Iodine Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⡇⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⢦⠀⠀⠀⠀⠀⠀⠀⠘⢲⠚⠓⠒⠒⠓⠒⠒⠚⠲⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢣⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⢸⠀⢠⢻⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠇⠀⠸⢸⠀⡇⡎⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⡇⡎⠀⡇⡇⠀⢀⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢠⠇⡇⠀⡇⢇⠀⣏⢦⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡇⢸⢸⠀⠀⢹⢸⠀⡎⡼⠀⠀⠀⠀⣀⣉⣋⣁⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢳⢸⢸⠀⠀⡏⡞⢰⢡⠃⠀⠀⠀⡏⠀⠀⡏⠀⠈⣱⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠸⡘⡄⢰⢡⠃⡇⡎⠀⠀⠀⠀⠉⠒⠒⠛⠒⠚⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠮⠾⡤⠛⠉⠛⢮⠤⠧⣇⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠇⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⢀⠀⠆⡠⢠⠀⠄⠠⠀⠠⠀⠆⡄⠠⠀⢠⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⣶⣿⣷⣄⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣝⢿⣿⣿⣿⣿⣿⣷⣍⣩⣍⢻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢟⣽⣷⣝⢿⣿⣿⣿⣿⣿⡿⣋⢾⣿⣿⣿⣿⡿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⣱⣿⣿⣿⣿⣷⢙⣿⣿⡿⣫⣾⣿⣷⣝⢿⣿⣿⣷⣌⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⣼⣿⣿⣿⣿⣿⣿⡘⠿⣋⢾⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣆⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣝⠻⣿⣷⣝⢿⣿⣿⣿⣿⡜⢿⣿⣿⣿⡌⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⣛⣻⣿⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣷⣝⣋⣝⠻⣿⡦⣹⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣜⣛⢻⣿⣛⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠟⢻⣷⣬⣾⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⣛⣸⣿⣛⣃⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⢸⣿⣿⡿⢫⣾⣷⡙⢿⣿⣿⣿⣿⢃⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⢸⣿⡟⣱⣿⣯⡙⢿⣎⢻⣿⣿⣏⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⡿⢰⣿⣿⣿⣿⡎⣿⡆⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣷⠸⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣁⣉⣹⣿⣿⣿⢸⣿⣷⣝⠻⠿⠿⢟⣫⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⡄⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠷⠀⣀⣀⠀⠤⠿⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠃⠀⣀⢴⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠄⠀⠀⠉⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡟⠰⠋⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠂⠀⠀⠀⠀⠀⠂⠶⠠⠼⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡷⢀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠐⠒⠁⠉⠉⠉⢉⣉⡙⢻⡁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢻⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠓⠒⠒⠒⠒⠒⠚⣫⠹⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⡶⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠐⠀⠀⠀⠀⠀⠀⠉⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⢆⡠⠀⢚⣥⢴⠃⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡮⢸⠄⠒⣉⡶⠾⡇⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠈⣧
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣳⠟⡤⠞⠉⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⣀⡀⠠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣯⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠄⢀⣀⣠⡞⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠹⠟⡈⣧⠀⠀⠀⠀⠀⠀⠀⢸⡴⠛⠹⢟⡸⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⡞⠁⣉⠀⠈⢩⣎⢳⠀⠀⠀⠀⠀⢠⡞⠑⡁⠀⠈⢽⡌⢧⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⢸⡇⡆⢁⣲⡆⢒⣶⢈⡇⠀⠀⠀⠀⢸⠁⡒⢻⣲⠄⣒⡒⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡧⡇⢤⠿⢽⠯⣿⢸⡇⠀⠀⠀⠀⢸⣰⠒⣨⠾⠷⠾⣿⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣷⡧⠿⣆⣾⣃⣿⣼⡇⠀⠀⠀⠀⢸⢹⠤⠷⣄⣺⠤⣿⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣗⣳⣇⡨⢙⣚⣻⢧⡇⠀⠀⠀⠀⢸⣼⣻⣅⡢⢙⣺⣿⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡏⣇⡝⣣⣈⠡⢽⡵⡇⠀⠀⠀⠀⢸⢜⣃⣟⣃⣉⠰⢾⢸⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣇⣿⣬⣿⢛⣯⣽⢊⡇⠀⠀⠀⠀⢸⣺⣷⣬⡿⢚⣮⣽⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡏⡷⣾⣧⣤⣿⣾⠹⡇⠀⠀⠀⠀⢸⣿⠶⣶⣍⣲⣿⣾⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡧⣿⣿⣿⣛⣿⣿⣶⡇⠀⠀⠀⠀⢸⢹⡿⣿⣯⣟⣿⣿⢸⠁⠀⠀⠀⠀⠸⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿
⠀⢠⠀⠀⠀⢸⡇⣿⣿⣿⢿⢿⣿⠈⡇⠀⠀⠀⠀⢸⠀⣛⣿⣿⢿⢿⣿⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⢀⠀⠀⠀⠸⣇⣛⠛⣛⣛⠛⣋⣰⠃⠀⠀⠀⠀⠸⣄⡛⠛⠛⠛⠛⣃⣸⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣡⣤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠋⠀⠀⠋⠐⠡⡀⠈⠀⠁⠀⠀⠐⡀⠄⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⡠⠆⠈⠉⠙⡀⠢⡀⠁⠐⠀⠀⠀⠀⠐⠀⠂⠈⠀⠀⠙⠀⣠
⠀⠠⠄⠀⠀⠀⠀⠀⣇⠈⠀⡠⠄⠀⠄⠀⠀⢀⠊⠀⠀⠀⢀⠀⠀⠂⠢⠀⢈⠀⠀⠀⡀⠂⠁⠒⢀⡄⠘⡀⡀⠀⠀⠀⠀⠀⠠⠀⠰⠐⠀⣷⠈⠀⠛
⠀⠀⠀⠘⠀⠀⠄⠀⠘⣆⠀⠂⠈⡂⠰⠀⠀⠈⠈⠀⠠⠀⠀⠈⠀⠀⠀⠀⠈⠠⠀⠘⠀⠀⠀⠀⠈⡄⠀⡆⠀⠂⠀⠀⠐⠀⠒⠀⢠⡄⠠⠈⢖⠑⢋⠄
⠀⠀⢄⠀⢠⠁⠀⠀⠀⠸⡄⠀⠠⠀⠀⠦⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢠⠐⡃⢀⠈⠁⢀⡁⠀⢈⢈⣀⢀⣀⡀⢀⡉⠀⠘⢀⠬⠄⡋⠀⠀⣤
⠀⠐⠀⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀⠄⠀⠁⠀⠑⠄⢀⠀⢄⠀⢀⠔⡀⡀⠀⠐⠁⢀⠀⠐⠁⣀⣑⡀⠀⠀⠀⠀⠀⠀⠀⠇⠒⠀⠀⠀⠀⠵⠂
⠀⠀⢀⠀⠀⠁⠠⠀⠀⠠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢘⠠⠄⠀⠑⠀⠈⠀⠁⠀⠀⠢⢀⠀⠈⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠙⠢⠄⠀⠠⠄⠋⠀⠀⠀⠁⠐⠠⠀⠀⠀⡠⠎⠱⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠁⠄⠀⠒⠊⠀⠈⠂⣤⠀⠀⠠⠀⠀⢠⣀⠀⠀⠀⠀⠀⠠
⠀⠀⠐⠂⠀⠀⠀⢀⠀⠀⠁⢢⡄⠀⠘⠀⢀⠀⠀⠁⠂⠀⠀⠰⠤⠄⠀⠒⠀⡀⠀⠀⠀⠈⠈⠀⠁⠀⢀⡉⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⢈
⠀⠠⠂⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⡂⠀⠈⠡⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⢀⠀⠈⠠⢀⢀⠀⠀⠀⠀⢀⠀⠄⡄⠀⠂⠜⢀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⡐⠀⠀⠉⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠄⠀⠁⠀⢀⡀⠀⠀⠈⡙⠄⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⢀⠀⢠⠄⠀⠐⠀⠈⠀⠀⠀⢀⣤⠀⢀⣤⣤⣄⠀⣤⣤⣤⡀⠀⢠⣄⠀⣤⡄⠀⣠⣄⠀⣤⣤⣤⠈⠀⠀⠈⠅⢠⠈⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠈⠈⠀⠀⢀⠀⠀⣔⠀⠀⠀⠀⣴⡆⢸⣿⠀⣾⠃⠈⣿⠄⣿⡇⠈⣿⠀⢸⡟⠀⣿⡿⣦⣻⣿⠀⣿⡤⠀⠀⣴⣦⠀⠀⠀⠈⠈⠂⠀⠀⠄⡀⠀⠠
⠀⠀⢠⡶⠆⠀⠀⠀⠀⠈⠀⠡⠀⢀⠀⠀⠸⣿⠀⠻⣷⣴⠏⠀⢽⣧⡶⠋⠀⢸⡇⠀⣿⡇⠙⣿⡟⠀⡿⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠀⠁⡀
⠀⢀⠚⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⣀⠀⠀⢀⣀⣀⠀⠀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣀⡀⠀⣀⣀⡀⠀⣀⣀⡀⠀⠈⠂⠀⣈⣤⠠⡦⢀⠀⠀⠰⠂
⠀⠀⠄⡠⠖⠀⠀⡀⠀⡔⠀⠁⠀⠸⣯⣄⡀⢠⣿⠛⢻⡇⢸⡿⠀⢸⣿⠀⣿⣉⣿⠀⢼⡿⠋⠁⠈⣏⣉⠁⠀⢿⣌⡀⠀⠁⢀⡀⠀⠈⢀⠁⢉⠉
⠀⠀⡀⠀⠐⠀⠀⢀⠠⠄⢀⠀⠁⠀⣀⣸⣿⠀⣿⣄⣼⠇⠘⣿⣤⣼⡟⠀⣿⠙⣧⡀⠸⣧⣤⡄⠀⣿⣉⡀⠀⣄⣈⣿⠀⢠⠹⠟⡀⠈⠀⠀⡆⠀⠀⡨
⠀⠀⠀⠔⠈⠁⠉⠀⠀⠀⠀⠀⠂⠀⠉⠉⠁⠀⠈⠉⠁⠀⠀⠈⠉⠉⠀⠀⠉⠀⠀⠁⠀⠈⠉⠁⠀⠉⠉⠁⠀⠉⠉⠁⠀⢀⡁⢀⡅⠀⠀⠀⠃⠀⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠃⡀⠢⠀⠀⠠⢀⠂⠀⠀⠂⡠⠚⠀⢀⠀⠦⠀⣠⠀⠒⠀⠀⠕⠀⠀⠀⠀⠀⠈⠀⡀⠉⠉⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠔⠁⠠⠔⠈⠀⠀⢀⠀⠀⠀⠀⠁⠒⠰⠀⠀⠀⢁⠀⠀⠙⠛⠈⠀⠈⠄⠀⡇⢈⠉⠉⡀⡰⠔⠫⠀⠀⠀⠠⠀⠀⠠⠄
⠀⠀⠀⠀⠀⡄⢀⠄⠀⠀⠐⠁⠘⠀⡀⠐⠂⠀⠀⠀⠀⠀⡠⠐⠉⠀⠂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠐⠠⠌⠘⠡⢀⠀⠀⠎⠀⠂⠀⠐⠀⠀⠀⠆
⠀⠀⠀⡘⠢⠄⡥⠚⠁⠀⠀⠀⠀⠰⠀⢄⢠⠀⠐⠢⠐⡀⠀⠀⠔⠀⠐⠂⠈⡀⠀⠀⠀⠀⠀⢀⠁⠀⠐⢚⢠⡅⠉⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣀⠉⡘⠌⠠⠓⠁⠀⠐⢀⣀⢀⣀⣀⣈⣀⠀⠀⠀⠀⢠⠄⡆⣀⠀⣐⠀⠀⠀⠀⠀⠀⠅⠀⠀⠁⣠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠅
⠀⠈⠀⠀⢔⡘⣁⠆⠀⠀⠀⡠⠁⠌⠀⠁⠀⠴⠀⠁⠀⡀⡈⢠⡁⠂⣊⠀⠈⠀⠒⠀⠀⠀⠀⠀⢀⠀⠃⠁⠀⠀⠀⠀⢀⠀⡁⠀⠀⠀⠒⠀⠒⠀⠀⠂
⠀⠀⠀⠈⠣⠚⡄⢂⠀⠤⠄⠀⠂⠘⠐⠃⠚⠒⠀⠀⠀⠀⠘⢌⠢⠉⣐⠁⠅⠀⠒⠀⠈⠀⠀⠄⠁⠁⠀⠀⡀⠄⠂⠀⢘⢃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢠⢘⢔⠱⢚⣄⠁⠀⠀⠀⠀⠘⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡆⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⠒⠒⠂⠀⠀⠐
⠀⠀⠰⠈⣎⠒⡩⡔⠀⠀⠀⠂⠄⠠⠀⠄⠀⠲⠀⠁⠐⠈⠀⠀⠀⡀⢀⡀⡀⠀⠈⠠⠀⠀⠀⠀⡀⠀⠇⠀⠀⡀⠂⠀⠀⠀⠀⡈⠢⠀⠀⢀
⠀⠀⠄⠂⠈⣈⡑⠁⠀⠈⠀⠎⡀⠀⠈⡀⢀⢾⠀⠀⠀⠀⢀⡈⢈⠉⠀⠁⠁⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠐⠀⠀⠀⠐⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⡀⢠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡛⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠀⠙⠻⠿⠁⠀⠀⢠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣭⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣩⣽⢫⡅⡿⣟⣭⢻⢫⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣾⣷⣾⣿⣿⣿⣾⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⢀⠀⡀⢀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠙⡤⠀⠀⠒⠒⠐⠒⠒⠒⠀⠀⠀⠀⠐⠀⠀⢀⡉
⠀⠀⠀⠀⠀⠀⡠⠖⢅⣢⡄⠀⠘⠲⢤⡀⠈⠀⠎⠀⠀⠀⠀⠀⠀⠈⠁⠿⢾⡇⢀⣴⣗⣴⡟
⠀⠀⠀⠀⡰⠎⠀⠀⠀⠄⠹⣦⠑⠀⠀⠉⠓⠲⠬⡶⠦⢄⣠⢶⠴⠶⠶⠶⡾⢿⣿⠟⣋⣀⣇⡀
⠀⠀⠀⡰⠀⠀⠀⠀⠀⢀⢉⣼⣧⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⡀⠀⠀⠀⠠⠂⠀⢨⠈⢫⣭⣿⡷⠶⠒⢲⣦⡀
⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠲⠒⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⠆⠠⠀⠀⠀⣠⢴⣻⣿⠃⠀⠀⠀⠀⠸⠿⣦
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠂⠠⠊⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢶⣷⠤⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⢿⣧
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠸⢋⣽⣿⡆⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣪⡷⠛⠋⠉⠉⠛⠿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣄
⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠈⠋⣉⣽⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣷⡀
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⡲⠯⡴⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠷
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢠⡍⠛⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠇
⠀⠀⠀⢘⣦⠀⠀⠀⡀⢦⣀⠟⠑⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⡿
⠀⠀⠀⠀⠻⣦⡴⢴⠻⠆⠤⠤⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⣿⢾⣏⣂⣱⣿⣶⣖⣦⣀⣠⠖⠲⣤⣤⡀⠀⠀⢠⠀⢪⡅⠳⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣾⡿⠁
⠀⠀⠀⠀⠐⣿⣣⠀⠙⠉⠙⠋⠉⠀⠉⠙⣆⠀⡀⠀⠉⢠⠀⣽⣡⡄⢀⡆⠀⠀⠹⣦⠀⣀⠀⠀⠀⡀⠀⠐⣶⣾⡟⠋
⠀⠀⠀⠀⠀⠻⡧⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⣴⣟⣹⡿⣷⣷⣈⡇⠀⣤⠀⡁⠙⢊⣄⠀⠤⠥⠼⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠹⣧⣁⡀⠀⠀⠀⠀⠀⠀⢀⣁⣀⣼⣿⣿⣿⡟⠶⣬⣻⣿⣽⣉⠁⠘⠄⢸⣟⣍⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣀⣨⣿⣧⣿⣽⡿⣿⣿⣿⢻⣿⡄⠈⠙⠿⣿⣿⣿⣚⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠉⠀⠀⣿⣿⣿⣾⣴⣧⣦⣤⣤⠦⠜⠛⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠰⣯⠈⠉⠁⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣀⡀⠀⠀⠤⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡉⠉⠉⠁⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠗⠀⠄⠒⠒⠈⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠧⢀⠀⠀⠀⠀⠀⠁⡄⠂

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⢀⡴⠃⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠙⠻⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠛⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠙⢢⣀⠀⠀⢙⡆⠀⠀⣤⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⡴⠋⠈⠳⠴⠋⠀⠀⠀⡿⠶⠶⠶⠖⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠞⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠒⠒⠒⠀⠚⠳⠦⢄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡞⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣼⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠶⢾
⠀⠀⠀⠀⠀⣠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⠃⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⡤⢄⡠⢴⢀⢀⡤⡄⡤⣄⢸
⠀⠀⠀⠀⠻⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠑⠋⠘⠊⠈⠈⠁⠁⠙⠂⢸
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠋


⠇⠽⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠺⠅
⠀⠀⠀⠀⠀⠀⠠⠄⢠⠄⡀⡀⢤⠄⢀⠀⡤⡀⡀⠀⡠⠀⢀⠤⠀⠤⠀⠤⡀⡠⢄⠀⢀⠀⠤
⠀⠀⠀⠀⠀⠀⢘⣀⣸⣀⠃⡃⣸⢠⠃⣂⣇⡁⣁⣀⣃⣀⢀⣑⡀⣀⢀⡒⡄⣄⣸⢘⢣⣀⣁⡀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⠐⡄⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠃⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⣀⡀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠗⠤⠛⠠⠛⠘⠘⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠐⡆⠐⡆⡄⡀⠀⢐⣦⡔⠄⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣻⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠑⠁⠀⠐⠁⠑⠁⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⣿⠀⠀⢠⣤⡤⢀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠿⢶⣄⠀⠠⢾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠐⢦⣴⠟⠸⢦⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠊⠀⡱⠀⣜⡁⠀⠀⢎⠀⡷⠘⠁⠹⠀⢠⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⣞⡀⢸⣀⡹⠀⡀⢀⡝⠁⠱⣀⡰⠀⠃⢺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⣿⣿⡇⣿⣿⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠸⠿⠿⠿⠇⠿⠿⠿⠿⠇⠿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢸⣿⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⢸⣿⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡿⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠃⠀⠀⠸⣿⡀⠀⠀⢿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⣸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣋⣘⣁⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⢀⣿⠇⠀⣿⣿⣿⣿⣿⠿⣛⣛⣛⡛⣛⣛⣛⡛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣷⠀⠀⠀⢠⣿⠁⠀⣼⡟⠀⠀⣿⣿⣿⣿⣏⢸⣿⣿⣿⡇⣿⣿⣿⣿⠆⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⣼⡟⠀⢰⣿⠁⠀⠀⣿⣿⣿⣿⣿⣦⣙⣛⣛⣃⣙⣛⣛⣩⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠈⠿⠆⠀⢰⣿⠁⠀⣼⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣃⡀⣠⣶⣿⣿⣶⣄⢀⣀⣙⡃⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣿⡄⠄⠀⠀⠀⠀⠀⢀⡼⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠑⢸⣿⣿⣄⢦⠀⠀⠀⡠⠋⡄⣻⣿⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⢭⣸⣿⣦⣄⣀⠀⠀⠀⠀⡴⢫⠖⡂⠸⣿⣿⣿⣿⣷⣮⠭⠀⠊⠐⣱⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠬⠷⠛⢉⣜⣧⡝⡻⢦⣀⢀⡠⠃⠃⠀⠄⠘⢿⣿⣿⣼⣿⠁⠀⠀⠀⣴⡛⠋⢛⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣠⠞⣉⣩⣍⢻⣦⣿⡟⠀⢠⡄⠀⠀⠀⠀⣿⣿⣿⣿⣧⡞⠋⠛⠁⠁⠘⠉⣿⣿⣿⣿⣵⣳⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠒⢲⡃⠨⢻⣷⠟⣻⠁⠀⠠⠿⣦⣀⠤⢤⣤⣿⣷⣿⣿⣿⣆⣀⠁⠐⠊⠄⡛⢿⣿⣿⣿⣿⣷⣿⣿⡄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⢐⠝⠉⣰⡏⣠⡄⢀⡀⠀⠨⢑⠤⡀⠉⠉⠙⢹⣿⣿⣷⣤⡄⠀⠀⠀⠀⠉⠒⠻⣿⣿⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠐⠁⠀⠀⠁⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⣺⣿⣿⡿⠿⣿⡆⠀⠀⠀⠀⠀⣐⣿⣷⣾⣿⣯⠅
⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⢀⣠⠄⡀⠀⠀⣀⠀⠰⠿⠛⡛⠓⠸⠯⠿⠽⠓⠒⠖⠻⡍⠛⠉⠉⠁⠐⠚⠒⠱⠶⠦⠆⠲⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⢀⡉⢀⢀⡀⠀⠁⠀⠉⠁⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠲⠂


