Birdcage Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣉⣋⣙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣰⠋⠉⠉⠉⠳⣌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢿⣄⠀⠀⠀⠀⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡶⣿⠟⠛⢿⣻⣿⣻⠟⢻⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢚⣩⡵⠞⠋⣁⡴⢞⣷⣦⣤⣤⣤⣴⠶⣿⣄⠙⢿⣝⠶⣌⡙⠳⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣡⠶⠋⠁⠀⣠⠞⢋⡴⢻⠟⡿⠀⢸⡆⢻⡀⠹⣌⠳⣄⠙⠷⣌⠙⠶⣄⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣠⠞⠁⠀⢀⣴⠞⢁⡴⠋⣰⠏⢸⠃⠀⢸⡇⠀⢷⠀⢹⡆⠈⢷⣄⠈⠳⣄⠈⠻⢦⡀⢀⣹⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠃⣨⡟⠳⢦⣄⣴⣏⣁⣰⣟⡁⣰⡏⣀⣿⣀⣀⣸⡇⠀⢸⡇⠀⢻⣄⢀⣙⣦⣀⣘⣧⡤⠶⠟⢿⡁⠈⠳⣄
⠀⠀⠀⠀⠀⣠⠟⠁⣰⠏⠀⠀⣠⡞⠁⢉⡽⠋⠉⢹⡟⠉⢙⡟⠉⠉⢻⡏⠉⠙⣿⠉⠉⢻⡏⠉⠉⢿⡀⠈⢳⡄⠀⠀⠙⣦⠀⠙⣧
⠀⠀⠀⠀⣴⠋⠀⣼⠃⠀⠀⣴⠏⠀⣠⡞⠁⠀⢠⡟⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢿⠀⠀⠈⣧⠀⠀⠈⣷⠀⠀⠹⡆⠀⠀⠘⣧⠀⠘⣧
⠀⠀⢀⡼⠃⠀⣼⠃⠀⢀⡼⠃⠀⢰⡏⠀⠀⠀⡾⠁⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡆⠀⠀⢹⡄⠀⠀⠸⣇⠀⠀⢿⡀⠀⠀⠸⣇⠀⠸⣆
⠀⠀⣼⢷⣄⣼⠃⠀⠀⣾⠁⠀⢠⡟⠀⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⠀⣻⣤⠶⢻⡄
⠀⢸⡇⠀⢹⡿⢦⣤⣼⣃⡀⠀⡾⠀⠀⠀⠀⣾⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⢸⡇⠀⢀⣻⣤⠶⠚⢹⡇⠀⠀⣿
⠀⣿⠀⠀⣾⠀⠀⢠⡏⠉⠙⢻⡟⠲⠶⠶⢶⣷⣤⣤⣤⣤⣿⣤⣤⣤⣼⣧⣤⣤⣼⣧⣤⣤⣤⣿⣤⡤⠶⠾⣷⠛⠋⠹⡇⠀⠀⠀⣷⠀⠀⢹⡄
⢰⡇⠀⠀⡏⠀⠀⡿⠀⠀⠀⣾⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢹⡀⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⢸⡇
⣼⠁⠀⢸⠇⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⠀⢻⠀⠀⠀⢻⠀⠀⠀⣿
⣿⠀⠀⣿⠀⠀⣼⠁⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⢿
⣧⣀⠀⡟⠀⠀⣿⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⢀⣼
⣿⠉⠻⣷⣤⣀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⣷⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⢀⣸⣧⠾⠋⢹
⣿⠀⢸⡇⠀⠉⡟⠛⠳⢶⣯⣀⡀⠀⠀⣿⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇⣀⣤⠴⣿⠛⠋⢹⡇⠀⠀⢸
⣿⠀⢸⡇⠀⠀⡟⠀⠀⢸⡇⠈⠉⠙⢻⡿⠲⠶⠶⠶⢾⡷⢤⣤⣤⣤⣼⣧⣤⣤⣤⣿⣤⣤⡤⠶⣿⠶⠒⠛⢻⡏⠁⠀⠀⣷⠀⠀⢸⡇⠀⠀⢸
⣿⠀⢸⡇⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸
⣿⠀⢸⡇⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡄⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣧⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸
⣹⠀⢸⡇⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⢠⡅⠀⠀⠀⣩⠀⠀⠀⠀⢈⡄⠀⠀⠸⣿⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢸
⢹⡄⢸⡇⠀⠀⡇⠀⠀⢸⡇⠀⠀⢀⣸⣇⣤⣤⡤⠶⢾⠷⠾⠿⠛⠛⢻⡟⠛⠛⠛⣿⠷⠶⠦⣤⣼⣇⡀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⢀⣤⢿
⢸⣦⣸⡇⠀⠀⡗⠀⣀⣼⡷⠖⠛⠋⢹⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡀⠀⠀⠀⠨⡏⠙⠛⠲⣿⣄⡀⠀⣿⢀⣀⣼⡟⠋⠁⢸
⢸⡇⢸⡟⢳⣶⣿⣏⣁⣘⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⣀⣀⣿⣤⡽⠷⣿⣍⠁⢸⡇⠀⠀⣾
⢸⡇⢨⡿⠋⠀⣿⠈⠉⠙⡿⠛⠒⠶⢾⡷⠶⠶⠶⠶⣾⡦⠤⡤⠤⠤⣼⡧⡤⣤⡤⣼⡷⠶⠶⠖⠛⡟⠛⠉⠉⣿⠀⠀⠀⣿⠈⠳⣾⡇⠀⠀⣿
⠸⣷⠟⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡿⣦⠀⡟
⢰⣿⡄⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⠸⣷⡇
⢸⡟⢷⣇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⣰⢿⡇
⠀⠻⣌⠻⢦⡀⣻⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡿⠀⣀⣼⠟⣡⠞⠁
⠀⠀⠉⠳⣤⡉⠻⢦⣄⡀⢸⡄⠀⠀⠸⡇⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⡏⠀⣀⣤⠿⠞⣉⣴⠞⠁
⠀⠀⠀⠀⠈⠙⠳⣦⣈⠙⠻⠧⣤⣀⠀⣷⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢀⣀⣧⣤⡤⠶⠛⠛⢉⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣍⣙⠛⠛⠶⠶⠦⡾⣦⣤⣦⣤⣤⠾⠷⠶⠶⠶⠾⠓⠛⠛⠋⢉⣀⣠⣤⠶⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣡⣄⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣅⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣮⣤⣤⣵⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⠟⢛⠿⡟⢽⠏⢹⠟⢩⠙⣿⡿⢿⡻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⣡⡞⠁⠞⠀⠊⠀⠌⠀⠀⣧⠘⣟⠂⠙⠢⡙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠂⡴⠛⠀⡔⠀⠀⡘⠀⠀⠀⠀⢹⡀⢹⡆⢣⠐⠘⢦⠹⣝⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⡜⢀⠁⠸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠧⠀⠆⡄⠀⢃⢹⡼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢱⠁⢠⠁⠈⠀⠃⡇⢈⠀⠀⠆⠐⠀⠀⠸⡇⠀⠀⠀⠘⣧⠀⠀⢾⣣⡾⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠛⢿⡟⠁⣉⠈⢁⠈⣀⠁⠉⠈⠁⠃⠘⡋⣁⠈⢋⠀⠈⡀⡀⢉⠀⠀⠉⢛⣋⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣶⡶⣶⢶⡶⣶⢶⡶⣶⢶⣶⡶⢶⣾⣾⣶⣾⣶⣿⡿⠷⣾⡶⠿⢶⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⡇⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢰⠀⠀⢸⡇⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⡇⠀⠀⢀⠀⡇⠀⠀⠀⠀⣴⣷⡄⠀⡇⠀⢸⠀⠀⢸⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⡇⠀⢸⠀⠀⢸⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⡇⢸⠀⢸⠀⠀⠀⠀⠀⣴⣿⣿⣷⣼⠇⠀⠈⠀⠀⢸⠀⠀⢸⡇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⡇⢰⠀⢸⠀⠀⠀⠀⠀⠛⣿⣿⣵⣿⠀⣀⣤⠀⠀⢸⠀⠀⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⡇⠘⠀⢸⣶⣦⠄⢰⡄⠳⣽⠛⢻⣿⢛⣤⣼⢃⡀⢸⠀⠀⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⡇⠐⠃⠉⠙⠋⠀⢸⡆⠀⢹⣷⡶⢛⣿⠛⢻⣧⣿⠀⠀⠀⢸⡇⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢾⠀⡇⠠⠀⢰⣠⡦⣼⣯⡃⠈⠉⣿⠇⢸⠏⠁⡈⠻⢿⣿⡄⠀⢸⠇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⠀⡇⠀⠀⠐⠘⠇⠀⢛⣉⣦⣴⣿⣠⣾⡀⠀⠀⠀⠀⠙⠇⠀⢸⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⠀⡇⠀⡆⢈⠀⠀⢀⣼⣿⡟⢁⣿⡛⢻⣷⣄⣀⠀⠀⡀⠀⠀⢸⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣷⠿⠒⠛⠚⠗⠚⠛⠟⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠿⠛⠒⠻⣿⣿⣻⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⠄⠶⠒⠒⠀⠒⠀⠘⠃⠐⠀⠐⠀⠶⠂⠀⠀⠀⠉⠂⠀⠀⠠⢠⣿⡿⠿⡇
⠀⠀⠀⠀⠀⢀⡸⠛⠛⠻⠗⠐⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊⠛⡒⠓⠙⠃
⠀⠀⠀⠀⠀⠈⡏⠙⠛⢻⠀⠀⠀⠠⣄⠀⢀⣥⡄⠀⢀⣀⠀⣠⡀⠀⠀⠠⠄⠀⣀⠠⠀⠀⠀⢸⠛⠉⠉⢹
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⣷⠛⠛⣤⣤⣤⡄⠈⠙⣯⡝⠀⠐⠃⠠⠆⢤⣥⠄⠂⠀⠀⢸⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢠⠿⠆⠰⣾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢺⣷⡀⠉⠩⢥
⠀⠀⠀⠀⢠⣅⡌⣁⣘⠒⠐⠂⠃⠀⠀⠀⠐⠀⠓⠀⠐⠓⠂⠘⠀⠀⠀⠁⠀⠀⠃⠀⠀⠂⠀⠀⣡⣤⣀⣀⣬⡄
⠀⠀⠀⠀⠘⠋⠟⠻⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣻⣛⣋⠩⠍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⡄⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣽⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣅⠀⠀⠀⠀⠀⠀⠀⠘⠘⠐⠘⠘⠀⠀⠘⠒⠂⠒⠀⠀⠀⠀⠀⠀⢠⣼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣟⣳⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣿⣷⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠋⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡀⠀⠈⠁⠆⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⡀⠀⢳
⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠒⠐⠂⠀⠀⠀⠐⠒⠒⠠⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠀⠁⠆⠀⢀⣀⣀⡑⠀⠐⠻⢦⣀
⠀⠀⠀⠀⠀⠀⢹⡤⡄⣀⠀⠀⠚⠒⠗⠚⠛⡀⢀⣀⣀⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣯⠤⣀⣀⣀⠈⢉⣙⣉⢉⢁⣀⣀⣀⣼⡿⠗
⠀⠀⠀⠀⠀⠀⢘⠂⠀⠈⠙⠒⢤⣄⡀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠈⠁⠉⠈⠉⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣅⠀⠄⠀⠂⠂⠊⠛⠛⠳⠶⠦⠠⠄⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠄⠤⠀⠳⢦⣴⣤⡄⠀⠀⠀⢤⠀⣤
⠀⠀⠀⠀⠀⠀⢻⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣥⣀⡀⠀⠀⠀⢤⣼⠷⣤⡀⠀⠀⠀⢘⡧
⠀⠀⠀⠀⠀⠠⣟⠉⢥⣚⡚⣠⣤⣆⣂⣠⣤⣤⣙⣾⣠⣥⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠢⣄⣖⣺⣦⣦⣶⣶⣶⣶⣥⣤⣿⣉⣈⣄
⠀⠀⠀⢠⣶⣷⣿⣀⣸⣿⣟⡛⡋⠉⠋⠙⣋⣿⣿⣿⡉⣸⣧⣾⣦⡀⠀⠀⠀⠀⢠⠀⢳⣿⣀⣿⣿⣿⣿⡛⠛⠟⠟⣟⣿⣿⣿⣿⢀⣟⣷⣦⡀
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⡿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⣿⣿⣿⣿⣿⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠟⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢿⡿⢿⣦⢀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣶⡄⢠⣴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠓⠘⠋⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠚⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⢾⣿⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡤⣾⡧⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠐⠺⡿⠯⠍⠉⣀⣈⣁⣀⣀⣀⡈⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⡀⢀⠠⠈⠁⠀⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠧⠤⠀⠄⠀⠀⠀⠀⡁⠠⠼⠿⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠀⠀⠠⠠⠀⠠⠀⠤⠠⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠾⣯⢐⡀⠀⠀⠀⠛⠛⠛⠛⠋⠁⠀⠀⡐⢺⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⡐⠂⠀⡀⠘⠟⢾⠛⠛⠇⢀⣀⡒⡘⣿⠖
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠓⢶⡒⠃⠒⠐⠒⠒⠚⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠉⠁⠉⠈⠁⠉⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡈⠑⠦⢤⡄⢀⠀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣄⠀⠀⠀⣀⡤⠤⠒⠀⢤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢀⠀⠀⢠⣴⡎⠁⠀⠀⠀⠀⠀⢀⣉⡷
⠀⠀⠀⠀⠀⠀⣼⢷⡒⣤⣤⣤⣤⣄⣄⣤⣤⣤⣤⣴⠘⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣍⣿⣷⣶⣿⡯⢿⣴⣴⣬⣶⣷⣷⣎⣉⡁
⠀⠀⠀⠀⣠⣆⣿⠈⣿⣿⣿⣿⣾⣶⣶⡾⣾⣿⣷⣿⡟⢀⣌⣶⣄⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠸⣿⣿⣿⣿⡿⣿⣿⣿⣷⣧⣤⣼⣿⣿⣿⣿⣿⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣤⡀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣥⣽⠿⠿⠿⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣦⣬⣿⣷⡾⠿⠏

