2020 Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡄⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⢀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠋⠁⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠁⠀⢀⣠⣄⠀⠀⠀⢿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡶⠖⠂⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⣾⣿⣿⠂⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢀⣀⣿⣿⠟⠀⠀⣠⣿⣿⣿⡇⠀⠀⢸⣿⣿⡿⠀⠀⢀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⣴⣶⣦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠸⣿⡿⠃⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⠁⠀⠀⠀⠸⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⠀⠀⠀⠉⢉⣿⣿⣿⡟⠁⠀⠀⠀⠉⠙⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⡀⠀⠀⠀⢿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⣰⣿⣿⡄⠀⠀⣼⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡏⠀⠀⠴⠿⠛⠋⠉
⣀⣀⣀⣼⣿⣿⣿⠁⠀⠀⣿⣿⡿⠁⠀⢀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣴⣶⣿⣿⡟⠁⠀⢀⣾⣿⣿⣿⡇⠀⠀⢸⣿⣿⡟⠀⠀⢸⣿⣿⣿⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⣽⣿⣿⣿⠋⠀⠀⠀⠉⠹⣿⣿⣿⣇⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣾⣿⣿⡿⠁⠀⢀⣴⣶⣶⣶⣿⣿⣿⣿⣄⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠖⠂
⠉⠉⠉⢉⣿⣿⣿⡿⠀⠀⢠⣾⡿⠟⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣸⣿⣿⣿⠃⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠻⠟⠿⠿⠛⠻⠿⠀⠲⠾⠛⠿⠻⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠒⠂
⠀⠉⠉⠈⠉⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⠀⣀⠄⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠀⠈⠀⠐⠂⠃⠀⢀⠠⠄⠐⠤⠄⠈⠡⠂⠠⠀⠂⠀⠀⢠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠴⡤⠈⠁⠀⡀⠢⡀⠀⢠⡀⢈⠀⠀⠀⠰⠀⠀⠄⠐⠀⠁⠀⢤⠁⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠱⠄⡔⠀⢠⠀⢀⠤⠆⠀⢰⡈⢂⠈⠉⠀⠂⢀⠄⠈⠀⠀⠀⡀⠀⠀⡀⠁⠀⢀⣦⠠⠈⠂⠄⠐⠁⠀⠀⡅
⠀⠀⠀⠀⠐⠂⢰⠀⠀⠤⠬⠤⡀⢀⠒⠪⠀⠁⣐⡀⠁⣀⠤⠀⡠⠀⢈⡉⠈⠃⠀⣠⢀⠀⠈⠀⠐⠒⠗⠒⠀⢈
⠀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠰⠀⠁⢀⡀⣄⠀⠚⠙⡁⠀⠀⣐⣖⢈⠀⢄⠊⠑⠂⠙⠉⠃⢠⠀⠀⠀⠀⠶⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⣔⡁⠔⠈⡂⢐⠀⠀⠀⠉⠀⠁⠀⠀⠈⣄⠀⠀⠀⠀⠀⠺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀⢀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⢀⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡀⠸⠀⡄⠀⠀⠀⠀⡀⠐⠀⠐⠀⠀⠀⠐⠒⠂⠄⢀⡀⠀⠀⠀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢇⠀⠒⠀⠀⠐⠄⠀⠀⢀⠖⠀⠠⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⣠⣴⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⢠⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣰⠟⣩⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡀⢿⢃⢇⣾⣿⣿⡿⣿⠀⠀⣠⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣎⢸⣾⣿⣿⣿⣧⣇⣀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠈⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⠉⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⡗⠁⠀⠀⠀⡿⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣠⣶⣿⠤⠾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠙⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣄⣸⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣀⡀⠀⠉⠉⠙⠛⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣀⣀⠀⠀⢄⠀⠁⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣌⣁⠀⠀⠂⠉⡀⠀⠐⣌⣱⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣰⣼⣦⡦⠶⠶⢶⡄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠉⠉⠀⠀⡀⠀⢀⣼⣿⣧⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⢈⣡⣿⣿⣿⣿⣿⣷⣾⡄⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣯⣭⣙⣿⣿⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣽⣿⣟⣻⣿⣦⣀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⣿⡷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠋⠟⠋⢻⣻⡏⠉⠉⣿⣾⣿⣿⡿⠻⢿⢟⣻⣯⠀⠈⠻⣻⣍⣽⣿⣼⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡀⠀⠀⠈⣹⠁⠀⠀⠉⠈⣠⢟⡭⣖⣵⣿⣿⣿⣄⣀⣀⡈⠺⡐⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⠏⢻⡟⣣⣤⡄⡾⠃⠘⠀⠀⢠⣊⣵⣿⣾⣿⣿⠿⠿⠿⠛⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠳⢻⡿⠁⠀⢤⣆⣩⡿⢁⣀⠄⠀⠀⣶⠽⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢠⡟⠀⠀⠀⠀⠈⠉⠁⠉⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢠⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠐⢻⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢟⡀⠀⠀⠀⠠⠰⣦⠄⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠻⣿⣿⣦⡀⠀⠀⠀⠀⠺⠿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣛⡙⠛⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢛⡋⠛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠙⣦⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠙⣦⢨⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⡇⠀⠀⢀⣠⡴⠾⠛⠛⠳⣤⣝⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⣠⣶⣿⡽⠞⠁⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁⠀⢀⣠⣤⠴⢦⣄⢙⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠃⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠀⠀⠀⠀⢀⣠⣾⣿⡿⠃⢀⣤⣾⣫⠥⠂⠀⠀⠈⢿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⠋⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡟⠀⢀⣤⣶⣿⣿⣿⠛⢀⣴⣟⣿⠟⠁⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠟⠉⠀⠀⠀⠘⢿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣾⣿⣿⡟⣡⣾⣿⣿⡿⠟⠋⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠋⢸⣿⣿⣿⣯⣀⠀⠀⠀⢿⣿⣿⣦⣤⣀⡀⠀⠀⠀⣠⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣖⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣶⣢⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣦⡀⠀⠈⠉⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠞⢭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠇⢯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⡄⢄⡀⠠⠤⠤⡀⠀⠀⡠⢄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠻⠗⠛⣺⡿⠟⠢⡐⠀⣴⡏⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⣀⠕⢩⢡⠭⠅⠈⡆⠀⠀⠸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠹⠟⠘⣦⠝⠀⢆⡴⡇⡖⢷⠀⣶⠁⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡀⠀⡅⠀⠀⠄⠁⠀⢸⠸⠂⣿⠰⠁⠜⣩⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⠿⡀⠀⠀⠀⠀⠀⡞⠩⠁⠀⡿⢀⣠⣤⣼⣿⠿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡍⢇⠈⣲⠃⠐⡴⣳⣖⠤⠈⢀⡉⢴⠚⠍⠒⠫⠭⠤⠴⣚⣛⣻⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⢴⡋⣄⢦⠁⠠⠠⠘⣰⠈⠃⣁⣀⣁⣈⣀⣀⣀⣉⣉⡉⠉⠩⣍⣉⣉⠩⢙⡻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠅⠀⠀⠻⠀⠻⠀⢁⠈⢣⠑⠀⠐⢉⣁⠐⠶⠶⠆⠀⠒⠶⠶⠤⠤⠭⠭⣍⣉⡙⠛⠮⣝⡲⢭⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⠤⠤⣦⠴⠋⠀⢀⠀⢤⡈⠒⠬⣍⣙⡓⠶⢶⣶⡷⠖⠚⠒⠒⠲⠶⠤⣍⠓⢦⡀⠙⢦⠳
⣠⠒⠉⠉⠙⢳⡶⢦⡤⠤⣤⠖⠲⡄⠀⠀⠀⠀⣠⡀⠛⠻⠦⡙⢿⣶⠀⠀⡽⡣⠿⠋⠐⠚⠉⣉⣭⡭⣝⡓⠦⣅⡢⡙⢦⠀⣇
⣷⣰⡋⠀⠀⠀⠁⠀⠙⠀⠈⠇⠀⠀⣠⠴⠒⠉⠛⠻⢷⡄⠀⠀⢰⠏⠀⣰⡕⣡⠶⠒⠒⠒⠒⠚⠛⢀⣠⣬⣅⣈⠙⢮⣞⡇⣿
⠀⠉⠑⠀⠀⠀⠐⣒⣒⡲⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⢀⣟⡠⠴⢷⡶⠾⠦⠤⠄⠀⠀⠀⠀⠒⢚⣋⡽⠃⠀⠀⢤⣵⣼⣿⣿⠏
⠀⠀⠀⠀⣰⡦⢿⣙⡖⢦⠀⠀⠀⠀⠀⠀⠤⢶⠶⠿⠭⣧⣄⣤⠤⠥⠴⠒⠒⠚⠻⣏⠉⢭⡉⢭⡀⠀⣐⠒⠶⠶⣬⡿⠟⠁
⠀⠀⠀⠀⠻⠖⢻⣧⣼⠫⠤⠒⠂⠀⠀⠀⠀⠣⠴⢧⣠⣗⠀⢸⡀⢀⣧⣀⣸⣧⣤⣿⣧⣤⣷⣤⣷⡤⠬⠿⠗⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡆⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⢀⣀⠀⠀⣼⠋⢀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠋⢹⡇
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠠⠆⢀⡿⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠈⢆⣸⠁
⠀⠀⠀⠀⠀⣉⠬⠛⢦⠀⠀⠀⡿⣧⡔⠀⠀⠷⠞⠳⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⠚⠋
⠀⠀⠀⠀⠰⠁⠀⠀⢀⠇⣀⣀⠐⣤⠀⠀⠲⢤⠀⣠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡈⠠⠤⠤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠧⠆⠀⡠⠀⠀⠀⠀⢀⣴⠁⠟⠈⣷⡤⠊⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⣠⡴⠶⠦⣤⣤⠸⠁⡀⡠⠐⠠⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⠷⠄⠀⢠⡾⠻⣿⢷⡄⠀⠀⠀⣀⣀⣸⣿⣻⣿⡿⠥⠤⠶⡤⠤⠤⠶⣯⣤⠀⠀⠈⠋⢠⠞⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠡⣄⡀⣠⠋⠀⠀⠈⠐⠿⢤⣤⡉⠈⠻⢻⣿⠿⢿⣷⠖⣷⣶⣿⣿⣿⣤⡾⡿⠃⠀⠀⠠⠼⠇⠀⠀⢀⠐⠠⠆
⠀⠀⠀⠀⢠⣄⡀⠀⡱⠧⣤⡀⠈⠁⠀⠀⠀⡄⠢⠀⣀⡀⡷⢄⡀⢂⣈⠶⠖⠁⠐⣿⣿⣿⣯⣌⠁⠀⣤⡶⠛⢃⠀⢰⠂⠀⠀⠛⠀⠀⣤⡀
⠀⠀⠀⠀⠺⣿⣿⣶⣶⣤⣄⣉⠀⠀⠀⠀⠀⠀⠀⢀⢈⡏⠐⠚⠁⠉⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣅⡠⠶⠖⣾⠎⢲⣄⣀⣀⣤⡴⣿⣿
⠀⠀⠀⠀⢸⣯⣹⣿⣿⣿⣿⣿⣿⣿⣶⣶⣴⠶⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣋⠘⠉⠉⠠⠴⣻⣿⠃
⠀⠀⠀⠀⢸⣿⣿⣛⣿⣿⣿⣿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⢿⣷⣶⣦⣸⣿⡆
⠀⠀⠀⠀⠈⠿⠿⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡞⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣜⢿⡇⣙⣿⣯⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠙⠶⣾⡿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠌⣋⢵⣦⣋⠹⣿⢿⣿⣿⡛⢿⣿⣿⣿⣿⢿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⣿⢿⣿⣶⣞⣷⣮⡟⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣛⠻⢿⣿⢌⣿⣿⢿⣿⣿⣷⣍⢳⣍⠿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣟⣛⠛⢻⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣷⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⣿⣿⣶⣮⣭⣝⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣟⡛⠿⢟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣯⠷⠲⡮⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣷⣄⢢⣮⣽⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣾⣿⣿⡿⣿⣿⣷⣿⣿⣾⣿⣾⡆⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⠻⣟⠃⠙⠻⣿⣿⡁⣿⠉⣿⡿⣿⣷⣿⣿⣿⣴⣿⣿⣏⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠟⢟⣿⠟⠀⢹⢿⡄⠘⠀⡜⢠⣿⢧⣟⠛⠶⣿⣿⣹⢟⡻⣿⣿⢿⡿⣿⣿⣿⣿⣿⣟⠿⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠉⠃⠙⠋⠁⠘⠋⠸⠀⠀⠀⠘⠏⠀⠰⠃⠘⠷⠄⠉⠀⠀⠀⠀⠙⢧⣴⡌⠑⠈⠑⠈⢿⠿⠏⠐⠿⠖⠂⠈⠟⠻⠟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢀⠀⢀⡀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠁⠚⠁⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⢿⣿⣿⣿⣶⣶⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠀⠀⠀⣠⣴⣾⣿⡿⠿⠛⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣷⣄⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⢴⣤⣿⣿⣿⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠿⠻⠿⠿⢿⣿⠿⠿⢿⣿⣿⣦⠀⠀⠀⢨⣿⡿⠛⠋⠉⠉⠛⢿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠉⠁⠀⠀⠀⠀⠀⠻⠶⠾⠟⠉⣻⣿⡆⠀⠀⣼⡟⠀⠀⠀⠀⠀⣴⣦⣬⡉⢻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⣀⣀⠀⠀⣠⣤⣴⣿⣿⣿⡟⠁⠀⠀⢸⣿⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠈⠉⠉⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⢠⣿⣿⣿⡟⠛⠛⠉⣿⣿⠋⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⢘⣿⣿⣿⣧⣀⠀⠀⣿⣿⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⣽⣿⠀⠀⠀⠀⢨⣿⣷⣶⣶⣶⡄⢸⣿⣿
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣟⣥⣴⣿⣿⣷⡄⠀⠀⣸⣿⣿⣦⣀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠁⢸⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⣠⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⡀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⢠⣾⠟⣡⡾⢋⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣠⣿⣿⣿⣿⡿⠄⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠈⠁⠘⠟⠁⠀⠉⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣷⡄⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠁⠀⠀⠈⠙⠻⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣦⡀⠘⠿⠀⠀⠀⠀⢀⣾⠂⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣿⣿⣿⣿⣦⡀⠉⠁⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡆⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⡟⠁⠈⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡉⠿⣦⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣌⣿⣿⣦⣀⣤⣾⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⣿⠁⠙⢿⡿⠉⠿⠉⠿⠋⠻⡿⠀⠀⠀⠙⠿⠿⠛⠙⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢋⢀⣁⡀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⣸⠀⠙⢿⡟⠛⠀⠋⣤⠞⢷⡄⠀⠀⠀⠀⢠⡿⣧⡀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⣿⠀⠀⢹⡇⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⣿⠃⠘⣧⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⢰⣿⣶⣶⠿⣷⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⣿⠀⠀⢠⡿⠁⠀⠀⠀⠙⣷⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣀⠀⠉⠁⠀⠈⠻⣶⡶⠟⠃⠀⠀⠺⠇⠀⠀⠀⠀⢀⢀⣀⣤⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣀⣿⠀⣴⣄⣤⠀⠀⣀⠀⣀⡀⣀⠀⢀⣀⣴⡖⢀⡀⢿⠉⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⡿⠻⣏⢸⡇⢸⠙⡆⢸⡉⣿⠉⠀⢿⣤⣾⠟⣭⠀⢸⣇⢸⡀⠘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⢸⡇⠀⣿⠘⠃⢸⠀⡀⢸⡇⣿⠀⠀⣼⠟⢻⡆⣿⠀⢸⣿⣄⣇⠀⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣼⠀⠀⣿⠀⠀⢸⠀⣦⠘⠀⣿⡄⠀⣿⠀⢸⡇⣿⠇⠀⠛⣿⣿⠀⢹⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡟⠀⠀⣿⠀⠀⣽⠀⣿⡇⢀⡏⠀⠀⣿⠀⢸⡇⣿⠀⠀⠁⢹⣿⠀⣿⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠾⠇⠀⢰⡟⢀⣀⡙⢠⣿⠀⢸⠃⠀⠀⣿⡀⢸⡇⣿⠀⠀⣹⠀⣿⠀⠺⠇⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠙⠃⠘⠿⠃⠘⠟⠀⠻⣦⣤⠀⣿⠇⠸⠇⠻⠷⠃⠈⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣿⣧⠀⠀⣸⡏⠀⣠⡾⠿⣶⡄⠰⠶⢶⣶⡾⠿⠟⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡟⢻⡀⠀⣿⠀⣼⡏⠀⠀⠸⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠹⣧⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⣿⠇⢸⡇⢀⡿⢰⣿⠀⠀⠀⠀⣿⠄⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⠁⠀⣻⣆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣸⡏⠀⠀⢡⣾⠃⢸⣏⠀⠀⠀⢠⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⠿⠛⠛⠛⢿⡄⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠰⡿⠁⠀⠀⢸⣿⠀⠸⣿⡀⠀⢀⣼⠇⠀⠀⠀⣿⡇⠀⠀⠀⣰⡿⠀⠀⠀⠀⠘⢷⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠿⣶⠟⠃⠀⠀⠀⠐⠿⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⢀⣀⣤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⣷⡀⢀⣶⡄⠀⣠⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣴⡄⠲⠿⢿⡏⠉⠀⠸⣧⣀⣴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣷⠈⢷⠟⢩⡇⢰⡏⠹⡇⠀⢿⣆⠀⣼⡇⠘⡟⠛⠛⠀⢹⣧⠀⠀⢹⡇⠀⠀⢸⡇⠀⠀⠀⣿⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣾⠃⠀⠀⠀⢘⡇⠸⠀⠀⡇⢠⡄⢡⣾⠃⣠⠐⡇⠀⠀⠀⢸⡇⢻⣄⢸⡇⠀⠀⢸⡇⠀⠀⠀⢹⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠰⠿⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⢼⡇⠀⠁⠀⣏⠸⡷⠤⠀⠀⢸⡇⠀⠹⣿⡇⠀⠀⢸⡇⠀⠀⠀⠸⣿⣷⠏⠁
