Illegal Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣴⣄
⠀⠀⠀⠀⠀⣀⠀⠹⣶⡶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣾⣷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢁⣽⡗
⠀⠀⠀⠀⣰⣿⡄⠐⢯⡀⠁⠝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⠠⣿⣿⠀⣠⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢋⣶⣿⣿⠟⠁
⠀⠀⠀⠀⣟⣻⣿⠀⠀⠙⢦⡀⠉⣹⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡟⠉⣐⣛⣛⡛⠁
⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢙⡾⠿⣷⡶⠂⠀⠀⠀⠀⠀⠀⠒⠶⢶⣿⣿⣿⣿⣿⣿⡶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠙⢃⡴⠚⠁⠀⡤⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠩⠍⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣋⣀⣀⣀⣀⣀⣘⣀⣈⣳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠈⠁⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠦⠤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡖⠛⣿⢦⡟⢲⣿⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⡁⡉⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠒⠒⠂⠤⠬⢭⡉⣙⣷
⠀⠀⠀⠀⢰⣗⠛⠛⠋⣉⣉⡉⠛⠛⢛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡈⠿⣿⣎⣁⣡⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠉⠉⠙⣡⣿⣿⣧⡙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⡀⠀⠈⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢀⣠⡀⣀⣀⢰⣦⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠈⠋⣠⠙⠋⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢿⣿⣿⣷⣶⣾⣶⣶⣶⣷⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣦⣄⣤⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠽⠟⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⠁


⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⢀⣴⣿⠋⠉⠁⠀⣉⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠉⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠉⠉⠀⠈⠉⠛⢿⣦⡀
⠀⠀⠀⢀⣾⠋⠻⣷⣄⠀⣦⣼⡟⠆⠈⢻⣆⠀⠀⠀⠀⠀⢰⡿⠉⢻⣦⡀⢸⡇⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⣰⡟⠙⢿⣦⣀⠔⢶⡂⠒⢢⠙⣷⡀
⠀⠀⠀⢸⡇⠀⠀⠈⢻⣿⣿⣿⠛⠂⠀⠈⣿⠀⠀⠀⠀⠀⣾⡇⠀⠘⣿⣿⣿⣿⣠⣶⠃⠀⠸⣷⠀⠀⠀⠀⠀⣿⠁⢀⡔⠛⢿⣦⣀⣿⣶⡿⠀⢸⡇
⠀⠀⠀⢸⣇⠀⠀⠰⣿⣿⡿⣷⣄⠀⠀⢀⣿⠀⠀⠀⠀⠀⢿⡇⠀⣤⣼⣿⣿⣿⣿⣧⣤⠀⢸⡟⠀⠀⠀⠀⠀⣿⡀⠸⣷⣄⣿⠙⢿⣿⡉⠀⠀⣸⡇
⠀⠀⠀⠀⢿⣆⠀⠰⠋⠉⠀⠈⠻⣷⣤⣾⠇⠀⠀⠀⠀⠀⠘⣿⣄⠀⠘⠛⠋⠙⠛⠻⣦⣠⣿⠃⠀⠀⠀⠀⠀⠘⣷⡀⠈⠛⠿⠿⠋⠙⢿⣦⣰⡿
⠀⠀⠀⠀⠀⠙⢷⣦⣄⣀⣀⣀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣀⣀⣀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⣀⣀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣛⣃⣀⣀⣀⣀⣘⣛⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⣉⣍⣉⣉⣉⣉⣍⣉⣉⣩⣭⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⣶⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⣠⣿⣿⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⡏⣾⠟⠋⣩⣍⢙⠻⣧⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⣰⣿⣿⣿⣤⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣃⣛⣛⣛⣛⡻⢟⣻⡇⣷⣾⣇⡛⢛⣼⣷⡿⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡆⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⢻⣿⠘⠛⠻⣌⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣛⡛⢛⠛⣛⣛⣛⣘⣛⣛⠛⣛⠂⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣶⣶⣶⣶⣶⣶⣶⢠⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣛⣣⣿⣿⣿⣿⣿⣿⣇⣻⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠀⠀⠻⠿⠿⠿⠿⠿⠿⠘⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡆⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠿⠿⠿⠿⠟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⠟⣛⠻⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣡⣾⣿⣿⣿⣿⣿⣿⣷⣝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢋⣵⣾⣿⣿⣦⡙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣡⣾⡿⣡⡙⣿⣿⣿⡿⣡⡙⣿⣷⣝⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢩⣾⣿⣿⣮⣉⣴⣿⣿⣷⡌⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣤⡄⣿⡇⢠⣤⣤⡄⣿⡇⢠⣼⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡻⠋⢭⣭⣭⣭⡭⠉⠟⣵⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣭⡀⣿⡇⢨⣭⣭⡅⣿⡇⢨⣽⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢸⣦⣭⣭⣭⣴⡇⣾⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣭⡁⣿⡇⢨⣭⣭⡁⣿⡇⢨⣽⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣌⢿⣿⣿⣿⡿⢡⡟⣩⣶⣶⣷⣦⣄⠀⠀⠀⠀⠀⠸⣿⡜⢿⣟⣁⣛⣃⣘⣛⣛⣁⣛⣃⣘⣿⠟⣡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣬⣭⣥⣶⡿⢸⣿⢿⣿⡿⢋⣽⡆⠀⠀⠀⠀⠀⠈⠻⣷⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡇⣿⣿⠿⠸⣿⣦⣉⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣝⣛⣛⣛⣛⣛⣛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠋⠁⠀⠀⠈⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⣿⠀⠀⣤⢀⣤⠀⠀⣠⡄⠀⠀⣤⣤⣤⠄⢀⣤⣤⣤⡄⠀⢀⣤⣄⠀⠀⣤⠀⠀⠀⢸⣧⣤⣄
⠀⠀⣿⠀⢸⣿⢸⡟⠀⠀⣿⠇⠀⢸⣿⣤⡤⢰⡟⠁⣤⣤⡀⢠⡾⠋⣿⡀⢰⣿⠀⠀⠀⢸⡇⢸⣿
⠀⠀⣿⠀⢸⡇⣼⣧⣤⡄⣿⣤⣤⢸⣧⣤⣄⠘⢷⣤⣤⡿⢠⡿⠛⠛⢻⣧⢸⣧⣤⡄⠀⢸⡇⢸⣿⠿⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⣿
⠀⠀⠛⠛⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⢸⣿⠀⣿
⠀⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣤⡄
⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⣿⢿⣿⣿⢿⣷⣄⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣷⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⢿⣿⣿⠿⠷⠿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠠⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠷



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣷⣶⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⣿⣯⣿⣿⣯⣿⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⢿⣿⣿⠿⠿⠻⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⣿⣿⡟⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣞⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡁⢸⡆⢄⣸⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢺⠁⠀⡟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣾⠀⢰⡇⣼⠁
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣿⠀⢸⡇⡟
⢀⣠⡤⠾⠓⠛⠛⠛⠒⠒⠤⣄⠀⠀⠀⠀⠰⣋⣠⡤⢷⣄⣤⠾⠏⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⣠⠤⣼⡇⠀⠀⠀⣿⠀⢸⡇⡇
⠋⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣯⡴⠾⣻⡭⢦⡀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⣼⠁⠠⠈⢻⡀⠀⠀⢸⡀⢸⡇⣿
⠀⠀⢰⡋⠉⠁⠃⠸⣀⣠⠴⠃⠀⠀⣠⠾⠙⠁⢡⢞⠁⠐⠲⠱⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⣸⠁⠀⠀⢸⡇⠘⡃⢿
⠑⠢⠼⡇⠂⠀⠠⠾⡇⠀⠀⣠⠤⣾⣇⡀⠀⢰⡏⣿⠀⠀⠀⡤⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣻⠀⠀⠀⠀⠂⠀⠁⢸⡆
⠀⡠⢖⡻⡆⠀⢠⡴⡇⢠⡏⠙⢚⢛⠧⠿⣄⢀⣹⡈⠁⠀⠚⠓⠙⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢠⠻⡍⢦⣬⣵⣻⢏⣟⢺⣇⠀⢰⡅⠀⠀⠌⠁⠀⡼⢄⣀⠀⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣆⣷⣠⣤⣶⠛⠛⠛⠛
⢸⠀⠘⡌⡿⠚⠙⠚⠘⣬⡟⠛⠛⡇⠀⣠⠔⣠⠞⠀⠀⠈⢑⡾⡄⢀⣀⣀⡀⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠙⠿⣿⠀⢹⡆⢠⠀⣀
⠀⡇⢨⠟⢶⠀⠀⠀⠀⢿⣧⠀⣼⠋⠑⠳⣾⠋⣷⠀⢀⠞⢸⢃⣿⣟⡻⣿⠛⠀⡀⠀⠀⠀⣿⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢸⡇⠸⣀⡿
⠀⣿⡈⢲⡞⠀⠀⠀⠀⡼⠼⡶⢭⣧⣴⡟⠋⠷⡮⡗⢫⠔⠀⠘⣿⣿⣽⣿⠀⠀⣀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⢀⠤⡄
⠀⠈⠙⣏⣀⢀⣀⣠⠤⣟⢸⠛⠉⠐⣟⠀⠀⣘⠇⣼⡾⠀⠀⠀⠈⣿⣟⠿⣆⠨⣁⡀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠘⠢⣄
⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⠀⠙⡶⢛⣝⠢⠜⠳⢶⣿⠀⢀⣶⣦⠀⣿⣧⣄⣹⣷⣬⠟⣆⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢒⠃
⠀⠀⠀⡇⠀⠀⠀⡞⠁⡿⠀⣼⣼⡏⠹⢿⣄⢀⡤⠺⢿⡯⠊⠀⠐⣿⣿⣀⣌⣀⣟⡷⣾⠃⠀⠀⠀⠀⠉⠛⠒⠲⠒⠲⠶⠤⣤⣤⠇⠀⣼⠀⢀⣾⡆
⠀⠀⠀⣇⠀⠀⢀⡇⠀⡇⣸⠀⢿⡇⢂⣰⣋⠤⢒⢆⣨⠃⠀⢀⡀⡇⢸⠁⠈⠁⢸⢦⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢾⣁⣀⣉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠀⠀⢀⣀⣠⠀⠀⠀⠰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠒⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠘⠛⠻
⠀⠀⠀⠀⠀⠀⠰⠀⠆⠰⠀⠆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠿⠿⠿⠷⣶⣤⡀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢻⣆⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⣔⠀⠀⠀⠀⣾⡟⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠰⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠈⠁⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠹⣧⡀⠀⣠⣿⠃⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⠀⠀⠀⠀⠙⢿⣷⣤⣀⡀⠀⢀⣘⣷⣾⠟⠁⠀⠀⢸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⡏⠸⣿⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⣿⠁⠻⡇
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠛⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⠀⠀⠀⠘⠿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⣠⣴⣶⣦⣤⡀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠁⠀⠀⠈⠻⣿⣿⣿⡿⠋⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⡀⠈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠀⠀⢸⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠘⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠃⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦
⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⢾⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠆
⠀⠀⠀⠸⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⠀⠈⠛⠋⠀⠀⠙⠛⠁⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣾⣶⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠉⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣻⣿⣿⣿⣿⣿⣿⠿⣿⣷
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⡿⠿⠛⠛⢉⠉⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣁⣀⣤⣴⣶⣾⣿⣿⠿⠿⠛⠋⣉⡁⢀⣴⠶⠾⣦⢸⡇⠀⠀⠀⠘⣿⣿
⠀⠀⢀⣀⣠⣼⣿⣿⣿⣿⡿⠿⠟⠛⠋⢉⣠⣤⡶⠀⣾⠛⢉⣭⡀⣿⠶⠞⢻⡄⢿⠶⠞⠛⣀⣹⣿⡇
⠀⢿⣿⡿⠿⠛⠛⠉⠉⠀⠀⣶⠀⠀⠀⠘⣷⠶⠶⠀⢹⣦⣤⡼⠇⠘⢃⣀⣠⣤⣶⣶⣿⣿⡿⠿⠟⠛
⠀⠸⣿⣧⠀⢺⡜⣧⠀⠀⠀⢸⣇⣀⣤⡤⠻⠶⠛⣋⣀⣤⣤⣶⣾⣿⣿⠿⠿⣻⣿⣿⣿⣇
⠀⠀⢿⣿⡄⠘⣧⢻⡶⠶⠞⢃⣉⣩⣤⣴⣶⣾⣿⡿⠿⠟⠛⠋⠉⠁⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠘⣿⣷⣀⣥⣤⣶⣶⣿⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⢻⣿⠿⠟⠛⢻⣿⣿⣿⣷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠿⠛⠛⠛⠻⠷⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⢀⣠⡤⠤⠶⠦⠤⢤⣀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⢛⡛⣛⠛⠛⢛⡛⠛⠛⣛⣛⣛⠛⣛⣛⠛⠛⢛⡛⠛⢛⠛⠛⠛⠛⢻⡇
⠀⠀⠐⢶⣶⡶⠶⠶⣿⠀⠀⢸⡇⣿⠀⠀⢸⡇⠀⠀⣿⠦⠁⢰⡏⣨⡅⢀⣟⣷⠀⢸⠀⠀⠀⠀⢸⡷⠶⠶⣶⣶⠖
⠀⠀⠀⠀⠙⠻⣦⡀⣿⣀⣀⣘⣁⣛⣛⣓⣘⣛⣛⣂⣛⣛⣓⣀⣛⣚⣁⣚⣁⣈⣃⣘⣛⣓⣀⣀⣸⡇⣠⡾⠋
⠀⠀⠀⠀⣠⡾⠋⠀⠉⠉⢿⡏⠙⢿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠏⠉⣿⠏⠉⠁⠈⠻⣦⡀
⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⣿⡄⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣼⡿⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠈⠙⠲⠦⠤⢤⠤⠤⠶⠚⠉⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣄⣀⣀⣤⣤⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣴⡄⠀⢀⣴⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⢠⣿⣿⣦⣾⣿⣿⣶⣿⣿⣇⣴⣿⣧⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠉⠉⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣶⣖⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣿⣿⠋⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠋⢉⣉⣉⣁⣠⣤⣤⣤⣴⡆
⠀⠀⠀⠀⠀⣀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠋⣉⣉⣁⣠⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠟⠛⠛⠛⢉⣉⣉⣁⣤⣤⣤⣤⣴⣶⣶⣶⣦⣼⣿⣿⣿⠿⠛⠻⢿⣿⣿⡿⠉⠉⢻⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣶⣾⣿⣿⡿⠿⢿⣿⣿⡟⠛⣻⣿⣿⡏⠀⢀⣀⣀⡟⠁⣠⣤⣄⣴⣿⣿⠇⠀⠀⠸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⢸⣿⣿⡇⠀⣿⣿⣿⡇⠀⠿⠿⢿⡇⠀⣿⠛⠛⠉⢹⡿⠀⢸⡆⠀⢻⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⢸⣿⣿⡇⠀⣿⣿⣿⡇⠀⣠⣤⣼⡇⠀⣿⣶⡆⠀⢸⠇⠀⠈⠀⠀⠈⣿⠀⠘⠋⠉⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⣺⣿⣿⡇⠀⢿⠿⠿⠇⠀⠛⠛⠛⣧⡀⠈⠉⠁⡀⣈⣀⣸⣿⣿⣧⣤⣿⣶⣶⣶⣾⣿⣿⣿⡿⠿⠿⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⠈⠉⠈⣁⣀⣀⣀⣠⣥⣤⣤⣶⣶⣿⣿⣷⣾⣿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠋⠉⢉⣉⣀⣀⣤⣤⣤⣴⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣶⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠉⠉⢉⣁⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠃
⠀⠀⠀⠀⠸⡿⠿⠿⠿⠟⠛⠛⠛⠉⠉⢉⣉⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉⢉⣿⣿⣿⣿⠿⠿⠓
⠀⠀⠀⠀⢀⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠘⠿⠿⠿⣻⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠿⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⣻⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠉⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠛⢿⣿⡇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠸⡿⠋⠀⢻⡿⠁⠈⠻⠏⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣽⣵⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣶⣶⣾⣿⣿⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠤⠤⣤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⠚⠉⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠉⠗⡢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠈⠀⣠⡀⠀⠀⠀⣤⣾⣷⣤⠀⠀⠀⢀⣄⠀⠑⠫⡲⣄
⠀⠀⠀⠀⠀⠀⢠⢞⠅⠀⠀⠐⢿⣿⣿⠟⠀⠀⢸⠿⢿⡇⠀⠀⠻⣿⣿⡿⠃⠀⠀⠪⡳⡄
⠀⠀⠀⠀⠀⣰⡻⠁⠀⣠⣄⠀⠘⠋⠛⠂⠀⡀⠠⠤⠤⠄⠀⡀⠐⠛⠙⠃⠀⣰⣆⠀⠈⠝⣆
⠀⠀⠀⠀⣰⠃⠀⠈⢻⣿⣿⡟⠁⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠈⢻⣿⣿⡟⠁⠀⠈⣂
⠀⠀⠀⢰⠃⠀⠀⠀⠘⠋⠙⠃⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠘⠉⠙⠃⠀⠀⠀⢸⡆
⠀⠚⠉⢉⡈⢉⣉⠉⠉⠉⢉⣉⠉⠉⠉⠉⣉⣉⣉⣉⠉⠉⣉⣉⣉⠉⠉⠉⣉⡉⠉⠉⢉⡉⠉⠉⠉⠉⢳
⠀⡇⠀⢸⡇⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣭⣍⠉⠀⣾⠋⢉⣛⡃⠀⢠⡿⢿⡄⠀⢸⡇⠀⠀⠀⠀⢸
⠀⡇⠀⢸⡅⢸⣿⣀⣀⣀⠸⣿⣀⣀⣀⠀⣿⣉⣁⣀⠀⢿⣄⣘⣿⠇⢠⡿⠷⠿⣿⡄⢸⣇⣀⣀⡀⠀⢸
⠀⢧⣀⣈⣀⣈⣉⣉⣉⣉⣈⣉⣉⣉⣉⣀⣉⣉⣉⣉⣀⣀⣉⣉⣁⣀⣉⣁⣀⣀⣈⣉⣈⣉⣉⣉⣁⣀⡼
⠀⠀⠀⠸⡆⠀⠀⠀⢠⣀⣀⡄⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢠⣀⣀⡄⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠹⡄⠀⢀⣼⣿⣿⣧⡀⠀⠂⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⢀⣼⣿⣿⣧⡀⠀⢠⠏
⠀⠀⠀⠀⠀⠹⣅⠀⠀⠀⠉⠀⢠⣄⣤⠄⠈⠁⠀⠒⠀⠂⠀⠁⠠⣤⣠⡄⠀⠹⠏⠀⢀⣨⠌
⠀⠀⠀⠀⠀⠀⠈⢮⡠⡀⠀⠠⢾⣿⣿⡦⠀⠀⢸⣶⣶⡇⠀⠀⣴⣿⣿⠷⠄⠀⢀⠐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⢀⢄⠀⠙⠁⠀⠀⠐⠛⢿⡿⠛⠂⠀⠀⠈⠋⠀⡀⣐⠅⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢤⣀⡀⠀⠀⡀⠈⠁⢀⠀⠀⣀⣀⡦⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠘⢶⣤⣼⣿⣏⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⡇⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠙⠃⠛⠱⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢾⣿⡀
⠀⠀⠀⠀⢀⣀⣀⣿⣀⣀⠀⢀⣀⡀⢰⡄⠀⠀⢰⣿⣿⣿⣿⢀⣠⣶⣤⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢀⣿⡂⠀⠀⣠⣿⠇
⠀⠀⠀⠀⢸⣿⣿⢽⣿⣿⣿⣿⣿⣿⣛⢶⠀⠀⢸⣿⣿⣿⡏⠜⠛⢻⡍⠛⠛⠂⠀⠀⠛⣿⣿⣏⠉⠀⣼⣿⣿⡆⠀⣿⠃
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠃⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⣿⣿⡏⠈⠀⠉⠉⠙⠀⠀⠋⠈⠁
⠀⠀⠀⠀⠘⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠃

⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⢠⢖⡒⢄⠀⢀⡀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢀⡄⠀⠀⣠⣤
⠀⠀⠀⣈⣻⣃⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣸⣿⣇⠘⠄⠀⠐⢘⣥⣌⣳⠀⠀⠀⢀⣹⣟⣠⡤⣾⡿⣶⡄⠛⢛⣥⡀
⠀⠀⣸⣿⡏⣿⣷⡀⠀⢠⡀⠀⣤⣤⢿⠂⠀⠀⣾⣿⣿⣿⣧⢠⠖⠑⢮⠀⠑⠃⠀⠀⢠⣿⡿⢹⡇⠀⠻⣷⣾⠿⠿⠿⣿⣧
⠀⠓⢿⣿⣷⣿⢿⡿⠿⠿⠿⠷⠿⠿⠿⠿⠷⠀⣿⣿⣿⣷⡿⠐⢦⣨⠞⠤⠠⣄⠀⠀⢸⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⣼⣿⣿⣼⠷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⠀⠆⠉⢲⠸⡘⠅⣸⠀⠐⠋⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢿⣿⣿⠻⣷⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠈⠦⣀⠼⠀⠉⠉⠁⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿


⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡀
⠀⠀⢠⣟⡙⡟⣁⠀⠀⢶⠄⠀⢰⡦⠀⠀⠀⠀⣠⣿⣿⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣄⠀⠀⠀⠀⠀⠀⡶⢶
⠀⠀⠈⠉⣧⣣⡿⡇⡴⣿⡷⢀⢾⣿⠆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠐⠄⠀⠀⡿⣿⣿⣿⡇⠠⠄⠀⠀⠀⣵⣶
⠀⠀⠀⠀⣿⣿⡇⡇⠙⣿⡟⠈⢻⣿⠊⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⠀⠀⡇⣿⣿⡿⣇⣤⣎⣿⣿⣉⣿⣿⣀
⠀⠀⠀⠀⣿⣿⡇⠤⠄⣿⠇⠤⢼⡿⠠⠀⠀⢸⣿⣿⣿⣿⡗⠃⠤⠀⠠⠀⠹⠎⠀⠀⠀⠀⣿⣿⡇⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇

⠀⣁⣈

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⢛⣛⣛⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣶⣦⢰⣆⠀⣬⡻⣷⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⢿⣿⢛⣿⡿⠿⠿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⢘⣛⣻⣛⣛⣻⡋⣿⢻⠀⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣟⠛⠛⡛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢸⣿⣧⣰⣭⣝⠃⣿⢸⠀⣠⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣤⡴⠃⣰⣶⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣝⠿⣷⣿⣾⡾⣱⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⣰⣿⢸⣷⡀
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⢫⣾⡟⠘⣵⣿⣸⣷⡀⠀⠀⠀⠀⠀⠀⣨⣭⣭⡅⠀⠀⣼⣿⣿⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⠿⠟⠋⠛⠁⠀⠸⠿⠿⠿⠿⠗⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⢀⡄⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⣷⠆⠀⠀⠀⠀⣾⣻⡆
⠀⠀⠀⠀⠀⠀⣐⣛⣣⣿⣭⣭⣭⣭⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠟⠀⣠⣄⡀⠀⠹⡿⠁
⠀⠀⠀⠀⠀⠀⣿⣷⢰⡿⠉⣿⡝⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣭⣭⡅⠀⠀⢿⡀⢀⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣾⣷⣶⣾⣾⢞⣭⣥⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⣿⣿⠆⣀⡀⢉⣉
⠀⠀⠀⠀⠀⠀⣿⢱⣶⣶⣶⣶⡏⣬⢭⡍⣭⢩⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⡑⠶⣟⠱⣿⣛⢫⣝⣛⣿⠆
⠀⠀⠀⠀⠀⠀⣿⡸⠿⠿⠿⠿⣿⢐⠶⣒⡲⣖⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⢋⣼⣿⣮⡝⢸⣏⣉⠁
⠀⠀⠀⠀⠀⠀⣿⣯⣭⣿⣿⣿⣿⣜⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁⢸⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⣫⢡⡄⣤⢩⡻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣲⣶⣶⣶⣶⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⣼⡇⣿⡇⣿⡇⣿⡜⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⣶⠶⢶⣶⢹⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣿⡇⣶⡄⣶⡆⣶⡆⣴⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠾⠇⣿⢸⡇⡶⣠⣶⢖⣴⡶⣢⣤⡀⠀⠀⠀⠀⢘⣿⣿⣌⠧⠻⡇⡿⠡⢏⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⣐⣛⣓⣈⣸⡇⠈⠉⠁⠉⠉⠀⠉⡍⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣶⣶⣶⣾⡿⢋⣵⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣭⣭⣭⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣿⣿⡏⣭⣭⣭⣭⣽⢡⣿⢿⣿⡿⢋⣽⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣭⣭⣭⣭⣭⣥⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⣿⣿⣷⣮⣿⣯⣯⣿⡜⣿⣷⣍⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠎⠛⠻⠟⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣤⣄⠀⠀⣰⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢏⣼⣿⡇⣿⡿⢮⡿⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣿⠟⣿⡇⠛⠋⠀⣨⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⣾⣿⣿⡇⣿⡿⠛⠶⠶⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⣿⣿⡿⠀⠀⠀⣼⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢱⣿⣿⣿⣿⣇⠺⠿⠿⠿⠷⠶⢧⠀⠀⠀⠀⠘⢆⣤⣤⡰⠇⠀⣿⣝⡿⠑⠒⠂⠀⣷⣬⡻⡇
⠀⠀⠀⠀⠀⠀⠘⣛⣛⡛⢘⣛⣛⣛⣿⢟⣛⠶⣛⣛⡛⢛⣛⠀⠀⠀⠀⠀⠸⣯⣾⠗⣠⣤⣭⡛⠀⢀⠀⠀⠀⠿⠦⠾⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢸⣿⣿⣿⣷⣿⣿⡇⣿⣿⡇⣿⣿⠅⠀⠀⠀⠀⠀⣈⢉⣾⣟⢿⣿⣿⡆⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⣿⣿⣿⡿⣿⣿⣷⣬⣫⣴⣿⣿⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣴⡟⡴⠆⢀⣾⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡸⣿⣿⣗⣿⣿⣿⣿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠛⠂⣿⣷⣿⣿⣿⠇⠀⢀⣾⣿⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠌⠻⠯⠿⠿⠿⠿⠸⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣴⡿⠀⠠⣿⣿⣿⣿⣿⣿⠆

