆⠀⠀⢀⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢠⣶⡄⠀⠀⢠⣶⡄⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠄⣀⠀⠀⠀⠀⠀⠀⢀⡀⠜⠀⠁⠀⠀⠀⠀⠈⠛⢡⣶⣶⡌⠛⠁⠀⠀⠀⠀⠀⠘⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣈⠉⠀⠈⠁⢁⣀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⢏⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠃⢀⡀⠀⠋⠈⠉⠀⣐⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠂⠁⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠄⠀⠀⠀⠀⠄⠱⠄⠀⠀⠀⠀⠀⠠⣤⠒⠉⠉⠉⠁⠀⠀⣀⠀⠄⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⢀⠃⠀⠀⠑⠚⠉⠁⠀⠀⠉⠀⠙⠀⠘⠋⠁⠀⠀⠀⢰⡆⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠉⠁⠨⠍⠁⠉⠉⠉⠉⠉⡉⠉⢉⠉⠉⠉⠉⠉⠻⠄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠉⠉⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠂⠈⠁⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠂⠀⠒⠒⠒⠒⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⡾⠛⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠋⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠀⠀⠉⠙⠿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠒⠀⡿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡿⠁⠀⠀⠀⢹⠻⢿⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠚⠋⠀⠀⠀⠀⠀⠀⠁⠀⢱⠙⢻⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠈⠑⠉⢹⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠛⠋⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⢤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢛⠻⢿⣏⣿⡿⣿⣶⣝⢿⣿⣿⣿⣷⡔⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣥⣶⣤⣤⣀⠁⠈⠙⢿⣿⣛⣿⣮⢻⣿⣿⣿⣿⣮⡳⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⣿⣙⣿⣇⣿⣿⣿⣿⣿⣷⡹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣦⠀⠘⢿⣯⣿⣾⣿⠏⢻⢿⣿⣷⢹⣝⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢿⣟⣿⡏⢾⡷⠞⢛⣿⡏⣯⣷⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢧⡼⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣽⣷⣎⠙⣉⠾⣿⣷⢻⣿⣇⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣨⡀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠸⣿⡿⠬⠿⣿⡺⠿⣿⣟⣿⣿⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡌⠁⢠⣤⣶⣶⠒⠀⠉⠛⠻⣿⣿⣿⣿⣿⣧⠀⢻⣶⣶⣤⡁⢰⣦⣿⣿⣿⢧⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⡄⠀⣿⠿⣿⣿⡿⠓⠀⠀⠘⢿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣇⢸⣿⣿⣿⣿⣿⣿⣿⢻⣽⣿⡿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢋⣾⣻⣼⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠠⣿⣿⣿⣿⡿⠋⠀⠀⢫⣾⣿⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⢲⡶⣘⣿⡉⠁⠀⠀⠀⠀⣾⣿⢏⢼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠁⠀⠀⠀⠀⠀⠀⣀⣴⣾⡟⣳⣦⣥⣶⡟⣷⠀⠀⠀⠀⢠⣿⡵⣫⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⣀⡀⠠⢤⣶⣿⣿⠟⠀⢻⣎⣻⣿⣛⣾⠃⠀⠀⠀⠀⠀⣿⣾⢟⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠁⠀⠀⠀⠙⢿⣾⠿⠛⡄⠀⠀⠀⠀⠀⣟⡵⣫⣸⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣾⣦⡀⠀⠀⠀⢯⣾⢣⣿⡿⣿⣿⣿⠇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⢴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⡤⢞⡯⠋⢀⣿⣦⡀⠀⡿⣡⡿⢿⣾⣿⣿⡟⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢾⣿⠋⢽⢧⡀⠀⣀⣀⠤⢖⣪⠵⠚⠉⢀⣴⣿⣿⢻⣿⣦⠘⣫⣶⣿⣿⣿⢟⡾⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠹⣿⡿⠁⠀⢈⣳⡬⡥⠴⠒⠊⠉⠀⠀⣀⣴⣿⣟⣻⣿⣿⣏⠿⣿⣿⣿⣿⣿⡷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣽⢻⡿⠇⠀⢀⣾⣿⣟⠀⠀⠀⠀⠀⣠⣾⣿⢛⣿⠟⣿⠿⢋⡤⠚⣡⣾⡿⣫⣷⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⣿⡆⠀⠀⠛⠟⠃⠀⠀⣠⣴⠿⣿⣬⣿⣿⡿⢓⡡⠞⣡⣴⣿⣿⣿⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⡿⣿⣿⣦⣤⣄⣠⣴⣾⣽⣿⢿⣿⣿⡽⠟⣩⠴⢋⣴⣾⣿⡿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠙⠧⣁⠛⠼⠯⢹⣯⡩⠯⠹⠟⢒⣉⣭⠴⢚⣋⣤⣶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠒⠒⠛⠛⠙⠒⠛⠋⠉⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠶⠾⠿⠿⠿⠿⠿⠷⣶⣶⣦⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠻⠿⠿⠶⣶⣶⣦⣤⣤⣤⣀⣀⣀
⠀⠀⠀⢠⣶⠟⠉⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠷⢶⣦⣄
⠀⠀⢠⡿⠁⠀⠀⠀⢼⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⣿⠃⠀⠀⠀⠀⠈⠛⠶⠦⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠷⠶⠶⠶⠶⢦⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⠟⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⣴⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⣿⠀⠈⠛⠶⠶⠶⠶⠶⠶⢦⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡟
⠀⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠛⠉⠉⠙⣿
⠀⢠⡿⠋⠛⠛⠛⣻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⣿⡁⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠙⣿⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⢠⠏⠀⠀⢻⠀⠀⠀⠀⢀⣠⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⢻⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⣼⠀⠀⠀⠀⠳⢤⡴⠞⠋⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⠦⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣰⠿⢶⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⢸⣧
⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣷⣤⣴⣶⣶⣶⣶⣦⣤⣶⣧⡴⣶⠀⠀⠀⣠⣿⣶⣾⠇⢀⣴⠟⠛⠷⣦⡀⣠⡾⠛⢈⣿⠆⠀⠀⠀⠀⠘⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣋⣉⠉⢉⣀⣠⣤⡟⠀⣀⣀⣀⡿⣯⠀⣿⠁⠀⠀⣿⡅⠀⣿⡁⠀⠻⣧⡀⠀⠈⠛⠋⠀⣰⡟⠃⠀⠀⠀⠀⠀⠀⣿⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣧⠀⣿⠉⢹⡇⣼⠋⠉⠉⠀⢻⠀⢸⡇⠀⠀⢸⡇⠀⢻⡇⠀⠀⠈⠻⣦⠀⠀⢀⣾⣿⠾⣆⠀⠀⠀⠀⠀⠀⢹⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡀⠀⣷⠸⣤⠶⢦⠀⢸⡇⠘⣷⠀⠀⢸⡇⠀⢸⣇⠀⠀⠀⠀⣿⠀⠀⠸⣇⢻⣄⡼⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣿⠀⢸⡇⠀⣿⠀⣴⠶⠟⠀⠘⡇⠀⣿⣄⣀⣸⣇⠀⠀⣿⣤⣤⣤⡀⣿⠀⠀⠀⣿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢿⠀⠀⠀⠀⠀⠀⢿⡉⠻⣦⣤⡾⠃⠀⣼⠇⠀⣿⡀⠹⣤⠤⠤⠶⣿⠀⠉⠉⠉⠉⢿⠀⠀⠈⠀⠀⠘⣧⢿⡆⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠘⢷⣦⣄⣁⣀⣤⡾⠋⠀⠀⢹⣧⣤⣤⣤⣤⣴⣿⣦⣶⣶⣶⠶⢿⣷⣶⣶⣶⠶⠾⠟⢸⣧⣤⣤⣿⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠋⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠛⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠀⢻
⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢸
⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸
⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣿⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸
⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⢀⣀⣠⠴⠋⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠟⠁
⠀⠀⠀⠀⠈⠛⠷⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠶⠶⠶⣶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⠶⠶⠿⠿⠿⠛⠛⠛⠉⠉⠁
⣿⣿⣿⡿⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠁⠸⠟⠁⠀⠛⠉⠉⠁⠀⠉⢹⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠈⠐⠀⠀⠀⠀⠐⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠘⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠋⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠊⠇⡂⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠴⠡⠂⠀⠀⢃⠀⣀⠀⠀⢀⡠⠞⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡆⠀⣢⠀⢀⢻⠀⠀⠙⠉⠉⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠰⠉⢲⡦⠀⠅⠈⠉⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠙⠃⠀⢉⢰⠍⠀⠀⠂⡀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡘⠄⠀⢄⠠⠘⠐⠀⠄⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢤⠄⣀⣀⡱⠁⠀⢈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠀⠀⠀⠀⠀⠊⡤⠀⢀⠠⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠐⠀⠀⠂⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⡴⣾⠛⠶⠂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡴⢾⠻⠦⠁⠾⣷⢂⡈⡙⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡀⢡⣼⣧⣦⡀⣭⣘⡃⣹⠟⢳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢣⠞⡻⢈⡁⠀⣙⡒⠛⡓⢻⡏⢉⡿⠲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢲⡤⠿⡌⠈⣟⢷⣷⠙⣏⡉⢉⡹⣞⠋⢙⠁⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡎⠑⣤⣀⢬⣙⣛⣁⣏⣤⠊⢙⣮⣵⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣿⠛⠀⠙⢩⡿⡆⢻⡏⢡⣾⣀⠉⠠⡤⠀⠈⢷⡠⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢖⣿⣿⠀⣾⣧⡦⠲⣶⢦⠞⠠⢞⠘⠋⠓⠀⠂⠠⣄⡀⠳⠂⠹⠆⡁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⢏⣀⣤⠈⡂⠿⠃⢻⠎⣶⣇⠈⣑⠢⠄⡈⠐⠤⡈⠑⢢⡈⠠⠀⠙⠦⡐⠤⠄⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢷⡹⣿⣻⠁⣿⠷⠈⠳⢭⡀⠠⠼⠷⠲⣄⡉⠐⠿⡤⢄⠐⠀⠉⣆⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣯⢁⡸⠟⠀⠀⠠⠀⡈⠀⠂⠀⣀⠈⠑⠢⠀⠐⢄⠑⠆⠀⠄⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠁⢄⠀⠀⠀⠀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠆⠁⠀⢠⡄⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠄⠀⣤⠀⠃⡀⢠⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡀⠀⠂⠃⠀⠀⠀⠀⡁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠠⠀⡇⠀⠘⠁⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢰⠀⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠀⠀⠐⠀⠘⠀⠀⢘⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣆⠀⠀⠀⠀⢀⡿⠈⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡰⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣿⣹⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣞⠻⣶⠟⠛⠛⣷⡶⠶⢶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣴⠟⠛⠻⢶⣤⣼⣯⣴⠞⠛⠷⣤⡀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢠⡿⠁⠀⠀⢰⡟⠁⠀⠀⠘⣷⠀⠀⠈⢷⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢀⡿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢿⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣼⠃⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠈⣷⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢠⡟⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢹⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣿⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⡟⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡆⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⣷⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣿⣄⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣠⡿⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠛⠳⠶⢤⣤⣤⡤⠶⠞⠛⠉⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠆⠘⠀⠛⠛⠛⠛⠂⠒⠠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢠
⠀⣀⣀⣀⣀⣀⣀⣀⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⢸⡏⢩⣍⠉⠉⢉⣍⠉⠉⠉⠉⠉⠉⠉⣭⡉⣿⠀⠀⠀⠀⠀⣠⣿⢳⣤⣤⣠⡞⣿⣄⣤⢶⡄⢀⣀⠀⣀
⢸⡇⠀⠀⠀⠀⢸⡏⣛⠶⣤⡀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⣾⠹⣿⢸⡇⣿⢸⡇⣿⢘⡏⣸⠃⢸⠙⣷⠏
⢸⠃⠀⠀⠀⠀⢸⡇⠈⠄⢀⡽⢳⡦⠀⠀⠀⢈⡀⠀⠀⣠⣼⣤⣿⣼⣥⣿⣼⣧⣿⣼⣧⣿⣠⣼⣼⣧⣼
⣼⠀⠀⠀⠀⠀⢸⣧⡴⠶⢋⣴⠟⠁⠀⠀⠀⢸⡇⠀⠀⢸⡟⢦⡀⠀⠀⣀⡴⢾⣍⣀⡴⢟⣉⣠⣤
⣿⠀⠀⠀⠀⠀⠘⠷⠖⠛⠉⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⣷⠀⠙⠶⠛⠉⠀⠀⠀⠉⣰⠿⢭⠍⠘⢳⣼
⢿⣄⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⣠⠟⠀⠀⠀⠈⠷⣄⡀⠀⠀⠀⠀⢠⣾⣭⣤⣤⣤⣤⣤⣽
⢠⡟⠛⢿⡛⢿⣟⠛⠛⣿⠛⢛⣿⠟⣻⡟⠛⣿⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⣇
⠈⢧⡀⠀⠛⢦⣌⡙⠛⠛⠛⣋⣡⡶⠋⠀⣠⡿⠀⠀⠀⠀⠀⣤⠶⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⢀⣀⣀⣀⣿⣀⣀⡀⠀⠀⠀⠀⠘⠳⠶⠀⠆⠰⠶⠛
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣭⣭⣭⣭⣭⣭⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠤⠤⠶⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠚⠓⠒⠒⠒⠚⡏⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀⠀⣼⣋⣙⣉⣛⣙⣋⣙⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⣿⣉⣉⣉⣉⣉⣉⣩⣇⠀⠀⠀⠀⠀⣸⢡⡴⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠤⠤⠤⠤⠤⠤⢤⡿⠀⠤⠤⠤⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠶⠶⠞⠳⠶⠶⠶⠶⠞⠁


⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⣀⡀⢀⢀⡀⠀⡀⣀⣀⡀⢀⣀⠀⣀⠀⣀⣀⡀⣀⠀⣀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⣇⢀⡇⣿⢳⣾⢸⡷⢾⡇⣿⠶⠂⣾⣹⡆⣿⠀⠈⡏⠀⣿⠶⣿⠈⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠀⠉⠈⠁⠀⠁⠉⠉⠉⠁⠀⠉⠉⠉⠁⠁⠀⠁⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣠⡞⠛⣦⣴⠛⢳⡄⡟⠛⣦
⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣯⣀⣈⣷⣶⣋⣹⣶⣾⣡⣷⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢶⣶⣤⣤⠖⠛⠛⣶⣤⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣛⣯⣭⣭⣽⣶⣶⣭⣙⢿⣧⡀⣾⣵⣿⣚⣭⣟⣛⣻⣭⣭⣟⣒⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠋⠉⢡⣤⣴⡶⠾⠿⠿⣿⣿⣩⣭⣭⣽⡿⠿⠿⡿⠿⠶⣤⣬⣭⣍⣛⠺⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣁⠀⣠⠴⢚⣉⣥⣶⣶⡾⠟⠛⣩⡵⡹⠁⡝⣌⢿⣿⣲⢬⡉⠛⠲⢮⣍⡛⠮⣝⢦⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣼⡵⠊⣡⣶⣿⣿⣿⠟⠁⢀⣴⣿⠋⣼⠁⠀⢹⡜⣆⠻⣿⣷⣮⡓⢄⠀⠉⠻⣷⣌⠳⣽⣦⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣾⡟⢁⡜⠉⢈⣿⡟⠁⢀⣴⣿⣿⠃⣸⠇⠀⣤⣼⣷⠸⡆⢹⡿⠻⣿⣎⠳⡀⠀⠘⣿⣷⡌⢿⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⡾⠁⣾⡟⢀⠎⠀⠀⣼⡟⠀⢠⠋⠀⢹⠇⢠⡿⠀⠀⠀⠈⣿⡄⣿⠈⠁⠀⠘⣿⣇⢹⡀⠀⠘⣿⣿⡌⣿⣧⠹⣆
⠀⠀⠀⠀⠀⠀⣼⠃⣼⡿⠀⡞⠀⠀⣸⣿⠁⢀⠇⠀⠀⣿⠀⣼⡇⠀⠀⠀⠀⣿⡇⢹⡆⠀⠀⠀⢹⣿⡄⢷⠀⠀⠸⣿⣷⠘⣿⡆⢻⡄
⠀⠀⠀⠀⠀⢰⠏⢰⣿⠃⡼⠀⠀⢠⣿⡇⠀⡼⠀⠀⢸⡇⠀⣿⠇⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⢿⣷⠸⡆⠀⠀⢻⣿⣇⠹⣿⡀⢿⡀
⠀⠀⠀⠀⢀⡟⢀⣿⡏⢰⠃⠀⠀⣼⣿⠀⠀⠁⠀⠀⣾⠀⢸⣿⠀⠀⠀⠀⠀⣿⣇⢸⣿⠀⠀⠀⠀⠸⣿⡆⢻⠀⠀⠈⣿⣿⡆⢻⣷⠘⣷
⠀⠀⠀⠀⣼⠁⣾⡿⢀⡏⠀⠀⢠⣿⡇⠀⠀⠀⠀⢠⡏⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⣿⣷⠘⡇⠀⠀⢹⣿⣷⠈⣿⣇⢹⣇
⠀⠀⠀⠀⠙⠶⣿⠃⣼⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⣸⠇⠀⣿⡟⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠙⣿⡆⣿⠀⠀⠘⣿⣿⣇⢻⣿⣾⠏
⠀⠀⠀⠀⠀⢀⡟⢠⡇⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⠀⠀⠀⠀⠈⣿⣅⢘⡅⠀⠀⢻⣿⣿⡜⣿
⠀⠀⠀⠀⠀⠈⠳⣾⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⢰⡟⠀⢠⣿⡇⠀⠀⠀⠀⠀⢸⣿⠸⣿⡇⠀⠀⠀⠀⠀⢿⣿⠀⣧⠀⠀⠘⣿⣿⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠛⣿⠀⠀⠀⠀⠀⣸⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇⢹⡶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣄⣀⣀⣿⣃⣀⡼⢿⡇⠀⠀⠀⠀⠀⣾⣿⠀⣿⣧⣀⠀⠀⠀⣀⣸⣿⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠙⠲⠦⢤⣤⣤⣿⡿⠞⠃⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠃⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⢾⣄
⠀⠀⠀⠀⠀⠀⠀⣰⢻⠻⡐⠀⠀⠀⢠⡟⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠰⣟⣲⠿⣋⣩⡝
⠀⠀⠀⠀⠀⠀⠀⠉⣥⣀⣤⡀⣤⣤⣠⠁⠀⣠⡮⣭⡁⠀⠘⡆⠀⢸⣿⣿⣫⣭⣽⡿⣻⡆
⠀⠀⡠⠒⠚⠓⠶⣄⠉⣿⣏⠉⣉⣿⠟⢠⠞⠉⠀⣠⠼⠶⠴⢧⡀⠐⣿⣷⣶⣶⣶⣿⣉⠀⠀⣸⢷⣤⡀
⠀⢸⡁⠀⠀⠀⠀⣽⡆⢻⣿⠿⣿⣿⠀⣮⠁⠀⣴⠃⠀⠀⠀⠀⠨⡀⢻⣞⣽⣟⣩⣿⠇⠀⢸⡼⣊⡿⢽⣄
⠀⠀⠇⠀⠀⠀⠖⢾⡀⢸⣿⡟⣿⡟⠀⠀⡏⣶⠻⡄⠀⠀⠀⠀⢸⡇⠀⣿⣯⣭⣿⡏⠀⢲⡟⣓⢻⡆⣽⣷⡆
⠀⠀⢰⣠⢀⣶⣤⡞⠁⠈⣷⡟⣿⠇⠀⠈⠳⣿⣤⢈⣖⠒⢘⣿⡿⠀⠀⢸⣟⠟⣿⡇⠀⠙⣷⣷⣦⣭⣽⣶⠃
⠀⠀⠘⣭⢻⣽⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠋⠀⠀⠀⠘⣯⡿⣿⠀⠀⠀⢻⣿⡿⣷⠺⡟
⠀⠀⠀⢆⣹⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣉⡏⠀⠀⠀⠀⠀⢿⠿⣿⠀⠀⠀⠸⣾⡷⢿⢿⠇
⠀⠀⠀⢸⠷⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣶⢛⠃⠀⠀⠀⠀⠀⢸⣶⡏⠀⠀⠀⠀⢹⣿⢾⡟
⠀⠀⠀⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⣰⡦⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠁⠀⠏⢀⣴⠞⠋⠉⠁⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠲⠶⠦⣄⡉⠀⠀⠀⢛⠃⠠⠀⠀⠈⠀⠀⠀⠀⠀⣤⣶⣾⢿⣦⡀
⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢿⡄⠀⢵⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢿⣇
⠀⠀⣾⡏⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡆⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⢀⣄⣠⣀⣤⣶⣶⣿⣿⣿⣿⣿⡿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⡟
⠀⠸⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣾⠟⠁
⠀⠀⢻⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠙⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⠏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠒⠀⠂⠂⠀⠀⠘⠀⠀⠀⠁⠀⠘⠀⠀⠀⠠⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠖⠦⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠋⠀⠀⠰
⠀⠀⠀⠀⠀⣀⠄⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠙⠀⠀⠈⠁⠀⠀⠄⠀⠀⢀⠀⠠⡀⠀⠀⠀⠀⠠⣰⡄
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚
⠀⠀⠀⠀⠀⠈⠃⠀⠐⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠂
⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠈⠀⠂⠀⠀⠀⠀⡀⠈⠀⠀⡕⢈⠀⠀⠀⠀⠀⡲⠤⣤⡀⢀⣀⠀⠀⠀⢀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠔⠀⠀⠀⠀⠉⢀⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠀⠈
⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠶⠾⡟⠙⠛⠳⣶⠶⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⣉⣙⣲⠀⠒⠛⠒⠀⠈⠰⢖⣂⢩⣍⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠡⠚⠉⠀⣠⠿⠒⢦⣤⠴⠀⣴⠒⠚⢧⡀⠀⠉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡾⠁⠀⠀⡼⠁⠀⠀⠉⠀⠀⠀⠙⣄⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠘⠀⠀⠀⢰⠁⠀⠀⠀⠀⢠⠀⠀⠀⠘⡆⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢀⣠⣾⠃⡞⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀
⠀⠀⢀⡴⠟⢩⡟⢸⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣆⠀⠀⢠⢸⡏⠛⢦⣀
⠀⡴⠋⠀⢠⢿⡇⡟⠀⠀⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢹⠀⠀⠸⡄⣿⢆⠀⠈⠳⡀
⡼⠁⠀⠀⡏⠈⢿⣧⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡄⠀⠀⣷⠇⠈⡆⠀⠀⠹
⣇⠀⠀⠀⣇⠀⠈⠻⣄⣀⣿⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣇⣀⠐⠃⠀⢀⠇⠀⠀⢀
⠹⣆⠀⠀⠈⢦⣀⠀⠀⠈⠙⠷⣤⣤⠤⢿⣦⣄⡀⠀⠀⢀⣀⣼⠧⣤⡤⠶⠛⠋⠁⠀⢀⡴⠋⠀⠀⢀⡾
⠀⠙⢷⣦⣀⠀⠈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⣀⣠⠴⠞⠉⠀⢀⣠⡴⠋
