Atomic Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣬⡻⠟⠁⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣼⡷⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⢀⣬⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠠⠤⠤⠤⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡿⢿⣿⣿⡿⣸⣿⡏⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⢰⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⠘⠻⠿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⡶⠿⠿⣿⣷⣦⡀⠀⠈⠙⠿⣿⣶⣄⠀⠀⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣼⣿⠀⠀⠀⢈⣿⣿⣿⡄⠀⠀⠀⠈⠻⣿⣷⣦⡻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣿⣤⣦⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣹⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⠀⠀⠀⠀⣿⣿⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣠⣴⣿⡿⠋⠀⠛⣵⣾⠶⣆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠛⠻⠿⠿⠛⠋⠀⣠⣴⣿⡿⠟⠉⠀⠀⠀⠀⢿⣷⣤⠞
⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⣆
⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣀⣠⣤⣶⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣤⢀⣴⣶⣦⡀⣴⣶⣾⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠸⣇⣈⣿⠇⠛⢫⣭⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠤⠄⣀⣠⣤⣤⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⣀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣦⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠀⠂⠀⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣀⣤⣤⣶⡶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠀⡀⠀⠀⠀⠀⢀⣨⣵⡶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠉⠀⠀⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⣀⣴⡾⠛⠁⢀⡔⠈⢳⣦⡀⠀⢹⡇⠀⢄⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡻⣷⣍⡁⠀⠀⠀⢾⣦⣤⣼⣿⡇⠀⢸⡇⠀⠀⢀⠶⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⣮⠁⠀⠙⠿⣶⣄⡀⠈⠻⠿⠿⠟⠁⠀⣸⡇⠠⠂⠀⠀⠈⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠁⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠛⠿⣶⣤⡀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠐⠀⠀⠀⠀⠀⠉⢡⣿⢰⣦⣤⣀⠀⢠⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠄⠀⠀⠠⠤⠐⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠉⠉⢷⣴⡿⠶⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣴⠟⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠙⠉⣴⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠿⠿⠿⣿⣶⣶⡤⣼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⣹⠓⠿⣿⣶⣤⣀⠀⠀⢀⣀⣠⡤⠼⣿⡖⠒⠒⠒⠒⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣉⣙⠿⣿⣾⣍⡁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣁⡤⠖⠋⠉⠀⠀⠀⠙⠻⣿⣶⣄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⣠⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⣠⠶⠋⠁⢸⡇⠀⠀⠀⢀⣤⣶⣶⣶⣦⡀⠀⠀⠀⠀⣿⣿⣴⣄⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠳⣅⠀⠀⠀⢸⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿⣿⠙⢿⣷⡀⢀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠈⢷⡀⠀⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⡇⠀⠀⡹⣿⣮⡛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠙⠇⣿⠀⠀⠀⠀⠀⠛⠿⠿⠿⠟⠁⠀⠀⠀⢰⣿⣇⣴⣿⠿⠊⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⠘⠋⠁⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠸⠟⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠦⣄⠀⠀⢀⣠⣤⣶⣿⠿⢃⣿⡏⠀⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⢀⣀⣠⣤⣬⣝⡶⣝⡛⠋⠉⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠠⠿⠿⠿⠿⠾⡟⠛⠛⠋⠉⠁⠀⠀⠉⠓⠦⣄⢠⣿⠏⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠒⠶⠤⣤⣤⣤⣤⣤⡈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣤⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠉⠉⠙⢲⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⡴⠖⠛⠉⠉⠀⠀⠈⢱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⠳⢦⣀⠀⢀⡤⠖⠋⠁⠈⣇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⡬⠛⢯⡀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⣀⣯⣤⣤⠤⣤⠶⠯⠶⠤⠤⠽⠶⣤⣤⣤⣀⣇⣀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡤⠴⠚⠋⠉⢩⡇⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢻⠉⠉⠙⠒⠢⢤⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⠋⠹⡄⠀⠀⠀⠀⢸⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⠀⠀⠀⠀⠀⡼⠁⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⡟⠀⠀⠹⣄⠀⠀⢀⢾⠁⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠈⡧⡀⠀⠀⡼⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠁⠀⠀⠀⠀⠘⢦⡴⠃⢸⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⡇⠹⢦⡼⠃⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⢠⡞⢳⡀⢸⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢠⡇⣠⠞⢧⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⡀⠀⠀⠀⢠⠏⠀⠀⠙⢾⡇⠀⠀⠀⠀⠀⠙⠛⠿⠛⠁⠀⠀⠀⠀⠀⢸⡷⠁⠀⠈⢳⡀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣠⡏⠀⠀⠀⠀⠘⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠀⠀⠀⠀⠀⢳⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠓⠲⠤⢤⣄⣧⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣀⣾⣤⡤⠴⠖⠚⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢹⡉⠉⠉⠻⣏⡒⠒⠒⠒⣫⠟⠋⠉⠉⢩⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⠙⢦⣤⠚⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣀⡤⠞⠁⠈⠙⠲⢤⣀⠀⣰⠃⢀⣀⣀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⠀⢀⣀⣠⡤⢿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠈⣹⠟⢲⣿⣿⣿⣇⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠘⠿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠒⠒⠒⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠖⠒⠒⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡶⢫⠤⠤⠤⠤⠤⠤⠌⠳⣄⠠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣠⠤⠶⠚⠋⠉⣉⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠉⠉⠉⠛⠲⠦⠄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⡇⠊⠉⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⢰⠇⠈⠲⢦⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠹⡄⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⢻⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⠉⠡⡄
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠳⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣰⠏⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣰⠃⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⢰⠏⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⡏⣀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢹⡄⠀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠉⠙⠲⠦⣤⣀⣀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣀⣠⡤⠄⣧⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠈⠳⣤⡰⠤⠤⠤⠤⢤⡤⠄⣠⠞⠡⠶⠶⠒⠛⠛⠉⠉⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⡀⣿⣿⣿⡆⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⢠⣖⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣚⡻⢦
⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠈⠉⠓⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡎⠉⠉⠒⠢⠤⣉⡒⠤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡠⣒⠥⠂⠉⠀⠉⣱⣖⣒⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢣⠀⠀⠀⠀⠀⠀⠈⠑⠢⢍⡓⠤⣀⠀⢀⢔⠿⠊⠀⠀⠀⠀⠀⠀⣹⠿⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣬⡙⢷⡁⠀⠀⠀⠀⠀⠀⠀⠀⢇⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠱⡀⠀⠀⠀⠀⠀⠀⢀⣀⣞⣕⣉⣳⣬⡲⢄⡀⠀⠀⠀⣀⣀⡜⣜⠁
⠀⠀⠀⠀⠀⠀⣀⡔⠒⢦⡤⠼⢆⠑⣚⡉⠩⠭⠤⠄⣶⡶⠂⠒⠒⠋⠉⠉⠒⢝⡳⣄⠒⠒⡺⠱⠓⠂⠭⠔⣒⣒⠤⢀⡀
⠀⠀⠀⠀⡠⣿⢻⡩⣽⣾⠟⠉⠉⠳⡈⢂⠀⠀⢀⢎⠟⠀⠀⣠⠴⠶⠦⣄⡀⠀⠙⠪⣑⡴⠡⠃⠀⠀⠀⠀⠀⠀⠉⠓⢬⡑⠢⢄⡀
⠀⠄⡔⠉⠀⡼⠳⠀⠀⠀⠀⠀⠀⠀⠈⢆⠑⣤⢣⠂⠀⠀⣼⠥⠀⣤⣀⢈⣷⡜⢴⠀⡸⡡⢟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⢹⢀⠂
⢇⠊⢧⡀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣌⠣⠀⠀⢀⢿⢟⢠⣌⣡⣴⢿⠇⠈⡴⡱⠁⠀⠑⢝⢦⡀⠀⠀⠀⢀⡠⠞⠁⣀⠔⠡⠊⠁
⠀⠀⠀⠈⠓⠤⣀⠙⠒⠤⢤⣀⡀⠀⠀⡰⠡⠉⠑⢕⢄⠑⠀⠙⠺⠿⠿⠟⠁⢀⢞⠜⠁⠀⠀⠀⣈⡳⠽⠤⢐⡊⠥⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠠⠤⠭⢍⣉⣉⣀⠒⠒⠓⠛⠢⠀⢀⠀⠀⢀⡴⣡⣻⠶⠬⠩⠉⠅⠐⠒⠈⠻⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠇⠀⠀⠀⠀⠀⠀⠉⠻⣿⢄⡠⡫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠾⢷⣄⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢰⠁⠀⠀⠀⠀⠀⠀⢀⣤⡪⠊⠀⠀⠀⠉⡎⠗⣹⣦⡄⡄⠀⠀⠀⠀⠀⠀⠀⢸⠂⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢸⡀⠀⠀⠀⢀⡠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠛⠿⢟⢝⣿⣿⣾⠤⢀⣀⣠⠤⠔⠁⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⠒⠒⣊⠕⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠊⠀⠉⠒⠒⠤⠤⢀⡀⠤⠖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⣒⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠂
⠀⠀⠀⠄⠢⢂⢤⠄⡠⣂⢔⠄⡠⢀⢈⣈⠔⡠⣢⢄⢔⠠⢂⢔⡠⡢⢂⠔⡠⡠⢀⢄⠄⡠⣠⢀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠘⠑⠙⠁⠊⠊⠔⠱⠁⠎⠰⠣⠊⠞⠡⢪⢊⠔⠵⡵⡪⠪⢔⠕⡩⣺⢕⡡⡪⡪⢔⢔⢔⢤⡄⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⢀⣤⡄⢀⠉⢀⣠⣤⡀⢀⣀⣤⠀⠀⣠⣤⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⣦⢰⣏⠀⠀⣼⠀⣿⠷⡚⠁⣾⠋⢸⠇⢾⡁⡀⢰⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠉⠉⠀⠁⠀⠈⠉⠉⠀⠁⠀⠈⠀⠈⠉⠁⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⣀⡀⠀⢰⠁⠀⠀⠀⠀⠈⡆⠀⢀⡠⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠑⡧⠀⠀⠀⠀⢀⣠⠼⠊⠁⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⠁⠀⠓⣄⡔⠋⠀⠀⡆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⣀⣸⢀⣠⡖⣁⣉⣦⣀⠀⣁⣀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⢏⠉⠀⠀⣧⠜⠁⠀⠀⠀⠀⠈⠢⣸⠀⠀⠉⡹⠓⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠣⣀⠔⡇⠀⢠⣾⣿⣿⣦⡀⠀⢸⠦⣀⠜⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⡠⠛⢄⡇⠀⠸⣿⣿⣿⣿⠇⠀⢸⡴⠛⣄⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄⢀⣀⡜⠀⠀⠀⡗⢄⠀⠈⠉⠉⠁⠀⡠⢺⠀⠀⠈⢢⣀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠉⠁⠒⡗⠀⠳⣤⠠⠄⡤⠞⠤⠾⠀⠒⠂⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢠⠀⠀⢈⡵⠮⡀⠀⠀⠂⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⣸⡤⠚⠁⠀⠀⠈⠑⠠⣀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠄⠒⠉⠀⢆⠀⠀⠀⠀⠀⠀⡔⠀⠈⠒⠤⠄⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠤⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣽⠆⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⠋⠁⠐⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠉⠛⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣀⠔⠀⢸⡇⠀⢠⠖⠛⢷⣆⠀⠀⠀⠀⠑⢄⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠻⢦⣀⢸⡇⠀⠸⣷⣴⣿⡿⠀⠀⠀⠀⣠⠾⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠙⠿⣧⡀⠀⠈⠉⠉⠀⠀⣠⣼⠟⠁⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢿⡛⠷⣦⣄⣤⡶⠟⠉⠁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⣀⣀⣼⣷⡶⠛⠋⠛⠳⢶⣼⣤⣀⣀⣀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠘⢷⣄⠀⠀⣀⠜⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠉⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⡿⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⣠⡴⠶⠛⠉⠉⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢷⠀⠀⠀⠀⠀⢀⡼⠻⣦⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠈⠓⠦⠤⠤⠶⠋⣠⡾⠋⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⣠⣾⣯⣤⣤⣤⣤⣤⣤⣀⣈⣿⣆⣀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⠶⠟⠛⠛⠉⣽⠟⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠻⣿⡛⠻⠷⠶⣦⣤⣿⣀
⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠉⣿⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢸⡟⠛⠻⢶⣤⣀
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⢹⡆⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢀⣿⠁⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠈⣿⣠⡟⠁⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠓⣦⠀⠀⠀⠀⠀⠈⢿⡄⣼⠇⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⢀⣿⠻⣧⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣼⠟⣷⡀⠀⠀⢀⡠⠶⠶⠦⣄⣾⠃
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣼⠇⠀⠹⣧⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣼⠏⠀⠸⣧⠀⢠⠎⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⢰⡿⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⢻⡆⢸⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣧⣄⣀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⣀⣸⣷⡾⢦⣀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠛⠛⠷⠾⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣤⡶⠶⠾⠛⠋⠉⣿⡄⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠶⠦⣄⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⠀⠈⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣀⣴⠾⠋⠀⠉⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠷⣦⣤⣤⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⠶⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣧⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠟⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⢄⣀⠇⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠤⠒⠉⠉⠀⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠒⠤⣀⠀⣀⠤⠒⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡠⠚⠉⠢⢄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⣀⡴⠊⠁⠀⢀⣀⡀⠀⠈⠓⣧⡀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣀⠴⠊⠁⡆⠀⠀⡜⠁⠀⠈⢢⠀⠀⡇⠈⠑⠤⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠲⢄⡀⠇⠀⠀⢣⡀⠀⢀⠜⠀⠀⡇⢀⡠⠒⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠈⠉⠁⠀⢀⡤⡟⠁⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⢤⣀⠔⠊⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠤⠒⠉⠀⠉⠒⠤⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣄⣀⡀⠤⠒⠊⠉⡆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠑⠒⠤⣴⣶⡄⠀⠀⠀⠀⠀⡱