⠀⠀⠀⠀⠀⠀⠀⣸⠏⡀⢀⠀⠀⠀⠀⠀⢾⡏⢶⣤⣼⢻⡾⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢸⡇⠀⠀⢹⡇⠀⠀⠘⠃⢀⢀⣤⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⣿⣉⣳⣾⠄⢾⠀⢠⠀⠀⢀⡈⠉⠁⠸⠇⠀⠀⠀⠐⠟⠰⣷⣶⣶⠆⠸⠇⠀⠀⡀⢠⡤⡦⠸⠇⢀⠀⣿⠀⢻⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⠑⢾⠄⣿⡄⠸⠀⠀⣸⠇⢰⠋⠠⡀⠠⠀⠀⢠⠀⣆⠀⣶⠆⢀⡶⠀⢸⡿⠇⢸⡇⠇⠀⠀⢿⠀⡟⢣⡌⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⣁⠆⠀⠀⣿⠄⠘⠀⢰⡿⠀⠀⠀⢸⠀⢻⡄⢹⡄⠀⠙⢤⡄⠀⠀⣿⡿⠀⠀⠀⠻⠆⠁⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠉⠠⡀⠄⠀⠀⢤⡀⢸⠀⢰⡄⠀⠘⡄⢸⡇⢸⡁⢀⡄⢸⣧⠀⢠⣿⠀⠘⠁⠀⠀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠦⠀⠁⠈⠃⠘⠓⠀⠈⠃⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⠶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⣤⣤⣤⣤⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡏⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣿⠘⠿⡿⣴⣷⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣀⣀⣀⡀⠀⠙⠒⠒⠁⠀⠀⠀⠀⠀⠀⠰⡎⡇⣠⣴⣧
⠀⠀⠀⠛⠀⠀⠀⢻⡏⠀⢻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⠿⣿⢀⡀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢉⣄⣿⣿⣿⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠺⢱⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⣿⣿⣿⢿⣿⡇
⠀⠀⠀⣧⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣧⣭⣭⣽⠚⠛⠛⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣶⠿⠿⠾⣿⣿⡇
⠀⠀⠀⡷⡆⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡉⠻⢿⠇⠀⠀⠀⠀⠀⣿⡏⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣴⡻⠿⠃⠀⣽⣿⡿⣿⣿⣿⠀⠀⢀⡀⠐⢿⡇⣄⣠⠴⠂⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣿⣿⣿⣿⠛⢳⢯⠙⣿⣿⣿⠿⡿⠂⠀⠀⠈⢉⡼⠛⠛⢻⣤⣤⣼⢿⣶⣿⡇⠛⢁⣠⡤⠀⠀⢰⡞⠉⣧⣤⠄
⠀⠀⠀⣿⣿⡿⠿⠇⢸⢸⣶⣿⣿⣿⠓⠦⢤⣀⡀⢠⠞⠀⠀⠀⢸⣿⣿⣿⠈⣿⣿⣿⣿⣿⣯⡇⠀⠀⠈⢡⣄⣿⣿⣤⣄
⠀⠀⠀⠉⠉⠀⠀⠀⢸⡞⢿⣿⠿⠟⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀⣼⣧⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⢸⢸⡇⠀⢹⡇⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣆⣿⣿⣿⠀⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠈⠁⠀⠐⣂⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠉⠛⠲⠄⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⡤⠤⠐⠀⠕⣹⣿⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣿⢿⡄⠀⣀⣀⠀⠙⠓⠔⠘⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣿⣿⣾⣦⠘⢿⢿⣷⡶⢦⣄⡀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢰⡏⣷⠷⣗⣺⣿⣿⣿⣦⠀⠀⠈⠙⢶⣬⣙⠷⣦⣤⣀⣀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⣷⠻⣄⠀⢿⣿⣿⣿⣷⠾⠗⠈⠙⠲⠿⣮⣕⠦⡉⠓⢾⣉⠛⡳⠶⢷⣄⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣶⣟⠁⠢⣥⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡓⠀⡨⠙⠻⣥⠄⠚⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣏⡍⠁⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢙⠫⣄⣠⣀⢌⡳⣴⠒⠛⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢸⣫⣿⣇⡈⠉⢲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢫⣳⣌⣶⠒⠲⠵⡀
⠀⠀⠀⠀⠀⠀⠈⢸⣴⣿⣟⠉⠛⢶⠎⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠰⣞⣧⠤⠀⠑
⠀⠀⠀⠀⠀⠀⣴⠿⢏⡉⠈⠉⠛⣾⡁⠰⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣆⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢾⡀⠀⠈⠑⡐⢲⣧⣷⠄⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⢸⡏⠉⠳⣷
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠢⣼⣿⣬⡉⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢼⠛⠂⠀⠸
⠀⠀⠀⠀⠀⠀⠈⠒⠤⣴⣇⠀⣩⣿⣾⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠻⠉⠡⡀⠠⠄⢠⣤⣤⣠⣬⡏⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢿⠟⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⢃⠀⠀⣴⠶⣤⡿⣶⠃⠈⢠⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠈⠉⠉⠉⢳⡌⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠈⠁⢨⡛⠀⣻⣴⣿⣿⠟⠢⠀⠀⣶⡂⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢶⠖⠂⠐⠆⠚⢻⡆⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠄⢠⣟⡿⢞⠃⠘⣿⢯⡀⠀⢁⣰⡏⠐⠁⠠⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⢀⢀⡀⠐⠛⢦⡀⠀⠈⠃⠀⠀⠀⢀⡀⢀⣀⡀⠤⢾⣿⣾⢟⣩⢾⠁⠀⠣⢀⠌⠁⠐⠞⠂⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠁⠀⠀⢀⡠⠛⠲⣄⡂⠀⠀⠄⣀⣀⠠⠴⠐⢋⣉⣭⠷⣏⠀⠀⠃⠀⣰⠁⢀⡠⢊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖⠊⠁⠀⣀⠄⠋⠉⠛⡷⠶⠶⠷⠒⠛⠛⠉⠉⡆⠀⠩⠀⣀⡴⠾⠏⠉⠉⠀⠘⠂⢢⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢯⣁⠀⢀⡠⠜⠀⠀⠀⠀⠀⠠⠀⠀⢐⠁⠤⠛⠉⠉⠉⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠂⠀⠐⠃⠀⠈

⣟⢿⣍⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⡶⠿⢟⣿
⡛⠉⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋⣉⣉⣀⢠⣴⣶⣶⣭⣓⢤⡀⠀⠀⠶⣶⣿⣿⣿⣿⣿⣿⡿⣻⣿
⠀⣀⣀⣘⣧⠀⠀⠀⠀⣀⣴⠾⠛⢉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣻⣿⢟⣿⠞⠸⣶⣿⣿⣭⣿⣿⣿⠿⠿⠾⠋⣸
⣾⣿⣿⡿⣿⠀⣀⣴⠞⠋⠀⣠⣴⣿⣿⠟⠉⠀⠀⠈⠛⠛⠛⠋⠉⠋⠁⠀⠀⢻⣟⠚⣿⣿⣿⣿⣷⡆⣀⣀⣻
⠉⠙⠛⠁⢻⣾⡿⠁⣀⣴⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⡏⠈⠿⣻⢿⣿⣿⣟
⡄⢀⠸⢻⣿⣿⣥⣾⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠴⠶⠟⠋⠁⠈⠙⠻⣿⣷⣾⣟⣧⣿⡿⠋⣻
⣙⢿⠜⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣉⣀⣠⣾⣷⣀⣤⣾⣿⣿⣿⣿⠙⠿⣿⣿⣇⣾⣿
⣿⣽⣤⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠻⠿
⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣻⢲⡖⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠄⢠⣀⡄⢀⣽⣿⣽⣾⣿⣿⣿⣿⣿⣿
⠀⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠃⠀⠃⠈⡁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠐⣶⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⢰⣶⣶⠀⠀⣰⣷⡀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠤⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣷⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⢿⣿⣿⣿⠿⠛⠋⠉⣹⡿⠅⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⠿⠿⠟⠛⠙⠋⠉⠉⠉⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⠀⠘⠀⠀⠄⠀⠐⠀⠀⠀⠀⢻⡀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣿⣿⣄⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠉⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣠⠄⠀⠀⠀⠀⠀⢨⣿⡿⠁⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⠟⠃⠀⠀⠀⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⣠⡶
⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣾⣿⣿⣿⡷⠀⠀⠀⣠⣾⣿⡟⠀⠀⣠⣾⣿⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣦⣄⡀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⢃⣠⣴⣿⣿⣿⣿⠁⣠⣾⣿⣿⣿⣿
⠁⠀⠙⢧⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢻⡿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⠏⠀⢻⠛
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠁⠀⠀⡹⢿⡏⣙⣿⣿⣿⠏⢁⡠⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⡟⠀⠄⠈⠛⣶⣏⣻⣿⣿⣿⣿⣻⡿⠇⣠⡜⣼⣿⣿⣿⣿⣿⡿⠛⠉⠀⣠⣾
⠀⠀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⡀⣸⣠⣿⣬⣿⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣴⡿⠟
⠀⠀⠀⡀⠀⠀⢫⡍⠙⠛⠿⣿⣿⣿⣿⣿⣷⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⢀⣴⡿⠋
⠀⠀⢠⣿⣇⠀⠀⠻⢦⣄⣀⣈⡙⢿⣿⣿⣛⠻⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡫⣾⣶⣴⣿⣿⠣⣄⣤⠈
⠀⠀⢸⡇⣿⠀⠀⠀⠀⠈⠉⠉⠉⢉⣿⡿⠛⣷⣤⣬⡿⢋⡀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⢈⣭⣿⣿⡿⠟⠋
⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠀⢀⡼⣿⡿⢡⢌⡏⣸⣿⠿⠟⠋⠉⠛⠻⢿⣿⠟⠁⠠⢰⡾⠟⠁⠀⠀⠀⠠⠂
⠀⠀⢸⠃⡷⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠿⠷⡃⠞⠁⠋⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠓⠋⠀⠀⠀⠰⣶⡂
⢠⡄⠸⠀⠀⠀⢻⡇⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠈⠀⠀⠀⢠⣿⣿⣾⣿
⡀⠿⠀⠀⠀⠀⠌⠀⠀⠰⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣴⣶⣦⣿⣿⣿⣼⡀⠀⠀⢰⣿⣿⣿⣿⣿
⠙⠛⠀⠛⠛⠋⠀⠀⠈⣼⠋⢧⣤⣤⣠⡀⢤⡀⢀⡄⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠤⠠⠀⠀⠀⢠⣿⣦⣀⣉⡘⣟⠿⣾⠿⠿⠶⠶⠿⠟⣿⡿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣠⣴⠦⠋⣠⣿⣿⣷⣍⣿⣷⣄⣠⣄⢒⡈⢁⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⣹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⠇⢰⣾⣿⣿⣿⣿⡟⠡⠙⠻⢿⣿⣯⠀⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠁⠀⢀⣾⣿⣿⣿⣿⣿⡛⠻⣿⣿
⠀⠀⣦⣴⡟⢉⠀⢰⣾⡿⠉⢠⡿⠋⠀⠀⡀⣠⢼⣿⡿⢀⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣇⠀⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠉⠉⠛⣷⡄⠀⠀⠀⣀⣀⣀⠀⠀⣰⡿⠋⠉⠉⠉⢻⣧⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⠀⠀⣽⡇⠀⣠⡾⠟⠛⠛⣷⡠⣿⠁⠀⠀⡀⠀⠀⣿⠀⢀⣴⡿⠛⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠶⠟⠀⢠⣿⠁⣼⡟⠀⠀⠀⠀⣿⡇⠙⠷⠶⠾⠃⠀⣼⡟⣠⡿⠋⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⢠⣿⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢀⣼⡟⣰⡿⠁⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⢡⣴⠾⣿⡀⠀⠀⢀⣾⠏⠀⠀⠀⢀⣴⡿⠋⢠⣿⠇⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠁⠀⣿⡇⠀⠘⣷⡀⣠⣾⠏⠀⣀⣤⣾⠟⠋⠀⠀⣿⡇⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣷⣦⣀⡀⠀⠘⢿⣶⣤⣶⣿⣟⣥⣶⣿⣿⣿⣷⣄⣀⠀⢰⣿⣷⣦⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠉⠀⠀⠉⢀⣰⡆⠀⠀⠀⣠⢀⣿⣿⣄⡀⠀⠀⠀⠀⢀⣴⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡈⣉⡁⢀⡀⡀⢀⢀⣈⣉⣉⣁⡀⢀⠀⢀⠈⢉⣈⣉⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⣴⡟⠰⢻⡇⠀⠓⠀⡠⠺⣿⣿⣯⢿⡇⢰⡾⡇⠐⢋⣷⡽⣻⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠊⠁⢀⡴⣿⡇⠀⠀⠀⢀⣤⣿⣿⡖⡀⡇⢸⣿⣷⣐⣿⣿⡷⢩⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣟⢳⣄⣇⠈⠀⣹⡇⠀⠀⡔⠁⣹⣿⣿⣍⣥⡇⢸⡿⣿⣿⣿⣿⣾⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣿⣹⣿⣷⣠⣾⣿⡇⠀⣀⣀⣼⣿⣿⣯⣿⣿⡇⢸⣿⣾⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⠤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣭⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣿⣶⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣦⣴⡆⠀⡀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡦⣿⣿⣿⣿⣦⣖⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣰⣿⠀⠀⣨⠔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣴⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤⡧⣶⠛⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⣿⠀⢀⣴⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣷⠠⠗⠀⠀⠀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡏⡄⣋⡀⠰⠊⠁⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠜⠋⠉⢹⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠘⠛⠛⠛⣉⠉⠀⠀⠀⠀⢀⣀⣤⣴⣾⣾⡏⣄⠀⠀⠀⢰⡄⣠⠐⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣄⠀⠀⠀⠀⢀⡀⠀⣂⣀⣠⣿⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⠟⢠⣿⡈⠒⠂⣾⡇⠀⡼⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⠀⢀⢤⣠⣷⠀⠀⠁⠀⣦⡀⣠⠤⠰⢦⣈⡙⣿⣿⣿⡿⣟⠿⡿⠿⣿⡟⠛⠀⠀⢼⣿⡆⠁⠋⣿⣷⠠⣤⠇⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⣧⣿⠶⠃⠀⣩⠗⠮⣂⠄⠘⡥⠈⠁⡄⢰⠈⣉⠭⠛⠩⠑⢊⠀⠊⢠⠈⠂⠒⠥⣢⡠⣀⠀⡌⠛⠜⠛⠿⡂⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠧⠊⠁⠠⠀⡠⢞⡄⢠⠀⠀⠁⡀⠀⠉⠀⠟⠃⠋⠀⠀⣀⡠⠂⠨⠂⠀⣾⡆⠐⢄⡀⠈⠉⠈⢸⠷⣦⠀⢂⠂⢘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠤⢊⠁⠀⢀⠰⠀⡀⢸⡀⣆⠀⠙⠒⠁⠀⠀⠀⢠⠀⠂⠀⠀⠄⠀⣿⢠⣿⣇⠀⠂⠀⡀⠀⣁⠤⠊⠀⣦⢸⠀⢸
⠀⠀⢰⠐⡀⠄⠀⠀⡀⠀⠀⠀⠀⠁⠈⣁⣀⣀⡛⠗⠉⠀⣠⡠⠄⠀⡀⠀⣾⡆⠤⣌⣀⣀⠼⠿⠾⠟⠛⠋⠉⡉⠔⠈⠀⠀⠐⠀⡏⢸⠀⠘
⠀⠀⠸⠀⠤⡌⠒⠠⠄⠀⠉⠑⠂⠀⠿⢿⣿⡭⠿⠇⠀⠀⠂⠀⠀⠈⠝⠐⠿⠧⡈⢁⡠⠂⠅⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⠈⠤⢆⡀⠰⢠⣦⣤⣄⡀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠛⠿⠃⠒⠀⠀⣤⣀⢀⠀⠀⠁⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠭⢃⣆⠶⠀⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⢀⠔⠀⠀⡤⠊⠁⠀⠠⠾⣖⣋⠉⠀⠀⠠⠂⠄⠸⡄
⠀⠀⠀⠀⠀⢄⡀⢠⣄⠲⢶⣶⡀⠁⠰⣶⠁⠔⢖⡋⠀⢶⣀⡀⠒⠒⠂⠒⠻⠫⣙⡃⠋⠠⡀⠒⠲⣄⠰⠒⠒⠒⠒⢲⡆⡀⡒⡆
⠀⠀⠀⠀⠀⠀⠑⠤⠀⠢⢀⣙⢣⠀⠀⠉⢧⠀⠈⠱⡄⠀⠀⠈⠁⠀⠀⠀⠀⠁⠏⡆⠀⠀⠀⠀⠉⠀⠉⠁⢠⠀⠀⠀⠐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⣀⡀⠈⠈⠂⠀⠀⠀⠈⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⢀⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠀⢀⠱⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠀⠑⠄⠘⠡⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡶⠀⡂⠀⠙⠒⣠⡀⠈⠢⢈⡆⠀⠀⢀⣀⠀⠀⠀⠄⠀⢀⣀⡀⡀⠀⢠⡀⠀⠀⠀⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠙⠒⠒⠒⠒⠚⠛⠚⠛⠛⠀⠀⣿⣿⣧⠀⠀⠐⠚⠃⠤⠉⠛⣠⠝⡳⠆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⡀⠀⠠⠁⠀⠀⠀⠐⠊⠀⠔⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠐⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⠀⢀⢈⠀⣠⣾⡇⠀⠀⢀⡀⠘⠠⠖⠁⠤⠒⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣴⣶⠆
⠀⠀⠀⠀⠐⣆⠀⠀⠀⢹⡟⠀⠐⡆⠈⠙⠁⠀⣠⢈⣇⣉⡉⠁⠁⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⣘⠠⠴⠾⠛⠁
⠀⠀⠀⠀⠀⠻⣷⣮⣤⡞⠉⣶⠀⠀⠀⠰⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⣦⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠀⢩⠒⢲⠈⠁⠀⠘⢀⠋⠀⠐⠢⠄⠀⠀⣀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠂⠈⠓⠋⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠓⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠉⠀⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠑⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣂⣀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠠⣤⡀⡀⠀⠈⡆⠀⠀⠀⠀⠠⠂⠀⢀⢰⢃⣊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢴⠠⣤⣤⣠⠀⢰⣿⠃⠀⠀⠀⠀⠀⠘⣳⣷⣦⡀⠘⠀⠀⠀⠀⠈⠀⠰⠈⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠐⣻⡿⠷⣇⠀⠀⠀⠀⠄⠀⠀⠰⠃⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⣿⠀⠁⠀⠐⣴⣿⣦⠄⠁
⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⢀⣠⣴⠶⠶⠒⠂⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠁⠀⠀⠀⠀⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠐⣴⣦⠀⠀⠰⠂⣀
⠀⠀⠀⠀⠈⠈⠂⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠼⠖⠀⠈⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠙⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⡇⠈⠉⠀⢠⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⣄⠀⣿⣿⣏⡄⠀⠀⣾⣿⣿⣿⣿⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢀⢻⣦⣿⡏⣽⡇⣴⡷⣽⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣌⠿⣷⣴⣝⠿⡟⣭⣿⡟⢡⣷⣮⠻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢩⢔⣿⡾⠿⣷⣿⣿⣿⣿⣇⢹⣷⣿⢦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣶⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡉⢙⣷⣿⡿⠿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣜⣀⣀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣟⣴⣾⣾⣿⣻⣿⣿⣿⣿⠏⠉⠉⢻⡀⠀⠀⠁⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⡿⣿⢿⣟⣿⣏⣾⣿⣿⣿⠀⠀⠀⢸⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⣄⠉⠙⢿⣿⣿⣿⡿⠋⠙⣠⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⠟⠛⣶⣧⣹⣿⣿⡤⣷⠞⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⣿⠏⠉⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣶⣦⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⠀⠹⢿⣀
⠀⠀⣴⣯⡁⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⣶⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷
⠀⠀⣿⣿⣿⡶⠤⠼⣿⢿⠇⠀⠀⢠⣤⣤⡀⠀⣿⣿⣏⢯⣷⡦⠀⠀⠀⠈⠉⣹⣿⠋⠀⠀⠀⠀⢰⡤
⠀⠀⣹⡟⣷⡀⠀⣴⣾⣶⡆⠀⠀⢸⣻⣻⡇⠀⢸⡿⣿⠛⠻⢇⠀⠀⠀⠀⢠⡿⢻⣦⣀⠀⢀⣼⣿⣷⣆
⠀⣴⠟⠁⣿⠃⣴⣿⣿⣿⣿⡄⠀⢸⣷⣾⡇⢴⠟⠁⣿⠇⣠⣾⣄⡀⠀⠀⢼⡇⠀⠈⠉⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠚⠛⠋
