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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠛⠁⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢤⣤⣄⡀⠀⣤⠀⠀⢀⡠⣤⣤⠠⠤⠀⠀⠀⠘⣿⡿⠟⠛⠙⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠛⠁⠉⢀⡇⠀⠀⠈⠀⠉⠉⠀⠀⠀⠀⠀⠀⠁⣴⣋⠉⠲⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡟⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠠⠤⠄⣶⣶⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⣀⣀⣀⡀⣀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠉⠙⠓⠦⢄⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⣶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠟⠋⠁⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠒⠲⢾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⣿⠿⢿⡿⢻⡿⠛⣿⣿⣿⣿⡿⠶⠂
⠀⠀⠀⠀⠀⠠⠚⠙⠿⠿⣷⣤⣿⡾⠿⠷⠾⠿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠸⣶⠶⠶⠯⣴⠾⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠛⠛⠻⠿⢿⣿⡿⠿⠛⠉⠉⠉⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠠⠋⠉⠛⠳⠦⠀⠀⠀⠀⢀⣀⣀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⣀⣀⣀⣤⣶⣶⣾⣷⣶⣤⣀⣀⣤⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⣿⡿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠶⠒⠚⠒⠲⠶⢾⣟⡛⠻⢯⡛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⢀⣠⠴⠖⣒⣠⠤⠀⠈⠙⢦⡀⠉⠳⣄⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣴⠞⠋⣀⡴⠚⢉⣀⣠⣤⣤⣤⣄⡀⠀⠀⠙⢦⡀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢀⣴⠟⠁⣠⠞⢉⣤⠶⣿⣭⣭⡷⠽⠷⠄⠩⢤⣄⠀⠈⢷⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⢠⡞⠁⢠⡾⠃⣴⣟⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⣧⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⣰⠟⠀⣰⠟⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⣇⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢠⡟⠀⣰⠏⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠘⣆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢀⡿⠀⢰⡟⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡘⣧⠀⠀⠘
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢸⠇⠀⣾⠁⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠈⢷⡘⣧
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⠟⠛⠛⠻⡄⠀⣼⣷⡘⣇⢸
⠀⠀⠀⠀⣾⠁⠀⠀⠀⢠⡏⠀⣾⠁⠀⣿⠒⠚⠿⢿⣿⣿⣷⡶⠀⠀⠀⠀⢠⠎⣡⠴⠒⠲⢤⡀⠀⠀⣿⣿⣧⠹⣾
⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⡿⠀⢐⢼⢿⠳⡬⣕⠀⠀⠀⠀⢠⠇⢈⠴⠪⠙⡟⠳⠀⠀⠀⣿⣿⣿⡆⢿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢀⡏⠀⠀⡇⠀⠀⠈⠓⠚⠁⠈⠀⠀⠀⠀⢸⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⣿⣿⢸⡇⠸
⠀⠀⠀⣾⠁⠀⠀⠀⠀⣸⠁⢸⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡍⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⠁⣾⠁
⠀⠀⢀⡿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡌⢰⡏
⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣇⡾
⠀⠀⣸⠇⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠃⠀⠀⢰
⠀⠀⣿⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⢮⣙⡓⠒⠒⠒⣛⡽⠋⠀⠀⠀⠀⣰⠏⢠⡏⠀⠀⠀⣾
⠀⢰⡟⠀⠀⠀⠀⠀⢸⡇⢀⡟⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣰⠏⠀⣼⠁⠀⠀⢰⠇
⠀⣼⠇⠀⠀⠀⠀⢀⡾⠁⣸⠇⠀⠀⠀⡇⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⢠⡏⠀⠀⠀⣾
⢠⡟⠀⠀⠀⠀⣠⠞⠁⢠⡟⠀⠀⠀⢰⡇⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⡏⣿⠀⠀⢸⡇⠀⠀⠀⣿
⠾⠁⠀⠲⠚⢿⣥⣤⡴⠋⠀⠰⣄⠀⣸⠃⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠚⠉⠀⢰⡇⠘⣧⠀⢸⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠷⣼⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠁⠁⠈⠉⠉⠉⠉⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣾⣿⣿⡿⠛⠉⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠿⠿⢿⣿⣷⣶⣷⡆⠀⠀⠀⠀⠀⠈⣉⣭⡿⠖⣚⣳⣶⣦⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣾⣿⣿⡿⠿⣿⠀⠀⠀⠀⠠⣾⢾⣿⠾⢿⡿⢿⠻⣿⡿⠛⠻⢿⣿⣿⣿⡿⠻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⢿⣿⣬⣿⠗⠁⠉⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠢⢤⣿⣷⣾⠟⠁⠀⠀⢸⣿⣿⡟⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⡯⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⢀⣤⠤⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠙⣻⣿⣿⣷⣿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⠉⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣏⣈
⠀⠀⠀⠀⠀⠄⢀⣙⣿⣿⣿⣷⡸⠀⠀⠐⠿⠟⠛⠃⠈⠁⠀⠙⠿⡶⠿⠆⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣭⣿⣿⣿⣿⣿⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠃⠀⠀⠀⠀⣠⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣨⣽⣿⣿⣿⣁⡜⢳⣝⡀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠀⣀⣴⣿⣿⣿⣿⡏⠉⣓⣚⣛⡿⠿⠿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⠉⠁⠀⠀⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⠟⠉⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠙⠻⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠟⠁⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠙⠛⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⣀
⠀⠀⠀⠀⠀⠀⣀⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠀⠈⠁⠄
⣀⠤⠀⠐⠊⠁
⠉

⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠻⠿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⠿⠟⠛⠛⠲⢤⣄
⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠶⠶⢦⣤⣀
⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⢀⣘⣳⣾⠟⠁⠀⠀⠀⠀⣠⣴⡶⠢⢄⡀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⣤⣶⠖⠤⣀⠀⠀⠀⠀⠙⢿⣶⣚⣀
⠀⣩⠟⠁⠀⠀⠀⠀⠀⣸⣿⣿⣧⣀⣀⣷⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⣿⣿⣿⣄⣀⣼⡆⠀⠀⠀⠀⠀⠙⢯⡁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡐⡊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠳⢢⢩⠓⣿⣯⣿⣿⣿⣿⢿⣿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀
⠀⠀⠀⠐⡀⠐⠩⠀⠞⠾⡆⣰⣻⣿⡿⣿⠟⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⡁⢐⢐⠞⣴⢺⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠄⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⠀⠀⢀⣀⡇⠀⣧⣇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⡖⣆⠒⠉⢹⣿⣧⣿⣓⣿⣯⣽⣿⣿⠤⣿⣧⣠⠇⢠⣾⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⣁⣿⢺⣮⣧⣡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⣿⣿⣁⠀⣠⣾⡇⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣤⣿⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣏⣶⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣰⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢡⣴⠿⣿⣦⠈⠿⣛⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠋⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣴⣿⣿⣿⡿⠟⠉⣿⣿⡏⣩⠇⠸⣇⣀⣼⡋⠁⠁⣤⣾⣿⡇⠀⠀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⡿⠟⠁⠀⠀⠀⢻⣿⣯⣤⡶⣆⠉⡛⢛⢡⣴⡖⣺⣿⣿⠇⠀⠀⠀⣀⣠⣶⣿⣿⠿⠿⣟⡛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⢤⣀⡀⠀⠀⠀⠈⢻⣿⣿⣷⣿⣾⣿⣼⣾⣿⣿⣿⣿⣏⣠⣤⣶⢾⠿⠟⠉⠑⠀⠈⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠀⠁⠈⠩⠋⡭⠟⠟⡳⣻⣿⢻⣿⡿⢿⢿⡿⢿⠿⠿⠿⠟⢟⠉⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠁⠀⠓⠘⠂⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⣿⠤⣤⣀⡀⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡇⡼⣈⣥⣴⣿⡿⠟⠷⣛⢁⣴⣆⣽⣷⡶⣮⣿⠿⠛⢛⡵⠷⠀⣤⣴⠆⡤⢰⢲⢦⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⡏⢠⠏⠥⢐⡛⠓⠻⢿⡟⠉⣽⠛⠢⠤⠜⠆⡤⠤⢼⠓⡾⡭⠉⣠⠑⢢⢸⡟
⠀⠀⠀⠀⠀⠀⢚⣿⠃⠰⢣⡟⠛⠻⢻⠿⠿⣷⠺⣿⠟⠁⠀⠀⠀⢀⣯⣭⠙⠃⠚⢉⣰⠟⡻⠛⠓⣶⠦
⠀⠀⠀⠀⠀⠠⣾⣿⡇⠀⣾⠁⠰⡜⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣛⣂⡭⠋⣰⣗⠈⠀⣿
⠀⠀⠀⠀⠀⠴⣸⣿⠀⠠⣿⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⡀⠱⠎⢸⠇⠀⢸⡿
⠀⠀⠀⠀⡀⢈⣿⡏⢀⣀⣾⠀⠀⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠁⢸⠀⢸⠆⠀⢸⡏
⠀⠀⠀⠀⠵⠸⣿⡟⡡⢃⣿⣼⡥⢿⡾⠂⠀⢹⠂⠀⣤⣀⣠⡄⠀⣸⠟⠀⠀⠤⠤⣾⣦⢼⣧⡆⣿⣿⣦⡀
⠀⠀⠀⠠⠁⠠⠈⢿⠷⠿⠧⣤⡀⠄⠐⠀⠀⠉⠁⠀⠛⠛⠛⠁⠁⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠟⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠃⠁⠀⠂⠘⠛⠛⠁⠀⠀⠀⠀⠀⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣂⣀⣀⣀⣀⣉⡋⠁⢠⣤⣤⣤⣤⣴⣶⣶⠿⠟⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠛⠛⠛⠛⠓⠒⠂⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⠀⠀⠀⢠⣤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣽⣿⣿⣿⣿⣿⣿⠟⣻⡿⠿⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡤⠿⡋⠀⠀⢀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣼⣫⣾⣿⣷⢿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠟⠤⠀⠈⠛⢿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣵⣿⣿⣿⡿⠛⡷⠛⠛⠉⠉⠁⠀⠀⠀⣼⠃⣠⣾⡿⠿⠶⣄⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⠏⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⣀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡻⠁⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣠⣶⢶⣤⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⡤⠀⠞⠁⣴⠞⠉⠁⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣽⣷⡀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠁⠀⠀⠼⠁⣀⣤⣤⣤⣄⣰⣤⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⡄⢀⡾⠛⠻⣦
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢰⡏⠀⠉⠛⠿⣿⠿⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢻⣿⣧⠤⠶⠀⠀⠀⠀⠀⠀⢻⡿⠕⠁⠀⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⠷⠀⠀⠀⢸⣿⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⢶⡆⢸⠇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣾
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡽⠟⠋⠁⠀⢀⠀⠀⠀⠀⢀⣶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠻⣤⡼⠷⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠟
⠀⠀⠀⠀⠀⠀⣀⠈⣿⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⣼⠛⠛⠻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⢻⡄⣿⠢⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⢻⡸⡆⣸⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠷⣦⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠘⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠟⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣠⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⣶⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣭⣤⣤⣤⣤⣤⣀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣥⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣻⡿⢟⣿⡿⠃⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡝⢛⣭⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢦⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⠛⠛⠁⠀⠉⠐⠳⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠑⢦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠃⠀⢀⣀⡤⠔⠚⠁⢠⢤⠄⣀⠀⠃⠀⠀⢀
⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡤⠶⢒⠉⠁⠀⠀⠀⠀⠀⠈⠑⣅⠀⡅⡇⠀⠀⢸
⠘⡄⠀⠀⢀⡀⠀⠀⠀⢀⣀⡄⠀⠀⡄⠀⢠⠆⡰⠏⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠀⡰⠀⠀⢀⡏
⠀⠹⡄⢱⠀⠑⣄⠀⠰⡏⠀⡆⢀⡜⠰⡴⠃⡾⠁⠐⠄⠀⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠃⣀⢀⡞
⠀⠀⠙⡄⢳⡀⠈⠳⣄⠙⢢⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡰⠄⢠⡆⢀⠏⠛
⠀⠀⠀⢹⡀⢳⣄⠀⠘⡗⠦⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠒⣤⠏⣧⠞
⠀⠀⠀⠈⠧⠞⠉⠳⣄⡇⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⢂⣀⣡⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣈⣃⣀⣀⣀⣀⣀⣉⣙⣒⡒⠶⣤⣤⣤⣤⣤⣤⣤⡽⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⣀⡠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣤⣤⣤⣶⣤⣤⡀⠀⣴⣶⡾⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⢿⣿⣿⡇⠀⠀⢀⡾⠋⠁⠀⣀⠀⠉⠉⠁⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠰⣄⠀⠹⣿⡇⠀⠀⠀⠀⠀⠈⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⡆⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣦⣀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠛⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣦⡀⠛⠉⠉⠀⠀⢀⣠⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣦⡀⠀⠀⠘⠛⢁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣶⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠛⠛⠛⠛⠛⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣄⠀⠀⠀⠀⠀⠀⣠⣤⠶⠖⠒⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣠⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣼⠟⠁⣠⣴⡾⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡟⠁⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠛⢷⣤⣼⠏⢠⣾⣋⣀⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⣿⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠿⠛⠉⠉⠉⠀⠀⠀⠉⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣤⣿⣦⠈⠻⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠉⠈⠙⣿⠟⠛⠛⠛⠻⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⣿⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣴⠾⠿⣦⡀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⢀⣤⣿⣷⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠀⠀⣀⣴⣿⡇⢸⣿⠀⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⣠⣾⠟⠁⠙⠁⣾⡏⠀⠀⠀⠀⠀⠀⢿⣟⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢠⣼⠟⠁⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠿⠗⠀⠀⠀⣴⡏⢠⣴⠟⠁⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⡀⢀⠀⠀⣿⣷⡿⠃⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠛⠛⣻⣿⣿⠋⠀⠀⠀⠀⢀⣴⡿⢋⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣅⢀⣴⠾⠿⠿⣿⣶⣤⣤⣴⣶⣏⣉⠴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⣶⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣁⣀⣀⣴⡟⠁⠈⠉⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⡿⠋⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠀⠀⣴⡟⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⣼⡟⠁⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⣇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⡇⠀⠀⠀⠀⣾⡟