⠀⠀⠀⠀⠀⠈⠑⠒⠒⠂⠻⠿⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠙⠛⠋⠁⠐⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⠉⠛⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠉⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣤⣀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣀⣀⣘⣷⣄⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣠⣶⣾⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠶⠾⢿⡟⠛⠛⠛⢛⣿⠟⠛⠋⠉⠙⠙⠛⠻⣿⡛⠛⠛⠛⢻⡿⠷⠶⣦⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠋⠉⠁⠀⠀⠀⠀⠈⢿⡄⠀⢠⡾⠁⠀⢀⣴⣶⣶⣦⣀⠀⠘⢷⡄⠀⢠⡿⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢀⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡟⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⢻⣶⡟⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁⠈⠙⣷⡀
⠀⠀⠀⠈⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣧⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⣼⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⣄⣀⡀⠀⠀⠀⠀⢀⣾⠃⠀⠘⢷⡄⠀⠘⠻⠿⠿⠟⠋⠀⢠⡾⠃⠀⠘⣷⡀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠶⢶⣾⢧⣤⣤⣤⣬⣿⣦⣤⣄⣤⣀⣀⣤⣴⣿⣥⣤⣤⣤⡼⣷⡶⠶⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠙⢿⡍⠉⠉⢩⡿⠋⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢀⣤⣴⣦⣄⣠⡾⠋⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⣀⣤⠾⠛⠿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣀⣀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⣄⡄⠀⠀⣤⢠⣤⠀⠀⠀⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠔⠎⡛⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠃⠇⠘⠂⠉⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⡀⢸⣳⡠⠔⠁⠀⠀⢀⡜⠓⢤⡀⢰⠋⠀⠀⠠⠀⠠⠀⡇⠀⣰⣦⡁
⠀⠀⠸⢇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⣼⣠⠏⠀⠀⠀⠐⠀⠈⠀⠀⠈⠀⠸⡆⢄⠀⢅⠀⣐⠼⠁⠀⡇⡜⠃⠀⠀⠀⣀
⠀⠀⠀⠘⡆⡜⠀⠀⠀⠀⣤⠀⡰⠂⠀⠀⠼⢾⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠒⠒⠟⣶⠛⠛⠉⠁⠀⠀⠘⡇⣇⠀⢀⡰⢶⣝
⠀⠀⠛⢄⢻⡇⠀⠀⣠⡞⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⢉⠦⠤⠤⠴⢠⡤⣧⡄⢼⠆⠀⠀⢰⠉⠂⠌⣐⠀⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢳⣈⠉⢉⣀⠜⠋
⠀⠀⠀⠈⢢⡇⠀⠀⢈⣧⡀⠀⣸⡄⠀⠀⡠⡄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠘⣆⠌⠐⠀⢁⠂⠈⠂⡼⠀⣠⣤⣄⠀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢳⠀⠀⠁⠘⠛⠙⡍⠀⠀⠀⠉⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⡽⠀⣀⡠⢤⠤⡬⠷⣤⠤⠤⠤⠖⠋⢀⢮⠞⠁⠀⡴⢹⢀⣤
⠀⠀⠀⠀⠀⢹⢠⠶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡿⠋⠀⢸⠁⠠⠀⠀⠀⠄⠈⢦⠀⠀⠀⠀⢸⠘⠒⣦⡸⠁⠓⠊⡼⠀⣾⣿⡆
⠀⠀⠀⠰⢆⢸⡏⠀⠀⡠⠤⠤⣀⣀⠴⠊⠳⡀⠀⠀⠀⠀⣀⡠⣤⠄⣀⣀⣀⠘⣦⠈⡀⠀⠀⠄⠀⣘⠇⠀⠀⠀⠈⠓⠊⠡⡷⠒⡆⣸⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠈⢾⡇⠀⠘⣅⠀⠀⡼⠉⠓⢦⣸⠁⠰⠀⠖⠉⠃⠀⠀⢀⠠⠐⠉⠉⠉⢷⠚⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⠖⢲
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⢦⡞⠁⠺⠇⢠⡏⢢⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⣤⣤⠀⢠⣤⡄⠀⣠⣤⡀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠠⠂⢼⠙⠲⢄⣠⠋⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣠⡇⠀⢸⢹⠀⠀⠀⡿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡴⠞⠁⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣾⠀⢸⣿⡇⠀⢸⢸⠀⠀⠀⣙⠇⢸
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀⠠⠍⡇⠀⠀⢀⢏⣌⢧⠀⠀⢸⣿⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣧⠄⢾
⠀⠀⠀⠀⠠⠀⡷⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠛⠀⡇⠀⠀⢸⢸⣸⢸⠀⠀⠘⠋⠀⠈⠛⠁⠀⠘⠛⠀⠀⠀⠿⠤⠼⠰⠆
⠀⠀⠀⠀⠀⠀⡧⡇⠀⠁⠀⠀⠀⠀⠀⠀⠐⡿⠷⢶⠂⠀⠀⢸⢰⡏⠙⢆⡇⠀⡆⠘⣆⣡⠞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⠀⠀⠀⠀⠀⠀⢀⠇⠂⠘⡆⠀⠀⠀⠶⠁⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⢠⣾⡁⢀⣀⣠⠔⠛⠓⢤⣠⠎⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⢹⡀⠀⠀⠀⢀⠀⢀⠀⠀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠈⠛⠿⡀⠀⠀⠀⠈⠋⠀⠤⠆⠀⠈⠁
⠀⠀⠀⠀⢀⡞⠤⣈⣆⠀⠀⠀⠀⠀⠀⣸⠃⠠⣕⠀⠀⢧⠀⠀⠀⠉⠀⣿⠀⠀⠀⢰⡆⠀⢇⣤⣤⣀⣀⣀⣀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡞⠈⠘⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⠀⣤⡄⠀⢶⠀⠀⠀⠣⠤⠄⣀⣀⣀⣀⣤⠀⠀⢠⣤
⠀⠀⠀⠀⢣⣀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⠐⣿⠀⠀⠀⢸⢿⠀⣿⡇⠀⣿⠀⠀⡤⡄⠀⠀⠀⣶⠀⠀⡷⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣠⡀⠙⠿⢿⡿⢿⡿⠿⠿⠿⠿⠛⠀⠀⠀⢿⠀⠀⠀⢸⢸⠀⡇⡇⠀⣿⠀⠀⡇⡇⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢖⣢⣀⣀⣹⣷⣤⡤⠼⢷⣦⠤⠤⠄⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⢸⢸⠀⡇⡇⠀⡿⡆⠀⡇⠇⠀⠀⢸⢸⠀⠀⣷⡇⠀⠐⢶
⠀⠀⠀⢀⣤⣤⠔⠊⠉⡿⠃⠀⠐⠖⠁⠀⠺⠊⠀⠀⠀⠀⠀⢤⡄⠤⠤⢾⠤⠶⠶⠾⠾⠶⠷⠷⠶⠻⠗⠦⠗⡖⠒⠒⣼⠛⠒⠲⠟⠓⠿⠆⠙
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠘⠈⠀⠈⠉⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠢⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⣤⡶⠶⠶⣦⣄⡈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⡾⠋⠁⠀⠀⠀⠀⠙⠳⣄⡀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠦⠖⠂⠀⠄⠀⢠⡾⠁⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣀⣤⠤⠶⠦⠤⢤⣄⡀⠀⠀⠙⠠⢀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⣀⡤⠴⠶⠶⠶⠤⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠞⠁⠀⠀⠀⠀⠀⠀⡼⠁⣰⠖⠢⢄⠀⠈⠐⠤⣀⡤⠞⠋⠀⣀⠤⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠘
⠀⠀⠀⠀⠀⠀⢀⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⡏⠀⠀⠀⠉⢳⣤⠀⠀⠀⠀⡔⠉⠀⠀⠀⠈⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⢀⡿⠀⣿⠃⠀⣀⣠⡶⠋⠀⣠⣄⣄⠀⠈⠒⢄⣀⣀⠀⢹⠆⠈⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⣧⠀⠀⢀⣀⢠⠤⠼⠃⠀⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠛⠃⠀⠳⠶⢤⣄⣀⠀⠀⢀⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢫⠀⠙⠒⠈⠉⢀⣀⣠⣤⠀⢰⣶⠖⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣶⣆⠀⢀⡤⣄⣀⠉⠙⠓⠚⠀⢰
⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⢴⡖⠋⠉⠁⠀⣿⠀⠘⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠛⠀⠈⡇⠀⠉⠉⠓⢶⠀⢰⡏⠁⠠
⠀⠀⢀⠴⠁⢀⡤⠞⠹⣆⠈⢷⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⣧⡀⠀⠀⢀⠃⢀⡿⠙⠢⣄
⠀⣠⠏⢀⡴⠋⠀⠀⠀⠹⣷⠈⢷⣀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢷⣄⠠⠃⢀⡾⠁⠀⠀⠈⠳⣄⠀⢢
⢠⡇⢀⡾⠀⠀⠀⠀⠀⠀⠹⣧⠀⠋⠀⣠⡄⠀⡇⠀⠀⠀⠀⠀⠀⣰⡋⠁⠀⠀⠨⠿⢧⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⠙⠁⣰⡟⠁⠀⠀⠀⠀⠀⠘⡆
⢸⡇⠸⣇⠀⠀⠀⠀⠀⠀⠀⡹⠃⠀⠐⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⣯⡔⠀⠀⠀⣶⣿⣺⠀⠀⠀⠀⠀⠀⠁⠀⣼⠇⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⡿
⠈⣧⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⢀⠶⣄⠈⠁⠀⡃⠀⠀⠀⠀⠀⠀⠘⣿⣶⡰⣶⣟⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢄⠀⢳⡄⠀⠀⠀⠀⠀⣼⠃⢀
⠀⠈⢷⡀⠙⢷⣄⠀⠀⠀⠀⢠⠂⠀⠙⢷⡀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⢢⠀⠹⡄⠀⢀⣠⠞⠁⣠⠈
⠀⠀⠀⠙⢦⣄⠉⠻⠾⠂⢠⡇⠀⠀⠀⠈⡇⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡇⠀⠀⠀⠈⢧⠀⠛⠖⠋⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠉⠛⣦⠀⠀⠉⠛⠶⢦⣤⣤⣷⠀⢸⣦⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣴⣶⠀⢰⡇⠀⠀⠄⠒⠉⠀⠀⢠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣾⠛⠷⠶⣤⣄⡈⡉⠀⠈⠛⠛⠂⠀⠑⠦⠤⠤⠤⠤⠤⠤⠤⠤⠞⠁⠐⠚⠋⠁⠀⠀⣀⣀⣀⡤⠶⠚⢻⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠐⠇⢠⡇⠀⠀⠀⠀⠉⠉⢹⡆⠀⢶⠶⠶⠶⣦⡀⠀⠀⢤⣤⣤⡄⠀⢀⣠⠤⠤⠶⠶⡄⠀⡾⠉⠁⠀⠀⠀⠀⠀⡇⠀⠗
⠀⠀⠀⠀⠀⠀⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷⠀⢰⡀⠀⠀⠀⠙⢦⣄⠀⠉⠁⣠⡶⠛⠁⠀⠀⠀⣸⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠈⡇⠀⢷⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢧⠀⢀⣠⡴⠚⢁⣠⣤⣀⠀⠁⠀⢀⠀⠀⢰⡇⠀⡞⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠻⣦⣀⣀⡀⣀⣀⣀⣠⠀⠀⠛⠉⣀⣤⡶⠛⠉⠀⠈⠙⠷⣄⣀⠈⠀⠛⠀⢨⡀⠀⠀⠀⠀⢀⣀⡴⠋⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡉⡉⠉⠉⢉⣉⣀⣤⡀⠐⣟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⣀⣀⡈⠉⠉⠉⠉⠉⠁⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠉⠀⠡⡀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⡰⠁⠈⠉⠉⠀⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⣀⣤⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣀⣤⡶⠾⠟⠛⠛⠿⠿⢿⣿⣶⣦⣤⣄⣀⠀⠀⠀⢀⣀⣤⣶⣶⡿⢿⡟⠛⠛⠛⠛⠛⠛⠻⢶⣄
⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠉⠛⢿⣿⣶⣾⣿⡛⠋⠁⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣠⣴⣾⠿⠛⠉⠙⠻⢿⣷⣦⣄⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠈⢧⣀⡀⠀⠀⠀⠀⢀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⢠⣾⣿⣿⣷⠀⢀⣴⣿⠟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣦⡀⠀⠀⢀⠎
⠀⠀⠈⠿⣿⣿⡟⣰⣿⠟⠁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣄⠙⢿⣦⡠⠃
⠀⠀⠀⠀⠀⢠⣾⢿⡁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠙⢿⣆
⠀⠀⠀⠀⣰⡿⠃⠀⠙⠦⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠈⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠂⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢤⣤⣤⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣴⠞⠛⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣀⠤⠴⠒⠚⠋⠉⠉⠉⠉⠙⠛⢦⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⢀⡀⠤⠒⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠘⣃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⢀⣤⠔⠊⠁⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄
⠀⠀⠀⠀⢼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠏
⠀⠀⠀⠀⠘⠻⠿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⣠⡶⢻⠁⠀⠀⣠⡴⣀⣐⢿⡷⡆⢠⠀⠀⠀⠀⠘⠂⢀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣀⣴⠟⠁⠀⣼⠀⠀⠀⣏⣾⣿⢟⣥⣬⡻⣿⡄⡀⠀⠀⠀⠀⠀⠑⢦⡀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢏⠀⠀⠀⠀⣿⠀⠀⣟⣤⣝⠇⣿⣿⣿⣿⢸⣥⡅⠀⠀⠀⠀⠀⠀⠀⣹⠾⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠘⠢⡀⠀⣿⠀⠀⠉⣼⢃⢿⣮⣽⡏⢵⣿⣿⠇⠀⠀⠀⠀⢀⠠⠊⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠑⢿⡀⠀⠀⠘⠘⢷⣯⡽⢼⡷⠏⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⡀⠄⠊⠉⠻⢶⣤⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠘⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣾⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠈⠛⠿⠶⠶⠶⠶⠶⠾⠛⠛⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠉⠙⠛⠿⠶⢶⣶⣦⣴⣶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠟⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⡀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⠿⡿⠻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⠿⠿⠿⠿⠿⢿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠿⠛⠛⠻⠿⠿⢶⣶⣶⣦⣤⣄⣀⠀⠀⢈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⢀⣠⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣷⣆⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣴⠿⠛⠉⠀⠀⠀⠀⣀⣤⣤⣶⡶⠶⠶⣶⣶⣦⣤⣄⡀⠀⠀⠙⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣤⡶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠠⡀⠀⠀⠀⠉⢻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⢀⣤⣶⣶⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣤⣀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣠⣤⣶⡶⠶⠶⠶⠶⣦⣤⣄⡀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠋⠉⠉⠉⠉⠙⠛⠷⣶⣄⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠛⠋⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣶⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⣖⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣚⡻⢦
⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠈⠉⠓⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡎⠉⠉⠒⠢⠤⣉⡒⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣒⠅⠂⠉⠀⠉⣱⣖⢒⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢣⠀⠀⠀⠀⠀⠀⠈⠑⠢⢍⡒⠤⡀⠀⢀⢔⠯⠊⠀⠀⠀⠀⠀⠀⣸⠿⣲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣬⡙⢷⡁⠀⠀⠀⠀⠀⠀⠀⠀⢇⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠱⡀⠀⠀⠀⠀⠀⠀⢀⣀⣜⣕⣉⣳⣬⡲⢤⡀⠀⠀⠀⣀⣀⡜⣘⠁
⠀⠀⠀⠀⠀⠀⣀⡔⠒⢦⡤⠼⢆⠑⣚⡉⠩⠭⠤⠄⣶⡶⠂⠒⠒⠊⠉⠉⠲⢝⡳⣄⠒⠒⡺⠱⠓⠂⠭⠕⣂⣒⠤⢀⡀
⠀⠀⠀⠀⡠⣿⢻⡩⣽⣾⠟⠉⠈⠣⡈⢆⠀⠀⢀⢎⠏⠀⠀⣠⠴⠶⠦⣄⡀⠀⠙⠪⣑⡤⡡⠁⠀⠀⠀⠀⠀⠀⠉⠓⢬⡑⠢⢄⡀
⠀⠀⡔⠉⠀⡼⠳⠄⠀⠀⠀⠀⠀⠀⠈⢆⠑⣤⢣⠃⠀⠀⣼⢥⠀⣠⣀⢈⢷⡜⢤⠀⡸⡡⢟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⢹⢀⡂
⢇⠊⢧⡀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣌⠣⡀⠀⣀⢿⢟⠡⣄⣡⣴⢿⠇⠈⡔⡱⠁⠀⠑⢝⢦⡀⠀⠀⠀⢀⡠⠞⠁⣀⠔⠡⠊⠁
⠀⠀⠀⠈⠓⠤⣀⠙⠒⠤⢤⣀⣀⠀⠀⡰⠡⠉⠑⢕⢄⠑⠀⠙⠺⠿⠿⠟⠃⢀⢞⠜⠁⠀⠀⠀⣈⡱⠽⠤⢐⡊⠥⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠂⠠⠤⠭⢉⣉⣉⣐⠒⠒⠓⣓⠢⠄⢀⠀⠀⢤⡴⣡⣻⠶⠭⠭⠌⠁⠐⠒⠈⠻⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠇⠀⠀⠀⠀⠀⠁⠉⠻⢿⢍⡠⡫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡋⠾⢷⣄⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⠁⠀⠀⠀⠀⠀⠀⢀⣤⡪⠊⠀⠀⠀⠉⡎⠗⣈⣇⡀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢸⡀⠀⠀⠀⢀⡠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠛⠿⢟⢝⣿⣿⣾⠤⢀⣀⣠⠤⠖⠁⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⠒⠒⣊⠕⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠊⠀⠉⠒⠒⠤⠤⢀⠀⠤⠔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⡒⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠕⠂
