Wardrobe Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠒⠛⠛⠓⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣭⣴⠶⠞⠛⠛⠳⠶⣦⣬⡛⢦⡀
⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠶⠦⠴⠶⠶⠋⣴⠟⢉⣀⣤⡴⠶⠶⢦⣤⣀⡉⠻⣦⠙⠶⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀
⠙⢿⣿⡟⠒⠒⠒⠶⠶⠒⠒⠒⠒⣒⣒⣒⣒⣂⣀⣀⣐⣒⣉⣥⠶⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠶⢬⣉⣓⣂⣀⣀⣐⣒⣒⣒⡒⠒⠒⠒⠒⠶⠶⠒⠒⠒⢻⣿⡿
⠀⠀⠈⠻⣟⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠛⠛⣻⠟⠁
⠀⠀⠀⠀⠙⢦⠀⠀⠙⣿⡿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢿⣿⠃⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⡀⠀⢀⣠⣄⡀⠀⢀⣠⣾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠟⢻⡟⠻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠃⠀⢸⡇⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣼⣇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⢸⡇⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⣀⣠⣤⣄⣀⡀⠀⠀⠀⢰⣿⢸⠋⢸⡇⠹⡇⣿⡆⠀⠀⠀⢀⣀⣠⣤⣄⣀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣇⣠⣶⠟⠉⠉⠉⠉⠉⠙⠻⢶⣄⣸⡿⢈⡆⢸⡇⢸⡀⣿⣇⣠⡴⠟⠋⠉⠉⠉⠉⠉⠻⣶⣄⣸⡏⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠀⣸⣇⠈⠁⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣦⣤⣤⣄⣀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣷⠀⠀⠀⣿⣿⠀⠀⠀⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⣿⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢠⣿⣿⡀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⢸⣿⣿⡇⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⣿⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡀
⠀⠀⠀⠀⠀⢀⡿⠋⠉⢩⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⡍⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⢿⡄⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⢠⡿
⠀⠀⠀⠀⠀⠀⣾⠃⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠃⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠂⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⠀⣀⠡⠀⠀⣍⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⣇⣀⣀⣀⣤⣀⣀⠊⠋⡀⢈⠙⢲⡇⠀⢀⠀⣿⢀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡞⠻⠚⠿⢷⠿⡶⡾⠻⢶⡟⣿⢻⣟⠛⠛⠛⠻⢿⡋⢉⠉⠁⠈⢀⢸⣷⠀⠈⠛⠛⢻⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡷⢶⣟⣶⢿⢦⣾⣩⠿⠞⢹⠉⣾⠋⠉⠉⠉⠉⠈⠉⠉⠁⠀⠉⠉⠙⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣿⡄⠀⠘⠃⠰⠀⠀⠐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⡈⠀⠀⠀⢸⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⠀⣦⣄⠀⠀⣠⣄⣐⡀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣶⡇⠀⠀⠀⢸⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⠀⠀⠀⣽⠂⠀⠀⠀⠀⠻⠄⠀⠀⠀⢀⡀⣧⠀⠀⠈⣿⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⠀⢸⠀⠸⠆⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⡀
⠀⠀⢀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⠀⠘⠀⠋⡆
⠀⠀⡸⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠃⠀⢃⣠⢶⠦⠠⡤⣤⡤⠤⢄⣀⣘⡂⠄⢸⠀⠀⠀⢸⠁⠀⠀⠀⠀⡄⢰⡇
⠀⠀⠰⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⣀⢀⡠⠀⠀⠀⠁⠀⢀⢰⣚⣚⣒⣧⣚⣃⡒⣀⣲⡷⢇⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⠁
⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⠀⠈⠃⠺⠤⠀⠀⠀⠀⠀⠸⠉⠉⠁⠁⠀⠉⠉⠉⠈⠁⠀⠙⠩⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠁⠀⠀⠀⠀⠀⡀⣀⡀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠤⣄⡀⠀⢀⠀⠀⠀⠀⠀⡇⠀⣀⣴⣾⡿⠦⡀⢠⠄⠀⠠⠤⣄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠋⠉⠉⠉⠛⠙⣷⣤⣤⣤⣤⣼⣇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠂⠀⠀⠀⠀⠐⠘⠛⠛⠛⠛⠛⣺
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠈⠉⠉⠋⠀⠀⠀⢄⣀⡀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠁⡀⠉⠀⡈⠍⢠⣄⣀⣀⣀⣀⣤⠀⠰⠞⡿⠧⠀⠀⣿
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⡇⠀⠈⠉⠈⡇⠀⠀⢀⣛⢀⢀⡀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢨⢸⣃⠀⠀⢀⢀⣆⠀⢫⢘⢘⢰⢸⡅⢸⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⡀⠀⠀⠀⠀⢁⣀⣀⣤⠀⠀⠙⠁⠀⠀⠀⠘⢳⡊⢆⠈⠀⡈⠻⢃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠁⣀⣀⡀⠙⠛⠛⠛⠛⠋⠁⠉⠋⠚⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠉⢉⠉⠈⠐⠂⠠⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⢀⡆⠀⠄⠂⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠐⣚⣻

⠀⠀⠀⠀⠀⠀⠂
⠀⠰⠂⣰⠀⠀⠲⠂⠲⠒⠒⢲⠒⠒⠶⢲⠒⠒⠒⠒⠒⠒⠶⠒⠂⠠⢦⠤⠤⢤⣤⠤⠤⢤⡤⠤⡤⠠⣄⣤⣀
⠀⢀⣴⠃⠀⠀⢰⡄⡈⡀⠀⣈⠀⠀⢁⣨⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⡀⠀⡆⠀⢸⠀⡀
⠀⢸⣧⠠⣿⠀⠀⠉⡋⠁⠀⠙⣧⠀⠁⣭⣿⠄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣄⠀⠃⠀⢸⠀⡇
⠀⠉⠀⠀⣿⠀⠀⢠⢷⠀⠀⢰⡌⠀⠀⠘⠟⠀⠀⠀⠀⠀⠄⡗⠀⠀⠠⠀⠀⢀⢸⡇⠈⠀⠉⠀⣄⠀⢸⡀⠁
⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⠀⢸⡯⠀⠀⢰⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠰⠂⠀⢸⢸⠇⠀⢠⡀⠀⠈⠀⢸⡅⢀
⠀⠀⠀⠀⣿⡄⠀⠀⠋⡄⠀⢸⡷⡄⠀⢸⠀⠀⠀⠀⡄⠹⠘⠛⠀⠀⠀⠀⠀⠀⣼⠀⡄⠸⡆⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠂⠀⡆⠁⠀⢸⡀⠀⠀⠀⠀⠘⢄⣀⠀⠀⠀⠀⠀⠀⠉⠀⢺⡄⠀⠄⠀⠀⢸⡇
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⠇⢀⠀⠀⠀⠀⠘⢻⡇⠀⠀⠀⠀⡀⠀⠀⢸⡁⠀⠀⠀⠀⢸⡗⡴⠂
⠀⠀⠀⠰⢸⠁⣠⠄⠀⢰⠀⢠⣽⠀⠀⡆⠀⠈⠀⠀⠀⡆⣿⣾⡜⡀⠀⠀⠀⢠⠀⠀⠸⡇⠀⠀⠀⠂⢸⣿⠀⠁
⠀⠀⠀⢰⢸⠀⢿⠀⠀⢛⠀⠀⠟⠀⠀⠇⠀⠀⠀⠀⠀⢃⠀⢈⢻⡇⠀⠀⠀⣠⠀⠀⠀⠀⠀⠑⠐⠁⢸⣏⡄
⠀⠀⠀⢸⢸⠀⢸⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠰⣦⠀⠀⠈⠿⠿⡎⡶⠀⠀⠀⡏⡀⠀⠀⠀⠀⠆⠀⠀⢸⣿⣓⠛⡀
⠀⠀⠀⠘⢸⡇⠀⢠⠀⢸⡆⠀⢠⠄⣠⡄⠀⠀⢋⠀⠀⠀⣀⣀⣇⠰⠀⢀⠀⠁⠇⠀⠀⠀⠀⠄⠀⠀⢸⣿⣬⡤⠃
⠀⠀⢰⣷⢸⡇⠀⠸⠀⠈⠣⠄⠘⠃⠿⠲⠀⠀⠘⣄⣴⡾⠿⠿⣿⠀⣷⠏⠀⠈⢂⠀⠀⡄⠀⠀⠀⡄⢸⡇
⠀⠀⡶⠀⢸⡇⠀⠀⠀⠴⠀⡀⠀⠰⡇⢰⠀⠀⠀⠀⢸⣧⣀⠀⢿⠀⣿⠃⠈⡄⡏⠀⠀⠁⢀⠀⠀⠀⢸
⠀⠀⢱⣦⠸⣇⣴⠀⠂⠁⠀⡇⠀⢀⡇⢸⠀⠀⢀⠀⢸⡿⠧⠟⢸⡆⣿⠀⠀⡷⢁⠀⠃⠌⠉⠀⠀⡇⢸
⠀⠀⠀⠀⠀⡏⣿⠂⣀⠀⠀⠀⠀⢸⡇⠸⠀⠀⠐⠀⠘⢃⠀⠀⢐⡇⣿⠀⠀⣷⡴⠀⠀⠀⠈⢠⠀⡇⢸
⠀⠀⠀⠀⠀⡇⠚⠁⠀⠀⠈⡇⠀⢰⣧⠀⠀⠀⠀⠀⠀⢻⠰⢷⠈⡇⢹⠀⠀⠻⡼⠀⢠⠄⠀⢿⠀⠁⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⡟⢸⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢦⡀⠀⢸⠀⠀⢸⡀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡿⡇⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠃⠀⠀⠀⡅⡀⠈⠀⣴⠦⠀⠈⣾
⠀⠀⠀⠀⠀⡇⠀⠀⣀⠀⠈⠁⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢘⠀⢐⡀⡀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠘⠋⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣠⣴⢠⠀⣿
⠀⠀⠀⠀⠀⡧⠀⠀⣠⠀⠸⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠄⠀⠀⠘⠃⣉⡏⢀⣿
⠀⠀⢀⠄⠈⠁⢠⣬⣿⣦⣢⣇⠀⠀⠀⠀⢴⡧⣤⡄⠀⢠⣇⢸⡆⠀⢰⣈⠀⠳⢦⠀⠁⢠⣶⠤⠀⡴⠙⠂⠄
⠀⠰⡗⠒⠒⠒⠂⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠌⠡⠄⠲⠁⠐⠒⠐⠒⠒⠒⠒⠒⠒⠒⠚⠛⠒⢲⡀
⠀⢎⣽⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟
⠀⠉⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠠⠐⠺⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣸⠁⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡄⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡱⢸⡇⣎⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣓⠒⢒⣒⣒⣒⣒⣒⣒⣒⢒⠒⠚⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠉⠉⠉⠉⠉⢩⢛⡍⠉⠉⠉⠉⠉⢸⠉⠉⠉⠉⠉⡝⣩⠉⠉⠉⠉⠉⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⢸⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠲⠤⠦⠤⠤⠤⠤⠤⠔⠒⠒⠒⠒⠒⠒⠛⠿⠿⠷⠞


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

⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⢰⣄⣀⣀⣤⣾⣇⢁⡄⢠⡀⣬⠁⣦⢀⡄⣠⠀⣄⢀⡄⢠⡀⣤⠀⡄⣠⠀⣤⢠⡄⢠⡀⣤⢠⡤⣠⠀⣧⠘⣄⢠⡄⢠⡀⣈⣳⣤⣠⣠⣤⣆
⠀⠀⠀⢸⣤⣦⣤⣾⣯⣹⣿⣏⣿⡛⠃⠀⢿⠀⢉⣭⡤⣉⣠⣥⣌⣁⣩⣄⠑⣨⢀⣥⢨⣥⣈⣄⢩⣌⣥⣤⡌⣥⠀⠘⣏⣽⣿⣿⣿⣿⣿⣶⣵⣾⣏
⠀⠀⠀⠐⠀⠀⠈⠝⠓⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠐⠒⠚⠋⠀⠀⠀⠋
⠀⠀⠀⠀⣆⣀⣀⣇⠀⠀⠀⢀⡀⠀⠀⠀⣠⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⣀⣴⠄
⠀⠀⠀⠀⢸⣉⣟⠀⣤⣄⢴⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠠⣤⣠⣤⡀⣻⣀⣇
⠀⠀⠀⠀⢻⣿⣽⢋⣿⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣿⡋⢿⣦⡞
⠀⠀⠀⠀⢸⠁⠸⡀⣀⣀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⣀⣀⡀⢸⡿⡇
⠀⠀⠀⠀⢸⡇⠀⠂⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢸⣿⡿⣿⡇⠸⠃
⠀⠀⠀⠀⢸⣷⢸⡂⣿⠅⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠈⣿⡇⠀⠇
⠀⠀⠀⠀⢸⣿⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⢸
⠀⠀⠀⠀⠸⡇⠀⠀⣤⣤⣤⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣤⣤⣄⡇⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⡉⣉⠉⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠈⡉⡈⢉⠁
⠀⠀⠀⠀⢠⠘⠀⠀⣶⢏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢤⡜⢷
⠀⠀⠀⠠⡋⠉⠉⡶⡇⠀⠻⠀⠀⠀⠀⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠙⡁⢸⢦⢉⣉⢁⠄
⠀⠀⠀⠘⢄⣀⣀⡜⡗⡏⢆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠷⢸⣿⠣⣀⣀⡠
⠀⠀⠀⠀⢸⠀⢨⡇⡆⠙⠻⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠻⢹⠀⠋⠈⠃
⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⠀⢠⠀⣠⡾⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢶⣾
⠀⠀⠀⠀⢸⣄⣰⡆⠍⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠸⠀⣄⠀⡄
⠀⠀⠀⠀⡌⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⢰
⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠚
⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⢀⠂⠀⠀⠀⠀⠠⠤⠒⠈⠈⠀⠀⠰⠇⠠⠐⢁⠀⠀⠆⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⡰⠁⡀
⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠠⠀⠀⡠⠈⠑⠄⣀⠠⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⠀⠀⠀⡐⢊⡤⠈⠀⠀⢀⠄
⠐⠉⠀⠀⠀⢀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⠁⠎⠀⠀⡄⢂⠠⠋⠀⠀⢀⠔⠊
⠀⢀⣀⡀⠀⠀⠀⠀⣀⡐⢄⡠⠒⠀⠀⠀⠀⠀⠀⡄⠀⢀⠂⡀⠴⠗⠂⠂⡀⠀⠀⠀⠀⢀⣼⠘⠀⠀⠉⠀⠀⠀⠠⠒⢀⢀⣀⣀⡀
⠀⠈⠀⠈⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⠊⠘⠀⠂⠀⠀⠈⠀⠀⣀⠤⠋⡁⣀⠀⠀⠠⢂⠀⢰⣿⣿⡿⠹⣿⣿⡷
⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣶⠆⣄⢀⡄⠸⠀⡋⠀⠈⠉⠀⠒⠂⠀⠠⠴⠊⠁⡠⠚⠀⣿⡀⠀⠀⣿⠂⠈⠛⠋⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠐⠀⠀⠠⠀⠀⢈⠀⠀⠄⠀⠈⠁⠒⠀⠤⠀⠀⠀⠀⣠⠔⠋⠀⠀⣼⣿⣷⣶⣾⣿⣧⠀⠠⠀⠀⠀⠀⠠
⠀⣀⠀⠀⠨⢀⠀⠀⠀⢀⠃⠀⡄⠀⠀⠰⠀⠠⢀⠄⠀⢀⣀⠀⠄⠤⠄⠄⣀⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠏⠀⠀⢂⠀⠀⡐⠀⠃
⠀⡀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⠈⠀⠀⠁⠐⠀⠀⠂
⠈⠸⣿⡿⠀⠀⠺⣿⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⡰⠀⢰⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⣿⣿⣿⡀⠀⡀
⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⢄⠀⠀⠀⠀⢀⠀⠀⠀⢇⠀⠀⠆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠛⠿⣿⣿⡿⠿⠇⠸⡀⠀⠀⠁⠀⠀⡸
⠀⡈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠐⠀⠈⠁⠎⠀⠀⠀⢀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠤⠀⠄
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠛⡀⠀⠀⠐⠀⠂
⠈⢀⠀⠀⠀⡇⠀⠀⠀⠐⠀⠁⠐⠂⠀⠒⠈⠠⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⡐⠀⠀⢂
⠀⠀⠈⠀⡈⠈⠀⠀⠀⣀⡀⠀⣴⣶⣆⠀⠀⠄⠀⠁⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠡⠀⠀⡜
⠀⡀⠀⠀⠀⢀⠑⠀⠻⣿⠿⠀⠈⠛⠁⠰⡁⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠢⠀⠂⠀⠀⠈⠀⠀⠒⠀⠐⠂⠀⠁⠀⠅⠀⡆⠀⠀⠀⠃
⠀⠑⠠⣀⠀⠙⠃⠠⡀⠀⠠⠄⠀⠀⠀⠐⠈⠂⢨⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠜
⠀⡄⠀⠈⠑⠠⠄⣀⠨⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠐⠀⠀⠀⠐⠁⠀⢀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠐⢠
⢀⠈⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⡀⠀⠀⡄⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠘⢃⠀⠀⠀⢁⠀⠀⣀⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠆⠀⠀⠀⢀⠆
⠀⠀⠠⡀⠀⠀⠀⠀⡌⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠆⠀⠀⠈⣀⡀⠀⠤⠄⠀⠀⡤⠠⠀⠤⠀⠀⠀⠂⠀⠀⠀⡄
⠀⠀⠂⠈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠆
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠃⠘⠀⠀⠀⠀⠁⠠⠠⠄⠠⠂⠀⠀⠀⠑⠂⠀⠘⠁⠀⠒⠈⠁⠀⠀⠀⠁⠈⠈⠀⠈