⣿⣾⡾⠿⠾⠾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣾
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⡀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⣧⣾⣷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡠
⠛⠻⠿⠿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣹⣧⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠞⠉⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣀
⡄⠀⠠⡤⠀⢉⠁⣀⠀⠙⢧⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢨⡷⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⢁⡆⠀⠀⠀⠀⠀⠀⢠⣷⢠⣦
⣷⡀⠀⢁⡤⠀⣀⣿⡀⠀⠀⠀⣸⣧⠀⣰⡀⠀⣀⣠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠾⠱⠦⠀⠀⠀⠁⠀⠀⠀⢠⡜⠃⠀⠀⠁⠘⢦⠀⢀⠀⠀⢸⣿⣿⣿
⣿⣇⠀⠸⠁⣰⣿⣿⣖⠀⠀⠀⠿⠿⢦⣿⡅⠠⣿⡍⠀⣞⠋⠘⠃⠀⢠⡀⠀⠸⠃⠤⠀⠀⠀⠀⢀⠀⢡⣀⠀⡴⠣⢀⡖⠀⠦⠐⠀⣄⢀⠀⠚⠀⠀⠀⠀⠿⠟
⠉⠀⡀⠀⠀⣿⣿⠇⠀⠀⠈⠁⠀⠀⠸⠿⠛⠀⠈⠁⠈⠉⠀⡆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠒⠛⠠⠆⢨⣤⣵⣄⣠⣄⣆⠰⠇⢱⣿⠈⢠⠂⣴⠀⢀⠀⣤
⢀⡼⠀⡀⢸⣿⣿⡡⠀⠀⠀⠀⠀⠢⣄⢠⣶⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠛⠁⠀⢰⣿⣢⣿⠀⠻⡆
⡜⠁⢠⣿⠀⢀⡀⠀⠀⠛⠷⠖⠆⠤⠀⠈⠛⠓⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠠⠄⠀⠙⠛⣿⣿⡧⠈⠡
⠀⠀⣿⠭⣅⠀⠹⠅⠀⢀⠀⠀⠀⣄⡀⠀⠀⠀⠀⢄⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⣤⣀⡁⠲⠤⠀⠀⠀⢤⡄⠀⠀⠀⢒⡀
⠀⢠⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣣⡀⢀⠀⠀⢤⣀⡀⠈⣿⣄⣀⡀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠀⠉⢭⣦⣀⣁⣴⣶⣶⢾⡇⠀⠀⠤⢀⣄
⠀⠬⠭⠍⠉⠀⠀⠀⣀⣤⣄⣀⣠⣤⣾⣷⠶⠴⠖⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠲⠿⢳⣤⣄⣀⠉⠀⠦
⠀⠀⡀⠀⠀⠀⠁⢀⣀⣤⡾⣟⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠶
⢀⣀⣤⣤⣤⣶⠿⠿⠟⠟⠛
⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⢻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣾⣦⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠁⠀⠹⣿⣿⣶⣄⠀⠀⠀⠀⢿⣿⣿⠟⠁⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⡿⠃⠀⠀⠀⠁⠹⣿⣿⣦⣀⠀⠀⠈⠛⠁⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⠀⠀⠀⠀⠀⠉⢿⣿⣷⡀⠀⠀⠀⠀⠀⣘⣿⣿⣁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠀⠀⠀⠀⠀⠈⠿⢿⣿⣦⣄⠀⠀⢨⣿⣿⣿⣯⣈⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠧⠀⣤⣽⣿⣿⣥⣼⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿
⣠⣤⣤⣆⣿⣷⣾⣾⣶⣿⣿⣷⣷⣷⣦⣶⣶⣾⣷⣶⣦⣤⣤⣶⣤⣤⣦⣤⣤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢒⣲⣖⣒⣲⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣧⣤⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣶⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢍⠙⡶⠊⣀⡤⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦⣶⣿⣷⡟⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠉⠉⠘⢶⣟⣺⣿⢦⣴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⢉⣿⠞⣹⠮⠿⠟⠂⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⣠⢽⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢈⣷⠞⢉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠖⠻⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠌⣿⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠺⠻⠿⣯⣟⢻⣿⣾⣿⣤⣊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠘⠛⠽⣿⢿⣽⣗⣖⣠⢄⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣄⣀⣀⣀⣀⣀⣰⣧⣤⣿⣥⣤⣭⣭⣋⣉⣀⣀⣰⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣝⣛⣩⣛⣻⣟⣽⣟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⢠⠆⠀⢀⠠⠤⢤⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠠⢀⠤⢲
⠀⢀⠀⠀⠀⠀⠀⢘⠉⠉⠉⡁⠀⠀⠀⠀⠀⢠⠕⠠⢀⡀⠀⠀⠀⠀⢠⠖⠉⠉⣐⣂⠤⠚⠚⠉⣰⠔⡇
⡠⠫⣻⠫⣀⠼⢟⣟⣀⣀⠀⠅⠀⠀⠀⠀⠀⠸⠀⠄⠀⠇⠀⠀⠀⠀⡄⠿⣴⡪⡽⢁⡠⢂⣲⡮⢔⡾⢃
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠧⠄⠀⠀⠀⠀⠀⠈⠲⢤⣉⣙⠚⠛⠕⢊⣉⣡⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⢀⣤⣄⡠⠊⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠰⢂
⠀⠀⠈⣷⠋⠀⢀⠀⠋⢂⠀⠀⠀⠀⢀⣀⠀⣦⣴⣋⣉⠖⢫⠀⢀⡀⠀⠀⠀⠀⢨⠛⠓⠚⠓⡀
⠀⠀⠐⡯⠒⠒⢒⠚⢓⢿⠇⠀⠀⠘⣄⣦⣏⣏⠁⠀⡘⡾⢞⢳⣿⡌⠀⠀⠀⠀⠸⣶⣦⣤⣾⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣽⢧⢭⣋⠽⠶⢺⣭⣠⠵⢊⡇⠀⠀⠀⠀⠀⡆⠀⠀⢠
⠀⠀⠀⠱⣄⡀⢀⠀⣀⡜⠀⠀⠀⠈⢫⡑⠒⠦⠵⡀⢀⠾⠗⢚⡽⠁⠀⠀⠀⠀⠀⠇⠀⠀⠆
⠀⠀⠀⠀⠘⠛⠋⠛⠛⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠿⠿⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⡆
⠀⠀⣠⢴⠒⠒⢲⣤⠆⣰⡶⣢⠀⠀⠀⠀⠘⠣⣄⣀
⠀⡘⠁⡐⣰⡾⠋⡉⣞⠋⢞⠉⠀⠀⠀⠀⠀⡖⠋⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠢⢄
⢰⣷⣚⢿⣽⠀⠮⣀⠴⢀⣼⡆⠀⠀⠀⠀⠰⠛⠒⠒⠒⠚⢢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁
⠘⣟⠻⠿⢶⣿⣶⡶⠿⠟⢻⠇⠀⠀⠀⠀⢸⠸⣄⠀⣀⠆⡀⠀⠀⠀⡤⠎⢠⡀⠁⠀⠀⠀⠀⢁⠀⠘⠄
⠀⠘⢿⣒⠤⠄⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠀⠑⢲⠛⡗⠊⠁⠀⠀⠀⠀⢇⠻⠿⠟⠀⠀⠀⠀⠰⠀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣤⣶⣶⣤⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⣿⣿⣿⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣶⣶⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠉⠂⠀⡛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠒⢤⡀⠘⣜⣷⣦⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⡿⢶⣄⠀⠀⠀⠉⢦⡈⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⣀⣼⣿⡻⢇⢤⡀⠀⠀⠀⠹⣄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⣛⠇⠫⠻⠏⠁⠀⠹⢿⣶⣄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⡀⠀⠀⠠⡤⣤⣸⡟⠋⠉⠛⢯⠈⠳⣄⡄⡀⣼⣿⣿⡆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠈⣡⣶⣺⡟⢶⣜⠦⣴⣿⡝⢯⣾⣺⣿⠇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣴⡾⠟⣿⣿⠷⢋⣙⣷⠉⠿⠻⡾⠉⠉⠀⠀⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⢀⣨⡍⠀⠜⠁⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⡀⠀⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡳⣄⠀⠈⢛⡞⠁⠀⠀⠀⠀⢠⠾⠋⡀⠄⣠⣾⣿⣬⣾⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⢠⡛⢦⣄⡀⠀⠏⠀⠀⠉⠀⣘⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⡀⢷⣄⠈⠙⠛⠓⠒⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀
⠀⠀⠀⢀⡴⠒⢋⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠐⠀⠀⠠⢹⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⠄
⠀⠀⠀⡎⠤⠠⠂⠲⠅⠀⣧⠀⠀⠀⠀⠐⣿⠽⠯⠿⠽⢿⡇⠀⠀⠀⠀⠀⠐⣏⠴⣠⣄⣰⡿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸
⠀⠀⠀⢿⠀⠘⠢⢴⡁⢀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⠀⠀⠸⡆⠡⠈⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣰⠁
⠀⠀⠀⠈⠓⠧⣤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠾⠃
⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢠⠖⡛⠉⠉⢉⡉⡙⢦⠀⠀⠀⠀⠀⠀⢀⣖⢒⣆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠚⠓⠲⣄⠀⠀⠀⠀⠀⠀⣴⣞⣿⣿⣿⡶⡄
⠀⠀⠘⣎⡔⠜⠓⢼⠵⢈⠟⠀⠀⠀⠀⠀⢰⡉⠤⡠⠭⣿⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣘⡇⠀⠀⠀⠀⠀⢿⣯⢿⢹⢻⣽⠇
⠀⠀⠀⠘⣄⣸⡃⣭⣰⠏⠀⠀⠀⠀⠀⠀⢸⠄⣀⡼⡀⣼⠂⠀⠀⠀⠀⠀⣷⣦⣤⣼⣧⣤⣴⡇⠀⠀⠀⠀⠀⢻⠛⠛⠛⠛⡗
⠀⠀⠀⠀⠘⣏⡵⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡤⠜⠁⠀⠀⠀⠀⠀⠈⠲⣀⣓⠉⢀⢛⡸⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠿⠂⠀⠀⠀⠀⢠⠋⣿⢁⣾⡇⢳⠀⠀⠀⠀⠀⡶⢷⣾⡶⣄
⠀⠀⢠⢴⠞⠛⠛⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⡜⠀⡟⠻⣴⡇⠘⡇⠀⠀⠀⠀⡏⢦⢻⣿⣿⣆
⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢴⡖⠚⣛⣛⡒⢲⡆⠀⠀⠀⠀⠀⣧⠀⣫⡴⢛⡆⠀⡇⠀⠀⠀⠀⡇⠀⠱⡻⣿⣿⣄
⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⢸⣤⣀⣀⡀⣀⣸⠃⠀⠀⠀⠀⠀⢣⠀⢿⠙⢺⡇⢠⠃⠀⠀⠀⠀⡇⠀⠀⠙⢝⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡏⠛⠛⠻⠟⢻⠀⠀⠀⠀⠀⠀⠘⣄⣸⣞⢋⣧⡼⠀⠀⠀⠀⠀⠧⠤⣤⣄⣈⣮⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⡏⠉⢳⠏⠉⢷⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢱
⠀⠀⠀⢠⣇⡀⣀⢆⣀⣾⠀⠀⠀⠀⢀⡤⠚⠉⠁⠆⠉⠑⢦⡀⠀⠀⠀⠀⣀⢔⠚⠿⠟⢷⢦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡾⡛⠁⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⢸⡀⠀⣸
⠀⠀⠀⢸⡀⠀⠀⠀⠀⡞⠀⠀⠀⠸⣽⡀⠀⠀⠀⠀⠀⠀⢀⣞⡶⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠸⡀⠀⡸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⠻⣖⡒⠒⣲⠾⠛⠁⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠓⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣀⣀⣀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣴⣾⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣶⣶⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣉⣤⠒⠒⢲⠒⠠⢌⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠐⣎⢦⠀⠘⡀⠀⣠⠜⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠃⣼⣁⠀⡀⠈⣆⠻⠀⡦⢊⠴⠊⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠂⠠⢤⣄⡉⠀⡀⠀⡋⠒⠒⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⣠⣞⡵⠞⣳⡌⢫⠉⠁⠀⢐⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠻⠙⢫⠋⠁⠈⠀⡇⢿⠉⢦⡱⡀⢀⠎⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠉⠀⡀⠀⢀⡸⠀⢀⡱⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠈⠈⣿⡉⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣀⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁⠀⠉⠉⠉⠁⠀⡀⠄⠂⠙⡗⠲⣤⣀⣀⣀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠙⢆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⢀⠔⠈⠁⠀⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⡈⠀⠠⠂⠁⠑⢄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡔⢃⠀⠀⣀⣀⣀⣀⠀⢦⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠈⢠⡡⣀⠤⠒⠉⠀⡀⠀⠀⡏⠉⠉⠉⠀⠀⢱⠀⠀⠀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⢀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⠓⠴⢦⡂⠀⠀⠀⠀⠀⡕⠸⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⡰⠁⠀⢠⠧⢄⡀⠀⢩⠁⠒⠒⠂⠐⠈⠁⠀⠀⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣁⠀⠰⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡀⠀⠀⠀⠈⠢⢀⡀⠀⢀⠄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⠙⠶⢤⣀⣀⣀⣀⡀⢀⣀⣀⡠⠴⠚⠋⣠⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⠲⠤⣤⣀⣀⣈⣉⣀⣀⣤⠤⠖⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⡾⡧⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡷⣥⡀
⠀⠙⠋⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣽⣯⡧⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠠⡷⠷⢚⠆⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣟⠐⠀⠀⠀⣀⠀⠀⠀⣴⣖⣉⡁
⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡷⡿⠉⠀⠠⠤⠴⠦⠄⠀⠗⡆⣖⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠤⠰⠀⠀⠀⢛⣻⣿⠒⠒⠒⠒⠶⠒⠂⠈⢷⡥⢶⠛
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⡤⠄⠀⠀⢀⠀⠀⠀⡼⠓⢘⡆⠀⠀⠀⣶⣾⠄⠼⢾⡗⢀⣸⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠰⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠨⠇⠀⠀⠈⢉⠀⠀⠰⠛⠃⡠⠷⠀⠀⠀⣟⠻⠀⡈⢿⡅⠸⠁⣇
⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠘⢶⡀⠀⠀⢀⡤⢾⠭⣧⢀⠀⣂⣀⡀⡰⠛⢠⣤⣼⣉⢛⣿⢻⠄⠀⢀⡇⠀⣧⠁⡞⣇⠐⡟⣿⠒
⠀⠀⠀⡇⠀⣼⣠⣸⣧⣷⣶⣶⣿⢶⠶⠿⠿⢷⣛⣛⣛⡙⠯⠭⢿⣷⢰⣾⠿⠻⣿⡿⣿⡟⣷⣾⣿⣴⣦⣻⣶⣯⣿⣅⣿⣿⣷⡆
⠀⠀⣠⣧⣸⡭⣟⣹⣿⣻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠛⠓⠦⠬⢭⣍⣀⣚⠛⠾⠿⢿⣷⣿⣿⣿⣿⠍⣿⣿⣿⣷⡽⠿⣷⠃
⠀⢠⠤⣗⣨⠶⣫⠽⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⢠⡀⠀⠀⠀⠀⠉⠉⠙⠓⠒⢶⣤⣤⣥⣉⡛⠺⠿⠿⣿⣿⣷⣶⣾
⠀⠀⢘⣩⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢯⣷⣦⣤⣄⡉⠁
⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣿⠈⢷⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣴⣦⣀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⣀⣀⡁⢤⠀⡄⢠⠀⠀⠒⢠⣄⢠⠀⠂⠀⠀⠀⠀⣿⠀⢹⡆⠘⢧⡀⠀⠠⠛⠶⠀⠀⠀⠂⠀⠀⠀⣠⠶⣥⣛⠶⣮⠃⠀⠀⠀⠀⠀⠀⢀⣰⢶⣄
⠀⠀⠉⠙⠉⠁⠘⣄⠏⠸⠀⠀⠰⠶⠶⠶⠶⠛⠀⠀⠀⠀⠻⠇⠈⠁⠀⣰⠇⠀⠀⠤⠰⠶⠔⢠⠀⠀⠀⠰⢫⣾⢱⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⣾⡗⣬⠟
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠦⠶⠞⠉⠀⠀⠀⠀⠀⠀⢄⠶⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠔⠢⡀
⠀⠀⠀⠸⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣄⠾⠂⢠⠀⠀⠀⠷⣿⣷⣿⠿⠃⠀⠀⠀⠴⠀⠉⠒⢄⢀⡀⠀⠀⠀⢰⣶⣶⣶⣾⣧
⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⢠⠖⠐⣮⠙⠃⠀⠀⠀⠀⢷⡆⠉⠀⡠⠋⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⣀⣾⠃⣀⡀⠙⠷⠀⠀⠀⠈⠉⣙⠿⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢰⠈⠁⠈⢁⡀⠐⠀⠀⠀⢀⡀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠶⠀⠰⠏⠁⣀⣉⣁⡀⠀⠀⠀⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠉⠛⠉⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠰⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠉⠁⠂⠀⠀⠀⢠⡀⠀⢛⠿⠀⣀⠀⠀⠀⠀⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣠⣄⢀⡀⠀⠀⢴⠁⠐⠀⠀⠀⠀⢸⡇⠀⠀⠙⠂⠀⠐⠀⠈⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠀⢉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠉⠁
⠀⠀⠀⠀⠀⢨⣯⡹⡭⠀⠀⠀⠁⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠟⡷
⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢐⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣦⠀⠀⠀⠀⠀⢀⡀⠲⡀⠚⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠛⠃⠀⡘⠛⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠀⠀⢁⡇⠀⠀⠀⠄⠘⠃⢀⡵⠚⠃⠀⠤⠀⠀⠈⠂⠀⠃⠀⠀⠀⡟⠻⠿⠿⠛⢋⢳⡆
⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⡄⠀⠀⣤⠀⠀⠀⠀⠀⢤⡄⠀⠠⢦⣤⠇⠀⠐⠲⠖⠲⠟⠁⠀⠀⠀⠀⠲⠐⠆⠀⢠⡤⠄⠀⠀⠀⠈⠢⠄⠀⠴⠋⠉
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⣮⠀⠀⠀⠀⠀⠈⢀⠀⠐⠀⠀⠀⠀⠀⢀⠄⠴⣿⡟⠀⠀⢀⠋⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠒⢠⡆⢠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠠⠀⢀⡤⠈⠀⠀⠀⠈⠀⣄⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⣄
⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠⠀⠀⠠⢾⣥⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠊⠀⠀⠀⠀⠀⠀⠉⠉⡆⠀⠀⠉⠀⠀⠀⠈⠀⠂⠰⣆⣶⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⠘⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠊
⠀⠀⠀⠀⠸⠷⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢟⡻⢿⣧⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠶⠶⠀⠀⠰⠾⠦⠀⠀⠀⢰⠆⢀⡛⡄⠀⡀⠀⠀⠀⣠⣶⣶⣴⡆⠐⠓
⠀⠘⢶⣶⣄⣤⣴⣶⡞⠀⠀⠈⠛⢧⣼⣿⡶⠟⠀⠀⠀⡀⠈⠁⠈⠀⡀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠂⠀⠰⠃⣠⢻⡿⠓⢾⠉⢓⡄⠘⠿⠶⣤⡞
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠞⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⢾⣤⠴⠛⠓⠋