⠀⢀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⡤⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄
⠛⠛⠟⠛⡟⠛⠛⠛⠛⠛⠛⠛⡻⠛⠻⠛⠛⠙⠛⠛⠛⠛⡟⠛⠟⠛⠛⠛⠛⠛⠛⠻⠛⠿⠛⠛⠛⠛⠛⠛⠛⡿⠛⡟⠛
⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴
⣦⣤⣀⠀⢰⣿⣿⣿⣿⣿⠀⢀⣠⣤⣤⣄⡀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⢀⣀⣠⡀⠀⠀⢰⣿⣿
⣿⣿⣿⣆⣼⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⣲⣤⣾⣿⣿⣿⣿⣿⣤⣤⣲⣿⣿⣿⣿⣿⣧⣀⣼⣿⣿⣿⣿⣄⣠⣼⣿⣿
⣿⣿⣿⠃⢈⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣿⣿⠞⠉⣿⣿⣿⣿⣿⣿⣧⠀⠚⢻⣿⣿⣿⣿⠋⠀⣿⣿⣿⣿⣿⣧⣀⣽⣿⣿
⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⠟⠀⢰⣿⣿⣿⣿⣿⣿⣿⢀⣰⣿⣿⣿⣿⣿⣇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣶⣶⣶⡆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠟⢭⣈⣡⣬⣿⣿⢸⣷⣿⣯⣟⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⠿⣿⣿⣿⠿⠶⠿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣾⣿⡟⠁⠚⠋⠁⠀⡀⢤⣠⣄⣩⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⣠⡏⠸⠋⠁⠊⠁⠋⠩⠋⣉⣿⠋⢀⢠⡆⣶⣶⣗⣿⣿⣿⣿⣿⣾⣇⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠋⠀⣠⣴⣶⣆
⠀⠀⠀⠀⠀⢀⡿⢀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡀⠀⣀⣤⣶⣿⠟⠉⠲⣶⣤⡀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢸⡇⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣶⣿⣿⣦⣄⢘⣿⣿⣿⣿⣷⡜⣿⣧
⠀⠀⠀⠀⠀⢸⡇⠀⣌⢁⣠⣤⣤⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠰⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠸⣇⠀⠙⠛⠉⠉⠉⠓⠀⠀⠀⠀⣿⣿⡿⢛⡿⠃⠈⠙⠛⣿⣿⣿⡿⣴⣤⣸⣦⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢤⣿⠀⠘⣿⣿⠶⠀⠀⠀⠀⢠⠀⠸⠿⠁⣾⢁⣴⣶⡄⢠⣿⣿⡿⣱⣿⣿⣿⣿⣿⣦⡙⠛⣿⣿⣿⣿⣿⣿⣿⣿⠏⣠⡀
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠈⠁⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠀⠸⣟⠋⠃⠘⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣆⢹⣿⣿⣿⣿⣿⡿⢃⣾⡿
⠀⠀⠀⢀⣴⡿⠁⠀⣄⠀⠀⠀⠀⠀⠯⣄⠀⠀⠀⠀⠀⠀⠠⠿⠀⡠⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟⠋⣵⣿⣿⣷⣆⡀
⠀⠀⠀⣾⠹⣧⡐⢶⣬⠇⢰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⣤⢶⡅⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⢰⣿⣿⣿⣿⣟⡁
⠀⠀⢀⣿⣦⠈⣿⠾⠷⠞⠛⢩⣋⡗⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣾⠇⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠸⣷⣿⣾⣿⣤⣤⡴⠶⠞⠉⣹⠇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⢸⡇⣿⠿⣿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣻⣄⡀
⠀⠀⠀⣿⣅⣀⣀⣀⣀⣤⡴⠶⢛⢹⡶⠀⠀⠀⣀⣤⡶⢄⣾⡿⠁⣠⡿⠃⣿⢠⣿⣧⣀⢠⠒⢤⣀⣀⣀⣀⣀⠀⣠⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠙⣿⣋⠉⠉⠉⣁⣠⡤⠾⢿⣦⣀⣤⠞⠋⠁⢠⣾⡿⢁⣴⡟⠁⠀⠋⣾⣿⣿⣿⣿⣄⠁⢀⡅⠀⠀⢙⡷⠿⣿⣷⣿⣿⣿⣿⡷⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠙⠛⢿⣟⣋⣉⢁⣠⠶⣿⠉⠉⠉⢀⡆⢀⣿⠟⣠⡿⠋⠀⠀⠀⠀⣿⣿⣿⠟⢻⣿⣿⠏⠀⢀⣴⠃⠀⢀⡄⠻⢿⣿⣿⣿⣷⣤⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢀⡄⢻⣧⣄⡐⠛⠀⠸⠃⣼⠟⠁⠀⠀⠀⠀⠀⢸⡿⣡⠀⠀⢰⠇⠀⢰⡿⠁⠀⣠⡟⠀⠀⣠⣿⣿⡻⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠁⢰⡏⠉⠻⣧⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⡼⠀⢠⡟⠀⠀⣾⠋⠀⢀⣼⠏⠉⠉⣿⡄⣌⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⡽⠃⠀⠘⠇⠀⠀⢁⡾⠉⠀⣀⣀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣏⢰⣾⣇⡀⣾⡏⠀⢰⣿⠃⠀⣰⠟⢹⣿⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣠⡞⠁⢸⠁⠀⡤⠀⡌⢦⠀⠞⢀⣠⠞⠋⠁⠀⠳⠀⠀⠀⣼⠃⠀⠀⠀⠈⠓⠛⢹⢄⣽⢿⠓⣆⣿⡧⣄⣼⣷⣄⣼⢏⣿⣿⡿⠁
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠄⢸⠁⠀⠀⠈⣧⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠘⠷⠟⠁⢷⠟⠁⠈⠙⢿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢠⣾⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠳⣦⡙⢿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣄⠙⢦⣤⣄⣠⡴⠶⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠺⣤⡈⠛⢠⠟⠀⠙⢦⡀⠀⣠⣾⣿⣿⣿⠟⠁⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠀⣰⠋⣁⡴⠋⠀⠀⠀⠀⠙⢶⣌⠻⣿⠟⠁⠀⠀⠀⠈⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠻⣮⡉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⣠⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠚⠛⡦⢹⣿⡿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⣠⣾⣿⣿⣦⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⣰⠃⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣤⣾⣿⣿⣿⠟⠁⠀⠀⠈⠳⣄⠀⠀⠀⣀⡼⠃⢀⡴⠳⣌⢛⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⢻⠶⠞⠉⠀⣠⠞⠁⠀⠈⠉⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⣿⣀⣠⡴⠻⣧⡀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣬⣭⣥⣶⣷⢠⡟⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⡿⠿⠿⠛⠋⠁⠈⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠉⠉⠉⠉⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣤⣀⣀⣤⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣆⠙⣧⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡈⢷⡀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣾⠏⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⠱⠄⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡾⠁⠀⠀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠠⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⡀⠀⠘⠀⠞⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⢀⠎⠀⠀⣀⠶⠋⠁⠀⡈⠂⠀⠀⠀⠨⡀⠀⠉⠳⢎⡀⠀⠠⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡞⠀⠀⠀⠁⠀⣀⢄⣈⠤⠐⠒⠒⠒⠠⠌⣐⢄⡀⠀⠈⢂⠀⢳⠀⡇⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢧⠀⠇⠀⡀⠂⠂⣡⠞⠁⠀⠀⠀⢀⢀⠀⠀⠠⠀⠑⠺⡀⠀⠈⢀⡞⢰⠃⠀⡰⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣄⠟⠁⠀⠀⠠⠶⠀⠘⠚⠀⠀⢰⠀⠀⠀⠘⢦⣀⡈⠀⠀⠀⠈⠒⠋⠉
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠈⠀⢀⣠⡏⠀⠄⢰⢴⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠈⡆⠉⠉⠉⠉⠉⠉⠀⠋
⠀⠀⠀⠀⠀⡸⠀⠀⠠⡀⠙⣄⠀⠉⠉⢠⠁⠘⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⢀⠔⠁⠀⠐⠈⢀
⠀⠀⠀⠀⠀⠉⠑⡴⠁⠀⠀⣽⠀⠀⠀⢸⠀⠀⠀⠛⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⡆⠈⠀⠀⣄⠀⠀⢠⠏⠀⠀⠀⠀⠀⢤⠀⡀⠀⡌⠀⠀⠀⠉⠒⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠘⡄⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⢤⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣠⠖⢊⣠⣤⡴⠶
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⢒⠀⢀⡏⠀⠀⠀⠈⢢⡀⠀⠀⠁⠀⠀⠓⠀⢀⠈⠀⠀⣠⠊⠀⠀⠀⠀⠀⢸⡅⢀⠀⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⣠⡴⠋⣠⠊⠀⢀⣀⠀⠀⢂⠜⠓⠤⠄⣀⣀⣁⣀⣀⠤⠔⠛⢄⢂⠀⠀⠀⢀⡀⠀⠑⢄⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠁⠀⢠⠏⠀⠀⠀⠀⠀⠁⠀⠈⠻⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠄⢰⡎⠈⢀⠀⢀⠈⠑⣄⠀⣠⠐⢚⣁⣠⠀⠀⠈⠀⠀⢘⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⣮⠀⠐⢸⡇⠀⠀⢸⡇⢰⡏⣄⠀⣤⠀⠜⣴⠛⠄⠈⠉⡁⠀⢠⡆⠀⢀⡄⠁⢀⠀⢀⠀⠀⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠸⡇⢠⡀⠈⠇⣼⢁⣛⡀⠘⠀⠾⠁⣾⠀⢘⠛⠁⠀⢸⡇⠀⢸⡇⢠⡟⠀⣶⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠁⠘⣷⡀⢰⣇⡜⠸⣧⣀⡄⣲⡀⠹⠶⠶⠶⠞⠀⢸⡇⠀⢸⡇⢸⠇⢀⣍⣠⣼⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣠⠈⠷⠴⠞⠀⠀⠀⠿⠤⠞⠷⠤⠖⠀⠀⠀⠀⠈⠻⠶⠤⠖⣿⣄⡀⠀⢠⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⣠⠶⠞⠛⣛⣛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣤⡴⠶⠶⠶⢮⣍⠛⠂⢀⣤⠶⠛⢛⣉⣉⣙⠛⠶⣄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠳⢶⣤⡀⠀⢈⣁⡴⠚⠋⠉⠉⠙⠛⠷⣦⡀⠈⢷⡀
⠀⠀⠀⠀⢠⡾⣰⠏⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⠋⠙⠻⣆⣀⣀⣀⠀⠀⠀⠈⢻⡄⠈⣷
⠀⠀⠀⠀⣾⢡⡟⢠⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢿⣷⣶⠞⣩⣭⣭⣉⡛⢶⣄⠀⠀⢻⠀⣿
⠀⠀⠀⠀⣿⢸⡃⣿⠀⣤⡤⠴⠶⢚⣩⣤⡀⠀⠀⠴⠾⠿⠶⠿⠿⠿⢿⣿⣦⡙⣧⠀⠘⠀⣿
⠀⠀⠀⠀⠻⠸⣇⠸⣦⠀⠀⠀⠘⣆⢻⣿⠷⡠⢂⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣭⣙⡻⢶⣤⡉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠙⢃⡸⠿⣦⣄⣀⣈⡤⢖⣫⠞⢋⡭⠶⠛⠉⢉⡉⣉⣉⠉⠛⠻⢿⣿⣷⣮⣙⢷⣄⠙⣦⡀
⠀⠀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠈⠉⢁⠴⠯⢍⠝⠉⠠⡆⢠⡆⢸⡇⣿⣽⠀⠀⢠⡄⠈⠻⣿⣿⣷⡝⢷⡈⢷⡀
⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⡇⠸⢿⠸⣷⠖⠀⢸⡇⠀⢀⡈⠻⣮⢻⣆⢻⡄
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣰⠶⡆⠃⠀⠁⠀⣾⠀⣷⠀⠀⠘⠁⠀⠏⣿⠀⠙⣧⢻⣆⢻⡄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡐⠀⠀⡇⢿⣀⡟⠀⠀⠀⠀⣿⢻⣿⠀⠀⠀⠀⢀⣴⣧⡆⠀⠸⣇⢻⡄⣧
⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⡇⠀⠈⠁⠈⠛⠁⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄⢿⠘⣧⢸
⠛⢠⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⢰⠛⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⢸⡆⣿⠘
⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠸⠶⡟⠀⠀⠀⠀⠀⠀⠀⢻⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⠇⣿
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡆⢀⡼⠁⠀⡀⠀⠀⠀⠀⠀⠘⣧⠅⠀⠀⠀⠀⠀⠀⣀⡌⠁⠀⢸⠀⡿
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣷⠀⠀⠀⡟⣻⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⠏⡇⠀⠀⠆⢀⠇⠘
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⢿⡄⠸⣧⠀⢠⡟⢳⠀⠀⣤⣤⣤⠀⠀⣠⠀⠀⢀⠶⠆⠘⠛⡟⠀⠀⠀⠘
⠀⠀⢀⠘⣯⣉⣉⣉⣉⣉⣉⡈⢿⣆⠹⣧⡈⠳⠞⠀⠀⠀⢰⠇⠀⣰⠃⠀⠀⠸⣆⠀⠀⠀⠃⡀
⠀⠀⠘⣧⠈⢻⣭⣭⣭⣭⣭⣥⣤⠻⣷⣌⢿⣦⡀⠀⠀⢀⡏⠀⠀⡏⢻⠀⢠⣤⠟⠀⢀⡠⠮⡭⠋
⠀⠀⠀⠈⠳⣄⠙⢷⣶⠶⠶⠚⠛⠶⣾⠿⢷⣯⣻⣶⣤⣈⠀⠀⠀⠻⠞⠀⠀⣀⣠⠖⣋⡴⢊⡕
⠀⠀⠀⠀⠀⠉⠳⠦⠉⠻⠃⠀⠀⠀⠘⣧⣀⣉⠛⠿⢿⣿⣿⣿⣷⣶⣾⣿⣿⠿⠞⣋⣥⠚⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣀⠤⠾⣷⣶⣒⣒⡂⠀⠀⠀⠉⣉⣉⣀⣤⠴⠖⠋⠉⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⠋⠉⠁⠀⠀⠀⠈⠉⠉⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢃⣄⠀⣜⡇⠀⠰⠆⠀⠀⠀⠀⠀⠰⡄⠀⣿⡀⠖⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢀⡽⠞⠋⠉⠭⢝⠢⡀⠀⣖⢲⡤⣄⡤⠒⠒⠛⢧⣞⣳⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⡤⢒⡶⢋⣀⣈⣽⣖⣧⡀⠀⠀⠀⠀⠉⠛⢄⠰⠚
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⢺⢀⡤⣚⣩⠿⠛⠉⠉⠉⠉⠉⠙⠳⢤⣈⢳⣄⡀⠀⠀⢸⡌⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⢋⡽⠋⣀⣴⠖⠛⠉⠉⢹⠛⠒⠤⡀⠈⠻⣟⠛⠲⢤⣀⣡⠇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢠⠏⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢣⡀⢠⠙⣅⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢳⡀⢇⠸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⣇⢸⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠂⠀⠀⠀⠀⠸⡊⢂⠀⠀⠀⠀⠀⠀⣀⣧⠀⢹⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡄⡇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⡼⢸⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⢇⡏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠱⣜⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⣼⢞⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠓⢍⣒⠤⠀⠰⠀⠀⠀⣀⡴⠊⠁⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣲⢤⣈⠉⠛⠒⠛⠋⠉⢁⣀⡤⢞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⠉⠙⠛⠛⠋⠉⠉⠀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⢱⣀⣤⠮⠀⠀⠤⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⠀⠀⠴⣤⣀⡎⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⠟⠉⠀⠀⠀⠀⠀⠈⠳⡀⠀⠨⢥⡬⠇⠀⢠⠞⠁⠀⠀⠀⠀⠀⠉⠛⢷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⣸⡇⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠠⣔⠩⠔⠐⠂⠉⠉⠁⠀⠀⠈⠉⠉⠐⠒⠤⠍⣲⠄⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⠄⢊⠗⠊⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠑⠺⣑⠠⣀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠔⠊⡠⠞⠁⠀⠀⢀⡠⠔⠂⠉⠀⠀⢀⠈⢁⡀⠀⠀⠉⠐⠢⢄⡀⠀⠀⠈⠳⢄⠁⠢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠖⠁⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠈⠢⣄⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠳⡀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣼⠄⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠃⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⡀⠁⠀⠘⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠘⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠒⠘⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠒⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠄⣀⠐⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡀⠰⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢏⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡹⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⣠⠞⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠈⠓⢄⡠⠀⠀⠀⠀⠀⠀⠏⠰⠀⠀⠀⠀⠀⠀⢄⡠⠚⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠠⠄⠀⣀⣀⠠⠄⠊⠁⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠢⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠔⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡔⠁⠈⠁⠒⠠⠤⠄⣀⣀⣀⡀⠠⠤⠄⠒⠈⠁⠈⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉


⣶⣶⣶⣶⣶⡾⣶⣶⣶⡶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⢷⠾⠷⠾⠷⣶⡶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢣⣰⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠞⣱⠃⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠈⢳⠦⡀⠀⠀⠻⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡆⣀⠀⠠⡧⠴⢤⠇⣠⣒⠛⢁⠎
⠀⠀⠀⠀⠀⠈⠫⣀⣨⠆⠀⠀⠀⠙⠃⠹⣸⠀⠀⠀⣠⣠⢄⠀⠀⠳⠛⠛⠀⠀⠀⠀⠀⠚⠉⠁⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢠⠔⠊⠙⠲⢍⠀⣴⣒⠃⠁⢈⣍⡷⠀⡠⠚⠉⠉⠉⢢⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⡇⠀⠀⠀⠀⣨⠇⠈⠈⡇⠀⢸⠀⠀⠀⠧⣀⠀⠀⠀⠀⡇⠀⠘⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠧⠤⠴⡂⢫⡤⠴⠒⢲⡃⣠⣼⠋⣱⠖⣲⠏⢨⡓⠒⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⢀⠠⢀⣀⢀⠀⠀⠀⢀⠀⠀⣘⡆⠈⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠄⠊⠁⠀⠀⠀⠈⡍⠁⠀⠀⠀⠀⠁⠀⢀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡔⠁⠀⠀⠓⠀⠀⠀⠀⣄⠀⠀⠀⠀⠒⠀⠀⠀⢢⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠏⡢⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢣⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡐⠄⠀⢠⠀⢀⠂⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠑⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠉⠂⢄⣀⠀⠀⠀⠆⠀⠀⠀⢘⠁⠐⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡴⠤⠶⠤⣄⣉⣉⣉⣛⡤⠬⠥⠦⣄⡄⢖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⡀⠀⠀⠀⢹⠃⠀⣀⠀⠀⣀⣴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢦⡀⠀⢸⠀⠸⣍⡷⠀⣿⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣸⣄⣀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⢁⣠⡤⣤⣀⠉⠓⢯⡀⠀⠀⠀⠀⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣦⣀⣀⣠⠎⠀⠀⣰⣫⠞⠉⠓⢮⣳⡀⠀⠘⢦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣋⣳⣀⠈⠛⠁⠀⢠⠇⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⠀⠈⢯⠉⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠛⠛⣿⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⢸⡶⠶⠶⠶⠶⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⢾⣙⡦⠤⠤⠤⠤⠤⢿⠀⠀⠀⡇⠀⠀⠀⢾⠆⠀⠀⠀⡇⠀⠀⢸⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀⣀⣀⣘⣆⠀⠀⡇⠀⠀⠀⠺⠀⠀⠀⠀⡇⠀⢀⡞⠀⢀⣀⡈⠉⠳⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠙⢦⡀⠳⠤⠤⠤⠤⠤⠤⠤⠴⠃⣠⠞⠛⠛⠻⠤⠗⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣳⠀⠀⠀⠀⠀⠙⡶⢤⣀⣀⠀⢀⣀⣠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⡍⠉⢹⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡞⢦⠀⢸⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢠⡇⠀⠙⠋⢠⠾⣆⠀⠀⠈⢳⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠼⠉⠁⠀⠀⠀⠈⠓⠋⠀⠀⠀⠈⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣷⣿⣿⣿⣿⣟⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠉⠉⠛⠛⠛⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠿⠿⢿⣷⣶⣶⠿⢿⣦⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣠⣤⣦⣤⣀⠀⢀⣿⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠋⠀⠉⠻⢿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⠶⡾⠷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣃⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠲⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⣆⡔⢦⡄⢤⡶⠆⠀⣤⢦⠔⡆⢄⡤⢠⣠⢠⠀⣤⠴⣤⠶⢤⠤⡇⠱⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣭⠏⠓⠋⠓⠛⠃⠀⠀⢸⠉⠓⠛⠛⠛⠛⠚⠐⠛⠋⠒⠙⠃⠘⠒⠃⠀⢘⣿⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠊⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⢠⢦⠀⠀⠀⠀⠀⢸⣿⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⣆⡧⡇⠀⣾⢹⢾⣉⢾⣃⡀⠀⡧⣇⣬⢽⢻⢸⢹⠀⠀⠀⠀⢸⣿⠀⠀⠉⠒⠂⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠈⠁⠈⠁⠉⠀⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⢤⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠃⠀⣀⢀⡀⢀⡀⢀⠀⡀⠀⡀⢀⡀⣀⠀⣀⠀⣀⢀⠀⢀⠀⡂⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠸⠉⠙⣊⣙⣈⣙⣉⣃⣉⣊⣑⣋⡚⠁⠛⠁⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠉⠙⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⣰⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆⠀⠀⢀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣏⣀⣛⣛⠁⣿⠀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⣿⠈⣛⣛⣈⣹⠉
⠀⠀⠀⠀⢠⣴⣄⣀⣉⣉⣉⣉⡁⣿⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣿⢈⣉⣉⣉⣉⣀⣠⣶⡄
⠀⠀⠀⠀⠈⠛⠋⠀⣭⣥⣭⣬⡁⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⢠⣭⣤⣭⣤⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⣶⣤⠄⣿⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⣿⠠⠤⣶⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠈⠻⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢀⢀⣤⣤⣤⣤⣤⣄⣀⡁⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⢻⡄⠀⠀⣪⠒⠉⠉⠀⣄⠀⠉⠉⠙⠺⢕⢦⠀⠋⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣁⠀⣸⠇⠸⠀⠀⠀⠏⠀⠀⢰⠀⠀⠀⣧⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣯⣯⣅⠢⡀⠀⡶⢦⢄⣈⡤⣖⣒⣿⣦⣦⣶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⢠⢫⣾⣿⣿⣿⣿⣯⡵⠃⣀⣈⣷⣾⣇⠾⣿⣿⣿⣿⣿⣷⡀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢆⠻⣛⣽⣿⣽⣿⠿⠿⣿⣿⣿⣯⣭⣙⣳⣾⢿⣿⣿⣿⣿⡽⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠈⢛⣽⠟⣫⣵⠞⠛⠋⠉⢹⠛⠛⢿⣟⠿⣷⣽⣿⣯⣟⡻⢃⠃⠀⠀⣀⣠⣤⠶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠂⢠⡞⣡⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⢻⣿⡝⣿⣿⡅⠀⠛⠉⠉
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢠⡟⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⣷⡈⣿⣿⡀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⣼⢡⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡇⢸⡇⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⢸⠀⡇⠁⠀⠀⠀⠀⠸⡋⡀⠀⠀⠀⠀⠀⠀⡀⣿⢸⣿⢸⡇⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠖⠛⠉⠉⠀⠀⢻⠸⡆⣇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡿⢸⡏⢸⡇⡇⠀⠀⠉⠙⠛⠲⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣇⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣾⢃⣾⡟⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠘⣦⠳⣜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣼⢃⣾⠟⠀⠀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣠⠄⠈⠳⣿⣷⣭⣒⠤⠀⠰⠀⠀⢀⣠⡴⢟⣥⣾⣷⠟⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣈⣿⢿⣯⣭⣽⣿⣿⣿⣿⣿⣿⢿⣟⠉⠀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⠀⡉⠉⠉⠉⠉⠉⠀⠀⠹⣻⡄⠀⠙⣆⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣰⠁⠀⠀⠀⣧⠀⠀⠀⠀⠈⠋⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠟⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⢸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢿⠿⢿⠿⠿⢿⡆⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠺⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠶⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣠⠴⠶⠚⠛⠳⠶⠶⣦⣄⡉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡿⠲⠶⣦⡄⠀⡞⠉⡇⠀⣠⡶⠞⠛⠛⠷⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⠴⠞⠁⠀⠀⠀⢹⡆⣇⠀⡇⣼⠧⣤⣤⣀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠛⠒⠳⠿⣤⣀⠀⠉⠛⠶⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣤⠾⣋⣤⠴⠶⠚⠛⠛⠛⠛⠻⢦⣄⠀⠈⢳⣄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡗⠚⠛⠋⣩⡶⢛⣩⠴⠖⠛⠛⠛⠛⠛⠶⣦⣄⠙⢷⣦⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⠟⢁⡴⠋⠁⠀⠀⢰⣉⡷⠀⠀⠀⠀⠉⠻⣦⡙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣾⠋⣰⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⠃⡼⠁⠀⢠⣞⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⣸⠃⠀⠀⠀⠈⠓⠦⣄⣙⣄⣀⠀⠀⠀⠀⠀⣀⣀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⢠⡴⡄⠀⠀⠀⠀⠀⠻⣄⡼⣄⠀⠀⠀⠀⠳⠞⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢿⠀⢿⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢳⡈⢷⡀⠀⠀⠀⠀⢰⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠘⣇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⡆⠀⠀⣠⠏⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠘⢆⠙⢦⣀⠀⠀⠀⠀⠀⠀⡤⣤⠀⠀⠁⠀⢀⡼⠋⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⠀⠙⠳⠶⣤⣤⣀⣀⣉⣉⣠⣤⣴⠞⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⢻⡄⠀⠀⠀⠀⠉⠉⠉⠁⣀⣠⣤⡶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠋⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠳⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⣴⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠛⠛⠛⠛⠋⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⡻⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⢟⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣅⣨⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄

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


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


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠚⠉⠉⠉⠉⠀⠀⠉⠉⠉⠙⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣞⡷⠖⠚⠛⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠳⠶⢤⡞⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢈⡦⠀⠀⡊⠉⡆⠀⠠⡊⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣀⠔⠊⢁⣀⣠⠠⠬⠯⠀⣀⣀⡈⠑⠢⣀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠤⠒⢫⠷⠒⠉⠁⣀⠀⣤⣤⢄⣤⢀⣀⠈⠉⠒⠴⣹⠂⢄⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⡠⠚⠁⣀⣔⠶⠙⠋⠉⠉⠁⣄⠉⠉⠘⠻⢷⣠⣀⠈⠓⢄⠈⠓⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡠⡪⠋⠁⢰⡆⠀⠀⠠⡆⠙⡦⠀⡄⠀⣤⠈⠙⢕⢄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⢮⠊⠀⠀⠀⢸⡇⢹⠀⠀⠃⠲⠆⡰⠀⢻⠀⠀⠀⠀⠑⣷⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⣯⠃⠰⢆⠀⡠⡈⠁⠈⠀⠀⠀⠀⢠⠁⠀⠉⠀⠀⠰⡄⣤⠘⣵⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣾⠃⠀⠀⢸⠸⣇⠇⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠮⠄⠀⠀⠘⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡏⠀⡀⢀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣜⣀⣀⣀⣀⣀⣀⡀⠠⡄⠀⠀⢿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣇⠀⠋⠈⠞⠀⠀⠀⠀⣀⣤⣶⠿⠿⠛⠛⠛⠛⠛⠛⠋⠈⠇⠛⠀⣸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⢠⠠⠀⢰⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠱⣧⠀⣠⠸⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢀⡀⠀⣼⠏⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠹⣧⡀⠀⠀⠀⠀⣩⠁⠀⠀⢀⠀⠀⠀⢰⡂⠀⠀⠀⠀⢁⣼⠏⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠘⢝⢄⠀⠀⣤⠁⠀⠀⠀⡟⡦⠀⠀⠀⢋⣄⠀⠀⡠⡮⠃⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠙⠯⣦⣄⡀⠀⠀⠀⠰⠆⠀⠀⠀⢀⣡⣴⠟⠊⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢢⢄⡀⠁⠚⠳⠶⠦⠶⠦⠴⠶⠚⠓⠉⠀⣠⡔⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢁⠔⠁⠀⠉⠑⠒⠤⠤⢀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠑⢄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⢠⣶⣿⠿⢟⣫⣽⠄⠀⠀⠀⠀⠰⣄⠀⣿⡆⢀⡴⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁
⠀⠀⠀⠀⠹⣧⣶⣿⠿⠛⠋⠀⠀⠀⠀⠀⣀⠹⣧⣿⣧⡟⢁⡀⠀⠀⠀⠀⠀⢸⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣺⢿⠟⠁
⠀⠀⠀⢀⣴⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣿⠿⢿⣟⡉⠀⠀⠀⠀⠀⠀⢸⣧⣤⣿⢰⣉⣰⠀⠀⠀⠀⠀⢀⣨⠿⡕⠁
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠘⠛⠋⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣴⣦⣼⠃⠀⠀⠀⠀⠀⠀⢀⣔⠐⠷⠖⣲⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⣀⠀⢶⣶⠤⠤⠤⠤⠄
⠀⠀⠀⢀⡴⠻⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⣹⣿⣹⠀⠀⠀⠀⠀⠀⠀⣿⡇⣤⣄
⠀⠀⠰⣿⣦⡀⠈⡻⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣑⠀⢠⣤⣴⠀⠀⠀⠀⠀⠀⠰⢧⣉⣉⣡⠿⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⡆
⠀⠀⠀⠈⠻⡿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠿⠿⠿⠇⠀⠀⠀⠀⠀⠐⡶⠛⠉⠓⡶⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃


⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠖⠶⢲⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣷⣖⠉⣉⣆⠀⠀⠀⠀⠀⠀⠀⢀⡷⠓⠛⢾⡁⠀⠀⠀⠀⠀⠀⢫
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣆⠨⢻⢯⡀⠉⣂⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⠸⡇⠀⠀⠀⠀⠀⠠⣶⣾⡇⠀⢀⡀⣀⣤⡤
⠀⠀⠀⣛⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⢶⡇⠄⠋⠀⠀⠀⠀⠀⠀⢸⡇⣭⠀⠶⠂⠀⠀⠀⠀⠀⠀⣨⣿⣷⣾⠿⠟⠉
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠸⠒⠒⠒⠓⠓⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢀⣀⡀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⡠⠊⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠾⡁⠀⠀⢀⣀⣤⡤⠆
⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢇⡀⢀⡼⢫⠂⡰⡆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣴⣶⡿⠟⠋⠁
⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠛⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢠⡞⢳⡄⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⠀⢿⣴⣇⠀⠀⢹⡏⠀⠀⣸⣦⡿⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡶⠃⠀⠀⠀⠈⣷⠀⢸⡇⠀⣾⠁⠀⠀⠀⠘⢧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣉⡿⠶⠶⣄⠈⠳⣄⠀⢀⣀⣿⠤⠼⠧⠤⣿⣀⠀⠀⢀⡴⠋⢀⡤⠤⢴⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠳⣄⣨⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣏⢀⡴⠋⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣾⢛⣷⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⣾⡛⣷
⠀⠀⠀⠀⠀⠈⠉⠙⢦⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⢸⡏⠀⠀⢹⡇⠀⠀⠀⠀⠈⣧⣤⣤⣤⣤⡴⠋⠉⠁
⠀⠀⠀⠀⣠⠶⣄⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⡴⠞⠓⠒⠒⠚⠳⢦⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣠⠶⣄
⠀⠀⠀⠀⠙⠶⠋⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⡇⠀⠀⢺⡗⠀⠀⢸⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠙⠶⠋
⠀⠀⠀⠀⠀⢀⣀⣠⠞⠛⠛⠛⠛⢻⡀⠀⠀⠀⣇⡀⠀⠘⠃⠀⢀⣸⠀⠀⠀⢀⡟⠛⠛⠛⠛⠳⣄⣀⡀
⠀⠀⠀⠀⠀⢿⣬⡿⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⢿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠞⠁⣹⠷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠙⢦⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣯⣸⠗⠒⠚⠁⣠⠞⠁⠀⠀⠉⣿⠒⢲⡖⠒⣿⠉⠀⠀⠙⢦⡀⠙⠶⠶⣾⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⡄⠀⠀⠀⢀⡿⠀⢸⡇⠀⢿⡀⠀⠀⠀⢠⠾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣾⠻⡏⠀⠀⣸⣇⠀⠀⢹⠟⣷⠀⠈⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠘⢧⡼⠃⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⠛⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢸⡃⠈⡷⠴⡏⠀⢹⠀⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⠀⠙⠛⠁⠀⠉⠛⠉⢀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⡀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⡟⠒⠶⠶⠶⠚⡏⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣻⠇⠀⢻⠉⣻⠿⡏⢹⠃⠀⢿⡳⠤⣄⣀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢴⡃⠀⠀⠈⣇⢳⣶⢋⡏⠀⠀⠀⣹⠄⣰⠋⠳⡄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦⣀⠀⢹⡞⠘⣿⠀⠀⣠⠞⠁⢠⡇⣾⠁⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢷⢰⢇⡴⠋⠁⠀⠀⠘⡇⠉⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠇⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣤⠴⠞⠙⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣒⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡫⠭⠭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠉⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢁⣴⣶⣶⣄⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⣀⣀⡀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣋⣹⣟⣁⣀⣀⣀⣀⣻⣟⣘⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠵⠧⠤⠾⠿⠿⠿⠿⠦⠤⠷⠝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡿⠿⢿⣿⣿⣿⣗⣺⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠀⠀⠀⢀⣠⣀⠀⠀⠀⠬⠅⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡅⠀⢀⣿⣿⣿⣗⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠠⠭⠤⠄⠀⠸⣿⣿⠂⠀⠠⠤⠥⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⡿⠀⢹⣿⣿⣿⣿⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⠩⠭⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠁⠀⠀⠈⠛⠋⠀⠀⠀⠨⠍⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣶⢰⡆⢰⡆⣶⠀⣶⢰⣶⢰⡆⢰⡆⣶⡆⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⢸⡇⣿⠀⣿⢸⣿⢸⡇⢸⡇⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣴⣦⣴⣦⣶⣴⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣴⡘⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣍⣍⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣿⣿⣿⢰⣿⣧⢹⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢲⣬⣙⠿⢸⣿⣿⠸⢟⣩⡶⢢⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡹⣿⣷⣼⣿⣿⣴⣿⡟⣡⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠩⣥⣤⣹⣿⣿⣿⣿⣡⣬⡍⢙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⡦⠈⢛⡁⣙⠉⠠⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⢛⣛⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣶⣿⠿⠟⣛⣛⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⣛⣛⣛⡛⢛⡛⠛⣛⠛⡁⢾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣦⡙⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠸⠿⠿⠿⠸⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣦⣝⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣶⣶⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⣿⡿⡿⣿⣷⡘⣿⣿⣿⣿⣿⣷⣝⢻⠏⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠿⠿⠷⠮⠛⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⢀⣀⣀⣿⣿⣸⣿⣇⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⣀⣀⢀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠰⡄⠇⡰⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠁⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠊⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠠⡀⡆⢠⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠀⠘⡶⠁⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠇⡠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡀⡇⢠⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢰⠂⠐⠂⠁⠬⠐⠀⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠂⠀⠱⣶⠃⠈⠂⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠂⠀⠀⡨⠠⠞⠀⠆⠇⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠦⠀⠅⠀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠈⡌⡔⠀⡤⠤⠤⢤⣀⠀⠀⠈⠀⢳⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠭⠭⣙⣒⠀⠸⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⠭⢝⣒⣂⠀⠰⠿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⣩⣍⣙⣓⣒⠦⠥⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣒⣒⣒⠭⠭⣓⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣒⣒⣒⣒⣒⡦⠭⢭⣒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠉⠙⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣉⣀⣉⣉⣉⣉⣓⣒⣤⣼⣤⣤⣐⣀⣀⣀⣀⣀⣀⣀⣈⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⠻⢷⣦⣀⣀⣤⣤⣤⣤⣀⣀⣠⣾⠟⠋⠉⠉⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⢀⣤⣶⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢿⣶⣤⡀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢹⣷⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠆⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠘⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣴⣶⣶⣦⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⣹⣷⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣦⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠙⠻⢿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⡿⠟⠋⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢄⠠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⣠⣄⡒⠤⣖⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠋⠩⠇
⠀⠀⠀⠀⠀⢔⡚⠶⣍⠏⠒⠊⠁⠀⠈⠉⠓⠺⠭⣛⡲⠤⣄⡀⠀⡀⠀⠀⠀⠀⠔
⠀⠀⢠⠄⣂⡥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠳⠏⠓⠘⠁⠉⣡⢦⡤⣀⣠⢀⣴⡃
⠀⢻⠶⣤⠭⣒⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡲⢧⢰⠆⠀⠀⠀⠈⡃⠘⣿⣿⡿⣟⡝⠃⠠⡀
⠀⠈⠑⢄⡁⠪⣙⠷⣂⠍⠐⠠⠀⡀⠀⠀⢠⠴⡘⣆⣹⠶⡄⠀⡠⠀⠀⠔⠁⠀⠸⠛⠋⢩⠙⢄⢀⢈
⠀⠀⠀⠀⠈⠒⢄⠁⠨⡙⠦⣌⣑⣄⠱⡇⢄⣶⠟⠋⠁⠀⡴⠊⠀⠰⠧⢴⡄⠛⠟⢑⠀⢀⡧⢘⠿⠟⣻⣗⣂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⠁⠪⣗⣿⣧⡿⠊⠁⠀⠀⠠⠊⢠⣾⣿⣶⠀⠀⢰⡆⠨⠄⠀⠈⢀⡠⠐⠈⠉⠁⢋⠔⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡈⠙⠫⢛⠦⣴⣖⣸⡗⠐⠞⠓⣻⡿⢇⠀⢸⣷⡴⠂⡀⠐⢁⠔⠀⢀⣴⠟⠃⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⣉⠎⠛⢠⣿⡶⠒⣏⣁⡸⠚⣶⡿⠏⠤⠊⣠⠖⠀⣠⣴⡿⠁⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣐⠒⠠⠊⠁⣲⠀⠀⣤⣾⣿⡴⠀⡤⠔⢠⠤⠴⠶⡽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡉⠈⠙⠪⣉⠻⢿⣟⣒⣲⠶⠓⣰⠶⠲⣲⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⠑⢠⡈⠛⢁⣴⣋⢡⡠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠈⢳⠉⡊⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣾⡴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚
⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠠⡐⠠⠄⠀⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠊⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠠⠀⠈⠒
⠀⠀⠀⠀⢊⣀⣀⣙⣀⠂⡖⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⣀⣀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡠⠪⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢗⠄⠀⠀⠐
⠀⠀⠐⠴⣠⠤⠤⣀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠂⠤⠐⢳⠁⠙⠚⠂⠤⠊⠀⠀⠀⠀⠀⠀⢎⡀⠁⢀⠀⡀⠋⠀⠀⠀⠀⠀⠀⢀⡴⢅⠀⠀⡠⠁⣄⠄
⠀⠀⠀⠀⠀⠌⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⣈⠖⠁⠙⡀⢀⡀⠀⠀⠀⠀⠀⠈⠲⣄⠑⠊⣰⠞⠁
⠀⠀⠀⠀⠐⠆⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠬⠀⠠⠤⠌⠥⠴⠀⠀⠀⠀⠀⠀⠀⠀⠱⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⡀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢆⠃⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠢⠁⠉⠁⠢⡀
⠀⠉⠀⠂⣴⠁⠀⢡⡄⠂⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⢄⠀⠀⠀⢄⠈
⠀⠀⠑⣂⡀⠁⠀⠘⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢈⡂⢤⠄⢠⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠃⡀⠀⠀⠃
⠒⠥⠀⠈⡄⢠⡄⢠⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢴⠀⡀⠀⠠⢡⠘⠁⠀⠀⠀⠀⠀⠀⢋⠀⠀⢨⠀⡠⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⠡⠀⠀⠐
⠀⠀⠀⠰⠔⠉⠈⠂⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡁⢒⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⠀⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢄
⠀⠀⠀⢰⠒⠒⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠌⢳⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠳⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡆⠶⡀⠑⢄
⠀⠀⠀⢸⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠀⢏⢳⢄⠃⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣐⡑⢆⠁⢀⡘⠷⠀⠀⠀⠀⠀⠀⠀⠐⣁⢖⡰⠀⠄⡄⢰
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠈⣉⠦⡑⠤⠇⡠⠊⠀⠀⠀⠀⠀⠀⢠⠳⠁⠀⢀⡧⠊⠀⠀⠀⠀⠀⠀⠀⠀⢆⠘⠀⠣⢀⠰⣀⠆⡠⠊
⠀⠀⠀⢠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠢⠜⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⣄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠊

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢀
⠀⠀⠀⠀⠀⠀⠠⡈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠘⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⢀⠔⢂⡀⠜⠀⠀⠀⠀⠀⠀⠀⠠⡢⠁⠨⠍⢁⠢⡀
⠀⠀⠊⠀⠀⠐⠨⠛⠘⠨⠙⠂⠀⠀⠀⠀⠀⠀⠀⠄⡠⠒⢄⠀⡈⠶⠀⠀⠀⠈⠒⡂⠈⢒⣒⣀⡔⠁⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄⠀⠄⠀⡇
⠀⠀⢈⠠⣀⣀⠠⡊⠀⠀⢰⠁⠀⠀⠀⠀⠀⠠⡀⢚⣠⡹⠊⠂⠀⠀⠀⠀⠀⠀⠁⢀⡄⢁⡔⠋⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠆⡶⠂⠀⡇
⠀⠀⠀⠈⠐⢒⠈⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠃⠃⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁⠇⠄
⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣈⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡽⠒⠚⠉⠉⠉⠉⠉⠒⢿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣞⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡵⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡝⠶⠛⠻⠿⣷⣶⣄⠀⣤⣤⣀⡠⣴⡬⠾⠟⠚⠳⠾⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⠀⣀⣤⡶⠾⢿⣿⠊⠛⣿⠿⢿⠿⠷⣦⣤⣀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣾⣿⣿⣿⣏⢻⡀⣨⣟⣁⡀⣴⣿⠄⣈⣳⣄⣼⡿⢛⣿⣶⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⠛⡿⣿⣿⣦⣿⠿⠖⢀⣀⣁⣀⣀⡀⠐⠿⣿⣥⣿⣿⠋⠛⢻⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⡹⣫⣿⣋⡴⠞⠋⠉⠉⠀⠄⠉⠉⠙⠲⢦⣝⢻⡙⢯⠒⠺⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣩⡾⠋⠀⠀⠀⠀⠀⣰⠺⠀⠀⢠⡞⠂⠈⠳⣄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣴⠋⠀⠀⠀⠀⠆⡇⠀⠈⠉⠀⢠⠽⠇⠀⠀⠀⡈⢧⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠃⠀⠀⣾⣷⣀⡀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢸⡋⠑⠈⣧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡟⠀⡀⣀⠉⠁⠙⠦⠦⣀⣀⢠⡏⠀⠀⠀⠀⠀⢀⡀⣆⢹⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⢰⠂⡿⠀⠀⠀⠀⠀⠈⣷⣾⠀⠀⠀⠀⠀⠀⢈⡇⠛⢸⣷⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⢿⣿⠞⠀⠀⣀⠀⠀⠀⠀⠠⠋⠈⠣⡀⠀⠀⠀⢀⡄⠀⠂⣼⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠙⢦⠠⠄⠿⠀⠀⠀⡰⠁⠀⠀⠀⠑⣄⡀⠠⠺⣧⣄⡖⠉⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠠⠃⠐⢆⠀⠈⡅⠀⠀⢠⡆⠀⠀⢨⣦⣀⣶⣿⠋⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣴⣦⡀⠼⣦⣤⣀⣀⣘⡋⢀⣠⣢⣾⡿⢟⠕⡊⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣮⣭⣓⣻⣿⣿⣛⣛⣩⣵⣾⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣹⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⢩⣭⡍⣟⣿⣿
⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⡿⣿⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⠳⡏⢿⠛⡿⢹⠞⣿
⠀⠀⠀⠀⠀⠀⢸⢸⣫⡍⣏⣿⣠⢀⣤⢀⣤⢘⡁⠀⠀⠀⠀⣿⣏⢷⢻⣿⡟⡾⣹⣿
⠀⠀⠀⠀⠀⠀⢸⣈⣒⣒⣉⣿⠀⠛⠁⠛⠁⢻⡁⠀⠀⠀⠀⣿⣿⣷⣶⣤⣴⡾⣫⣽⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣾⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⠱⠶⠶⢾⣷⣿⣿⡿⣫⣷
⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠆⠤⠤⠤⠤⠼⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⡻⢿⣾⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⣴⣴⣦⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⣶⡄⣤⠏⠹
⠀⠀⠀⠀⠀⠀⣴⣿⢧⣿⣿⣿⣯⠿⣭⣥⣶⠄⠀⠀⠀⠀⠀⠀⠀⠓⢸⣿⣾⠇⠀⢀⡾⢧
⠀⠀⠀⠀⠀⣸⣿⡿⣾⣿⣿⣿⣹⠿⣿⣷⣶⡆⠀⠀⠀⠓⣤⣤⡔⠀⢾⣽⠏⠒⠂⢸⣮⢻
⠀⠀⠀⠀⠀⣭⣭⣅⣭⣭⣽⣯⣭⣍⣭⣭⣭⣭⠀⠀⠀⠀⠳⠿⣵⣶⣶⣅⢠⠄⠀⠘⠓⠛
⠀⠀⠀⠀⠀⠹⣿⣿⢻⣿⣿⣿⣿⣯⡻⣫⣿⣿⠀⠀⠀⠀⠲⢸⣿⣾⣿⣻⣆⡤⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠙⢿⣎⢿⣿⣿⣿⣿⡏⣿⣿⣿⠀⠀⠀⠀⠒⠊⣿⣽⣿⣿⠀⢀⣾⣇⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠛⠛⠛⠃⠛⠛⠛⠀⠀⠀⠀⠀⠀⠙⠿⠾⠃⠀⠻⠿⠿⠿⠟⠂

⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣰⣧⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⣿⣿⣷⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣼⣾⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣿⣛⣿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠘⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⡟⣫⢷⣿⡿⡽⠿⢎⢿⣿⡾⣝⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡿⣿⣯⣟⣵⣿⡿⡿⣿⡟⣻⣯⠀⠀⠀⠀⠀⠀⠀⠘⢀⣠⣽⣿⣿⣟⣻⢯⣿⣿⡿⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⢿⣿⢳⠛⠋⢀⡿⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⢯⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠈⠃⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⢀⣠⣤⢧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠎⢰⠈⢢⣼⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⢶⡶⣶⢶⢿⣿⣿⣿⣿⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⣿⣿⡿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⡟⡗⡿⡺⣿⣿⣗⣻⣿⣿⣷⣿⣷⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣽⣿⡖⠀⠻⣿⣽⠿⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠐⢿⣿⣿⣿⣿⣿⣿⡏⠏
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⣀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠉⠛⠿⡽⠟⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣿⣿⣶⣶⡶⠂⠀⠀⠁⠐⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⡖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠁⠀⠀⠀⠈⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠖⠊⠉⠙⠒⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⠿⣧⣀⡄⠀⠀⠀⠈⠁⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠯⠿⠧⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⡀⣠⣾⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡄⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠰⠶⠐⠃⠁⠀⡠⢤⣠⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡀⣀⠴⠛⠶⣤⡀⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⡀⠈⢹⠘⠋⡏⣉⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣯⣿⠧⠴⠭⠭⠤⠭⠭⠯⠬⠽⠿⠭⠅⠀⢸⠀⠀⡡⠯⠭⠭⠭⠭⠭⠭⢭⣯⣭⣟⣛⣿⠛⣶
⠀⠀⠀⠀⠀⠀⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣏⢸
⠀⠀⠀⠀⠀⠀⢿⣏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡸⠀⣀⣃⡀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⣿⢹
⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡷⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣻⡆⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⡇⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣽⡇⠀⠀⠸⠀⠀⠀⡀⠀⢀⡉⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡠⠠⠄⠀⠠⠄⠀⠀⢸⢣⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠿⡇⠀⢰⠀⢸⠀⠀⡇⠀⠀⠐⠀⢸⠀⠀⢀⠀⠀⢁⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢃⡇⠀⢸⠀⢸⠀⠀⣇⣀⣀⣀⣀⣘⡀⠀⢸⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⡸⠀⣿⣻
⠀⠀⠀⠀⠀⠀⠀⣿⣣⣷⠀⠈⠀⠁⠀⠶⠿⠿⠿⠿⠿⠿⠵⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠏⠐⠀⠀⢀⣧⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡡⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡃⣸⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣗⣶⣭⣷⡒⣒⣶⡶⢦⡶⠶⠶⠶⠶⢶⠶⠒⡒⠒⣶⣶⢶⡶⠒⠲⣶⢶⣶⡶⢒⠟⢹⡇
⠀⠀⢸⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠚⠛⠚⠛⠛⠒⠛⠒⠛⠒⠒⠒⠒⠒⢶⠄
⠀⠀⠀⠉⠉⠓⠓⠓⠚⠛⠓⠒⠛⠒⠚⠛⠓⠚⠚⠓⠛⠚⠚⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠦⠴⠤⠤⠞

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣠⣀⣤⣤⣤⣀⣠⣀⠀⢀⣀⣤⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣾⠏⠙⠁⠙⣉⠛⠋⠙⠿⠛⠋⠉⣿⣄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠁⠀⠾⠂⠀⠛⠀⢠⡆⠀⠀⠺⠆⠈⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⣿⠀⠶⠀⠀⠀⠛⠀⠀⠀⠰⠖⠀⠀⠀⣿⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠛⠂⠀⠶⠀⠀⠀⠀⠶⠀⠀⣿⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⢠⡆⠀⣀⡀⠀⠀⣀⠀⠀⠀⣀⠀⣿⠀⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣶⡾⢿⣿⠀⣿⣀⣀⣀⣀⣈⣁⣀⣀⣉⣀⣀⣀⣉⣀⣿⠀⣿⡟⠹⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠋⣀⠘⠁⢸⣿⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣿⡇⢠⡄⠸⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠋⠀⣠⡈⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⢁⣄⠀⠀⠈⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠰⠆⠉⠀⢀⣀⡀⠘⠃⠀⣀⡙⢃⣀⣀⣀⢀⣈⠛⠀⣤⠀⠛⠀⠀⠉⠀⠛⢀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⣤⣄⣀⣤⡿⠛⣿⠛⢻⡿⠛⠻⠛⠙⠋⠙⠛⠙⣿⣄⣀⣠⡶⢶⣆⣀⢀⣀⡈⠉
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠉⠙⠏⠀⢠⡄⢀⣀⠀⠀⠀⠀⠶⠀⠾⠀⠀⠈⠙⠋⠹⠇⠀⠉⠹⠿⠉⠻⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠾⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠟⠛⠛⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣀⣠⠤⣴⣲⣽⠇⠀⠀⢀⡀⠀⠀⠀⠠⣤⠖⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⣠⠤⣒⣺⣭⣷⣚⣻⣭⣥⣤⣤⣤⣤⣟⣙⢻⡆⠀⠀⢠⣴⡶⠶⢦⣄⡉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠘⣯⠵⠞⢋⣩⡼⠟⠋⠉⠀⠀⣀⣀⣀⣀⠀⠈⠉⠙⠳⢦⣄⠈⢷⡱⡄⠀⠈⠛⣦⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⣠⠞⠋⠁⢀⡠⠴⣒⣯⣭⠭⠶⠦⣬⣭⣙⡒⠦⣀⠀⠈⠳⣦⡻⣜⣆⠀⠀⠈⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠟⠁⢀⡠⣚⡥⠖⠋⠉⠀⠀⢠⡆⠀⠀⡤⢨⣝⠳⣬⡳⣄⠀⠈⢳⣿⣮⢦⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⡴⣫⠞⠉⠀⠀⠻⠂⠀⠀⠀⠀⠀⢈⡁⠶⡟⠀⠈⠻⣎⢣⡀⠀⠹⣏⢷⡳⡀⠀⣸⠀⡖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⢏⡼⠃⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⠀⠀⠀⠀⣀⡘⢧⠹⡄⠀⢹⡎⢳⣝⣤⠏⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⢣⡞⠁⠲⠞⠋⠀⠀⠀⠀⠀⠀⠀⢰⣏⡾⠁⠀⠀⠀⠀⠉⠁⠈⣧⢹⡀⠀⣿⠀⠙⠁
⠀⠀⠀⠀⠀⠀⣸⠃⠀⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠠⡇⡇⢀⣠⠀⠀⠀⠀⢠⡾⠋⠛⣲⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢈⡇⡹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⣧⠀⠁⠀⠀⠀⠀⠀⠛⠶⠛⠁⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⢇⡇⠀⣼⠁⡶
⠀⠀⠀⠀⠀⡀⠸⣇⠀⠸⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⠀⢠⡟⡼⠀⢠⡟⠀⠃
⠀⠀⠀⠀⠀⠹⠀⢻⡄⠀⢳⣻⡄⠀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⣰⡿⢣⡟⡼⠁⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠳⣽⣦⡀⠀⠀⣤⣶⠦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢉⣵⢫⠞⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡻⣄⠀⠈⠻⣿⢦⣀⠻⠽⠂⠀⠰⠇⠀⠀⠘⠃⣀⣠⠶⣋⠕⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠐⣯⣀⣽⠟⠻⣦⡀⠈⠙⠯⣟⣓⠶⠶⠶⠦⠤⠶⠖⣛⡭⠔⠋⠁⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠷⣤⣀⡀⠈⠉⠉⠉⠛⠛⠋⠉⠀⣀⣠⡴⠞⠉⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠶⠶⠶⢺⣿⠭⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠉⠉⠉⠀⠀⠀⠈⠉⠉⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⢀⣠⣤⠶⠚⠛⠋⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⣠⡤⣤⣄⠀⠀⠀⠀⠀⠉⠛⢦⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢰⣟⠁⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣠⣄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣰⠏⠈⠳⣄⠀⠙⢷⣄⠀⠀⠀⣠⣬⣷⠀⠰⣯⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣰⠏⠀⠀⠀⠈⠳⣄⠀⠙⢷⣤⡞⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠙⢦⡀⠀⠀⠀⠀⣶⡀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢙⣷⣄⠀⠙⢦⡀⠀⠀⣿⣧⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣼⠁⡟⢷⣄⠀⠹⣦⡀⣿⠸⣆⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠙⢷⡀⠈⠻⣿⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⠛⠛⠳⠶⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣙⢦⡀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⣠⠞⠉⠀⣀⣤⣤⣤⣤⣄⡀⠈⠙⢦⡀⠀⠀⠀⣧⠀⢸⡏⢹⠀⠙⣦⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢠⡟
⠀⠀⠀⠀⣼⠋⠀⣠⠞⠉⠀⢠⣆⠀⠈⠙⢷⣄⠀⢻⡄⠀⠀⣿⠀⢸⡇⢸⠀⠀⡏⠙⢦⡀⠈⠳⣄⠀⠀⠀⠀⣼⠃⠀⠀⢀⡾⠁
⠀⠀⠀⢰⡇⠀⣼⠋⠀⢀⡾⢻⡟⠛⣦⠀⠀⢹⡆⠀⢿⠀⠀⣿⠀⢸⡇⢸⠀⠀⡇⠀⠀⠙⣦⡀⠈⠳⣄⣠⡾⠁⠀⠀⢀⡾⠁
⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠛⢾⣧⣄⠀⠀⠀⠀⣿⠀⢸⡆⠀⣿⠀⢸⡇⢸⠀⠀⡇⠀⠀⠀⠈⠻⣦⡀⠈⠋⠀⠀⠀⣠⡞⠁
⠀⠀⠀⢸⡆⠀⢿⡀⠀⠀⣆⠈⡇⢉⣷⠀⠀⢰⡏⠀⣼⠁⠀⠙⠶⠛⠀⠘⠳⠞⠁⠀⠀⠀⣀⣤⠞⠃⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠈⢷⡀⠘⢷⣄⠀⠈⠻⡿⠛⠁⠀⣠⠟⠀⣰⠏⠀⢠⣤⣀⣀⣀⣀⣀⣠⣤⡴⠶⠛⠉⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠈⢳⣄⠀⠙⠳⠶⣤⣤⡴⠶⠛⠁⢀⣴⠏⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⢀⣀⣤⠴⠛⠁⠰⢤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠠⠀⠀⣀⠔⢊⠩⠔⠒⠃⠈⠘⠒⢄⡈⣧⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠈⠀⠰⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⡆⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⣠⣡⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣷⡀⠀⠤⢠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠉⠁⠀⠀⠀⠈⠢⡀⠀⠀⢀⠤⢤⠞⠉⠁⠀⠀⠉⠲⢤⡌⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⣿⡇⠀⣀⣠⣤⢶⣶⣾⣿⡷⠀⠀⠀⠑⢲⡗⢤⣀⠀⠀⠀⠀⠀⠈⢑⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣆⢹⡘⣿⡽⠟⢶⢲⠲⠶⢶⣿⣶⣤⣬⣱⣦⣄⡀⠀⠀⣔⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⡿⢿⣛⡭⠖⠛⠋⠉⢉⡍⠛⠒⠯⣽⢿⣷⣿⣿⠛⠻⢿⡒⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠊⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣟⢻⣅⠀⠀⠀⠙⢦⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡴⠣⡄⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠐⠃⠈⢣⡙⣆⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⡧⠄⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢑⢻⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠅⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢸⢠⡄⠀⠀⠀⢀⣠⣴⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡇⣸⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣼⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢠⡇⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢷⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡖⣠⡎⡀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠴⢿⣷⣋⡄⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢳⣌⣒⣤⠼⣀⣀⣀⠠⠤⢤⠐⣐⣩⣶⠞⣋⣉⠁⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣤⣮⠟⠉⠛⠶⢿⣶⣾⣾⣿⣿⡻⠟⠛⠒⣰⠁⠿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⡄⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⣻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⢄⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢴⡁⠀⠀⠀⠈⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠤⠄⠠⠄⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠁⡠⠐⠢⡀⠙⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⢠⡀⣀⠄⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠢⢀⣈⣁⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠐⠠⣤⠀⠀⠤⠔⠚⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠳⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⠤⠄⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣭⠆⠀⠈⢩⠋⠝⣋⡛⢍⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢁⢠⠸⡘⢅⠸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢀⠔⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⣤⡄⠊⠀⡈⢆⠢⢍⣒⣒⡡⠊⠳⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⢀⠀⠘⡌⢆⠀⠀⠉⠢⡀⠀⠈⠢⡀⠀⢀⠊⠀⠀⠀⢠⠊⠉⠑⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡼⠠⠁⠀⠀⠀⠑⠃⠀⠘⡌⢆⠀⠀⠀⠈⠦⡀⠀⠈⢦⡌⠀⠀⠀⠀⠐⢄⣀⡠⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⢡⠃⠀⢠⡆⠀⠀⠀⠀⠀⠘⡜⡆⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣦⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⣀⡀⠀⢤⣄⣀⡠⠤⠿⠐⠒⠺⠄
⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⠱⡘⣄⠀⣀⣀⡀⠤⢤⣗⠒⠦⠭⠙⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢰⢃⠂⠀⢠⢢⠀⠀⠀⠀⠈⠛⠁⠀⡠⠠⡀⠱⡘⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢠⢃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠡⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣣⣤⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡀
⠀⠀⠀⣠⠈⣳⢖⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣾⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢫⣿⡂
⠀⠀⢠⢿⡇⠘⢦⡈⠝⣦⠄⠀⠀⠀⠀⠀⠘⢿⣦⣿⣧⣾⠟⠀⠀⠀⠀⠀⠠⣾⡿⠣⣼⠿⠋
⠀⠀⠈⠛⠁⠀⠀⠙⠞⢿⡶⠀⠀⠀⠀⠀⠒⢾⣿⣿⣿⣿⠶⠂⠀⠀⠀⠀⠈⣋⠔⠉⠩⢉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠚⠓⠒⠒⠒⠒⠒⠚⠓

