Igloo Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⢒⣒⣒⣒⣲⣦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⠀⠀⠀⡠⠊⠀⠀⠀⠈⠳⡀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠢⢄⣀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⣙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢀⠜⠉⠉⠛⠒⠒⠒⠲⡆⠀⠐⠒⠒⠓⠒⠺⡉⠉⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢀⣞⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⣨⣇
⠀⠀⠀⠀⠀⡾⢀⠛⠢⢄⣀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⡠⠴⡊⠀⠘⡆
⠀⠀⠀⠀⢰⠇⡘⠀⠀⠀⠀⠈⠉⠑⠒⠒⢲⠂⠀⠀⠤⠤⠴⠦⠤⠤⠄⠰⡖⠒⠊⠉⠉⠉⠀⠀⠀⢱⠀⠀⢻
⠀⠀⠀⠀⢸⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⣀⣸⡀
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠞⠋⠙⠤⢄⡀⠉⠙⠲⣄
⠀⠀⠀⠀⢸⠬⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠋⠀⠀⠀⠀⢀⣠⠜⠟⠛⠓⠺⣧
⠀⠀⠀⠀⢸⠀⠈⠙⠒⢦⠤⣀⣀⣀⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣞⡀⠀⠀⠀⠀⡴⠋⠀⢀⣥⠴⠦⡄⠸⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠁⠉⠉⠉⡟⠀⠈⠑⠂⢤⣞⠁⠀⡰⠋⠀⠀⠀⢸⡤⡇
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡾⠀⠉⣹⠁⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⡇⠀⠀⡟⠙⠒⠲⠶⠤⠼⠧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠶⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠿⣄⣉⠐⠀⢼⡧⠔⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠾⠷⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠊⠉⠉⠉⠉⠉⠉⠒⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠓⢦⡀
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⡏⠈⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠀⠀⠀⠘⡆⠀⠀⠙⢆
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠱⡀
⠀⠀⠀⡜⠉⠑⡖⠤⠄⣀⣀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⡸⠤⠔⠒⠉⢻⡄
⠀⠀⡜⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠐⡖⠒⠒⠒⠒⠛⠒⠂⠀⠉⠉⡏⠉⠀⠀⠀⠀⠀⠀⠀⠸⡘⡄
⠀⣸⡀⠀⠀⡎⠀⠀⠀⡠⣴⠒⠈⠉⠉⠉⠒⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡀
⢀⠃⠉⠒⠴⢅⣀⣰⠟⠋⢫⠉⠉⠉⠑⠢⣄⣀⠤⠑⡄⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣀⡠⠤⠚⠉⢃
⠸⠀⠀⠀⠀⠀⡼⠁⠀⠀⢘⣄⣀⡀⠀⡴⠉⢦⠀⠀⠘⣄⣀⣀⣀⡤⠤⠤⠤⠷⠖⠒⠒⠉⠉⡇⠀⠀⠀⠀⠸
⣇⠀⠀⠀⠀⢰⠓⠢⢄⣞⣁⠤⠤⣿⣾⠀⠀⠈⡆⠀⠀⢧⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀
⡇⠙⠢⢄⡀⢸⠀⠀⡼⠀⠀⠀⠀⣿⣿⡇⠀⠀⡇⣀⠔⢻⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⠤⠚⣇
⠈⠢⣀⠀⠈⢹⠤⣀⣇⢀⣀⠠⠤⣿⣿⡷⠒⠒⡟⠁⠀⢸⠀⠀⢣⠀⠀⠀⠀⠀⣀⣀⡠⠤⠔⠊⠉⠀⢀⡠⠋
⠀⠀⠀⠑⠢⣼⠀⠀⡏⠀⠀⠀⠀⣿⣿⡇⠀⠀⡇⠀⠀⢸⠒⠒⠚⠒⠈⠉⠉⠉⠘⡄⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⢸⠤⣀⣇⣠⠤⠒⠉⣿⣿⡇⠀⠀⣇⡠⠖⢹⠀⠀⠀⠀⠀⠀⢀⣀⣀⠧⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠘⠤⣀⣇⣀⠤⠒⠋⠉⠉⡟⠉⠁⡏⠀⡠⠞⠒⠒⠚⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣧⣤⣤⣷⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⠤⠒⢊⠝⠋⡍⠙⢏⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠂⡩⠛⠉⠉⠫⡑⠒⠤⡀
⠀⠀⢀⡴⠋⠀⠀⡰⠁⠀⠀⡇⠀⠀⠳⡀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⢀⠜⠀⠀⠀⠀⠀⠈⠆⠀⠈⢢⡀
⠀⢠⠎⠈⠉⠉⡸⠓⠒⠒⠒⡗⠒⠒⠂⢳⠉⠉⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠁⠀⠚⠒⠒⠒⠲⠒⠒⠂⠘⡆⠉⠀⠘⡄
⠀⡎⠀⣀⡤⠴⠧⡤⣀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢰⠁⢀⣠⠤⠼⠤⢄⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠸
⢸⣧⠾⢛⣓⣒⡚⢄⡠⠓⣄⡇⠀⠀⠀⠀⢸⢀⣀⠤⢺⠀⠀⠀⠀⠀⠀⣆⡴⠗⠚⠒⠒⠢⣀⠈⠢⡀⠀⠀⠀⠀⠀⡄⢀⡠⠔
⣸⢣⣞⡽⢯⣿⣿⢦⠱⣀⠸⡗⠒⠒⠒⠈⢹⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⡟⠐⠁⠀⠀⠀⠀⠈⢆⠀⢿⠂⠒⠒⠀⠉⠋
⡇⣾⣀⣤⣤⡶⣿⣸⡆⡇⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⣿⣮⣿⣾⣟⣻⣍⡇⡏⠀⡇⠀⠀⣀⣀⡸⠤⠒⠉⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⣀⣀⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠴⠒⠋⠉⠉⠑⠛⠓⠢⣄⣀
⠀⠀⠀⠀⠀⠀⢠⢊⡥⢀⣠⡤⢀⠀⢀⣀⣠⢄⡠⠦⠤⠀⣈⣙⣄
⠀⠀⠀⠀⠀⣀⠼⠓⠦⣴⠯⢤⣄⣒⣊⣍⣀⣀⣀⣀⡔⡩⠥⠤⠚⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣼⠁⠀⠀⡔⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⣆⣀⣠⠴⠒⠚⠉⠋⠉⡉⠳⠦⡄
⠀⠀⠀⡸⠁⠈⠁⡨⠓⠒⠐⠤⠀⠤⠴⠣⠤⠤⠤⠀⠦⠔⢣⠀⠃⠉⠉⠁⠀⡜⠉⠉⢹⠒⠚⠉⠋⠀⢧
⠀⠀⢰⣇⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠇⢀⣀⡬⣀⣀⣀⣀⠀⢿
⠀⢀⡟⠀⠈⠙⠃⠐⠒⠦⠤⠀⠠⠤⠄⠤⠤⠀⠤⠤⠤⠤⠔⠂⠚⠙⠉⠉⠉⡆⠀⠀⡇⠀⠀⠁⠀⠀⢸⡀
⠀⢸⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣇⠀⠀⣀⢀⡀⠉
⠀⠛⠠⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠈⠉⠉⠀⠉⠀⠀⠉
⠀⠀⠀⠉⠀⠀⠀⠒⠀⠀⠀⠤⠒⠀⠚⠀⠀⠈⠉⠒⠀⠀⠀⠀⠁⠐⠂⠀⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⠒⠒⠒⠂⠠⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠩⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠉⠉⠚⠓⠒⠚⠓⠉⠉⠉⠛⠒⠲⠞⠳⠦⣬⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠐⠢⠤⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣀⠤⠤⠖⠛⠉⠀⠀⠀⠀⠀⠀⣢⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠓⠛⠒⠒⠓⠋⠙⠢⣀⣀⡀⠀⠀⢸⡄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⣀⣀⣀⡤⠊⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠉⠉⠉⠉⠉⠙⢦⣀⣀⣀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⡄⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠚⠉⠉⠉⠉⠛⠒⠦⣄⡀⠀⠀⠀⠀⠉⠀⠈⠣⣀⣳
⠀⠀⠀⠀⠀⠀⠀⡴⠉⢻⠀⠀⠀⠀⢀⣀⣤⠖⠁⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢤⡀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠠⢈⡞⠁⠀⠀⠀⠸⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢡⠀⠙⣄⠁⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢀⡷⠖⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠻⡄⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡎⠀⢠⣴⣶⣿⣿⣿⣿⣿⠛⠲⡄⠀⢸⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⢰⡃⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⢸⠀⠀⠀⣇⠀⠀⠀⠀⣆⠀⠀⠸⡄
⠀⠀⠀⢀⡟⠲⠔⣟⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠀⢸⠀⠀⠀⠸⡄⠀⠀⠀⢻⡀⢀⣤⠃
⠀⠀⠀⣸⠀⠀⠀⠈⠑⠒⠦⠤⣤⡴⢻⡀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠇⠀⢸⡆⠀⠀⠀⠈⠓⠒⠛⠁⠉⠁⠈⣇
⠀⠀⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⢧
⠀⠀⡇⠙⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⡟⢀⡐⡛⢦⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡠⠋⢸
⠀⠀⢣⡀⠀⠙⠢⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠚⠒⡁⣸⢻⠇⠀⠀⠀⠀⠀⢀⣧⡤⠔⠋⠀⠀⡼
⠀⠀⠀⠙⠦⣀⠀⠀⠈⠉⠒⠢⠤⣀⣀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⢸⣿⠿⠛⠛⠛⠻⠿⡄⠀⢸⠁⠈⡄⠤⠤⠔⠒⠊⠉⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠙⠑⠒⠒⠒⠒⠶⠖⠖⠶⠶⢧⡀⠸⠁⠀⠀⠀⠀⠀⠀⠑⣤⣸⡀⠀⡇⠀⠀⠀⠀⢀⣀⠤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠔⠒⠒⠒⠀⠈⠉⠉⠁⠀⠀⠈⠉⣑⡖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠓⠤⠤⠤⠄⠀⠐⠒⠒⠂⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠒⠉⠉⠉⠓⠒⠦⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣠⣤⣤⣀⣀⡀⠀⣠⣤⣄⣀⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢉⡏⠁⠀⠈⠉⠉⠙⠛⠃⠀⠀⠉⠉⣙⡄⠀⠀⠀⡜⠛⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⡟⠛⠛⠺⠲⠶⠶⠶⠶⠒⠒⠚⠋⠉⠁⠈⠄⠀⢻⣟⣛⣓⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠓⠚⠋⠛⠛⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⡀⢧⠈⢭⠉⡉⠉⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢛⣿⣿⣦⠀⠀⠀⣠⣠⣀⣈⣳⠄⠠⢴⠤⠴⠖⠒⠓⠛⠒⠾⣷⡚⢿⡛⠛⣦⣴⣧⣸
⠀⠀⠀⠀⠀⠀⣠⠤⠤⡴⠖⠛⠋⢱⡏⣾⣿⣿⠘⣷⠛⠋⠀⠀⠈⠈⢩⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢈⢈⡌⠉⡤⠶⠦⢤
⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠘⠷⣿⣟⣻⠀⣾⠀⠀⠀⡆⠀⢀⣸⣀⣀⣘⣀⣀⣀⣀⣠⣤⠴⠞⢧⠀⠀⠀⠀⣐⡀⠀⠀⠀⠑⢄
⠀⠀⠀⢰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣿⠶⠶⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠐⠉⠉⠉⠉⠉⠉⠉⠉⠃⠀⠀⣸