⠀⠀⠀⢀⣤⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⢿⠆
⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠤⣴⡄⠀⠀⠀⠀⠘⠿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢷⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖⢠⡀⠀⠀⠀⠀⠀⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠄⢀⡤⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢲⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠓⠇⠀⠀⠄⠀⠀⠀⠙⠷⣶⣶⠶⠟⠁
⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠆⠈⠄⠀⠑⠄⠀⠀⠀⠀⠀⠀⣀⠀⠠⠔⠀⠀⠀⣤⣙⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣧⡄⠀⢀
⠀⠀⠀⠴⠆⠀⣤⣾⠃⠀⠀⠤⠤⠤⠌⠉⢫⠀⠀⠀⠀⣤⣹⣿⣧⣄⠉⠀⠀⠀⠀⠀⢠⡄⠐⢳⡄⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢠⡤⠲⡦⠴⠶⠀⠀⠂
⠀⠀⠀⠤⠄⠻⣄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⠋⣻⡑⢫⣧⠀⠀⠀⠀⠤⠸⠃⠶⠊⠀⠀⠀⠀⠿⠟⠀⣀⢠⣦⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠂⠐⠛⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠉⠀⠀⠀⠈⠤⢀⣀⣀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠉⠉⠉
⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡦⠄⠀⠀⠀⣀⠐⠂⠀⢠⡤⠀⠀⠀⠀⠠⠶⠶⠦⠀⢀⠀⠀⠀⠀⠛⠏⠀⠀⣤⢶⠒⣄
⠀⠀⠀⣤⣴⣶⡆⠀⠀⠀⠀⠰⠀⠀⠀⠹⠀⠀⠀⠀⣴⠿⠛⠉⠀⠀⠀⠘⠀⠀⠁⠀⣄⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡌⠄⠀⠀⠀⠁⠂⠂⠀⠐⠫⠜⠋
⠀⠀⠀⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠟⠃⠀⠀⠀⢈⣿⡷⠲⠶⠿⠓⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⡿⢿⣿⡿⠛⢿⡿⣿⡿⣿⠿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣥⣤⣼⣷⣠⣾⣠⣴⣷⣬⣴⣿⣥⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣆⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⢻⣿⣿⣿⣿⣿⣿⡿⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⠀⠀⠀⠀⢀⣼⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢡⣾⣿⣷⡘⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⡟⠁⠀⠹⣆⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠈⠉⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣴⣦⣤⣴⣦⣤⣴⣦⣤⣴⣦⣤⣴⣤⣤⣤⣤⣤⣶⣤⣤⣦⣤⣴⣦⣤⣴⣞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⢀⣴⣄⡀
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⢿⣿⣿⣿⣿⣿⠟⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣄⠀⠀⠀⠀⢀⣾⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣾⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⠟⠛⠻⣄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⢡⣶⣶⣦⢹⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣏⠀⠀⢀⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠶⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠐⠀⠀⠀⠤⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⡰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢿⠧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⢀⠠⠐⠂⠉⣀⣀⣀⣀⡀⢁⠶⠭⣓⠦⡀⠀⠙⠔⢅⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠖⠀⠠⠖⠛⠉⢉⡔⠣⢌⢙⠣⣍⡒⢌⠳⡄⣄⠀⠀⡀⡀⡢⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌⠀⠴⠐⠁⠀⠀⠀⠀⠀⠑⠰⢯⡈⠑⠠⣁⠡⡄⢉⠀⢣⠙⣌⠳⣄⠀⠘⠄⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠑⠈⠢⠑⡇⠘⡄⠠⡀⠑⢬⣑⠲⣄⠀⠂⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⡀⠀⠘⡀⠡⡘⢤⡘⠂⡀⡀⠂⢤⡑⣌⢣⠁⡀⠈⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⢀⠂⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡁⠑⣄⠳⡜⢆⠱⢌⠢⢤⡂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⡙⢅⠈⢢⠙⠀⡑⠦⡽⠊⠀⢦⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠣⡀⢡⠈⠢⢌⡀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⡐⠀⠌⠀⠀⠀⠈⠃⠠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢆⡈⠑⢢⠍⠀⠀⠀⠀⠐⠀⠑⡀
⠀⠀⠀⠀⠀⠌⠀⠘⠄⠀⠀⠀⠀⠠⠀⢳⡀⢀⠀⠤⢤⠤⠂⠀⠲⠂⠰⠂⠐⠢⠄⢀⣀⠀⠀⠀⢠⠏⠢⠌⠛⠀⠀⠀⡀⠀⠀⠀⠀⠄⠈⠄
⠀⠀⠀⠀⠌⠀⡐⠀⠘⡀⠀⠀⠀⠀⡡⠤⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠰⣏⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠈⢂⠈⠂
⠀⠀⠀⠐⠀⡐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠈⢄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠛⢛⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣬⣉⠙⢀⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⠥⡬⠀⠥⠬⠼⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠊⠀⠰⠀⠀⠀⠈⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢰⠞⠉⠉⠉⢳⡀⠀⠀⢀⡞⣁⣀⣀⡠⠤⠤⠖⠒⠀⠒⠢⠐⠄⠳⢤⣙⣧
⠀⡴⠖⠛⠂⠀⠀⠀⣸⡃⠀⣴⠛⠏⠠⠀⠀⡆⠀⠈⠀⠀⢘⠀⢰⠀⢠⠀⠘⡀⠉⣧
⢸⡇⠀⠀⠀⠈⢢⠀⠈⢿⡀⡏⢠⠀⢈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⡄⠀⠀⠀⡇⠘⣸
⠀⣿⠂⠀⠀⡀⠈⠀⠀⣸⡿⠛⠀⢤⣚⣩⣭⣩⣒⡉⠉⠀⠴⠖⠒⢆⣀⡐⣖⣳⠂⠉⢳⡄
⢀⣸⡇⠀⠀⠁⠀⣤⠞⢹⠃⠀⠀⢰⠀⢀⠊⣴⡿⢷⠀⠀⠀⡴⢷⣦⠑⡄⠈⡇⠀⠀⠈⣷
⡟⠀⠀⠀⣆⠀⠀⡹⠶⠛⠠⡀⢀⡸⢠⣬⣄⣛⣿⠞⣠⣤⣄⢻⣿⠟⡠⣁⢀⡃⠀⠀⠀⣿
⠻⢤⣠⣤⠁⣀⢀⣠⡶⠓⠲⣦⣠⣾⣻⠟⠟⠩⢋⠕⠛⣏⠙⢌⠫⠑⠟⢿⢙⣷⣤⡤⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠁⢀⣴⠞⠋⠉⠙⡇⠀⠀⠀⠀⠀⠚⡍⠂⠀⠀⠀⠀⠀⡋⠉⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠔⠊⠘⠀⠀⠀⠀⠀⠀⠜⠙⢆⠀⠀⠀⠀⠀⡈⠉⠑⠤⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣀⣀⡀⢳⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠄⠊⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⠒⠂⢲⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠤⢽⡇⠀⠐⠒⠀⠤⢼⠀⠀⠀⠀⠀⠀⡸⠄⠒⠊⣿⣤⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠁⠀⠐⠒⠂⢎⣰⣣⣆⣴⣰⣠⠗⠒⠈⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣛⣦⣤⣀⣀⣀⣤⣐⣿⣾⣓⣠⣄⣀⣀⣠⣴⣞⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠀⠈⠀⣼⣿⡀⠈⠀⠀⠀⠈⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⢷⡳⢄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣯⣓⢦⡀
⠀⠀⠀⢀⡴⢋⠡⠈⠄⡀⠍⠙⣦⠀⠀⠀⠀⠀⣰⣒⣚⣛⣛⣶⡀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⠀⠀⠀⠀⣠⢞⣩⠴⣿⣼⣚⣍⣳⣽
⠀⠀⠀⢸⡧⠬⠤⠤⠤⠤⠤⠤⣼⠆⠀⠀⠀⠘⣧⡤⠤⠤⠤⣤⠇⠀⠀⠀⠀⠀⣞⡉⠉⣧⠀⠀⠀⠀⣼⢡⢏⠤⡅⡏⠿⢟⣫⡵⣆
⠀⠀⠀⢸⡗⠛⢒⣒⣛⣒⠒⠓⢺⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠸⡇⢹⡇⣸⠀⠀⠀⠀⣇⢸⠘⢒⠋⠷⢚⣭⡀⡇⣿
⠀⠀⠀⠀⢿⣙⣯⡉⠉⢉⣽⣋⡟⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡟⠉⠀⣿⠀⠀⠀⠀⠹⡜⢧⡻⠗⣡⣌⣋⡼⣡⠏
⠀⠀⠀⠀⠀⠉⠉⠙⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣸⠃⠀⠀⠀⠀⠀⢠⣇⣀⣀⣻⠀⠀⠀⠀⠀⠙⠦⣍⣛⣓⣛⣩⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⣠⣞⢻⡏⣿⢓⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢿⡳⣄
⠀⠀⠀⢸⠷⠾⠮⠷⠧⠾⠶⣿⠀⠀⠀⠀⠀⠀⣤⣤⣯⣯⣤⣄⠀⠀⠀⠀⠀⣠⠖⠋⡀⢀⠐⡈⠓⢦⡀⠀⠀⠀⠀⣰⢋⠀⡁⢈⠙⣿
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⢛⡧⠀⠀⠀⠙⣶⠓⡶⠚⠲⡖⢳⡞⠁⠀⠀⠀⠀⣏⠠⠀⠄⣠⡼⠁
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣀⣿⣀⣀⣀⣀⣟⣋⣀⠀⠀⠀⠀⢸⡄⢹⠀⢸⠃⣼⠀⠀⠀⢀⣤⠞⣩⠷⠖⠛⠁
⠀⠀⠀⠀⠸⢤⡤⠤⠤⢤⠼⠀⠀⠀⠀⠈⠧⣄⣀⣀⣀⣀⣀⣀⣤⠟⠀⠀⠀⠀⠀⠳⣼⣧⡾⡴⠃⠀⠀⠀⠘⢦⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⣳⣆
⠀⠀⠀⠀⠀⢀⣀⡤⠞⣧⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣄⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⡄⠀⠀⠀⣸⣥⣤⣽⣧⣬⣧
⠀⠀⠀⠀⣰⠯⠤⠤⠤⠬⢷⡀⠀⠀⠀⣰⠋⠀⠀⠏⠀⠀⠟⠁⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣠⡏⠙⠓⢿⠀⠀⠀⠹⡞⠛⢚⣷⢶⡏
⠀⠀⠀⠀⡷⠶⣤⠴⢤⡴⢶⡇⠀⠀⠀⣟⣶⣒⣲⣒⣲⣒⣒⣖⣒⣶⣻⠀⠀⠀⠀⠀⢀⡞⠉⠙⠦⣄⣸⠁⠀⠀⠀⢻⣚⣩⡴⡾
⠀⠀⠀⠀⢹⡀⢸⠀⢸⠀⣸⠁⠀⠀⠀⣿⠦⠭⡽⠯⢥⡬⠵⠮⡭⠴⣻⠀⠀⢀⣠⢾⡉⠙⢦⡀⠀⢈⡗⠀⠀⠀⠀⠈⣏⣡⢴⠃
⠀⠀⠀⠀⠈⣧⣸⣄⣼⣠⡇⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠚⠛⠛⠉⠁⠀⠀⠘⠲⢤⣷⣀⣠⠿⠖⠋⠀⠀⠀⠀⠀⠀⠹⣄⡟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⢶⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⢀⣤⡀⠀⠀⠀⠀⢠⡤⠤⣤⢀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡿⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣧⡾⡶⣴⢧⡀⠀⠀⣴⠋⣄⠙⣶⠋⣄⠙⣆⠀⠀⠀⠈⡏⠉⡇⠸⡷⢾⡇⠀⠀⠀⠀⠀⠀⣠⢞⡼⣳⠋⠈⢹⡄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠶⢿⣹⠋⠀⠀⠰⡇⢀⡼⠀⢸⢀⡞⠀⢸⠀⠀⠀⢀⡇⠸⣧⡞⠁⠈⢳⡆⠀⠀⢀⡴⢞⡵⣫⠟⠁⠀⠀⣸⠃
⠀⠀⠀⣴⠋⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠀⢳⡈⠃⣠⢾⡈⠇⣠⠟⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢿⢚⡿⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠘⢦⣀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠙⠶⠃⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠈⠻⣄⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠲⠚⠓⠒⠶⠚⠃⠀⠀⠀⠀⠈⠉⠉

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

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣠⢞⡥⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⡖⢋⣩⡼⠷⡄⠀⠀⠀⠀⠀⠀⢠⢞⡭⠂⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⡞⢡⠘⣿⡆
⠀⠀⠀⠀⣏⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣼⠛⣉⣥⠴⠾⡇⠀⠀⠀⠀⠀⠀⡏⡞⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⡇
⠀⠀⠀⠀⠹⣌⠆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢼⣋⠉⣤⠶⡟⠁⠀⠀⠀⠀⠀⠀⣧⢣⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡧⢸⠀⣿⡇
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣤⣿⣤⣧⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⣯⣼⣤⣿⡇
⠀⠀⠀⠀⠀⠈⣳⣶⣶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠐⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿

⠀⠀⠀⠀⠀⣀⢀⣤⢠⣄⢀⡀⠀⠀⠀⠀⠀⠀⢠⠖⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣤⣶⣤⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣰⣞⠛⣶⡀
⠀⠀⠀⢀⣘⠋⢾⡏⠉⡽⠎⢛⣀⠀⠀⠀⠀⠀⠙⠦⣀⣬⣬⡃⣤⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⡿⠉⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡙⠿⠾⠏⡁
⠀⠀⠀⠾⠿⠸⡿⠀⠀⢛⡿⠸⠿⠆⠀⠀⠀⠀⠀⢠⣿⣿⡸⢞⣵⢟⣀⠀⠀⠀⠀⠀⣧⣿⣿⡇⠀⠘⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡸⠁
⠀⠀⠀⢶⣶⠰⣿⡶⢠⣾⡷⢰⣶⠆⠀⠀⠀⠀⠀⠘⣿⣿⣧⣌⠡⠿⡋⠀⠀⠀⠀⠀⠹⣿⣿⣷⡗⣰⣿⣿⠇⠀⠀⠀⠀⠀⣠⠀⣾⣮⣩⣶⡆⢠⡀
⠀⠀⠀⠈⢩⣄⢶⣷⢻⣶⢆⣬⠉⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⢻⡂⢯⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⢸⣿⠀⠿⠿⠿⠿⠇⢸⣿
⠀⠀⠀⠀⠀⠈⠀⠛⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠑⠻⢦⡀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⣀⠀⢀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⣿⣿⣷⣿⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⢸⠟⢉⣉⠛⣿⠀⢠⢀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠻⣿⣿⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣄
⠀⢀⡄⣿⣿⣿⣿⢰⣀⣠⣶⢾⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⣴⣴⠀⠀⠀⠀⠀⣿⣿⡈⠀⢁⣿⣿⠀⠀⠀⠀⠀⡞⠀⢠⡀⢙⣶⣄⠀⠀⠸⡆
⠀⠘⠇⣿⣿⣿⣿⢸⣿⠶⠶⢸⣿⠀⠀⠀⠀⠀⠀⣿⣶⣷⣾⣷⣿⠀⢿⢿⠀⠀⠀⠀⠀⣿⣿⠟⠛⠛⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⠀⠀⢀⡇
⠀⠀⠀⣿⣿⣿⣿⠸⣿⠛⢻⢸⣿⠀⠀⠀⠀⠀⠀⠙⠿⣿⢿⠿⠟⠀⢸⢸⠄⠀⠀⠀⠀⣿⣿⣈⣠⣀⣿⣿⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠉⠉⡇⢀⡼⠁
⠀⠀⠀⠛⠋⠩⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣄⣀⣠⣎⡞⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣤⠼⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⠀⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⢫⣻⠇⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⢀⠰⠚⠋⠁⠸⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⠀⠀⠀⣾⣭⣶⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⣤⡤⢸⠩⣭⣭⡭⢻⠠⣤⡀
⠀⠀⠀⠉⠛⠗⣶⣶⣤⣀⡀⡶⠒⢦⠀⠀⠀⠀⠀⠀⢨⡿⢿⡅⠀⣴⡟⠉⢻⣆⠀⠀⠀⣶⡆⠀⢬⡥⠀⢠⣥⠀⠀⠀⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡆
⠀⠀⢠⣶⣶⣴⣬⣿⣿⣿⣿⣿⣾⣿⣷⣦⣄⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⣾⣧⠀⢸⣷⠀⠀⠀⠿⠟⣵⡎⠿⠿⠿⠿⠏⣶⠹⠿⠃
⠀⠀⠘⠿⣟⢙⡿⠿⠿⠿⠿⠿⢿⡟⢙⡿⠿⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⡟⠀⠀⠀⣿⡇⠀⣿⣿⠀⢸⣿⠀⠀⠀⢸⣿⣮⣴⣿⣿⣿⣿⣷⣭⣾⣿
⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠉⠀⠀⠉⠀⠉⠀⠀⠀⠃⠀⠀⠀⠙⠁⠀⠙⠋⠀⠈⠋⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⢶⣤⣠⢤⣤⠤⢤⣤⡄⠀⠀⠀⣤⡤⠀⠒⠶⣆⠀⢂⢶⢶⣖⣀⣶⢶⣶⣆⣲⢲⡖⢂⠒⣶⡆⠀⣠⠄⠀⣲
⠀⣆⠁⠀⠀⠉⠛⠋⠀⠻⠦⣤⡶⠏⠀⠀⠀⠀⠈⠙⠓⠁⠀⠘⠙⠁⠈⠳⠯⠵⠋⠙⠃⠎⠀⠙⠲⠋⠀⠸⣿
⠀⡟⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾
⠀⣷⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡻
⠀⠣⢑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠿
⠀⠶⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⣛⡆⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣼⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤
⠀⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠈⠻⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠶⠖⠒⠚⠉⠉⠉⠛⠛⢚⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⢰⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢉⣙⣛⣛⣓⡚⢛⠚⣒⣙⡉⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⢸⠻⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⣯⣻⣏⣏⣿⣹⣹⣏⣿⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣦
⢨⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡉⢩⠛⠉⠉⠉⣿⡉⠉⡇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡶
⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⣇⠐⢳⡐⡂⣰⢻⢁⠄⡇⠐⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣦
⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡧⡬⡥⡾⡾⡶⣿⢷⣶⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿
⠀⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠑⠧⢧⣧⣧⣇⣿⣾⠿⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⣭
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠶
⠀⣶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢅⢀
⠀⣮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⢿
⠀⡷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣬
⠀⣿⡄⠀⣠⠦⣄⠀⡠⠠⣄⣠⢖⣲⢦⡀⢀⣄⡄⠀⢀⢤⣄⡀⠀⠀⠀⠀⣰⠾⠛⠲⣦⠀⣠⣤⣀⠀⠀⢀⠹
⠀⠯⠀⠐⠋⠀⠸⠿⠤⠁⠼⠧⠿⠽⠿⠷⠿⠉⠽⠷⠷⠩⠀⠹⠶⠄⠀⠚⠛⠀⠀⠀⠘⠛⠓⠒⠛⠓⠋⠙⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢒⣁⡤⣠⣦⣤⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠛⠛⢻⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⣤⠀⡏
⠀⠀⠀⠀⠀⠀⢠⡲⣿⣿⡆⠀⠀⠀⣠⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⣀⢰⣿⡄⠀⠀⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠟⠁⢀⣀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⠀⠀⡀⠒⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣇⠂⠀⠀⠀⠁⡴
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⡻⣿⣿⣷⣄⡀⠶⣾⡁
⠀⠀⠀⠀⠀⢀⠖⠛⢢⡀⠙⢮⡛⠿⠿⢿⣿⣿⡻⡢⡀
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠹⡄⠀⠱⡄⠀⢰⣿⣿⣿⣇⢡⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢹⡀⠀⠙⣄⠀⢹⣿⡿⣿⣿⣇⠀⠀⠀⠀⢾
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢧⠀⠀⠈⠉⠈⢿⣷⠘⠿⠸⡄⠀⠀⡰⠛
⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢱⠀⡸⠁⠀⠘⢿⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣰⠁⠀⣤⠀⢺⡏⠁
⠀⠀⠀⡇⠀⠀⠀⢀⠀⢾⣣⡟⠀⠀⠀⢀⣀⣠⠀⠁⠀⠈⢁⡠⠂⠀⠙⠛⠛⠀⢀
⠀⠀⠀⢡⠀⠀⠀⡡⢖⢛⡷⠒⠘⠉⠉⠁⠀⢰⠀⠀⠀⠀⢸⣽⣇⠤
⠀⠀⠀⠘⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠖⠒⠉⢸⠐⠄⣾⣛⠛⠛⠛⣛⠛⠃⡄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠏⠀⢀⠀⠀⢸⡄⡆⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⠲⠆⠶⢶⠒⢚⠋⠍⠅⠤⢨⡧⠷⠶⠶⠾⠶⠶⠞⠣⠶⠀⠀⠀⠀⠀⠠⠀⠼⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡆
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⣿⣿⠟⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡠⠀⡀⠀⢀⠢⢤⡀⠀⠀⡏⠀⢻⢀⡤⣠⠊⠉⠀⣀⠤⠄⠀⢞⢡⠀⠑⢆⠀⠐⠤⡀⠀⠀⡞⢳⡄⠀⢠⠟⠓⠦⣔⠒⠓⠦⣄⣀
⠀⠀⢀⠰⢀⠑⣄⢻⠀⡠⠓⠀⠀⠈⠓⠁⠸⡀⣀⠇⠀⢸⣇⠀⠀⠀⠈⠉⠁⠉⠁⠤⠤⠒⠳⡄⠀⣧⣸⠃⢀⡟⠀⠀⠀⠈⠙⠒⠶⢤⡈⠉⠉⢱⡀
⠀⠀⠀⠂⢢⣽⣿⡻⠋⡠⠔⠒⠉⠉⠓⠦⣀⠀⠀⠀⠀⢸⣧⠤⢄⣀⡀⠀⠀⠀⠈⢉⣀⡈⢽⡃⠀⠼⠁⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⢀⡟⡄
⠀⠀⠀⢊⢀⠔⠁⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⡅⢀⡀⠀⠈⣈⡉⢉⣉⣀⠀⠀⣰⠇⠀⠀⣠⠞⠀⠀⠀⠀⠁⠀⠀⣠⢎⡵⠁⠀⡼⡸
⠀⠀⠀⠀⠁⢀⡎⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⣇⠀⠙⠒⠦⠬⠤⠤⠄⠀⢀⣀⠠⠤⠘⠁⠀⠀⣧⣀⠀⠀⠐⠀⠀⠀⣰⢣⠏⠀⠀⢠⢇⡇
⠀⠀⠀⠀⠀⠸⣇⠠⡀⠀⠀⢀⡞⠁⠀⠀⠙⢦⠀⠀⡿⠀⡴⠾⠶⡆⠀⠀⠀⠀⠀⠠⢤⣄⣀⠀⠀⠀⠘⠫⢽⣲⡦⣄⠀⢠⠇⣾⣀⣀⣠⡼⢸⠁
⠀⠀⢀⣤⣄⠀⠙⢦⡀⠀⠀⠘⣇⠀⡀⢀⡄⣸⠀⢠⠇⢸⠁⠀⣰⠏⠀⢠⠒⠒⢦⡀⠀⠀⠈⣧⣀⣀⣀⣀⣀⡈⠙⠲⣝⣫⠞⠓⠶⠶⠶⠖⠋
⠀⠀⣾⠁⠈⠳⣄⠀⠉⢦⡀⠀⠈⠳⠬⠵⠞⠁⢠⠟⠀⠘⠒⠚⠁⠀⠀⠘⢮⣀⣠⡇⠀⢰⣯⣽⣏⣀⣠⣤⠤⠽⠇⠀⠀⠀⠀⠀⢠⠖⠒⠲⡄
⠀⠀⣿⡄⠀⠀⠈⠑⢤⣀⠑⢄⡀⠀⠀⢀⣠⡴⢋⣠⠴⠖⠒⢛⡓⠒⠲⢦⡀⠀⠀⠁⠀⢸⠀⠀⢹⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏⠀⠀⠠⢹⡀
⠀⠀⠸⣷⡀⠀⢀⣤⣤⡈⠑⢦⡈⠉⠉⠉⠁⡴⢯⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠲⢄⡀⠀⠸⣄⠤⠤⢷⡤⠤⠤⣤⡇⠀⠀⠀⢠⣾⠀⠀⠀⠀⠃⢦⡀
⠀⠀⠀⡟⣧⠀⢻⠀⠀⡝⣦⠀⢹⡄⠀⠀⡞⠀⠈⣧⠈⠀⠀⠀⠀⠀⠀⡎⠀⠀⢠⢙⡆⠀⣟⠒⠒⠚⣷⠚⠛⢉⡇⠀⠀⠀⠸⡍⢙⠋⠙⠉⠉⣩⠁
⠀⠀⠀⢿⠘⣇⠀⠳⢤⣤⠟⠀⢰⠇⠀⢰⠳⣕⠀⠘⣆⢀⣒⣒⣀⣀⣸⣇⣀⠐⣡⠞⢳⠀⢻⠀⠀⠀⠸⣆⠀⢸⠃⠀⣀⡀⠀⠙⠾⣷⣤⡤⠞⠁
⠀⠀⠀⠈⢧⣈⠳⢤⣀⣀⣠⣴⠟⠀⠀⢿⣐⣨⠷⠞⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⢧⣀⣸⠇⠈⢧⣀⣀⣀⣉⣠⠜⠀⢯⠉⢧⠀⢀⣠⡴⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠋⣁⣀⣀⡀⠀⢉⣁⡀⠀⠀⠀⠸⡳⣶⣄⡇⠀⠀⠀⠀⠈⠁⢀⣴⣦⣍⡉⠉⠀⠀⠀⠀⠈⠓⠻⣄⣈⡓⠶⠴⠶⠞⠃
⠀⠀⠀⢰⡒⠦⣴⡒⠋⢉⣉⣀⠠⠀⠉⢶⡁⠀⡝⠀⠀⠀⣠⠽⠾⢿⣧⣠⠤⠤⣄⡀⠀⢸⣝⡻⠷⣯⣭⣍⣉⣉⣉⣛⠲⢤⣀⠈⠉⠓⠦⣄
⠀⠀⠀⠘⣧⠀⠱⢭⣒⣤⣀⠀⠠⠄⠀⠀⠱⡀⢨⠆⠀⣼⢡⠊⠈⠀⠈⠁⠀⠀⢠⠹⡄⠈⠻⢿⣷⣶⣶⣯⣭⣭⣽⣛⡻⠶⣌⢳⡄⠀⠀⠘⣆
⠀⠀⠀⢰⡿⡦⢤⡘⣿⣭⢻⢷⣦⣄⡀⠀⠀⠑⣌⠆⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠶⠤⢤⣉⣉⣉⣉⡉⠛⠻⢷⣼⠋⠀⠀⠀⠀⣿
⠀⠀⢀⣀⠓⢬⣒⢵⡀⠛⠉⣟⠀⠛⢿⢗⠦⣄⡈⣧⠀⠸⡆⠀⠀⠀⠏⠘⠆⠀⠀⢰⠁⠀⢸⠭⠶⠶⠶⠶⠶⠦⣭⣍⡓⠤⡉⠀⠀⢰⠋⠉⠉⣹⠂
⠀⠀⠀⠈⣳⡀⠈⠑⠮⣓⢦⣈⠉⠳⡀⢀⡙⠲⠽⠇⠀⠀⠹⣔⠤⡀⠀⠀⠀⠄⣠⠏⠀⠀⠘⠿⢿⣒⣒⣒⣒⣒⠢⠌⡙⣾⠀⠀⢠⡟⡶⠒⢲⣿
⠀⠀⠀⠑⢿⣧⣤⡠⠴⡌⠑⢬⡓⢦⣉⣁⡸⠂⠁⢀⣀⠀⠀⠈⠓⠒⠛⠓⠒⠛⠁⠀⣀⣤⠴⠶⠲⠶⠦⣄⡀⠈⠙⢦⣄⣷⠀⠀⠸⠶⢷⣬⣿⣸
⠀⠀⣠⢾⡿⠋⠛⠒⠋⠀⠀⠀⠈⠓⠖⠋⢀⡠⠞⠉⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⢉⢳⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠈⠁
⠀⠀⠉⢁⡤⠤⠤⠄⢤⣄⣤⠤⠤⡀⠀⠀⡿⠤⠤⠤⠤⢤⡴⢦⣄⣈⣓⠲⢄⠀⢸⢿⠀⠀⠀⠀⠀⠀⠀⠀⡰⡠⠃⢀⡴⠞⣻⠋⠉⠉⠉⠓⢦
⠀⠀⡎⢹⠉⣿⠛⠛⠛⠛⠛⠛⠛⡏⠀⠀⣛⣷⠀⠀⣀⠀⠙⠋⠀⠀⠉⠉⢻⠀⠈⢧⡳⢤⡀⠀⠀⠀⠀⠀⢸⠁⢀⡞⠀⢎⠉⣠⠶⢶⣄⠀⠈⢧
⠀⠀⠹⣼⣧⠘⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣯⡀⠀⠸⠽⠂⠀⣠⣤⣄⠀⠀⢳⠀⠀⠀⠙⣆⢷⠀⠀⠀⠀⠀⠘⡇⠸⡇⣠⠊⠘⣇⠀⣸⣽⠀⠀⣸
⠀⠀⠀⢠⡛⣦⣀⡀⠀⠀⠀⣠⠗⡄⠀⠀⣾⠁⠀⠀⠀⠀⠀⠛⠶⠟⠀⠀⢸⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⢹⠀⢷⡸⢄⣠⣌⢉⣩⣁⡖⣲⠏
⠀⠀⠀⠀⠉⠛⠿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠙⠒⠒⠒⠒⠒⠚⠙⠒⠒⠒⠒⠚⠀⠀⠀⠀⠈⠳⣧⣄⣀⣠⣤⡤⠞⠀⠀⠙⠶⣤⣈⣉⣠⡭⠞⠁

⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⠀⠀⠀⠀⠀⣴⣿⣿⣷⣄
⠀⠀⠀⠰⣿⣷⡀⠶⠾⠷⠶⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⠀⠀⠀⣴⣿⣿⣿⣿⣿
⠀⢀⣤⣶⣿⣿⣿⣶⣤⣤⣴⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢋⣴⣶⣦⡙⣿⣿⣿⣿⣿
⣰⡿⠟⠛⣛⠻⢿⣿⠁⠀⢹⣿⠟⢛⠛⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⠿⠁
⡟⠀⠀⠀⠹⣇⠀⢻⣷⣶⣿⠁⣠⡟⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣶⣶⡄
⠁⠀⠀⠀⠀⠙⠗⣸⣿⣿⣧⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⠆
⣶⣶⠶⠿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡤⣀⠉⠀⠀⠀⠀⢀⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⠀⠀⠁⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⢠⠇⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠠⣀⣠⠖⠄⠀⢃⠀⠀⡇⠀⠀⠀⠀⠀⡠⡈⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠔⠀⠀⠀⠀⢳⠈⠀⣰⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣦⠀⠂⠁
⠀⠀⣀⠀⠀⡀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣤⡐⠀⠃⠀⠁⠀⠀⠀⡀
⠀⠀⠸⢄⠀⣀⣤⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⣀⣙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡙⠈⠉⠉⠉⠉⠀⠄
⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣻⣟⣟⣿⣿⣟⣿⣻⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠂
⠀⠀⠀⢐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣬⡉⠐⠒⠀⠀⠤⠠⠄⣀⡀⠀⢀⠁⠀⠈⠛⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⣰⡇⠀⠀⠠⠤⠀⠀⠚⡗
⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠐⠀⠀⢀⣤⣾⣿⡀
⠀⠀⠘⢿⡿⣿⣖⠀⠀⡀⠀⣀⣀⡰⠖⠉⠉⠉⠀⠀⠀⠀⢠⡀⠀⢠⡆⠀⠀⢴⠆⠀⠀⠀⠀⠂⠀⠀⢙⡌⠳⠴
⠀⠀⠀⠀⠳⢌⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠤⠻⠥⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠛
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠇
⠀⢠⠂⣈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇⠀⠀⡀
⠀⠀⠘⠌⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠀⠃⠀⠀⠀⠈⠢⢀⡀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣟⣿⣟⣛⣻⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡤⣤
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠲⡟⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡀⣤⣋⡷⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⠤⠞⣻⡥⣤⡷⣄⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⠦⣤⠴⠚⠉⠉⡇
⠀⠀⢀⡼⢣⡤⣼⣆⠀⠀⠙⠢⣄⣀⠀⠀⣀⠀⠀⠀⢸⣡⢇⣴⣏⣭⣷⢯⡽⠛⡆⠀⠀⠀⠀⠀⠸⣅⠀⠀⢀⡤⠶⠦⣄⠀⢳⣀
⠀⠐⠛⠲⡏⠀⠀⠉⠳⣄⡀⠀⠀⠈⠉⣻⣹⠀⠀⠀⢾⠻⠿⢾⣁⣄⣽⡞⠁⠀⢻⠀⠀⠀⠀⠀⠀⠈⡷⠀⡏⠀⠀⠀⢸⡆⠀⠈⢳⡀
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠙⠛⠻⠿⣏⣁⡍⠀⠀⠀⠘⣆⠀⠐⠂⢠⠏⠐⠀⠀⠈⣇⠀⠀⠀⠀⣠⠞⠁⠀⠳⣄⣀⣀⡼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠙⠦⢤⣀⣀⣀⣀⣀⣼⣁⡇⠀⠀⠀⠀⠘⢦⠀⢠⠏⠀⠀⠰⠀⠀⠸⡄⠀⠀⠀⡇⠀⠀⠀⠀⣀⣉⣁⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⢳⠇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠙⠲⠶⠒⠋⠉⠁⠉⠉⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡀⠀⠘⠀⠀⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⣄⡀⠀⣄⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡽⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⠧⣽⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⢠⡤⣼⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⡀⠀⠀⢠⡞⣠⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⢳⡼⠛⢦⡼⢾⠙⢶⣳
⠀⠀⠀⣠⠶⠶⠶⠖⠛⠛⠒⠒⠒⠒⣦⠀⠀⠀⠀⠀⣾⣛⣚⣛⣛⡇⠀⠘⣇⢳⢀⣤⣄⣴⡿⠋⠀⠀⢸⣛⡛⠳⣴⡋⡇⣾⢧⠈⢻⡀
⠀⠀⠀⠙⣟⠉⠉⠉⠉⠉⠉⠉⠉⡿⠉⠀⠀⠀⠀⣠⠟⠉⠉⠉⠉⠓⢦⠀⠟⣀⣻⡀⠈⢿⠄⠀⠀⠀⠈⣠⣸⣳⡀⠙⣧⡏⣸⣇⣠⣷⠆
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡞⠉⠙⢿⠳⠛⠀⠀⠀⠀⠀⠘⣏⠉⠉⠛⠋⠉⠉⠁⠀⡟
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡾⠶⢿⡶⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⡤⣤⣷⣠⣁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣹⠀⠀⠰⣇⠐⠀⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠚⠛⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⢛⣿⡿⠲⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⡤⣤⠤⣀⣠⣀⣠⣀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⣠⠟⠛⠉⠛⠲⢤⡿⠶⣿⣶⠦⣄⠀⠀⠀⠀⠀⠀⣠⠶⡛⠉⠉⠯⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⢸⠾⠧⣤⣠⠆⠀⠀⠉⠳⣄
⠀⠀⠀⣼⠁⠀⠀⠀⠰⠀⠀⠙⢾⠓⣼⡧⣌⢧⠀⠀⠀⠀⡾⢡⠆⠙⠀⠛⠀⠘⢁⣤⠹⡄⠀⠀⠀⠀⠀⣼⠹⢦⠀⢈⡙⠲⢦⡀⢀⡀⠳⣄
⠀⠀⠀⠹⣄⠂⠀⣤⠀⠀⠈⠀⣌⢳⡘⢾⣻⣸⠄⠀⠀⠀⢹⡦⠤⠤⢤⣼⣭⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⣿⣃⡾⢀⡴⡒⣒⡶⣝⢻⣄⡤⠿
⠀⠀⠀⠀⠈⠳⢤⣀⠀⠠⠆⠀⠀⠀⢧⣸⣹⣾⠀⠀⠀⠀⢼⡳⠒⠛⣲⣶⣖⣛⣓⣶⣞⣹⠆⠀⠀⠀⠀⣟⠡⠀⣟⢾⣹⣽⠽⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠃⠀⢸⡏⡽⠃⠀⠀⠀⠀⠈⢿⡛⠉⠉⠉⠛⠯⣍⠉⢉⡟⠀⠀⠀⠀⠀⡏⣙⡦⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠚⠛⠋⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⠶⢶⣄⣀⠀⠀⠀⠀⣀⡶⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠉⠙⠷⠲⢦⡼⢧⡀⠦⢄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠔⠲⡤⢄⢀⠀⠀⢀⣈⢥⠀⢈⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⢀⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠒⠃⠀⠀⢠⠀⠀⠀⠀⢀⣄⣼⠃⠰⠚⠁⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⣀⡴⢿⣰⠀⢀⣘⣿⣄⣠⣄⠀⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⠗⠚⡅⠐⠂⢹⡆⣸⠋⠁⠀⠀⣬⡟⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠓⠋⠀⠀⢠⠄⠀⠀⠀⢻⣧⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠫⠇⠀⠀⠀⠀⠀⠀⡀⢺⡧⠤⢤⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡆⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣿⡗⠒⠚⢁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠒⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡦⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣑⣑⣚⣘⣃⣄⣤⣤⣤⠤⠤⠬⠶⠮⢕⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣿⣿⣟⣦⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⡀⠀⢰⣷
⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⣾⣿⣿⣟⠉⠻⢿⣗⣦⡀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⠀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣾⣿⣼⡇⢀
⠀⠀⠀⠀⣧⣿⣧⣦⣾⣿⣻⣛⣽⡿⢷⣄⣀⣰⣼⣿⣷⣾⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠾⣿⣭⣉⡙⠛⠿⠟⠙⢺⣿⣿⣷⣿
⠀⠀⠠⣤⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠈⠉⠛⠛⠿⣿⣽⣿⣿⡷⠦⠀⠀⠀⢀⣚⣏⣽⣿⠿⠿⠿⠿⠷⠶⠤⠤⣤⡤⢤⠴⠶⠿⠿⠿⠿⠿⠿⢦



