Strolling Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣠⣶⣿⣿⣿⣷⣤⣴⣾⣶⣤⣶⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠘⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠚⠁⠙⢦⠀⠀⠀⠀⡜⠁⠈⢣⡀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⠠⠆⠀⣸⠀⠀⠀⠀⡅⠶⠀⢸⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠶⢄⠰⠋⠀⠀⠦⣀⠱⢄⡠⠎⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠿⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣦⣄⣀⣀⣀⣤⠴⠞⠉⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠻⢿⣿⠿⣿⠇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣠⡼⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠼⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠉⠀⠀⠀⠈⠙⠛⠿⠶⠦⣤⣤⣤⣤⡤⠶⠿⠛⠻⣷⡀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠉⠁⣏⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣨⡗
⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡴⠞⠛⠙⠛⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡉⠙⢦⡀
⠀⠀⠀⠀⠀⣠⡟⠁⣠⠟⠁⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠙⣦⡀⠛⠶⣄⡀
⠀⠀⠀⣠⠴⠋⢀⡼⠋⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⠶⢤⣀⠉⠳⠦⣄⣀⡀
⠀⢠⠞⠁⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠈⠙⠶⢆⠀⠀⠉⠛⢦⣀
⣶⠋⢀⣤⣄⢀⡟⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⢀⣤⣀⢉
⠈⠻⠟⠁⠘⠛⠁⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡾⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣉⠉⠻⣷⡀⠀⠀⠀⢀⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⣦⣤⣈⠀⢹⣿⣄⠀⠀⣠⣴⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⠉⣿⠿⠿⠛⢿⣶⢿⣏⠁⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠁⠀⣀⣴⡟⠁⠀⠀⠀⠀⠁⠀⠻⣆⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠟⠛⠁⡀⣠⣴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⡁⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣶⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⣴⣿⠇⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⢰⣿⣿⣤⠀⠀⠀⣿⣿⡆⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⣿⣿⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣿⣿⠙⠂⠀⢠⠋⢾⣿⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⢸⣿⡗⠂⠀⠀⠸⣿⣿⡀⠀⠀⡌⢸⣿⢢⠀⠀⠀⢸
⠀⠀⠀⢠⡟⢻⡆⠀⠀⠀⠀⣾⢻⡄⠀⠀⠀⢨⡿⣷⠀⠀⠀⢸⡿⣷⠀⠀⠀⠀⠹⣿⠀⠀⠀⢀⣾⢻⡆⠀⠀⠀⢸
⠀⠀⢠⠏⠀⠘⡇⠀⠀⢀⡾⠁⠈⣧⠀⠀⣠⠟⠁⢻⡀⠀⢀⡟⠀⠹⡄⠀⢀⡤⠞⣿⠀⠀⢀⡞⠁⠀⢷⠀⠀⠀⢸
⠀⠀⠛⠆⠀⠀⠻⠒⠀⠘⠇⠀⠀⠙⠃⠀⠛⠀⠀⠘⠓⠀⠘⠂⠀⠀⠛⠂⠈⠃⠀⠛⠆⠀⠛⠆⠀⠀⠈⠓⠂⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣴⣿⠟⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣿⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⣷⣿⣿⣿⣿⣿⡿⠛⠉⠁⠁⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⣿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹⣿⣿⠇
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠺⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠇⠀⠸⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⡃⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⡆⠀⣿⣧⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣷⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⠃⠀⠰⠎⢻⡆
⠀⠀⠀⣴⠟⢉⣉⡉⠻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠁⠀⠀⢠⣄⣾⠃
⠀⠀⢰⣿⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⢿⣆⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠈⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⣤⣤⣤⣤⣤⣤⣤⠴⠞⠋⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠉⠛⠛⠻⢷⣶⣶⣶⣶⠶⢿⣻⣿⠟⠁⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢿⡄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⣄⠀⠀⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠘⣷⡀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⠏⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠉⠻⢶⣤⣤⣤⣴⠶⠟⠛⠉⠹
⣠⡶⠶⠶⠶⠶⠾⠛⠃⠀⠀⠀⣠⡾⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠻⢶⣤⣄⠀⠀⠀⢠⣤⠶⠛⠋⣽
⠈⢿⡶⠶⠶⠶⣶⣆⠀⠀⠘⢷⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣾⡟⠓⠶⠟⠋
⠀⠈⠻⣦⣴⠾⠛⠻⣧⡀⣀⣼⠇⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⢷⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠷⣤⡀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠈⠙⠷⣦⣌⢻⣦⡀⢀⣤⡾⠛⠉⠹⣧⠀⠀⢀⣠⡶⢾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠾⣿⡉⠀⠀⠀⠀⠀⠀⣀⣽⠟⠉⠻⣿⡁⠀⠀⠀⠀⠙⣷⣴⠟⣷⣮⣿⠟⠋⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⡆⠀⠈⢷⣤⢄⣤⣠⣴⠾⠋⠁⠀⠀⠀⠘⢷⣄⠀⠀⠀⣠⣿⡇⣪⡾⠋⠁⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣄⠀⠸⢧⣼⣁⡼⠻⣶⡄⠀⠀⠀⠀⠀⠀⠹⣧⣴⠟⣩⣫⡾⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⠦⠤⠤⠤⠭⠧⠤⠤⢿⡄⠀⠀⠀⠀⠀⠀⢸⣧⢚⣵⠏⠀⠀⠀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⣀⣠⡴⠟⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣧⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡟⠉⠙⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠶⠖⠒⠂⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢆⠈⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣰⡀⠹⡀
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢼⠀⢆⠀⢰⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢣⠘⠀⡇
⠀⠀⠀⠀⠀⢀⠁⡀⢀⠔⢀⠠⡄⢀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠛⠈⠒⠂⠛⢠⠉⢆⠀⠀⠀⠀⠀⠀⠀⢀⠐⡄
⠀⠀⠀⠀⠀⠀⠈⡦⠀⠈⠀⠀⠈⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠰⠆⠀⠀⠁⠀⠠⡀⠀⢀⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡤⠀⠀⢠⡄⠀⠸⠆⠐⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⡀⠚⣧⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠊⠀⢰⡀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣀⡤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠦⡤⠄⠲⢧⡀⠀⠀⠀⢀⡙⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⡠⠤⠺⠛⠭⠥⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣃⣝⠀⡀⠀⠀⠠⡀⠀⠀⠓⠊⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠁⠀⠰⠀⠀⠐⣄⡃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣀⠜⠀⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢄⢰⡉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁⠰⠠⣀⠀⠀⠀⠀⠀⠀⡀⡔⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⠀⠀⠈⠀⠀⠐⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡄⠰⠦⠄⠀⠀⠀⢠⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢠⡀⢄⡒
⠀⠀⠀⠀⠀⢠⠈⠉⠀⠀⠀⠀⠀⣇⠀⣀⠀⠀⠀⢤⠀⢀⠃⠀⡀⠀⠀⠀⠀⠀⠀⡐⠲⠰⠄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠈⠂⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⢀⠎⠀⠀⠀⠟⢠⠂⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢒⣒⡲⢤⠄⠀⢀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠏⠀⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢇⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢐⠄⣸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣸⠀⠀⡷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⢀⠃⠡⠀⠀⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⡇⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠄⠀⠀⠡⡀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣟⠳⢄⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠌⠀⠀⠀⠀⠱⡀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿⠀⠀⠙⠢⣄⢸⠀⠀⢀
⠀⠀⣀⠠⠤⢄⡀⠀⢀⡴⠁⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⣎⠉⠀⠀⠈
⠀⠘⢃⠺⢦⣄⠙⠐⠧⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠈⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⢸⠀⠀⠀⠀⡿⠀⠀⢀⠆
⠀⠀⠀⠑⠀⠑⠳⣄⠀⠀⠀⠑⢆⠈⠀⠀⠀⠀⠀⡠⠔⠚⠀⠀⠀⢠⣢⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠊⠀⠃⠉⡆⠀⠀⡼⠁⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠐⠀⡉⣂⡀⣠⠋⠀⠀⠀⠀⠀⠸⣠⣤⣂⣀⣡⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠙⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⡏⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠛⠿⣿⣻⣿⣿⢿⣿⣿⢋⣾⡇⣼⣿⣿⣿⡇⠀⣾⠃⣿⣿⠀⢸⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣦⡖⠞⢉⠹⠀⣅⣤⣀⣦⣀⠉⠣⣉⣳⢸⣿⣿⣿⣿⣿⣿⣧⠀⣿⢀⣿⣿⠀⣸⣿
⠃⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣶⣾⣷⠿⠿⠛⣋⣁⡀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⠈⣿⣿⢀⣼⠟
⡀⢠⣿⣦⣴⠄⠀⠀⠀⠀⠀⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢃⣀⣬⣽⡟⣿⣿⣿⣿⣿⣽⣿⣿⣿⣏⣿⢀⣿⣿⠏⠁
⡇⠘⣿⣷⠛⠀⠀⡄⠀⠀⠀⠀⠀⠸⢹⣿⣿⣿⣿⣿⣿⠛⢁⣀⡀⠈⠹⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠠⣴⣶
⢡⡄⠻⣿⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠙⢙⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠘⠉⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⢟⡏⠉⣨⣿⣾⣿⣿
⠸⡷⡄⠈⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⣿⣿⣿⣿⣤⡄⠀⢀⣯⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿
⢾⣿⣯⠀⠀⠂⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣬⣽⣿⠛⠛⣡⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠢⣿⣯⢰⠀⠀⠀⢄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⡄⠘⢋⡄⠘⠸⢿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿
⠸⢿⣯⣹⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣫⣴⣿⡇⠀⠻⡔⢿⣿⡿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠛⣋⠀⠀⣧⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣿⡿⣿⣽⣿⣿⠁⠀⠀⠉⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣽⣧⣀⣿⡀⠀⣿⠸⡇⠀⠀⠀⢀⣀⣀⣴⡦⣶⢺⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠠⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠿⣷⡿⣏⢉⣿⣿⠀⠃⠀⣤⣴⣿⡏⡀⣰⠞⡋⢨⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆⠀⠀⠠⠒⠂⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⠋⢹⡏