⠀⠀⠀⠀⠑⠲⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⢀⣠⡤⠤⠤⢤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠶⠒⠄⣀⣀⣀⣀⣠⣤⠤⠤⠶⠒⠒⠒⠒⠒⠒⠛⠛⠋⠉








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⢺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡧⠀⢰⡿⠛⠁
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⢾⣿⣷⠂⠈⠀⡃⠀⠀⠀⢰⡄
⣶⡀⣴⠀⣾⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡀⠁⠀⠲⣼⡟⣷⠃⠀⠚⡿⢀⣾⣀⣿⣷⢀
⣿⣵⣿⣭⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⣷⢀⠀⢠⠎⠀⠸⠓⠀⠀⢳⡿⣿⣿⡿⠁⠀⠇⣼⣿⣿⣿⣿⣿
⣿⣤⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⠀⠎⠆⠀⠀⠡⠀⠈⢻⣼⡟⡼⣇⡤⠀⠀⠛⠙⠙⠛⠛⠻
⠉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠈⢳⡾⠋⣿⢷⢹⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠛⠿⠚⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠉⠉⡍⠀⠀⠀⠉⢩⡉⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠤⢄⣀⣀⠀⡸⠀⠀⠀⠀⠀⢀⣳⣀⡤⠤⢾⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⠀⠈⠉⠉⠉⠉⠉⠉⡍⠉⠀⠀⠀⠀⠀⢣⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠡⠤⣀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣀⣠⠤⠒⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⢀⠇⠀⠉⠉⠉⠛⠒⢲⠒⠒⠒⠒⠒⠒⠒⠋⠉⢹⠉⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢠⣏⡀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⢷⡀
⠀⠀⣠⢶⡶⣿⡛⠒⠺⢷⡶⠥⣤⣀⣀⣀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀⣤⣤⠴⠒⠋⠁⠘⣧
⠀⣼⠃⠈⣇⠀⠉⠳⡀⠀⠙⢶⠒⠋⠓⣌⠉⠉⠉⠉⠉⠉⠉⠉⠹⡏⠉⠉⠉⠉⠉⠀⠀⠀⢸⡀⠀⠀⠀⠀⢻
⢰⡇⢠⡾⠫⣿⣦⠀⠹⣄⣀⣀⢧⡀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⢸⡗⣾⢷⡶⣾⢿⣷⠋⠹⡄⠀⠀⢳⠒⠉⠉⣇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸
⢸⡇⣿⢹⣿⣿⣿⢿⡄⠀⢹⡀⠀⠘⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣼
⠈⠛⠿⠿⠿⠛⠛⠻⣗⠒⠺⡷⠒⠉⣇⠀⢀⣸⣤⣤⣀⣀⣀⣀⣀⣿⣠⣤⣤⣤⣤⣤⣤⡴⠶⠿⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣀⣷⣀⣤⠿⠟⠋⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠄⠀⠂⠠⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⢛⠭⠋⢹⠟⠙⢻⠚⠛⢏⠉⠉⠛⠿⢶⡦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⣹⠷⠶⠾⣥⣀⣴⣁⣀⠀⣘⣀⡀⣀⣑⣀⡀⣠⡤⠬⠲⢝⠶⣄
⠀⠀⠀⠀⠀⠀⡰⠋⠀⣠⠊⠀⠀⢀⡾⠁⠀⠀⠀⡿⠀⠀⠀⠀⢹⠀⠀⠀⠈⢳⡀⠀⠀⠁⠪⣑⣄
⠀⠀⠀⠀⢀⡼⠷⣤⣔⠁⠀⠀⢠⡟⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠹⣄⣀⣠⣄⡬⠟⢟⢆
⠀⠀⠀⣠⠋⠀⠀⠀⢀⡟⠉⠁⠉⠛⠛⡿⠒⠒⠚⠛⠛⠻⡗⠒⠒⠛⠓⠚⠻⡟⠛⠉⠉⠉⠻⡄⠀⠀⠪⢳⡀
⠀⠀⡼⠻⢄⡀⠀⢀⡟⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠹⣄⣀⣠⠟⢷⡀
⠀⣼⠁⠀⣰⠋⠁⠚⠛⠲⡦⠤⠤⢀⣿⣤⣄⣀⣀⡀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣘⣷⡾⠿⣯⣉⣐⣛⣩⣈⣄⡈⢷
⠀⡿⣄⢠⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠁⠀⣤⣞⠁⠀⠀⢀⡴⠟⢯⠉⠁⢹⠛⢻⣆
⢸⢸⠁⠉⠛⢲⠦⢤⣀⣿⣀⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⠋⠉⠒⣴⣏⠀⠀⢈⣄⣤⣏⢀⡾⠋
⣿⣿⡀⠀⠀⣤⠀⠀⠀⠀⠉⠉⠙⡟⠓⠒⠂⠚⠒⠒⠒⠲⡖⠒⠒⠒⠒⢺⡿⠀⠀⢸⠇⠀⠑⣴⣿⣿⣿⣿⣿⠀⢀
⣷⠀⢹⠳⠤⣿⣄⣀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣧⡀⠀⣿⡀⠀⢠⣿⣿⣿⣿⣿⣿⠖⠉
⠻⢄⡸⠀⠀⠀⠉⠉⠛⠙⠒⠒⠸⠷⠶⠶⠤⣤⣤⣤⣤⣤⣿⣤⡤⠤⠤⡬⣿⠉⠓⣿⠟⠒⢺⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠒⠦⣄⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⣿⠀⠀⣿⠀⠀⢸⠉⠛⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠷⠶⢶⣤⣤⣄⣀⣄⣀⣀⣀⣀⣀⣀⣤⣾⣿⣿⠿⣆⠀⢸⠀⠀⣸⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠀⡄⠀⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠓⠂⢠⠄⠀⠐⠓⠀⠘⠶⠂⠉⠉⠉⠁⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⡄⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠅⣴⡀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠁⠷⠄⠄⢠⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠁⢲⠀⠀⠀⠈⠁⠀⠐⡖⠀⠈⠉⠉⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠃⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⡐⠁⠀⢄⡐⠀⠁⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠨⢤⠓⡄
⠀⠀⠀⠀⠀⢠⠁⢀⠀⠌⠁⠀⠈⠉⠙⠒⠀⢤⡄⠤⠄⠄⠠⠶⠤⠤⢠⡬⠄⠀⠀⠛⠒⠒⠀⠐⡂⠀⠁⠀⠀⠀⡀⠆⡸⡄
⠀⠀⠀⠀⠀⠃⠀⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⢉⠜⠋⠉⡁⠀⠀⠀⠒⣣⣱
⠀⠀⠀⠀⠠⠄⡐⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡨⠅⠀⠠⣠⠁⠀⠀⠀⠀⢀⠔⠀⡀⠀⠀⢈⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠢⠤⠀⣀⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠃⠀⠀⠀⠀⠉⣙⠀⠒⢳⠆⠀⣤⣚⣯⣗⣦⣠⣈⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⡟⠀⠀⠀⠈⠉⠀⠘⠄⠄⠀⠀⠀⢀⡏⠀⡀⡈⣨⣅⣍⣥⣽⣿⣿⣯⡧⣷⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠈⢁⠀⠂⡦⠤⠻⢧⣠⣅⣄⣠⢬⣭⣭⣭⣭⣍⣭⣠⣿⣿⣿⠿⠿⡿⠿⠷⠶⠤
⠀⠀⠀⠀⠀⠀⠙⠂⢄⡀⠠⠠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠈⡀⡀⡈⠀⠀⡇⠀⠀⡄⠀⠈⠁⠀⠀⡏⠢⠑⠜⠆⣝⠀⡯⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠲⠦⠤⢤⣤⡀⠀⢀⣘⣀⣰⣀⣈⣀⣀⣀⣁⡠⢇⣀⣄⡣⠀⠈⠀⠀⠀⠀⡐⠀⡏⠉⠛⠒⠢⢧⠠⠏⠁⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠂⠂⠉⠋⠛⠒⠐⠒⠬⣤⠤⠤⠧⠤⠀⠈⠀⣀⠀⠠⠄⠄⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⢛⡿⠿⠿⠿⠟⢿⡋⠛⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⠋⠁⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠉⢛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⣹⠗⠶⢤⣴⣇⣀⣀⣀⣀⠀⠀⣀⣀⣹⣤⣤⠴⠖⠛⠉⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠈⠉⣹⠉⠉⠉⠉⠉⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢳⣤⣀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢀⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢠⠏⠀⠈⠛⠺⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣀⡤⠶⠚⠉⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⠏⠀⠀⠀⠀⠀⠀⠀⢨⠏⠙⠓⠒⠶⠶⠤⠿⠤⠤⢤⡤⠤⠤⠴⠖⠒⠛⠋⠉⠉⢷⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠳⣦⣄⣠⣤⣀⣀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⡤⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠷⠒⣋⣩⣥⡶⠖⠛⠋⠛⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠴⠖⠋⢹⡄⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡴⠋⠉⠉⢹⠟⢯⡀⠀⠙⣆⠀⠀⠀⠀⠈⢳⡶⠦⠤⣤⣤⣤⣤⣤⣄⣠⣿⣤⡤⠤⠶⠶⠲⡞⠋⠉⠁⠀⠀⠀⠀⠘⡇⠀⢸
⠀⠀⠀⠀⠀⣰⠛⢦⠀⠀⠀⢸⠀⠀⠙⣆⠀⠈⢧⠀⠀⢀⣀⣠⣽⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣸⡆
⠀⠀⠀⠀⢠⡇⠀⠈⣳⡤⠤⣿⡀⠀⠀⢘⣦⠤⠾⠓⠛⠉⠁⠹⡄⢻⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⣠⠿⠋⢹⡇
⠀⠀⠀⠀⢸⣧⣀⢠⡏⠀⠀⠈⢳⣄⡴⠋⠹⡆⠀⠀⠀⠀⠀⠀⣧⠈⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣀⣠⡴⠞⠋⠁⠀⠀⢸⠇