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣄⣀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢀⣠⡀⢀⣴⣷⣦⣤⣤⣴⣿⣵⣾⣿⣿⣿⣦⣀⡀⣀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣄
⠀⠐⠛⢿⣿⣿⠉⠙⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠉⠋⣹⣿⣿⣿⣷⠴⠀⠀⠀⢀⣾⣷⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⣷⣄
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⡠⠔⣁⣻⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠏⠱⣴⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣛⣋⣳⣄⠀⠀⠀⠔⠋⠹⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠹⡶⡀⠀⠀⠀⠀⠀⠘⠁⠀⠂⠀⠀⡠
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠶⠀⠠⠞⠃⠀⠰⠧⠄⠀⠀⠀⠀⠀⢀⠀⢀⣀⣴⣂⢀⣤⣄⣀⣀⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀⣰⠋⢳⡄
⠀⠀⠀⠀⢀⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⣩⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣠⠤⣄
⠀⠀⣤⠐⠿⡞⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡞⠛⣡⢼⡛⠷⢯⣷⡀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠒⠚⠉⠀⠀⢸⡆
⢀⡞⣹⠷⠒⠿⣄⠀⠀⠈⠓⠦⢤⣄⣀⡄⠀⠀⠀⣸⣴⣣⢾⠷⠶⢿⢻⡞⠁⢧⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⡴⠛⠛⠳⣄⠀⠳⣄
⠉⠉⢻⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⣀⠿⣽⠀⠀⠀⢻⠙⠛⠻⠶⠶⢾⠏⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡼⠀⢸⠀⠀⠀⠀⣹⠀⠀⠈⢳⡀
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠉⠛⠛⢦⣴⡆⠀⠀⠀⠈⢦⠀⠈⠁⣰⠋⠈⠀⠀⠀⢻⡀⠀⠀⠀⣰⠋⠀⠀⠈⠳⢤⣤⠴⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣤⣤⠤⢿⣸⠇⠀⠀⠀⠀⠈⢧⡀⣰⠃⠀⠀⠘⠀⠠⠈⣇⠀⠀⠀⢿⡀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢀⠀⠀⠀⢸⡆⠀⠀⠀⠙⠓⠚⠉⠀⠀⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠤⠀⠈⠀⠀⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠙⢦⣀⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢰⢡⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⣾⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⣠⠤⡎⢉⡆
⠀⠀⠀⠀⠀⠀⠀⡟⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⣀⠀⠀⢀⡞⣠⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠙⣦⠟⠳⣦⡟⣏⠹⣞⣦
⠀⢠⠶⠶⠶⠶⠚⠓⠓⠒⠒⠒⢲⡄⠀⠀⠀⠀⠀⣞⣛⣛⣛⣛⡇⠀⠈⢇⢹⢀⣤⣄⣠⡿⠋⠀⠀⠀⣏⣛⠛⢦⣞⢹⢠⡿⣆⠘⢧
⠀⠉⢻⠉⠉⠉⠉⠉⠉⠉⠉⢉⡏⠁⠀⠀⠀⠀⡴⠛⠉⠉⠉⠉⠑⢦⠀⠞⢀⢸⣄⠈⢹⡇⠀⠀⠀⠀⢀⣈⡟⣆⠈⢻⣸⠃⣼⣄⣨⣷
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⣞⠉⠙⢿⡳⠞⠁⠀⠀⠀⠀⠀⢻⠉⠉⠙⠛⠛⠉⠉⠁⢠⡇
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡼⠶⢿⡶⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣤⣀⣷⣠⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⡏⠐⠈⠙⣏⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠒⠒⠒⠶⠟
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠻⠶⠶⣶⡿⢦⣧⡴⠋
⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠓⠲⠶⢤⣄⡀
⠀⠀⠀⣰⠿⢞⠳⠮⣍⡹⣯⣽⣇⣤⣀⠀⠀⠀⠀⠀⠁⠀⢀⣤⠶⠒⠒⣖⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⣀⡀⣠⠀⠈⠉⠳⢤⡀
⠀⢀⡞⠁⠀⠈⠀⡄⠀⠙⢦⡸⢭⣳⣈⠱⡄⠀⠀⠀⠀⣴⢋⡀⠳⠀⠶⠀⠡⠆⢈⠳⡄⠀⠀⠀⠀⠀⢸⡷⣤⡈⠉⠓⠦⣤⡄⠀⠀⠙⢦
⠀⠸⡆⠠⠀⠀⠀⠀⠀⠂⠀⠹⡄⢿⣽⣇⢹⠀⠀⠀⠀⢷⣌⣁⣀⣀⣀⣖⣀⣈⣋⣁⣹⠀⠀⠀⠀⠀⢸⡷⢈⡇⢀⣩⡤⣤⣙⠶⣞⠀⣈
⠀⠀⠘⠦⣀⠈⠁⠀⢀⠀⠈⠃⠹⡄⣹⢾⣼⠀⠀⠀⠀⣸⣭⣭⠭⣍⣉⣩⠽⢯⣍⣩⠿⡄⠀⠀⠀⠀⢸⠟⠋⢠⣯⡴⡖⣟⣮⠷⠞⠋⠉
⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠠⠄⠀⢷⡷⢺⠃⠀⠀⠀⠀⠹⣦⠤⠶⠶⠿⢶⣶⠶⠶⠖⢺⠁⠀⠀⠀⠀⢸⠶⣄⢘⣦⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣄⣸⡴⠃⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣈⣹⣦⡴⠋⠀⠀⠀⠀⠀⢾⠴⠛⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⢀⣤⣀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣴⣿⣿⣿⡆⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠛⠿⠿⠛⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⡼⠀⠀⠀⣀⣤⡤⠤⠶⠤⣄⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⢺⡆⠀⠀⠀⠀⠀⢨⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠐⠒⠒⠒⠒⢠⡀⠀⠀⠀⠀⠀⢀⡤⠼⠧⠤⠤⠤⠤⠤⠼⠼⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠃
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⢀⣀⡠⠤⠔⣊⡅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂
⠀⠀⠀⠐⢬⠭⠉⠉⠭⠭⠁⠤⠐⠂⠉⠀⡂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠘⡄⢀⣀⣀⡀⠤⠤⠀⠐⠒⠈⢹⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⢀⣀⣀⣀⠀⠻⣃⣟⠘⠃⠀⣠⠒⠢⠤⠭⠍⠉⠐⢲⠋⡇
⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⣤⣦⣄⠀⠀⠀⢸⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠐⠒⠒⠒⠒⠒⠐⠀⠐⠃⠀⠀⡇⠀⠀⠢⠠⠐⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⢱⠀⠀⠰⣿⢿⣽⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠁⢀⡀⠀⠠⠤⠤⠄⠚⠃⢰
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⠻⠟⠋⢀⣀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⣔⡉⣑⢏⡠⢶⠀⠀⠀⠀⠘⠁⢠⢰⡴⠄⠶⠦⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠐⢖⠒⠒⠂⠈⠉⠀⠀⠀⡄⠀⠀⠀⡠⠠⠤⢊⠭⠠⠕⠀⣈⠔⠨⠥⠄⠒⠤⠀⠀⠀⠀⢰⣈⡡⡀⠤⠤⠤⠤⠒⠃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⠀⠀⣀⡠⠄⠊⠀⠀⠀⠀⡗⢒⡒⢁⠔⠒⠊⠀⠀⠀⠀⣀⠀⠀⡀⡀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢰⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠓⠂⠑⢊⣐⢂⡮⠭⠱⢞⢄⡨⠮⡀⢴⠀⠀⠀⠀⠆⠀⠐⠒⠂⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠐⠠⣀⠀⠀⠀⠀⠀⢸⣁⢉⡈⡡⠤⠤⠊⠑⠒⠉⠑⠀⠒⠢⠼⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀⠤⡐⠜⠋⠁
⠀⠀⠀⠀⢀⠴⠈⠠⠤⠤⠄⠤⠤⠤⠤⠄⠤⠤⠤⠬⠷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠧⠤⠶⡈⢁⡀
⠀⠀⠀⠀⠘⣷⣮⡴⡤⠰⢶⣤⣤⣄⣄⣠⣡⣤⣐⣲⣶⠃⠀⠀⠀⠀⠀⠀⠀⣔⠈⡡⠄⠰⡀⠠⡤⢀⣐⠢⠤⣄⡉⢲⡄
⠀⠀⠀⠀⠀⢏⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⢋⡁⠃⠀⠀⠀⠀⠀⠀⠀⢙⠛⠠⠬⢵⣏⣘⣏⣁⣬⣡⠦⠶⠚⢛⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⠀⠀⠀⠀⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣀⠤⣴⠒⠊⠉⠙⠩⢛⣱⣄
⠀⠀⠸⣽⠶⠤⠤⠤⠔⠒⠊⠉⠕⢿
⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠠⣴⠄⢰⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠻⣤⡀⠀⠀⠀⠀⠠⡾⠀⠀⡀⠀⠀⠀⠀⠀⠀⢛⡁⠀⠀⠀⠉⠀⠰⡄⠀⠀⢰⠀⠈⠁⠀⡀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⡊⠁⠀⠀⠀⢀⡤⠊⠀⠀⢠⡏⣞⠤⠒⠐⠂⣀⠉⠁⠀⠀⣀⢔⣠⠤⠀⠀⠀⠀⠀⢀⠀⡎⣠⠒⠀⠀⠴⠂
⠀⠀⠀⢠⡊⠉⠁⠀⠀⣂⠟⠀⠀⠀⠀⠈⢳⣿⡼⠉⠀⠈⠀⠀⠒⠀⢠⣏⣞⡵⠞⠛⠙⠐⠆⠀⠐⡤⣤⡇⠄⠀⠶
⠀⠀⠀⢟⣂⠀⣠⠒⢋⡝⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⡟⠋⠛⠁⠀⠀⠀⠀⠀⠈⣿⠃⠐
⠀⠀⠀⠈⠿⣭⣑⡂⠉⠙⣆⠀⠀⠠⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⢠⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠉⠓⠿⢠⡰⠃⢀⠠⠾⠿⢿⡆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⣾⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠾⠛⠋⠉⠉⠉⠂⠀⠀⡄⠀⠀⠉⠁⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡶⠒⠒⠒⠒⠒⢲⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⡇⠠⠴⠆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠒⠒⢒⣒⡒⠒⠒⠒⠒⠒⠚⠃⣠⠗⠒⠒⠒⠒⠒⠚⢧⡀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣠⡽⣛⡛⢦⡀⠀⠀⣞⢓⣒⣒⣒⠒⣒⣒⡒⣒⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠞⠹⡟⠁⠉⢻⠇⠀⠀⠉⠉⠉⠉⠉⣩⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⡶⢶⣏⣀⠀⠀⠀⠀⠀⢯⠀⢸⠆⢸⡄⠀⠀⠒⠒⠲⡖⠒⢲⡖⠒⢶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠉⠉⠀⠈⣷⠀⠀⠀⠀⢘⣀⣤⣄⣘⠁⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⡴⠀⠀⡶⢳⢄⣿⡀⠀⣀⣰⠯⠥⠤⠤⠭⢷⣀⡀⠀⠀⠀⡇⠀⣸⣇⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣅⠀⠘⠳⢦⣠⣟⡏⢠⠈⠓⠶⠯⣿⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⡞⢹⣆⡏⢹⢰⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⡶⠴⣇⣹⠇⢺⠶⠶⠶⠶⢾⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠙⠚⠙⠓⠚⠈⠓⠋
⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠳⠶⠶⠶⠶⠞⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⢸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠶⣶⠶⠶⣶⣶⡶⠶⢶⡶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⣿⠸⠭⠇⠸⠭⠭⠿⠀⠯⠟⢸⠀⠘⠚⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢛⣛⣛⣛⣛⣛⣛⣛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣈⣉⣉⣉⣉⣉⣉⣉⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠁⠀⠀⠐⢀⠠⣄⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠀⠀⠀⠀⠈⠁⠈⠃⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠩⠤⠰⢒⣲⣲⣟⣉⣡⣆⣖⣶⡶⡤⢄⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⣿⡷⠴⠛⠻⠿⠿⢿⢟⢻⢛⡛⠛⠃⢠⣤⣾⠃⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠂⠈⠑⢾⣿⢿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡤⠊⠀⡠⣿⡟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⠕⠶⣤⣀⡀⠙⠻⣿⣿⣿⣹⣿⣿⣿⣿⠿⠋⠀⣀⣬⣤⢟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡿⣿⣶⣤⣀⡉⠛⢷⣆⡈⠛⢿⣿⣽⡿⠛⢡⠴⠶⠛⢉⣩⣴⠞⠋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣴⣬⡿⠀⠀⠉⠛⠛⠿⢷⣬⣥⡶⢤⣽⢫⣴⢶⣀⣤⡴⠶⠛⠉⠀⠀⠈⠁⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣱⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⠊⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠠⠟⠻⢏⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢎⢽⡄
⠀⠀⠀⠀⠀⠀⣗⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠻⡀
⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡰⠀⠀⠈⠥⠀⣧
⠀⠀⠀⠀⠀⣿⠘⠄⠄⠈⠳⢘⠲⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠋⢻⡤⠂⠐⠠⢰⣽
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠐⢮⢀⣦⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⠞⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡿⠃⠁⠀⠀⠀⠀⣴⣆⣿⡜⠇⢠⡐⡠⣠⠀⠘⢁⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢻⣾⡿⠄⡖⡶⠆⠀⠀⠐⣿
⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠚⠟⢻⡿⣟⣿⣒⣛⣛⣈⣀⢀⠏⢐⢄⡟⠁⡽⠋⠉⠱⣖⠶⠺⡗⡻⢿⡇⡇⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢇⡗⠀⠐⠠⠶⢻⣿⡝⠫⠍⠀⢹⠟⠛⠇⡮⠀⠴⠮⠒⠀⠀⠀⠀⠈⠁⠀⠀⠫⠞⢹⡗⠦⠄⠀⠀⠀⠀⢰⡾
⠀⠀⠀⠀⠀⢸⠜⠉⠀⠀⠀⠀⠸⠋⠙⠀⠁⠀⠀⠀⠀⠀⢳⣦⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣽⢿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢺
⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡑⢀⡻⣿⠛⣛⠩⣶⢤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡄⡀⠀⠀⡀⠄⠀⢨⡴⣟⣫⣻⡿⡖⠱⠘⠈⠁⠹⠃⢽⡠⣢⣠⣔⠀⢀⡀⠀⠀⠀⠀⣧⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⡲⡶⣿⡳⠛⠟⡿⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⡙⣲⠤⢆⣠⣤⣶⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢩⠏⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉⠉⢡⢁⣞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠋⠄⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠈⠀⠚⠛⠛⠒⠒⠒⠠⠒⠚⠟⠉⠉⠛⠊⠉⠁⠀⠀⠀⠀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠄⠄⠖⢦⠐⢶⡶⠶⣶⣠⢶⣶⣶⡶⠲⠹⠟⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⡆⢀⠀⠲⠒⢲⡶⠶⠶⣶⠶⠖⠠⢀⣀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠓⠀⠈⠀⡀⠈⠀⠀⠀⠈⠀⠀⠀⠤⠀⠀⠰⠀⠈⢩⣟⠻⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠞⠃⠉⡀⠀⠀⠊⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠅⠀⠐⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠤⠀⠀⠀⠙⠁⣽⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢤⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣘⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠐⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣟⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣷⣤⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢛⠿⢶⣦⣤⣀⣀⣀⣀⣄⣀⣀⣠⣤⣀⡀⠀⢀⣀⣀⣀⣀⣀⣤⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⠿⣟⣛⡭⣉⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣷⣦⣈⠉⢉⣛⣛⠛⢛⣻⣟⠛⠟⠻⢿⢿⣿⣯⣿⣯⣍⣭⣭⠽⠈⢦⡉⠁⢀⣰⡟⢉⠾⠋⢗⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠰⠚⣉⡙⠿⠿⠍⣉⣿⣿⣿⡟⢻⣿⣿⣦⣤⣤⣴⠿⣷⠾⡯⠟⠛⠿⢿⡴⠋⠳⢍⣻⡿⠻⠷⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣭⡥⣄⡀⠀⠋⠙⠿⠻⠿⠛⢻⠿⢻⡿⡯⠋⠀⠋⠉⠀⠐⠂⠒⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠧⣿⣿⣷⣶⣏⡋⠉⠀⠀⠒⠓⠀⠊⢀⣠⣤⣴⣶⣶⠶⣶⣶⣶⣾⣿⣿⣿⣿⡿⢛⣽⣿⣇⣼⢧⣼⡏⣨⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣌⢙⣧⣄⡙⢟⣷⣄⠀⠀⢀⣠⣾⣿⣯⣤⣶⣿⣭⣼⡿⢃⣾⠏⢀⣾⠿⣿⢹⣟⠉⣸⣿⣏⣾⡯⣴⡏⢀⡀⠘
⠀⠀⠀⠀⠀⠀⠂⠈⠉⡃⠛⢮⠛⠿⢙⣛⢻⣿⣧⣴⣿⣿⣛⠉⢻⢿⣭⣼⣿⣋⣾⡿⠿⠿⠿⢛⣴⣟⣨⡿⣿⣿⠿⡿⡗⠓⢄⣁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠶⡥⢄⡠⢀⣀⠀⠀⣈⡉⣉⣉⣤⣤⣞⡀⠒⡢⠈⠟⠀⢐⠊⣀⣀⣉⣡⣽⣄⣤⣤⣏⣀⣿⡿⠶⠾⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠙⠛⠻⠙⠿⠟⠋⠛⠻⠗⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠁⠉⠹⡟⠀⡀⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣄⠀⠀⠀⣰⣀⣼⣦⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠤⠤⠤⠤⣤⣤⣤⣄⣤⢀⣀⣠⣤⣄⣴⣶⣴⣤⠤⠴⠾⠿⠿⠒⠶⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣴⣾⣿⣿⣷⣤⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣀⣀⣤⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⢤⡴⠞⠋⠉⠁⠀⠀⠉⠙⠳⣦⣠⡴⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⢴⣿⡁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⣤⣴⠒⠂⠀⢸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣧⠴⠤⠤⠄⠀⠀⠀⠀⢀⣠⡤⠶⢤⣶⡋⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠷⠀⠀⠉⠉⠉⠙⠳⣦⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⠴⠲⢯⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢉⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿⣿⢿⣏⠁⠈⠛⠀⠀⠀⠀⠒⠚⠛⣷
⠀⠀⠀⠀⠀⠀⠙⢳⣦⣤⣤⠄⠀⠀⠀⠀⠀⠀⣿⠁⠈⢻⣿⣿⣷⣄⣤⡄⠀⠀⠀⣤⣄⣴⣿⣿⣿⠋⠀⢹⡆⠀⠀⠀⠀⠀⠀⠻⣷⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢺⣿⣿⣿⣿⡿⠁⠀⠀⠀⠹⣿⣿⣿⣽⣿⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⣀⣈⣩⣴⠋⠀⠀⠀⠀⠀⠀⠈⢷⣭⣉⣀⣀⣠⣾⠇⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⣀⠀⠀⠀⠀⠀⡀⠈⠙⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠈⠉⠛⢻⠟⠻⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠛⠛⠛⠛⣿⠁
⠀⠀⠀⠀⣠⣴⣯⡁⠻⠶⢤⡀⠀⠀⣠⠏⠀⠀⠈⢻⡶⠶⣤⣤⣤⣤⣤⡾⢿⡷⠶⠶⠶⣦⡴⠶⠶⠶⠛⠉⠘⣆⠀⠀⠀⠻⠶⠛⢻⡇
⠀⠀⠀⠀⢀⡴⢋⣠⡤⠒⠁⠀⠀⣴⠋⠀⠀⠀⠀⠿⠦⢤⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⢀⣀⣀⡀⠀⠀⠀⠸⣆⠀⠀⣀⠀⠀⣸⡇
⠀⠀⢠⣾⡵⠚⠉⠀⠀⠀⠀⢀⣾⣁⣤⢤⡄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢸⡇⠀⠀⠰⠟⠋⠉⣹⠃⠀⠀⠀⠀⠹⡶⠟⢻⡀⠀⣿⠁
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡞⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⢠⡏⠀⠀⢰⡄⠀⠀⠀⠀⠘⣇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡴⠞⠁⠀⠀⠀⠀⡄⠀⠀⠀⢾⣥⠶⠚⣿⠀⠀⣼⠀⠀⠀⢸⠙⠛⣹⠃⠀⠀⢻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣀⡴⠋⠀⠀⠀⠀⠀⢠⠞⠷⢤⡄⠀⠀⠀⠀⠀⢿⠀⢰⠇⠀⠀⠀⠾⢲⢰⠇⠀⠀⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠞⠉⠀⠀⠀⠀⠀⠀⢴⠗⡶⢠⠞⠀⠀⠀⠀⠀⠀⢸⠀⡾⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠃⠀⠀⠀⠀⠀⠀⠀⢸⢠⠇⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡤⢀⠴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠉⠠⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠳⣤⣤⣤⡤⠴⠚⢁
⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢀⣀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠛⠀⠑⠒⠒⠒⠛⠋⠉⡠⣄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠒⠚⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠶⠶⠖⠒⠒⠒⠒⠶⠤⢬⣽⠓⠦⢤⣤⣀⣀⣀⣀⣀⣤⠴⠶⠛⡏
⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠀⣤⠀⠹⠓⠒⠶⠦⠤⠤⠤⠤⠶⠒⠚⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢠⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠶⠀⠀⠀⠀⠿⠀⠀⠀⠘⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠰⠆⠀⠠⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣿⢿⡿⠿⢿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠂⠀⢀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⠳⡖⢋⠕⢊⣾⠟⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⢀⣿⠃⠀⠉⠉⢠⣿⠇⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢧⠀⠀⠀⣾⣿⣀⣀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⣧⣀⣼⡇⢀⣿⣿⣿⣿⣿⣿⠀⠀⠸⣄⠀⣀⣠⣤⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣟⢶⡀⠀⠀⠀⡼⠁⠀⣰⠋⠉⠁⢸⡇⢐⣿⣿⣿⣿⣿⣿⠀⠀⠀⣽⢫⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢠⠿⠀⢹⡆⠀⡰⠁⢀⡜⠁⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣇⢀⣠⠗⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣦⡿⣄⢀⣀⡼⢀⠔⠁⠀⡎⠀⠀⠀⠀⠀⠈⡇⢸⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⢷⡿⡆⣤⠀⢁⠀⡀⢰⠃⠀⠀⠀⠀⠀⣰⠃⣬⣿⡿⠟⠋⠉⠀⠀⣀⣤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢹⡿⣔⣿⡻⠛⢡⡀⠀⠀⠀⠀⡴⢁⣾⣿⣿⡄⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⠀⠈⠁⠙⠉⠳⠤⠏⢹⣄⣀⢀⣜⡡⢟⣫⣥⣾⣤⣤⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠀⢀⣺⣷⣤⣦⡀⠀⢀⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠸⣿⣏⠀⠉⣷⢀⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠓⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣙⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡠⠶⠒⠛⠉⠉⠉⠉⠛⠒⠶⢤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠋
⠀⠀⠀⠀⣰⠋⡠⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠟⠀⢴⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠆
⠀⠀⠀⣰⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⡄⠀⠔⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣠⠉⢠⡀⠀⠀⠠⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂⡀⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⢠⡀⡀⠀⠀⡰⡛⠁⠀⠀⠀⠀⠀⠀⢘⡄⣀⣡
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢆⠀⢁⠜⡝⠁⠀⠀⠀⠀⠀⢀⠤⠚⠛⠛⠋⠀⠀⠀⠀⡀⠀⠀⠱
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣿⢠⠃⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠀⠀⠀⠐⠀⠀⠂⠁⠀⠀⠀⡀⠠⠀⠀⠒⠂⠀⠀⠀⡔
⠀⠀⠀⠀⠈⢦⠀⠀⠀⢀⡇⡞⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⢀⣤⣴⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⣜⠣⢄⡀⣼⣸⠁⠀⠁⣀⡖⠋⠁⠀⠀⠀⡴⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢀⠤⠂⠈⠁⠀⠁
⠀⠀⠀⠀⠀⣍⠳⠄⢀⠀⠈⠉⠉⠉⢑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⡡⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢸
⠀⠀⠀⠀⠈⠆⠉⠒⠢⠄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⠀⡄⢪⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠑⠢⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢢⣄⣀⣀⣀⠀⢀⠔⠒⠀⠀⠀⠀⠀⢐⣉⣶⠆⠀⣀⢬⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⣠⠴⠛⠉⢀⡤⣚⡭⠟⠀⠀⠀⣰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⡠⢚⡥⠋⠁⠀⠀⠀⠀⠐⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢢⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠛⣓⡶⠀⠀⠈⠑⣣⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠐⢁⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣸⠞⠁⠀⠀⢀⠔⠋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡰⠂⠀⠰⠤⠀⠀⠠⠂⠀⠀⠀⣠⠛⡀
⠀⠀⠀⠀⠚⠢⠀⠈⠁⠀⢀⡴⠊⠁⠀⠀⡠⠚⢁⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣤⡄⠀⠀⢯⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠙⠲⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡴⠋⠀⠀⠀⣀⠬⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⡠⠔⠊⠁⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣀⠀⠀⡀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢀⠞⠁⡠⠔⠉⠀⢀⡴⠒⠁⠀⠀⠀⠀⠀⠀⢰⠀⠈⠂⠐⠋⢹⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⣰⣫⠖⠉⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⣀⣀⠀⢨⠀⠀⠠⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢀⡾⠋⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⢆⢀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠉⡄⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢲⡀⠀⠂⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠴⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠴⠛⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠁⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠠
⠀⠀⠀⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⡀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠴⠀⠒⡁⠀⠀⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠐⠒
⠀⡀⠀⠀⠀⠈⠤⠠⠴⠒⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠉⠁⠀⠀⠀⣤
⠀⢁⠀⠀⠀⠀⠀⢀⡀⠄⠠⠀⠀⢉⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔⠈⢀⡂⠀⠀⠀⠀⡅⠀⢀⡀⢀⡤⢠⠠⠀⠁
⠀⠈⠀⠬⠀⠈⠀⠃⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠘⡀⠀⠁⠀⠀⠀⠇⠀⢀⣤⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠛⠁⠀⠀⠄⠀⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⢛⠉⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⢀⡀⠀⣁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢨⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠁⠈⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠄⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠲⠠⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠘⠆⠘⠃⠂⢂⠄⠀⠀⠀⠀⠀⠤⠄⢙⣐⣒⣉⠠⠤⡆⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⡀⠀⠀⠀⠀⣀⠤⠀⠀⠠⠤⢀
⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠸⡀⢆⢀⣄⠲⠀⢂⠀⡇⠀⠀⠀⠀⠰⢠⠂⠈⠲⠂⠀⠀⠀⠀⠀⢀⠈⠀⣀⡀⠁⡁⣀⣀⢱
⠀⠀⠀⠀⠛⠄⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⢀⠜
⠀⠀⠀⠀⠁⠀⠈⠠⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠒⠒⠒⠒⠒⠁⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⡠⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⢀⠠⢐⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⣀⣢⢠⣈⣀⣥⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⡿⠂⠀⠀⡐⢁⢲⡀⠀⠸⠀⠀⠀⠀⣠⠤⠶⠶⠿⠿⢶⡄
⠀⠘⠈⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠾⠀⠀⠀⠀⠀⠛⢾⡡⠀⢁⡆⠀⠀⠀⢹⠀⠀⠀⠄
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠁⠠⠀⠀⠀⠀⣶⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢠⠀⠀⠀⢨⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡄⠀⠀⠐⠀⠀⠀⠐⠒⠘