⠾⠿⢿⡿⠿⠿⣿⣄⢱⣄⣉⣿⡏⠀⢠⡏⠂⣰⣿⣿⢿⣿⣿⣿⠟⣻⢟⡟⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿
⣿⣇⡀⡹⠿⠆⠹⠯⠘⠑⣾⣿⡇⢊⣾⡧⢼⣿⡿⣿⢿⣿⣿⣏⡴⢋⡞⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣄⡸⣿⣿
⠉⠉⠁⣻⡯⣴⣶⣏⡙⠓⢾⣿⣿⣾⣿⢀⢰⣿⣍⠻⣮⣿⡿⣋⢔⡿⠁⠀⠀⢠⠞⠀⠠⠀⠠⠀⠀⠀⠀⠂⢀⣠⣌⣙⡷⣿⣿
⠲⠖⠋⣠⣀⡿⠿⠉⠉⠛⣿⣿⣿⣿⠋⢣⣿⡿⠛⢃⡾⣫⣾⣿⡿⠁⠀⠀⣴⠋⠀⠀⠀⠀⠂⡀⣠⣾⣧⣀⠀⠀⠉⠉⠿⣆⡉
⣿⣿⣵⣿⡟⠋⠀⠀⠀⣼⣿⣿⣿⣿⢘⣿⣟⣴⣾⣿⣿⠁⠀⣸⡇⠀⣠⣾⠇⠀⡐⠀⠀⡀⣀⣴⣿⣿⣿⣿⣿⣄⠈⠳⠦⡈⣿
⣿⣿⡿⠏⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⠿⡜⠀⢰⡿⠀⣼⣿⡿⠒⠂⠀⢊⣼⡏⣿⣿⣿⣿⡿⢿⣿⣿⣷⣤⡀⠐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⢀⡇⢀⣿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣶⣮⣝⡻⠿⠿⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⣿⣿⣿⣿⣸⣇⣾⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣝⠻⣿⣿⣷⣶⣤⣾
⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⠀⣿⢹⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣼⣿⣿⣿⣿⡏⢁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⠻⣿⣿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣡⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣤⡀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣠⣶⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⣿⣿⡇⣶⣄⣀⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⢸⣿⣿⣿⢺⣿⣷⣄⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⣸⣿⣿⡿⠈⠿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣿⠏⣿⣿⣿⡏⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⣿⣿⡿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣝⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠎⠻⣿⣿⡇⠀⠀⢀⣤⣤⣤⣤⣤⣾⣿⣿⠏⠈⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⣿⣿⣧⠀⠀⠘⠿⠿⠿⠿⠿⠛⠛⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣍⠁⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⢠⠂⣀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠻⠆⠀⠳⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⠀⣀⡤⠊⠠⠆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⠀⠠⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡾⠀⢠⢸⠸⣿⠀⠀⣤⠂⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⡾⣇⠐⠒⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠼⠿⢶⡴⠾⡞⢁⣶⡀⠚⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⡛⠀⠑⣭⣭⠉⠹⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⡌⠀⢻⣤⣬⣿⣠⡿⣆⠀⠀⠀⢀⡴⠛⠳⢦⣠⠞⠁⠀⠀⠀⠀⠀⠀⡄⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠲⣤⡼⠁⠀⠀⠀⠨⡿⠈⡵⣿⠀⠀⢀⡾⠁⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⢀⡼⠁⣿
⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⡴⣿⠀⠀⠀⠀⢨⢼⠇⣼⢥⠇⠀⠀⢸⡁⠀⢸⣽⡇⠀⠀⠀⠀⠀⠀⢠⣾⠖⠲⡇
⠀⠀⠀⠀⠀⠀⡿⠀⢀⡏⢸⣿⡂⠀⠉⠀⢀⡟⣼⠁⣎⠀⠀⠀⠘⣇⠀⠀⣿⣁⠀⠀⠀⠀⠀⠈⡼⡿⠀⠀⡇
⠀⠀⠀⠀⠀⢠⡇⠀⡜⣠⡾⠧⢭⣓⡲⠤⠾⣴⢣⠀⠈⢣⠀⠀⠀⢻⡄⠠⡇⢈⡟⠓⠒⠒⠒⡾⠁⢹⠀⠀⡇
⠀⠀⠀⠀⠀⣸⠃⢸⣿⠏⠀⠀⠀⠀⠉⠉⠉⢻⣄⠱⢄⡀⠧⣀⠀⢸⡁⠀⡹⡏⠀⠀⠰⡄⢰⠇⠀⢸⡄⢸⡅
⠀⠀⠀⠀⣤⠿⢀⣼⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠳⣠⡤⢷⡌⠳⣆⡏⠁⢀⠀⠀⢹⡞⠀⠀⢸⢁⣀⣹
⠀⠀⠀⠀⠈⢹⡟⠁⠀⠀⠀⢸⠀⠀⠘⠦⠀⠀⠀⠀⠹⣄⠈⠁⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢷⡀⠀⠘⠂⠻⠏
⠀⠀⠀⠀⠀⠘⠷⠦⢴⣶⠶⣾⣀⣀⣤⣤⣤⡤⠶⢺⡿⠛⠀⠀⠀⠀⢠⡟⠻⣦⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⣸⠃⠀⠈⢷⡀⠀⠈⣧⠀⠀⠀⠀⢀⡾⠃⠀⠹⣇⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⣠⠃⠀⠀⠀⠀⢻⠈⠀⢸⠀⠀⠀⢀⡿⠀⠀⠀⠀⢹⣦⠠⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⢸⡀⠀⠈⡇⠀⣰⠿⣧⡀⠀⢀⡴⠋⢸⡄⢀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⢀⡟⠄⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢹⣠⠏⣀⡀⠙⢛⡟⠀⠀⠀⣷⡜⠄⢀⣠⣷⡄
⠀⠀⠀⠀⢀⡾⠙⢒⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡊⠝⢿⠣⣄⠀⠠⠏⠀⠀⠀⠀⠘⣿⡔⠋⠀⢀⣻⡄
⠀⠀⠀⠀⣿⢣⠤⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢈⠷⠶⢛⡽⡄⠀⠀⠀⠀⠀⠘⢿⣅⡴⢋⠟⢻⢟⡷
⠀⠀⠀⠀⠙⣦⠄⠿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⠁⢴⡗⠿⣤⠽⠀⠀⠀⠀⠀⠀⠀⠙⣷⠚⣰⣧⠞⠁
⠀⠀⠀⠀⠀⠙⠦⠴⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠝⠁

⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⢠⣶⣶
⠀⠀⠀⠀⠀⠀⢀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣅
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠻⢧⣽⣿⣿⣿⠛⢻⣆⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠉⢻⡄⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⠋⠁⠀⠀⢸⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠐⠛⣿⣿⣿⡿⠀⠀⠁⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⡇⠀⠀⠀⠀⠀⣸⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠟⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣿⡇⠀⠀⠀⠀⠀⢹⡟⠀⠘⣿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣓⡂⠀⠀⠀⠀⠀⠀⢸⡧⠀⣿⣅⠀⠀⠀⠀⠀⢸⣧⠀⠀⢹⣷⣄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡹⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢻⡇⠀⠀⠀⠀⠀⠀⢿⣿⣧⣤⠀⠀⠀⠀⠀⠀⢸⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠚⠃⠘⠃⠀⠀⠀⠀⠀⠀⠙⠛⠓⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠐⠒⠒⠠⢤⡴⠋⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢍⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢅⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⡀
⠀⠀⠀⠀⠀⢰⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠶⠖⢦⡠⠊⠙⠲⢄⠀⠀⠀⢀⠜⠁⠀⠀⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⠀⠀⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⡔⠒⠓⡤⣤⣀⣀⠀⠀⠀⣀⡀⢀⣀⡤⢶⠀⠀⠙⠴⠊⣷⠀⠀⠀⢑⡆⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠝⠦⣀⠀⠀⠀⠀⠀⡿
⠀⠀⡠⠚⠁⠀⢸⠤⠄⠀⠈⠉⠉⠉⠁⠉⠉⠀⠀⠇⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠈⢹⠛⠋⠓⠒⠂⠉⠁⢈⡉⠀⠀⠀⡏⠳⣌⠀⠀⢠⠇
⠀⠀⢹⠋⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠇⠈⠓⣄⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⡇⠀⠈⣣⣀⠞⢤
⠀⠀⠘⠒⢲⢖⢽⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠤⡀⣤⢆⣠⠊⠀⠀⠀⠈⢣⠀⠸⡀⠀⠠⡀⠀⣀⠀⠀⠀⠀⠀⠀⠳⠤⠾⡹⣏⠀⠀⠱⡄
⠀⠀⠀⠀⠈⠁⠀⠱⣄⠀⠑⠐⠉⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⡸⠀⠀⠀⠐⢀⡼⠂⠀⠑⢤⡀⠙⠋⠉⠀⠀⠀⠀⠀⠀⢀⡴⡞⠁⠘⣄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣲⣤⣄⣀⣀⣤⣤⣤⡖⠊⠑⠲⢤⡧⢻⠀⢀⣾⡸⠀⠀⠀⠀⣠⠬⢵⡶⡤⠠⢤⣴⠖⣲⠮⠥⡒⠑⢆⠀⠘⠶⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⣵⠉⠉⢀⣤⣇⡙⢖⠐⠒⠢⣷⣤⡳⠋⠀⠀⠀⠀⠀⢰⠇⠀⠀⡇⠈⠙⠉⠀⠀⣇⠀⠀⠘⢇⠙⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡴⠃⠀⢰⠫⠀⠈⢱⠀⠙⢦⡀⢈⠻⡀⠀⠀⠀⠀⠀⠀⢸⡆⢀⣴⠁⠀⠀⠀⠀⢃⠹⣄⣀⡠⠼⠈⣏⠡⠜⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠐⡇⣾⠋⠀⠀⠀⢸⠀⠀⠀⠀⠣⡀⠀⢱⡀⠁⠹⡀⠀⠀⠀⠀⠀⠀⢇⢻⡝⠀⠀⠀⠀⠀⠈⢄⠙⢳⠄⠀⠅⠹⣀⡶⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⣼⠗⡖⠀⢡⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⠀⠏⠁⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠁⡸⠀⢀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡀⠀⠀⠀⣀⡠⠊⠀⠀⡘⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠧⣉⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠩⠍⠙⠛⠙⢲⡣⢠⡴⠤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⡶⠲⠤⠤⣤⠤⢴⠚⠿⢍⣓⠎⣁⠖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢦⣤⠤⢤⡀⠀⢈⡗⠫⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⢀⡏⠀⡼⠀⢈⠁⠈⠙⢦⡀⠐⠀⠠⡤⠂⠀⠀⠀⠀⠀⡴⠁⠀⠀⢸⡴⠋⢀⡼⡠⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣍⠳⢄⣀⣀⡼⢦⡞⣁⡴⠛⡄⠀⠀⠀⢳⠤⠘⠃⠀⠀⣤⠤⠤⠤⣟⢄⡀⠀⢀⡇⠙⣖⠉⠂⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢠⠞⠉⢧⠀⠈⡹⠂⠀⣘⠀⠀⠘⣧⣀⡤⠃⠀⠀⠀⢸⠀⠀⠀⢀⡞⠀⠀⠀⣀⠏⠂⠉⠚⣻⠀⠀⢈⠁⠰⠀⠀⠀⠀⡾⢀⢀⡀⠀⢄
⠀⠀⢀⠄⠒⡏⠀⠀⠀⠉⠉⠀⠀⠀⡸⠀⠀⠀⠳⡀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡀⠀⠀⠀⢀⠞⠀⠀⡄⠉⠓⡈⠠
⠀⠀⠆⠀⠀⠘⣄⠀⢀⡀⡀⠀⢀⡠⠃⠀⠀⠀⠀⠙⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠙⠢⣄⣀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠙⠢⠤⠖⠋⠀⠀⠀⠀⠐⠀⠹
⠀⠀⠀⠂⢄⠀⠀⠉⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠂⠀⠀⠰⡀⠤⠀⠀⣠⠁⣠
⠀⠀⠀⠀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⡁⠰⠀⠀⠀⠀⠀⣀⠈⠠⠀⠀⠀⠀⠢⠄⠀⡀⡀⠀⢈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠌⠉⠀⠀⠀⠀⠤⣈⠁⠀⠀⠀⠠⡀⠀⠀⡀⠀⠈⠀⠀⠒⠀⠀⠀⠈⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇⠠⠞⠁⠀⠀⠀⢠⠊⠀⢀⠘⠀⠄⠀⠀⠈⠀⠀⠺⠂⡤⠀⠘⣁⠀⠀⠀⠀⠀⢠⢾⡁⠰⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠤⢀⠀⠀⠀⠀⠀⠀⠉⠠⠄⣠⠀⠀⠀⣀⠀⢪⠀⠠⡄⠀⠈⠀⠀⠀⠀⠀⠀⣰⠊⢀⡄⠀⠉⠀⠀⠀⠀⢀⣸⠀⠀⡠⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠌⡱⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⡠⠿⢦⠒⡡⣌⠀⠀⢀⡔⠉⡀⠙⠦⠚⡄
⠀⠀⠀⠀⡠⢒⠀⠀⠀⢠⠲⢄⠁⠀⠀⠀⠸⠒⠀⢠⠘⠀⠀⠀⠀⡄⠀⠀⠐⠀⠀⠑⠂⠀⣠⠞⢢⠔⠀⠉⡟⣽⠀⢰⠃⣸⠋⠀⠀⢀⡀⡁
⠀⠀⠀⠀⠀⠀⡖⠀⠉⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠄⢀⢠⠆⠈⢄⠀⠀⠀⠀⠀⠂⠤⢤⡏⠐⣹⠀⠀⢘⡎⠁⠀⢰⠀⡿⣀⠀⢀⠸⠁⠁
⠀⠀⢠⠄⠀⠀⡄⠀⠀⡤⠀⠀⢠⠲⠄⣀⣀⠀⢀⠀⠴⠈⢣⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡾⠁⠉⠀⠹⠅⢄⠂⡀⡇⣱⠀⠠⡸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⢦⠝⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠀⢀⠀⠀⠀⠀⣢⠈⠁⢸⣇⠀⠀⢧⠀⠓⠂
⠀⠀⠀⠀⠀⡠⠃⠀⠐⠑⢄⠀⠀⡀⡀⡀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡽⠉⡓⠙⢏⠉⡃⠀⠀⡜⠘⢆⠀⠀⣇
⠀⢀⡠⢞⣄⠀⠀⠈⠀⠀⠈⠀⠀⡌⡴⠀⠀⠀⠠⠀⠀⠀⠣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠔⠀⠀⠈⠀⠡⢀⠾⣀⣠⠜⡆⠀⢹
⠀⠀⢻⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠎⠀⠀⠀⠀⠡⡔⠿⢀⡀⠁⠀⠸⡄⠈⣣⣀
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠸⠆⠀⠀⠀⠀⠀⠱⢨⡽⠥⠀⠀⠀⠙⢦⣩⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢲⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠐⣠⠁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⢈⣵⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⠇⢸⣧⠐⠀⠀⠀⠀⠀⠀⢀⡀⠈⠙⠋
⠀⠀⠀⡠⢾⣿⣿⣿⣿⣷⣶⡤⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣀⣠⣤⣄⣸⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢠⣗⣷⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣿⣿⣿⣿⣟⣿⣿⣿⡄⠀⠰⣿⣿⣿⣯⣿⢱⣿⣿⣿⡄⠀⠀⠀⠁⠬⢦⣤⠆
