Housekeeping Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⢹⣿⣿⣿⣿⡿⠟⠛⠻⠿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠁⠀⠀⢀⣀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣨⡾⠋⢀⣤⣶⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⡾⠋⣠⣾⣿⡿⠿⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⡼⠋⢸⣿⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⣶⣯⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣟⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠛⣿⣿⣿⡇
⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⢀⣿⣿⣿⠃
⠀⠀⠀⡸⠤⠤⠄⠠⠤⠤⠤⠀⠀⠀⠚⠃⠐⠒⠂⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡿
⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠢⠦⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠂⠀⠄⡀⢸⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡟
⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⠋⠉⠉⢤
⠈⡠⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠁⢠⠄⠢⡀⠑⡀
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠀⠒⠀⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⡅⠀⢀⡇⠀⡁
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⡱⠆⠄⠀⠙⡄
⠘⠳⠶⠤⠤⠤⠤⠤⣤⣤⣤⣶⣶⣾⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣴⣶⣿⠘⣿⣄⡨⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠋⠉⠉⠉⠉⠉⠉⠉⠉⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠁⡀⠔⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣙⣏⣨⣉⣹⣍⣋⣽⣋⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⢠⣠⣴
⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⢰⣤⣀⠀⢿⣿⣿⣧⡆⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⢰⡿⠋⠉⠳⣄⠀⠸⣿⣿⡆⠸⣿⣿⣿⡇⢰⣿⣿⠉⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⠳⣄⠘⣿⣿⣦⣹⣿⣿⡇⣾⣿⡇⢠⢇⣀⣶⣿⡑⢆⠀⢀⠴⠃⠀⣠⠞⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣋⠀⠀⠙⢷⣽⣿⣿⣿⡿⠿⠿⠿⠿⢷⣾⣿⡿⢿⣿⣿⠷⠖⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠛⠇⠈⠙⠲⢤⣠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠃⠀⠀⣠⠞⠁⠀⣠⡀
⠀⠀⠀⠀⠀⣾⣤⣬⣶⣤⣀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠠⠞⠁⠀⣀⣼⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⠚⠉⠁⠀⠉⠙
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣿⣉⡩⠟⠋⠉
⠀⠀⠀⠀⠸⠉⠉⡟⠛⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣤⡄⠀⣴⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣏⠀⠙⣦⠉⠛⠿⣿⠿⠃
⠀⠀⠀⠀⠀⣀⠀⣀⠀⣤⣠⠞⠃⣀⣼⣿⣿⣷⣶⣶⣶⣶⡶⠿⣿⡁⠈⠻⣝⠖⢃⣤⡀
⠀⠀⠀⢠⡞⠉⠳⣄⣠⠞⠁⢠⢿⣿⣿⣿⣿⣧⢸⣿⣿⢿⠀⣠⣿⠷⠶⣤⣙⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠙⢦⡐⠿⠳⣄⠀⠈⢯⢿⣿⣿⡿⣟⠋⡟⢡⠋⢰⠏⣀⣀⣀⣸⣿⣷⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⢈⠇⠀⠈⢯⡯⠗⠋⠁⠀⠱⣼⣄⠈⠶⣿⣿⣿⣿⣿⠛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡷⠄⠀⠋⠉⠉⠉⠙

⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠤⠀⣒⠠⠄⠈
⠀⠀⠀⠀⠀⠂⠀⠀⡁⠖⠲⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠒⠀⠃⠀⠤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠶⠶⢿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠀⠒⠀⢀⠀⠐⠤⣤⣤⣀⣀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣩⣴⡖⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢙⡻⣿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣮⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠁⠀⠀⠐⠀⠀⠆⠈⡿⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣾⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠒⠀⠠⠤⠀⢀⢀⣩⡄⢸⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠐⢀⠞⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡇⣾⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⡾⣴⣾⣿⣿⣣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⣿⣿⣿⢺⡇
⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⢧⣿⡄⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⢸⡇⢀
⠀⠀⠀⠀⠀⣼⢸⣿⣿⣿⢸⣿⡇⠀⠀⢀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡗⣸⣿⣻⡟⠯⠿⠇⠈⡈
⠀⠀⠀⠀⠀⢹⢸⣿⣿⣿⣧⣭⡅⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠉⠉⠀⠀⢠⠀⠀⠁⠠
⠀⠀⠀⠀⢀⠸⣞⠿⠿⠿⠿⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠴⠀⢀⡀
⠀⠀⠀⠀⠈⠉⢉⠛⠛⠛⠛⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⡄⠀⠀⣀⡀⢀⣄⠀⠈⣀⣠⣤⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠿⠘⠃⠉⠉⠀⠀⠀⠀⠀⠉⠛⠛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠂⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠠⠃⠐⠁⡌⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣠⡤⠀⡀⠀⠀⠀⠀⠀⠀⠰⠃⠈⠀⣀⠄⢲⠁⠋⠀⠀⠀⠀⡠⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡃⠀⠁⠈⠁⠦⢀⡠⢠⠄⠲⠘⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⣀⠠⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠶⠦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⡀⠉⠁⠀⠀⠈⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⣀⠀⠐⡢⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡤⡄⠀⢧⡴⠓⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠋⠀⠀⠀⠀⣦⠀⢉⡆⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠀⠀⠀⠀⠙⠛⠯⡀⠀⠀⠀⠀⡠⡻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠴⡚⢧⣘⠄⢠⣿⣾⣶⡴⢺⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣠⣴⣆⣠⣾⣿⣿⣯⡋⣼⠀⠀⠀⢀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⢟⠞⢰⡀⠉⢲⡊⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡻⣿⠿⢿⡿⡣⢯⡖⢋⡙⠦⠴⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡜⠁⢀⣜⠝⢲⠉⠻⢦⣄⠀⣸⠃⠀⠘
⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⣀⡀⠀⠀⠈⢧⠘⣦⠴⡿⠃⠀⣸⠀⣠⣾⣿⣆⡏
⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⢻⡡⣴⠆⠀⡰⠁⣠⣽⡿⠃⠈
⠀⠀⠀⠀⢰⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡠⡪⠛⠁⢀⡜⠁⠀⣻⣿⠃
⠀⠀⠀⠀⣿⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢀⣔⠕⠁⠀⢠⣾⣤⣄⣿⣿⣿
⠀⠀⠀⠀⢻⠀⢸⡇⠀⠀⠀⢀⡠⠄⣠⡻⠋⠀⠀⠰⠿⠿⡟⠛⣿⠛⠻⡄
⠀⠀⠀⠀⠈⠢⠈⠃⠀⠀⡐⠁⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢃⠀⡿⢂⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡀⡇⠘⡄⢸
⠀⠀⠀⠀⠀⠤⠒⡉⠀⡈⠐⢀⠄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢃⢸
⠀⠀⠀⠀⠀⠈⠲⢖⢃⢄⣢⡵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠁⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⢸⣿⣿⠁⠀⠈⣿⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⢸⣿⣿⣷⣦⣾⣿⣿⡇⢀⣴⣾⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣤⣤⣴⡎
⠀⠀⠀⠀⠀⠀⣠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⣴⣒⣦⣄
⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣯⣷⣿⣿⣿⣾⣽⡲⢤⡀
⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⣤⣤⣤⣤⠠
⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢹⣿⣿⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠁⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀
⠀⠀⠀⠀⠀⠀⣀⣿⣇⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⢈
⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢠⠀⠸⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠎⢑⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⡀⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡄⠀⠀⠀⠀⡂⠈⢦⣶⣶⣶⡆⠀⠹⣿⣿⣿⢻⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣶⠀⠀⠀⡀⠀⢸⣿⣿⣿⡇⠀⠀⠈⠉⠀⠘⠃⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⣄⢼⣿⣿⠁

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢰
⠀⠀⠀⠀⠀⠀⡀⣀⠀⣠⠀⣀⣀⠀⠀⣀⠀⣠⠃⠇
⠀⠀⠀⣴⠽⠉⠉⠁⠀⠀⠂⠀⠠⠹⠝⠿⠄⡏⣜⠀⠀⢠⠤⣄⠸⣭⡍⡭⠛
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠐⠀⠙⣀⡖⢙⠀⢸⢚⣩⠇⢁
⠀⠀⠀⠀⠀⢘⣆⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠎⠁⠈⡇⠀⠸⠇⠀⠈⠐⡲⠀⠶⠒⠒⠲⠖⢊⠚⠂⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠦⣀⠀⠈⠉⠈⡩⠀⠄⠀⠀⠰⠀⠀⠀⠀⠆⠰⠀⠐⠀⠀⠀⠀⠀⠙⢧⡉⠒⠦⠤⣁⡐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⢜⣉⡀⠏⠓⠨⢅⡒⠠⢄⡈⠀⠀⠀⢀⣀⣘⣀⣀⣀⣀⣘⣐⣀⡀⠶⠊⢭⠉⠫⠏⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠂⠀⠀⠀⡁⠊⠐⢢⠭⠭⠤⠤⠄⠀⠀⠐⠒⠒⠒⠒⠒⠲⠒⠓⠛⠀⠆⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠉⠉⠉⠉⠀⠉⠙⠒⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠂⠀⠀⠀⠀⠀⠀⠐⠚⠂⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢸⢸⠀⠀⠀⠀⠀⠀⠀⡇
⢀⣀⣠⠤⠤⠄⠒⠤⠧⠀⡀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠤⠤⠤⠤⠴⠖⠒⠒⢒⣚⣻⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠳⡄⢀⠀⠠⠀⠀⠀⠠⠐⣃⠄⠀⠀⠀⢀⢀⢸⠒⢂⣀⠈⠉⠛⠉⠉⠉⠁⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢠⠀⠀⠀⠀⢒⠘⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⠖⠀⠠⡀⢑⠊⠈⠉⠁⠐⠄⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡄⡄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣤⡧⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢀⡀⢸⢸⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠂⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣠⣸⣀⣀⢀⡈⢀⡈⠠⠤⠤⠀⠴⠊⠛⢾⡉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠤⣈⠙⠖⠒⠒⠀⠚⠋⠛⠩⣉⣁⣀⡀⠠⠤⠤⠤⠤⠄⠒⠆⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠁⠓⠒⠘⡇⠀⠠⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠎⠉⣿⣿⣤⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡼⣿⡿⠿⢋⣵⣾⣿⣿⣿⣿⣿⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣴⣿⣿⡿⢿⣿⣿⣟⠛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠈⠉⣉⣤⣶⣿⣿⣿⣿⡄⡌⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠀⠀⢈⡿⠹⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⣿⡇⠀⢀⡾⢡⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⢀⡼⢡⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣿⣴⣿⢡⣿⣿⣿⣿⠟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠟⡿⢡⣿⣿⣿⠟⠁⠀⢹⡿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⣸⢣⣿⣿⡿⠋⠀⠀⠀⠈⣷⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢹⣸⣿⣿⠁⠀⠀⠀⠀⠀⢹⣇⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⡞⣿⣿⣆⠀⠀⢠⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⣿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣌⢿⣿⡆⢀⠘⣿⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣛⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⢻⣿⡄⠤⣿⡇⣿⣿⣿⣿
⠀⣀⡤⠴⠒⠛⠋⢉⣿⠉⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠛⣿⡄⢻⡇⣿⣿⣿⣿
⠀⠹⠿⢿⣿⣶⡾⠿⠛⠁⠀⠀⠀⠀⣀⣠⣴⠾⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⣶⣿⣯⣿⣿⣿⢸⣧⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣠⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠃⣴⣿⣿⣿⣿⣿⡿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢠⣿⣿⣿⠿⠋⠁⠀⠈⢻⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣄⠀⠀⠀⢀⣠⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡅⠀⠀⠸⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⣀⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⢦⣤⡀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡋⠀⠀⠈⢻⣦⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣷⣦⡀⢸⣿⣿⣦⣼⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠘⢿⣿⣿⠿⠿⠿⠿⠛⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣿⣿⣿⡿⠃⠀⠀⠈⢿⣿⣿⣿⣇⠀⠀⢠⡤⠘⠛⠋⢀⣾⡿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⢠⣿⡃⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠇⠀⠀⠰⢿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⣀⣀⣀⣾⣿⡇⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⣷⣶⣶⡆⣀⣀⡀⠀⠀⠘⠿⠋⠉⠉⠉⠀⠹⣿⣄⠀⠀⠀⠀⢿⣀
⠀⠀⠀⠀⠀⠀⠈⣿⣾⠃⠀⠀⠀⠀⣰⣿⡿⠁⠀⠈⠉⠁⠲⠶⠶⣶⣶⠶⠖⠀⠀⠀⠉⠻⠿⢦⣄⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⢠⠙⣿⣿⡶⢤⣶⣦⣤⣤⣀⠀⠈⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠘⣿⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⡿⠀⣿⠃⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠸⢧⡀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠘⣿
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⢰⣿⣿⣿⣿⣿⠀⢀⣠⣤⣼⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⢸⣷
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠛⠁⠀⣿⠀⠀⣿⣿⣿⣿⠏⠈⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣿⡄⠀⢻⣄
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⣿⠀⢰⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⢻⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿⡇⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠈⢿⣧⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣇⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⡄⠀⣿⡇⠀⠀⠸⣿⣄⠀⠹⣶⡤⣀⡀
⠀⠀⠀⠀⠈⠰⠶⣦⣀⡀⢨⡿⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⡾⠃⠀⠀⠸⡁⠀⣿⡇⠀⠀⠀⢻⣿⠀⠀⠀⠀⠉⣻⣦⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣉⠁⠛⠁⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⢹⡇⠀⠀⠀⠘⣿⣇⠀⢶⣿⠶⠟⠉⠀⠈⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣶⣶⣤⣤⣤⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣇⠀⠀⠀⠀⠘⣿⣷⠀⠀⢀⣀⣀⣀⣀⣀⣤⣄⠈⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⡏⣠⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣿⣀⠀⠀⠀⣀⣀⣠⣤⣴⡿⢻
⠀⠀⠀⠀⠀⣼⣿⡏⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠻⠿⠟⠛⠛⠉⠀⠀⣿
⠀⠀⠀⠀⣼⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠠⣦⡤⠀⠀⢠⣿
⠀⠀⠀⢰⣿⡟⢠⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⢠⣿⡿⢁⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⣿⡿⢁⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⣿⠇
⠀⣰⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠻⠿⠀⠀⠀⠀⣸⡋
⠀⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣧⠀⠐⢶⣶⡶⠿⣯⡀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⣾⣷⣾⠿⠟⠁
⣤⣄⠀⠈⠙⠿⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠈⠛⠻⢷⣦⣤⣀⡀⠉⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠉⠛⠛⢷⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣛⣛⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⣉⣿⣿⣿⣿⣿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣶⣶⣶⣤⣉⣉⣉⡉⠛⠛⠛⠛⠛⠛⠿⠛⠁⠀⠀⣀⣀⣀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⢿⡿⢶⣶⣶⣶⣶⣶⣶⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣿⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢀⣿⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠸⡇⢘⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠀⢸⣧⠈⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⢸⣿⠈⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣤⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠰⠆⠀⢤⠀⣀⡀⢀⣀⠀⠋⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢰⣿⣶⣿⣿⣌⠧⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠵⠄⢸⣿⠀⢀⣰⣾⣷⣤
⠀⠀⠀⠂⢀⣴⣶⣴⣿⣿⣿⣿⣧⣭⣄⣠⣤⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⠀⠈⠈⣿⣿⣟⠙⠷⠄⠀⢀⠀⠀⠁⠂⢸⣿⠀⠚⠻⣿⠟⠛
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⢘⣿⡇⠀⢰⣿⣿⣿⠀⠀⠀⠀⣾⣷⣾⣦⡄⢸⣿⠀⠄⠀⣿⠀⢠
⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠠⠀⡇⠀⣿⣟⣿⣿⣿⣿⣿⡏⠀⢸⣿⠃⠀⣾⣿⣿⣿⣷⣤⠀⠘⢿⡿⠿⡟⠁⢸⡇⠀⠀⠀⣿⠀⢸
⠀⠀⠋⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⠃⠀⠀⢿⣿⣿⣿⣿⣿⣿⢃⠀⠸⠿⠀⢠⣿⣿⣿⣿⢼⣿⡇⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠄⠀⣾⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⢴⣦⠀⢸⣿⣿⣿⣿⣾⣿⡃⢀⡀⠀⠀⢀⣾⣿⣿⣿⣆⠀⠛
⠀⠀⠀⠀⠁⠀⠀⠙⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠀⣿⠀⠀⢤⠀⢿⣿⣿⣿⠿⠏⠀⠀⠈⠁⠀⠘⠛⠛⠛⠛⠛⠛⠀⣿⣽⠀⠀⠛⠛⠛⠛⠛⠛
⠀⠀⠊⠃⠀⠆⡀⢀⣾⣶⣀⢰⣿⣿⣆⣀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⠠⠀⢠⣤⣄⠀⠀⠀⠰⠀⠀⠀⣿⣿⠀⠀⠓⠒⠂⠀⡀⢀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⢠⠀⣿⠀⠐⠀⠄⠀⠀⣠⣿⠃⠀⢀⡏⢠⣾⣿⣿⢶⣄⡀⠀⠴⠄⠀⢹⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⠀⠀⣠⣶⣿⣿⣿⣧⠀⠀⢈⠀⢸⣿⣿⣿⣄⣹⣷⠀⢀⣠⡤⢸⡇⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⣼⣿⣿⣿⣿⣿⣿⠇⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⠈⠻⣿⡇⢸⡇⠐⢀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⠁⠀⢰⣿⠀⠀⢉⠛⠿⣿⣿⠟⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⠀⠀⠈⠀⣸⣧⠀⠌⠀⠈⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠰⣿⣿⢿⡿⣿⣿⣿⣿⣿⡿⠿⠟⠠⡎⠀⠀⢸⣿⠀⠀⠈⢈⠐⠀⠉⠀⠀⠀⠛⠋⠀⠉⠉⠉⠉⠉⠁⠀⠀⠄⠀⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⢿⣿⠀⠲⠆
⠀⠀⠀⠈⠁⠀⠆⠘⠿⢿⣿⡏⠀⠀⠀⠀⣀⣀⣀⣸⣇⣀⣀⣀⠀⠀⠀⠃⣠⢾⣿⣿⣿⣷⣦⠀⠀⣘⣿⡀⠀⠈⠀⠀⢹⡿⠀⠠⠀⠀⠸⠿⠿⣿⣿⠇
⠀⠐⣀⠀⣀⣀⠀⠀⠀⢸⣿⡇⠀⠘⠁⢠⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⡀⣼⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⡇⠀⠀⠀⠀⢸⡏⠀⠘⠀⠀⠀⠀⢤⠠⡄
⠀⠀⠉⢀⣿⠿⣿⡆⠀⠘⠿⠟⠀⠀⠀⠈⠉⠈⠈⠉⠉⠋⠉⠁⢁⠘⡀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⣿⣿⣆⠀⣀⠀⢸⣧⠀⠀⢀⡀⢠⡹⣎⣧⣿⠀⡄
⠀⢀⡀⠀⠀⢼⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⢁⠀⠅⠹⣿⣿⣿⣿⣿⣿⠟⠀⠹⠿⠿⠿⠃⠀⠀⢰⣿⣿⡆⠘⢃⡁⠈⢻⣿⣿⣿⣧⣿
⠀⠐⠀⢀⣤⣾⣿⡇⠀⠄⠀⠀⣠⣤⣶⣾⣿⣿⣿⣶⡟⠀⠀⠀⠅⠳⠀⠀⣀⠉⠛⠛⠉⠁⣤⣤⣤⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠙⢿⣿⣿⣿⣾⣧
⠀⠀⢰⣿⣿⣿⣿⡇⠀⠁⣠⣾⣿⣽⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠘⠀⠠⡦⢠⣶⣶⣦⣄⠀⠙⠿⠏⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⢤⢶⣾⣿⣿⣿⣿⡏
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠘⢿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠟⠙⠛⠛⠷⠶⠀⠤⠘⣿⣿⣿⣿⣧⠀⠀⣀⣤⣴⣶⣶⡄⢰⢿⣿⣿⣿⣿⣿⡆⣠⢶⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠈⠉⠉⠉⠉⠀⣄⠀⠀⠙⢿⣿⣿⠿⠿⠿⢻⡇⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⡿⠟⠀⠀⣿⣿⡿⠟⠋⠁⠸⠟⣿⣿⣿⣿⣿⠇⠰⠟⡿⡿⠋⠈⠁
⠀⢠⠋⠂⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠁⢀⢀⡄⠀⠀⢠⣶⣶⠂⠂⠀⢿⣿⣿⣿⣿⡟⠀⠀⣀⠀⠉⠁⠄⠀⢁⡀⠀⠀⣉⠉⠁⠀⠀⢀
⠀⠀⢀⣴⣶⣾⣿⣿⣾⣷⣶⣦⣄⠀⠀⠙⠂⠀⢰⣿⣿⡧⠉⠀⠀⠀⠀⠈⠉⠛⠋⠀⠀⣼⣿⠗⠀⣀⣰⣿⣿⡿⠀⠴⠋⠀⠠⠀⢈⠋⠀⠀⠀⠀⣠⣶⣷⠄
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠘⠛⠛⠁⠀⠀⢀⣼⣯⣭⣴⣶⣶⠆⠀⠉⠁⣶⣿⣻⣿⠏⠁⠀⠀⣀⣥⣤⣤⣤⣴⣤⣅⠀⠀⢸⣿⣿⣧
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠶⠶⣶⣶⣦⣄⠀⠀⠀⠻⠿⠿⠟⠛⠛⠀⠀⣀⠘⢿⣿⠿⠋⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠸⠟⠿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠋⠀⠀⠀⠀⢀⣸⣿⠇⠀⠀⠔⢈⣤⣄⠀⢸⣯⣭⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡇
⠀⡀⠀⠀⣠⡀⠀⠤⠀⣠⣶⠿⠿⠟⠛⠛⠛⠛⠛⠛⠉⠁⠀⡄⠀⣀⣸⣿⣿⠀⢸⣿⣿⣿⣿⠀⢰⣿⣿⣿⠀⠀⠀⠘⠿⠿⠿⠟⠛⠛⠁⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⢀⠈⠀⠀⠀⠀⠿⣷⣤⣤⣤⣤⣶⠶⠶⠶⠶⠟⠛⠛⠛⠛⠛⢻⣿⣿⠀⣸⣿⣿⣿⣿⠀⠸⣿⣿⣿⠄⠀⣀⣀⣀⣀⣀⣀⣤⣶⣶⣶⠀⠘⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠐⠔⠈⠛⠋⠀⠈⠛⠛⠛⠋⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⢈⣄⠡⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠅⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠗⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠲⠶⠒⢂
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠂⠀⠀⠃⠉⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠉⢙⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⢰⡀⠀⡀⠠⠤⡤⡦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⡀⠀⠀⠀⢀⡄⠀⠀⠀⡀⠠⡀⠀⠐⠂⠐⠒
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⠉⠃⢀⠀⠈⠀⠒⠁⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃
⠀⠀⢀⠀⠠⢀⡁⢈⣀⣈⡀⠀⠀⡄⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡄⡀⠐⠶⣤⢤⡄⠀⠀⠀⠀⠀⠀⠂
⠀⢀⠖⠂⠆⠀⠀⠀⠀⠀⠀⠀⡀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠈⢃⠇⠻⠛⠁⠒⠲⠀⠀⠀⣀⣀⢠⣤⠀⠀⠠⡐
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⣿⣿⣿⣿⣿⡇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣀⢀⣀⣀⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣀⠈⢳⡀⠀⣿⡿⠟⠛⠛⠁⠀⠐⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠺⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⢀⠀⣞⣾⣿⣧⠀⣧⠀⣿⠁⠀⠀⠀⠀⠀⠀⣽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡴⢓⡀⣀⣀
⠀⠀⠀⠀⢸⠀⣿⣿⣿⡇⠀⣿⠀⣿⢠⣴⣶⣶⣿⠀⠘⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡘⣷⡄
⠀⠀⠀⠀⠀⠀⢳⣻⣿⠏⢀⡏⠀⣿⡞⣿⠿⢛⣿⣤⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⣿⣇⢿⣯⣇⣓⣿⣿⣿⡇⠀⠀⠀⠀⣼⠆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⡘⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠛⠉⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡘⣿⡄⠀⠀⠀⠀⠀⠀⣹⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⡈⣷⠀⠀⠀⠀⠀⠀⢸⠾⠇
⠀⠀⠀⠀⠠⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠀⠀⠀⣈⣉⣭⣽⣿⣍⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠐⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⢟⣛⠀⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣯⢸⡇⠀⠀⢸⡇⠀⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⣷⣶⣶⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠸⠀⠀⠀⠀⡁⠀⠀⠀⠀⠂
⠀⠀⢀⣠⣤⣤⣴⣶⣶⣶⣿⣧⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡀⠀⠚⢩⡀⠀⠀⠀⠀⠀⠉⠀⠂⠄⡀
⠀⠀⠘⡿⠿⠿⠛⣿⣿⣿⠿⠿⠿⠛⠇⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡆⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠘⠛⠿⣿⡿⡿⣿⣿⠻⠗⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸⠿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠄⠀⢠⡶⠖⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣾⣶⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⠀⠀⠀⠀⠀⠐⠀⣀⣀⣀⡀⣀⣀⣀⡀⠀⠀⠀⠀⣶⡄⠀⠂⠀⠀⣿⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠚⠛⠚⠿⢿⣿⣿⣷⣀⡀⠂⠀⠀⠠⠀⡀⠀⢀⣴⠿⠿⠛⠃⠀⠀⠀⠨⠀⠰⠭⣭⣭⣭⠭⠵⠂⠸⣆⠀⠀⢿⡄⠀⡀⠀⠀⢿⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠓⠀⠀⠀⠀⠀⠀⢀⠀⢬⣙⣛⣛⣛⣛⣫⡅⠀⢻⣦⠀⠀⠙⠋⠀⠀⠀⠀⠻⣷⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡲⠶⠿⠷⠶⣆⠙⠛⠛⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢻⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⡇⠀⢀⣶⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⠇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⠇⠀⠀⠀⠘⣿⣿⡿⠿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡏⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢉⡿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇⠀⠀⠀⣾⠁⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠙⠻⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡦⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⣴⣶⣶⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⣀⣷⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁
⠀⢸⡟⢻⣟⣻⡆⠀⠀⠀⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠘⠃⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛

⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢰⣷⣶⣶⣿⡆⠀⠀⠀⠀⠀⢰⢻⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⢀⣰⣾⣷⡄⠀⣠⡀
⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣾⣿⢀⣀⣃⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⠇⢞⠁⣹
⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠛⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠙⠁

⠀⡀⠀⠀⢠⠄⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⣿⣧⣠⣟⣴⣇⠀⠀⠀⠀⠀⠀⢸⣦⣽⣥⣾⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣷⠀⠀⠀⠀⠀⠀⠀⣴⠛⠙⠻⣆
⠈⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⢛⣛⡛⠃⠀⠀⠀⠀⠀⠀⣷⡀⠀⢀⡿⠘⡿
⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠃

⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢸⣿⢛⡻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢓⡒⢲
⠀⢸⣧⣸⣋⣿⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⣿
⠀⠘⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠓

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⢀
⠀⣀⣠⣀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠻⡿⠃⣼⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢿⠀⠀⠀⠀⠀⠀⣠⡀⠰⣶⣶⡶
⠂⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠄⢭⣯⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣯⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⠾⠉⠀⠈⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡻⠃⢠⣿⣤⣤⣶⣶⣶⣦⣤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣿⣟⣿⣿⣾⡁⢠⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣯⣾⣿⣿⡿⢟⡙⠃⣾⡟⠉⠉⠀⠀⠈⠙⠻⠿⣿⣿⣿⡄⣀⣀⣀⣀⣠⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⣾⢟⢸⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣧⢷⢸⡇⣼⢠⡇⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢸⣿⢿⢸⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠺⣿⣿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⢴⡎⠛⠿⠷⠻⣷⢸⠀⡌⠻⢿⣷⡄⠀⠀⠀⠀⣴⣿⠿⠋⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠠⣞⣶⣴⠋⠓⠦⣄⠀⢸⠀⡇⠀⣀⡀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢨⠗⠋⠀⠀⠀⠈⠉⢸⣠⡗⣿⣿⣷⣤⣀⣀⣀⣠⣾⣿⣿⣷⣶⣤⣛⠿
⠀⠀⠀⠀⠀⣀⡟⠛⢛⠛⠛⠛⠛⢃⣼⣿⣧⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠸⣅⣷⠴⠛⠁⠀⠀⢀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠘⣿⣸⣿⢹⣿⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠸⠄⣴⡄⣤⡄⣤⡻⣯⢿⣞⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠈⠑⠙⠺⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢀⡀⣀⣀⢀⡀⢀⡀⣀⠀⡀⢀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠑⠚⠓⠂⠚⠛⠃⠘⠑⠛⠀⠋⠘⠁⠛⠀⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢄⠀⠀⠀⠊⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠁⠈⠄
⠀⠀⠀⠀⠀⠰⢤⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠘⠧⣄⠀⢀⣀⡀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⡀⠀⠀⠀⣶⣄
⠀⠀⠀⢰⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡀⠀⠈⡿⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣾⣿⣿⣿⡿
⠀⠀⠀⢸⠰⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⢳⠀⠐⢳⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⠀⢃⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡀⠀⣁⠠⠤⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⡿⢿⡉⠁
⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠀⠀⠀⠀⠙
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⢜⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠔⢠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣷⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢺⡶⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢿⡦
⠀⠀⠀⢨⠀⢦⣄⡀⠀⠀⣠⠖⠁⠀⠀⠁⠀⠀⠀⠀⠀⡇⠐⣤⣀⠀⠀⢀⣤⠞⠡⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣤⣀⠀⠀⢀⣴⣾⣿⠀⠈
⠀⠀⠀⢸⠀⠀⠈⠙⠓⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠉⢻⠶⠏⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣶⣿⣿⣿⣿
⠀⠀⠀⣾⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠐⠛⠋⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢛⠛⠉⠋⠉⠉
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠉⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢀⠀⡀
⠀⡿⠿⠷⡿⠿⠸⣤⣤⣤⡆⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠇⢤⣤⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⢿⠿⠇⣦⣤⣤⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣴⠶⠶⣤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠁⠀⠀⠀⠀⠀⠀⢰⣦⡦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢠⣤⡄⢀⡀⠀⠀⠀⡆⠀⠈⠁⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠼⠗⢻⣧⣤⣶⠟⠀⠀⠀⠀⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠋⠀⠀⠀⢀⣀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠛⠛⠳⠦⣀⠀⢻⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣴⡆⠀⡀⠀⡀⠀⠀⠻⢷⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢁⢀⠙⠃⠀⠊⠀⠀⠀⠀⠀⠙⢷⣍⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠿⣿⡿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠙⠇⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣍⣳⡌⡿⣿⣿⣿⣷⣶⣤⣤⡀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣍⣿⣿⠃⡘⠀⠀⣴⡄⣤⣾⡆⠀⠀⠀⣯⡍⠉⣭⡷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡻⠿⣤⣸⣿⣿⣿⣿⣿⣿⣿⠏⢠⠀⠀⠀⡁⠀⠈⠁⣷⣦⣶⣶⡟⣿⢻⣟⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣎⣻⣿⣿⣿⣿⣿⠀⠀⠰⣃⠀⢀⢺⣷⣀⣀⣀⣿⣿⣿⣷⢿⣯⣉⣿⣿⡟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣤⣤⣿⠟⠋⢭⣭⣽⣿⣯⣧⣬⣥⣯⣭⣥⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣾⡿⢿⡟⠙⠋⣠⣤⠿⠟⠋⠉⠀⠀⠀⠀⠠⠲⣤⡾⠛⠉⠉⠉⠙⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣿⣿⡿⣩⢐⣯⡁⠀⠀⠁⠀⠀⠀⣀⣤⡶⠛⠻⠶⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⢈⣽⣿⣿⣿⣿⣤⣏⠻⣽⣷⠀⢊⣤⣤⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢀⣴⡟⢂⣼⣿⣿⠘⣿⣿⣿⣿⣀⣿⣿⣿⣼⣿⠟⠛⠙⢿⣄
⠀⠀⠀⠀⠀⠀⣟⠀⣦⢾⣿⣿⠏⠀⠙⢿⣿⣿⣿⣶⣿⣿⣿⣿⠀⠀⠀⠀⠛⣷⡀
⠀⠀⠀⠀⠀⠀⢹⣶⣿⣾⣿⣿⠀⠀⠀⠀⠀⠉⠻⠛⠛⠋⠁⠁⠀⠀⠀⠀⠄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⣸⠟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡄
⠀⠀⠀⠀⠀⢠⣿⠃⣨⣿⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⢀⡄⠀⠀⢾⣿⡀
⠀⠀⠀⠀⢀⣼⢫⠄⢻⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⠀⠈⠉⣷
⠀⠀⠀⠀⢨⣏⠂⠰⣿⡷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⣸⡿⠀⣸⡏⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠀⠀⢸⣿⡀
⠀⠀⠀⠀⣾⠃⠀⣿⣷⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠇⠀⠀⣠⣿⡇
⠀⠀⠀⢠⣿⣧⠰⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠄⠀⠀⣻⢄⠁⠀⢸⣿⠃⡀
⠀⠀⠀⢸⣏⡟⢠⣿⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢻⣼⠀⠀⠈⡏⠠⣇
⠀⠀⠀⣾⣿⠃⣸⡿⢸⡇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣵⠀⣿⡆
⠀⠀⢠⣿⣼⣷⣿⢳⣿⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠋⠀⢹⣧
⠀⠀⢸⡏⠙⠉⣿⡈⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣧⠀⠀⠀⣿⠀⠠⣿
⠀⠀⣾⠁⠁⢘⣿⠇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡇⠀⢸⡇
⠀⠀⣿⠂⠠⠈⣿⠀⣾⣿⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢸⣷⠀⠘⡇
⠀⠀⣿⠀⠘⣿⡏⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢻⡀⠀⣷
⠀⠀⣿⠀⢰⣿⡇⢰⠟⣿⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣿⣄⠀⣷
⠀⢰⡿⠀⠐⢿⣧⣾⠀⣿⠂⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⢨⣿⠀⡷
⠀⢸⡇⢀⠐⣿⠉⣿⠇⣸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡀⠀⠀⠸⠇⠈⠀⠀⠈⣿⠀⡇
⠀⣾⠇⡈⠀⢿⠀⡝⠿⣿⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⡇⠀⡇
⠀⢻⡞⠃⠣⣼⠀⠙⢳⣿⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⢠⡇⢸⡇
⠀⣿⣧⣤⣷⣿⠇⣶⡀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠁⢸⡇
⠀⡏⢠⡟⠁⣿⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⡃⠀⠀⠀⣿⠀⠀⠁
⠘⠳⠬⠿⠆⣹⡃⢾⣿⣿⣷⣤⣤⣆⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢰⣿⣷⣾⡷
⠀⠀⢰⣶⣿⣿⣧⣄⣉⡙⣿⣟⣓⣺⣿⣿⣿⣟⣻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⡇
⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠷⣶⣭⣷⠆

⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠶⣦⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⡿⠛⢻⡟⠛⠛⠛⢿⠟⠛⠛⠻⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠰⣿⣿⠀⢸⠶⢿⡇⠀⡽⠶⣿⠀⢈⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣷⣤⣼⣧⣤⣤⣤⣿⣦⣤⣤⣴⣿⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⢻⣿⡏⢻⣿⡟⢹⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⡇⢸⣿⡇⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣿⡇⢸⣿⠇⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⡇⢸⣿⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⡇⢸⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣾⣷⣾⣷⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠉⣩⣉⣹⡏⠉⣹⣿⣯⠉⢩⣉⣹⣿⠋⠉⠹⣿⡏⠉⠙⢯⠉⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡀⠚⠉⣹⡇⠀⠛⢻⡿⠀⠐⠋⢹⡏⠀⡄⠀⢻⡇⠀⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⣺⠄⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⣀⠘⠈⠁⠀⡀⠀⠀⢸⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡠⠤⠴⠧⠄⣀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠞⡡⠔⠀⠘⣟⠀⢢⣍⠺⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣠⠔⢀⣁⢀⣉⣛⡑⠠⢀⠀⠐⡑⡼⠁⡷⢌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠒⠲⠀⠀⠐⠡⡊⠀⠀⠀⡄⡁⠉⠑⠢⣄⠀⢬⡃⢠⡧⠜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠒⠀⠀⠃⠀⠀⠀⠀⠀⢡⠡⠀⠀⠀⠀⠙⣦⡑⢏⠈⠈⠀⠀⠖⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢃⢣⡀⠄⠂⡜⠁⡟⢢⠑⡀⢀⡙⢢⠔⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣓⣋⣀⣐⡂⠤⣄⡀⠑⡑⢄⠜⢀⡜⠀⠀⢱⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠊⠉⠈⢆⠀⠉⠁⠂⢍⡓⠾⡛⢤⡾⣀⠀⠀⠀⡇⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠈⢌⣦⣠⡴⠒⠚⠳⢶⡈⠺⡏⠢⠍⣁⡉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⡄⠀⢀⡼⠓⠢⢍⡓⢤⡀⠀⢹⡑⢌⠢⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣘⡾⠖⠋⠀⠀⠀⠀⠈⠲⣝⢦⣸⠇⠀⠣⡘⡄⢀⡆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡋⠁⣀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢦⢹⡀⠀⠀⠐⠘⡄⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⢾⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠀⡀⠐⡡⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠙⢦⡀⠀⣠⠞⠁⠀⠀⡰⠃⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠙⠺⣯⢋⠄⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣳⠴⠋⠀⠐⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⢀⣠⣤⣤⣤⣀⠀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⡴⠞⠉⠀⠀⠀⠀⠉⠙⠶⣄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣇⣴⠋⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⢷⣀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣽⠁⠀⠀⠀⠀⠀⠀⣠⠞⢳⡄⠀⠀⠀⢻⠁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣿⠀⠀⠀⢀⣀⣤⠞⠉⠀⠀⠙⠦⣄⡀⢸⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡟⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣧⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣀⣀⣹⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣀⡀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁⠀⡯⠉⠓⠦⣤⣠⣤⠶⠋⢹⡇⠀⠉⠉⢙⡋⣰⠖⠒⠛⠓⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡀⠀⠀⠀⠀⠀⠀⠀⣨⣧⡀⣠⠞⠋⠙⠿⠀⠀⠀⠀⠀⣿⠶⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⡞⠻⣇⠈⠙⠶⣄⡀⠀⣠⡴⠛⠁⢠⡟⣻⠗⠦⣄⠀⠀⠀⠀⠀⠀⠁⣠⠴⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⣷⠀⠀⠹⣆⠈⠳⠦⣤⡤⠿⠻⠥⣤⠤⠖⠋⢠⠏⠀⠀⢸⠷⡄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢹⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣼⠀⢹⣄⣀⣠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⡟⠀⠀⣯⠉⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢰⡇⠀⠀⢹⢶⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⢸⡇⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⢸⡇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠒⠒⠒⠒⠒⠒⠚⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠚⠚⠛⠚⠒⠒⠛⠛⠛⠚⠒⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⡿⡿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡦⣜⢷⣶⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣿⣭⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠚⠉⠁⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠖⠚⠉⢁⣰⡏⢳⡴⢦⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⡀⠸⡄⠀⠀⠀⠀⣯⠈⣇⠈⡇⠘⡗⠓⡆⠘⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⠇⢹⢀⢷⠀⠀⠀⠀⢸⡀⢹⠀⢻⠀⢿⠀⢷⠀⠹⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⠀⢸⠟⠈⣇⠀⣠⢤⡈⡇⠘⠇⠘⠃⠸⠆⢸⡄⠀⢻⡀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡏⠀⠘⡇⠀⠘⣆⢳⡀⠹⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠁⠀⠀⠹⡄⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⠴⠿⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢷⠀⠙⢦⣀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⡴⠚⠸⠶⠖⠒⠈⢹⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣟⠙⠋⠙⠋⠙⠋⠙
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⣏⠀⢀⠀⠄⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠶⠓⠒⠒⠊⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠃⡀⡄⠀⠀⠀⠀⠀⠈⠓⠒⢦⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠀⠈⠀⢄⠃⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠁⢠⠡⠙⢆⠀⠀⠘⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠗⠀⠀⠀⠀⠹⡖⠼⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠀⢀⠀⠓⠒⡤⠂⠈⠀⠀⠹⢶⠼⡦⡴⠀⠀⠠⢂⠆
⠀⠀⠀⠀⠀⠀⠀⢦⠀⢰⡀⣠⢀⠀⢤⠀⣠⠀⠁⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡰⣂⣽⢦⡀⠀⣡⣶⡄⡀⢀⣠⣌⠀⠉⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠁⣃⣠⡈⠉⠁⠀⡐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡁⠀⠂⠂⣀⠙⠋⠙⣀⠈⠛⠋⠀⠠⠃⢦⡦⠤⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⡀⡁⠀⢀⢀⡄⠈⢀⡄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢠⠂⠀⠈⢻⣷⣠⠀⠀⢀⡀⠀⠀⡄⠐⡌⢢⡃
⠀⠀⠀⠀⠀⠀⠘⠀⢀⢤⣳⠥⠬⠦⠥⠴⣫⢖⡄⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⢄⡣⡧⠬⠴⠌⣴⡥⣶⡀⠀⡸⠁⠀⠈
⠀⠀⠀⡀⠀⠀⠀⢀⠮⣗⠯⠭⠭⠭⠭⠽⠉⡷⢖⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠫⡳⢻⠭⠭⢽⣿⣿⡟⢔⡻⠄⠈
⠀⠀⠀⠀⠆⠀⠀⠶⡽⠃⣏⣉⣉⣋⣉⣙⠛⢭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠳⠝⠀⣬⣭⣭⣭⣭⣭⣇⠈⢪⠽
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⠉⢉⣩⢉⣤⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⠀⠓⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠤⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⡄⠂⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠰⠟⠀⠨⠅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠰⠠⠳⠜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠅⠽⠈⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠉⠛⢶⣤⣀⣀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣿⣶⣦⣤⣤⣤⣤⣤⣤⣼⣷⡄⠀⠀⠹⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣾⣯⣤⣤⣤⣤⣤⣴⣶⣾⣧⣤⣤⣤⣤⣤⣤⣤⣄
⢀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⢻⣆⠀⠀⠀⣾⠃⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣄
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⠶⠶⣶⣶⣶⠶⠶⠶⠶⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⣠⣄⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⣾⠋⠙⣷⡀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠃⠀⠀⠘⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣿⡀⠀⠀⢀⡿⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠈⠛⠷⠞⠛⠁⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣤⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠙⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠟
⣶⣶⡶⠶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠉⠉⠉⠉⠰⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠉⠁⠒⠢⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣀⣠⠶⠛⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠁⠀⠉⠙⠁⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⡀⠀⠀⣀⡠⠦⡤⠀⠀⠀⠀⠀⠀⠀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⠉⠀⢀⠀⠀⠀⠀⠀⢀⠃⠀⠉⢢⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡴⠒⠒⠀⢀⡞⠀⠐⠒⣤⠲⠋⠀⠀⠀⠀⠑⣌⠱⡀
⣦⣀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠉⠀⠀⠀⣿⠀⠀⠀⠰⠀⠀⠀⠘⣧⠉⡄
⠀⠀⡀⢃⠀⠀⠀⡄⠉⠀⠈⠁⡄⢀⠆⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡟⠀⠀⠀⠀⡄⠀⠀⠀⠀⢇⡸
⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⡀⠠⢹⡎⠀⣀⣤⠘⠀⠀⢀⣀⣀⣴⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣄⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢡⠋⡀
⡆⢸⡀⠀⠀⠀⣀⡠⠃⠀⠋⠠⢼⣇⠚⠉⠀⠀⠀⢂⣀⣀⣀⡀⠉⠉⠙⠂⢤⠀⠀⠀⠀⠀⠘⠁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠑⠂⢠⡄⠐⠊⠉⠀⠐⠒⠒⠒⠂⠻⣧⠄⠀⠀⢀⠚⠉⠉⠉⠉⠉⠀⠐⢶⠞⠀⠀⠀⠀⠀⠀⢰⠀⠀⢛⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀
⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠶⠶⠾⣤⣤⣀⢀⠀⣀⣀⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⡰⢂⡌⠀⠀⠂⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠗⠀⡏⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠆⠀⣀⣈⡉⠉⠀⠲⠶⠤⠾⠋⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⢤⡴⠊⠀⠀⢀⠀⠀⠀⡈⠉⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⠿⠷⠶⣶⣶⣶⡶⠶⠶⠿⠿⠗⠀⠉⠙⠛⠛⠉⠉⠉⠁⠀⠀⠀⢠⠁⠀⠀⢰⡄⣀⡀⠀⠤⠀⠁⣀⡤⠞⠁⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⠒⠂⠀⠀⠒⠒⠒⠀⠠⠤⠤⠤⠀⠀⠀⠠⠴⠀⠀⠀⣀⠎⠀⠀⠀⣀⡠⠄⠒⠁⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠛⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⡖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠛⠉⠙⠳⡶⠶⠶⠦⢤⣀⡀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠈⠛⡶⠶⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⡆⠀⡌⠀⠀⠀⠀⠀⠀⢀⠘⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢁⡴⠀⠀⠀⠀⠀⠀⠀⠂⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠠⠈⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠃⠀⠀⠀⠀⠀⠈⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢧⠀⠀⠀⠀⠙⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠧⠀⠀⠀⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢸⠀⢁⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠈⠀⠀⣷⡀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⡆⢃⡀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠁⠈⢃⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⠀⢤⠀⣆⠀⠀⡄⠀⠀⠘⠀⠀⣀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠲⠷⠶⠆⣤⣤⡀⠉⣄⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠆⠉⠀⠀⠀⠀⣀⠰⠆⠀⠁⠀⠀⠀⠙⠋⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠈⠉⠁⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠌⠉⠉⠉⠪⢢⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠌⠀⠀⠀⠀⠀⠡⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠸⢁⠀⡠⡄⠀⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠂⣠⠀⢤⠲⡏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠂⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠸⠖⠒⠒⠒⠒⠒⠒⠶⠞⠀⠀⠀⠈⣥⡛⡟⡛
⠀⠀⠀⠀⠀⠀⠰⡟⢢⠤⢄⠀⠛⢤⡀⣀⣀⣀⣀⡠⠂⠀⢸⠁⠀⠀⠈⡇⡗⠲⠶⢶⢠⣤⣤⣄⣀⣀⣤⣶⠘⠓⠒⠒⠒⣤⠾⠿⣏⡛
⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⠜⠁⠂⣤⡀⢁⡅⠀⢠⡀⠀⢀⠸⠶⠶⠶⠶⠇⡇⠀⠀⣾⡌⣿⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢠⠊⠀⠀⠀⠐⣴⠁⠀⢀⣀⣀⡀⠀⠘⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⣈⠉⠈⣅
⠀⠀⠀⢠⠀⠀⠚⠁⠀⠀⠈⠉⠈⠙⠉⠂⠐⠂⠂⠀⠓⠐⠓⠀⠀⠀⠀⠘⠃⠛⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⡀⢸⡇⠀⠀⢹⠄
⠀⠀⠀⠀⢹⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⢻⡟⠛⡟⠛⣟⠛⣿⠛⣻⠛⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡄⢀⡖⠓⠒⠒⠚⢲
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⢸⠀⡇⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⢁⠀⡀⠀⠀⠈⢀⠀⣲
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⡇⠀⢸⠀⡇⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⡀⢸⠉⢰⡖⠘⣿⠒⣶⠂⡒⡎⢳⠀⢸⡇⠀⣧⠀⣸⣠⣧⣤⣷⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠄⠀⠠⠦⠀⠀⠠⠄⠀⠶⠇
⠀⠀⠀⠀⠀⡃⠨⠄⠸⡇⠀⠷⠀⢿⠀⠅⠇⠸⠀⠸⢧⡤⠿⠠⠼⡿⠓⠂⠛⢻⡿⠶⠤⣤⠠⠤⠤⠤⠤⠠⠄⠐⠒
⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡃⠀⠀⠀⠀⠀⠀⠀⣺⢴⣶⣶⠿⢶⣿⣷⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠐⠦⠾⠤⠤⠴⠴⣏⠀⠀⠉⠉⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠴⠤⠶⠖⠚⠋⠉

⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠉⠐⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⢤⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢤⣤⡄⢠⢀⣀⣠⢤⠤⠳⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⠁⠀⠀⠀⣁⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⡠⣉⠀⠀⠀⠀⠀⣈⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⡈⠀⠀⠀⡀⠀⠈
⡠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠹⠂⠀⠈⠁⠀⠰⠀⠀⠱⡐⡀
⠀⠀⠀⠂⠀⠔⠀⠀⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆⠀⠀⠑⡐
⡀⡄⠀⠀⠀⠂⠀⠄⣠⡇⡴⠂⠀⡠⠀⠀⠒⠖⠤⡀⠀⠀⠀⠀⠃⠀⠰⠀⠀⠀⡇⠀⠀⠀⠐⡡
⠁⠅⡄⠒⠉⠀⠒⠀⠈⢿⠤⠀⠀⡘⠉⠉⠁⠐⠒⠖⠃⠀⠀⠀⠈⡀⢐⣃⠀⢀⠀⠀⠀⠀⠀⠐⡀
⠀⢰⠀⠀⠀⠀⠀⠀⠀⡸⢐⠑⠚⠂⠠⠤⠤⠤⠤⠼⠒⠒⡆⠀⠀⠁⠠⠀⠀⢈⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠰⠉⠀⡀⠐⠀⠤⢤⠖⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢁⡠⠀⠀⢀⠀⢀⠛⣄⠀⠀⠀⠀⠀⡠⠞⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠛⣓⡲⠶⢖⡒⣚⢛⣃⠈⠉⠉⠉⠉⠉⠀⠀⠄⠀⠀⡥⠤⠀⠐⠈⡠⠖⠉⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠒⠈⠀⡀⠾⠤⠤⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡯⠥⠤⣄⣀⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡈⠀⠀⠈⠉⠐⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠂⠀⠀⠀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠎⠀⠀⠀⡠⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡏⠀⠀⠀⠈⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⡆⠀⡄⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⠀⡘⠀⠠⠀⠠⡀⣀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠂⠛⠰⠶⠄⠨⣄⡀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠈⡁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢸⣿⡿⠹⠁⠘⡿⢀⣿⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⡇⠈⠁⠀⠀⢀⡀⠀⠈⠻⢻⠏
⠀⠀⠀⠀⠀⠘⢟⠠⠛⠐⠀⠈⠁⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⢈⡁⣠⣔⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠁⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠀⠀⠀⠀⣰⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣊⠑⣾⠀⠀⣠⡎⠣⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣤⡠⠤⠲⢾⣟⣶⣤⣀⡀
⠀⠀⠀⢰⢁⠿⢷⠂⣯⢜⠒⠊⡠⠜⠶⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣷⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⣀⠭⠷⡶⠧⢤⣎⣀⣿⠀⢀⣀⣠⣾⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠙⠉⠀⠀⠀⠀⠙⣻⣿⣿⡇⠀⠀⠀⠀⢹⡿⣿⣿⣯⡟⠒⠠⢤⠴⣰⡆
⠀⠀⢸⣀⡠⠤⠿⠾⠤⠤⣀⣨⡿⣭⣿⣿⣷⣿⣴⢶⣲⡀⠀⠀⠀⢹⣿⠀⠰⠆⠘⢃⠀⠀⠀⠀⢉⡹⠁⠀⠀⠀⣀⠴⠋⠛⠛⠋⠉⠀⠒⢚⡶⣸⠃
⠀⠀⢹⡀⠈⠉⠁⠀⠉⢉⠔⢹⣟⣿⣿⣿⣿⣾⣧⣻⣿⠇⠀⠀⠀⠀⢋⡂⠰⠆⠀⠘⠁⠁⠀⣠⠎⠀⠀⣠⠴⣊⢃⡆⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋
⠀⠀⠀⢳⠶⠶⠦⠔⠊⠁⠀⡿⠽⠾⠳⠿⠷⠛⠉⠙⠁⠀⠀⢀⠤⠠⡴⢍⣄⣀⢀⣀⠡⢴⠊⠁⠀⡠⠋⠀⡴⣻⠞⠁
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⣕⡂⠉⠀⡎⠢⢄⠀⠀⡐⠣⡄⣠⠞⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠑⠂⠤⠤⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠻⣧⣠⠂⡤⢀⡀⢁⡶⠉⡦⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠈⠀⠀⠀⡨⠂⠈⠀⡎⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠈⠹⠵⣢⡠⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠖⠂⠀⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⣀⠐⠷⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⡁⠀⠀⠠⠊⢀⠄⠑⠢⣀
⠀⠀⠀⢄⣀⠀⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠈⠀⠀⠀⠌⠅⠉⠢⢄
⡀⠐⠀⠀⠈⠀⠤⠁⠊⠣⠲⢆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠉⠢⢄
⠑⡏⠉⠉⠉⠉⠉⠉⠉⠉⠩⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⠢⡀
⢠⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠈⢦⣀⣀⣀⠠⠀⠈⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠰⠁⠀⠀⠀⠀⠀⠔⠀⢀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠠⠊⢀⠠⢤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⣂⣀⣀⣀⣀⣀⣘⣀⣀⣉⠀⠀⠀⠀⠀⠀⡀⢊⠀⠔⠁⠁⠀⠠⠔⠀⠈⠀⣄
⠀⠀⠓⠤⠤⠤⠀⠠⠤⠄⠊⠀⠀⠔⠂⣋⠉⠀⠛⢛⠻⠛⠛⠛⠛⠛⠛⠋⠀⢀⠈⠬⠒⠀⠁⠀⠖⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡰⠀⠉⠁⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣥⢹⡃⠀⠀⠀⠀⣛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠘⠃⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣦⣈⣤⣾⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡗⠾⠿⠿⠟⠛⢋⣿⣿⡙⢢⡤⠤⠤⠤⢤⡞⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠰
⠀⡐⢦⣶⡶⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠘⢄⠀⠀⠀⡀⠠⢠⠎⠀⢸⣇⠀⠀⣀⣀⣤⣬⣽⣿⣡⡾⣹⠆⠀⠀⠛⠢⠀⠀⠀⠀⠀⡀⠀⡎⣌⢢⠇
⠀⠀⠘⠛⠉⠀⠀⣀⠀⠀⠀⠀⠀⠱⡀⠀⠈⢆⠉⠀⠀⣠⢷⣤⣀⢸⣿⣾⣿⣿⣿⣿⣿⣿⠿⠇⠉⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⡸⠁⡿⣏
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠱⡀⠀⡈⢆⠀⡰⠛⠳⣬⣍⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣷⡼⠀⠈⠉⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠱⡐⠂⠈⠋⠀⡙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠹⡸⣿⣷⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣙⣛⢻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠉⠁⠉⠛⠿⠟⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣤⡆⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠉⠁⠀⠀⣿⠛⣷⠄⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠒⣤⠀⢀⣾⠇⠀⠋⠀⣽⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠇⠠⠄⠀⠀⠹⣿⡎⣀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠃⠀⠀⠻⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠠⠄⠀⠉⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠘⠀⠰⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⣈⣹⠇⠀⠀⠀⠀⢀⣠⣾⣿⡏⣰⠟⠀⣰⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡙⢿⡄⠀⠀⠀⢴⠿⠋⠁⠀⠀⠋⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⣴⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⣿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⠀⢺⣿⣿⠁⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡏⠠⠀⣿⡏⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢠⡇⢰⣿⡟⠀⣼⡟⠁⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣁⣼⣿⠁⠐⠃⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⣿⣿⡟⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⡸⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣇⠀⠀⣿⣿⠇⣿⡏⠀⠀⠀⠀⠀⠀⣾⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠰⠏⠀⣼⡿⠃⢸⡿⠃⠀⠀⠀⠀⠀⠀⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢠⡄⠁⠀⣰⡿⠁⠀⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠙⠂⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⢰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣤⣦⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣠⡀⠀⣄⠀⣠⣿⣀⣤⣤⣴⣶⣤⣤⣤⡖⠈⠉⠁⠀⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠉⢠⠟⠛⠛⠉⠉⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⠀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠛⠓⠠⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢟⣤⢀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠒⠀⢀⡀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠻⠞⠣⢿⣆⢠⡆⢰⣴⣷⠀⠀⠈⠁⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⠸⠏⢻⣤⢀⣼⣧⡀⣴⠀⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠙⠟⠹⠇⠻⠟

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠿⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠲⡄
⠀⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⢪⠐⠇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠛⣠⠞
⠀⠀⠀⠀⠀⠀⡞⠓⢤⡜⠀⠀⠀⠸⡆⠀⠈⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢸⠃⠀⠘⠦⣄⡀⠀⠀⢳⠀⠀⢻⠀⡏⠀⠈⠉⠉⠙⠛⠲⣤⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠙⡆⠀⠈⣇⠀⠸⡆⢹⠀⣰⢲⠶⠤⣤⠟⠁⣀⡴⠋
⠀⠀⠀⠀⠀⡇⠀⠀⡠⠐⠢⠄⢸⣀⡠⠞⠉⠙⢧⡏⢠⡏⢛⡶⠞⢁⡴⠮⡉⠳⡀
⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⠀⠈⠄⢫⠀⢠⠖⠉⠁⠀⠈⠳⣏⡠⠴⠿⠓⠒⠛⠉⠉⠱⡄
⠀⠀⠀⠀⠀⠈⢧⠀⢇⠀⠀⠀⢸⠈⢆⡜⡔⠉⠉⠒⠢⠀⢸⠀⣀⠤⠴⠒⠒⠒⢄⠀⣧⡤⠖⠲⢤⣀
⠀⠀⠀⠀⠀⢀⣸⡄⠈⢦⠀⠀⣸⠀⠘⣤⠁⠀⠀⠀⢠⠀⢸⠀⠇⠀⠀⠀⠀⠀⠈⡄⢏⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠰⣅⠀⡱⠀⠀⠉⠉⠁⠀⠀⠈⡀⠀⠀⠀⠸⢀⡏⡀⢀⠀⠀⠀⠀⠀⠀⠇⢸⠀⡠⠖⢆⠐⡆⢳
⠀⠀⠀⠀⠀⢹⠂⠉⡑⠒⠒⠒⠒⠒⠠⠤⠼⠤⠤⠤⠧⠼⠽⠧⠼⠤⠤⠤⠤⠤⠤⠼⠤⠯⠔⢀⡸⠀⠇⠈⣇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠉⠉⠙⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⢸⠀⠸⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣧⠀⢸⢦⣠⠇
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠶⢻⡀⡸⠀⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡤⢀⡤⠐⠈⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠙⠀⠠⠀⡹⠫⠤⢀⠀⠀⠀⠲⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡀⠀⠸⣇⡠⠂⠠⠄⠒⢀⠈⠀⢶⢄⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡄⠀⣀⣒⡀⠀⠀⠠⠰⡁⠀⠂⠀⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠁⡀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠘⡤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠆⠋⠀⠄⣤⡄⠀⠀⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠜⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠊⠀⠠⠁⠤⠀⠀⠀⠀⡠⠀⠀⠀⠌⠁⠒⣣⠐⠀⠀⠈⣀⠨⠂⠁⠀⠀⠀⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠒⠂⠉⠓⠛⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⡀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠉⠀⠂⠠⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⡄⠰⠽⠂⡀⠰⠀⠀⠊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⡇⠀⠀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠐⡀⠀⠤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣆⢤⠃⠀⠀⠀⠀⢂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡱⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠞⠃⠀⠈⠉⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⣒⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠖⠉⠉⠈⡆
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠠⠀⠀⠀⠀⠀⠀⠀⠉⠁⡰⠁⠀⠀⠀⢀⠇
⠀⠀⢘⠏⠀⠀⡀⠂⢠⣠⢻⢃⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⣠⣤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⢤⠔⠃⠀⠀⠀⢀⡞⢠
⠑⠶⠊⠀⡤⡲⠣⠐⠅⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠂⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠈⢦⠃
⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉⡀⠀⠀⠀⠀⠐⠠⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠀⠀⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡏⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣠⣦⡈⠻⠁⠀⠀⠀⢀⡴⠁⢀⠞⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠠⠤⢄⡀⠀⠀⡇⢸⠀⠈⠟⠁⠀⠀⠀⠀⡠⠊⠀⡴⠃⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡇
⠀⠀⠀⠀⢀⠞⣡⢢⠖⠀⠀⢠⣀⠙⢆⠀⣷⣾⠀⡤⠤⠄⠐⠒⠀⠾⡅⣠⠊⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢠⠀⡇
⠀⠀⠀⠀⡜⠠⢣⠃⠀⠀⢀⠎⢀⡩⢾⠀⣿⣿⠀⢷⡏⠀⢀⢀⠈⠀⠟⠁⠀⠀⠀⠀⣴⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⡔⠁⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⡖⠒⠁⠀⠀⠐⠸⡄⡟⢻⠀⢻⣇⣄⡿⣿⣷⡄⢰⠀⠀⠀⠀⠀⠘⡇⢸⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⡠⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠸⡀⠀⠀⢱⠀⠀⠀⠀⣀⣰⠁⡇⢸⠀⣹⡇⢎⠀⢀⠝⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢀⠔⠁⢠⠞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⠎⢹⠒⠢⡇⠀⢀⣀⣀⡠⠃⠀⡇⢸⠀⣟⣣⠳⢴⠁⠀⠀⡄⠀⡀⠠⣿⠆⠀⡇⢸⠀⠀⠀⠀⡰⠋⢀⡔⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⢸⠀⢀⡇⣀⡧⠤⢄⣀⠀⠀⡇⢸⠀⣸⠃⠀⡎⠉⠉⠉⠀⠲⣿⠆⠈⠀⠀⡇⢸⠀⠀⣠⠎⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡸⢀⣞⢞⢱⠈⢱⢂⠇⠀⠈⠉⠒⠳⠼⡤⠃⠀⡜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⢸⠀⠐⠁⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠛⠒⠉⡜⡘⡜⠀⡇⡜⠀⠀⠤⣀⡀⠀⠀⠀⠀⡰⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠧⠤⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠤⠄
⠀⠀⠀⠀⢀⠃⠃⡇⢰⠀⡇⠀⠀⣠⢿⠈⠉⠒⠂⠔⠁
⠀⠀⠀⠀⢸⠀⠸⡇⢸⠸⢀⣀⠔⠁⠈⡆
⠀⠀⠀⠀⠸⡀⢸⢉⠊⠉⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢣⢸⢸⡇⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢿⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⡶⠟⢉⣉⣙⣲⡊⠙⠆
⠀⠀⠀⠀⠀⠀⢠⢞⣥⣾⣿⡿⡁⢀⣰⣾⣟⣿⣥⣖⣃⣀⡠⠤⢄⣠⣴⠎⣻⣶⢄
⠀⠀⠀⠀⠀⠀⠏⣾⣿⣿⣿⢿⠏⡻⢃⠋⠀⠀⠀⠀⠈⠁⢀⠄⣿⣿⣿⠆⠉⠋⣸⡆
⠀⢠⣶⣖⣼⡛⢸⣿⣿⣧⡗⣎⣾⡟⢧⡀⠀⠀⠀⠠⠤⠒⠁⠀⣿⢩⠫⠷⢀⣠⣿⠃
⠀⣼⣾⠟⠚⡛⢺⣿⣿⣾⠄⡼⣼⣉⢦⡀⠀⢀⠀⠀⠀⠀⠀⢀⣏⡰⣤⡀⣽⡿⠁
⣼⣷⣯⣄⠀⠀⢼⣿⣯⣌⠈⣍⠄⢼⡟⢭⠚⠻⡝⠙⠛⠲⡶⠛⠉⡍⠁⠀⢨⡇⠀⠀⠀⣀⣤⣤⣄⣐⡲⢤⣀
⢿⣿⣿⡜⠷⡀⠀⣽⠿⠷⠀⢻⣃⠈⣶⠨⠷⡂⡆⠀⠀⢀⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⣿⣿⣷⠟
⠈⠻⠿⠇⠀⠱⠀⢻⡖⢶⡆⠏⠀⢠⣷⠠⠿⠳⠣⠀⠀⢠⡀⠀⢸⠀⠀⠠⠄⣀⠀⠀⠀⢼⣿⡿⢭⡉⢹⢠⣿
⠀⠀⠀⠀⠀⠀⢑⣀⣗⠀⠳⠚⣟⣌⢡⣦⢰⠿⠹⡄⠀⠀⡇⠀⠈⠀⢀⣠⠄⠒⠉⠉⠁⢘⣿⡅⠀⠉⢛⣿⠋
⠀⠀⠀⠀⠀⠀⢿⣿⡿⣽⢹⠥⠹⣚⣼⣶⣾⣿⣤⣾⡀⠀⢣⠀⢈⡀⠀⠀⠀⠨⠔⠒⠠⢀⣹⣧⣤⣤⣺⡟⠀⠲⠦⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣚⣾⣶⣾⣿⣿⣿⣿⣿⣷⠚⢇⠀⠀⢇⠈⠉⠭⠗⠒⠀⠀⠒⠐⣿⣿⣿⢿⣿⣿⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣿⣻⣫⣵⣾⣿⡿⢻⣿⡖⠈⠢⠀⠈⠄⠀⠀⠔⠂⠁⠀⠀⠁⠛⠛⠿⠿⠙⠛⢦⠂⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣻⣯⣷⢾⢿⣴⣶⣦⣀⡑⡄⠐⠤⠴⠶⣶⣶⣶⣦⣤⡀⠀⢀⣀⠀⡿⠼⢂⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠛⠛⠛⠋⠀⠀⠈⠉⠙⠛⠿⠿⣦⣶⣦⣄⣬⣵⣿⣿⣿⣷⣀⡻⠛⠪⢁⡊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣑⣛⣛⣣⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡆⣿⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡋⠋⠛⠛⠛⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣿⣿⣿⣿⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⣿⣿⣿⡿⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⡉
⠀⠀⠀⠀⠀⢠⣾⣷⣆⠸⡆⠀⡿⠁⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⡏⣿⣿⡏⠀⣷⠀⡇⣀⣤⣤⣶⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣿⣶⣧⡘⣧⡀
⠀⠀⠀⠀⠀⠸⣻⣿⡟⢀⡇⠀⣿⢻⣿⣟⣻⣄⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡶
⠀⠀⠀⠀⠀⠀⠈⠁⣠⠞⠀⠀⣿⣸⣗⣷⣹⣽⣿⣿⠁⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣇⢠⡇
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡘⣿⡄⠀⠀⠀⠀⠀⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠙⠛⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡘⣷⠀⠀⠀⠀⠀⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⡘⡇⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠓⠒⠶⠶⠶⢤⣤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡈
⠀⠀⠀⠀⠤⠤⠶⠶⠶⠟⠛⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏⣾⠂⠀⠈⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⡏⠀⠀⠀⢃⠀⠀⠁
⠀⠀⢀⣀⣤⣤⣤⣤⣴⣾⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠚⢃⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠸⠟⠿⠟⣿⣿⠿⠿⠿⠟⠻⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⣀⣾⠁⠀⠀⠀⠈⠛⠿⢿⠿⠿⣿⠟⠛⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⠠⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⢁⠀⠛⠋⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀⡀⠀⠀⠀⠀⣿⣟⠀⠐
⠀⠀⠀⠀⠠⣤⣤⣼⣶⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⡄⠀⠀⠀⠀⠀⣒⣶⠶⠶⢶⣖⠀⢠⠀⠀⢸⡏⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠴⠶⠾⣿⣿⣿⣿⣶⣤⣤⣶⣴⣧⣤⠾⠯⠍⠀⠀⠀⠀⠀⠀⠀⡪⠭⠭⠭⠭⠵⠀⠈⢧⡀⠀⠻⠶⠂⠀⠀⠹⣿⣄⣠⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⣽⣯⡭⠭⠳⠶⠞⠓⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢿⣿⣿⡷⠖⠂⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⣀⣀⠤⠚⠁⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⢠⣿⡆⢀⣿⣿⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⢀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠈⠙⢿⡟⠋⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣸⣿⡏⠀⣤⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⣿⣿⡇⣿⣿⣧⣿⣿⠃⣸⣿⠇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⣿⣿⣷⣿⣿⣿⣿⡟⢰⣿⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣿⣿⣷⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣀⡤⠤⠶⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢀⣠⠼⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠋⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⣠⣼⠽⠿⠦⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠛⡍⠀⢠⠭⠁⠀⢐⢢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠬⠶⡄⠀⠈⠚⠁⣀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣞⠢⣄⣀⣀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠉⠈⠊⠓⢾⣆⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣖⠽⠃⠀⠀⠀⠀⠀⢀⡀⠁⠀⠙⣆⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡞⠁⠀⡤⣶⠻⠀⠀⡔⠁⠈⡆⢠⠀⠸⡄⠀⠀⠀⠸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠳⡀⠀⠳⡇⢀⣀⣼⠔⠂⠀⡇⡾⡁⠀⠽⠦⠄⣀⣠⣤⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠹⣄⣀⠷⢚⡄⠸⢄⣀⡴⠁⡿⣜⡂⠤⢀⡀⢡⠊⣄⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢴⡋⣁⣤⢴⠟⠲⣄⣀⣀⣀⢧⠀⠉⠙⠲⢆⡽⠤⢿⣤⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡝⠋⠉⢸⠀⢰⡏⢁⡄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠘⠁⠀⠀⠈⠳⡈⢣⣾⠆⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢰⢹
⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⣶⣄⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠏⡄
⠀⠀⠀⠀⠀⣠⠞⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢳⠈⡓⠀⠀⠀⣸⠀⠀⠀⠀⠀⢓⢡
⠀⠀⠀⢀⡞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣇⠀⢀⡌⢠⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠘⣸⠀⢀⠀⣄
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣾⢀⡏⣀⣀⣀⣐⣳⠐⢯⠍⣰⠖⠀⢱⠀⠈⢟⣉⡓⢲⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣛⣳⣜⠯⣝⣓⠠⠤⠬⣓⡦⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⠴⠋⠀⠀⢈⡉⣉⠍⢥⡀⠀⠀⠀⡤⠶⠋⠛⠒⢲⡄
⠀⠀⠀⠀⠈⠙⠲⢤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⡅⠙⠄⠀⣀⠅⠀⠀⠘⠓⠒⠢⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠐⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠚⠚⠋