⠀⠀⠀⠀⢤⠉⠀⠀⠀⠀⠀⢀⣠⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢘⠆⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⠰⣆⠀⡀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⡄⠄⠀⠀⠀⠀⠸⠁
⠀⠀⠠⠂⠀⠀⠂⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠲⢎⠣⣄⠀⠀⠈⠀⡃⠈⠠⠈⠀⠠⠀⠑⠠⠀⠀⠀⢀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⢙⡧⠄⠀⠀⠀⠀⠐⠨⠤⠒⠀⠡⠀⠀⠀⠀⡁⢈⠀⠠⠋
⠀⠀⠈⠀⠈⠁⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠠⠊⠀⣀⣤⣤⣀⡈⠀⠀⠀⠀⣿⢿⡄⢸⣟⠖⠁⠀⠀⠀⠀⢀⡨⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠐⠀⠂⠀⡰⢛⡅⠈⠀⠁⠀⠀⠰⠀⠀⢨⠉⠃⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⢋⣀⣠⡠
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠨⡀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠠⡬⠀⠀⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠚⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢀⠀⠀⠀⠀⠀⢀⣠⠄⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢀⠂⠈⠂⠀⠀⠀⢠⡏⠃⠀⠀⠀⠙⡆⠀⠀⠀⡀⠔⠒⠂⠀⡀⢀⠠⠀⠀⢀⠔⠀⠀⠀⠑⠄
⠀⣀⠀⠀⠐⠚⠛⠛⢲⠇⠀⠀⢀⠈⠀⠂⠂⠀⠁⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⢈⠀⠀⠒⡄⠶⠀⠒⢄⡜⢂
⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢆⠤⠁⠔⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠘

⠀⣀⠀⠀⠀⠀⡀
⢀⠀⠀⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⣰⣤⣤
⠀⠀⡀⠙⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠃
⠀⠀⡇⠀⢣⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⢀⡜⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⠦⡀
⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠶⠒⠃
⠀⠀⢀⠀⠸⠀⠀⠠⠀⠘⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠄⠄⠤⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⣴⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⣴⡿⠃⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠘⢿⣦
⠀⠀⠀⠀⣼⡟⢁⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⡈⢻⣧
⠀⠀⠀⣼⡿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⢿⣧
⠀⠀⢠⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣀⣠⣄⣀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⡄
⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⣾⣿⣿⣿⣿⣷⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⢸⣿⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠙⠋⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⡀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣄⣉⡉⠛⠛⠛⠛⠛⠛⢉⣉⣠⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⣸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣉⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⡿⢿⠙⠶⠖⠀⠀⠈⢛⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣌⣀⠠⠀⠀⠀⠀⠈⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⣄⠤⠀⠠⠀⠀⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣠⠆⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⢷⣶⠾⠛⣉⡤⠞⣡⣿⡿⠛⠷⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢡⣾⡟⠁⠀⠿⢤⡼⠥⢤⣴⣿⡿⠁⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⠀⠀⠀⣾⣿⠁⠄⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣯⡀⠀⠀⠀⠀⢀⣼⣿⡏⠀⠀⠀⠀⠀⠈⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠈⢷⡀⠀⠀⠀⣸⡿⠛⢿⣶⣤⣤⣤⣤⣤⣼⣿⣿⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⣷⡀⠀⣠⣿⡇⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⣷⡀⠀⠀⢀⣀⣀⣀⣀⣿
⢀⠀⠀⠀⠀⠀⠀⣾⣆⢠⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⠞⠿⠷⠛⠉⣿⠃⠀⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣷⣴⠞⠋⠉⠉⠉⠙⣿
⣼⡄⠀⠀⠀⠀⠀⢻⡜⠛⣧⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣻⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢰⡿⠹⡆⠀⠀⠀⠀⠀⢿
⣿⣧⠀⠀⠀⠀⢠⡜⡇⠀⠸⣷⡇⠀⠀⠀⡼⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣾⡧⠚⠁⠀⠀⠀⠀⠀⣸
⣿⣿⣆⠀⢰⡄⠸⠀⠀⠀⠀⣸⠇⠀⢀⡼⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡿
⢹⡟⢿⣆⣺⢷⡜⢦⣀⢀⣠⠟⢀⡴⠋⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠈⣧⣈⣿⣯⣠⣿⡆⠈⢉⠁⠀⢸⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠛⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁
⠀⠘⣿⣇⠙⢿⠙⢿⣀⣿⡀⢀⠘⡴⢀⣴⢆⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⣶⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⡏
⠀⠀⠈⣿⡄⠀⠀⠘⣿⡿⣇⢸⣷⡙⡟⠼⠋⢁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⢠⣾⣿⣿⣿⣡⡄⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡀
⠀⣧⠀⢹⣷⡀⠀⠀⢻⡇⠹⣾⡏⢿⡆⠀⠀⣼⠿⣆⠀⠀⠀⠀⠀⢀⡾⠁⣤⣿⡿⠿⠛⢉⡝⠀⡔⣠⢀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⢳⣾⣿⡇⠀⠀⠀⠀⡀⠈⠀⠀⠙⠳⠛⠋⠀⢹⣧⣀⣀⡀⢠⣞⣡⠶⠋⣁⣤⣶⣿⣿⣤⣾⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢿⡀⠈⠋⠋⠀⠀⠀⢦⣿⣦⡀⢠⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠘⣧⠀⠀⠀⠀⠀⣷⣾⣿⡯⠻⢿⣿⣦⠀⠀⠀⢸⣿⣿⡿⢟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠘⣧⡀⠀⠀⠀⢻⣿⣿⡁⠀⠀⠈⣿⣿⣀⣤⢿⣛⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⢷⣤⣤⣀⠹⠿⠿⠶⣖⣒⣭⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⢻⡉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠙⠷⣤⡀⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣭⣷⣶⣾⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠂⠁⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣬⣵⣶⣶⣶⣤⡀⠀⠉⠋⠙⠿⠟⠉⠀⠀⠀⠀⢀⢀⣠⣠⣀⣀⣤⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠴⠶⠿⢟⣛⣫⣿⣿⣿⠛⠻⡹⣄⠀⠀⠀⠀⢢⡄⣀⣀⡤⠼⠛⣻⣿⣯⠿⠯⠽⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣿⣍⣿⠶⠶⡶⠲⢶⡿⠿⠏⠁⠘⠿⠀⠀⢹⡘⣦⠀⠀⠀⢸⣿⣿⣽⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⣀⣶⡿⠁⠀⡃⠀⢂⠀⢀⠊⠀⠀⣸⠀⠀⡀⠀⠀⠟⠻⣿⠀⠀⡾⢸⠟⠉
⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⡼⠛⠋⠁⠠⡄⠀⠠⠈⠀⠐⠀⡀⠰⢧⠀⠀⠇⠀⣤⡠⠖⠉⠴⢛⣵⠋
⠀⠀⠀⠀⠀⠀⡀⠀⣰⠏⠀⠀⢠⡀⡀⠳⠄⣀⡦⠠⠀⠐⠀⢀⠠⠤⠐⠂⢉⣀⣠⣤⣴⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠃⠸⡅⠀⠝⠐⠅⠀⠈⢀⠀⣘⣂⡀⠔⠋⡩⠔⠲⠾⠞⢙⣩⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠻⣶⣶⠐⠀⠀⠀⠐⢁⠜⣁⣴⠂⠀⣠⣠⡤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣤⠶⢋⣤⣚⣥⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠐⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣤⣤⣄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⣀⣀⠤⠄⠒⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡔⢲⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠓⠲⢄⠀⡀⠤⠒⠒⡲⠦⣀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠋⠀⠑⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⠁⠀⠀⠀⠀⠀⠘⣦⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⢠⠀⠀⠀⢻⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠠⠀⠛⠺⡄⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣽⠀⠀⠀⠀⠀⠀⠀⠀⣈⠃⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⡀⣀⡀⠻⠗⠀⣣⠈⠀⠀⠀⠑⠄⣀⢀⠀⠀⠀⠀⠀⣴⠋⠊⢉⠉⠣⣀⠀⠀⡀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠐⠁⣠⢄⠹⢶⠤⠾⠃⠦⡀⠀⠀⠀⠀⠀⠈⠑⠂⠄⣰⣾⡀⠀⠀⠈⠀⠀⡤⠑⠛⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠠⣀⡩⠞⣼⠁⠀⣠⠀⠘⢂⡤⠐⠒⠀⠐⠒⠠⢄⣹⡞⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠄⠠⣄⠸⡀⣶⢌⢁⣠⡞⠉⣠⠀⠐⠀⡒⠀⠀⠀⠉⢧⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢸⡆⢧⣅⣀⢀⡀⠀⠀⠈⠣⠈⠀⠁⠔⠁⠀⠀⠈⣳⣦⠤⢀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠥⠎⠀⡟⠁⠀⠈⡇⠀⣀⡈⠁⠀⠀⠀⠐⢂⠨⠓⠝⠋⠀⠀⠀⠀⢘⠀⡠⠂⠀⠀⠉⠒⠤⠤⣀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⣇⠸⠖⠀⣣⠀⠀⠤⠂⠀⡐⠠⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⡶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠾⠀⠓⠒⠒⠙⢄⠀⢂⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⣡⠀⠀⠀⠤⠐⢞⠈⠑⠤⣀⣀⠀⣀⣳⡀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢿⠁⠁⠀⠠⣄⠀⠀⠀⠀⠀⢨⠀⠀⠈⠢⠵⠀⣠⣄⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⡀⠀⠉⠉⠳⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠈⠐⠂⢤⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⢀⣀⠤⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠉⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠐⢄⣀⣀⣀⠠⠜⢗⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣁⠠⠊⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⣠⠴⠖⠒⠉⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉⠉⠁



⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣶⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡠⠀⣀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡶⠁⠉⠉⠀⠐⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠎⠅⠐
⠀⢀⠌⠀⠀⠀⠉⠀⠀⢸⠷⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠝⠀⠀⠀⠀⢀
⠀⠀⠆⠀⠀⢉⡁⠐⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠟⠀⠀⠀⣀⠔
⠀⠀⣆⠂⠠⣈⠁⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⠀⢰⠀⠀⠀⢸⠞⠄⠁
⠀⠀⠋⠀⠀⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠀⢸⠀⠀⢠⠇⠈
⠀⠀⢰⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⡎
⠀⠀⠘⡄⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠸⠤⠼⠳⠶⠚⠛⠛
⠀⡩⡉⣷⡏⠷⢶⣿⠀⠒⠀⠒⠚⠛⠶⠾⠓⠒⠛⠓⠒⠚⣒⠚⠐⠒⠒⠚⠒⠒⣛⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⡴⣱⣶⣿⣶⣤⣴⣿⢢⡤⡀⠐⠂⠀⡀⠀⡀⣄⣀⣀⠀⠉⣠⣾⣯⣠⡀⣀⣤⣀⣤⣦⠤⠂
⣗⣿⣿⣿⣿⣿⣿⡗⢰⣯⣴⣧⠄⠀⠀⠀⠁⠈⠉⠈⠙⠛⠛⠛⠛⠟⠛⠨⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒
⣿⠿⣿⣿⠟⠛⣿⠇⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠁⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⢊⣿⣦⣀⠀⣠⣤⣴⠏⠁⠠⣟⣿⣿⠟⣦⠤⢤⣀⡠⠚⠛⢩⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢐⣩⣿⣯⣭⠿⠗⢃⣀⣠⣤⣄⠈⠻⠿⠅⠀⣠⣿⣿⣟⡢⠀⠻⡿⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠛⠋⠉⠀⠀⠐⠋⠉⢠⣾⣿⠧⠀⠀⠀⠘⠻⠿⣏⣻⡯⠀⢀⠀⠀⣾⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢶⠄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⠜⠂⠁⠀⠘⠿⣿⣿⣟⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣆⣀⣠⣠⣠⣀⣠⣸⠀⠀⢀⣠⣤⣄⣀⣠⣤⣀⣀⣀⣀⣀⢀⣰⣷⣀⠀⠀⡀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉⠉⠉⠹⠯⠭⠭⠽⣛⣛⣿⠿⣟⡛⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠻⠯⠭⠭⢝⣓⡚⠻⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒⠒⠪⠭⠭⣛⡳⠦⢄⡀⠀⠀⢈⣉⡉⠉⢉⣍⣑⣒⡂⠤⠄⣀⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠉⠹⠗⠶⠎⠉⠒⠀⠀⠀⠈⠉⠉⠉⠛⠛⠭⠍⣉

⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⡄⢀⠐⠁⠔⢸⡄⠀⠀⠀⠀⠀⠀⠁⠀⣵
⠀⠀⠀⢀⡠⠈⠋⠆⢰⠆⣝⠀⠀⠀⠐⠐⠶⠚⠙⠿⠿⣖⡂⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⢀⡼⠎⠀⠀⡀
⠀⠀⠈⠐⠂⠀⡧⠐⠒⡠⠡⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠚⣿⡷⠀⣤⢠⠌⣍⢀⣀⣈⣉⡀⢀⡀⠀⠀⠈⠙⠒⠒
⠀⠀⠀⠤⡀⢶⠸⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣴⣿⡇⠀⠈⠻⠴⣧⣰⣄⣤⣠⡷⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠃⠀⢰⡄⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣷⣄⣤⣾⣿⣿⡿⠀⠀⢀⡀⠡⠔⠤⠀⠀⠀⡇⠀⠀⠀⠐⠒⠀⠡⡀
⠀⠀⠀⠀⠀⡠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠙⠁⠀⠀⠀⢀⣀⠀⢐⠇⠀⢠⠃⠀⣄⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠸⠀⠀⠹⡇⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠁⣀⣴⠋⠀⠀⠘⢄⣀⢀⣠⣠⣶⡞
⠀⠀⠀⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⢘⣁⠀⠀⠀⠀⠻⣷⣶⣾⣿⢃⡖⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠨⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠖⠋⠀⠀⠀⠔⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⡀⡇⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣸⡀⠀⠀⠀⠀⠀⠀⢂⣿
⠀⠀⠀⠀⠀⠑⠀⡾⠀⠀⠐⢦⣀⣧⣀⣴⡷⠃⠠⢂⣱⣯⢧⠷⠾⣷⣄⠀⢐⠀⣏⠀⠀⠈⠻⣤⣀⣀⣀⣰⣦⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠃⠁⠈⠉⠆⠀⠀⠈⢻⡆⠀⠀⢹⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠡⠀⠀⠀⢴⡇⠀⠈⠒⢤⣄⣠⣼⣶⠿⠋⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣤⣀⣠⡶⡀⠀⠀⠇
⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⣾⣎⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⣄⠀⠀⠏⠀⠀⠀⠀⠠⠟⠃⠙⠁⡧⠠⠄
⠀⠀⠀⠀⠰⡿⢿⣿⣿⣿⣿⣟⠓⠀⠈⠳⣦⣄⣀⣀⣀⣀⣀⣴⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣶⣦⣤⡀⠀⠀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠘⢿⡿⠟⠋⠉⠉⠉⠉⠙⠻⢿⡿⠃⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡀⢀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠟⠋⠉⠀⢀⣤⣉⠉⠛⢿⣿⣿⣿⠋⠀⠈⠻⣿⣿⣿⡿⠛⠉⣉⣤⡀⠀⠉⠙⠻⢿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⢰⣶⣦⣴⣶⡆⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣤⡀⠀⣿⣿⣿⣿⠀⢀⣤⣮⣬⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡟⢀⣿⣿⣿⣿⡀⢻⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⣶⣾⣿⣿⠿⠛⠉⠀⠀⠉⠛⠿⣿⣿⣷⣶⣤⣤⣤⠤