⣤⡾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣿⠀⠀⢀⣾⡟⠁
⠻⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢀⣿⣤⡾⠟⠁
⠀⠈⠛⠿⢶⣶⣶⡶⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⡇⣿⡇⠀⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢀⣼⡟⢀⣿⣇⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣦⣤⡀⠀⠀⠀⠀⣸⡿⠿⠿⠿⠿⠟⠋⠀⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣅⠉⠉⠙⠿⠆⠀⠀⢹⡇⠀⢀⣤⣤⠶⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠸⣷⡞⠛⣤⣴⢀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⣇⠀⣠⡿⣡⣾⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣧⡀⠀⠀⢀⣿⡿⠋⠶⠁⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣄⠘⢧⣄⣰⠿⠋⠀⠀⢀⣼⠿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢹⣷⣼⡿⢋⣴⣤⡀⢠⣿⠁⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠟⣥⢠⣾⠻⠏⢠⡾⠋⠉⠻⣟⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢀⣶⠋⣶⠉⢀⣰⠟⠀⠀⢀⣴⠛⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⣠⣾⣁⣰⡟⢀⣼⠃⠀⠀⣠⡿⠁⠀⢀⣼⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⢷⣶⡿⠛⠉⠙⠻⠿⣿⡁⠀⢠⡾⠏⠀⠀⢀⣾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣓⠈⠉⠛⢷⣦⣄⠀⠀⠈⢿⣷⣿⠃⠀⠀⣠⣿⠁⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣿⡿⢶⣿⣿⣷⣄⠀⠈⢻⡿⢷⠄⣴⡿⠃⠀⠀⢸⡟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⢀⣽⡿⠻⣦⡀⢸⡇⠀⣶⡟⠀⠀⠀⠀⠈⢁⡸⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢀⡿⠁⠀⠈⠻⣾⣅⣼⠏⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣀⣠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠾⠛⠉⠁⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠻⠿⠿⠷⠶⠶⠶⠿⠿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠉⠛⠛⠿⢷⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⠀⢹⡏⠀⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⡖⠛⠛⠛⢋⣉⣁⠀⣾⣇⠀⠈⠻⠤⣴⣦⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠾⠛⠋⠁⠀⠉⠷⠶⠶⠞⢉⣉⡉⠶⠾⠿⠟⣉⡰⠶⢄⣀⣼⠿⠛⠳⠶⢶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠖⠉⠉⠁⠀⠀⠀⠀⠉⠁⠲⠞⣉⣁⠀⠀⠀⠀⠀⠉⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⣤⣤⠶⠾⠿⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⢦⣤⣤⡀⠀⠀⠀⢹⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣷⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣟⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣌⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⠏⠁⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠿⣿⢟⣿⣿⣿⠟⢛⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣠⣾⣿⡿⢡⠞⣵⡿⠟⠋⢀⠔⡫⠟⣯⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣤⣼⠏⠀⠠⢀⠔⠀⣠⣾⠿⣛⣽⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⣿⢟⣽⢟⣵⡾⣋⣴⠿⣫⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡁⢀⣴⡿⢛⣥⣾⣿⠟⣻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣯⡾⢡⢋⣵⠟⣡⡾⢛⣥⡾⣫⠞⠋⠗⣛⣭⠶⣻⣿⣿⣿⣿⣿⣿⣿⣏⣴⡿⠟⣫⣴⡿⢟⣭⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣰⣾⠿⡏⠛⠻⣿⣦⣾⡇⡼⢋⣴⠟⣡⡞⣡⢊⣴⠿⣋⣵⣾⡿⢟⣻⣿⣿⠿⣿⣿⣿⣿⣶⠟⣋⣥⣾⢟⣫⣿⣿⣿⣿
⠀⠀⠀⢀⣾⡟⠻⡇⠁⠀⠀⠀⠙⢿⣿⣧⣿⣥⣾⣿⣴⣷⣿⣿⣾⡟⣩⣴⡾⢟⣋⣥⣶⣿⢿⣿⣿⣿⣿⣿⣟⣭⣶⡿⢟⣩⣾⣿⣿
⠀⠀⣠⡿⡹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠉⠉⠉⠙⠛⠻⢿⣷⣾⣿⠿⣛⣥⣶⢟⣫⣿⣿⣿⣿⣿⣯⣥⣶⣿⣿⣿⣿⣿
⠀⢰⡿⢇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣯⡶⢟⣫⣽⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⣫⣽⡿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿
⢀⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣩⡴⢞⣻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⣿⠤⠁⠀⣠⣤⣤⣦⣄⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠈⢿⡷⠛⠋⠩⠕⣒⣻⣭⣿⣿⡇
⠀⣿⠶⠶⠄⠀⠈⠉⠛⣿⣿⠃⠀⠀⠘⣷⣦⣤⣀⣠⣤⣶⣶⠿⠿⠟⠋⠉⠛⢻⣷⠀⠀⠀⠸⣿⡄⢀⣒⣫⡭⠽⣿⣿⣿⡇
⠀⣿⣦⣤⣤⣤⣤⣤⣾⣿⠃⠀⢀⠀⠀⠈⠛⠛⠻⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⣿⣏⠉⣩⣤⣴⣿⣿⣿⣿⠇
⠀⢸⡷⣬⣩⣟⣹⣿⣿⠇⠀⠠⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡾⠿⠿⣿⣿⣿⣿⣿
⢀⣿⠀⠈⠛⠿⢿⣿⡏⠀⠀⠆⠀⠙⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⢰⡿⢋⣤⠀⠛⠣⢹⣿⣿⠇
⢸⡇⠀⠀⠀⢀⣿⣿⠁⠀⣆⠀⠀⠀⠀⠙⠷⣬⣙⠛⠿⢶⣶⣶⣶⣶⣶⡿⠛⠋⠁⠀⠀⠀⠙⢱⣿⡧⢂⣄⣠⣿⣿⠏
⣿⠁⠀⠀⠀⢸⣿⠃⠀⠰⠈⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣼⣿⡇⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⣰⣿⣿⡿⠋
⣿⠀⠀⠀⢠⣿⡿⠀⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣼⠿⠛⠁
⣿⡆⠀⠀⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠁
⢹⡇⠀⣠⡿⣿⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠘⣷⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⢻⡇⠙⣦⣿⡀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠈⣿⡆⠙⣿⣧⠀⠀⠀⢀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⢹⣷⠀⠀⠹⣧⣀⣴⣿⣅⠘⢿⣷⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⣿⡄⠀⠀⠘⢿⡟⠳⡙⢦⡀⠙⢿⣬⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⢸⣧⠀⠀⠀⠀⠻⠄⠀⠀⣉⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⢰⣷⣾⣿⣿⣿⣶⣶⣿⡇⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⣾⡿⣿⢿⣿⣿⡿⠻⠋⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⢿⡄⠀⠀⢿⣿⣿⣿⣾⠿⠛⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠇⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠘⣟⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠻⣤⣀⣠⣴⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣴⣶⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣾⣿⡿⢉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣸⣿⣿⠁⡈⠀⠏⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣤⡀⠀⠀⠀⠸⠹⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⡆⢰⡁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡇⢠⣼⣿⣿⣿⣿⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣦⠠
⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣿⣿⣿⣷⣤⣀
⡌⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠺⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠀⠘⣿⣿⣿⣿⣿⣿⣍⣳⣦⣄⡀
⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣾⠿⠛⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠙⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⡳⡀
⠀⠀⠙⢯⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⡀⣀⣤⣴⣿⣿⣿⠏⠀⣰⣮⡟⢋⠝⠫⠟⠉⠛⣋⠀⢂
⠀⠀⠀⣨⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣍⠋⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣯⣶⣿⣿⣿⠷⠏⠀⠀⢀⡰⣟⠁⠚⢋
⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠋⠕⠛⠻⢿⣿⣿⢻⣿⣿⣿⢻⠇⢀⣠⠴⣂⡛⠰⠀⠀⠰⣿
⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⢻⡿⢿⠛⠋⠞⢰⣿⠃⣾⣿⠿⢿⣷⣦⡀⣽
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⢻⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣀⣼⣇⡼⠁⠿⠁⠂⠀⠀⠈⢻⣿⠈
⠀⠀⠀⠀⣌⣉⣉⣥⣤⣶⠒⠻⣿⡛⠿⣿⣿⠀⠀⢻⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⡴⠂⠀⠈⡐⠀⠀⠀⣸⡏⡄
⠀⠀⠀⠀⣿⡘⣿⣿⣿⣿⣷⡀⠹⣿⠃⠀⢙⡄⠀⠈⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣿⠿⠋⣀⣀⣀⣀⣴⣤⣶⣶⣿⣾⣷
⠀⠀⠀⠀⣿⡿⣾⣥⣌⠉⠛⢿⡆⢭⣄⠺⣿⣿⡄⠀⠀⠙⣧⣶⣎⢻⣶⣦⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢃⠔⠋⡍⠉⠛⠋⠉⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣧⣴⡤⠀⠀⠉⠁⠛⠓⠋⠓⠋⠉⠁⠀⠀⠀⠈⠻⠟⢸⣿⣿⣷⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⢣⡴⣣⣤⣤⡄⠤⣀⡀⠻⣯⡙⣿⣿
⠀⠀⠀⢰⣿⣿⣯⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣷⣤⢸⣿⣿⣿⣿⣿⣄⠙⢿⣦⡉⢻⠟⠁⠰⡾⣟⣷⣾⣷⣌⢻
⠀⠀⠀⢸⡿⠛⠛⠻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⢿⣧⠙⣿⣸⣿⣻⣿⣿⢿⣿⣇⡾⣿⣿⣆⡀⢀⡀⢈⢯⣼⡏⢿⣿⣿
⠀⠀⣰⣾⢿⡟⣫⠄⢀⡤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⡧⢈⣿⡏⡽⣿⣤⣈⣿⡿⠀⢸⣿⣿⣷⡈⣷⢸⣷⡽⣇⢸⣿⣿
⠀⢀⣿⣷⣟⡽⠧⢀⡌⢀⣀⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢠⣟⢻⡟⠰⡄⣿⣿⣿⣿⠃⠀⠀⠻⣿⣿⣷⡀⠘⡇⢻⣽⣎⣿⣿
⠀⢸⣿⣿⣿⠂⣠⣾⣷⣿⡿⠛⠁⠀⣠⣴⣶⣾⣿⣿⢿⣇⣀⠠⣶⣶⣿⣶⣤⣼⣟⣫⣩⣿⣿⣿⣿⡟⠀⣠⣤⣄⣉⣿⣿⣇⣠⣿⡀⢘⣿⣿⡏
⠀⠀⢿⣿⣯⡄⢹⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣉⣁⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠃
⠀⠀⠈⢿⣿⣿⡜⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠻⣿⣷⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠻⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢛⣿⣿⣿⠿⢻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣀⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣦⣴⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠀⣿⣿⣿⣿⣿⢟⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠛⠛⠋⠉⠉⠉⠙⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣖⠒⠲⠤⣄⡀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣈⣙⢦⡀⠀⠙⠪⣧⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡉⠛⠂⠀⠀⠈⠃⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠦⠤⣀⣀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⣂⡀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠎⠀⠀⠀⠀⡀⠀⠀⡀⠀⠙⠦⡀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢦⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠣⡀⠀⢠⡠⣄⡠⠴⠃⠀⢠⠖⠉⠀⢀⠲⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠆⠀⠀⠀⠀⠈⠒⢢⠀⠀⣘⡚⢣⠉⠋⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠄⠒⠂⢊⠔⠤⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢢⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣨⡘⡻⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠓⠒⠀⠀⠂⠬⣐⠒⢒⣛⢆⢳⡎⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢉⠆⠐⠎⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣯⠋⠉⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⡏⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣰⡟⡹⢽⡲⢒⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⠀⠒⠊⠉

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




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣀⡀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠉⠉⠉⠉⠙⠓⠲⢤⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠇⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠛⠛⠛⠿⣿⣿⣿⡆⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⢈⡙⠀⠀⢈⣡⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡿⡿⠀⠀⠉⠁⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⠀⠀⠀⠀⣀⡀⠀⡄⢠⣧⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⠛⠚⢃⣾⣿⣿⣀⠹⣦
⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠁⠒⠦⢴⣾⣿⣿⣿⣿⡆⠈⠇
⠀⠀⠀⠀⠸⠁⠀⢸⣿⣿⢿⣿⠟⠀⠀⠀⠀⠀⢸⢍⡙⠻⣿⡏⠀⠀⡘
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠙⠟⢆⠈⢇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⡸⠁⠀⠐⠶⣄⠀⠀⠘⢦⡀⠀⠀⠀⠀⠘⡈⢆⠈⡆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⠓⢄⠀⠀⠈⠢⡀⠀⠀⠀⢃⠀⡜⢧⠃⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⡸⢀⠴⠒⠒⠒⠢⢄⣀⠷⡀⠀⠀⠀⠐⢤⡀⡌⠀⢳⡜⠀⠀⠔⠀⡐⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⢧⣳⡀⠀⠀⠀⠀⠀⠈⢶⠱⡀⠀⠀⣠⣦⠙⠀⠉⠛⢣⠀⠈⠀⠀⠀⠠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠈⠛⣄⠀⠀⠀⠀⠀⠀⠢⠙⠂⠈⠀⠸⠀⠁⠀⠀⡸⢀⠀⠀⠀⡀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠐⠂⠀⠅⠈⢣⡀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠂⠀⣠⠔⠗⠁⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⢀⡆⠀⠀⠙⢄⠀⠀⠀⠀⠀⠐⡀⠀⠀⠠⢃⠜⠁⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠤⢪⠐⢤⡀⠀⠀⠑⠄⠀⠀⠀⠀⠈⠄⡠⠋⠁⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠑⢄⠈⠑⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⢀⡠⠊⠀⠀⠂⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣤⣤⣤⣤⣴⣶⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⢀⠄⠀⠀⣀⠐⠈⠀⠀⠀⠀⠀⠀⠁⠒⣅⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣦⣤⣄⣠⣶⣶⣶⣄
⠀⠀⠀⠐⠁⠀⠀⠀⠨⠆⠀⠀⠀⢀⠀⠀⠸⠀⠀⠀⠌⢂⠀⠀⠀⢠⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣿⡇
⠀⠀⠀⡇⠀⠀⠚⠀⠀⠀⠀⠀⠀⠜⠂⠒⠂⠒⠒⠀⠤⢈⡔⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆⢹⣿⠃
⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⢄⢘⠀⠀⠀⠀⣿⣿⡏⣿⣿⣿⣿⣿⣿⠉⠁⢀⠀⠀⠀⠀⠐⢡⠘⠁
⠀⠀⠀⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠘⠑⠀⠀⠀⠈⢙⠀⠀⠀⠀⠘⠿⣇⢿⣿⣿⣿⡿⠃⠀⠠⣿⡆⠀⠀⠘⠿⢾
⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⠀⠀⠀⠈⠀⠁⠀⠔⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢀⡀⠀⠀⢀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢄⡀⠀⠀⠀⠀⠐⠂⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⣈⠣⡀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣒⠖⠲⡒⠒⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡀⠀⠀⠀⠀⡈⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣶⣿⣦⡀⣱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⣿⣿⣿⣿⣿⠣⢌⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠘⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡌⢸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠈⠉⠁

⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡂
⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠛⠃⠀⠀⠀⠀⠀⠀⠐⣁⡐⠀⠖⢂⠀⠀⠀⠀⠀⠀⢀⢐⣚⠤⠖⢩⡄⠉⠀⠀⠀⠀⢀⠀⠦⣤⠀⠤⠖⣠⡄
⠀⠀⠀⠀⠀⠳⠖⠀⠀⢴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⠉⢁⡀⠀⠀⠀⠀⠀⠈⠁⠀⠑⠄⠀⣀⠀⠀⠀⠀⠈⠉⠁⠀⢘⠁⠉⡀
⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠠⡄⠀⠀⠀⣀⡏⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣀⣠⡏⠀⠀⠀⠀⠀⢄⠀⠿⠎⠠⠜⠟⠁
⠀⠀⠀⠀⠙⠀⠀⠀⠀⠘⠋⠉⠉⠉⠀⠀⠀⠀⠀⠹⠶⠾⠿⠟⠀⠀⠀⠀⠀⠀⢀⡽⠟⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠼⠁
⠀⠀⠀⠀⢀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣤⠶⠄⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠿⠍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠶⠶⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣴⣤⠀⢹⢿⣦⡰⠀⠀⠀⠀⠀⠀⠀⣀⡀⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠰⠆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣴⠀⢈⣉⣄⣀⣍⣉⠁⢦⡀⠀⠀⠀⠀⠄⠛⠥⢁⠠⠉⠁⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠙⠛⠛⠛⠋⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠴⠛⠛⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠐⠛⠆
⠀⠀⠀⠀⠠⠶⠃⠀⠀⠘⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠
⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⢀⣀⣀⣍⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠭⠉⠿⢠⠀⠀⠀⠀⠀⠀⠰⣿⣶⣶⣷⠄

⠀⠀⠀⠀⠰⢦⡀⠀⠀⢀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠛⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠘⠇⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠷⠀⠀⢠⡄⠀⠀⠀⠀⠀⠰⠛⣉⠀⠀⠛⠳⠦⠀⠀⠀⠀⠀⠀⠈⠁⣄⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠐⠀⠀⠐⣂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠈⠁
⠀⠀⠀⠀⢰⡿⠀⠀⠀⠐⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⣀⣀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠗

⠀⠀⠀⢀⡤⠒⠚⠉⠉⠙⠒⠲⢤⡀
⠀⢀⡜⠁⠀⠀⠀⢀⡤⢤⡀⠀⠐⠈⢣⡀
⠀⢸⠀⠀⠀⠁⣰⠃⠀⠀⠙⢦⠀⠈⠆⢻⡀
⠀⢸⠀⠀⠓⠒⠉⠙⡄⠀⠀⠘⡆⠀⠀⠀⢧
⠀⠘⢆⠀⠀⠀⠀⣀⠟⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠈⠙⠊⠉⠉⠀⠀⠀⠀⢀⡇⠀⡀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡂⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⡠⠖⠋⠓⢦⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢱⠀⠀⠀⠈⠧⠐⠒⠒⠉⠋⠀⠈⠉⠉⢙⠒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠫⠭⠍⠉⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠀⠀⠀⠀⢀⠀⠈⠈⠓⢤
⠀⠀⠀⠠⠖⠒⠊⠀⠀⠁⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠱⣄
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠔⠣
⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⡀
⠀⠀⣀⠞⠀⠀⠀⠠⠠⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡
⠀⠀⠣⠶⣈⣩⡅⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⢤⡚⢢⡀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⡖⠚⢀⠀⠀⠙⢯⠁⠒⠒⠒⠒⠒⠒⠒⠹⠁⠀⠱⠀⢀⠼⠀⠙⣆⠀⠀⠘⡆⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⣀⣤⡴⢥⣤⠀⠀⣔⠁⠀⠀⢀⢸⡆⠀⠃⠘⣉⣁⣀⣀⣀⣀⠈⠑⠀⠄⠠⣄⣀
⡠⣀⣠⠖⠉⠒⠤⠤⠔⠀⠂⠤⠇⠄⠈⠀⠠⠞⠒⠒⠒⠒⠶⠶⠒⠋⢀⣠⣴⣆⠀⢀⠀⡥⢀⣈⣉⡡⠤⠻⠀⠀⠠⠇⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀⣀⡀⠉
⢧⠄⠒⠒⠉⠁⠒⠒⠒⠀⠒⠀⢖⣦⣦⠤⠜⠈⠁⠂⠉⠑⢲⠄⡠⠚⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡤⢂⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡴⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⡰⠉⢀⣀⡀⠈⠄⠀⠀⠀⠀⠘⠀⢀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⠁⢀⠔⢀⡆⠀⠀⠀⢀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⡄⠈⠉⠁⠀⠀⢀⡀⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⠈⠑⠂⠀⠀⠂⠁⠀⢀⢠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣄⣄⣀⣀⡦⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠋⠉⠉⠉⠉⠙⠛⠛⠿⠟⠛⠉⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⢠⠟⠉⠛⠻⢶⣤⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣷⣶⣤⣤⣶⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⣿⣿⠿⠿⠟⠛⠉
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠊⠉⣉⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠐⠢⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⠿⠿⠿⠛⠛⠛⠋⣉⣉⡉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⢙⣛⠶⣦⣀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⣴⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⡠⠴⠚⠋⠉⠉⠉⠉⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠆⡐⠀⣰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠳⠤⠖⢊⣉⠉⠉⠲⣄⠀⠠⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠁⣰⣿⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠊⠉⠉⠉⠉⠲⣌⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⢀⣿⡏⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣄⣀⣀⣀⡠⠜⠃⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡆
⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠁⠀⠀⠀⢀⣠⠞⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⠇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠋⠛⣋⣉⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⡠⠂
⠀⢀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⡄⠂⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣋⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣴⠟
⠀⠈⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⠞⠁⢀
⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⣀⣤⡇⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠕⠊⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⣴⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠄⠈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠷⠶⣦⣀⠀⠀⠀⠸⣆⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢷⣦⣀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠈⠛⢷⣦⣘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠿⠛⠉⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⢠⣶⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠙⠿⠿⠛⠁⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡻⣿⣿⡖⣷⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡷⣿⣿⡟⡁
⠀⠀⠀⠀⠀⠀⠰⣦⢹⡟⠛⢿⣿⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣿⣿⠟⠻⣟⢥⠶
⠀⠀⠀⠀⠀⠀⠛⢶⠟⠀⠀⢸⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⠀⠀⠹⣶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠶⠦⠽⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠙⠻⠿⡿⠿⠿⠿⠷⠶⠶⠒⠚


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣶⣦⣤⣤⠀⠀⣠⣤⣴⣶⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣉⠙⠛⠻⠟⠛⠿⠛⠛⢉⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣤⣤⣤⣤⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢆⣉⣉⣩⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠶⢛⣫⣥⡴⠦⣬⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⡴⠞⠋⣉⣤⡶⠶⠟⠛⠛⡿⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣡⠞⠋⣀⣴⠟⠋⠁⠀⠀⠀⠀⠀⡇⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞⠁⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⢀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠙⢿⣿⡆⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⡀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⣸⣅⣠⣤⣤⣼⣿⡿⠿⣿⣿⣿⠿⢿⠟⣿⠟⠛⢿⣿⡿⠟⠋⢻⡟⠋⠀⠀⢀⡿
⠀⠀⠉⠉⠉⢹⠁⠁⠀⢉⠀⠘⣧⡀⠈⠉⣁⣴⠏⠀⡟⣧⡀⠀⠉⠉⣀⡴⢻⡇⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠉⠛⠉⠀⠀⠀⠰⢄⠙⠛⠚⠛⢉⡴⠋⠀⠀⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⠁⠀⠀⠀⠀⠀⠀⣼⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠘⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠁⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢹⣿⡛⠛⠛⠷⢶⣤⡀
⠀⠀⠀⠀⠀⢀⣠⣤⢤⣾⣿⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠘⣿⣷⠶⣦⣀⠀⠙⢿⣦
⠀⠀⠀⣤⠟⠋⠉⣀⣠⣿⣿⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⣿⣿⠀⠈⢻⣆⠀⠀⢿⣇
⠀⢀⣾⠏⠀⣠⡿⠋⢹⣿⡏⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⢈⣿⠀⠀⢸⣿
⠀⢸⣿⠀⠀⣿⠀⠀⣾⣿⠃⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣤⣤⣴⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣸⡟⠀⠀⣾⡟
⠀⢸⣿⠀⠀⢿⡄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⠟⠀⢀⣼⡿⠁
⠀⠀⢿⣇⠀⠈⠻⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣠⣴⣿⠟⠁
⠀⠀⠈⢻⣧⣄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠻⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⣶⣴⡾⠋⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⣀⣀⢀⣀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣴⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤
⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣴⣶⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣠⣤⣤⣤⣤⣤⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠛⠛⠛⠛⠛⠀⠀⠀⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⣀⠀⢀⣀⠀⠄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠈⢷⣦⣤⣀⣀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢈⠟⠛⢛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⣶⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⢿⣑⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⠟⠛⠋⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⡄⠀⠀⢰⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢧⠀⠀⡎⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⢤⣄⣀⣀⣠⠷⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢯⣉⠙⠛⠳⠦⣤⡀⠠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠳⣦⡀⠀⠀⠙⠳⣜⢦⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣟⡓⠶⢽⣦⡀⠀⠀⠈⠙⢧⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠥⠆⠀⠀⠈⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⠖⠋⠙⠲⢄⠀⠀⠀⠤⠄⠈⠙⠦⣄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣰⠁⠀⠀⠀⠀⠈⢳⡤⠚⠑⠲⡀⠀⠀⠈⠑⢦⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⣄⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⣄⠀⠀⠀⢀⡀⡇⡀⠀⠀⣰⡃⠀⠀⠀⣠⠴⠒⠈⠁⢠⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢈⡒⠤⣤⡼⠟⠛⣿⠞⠉⠁⠀⠀⠀⣜⣀⡤⠤⣄⣸⢦⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⣄⠀⠠⢀⠂⠀⠀⡔⠊⠙⢢⠀⠀⢀⣈⣉⠙⢧⠀⠁⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣆⠀⠀⣠⠖⢤⠤⠧⠦⠴⢃⡴⢄⡀⠉⠈⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠉⠑⠋⣠⠋⠀⠀⠀⣠⠏⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢊⠑⡄⠀⠀⠀⠀⠀⢹⣀⣀⠀⣧⡀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠙⠁⣄⠈⠀⠘⠲⣄⠀⠀⠀⠀⠀⠈⡆⢣⠘⣆⠀⡼⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠦⣤⢤⡤⣄⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸⠘⡴⡍⠋⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣃⣸⡀⣘⡂⠉⣉⠑⠢⢄⣀⠀⢀⣘⣯⠇⢱⢄⡴⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢬⣉⣁⠠⣴⣸⡈⠿⢆⠀⡐⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡷⠐⠚⠷⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⡀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢸⠙⠉⠉⠉⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣸⠗⠀⠀⠀⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢺⣤⣤⣧⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠰⢭⡃⡴⠧⠭⡿⣉⣉⡩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠥⠭⠭⠥⠤⠕⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠗⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠈⢻⣿⣿⣿⠟⢀⣤⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⢸⡇⠉⠀⠀⢸⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠟⠀⢸⣷⣷⣴⢀⡈⠻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⣭⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠛⠿⠛⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣄⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⣿⣯⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠏⠀⠀⠀⠀⣼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣤⡀⠀⢰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠇⢀⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⢿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⣿⣿⣿⣿⣿⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣾⡿⠃⠀⠀⠀⠈⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠀⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠴⠒⠒⠚⠒⠒⢦⣄⠀⠀⣠⠔⠒⠚⠙⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠉⠙⢿⡿⠋⠉⠉⠉⠁⣹⣇⠀⠘⠻⠋⠈⠉⠉⠉⠈⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢀⣴⠏⠻⣆⡀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣤⣤⠶⠛⠁⠀⠀⠈⠛⠶⣦⣤⡴⠖⠋⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠤⠖



⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⢿⡀⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⢦⠀⠀⠀⠀⠀⠉⠙⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣶⣤⣀⡀⠀⡘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡶⣿⣿⠿⠿⠿⠿⣿⡟⠲⢧⡀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠛⠉⠁⠀⠀⢰⣿⡡⠤⣤⣤⣤⣈⣿⡄⠀⠙⠓⠾⣆⠀⠀⠀⠀⠛⠦⣤⣤⡀
⠀⠀⠀⣠⣴⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠛⠿⠃⢀⣿⠃⠀⠀⠀⠀⠙⠶⣄⣀⠀⠀⠀⣤⠞⠁
⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢶⣦⣴⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⠳⣄⡀
⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⠉⠉
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⣿⣷⡄
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⢻⣿
⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣠⣾⡟