⠀⢡⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⢿⣿⣿⣳⣽⣿⣿⠀⠀⢘⣿⣟⣿⣿⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣈⢀⢐
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣿⡟⠁⡀⡀⠀⣿⣿⣿⣾⠂⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠠⠀⡆⠀⠀⠀⢀⠹⡆⠄⠆
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⡇⠀⢹⣿⣿⣿⡀⠀⠀⣿⣿⠋⠈⠉⠀⡉⠀⠀⠀⠀⣬⠀⠀⠀⠀⠈⠀⢾
⠘⢿⣿⢿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠿⠃⣿⡇⠁⠀⠁⠀⢸⣿⣿⣿⡇⠀⠐⣿⡏⠀⠀⣐⣲⡅⠀⠀⠀⠀⡆⠛⠒⠒⠒⠒⢻⡟
⠀⠀⠀⢘⢿⣿⣿⣿⣿⣷⢿⣿⣇⠀⠀⣿⣿⣤⣶⣶⣶⣾⣿⡿⣿⣷⠐⣸⣿⣷⠀⠀⠀⠂⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡿⢡⡀
⠀⠀⠀⠠⠿⡿⣿⠿⢿⠿⣾⣿⡿⠀⠀⠸⣾⡿⢿⠙⡟⠻⡟⠂⢹⣿⠀⢿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠠⠆⣰⢶⣦⣶⢶⣾⣷⣷
⠀⠀⠀⢠⢀⡀⢠⣆⢰⣿⣿⣵⠇⠀⠀⠀⠛⣧⣰⠠⡟⠻⠃⠀⢘⡟⠀⢹⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠘⣇⣚⣅⠀⠚⠣⠝⠂⠀⠀⠀⠀⣐⣉⠀⠙⠀⠁⠀⠘⠁⠀⠘⣿⣿⠉⣿⣿⠃⠀⠀⠀⠀⠀⠹⣷⡍⢛⠛⢻⣗⠃
⠀⠀⠀⠀⠁⠛⠚⠃⢀⠀⠀⠄⠀⠀⠀⠀⠀⠘⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⢻⢿⠀⢿⠏⠀⠀⠀⠀⠀⠀⠀⣒⠂⠐⠀⠸⠡
⠀⠀⠀⠀⠀⠐⢷⠀⠀⠀⠤⠴⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⢈⠀⡠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠂⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢨⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠁⠀⠒⠒⠀⠀⠀⠀⠀⠀⢻⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠒⢶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠯⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠈⠉⠉⠉⠉⠉⠁⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡖⠐⠒⠢⠤⠄⠀⣀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠖⠲⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠢⢄⣀⣀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠀⠀⠀⠀⠀⠀⠻⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⡟⠉⠋⠀⠈⠑⠢⣄⠀⠀⠀⣠⠜⢻⠰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠉⠁⢸⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠇⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⡀⠠⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢠⡀⢠⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡌⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⡎⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⠀⠀⠀⡀⢾⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢐⣇⢀⡤⠀⠒⠂⢉⡧⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠉⡶⠶⠁⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠆⠀⢀⡤⠚⠉⠀⠀⠀⣀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢀⡸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⣄⠀⢀⣀⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⡜⠁⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⠁⢀⠾⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⡆⣀⡜⠀⠀⠀⠀⠀⠀⠸⡆⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣤⠋⠀⣉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠓⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠸⠂⢀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠸⣄⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠎⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⣠⠜⢧⡀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠀⠠⠔⠒⠈⠣⣄⡠⠷⣀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠫⡄⠀⠀⠀⠠⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢺
⠀⠀⠈⢦⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⠠⠟
⠀⠀⠀⠀⠱⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠂⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣀⡀⠤⠤⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⢀⡀⢀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣷⣷⣶⣀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⢛⣋⣥⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣠⣿⣿⠿⢋⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿
⠀⢀⣾⣿⢛⣵⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⣦
⢀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⢺⣿⣿⣿⣿⣿⡿⠋⠁⢰⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢷⠈⢷⡀
⢿⣿⣿⣿⣿⠟⠁⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠈⢷⠀⣀
⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⠞⠋⠙⠳⣄⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⡄⠘⣟⠉⢳⡄
⠀⠈⢿⣿⣿⡄⠀⠀⠀⣼⠁⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠘⠆⣈⠗⠀⠀⢹⣤⡾⠃
⠀⠀⠘⣿⣿⣿⡄⠀⠀⢿⠀⠀⠀⠈⠛⠀⡟⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠘⣿⣿⣿⡄⠀⠈⠳⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣿
⠀⠀⠀⢰⠟⠍⠩⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⢀⡼⠁⢈⡆⠀⢰⡏
⠀⠀⠀⠸⣄⣒⣀⡴⣆⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠙⠓⠛⠋⠀⣠⠋⠀⢀⡾
⠀⠀⠀⠀⠈⠉⠁⠀⠘⣧⠀⠀⠀⠀⠈⢳⡦⠄⣀⣀⣀⣠⡴⠚⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢸⣿⣷⣶⣾⠉⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠘⢦⣈⣉⣿⡀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⠀⠀⠀⠀⠀⠀⢀⡴⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣉⣥⡝⠷⡦⣤⣤⠴⠞⠫⠑⣤⣽⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠉⣉⠆
⠀⠀⠀⠀⠀⠀⢀⡜⡿⠋⠁⠈⠉⢸⡇⠀⠀⠒⠀⠀⠈⠁⠀⢹⡍⠻⢿⡶⠤⠤⣤⣤⣤⠞⠁⣀⣠⣴⢺⡏
⠀⠀⠀⠀⠀⢠⡏⣼⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣀⣀⣀⣧⠄⠀⠉⠑⠲⠤⢤⡄⢠⡞⠉⠉⠉⠋⠁
⠀⠀⠀⠀⠀⡾⠀⣏⠀⠀⠀⠀⠀⣿⠀⠀⢀⢤⡲⠙⠓⠀⠉⠩⢰⡞⠡⢆⣄⠀⠀⢸⡇⠘⢧⡀
⠀⠀⠀⣠⠟⠁⠀⠈⠳⣄⠀⠀⢰⡿⢕⡎⡔⠁⠀⠀⠀⠀⠀⠀⠺⡗⠀⠀⠙⠵⡄⠀⢷⣀⡼⠃
⠠⠀⢴⡇⠀⠀⠀⠀⢢⡼⣤⢲⢵⣟⡇⠹⠿⠺⠗⠿⠺⠟⠿⠛⢻⡇⠀⠀⠀⠀⡼
⠈⠒⠚⢇⣰⡄⢀⣄⢈⡷⠚⢵⡄⢻⡇⢦⡀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠁⠙⠋⠉⠋⠁⠀⠀⠈⢹⡵⡛⢦⣤⣄⠀⠀⠀⠹⣧⡠⠀⠀⠀⠀⣐⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡀⠈⢳⣿⣿⢿⣄⠀⣠⣼⠟⢀⠀⣴⡞⠋
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠷⠾⠁⠁⠋⠱⠞⢻⡿⣷⡉⠙⠈⠀⠀⠀⠀⣠⣴⠶⢶⢶⣶⡄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⠻⣄⠀⠀⢀⣴⠞⢻⡤⣖⠝⡺⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢏⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡘⣷⣸⣿⠿⣤⢎⢟⡵⠋⠁
⠀⢸⠀⠀⠀⢠⡞⣏⠉⣷⣿⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣤⠟⢡⣾⠕⡷⠋
⠀⣠⠀⠀⠀⠈⢷⡹⣄⣛⣙⣁⣬⣅⣈⣹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⣣⡾⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠐
⠀⢰⠀⠀⠀⠀⠈⠓⠒⠒⠲⠒⠖⠒⠓⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣮⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠐⠋⠁⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⣄⠀⠀⠀⢈⣍⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠻⠟⡀⠀⠀⢸⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠘⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠘⠓⠛⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢶⣶⠒⢚⡿⢿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢯⣹⢟⡭⠖⠛⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⠀⣾⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⡇⠀⡏⠀⠀⠀⠀⢤⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠁⢸⡅⠀⡇⠀⠀⠀⠀⢸⠙⢦⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠛⠁⢀⡤⢾⡇⠠⡇⠀⠀⠀⠀⢸⡆⠈⢧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢈⡽⠶⠚⠉⠀⢸⡇⠀⡇⠀⠀⠀⠀⢸⡇⠀⠘⢧⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢀⣸⡇⠀⠀⢠⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠉⠉⠉⠉⠉⢹⠇⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡶⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡞⠁⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡼⠙⢦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡀⣰⠃⠀⠀⠳⢴⡎⠙⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡟⠉⠀⠀⠀⠀⠀⣹⢶⠞⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣺⣁⣼⠁⠀⠀⠀⠀⠀⠀⡿⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⣀⣉⣉⣹⠀⠀⠀⠀⠀⠀⠀⢧⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣤⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⢀⣤⣶⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⡆
⠘⠿⠟⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠿⠿⠿⠿⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿⣧⠉⠉
⠀⢀⠴⢿⡿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠓⣄
⢰⣿⣧⠬⢻⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⣿⣿⡏⠣⣾⣿⡆
⠈⠉⠁⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⢸⣿⡇⠀⠈⠉
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡯⠄⢱⠀⠀⠀⠀⠀⡧⡌⠃⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠑⠤⠃⠀⠀⠀⠀⠀⠳⠴⠃⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⠀⠀⠀⠀⣀⡴⠃⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠛⠉⠁⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠋⠁⠀⠙⠶⠶⡤⢤⡤⡤⠤⠶⠞⠛⢿⡈⠳⡄
⠀⠀⠀⠀⠀⢀⡴⠋⠉⣈⣷⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣿⡀
⠀⠀⠀⠀⢠⠞⢀⡴⠋⠁⠀⠉⣿⠀⠀⠀⠀⡴⠒⠒⠤⠊⠉⠑⡆⠀⢹⡷⡄⠱⢄⡀
⠀⠀⢀⡰⠏⣠⠞⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⣰⠃⠀⠘⣧⠈⠳⢤⣉⠓⠦⣀⠀⡀
⢀⡴⠉⠀⢠⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠙⠢⣄⣠⠞⠁⠀⠀⠀⣿⡀⠀⠀⠈⠑⠲⡄⠈⠉⠓⢤
⢯⣠⡴⢦⣼⠁⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣆⣰⠶⣤