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠖⠀⠀⠀⢀⣿⡇⠀⠈⣷⠴⠦⡀⠀⠀⠀⠀⢰⣿⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢲⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣷⣾⣷⣄⡀⠀⠀⠀⠀⠐⢢⢠⣼⣷⣤⣀⠀⢢⣐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⡿⠿⠝⠛⢷⣦⣄⣀⢀⠀⠀⠹⡏⠈⠙⢿⣿⣆⠹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣤⡈⠂⡀⠀⠙⣿⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠻⢿⣷⡤⡀⠀⠈⠙⠛⠁
⠀⠀⠀⡔⠀⣀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠴⣢⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢀⣠⣶⣄⡁
⠀⠀⠀⠀⠀⠈⢩⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠠⠄⠀⠐⠄⠐⠛⢉⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠏⠠⠁
⠀⠀⠀⠀⠂⠈⠙⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠂⠹⣿⣿⣿⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠂⠿⠧⠴⢶⡶⣷⣶⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⠁
⠀⠀⠀⠈⠐⠒⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠰⠶⠒⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣿⣷⣠⣾⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣈⠋⠈⢛⣛⣋⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠋⠀⠈⢻⣿⠟⠛⢿⣄⡀
⠀⠀⠀⠀⠠⣷⠒⣠⣄⣀⠀⣀⡤⣢⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠄⠀⣀⣠⡤⠀⠀⠀⠀⢨⣿⣅⣀⡀⣈⡿⠗
⠀⠀⠀⠀⠀⠘⠆⠉⠊⢙⠛⠻⠿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠀⠐⠛⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠚⡸⢛⣧⣴⣧⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠄⠀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠘⡤⠔⠀⠉⠉⠉⠉⠉⠁⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⣀⢀⣀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠏⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣠⡟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢸⠀⠀⠀⠀⠠⣄⡀⠀⠀⠤⠄⠒⣩⠏⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢻⠀⠀⠀⠀⠀⢀⠶⢆⡀⣀⠰⠚⠁⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⣷⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠖⠓⠀⠤⠤⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⢌⡦⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠻⣦⠈⠡⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⢶⠶⠶⠶⠆⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠘⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠔⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢋⡆⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡺⠏⠦⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠰⠀⠳⣄⠀⠀⠀⣀⢠⠔⠒⡝⠁⠈⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠈⠷⠐⠊⠁⠀⢀⡠⠟⠲⠦⣤⣀⡀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠈⠱⣤⡜⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⡠⠛⠻⠉⠉⠉⠁⠀⠀⠈⠒⢄⡀⠀⢀⠞⠀⠈⠙⠃⠤⠤⠌⠿⠿⣧⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠙⠓⠤⣀⣀⣀⣀⣀⣀⣀⡀⠀⡨⠛⢷⣄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢈⣳⣤⡚⢠⡄⠀⠀⠀⠀⠀⣤⣤⣤⡀⣀
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠒⣒⣖⡚⠛⣻⣿⡛⢻⣿⣖⠒⠒⠖⠲⠚⠷⠲⣶⣶⡶⠶⣶⣶⡿⢵⣶⣤⢶⣤
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠘⠋⠉⠃⠈⢉⠉⠃⢹⣏⣹⠃⠀⠀⠀⠀⠀⠰⣿⣽⣿⢺⣿⣹⡿⢺⣯⣽⣿⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡦⠤⠴⣦⠤⠤⠤⣤⡤⣥⣤⣤⣤⠤⠤⣤⣤⣤⣬⢭⣥⣤⣭⣭⣥⣬⣭⣉⣠⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡀⠀⢠⣀⣀⣠⣘⣀⣀⣀⣀⣀⣀⣀⣀⣛⣛⣛⣛⣛⣛⣻⣛⣛⣛⣻⣻⣯⢸⡇
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠐⠒⠚⢦⠔⡄⠤⠀⠀⠀⠤⠄⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⠀⠀⠈⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠰⣆⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⢸⠀⠀⠀⠀⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠍⢹⠀⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⢘⣄⠀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠭⠥⠤⠬⠤⠤⠤⠤⠤⠤⠤⢴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠓⢤⣤⣇⠀⠱⢦⣄⠀⠠⣄⡀⢀⣀⣀⡀⢀⣀⣀⣀⣀⣀⢀⣀⠀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣀⢀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛⠓⠒⠚⠛⠛⠛⠒⠛⠓⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣠⠤⠶⠶⠶⠶⠶⠶⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠉⠉⠉⠉⠙⠋⢁⣀⣀⣠⣀⣀⡀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⢀⣀⣀⣀⠀⣠⠖⠛⠉⠉⠀⠀⠀⠉⠙⠳⣤⡀⠀⠀⠳⠶⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠛⠁⣠⠴⠚⠉⠁⠀⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⢀⣀⣤⠾⢥⣄⡀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠛⠲⢦⣄⡀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⡤⠞⠉⠀⠀⠀⠀⣀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠸⠧⣤⣀
⠀⠀⠀⠀⢀⣠⠏⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⠶⢤⣄⠀⠉⢷⡄
⠀⠀⢀⡴⠋⠀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⠳⡄⠀⢻⡄
⠀⢠⡟⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣹⠆⠀⠀⠀⠀⠀⢹⡄⠈⣷
⠀⣿⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣿
⠀⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠛⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡿
⠀⠙⣦⡀⠘⢦⣀⣀⣀⣀⣤⠖⠋⠡⠼⠏⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⡞⢀⣼⠃
⠀⠀⠈⠛⠶⣦⡌⠉⢹⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠘⠋⠉⡿⠶⠤⠤⠶⠚⠉⣠⡞⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠴⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠀⠀⠀⣀⣠⠼⠁⠀⠀⠰⣶⠶⠟⠉
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠙⠓⠶⠦⣤⣤⣤⣤⣤⣤⠴⠛⠲⣤⣀⣀⣀⣀⣀⣀⣤⠴⠚⠉⠀⣨⣍⣉⣉⣉⣀⣀⣠⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⢹⡏⠉⠉⠉⠀⡾⠤⠴⠖⣛⣡⣤⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣤⣤⣤⣤⡴⠾⠛⠁⠀⠉⠛⢳⡟⠀⠀⠀⠀⠸⠷⠖⠛⠋⠉⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣀⣠⣤⠶⡆⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠚⠋⠉⠀⠀⠀⡇⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠊⠁⠯⠽⠭⠇⠤⠀⠿⠿⠼⠵⠸⠥⠁⠸⠿⠫⠼⠯⠼⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠁⠀⠀⠀⠀⠈⠶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⠂⠀⠀⠀⠢⣤⣀⣠⣴⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠙⠢⢄⣀⡀⠀⠀⢀⣀⣠⠟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⠅⠠⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢁⡄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡊⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠒⠂⠀⣀⠄⠀⠀⣀⡄⣀⡤⠞⣹⠁⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠣⠀⠈⠉⠁⠒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⣉⠙⠛⠛⠉⠁⠉⠁⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢅⣀⠤⣤⢤⣤⢤⡤⠼⢑⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⡄⠀⠜⣀⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠼⡇⠂⠀⠒⠒⠒⠒⠂⠀⠐⡁⠀⠠⠀⠀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⠀⠰⠀⠀⠀⠣⠜⠉⡤⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠂⠉⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠢⣀⡠⠆⠀⠀⠀⠀⠀⠀⡠⢶⠁
⠀⠀⡀⠀⠠⠀⠀⠀⠀⠇⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢⠀⠀⢀⣀⠀⠀⠀⠀⠑⠢⢄⣀⣀⣀⣀⣀⣤⣤⡲⠯⠖⠋
⡀⢀⡍⠀⠐⠒⠒⠂⠀⢸⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠀⢀⠀⣤⢯⡑⢢⣀⠀⢀⡔⠙⣏⡙⠉⢹⠏⠀⠀⠙⣄⠀⠀⡀
⡇⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡦⢀⢀⡀⠀⠀⡀⠀⣀⠠⠤⠌⠀⠠⠀⠘⢯⣻⠄⠈⠒⠚⠳⠆⠀⠀⠉⢹⠀⠀⠀⠀⡇
⡇⠠⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⠠⠀⢀⠀⢀⡀⠀⢀⠤⢄⠀⠀⢀⡤⠴⠖⠊⠷⠀⠚⠀⠘⠒⠲⠤⢄⣸⠒⠤⣀⡴⠁
⡇⠀⠈⠦⠤⢤⠤⠤⠴⠏⣄⠀⠀⢩⠀⠀⠀⠁⠀⡯⠉⠐⠅⢤⣀⠀⠀⠘⡄⠀⠈⠁⠠⠠⠠⠄⠀⠀⣀⡤⠞⠋⠀⢠⣧⡴⢲
⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣟⠀⠀⠀⠐⡬⠤⣈⡤⠼⡆⠁⡀⠀⠀⠀⣺⠌⠉⠀⠀⢀⣠⠔⠉⢹⣤⠶⠥⠤⠤⠤⠤⠤
⡇⠁⠈⠁⠀⠉⠁⠉⠉⠉⠉⠁⠈⠈⠉⠁⠁⠁⠁⠉⠐⡀⠀⡌⠁⠉⠉⠁⠉⠁⠐⠠⠄⢀⣀⠳⣊⣤⠤⠚⠉⠀⠀⠀⠸⡎⡅⠉⠁⠉⠉⠈⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡖
⡇⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠉⢹⠃⠀⠒⠂⠒⠁⢧
⡇⠀⠀⠀⠀⠀⠈⠉⠒⠂⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡄
⠧⠤⠤⠤⠤⡌⠀⠀⠀⠀⢨⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡔⠉⠑⠲⠤⢤⡼⠀⠀⠀⠀⠀⠀⠀⢣⠠⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢣⡀⠀⠀⠀⢀⠷⠤⣄⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠐⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⣄⠀⠀⠀⠉⠉⠐⣒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠂⠉⠁
⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢧⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⣴⡿⠖⠃⠀⠀⠀⣤⡄
⠀⣠⣴⡾⠳⢂⡘⢻⡛⠁⣠⣆⣘⡳⣶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⣴⣿⣿⠟⡄⠀⠀⠀⠻⠇⠀⠀⠀⢀⣀
⠀⠀⠀⠶⠛⡻⠛⠛⠛⠀⠀⠀⠈⠃⠊⠁⠀⠀⠀⠀⣠⡄⠰⣏⠼⢛⣱⣿⣿⢿⡟⢀⡖⠀⠀⠀⠀⠀⠀⠠⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢟⡟⠁⡀⠀⠈⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣞⠀⣸⡿⠞⢀⠀⠀⢠⣤⠂
⠀⠀⠀⠀⠀⠀⣰⡟⠟⣴⣿⡷⠷⡄⠀⠀⠀⠙⢇⠀⠸⠅⠀⣸⣆⣴⣤
⠀⣠⠄⢴⠇⠞⣛⣵⣿⣿⣿⠇⢠⠀⠀⠀⢠⠀⡀⠶⢤⣰⣄⡿⠟⡏⣀⠀⠀⠀⠀⠀⢀⡠⠢
⠀⣀⠀⡆⠀⢠⢿⠋⠀⠐⠁⠀⣼⣇⢰⣦⣿⣦⣿⣿⣤⠿⠿⠷⣾⡿⠋⠀⠀⠀⠀⢠⠃⠀⠀⢁
⢀⣿⣴⣷⡄⢀⣄⣼⣧⣦⣾⣷⣿⡿⠻⠿⠟⠋⠉⠀⠀⠴⠟⠛⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢆
⠈⠉⠉⠙⠓⠪⢥⣀⡀⡐⢨⡉⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠃⠀⠀⠀⠀⠀⠈⢢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠺⠷⠤⠿⢷⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠁⠀⠀⠀⠀⠀⢂⠤⠊⠀⠀⠀⠀⠀⠀⠈⠑⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⢀⡘⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⠀⠐⠂⠰⠀⠀⢀⠀⠀⠀⠐⠂⢀⠀⠀⠀⠤
⠀⠀⠀⠃⠀⠀⢀⠀⠠⠀⠀⠀⠒⠀⠀⣀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠰⠀⠈⠠⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠰
⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠔⢠⣴⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⣐⣦⣀⣤⣦⠤⠀⠈
⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⢀⠄⣤⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣖⣀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⡐⢡⣾⣿⣿⣿⣿⣿⡀⠀⢀⣴⣤⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣛⠋⠁
⠀⠀⢠⠀⠀⠀⠀⠀⠀⡐⠐⣋⣯⣿⣿⣿⣿⣿⠃⢀⣼⣿⡿⠿⠿⢿⣿⡿⠿⠿⠉⠉⠋⠻⠿⠁⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠁⠀⠀⠠⢀⣾⣿⣿⣿⣿⣿⣿⣿⡦⠈⢭⠻⣈⡀⠈⢠⠏⣠⣤⣤⣤⡈⢢⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⠁⠀⠀⠈⠀⣨⣶⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⢑⢘⡟⣣⣴⣼⣿⣿⣿⣿⣿⣄⡄⠀⠀⠀⠃⠀⠀⠀⠀⠰
⠀⠘⠀⠀⠀⠀⠀⠐⣷⣿⣯⣉⣽⣿⣿⣿⣿⣿⣾⢅⡠⠼⠿⠟⠛⠛⠋⠘⣿⣿⣿⣿⣿⠈⢄⠀⠀⠀⠠⠀⠀⢠
⠀⠠⠀⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⣌⠁⠄⠀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⣮⡆⠀⠀⠀⠉
⠀⠀⠆⠉⠀⠀⠀⠠⡀⠀⠈⠻⣷⣦⣍⣻⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠻⣣
⠀⠀⠀⠈⠁⠀⠀⠀⢠⠈⠁⠀⠈⠳⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢩⣗⠙⠴⠃⡆⠀⠀⠠
⠀⢀⡀⠀⠀⠀⠀⠃⠀⠀⠀⠠⠀⠀⠠⠌⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⢛⡟⠄⠀⠀⣇⠀⠀⠀⠈
⠀⠀⠀⡄⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠈⢂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠓⠀⠀⢀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠈⣿⣰⣵⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡼⣥⣬⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣶⠾⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⠻⠿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⣉⡉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠺⠟⢀⣝⠁⠺⠗⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠶⠃⠹⣀⠀⣀⣄⡀⣺⠙⡄⠀⡠⣼⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠉⠛⠂⠀⠉⠁⠀⠙⠒⠃⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⠖⢒⣲⠀⡔⢲⡀⢠⠶⡄⡔⠒⠒⠲⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠘⠦⢌⠓⣤⠁⠇⢣⢸⠀⣇⡉⡇⢸⠉⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣴⣒⣚⡠⣏⡰⢧⣈⣾⣀⣀⡙⣇⣸⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠤⠤⢤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠄⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠲⠤⢄⡤⠴⠒⠒⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡆⠀⠀⠀⠀⠀⣠⢋⡜⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⡴⠒⢄⢀⣠⠗⠉⢀⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠑⠒⣺⢡⠃⠀⢀⡎⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠊⠑⠤⠖⠁⠘⠦⠴⠋⠳⠔⠃⠘⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡟⠀⡠⡄⢠⣤⠀⠀⣠⡉⠀⠀⣠⡦⢦⠀⠈⣿⡔⠒⠆⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠁⠈⠀⠈⡏⠀⠀⢀⡷⠁⠀⠀⡸⠀⡼⠀⢀⠇⠘⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠣⠔⠂⠀⠣⠤⠴⠊⠳⠴⠚⠐⠃⠀⠘⠒⠊⠀⠀⠉⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠦⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠁⠀⠈⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠐⢀⠄⠂⠈⠁⠈⠉⠀⠢⢄⠀⠈⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⢠⠀⣆⣔⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠎⠁⠀⢈⠆⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠄⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠴⠠⠀⠀⠀⠊⠄⠀⠀⠀⠐⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠆⠂⠀⢸⠀⠠⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣽⠟⢻⣏⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢁⢠⢴⠊⠐⢀⣨⠉⠕⡢⢦⢬⡉⠁⠀⢁⠅⢤⣀⠀⢂⢤⣄
⠀⠀⠀⠀⠀⢀⠌⢠⠔⠚⣉⡀⠀⢠⡖⠗⢒⣦⣐⣂⣁⢄⣩⡆⣤⢷⠖⠢⡆⣐⣀⢌⠑⢿⣆
⠀⠀⠀⠀⠀⡈⠀⣎⠀⠞⣁⡅⢀⠏⠘⢠⠋⠀⠀⠀⠀⠀⠀⠀⡉⠉⢹⡇⢡⠀⢈⠽⠂⢈⠈
⠀⠀⠀⠀⠀⠠⡀⠈⠓⠤⠤⣉⣀⢤⣀⡆⡄⠉⠙⠀⠀⠀⠐⢡⡴⠾⢼⡧⣤⣭⢥⣤⣴⣶⣷
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠂⠠⠀⠼⠯⣈⡻⠓⠲⣒⠲⠶⠴⠲⢾⠷⠶⠊⡀⠤⠄⠰⠼⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠐⠉⠠⠾⠅⠠⠀⠀⠢⠸⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠄⠂⠐⢀⠀⢠⠀⠀⡐⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⠈⠀⠌⡂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠀⢁⣀⡔⢐⠨⡀⠐⠒⠑⠀⢸⠀⣀⡄⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠉⠚⠛⠉⠄⠂⠔⠨⢘⠳⡦⠄⡠⠐⠜⠡⢐⠠⠀⠁⠈⠪⠐⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣤⣀⡤⠤⢤⡀⠀⠀⠀⠙⠆⣰⠿⠃⠀⠀⠀⢀⣶⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢱⠀⠀⢀⡀⠀⠀⢠⡞⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠞⠀⠀⢸⣧⠀⢰⡿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⣦
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⢧⡉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⡴⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⠃⠀⠀⠀⠚⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⠲⠤⠞⠋⠛⠿⠿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠈⢧⣄⣀⣠⡞⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⢀⣤⠀⠀⠀⢸⡏⠉⠉⠉⠉⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⡂⠀⠀⢸⣧⣤⣤⣤⣄⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣄⡀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⠇⠀⠀⢸⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⡿⣿⣿⣿⣷⡀⠀⠀⠛⠻⠛⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣇⣽⣿⣿⣿⡇⠀⣀⣀⣀⢀⣀⣀⠀⠨⣿⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡿⠁⠀⢨⡟⣱⣿⡇⠀⢿⣿⣿⣿⣿⣿⠁⠈⣿⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣷⠀⠀⠀⠁⠙⢿⡇⠀⠀⠛⠻⠿⠛⠁⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢠⣾⡇⠀⢀⠀⠀⠤⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⡇⠀⢈⠀⢁⠀⠆⠀⣠⣴⣿⣤⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠄⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⢠⣄⣀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠙⠻⠟⠛⠁⠀⠀⠀⠙⠛⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⣤⣤⡄⠀⣿⣿⠀⣤⣤⡄⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠿⠟⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆⣸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣶⠾⠿⠟⠋⠉




