Leg Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⢿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣫⢟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡫⢋⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡟⡟⠝⡟⢹⣇⠙⢿⣯⠻⣿⣿⣿⣿⣾⣿⣿⡱⢋⢞⡵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡫⢡⠀⠈⠀⠈⠁⠀⢻⣦⡅⢻⣷⣿⣿⣿⣿⣿⣿⣿⣵⣫⢊⠞⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡏⠃⠁⠀⠀⠀⠀⠀⠀⢀⡀⣿⣧⢠⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣏⢜⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠇⡇⠙⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡞⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠳⡤⠀⠀⠀⠀⠀⡄⣤⡯⣰⣏⢹⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠰⣇⣴⣿⡴⢴⣿⣾⣿⣿⣿⣿⣿⢺⣿⡿⠿⠿⠿⠿⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢿⣿⣼⠁⠈⣿⣻⠧⣿⣿⡿⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠞⣸⡟⣧⠀⠙⠿⢷⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠟⢸⣝⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣞⠀⠀⠀⠀⠰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢿⠄⠀⠀⠀⣸⣿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢺⠀⠀⠀⢠⢿⡿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⠃⣄⠐⡱⡷⢻⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠃⢊⢈⢄⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠸⠀⠰⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⠀⠀⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢸⠀⡄⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢰⠇⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⡼⠀⣧⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⠀⠀⠁⢸⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠐⣇⢻⡆
⠀⣀⣀⣠⣴⣾⠻⠋⠀⠀⠀⠀⣀⣀⡀⠀⠰⠊⡋⠘⣷
⠘⢯⣭⣅⣈⢀⢀⣤⣤⣴⠿⠷⠻⠿⠿⢿⣶⣦⣤⣴⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⢠⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⣼⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⢾⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⢻⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⢘⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⢠⠶⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⠀⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠋⠁⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠃⠶⠤⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠧⠄⠀⠐⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡟⠕⢁⡴⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡄⢠⣴⠲⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠺⠵⠷⠤⠤⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠧⠼⠦⠗⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡔⢆
⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⡇⣿⠇⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠾⢿⣿⣿⣿⣿⣿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣿⣿⣿⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠛⠈⠻⣷⣔⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⣤⠋⢀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠦⣄⠀⠀⠀⠀⣿⣿⣿⣿⡇⠈⠁⠐⠋⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⡂⢰⣤⣠⣿⣿⣿⣿⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣹⣿⣿⣿⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠈⠙⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡿⣋⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠛⠛⠛⠛⠛⠋⢉⣼⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⢄⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢀⠀⠐⡀⢈⠀⢂⠔⢂⠔⠡
⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⢈⣤⠄⠂⠓⠶⠶⣤⡀⠀⠀⠀⠠⠊⠀⠔⠁⠀⢀⠀⠀⠀⠊⠀⠀⠐⢀⠊⣤⣁⠀⠘⠁⠀⠀⠀⢠⠋⠀⣮
⠀⠀⠀⠀⠸⣧⠀⣀⠠⠰⠴⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⡴⠤⣀⠀⠀⠀⠀⠀⠀⠤⣀⡠⠔⠒⠉⠀⠀⠉⠳⠶⠶⣠⠴⠴⠃⠠⠎
⠀⠀⠀⠀⠆⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⢤⡁⠓⠲⠦⠤⠄⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣀⣀⣁⢰⡄
⠀⠀⠀⢰⣦⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣠⠄⢤⠀⣤⣤⣈⣄⣀⠀⠀⠀⠀⠤⠐⠒⠈⠉⠀⠈⠐⣠⠃⢀⠞⠀⠀⢸⣾⠰⠋⠁
⠀⠀⠀⠀⣿⡄⢻⡇⠀⢀⠤⠄⠐⠒⠊⠉⠉⠉⠀⠀⡀⢀⡀⢀⡀⢠⠀⣠⡀⣀⢉⡉⣩⢱⣄⡀⢠⣄⣄⢀⡀⢠⡀⠀⢠⠂⠀⢀⠀⢻⠏
⠀⠀⠀⠀⢹⣷⠘⣿⣷⡀⠀⢠⡄⣠⠠⡄⢠⣤⣶⣄⣿⣼⣿⣸⣷⣾⣷⣟⣧⢽⣾⣿⣿⣿⣏⣿⣿⣿⣿⣼⣷⡾⠁⡰⠁⠀⢀⡏⢀⠎⠀⣰⠃
⠀⠀⠀⠀⠈⣿⡆⢿⣿⣧⠀⠸⣷⣿⣶⣷⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⡄⠀⠀⠀⢈⢇⠎⠀⡀⢯
⠀⠀⠀⠀⢀⣿⡁⠘⣿⣿⡄⠈⠉⢉⠁⠁⠀⠀⠀⠀⠀⠃⠉⡙⠙⠻⣿⣛⡛⠛⠛⠛⣻⢛⡉⣉⠙⣉⡿⢋⠔⠉⠀⠀⠀⢀⠾⠊⠀⠀⠄⢸⡆
⠀⠀⠀⠀⠀⢹⣿⡸⣿⣿⣷⣄⠀⠀⢀⠀⣄⠀⣀⡠⠤⠤⠤⠝⠛⠛⠁⠛⠛⠋⠁⠀⠁⠈⢀⣀⣀⡜⡡⠁⠀⠀⠀⠀⡰⠋⠀⠀⠀⢀⡔⠙⠻⡆
⠀⠀⠀⠀⠀⣺⠿⠇⢿⣿⣿⣿⡄⠀⢠⣤⣾⡇⠀⠀⠀⠀⣀⠠⠤⠀⣀⠠⠄⠒⠒⠊⠛⠉⠋⢉⠏⠀⠅⠀⡀⠀⢀⠞⠁⠀⢀⡤⢤⡎⠀⠀⡴⠇
⠀⠀⠀⠀⠰⡇⠁⠀⡀⢻⣿⣿⡇⠀⠀⠞⢛⣣⣤⣀⠀⠈⠁⠀⠀⠚⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠰⠷⠿⠞⠁⠀⠀⡔⢡⠀⠘⠳⡦⢀⣄
⠀⠀⠀⠀⠒⣶⠀⠎⠁⣈⣿⡿⠟⢠⠀⣾⣿⣿⣿⣿⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠒⠒⠒⠒⠂⠰⢤⣀⣀⣺⠣⠾⠶⣿⣼⣿
⠀⠀⠀⠀⣦⣜⠂⠘⣩⣥⣤⣴⣶⡾⢶⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠂
⠀⠀⠀⠀⣿⣿⣿⣅⠀⢀⡀⢤⣿⣇⠀⣬⣿⣿⣿⡟⠹⣿⣄⠀⠀⠀⠉⠉⠛⠓⠲⠶⡦⣤⡄⢰⡄⢀⡀⡀⠀⠉⣉⠁⠀⠀⢀⡢
⠀⠀⠀⠀⣿⣿⣿⣿⣦⠈⢷⡘⢻⣿⡀⠔⣸⣿⣿⣷⡀⠀⠙⠿⣇⠀⠀⠀⠀⣤⠀⡀⠀⠀⠀⠸⠛⣿⣿⡿⣷⣷⣿⡠⢀⡀⠀⠉⠀⠐⡦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡈⠹⣷⢿⠟⢳⣿⣿⣿⣿⣧⠀⠀⠀⠙⢿⣆⠀⠀⠁⢸⡇⠀⠀⠀⠀⡀⠉⠉⣿⣿⣿⠛⣓⡶⠈⠱⠀⡀⠀⠂⠤⣤⡀
⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣷⠀⣩⠏⠤⢶⡾⠿⣿⢻⣿⡄⠀⠀⠀⠀⢿⣷⣮⣲⣿⡇⠀⠀⠐⠆⠘⣺⢻⣤⡤⠛⠉⠕⠀⠀⠰⣁
⠀⠀⠀⠀⣿⡌⣿⣿⣿⣿⣿⣧⡯⣤⡖⢋⠀⠀⢹⣆⣾⡗⠀⠀⠀⠀⠀⠛⣿⣿⣿⣇⡇⠀⠀⠀⠣⡌⢷⣼⣧⠀⠀⠐⠤⣀⡀⠀⠉⠉⠛⠒⠒
⠀⠀⠀⠀⡃⠁⢻⣿⣿⣿⣿⣿⡛⢿⣤⡘⣷⠀⠘⣿⣿⠃⠀⠀⠐⠂⠀⠀⢿⣿⣿⢿⣇⠀⠀⠀⠀⡠⠲⢠⣽⡘⣷⢦⣄⢀⡌⠁
⠀⠀⠀⠀⡃⠀⠸⣿⣿⣿⢿⣿⣿⣌⠋⢿⣆⠑⠄⢸⣿⣆⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⢻⡀⢀⠄⠀⠐⡖⠢⠹⣧⡹⣞⣿⣛⣿⣶⣿⣄⣀⣀⡀
⠀⠀⠀⠀⠐⢶⣤⣙⡿⠿⢸⣿⣿⡘⣆⠸⣿⣧⠀⠈⣿⣿⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣆⣷⡄⠀⠀⠂⠀⠀⠀⠋⠹⡿⢿⣿⠟⣿⣿⣿⣿⣟⣿⡄
⠀⠀⠀⠀⠀⣠⣽⣿⣷⠤⠾⣿⣿⣧⠹⡄⢹⣿⣧⣀⣸⣟⠸⣧⠀⠀⠀⠜⣿⡿⠟⠻⣿⠋⠁⠀⠀⠰⠴⣤⣀⠀⠀⢠⣀⢠⣤⡍⠿⢿⣿⣷⣿⠁
⠀⠀⠀⠀⠸⠟⠋⠉⠉⠑⠦⠈⠛⢿⣄⠀⠘⣿⣿⣿⠉⠛⢿⣾⣇⡀⠀⢿⣿⠇⠀⣠⢟⠀⠀⠀⠀⠀⢀⣧⢉⡄⠀⢸⣏⢹⣿⣀⣿⣽⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⣿⣧⡄⢻⣿⣿⡇⠀⠀⠘⢿⣿⣆⣸⡟⠀⡐⢣⢹⠀⠀⠒⠁⠀⠘⠻⢈⣇⣼⣺⠿⠈⠋⠙⢻⠧⣻⣿⣿⡇
⠀⠀⠀⠀⣄⣠⣴⣾⣿⣿⣷⡄⢻⣿⣿⣿⣷⡈⠛⢿⣟⣀⣀⠀⢀⠉⠻⠿⣷⣖⠀⢐⣾⡇⠀⠀⠀⢺⡤⣠⡄⠎⠉⢷⠀⣾⡄⠘⣿⡶⡯⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣷⡶⠋⠉⠙⣿⣿⣷⣄⠀⠀⢾⣿⣷⡀⢿⡇⠀⠀⠀⠀⠀⠁⠋⠀⢣⣤⢤⣽⠛⡶⡟⣳⣤⣤⡿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠢⠂⠝⢿⣿⣿⣷⣶⣿⣿⣿⣷⣸⣇⣀⡀⠀⠀⠢⠀⠄⠀⠶⠋⠘⠈⠐⡶⣯⡅⢡⣉⢱⠇
⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠂⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠟⠿⠿⠿⢿⣷⣤⠀⠀⠀⠓⠀⠁⠁⢤⠁⠼⠰⠌⠒⠻⢿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⣾⣻⣭⣥⠴⠶⣦⣹⣦⣄
⠀⠀⠀⠀⠀⢰⡿⢃⡀⠠⠤⠶⠾⣿⠇⣹⡀
⠀⠀⠀⠀⠀⣸⠃⢏⣀⠀⠤⠤⣤⣾⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠶⠒⠒⠒⠒⠀⠀⣀⡤
⠀⠀⠀⠀⠀⣿⠀⠈⠁⠐⣦⣤⡤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠐⠦⣤⣤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠙⣧⣴⢟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⣷⠀⣶⠀⠀⠀⢸⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣿⠀⠛⠶⠶⠶⢋⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡴⢒⡦⠖⣻⠓⠆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠻⡻⣌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣿⣤⣘⣵⣋⡉⠉
⠀⠀⠀⠀⠀⠀⢻⡀⢢⡀⠀⠀⠙⣦⠙⣧⡀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⢿⣟⢷⣄⠐⢶⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⡀⠀⠀⠈⠃⠈⢷⣄⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣆⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⠃⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣧⣄⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠛⠦⣌⠙⠻⠷⢶⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢠⡿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢸⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠸⣄⠀⣿⣿⢧⣀⣀⡀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸⡇⣸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡎⠒⠀⢹⡄⠑⠚⠒⠊⠀⠀⠀⠀⠀⣿⣿⢯⠐⠦⣤⣤⣤⣴⠾⠋⠀⡠⣞⠄⠀⢸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⣤⠾⠃⢹⡄⠀⠀⠀⠀⠀⠀⢄⠀⡼⡼⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⠷⠶⢶⣤⡤⠤⠄⠀⠀⠸⠃⢰⡄⠀⠀⠀⠉⠢⣭⣥⣷⠄⠀⢀⣾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⠋⠀⠙⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠙⠦⣤⣤⠤⠀⠈⠁⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠉⠛⠛⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⣀⣠⣴⠾⠛⠛⠋⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠄⣀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⡠⣍⠈⡤
⠀⠀⠀⠀⢠⠛⠁⠀⣆⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⡽⠷⢤⡑⠃
⠀⠀⠀⠐⠁⠀⠀⡴⢯⡰⠃⠀⠀⠀⠀⢠⠔⠂⠀⠀⠀⠀⠀⠠⠒⠒⣾⡀⡄⢈⠀⠀⠀⢸⣇⡀⠀⣇⣀⢀⡀
⠀⠀⠐⠀⠀⠀⣼⡁⠀⠇⢀⡀⠎⣀⢠⣞⢠⠊⠁⠀⠀⠀⢀⠀⠀⢸⠋⢹⡅⠈⠀⠀⠀⠈⣀⣀⠀⠡⣄⠁⠃⠀⠀⠀⠀⠀⠀⠀⠊⢉⣤
⠀⠀⠰⠀⠀⣴⢗⡇⠈⣨⢯⠗⢸⣁⡈⢹⠦⠃⠀⠀⠀⠀⠌⠀⢰⡏⡆⠈⡀⠀⠁⠒⠒⠿⡦⠽⠿⠒⠀⠫⠟⠀⠀⠀⠀⡔⠒⠾⣷⠯
⠀⠀⠀⠉⢓⠗⠲⢝⡒⠀⠀⠀⠆⠡⠁⢀⢀⡀⠀⠀⠀⠀⠀⣀⣸⠿⠛⣄⠀⠀⠀⠀⠀⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠁⠸⠇
⠀⠀⠀⠀⠀⠂⠀⢀⠀⡔⠀⠀⠈⠘⣂⠈⣹⣄⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⠃⠤⠀⠀⠀⣈⣣⡀⢀⣘⠀⠀⠀⠀⠀⠀⠸⠀⡌⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠼⠀⠇⠀⠀⠀⠀⠘⣭⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⢁⠌⠈⢇⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠘⠆
⠀⠀⠀⠠⠎⠀⠀⠀⣆⡀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢹⠅⠁
⠀⢀⠔⣤⠂⠀⠀⡐⢡⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠂⠀⠀⠀⠀⠀⠀⠀⠔⠢⡀⠀⠀⠈⡖⠳⠄⡄⠀⠀⠀⡠⡔⠰
⠀⠀⠁⠄⠀⠀⠀⠉⠦⠈⠐⠣⡀⠀⠀⠀⠀⣍⡀⠀⠀⠀⠘⠆⠀⠀⠀⠧⠤⠶⠀⠈⠒⠒⠒⠁⠀⠀⠘⠚⠉⠀⠀⠀⠀⠠⠁⢀⡛⢄

⠀⠀⠀⣠⡖⠀⠀⠀⠀⠀⠀⣠⠞⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⣀⡀⠀⢀⣠⡤⠄
⠀⠰⠊⠁⢠⢀⠀⠀⠁⢠⣞⣁⢤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢚⣛⣒⣄⠀⢶⠎⠁⢸⠷⠟⠛⢻⡶⠟⠋
⠀⠠⠔⠀⠈⢋⡥⠂⣰⣿⣡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠞⠉⠀⣀⣼⣯⠀⠀⠀⣠⠔⡶⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⠁⣈⡁⡤⢸⡇⠈⢿⣄⠠⡤⠌⣬⣾⣶⣤⠀⠀⠀⠀⠀⠀⡠⠎⢀⣀⢚⣿⡅⠀⠀⠀⢀⠀⡀⠞⡁⠚⡁⠀⠀⡄⠀⠀⠀⠀⠀⣴⣿⣿
⠘⣳⢤⣍⡠⡷⠃⣸⠇⠀⠀⠉⠑⢦⠀⠀⠉⠟⠊⠹⠓⠀⠀⠀⠀⠀⠘⣧⠎⠈⠁⠤⠀⠠⡈⠁⠀⠀⡌⠈⠉⢗⢢⣿⠄⠀⠀⢀⣲⣿⣿⣿⣯
⠈⠛⢳⡙⣐⣸⣾⣧⠄⠀⠐⠉⠀⢼⡏⠁⠈⠢⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠁⠀⠉⠁⣰⣶⣿⣿⣿⣿⣿⣷