⠀⠀⠀⠀⢸⡇⠈⢻⠛⠛⠋⠉⠉⢻⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢻⠀⣿⠶⠤⢤⣿⣤⣤⣤⣤⣤⣤⡤⠤⠶⠖⠚⠛⠉⠁⠈⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠙⠲⣼⣆⣀⣠⡤⠴⢾⡆⠀⠀⢸⡀⠀⠀⠀⢀⣀⣼⠶⢿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⡟⠛⠒⢾⡷⠒⠛⢹⡏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⠀⢸⡇⠀⠀⢸⣇⣠⡤⠖⠋⠙⠓⠶⠶⠶⠤⠤⠤⠼⠤⠤⠴⠶⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⣧⠴⠖⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡁⠀⠀⣠⡾⠋⠁⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⠎⠁⣠⡾⠛⠛⠶⠶⢾⣧⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⣀⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠰⠃⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠙⠛⠛⢻⡟⠛⠛⠛⠛⠛⠉⢻⡏⠉⠉⠉⢷⡄⠀⢀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢻⡄⠀⢳⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡄⠀⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠛⠛⠳⠶⣶⢦⣤⣤⣿⣄⣀⣀⣀⡀⣀⣠⣤⣤⣿⣄⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠈⠉⠉⢉⣿⠟⠋⠉⠀⠀⠀⠈⠉⠛⢶⣤⣤⣤⣴⠿⠶⠶⣿⠛⠉⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢹⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⣠⡶⠟⠛⠛⠻⢶⣄⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⣷⠶⣦⣤⣿⣀⣀⣀⣀⡀⢀⡿⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠈⣷⠀⠀⠀⢸⡇⢀⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠛⢻⡇⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠟⠛⠛⠛⠛⠋⣿⠅⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣿⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⠶⠶⠶⣶⣶⣶⣶⣿⣄⣀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⣿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⣀⠀⡇⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠜⠀⡇⢾⠀⣹⢸⡁⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠇⠽⠂⠁⠈⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠂⢛⡛⠯⠭⠭⡙⠚⠒⠒⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⢉⣒⣄⣀⣀⡀⠁⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⡡⠚⠓⠠⠄⣀⠀⢀⠠⠐⠈⠁⠀⠀⠀⠈⠢⢄⠈⠉⠐⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠠⠊⠀⠀⠀⠀⠀⢀⠝⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣀⡴⠁⠀⢀⣀⠀⠀⠔⠁⠀⠀⠀⠑⢄⢀⡠⠔⠂⠉⠀⠀⠀⠈⢇⠉⠐⠢⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠏⠉⢀⡌⢀⡀⠀⠀⠀⢈⠛⠠⡀⠀⠀⠀⡠⠖⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢡
⠀⠀⠀⢀⡀⠤⠴⢂⣉⠉⠀⠀⠀⠈⠉⠒⢆⠀⠀⠈⠢⣠⠊⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⣀⣀⣈⣆⠀⠀⠀⡆
⠀⢀⠔⣁⠤⠀⠤⠄⣀⠉⠢⡀⠀⠀⠀⠀⠀⠣⠀⠀⠔⠑⡄⠀⠀⠀⠀⢈⡤⠔⠂⠉⠁⠀⠀⠀⠘⡉⠁⠂⢤
⢀⠂⡜⠁⠀⠀⠀⠀⠀⠑⠄⠈⢄⠀⠀⠀⠀⠀⢡⠎⠀⠀⠘⡄⠀⡠⠒⠁⠘⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈
⡈⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⡄⠀⠀⠀⠀⠀⠆⠀⠀⢀⠴⠈⠀⠀⠀⢀⣡⠤⠐⠂⠀⠀⠀⠀⠺⠦⢄
⡇⡇⠀⠀⠀⠀⢀⣀⣀⠠⠤⢤⠀⠁⠀⠀⠀⠀⠀⢸⠀⡠⠋⠀⠇⣀⠄⠊⠁⠘⡀⠀⠀⠀⠀⠀⠀⠀⡃⢀⡨
⠓⠓⠒⠈⠉⠁⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⣀⡀⠤⠮⠤⠤⠤⠾⠀⠠⠤⠤⠤⠧⠄⠀⠐⠒⠒⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⢛⡿⠿⠿⠿⠿⢿⡛⠓⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠋⠉⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠈⠙⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠈⣩⠗⠶⠤⣴⣇⣀⣀⣀⣀⡀⣀⣀⣀⣷⣤⣤⠶⠒⠋⠉⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠉⣿⠉⠉⠉⠉⠁⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢛⡶⣄⡀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢀⣨⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡞⠀⠀⠉⠛⠷⢤⣄⣀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣤⠴⠚⠉⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⡼⠁⡞⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠙⠛⠒⠒⠒⠷⠶⠶⢶⡶⠶⠶⠒⠚⠛⠉⠉⠀⠘⡆⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢠⣧⣼⠁⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⢉⣳⣦⠤⠤⢤⣄⣀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⣠⡴⠚⢳
⠀⠀⠀⢀⣠⣴⣞⣋⣩⡤⠶⣖⠛⠉⠁⠈⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠗⠛⠉⠈⣇⠀⢸
⠀⢀⣴⠋⠁⠀⠀⡏⠙⢦⡀⠈⠳⡄⠀⠀⠀⠀⠙⣟⠓⠒⠲⠶⠶⡶⠦⠤⠴⠷⠶⠒⠒⠛⠋⢹⡉⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸
⢀⡞⠙⢦⠀⠀⢀⡇⠀⠀⠹⡄⠀⠹⣄⣀⣠⡤⢴⠞⣇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⣀⡴
⢸⠀⠀⢨⠗⠚⠚⢧⡀⠀⣠⢿⡖⠋⠉⠁⠀⠀⠈⡇⠸⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣀⡴⠞⠉
⣼⠳⣤⣾⣀⣀⣤⣤⢷⡞⠁⠀⢷⠀⠀⠀⠀⠀⠀⢻⠀⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣠⡤⠶⣾⠋⠁⠀⠀⠀⢀
⠻⣄⠀⣯⠀⠀⠀⠀⢈⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⠒⠛⠒⠒⠒⠒⡖⠒⠚⠋⠉⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸
⠀⠈⠓⠿⠤⠖⠒⠛⠉⣧⣀⣀⣀⡇⣀⣀⣤⡤⠶⠚⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⡏⠉⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⣀⣼⢤⣀⣀⣀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣠⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⡇⠀⣀⣤⠷⠚⠋⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢉⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢂
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠴⠶⠒⠒⠒⡲⠖⠢⣄
⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢣
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⣠⣤⡀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⣿⣿⣿⠀⢰
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⡄⢸
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠂⠀⠀⠀⠀⠀⢀⡃⢀⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠉⠐⠂⠤⠤⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣅⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠒⠒⠲⠖⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠒⠛⠓⠚⠛⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠉⠉⠓⢲⠖⠦⠤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠒⠒⠚⠙⡏⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣁⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢹⠓⠒⠦⠤⣤⣀⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠷⠒⠊⠉⠀⢸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢧
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢸⠁⠀⠀⠀⡠⢴⠖⠈⠉⠉⠉⠉⠚⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⣇
⠀⠀⠀⠀⠀⠀⢀⠇⠉⠒⠤⣇⡀⢀⣴⠟⠊⢻⠉⠉⠉⠉⠒⢤⡀⣀⣀⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⣀⡠⠜⠋⠹⡄
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢈⡟⠁⠀⠀⠘⡆⠀⠀⠀⠀⡴⠛⣄⠀⠀⠘⡄⠀⠀⠀⠀⣀⣀⣀⣀⣣⠤⠤⠤⠒⠒⢻⡋⠁⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡸⢤⣀⠀⡰⠚⠉⠉⣻⣦⡞⠀⠀⠸⡀⠀⠀⢹⠉⠉⢿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡟⠦⣀⠀⠀⠀⡇⠀⠈⣹⠋⠉⠉⠉⢹⣿⣷⠀⠀⠀⡇⠀⢀⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣴⢻
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠙⠲⢤⡇⠀⠀⡇⠀⠀⠀⠀⣸⣿⣿⣇⣀⣀⣧⠖⠁⢸⡄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠦⠒⠋⠀⢸⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⡏⠑⠢⡧⠤⠒⠚⠉⢹⣿⣿⡇⠀⠀⡇⠀⠀⢸⡃⣀⣀⣇⣀⣀⣀⣀⠤⠤⡴⠒⠚⠉⠁⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡇⠀⠀⡇⠀⠀⠀⢀⣸⣿⣿⡇⠀⠀⡇⠀⠀⣸⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠢⢄⣧⠤⠒⠋⠁⣸⣿⣿⡇⠀⠀⣇⡤⠊⢹⡂⠀⠀⠀⠀⠀⣀⣀⡠⠤⠼⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣀⣇⡠⠔⠒⠉⠀⠀⠉⡏⠉⠉⡏⠀⣠⠞⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⡷⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⢶⣶⠶⠲⢶⣶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣉⣀⣀⣴⡋⠀⠀⠀⠀⣈⣳⣄⣈⣩⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⢨⡟⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠙⣧⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⢀⣴⣯⣄⣀⡀⠀⢠⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣧⠀⢀⣀⣠⡿⣦
⠀⠀⠀⠀⢀⡾⠁⠀⢈⡿⠉⠛⠛⠛⠓⣶⠶⠶⠶⠾⠷⠶⠶⠖⢶⡞⠛⠛⠛⠉⠹⣇⠀⠈⢷⡀
⠀⠀⠀⢀⣾⠁⠀⠀⣾⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢻⡄⠀⠈⣷⡀
⠀⠀⠀⣼⠛⢻⡶⠶⢧⣤⣤⣄⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣠⡴⠶⠚⠛⠛⠻⣿⣧⣄⣀
⠀⠀⠀⡿⠀⣾⠁⠀⠀⠀⠀⣹⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⣩⠟⠙⠛⠲⠶⣤⡶⠛⠉⠉⠉⢹⡿⢦⡀
⠀⠀⢠⡇⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⡾⠋⢷⡀⠀⠀⢀⡾⠀⠈⢻⡄
⠀⠀⠈⡟⠳⠿⢦⣤⣤⣀⣀⣿⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡾⠁⠀⠀⠀⢠⡟⠀⠀⠘⣧⣤⠶⢾⣇⠀⠀⣠⣿