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

⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢄⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠠⡏⠀⠀⠀⠀⠈⠉⠑⢆⠀⠋⢰⠋⡠⠄⠀⠓⣄⠀⠀⠀⢘⠁⣀⠥⠀⠀⣌⢮⠑⠁⢀⠀⠀⠀⠊⢄⡊⠀⠠⠀⠁⠄⢱⠀⢛⣌⠂⠀⢀⡠⠤⣤⣄⡁
⠀⠀⠀⠀⠀⠀⡀⣀⠠⢤⣋⠃⠀⠘⡄⠈⠂⡄⡀⠈⠆⠀⢠⠁⠀⣁⠀⡸⠀⠀⠀⠐⡂⠀⠐⠀⠠⠀⠀⠀⠆⠀⠀⠐⢀⠆⠀⣀⠑⠃⠰⡯⡤⠄⡹⢺⠁
⠀⠀⠀⠌⡄⠡⠀⠀⠔⡠⠌⡳⠀⠀⠘⢆⡀⠠⣈⠢⣘⠀⠀⠙⡝⡣⢬⠁⠀⠀⠀⠠⠅⠀⠀⠸⠀⠀⠠⠁⠀⠀⠑⡠⠋⠀⠀⠉⠓⢄⠀⡄⠀⠀⠀⢸
⠀⢀⡀⠰⣉⣉⡀⠤⠂⠂⠉⣀⣀⢀⣀⡀⠈⠒⠈⠔⠀⢠⢶⠀⢛⠽⣳⠀⠀⠀⠡⡡⠊⠘⢄⢀⠰⢀⠅⡀⠀⢂⠜⠁⠈⠦⢇⣀⣀⡘⠀⠃⠠⠤⠔⠈
⠀⠈⠇⠀⠀⣤⠀⠀⡠⠔⠋⠃⢠⠃⠌⠠⠉⢲⡀⠀⠀⣿⠙⠀⠈⣞⡜⠀⠠⠬⡶⠁⠀⠀⣸⡀⢀⠐⢄⡩⠔⢁⠄⡀⠀⣀⠀⣀⠌⠀⠄⢠⡀⠀⠀⢒⠄
⠀⠀⢀⣄⡀⠁⠀⡈⢁⣀⣎⡀⠦⠬⠐⡦⣮⠵⡎⠀⣀⣤⣄⢀⠀⢻⡃⠀⠀⠀⠙⡂⠐⡦⠛⠬⠂⠀⠀⠀⠀⠌⡔⢀⣀⠀⠉⠉⠀⠐⠂⠁⠀⠆
⠀⠀⢿⠀⢹⡄⠘⠰⠒⠠⠐⠚⠒⠒⣉⣀⡤⠞⠀⢘⠇⠀⠀⢉⡗⠀⢠⠄⢠⠀⠀⡀⠼⠀⠀⠀⠀⢀⣤⣄⣤⣎⠀⠈⠋⠀⠊⠑⠀⠀⠈⠀⣀⡀⣀⠂
⠀⠀⢸⢀⠤⠑⡄⠀⠉⠓⠃⠉⠉⠉⠉⠀⠀⠀⠐⡼⠀⠀⠀⢸⡇⠀⢠⠰⠀⠀⠎⠁⠃⠀⠀⠠⣿⣿⣝⣿⣵⣿⣷⠀⢠⠄⠀⠀⡆⠀⠤⣼⡁⢠⠍
⠀⠀⢱⡈⠒⢀⡇⠀⠀⠀⢳⠚⠒⠀⠤⠄⠒⢶⠂⠈⠃⠒⠶⠃⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢠⡀⠀⡼⡅⠀⠀⠀⠀⢃⠀⢄⣀⡠⠃⠀⠀⠉⠉
⠀⠀⠀⠙⠙⠋⢀⠴⡀⡀⢄⠉⠀⠐⠒⠒⠊⠁⠀⢀⢠⡠⢂⡀⠀⠀⣀⠁⠀⠀⠒⠒⡶⢦⠀⠉⠉⠀⠀⢘⠻⠀⠉⠁⠀⢀⠇⠀⠀⠀⡀⠀⢣⠈⠂⢢
⠀⣰⠂⢐⠀⢠⠃⠀⠉⠁⢈⠀⠀⠀⠠⠄⠤⡄⡖⣆⡿⠿⢉⡕⢀⡔⢱⠊⠉⠌⠉⠉⠐⠁⠀⠈⠒⠀⠤⠀⠀⠐⠀⢠⠌⠁⠀⠁⠀⠞⠁⠀⢀⠗⠲⣎⠄
⠀⠈⠉⠀⢀⠆⠀⠀⠀⠀⠄⢠⢤⠀⢲⠒⠀⠛⠁⣀⠠⠔⢋⡔⠹⠃⠈⣰⣹⠟⠉⠀⠒⠀⠒⡄⠀⠶⠀⠀⣠⣀⡀⣠⡄⠀⣀⠀⢰⠁⠂⣰⠁
⠀⠀⠀⢀⠎⠀⠀⠐⠀⠈⠀⠀⠁⠀⠈⠁⠀⠈⢉⡀⠀⣠⣈⡥⠴⠖⠚⠋⣡⣄⢠⠀⠀⠀⢀⠇⠀⠀⣤⢒⠇⠨⠄⠤⠀⠅⠘⠠⡀⠙⢁⠼⠂⡀⣀
⠀⢀⠀⠰⢈⠋⠣⣀⠄⠀⠀⢀⡂⠀⠤⠄⠀⠀⠙⠁⠀⠁⠀⣤⠀⠀⠀⢞⣭⠍⢸⡀⣀⠔⠁⣀⠀⠀⡀⠀⠠⡈⠉⠉⠁⠉⠉⠁⡁⠀⠀⠀⡘⠀⠀⢁
⠀⠘⢤⡀⠀⠈⠁⣁⡀⠀⠀⠰⠺⣾⠪⠀⠁⢠⣶⠶⠒⠒⠠⢟⡦⣄⡀⠘⠁⡠⠀⠉⠀⠀⠈⠊⠀⠀⡆⠁⠀⠀⠀⠉⠀⠁⠀⢹⠁⠀⠀⠀⠇⠀⢀⡘
⠀⠀⡀⠄⠲⢈⠜⠁⢀⠀⠀⠀⠀⠑⠄⠀⠠⠋⠀⠀⠀⠀⠂⡈⠉⠉⠀⠐⠏⠄⢀⠀⠄⠠⠔⢈⣆⠈⠑⠂⠠⠤⠤⠤⠤⠤⠔⢚⠳⠀⠀⠰⠈⠀⠁⡇
⠀⢨⠁⠙⠀⡁⠆⠀⡠⢀⡠⠄⠤⢄⠀⠀⠁⠀⠀⠀⠀⢈⡄⠀⠔⠈⠉⢂⠀⠈⠁⠐⠀⠄⠂⠉⠀⠀⢀⡄⡄⠀⠀⠀⠀⠀⠈⠀⢀⣎⡆⠐⠤⢤⠰
⠀⠀⢅⢐⡂⠠⠔⠁⠠⡃⣰⠉⢱⠀⢧⠀⠛⠀⠀⠐⠚⠛⠁⠜⠀⠀⠐⠸⡄⠐⡄⠠⠜⠀⢂⠉⠆⠀⠏⠐⡀⠄⠀⠀⣀⠐⠠⠀⠀⠓⠇⠀⡇⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢱⠈⠈⢁⢀⡠⠃⣾⠑⠒⠒⠄⠠⠊⠈⠀⠀⠀⣸⠃⢀⠀⠀⠀⠀⠘⠁⠀⠀⣀⠼⠒⠀⠀⠒⠺⡀⠀⠀⣈⡆⠀⠀⠀⠀⡀
⠀⠀⡁⢨⠽⢹⣀⢓⡀⠈⠢⠚⠁⠀⠀⢰⠁⣀⡤⠊⠀⡇⡀⠰⠿⠀⠃⢸⠀⠈⠄⠉⡁⠀⠀⢀⠴⠊⠁⠀⠀⢀⠔⠂⠀⠉⡹⠊⠀⠀⠀⢾⣟⣀⠤⠂
⠀⠈⠁⠀⠠⡀⠸⢘⡀⠀⠰⣉⣈⡳⠀⠀⠁⠘⠘⡄⠀⠡⡐⠀⠀⢈⣠⠎⠀⠀⠀⠈⠁⢀⠄⢃⠀⠀⠀⠤⢀⠀⠀⠀⢀⡼⠡⡮⠀⠀⠒⠊⠉
⠀⠀⠀⠓⠀⠃⣉⠉⠁⠀⠀⠁⠀⢠⠄⠀⡄⠀⠤⠂⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠨⠦⠀⣑⠬⠤⣉⣒⣒⠚⣥⢊⡛⠋⠥⠐⠊⠁⡄⠄⠀⠄⠀⠀⢀⡀
⠀⠀⠀⢀⠀⠁⢈⠀⠀⠀⡉⡕⢀⠈⠒⠬⠀⢀⣤⡀⢰⡋⠉⠉⣷⠀⠉⢱⠀⠀⠃⠀⠀⠀⡀⠀⠀⢀⡠⣤⠀⡠⢦⡀⣀⠤⡀⢀⠉⠀⠀⠀⠀⠀⠈⠖
⠀⢠⠀⡀⠢⠀⠠⠐⠀⠀⠱⢡⠞⠁⠀⠀⠀⠱⡤⠜⠀⠇⠀⢸⠇⠀⠀⡆⠀⠈⠢⠀⣠⡔⠁⠀⠀⠈⠠⠋⠈⠄⠎⠀⠅⠠⠁⠀⣴⣤⢂⡘⠃⠁⢆⣀⡄
⠀⠀⠉⣀⣤⠐⠤⡀⠀⠀⢀⢰⠀⠀⡠⠒⠒⡄⠀⠀⠀⠉⠉⠉⠉⠀⠈⠀⠰⠆⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢇⠠⠤⠀⠀⠀⠀⣠⠃
⠀⠀⣼⢳⣽⣐⡟⠉⠀⠼⠄⢈⠔⠉⠀⠀⠂⡄⠀⢀⠔⢁⡀⠤⠵⠆⠀⠄⠠⠄⠂⡚⠀⠀⠀⡀⠔⠈⠈⠉⡶⠻⡿⠥⠀⠀⠀⠀⠈⠒⠂⠤⠤⠤⠔⠁
⠀⠂⢻⡶⠒⠉⠀⠀⡄⠀⠀⠸⢀⠀⣀⡠⠊⣤⠀⡇⡰⠋⠀⢠⢤⠀⠑⠒⠒⠒⠊⠀⠀⠐⠀⣀⣀⠤⠤⠔⠁⠀⠈⠢⣀⠀⡀⠀⠠⠄⠁⠀⠀⣀⣀⢺⠃
⠀⠀⠀⠈⠐⠐⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠀⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠋⠉⠉⠉⠛⠓⢲⢤⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⢠⣦⠰⣿⠇⠀⣭⡓⠦⣄⡀⠀⠀⠀⠉⢳⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⠼⠃⠀⢸⣻⣮⠟⠛⠙⠲⢼⡟⠁⠀⡀⠉⠓⢦⡀⢀⡟⣦⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⡄⢀⣤⢀⣸⠃⠀⠀⠀⠀⠀⠓⢤⡿⠋⠀⠀⠀⣙⡾⠀⣇⣀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠖⠃⠀⢹⡏⠀⠽⠂⠀⠀⠠⢶⡆⠈⢣⡴⠶⠂⠈⠉⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣯⡿⠁⣠⣄⣠⣄⣀⣠⡀⢀⡀⢸⠀⠀⠀⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠳⡀⠀⠀⠁⠀⠉⠀⠈⠉⠀⢈⡷⠤⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠤⢤⣤⣄⣠⡤⣶⡿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠋⠙⠲⢶⣶⠶⢾⣋⣉⠀⢸⣧⠳⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣨⣧⣄⣠⣼⣧⣀⣀⣈⣉⣠⣼⡇⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⣿⢻⠀⢠⡄⠀⣄⠀⠀⠸⠀⠀⠀⣴⢃⡜⠉⠛⠒⣶⠶⢭⡉⠁⠀⢸⡇⠀⢳⡀⢘⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⡇⠘⡆⣸⢧⢠⢿⡀⠀⠀⠀⣠⠂⣿⢿⡁⠀⠀⢸⠁⠀⠀⠙⠒⠦⣼⡇⠀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣳⠈⢠⡄⠙⠃⠘⠋⣸⠁⠀⠀⠐⠇⠀⢁⣸⠧⠤⠤⣼⡀⠀⠀⢰⣧⡖⠃⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠀⡼⠙⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⠒⣾⣲⣶⣏⣙⣛⣻⣛⣲⠭⣟⡷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⠤⣾⠿⠭⢞⡿⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣧⣀⣀⣠⣧⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢡⢾⠉⡏⢇⠙⣍⠪⣝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣴⠃⣰⠖⢾⡘⡶⠋⣳⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠉⠒⠋⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠒⡒⢲⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⣸⣤⣤⠀⣉⣈⣃⣀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢦⣤⣤⡄⢀⣀⣠⡩⢵⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢘⣻⣿⠧⠤⡤⠼⣿⣾⣻⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣹⠋⠁⠀⠀⠀⠀⠉⢭⣇
⠀⠀⠀⠀⠀⠀⢠⠓⡇⠀⠀⠀⠀⠀⠀⠀⠘⢟⣆
⠀⠀⠀⠀⠀⢠⠏⠀⠀⡀⠀⠀⠘⣷⡦⣤⡁⠐⠿⢄
⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⢉⠉⠈⠩⢷⣦⡘⡄
⠀⠀⠀⠀⢸⣷⣤⡀⠀⠀⠀⠀⡀⡞⠀⠀⣨⣒⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠞⢼⢢⡦⢀
⠀⠀⠀⠀⠘⣿⠙⠟⠱⠶⣶⣟⣾⣟⣵⣿⣿⣿⣟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠌⠀⠫⡀⠄⢠⣀
⠀⠀⠀⠀⢠⢗⠀⠀⠀⠀⠀⠀⡉⣛⢠⠮⣹⣿⡏⢹⣿⠀⠀⠀⠀⠀⠐⠀⠊⠁⠀⠀⠀⠀⠀⠁⠁⡀⠰⢦⣄⠀⡂⢠⡀
⠀⠀⠀⠀⠘⡜⣴⠀⠀⠈⠀⢀⣈⠉⠙⠳⣿⣻⢉⣾⠏⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠨⠨⣆⣄⠁⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣆⣀⣀⣼⠡⠀⣠⣿⣿⣻⡾⠋⠀⠀⠀⠈⠀⠁⠀⠀⠄⡄⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠌⠒⡍⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡯⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠂⠠⢠⠠⠁⠀⠀⠀⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣻⣿⣷⣖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣍⣚⣿⣯⣥⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣚⣿⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⢿⣿⣿⡷⢞⣫
⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⡷⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠠⣄⣛⣿⣿⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠠⣙⣻⣿⣿⣟⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠢⡀⠀⠀⠀⠀⠀⡉⣽⣿⣿⣿⣿⣿⣏⣡⣤⠄
⠀⠀⠀⠀⠀⠀⠠⣭⣿⣿⣿⣿⣯⠔⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⣌⠛⢷⣿⣿⣿⣿⣿⣿⠟⢁⣀
⠀⠀⠀⠀⠀⢀⣄⣾⣿⣿⣿⣏⣥⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣞⡉⠻⣶⣿⣿⣿⣿⣿⣿⡿⠛⣩⡀
⠀⠀⠀⠀⠀⢠⣭⣿⣿⣿⣿⣷⣦⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠬⣿⣿⠾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣷⠶⠛⠓⠛⠒⠒⠐⠒⠒⠛⠉⠀⢀⣄⣰⣷⣿⣷⣞⣙⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⡥
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣯⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠈⠩⣽⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣽⣶⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠤⠴⢾⢿⣿⣿⣿⣿⣿⣶⣤⡤⠖⠀⠀⣆⠀⡄⠀⠀⠀⡄⢰⣇⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⣯⣍⡀⠀⠀⢰⡗⢠⡇⠀⡗⠀⣯⢸⣷⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣤⣿⢀⣟⣾⣿⣭⣷⣿⣿⣤⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⣸⣏⣟⣿⣫⣋⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠅⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣼⣿⡿⢠⠀⠿⣿⣿⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠑⢀⣾⣿⡁⣽⣿⡇⠀⠠⠀⠈⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃⠀⢠⣿⣿⣿⣶⣿⣿⣧⠀⠀⠖⠀⠀⡀⠀⠀⠀⣤⣶⣯⠀⠀⢀⡀⣠⣴⡾⠋⠀⠆
⠀⠀⠀⠀⠀⠀⣀⣴⠀⡀⠀⠀⣠⡞⠀⠰⣦⣾⣿⣿⣿⠃⣴⣿⣿⡿⣿⣿⣽⣿⣿⣦⣀⠀⠀⠀⡀⢀⠀⢀⣼⣯⠏⠀⠀⠋⠈⢹⡾⠃⠀⠐
⠀⠀⠀⠀⢰⢿⣿⣇⠀⢀⣴⣿⣿⡟⠀⠀⠀⠙⣿⢻⣿⡖⠻⣻⡿⡃⠛⠋⡵⢟⠿⢿⣿⣿⣶⡀⠀⠀⣀⡈⠼⠟⢠⢤⡼⠁⢀⠘⠃⠀⣰⡄
⠀⠀⠀⠀⢙⢿⠟⠻⢠⣾⣻⢹⣋⡷⢸⢁⢸⡆⠘⣇⠹⠏⢤⡁⠠⣸⡄⢀⠀⠀⠀⠈⠁⠛⠹⣿⢶⣦⣌⢁⠋⣰⡎⣸⣯⡀⢸⣧⢰⣦⢼⣿⣀
⠀⠀⠀⠀⢀⣆⣶⣶⣿⣿⣶⣾⣷⣷⣿⣾⣼⣷⣶⣿⣧⣿⣿⣧⠀⣽⣿⢠⡆⠀⠀⠀⠀⠀⢀⣀⣴⣹⣇⣨⣤⣿⣇⣸⣿⣧⣶⣿⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣀⣁⡒⠺⠿⠿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡋⡿⣿⣿⣿⢿⣿⣯⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠆⠀⠀⠈⠙⢻⣿⣿⣾⣯⢻⢹⢸⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣾⣿⣿⣷⣿⣿⣟⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠉⢀⣄⠀⢀⠀⣷⣸⢫⠻⢿⣄⣾⣌⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣡⣿⢿⡿⢻⣟⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⡄⠀⠀⠀⠀⠀⣰⡀⣸⣿⣦⣴⣇⣿⢾⣷⡦⠊⠉⢡⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣏⢹⢸⣷⠋⣿⣿⣿⣿⣯⡀⠀⣾⣆⠀⠀⢼⣯⠀⠀⠀⣤⣤⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣇⣤⣽⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣸⠉⢾⣬⣿⣿⣿⣭⣠⠌⣿⢗⡀⢈⣾⣿⣞⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣶⣦⣝⣿⣿⣿⣋⡥⢒⣿⡿⠃⡀⣹⣟⣯⡆⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⢃⣿⣿⣯⣶⣃⡄⠀⠀⠀⣾⣟⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣦⠀⠰⣶⣿⣿⣵⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠴⠲⣾⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣷⣆⣹⣾⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠈⠉⠉⠉⠉⠉⠰⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢁⡄⢶⠋⠉⠉⣙⢻⠋⠑⠶⣤⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡾⠋⠹⣄⢳⡀⠀⣿⢸⠀⠀⠀⣠⠟⣦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢰⠏⠀⠀⠀⠙⣆⠙⣦⡁⠸⠀⡴⠚⣡⠞⠁⢻⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡤⠄⠀⠲⠶⠒⠒⠒⠒⠒⠒⠒⠲⠂⠀⠀⠄⢠⡏⠀⠷⠠⠤⢤⣤⣤⣬⣧⠘⠁⠘⢁⣀⠊⠁⠀⠀⠀⢳
⢀⣠⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠗⠒⠲⠢⠤⣤⣤⡀⠀⡀⠀⠈⣉⠉⠐⠒⠒⠂⣾
⣿⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣴⣋⣤⡾⢻⣆⠙⣏⠉⠉⠉⠉⠀⢶
⣿⠀⠀⠀⠉⠙⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣵⣤⣵⡦⢲⡶⠛⡩⠛⠉⡄⢸⡟⢶⡈⢷⡀⠀⠀⢰⠇⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠘⢂⡀⠈⠀⠀⠀⡇⢸⡇⠀⠙⣎⡱⣄⣰⠋⢠⠟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⠤⣀⡀⠀⣧⢸⡇⠀⣀⣨⡷⠉⢁⡴⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢈⠉⠁⠘⢻⡟⠈⣉⣁⣤⠖⠉
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢺⠟⠀⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⣷⢠⡞⠙⠛⠦⠴⠦⠤⠖⠛⠛⠉⠉⢉⣛⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠛⠋⠉⠙⠛⠶⣄⠀⢠⡏
⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁⠀⠈⢣⠉⠛⠳⠶⠶⠶⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇
⠀⢻⠀⣧⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠀⣠⠂⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠇
⠀⢸⡆⢿⠀⠀⠀⠀⠀⣾⠀⢀⣀⠴⠚⠉⢣⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿
⠀⠀⡇⢸⡄⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⣷⠈⡇⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⠶⠖⠒⠐⠀⠒⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⢹⠀⣧⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡧⠀⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⡇⣼
⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠈⢷⠘⢦⡀⠀⣀⠴⠞⠁⠀⡧⠀⠀⠀⡟⠉⠉⠉⠁⠀⠀⠈⡇⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠑⠶⠞⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⢠⡏
⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⢟⠀⡀⠀⠁⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢀⡏⢸⡇
⠀⠀⠀⢻⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠉⠳⢮⣵⣏⠀⠀⠀⠀⠀⠀⠀⢸⠅⡇⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠘⡆⢿⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢐⣇⡀⠀⠀⠀⢹⡗⠶⠠⠤⠤⠀⠀⠀⠶⠁⠀⠀⠀⣼⢠⡇
⠀⠀⠀⠀⣇⠸⡇⠀⠀⠀⠀⠀⢰⠋⠀⠀⢠⠏⠀⠉⠳⢦⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠇
⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⢃⣦⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⡄⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠇
⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⣀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇
⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠁
⠀⠀⠀⠀⠀⠀⢻⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢙⡆⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠞⠋⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⢦⣀⠈⠙⠛⠛⠲⠶⠶⠶⠶⠶⠶⠖⠛⠛⠉⠁⣀⡤⠾⠋⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠉⠛⠳⠶⠦⢤⣤⣀⣀⣀⣀⣤⠤⠴⠶⠛⠛⠉⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠦⢀⣀⣀⣀⣀⣀⣀⣀⡤⠶⠶⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣟⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣷⡿⡿⣿⣿⣿⣻⢿⣻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⡤⠤⠴⢖⠛⠛⢋⠙⣙⠛⠓⡂⠳⠾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠀⣠⣄⣃⣀⣡⣈⡂⣐⣂⣀⣀⡒⣀⣡⣀⠀⠶⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣐⣈⣉⣭⠉⠋⠙⠉⠉⠭⠉⠹⠉⠀⣉⡀⠀⠀⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡏⣿⣿⡟⣿⣿⣻⣿⣿⣻⣿⣟⣿⣿⡇⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣷⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡏⠁⣴⠟⠛⠯⠉⠉⠉⢩⣿⣭⡍⠉⢹⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡇⠀⣘⣷⣤⡄⣠⡀⠀⣟⡿⣿⠖⠀⣼⠀⠈⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡀⣿⣾⡉⢀⣿⣼⢡⡟⣰⡟⢡⣏⢰⠇⣿⠀⡀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠂⣿⣈⣩⣯⣥⣬⣯⣽⣯⣽⣯⣽⣯⣤⣿⠀⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢹⣿⡏⣿⣿⣇⢸⣿⣿⢸⣿⣿⢻⣿⣿⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠳⠾⢿⣿⣼⣿⣿⣼⣿⣿⣼⠿⠋⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠀⠀⠀⠀⡄⠀⠈⡀⠠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⠀⠀⠀⠓⠀⠀⠀⡀⠒⠀⠀⢀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢉⠙⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⠂⠀⠀⠘⠒⠀⠤⡀
⠀⠀⠀⢀⣠⡤⠟⠓⠛⠉⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⢀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⡞⠀⠀⠀⣀⠤⠴⣶⣻⡯⠥⠼⠥⠀⠀⠉⠉⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⢧⠀⣀⡤⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⠟⠁⠀⠀⠀⢀⣀⣤⡤⠤⠐⠒⠐⣒⡛⠩⠤⣄⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣿⠀⠀⠀⢰⣼⡿⠷⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⡖⠀⠂⠄⡄⠀⠀⠲⠀⠀⠻
⠘⡷⠖⠋⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⢸
⣾⠁⠀⠀⣀⣠⠤⠶⣖⣛⣫⣭⠥⠦⠔⠂⠀⠈⠑⡄⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠺⡈⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⢻⡀⠀⠊⢉⡴⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡠⠃⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⣹⢗⣽⠋⠀⠀⢀⣀⣤⠤⠶⠶⣲⠾⠿⠯⢥⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠃⢸⠇⠀⠀⢰⢟⠉⠈⠙⠻⣼⠃⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡰⠁
⢸⠀⠈⠢⠤⠴⠋⢸⡀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⡰⠁
⣼⠄⠒⠚⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⢀⠇
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸
⠀⠙⢆⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠂⠒⠀⠚⠀⠀⠀⠀⠤⠧⠠⠴⠒⡶⠁
⠀⠀⠀⢻⠛⠉⠉⠛⠉⠉⠙⠯⠉⠙⠯⠉⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣙⣓⠀⡇
⠀⠀⠀⢸⡆⠀⠀⠤⠤⠤⠤⠤⠤⠀⠄⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⠠⠤⠤⠤⠤⢤⣤⠶⡶⠶⠀⡇
⠀⠀⠀⠺⡉⠀⠰⠶⠶⠶⠒⣒⣒⣛⡫⠁⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⠀⠀⠀⠀⠠⠤⠤⠴⣶⣾⣿⠀⡆
⠀⠀⠀⢺⡅⠐⠶⠶⠶⠚⠛⣛⣚⣓⡭⠁⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⠀⠀⣀⣀⣀⣀⣠⣵⣶⣶⢶⠀⡇
⠀⠀⠀⢾⡅⠀⠶⠶⠶⠒⠚⢛⣒⣛⠻⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⣤⣤⡶⣶⡃⣠⠇
⠀⠀⠀⠈⠳⣄⡀⠒⠒⠒⢚⣓⣛⣾⣥⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠈⠉⠭⠋⠔⠁
⠀⠀⠀⠀⠀⠀⠻⡍⠉⠉⠉⡭⢉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣛⣛⣿⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣧⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⢦⡔⢀⠎⠀⣤⢀⠄⠀⡱⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⠀⠁⠊⠀⢀⠁⠈⠀⢀⠅⡀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠀⠀⢤⣡⠎⠀⢀⠀⠄⠀⢀⢎⠜⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⡄⠀⢀⡖⠀⠀⠀⡠⠂⠀⢀⠔⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡃⢐⠎⠀⢀⠊⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠐⠀⠀⠀⠀⠁⠀⠠⠔⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⡀⠀⠀⠀⡁⠀⢀⢂⠅⢄⠈⠒⠀⢠⠄⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⡀⢖⠁⠄⠀⠀⠁⠀⢨⡌⠓⠃⠀⠀⠀⠈⠡⠂⠀⠀⠇⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠙⠸⠒⢁⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⡄⠤⢉⢀⡔⠘⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠈⡀⠀⠀⣰⠆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⡀⢉⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⢀⠄⠉⠁⠻⣚⠠⠧⠄⣄⠀⠘⢆⠀⠀⠀⠀⣀⠀⠁⠈
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⡠⠇⠀⠀⠀⡁⠀⠀⢄⡀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠴⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡇
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠍
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠎
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⢄⡀⣀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣿⣿⣻⣿⣿⣟⣻