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠐⢿⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣇⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⡀⠀⠀⠀⠀⠀⠘⣦⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡙⠛⢷⡀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠶⣤⣄⣘⣿⣄⢂⣡⠶⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⣠⠏⠙⠁⠻⠛⢷⡀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⢁⣠⡤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠻⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⡞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣽⣄⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣮⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠐⠋⠁⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⣄⠀⠀⠀⢈⣍⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠻⠟⡀⠀⠀⢸⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠘⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠘⠓⠛⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢶⣶⠒⢚⡿⢿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣹⢟⡭⠖⠛⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⠀⣾⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⡇⠀⡏⠀⠀⠀⠀⣤⡄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠁⢸⡇⢀⡇⠀⠀⠀⠀⢿⠙⢦⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠛⠁⢀⡤⢾⡇⢈⡇⠀⠀⠀⠀⢸⡆⠈⢧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣯⢈⡽⠶⠚⠉⠀⢸⡇⠀⡇⠀⠀⠀⠀⢸⡇⠀⠘⢧⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢀⣸⡇⠀⠀⢠⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⢹⠇⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡖⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡴⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⡞⠁⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡼⠙⢦⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡀⣰⠃⠀⠀⠳⢴⡎⠙⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡟⠉⠀⠀⠀⠀⠀⣹⢶⡞⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⣻⣁⣼⠁⠀⠀⠀⠀⠀⠀⡿⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣈⣉⣽⠀⠀⠀⠀⠀⠀⠀⢧⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠒⠒⠒⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠊⠁⠀⣀⣀⠤⠤⠤⠤⠤⢀⣀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠀⢀⠔⠋⠀⠔⠀⠀⠀⠀⠀⠐⠂⠑⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢸⠀⠀⠀⠀⢔⡠⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣈⡁⠀⠀⠀⠈⠀⠀⠀⠀⠤⠊⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠄⠀⠀⠀⠀⠀⢀⣀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣋⠀⠀⠈⠉⠉⠉⠉⠉⠀⢱⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠐⠉⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠒⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠱⠔⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠛⠒⠒⠒⠒⠒⠒⠒⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⢀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠑⠲⡄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⢳⠶⠿⡦⣞⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⢚⣩⠴⠒⠁⠀⠀⠀⠘⢆⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣚⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⢀⠂⠀⠀⠀⠠⠐⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠳⠄
⠀⠀⠀⡀⠀⠸⠀⠀⠀⠀⠂⢀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⢀⠀⠐⠄
⠀⠀⠀⠃⠀⠀⡆⠀⠀⠇⠀⠀⡄⠄⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠘⠀⠀⠀⡇⠃⠈⠀⡄⠀⢰⡄
⠀⠀⠀⢳⠀⠄⢱⠄⢦⠀⠀⢸⡃⢰⠰⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⡀⠀⢀⠃⠀⢓⣤⠈⢠⠐⠘⠣
⠀⠀⠀⠀⢧⠀⠈⡆⢰⠀⠀⠞⠋⡆⠀⠀⠠⠃⣀⠀⠀⠀⠀⠀⠀⠀⢣⡼⡆⠀⠘⠘⡇⠀⡟⠐⠀⠀⡟⠀⠀⠈
⠀⠀⢀⠀⣾⠊⠀⡟⠀⠀⠀⠀⠀⡇⢠⡕⡣⠟⠉⠁⠋⠀⠀⠀⠀⠘⠊⠙⠠⠀⢸⠀⠀⡀⠁⠀⠀⢠⠁⣆
⠀⠀⢸⠀⡇⠀⠀⣀⡀⡃⣾⠀⡇⢰⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢪⡄⡇⡇⢷⠀⠀⠘⠀⠈⢧⠀⢀
⠀⠀⢸⠸⢃⡇⠀⡿⠀⡇⣼⣤⠾⠣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⢋⡇⠻⠀⢀⠀⡆⠲⣤⠘⣿
⠀⠀⢻⢠⠼⡇⢸⣷⡄⣡⣸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⣡⣌⢰⡗⠀⣰⠀⣁
⠀⠀⢸⢸⡄⡇⠐⠐⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⢎⠐⠃⠀⢿⠀⣽
⠀⠀⢼⢸⠃⠀⢊⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠘⣵⠂⢿
⠀⠀⠁⠔⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⣄⣱⠇⠈⠄
⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠿⠛⠛⠛⠛⠻⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⡉⡀⠀⠀⠀⠉⠀⠀⠀⠸⠕⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⡀⠀⠐⠁⢀⢄⠀⠀⠀⢐⠂⠀⠀⠀⠊⠁⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠁⠀⠀⣀⣈⣈⣠⣤⣤⣤⣤⣶⡶⠶⠶⠶⠶⣶⣤⣤⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⣀⣤⡶⠶⠿⠛⠛⠉⠉⣉⣁⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣈⣉⠙⠛⠳⢦⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠛⢉⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠶⣦⣌⡙⢷⣄
⠀⠀⠀⣠⣴⠟⢋⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⢶⣿⡄
⠀⢠⡾⢋⣠⡶⠟⢩⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⠃
⢠⣿⣰⠟⠉⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡀⠀⠀⣠⡾⠋
⠸⣿⠋⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣿⣿⣿⣿⡗⢺⣏⣀
⠀⢹⣦⣴⠙⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⢸⣿⣿⠿⡅⣸⣿⣿⡆
⠀⢿⣿⣿⣄⣽⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠉⠁
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣘⠆⠀⠀⠀⠀⠀⢰⠏⢈⣳⠀⠀⠀⢸⣿⡟⢁⣼⣦⡀
⠀⠀⢠⣾⣇⠈⣿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⣩⠆⠀⠀⠀⠀⠀⠸⣆⣈⡿⠀⠀⠀⣸⡏⠁⠻⣿⣿⠇
⠀⠀⠻⣿⠿⠓⠁⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡄⠀⠈⠉⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠋⠉⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⢿⡷⢿⡟⠉⣁⣭⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⣟⠀⠀⠀⠀⠀⠉⠋⠑⠒⠒⠒⠚⠉⠀⠠⠉⠋⠙⣷⡀⢈⡷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⠶⠚⠙⠓⢶⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡋⠙⢦
⠀⠀⠀⠀⠀⣼⠃⢠⡾⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠀⠀⠀⠀⠈⣿⠙⢦⡀⠓⠦⣄⡀
⠀⠀⢀⡤⠚⠁⣴⠋⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠀⠀⠀⠆⠀⠀⠀⢿⡇⠀⠙⠓⠤⣀⡉⠓⠦⠤⣀⣀
⠀⣴⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠈⠙⠓⡄⠀⠈⠉⠳⣄
⢿⣁⣠⠶⣆⣸⠃⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠱⣆⢀⡶⣦⣤
⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⢠⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠿⠶⠶⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⡶⠾⠷⢿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠙⠛⠻⣧⡀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣄⡁⠂⢹⣷⡀⠀⢂⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠉⣻⠷⣾⠟⣿⣶⣿⡋⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠛⠋⠁⢀⣠⡾⠋⠀⠀⠀⠈⠁⠈⢷⡄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠛⠉⢁⣀⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣶⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣤⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣈⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣦
⣾⣶⣾⣶⣾⣷⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣷⣶
⣿⣿⣿⡇⠀⠉⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠙⠻⠿⠿⠿⠟⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀⠀⣿⡇⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠆⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠻⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⡀⢀⣀⣀⠠⣶⡿⠉⠉⠀⢠⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣴⣾⠛⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⠿⠿⠂⢀⣀⣀⣀⡀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠛⠋⠈⠙⠻⣦⣤⠀⠀⠀⣤⣤⡴⠛⠋⠀⠀⠀⠀⠈⢻⣿⠿⢿⣶⣦⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠓⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠉⠉⠉⠉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⣿⠇⠀⣿⣤⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢴⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⣿⡇⠀⣿⠟⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠉⠀⠀⠈⡇⠀⢠⣿
⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠛⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠸⣿
⠀⠀⠠⣾⣿⣿⣇⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⣷⡆
⠀⠀⠀⠉⠁⠙⡿⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⠀⠀⠙⢇
⠀⠀⠀⠀⠀⢰⣿⠀⠀⣠⣤⡾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⢀⣤⣤⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡖⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠙⣥⡀⠀⠀⠀⢀⣶⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⠟⠁⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⣸⡿⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⠀⠙⠿⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠛⣄⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢈⡿⠀⠀⠀⠀⠀⠘⠿⣆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⠀⠀⠀⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠋⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠿⠋⠀⠀⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠉⠁
⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠛⠋
⣀⣀⣉⣉⣉⣈⣹⣿⣿⣿⣿⣋⣁⣠⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣷⣶⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠁⢻⡇⠈⠉⠛⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⡿⠁⠸⠿⠛⠩⠿⠛⠉⠀⠀⠸⠃⠀⣀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⠀⢀⠤⣄⠀⠀⠀⣿