⠀⠀⠀⣇⠀⠀⠀⢿⠀⠀⠉⠉⠉⠉⢻⠋⠛⠛⠛⠛⠛⠛⠛⣻⠛⠶⠶⢤⣤⣿⣀⠀⢀⡾⠛⠛⠲⠶⠿⣷⠟⠁⢸⡄
⠀⠀⠀⣿⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠋⠉⢛⣿⡁⠀⠀⠀⠀⠀⢹⡄⠀⢸⡇
⠀⠀⠀⣿⠙⣿⠶⠾⣧⣤⣀⣀⣀⡀⣸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⢸⡏⠉⢹⡟⠛⠶⠶⣾⡷⠞⢻
⠀⠀⠀⢹⡄⢹⠀⠀⠀⠀⠈⢹⠉⠉⠉⠛⠛⠛⠛⢻⡟⠛⠛⣿⠓⠶⠦⣤⣿⣤⣤⣼⡇⠀⢸⡇⠀⠀⠀⣾⠀⠀⣾
⠀⠀⠀⠘⢧⣸⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢸⣧⠤⢼⣧⣤⣀⡀⣿⣀⣤⠟
⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⢦⣼⣧⣤⣄⣀⣀⣀⣀⣸⣇⣀⣀⣿⡀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠈⠉⠛⠲⠶⠿⠴⠶⠛⠁

⠀⠀⣀⡸⠓⣧⣎⠀⡀⢀⠁⡠⠈⠀⣨⠻⢈⣠⣀⡐⠫⠓⡦⢲⡖⣔⡀⠀⠀⡧⡤⢀⠪⢇⡠⠽⣿⢸⡸⠈
⠀⠀⢣⠠⢀⡇⠛⠠⢯⣧⣟⣉⣶⣴⡐⠚⡩⢤⠈⣙⡏⠙⢏⠁⠠⣬⠛⣫⢬⠅⠆⢹⣡⠀⢠⢂⡈⠧⢀⠉
⠀⠀⠠⠂⢸⠓⣌⠘⠳⡞⣽⡟⢒⠯⡇⠀⢠⡀⠈⠋⢹⠼⠘⡠⠔⠃⠀⠘⠰⣎⠀⠘⠂⠂⢉⣁⣀⡠⠴⠦⢴
⠀⠀⣠⡀⠆⠈⢀⠄⠀⠁⠉⠙⠚⠦⡼⠀⠲⡷⡤⢤⠬⢶⠛⠀⠀⠀⠀⠀⠀⠈⠡⣿⣿⡿⣿⡛⣥⣀⡂⠐⠨
⠀⡆⢩⠘⢠⢚⠀⢂⣐⡂⣐⠤⣄⠀⡘⡄⣟⣍⢍⣿⣿⡏⠀⡀⠴⠀⠁⠈⠀⠐⠀⢻⣈⣹⡇⣷⢏⠁⠸⡈
⠀⠓⠐⠋⣶⠀⣤⣭⣶⠀⠀⠀⣤⡑⡆⢹⢤⣿⠀⣿⣿⠱⠀⠿⠿⠇⠀⠀⠿⠉⠆⢸⡿⡿⣿⣝⠀⡁⠈⠀⠢
⠀⠀⠰⣦⢸⠟⠙⠘⠋⠃⠀⠺⠯⠿⠇⢫⠒⠛⠃⡿⠿⠀⣇⠠⠄⠀⠀⠀⠀⠠⢀⣬⣼⣥⣬⣧⣼⠁⢸⢅
⠀⢐⢾⣷⠀⠀⠓⢄⠀⠀⠀⠀⡠⠒⠀⠀⢀⠐⠀⢠⠄⠉⢘⠀⠀⠐⠀⠀⡞⠀⢽⢯⣤⣤⣀⡀⢉⣠⣨⡀
⠀⠘⢰⢸⡉⠀⠀⢀⠍⣲⢦⡀⠀⠀⠀⢸⠊⢀⣂⡉⢸⠀⣇⠤⢀⣠⣀⡀⠘⠄⡍⠐⠈⡗⠮⢤⢠⠀⢠⠀⢀
⠀⠀⠰⢸⠀⢠⡪⠑⠦⣤⣜⠟⠳⠀⢠⠆⢰⠀⢽⠏⢠⠊⠁⣦⠉⠀⠈⣰⠆⠀⠘⡆⣇⣁⣿⠀⢸⠀⠰⠤⠉
⠀⠀⢥⠿⡄⢠⠀⠸⡶⢣⢝⣲⢾⢰⡇⠘⠘⠂⠙⠀⢨⢠⠀⠁⠀⡀⠀⠘⠀⢠⠄⢿⡍⣉⣉⣉⣙⡂⡤
⢀⡀⢠⡄⠀⢺⠀⠱⢨⢋⠦⢱⡾⡟⠀⡀⠀⠀⠀⠀⠙⢿⢘⠀⠀⠀⠀⢀⠀⠈⠄⠗⠒⢰⣐⠀⠈⠃⠰
⠙⠋⠶⡄⢻⠸⠀⡖⠢⠍⠖⠸⠀⠓⡄⢠⢀⠰⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⣀⣀⣈⣂⠂
⠀⠀⢬⣀⡌⠁⡰⠀⠀⠈⡁⠀⠉⣁⢱⢸⡂⠀⠀⠀⠀⡸⠀⣀⠀⠀⠀⠈⠀⠀⠀⢀⠁⠲⠤⠽⣽⠀⠈
⠀⠀⠛⢉⡖⠐⠀⠀⠠⠀⠀⠸⠀⠘⡂⠃⠃⠀⠀⢀⣖⡃⠐⠁⠁⢀⡀⡀⠀⠤⠄⠈⣉⠛⢛⢛⢃⠀⠈⠉
⠀⠆⠽⢸⠀⡤⠀⠀⠀⠂⠀⠀⠀⠀⠇⠸⠷⠶⠦⠉⢯⡄⢀⠒⠄⠉⠀⠓⠀⢀⡄⠀⠂⠀⠀⠻⠾⠓⢾⢾⡀
⠀⠆⡽⢸⢧⡀⠀⠀⠀⠀⠀⣄⠀⠢⠄⢖⡁⠀⠀⠀⠀⡃⠉⠀⠀⠃⠀⠰⠠⡄⠀⣴⠠⠀⡀⠀⠘⠂⢨⠀⠈
⠀⡄⢠⡼⡾⠡⠄⣀⠀⠀⡔⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠈⡀⠙⠙⠁⠀⠆⠀⠀⢀
⠀⠉⢠⣐⠢⠲⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠿⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠄⠒⠀⠃⢀
⠀⠀⠘⢉⠈⠐⠲⣦⠀⠀⣠⠐⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠾⠀⢐⠀⢸
⠀⠀⠀⠈⠀⠀⠀⠉⡀⠈⢹⠘⡆⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠉⢆⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣿⣿⣾⠟⠉⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠃⢿⡿⠛⠁⢠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠡⠙⠿⣦⣶⣤⡀⣠⡀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣤⠀⠀⢀⣹⣿⣿⣿⡿⣿⡿⠆⠀⠤⣄⠀⠤⡀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠼⠟⠋⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⢤⣀⣤⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⡃⢠⣤⣤⣤⡄⠀⠀⠶⣲⣷⠒⠒⠾⣿⡏⠀⢠⣿⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⢿⣿⡿⠑⡄⠀⠀⠀⠿⠿⠀⠀⠀⠉⠃⢠⣿⡟⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠋⠏⠀⠀⡴⠁
⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠾⣷⣦⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣴⣮⡄⠈⠁⠀⡸⠁
⠉⠁⠀⣀⠿⡇⠀⠐⣧⡀⠀⠀⠀⠀⠙⠿⠧⢀⠸⣿⣿⣦⣀⠀⠀⠶⠖⠀⠀⡠⣺⣿⣿⡇
⠀⠀⠀⠡⠤⠀⠀⠀⢈⣽⡗⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡆⠙⣶⠦⠄⠀⠀⢰⡈⠳⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘⠋⠀⠀⣰⣆⣀⠀⠈⡡⠊⠉⠉⠉⠛⠛⠅⠀⠀⠀⠈⣿⡀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣶⣤⣄⡄⠀⢀⡀⣤⣴⣶⣤⣶⣄⣐⣥⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠷⢶⡶⠾⠿⠿⠛⠋⠻⢿⣿⣿⣿⣷⣾⣿⣶⣿⣿⣸⣿⣿⣿⣿⣿⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠃⠈⠙⠛⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠈⠙⠛⠲⣄⣀
⠀⠀⠀⠀⠀⠀⣸⠃⠀⣀⣼⣦⣄⡀⠈⢻⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣀⣼⣦⣄⡀⠈⢹⡀
⠀⠀⠀⠀⠀⢰⠃⠀⡜⢛⠛⠻⠛⠿⡄⠈⢷⠀⠀⠀⠀⠀⢠⡇⠀⡼⢿⠟⠿⢿⠿⡆⠀⢷
⠀⠀⠀⠀⠀⢨⢧⠀⢣⠀⠀⠀⠀⢠⠃⢠⣯⠀⠀⠀⠀⠀⢨⢷⠀⢣⠀⠀⠀⠀⢠⠃⢀⣮
⠀⠀⠀⠀⠀⣸⣄⠳⠀⠑⠲⡴⠒⠁⠰⢋⣼⡀⠀⠀⠀⠀⣸⣄⠳⢼⢳⠲⡴⢲⢻⠰⢋⣼⡀
⠀⠀⠀⠀⣸⠁⢈⠓⠦⢤⣤⣧⣤⠤⠖⡋⠀⢹⠀⠀⠀⢸⠁⠈⡓⢾⠼⣤⣧⡼⢼⠖⢋⠀⢹⡀
⠀⠀⠀⠀⠸⣄⠀⠹⠀⠀⢸⣝⠀⠀⠸⠁⢀⡼⠀⠀⠀⠸⣄⠀⠹⠹⠻⢘⣽⠸⠻⠹⠁⢀⡼
⠀⠀⠀⠀⠀⠈⣟⠳⠤⠤⠼⡾⠤⠤⠴⢛⡏⠀⠀⠀⠀⠀⠈⣟⠳⠤⠤⠼⡾⠤⠤⠴⢛⡏
⠀⠀⠀⠀⠀⠀⠈⢹⣶⣞⣻⢹⣓⣷⣾⠉⠀⠀⠀⠀⠀⠀⠀⠈⢹⣶⣖⣻⢹⣓⢷⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠽⠸⠥⠤⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠤⠼⠸⠤⠤⠼⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣍⢺⣷⣶⣖⢲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣼⠆⢼⡃⠀⠇⢙⣷⠈⢱⣤⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⣿⠻⣾⣷⣆⣷⠟⠈⠓⢶⡒⢧⠀⠀⠀⠀⠐⣦⡜⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢿⡶⢻⣻⠉⠉⠀⠰⢣⣾⢏⡏⢡⠀⠀⠀⠀⢾⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠺⡿⣶⣯⠓⣒⣄⠀⠉⠁⢸⡏⡈⠀⠀⠀⠀⢀⣹⣿⣷⣜⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⠛⠻⢮⢿⣿⣟⠁⠁⢀⣠⠀⣧⠏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣝⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠻⣿⢿⠷⢦⣬⣤⠂⠈⠄⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣝⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢰⡆⢚⣏⠐⣾⣟⠛⣷⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡝
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣷⠀⣿⡧⠤⠿⢧⣌⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠯⣀⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⢷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠛⠛⠿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠃⠀⠀⠀⠀⠈⠙⠃⢁⣡⠙⠻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠔⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠙⠋⠉⢽⣿⠋⠀⠀⠀⢀⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣤⣤⡀⠀⠀⠀⠈⢻⣿⠟⠋⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠙⠿⣦⣀⣴⣾⣿⡟⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣇⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢢⡙⠖⠠⠀⠀⠀⠃⠀⠀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠂⡁⣤⠆⠀⠀⠠⠁⠀⠀⠀⠀⠠⣦⡀⢰⣿⣷⣀⠀⠀⠀⠀⠀⣀⣀⡀⠶⠤⠴⠛⠛⠛⠋⠉⠉⣉⡀⠀⢠⡄
⠀⠀⠀⠀⠀⠃⠠⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⠿⠤⠋⠙⠛⠂⠉⢉⣉⡀⠀⠤⠄⠀⠐⠒⠀⠀⠉⠉⠀⠀⠀⠀⠀⠜⠁
⠀⠀⢀⣀⣀⣡⠤⠥⠤⠀⠒⠒⠂⠈⢉⣉⡀⠀⠠⠄⠀⠐⠒⠂⠀⠉⠉
⠀⠀⠀⠀⡀⣄⠀⠠⠐⠂⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠁⠀⠈⠀⠀⠃⢤⡰
⣠⡤⠤⡀⢒⠀⢠⢤⣀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠟⠀⢀⣠⡽⠛⠋⠉⡾⠏⠄⠀⠀⠀⠀⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠀⠀⠒⠈⠉⣉⠁
⡿⢛⠋⠁⠀⠀⣰⠏⣈⠒⡀⢀⠀⠀⠀⠀⠊⠀⠀⠀⡀⢀⣀⠀⠀⠤⠄⠐⠒⠂⠀⢉⢉⠀⠀⠄⠀⠀⠒⠀⠀⠉
⠋⠀⠀⠀⠀⢸⡏⠀⢉⣀⣂⢤⢤⠥⠖⠒⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⣖⣦⣤
⣄⣶⣶⣶⣶⣿⣍⣹⣿⡟⣿⣇⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⣿⣿⣾⣿⣿⣼⣆
⣿⣿⣿⣿⣶⣿⣿⣿⠿⢿⣿⢃⣿⠛⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠉⠛⠻⢿⣿⣿
⠉⠀⠈⢻⣿⣯⠉⠁⠀⠀⠾⠛⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⠳⠆⠘⠀⠀⣀⣦⣼⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣾⣿⣿⣿⣿⣀⣠⣦⣈⠀⢀⠀⠀⠀⠀⠀⠈⣻⣿⣶⣿⣷⣦⣤⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⣀⣀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠟⠋⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣽⣿⣿⠃
⢘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠛⢿⠃
⠈⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⢹⣿⣦⡀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⠋
⠀⠀⢀⠀⠀⠉⠈⣿⣿⡿⠟⣋⠍⠋⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀
⠀⢦⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⡈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠉
⣀⣴⠀⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠁⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⢠⠀⢠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⣿⣿⣦⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢨⠉⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⢸⣿⣿⠇⠀⠙⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠷⠿⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣤⣴⣦⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣾⠁⠈⠉⢻⡦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢣⠠⠄⢄⠜⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉⠁⠠⠙⠂⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⣠⣤⣄⡀⠸⠉⠉⡀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣯⡧⣾⡕⠦⢀⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣴⣶⣀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⣿⠟⠀⣀⣀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⡏⣁⣤⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡌⠏⠀⠈⠀⠉⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠙⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉⠻⣿⣷⠄⠀⢠⡔⠀⠈⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠄⡁⠀⠀⢢⠀⠀⠀⠀⢔⡀⠀⠀⠀⠂⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠁⢀⣡⠀⠀⠈⢷⡀⠀⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠠⠼⠃⠀⠀⠀⠰⡌⢿⣿⠤⠄⠀⢹⠀⢀⢀⢠⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣦⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣽⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣱⢸⡆⠀⠀⠀⠀⢧⣤⣸⠸⡇⠀⣠⠀⠀⠀⠀⠐⢦⠄⣚⡉⠧⢠⣤⠤⠾⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠻⣆⣦⡀⠀⠀⠀⠀⣀⡐⣦⣦⣀⣭⣾⣧⠀⠀⢀⣶⠛⠻⠀⢛⣧⠤⠌⠀⠀⠀⠀⠀⠑⠲⠀⠠⠚⠹⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠿⢶⡀⠀⠰⠶⠳⢻⣟⠻⡟⠛⠍⠙⢢⡉⠉⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡄⠀⠀⠀⠀⠲⢄⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⡐⠂⠀⠒⠲⠶⠶⠻⠿⠀⠀⠘⠣⠴⠾⡧⠈⠁⠀⠀⠀⠖⣶⠲⡆⠀⠀⠀⠀⠀⠀⡤⣮⠜⡷⢀⠘⣹⠄⠚⠀⣡⠶
⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡀⠩⠀⠓⢀⣀⡀⣀⡀⣈⠉⠁⡾⠆⠄
⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣃⠀⠀⠀⠈⠝⠉⠉⠁⠀⠈⢱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢗⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡤⠤⠚⢀⣴⣦⠄⠾⠛⣅⣀⣀⠻⣧⢤⡰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠀⠀⠀⠀⠀⠉⣩⠇⠟⢠⡟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠐⠂⠀⠀⠀⠀⠀⠒⠂⠃⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢒⡾⠟⢋⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⢁⢤⣾⡯⠖⢛⣩⠝⠛⠉⠉⠉⠉⠉⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⠟⠁⠀⠀⠁⠾⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣀⡾⢋⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⠶⠾⠝⠻⠷⠶⠦⠽⣶⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠋⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⢀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⡆⢿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⣨⠄⠒⠒⠒⠒⠢⢬⣁⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠡⠈⢧⡸⡀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠈⠉⠉⠉⠲⣄⣂⠤⢀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⢀⡼⣣⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠁⠈⠳⣷⡀⠀⠀⠀⠀⢸⡇⠀⣠⠖⠒⠲⢤⡀⢸⡇⠙⢦⡀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠶⠟⠛⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣤⡀⡞⠳⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢱⢠⠇⠀⠀⠙⢦⡀⠀⠀⠀⠀⣀⠀⠀⠀⡀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡎⣹⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡀⠀⠀⠀⠈⣝⠳⣄⠀⣀⣈⠁⠈⠉⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣧⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠈⢀⠀⠀⡀⠀⠀⠸⡆⢠⠞⢉⡉⢳⣀⣠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣷⠒⠒⠶⢦⣤⠀⠀⠰⠞⠋⠀⢄⣴⣾⣿⠛⣟⣿⠋⠀⠀⠁⠈⢀⠊⡀⢸⣏⢢⡘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣶⣯⣴⣀⠀⠀⠀⠀⠀⠀⣸⣏⢹⣿⣿⠇⡾⠃⠀⠀⠀⠀⣠⠀⡠⠁⡼⢻⡄⠱⡀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣏⠈⡉⣿⣷⣄⠀⠀⠀⠀⠛⢈⣲⣬⡵⠎⠀⠀⠀⠀⠀⠀⢿⠀⢀⡼⠁⠀⠹⣆⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣵⣬⠿⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⠙⠀⠀⠀⠘⢦⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⢣⡀⠀⠀⠀⠀⠳⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⡜⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣧⡀⠱⣄⠀⠀⠀⠀⢡⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠀⠀⠀⡀⠠⡶⠃⠀⠀⠀⠀⢀⣤⠞⠁⠟⢻⡏⠙⠷⣤⣅⣀⠀⠀⠀⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣍⠑⠆⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠈⣿⠀⠀⠈⠻⡙⢿⠶⣤⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠀⢳⡀⠹⡇⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⢇⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣤⠤⠀⠀⠀⠀⢰⡇⠘⣷⡀⠀⠀⠀⠀⢸⠀⢹⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⣾⠁⢰⢸⠛⢦⣄⠀⠀⢸⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⠈⠀⠀⠙⣧⠀⠀⠀⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣧⠸⠀⠀⠀⠀⢸⣆⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⡠⢒⡽⢷⡀⠀⠀⠀⢸⣏⡴⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢠⣿⢙⣁⣏⣹⠂⣐⠠⣀⣠⣴⡟⠀⠀⠐⠉⠀⠈⢷⠀⠀⠀⡼⠋⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⣿⠁⣀⠀⠀⢉⣼⠾⠛⠉⢁⣿⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⡜⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠋⠐⠐⠐⢺⡏⠀⠦⣠⡶⠋⠁⠀⠔⠃⣸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣷⣮⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⢀⣾⣤⠶⢻⡿⠁⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⡀⠀⢀⣠⠾⠛⠁⠀⠀⣾⠁⠀⠀⠀⠀⣠⡟⠁⠀⢀⣀⣤⡴⠞⢋⡼⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠁⢠⣴⠟⠁⠀⠀⠀⠀⣼⠇⠀⠀⠀⣠⠾⢋⣠⣶⣞⣫⣭⡴⠶⣿⠟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣤⡞⢁⣀⣀⣀⣠⣤⣞⣁⣀⡠⠶⠛⢛⣋⡽⠟⠉⠉⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠤⠤⠤⠤⠤⠄⠤⡤⠤⢤⠤⣤⣤⣤⠤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⢀⣠⠄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣐⣲⣟⣌⣀⣠⣤⣴⣩⠂⡈⡓⣀⡐⢶⠰⡏⢻⠁⠀⠈⠉⠉⢁⡈⣈⠁⠀⣀⣤⣄⡗⠚⣒⠀⣠⢤⠤⠄
⠀⠀⠀⠀⠀⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣛⢉⣶⣾⣷⣾⣿⣿⣿⣿⣶⣿⣿⣧⣤⣶⣾⣿⠿⣻⣻⡏⣀⣙⠓⣭⣦⣦⢽⣿⣿⠂⡀
⠀⠀⠀⢘⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣾⡽⢿⣿⡟⠊⣻⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⣽⣿⣾⣊⣙⣿⣿⣄⣀⣀⣀⢀⠀⠄
⠀⠀⠀⠰⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣷⣦⣀⢠⡀⣉⣠⡄⠀⣈⠀⣤⣉⡉⠉⣛⡛⠿⣿⣿⣿⣿⣿⣿⣿⣟⣤⣬⣥⣬⣭⢁⣀⣀
⠀⠀⠀⠐⠀⣴⣿⣿⣿⡷⣿⣿⣿⣿⣿⠟⠟⠻⡝⠿⢿⠿⡿⠾⣿⣁⣖⣢⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡄
⠀⠀⠀⠁⣛⣿⣿⣷⣿⢿⣷⣮⡿⣝⣿⣶⣾⣿⣿⣷⣵⣤⣦⣾⠿⠛⠻⠟⠛⠛⠛⠛⠛⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣧⡀
⠀⠀⠀⣥⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠈⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠡⠙⠻⠛⡟⠛⣿⡟⠁
⠀⠀⡍⠁⠘⠛⠒⡒⠂⠀⠀⠠⠐⠂⠐⠛⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡈⠂⠀⢹⣿⣿⣿⣿⣿⣿⡟⢁⣾⣦⢺⣷⣷⠼⠫
⠀⠀⠻⠤⠌⠠⠼⠇⣰⣖⠠⠤⠀⠀⠀⠠⢀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⡼⠇⡹⠄⠀⠿⠦⠁
⠀⠀⢲⣶⠶⣖⠖⠚⣿⣇⠠⢤⣄⡀⠀⢹⣿⣿⣿⡀⠀⣀⠠⡒⠀⣿⡆⠀⠀⣤⣤⣄⠀⣷⣶⢀
⠀⠀⠘⠀⠐⢀⠀⠀⠈⢉⠃⠀⠀⠉⠻⣦⣿⣿⣶⣠⢄⠈⠈⠉⣶⣿⣿⣧⣼⣿⣽⣿⣶⣿⣿⠀⢸⣶⡄⣸⣿⣿⣷⢠⡶⣶⣦⣰⡖⣶⠀⣶⣦⡀
⠀⠀⢿⣿⣿⣿⣿⣧⣤⣷⣾⡶⠆⢠⣿⣿⣿⣿⣿⣿⣾⣷⡀⠀⠻⣿⣟⢻⣿⣿⣟⣽⣿⣿⡋⢀⣾⣿⣿⣿⣿⣿⣏⣿⣿⣿⡿⣿⣿⣯⣶⢿⣿⣿⢀
⠀⠀⢉⡀⠀⣠⣠⣤⣎⣿⣿⣦⣤⣿⣿⣿⣿⣿⣟⢿⣟⣏⣡⣴⣆⣿⣷⡄⠸⣿⣿⣿⣿⣯⣧⣙⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌
⠀⠀⠠⠀⠈⠛⠲⣿⣿⣿⡿⣿⠯⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣿⣏⠛⠏⣿⣿⣿⣿⣿⠛⣛⣸⣿⡛⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⠿⠿⢿⣣⠹⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⡄⢴⣿⢸⣿⣿⣿⣿⣿⣿⢻⣻⣿⣿⣿⣿⡊⠢⣿⣿⣿⣿⣿⣦⣭⣽⣯⣼⣿⣽⣿⣿⣿⣷⣿⣫⣿⣸⣿⣤⣄⣀
⠀⠀⢀⣀⣰⣦⣤⣤⣄⠀⠀⠀⠀⠘⠹⢿⣿⣿⣿⣿⡛⠀⠀⠀⢨⣁⠀⠀⠀⢻⣟⡿⠙⢿⣿⣏⠉⢯⣿⣿⡉⣹⣏⢽⡿⢿⣿⣿⣾⣿⠋⠉⠉
⠀⠀⠘⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⢠⣤⣤⣬⣭⣥⣄⣀⣸⣿⣄⣀⡀⠈⠻⠇⠈⠉⢽⠿⡿⣿⣿⡿⠿⠾⠽⣿⣿⣤⡄
⠀⠀⠉⠹⠿⠿⠟⠿⠿⠿⠷⠾⠷⡶⠀⣼⣿⡿⢻⣿⣿⡆⠀⠀⠈⠉⠍⠉⠩⢻⣷⣀⠠⣤⣤⡶⠤⠀⣴⠶⣶⣶⡿⠷⠲⠶⠖⠂⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣿⣧⠄⠻⣿⣿⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣵⣄⣤⣿⣿⣭⣿⣶⣴⣾⣾⣿⣿⣤⣤⡀⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠛⠛⢻⣿⠛⠋⢉⣿⡿⠛⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣿⠛⢿⡫⠝⠛⠉⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠿⠛⠛⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣶⣶⣾⣭⣉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣉⡿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠁⠁⠁⠈⠉⠛⠻⢿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠉⣍⠻⣿⡟⠁⠀⠀⠀⠓⠦⣄⡀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢀⠆⠃⠀⠀⠀⠀⠙⣷⣦⣌⡙⢦⠀⢀⠴⠃⢸⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠘⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⠂⣠⣶⢶⡌⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠿⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣂
⠀⠀⠀⣀⣀⣄⣾⡇⠀⣶⣾⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣄⡀
⣴⣶⣿⣿⣿⢿⣿⣷⡄⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠁⣿⣿⡿⢿⣶⣶⣤⣀⡠
⡿⣻⡿⠋⠀⡘⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⠁⣀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠐⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⢹⣦⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⢰⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣥⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠸⣿⣿⣿⣿⡿⣱⣮⣝⣛⣛⡛⣫⣭⣶⣿⡿⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣩⣭⢩⣵⣾⣿⣿⣿⣿⣿⣿⣶⣭⢉⣭⡁⠀⠨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣾⡀⠀⠘⣿⣇⠾⠇⠿⠇⠿⢆⣿⡟⠀⠀⠀⠀⢠⡀⢰⡆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂⠈⣿⡀⠀⠘⠿⣿⣿⣿⣿⣿⡿⠟⠀⠀⣀⠀⣷⠈⠻⣾⡷⠋⢀⡿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡦⠈⠛⠇⠀⠀⢠⣭⣭⣭⣥⠀⠀⠀⠀⣉⣻⣿⣇⡀⢸⡇⢀⣼⣷⣟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀⢴⣾⣿⣿⣿⣿⣷⡄⣠⢈⣉⠉⠀⠈⣹⣿⣿⣋⠁⠈⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣦⣙⠿⣿⣿⠟⣡⣾⣿⡦⢉⣻⣿⡟⠉⢸⡇⠙⢻⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣧⣫⡅⡿⠀⣠⣾⣷⣄⠈⣷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠦⠘⠁⢸⡇⠙⠃

⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣽⣽⣿⣿⣷⢿⣹⣗⡠⠀⠀⠀⢀⠀⠀⡀⠀⠀⢀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⢈⡅⢩⣿⣿⣿⠿⣿⣿⣟⢻⣿
⣿⣿⣟⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢽⣿⣿⣿⡿⣿⣿⠛⣿⣽⡿⣿⣤⣥⣌⣛⡢⣽⠷⠤⠚⠀⠀⠚⠷⣚⡃⣎⣧⠴⢿⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣩⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠈⠓⠒⠦⠤⣤⣲⡀⠀⠤⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠐⢦⣄⣀⣀⡈⠀⠉⠉⢀⣀⠀⠀⠈⠤⠉⠬⢭⡿⠿⠿⠿⣶⣿⣿
⠙⠂⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⡀⠀⠈⠉⠐⠠⡎⠛⢩⣿⣟⠓⣄⠀⠤⠤⠄⠠⠤⠤⠐⠂
⠀⠀⠀⠀⠀⠀⣀⡄⢀⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣓⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠰
⠀⢀⣤⠀⠀⠀⠘⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣤⣶⣟⣋⣀⡀⠀⠀⢀⣤⣄⣾⣿
⣿⣿⣿⣷⣶⣤⣀⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣩⣷⣿⣿⣿⣟⠛⠛⠛⠀⡠⣿⡗⣿⣻⠿
⡿⠋⠉⠉⠙⠛⠛⠛⠿⠟⢛⣿⣿⣿⣿⣿⣿⣤⣄⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⣰⣾⡟⠋⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠀⠀⠙⢻⣿⣿⣿⣿⣿⣯⣿⣿⣷⣾⣿⣅⡲⠶⠶⠶⠿⠿⠷⠶⠿⢾⣿⠟⠛⠉⠁⠁⠈⠉⠘⠀⣀⣀⣤⡀⣽⣥⡌⠀⠐⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣸⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⠈⠈⠀⢸⣿⣷⣤⣀⣠⣿
⣿⣿⡶⠶⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠅⠙⣿⣿⣿⣿⣿⣿⡿
⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⢰⣿⠿⢏⢉⣠⣤⣴⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣴⣤⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⠇⠀⠰⠀⠀⣰⣿⣿⣿⣿⣟⠀⠀⠀⠈⢹⣤⣸⣯⣿⡏⠉⠹⣿⣿⣿⣿⣿⣷
⣿⣿⣯⣿⣿⣟⠸⠇⠀⠀⠀⠀⠐⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠈⠀⠀⣼⣿⣿⣿⣏⢿⣷⣶⣴⣶⠤⣾⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠐⠂⢺⣿⣿⣿⣿⣿⡷⠀⢿⣿⣿⣿⣿⣿⣿⠃⠀⠄⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⡉⠉⠁⠀⢀⣼⡏⢉⣭⣶⣶⣦⠉⢺⣿⣿⣿⣿
⣷⣿⣿⣿⣿⣿⣿⡤⠆⠀⠀⠀⠀⣸⣿⠿⠏⠹⠏⡚⢀⣼⣿⡇⠈⠉⠛⠉⠀⢠⠀⠀⠀⠀⠀⣿⣉⣿⣿⣿⡀⠀⠀⠀⠐⠛⢿⣿⣿⠿⣿⣿⡟⢃⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢿⣿⡇⠀⠀⠀⠀⠀⠀⢨⠿⠐⣿⣿⡷⢺⣿⣿⠘⣿⣿
⣿⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠇⡟⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣿⡇⡀⠐⠲⠀⠦⢰⣿⣿⣿⣿⣿⣿⣼⣿⡇⢸⢿⣿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠖⠰⢿⡿⠂⠀⠀⣠⣴⡆⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣶⢶⡀⠐⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⠟⠀⠀⠨⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠦⣽⣿⣿⣷⡀⣀⣄
⣿⣷⣶⣶⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢈⣶⣿⣿⡿⠿⠿⠛⠉
⠋⠀⠉⢉⣽⠟⠙⠛⠛⣿⣿⡿⣿⣶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠈⠹⠄⡀⠀⠀⢾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤
⣦⣴⣸⣟⣁⣀⣠⣤⡟⠋⠀⠀⣠⣾⠟⠁⠉⠉⢹⡿⠿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠐⠐⢒⡀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠼⠏⠉⠙
⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣷⣷⣾⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⡄⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡠⠂⠀⠀⠣⠖⠚⠋⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠀⠘⣄⠀⠀⡀⢊⢻⡅⣀⣠⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠆⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠐⡄⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⢀⡐⠖⠒⠀⠀⠈⠀⠀⠀⠀⠈⠙⢧⠀⡄⠀⡠⠐⠉⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠉⠑⠖⠒⠆⠙⠁⠀⠀⠐⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠲⠂⠁⠀⠀⠀⠀⠀⠒⠐⠒⠒⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⡀⠀⠀⠀⣅⠀⠈⠀⠀⠀⠠⣶⡀⠀⠀⠀⠉⠡⠘⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡘⠂⠀⢀⡀⠀⠀⠙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⣌⣻⣶⣶⡄⠀⠀⠀⠀⣀⠚⠤⣀⠇⠀⠀⠈⠀⣀⣀⠀⠁⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣔⣋⣀⣀⡄⠋⡀⢲⡶⠛⠈⠉⠉⠀⢂⠄⡠⠞⠀⠀⠀⢈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠐⠠⠒⠀⠤⠐⠂⠀⠀⠀⡀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣛⠛⠉⠀⠀⠀⢀⡀⠀⢠⡄⠀⠀⠐⢆⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⠀⠀⠀⠀⠀⡠⠄⠂⠠⠀⢲⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⠀⠀⠈⠂⠉⠁⠀⠀⣴⠀⠀⠀⠈⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠀⠀⡾⠀⠀⢀⠂⠀⠀⠳⡀⠀⠀⠘⠷⠄⣀⡐⠠⠀⠀⠴⠀⠇⠀⠀⠀⠀⠀⠈⠲⡀⣼⠇⠀⠀⠀⠀⡇⠀⠀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠐⠄⠤⢶⠀⠀⠠⡏⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢉⡅⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⠠⠀⠀⢰⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣄⠀⠀⠈⡄⠀⠓⠀⠐⠀⠀⢀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⡠⠂⣽⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠄⠒⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠦⠀⠀⢁⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠉⡔⠀⠀⡠⠊⠐⢀⡆⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠤⠶⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⠄⢀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠘⠆⢸⡅⠀⠄⣸⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⢀⡠⠉⠀⠓
⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⠁⠀⡶⢋⠀⠀⠀⠀⠀⠀⠒⠀⠀⠤⣦⠤⣤⢤⣤⣤⣤⠀⠐⠛⠒⠈⠉⠉⠉⠉⠉⠉⠳⢾⡃⡟⠀⠉⠁⠀⠀⠀⢀⠦⠠
⠀⠀⠀⠀⠀⠀⠀⠐⡀⢈⣠⠀⡃⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⡀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣙⠷⣏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠸⠄⠈⠂⠈⠠⣤⢵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡀⠈⠂⠤⠀⠀⠀⡄⠀⠈⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠹⣦⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣦⠀⠑⠄⠀⢰⡃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠀⠀⢀⠀⠀⠀⣄⡀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠒⠀⠘⢦⡀⢀⠞⠉⠉⢰⠶⠆⠠⣤⢄⣰⡒⠀⢹⣄⠈⢢⠈⠓⠤⠤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡕⢤⡀⠀⠆⠀⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⢀⡆⢄⠁⣀⠴⠀⠄⠀⠹⡄⢀⡀⢀⡀⠛⠀⠀⠀⢦⠁⠁⠤⠉⠙⣦⠀⠱⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠬⢒⠠⠦⠄⠀⠀⠀⢠
⠀⠀⠀⠀⡐⠒⠣⠠⠔⠂⠀⠀⠀⠀⠀⠀⠈⠲⡙⠂⠑⠂⠀⢀⢀⠀⠱⣄⠀⠤⠀⠀⢳⣄⠀⠢⡀⠙⠦⠠⠐⣄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄⠐⠆⠤⠀⢀⡼
⠀⡀⠀⠀⠁⠉⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢢⡀⠀⣀⠺⠄⠀⠀⠈⢢⡀⠀⣀⣸⡜⢣⡀⠀⠄⡀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢰⠀⠀⠀⢀⠈⠙
⢈⡁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠙⢶⡁⠀⠀⠁⠔⠒⠀⠙⠦⠀⠬⠥⠽⠽⠶⠠⠘⠖⠐⠚⠓⠒⠋⠉⠁⠀⠀⠀⠀⠀⠸⠀⠐⠀⠀⠙⢤
⢸⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡁⠂⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠄⠀⠀⡀⠀⠀⢀⡀⠀⠀⠃⢠⡀⠀⠀⠀⢰
⠰⠃⠶⡤⠀⠀⠈⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⡕⢶⡀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠒⠀⠁⠐⠁⠀⢀⣂⡈⠓⠺⠛⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣳⠄⠁⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⣆⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣴⠶⠚⠛⠉⡉⢹⡇⡀⠀⢐⢀⡴⢀⠴⠛⢆⠀⠀⠀⠘
⠠⠀⠀⠀⢂⡑⣾⠃⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠀⠀⢘⣆⢠⡙⣢⣤⣴⣴⠖⠒⠉⢉⢛⣄⡀⢠⣀⡈⠀⢰⡂⣈⠙⣻⣰⡀⠁⠀⠀⠀⠈⢂⡀
⠀⠈⠀⠁⠈⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣴⣾⢃⠀⠀⠀⠀⠀⢀⢸⠀⢠
⠀⠀⠀⠀⠀⠀⣳⣄⠀⠀⠀⠀⣇⢸⡁⠀⠀⣻⣿⡟⣡⣴⣼⣧⠀⠸⠈⣤⢸
⢀⠀⠐⠃⠀⠀⢻⣿⠀⠀⠀⣎⢿⢼⣧⠰⠀⣿⣽⣿⣿⣿⣿⡟⡀⠒⢀⢿⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⡀⠁
⣸⣿⠀⠂⠀⠀⡏⣗⠀⠀⢸⣿⣿⣾⠛⠘⠀⠁⢹⣿⣿⣿⣿⡏⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⠘⣳⣶
⣿⣿⣄⠀⠀⠀⠁⠃⠀⠀⠀⢹⡿⠋⠀⠀⠀⢀⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠃⠀⠀⣽⡟
⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠟⠋⠁
⠸⠟⠿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⢽⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠽⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢝⢦⡀⠀⢀⡤⡺⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⣽⣿⡵⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⢔⡫⠚⠛⢮⡢⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣠⣾⠗⠉⠀⠀⠀⠀⠙⢮⣢⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣯⣄⡉⠁⠀⠀⠀⠀⠀⠀⠙⠪⡲⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠨⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠈⠪⡳⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣈⠪⡳⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⣇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⡿⡟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣤⣶⣵⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠚
⠀⠀⠀⠀⠀⠀⢰⡿⣻⣯⣵⣿⣿⡏⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣠⣃⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠻⣿⡿⣿⣿⢃⣾⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠏⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡾⠿⠿⠿⠊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠴⠃⠈⠓⢢⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢁⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡟⡗⠚⡇⠀⠀⠀⠀⣠⡶⠶⠂⠐⣧⢒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣏⠸⣇⣇⣀⣙⣲⡄⢸⣾⢟⣶⣄⣀⠚⢉⣀⣀⣀⣀⣀
⠀⣁⡀⠤⠒⠚⠃⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠀⠸⠖⠠⢀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⠀⡀⠀⠀⠸⣿⠀⣚⣸⡇⠕⣒⣿⣿⣴⣟⣀⣀⢠⣤⢼⣤⣴⣽⣿⣧⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠆⠀⠀⡇⠀⠀⠀⣿⠀⢨⣼⣿⣿⠿⣿⣿⡾⠛⠓⠚⠋⠉⠙⣿⢩⡿⣿⣿⡇⢸⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠰⡀⠀⣿⠀⠀⢸⠁⠀⠀⢱⣏⢸⣷⣿⣿⡇⠀⠀⠀⠀⢀⣸⣟⣿⢧⣿⠙⣷⠸⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠑⡀⣿⣆⠀⠇⡐⠀⠀⡸⣳⠟⣻⣿⣿⡉⣹⡏⠋⢋⡹⢤⣿⢟⣸⡟⡼⡟⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠻⣿⣶⠗⠀⠠⠼⠃⠁⠐⠋⠀⠙⠳⠄⠀⠀⠀⠴⠿⠵⠾⠯⠾⠟⠀⠀⠀⣰⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠛⡟⠛⢒⠛⣿⡦⠼⣿⣿⣿⣿⣿⣿⣿⢏⣾
⡀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠙⢻⣿⣿⣿⣇⠙⢿⣆⠑⠤⠤⠤⠼⢷⡄⠸⣿⣿⣿⣿⣿⡏⠞⠁
⠀⠀⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⡀⠰⢿⣏⠙⠿⠷⠦⠙⠂⠀⠀⠀⠀⠀⠈⠀⠨⢿⣿⡽⠛
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣶⡤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⠉⠒⢄⡉⠁⠒⠠⠤⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⡄⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣦⠀⠀⠈⠢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠉⠉⠉⠁⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠿⢶⣌⠉⠁⠐⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢰⠟⣱⣦⡀⠀⠀⠀⠀⠀⠀⢈⣄⡀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⣇⣴⠟⠉⢳⡄⠀⠀⠀⠀⠀⠀⠙⢇⢀⣀⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡇⢸⣃⣤⠶⠶⢿⣆⠀⠀⠀⠀⠀⠀⠈⢫⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⢻⡟⡀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠐⣶⡆⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢇⢸⣷⠻⢷⠀⠘⠻⠿⣧⠀⠀⠀⠀⠀⠈⣟⠀⠀⠀⠀⣷
⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠘⣷⣄⠀⣀⣀⡀⠀⠈⡆⠀⠀⠀⠀⣀⡹⡇⠀⠀⠀⣿
⠀⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠈⣇⠘⣿⡇⠬⣥⡤⠾⣤⡇⠀⠀⠀⠀⣽⠋⡿⠀⠀⢠⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠘⣆⠈⠛⠶⣤⣄⣠⣿⠀⠀⠀⠀⢰⠎⠈⠦⣄⣠⡾
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠏⣄⠈⢦⣀⠓⣤⡤⠉⠙⠃⠀⠀⠀⢻⣶⠳⠶⢦⣈⡷⣦⡀
⠀⠀⠀⠀⢰⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠀⠀⠘⢦⣄⠽⣇⢀⣀⡀⠀⠀⠀⠀⢰⠾⠁⠀⠀⠀⡨⠟⠻⣿⣆
⠀⠀⣀⡀⢸⡇⣷⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠐⠛⠋⠀⠉⠀⠛⢻⠇⠀⠀⠀⠀⠀⠀⠉⠀⣹⠟⠀⠀⠀⠔⠛⠀⠀⠀⢀⠎⠀⠀⠀⠈⢿⣀⣀⡀