⠀⠀⠀⠉⠻⢿⣶⣤⣄⣀⠉⠙⠛⠓⠶⠶⠤⢤⣤⣤⣤⣤⠤⠤⠶⠖⠚⠋⠉⠀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠷⣶⣶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣤⣶⡶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣶⣟⣻⣿⣿⣿⣿⣿⣿⡷⠶⠶⠆⠀⢀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠿⢛⣩⣥⣶⠾⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣹⠟⣡⣾⣿⣿⣉⣤⣴⡿⠿⢟⣛⡛⢻⡛⠛⠛⠛⠛⠛⠛⠻⠷⠶⣦⣤⣄⡀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⡄⣿⢸⣿⣿⣿⡿⢛⣩⣴⠀⢭⣽⣷⡂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠠⣌⡙⠻⣷⡄⠀⡀⠀⢰⣯⣻⣄
⠀⠀⠀⠀⠀⠀⢀⣾⡏⢠⡄⠳⣝⠿⣿⣿⣧⣶⣿⣿⣿⠀⢭⣿⣟⡂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠨⣿⣿⣷⣮⣿⡾⣃⠴⠋⢈⣁⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⡀⠀⠈⠳⢮⣽⣻⡿⣿⣿⣿⣀⣭⡿⠓⠂⠘⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣨⣿⡿⠿⢛⣉⣬⣷⣾⣿⣿⣿⡟⣸⡇
⠀⠀⠀⠀⠀⠀⣾⣴⣌⡻⣷⣄⡀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⣫⣾⡟⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠶⣽⣻⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⢿⣿⣶⣄⣤⠾⢟⣫⣵⣾⣿⣿⣷⣿
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⣹⣿⡷⠾⢯⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⡀⣠⣤⣤⡤⠴⢖⣛⣛⣭⣥⣶⠞⠛⢹⣿⡏⣿⣿⣿⣼⣆
⠀⠀⠀⠀⣼⢿⣿⣿⣿⡟⠀⠀⣿⣿⡇⠀⠀⠀⠈⠉⠉⠉⠛⠀⠘⠛⠛⠛⠻⣷⡶⠒⠒⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⡄⠀⢸⣿⣧⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣿⠸⣿⣿⣿⡇⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⣛⣻⣻⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⢇⣿
⠀⠀⠀⠀⠘⣧⡙⢿⣿⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣛⣿⣿⣛⣻⣏⠀⠀⢙⣛⣿⣿⡿⢫⣾⠃
⠀⠀⠀⠀⠀⠈⠻⢦⣉⠻⢦⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⣀⣸⡿⠟⣉⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣬⣉⠛⠶⢦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠟⠛⣉⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠶⣬⣭⣍⣛⣛⣛⠲⠶⠶⠶⠶⠾⠿⠷⠶⠶⠶⠖⣛⣛⣛⣩⣭⣥⠴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣄⡀⢀⡠⠤⣄⡀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⣉⣉⣉⣙⡒⢬⡻⡄⠀⢠⠟⢁⡤⢒⣊⣉⣉⣓⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠖⠉⠀⣠⠔⠚⠉⠉⠙⡿⠊⠉⠑⢿⡉⠉⠉⠓⢤⡈⠉⠒⢬⡓⢄
⠀⠀⠀⠀⠀⠀⢀⠞⠀⣠⠞⠀⠀⠀⡼⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠙⢦⠀⠀⠙⣌⠳⡄
⠀⠀⠀⠀⠀⢠⠏⠀⡴⠁⠀⠀⠀⡸⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢣⠀⠀⠈⢦⠘⢆
⠀⠀⠀⠀⢀⡎⢀⡜⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢣⡈⢧
⠀⠀⠀⠀⡞⢀⡞⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢳⡈⢧
⠀⠀⠀⡼⠀⡞⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢳⡈⢧
⠀⠀⢸⠁⡜⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢳⠈⡇
⠀⠀⠘⡾⠁⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢷⠃
⠀⠀⣰⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⣇
⠀⠀⣇⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⣸
⠀⠀⠈⠑⠒⠒⠒⠒⠒⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣸⠒⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⣀⣀⣀⡤⠤⠖⠛⢄⡀⠀⠀⠀⠀⠀⢀⣠⠜⠓⠦⠤⣄⣀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⡠⠀⠀⠉⠒⠀⠤⠄⠀⠤⠄⠀⠀⠀⠀⢄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⡄⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠀⠀⢸
⠀⠀⣇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠸
⠀⠀⡇⠐⠆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⡇⠀⠐⠅⠀⠀⠀⠈⠁⠀⠀⠀⠀⠋⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⡇⠀⡖⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠡⠈⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣄
⠀⠀⡇⠁⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠱⢄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠠⠀⠀⡈⠀⠀⡰
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠂⠀⠀⠀⠠⠁⠐⠣
⠀⠀⢁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⡠⠀⠀⠀⠀⠀⣀⠀⠀⠠⠞⠈⠆
⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡤⠂⠀⠠⠤⢤⠤⠐⠂⠈⠁⠀⠀⠀⠊⢀⠀⣀⠀⠀⡜⠀⠀⠀⠀⠀⢀⢸
⠀⠀⠀⠀⠒⠤⣀⡀⠀⠀⠀⠀⠀⢀⣀⠢⡑⠂⠐⢢⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠈⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠈⠀⠀⠈⠉⠀⠈⠂⠀⠒⠉⠉⠑⠀⠊

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀
⠀⠀⠀⠀⠀⠀⡏⡇⠈⠙⣒⠤⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⡇⠑⠖⠋⠉⡷⠀⠀⠉⣉⡒⠲⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⠀⠣⣤⠴⠋⠀⠱⣀⡠⠔⠒⢦⠉⠉⠙⣒⣶⠦⠤⠤⣄⣑⣄
⠀⠀⠀⠀⠀⠀⢧⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠑⠒⠉⠀⠀⠙⠒⠶⠤⠖⠛⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⢾⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠒⣈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠏⢸⠀⠀⢱⢄⠀⠀⠘⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡂⠀⡟⠀⢸⠀⠀⠈⠚⠝⡲⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣆⣇⣂⡇⠀⠸⠁⠀⠉⠑⠄⠘⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣇⣆⣲⣿⡿⠉⠀⠀⠀⠀⠀⡀⢠⠀⠀⡇⢸⣶⠀⢀⠀⢀⠀⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⡶⠒⡟⣷⣿⡟⠀⠀⣀⠤⠀⠀⠀⡠⠋⠀⠜⠀⡿⣒⣾⠒⢀⣰⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣿⣷⣶⣿⣿⡇⣠⠞⠁⣀⣠⠔⠊⠁⠐⠋⠐⠚⡇⢻⣿⣶⣸⠘⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠛⣿⣧⣄⣿⡉⠉⠉⠉⠁⠀⠀⠀⣠⠀⠀⠀⠀⡰⠁⢀⣿⣿⣿⣺⡛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡧⢄⣿⣟⣀⣀⣠⣤⡴⠖⣋⠠⠀⠀⣠⠞⠁⢠⢿⠹⣾⣾⠻⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠍⠟⢿⡁⠀⠀⠀⠉⠀⠀⠀⠀⢀⣠⠞⠁⢀⡴⠃⣸⠀⣹⠻⢿⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣥⣿⣃⠀⠀⠀⠀⢀⣀⠤⠖⠋⣀⣤⠖⠋⢀⡴⠋⢀⣿⣦⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⡌⠉⠙⠛⠛⠻⠶⠶⠒⠚⠋⠉⣀⡤⠖⠋⠀⢀⡞⠀⡏⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢽⣇⠀⣀⠀⠀⢀⣀⣀⡤⠖⠋⠉⡁⠀⠀⢀⡠⠋⢀⡼⠁⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣿⠻⢭⣝⣲⠶⠖⠒⠚⠈⣀⣠⣄⣒⡯⠶⠚⠁⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⡄⠀⠀⠉⠉⠉⠉