⢀⡀⡀⠀⠀⣸⣿⣿⣿⣿⡏⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⡱⠀⡇⠀⠀⠀⠀⢈⠆⢸⠀⠀⢀⡟
⡏⠀⠩⠆⢀⣿⣿⣿⣿⡟⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠠⠀⠈⠓⠋⠀⠀⢸⡇
⠘⣦⣀⣀⣼⣿⣿⣿⠋⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⣿
⠀⠈⠙⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⣤⠿⣆⠰⡄⠀⠀⠀⠀⠀⠑⠒⠤⠤⠤⠒⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠘⠛⢶⣄⣀⠀⠀⠀⠀⠀⣀⣠⣴⡟⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠚⢇⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠁⠈⢷⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢀⡴⠚⠛⠒⣶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡊⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠏⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⠦⣈⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣼⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠲⢤⡉⠐⠒⠦⣀
⠀⠀⠀⠀⠀⢠⡞⣀⣤⣀⡟⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠸⣄⢀⣤⣈
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⣾⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠊⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠛⠓⠒⠶⠶⠶⠶⠶⠚⠛⠛⠛⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣀⠀⠀⠀⠀⠀⠀⠲⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⣧⡀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠶⣤⣁⣸⣷⡄⢀⣤⠾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠃⢀⣴⠋⠙⠉⠛⠛⣧⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⢁⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠈⢷⡀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⡞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣬⣷⣤⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣧⣙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠇⡁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁⢸⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⢸⠀⠀⠀⠀⣽⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⡁⠘⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢹⣿⡿⠛⠁⠆⠀⠀⠀⠀⢹⣿⣿⣿⣿⠗⣹⣿⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢪⣿⣿⣆⢰⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⠚⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢘⠀⠀⠀⠀⠀⠀⣿⡏⠀⠠⠀⣾⠟⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⢸⠀⠀⠀⠀⠀⠀⣿⠁⠀⡧⢠⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⢸⠀⠀⠀⠀⠀⠠⣿⠀⠀⢧⣸⣿⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠹⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⡆⠀⠀⠀⠀⠀⠀⢻⡆⠀⢨⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠘⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⣿⣿⣿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⢀⣀⣀⣀⣀⣈⣿⠀⣿⣟⣋⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⣗⣰⠞⢿⠓⠂⣿⣿⣿⡇⠀⠀⠀⠀⠉⠙⠛⠛⠙⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠟⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣉⣌⣹⣥⣬⣉⣹⣏⣩⣿⣫⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠉⠉⠿⢿⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠠⢶⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢠⡔⢲⠈⢀⡀⠀⠀⡠⠤⣁
⠀⠀⠀⠀⠀⠀⠀⠘⢲⠖⠀⠲⣉⠀⠀⢳⠁⠈⠱⣮⡄⠀⢸⠀⣀⠤⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠢⡰⠒⠢⢵⢄⠈⢦⠆⠀⢸⠤⠔⠚⠈⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⠷⡈⢒⣒⠊⠀⠀⠉⠒⠐⢲⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠒⠓⢂⠀⠹⡀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁⠱⡀⠈⢄⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠁⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠧⡀⠠⢧⡀⠀⠀⠀⠀⠀⠀⢀⠒⢥⢋⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠬⠳⡀⠀⠀⠀⠀⠀⠉⡇⠀⠀⢠⡄⠀⠀⡠⠎⠛⢫⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡠⠊⡡⠔⠚⠧⠔⠊⠁⠀⠀⡀⠙⣄⠀⣮⣉⡉⠩⠔⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠛⡄⠀⠀⠀⠀⠀⡇⠀⡆⢀⠈⠘⣇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣀⠀⣺⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢖⠒⠒⠉⠙⢝⢎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠈⠉⠳⠀⢠⣴⡒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣄⡀⠀⠀⠀⠹⣾⠧⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⢎⡉⠀⢙⠏⠉⠢⢄⠀⢠⣿⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡘⢿⠓⠚⢤⠀⠀⠀⠉⡶⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⠤⠞⠀⠀⠀⠀⠁⠀⠘⠛

⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⠀⠀⠀⠀⢀⣠⣿⣤⡀
⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣻⣿⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⠍⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣆⠀⠀⠀⠀⡾⣾⣿⣿⣯
⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠺⢸⡿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠃⢻⣿⣿⠛
⠀⠀⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡇⢿⡇⠀⠀⠀⠀⠀⢸⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡃
⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡀⠀⠀⠀⠀⠀⣠⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⣿⠙⠲⠄⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠘⢿⣿⡿⠟⠷
⠀⠀⠀⠀⣼⠟⠈⢿⡄⠀⠀⠀⠀⢠⡿⠁⠙⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣴⣿⠟⣿⠀⠀⠀⠀⠀⠀⣿⡿⣷⡀
⠀⠀⠀⣸⠏⠀⠀⠈⠻⣤⣤⠀⢠⡿⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⢠⡟⠁⠀⢻⡀⠀⠀⠀⠀⣰⡿⠀⠙⣷⡀
⠀⠀⠀⠛⠓⠀⠀⠀⠀⠉⠀⠀⠘⠳⠂⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠛⠓⠀⠀⠈⠉⠁⠀⠀⠐⠛⠁⠀⠈⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢀⡤⠐⠒⠀⠀⡑⠀⢀
⠀⠀⠀⠀⠀⠄⢀⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⠀⠀⠀⠀⣀⡀⠀⠀⠠⠀⡀⠠
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠠⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠀⠀⠀⠀⠘⠒⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠉⠁⢀⡖⠀⠐⢠⡀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠀⠀⠀⢀⡄⠂⠀⠰⡀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠐⢀⠠⠛⠒⠂⡶⠐⠂⠈⠉⠑⠂⢶⠐⠒⠛⠄⡀⠂⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⡆⡀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⢰⣦⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡉⣁⠀⠀⠸⡍⠁⢰⠀⠀⡂⠈⢉⠇⠀⠀⣈⢉⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠉⠂⠀⣀⠈⠀⠀⠀⠀⠀⠀⠁⣀⠀⠐⠉⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠀⢀⠐⠠⢀⣀⡀⡠⠄⠂⡀⠀⢻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢂⠀⡠⠀⠄⢄⠀⡐⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠲⡀⢄⡠⢀⠖⠁⠀⢀⠠⠤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢈⡒⢒⡉⠀⠀⠈⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢱⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡠⢄⣀⡰⠤⠤⢆⣀⡀⠔⠁⠀⣸⢀⡆⡠⠤⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡎⡀⡀⡀⠀⠈⠁⠀⠀⠀⠀⢠⠈⠀⡠⠋⠀⠐⠀⠂⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣈⡩⢀⠄⠀⠀⠀⠀⠀⠀⠀⠅⠄⠔⢁⠄⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⠀⢀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡀⣀⠰⠀⠀⠀⠀⠀⢇⠀⠀⠀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠖⠚⠛⠋⠉⠉⠉⠉⠛⠛⠓⠒⠶⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠶⠶⠶⠶⠦⠤⣄⣀⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠁⠀⠀⢀⡤⠖⠋⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠉⠓⢦⣸⡇
⠀⠀⠀⠀⠀⠀⣴⣯⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠞⠉⠑⡄⠀⠀⠀⠀⠀⠀⡴⠋⠙⢦⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⡷⠀⠀⠀⠀⠀⠀⡁⠤⠀⢀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠳⣤⠴⠃⠀⠀⠀⠀⠀⠀⠙⢤⡤⠎⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⡛⢷⠾⠋⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⣀⣠⠴⠚⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠿⠦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡏⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⢿⡄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢀⣠⣧⣀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⡰⣇
⠀⠀⠀⠀⠀⢀⡾⠁⢠⡞⠉⠁⠀⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠈⠳⣄
⠀⠀⠀⠀⣠⠟⠀⣴⠋⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⣤⡈⠙⠲⣤⣀
⠀⢀⡴⠛⠁⠀⡞⠁⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠉⠓⠶⣤⣉⠛⠒⠶⢦⡀
⣠⠞⠁⣀⠀⢠⠇⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠙⠲
⠳⠶⠾⠋⠣⠼⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣦⣸⠟⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⣦⣤⣤⡀⠀⠀⠀⠀⠘⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠠⢀⠙⣿⡄⠀⠀⠀⠄⢈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠻⠶⣦⣄⣹⣿⣦⢈⣤⡶⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⣠⡾⠉⠛⠀⠻⠟⠻⣄⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⣀⣠⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⠚⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⣶⣿⣶⣦⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⡤⠤⠤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠻⠷⠶⠦⣤⡄⠀⠀⠀⠀⠀⠉⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣀⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⡏⠙⠛⢿⣿⠶⠶⠦⠀⠀⠀⠀⠙⣷⠒⠶⢠⣤⣀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣄⠁⠀⠀⠀⠀⠙⠛⠶⠤⢀⣀⣀⠀⠈⢳⡀⠀⠀⠈⠙⢷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠚⠛⠀⠀⠀⠀⢀⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠘⠇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡄⠀⠀⡠⠀⠐⠒⠀⠀⢠⡄⠆⠀⠀⢠⡄⠀⠐⠒⠂⠀⠀⠀⠀⠀⠈⣿⣀