⠀⠳⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠑⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠳⠦⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣠⣠⣠⣀⣀⣄⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠶⠶⠶⡶⠶⠶⢶⠄⠀⠀⠀⠀⢀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡗⠀⢠⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠷⠄⠀⠰⠟⠁⠀⠘⣯⡀
⠀⠀⠐⠛⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡆
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠚⠒⠛⠒⠓⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠐⠤⣄⡤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠀⣀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⢠⡆⠀⠀⠀⠀⠠⣤⣠⡤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠁⠉⠉⠉⠉⠈⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣿⣋⣀⣀⣀⣀⣈⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣁⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⢋⣤⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡟⣛⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠛⠻⠿⣿⡿⠿⠿⠿⢶⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⣿⠋⠉⠉⢳
⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⣠⡞
⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⡄
⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠲⠖⠀⠁⠉⠛⣿⡿⠉⠉⠉⠙⠃⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠀⠀⠀⠀⠀⠀⢰⠸⠇⢀⠀⠄⠐⡀⠀⠀⡀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠁⡰⠃⠈⠀⠀⠀⠑⢀⡀⠠⠀⠠⠀⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⡀⠀⠀⠀⠃⠀⢁⠀⠀⠀⠆⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⣤⣦⣤⣤⢀⢀⡄⠠⠄⢠⣀⣠⣄⣠⣤⣀⢦⣀⣤⢰⣀⠄⣰⣄⣴⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠙⠋⠛⠛⣛⣛⣛⣛⣻⣿⣻⣟⣛⣟⣻⣏⣿⣿⣿⠛⠉⠙⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠛⠟⠛⠛⠋⠉⠉⠁⠁⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣆⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣶⣏⣛⣀⣀⢈⡉⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠷⠾⠿⢾⣷⣾⣆⣐⣲⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠘⠛⠓⠒⠚⠒⠒⠒⠒⠒⠶⠖⠒⠒⠶⠶⠒⢺⡏⠛⠉⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⢀⣶⣾⣿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⢀⣄⠀⣀⣀⣀⣄⣨⣄⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⡅⢀⠀⡀⠠⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠿⠿⠿⠟⢻⠋⠙⠉⠉⠙⠁⣽⣀⡀⣠⣼⡅
⠀⠀⠀⠀⠀⣇⢸⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⢸⡃⠰⠀⠀⠀⠀⢿⣧⠀⣺⣬
⠀⠀⢀⣀⣠⣯⣼⣀⣇⣷⣤⣶⣶⣾⣶⣿⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣶⣿⣶⣾⣿⣄⣿⣿⣦⣠⣄⡀
⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠛⣿⣭⣯⣭⣭⣯⣽⣿⣿⣿⡿⢿⣿⣿⣿⣽⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣶⣦⣤⣤⣌⣿⣢⣤⣀⣀⣠⣤⣠⣤⣭⣭⣭⣉⣽⣀⣤⣠⣤⣤⣤⣴⣿⣷⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠒⠚⠛⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠛⠛⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⢀⣰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡁⠠⢊⠋⠌⠉⢹⠉⠉⠌⠉⠂⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣀⣇⣸⣀⣆⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡁⢘⠀⡇⢸⠀⡇⠈⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣶⣶⣤⣅⣘⠀⣧⣾⡀⠇⣠⣤⣶⣶⣶⡇⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠛⣿⣿⣿⡟⢛⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠾⣶⠾⢿⣿⣷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⠂⢸⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣴⣤⣦⣴⣿⣦⣼⣤⣾⣶⣤⣦⣴⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠉⠁⠁⠈⠉⠉⠈⠀⠉⠈⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣧⣤⣾⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣿⠟⡫⠟⢡⠏⠀⢳⠙⢿⡻⣟⡷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡿⠋⡵⠊⢀⡞⠁⠀⡞⠀⠀⠈⡇⠀⠹⣄⠙⢦⠙⠿⡗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡵⠋⢀⡞⠁⢀⡞⠀⠀⢰⠃⠀⠀⠀⢹⠀⠀⠹⡄⠀⢳⡀⠈⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡞⠁⢠⡏⠀⠀⣼⠁⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⢳⠀⠀⢹⡄⠀⠱⡄⠙⣆
⠀⠀⠀⠀⠀⠀⢠⠏⠀⡞⠀⠀⡾⠀⠀⢀⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠘⡆⠀⠀⢷⠀⠀⢹⡄⠈⢧
⠀⠀⠀⠀⠀⢀⡏⠀⢸⠁⠀⢰⠇⠀⠀⢸⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⡆⠀⠀⣧⠀⠘⡆
⠀⠀⠀⠀⠀⢸⠀⠀⣼⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⣿⠀⠀⢳
⠀⠀⠀⠀⠀⢸⠀⠀⣻⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⢿⠀⠀⢸⡀⣀⣀⣸⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣇⣀⣀⣀⣻⣀⣀⢀⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠒⠛⣿⠿⠿⢿⠿⠿⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠿⠿⢿⡿⠿⠿⣿⠛⠒⣺
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣾⣶⣶⣶⣾⣶⣶⣶⣴⣷⣤⣤⣤⣤⣴⣷⣶⣶⣶⣿⣶⣶⣾⣷⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠼⣄
⠀⠀⠀⢀⣠⡤⡶⢞⠗⠗⢶⡦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⣦
⠀⢠⠖⡹⠋⡜⠀⠘⠀⢸⠀⠑⡌⠫⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣬⣿⣷⣄⠀⠀⠀⡀
⢠⣃⣼⡀⣼⠤⠤⡧⠤⠤⡤⠤⢵⡀⣱⣈⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣴⣿⡟
⡟⠲⡧⠤⡤⠤⠤⡧⠤⠤⡧⠤⠬⡤⠤⡗⣚⡦⠤⠤⡤⠄⢠⠆⠀⢤⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠇
⡇⠀⡇⠀⡆⠀⠀⡇⢰⠊⠁⠈⠉⠉⠉⠉⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⣡⣾⡿⠟
⡇⠀⡇⠀⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⣾⣿⡿
⡇⠀⡇⠀⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠉⠁
⡇⠀⡇⠀⡃⠀⠀⡇⢸⡀⠀⣀⣀⣀⣀⣀⠤⡗⠒⠒⠓⠀⠈⠁⠀⠉
⡧⠴⡗⠒⠗⠒⠒⡗⠚⠛⡟⠓⠚⠗⠒⡧⢤⡇
⡏⠉⡇⠐⡓⠒⠒⡗⠒⠒⡗⠒⠚⠓⠈⡏⠉⡇
⡇⠀⡇⠀⡃⠀⠀⡇⠀⠀⡇⠀⠀⡀⠀⡇⠀⡇
⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡖⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣶⢶⡾⢻⠻⣶⢶⣶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⡴⠋⢡⠏⠀⢸⠀⠈⢧⠈⠳⡌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⡾⠁⢀⡟⠀⠀⢸⠀⠀⠈⣇⠀⠙⣆⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣼⠁⠀⢸⠃⠀⠀⢸⠀⠀⠀⢻⠀⠀⢹⡄⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⡇⠀⠀⣼⠀⠀⠀⢸⠀⠀⠀⢸⡄⠀⠀⣇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⢼⡧⠤⠤⣿⠤⠤⠤⢼⠤⠤⠤⢼⡧⠤⠤⣿⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⢺⡗⠒⠒⣿⠒⠒⠒⢺⠒⠒⠒⢺⡗⠒⠒⣿⠒⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⠿⠿⠶⠶⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⡀⠀⣀⠀⢀⡀⠀⣀⠀⢀⠀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⡀⠀⣀⠀⠀⠀⠀⣀⠀⢀⠀⢀⢀⣀





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠿⢿⢿⢿⣿⡶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⡟⡉⡀⢸⢸⠁⡗⢳⢿⣝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡙⠰⠀⠀⠃⣼⢸⠀⢀⠂⡌⣏⣯⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⡆⢰⢸⠀⡟⠘⠀⠸⠰⠀⠏⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠁⠀⡇⢸⣿⡏⡏⠈⠈⣿⡯⢸⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡇⢸⣿⡇⡇⠀⠀⣿⡇⢸⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡀⠀⡇⢸⣿⡇⣧⠀⠀⣿⡇⢸⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⠀⠇⢸⣿⡇⣿⠀⠀⣿⣷⢸⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⡗⠖⡖⢾⣿⡿⣿⠲⠶⣿⣿⠶⠶⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠃⠀⡇⢸⣿⡇⡏⠀⡀⣿⣿⠨⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣤⡄⡇⢸⣿⡇⢹⠀⡇⣿⣿⠀⡄⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠛⠃⡇⢸⣿⡇⡄⠀⠇⣿⣿⠀⠃⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⣤⣄⣇⣼⣿⣧⣧⣤⣤⣿⣿⣬⣤⣽⣽⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣦⡀⠀⣾⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣺⡦⠴⠶⠖⠶⠶⢶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⡁⢦⡐⠢⡙⠊⠁⠈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⡄⢒⠀⠑⠀⠁⠀⠈⠂⠀⠀⠀⠀⠪⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠋⠉
⠀⠀⠀⠀⠀⠀⣰⣿⡴⠸⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⢠⣿⣄⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⠘⠋⠉⢿⡆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⠀⠄⢀⠀⠂⠀⠀⠀⠀⠄⠀⠀⡤⠀⠧⠀⠡⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⡀⠠⠀⠄⠀⠀⠀⢠⠠⠊⠄⠀⠹⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠸⣯⠄⢂⡀⣄⠘⠄⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠓⠀⠀⠀⠀⠈⢿⡄⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠹⣿⠊⢃⠈⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢾⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠄⢦⣸⣄⣐⣂⠀⢒⠀⠸⡬⠐⠀⠀⠀⠄⠀⢣⠀⢂⠙⡀⠜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡈⢄⠀⡀⠐⡀⠀⠀⠀⠀⠄⠀⡡⠀⠘⠀⠀⠂⠀⠀⠈⡈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣎⡈⠂⠐⡀⠠⠄⠈⡦⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⡈⠽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣔⡄⣆⠹⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣤⣴⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⡈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⠈⠄⠡⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⡈⡀⢠⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⢛⣉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⠷⠶⠖⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠠⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣶⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡖⠚⠛⠿⠿⢶⠃⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠊⢀⠆⠀⢀⠀⠀⠀⢀⠠⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⢀⡇⢀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⠊⠐⢠⠃⡐⠀⢸⠀⠀⠀⠀⠑⡈⠢⡀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⢀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢰⡀⠇⠰⣇⢠⣼⣠⠴⠢⣄⣢⠲⢀⣰⢄⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣠⡿⠤⠥⢨⡍⠫⠏⠈⠭⠙⠃⠿⠏⠭⠄⢬⠿⢿⡴⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⢸⠁⠘⠃⠘⠓⠒⠛⠒⠒⡟⠒⠚⡇⠘⡏⠉⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⡇⠀⠀⠐⠂⠀⢲⠒⠒⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⡇⠀⡠⠀⡀⡀⣸⡀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡄⠀⡄⠀⠀⡇⠀⠀⡇⠀⡇⠀⡇⠀⠀⠁⢸⠀⢨⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⠀⠀⡇⠀⡇⠀⡇⠀⡀⡀⢸⠀⠈⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠇⠀⡇⠀⠀⠃⠀⠀⡇⠀⡇⠀⡇⠀⠁⠉⢹⠉⠙⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢰⡇⢀⡧⠠⡤⠤⠄⠄⠼⠤⠴⠤⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠃⠀⠀⢀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢀⠀⠘⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⣸⡀⢀⡄⠀⣤⠀⣧⠀⢠⡄⠀⣼⢀⢸⣀⣸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⣮⣃⣈⡀⠀⡀⠀⢀⠀⢳⠀⢠⠀⠀⣚⠀⢸⠁⣹⠉⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠽⣢⣦⠾⢯⡭⣉⣉⣭⣍⣰⣬⣉⣭⣭⣩⣬⢥⡴⠳⣾⠀⠀⠀⢀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠈⠉⠉⠒⠛⠒⠒⠛⠁⠤⠤⠀⠘⠛⠓⠊⠉⢀⠘⠀⠀⠙⠿⠿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠶⠶⠶⠶⠶⠶⠒⠒⠘⠃⠉