⠀⠀⠀⠄⠢⢂⢤⠄⡠⡂⢄⠄⡠⢀⢈⣈⠔⡠⣢⢄⢔⠠⠂⢔⠠⡢⢂⠔⡠⡠⢀⢄⠄⡀⡠⢀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠐⠑⠙⠁⠊⠊⠔⠱⠁⠎⠰⠣⠪⠞⠡⢪⢊⠔⡵⡕⡨⠪⢔⠕⡩⣊⢕⡡⡪⡪⢔⢔⢔⢤⡤⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠟
⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⢀⡴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡟⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⠙⢲⣤⣴⠚⠉⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⡠⠖⠊⠁⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⣀⣀⣀⡀⠛⢄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠣⡀
⠀⣼⣿⣿⣿⣷⠀⠀⠙⠢⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠈⢠
⠀⢻⣿⣿⣿⡿⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⡞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡸⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⢀⡼
⠀⠈⠓⠶⢤⣤⣄⣀⣀⣀⣤⡤⠤⠴⠒⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠗⠲⠤⠤⣼⣿⣿⣿⣷⣤⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠹⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⠞⠁

⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⠂⠐⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣀⠀⠀⡥⠀⠀⠀⠀⠤⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢠⡎⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢃⠀⠀⠊
⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠐⠂⠄
⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⡃⠀⠀⠀⡤⣖⡧⠀⣂⣀⠔⢒⠄⢀⣠⠀⠀⠀⢠⠀⢀⢢⣄⣐⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠁
⠀⠀⡀⠀⠠⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠨⢰⡌⢇⠀⢀⣼⠀⢠⠔⠀⢠⡜⢸⢀⠇⢀⡠⠮⠂⠀⠀⡇⠀⠀⠙⠦⠀⠙⠁
⠀⠀⠃⠃⠉⠁⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠋⠁⠚⠁⠐⠚⠀⠙⠀⠋⠀⠀⠊⠁⠛⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠆⠀⠀⠁⠶⠦⠠⠶⠦⠼⠿⠧⠷⠄⠴⠶⠴⠄⠠⠶⠶⠦⠿⠇⠀⠘⠀⠀⣖⠀⠤⠀⢐⡀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣀⠀⠀⠈⢠⡄⠀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠉⢀⣀⣀⡀⠊⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠉⠙⠳⢦⣄⠀⠀⠀⠀⣀⡴⠞⠋⠉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢻⡆⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡟⠁⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣠⣤⡤⠴⣶⠿⠶⠶⠶⠶⠶⠶⠿⣶⡤⣤⣤⣄⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⢻⡏⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢹⡏⠛⠳⣶⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⢻⡀⢀⡾⠁⠀⠀⢀⡴⠶⠶⣤⡀⠀⠀⠈⢷⡀⢀⡿⠀⠀⠀⠙⠶⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡾⠁⠀⠀⢰⡏⠀⠀⠀⠀⢻⡄⠀⠀⠈⢷⣾⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⠀⠀⠀⠸⣇⠀⠀⠀⠀⣸⠇⠀⠀⠀⣼⢷⡀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⣼⠁⠘⢧⡀⠀⠀⠙⠳⠶⠶⠞⠁⠀⠀⢀⡼⠃⠈⣧⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣄⣰⡇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢸⣆⣠⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠛⠓⠲⠶⠿⣦⠤⠤⡤⣤⠤⠤⣴⠿⠶⠖⠛⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢀⣀⣀⣴⠟⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣀⠀⣀⣠⣿⣀⣿⠁⠀⠀⠀⠉⠳⢦⣄⣀⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠓⣤⠤⠤⠤⠤⠶⢍⡀⠀⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠓⢄⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⡤⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢚⠦⣀⢀⠓⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢸⠀⠀⠁⠀⢱⠀⠑⠂⠔⠂⠤⠤⠤⠴⢂⠀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣂⣀⡀⢀⡤⠐⠒⢷⡦
⠀⠀⠀⠀⠀⠀⠘⠴⡀⢠⢧⡄⢆⠰⠤⣀⠀⡷⠶⡏⠁⢠⠀⣹⠹⣄⠀⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠳⡳⠈⠉⠀⠀⠘⠦⠤⠔⠣⢀⠌⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠽⣝⠉⣝⢉⡎⡃⠀⢤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠛⢆⠈⣷⡱⣤⠜⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢄⠀⠀⠀⠀⠉⠀⠀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⡦⠤⣀⠀⠀⣀⣰⠖⠉⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠤⠇⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⡑
⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠜⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢸⠀⠀⣀⣀⠤⠄⠃⠀⠰⡤⠤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⡩⢷⠈⠉⠉⣹⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠀⠀⢀⠔⠊⢆⡀⠑⠒⢶⡁⠉⣉⠽⣄⠀⠀⠀⢀⠄⠀⢇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⠈⠀⠀⠀⢋⠉⠉⣷⠀⠙⠤⠤⠋⠳⡀⠐⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠑⠊⠁⢇⠀⠀⠀⠀⡰⠁⠃⠀⠀⠀⠀⢀⠇⠐
⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠊⠀⠑⢄⡀⠀⠀⠀⡠⠊⠒⠒⠒⠉⠀⠀⠈⠢⠄⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠿⠿⠿⢿⣿⣿
⠀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⣸⣿⡿
⢸⣿⣿⣿⡿⠛⠛⠛⠛⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⣴⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋
⠀⠻⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⢁⡴⠟⠛⠻⢶⣄⠀⢰⡇⠀⠀⠀⠀⠀⡀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣠⣴⠾⠛⠉⠀⠀⠀⡟⠀⠀⠀⠀⠀⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠷⣤⣀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⣸⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠈⠒⠀⠐⠚⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⢀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⠤⣴⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠛⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠛⠉⠛⣷
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠳⣦⡀⠀⠀⢀⣴⠞⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⠛⣦⣴⠋⠁⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡾⠋⠙⢷⣄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣷⣤⣤⠴⠶⠶⠶⠶⣿⠶⣶⡿⠛⠛⠛⠛⠛⠛⢿⣶⠶⣿⠶⠶⠶⠶⠦⢤⣤⣿⣅⣀⣀⡀⠀⠀⢀⣠⣀⡀
⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⠋⠉⠉⠀⠀⠙⢧⡀⠀⠀⠀⠀⣿⡾⠋⠀⠀⢀⣀⣀⡀⠀⠀⠙⢷⣿⠀⠀⠀⠀⢀⡼⠋⠀⠀⠉⠉⠙⠛⢳⣿⣿⣿⣿⡀
⢀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⣠⡾⡏⠀⠀⠀⣴⣿⣿⣿⣿⣦⠀⠀⠀⢹⢷⣄⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠟⠙⠷⣤
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣏⠀⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⠀⣹⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠙⢿⣇⠀⠀⠀⠘⠻⢿⡿⠟⠃⠀⠀⠀⣸⡾⠋⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠉⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⢶⣾⣯⣤⣤⣄⣀⣀⣀⣿⣀⣙⣷⣄⣀⡀⣀⣀⣠⣾⣋⣀⣿⣀⣀⣀⣠⣤⣤⣽⣷⡶⠶⠖⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠉⠉⠉⣿⠉⠉⠉⠙⢿⣍⣩⡿⠋⠉⠉⠉⣿⠉⠉⠁⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢀⣤⠟⠻⣄⡀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⡴⠟⠁⠀⠀⠈⠻⢦⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣤⣤⣄⣠⡶⠛⠁⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⣤⣤⠶⠟⠛⠿⠿⠟⠁⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣤⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⣭⠙⠒⠒⠒⠒⢉⠙⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⠉⠉⠙⢢⣄⣀⣤⠬⣁⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢃⡼⠁⡏⠀⠀⡀⠈⢻⢶⣧⠀⠀⢀⣠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠋⠩⡀⠀⣸⣤⣀⣉⡰⠃⠸⣄⡤⠖⣉⡽⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠀⠀⠀⠑⠦⣝⡋⠀⠀⠀⠀⠀⠀⢀⣀⣤⢚⡡⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣞⠣⣄⠀⠀⠀⠀⢀⡎⠉⣿⣒⢒⣺⠋⡹⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠗⠀⠏⠑⠠⡔⠉⡇⢀⠂⠀⢀⠃⣞⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠳⠞⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⡘⢠⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠒⠤⢀⡀⢴⢰⠃⠀⠀⠀⡇⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡠⠴⠋⠀⠀⠀⠀⡆⡃⠀⠀⠀⠀⢠⠔⠋⠂⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⠀⠀⠀⣀⠴⠥⡀⠀⠀⠀⡠⠃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡶⢄⣈⡷⠀⠀⠈⣢⠤⠒⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠈⢳⠧⠶⢊⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠃⠀⠀⠀⣴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⡟⠀⢦⣀⣀⠀⠀⠀⣀⣀⠠⠜⢄⠀⠐⡠⠀⠀⠀⢢⠉⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢧⣀⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠈⠉⠒⠒⢲⠁⠀⠀⠀⠠⠀⢀⠀⠀⡇⠀⠀⢸⠀⠈⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡞⠒⠦⢄⣀⠀⠀⠀⠀⠀⢱⠀⡠⠊⠀⠀⠑⢄⠀⠀⠌⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⢀⠂⠀⠐⠒⠩⡉⠉⠉⠉⢣⡀⠀⠀⠀⠈⠉⠉⠙⣳⠃⠰⡀⠀⠀⠀⠀⠀⡆⢠⠂⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠸⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠊⠉⠁⠀⠈⢆⠀⠀⠀⠀⡰⢁⠆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⢨⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠙⠿⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠈⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣀⣀⣠⣤⡤⠤⠴⠶⢶⡶⠛⠒⠒⠒⠒⠒⠒⠛⢶⡶⠶⠦⠤⢤⣤⣄⣀⣀⠀⢨⡟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠞⢻⡏⠉⠁⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠉⢙⡿⠳⠶⣤⣄⡀
⠀⠀⠀⠀⣰⣿⣿⠏⠉⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣠⠞⠁⠀⣠⣾⣭⣦⣶⣶⣤⣭⣷⣦⡀⠈⠳⣄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⠙⠳⣦⣄
⠀⠀⣴⡿⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⡴⠃⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠈⢧⡀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠞⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠻⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡼⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠹⣄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣠⠏⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢠⣤⣴⡿⠋
⠀⠀⠀⠀⠈⠛⠷⢦⣄⣀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠈⢳⡀⠀⠙⠻⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⡼⠁⠀⠀⠀⠀⠈⢳⡀⠀⠀⢀⣠⣴⠾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢲⡾⢧⣤⣀⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠉⠉⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣀⣀⣤⡤⢷⡖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡇⠀⠀⠀⠉⠉⠉⠙⠛⠛⠚⠛⢶⡒⠒⠒⠒⠒⠒⡶⠟⠓⠚⠛⠋⠉⠉⠉⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣀⡀⠀⢀⣀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣄⡀⠀⢀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠁

⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣶⡞⢻⣿⠀⣼⡎⢉⣁⣀⣉⢉⡤⠄⠐⣒⡒⠒⠒⠂
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢠⣿⠻⠦⠤⠤⠤⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢶⠾⠷⠾⠎⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀
⢸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡯⢿⠿⢿⡷⠠⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣶⣤⣶⣶⣴⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣤⣤⣶⣿⣿⣿⣿⣷⡄
⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣼⣧⣴⣿⣯⣽⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡉⠙⠻⣿⣿⣿⣿⣿⣷
⢸⣿⣽⣿⣿⣿⣿⣷⣄⣸⣿⣿⣧⣤⠤⠄⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣟⡥⢃⣴⢿⣿⡿⣿⣿⣿
⢸⡏⠉⠤⠈⣤⣼⣿⣤⣿⠏⠉⠛⠛⠛⠛⠘⠘⢀⣤⣤⡀⠀⣀⣤⣤⣶⣶⣦⣬⠛⣿⣿⣿⣿⣿⠶⣴⣯⢀⣿⣿⡿
⢠⡧⠠⠤⠖⠱⠶⣴⣦⣴⣦⣔⡢⢤⣤⢤⣤⠏⣀⠙⠿⡟⠀⢹⡿⠿⣿⣿⣿⣿⣷⣿⣿⣿⡿⢳⣿⣞⣿⣿⣿⣿⡇
⢸⣿⣚⠀⠰⠧⠀⠛⠒⣻⣿⣿⣟⡛⠀⠀⠈⡄⠉⠀⠀⠀⠀⣿⠓⣄⠀⠀⠀⢹⣿⣿⣿⣿⡷⣾⣿⣿⣿⣿⣿⣿⡀
⠈⣉⡉⠀⠀⠘⢃⣸⣭⣿⣿⣿⣯⣥⠄⠀⠤⠀⡤⠋⠉⠶⣶⣿⣿⡿⠀⠀⣶⣿⣿⣿⣿⣟⣿⡿⣿⣾⣿⣿⣿⣿⡇
⢠⡄⡄⡄⠀⣀⣀⣉⣉⣭⠈⠁⠀⠉⠀⠒⠀⡐⠃⠀⢸⣧⣴⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⢸⡟⣿⣟⠼⣏⡈⣻⣭⡅⠀⠀⠀⠀⠀⠀⠈⠉⢁⣤⣾⠏⠙⠛⠋⢻⣿⣿⣻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀
⢸⣧⣿⣯⣤⣤⣿⣷⣟⠃⠀⠀⠀⠀⢀⠀⠐⠒⠀⠀⢀⡄⠀⠀⠀⠀⠩⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⢸⣿⡿⡯⣿⣿⣿⣿⣖⣂⠐⠒⢦⣤⡌⠁⠀⢠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠉⢠⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⣿⣟⡿⢿⣿⡿⢿⣿⣁⣠⣷⣼⠏⠀⠀⠀⢸⡇⠀⣠⠀⣤⢦⠀⣠⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣿⠟⠁⠀⠘⠟⠀⢲⡀⠀⠀⠀⠄
⠸⡟⢩⢷⣼⠿⠄⠀⠀⠀⠤⡔⠁⠀⠀⠀⠀⠀⡇⠀⠻⣤⣷⣏⣸⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠿⠋⡠⠁⠀⣽⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⡄⣤⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣟⣷⡿⠟⠁⣀⡾⠁⠀⠀⠏⠀⠀⠀⠀⢀⣾⣿⣿⠇
⢠⣅⣽⣧⠉⠡⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣿⡿⠋⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋
⢸⣿⣿⣶⣶⣶⣗⣂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⢀⠔⣻⣿⠃⠀⠀⠀⠀⠀⣠⡖⠈⠁⠈⣻⣿
⢸⡿⣿⣿⣿⣭⣽⠿⠆⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢋⠁⠀⠀⢀⣴⣿⣶⣿⣇⠀⣀⡴⠟⠃
⢸⣿⣿⣿⣿⣿⣿⣿⢾⣛⠼⡃⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⣹⠟⢡⠆⣠⠀⢠⣿⣿⣿⣿⣿⡿⠋
⢸⣏⣟⣿⣿⣿⣽⣿⡂⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⠏⢠⡏⠀⣿⣿⣿⡿⠋⠁
⢠⣍⡋⣛⣹⣿⣟⣉⠉⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣾⠟⢀⣿⣏⣤⡿⠟⠁
⢸⡏⠉⠉⠉⢉⣹⣟⣛⠂⠀⠀⠀⠀⠒⢶⢂⠀⠀⠀⠀⠀⡀⢀⣀⣤⣴⣶⣿⡽⠟⢁⣤⣾⣿⡍⠁
⢸⣧⣶⣧⣾⣿⣥⣿⣿⣷⣦⣤⡄⣀⣠⣤⣈⣉⣉⣡⣶⣿⣿⣿⣿⣶⣿⣾⣷⣶⣾⣿⣿⣿⣿⣷⣦⣄
⢸⣯⣟⣻⣉⣍⣉⣡⣤⣶⣼⣿⣏⣉⣀⣼⢯⣉⣉⣻⣟⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠘⠛⠛⠛⠛⠛⠙⠋⠙⠋⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⣙⠷⠶⢶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠃⠠⠀⠤⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⣀