⠀⠈⠙⠿⢷⣶⣶⣤⣤⣤⣤⣴⣶⣶⢶⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠛⢻⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⣀⡤⠴⠶⠶⠒⠒⠒⠒⠒⠶⠶⠶⢾⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⡀⠀⠀⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡟⠀⠀⠈⢉⣿⡏⠉⢉⡽⡿⢯⠉⢉⡿⣏⠉⠉⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠁⢸⡿⠀⠀⠀⠀⢸⠇⠻⣤⠞⢠⠇⠈⣷⠞⠁⣿⠀⠀⠀⠀⣤⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀⠀⣿⠇⠀⠀⠀⠀⣿⠀⠀⠙⢦⣬⡤⠞⠁⠀⠀⢿⠀⠀⠀⠀⢻⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⡿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣴⠟⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡏⠰⠆⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⢰⣿⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⠀⠀⣸⡀⠀⠀⠀⢀⡇⠈⠁⠀⣀⣀⣸⣇⣀⣠⣤⣤⣴⣿⣾⡇⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⠿⢿⣿⠿⠿⠿⠿⠿⠿⠿⠛⢻⣿⠛⠛⠋⠉⠉⠉⠹⢷⣶⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠙⠉⠙⠛⠛⠿⣄⠀⠀⠀⠀⠀⠀⣠⠟⠛⠋⠉⢙⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠶⠶⢿⣀⣹⡆⠀⠀⠀⠀⢰⣏⣠⡷⠶⠦⣼⣤⡴⠿⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠁⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡄⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⣄⣀⣈⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⠉⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠚⠃⠉⠉⠉⠉⠙⠳⢆⣤⡠⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⡄⠀⡦⢘⡇⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⡩⠒⠙⢦⡀⠘⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⡇⠀⡷⣾⠀⠀⠀⠀⢠⠃⢀⣤⠀⠀⠀⠀⠠⠤⣄⠀⠀⠀⣎⣀⣤⡄⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⡀⠀⠀⢀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢄⠀⢧⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⠈⡇⠀⠀⢸⠀⠀⠀⢰⣶⡀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠁⠀⢻⡀⠀⢸⠀⠀⠀⠸⠿⠃⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣧⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠍⠁⠀⠈⣧⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠂⠀⠀⠀⠀⠀⢠⠃⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠙⣦⠀⠀⠀⣀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⡒⢶⣶⣤⡄⢹⡄⠀⠈⠓⢦⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣶⣿⣿⣿⡶⠚⣷⡗⠂⣄⠀⠀⠀⠀⠈⠙⠲⠦⢤⠤⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠜⠛⠿⠿⠿⠧⠤⠿⠛⠀⠈⢧⡀⠶⠶⠶⠶⠶⠶⣾⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡧⠀⠀⠀⠀⠀⠀⠗⠀⠈⠛⠲⠤⣄⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⡛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⣄⠀⠀⠀⣠⠾⠋⣠⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡈⠢⣈⣷⣶⣿⠥⠤⠶⣤⣥⣀⡈⢳⡀⢀⣀⣤⠴⠒⠋⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣀⣀⣀⣸⢃⣠⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢯⡉⠀⡀⠄⠂⠘⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠤⠀⠀⢀⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡀⠀⠐⣟⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠂⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣿⢣⡀⠀⢀⡤⠞⠉⠉⠀⣈⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⣧⡈⠀⠀⣯⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠄⠀⢀⠀⢸⣄⠙⢦⠎⢀⣠⠶⠛⠛⠙⠛⠳⢦⣷⠀⠀⠀⠀⠀⠀⣿⠑⠀⢀⣀⠉⠓⠦⣤⣀⡀
⠀⠀⠀⢠⡴⠚⠋⠉⢀⡀⠠⠄⠐⠂⠀⠉⠳⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠐⠤⠀⣠⠟⠃
⠀⠀⠀⠀⠙⢦⣀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣤⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⢀⣼⡇⠀⢀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⣼⠁⠰⡀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠆⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠶⠶⠒⠚⣉⣤⠞⠁⠀⠀⠀⠀⠀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢊⠵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡶⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠄⠘⣷⡀
⠀⠀⠀⠀⠀⠰⠯⣄⡈⠐⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠤⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢁⣠⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠙⣛⣿⡿⢯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠌⠀⠀⡀⠤⠀⠀⠂⠀⣼⠏⠀⠀⠀⢹⡄⠀⠀⠜⠑⠀⡀⠀⠁⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠐⠂⢀⣀⣀⣀⡀⠀⠘⠀⣿⣄⠀⠈⢀⣼⡇⠀⠊⠀⣀⠀⠀⠑⠲⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠖⠚⠋⠉⠉⠁⠀⢻⡄⠀⠐⠙⣯⣝⣛⣭⢿⣌⠀⠀⣼⠋⠉⠙⠓⠲⠶⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣴⠟⠉⠀⠉⠀⠀⠉⠻⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠦⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠠⢤⣄⣀⣉⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡁⠀⠀⠀⠀⠀⠀⠈⠛⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⢀⡤⠶⠀⠀⠰⣒⡒⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣀⡀⠘⠉⣰⣀⡀⠉⠉⢀⣿⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢞⠝⣩⠧⡄⢈⣉⡀⢀⠔⢁⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠁⠋⠀⢀⣹⠀⠶⠀⡨⠛⠉⠁⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⢣⠀⠀⠀⢸⣯⣴⠶⠾⡇⠀⠀⢀⡞⠳⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠀⢀⠜⠳⢤⣀⡾⢿⡁⠀⣠⠷⡄⢀⡸⢷⡀⠀⠀⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⡶⠛⢷⣤⡀⣩⡃⢀⡿⢯⡀⢀⣽⣅⣠⠾⠓⠶⣄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣠⡾⠋⠀⠀⠀⠈⢻⠉⢷⡾⢧⣠⡷⠛⠰⡏⠀⠀⠀⠀⠈⢧⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡎⢠⡾⠋⠀⠀⠀⠀⠀⠀⢠⡇⠀⠁⡼⢹⡇⠀⠀⢳⠀⠀⠀⠀⠀⠀⠳⣄⠀⢱⡀
⠀⠀⠀⠀⠀⠀⡼⣴⠏⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠇⠈⣧⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⢿⣶⣧
⠀⠀⠀⠀⠀⡸⠹⡿⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢸⠀⠀⣿⡄⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠈⢷⠹⣇
⠀⠀⠀⠀⢠⡇⠀⠁⠀⠀⠀⠀⢀⡤⠞⢹⡆⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⣿⠙⢦⣀⠀⠀⠀⠀⠈⠇⢹
⠀⠒⠒⠒⠚⢧⡀⠀⠀⢀⣠⣖⣉⣀⣀⣬⣧⣤⣤⠴⠧⠤⠤⠾⠷⠤⠤⠤⠷⠤⠤⠽⢷⣤⡀⠀⠀⠀⣸⣧⣤
⠀⠀⠀⠀⠀⠀⠙⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠟⠛⠋⠉

⣿⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠟⠛⣦⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣀
⠀⠀⠀⠀⠀⢀⠀⢊⣀⡂⢱⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡀⣈⡀⠌⠠⡀
⠀⠀⠀⠀⠀⠢⠀⠈⠁⠂⠊⠉⠉⢣⠀⠀⠀⡆⠀⠄⠐⢀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⢢
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠢⠄⣀⠠⠊⠀⠀⠀⠁⢸⠀⠀⠀⡀⠀⠀⠘⠤⠤⠤⠀⠒⢁⠄⠀⠀⢈
⠀⠀⠀⠀⠀⢾⠀⠤⠦⠤⠖⡇⠀⠀⠀⠀⠀⠐⠘⡄⠀⠀⡆⠇⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠼
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠢⠀⠀⡀⠀⢸⠀⠈⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠀⢐⠀⠈⠐⠁⠀⡆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⡀⠐⡤⠠⢤⠔⠃⠀⡄⠀⠀⠀⠘⠢⡤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠢⣀⣀⣀⣠⠄
⠀⠀⠀⠀⠑⠀⠠⠴⠀⠢⠀⠀⠘⠀⠀⠀⠀⠀⠂⠠⠬⠥⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠒⡄
⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠤⡀
⠀⠀⠀⠀⢐⡄⡄⠈⡀⠀⠀⠀⢀⢀⠠⠀⠀⣠⠖⠀⠾⡀⠀⠀⢀⠀⡂⠀⠀⠀⠀⡐⠡⣤⢠⡄⠘⡀
⠀⠀⠀⠆⠀⠈⠁⠀⡅⠀⠀⠀⠀⠀⠀⠀⢀⢉⠀⠤⠐⠋⡄⠀⢀⠁⠀⠀⠀⠀⠀⠅⡈⠉⠈⣣⠀⡇
⠀⠀⠈⠀⠀⠘⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⢈⠘⣷⣶⣶⡇⡇⠀⠁⠀⠀⠀⠀⠀⠀⡄⠈⠈⠍⢤⡀⡇
⠀⠀⢠⡀⣠⠂⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠉⠟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠤⠄⠠⡇
⠀⠀⢰⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠄⢘⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⢀
⠀⠀⠀⢠⠤⠤⠤⡔⠀⠀⠀⠀⠀⠀⠀⠁⠀⢦⠄⠀⠠⠤⡇⠠⠐⠀⠀⠀⠀⠈⠀⢒⢢⠤⠤⠤⢔⠃⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠐⠄⠊⠀⠀⠀⠀⠂⠀⠀⠤⠃⠀⠂⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠀⡀⢠⠀⡄⢠⠀⠆⢠⠀⠀⢠⠀⡄⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣇⣤⣤⣤⣤⣤⣄⣶⣴⣤⣤⣴⣦⣦⣸⣿⣆⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠉⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡀⢀⣀⣀⣠⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠉⠻⡍
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢻
⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡼
⠀⠀⠀⠀⢠⠏⠀⠀⠀⣿⠀⠀⠙⠦⣀⠀⠀⣀⡠⠖⠛⠉⠉⠓⢦⡾⠁
⠀⠀⠀⠀⢸⡄⠀⠀⣸⠏⠣⣀⠀⠀⣈⡷⠚⠁⠒⠀⠀⣀⣀⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠻⣤⣤⡏⠀⠀⠈⢉⡽⠋⠀⠀⣠⠤⠤⡀⡤⠒⠦⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠋⠀⠀⠀⢸⠁⣶⣂⣿⣥⣞⡄⣣⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠘⠦⣬⠿⠃⠙⣯⡴⠃⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢀⡤⠶⠒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠙⠒⣦⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠘⢦⡀⠀⠀⣄⠀⠀⠀⠀⢀⡀⠀⠀⢠⠏⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠛⢦⣀⢸⠀⠀⠀⠀⡏⠀⠀⣴⠏⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣤⣤⣽⠴⠛⠁⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠶⢤⣄⣀⣀⣀⡀⠈⠙⢋⣁⣀⣀⣠⣤⡶⠞⠻⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡆
⠀⢀⡤⠤⣄⡀⠀⢸⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠘⡆⠀⠀⠀⠀⠀⢹⡀
⡼⠁⠀⠀⠀⠙⢦⡿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣷⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⢀⡠⠶⠒⠓⠲⣄⠀⠀⠀⠀⠀⢀⠂
⣇⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⡷⠋⠀⠀⠀⠀⠀⠈⣧⠀⠀⢀⠔⢁⡞
⠸⣄⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢸⠁⠀⠀⠀⣼⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⢀⡟⣀⠔⠁⣠⠊
⠀⠙⣆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⣠⠞⠁
⠀⠀⣈⣳⣄⣀⠀⠀⠀⠸⡆⠀⢀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣀⡴⠒⠳⢤⡀⠀⠀⠀⠀⣠⠞⠋⢀⡤⠞⠁
⠀⡾⠁⠀⠀⠉⠉⠑⠂⠀⣷⠞⠙⠓⠀⠀⠀⠀⠛⢦⡀⣠⠖⠛⠀⠀⠀⠈⠛⣦⠀⠀⠀⠙⣦⣴⣚⣫⡥⠖⠚⠉
⠀⠻⠤⣇⠀⠀⢸⣇⣀⣰⢿⣄⣰⠂⠀⠀⢰⡄⠀⢸⡟⣟⠀⣴⠀⠀⢠⠀⠀⣿⢀⡄⠀⠀⣸⠏⠁
⣀⣀⣀⣉⣉⣉⣉⣉⣉⣀⣀⣀⣙⣓⣒⣒⣚⣋⣉⣉⣁⣉⣙⣻⣶⣶⣟⣛⣋⣉⣙⣦⣶⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⣻⣷⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣙⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣿⣿⢿⣿⠾⠟⠋⢿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣶⣶⣙⣻⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣉⠿⠟⣛⣯⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣯⣿⣿⣿⣿⡿⢿⠿⣿⣿⢿⣉⢛⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢹⡆⣿⣿⣷⣾⣿⣿⣻⣿⠿⣿⣿⣷⣯⡴⠀⢠⡽⣿⣿⣿⣦⣹⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⣿⡿⠙⣩⠷⠀⠉⣽⣿⡏⠙⢿⣿⣿⡟⠁⠀⣿⣿⡈⠙⠷⣾⡉⠀⠀⠉⢿⣿⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⣿⣿⠿⠛⠛⠋⢙⣋⣉⣠⣤⢁⣀⣀⣀⡛⠿⠿⣄⠀⣿⣿⡀⠀⠀⢛⣋⣡⣤⣤⡤⠤⢤⣤⣄⣀⠉⠉⣙⣻⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣷⢸⣿⣿⣿⣿⠋⣰⣿⣿⣿⣿⣿⣿⣦⣄⣀⣉⣉⣀⡠⠞⣡⣾⣿⣿⣿⣿⡀⠀⣰⣿⣿⣿⡜⣿⣿⣽⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣿⡇⢼⣿⣿⣿⠿⠿⣿⣿⣿⣿⡟⢻⣿⣧⣿⣿⣿⣿⣿⡇⢸⣿⣯⣀⣼⣯⢻⡟⠀⢸⣿⣿⠟⠃⣸⣿⣿⠟
⠀⠀⠀⠀⠀⠸⣿⣿⡇⢸⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣼⣿⣿⣿⣿⣯⡾⠃⠀⣰⣿⣿⡇⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⡸⣿⣿⣿⣂⠀⠘⠿⣿⣿⡿⠿⠛⣩⣟⠁⠀⠀⠄⠹⣧⠉⠉⠛⠋⠉⠁⠀⣠⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣏⡹⣿⣿⣶⣤⡄⠀⠀⠀⣠⡼⠻⣇⣤⡄⣀⢠⣤⡼⠷⣄⡀⠀⠀⠀⢾⣿⣿⠿⣡⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡷⠹⣾⣭⣭⣤⡤⠶⠒⠋⠁⠀⠀⠀⠙⢧⣿⣼⠃⠀⠀⠀⠙⠛⢶⣶⣾⣷⣶⣋⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⢿⣿⣿⣿⡆⠀⠒⠀⠂⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢀⠠⠍⣠⣿⣿⣿⡿⣟⣾⣿⣾⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⣿⣿⣷⣂⣱⡶⠊⠁⠀⠀⢠⣿⣧⡀⠀⠀⠀⣀⣀⣰⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⣿⣿⣿⡿⠿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣷⣄⠁⠈⠙⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠻⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣿⣿⣷⣦⣤⣤⣴⣦⣤⣶⣿⣿⣿⣿⠏⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⠛⣿⣿⡿⠛⠿⠋⢈⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠀⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠽⠿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⠕⠶⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⣠⣤⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣾⣿⣿⣿⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⠟⠁⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⠋⠀⠀⠀⠀⠘⣿⡆⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠇⢹⣎⣿⡏⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⣷⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣄⣸⣿⣿⠁⢠⣴⡶⠛⠛⠃⠄⠙⣿⣇⢠⡏⡀⠀⠀⠀⣼⠀⡀⡼⢸⠀⣿⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠈⣽⣶⠂⠀⣀⣀⣀⣿⡇⠀⢻⠃⣀⠀⣰⣿⣸⣷⡇⣾⣤⣿⣿⡇⠀⠀⡄⠀⢰⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠸⣿⣿⣿⣿⠿⠏⣸⣦⣤⣄⣾⣿⣿⡿⣿⣿⣿⢃⣿⣿⣿⣿⡇⣼⢠⣿⡇⢸⢀⠀⢀⢀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠈⠉⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣣⣞⢻⣿⡿⣼⣿⣿⣿⣿⡇⣿⣾⣿⣇⣿⢸⣦⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⣔⠂⠀⠀⠀⠀⠀⠀⠀⢠⣿⢹⣿⣿⣿⡿⠋⠴⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢋⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⢿⡟⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡦⠻⠶⡄⠀⠀⠀⠀⠀⠀⢻⣆⠘⣿⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⣾⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣀⣤⡾⣣⣾⣿⣿⡟⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠤⠶⣾⣿⣿⡿⣿⣿⣿⣿⡏⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⡟⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣿⠁⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠋⠀⠀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣧⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠛⠛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⢠⡶⠿⢷⣦⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠘⠃⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠒⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⠴⠒⢦⡄⢀⡴⠛⠛⠳⣄⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠠⡷⢤⣤⣤⣿⢸⡧⣤⣤⣤⢽⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⡉⠀⠀⠀⠀⢳⣌⣉⣽⠃⢈⣧⣌⣛⣻⠟⠀⠀⠀⠀⠘⢛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡙⠀⠀⢀⣴⠞⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠷⣦⡀⠀⠀⠀⠸⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠙⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⡇⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣷⣶⣶⣶⣾⠿⠛⠁⠀⠀⠀⣿⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⡇⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠀⣼⡆⣸⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⢷⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠛⠋⢰⠏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢻⡞⢩⠗⡞⠉⢷⠳⣄⣠⡤⠤⠤⢤⣄⣀⣀⣀⣠⠏⠀⢀⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣼⠀⡟⢀⡇⠀⢸⠀⣿⠀⣀⡤⠖⠋⠉⠀⠀⠀⠉⢀⣠⣾⠙⢎⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢿⣀⡇⢸⡆⠀⣼⢀⡿⢾⣥⣤⣤⣤⠶⠶⠶⠟⠛⠋⢹⡿⢦⣈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠙⠛⠳⠶⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠷⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡟⠀⠀⠉⠉⠉⠉⠉⠉⠙⣟⠉⣿⠁