⠀⠀⠀⠀⠀⠀⢠⡞⠉⢹⡏⠀⠈⠃⠀⠰⢶⣶⠄⠈⠀⠀⠀⠀⠀⠀⠀⠒⣾⡶⠀⠀⠀⠀⠀⠈⣿⠉⠓⣄
⠀⠀⠀⠀⠀⠀⢾⠀⠀⠸⡇⠀⠀⠀⠀⠀⠴⠿⠁⠀⠀⠀⠂⠀⠀⠈⠁⠔⠻⠟⠁⠀⠀⠀⠀⠀⡇⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠈⠇⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠀⠀⠀⠀⠀⠀⢀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢶⡿⣧⠉⠘⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⡿⠁⠉⠉⠁⠈⣰⡆⠋⠀⠀⠉⠁⣠⣾⢳⠏⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠈⢃⣾⠀⠀⠀⠚⠁⠙⠄⠀⠀⠀⠊⣿⣹⠇⠀⠀⠀⠀⠀⣰⠏⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⡟⠀⠀⠸⠃⣦⠀⠀⠀⡆⠀⠀⠀⠀⣠⢞⡷⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢰⡇⠀⠀⠀⠻⡈⢧⣀⡀⠁⠀⣀⣠⡞⣡⠞⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⠁⠀⠀⠀⠀⠙⢦⡉⠛⠒⠚⣉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣄⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⣸⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠠⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⠑⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⠙⠷⠶⠀⠸⠁
⠀⠀⠀⠀⢠⣿⠦⠛⠋⢻⣦⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⢠⣞⣁⡶⢸⠀⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠉⠙⠿⣧⣼⣯⣸⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⢦⣤⣤⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⢿⠃⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠷⠶⣶⣄⡈⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣘⣻⣷⣦⣴⣟⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠿⠛⠇⠀⠀⠀⠀⠀⠀⠉⠛⠒⠷⠒⢊⣉⣉⡉⠉⢉⠙⣿⣿⣇⣶⣼⠟⠁⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⢠⣦⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠛⠛⠛⠛⠛⣛⣛⣉⣙⣿⣻⣿⣴⣶⢾⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠘⡏⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡾⢿⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⢸⠁⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠤⠤⠤⠤⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠐⠒⠒⠂⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣄⣠⣤⣤⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣧⡴⠆⣈⣋⣀⠴⣤⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣉⣉⠉⣉⣉⣻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠗⠊⠉⠉⠀⠐⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⢴⣶⣶⣶⡤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣶⣿⣶⣶⣾⣿⣿⣶⣶⣾⣷⣆⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣤⣾⣏⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⣠⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠈⠉⠙⠻⢿⣿⣿⣿⣿⣿⡿⠛⠉⠙⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⠀⣿⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠀⠛⡟⣦
⠀⠀⣠⠞⣉⣙⠻⣿⠇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠉⠁⠀⢠⣤⡿
⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠇⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠈⠳⠶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⠤⣀⣀⣀⣀⣠⠤⠞⠁⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠛⠓⠶⣶⣶⣶⣶⣚⣿⠟⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠠⡄⣠⠿⣧⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢘⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠈⠳⣦⣀⣀⣀⣠⡤⠶⠲
⣠⠶⠶⠶⠶⠾⠋⠀⠀⣠⡴⠋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⠉⠉⠁⠀⠀⢀⣠
⢿⣀⣀⡀⣀⠀⠀⠀⣼⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠙⠳⢶⡄⠀⠀⣴⣚⣋⣹
⠀⢻⣍⣉⣽⢿⡄⠀⢘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠸⣇⣀⣰⠏⠉⠉⠁
⠀⠀⠉⠉⠀⠀⠙⠛⠛⠁⠀⠈⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡿⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠳⣦⡀⠀⠀⠀⠀⠙⣷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠛⢶⣌⡻⣦⡀⢀⣠⡴⠛⢷⡀⠀⢀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣿⠉⠀⠀⠀⠀⣨⡿⠋⠻⣿⠁⠀⠀⠈⢷⡴⢻⣧⣿⠟⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⣯⠱⡀⠈⢷⢲⠆⣴⣞⠉⠀⠀⠀⠈⢳⡄⢀⣤⣾⣟⡽⠋⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠱⣀⣈⣙⣺⣃⣙⣷⡀⠀⠀⠀⠀⢹⣏⢑⡵⠋⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠤⠤⠤⠤⠤⠾⠃⠀⠀⠀⠀⠀⠻⣾⣁⣠⡴⠞⠁
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣠⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠄⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀
⠀⠀⠈⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠀⡈⠀⢀⣀⠆⢠⠀⠀⠀⠀⠀⡰⢹⡏⢀⡠⠊⠑
⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢄⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢤⡀⠚⠀⠀⠀⠀⠀⠀⠀⠬⠔⠋⠠⡚⠀⠀⠀⡀
⠀⠀⢀⢀⡞⠉⠹⡅⠀⠀⠀⠀⠀⠀⢀⠀⡷⠓⠄⢄⠈⠀⠀⠀⠀⠀⠀⠚⠁⠀⠒⡎⢀⡀⠀⠀⠀⠀⢷⡿⠉⡀⢴⣴⠃⠀⠀⠀⠀⠀⠀⡀⣤⠇⠤⣤⡘
⠀⠀⠄⢎⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠓⠼⠀⠐⢊⠛⠀⠀⠀⠀⠀⠠⣗⠠⡦⡄⠀⢩⠔⠃⠀⠀⠀⡀⠊⣀⡄⠀⠀⠘⡁⠀⠀⠀⠀⠀⡘⢠⢻⠀⠹⡘⠃
⠀⠨⠀⢰⠋⠀⠀⡰⠎⠆⠀⠀⠀⠁⠀⠐⠛⠋⠉⠀⠀⠀⠀⠀⠀⠂⣼⡛⠁⠁⠀⡀⠀⠀⠀⠀⠀⠲⢨⠃⠐⠂⠀⠀⠁⠈⠚⠀⠀⢰⠁⠈⡦⠓⠀⠀⠐
⠀⠀⠀⡤⠀⠠⢄⢀⠄⠀⠀⠀⠀⡰⠺⠀⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⡠⠬⠢⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠋⠀⠀⠀⠀⠀⠀⠁⠘⡤⠀⡤⠤⢐
⠀⠀⠀⠀⠁⠀⠀⠀⠢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⡵⠠⠀⠀⠀⠀⠓⠑⠀⠀⠀⢤⠒⠄⠀⠀⠀⠀⠀⠊⠖⠘⠠⡭⠀⠀⠀⠀⠀⠀⠀⠊⡛⠁⠀⠐⡮⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠤⠄⠄⣠⠖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠉⠲⢂
⠀⠀⠀⠀⢀⣀⠴⠊⠀⠀⠀⠀⠀⣰⠁⠀⠈⢦⣀⠀⠀⠀⠀⠀⡠⠚⢅⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢱⠀⠀⢉⡆⠀⡤⠘⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡤⠼⣄⣀⣀⠀⠀⠀⠀⠀⢀⢠⠄⠀⢤⠔⠏⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⡀⢙⠦⣀⠀⠀⠀⠀⡇
⠀⢠⣞⠁⠀⡧⠀⠈⠁⠀⠉⡉⠉⠁⠜⠀⠀⠀⠀⡤⡀⠀⠀⠙⠛⠋⠒⠂⠉⠈⣍⠀⠀⠚⠳⣄⠀⢠⠃
⠀⢀⠃⣀⣀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠇⠀⢀⠀⡠⠁⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⡨⣧⠋⠢⡀
⠀⠀⠀⠸⠊⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡫⢛⠁⠀⠀⠀⣠⠀⠡⡀⠐⠆⠂⠀⠀⠀⠀⠀⣩⡕⠙⡄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠈⣓⢤⣤⣀⣠⡤⣤⠒⠙⠢⣼⠻⠀⡰⡴⠀⠀⠀⡠⠰⣶⣤⣤⡤⠐⡶⠮⣖⡓⡄⠘⠢⠔⠁
⠀⠀⠀⠀⠀⠀⡰⠁⠀⡷⠀⣠⠒⠦⡙⢍⠈⠚⣶⠏⠀⠀⠀⠀⠀⠁⠀⡿⠀⠀⠀⠀⡇⠀⠈⡂⣹⠆⢠⡄
⠀⠀⠀⠀⠀⠀⡇⡤⠚⠀⠀⡇⠀⠀⠠⡀⠙⡄⠈⢆⠀⠀⠀⠀⠀⡆⣞⠁⠀⠀⠀⠀⡘⠶⡈⠡⠸⠈⣤⠃
⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠑⠤⡀⠀⠀⣠⡶⠐⠈⠄⠀⠀⠀⠀⠈⣻⠀⠀⠀⠀⠀⠀⡐⠀⠘⠈⠈⢹
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⣠⠃⠀⢀⠃⠀⠀⠀⠀⠀⠳⡤⠤⢤⣤⣤⣀⠀⢀⠃⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠑⠂⠤⣤⣀⣀⣀⣀⡠⣶⡻⣑⠒⡁⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠀⠀⠸⡵⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⡟⢀⠇⢐⠉⠙⢮⡀⠀⠀⡀⠀⠀⠀⠀⡠⠉⠀⢹⡄⠊⡠⠉⠑⠠⡀
⠀⠀⠀⠀⠀⡠⡴⠣⣌⡢⠤⡼⠳⣮⠔⠉⡄⠀⠀⢱⠀⠁⠀⡰⠖⠆⠺⠱⢄⣀⡆⠘⡯⠀⠀⠀⠀⢸
⠀⠀⠀⠠⡎⠀⠑⠄⠈⠁⠀⡄⠀⠘⡓⠊⠀⠀⢀⡜⠀⠀⢘⠀⠀⠀⠀⠀⠀⢰⠀⠀⡆⠀⠁⠀⢠⠋⠀⠲⠄⠂
⠀⠀⠀⠀⢣⡀⢀⢀⠀⢀⡰⠀⠀⠀⠙⠢⠤⠒⠉⠀⠀⠀⠈⠢⢄⡀⠀⠀⢀⠎⠀⠀⠈⠢⠤⠔⠁⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣟⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⢉⣙⠻⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⠀⡀⠀⠀⠀⠀⠀⠈⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠏⠀⠁⣶⠊⢠⠆⠀⠠⢶⡕⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢭⣻⠃⠀⠀⠀⠀⠰⠋⠀⠀⠀⠉⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠐⢦⣠⠋⣠⡠⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣰⡏⡠⠀⠀⠀⠀⠦⣤⣄⠀⠀⠀⠀⠀⠀⣿⣟⢶⡀⠀⠀⠀⡄⠘⣇⠀⠉⣴⠿⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢶⠢⠤⣀⣀⡀⠀⠀⢀⡀⣀⢠⠀⣿⠀⢸⠇⠀⠀⠰⡅⠀⠸⠂⠀⠀⠖⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢳⣶⣤⣤⣤⣄⣀⣠⠴⠃⠀⠀⣿⠒⠋⠀⠀⠴⠊⠳⡀⣀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠈⣙⠛⠛⢛⡉⠁⠀⠀⠀⣼⠏⣠⠴⠂⠀⠢⡀⠀⣹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⡾⠟⠋⠀⠀⠀⠀⣀⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⢤⣤⣤⣤⣴⡶⠛⠉⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡀⠀⠀⠰⢋⠀⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⣇⠀⡽⠖⠶⢣⣼⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠈⠈⠉⠀⠀⠃⠈⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⣴⠀⠀⠀⠀⢸⠀⢀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣤⠾⡇⠀⠀⠀⠀⡆⠀⠈⠆⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣇⠀⢀⡾⠁⢸⡇⠀⠀⠀⠀⡇⠀⠀⠘⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢅⠈⢡⠋⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⢱⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢙⠁⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⢘⢦⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⢤⣄⡀⠀⠀⠀⠳⣬⣭⠇⣾⠀⠀⠀⠀⠀⢰⠟⠻⠷⠶⠷⢦⡀⣠⠿⠾⣤⡀
⠀⠀⠀⠀⠀⢀⡴⠋⡁⠐⠁⠀⠀⠈⠛⢶⣄⢀⣼⡟⠈⠂⠉⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢀⡴⠋⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⡒⠲⢤⡀⠀⠀⠀⠙⣷⡀
⠀⠀⣰⠋⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⣷⠀⠀⠙⢦⡀⠀⠀⠈⢿⡄
⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢫⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠈⢿⣄⠀⠀⠘⣷
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⡆
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⠶⠟⠁⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⢠⢲⣤
⠈⠣⣄⠀⠀⠀⠀⠀⠀⠄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⠛⢦⡀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣼⠁⣀⣤⠾⠤⠴⠋⣼⠇
⠀⠀⠈⠓⡶⢋⣙⠲⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⠙⠲⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠁⠀⠀⣠⣠⡴⠋