⠙⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣓⠶⠤⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠉⠁⡀⠒⠂⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠂⣀⠐⢀⡀⠐⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠐⠀⣀⠐⢂⣀⠐⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⢁⣀⠒⢂⣀⠉⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣁⠤⠒⢂⠤⡉⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣁⣉⣉⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠂⠀⠒⠒⠀⠀⢠⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠘⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠂⠀⠀⠀⠀⠒⠄⠀⠑⡆⠀⠀⠐⠉⠀⢀⠀⠀⠉⠂⠀⢈⠆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢠⣰⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⣤⠀⠀⠴⡆⠀⠠⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⣦⣀⣀⡀⢀
⠀⠀⠀⠀⠀⣠⣶⢿⠀⠀⠀⣀⣀⢀⢀⢀⣀⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠋
⠀⠀⠀⢀⡼⠋⠀⢸⠀⠀⠸⠛⠧⠾⠾⠿⠿⢋⠆⠀⠀⡇⠀⠀⠀⣤⡤⡄⡄⣤⡄⠀⠀⠀⡇
⠠⠀⠀⠘⠁⠀⠀⢸⠀⠀⢸⠿⣿⢻⠙⡏⣯⠿⠄⠀⠀⡇⠀⠀⠘⠛⠓⠓⠓⠚⠀⠀⠀⠀⡇
⠠⠔⠂⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠊⠓⠁⠀⠘⠃⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠸⠀⠀⠘⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠠⠀⢀⠀⠀⡄⠘⠔⠀⠀⡠⠒⠐⠂⠄⠀⣀⠀⠀⠀⡀⠀⠠⢒⠄⠀⠀⠀⠀⠀⠚
⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠇⠀⡄⠀⠀⠀⠰⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠀⠂⠂⠀⡀⠀⠀⠠
⠀⠁⠀⠀⣠⢂⢶⡀⠀⠀⠀⠀⠀⠀⠈⠐⠔⠀⠁⠀⠀⠄⠈⢀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠈
⠀⠀⠀⠀⠅⠬⠋⠀⠀⠀⠀⠄⢀⡀⠠⡀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠁⡂
⠀⠀⢀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠚⠀⠂⠀⢠⠉⡄⠠⠀⠀⠀⠀⠂⠁⠠⠀⠐⣢⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠⢴⠰⡄⢀⡐⠀⠀⠈⠀⠀⠀⠀⠀⡀⢤⡶⠄⠘⢊⣁⠠
⠀⠀⠀⠂⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠁⠉⠀⠀⡀⠤⢀⠀⠀⣼⡟⠁⠀⠀⠀⠈⠀⠠⡴⢡⢿⡄⠈⠂
⠀⠀⠀⠀⠐⠀⠧⠰⢺⠀⠀⢀⡀⠀⠐⠀⠀⠀⡊⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠒⠒⠁⠀⠀⣀
⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣮⣹⣲⣲⠀⠀⠱⢄⣀⡠⠜⠀⠀⠄⠃⠀⣯⣩⡣⢭⠀⠀⠀⡀⠄
⠀⠢⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⡨⠟⠀⠀⠄⠀⠀⠀⣀⠰⢆⠠⡀⠠⠀⠀⠀⠀⠀⠀⠀⢨⠁⢀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣇⠊⠀⠀⠀⠀⠀⡀⠀⠃⠀⠈⠀⠃⠀⠀⡄⡐⡠⠀⠀⢁⠈⠀⡌
⠀⠀⠠⠀⠀⢰⠃⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡆⠀⠄⠐⠀⠘⠒⠓⡇
⠀⡄⠀⠐⠤⠤⢴⠂⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀
⡐⢀⡂⠀⢠⣀⣼⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠀⠐⠂⠂⠀⢬⡣⣁⣸⠀⠀⠐⠀⡂⠀⠀⠐⠂⠀⣘
⣿⡫⠁⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⡻⣥⡆⠀⠀⠅⠀⠀⠀⠀⠀⠀⠄⠸
⠎⠋⢀⣴⠿⠝⠛⠁⠀⠀⢀⠀⠀⢀⣾⣦⡄⠀⠀⢨⢀⠠⠶⠀⠀⠉⠊⠀⣀⣀
⣄⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⡂⠀⣾⣿⡿⠟⠁⠀⠈⠠⠄⠊⠀⠀⠀⠰⠎⢲⡻⠄⠀⠀⠠⠠⠀⠀⠠
⠄⠀⠀⠆⠀⠀⠀⠀⣏⡀⠐⡆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢄⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠐⠂⠀⠀⠀⠀⠤⠀⠀⠀⠉⠀⠨⠁⠸⠆⠀⠀⠀⠰⠂⠠⠀⠀⠡⠀⠄⠀⠀⠀⠔⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠶⢶⣒⡢⣤⠒⠒⠒⡤⠖⢒⣒⡦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠔⠒⠒⣒⠶⠭⠾⣧⡠⠾⢤⡴⠭⠶⢶⡒⠒⠤⢍⡲⢄
⠀⠀⠀⠀⠀⠀⢠⠊⠀⡰⠋⠀⠀⡰⠋⠀⠀⠀⢠⠏⠀⠀⠀⠙⡆⠀⠀⠈⠳⣄⠀⠙⢦⡑⣄
⠀⠀⠀⠀⠀⣰⠃⢀⠞⠀⠀⠀⡸⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⠱⡌⢦
⠀⠀⠀⠀⢠⠃⢠⠋⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⢇⠀⠀⠙⡄⠳⡀
⠀⠀⠀⢠⠇⢠⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄⠀⠀⠘⡄⠱⡀
⠀⠀⢀⠎⢠⠃⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠘⡄⠱⡀
⠀⠀⡜⢠⠃⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠘⡄⣣
⠀⠀⢈⠏⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠸⡃
⠀⠀⡞⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢱
⠀⠀⢧⣀⣀⡀⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⠼
⠀⠀⠀⠀⠀⠉⠀⠀⠹⡀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁⠀⠉⠒⠤⠤⠤⠤⠤⠒⠋⠀⠈⠉⠙⠓⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠚⠛⠛⠛⠛⠛⠷⠶⠶⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠒⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠉⠛⠒⠒⠒⠲⠶⠤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠒⠲⠶⠶⠦⠤⠤⠤⢤⡤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠒⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠙⠛⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠘⡇⠀⠀⢀⡴⡿⠀⠀⠀⠀⠀⠀⢲⣄⠀⠈⣿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠙⠒⠚⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⢦⡀⠀⠀⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⣤⣤⣬⠷⠶⢶⡶⠶⠶⢶⡷⢶⡆⠀⢰⡷⢶⡇⢰⡟⠙⢷⣤⡴⠋⣸⠇⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠷⠶⣦⢰⡖⢺⣇⡶⠲⠚⣧⠸⡇⠀⢸⡇⢸⡇⠀⠛⢦⡀⠀⢠⣾⣥⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡇⢸⡏⠧⢶⡀⢸⠀⣷⠀⢸⡇⠈⡇⠀⠀⢸⡇⠀⢸⠻⣬⠃⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⢶⡄⢀⡿⠀⣿⠈⡇⢺⡉⣁⣸⡆⠻⠶⠾⡇⠀⠿⠖⢻⣼⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠹⣦⣉⣉⣠⡼⠃⠀⣷⣬⣭⣥⣼⣧⣤⣤⡤⢿⣤⣤⡤⠶⠏⣷⣤⣼⠟⣆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠛⠀⠀⠉⠙⢦⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⡜⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠁⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠲⠶⠶⠶⠶⠦⠤⠤⠤⠤⠶⠶⠶⠶⠶⠾⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢉⠀⠳⣄⡀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢢⡀⠀⠈⢳⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢀⢤⠀⠤⢀⣀⡁⢐⢿⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠁⠛⠁⠉⠙⢒⠉⣠⠞⠉⠀⠑⣄⣀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡞⠀⡇⠀⠀⠀⠀⡟⢻⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠠⠴⢃⣀⣠⠴⠚⠀⣘⡇⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡖⠒⠴⠒⠒⠂⠀⣀⡤⠖⠋⠉⠉⠀⣀⣤⣴⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠲⢤⣤⣤⣴⣿⣅⣀⣀⣤⣶⠶⡿⠛⠝⣃⢈⣨⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢤⣀⡀⢀⠀⠀⠈⠀⠈⠈⠎⠁⡡⠀⣀⣢⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⡌⡏⢹⡗⢺⠓⡟⠛⡟⢻⠿⣿⠛⡏⢹⣡⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠑⢹⡈⣇⠨⡇⢳⠠⡇⢸⠀⡇⢰⡇⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣾⡇⣿⠀⣧⣸⢰⣇⣸⣀⣧⣼⣹⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⡇⣿⣿⣝⡟⣿⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢉⣭⣭⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠀⡂⠂⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠊⠑⠛⠉⠀⠀⠀⠀⠂⠔⠀⠐⠲⢀⠀⡀⠒⠃⠀⠀⠀⡊⣀
⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠐⠒⠃⠉⠁
⠀⠀⠀⠄⠀⠀⠀⠈⠀⣀⠠⠒⠀⠀⠀⠀⠐⠠⢌⠀⠂⠀⣀⠔⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠁⠀⠀⠀⠀⡀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠩⠔⠁⠀⠀⠀⠀⢸⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡄⢀⡄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠔⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠀⠒⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠄⢀⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠰⡀⠀⠀⠀⠀⠀⢄⠀⡖⠁⠀⠀⠀⠀⢶⠀⠀⢀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡒⠀⠐⠀⠀⠀⠐⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠒⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠐
⠀⠀⠀⠂⡘⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠠⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠈⠑⠌⠁⠈⠈⠁⠀⠀⠀⠀⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠐⠂⠀⠴⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁
⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠒⠀⠁⠀⠤⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⠀⠉⠁⡀⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠄⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠈⠉⠁⠰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠐⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠄⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠈⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠁
⡔⠈⠒⢄⠠⡀⠀⠀⠀⡀⠀⠄⠀⠀⠀⢀⠈⠀⠠⡀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠒
⠁⠀⠀⠀⠢⠈⠁⢀⠉⠀⠀⠜⠀⢀⠔⠈⠈⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠈⡄⠀⠀⠀⠠⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠒⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠑⠠⠀⠀⠠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡈⠡⡘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠈⠥⢉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠤⠤⠄⠀⠀⠈⠀⠀⠀⠀⠂⠆
⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠀⠠⠶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠩⡕⠀⠀⠀⠐⡀⠀⠹⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠶⢭⡉⠂⠀⠀⠀⠀⠀⠈⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢶⣤⠖⠛⠛⠛⢶⡴⠚⠛⠛⠲⣴⠞⠛⠛⡓⢦⣴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠶⠛⠛⣾⠑⠟⠉⠙⢃⡟⠀⠞⠉⠙⠃⠘⣧⠞⠋⠙⠓⠹⡞⠛⠳⠌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡿⢠⠀⠀⢠⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣼⠁⠋⠀⠀⣼⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⢸⠃⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣼⢠⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢀⡇⠸⠇⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣾⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿
⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⠻⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠾⠷⠶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠟
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⣀⣴⣞⣡⡄
⠀⠀⠀⠀⠀⠀⡔⣿⣿⣮⢞⡵⣫⠝⠁
⠀⠀⠀⠀⢠⢖⡣⣞⡿⠁⢀⣜⣡
⠀⠀⠀⠀⣎⣿⣿⣿⣷⣠⠜⠻⡀⠀⠀⢀⡀⠀⠀⠀⠈⠁⢀⣀⡀
⠀⠀⠀⠀⠙⠿⣿⣿⠿⣇⠀⠀⠱⣄⠀⢉⡤⢤⣀⠀⢀⠞⠁⠀⢸⠂
⠀⠀⠀⠀⠙⠓⣼⣅⠀⠀⠑⢢⡄⠀⠙⠺⣆⠀⠈⠳⣼⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⡎⢻⣷⣄⠀⢀⣭⡓⢤⡀⠈⢣⡀⠀⠈⢳⡭⠥⠤⠤⠤⠤⠔⠒⠚⠉⠁⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠳⢿⡈⢮⣑⢮⡀⠈⠙⢿⣦⡀⠙⠲⢤⠤⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⢀⡤⢔⡛⠿⠆⠀⠀⠀⠃⠀⠀⢹⡉⠁⠐⠒⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠘⠋⣉⠴⠚⢇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠻⠻⡖⠚⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢣⠀⠀⠙⠛⠓⠿⣶⡄⢀⣤⣶⠿⠛⣁⣸⠾⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠹⡭⠞⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠧⣄⠀⠀⣾⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠷⢖⠤⣈⣋⢿⠢⣄⠀⠀⠀⠀⠀⠀⠀⠸⡁⢠⡼⠂⠀⠙⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠔⣊⡥⠄⠒⠂⠀⠀⢲⡀⢱⠀⠁⠀⠱⢬⡑⢤⡀⠀⠀⠀⠀⠀⣈⡦⡇⠀⠀⠈⠳⡞⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣈⠳⢄⡠⢊⡵⠊⠁⠀⠀⠀⠀⠀⢰⡣⠷⠞⠀⠀⠀⠀⠈⢯⠓⢬⡒⠒⣒⡩⠥⠒⠊⠹⣤⣀⡀⠠⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠑⣒⣻⡀⠀⣀⠉⣉⣉⡹⠂⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡩⠕⠊⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣤⣀⣀⣀⣀⣀⣀⢤⠤⣴⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⠧⡔⠈⢋⠫⠏⠉⢖⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣍⠋⢀⢿⣾⣿⡛⠺⣧⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣶⠷⠞⠿⢳⣾⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⡴⢩⠃⠀⠀⣠⠎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⡀⠈⣛⣦⣀⡀⣀⣨⠇⣴⡉⠀⠀⢉⣑⣾⣖⠒⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣏⡫⢟⣽⡩⠊⠀⠀⠉⠒⠭⣓⠿⣏⡩⣟⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠽⠷⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⢦⠀⠀⠀⠀⠀⠀⡠⠔⠒⢤⡀