⡴⢦⣀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠶⢶⣾⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠶⠿⠶⣶⣤⣄⡀
⢳⡄⠉⠛⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆
⠀⠙⠳⠶⠤⣤⡴⠶⠶⠶⠶⠶⣦⣤⣤⣤⣾⠇⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⢀⣀⣠⣼⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⠲⣤⣤⠒⠚⠉⠉⠈⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠛⠦⠤⠤⠤⠤⠤⠮⠽⠯⠥⢤⠭⠽⠤⠤⠤⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡴⠟⡋⠈⠉⠉⠈⠃⠒⠶⠤⢄⣀⠀⠀⠀⠀⠀⢀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠐⣿⡇⠀⢀⠴⠦⠀⠀⠀⠀⠤⣠⣭⠉⠙⢿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢷⣄⣰⢋⡙⠢⠀⠀⠰⢋⣭⢱⡀⡠⡿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⠀⠳⣝⣣⠜⡏⠈⠛⢬⠭⠞⡁⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠞⠃⠀⠘⠦⠖⠃⠀⠀⠒⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠠⢤⣄⣀⡤⠔⠂⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠈⠉⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⠉⠉⠓⠒⠉⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⡇⠘⢦⡀⠀⠀⡠⠃⣸⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠘⡄⠀⣹⢖⠚⡅⢰⠃⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠦⠀⠀⠀⠀⠀⠘⣤⣯⡙⢻⣧⡎⠀⠀⠀⠠⠆⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢾⣏⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡐⠄⢹⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡇⠐⢼⣈⡀⠀⠀⠀⢰⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⢦⠈⣿⡇⠀⠀⠀⠈⡇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡷⢄⠑⣼⡇⠀⠀⠀⠀⢹⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣇⠀⠑⣼⡇⠀⠀⠀⠀⠈⡄⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣭⠑⢦⣸⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠶⠶⠶⠶⠦⠾⠦⠶⢤⣤⣤⣷⣤⣤⣤⣤⣤⣿⣵⡶⠿⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠿⠶⠶⠖⠒⠒⠒⠒⠛⠺⠶⣤⡄
⠀⠀⠀⠀⠴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⢰⡶⢶⣦⠀⣀⡀⠀⢀⣀⣀⠀⢀⣀⣶⠀⢀⣀⠀⢀⣀⠀⠀⠀⣴⡶⣶⠄⢀⡀⣀⠀⠀⢀⣶⣶⣶⣶⢰⣶⣀⠀⢠⡶⠀⢀⡀⠀⣶⠆
⠀⠀⠀⠀⠀⠀⣾⣧⡾⠏⣾⠋⣿⠀⣿⠋⣿⢲⡟⢹⡟⢰⣿⣽⡇⢸⡟⠁⠀⢸⡿⢀⣿⠀⣿⠋⣿⠃⠀⠀⢁⣿⠁⠁⣸⡏⣹⠇⣸⡇⠸⣯⣙⠠⠟
⠀⠀⠀⠀⠀⠰⠿⠂⠀⠀⠿⠾⠋⠰⠿⠀⠿⠸⠷⠾⠇⠘⠷⠞⠀⠿⠇⠀⠀⠘⠷⠾⠋⠰⠿⠐⠿⠀⠀⠀⠼⠿⠀⠠⠿⠂⠿⠀⠿⠀⠷⠾⠟⠰⠂

⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⢀⡤⠶⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠉⠻⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⠛⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠘⢶⣤⣤⣴⣶⣿⣿⣿⣿⣯⣭⣩⡀⡄⢀⠀⠀⠉⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠋⠁⠀⢀⠀⣄⣨⣩⣭⣿⣿⣿⣿⣿⣶⣤⣤⣴⠞
⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⣾⣷⡤⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⢀⣴⣷⣾⣿⣿⣿⣿⣿⠛⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⢸⣿⡿⢻⣿⣿⣿⡿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⣿⣿⣿⢻⣿⣿⠀⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠻⣿⣿⣿⣿⠟⠁⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠙⢿⣿⣿⣿⡿⠃⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣭⣭⣤⣤⡤⠄⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠤⣤⣤⣬⣭⣵⡶⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⠀⠘⠃⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠌⠛⠛⠛⠻⠿⠿⠷⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⣀⡀⠒⠂⣠⣤⠿⠿⠿⣷⣄⠀⠀⠀⠀⠣⡀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠴⠉⠀⢀⣶⠏⠉⠀⠀⠀⠈⠹⣷⡀⠀⠀⠀⠉⣄⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣶⡀⠀⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠛⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠔⠛⠻⠟⠉⠙⠛⢻⡆⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠻⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣠⣤⣾⣤⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠉⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠁⠀⠀⠀⢀⠈⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠁⠀⠀⠀⠐⠁⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⣀⠀⠰⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠁⢠⣤⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠘⠁⠀⢰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣘⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠁⠈⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣀⣀⣀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠙⠉
⠀⠀⠀⠀⠀⠀⢀⡿⢷⣄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠛⠛⠠⣤⡀⠀⠀⠀⣤⣤⣶⡆⠀⢤⣤⣤⣤⠀⣠⡾⠋⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠶⠛⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡆⠀⠰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⠿⣄
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠘⠀⠀⠀⠀⠈⠁⢰⣿⣿⣷⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⡿⠀⢠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠏⠀⣀⠀⠀⠀⣿
⠀⠀⠀⢴⣶⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠉⠉⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠁⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⢉⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⢀⡀⠤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠁
⠀⠀⠀⠀⠙⢿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⠃⣀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣤⡿⠃
⠀⠀⠀⠀⢰⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠛⠛⢠⠀⠀⠀⠀⠄⠀⠀⠀⣠⣤⡿
⠀⠀⠀⠀⠀⠀⠙⠛⠳⣤⣄⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠗⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠘⠻⣦⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⣼⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠀⠀⠀⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⢀⣴⡞⠁⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⡶⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣴⠟⠁⠀⢠⡴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⣾⠁⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣦⠉⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⣿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⣿⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⡀
⠀⣴⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠙⣤⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠉⢷⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠙⢳⣤⡀⠀⠈⠛⢦⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠡⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣤⠐⠛⠓⠚⠛⠛⢤
⠀⠀⠀⠀⠀⣀⠾⠃⠈⠁⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢾⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⠉⠁⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⢠⡌⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⢶⡟⠁
⠀⠀⠀⠀⠀⠀⠘⠇⣀⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣴⣦⣤⣀⣠⡟⠛⠀⠛⠛⠁⠀⠘⢻⣤⣤⣤⣤⣤⣤⣤⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠒⠚⠋⠉⠉⠉⠉⠛⠛⠒⠲⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⠀⠉⢩⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⠀⠈⠁⠀⠀⠀⠉⠉⢛⣳⣒⠀⠀⠀⠀⠀⠀⠐⢶⡚⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠢⣄⠀⠉⠉⡿⠉⢉⡟⠀⣀⣤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⣠⡤⣷⣤⠴⠶⢦⣄⠘⠀⠀⡼⠁⢠⡏⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢉⣿⡋⠀⠀⠀⠀⠸⡆⢀⡾⠁⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠛⠛⠲⠶⢤⣤⣄⣀⠀⠉⠛⠛⠋⠉⡟⠃⠀⠀⠀⠀⢀⣷⠏⠀⠀⣼⣿⣀
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠒⢿⣄⠀⠀⠀⢀⡼⠁⠀⠀⣰⠟⣉⣁⢹⡄
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠈⠫⡓⠶⠶⠛⠁⠀⠀⠀⠁⠸⠽⠋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡁⠐⠒⠠⠤⠀⠄⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢷⠶⠶⠶⠶⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠴⣚⡉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢉⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠟⡱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡙⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢉⣉⣙⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠞⠀⢀⡾⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠀⢀⣉⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣠⠟⠀⠀⠀⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡌⠉⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡼⠁⠀⠙⢦⡀⠒⠦⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣠⡞⠁⠀⠀⠀⢤⡀⠀⠀⠀⣸⣿⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⢀⣤⠤⠤⢶⡟⠳⣄⠀⠀⡴⠋⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠙⢦⣀⡴⠿⣏⢀⣠⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠘⠓⢺⣿⡆⠀⠺⠞⠉⣾⢕⡾⠃⠈⠑⢦⣀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠤⢾⣿⠇⠀⠀⣀⡤⣿⠘⣇⣾⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠤⠖⠋⠁⠀⣿⠀⡇⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣶⠖⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠴⠶⣶⣶⠶⣶⣒⣂⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⡦⠬⣀⣸⣦⣀⣀⣀⣀⣀⣀⣤⣤⣴⡖⣻⢛⡙⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠷⠾⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠿⠻⠿⠿⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠿⠿⠻⠷⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠉⠉⠉⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉⠉⠉⠉⠛⠿⣦⣄
⠀⠠⣀⣀⣤⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⡠
⠠⢤⣴⣿⠋⠀⠀⠀⠀⠀⠀⣴⣾⣿⠉⠉⠳⡄⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⣾⣿⡋⠉⠑⣄⠀⠀⠀⠀⠀⠈⢻⣷⣤⠤
⢀⡤⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣴⣿⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⣿⣿⣿⣷⣤⣤⣾⡇⠀⠀⠀⠀⠀⠀⠙⠧⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡰⠀⡴⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢡⡇⡗⠒⠂⠤⠀⠀⠀⠐⢈⣹⢪⡧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⠇⡿⡿⢻⣿⠦⠀⠀⠀⠘⠻⠟⠐⢄⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢐⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡵⠊⠉⠉⠑⢄⠈⠇⠰⡀⠀⠀⠀⠐⢾⣿⠗⠀⢀⠊⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⡜⠀⠀⠀⠀⠀⠘⡄⡆⠀⡇⠀⠐⢄⡀⠀⠀⡠⠀⠘⡄⠀⠀⠀⠀⢱⡦⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡇⠀⣠⡇⠀⠀⠀⠀⢉⠏⠀⠀⠡⠈⠲⢄⠀⠀⢸⡟⣿⠤⣶⡄
⠀⠀⠀⡜⠁⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⣇⡴⣹⠀⠀⠀⠀⠀⡎⠀⠀⠐⠢⠤⢄⡀⢱⠢⣼⡇⣿⠀⢹⡇
⠀⠀⠸⠀⠀⠀⠀⠀⡴⠁⠀⣀⠀⠀⠀⠀⠀⠀⡟⠐⠁⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠈⠙⠚⢿⣼⣿⢦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣀⡠⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡰⠇
⠀⠀⠀⠀⠀⠒⠊⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢰⢠⠃⡠⢎⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠀⠀⢠⡤⠮⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣶⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠋⠉⠙⠻⣷⡄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠈⠙⠿⣷⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣤⡀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠐⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠋⠀⠀⠀⠀⣠⣾⣿⠀⣿⣦⡀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠟⠀⠻⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣤⣤⣤⣤⣿⣿⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣼⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣶⣾⣿⡿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠛⢫⡉⠏⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⣥⠵⠁⠁⠀⡀⡈⡄⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣄⣤⣤⣶⡾⡻⡙⡑⠇⢣⠈⠎⠴⣣⣾⣡⣆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣟⢻⢟⡽⡝⣯⢭⢗⠃⠈⠘⢂⣠⣤⣶⣿⣟⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣽⣿⡋⠝⠸⠹⡙⠸⢜⡈⠄⠴⠾⠿⠟⠻⢿⣿⣿⣽⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣿⡷⠟⢿⠏⠲⢀⣤⣾⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⣿⠙⣫⣤⠒⠉⠙⢴⣿⠆⠀⠀⠀⠀⠀⢀⠂⠀⠁⢠⠀⠘⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⡟⢛⡴⠞⣿⠁⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⢾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢯⡹⠟⣱⣿⣀⠀⢿⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⣾⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⣿⣿⣷⡍⣶⣿⣿⣯⣥⣾⣄⠀⠀⠀⠀⢀⡕⠀⠀⠀⠀⠀⠈⠐⠖⠀⢀⣸⡿⢟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣝⣡⣹⣿⣿⣾⣿⣿⣿⣿⣿⡋⠉⢓⡶⠂⠀⢸⠏⠀⠀⠀⡃⠀⠀⢀⣶⣦⣼⢟⡕⠍⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⡿⣽⣾⡟⣽⡿⣿⣿⣿⣿⣧⢄⣼⡧⣤⠤⣾⣧⣤⣤⣾⣷⣶⣶⢾⣿⣿⠻⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡿⣻⢻⢯⣾⢿⡿⣿⣾⣿⣟⢟⠩⡒⠋⡗⣲⡟⢹⡿⣿⢺⠳⡙⢢⢃⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢭⢯⣿⣷⣾⣧⣷⡿⢸⣀⠧⠀⢲⠡⢿⣷⡇⣆⡗⢎⢎⠇⠡⢇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣯⣸⡸⣴⣬⢂⣿⣿⢯⣿⣻⢺⢡⠈⡃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣭⣽⣿⣨⣼⣼⣮⡯⡹⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠘⢁⣼⠃⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣿⠀⠀⢸⡅⠀⠀⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⠀⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⠀⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠟⢧⣀⣀⡽⠛⠿⢿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣞⣁⣀⣀⣀⣉⣉⣀⣀⣀⣀⣈⣧⠀⠀⠀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⢉⡉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠉⠉⠉⠉⠉⠉⠉⠙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣖⣒⣒⣒⣒⣒⣛⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⠸⣿⣿⣿⣿⡇

⠀⠀⠀⠤⠠⠄⠤⠤⠤⠤⠤⠤⠤⠀⠀⢀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠖⠂⠀⠀⠀⠀⠐⠒⠂⠒⠀⠀⠀⢇⠆⡀⢠⠀⠀⠀⠀⠀⠀⢡⠀⢀⡀⠀⠈⠀⠀⡡⠞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⡠⠀⠀⠀⠀⠀⠀⠀⣀⢸⠃⠀⠀⠀⢤⣬⣀⡀⣀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⢄⠀⢾⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠑⠐⠂⡤⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣠⣄⠀⠈⠳⣅⠈⠀⠀⠀⠀⠀⠀⠀⠰⠃⠐⡋⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠈⠁⠠⠀⠨⡒⠀⠀⠀⠀⠀⠀⠀⠀⡟⡀⠀⠈⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⣉⡁⢋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢁⠀⠀⠀⠀⡀
⠀⠀⠀⡬⠹⡄⠀⠀⠀⠀⠀⠀⠂⢀⡡⠀⠀⠡⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣃⡈⠂⠐⠓⠠⠀⠂⢠⠂
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢱⡀⠀⡠⠂⠐
⠀⠀⠈⠀⠀⠈⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⣶⣾⢋⣈⡯⠥⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡄⠀⠀⠁⢀⣀⠤⠟⠛⠫⢀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⠋⠀⢀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡄⠚⠁⠀⠀⠀⠀⠐⠑⠀⡂⠀⠀⠀⡅⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠄⠁⠀⢤⡀⢘⣁⣀⠈⡅⠂⠐⠋⠀⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠐⠒⡺⠋⠀⠀⠐⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠟⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢛⣡⣤⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠉⢿⣿⣿⠀⠀⣿⣿⣿⠋⠁⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠀⠀⠀⣼⣿⣿⠀⠀⣿⣿⣿⡀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣶⣿⣿⣿⡏⠀⠀⢹⣿⣿⣿⣷⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⡿⠿⠟⠉⠀⠀⣀⣀⣀⠀⠉⠻⠿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠈⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠛⠛⠛⠛⠛⠛⠛⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⡋⠁⠁⠠⠈⠉⠭⠑⠒⡛⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⣸⡃⠀⠀⠀⣌⢀⡤⣌⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⠃⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠰⠀⠀⠸⣇⢹⠀⠀⠀⠈⠁⠀⣀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣧⠼⠀⠀⠀⢜⣩⡤⠤⠔⠊
⠀⠀⠀⣠⠴⣒⠲⢄⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠠⠴⠃⠀⠀⠀⢸⠇⠀⠀⠀⠀⣬⡅
⠀⠀⠀⣇⣸⠉⣹⢀⠆⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⢀⣞⡵⠋⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠁⣐⡒⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⢡⡦⠀⠀⠀⢀⣀⠤⠔⠒⠚⡳⡦⠀⣬⠀⢀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠇⢇⣀⠬⠀⠸⢸⠧⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⡄⠀⠀⠀⠀⠀⢰⠋⠁⠀⠀⠈⢸⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⢦
⠀⠀⠀⠀⢀⣏⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡝⠀⠀⣺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣇⢠⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢠⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢹⡌⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⢸⠀⠀⠀⠻⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⠉⠉⠀⢘⣗⡿
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠉⠉⠙⠓⠲⠤⢄⣀⣀⣧⠀⠀⠀⠀⢧⠠⢤⣸⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠘⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠉⠛⢳⡖⠒⠒⠒⠲⠶⠶⠤⠤⠤⠤⢿⡄⠀⠀⠀⢸⡥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠘⠂⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣴⣿⠿⣿⡆⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠉⠉⠀⠈⠁⠈⠛⢿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠿⠋⠚⡘⠄⠀⠀⠀⠀⢠⠈⢉⠋⠛⡿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠋⠁⢀⡀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠈⣠⣀⣀⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣬⣤⡄⠀⡀⠀⠀⠀⠀⠀⡀⠀⢠⢎⣥⣽⣿⣿⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣶⣿⣯⣿⢟⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⢼⡿⠛⢻⡆⠀⠀⠄⠐⡖⢀⢸⣟⣷⣴⣾⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠓⠴⠞⡉⠀⠁⠀⠀⠀⠠⢦⣾⣿⠿⢿⣿⣿⣿⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡵⠶⣽⣤⠁⠰⠆⠀⠀⠸⠟⣣⡾⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣶⠆⠏⠹⢮⡀⠀⠀⠀⣠⢎⣥⣶⣍⠫⣿⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⢼⣿⣿⡤⣤⣴⣿⣿⣷⣭⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠻⢿⠿⠟⣟⣠⣧⠽⢽⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠋⠝⣽⣯⣰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀
⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣼⣷⣭⣿⣥⣥⣯⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣾⣷⣿⣿⣿⣷⣦⣍⣼⣿⡿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠈⠉⠛⠀⠈⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣷⡿⢠⣿⣿⡿⣀⠙⢿⣯⣿⠿⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣴⡿⠋⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢿⣿⣿⣿⣻⣿⣿⣯⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⠀⠀⠀⠻⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⣀⣀⣀⣀⡀⠀⠀⣠⡴⠶⠶⢾⡽⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣿⣿⣿⣿⠉⠐⠊⣁⣈⣉⣻⠇⠀⢰⣧⣶⣿⠿⣿⠃⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠄⢸⣠⡟⢿⣿⠇⠘⢛⣿⣿⠟⣛⡧⠀⠀⢻⡿⠿⠖⠉⢧⠀⠀⠀⠠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⠈⢻⣇⠸⣿⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠑⢄⠀⠀⠘⡇⠀⠀⠐⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠃⠀⠈⢿⢳⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠞⠀⠀⢰⡇⠐⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⢺⡀⠀⠈⣷⣍⢻⣄⠀⢸⡄⠀⠀⠀⠀⣀⣴⣿⣾⣷⡦⡼⠀⠀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣯⣷⡄⠀⠈⢿⣿⣟⢦⡀⠁⠀⠀⠀⠀⠉⠛⠻⠛⠋⣰⠃⠀⢰⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣿⣿⡇⠀⠐⠶⠽⣿⡀⠙⣳⣦⣀⡀⠀⠀⠀⠀⠀⢰⠏⢀⡄⣽⣿⠇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣽⣴⡄⠀⠀⠀⠀⣿⠃⠀⠀⣹⣟⠛⠿⣶⠶⠶⠶⠟⢀⣿⣗⣿⠃⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢨⡿⠻⠂⠀⢀⣀⣿⠀⠀⠀⠈⠁⠀⠀⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⠂⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠁⠀⠀⢀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⢹⣿⣇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⠀⣠⣶⣿⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢇⣀⣿⠛⢿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣶⣿⡿⠃⠀⠉⢓⣚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣽⣿⣶⣤⣧⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠋⠀⠀⠠⠀⠀⠒⠈⣥⣤⣬⡽⠿⠾⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣘⣛⣛⣉⣉⣠⣽⣿⣻⣟⣿⣿⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠊⠉⠐⢢⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⢔⠑⢆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡔⠁⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠠⢄
⠀⠀⠐⠀⡀⢀⠀⠄⠀⠀⠀⠈⠂⢩⠒⠉⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠂⢄⠀⡇
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠦⡐⣃⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠐⠤⢚⡤⠞⣠⠇
⠀⠀⠀⠀⢠⠋⡢⠀⠀⠀⠀⠀⠀⠀⢨⣀⣨⠇⠀⣀⠑⠀⠀⠀⠀⠀⠀⠈⠐⢀⠄⠀⠀⠀⠀⠀⠀⠀⠘⣄⠠⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠠⠄⠂⠀⠈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡠⠄⠀⠀⠠⠁⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⣀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⠀⡐⢢⠈⠄⠀⠀⠀⠀⠐⢂⠀⠀⢀⠔⡀⠌⠂⠀⠀⠀⠈⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⢆⢁⠊⠀⠀⠀⠀⠀⠀⠈⠓⡀⠈⠢⠥⠀⠀⣠⡤⠄⠠⠄⠄⠁⠀⠀⠀⢀⠀⠀⡂⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠒⠊⠀⠘⠍⠧⡄⠀⠀⠀⠀⢀⡠⢠⡀⡓⠊⠀⠀⠀⠀⠀⠬⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠠⠀⠤⠀⠀⠀⠀⠀⠀⡐⠀⠀⢆⠀⠀⠀⠀⠈⠀⠱⣠⠉⢒⠁⠀⠀⠀⠀⠁⠤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠂⢀⠄⠀⠀⠀⠀⠊⠐⠲⠚⠃⠀⠀⠀⠀⠀⠀⡨⠃⠉⠀⠀⠀⠀⠀⠐⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣠⠈⠀⠀⠀⠀⠀⠀⠐⢀⠀⠠⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⡠⠀⠂⠄⠀⠀⠀⠀⠀⠀⢐⢀⡆⠀⠀⠀⠄⠀⠀⠰⠐⡤⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⢄⡀
⠀⠀⠀⠀⠀⢀⡐⢑⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡆⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢯⣰⠖⡆⢹
⠀⠀⠀⠀⠀⠀⠨⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠄⠒⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⢉⠕⠋
⠀⠀⠀⠀⠀⠀⠠⡉⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠖⠉⠡⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⢢⢈⠀⠀⠀⠀⠀⠀⣀⡂⠒⠂⢤⡠⠐⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢀⠊⠀⠀⠀⢠⠂⠁⠀⠀⠙⠄⢸⠇⠁⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡆⠀⠀⠀⢧⢀⣀⣀⡀⠀⠈⣩⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠉⠐⠀⠻⠟⠩⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠶⢦⠠⠀⣤⠀⣤⣤⢤⢤⢤⣤⡄⣠⣀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠆
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢨⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⡀⢀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⠀⡞⠁⠀⢹
⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠘⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⠀⠀⢠⠀⣸⡁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠐⠀⠐⠂⠀⠀⢀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣒⠂⠀⠀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠟⠃
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠠⠀⠙⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠉⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣴⠃⠈⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⠘⠿⠟⠛⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠂⣀⠀⠀⠀⠛⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠂⠀⠀⠀⠀⢈⣿⡦⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣠⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⣔⣲⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣝⡻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣯⣟⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣤⠀⠀⢸⣿⣿⡿⣛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⡿⠛⠉⠀⠀⠀⠀⠈⠉⠙⠋⠀⠀⠀⠰⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠈⠻⣿⣿⣿⠀⠀⠀⠀⠉⠀⠀⠀⢶⣴⣶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣭⣦⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⣿⣿⣇⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠉⠙⠂⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣴⣾⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠿⠟⠋⠉⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⢠⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⢠⣤⣶⡟⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠈⠉⠋⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠢⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠻⠿⢿⣿⠿⣿⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⢙⡁⣤⣶⣾⣾⣷⣶⣬⡄⠀⠀⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠈⢉⡽⠟⠛⠉⠁