⠀⡀⢢⡙⠷⠾⣽⣝⠂⠀⠒⠗⠀⠈⢷⡀⠀⢀⠀⠁⠀⠠⠒⠉⠈⠁⠀⠀⠀⢠⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢚⠀⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠐⢄⠀⠘⢦⣀⠈⠃⠀⠀⠰⠑⢄⠀⠀⠀⠑⣳⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠚⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⢿⣶⣄⡈⠁⢄⠀⠀⠀⠀⡘⣆⠀⠀⠀⠙⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠈⢀⡙⡉⠻⠶⢄⠉⠀⢄⠀⠡⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠⠃⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⣄⠀⠁⠈⠊⠙⠓⠍⠙⠪⠳⠀⠀⠀⠘⢁⠢⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠉⢀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠸⣯⡳⣄⠀⠀⢀⠀⠀⡤⠂⢀⡀⠠⠈⠀⠀⠀⠋⠲⢦⡤⡀⠠⠄⠚⡁⠈⠀⠀⠀⠀⠠⠀⢀⣀⣁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢀⠙⣿⣜⣖⢈⠀⠀⢸⡇⢆⣼⠀⠀⠀⠐⠂⣀⠂⠀⠀⠹⣤⡄⣤⠀⠀⡀⠀⡆⠀⠈⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠸⣷⡄⢙⣿⣗⣥⠴⠖⠿⣿⡿⢦⡀⠀⠀⠸⣿⣤⡄⠀⢀⣼⠑⢿⣷⣤⣷⣌⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠹⣷⣄⣝⠻⠁⠀⠀⠀⠈⠁⠀⢧⠠⡴⠛⠛⣉⡝⢹⡁⠉⠂⢰⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡟⠻⢿⣿⣿⣿⡇
⠀⣻⣿⣿⣷⣿⡀⠀⠂⢀⢰⣶⡧⠚⢻⠁⠀⠀⢉⣧⣸⣧⠀⠀⠺⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡠⢋⠞⡡⠀⠙⣿⣿⡇
⠀⠉⢀⠁⠈⢙⡁⠀⠀⠀⠙⢿⢁⣿⠓⡆⠀⠲⢴⡟⡧⡌⠁⠀⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⠉⡉⠉⠉⠛⠛⠋⠉⢙⠿⠛⡁⢮⣐⠋⢦⡙⣻⠃
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢰⢿⡸⠏⡸⠛⠀⠀⠹⠿⠀⡀⠀⠀⠴⠶⠾⠿⠷⠾⠿⠿⠿⠟⠛⠚⠓⠾⠿⠟⠷⠶⠆⣰⣦⣤⣄⡈⢷⣶⡟⣮⠃
⠀⣷⢀⣤⠀⠈⠉⢸⣷⣾⣿⣿⣵⡀⠱⢤⣤⣤⣴⣶⣦⠍⠀⢀⠁⠬⠀⠀⠀⠀⠀⠁⠢⠆⠦⣄⡀⠀⢾⡇⠀⢀⠾⠿⣿⠿⠿⠤⣽⢿⣯⣾⡷
⠀⣿⣿⣿⣧⠀⠀⢘⣿⢿⡿⡿⢹⠁⠀⠸⣿⡟⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠄⠀⠀⠀⠙⡄⠂⢠⣖⠃⠤⢠⢆⣶⣶⠂⣨⣷⣽⣿⣇
⠀⣿⣿⣿⣿⡉⠀⢨⡇⠨⠀⣿⡬⠀⠀⡔⠙⠁⢹⢿⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⢬⣄⠀⢸⣶⡟⠋⠀⠀⠀⠙⢻⣿⣾⣿⣿⣿⣿⠇
⠀⠻⠿⠟⠋⠀⠀⠸⢉⠅⠀⡧⢾⠀⠘⢑⠆⠀⠘⢸⡄⢤⠆⠀⡄⡤⣤⣀⠀⠀⠀⣀⠤⠴⠡⣾⣛⠀⢻⣟⠉⠒⠉⠘⠀⠀⠀⠙⠛⠟⠛⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠀⠐⠀⠀⠀⠀⠐⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⠤⢶⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡞⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠀⠀⠀⣷⠷⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⣂⣄⠀⠀⢀⠄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠈⢣⣆⣴⠁⠂⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣝⠆⠄⠀⢼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠻⣧⡄⠘⡰⣒⣾⢒⠀⠀⣶⡒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠌⠉⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠦⠀⠀⢹⣷⣀⣹⠹⣿⡄⠱⣂⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣗⣹⠆⢻⣿⣈⣻⡿⣷⡰⢏⡿⠐⣁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣶⢇⠀⠀⠀⠀⡀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣋⣿⡿⣿⣛⡿⣷⡯⡓⣻⣀⡯⣎⠂⠈⠂⠀⠸⡁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡠⠞⠑⠋⠙⠒⠂⠛⠃⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣌⢳⢡⡙⣷⡿⣿⣢⣽⠋⠁⠈⣿⢂⢆⡀⣸⣤⡄⠀⠀⣼⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⡌⠀⠀⠀⠀⠀⢀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠭⢿⣦⡝⠳⣯⣟⣨⣿⣨⣌⣻⡍⢛⣥⠓⠯⠀⣲⣴⠄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠄⠀⠐⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡔⠂⠀⠺⣷⣎⡟⣭⣦⠏⣫⡽⠃⠚⣠⠶⠙⢤⠀⢼⣻⣦⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠰⡆⠀⠀⠁⡏⠓⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⢨⠽⡲⢞⡽⣟⡁⠈⢀⠎⢁⠀⠀⠘⢤⣿⡃⠮⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡧⡂⣼⠖⢷⢼⡇⣁⣠⠟⢱⡆⣿⢰⡰⢰⣈⣿⠇⡆⣼⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠿⣏⢿⣧⣸⢾⣿⣿⣿⢿⣸⡇⠿⠼⠃⠘⠘⠉⠀⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠙⠸⠓⠈⠙⡛⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⠈⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠓⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢧⠀⠀⡄⠀⡏⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⣇⠀⠀⠀⢸⣖⡒
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡸⠟⢀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠠⣀⡠⠊⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒

⠀⠀⠀⢰⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱
⠀⢠⠒⢺⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡗⠒
⠀⢼⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⢀⣠⢤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⢤⢄⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡶
⣤⢸⡀⠀⡇⠀⠸⣒⡒⢒⣒⣒⣇⢠⡠⠀⢗⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⠂⡤⢠⡁⠀⠀⠀⢈⡇⠀⢸⡇⠀⡇
⠛⠀⡇⠀⡇⠀⠀⠁⠀⠀⠀⠀⠙⣆⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣴⠋⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡇
⠀⠒⡇⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⣀⡤⢤⣤⡄⠀⢀⢤⡖⠲⠤⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠉
⠀⠀⠣⠤⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠘⠉⠀⠈⢑⠈⠉⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⠒
⠀⠀⠀⠀⠣⠄⠤⠃⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⣀⠀⠎⣤⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠓⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠀⠀⠀⡄⠘⠛⠀⡆⠉⠁⢀⠃⠀⠀⠈⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⠀⡠⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⡀⡀⠐⠀⠀⠀⠀⠀⡀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠀⠀⠀⠀⠀⠀⠖⠂⠀⠃⠠⠃⠀⠔⠀⠀⠀⠀⠀⠀⠼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⣀⡉⠦⠀⠀⠀⠐⠤⠀⠀⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣀⢀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡤⡀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠑⠒⠒⠒⠉⣿⠀⢹⡏⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢂⠀⢀⡹⠂⢎⡀⠈⡐⠉⠈⠲⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣐⣪⡥⠶⠶⠒⠒⠒⠒⠶⠶⢬⣕⣂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣪⠴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠉⠛⢦⣕⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢱⡄⠄⠀⠀⠀⠀⠈⠳⣮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣡⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠖⠚⠁⠀⠀⠀⠀⠈⠳⣝⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣴⠋⠀⠀⠀⠀⢀⣴⠗⠢⡀⠀⢰⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣸⠃⠀⠀⠀⢀⡴⣪⠞⠀⢰⣿⣿⣯⠬⣹⣿⣿⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢱⡏⠀⠀⠀⣰⣯⡞⣿⠀⢰⣿⣟⣿⡴⢚⡿⣿⣿⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⣾⠀⣠⢤⣾⠟⠁⠀⠙⠀⢻⣿⣿⣿⣥⡾⠟⣿⣿⡇⢠⡟⠈⠐⠢⣄⠀⢠⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠞⠁⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣥⡆⣿⣿⡇⠈⠟⠀⡀⠢⢌⠉⣿⠦⢤⣀⡀⢀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⣰⡂⠀⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⢸⣿⣿⡄⠀⠀⠈⠂⠀⣼⣿⣇⣼⡄⣯⣩⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣦⣽⡃⢘⣿⡄⠀⠀⣸⡘⣿⣿⣿⣿⣿⡿⢁⣿⢿⣷⠀⠀⣰⡄⣰⣿⣿⣿⣿⣿⣿⣿⡿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⣷⣾⣿⣿⠀⠀⣿⣿⣧⣄⣹⣿⣿⣶⣿⣷⡾⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣅⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⠟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⢟⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢝⠻⢿⣿⣿⣿⣿⠃⢠⣿⣿⣿⠀⢸⣿⣿⣿⣿⠟⡫⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠩⢝⡛⠻⠤⠾⠿⠿⠧⠠⠞⢛⠋⠍⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣹⠳⡀
⠀⠀⠀⠀⠀⢀⡞⠁⡟⠀⢠⠃⠀⡇
⠀⠀⠀⠀⠀⡼⠀⢸⠃⠀⡎⠀⠀⠃
⠀⠀⠀⠀⢰⠇⠀⡟⠀⢰⠃⢀⢰
⠀⠀⠀⠀⣸⠀⢰⡇⠀⡾⠀⣾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠤⣀
⠀⠘⠀⠠⡿⠀⣸⠀⢀⡇⢠⡿⠧⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡸⠀⢸
⠀⠀⠀⠀⡇⠀⡿⠀⢸⠁⢸⣿⠃⠐⠂⠀⠨⢍⠋⣰⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⡇⠀⣸⠑⣆
⠀⠀⠀⠀⡇⠀⡟⠀⢸⠀⢸⢋⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠐⠚⠀⠬⢐⡂⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠜⡤⠀⡼⠀⢠⡇⠀⡏
⠀⠀⠀⠀⡇⠀⡇⠀⣾⠀⠀⠆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠤⣤⡉⣿⠆⠀⠀⠀⠁⠒⠇⡀⡐⠀⠄⠀⠀⡜⣼⠇⢰⠃⠀⡾⠀⢰⠃
⠀⠀⠀⠀⢃⠀⡇⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢄⡉⠈⠛⠑⠀⠈⠑⡄⠀⠀⢀⡴⠀⠀⠈⠑⠂⠨⣙⡟⢀⠏⠀⣸⠃⢀⡟
⠀⠀⠀⠀⠈⠉⢧⠀⢧⢠⠃⠀⠀⠀⠀⠀⠠⡀⠀⠀⠙⢄⠀⠀⠀⠀⠠⣼⡤⠔⠋⠀⠀⠀⠀⠀⢀⢱⡿⢀⡞⠀⢰⠏⠀⡼⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⢶⠢⢲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠞⠁⡜⠀⢠⠏⠀⡼⠁⠂⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠉⠉⣽⣦⣄⣀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡼⠁⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠻⠿⣿⣿⠿⡇⠑⣄⠀⠀⡀⠈⠀⡎⠀⡜⠀⣠⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⣿⣦⣤⣀⣀⣴⡇⠀⠈⡠⠊⠀⠀⠀⣇⠎⢀⡴⢇⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡈⠙⠻⠿⢻⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠷⠶⠞⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⣴
⠀⠀⠀⠀⣀⣄⣀⣾⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣤
⠀⡇⠀⢸⣿⣯⣩⣿⣿⣤⡇⣇⢀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡶
⣤⣧⢰⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣇⣿⣦⣦⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣠⣴⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣦⢀⠀⣤⣰⠀⠀⠘⣿⣿⠀⣿⡟⣸⣄⣴⣶⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⠀⠀⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⢤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⣍⡓⠶⢤⣀⢰⠏⠀⠀⠀⠙⠓⠚⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠃⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⠀⠀⠀⠀⠀⠀⠀⡀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢉⠀⠀⣈⠀⠙⢀⠀⠰⠀⢠⡞⠀⡀⠘⠂⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⢀⡀⠐⠂⠀⣀⡴⠃⠀⣠⠣⣼⠀⠀⢀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⣷⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⠀⡀⠘⣀⣠⡤⠴⠖⠛⠁⠀⢀⡴⠃⠀⠙⣄⠀⣾⠀⠀⠛⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⣿⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠐⠉⠀⢀⣀⣈⡤⠶⠛⠉⠀⠀⠀⢀⡤⠶⠚⠋⠀⠀⠀⠀⠈⢧⣿⠀⠀⡴⠀⠄⠀⠈⠀⠀⡀⠉⠀⠀⢠⡄⢻⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢤⣄⣀⣀⡤⠶⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢠⡇⠀⠀⠐⠂⠀⠀⠁⠀⠁⠀⠀⠀⢸⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣷⠀⠀⠀⢀⠈⠃⠀⠀⢀⠁⠘⠃⢸⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢀⡇⠀⠐⠂⠀⠈⢀⡀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣘⡇⠀⠀⠀⠀⠀⠈⠁⠀⣾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⣠⡤⠶⠒⠒⠀⠀⠠⡄⠀⠀⠈⠉⠓⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⡴⠋⠁⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠘⠂⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠔⠋⠀⣼⠃