⠀⠀⢀⠔⠋⠉⡉⠑⢄⠀⠀⠀⠀⣀⣀⣠⣏⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⠀⢰⠁⠀⠀⠀⡇
⠀⠀⢸⠈⠠⡁⡨⠀⢸⠀⠀⠀⠀⢸⠉⠉⠉⡉⡟⠀⠀⠀⠀⠀⠱⡞⠞⠃⡿⠁⠀⠀⠀⠀⠈⢣⠀⠀⢠⠃
⠀⠀⠈⠣⣄⣈⣐⡰⠋⠀⠀⠀⠀⠈⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢳⠀⢪⠃⠀⠀⠀⠀⠀⠀⠀⠳⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠞⠀⠀⠀⠀⠀⠀⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽
⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣖⠉⢁⡀⣠⠈⢱⠄⠀⠀⠀⠀⣠⠾⠹⢦⡀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠲⡄⠀⠀⠀⠀⢰⣿⣿⣿⡿⣶
⠀⠀⠘⡎⢫⠅⠇⢡⠎⠀⠀⠀⠀⢀⠗⠀⠂⠘⡇⠀⠀⠀⠀⢸⣆⠲⢶⡶⠶⡾⠀⠀⠀⠀⢸⣿⣿⣼⣿⡏
⠀⠀⠀⠙⡖⢂⣻⠋⠀⠀⠀⠀⠀⠈⢯⣐⣉⡾⠃⠀⠀⠀⠀⠼⡏⠟⠟⠻⠟⡿⠀⠀⠀⠀⠈⡇⠀⠀⢸
⠀⠀⠀⠀⠘⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠉⠀⠀⠀⠀⠀⠀⠓⠶⠖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠃⠀⠀⠀⠀⡎⣽⣠⢿⠈⡆⠀⠀⠀⢰⡛⣿⣗⣆
⠀⢀⡶⠋⠉⠀⠉⠙⢷⣄⠀⠀⠀⣀⣀⣿⣇⣀⣀⠀⠀⠀⠀⢰⠀⢨⢙⣾⠀⡇⠀⠀⠀⢸⠘⢜⣿⣿⣆
⠀⣼⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⠈⡏⠁⠀⠀⠀⡟⠀⠀⠀⠀⠸⠂⢸⢯⣴⠀⡇⠀⠀⠀⢸⠀⠀⠪⢿⣿⣆
⠀⠈⠉⠉⠉⠉⠐⠒⠉⠉⠀⠀⠀⢻⠻⠿⠶⠿⡇⠀⠀⠀⠀⠀⢇⢸⡴⢻⣀⠇⠀⠀⠀⠸⣀⣀⣀⣑⣹⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠈⠱⠋⠈⠀⠀⠀⠀⠀⠀⣀⣀⣀⣐⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⡃
⠀⠀⠀⡖⠂⠀⠂⠐⡆⠀⠀⠀⡰⠊⠀⠀⠁⠀⠙⢆⠀⠀⠀⣠⡾⠉⠙⠉⠛⣦⠀⠀⠀⠀⠀⢸⠅⠀⡅
⠀⠀⠀⡇⠀⠀⠀⢀⠇⠀⠀⢤⡇⠀⠀⠀⠀⠀⠀⢸⣀⠀⢸⠇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠸⠂⠀⡇
⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠈⠛⠶⡤⠤⠤⣤⠶⠟⠁⠀⠘⠦⠤⠤⠤⠤⠤⠔⠋⠀⠀⠀⠀⠀⠓⡖⠁
⠀⠀⠀⠘⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠈⠙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠻⠯⠤⠤⠤⠤⠭⠭⠭⠽⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣭⣙⣒⣒⣒⣒⣒⣒⣒⣒⣚⣩⣬⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠁⡏⢹⠉⡏⢹⠉⡏⢹⠉⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠾⣦⣧⣼⣤⣧⣼⣤⣧⣼⣤⠷⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⠶⢶⡶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠻⠶⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⢀⡴⡖⢶⡄
⠀⠀⠀⠀⠀⢀⡴⠶⣤⣤⣄⡀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⣿⡀⠀⠀⣴⠟⢮⣁⣼⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣸⠐⠚⣻⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠆⠘⣧⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠋⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣉⣸⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠉⠉⠉⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠈⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣀⣛⣛⡛⠻⠿⢯⣛⠿⢦⠀⢦⣼
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⢁⣀⠀⠀⠚⠋⣉⣩⣭⣟⠛⠶⠀⠈⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⣿⣡⠴⠀⠀⠸⠟⣦⣀⣀⣠⣴⣶⡄⠀⠀⠀⠀⣠⡶⠶⣤⠀⣿
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢠⡟⣡⠾⠋⠀⠀⠀⠀⢉⣩⢴⣿⡽⠂⠀⠀⠠⡾⠋⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢸⠃⣠⠆⠀⠀⣀⣾⣯⣶⣯⡵⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢻⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣴⠶⣠⣾⠃⠀⠀⠀⣉⡵⢞⣽⠏⠀⠀⠀⠀⠀⠀⠀⠿⠶⡶⠿⠦⠀⠙⢷⣄⠀⠀⣠⣄
⠀⠀⠀⣀⡴⠿⠀⠀⣠⢴⡟⠀⢸⠇⡿⠀⠀⠀⠀⠉⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⡟⠉⠻⢶⣤⡶⢶⣄⣀
⠀⠀⣾⠋⣰⠟⢡⣾⡟⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠀⠀⠀⠀⠙⠷⢾⣧⣽⡿⣦⠀⠀⣠⠶⡆
⠀⠸⣏⠀⠿⠀⡿⠘⠇⠀⠀⠰⣄⡀⠀⠀⠀⠲⣦⣀⣀⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢿⣶⣞⣡⠞
⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣦⣤⣴⢿⣯⣩⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣙⠶⢤⣤⣤⠶⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢧⣤⣤⣤⡤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡖⢀⡟
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⠻⠶⠖⠂⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠿⣤⡌⠻⠓⢶⣦⣤⣀⡀⠀⠀⠀⢠⡏⢀⡾
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣴⠏⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡅⠀⠀⠈⢻⡄⠀⠀⠓⠀⠈⣹⠟⠲⠦⣼⣁⡾⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣶⣶⠶⠛⠁⠀⣀⣼⣄⣠⣄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⣿⠆⠀⠀⢀⡾⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠉⠛⠉⠈⠙⢷⣶⣶⣤⣴⣟⠉⠙⠛⠓⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠄⠶⢦⣤⣄⣀⠀⠀⠙⠿⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠙⠻⠶⣤⡀⠀⠸⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⠶⠛⠂⠀⠀⣀⠀⠀⠈⠙⢦⠀⣤⣿⣧⡀⠀⣀⡠⣄⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠴⠋⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⢶⣶⡉⣿⠟⢋⣡⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠗⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⠀⠈⢻⣽⡏⠀⡈⢻⡿⠳⠶⢦⣤⣤⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣿⠛⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠛⣅⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⣼⠂⠀⠀⠀⠀⢿⣿⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣗⠀⠈⠙⠷⣦⡀⠀⠀⢀⣴⣿⠋⠀⠚⠁⠀⠀⠀⠀⠀⠈⢿⡉⠉⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠛⣷⠀⠈⠙⠆⣴⡟⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠙⠳⢶⡶⣦⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⣰⠿⡅⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⠀⠀⠀⣼⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢿⣀⡞⠁⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⢤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⠩⣓⠦⣄⡀
⠀⠀⠀⠀⢀⠴⠚⠩⠁⠀⠀⠀⠄⠀⠀⠀⠀⠉⠙⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢔⠋⠀⠀⠀⠀⠈⠁⠪⢉⡓⠦⣄⡀