⠀⠀⠀⣠⡶⠶⢦⣤⣤⣾⣿⣶⣤⣤⠶⠶⢦⡄
⠀⠀⠀⠻⣤⣤⣤⣤⣴⣶⢶⣶⣤⣤⣤⣤⣼⠏
⠀⠀⠀⠀⠀⠁⢀⣠⣤⣿⣤⣿⣤⣀⡀
⠀⠀⣀⣀⣀⡾⠋⠁⠀⠀⠀⠀⠀⠈⠙⣦⡀
⠀⢸⡏⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠛⠳⠶⣤⡀
⠀⢸⣇⣀⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢻⡄
⠀⠈⠉⠉⠉⠷⣄⣀⣀⣀⣀⣀⣀⣀⣴⠟⠙⠛⢷⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣻⣄⣀⣄⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠹⠟⠟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢛⢙⡛⡛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣉⣉⣉⣉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⢨⢬⣥⡥⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢦⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡟⠀⢀⡴⠛⠙⢷⣴⠞⠳⣆⠀⠀⢠⡶⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⡄⠀⠉⠁⢀⡴⣋⣤⣤⣴⣏⠁⢀⣴⣯⡀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⣰⠞⠋⠛⣫⣤⣀⠀⠉⠛⠋⠛⣿⣿⠀⠘⠷⣤⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣸⣧⡾⢻⡾⠛⢾⣇⠀⠙⢷⣴⠛⠛⢦⡘⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣿⢻⣇⠘⣧⣀⠀⠙⢷⣄⠀⠙⢷⣄⠀⠙⢦⣄⠛⢶⡄
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢹⡿⠁⣼⠋⠛⢦⡀⠀⠙⢷⣄⠀⠙⢷⣄⠀⠙⢧⣈⣿
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⣇⠀⠹⣦⡀⠀⠙⢦⡀⠀⠙⢷⣄⠀⠙⢷⣄⠀⠙⢿⡄
⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠛⠉⠳⣶⠈⠻⣦⡀⠀⠙⢦⡄⠀⠙⢷⣄⠀⠉⠀⠀⠀⠙⢦⣄
⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠈⠻⣦⡀⠀⠙⢶⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡶⢶⣌⡻⣦⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⢋⡵⠛⠛⠛⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⡴⠟⠉⠉⠁⠀⠈⠻⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠙⢷⣄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠙⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡴⠋
⣶⣤⣶⣤⣦⣦⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣼⣿⣴⣦⣴⣦⣤
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠺⠿⠿⠗⠢⣀
⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⣰⣾⡿⣿⣿⣿⣶⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⢼⣿⣿⠿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⣠⣾⣿⣿⣿⣷⣹⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣾⡿⠛⠛⠻⢿⡆⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⣀⠠⠄⠀⠀⠀⠀⠠⠄⣀⡇
⢼⣻⢿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣻⠇⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠋⠀⠂⠛⠂⠈⠃⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠘⢳⣶⣿⣿⣦⣤⡾⣿⣶⡞⠋
⠀⠈⠙⠾⣽⡿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣶⡙⠀⠘⠛⣩⡆⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠳⢽⡻⢿⣿⡿⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣾⣿⡇⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠁⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠾⠿⠾⠿⠷⠗⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣤⣀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⣴⣷⣤⡙⣷⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⠀⢻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠈⠻⢿⣿⣾⣜⢟⡿⣦⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣮⣭⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣴⣶⣒⣰⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⠿⠂⠀⠀⠀⠀⠙⡿⢩⣶⣍⢿⣿⣿⣿⣿⣿⡿⣃⣤⣤⣼⣿⣤⣤⣄⡀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠈⠉⠉⠉⠉⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⣴⣶⣤⡀⠀⠈⣻⣿⡿⠟
⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⢿⣶⡶⠿⠃
⠀⠀⠀⠀⠀⣘⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿
⠀⠀⡇⠀⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡈
⠀⣴⣷⣼⣿⣿⣿⣿⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠈⡏⠉⠁⣿⣿⣿⠀⠈⣛⠟⠀⠀⠀⠀⠀⢰⣿⣿⡟⠻⣿⠿⠿⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⢹⣿
⠀⠀⡇⠀⢀⣿⣿⣿⡀⢾⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⢇⠀⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⣘⢋⡀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⣀⢤
⠀⠀⡇⠀⢸⣿⠙⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠉⠒⢤⡀⠇⠀⠀⠀⠀⠀⠀⢀⣍⣨⣥⣦⣷⣾⣾⣾⣷⣷⣷⣿⣾⣿⣿⣆⣌⣛⢀
⢠⣤⣧⣤⢸⣿⡄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣴⣶⣦⣄⠀⡱⢸⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿
⠈⠉⠉⢹⣾⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⢿⣿⣿⣿⣯⣤⣼⣦⣤⠀⠀⠀⠀⠀⠙⠋⠋⠋⠛⠙⠋⠛⠙⠋⠛⠛⠙⠋⠙⠙⠝⠛⠉

⠀⠀⠀⢀⣤⣤⣤⣤⣀⣀⣀⣀⣶⠟⢻⣆⣀⣀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⢰⡟⠁⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠙⣷
⠀⠀⠈⢿⣤⣤⣤⣤⣴⡶⠶⢶⣶⠿⢷⣶⠶⠶⣶⣤⣤⣤⣤⣴⠟
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣟⠀⢈⣿⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠶⣦⣄
⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⣾⠟⠛⠛⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠷⠶⠶⣦⣤⣀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠈⠉⠻⢶⣄
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠿⠷⠶⠶⠾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡾⠋⠀⠀⠀⠈⠙⣷⡄⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠛⠛⠛⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣀⣄⣀⣤⣀⣤⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⢩⣯⢹⡏⢹⡏⣽⡍⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠉⢻⣦⣁⢀⡀⢨⡀⣨⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢠⣿⠋⠘⠃⠘⠃⠙⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡶⠟⠹⠗⠸⠇⠸⠇⠿⠂⠿⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⢠⡆⢠⡄⣴⠄⣶⠀⠀⠀⣸⡟⠉⠉⢻⣆
⠀⠀⠀⠀⢀⣴⠾⠛⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⡶⠟⠛⠛⠻⢶⣤⣠⣀⣤⡀⠀⠀⠹⣧⣀⣀⣼⠏
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠙⠻⠋⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠙⢷⣄⠀⠈⠉⠋⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣠⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀
⠀⠀⠀⠀⠘⢷⣄⣀⢀⣀⣴⡟⠁⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⣠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡟⠁⠈⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠘⣷⣄⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣧⣤⣀⡀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢷⣶⡟⠉⠉⠙⢷⣄⠀⠀⠀⠀⠈⠛⠷⠶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠈⠻⣦⡀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣀⣠⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠈⠻⣦⡀⠀⠈⠻⢷⣄⠀⠀⠀⣠⣶⢶⠶⠟⠋⠁⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠻⣦⡀⠀⠈⠻⣷⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠻⣦⡀⠀⠈⠙⢶⣤⣤⣤⣤⣤⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠻⠶⠶⠶⠾⠿⠿⠿⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣶⣶⣦⣶⣶⣦⣤⣤⣤⣤⣦⣶⣤⣶⣴⣤⣤⣤⣤⣴⣶⣶⣶⣤⣶⣴⣤⣤⣤⣤⣤⣤⣴
⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠻⣿⣷
⠀⢀⣶⣿⣿⣿⢻⡿⠦⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠸⢻⣿⣧⡀
⠀⠈⢿⣿⣿⣷⠈⡂⠀⡀⠀⠀⠀⠁⠠⠤⠤⠤⠴⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠸⢷⠆⣼⣿⣿⠇
⠀⠀⠀⠉⠉⠁⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡍⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣦⣙⡀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⢠⣿⡏⢸⣿⠃⠀⠀⠀⠀⣠⡤⠦⢤⡀⠀⠀⠀⢀⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⠀⠀⠀⠀⠀⣞⢹⣇⣀⡏⡣⠀⠀⠀⢰⡇⠀⠀⠀⣳⠀⠀⠀⢸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡆⠀⠀⠀⠀⠈⠈⠉⠉⠁⠁⠀⠀⠀⠈⠳⣄⣀⡤⠋⠀⠀⠀⢸⣿⠇⢹⣷
⠀⠀⠀⠀⠀⠀⣸⡇⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁
⠀⠀⠀⠀⢀⣴⣾⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⣄
⠀⠀⠀⣰⣿⣿⣿⡘⠏⠉⠀⠀⠀⠀⠀⠀⢰⣿⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢻⣿⣿⣷⡄
⠀⠀⠀⠙⢿⣿⣿⠃⢠⣀⡆⠀⠀⠀⠀⢠⣾⣿⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠈⢹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠘⠿⢿⣛⡃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢰⢍⣛⡀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⣴⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠸⣿⡆⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⣅
⠀⠀⠀⣠⡄⠀⠀⣿⡇⠀⣿⣇⠀⠀⠀⠀⢀⣿⢹⣿⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣼⣿⠀⢸⣿⠀⣀⣀⡀
⠀⠀⠀⠉⠷⠆⠀⠛⠃⠀⠙⠛⠀⠰⠖⠂⠘⠃⠘⠋⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠛⠋⠀⠘⠛⠀⠻⠿⠿⠶

⣿⡏⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⢀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡤⠓⠋⠑⠊⠒⠒⠤⣀⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠄⠀⠄⠀⠀⠆⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⢉⠑⢄⠣⣄⡐⠒⠊⣀⡠⠴⠊⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡇⠀⠄⠐⡦⠀⢸⠔⠊⢉⠭⠕⠚⠁⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⢀⡀⠔⡹⠒⠎⠁⠀⢀⡠⠤⠒⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⡠⡄⠀⢀⠝⢉⠇⠀⠀⢻⣀⡸⠒⠉⢧⠀⠀⠀⠀⠀⢀⣘⡀⠀⠀⠀⠀⠀⢀⣀⠠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣸⠀⡇⣀⢸⡀⡸⡄⠀⠀⠀⢈⠆⠀⠀⢸⠖⠒⠒⠈⠉⠁⠀⡇⠀⠀⠀⠀⡴⠉⡗⠚⢢⠈⢣
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠙⣄⣹⣛⠒⣳⢄⣀⡤⣺⠽⡆⠀⢸⠀⠀⣠⢄⠀⠀⠀⢹⠀⠀⠀⠀⣇⠀⡇⠀⠀⠇⢸
⠀⠀⠀⠀⠀⡴⠽⠤⠤⠬⠭⠤⠤⠽⠤⠬⠷⠥⠤⠷⠋⣡⡄⡇⠀⣌⣀⣀⠇⠈⣆⣀⠠⢼⠀⠀⠀⠀⠈⠢⠧⠀⠤⠞⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣠⠇⡇⣀⠗⠤⡏⠀⠀⢈⠗⠒⡻⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡠⠚⠘⠒⠒⠚⠀⠀⠀⠓⠒⠊⠁⠀⣰⢩⣒⡆⣖⡝⡆
⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠀⠴⠀⠤⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⣖⡤⠤⢴⡢⠕