⠀⠀⠀⣠⣤⣶⣿⡦⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣻⠀⣛⠀⣻⠀⠀⠀⠀⠀⢸⣿⠀⣀⡀
⠀⠀⢠⣿⣿⣿⣿⠁⠀⠀⠀⠀⣀⡸⠿⠿⢿⣀⡀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⠀⠀⠀⠀⠘⠛⣷⣿⣿
⠀⠀⠻⡿⠟⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠿⠀⠻⠀⠻⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠒⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣶⢶⣶⣶⣶⢶⡄⠀⠀⠀⢀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⠤⡷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠻⠿⠃⠀⠀⠀⠠⠿⠿⠯⠽⠿⠷⠀⠀⠀⠀⠀⢸⣿⣿⣿


⠀⠀⣴⣶⣶⣶⣶⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡟⢀⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠘⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢾⠷⠋


⠀⠀⣀⣖⣒⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢰⠒⠒⢶
⠀⠀⣿⡿⢿⢿⣿⡇⠀⠀⠀⠠⣷⣶⣾⣷⣤⣀⡀⠀⠀⠀⠀⠀⢀⣰⣿⡿⠋⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠉⠉⠙⠛⠉⠁⠀⠀⠀⠀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢠⡀⠀⠄⠀⠀⣠⠀⠀⠚⠉⠛⠛⠒⠢⠀⠉⠑⠉⠈⠈⠈⠀⠉⠡⠀⠒⠚⠛⠋⠛⠂
⠀⠀⠀⠀⠀⠀⣵⣶⣶⣶⣽⡅⠀⠀⠀⠀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⠀⠀⠰⣏⡯⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⢠⣿⣷⣦⣻⣷⣿⠻⠿⣷
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⣨⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠙⠛⠻⠿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⣀⠤⠤⣄⠀⠀⠈
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠸⣟⡷⢰⡷⣀⠃⠀⠀⡞⠁⠀⠀⢸⡆
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢰⣆⠀⠀⠀⠙⠃⣼⣧⣿⠁⠀⠀⣇⠀⠀⢀⡸⠃
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢿⣆⠀⠀⠀⠀⣿⠉⠃⠀⠀⠀⠈⠓⠛⠉⠀⠀⢰⠀⣅
⠀⠀⠐⠿⠿⠛⠋⠙⠟⠛⠉⠁⠀⠀⠀⠀⠈⠁⢀⡤⠶⣦⡀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠺⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⢚⣂⠀⠀⠀⣾⠀⠀⠀⣷⠀⠀⠀⢰⣿⠻⡀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣾⣿⣆⠀⠀⠙⣦⣠⠴⠃⠀⣠⣤⣾⠇⠀⠇⠀⢤⣤⣾⣷⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⣿⠁⠀⠀⠚⠛⠉⠁⠀⠀⠀⠀⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⢸⣓⠆⠀⠀⠀⠀⠀⢀⣴⠿⠒⠺⢧⣄⠀⠀⠀⠀⠀⠀⣶⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣄⣀⠀⠀⠀⠀⠀⢀⡴⠶⠶⢄
⠀⠀⠀⡰⠛⠉⠛⢦⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢀⣿⡀⣸⣇⠀⠀⠀⠀⠀⢠⠞⠁⠀⠈⢻⡟⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⣧
⠀⠀⠀⣇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠻⣧⣀⠀⢀⣠⡿⠀⠀⠀⠀⠀⢸⣛⡧⣟⣻⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠻⣄⡀⢀⣠⠇
⠀⠀⠀⠈⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠓⠆⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⠀⠀⠀⢸⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⣠⣽⣧⡀⠀⠀⠀⠀⠀⢀⣠⣟⣉⣉⣳⣄
⠀⠀⠀⠀⢰⣧⡎⠀⠀⠀⠀⠀⠀⠘⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⣀⡠⠄⠊⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⡀⠀⠀⠀⠀⣘⠛⠿⠧⠼⠿⠛⣃
⠀⠀⠀⠀⣮⣿⠇⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠘⢍⠝⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠟⠁⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡆
⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⢰⣶⣦⣤⣤⣶⣶⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡐⠉⠰⡄⣤⠊⠑⢆⠀⠀⠀⠀⢠⠖⠛⠛⢦
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠸⠟⠋⠉⠙⠛⠟⠀⠀⠀⠠⣦⣭⣝⣛⣛⣭⣥⡦⠀⠀⠀⠑⠤⠔⠀⠘⠠⠴⠃⠀⠀⠀⠀⢿⡀⠀⠀⣸⠇
⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⣀⣀⣀⡀⣀⣀⣀⠀⠀⠀⢀⣤⠤⢤⣀⣤⠤⢤⡄⠀⠀⠀⠀⡰⠚⠛⠓⢦⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹
⠀⢸⣿⣿⡿⠁⢿⣿⣿⡇⠀⠀⠘⣧⣀⡼⠉⠳⣄⣠⠇⠀⠀⠀⠐⡗⠀⠒⠀⢸⡇⠀⠀⠀⠀⠀⠐⣿⢊⣾⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠙⣦⣤⣴⠋⠀⠀⠀⠀⠀⠀⠰⢿⣷⣿⠿⠀⠀⠀⠀⠀⠀⠀⣷⡖⢶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁

⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣄⠀⠀⠀⠀⢰⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⣃⣀
⠀⠀⠀⠀⠺⠻⣿⣿⡿⠉⠀⠀⠀⠀⢠⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠘⢿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡃
⠀⠀⠀⣴⣶⣴⣿⣷⣤⣄⠀⠀⠀⠀⠀⣀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣁⣻⣷⣤⣤⣤⠀⠀⠀⠀⢀⣀⣴⣿⣷⣶⣶⣦⣄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⢁⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⢿⣧⡀
⠀⠀⣸⡟⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠏⠙⠉⠉⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⣿⣿⣿⣿⣿⢿⣧⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⠀⠀⠙⣿⣦
⠀⠀⣿⡃⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⠆⢿⡇⠀⠀⠀⠀⠀⢀⣴⣿⣥⣤⣽⣿⣿⣿⠇⠀⣿⡀⠀⣸⡟⠈⡿⠿⠿⠟⠀⠀⢀⣾⠿⠋
⠀⠀⢹⣇⠀⠈⣩⣽⣿⣿⡆⠀⠀⠀⣰⠃⡀⠓⠀⣼⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⢈⣻⣿⣿⠋⠀⣠⣿⠇⢰⣿⠃⠀⠋⠹⠿⣷⣶⠞⠋
⠀⠀⠀⢻⡆⠐⣿⣿⣿⣿⡇⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠢⢄⡀⠀⠀⠀⣠⣿⣿⣿⣿⣶⡟⠋⠁⢀⣿⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣺⣧⠀⢨⣿⣿⣿⣿⠀⠀⠀⠃⠁⠀⠀⠀⣿⠀⢄⠀⠀⠀⠉⢻⣶⢰⣿⣿⣿⣿⣿⣿⠁⠀⢀⣾⠁
⠀⠀⠀⠀⠉⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠉⠻⠆⠀⠢⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⠀⠆⠀⢳⡀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡇⠀⠀⠀⠀⠂⠀⢘⣧⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠁⠀⠀⠀⢀⣿⣷⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠀⠀⠀⠀⠘⣠⣾⠿⢿⡂⠤⠶⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠘⣿⠟⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⡿⣿⡏⠀⠀⠀⠀⠀⣸⡟⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⢀⡿⢁⣿⠁⠀⠀⠀⠀⢀⡿⠁⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⡇
⠀⠀⠀⢀⣾⠁⣸⡏⠀⠀⠀⠀⠀⣼⠃⢰⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⡇
⠀⠀⢀⣿⠇⡸⣿⡇⠀⠀⠀⠀⠀⣿⠱⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣷
⠀⠰⣿⠏⠀⠀⣾⡁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡿
⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣈⣙⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀
⠀⢠⠖⠉⠙⠛⠋⠉⠒⡄⠀⠀⠀⠀⣫⠹⠙⠩⣹⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡿⣟⡽⢿⠀⠀⠀⠀⠀⠀⢠⡶⣶
⠀⠈⠣⡄⠀⠀⠀⢠⠴⠁⠀⠀⠀⠀⠀⠀⢰⠀⢘⠀⠀⠀⠀⠀⢠⡏⠉⠉⠉⠉⣧⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⢸⠀⠀⠀⠀⠀⠸⣄⣘⠦⢮⣀⡼⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢷⣈⢙⣶⡀
⠀⠀⠀⠳⠤⠤⠤⠞⠀⠀⠀⠀⠀⠀⠸⠀⠿⢄⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⣠⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⣀⡴⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠙⡄⠀⠀⠀⠀⠀⠀⢰⠖⣶⣶⠲⡆
⠀⢰⡋⠁⠙⠩⠛⠈⠙⡆⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⠶⢖⢲⠀⠀⠀⠀⠀⠀⡠⢶⣄⣠⠧⣄⠀⠀⠀⠀⠀⢸⠔⠃⠘⠢⡇
⠀⠀⠛⡇⠀⠈⠀⢸⠛⠀⠀⠀⠀⢏⣁⣉⣀⡉⡲⣆⠀⠀⠀⠀⠀⠘⠈⢀⠀⠘⠀⠀⠀⠀⠀⢰⢀⠀⠈⠀⢀⠸⡀⠀⠀⠀⠀⢸⠃⢸⡄⢸⡇
⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⠒⠊⠒⠚⠀⠀⠀⠀⠀⡇⣼⣠⠆⠠⣸⡇⢇⠀⠀⠀⠀⠸⠁⣸⠇⠈⡇
⠀⠀⠀⠓⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠹⠥⠤⠤⠽⠹⠼⠀⠀⠀⠀⠘⠒⠋⠙⠒⠃

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⡯⣿⣿⠀⠀⠀⠀⠀⣶⣦⣶⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣤⣴⣶
⠀⠀⠰⠮⣭⣭⣭⡵⠖⠀⠀⠀⠀⠀⡿⢿⣟⡿⠿⠀⠀⠀⠀⠀⠾⠿⠟⠛⠿⠿⠿⠁⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋

⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⣸⣿⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⣰⠛⠛⠛⢳⡀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣠⣀⡀⠀⠀⠀⠀⣾⣿⡧⢾⣿⡷
⠀⠀⠀⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⡇⠀⢀⠀⠀⣇⠀⠀⠀⠀⠹⣿⡿⠋⠹⣿⣿⠃⠀⠀⠀⠀⢿⣿⡇⢺⣿⡿
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠸⠧⠤⠾⠦⠤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣾⠆⠸⣶⠇
⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡟⢰⢯⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⡇⣿⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⣿⣾⣿⣿⣿⣿⣿⣿⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣿⡿⣿⡿⠋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣷⣷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢟⡽⣻⣯⣾⠙⣿⣿⣮⡙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⢁⡾⣱⣿⣿⠃⠀⠈⠻⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⡟⢀⡿⣱⣿⠟⠁⣀⡀⠀⠀⠘⢿⣟⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣾⢀⣾⣿⡿⠃⠰⠋⠁⠀⢀⠀⠀⠸⢃⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⡇⣼⣿⡟⠀⠀⠀⠠⠶⠞⠃⠀⠀⠀⠘⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⢀⣤⣤⣄⠀⠀⠀⢠⠎⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⡀⠈⠛⠛⠃⠀⢀⠔⠁⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⢌⠢⣄⣀⠠⠐⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⡦⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠐⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⣠⡸⠋⠁⠀⠀⠀⠉⠀⣉⢐⡾⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠁⠀⠠⡆⠀⠀⠀⠀⡈⢁⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢘⡁⣲
⠀⠀⠀⡐⢀⠀⠀⠈⠑⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⣭
⠀⠀⠀⠹⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠚⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣖⣤⡄⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢈⡁⠐
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣼⡟⠉⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄⠒
⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠂⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣮⣁⠀⠠⠀⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢘⡁⡤
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⣿⣿⣿⡿⠨⣼⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⠀⠀⠀⢀⡁⢠⡄⠄⢀
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣛⣛⡿⠇⢀⠏⢹⢿⠀⠀⠀⠀⠰⠀⠀⠀⠈⠀⠘⣿⣃⡠⠂⢸⡏⠩⠁⠖
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠙⢫⡂⡦⢰⣟⡂⠀⠀⠀⠀⠊⠀⢀⠀⠀⠀⠀⢿⣯⡖⢲⢻⡍⢸⡍⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠙⣾⠛⢷⠒⣟⢂⠐⡀⠀⢘⢂⡀⠰⠟⠾⠛⠛⠚⠈⠃⠘⠀⠋⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⣶⠀⠈⢄⠉⠀⠚⢻⢻⡁⣀⣀⣀⠀⠀⠀⠀⠀⠐⢛⠄⢐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⢸⡏⠙⢶⣘⠀⠀⠀⠀⠀⠀⠉⣩⡯⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⢀⢅⣾⡥⠆⠀⠙⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡼⣁⣾⠚⡞⠛⠀⠀⠰⣶⣄⡀
⠀⠀⠀⠀⠀⣰⠷⠹⢹⡆⠀⠀⠀⠀⠀⣈⢳⣳⢂
⠀⠀⠀⠀⠀⢸⢊⣏⣼⠃⠀⠀⠀⠀⠀⠈⣮⡉⣼⢣
⠀⠀⠀⠀⠀⠀⡘⠙⠑⠀⠀⠀⠀⠀⠀⠀⠻⠌⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡒⣁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣯⣿⣽⣷⡀⠀⠀⠀⠀⠀⠀⢀⢾⢹⣹⣽⣿⣵⢞⡃⠀⠀⠀⠀⠀⠀⡰⣷⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠠⡎⠀⢘⣿⣿⣿⠇⠙⣆⠀⠀⠀⠀⠀⡜⠀⣯⣿⣿⣟⣵⡿⠀⠀⠀⠀⠀⠀⡐⠀⢹⣿⣿⠇⠈⢄
⠀⠀⠀⠀⠀⠀⠀⢣⣀⣾⣿⣿⣿⠀⠀⠹⠀⠀⠀⠀⠀⡁⢀⣾⣿⣿⣿⡏⠡⠀⠀⠀⠀⠀⠰⡁⢠⣿⣿⣿⣦⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⢳⠀⠀⠀⠀⠀⠀⠀⠑⣾⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠘⡎⢇⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⣯⢹⣿⡿⡄⠀⠀⠀⠀⠀⣰⡿⣿⠋⠻⣾⣿⣆⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠘⠘⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠃⠀⣀⡜⠀⠀⠀⠀⠀⢰⡟⢱⡏⠀⠀⠈⢿⣿⡄⠘⠷
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢀⡔⣩⢼⡇⠀⠀⠀⠀⠀⣿⣡⠈⢹⡏⠐⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⢻⠀⡀⢐⠀⠀⠸⡇⠀⠀⠀⠀⠘⢛⡟⠀⢸⡇⠐⠊⢱⡸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣅⠀⠀⠀⠀⠂⠇⠀⠀⠀⠀⠀⣼⢘⡞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢳⠀⢸⡇⠀⠀⠀⠹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠨⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣷⢻⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡀⠀⣇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠢⠀⠀⠃⠄⠀⠀⠀⠀⠀⠏⠀⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⢹⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⢁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣀⢀⣠⣤⢠⣄⣤⢠⣤⣤⣠⣤⣄⡤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠚⠂⠀⠀⠘⣿⣤⣤⡿⠇⠀⠀⠐⠃⠀⠈⠁
⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⠀⢀⣀⡀⠀⠀⢀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢱⠶⠶⠶⠶⠶⠶⠦⢦⠶⠶⠲⠶⠶⠶⠶⠶⠆⠀⠀⡀⡀
⠀⠀⠀⠀⢿⣏⠉⠀⠀⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⢊⣷⡄⠀⠀⠀⠀⡠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣮⠙
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⢤⡄⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣡⠾⣧⣤⡤⠦⢠⣤⡀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠉⠀⠀⠀⢀⡤⠈⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠲⠛⡟⠓⢀⠀⠛⠁⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⢠⠀⠀⠀⠸⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠈⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣿⡄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⣿⡆⠀⢰⡀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠟⡇⠀⠸⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢲⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠃⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠛⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠁⡳⠀⣈⣀⣁⣀⣀⣀⣠⡤⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢃⣺⠋⠀⠀⠀⠀⠀⠛⠉⢁⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⡉⣹⣄⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠆⠀⠀⠀⠀⢋⠀⠀⠀⠒⡀⠛⠚⠛⡹⢿⣻⣿⢿⣿⣿⣿⣟⠛⣛⡻⢽⣯⢿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⢠⣀⡴⣂⣈⠗⢀⢠⡔⣤⡀⢠⡀⠀⣿⠀⠀⠀⠒⠓⠚⠛⠛⠛⠛⠛⠃⣈⠃⠈⠻⠒⠛⣻⠟⠋⡽⠿⠿⠆⠹⠟⠛⠱⠶⠾⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⡟⠁⢻⠇⢨⣷⡟⣡⣶⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⣀⢀⡷⢠⠾⠧⠴⠇⠠⠤⠿⠡⠾⢿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠜⠋⠀⠀⠀⠀⠀⡤⠀⠐
⠀⠀⠀⠀⠀⠀⣤⣷⠐⢦⣤⡄⢠⣄⢠⡀⡄⢠⣠⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⢸⡿⢹⡏⣿⣭⠟⣴⠞⣰⣿⣿⡿⣿⣿⣟⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠶⠠⠄⡰⠷⢌⣴⠯⣵⠿⠥⠀⠧⠄⠮⠇⠀⣦⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠉⠀⠉⠈⠉⣉⡉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠈⠉⠁⠈⠉⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⡠⠀⠊⠁⠀⢉⡀
⠀⠀⠀⠀⠘⡧⠀⠀⣀⡤⣿⠷⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠙⠚⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣠⡶⠟⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⣭⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⠀⠙⡿⠗⠲⢲⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⢹⣿⠷⢾⣶⣶⣾⣿⣿⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡘⣷⡀⠀⠀⠀⠀⠀⠰⢄⡲⣭⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢷⡀⠀⠀⢀⣴⠟⠲⣽⣿⣿⣿⡄⣿⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣦⣀⣾⣿⣦⣀⡀⠑⠫⣛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⣶⣤⣙⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢳⣏⡿⠸⡷⢈⠛⠛⢻⣟⠿⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡿⣿⠁⠄⠀⠘⠀⢀⠏⠉⢳⣌⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢀⡾⣰⠃⢀⣤⣴⣶⣶⣴⣄⠀⠀⠙⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠞⠁⠏⢠⣾⢻⣿⣿⠟⢿⣿⣷⣤⡀⢿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠚⠀⡼⠀⣿⣿⠀⢿⣿⠀⢰⣿⣿⣿⡽⠜⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣴⠋⠀⠀⢠⠀⢠⣿⣿⡇⠘⣿⡄⢸⡿⣿⣿⣧⡚⢿⣿⣿⡿⣿⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠘⠇⠀⠀⢠⡄⠀⣸⣿⣿⣿⡄⢸⡇⣾⣿⡏⢿⣿⡇⠀⠹⣿⣧⡹⣧⠀⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡇⠀⢠⠟⠀⣰⣿⣿⣿⣿⠷⢼⣷⣿⣿⡀⢸⡟⣧⡀⠀⢻⣿⢿⢿⣷⡀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠂⠀⣾⢱⣿⣿⣿⡟⠀⢸⣷⣿⢻⡇⠘⣧⣿⣦⡀⢸⣏⠈⢂⡙⣿⣆⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⡿⢜⣧⠀⣿⣿⡙⢷⡌⣿⡄⠀⠱⡸⢿⣿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⣾⣿⠇⠘⣿⠀⣿⢿⡇⠀⢻⠛⢷⣤⠀⠘⣼⣿⡅⠹⣷
⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢰⣿⣱⣿⣿⣿⣿⣿⣷⠀⣿⣿⠀⠀⣧⢀⣿⠘⣧⠐⠘⡀⠀⠙⣷⡄⠘⣿⣇⠀⣿⡆
⠀⠀⠀⠀⠀⠛⣧⠀⠀⠀⠀⣠⡾⢋⣿⣿⣿⣿⣿⣿⣿⡆⢸⡇⠀⢰⠇⢸⡟⠀⢹⡆⠀⢁⠀⠀⠈⣿⡄⢸⣿⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠘⠛⠶⢦⣾⣏⣴⣿⣿⡿⢹⣿⣿⣿⣿⢷⣸⡏⠀⢿⠀⠀⡇⢠⠀⣷⠀⠈⣄⣀⣀⣘⡇⢀⣷⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣷⣿⣿⣹⣿⡏⣼⠉⣷⠀⠀⠀⠀⣷⣶⣾⣿⣧⡀⢻⡟⠛⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠻⠿⠟⢻⣄⡿⠂⢹⡷⠶⠾⠾⠟⠉⠉⠛⠛⠛⠛⠁⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢴⠾⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣉⣋⣉⣉⣩⣏⣉⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⠀⣰⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠴⢿⣿⡟⠋⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⠀⢀⣽⣶⡶⠆⠀⠀⠀⠀⠐⠞⣿⣿⠿⠂
⠀⠀⠀⢨⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠀⠀⠀⠀⠀⢈⣿⣄⡀
⠀⣼⣿⣿⣿⣷⡆⠀⠀⠀⢠⣶⣾⣿⣷⣶⡄⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣦⠀⠀⠀⢰⣿⣿⣿⣿
⢠⡟⢿⣿⣿⣿⡇⠀⠀⠀⣾⠛⣿⣿⣿⠙⣧⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠙⢷⡄⠀⠈⢿⣿⣿⣿
⢸⡇⠘⣿⣿⣿⡇⠀⠀⠀⣿⠀⣻⣿⢛⡀⣿⠀⠀⠀⣸⡇⢹⣟⣛⠁⣠⡴⠛⠁⠀⢠⣾⣿⡿⢿⡄
⠀⢻⡀⢼⣿⣿⣧⠀⠀⠀⠘⠷⢀⠉⠐⠄⠇⠀⠀⢰⡟⠀⠀⠈⠉⠘⠉⠁⠀⠀⠀⣸⣿⣿⣷⠈⣷⡀
⠀⢸⡇⢨⣿⣿⣿⠀⠀⠀⠠⠂⠀⢄⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣷⡈⢠⣶⠀⢀⣿⣿⣿⣿⡄⠙⣇
⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠑⡆⠀⠀⠁⠀⠀⠘⠇⠀⠀⠀⠀⠠⣿⣷⣸⠃⠀⣼⣿⣿⣿⣿⣿⠀⠙⠁
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⢻⣿⣿⣿⠿
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿
⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⣸⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣿⢹⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⣴⠇⣼⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄
⢸⠟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠴⠒⠒⠛⠉⠉⠙⠲⡖⠚⠛⡟⠲⢤⣀
⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡧⠤⢤⡿⣦⣤⣈⣙⣲⣄
⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠐⠒⠉⢇⣀⡠⠤⢼⣅⣀⣀⣀⠀⠉⠉⡇
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠂⠉⢁⣘⡠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠓⢦⣀
⠀⠀⢀⡜⠢⣄⠀⠀⡰⠋⠁⣀⠽⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠴⠖⠻⡄
⠀⠀⢸⡁⠀⠀⠙⠲⣇⡀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠶⠒⠒⠛⠉⠉⠁⣀⣀⣀⣤⠤⠤⢼⠁
⠀⠀⠀⣳⢄⡀⠀⠸⠀⢹⠲⣄⡀⠀⠀⠀⣠⠴⠒⠋⠉⢁⣀⣀⣤⠤⠴⠒⠒⠛⠉⠉⠉⣀⣀⣤⠤⠖⢺⡄
⠀⠀⢠⡇⠀⠙⠲⣴⠀⠀⠀⠀⠙⠦⣄⢰⠃⠀⠖⠚⠉⠉⢁⣀⣀⣀⡤⠤⠖⠒⠛⣉⣉⣉⡤⠤⠴⠖⢺⡀
⠀⠀⠀⠳⣀⠀⠀⡜⠙⣶⣄⡀⠀⠀⠀⢙⡷⠲⠒⠚⠛⠉⢉⣉⣠⠤⠴⠖⠒⠛⠉⠉⣀⣀⣤⠤⠴⠶⣏
⠀⠀⠀⠀⠈⠓⢦⡇⠀⡇⠀⠉⠲⢤⡀⢸⠀⠀⠀⠒⠛⠉⠉⢁⣀⣠⡤⠴⠖⠚⠋⠉⢁⣀⣠⡤⠤⠖⢯
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣇⠀⠀⠀⠀⠈⢛⡷⠤⠶⠖⠒⠛⠉⠉⣁⣀⣤⠤⠶⠒⠚⠋⠉⠀⢀⣀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀⠀⢸⠀⠀⣤⠤⠶⠒⠛⠉⠁⣀⣀⣀⡤⠤⠶⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣦⣀⣤⠤⠶⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣎⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⡹⠶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢐⣋⣘⣸⣉⣀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢀⠂⡆⢡⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⠀⠀⠀⠀⣍⢸⠀⠀⠅⠈⠀⠀⠀⠀⠳⠂⠄⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢰⠚⡔⡇⠜⢸⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠀⠘⠒⠂⠃⠂⠘⠊⠀⠀⣾⣷⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡴⠢⠆⠀⠲⣄⡀⠀⠀⡘⠉⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⡔⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⠀⠀⠀⢀⢸⠃⠁⠀⠃⠀⠀⢀⠡⠀⠀⢠
⠀⠀⠀⠀⠀⣥⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⢀⠈⠀⠀⠀⠀⢀⠀⠉⢉⡱⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒
⠀⠀⠀⠀⣄⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠷⠋⢯⣄⠀⠀⢠⣦⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠍⠿⠹⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢼⠀⠠⣻⠈⢌⡆⠀⠀⢰⣦⢤⢠⠄⠀⢠⢤⡤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠊⠀⠀⠀⠈
⠀⠀⠀⢹⠈⡌⣤⡺⡃⡡⠐⠀⠀⠀⠘⠛⠂⠈⠇⠀⠛⠾⠥⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣸⡔⡏⣟⣓⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣯⡧⡏⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣸⠼⠽⠼⣌⣧⠀⠀⠀⣗⢰⡄⣀⡀
⠀⠀⠀⠉⠙⠻⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢤⣦⡆⠀⠀⠀⠀⢰⣦⡤⣦⠀⠀⠀⢠⣯⠃⠀⠀⠀⢠⣽⡀⠀⠀⠉⠀⠷⣿⡅⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⡀⠄⠀⠀⠀⠤⠄⣀⡀⠀⠀⠀⠦⠤⠤⠆⠀⠀⠀⠀⠸⠤⠤⠴⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⢳⠀⣨⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠄⠠⠁⠀⢀⠛⠱⡶⠁⠀⠀⠀⠰⠶⠶⠟⠀⠀⠀⠀⠹⠶⠶⠾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣮⢭⣵⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠤⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢺⣭⣻⢶⣒⣫⣵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠓⠋⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡏⠀⢠⡷⠶⠴⣏⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠠⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠇⠀⡇⠄⠌⢷⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣂⣅⣠⠸⡄

⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀
⠀⠀⠀⣸⣿⣷⡀⠀⠀⠀⠀⠀⠘⣿⣿⣽⣝⣿⡿⠀⠀⠀⠀⠀⠸⣿⡆⣴⣿⠇⠀⠀⠀⠀⠀⠻⣿⠟⠺⢿⠿
⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣤⠤
⠀⠀⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⠀⠀⠼⠿⠃⠙⠿⠧⠀⠀⠀⠀⠀⠀⠀⠘⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣀⣆⡀⠀⠀⠀⠀⠀⠀⢰⣲⣾⣤⣿⣗⡆⠀⠀⠀⠀⠀⠐⢗⣣⢜⡾⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⣾⢧⣷⢿⡄⠀⠀⠀⠀⠀⠀⠉⡏⡇⡏⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡾
⠘⠛⠛⠋⠉⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⢀
⠀⠀⠛⢷⡄⠀⠀⠀⠀⠀⠀⠰⢂⡐⢪⠴⠲⢤⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠑⡄⠀⠀⠀⣸⣿⣗⠀⣀⣀⡀
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠉⠟⠉⠀⠀⠀⢸⠄⠀⠀⠀⢸⠀⠀⡆⢰⠀⠀⡇⠀⠀⡞⠁⠀⠈⢳⡏⠙⢿
⠀⠀⠀⠈⣿⣿⣦⡄⠀⠀⠀⠀⠀⠈⠀⢀⠀⢀⣸⠀⠀⠀⠀⠈⠳⣄⣀⣀⣠⠜⠁⠀⠀⠳⣀⠀⢀⡼⠑⣄⢈
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢀⡀⣐⠀⠀⠀⠀⠀⠀⣴⣆⣾⣿⠀⠀⠀⠀⠴⣾⣿⣿⣿⣷⣦
⠀⠀⢻⣷⣄⣀⢠⣤⡀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡟⠀⠀⠀⠀⠀⠀⢹⣟⣿⡇⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⡿⣧
⠀⠀⠸⠀⠙⢿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⡆⠀⠀⠀⠀⠀⠸⣿⡿⠃⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣠⣤⣤⠤⣤⣤⡀⠀⢀⣀⣤⡤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠐⢯⠉⠉⠀⠀⠀⢠⠇⠀⠀⢣⡀⠀⠀⠀⠀⣴⠁⠀⠈⢳⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠤⠠⠤⠠⣸⠀⠀⠀⠈⡗⠂⠐⠂⠀⣿⠀⠀⠀⠸⠆⠔⠐⠤⠒⣼
⠀⠀⠀⠀⡤⠴⠒⠒⠚⠃⠐⠀⠐⠒⠚⠒⠒⠒⠚⠃⠀⠀⠀⠁⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⣉⠱⡄
⠀⠀⠀⠀⢹⠀⢐⠆⠀⠉⠁⠀⠀⠀⠘⠉⠉⠉⠈⡃⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠀⡇⠀⠇⢰⠃
⠀⠀⠀⠀⢸⠀⠈⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡄⢸
⠀⠀⠀⠀⠸⡇⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⡇⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡇
⠀⠀⠀⠀⠀⡇⠀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⡇⠀⡆⠀⢸⠀⠀⠀⠀⠀⡆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠃⢸⡇
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡷⠀⣰⠄⠀⠧⡀⠀⢀⣰⠂⠀⠐⠂⠲⡀⠀⠀⠀⢠⠇⠀⠀⠸⡀⠀⠀⠀⡸⠁⠘⠀⡀⠳⡀
⠀⠀⠀⡠⠞⢁⡜⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠁⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠘⣄⠘⠦⡄
⠀⠀⠰⣁⣀⣈⣁⣈⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣈⣉⣉⣉⣉⣩⣭⣭⣭⣥⣤⣤⠤⠬⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢰⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⢠⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⣾⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣧⣟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⠟⠋⠋⡿⢸⣿⡏⢻⣿⡈⣿⣯⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⣹⢿⣿⠟⡥⠀⠀⢠⡇⠀⢿⡇⠈⣿⡇⠘⣿⣇⠈⢻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣾⣣⠏⠀⡼⠁⠀⠀⣼⠐⠀⠸⡇⠀⢸⣿⠀⠘⣿⡆⢻⡷⠈⠻⣯⡕⠢⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡾⡿⢣⣤⠎⠀⠀⠀⢀⢇⡇⠀⠀⣧⠀⠘⢿⡇⠀⠈⠙⠀⠙⠀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠜⠁⠘⠃⠀⠀⠀⠀⡈⣼⠃⠀⠀⣿⠀⢀⡈⠃⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠁⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠈⠂⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⣾⣿⣿⣶⠤⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠐⣛⣉⣤⡀
⠀⠀⠀⠀⠋⢻⣿⡿⠀⠀⠀⠀⠀⣾⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⠋⠁
⠀⠀⠀⢀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣻⣯⣀⣀⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⣴⣾⣿⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡆⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⣾⡿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠛⠿⠿⠰⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠙⢿⣦
⠀⢀⣿⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡜⢿⡇⠀⠀⠀⠀⠀⣠⣿⣃⣈⣿⣿⣿⡿⠁⣿⡀⠀⣸⡏⢹⣿⣿⣿⠁⠀⢀⣴⡿
⠀⠈⢿⡄⠈⢛⣿⣯⣿⠄⠀⠀⢀⡴⢀⠘⠄⣸⡇⠀⠀⠀⠀⠐⠛⠛⠉⠙⣿⣿⡿⠁⢀⣼⠇⢠⣿⠁⠸⠛⠷⣖⣤⡶⠛⠁
⠀⠀⠈⣷⠀⢾⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⣿⠀⠀⠐⠤⣀⠀⠀⠀⣠⣿⣿⣿⣧⡾⠛⠁⢀⡿⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣿⡄⢀⣿⣿⣿⣿⠀⠀⠘⠘⠀⠀⠀⣿⡀⠄⠀⠀⠈⠙⣷⢰⣿⣿⣿⣿⣿⠇⠀⢀⣾⠁
⠀⠀⠀⠈⠁⠘⣿⣿⣿⡿⠀⠀⠀⠀⠀⢀⠀⠈⠙⠀⠈⠢⣄⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠈⠃⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⡀⢠⣤⣿⡀⠀⣀⡤⠂⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⢠⣿⡟⠀⢤⣴⣤⠄
⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⢰⣿⣏⠙⠀⠉⠁⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⠀⠻⠅⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣰⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣸⠏⣿⠇⠀⠀⠀⠀⢸⠏⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡿
⠀⠀⠀⣠⡏⣠⣿⠀⠀⠀⠀⢀⣿⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿
⠀⠀⣰⡿⠠⢻⡿⠀⠀⠀⠀⢸⡏⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡄
⠀⠘⠟⠁⠀⠸⣇⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠃

⠈⠀⠐⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠠⠾⠤⠖⠄⠀⠀⢠⣥⣷⡦⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠞⠤⠒⠄⠀⠀⢀⣯⣴⡆⠀⠀⠀⠀⠤⠂
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡽⣿⠂⠀⠀⠠⠀⠠⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣽⣿⠀⠀⠀⡰
⠉⠀⠀⠀⠀⠀⠈⠐⠀⠁⠀⠘⠛⠛⠉⠀⠀⠄⠀⠀⠀⠠⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠈⠐⠀⠁⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣄⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣄⠀⠀⡀⠀⠀⠒⠒⠒
⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠸⠂⠸⠀⠀⣷⢾⠀⠀⠐⠒⡖⠀⠀⠀⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠈⠇⠂⠈⠀⠀⣷⢶⠀⠀⢰⠂⡖
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⡇⠀⠴⠚⡤⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⡇⠀⠴⠛⡤⠀⠀⣴
⠛⠂⠀⠀⠀⠀⠀⠀⠀⢀⠐⠶⠂⠀⠀⠉⠉⠀⠀⢲⠀⡇⠀⠶⠿⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⢀⠐⠶⠂⠀⠀⠉⠉⠀⠀⢲⠀⡧⠀⠶⠿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠉⢹⡀⠀⡀⢀⠀⠀⠸⠄⠇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠉⢹⡀⠀⣀⢀⠀⠀⠹⠄⡇
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠂⠃⠀⠉⠉⠀⠀⠀⢀⡶⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠀⠀⠂⠒⠒⠃⠃⠀⠉⠉⠀⠀⠀⢀⡴⡄
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⡘⠾⣷⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡘⠾⣷⠀⠀⠀⢀⣀
⢶⠖⠀⠀⠰⠖⠶⠂⠀⠀⠀⠂⠀⠀⠘⢿⣦⠀⠀⠀⠀⣾⣿⣆⠀⠀⢶⠖⡀⠀⠰⠖⠶⠂⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⢀⣾⣿⣦
⠈⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠰⠾⠀⠲⠆⠀⠀⢠⣏⣳⡆⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠀⠔⠆⠀⠀⢀⣧⣾⡆⠀⠀⠀⠀⠠
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⡽⣻⠂⠀⠀⠰⠀⠠⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣽⢿⠀⠀⠀⡐⠀⠤
⠁⠀⠀⠀⠀⠀⠈⠐⠀⠁⠀⠈⠙⠛⠉⠀⠀⠄⠀⠀⠀⠠⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠈⠐⠀⠁⠀⠈⠛⠋⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣄⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣄⠀⠀⡀⠀⠀⠒⠒⠒
⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⠃⠩⠀⠀⡷⢶⠀⠀⠰⠀⡖⠀⠀⠀⠉⠉⠈⠈⠁⠀⠀⠀⠀⠀⠈⠇⠁⠈⠀⠀⡷⢶⠀⠀⠰⠀⡖
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⡇⠀⠴⠛⡴⠀⠀⣤⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⡇⠀⠔⠋⡤⠀⠀⣤
⠛⠂⠀⠀⠀⠀⠀⠀⠀⢀⠲⠶⠆⠀⠀⠉⠉⠀⠀⢲⠀⡇⠀⠶⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⠰⠶⠆⠀⠀⠉⠉⠀⠀⢲⠀⡧⠀⠶⠿
⡐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠉⢹⡀⠀⣀⢀⠀⠀⠸⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⢹⡀⠀⣀⢀⠀⠀⠹⠄⠇
⢠⠀⠀⠀⠀⡀⠀⠀⠀⠂⠒⠒⠃⠃⠀⠉⠉⠀⠀⠀⢀⡶⡄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠂⠃⠀⠉⠉⠀⠀⠀⢀⡶⡄
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠾⣷⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⡘⠾⣷⠀⠀⠀⢀⣀
⢶⠖⡀⠀⠰⠖⠶⠂⠀⠀⠀⠂⠀⠀⠘⢿⣆⠀⠀⠀⠀⣾⣿⣦⠀⠀⢶⠖⡀⠀⠰⠖⠶⠂⠀⠀⠀⠂⠀⠀⠘⢿⣦⠀⠀⠀⠀⣾⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠃⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠂⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⠀⣀⠡⠀⠀⣍⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⣇⣀⣀⣀⣤⣀⣀⠊⠋⡀⢈⠙⢲⡇⠀⢀⠀⣿⢀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡞⠻⠚⠿⢷⠿⡶⡾⠻⢶⡟⣿⢻⣟⠛⠛⠛⠻⢿⡋⢉⠉⠁⠈⢀⢸⣷⠀⠈⠛⠛⢻⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡷⢶⣟⣶⢿⢦⣾⣩⠿⠞⢹⠉⣾⠋⠉⠉⠉⠉⠈⠉⠉⠁⠀⠉⠉⠙⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣿⡄⠀⠘⠃⠰⠀⠀⠐⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⡈⠀⠀⠀⢸⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠀⠀⠀⣦⣄⠀⠀⣠⣄⣐⡀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣶⡇⠀⠀⠀⢸⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⠀⠀⠀⣽⠂⠀⠀⠀⠀⠻⠄⠀⠀⠀⢀⡀⣧⠀⠀⠈⣿⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⠀⢸⠀⠸⠆⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⡀
⠀⠀⢀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⠀⠘⠀⠋⡆
⠀⠀⡸⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠃⠀⢃⣠⢶⠦⠠⡤⣤⡤⠤⢄⣀⣘⡂⠄⢸⠀⠀⠀⢸⠁⠀⠀⠀⠀⡄⢰⡇
⠀⠀⠰⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⣀⢀⡠⠀⠀⠀⠁⠀⢀⢰⣚⣚⣒⣧⣚⣃⡒⣀⣲⡷⢇⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⠁
⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⠀⠈⠃⠺⠤⠀⠀⠀⠀⠀⠸⠉⠉⠁⠁⠀⠉⠉⠉⠈⠁⠀⠙⠩⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠁⠀⠀⠀⠀⠀⡀⣀⡀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠤⣄⡀⠀⢀⠀⠀⠀⠀⠀⡇⠀⣀⣴⣾⡿⠦⡀⢠⠄⠀⠠⠤⣄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠋⠉⠉⠉⠛⠙⣷⣤⣤⣤⣤⣼⣇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠂⠀⠀⠀⠀⠐⠘⠛⠛⠛⠛⠛⣺
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠈⠉⠉⠋⠀⠀⠀⢄⣀⡀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠁⡀⠉⠀⡈⠍⢠⣄⣀⣀⣀⣀⣤⠀⠰⠞⡿⠧⠀⠀⣿
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⡇⠀⠈⠉⠈⡇⠀⠀⢀⣛⢀⢀⡀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢨⢸⣃⠀⠀⢀⢀⣆⠀⢫⢘⢘⢰⢸⡅⢸⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⡀⠀⠀⠀⠀⢁⣀⣀⣤⠀⠀⠙⠁⠀⠀⠀⠘⢳⡊⢆⠈⠀⡈⠻⢃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠁⣀⣀⡀⠙⠛⠛⠛⠛⠋⠁⠉⠋⠚⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠉⢉⠉⠈⠐⠂⠠⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⢀⡆⠀⠄⠂⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠐⣚⣻

⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⢶⣶⡶⠶⠶⠶⠶⠶⠶⢶⣶⢶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣤⣿⠇⠀⠀⠀⠀⠀⠀⠘⢋⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣧⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣠⣾⠟⠛⠛⢿⣯⣀⣽⡟⠛⠛⠻⣿⣟⠛⢻⣯⣀⣽⡟⠛⠛⠻⣷⣄⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⣠⣾⠟⢁⣤⡀⠀⠀⠉⠛⠉⠀⠀⢠⣤⡈⠻⣷⣄⠉⠛⠉⠀⠀⢠⣤⡈⠻⣷⣄⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠻⣷⣶⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣶⡿⠋⠀⠀⠀⠀⠀⢸⣿⢿⣶⡿⠋⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠈⠉⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⢿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠶⠶⠶⠶⠶⠶⠶⠶⢾⣿⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡄⠀⠀⢻⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣾⡏⠀⠀⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⣿⣍⣍⣩⣉⣉⣉⣩⣍⣍⣽⡗⠀⠀⢿⣯⣉⣉⣉⣉⣉⣉⣉⣩⣽⣿⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣇⣀⣀⣀⣿⣋⣉⣉⣉⣉⣉⣉⣉⣉⣻⣷⣀⣀⣾⣏⣉⣉⣉⣉⣉⣉⣉⣉⣹⣿⣀⣀⣀⣸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣿⣷⡆
⠀⠀⠀⠀⠀⠀⣾⣷⣤⣤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣾⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠂⠀⡄⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⣠⣤⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠂⠀⠀⠀⠹⠉⠛⠋⢁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡞⠻⣤⣶⣿⣁⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⣤⠀⠀⡄⣄⡝⢃⡼⠷⠃⠈⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣧⠄⠠⠀⠹⣶⠁⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠘⠡⣀⠀⢠⣿⣟⡀⠠⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⠂⠀⠀⢸⣏⢷⠈⢢⣜⣳⢆⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡀⠀⢰⣿⡇⠀⠀⠈⣿⢸⡇⡄⠙⢣⣴⠋⠑⢆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⣾⠿⠃⠀⠀⠀⠈⢸⠇⡇⠀⢈⠁⢀⣉⡋⠓⠀⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠙⢷⣼⣥⠀⠀⠀⠀⠀⠀⣸⠀⡇⢀⣼⠟⢩⣤⣿⠓⠀⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠌⢻⡀⠀⠀⠀⠀⢀⡇⠀⠁⢹⠿⠉⠉⠛⠋⢦⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡘⠀⡀⢠⠌⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⣤⠠⠉⢹⣰⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⣀⣀⣀⠴⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⠆⠀⠄⠀⡤⠤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⡁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠀⠸⢇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠆⠹⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠉⠋⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠦⠀⢠⡤⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⢰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⡀⣀⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢠⡤⡴⢿⣏⣱⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣍⠀⢂⡝⢌⡀⢾⣭⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⠶⣺⠀⢻⣬⡳⣝⣾⢮⣍⣣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠜⡁⢯⣰⣈⣿⣽⠿⣎⠳⢯⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠘⠒⠾⣽⣿⣳⣦⠈⢧⠻⢿⣷⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⣻⣻⣵⡻⠼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡱⢿⣿⣯⣹⣾⡬⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢠⣭⣷⣽⡪⢽⣏⢼⠨⣕⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣦⣾⣽⣵⣿⣮⣷⣔⣵⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠶⣥⣭⣍⣿⡟⢿⣫⣟⠿⡻⣷⣚⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡛⢲⠆⢸⣹⣮⢺⣿⡻⣕⢮⡛⢍⠓⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣿⡅⢸⡟⠟⣻⣿⣝⣼⡻⣿⣷⣏⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢐⣶⢿⠇⣸⣠⣮⣿⣿⣫⣏⣗⡟⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣿⠯⠞⣸⠇⣭⣝⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠘⣏⣒⡷⠶⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡿⣅⢯⣭⣉⣻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣍⣷⠷⢶⠶⢬⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⢻⣿⢿⡷⠒⠺⠖⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢺⣿⣏⣿⢛⣽⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⣋⡟⣿⡄⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠧⣾⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠉⠻⢫⡀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠁⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠐⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣾⠀⠸⣿⣶⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⢨
⠠⠤⠤⠠⠤⠤⠤⠤⠤⠠⢤⣄⠀⢀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠉⠉⠉⠉⠉⠙⠉⠈⠉⠉⠉⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠄⠀⠀⠀⠀⠂⠄⠤⠀⠄⠠⠄⠰⢄⠀⠀⠀⠀⠀⠀⢀⠠⠤⠸⠀⠧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒⠂⠒⠒⠒⠛⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠻⠐⠋⠙⠈⠁⠀⠀⠀⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁
⠐⠐⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢈⣥⠀⣀⠀⠀⠀⠀⡀⠀⣀⠂
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠈⠁⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⢤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠐





⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠰⣉⣟⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣸⠶⢤⠀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⢒⣟⣳⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠖⠋⠉⠉⠉⠉⠉⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠇⠀⠙⢟⢤⠚⣽⠎⣒⣏⣹⣒⡁
⠀⠀⡠⢚⠭⠟⠛⠛⠛⠯⣕⢦⡀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡇⢀⡤⠴⠒⠒⠒⠒⠒⠦⡄⢹⣽⠀⠀⠀⠀⠀⠘⢄⠀⠈⣼⡆⢠⡿⠓⠒⠒⠒⢽⣆
⠀⠀⣧⠇⠀⠀⠀⠀⠀⠀⠈⢳⣱⠀⠀⠀⠀⠀⠀⠀⢸⣤⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣏⢨⠁⠀⠀⠀⠀⠀⠈⠙⠚⠋⠀⠸⣄⠀⠀⠀⠀⢀⡟
⠀⡠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣠⡏⣳⠦⣤⠤⠤⣀⣤⠾⡇⣸⠏⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠈⠙⠛⠛⠋⠁
⢀⡗⢁⡐⠀⠀⠀⠀⠀⠀⠩⠂⡹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢯⣀⣿⣤⡀⢸⣺⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⡜⠁⠘⣰
⢸⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣍⣻⠟⠉⠀⠀⠀⠀⠀⠀⡤⣤⣄⠀⠉⣀⡼⠃
⠀⠳⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣨⡄⠀⠀⠀⠀⠀⠀⢸⡈⠀⢙⡀⢰⢏⣀⡀
⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⢇⠀⠀⠀⠀⠀⠀⠀⠻⣦⡈⢁⠞⠾⠂⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣸⠀⠀⠀⠀⠀⠀⢀⣀⣸⣷⠸⣄⠀⠀⢚⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢠⠀⠲⣄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠻⠿⠿⡀⠀⠀⠀⠀⢠⠋⠀⠻⠟⣆⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⢀⢶⡋⣉⡩⠓⣯⣽⠞⢯⣉⣉⡷⡄⠀⠀⠀⠀⢠⣁⣀⣀⣀⡇⠀⠀⠀⠀⠐⢄⡀⠀⣼⠟⠀⠀⠀⠀⠀⠐⡖⠒⠒⠂⡖⡆
⠀⠀⠀⠀⠀⢀⡴⠃⡼⢁⣽⣠⣞⣋⣉⣻⣄⣩⡈⢱⠹⡄⠀⠀⠀⢸⠛⠛⠛⠛⢳⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⣧⣀⡀⣀⣅⡁
⠀⠀⠀⠀⢠⡮⠤⠾⠀⢤⣬⣭⣭⣭⣭⣭⣭⣥⣤⣤⣧⣼⡄⠀⠀⣦⣤⣤⣤⣤⣤⣇⠀⠀⠀⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠉⠁⢸⠁
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⠉⠁⠀⠀⠀⠀⢹⠄⠀⢸⣞⣟⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠦⠖⢺
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢲⣶⣶⣶⣶⣶⠋⠀⠀⢈⡿⠿⠿⠧⣄⠀⠀⠀⠀⠀⠀⠀⣧⢀⡀⡀⣸
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠱⣄⣀⡴⠁⠀⠀⡴⢡⠂⠉⠙⠒⢄⠳⡄⠀⠀⠀⠀⢀⡇⠀⠉⠁⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⢸⡆⣷⠀⠀⠀⠀⣧⣷⣶⣶⣶⣾⣼⡆
⠀⢀⡤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⡤⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠⡄⠀⠀⣠⠜⡴⠃⠀⠀⠀⣾⡿⢋⠀⠀⣠⣍⢻⣷⣠
⢠⡏⡶⠈⢁⡈⠛⣶⡙⢿⠉⣤⠀⣀⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠤⠴⠶⠂⠀⠀⠀⠀⢸⣿⠇⠘⠻⠾⠟⠁⢸⢿⣿
⠘⣇⠠⡦⢈⣅⢉⣹⣬⡿⣀⣶⢀⡉⠸⠇⡟⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡣⡀⠀⠀⠀⣠⢪⠏
⠀⢻⡇⢀⣌⣩⡜⡻⠁⠘⢏⣓⢾⣁⣴⡞⠁⠀⠀⢀⡴⢚⣩⣼⠗⠀⠀⠀⠀⠀⢀⠴⣒⣢⡀⠀⣠⣒⠢⡀⠀⠀⠀⠀⠀⠀⡟⣮⣤⣶⣭⢶⣹⠃
⠀⠀⠙⢯⣍⣉⠟⠀⠀⠀⠀⠙⠺⠯⠚⠀⠀⢀⡴⣫⢖⡯⡴⠁⠀⠀⠀⠀⣠⣞⠕⠈⢸⣇⠇⠀⣏⡇⠑⠻⡦⡀⠀⠀⠀⠀⠙⡇⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠞⠁⢸⢰⠃⠀⠀⢀⡠⢊⣜⣁⣀⣀⠈⠁⠀⠀⠈⢁⣀⣀⣘⣎⡢⡀⠀⠀⠀⡗⠚⠉⠓⢺
⠀⠀⠀⠀⠀⢀⡠⠔⠒⣿⡀⠀⠀⠀⣠⢞⡵⠛⠲⢄⠘⢸⠀⠀⣠⠟⣻⣿⣶⣿⣷⣶⣽⣆⠀⡔⣭⣶⣾⣿⣶⣶⣭⣻⣦⠀⠀⣅⣀⢤⣀⣸
⠀⠀⠀⢸⠉⠀⠀⠀⠀⠘⣷⣄⣠⠞⣡⠎⠀⠀⠀⠈⢧⢸⠀⠀⠳⢬⢿⣿⣿⣿⣿⣿⣿⡏⠛⢹⣿⣿⣿⣿⣿⣿⡿⣣⠎⠀⠀⡇⠀⠀⠀⢸
⢷⡾⠿⡿⣶⠶⠶⠶⠶⠶⠛⣋⠴⠋⠀⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⠈⠳⢝⡛⠿⠿⠿⣟⠷⠉⢯⡻⠿⠿⠿⢛⡫⠔⠃⠀⠀⠀⡷⠒⠒⠂⢸

⠀⠀⠀⠀⢶⣶⡀⠀⠀⠐⢿⣿⣿⡿⠀⠀⠀⠸⠛⢿⡇⠀⢰⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣷⣿⣿⠅⠀⠀⠘⣿⠧⢴
⠀⠀⣶⣶⣾⣿⠇⠀⠀⠀⠀⠈⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠉⠀⠆⠀⠠⠓⣤⠂⠀⠈⠉⠉⠃⠀⠀⠘⠟⠉⠃⠀⠀⠀⠀⠿⠀⠘
⠀⠀⠈⠈⠋⠁⣤⠀⠀⠀⠀⠀⠀⣹⣷⣦⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⢀⢀⡀
⠀⠀⠀⠀⢐⣾⣿⣿⣧⡀⠀⢸⠀⣿⣿⣿⡀⢀⣀⣀⣄⡀⠀⠀⢀⣀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⢀⡇⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠃⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⣧⢸⣿⣿⣿⡇⠀⠀⣸⣿⣇⠀⠀⡿⡛⠢⠀⠀⠀⠀⠀⣀⠀⢠⣿⣿⣿⠁⢀⠀⠀⢙⡟
⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⡿⢸⣿⣿⣿⣅⠀⠈⢿⡟⡛⠀⠈⠁⠇⠀⠀⠀⠀⠀⠃⠉⠠⣼⣿⣿⡟⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠠⠌⢋⣿⡇⠀⠀⠀⠀⠐⠋⣿⡇⠀⢻⣿⣿⡿⠀⠀⢈⡁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣧⢰⡆⠀⠀⠀⠀⣼⣧⠀⠀⢯⣿⠡⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⠇⠀⠀⠀⠀⠈⣿⡀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠻⡆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣸⠁⠀⠀⠀⠠⠀⣀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠃⠀⠀⠀⠀⠀⠀⠿⢧⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⡛⣋⣛⣛⢁⣛⢛⣛⡛⡃
⣶⣶⣶⣶⣷⣿⣿⣿⣶⣾⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⡂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⡇⠈⠉⠀⠁⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆
⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⠠⠤⠤⠤⠤⠤⠦⠤⠄⠠⠤
⡗⠒⠒⠂⠲⢦⠖⠂⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠄⠲⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠒⠀⠀⠀⠂
⡇⠀⠀⡠⠀⠒⠠⢀⠀⢀⢀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢈
⡇⠀⠊⠀⣀⠀⠀⢀⠃⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠤⠀⣰⡊⠂
⡇⠀⠄⢘⡛⢃⠘⢿⠀⢰⣶⡶⢰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠂⠀⠀⠀⠀⠤⠄
⡇⠀⠀⠀⠀⠘⠃⠀⠂⣍⠁⠸⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠤⡀⠉⠀⠀⡄⠈
⡇⠀⠀⠀⠈⠈⠘⠁⠀⢀⠱⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⢀⠈⠀⠀⢹⠀⠂
⡧⠤⣄⠠⠸⣀⡤⡑⠤⣀⣀⣻⣁⠥⠖⢈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠠⠠⡤⠤⣜⡙⠃⠀⠀⠀⠈⠆⠀⠀⢠⠀⠀⡀
⣇⠀⠈⠉⠉⠁⠘⣄⠀⠀⠈⠁⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠁⣄⠜⣀⠀⠁⠁⠀⠀⠐⠀⠀⠈⠀⠀⠐
⠏⡁⠀⠀⣀⡆⢸⠓⠄⠁⠀⠄⠁⠀⠀⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⡵⠊⠀⠒⡄⠀⠀⠀⠀⢂⠀⠀⠀⠄⢢
⠀⠀⠀⠁⠐⠁⠈⡄⠀⠀⠀⣤⢥⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠢⡀⢈⠍⠀⢣⡀⠀⠀⠁⠀⠈⡆⠀⠀⠈⡀⠀⢀
⠀⠀⠈⠀⠀⠀⢀⣀⠀⠀⠀⠂⡀⠐⠊⠀⢑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠵⠁⠀⢀⠌⠑⠈⠄⠀⠀⡀⢣⠀⠀⠀⠐⠈
⠀⠀⠀⠀⣀⡀⢴⢃⠀⣂⠂⢀⡤⢒⢂⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⡀⠔⠁⠀⠀⡆⠰⠀⠀⠹⣼⡀⠀⠀⠀⠀⠁
⠁⠀⠁⠴⠸⠁⡇⠈⠇⠑⠂⠘⠀⠁⢠⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠈⠃⠀⠀⠀⠠
⠂⠈⠑⡤⡀⢀⡱⡎⠂⠉⡶⢄⠀⠔⠱⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢸⠀⠈⠀⡘⠉⠀⠘⡆⠀⠀⠀⠀⣸⡉⢢⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡓⠂⠚⡀⠀⠀⠀⠀⡄⢆⡀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠁⢰⠀⠀⠀⡀⠀⠀⠀⡀⠠⠰⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠩⠛⠀⠃⠀⠀⠀⠀⠁⠘⠾⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠒⠲⡄
⠀⠀⠀⠀⡀⠀⠀⠀⠤⠁⠀⡠⠃⠀⠀⠀⠀⠀⠀⠈⢤⡄⠀⠀⠀⠀⠤⠀⡇⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠘⣄⠐⡀⠂⢠⠞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠤⠀⡇⠀⠀⠃⢠⠄⠀⠐⠀⡇⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠈⢣⡁⠀⠁⠀⠀⠀⠘⠀⠀⠀⠰⠀⠒⠀⠀⠄⠀⠀⠠⠔⠀⡄⠀⠀⠀⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⡀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠈⠀⠀⡐⢄⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠄⠄⠀⠀⠀⠒⠒⠒⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠃
⠀⠀⠀⠀⢀⠟⠛⠤⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠐⠉⠀⠀⢀⣄⡠⠄⠂⠀⠁⠀⠀⢀⠄⠁⠀⠀⠐⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢀⡀
⠀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠒⠛⡀⢢⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠁
⠀⠀⠐⠂⠈⠀⠀⡵⠞⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠋⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠁⠀⡠⠀⠀⠘⠀⠀⠀⠀⠰⠉⠀⠀⠀⣀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆
⠀⠀⠆⠀⠀⠀⠐⠒⠒⢂⣀⣉⠁⠄⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣤⣃⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠠⠒⠀⠠⠶⠖⠠⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⣱⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⡴⠊⠉⢢
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠳⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⣄⣀⠸⡄
⠀⢠⡤⠀⠀⠰⡇⠰⠆⠀⠈⠁⠀⠆⣐⢢⠆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⠇⣸
⠀⠈⢡⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⢠⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡴⠾⠃⠐⠒⡆⠀⠀⠀⣾⣫⡿⢀⠃
⠀⠀⠈⠀⠀⠀⣇⠈⠀⠀⠀⠀⢀⠂⠀⠄⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⠇⠂
⠀⠀⡃⠀⠀⠀⢻⡄⠀⠀⠀⡰⠃⠀⠀⡂⠀⠀⠀⠄⠀⠀⠀⠠⠀⠐⠚⠀⠀⠀⠄⠀⠒⠐⠀⠂⠆⠀⡟⠀⢸⠆
⠀⠀⠈⠒⠀⠀⠀⠙⠓⠒⠚⠀⠀⠐⠂⠓⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠉⠀⠀⠉⠁⠁⠀⠳⠴⠋

⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠂⡄⢠⠀⡀⢠⠆
⠀⠀⠀⠄⠒⡃⡀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠐⠁⠂⣠⢀⠄⡀
⠀⠀⠀⠤⠤⠴⠐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠐⡰
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠞⠲⠦⠄
⠀⠀⠀⠀⠀⡾⠋⠉⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢀⠀⠀⠸⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⢠⠟⠂⠀⠐⢿⠀⠀⠀⠀⢀⠀⠀⡜⠀⢀⡀⠀⠀⣀⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠸⡃⠀⠀⠀⢸⠄⠀⠀⠀⠈⠀⠈⠀⠀⠉⠉⠈⠁⠀⠹⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠘⠀⠉⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠤⣵⠀⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠳⠃⡂
⠀⠀⠀⠀⠀⠀⡀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠖⠰⠶⠶⡄⡀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠔⠁
⠀⠀⠀⠀⠀⠀⣇⠀⠃⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠄⢠⠋
⠀⠀⠀⣀⣀⣀⠈⠀⠀⠀⠤⠴⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠐⠑⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠂⠀⠉⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠒⠀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠒⠁

⠀⠀⠸⠿⠿⠿⠃⠀⢀⠀⠀⠀⠀⣀⠀⡀⣶⣶⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠒⠓⠐⠒⢐⡒⠒⠒⠐⠒⠀⠀⠀⠈⢭⡘⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢉⣀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡞⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡰⣿⣿⣷⣼⠏⠃⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣥⣿⠛⡁⣥⣴⠆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡯⣾⣧⣿⠈⣵⣼⣟⢹⡟⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣩⡶⠟⠀⣴⣷⣿⣽⡏⠉⢠⡞⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠀⢈⣼⠟⣋⣤⠞⠻⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠉⠻⢿⣿⣿⣿⣿⡛⢻⣿⣟⣿⣿⢿⣷⠀⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠋⠀⠘⠛⠻⣷⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡀⠀⠀⢄⠀⠀⠀⠀⠙⢳⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣴⡞⢀⠀⡀⠈⠀⠄⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠈⣏⣛⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⢠⠈⠁⠉⢁⣀⣴⠏⠀⠸⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣤⣶⣶⣶⣿⣿⡿⠁⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠘⠃⠀⠀⠀⠶⠄⠶⠰⠀⠁⠀⠘⠋⡴⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠰⠶⠒⠂⠒⠒⠀⠖⠒⠀⠄⠂⡘⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣿⢾⣿
⠀⠀⠀⠀⠀⠀⠐⠛⠈⠛⠀⠒⠒⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢽⣯⣿⣄⣀⣠⣤⡄⠀⣰⣾⣿⣿⣿⣿⣿⣿⣯⣼⣿
⠀⠀⠀⠀⠀⠈⠁⢉⢀⡐⠉⡁⠀⡈⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢿⠛⠉⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⢿⢿⣿
⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿
⠀⠤⠂⢸⡟⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡄⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢤⠀⠈⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠸⠁⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠲⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠃⢀⣀⣀⢀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠉⠀⠁⠀⠙⠛⣿⡿⡟⠻⣿⠿⡶⠀⠿⠀⡶⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣟⠁⠲⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⠁⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣷⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠿⡿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⢿⣿⣿⣿⣟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣮⣿⣿⣿⣿⣿⣿⡿⣿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣋⣱⣾⣿⡿⠏⠙⠛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣾⡿⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣾⣟⣿⡟⠙⣻⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣭⢠⣿⣿⣿⣟⡏
⠀⠀⠀⠀⢠⣾⣿⣾⣿⢱⣿⣿⣿⣿⣿⣿⠟⠉⣁⣿⣿⣷⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⠏⣿⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣿⡜⣿⣿⣿⣿⣿⣴⡶⣾⡿⣡⣿⣿⡿⠿⢿⡿⠸⣿⣹⣿⢍⣀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠿⣿⢷⣼⣿⡟⠻⢷⣿⣄⣿⣿⠿⠛⣁⣀⣀⣉⣀⣤⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠻⠄⠀⢠⣿⣿⡿⠶⣾⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣇⠀⠙⢉⣿⠛⢿⠿⢿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠿⢿⣿⣦⠴⠟⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠭⠉⠉⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠿⠏⠉⠋⠹⠿⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠿⠋⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣀⠀⠙
⠀⠀⠀⠀⠀⠈⠹⣶⣄⣀⣀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠀⣴⣷⣄⣀⣀⣤⣶⠟⠀⠀⠀⠀⠀⠀⢾⣿⣷⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣧⡤⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠃
⡿⢿⢿⠿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠉⠑⠲⡄⠀⠀⠀⠀⠈⠀⠒⠠⠀⢀⣰⠀⠀⣈⣉⣉⣉⣉⣉⣉⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠈⡇
⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠶⠶⠶⠶⠶⠖⠄⠉⠉⠉⠉⠉⠉⢉⠛⠯⠭⠍⠛⢛⣛⡇⢰⠁
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⠀⢸
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡰⠀⠠⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣆⠀⣀⣀⣀⣀⣦⣀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⢠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠼⡀⠀⠁⠗⠐⠂⠀⠀⠐⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⠖⠀⠒⠒⠀⠀⠐⠂
⠀⠠⠀⠀⠀⠀⠀⠀⡆⠣⢀⠔⠉⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⠉⠉⣉⣻⣦⠀⣔⠄⢠⣠⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⡀⠀⠀⠀⠀⣀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣽⠍⢈⢿⣇⢘⣷⡀⣻⠂⣒
⠀⠀⠀⠀⠀⠀⠀⠐⡒⡈⡀⠈⡏⠀⢠⠂⡃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠼⢸⢸⠀⡇⡏⠸⢘⠹⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢃⠘⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢀⣉⣻⢸⠀⠀⠁⠀⠈⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⡅⠀⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⢻⣿⠘⠀⡀⠀⠠⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⢀⠀⡠⢤⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⠀⠀⠀⠀⠀⠀⠀⠐⢠⣤⣴⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠑⢀⠉⠠⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣶⣶⣇⠀⢸⡿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠈⠐⠁⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠙⠛⠉⣉⣬⡀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⠀⠀⠤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣶⡄⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠉⠈⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⢸⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠉⠋⠈⠀⡂⠈⠈⠉⠉⠉⠁⠙

⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣿⣿⣄⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣤⣤⣤⣤⣴⣿⣿⣦⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣴⣿⣿⣦⣤⣤⣤⣤⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣶⣷⣾⣷⣿⣿⣿⣶⣶⣶⣾⣿⣿⣷⣿⣿⣿⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⣄⣀⣀⣀⣀⣀⠶⠶⠶⠶⠶⢶⣶⠶⢶⣤⣤⡄
⠀⠀⠀⠖⠀⠀⠀⠀⠀⢀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠉⠉⠉⢩⡟⠁⠀⠀⠀⢠⣇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⢰⡏⠂⠀⠀⠀⠸
⠀⠀⠀⠀⡀⠰⠐⠊⡙⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠸⠷⢆⡀⠀⣀⡸⠷⠆⠀⢀⢀⡀
⣀⠀⠀⠀⠀⠄⠃⠈⠆⠀⠈⠁⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠶⠶⠶⠶⠒⢾⡀⢀⡀⠀⠀⢀⠀⠀⠀⠁
⠁⠶⡀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⣿⡃⣸⡇⢻⡅⢻⠀⣺⠄
⠀⠀⠈⠑⠁⢂⡠⡀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠞⡁⠀⠇⡇⠻⠙⢇⠈⠃
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣴⡃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡽⡇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⢦⣄⢀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⡀⡉⢉⢉⠉⡇⠀⠀⠀⠠⠤⠂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠔⡟⠠⠁⠃⠂⠦⠤⡀⣈⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⡆⠀⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠙⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣽⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠐⠊⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⢀⣀⠀⡀⣀⠠⠄⠒⠂⠉⠉
⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠚⠓⠂⠠⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⡿⠿⠛⠋⠉⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡳⣦
⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣵⣿⣏⣿⣄
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢀⣠⣴⣾⣿⣛⣯⠭⠶⠶⠶⢶⣶⣯⣭⣻⣷⣦
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣴⡿⣟⣿⣿⠋⠁⠀⢀⣠⣴⠶⣟⣻⣯⣭⡭⣭⣿
⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⣩⢾⣿⣿⡿⣷⣦⣴⢟⣯⣾⣿⡫⣿⣿⣿⣶⡿⠟
⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢟⣵⣾⡷⣄⣸⣿⣟⣾⡿⣱⣿⢿⣷⣿⣿⣿⣍⠉
⠀⠀⠀⠀⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣡⣿⣿⣿⣿⣷⣯⣭⣿⡿⣰⣯⣿⣿⣿⣿⣿⠿⠿⣿⣦⡀
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣟⣿⣿⡿⣿⡿⢻⣿⢿⡇⣿⣷⣿⣿⣌⠛⢁⣴⠆⢨⡟
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⡿⣻⣿⡘⣿⡀⢷⡸⣷⡿⣻⣿⣿⠋⠛⠶⡾
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣯⣼⢿⣿⡻⣿⣷⡜⢿⣾⣷⣿⣿⣿⣽⢻⣷⣄
⠀⠀⠀⠀⠀⣿⣿⣠⣤⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡿⢣⣶⣮⣅⣹⣟⠛⣮⣻⣿⣿⣿⡿⠿⣟⣽⡟
⠀⠀⠀⠀⠀⡿⣿⠿⠟⠋⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣱⣿⣿⣿⣿⣧⣼⡷⠛⠛⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⣸⢰⢿⣿⠛⣇⠛⠃⠀⠀⠀⠀⠀⠀⣠⣏⢱⡀⠀⢸⣿⡟⢿⣿⣿⣿⡇⠉⠁
⠀⠀⠀⢠⣿⣢⣾⣿⣄⣻⣿⠁⠀⠀⠀⠀⣠⡾⣿⣿⣼⠇⠀⢸⣿⣏⣸⣿⣿⣿⡇⢀⣠⣴⣶⣦⣄⡀
⠀⠀⠀⣼⡐⣧⣿⡏⣹⣿⣇⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡿⣻⣽⣿⣿⣿⣇
⠀⠀⣼⣿⣷⡟⠛⠻⠟⠛⠉⢉⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣾⡟⣿⢻⣿⢻⡿
⠀⠰⢿⣿⡿⠀⠀⠀⠀⠀⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢳⣿⣟⣾⣟⣼⢃⡿⠁
⠀⠀⢘⣿⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⣿⣿⣿⣿⡿⢃⡾⠁
⠀⠀⣼⠃⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⡄⣿⣯⣛⣋⡴⠟
⠀⢰⣇⣴⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⡏⣿⡿⠋⠁
⠀⡾⠛⢻⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣻⣿⡇
⠀⠁⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⡇
⠀⠀⢠⡟⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣼⠃⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⢰⡟⣀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡄
⠀⣿⣿⣿⣶⣶⣦⣾⡟⠛⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣿⠋⠀⠈⠉⠛⢫⡇⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣄⣀⡀⠀⣀⣿
⠜⠁⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⡿⢋⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠂⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⠀⠠⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠸⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠘⢿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⠀⠀⠉⠻⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⠿⣿⣿⣷⣎⣙⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠛⣿⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣾⣿⣿⡛⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣤⣾⡿⢋⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣠⣿⢿⣿⡟⠀⣫⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣾⣿⣿⣿⣫⡇
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⡿⠛⠈⣁⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⡿⢹⡏
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣴⣶⣾⣿⢩⣿⣾⣿⣿⢿⣿⣯⢸⣿⢹⣿⣍⣁⣠⣿⣻⠆
⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣷⣍⣿⣟⠻⣿⣯⣿⡀⣿⣿⣿⠿⠛⠁⠀⠐⠋⢀⣀⣿⣿⣿⣷⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠰⣏⡀⠁⠈⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⣿⣿⡏⠀⠙⠿⠾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⣿⡆⠄⣠⡾⠏⠀⠘⠁⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠠




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣯⣤⣤⣧⢡⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣤⢀⣰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⣾⣿⠁⢸⣿⡧⠀⠀⠀⠀⠀⠀⣀⣤⡶⠀⠠⣦⣄⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠘⠿⡷⠾⠯⠟⠲⠄⠀⠀⠀⣾⣿⣿⣧⠀⢠⣿⣿⣿⣿⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠀⠀⠀⠀⠀⠰⠀⡀⠀⢰⣿⣿⣿⡿⠳⢸⣿⣿⣿⣿⣧⠀⣆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⠀⠀⠀⠀⠀⣄⠀⡏⠀⢼⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⠀⢿⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢙⡄⡀⠀⡄⢀⣆⣿⠠⡆⠀⣾⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⡆⢸⣿⡀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠄⣺⣷⣼⣶⣷⣾⣿⡿⣇⢻⣸⣿⡏⡹⠛⠻⠹⠟⢯⣿⣿⣿⣿⣅⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢁⣴⠏⣠⣶⣾⣿⣿⣿⣿⠹⣆⢹⡿⠓⠓⠀⠀⡄⠀⠘⠻⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⠟⠙⢀⣼⣿⣿⣿⣿⣿⣿⡇⠸⣦⣁⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⠁⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⠃⠸⠋⠀⠐⡃⠀⢀⡷⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣀⠀⢀⠀⡄⡱⠄⠸⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠉⠁⠀⠀⠈⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁⢿⣽⣿⠀⠁⠀⢸⣆⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⢰⣿⢟⣿⠀⠀⠀⢸⣿⡄⠰⠇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⢀⣿⣿⣿⡀⠀⠀⢸⡏⣷
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠳⠀⠀⢀⡀⠀⠀⠀⠀⠰⣿⣿⣿⣿⢸⣿⣿⣿⣿⡇⠀⠀⠀⣷⠸⣇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡾⠛⠉⠀⠀⠀⠀⠀⠀⣿⣿⠸⣿⣼⣿⣿⣿⣿⡇⠀⠀⠀⠏⠂⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⣿⣿⣿⣿⡇⠀⢀⣤⣤⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣸⠀⢱⣿⣿⣿⣿⠿⠛⠁⠀⢹⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⢸⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⣈⣆⠀⠀⠀⠀⠀⢀⡞⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠃⠀⠀⠀⠈⠿⠏⠀⠀⠀⠀⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛

⠀⠀⠀⠀⠀⡀
⠀⠀⠰⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡆⠀⠀⠀⠀⠁⠀⢀⡀
⠀⠀⢠⣴⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⣸⣅⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⣆⠁⠀⠁⠀⠻⡀⠀⠀⠹⣛⡻⠏⠀⠀⠀⠀⠐⠚⠂
⠀⠈⢿⢀⠀⠀⠠⠆⠁⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠂⡄⡊⠔⠀⠀⠀⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣹⣿⣛⠛⠷⣤⠀⠀⠀⠠⢾⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣟⠁⠀⠙⠄⠘⠇⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⣤⣛⣿⡿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠰⡤⡀
⠀⠀⠀⠀⠈⡀⢀⠀⠘⠁⠀⠀⠀⠀⠘⡛⠿⢷⣬⡷⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡄⠈
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣏⠉⢀⡍⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⠌⠄⠀⢀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠂⠀⠋
⠀⠀⠄⠒⠀⠆⠰⢀⡀
⠀⠈⠀⠀⡠⠢⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠤⠤⠮⠤⡴⠀⠀⠀⠀⠀⠀⠤⢤⠀⠀⠀⠄⣤⠀⠀⠀⠉⠜⠀⠀⡀⠀⠀⠀⠀⢀⡀⠉
⠀⠀⠀⠀⣶⢓⡦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠀⠀⠄⠀⠐⠖⠀⠀⠀⠀⣾⡟⠛⣷
⠀⠀⠄⠀⠈⠋⠄⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠹⢷⠾⠟⢠⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠉⣂⠢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠤⠢⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠂⡀⠀⣐⡬⠇⠈
⠀⠀⠀⠀⠈⠠⠈⠀⠣⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⢐⠀⠀⠰⠄⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠤⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠂⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⠀⠀⠀⡾⠀⢀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠄⠀⠆⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⣷⡎⡙⢦⡗⠀⢀⣧⠀⠠⢤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠏⢀⠀⢰⡍⠂⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠳⠄⠀⠈⠇⠀⠠⠀⠀⠨⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⠁⠠⠀⠀⠀⠀⠀⠠⣀⠐⢡⠐⠀⠟⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠚⠀⠈⠄⠀⠀⠈
⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠰
⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⢤⣤⡬
⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠋⠉⠉⡩⠅⣂⠀⠢⠄⣉⡀⠡⣤⣍⡄⠀⠀⠀⠀⠀⣠⣀⣀⣀⣄⣤⠀⠀⠀⠠⢌⠳⠤⠖⠠⠀⠀⢀
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡉⡉⡁⠀⢈⠉⠈⢁⡀⠀⠀⡉⠙⠀⠠⠀⠀⠀⠈⠀⠀⠈⠂⠀⠀⠀⠐⠀⠀⠀⠉⠈⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⢲⠒⠒⠒⠒⠒⠒⢲⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠦⢦⣄⠀⠀⠀⠀⠀⠀⠀⢸⢉⡽⣍⠉⡽⣍⢹⣀⣤⣤⣤⣤⣤⣸⡇
⠀⠀⠀⠀⠀⠀⣠⠞⢤⠤⠖⠻⣌⡆⠀⢀⡀⠀⠀⠀⢸⢸⡇⢹⢠⡇⢸⢸⢹⠟⠛⠛⠛⢻⠻⡇
⠀⠀⠀⠀⠀⠀⢳⠀⢘⣆⣀⡴⣫⠃⣀⣼⣦⣀⠀⠀⢸⢸⠃⢸⢸⡇⢸⢸⣏⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠈⠳⠉⢑⠟⢾⡁⣻⡏⢧⡸⢹⡟⠀⢸⢸⠀⢸⢸⡇⢸⢸⠀⣀⣤⣷⡄⡀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠃⡄⠘⣆⡆⠉⢹⠀⠈⠳⣤⡇⠀⢸⢸⠀⢸⢸⡇⢸⢸⠛⣿⠀⠉⠀⣟⢳⡆
⠀⠀⠀⠀⠀⠀⢠⠇⠹⣄⠘⢯⡒⢛⡧⠤⠤⠼⡇⠀⢸⠸⠤⠼⠸⠦⠼⢸⠀⡇⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⢳⢄⣹⢮⠀⠀⠀⠀⢧⠀⢸⠒⠒⢒⣒⠒⠒⢺⢰⠇⠀⠀⠀⠸⣼⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢰⠃⠸⠄⠑⠞⠁⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢸⠿⠤⠤⠤⠤⠴⢿⡇
⠀⠀⠀⠀⠀⠀⣧⠀⢸⣇⠀⢶⢸⠇⠀⠀⠀⠀⠘⡇⢸⠋⠙⠫⠭⠉⠙⢻⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⢿⡀⢸⣾⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⢠⡤⠬⠤⠤⠤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⢸⡇⢸⡟⠦⠔⠦⠴⠢⠤⠇⢸⠉⠉⠩⠽⢸⡏⠉⠉⠉⢹⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⠀⡾⢸⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⢸⡯⠭⠭⠭⢽⡇⣚⣛⣻⡇
⠀⠀⠀⠀⠀⠤⠿⠶⠧⠼⠷⠾⠦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⠷⠶⠶⠶⠾⠧⠤⠤


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢤⡉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⢁⢸⣿⣤⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠸⢸⣿⣟⣓⠀⠀⠒⢺⣿⡟⠟⡟⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣽⣽⣍⣭⣭⡍⠉⣡⣬⠉⣭⣍⣠⣍⣁⣉⢹⣟⠙⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣉⣩⣅⠈⢉⣿⢫⣭⣿⣍⣿⣋⣿⣯⢻⠿⢺⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣭⡭⠉⠸⣿⣿⣗⡉⠉⣻⣯⣠⡽⠋⠙⣿⡏⠛⠛⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⡤⠤⢿⣿⠌⠛⢙⡿⣿⣾⣥⡀⡆⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡓⠛⠓⠒⣾⣿⠂⢀⣼⠀⠸⣿⡌⠁⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣁⣉⣁⣀⣿⣿⢠⡌⠀⠀⠰⢾⡇⠀⠁⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠋⣉⢉⣿⣿⡿⠁⠀⠀⠐⢸⡷⠀⠰⠀⠀⠀⠀⠀⡇⠀⣀⡀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⢶⣭⠤⣽⣿⡀⠀⠀⠀⠀⠉⣇⠀⠀⠰⢆⠀⠀⠀⡆⠉⢻⠀⠝⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⢸⣿⣴⣿⣤⣴⢿⣿⡄⠀⠀⠀⠀⠀⣿⡀⠀⠀⠘⠀⠀⠰⡇⠀⢸⣇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⢸⣯⣹⣧⠀⠀⢸⣿⡅⠀⠀⠀⠀⠀⣹⠁⠀⢤⣤⠦⠀⢐⡁⠀⠿⣿⠧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣏⠉⠐⠂⢺⣿⣄⠀⠀⠀⠀⠀⠸⠃⠀⢀⡟⠀⠀⢸⡃⠀⡎⣿⠀⡄⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢧⣤⣴⣤⣿⣿⠀⠀⠀⠀⠀⠀⡄⠀⠈⠇⠀⠐⣾⠇⠀⢤⠟⠤⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠒⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⠉⠀⠀⠠⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣟⠗⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠥⠤⠀⠀⠈⢼⣇⠀⠀⠀⠀⠀⠀⢰⣦⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡀⣄⠀⣺⡛⣤⠇⠀⠀⠀⠀⠀⠀⠘⣻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠁⠁⠉⣹⠋⣿⣿⠆⠀⠀⠀⣠⠒⣎⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣏⡙⢻⠃⠀⢿⣿⣃⡴⠒⠛⠀⠀⢼⠿⡆⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣾⣿⣿⣤⠤⠤⠀⠀⠈⠀⠈⢻⡗⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⣿⡇⠀⠀⠠⠀⢿⣯⣿⣷⣤⡀⠀⠀⠀⠀⠀⣻⣅⣀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿⠏⠒⠲⡖⠐⣚⠉⣿⣭⠭⠥⠨⠥⢼⣿⣈⣉⡈⠉⢴⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⢿⠠⣟⢺⣿⣨⣧⠀⣿⣿⠆⠀⠀⠀⠀⣿⣟⡉⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣓⠀⠀⠀⠀⠀⢸⣷⣭⣄⣽⣛⣙⣀⣛⣟⠛⣛⣛⣛⣛⣉⣙⣏⣙⣛⠺⣿⢿⣷⣄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢲⣤⠛⠃⢸⡏⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣾⠁⠀⠈⠙⠓⠦⣄⠀⠀⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⣿⡄⠀⠐⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣛⣙⠆⠀⠀⠀⠀⠀⠀⠀⠘⠻⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣋⣛⣻⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣭⣉⣽⣿⣿⣿⣉⣟⣋⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠈⠀⠀⡀⠀⠀⠈⠀⡐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣶⡖⢈⣀⡀⠀⠀⠀⠀⣐⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣏⣀⣤⡤⠖⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡿⡇⢰⣄⣯⣻⡂⡀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢹⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⢱⠃⠈⢿⣿⡧⠽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⠿⡆⠀⡴⠞⠉⠀⠀⠉⠐⠯⢠⡈⣺⡿⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢏⠻⠿⣤⡫⢿⣇⠀⠀⣂⠒⠀⢐⣨⣿⣿⠟⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠹⠀⠸⣿⣿⣶⣶⣶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⡀⠀⠒⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡀⣹⠀⠀⠀⠀⠀⠀⢰⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠰⠉⠉⠉⠉⡆⠀⠀⠀⠀⠀⠀⡦⢴
⠀⠀⠐⠈⠁⠀⠀⠈⠉⠉⠙⠆⠀⠀⠀⠀⠀⡰⠁⠁⠀⢂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠼⢀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠢⡈⠑⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢠⠒⠒⠒⠀⠐⠒⠒⢆⠀⠀⠀⠓⠒⠒⠒⠒⠒⠂⠚⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⢠⡁⣄⣈⣤⣀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⣀⣀⣀⣀⠀⡼
⠀⠀⠀⠀⢀⠊⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠶⠔⠒⠒⠀⠀⠀⠀⠀⠀⠔⠋⠉⠉⠑⢂⠀⠀⠀⠀⠀⠀⠛⠛⢻⠙⠓⠃
⠀⠀⠀⠐⠁⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⡠⡴⠚⠒⠢⢄
⠀⠀⢐⠀⡀⠀⠤⠤⠀⢤⠀⠀⠀⠀⠀⠀⡠⠒⡚⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠌⡐⠀⡆⠀⠡⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠰⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢅⣀⣅⣀⣀⣄⠼
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢠⠀⢤⡀⣀⡀⣠⣀⣸⠀⠀⠀⠀⢰⣒⣀⣀⣀⢰⣿⣶⠀⠀⠀⠀⠤⠄⢐⡲⠈⢰⠨⠆⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⠚⠀⠐⠒⠚⠀⠀⠀⠀⠈⣄⣂⣐⣈⢹⠟⠉⠀⠀⠀⠀⡂⠀⠀⣀⡀⠸⠧⠀⣔⣀⠀⠄
⠀⠀⡰⠀⠀⢠⡀⠠⠐⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⠪⠁⠀⠀⠀⠀⠀⠀⠤⠒⠄⣀⡀⠀⠀⠀⢀⡠
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡜⠁⠉⡦⠀⠀⢴⠁⠈⡃⠀⠀⠀⠀⢀⠀⠀⠰⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢠
⠀⠀⠗⠒⠒⠒⠒⠒⠒⠂⢳⠀⠀⠑⠀⠐⢁⣀⣀⠀⠙⠻⡄⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⡺⠋⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠤⠬⠆
⠀⠀⠒⠤⠤⠤⠤⠄⠐⠒⠚⠀⠀⠀⠀⡔⠑⠒⠚⠓⠀⢠⡇⠀⠀⠀⠀⠀⡀⠀⠛⠂⠀⡸⠀⡀⠀⠀⠀⠀⠀⡘⢠⠠⠴⠶⠛⣣⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠖⠁⠠⠁⠀⠉⠁⠉⢄⡀⠈⡀⠀⠀⠀⠀⠤⢤⣤⠔⠒⠋⠁⠀⡱⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠎⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡚⠁⠀⠀⠀⠂⠚⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠈⡇⡸⠀⡇
⠀⠀⠀⠈⠀⠀⠀⠀⢠⢅⡀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠂⠁⠈⠚
⠀⠀⠀⡤⠒⠦⡀⢠⠧⠄⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢼⠛⠛⠚⠒⠚⢹⡇⠀⠀⠀⢀⡀⠀⠀⣠⡀
⠀⠀⠀⢋⡂⢈⡡⠘⠀⢂⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠉⠉⠀⠀⠄⠀⠀⠠⠇⠀⠓⣀⠀⠀⢀⣀⣘⣀⣀⣀⣀⣠⣤⣧⣀⠀⠀⢸⣿⣷⣾⣿⡇
⠀⠀⠀⠂⠡⠎⠀⠀⠉⠉⠀⠀⠀⠀⢰⠉⠑⣄⢀⠀⠈⢉⠀⠀⠐⠀⡀⠈⠉⠉⠐⠒⣐⣂⣀⣀⡴⠃⠀⠀⣁⠀⠈⠀⠉⠉⠉⠁⠀⠀⠀⠁⠀⠈⠛⠁⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⡠⠤⠀⠀⠀⡆⠀⡙⠚⢅⠀⠠⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠀⣳⣰⠀⠀⡂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡻⡿⠻⢷⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⢠⣄⡀⠀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⣿⣦⣤⣬⣶⡶⣾⣷⣤⡄⠀⠀⠀⢀⢿⣀⣸⢆⠀⠀⠀⠀⠀⡬⢅⡈⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⢂⠀⠀⠈⠒⠤⠤⣄⣐⣁⣈⣍⣨⠥⠟⠊⠀⠀⡴⠁⠂⠀⠀⠀⠑⡄⠀⠰⣊⡀⠊⠑⢄⣀⠄
⠀⠀⢀⠄⠀⠁⠄⠠⠃⠀⠀⠁⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⢶⢶⠛⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⣀⠀⠠⠤⠤⠤⢼⢄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠠⡐⠈⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⡉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⣀⣀⣀⣀⣀⡘⠠
⠀⠀⠀⢲⣐⣒⣒⣒⣈⠭⠕⠊⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠊

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⡀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⢇⣴⣿⣿⣿⡟⣛⣻⣿⢿⣿⡿⠥⠁⠀⠠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛⣿⣿⣿⠏⢼⣿⣿⡟⠛⠈⣉⠉⢉⠀⠈⢀⣴⣦⣀⣤⡀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠘⠛⣴⣶⣶⣌⢧⣈⠻⢁⡁⣀⣀⣀⣀⡀⠀⠀⠘⠿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠹⣿⠿⠋⢠⡿⣀⢺⠛⣿⣿⣿⣿⣿⣿⠀⠀⠠⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠧⢠⡇⣤⣀⣴⣾⠿⠸⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠰⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠙⢿⣷⣶⠶⠚⠻⢿⠿⢬⣽⣧⣀⠀⠀⠀⠹⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠋⡤⢈⡿⠁⠠⣤⢀⠁⠀⠀⢈⢙⠻⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⡇⡽⠟⠁⠐⠀⠀⠠⠀⠀⠀⠀⠘⢃⠂⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡎⠃⣀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠐⠀⠀⡀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢹⡇⠀⣿⡗⠀⠀⠀⠓⢦⣄⣤⠀⢴⣦⡄⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⣿⡧⢦⣠⣀⣀⠀⠀⠀⢰⣾⣿⠙⢀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⣴⣿⣷⣶⣿⣯⠀⡷⠶⣾⣿⣶⣿⡖⢀⢾⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡿⢾⣷⠽⠟⣿⡿⣿⠟⠒⠒⣿⣿⣿⣿⣿⠐⠍⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢰⣶⣶⠀⣌⢑⣯⠁⢀⡈⠉⠉⠛⠃⠿⢀⣄⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣀⣉⠁⠜⠉⠈⠁⠀⠈⠉⣗⠀⠀⠀⢀⡽⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣶⣶⡦⠄⠀⠀⠰⢶⣿⠿⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠘⠻⢿⠿⠿⣿⣿⣿⣿⡿⠿⡿⣿⠿⠿⠿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠔⠀⠀⠐⠚⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠒⠖⠠⠤⠶⠐⠚⡛⠂⠐⠂⠀⠀⠀⠀⢀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⡉⠉⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠐⠃
⠀⠀⠀⠀⠀⢀⡀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠊⠀⠀⠁⠘⠷⠀⠀⠀⠀⠀⠠⠤⠄⠤⠤⢠
⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⠀⠚⠃
⠀⠀⠀⠑⣄⢀⠀⠀⡠⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢦⣟⡋⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠣⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉
⠀⠀⠀⠈⠆⠀⠀⢀⠄⠀⠀⠀⠀⠀⠛⠐⠘⠉⠀⠀⠀⠀⠀⠁⠀⠀⣰⠀⠀⠀⢀⠀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠘⠰⠁⠀⠀⠀⣇⣠⣤⣤⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠂⠀⠐⠢⠴⠂⠀⠀⠀⠘⠛⠧⡼⢳
⠈⠢⠄⣀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠐⠊⠁
⠀⠀⠀⠠⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡠⠄⠀⠀⠀⡠⠐⠀⠈
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⡀⠀⠀⠀⠀⠀⠀⣾⣃⠀⠀⠀⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢠⡇⠀⠀⠀⠀⠀⠀⣧⣤⠀⠀⠀⠀⠀⠰⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣀⠐⠋⠀⠈⢁⣀⠀⠀⠀⣴⠚⠛⠛⣷⣦⠀⠀⣀⣰⡿⣇⡀
⠀⠀⠀⠀⠀⡠⣶⣯⣿⣿⣶⢶⣿⣿⡷⡄⣼⣿⣿⣷⣾⣿⣿⠀⢸⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⠟⠛⢸⣿⡿⠃⢻⠇⠐⣻⡿⢻⣿⣿⣿⢹⣷⣸⣷⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠙⠂⣰⣾⣿⣯⡐⠃⠀⠸⠟⢻⣏⢿⢿⣿⣟⠻⠛⣿⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⣿⣇⠀⠀⠀⠀⠋⠸⣿⣿⣿⡿⠁⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡻⡯⡿⠀⠀⠀⠀⢀⠀⠈⣷⣿⣧⠀⠀⢹⡿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣴⡇⠀⠀⠀⠀⠀⢧⠀⢳⣿⣿⠀⢀⠖⢞⣾⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⠀⠀⠀⠘⣷⣸⣿⡇⠀⢸⡿⡿⠓⡝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠲⢾⣿⣿⣿⣥⣷⣿⡄
⠀⠀⠀⠀⠀⠀⢀⡠⠊⢹⣿⡿⣿⠎⠂⠀⠀⠀⠀⡇⠁⢿⠀⠀⠀⣾⢨⣿⣿⣿⢿⡆⣧
⠀⠀⠀⠀⠀⠀⡼⡁⠀⡎⢀⠀⠇⠀⠀⣝⠒⢤⡠⣷⣦⠘⠄⠀⠀⣿⡜⣷⡟⣿⠘⢠⢸⡄
⠀⠀⠀⠀⠀⣼⠁⠀⣸⠁⢸⠀⢰⠀⠀⠈⢧⠹⡗⣾⣿⡀⠀⠀⢠⠃⠃⠛⠀⢹⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠃⠀⢠⢇⠀⠈⠀⢸⡆⠀⠘⢿⣷⣇⠈⣿⢷⡀⡆⠀⠀⠀⠀⠀⠘⠀⠀⡆⢸⠀⠀⠀⢀⡔⢁⠞
⠀⠀⠀⠀⠀⠀⠀⡎⡸⠀⢰⣀⠘⣷⠀⠀⠈⠿⠻⠁⣿⢸⣿⡇⠘⠇⠀⠀⠀⢠⠀⠀⠉⠀⠇⢀⡀⠉
⠀⠀⠀⠈⠚⠄⣾⠁⢧⡆⠸⡿⠄⠻⠀⠀⠀⠁⠀⠀⠋⠈⠿⡓⠀⠀⠀⠀⠀⠈⠰⣆⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣥⣾⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉

⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⣤⣤
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠤⠀⠀⠒⠒⠒⠒⡀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠁⠀⠀⠠⠠
⠀⠀⠀⠀⡸⠆⠀⠬⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠠⠀⠂⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠱⠄
⠀⠀⠀⠀⣠⡶⠖⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠸⠟⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠤
⠀⠀⠀⠈⠀⠀⡠⡘⠁⠀⠀⠀⢀⢼⢄⠠⡉⢜⠸⠔⡗⠡⠊⠅⠂⠊⠘⠀⠀⠀⡃⠀⠀⠀⠀⠈⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂
⠀⠀⠀⠀⢸⡾⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⡄
⠀⠀⠀⠀⠈⠐⢳⣞⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⣷⢀⠀⠀⣮⡇⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⣀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠈⠡⠤⠀⠠⠍⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠈⠛⠃⠀⠉⠀⠀⠀⠀⠀⠀⡄⠄⠄⢠⡤⢄⣠⠄⣤⣨⣩⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⢠⣤⠄⠀⠀⠀⠀⠀⠀⠹⣚⡛⠀⣂⠀⠀⠀⡆⠂⠆⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠉⠀⠈⠅⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣤⣤⣿⣾⣿⣿⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣫⣿⣝⣿⢹⣿⣿⢿⡯⡿⡻⣭⠟⢶⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣶⡌⠐⠘⢷⣰⣀⢞⠹⣷⡗⢸⠡⢴⣃⣴⠂⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠈⣿⣿⣿⡷⢒⣗⡳⢫⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣁⣽⣵⢿⣾⢱⣯⣟⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣮⣿⣵⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⡇⣹⡻⣭⣾⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠻⢿⣟⠳⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠇⢙⡟⠉⠐⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠯⣙⣃⣺⣿⣘⣓⠐⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠯⡟⠛⠶⢶⡿⠼⢰⠣⡍⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡽⠨⡿⣿⣽⣛⣻⡡⢼⠠⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣉⣯⣰⣉⣿⣿⢫⣘⣂⣀⣀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣴⢃⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣿⣿⣿⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⠃⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠉



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

⠀⠀⠀⠀⠀⠀⠥⠂⠒⢛⡷⠒⠐⠚⡷⠂⠒⠒⠒⠚⡿⠒⠀⠒⢛⠗⠒⠒⠚⡷⠂⠐⠒⢛⡷⠒⠐⠒⠒⢫⠗⠐⠲⠨⠁
⠀⠀⠀⠀⠀⠀⠀⡠⢤⠈⢁⣤⡄⠨⠅⣄⢀⢀⣠⣨⣅⣤⣀⡄⠉⠠⣄⣤⠈⠁⣄⣀⣄⠈⢁⣄⣀⣀⠀⠉⠀⣀⣀⣀⡀
⠀⠀⠀⠀⡀⠒⠁⠀⡤⠒⠈⠀⠠⣠⣤⠀⠈⠀⠀⡩⠟⠋⠁⣠⠶⢊⡇⠻⢿⠭⡟⢸⠛⠯⡍⢸⣿⡿⠭⣟⠉⢿⠙⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢸⠁⠂⠀⠀⠀⢹⠛⠋⠉⠉⢙⡽⠋⠙⢛⠻⠁⠘⢿⡭⠽⠃⠸⢏⠉⢋⠀⠀⠑⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢠⠂⠰⠀⠀⠀⠈⠀⠀⠀⡀⠀⠂⠀⢠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡇⠀⠐⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠉⠉⠑⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣤⣤⣤⢤⡀⠀⠀⠀⢰⠠⠠⠤⠤⠀⠠⠄⠀⠀⠀⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠐⠐⠄⠀⠀⠨⠤⠀⠄⠄⣬⠀⠀⠀⡈⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠉⠉⠉⠉⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⣠⢃⣀⣀⣀⣀⠀⠂⡀⢀⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⣐⣉⣉⣈⣁⣁⣒
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠘⠂⠓⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠂⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⢶⣶⣶⡶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣭⠴⠚⠋⠉⠉⠉⠉⠙⠓⠶⢭⣝⣓⣶⣦⣤⣤⣤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠭⠔⠚⠉⠀⠀⣀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠉⠉⠉⢹⡏⠙⠓⠊⢭⣍⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠠⠖⠋⠉⠀⠀⣿⠀⠀⠈⠉⠓⠲⠤⠤⠄⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠉⠐⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣶⠀⠀⠀⠀⠀⠀⡆⣿⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⣿⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⡇⢸⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⢀⡀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⠀⠀⠀⠀⢸⠀⡇⠀⢸⢼⠁⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠈⠀⡇⠀⢸⡀⠁⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡇⢰⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡇⠸⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣀⡀⠀⠀⠀⠀⡇⠀⢰⡇⠀⠀⠀⠀⠀⠀⣸⠀⠀⢰⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣤⣤⣈⣉⠉⠛⠚⠃⠀⠸⢧⣄⣀⣀⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢭⣙⡛⠓⠶⠦⠤⣄⣀⡀⠉⠉⠙⠛⠛⠀⠀⠀⠀⠀⢀⡴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠶⠶⠷⠾⠶⠦⣤⣀⠀⠀⠀⠀⠡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣭⠉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⠀⡀⠀⠀⠀⠀⠀⠀⣿⢻⠽⠇
⠀⠀⠀⠀⠀⢀⣠⣙⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣻⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⢀⣁
⠀⠀⠀⠀⠀⠘⣿⣿⣸⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣼⡉⢿⠀⠀⠀⠀⠰⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⢸⡸⣿⣿
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⡏⠛⠀⠀⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣰⡄⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣿⣿⡇⠀⠀⠀⠀⠀⠸⠿⠟⠋⠹⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠿⠇⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣀⠀⡀⠀⠤⡀⠀⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⢰⠃⠀⠒⢒⣚⠒⠒⠀⠀⣠⡃⠀⠀⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠴⠂⠀⠁⠀⠦⠖⡩⢑⡘⢒⣶⠢⠑⣀⠵⣠⢀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠙⡲⠊⠉⠈⠢⣀⣈⠂⠀⠀⠀⠀⠖⠅⠊⠀⠀⢡⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠄⠀⠀⣃⠀⠠⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡅⡇⠀⠀⠀⢸⠀⠀⠀⠰⠀⠀⠠⠉⠀⢀⠀⠀⠸⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡴⡇⡃⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠶⠦⠀⡈⠀⢀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡇⠃⠀⠀⠀⣄⠀⠀⠀⢐⠂⡌⣀⠀⢀⠃⠒⡞⠀⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢃⡧⣀⡀⢴⠀⠀⣀⡁⠒⡚⠇⠉⠀⠈⠀⠨⠁⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⠛⠒⠒⠚⠀⢀⠂⠀⠀⡘⠀⠀⠀⠀⠀⠂⠀⠘⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣀⠀⣄⠀⠀⠀⠀⢀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠰⠀⠀⠄⠸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⠙⡀⠀⠀⠀⠸⠀⠠⠤⠬⢀⠀⠀⠀⠀⡀⠀⠰⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠂⠤⠀⢴⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠉⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⡀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡃⠀⢀⢘⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠙⡀⠈⠈⡏⠉⡇⠈⠀⠀⠀⠀⢨⠉⠉⠉⠉⠉⠉⠉⠩⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠈⢄⠀⢁⢀⠑⠂⢄⠂⡀⠀⢘⡀⠀⠀⠀⠀⠀⠀⢐⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⡁⠞⠛⠓⠚⠛⠛⠁⠒⠚⠀⠀⠀⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣴⣿⣿⡛⢶⣿⣶⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡄⠈⢯⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠈⠙⠻⣿⡦⠀⢿⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡀⢀⡀⡀⠀⠘⣿⣿⣶⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠙⠛⠁⠀⠙⣷⡿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⢰⣿⢡⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣸⡀⠣⠀⣼⠇⠀⢀⠀⠀⠸⣿⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡀⠀⠀⠶⠘⠏⠀⠀⠀⢻⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠉⠀⢰⠆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢹⡆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠀⠀⢸⡧⣰⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠈⠉⠙⠻⢿⡿⢿⡿⠿⠿⠿⠛⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⠇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⣾⠀⠀⠀⠀⠀⡰⠀⠀⢠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⣾⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠈⠁⠂⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⣀⠀⠀⠀⢸⣯⣿⣿⠿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣷⣶⣤⣤⣈⣉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠙⠿⣿⣿⣿⣿⣿⣿⡟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠺⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⣤⠀⠘⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⣘⣩⣽⣿⣿⣶⣤⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⣿⣿⣿⣟⣷⡄⠀⠀⢻⣿⣿⣿⣯⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⢠⣀⣸⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣻⣿⣷⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠒⠒⠒⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡟
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⣀
⠀⠀⢠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⣿⣷⣤
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣬⣧⣾⣿⣿⣿⣽⣿⣿⡿⠿⡀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢇
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⢀⡀⠀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢬⣿⣾⣧⠀⣠⣅⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⣿⣿⡟⣰⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢸⣿⣿⣟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⠃⠉⢰⣿⣿⣿⢿⡿⠻⢡⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣟⢁⣀⣐⣀⣿⡟⢁⣿⣁⠀⣀⣸⣿⣿⣿⣿⣿⣿⣟⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⣿⡿⣋⣛⡿⠿⠿⠿⠿⠿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣼⣷⣶⣶⣶⣶⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⢀⣼⣿⣿⡷⠆⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠂⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦
⠀⠀⠀⠈⣿⠟⠀⠀⠀⠀⠀⠀⠀⠤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⢠⣶⣶⣿⣦⣤⠀⠀⠀⠀⠀⢀⠀⠿⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣀⡀
⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⡿⠀⠀⠂⢠⠀⠀⣀⣤⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⢿⡆
⢀⡿⠘⣿⣿⣿⣿⠀⠀⠀⢰⡏⠀⠀⢀⠐⠻⣇⡼⠃⠀⠀⠀⣿⢻⣿⣿⡿⠃⠀⠀⠀⠀⢰⡟⢹⣿⣿⡟⠘⣧
⠀⢿⡀⠘⣭⣿⣿⠀⠀⠀⠈⢷⡀⠀⠀⢱⡏⠉⠀⠀⠀⠀⠀⣿⡀⢿⣿⣷⣤⠀⠀⠀⠀⠸⣧⡀⢿⣏⠡⢄⡿
⠀⠈⣇⠘⢿⣿⣿⡇⠀⠀⠀⠀⠳⣤⠀⠀⠇⠑⠀⠀⠀⠀⠀⠹⣇⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠗⠠⠀⠐⠈⠃
⠀⠈⠻⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠐⠀⠀⠄⢀⠀⡀
⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⠀⠀⠀⠈⠹⡆⠀⠈
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠂
⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⣼⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢠⠏⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⣰⡟⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠰⠟⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿

⠀⠀⡂⠒⠀⠂⠂⠀⠀⠀⠀⠬⠭⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠐⠀⠒⠒⠐⠂
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠛⠛⠙⠛⠀⠀⠀⠀⠀⠸⠯⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⠶⢴⣤⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠀⠃⠀⠋⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠠⠤⠀⠀⠤⠤⠀⠀⠀⠀⠅⢁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠐⠐⠀⠀⠐⠐⠀⠀⠀⠀⡃⢠⣠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢷⠁⢱⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⣀⣀⣀⣀
⠀⠀⡒⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢠⢐⠰⠀⡄
⠀⠀⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⡂⠀⠀⠀⠀⠈⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