⠀⡀⢀⣤⣴⣶⣦⣀⣀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡾⢥⣤⣤⣤⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡄⠀⠀⠈⢿⣿⣿⣿⠿⠟⠻⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿
⠀⠉⠉⠻⠛⠻⣿⣿⣿⣿⣿⣿⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣷⡄⠀⣾⣿⡿⣱⡾⢷⣎⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⡟⠛⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠀⠘⠷⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣀⣸⣆⣠⣾⣧⣀⣀⡀
⠀⠀⠀⡇⣿⠀⢀⠀⡀⢸⣿⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⣀⣷⣿⣶⣿⣾⣷⣿⣿⣿⣿⣼⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡿⠗
⠀⠀⣿⡿⢿⡿⢿⠿⣿⠿⡿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⡦
⠀⠀⣿⡗⢺⡗⢿⠒⣿⠚⡟⢺⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣤⣄⣀⣀⣀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⣶⣾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⣟⠿⠿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠿⢋⣫⣬⢙⣿⡗⠀⣀⣤⡾⣿⣿⣿⡿⠟⠛⠻⡿⠿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠺⡿⠋⠉⢻⣿⣷⣾⣿⣿⣿⡿⡿⣿⠤⠼⠓⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠛⠻⣿⠾⣧⡤⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠓⢢⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣅⡻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠷⣦⣶⣶⡄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⢹⡿⣫⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⣼⠏⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⠃⠀⢀⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡆⠀⣴⣿⣿⣽⡟⠁⠀⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡈⣿⣿⡿⣿⠁⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠇⠘⠀⠀⠀⠀⠀⠸⠿⠈⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⢀⣀⣴⣞⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡷⠉⠛⣿⣿⠋⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⢽⡿⠀⠀⠀⠈⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠻⠻⠿⢩⣵⠿⡏⡟⠀⡇⢹⡏⢛⣦⠉⠙⠌⠛⠿⠟⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⠿⡻⠁⠐⠀⣰⣿⠹⣴⣗⡇⢼⣇⡟⢀⡿⠛⣿⡟⠠⠀⠀⠀⠔⠈⠟⡿⣿⣯⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣋⣿⣿⡟⠋⠁⠀⠀⠀⠀⣟⣿⣼⣿⣹⣧⣟⡼⢁⡿⢃⣴⡿⡿⣄⠀⠀⠄⠀⠀⠀⠀⠉⢛⣿⣿⣯⡛⢛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣼⢯⠉⠀⠀⠀⢀⠀⠂⠁⠀⣿⣿⣿⣧⣿⣟⡞⣡⠟⣠⡾⠫⣺⣿⡏⠀⠀⠀⠀⠙⠦⢀⠀⠀⠀⠈⣹⢿⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠙⠈⢈⠿⠋⠉⠀⠀⠈⢦⡀⠀⠀⢠⣿⣿⣿⣿⣿⣫⡾⣣⡼⣣⣾⣿⡿⢋⣷⣤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠻⢏⠀⠐⢙⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠛⠂⣠⠴⡄⠠⠤⠤⠤⢤⣤⣤⣌⣙⣿⣿⣿⣝⢷⣾⣿⣿⣿⣯⣼⣛⣯⣾⣿⠿⢿⡇⠀⠄⡀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠨⠷⢠⡀⠙⠿⣿⣿⣿⣿
⣿⣿⣿⡧⡁⣠⣾⡉⠀⠀⠀⠠⠀⠀⠀⠀⠈⢹⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠚⢺⣿⠷⢨⣿⣿⣷⣦⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣹⣦⠀⣡⣿⣿⣿
⣿⣿⠿⠋⠡⠝⠉⠉⠀⠠⠃⠀⠀⠀⠀⠀⠀⠈⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣼⠁⢀⣸⡏⠸⣿⣿⣿⣦⣀⠀⠀⠀⠀⠣⠀⠈⠈⠋⢥⡉⠹⣿⣿
⠿⢿⣤⣾⣅⡀⣀⠔⠃⠀⠰⣦⣠⣄⠠⢮⡦⠐⣿⣿⣿⣿⣿⣿⣿⡿⢿⢟⣿⣿⣿⡟⣻⠏⠁⠀⢀⣿⡇⢰⣿⣧⣋⡻⣿⣷⣄⠀⠀⠀⠈⠓⣄⠀⣀⣿⣤⣾⠿
⢀⠊⠉⠙⠓⠻⠟⠅⠀⠀⠀⠸⢿⡿⠀⠈⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⠀⢀⣰⣾⣿⠁⣼⣿⠟⣿⣯⣦⣬⣿⣷⣤⡀⠀⠀⠙⠽⠒⠛⠉⠉⢀
⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣼⡆⠠⠀⣀⣤⣶⣿⡆⢿⣿⣿⣿⣿⣾⣿⣿⣷⢿⣿⣾⠏⠀⢸⣿⣿⢟⣾⣿⣿⣿⡿⠋⠉⠉⠛⠻⣿⣻⣶⣄⠀⠀⠐
⠁⢒⠠⠄⣀⠀⠀⠀⣠⣶⠟⣁⣤⣶⣿⣿⢿⢿⣿⣿⠈⠁⠀⠀⢉⣿⣿⣿⣿⣯⣿⣿⡀⡀⢸⣿⠋⢈⣿⣿⡏⠉⠀⢀⠀⣤⣴⣄⣿⣿⠿⡻⠷⡄⠀⠀⠤⡐⠊
⠀⠀⠀⠀⠀⣀⣴⣿⣽⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⠀⠀⠀⠸⣿⣿⣿⣿⣿⣾⣿⣷⣷⣟⣋⣴⣿⣿⣿⣷⣶⣧⢯⣝⣿⣿⣯⣿⣿⣼⡶⠂⢿⡆
⠀⠀⢀⣤⣾⠿⠟⠛⠛⠋⠉⠉⠉⢩⣿⠛⠛⠻⠿⣿⣿⠀⠀⠀⠀⢹⡿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣣⣯⣿⣿⣿⣿⠿⢁⣶⣶⠿⣿⠀⠀⢀
⠀⢸⠙⠥⠧⠤⠤⠠⢄⠂⠀⠀⠀⢸⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⣁⣴⣾
⠀⠘⣇⢠⡶⡶⠶⣞⣛⣲⣒⣂⢀⠔⠛⢿⣿⡽⣷⡄⠀⠀⠀⠀⠀⠘⣯⣿⣿⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⣿⠋⣻⣿⣿⣿⡧⡀
⠀⠀⣿⣷⣶⣿⠾⠟⠿⣿⣿⣭⣤⣤⣿⠃⠀⠀⢸⣿⡄⠀⠀⢀⣤⣾⣿⣿⣿⡁⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣴⣾⣷⣿⣿⣿⣿⣿⣾⣼⣦⣤
⠀⣠⡻⣯⣿⣧⣀⣀⣀⣾⣟⣉⣬⣿⣿⠀⠀⠀⠈⢿⢿⡄⠀⢿⣻⣾⣿⣿⣿⢃⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣤⣶⣷⣿⣿⣿⣿⣿⡿⠿⣿⣿⠿⠿⠛
⠀⠙⢿⣗⣚⣒⣒⣚⣛⣓⣒⣒⣒⣚⣶⣦⠀⠀⠀⠈⠃⠈⠀⠀⠈⠉⠉⣱⣷⣯⣯⣽⣿⣿⡇⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠻⠾⠿⠿⠿⠁⠀⠀⠀⠀⠀⠉⠛⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⠀⠀⠀⠀⠀⣤⣤⣀⣀⠀⠀⠀⠀⣶⣶⢦⣤⣤⣄⡀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡆⠀⠀⠀⢀⣀⠀⠉⠻⣿⡄⠀⢰⣿⡏⠀⠙⣿⣿⣿⣦⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡿⠂⢛⣿⣮⡛⢿⣆⠀⢸⣿⠀⣾⠟⠀⢀⡀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣛⣴⣿⣿⣿⣿⡁⠀⠉⠳⡸⣿⢸⠏⣴⣿⣿⢿⣆⢿⣿⣿⢏⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣿⣿⣿⣿⡿⠁⠰⣶⣦⣜⣿⡼⢀⣿⠋⠉⣸⠟⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣹⡿⠟⢋⣀⣀⣀⣿⣿⣿⣿⡇⣼⣿⣧⣸⡏⠀⣼⠟⠛⠛⠛⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠘⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢀⡴⠋⠛⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣠⣤⣴⠻⣿⣿⣿⣿⢿⣿⣛⣿⣿⣿⣦⣄⠀⣸⡧⢀⠐⣿⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠞⣿⣿⣦⣀⣴⣄⠀⣸⣿⠟⣿⣿⢿⣿⣿⣿⡇⠰⢿⣿⣿⠙⣦⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⣿⣿⣿⣿⣿⣷⡜⣿⠀⢻⣿⣿⣿⡟⠃⢉⠶⢦⡉⢀⣸⣿⣟⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣻⣷⣼⣿⣿⣿⣿⡏⢿⡏⠀⠀⢠⣤⣤⣄⡀⠀⠀⡈⣙⠹⠛⠛⠁⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⡇⠘⢷⣞⠛⢿⡝⢿⣿⣿⣦⣼⣿⣥⡘⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠋⠉⢻⣾⣿⣿⢃⣴⣿⣿⣤⣬⢧⣤⣽⣿⣿⣿⡿⣿⣿⣦⠙⠇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡈⣿⣿⣿⣿⡻⣧⡠⣥⡼⠈⠻⣿⣿⣿⠿⣿⣻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣻⣾⡿⣿⣿⣿⣿⣿⣧⠀⠀⠡⣤⣤⣤⣼⣿⣿⣿⣎⣿⣿⡿⣿⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣷⣾⣟⠿⣿⣿⢿⣿⣿⡆⠀⠀⢬⣝⡛⠿⢿⣿⣿⣿⣿⡟⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⡜⣿⠄⣿⡿⠁⢀⣴⡟⠹⣿⣦⣄⠀⠉⢻⡿⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⠛⢿⣿⣿⡟⠀⠛⣵⣾⣿⣿⠃⠀⠘⢿⣿⣇⠀⠀⣇⢸⣿⣿⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠄⠀⠈⠻⠏⠼⠿⠿⠿⠟⠃⠀⠀⠀⠀⠻⠿⠀⢸⠧⡾⢿⠿⠇⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡴⢶⡽⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠄⢀⡀⠀⠀⠀⢀⣠⣠⣀⡀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⢇⡄
⠀⠀⠀⠀⠨⠵⢞⣳⡟⠻⠂⠀⠀⠀⣠⣀⣤⡀⠀⠀⠀⠈⢢⣄⣠⡄⠃⠀⠀⠀⣿⣿⡟⠁⠀⣀⣾⣿⢋⠈⠀⠀⠀⠀⠀⠀⠐⣯⠁⡄
⠀⠀⠀⠀⠀⠀⠁⢻⠀⠀⠀⠀⠀⠰⣿⠿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡇⠠⠔
⠀⠀⠀⠀⠀⠀⠀⡎⠰⠃⠀⠀⠀⠈⢾⣿⣽⡞⡗⠀⠀⠀⠀⢲⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⢀⠀⠀⠀⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠉⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⣤⠀⢘⣦⠾⠁⠀⠠⠐⣦⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⢠⣤⢞⠀⠀⢀⠈⡄⢠⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠔⠋⠀⠊⠀⠀⠀⠀⠀⠀⠅⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡟⠋⠉⠁⠀⠀⠈⠀⢑⡺⡍⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡇⠀⠀⠀⠀⠀⠈⠰⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢀⡄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣨⠁⣉⠀⠀⠀⣶⡶⠀⠀⠀⡇⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⡀⢉⠀⠀⠀⠀⠆
⠀⠄⠀⠀⠀⠀⣀⠀⡦⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡃⡿⣃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⠀⠀⠀⠾⡷⠄
⠀⠂⠀⠀⠀⠀⠈⢰⡖⠚⠀⠀⠸⠿⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣾⠆⠄
⠀⠂⠀⠀⠀⠀⠀⠘⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉
⠀⡂
⠀⠁
⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣾⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣻⣿⣿⣻⡿⣿⣿⣿⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣛⣿⣿⣿⣻⣿⣿⣿⠀⠀⢬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⠟⣿⠋⢀⠹⢧⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣽⡟⠁⣠⢖⣾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡄⠀⠀⣿⠸⢊⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⠉⢸⡇⣿⣵⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢽⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣸⠇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⢠⡀⠀⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡯⣿⣷⠀⠀⠙⢦⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⣶⣆⠀⠀⠀⠀⠀⣨⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢯⡿⠇⠀⢀⣤⣤⣤⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣷⢫⡟⢀⣴⣄⣠⡾⠟⢁⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣦⢐⣞⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣾⠟⠉⠛⠋⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢐⣗⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⢰⣿⡀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣷⣾⣿⣶⣶⣷⣷⣯⣿⣾⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⣶⡄⢸⣿⠁⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣀⣿⣇⣸⣿⣄⣸⣿⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣿⠉⠙⡟⠻⠟⡟⠟⠻⡟⠻⠟⠿⠻⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠃⠀⠰⠀⠘⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠠⡐⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠀⠀⠀⠀⡶⠀⠀⣠⠖⠒⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣀⠀⠠⠔⠂⠂⠄⢠⠆⠀⠀⠀⠀⠁⢣⠴⠋⠁⠀⠀⠀⠀⠀⠀⣠⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠲⢠⡄⠂⠀⠈⣀⡀⢀⡤⣄⠀⠁⠀⡀⢀⡀⡀⢀⡐⠀⠂⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⡇⠉⠲⠐⠀⠆⠈⠀⠠⠘⠒⠃⢄⣀⡀⠀⠈⠁⠚⣍⠙⠒⠊⠄⢈
⠀⠀⠀⠀⠀⠀⡰⠁⠉⠒⠀⠀⠀⠀⠀⠀⠈⠃⠀⠁⢸⡴⠷⠖⠀⠀⠊⠀⠀⠀⠍⠙⠲⢄⣐⠀⠰⠀⡆⠀⢄⠀⠈⠑⢆⠁
⠀⠀⠀⠀⠀⠀⠦⠀⠀⢀⠀⠀⠠⠀⢀⠀⠐⠀⠀⠀⠔⠃⠐⠀⠢⢄⠈⠀⠈⠀⠀⠀⠀⠀⠙⠀⠀⡜⠀⠀⠀⠀⠀⠀⠠⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠶⠀⠀⠀⠁⢀⡁⠔⠖⠃⠊⠀⠀⠡⠀⠀⢐⠤⣖⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢐⣀⠄⠀⠀⠀⢠⢿⡤⠓⠀⠀⠂⠁⠀⠀⠀⠀⠀⠁⠆⠃⠀⠀⠀⠀⠀⠀⡀⡄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠃⠀⠠⠄⠀⠘⠙⠀⠀⠀⡀⠤⠂⠀⠀⠢⠀⣨⠄⠈⠀⠀⠀⠀⠀⠀⡰⡠⢠⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠂⣤⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⢣
⠀⠀⠄⠀⠀⠀⡀⠚⡓⢂⠀⢶⠀⠀⢸⣆⡤⠖⠚⠉⠣⠄⠀⠐⡱
⠀⠀⠈⡄⠀⠂⠀⠀⠀⠈⠖⠈⡤⠀⡜⠉⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠘⢄⠀⠁⢀⠀⠈⠀⠔⠀⡘⠀⠀⢀⡠⠖⠋⠉⠐⠀⠂⠀⠀⠐⠳⣄
⠀⠀⠀⠀⠈⠁⢀⠀⠑⠀⢐⢀⠆⠀⢀⣴⣭⢀⡁⠉⠐⠄⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⢠⡊⠁⠀⠠⠀⠸⡌⠀⠇⢠⠋⠀⠀⠠⠄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢗⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⣆⢀⡇⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⠆⠀⠀⢀⣀⠀⠀⠀⡀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠈⠀⣈⢰⠦⢶⠋⣱⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣾⣀⠀⢀⠁⣾⢀⡴⢋⢀⠀⠠⠀⠐⣮⠛⢶⣦⣿⡉⢿⠁⢸⣇⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⢿⣿⠀⠀⠀⢰⣾⣱⣸⣿⣆⠀⠀⠘⢿⢦⣈⣿⣛⣿⢯⡵⠎⠉⣤⠥⡼⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣇⠀⢡⣌⣿⡻⣿⣿⠏⠀⠀⠀⢘⣦⣴⡎⠉⢳⠀⣖⡀⠈⠀⠸⠯⣬⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡟⠨⡭⠝⣻⣦⡀⠓⠌⠛⠾⠿⠿⠀⠀⣠⣾⢿⠷⢾⠲⣦⣤⠃⠋⠀⠀⠰⢇⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⢈⠀⠀⢀⡴⢋⣿⡟⠦⣄⣀⣀⣠⠤⠔⠚⠁⠀⠀⠀⠀⢱⢦⣿⡆⢳⡀⠗⣀⡪⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢴⣍⣠⡤⣂⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡽⠈⢠⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠉⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠄⣀⡈⠁⠐⢁⡐⠃⠀⠀⠀⣀⣀⣀⠀⣶⠀⠀⠻⠇⠀⠀⢘⣤⣬⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠏⠀⠀⠀⠀⢏⠙⠦⠴⠿⠉⠀⠀⠀⠀⠀⠀⢷⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠆⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⢁⡄⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠞⡀⢀⣀⣀⠀⣀⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⣠⣾⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡏⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠈⠀⠀⠀⢰⠋⢠⡎⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣦⣨⣄⠘⠁⣊⠳⣄⠀⠀⠀⠀⠀⠀⠹⣦⣤⣀⣀⣀⣀⣀⣤⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠩⠿⢳⣽⡷⠀⠁⠄⠀⠐⢶⣤⠀⠀⣀⠀⠘⠛⠋⠁⠈⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢻⣥⡀⣖⢘⣧⣉⠁⠘⢷⡄⠀⡀⢹⠀⠘⡿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⠿⠋⢹⠆⠊⢻⡶⠺⠫⣴⠀⣤⡤⠀⠀⠀⠀⠀⠉⠙⠛⠶⠤⢠⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠌⠀⠀⢄⠸⠋⠑⠆⡐⠂⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠈⣱⠂⡤⠠⢣⠀⠲⠀⠈⠘⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⡾⠁⠀⢰⠀⡿⡷⠦⠖⢶⠦⠀⠴⡀⠀⠉⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⣤⡆⣼⠆⠠⠆⠈⠉⠄⣀⣨⠟⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣈⡝⠀⣤⠄⠴⠛⢤⠻⣥⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣮⡍⣠⠶⠂⠲⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣯⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⢳⠗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣾⠛⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⢶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣰⣦⠠⠷⡎⠿⡂⠿⠷⢆⢠⡦⢄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢫⡟⢐⠀⠀⠀⠀⠀⠀⠀⠀⢰⡡⣿⣿⣿⢷⣇⠘⠂⠑⠀⠀⠈⠡⠐⠏⢳⠀⠀⠀⠀⠀⠀⠀⠊⠖⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠠⣠⣿⣿⣿⡇⡽⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⣿⣿⡟⠈⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣀⠀⠀⠀⠀⠀⠀⣀⡀⠸⣵
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣼⠃⠈⠀⠀⠀⠀⠀⠀⠀⣼⣿⠙⣿⣿⣿⣟⣦⡞⠗⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠄⠈⣽
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⠐⣿⣽⣿⣿⡻⣷⢊⠊⠇⠠⠀⠀⠀⠀⠀⠀⠀⡴⡇⠀⠀⠀⠀⠀⠀⠘⠂⠝
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣰⡿⣿⣿⣿⣟⡻⣦⠈⠠⠶⢀⡖⠂⠀⠈⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⢉⡛⣿⣿⡀⠑⠙⡿⣏⣿⣿⣍⢷⣀⣦⡀⣠⠀⡃⢀⣾⣯⠅
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣼⡇⠄⠀⠀⠀⠀⠀⠠⢠⣿⣶⣿⣿⣦⠀⠷⢮⣽⣿⣿⣿⣿⣷⣿⠃⣠⣷⣾⡿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣾⢿⣿⣷⣄⡘⠿⠿⠿⠻⠻⠿⠿⢿⠿⢾⡧⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢰⣟⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣻⣝⢿⣿⣿⣷⣷⣶⣶⣶⣶⣶⠶⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰
⠀⠀⠀⣻⣧⣿⣿⣿⣿⣧⣿⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣈⠻⠾⣳⣼⡟⠻⣿⡋⢻⠅⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠎⠇⠀⠐⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⣿
⠀⠀⠀⢿⣯⣵⣿⣿⣿⣿⣿⣖⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣺⡏
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢼⡏⠁
⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣝⠢⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣳⣄⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⠶⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣝⢠⣀⡀⣀⠀⠄⠀⢀⣀⠀⠀⠀⠀⢀⣀⣈⣴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣤⣉⣷⡶⠀⠈⣁⣀⣶⣶⣾⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠚⠻⠟⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣛⣓⣃⣐⣋⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠸⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠡⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⡁⠀⡀⠈⠢⠀⠀⡄⠀⠀⠀⢀⠎⢠
⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⡠⠖⠋⠁⠐⠀⠀⠀⠈⠑⠢⣀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⡆⢠⠃⠀⠀⠀⠘⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠎⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠔
⠀⠀⠤⠤⢄⣀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠠⠤⠶⠒⠊⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⢇⡀⣀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠠⠂⠐⠈⠀⠀⠀⠈⠂⢄
⠀⠀⠐⠚⠛⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠆⠀⠀⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠁⠀⠀⠀⠤⠀⠀⢈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⢄⣀⣀⣀⣀⢠⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠆⠠⠀⠤⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⡂⢀⠀⠀⠀⠀⠀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠂⠀⠒⠒⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠈⠀⠀⠀⢁⠐⠐⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠶⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⢀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠤⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡠⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡗⢄⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠐⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠦⣠⠞⠂⠁⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠁⠳⣄⠴⠁⠀⠀⠀⠀⠀⠀⠐⡈⠁⠃⠉⠀⠐⠂⠒⠂⠀⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠆⠄⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠤⣄⣀⠁⠀⠀⣇⢀⣠⠤⠈⠁
⠀⠀⠀⠀⠀⠀⢖⠀⠀⠀⡠⠀⠀⠀⠄⠀⠂⠀⠀⠐⢄⠀⠀⠀⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⡀⡄⠀⠀⠀⠀⠀⢠⢀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⡄⠀⠀⠔⠀⠀⢀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⡀⠀⠀⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣃⣿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⢸⡟⡼⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣽⢿⣟⠐⠂⢴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡎⣿⠁⠷⣟⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡶⠚⠙⠻⢇⣴⣛⣯⣭⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⣰⣿⡿⣿⠇⡄⠀⢡⡌⠉⠛⠻⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠄⢠⣿⣯⡗⠀⣹⣄⣓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢧⡟⠀⢀⠟⠋⣴⣶⡿⡷⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄⠀⠀⣶⣆⡾⠶⠷⠄⠛⠋⣐⣘⣉⣀⠈⠉⠹⣿⣿⣶⣯⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣹⣷⠆⣿⣟⡈⣚⠀⡈⠁⡀⠙⣿⣿⣭⣛⣷⣄⠀⠙⠻⠿⢿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢀⡀⣶⠿⠋⠤⢀⣹⡷⠋⠉⢀⣀⡘⣇⢯⣀⢀⣰⡅⠀⠈⠻⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣛⠛⢆⡀⠂⠀⣧⠀⢻⣿⣦⣿⣦⣟⣿⣯⠝⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠿⠿⠸⢎⣷⣴⢖⠏⠀⠀⠈⠉⠈⠻⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣭⠻⢏⡤⠶⣚⣿⣯⠿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⣻⣿⠿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⣀⣀⠀⠀⣶⣿⣦⣀⣀⣀⣀⣀⣀⣶⣿⣦⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢿⣿⣿⠆⠀⠻⠿⠛⠀⠀⠀⠀⠀⠀⠻⠿⠏⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡞⠁⠀⣀⣀⡀⠀⠉⠉⠙⠦⣄⣀⣀⣀⣀⣀⣀⣀⣴⣶⣦⠀⠀⣼⣿⣿⡆⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠐⣿⣿⣿⠖⠲⠦⣄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠻⠿⠟⠀⣠⠞⠛⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⢱⡆⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠞⠁⣰⣾⣷⡄⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⢀⣴⣶⡄⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⢀⣴⣶⣇⠀⠀⠀⠀⠘⢿⠟⠁⣴⣶⣶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠀⠘⢿⡿⠿⣄⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠘⠿⡿⠏⠀⠀⠀⠀⢠⠞⠀⠀⠻⠿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠓⠢⣀⣀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣼⣿⣷⡄⢀⣤⣤⡀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠉⠛⠉⠀⠀⠀⠀⠙⠛⠛⠀⢸⣿⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢋⣤⣦⡄⠀⠀⢠⣶⣶⣄⢀⣀⣠⠞⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠘⠿⡿⠏⠉⠉⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⢤⣄⣀⠀⠀⠀⢠⠞⠛⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⡏⣿⣎⠻⣧⣹⣷⣌⢷⣌⠛⠿⣶⣦⣤⣽⣿⣿⣿⣿⣿⣿⠏⣠⣾⣿⣷⠟⣋⣵⣿⣴⣾⣿⠟⣀⣾⣫⡾⠋⢀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠘⢿⣦⠀⠙⣿⣿⣷⣿⣷⣦⣄⠈⠉⠻⣿⣏⠛⣿⣿⣿⣸⣿⣿⣫⣶⠟⣋⣭⣿⣿⣿⣷⣿⣿⣿⠋⣀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣇⠘⢿⣷⡀⢸⣿⡏⠉⠛⠻⢿⣿⣦⡄⢹⣿⢀⣿⣿⣿⣿⣿⣿⡟⣡⣿⣿⣿⠿⠛⠛⠉⢻⣿⣣⣾⣿⠿⢋⣾⡟⣸⡿⣿⣿
⣿⣿⣿⣿⢿⣿⣆⠈⢿⣧⣸⣿⢰⠟⠓⢶⣄⠈⠛⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢁⣠⠶⢛⢻⡜⣿⣿⡿⠁⣠⣾⠟⢠⣿⢣⣿⣿
⣿⣿⣿⣿⣯⠻⣿⣦⠘⣿⣿⡇⣾⢹⢿⠴⣬⠳⡄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⣠⠏⢥⣾⢆⠙⡇⣿⡿⣠⣾⡟⠉⢀⣾⣣⣾⣿⣿
⣿⣿⡜⣿⡙⣷⣝⢿⣷⣹⣿⡇⣿⢸⣌⠽⠎⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣔⠽⣯⡯⡄⡇⣼⣿⡿⠋⢀⣴⣿⣿⡿⣿⣿⣿
⣿⣿⣷⡸⣷⡈⠻⣿⣿⣿⣿⣷⢻⡸⣫⣻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣚⣟⢡⣧⣿⣿⣤⣾⣿⣿⠟⣫⣼⣿⣿⣿
⣿⣷⣽⣷⣿⣿⣤⡈⠻⣿⡿⢿⠿⠳⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠞⢿⠿⢿⣿⣿⣯⣵⣿⣿⢟⣵⣿⣿
⣿⣯⡙⠛⠿⣿⣿⣟⠛⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠛⣿⣿⣿⣶⣿⣿⣿⣿
⣿⡤⠶⠶⠶⣶⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⡾⠛⠛⣷⢦⡀⠀⠀⠀⠀⠀⢀⡴⣾⠛⠛⣦⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣷⣶⣦⣼
⣿⣶⣾⣿⣿⣿⣷⣶⠶⠭⠙⠂⠀⠀⠀⠀⠘⡇⠀⢸⣿⠀⢳⠀⠀⠀⠀⢠⠏⠀⣿⡀⠀⣸⠀⠀⠀⠀⠀⠀⠩⠽⠿⣿⣿⣿⣿⣿⣿⣶⣿
⣿⡿⢛⣽⣿⣯⣥⡶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⢸⡿⠀⣸⠀⠀⠀⠀⠸⣆⠀⣿⠁⡰⠃⠀⠀⠀⠀⠀⠀⠀⠐⠲⣶⣭⣿⣷⡦⣍⣛⣿
⣿⣴⠿⣫⣿⣿⠋⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠓⠋⠁⠀⠠⡤⢄⠀⠈⠑⠛⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠈⠻⣿⣿⣿⣶⣍⣻
⣿⣵⣾⣿⣿⢃⡴⠞⣩⣵⠿⠛⠋⣻⣷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣟⠉⠛⠿⣯⡛⠳⣤⡹⣿⣿⢿⣿⣿
⣿⣵⣿⣿⣿⡟⠁⣴⠟⠁⠀⢀⣴⠟⡟⠀⠈⠙⠳⢦⣄⣀⣀⣀⣴⣿⣄⣀⣀⣀⣤⠶⠟⠋⠁⠀⣟⠻⣆⠀⠀⠈⠻⣆⠈⣷⣿⣿⣾⣿⢿
⣿⡿⣿⠉⣿⡇⢸⠏⠀⠀⠀⠋⠀⣴⣷⠀⠀⠀⠀⠀⠀⠈⠉⠓⡞⡟⣶⠋⠉⠁⠀⠀⠀⠀⠀⠀⡿⡄⠈⠑⠀⠀⠀⠈⠀⢸⣿⣿⣾⢃⣾
⣿⣷⣿⠀⢻⣧⠈⠀⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⠀⠀⠀⠀⡾⢀⣧⠸⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⢿⡏⣿⣿⣿
⣿⣿⣿⣧⠈⢻⡆⢸⣄⠀⠤⠴⠞⢫⣿⡇⠀⠀⠀⠀⠀⠀⡼⢁⡾⡟⢧⠙⣆⠀⠀⠀⠀⠀⡀⣿⣯⡙⠓⠦⠀⠀⣰⡇⣸⠏⠚⢀⣿⣿⣿
⣿⣿⣿⣿⣷⣀⠙⠘⣿⠳⠶⢴⣶⡿⠋⣷⡇⠀⠀⠀⠀⡼⢁⡞⠀⡇⠈⢧⡘⣆⠀⠀⢀⠀⣷⡿⠹⢿⣶⠤⠶⢚⡿⠐⠋⠀⣠⣾⣿⣿⣿
⣿⣿⣿⣿⠿⣿⣷⣶⣾⣿⣾⣿⣥⣤⡀⠸⣧⡜⠀⠀⣸⣡⠏⠀⠀⡇⢀⠈⠳⡼⡆⠀⠀⣧⣿⠃⣀⣤⣽⣿⣿⣿⣶⣶⣶⣿⢿⣿⣿⣿⣿
⣿⣮⡛⣿⣧⠀⢈⣽⣿⣿⣿⣷⣦⡈⢻⣆⢹⣇⠀⠀⣿⣿⠀⠀⠠⡷⠀⠁⢠⣿⡇⠀⠀⣼⠃⣴⠏⣡⣴⣾⣿⣿⣿⣭⠁⠀⣾⡿⢋⣾⣿
⣿⣿⣷⠈⢿⣦⣾⣿⡏⣿⣏⢻⣿⣿⣆⣿⢸⡟⢦⡀⢸⣿⡆⠄⠄⡄⠈⡀⢸⣿⠃⣠⠞⣿⣰⣏⣼⣿⣿⡏⣾⡟⢹⣿⣧⣼⡟⢠⣿⣿⣿
⣿⣿⣿⡇⠈⣿⣿⢿⣷⣽⣿⣾⡿⠘⣿⣿⣿⠇⠀⠹⣄⣿⣧⠀⠐⠙⢀⠀⣿⡟⣴⠃⠀⢹⣿⣿⣿⡉⣿⣷⣿⣧⣾⠿⣿⡟⠀⣸⣿⣿⣿
⣿⣿⣿⣷⠀⢸⣿⡘⢹⣿⡏⢻⣿⣾⣿⣿⡟⠀⠀⠀⠹⣿⣿⠀⠉⠀⠑⢠⣿⣿⠃⠀⠀⡈⢿⣿⣿⣶⡿⡯⢻⣿⡇⣹⣿⠇⢀⣿⣿⡟⢸
⣿⡘⢿⣿⣆⠈⢿⣿⣿⣫⣄⣱⣦⣾⣿⣿⡇⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⢸⣿⣿⣷⣮⣐⣮⣹⣿⣿⠟⢀⣼⣿⡿⢁⣼
⣿⣿⣾⣿⣿⣷⣦⣭⣝⣛⡻⠿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⡿⠿⣛⣋⣭⣤⣴⣿⣿⣿⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢍⡱⠋⠉⠀⠀⠉⠒⢰⠆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⠻⠿⠓⢟⣀⠬⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⣳⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢿⣠⣤⡀⢼⣇⠀⢀⣞⠀⢲⣾⣿⣶⠛⠃⠀⣤⡙⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣧⣶⡿⠊⣡⣤⣤⡐⢀⣀⡤⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣏⢹⠉⠁⣞⣵⣷⣤⡎⣿⣿⣿⣿⣿⠯⠃⠉
⠀⠀⠀⠠⠤⠤⡄⢸⠀⠀⣆⡖⠀⠁⠃⣸⠘⣿⢻⡋⣰⡏⠻⠛⡌⣽⣿⣿⣿⡅⠎
⠀⠀⠀⠀⠈⠂⠀⢸⣇⠀⣡⠁⠀⠀⠂⠘⠆⢻⣆⣿⣿⣧⢀⣰⢧⣻⣿⠯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠛⠗⠀⠀⠀⠀⠐⠂⠹⣦⣽⣿⣮⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠓⠀⡤⠊⠁⠀⠀⠀⠀⣄⣾⣿⣿⣿⠒⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢿⡧⠄⠀⠤⠀⠀⠀⠀⠀⠘⢛⣿⠛⣿⣽⣿⡿⠘⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣤⣠⣼⠷⠶⠓⠒⠀⠀⠀⠀⠀⢀⡟⠙⢯⣜⣥⣿⢥⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠂⠈⠉⠙⠙⠆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠐⢀⠸⡇⠈⠑⢄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠳⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡈⠃⠀⠀⠤⠀⠀⠈⠀⢀⠀⠱⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄
⠀⠀⠀⢀⠐⠂⠐⠂⡃⢸⠀⠀⠈⠀⠀⢀⢀⡀⡀⠀⠀⠙⠢⣽⣀⡌⠀⣀⠶⡤⢤⣀⠀⠀⠀⣶⢰⠟
⠀⠀⠀⠀⡠⠐⠀⠀⠁⢨⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣣⠏⠀⡄⠀⠀⠀⠀⠤⣴⣯⣀⢀⡀⠀⢀⠶⡃
⠀⠀⠀⠀⠀⣠⠖⣃⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣯⣭⣭⠛⠀⠁
⠀⠀⠀⠀⠀⠨⠁⠙⡂⠀⠈⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠛⢿⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠂⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠢⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠶⢦⡴⢶⣖⣛⠛⠉⠙⠛⠛⠳⢦⣄⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠹⡆⠀⠀⠀⠀⠀⠉⢷⡀⢠⠾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⣩⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠙⢣⠠⢤⣤⣤⣄⡀⢻⡎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢁⣴⡟⣹⠞⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠸⣄⡀⢿⠀⠀⠙⠾⠁⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⣾⣥⠟⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⠙⣻⣇⣀⡀⠀⠸⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣼⠏⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠦⣤⡾⠋⣀⣠⣤⠖⠲⣌⡑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⠛⠉⠉⠉⠀⠀⠀⠈⠙⢦⡙⢦⡀⠀⣠⠄⠀⠀⠠⡄⢀⡼⣡⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠾⠻⣦⠱⣴⠃⠀⠀⠀⠀⢹⠏⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⢴⠤⣤⣤⣀⡀⠀⠀⠀⠀⠈⢳⡉⡄⠀⠀⠀⠀⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⣀⠀⠀⢻⡜⡄⠀⠀⠈⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢆⡀⠀⢿⣰⠀⠀⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⡇
⠀⠀⠀⠀⣀⣠⠶⠋⠀⠀⠀⠀⠀⠀⢀⠀⣤⣄⣀⣀⠀⠀⠈⠀⠀⠀⠀⠈⢣⡀⠀⢇⢇⡀⠻⠒⠂⠀⠀⠀⠀⠀⠀⣠⡶⠊⠉⠀⠀⣿
⠀⠀⠀⠚⢏⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠂⠈⠉⠀⢀⠀⠛⠒⢤⣀⠀⠈⢷⡀⠸⡘⢻⠆⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠉⠓⢿⣍⡉⠙⠳⣤⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠈⠓⢦⠘⣧⠀⠇⢸⡄⠀⠀⠀⢀⣤⠋⠀⠀⠀⠀⣠⡄⠀⠀⢠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣼⡄⢸⠈⡃⠀⠀⣤⠎⠁⠀⠀⢀⢀⠞⠁⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⠐⣦⣀⣀⠀⣤⣀⣰⡶⠛⠛⠉⠀⠙⠷⣄⡀⢳⠀⢰⡇⠀⠀⡀⠀⣸⠋⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⡀⣿⠀⠀⠀⠹⣮⠃⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠉⠉⠙⠿⣄⣀⣀⣠⣉⡀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠙⠳⣄⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠙⠙⢦⣸⢦⣷⣤⠶⠞⠛⠛⠙⠛⠶⣄⠀⢀⣀⣀⣀⣀⠈⢳⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠛⠉⠉⠉⠉⠉⠉⠛⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡴⠞⠁⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠋⠁⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢤⣰⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⣴⠤⣝⣋⠩⣙⣻⠇⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⡇⠀⠀⠀⣴⠒⢦⣄⠀⠀⣰⣿⠀⣿⡀⠿⠃⢀⣼⢿⣔⢇⡦⠞⠁⠀⠀⠀⠀⠛⠷⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠻⣄⣀⣴⡟⠀⠀⠈⣳⡾⠃⠹⣧⣌⠙⠳⣤⡏⠁⠀⠹⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⣼⠋⠀⠀⠀⠀⠙⢷⣄⠟⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣀⠠⠤⠤⣀⣾⠁⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⡴⠶⠤⠶⢤⡀⠀⠀⢀⣤⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠋⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠘⡇⠀⢀⣴⠏⠀⣠⡴⢶⠉⠉⠀⢀⡿⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣠⡤⠤⠤⠚⠋⠁⢀⡼⠋⠀⠘⠳⢄⣀⡼⠃⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠷⢦⣄⣀⣀⣴⠞⠉⢩⠟⠁⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠶⠦⣄⣀⠀⠀⠀⠀⠀⠀⢈⣽⠟⠁⠀⠀⣠⡶⣿⠖⠴⠶⠛⠛⡉⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠋⠉⡜⠁⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠷⣤⣀⣀⣀⣀⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⠉⠉⠉⣉⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⣀⣀⣠⡶⠏⠥⠤⠤⢤⣬⡽⢶⣤⣀⣀⣀⣀⣀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣴⣾⣿⣿⣷⣦⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠒⢎⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠘⠂⠊⣘⣿⡿⣶⣦⣄
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣶⣶⣀⠀⠀⠀⢈⣛⡛⠋⠹⠿⠿⠿⠿⠿⠿⠿⢿⣿⡸⡇⠴⠀⠋⠻⠀⠼⠟⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⡄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠻⢀⣴⣶⣶⣄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣿⣾⣶⣾⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣾⣷⠀⠀⠰⠀⠀⠀⠀⠀⠀⣿⡯⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠻⣿⣿⣟⠁⠉⠁
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⣶⣼⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠈⠙⠛⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠀⠀⣨⣤⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣾⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠯⢧
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣳⣖⠒⠚⠛⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢐⢈
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠸⡄⢀⡼⣾⢿⣋⣤⣀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣤⣶⣶⣶⣶⣿⣭⣍⣛⠿⢃⣠⣴⣶⣤⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠛⠛⣿⣿⡇⢰⡆⣄
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⢿⣿⣿⣻⣿⣿⣿⣿⣿⣷⠟⠉⠀⣀⠈⠛⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣃⣀⣀⣿⣿⡇⠀⠁⢸⣦
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣾⣿⡇⣿⣿⣿⣿⣿⡿⠋⢀⣶⣿⣿⣿⣦⠈⢿⣄⡀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⡀⠘⣿⣧⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠋⠀⠻⠟⠻⢃⠙⠁⢠⣿⣿⣿⣿⣿⣿⣧⠀⢠⡄⠀⠀⠁⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠁⠀⠸⢋⡄⠀⠁
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⡄⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣾⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠘
⠀⢰⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⣿⣿⣶⣾⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣴⣶⣶⣾⣿⣿
⠀⢸⡿⢿⠿⢿⠁⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠈⣿⡄⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠠⣾⣿⣿⣿⣿⠃⠀⣾⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⡀⢹⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠋⠀⠀⠙
⠀⠀⠀⠀⠈⠉⠉⠀⠁⠀⠉⠉⠉⠉⠉⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢺⠟⢟⠀⠤⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⡀⠒⠒⠄
⠀⠀⠀⠀⠀⠀⢀⠈⠂⠈⠆⠀⠀⠀⢀⠅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡟⡉⠉⠂⢰⠤⠐⠓
⠀⠀⠀⠀⠀⠀⠈⠁⠸⢄⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⠃⠀⠠⠈⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⢀⠀⠀⠀⠠⡀
⠀⠉⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀
⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡄⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡠⠀⠤⠶⠈⠊⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⡆⢀⡀⠑⠠⠐⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢦⠧⢄⠉⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢀⢤⣄⡧⣩⣴⣥⣍⣄⢁⠒⠀⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⢨⣐⡲⢦⢹⣿⡿⠧⠼⠶⠎⠾⠾⠿⠫⠶⠶⢂⣄⣤⣀⡀⠄
⠀⠀⠀⠀⠀⢀⣐⢶⡸⣜⠎⠓⣑⢴⠦⠀⠀⠀⢠⣤⣤⡄⠀⠠⢀⡈⠉⠉⠢⣦⣴⠆⣚⢁⣂⡀
⠀⠀⠀⠀⡈⢀⠡⠙⠃⠛⠛⠉⢀⠄⠀⠀⠀⠀⠚⠛⠛⠃⠀⠰⠊⡁⠾⠋⣗⠒⠒⣒⠮⠍⠁
⠀⠀⠀⢐⠨⠲⠥⠖⠴⠶⠶⠦⢁⣠⢖⢄⣤⣤⣤⣤⣤⣤⣤⣔⣦⣉⣀⡐⣙⣛⢉⣥⡓⢈⡃⢒⣒⡂
⠀⠀⠀⠀⢁⣴⠳⡎⣿⡡⡋⠛⠛⠪⠓⢉⠨⣻⣾⢛⠛⡹⡛⣟⣒⠒⣲⠭⢤⣔⢮⢲⡆⣷⡽⣎⡛⠟
⠀⠀⠀⠀⠈⠉⠑⠛⠣⢪⡪⣉⠋⠧⣒⡵⣪⠵⠟⣸⠻⡝⣦⣤⡦⣿⡦⢹⡏⣿⡜⣦⢹⣬⣁⠈⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢇⡟⣾⡞⣵⠊⣵⡇⣿⡿⢃⣧⢙⡏⡗⢿⣁⣿⣇⣿⡇⣹⣧⣉⡭⢳⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢻⡇⢻⣆⢹⣧⠘⠿⠸⠿⠿⣤⠃⣮⡙⠛⠉⠉⠁⠉⠉⠉⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⠺⢉⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⠛⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⠀⣠⠴⠚⠛⠳⡤⠶⠒⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠚⠋⠀⠀⠀⠀⠀⠉⠻⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⢿⣤⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⠯⠖⠚⠳⡄⠀⠀⠀⠀⢩⡟⠁⢀⣤⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠛⢷⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⢀⣠⡿⠒⠒⢶⡀⠈⠻⠶⠟⠉⠉⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢰⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠓⠀⠀⠀⠀⠀⠘⠷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⢶⣶⣤⣀⠀⠀⠀⢀⣀⡀⠀⠀⢻⣷
⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⢀⣤⣾⠿⠿⠿⠟⠀⠀⠀⠀⠘⠿⠿⠿⠿⣶⣄⠘⣧⣀⡀⠀⣀⣿⠈⠙⢷⣄⣰⡟⠙⣇⠀⠀⠘⣿⣇
⢰⣿⠇⠀⠀⠀⣠⣤⣤⡀⢺⡟⠃⠀⠀⠀⠀⠀⠀⠈⠁⣠⠴⠒⠒⠲⢤⡀⠀⢀⠤⠒⠒⠒⢦⣀⠉⠁⠈⠙⠛⠛⠛⠁⠀⠀⠀⠉⠉⠀⠀⠻⢶⣶⠾⢿⣿
⢸⣿⠀⠀⠀⠿⠯⢤⡈⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠙⡴⠁⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣾⣿⣤⣀⣀⣀⣀⠀⣿⢀⣀⠀⠀⢀⣴⡿⠛⠛⠻⣾⡇⠀⣀⣀⢀⣀⣀⣀⣇⡀⣀⣀⣀⣀⡀⠀⢹⠀⣴⠞⠉⠉⠉⠓⠦⣤⣤⠤⠤⣄⡀⠀⠀⠀⠀⠀⣿
⣿⡿⠈⠉⠉⠉⢻⡇⠉⠉⠉⠉⠙⣾⣿⠀⠀⠀⠀⢘⣿⠁⢹⡿⠀⠀⠀⠈⡇⠀⠀⢼⡿⠀⠀⠁⡿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠹⣿⣀⠀⠀⠠⣂⢽⣦⡀⠀⠀⠀⢀⣴⢻⣄⠀⠀⠀⠀⢀⣼⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢶⡄⠀⣿
⣿⣿⠀⠀⠀⠀⣿⠠⣶⡧⠀⠀⠀⠀⠉⠛⠿⢿⣶⡕⠊⠈⢿⣶⠤⠶⣻⠋⠀⢻⣳⠶⠴⠶⣻⠏⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢀⣧⠀⣿
⢸⣿⡄⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣦⣀⠀⠙⣷⡋⠁⠀⠀⠀⠉⠛⠒⠋⠁⠀⠀⠈⠻⣦⣤⣤⣤⡀⠀⠀⠀⠀⠚⠋⠀⠀⠀⣸⠇⢠⣿
⠈⣿⣇⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠈⠛⢷⣄⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⢻⡄⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⣿
⠀⢹⣿⡄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⢀⡽⠷⠶⠿⠷⠴⠶⠶⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣾⠋⠀⠀⢠⣿⡏
⠀⠀⢻⣿⡄⠀⠀⠀⠀⣿⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⡟
⠀⠀⠀⢻⣷⡄⠀⠀⠀⢻⡄⠀⠀⣼⠃⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢰⡟⠀⢠⣿⡟
⠀⠀⠀⠀⠻⣿⣆⠀⠀⠘⣷⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣤⣴⡿⠁⣰⣿⠟
⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠈⠻⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠚⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠓⣦⢀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣾⡋⠀⠀⠙⠲⠤⠤⠤⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⣿⡿⠟⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⡄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠁⢠⠀⡀⢠⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠠⠃⠈⠑⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡶⠀⠀⣀⣤⠆⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡻⠻⠆⠀⠀⠀⠀⠀⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⠀⠀⢀⣠⣆⣠⠀⠀⠃⠀⢀⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⡿⠁⠀⢠⡋⠉⠹⢃⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠶⠆⠂⠀⠺⢿⢇⡀⡀⠀⡸⠿⠛⢀⡿⢶⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⢻⡃⢠⠇⠀⢠⠂⠀⠠⠀⠀⠀⣴⣿⠟⠲⣄⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⠏⠀⠀⡇⠀⢠⣿⠟⠻⠀⠀⢹⠆⠀⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢠⣄⠀⠘⢀⡄⠀⠃⠀⠉⠉⠀⡴⠞⠀⠀⠀⠀⢠⡆⠈⢢⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⠀⠀⠀⠞⢷⡛⠱⠄⠀⠀⡴⠀⠀⠀⠀⠀⠀⠐⠿⠃⣀⣸⡟⠦
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡄⢲⡆⠀⢠⣷⢀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⢸⡋⠁
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡎⠑⠇⠀⢘⠉⠘⡇⢢⣿⡶⠊⠀⠀⠀⠀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠃⠀⠠⡘⢾⡄⠀⣏⡼⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⣷⡄⡟⢀⣶⣇⠀⠀⠀⠲⢀⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⡀⢠⡄⢱⠙⡄⢸⣿⢋⡀⣘⣀⠀⠘⠛⠛⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠱⣾⡇⢸⡄⠁⠀⡇⢈⠷⠛⡟⢃⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⣤⣾⣶⡷⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠀⠈⡇⣸⣷⣷⡿⠟⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⢳⡏⣿⠥⣶⣶⣽⣿⡴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠠⣄⡀⠀⠀⠀⣸⠇⣿⣠⣨⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠙⢦⡀⠀⣿⡔⡿⠃⠛⠛⢿⣼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢿⡆⢿⡿⡇⠀⢀⣀⡀⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣿⠾⠃⠀⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠦⡀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠉⢠⣄⣀⠀⠀⠀⠀⢱⣼⣷⣾⠄⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡀⢤⣚⣿⢿⡟⠀⠀⠀⠈⠀⠀⠀⠑⠄⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠑⠤⣄⡀⠀⠀⠀⠀⠀⢈⡄⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣿⣿⣻⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⡠⡀⠀⠀⠂⠄⠐⠰⢄⣀⢀⡠⠀⠠⠉⠀⠐⠀⠀⢹⢿⣿⡿⠄
⠀⠀⠀⠀⠀⠸⠅⠀⠀⢀⠄⠉⠀⠀⢰⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠏⠃⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠁⠀⠀⠀⠸⡺⣄⠀⡀⠀⠀⢃⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠤⠄⣀⠈⠻⢿⣾⣷⡤⠂⠁⠀⠀⠸⠁⠀⣤⣀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠀⡠⢮⡿⠁⡀⠀⠠⠬⠿⢻⣿⣿⢷⡦⠤⠖⠚⠁⠙⢷⡏⠿⠛⠀⠀⡁⠀⠐
⠀⠀⠀⠾⠁⠉⠀⠀⠈⠁⠀⠀⡄⢀⣨⠶⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠜⠃⢨⠀⣠⠀⠀⠂⠀⠀⠀⠀⠀⠀⣰⠌⠀⢀⡏⢓⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠠⣿⡃⠀⠀⠑⠒⠀⠒⠂⠁⠀⠀⠀⠩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⡙⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⡶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡙⢿⣡⣌⣷⡀⠀⠀⠀⣤⣤⣤⣸⣿⠀⠀⠀⠈⠉⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣁⣨⣿⣿⢠⡴⣞⠭⠃⠁⠀⢁⣀⣠⣶⣿⣿⣿⣿⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⡿⢿⣇⠀⢀⣀⠤⣾⣿⢿⡟⠿⣯⣿⣿⣇⠀⣿⣍⣿⣛⣡⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠘⡇⢰⣿⣿⣟⣡⣾⣿⣽⡟⠙⠷⠿⣿⣿⣿⡟⢠⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⡿⠛⠟⢿⠛⢿⣿⣿⣿⢏⣡⣤⣄⠘⢿⣿⣿⣿⣿⣿⣿⡐⠰⠄⠀⠘⠙⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣼⣿⢿⣿⣿⣩⣦⠀⢘⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠛⠛⣻⠟⠉⠙⠻⠿⠿⠿⠿⢿⣾⣿⣿⣿⠃⠀⢀⡀⣀⣠⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠚⠃⣁⣠⣀⠀⠀⢀⠀⠀⠶⠶⢾⣿⣷⣿⣿⣄⡀⢈⠁⠈⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠏⡙⠛⢷⡾⣟⢿⠷⣦⣄⠀⣿⣿⣿⣿⣿⠓⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠦⠀⠀⠀⠀⡀⠉⣀⡀⡉⣿⣿⡟⠉⡿⢻⣏⠙⢦⣤⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⠘⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⠆⠀⠀⠀⠀⣀⡤⠶⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣉⣉⠑⠦⢄⠀⣼⣿⡏⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⣴⡾⠋⣰⠟⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⡉⠁⠀⣠⠴⠀⣿⣿⣷⣾⠏⣿⣿⣇⣤⣤⡶⠂⠀⣿⣿⡇⠀⠸⠇⡀⢨⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⠀⠀⢸⡿⠀⢀⣿⣿⣟⣳⣷⣿⣿⣿⠋⠉⠁⠀⢸⣿⢇⠀⢰⠏⣼⢡⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠈⣧⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢸⣿⠷⡀⣼⡗⣋⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡏⢹⣷⣥⡂⢙⣿⣎⣻⣿⣿⣿⣿⡿⣿⠿⣧⠹⣿⡄⠸⣿⡎⠠⣿⡶⣅⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣿⢷⣄⣸⣿⣷⣿⡧⠾⣿⣿⣿⣿⣿⡟⠀⣿⠀⠈⠀⠹⣿⡆⠙⣿⣼⣿⣷⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⣿⣿⣿⠿⠿⠏⠉⠉⠁⠀⠀⠀⢀⣶⡢⢰⣿⣿⣤⡸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠿⠟⠋⠹⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⣾⣿⣿⡇⢸⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣭⡙⠻⠦⣄⡀⠉⠿⠿⢿⣿⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣶⣄⠈⠙⢿⢶⣄⠀⣿⠙⢿⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⡌⠹⣷⡀⠀⠀⢹⣧⠀⠀⠀⠙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡂⠈⢻⣶⠀⠀⢸⡆⠀⠀⠀⠈⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣄⡘⣿⣿⡄⠀⢯⠀⠀⠀⠀⢹⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⡈⠻⣿⣄⢸⣆⠀⠀⠀⢸⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣶⣯⣿⣿⣿⣦⡀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠟⠉⠙⠻⢿⣷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⣿⣛⣻⣿⣿⣻⣛⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⣷⣿⣿⣉⡿⣷⣟⢹⣛⠿⣾⣿⣻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣬⢿⣯⣟⢻⡹⣿⢷⣻⡿⣽⣿⣻⣾⢛⣯⣯⣙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣹⡟⣻⠟⣭⢯⣷⡿⣿⣿⣩⣷⣿⣿⣿⣿⣿⣵⢿⣜⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣼⣿⣛⣿⣟⣟⣿⣶⣿⢽⣿⢿⣾⣏⣥⣽⣿⣿⣭⣯⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣥⣭⣿⠿⣿⣿⡞⢿⢿⣿⣻⡿⢿⣽⣿⣿⣟⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣯⣿⣾⣿⣶⣿⣟⣿⣿⣶⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣄⠀⡰⠿⣆⠀⠀⢀⣠⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣴⣿⣾⣿⡷⢡⢰⡿⢆⣰⣿⡿⡿⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠲⣶⣤⣄⠀⣿⣴⡼⠛⠟⠛⠳⢧⠀⡾⠋⠉⠉⠛⠛⣧⣿⠀⣴⠇
⠀⠀⠀⠀⢲⣶⣖⣦⣽⣯⣿⣿⣟⠁⠀⠀⠀⣀⣀⣘⣿⠁⠀⠀⠀⠀⠀⠀⣹⣿⣃⡖⣶⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠳⣿⡿⠟⠉⠉⠉⠉⠒⣤⣶⠟⠋⠉⠉⢻⡷⠶⠿⠷⣦⣀⣴⠋⠉⠙⠛⢿⣿⡛⠋
⠀⠀⢀⣀⣀⣀⠙⣇⠀⠀⠀⠀⡠⠎⠁⠁⠀⠀⠀⡀⠸⠁⠀⠀⠀⠈⢿⣏⡀⠀⠀⠀⠀⣙⣀⣀⣀⣀⡀
⠐⠻⣿⣭⣿⠿⠿⠛⠓⠤⣤⣾⠁⠀⠀⠀⠀⡀⢀⣀⣔⣠⣀⣀⠀⡴⠋⠛⠻⣷⣄⣠⠖⠉⠙⣿⡿⣿⣿⠟
⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⣽⠿⣆⠀⢀⢠⣼⣿⡿⣟⣻⡟⢻⣷⣮⣐⡀⠀⠀⢈⣿⠁⠀⠀⠀⠈⢻⠾⠃
⠀⠀⢀⡰⣤⡀⠀⠀⠀⢺⣟⠀⠀⠁⢢⣿⡿⡳⢎⡶⠋⢾⣼⣯⣿⣿⣧⠤⠒⠛⠛⢷⣤⣤⡤⠴⠛⠄
⠐⠾⣿⣅⠀⣉⣳⡶⠤⠼⢯⣁⡀⠀⢼⣿⢿⢧⣪⠀⣶⠀⠱⡿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣯⣀⠀⢠⣶⡞⣷⣦
⠀⠀⠈⣉⡽⠃⠀⠀⠀⠀⣼⠟⠋⠉⢹⣿⣖⣿⣋⠒⡄⠠⡞⣽⣟⣿⡇⠀⠀⠀⠀⢀⡏⠀⠙⢿⣿⠏⠈⠉⠁
⠀⢀⣾⣿⠅⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢹⣿⣿⣵⡿⣿⣾⣿⣟⣿⠏⠉⠙⠓⠶⡶⠞⠁⠀⠀⠀⠿⢉⠲⣄
⠰⠿⠿⣯⣄⣀⡀⣀⣠⠼⣿⡀⠀⠀⢀⡔⠈⠙⠙⠿⠿⠟⠛⠉⢱⡄⠀⠀⠀⣸⠷⠦⡀⠀⠀⠀⢤⣿⣿⣾⣧
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠸⢷⣦⠤⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⡠⠞⠁⠀⠀⠈⣦⡀⠠⠶⠛⠋⠉
⠀⠀⠀⠀⠀⣔⠄⠀⠀⠀⠀⣼⠃⠀⠀⠉⠳⣦⣄⣀⣧⣀⣀⡤⠞⠁⠙⡆⠀⠀⠀⢀⣠⠷⡄
⠀⠀⠀⠀⠺⠓⠖⠒⠶⠚⢛⣿⠀⠀⠀⠀⠀⠀⢸⢿⣿⠉⠀⠀⠀⠀⠀⠘⣷⠤⠶⣊⠳⣷⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡽⢧⣤⣤⣀⣤⣤⡏⠈⠛⢳⣄⡀⠀⣀⣀⡴⢟⡀⠀⠀⠈⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣏⣿⣛⣽⣿⣿⡇⠀⠀⠀⢸⠟⡟⣿⡏⣷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⠁⢽⡋⣿⠟⣄⠀⣠⠋⠀⠹⣜⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠈⠿⠃⠀⠀⠀⠙⠏

⠀⠀⠀⠀⠀⠀⠀⣤⠶⠶⣶⣶⣦⣄⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢿⣳⣤⣖⡿⠿⠛⠉⠉⠉⠀⠐⠒⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠙⢷⣄⡀⠀⠐⢶⣶⣶⣶⣶⡿⠿⠓⠻⠷⠶⡤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠁⠀⠀⠀⢀⡀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⢦⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠈⠙⠺⠿⠶⣶⠶⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢧⣀⣀⠀⠀⠀⠈⢻⡧⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠟⠛⢦⡀⠲⠀⠀⠃⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣤⢶⣻⠗⠁⠀⠀⠀⠙⢦⠀⠈⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠓⠰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡆⠀⠀⠀⠀⠀⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⣶⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡜⣿⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠁⠀⠹⡘⢰⣃⡤⠶⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡟⠁⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⡀⣟⡾⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣇⢸⢻⡄⠀⠀⠀⣀⡀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠻⣧⢻⣄⠀⠀⡿⠛⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠘⠳⣝⢷⡼⠁⠀⠀⠀⠀⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢻⣿⡄⠀⠀⠀⣸⣳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⢘⠹⣿⡀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⢿⣇⠘⠂⢹⡇⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣄⡻⣷⣧⠀⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢦⣉⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢉⠣⠁⠀⢀⣴⣶⡿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣒⣯⠽⠛⠉⠉⠉⠓⢌⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠢⠴⠣⠂⠀⠂⠉⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⢆⢀⡔⠊⠀⠀⠀⡠⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠌⢁⠀⠀⠀⢀⠃⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠠⣂⣠⣀⡁⡠⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡀⡀⠠⣀⡀⠀⢀⡀⢠⢤⠶⢠⠊⡇⠈⠃⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠡⠄⠠⠥⠥⠴⠥⠤⠍⠈⠀⠄⠐⠂⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⢴⣾⣿⣿⠀⠘⠈⠙⠓⠀⠀⠒⠀⠀⠀⠐⠁⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⡿⠀⡀
⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⠀⠀⠀⠄⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠋⡟⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾
⢀⠔⠈⠁⠈⠁⣀⠀⠀⠀⠀⢠⣖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢀⣾⣏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠤⠀⡴⠊⠉⠉⠐⢄⠀⠀⠀⠁⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣀⡀⢀⣤⣿⣿⣟⡡
⠁⠉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠦⣄⡠⠲⠒⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⡀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠑⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡻⣿⣿⣷
⠙⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠴⠊⠁⠀⠀⢀⣿⡟⠃⠀⠃⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀
⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠈⢆⠀⠉⠁⠀⢄⠵⠶⣦⠀⠀⠀⠀⠀⢀⠀⠀⢽⣿⣿⡇⠀⣾⠇⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⣀⢀⡀⠉⠉⢻⣿⣿⣧⢠⣿⣷⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠻⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠳⡀⠀⣰⠁⠀⠠⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠃⠀⠁⠈
⠀⠐⢄⢿⡄⡀⠀⠀⠀⠀⠕⠀⠄⠀⠀⠀⠀⠀⠀⠒⢂⣩⡴⠋⡉⠀⠙⢷⡀⠈⠀⠀⠐⠒⠒⠄⠠⠄⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠆
⠀⠀⠀⠀⠙⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠋⠅⡔⠁⠀⠋⠀⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣴⣶⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠖⠀⠀⠀⠀⠀⠀⠁⠰⣀⠀⢀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠈⠙⠙⠓⠚⠠⢄⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠉⠉⠒⠆⠒⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣦⢠⠀⢀⣴⣿
⠀⠀⠀⠀⠇⠑⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⣿⣿⣿⠏
⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠸⠁⠀⠀⢀⣤⣞⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣟⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⠀⠀⠀⠈⢻⣿⣷⣀⢠⠀⠀⣤⣿⣿⠋⢀⠀⠠⠠⠀⠠⠄⠀⡸⠉⠐⠙⠧⣐⠀⠤⠤
⠲⢀⠈⠁⠀⠁⠀⠀⠠⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠈⠻⣿⣿⠛⠃⢸⣿⣿⡡⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠒⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣷⣶⣦⣉⡀⠃⢌⣿⣟⡡⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠐⠚⠉⠉⠉⠻⠛⠛⠛⢻⡿⢿⠁⣠⡆⢫⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⣄⢄⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣒⠯⠥⣸⣦⣀⢇⣤⡄⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣼⣿⣿⠞⠒⢦⣿⢟⢿⣿⠉⢁⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠸⣿⠋⠀⠀⠀⠀⢙⠃⠀⠌⡃⢻⣷⡿⡋⠓⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⢀⠀⣀⣞⣱⣿⣿⣾⣿⣷⣷⡿⢋⣀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠠⢬⣟⣛⣻⣯⣉⣭⣥⣽⡿⠿⠟⢋⡴⠛⠉⠀⠈⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⣤⣬⣤⡿⠿⣿⣿⣝⠢⣄⣀⣀⠐⣯⡾⠋⠁⠀⣠⣶⠖⣰⡿⠿⠵⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠷⢤⡃⠀⠀⠄⣴⠟⠡⠿⠶⣤⣄⠉⠀⠀⠁⣤⣤⡠⠌⠁⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⣄⣀⣤⣤⣤⠤⡤⡤⠤⢤⣄⡀⠸⡆⠀⠀⠀⠈⢿⣿⣣⣤⠞⠉⠀⢀⣴⠷⠚⠋⡹⠟⠉
⠀⠀⠀⠀⠰⠞⠛⠛⠛⠒⢦⣶⡂⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀⠀⢀⣾⣿⣫⣄⣤⣖⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣴⣶⣶⠴⡟⠛⠙⢻⡟⠉⠁⠀⠀⠉⠁⠂⠀⠀⣡⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡾⠋⡽⣫⡵⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣠⠴⠛⠁⠀⠀⠀⠀⠀⡸⠀⠀⠀⣀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣶⠾⠿⣿⠟⠛⠓⠳⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡬⠀⠠⡐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡼⠧⣀⠀⡵⡖⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣐⠂
⠀⠀⢠⠄⣄⠀⠀⠀⠈⠀⢀⡀⠁⠰⢒⠀⠀⠀⠀⠀⠀⠀⡱⠁⢈⠃⠒⡄⣹⣤⡗⠋⠦⡂⠀⠀⠀⠀⣠⠀⠀⠀⡠⢤⠧⣄⡡⠔⠋⠙⢺⣄
⠀⢐⣁⢠⠀⢄⠀⠀⠀⠸⣸⣁⡎⠀⡸⠀⡐⠀⠒⢀⠀⢸⡇⠀⢀⠀⢰⣾⠋⡎⠹⣄⡠⠇⠤⢢⠀⠃⠁⠀⢂⣸⠀⠀⠀⢸⠃⠀⠀⠀⠀⣸⠀⡀
⠀⠣⡑⠄⢁⡢⣄⠀⣀⣶⢿⣥⣇⡠⠀⠀⢂⠀⠘⢄⠀⠈⢳⣀⣠⡤⠺⢃⠌⣀⣤⡞⣆⠀⠀⢨⠀⠀⠐⢸⠟⠁⠀⠀⢔⠸⡄⠀⠀⠀⢰⣁⠚
⠀⠀⢿⡃⠶⢽⣣⣲⣷⠾⠶⠒⠊⢀⠀⠀⠂⠀⠀⠀⠑⠂⠈⠻⣵⣷⣴⣯⡔⠋⠁⠴⠁⣀⠔⠁⠀⠀⠀⠀⡳⣌⡢⣤⣘⢦⠀⢀⣄⣠⡾⠁⡀
⠀⠀⢀⡀⠄⠀⠜⣿⢿⢄⠀⠀⣢⡄⠅⠠⠀⠀⠀⠀⠀⢀⣤⢾⠛⣽⠛⡟⠿⠻⡟⢧⡁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢲⡿⠟⢿⡿⠿⠥
⠀⠀⠀⠁⠀⠾⣠⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⡇⠀⠈⠉⠙⠋⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⡸⠋⠋⠁⠀⠈⡇⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⡿⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣴⠏⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠚⠛⠋⠉⣉⡤⠚⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⡠⠒⢂⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⡆⠀⠀⠀⠀⣸⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡞⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁⠀⠀⠀⠸⡄⠀⠀⣠⠏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⣁⠀⠀⠀⠀⠀⢰⡧⠖⠚⠁⠀⠀⠀⠀⢸⠀⢀⡤⠖⠒⠚⠁⠀⠈⡇⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠁⠀⠀⠐⠈⣁⠤⠄⠒⠀⠰⠋⠀⠀⠀⠀⡞⠀⠀⠀⢸⡷⠋⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⡆⡜
⠀⠀⠀⠀⠀⣠⠊⠀⠈⠑⠀⢀⣀⣠⣤⣤⡤⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⡴⠀⠀⠀⠀⠀⣀⣤⠤⠴⠒⠋⠀⢀⡴⠚
⠀⠀⠀⢠⠞⠁⠀⠀⣀⣴⢚⠟⠉⢁⣠⢬⣗⠢⡀⠙⡌⢙⠲⣤⡀⠀⠀⠀⠀⠀⠐⢣⡄⠀⠀⢠⠋⠁
⠀⠀⢠⠏⠀⠀⣠⡞⠁⠈⠀⢀⡴⠋⢰⡒⠪⣑⣁⢀⡈⠪⡱⠒⠛⢦⡀⠀⠀⠀⢠⠊⠀⠀⠀⠸⣄⣀⡴⢧
⠀⢀⡏⠀⢀⡼⠑⡄⠂⠀⠀⠑⠒⢦⠏⠀⠈⠀⡖⠐⠛⠀⠀⠀⠁⠇⠹⣆⠀⠀⡌⠀⠀⠀⠀⠀⠈⠉⣠⠃
⠀⢸⠀⢀⡞⠀⠀⢇⢠⠄⠀⠀⠀⠀⠀⠀⠠⡂⠁⠀⠀⠀⠀⠀⠀⠀⣀⣹⡆⠀⠳⠒⡆⠀⠀⢀⡠⠚⠁⠀⠀⠀⢀
⠀⡟⠀⣼⠀⠀⠀⠘⣌⡃⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⡩⢿⠀⠀⡴⠀⣠⠖⠉⠀⠀⠀⠀⠀⢀⡸
⠀⡇⠀⡇⠀⠀⠀⠀⠙⡏⢧⡖⡉⠀⠀⠱⠅⠀⠀⠀⠀⠀⠀⠀⠀⣤⠼⠀⣸⡇⠀⠀⢰⠁⠀⣀
⠀⡇⠸⡇⠀⠀⠀⠀⠀⠀⠘⣬⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⢚⣹⠇⠀⠀⠘⠶⠚⡅
⠀⢷⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⠙⣲⠦⠤⢤⡀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡾⠀⠠⡀⠀⠀⡰
⠀⠘⡆⠸⣆⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⠁⠀⠉⠉⠳⣄⠀⠀⠀⡇⣄⠀⣸⠃⠀⠐⠀⠀⡰⠁
⠀⠀⠹⣄⠙⢆⠀⠀⠀⠀⠈⢦⠁⡀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠓⢒⡾⠃⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠘⢦⡈⠳⣄⡀⠀⠀⠀⠑⠦⣁⠀⠀⠀⢀⠔⠀⠀⠀⢀⡴⠟⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠹⢦⡀⠙⠲⢤⣀⡀⠀⢸⠇⠀⢀⠃⢀⣀⣤⠖⠋⠀⢀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣄⣈⣉⡙⠛⠛⠛⣛⣛⣉⣩⡤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠈⠡⠀⢷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣤⣦⣤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠢⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢏⡜⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠂⠈⠙⡆⠈⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡼⣄⣀⡂⠀⠀⠀⣇⠀⠀⠠⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣡⠤⠶⠚⣟⠁⠀⠀⢠⠝⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠾⠵⠒⠶⣅⣀⢸⡆⡀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠁⠈⢣⡀⠀⢰⠀⠈⠉⠑⢢⡤⠂⠀⠀⠀⡤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢦⡤⠀⠀⢀⠀⡀⠈⠳⡀⠀⠀⢸⡆⠀⠀⢁⢀⠀⠀⠇⠀⠀⠀⡔⡠⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣠⠀⠀⠀⠀⣀⣀⡀⠐⠄⠀⠀⠙⠀⠐⢌⡴⢂⠀⠀⠀⠀⣐⣎⠋⠀⠉⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠈⠢⠀⠀⠸⡄⠀⠐⠀⠀⠈⢢⠹⢦⣾⠠⠂⠀⠀⠠⠀⠐⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠁⢀⡊⠀⠀⡄⠤⠆⠀⣤⡀⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠀⢠⡀⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁⠠⠶⠂⠒⡻⢆⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠂⣷⡋⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠐⠚⠞⠁⠀⣈⣀⠤⠤⠬⢶⡀⠀⠀⠀⠀⣤⢤⡀⠀⠀⠀⠀⠆⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⣀⡤⠂⠉⠁⠀⡀⠠⠀⠐⠳⠆⣀⠀⠀⡇⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⢀⠄⡜⠱⠶⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢄⡄⠤⣠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠖⠉⢶⠞⠁⠀⠀⠀⠀⠀⠀⠡⡀⠀⠙⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⠦⣴⡟⠠⠄
⠀⠀⠀⠀⢀⠀⠂⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠆⠀⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⡈⣤⠀⣤⣄⡂⠀⠀⠀⠀⠀⢀⢤⠔⠳⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢰⠀⠹⠀⢀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠓⢄⣰⠊⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⠀⠀⠀⢸⠇⠀⠀⠀⡰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠐⢼⣤⣆⡠⠖⠓⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠩⡷⣤⢏⠒⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠀⠀⠀⠀⣠⠃⠈⠀⠹⠒⠤⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠄⠀⠈⠉⠁⡨⠋⢹⠀⠀⠀⠀⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠑⠀⣰⠁⠀⡀⢧⠀⠀⡜⡟⠲⡄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⣧⠀⡈⡐⠀⠑⠊⣀⣇⢀⡞⠁⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠈⠈⠙⣆⠀⡁⠀⠰⣉⠀⢈⡿⢽⠊⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠙⢧⣇⠀⠀⠀⢉⣟⠀⢨⠓⠒⢒⣃⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠘⠛⢄⡀⠀⢸⡘⠖⠁⡠⠊⠁⠀⠉⠲⣟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡇⣿⣄⠾⠟⠁⠀⠀⠀⢀⣟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣇⢙⣧⣶⠖⡀⠠⣤⠄⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⠈⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠐⡀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠁
⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠴
⠀⠀⠀⠀⢀⠀⠀⠀⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠈⠂⠠⠀⠍⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⡁
⠀⠀⠀⠀⠀⡄⠅⠂⠦⠩⠒⡐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠞⠀⠂
⠀⠀⠀⠀⠁⠸⠀⠠⠉⠓⣄⠸⠃⠨⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠚⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣱⡆⢀⠄⠐⠂⡄⢠⠀⣉⠀⠀⠀⠀⠀⡀⠁⠁⠀⠢⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⢟⠙⠀⠀⠀⠀⠹⠞⣀⡉⠄⣀⣠⣤⠜⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠊⠈⢦⡀⠀⢀⡁⠐⠀⠤⠔⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠄⢠⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⣀⠀⠠⢄⣀
⠀⠀⠀⢠⣄⣤⡾⣆⣾⣟⣉⠳⡖⠉⢁⡄⠀⢾⣠⣄⠀⠰⠤⣄⣉⢳⠉⢱⣮⡙⣮⣷⠀⠀⠀⢀⣀⣀⡴⠖⠲⢦
⠀⠀⠀⠀⠉⠛⠿⠛⠛⠻⣷⣶⣶⠎⣿⣧⣼⣶⡉⢉⠉⠀⠀⢺⣿⣼⡷⠘⢻⢳⣿⢧⠀⠀⣴⠋⠀⠋⢷⠀⡁⢸⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣋⡁⠙⢻⣿⣷⠍⣐⡎⡝⢀⠉⣾⢏⠙⢿⣌⡀⢸⠃⠀⠀⢄⣸⠀⠀⣺⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢶⣿⠋⠽⠿⠿⢉⡙⢯⣿⡿⠿⠏⢙⣻⡇⣹⡇⢂⣾⡀⣇⣿⠀⠀⠀⠀⣿⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⢀⣠⠤⣿⡗⠀⡞⠃⢀⣤⣷⣿⣿⣿⣶⣴⣦⠴⣶⠟⠿⠞⠛⠿⡋⠀⣗⣾⣿⠀⠀⠀⠀⣿⠀⢠⡇⢸⠇
⠀⠀⠀⠀⠀⣿⣯⣍⣈⣀⣽⣿⣾⣿⡟⣻⣿⣿⣿⣿⣷⣦⣶⣿⣶⣤⡤⠚⠻⠿⣶⠿⠃⣿⠀⠀⠀⠈⣿⠀⢸⡇⣿
⠀⠀⠀⠀⢀⣙⣋⡀⠀⣀⣯⣽⡁⢈⣻⣉⣋⠀⠙⠿⠦⠽⣿⡟⢓⣨⣶⣞⠕⢀⡴⠖⠶⣿⠀⠀⠀⠀⣿⠀⢸⡇⢿
⠀⠀⠀⠀⢸⡯⢥⡏⠀⢿⠶⢾⠁⠘⣷⠶⡏⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡿⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⢸⡇⢸
⠀⠀⠀⠀⢸⡇⠀⡇⠀⢸⡆⢸⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢸⡇⠀⠀⠠⢠⡇⠀⠀⠀⢸⡆⠈⠃⢸⡀
⠀⠀⠀⠀⢸⡇⠀⣇⠀⢸⡅⢸⡆⠀⡇⠀⡇⠀⠀⠀⠀⣀⣠⣴⣾⡿⠛⡏⠸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇⠘⡇⠀⡗⠀⣷⢠⣤⣶⣿⠿⠛⠋⠁⠀⠀⡇⢰⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣼⠀⠀⢸⡀⣸⠀⠀⣷⢠⡇⠀⢹⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣠⡴⠶⠶⠶⠒⠛
⠀⠀⠀⢀⣿⣀⣀⣈⣇⣿⣀⣀⣸⣸⣇⣀⣸⣇⣸⡇⠀⣀⣀⠀⣀⣀⠀⡇⢸⡇⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠈⠜⣻⡇⠀⢿
⠀⠀⣸⣟⣋⣉⣉⣭⣭⣽⣭⣭⣽⣭⣽⣯⣭⣭⣽⣧⠀⢿⣿⠀⣿⡏⠀⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇
⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⡏⠀⣤⣤⠀⠀⢠⣤⣄⠀⠀⣠⣤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⡇⠀⣿⣹⠀⠀⢸⣧⣿⠀⠀⣿⣼⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠀⠀⢸⠀⠀⠀⣀⣀⠀⠀⢸⡇⠀⠀⠀⠈⠛⠦⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⠇⠀⣸⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣯⢽⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠲⣞⣀⣠⡟
⠀⠀⠀⣿⣞⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣄⣀⣠⣧⣾⣄⣀⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠯⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠟⠛⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠶⡻⢟⢛⣻⢿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⣠⠶⣒⣿⣴⠿⠟⠻⢷⣼⣯⣛⢦⡀
⠀⠀⠀⠀⠀⢰⣿⣀⡀⠈⠁⠀⢴⣿⣿⣿⣿⣿⣷⣥⡥⣿⣧⠀⠀⠀⣼⢃⣾⠋⣡⡿⣶⢶⣗⡾⠷⣤⠙⣷⢻
⠀⠀⠀⠀⠀⠈⣷⡠⣤⣴⣤⣀⣼⣿⣿⣿⣿⣿⣭⣤⣜⣩⠏⠀⠀⠀⣿⣼⠇⣼⡟⢠⠊⠉⠉⠙⣦⣸⣧⣼⣿
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣧⡀⠀⠀⠀⣼⠛⣻⡋⢻⣣⠀⠈⠀⠀⡼⣽⢸⣿⠘⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⢻⡄⠘⢷⣌⣷⡛⠲⡖⠋⣹⡷⣾⣋⣴⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣟⢽⣿⣿⣿⣻⢧⣿⣿⣿⣿⠿⠿⠛⠁⠀⠀⠀⠀⠳⣤⣠⠿⣌⣙⠛⡷⠛⠉⣠⢏⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠻⣧⡺⢿⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⣉⣉⡿⣾⣛⡭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡁⠠⠀⡍⡳⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⢦⣼⡄⣷⣄⢸⡶⠋⠀⠈⢹⡄⠀⠀⠀⠀⢠⡞⠉⠀⠀⠙⢦⡀⠀⢂⣿⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠐⣴⣼⣿⣿⣿⣿⣶⣶⠀⢀⣼⠃⠀⠀⠀⠀⢼⣠⡤⠤⢤⡀⣸⣇⣀⣼⠋⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣈⣱⣤⣼⣿⡿⣻⡛⣿⣿⣷⠶⠿⢥⡀⠀⠀⠀⠀⡼⠡⢘⠌⡐⣽⠉⠀⠈⢳⣀⣄⣤⠟⠹⣆
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠐⣾⣿⣷⣬⣵⣿⣿⡿⠓⠀⠀⢹⡄⠀⠀⠀⡇⠄⠀⣨⠷⠿⣄⣀⣀⣾⠫⣀⠀⠐⣀⡿
⠀⠀⠀⠀⠀⠀⠹⢦⣀⣀⣤⢾⣿⣿⣿⣿⡏⠳⡶⠤⠶⠟⠀⠀⠀⠀⠙⢆⣰⠁⠀⠀⣸⠫⠍⠙⢦⣠⣅⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠘⠟⢹⡏⠋⠁⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢹⡔⠐⠀⠀⠘⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣰⠃⠙⢤⡀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠟⠶⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠈⠰⢧⡤⢣⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣋⠝⢃⠂⡀⠀⣀⣀⢴⡠⠀⠠
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠤⠤⠀⠁⠀⠈⣴⢤⡀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⡠⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠱⡛⠣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣉⣤⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠊⠁
⠀⠀⠀⠀⢀⠈⠉⠀⣬⠧⠀⠀⢤⡄⠀⠀⣠⡄⠀⢀⡀⠀⡀⠀⢤⡄⡄
⠀⠀⠀⠀⠀⠀⠔⠂⠉⢷⡀⠢⠾⠷⠫⠞⠿⠅⣤⢛⠉⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣄⣠⠴⠈⡈⠀⠄⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡋⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⣤⣤⣭⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠖⠒⢦⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠸⡄⠀⠀⢠⠀⠀⠐⠒⠀⣇⠀⠀⢀⡇⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⢸⣁⡇⢠⣄⠀⠀⠀⠇⠀⠀⠀⠁⠀⠀⠈⠇⠀⠀⠀⡠⠈⠓⠒⠋⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠰⠏⠀⠀⠀⣀⣠⡤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠖⠉⠁⣀⣀⣠⣄⣀⣀⠈⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⡄⠀⠀⠐⠋⢀⡠⠖⢋⣩⢇⣠⠤⠞⠈⠉⠲⢄⡀⠛⠇⠀⠀⢠⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⢸⢡⡞⡆⡇⠀⠀⠀⣠⢋⡖⠉⠉⠀⠸⠀⠀⠀⠀⢀⣀⣀⠙⣄⠀⠀⠀⢸⢸⢹⡎⡇
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⡇⠀⠀⣰⠗⠋⠀⠀⢰⣊⣁⣀⣀⣀⠀⠳⢤⣬⠷⠈⣆⠀⠀⢸⢸⢸⡇⡇
⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄⠀⠀⢠⢤⣸⠀⠀⢸⢸⢸⡧⡇
⠀⠀⠀⠀⠀⠀⢸⢼⡇⡇⣇⠀⠸⡇⠀⠀⢠⠞⠉⠉⠉⠙⠒⠚⠉⠀⠀⠀⢸⠆⢹⠇⠀⣸⢸⢸⡇⡆
⠀⠀⠀⠀⠀⠀⡞⢸⠃⢳⢸⡄⠀⢧⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⡼⠀⢠⡇⡞⠘⡇⢳
⠀⠀⠀⠀⠀⠀⠱⣌⣓⡋⠘⠀⠀⠈⢧⠀⠀⠑⠦⡄⠀⠀⠀⠀⠀⠀⠀⢀⡏⡴⠁⠀⠀⠣⣍⣛⡁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠈⢳⠀⠀⠀⠀⢀⡸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⡀⠉⠓⠦⠤⠬⠧⠤⠴⠚⠉⢀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣤⣀⣀⣤⡤⠴⠚⠉



⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢜⠈⠽⢦⢴
⠀⠀⠀⠀⠀⠀⠀⣜⡁⠀⠀⢈⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠦⣂⠶⢨⢚⠀⠀⠀⠀⠀⢤⡆⠀⡺⠻⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢢⢪⠢⡀⠀⠀⠀⡌⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠚⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡑⣄⠀⡸⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠙⣌⠲⠁⣰⠁⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠊⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢧⣈⢆⡸⠁⠀⠀⠀⠀⠀⠀⠈⢁⣄⣦⠄⡇⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡚⢧⣚⣉⣁⣄⠀⠀⠀⣤⡤⣉⠙⠃⣄⣧⡏⡇⠢
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⣴⣿⣶⡿⣿⣿⣿⣷⣶⣴⣶⣠⡈⢍⠀⠈⠐⠚⠛⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⣿⣿⣭⡀⠀⠬⠻⠿⠿⠿⣿⣿⢻⣿⣌⣤⡘⣿⣿⡾⠾⣯⣢⡀⠀⠂
⠀⠀⠀⠀⠀⡠⠁⠀⣴⣿⡟⣿⣿⣿⠃⠀⠘⠀⠀⢴⣿⣿⣿⠻⠿⢍⣻⣵⣿⣿⣷⣦⣸⣿⠑⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡁⠈⢠⣿⠀⠀⠀⠀⠤⠈⠙⢻⡟⠀⢀⣨⣿⣿⣿⡿⠿⠟⢛⡧⡠⢼⡄⠀⠘
⠀⠀⠀⠀⠄⠀⢰⣿⣿⣿⡇⠀⠈⠉⠓⠢⠤⠔⢒⠶⣶⣦⡀⠀⢈⣹⣿⣿⣿⣾⠿⠚⢋⣉⣠⡏⢰
⠀⠀⠀⠠⠀⠀⢻⣏⠙⠩⣶⣦⣤⣄⣀⡠⠤⠤⠅⠀⠛⠿⠟⢡⣾⣿⠛⠛⠉⢀⣴⣶⣿⣿⠏⠃⠀⡇⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠃⣿⣧⣤⣤⣤⣄⣀⡕⠀⠀⠀⠀⠀⠀⡀⢀⣿⣿⡿⠿⠟⠛⠛⠿⣟⣼⣿⠀⠀⢀⡇
⠀⠀⠀⢀⠀⠀⣇⣀⣭⣍⣻⣿⣿⣾⣇⠀⠀⣴⣶⣾⣿⣧⣽⣿⣭⣤⣶⣶⣤⣤⣴⣿⣿⡏⠀⠀⣿⡇⠀⠀⡀
⠀⠀⠀⠈⠀⠀⢹⣿⣿⣿⣿⢫⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣼⣿⠃⠀⢠
⠀⠀⠀⠀⠠⠀⠈⢿⣿⣿⣿⡆⠘⠿⣿⣿⣷⣶⣍⣅⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣶⠮⢹⣿⣿⣿⠛⠛⠀⠉⠛⠛⠉⠈⠉⠉⠉⠉⠻⠿⠛⢻⠟
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣯⣛⠿⡷⠀⠸⣿⣿⣧⣤⣀⠀⠀⠀⠀⢀⠤⠤⠄⠠⠤⢒⠄⣼⠋⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠻⣇⠀⠀⠀⠈⠉⠛⠿⠛⠉⢉⡓⠰⡧⠔⠒⠒⠖⠒⡫⠞⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠑⠠⢀⡀⠀⠀⠀⢀⣀⠻⣿⣿⡷⠀⣀⠤⠖⠉⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠉⠀⠀⠚⠛⠛⠛⠛⠈⠉⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⣻⣿⣭⣟⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣵⣾⠟⢻⢿⡿⡟⠻⢷⣮⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⢄⠀⠀⠀⢀⣀⣀⣰⣋⣾⠏⠀⠀⢈⣲⣖⡁⠀⠀⠹⣷⣹⣆⣀⣀⡀⠀⠀⠀⡠⣖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⣦⣶⡿⠟⠛⡛⠛⠛⠿⣷⣤⠺⡱⡟⢻⢎⠗⣤⣾⠿⠛⠛⢛⠛⠻⢷⣶⣴⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢻⣶⡢⡀⢃⢸⠀⠀⢸⣳⠏⠀⢎⡱⠀⠹⣞⡇⠀⠀⡆⡘⢀⢔⣶⡟⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠈⠪⣿⣿⡘⡈⣠⣜⣰⣯⡴⠞⠛⠛⠳⢦⣽⣆⣣⣀⢁⢇⣿⣿⠕⠁⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠈⣁⣐⡒⠪⡽⣻⠥⢭⣿⢏⣤⠲⣛⣛⠶⣤⡹⣿⡭⠬⣝⣿⠅⢒⣒⣉⠁⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣫⣽⣿⡄⠀⠀⣀⢾⡅⣇⣯⢟⡇⠿⠱⢏⡡⢌⡹⠎⠿⢸⡻⢿⣸⢨⡷⣀⠀⠀⢠⣿⣯⣝⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⣿⠋⠁⠙⣿⢶⣲⡴⣾⡿⢟⣩⡍⡆⠉⡉⢸⠀⠀⡇⢉⠉⢠⢩⣭⡻⢿⣶⢦⣖⡶⣿⠋⠀⠙⣿⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⣿⣁⡀⢀⢞⡢⠋⠁⣰⠏⢠⣟⡵⢦⣃⣈⣮⣿⡦⢴⣿⣷⡁⣘⡰⢮⣛⡄⠹⣆⠈⠙⢔⡳⡀⢀⣈⣿⣞⣧
⠀⠀⠀⠀⠶⡷⡷⠶⡷⣾⣿⡷⣸⢾⣿⣉⢎⡱⣿⠀⣿⢺⡁⢎⠀⠀⢈⣏⢾⡷⣹⡁⠀⠀⡱⢈⡇⣿⠀⣿⢆⡱⣋⣷⡷⡏⢾⣿⣿⢾⠶⢾⠾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⣿⡉⠁⠈⢮⠕⣄⡀⠹⣆⠘⣭⡳⠎⡍⣉⢿⣿⠗⠺⣿⡿⡉⢩⠳⢞⣭⠃⣰⠟⢀⣠⠮⡵⠁⠈⢉⣿⢯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⣿⣄⡀⣠⣿⠾⠽⢳⢿⣷⣮⣛⣃⠃⣀⣁⢸⠀⠀⣇⣈⣀⡘⣸⣛⣵⣾⠿⠞⠯⠷⣿⣄⠀⣠⣾⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣝⣻⣿⠃⠀⠄⠉⢾⡃⡏⣟⣎⡇⣶⡰⣎⡑⢊⣱⢆⣶⢸⣱⣻⢹⢘⡷⠉⠀⠀⠘⣿⣟⢫⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢀⣉⠩⠥⢔⣷⣽⡒⣚⣿⣎⠛⠼⣭⣭⠧⠛⣱⣿⣓⢒⣫⣾⡢⠬⠍⢉⡀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⢀⢔⣿⣿⢱⢁⠙⢫⠸⣟⠳⢦⣤⣤⣴⠞⣻⠏⡝⠋⡈⡎⣿⣿⡢⡀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣼⠿⠕⠁⡎⠸⠀⠀⢸⡽⣆⠀⢌⡱⠀⣰⢯⡇⠀⠀⡇⢁⠈⠪⠿⣧⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⠟⠿⢷⣦⣤⣥⣤⣤⣶⠿⠛⢴⡱⣧⣽⢎⡦⠛⢿⣶⣤⣤⣬⣤⣴⡾⠿⠻⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠊⠀⠀⠀⠈⠉⠉⠹⣍⢿⣄⠀⠀⢈⡽⠯⡁⠀⠀⣠⡿⣹⠏⠉⠉⠁⠀⠀⠀⠑⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡻⢷⣦⣼⣾⣷⣧⣴⡾⢟⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣽⣹⣟⣯⠷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⠿⣯⢧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣶⣮⣵⣶⣿⣛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢛⣽⣿⣿⣿⠟⣿⡿⠿⢿⣿⣯⣿⣿⣿⣿⠋⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣵⣿⣿⣇⣠⣼⣿⣿⣿⣿⣿⣯⣍⣉⣓⣲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠉⠉⠉⠙⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⣿⣫⣿⣿⣿⣹⣽⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢻⡟⠛⠉⠙⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⠁⠀⠀⠋⠁⠀⠜⠓⣿⢿⣿⡟⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠈⠁⠈⠁⠁⠘⢿⣿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠃⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⣽⡆⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⢀⣋⣀⣀⣀⠈⠉⠙⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡴⠛⠉⠉⠉⠉⢉⣭⣙⣿⣿⣿⣿⣿⠁⡛⠳⣦⡀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⣰⠋⡀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⡗⠀⢐⠈⢳⡄⠀⢳⡀
⠀⠀⠀⠀⠀⢰⠃⠀⣸⠃⢂⠐⠊⢐⢀⠀⠀⢀⣾⣿⣿⣿⣿⡟⣛⣸⣿⡀⢀⣆⠹⡆⠀⢳
⠀⠀⠀⠀⠀⡾⠀⢰⣿⠀⠀⣠⣶⣶⣶⣦⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣦⣿⠀⠘⡇
⠀⠀⠀⠀⠀⠇⠀⢸⣿⡇⠀⠻⢿⣿⡿⣿⣿⣾⣿⣿⣿⠟⠩⠀⠀⠀⠈⠈⡉⢿⣿⣿⡇⢀⢷⡤
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡀⠠⠀⠀⠀⠀⢀⠀⣾⣿⣿⡇⠈⠻⠃
⠀⠀⠀⠠⢾⡷⠀⠸⣿⣿⣿⡿⢛⠛⡉⠉⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣼⣿⣿⣿⠁⠀⡞
⠀⠀⠀⠀⠀⣇⠀⠀⢻⣿⣿⣿⣴⡙⠁⠣⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⣹⣿⣿⣿⠏⠀⣸⠁
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠻⣿⣿⣿⡿⠞⠔⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠙⢿⣿⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠈⠛⠿⣿⣿⠻⡟⠻⠛⣟⢻⣛⡽⠿⠋⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡀⠀⠀⠈⠉⠉⠉⠉⢉⠉⠁⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠶⠂⠰⢿⡒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠰⡒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣽⣾⣷⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡄⠛⠍⢙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⡀⣰⢀⠁⠁⠘⢧⠴⢦⣀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⠟⣄⠠⠁⢠⠀⣠⠃⠀⠙⣴⣯⣾⣷⣡⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣽⠷⠾⠓⠠⠼⣋⣁⣁⠠⠔⠉⠹⢿⠛⠭⡛⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⢀⠄⣀⡀⠀⣻⣗⣀⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠱⡐⠋⠉⠠⠆⠉⠿⢿⣃⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⢿⣀⠀⠈⣁⣠⡶⣶⣶⣏⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⡽⠿⠋⠻⣽⣟⣽⣠⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠲⣤⣶⠘⠟⣧⣷⣚⠁⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡿⣭⣀⡀⠀⠀⠈⢻⡄⠆⢹⡟⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠄⣈⡁⠒⢦⣆⣻⠈⠀⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠛⣷⣰⣿⣷⠶⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣈⣽⣿⡽⠐⡏⠉⠃⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡻⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⡈⠁⠈⠀⠀⠀⠀⡀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠞⠋⡁⠉⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⢰⠀⠀⠀⣶⣞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡐⠄⠀⠂⠀⠀⠳⠄⢄⡄⢠⣎⣥⡶⣬⡀⠠⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣰⣦⡴⠟⡽⣾⠟⣧⣰⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠲⡄⠀⣼⠁⣹⣇⣐⡈⠀⢀⡄⠵⠀⢸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⢀⠰⣟⢫⣿⠏⠹⠆⠀⣿⡀⠘⠃⣆⠉⠀⠈⠁⠀⠀⣿⣸⠇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣀⠀⢁⣾⡀⠂⠀⠀⠀⡄⠀⢤⠀⠙⢆⢠⡏⠀⠀⠀⣰⡆⣠⠛⡟⠀⢁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠐⣿⠾⠂⠀⠀⠀⠀⠀⠀⠀⠐⠦⠼⠖⣿⠿⠥⣀⠀⠀⠀⠈⢦⣤⣀⣀⡆⡺⣶⣤⣿⣿⡿⢿⣡⠄⠀⣠⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡧⠈⠀⢀⠃⠀⡠⠄⠄⠀⣠⣄⡀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠑⠠⡈⠁⠨⣽⠿⠻⠿⢷⡾⠀⠀⠀⠀⠀⠉⢱⣾⡻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠀⠰⡀⠀⠠⠊⠀⠐⠂⠄⠀⣨⡹⢢⡀⠀⠀⣦⣀⠀⠀⠀⠀⡠⣶⣞⣾⡤⢤⣄⠐⠂⢠⣴⣦⡀⠀⠀⠗⠀⠀⠉⠀⠭
⠀⠀⠀⠀⠀⣀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠁⣶⣄⠈⠙⠀⣰⣶⣆⣼⠅⠀⢻⠀⢐⣿⡏⣀⣰⣦⡋⣡⣾⠨⣛⠳⣦⠈⠳⢦⠀⠘⣻⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠨⠷⡄⡀⠈⣿⣿⣿⣧⡀⠀⢸⡋⠣⣦⡰⠀⠠⣅⠀⠀⠀⣸⡆⢀⣹⢋⡅⣭⣭⣷⡛⢟⡉⣤⣰⣎⠘⠐⠈⢡⠀⠈⡘⢹⣿⣷
⠀⠀⠀⣠⡄⠠⠄⣶⠥⠀⢻⣿⣿⣿⣟⣁⠂⢻⡲⠆⠈⠁⠀⠙⠷⣴⡖⢋⠃⢨⣵⣎⣿⠇⣸⣾⠃⣠⣿⣝⡁⣩⠄⠐⡦⣿⡀⠀⠀⠙⡿⠋
⠀⠀⠀⠹⠀⠀⠺⣧⣰⣷⣌⠛⠻⣿⣿⡯⠀⠀⣦⣄⣀⣦⣤⣀⠘⠭⠿⣽⣧⡹⡏⢿⣿⣌⡥⣾⠞⠋⡰⠋⠀⠀⠠⢀⣡⠟⠳⠀⣶⡀
⠀⠀⠀⠘⠀⢄⢀⣴⠹⠃⠈⠉⢺⣿⠋⢠⡷⠀⠹⡄⠉⠀⠈⠉⠐⣻⣿⠏⢸⣧⣵⢿⣭⢬⣿⣁⣠⡆⠀⠀⡀⣰⠗⢻⡷⠂⠀⣼⡟⠁
⠀⠀⠀⠉⠉⠉⠋⠀⠀⠀⠀⠐⣿⠁⠀⢻⣴⠄⠀⣤⣾⡿⠻⣿⢿⢹⣵⣯⣾⣿⢫⡲⡦⣴⡀⠈⠉⢄⣀⠸⠛⠿⠀⡼⢁⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁⠙⠀⠰⠃⠈⣠⣿⡷⣿⡟⠠⣝⡷⠾⠿⢶⡶⠛⠉⠀⠀⣰⠞⠀⣿⣿⣿⣿⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠔⠀⠀⢰⣿⣿⣿⠻⠁⠀⠉⠃⠀⠀⠂⢤⠿⠗⠋⠙⠁⠀⠀⠀⢨⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣶⣯⠀⠀⠀⣽⠿⠛⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡔⠀⢈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⡀⠄⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠧⠓⡐
⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠠⠂⠀⠰⠂⠀⠠⠄⡀⠀⠉⠀⢘⡀
⠀⠀⠀⠀⠀⠂⡀⠀⠄⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢢⠔⣶⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠳⡠⠤⣤⣈⡐⠄⠀⠲⣻⠆⡁⠈⠍⢲
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢐⡀⢠⣿⡿⠀⠁⢄⡀⠀⠖⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠂⠀⠀⣁⡀⠀⠀⠃
⠀⠀⠠⠀⠀⠄⠈⡀⡠⠛⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⡿⠀⠈⠄⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠐⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⡸⠀⡀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠠⢤⣵⣠⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠼⢿⠟⠁⠁⠄⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄⠀⠀⠀⢄⠀⠄⣎⠁⠀⣰⠛⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠰⠀⣈⠈⠀⡆⠈⣢⡴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣱⡶⣄⢀⡠⠂⠀⡀⠉⠹⡇⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠦⠙⢤⣤⣀⠀⠨⣿⣤⣿⡄⠀⠀⠈⠃⠖⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠈⠨⠹⠀⣠⠀⠘⠉⡀⠙⠄⣀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠠⠐⣿⠁⠂⠰⠀⠍⠁⠀⠀⠢⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣶⡆⠉⢡⣀⣴⡄⠀⢀⠔⢲⣶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡟⣿⣿⡦⡤⠄⠠⣼⢀⡎⠀⣠⠟⠒⠲⢬⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢸⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠁⡀⠀⠀⠀⢹⡿⣏⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⣝⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⡏⠀⠀⠀⣸⠇⣹⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢤⠈⣽⣿⣿⣿⡾⠚⢻⣿⣿⣟⣿⣿⡿⠀⠐⢾⡁⠀⠀⠀⠉⢳⡟⠀⣀⣼⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⢸⣿⣏⣿⣿⣿⣟⣠⣤⠬⣧⡀⠀⠀⠀⢈⣇⠀⠈⠙⣷⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⣦⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⡟⣾⣿⣿⢻⡏⠀⠀⠀⠀⢀⣤⡴⠶⠾⠿⢄⡀⠀⠸⣇⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠈⠙⠲⠾⠿⣿⣿⣿⠀⠀⢸⣱⣿⣿⠁⢨⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠂⠀⣿⢰⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠖⣀⡚⠚⢷⡤⣼⡇⠀⠀⢰⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⡟⠁⠀⠀⠀⠙⠃⣤⡈⠳⡀⠀⢸⣿⣿⣿⠀⢸⡷⠶⠶⠒⠚⠾⠅⠀⠀⠀⠀⢀⣦⡤⠟⠀⢠⡾⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡛⠀⡰⠀⠐⢤⠀⠀⠀⠁⠀⠙⣦⡼⠿⢿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠻⠛⠓⠀⠀⠀⢿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠾⢄⠔⠁⠀⣡⡀⠀⠀⠀⠰⣿⡀⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⡀⢀⣠⠇⠀⠀⠀⢀⠀⠀⠈⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠠⡖⠁⠀⠐⠀⠀⠀⠀⢹⣧⠀⠀⢻⠀⢸⡇⠀⠀⠀⠀⢀⡿⠚⠁⠀⠀⠉⠋⠉⠉⠓⣆⠀⠚⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠰⣀⠤⠐⡇⠀⠀⠀⠀⠿⠀⠀⠸⠀⢸⣿⡶⠶⠶⠶⣿⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⡇⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⣀⠀⠀⠇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠈⠙⠶⠦⠤⠤⠤⠞⠀⢠⣤⠟⢰⡆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠸⠟⣟⣹⡆⠸⣀⡅⠀⢰⠀⠀⠀⢸⡇⠀⠀⠀⢸⣦⠀⠀⠀⢀⣄⠀⠀⠀⢨⣿⡀⠘⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⡿⣿⣿⡇⠀⠉⠀⠀⢸⠀⢀⠀⢸⡇⠀⠀⠀⣸⠏⠙⠛⠛⠛⠉⠛⠚⠚⠋⠙⢿⣞⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠐⠧⣄⣹⠃⠀⢠⠀⠀⣼⣴⣾⣧⣸⡇⠀⠀⣼⠏⠀⠀⣰⣄⡀⡴⠖⠛⠳⣄⠀⢈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣄⣀⡜⠳⢤⡝⠀⢀⡄⠀⢀⠟⢿⣿⣿⣿⣇⣠⣿⣏⢀⣠⠴⠋⠈⢻⡅⠀⠀⠀⢹⣶⡾⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠻⢦⣾⠁⣠⡿⢀⢀⣾⣆⣼⣿⣿⣧⡟⠁⠀⠉⠉⠁⠀⢀⣰⣾⠃⠀⠀⢐⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣙⣣⣏⣴⣿⣿⣿⣿⣿⣮⣻⡿⡇⠀⢀⢀⣹⣷⡾⠿⠋⠁⠀⣰⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣾⣟⢿⡿⣿⣿⣿⣿⠃⣟⠒⠛⠋⣉⠀⠀⢀⣆⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠽⢿⣿⠟⠃⠀⠘⢦⣤⣴⣿⣤⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⡾⠟⠀⠀⢸⣟⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣀⠀⠀⠀⢸⣧⠀⠀⠀⣾⣿⣯⡀⠀⠀⠀⠀⢰⡿⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠷⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠿⣶⠄⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⡀⠀⠀⠛⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠻⣶⡾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⢀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢃⠠⠃⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠂⠄⠁⢡⠈⠀⡀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠘⡆⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣿⠃
⠀⠀⠀⠶⣦⣄⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠈⠀⠠⠃⠀⢤⡂⡀⢱⠈⢡⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⡶⠟
⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠘⣄⠈⠢⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⣤⠀⠀⣀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠐⢂⠈⠁⠀⠀⠀⠀⠀⠀⢠⡶⠷⢶⣄
⠀⠐⣿⣀⣠⡿⠀⠀⢀⠀⡇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣾⢠⡿
⠀⠀⠈⠛⠋⠁⠀⠀⠀⠉⢅⠸⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠤⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⡆
⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠛⠻⠇
⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⡡⡀