⠀⠀⢙⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠺⡶⠿⠿⡷⠀⢠⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢠⠄⠤⣀⡤⠔⠢⡀⠀⠀⠘⡷⠤⠒⠉⠀⡀
⠀⠀⠀⠈⠁⡀⠀⡿⠀⠀⡇⢀⡀⠀⠀⠀⢀⡎⠉⠑⡍⠀⠀⠀⢀⣧⢠⡼⠵⢤⣀⣱⠀⠄⠊⠀⠀⢘⠉⢵⠈
⠀⠀⠀⠀⠀⠁⡰⠥⠀⠀⢣⠀⠁⡠⢤⣴⠟⠻⠀⢀⣼⠀⠀⠐⣅⠈⢲⡀⠀⣤⠑⣾⠉⠛⢆⠀⢠⢬⠉⢁⣤⡀
⠀⠀⠀⠀⠀⢠⠃⣠⣀⣠⣌⣆⠀⠣⡴⡅⠀⠀⠙⡿⠿⠀⠀⠀⢈⡉⢶⠬⠦⣄⣠⡏⣦⠤⠚⠀⡈⢊⡴⣋⠜⠁
⠀⠀⠀⠘⢁⣿⣿⣿⣿⣿⣿⣿⣆⠀⣷⠹⡄⠀⣐⣒⣒⡄⠀⠀⢀⣸⠈⢆⣀⡤⠛⠒⠊⠀⠀⠀⡤⣋⠜⢁⣀⡀
⠀⠀⠀⠀⢎⢻⣿⣿⣿⣿⣿⡿⢈⠀⢸⠖⠚⠚⠒⠒⠒⠚⣆⠀⠀⠙⠀⠀⠀⠀⣠⣴⠶⢒⡚⠿⣿⠁⠀⠀⢸⣏
⠀⠀⠀⠀⣈⠑⠒⠂⡒⠒⠒⠈⠁⠉⠘⠒⠦⠤⠤⠦⠆⠒⠃⠀⠀⣤⡆⠀⠀⡾⢋⠖⡒⠒⡌⢢⠙⡄⠀⠀⢸⣍⠁
⠀⠀⠀⠈⠚⠁⠀⠀⠻⠯⠀⠀⠀⠀⠀⠄⠀⠀⠀⠔⠒⠄⠴⠀⠀⠈⠁⠀⠀⡇⠸⣀⡛⠘⡜⡸⠀⡇⠀⡆⢸⠮
⠀⠀⠀⠀⢴⢀⡶⠚⢲⡀⠀⠓⠺⠰⠀⠀⠀⠀⠀⠊⠒⠤⠴⠂⠀⢠⣤⠀⣄⠹⣤⠈⢉⠉⠐⣡⠞⠀⠀⠧⠿⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠀⠀⢠⠴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⣠⠞⠛⠳⡄⠀⢰⡆⠉⠂⣴⠶⠶⠄⠈⢀⡀⠀⢀⡄⠀⣠⠤⢤⡒⠀⢸⣂⣀⣀
⠀⠀⠀⠀⠐⣆⠀⠀⢀⡇⢰⡇⠀⠀⠀⠉⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⣾⢧⠀⢸⡇⢰⡇⠀⠈⠁⠀⢸⠉
⠀⠀⠀⠀⠀⡌⠛⠒⠋⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣟⠛⠛⠂⠀⡟⠈⣇⢹⠃⢺⢁⠀⠀⢀⡀⠸⠶⠖⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢧⣀⣀⣠⠄⠀⣸⡇⠈⠁⣿⣤⣤⡄⠸⠃⠀⠘⠟⠀⠈⠦⠤⠖⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⢒⣤⣤⡄⢀⡴⣊⠥⢄⠉⢆⠲
⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠠⡆⠀⠀⠘⠀⠈⢫⡂⢠⠛⠒⠠⢼⡀⠛⠑⠙⠀⠀⣀⠁⡀⡄⢠⣾⡔⠁⠀⢀⠃⣼⢠⣤
⠀⠀⠀⠀⠀⠁⢰⠭⡆⢰⠯⡆⠀⡟⡇⠀⠀⠀⡸⠡⠀⠀⠠⠀⠈⠟⢠⣄⣤⠀⠳⣄⢀⡗⠋⠀⠀⠀⢴⣧⡴⠃⢸⠏
⠀⠀⠀⠀⢶⣾⣯⡿⡽⠽⠯⡿⠿⠿⣿⠟⠀⠘⢇⠀⠀⠀⠀⠀⠀⡆⠈⠻⠃⠀⢂⡬⠛⠬⡀⠀⠀⠀⠔⠈⠀⢻⣮⡄
⠀⠀⠀⠀⢸⣿⠀⣶⡇⠀⣿⡇⢠⣿⣿⠀⠀⢀⡠⠣⡀⠀⠐⣂⣀⡇⠀⠀⠀⡰⠉⠀⠀⠀⠉⢦⠀⠀⠀⠀⣦⣠⠟⠁
⠀⠀⠀⠀⢸⣿⠀⣿⡇⠘⠿⠇⠀⠿⣿⡆⠀⡉⣀⢀⠙⠉⠀⠀⠀⠀⠐⠀⠙⠁⠀⠀⠀⠀⠀⠸⠟⠄⠀⠀⠸⠛⠋
⠀⠀⠀⠰⠿⠿⠿⠯⠿⠿⠯⠿⠽⠽⠽⠿⠃⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣸⣿⣷⡀⠀⠀⢀⠔⠋⠉⠙⢢⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⡰⠚⠉⠉⠲⡄⠀⠀⠀⢀⣿⣁⠀⠀⠀⠀⠀⢰⣿⣿⣦
⠀⠀⠀⣘⣿⣟⡀⠀⠀⠘⣌⠁⠉⢈⡰⠃⠀⠀⠘⢿⣿⠛⠀⠀⠀⢧⡉⠈⠀⣁⠆⠀⠀⢰⡎⠉⢹⡦⠀⠀⠰⣶⣿⣿⣿⠇
⠀⠀⠿⠿⠇⠿⠿⠇⠀⠀⠘⠋⠉⠉⠀⠀⠀⠸⠷⠜⠏⠾⠷⠀⠀⠔⠋⠉⠉⠁⠀⠀⠠⠾⠷⠤⠾⠷⠆⠀⠀⠉⠛⠏⠁
⠀⠀⢀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠠⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⡇⠀⣄⠀⠀⡇⠀⠀⡏⠩⠭⠍⢹⠀⠀⠀⡏⣩⣬⡉⡆⠀⠀⡔⢒⠛⠓⠲⡆⠀⠀⣴⡄⠁⠀⠀⡁⠀⣠⣇⣀⣀⣀⣧⡀
⠀⠀⣧⣤⣬⣤⣤⡇⠀⠀⣇⣀⣀⣀⣸⠀⠀⠀⡟⠿⠿⠋⡆⠀⠀⡇⠘⠈⠛⠀⡇⠀⢸⣿⣯⣿⣿⣽⣿⠀⣿⡿⠿⠿⠿⣿⡇
⠀⠀⠀⠠⠿⠧⠀⠀⠀⠰⠿⠿⠿⠿⠿⠆⠀⠀⠳⠰⠦⠶⠁⠀⠀⠙⠻⠛⠛⠟⠁⠀⠈⠿⠏⠛⠛⠛⠋⠀⠀⠣⠤⠤⠤⠃
⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⢠⠤⣄
⠀⠀⣿⡷⢀⡲⡝⡄⠀⠸⠿⠉⠉⡹⢿⠆⠀⢠⡏⠀⠀⠈⣆⠀⠀⠀⡇⠀⣀⡇⠀⠀⠀⢰⢸⣤⣧⡦⠀⠀⢀⣀⣤⠴⣒⣺
⠀⠀⠙⢷⣄⣠⣤⡀⠀⠀⠀⢘⣁⡇⡆⠀⠀⠘⠋⠀⠀⢀⠟⠀⠀⠀⣇⣚⣛⡇⠀⠀⠀⠈⠾⣛⠯⠃⠀⠀⠘⢻⣻⠿⢿⣿
⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠘⠒⠒⠒⠃⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠓⠓⠓⠃⠀⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⠤⠤⡀⠀⠀⠀⡴⡀⠀⢀⢦⠀⠀⠀⣀⡤⠤⣀⡀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⡠⠤⢄⠀⠀⠀⠀⢀⡤⠤⣀
⠀⠀⡾⠟⠁⢠⣿⡆⠀⠀⢧⠇⣛⠸⣸⠀⠀⣴⠉⡜⠛⡌⣳⠀⠀⠀⣿⣦⡝⠎⠇⠀⢰⠊⠀⣀⠀⠈⡇⠀⢔⠉⠀⠀⠀⢙⠆
⠀⠀⢧⡰⣷⠆⣸⠇⠀⠀⠀⣰⣟⣆⠀⠀⠀⠸⣹⡖⠖⣟⡝⠀⠀⠀⢈⠻⠿⠆⠀⠀⠀⠉⢡⣿⡇⠉⠀⠀⠈⠉⢺⣿⠊⠁
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠈⠛⠘⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⢠⣒⣒⣒⣀⠀⠀⠀⡔⠒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠦⢤⠀⠀⠀⢠⣴⡶⣦⣄⠀⠀⠀⣀⣀⣠⢄
⠀⠀⢸⣿⠋⢻⣿⠀⠀⠀⡇⠍⠅⠄⢹⠀⠀⢨⣛⢿⣿⢟⣥⠀⠀⣜⠨⠭⠭⢈⡇⠀⢠⣿⡵⣯⢲⣻⡆⠀⠀⠒⠒⠲⠔⠁
⠀⠀⢸⣏⣀⣉⣿⠀⠀⠀⡇⠭⠭⠅⣸⠀⠀⢸⣫⣷⣶⣯⣻⠀⠀⡿⣻⣮⣶⣛⡇⠀⠀⠻⣬⣾⣷⡿⠁⠀⠀⠾⣿⠶⠿⠆
⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁
⠀⠀⢠⡤⠤⠤⠤⡄⠀⢀⠔⣒⠒⠒⢦⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⣀⣤⣤⣤⣤⡀⠀⠀⣠⠖⠒⠲⣄⠀⠀⢠⣴⣤⣤⣤⣦⡀
⠀⠀⠹⣐⣂⢒⡀⡅⠀⢸⠀⢬⡙⢦⠀⡇⠀⢰⣷⣖⠉⠉⠉⠀⠀⣿⣙⣉⣩⣹⡿⠀⢰⡧⠘⠶⠊⠬⡇⠀⢈⣉⡏⡏⡏⣿⡇
⠀⠀⣈⠻⠿⠈⠇⡇⠀⠸⣄⠃⠙⢘⣢⠃⠀⠀⠉⠀⠙⣿⣷⠀⠀⠈⠹⠟⠋⠋⠁⠀⠀⠳⣄⣀⣠⠜⠁⠀⢸⣇⣇⣇⣇⣉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠉⠻⢿⣦⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⠋
⠀⢀⣠⣤⣄⡀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣇⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀
⢰⣿⣿⣿⣿⣿⣿⠛⠋⠉⠉⠉⠉⠉⠙⠛⢻⣿⠿⢿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⠿⢿⡛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠻⢿⣷⣤⡀
⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠉⠙⠛⠿⣷⣶⣤⣀⢶⣾⠿⠟⠋⠉⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄
⠀⣿⡿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⣠⣶⣿⣷⣦⣤⣶⡿⠛⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢰⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠙⠻⣿⣦⣄⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣠⣼⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣄⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠉⠀⠉⠛⠋⠁⠀⣠⣤⡠⣤⢀⣀⡀⠀⠀⠀⠀⠀⠉⠻⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⠀⠀⠀⠀⠀⠀⢠⠾⣣⣾⣽⣫⣶⣯⠿⣻⣶⣦⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⣴⣾⠟⠁⢸⣿⠀⠀⠀⠀⠀⣾⢯⣶⣷⣯⣿⣝⣻⣭⣿⣿⡮⣿⣶⡄⠀⠀⠀⠀⣿⣿⠈⠻⣿⣦⡀⢀⣀⠀⣰⣿⠋
⠀⠀⠀⠀⠈⢻⣷⡄⣠⣿⠟⠁⠀⠀⣿⣿⠀⠀⠀⢀⣶⡷⣬⣻⢟⣻⡿⣭⡟⣯⣶⣯⢷⣯⣹⣅⠀⠀⠀⠀⢸⣿⠀⠀⠈⣻⣿⣮⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⣩⣾⢟⡁⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣻⣯⣹⣟⣿⣿⣿⠹⣯⣽⡿⢟⣭⣟⣽⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⡿⣿⣟⢿⡇
⠀⠀⠀⠀⢀⣼⡿⠋⠻⣿⣦⡀⠀⠀⣿⣿⠀⠀⠀⠐⢿⣗⡻⣿⡎⣽⣷⣿⢿⣿⣗⣽⢿⡿⢻⡿⠀⠀⠀⠀⢸⣿⠀⠀⢀⣽⣿⣿⣮⣿⣿⡁
⠀⠀⠀⢠⣾⠟⠀⠀⠀⠈⠛⣿⣦⡀⢹⣿⠀⠀⠀⠀⠀⣿⣿⢇⣿⣿⣭⣽⣾⡥⣿⣿⠷⣾⡷⠀⠀⠀⠀⠀⣿⡿⢀⣴⣿⠟⠁⠈⠉⠀⠹⣿⣄
⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⠀⠀⠀⠀⠀⠈⠁⢿⣷⣭⣶⣟⣛⣵⣦⣷⡿⠉⠁⠀⠀⠀⠀⠀⣻⣵⡿⠋⠁⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣀⠀⢀⣀⡀⠀⠉⠈⠻⠟⠙⠟⠃⠈⠀⠀⠀⠀⠀⣀⣴⣾⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠙⢿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣄⡀
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠋⠁⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠈⠿⣿⣿⣿⠿⠛⠿⣷⣦⣄⣾⠿⠛⠉⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠘⢿⣧⡀⠀⠀⠀⠀⢀⣤⣶⣦⣄⡀⠀⢿⣷⠀⠀⠀⢀⣠⣤⣶⣾⠿⠉⠛⠿⢷⣶⣤⣄⣀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠿⠿⠋
⠀⠀⠀⠙⢿⣷⣦⣤⣠⣿⣿⣿⣿⣿⣷⣤⣼⣿⣶⣾⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⢶⣿⣧⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣿⣿⣿⡿⠋⠉⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⡟⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠋⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⠿⠟⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⢠⣖⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣚⡻⢦
⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠈⠉⠓⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡎⠉⠉⠐⠀⠄⣉⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣚⠅⠂⠁⠀⠉⠱⡖⠀⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢣⠀⠀⠀⠀⠀⠀⠈⠑⠢⢍⡒⢤⡀⠀⣀⢔⠿⠊⠀⠀⠀⠀⠀⠀⣩⠿⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣬⡙⢷⡁⠀⠀⠀⠀⠀⠀⠀⠀⢇⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠱⡀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡕⣉⣳⣬⡲⢤⡀⠀⠀⠀⣀⣀⡜⣜⠁
⠀⠀⠀⠀⠀⠀⢀⡔⠒⢦⡤⠼⢆⠑⣚⡉⠩⠭⠤⠄⣶⡶⠒⠒⠒⠋⠉⠉⠲⢝⡳⣄⠒⠒⡺⡱⠓⠂⠭⠕⣂⣒⠤⣄⡀
⠀⠀⠀⠀⡀⠦⢺⡩⢽⣘⠏⠈⠉⠣⡈⢆⠀⠀⢀⢎⠟⠀⠀⣠⠠⠢⠤⢄⡀⠀⠙⠪⣑⡤⡡⠃⠀⠀⠀⠀⠀⠀⠉⠓⠬⡁⠢⢄⡀
⠀⠄⡔⠉⠀⡜⠃⠄⠀⠀⠀⠀⠀⠀⠈⢆⠑⣤⢣⠃⠀⠀⣼⢧⠀⣠⣀⢈⣷⡘⢶⠀⡸⣡⢟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⢹⢀⡂
⢇⠊⢧⡀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣌⢣⡀⠀⢀⢿⣟⢡⣈⣡⣴⢿⠇⠈⡴⡱⠁⠀⠑⢝⢦⡀⠀⠀⠀⢀⡠⠞⠁⣀⠔⠣⠊⠁
⠀⠀⠀⠈⠓⠤⣀⠙⠒⠤⢤⣀⣀⠀⠀⡰⠡⠉⠑⢕⢄⠙⠀⠙⠺⠿⠿⠟⠃⢀⢞⠜⠁⠀⠀⠀⣈⡳⠽⠤⢐⡪⠥⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠂⠠⠤⠭⢍⣉⣉⣀⠒⠒⠓⠓⠂⠄⢀⡀⠀⢀⡴⣡⣻⠶⠬⠩⠍⠅⠐⠒⠈⠻⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠃⠀⠀⠀⠀⠀⠀⠉⠻⣿⢅⡠⡫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠾⢷⣄⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⠁⠀⠀⠀⠀⠀⠀⢀⣴⡪⠊⠀⠀⠀⠉⡞⠗⣈⣇⡀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢸⡀⠀⠀⠀⢀⡠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠛⠿⢟⢝⣿⣿⣾⠤⢀⣀⣠⠤⠔⠁⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⠒⠒⣊⠅⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠊⠀⠉⠒⠒⠤⠤⢀⠀⠤⠖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⡒⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠂
⠀⠀⠀⠄⠦⢂⢄⠀⡠⡂⢐⠄⡠⢀⢀⣉⠔⡠⣢⢄⢔⠠⠂⢔⠠⡢⢂⠔⡠⡠⢀⢄⠄⡀⣠⢀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠐⠑⠙⠁⠊⠊⠔⠱⠁⠎⠰⠣⠪⠞⠡⢪⢊⠔⡵⡵⡪⠪⢔⠕⡫⣪⢕⡡⡪⡪⢔⢔⢔⣤⡤⣀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⣠⠄⠈⡁⠀⠘⠀⠾⠂⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣦⡠
⠀⠀⠀⠀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣷⡀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⠷⠾⢓⣆⠀⠂
⠀⠀⠀⢠⡤⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣦⠖⣾⣯⡉⠀⠀⡀⠀⣴⣷⣿⣼⣿⠗⠀⠀⡀⠀⠀⠀⠀⠀⠟⣷⢿⡏⠉⠂
⠀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⠐⣴⣷⣏⣩⡅⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⣄⠀⠀⠀⠀⠀⢤⠿⠙⠍⠋⠈⠀⠀⠀⠀⣠⣤⢰⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⡀⢰⡤⠀⡀⠀⠀⠀⠀⢺⡃⢰⡆⠀⠈⡀⠀⠀⠀⠻⠀⠀⡅⢻⠋
⠀⠀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠰⠆⠀⠀⠈⠁⠀⠙⣦⡀⣿⠀⠀⠀⠘⢶⡰⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢠
⠀⠀⠀⠰⠂⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢶⣴⣇⣀⠀⠀⠀⡀
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⣾⣭⣻⢻⡷⡔⠚⠁
⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠈⠀⠀⠐⣟⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⣼⣩⡄⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢿⡿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠁⣼
⠀⣀⠀⠀⠀⠰⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠐⢀⣠⣞⠁⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠘⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠿⠀⠀⠀⠀⠄⠉⠙⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢈⠀⠀⠀⠀⠰⠦⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠈⢘⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠁⡄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣲⣾⣿⣞⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡚⢲⡄⠀⠀⠘⠟⢻⣿⠛⠃⠀⠀⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠎⣰⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠉⠳⡖⠀⠀⢠⠆⠀⠈⠀⠂⠀⠀⠀⡀⠠⢥⣮⣿⣿⣷⡀⠀⠠⠄⠀⣇⠀⠉⢸⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⢠⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠻⠻⠟⠑⠦⠀⠀⠀⠉⠀⠀⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣴⠆⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠒⠀⠀⠠⣤⠉⠀⠀⠄⠀⢨⠀⠀⡀⠀⢼⡴⠁⠀⣀⣀
⠀⠀⠀⠀⠁⠀⠀⡼⠀⠀⠀⠀⠀⣀⡴⠒⣫⡟⠁⠀⠀⠀⠀⢰⡀⣠⡔⠀⠀⠀⠛⠀⠀⠀⠀⠀⠗⠀⠞⠀⠀⠈⠁⠐⣴⠛⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠞⣁⡴⠚⠁⠀⠀⠀⠀⠀⠐⠘⢿⣿⣆⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢄⣸⣿⣯⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣸⣧⠖⠚⠙⠋⠉⠁⠀⠀⠀⠀⠀⠙⠲⠦⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢴⣶⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢰⡖⠤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡶⠋⠀⢀⢸⠀⡠⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡴⠿⠛⠛⢛⠛⠛⠋⠈⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠖⠉⢰⡴⠚⠁⠀⠀⠀⠘⠾⠊⠀⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠐⡀⣰⠁⠀⠀⡠⡀
⠀⠀⠀⠀⡀⠄⢠⣤⠔⠉⢀⡤⠞⠁⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠳⠿⠥⠀⠀⠶⠃⠸⠀⠀⠈⠣⣀⠀⠀⢀⣨⠔⠁⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⠋⠀⠀⠀⠀⠶⠂⠀⢠⠏⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⡤⡸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠀⠸⢦⣾⣥⢤⣄⠀⠀⢆⠠⢴⢀⢀⠎⠀⠉⠒⠢⢄⣀⠀⢀⣀⠀⠀⠀⠀⠙⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠋⠀⢰⣖⡒⢤⡀⠀⣄⢠⠀⠀⠀⣾⢁⣀⣠⣟⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠟⠢⠀⣠⠀⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠉⢹⠗⠋⡀⠐⠂⠂⠀⠀⠉⠛⢡⠏⢀⣽⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⢀⠤⠐⠋⠁⢀⠏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠋⠉⢳⡄⠀⠀⠈⠢⡐⠀⠀⠠⢀⡸⠀⠀⠀⠀⢀⠀⣹⠧⢒⠆
⠀⠀⠀⢰⣿⣶⣤⣄⡀⠘⢄⠀⠀⡆⢀⠀⠀⢠⡄⠀⠀⠀⠸⣅⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠒⠊⠉⠀⠀⠀⠀⠀⢺⡷⠛⠊⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⡼⠀⠀⡏⠙⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡞⠙⡆⠀⠼⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢄⣀⠀⢠⣤⡌⢀⡄⠀⠀⠀⠸⡿⠀⠀⠀⢀⣼⠙⡣
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠛⠃⡀⣀⠀⠀⠀⣞⣽⣿⣿⣿⢶⠚⠁⠘⠀⠉⠀⠁⠀⠀⠀⠀⢿⠀⢀⣴⠟⠊⢹⡅
⠀⠀⠀⠀⠄⠸⣔⡽⠀⠀⠠⢀⣘⣧⣤⠤⠴⠞⠁⠀⠀⠀⠀⢹⢹⠁⣀⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠭⡟⠁⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢳⣏⠉⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠉⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠹⣧⠀⠀⠂⢉⣻⣦⣤⡼⠀⠀⣀⡀⠀⠀⠀⠀⢃⢿⣿⠀⣿⠀⠀⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠸⣍⣀⠈⡧⠀⠀⠘⠉⠀⠃⠀⠀⠄⠸⡸⣿⡇⢻⡇⠘⠋⠉⠻⡁⢘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠧⡡⠃
⠀⠀⠀⠀⣀⡀⠀⠀⢈⡴⠒⠛⠻⣏⠃⠀⠀⠀⣀⡠⢠⡀⠀⠠⠀⡇⢿⣿⣘⣿⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠑⠁⠈
⠀⠀⠀⠀⠛⠁⠀⠀⠾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢼⣟⡩⠋⠀⠀⠀⠰⠃⠀⠳⣄⠀⠀⣠⣐⣒⣒⢒⡄
⠀⠀⠀⠀⢞⣹⠀⢴⡦⠀⠀⠐⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⠆⠀⢀⡠⠖⠉⠀⠀⠈⡟⠛⠛⡟
⠀⠀⠀⠀⠀⣀⣶⠶⠦⢤⣤⡤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠙⠙⠈⠂⠾⠃⠀⠀⢰⣹⠆⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⣀⣤⣤⣤⣤⣖⠚⠉⠉⠉⠉⠉⣷⠴⠶⠶⠶⠶⠚⠃⢀⠀⠀⢠⢊⣁⣀⣨⣌⣲
⠀⠀⠀⠀⡀⠸⣄⣀⣀⣠⠇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠸⢿⣿⣿⣿⣿⠟⠰⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠘⠒⠒⠒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⠓⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠋⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣙⣦⣀⣤⣤⣤⣤⣤⣤⣤⡾⠯⠤⠤⠤⠤⠽⣷⣤⣤⣤⣤⣤⣤⣤⣀⣴⣏⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠚⠛⠛⠉⠉⠉⠙⣧⡀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⢀⣼⠋⠉⠉⠉⠛⠛⠓⠶⠶⢤⣤⣀⡀