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠐⠒⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡰⠠⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠘⡆⠀⠀⠀⠀⠲⡀⢸⡰⠃⡠⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⠈⠉⠉⢳⠁⠞⠁⠔⢀⡤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣆⠔⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⢲⣥⣤⠤⠖⠊⠀⠀⠀⠀⢈⢀⡀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⢀⢀⠀⠀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠉⠑⡄⠀⠀⠀⠀⠁⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠫⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⢰⠁⠀⠀⣀⠄⠈⠀⠈
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⡉⠗⠊⠁⡀⠀⠊
⠀⠙⣍⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠈⠀⠀⠀⡀
⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠐⢀⡖⠱⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠧⡀
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣼⠀⠀⡀⠀⠱⣴⣥⠀⠢⣀⣠⠴⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⠰⣇
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠰⢎⣉⠉⠑⡆⢿⠄⡞⠀⠀⢺⠝⠊⠀⠀⠈⣡⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠊⢡⠤⠂⠀⠃⠀⡜⠀⠀⠀⠁⠀⠀⠀⠀⠈⣓⡄⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⠀⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠒⠀⠠⣤⠄⠨⣭⠀⢀⣒⣀⣐⡀⣀⣉⣩⠤⠘⣆⣠⠀⠀⠀⠄⠐⠂⠈⠉⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠚⠀⠀⢳⠀⠀⠀⠀⣻⠀⡎⠉⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢹⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠚⢻⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠼⣤⣀⣀⣸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠻⠭⠤⠬⠿⠯⠭⠭⠭⠿⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢳⡤⠖⠻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⠀⠀⠾⠷⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢑⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⣯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⠀⠀⢠⡄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠟⠀⠀⠀⠉⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠘⠃⠀⠁⠀⠀⠀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⡀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⠛⠛⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠀⠀⠀⣀⡀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⡶⠀⢁⣤⠔⠈⠆⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⠀⠀⠀⢄⠀⠀⠀⠀⢸⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠂⢰⠀⠈⡟⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡠⠄⠀⠀⠀⣰⠇⠚⠠⡀⠠⠌⠀⠈⠀⠀⠃⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⠊⠀⠀⠐⠀⡴⠙⠀⠀⠀⢠⣷⣄⡀⠀⡇⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣠⡿⠋⠈⠢⡀⠀⠀⡴⢆⠀⠀⠀⡀⠛⣿⡏⠙⠲⡄⠠⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢉⡛⠀⠈⢦⣰⠌⠀⠀⣿⣧⠀⠀⢠⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⠀⠀⠀⣿⣿⡆⠀⠈⠁⠀⠂⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⣠⠀⠀⢸⠖⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠛⢉⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠉⠀⠀⢸⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⡆⠀⠸⢷⣄⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡏⠀⢃⠀⠀⠀⠙⠻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣨⣷⣤⡀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⣰⣦⣀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⡟⠉⠓⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⣀⡀
⠀⠀⠀⣀⣀⣀⡀⢀⣀⠰⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠉⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⢶⣶⣶⣶⣶⡆⠀⠀⢴⣶⣶⣶⣶⡆⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠎⢨⠙⢻⠉⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠉⡝⠛⢅⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠢⡀⢸⠀⠀⠀⠀⠛⠛⠁⠀⡄⠀⠀⠘⠛⠋⠀⠀⠀⠀⡇⢀⡜⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⣌⣸⠀⠀⠐⠴⠢⠀⠀⢠⠁⠀⠀⠀⠀⠰⠄⠆⠀⠀⣆⣁⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠒⢀⣀⢤⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡆⠀⠀⠀⠀⠀⠀⠀⠫⠅⠤⠊⠀⠀⠀⠀⠀⢠⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢤⠄⢀⣀⣀⣀⣀⡀⢠⡤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣥⣬⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢶⣛⢺⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣟⣛⣛⣿⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡷⠾⠭⠭⣭⣽⠛⠛⣉⣉⣥⣭⣉⣡⣴⣶⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⣯⣍⣡⣄⣀⠀⠀⢚⡔⢩⣿⣿⣿⣿⣹⣗⢜⢿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣷⣯⣐⠈⣟⠿⣿⣿⣶⣄⠀⡞⢀⣿⣿⣿⣿⣿⣿⣿⣦⣩⠉⢱⡛⠙⠋⢿⣿⡿⣛⢿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⡿⡿⡇⣧⣌⣿⣶⡄⠈⠁⠀⡇⢼⣛⣿⣿⣿⣿⣿⣿⡿⣌⠒⣫⣶⣿⣿⣿⣿⢹⣿⣿⣿⡇
⠀⠀⠀⢸⠿⣿⣿⣿⣿⣷⣧⣿⣿⣽⣿⣿⣾⣷⠇⡼⢿⣿⣿⣿⣿⣿⡷⠆⢺⡾⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢀⠀⠀⠀⣴⡆⠉⣽⡎⢻⣷⡂⠀⢀⣤⠀⢀⠀⠐⠒⠳⠶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⣤⠀⣀⠀⣀⠀⣠⠟⠉⢻⣦⡀⠀⣰⡟⠘⢿⣦⡀⡠⢻⡤⢤⠀⣛⠓⢲⣿⡀⢸⣿⣤⣤⢸⣿⠀⠀⠀⣀⠤⣶⣤⣄⣈⠉⠛⠛⠗⠒⠈⠀⠀⡀⠔⠒
⠀⠀⢻⣤⠙⠃⢩⡾⠋⠀⠀⠀⠘⣿⣤⡿⠀⠀⠠⠝⡻⠁⠈⣧⠛⢛⣋⣤⣼⣿⣥⣼⣿⣌⡀⠀⠀⠠⠔⠉⠀⠀⠀⠈⠉⠛⠿⣷⣦⣄⡀
⠀⠀⡨⣶⡄⢰⣿⢉⣈⠀⣀⠀⠀⠘⢋⣀⣤⢀⣤⠘⠀⠀⠀⡽⠞⠋⠉⠉⠀⠀⠀⠉⠉⠛⠿⣷⣦⣀⠀⠀⠀⢀⠠⠰⠶⣶⣄⣀⠉⠉⠁
⠀⠀⣇⡛⠁⢸⣿⠸⣿⡄⣿⠀⣷⡄⢸⠿⠛⢸⣿⠀⠀⠀⠊⠀⠀⠀⣠⡴⠶⠿⠛⠻⠿⢷⣶⣤⣙⠻⣷⣦⡀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⠠
⢰⣿⢉⣁⡀⠸⣿⠀⠀⠀⣿⠀⠁⣠⣦⣀⡐⢺⣿⠀⠀⠀⠀⠀⡠⠊⠁⢀⠤⠖⠚⠛⠛⠿⣾⣿⣿⣷⡈⠙⢿⣦⠀⠀⠀⢀⣠⣀⠀⠀⠉⠀⠀⢀⠠⣤⣄
⢸⣿⠸⠿⠃⠀⣿⡇⣶⡆⣿⣷⣦⢸⣿⠿⣿⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠙⣿⣿⣿⡄⠈⢻⣷⡀⠀⠀⣭⣿⡿⠶⣦⠤⠞⠁⢸⣿⠉
⢸⣿⣠⠤⣄⠀⣿⡇⠻⠃⣿⣿⣿⢸⣿⠀⡄⠀⣿⣷⠀⠀⣰⠀⢀⠀⠀⠀⠀⠀⢀⡴⠊⢹⡿⠀⢸⣿⠹⣿⡀⠀⢿⣧⣰⡇⢸⣿⣿⣆⢀⣀⢀⠀⢸⣿
⢸⣿⣿⠀⢿⡀⠸⣻⣤⣄⣿⠇⠀⠈⣿⡀⣷⠀⣿⡟⣦⠀⣟⣀⣇⣀⣀⣀⡠⠞⠁⠀⠀⣸⡇⠀⠠⡿⠀⢿⡇⠀⢸⣿⣿⡇⠘⠸⣿⣿⢸⣿⣾⡇⢸⣿
⢸⣿⡬⣤⣾⡃⠀⢻⣿⠙⠿⡶⣤⡀⣿⡇⣀⣠⣿⣷⣿⠿⢻⣿⣿⣏⠉⠛⠛⠲⢶⣤⣀⣿⠀⠀⠰⠃⠀⣼⠃⠀⢀⡟⣿⡇⢀⠀⣿⣿⣸⣿⣿⡇⠸⣿⡄
⡾⠋⠉⢿⣧⢰⡄⠈⢿⣇⠀⠀⠀⣩⣿⡿⠛⠛⣿⠃⣻⣆⣼⠇⠈⢿⣧⡀⠀⠀⠀⠘⣿⡏⠀⠄⠀⠀⡰⢣⣤⣶⣾⠁⣿⡇⠠⠀⢸⣿⣀⣼⢿⡏⠀⣿⡇
⡁⣄⠀⢸⣿⠈⠁⠀⠘⣿⣆⣠⣾⠟⠁⠀⠀⢰⡟⠀⠘⠿⠋⠀⠀⠈⠻⡷⠄⠀⠀⠀⢻⣿⠀⣀⡀⠀⠀⢻⣿⣿⣿⡀⣿⡇⢀⡇⢸⣿⡁⠄⠚⠃⠀⣿⡇
⡇⠉⠀⢸⣿⡀⣿⡇⠀⠘⣿⣏⠁⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⣀⠤⣴⣶⣶⣆⠀⠀⢸⡿⠈⠉⠀⠀⠀⢸⣿⣿⣍⣉⣿⡇⢸⠃⢸⣿⠆⠀⠐⠂⠐⠛⠁
⡇⢼⣧⠀⣿⡇⠈⢀⣀⣀⣸⣿⠀⠀⣠⣴⣶⣏⣻⠀⠀⠀⠀⢸⣿⡄⠿⠿⠛⠃⠀⠀⣿⣇⠒⠒⠀⠀⠀⠘⠛⠛⠋⠉⢿⣇⢘⠀⢸⣿⡇⠀⠘⣿⡆
⡇⠈⠉⠀⠿⠋⠉⢹⣿⡀⠈⣿⡇⠸⠿⠾⠛⠋⠹⣇⠀⠀⠀⠈⣿⣇⠀⠀⠀⡀⠄⣺⡿⠁⠐⠂⢀⠀⠀⠀⠀⢀⡀⠤⢾⣿⠚⠃⢻⠛⢻⠀⠀⣿⣇
⠇⠀⠀⠀⠀⣴⡖⠚⠛⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⣿⡂⠀⠀⠀⢠⣿⠇⠀⠀⢀⣠⠄⠒⠈⠀⢹⣿⠀⠘⣿⠀⠀⢸⠀⢸⠀⠀⢹⣿
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠈⣿⣇⠀⠀⠐⠀⠀⢘⣷⡀⠀⠀⢤⣿⠇⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⣤⡄⠀⣸⣿⠻⣷⣾⠀⠀⣸⡀⣸⡀⠀⢸⣿
⠀⠀⢰⣿⠉⠛⠉⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠉⠻⣷⣄⣾⠏⠀⠀⠀⠀⣿⡇⠠⠤⣤⣼⣿⠀⣿⡇⠀⢸⣿⠄⣿⣿⡇⢸⣿⠀⢹⣿⡆⢸⣿
⠀⠀⢸⣿⠀⠀⠀⢸⣷⠀⠀⠠⠀⠙⢿⣶⣄⡀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⣿⠇⠀⠀⠀⠈⢻⣷⡟⠃⠀⠘⠟⢠⣿⢳⣿⡌⠙⠁⣸⡿⠁⢸⣿
⣀⣤⣼⠿⠀⢀⣴⢟⣿⠀⠀⠀⠀⢀⠔⠉⠛⢿⣦⣄⡀⢀⣤⠾⣿⠟⠛⠛⣳⠾⠋⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠶⠤⠼⠿⠧⠙⠛⠲⠞⠉⠀⠀⠘⠛⠀⠤
⣿⠀⠀⠀⢀⡾⣿⠀⣿⡇⠀⢀⣴⠋⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⡀⠀⠀⠈
⠛⠁⠀⣠⠟⢰⡿⠀⣿⡇⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠒⠲⢦⡄
⠀⠀⣴⠋⢀⣿⠃⠀⢻⣷⢸⣿⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢻⡀
⠀⣼⣧⣀⣼⡏⠀⠀⢸⡿⢸⣿⠀⠀⠀⠀⠀⠋⠀⠀⠀⣠⡞⠁⠀⠀⠈⣿⣧⠀⠈⡇
⠀⠀⠈⠉⣿⠓⠋⠉⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⡀⠀⣀⠸⣿⡀⠀⣿
⠐⣿⡟⠾⠷⠶⠤⣶⡆⠀⠀⣿⡇⠀⠀⢀⠀⠀⠀⠀⠀⣿⡷⣿⡿⠊⡿⠁⣿⡇⠀⢻
⠀⠛⣡⣴⣄⡀⠀⢉⣤⣄⣀⣻⣧⠀⠀⢸⡆⠀⠀⠀⠀⢿⡇⠹⢿⡾⠁⠀⣿⡇⠀⢸⡄
⠖⠈⠁⠈⠉⠉⠉⠀⠀⠉⠉⢹⣿⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⣇⠀⢸⣇
⠀⢠⠾⣷⣄⡀⠀⣠⣶⣄⣀⣀⣿⡆⠀⠀⣷⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢦⣷⡄⠀⢀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡄⠀⠀⠀⠀⠀⠀⠀⠤⢾⣿⣿⡸⡈⠃⢰⢽⠆⠀⠴⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣀⣤⠴⠞⠒⠒⠊⠟⣛⠋⠀⠀⠈⠁⠀⠊⣉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠈⠀⠀⠒⢬⢴⡄⠀⠈⠱⣦⣄⠀⠀⣠⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⢠⡟⠃⠀⠠⡝⢋⡘⠀⠘⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠇⠀⢾⠘⠇⢀⡆⢈⡁⠘⠁⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣞⠀⢇⠈⡿⠄⠙⠐⠷⠦⡄
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢀⠈⠀⠀⠄⠀⢀⢳⢺⠁⢠⡀⠠⡀⠀⠑⠂⠀⠤⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠘⡘⢌⡌⢎⢧⠈⠇⠀⠘⠀⠀⠀⠀⠀⠀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄⠛⠬⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠻⡯⠵⠖⠂⠰⠋⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠤⣀⡀⡿⡾⠿⠛⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠀⠤⢀⠀⠀⠈⠹⡓⠒⠻⠀⣾⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠤⢷⠄⡤⠿⠨⠻⢄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠚⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⣿⣿⣿⣿⠛⢶⣄⠀⠹⣿⣿⡿⡿⠯⠄⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⡟⠀⠀⠙⣷⡀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠿⣭⣄⣀⣀⣀⠘⢷⠀⣻⡇⠤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⠉⠉⠉⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠙⠿⠿⠿⠿⠟⠀⠈⢻⣿⣿⡿⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣤⣤⣄⣀⣀⣀⣠⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣦⣴⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣴⣶⣾⣋⡉⠉⠀⠀⠈⠙⠷⣤⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠚⠋⠉⠛⠿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣁⣀⠀⠀⠈⠻⣦⡀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⢉⡉⠉⠀⠻⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣶⣶⣦⣤⣤⣤⣤⠶⠟⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⣦⣤⠤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠘⠉⠀⠀⠀⠈⠛⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡀⢹⡀⠀⠀⠀⠀⠀⣠⠀⠙⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠉⠉⢻⣿⣎⣳⡈⠃⠀⡔⠋⠁⠀⢠⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⢌⣙⣿⣿⣶⣬⣟⢠⡀⠀⠂⠆⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠿⣽⣯⣹⣾⠟⠁⠀⢰⣆⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⠀⠀⠀⠐⠀⠀⠀⣰⠃⣽⠣⠀⠀⠀⠙⢻⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣈⣂⣤⡀⠀⠀⠁⠀⡇⠀⢀⣦⢒⣤⠋⠀⣯⡀⠀⠀⠁⠐⠀⠀⢂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣛⣁⣒⣈⢉⣛⡿⠷⠦⠀⠀⣈⣤⣾⣿⣿⣯⣤⢤⣾⣿⣦⣤⣤⣤⣴⣶⣼⣿⣶⣦⣤⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⢀⠁⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⡇
⠀⠀⠀⠀⠀⠈⠀⡌⢞⠀⢉⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢀⡴⠀⠀⠀⣀⢉⣈⣹⣉⣉⣹⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⡿⠧⠼⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠈⠲⠊⡉⠁⠈⠐⠦⢼⣄⣠⣤⣴⣾⠋⠀⠀⠀⠈⣨⡟⡉⣹⣿⣿⣻⣿⣩⣿⣿⣿⣩⣿⠉⠉
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠁⠀⠛⠒⡶⠤⠙⠛⡛⣿⡏⠀⠀⠀⣠⣾⣿⠂⠀⣿⣿⣷⣿⣷⡋⣽⣿⣿⣿⣿⠧
⠀⠀⠀⠀⠀⠈⠉⠰⠠⠤⠤⣤⣄⣀⠀⠀⣀⢴⣿⣿⡿⠁⠀⣠⣼⠿⠿⠟⡡⣾⣿⡭⣿⣟⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣶⣿⣿⣿⠋⠁⠀⠸⢿⣧⣴⣴⣾⣧⣿⣭⣠⡬⠻⣿⠏⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣤⣧⠿⡝⠿⢶⠤⣤⣌⣉⠿⠿⢿⣿⣿⡷⠀⠉⠁⠰⣿⣄⣀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠘⠉⠀⠀⠀⠀⠀⠈⠉⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀
⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠁⠉⠛⠆⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣻⣿⡂⠀⠀⠀⡰⠋⢁⣠⣤⣄⠈⠻⡄
⠀⠀⠀⠀⡜⠁⠀⡤⠶⠶⣄⠀⠀⢸⠀⠀⠀⠀⠀⢸⢿⡟⠁⠀⢹⣩⣹⠃⠀⠀⠀⠁⠠⡃⠀⠀⠈⠇⠀⠘
⠀⠀⠀⠀⡀⠀⡌⠀⠀⠀⣸⠀⠀⡸⠀⠀⠀⠀⠀⠸⣾⣷⢀⣴⣿⡿⠋⠀⠀⠀⠀⢦⣤⠵⠄⢀⣀⠞⠀⢠
⠀⠀⠀⠀⠙⠚⠁⣤⠖⠚⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢡⡻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⢉⣁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⡇⢰⠚⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⠇⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⢸⠉⠁⠀⣀⣤⣄⡀
⠀⠰⠊⠉⠑⠄⠀⡇⢸⡀⠀⠀⠀⠀⠘⠉⠙⢧⠀⠀⠀⢹⣿⠦⠀⠀⠀⠀⠯⠈⠙⠄⠀⠸⢬⠆⠀⠸⠅⠀⠀⢙
⠀⠀⠀⠀⣠⠆⠀⠘⣒⣃⠀⠀⠀⠀⢀⡤⢤⡾⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⣀⠄⠀⠰⣋⠀⠀⠀⠀⢀⡤⠎
⠀⠀⠐⠉⠁⠀⠀⠀⣁⣨⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢈⣛⣁⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠘⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠰⠀⠀⠀⣠⠶⠛⠉⠉⠉⠛⠗⢦⡀⠀⠁⠀⣀⣤⣶⣶⣶⣶⣦⣄⠉
⠀⠀⠀⠀⢠⠞⠋⠉⠉⠉⠛⢷⠀⠀⠀⠀⠀⢸⠁⢠⣾⠛⠲⣤⡀⠀⠀⠁⠀⠀⣼⡿⡟⠛⠛⠛⢿⣏⠙⡆
⠀⠀⠀⠀⢫⣀⡴⠚⠛⢆⠀⢈⡁⠀⠀⠀⠀⠘⢤⣠⡜⠀⠀⠀⡇⠀⢀⡖⠀⠀⢼⣟⣿⠀⠀⠀⢸⣿⣴⠃
⠀⠀⣀⣄⡀⠀⠀⠀⢀⠎⣰⠟⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠉⠁⠀⠀⣴⣿⠟⠁⢀⣀⡀
⠀⢨⣀⠀⠙⠀⠀⠀⡞⢸⠁⠀⠀⠀⠉⠉⣻⠀⠀⠀⢀⠴⢋⠔⠊⠁⠀⠀⠊⠙⠢⠀⠀⠀⢸⡟⠁⠀⠀⡁⠀⠈⡆
⠀⠀⢀⡤⠖⠀⠀⠀⢻⡞⠀⠀⠀⠀⢸⠉⠉⠀⠀⠀⠘⢶⠃⠀⠀⠀⠀⠈⢁⠤⠎⠀⠀⠀⣘⡁⠀⠀⠀⠀⠊⠉
⠀⠀⠠⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠸⣌⠇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢰⠇⠀⠀⠀⠀⠀⠀⢾⠋⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡄⠀⠀⠀⠳⡘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠻⣧⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣄⢀⠀⠀⠀⠀⠀⠾⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢠⡿⠀⠀⠀⠀⠀⠀⣠⡤⠶⠿⢋⣉⣙⣭⢨⣟⡹⢿⢷⡀⠀⢿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢀⣾⣿⣷⠇⠾⠟⠛⠛⠛⠋⠋⠀⠀⠘⣧⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡀⠀⢠⡿⣡⣿⠋⠀⠀⠀⠀⣀⠠⠀⠀⡀⠤⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠙⢦⣄⠀⣄⠀⠀⠛⠋⠀⢸⡇⢹⣯⠀⠀⠀⠀⠩⠀⠀⠀⢀⡀⠀⣀⣀⣽⠀⠀⠀⠀⠀⠖⠋⠁
⠀⠰⢶⣄⠀⠀⠀⠈⠳⣌⠃⠀⠀⠀⠀⢸⣿⣤⣼⣦⠀⠀⣠⣴⣶⣶⠖⠀⠀⠙⣻⣭⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠖
⠀⠀⠀⠉⠛⢦⣀⠀⠀⠈⠳⣄⠀⠀⠀⠸⣾⣾⣿⡛⠁⠐⠋⣴⠿⠷⠂⠆⠐⣎⠫⠥⠀⣧⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠛⠉⠉⠁⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠁⠀⠀⠀⣻⠿⡿⠗⠀⠀⠀⠀⠋⠉⠀⢀⠀⠘⣦⠀⠀⢹⠀⠀⠀⠀⠀⠀⣀⣀⣀⣖⣒⠿⠭⠭⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⢿⠉⡇⠠⠀⠀⠀⠀⠀⠀⢀⠐⠒⠚⠉⠰⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡛⠀⠀⠀⠀⠀⠀⢰⢃⣠⠤⠴⠖⡂⠜⢸⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⢤⡤⠀⠀⠀⠀⠀⠙⢶⢇⡀⠀⠲⣄⠉⠈⠀⢰⠞⠉⡑⠢⣼⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⠀⠀⠀⢀⣼⣏⢳⣤⡀⠙⠂⠀⣴⠫⠴⠚⠉⠑⢿⠀⠀⠀⠀⠀⢦⣀⣀⣀⢀⠠⣤⠤⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠰⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠑⢍⠻⢿⣶⣴⡇⠀⠀⠒⠋⠉⠀⣳⠀⠀⠀⠀⠠⣄⣀⡀⠈⠈⠉⠉⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⣧⠀⠀⠙⠲⢄⣽⠀⠀⠀⠚⠁⠘⠉⢈⣧⠀⠀⠀⠀⠒⠂⠉⠉⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣇⠹⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠐⢡⠟⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠛⠋⠻⠿⠟⢛⠿⠛⠛⠁⠀⢻⣆⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⣿⠀⠉⠀⠀⠉⠉⠙⠻⠷⣶⣤⣀