⠀⠀⠀⡴⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⣲⠽⣶⡤
⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⢤⣄⣀⠀⠀⠀⠀⠁⠀⢹⡀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠒⠈⠁⠀⡀⠀⠸⢿⡄
⠀⠀⣿⠈⠀⠐⠁⢀⠠⠈⠓⠶⠤⠤⠖⠛⠁⠀⠀⠠⠀⠀⠀⢨⡇⠀⣴⣫⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⣀⠀⢀⠶⠚⠈⢳⡀
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠐⠀⢈⡇⠀⠸⡄⢀⡀⠀⠀⠀⠄⣀⠀⢀⠄⠀⢠⠖⠁⠈⠁⠀⢯⠁⠘⠃⡠⠒⠋⣹⠄
⠀⠀⠘⢆⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⢃⡞⠀⠀⠠⣏⠑⠲⠶⠋⠐⡞⠁⠈⠁⢀⠡⢋⣴⠆⠂⢘⣅⣐⣫⡭⠵⠶⠒⠋⠁
⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠈⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠸⡧⠖⠁⠠⣀⣒⣀⣻⡾⠤⠷⠒⠚⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⣀⣀⣀⣀⡤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠯⡍⣱⡎⠉⢃⡀⢀⣠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⣰⠋⠉⣿⡄⠁⣿⣣⠂⠋⠁⢩⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠋⠩⠉⠉⢠⠀⠀⠀⢩⡉⠉⢳⠶⢤⡀⠀⠀⠀⠀⠀⢀⣤⡇⠈⣿⡿⠀⠴⣯⣿⡶⠉⢹⣄⠰⠀⠈⠀⠆
⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠹⣆⠀⠀⠀⡠⢾⣄⠈⠲⠀⠙⡒⠀⠹⣻⣿⠀⠀⠸⡄⠀⠀⠸⣆
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣹⠀⠀⠘⠦⡌⠻⣇⠀⠀⠀⠁⠀⠀⠡⣹⣄⠠⠀⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⡇⠀⠀⠀⠈⠳⣄⡀⠑⡀⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢠⡞⠙⠂⣦⣀⣀⢀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠠⠄⠖⢊⡟⠀⣺⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠘⡆
⠀⠀⠀⠘⣇⠀⠀⠙⠲⡮⠥⠬⣭⣩⠭⢭⣩⠍⠠⠀⠔⠶⠿⠁⢁⠼⣧⡀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⣸⠋⠀⡢⠄⠉⠂⠈⠓⢀⣤⠤⠀⠔⠀⠉⠐⡴⠂⠒⡡⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠳⡀⠞⠙⠒⠤⢴⣤⣎⠤⠬⠛⠒⢄⠀⠠⠚⠓⠒⠋⠁⡤⢾⠉⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢫⡀⠀⡀⣀⣀⠀⠤⠶⠤⢀⠀⠀⣈⠤⠖⠤⠄⠐⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⠀⠀⡀⠀⠀⠈⢉⠀⢀⣠⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣤⣤⠤⠤⠤⠖⠒⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠴⠾⢯⣍⣛⠓⠦⣴⠚⠋⠉⠛⢲⡤⠖⠛⠛⣛⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠤⠤⠤⠤⣌⣉⠲⣌⢷⡀⠀⣰⡏⠀⣠⢖⣋⣡⣤⠤⠤⣤⣈⡉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⡴⠚⠉⠀⠀⣠⠶⠚⠉⠉⠉⠙⢻⠟⠋⠉⠉⠛⢿⡉⠉⠉⠉⠓⢦⡀⠀⠉⠛⠦⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡴⠋⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠉⢳⡀⠀⠀⠈⢷⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠞⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠹⣄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣰⠋⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⣦⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⡼⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⢧⠀⢳⡄
⠀⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⢧⠀⠹⡄
⠀⠀⠀⠀⢠⠏⠀⡼⠁⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⢧⠀⠹⡄
⠀⠀⠀⢀⡟⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⢧⠀⢻⡀
⠀⠀⠀⣾⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠈⢧⠀⣷
⠀⠀⠀⠙⣶⠃⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣶⠏
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠈⠛⠒⠲⠶⠶⠒⠒⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠒⠒⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⡼⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠙⠲⢤⣤⣀⣀⣀⣀⣤⠤⠖⠋⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉






⣿⣿⣶⣶⣷⡿⣾⣾⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣾⡿⣶⠿⠶⠶⠾⣶⡾⠷⢶⠶⠶⠿⠷⡶⠷⠶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣯⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢻⣷⣸⢙⠢⣄
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢹⣌⢿⣧⣾⠀⣿⠛⢦
⠀⠀⠀⠀⠀⠀⠈⢿⣶⣤⣄⣀⣠⣤⣼⠿⣿⣿⣿⣷⣿⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡆⠀⢻⣿⣿⣿⣷⣄⡇⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⠈⠉⠀⠉⠁⠂⠀⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⡀⠒⣿⠟⠁⠈⠉⠙⠒⠲⠤⢤⡀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠟⢁⠔⠀⠴⢶⣶⣿⣓⠦⣤⣧⣾⣷⣦⣀⣤⡶⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠃⣰⣿⣵⣿⣶⠾⣿⣿⡿⢿⣿⣿⣿⡿⢛⣽⣿⣤⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠠⠀⣿⣿⣿⣿⣿⣷⣌⢻⣾⣿⣷⣿⣧⣴⣿⣷⣿⣿⣷⡀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠉⠙⠛⠻⢯⣟⣿⣷⣝⡻⢿⣿⣿⣿⣻⣿⣿⣿⣿⢿⡶⠃⣴⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢤⠾⢿⣿⣶⣿⣿⣿⣿⣿⡿⠟⠛⠉⣰⣿⠟⠁⠀⠞⢠⣾⣿⡿⠟⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣤⣴⣶⣾⣷⣿⣧⠈⣻⡿⠛⠋⠉⠉⠀⢀⣀⣠⣼⣿⠏⠀⠀⣸⣿⣿⣿⣿⠃⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠉⠙⢿⣿⣿⣿⣷⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠃⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠛⠛⠁⠀⢀⠁⢠⣿⣿⣿⣿⡿⠀⠀⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣀⣂⣴⣿⣿⣿⣿⠟⠁⠀⠀⢠⡟⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣦⡀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⢻⣿⣿⠟⠋⠉⠁⠀⠀⠀⣠⣷⣶⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⡏⣿⣿⡄⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠋⠉⠙⢻⣟⣿⣿⣿⣿⣿⣦⣄⡀⠀⢀⡴⠊⠁⠀⢸⡇⣿⣿⣿⣶⣶⣿⡟⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠃⠀⠀⠀⠀⣧⢨⣿⣿⣿⣿⣿⡃⢠⡟⠻⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⣿⣿⠛⢻⣿⠟⢈⣹⠟⠁⠀⠀⠀⠀⠀⠘⢶⣽⠙⣿⣿⣿⡳⣼⠃⠀⠀⠀⠈⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⣠⡾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⡏⠙⣿⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣼⠟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠸⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣥⣬⣧⣴⣤⣽⣮⣼⣯⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