⠀⢀⣤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢀⡾⠃⠀⣴⣾⣿⣶⣶⣿⣷⣦⠀⠘⢧⡀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠟⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⢻⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⣦⠀⠀⠀⠉⢻⣿⣿⣿⣿⡟⠉⠀⠀⢀⣴⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⠛⠶⣤⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠈⢷⡄⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⢠⡞⠁⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣷⣤⣤⣀⣀⣀⣠⡟⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠈⢳⣄⣀⣀⣀⣤⣤⡤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠉⣹⠟⠉⠙⠛⠛⠛⠛⠛⠛⢿⡖⠒⠒⠒⠒⣲⡟⠛⠛⠛⠛⠛⠛⠋⠉⢻⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢉⣵⠟⢹⡏⠻⣮⡉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⡞⠁⠀⢸⡇⠀⠈⠻⣄⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⢸⡇⠀⠀⠀⠹⣆⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡆⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠗⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⣿⠤⠤⠤⠤⠤⢼⡧⠤⢀⣴⠞⠛⠛⠛⠶⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇⣰⠟⠀⠀⠀⠛⠳⣄⠈⢷⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠸⣇⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⢻⡄⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠻⣆⠀⠀⢸⡇⠀⠻⢦⣄⣀⣀⣀⣤⠾⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠙⢷⣄⢸⡇⣠⡾⠃⣈⣭⠉⠉⠁⠀⢼⡋⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠿⠿⠿⠿⠚⠛⠉⠀⠀⠀⠀⠀⠈⠙⢦⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢀⣀⡀⣀⡀⢀⣀⠀⣀⢀⠀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡿⣾⠀⣿⠀⣿⡂⢸⢾⡃⣿⣾⠃⣿⡂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠈⠀⠉⠁⠈⠀⠁⠁⠈⠀⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠂⠠⢄⣀⣀⠠⠤⠶⢀⠀⠤⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠒⠻⢤⣀⠀⣀⣐⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⢠⠏⢿⠁⠀⠉⠙⣶⣦⡀⠀⠀⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠊⢅⠀⢘⢦⣀⣁⡼⠁⢇⣠⠔⣉⢽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠀⠀⠀⠈⠑⠬⣙⡁⠀⠀⠀⢀⣀⢠⠤⠎⠡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡾⣝⠢⣀⠀⠀⢀⡼⠀⡝⠭⠭⡟⣁⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⢈⢀⠃⠈⠐⠁⢰⠀⠃⠀⠰⢀⠇⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢱⡀⠀⠀⠀⡘⡠⠀⠀⢠⡆⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡜⠀⠈⠁⢉⠁⠁⠀⠀⠀⠃⡠⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢈⠄⠀⠀⢀⣠⡊⠀⠀⠀⢀⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⢄⡀⡯⠀⠀⠑⣄⠠⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠏⠀⠈⢳⠤⠖⡊⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠣⠄⠀⠀⠀⢠⠄⠒⠁⠤⠀⠤⠀⠀⠀⠳⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠤⢀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⡀⠀⠀⡈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠠⠀⠀⠉⠒⠒⠠⠐⢦⠀⠈⠉⠒⠒⠂⠤⢤⡄⠐⠁⠀⠀⠀⠈⠀⢀⠄⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠒⠉⠀⠨⡀⠀⠀⠀⢀⠂⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠁⢠⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣟⣟⣿⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣻

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠁⠉⠉⠙⠛⠻⢵⣿⣷⣿⣿⣿⣿⡹⠿⣿⣿⠇⣐⠺⠁⠛⡌
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠉⠻⢿⣍⣿⣷⣾⣟⣷⣻⣷⣸⣳⣦⠴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠙⢿⣿⣟⡷⢿⣿⣿⡿⡿⠛⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⣴⣿⣿⣿⣿⡿⠿⠿⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣄⠈⢿⣯⣤⣶⢷⣾⠷⢿⣧⠳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄⠈⢿⣽⣭⣯⣿⡿⢼⣯⡻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣷⡀⠈⢿⣿⣿⣦⣝⣸⣿⣍
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⣿⣿⣿⣿⣿⡆⠈⢿⣦⢿⣷⣮⡏⡛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⠯⡝⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⢸⡿⠿⣭⣾⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⠚⠈⡌⠀⢀⣤⣦⡾⠻⣿⡋⠀⠀⢀⣿⣶⣿⣟⣷⢤⣤⣫⣿⣿⣿⢿⣧⣿⠂⢸⣿⣟⣿⣽⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⡀⣄⠀⠀⠀⠀⠠⠜⠻⠋⠀⠀⠀⠘⠟⠛⠓⠒⠛⠃⠈⠛⠁⣿⣿⡇⠿⡟⠀⣼⢾⣽⢽⣷⢿⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣧⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡌⢰⠇⢠⣿⣾⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢳⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣼⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢰⣿⣽⣿⡆⠀⠀⠜⠑⡲⢤⡀⢀⣠⣼⣇⠀⠀⠀⠀⢠⣠⣿⣿⣿⣿⣧⠀⢹⣿⣟⣿⠻⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⠻⣽⣿⡃⣀⠔⢈⣁⠄⠄⠛⠛⠿⣛⡻⠷⢶⢶⣾⣿⣿⣿⡿⠈⣿⣏⠀⢀⣿⣿⣿⣶⣿⣃⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣽⡄⠀⡶⠛⠛⠁⢰⣯⠽⣉⣀⣩⣍⣑⣒⣛⣿⠖⠢⢿⣿⣿⣿⣿⣦⣿⣿⡀⠈⣿⣿⣿⣿⣏⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠋⠉⠉⠃⠀⠀⣿⠃⢾⣿⣿⣿⣿⣿⡇⠀⣿⣿⢿⣷⣿⣽⣷
⣿⣿⣿⣿⣿⣿⣿⣿⠁⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢁⣿⣿⣿⠇⠀⡙⣿⣿⣟⣽⣯⣙
⣿⣿⣿⣿⣿⣿⣿⣿⠀⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠑⣾⣿⡏⠁⠀⣴⣧⣿⣽⣿⣿⣻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⣄⣹⡐⣿⣾⣿⡇⠀⣰⣷⣾⣿⣿⣯⣷⡿⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⣿⠁⣀⡆⠀⡀⠀⠀⢠⣤⣦⣶⣦⣁⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⠃⣠⣯⣿⣿⣿⣿⣿⣿⣿⣔
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠙⢷⣻⣷⣼⢿⣿⣦⣾⣿⣿⣿⣿⣿⡻⣗⣴⣶⣾⣟⡿⠟⠛⠉⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠳⣍⢙⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⢀⣠⣶⡿⣿⣿⣿⡏⣿⡗⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠈⠙⠙⠻⠿⠟⠛⠛⠉⠉⠁⠀⢀⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⢀⣴⠟⠛⠛⠛⠙⠛⠛⠛⠿⠶⣶⣤⣄⣀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣠⣤⡶⠶⠿⠛⠛⠛⠋⠉⠛⠛⠿⣦⡀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡿⠷⣶⣤⣀⡀⠀⠀⠀⢀⣀⣤⡶⠾⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⢛⣷⣶⣾⡛⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⡟
⢈⣿⠶⠿⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢀⣠⣴⠾⠛⠉⠀⠉⠛⠷⣦⣄⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠙⣷
⣼⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠟⠋⢀⣤⡶⠟⠛⠛⠿⢶⣄⡙⠻⢾⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣼
⠸⣷⣄⣀⣠⣾⡃⠀⠀⠀⠀⢀⣤⡾⠛⣿⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠉⢿⣄⠀⣿⠛⢷⣤⡀⠀⠀⠀⠀⢈⣿⣦⣤⣤⡾⠟
⠀⠈⠉⠉⠉⠈⠻⣶⣄⣠⡶⠟⠁⠀⠀⣿⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⠀⠀⠉⠻⢶⣄⣠⡶⠟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣴⠟⢿⣤⡀⠀⠀⠀⣿⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⣿⠀⠀⠀⢀⣴⡿⠿⣦⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠈⠻⢶⣄⡀⣿⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⣿⢀⣠⡾⠛⠁⠀⠀⠈⠻⣦⡀
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣤⡀⠀⠻⢷⣄⣀⣀⣀⣀⣤⡾⠋⢀⣤⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠛⠷⣦⣄⡉⠙⠛⠛⢋⣡⣴⠾⠛⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠙⢛⣷⣶⣾⡛⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⡶⠾⠛⠉⠀⠉⠛⠷⣶⣤⣀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠹⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠿⠶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣼⠏
⠀⠈⠙⠛⠛⠛⠟⠛⠛⠛⠋⠉⠁⠀⠀⢀⣻⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠈⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣴⠟⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⡄⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⣀⣴⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⠂⠈⠁⠒⠌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡐⠁⠀⠀⠀⠀⠈⢂⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠆⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢉⠄⠀⠀⠤⢁⡀⠂⠄⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⢀⠠⠐⢀⠈⠥⠀⠀⠠⡉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠁⠂⠄⡁⠠⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⠄⢈⠠⠐⠈⠀⠀⠀⠀⠀⠀⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡈⠢⢄⠀⠀⠀⠀⠀⠀⡠⠔⣁⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠠⡁⠢⡀⢀⠔⢈⠄⠊⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠆⠀⠀⠀⠈⡰⠈⠁⢆⠁⠀⠀⠀⢰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠡⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⠠⠊⠠⠮⠱⠄⠑⠤⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠜⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠂⠑⢀⣀⠀⠭⠥⠀⠄⠠⠂⠶⢀⠔⠀⠀⠀⠀⠀⠀⠢⡀⠦⠰⠄⠄⠀⠤⠍⠀⣀⡀⠈⠐⠥⠀⡀
⠀⠀⠀⠀⠀⢀⡠⠄⢂⠉⠤⠀⠒⠈⠢⡈⢄⠀⠀⠀⠀⠀⠘⢃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡘⠃⠀⠀⠀⠀⠀⡠⢁⠔⠁⠒⠀⠄⢉⡐⠠⢀⡀
⠀⠀⠀⢀⠔⡁⠔⠈⠀⠀⠀⠀⠀⠀⠀⠑⢄⠢⡀⠀⡠⠃⠐⠁⠀⠀⠀⠀⣀⠠⠄⣀⠀⠀⠀⠀⠀⠂⠐⢄⠀⢀⠔⡠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠢⢈⠢⡀
⠀⠀⢀⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡐⠊⡠⠆⢀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠰⢄⠑⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⡄
⠀⠀⠘⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⢄⠑⠆⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠰⠊⡠⠌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠃
⠀⠀⠀⠈⠢⡁⠢⢀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠔⠁⠀⠑⠄⠀⡀⠀⠀⠀⠀⠉⠐⠂⠁⠀⠀⠀⠀⢀⠄⠠⠊⠀⠈⠢⠐⢄⠀⠀⠀⠀⠀⠀⠀⡀⠔⢈⠔⠁
⠀⠀⠀⠀⠀⠈⠁⠂⠌⣁⠀⠀⠤⢀⠔⢁⠊⠀⠀⠀⠀⠀⠠⡌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⡄⠀⠀⠀⠀⠀⠑⡈⠢⡀⠤⠀⠀⣀⠡⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢒⠄⡀⠈⠉⠀⣐⠒⠀⠈⠘⠆⠶⠈⠠⠀⠀⠀⠀⠀⠠⠔⠁⠶⠠⠂⠁⠀⢐⣂⠀⠉⠁⢄⠠⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡐⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠒⢄⠐⢆⡲⠂⡠⠂⠀⠀⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⢂⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠇⠀⠀⠀⢀⠵⢀⡀⠎⡀⠀⠀⠀⠸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⠐⡡⠔⠁⠈⠢⢈⠂⢄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢉⠔⠊⠀⠀⠀⠀⠀⠀⠑⠢⡈⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⡀⠄⠂⡁⠔⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠂⢈⠐⠠⢀⠀⠀⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢈⠂⠀⠀⠒⡈⠁⠄⠒⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠀⠀⠈⠐⠀⠈⠐⡒⠀⠀⠐⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠰⠀⠀⠀⠀⠀⠀⠀⠀⠄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠡⡀⠀⠀⠀⠀⢀⠌⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡐⠤⣀⡀⠤⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⡤⠶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠒⠒⢶⡿⠶⢤⣤⣄⣀⠀⠀⠀⢀⣀⣠⣤⠴⠾⣷⡖⢺⣧⣴⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠈⣉⣿⠷⢿⣏⡉⠀⠀⠀⠀⠹⣧⠀⠛⠿⠿⠟⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣀⣤⠶⠛⠉⠀⠀⠀⠈⠙⠓⢦⣄⡀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀⠀⠀⠀⠀⠀⢀⣿⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣼⣧⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⣎⠉⣿⣿⣧⠀⠀⠀⣠⣾⡏⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠀⣠⠞⠁⣸⡇⠀⠀⠀⣴⠋⠉⢻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣿⡀⠙⢦⡀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣦⡾⠁⠀⠀⣿⡇⠀⠀⢰⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⡇⠀⠀⠹⣦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢷⡀⠀⠀⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⡇⠀⠀⣰⠟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠙⢦⡀⢸⡇⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣿⠀⣠⠞⠁⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠙⢿⣇⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⣿⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢻⡟⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⠛⠶⣤⣀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢀⣉⣿⡶⢾⣏⣁⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠤⠤⠿⣷⠶⠚⠛⠋⠉⠀⠀⠀⠈⠉⠛⠛⠲⢶⡿⠧⠤⠴⠶⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢸⣧⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⣤⡾⠿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⣶⠿⠛⠛⣿⠛⠛⠿⣶⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⢀⡴⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢦⡀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣼⠃⠀⠀⠀⣀⣀⣀⣀⣿⣀⣀⣀⡀⠀⠀⠀⠘⠆⠀⠀⠀⠈⠻⣄
⠀⠀⢀⡞⠁⠀⠀⠀⣀⣼⠧⠶⠚⠛⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠙⢛⣤⡶⠿⠛⠛⠛⠛⠿⢶⣤⡀
⠀⢠⡟⠀⢀⣤⠶⠋⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⡾⢠⡶⠋⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⢸⣷⠋⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⣿⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠤⢤⣤⣤⣤⣤⣼⡧⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣿⢤⠌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⢹⣧⡀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏
⠘⣏⠳⣄⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣄⣠⣤⣤⡶⣿⣿⡿⢋⣤
⠀⠹⡆⠈⠛⢦⣄⡀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠍⢉⣩⠴⠂⠈⢡⣾⣿⣿⣷⡄