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡄⢄⡀⠀⠐⢄⠀⢆⠀⠀⠀⠀⠀⠀⡄⠀⢦⠰⢄⠰⠀⡆⠆⡀⠄⠠⡀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣄⣀⡀
⠀⠀⡠⣾⣷⡀⠀⠀⢠⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠐⢋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡿⠓⣀⠀⠀⣬⣇
⠀⠀⢈⣼⣷⣄⣿⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⣽⡇⠀⣼⣿⠀⡀
⠀⠀⣼⣿⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⡇⠉⠉⠉⠉⠉⠉⠁
⠀⠸⢧⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠖⣩⣿⣿⠀⠀⠀⠀⠀⠀⠐⢣⣿⣿⣿⡄
⠀⠠⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏⣿⡿⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠿
⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠻⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⡀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡀⢠⣾⣦⣄⣀⡠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⣀⣀⡀⠀⠀⠀⠀⡐
⡇⢸⣿⣇⣉⡉⠉⠉⣿⡋⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⡍⠻⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⠐⠚⠛⠓⢺⡟⠛⠓⠒
⠥⠬⢿⠿⠿⣿⣆⢰⠃⢣⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠹⠆⠇⣠⣶⣇⠐⣶⣶⣶⡶⠀⠀⠀⢸⣀⣻⣿⣿⣿⣆⠀⡏⢱
⠀⢀⠎⢂⠀⠈⢿⡏⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢸⠘⣧⠀⢸⣍⠛⠛⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠠⢣⠀⠈⠻⣿⠁⠀⡇
⠀⠀⠀⠈⠀⠀⠼⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠂⠘⠿⠗⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠁⠀⠂⠀⢠⠇⠀⠀⢸⡄
⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠐⠀⠂⠀⠀⠀⠐⠐⠀⠀⠀⠀⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⢳⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣰⠏⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢰⡿⠶⠞⢛⣛⣋⣉⣭⣭⣭⣭⣭⣭⣉⣉⣙⣛⡛⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠃⢠⡿⠶⠟⠛⢋⡟⣯⣩⠴⢤⠀⠀⢀⣀⣉⡭⣍⠉⠙⠛⠳⠶⣤⣉⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣱⡏⢀⣾⡥⢤⡴⠚⣋⣻⣯⡷⠒⢾⣤⠞⡉⠉⠙⢦⣤⣅⠀⡞⠛⣦⣀⣿⠂⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠀⣼⡛⠷⢦⣤⣼⣅⣈⣿⣀⣀⣀⣠⣎⣉⣙⣦⣀⣤⣼⣷⠷⠶⠛⣋⣡⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣸⣏⡙⠛⠶⠶⣤⣭⣭⣍⣉⣉⣉⣉⣉⣉⣉⣉⣥⣤⣤⣤⠶⠞⠛⠉⠁⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⣿⡛⢿⣶⣤⣤⣄⣀⣀⣈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣀⣀⣀⣤⣴⠶⣾⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⡟⢸⡧⠀⠻⣦⡄⠈⠉⠉⠛⠛⠻⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⢠⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⣾⠁⠘⣁⡀⠀⠈⠉⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸⠇⠀⠀⢹⡁⠀⠀⠀⠀⠉⠛⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠿⠛⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣠⡏⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠷⠶⣦⣤⣤⣤⣤⣤⣾⣏⣿⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣉⠉⢙⡧⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢙⡿
⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣴⣟⣉⣉⠉⠛⢿⣇⠀⠀⠀⠀⠘⣷⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠹⣇⠀⣀⡼⠋⠉⠉⠙⠻⣄⠀⢀⣿⠶⣶⢾⣿⣷⣦⣿⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⣀⠀⠀⠀⣉⣿⠟⣠⠾⠛⠻⠷⣦⣈⡛⠛⢁⣼⠃⣼⣿⢹⣿⣿⣿⠿⢶⣦⣴⠿⣇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⠇
⣿⠛⠷⠛⠋⣡⡾⢋⣠⣤⣤⣀⡀⠉⣹⣿⠟⠁⣴⠟⣿⡌⣿⡋⠁⠀⢀⡙⠷⠶⠟⠉⣉⣉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡿
⢻⡟⠶⠶⠛⠉⣠⡟⠉⢀⣀⠉⣹⡟⠋⣁⣤⠞⢁⡼⢻⡇⣿⢿⡶⠶⠛⠻⢦⣄⣤⠞⠉⠉⢻⣤⣿⣀⠀⠀⠀⠀⢀⣀⣀⣤⠶⠋⠑⠟⠳⠚⠴⢂⡀
⠈⠻⣤⣄⣠⡴⠏⣠⡾⠋⠉⣹⡟⠀⣼⠋⢠⣾⠋⣰⠟⢡⡟⢠⣿⡟⠛⢶⣤⣤⡴⠞⠛⣦⡼⣫⢍⡉⣋⠿⣋⠟⣩⠟⢉⡴⠚⣠⠞⣠⢞⡴⠀⠈⠴
⠀⠀⠈⠻⣧⣄⣴⠏⢀⣴⠟⢻⡇⠀⣿⠀⣿⣿⠀⢿⡀⢸⣇⠸⣯⣳⣦⣤⣤⣤⣄⣤⠞⠋⠀⠀⠀⠀⠁⢘⣡⠞⣁⡴⠋⠀⠚⠁⠘⠁⠈
⠀⠀⢀⡴⢆⣉⡻⢶⣾⣅⣠⠟⢿⣄⣿⣤⣷⣼⣧⣌⡻⣦⣝⠷⣬⠙⠛⣦⠀⠀⣩⣄⡴⣋⣤⡤⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠐⠉⠛⠵⢂⡤⠉⢡⣶⡄⠉⠉⠀⣠⢶⡤⣨⠙⠛⠻⠶⠾⠛⠛⠛⠁⠀⠁⠉⠘⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⠿⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢹⣿⣷⣾⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠰⣶⣄⠀⠀⠙⠻⠿⠿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡤⠄⠀⠀⠀⠀⠀⠀⠀⣶⣦⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣶⣿⣋⣠⣴⣿⣿⣿⣿⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣻⣿⣷⣶⣤⣀⠀⠀⠀⠀⢀⣀⠀⠐⣧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠂⠉⠁⠀⢀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⢀⣠⡎⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣤⡞⠛⠃
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣜⣻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠈⢉⠛⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⡠⠀⢠⣤⡿⠿⢿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⠟⢛⠛⠿⠛⠙⠿⢏⣽⣷⣦⣠⣴⣉⣠⣾⣇⠉⢱⡆⠸⣿⣿⡏
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠘⠳⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⠃⠀⢿⣿⠃
⠀⠀⠀⠈⠻⣿⣿⣿⣿⡃⠀⠀⠀⠀⠙⢦⡀⠀⢀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢰⣿⣿⡆⢸⣿⡆
⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣦⡄⠀⢀⣀⠉⠀⠜⠛⢉⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣿⣿⣿⡇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠒⠀⣽⠿⠀⠀⠺⣋⡙⠋⢹⣿⣟⠉⣿⡿⠀⢸⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⣠⠇⠀⣀⣠⣤⣴⣿⣿⠋⠁⣠⣤⣤⣤⣄⣼⡇⠀⠀⢀⡀⠀⢷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠔⠋⠀⣾⣿⣿⣿⣿⠿⠛⠓⠀⠀⠈⠉⠉⠹⡏⠀⠀⠙⠋⠉⠁⢸⡇⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠈⠙⣿⡏⢤⡶⠠⠤⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠉⠉⠀⠸⠇⣴⡄⠀⠤⠤⠄
⠀⠀⠀⠀⠀⢀⣤⡏⠀⠀⠀⠀⣠⡄⠀⠚⢁⣴⠷⠶⠶⠖⠀⠲⠶⠆⠾⠿⠀⣀⣀⣀⣤⣤⣀⣀⣹
⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⢠⣾⡏⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠀⠐⠛⠻⠿⣶⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⣿⡟⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠙⠷⠶⣶⣿⣿⡿⠃⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢹⡟⠀⢠⣿⣿⡇⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢠⠀⣼⠃⠀⣿⣿⡿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⣧⣸⣷⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⢹⣿⣿⡏⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⡄⠀⠸⣿⣿⣿⣟⠀⠂⠀⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣶
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢉⣹⣿⣷⣶⣤⡀⣀⣀⣤⣴⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⡀⣠⡟⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣧⠀⠀⢴⣦⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠿⠿⠏⠀⠀⠸⣿⣧⡷
⠀⠀⠀⠀⠀⢠⣿⠀⣀⣠⣄⠀⠀⢀⣠⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣥⠀⣄⠀⠀⠀⠀⠀⠘⢻⣿⣧⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠉⠹⡟⠀⠀⠈⠀⠻⣿⣿⣿⣿⣿⡿⣿⠛⣿⣿⣿⡿⢻⣿⣿⣿⣿⢻⣿⣿⡿⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠸⠛⣿⡇
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠰⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣼⣶⣦⠀⣹⠿⣿⣷⣾⠁⠹⠉⡟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠺⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣧
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⠆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⠄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⣿⡷⠛⠇
⠀⠀⠀⠀⠈⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⣤⣤⣤⣤⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠻⢤⡀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⢀⣴⣾⠛⢿⣿⣿⣿⣿⣿⣿⠟⢹⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⢿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣠⣤⣴⣶⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣦⣄⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣇⢾⣿⣿⣿⣿⡿⠋⣹⣿⣿⣿⣿⣭⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⣹⣿⣧⣾⣭⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠿⠟⠋⢿⣇⠸⢿⣿⣿⡿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠘⠻⣶⣾⠟
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⢼⣿⣿⡿⠛⠿⠟⠉

⠀⠀⠀⢠⡤⠄⠀⠘⠟⠋⠄
⠀⠀⠀⠀⠙⠡⠽⣴⣦⣠⣰⡄
⠀⠀⠀⠀⠀⢲⠤⣀⣁⠉⠉⠁
⠀⠀⠀⠀⠀⢈⠲⡤⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠁⠁⡇⠀⠀⢀⠀⠀⠀⠀⢀⣾⣷⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠳⠄⠙⠋⠉⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡴⠄⣄⠀⠀⢠⣾⣿⣯⣤⠶⠔⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣎⠁⠀⠙⠂⠺⣿⣿⡿⠇⢀⣴⣾⢉⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠛⠛⢦⣄⡁⠀⢸⣧⠄⠀⣼⣿⠁⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⠀⠀⠀⠈⠋⠁⢀⡈⠻⢴⣤⣿⢋⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠘⠀⠀⠀⠀⠈⠑⢾⣷⣤⡀⠉⣼⠷⢠
⠀⠀⠀⠀⠀⢀⡄⠰⣧⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⢿⡿⢿⣦⣿⣶⣾⡀⠀⠀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠈⠁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣧⠉⠁⠀⠙⢳⣆
⠀⠀⠀⠀⠀⠀⠀⢹⡇⢴⣏⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⡆⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡼⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⠿⣿⠀⠀⠀⡠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢱⢠⡏⠡⠀⠀⠀⢿
⠀⠀⠀⠀⠠⠶⢤⠶⣿⠀⡄⢸⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣼⣇⡆⢡⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣷⢸⣷⣦⣾⠀⠂⠀⠀⠀⠀⠀⠀⠀⡿⢿⠸⣿⣇⠀⣧⢰⣿⠂
⠀⠀⠀⠂⠀⠀⠀⠀⣿⣿⣟⣼⣿⣿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⣸⢆⣯⣿⡁⢹⣎⡉⢉⢀⢀⣐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⢠⠀⠀⣿⣿⣬⣿⣿⠿⠿⠿⣛⠟⠛⠻⣿⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⣿⣿⣿⣿⠀⢸⡁⠀⠀⠀⠀⣿⠀⢠⣿⣿⣿⣿⣿⣀⣀⣠⣿⠒⠦⣤⣿⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣧⡿⣿⣿⣿⡇⣀⠘⡇⠀⠀⠀⠀⣿⠀⠀⢹⣿⣿⣿⣿⠋⠁⢀⣿⣷⣄⣤⣹⡇
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣯⡡⢹⣿⣿⡏⢿⣃⣀⡇⢀⠀⢹⣀⣀⣨⣿⣿⣿⣿⠛⠛⠁⡿⠉⣧⡀⠸⠂
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣾⣿⣿⠃⢴⡿⣿⣿⣿⡶⢾⡟⠛⢽⣿⣿⣿⣿⠀⠀⣠⠖⠚⣉⣠⣴⡟⣇⡀
⠀⠀⠀⠐⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢺⣧⢻⣿⠃⠀⠸⡄⠈⣿⣿⣿⣿⣿⣤⣶⡿⣿⣿⣋⣉⠀⠉⠙
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣭⣼⣟⠀⠀⠀⠂⠸⣿⣿⣿⣿⣻⡆⠛⠛⠛⠟⠉⠉
⠀⠀⠀⠀⣰⠿⠛⠛⡿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⠊⠁⠀⠀⠀⠀⣀⡼⠋⠛⠛⡿⠿⠛⠻⠿⠟⠛⠛⠛⠛⢿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⡙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⣾⣏⠈⠉⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣿⠿⢟⣛⣿⣮⣛⠻⠿⠟⣿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣤⣾⣿⡟⢡⡾⣫⡿⣿⣿⢿⡿⣿⡿⣯⣻⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⣳⣤⣽⣿⡿⢱⣿⣿⡟⣰⢏⣿⣹⣗⣿⢷⡙⢿⣮⣛⠶⣿⣤⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣩⣽⣿⣿⣿⠃⣼⡿⣹⡇⣿⣿⢿⠙⣿⣽⣆⣿⢦⡙⢿⣷⣮⠻⣿⣷⡶⣆⠀⠘⠻⠶⠶⠾⠟
⠀⠀⠀⠀⠀⣠⣤⠖⠛⠉⢱⠞⢻⣿⡽⢟⣡⠞⣽⣿⣿⡇⣿⣿⣿⡆⣿⣿⢿⣧⣾⠻⣆⠹⣎⢷⠘⣯⠻⣟⠛⠛⢶⡄⠀⣤⠶⠖⠛⠻⢦⣄
⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠘⠻⣯⣭⠶⣻⣿⠾⢋⡾⣡⣿⣿⣏⣼⠇⣿⣿⡘⣏⢹⣧⣹⠌⣿⢿⡆⢿⡛⠋⠀⣴⣟⣁⠀⠙⠛⠛⠛⠛⠛⠋
⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⣰⣟⣫⡾⢿⡏⣿⣿⡟⣰⡏⣿⣇⢿⡿⣿⣿⠀⣿⣌⢳⣼⣿⡆⠀⠀⠀⠉⢻⡆⠀⠰⣾⣿⠆
⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⢻⡆⠀⢾⣿⣧⠾⠷⠞⠻⣷⢿⣟⣱⢿⡇⢿⣿⣎⢻⣿⢿⡶⣷⠾⠀⠉⠋⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣀⣀⣀⣹⣾⠟⠋⠀⣿⣼⠁⣿⢻⡇⠘⠷⠛⢀⣀⣠⣤⡴⠶⠛⢉⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠈⠛⠓⠒⠛⠛⠛⣋⣉⣩⣉⠛⠦⣤⣤⣤⣬⣅⡀⢙⣋⣡⡤⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣤⠴⠟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡲⠒⠤⣀⢤⡀⠀⠀⠑⠺⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡰⠃⠀⠈⣷⣀⠀⠀⠀⡏⠉⢩⠟⡖⠚⠒⠓⡞⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢅⠀⢀⠞⠀⠈⠑⠢⣤⣣⠔⠁⢀⣇⠀⠀⢀⣇⠀⠀⠀⠀⢀⣀⠿⠭⠭⠭⠭⠭⠽⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠈⡇⠀⡠⠊⠀⠉⠉⠁⠀⠳⡄⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⣀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣡⣤⣤⣀⣀⣠⣔⣹⣄⣸⣇⣀⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢰⡋⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡆
⠀⠀⠀⠀⠀⠹⣭⠭⠭⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣭⠏
⠀⠀⠀⠀⠀⠀⣷⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡞⢳⠀⢰⠋⡆⠀⡖⢳⠀⢠⠊⣦⠀⢰⠑⡄⠀⡎⢲⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⡇⠀⠇⢸⡀⢸⠀⣿⠀⠀⠀⡇⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⣿⠀⠀⠀⡇⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⣿⠀⠰⠀⡇⠀⡇⣸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢹⠋⢱⠒⡖⠒⣾⡻⡆⢲⡶⢶⠒⢶⠶⡖⠋⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⠀⠀⠀⠀⡃⠀⡇⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⢸⠀⡇⠀⣾⠁⡇⢸⣷⣾⠀⢸⠀⡇⠀⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠦⣿⠀⠰⠚⡇⠀⡇⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣧⠀⣇⢸⠀⠸⡖⡇⠀⢿⢀⠇⠸⣇⣿⠀⢸⢀⡇⠀⣇⣸⠀⢸⡀⡇⠀⣇⢸⠃⠸⡀⡿⠀⢸⢀⠇⠀⣇⣻⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢩⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⢤⣀⣀⣴⠔⠉⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⠀⠀⠀⠈⡈⠈⢀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠐⢈⣇⠐⠁⣠⡔⠀⠈⠂⠋⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⢠⠊⠀⠀⠁⠂⢤⢊⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣘⡆⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⣦⡔⢽⢢⠆⠸⠧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠺⣄⠀⠀⠀⠀⠀⢠⢤⡜⠀⠀⣠⠄⠰⠼⣀⣀⣈⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⡀⠀⡖⠀⠀⠈⡟⠀⠀⠁⠀⠈⠉⢁⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢹⠃⡀⠀⢀⡀⢰⠁⠀⠀⠀⢁⠈⠀⢀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠐⣐⡄⠊⠉⠑⠀⠀⣀⠀⢸⢀⡀⠦⠤⠶⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⡰⠇⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⣸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠉⠑⣆⠀⠁⠀⠀⡄⠀⣗⠓⢂⠂⠁⠀⠀⠗⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡂⠀⠀⠀⠹⣄⠸⡀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣿⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠈⠑⠀⠀⣿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⢤⣄⠀⠀⠀⣀⣀⣠⠀⠀⠈⠈⢀⠙⠠⠩⠁⢘⡙⠁⠐⠐⠊⠡⠏
⢀⣀⣀⣀⣀⠀⡀⣀⣄⣄⠴⠹⠄⢀⢀⡥⣀⠙⡷⡄⠀⠐⠋⠁⢊⡀⠂⠀⣤⠄⠀⠀⠀⠐⠂⢰⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠖⠋⠉⢀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠒⢢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄⠛⠀⠈⠁⢑⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠌⠀⠀⠀⠀⠀⠀⢊⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠀⠀⠀⠀⠀⠀⠀⠘⣰⠀⢀⡤⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡧⠒⢠⡌⢤⡒⠮⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠀⠀⠀⠀⠀⠐⢢⠀⠀⢀⠀⢀⢀⠀⠀⣤⡴⠶⠆⠗⠒⠒⠒⠒⠒⠒⠒⠶⠁⠀⠀⠀⢩⠖⡿⢛⠇
⠀⠀⠀⠀⠀⠀⠰⡟⢂⠤⢤⡀⠛⢆⡀⣀⣀⣀⣀⡠⠞⠀⢐⠁⠀⠀⠀⡇⢸⠺⠶⢶⠄⣦⣤⣤⣠⣤⣠⣶⡂⠛⠚⠒⠒⢢⠜⠿⣿⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢇⢀⠜⠁⠂⢼⡄⠈⣄⣀⣨⡄⠀⢀⡨⠶⠶⠶⠶⠧⣼⠀⠀⢼⣧⢻⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢨⠀⠎⠀⠀⠀⠈⢤⠋⠀⠀⠀⡀⠀⠀⠈⢂⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠀⡆⠇⠀⠀⠀⠀⠀⢸⠀⠎⠁⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣰⠉⠉⢹
⠀⠀⠀⠀⠂⠀⠚⠃⠀⠀⠈⠉⠉⠉⠉⠉⠐⡆⠀⠀⠚⠀⠓⠀⠀⠀⠀⠐⠃⠘⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠠⡿⠀⠀⠈⡇
⠀⠀⠀⠀⢹⠛⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⣟⠛⢛⠛⠛⠛⠛⠛⢛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⡔⠒⠂⠀⠀⠓⡄
⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⠀⠀⠁⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠀⠁⠈⠉⠈⠉⠁⢉⡆
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⠀⠀⠀⠆⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡇⠈⢹⠙⠻⠉⡿⠉⣷⡆⢲⠂⠉⡇⠀⢻⠀⠸⠀⣀⣤⣦⣤⣼⣆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠰⠠⠀⠶⠀⠤⠠⠤⠀⠤⠜
⠀⠀⠀⠀⠀⡇⠀⡸⠘⡶⠀⠇⠀⢿⠇⢸⠀⢠⡇⠀⡾⡰⠴⠤⠮⣿⠧⠄⠘⠫⡿⠶⠶⢤⠄⠀⠀⠀⠀⠠⠤⠤⠀⠊
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠉⠉⠉⠉⠉⠉⢹⣥⣶⣾⣿⣶⣿⣷⣶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠜⠆⠾⡠⠆⠴⠤⢿⠀⠀⠉⠉⠉⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⡤⠴⠒⠒⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣁⣛⣻⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣯⣭⣽⣭⣿⣿⣿⣽⣍⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⢤⡀⢀⡠⠐⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣇⠀⣠⠀⠀⠀⢠⣤⠀⠀⠀⠔⠀⠉⠁⠢
⠀⠀⠓⠃⠀⠀⢰⠁⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⢁⣀⡀⠀⠉⠉⠀⠀⢀⡀⠈⠀⠀⢘⢀⡄⠀⠀⠀⠀⡔⠁⠉
⠀⠀⠀⠀⢠⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢀⠀⠀⠀⠀⢀⡜⠢⡠⠚⣄⠀⠀⢀⡴⠥⠄⠠⠤⢄⡀
⠀⠀⠀⠀⣈⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠥⢀⠀⠀⠀⠀⠀⠃⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠒⠈⠀⠀⠀⠀⠒⠎⠀⠀⣠⣒⡒⠦⠔⠒⠢⠤⡄⠀⠈⠑⢄⠒
⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡇⠰⡞⡵⣶⡦⠀⠀⣬⣭⣁⠁⠀⠀⠀⣸
⠀⡏⢉⠉⡉⠉⡉⢉⠉⣀⣀⡄⠬⠤⠄⢀⡀⠀⠀⠁⠀⠇⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⢹⢶⠀⢏⡻⡇⠀⢸⡸⣿⠁⢃⠒⣄⡴⠃⡖
⠀⡇⠀⠀⢀⢸⡀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠛⢭⠭⠍⠀⡥⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⢸⠈⢆⠀⢰⡥⠤⣴⠉⠁⢀⡌⢴⠉⠀⠆⡗
⠀⡇⢀⠀⡰⠊⠀⠀⠤⠤⢤⣄⠀⠐⠒⣠⢄⡀⢣⡠⠀⡇⠰⠐⠒⠐⠂⢲⠒⠂⠀⠀⢸⠀⠨⠷⡦⢭⣨⡥⡴⢚⠉⡹⢋⢧⣁⣤⣷
⠉⠀⠉⠀⡇⠀⠀⠀⠉⠐⣚⣧⠀⠀⢴⣻⡄⢃⡿⢤⠀⡇⢀⠀⠀⠀⠀⢸⠀⢀⠀⠖⠁⠁⠐⡞⠀⢰⠀⠅⢀⢸⠊⠐⠁⠈⡇⡐⡒
⣀⡄⠉⠁⠳⢦⠃⡵⠇⠀⣌⢫⠇⠀⣩⡛⡣⢸⠃⠜⠀⠃⢸⠀⠀⠀⠀⢸⠀⠈⠀⠐⠂⠈⠀⠳⠄⣨⡀⢘⢇⣀⠉⠒⠒⠘⣧⠆⢇
⢀⣠⣤⠦⠄⠠⠛⢦⠀⠀⠈⡵⢦⣤⣷⠍⠀⠌⠀⣁⠀⡁⢸⠀⠀⠀⠀⢸⡔⠁⠠⠔⠀⣰⢯⢻⡄⢙⡷⢡⠞⡀⠈⠑⠒⠒⠙⡀⢸
⠈⡆⠀⡬⢤⢰⣪⠞⠓⢤⣀⠘⠂⢐⣋⣤⠞⢋⡡⠜⡇⡀⢸⠀⠀⠀⠀⢸⠐⠤⢄⣀⣄⣀⡄⠀⣠⢈⣱⡁⠀⣣⣀⣀⣀⠠⠔⡇⠸
⣀⣿⠀⠇⠲⡼⡅⠀⠀⡇⠉⢛⠫⠚⡩⠔⠊⠁⠀⠰⠁⡀⠗⠠⠦⠠⠄⣸⠄⠂⠌⠣⠸⠄⢤⠔⡹⡏⢸⠁⠀⠀⢳⠀⠀⠀⠀⣭⢉
⠤⠘⠀⡇⡘⢰⣀⡀⠠⢼⢠⠋⡝⠁⠀⠀⠀⢀⠠⠺⣒⣊⡉⠔⠓⡄⠢⠔⠢⡀⠈⡉⠤⠶⣭⠊⠀⡆⢸⠀⠀⠀⢸⡇⠀⠀⠀⢸⠄
⠒⠉⠀⢹⠁⠈⠲⠒⠛⢋⡇⠸⠀⢀⡠⠔⢊⡡⢐⠩⠐⠉⡠⠔⠊⢳⠴⠒⠛⠩⠀⠀⠀⠀⣿⠈⠀⠉⢸⠀⠀⠀⡞⠹⡄⠀⠀⢸
⠀⠶⠄⠱⢤⣀⡀⠑⠢⢄⡑⠦⣮⣅⠴⠮⡔⠊⣀⠤⠂⠁⠀⠀⠀⠸⡄⢀⣀⠀⠀⠀⡐⠀⢸⠠⠶⠄⠸⣽⣖⣱⠇⠀⢳⡁⢮⡼⢀
⠀⠀⠀⡐⠁⠀⠈⠉⢲⠚⠻⣄⣠⣏⠀⠂⠓⢎⠀⠀⢠⠊⣿⠽⠂⣄⢗⠝⠁⠀⠀⠠⠀⠀⣴⠀⢀⣄⢰⠃⠄⢇⠀⠒⣂⢇⡀⣇⠈
⠀⠀⠀⡇⠀⠀⠀⢰⡟⠈⠀⠘⢟⠋⠀⠀⠀⠈⠄⠀⠠⢡⠀⢀⡶⣼⠚⡤⠖⠤⠒⣦⣐⠊⡿⠴⣥⠤⠥⠤⠤⢬⡖⠛⠚⠀⠀⢈⡆
⠀⠀⠀⡇⠀⠀⠀⠘⢹⠀⠀⠀⠘⡣⡀⠀⠀⠀⠈⠀⠘⢯⣉⡜⠂⠁⠀⡛⢄⢠⡾⠜⣹⡦⣤⡔⠉⠉⠈⠈⠁⠉⠓⠀⠈⠉⠓⠒⣁
⠀⠤⡄⡇⠀⠀⠀⡇⠈⡆⠀⠀⠀⡗⠓⢄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠜⠉⠛⠉⠙⢎⠀⠀⢙⢆⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⡧⠤⠤⢤⡇⠀⢸⡄⠀⠠⡇⠀⠈⢢⡀⣀⠀⠈⡄⠀⠀⠀⠀⠎⠀⡘⠀⠀⠈⡈⢣⠀⠀⠁⢻⠉⠙
⠀⠀⠀⢹⣁⣹⣉⡀⠀⠀⠹⠤⠽⡥⢄⠀⠋⣕⢄⠱⡀⠘⣄⣀⠤⢸⠀⢠⠀⠀⠀⠀⠰⠈⡆⢀⠈⠈⡆⠀⠀⠀⠀⠀⠐⠓⠶⠳
⠀⠤⠤⢾⣀⠀⠁⣈⣧⠒⢷⢄⣀⣀⠠⡧⠡⣑⣀⡤⠋⠀⠉⠀⠀⢸⠀⠘⠄⠀⠀⠀⠀⠀⢡⣀⣙⣀⣱⣄⡀
⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣐⣲⣆⣀⣀⣀⣀⣀⣘⣙⣁⣘⣒⣒⣔⣒⣒⣚⣒⣒⣚⣃⣀⣀⣈⣁⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⠙⢟⠁⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⡈⠁⣾⡷⠦⠀⠀⠘⣅⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠂⢡⡞⠿⣁⠀⠸⣦⢠⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠠⠆⠀⣀⡄⠀⠀⢸⡧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣯⣦⠀⠋⠀⠀⣀⣐⣸⠇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⡴⠺⣿⢛⣿⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠹⡿⠀⠀⠀⣰⠃⢸⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⡇⣀⠀⢠⠯⣴⡞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢀⡟⠀⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢀⡞⠀⢠⣧⠊⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡞⠀⠀⠸⠟⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠇⢀⣸⡆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣰⠃⠀⠀⢀⠀⠀⡀⠀⠀⠀⡄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣰⠃⠀⠀⠀⣿⠀⠀⡇⠀⠀⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⢰⣿⠀⠀⢀⣼⣷⡄⢀⣧⢀⠀⣾⣿⢸⡄⠀⠀⠀⠀⠀⣀⣀⣠⣀⣠⣤⣤⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠀⠀⢰⣿⣿⡇⢸⣿⡌⠀⣸⠟⠚⠁⠀⠀⠀⠀⠈⠿⡉⠻⣽⣿⡭⠥⠆⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠅⠀⠀⣸⣿⡿⠇⠘⠛⠷⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠈⠛⠿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⡆
⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢾⣿⣷⢄⡠⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠏⠸⢛⣾⣦⣄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⣾⣿⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙


⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠶⢤⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠤⠶⡦⠠⠶⠀⠀⠄⣀⣀⡀⣲⢐⠀⠀⢰
⠀⠀⠴⠚⠉⠁⠸⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⢀⠠⢤⠄⠤⠰⠒⠒⠤⠠⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⣦⣀⠠⠤⡆⠀⠀⠰⢂⠁⠀⠸⢉⠦⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⢀⣀⢠⡿⠂⠂
⠀⠀⣀⣂⣤⣴⠁⠀⠀⠀⠠⠐⠃⢦⣸⠄⠀⠀⠾⡿⠇⢺⢿⣷⠞⠀⠀⠀⠀⢀⣇⡠⠂⠉⠁
⠀⠀⠿⠿⠿⣿⠀⠰⢄⡁⠀⠀⠀⠀⠙⠤⢤⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡼⢫⣤⡄
⠀⠀⠠⠀⣀⡄⠀⠀⡄⠀⢀⡆⠀⠀⠀⠀⠀⠉⠉⠢⠀⠓⣤⠔⠢⡘⠉⠉⡔⠀⠈⡟⠀⠀⠉
⠀⠀⠀⠀⢀⠃⠀⣦⠂⡐⠈⡇⢠⠀⠶⠆⠀⠀⠀⢸⡶⣄⠀⠀⠒⠂⠰⡴⣤⢠⡖⢶⣄⠀⢤⡤⣤⣤⣤
⠀⠀⠠⠄⡾⠀⠀⣀⣀⠀⢰⡇⢌⠆⠄⡠⠀⣐⠁⢰⠀⢸⠀⡦⠀⠀⢀⠃⠀⢸⣷⣬⣿⠆⠄⠉⠉⠉⠉
⠀⠀⠀⠀⡆⠀⠈⢠⠌⠂⣸⠇⠊⢀⡶⠇⢀⣿⢈⢸⠀⢸⢈⣿⠀⠀⠘⠀⠀⣴⠲⠌⢀⡢⠀⢀⠂⡀
⠀⠀⠀⠀⠀⠀⠈⠚⠀⠂⢻⠶⢡⣌⠝⠀⠁⠹⠠⢸⠀⢸⠀⣿⠀⡆⠀⠀⢘⣮⠆⠐⡏⡀⡲⠎⠐⣠⡄
⠀⠀⠀⢸⠀⠀⡏⠉⠉⠉⡟⢁⣀⡔⣠⣁⣀⠠⠀⠰⠀⢄⠀⣽⣀⠓⠂⠀⠀⠇⠐⠁⣴⠁⠀⢰⢰⠙⠹
⠀⠀⠀⡄⠀⣸⣧⡠⣤⡖⠿⢷⣤⣿⡅⢴⡃⢦⣭⢹⠀⠈⠂⠀⠈⠃⠀⠀⠀⣏⠀⡄⠩
⠀⠀⢀⠁⠀⡭⠭⠅⠾⠁⠐⠚⠉⠁⠃⢜⡁⠔⢢⠀⠀⠀⠀⠀⢀⡷⠋⠁⠀⢻⠀⠛⡀⠐⠀⠘⠊⠁
⠀⠀⡌⠀⠀⠀⠀⠀⠀⢀⠀⢀⢠⡀⢠⡾⡇⠸⡧⢸⠯⠸⠤⠒⠁⣀⠄⠀⠀⠀⠆⠀⠀⢶⠄⠀⠀⠠
⠀⠀⡆⠀⢀⣀⣠⡄⠀⠸⠟⠋⠉⡅⢠⣠⣤⠄⢠⠘⠀⠀⠀⠀⠔⠀⠀⠀⠄⠀⠀⢡
⠀⠀⠃⠀⠭⠑⠈⠁⠀⠀⢀⣠⡤⡛⢀⣉⢥⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠽⠀⠀⠲⢀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⣠⣇⡘⢀⣠⣄⡤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠦⠀⠀⠀⠀⠾⠷
⠀⠀⠀⠤⡶⠛⠋⠁⠀⠠⠐⠘⠛⠙⠉⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠰⠀⠀⠀⠒⠒⠂⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠑⠂⠆



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠂⠐⠂⠀⠒⠀⠀⠀⠒⠀⠐⠂⠀⠐⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢤⢎⠒⢠⠀⠀⠀⠀⠀⢠⡀⠄⠀⠀⠀⠀⢁⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠐⠀⠀⠀⠐⠖⠀⠀⠀⠈⠃⠘⠤⠈⠀⠀⠀⠀⢈⡁⣀⣠⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⡈
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈

⠀⡀⠀⡠⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⢀
⠀⢀⠁⠈⠉⠑⠀⠀⠀⠀⠀⣦⠂⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠠⠄⡈⡀⠤⠀⠀⠀⠀⠂⠂
⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠽⠚⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠉⠑⠂⠀⠀⠀⠠⠀⠀⢀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⣀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠈⠀⠀⠄⠠⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠠⠉⠉⠁⠀⠀⠀⠐⡄⢠
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠢⡈⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠠⠈⠃
⣤⣤⢦⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣷⣤⣤⣤⣤⣤⣤⣶⣤⣤⣶