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢀⡔⣻⢭⡟⣗⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⡙⣿⡀⠀⠀⠀⠀⠀⡞⠀⠈⠉⠍⠙⠒⡆
⠀⠀⠀⠻⠭⠍⣭⣭⠩⠭⠟⠀⠀⠀⠀⠀⢠⡦⠙⠻⠦⠴⠟⠁⠀⠀⠀⠀⢰⣧⡄⠀⠀⠀⢀⡀⡇
⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠛⣀⠛⢸⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⡹⢿⡿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠘⠿⠿⠶⣤⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢀⡴⠞⠛⠛⠳⢦⡀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠛⠲⣤⡀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠶⣄
⠀⠀⢰⠏⠻⣦⣀⣾⣗⠀⢹⡄⠀⠀⠀⣰⠋⢳⣄⣰⡀⢀⠈⢳⠀⠀⠀⠀⡼⠉⢳⣤⠐⢆⠐⡌⢷
⠀⠀⢸⠀⠀⢰⣿⣿⡁⠀⠀⡇⠀⠀⠀⣿⠀⣘⣿⣿⣷⣏⡀⢸⡇⠀⠀⠀⡇⢰⣄⢸⠷⣾⠟⠁⢸
⠀⠀⠘⢧⡀⠜⠉⠀⠹⣦⡼⠃⠀⠀⠀⠸⣆⠈⠙⠛⠛⠻⣤⡾⠀⠀⠀⠀⠻⣄⠙⠻⠖⠉⢳⣤⠟
⠀⠀⠀⠀⠙⠲⠦⠴⠖⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠦⠶⠞⠋⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠶⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⡻⠛⠇⠚⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣎⠂
⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⣀⠀⠀⢠⣠⡆⠴⠦⣀⠀⠤⠀⡇⡆
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢺⡄⠀⠘⢡⣀⠀⠇⠘⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣙⠀⠄⢾⠉⠳⡜⠚⣧⠀⠀⣻
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠉⠀⠀⠠⠈⠚⡗⢺⡣⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠺⠯⠭⢽⣷⣒⣒⣒⣖⠀⠀⠀⠬⡔⡒⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠉⠉⠉⣿⠅⠓⠂⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠠⣀⠤⢤⠀⢐⠳⠶⠶⢴⣤⠞⠋⠀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢤⣄⠲⣙⡟
⠀⠀⠀⢀⠜⠟⠀⠂⠌⢩⢉⢛⢃⠓⠍⠀⡀⠀⠀⠀⠁⢩⡁⠇⠀⠀⠡⡀⠀⠀⠄⠀⢀⣠⠽⠛⠋⣟⠮⢤⡄⢰⡆
⠀⠀⢠⣊⠦⠠⠀⠅⠃⠍⠅⠔⡸⠸⠭⠈⠀⠀⠐⡒⠒⠘⠧⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠼⠷⠂⠐
⠀⠀⠀⠙⠛⠧⠀⣀⠀⠀⠀⡀⠀⠈⠀⠁⠀⠀⠒⠎⢀⡇⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠭⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠐⠒⠴⠦⠤⣤⣤⡾⠃⠤⠤⠆⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠇⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡀⠀⠀⠠⠀⠣⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢄⠀⠉⠑⠂⠄⡁⠣⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢢⡀⠠⠐⠘⠀⠙⠃⠀⠀⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣅⣒⠒⢦⡔⠀⠄⠈⠈⠢⡀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠤⠤⠀⢀⣀⡀⠀⠀⠀⢠⣧⣈⠄⠀⡌⢀⠎⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌⡃⠠⡀⠀⣄⡨⠄⠀⠀⠀⠀⠘⡆⠀⠀⠀⡀
⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌⠀⠀⠀⠀⠢⠤⠐⡑⡙⡤⢄⠀⠀⠀⢀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠁⣀⡐⠌⠀⠀⠁⠀⠀⢀⠀⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⡤⣴⣈⡀⢀⣈⡙⣄⠀⠀⠀⠀⠀⠀⡌⠀⠘⡓⠈
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠸⡿⠗⢓⠉⡡⡴⠮⠉⠈⠑⠠⠤⠀⠀⠐⠀⠀⠀⠤⠀⢀
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠡⠰⠆⠙⠢⠄⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠘⢳⠊⠀⠀⠀⠀⣴⠛⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⡧⠀⠀⠀⠀⠀⢫⠂⠀⢨⠀⠀⠀⢸⠀⠀⢰⡆⠀⠈⡟⠁⠀⢱⠁⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⢸⠀⠀⢐⠀⠀⠀⣇⠀⢰⠁⠀⠀⠀⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⣁
⠀⠀⠀⠀⠀⠀⡴⠲⣄⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣤⣼⣿⣤⣤⣤⣼⣷⣤⡄⠀⠀⠀⠀⢸⣏⣉⣉⣛⣀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⡄⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡤⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠁⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠴⠋⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⢸⡗⠒⠒⠪⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⠔⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠶⢶⡶⠶⣶⠶⠶⢶⡶⢦⠶⢶⡶⠶⣶⠶⢶⡶⠦⠼⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠀⢸⠀⢸⠀⢸⡄⠀⣷⠀⠘⢄⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣹⠦⠤⠤⠴⠃⠀⠀⠿⠂⢰⠁⠀⢸⠀⠈⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠸⣅⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⢠⠞⢦⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⠾⠿⠿⠿⠿⠷⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⢉⣁⣤⣤⣶⣶⣶⣶⣤⣤⣈⡉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢁⣴⣾⠟⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⠿⣷⣦⡈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢀⣴⡿⠋⣠⡀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠙⢿⣦⡀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣠⣿⠋⠀⠀⢘⣿⣿⡛⠉⠉⠉⠉⠉⠉⠛⢿⣦⡀⠀⠀⠙⣿⣄⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢰⣿⠃⠀⠀⢀⣿⠏⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠘⣿⡆⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣿⠇⠀⠀⠀⢸⣿⣤⣤⣬⣿⣷⣤⣤⣤⣤⣤⣤⣤⣿⡇⠀⠀⠀⠸⣿⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠉⣩⣽⣭⣍⠙⣿⣿⣭⣍⠉⣉⡉⠉⠀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⢀⣀⣉⣁⣉⣛⣟⣛⣹⣿⣟⣻⣛⣁⣀⡀⠀⠀⠀⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⣿⡆⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⡛⠛⣿⡇⠀⠀⠀⢰⣿⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠸⣿⡄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣿⡇⠀⠀⢠⣿⠇⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠹⣿⣄⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡄⠀⣠⣿⠋⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡈⠛⢿⣦⣤⣀⣀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠛⢁⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣈⡉⠛⠛⠿⠿⠿⠿⠛⠛⢉⣁⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢶⣶⣶⣶⣶⡶⠿⠟⠛⠉

⠐⠒⠒⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢄
⠀⡀⠠⠌⠳⠀⡘⠀⠀⠀⠀⠀⢈⠠⠂⠁⠄⠀⢀⠀⠀⠀⠀⠀⠈⢃⠢⠅⠠⢂⠍⠀⠀⠀⠀⠀⠀⠀⠔⠃⠃⢔⠕⠀⠀⠀⠀⠀⠀⠀⠠⠄⢁⠄⠁
⠀⠀⠉⣀⠤⠀⡤⠀⠀⠀⠀⠀⠀⠀⡒⠐⠒⡂⠁⠀⠀⠀⠀⠀⠀⠆⠁⢀⠀⡁⠆⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠘⠐⠀⠐⢘⠀⣀
⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⡴⠈⠁⠀⠀⠀⠀⠀⠀⠀⢰⢐⠀⠂⠂⢐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣄⠀⠀⠀⠀⠀⠀⠠⠒⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉
⠀⡀⠀⢄⠀⠔⠈⠀⠀⠀⠀⠀⢐⠀⢀⠴⢀⠸⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⠆⠀⠀⠀⠀⠀⠀⠀⠩⠘⡁⠐⢄⡲⠀⠀⠀⠀⠀⠀⢠⠂⠁⠈⠢⡀
⠈⢄⠈⠂⠅⠰⠊⠀⠀⠀⠀⠀⢠⢀⠨⠙⠨⢸⠐⠀⠀⠀⠀⠀⠀⢠⡧⠔⠁⠀⠀⠀⠀⠀⠀⠀⠨⠜⠣⠀⡐⠀⠙⠁⠀⠀⠀⠀⠀⠁⢠⢌⡀⠈⡁
⠀⠒⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠂⠀⠒⠘⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠁⠊


⠀⠀⡔⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠂⢆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⢀⠀⠀⢅⠀⠀⠀⠀⠀⠈⠋⠘⢁⠄⠠⡀⠀⠀⠀⠀⠀⠀⢇⢀⠀⠄⠈⠈⡀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⢀⠇⠀⢀⠈⠀⢀
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣜⠀⠠⠃⠐⠐⢋⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠎⠀⠵⠠⠆⠀⠀⠀⠀⠀⠐⠀⡂⠄⡂⠕⠊
⠋⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠙⠊⠐⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐


⠀⠀⠂⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶
⠀⢰⠠⡀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⠠⠀⠀⠀⠀⠀⠀⢠⠎⢃⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠐⠌⢂⠀⠀⠀⠀⠀⠀⠀⠂⠐⢄
⠀⠤⢕⠀⠪⠠⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠨⠀⡇⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⡆⢒⠀⢒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢀⣠⣤⣤⡀
⢀⣴⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⠀⢠⣿⠋⠉⠙⣿⡆
⣾⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠘⣿⣦⣀⣠⣿⠃
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣶⣶⣶⣶⣾⣿⠁⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⠉⠉⠻⣷⣤⣶⡿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣤⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡏⠀⠈⢻⣷
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣷⣤⣤⣾⠏
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⢿⣷⣄⣀⣀
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⣿⡆
⠙⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣿⣇⠀⢀⣼⡟
⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣈⣻⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠻⢷⣶⣤⣴⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣤⣦⣠⣤⣤⣤⣬⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠑⠂⠀⠀⣸
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠒⠒⠲⠶⣤⣤⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⢻⣿⣿⣿⣿⡟⠩⠐⠆⢻⣿⣿⠇⠀⠀⠀⠐⠛⠻⢷⣤⣄⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⣆⣴⠀⠸⣿⣿⠀⠀⠀⠀⠀⠲⠆⠀⢸⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢸⣿⣿⣿⣿⡆⠸⠇⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠙⠻⢿⣿⣿⣿⡿⠠⡄⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠆⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡟⠲⠄⠀⠀⠀⠀⠀⠀⣈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⠀⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢞⣯⣭⣭⣓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠟⣛⣛⣛⣛⠛⠿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣫⣴⠾⠿⠿⢿⣿⣿⠿⠿⠶⢌⡙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣶⣿⣿⣶⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣻⣿⣿⣿⣿⣿⣿⣯⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣛⣛⣛⣛⣛⣛⣋⣛⣛⣛⣛⣛⣛⣛⣛⣓⣂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠙⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠣⠤⠤⠤⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠠⠄⠀⠐⠀⠉⠍⠀⠀⠀⠐⠀⣆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠉⠀⠀⠀⠈⠀⠣⠀⠀⠀⠀⠀⠀⠀⣀⠴⠆⠀⠀⠀⠀⠀⠀⠀⠐⠘
⠀⠀⠀⠀⠀⠀⢀⢀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⢀⠔⠈⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠠
⠤⠄⠒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠥⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⢀⠄⠀⠠⣄⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠤⠂⠀⠀⠀⡀⠈
⠠⡀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠐⠈⠁⠀⠐
⠁⠐⠀⠀⠑⢀⣠⠄⠊⠀⠀⠀⢀⠀⠀⡐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠐⡀⠀⠀⠀⢠⠀⠂
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠀⠒⠋⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⣀⡀⠀⠀⠀⠀⠐
⢆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠠⠎⠁⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠐⠤⠀⢀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠣⠀⠀⠀⠀⠁⠀⠀⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣄⣂⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣆⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠁⠀⠀⠀⠀⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠉⠉⠛⠶⣄⠀⠀⠀⣴⡶⢦⡀⠀⣠⡴⠖⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⢠⣾⡟⠉⠀⠀⠹⣄⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠙⠂⢀⣠⡴⣞⣩⡤⠶⠒⠚⠛⠛⠓⠒⠶⢤⣄⡈⠳⣤⡀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠶⠚⣩⡵⠛⣩⡤⠶⠚⠛⠉⠉⠉⠉⠙⠓⠲⢤⣍⠓⢦⣝⠲⣄⡈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣌⠳⣄⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠹⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡇⠀⠀⢹⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⣠⣾⡿⠟⣿⠟⠀⠀⠀⠀⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢿⣷⣤⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡄⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢧⡀⠹⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣄⠘⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠓⢦⣙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢦⣌⡙⠲⠦⢤⣤⣤⣤⣤⠤⠴⠖⢋⣡⢴⣞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡴⠛⠉⠉⠛⠒⠶⠶⠦⠴⠶⠶⠒⠛⠉⠀⠀⠈⠳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣄⡀⠀⢀⡀⠀⢻⠁⠀⠀⠀⣼⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢸⡇⠰⣏⡽⠀⢸⠀⠀⣠⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣇⣀⣼⣀⡀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠆⠀⠀⠀⠀⣸⠗⠋⠁⣀⣀⣀⡀⠉⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠞⠀⠀⢠⡾⠋⠉⠉⠻⣦⠀⠀⠈⢦⣀⣀⣠⡖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠉⢩⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢧⠀⠀⠙⠋⢀⣰⣋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠶⠶⠶⠶⠶⣾⠀⠀⠀⢰⣿⣷⣶⣶⣶⣶⣿⣷⡄⠀⠀⢸⡟⠛⠛⠛⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⠀⠀⠀⢸⣿⣿⣿⡁⣹⣿⣿⣿⡇⠀⠀⢸⠧⠤⠤⠤⠤⠤⣞⣹⠄
⠀⠀⠀⠀⠀⠀⠀⣴⠓⡶⠋⠉⣀⣀⠀⠘⣆⠀⠀⢸⣿⣿⣿⣇⣿⣿⣿⣿⡆⠀⢀⣞⣀⣀⣀⣀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠧⠼⠛⠛⠛⢦⡀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠁⣠⠞⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣤⣀⣀⠀⢀⣀⣠⠴⡞⠁⠀⠀⠀⠀⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠉⠉⣽⠀⠀⡗⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢸⠀⢠⠞⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢲⠋⠀⠀⢀⡾⢦⠈⠛⠁⠀⡇⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠉⠹⠤⠏