⠀⠀⠙⣄⠀⠀⠈⠙⠳⢷⣤⣀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⣤⡿⠚⠋⠁⠀⠀⣰⠊⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠈⢧⠀⠉⠉⠛⠛⠓⠒⠒⣿⠒⠒⠚⠛⠋⠉⠉⢀⡾⠁⠀⠀⠀⢀⡾⠁⠀⠀⠘⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠻⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⠏⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣬⣿⣦⣄⣀⠀⣿⠀⣀⣠⣴⣫⡥⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢰⣿⣿⣷⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⣀⣷⠴⠒⠋⠙⠿⡿⠟⠁⠀⠉⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠙⠻⠿⣿⣶⣤⣔⠊⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣤⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⡀⠄⠊⠉⠛⢿⣿⣶⣄⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣄⡀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣄⠀⠀⣠⠔⢹⣿⡇⠀⠀⣴⣾⣿⣿⣷⣆⠀⠀⠀⡟⠻⣿⣶⣄⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⢎⠀⠀⢸⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⡇⠀⠀⣹⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠙⠢⣸⣿⡇⠀⠀⠹⣿⣿⣿⡿⠏⠀⠀⠀⣥⣴⣿⠿⠋⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⣿⣧⢀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠁⠢⢄⣀⣤⣶⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣠⣤⣶⣿⠿⠛⠫⢄⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⡀⣀⣀⣀⣠⣤⣴⣶⣿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠈⠉⡿⠢⠤⣄⣀⣀⠀⠀⠀⠀⢀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⣠⣶⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢀⣼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠾⠛⠁

⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢰⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠠⠁⠠⠬⠐⢂⠀⠀⠀⠀⡦⠀⡐⠉⢂⢀⡀⠀⠀⠀⣀⣧⡵⠖
⠀⠀⠀⠀⠀⣥⠜⠁⢢⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠸⡀⠁⠐⠀⠂⢉⡜⠀⠀⠀⠈⢢⡠⣯⡟⣍⣈⠏⠀⠀⢼⠁⢈⡇
⠀⠀⢀⠀⡀⠀⠀⠰⢀⠀⠀⠘⡁⣤⠁⠠⡀⢀⠀⠀⠑⠤⡄⠀⡠⠎⠀⠀⠀⠀⠐⣁⣸⠨⢵⡋⠑⡀⢀⡴⠚⢯⠉
⠀⠀⠈⡉⠅⠀⠠⢂⠌⠀⠀⠀⠐⢄⠀⠀⠐⣄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠈⠉⠀⠀⠈⠉⠚
⠀⠀⠀⢰⣶⣿⣷⣷⣀⠀⠀⠀⠀⠈⠰⢤⡾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠤⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⠑⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⡈⢀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠠⠘⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⡠⢊⠆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠉⡇⠠⠀⠀⠀⠀⠀⢻⣷⠄⠀⠠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡦⢄⠤⠖⠁⠀⠀⠀⠀⠀⠠⣠⣶⣶⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠢⠄⠄⠊
⠀⠀⠀⠀⠀⠠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡡⠀⠀⠀⠀⠈⢤⡐⢀⠆
⠀⠀⠀⠀⢠⠁⠏⠀⠀⠀⠄⢀⠴⠀⠀⠀⢠⠿⢿⠿⣿⣿⡿⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠂⠐⠔⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠒⢀⠸⠈⠂⠀⠀⠀⠀⠀⠀⢂⡀⠀⡰⢀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣈⠁⢈⣤⣤⡗⠀⠀⠀⠀⠀⢠⠊⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡍⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⡆⠀⠄⣀⠀⠀⠀⠀⠀⠀⡠⢐⣤⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠐⠁⠀⠑⢩⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇⣀⣀⠀⡄⠀⠀⠀⢰⡸⠏⠟⢀⢰⠀⠀⠀⠈⡀⣬⢢⠀⠀⡠⡂⠥⠀⠠⠁
⠀⠀⠀⡁⡄⠀⠐⠒⠄⠅⠀⠀⠀⠀⠀⣠⣴⡾⠛⠻⢿⠃⠀⠀⠀⠀⠀⠃⠈⠄⢉⡀⠂⠀⠀⠀⠐⠠⢙⢠⣌⠜⢁⠠
⠀⠀⠀⠑⠈⠀⠒⠨⠈⠀⠀⠀⠀⠀⠠⡈⠛⠃⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⣀⣰⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠓⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠈⣴⣶⣾⣓⣖⣦⠈⠉⠀⢉⣄⠹⣳⣍⣐⢦⡀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢻⣿⠿⠿⣿⡋⠀⠀⠀⢠⡟⢁⣴⠾⣿⣀⡹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⣾⡿⣶⠶⢸⣿⠀⠀⠀⠀⢳⣤⠙⣷⣝⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⠓⣼⣿⣧⠀⠀⠀⠈⠈⠉⠉⡉⠻⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠙⠋⠉⢿⡋
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⢷⢠
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⣠⣾⢿⣿⣿⣿⣿⡿⠛⠘⢳⡶⠎⢏⣹⢾⣷⡄⢀⠀⢀⣀⣴⣯⠾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⣿⣿⣠⠖⢻⣟⣿⡇⢀⠺⠈⠡⠂⣱⠻⣿⣅⣸⣏⠋⢿⢛⡽⠛⠋⠉⢣
⠀⠀⠀⡀⡶⠠⢄⢠⢞⣥⣱⣿⡥⠋⠀⠀⠘⣿⣿⠶⡍⠀⠂⠀⠠⡏⠀⠀⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠈⢂
⠀⠀⠠⣿⠳⡶⠠⠿⠛⣉⡁⣤⣶⠶⠦⠀⠀⠈⠙⣶⣦⡀⠀⠀⢀⠀⣴⡀⠀⠀⠲⠾⠅⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠐⠃⠀⠀⢀⠀⢊⣿⣶⣤⠤⠴⠖⠂⠀⠄⠀⣙⣿⡿⣦⣄⠀⠀⣿⣷⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠄⡰⣀⣆⠀⣠⠀⠈⠁⢀⣉⣁⣴⡶⠖⠀⠠⢤⣬⣿⣿⣾⣿⠿⠚⣿⣿⣶⣖⣶⡈⢹⡷⣦⡄⠀⠀⠐⠂⠀⠘⡄⡤
⠀⠐⠓⠀⠀⣤⠁⣍⡶⣏⣉⣉⠾⡍⠀⠀⠀⠀⠀⠉⠠⣤⣬⣋⡑⠂⠛⠙⢿⣿⣦⣄⣸⣀⣹⣏⣸⣤⣀⣤⢤⡀⡼⡠
⠀⣿⣷⣾⣧⡙⠀⠘⠧⣌⡉⢛⠷⣶⢶⣦⣤⣀⣠⣴⡶⢶⣿⣿⣿⣿⣿⣭⣉⣿⣿⣼⣿⡿⠉⠉⣁⠀⣈⣩⣤⣦⣌⠱⡇
⠰⢿⡿⠋⠿⢥⡆⠀⠤⣚⠃⢨⢧⣙⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣼⣦⣿⣛⡿⢋⡶⣿⠇
⠀⠀⠡⢄⠀⠀⣈⣉⣤⣽⣾⣷⣿⣿⣿⣿⢿⣛⠿⣶⣶⣻⣽⡿⣿⣷⣾⣿⣿⣭⣿⣿⣿⣿⣷⣿⣿⡿⣿⣽⣿⣿⣦⣄⡀
⢀⢐⣶⣀⣶⣿⣿⣿⣃⣀⣀⣸⣟⣿⣿⣗⣐⣉⣀⣈⣙⣿⣿⣿⣟⣛⣛⣋⣺⣙⣿⣿⣽⣯⣽⣿⣿⣿⣭⣽⣿⣬⣯⣭⠃
⢸⠉⠉⠉⠉⣭⣭⣭⡭⢉⡉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⡀⠀⠀⠀⢀⣿⣃⡀⠀⠀⠀⢸
⢸⠀⠀⣀⣠⡾⢸⡿⠗⠀⠃⣄⠀⣿⣟⢈⠛⢢⠀⠉⠉⠉⠉⠛⠉⠉⠀⢀⠀⢯⢿⡏⠀⠀⠀⢸⣧⣿⣿⣷⠀⠀⠀⢸
⢠⢰⣚⣛⣃⣀⣚⣓⣀⣀⣀⣘⣓⣿⣿⠀⠀⠁⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⢸⢻⣤⣶⣶⣟⣷⣟⣘⣛⣂⣀⣀⣸
⢸⠈⠉⠉⠉⠉⣉⣉⣉⠉⠉⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠉⠉⢉⡉⡉⠹⡏⠉⠉⠉⠉⢹
⠸⠤⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠙⠁⠀⠈⠐⠷⠀⡵⠆⠂⠀⠀⢸
⢰⠚⠛⠛⠛⣛⣛⣻⡛⠛⠛⣛⠛⢻⣿⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢺⣿⣿⣿⣿⢿⣷⣾⣿⣾⣷⣶⣾
⠸⢦⣠⠤⢤⣿⣿⣿⣿⣥⡄⠈⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣻⣍⢉⣉⢀⣀⣀⣀⣀⣉⣹⣿⣿
⠸⢠⣤⣤⣤⣤⣤⣤⣴⣤⣤⣴⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣟⣿
⢸⡀⢸⣦⣴⡇⢴⣆⡀⠘⣤⠀⡄⣺⡟⠀⠀⠀⠀⠰⠤⠤⠀⠠⠤⠤⠄⠀⠀⣾⡇⢸⣿⢻⡿⡿⡿⣿⣿⢿⡿⣿⡿⣽
⢸⢂⣼⣿⣿⣧⡀⢻⣥⣴⡿⢋⣮⣿⣿⣶⣦⡄⠀⠀⠠⠀⠀⠄⠠⠀⠀⢀⣴⣿⣷⣿⣸⡘⢹⡇⠳⡬⠎⢸⡇⠙⢃⣽
⢸⣿⣿⢏⣤⣽⠟⡟⢇⠛⣷⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⢿⡝⣿⣿⣶⣿⡏⣽⡿⣿⣿
⢸⢭⣿⣿⣿⣿⣧⣆⡺⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⡘⢷⣬⣽⠛⣛⣴⣿⡇⣿⣿
⠘⡗⠀⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡷⢿⣶⣾⡿⠿⢿⢿⣷⣿⣷⡾⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⢸⣿⣿⣿⣾⣿⣿⣍⡙⣿⢛⣉⣿⣿⣷⣾⠟⣿⠀⢸⣿⡿⠇⢰⣼⣽⣿⣿⢿⣧⣿
⠀⡇⠀⣼⡖⢂⠀⠠⡴⠲⣶⣶⣶⢸⣿⡇⠀⡏⠉⠈⠉⠉⠉⠉⠉⠁⠀⠀⢹⣧⣹⠀⢸⣧⠀⣠⣀⣀⣄⡄⠀⣾⣿⣿
⠀⣭⠇⠀⠖⣤⣄⣈⣙⣾⣇⡽⠷⣼⢸⠿⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⣿⠘⠛⠋⣴⣿⣿⣿⣿⣿⡟⠛⠛⢿
⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⢰⡶⠿⠶⠶⠿⠟⢠⠛⣤⣬⣍⣥⣬⣴⠶⡾⠀⣈⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⡘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⡹⠛⡟⢯⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢅⡀⡼⠁⠀⡇⠀⢳⣀⣠⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢹⠛⠒⠲⡗⠒⠛⣏⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⢺⠒⠒⠲⡗⠒⠒⣿⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣀⠼⡖⠒⠚⡟⠂⢶⠦⠤⣄⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁⠀⢳⡀⠀⡇⠀⠸⡆⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣙⣤⣧⣞⡥⠳⠋⠲⣌⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⣤⠶⠶⠶⢤⣀⠀⡟⠛⠛⠖⠒⠲⠶⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣠⣼⡏⠀⠀⠀⠀⠉⣧⢿⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠦⠤⠤⠤⠴⠦⠤⣷⠀⠀⢰⠇⠀⠈⠉⠙⠓⠒⠾⢥⣸⡄⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⢀⡟⠀⠀⠀⠀⠀⠠⠀⠀⡘⣿⣯⠉⠉⠉⠉⠙⠉⠉⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠈⣏⢹⡁⠀⠀⠘⠒⠂⠀⠀⠠⠀⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠒⠒⠢⠶⠶⠤⠤⠤⢿⠈⡃⠉⢉⣉⠁⠒⠂⠤⠀⢍⣭⣭⣇⡀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠦⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⢀⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢣⠞⠋⠀⠉⠓⢄⠀⠀⢸⡏⠘⣿⠀⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠈⣄⣰⡟⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠘⠳⢦⡄⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠀⠻⠶⠶⣶⣶⡶⠶⠶⠶⣾⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠶⢤⣄⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⠔⠒⠂⠤⣀⢀⡿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠐⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣀⡀⠀⣈⣿⠃
⠀⠀⠛⠒⠂⠀⠀⠀⠀⠹⣄⣀⡀⢈⣠⣞⠁⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠋
⠀⠀⠀⣤⠄⠀⠀⠀⠀⢸⠃⠉⢹⠉⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡄⢸⠀⣦⠀⢿⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢦⠀⠈⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢸⠇⠀⠀⠟⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠒⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣼⡀⠀⠀⢰⡾⠂⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣄⠀⠈⠁⢀⡤⠤⢤⡀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⣸⠹⣆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⠈⠀⠀⠀⠀⣀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⡾⢁⠀⠀⢳⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢠⠏⠀⠸⡆
⠀⠀⠀⠀⠀⢀⡟⠀⢀⡇⢹⠀⠀⠀⢸⢁⠹⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⣄⠀⠀⠀⠻⣄⣀⣤⠋⠙⢯⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣾⠁⠀⣼⠔⢹⡆⠀⠀⠈⠞⠢⣷⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠳⢤⡀⠙⢦⡀⠀⠀⠀⠀⣴⠃⢀⠀⠀⠀⣿
⠀⠀⠀⠀⣸⠇⠀⠀⠈⠙⠾⠁⠀⠀⠀⠖⠊⢀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡴⠫⠀⠀⠀⠙⢦⡀⠙⢶⣀⢀⡾⠁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⢠⡟⠀⠀⣟⣹⡦⡀⠀⠀⠀⠀⡠⢴⡉⢃⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣹⠏⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⡾⠁⠀⠘⢍⠀⠈⢻⠀⠀⠀⠀⣛⠀⢈⡾⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢂⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⣸⠇⠀⢠⣄⠀⠙⠦⣴⠀⠀⠆⠀⠹⠔⠉⢀⣤⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢯⡙⠲⢤⣀
⠀⠀⠿⣄⠀⣿⣠⠟⢦⡀⠀⠀⠀⡆⠀⠀⢀⠴⢯⡘⡇⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠐⢀⣠⡶⠟⠋⠀⠀⠠⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⡿⠀⣠⠾⢧
⠀⠀⠀⠈⠻⠋⠀⠀⠀⠙⡆⠀⠀⡄⠀⠀⡇⠀⠀⠙⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⣖⢛⠛⠍⠐⠁⠔⢀⠄⠀⠀⠀⣀⣠⢶⣟⣁⣀⡴⣚⡵⠛⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠶⠶⠶⠒⢻⡟⠉⠀⠀⠉⢉⣡⠞⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⡇⢀⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠙⠓⠶⡴⠋⠁⠀⠀⠀⢀⣀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣯⣠⠤⠶⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁⠀⠀⠀⠙⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⣇⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⠿⠛⠛⠋⠉⠉⠉⠛⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⣠⣤⣶⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⢶⣤⣤⣇⣀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⢠⣾⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠙⠛⠲⢦⣤⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣴⠟⠉⠀⠉⠛⠲⠤⣄⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏
⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⡴⢶⣶⣦⡻⣿⣿⣿⣶⣄⠀⠀⠈⢹⣿⡄⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢠⣿⣄⣠⣿⣿⣿⣽⣿⣿⣿⣿⣷⡀⠀⢸⣿⠀⠁⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⡀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⡀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢸⣿⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠛⠛⠋⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠙⠻⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⣤⣤⣾⣁⣀⣀⣀⣼⡏⠀⠀⠀⢀⣀⣀⣤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠈⠉⠉⢩⣿⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠟⠁
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⢛⣿⡿⠋⠉⣿⠉⠙⠻⣿⡛⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⢠⣿⠋⠀⠀⠀⣿⠀⠀⠀⠙⢿⣄⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣋⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣘⣿⣀⣀⣀⣀⣀⣘⣿⣆
⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⠛⠛⠛⠛⢻⡿⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⢻⣟⠛⠛⠛⠛⠛⠛⢿⣆
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⠇
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣷⣶⣤⣈⡁⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⡉⠉⠉⠉⠉⢻⣏⠉⠉⠉⠉⠉⣿⠉⠉⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠘⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⠈⢻⣦⡀⠀⠀⣿⠀⠀⢀⣴⡟⠀⢹⣿⣿⠟⠙⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢶⣿⣿⣶⣤⣿⣤⣴⣿⣿⣶⠷⠀⢿⠏⠀⠀⠀⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁


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