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⢀⣿⠀⠀⠀⠀⠀⠀⡠⠔⠒⢆⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠕⠋⠉⠝⢆⠀⠀⠀⠀⠀⡼⢋⠻⡄⠀⣽⠀⠀⠀⠀⢠⣯⣄⠀⠀⠀⣭⠆
⠀⠀⠀⢀⡴⠟⢦⣀⠀⠀⠀⠀⠘⢿⡀⠒⢒⡮⠗⠀⠀⠀⢸⠁⢻⠃⣷⠀⣿⠀⠀⠀⠀⠸⠿⣿⣿⣿⡿⠛
⠀⠀⠀⠙⠦⠤⠦⠋⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⢸⡒⠚⠒⣿⣠⣿⣄⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⠉⠉⠛
⠀⢠⠖⡩⠍⡑⢢⠤⠤⠄⠀⠀⠀⠀⢀⡤⠔⡢⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣖⣒⣺⢷⣶⣶
⠀⢸⠐⠀⠁⠀⢸⠀⢸⠀⠀⠀⢠⠊⠡⢢⡟⠟⠈⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣤⠠⠄⠄⡏
⠀⢸⠀⠀⠀⠀⢈⠀⢸⠀⠀⠀⢸⡀⠢⡥⢐⣨⠖⠁⠀⠀⠀⠀⣀⣠⣧⡤⠤⣄⠀⠀⠀⠀⢸⣿⠀⠀⢡⡇
⠀⠈⠲⠤⠤⠔⠋⠉⠉⠀⠀⠀⠀⠘⠓⠛⠃⠀⠀⠀⠀⠀⠀⣿⢄⣈⣩⣥⡴⠟⠀⠀⠀⠀⠘⠿⠤⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⡶⣏⢿⣿⣷⡀⠀⠀⠀⠀⠀⢀⡠⠔⠓⠦⣄⠀⠀⠀⠀⠀⢀⡠⣴⡶⠀⠀⠀⠀⠀⠀⠀⠰⡛⢻⠂
⠀⠀⣸⡜⠂⠀⠹⣧⠀⠀⠀⢠⡒⠁⠀⠀⠀⠀⡈⢽⡂⠀⠀⢸⢳⠟⠛⠉⢉⣷⡀⠀⠀⠀⣀⠔⠋⠙⠢⡀
⠀⠀⠓⡤⡀⠀⠀⠣⡀⠀⠀⢸⡀⠀⠂⠠⠀⠈⣠⡾⠁⠀⣸⣍⡚⢀⣠⢞⡿⠛⠁⠀⠀⢸⠁⠀⠀⠀⠀⢱
⠀⠀⢀⡇⠈⡄⣀⣠⠗⠀⠀⠈⠙⠷⣤⣤⣤⠟⠁⠀⠀⠀⠓⠿⣿⡯⠟⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠊⠉⠁
⠀⠀⠀⠀⡤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⢷⣴⡟⠅⠀⠀⠀⢾⠙⡦⠤⠤⢤⡤⣤⠚⡆⠀⠀⣴⣊⡉⠉⢉⣲⡄⠀⠀⠀⠀⡼⠉⠻⠯⠿⡆
⠀⠀⠀⣠⠞⠀⢆⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠘⡶⠃⠀⠀⢸⣌⡉⠉⣉⢼⠁⠀⠀⠀⠀⣧⠀⠀⠀⠀⣴
⠀⠀⠀⣏⠉⠀⢸⡆⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠚⠁⠀⠀⠀⠀⢧⣍⣉⣠⠎⠀⠀⠀⠀⠀⡯⢀⣀⣀⡀⢽
⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠠⠀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠒⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠄⠁⠀⠈⠑⠠⣀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⢀⠔⠁⠀⠀⠀⠁⠉⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡐⠠⢤⣀⠀⠀⠈⠑⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⢀⠀⠈⠁⠂⠠⠱⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⢀⡾⠃⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⣸⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢥⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠘⠃⠀⠀⠀⠀⠰⣷⠀⠀⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠁⠀⠐⠶
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠀⠁⠀⠀⠀⠀⢀⡤⣀⠀⢡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌
⠀⠀⠀⠀⠐⣇⡀⠠⠀⣀⠁⠠⣀⡀⠀⠀⠀⣀⣠⠐⠁⢀⡀⠀⠀⠀⣀⡀⠀⢘⣦⠀⠀⢀⡛⢄⣂⡤⠀⣀⣀⣀⠠⠐⢻⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠂⠠⠄⠈⠁⠙⠂⠈⠓⠂⠀⠤⠀⠐⢈⣈⠂⠀⠀⢸⡆⠀⡌⠁⡑⠀⣀⠱⠈⠀⠀⠀⠑⠢⢀⠀⠀⠀⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠀⠀⠀⠀⠀⢀⡀⣰⠀⠲⠯⠤⠴⠄⠴⠾⠷⠤⠖⠲⠆⠀⠀⠉⠛⠀⠀⠀⠀⡀⠈⠓⠀⠠⠄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠈⠉⠀⠒⠂⠈⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠐⠀⠀⡀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⢀⡀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡀⠀⠈⠆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡘⠐⠰⠂⠀⠆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡀⣴⠀⠐⡄⠒⢴⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⡐⠀⡠⠀⠀⠀⠈⢢⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠕⠆⠦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⡏⢡⡀⠑
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠉⠒⠉⠉⡠⠂⠀⠀⢀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢐⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⢠⠂⠂⠒⠀⠀⠀⠀⠀⢤⢀⡗⠈⠢⠤⠄⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠉⢀⠽⡽⠋⠒⠂⠀⠀⠀⠀⠤⠐⡇⠀⠀⠀⠀⠀⠀⠈⠆⢄⠀⢄⢀⡀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠲⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⢠⡃⠀⢱⠂
⠀⠀⠀⠀⢬⠇⠀⠀⠀⠀⠀⢸⠌⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡶⢫⣜⠒⠊
⠀⠀⠀⠀⠂⠲⢁⠀⠀⠀⠀⠇⠀⠠⠒⠎⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡰⠡⠃⠀⠉⡆
⠀⠀⠀⠀⠀⠃⠈⡄⠲⠀⠴⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠰⠖⡐⠠⠁⠀⠀⠠⠁
⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠜⡰⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠏⡐⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⡠⠄⠒⠀⠈⠀⠈⢸⠄⠀⡠⠂⢸
⠀⠀⠀⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠀⢀⠼⠠⣀⣀⡀⠀⢠⡁⠊⠆⠀⠠⠀⠀⣉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⢦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠹⠿⠷⢮⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠈⢻⣧⠀⠀⠀⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠆⠀⠀⠀⠀⠀⠂⠋⠉
⠀⠀⠀⢀⣠⡶⢛⣵⠏⢙⣱⠃⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣾⡟⣴⣿⡏⠀⣾⠉⠀⠠⣤⣀⣀⡤⣆⣀⣀⠄
⠀⠀⣾⣿⣴⣿⣿⡇⠀⣷⣤⠀⠀⠀⠀⠀⠀⠐⠛⠋⠀⠂
⠀⠀⢸⣿⣿⢿⣿⣠⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠛⠛⠶
⠀⠀⢸⣿⣿⣿⣿⣿⢷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⡀
⠀⠀⢸⣾⣿⣿⣿⣿⢸⡏⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠠⠤⠤⠤⠶⠖
⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⡶⠦
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣦⣤⡀⢤⣤⣀⣔⠒⠀⠀⠀⡄⠉⠉⠉⠉⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡁⠀⠀⠀⠉⠉⠉⠀⠀⠁
⠀⠀⣺⣿⣿⣿⣿⣿⠿⠟⠋⠉⠉⠛⠛⠛⠋
⠀⠀⠀⠁⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣝⣄⡠⣤⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⢸⣽⠀⠀⠀⠀⠀⣠⠔⠊⠉⠈⠳⢄
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢰⡉⠀⠀⠓⠵⣄⠀⠀⠀⠀⢀⡞⡁⢈⢳⠀⢸⢸⠀⠀⠀⠀⠀⣿⣶⣄⣀⣀⣀⣴
⠀⠀⣠⠞⠛⠳⣄⠀⠀⠀⠀⠀⠘⠺⣄⠛⢁⡵⠚⠁⠀⠀⠀⢸⠀⡑⠀⢸⡄⢸⡎⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⠟⠉
⠀⠀⠙⠶⠤⠖⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⡛⠒⠒⢺⣥⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠉⡙⠛⠛⠛
⣠⠖⢒⡒⠒⢤⣀⡠⡄⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠶⢦⠤⢤
⡧⠀⠀⠈⠁⢸⠀⠀⡇⠀⠀⠀⢀⠔⢊⣥⣢⢟⠎⢱⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⡉⢡⠈⡅⡏
⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢟⠠⡀⠘⢉⠄⢂⡽⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⢀⡇
⢧⡀⠀⠀⢀⡴⠤⠔⠃⠀⠀⠀⠸⢦⣄⣯⡴⠚⠁⠀⠀⠀⠀⠀⣤⠖⠪⠿⠿⠀⢙⡇⠀⠀⠀⠀⠘⡇⠀⠂⠈⢸⠇
⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠵⢤⠶⠖⠛⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠰⡻⣎⢻⠿⢿⣦⠀⠀⠀⠀⠀⠀⡠⠔⠚⠁⠑⠲⣀⡀⠀⠀⠀⠀⣠⢴⣺⠽⣃⣀⠀⠀⠀⠀⠀⠀⠀⢫⣉⡼
⢀⣵⡈⠀⠀⠸⣿⠄⠀⠀⠀⡶⢉⠀⠀⠀⠀⠀⡀⠔⢿⠀⠀⠀⢸⠙⡟⠋⠉⠀⣨⡷⡀⠀⠀⠀⢀⠔⠋⠉⠉⠢⡀
⠈⠲⡄⠀⠀⠀⠹⡄⠀⠀⠀⣯⡀⠈⠐⠠⠄⠈⢀⣴⠟⠀⠀⣸⣍⡙⠁⣠⡴⣻⠿⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱
⠀⢸⠀⠙⡀⠀⣀⡼⠀⠀⠀⠉⠻⢶⣤⢠⡠⣲⠞⠁⠀⠀⠀⠓⢿⣟⣟⡵⠚⠁⠀⠀⠀⠀⠀⠰⡅⠀⠀⠀⠀⠀⢠
⠀⠈⠛⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠋
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠐⣇⣀⣼⡏⠀⠀⠀⠀⡴⢢⣀⡀⠀⠀⠀⠀⢀⢤⠀⠀⠀⢀⡤⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⢈⡟⢻⣧⡶⣦⡀
⠀⠀⠀⣹⠛⡇⠁⠀⠀⠀⠀⢳⣴⠏⠉⠉⠉⠋⠉⣧⢸⠁⠀⠀⢻⠒⠶⠶⠶⠶⣾⠃⠀⠀⠀⠀⡸⠀⠀⠀⠉⠉⢇
⠀⢀⡎⠁⠀⢱⡄⠀⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠈⡗⠤⠤⠤⠚⡏⠀⠀⠀⠀⠀⡗⠉⠉⠉⠉⠉⣿
⠀⢸⣇⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠞⠁⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠐⣺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠓⠦⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⣝⠻⠖⣲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⠀⠀⠀⣠⢾⡷⡾⢀⡜⠙⣆⠀⠉⢫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣃⠀⢀⡠⢞⣁⠀⠀⠷⠋⢀⣀⠘⢆⣦⢀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⡹⡴⠋⠀⠈⠈⠁⠀⠀⠀⠈⠈⠀⠈⠙⡶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡇⠀⠀⠀⢠⠄⠀⠀⠀⠰⡦⠀⠀⠀⡟⠱⡄⠀⢀⠜⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣇⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡷⠚⢁⡴⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠢⣤⠔⠀⠀⠀⢀⡴⠓⠋⠻⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣤⣀⣀⣀⣀⣠⣤⡖⠻⡇⠀⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢙⠲⠋⠀⠳⠋⢘⡶⡹⠓⠒⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢸⡀⠀⡠⠔⠶⣯⢾⣁⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠙⠲⣇⠀⠀⣸⠘⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡏⠓⢤⣀⠜⡽⠋⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⣄⣀⠀⠐⣏⠙⠒⢢⠀⠀⠀⠀⠀⣿⣷⡶⣫⣾⣶⣶⣶⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠰⢧⡐⢥⡄⣰⠇⢀⣤⢖⠒⠒⢿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⢀⣠⠇⠀⠉⠁⡴⠋⠘⠦⣳⢄⠀⠹⡟⢻⠛⠉⠛⢿⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠱⠤⠞⠁⠀⢀⡤⢊⠀⠀⢀⠀⠉⠻⣷⣤⠇⢸⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣔⠋⠀⠀⠀⡸⡏⠉⢹⡀⠀⠀⣸⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠴⣗⣋⣉⣹⠇⠀⠀⠿⢝⣛⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠐⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠃⠀⠋⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠈⠉⠀⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠀⡈⠃⡀⠀⠄⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈
⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠐⠒⠒⠠⠄⠤⣀⠠⠤⠤⠐⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢡⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⠤⠀⠒⠒⠒⠂⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠀⠀⠈⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⡄⠠⠄⠀⠤⠄⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡅⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⢘⡂⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠐⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⣿⡿⢿⣿⠿⣿⡿⡿⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⣀⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣹⠿⠿⢿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⡶⠟⢻⡿⣿⡟⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢁⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⡀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⢀⣴⠞⢋⣽⣿⣿⢟⣛⠉⠉⠉⠙⠛⠛⠿⣿⣷⣍⠳⢤⡀
⠀⢠⣾⢁⣴⢋⠊⢸⡿⢷⣌⠉⠁⠒⠂⠠⠤⢀⣀⠉⠻⣷⣄⠻⡆
⠀⢸⣇⣾⡣⠃⣠⡟⠁⠀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠈⠐⠨⣿⣦⡿
⢀⣴⢿⣿⠷⠞⠋⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣤⣤⣤⣴⡶⠟⣿⠷⣄
⣿⠁⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿
⠻⣆⡈⣿⡄⠀⠀⠀⣠⡅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣠⡟
⠀⣨⡿⠿⣿⣆⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠉
⠀⣿⠐⠦⣈⡻⢷⣤⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣀⣴⢾⣿⣤⡀
⠀⢙⣿⣶⣶⠾⣷⣎⡙⢿⡶⠶⣶⣶⠶⠶⣾⡟⠉⠀⢀⣿⠉⠻⣦⡀
⢠⡟⢡⣄⣹⡆⠀⣿⢸⢸⣧⡀⠈⢙⣿⢿⡟⢷⣤⣴⢿⣿⠀⠀⠈⢻⣦
⠈⠷⣤⣉⣩⣤⡾⠃⣋⣾⠏⠻⠿⠋⠙⢹⡇⠀⠀⠀⠀⠹⣆⠀⣀⣾⠿⣦⡀
⠀⠀⠀⠙⢻⣿⣿⡟⢉⡿⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠹⣷⡟⠁⠀⠈⠻⣦⣀
⠀⠀⠀⠀⣼⡿⠉⠛⣿⢇⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⢀⣾⡟⠁⠀⢰⣿⡴⠀⠀⠀⠀⠠⡶⣦⠀⠀⠀⠄⣀⣠⠿⣿⣿⣷⣄⠀⠀⠀⠈⢹⣧
⠀⠀⣾⡿⠀⠀⢀⣾⠋⠛⠛⠓⠶⠶⠶⠶⠿⠷⠒⠛⠋⣉⣠⡴⠟⠻⣷⡙⢿⣄⠀⠀⢸⡇
⠀⢸⣿⠁⠀⣰⡟⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠿⠛⠛⠋⠁⠀⢀⡀⠈⢻⣾⡇⠀⠀⣿⡇
⠀⣿⠃⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⣤⣤⣶⣾⢿⣧⠀⠀⢹⣷⡀⢰⣿⠇
⢸⡟⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⢻⠙⣿⠀⢿⠈⣿⠀⠀⠀⣿⣿⠿⢿⣦
⠸⣇⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠘⡆⢹⡄⢸⠀⣿⠀⠀⠀⢸⣿⠀⠀⣿⣇
⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢣⠀⢇⠀⣿⡏⠀⠀⠀⣼⣿⡶⠾⣿⣝⢳⣤⣴⣶⠶⢶⠶⣤⣄⣀
⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣆⣿⣼⠟⠀⠀⠀⣴⢿⠙⣷⡄⠀⠉⣻⣿⣿⡿⢿⣮⣑⠒⣈⡿
⠀⠀⠀⣿⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣀⣴⢾⠏⠁⣠⠟⠁⠀⢠⣿⠹⣿⠿⠷⣿⡙⣿⣏
⠀⠀⠘⠻⢿⣬⣯⣽⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠶⠿⣝⣉⣥⡴⠛⠁⠀⠀⠀⢸⡟⢆⢹⣷⢄⠙⣯⣍⣹⣷
⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⠻⠿⠿⠿⠿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠸⣇⠬⢸⣿⠀⣌⢿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⡟⠳⣦⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡇⡇⢸⠀⠀⠀⠀⠀⠀⠀⡰⢻⣤⠉⠁⠀⠀⠈⠉⢠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡿⠀⡟⢻⠡⠦⣤⣦⡄⢠⡄⠁⢸⢱⠟⡇⣠⡆⢠⢀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⣿⠀⠀⠀⠀⣼⡃⠀⠃⠸⠀⠒⠙⠉⠈⠛⠂⠀⠛⠈⠀⠙⠛⠓⠛⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠷⣶⣿⣀⣴⠞⠛⠛⠛⠷⣄⠀⠀⠀⠦⠤⠄⠰⠤⠤⠄⠤⠰⠤⠴⠰⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⠀⠀⠀⠀⠀⢘⣿⡟⢁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠶⠶⠶⠶⠶⠾⣿⡟⠟⠛⠛⠛⠛⢛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⡶⠾⠚⠛⠛⠛⣿⡀⡇⠀⠸⡆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣿⡇⡁⠀⠀⠀⢀⣀⣿⡀⡇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠘⠛⠛⠛⠋⠉⠉⠀⢻⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⣟⡇⣴⣦⣦⣶⣶⣶⣶⣗⢸⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⣟⡇⢿⡿⠿⠿⠿⠿⠟⠻⢸⡄⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⢸⠀⠀⠀⡴⡳⡄⠀⠀⠘⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠦⡄⠆⠀⢀⣸⠄
⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⣿⢸⠀⠀⡰⠱⣧⣌⢆⠀⠀⡇⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠥⠅
⠀⠀⠀⠀⢿⠀⠀⠀⢰⣿⣿⢸⠀⣰⢁⣀⣈⣻⡀⠷⠀⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣀⠤⠄⠀⠀⠀⣸⠩⣩
⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⢸⠀⠈⣤⣤⣤⣤⢶⠆⠀⢹⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⡜⠘⠠⠠⠤⠤⠞⠻⠓
⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⠸⡀⠀⠹⠿⠏⠿⡘⣃⡀⢸⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠉⠉⠀⠉⠁
⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⠀⡆⣿⢽⣸⣿⣿⣿⢿⣧⠸⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢸⡀⠀⠀⣿⣿⣿⠀⡇⠉⠈⠉⠉⠀⠁⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⢸⡇⠀⠀⣿⡿⣿⡀⠣⠄⠐⠐⠒⠒⠂⠀⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⣴⠄⠀⣤⠄⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⡇⠀⠐⣿⢄⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⡇⠀⠀⣿⠿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢀⣷⣀⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠉⠉⢀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢰⣿⡿⠿⠛⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣇⣀⣀⡈⠿⡟⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢯⣰⣶⡌⢦⣈⠻⣗⠲⢦⣀⡄⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠈⠻⣾⡛⣄⣿⠿⠺⣿⣟⠺⣿⣷⠄⠀⠀⠈⠈⢻⣿⣿⠏⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠈⠛⠀⢀⣀⣀⣀⣤⣬⠄
⠀⠀⠀⠀⠀⠈⢷⣴⡹⠶⠤⣌⣐⢫⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢐⢞⣻⡧⢀⣔⢶⣲⠟⡟⣿⣿⣡⢱⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠓⠒⠠⠍⠁⠀⠀⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⠸⢿⣿⣷⣹⠿⠿⠿⠚⠛⠋⠉⠉