⠀⠀⠀⣠⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣿⢦⣾⡟⠁⠀⠀⠀⠀⢀⡼⢻⡆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢿⡳⡄
⠀⠀⢠⣿⠏⢳⣄⠀⠀⠀⢀⡀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⣸⠞⠁⠈⠓⢤⡀⠀⣠⣾⠀⢸⣇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠸⣧⢹⡄
⠀⠀⣾⠏⠀⠀⢻⣆⠀⠀⢸⠃⠀⠀⠀⠀⠙⠿⠂⢀⣴⣾⡧⢄⡀⠀⠀⠀⢳⣴⣿⣿⣇⠀⣿⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡄
⠀⣸⡿⠀⠀⠀⠀⢻⡆⠀⢸⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠙⣄⠀⠀⠀⢻⣿⣿⣿⡄⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠛⠁⠘⣷
⠀⣿⠇⠀⠀⠀⠀⠈⠿⠀⡿⠀⠀⠀⣀⣴⣿⣿⠿⠁⠀⠀⠀⠀⠀⠘⢆⠠⠟⣠⣿⣿⣿⣧⣿⠀⠀⠀⠀⣸⠁⠀⣀⠀⠀⠀⠀⠀⠀⢸⣇
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠏⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣿⡄
⣾⠇⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢚⣥⡾⠸⠟⢻⣿⣿⣿⣿⠀⠀⢠⡟⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢹⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠠⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠈⠉⠈⠢⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠈⠄⠠⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠠⢄⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⣈⠢⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢪⠖⢺⠃⠀⢀⡁⠀⠀⢡⡀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠄⠀⡀⠀⠈⠄⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃⠠⡏⠀⢀⠂⠠⠀⡀⠒⠈⢢⡀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠀⠀⠓⡀⠚⢂⠀⠀⢈⠄⠀⠄⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡉⠹⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠈⠆⠀⠈⠀⠢⡷⣤⣄⠤⢎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠈⢿⠘⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⡬⠐⠂⠉⢙⠈⠀⠀⠀⠀⠈⠸⠑⠂⠀⠀⠀⠀⠀⠀⠄⠂⠉⠀⠀⠐⡴⠃⠁⠈⠂⢀⠌⠉⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⠁⢄⡀⠀⡀⠔⠁⠀⠀⠀⠀⠐⡀⠀⡈⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠜⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠈⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⡄⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠑⠠⠄⠤⠐⠈⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠄⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠏⠐⠂⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⡀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⠂⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠉⠁⠒⠂⠄⠀⢀⠀⢂⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⢀⡀⢔⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠈⠀⠀⠒⠒⠒⠒⠀⠀⠄⣀⠀⣂⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⡇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢻
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⠿⣶⣶⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡟⢦⡈⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⢶⣿⣿⣯⣿⣿⣿⣿⣿⣜⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣼⣿⣿⣦⣿⡿⢿⡟⢿⣿⣿⣿⣿⠿⣌⠙⠋⡿⠘⣿⣿⣿⢹⡿⢿⠏⠀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠟⠻⣿⣿⣏⡉⢃⠀⢿⠀⠉⠙⠟⠙⠂⠙⠃⠘⠀⠀⠘⠿⠇⠸⠃⠀⣿⣿⡿⣿⣀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣾⣿⣤⡤⢄⣀⢙⣧⣄⣹⣶⣾⣷⠶⣶⡶⣴⣷⣶⣶⣾⣶⣤⣆⣴⠄⢿⡏⠀⠘⢿⣿⣷⡄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣯⣥⣥⣤⢀⣉⣾⣿⣿⣿⠃⠈⠿⡷⠿⣷⣹⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⠿⣿⣤⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⣩⢷⡆⠰⠂⠈⠛⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠉⠙⠙⠿⠻⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡝⢷⡀
⠀⠀⠀⣿⣿⣿⣿⡟⢛⣿⣩⡾⠟⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣷
⠀⠀⠀⣿⣿⣿⣿⣷⡿⢿⡁⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⣠⠼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣫⡅⠘⣉⡴⠋⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣤⡶⢿⡏⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠸⣿⡿⣿⣿⣿⣷⣿⣿⣿⠟⠉⣰⢞⣵⠂⠀⠀⠀⡰⠚⠀⣀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣾⡿⠛⢻⡇
⠀⠀⠀⠀⢿⣷⣿⣿⣿⣿⣿⣿⡿⣷⣾⣽⠟⠁⠀⠀⠀⠀⢀⣴⡾⠿⣿⣿⣿⣿⣿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣉⡀⠀⠻⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠙⢛⣿⣿⣤⣤⣀⠀⠀⠀⣴⠟⣻⠞⠻⣿⣿⡇⢰⡆
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢀⣠⣶⣿⣿⣿⡍⠛⠻⠉⢻⡆⠀⠀⠀⠈⣁⠀⠀⠉⠁⠀⠘⡇
⠀⠀⢠⡶⠊⠉⠉⠛⠛⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣿⣁⡀⠹⠛⠃⠀⠀⢀⠂⠈⠀⠀⠀⠀⡀⠉⠓⠶⠀⠀⠀⠀⢹
⠀⢠⠈⣠⣶⣶⣤⣀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣀⠀⠀⠄⠁⠀⠀⠀⠀⢠⣄⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠸
⠀⠸⠇⠿⢿⡀⠉⠛⢿⡀⠈⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠐⣦⠈⠇⠀⢼⣿⡇⠀⠀⠘⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠱⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠆⣴⣀⠀⠀⢿⣦⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠃⠀⠀⠁⠀⠀⠀⠀⣨⣦⡄⠀⠀⣸
⠀⠀⠀⠀⠈⢧⠛⢷⣄⠀⠈⠁⠰⣀⠰⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠈⢿⣤⣴⠶⠶⢶⣤⣀⣴⠋⠀⠹⣷⠀⢹
⠀⠀⠀⠀⠀⠀⠓⠤⣍⡀⠀⠀⠀⢻⡄⠰⠀⠣⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⣿⣿⣿⡄⠀⠀⠀⠈⠓⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡀⠀⠀⠀⠀⠀⠋⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⠉⠉⢠⣀⣠⡤⠀⠀⢤⣤⣄⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⡿⠋⠀⠀⠀⠀⠉⠛⠦⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣰⣷⣤⣤⣤⡄⠀⣀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⣿⠃⠘⢿⡉⠙⢿⣿⣿⣷⣦⣤⣄⠀⠀⣠⡾⠋⠀⠀⠰⠶⠶⠾⠛⠉⠉⠀⠀⠀⠀⠀⠉⡆⠹
⠀⠀⠀⣀⣀⣀⣠⣾⣿⠟⠁⠺⣿⣿⣿⡆⠤⠀⠈⠳⣤⡙⠛⢿⣿⣿⣿⣤⣰⣿⠃⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣷⣀
⢀⡴⠚⠛⠛⣿⣿⠛⠁⠀⠀⠀⠙⠻⡟⢳⠄⠀⠀⠀⠀⠙⢶⣄⠈⠻⣿⣿⣿⡇⢀⣶⠆⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠁⠘⠛⠿⠿⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢏⡦⠀⠀⠀⠀⠀⠀⠈⠻⠦⣈⣿⣿⣧⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⢄⡀⠀⠀⠀⣴⣿⣿⣿⡟⢿⠇⠀⣠⠁⠀⠀⠀⠀⠀⠉⠛⠛⠳⢤⣤⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡄⠀⠘⣷⣀⣼⣿⣿⣿⣿⡏⠀⠄⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⠀⠀⠀⠀⠙⢶⣄⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⢀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⠙⢧⣤⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠐⠿⢶⠶⠶⠛⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡿⠧⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠁⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢣⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠶⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣿⢱⣿⣀⣴⠏⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠿⣿⣿⠛⣿⠇⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠚⠋⢿⣶⡇⣿⣿⣸⡿⣴⠋⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⣼⠀⣿⡅⣿⢻⡿⠸⡿⠀⡎⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠘⣤⠿⣿⡇⠘⠷⠀⠁⠄⢡⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠛⠉⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠤⠂⠉⠉⠉⠐⠢⢄
⡌⠀⠀⠄⠒⠠⢀⠀⠀⠑⡀
⢇⠀⠸⠀⠀⠀⠀⢡⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠌⠀⢀⠃
⠀⠀⠀⠀⡀⠊⠀⠀⡠⠊
⠀⠀⠀⠘⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡙⢄⠀⠀⡔⠄⠀⢀
⠀⠀⠀⢠⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠄⠁⠀⠀⠁⠀⠀⠀⠃⠈⠰⠁⢆⠰⠄
⠀⠀⠀⠈⢄⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠨⠀⠢⠀⠀⠀⠀⠀⠀⠀⡠⡶⢠⠄⠀⠤⢀⠀⠀⠀⢀⡀⠠⢤⡀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢰⡄⠂⠀⠂⠀⠀⠐⠂⠐⠠⠀⠀⠀⠀⡠⢤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣸⢠⣖⠒⠢⡀⠀⢠⣄⠠⢄⠀⠀⠐⠎⡠⠂⡆⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⢄⡀⡁⢻⠘⠥⠀⠔⠇⠀⠫⣉⣀⠜⠀⠀⠀⠀⠑⠀⠇⣠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠀⢀⠈⠙⢸⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠸⠀⠀⡀⠀⠀⠁⠄⠤⠀⠀⠀⠀⠀⠀⠀⢘⡁⢀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠁⠀⠀⢃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⢠⠁⠐⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⠠⡀⣀⠀⠤⠀⠈⠀⠀⠀⠀⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠈⠐⠀⠤⠄⠤⢴⠀⠠⡀⠀⠀⠀⠀⠀⡠⠂⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⡰⠤⡒⠉⠀⠀⠀⢀⠄⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⠀⠀⠀⠀⠀⠐⡠⠚⢠⢀⡘⢆⠀⠠⠂⠁⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠂⠀⠀⠒⠀⠀⢀⠂⠑⢄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠊⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠚⠀⠢⡀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⡀⠀⠀⠇⠀⠀⠀⢈⠀⠈⢢⣤⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡘⠀⠀⠀⠀⠀⠂⠄⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠄⢤⠀⠀⢠⠀⠀⠀⠀⠀⠄⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠾⠿⠿⠿⠿⠿⠿⢶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⢀⣠⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⣠⣿⠋⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⣰⡿⠁⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⢶⣦⡄⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⣰⡿⠁⠀⠀⢰⡿⠁⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠉⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠛⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣼⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠹⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡾⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣛⣿⣿⣷⣶⣶⣿⣿⣟⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠤⠆⠀⠀⣀⠀⠞⡇⠀⣄⠀⠈⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠈⠀⠀⠀⠊⠙⠛⠀⢽⠀⠀⠁⠀⠉⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣴⣿⠞⡁⠀⢀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣭⣷⣿⣵⣛⣻⣻⣿⣷⣤⣶⣤⣄⢀⣤⣶⡭⣭⣽⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠰⣾⡻⣻⡿⡛⠿⣿⠟⠻⠏⢻⣿⣿⣿⠋⠸⢙⡾⣿⢟⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢿⠔⠉⠉⠉⠉⠀⠀⠀⣀⠛⠈⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣽⣤⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⡇⢻⠕⠀⠀⠀⠀⠀⠀⣀⣀⠈⠋⢀⡠⠀⠀⠀⠀⠀⠀⠠⠦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡂⠀⠀⠀⠀⠀⠀⠈⠉⢉⡈⠝⢂⡞⠓⠄⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⣤⣠⣤⣤⣆⠀⠀⠀⢀⣠⣾⣿⣿⣷⣤⣤⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠉⢹⣿⡾⠋⠉⠁⠀⠈⠉⠉⣟⠉⠉

⠀⠀⠀⠀⠀⢀⣠⠖⠚⠛⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣗⠖⣦⡀
⠀⠀⠀⠀⢰⠋⡃⠀⠀⠀⠀⠀⡿⢆⠀⠀⠀⠀⠀⠀⠀⢀⡴⢯⣀⣒⣪⠳⣄⡀⠀⠀⠀⠀⠀⠀⢠⡟⠿⠿⠟⠉⠇⡿⣷
⠀⠀⠀⠀⡸⡏⡁⢤⠀⡄⢤⡀⠙⣾⡀⠀⠀⠀⠀⠀⣠⢊⡶⠓⠋⠉⠙⠢⣄⢳⡀⠀⠀⠀⠀⠀⢸⡇⢀⡀⣄⠀⣀⠀⣿
⠀⠀⠀⠀⢹⡍⢧⠀⠰⣅⠀⢀⡼⡿⠀⠀⠀⠀⠀⠀⢳⣿⠀⠰⣤⠠⢤⡄⢸⢸⠃⠀⠀⠀⠀⠀⢾⣷⠀⠁⡟⠀⠉⢰⣾⠇
⠀⠀⠀⠀⢸⠅⠀⠀⠰⠶⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢠⡀⠀⠁⢲⡟⠀⠀⠀⠀⠀⠀⠈⢻⡓⠀⠭⣥⠀⢫⠎
⠀⠀⠀⠀⠸⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡂⠨⠭⠄⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⣄⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠙⠶⣄⠦⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢵⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢠⣾⣿⡇⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠙⡟⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠙⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣀⠀⠀⣀⠀⢘⣇⠀⠀⠀⠀⠀⠀⢸⣼⡿⠿⠛⠁⠸⢿⣿⡆
⠀⠀⠀⠀⢾⣿⡡⣤⢀⠀⢤⠈⣿⠇⠀⠀⠀⠀⠀⠀⠀⡾⣷⠈⠃⡇⠈⠁⠈⣿⠇⠀⠀⠀⠀⠀⢈⣿⠐⣦⢢⠐⣦⠀⣿⠄
⠀⠀⠀⠀⠘⢿⣅⠀⠨⠄⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣯⣷⣦⣾⠋⠀⠀⠀⠀⠀⠀⠈⢿⣆⡀⢨⡄⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠙⠧⣈⣉⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣐⠓⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋⢀⠀⠀⠈⠛⠋⠙⢿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠀⡻⣿⠛⠃⢶⣿⡿⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠉⠁⣰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⣠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⡄⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡤⢶⠲⡋⠁⠀⠙⣁⣂⠐⠟⠛⠓⡤⢄
⠀⠀⠀⠀⠀⠀⣠⣴⣉⣈⠀⣀⢀⡃⡀⠀⠀⠱⡄⠒⠶⣼⣤⣧⠦⠬⣿⠲⡤⣀
⠀⠀⠀⠀⠀⣸⠉⠛⠙⠛⠛⠛⠻⠛⠟⠟⠛⠛⠁⠀⠉⢡⠀⡇⠀⠀⠈⢳⡀⠈⡆
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⠂⠀⠀⠠⢼⣠⣧⣀⣀⡀⠀⠱⡄
⠀⠀⠀⠀⠀⠁⠀⢀⡤⠴⠲⠖⠺⠒⠒⠲⠋⠀⠀⠀⠀⠛⠉⣿⠙⠻⣿⣦⠀⠙⣦⡀
⠀⠀⠀⠀⣴⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⢹⠀⠀⣿⣿⡆⠀⢹⠇
⠀⠀⠀⠀⡏⡀⢀⠀⣀⣁⣠⣀⣤⢀⣄⣤⠀⢄⠁⠀⠒⡇⢠⢸⠀⠀⠀⢻⣿⡄⢸
⠀⠀⠀⢰⣏⡁⡀⠀⠠⡦⠠⠤⠿⠠⠶⡇⠀⠀⠀⠀⠀⠐⡂⢺⡆⢤⣴⠸⢿⣧⣸
⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠄⣡⠸⡇⠀⢸⡀⠘⣿⣷
⠀⠀⠀⢰⡏⣉⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠱⠀⠀⠄⠀⠀⠈⢤⡇⠀⢸⣟⠠⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣼⣏⠙⠚⣿⡆
⠀⠀⠀⠈⣧⣀⡀⣀⠀⠄⢤⢠⡦⣤⠤⡤⣤⠄⠐⠂⠀⠀⠠⠄⡆⠸⠉⣿⣠⡀⢿⡁
⠀⠀⠀⢀⡏⠉⠉⠋⠛⠁⠉⢘⠃⠛⠛⠏⠈⠁⠀⠀⠄⠂⠀⠀⠆⠀⠀⣁⠈⢿⣦⡃⠀⡀
⠀⠀⠀⢸⣶⡒⣶⢂⣺⣀⣀⣸⡀⣄⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢿⠠⣾⠈⢳⡀⢱
⠀⠀⠀⠘⠉⠀⠁⠀⠀⢀⡀⣨⢀⣅⢠⡅⢈⣁⠀⣀⠀⠀⢀⠀⠈⡐⠂⠁⠀⣿⠑⠦⠉⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠁⠀⠀⠁⠈⠀⠈⠉⠃⠀⠀⠀⠀⠀⣄⠈⠳⣇