⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⢠⠀
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⣿⣿⡇⠀⠀⠀⠀⠂⠀⢸⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡿⠟⠃⠀⠀⠀⠀⠀⡀⢸⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢎⠑⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠂⠀⣵⣿⣿⣿⠿⠀⠀⠀⠀⠠⠁⠀⡌⠀⡰⠁⢀⣤⣶⢫⠀⠀⠀⠀⠀⡇⠸⠀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⡿⢠⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄⠸⠟⠛⠃⢀⡀⠀⡄⠀⠀⠀⠀⢘⠀⠀⢃⠀⡼⣛⣼⠀⠀⠀⠀⠀⠠⢀⠼⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⡿⡁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⢠⣤⣶⣾⣿⣿⠃⣰⠁⠀⠀⠀⠀⠀⢀⣨⡿⠀⢼⡿⠋⠀⠀⠀⠀⠀⣀⣾⢂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢙⠭⡽⡿⡇⢀⢠⡛⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣪⠃⢸⠃⣨⣭⣽⣯⡝⠁⣼⠏⠀⠀⠀⠀⣰⠁⢞⣷⠂⠈⠀⠀⠀⡠⠀⠀⠀⠀⣾⠏⣼
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠟⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠒⠀⠆⠸⠛⠻⠙⠉⠀⠳⢣⠀⢀⢄⡔⣰⡇⠴⠋⢁⢀⠀⠀⡄⢀⠁⠀⢠⠀⠰⠋⢠⣿
⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠀⣚⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠴⢀⣄⠁⣰⣽⢟⣼⡏⣰⠄⣠⡶⣴⣶⣆⡴⠂⠀⠀⠏⠀⠀⣠⣿⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⢀⡀⠀⠀⢤⡀⣴⣣⢖⣵⣾⣿⣷⣿⡟⣼⣿⣷⣹⣿⣟⣲⣴⣼⡏⠀⢀⡀⠀⣴⣿⣿
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣜⡄⠸⣿⣾⣿⣾⣟⣴⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠘⠁⠀⣿⣿⣫
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⡟⣤⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡹⡻⣿⣿⡏⠀⠀⡾⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣧⡸⢍⣿⣷⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢛⡿⠁⠀⠀⡇⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠁⠀⠀⠀⠁⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢫⣽⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⠀⣦⣄⢘⠻⣿⣻⠿⠻⠿⠿⣿⣿⣿⣛⢉⣒⣢⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⡇⡄⣿⡿⣾⣿⣶⣝⠷⢶⣾⣤⣴⣿⣯⣿⣿⣦⣦⣤⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣷⠇⢽⡇⣿⣿⣿⣿⣿⣶⣉⠙⠿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢿⣿⣿⡿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⡀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠉⠛⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⣧⠈⣷⢻⣿⣿⣿⣿⣿⣿⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⢻⢸⣿⣿⣿⣿⣿⠟⠁⣿⣗⢠⠀⠀⠀⠀⣀⡣⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⣀⡴⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠘⠟⡿⠿⠟⠋⠁⢐⡡⣿⣿⡆⠀⠀⠀⠀⠸⣿⣼⣿⣿⠀⠀⠀⢰⣭⢀⠀⠀⠀⠀⠀⠘⣿
⣤⣶⣾⡆⠋⠀⠀⢠⢂⣴⡇⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣼⢅⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⡏⣿⣷⣀⠀⠀⠀⡀⢻⣿
⣿⡿⠛⢁⡠⣴⠓⣱⣿⣿⢧⠀⠀⠀⠀⢸⣻⠀⠀⠀⠀⠀⢰⠀⠀⣀⠀⠀⢀⠀⠀⠀⣠⣾⣿⣏⠞⣱⣿⣿⣿⡐⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠸⣸⣿⣿⣿⣦⡀⠀⢹⡀
⣠⣴⣿⣿⠇⢡⣾⣿⣿⣿⣻⡄⠀⠀⠀⠈⣭⠀⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⣿⣿⢣⣆⠈⠻⣿⠳⠋⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹⣷⢿⣿⡄⠀⠀⣱⣿⣿⣿⣿⣿⣷⠀⠀
⡿⣛⣿⠗⣴⣏⢿⣿⣿⣿⣽⡇⠀⠀⠀⢠⣙⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢧⣿⣿⣆⠳⡄⠛⢢⣿⣿⣿⣿⣷⢨⠀⠀⠀⠙⣩⣾⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀
⢹⡿⢋⣾⣿⣿⣎⢿⣿⣿⡎⡇⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠐⠀⢸⣿⣿⣿⣾⡿⡟⣿⡜⠙⢼⣿⣿⣿⣿⣿⣿⡆⠃⠀⠀⢸⣿⣿⠘⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢠⣿⣿⣿⣿⣿⣮⠿⣿⡇⣧⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⡉⢫⠾⠋⠘⠷⣊⣵⣿⣷⣽⢿⠿⣟⣋⣭⣦⠈⠀⠀⢸⣿⠇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⡏