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠎⢀⢶⠆⠀⢀⣀⠀⠀⢶⢦⠈⢎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠞⡎⠀⣴⣿⣿⡿⡆⠈⡏⡆⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡹⡀⢹⢡⠀⠙⣿⣿⡾⠃⢠⢣⠃⣸⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣣⡀⠓⠃⢀⠏⠉⠹⡄⠑⠃⣠⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⡿⣦⠀⠀⠀⠙⠃⠀⢀⡎⣰⣿⡆⢳⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣴⣿⣿⡿⡫⣪⣾⣿⣻⡆⠀⠀⠀⡼⢀⣿⣿⣿⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢿⣿⣫⣪⣞⢟⣕⣕⣵⠇⡀⠀⡰⠁⣾⡿⠀⢻⣧⠈⣆⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠈⠉⠉⢉⡉⠉⠉⠀⠀⠀⢠⠃⣼⣿⠁⠀⠈⣿⣇⠘⡄⠀⠀⠀⣠⡶⣿⣿⣿⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢰⣿⠃⠀⠀⠀⠘⣿⡆⠹⡀⠀⣇⠻⣾⣿⣿⣫⣫⡾⢣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⣿⡿⢯⣭⣭⣭⡭⢿⣿⡄⢳⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣾⣿⣤⣤⣤⣤⣤⣤⣤⡤⣬⣿⣧⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⣿⣿⣱⣰⣹⣜⣞⣽⣼⣰⣹⣹⣿⣧⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣸⣿⠟⠉⠉⢉⡽⣋⠭⣙⢯⡉⠉⠉⠻⣿⣆⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⣻⡟⠀⣀⠔⡩⠚⠁⠀⠈⠑⢍⠢⣄⠀⢻⣟⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⢣⡿⢠⢞⡥⠊⠀⠀⠀⠀⠀⠀⠀⠑⢮⡳⣄⢿⡞⡌⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣟⣿⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣾⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠾⠾⠿⠿⠷⠿⠿⠿⠴⠾⠷⠶⠶⠿⠿⠶⠖⠛⠶⠶⠶⠿⠚⠛⠛⠛⠛⠓⠚⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡿⠂⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠿⢿⣿⣿⣶⣶⡤⣸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⢱⡇⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣿⡇⠶⠶⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⠉⠛⢿⣿⣶⣤⡒⠛⠉⠉⠁⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣀⣤⠶⠚⠃⠈⠙⠻⣿⣷⣤⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣀⡴⠞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣤⠞⠉⠀⠀⣼⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⣾⣿⢹⣦⡀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠳⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⠈⠻⣿⣦⡀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠘⢷⣄⠀⠀⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⠀⠀⠈⠻⣿⣮⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠙⢷⣀⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠟⠀⠀⠀⠀⢰⣿⡟⣀⣴⣾⡷⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⡿⠟⠉⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢁⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠉⠳⢦⣀⢠⣤⣶⣾⣿⠿⠛⠉⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⠀⣀⣀⣀⣀⣀⣿⢠⣤⣴⣶⣾⣷⣭⠳⢮⣍⡉⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠋⠰⠿⠿⠿⠿⠿⢸⡜⠛⠋⠉⠁⠀⠀⠀⠀⠈⠙⠳⢦⡄⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠲⠶⢤⣤⣄⣀⣀⣀⣤⡈⠛⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢀⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣿⡿⠟⠋



⣿⣷⣶⣶⣷⡿⣾⣿⣿⡿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⠿⣷⠶⠶⠶⢶⣾⠷⠶⢾

⠀⠀⢀⡔⡾⠻⡗⢦⠀⠀⠀⠀⣠⠞⠳⣶⠀⠀⠀⠀⠀⢠⢞⠟⠓⣄⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⣠⣾⣷⡀⠀⢀⣠⠚⠉⠓⠤⣄
⠀⠀⡞⠉⡟⠉⡟⠉⡇⠀⠀⠞⣡⣾⣷⣌⠳⠀⠀⠀⠀⢸⡈⠀⢀⡝⠀⠀⢾⣏⣿⣹⣏⣿⡇⠀⠠⣤⡻⣿⣿⣿⡿⠀⠀⢻⣀⣀⣀⣀⣀⡼
⠀⠀⠹⣏⣯⢉⣏⡽⠁⠀⠀⢸⣿⠉⠉⣿⡇⠀⠀⢠⣾⡟⠉⠉⠉⠀⠀⠀⢨⠿⠿⠿⠿⠋⠀⠀⠀⠙⠻⠷⡿⠛⠁⠀⠀⠀⠀⠸⣿⡷
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⢠⠀⢀⣀⠀⠠⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⢰⣝⢿⣿⢟⣵⡄⠀⠀⠰⠋⠭⠭⠹⠆⠀⠀⢀⣾⠫⣗⢪⢻⡆⠀⠀⠘⠀⠈⣟⠀⠰⠁⠀⠀⣠⣉⠙⢷⣄⠀⠀⠀⠀⣠⣄⠴⠚⠛⠁
⠀⠀⠸⣫⣶⣶⣮⡻⠅⠀⠀⣿⡦⡉⢉⢴⣿⠀⠀⠈⢿⣑⡭⣬⣾⠇⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⢀⡉⢻⡄⢹⡆⠀⠀⠀⠙⠋⠳⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠞⠛⠛⠳⠉⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠐⠋⠉⠙⠀⠀⠀⠀⠙⠁⠀⠋⠀⠋⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠌⡀⣐⠀⠀⠀⠀⠀⠀⠀⠁⣷⠉⠀⠀⠀⠀⠀⠀⣴⢂⢶⡄⠀⠀⠀⣀⡀⢀⣼⣇⣀⡀⠀⠀⢸⣿⣿⣷⡆⣿⡆⠀⠀⣀⣠⢿⣆⣀⡀
⠀⠀⠀⠁⠸⣿⣷⠂⠀⠀⠀⠀⣦⣿⣿⣶⠀⠀⠀⠀⢠⣦⣙⣰⡆⠀⠀⠀⣿⡇⣿⣿⣿⣿⠃⠀⠀⠿⢿⣿⡿⠇⠿⠇⠀⠈⠻⣷⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠉⠙⠷⠄⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠈⠿⣿⡿⠃⠀⠀⠀⠛⠃⠛⠛⠛⠛⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠰⠟⠛⠻⠇

⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⢀⣴⣶⣶⡆⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⢰⣶⣤⣤⣄⡀⠀⢠⣶⣦⣠⣶⣆⠀⠀⠀⠀⢀⣴⣶⣶⣄
⠀⠀⣶⣰⣿⣾⡇⠀⢢⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢸⣿⠟⣩⡛⣿⣷⠀⠀⡖⠚⠚⣚⠒⣾⡏⠀⠀⣽⣯⣬⣿⡅⠀⡀⠀⠀⣿⣿⠈⠙⣿⡇
⠀⠀⠉⠈⠻⣿⡇⠀⠂⠀⠀⢠⣾⡇⠸⡿⠇⠀⠀⢸⣿⣎⣛⣣⣿⣿⠀⠀⣧⣾⣷⣤⣦⡟⠃⠀⠀⣿⣿⣿⣿⡇⢿⡇⠀⠀⠻⣿⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⣤⠤⠤⠤⠤⠤⡄⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⢠⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠀⠀⠀⠀⡤⠤⠤⢤⣄
⠀⠀⣿⠀⠰⠦⠀⠀⡇⠀⠀⣿⠀⠰⠆⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⣶⢸⣿⣷⣶⣶⣶⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⡇⣉⣉⣨⠙⡆
⠀⠀⠻⠶⢶⣶⠶⠶⠇⠀⠀⣙⣶⣶⣶⣶⣏⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⢹⢸⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⡇⣒⣒⣒⠀⡇
⠀⠀⠀⠰⠾⠿⠶⠀⠀⠀⠐⠿⠿⠷⠾⠿⠿⠂⠀⠀⠘⠿⠟⠿⠟⠀⠀⠀⠘⠾⠿⠿⠿⠿⠟⠀⠀⠘⠿⠿⠿⠿⠃⠀⠀⠀⠳⠶⠶⠶⠶⠃
⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⣠⣮⣀⣀⣀⣧⡄⠀⠀⠈⣧⣤⣤⣤⣤⠄⠀⠀⢀⣀⣀⢻⣽⠇⠀⠀⠀⠀⣠⣾⠅⣸⠂⠀⠀⢠⠞⠋⠋⠲⡄⠀⠀⠀⢰⠁⠀⠈⡃
⠀⠀⣿⡿⢿⣿⡿⣿⡇⠀⠀⠀⢹⣿⡿⠿⠟⠀⠀⠐⣾⠟⢻⣷⠀⠀⠀⠀⢠⠞⣟⣡⣾⠇⠀⠀⠀⢿⡀⠀⠀⢀⡟⠀⠀⠀⠀⢣⡀⡰⠃
⠀⠀⠉⢣⣘⣓⣠⠟⠁⠀⠀⠀⠠⡶⠠⣶⠀⠀⠀⠈⠻⡷⢿⠛⠀⠀⠀⠀⠸⢤⣬⠟⠁⠀⠀⠀⠀⠈⠓⠴⠶⠚⠁⠀⠀⠀⠀⠀⣿⠇

⠀⠀⠀⠀⠀⠀⠈⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢜⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⡒⠶⢶⠤⢄⠀⠀⠑⠊⠀⠀⠀⠀⠀⢀⠔⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠠
⠠⢀⠀⡇⠀⠀⠀⡠⢤⣶⣆⢀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡾⠀⠀⠀⠀⠱⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡸
⠀⢀⠆⠀⠀⠀⠀⠀⠉⢄⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠙⠋⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠸⠀⠀⠠⠂⠀⠀⠀⠀⠑⠇⡀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠐⠄⠉⠉⠉⢠⠁⠀⠀⠀⠈⠳⣤⣀⣀⢴⢣⠀⠀⠀⠀⠀⠠⠎⠉⠑⡄⠀⠀⠀⠀⠸
⢀⠇⠀⠀⠐⠄⣀⡀⠀⠀⢀⡀⠈⡀⠤⣄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡼⠶⣄⠀⠰⠋⠙⠳⡄⠀⠀⠈⠧⣀⡴⠁⠀⠀⠀⠀⠠⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢺⡀⢀⡸⠀⠘⢤⣀⠴⠃⠀⠀⠀⠀⠀⡶⠖⠢⡀
⠀⡠⠶⢤⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡞
⠀⣇⠀⣠⡦⣀⣀⢄⡀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⠚⠁
⢀⣈⠉⠁⡸⠋⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠁⠈⢳⠄⣇⠀⠀⠀⢠⠆⠀⠀⠀⢀⠄⢤⡀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡠⠀⠈⠀⠀⠀⠘⠀⠙⢄⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⠀⠀⣀⠠
⢤⠴⠋⠀⠈⠲⠤⠖⠃⠀⠀⠀⠀⠹⢄⣀⠇⠀⠀⠀⠀⢀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢣⡀⠀⠀⠆⠻⠟⠀⠚⠂⣤⠄⠀⠀⠀⠀⠠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠁
⢀⣤⠆⢱⠀⠀⠀⠀⡀⠤⡀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠠⠉⠐
⠉⠀⠀⠀⢢⠀⠀⠘⢅⣀⡼⠓⠢⢼⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠄⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠾⢁⡤⠦⠠⣄⠀⠀⠀⡠⠄⢄⠀⠀⠀⡠⠲⠶⠟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⢄⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠈⡧⠀⠘⢅⣀⡼⠀⠀⠀⠈⡂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠙⠢⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠢⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠁⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠁
⠀⠀⠘⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢠⠋⠀⠀⠀⠉⢣⠀⠀⠀⠀⢠⠀⠲⡎⠉
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⡄⠀⠀⠘⢄⡀⠀⠀⣠⡎⠀⠀⠀⠀⠸⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠃⠀⠀⠀⣸⣙⠉⠉⢱⠼⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠲⢴⠘⠣⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⢱⠀⢣⡀⠀⣨⠃
⠀⠀⠀⠀⠀⢠⠒⠒⠄⠀⠀⠀⠀⠀⣇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠡⠀⢀⠌⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠀⡄⠀⠀⠠⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢘⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠛⠉⠛⠛⠒⣾⠦⣤⣀⠀⠀⠀⢀⣠⣾⡁⢨⡗⠛⠋⠉⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣈⡽⠶⠾⣍⡀⠈⠙⣏⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢸⣇⡴⠚⠉⠀⠀⠀⠈⠙⠲⢄⣿⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢀⡴⢿⠁⠀⠀⢀⡤⠴⠤⣄⠀⠀⠀⢹⠷⣄⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠞⠉⠀⣿⠀⠀⢀⡏⠀⠀⠀⠈⣇⠀⠀⢸⡇⠈⠙⣦⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠳⢄⡀⢽⠀⠀⠀⢧⡀⠀⠀⣠⠇⠀⠀⢸⠇⣠⡴⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠉⠙⠋⠁⠀⠀⢀⣼⠟⠁⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⡏⠙⠢⣄⡀⠀⠀⣀⡤⠞⠉⣿⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢠⠷⣄⣀⣠⠽⠛⠻⢥⣀⡀⢠⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⠤⠴⠶⢯⣀⣸⠋⠀⠀⠀⠀⠀⠈⠉⣻⠓⠶⠶⠤⠤⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠴⠞⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢻⡄
⠀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀
⣼⠏⠁⠀⠈⠉⠛⠻⢶⣤⣀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣀⢀⣠⣴⠾⠛⠋⠉⠀⠀⠉⢿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⣀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠈⠙⢷⣄⠀⢀⣴⠟⠉⠛⠿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣩⡿⣯⡁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋⠀⠈⠻⣦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⣿⢀⣤⡞⠁⠀⣀⣀⣀⣀⠀⠙⢶⡀⢻⡇⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠈⢻⣦⣠⣤⣀⠀⣿⡿⠋⢀⣶⠟⠉⠉⠉⠙⠻⣦⡈⠻⣿⡇⠀⣀⡀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⢰⡟⠉⠈⠙⢿⣿⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠘⣷⠀⢸⣿⠟⠛⠛⢿⡁
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢀⣼⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣽⠿⠿⠛⠻⣿⡀⠈⢿⣄⠀⠀⠀⠀⠀⠀⣰⡟⠀⣸⡿⠷⣦⡶⢿⡅
⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⣿⠿⣆⡀⠙⠷⣤⣤⣤⣴⠾⠋⣠⡾⢿⡇⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢿⡄⠉⠳⣤⡀⠀⠀⠀⠀⣠⡾⠋⠀⣼⠇⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠻⣦⡀⣠⡾⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣼⠿⢿⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⢷⣶⠟⠁⠀⠀⠙⢷⣤⡀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⢶⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠹⣧⣄⣀⣠⣤⣶⠾⠛⠉⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠈⠙⠻⢶⣦⣤⣀⣀⣤⡿
⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⣠⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢋⡽⠟⢻⠛⢿⣍⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⣰⠟⠁⠀⢸⠀⠀⠙⢷⡀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⣼⠃⠀⠀⠀⢸⠀⠀⠀⠀⢻⡄⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣸⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢻⡄⠀⠀⠀⢸⠀⠀⠀⠀⣼⠃⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠹⣆⠀⠀⢸⠀⠀⢠⡾⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣈⣳⣤⣸⣠⣴⣋⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣧⡇⢹⡏⣾⠍⢸⣽⠂⣷⣾⢸⡯⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠁⠈⠁⠉⠉⠈⠈⠁⠁⠉⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⣦⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣿⣿⣇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡃⢛⠀⣼⡳⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣧⠞⠒⠋⠙⡧⢰⢛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠑⢤⣀⢀⠠⠇⠂⠈⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠒⠂⠁⠀⠸⠔⢢⡀⢀⠐⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⡀⠀⠁⢀⠹⠂⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠠⠀⠒⠉⠃⢄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡜⠫⡀⠀⠀⠀⠠⠀⠀⠁⠠⠼⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⡀⠀⠀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠈⡹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⣀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⡌⠛⠛⣱⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠄
⠀⠀⢀⠃⠀⠀⠀⢀⣀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⡸⠀⠀⠀⡎⠘⠛⠉⠐⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡇⠀⠀⠀⠡⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢠
⠀⢸⠀⠀⠀⡀⠀⠑⢄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜
⠀⡇⠀⠀⠊⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⠈⠉⠁⡄⠀⠀⠀⠀⠱⢤⠤⠴⡟⡀⠀⠀⠀⠀⡜⠁⠈⡆⠀⠀⠀⢠⠃
⡜⠀⠀⠀⠒⠠⠤⠀⠀⡤⠠⠌⠒⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠚⠓⣄⠀⡏⠈⠙⡆⠀⠀⠘⠢⠾⠁⠀⠀⠀⠀⠢
⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠦⠤⠃⠀⠙⠒⠊⠀⠀⠀⠀⠀⡜⠉⠑⡄
⢰⠋⠈⣷⠀⠀⠀⠀⠈⠙⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠴⠃
⠀⠑⠒⢉⡞⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠉⠙⣦⢸⠀⠀⠀⢠⠂⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠂⠈⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠤⠔⠉⠈⠳⠤⠖⠁⠀⠀⠀⠘⢄⣠⠃⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠘⠛⠃⠓⠀⡤⠀⠀⠀⠀⠠⣶
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⣤
⡤⠞⠁⢆⠀⠀⢠⡔⠒⣄⢀⣀⡴⠛⠲⡄⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠀⠀⠈⢂⠀⠈⠓⠒⠉⠀⠘⣄⠀⠀⣽⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⡀
⠀⠢⠀⠀⠀⠱⣀⡀⡀⠀⠀⠀⠈⠉⠉⠀⡞⠁⠀⠱⣄⡀⣞⠁⢘⠄⠀⠀⢢⠈⠀⢸
⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠙⠦⣄⠴⠁⠀⠀⠁⠁⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠁⠈⡄⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⡀
⠀⠀⡇⠀⡀⠀⠐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡞⠉⣢⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢰⠀⠀⠀⢀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠰⣄⡠⠇
⠀⠀⠰⡀⠁⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡎⠀⠀⠀⠈⡆⠀⠀⠀⠐⠒⠚⡄
⠀⠀⠀⠈⠂⠤⢀⣀⡀⠄⠊⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠀⠀⠑⢤⣤⠤⠞⣀⠀⠀⠀⠈⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⣧⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⢀⠔⠛⢄⠀⡎⠉⠉⡆
⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⡏⠀⠈⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠠⠞⠀⠘⠒⠚⠁