⠀⠀⠀⠀⣇⠳⠼⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣄⡀⢈⣉⣑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠚
⣀⣀⣀⣀⣈⣳⣶⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣷⣶⣶⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣟⣟⣟⣫⣙⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣭⣭⣿⣯⣭⣭⣭⣭⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠀⢠⡀⠀⠀⠀⠀⠀⢠⡴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠠⠒⠀⠸⠇⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⡀⢰⡆⠀⠀⠀⠀⠀⠀⠶⠀⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁⠈⡇⠀⠖⠐⠀⠄⠀⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠉⠠⡄⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⡴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⣄⡀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⢀⡀⠀⢰⣦⣴⡂⠀⠀⠀⠀⠀⣼⠁⠀⠻⢯⣥⠅⢩⡐⠈⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢈⡹⢷⠘⣯⠽⠅⠀⠀⠀⠀⠀⣛⠀⠀⠍⣛⠥⠈⢤⣇⣀⢠⣶⠆⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⣷⣲⣯⢽⣦⠹⣿⡅⢀⡀⣠⡄⣀⣿⡤⢀⡳⠎⣃⣐⣿⣓⡘⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢒⣯⠄⠼⣏⣉⣽⣞⣛⣳⡆⢊⡀⣏⡜⢦⠚⣽⡹⠯⣭⠟⢩⣭⣿⣤⡝⠚⣂⣉⣀⡂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣴⢊⣉⣁⣹⡖⠒⠛⣥⠄⠛⣭⢫⣾⠿⠿⣷⣾⠇⣿⡌⣭⠿⠭⣭⠿⠆⣼⣉⡉⣉⣙⣍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢯⡭⠭⠽⣏⣉⣩⣿⣉⣹⣿⣿⣛⣾⣁⣹⣟⣁⢐⣿⣉⠙⣷⣉⡉⠻⠇⠀⣀⡭⠽⠦⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⢃⡀⠲⠾⢦⠤⠾⢥⠤⠐⠳⣤⠄⠿⠤⡤⠷⠤⢤⠽⠦⢤⠾⠷⠤⢦⡀⠘⠓⠰⣶⠂⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠯⣤⠤⠴⢧⡤⠤⠾⠠⡤⠤⠧⠤⠤⠤⠕⢤⡤⠼⠦⣤⡬⠧⠠⠤⣤⠵⠶⠶⠶⠾⢛⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠈⠀⠀⢀⡄⠀⠀⠀⣀⠀⠀⠀⠀⢳⡀⢀⣈⣳⣄⣀⢀⣀⡳⢦⠀⢤⢤⣬⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠀⠀⢀⠜⠉⠀⠀⡝⠉⠁⠈⢉⡏⠉⠀⠉⠉⢻⠉⠀⠀⠙⠢⡀⠀⠀⠀⠑⢤⠀⠀⠀⣀⡲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣐⠖⠀⠉⠀⢰⠂⠈⠉⠁⠈⠏⠈⠀⠀⠈⠫⠀⠉⠉⠉⠏⠉⠉⠉⠀⠀⠀⠀⠀⠙⠧⣀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣷⣦⣴⣾⣿⣿⣿⣶⣴⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣤⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠻⣿⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⢸⣿⣿⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣴⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢷⡀⠀⣿⡏
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⢛⡛⣷⡀
⠀⠀⠀⢀⣼⠿⠿⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏⠀⠀⠈⠁⣸⡇
⠀⠀⢀⣿⠁⠶⠶⠆⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛
⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠘⠻⢦⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠃⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡿⠷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣴⣶⠟⠋⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠉⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠁⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣠⡾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠘⠻⣦⣀⠀⠀⣀⣀⣀⣤⣶⠶⢶
⠀⣀⣀⣀⣀⣀⣠⣴⠟⠁⠀⠀⠀⣠⣶⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠁⠀⠀⢈
⣾⠏⠉⠉⠉⠉⠉⠀⠀⠀⠀⣠⡾⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⢿
⢻⣦⣀⣀⣀⣀⣀⡀⠀⠀⠰⣿⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠛⣿⠆⠀⠀⣾⣿⣯⣥⣤⡿
⠀⠸⣿⡉⢉⣩⡿⣿⡄⠀⠀⢻⡆⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣄⠀⢀⣿⠁⠀⠉⠁
⠀⠀⠈⠛⠛⠉⠀⠈⠻⣶⡶⠟⠁⠀⠀⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠇⠀⠀⠀⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠻⢶⣄⡈⠻⣦⡀⠀⠀⠀⣠⣴⠿⢷⡄⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠋⠁⠀⠀⠀⠉⠛⢷⣾⠿⣦⣴⠞⠋⠁⠀⠈⢿⣆⢀⣴⡞⣫⣝⣿⡿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠉⢻⣦⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠈⢿⣆⠀⠀⠀⠀⠀⣿⣟⡵⣫⡿⠛⠁⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⢳⠀⠀⢹⠟⣹⠟⣺⣿⣉⠀⠀⠀⠀⠀⠀⠙⢷⡄⢀⣴⣾⡿⣫⡾⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢣⡀⠈⠛⠛⢶⡋⠈⠻⣧⠀⠀⠀⠀⠀⠀⠈⢻⡟⣡⢮⣾⠋⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣵⡟⠁⠀⢀⣤⡾⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⠉⣉⠉⡉⢉⡉⢉⣉⣉⣉⠀⠀⡀⣀⠀⣀⣀⣀⠈⣻⣷⣶⡿⠛⣁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠛⠛⠛⠛⠓⠒⠦⣤⡀⠀⠀⠀⠀⢀⣠⠶⠚⠛⠛⠛⠓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡠⠒⠛⡖⢦⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⡏⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⣠⠇⠈⡇⠀⠀⠀⠀⠀⢸⠦⠤⠞⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⢍⠉⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠳⣀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠀⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠻⡉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⢶⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠒⢒⡲⠖⠒⠒⠒⣒⠖⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠑⠦⣄⣀⠀⠀⢠⠖⠉⠉⠉⠁⠀⢀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠒⠒⠒⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠋⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢠⡾⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣧⠀⠀⠈⠓⠦⣤⣄⣀⣀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⡟⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⣠⡴⠋⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣾⠁⠀⠀⠀⠀⠀⠈⠛⢦⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣤⡤⢤⣤⠄⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣾⠁
⢀⣀⣤⢦⡞⠓⠦⣤⣤⣼⠆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠀⣸⡇
⣾⡉⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣶⣿⣿⣷⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣄⠀⢄⣀⣀⡀⠀⠀⣀⣀⣴⣶⣿⣿⣿⣿⣾⣦⣤⣤
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣉⣹⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠉⠉⠉⠁⠀⠉⠉⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠃⡠⢤⠄⠀⠀⠀⠠⠤⢤⡀⠀⠈⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠆⠘⠓⠞⣀⠀⠀⠀⠰⢦⣤⠴⠀⠀⠘⠿⢋⡵⢌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠷⣀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠎⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠠⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢶⡶⠖⠛⠙⣿⡇⠲⢢⡶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠸⣥⡤⠒⢈⡤⠋⣡⠿⣦⡉⢶⣽⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⢀⣶⠀⣤⣤⠟⢀⡾⠁⠀⠀⢹⡄⣿⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⣼⠃⡜⠉⠉⠁⣛⠛⠻⠤⠾⠑⢦⠀⠀⠈⣧⢼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠙⠯⣷⣾⡏⣼⠁⠀⠀⠀⠈⠺⡂⠀⠀⠀⠀⡀⠀⠀⣿⣼⠀⣜
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣷⠄⣀⡀⠈⠛⣿⡆⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⢙⣦⠀⣿⡏⢠⠟⣦
⠀⠀⠀⠀⠀⠀⠀⣼⢉⡷⠀⠉⠉⠙⠛⣿⠇⠀⠀⠀⠀⠀⠀⠙⡇⠺⣹⣶⠎⢩⡄⠻⣁⡿⢀⠏
⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⢿⣦⠀⠈⢱⡄⢿⠀⠸
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠇⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣿⠀⠀⠈⢢⣼⣠⠇
⠀⠀⠀⠀⠀⠀⠰⣇⣸⡀⠀⡠⢦⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠸⡄⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠛⣁⡼⢸⡇⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡟⠃⠛⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⡙⠲⢦⣤⣤⣄⣠⣤⣤⣤⣴⠶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠳⢶⣯⡁⢀⣀⣁⣁⣤⠟⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⠀⢀⡀⠈⠁⠉⠳⠦⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⡇⠀⠉⠉⠀⠀⣠⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣾⠁⠀⠀⠀⠀⠀⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣰⡟⠀⠀⠀⠀⠀⢀⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⣠⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣴⠃⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠸⠆⢠⡟⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡎⠻⡄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠠⡀⠀⣠⠆⠀⢳⠄⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣜⠁⠀⠀⢰⠏⠀⠀⠀⣄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣶⣅⣀⡀⠀⡌⠀⠀⠀⠀⠸⡆⠀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣆⠃⠀⠀⠀⠀⠀⢉⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢰⢶⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⡓⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⠆
⠀⠀⠀⠀⠠⠐⠐⢃⠀⠀⠀⠀⠀⠀⠀⢤⡟⠐⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠑⠆⡀⠀⠀⠀⠀⠀⠐⢈⣤⡆⠀⢀⡆
⠀⠀⠀⠀⠀⢠⢨⡟⠆⠀⠀⠀⠀⠒⣲⠁⣹⠁⠀⠀⠀⠀⠀⠀⠙⡉⣇⣀⠀⠀⠀⠀⠀⠀⢀⢾⠀⠈⠁⢌⠂⠀⠀⠀⠀⢈⠋⡆⢨⣐⠕
⠀⠀⠀⠀⠐⢲⠁⠘⠁⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⢦⣀⣋⠑⠑⠀⠀⠀⠠⠦⠕⢺⠀⣠⠀⠀⠚⠃⠀⠀⠀⠘⢾⠀⢠
⠀⠀⠀⠀⠀⢘⠐⠚⠀⠀⠀⠀⠀⠀⢀⠂⠸⠀⠀⠀⠀⠀⠀⠀⠘⢯⠝⠲⠄⠀⠀⠀⠀⠀⠀⠈⠇⣰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠙⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢤⠀⠀⠀⠀⠀⠀⠀⢰⢀⣇⣐⠂⠷⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⡌⠑⡈⢄⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⠀⠈⠫⡂⠀⠀⠀⡰⢁⠴⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠑⠃⠃⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠰⠤⠀⠀⠀⠀⠀⠀⠐⠞⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠅⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣿⠀⠀⠀⠀⠀⠀⢸⡁⡍⠀⠀⠀⠀⠀⠀⡎⢻⠟⠀⠀⠀⠀⠀⠀⠀⠞⢻
⠀⠀⢀⡀⠀⠑⠜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣿⣧⠀⠀⠀⠀⠀⠚⢻⠓⠄⠀⠀⠀⠀⠀⢉⠿⡀⠀⠀⠀⠀⠀⠀⠀⡑⠙⠄