⠀⠀⠀⠀⠀⠀⡼⠁⢠⡞⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⣄⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠈⠛⠲⠦⠤⠴⠚⠁
⠀⠀⠀⠀⠀⢰⠃⠀⡾⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠀⠀⠀⠈⠁⠀⢠⡄⠈⠳⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⣿⠒⠦⢤⣄⣀⣀⡴⢞⡭⡍⠳⣄
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠋⠀⠀⠰⠄⠀⠀⠀⣴⠃⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠐⠒⠀⠀⠀⠀⢚⣀⣠⠤⠀⠀⠀⣸⡇⠛⠛⠓⠒⠲⣦⠰⣄⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠈⠐⠒⠒⠶⠶⠶⠤⠤⠤⠶⠶⠶⠶⠖⠛⠛⠉⣁⣠⣤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠈⢷⣌⡙⢋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠤⠤⠤⠤⠤⠤⠶⠶⠶⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣞⡋⠉⡉⠩⠉⡭⠉⠉⠉⣐⠀⢰⠄⠠⢠⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡖⠋⠉⠀⠐⠀⠂⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢿⠟⠉⠋⠅⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡜⡡⢌⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⢣⠀⠀⠀⠃⣧⡟
⠀⠀⠀⠀⠀⠀⠀⣾⠰⡈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢰⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠘⠠⠀⠀⢸⡄⣾
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠂⠀⠀⣢⡿⢱⠇
⠀⠀⠀⠀⠀⠀⠀⢹⡂⠀⠀⠀⠀⠀⠀⢀⡐⣶⣶⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⣱⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⣬⣼⡟⠁⠀⠹⣷⠀⠀⣀⣀⣀⢀⣈⠥⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠻⣷⣞⣥⠶⠖⠏⠀⠊⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣯⢦⡄⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⢀⡟⣀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠼⠿⠿⠚⠉
⠀⠀⠀⠀⠀⢀⣞⣵⠃⠀⢸⣿⡏
⠀⠀⠀⠀⣰⠿⠟⠉⠀⡀⢾⣿
⠀⠀⠀⠀⣧⠈⠀⠰⣶⡇⣀⣿
⠀⠀⠀⠀⠈⢳⠶⢶⡠⢠⡿⠃
⠀⠀⠀⠀⠀⠈⠙⠓⠒⠋

⠀⠀⠀⠀⣠⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⣴⣴⠀⠀⠀⠀⢠⣤⣄
⠀⠀⣠⣿⣿⣄⣄⣼⣿⣧⡄⠀⠀⠀⣰⠛⢻⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠿⣿⠁⠀⢸⡿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣶⣄⣀⣀⣰⣶⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⡇⠀⠀⠀⠀⠀⠀⢠⣥⡄⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⣤⡄⠿⠏⢉⣯⡹⠿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢀⣹⣏⣀⣀⣀⣀⣀⣀
⠀⠀⣿⡟⠛⠛⠻⣿⠅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡏⠉⠉⠉⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠘⢷⠂⠀⠀⢠⡾⠏⠀⠀⠀⠀⣀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣀⠄⢀⠤⣀⢰⣄
⠀⠀⣤⣾⣶⣶⣶⠾⢧⡀⠀⠀⠀⠈⣿⠀⠉⠉⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⢿⣯⣼⣿⣿⣄⣠⠇⠀⠀⠀⠸⠿⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⣤⣤⣤⣤⣿⣇⠀⠀⠀⠀⣿⣿⣤⣄⣾⣿⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣄⠀⠀⠀⠀⠙⢻⠉⢹⣿⡍⠉⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣄⣅⣸⣿⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⠀⡇
⠀⠀⢶⣶⣶⡶⡶⣶⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣍⠀⠀⠀⠀⠀⠼⠄⠨⠿⠄⠤⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⢤⣤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠐⣄⣰⣉⣷⡀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠁⢀⣀⣀⣀⣤⣤⠖⣛⣋⣿⣿⠧⣀⣀⣸⡟⢻⠶⠖⢺⣗⣚⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠉⠹⢿⠐⣿⣿⣿⣿⣟⡢⢄⠙⢿⡀⢣⡍⠀⠀⣌⣫⡙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⣄⠀⠀⠀⠀⠀⠎⠀⣿⠟⠁⠈⠈⠀⠀⠑⣄⣷⣀⡀⠀⠀⣨⠉⠻⣤⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠥⠀⣽⣿⣷⡴⠀⡼⢫⠂⠀⠀⠀⠀⠀⠠⣿⣿⣿⡧⣀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣁⣤⡤⠒⠊⠉⢵⢢⠁⣜⡷⠁⠈⢷⣄⠀⠀⠀⣴⡿⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡠⠚⣹⣿⠀⠀⠀⠀⠀⡎⠰⡎⠀⠀⠀⢀⣻⣦⡀⠀⠈⠳⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⡞⠁⡄⣿⡏⣀⠄⠀⠀⢰⣷⢳⣶⠞⢀⡰⢿⣝⠿⠆⠐⠲⣾⠿⡁
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣈⠁⢀⢸⣿⠏⠁⣀⠀⣤⣼⢿⡿⠥⠒⠁⠀⠁⠉⠀⢰⣤⠀⢹⡄⢇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠘⣾⣿⠾⠋⠀⠀⠘⢿⣮⢻⡄⠀⠀⠀⢠⣦⠞⠁⠘⣧⠀⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⢀⠖⡁⠀⠀⠀⢹⢸⠗⠀⠀⠀⢻⡏⠀⠀⠀⠻⣷⡆⠀⠀⠘⠿⣦⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡸⣿⣿⣿⣀⡘⠁⡠⠋⠈⠀⠀⢤⣤⣸⣷⣄⠀⠀⠀⠉⠻⢄⣀⠀⠀⠈⠛⠷⣤⣤⠐⠧⠒⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢙⠻⢿⣟⣡⣾⡁⠀⠀⠀⠀⢠⣿⡏⠀⠙⠯⣕⠢⣤⣀⡀⠈⠉⠒⠢⠤⣄⣀⠙⠷⣶⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠘⠋⠉⠉⠉⣽⣭⣍⣩⡴⠎⠀⠀⠀⣸⠋⠉⠀⢙⣷⡄⠀⠀⠀⠀⠈⠙⠲⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠠⣿⣿⢻⣿⣇⠀⠀⣀⣴⠟⠀⠀⣀⡘⠏⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠐⠫⢻⡆⢻⣿⣧⣾⠟⣿⠺⢿⡛⠉⠉⢰⡏⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡁⡆⢘⠛⠿⠄⠁⠀⢠⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠁⠃⠘⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣤⣤⡴⠒⠀⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⣿⠁⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢘⣦⣘⣿⡏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢠⡾⠛⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠱⣿⠷⣤⡾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣼⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣷⣄⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠐⠉⠩⠁⢀⣿⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⠀⢀⣀⣤⣶⣾⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠤⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢨⣿⣇⣿⠀⠀⠛⠻⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠁⢸⠀⠀⠀⠀⣨⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡎⠴⡶⢷⣾⡴⠾⠿⠿⠿⠸⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⣦⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⣉⡉⠍⠂⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⠈⢡⡤⠴⠶⠶⠞⠋

⠀⠀⠀⠀⠀⣀⣠⢀⣆⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⣰⣀⣀⣀⣀⣀⣶⡀
⠀⠀⠀⠀⠀⠙⠶⣾⣷⡶⠞⠁⠀⠀⠀⠀⠀⢰⣿⠻⠦⠄⣴⠂⠀⠀⠀⠀⠀⠠⠞⣿⢦⣠⢀⡿⠀⠀⠀⠀⠘⠐⣆⡶⣠⠆⠛
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⠀⢶⣭⣷⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣸⠟⢶⡀⣸⠇⠀⠀⠀⠀⠀⠀⢘⣿⡏
⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠶⠿⠻⣗⠀⠀⠀⠀⠀⠀⢠⣭⣥⣴⣾⣶⡿⠀⠀⠀⠀⠀⠀⠀⣼⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢉⡳⢄⠀⠀⠀⠀⠀⠀⠀⠐⢲⡂⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣃⣀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡄⠀⠀⠀⠀⠀⠀⢈⣭⣀⣠⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣥⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠟⠂⠀⠀⠀⠀⠀⠀⠻⣿⣥⣄⠈⣻⡇⠀⠀⠀⠀⠀⣠⠿⣿⣿⢷⣄
⠀⠀⠀⠀⠀⣤⣬⣿⣏⣿⣤⣄⠀⠀⠀⠀⠀⠀⣾⠁⢻⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡟⣷⣿⡇⠀⠀⠀⠀⠀⠘⣿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢠⡴⣿⣿⠦⡄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣷⣆⠀⠀⠀⠀⠀⠀⠠⡾⣿⢧⡀⢀⡾⠃⠀⠀⠀⠀⢶⠎⣷⣦⣤⣄
⠀⠀⠀⠀⠀⢰⠀⣿⣿⠀⡰⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣮⡅⠀⠀⠀⠀⠀⠀⢹⡟⢳⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⡙⣷
⠀⠀⠀⠀⠀⠈⠢⡌⣿⠴⠁⠀⠀⠀⠀⠀⠠⠴⠚⠉⠀⠸⠁⠀⠀⠀⠀⠀⠀⢸⠃⠼⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠹⣄⡀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⡇⠀⠀⠀⠀⠀⠀⠰⡶⣻⠄⢀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣷⡘⠟
⠀⠀⠀⠀⠀⠀⠸⢿⣟⢻⠆⠀⠀⠀⠀⠀⠀⠀⠸⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠐⠿⣷⣾⡿⢻⡄⠀⠀⠀⠀⠠⣔⣛⡯⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⢀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠉⠉⠉⠉⠉⠉⢛⡆⠀⠀⢀⣀⣈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣀⠤⠒⢉⡠⠖⠊⠁⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠄⣀⣀⣀⡀⠀⠀⠘⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣠⠔⠋⢁⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡼⠁⠀⡰⠋⠀⠀⠀⣀⣀⡀⠤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⡸⠁
⡇⠀⢰⢀⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣰⠁
⡇⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠲⠢⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠱⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠒⠒⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⡇⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⢣⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠦⡀⢳⡄⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠙⣄⠙⣄⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣆⠈⠳⢄⡘⠲⢄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠣⣀⠀⠈⠉⠉⠀⠈⣣
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠬⢦⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠉⠀⡕⠾⠛⢹⡞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠔⠊⠀⠰⠖⠋⢹⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⡀⠀⠀⢀⡤⡀⠀⡇⠀⠀⠀⠀⠶⠋⠙⣯⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⢉⡟⠒⢼⠀⢈⠀⠇⠀⠀⠀⠀⠀⠰⠟⢀⡽⠛⡟⠟⠉⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠈⠉⠉⡘⠛⢺⣂⠆⢸⠦⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠋⡀⢠⠄⠂⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠛⠓⣏⡙⠀⠈⣄⣀⣈⣱⣤⠤⢤⣤⣀⣴⣉⣀⣄⠠⡴⢺⠝⠁
⠀⠀⠀⠀⠀⠀⢀⠔⢃⠀⠀⠀⠀⠀⠒⠒⢲⣋⡇⠀⠀⣏⣭⡠⠤⠤⠐⠉⠙⠒⠒⠐⠂⠀⠉
⠀⠀⠀⠀⠀⢠⡾⣄⡀⠡⠀⠀⠀⠀⠴⡶⠾⣽⡇
⠀⠀⠀⠀⠀⠙⠲⢤⡉⠺⢦⡀⠀⠀⠀⢴⠶⠳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⢽⢦⡀⣀⠀⠀⠞⠙⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢝⢦⣈⠀⠂⠤⢀⡀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢗⣲⠤⢤⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉



⡟⠙⣿⣻⣻⣛⣻⣟⣟⣛⣿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⣀
⠀⡀⣿⢠⠀⢀⢠⣇⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⡇⣾⣼⠆⢸⣸⣿⠀⣦⣿⠀⠀⠀⠀⢀⠀⠀⠁