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

⠀⠀⠀⠀⢀⣀⠠⠤⣄⠒⠀⠀⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⢀
⠀⠀⠀⠀⠈⢉⠄⣡⣀⣀⠂⠀⠀⠀⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⣀⣀⢀⠀⠀⢀⣀⣠⠤⠼⡛⡇
⠀⠀⠀⠀⠀⠀⠀⠆⡤⠀⠤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠩⣁⢀⣒⡒⣞⢋⠉⡽⠃⠀⠑⡘⠁⣧
⠀⠀⠀⠀⠀⢰⠆⡇⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢒⣂⠠⠁⢠⡀⢹⣀⠈⢀⢀⠀⠈⡰⠃⢸
⠀⠀⠀⠀⢘⠸⠀⡇⠀⠀⠀⢀⡴⠋⠙⠛⠷⣄⠀⠀⠀⠀⠀⠀⠂⠆⠈⠀⢸⠊⡀⡠⠐⢀⠔⠁⠄⠸⡆
⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⣸⢰⣯⠝⢣⢀⢹⡆⠀⠀⠀⠀⢸⣙⢛⢰⡄⢸⠔⠀⠈⠐⠋⠄⠂⠜⢄⡇
⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠸⣇⡩⠭⠀⠠⣿⠉⠀⠀⠀⠀⠀⠉⡛⣿⠃⢸⠁⢀⠀⠀⠀⠀⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠝⡚⡮⠔⠀⠀⠀⠀⠀⠀⠀⠛⠛⢠⢸⠀⠘⠀⠀⠀⡀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⢄⠀⠀⠀⠀⠀⡀⢰⠂⠀⠀⠀⠀⠀⠀⠄⠑⢹⠀⠀⠀⠀⠀⠀⠁⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣦⣱⣀⡀⣖⣡⣴⢟⠡⠀⠀⠀⣐⠚⡗⠀⠀⢐⠄⠂⠀⠀⠀⠄⠀⠂⢰⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠰⣿⣭⣻⠹⣿⡍⠀⠀⠀⠀⠀⠀⢀⡫⠀⠁⡥⠀⠀⢈⠀⠀⡀⣔⡔⠕⡇
⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠻⣿⢿⡾⠗⠃⠀⠀⣠⠢⠁⣤⣸⣔⢀⠀⡧⢀⡀⣑⢄⠀⠈⠐⢈⣼⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠤⢀⠀⠘⠘⣿⠀⠀⢠⢞⠁⠀⠀⣤⣍⡚⡧⠃⠗⠅⠂⠂⠂⠀⠄⠠⠈⣸⣗⠂⠀⠀⠒⠂
⠀⠀⠀⠀⠀⢥⣣⠄⠀⠀⠀⠀⠀⠉⠧⡽⠧⠴⢓⠁⡀⠀⡀⠀⣯⣻⠇⠀⡃⡀⢀⠔⠂⠀⠁⠈⣠⡿⡛⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠠⠀⣀⡀⢨⣴⠖⡽⡺⠹⠻⠲⠾⢇⢈⠵⣠⠏⠀⠉⡁⠀⢀⣠⡞⠋⠁⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠛⠏⠥⣠⣄⠀⡠⠈⠐⢉⠵⣫⢘⠑⢁⢀⣠⡶⡋⠉⠉⣁⠛⡣⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠺⠥⢼⣯⣸⣬⣶⣿⣝⣛⣤⠅⣭⠭⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠀⠄⠒⠊⠉⣿⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⢀⣀⣠⡅⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢻⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣡⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⠟⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣻⣿⣿⣿⣷⣾⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠁⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠿⠿⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣹⣁⣼⢿⣿⡿⠛⢩⢿⠟⠛⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠻⠿⣅⣛⣋⡹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢦⣤⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣴⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⣻⠏⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣶⣄⡀
⠀⠀⠀⣠⣤⡠⣲⡿⣫⡾⢁⣾⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢸⡟⣼⢇⡄⡀
⢀⣠⣾⣿⣿⢃⣯⣾⠏⣠⡿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣶⡿⣖⣤⣾⣿⣿⣿⣿⣿⣧⣿⡸⣧⣿⣤⡀
⣿⣿⣿⣿⣿⣼⣧⣿⢸⣿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡜⣷⡿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣿⡼⣧⢶⣄
⣿⣿⣿⣿⣿⣿⡟⣇⣿⣟⢞⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⡙⢾⣙⢶⣽⣿⣿⣿⣿⣿⣾⣿⣟⢿⣾⣷⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡱⣟⢿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡨⠦⠡⡀⠀⠀⢢⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠢⣢⠄⠀⢰⠦⠀⢀⢀⡝⣄⠛⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⡇⠀⠀⠀⠀⢀⠃⠀⠀⠀⠈⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠤⠄⠈⠁⢠⡐⢄⡈⠀⠀⠀⠀⢸⡎⠙⢢⡐⠄
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⡀⠀⠨⠈⠀⠤⠤⠤⠤⠄⠀⠆⢸⡦⣕⢄⠀⠀⠀⠸⠀⠌⣲⠉⠈⠀⠄
⠀⠀⠠⠤⠀⢀⣀⡃⠀⠀⠀⠀⠀⢰⠄⠀⠠⠤⠤⠤⠤⠤⠤⠄⠄⠀⡌⠩⣀⠷⣥⠄⠀⡆⠀⠀⡆
⠀⠀⠐⠠⡀⠀⠀⠀⠈⠉⠉⠐⠂⠒⠤⠤⣄⢀⣀⠀⠀⠀⠀⠀⠠⠀⠉⢄⠈⠳⡜⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠑⢌⠉⠉⠉⠉⠁⠐⠒⠒⠂⠤⠄⣉⡉⠓⠂⠤⣀⡀⠘⢀⠨⠑⢄⠈⠢⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⠌⣉⠑⠢⠬⣠⠑⢄⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠉⠐⠂⠠⠤⢈⣁⠒⠠⢍⡓⠷⣤⡈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠈⡉⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠬⠉⠒⡂⠭⠒⣩⠶⣌⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠁⠄⣐⠒⠠⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣆⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⣀⣘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠋⣃⢸⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⡐⠴⠖⣲⠚⠋⠋⠛⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⠨⢯⣀⣀⣤⣤⠤⠶⠞⡒⡉⡩⠻⠢⠊⠉⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⣀⣀⣀⡀⡁⠀⠀⠀⠩⠛⠉⠀⠀⢀⠤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⢀⡀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣃⣠⣤⣀⢀⠀⡀⣖⣿⣭⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣷⣶⣮⡝⣧⢾⣱⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠛⣿⡮⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣿⣽⣟⣴⣿⢷⡿⠿⠋⡏⠈⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣿⣿⣿⣏⠉⠉⢁⣤⣶⣤⡀⡇⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⣿⣿⡄⠀⢿⣷⣷⣿⡟⡃⠀⣸⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⣤⠁⢸⣍⣻⣦⣌⠙⠋⠁⢀⣧⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣤⠾⣻⣇⠘⡿⢷⣴⠖⠿⡏⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣋⠠⡿⢺⣾⣁⠀⠀⠀⢠⡇⢀⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠸⠟⣩⣧⣥⣬⣤⡔⠀⢀⡊⣃⡤⠋⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⣿⡿⢿⣿⡷⠀⣸⣿⣿⣟⣦⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣟⣿⡟⠻⠿⢯⡿⠿⠶⠄⠿⠝⠿⠷⠏

⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠚⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠙⠛⠓⢦⡀
⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⣰⣞⣁⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸
⡇⠀⠀⢠⡖⠒⠒⠒⠒⠒⠚⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠀⠈⣿⠀⠀⢸
⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸
⣧⠀⡴⢺⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⣿⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸
⣿⠀⢇⣸⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠚⠓⠲⣤⡀⠀⢸⡇
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡠⠶⠒⠶⣄⠈⠹⡄⢸⡇
⢿⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡎⡴⠋⠉⠙⣎⢳⠀⣷⢸⡇⠀⢀
⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⣇⢧⣀⠀⣀⡟⣸⠁⡟⢸⡇⠀⢸
⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠳⠭⡭⠥⠞⣡⡾⠁⢸⠀⠀⢸
⢸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠶⠖⠛⠁⠀⠀⢸⠀⠀⢸
⢸⡇⢀⣤⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠸⡇⢸⡇⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⡇⠘⡇⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⣿⠀⠓⢻⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⢸⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⢸
⠀⢸⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠘⠧⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠶⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢰⣿⠁⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠟⠁⣴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠻⠋⠀⠀⠀⠀⠀⠛⡛⠛⠃⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣈⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠿⠇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡴⢾⣧⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣇⡏⠘⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡀⢸⠛⣦⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⠷⣿⡴⠏⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣀⠀⠀⠐⠃⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦
⠀⠀⠀⠀⠀⢀⣾⣿⠻⢿⣷⣶⣤⣤⣤⣴⣶⣿⡿⠟⠻⢿⣿⣶⣦⣤⣤⣤⣶⣾⡿⠟⢿⣷⡀
⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠈⠉⠉⠛⠋⠉⠉⠀⣤⣤⣤⣤⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠈⣿⣷⡀
⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣿⣿⣿⣿⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⣠⣤⠄⢠⣤⡀⢠⣤⡤⢠⣤⡤⣤⣤⡤⣤⡀⣤⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⣛⣷⡆⣾⣿⣧⢸⡷⠖⢸⡗⡂⠀⣿⠀⠈⣿⠃⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠉⠉⠀⠉⠀⠉⠈⠁⠀⠈⠉⠉⠀⠉⠀⠀⠉⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⢸⣯⡅⣿⣾⣽⡇⢿⣍⠉⣿⠁⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠘⠃⠀⠛⠙⠈⠓⠛⠋⠀⠛⠀⢀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣤⣤⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢱⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢉⠀⠀⠜⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣨⠤⠉⠐⠊
⠀⠀⠀⠀⠀⢇⠂⠔⠊⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⣀⠀⠀⠀⠀⠀⠀⠀⠎⢦⣀
⠀⠀⠀⢀⡀⠊⠀⠀⠐⠣⣀⠀⠀⣀⠏⠂⠐⡦⠀⠀⠤⢊⠀⠠⣫⠀⠀⠀⠀⠀⠰⢥⢀⣐⣱⠄
⠀⠀⠀⠀⠀⠀⠱⡄⠌⠀⠉⠀⠀⠉⢢⡀⡼⠅⠀⠀⠀⠸⠎⠀⠁⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠉⢁⡀⠀⠀⠀⠀⠀⠉⠀⡀⠤⠄⠠⠄⡀⠀⠀⠀⢰⠲⢄⢀⣤⠆⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡰⠘⠀⠰⡄⠀⢀⠔⠁⠐⠀⠀⠀⠀⠈⢦⠀⠀⠀⠄⠀⠁⠮⢀⠀⠀⡀⠜⠀⠱⢤
⠀⠀⣀⠀⠀⠀⠈⠓⢦⠀⠀⡞⠀⠀⠆⠀⠀⢀⡠⠖⠲⠤⡀⠀⢳⠰⠧⠠⠀⡠⠄⠒⠃⠈⠑⡆⢠⡈⣘
⢀⠼⠁⠓⡖⠀⠀⠀⠸⠴⠉⠓⠀⠀⠀⠀⢀⡞⠀⣀⣀⣀⣸⠀⠀⡀⠀⠀⠢⠁⠀⠀⠀⠀⠀⠖⠈⠀⠈
⠉⢰⠐⠒⠚⢀⠔⠒⠂⠐⠒⠒⠃⠈⠁⠀⠘⠉⠉⠉⠉⠉⠉⠄⠀⠉⠓⠒⠒⠀⠀⠄⠤⠀⠀⠀⡖⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢬⠀⠀⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔

⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠀⡇⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡄⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⡧⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢳⠇⠀⠀⠀⠀⢀⡾⣫⠷⠶⠾⣝⢷⡀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⣼⢹⡇⡎⠉⢱⢸⡏⣧⠀⠀⠸⣿⡏⣿⡿⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠸⣞⢷⣕⣒⣪⡼⣳⠇⠀⠀⠀⠈⡇⡏⠁⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠙⠳⠿⠭⠿⠞⠋⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⡗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⢸⢸
⠀⠀⠀⠀⠀⠀⣇⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞⣸
⠀⠀⠀⠀⠀⠀⠈⢹⠉⠉⠉⢉⣹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣏⠉⠉⠉⠉⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠴⠃

⢰⠒⢖⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⢤⣀⡀
⢸⠀⠀⠒⠉⢑⡲⠤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠈⠉⢒⣢⣤⣀⡀
⢸⠀⠀⠀⣠⢞⢍⡟⠂⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⣉⢽
⢸⡇⠀⠀⢀⠄⠁⠀⠀⠀⡇⠀⡤⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣿⠀⠀⡄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⣨⠿⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⡇⠀⡧⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠷⣤⣤⠄⡇
⠀⡇⠀⠀⠀⠀⠀⣾⣡⠀⡇⠀⡇⠀⠀⢠⡏⠀⢀⡞⢀⣷⠊⠁⠈⣽⢲⡀⠀⠻⡄⠀⠀⢠⣿⢠⣤⡇
⠀⡇⠀⠀⠀⠀⠀⠊⢸⠀⡟⠛⣧⠀⠀⢸⡁⠀⠘⣏⠷⣧⡀⠀⠀⢽⡽⠃⠀⠀⡇⠀⠀⢸⡟⠀⢸⡇
⠀⠃⠀⠀⠀⠀⠀⠠⢸⠀⡇⠀⣿⠀⠀⠀⢳⡀⠀⠀⠀⠀⠉⠻⠿⠏⠀⠀⠀⡾⠁⠀⠀⠀⡇⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⣧⣄⣾⡄⠀⠀⠀⠙⠢⠄⣀⢀⡀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠙⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⢸⠃⠀⠊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⢸
⠀⠐⣤⣀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠒⢺⡀
⠀⠀⠈⠛⠿⠶⣤⣀⠼⠀⡀⢀⢘⠇⠀⠀⠀⠓⠒⠒⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣓⣖⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇⣙⠮⠤⡤⠤⠤⠤⠀⠀⠒⠒⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠛⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⢭⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⢰⣮⠀⠀⠲⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠁⠀⠐⠉⣠⣾⣯⣼⡤⠀⠀⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⡙⠓⢴⡄⢀⡴⠞⠁⠁⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢱⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢉⠀⠀⠜⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣨⠴⠋⠐⠊
⠀⠀⠀⠀⠀⢇⠀⠔⠈⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⣀⠀⠀⠀⠀⠀⠀⢀⠎⢦⣀
⠀⠀⠀⢀⡀⠊⠀⠀⠐⠧⣀⠀⠀⣀⠏⠂⠰⡦⠀⠀⠤⢊⠀⠠⣫⠀⠀⠀⠀⠀⠰⢥⢀⣀⣱⠄
⠀⠀⠀⠀⠀⠀⠙⡄⠌⠀⠉⠀⠀⠉⢲⡀⡼⠅⠀⠀⠀⠸⠊⠀⠁⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠉⠀⡀⠤⠤⠠⠄⡀⠀⠀⠀⢰⠒⢄⢀⣤⠆⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠰⠘⠀⠰⡄⠀⢀⠔⠁⠐⠀⠀⠀⠀⠉⢦⠀⠀⢀⠄⠀⠁⠮⢀⠀⠀⡀⠜⠀⠱⢤
⠀⠀⣀⠀⠀⠀⠈⠓⢦⠀⢀⡎⠀⠀⠆⠀⠀⢀⡤⠖⠒⠤⡀⠀⢳⠰⠧⠠⠀⡤⠄⠒⠃⠈⠑⡆⢠⣌⣈
⠀⠘⠁⠓⣖⠀⠀⠀⠸⠴⠉⠛⠀⠀⠀⠀⢀⡞⢀⣀⣀⣀⣸⠀⠀⡀⠀⠀⠢⠁⠀⠀⠀⠀⠀⠒⠉⠀⠈
⠉⢰⠔⠒⠚⢀⠔⠒⠀⠒⠒⠒⠃⠈⠁⠀⠘⠉⠉⠉⠉⠉⠉⠄⠀⠁⠓⠒⠒⠂⠀⠠⠤⠀⠀⠀⠖⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢨⠀⡀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⢈⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢉⣀⠙⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣼⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣻⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⣧⢸⡅⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣿⠷⡿⢼⡷⢷⡼⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⠟⢋⣡⠖⢁⠜⠀⢸⠃⠘⢧⡈⠛⣭⡙⣳⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⣠⣴⠶⠟⣋⣭⡭⠛⢛⡿⠟⠛⠓⡾⠟⠓⠓⢺⡿⠓⠛⠻⣿⠉⠙⠻⣤⣉⠛⠻⢯⣭⣍⠉⠛⠷⢶⣦⣀
⠀⠀⣠⡶⠟⠉⠀⣠⠞⠉⠀⠀⣴⠋⠀⠀⠀⣸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⡀⠀⠀⠙⢷⡀⠀⠀⠉⢳⣆⠀⠀⠈⠙⢷⣆
⢠⡾⠁⠀⠀⠀⣼⠋⠀⠀⠀⢠⡟⠀⠀⠀⢰⡇⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠸⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣿
⢸⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⣧⣀⣀⡀⢀⣀⣿⣀⡀⠀⣀⣀⣧⣀⣀⣀⣀⣹⣄⣀⣀⣀⣸⣇⡀⠀⠀⢠⡿
⠈⠻⣦⣀⣠⡤⢻⣧⠘⠛⠛⠙⡟⠀⠉⠃⠈⢿⡄⠀⠀⠀⠀⣿⠁⠈⠉⠀⢈⣏⠀⢀⣀⣠⣍⣁⣈⣀⣨⣏⣁⣙⣷⣾⡏
⠀⠀⠉⢹⡏⠛⠛⢻⡞⠛⠛⠛⢻⡟⠛⠛⠛⠻⣷⠿⠟⠛⠒⠛⡟⠛⠛⠛⠻⡟⠈⠉⠉⢹⡏⠉⠉⠉⢹⡏⠉⠉⣿⢹⡇
⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⢀⣀⣀⣿⡤⣤⠤⠤⢤⣧⢤⣤⠄⢰⣧⠀⠠⣤⣼⣷⡄⣀⣀⣸⣇⠀⠀⣿⢸⡇
⠀⠀⠀⠸⣇⢀⡄⢸⡷⠒⠀⠀⠙⡏⠁⠈⠙⠻⣿⣧⣶⣶⣦⣤⣧⣀⡀⠀⢠⣯⠀⠀⠀⠀⣧⠀⠈⠉⢹⡟⠛⠒⢿⣾⡇
⠀⠀⠀⠘⣿⣥⣄⣸⣇⠀⠀⠀⢸⣧⠀⠀⠀⠀⣾⠀⠉⠙⠛⠻⣿⠻⢿⣷⣾⣿⣄⡀⠀⠀⢹⠀⠀⠀⢸⣇⠀⣀⣼⣾⡿
⠀⠀⠀⢸⣧⠀⠈⠙⣟⠛⠛⠳⠾⣿⠶⠶⠶⠶⣿⡶⢶⡶⠤⢠⣿⡤⠤⠤⠾⣿⠿⠿⠛⠛⢿⠛⠛⠛⢻⡟⠋⠉⢹⠄⣿
⠀⠀⠀⠸⣿⠀⠀⠀⣿⠀⠀⠀⢰⣧⠀⠀⠀⠀⣿⡀⣠⣾⠟⠠⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⣿
⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⢹⡀⠙⢣⡀⠀⢸⠓⠤⣄⠀⢿⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⢿⡄
⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠠⣿⠀⠀⠀⠀⢸⡅⠀⠈⢿⠀⢻⣦⣀⠈⢳⣸⡇⠀⠀⠀⢸⡇⠀⠀⠘⣧⠀⠀⠸⡇⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠸⣆⣸⡆⠙⠓⣀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⣟⠀⠀⠘⡇⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢿⣯⡄⠀⠛⢻⣯⣄⠀⠀⢸⡷⠀⠀⠀⣿⠀⠀⠈⣏⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⢀⡀⢈⡇⠀⠀⢀⠀⢸⡏⢹⣿⡿⢿⣿⡿⣧⣀⣀⣧⡀⠀⠀⢻⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⣿⠆⢀⡀⣼⡧⠶⠶⠀⢿⣿⢛⠉⢛⣻⡟⠀⠀⠀⠀⢸⣷⢿⡇⢈⣿⡟⠷⣿⠁⠈⣿⠀⠙⠃⢻⡷⢶⣴⣷⣸⡇
⠀⠀⠀⠀⢿⣶⣁⣀⣸⣆⠀⡀⠠⢾⣿⠏⠁⠈⣿⡇⠀⠀⢠⣼⣿⣃⣿⡻⠟⠛⣿⡔⢣⠀⠀⣿⠀⠀⠀⢸⡇⢀⣙⣷⣿⡇
⠀⠀⠀⠀⣸⡏⠉⠙⢿⡉⠛⠁⠤⣼⡿⣼⣿⣤⣿⣧⣤⣸⣿⣿⢻⣯⣭⣁⡀⣤⣿⣶⣤⣶⣤⣿⡤⠶⠶⢿⡟⠋⠉⣧⢸⡇
⠀⠀⠀⠀⢿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣷⠘⢿⠀⣿⡿⠛⠉⠉⠁⢸⣇⣀⠈⠀⢀⣀⡀⠀⠀⣀⣀⡀⣀⡀⣼⣧⠴⠺⣷⢸⡇
⠀⠀⠀⠀⣸⡇⠀⠀⢸⡇⠀⠀⠀⠈⣿⡇⣿⠂⣿⣿⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⣿⠉⠉⠉⠉⣿⠉⠁⠁⢸⡄⠀⠀⢻⣿⡇
⠀⠀⠀⠀⢿⣇⠀⠀⢸⡇⠀⠀⠀⢀⣟⠉⠛⢶⢙⣿⣀⣀⣀⣤⣼⣿⣄⣤⣤⣀⣿⣄⢄⣠⠀⣿⣤⣤⣤⣼⣧⣀⠀⢸⣧⣿
⠀⠀⠀⠀⢨⡟⠀⣀⣰⣷⠖⠀⠛⠛⣟⠉⠁⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠀⢨⡏⠈⠛⢿⣅⣿⡀
⠀⠀⣀⣰⣿⣿⣯⣁⡀⠇⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠀⠀⠀⢸⣇⠀⢤⣈⣽⣿⣿⣿⣦
⢀⣾⣏⢿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣤⣤⣤⣤⣤⣤⣥⣤⣄⣠⡄⢀⣼⡀⢀⣀⡀⠿⠤⢤⣤⣤⣤⣤⣶⣿⣯⣤⣾⣿⣿⣿⠿⢃⣾⣿
⠘⠿⣿⣷⣯⣝⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣋⣥⣶⣿⠟⠃
⠀⠀⠈⠉⠛⠿⣿⣿⣶⣶⣤⣭⣭⣭⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠛⢛⣋⣉⣉⣭⣭⣽⣶⣶⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣷⣮⣍⣉⣙⣻⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⠿⣟⣻⣭⣭⣷⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⠶⠶⠚⠛⠛⠻⠿⠶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠶⢤⣀
⢠⣤⣶⣚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠋⠉⠙⡆
⢸⡀⠉⠙⠻⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣧
⠸⡇⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠛⠉⠁⠀⠀⢀⣀⣠⣾⠷⠿⠛⠛⡇⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠋⠉⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢿⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠄⣀⠤⠴⠖⠚⠋⠉⠁⠀⠀⣀⣠⠤⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⣿
⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠶⠦⣄⠀⣿⡇⠀⣿
⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣡⠴⠲⠶⣾⣆⣿⡇⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⣸⢣⡏⠀⢀⣌⣿⣿⡇⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠇⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠅⣿⣸⣷⠀⠸⡿⢸⣿⡇⠀⣿
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣅⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣧⡐⠘⢷⣝⡳⢦⣶⠄⢻⡇⠀⣿
⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⡀⠉⠙⢛⡡⠀⣾⡇⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⠿⠋⠀⠀⣿⠀⠀⢻
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠀⡶⠀⠀⠀⠀⠀⢀⣠⡴⠿⠁⠀⢸
⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢠⠆⠀⠀⠀⠀⠀⠀⠀⣀⣔⣾⣉⣁⣶⠿⠞⢋⣀⣠⣴⠶⠛⠉⠀⠀⣀⡀⠤⡿
⠹⣮⡛⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠋⣉⣩⣤⣶⠿⠟⠋⣉⣤⣴⣶⣶⠿⠛⠉⣀⣴⠃
⠀⠙⠿⣦⣉⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢧⡸⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠛⠋⣉⣤⣶⣾⣿⣿⡿⠿⠿⠷⢷⠿⠛⠉
⠀⠀⠀⠈⠙⠛⠋⠉⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡀⠀⢀⣀⣤⠶⠟⠋⠉⢀⣠⣴⣾⣿⣿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣤⡂⠀⠀⠀⠀⢸⡇⠀⠀⠸⠧⠞⠋⠁⠀⣀⣤⣴⣾⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢦⣀⠀⠀⢸⠇⠀⠀⠀⠀⣀⣤⡶⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣉⠓⠦⢈⠀⢀⣠⠴⠚⢉⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⣶⣬⣋⣡⣴⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠀⠀⡀⠠⣤⣔⡱⠎⣣⣤⠄⢀⠀⠀⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠶⢪⠛⠧⣫⡊⡀⢕⡪⢀⢑⣝⠼⠛⡕⠶⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⡶⡋⠡⣢⣇⡆⡥⠵⣼⣟⣻⣧⠮⢌⢰⣸⣔⠌⢙⢶⡂⡀
⠀⠀⠀⠀⠠⢀⡤⣄⡼⠫⢱⠖⣁⣻⣽⡿⣗⡤⣯⣍⣩⣽⢤⣲⢿⣭⢟⣈⠲⡎⠝⢧⣠⢤⡀⠄
⠀⠀⠀⠀⠀⠘⡨⣃⠡⣢⣧⣸⣅⣹⣵⡶⠏⣧⣿⡮⢵⣿⣼⠹⢶⣮⣏⣨⣇⣼⣔⠌⣐⢇⠃
⠀⠀⠀⠀⠀⠠⠿⡄⠩⠽⣷⡿⢱⡿⣴⡿⣧⣞⡞⢽⡯⢳⣳⣼⢿⣦⢿⡎⢿⣾⠯⠍⢠⠿⠄
⠀⠀⠀⠀⠀⣦⡫⠺⢅⠎⠙⡽⠯⣥⣩⢿⣅⣽⣾⣿⣿⣷⣯⣨⡿⣍⣬⠽⢯⠋⢱⡨⠗⢝⣴
⠀⠀⠀⣠⢔⡙⢄⢌⣶⢿⡟⢿⡺⡿⣞⣍⣾⣿⣿⣿⣿⣿⣿⣷⣹⣳⢿⢗⡿⢻⡿⣶⡡⡠⢋⡢⣄
⠀⠀⠀⠙⠪⣡⠊⢊⠿⣾⣇⣾⢵⣷⢯⣃⢿⣿⣿⣽⣯⣿⣿⡿⣙⡽⣾⣮⣷⣸⣷⠿⡑⠑⣌⠕⠋
⠀⠀⠀⠀⠀⠞⣕⢴⡊⣇⣠⣳⣖⡛⣙⣾⡋⣻⢿⣿⣿⡿⣟⢙⣷⣋⢛⣲⣞⣄⡰⢑⡦⣪⠳
⠀⠀⠀⠀⠀⠐⣶⠃⣐⣲⡿⣷⡰⣷⠻⣷⡟⢯⢥⣺⣗⡼⡽⢻⣾⠟⣾⢆⣾⢻⣖⣂⠘⣶⠂
⠀⠀⠀⠀⠀⢠⢱⠍⡐⡝⡟⢹⡋⣹⡻⠷⣆⡟⣿⢗⡺⡿⢻⣰⠺⢟⣏⢙⡎⢻⢫⢂⠩⡊⡄
⠀⠀⠀⠀⠐⠈⠓⠋⢳⣔⡸⠦⣉⣵⣻⣷⡯⠓⣟⣋⣙⣻⠚⢽⣾⣟⣮⣉⠴⢇⣢⡞⠙⠚⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠷⣅⡒⠝⡏⠇⡑⡲⢻⣯⣽⡟⢔⢚⠸⢹⠫⢂⣸⠾⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠖⢜⣤⡖⣝⢅⠁⡪⢕⠈⡨⣫⣲⣤⡣⠢⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠀⠁⠐⠛⠫⡰⢆⠝⠙⠂⠈⠀⠀⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠞⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡓⢋⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡿⠾⠏⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣷⣶⣿⠙⣧⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠛⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣙⣿⣿⣽⣿⣿⣯⣽⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⡵⠋⡙⠂⠀⠀⠛⡍⠳⣍⠛⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣬⣿⣿⣿⡛⠶⠙⡳⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡴⠋⠀⢠⠃⠀⠀⠀⠀⢡⠀⠈⠳⡀⠙⢦⠀⢀⣴⡶⣿⣿⣿⠉⠉⠉⣿⣶⣿⣿⣿⣟⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡾⠁⠀⠀⡘⠀⠀⠀⠀⠀⠘⡀⠀⠀⠹⡀⠀⢣⡀⠽⣿⣿⣿⣿⣧⣀⣼⠟⢻⣿⣡⣿⡟⢋⣿⢻⣆
⠀⠀⠀⠀⠀⠀⢰⠁⠀⡘⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⡀⠀⢶⠐⢛⣿⣿⣿⡿⠛⠁⠀⠘⣿⣿⣿⣅⣿⢉⡾⢿⣷⡀
⠀⠀⠀⠀⠀⠀⣆⠄⢠⡇⣤⢠⠄⣠⠃⡤⠀⣤⡄⢠⡄⢧⡀⢤⢠⡈⣧⠠⢠⡄⠈⠹⠟⠁⠀⠀⠀⠀⠀⠈⠈⢻⣿⣿⣾⣧⣾⡏⢻⡄
⠀⠀⠀⠀⠀⠀⣿⠶⢤⠦⠶⢤⠦⣴⠧⡴⠤⢴⠤⠶⡤⢶⡤⢦⠴⠤⠦⠤⢬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠻⠿⣿⡟⣿⡿⣆
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢰⠀⠀⠀⢰⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠸⠀⠰⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡟⠛⢻⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⢻⠛⠛⠛⠛⢻⠛⠛⣷⠛⠛⠛⠋⠈⠁⠅
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⢺⣴⠦
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠂⣉⣤⣷⡀
⠀⠀⠀⠀⠀⠀⡿⠒⠺⠒⠓⠚⠒⠿⠒⠗⠒⠺⠒⠒⠗⠺⠃⠺⠲⠒⢿⠐⢺⡟⠲⠶⠒⠂⠤⢠⠄⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢨⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢘⠀⠀⠀⠘⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢠⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛





⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢀⡠⣴⡖⢲⣦⢄⡀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣰⠻⣷⣘⡧⢼⣃⣾⠿⣆⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⡿⠷⢾⠁⠀⠀⠈⡷⠾⢿⡀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠿⢻⣄⠀⢀⣠⡛⠻⣿⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠘⢾⡟⣰⡏⢹⣆⢻⡷⠃⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠈⠙⠓⠚⠋⠁⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣴⠟⠉⠉⠉⠉⠉⠉⠉⠉⠻⣦⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⢸⣿⣿⣿⢶⣴⣤⣶⣠⡲⣽⣿⣿⡇⠀⣼⣿⡯⢶⣤⣤⣴⣴⡶⣽⣿⣧⠀⢸⣿⣿⣯⢖⣤⣤⣦⣤⡶⣟⣿⣿⡇
⠀⢸⣿⣿⣧⣿⠋⣤⣌⢹⡇⣽⣿⣿⡇⢸⣿⣿⡇⣹⡏⣠⣄⢹⡟⢸⣿⣿⡇⢸⣿⣿⡯⢸⡏⣡⣄⠙⣿⢸⣿⣿⡇
⠀⢸⣿⣿⣟⣿⣦⣉⣁⣼⡇⣿⣿⣿⡇⢸⣿⣿⡏⢽⣧⣈⣁⣼⡗⢸⣿⣿⡇⢸⣿⣿⣏⢸⣧⣈⣉⣴⣿⢸⣿⣿⡇
⠀⢸⣿⣿⢽⣮⢭⣩⣍⡭⣿⠿⣿⣿⡇⠈⢿⣿⡿⣮⢍⣭⣭⡭⣽⢿⣿⡿⠁⢸⣿⣿⡿⣿⢭⣩⣭⡩⣵⠧⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠘⠋⠛⠛⠛⠛⠛⠛⠉⠋⠛⠛⠛⠁⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠛⠋⠉⠁
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⢴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡶⢀⡀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⣠⠦⣀⡀⠰⠎⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⢰⣦⡀⠀⠀⠀⢀⠀⠀⢀⡴⠀⡀⠀⠀⠁⢕⡂⡴⠃⠀⠀⢸⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣉⢀⡀⠤⢐⠈⢻⠁⣼⣿⠃⠀⠀⢀⡌⠀⢠⣾⣿⣿⣦⡀⠉⠁⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠈⠢⡀⢥⠀⠔⣁⣠⣤⠀⡴⠉⠀⠀⣿⣯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠀⠀⠠⠆⠀⠀⠀⠁⠨⣶⣿⣿⣿⠑⠿⠀⠀⠀⠀⠈⠓⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠀⠀⠀⠀⠀⢠⣤⣞⣿⣿⣿⣿⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢠⡄⠀⠀⠀⠀⠀⡿⠀⠉⠻⠿⠟⢻⡇⠈⠀⠀⠀⠉⢢⣀⣀⡀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠢⣄⠀⢠⣴⠞⠛⠁⠀⠆⠀⡄⠀⣸⠃⠀⠀⠀⡀⠀⠀⠙⠈⠀⠀⠀⠈
⠀⠀⠀⠠⣄⠀⠀⠀⡀⠀⠀⠑⢄⠀⠁⠀⠙⢦⡀⠘⠰⣿⡿⠠⠄⠈⢳⣄⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠛⠀⠀⠀⠁⠀⠀⢶⡒⠀⠀⠈⠀⢰⡇⠀⠃⠀⠲⠀⣠⣤⠶⠛⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⡧⠶⠞⢦⣀⢠⡟⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢚⠉⠙⢋⡿⠁⠀⠀⠐⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠈⡄⢘⡆⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⢲⠷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣩⠞⢙⠏⡎⠈⢂⠘⠻⢻⡅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣸⢏⠎⡞⢰⠁⠀⠸⠀⠀⢠⡣⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⡀⣰⠃⠀⢰⠁⠸⠀⠀⠀⠀⠆⠀⣇⠱⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣰⢁⡏⠐⠀⠀⠀⡇⠀⠀⠐⠀⢸⠀⠹⠀⡆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠃⢸⠁⠀⢠⠀⠀⡇⢠⠀⠀⠀⢸⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠐⠀⡿⠀⠀⢸⠀⢠⠁⢸⠀⠀⠀⠈⠀⠸⠀⠸⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⢀⡇⢀⠀⠈⠀⠀⠀⢸⠀⢀⠀⢸⠀⠈⠀⢰⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⢀⠀⠸⠀⢸⠀⢠⠀⢨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⢸⡇⡸⠀⡇⠀⠀⠀⠀⠀⠂⠀⠈⠀⡞⠀⢸⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢸⠇⠁⠀⡃⢀⡁⣀⣀⢀⡇⠀⡃⢀⣇⠀⡘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠅⢸⠂⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⢤⡁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⡄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠒⡤⠿⠄⣀⣀⣀⣀⣀⣀⣇⣀⣀⣠⣇⣀⣠⡤⠧⠖⠊⠙⣁⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠉⠑⠲⠒⠒⠒⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠲⠤⠤⢴⣒⣐⣚⣀⣀⣀⣀⡠⣤⠤⠶⠾⠟⠛⠉⠉

⠼⣆⣄⣿⣿⣿⡏⣿⣽⣿⣿⣿⣷⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣿⣿⣿⣿⣿⣿
⢀⣿⣹⣿⣿⣿⣿⣿⣿⣿⠃⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⣸⠃⢹⣿⣿⣿⣿⣿⣿⣏⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣻⣿
⠟⢀⣾⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣿⣿⣿⣿⣿⣿⣿
⢠⣿⣿⣿⣿⣿⣿⣿⢋⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠟⠛⣛⡛⠻⢿⣿⣿⣿⣿⣿⣯
⠏⣿⣿⣿⣿⣿⣿⣿⣾⣇⣠⣤⣶⣾⣷⣦⣤⡀⣀⣀⣀⣀⣀⠔⢋⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠿⣋⣩⣭⣤⣬⡉⠛⣿⣿⣏⢀⣀⡹⠇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠀⣹⣿⠉⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿
⣸⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⡀⡹⣿⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣿⣿
⢾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⠃⣿⣿⡿⠷⢿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣿⣿
⠼⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣤⠄⣼⣿⣿⣴⣤⣾⣿⣿⣷⣭⣝⣻⣿⢋⣩⡴⠿⣿⣿⣿⣿⣿
⠠⣶⣿⣄⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠟⣃⣼⣿⣿⣿⣿⣿⣿⠟⣿⣿⠏⠈⢈⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠁⡟⢱⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⡶⣾⠿⠛⢷⡿⠇⠈⠉⢉⠀⠈⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠰⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⠀⠀⠀⠀⢀⣠⣬⣽⣶⣦⣤⡀⠀⢸⡇⠀⠀⠀⣺⣿⣿⣿⣿⣿
⠀⠀⠎⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⣰⣿⣿⠿⠿⠛⣛⣿⣿⣷⣸⡇⠀⠀⠀⡼⣿⣿⣿⣿⡿
⠀⠀⠀⠈⢿⣿⡹⢟⠛⣿⣿⣿⣿⣧⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇⠀⠀⠀⠀⣰⣿⠿⣿⡟
⠀⠀⠀⠀⢠⠿⢀⡤⠈⠉⡿⣿⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠛⠁⢸⡇⢀⡀⣨⣾⡏⠉⠀⠸⡗
⠀⡀⠀⠱⣤⡀⣴⠀⠀⠀⣥⣟⠈⠻⣿⣶⣤⣄⠀⠀⠀⣴⠛⠛⠉⠀⠀⠀⢸⡇⣩⣾⣿⣽⠁⠀⠀⠀⠐
⠔⠀⠀⢆⣼⣿⣿⠀⠀⢰⣿⣿⣷⣶⡾⠿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⡀⣀⣼⣷⣿⠋⠟⠁
⠀⠀⠀⣼⣿⣿⠏⣀⡀⣼⡿⡿⢻⣿⡇⠀⠈⣿⣿⣿⣯⣄⣀⣀⣠⣴⣾⣿⢿⡏⠁⠀⠀⣾⡀
⠀⠀⠀⣿⠛⠈⣾⣿⣿⡏⠀⣧⣼⡿⣷⠀⠀⠘⠟⠉⠛⣿⣿⣯⣥⠠⠞⠁⢸⡇⠀⠀⢀⣿⣷⣀⡀
⠀⠀⠀⠀⠤⠺⣿⣿⠏⠀⠀⣿⢿⣤⢽⡀⣾⣾⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣿⣿⣿⡟⢂
⠀⠀⠀⠀⠠⠂⡜⣡⣶⡄⠉⣿⠪⢤⣼⡿⣿⡷⣤⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⢰⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⡿⡡⠋⣿⠀⠈⠛⢀⡙⠁⠀⠀⢰⣤⡄⠀⠀⠀⢀⣿⢻⡇⠠⠀⠸⣿⠿⣷⣞⠳⠄
⢉⡉⠛⠉⢙⡒⠟⢋⣀⣠⣴⣿⣴⣶⣾⣿⡿⢿⣷⣛⢋⡉⠀⠀⠀⠀⣈⡁⢸⣇⣴⣶⠞⠋⣀⠿⡛⣿⣤
⣦⣷⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣧⣿⣿⣿⣷⣄⠀⠀⠀⣹⣿⣿⢾⣿⣟⣿⣶⡀⢇⢸⣿⡏⢻
⢿⣿⣿⣾⣿⣿⡿⣿⡟⣉⢻⣿⢹⣿⡛⠹⣛⣯⣽⣿⣿⡿⠀⠀⢆⣹⣿⡿⢿⣿⣧⣧⣘⡿⣦⠺⣿⣿⣿
⡸⠙⢘⣿⣟⣿⡿⣟⣷⡿⢸⡿⣸⣿⡿⠰⣶⣿⣿⣏⡁⠀⠀⠀⣾⣿⡷⢶⣿⣿⣿⡋⠘⢿⡿⠛⠛⠛⠛
⠁⠀⣿⣧⡬⣿⣿⣾⠉⠛⠺⣏⢹⣧⢴⣿⣿⣿⣯⣅⣿⣾⣆⠰⠽⣿⣿⡔⠿⡿⢿⡿⣾⣷⡶

⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⣀⣀⣀
⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁⠀⣿⣿⣿⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⢰⣦⣤⣤⣤⣴⡆⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⢀⣤⣤⣬⣥⣤⣤⣤⣬⣥⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠠
⠀⠀⠾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡇⠀⠀⣶⣶⣶⣶⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧
⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⡆⠀⡀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⡇⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠿⠾⠿⠿⠿⠿⠿⠇⠀⠀⠀⢀⣿⣿⣿⣿⠀⢀⠀⢀⡀⢈⡉⢉⡉⢀⡀⠀⢀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣶⣾⣿⣿⣿⣿⠀⠠⠀⢀⠀⢀⠀⠀⡀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣾⢿⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢯⠀⠀⠙⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠦⣿⣷⠶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣪⠽⠋⣩⢶⣿⡶⢲⢶⡺⡧⣙⢯⡳⢍⡓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡱⠊⢁⡴⢊⡔⣱⢫⡇⢸⠀⢧⠹⡌⢣⡉⢢⡙⠢⣈⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⠎⠉⡳⠛⣶⠟⢲⠗⢾⠒⢺⠶⢺⡆⠻⡖⠛⣖⠻⣏⠉⠣⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢠⠋⢀⡜⠁⡜⠁⢠⠏⠀⣼⠀⢸⠀⠀⡇⠀⢧⠀⠘⡄⠘⡆⠀⠹⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⣠⠃⢀⡞⠀⡼⠁⠀⡎⠀⠀⡇⠀⢸⠀⠀⡇⠀⢸⠀⠀⢳⠀⢹⡀⠀⣣⠴⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡏⠙⡾⠲⢴⠧⣄⣠⣇⣀⣠⣇⣀⣸⣀⣀⣇⣀⣈⣇⣀⣸⡤⠴⡟⠉⢸⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠇⢰⠃⠀⣾⠀⠀⣼⠀⠀⢸⠁⠀⢸⠁⠀⡏⠀⠀⡇⠀⠀⡇⠀⢻⠀⠈⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⣼⠀⠀⡇⠀⠀⡏⠀⠀⢸⠀⠀⢸⠀⠀⡇⠀⠀⢧⠀⠀⡇⠀⢸⠀⠀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠦⣿⣀⡟⠀⢰⠃⠀⢀⡇⠀⠀⣿⠀⠀⢸⠀⠀⣷⠀⠀⢸⠀⠀⣷⠀⢸⠀⠀⣇⡤⢾
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠈⡟⠓⢾⠤⣄⣸⣀⣀⣀⣏⠀⠀⢸⡀⠀⣻⠀⠀⣸⣀⣀⣿⠤⠶⡗⠋⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⠀⢸⠀⠀⢸⠀⠉⠉⡏⠉⠉⢹⡋⠉⢹⠉⠉⢹⠁⠀⣿⠀⠀⡇⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⢸⡆⠀⢹⠀⠀⡇⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⡇⠀⢸⣀⣀⣾⠤⠴⠲⡗⠒⠒⢺⡓⠒⢺⠒⠲⠼⣧⣀⣸⠀⠀⡇⠀⣇⡤⢾
⠀⠀⠀⠀⠀⠀⠀⠘⡏⡷⣤⣷⣚⣻⠁⠀⣾⠀⠀⠀⡇⠀⠀⢸⠁⠀⢸⡄⠀⠀⣇⣀⣹⣳⠶⣷⡚⡏⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡟⠁⡇⠀⢹⠉⠉⣿⠙⠒⠚⡗⠒⠒⢺⡓⠒⢺⡗⠉⠉⡏⠁⢹⠀⠀⡇⠙⣧⡀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠀⡇⠀⢸⠀⠀⢹⠀⠀⠀⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⢸⠀⠀⡇⠀⡇⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢯⡻⣄⣻⠀⢸⡄⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⡇⠀⢸⠀⠀⡇⣀⡷⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣝⠲⢤⣇⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⢸⡇⠀⠀⡇⣀⣼⠤⢖⣫⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢭⣛⡓⠶⠶⠷⠴⠤⠾⠶⠶⠞⠓⣒⣉⡭⠥⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣠⣾⣇⠑⠄
⣾⣿⡀⠐⡀⠀⠀⠀⠀⠀⠀⢀⠐⡀⣀⣴⣶⣶⣿⡿⣿⣿⣿⠀⠀⠀⣰⣤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠰⢀⣶⣦⡀
⣿⣿⣧⣤⠀⠁⡀⠀⣀⣀⠠⠀⣪⣾⣿⣿⣿⢿⣿⡇⠈⢿⣿⣷⣴⣼⠋⠻⣿⡀⠀⠁⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣖⠀⠀⠊⠁⣰⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣷⡀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠉⠃⣀⣄⡙⢿⣿⣿⣦⣶⣿⣿⢷⣦⣄⠁⠔⠀⠀⠀⠀⢈⣴⣶⣿⣿⣿⣯⣀⣀⣈⣽⣿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⣯⠙⢿⣿⣷⣀⣴⣾⣿⣿⣿⣮⣍⡻⠿⣿⣿⢹⠂⢈⣿⡿⢦⣀⣨⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠿⣿⣿⣿⣿⣿⣿⣆⢀⣠⣌⠻⣿⣷⣦⣼⠙⢿⣿⣿⡿⠛⣹⣿⠻⢿⣤⡬⣿⣿⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⣀⣴
⣀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⣉⡻⣿⣿⣦⣽⣯⣴⣾⣿⣿⣿⣶⣄⡀⢀⣹⣛⣿⠉⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣠⣾⣷⣾⣿⣿
⣿⣿⣦⠙⠸⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣌⡉⠋⠻⢿⣿⣿⣿⣿⠟⠁⠀⢀⣼⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣌⣽⠻⣿⣟⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡘⣿⠟⢁⣴⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣸⠸⠯⢸⠹⠽⠇⢻⢸⠿⠇⠘⠘⠛⠙⠿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣡⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣌⡐⠈⠄⠠⠀⠂⠀⠄⠀⠀⠀⠀⠁⠺⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣦⣀⣁⠐⣈⣤⣤⣤⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡠⠔⠢⡄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠱⡀⠀⠀⠉⠀⠀⢀⡜⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⣀⠀⠀⠱⡦⠦⣀⠀⣠⠎⠠⣇⡀⢀⠠⠤⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⢀⡠⠔⠈⠀⠀⢀⡀⡖⢼⠄⠁⠀⠀⢣⠀⠈⠻⣄⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⢐⠈⠀⢀⡀⠀⠀⢐⠐⠬⠈⠁⠀⠀⠀⣀⠤⠃⠀⠀⠀⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⢃⠀⠘⠑⡢⠀⠈⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢆⠀⠈⠁⢀⡠⠔⠈⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠈⢦⠄⠺⠁⠀⠀⠀⠀⠀⢀⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠤⢄⣀⣀⠆⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠧⠤⠒⠒⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠐⠒⠒⠐⠒⠂⠈⠀⠀⠉⠁⠐⠒⠒⠂⠁

⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⡀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣦⣠⣰⢊⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⢀⡔⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⢷⠤⠄
⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠳⡀⠀⠉⠑⠒⠂
⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⢾⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠵⡿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠶⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠆
⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡜⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢱⠀⠀⠀⢠⠃⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠘⡆⠀⠀⡜⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢳⠀⢠⡇⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠸⡄⣸⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣷⡷⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠁⠀⠀⠀⢀⡇⢠⡎⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠤⠶⠄⠀⠀⠀⠺⠃⠊⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣀⠀⠀⠀⢠⣦⢀⠀⡤⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣉⣉⣀⢀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⢭⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⢀⡀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠇⠐⣯⠙⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⢢⣬⠗⠀⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⣤⣄⣁⡀⣸⣿⣿⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠘⣿⣿⣿⣿⣷⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⡶⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢀⡤⠶⠷⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢴⡶⢤⣤⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠸⣇⣀⣀⣀⣼⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣸⣇⡀⠀⠙⢷⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠈⠉⡿⠉⠀⠀⠀⠀⠀⢠⡟⠀⢀⡴⠛⠉⠉⠙⢷⣄⠀⠹⣆⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣧⣤⣾⠁⠀⠀⠀⠀⠀⢿⣤⣤⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣄⠀⠀⠀⠀⢀⡿⠀⢀⡿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢀⣤⠶⠿⢦⣄⠀⠀⠀⠀⠀⢻⣄⠀⠙⠳⢶⡶⠾⠋⠀⢀⡾⠁⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢸⣅⣀⣀⣀⣽⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣸⣇⣀⣤⠶⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠈⠉⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠙⠳⠶⢶⡶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⣶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣴⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⢄⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠈⠂⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠐⠈⠀⠀⠀⠈⠀⠀⠂⠈⠐⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠏⠑⠂⠄⡀
⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠁⢂⠀⠈⠀⠀⡠⢲⠒⠤⠠⣄⡀⠀⢀⠢⠀⠀⠀⢀⠔⠒⠉⠀⢢⠑⠒⠒⠄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⡞⠀⡀⠀⠈⢢⡀
⠀⠀⡠⠊⠰⠄⠈⠑⢄⠀⠀⠀⠀⢂⡤⠚⠁⢀⠇⠀⠀⠀⠑⢠⠔⠫⡐⠀⠴⡁⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠠⡀⠀⢀⠤⠔⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠀⠙
⠄⠊⠀⢀⠆⠀⠀⠀⠀⠈⠑⢤⡚⠉⠀⠀⢡⡏⠀⠀⢀⡤⠀⠀⠀⠀⡥⠀⠀⠈⠒⠤⣀⠘⢦⡀⠀⠀⠀⠀⠀⠸⠖⠁⠀⠀⠀⠀⠀⠀⠀⡰⠊⠠⡀
⠀⠀⠀⠀⠓⢄⠀⠀⠀⠠⠀⠀⠉⠑⣄⢠⠞⠀⢀⠔⠉⠀⠀⣠⠤⠞⠁⠀⠀⠀⠀⠀⠈⠳⢄⠙⡄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠈⠁⠄⠪⠁⠀⠀⠀⡞⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠙⢇⢅⠄⠄⠀⢀⠀⠀⠀⠀⠀⠀⢰⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣬⡷⢶⣧⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢟⠟⢋⡜⠁⣼⠀⠀⢧⠉⢯⡛⢽⡻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⠝⢉⡔⠁⢠⠏⠀⢰⡇⠀⠀⠸⡄⠀⢳⡀⠙⣄⠙⢮⡓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡴⠃⢠⠏⠀⢀⠏⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢳⠀⠈⢧⠀⠙⢆⠙⢆
⠀⠀⠀⠀⠀⠀⢀⠞⠀⡞⠀⢠⠏⠀⠀⡼⠀⠀⠀⡟⠀⠀⠀⠀⢱⠀⠀⠈⡇⠀⠈⢇⠀⠈⢧⠈⢳⡀
⠀⠀⠀⠀⠀⢀⠎⠀⡼⠀⠀⣼⠀⠀⢀⠇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢹⠀⠀⢸⡄⠀⠘⡆⠀⢳
⠀⠀⠀⠀⠀⡼⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠈⡇⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⣀⣇⣀⣀⣇⣀⣤⣼⣤⣤⣤⣤⣧⣤⣤⣤⣤⣼⣤⣤⣤⣼⣤⣄⣀⣇⣀⣀⣿⡀⠀⣿
⠀⠀⠀⠀⠀⣯⣉⣙⣟⡛⢛⡟⠛⠛⢻⠟⠛⠻⠻⡿⠿⠿⠿⠿⢿⠟⠛⠛⢻⠟⠛⠛⣟⠛⣛⣿⣉⣉⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁

⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⢄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐
⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰
⠐⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠄⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠆⢀⡶⣶⠀⠀⠀⠀⠀⠀⠀⢸
⠛⠿⠶⠚⠓⢆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⣀⣸⠏⠀⠀⠀⠀⠀⠀⢠⣞⢀⣀⣀⣤⠴⡒⠒⠊⠉
⠀⠀⠀⢠⠀⠉⠁⠹⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠡⠀⠀⠉⠠⠤
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡄
⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣄⠀⠀⠛⠻⠖⠀⠀⠀⠀⠠⣴⠀⠀⠀⠀⣷⡀
⠁⠀⠀⠀⠀⠀⣀⠀⠤⠐⠒⠉⠁⣠⠎⣤⣿⣷⣦⠀⠀⠀⠄⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⣿⣿⣶⣤⣄
⠠⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠠⢁⠀⢸⣿⣿⣿⣆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣿⣿⡄⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⣿⡇⠹⣿⣿⠀⠀⢸⡆⠀⠀⢆⣴⡄⠀⣾⣿⣿⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡄⢿⣿⣿⡆⣠⣿⡇⠀⠀⠘⡇⠀⠀⠘⣿⣧⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⣿⡿⣿⣿⡟⠁⠀⠀⢠⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠓⠁⡁⠘⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠘⠚⠀⣶⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠃⠘⠛⢀⠛⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠘⠩⠏⢫⠵⢪⡝⠭⠥⠥⢸⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡶⠼⠧⠇⢠⣶⠄⠀⣈⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡅⠀⣇⣄⣠⣀⠸⣶⡆⠰⣍⠐⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⢤⡄⡠⣤⢤⡄⣀⡀⠸⠈⠈⠻⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠠⣤⣤⡤⡤⡤⣤⡄⠀⣿⠀⠀⣈⡇⣿⢸⠀⣽⠀⢰⣖⣄⣀⣠⣠⠐⠐⢒⣖⣠⡀
⠀⠀⠀⠀⠀⠀⠐⠋⠈⡴⡶⢵⠆⠈⠀⠛⠒⠀⠛⠓⠛⠛⠀⠛⠛⠀⠀⠀⡈⠟⠿⠻⠀⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⣀⣀⡀⠠⠀⡴⡆⢠⠴⠮⠿⡲⡄⢴⣷⠐⠀⠀⢣⣆⠤⠰⠀⣸⠀⠘
⠀⠀⠀⠀⠀⠰⡀⣘⠒⣠⡄⢤⢀⠙⠀⣛⡃⠈⢹⠀⠀⠀⠃⣰⣷⠀⠉⢁⠄⠆⢰⠂⠆⠁
⠀⠀⠀⠀⠀⠀⡅⢱⠀⣁⡁⣨⢠⠀⠈⠉⠁⠀⠠⠀⠀⠰⠀⠀⠀⠀⠀⠰⠄⠦⠰⠤⠦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠋⠘⠀⠉⠁⠉⠀⠰⠤⠄⠀⠠⠤⠤⠄⠴⠂⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡞⠛⠁⠒⠲⠤⠤⣀⣀⣀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡟⡇⡄⢐⠂⠆⠠⠀⡀⡀⠉⠉⠓⠒⠲⢇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⡇⠈⠰⠀⠀⠀⠀⠀⠀⡈⠈⠁⠂⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠂⢈⠀⠀⠀⠀⡰⠋⠈⠳⡀⠀⢘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠂⠰⠀⠀⠀⢠⠃⡔⡹⠉⢳⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡆⢘⠀⠀⠀⢸⠀⡃⣧⠋⡾⢆⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠂⠠⠀⠀⠀⢻⠀⡅⣯⠀⡇⡸⠭⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠡⢘⠀⠀⠀⢸⠀⡇⣏⠒⢻⠁⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠠⠀⠀⠀⠈⣇⠈⠚⢒⡏⠀⠅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡉⢐⠀⠀⠀⠀⠈⠢⠤⠊⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡁⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣇⡴⠋⡣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠂⠐⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⡤⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠄⢈⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠰⠀⠀⠀⠀⠘⢆⣀⣀⣀⡞⢩⡟⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡂⢈⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣼⠍⠒⠦⠬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡇⡇⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠄⢈⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⢉⢉⡇⡇⠄⠠⠀⠀⠀⢀⠀⠄⠂⣈⣤⢖⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⣇⡇⢀⠘⠄⠂⢈⣠⠤⠒⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣥⡠⠔⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠶⠶⠖⠛⠛⠋⠉⠉⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣏⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠙⢷⡀⠀⢸⡇⠀⠀⠀⢀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡈⢻⡄⠈⣇⠀⢼⣁⡟⢹⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⢻⠀⣿⠀⠀⠉⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠶⠿⠶⠶⠶⠶⠶⠶⠾⢧⣄⡀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠞⠋⠙⠛⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿

⠀⣀⡀⣠⡤⠐⠒⠂⠀⠉⠉⠉⠉⠉⠁⠀⠒⠂⠀⠀⠀⠠⠤⠤⠀⠄⣀⣀
⠏⣅⠂⡘⠓⠐⠤⠄⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣤⡶⣶
⠁⡣⢊⠏⣉⣁⡈⠀⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢔⡒⠂⠄⠭⢭⡼⠖⡲⣻⠟⢙⢄⢌⠞⣼⡄
⠐⣽⢸⠰⠒⠒⠀⠀⠉⠀⠁⠐⠒⠒⠂⠀⠤⠤⠀⠄⢈⣑⠊⡛⡀⣷⢄⠪⠪⠠⡢⡠⠊⡔⠨⡇
⡆⡅⠸⠀⠀⠀⠀⠀⣠⠴⠶⠶⣦⡀⠀⠀⠀⠀⠀⠐⠢⢸⠀⠒⠅⢰⠕⠅⢔⠔⠕⢀⡞⠀⠁⣷
⠃⡅⠀⠀⠀⠀⠀⣰⢁⡴⠶⣄⠈⢻⣄⠀⠀⠀⠀⠀⡤⢵⣄⣄⠀⠸⢢⠪⠪⠢⣶⢏⡤⢊⡔⢹
⡇⠅⠀⡀⠀⠀⠀⢿⣿⣿⡷⢓⠀⣶⣿⠀⠀⠀⠀⠀⠱⢦⢶⣽⠇⢸⠕⠔⢔⠎⡀⠔⢁⠈⢪⣺
⠇⠀⠀⠇⠀⠀⠀⠈⣿⣯⣭⣁⣴⣽⡽⠀⠀⠀⠀⠀⠀⢸⣬⣿⢠⢸⡠⣠⠔⢀⠈⠂⠠⠑⢝⢼
⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⠛⠻⠓⠉⠀⠀⠀⠀⠀⠀⠀⢸⠉⢁⠜⣸⠀⠪⠢⠀⠀⠄⢀⠂⠢⢱
⠀⠀⠀⡆⠀⣧⣙⠢⡀⢀⣀⠀⡀⠄⢪⣯⠄⠀⠀⠀⢀⣸⠐⡠⡋⣴⠑⢐⠀⠠⡂⡠⡑⡑⣰⢻⠄
⠐⠀⠀⠃⠀⠠⣛⢷⣼⣦⣄⣯⣔⣶⡏⠥⠀⠀⠀⢀⡌⢹⢋⠀⡂⣫⠪⠀⢁⠄⠔⢄⢌⠊⢢⢿
⢸⠀⠀⡆⠀⠀⠈⠘⣿⣤⣻⠸⣿⣆⠀⠀⠀⠀⢀⠀⢐⢸⠕⠐⠅⠗⢀⢈⢢⣪⡢⣢⣳⣵⡽⣸
⢸⠀⠀⠁⠀⠀⠀⠀⠙⢶⢷⣾⠓⠃⠀⠀⢀⡔⠂⢠⣷⣽⣣⢀⠁⡧⣣⠠⠀⠔⠄⣤⣼⣿⣽⣿⣟⣷⣦⡀
⠈⠀⠀⠈⠓⠹⠤⣀⡀⠘⡘⣿⡀⠀⠠⡴⡋⡠⡨⢺⣬⣕⣚⡆⠂⡗⢝⢯⠫⠪⡙⡝⠉⠼⢡⣿⣟⣉⡙⠛⠛⠂
⠸⣼⡢⡀⠀⠀⠀⠀⠈⠈⠧⣽⢧⢠⣺⡮⢌⠊⢐⢁⠔⣏⣿⠇⠀⡣⡓⡁⢕⢍⠊⠠⡐⣐⣽⣿⠥⠈⠉⠁⠠⠀⠤
⠀⠀⠉⠑⠢⠤⣠⡀⡀⠀⠀⢀⣵⡶⣾⢧⠻⡳⠶⠥⣖⣇⠍⣴⣰⠝⢌⠪⡢⡀⡐⣅⡴⠟⡁⠤⡄⠠⠤⠦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠊⠾⠟⣛⢅⢔⠜⢕⠁⢁⠕⢻⡿⣪⠺⠪⡑⡕⢁⣤⣾⡛⠚⠀⢶⣶⡖⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠛⠴⠷⣾⣫⣲⣟⢸⣔⣼⣾⡿⠏⢉⣉⡉⠙⠃⡘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣘⣒⣒⣒⡒⠒⠒⠂⠊⠀⣀⣀⣀⣀⡁⠀⠀⠀⢀⠉⠈⠁⠈⣀⣀⣀⣀⣠⣤⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢻⠉⠩⡄⢸⣿⣿⣿⡿⠛⠛⠉⠉⠁⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡍⠹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠠⠁⢸⣿⣿⣿⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⢸⣿⣿⣿⣟⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⢸⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠈⠲⠂⡶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠃⢸⡿⢉⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠄⢀⡈⠛⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠆⠀⠈⠋⠁⡎⠀⡀⠀⠀⠀⣠⡀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡆⢠⠀⠀⠀⢠⠀⣧⠀⠀⢠⠛⠳⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⣤⣴⣶⣎⠀⢿⠀⠀⢸⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⢸⣿⢿⣿⣿⠀⣿⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠁⢸⣿⣿⣿⣿⠀⣿⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⢀⡀⠀⠠⣄⠄⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠄⢾⢹⠏⠉⠉⠀⣛⠠⠀⡠⠄⠀⠀⠀⣇⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡂⢺⢡⣄⠀⣤⠀⣿⠀⠀⢠⡀⠀⠀⠀⠘⢶⢓⣀⡠⠄⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⡇⢨⡅⠠⠀⠀⠀⠿⠀⣷⠀⠀⢸⠋⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠘⣗⢠⠀⠀⠀⢠⠀⡷⠀⠀⠀⣀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⡇⡄⣿⢈⠀⡀⠀⠈⠀⡇⠀⠀⠀⠉⠀⠂⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⡇⢱⠀⣢⠈⠉⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⣸⡇⣸⡀⢼⣦⣤⣤⡄⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣀⣈⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⡿⣸⡇⡏⠃⢸⣿⡁⠀⠉⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⠐⠒⠈⠉⠀⠀⠀⠀⠀⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⣙⣷⣼⡇⠀⡅⢸⣿⣷⣶⣦⢀⣤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣽⡇⠀⠰⠮⠍⠉⢩⣿⣈⢿⡀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣶⡿⠿⠿⠭⠭⠭⠀⢀⣀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣤⣶⣶⣶⡖⢀⣉⣭⣙⣿⣿⣯⣭⣯⣿⣏⣒⣚⣻⣿⣛⠈⠉⠋⠙⠁⣉⣙⠛⠛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠁⠈⠙⠛⠃⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠴⠦⢤⣀⡞⠀⠈⢳⡀⡴⠚⠒⢦
⠀⠀⠀⢠⠎⠉⢢⡞⠁⠙⡦⠒⠲⣄⠀⠀⠀⠀⠀⢠⡟⠉⠉⢻⡆⠀⠀⠀⢰⡇⠀⠀⠀⠉⠁⠀⠀⠀⠻⠁⠀⠀⠀⢣
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠓⡄⠀⠀⠘⢷⣤⣤⠟⠁⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢉⡤⠦⠄⠀⡄⠀⢠⠀⣠⣤⣤⠀⢰⣦⠀⠀⣴⠲⢤⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⠀⠀⠀⠀⣇⣀⣸⠀⣿⠀⠀⠀⣸⠘⣇⠀⣿⠀⣸⠇⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣇⠀⠀⠀⠀⡏⠉⢹⠀⣿⠓⠀⠀⣿⣤⣿⠀⢸⠛⠁⠀⠀⠀⠀⢀⣹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠦⣤⡄⠀⠇⠀⠸⠀⣿⣠⣤⠀⡇⠀⠸⠇⠸⠀⠀⠀⣀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣴⣴⣶⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣀⣴⠾⠋⠉⢰⠶⠄⢠⡀⠀⡤⠀⡶⢤⡀⢰⠶⠦⠀⣄⠀⢠⡄⢠⠶⠀⢰⡆⢠⠀⠀⡤⢠⡶⠆⠀⠀⣿
⠀⠀⠀⠠⣾⡋⠀⠀⠀⠀⢸⣀⡀⠀⢳⡾⠁⠀⣇⣀⡷⢸⣤⠄⠀⣿⡄⢸⡇⠸⣄⠀⢸⡇⢸⡀⢰⠇⢸⣇⡀⠀⠀⣿
⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⢾⠉⠁⠀⡼⢳⡄⠀⣏⠉⠀⢹⠀⠀⠀⣧⠹⣼⡇⠀⠘⡇⢸⡇⠈⣇⣾⠀⢸⡏⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠿⠤⠦⠸⠃⠀⠛⠀⠋⠀⠀⠘⠒⠒⠀⠛⠀⠘⠃⠐⠚⠁⠘⠇⠀⠘⠇⠀⠸⠧⠤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇

⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡇⠀⠀⠀⢀⡔⠒⠒⠒⠒⠒⠒⠒⠲⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣸⡁⠀⠀⠀⢺⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡿⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣴⠃⣰⡆⣩⠍⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠸⠇⡆⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢰⣏⢸⣿⣸⣇⠀⠀⢨⣿⢸⠀⠀⠀⠀⠀⠀⠛⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢿⣄⠻⣷⣽⣓⣺⡿⢋⡾⠀⠀⠀⠀⣄⣀⣀⣐⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣭⣩⣥⠴⠋⠄⠀⠀⠀⠀⠈⠙⣿⣭⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠈⠁⢰⡊⠑⡄⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢇⣀⠼⡀⡄⣀⡜⣹⡞⡁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢨⠶⣌⣿⢟⣴⠿⡅⠀⠀⣠⠄⣄⠀⠀⢸⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢸⠃⠀⠀⠀⠀⢀⣀⣀⣯⡤⢀⡜⠁⠀⠘⣯⣤⣶⠶⠶⠟⡀⠀⢼⠅⢉⡑⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⢸⠀⠀⠀⠀⢰⣏⠀⣹⣴⣶⣎⣹⢦⣤⡞⢿⡟⠉⠈⠀⠀⡀⠀⠀⠀⠤⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠛⠙⠟⠛⢙⠃⢸⡟⠉⠲⣍⠳⢤⡀⠀⠛⢿⣿⡖⠚⢃⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣿⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠂⢻⡄⠀⠀⠘⠧⠴⠃⠀⠀⠀⢸⣿⡿⠛⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣆⡠⣿⠀⠀⠀⠸⣆⣀⣀⣀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⠾⠟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⠶⠶⠶⣶⣶⣶⢶⣶⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠀⠀⠈⠉⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣾⣿⣶⣿⣿⣷⣾⣿⣦⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⠿⣛⣩⣭⣴⠶⠶⢶⣤⣭⣙⡻⢿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⢋⣵⠾⣋⣭⣶⣶⣾⣿⣶⣶⣦⣍⡛⢷⣌⡻⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢏⣴⢟⣵⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣷⣝⢷⣌⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⡾⢣⣿⡏⣿⡏⣿⡏⣯⢋⡏⣿⡏⡏⠤⢹⡙⣹⣧⡹⣇⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⣼⣣⣿⣿⣧⣭⣷⣬⣵⣧⣦⣷⣬⣴⣧⣿⣼⣧⣿⣿⣧⢹⡆⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⣥⣴⣶⣶⣶⣬⣭⣭⣙⣛⠻⠿⣿⣿⣿⣿⣿⠙⣃⣙⢉⣿⡌⣧⢹⣿⡆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⢿⣿⢛⣛⢻⣿⡿⢟⠿⣿⣿⣷⣦⣭⣙⡻⠿⣧⣭⣥⣾⣿⠇⡿⢸⣯⠸⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠸⢇⣿⢈⣋⢰⡏⣼⣿⣷⢸⡏⣶⣬⢻⣿⣿⣿⣶⣦⣭⣭⣉⣚⣃⣛⣛⣁⣂⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡆⣿⣿⣿⣸⣿⣿⣼⣿⣾⣿⣮⣝⣥⣾⡇⠿⠟⣸⡇⣿⣿⠀⣿⢟⣛⡻⣿⠿⠿⣿⣿⣿⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣿⣿⡿⠿⠟⣛⣛⠛⣛⣛⡻⠿⠿⣿⣿⣿⣿⣿⣧⡙⠟⣡⣇⢿⣿⣿⣿⣿⢰⣿⣿⣿⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣉⡲⠾⣿⡄⠿⠿⣷⣝⠿⣬⡛⠿⣶⣮⣍⣛⠶⢭⣙⡛⠿⣿⣶⣭⣴⣿⣿⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠁⠀⠀⠀⠀⢻⡿⠿⣦⣍⣛⠲⠶⠮⠽⠿⠷⢆⣉⣙⡛⠶⠮⠭⣭⣭⣭⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⢻⣿⡿⠻⣿⡿⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠔⡄⠀⠀⢀⡤⠀⠀⠀⠀⢀⣀⠤⠄⠒⠀⠉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠌⠀⠐⠦⠂⠀⠘⠄⠊⢁⡠⠦⠐⡊⠉⠀⠀⠀⠀⠁⣢⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠠⠠⣀⠆⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⢊⡉⠀⠀⠀⠀⢁⣀⠂⢀⠀⠆⠸⠄⡀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⢀⠔⠀⠀⠘⠀⠉⠐⠆⠑⠤⡀⣄⠈⡆⠀⢡
⠀⠀⠀⠀⠀⠀⢠⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠇⠈⠀⠰⡄⠉⠁⠸⠂⠀⠀⠄⠀⢁⡀⠃⠀⠀⠆
⠀⠀⠀⢀⠀⠤⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠐⠒⠂⠹⠀⠸⠀⡁⣄⣰⠀⠀⠑⠒⠊⢀⣀⣄⠤⠞
⠀⠀⠀⠀⠠⡀⠐⢒⣢⣤⡀⠀⠀⠀⠀⠀⣔⢸⣄⠚⠀⠀⠀⠀⣀⣀⡀⡔⠁⠀⠀⢀⣀⡤⠔⠒⠋⠏⣁⡀
⠀⠀⢀⡀⠤⠃⠀⠘⠋⢀⣿⣤⠤⣤⣤⣄⡁⠀⠈⠳⣄⠀⠀⠀⠀⡀⠈⠤⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⢈⠝
⠀⠈⠁⢄⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠈⠉⠳⢦⡀⠈⠳⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣁
⠀⠀⠀⢀⠑⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡖⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠶
⠀⠠⣔⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠳⣜⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁
⠀⢠⠖⠛⠛⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠻⠓⠁⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⢿⡘⠶⠀⠀⠀⠀⠬⠿⠇⠀⠀⢀⣸⠆⠀⢀⣠⣴⣤⣄⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠊
⠀⠈⣿⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣴⣿⣿⣿⣿⣿⡆⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⢸⡇⠀⠀⠀⠀⠀⣤⣶⣶⣶⣞⠛⢹⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠉
⠀⠸⣇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠻⣆⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠉⠁
⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠤⣸
⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠦⢤⣤⠤⢶⠶⢞⡻⠫⠀⠀⢀⠤⡀⠀⠀⡜⠢⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠁⠀⠀⠱⠤⠃⠀⠈⠢⡐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠼⠷⡄
⠀⠀⠀⠀⠀⢠⠖⠛⠓⠋⠉⠉⠉⠉⣉⢉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠛⠛⠻⠭⠭⠭⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠼⠤⢤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠘⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣄⡀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡎⠀⠘⠀⢀⠟⢦⠀⠀⠀⠀⢸⠀⢸⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⡎⠉⠳⠀⢀⣈⡄⠀⠀⠀⢸⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠓⠤⠚⠀⢀⡼⠀⠀⠀⠀⢸⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠴⠚⠉⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⢸⠇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠑⠒⠒⠒⠒⠀⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠹⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⡉⠉⠉⠒⠒⠒⠒⠒⠒⠋⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣉⡏⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠠⠦⢄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠠⢀⣀⣤⣤⣬⣥⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⢛⣲⢾⣷⣦⣄⡤⠤⠤⢄⣀
⠀⠀⡜⣹⠿⠿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠀⣈⡟⠛⠉⣇⠀⠉⠓⠺⢖⣀⠉⠒⠤⣀
⠀⠀⣦⣼⠀⠀⠀⡽⣿⣿⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢀⠀⠈⠐⠦⣄⣀⠀⠈⠙⠒⠦⣤⣭⣶⡄
⠀⠐⠿⠏⠀⠀⢠⣇⢹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠘⠳⣶⡦⠀⠀⠈⢇⠑⠲⢤⣄⣀⠈⠙⠛⠄⣀
⠀⠀⠀⠀⠀⠀⣸⡿⣸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣿⡇⠀⠀⠀⢸⡀⠀⠀⡇⠈⢹⡀⠢⡤⣀⣃
⠀⠀⠀⠀⠀⠀⡌⡇⣼⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⢀⣀⠀⠈⠃⠀⠀⢁⠀⠀⡇⠀⢃⠈⢻⡀
⠀⠀⠀⠀⠀⠀⡇⠙⠛⠦⠤⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠒⠒⠒⠒⠂⠀⠀⢸⡇⣿⠀⠀⠈⠉⠉⠉⠒⠚⠀⠀⠁⠀⠘⣀⣾⡇
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⢸⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⡇⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠂⠂⠀⠉⠁⢀⠀⢀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣠⣴⡀⠀⣇⣿⢀⠀⠀⠀⠀⠀⠀⣉⣩⣭⣥⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣥⣤⣤⣾⣷⣶⣶⣶⣶⣿⣿⣿⣿⡇⠀⠃⢿⠐⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⢨⡿⡇
⠀⠀⠀⠀⠀⢸⠫⠝⠛⠋⠛⠿⠛⠋⠙⠛⠛⠛⠉⢉⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⡀⣴⣶⣄⠀⠀⠀⠀⠀⠀⢀⠀⣿⡾⠷⠋
⠀⠀⠀⠀⠀⠸⣿⣿⡷⡄⠀⢙⣿⣁⠀⠀⠀⠀⠀⠛⢿⡋⠁⠀⣴⣿⣿⣿⣿⡟⠉⠀⠀⢸⣿⣿⣿⣿⠀⠄⠐⠒⠈⠁⠀⢸⣿⠃⠀⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠀⢀⣛⣓⣀⣈⣛⣀⣐⣛⣛⣀⡀⢘⣛⣛⣛⣛⣛⣓⣚⠀⢰⢸⣿⣿⣿⣿⠄⠀⠀⠀⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠴⠤⠬⣥⠭⠭⠭⠍⠭⠭⠉⠉⠐⠒⠒⠖⠒⡂⠀⠀⠐⠂⠈⢘⣿⣿⣿⠏⠀⡸⣟⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠒⠒⠒⠓⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣯⡄⣠⣴⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠿⠿⠥⠤⠤⠤⠤⠤⠤⠀⢀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠔⠚⠉
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣶⣾⣿⢿⡏⣹⠍⠻⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢩⣴⡟⡁⢪⠏⡸⠀⠃⢸⠀⢹⢻⡻⣷⡌⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡴⣷⠏⡼⢣⠏⢠⡇⠈⠀⠘⡇⠀⡀⠡⡘⣏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠊⡼⠃⠜⠁⡾⠀⢸⠀⠀⠀⠀⠇⠀⡃⠀⢹⡌⢧⡑⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⡠⠁⡼⠁⠀⠀⢸⠇⠀⡾⠀⠀⠀⠀⡆⠀⢷⠀⠈⣇⠀⢧⠐⡀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢰⠇⢸⡇⢠⠃⠀⠊⠀⢀⡇⠀⠀⠀⠀⣿⠀⢸⡄⠀⢻⠀⢸⡀⠀⡀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⠏⢀⣿⠀⠘⠀⢠⠀⠀⢸⡇⠀⠀⠀⠀⡏⠀⢸⡇⠀⠈⠀⠀⡇⠀⠰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡘⠀⢸⡇⠀⠀⠀⡼⠀⠀⢸⠃⠀⡇⠀⠀⡈⠀⠸⡇⠀⢰⠀⠀⡇⠀⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⣸⠃⠀⠀⠀⡇⠀⠀⣸⠀⠀⡇⠀⠀⡗⠀⠀⡇⠀⢸⠀⠀⢿⠀⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⡟⠀⠀⠀⠀⡇⠀⠀⠏⠀⠀⡇⠀⠀⡇⠀⢠⠁⠀⢸⠀⠀⢸⠀⠀⣹⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⡇⠀⡄⠀⠀⠀⠀⠰⠀⠀⢸⡇⠀⠀⡄⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⡿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⢈⠀⢸⡇⠀⡇⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡇⠀⢸⡄⠀⢸⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣄⠀⢸⡇⠀⡇⠀⢸⠀⠀⡄⠀⠀⢰⠃⠀⢸⠀⠀⢸⠃⠀⣿⠀⠀⢸⠀⢰⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠃⠀⢸⡇⢠⡇⠀⣾⠀⠀⡇⠀⠀⢸⠀⠀⡈⠀⠀⢘⠀⠀⡟⠀⠀⢸⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡇⠀⢸⡇⢨⠁⠀⡙⠀⠀⡇⠀⠀⡀⠀⠀⡇⠀⠀⡛⠀⠀⡇⠀⠀⡸⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⣸⣄⠠⠄⠐⡗⠀⠈⠉⠉⠉⡏⠉⠉⠉⠉⠈⡉⠈⠉⠉⠉⠀⡗⠀⠤⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡠⠃⠀⣽⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⣿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⣄⣀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⢀⣃⣀⣤⠴⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡭⣭⣍⣛⠒⠒⠒⠒⠒⠒⠲⠲⠶⠷⠤⠴⠖⠒⠚⠓⠒⠒⠒⠛⠋⠉⠁⠀⠀⠀⣵⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⣅⡀⠀⠉⠉⢭⠙⠋⠉⠉⠛⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣍⣁⣒⡒⠠⠤⠤⠄⡤⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⡶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠙⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀⠀⠀⠀⠀⠐⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠈⠁
⠘⢄⠈⠒⠀⠀⠀⠐⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠉
⠂⠉⠁⠀⡀⠀⠀⢄⠀⠀⢠⠊⠈⠳⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠀⠐⠐
⠀⠈⣂⠀⡀⠀⠀⠀⠦⠀⢘⠣⡀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣄⡬
⠛⠢⠄⡀⢈⣆⡀⠘⠄⠀⠈⢥⠙⡈⠍⠓⠦⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⣀⠤⠀⠀⡤⠀⠀⣀
⠀⠀⠀⠈⠁⠘⠀⠀⠀⠐⢄⡦⠂⠐⠲⠄⠀⠀⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢢⠊⠀⢸⢀⠀⢁⠆⠀⣠⠋
⠀⠀⠀⡀⠀⠸⠀⠀⢠⢐⠿⠛⠁⠉⠀⠀⠄⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢸⡄⠀⢸⡌⢈⢈⠄⠀⠉
⢠⣀⠴⠶⠄⠀⠠⡔⠀⠅⡠⠀⠀⠈⠁⠀⠀⠀⠈⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⠀⠀⡴⢧⡀⠘⢇⡎⠈⠑⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠉⣐⠒⠊⠉⠉⠍⠭⠅⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡆⠀⠸⠝⠿⠶⠿⣷⡆⠀⣾⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⢤⣤⣤⣞⣉⣤⣦⠤⢤⣄⡀⠀⠀⠁⣀⣼⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⠥⠂⠀⠀⠀⠀⠢⠶⣦⣄⣀⣈⣉⣁⣀⣈⠅⢀⣨⣞⠕⠋
⠀⠀⠀⠀⠀⠀⢀⡞⡴⠁⠀⣤⢖⣀⡀⣀⡀⠀⠀⠀⣈⠩⠟⣉⠥⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⡞⠘⠁⠀⠀⠙⠿⠖⠋⠭⣻⡆⠠⠊⣡⠖⡩⠄
⠀⠀⠀⠀⠀⠀⠉⠒⠒⠪⠭⢑⣒⠤⢄⡈⠁⠠⠁⡼⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠤⢄⡼⢰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠰⠍⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡛⠳⠦⠨⠤⠶⠊⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⢯⣽⣲⣠⡤⠖⣀⢎⢷⢫⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣰⢡⠏⡼⡹⢹⠹⠏⡏⡎⢆⢣⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣧⡯⣴⣷⡷⠾⣶⡦⢧⣼⣼⣮⣧⣷⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⣿⣡⣾⣧⢲⣿⣯⢿⣹⣯⠞⣿⣟⡯⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣮⣭⢑⠛⠻⠉⠏⠿⠏⠿⠛⠛⢚⣩⠵⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠾⡟⣿⠷⣸⣟⣳⣆⢾⡭⣿⠲⣿⣚⡼⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣳⢲⠤⣛⣛⠀⠿⠿⠿⠯⠿⠷⠛⢲⣾⠷⣶⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇⢸⠀⡇⢸⠁⢸⠃⢸⠋⢹⡏⠈⡇⢸⡄⣯⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡇⢸⠀⣇⣸⣀⣸⣀⣰⡀⣈⣇⣀⣷⣸⡇⢿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣷⣾⣺⠿⢿⣽⢟⡿⢹⣥⣟⡛⣭⣿⡻⡟⣿⣿⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣙⣻⣿⣏⣿⣿⣿⣤⣾⣥⣿⣶⣿⣾⣿⣟⣿⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣭⣛⡛⠿⢿⣟⣛⣙⣿⣇⣤⣤⡴⢟⣛⣯⣵⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⣻⣶⡿⣶⡶⢶⣖⣲⢞⣛⣿⣿⣻⣿⠿⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢍⠙⣾⡿⣿⣭⣿⣿⣿⣿⣿⣾⠿⠾⣿⡿⠗⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠓⠛⠛⠚⠒⠉⠁