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠉⠙⠉⠉⠠⠦⠄⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠈⠉⠙⠉⠛⡀
⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠰⠧⠤⠄⣀
⠀⠀⢀⣠⡄⠒⠛⠓⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠑⢢
⣀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠛⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣌⠳⠆⠈⠉⠉⠉⠉⠠⠾⢋⡴⢆⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠖⠉⠁
⠀⠀⠀⠀⣿⠛⡟⠀⠠⠤⠤⠤⠤⢀⡴⣛⡾⠁⣀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠠⠴⠶⢌⡷⡄⠚⠛⠋⠁⠈⠻⢄⣘⡷
⠀⠀⠀⡌⢿⠏⠀⠀⠀⠀⠀⠀⢀⢮⡙⠁⠀⠀⠀⠀⠀⢰⠄⠀⢴⠄⠀⠀⠀⠀⠀⠈⠳⣡⡄⠀⠀⠀⠀⠀⠀⠹⠇⣴
⠀⠀⣴⠃⠘⠀⠀⠀⠀⠀⠀⢠⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⡸⠂⠀⠀⠀⠀⠀⠀⢸⠉⡆
⠀⠀⣅⢠⡆⠀⠀⠀⠀⠀⠀⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣴⠀⠀⠀⠀⠀⠀⢸⠀⡘
⠀⠀⠈⠲⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠻⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⢰⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⢴⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠃⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢾⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠓⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣀⠀⠀⠀⠀⠙⠿⣦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⣴⠷⠋⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀⠀⠀⠀⠈⠛⠛⢷⡾⠷⠾⠟⠛⠉⠀⠀⠀⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠂⢤⣤⡄⠀⢸⡂⠀⠀⣿⠂⢠⣤⡤⠰⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢹⠃⠀⠀⣟⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣼⡁⠀⠀⠉⠉⠉⢹⡇⠀⠀⣿⣉⡉⠉⠉⠀⠙⣧⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢸⠉⠳⣄⠀⠀⠀⠘⣇⠀⠀⠟⠈⢡⠀⢀⡴⠛⠋⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⡌⠀⠀⠈⢳⠄⢀⡜⠛⠳⠤⠀⠀⠈⠾⠯⠤⢀⣸⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⣧⣤⡤⠰⠛⠲⠞⠙⠲⠀⠀⠀⠀⠸⠀⠀⠀⠈⠉⠑⠒⢠⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠷⠶⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠶⠆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡦⠤⠤⠀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠈⢤⠤⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀⠀⡀⠀⠀⣇⣀⣀⣀⡈⠉⠿⢂⠒⠒⠀⢤⣤⣄⣀⡀⡀⢀⡀⢀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠉⣟⠉⠉⢻⠋⠹⠟⠛⢩⣽⠛⠏⣭⡅⠴⡉⢹⠖⠒⠐⡟⠈⢿⡁⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠒⠋⢉⠉⠹⢇⠿⠲⠶⢶⡄⠀⠀⣰⠴⠶⢧⣸⠇⠀⠈⠙⠆⠛⢃⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣄⣠⣾⡦⢾⣿⣦⡀⠀⠀⠀⣠⡟⢹⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠠⣿⡟⠛⠿⠀⠙⠁⣠⡛⠀⠱⠀⠀⠀⡛⠁⡾⢷⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣥⠄⠒⠒⠈⢀⣍⠉⠀⢿⠀⠀⣠⠃⢸⠀⠀⠳⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠲⠂⠈⠉⣹⠏⠉⢹⣿⠀⢠⡋⠀⠿⢢⡄⠀⢣⣀⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠛⠉⠠⠷⠤⠰⠾⠛⠳⡄⠙⠾⠆⠾⠇⠀⠈⠁⠖⠁⠶⠶⠄⠀⠀⠀⠀⠠⠶⣷
⠀⠀⠀⠀⠀⠀⢰⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠
⠀⠀⠀⠀⠀⢀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄
⠀⠀⠀⠀⠀⡌⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢡⡀
⠀⠀⠀⠀⠀⠏⠀⠀⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⢃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠧⠤⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⡀⠐⠀⣠⣄⣠⣀⣀⣀⣀⣀⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⢞⣛⡿⢿⠿⣛⣛⣫⣿⣬⡁⣿⡏⣨⣿⣯⣛⣛⣛⠿⠿⢿⣛⠢⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠶⢂⣀⣀⣀⣀⡰⠶⠬⠓⠉⠘⠫⡴⠖⣀⣀⣀⣀⣀⠶⢦⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣟⣭⣧⣽⣶⣾⣷⠈⠉⠑⠚⠵⡢⢄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣏⢻⣿⡿⢿⣿⣿⣿⠀⠀⠀⣀⠀⠀⣴⠃⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣶⣿⣿⣷⡻⣆⢻⣿⣶⣾⣿⣿⠀⠀⠉⠉⠀⡼⢃⠎⠀⠀⠉⢙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣮⢿⣿⣿⣿⣝⢦⣝⠻⣿⣿⣿⠀⠀⣀⡤⠞⡡⠋⠀⠀⠀⡔⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⣿⣿⣿⣿⣷⣽⡷⣦⡙⣿⢠⠞⢋⠤⠊⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣾⠁⡔⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡅⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠂⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⡏⣿⢯⡽⣿⢹⣿⡧⢲⢴⠂⢰⡆⠀⢰⣇⣑⠆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣾⣿⣷⣷⣿⣷⣾⣶⣿⣶⠉⠀⠉⠈⠁⠀⠈⠁⠈⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⣿⠟⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠖⢀⠀⠀⠀⠀⠀⠈⠁⠀⠻⣾⣿⣦⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⣷⣿⠀⠀⠀⠀⠈⠁⣤⡀⠰⢷⣾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣥⠀⠀⠀⠀⠀⠀⠀⠉⣲⣤⡤⠶⠾⠇⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠂⠉⠀⠀⠀⠀⠆⠘⠟⠋⠁⠀⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠷⡄⠀⠠⠄⢀⣤⡀⠀⠀⠀⠘⠃⠈⠀⠀⡄⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣾⠶⣃⠀⠀⠀⣈⠋⠀⠰⠆⠀⠀⠀⢀⣄⠀⢀⡀⠀⠀⠀⢀⠴⣶⣄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠋⠘⠀⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⢿⡟⠀⠀⠀⠻⣶⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠶⢨⡕⠶⢌⡁⢀⠀⣴⡄⢠⣾⡟⠓⡀⠀⠀⠀⠀⠐⠰⠆⠘⠦⠌⠉⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⠀⢶⠀⠀⢀⠀⡅⠙⠁⠦⢀⠀⠡⠼⢶⣥⠀⠀⠀⠀⠐⠆⠘⠲⡒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠛⠿⠛⣿⣿⣿⣶⣿⣿⣿⣶⣿⣿⣤⡈⠅⣤⡐⠊⠀⣤⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠣⡀⠀⠀⠀⠀⠈⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⠀⠀⠉⠀⠤⠀⢠⡄⠀⠀⠠⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡠⠦⣤⣤⡶⠚⠛⢦⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣷⣬⣿⡶⠆⠁⢀⡀⢀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠘⢿⣷⡄⠀⠀⠙⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣿⡏⣀⠿⡆⠛⠁⠀⠆⠀⠴
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠈⠳⣄⠙⢄⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⣿⣿⣿⣟⣀⡁⠀⠘
⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠈⠳⣄⠑⢄⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣷⣍⠋⠄⠀⠁
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⠈⢷⣄⠳⣄⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⡛⠃⡀
⠀⠀⡟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠙⢷⣌⠳⣄⣤⠶⠛⢉⣿⣿⣿⡇⣶⠄⢀⠈
⠀⠀⣧⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠹⣷⡇⠀⠀⠀⢸⣿⣿⣿⡆⠻⠀⠉⠁
⠀⠀⠹⣄⠀⠈⠓⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⡆⠀⢀⣠⠬⠟⠀⠀⢀⣼⣿⡿⢡⠀⠀⠀⠁
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠁⠀⠸⠟⠋⠉⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠂⠀⠀⠀⣀⡀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠤⠤⠤⠀⠀⢀⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠩⠉⢉⣁⣀⠀⠠⢼⡯⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠉⠐⢄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⣒⡒⠒⠆⠀⠀⡜⠧⡀⠰⠀⠀⠀⠀⠀⠌⠀⠀⠁⠀⠀⠄⠀⠠⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠤⠀⠤⠭⠭⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠁⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠉⠉⢁⣐⡒⠢⠠⠭⣁⣺⢀⣀⠀⠀⠀⡂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⡒⠒⠢⠬⠍⣉⣒⠒⠼⣿⠉⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠤⠤⠤⠭⠍⣁⣒⠂⠠⠬⢹⠿⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠙⠒⠂⠤⢤⡨⣄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⢴⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡨⡀⢀⢁⠀⣠
⠀⠀⠀⠀⠀⣴⣯⣿⣟⣻⣷⣯⣷⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠁⣡⠞⣋⣑⠷⡁⠈
⠀⠀⠀⠀⠀⢸⣿⣿⠈⠅⢹⣿⡟⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢧⢾⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣌⠻⢟⣴
⠀⠀⠀⠀⠀⠀⠀⠙⠯⡽⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠊⠙⠉⠑⠁⠾


⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⡇⠀⠀⠀⡔⠲⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⢠⣤⣤⣤⣤⣤⣤⡄⢸
⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⠇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠠⠤⠤⠤⠤⠄⠀⢸⠀⠀⠀⠀⡇⠀⠈⠉⠀⠈⠉⠁⠀⢸
⠀⠀⠀⠀⠷⠤⠤⠤⣤⣤⡦⠤⠤⠼⠀⠀⠀⠀⠷⠤⠤⢤⣤⣤⡦⠤⠤⠾⠀⠀⠀⠀⠷⠤⠤⢤⣤⣴⡤⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⣶⣶⣶⣶⢲⡆
⠀⠀⠀⠀⢰⣶⣶⣶⣆⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⣾⣗⣶⣶⣶⣶⣺⣇⡀⠀⠀⠀⠀⢰⣶⣶⣶⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠉⠁⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⠉⠉⣿⣿⣿⡇⠀⠀⠀⠀⢸⣷⣤⣥⣴⣦⣬⣤⣾⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⠟⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠏⠀⠀⣰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⢨⡅⠀⢠⡉⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠉⠉⠀⠀⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⣤⣀⣀⣤⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣤⣤⣾⣟⣛⡒⠶⣤⡀⣤⣀⣐⡄⣰⠞⠉⣁⣤⣶⣶⣿⣿⣿⣁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣷⠞⠃⠉⢻⡏⠁⠈⠓⠾⣿⣿⣿⣿⣿⣿⣿⣟⢦⠈⢳⡄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⣰⠃⣰⠃⣿⣿⣿⣿⣿⡿⢋⣠⡴⢶⣚⣻⣿⣿⣿⣿⣶⣦⣭⣿⡿⣿⣿⣿⣿⢸⡇⠘⡇⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⣿⠀⢿⣿⡿⢛⣿⠟⣫⣴⣾⠿⠟⠛⣛⠛⠛⠛⠿⣿⣿⣿⣷⣤⡉⠛⠿⣼⠇⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠹⣤⠞⠋⣠⠋⣡⣾⠿⢋⡀⠀⠀⠀⣿⠀⠀⠀⠠⠆⠉⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣾⡿⠁⠀⠈⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢈⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⡟⢠⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⣿⣧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣾⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⣿⠀⣀⡀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠿⠇⢸⢿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣼⣿⡄⠉⠁⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡆⠈⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⢃⣿⠇⢀⡟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢻⡄⠘⣿⣿⣷⡜⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣾⠏⢀⡞⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠻⣄⠈⢿⣿⣿⣦⣀⠀⣤⠀⠀⠀⣠⡀⠀⠀⠐⢂⣤⠞⣱⡿⠋⠀⠋⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠙⠿⣿⣿⣷⣦⣤⣀⣀⣘⣁⣠⡤⠶⣋⣵⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢫⣿⠿⢿⣿⣿⣿⣿⣿⣶⣾⠿⠟⠉⠹⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠈⠉⠉⠉⠀⠠⠤⠖


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠋⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡇⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⢳⡀⣈⢷⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⡞⠀⠀⠜⠃⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⢷⣄⠀⠀⠘⣆⠱⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⠏⠀⠀⠀⠀⣠⡾⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠙⢷⣄⠀⠈⠣⠐⠜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡴⠃⠀⠀⢀⣴⠞⠉⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⠊⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⡇⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡜⠀⣇
⢸⠁⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡐⠀⠀⢻
⣟⣀⣀⣀⣠⣸⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣸⣖⣀⣀⣀⣘
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⢀⣤⣦⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢶⣄⣿⣧⣴⠆⠀⠀⠀⠀⠈⢹⠿⣿⠿⡏⠁⠀⠀⠀⠀⢀⣴⡿⢻⣼⡟
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢤⣿⣿⣿⣯⠤⠀⠀⠀⠀⢀⣸⣶⣿⡶⣃⡀⠀⠀⠀⠀⠺⣯⡽⣿⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠙⠟⠈⠉⠁⠻⠋⠀⠀⠀⠾⠯⠤⠤⠤⠬⠟⠆

⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣄⠀⠀⠀⠀⠀⢸⢈⠀⢀⣤⣶⡆⠀⠀⠀⠀⠘⣶⣶⣶⡟
⠀⠀⠀⠀⠀⠈⠻⣿⠟⠹⣦⡀⠀⠀⠀⠀⢠⣾⣿⣾⠋⠁⠀⠀⠀⠀⢂⢁⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣀⢸⠇⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⠹⠋⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⡴⢏⡠⢆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣼⣟⣦⠀⠀⠀⠀⠀⠈⠲⠒⠒⠊
⠀⠀⠀⠀⠀⠠⢮⣑⣼⣷⣄⠀⠀⠀⠀⠀⠿⠋⣿⣏⠻⠇⠀⠀⠀⠀⠀⣴⢿⣤⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⠃⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋

⠀⠀⠀⠀⠀⠀⠀⢈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⢶⡆⠀⠀⠀⢠⣄⣀⣀⣤⠀⡀⠀⠀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⢠⡮⣬⣦⠀⠀⠀⠀⠀⣠⠞⣩⠛⡿⣳⡎⠁⠀⠀⠀⢘⠉⠉⠉⠉⠁⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⠀⣸⣿⠀⠀⠀⠀⠀⢻⡸⠇⢀⡟⢸⠇⠀⠀⠀⠀⢸⣿⣷⣦⣄⣠⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠀⠀⠀⠀⠀⠀⠉⠙⠿⠴⠋⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⠤⠼⠿⠿⠿⠿⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡖⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢈⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠐⠁⠈⢠⠀⠀⠀⠀⠓⠈⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠀⠠⠈⠀⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡌⠃⠀⠀⠀⠀⠀⡮⡄⠆⠀⠀⠀⢠⠄⠂⣠⠀⠀⠀⢀⡄⠀⠂⡄⠀⠀⠀⠐⢣⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠙⢁⠆⠀⠀⠐⠆⠠⠀⠆⠀⠀⠀⠀⠊⠀⠂

⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⢀⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠃⠘⠀⠌⠀⠀⠀⢈⡠⠔⢙⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠁⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⡉⠭⠍⣁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠁⠀⠀⠀⠀⠉⢢⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡄⣂⠉⠉⠉⠉⠉⠉⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⢌⠢⢄⣀⣀⠤⢊⠜⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣁⠂⠀⠀⢂⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⠉⡉⠉⢹⡁⠀⣀⠀⠀⠀⠀⢸⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠀⠘⠋⠀⠀⠈⠁⠢⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠂⢀⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠱⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⣀⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡎⢐⠄⢀⠠⡀⠀⠀⠀⠘⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁⢀⢀⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃⠠⡄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠉⠰⠒⠉⠑⠂⠀⠁⢠⡍⠉⠒⠊⠉⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢎⠈⡳⠸⠏⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠋⡀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠋⠑⢌⣉⡥⠚⠀⠤⢄⡀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠘⠤⢂⢌⠔⠀⢸⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢀⠌⠀⠀⠀⠀⠀⠈⠑⢠⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠈⠢⣑⠤⣀⠀⠀⢎⡰⠀⢀⡠⢐⠥⠊⠀⠀⢸⠀⠀⠀⠀⢠⣎⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⡀
⠀⠀⠀⠀⠀⠘⡤⠤⠤⠤⣬⠃⠀⠀⠀⠁⠒⠨⠭⠬⠭⠭⠐⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠪⣓⢦⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠱
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠇⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠈⢁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀
⠀⠀⠀⠀⠀⡰⠡⢤⠬⠥⡌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠂⡶⡰⣒⠂⠲⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⡜
⠀⠀⠀⠀⡰⠱⠁⠀⠠⡄⠀⠌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⠃⡣⡘⣢⡀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠤⣀⠀⢀⠔⠁⢀⠜
⠀⠀⠀⠰⠡⠃⢀⡀⠀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛⢊⡁⠈⠓⠤⣀⠜⠁⠀⠀⠀⠀⠀⠀⠑⡁⠀⡠⠊
⠀⠀⢠⢁⠂⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠲⢄⡀⠸⠀⠀⠀⠔⠁⢑⠀⠀⠀⢰⠈
⠀⢀⢂⠂⠀⠀⠀⠀⠰⣉⡁⠀⠀⠑⡐⡤⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠸⠤⠤⠀⠤⠄⢤⠀⠀⠀⢙⠀⠀⠀⠈⠐⠈⠀⠀⠀⢸
⢀⢂⠂⠀⠘⠃⠀⠀⠀⠀⠀⠰⠉⠆⠱⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣆⣘⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣁⣵⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠐⠂⡀
⠀⠀⠀⠀⠀⢀⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢸⣿⣷⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠌⠷⡄
⠀⠀⠀⣄⣰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢾⣿⣿⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣛⣛⡛⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠄⠀⢳
⠀⠀⣀⣼⠿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡿⣿⣿⣿⡾⣦⣌⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢼⢃⠊
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣾⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣀
⠀⠀⢳⣤⢸⣿⡇⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠺⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣻⡇
⠀⠀⠲⣿⣿⣿⣿⣿⣥⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣿⣆⠻⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢆⣤⡀⠀⠈⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠬⠿⠿⠿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣙⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⢀⡀⠀⠀⠀⠀⠀⠀⡁⢀⠇⡰⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠶⢠⠃⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷
⠈⠙⢿⣶⣤⣀⠀⠸⣈⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠁⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⡟⠋⠀⣀⣄⡀⠉
⠀⠀⠀⠙⠿⣿⣿⣦⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣇⠑⠦⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⣶⣶⣶⠖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠤⣿⣿⡗⠋
⠀⠀⠀⠀⠀⠈⠛⠫⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄
⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡁⠫⠢⡕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣶⢀⣀⡀
⠀⠀⠀⣾⣿⣿⣿⣧⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⢉⡉⢉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠢⣢⠈⠩⢢⡑⢄⠀⠀⠀⠀⠀⠀⠀⠸⣯⡾⠃⢾⡏⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠣⡤⡈⠪⠎⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢶⡎⠻⠞
⠀⠀⠀⠻⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠿⠿⠿⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣿⠉⠉⠙⠛⣿⣿⠿⢿⣿⣿⣿⣧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣄⠀⠀⠈⠛⠷⠟⠋⣹⡟⠛⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣴⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦⠶⠶⠶⢶⣤⡀⠀⠀⠀⠻⠿⠋⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣤⣤⣤⣤⣌⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⢛⡛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢫⣶⣿⣿⣷⣦⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢰⣿⣿⡻⠿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⣿⣦⣭⣭⣄⢮⡟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣎⢻⣿⡇⣿⣿⣿⣿⣿⣿⢳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣾⣿⣇⢻⣿⣿⣿⡟⣽⡟
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣯⣭⣭⡞⢜⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣷⣄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⡄⣿⣿⣿⣿⣿⡿⢠⣿⡟⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣼⢇⣾⣿⣶⣤⣄⡀
⠀⠀⠀⣸⣇⣿⣿⣿⣿⠻⠇⣼⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣟⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⢀⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢰⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣿⣿⣿⣿⣿⠿⠃⣼⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣿⣿⣿⣿⣶⣶⣦⣬⣍⣙⡛⠇⠹⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⡙⢿⣿⠇⣿⣿⣿⣿⡏⣼⣿⠿⠇⠀⠀⠀⠀⠀⢀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣈⣉⠘⣿⣿⣿⣿⣱⣿⣿⣿⣶⣦⣤⣀⡀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢉⣛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠃⠉⠛⠻⠿⠿⢿⣿⣿⣿⣟⣛⣛⡻⠇⣼⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣶⣦⣤⣭⣭⣭⣭⣭⣉⣁⣀⠀⠀⠀⠀⣐⠒⠮⢝⣿⡿⠿⠿⠿⠇⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡇⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣶⡇
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠹⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣝⣿⣿⣿⣼⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠨⣽⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣿⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⣿⣇⣄⠀⠀⠐⣶⣿⣿⣿⣿⣿⣯⡤⠀⠀⠀⠀⠀⠀⢀⡀⣾⣼⣿⣿⣿⣿⡯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⣿⣿⣿⣿⣿⣴⡄⠀⣙⣿⣿⣿⣿⣿⣟⡁⠀⠀⠀⠀⢠⣄⣿⣿⣿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣧⣤⠹⣿⣿⣿⣿⣿⡿⠃⠀⠀⣰⣷⣿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡖⢀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠠⠄⣄⣠⣄⣢⣄⣤⣠⡀⣀⠀⠸⠿⢿⣿⣿⣿⣏⣤⣿⣿⣿⣥⡄⢸⣿⣿⣿⣿⣿⣿⣿⡍⠁
⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣻⣿⣿⣷⣾⣿⣿⣄⣻⣾⣿⣿⣿⡻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠉⠀⠸⣿⣿⢿⣮⣽⣷⣾⣶⣦⣴⣤⣦⣴⣦⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠟⠻⠿⡿⠟⡿⠿⠃⣐⡶⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠓⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠀⣿⣷⣧⣝⠙⠟⠻⠻⢿⠻⠟⠻⠛⠛⠙⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⠃⡀⠸⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⠋⠀⠀⠃⠀⠈⠟⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠩⠅⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠛⠤⠰⠾⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠯⣄⠀⠐⢶⡔⡿⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠍⡀⠀⠀⠀⠀⠉⠁⠠⡀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡴⠁⠀⠀⣑⠢⡄⠠⠆⠉⠉⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⢠⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢤⠘⠀⠀⠀⠀⠀⢀⠂⠀⢣⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⢀⠐⣀⠀⠀⠂⡼⠁⠀⠀⠂⠀⡐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠄⠀⠀⡰⠂⡀⠈⠂⠀⠀⠀⠀⠄⢤⢀⡎⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠢⠤⢴⠁⠀⠀⠀⠀⠀⢁⠀⠀⠉⠉⠀⠢⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢛⠣⡀⠶⢴⠁⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⢀⣀⠳⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⡦⢬⡂⠈⢅⠀⠀⢀⠀⠀⠀⠀⠈⠙⠶⡬⠖⠁⣠⠃⠐⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠈⠀⢘⠀⠑⠀⠀⠀⠀⠀⠉⠲⠽⣞⠕⠁⠀⠠⠀⠁⠒⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢰⢄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠀⠀⠀⠀⠀⡰⢧⡌⠉⠂⠌⡒⠄⠀⠀⠁⠀⠀⠚⠀⠀⠉⠁⢀⡤⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡵⢸⠉⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⠂⣁⠀⠀⠠⡄⠀⠀⠀⠐⠂⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⢀⠈⠣⣀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⢄⠀⠀⠀⡼⠀⠀⢸⠀⠈⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂⠁⠀⢰⠃⠀⠀⠈⠁⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠒⠒⠐⠀⣠⣀⢘⠀⠀⠀⠀⠊⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⣠⡦⢀⠠⣤⣤⣬⡏⠤⠌⠀⠀⢀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠷⣚⣧⠴⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⢿⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⠀⢰⡇⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⠀⢸⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣾⠀⣾⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢼⡀⠀⠀⢿⠀⢻⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⣧⠀⠀⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠘⠃⠈⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢷
⠀⣴⡞⠋⠱⠀⢀⠄⠀⡤⢄⠀⠀⠀⡖⠒⠀⠙⢀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⢀⡤⠴⠶⠶⠶
⠀⠸⠛⠀⢀⣾⢀⠀⠀⡃⠈⡂⠀⠀⠠⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠾⣿⡀⣾⡀⠀⢣
⠀⠀⣱⣄⣠⠉⠀⠢⣄⣷⣀⠕⢀⡄⠀⠂⢄⣆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣺⡇⠀⢸
⢰⢡⠱⠃⢕⣅⢠⠴⠎⣉⣍⠚⠀⢤⡆⠈⣳⠑⡗⠈⡱⠦⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢘⠁
⠨⠡⠐⡀⠀⠇⠀⠀⢧⡀⢈⣆⣀⣈⡄⠀⣨⠘⣇⣰⠁⢀⠇⡀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣸⠁⠀⠐
⠈⠦⣁⣐⣀⣤⡙⠀⠀⣱⡤⠀⢤⠗⢓⠀⠀⢱⣭⠛⠲⠏⢁⡉⢱⡄⠀⠈⠓⠦⢤⣤⣤⣀⣀⣀⡀⡿⠀⡏
⠀⠀⡉⠉⠉⠝⠉⠒⠠⠒⠁⠀⠈⢁⡘⢠⠐⣼⡏⠀⠠⣈⠠⠘⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣁⣀⠃

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦
⠀⠀⠀⠀⢠⠶⢴⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠶⠶⢶⠃⠀⠀⠀⠀⠀⠀⠀⢯⣩⢏⡩
⠀⠀⢠⢞⠁⣰⠞⠁⠀⠀⠀⠀⠀⠀⢀⢔⡁⠀⣨⠂⠀⠀⠀⠀⠀⠀⣎⣡⣔⢗⡄⢹⠀⠀⠀⠀⠀⠀⣶⣀⣽⣿⢆⣶⡄
⠀⢰⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣿⠞⣥⡤⡀⠀⠀⠀⠀⠀⡏⣭⣴⣶⡎⢹⠀⠀⠀⠀⠀⠀⠀⢈⡵⢺⣁
⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠫⠾⠃⠀⠀⠀⠀⠀⠱⠬⠭⠿⠬⠞⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠈⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⠾⢦⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⢶⣾⣦⣿⣼⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⣵⣾⠿⠿⠀⠀⠀⠀⠠⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠮⣱⣄⣠⠃⠸⢆
⠀⣀⣉⡈⢉⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢀⣀⣀⡀⠀⠀⠀⠀⠀⠙⠟⢄⠀⠀⠈⢆
⣾⣤⣶⡘⣣⣦⣤⡇⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠸⣿⢿⣿⢫⣤⣤⠃⠀⠀⠀⠀⠀⠀⣧⣤⣵⣠⣤⣌
⠈⠛⠛⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠐⠿⠿⠟⠘⠛⠟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁
⡷⠶⠶⠶⠶⠶⢾⠀⠀⠀⠀⠀⠀⣄⡀⠀⣿⣿⠀⢀⣠⠄⠀⠀⠀⠀⣠⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠏⢀⠔⠢
⡇⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠘⢿⣦⣹⡿⣴⡿⠋⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢳⣾⡞⠁⠀⢠
⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠠⢴⣶⣿⣿⣟⣿⣶⣦⠄⠀⠀⠀⠀⠀⢰⣶⠠⣶⣶⡀⠀⠀⠀⠀⢀⢰⣿⣿⣯⣷⠠⠃
⣇⣿⣿⣿⣿⣿⣻⠀⠀⠀⠀⠀⠀⠀⢠⠾⠃⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⣾⠉⠀⠀⠀⠀⠀⣅⣈⣙⣛⢛⡁⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⢷⡄⠀⠀⠀⢀⣠⠤⠀⠀⠀⠀⢸⣿⣿⠀⢠⢷⠀⠀⠀⠀⠀⠀⠀⠀⢘⣙⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⢷⣶⣶⡇
⠈⠳⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⢴⣶⡦⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣮⣿⣿⣷⣿
⣶⣿⣶⣀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣯⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠏⣘⠀⢹⡉⢶⡀⠀⠀⠀⠀⠀⣠⣤⣴⠓⠤⠅⠻⣄⠀⠀⢀⡴⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡁⢀⡃⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠠⢸⣇⠀⠀⠀⠀⢸⠀⠀⣸⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠈⡿⡉⠉⢹⠉⠉⡿⡋⠁⠀⠀⠈⠉⠉⠉⠉⡙⠲⠦⢴⠋⠀⢠⠉⠃⢸
⠀⠀⠀⠀⠀⠀⢠⡴⢅⣠⠄⢸⠀⠴⣀⠵⣤⠀⠀⠀⠀⠀⢠⠊⠀⣰⠆⠀⠳⡄⢸⡄⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠋⢸⠈⠳⠦⠴⠋⠀⠀⠀⠀⠀⢸⡄⠀⣝⡶⠀⡀⡗⠀⠳⠤⠴⠯⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠉⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠙⠦⠬⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣢⣵⣼⣵⣊⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢲⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢁⠒⠒⠒⠒⢲⠃⢠⣒⡒⢄⠀⠀⠀⠀⡟⠂⠉⠀⢀⠴⢢⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠙⠟⠉⣹⠀⡇⠞⢳⠎⠀⠀⠀⠀⡇⢀⡤⠶⠋⠀⡀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⡉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⡇⠀⢳⡀⢰⣎⠹⡅⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡏⢠⣉⡤⠿⠒⢒⡟⠀⠀⠀⠀⠀⡀⢀⠠⠽⢥⡌⢢⡘⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠶⠒⠒⠓⠤⠤⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠶⠃⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢳⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠚⠉⠀⠀⠀⠉⠑⠦⡀⢔⠉⠉⣉⣩⠆⠀⣠⠖⠋⠁⠀⠀⠈⠙⠲⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠂⠉⢹⠁⠀⠀⠸⣅⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣠⠴⠊⣁⣠⠤⠤⠼⠦⠤⠤⢤⣀⡉⠓⢤⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢀⡤⠖⣻⠿⠓⠉⠀⣀⣀⡤⠤⠤⠤⣀⣀⡀⠈⠑⠺⢿⡓⠢⣄⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⡤⠚⠁⣀⠤⠒⠉⠁⠀⠀⠀⡀⣀⠀⠀⠉⠑⠲⢄⡀⠉⠲⣄⠉⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣠⠞⠁⠀⠀⢀⣤⠀⠀⠀⣏⣹⠃⠀⠀⣀⠀⠀⠙⢦⡀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠞⠁⠀⠀⠀⠀⠠⣿⠀⠀⠀⠉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠙⢆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢀⠎⠀⠀⣰⡔⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⠀⠈⢧⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡞⠀⠀⠀⠙⠳⠾⣀⢰⣦⣄⠀⠀⢰⠇⠀⠀⣀⣤⣶⠟⠻⠷⠀⠀⠘⡆⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⢀⡤⡄⠀⠀⠈⠉⠛⠿⣿⣦⣼⣤⣴⡾⠟⠉⠀⠀⠀⠀⢠⡀⠀⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⠀⠈⠓⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣘⠇⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢣⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣆⠀⠀⢠⠇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⢧⠀⠀⠘⠛⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠁⠀⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠳⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣴⡀⠀⠀⠀⢙⣳⠀⠀⠀⡰⠋⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠷⠏⠀⠀⠀⠈⠁⢀⡠⠊⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠈⠙⠲⠤⣀⣀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣹⠶⢤⣀⡀⠈⠉⠉⠉⠉⠉⠀⢀⣠⠤⠾⡉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡜⠁⠀⠀⠀⠈⠉⠉⠓⠒⠊⠉⠉⠁⠀⠀⠀⠙⣌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣺⣭⠿⠷⠶⠶⠾⠯⣿⣲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡫⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⠔⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠘⣯⢧
⠀⠀⠀⠀⠀⠀⢀⡴⣞⣯⣾⣷⠶⠶⠶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣶⣄⢸⡾⡆
⠀⠀⠀⠀⠀⢀⣏⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣉⡁⠀⠀⢻⡿⢻⣿⢣⠇
⠀⠀⠀⠀⠀⠘⣎⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠛⠀⠀⠈⣿⡾⣿⣘⣿⡲⣄
⠀⠀⠀⠀⠀⠀⠈⠑⠲⡯⣿⠀⠀⠀⠀⠀⠀⠀⠒⠛⢡⣿⡅⠀⠀⠀⠀⢸⣿⡿⡡⢀⢹⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⡇⣴⣿⣿⡶⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⣿⡿⠈⣼⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⡇⠈⠁⢸⠇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠠⠃⠀⣼⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⣆⡇⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡼⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⡇⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣷⠀⠀⠀⡤⠘⠃⠂⠐⠛⠉⢙⠢⡀⠀⠀⠀⢀⡀⣷⣬⣭⣭⣟⢦
⠀⠀⠀⠀⣠⢖⣻⣭⣽⣗⢿⢻⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢆⠙⣤⠄⢲⡀⠠⠤⠤⠄⠀⣿⣸
⠀⠀⠀⡼⡷⢩⠋⠀⠂⡍⢷⡤⡽⣦⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢦⠈⠓⠸⡄⠀⠀⠀⠀⢠⡟⣿
⠀⠀⠐⣷⡇⢠⠀⠀⠀⠐⢸⡇⡿⢮⡳⣤⡀⠀⠀⠀⠘⢧⠀⠀⢘⠳⣄⡀⠙⠀⠀⠀⢀⠞⢳⡞⡆
⠀⠀⠀⢳⡳⣄⠑⢦⠶⣣⡾⣳⠃⣰⢣⣿⣿⣶⣤⣀⣀⣀⣳⡀⠈⣆⠀⠉⠐⠛⠓⠛⠁⡴⠻⣞⢧
⠀⠀⠀⠀⣹⢞⣹⡧⣾⢻⣜⡇⠀⠈⠓⠮⣽⢛⣿⣿⣿⣿⡏⡁⠀⣏⠁⠀⠀⠀⠀⠀⢀⡆⠀⠘⣯⢣
⠀⠀⠀⢸⡷⣏⣡⡕⢸⠈⢧⡳⣤⢤⣖⣫⣶⣿⣿⣿⣿⣿⣟⣇⠀⠹⡦⠀⠀⠀⠀⠀⢨⢳⡀⠀⠘⣯⢧
⠀⠀⠀⠐⡗⣯⡤⢮⠸⠀⠀⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣏⠠⠀⠙⠦⣄⡀⠀⠀⠀⠀⡅⠀⠀⢸⡎⡆
⠀⠀⠀⠀⢯⢻⡄⢛⣶⠀⠀⠈⠉⠛⠟⠀⠈⠛⠻⠿⠛⠉⢰⠘⢄⣀⡠⠀⡀⠉⠉⠒⠒⠶⡆⠀⠀⣼⣿⣸⡀
⠀⠀⠀⠀⠀⠳⣽⣟⢻⠁⠙⠺⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⡼⠃⠀⡰⢻⣿⣧⢧
⠀⠀⠀⠀⠀⠀⠀⢸⢹⡄⠀⡜⠀⠀⠀⠀⢠⢞⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠁⠒⠛⠀⠖⠋⠀⣿⣿⣿⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢯⡳⠾⠶⣶⣶⣶⠒⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⣿⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⢟⣾⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢰⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠴⣺⣵⣾⣿⣿⡟⣠⠁⠀⠀⠋⠀⠀⠀⠀⠀⠠⡀⠀⠀⠄⠀⠀⠤⠆⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣎⢿⣿⣿⡿⢟⣉⣏⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⣿⣿⣿⡏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠮⠭⠭⠖⠉⠳⣝⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣿⣿⣿⣿⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠯⣽⣟⣲⡶⠦⣤⣤⣬⣥⣤⣴⣦⣭⣤⠤⢶⡖⣿⣿⡿⣣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠒⠒⠒⠛⠛⠋⠉⠙⢾⣫⠞⠁