⠈⠻⢿⣿⡇⢸⣿⣿⢀⣿⣿⠀⣧⠀⣆⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⡄⡄
⠀⠀⠀⠀⠿⠙⠻⠿⢼⣿⣿⢴⣿⢀⣿⣸⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠻⠗⠃⠃⠆
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠺⢷⣿⢠⡄⠀⠀⠀⠀⠀⠀⠁⠈⠈
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡂⠀⠀⠀⠀⢸⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⣿⠿⣄⡀⠀⠀⢸⠀⣧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠁⢰⣣⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⣀⠀⠀⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⠀⠀⠰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⡄⠀⢀⢂⡀⠸⠀⠀⠐
⠀⡇⡴⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⠉⢠⠠⠇⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⣨⡅⢀⠉⠅⢰⢢⡑
⠀⣅⢁⡤⡆⠌⠁⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡆⡄⢖⠀⠉⡎⡞⣌⣟⡍⠅⠋
⠰⠏⠨⡿⡶⢶⢾⠃⡶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡄⠀⣀⣀⣟⣀⣯⣽⣶⣼⣟⣹⡷⠤⠸⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣠⣴⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢙⠶⠿⠋⠉⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣽⣏⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠈⠙⢦⡄⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⢀⣼⣷⠀⠀⠀⠀⠀⠀⠈⠈⢶⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⢠⣿⡿⠛⠀⠀⠀⠀⠀⣀⣀⡜⣧⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⢿⠀⣾⣿⣿⠠⠀⠀⠀⠀⢲⣼⠋⢀⣬⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢸⠀⣿⣿⣿⣤⠀⠀⠀⠀⠈⠹⣆⣿⡿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⠀⠀⠀⣿⣿⣿⠶⠀⠀⠀⠀⠀⢋⣤⣾⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⣼⠏⠀⠀⠀⣿⢸⡏⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣳⡀⠀⢸⡿⠋⠀⠀⠀⠀⡏⣼⠀⠀⠀⠈⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡍⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⣿⣿⣯⣠⣿⡿⣤⢀⠀⠀⢘⢇⡇⠠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⠿⣿⣿⣋⡭⠖⠲⠏⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢿⣿⣷⣿⡿⠛⠉⠩⠁⠀⣀⢢⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠿⠟⠻⠁⠀⠉⠀⠂⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣆⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣙⣵⣿⣿⣿⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣽⡿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠟⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣷⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣏⠻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠈⠛⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠰⠶⣤⣹⣿⣿⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⣿⣿⣦⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠿⡟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⣿⣿⡁⠙⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣾⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢻⣿⠛⣠⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣸⣿⣶⣿⣷⣶⠆⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⣿⠀⠸⠿⢿⣄⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣄⣀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠛⠻⠂⠰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣦⣾⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠉⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠆⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠟⠛⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣤⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢀⣶⣿⣿⣿⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠿⠿⠿⠟⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⢿⣶⣿⣷⣤⠀⠀⣀⣀⣀⣀⣴⣦⠀⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣄⠀⠉⠀⠈⠉⠀⠀⢘⣿⣿⡏⠁⠀⣠⣤⣤⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⣿⡿⠟⠿⠛⠓⢶⠖⠈⠻⠿⠃⠘⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣄⡀⠀⠀⢀⠀⠀⠉⠉⠉⠀⠉⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠙⠷⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠉⠉
⠀⠀⣶⡿⠖⢶⣶⣿⡿⢠⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠠⠀⠀⠀⠀⠀⠲⠗⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀
⣶⣾⣿⣶⣶⣾⣿⣿⢁⣿⣿⣟⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣠⣶⣷⣾⣿⠁⢠⣶⣤⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⣿⣿⣷⣶⣶⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋⠀⠈⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇
⠉⠹⣿⣿⣿⣿⣿⣷⣶⣶⡟⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⡛⠛⣁⣤⣄⠀⢀⣠⣶⣶⣶⣦⡀
⠀⠀⠀⠉⠉⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉⠉⠉⠁⠀⠀⠉⠉⠁⠉⠉⠁
⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣿⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⠀⢀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⡤⣖⣛⣙⢧⡀
⠀⠀⠀⠀⡿⣉⠉⠤⠼⣿⢹⡆
⠀⠀⠀⢸⠇⠵⠄⡒⠒⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⢋⡉⠉⣁⣠⠔
⠀⠀⠀⢸⠀⠀⠐⠻⣭⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⢸⠀⣆⠀⢀⣷⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡀⠀⣀⡀
⠀⠀⠀⠘⣆⠈⠩⢽⣞⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣀⢘⡝⠻⠊
⠀⠀⠀⠀⢹⠀⡄⠀⠘⢦⠳⡄⠀⠀⠀⣠⠞⠉⠁⠀⠈⠉⠙⠿⡷⣆⠐⢦⡀
⠀⠀⠀⠀⢸⠀⠸⡄⠀⠀⠀⠙⢦⡀⣼⠃⠀⡀⠀⠀⠀⠀⠀⠀⠙⠘⢧⡈⠳⣄
⠀⠀⠀⠀⢸⡄⠀⢹⡀⠀⠀⠀⠀⠙⢧⣆⢸⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⠲⢌⠛⠶⠦⣤⣤⠄
⠀⠀⠀⠀⢸⡇⡀⠸⣧⡀⠀⠀⠀⠀⠈⢿⠻⡄⠀⠀⠀⠀⠀⣰⠃⠀⠀⡇⢰⠈⠇
⠀⠀⠀⠀⠀⢷⠹⠀⡏⠻⠶⠖⠀⠀⠀⢸⣶⡇⢤⣤⣤⣤⠞⠁⣠⡆⠀⣧⡏
⠀⠀⠀⠀⠀⠘⣧⠀⠻⣄⣀⠀⠀⠀⠖⠋⢹⠀⠀⠀⠀⢀⢄⠼⡋⠀⢀⣿⡅
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠨⡉⠉⠉⠉⠀⢀⣈⡴⠃⠀⠀⠀⠉⠉⠃⢀⡼⠋⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣈⠲⠤⠤⠀⠉⣀⣀⣀⣀⣀⣤⣤⠴⠞⠋⠀⠀⠀⠈⠙⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⡉⠙⣌⡋⠉⣿⣷⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⢟⣫⣴⣿⣿⡿⢁⠱⣄⡿⣷⠧⠟⢉⠹⠽⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣛⣭⣷⣿⣏⡀⣧⣼⠁⣬⠀⣾⣿⠋⠙⡇⠀⠀⠐⠁⠘⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⡿⢟⡛⣋⡇⣫⣩⠌⠙⠃⣈⢴⣟⣻⣿⠂⣤⠀⠀⠀⠀⠈⠀⠀⠀⠉⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⠋⠁⣹⠿⣋⢸⠁⠋⢀⡦⣞⡿⣻⣯⣭⣿⡄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣩⡀⠦⢨⠟⠐⣯⠈⣠⡄⢸⠘⣿⣿⡿⣿⣧⣿⣷⡄⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠝⢻⢶⣤⣤⡤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢿⡟⣯⣅⣼⢄⣠⡐⠫⡬⠟⠟⢀⣼⡷⠃⢀⣤⣿⣾⣿⣿⣿⣶⣄⢀⣄⠄⡀⣄⣴⣾⣤⣀⡁⢉⡤⡅⠀⠁⢀⣴⣿⣻
⠀⠀⠀⠀⠀⠀⠤⠂⠠⠁⠂⠀⠻⣤⣍⣶⣾⣿⣣⢒⡖⠆⠚⢺⢷⠇⢠⣿⠋⠋⠛⣿⠿⠿⠿⠛⠿⠿⠿⠿⠛⠛⠛⠻⢿⣷⣦⣼⣻⢖⣶⣹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠚⠀⠆⠀⠈⢠⠀⠈⣿⡛⠑⣟⡻⣿⣿⣿⠷⠦⣿⣤⣿⡏⠄⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠉⠀⠀⢀⡴⠃⠀⠀⠀⢿⣷⣤⣜⣃⣿⣦⡄⣀⣼⣿⡿⠛⠀⠀⠀⠀⡸⠀⠀⢀⠔⠃⠀⠀⠀⠀⢠⠅⠠⠤⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠙⣄⠀⠀⠣⣿⡿⢯⣼⣭⣿⣿⡟⢩⠀⠀⣴⣦⡤⠖⠁⠀⠀⠁⠀⠀⢀⠄⣀⠔⢁⣠⠄⢀⣀⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣴⣟⠆⠀⠀⠈⠛⠿⣟⣯⣯⣷⢶⣿⣦⣪⠖⠋⠀⠀⠀⠄⡀⠀⡠⠔⣉⡼⠷⡾⣋⣛⣋⣉⡁⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣁⣀⣤⣤⣤⣄⡠⠈⠉⠛⠿⠿⠿⠟⠋⠀⠀⠀⢀⣎⣜⠵⠚⢋⣛⣁⣀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠈⡁⣸⣿⠃⠀⠀⠠⢦⣄⠀⠀⡀⠀⠀⠀⠼⠿⣟⣛⣿⣿⠿⠿⠷⠈⠙⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁⠀⠰⠆⠤⠀⠠⡄⠀⣁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⣰⣶⣀⣀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣀⠀⣴⢶⣠⣰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⣾⣿⡗⣸⣿⣿⡏⠀⡯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⣻⣿⠿⣭⣿⣟⠀⣼⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⣿⣷⣿⣽⣿⡏⣿⣿⠃⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣿⣿⡿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⡿⠳⠆⢻⣆⠀⠀⠀⠀⠀⠀⡐⢡⠘⢶⣤⣹⡇
⠀⠀⠀⠀⠀⢸⣧⠀⠀⣠⣾⢲⠀⠀⠀⠀⠀⢠⢙⣿⣿⣿⣷⣀⠀⠁⠀⠀⠀⠀⡠⠊⢀⠏⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣴⣾⣿⠇⠀⠻⣄⠀⠀⢀⠆⢸⣿⣿⠏⠻⣿⣤⠀⠀⠀⢀⢎⡔⠀⠀⠀⣆⠀⢺⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠐⢄⠈⢳⣦⡈⠀⣼⣿⡟⠀⠀⣿⣏⠠⣠⠴⢩⡞⠀⠀⠀⢸⣿⣷⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⢷⣶⡤⠙⢦⣻⠳⣼⣿⡿⠁⠀⠀⣿⣗⡘⠁⠠⠋⠀⠁⠀⠀⡼⠋⣱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣍⢻⣿⣿⠿⠶⢿⣟⠁⠐⠀⠈⠩⣿⣇⠀⠀⣠⡾⠓⢋⡥⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⠶⣊⡙⠛⠿⣿⣿⣿⣿⣿⣶⣤⣤⠚⠉⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣿⣿⣿⣿⣶⡈⢿⣿⣷⣦⣤⣤⣬⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹
⣀⣀⣠⡇⠘⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠃⢸⣄⣀
⠛⠛⠻⡇⠀⠃⠀⠠⢾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠄⠀⠘⠀⢸⠟⠛
⠀⠀⠀⣇⢀⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⡀⣸
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠘⢆⠀⠰⢄⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⡠⡄⢀⡴⠃⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⡈⢲⡶⢄⡠⠀⠀⢄⡠⢶⡖⢁⣴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⠀⠓⢀⡀⠚⢠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣀⠟⢲⠿⣅⡀⢀⡨⢿⡾⠃⢀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢼⡟⢦⠀⠀⠈⠉⠉⠀⠀⠀⡴⢻⡧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠸⠓⠦⠄⠀⠀⠠⠴⠚⠇⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠀⡂⠀⠀⠀⠀⢐⠀⠿⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠃⣇⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⡼⠁⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⡤⠀⠀⠀⣸⠁⠀⠀⠀⢧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⠇⠀⠀⢰⠃⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠃⠀⠀⣏⠀⠀⠀⡇
⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢰⠃⠀⠀⠀⠈⠁⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢸⡇⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢸⠁⠀⠀⠀⠀⠈⠓⠦⣤⣀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠳⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣀⣃⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣈⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⡶⠶⠶⡶⠛⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡤⠴⠶⢖⠚⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠯⠁⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⣠⣾⠟⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⢀⣴⡟⠁⠀⠀⢠⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⢀⣰⣶⣿⠋⠀⠀⠀⣰⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣶⣶⣿⠿⠛⠩⠤
⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⢰⣿⣿⡿⠃⠀⠀⢠⡾⠁⠀⠀⢠⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⣊⣵⣿⡿⠛⠉
⠀⠀⠀⠀⢀⣴⠋⠀⢀⣰⡿⠋⡞⠀⠀⢀⣴⠏⠀⠀⠀⣰⡿⠋⠀⠀⠀⢀⣀⣀⣄⣠⡀⣠⣀⣠⣤⣾⣶⣾⠿⠟⠋⠁
⠀⠀⠀⢀⡾⠁⠀⣠⣾⠏⠀⣼⠁⠀⣾⣿⠃⠠⠄⣀⡴⠋⠀⠀⢀⣤⣶⣿⡿⠿⣿⡿⠿⠿⠿⠟⠛⠉⠁
⠀⠀⣰⠟⢃⣄⣴⡿⠁⠀⣸⠃⠀⢸⣿⠃⠀⢠⣿⠟⠀⠀⢠⣴⣿⠟⠉
⠀⠀⢿⣤⣿⡿⠋⠀⠀⢠⡏⠀⠀⣾⠏⠀⢀⣾⠏⠀⢠⣴⡿⠋⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⡞⢀⠀⣼⠏⠀⠀⣾⠃⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢰⡟⠀⢀⣾⠃⢂⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠢⢠⡿⠒⣤⣾⡧⠴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣰⡿⢧⡨⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡖⣲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠹⣟⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⣷⣽⢼
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⣶⣿⣿⣿⠃⠀⢀⣀⠀⠀⠀⠀⠀⢠⡏⠉⣾⣾⣳⣸
⢄⡀⠀⣤⣶⣦⡀⠀⠀⠀⠀⠀⡇⠀⢀⣼⣿⣿⣿⣿⡆⠀⢸⣹⢧⡀⠀⣠⢤⢾⡇⠾⣿⣿⣯⡼
⠀⠳⢄⣸⣶⣿⠛⠀⠀⠀⠀⠘⡟⠛⠉⣸⠛⠛⢻⡿⢿⣴⡋⣽⣇⡇⠀⠉⢀⡿⣭⠉⢻⡏⢹⠃
⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⢀⡇⠀⢀⠇⢠⠈⠡⡇⠘⡻⠤⢧⡟⡇⠀⢀⡞⠁⢿⡀⢨⣏⠀⠁
⠀⠀⢀⣾⣡⡄⢛⡄⠀⠀⠀⢸⠀⠀⢾⣆⡸⣤⣤⠇⢠⠃⡆⢸⠋⠀⠀⢠⢟⠶⠟⢲⢶⠿
⢠⡀⡾⠁⠀⠈⠉⠻⣦⢤⠀⡞⠀⠀⣨⡞⠀⠀⢫⠃⠈⣿⢛⡟⢀⡀⢀⣞⠏⠀⠀⠀⣧⡄
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠳⢆⡗⢳⣶⡿⠀⠀⢀⣀⣿⠾⠿⠾⣧⡈⠫⠻⠏⠀⠀⢀⣤⣞⣧
⠐⠒⠚⠛⠉⠉⠻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠛⠛⠛⢛⡏⢉⣉⠭⢭⣝⣛⠛⠓
⣿⣟⣛⣛⣛⣻⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⣠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠘⣈⡔⠁⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡞⢡⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠠⠖⢄⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠓⠂⠁⠀⠀⠀⠀⠆⠀⡀⠀⠀⢀⢂⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⡐⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢀⠊⠀⠀⢀⠄⠘⢠⠂⠀⡀⠀⠎⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⡔⠁⠀⠀⠀⡌⠐⠁⡌⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠎⠀⠀⠀⠀⡈⠰⠂⠀⠃⠀⠀⠀⡀⠐⠒⠀⠀⠀⡄
⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠁⠀⢀⡏⠀⠀⠀⠀⠰⠀⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡰
⠀⠀⠀⠀⡇⠀⠰⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⠁⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠐⠰
⠀⠀⠀⠀⡇⠀⠀⠘⢐⠀⠀⠀⠀⢠⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇⢰⠀⠀⢸⠀⠀⠀⠀⠀⡀⠉⠄⠀⠀⠘⠁
⠀⠀⠀⠀⡇⠀⠀⠐⠠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠆⠀⠀⠀⠀⡁⠀⠀⠀⠘⡄⠀⠀⠀⠌⠐⠀⠀⢀⡜
⠀⠀⠀⠀⢸⡀⠀⠘⠀⠁⡄⠀⠀⠠⠀⠐⠀⠀⠀⠘⡄⠀⠀⠀⡆⠀⠆⠀⠀⢇⠀⠀⢰⠂⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢧⡀⠀⠀⡀⠉⢀⠀⠀⢠⠀⠀⠀⠀⠀⠘⣄⠀⠀⢘⠀⠀⡀⠀⢸⡄⠀⢀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠧⠠⠀⠁⠐⠄⠀⠀⠀⠀⠀⠲⠄⠀⠀⠈⠢⠀⠀⠂⠀⠇⠠⠀⠻⠄⠸⠀⠀⠀⠠⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢳⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⣰⠃⢻⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⡆⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠃⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣰⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡴⠃⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠲⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⢢⡐⣅⡦⡵⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⢲⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⡀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣠⣆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⡆⠀⠄⠀⠀⠀⠀⠀⠀⠠⣴⡆⠠
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⢀⣀⣀⠀⣀⣀⣠⣼⣿⢁⣸⣿⣷⣀⣴⣶⣄⣤⡤⠀⠂⠀⠁
⠀⣾⣶⣦⣀⣠⣤⣤⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠙⠛⠿⣿⣿⣿⣿⣶⣮⣤⣶⣿⡄⢀⣤⣾⣷⡀⠀⣴⣶⣤⣠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⢠⣤⠄
⠀⢀⡀⠀⣤⣌⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠿⠇⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡄
⢀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⢛⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠋⠉⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⢠⡀⠀⠀⠀⠀⠙⠃
⠀⢾⡿⠿⢿⠿⢿⣿⢿⡿⣿⣗⠷⠟⠛⠉⠁

⠀⠀⠀⠀⢀⢰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣆⣀⡆⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠐⢻⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢾⡷⡴⠂
⠀⠀⠀⠀⠀⣞⡷⠀⠀⠀⠀⠀⢀⣯⣍⠁⠀⣻⣇⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⢀⣀⡿⣇⢣
⠀⠀⠀⠀⡀⠘⠃⢀⠀⠀⠀⠀⠘⢿⡿⠃⠀⢏⠉⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⢠⡄⢠⣼⣼⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣻⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡟⠀⠀⠀⠀⠀⠀⡞⣿⡦⠓⠀⠀⠀⠀⠀⠀⣠⡾⣷⡄
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⣰⠿⣬⠁⠀⠀⠀⠀⠀⠀⠓⣿⡄⠀⠀⠀⠀⠀⢀⡼⢻⡄⠸⣿⡀
⠀⠀⠀⠀⠰⠁⢸⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠰⠁⡇⠀⠀⠀⠀⠀⠏⠀⠞⠀⠀⠻⠇

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢈⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⣤⣾⣿⡇⠀⠀⠀⠀⠀⣀⣤⠟⠻⢿⠀⠀⠀⠀⠀⠀⠼⣿⠄⠀⠀⠀⠀⠀⣶⣾⣷⣶⣶⣦
⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠾⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢶⠠⢠⡄⡄⡦⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠙⢾⡗⠋⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠁
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⡎⠋⡇
⠀⠀⠀⠀⠀⡃⠃⠀⠀⠀⠀⠀⠀⣸⠀⠀⠓⠂⡄⠀⠀⠀⠀⠀⡞⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀

⢀⣤⣶⣦⡤⠶⡦⠀⠦⠀⠀⠀⠀⠀⢠⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣤⣤⣤⣤⢀⡄
⠸⣿⣿⡿⠀⠂⠀⠀⠸⠧⠀⠀⠀⠀⣸⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡆⣾⠀⣼⠆⣿⣿⣿⣿⠋⠀⣴⡀
⠀⢻⣿⣷⣿⣿⡇⠀⠀⣶⢄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣏⠀⣿⠀⢸⠃⢿⣿⣿⣿⡆⠀⣾⡆
⠀⠀⣿⡉⣸⣿⣼⠀⠈⢷⡄⠀⠀⣧⡅⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⡿⠀⢹⠀⢸⠀⢸⡿⣿⠸⠇⠀⢸⣁
⠀⠀⠈⢉⡿⣿⡉⠀⠘⠟⠀⠀⠀⠋⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣌⢿⡇⠀⣿⠀⠘⠀⢈⠁⢫⣶⣤⠀⢸⣿
⠀⠀⠐⡿⣿⣿⣿⠀⡰⠛⠆⠀⠀⠀⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠀⠐⣤⠛⠀⠀⠇⠀⠀⠀⠻⣿⡀⠈⢾
⠀⠀⠀⠛⠹⢟⣿⣠⠋⠀⠀⠐⣁⣴⢾⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⢹⡇⡀⠀⠠⠀⠀⠀⠀⠹⡇⠀⢿
⠀⠀⠀⠘⢷⡌⣿⣧⣤⣀⣰⣿⣿⣿⠟⡧⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡿⠀⠋⠀⠲⠈⠀⠀⠀⠀⠁⠀⠸
⠀⠀⠀⠀⠀⢣⣼⣿⣿⣿⣿⣷⣿⣿⣷⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡘⡁⠀⠂⠀⣤⣾⠆⢀
⠀⠀⠀⠀⠀⣾⡟⣽⣿⣿⣿⣿⡏⠹⣿⠂⡀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠙⣾⣧⣿⡄⠀⣈⠻⣯⡄⠙⡇⠀⢀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠻⣿⡇⢨⣿⠉⣁⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠺⣿⠉⣿⠇⠀⣾⡗⣿⡇⠀⢿⠀⢸
⠀⠀⠀⠀⠐⢿⣍⠛⣿⣿⠁⣙⢃⠈⣿⡆⡟⣸⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣸⣿⢿⡄⠀⢯⠌⠉⠱⢤⡀⢠⣼
⠀⠀⠀⠀⣠⣿⣿⠇⣿⡇⢸⡇⠋⢰⣿⣷⠀⢹⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣀⠀⢿⡇⣿⡇⠀⡇⠀⠄⠀⠀⠀⢸⣾
⠀⠀⠀⠀⡿⣾⣿⢸⣿⡇⣄⢰⠷⠘⣿⣿⠀⢘⣿⡄⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠹⣿⣇⢠⡏⠀⠀⣠⣀⣀⣸⡿
⠀⠀⠀⠀⣷⣾⣿⣦⣿⣷⠛⠸⠆⣰⣿⣿⡷⢈⣿⣷⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣶⣤⡘⠃⠰⠇⠀⠀⢨⣿⣿⠟⠁
⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣥⣶⡄⣿⣿⣿⣷⣾⣿⣿⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠀⠁⠈⢻⡿⠻⠒⣶⣞⣠⡙⠟⠙
⠀⠀⠀⠀⢰⣿⣿⡿⢻⣿⣿⣿⡇⣾⣿⣿⣿⣿⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⢠⣿⣤⣴⠀⢁⠻⣧⣽⠂⠁
⠀⠀⠀⠀⠈⢿⣿⣧⠈⣿⣿⡀⡄⣿⣿⣿⡄⢻⡷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠘⣿⡁⣿⠀⣼⡆⠘⠏
⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⣿⣷⡞⢿⣿⣿⡇⢈⣷⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⢻⣯⣿⡇⢸⣿⣶⡆⢠
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⣿⣿⡇⣿⣿⢻⠀⢠⡗⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⢺⣽⢟⡁⣸⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣬⣿⠉⣿⣿⣿⣿⠀⣻⢻⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣼⡿⣻⡇⢻⣏⣿⣯⣽⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡜⣿⠀⣇⣿⣷⡟⠀⣿⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⢹⣿⣏⡇⢸⣿⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⠀⡏⣿⡿⠃⢰⣇⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡆⠸⠛⠙⡇⢸⣿⢶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⡟⠀⣸⡏⠀⠀⢸⡟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⣀⡇⢸⣿⣿⣿⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡿⠀⠈⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢿⡇⢸⣿⣿⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠐⠀⢀⠀⠀⢿⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣷⠘⡇⠀⢩⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠋⠠⡇⠈⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣆⣥⠈⣸⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣼⢻⡆⠀⢰⡿⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⡿⠂⢹⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠀⠛⠘⠃⠀⣸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢉⡀⣼⣿⣥⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣠⠖⠀⠀⠀⠙⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠘⠃⠻⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⣿⠀⡖⠀⠀⠂⢡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⣸⡀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⠘⣧⣄⠀⠠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠚⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⡇⢸⡝⢿⣿⣦⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⢀⣿⣧⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⠣⣜⣂⣠⣿⠛⣦⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⠀⠈⢉⣿⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣩⡔⢫⣿⣟⣠⡄⠙⣷⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣟⡀⠀⢸⣿⡆⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⣿⣿⣾⣩⣶⢰⣿⠟⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣅⠀⠂⣿⡇⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⣿⢿⡀⣧⠸⡇⢻⣀⢄⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⡿⣿⣁⡀⢁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢸⡟⣼⠃⢻⡄⢿⡄⣿⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⢨⠋⣠⡌⣿⠟⣻⣀⣷⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣷⡏⠀⢬⠀⢼⠀⣽⠷⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠏⠛⠁⠄⣸⣿⠄⠀⠀⠛⢻⡿⢫⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠘⠿⠁⠀⣸⠀⢀⠀⢹⠆⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠟⠓⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠻⣞⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈⣶⣇⠀⣿⠀⣾⡁⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢽⡧⠀⠀⠀⠀⠀⣯⡝⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢉⣿⠀⢸⡇⠸⣷⡌⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⣤⠀⠀⢀⡀⢾⣷⣶⣦⣀⣰⣦⣧⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠸⢿⣦⣸⣧⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠿⣷⣄⡙⢿⠛⠛⠛⠛⠁⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣦⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣴⣦⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠲⠶⣖⡒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠤⠤⣀⣀
⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠢⣤⡤⠋⠙⣆⠀⠀⠀⣀⠤⠲⡄⠈⣉⡷⠶⣄
⠀⡠⠚⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠀⣻⠽⠛⠉⠀⢀⡠⠟⠊⠁⠀⠀⢘⡝⠢⡀
⡼⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⣋⠤⠛⠉⠀⢀⡠⠔⠊⠁⠀⠀⠀⣀⣤⡶⢭⡄⠀⠹
⣇⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡤⠖⠋⠁⢀⡠⠔⠊⠁⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⣇⠀⢀
⢻⣦⡀⠀⠀⠈⠓⠦⠤⢤⣤⡤⢲⡟⠋⠁⠀⠀⣴⠊⠁⠀⠀⠀⣀⠤⠶⠯⠤⠤⠤⢤⠤⠴⠒⠒⠋⢀⣾
⠀⠳⢝⠢⣄⠀⠀⠀⠀⠀⠀⠉⢹⡿⣄⣠⠴⠚⠙⣆⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡤⢚⠵⠃
⠀⠀⠀⠙⠲⢬⣑⡲⠤⢤⣀⣀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⢒⣒⣒⠶⠾⣒⣫⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣟⡒⠲⣬⠭⣍⣉⣛⣒⣒⡒⠒⠒⢒⣒⣒⣛⣉⣭⠭⠴⡖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢬⣑⢤⡈⠳⡄⠀⠙⢦⡀⠀⠀⠀⠀⣠⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣌⢦⠀⠀⠉⠒⠦⠤⠚⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡠⠔⠋⠁
⣄⣀⣄⣠⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣞⣉⣀⣀⣀⣀⣀⣠⣀⣤⣠⣀⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣂⡉⣁⣒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡴⠊⠁⠀⠀⠀⠀⠉⠢⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠆⠀⡠⠤⠀⠀⠀⠀⠤⡀⠀⡆⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⢦⡇⠀⠀⠀⠀⠀⠀⣧⢎⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣔⡺⠿⠜⠛⠓⠚⠓⠼⠿⣒⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣖⠽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢗⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠊⠁⠀⢀⡠⠴⠒⠊⠉⠉⠉⡍⠉⠉⠒⠲⠤⣀⠀⠀⠉⠪⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡫⠊⠀⠀⣠⠖⠉⠀⠀⠳⠀⠀⠀⠀⡇⠀⠀⠀⡼⠀⠈⠑⢦⡀⠀⠈⠪⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡝⠁⠀⢠⠎⠀⠲⣄⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣠⠘⢦⠀⠀⠙⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡟⠀⠀⡰⠃⠀⠀⠀⠈⠀⠀⡠⠔⠒⠉⠙⠹⠑⠒⠤⡀⠀⠀⠋⠀⠀⠀⠳⡀⠀⠘⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡹⠀⠀⢰⠃⠒⣄⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡠⠄⢳⠀⠀⠸⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠇⠀⠀⡌⠀⠀⠉⠛⢭⣉⠓⠒⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⡆⠀⠀⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⢀⣀⡀⠀⠐⣯⡵⠢⣀⠀⢸⠞⠁⠙⢆⡤⣀⠀⠀⢀⣇⣀⡀⢀⣤⡄⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠐⡟⠛⠀⠀⠉⠁⠢⣀⡠⢾⠀⠀⠙⠢⢌⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⢃⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠂⡤⠍⣲⣤⣀⠀⢀⠃⠀⠀⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡹⡀⠀⠘⡄⠚⠁⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠈⠁⡜⠀⠀⡸⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡳⠀⠀⠘⣄⠀⠀⠀⣠⠀⠀⠉⠒⠦⠤⠴⣧⠤⠖⠊⠁⠀⠀⢦⣀⠀⢀⡜⠀⠀⢰⡳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡡⡀⠀⠈⠢⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠈⡠⠊⠀⠀⡰⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⢆⠀⠀⠈⠢⢄⠀⠀⣰⠀⠀⠀⠀⡇⠀⠀⠀⠈⠃⢀⠤⠊⠀⠀⢀⢞⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⡦⡀⠀⠀⠉⠒⠤⠤⠄⣀⣀⣁⡀⠤⠤⠖⠊⠁⠀⠀⡠⣶⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⢗⡤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣖⠯⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠹⠷⠖⢲⣶⣶⠒⠲⠽⠑⠊⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠨⢙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢄⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠀⢘⢻⣄⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢚⣳⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣀⣤⣬⣽⡇⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛⠀⢄⣀⣴⣮⠾⠛⠉⢩⣿⣿⡃⠔⡠⠖⡄⠀⠤⠀⠀⠄⠀⠀⠀⡿⠀⣴⡒⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢁⣼⡿⠋⠀⠀⠀⠀⠀⣿⣿⣷⡺⠀⠓⠘⠤⠐⢀⡐⡆⠀⢀⣾⠁⠀⠻⢉⣒⡶⠚⠁
⠀⠀⠀⠀⠀⢸⣿⣄⣠⣄⣤⣰⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠛⢻⣧⠆⣀⣠⠐⠅⠔⣰⡎⢿⣽⡇⠀⠀⡀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠛⢁⢀⡀⠀⠰⣾⡆⣿⣷⣦⢦⡀⠹⣿⣿⣿⣿⣿⣾⠿⠟⠋⠉⢀⠤⠄⣤⣕⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡞⢻⣾⣧⣴⣾⡿⣳⠃⢹⣽⢃⢳⠀⠙⠛⠛⠋⠁⠀⠀⣀⡀⢠⣜⣛⣋⣩⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣝⣛⣛⡫⠊⠈⣶⡿⢣⠇⡸⠀⠀⢀⡴⠶⢾⣛⢺⣯⣕⣒⣻⣟⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⡀⢆⣀⠓⠚⠛⠋⠀⠀⠀⠠⠲⠿⠋⠐⠖⠋⠉⠉⠥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠁⠆⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⣿⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣸⠀⠀⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⡀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⣾⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡄⠘⣿⣿⣿
⠀⠀⠀⢰⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⡄⢿⣿⣿
⠀⠀⠀⣼⣿⣇⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢾⠇⠅⠸⣿⣿
⠀⠀⠀⢹⣿⡟⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣸⡄⠀⢸⣿⣿
⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿
⠀⠀⠀⠀⠸⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⡀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠹⢌⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣠⢞⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⣀⣀⣀⠀⠀⢀⣀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣩⣿⣿⣽⣿⣟⡉⡩⡟⣻⢶⡋⠉⣀⣤⢉⡟⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⡁⠀⠘⣿⣇⠁⠀⣹⣿⣶⣧⣷⠾⣟⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠀⠣⠇⠀⠠⡈⢿⣷⡋⡏⠁⠸⣿⡆⣤⣾⢇⣳⣬⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⣠⡀⠑⢄⡻⣶⣇⠀⠀⡻⣷⡞⠛⠛⢻⣯⡀⠈⠋⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⢷⣿⣿⠦⣶⣤⣿⣷⣼⣦⠀⠐⠻⣷⣄⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢠⣶⣦⡈⠷⣠⡜⠙⠿⣷⡈⠙⢿⣻⣿⣿⣇⡀⠂⢀⣽⣆⣘⣿⠆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⠃⠀⠀⣀⡢⡦⢘⣿⣿⣿⣦⡙⠿⣦⠀⠘⣾⠆⠹⣿⣍⣇⠢⡟⢶⣿⢿⣿⣟⣏
⠀⠀⠀⠀⠀⠀⡾⠍⠉⠀⠿⡠⣄⣱⡾⢿⣿⡿⠿⠿⠿⠿⠿⠿⣮⡓⢬⣉⣁⣶⣏⠸⣿⠇⣼⣿⣿⠛⠲⣷⡿
⠀⠀⠀⠀⠀⠸⣅⣈⡁⢀⣶⣾⡿⠋⠀⠀⠀⠀⢀⣀⢢⣀⣤⠀⡈⢯⠑⢶⡄⢈⣽⣿⡷⣾⠛⣿⢿⢦⣤⡟
⠀⠀⠀⠀⠀⠀⠙⢿⣷⡾⠿⠋⢀⣠⢤⣤⠐⠈⣻⣿⢏⡿⣿⡿⡞⢦⠣⡀⢱⡌⢻⡌⢳⣷⣞⣵⣋⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣹⣿⣿⣶⣾⣿⢟⢼⠁⢸⣿⢧⠈⡇⠈⠙⠿⣼⣇⠀⠊⠛⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣮⣟⣛⡫⠕⠋⠈⠾⡿⢯⠞⣠⠇⠀⣤⡀⠈⠙⠻⢶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠖⠶⣄⣀⡀⠀⠀⠀⠀⠀⠓⠚⠛⠋⠁⠶⠤⠀⠡⠀⢉⣀⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢷⠻⢮⡻⣗⠶⡤⣀⣠⣔⠠⣈⠱⠀⠀⠀⠘⠀⠰⣾⣆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⠿⣕⣳⣿⣦⣍⡓⠾⣷⣥⣄⣀⠀⣀⠀⠙⠷⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣽⡧⠶⠖⠛⠛⠛⠋⠀⠀⠀⠉


⣶⣶⣶⢶⠶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⡶⣶⣶⣶⡶⡶⢶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦⠘⠛⠁⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⢿⣿⠏⠀⠻⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣷⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⣄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣼⣶⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠃⠀⠀⠁
⡖⠒⢲⠖⠶⠶⠶⠲⡖⠲⠶⠶⠶⠶⠖⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⡶⣶⣶⣶⠶⠶

⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡛⢶⣄
⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣆⢻⣧⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄
⢶⣆⣚⠛⠛⠻⠉⠙⢻⣿⣿⣿⡇⢸⣿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣷⡌⣷⡄
⠀⠀⢸⣿⣿⣿⣤⣾⣼⣿⣿⣿⡇⢸⣿⠀⢠⠋⠻⠿⠿⠿⠶⢶⣶⣶⣶⣤⣤⣤⣤⣀⣤⠚⢿⣿⣿⣿⣿⣿⣷⠸⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣧⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⡁⠀⢯⣀⣈⣿⣿⣿⢀⣿⣦⣤
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⣡⣿⠏⠀⡇⠀⢳⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⠀⠀⠀⠀⢷⠀⠘⣿⣿⣿⣿⡿⣸⡟⠉⠁
⠀⠀⠀⠀⠈⠛⠿⣿⣯⣵⣾⠟⠁⠀⠀⢸⠀⠀⢳⡄⠀⠀⠀⣾⣿⡟⠁⡤⠙⡄⠀⠀⠀⠸⠁⠀⣿⣿⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠻⣄⡀⠀⣿⠋⠁⠀⠁⠈⣧⠀⠀⢀⣇⡀⠀⡝⠻⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠱⣄⣘⡇⠀⣀⡀⢈⣧⣠⡔⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣀⡀⠉⠑⠲⣤⡙⠛⢷⣄⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠛⠛⠆⠀⠀⠸⣷⡀⠀⠙⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⡀⠀⣿⣿⣿⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡿⠟⠉⠀⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⢰⠂⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣠⣾⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠰⣤⣀⣀⡀⠀⢠⠛⠙⠒⠉⠉⠀⢸⡀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⡸⣇⠀⠀⠉⠗⠁⠀⠀⠀⠀⢀⡴⠻⣓⢀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡏⠉⠉⠳⢤⡀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠿⠿⠟⠉⠉⠉⠉⠉⠓⠶⠶⠊⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠊⠉⠉⣳⠤⢤⠶⠲⢦⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠂⠉⠉⠉⠁⠀⠀⠀⡠⠖⠩⠆⡔⠁⠰⠆⠀⠃⠰⠆⠙⢢⡀⣀⣀⣀⢀⡀⣀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢀⣠⠤⠄⠚⠥⠀⣀⡀⠀⡀⢀⣀⣀⠤⠐⠒⠊⠉⠁⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⡿⠟
⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⢀⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠈⠉⠉⠀⠐⡒⠒⠒⠒⠒⢍⠉⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣤⡮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⡰⠊⠉⠙⡎⠀⠈⠆⢺⠂
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠰⠀⠀⠀⢠⡁⠀⢀⣇⠼⠤⣀
⠀⠀⠀⠀⠀⢣⠀⠠⠀⠀⠸⣟⠁⠀⣠⠖⠋⠉⠓⠢⣄⣠⠆⠀⠀⠀⠀⠘⠄⡀⠤⠋⠉⠋⡛⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠹⡀⠀⡇⠀⠊⠙⠙⠂⠀⠀⠀⠀⠀⢠⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣠⠊⠉⠉⠛⠦⢄⣀⠤⠖⠋⢢⡀⠀⠀⠀⣿⡗⢄⠀⠀⠀⠀⠀⠀⠀⠈⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⠑⢄⠀⠀⡿⠳⠀⠉⠒⠤⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⠀⠀⠀⣷⣦⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⡠⠔⠒⠒⠮⠉⢦⠀⠀⠀⠀⠀⠀⣼⣒⣭⠀⣀⣠⠛⠻⣇⣉⠑⢶⠤⠤⢤⣤⣤⠤⣤⡒⠁
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢑⠄⠀⠀⠀⠀⠯⠭⠚⢦⡱⠈⠒⠒⠛⣬⡁⢸⡄⠀⠀⠀⠀⠘⢀⠇
⠀⠀⠀⡆⠀⢀⣶⣧⠀⠀⠀⠀⢨⡗⠀⠀⠀⡠⠔⠚⢦⠀⢻⠂⠄⠀⠀⢇⠸⡀⡇⠀⠀⢀⠤⢦⡼⠒⢄
⠀⠀⠀⢇⢼⡛⢫⠑⠀⠀⠀⢀⣸⠃⣠⠖⠋⠀⢀⣠⡼⢀⡼⠀⠀⠀⠀⠹⣀⠉⠧⠦⠶⡇⢄⣀⡹⠤⢎
⠀⠀⠀⣨⢾⣿⠁⠀⠀⠀⡠⢻⡀⠹⡀⠀⠀⢉⣿⠴⠒⠉⠀⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠆⠀⠻⠤⢤⡞
⠀⠀⢸⠯⢞⠝⠒⠀⠀⡞⠀⣔⠉⢁⠈⢑⡒⠚⠛⡄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢢⡐⠉⠓⠲⡗⢶⠊
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠱⡀⠈⠉⡇⣀⠞⡞⠑⠉⢽⠛⢳⠖⠻⠿⠿⠧⠐⠒⠋⠁⠈⣰⢦⠀⢸⠘⣀⢤⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣠⡺⢿⣤⣀⣀⢈⡏⠀⢀⣀⡀⣀⡀⠀⣠⣶⣿⣷⣡⣖⡎⠞⢡⢀⡄⠀⣸⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢩⠽⣤⠹⢷⣮⣭⣄⣀⣀⣸⠉⢉⠡⢞⠝⣹⣏⣈⠷⡷⣻⣰⠾⠉⠀⡴⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⢺⣶⡾⢷⣄⣭⠛⠀⠀⠀⠀⠀⠘⠊⠁⠀⢳⠛⠃⢼⡶⣃⠄⠀⡠⢊⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠘⢮⡛⠶⣾⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⣀⠄⠙⢉⡠⢊⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢧⠀⠀⠈⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡠⠜⡮⣵⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠛⣼⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠒

⠀⠀⢠⣾⣿⣿⣦⠀⠀⠀⠀⢠⣾⣿⣷⣆
⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧
⢀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧
⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀
⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡀
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⣠⣼⡿⠟
⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣀⣀⣀⠀⠀⠀⢀⡔⠁⠀⠀⢀⣾⠟⠁⠀⢀
⠀⠘⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠉⠀⢀⣈⣷⣄⡴⠋⠀⠀⢀⢔⠝⠁⠀⠀⡰⠋
⢠⠞⠻⠿⠿⠟⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡠⠒⠁⠀⠐⠋⠀⠀⠀⢐⠕⠁⠀⢀⡴⠋
⠸⡄⠀⠀⠀⠀⣠⠾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡜⠁⠀⠀⠀⠀⠀⠀⣠⠖⢁⣠⠴⠚⠁
⢀⡏⠉⠀⠈⢉⣀⣀⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠔⢋⡤⠚⠉
⢸⢀⠀⠀⠈⠀⠀⠉⠉⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⠿⠊⢁⡤⠚⠁
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁
⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠙⠻⠷⣖⣶⠖⠐⠖⣒⡮⠽⠚⠁
⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⡀⠀⠀⠈⠉⠉⠁⠉⠀⠉⠉⠉⠁⠉⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⣉
⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⠓⢯⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⣿⣼⡵⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⠻⠅⠉⢧⡀⠈⠳⢾⡯⣳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡂⠀⠀⠙⠻⣿⣽⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣆⠀⠠⠶⣦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⢀⣹⠿⠯⢛⣿⢦⡀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⢸⠀⠀⠀⠀⠀⡞⢮⣷⡛⢀⣿⠀⣬⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⣴⣯⣤⠴⠶⠌⣛⣒⣚⢶⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⣲⣙⣿⣦⣿⡧⣁⠄⠿⠂⣈⠑⠶⣤⣴⣤⠔⠒⠀⠀⠀⠀⠀⣀⣤⡾⠾⠽⡲⢦⣿⣃⡽⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠈⢿⠒⡛⢿⢿⣿⣿⣀⠉⠒⠵⣩⠄⠶⢸⠛⣌⢦⠀⣀⡠⣴⢶⠉⣿⡄⠘⠛⠀⢀⣹⠞⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡌⠁⢸⠌⣿⡿⣿⣷⣦⡀⠀⠉⠢⡟⢸⠈⢦⡹⣟⢻⡦⠚⠃⠠⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠛⠘⡇⢹⣿⢻⡝⢿⣿⣷⣄⠀⢠⠇⠀⠀⠑⢮⡳⣤⣶⠗⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⠘⠂⣿⠸⣿⣸⡇⠀⠙⣿⣋⣆⡞⢀⠀⠤⢤⣜⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⡀⡄⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡁⠦⢸⡀⢿⣿⠇⠀⠀⠘⢯⡞⡡⢖⠚⣿⡏⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠒⣿⣿⣅⣠⣄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢃⢠⠈⡇⢸⡟⠀⠀⠀⠀⠀⢿⠀⢙⡃⢹⡇⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⣿⣼⣿⡄⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡀⣀⢡⢼⡅⠀⠀⠀⠀⠀⠸⡄⠘⡛⠸⡇⢘⣿⠀⠀⠀⠤⠐⠒⢈⣥⡟⣰⣀⣾⠿⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣥⣁⢸⣾⡇⠀⠀⠀⠀⠀⠀⢇⠀⢋⠠⡇⢀⣿⡎⠀⠀⣀⣔⣮⡟⢽⣧⠟⠻⣅⣀⠘⢿⡆⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠤⠛⣏⢯⠉⠉⣿⣾⣾⡾⡁⠀⠀⠘⡀⠛⡵⡣⢿⠿⡇⣶⢽⣿⣭⣻⠿⠋⠀⠀⠀⠀⠉⠁⠊⣿⠀⠀⢸⣿
⠀⠀⠀⢸⠀⠀⢀⡴⢐⣁⣤⣾⡿⠾⣷⣾⣿⠇⠁⠃⠀⠀⠀⠀⣖⡴⣪⢀⡿⣀⣇⠘⣿⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⠆⠀⢸⣿
⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⢠⣿⠃⢉⡹⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠘⠻⢷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣦⣤⣀⡀⠀⠀⠀⠀⣀⣠⣤⡶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⠾⠟⠛⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠛⠛⠳⢦⡄
⠀⠀⠀⢀⣠⡶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣿⡆
⣼⠃⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠛⠛⠋⠉⠉⠀⠸⣇
⣿⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣤⠀⠀⠀⢸⡆⢸⡆
⢸⡇⠀⠀⠀⣀⣤⣤⠿⠀⠀⠀⠀⠹⠂⠀⠀⠈⠁⠘⣧
⠀⢷⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢿⡀⠀⠀⠀⢀⣠⡴⠶⠞⠛⠶⠶⣤⡀
⠀⠘⢷⡀⠈⠷⣤⣄⠀⠀⠀⠀⢀⣤⡴⠞⠛⢉⣿⠀⢸⡇⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠙⠢⣤⣤⣿⡀⠀⠀⠀⠀⠹⢦⣤⣤⠞⠃⠀⣿⠀⠀⣼⠃⠀⠀⢀⣤⠶⠶⢦⣄⠀⠀⠈⢿⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⡟⠀⠀⠀⣾⠁⠀⠀⠀⢻⡆⠀⠀⢸⡇
⠀⠀⣠⠀⠀⠀⠀⠀⣠⠿⠳⢦⣤⣤⣤⣤⡴⠾⢯⡀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⣀⡾⠁⠀⠀⢸⡇
⠀⣼⠋⠀⠀⠀⢠⡾⠁⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠙⣦⡀⠀⠹⣆⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⣠⡟⠀⠀⠀⣸
⢰⡏⠀⠀⠀⢠⡟⠀⠀⠀⣠⡾⠛⠉⠉⠉⠙⠷⣤⡀⠘⣧⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠉
⣼⠁⠀⠀⠀⢸⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⡀⠀⠀⠀⠈⢹⡟⠛⠛⠛⢻⡍
⣿⠀⠀⠀⠀⢸⡇⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⣨⣿⣄⣀⡀⠀⢸⡇⠀⠀⠀⠸⣇
⠉⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⡀⠀⠀⠀⢀⣴⠏⠀⠀⣴⠏⠁⠈⠉⠙⢷⣼⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⢦⣄⣠⡾⠁⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⣰⠿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠙⢷⣶⣤⣀⠀⠀⠀⢰⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡇⠈⠉⠛⠳⠶⣼⣧⣀⣤⠾⠃
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠶⠶⠶⢾⡃
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣤⣤⣤⣤⣤⣼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠤⠀⠀⠠⢄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠡⠤⠄⠡⣤⠤⠠⠀⠀⡤⠈⠁⠉⠂⠀⠀⠄⠰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠉⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠠⠄⡀⠀⠀⠀⡀⠀⡀⠸⠦⢴⠂⠻⣬⡇
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠐⠠⣀⣀⡠⠀⠰⡏⢆⡀⠀⢣⣈⡈⠡⠀⠈⠀⠀⠀⢠⠉⠰⡆⠸⡀⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡀⠀⢘⠈⠁⠀⢠⠀⠀⡇⠀⠀⠀⠈⠘⠇⠠⠀⠀⣦⠹⣏⠄⠤⡈⠤⠀⠀⠀⠀⠀⠀⠡⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣜⠀⣀⢀⡽⠁⠀⠃⢠⠀⠀⠀⠀⢸⡇⠚⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠙⣆⠙⠮⢈⡃⢢⣃⡶⠀⠀⠺⡠⠀⡜⢠⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⢇⣠⠼⠲⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠘⠙⠋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠞⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠤⣀⠀⠀⠀⠀⠀⢃⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠢⠀⢀⣔⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠂⠀⢀⣺⠟⠋⢉⠛⡞⣵⠀⠐⠢⣀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠀⢠⡄⠀⢀⠉⠉⠋⠉⠉⠀⠀⠀⠀⠈⠒⠂⡀⢠⠄⡟⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠨⢁⡃⠀⢸⠀⠀⠀⡀⠀⣀⠀⠀⡂⣀⠤⡖⡲⠒⢤⡸⠕⠒⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠀⠘⠀⠂⠒⠒⠚⠀⠀⠀⠈⠉⠉

⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣾⣿⣿⡇
⠀⠘⢿⣿⣿⡟⠻⣏⡉⠉⠉⣉⠉⠉⢉⣩⡟⠹⣿⣿⣿⠏⠁⠈⢻⣿⣿⣿⠙⣯⣉⠉⠉⣉⡉⠉⠉⣩⡿⠉⣿⣿⣿⡏⠁
⠀⠀⠈⠛⠛⠁⠀⠙⢿⣦⣸⣿⣆⣴⡿⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠈⢻⣷⣤⣿⣧⣤⣾⠋⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢟⠃⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣖⣮⣭⣭⣭⣒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠟⠛⠛⠛⠙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⣶⣩⡄⠀⢹⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢿⣿⡟⠉⢙⠛⠋⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣁⣀⣭⠆⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢈⣙⣡⡤⠤⠄⠀⢀⢮⡄⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠛⢡⣤⣶⠀⠀⢠⣃⣾⣧⣦⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⢀⣴⣿⣿⡿⠛⠛⠛⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⣹⣿⣿⣿⣿⣿⡛⠁⢀⣀⣀⣀⠀⠘⢿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠈⠀⠀⠀⠀⢀⣽⡿⠛⠋⠉⠙⠿⠂⠈⢇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠀⠀⠀⢈⠙⠣⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣜⢦
⠀⠀⠀⠀⠀⣀⣀⣤⡤⢤⣤⣄⣀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⣠⠏⠀⠀⠀⠙⠦⣤⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠱
⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠈⠙⠻⣦⡀⠀⣰⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠓⢐⢃⡒⠊⠖⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⣼⠋⠀⠀⣠⣴⠶⠶⠶⢦⣄⠀⠀⠀⠛⠢⢿⣦⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢥⡀⠡⢀⡤⠀⠀⠀⠀⠀⠀⠀⣾⡇
⣼⠃⠀⠀⣼⠋⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠁⠀⣻⣿⠃⢠⡄⢀⣀⠀⠀⠀⠀⠀⠈⢳⣵⡾⠃⣀⠚⠁⠀⠀⠀⠀⠀⠀⣰⣿⢃
⡏⠀⠀⠘⠇⠀⠀⡏⢉⡇⠀⠀⣿⡇⠀⠀⠀⣾⣇⡛⠀⣘⣿⣿⣿⣧⡀⠀⣀⣠⣴⣿⣯⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⣴⣿⠏⡼
⣧⠀⠀⠀⠘⣄⠀⠈⠉⠀⠀⣰⣿⠃⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠃
⠸⣆⠀⠀⠀⠈⠑⣂⣤⣴⡾⠟⠁⠀⠀⠀⣼⣿⣿⣧⣰⣿⣿⣿⣿⣿⣿⠃⠀⠀⡇⣿⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠
⠀⠹⣦⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣟⠿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣸⣾⡿⠃⠀⠉⠉⢻⡆⠀⠀⠀⠀⣸⡇⣸
⠀⠀⠈⠻⣷⣦⣤⣀⣀⣀⣠⣤⣾⠶⣿⠿⣿⣿⣿⡿⠧⠀⡀⠀⢀⣉⣉⠉⠉⠉⠉⠀⠀⠀⠀⡆⠀⢸⡇⠀⠀⠀⢠⡿⢡⠃
⠀⠀⠀⠀⠀⠉⠛⢟⡛⠛⠋⠉⠀⠀⣨⠟⠢⣝⡳⢤⣀⠀⠀⠉⠋⠉⠁⠘⠋⢶⣀⠀⢀⣀⠀⠇⠀⠀⢠⠀⠀⣰⡟⠡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠋⠁⠀⠀⠈⡙⠶⣬⡙⠲⢦⣤⣄⣀⠴⠲⠦⠉⠑⠛⠀⠀⠀⠀⣠⡟⢠⣾⣟⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠙⠳⢶⣄⠉⠛⠿⠷⠶⢶⣤⣤⣤⣶⣶⣿⡿⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠓⢶⣶⣶⣾⣿⣶⣤⣙⠲⠦⣤⣀⠀⠀⠀⠀⣰⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠉⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣿⡿⣷⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣠⡟⢸⢹⠇⠘⢿⣿⣿⣷⣄⠉⠻⢿⣿⡆⠀⠀⠀⠙⣿⣿⡿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠳⡝⢧⣿⣸⠀⠀⠈⠻⣷⡈⠻⣷⣦⡀⠉⠛⠀⠀⠀⠀⠈⢿⣧⠘⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣷⢄⡹⡌⣿⣿⠀⠀⠀⠀⠈⠛⠀⠀⠉⠛⠦⠀⠀⠀⠀⠀⠀⠘⣿⠀⢻⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢻⣿⡞⣷⣹⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⠷⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣼⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡆⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣾⠁⢠⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣦⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠄⠀⠀⠘⣆⠀⠀⠀⣾⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡓⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣀⡀⠀⠠⣄⠢⡌⢢⣈⣢⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠛⠓⠋⠋⠈⠁⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣆⣀⣀⣀⣀⣀⣀⣀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⡿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠘⣿⣧⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣤⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⠀⠋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡌⠀⢢⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣛⠹⠉⠉⠉⠉⢹⡙⠋⠉⠉⠉⢹⡛⠏⠉⠁⣴⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⢃⠀⠀⠀⡖⠉⢱⠀⠀⠀⢠⠉⠙⡆⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠀⠀⠀⠀⠙⠖⠎⠀⠀⠀⠀⠖⠾⠁⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣶⡄⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣾⣿⡇⠀⠉⠻⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣸⣿⢿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠀⠈⠛⠛⠛⠉⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣇⣠⡀⣀⣄⡀⢀⣀⡴⠂⠐⠶⠆⠀⠀⢀⣤⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⣿⣿⢿⣿⣿⣷⣦⣤⣶⣶⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣀⣿⣿⣧⣤⡍⠉⠛⠿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠻⣿⣿⣦⠀⠀⠈⢿⣽⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠉⣿⣿⠀⠀⠙⢿⣿⡇⠀⠀⠀⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠙⣿⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⣿⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢿⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢸⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢾⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⣠⣴⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⣴⡿⠃⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣉⣠⣤⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⢻⣿⡟⢀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢸⣿⡇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣷⡄⠉⢁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠹⣷⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢻⣆⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣤⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠘⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⠸⡇⢀⣿⠇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠁⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢹⡇⠀⣼⡟⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⣿⣧⡈⠁⠸⣿⠀⣰⠋⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢹⡇⠀⢸⣿⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⡇⠀⣸⠛⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⢰⡎⠀⢰⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠘⠋⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠘⠁⠀⡀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⣠⠀⢀⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⣿⠀⢸⣧⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠹⠿⣷⣬⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⣴⡎⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠻⢿⠟⠻⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣀⣰⣾⡿⣷⣶⡌⠻⠆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⢉⣿⣿⣿⣿⣿⣤⣼⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠛⠋⣹⣿⣿⣿⣿⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⢠⣄⠀⢀⠼⠛⠁⠀⠀⢠⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠾⢿⡶⠂⠀⠀⠀⠀⢠⣼⠛⣿⣿⠉⣻⣿⠏⢸⣿⡏
⠀⠀⠀⠀⣶⣾⠇⠀⠀⠀⠀⣾⣶⠘⠃⣀⡟⠃⠀⣿⣿⠀⢈⣿⡷
⠀⠀⢠⣤⣿⠇⠀⠀⠀⣀⣈⣉⠛⠀⠀⢹⡇⠀⠠⠚⣿⠀⠘⠛⢁⡀
⠀⠀⠙⢿⠏⠀⠀⠀⠀⠙⠿⠛⠀⢰⣆⠀⠁⠀⣶⣤⡟⠁⠰⣷⣾⠃
⠀⢀⣿⠇⠀⠀⠀⠀⣴⣶⠆⠀⢀⠘⠿⠟⠀⠀⣽⣭⠀⠀⠀⣈⡉
⠀⠀⠀⠀⠀⠀⠀⣀⠙⠛⠀⠀⠸⣧⡄⠀⠀⠀⣿⣿⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢿⣶⠀⠀⠀⣰⡿⠃⠀⠀⢰⣿⠋⠀⠀⢘⣿⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣇⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⣤⡈⠋⠀⠀⢠⡿⠟⠀⠀⠀⢠⣿⠃⠀⠀⠀⣿⡏
⠀⠀⠀⠀⣼⡿⠃⠀⠀⢀⣾⡟⠀⠀⠀⠀⢨⡏⠀⠀⠀⢰⣟
⠀⠀⠀⠘⠋⠁⠀⠀⠀⣀⠉⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠃⠀⠀⠀⠀⠘⠋
⣤⣶⣤⣤⣤⣦⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠉⠀⠀⠉⠁⠈⠁⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣀⣠⣀⣀⣀⣀⣀⣤⣄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⣿⣿⣛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⢤⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣄
⠀⠀⠀⣴⣮⣿⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢿⣯⣶⡄
⠀⠀⢸⠉⡇⢸⠀⣿⣿⣿⣿⣷⡀⠤⠤⠤⠐⠒⠒⠒⠒⠒⠒⠠⠤⠤⠄⣰⣿⣿⣿⣿⠇⢸⠀⡏⢹
⠀⠀⣸⠀⡇⢸⠀⢸⣿⡟⣧⣴⣶⠶⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠷⢶⣶⣤⡟⣿⣿⠀⢸⠀⡇⢸⡀
⠀⠀⢳⡆⢡⠀⡆⠈⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⡏⠀⡆⢠⠁⣾⠃
⠀⠀⠀⠱⡈⣆⣰⣠⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣰⣀⡎⡰⠁
⠀⠀⠀⠀⠹⣿⣿⣷⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠋⠁

⠀⠀⠀⠐⡆⠱⠀⠀⡶⢡⠐⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣠⠄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢏⠁⠈⣾⢁⡞⠀⣷⡟⠀⠀⠀⠀⠀⢀⣠⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⠔⠂
⠀⠀⠀⠀⠸⣧⣀⣾⠋⠀⣠⣿⠇⠀⠀⣠⣶⣾⣯⣭⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀
⠀⠀⠀⠀⣀⣿⣿⠿⠟⠛⠋⠁⠀⠀⠀⠉⠀⠀⠉⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢠⠆
⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣀⣀⣀⣠⣴⣿⠴⠶⠒⠃⠈
⠀⠀⠀⠀⠀⠀⠀⣠⠾⢶⡚⠁⡠⠂⠀⠀⠀⠀⣞⡀⠀⠀⠀⠈⠛⠋⠉⠉⠉⠀⡀⢠
⠀⠀⠀⢀⣠⣴⡿⠏⠀⠀⠈⠙⠒⠊⠀⣀⣠⡴⣭⣄⣀⠀⠀⠀⠀⠀⠀⣴⠀⢰⣧⡷⠂
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡶⠟⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣤⣺⠿⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠟⣋⣤⡶⢶⣶⡶⠶⠤⠞⠒⠒
⠀⠀⣀⣀⣾⡿⠛⢿⣍⣛⣡⠤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣾⣿⣶⣋⣉⣀⠀⠀⠉⠙⣖
⠀⠠⠟⠛⠉⠀⠀⠀⠙⠛⠒⠒⢀⣀⠔⣀⣤⠚⠀⠀⠀⠀⠀⠈⠀⠀⠈⠩⢳⠦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣶⡯⠕⠊⠉⠀⠉⠀⠒⠯⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⣀⠀⠀⠀⠀⣴⣶⠿⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢯⣤⠴⠚⠀⠀⠀⡀
⠀⠀⣿⣷⣦⣴⡾⠟⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣄⣠⣤⣾⣿⣇⠀⠈⠀⢀⣠⠶⠖⣁
⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣀⣀⣀⠾⠀⠀⠀⠙⠟⠉⠀⠀⠈⠛⠓⠶⣿⣁⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⠄⣴⠋⠉⣀⠈⠉⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠈⠉⠁
⠀⠀⠀⢀⣠⣴⣾⣿⢷⣦⣤⣤⣴⣿⡶⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡶⠃
⠀⢤⣶⣿⡿⠋⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣶⣶⣦⣤⣼⠷⠶⠤
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉

⠀⠀⠀⢠⡆
⠀⠀⠀⣸⡃
⠀⠀⢀⣿⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⡜⠓⠀⢐⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⢠⢿⣀⠉⣽⣻⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠻⣍⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣋⣾⣷⣦⣄⡀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠠⠶⢾⣼⣟⣻⢿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠗⠀⣀⢴⣾⡿⠟⠁⠀⠀⠀⠀⣤
⠂⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⢡⣾⡿⠛⠃⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠉⠀⠚⠁⠺⠽⢦⠄⠀⠀⡀
⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠇⣠⡀⠀⠀⠀⠈⠳⣄
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠀⢻⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣶⣶⣶⠶⠤⠴⠐⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠉⠛⠋⠛⠻⢷⣦⣀
⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣀⡀
⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠄⠀⠀⡀⠀⠈⠙
⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠉⠂⠀⣤⡖⠀⡀⡀
⠀⢀⠀⠀⠲⠄⠉⠀⠀⠄⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