⠀⠀⣿⠸⣿⠙⡷⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠻⢶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠁
⠀⠀⠀⠀⢀⣴⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠸⣿⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⡶⣦
⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡄⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⣤⠀⠀⣀⡀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⣴⡿⠻⣶⡀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣴⡾⠃⠀⠀⠙⠛⣿⠛⠃⠀⠀⠀⣿⡀⠀⠀⠀⠹⣧⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠉⠁

⣿⣿⣷⣄⠀⠀⠙⣆⢷⡜⠓⠒⣏⣴⠨⢙⡪⠿⣿⣵⣿⣿⣽⡽⣿⢉⣿⣯⣭⡝⠃⠀⠈⡿⠁⣸⠀⠹⣷⠈⠻⠇⠹⠋⢿⡴⣆⣼⠟⠈⠴⡿⠁⠈⡛
⣿⣿⣿⣿⣿⣿⣯⣄⡀⠿⠛⠀⠀⠷⢦⠀⢹⡦⣺⡿⣥⣿⣿⣾⡿⡿⣟⣋⡯⣥⣰⠐⠒⢴⠆⠀⠤⢠⡁⠀⠀⠀⠀⠀⠘⣿⠏⠀⠰⡄⠈⠆⢀⡰⢯
⣿⣿⣿⣿⣿⣟⠻⢿⣍⡉⣶⣄⢀⠀⠈⠀⡘⠀⣡⢐⣦⣶⣿⣿⢧⣼⣬⣛⡹⠫⠍⠀⣀⣾⣷⡳⣴⡆⠁⠀⡀⠀⠀⠀⠀⠻⠀⢀⡤⠀⠹⡞⠉⣠⣮
⣿⣿⠟⠛⠛⣛⡷⢻⣿⠃⣛⢋⠟⠉⠀⠀⠀⠀⣿⠛⠳⠄⠈⠉⠀⠀⠈⠉⠉⡀⠀⠘⠯⠉⢹⡿⡹⢣⠄⣀⠋⠤⢀⠈⠓⠂⠄⠀⠀⣠⣶⡿⠞⠛⠁
⣿⣿⣿⣿⣿⣿⣀⣼⣿⡾⠿⣛⣤⣔⣎⠀⠴⠷⠖⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠟⠀⠀⣴⠂⠘⠃⠃⠀⣴⡄⢀⡀⠈⠢⢤⣦⡀⢠⠀⢀⡟⠀⠃⠤⢴
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣞⣛⡱⡄⡿⠛⠉⠁⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠞⢓⠦⡆⢈⣏⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⡏⢁⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⣶⢾⢚⣿⣼⡇⠀⠀⠀⢴⠆⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣾⣧⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⡿⠙⡇⣶⡾⣻⠿⣞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣼⡗⠀⠀⠀⠀⠘⢉⠁⠀⠁⠀⠀⠀⣀⠀⠉⠀⣀⠲⠚⠛⠘⣷⠐⠉⣹⠛⠀⢀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠵⠂⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⣤⠀⠘⠃⠄⠀⠀⠡⠴⠃⡟⠀⠀⠁⠀⠀⠀⠘⠻⣆⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡶⣢⡶⠖⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠲⠶⠙⢦⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⢏⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠚⠰⠮⠁⠀⠀⣀⣀⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣵⣛⠁⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠿⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣏⣾⡁⠀⠀⠀⠀⠨⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⣿⠆⠀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⢰⡹⠳⣷⡶⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠔⠏⢠⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⢘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⣐⡛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⢸⡶⣂⣒⠀⠀⠐⡯⡷⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠝⣰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠾⠿⠓⠂⣼⢊⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡟⢉⣽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠂⣄⢀⡀⠀⢿⠄⠙⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⠾⢸⡇⣾⢿⣻⣅⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠳⠶⣼⣿⣻⣤⣽⢿⠓⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢝⡢⠿⢟⠓⣡⣯⣟⡉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⠀⠙⡃⢿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠲⣏⡝⠁⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣀⡀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⢉⡴⠋⠸⠉⠣⣾⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⣤⣀⠄⠀⢠⡎⠀⠀⠀⠀⠀⠸⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⢆⠀⠤⢄⢠⡏⢀⠈⣻⠛⠁⠀⠀⡆⠀⠀⠠⠦⠤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⢄⠀⠀⠀⠈⠁⠀⢳⡷⠀⠀⠀⠀⠀⠀⢊⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⠂⠀⠀⢠⠀⠀⡐⠀⠀⠀⢠⡎⢧⣀⠀⠀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠰⣸⣂⣾⣁⠀⠀⠀⠀⠃⡌⢳⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠽⢏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠐⢾⡿⠟⠃⠀⠀⢤⣰⡴⠁⠘⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣢⣿⣿⠏⠀⠀⠀⠹⣄⠀⠀⠀⠘⣿⡇⣀⠄⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠢⡀⠀⠀⢀⡈⣳⣤⣄⡀⠘⢓⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠈⢓⡄⠘⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⠜⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡲⠚⠉⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠞⠁⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠁⠀⠀⠁⠀⠀⠀⣤⠀⠂⢰⡆⠀⠛⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢄⢠⠁⠀⠐⠘⢣⣄⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠑⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠠⠋⠀⠘⠲⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣠⠀⡠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣶⣿⣿⡿⠿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠃⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⠟⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠙⣿⡟⠁⠈⢂⠰⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠞⠀⠀⠀⠀⢀⡼⠋⠻⠀⠀⠹⠓⢄⠀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠁⠀⠀⠀⠀⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⡄⣸⠀⠀⠀⠀⠀⠀⠀⠀⡠⣱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠤⣀⣶⣾⣾⡅⡲⠇⢀⣀⡴⠀⡠⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⡹⢿⣿⠿⠋⠐⠀⠀⠙⠀⡜⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⣀⡴⠉⢧⡖⠘⠂⠀⠀⠔⠀⢳⣀⣤⡤⠂
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⣀⠀⡎⠀⠀⠀⠀⠘⢄⣀⠀⠀⠃⢀⡜⠁⠀⠀⠀⢀⡀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠐⢄⠎⠈⠑⢌⠀⠀⣀⣀⠀⠘⠘⣆⢀⠔⠋⠀⠠⣎⠀⠀⠀⠀⠨⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠁⠀⠀⠀⠀⠈⠓⠒
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⢰⢇⠀⠣⢀⡴⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠋⠙⠄⡑⢺⠁⢠⡀⠀⡴⠀⢠⠎⠁⠀⠀⠀⠀⠀⠊⠡
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠢⣄⡨⠓⠠⢀⡉⠁⠀⣰⠋⠀⠠⢄⠀⠀⠀⠀⠠⠀⠐⡠⠒⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠈⠁⢋⡏⠀⠔⠠⢪⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠈⠦⣠⠄⠼⡤⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⡀⠀⠀⠐⠀⠀⠀⢳⡀⠐⠤⣸⡴⢀⠀⠒⠛⠴⡒⠓⠆⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣠⠞⣡⠐⠀⠀⠂⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠞⠁⠀⠀⠀⠀⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⣛⣛⣒⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠻⣭⠀⠀⠀⠀⠈⠈⠉⠙⠓⠚⠋⣛⣶⣤⣀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⠷⠤⣄⡀⠀⠀⠀⢀⣠⣤⠞⠋⠉⠉⠀⠈⠛⢦⡞⢁⣠⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⠀⠀⢾⣫⡴⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⢸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠠⣶⣆⢀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠘⠓⠶⣤⣀⠀⠀⠀⢸⡇⢸⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠘⠿⠿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⡇⢸⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠾⠀⠀⠀⢸⡇⢸⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⠚⠁⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠀⠀⠀⠀⢠⠶⢾⡇⢸⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⡇⢸⡦⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⠀⠀⠀⠀⠀⠀⠀⡗⢸⡇⢸⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠛⠙⠛⢶⣾⠛⠛⠛⠛⠛⠶⣄⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢀⡼⠁⢸⡇⢸⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢠⡏⠁⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠙⢦⡀⠀⢠⡏⠀⣠⠞⠃⢸⠀⢸⣿⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢷⡀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⢳⣄⡼⠁⣰⠏⠀⣴⠛⠀⠘⠳⣄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠳⢤⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡉⠳⣄⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⡄⠀⠳⣄⣀⣀⡴⠋⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⡳⠶⠇⠀⡀⠀⠀⠀⢀⣤⠶⠚⠉⠀⠀⠙⢦⣄⡀⠈⠁⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⣿⡆⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡿⣇⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠋⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠛⠛⠛⠉⠉⠉⠙⠳⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⣋⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠹⢦⠴⠶⠶⠶⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡰⣠⠔⠠⠉⣀⣸⣷⣶⣶⣶⣶⠀⠀⢨⡄⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡉⢟⠉⠐⡀⢠⣾⡿⠟⠋⠙⠛⠿⠋⠀⠀⢿⣿⣟⠂⠀⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠠⢊⡈⠀⣒⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠄⡰⢀⠠⠿⣿⣧⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠰⠣⠉⠉⠀⠁⠂⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠋⠀⠀⠀⠀⠀⠈⠧⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⢴⡡⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⢀⣸⣿⣿⣿⣿⣿⡤⢹⡀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠈⠉⠉⠀⠀⠀⡅
⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⢸⡀⠀⠀⢰⣿⡏⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⡧⡀⠀⠸⣿⡧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢣⢹⣦⣄⣠⣄⡘⠒⠛⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣄⣈⣉⡸⠁
⠀⠀⠀⠈⡎⢿⣿⠿⠟⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣰⣿⣿⣿⣿⣿⣿⡇⢠⠃
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣄⣀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢠⠃
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣾⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠉⠆⣠⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠂⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⡐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣉⣉⡙⣁⠀⢈⠨⠀⡁⠀⠑⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠄⠀⠀⠐⠂⠈⠀⠀⠈⠒⠴⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⣛⣛⠿⠿⠛⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣠⣾⡿⠀⣴⣶⣶⣶⣶⣾⣿⣿⣶⡄
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⠀⠀⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠟⠀⠈⠻⠄⢀⣀⣤⣶⣶⣄⣴⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣵⡄⠀⠀⠒⠒⡒⠶⠶⠸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠛⠀⠀⢠⣾⣶⣿⠂⣴⣶⣾⣍⢃⣤⣬⡉⠿⣟⣋⡀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁⠀⢸⣿⣿⣯⠘⠷⠾⠃⣺⣿⣿⡿⠀⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⢉⣾⣷⣼⣿⣿⣿⣿⠁⠀⠟⠀⢀⣠⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡷⠉⢴⡄⣀⡀⠀⢺⣿⣿⣿⣿⣿⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠿⠿⠛⠒⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣷⡌⠿⠿⠻⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣾⣿⣷⣦⣤⣤⠀⠀⠀⣸⣿⣿⣿⡅⣰⠞⢶⡀⣽⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠙⠿⢟⣻⣅⠙⠳⠛⢠⣬⣟⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠛⢉⣉⠻⠟⣡⣴⣶⣶⡄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠛⣛⣟⡋⢰⣟⢙⣷⢈⣿⣿⣿⣿⠁⢨⡛⠻⣿⣿⡟⣿⣿⡿⠋⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⢠⣿⣿⣿⣯⣤⡙⡋⢁⣝⢿⣿⣿⣿⠀⣾⡇⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠉⠁⢰⡿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣄⡘⠿⠟⠹⣿⣿⣿⣿⡿⠇⠀⢠⡿⠁⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣮⣝⡻⢦⣄⠀⠈⠉⠀⠀⠀⠀⢠⡿⢁⣠⢾⣿⣿⣿⣿⣿⣿⣿⢟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠿⣦⡙⣧⠀⠀⠀⠀⠀⣴⠏⣠⢞⣵⣿⣿⣿⣿⠿⢛⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣸⡆⠀⠀⣠⠞⢁⡞⠁⠛⣉⣈⣭⠥⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣷⣄⠈⢁⠴⠋⠀⠀⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠁⠀⠀⠀⠙⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⡶⢶⠶⠶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⠶⠶

⡟⢿⣷⠀⠀⠀⣾⣿⣿⣿⡗⠀⠛⠛⢿⢿⣟⣿⣿⣿⣿⣁⣀⠀⠈⣿⣿⣯⣹⣷⡀⠀⠀⠀⠀⠀⠀⡿⡟⢻⣿⣿⢽⣧⣼⣣⣄⣀⣆⠀⢀⣀⡀⠀⠀⢀⡀⠁
⠀⠀⠈⠣⣄⣾⣿⠟⠋⡉⠙⠃⠀⠀⠀⠈⢿⣿⣿⣿⡿⢯⠈⢹⣴⣽⣿⣷⡾⣿⣇⠀⠀⠀⠀⢀⣼⣷⢿⣿⣿⣉⢼⣻⣿⣿⣿⣿⣿⣿⠿⣿⣤⣤⡤⣾⢁⠤⠤
⠀⠀⠀⠀⠸⡿⣯⣘⠂⠤⠘⠠⡴⠲⡤⢄⡀⢹⣿⣿⣾⣦⣵⣾⡿⣿⣿⣟⠻⣿⡇⠘⠀⠠⠻⢾⣿⣇⣴⣿⣿⣻⣿⣟⠻⣿⣿⡿⠳⢈⢁⠻⢿⣿⠰⠀⠛⢷⣿
⡷⣦⣤⡄⡤⢀⣯⣭⣿⡶⠖⠉⠠⠐⠠⣐⢴⣿⣿⣿⣿⠻⣿⡿⠀⠉⠃⠈⢶⡽⣧⡄⢀⢀⡀⠀⢬⣿⣿⣯⣿⡿⠳⠈⢡⣨⠈⣇⡠⣙⢻⢻⡿⢿⠟⢈⠁⣘⣿
⣿⣿⣽⣿⣵⣾⣿⣿⣿⣿⣷⣿⣷⣌⣠⡿⢏⢉⣣⠔⣿⠀⢸⡇⠀⠀⠀⠀⠘⣱⡯⠀⢿⣯⠀⢴⡛⠛⢿⡟⠠⡤⠆⠀⢺⠃⠀⠛⠃⠘⠂⣼⠞⠲⡀⠞⢣⣅⣿
⠛⠛⠋⠙⣾⣿⣿⣿⡿⢿⣿⣮⣙⣿⠻⢿⣾⣯⠉⣴⣿⣄⣼⣷⡄⠀⠀⠀⡸⠋⢁⠐⢪⣨⡔⠆⠻⠀⠀⢈⣀⢀⣠⡀⠘⡄⠀⢆⠀⠀⠑⢇⡘⠓⠷⣢⡏⢥⣿
⠀⠀⢀⠙⣿⠃⠻⡌⢻⣸⣿⣿⣦⢽⣯⢸⡟⠛⠻⢨⣄⣿⠿⡏⢿⡀⣠⣾⠁⠀⠈⠡⠜⣯⡉⣀⡿⠶⢚⣿⣿⣷⣾⡛⢦⡙⢰⡘⣆⠀⣾⢀⢡⡜⠒⠋⣴⣁⣸
⣄⠀⠸⢤⣿⣄⠀⠱⡄⢣⠌⠟⠻⡼⢛⠃⢦⣴⡀⣼⣯⣉⢴⡿⣦⢠⣌⣈⣤⣠⣜⠛⠋⣨⡉⠩⠛⢀⠀⢛⡀⠰⣬⣽⣷⣽⣿⡿⡾⣿⣝⣿⣿⠗⣯⠂⠉⠻⣿
⠻⠁⢿⣆⠛⣻⣖⠋⠳⣝⢛⣂⡉⣼⠠⠞⠈⠩⠀⢨⠎⢻⠄⣶⣯⣴⠁⢹⡿⣿⣿⣦⠾⠂⣠⡸⠆⢠⡀⠀⠀⢠⠔⠋⠉⠉⠙⠻⢾⣟⢓⣾⠇⠈⠀⠀⠀⠀⠈
⣄⣀⣀⡿⠓⠿⠿⣾⡷⠛⠛⢙⣓⢾⣷⣦⠋⢣⡄⢺⣷⢔⡀⠿⠿⠛⠲⡉⠻⢿⣿⣿⣀⠀⠉⢹⣷⣾⣥⣄⣚⠓⠂⠠⢀⣴⡀⣀⣠⣾⣿⣿⠀⠀⠀⠀⢠⣤⣄
⠻⠗⠉⠁⠀⠀⠀⠛⠀⠀⠀⠞⠈⠛⢻⣇⠉⠻⣴⠶⣤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⡀⣁⣼⣿⣿⣿⣿⣿⡋⢀⣑⡦⣽⣿⣿⢻⣯⠻⣿⠀⠀⠀⢲⡜⠋⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣾⣦⣽⣴⡁⠀⠀⢠⠂⢀⠆⠀⠀⡀⠀⠀⣿⣿⡿⠿⠿⢿⣿⣿⣿⣤⡤⢴⢾⣿⣿⣿⣶⣿⢮⣿⡧⡀⠰⣮⠁⠀⣤
⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⢀⠀⣠⡷⠖⠟⠻⢿⣇⠓⠀⠀⠀⠀⠀⠀⢸⣷⣬⠾⠛⠛⠊⠀⣀⠀⡈⠉⢫⣴⣾⣿⠛⠁⠻⡿⠈⠙⠓⠟⠀⠀⢻⠀⠀⣸⣷
⠀⠙⠳⣤⣀⠀⡀⢠⣇⣀⠇⢰⡉⠻⣏⡱⠶⠄⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡀⠀⢀⣀⣤⠁⠈⠄⠀⢀⣿⡟⠃⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⢟⠘
⠀⠀⠀⠈⠻⣿⣦⣾⣿⣿⡴⠻⠁⠀⠸⠀⠀⢀⣀⢼⣿⠀⠀⠀⠀⠀⠀⠀⢐⠏⢠⣶⠛⠉⠁⠈⠁⠀⠴⠶⠿⠿⣧⡀⣀⠀⠀⡸⢢⣸⣧⠀⠀⠀⠀⢀⣔⠁
⢀⡤⠂⠀⠀⣻⣿⣟⣵⡿⠀⠀⠀⠀⣰⡧⠀⠘⠉⠀⠈⢳⡄⠀⠀⠀⢀⣾⢳⣾⡿⠃⠀⡀⠀⠀⠸⠷⣦⣄⠀⢀⣀⠽⣿⡶⠋⢀⠼⠋⠙⠇⡀⠀⠀⢘⡿⡧⡀
⡿⠄⠀⠐⠚⠛⠉⠉⢸⠇⠀⠀⢀⣠⣿⡇⠀⠀⠐⠢⠶⢶⡿⣦⣨⠤⢨⡝⢜⣛⢶⡶⠋⠀⠀⠈⠀⠀⠀⠌⠀⠉⠉⢀⣈⣀⣠⣴⣾⡀⠀⠀⠈⣽⢾⠟⣅⠀⠉
⠢⠀⠀⢈⡁⠀⠀⢀⡜⠒⣠⠚⠋⠉⠙⠳⣀⠈⠳⡄⠀⠀⠓⠁⠣⢍⢕⡪⣝⠪⠷⢥⠄⠒⠀⠀⠀⢀⠴⠄⢀⡧⠀⠈⣩⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠠⡄
⣆⣀⣀⣀⣬⡾⠷⣷⣿⣆⠀⠀⠀⠀⠀⠀⣈⣿⡿⠛⠦⣀⣀⠀⠀⠀⠶⣙⣏⣟⠆⠚⠒⠂⠀⠐⣮⣗⠀⠀⠂⠀⠀⠠⣿⣿⣧⣄⠀⠀⢀⣴⣦⣤⡄⠀⠀⠃⣤
⣴⣾⠿⣧⣤⣶⣃⣿⡿⢿⣷⣶⣄⢀⡴⠚⠁⠈⠁⠀⠀⠉⠿⠷⠂⠤⡀⠖⠤⢠⠤⢤⣴⡶⠲⢾⣷⣦⣴⣾⣁⣠⣴⣾⣾⠏⣀⠁⠰⣷⣿⣿⣿⣿⣷⡀⠀⠆⢀
⣾⡿⠉⣿⠟⢯⡽⠁⠈⠀⠙⣿⣿⣿⠁⠀⠈⢀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠄⡀⠀⠀⠀⠀⠁⠀⠀⠤⠈⠉⠻⣟⢩⠟⡩⢉⡱⣿⠀⠀⣿⣿⠏⠉⡛⢿⣹⣶⠂⣤
⢛⠃⢠⠤⡌⠋⢷⠄⢀⠀⠀⣿⣋⡁⠀⠀⠀⠉⠁⠀⠠⢀⣾⡟⠋⠁⠀⠀⠈⠳⢦⣄⣀⠀⣠⣞⠃⠀⠀⠀⠉⣤⣳⣬⣦⣱⣾⣧⣰⣿⠛⡀⢀⣈⢁⢛⣿⣪⣅
⠈⠁⢸⠆⣤⣰⠮⣤⡘⣆⠈⠁⢹⡁⣴⠇⢀⡀⠀⠶⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢈⡿⠁⠉⠀⠀⠀⠀⠀⠀⣾⡟⢻⠛⠟⠿⣿⣿⣿⡄⠀⠀⢋⡸⣯⡀⢨⡙
⣄⢈⠻⣜⡦⠍⠦⠈⢳⣾⡏⠀⣊⡀⠀⠀⣈⠀⢀⠿⢍⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣷⠄⢀⠉⠹⠿⠁⢱⣤⡽⠁⠈⠙⡎⠁
⠰⣦⡏⡐⠄⣰⡈⢛⣶⣯⡰⢙⠋⢀⣤⡈⠁⠐⠄⣴⣿⣽⡿⠆⠀⠀⠀⠀⠀⠀⠀⣚⣏⡄⠀⠀⠀⠀⠀⠀⣠⠞⢹⣿⣇⣂⣨⡂⣄⠀⠀⢤⢿⡇⠠⣣⠄⠁
⠦⢈⣙⣷⣴⠿⠋⣨⣿⡦⢚⠛⠋⠀⠀⠈⠻⠂⢀⣼⣿⡿⠤⢀⠀⢀⣀⣀⣀⣠⠨⣿⣿⠀⠀⠀⠀⠖⠻⡛⠈⢀⡿⡋⣿⣏⣉⣙⣦⣄⣀⣀⣾⣿⣻⣗⡢⢠⡖
⠚⠉⠉⠻⢷⠴⣾⢡⠍⠁⢉⣠⠛⠒⠀⠠⣤⣴⠋⢻⣿⡧⣄⣀⠀⣼⣿⡿⠋⠀⠀⠀⣸⡂⠀⠀⢀⠀⢂⣀⠐⠚⠃⠁⠙⠛⡛⠿⣿⣿⠟⠋⠉⠙⠋⠀⡙⢾
⠂⠠⠴⡾⠃⠈⢋⠈⠠⡖⠉⠉⠀⠀⠀⡀⠙⠃⠄⠸⣿⣿⣿⣿⣿⣿⣿⣁⣤⠀⣀⣀⠙⣷⣰⣶⣯⡄⠻⠋⠀⠀⠀⠀⠀⠸⣛⢿⣿⡏⠀⡀⠀⡃⠀⠀⠘⠄⠃
⠀⠀⠀⠀⢀⡴⢰⡀⠀⠀⠐⠄⠀⠀⠀⠈⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⢿⣷⢿⡉⠀⣷⣿⣯⣄⠈⣷⣷⣤⡀⠀⠀⠀⢰⠘⢿⣾⣿⠃⢀⣵⡄⠀⠀⠀⠀⠈⠃
⠀⢀⣴⠁⠈⢁⣼⡇⠀⠀⠀⢰⡈⠢⡀⢀⣀⣄⣰⠿⠿⠿⠿⠻⠛⠿⢢⣄⠓⣼⣧⣞⣁⣦⣤⣠⣽⡛⢛⡉⠁⢰⠀⠀⠀⣻⡾⣌⡿⠗⠋⠙⠓⠀⡀⠀⣤⣄⠠