⠀⠀⠀⠀⠘⡀⢀⡅⠀⠀⠀⠀⠉⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⠢⠄⠀⠀⠀⠀⡄⠩⡒⣂⠥⠀⠣⠄⠚⢃⠀⠀⣄⠀⠙⠀⢸⠀⠀⠘⡇⠀⠀⠀⠀⡇⡇
⣈⠔⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠂⠉⣀⠠⠄⠒⠁⠀⠀⠙⠐⠒⠒⠁⠀⠀⠀⡝⠀⠀⠒⣊⡡⡣⠀⠀⠠⡀⠴
⡉⣦⠄⢻⠂⠀⢀⠔⣊⠩⠤⠤⠍⣑⠢⡀⠀⠀⠑⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠘⠓⠛⠋⠉⠁⠀⠀⠀⣵⠀⡦
⢡⠠⠄⡜⠀⠰⡡⡪⢒⠩⠥⠬⢅⠢⡑⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠋⠉⠉⠀⠑⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢣⡠⠰
⣢⡴⠔⠀⠀⠀⠘⠔⡡⢊⠉⠉⡒⢝⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢨⡇⢀⣋⣀⣘⠀⠀⠀⠀⠀⢠⡐⠊⣩⡞⠉⠀⠀⠀⣠
⠉⠀⠀⠀⠀⠀⠀⠀⠉⢡⠂⢲⠈⠁⠀⢀⠤⠠⠔⠒⠚⠉⠉⡆⠀⠈⠋⠀⠈⠀⠰⠄⢠⠄⠀⠀⠀⠀⠀⠀⠈⣏⢔⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⢰⠀⠀⢀⢀⠂⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠤⠐⠢⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠂⢖⠏⠫⠆⠀⠀⠀⢀⡠⠄
⡐⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠌⠂⠄⡌⠉⡄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡠⠄⢒⡢⠉⢰⠊⠙⣞⠀⠀⠰⡄
⠈⠀⠀⢸⠄⠀⢠⡋⠉⠒⠘⠀⠀⠀⠀⠀⠀⠀⠚⠓⠊⠉⠉⢀⢀⠤⡖⡀⡰⠠⣀⠀⠀⠈⡄⠩⢐⡂⠁⠀⠦⠤⠘⠇⠀⠀⠇
⠀⢤⠉⠀⠁⠀⠀⠉⢀⡐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣁⣤⠰⡊⢢⠠⠘⠥⠀⠀⢡⠈⠒⠂⠈⣀⡠⠄⠒⠁⠀⠀⢃⠒⢀
⠂⠉⠛⠀⠀⠀⡔⠉⡍⠑⣤⠀⠀⠀⠀⣀⣀⠀⣀⠀⠀⠀⠀⠱⡀⠴⠖⢙⡀⡥⢤⠀⠀⠀⠀⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⢰⠀⠀⠃⠀⠏⠀⠀⣪⠓⠀⠁⠀⢾⣇⠀⠀⠀⠀⠑⢄⡜⠣⣱⠄⠀⠀⠀⠀⢀⠠⢐⡂⠀⢐⡂⠄⡀
⠤⠀⢠⡀⠀⠈⠑⠂⠤⠚⠀⠀⠀⣹⠀⠀⣠⠀⠸⡹⡀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⡔⡡⡪⠕⢐⣒⠒⠬⡑⢌⡢⠀⠀⠀⣀⡠
⠀⡤⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠁⠀⠀⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠔⡩⠔⣒⠨⢕⠬⠂⠀⠀⠀⡐⠁
⠐⠃⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⣀⣀⠤⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⡡⠤⡉⠒⠀⠀⠀⠀⠀⡤⠥⠈
⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⡠⠴⠶⠛⠃⠀⠀⣀⡀⣀⡀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠃⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣒⣒⡒⠤⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⣤⠐⢒⢤⢄
⠤⠤⠄⠔⠀⠀⢠⢊⢔⡨⠐⠀⠐⠠⢅⠢⣑⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⢀⢇⢀⡕⢣⠆⡦⠴⠄⠀⠀⢠⠀⠄
⠂⠀⠀⠀⠀⠀⠈⠡⡡⢊⠥⠀⠠⢍⡢⠕⠀⠀⠀⠀⠀⠀⡀⠙⠻⡛⠹⢉⡙⠙⠀⠂⠀⠀⠀⢰⣀⣨⣁⡛⢰⠇⢀⡡⠀⠀⢸
⡀⣀⣀⠂⡄⠀⠀⠀⠀⠣⢊⠥⢍⠓⠃⠀⠀⠀⠀⠀⠀⠘⠢⠤⠴⠥⠤⠤⠁⠀⠀⠘⠁⠀⠀⠈⠧⣘⣸⠕⢣⡠⠊⠃⠀⠀⢸
⠇⢉⠉⠁⠐⡄⠀⠀⠀⠀⠘⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⢯
⠀⠘⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉⠉⠉⠀⠀⠀⣀⠠⡄⢒⣂⡲⡄⠀⠀⠓⠒⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⣰⢶⠀⠀⠀⠀⣀⡠⠤⠐⠒⢏⠏⠯⢇⠀⠀⠀⠀⠉⣡⠘⠀⠀⠀⣇⠁⠀⠀⠀⠸⣽
⠀⠀⠀⠀⠀⠀⡀⠈⠢⡤⢪⠃⠈⠀⠀⠀⢸⡀⠤⣐⠈⠉⠰⠈⠉⢶⠀⠀⠀⠰⠁⣼⠈⢻⠄⠀⢹⠀⠀⡞⠄⠀⣿⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣴⣶⢻⣥⡤⠗⠐⣭⣼⠟⣿⣿⣿⣿⣶⡄⠀⠀⢲⣦⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣭⣀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣾⣾⣿⣦⡀
⠙⠩⣤⣤⣄⡂⠉⢁⣤⠞⢫⣹⠟⠉⠛⠁⠀⢰⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡃⢺⡍⢦⠀⣄⠀⣦⣄⣠⣤⣬⣻⣿⣿⣷
⠶⠊⠭⠄⢀⣤⠾⠋⠁⣶⣾⢋⡀⠀⠀⠀⣠⣾⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠻⢿⣇⠸⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⣟⡟⠋
⠀⣀⣤⣾⠛⢉⠀⢰⣰⠟⠁⠀⠀⠀⢀⣶⣿⣿⣿⢿⠯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⢻⣿⣷⣿⠛⣩⣿⣿⣿⣿⣿⢷⣙⣫⡽⠿
⡶⠟⠋⠀⠀⢀⣤⠿⠃⠀⠀⠀⢀⢴⣾⠯⣾⣿⡁⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠘⣿⣿⡿⣿⣿⣿⣿⣟⡱⣿⣼⣾⣾⣿⣿⢑⡶
⠀⠀⠈⠁⣠⣮⡮⠀⠀⠀⠀⠀⣉⣞⣃⣾⣿⠿⠷⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠸⣿⡀⠿⣿⣿⣿⣽⣿⡿⠟⢛⣿⣿⣿⣧⡘
⠀⠈⣀⢋⡉⠩⢾⡟⠀⠠⣾⣦⠂⣀⣾⢿⠋⠉⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⣿⡆⠀⠀⠉⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣯⣼⣿⢿⣟
⠀⡴⢫⠅⠈⠀⠈⠀⠴⠀⣽⡇⢾⣾⣟⡎⠀⢾⣿⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⢿⡷⡄⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣟⠉
⠋⡠⠁⠀⠀⠀⡀⠂⠀⣴⠁⠄⣼⡟⡛⠀⡈⠉⢛⠏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⢿⠘⣀⠀⠀⠀⠀⠀⠀⢱⣟⣿⣷⡍⠁
⠞⠀⠔⠀⡠⠊⠀⢰⠟⠃⠀⣼⡟⠠⠀⠈⠀⡐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠆⠿⣇⣿⣷⣶⡀⠀⠀⠀⠀⠻⣌⢿⣿⣦⣠
⠀⠉⠀⠌⠀⠀⣰⢋⠀⢀⣼⡿⢀⠂⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡬⢿⣏⣥⡄⠀⠀⠀⠀⠹⣿⣻⣿⣿⡔⠒⠉
⠀⣴⠃⡀⢀⡾⠻⠁⠀⣼⡿⢀⡌⠀⠀⠀⣀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢷⠛⢯⣛⠃⠀⠀⠀⠀⠀⢜⢿⡙⣿⣿⡄
⠊⠀⠂⢁⡞⠁⠉⠀⣼⡿⠇⡜⠀⠀⠀⢦⢮⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⠀⠀⠀⡀⢠⣸⣟⠽⣿⣟⣃⠀⠀⠀⠀⠀⢀⣳⣯⢿⣿⣶
⣦⡀⣠⠏⠀⠀⠀⣼⣿⠃⣸⠁⠀⠂⢀⣞⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠿⠅⠀⠀⠀⡄⢚⡏⢻⣄⣻⣿⣯⡁⠀⠀⠀⠀⠀⠹⣿⡘⠿⣿
⣠⣼⢿⠆⠀⢁⣾⣿⡏⢰⠿⠆⠀⢠⠈⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⢀⣵⣼⣿⡙⢿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠷⣦⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠨⠅⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⢢⣔⢁⠀⠔⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠦⠤⢤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢐⠄⠈⠩⠩⠉⠉⠉⠀⠐⠀⠀⠈⠀⠰⠆⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠱⢚⠃
⠀⠀⠀⠂⠀⠀⠀⠀⠉⢸⣳⠀⢿⠁⣼⠉⢧⢰⡇⠀⢰⠏⢳⡄⡞⠉⡃⠈⢧⣰⠃⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠂⣼⡀⢿⣤⡞⠘⣧⣄⠸⣄⡼⠃⢧⣀⡟⠁⢨⠇⠀⠀⠈⠐⠀⠐⠂
⠀⠀⠀⠀⢠⠀⠠⠐⠀⡀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠄⠀⠀⠈⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⢠⠀⢀⠀⡀⠀⠀⠠⡀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠐⠈⠀⠀⠐⠄⠂⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠐⠀⠀⠐⠋⠀⠑⠂⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢻⣦⣤⣀⠀⠀⠀⠀⢠⡴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠁⠀⠈⠳⣄⠀⠀⢻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⢹⡷⢦⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠉⠉⠉⢷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⣄⠀⠈⠙⠋⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠸⣧⣤⡟⠀⠀⠀⠀⢀⣸⣀⣀⣀⣸⣀⣀⣀⣸⣇⣀⣀⣸⣃⡀⠀⠀⠀⠀⠘⢷⣤⡿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⣻⠀⠀⠀⡇⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦⠀⠀⠀⠀⠀⡇⠀⠀⢼⡋⢙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠀⠀⠀⠓⠒⠒⠒⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠃⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣨⡷⠒⠒⠒⠒⠒⠒⠒⠲⣿⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡘⠃⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⣿⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣻⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣀⣀⠀⠀⠀⠀⢿⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠸⠧⠤⢴⡏⢉⣷⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣦⣀⣀⣠⠞⠁⠀⠀⣾⠋⢹⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠀⠀⠁⠀⠀⠀⠀⠘⠛⠛⠁⠀⠈⠳⣄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠺⣇⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠟⣫⣴⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣌⠻⣿⣿⣿⠛⣡⣾⣿⣿⡿⢋⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣦⠉⠁⠚⠻⠿⠿⠏⠐⠛⢋⣡⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⡿⠃⣴⣿⣿⣿⣶⢂⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠹⠟⢀⣾⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⠏⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠺⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⠏⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢀⣶⣤⡈⠻⢿⣿⢸⣿⣿⣿⣿⣿⠟⠁⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⣿⣿⣿⣶⣤⡁⠙⠻⢿⡿⠟⢁⣴⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠃⠟⢛⣀⣤⣦⣬⣭⣭⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⢴⣶⢰⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠸⡇⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⢰⡿⠋⠉⠉⠛⠛⠳⢾⣯⣀⠀⠀⠀⣀⣤⣿⠶⠟⠛⠛⠉⠙⠛⣷⡀
⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⣸⠇⢀⣡⠶⠛⠉⠁⢸⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣤⠐⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣠⠞⠁
⠀⠀⠀⠀⠀⣴⣶⡎⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠉⠳⣄
⠀⠀⠀⠀⣠⠟⠋⠀⠀⠀⠈⢹⠳⢤⡀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠘⠀⣀⠌⠓⠢⢄⡀⢸⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠈⠷⣤⣤⣤⣤⠶⠖⠚⡏⠀⠀⠀⠀⠀⠀⡿⠐⠢⠤⢤⣤⣤⡴⠟⠀⠀⠀⠀⠀⢀⡴⠚⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠶⠋⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠁⠒⠂⢰⡏⠀⠀⠀⠀⣀⣠⣴⠶⠞⠛⠋⠉⠉⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣼⠇⢀⣤⡶⠟⠋⣁⡀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⣀⣿⠿⠛⠁⠀⠀⠸⢿⠿⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⡾⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠻⣦⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠙⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⢿⡌⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⢀⣀⣀⠠⢸⡇⠀⠀⠀⠈⠙⠛⢶⣦⣤⣀⣀⣀⣀⣀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣆
⠀⠀⠀⢀⣀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣠⣿⡟⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣶⣦⣄
⠀⠀⠀⠘⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⢦⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠻⣿⣿⣿⣿
⠀⠀⢰⣶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡋⠐⢷⡩⢷⡄⢠⣀⡀⢰⣦⣄⢀⣤⡀⣤⣈⣻⣿⣿
⠀⢠⣶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣟⣿⣶⠋⣹⣷⣿⣾⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⣿⣿⣿⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠈⣭⣿⣦⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣆⡽⢋⣹⡷⡾
⡟⢻⣿⣿⠿⡿⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢙⣟⣿⣿⣿⠟⢉⣠⣾⣿⣿⣿⣿⣮⣮⠿⣟⣬⣙⢶
⣠⣿⣿⠁⠰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿⣻⣿⣾⣿⣭⣾⠋⢥⢿⣧⣸⣾⣖⣻⣿⣿⣍
⣿⢿⣿⣏⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣶⠀⠀⠀⠘⣿⣿⡏⢈⣿⣿⣿⣿⣿⠿⣳⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷
⣯⣞⣛⠉⢻⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣿⣿⡄⠀⠀⠀⠈⣿⡗⠈⠛⢻⣾⣿⣿⣿⣩⣿⠟⣋⣁⣠⣼⣿⢿⣿⣿
⡗⠀⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢿⣿⡄⠀⠀⠀⠘⠀⠀⠀⠀⠻⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣌⣿⣿⣿
⠃⢈⣶⣿⣇⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣾⣿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⣀⠫⣿⠯⡇⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⡁⣿⣏⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣻⣿⣿⢿⡟⣿⡿⠆
⡠⠀⠀⠈⡘⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠁⢿⡄⢹⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡮⠿⣿⣷⣄
⠁⠈⠐⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣶⡶⢰⡞⣇⢸⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⣿⣿⣦⡤⣀⡀
⠀⠈⣥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⣥⣹⡎⠻⣿⣟⠛⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣽⣿⣿⣿⣮⠉⣉
⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⠛⢭⣏⢷⣿⣷⠀⠀⠀⠀⠀⠀⠀⠹⣿⣬⣿⣿⡿⣯⠁
⢀⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠘⣇⠚⢻⣮⣋⠉⠀⠀⠀⠀⠀⠀⠀⠱⡙⢿⡀⠻⣿⣿⣧
⣠⢇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡧⠀⠀⠀⠀⠀⠀⠀⢈⡀⢻⣶⢖⣻⣿⣷⠿⠀⠀⠀⠀⠀⠀⠀⠈⢌⢻⣴⡟⣿⣿
⡃⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⠦⠀⠀⠀⠀⠸⠂⠉⣽⠮⣿⡁⠀⣿⣿⣶⢶⠄⠀⠀⠀⠀⠀⠀⠈⢦⣿⣿⢍⣿
⡁⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠻⠛⡚⠀⠀⠀⠀⠀⡁⢈⣸⡆⠸⣷⣬⡽⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠌
⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⠀⠀⠀⠀⠀⠀⢈⣥⣴⣿⣟⠉⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⣦
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢷⢷⠦⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⣿⣆⠀⠙⢿⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟

⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠚⣩⡤⠶⠦⣬⡙⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠁⠴⢚⠲⠄⠁⠀⠀⠀⠀⠀⢀⠂⠀⠋⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣇⣀⣠⣿⡇⠀⠀⠀⢀⠀⠀⠤⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠽⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠽⠿⠿⠀⠒⠐⠃⠐⠂⠿⠿⠿


⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣒⣒⠴⠶⣂⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⣲⠁⢈⠆⢘⣒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣍⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢔⣀⡢
⠀⠀⠀⠼⠷⠤⢤⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣤⡆⠀⠀⠀⠀⠀⢸⣿⣟⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠪⠭⣕
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠿⠛⠛⠃⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⣶⣤⠚⣿⠳⣴⣶⠀⠀⠀⠀⣀⣤⣶⣶⣄⣀⠀⣤⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⣶⠖⣿⣿⡿⢶⡶⠀⠀⠀⠘⠿⠿⢻⠟⠿⠟⠈⠛⠛⢻⡟⠛⠛⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠸⠛⣿⣧⣠⣄⣀
⠀⠀⠀⠀⢠⣤⣭⣄⡄⡀⠀⠀⠀⠀⠀⢀⣐⢴⠀⠀⠀⠀⠀⣀⣲⠀⠀⠀⠀⠀⢸⣦⣍⣉⣵⣿⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⢿⣿
⠀⠀⠀⠄⢤⢤⡤⡤⡌⠄⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠀⠀⠀⠁⠭⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⡩⡀⠀⠀⠀⠀⠀⠾⡷⠤⠤⢬⣷⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⣄⡟⠙⢂⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣙⢒⠂⣾⣿⠈⢳⠀⠀⠀⠀⠀⣖⢂⣐⠁⢤⠤⠐⢂⡁⠇⠀⠀⠀⠀⢸⠐⠩⠃⡇
⠀⠀⢀⣴⣆⣿⣿⣿⣿⣴⣦⠀⠀⠀⠀⠀⠀⠲⣒⣆⢇⠀⣿⡟⠀⠀⠀⠀⠀⢄⠀⡶⠔⠒⠀⢀⠁⡦⠆⠀⠀⠀⠀⣸⢖⠤⡖⠇
⠀⠀⠀⠙⠁⠈⠙⠋⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠈⠒⠚
⠀⠁