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⠿⢿⣿⠋⣽⠍⢻⣿⣿⣭⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⡿⠀⣿⣧⠸⡟⠙⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠿⣿⡙⠄⠸⣿⡀⡇⣠⡿⠋⢘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣟⡿⡀⠛⠑⢨⡿⣄⣴⠟⠉⢋⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢿⡧⣜⣦⣤⣼⡇⣽⣷⣀⣴⠞⢉⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣼⢸⣿⢾⢷⣿⢻⣿⣏⣴⡿⠋⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⣿⢹⣿⡿⣨⠟⣿⡾⣿⡇⣾⡟⣿⣿⣠⣾⠛⣰⡿⣿⣦⡀⠀⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢯⣼⣿⣿⣿⣿⢺⡇⣷⣿⠀⣿⢇⣿⣿⣿⢅⣼⠟⢡⣿⡿⣿⡄⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢞⣵⢋⣬⣭⣵⡿⣿⢸⡇⣿⢹⠀⣿⢸⣿⢹⣿⣾⣿⣤⣿⠏⣰⣿⣿⠀⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡷⣫⡾⠛⣉⣩⣥⣦⢸⡏⣧⢹⣸⡄⣿⣿⣇⣿⡿⢻⣿⢸⣯⣰⣿⣿⣿⣇⠀⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⡾⠋⠚⠛⠛⠉⠉⢻⡟⡇⣿⡸⣿⣿⣿⣿⣹⣿⢡⣿⣿⢸⣿⣹⣾⣿⡏⣼⠃⠀⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡹⢻⣿⣿⣿⣿⣿⡿⢿⠇⣼⢠⡇⢸⣇⣽⣿⣿⢇⣿⡿⣼⣿⡏⣿⣿⡏⣿⣿⣿⣿⣶⣾⣼⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢋⣁⣠⣤⣤⠶⣶⢸⡇⣸⣇⣼⡿⢿⣿⢏⣼⣿⢣⣿⣿⢳⣿⣿⡇⣿⣷⣷⠹⣯⢷⣻⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡿⠿⠟⠛⣫⣁⡶⣿⡼⠧⠛⠉⡷⣾⡿⢫⣾⡿⢃⣾⣿⣿⢸⣿⣿⡇⣿⣿⢿⣧⣿⡞⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣀⣠⣭⡿⠟⠋⠁⠀⠀⠀⠀⢳⣌⣴⣿⠟⣡⣾⣿⣿⠇⣸⣿⣿⡇⣿⣿⡜⠃⣿⣷⢹⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⡉⣁⣴⣿⣿⡿⢃⣼⣿⣿⣿⡇⢸⣿⣷⠈⣿⣿⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠿⠟⠉⢠⣾⣿⡿⢃⣿⣷⢸⣿⣿⡆⣿⣿⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⢠⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢀⣴⣿⣿⠏⣠⣾⣿⣿⠈⣿⣿⡇⢸⣿⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⣴⣶⣶⣼⣥⣤⣧⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣷⣶⣿⠟⠋⣡⣾⣿⣿⡿⠃⣸⣿⣿⡇⣼⣿⢻⣿
⠀⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠘⢿⡋⢠⣾⣿⠟⠋⢁⣠⣴⣿⣿⡿⠀⣿⣿⢸⣿
⠀⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢻⣧⡀⠹⡿⠟⢁⣴⣿⣿⣿⣿⣿⠟⠁⣼⣿⡏⢸⣿
⠀⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠙⢿⣦⣀⣰⣾⣿⡿⠟⠋⠉⠁⢀⣼⣿⠏⣠⣿⣿
⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠉⣿⡿⠋⠁⠀⠀⣠⣴⣾⣿⠟⢁⣴⣿⢿⣿
⠀⣿⣇⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣙⣷⣄⣀⣀⣀⣀⣠⣾⣿⣛⣋⣁⣠⣿⣛⣁⣸⣿
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⢀⣀⣀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠈⠙⠻⠿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣿⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠖⠋⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠆⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⠿⠋⠀⠀⠎⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠌⢿⣿⣿⡿⡿⠋⠁⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠶⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⢿⣿⣿⣿⣿⣿⠀⢸⣿⠿⠿⠿⠋⠀⠟⠱⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⣿⣿⡿⠿⣿⡀⠀⠀⠀⠀⠀⢀⣈⡀⢐⢸⡇⠀⠀⠀⢀⣤⣤⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⣿⣷⡄⠀⡅⢰⣶⣶⣾⣿⣿⣿⡇⠀⢸⡇⢀⠄⣀⢸⢱⣿⠳⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠹⣧⠀⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⢨⣾⣍⣽⠷⠟⢠⢾⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠁⢸⣿⡿⣿⣿⣿⣿⢣⡄⠈⡇⠀⠙⣿⠇⠀⠀⢁⣸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⡄⠀⠁⠘⠋⠀⠀⠙⢿⣿⡏⠀⠀⡇⠀⣸⣿⠀⠀⠀⠙⠸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢳⠀⡇⠀⠀⠀⠀⣀⣬⣿⡇⠀⠀⡇⠀⠏⠃⠀⠀⠀⠰⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⡇⣴⠂⠀⠀⠉⢿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⡇⠓⠀⠀⠀⠀⠀⣿⡇⠀⠀⡇⠀⠀⠀⠰⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⡇⠀⠀⠀⠀⠀⢀⣀⡵⠦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⣇⣀⣤⣴⠖⠋⠉⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⣿⢿⡿⣿⢿⡿⣿⣿⡿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡝⢣⢛⡜⢧⢛⡜⣤⢣⡜⣼⣿⣿⣿
⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⡇⠀⠀⣷⣿⣿⣿⣿⣿⠀⠀⣺⣿⡶⢆⢾⣿⣿⣿⣿⣿⣿⡰⣸⣿⣿⣿
⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠐⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⢥⡌⣼⣿⣿⣿⣿⣿⣿⢡⢹⣿⣿⣿
⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣹⣿⣃⡘⣸⣿⣿⣿⣿⣿⣿⢃⢻⣿⣿⣿
⣿⣿⣿⣿⠀⠀⢸⣷⣶⣷⣶⣾⡇⠀⠀⣿⡇⠀⠀⣿⣶⣶⣶⣶⣿⠀⠀⢻⣿⡏⠱⢹⣿⣶⣷⣶⣾⣿⠎⠶⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠐⣿⡇⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⢺⣿⡝⢣⢛⡝⣩⢫⡝⣭⢫⡜⣼⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⡾⢆⠶⡰⢆⢶⡰⣂⢖⡐⣐⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢥⡌⣬⢡⡍⣬⢱⠉⡬⢡⢩⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣃⡘⣐⢃⡚⣘⢃⡛⣘⢃⢻⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏⠱⠙⠎⠰⠱⠎⠶⠱⠎⠶⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡝⢡⠉⡜⣥⢩⣜⣥⢫⡌⣼⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠴⠦⠄⣳⣶⣄
⠀⠀⢀⠄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡟⠁
⠀⢨⠈⢠⠈⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢚⡇⠀⠀⠀⠐⠆⠒⠞⠛⢿⠿⠂
⠀⠈⠁⢆⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠛⠂⠒⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⢀⡀⠸⠇⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠂⠠
⠀⠀⠀⠀⠀⠀⠉⢲⠖⠛⠀⠀⠀⣴⡧⠀⠀⠀⠀⠈⠀⠀⢀⠧⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠲⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠲⠆⠀⠀⠀⠀⠀⠈⢩⡭⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠈⢠⢶⡀⠀⠀⠀⠀⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢶⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣦⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣝⡔⣀⣀⡀⡔⠀⢆⡔⢱⣀⢀⣀⣀⣀⠀⡀⠀⣁⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠉⠉⠉⠀⠀⠈⠀⠉⠉⠉⠉⠈⠉⠀⠙⠈⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢈⢀⣴⣂⡀⡰⣂⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠉⠈⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⡇⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢰⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠘⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠛⠛⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠉⠻⣦⣴⡟⠉⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⣴⠿⠿⣦⣀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣀⣀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢠⠷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠘⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠘⠲⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠑⠦⢤⡤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⢤⡤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