⠀⠀⠀⠻⡂⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⠂⠟⠃⠀⠀⠀⠀⠐⡆⣛⢤⠂⠀⠀⠀⠀⠐⡿⣠⢧⠀⠀⠀⠀⢉⣅⠘⠇⠐⠤
⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠉⢂⠀⠀⠀⠀⠀⢠⡃⠘⢑⠂⠀⠀⠀⠀⠰⡅⡀⡀⠆⠀⠀⠀⠀⠁⢚⠀⡰
⠀⠀⠀⠀⠀⣹⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢂⠀⠀⠀⠀⠐⠇⢰⠈⠀⠀⠀⠀⠀⠀⡗⢀⠂⠀⠀⠀⠀⠀⠀⠈⠘⢳⠁
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣠⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⡌⢂⠀⠀⠀⠀⠀⠀⠸⡀⠘
⠀⠀⠀⠀⠘⠀⢆⠐⡀⠀⠀⠯⡤⠮⠤⡴⠀⠀⠀⠉⠈⠫⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠙⡌⢆⠀⠀⠀⠀⠀⠀⢡⠀⢃
⠀⠀⠀⠀⡀⠐⠀⠣⣐⠄⠀⠠⣶⠔⢆⠧⠄⠀⠀⠀⣆⠀⢡⡄⠀⠀⠀⠀⡄⢸⠀⡀⠀⠀⠀⠀⠈⡠⠀⠈⡼⠂⠀⠀⠀⣠⢀⠂⠠⠀⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠠⠒⠉⠈⠉⠐⠂⠐⠦⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣉⠉⠀⠀⢀⡴⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠉⠙⣦⣠⠧⠐⠀⠉⠀⠉⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣀⠔⠊⠙⡇⠀⢀⣠⣤⣤⣤⣄⡙⢧⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡠⠊⢤⠒⠁⣰⠋⡶⠋⠁⠀⠀⠀⠈⠻⣎⠃⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣎⡀⠀⠀⣀⡴⠛⣾⠁⡠⠖⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⢉⡽⠯⣅⠀⠀⢿⠀⠉⠐⠀⠀⠀⠀⠀⣼⡀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢺⡀⠀⢸⠀⠀⠈⢧⡀⠀⠀⠀⠀⣠⠞⠉⠛⢦⣠⠃⠀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠑⠂⠀⠀⠀⠀⠀⠉⠛⠒⠚⠋⣁⡼⠁⠀⠀⠝⠓⠒⠲⠮⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⢊⡿⠁⠀⠀⠀⠐⠶⣭⠶⠄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠈⠉⠉⠹⡍⠈⢢⣠⠏⠀⣀⠄⠒⠢⠤⡄⣀⣀⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠱⠴⠚⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣦⡀⠀⠀⠀⣠⢚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡟⢲⡀⠉⣁⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠂⠁⡘⣅⣠⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⢼⢨⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠉⠁⣅⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⣀⠀⠀⠀⠀⠀⡏⡏⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣌⠀⢸⢹⠋⠉⢉⣘⡸⠓⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠒⠒⠘⣿⠟⠛⠒⠀⠀⢀⠩⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⢿⡀⠀⠀⠀⠀⡠⢔⠛⢦⠀⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣈⣉⣉⣿⠁⢀⣠⣔⢊⣠⣓⣤⢸⡇⡇⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣒⡶⠟⠈⠁⠀⠀⡏⠐⢆⢊⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣴⣿⣿⣷⣦⣶⣦⣤⣶⣿⣶⣄⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠻⡿⠛⢛⠻⣿⠛⢻⠿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⡿⠋⣉⣉⣴⣿⣷⣾⣿⣷⣴⣿⣿⣷⣤⣶⣤⣽⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⠋⣩⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢡⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠁⠀⠀⠉⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢋⣁⡈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣮⠻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⢀⣾⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⣇⣀⢻⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⠦⣠⡾⠛⠁⠉⠙⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣀⢙⡇⠘⣿⡉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⠀⠠⠤⣴⠏⠀⠀⠀⠀⢀⡀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢀⣁⠎⠀⠀⢸⣇⢀⣼⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⠿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠹⢦⣀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡾⠃⠀⢀⡿⠛⠉
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⡀⠀⣁⣴⣿⡄⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡴⣄⠀⣠⠟⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠉⠘⢢⣤⢀⡀⠀⠀⡾⠙⠿⣀⡸⠟⠁⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢸⣿⣬⣉⠒⠒⣿⠖⠂⠉⠁⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠛⠛⠋⠀⠹⣧⠀⠀⠀⠀⠀⠘⠟⠿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠳⣄⡀⠀⢹⣧⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠈⠁⠚⠛⠋⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⢉⣀⣤⡟⠷⢶⣦⣤⣤⣶⡶⠟⠋⠀⢶⣄⣹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣽⠟⠋⠉⠛⠛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠻⣿⣦⣤⣀⣀⣀⣀⠀⢀⣀⣴⠋⠁⠀⢀⣀⣤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⢁⡴⠋⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠛⠷⣭⣍⡉⠉⠙⠉⠉⠀⢀⣴⠾⠿⣿⣥⡼⠃
⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⣴⠋⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠈⠛⠛⠲⣶⠀⢰⡟⠁
⣠⡶⠛⠋⠉⠉⠉⠁⠈⠉⠁⠀⢴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠘⢳⡄
⢟⣀⡀⣀⣠⣤⣶⡖⠚⠛⢷⣄⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⡿
⠈⠉⠛⢿⣉⣼⠏⠀⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢰⣿⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠟⠁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⢷⣄⡴⠏⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣧⣄⣀⡀⠀⠀⠀⠀⠀⠰⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠉⠉⣿⢷⣄⡀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣄⡀⠀⠀⣼⠋⠀⠻⣧⡀⢀⣠⣶⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⣿⠿⠾⠏⠀⠀⠀⠘⠻⠟⠻⣿⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡈⢷⣄⠀⠀⠀⠀⠀⣀⣴⠿⡏⠉⣠⡜⣧⢜⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠹⣧⡀⠀⣤⣾⣿⠁⠀⣼⣏⡵⠊⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠛⠣⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣈⠿⣆⣿⣿⠛⣡⢞⡥⠋⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⠉⠙⣷⣾⣧⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⣠⡾⠛⢡⡾⣧⠏⡠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠙⣆⠀⠿⠿⠟⠉⢉⡋⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣴⢋⣼⡭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡘⠒⠒⠒⠶⠒⠚⠓⢲⠖⠒⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣤⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⠿⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⡀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣠⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⡁⠀⠀⠀⠀⠀⣿⣯⣤⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⠀⣼⣻⣿⣿⡿⣦⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⡏⢧⠀⠀⠀⠀⢹⣿⣿⣷⠞⠁⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠙⣿⣿⣿⡀
⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⠈⢹⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⠀⠀⠀⠈⣿⢿⡏⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⢹⡟⢿⣇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣾⡟⠹⣷⠄⠀⠀⠀⠀⠘⡟⢸⡇⠀⠀⠀⠀⠀⠘⡿⢿⡿⠇⠀⠀⠀⠀⠈⣿⠘⣿⠀⠀⠀⠀⠀⠘⣷⠈⣿⡄
⠀⠀⠀⠀⠻⠿⠆⠀⠀⠀⠀⠸⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠨⠗⠀⠀⠀⠀⠀⠀⠇⠸⠆⠀⠀⠀⠀⠀⠀⠿⠃⠿⠀⠀⠀⠀⠀⠰⠿⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡀
⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⡿⠃
⠀⠀⠀⠀⠀⣿⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣄
⠀⠀⠀⠀⠀⠷⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠿⠄⠀⠀⠀⠀⠀⠀⠀⠘⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠘⠆



⠀⠀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄
⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⢀⣶⣿⣿⠆⠀⠀⠀⠀⠀⢀⣼⣿⣦
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠄⠀⠀⠀⠀⠀⠈⣿⣿⡅⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⢿⣏⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⢸⡟⢸⡇⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠘⣿⠿⣿⠋⠁
⠀⠀⠀⣼⡇⠈⣷⠄⠀⠀⠀⠀⠀⢿⣹⡀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⠀⠀⠀⠀⢀⣿⠀⣽
⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠈⠃⠙⠀⠀⠀⠀⠀⠉⠀⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡤⢄⣀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡦⠀⠀⢰⣶⣶⣶⢶⠆⠀⠀⠺⠷⠶⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⢣⠒⠀⠀⠀⠹⣿⠿⠻⠟⠀⠀⠐⠲⣶⢶⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣠⣮⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⠈⡇⠀⠛⡧⢿⡄⠀⠀⡘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠲⢤⣼⠁⠀⣠⢇⠳⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠻⢿⣿⣽⣿⣿⠿⣿⠛⠃
⠀⠀⠀⠀⠀⢀⣤⣼⣿⣯⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡄⠀⣀⣶⣀⠀⣼⣷⣿⠀⠀⠈⠣⡀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠈⡏⠉⠉⠀⠀⣼
⠀⠀⠀⠀⢐⠈⡭⠥⡀⠩⠩⠈⠀⠁⡁⣩⢉⠋⡉⠁⣠⣅⣸⡄⢹⣿⡏⠀⢹⣿⣿⠀⠀⠀⠀⠉⢀⣄⣾⠟⠉⢀⣤⣶⡶⠰⡇⢰⡄⠀⠠⡃
⠀⠀⣾⣶⣶⠟⠛⠳⣾⣷⣶⣶⣶⣾⣿⠟⣯⡖⣈⢀⣷⡿⢿⡇⠸⣷⢧⠀⢈⣿⣿⠀⠀⠀⠴⠋⠛⠿⣏⠀⠀⠘⢿⣁⠸⠀⡾⠚⠓⠀⢴⣩⡄
⠀⠀⠿⣿⣿⣿⣿⣿⢿⣿⠟⠿⢿⣸⣏⠀⣸⠿⡻⢟⣻⡇⢀⡿⢫⠟⠿⢧⠈⠛⠋⠀⠀⠀⠀⣶⣾⣶⣽⠷⡄⠀⣾⣿⠆⢠⣄⣴⣿⣷⠬⠟⠓
⠀⠀⠀⠙⠿⠿⠃⠤⠶⠤⣄⣀⣀⣿⣿⣷⣇⣤⣤⣈⣿⣿⣾⣷⣏⣀⣼⣿⣤⣤⣤⣤⣤⣤⣼⣿⣿⣭⣥⣾⣿⣼⣯⠿⠿⠶⠛⠒⠻⠋⠲⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡵⠆⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⣉⣉⣳⢲⡷⠀⠀⢀⠀⠹⠋⣁⡀⢺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⣀⠀⡈⢹⡁⣹⠋⠉⠷⣀⢠⠀⠰⣋⠭⠭⠁⠀⡙⣦
⠀⠀⠀⠀⠀⣦⣠⡀⠀⠀⠀⠹⣻⡗⠿⣧⣿⣷⣾⡟⠁⠘⠻⠓⠀⠙⠆⠀⠈⠋⣵⣦⣤
⠀⠀⠀⠀⠈⠋⡃⠀⠀⠀⠀⠀⠉⠙⣦⣠⣈⣏⣡⠏⡀⠀⢠⣴⣾⡗⠀⡴⠀⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠁⡠⣀⠂⠀⢀⠀⠈⠇⠙⠹⣾⠷⠊⠀⠀⠀⠀⠐⢀⠀⣄⢴⣾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⣤⠀⠀⠉⠳⠇⠀⠀⠀⠀⡀⠀⠀⡀⢀⣌⡨⢇⣻⡣⠔⠁
⣀⣤⣅⠀⣤⣤⣎⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣤⣤⠸⠶⢿⣷⢆⣤⠏⠉⠻⣇⠘⠿⢻⣶⣤⣤⣄⣀⢠⣷
⢀⢀⠀⠌⠿⠈⣿⣿⣻⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡇⠂⢸⡿⠴⠃⢀⣠⣮⢁⡹⢧⣾⣿⡏⢋⢋⢬⣀⢡
⢀⡴⠀⠂⠈⢷⢾⣿⣿⣿⣿⣿⡿⢾⣿⣿⣿⡿⢿⣪⠠⡿⢿⣶⣾⣿⢿⣿⣾⣿⣿⣾⣿⢓⣿⡎⠁⠂⡟