⠀⠀⠀⠀⢸⡇⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⡤⠀⠀⠀⢸⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⢯⡀⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠀⣧⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢾⡗⠤⢀⣰⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡞⠓⠶⢶⣌⠁⠀⣾⡃⠀⠀⠀⠐⠒⠛⠛⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣽⣇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠁⠀⠀⠀⠋⠀⢠⡏⠀⠙⠢⢄⡤⠴⢴⡷⣼⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣀⠀⠀⡀⠀⠀⠀⢠⡄⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⢠⣶⢯⣸⡏⠻⣦⡛⠻⣷⣤⣀⠤⠶⣶⡻⢶⠚⠲⠛⠣⢍⡀⢀⠃⠀⠀⢠⠏⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⡞⠁⠀⠙⢷⡄⠈⠙⢿⡉⠙⢿⣄⠀⠀⠑⢎⠻⡄⠀⠀⠀⠁⢿⠀⠀⠀⢸⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣇⣀⣀⣀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠹⠀⠀⠀⣼⡇⢀⠀⠀⣿⡀⠀⠙⢷⡀⠀⠈⣷⠸⡄⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡞⠁⠀⠀⠉⠘⠂⠀⠀⠀⠺
⠀⠀⠀⠀⠳⣀⣰⣿⡧⣸⣄⢀⣿⣇⠀⠀⢾⡇⠀⠀⠘⡇⢼⠀⠀⢰⡇⠀⠀⠀⡿⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡏⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⢹⣿⠋⠀⢹⠛⠿⣤⣿⠀⠀⠈⣿⣶⣦⣤⣷⣾⠿⠟⠟⠁⠀⠀⢸⠇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠋⠭⠞⣳⢤⡀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠈⢷⣄⠸⡇⠀⠀⣼⠋⠉⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠰⠧⣤⠀⣀⡀⠀⠉⠲⣝⢢⡀⠀⣘⠁
⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⡀⠈⠳⠼⢶⡞⠉⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⠀⢰⣧⣀⠀⠀⠀⠀⠀⠀⢰⣋⠀⠀⢀⣠⡀⠉⠙⠦⡀⠀⠈⢷⡙⣆⠃
⠀⠀⠀⠀⣀⠼⠿⢯⣽⠶⣿⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⡟⠀⡏⢸⠈⢳⣀⣀⣀⣀⠀⣀⡽⢛⡶⢫⡿⠀⠀⠀⠀⠙⣆⠀⠀⠹⣿
⠀⠀⠀⢾⣅⠀⠀⠀⠀⠀⠉⠘⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⢀⠇⠀⡇⠸⡀⠀⠉⠉⠉⠁⣀⠙⠷⠏⠀⣼⡗⠲⣤⡀⠀⠀⠉⠙⣆⢀⡼
⠀⠀⣤⠊⠀⠀⠀⣠⠖⠒⠒⢦⣤⣫⡟⠃⠀⠀⠀⠀⠀⢰⡏⠣⢤⣤⡤⢀⣾⠀⣰⠁⣴⠁⠀⠀⠀⠀⡰⠛⠃⠀⠀⠀⠉⢧⠀⠘⣿⣄⠀⠀⠀⠀⠟⠁
⠀⠀⡹⠃⠀⠀⣼⠃⠀⠀⠀⠀⠉⠉⠁⣠⠤⠒⠒⠒⠒⠾⣦⣤⠤⠥⣤⣤⣬⢁⠙⡾⠁⠉⣶⡄⠀⣸⠇⠀⣀⣠⠀⠀⠀⠿⣤⡾⠋⠛⢧⣤⣤⣤⣤⣤⡀
⠀⠀⣧⣤⠀⠀⠹⣆⡀⠀⠀⣀⣠⡴⠚⠁⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠈⠛⠂⠉⠀⠾⠿⠛⠀⠀⠈⢺⣿⢶⡤⢤⡤⠀⢤⠌⠈⠉⢃⡀
⠀⠀⠙⡄⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⠢⡀⢀⣷⠀⢀⡀⠀⠀⠀⠀⠰⠇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠙⣷⠀⢷⠖⠀⣠⠒⢀⣤⣴⡾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻
⠀⠀⠀⠀⠁⠚⣿⡷⠋⠱⢀⣷⠀⣼⠟⠇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠷⣤⣤⣼⡿⠇⠺⠛⠉⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣶⡆⠉⠉⠁⠚
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠠⠆⠀⠀⢻⣶⠚⠃⠙⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣰⠏⠋⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠂⠀⢤⣼⡟⠀⠀⢠⠇
⠀⠀⠀⠀⠀⢠⣿⡀⢀⡀⠀⠤⠀⠀⠀⢤⠰⡇⠀⠀⣾⠓⠸⡇⠀⠹⣷⣀⠀⡄⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⡸⠁⠀⠀⢸⣿⠇⠀⠀⡿⠂⠂⠀⠄⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⣿⠀⣴⠃⠀⢠⡟⠛⢿⣧⠀⠀⠀⠀⠸⣿⠁⣤⣴⡾⠁⠀⠀⠀⢼⡏⠀⠀⢸⠇
⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠹⡄⠻⡄⠀⢸⡅⠀⠀⢻⡄⠀⠀⠀⠀⣿⣴⠟⠁⢻⡀⠀⠀⠀⢸⠀⠀⠀⣸⠃
⠒⠀⠈⠉⠀⠻⠿⠇⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢳⣄⠘⢶⡀⠻⣄⠀⠈⣧⠀⠀⠀⠀⡟⠁⠀⠀⠘⢇⠀⠀⠀⡜⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢨⡆⠀⡄⢹⣆⠀⢹⣦⣼⣷⣄⣼⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⠀⠂⠀⡇⠀⠀⢿⡀
⠀⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⠚⠋⠀⠀⠀⠉⠉⠀⠀⣦⠀⠀⠺⡇⠀⠀⠀⠀⢸⡀⠀⢸⣇⣀⣀⣈⣿⣆
⠀⠀⠀⠈⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢰⠇⠀⠀⠀⠀⠸⡇⠀⣼⡏⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠸⡇⠀⠀⠀⠀⠀⠂⠀⠘⣷⠴⠶⠶⠋⠁⠀⠀⠀⠀⠉⠉⠁⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢿⣀⠀⠀⠀⠀⣷⡀⢀⡸⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠶⠂⠈⠉⠙⠒⠾⡄⠀⢈⡻⣏⠉⠉⠁⠈⠻⢄⡷⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠐⠶⠂⠀⠶⠖⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣤⣄⡿⠟

⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣤⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⢋⡀⠅⣨⣿⣿⡿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢠⣿⣤⣼⣮⠁⠀⠀⠈⣻⠃
⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣶⣾⣿⣿⣿⣿⣶⣶⣤⣴⠃
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀
⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠙⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿
⠰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡎⠀⢀⣴⣶⣦⠀⡇⠀⡠⠊⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠒⢏⠀⠀⠀⠀⢅⠀⠸⡟⠋⠉⠉⠁⠒⠿⣿⣷⡀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛
⡇⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠑⠲⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠇⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⠦⠤⠊⠁⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⡈
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠉⠒⠀⠒⠐⠁⠀⠀⠀⠀⠀⠀⢀⡤⠊⠉⠀⠀⠀⠀⠀⡇
⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣤⣀⡀⠀⠀⠀⣀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠉

⠀⠀⢀⡽⢛⣽⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣠⢀⣟⣴⣷⠆⠂
⠀⠀⠈⣵⣟⡇⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃⠀⠀⠀⠀⠀⣀⣴⣾⠖⠀⠀⠋⠀⠀⠀⠜⢡⡢⢟⣿⠒⠋⠀⠀⠀⠠⠄
⠀⠀⠀⠐⣷⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⣀⣄⣠⣂⡘⢁⣠⠞⠁⢠⡄
⠀⣄⠀⠐⢉⡿⣿⣷⣄⣤⣄⣠⢄⠞⠋⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⢾⠟⠁⠀⢠⣇⠀⠀⠛⣿⢫⣉⠴⢶⠞⠁⠀⠀⠀⠁⠠⣦
⠀⠈⠀⠀⠀⠰⣻⣿⣫⠟⣿⣧⣼⣤⣀⡄⠀⢠⢀⡀⡤⡒⢂⠒⠀⠀⠁⠈⡠⢀⣦⣄⣑⡀⣀⣰⣵⣊⡛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠂
⠀⠀⠀⠀⠈⠀⠀⠌⠴⠚⣉⣔⣿⡿⣠⡐⠛⣌⢿⣿⣇⡇⡘⠦⢠⣶⣶⡈⢫⡾⣟⣭⢿⣿⡿⠛⣹⣶⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⠀⠀⠀⠀⢀⡀⠀⠀⠈⠙⠛⠉⢡⠈⠳⠛⢠⡿⡿⠇⠝⠩⢏⣄⠦⠾⣯⣡⣟⡫⠷⠚⠋⢀⣦⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡌⣠⠔⠊⢀⠐⠁⢤⡃⡴⣼⠋⠀⠀⠀⠚⠉⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⣼⠂⡝⡡⠋⠻⠽⠹⡒⢄⠈⢤⣿⠀⠀⠀⠂⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⠋⢀⡄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢠⡄⠸⠇⠠⠃⠃⢨⠀⢸⡍⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠘⢀⠀⡑⢮⠀⡀⠑⡞⠁⣻⠔⠁
⠀⠀⠀⠀⠛⢄⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⣄⢧⠰⠶⠖⠶⢀⡎⠀⠼⡅⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣈⡧⠷⣃⡄⡗⠁⣀⠌⣄⠈⢮⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠚⠣⡜⣶⣂⠀⠀⠜⢢⠀⡶⠉⠀⠘⠛⠀⠀⠀⠀⠀⠀⡠⠾⠋⠉⠂⠙⠀⡇⢰⠋⡠⠅⣶⠋
⠈⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠃⡄⠁⠈⣻⡀⠓⣶⣼⣹⠀⠄⠀⠠⠠⡇⠀⠀⠀⠀⠄⠀⠀⠄⠀⡀⠀⠀⠀⣀⣀⢤⡤⢅⡦⣴⠀⠀⠰⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⣶⣿⣷⠀⢈⠋⢻⠛⠲⠃⠀⣑⢠⡶⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢣⣭⢀⠃⠀⠀⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⣀⣤⡤⠀⠀⠀⠀⠀⠑⢪⠀⣀⡀⢀⠦⠊⠉⠉⠋⠉⠀⠀⠀⠀⢤⣄⣀⣀⣀⠀⠀⠀⠀⣤⣀⣘⣿⢀⠀⠀⠤⠀⠈⠁
⠀⠀⢀⣀⠀⠀⠀⠘⠋⠉⠉⠀⢰⣂⡀⠉⠀⠀⠄⠽⠥⠬⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⣺⣿⡿⠇⠀⠀⠠⠄⠀⠙⠛⠓⣺⣿⣿⠹⠤⠛⠒
⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠄⠈⠁⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⠛⠀⠈⠉⠁⠐⠉
⠀⠀⠀⠀⠀⠐⣀⠐⠂⠀⠀⠀⢀⡢⠀⠀⠀⠀⠀⠀⠑⠂⠀⠁⠀⠈⠉⠉⠁⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣄⣀⡀⠀⠀⠀⠢
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠚⢯⠧⠤⠄
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣦⣵⣶⣤⣾⣵⣿⣽⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡅⠢⠄⠀⠔⠁⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⡀⢤⡝⣦⠉⠁⠀⠀⢀⡘⢧⡛⠂⠀⠀⠀⠀⠀⠈⠑⠒⠂
⠀⠀⠀⣸⡱⡀⣀⠎⠀⠀⠀⠀⠀⠀⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠃⠺⣂⢠⡄⢨⣥⠤
⠀⠀⡀⠹⣱⠡⠃⠀⠀⠀⠀⢀⠀⠀⠀⠨⣄⣷⠄⠒⠀⢲⢏⣄⠀⠀⠀⠀⠀⠀⣽⣍⠦⠀⠮⣶⠻⠀⠀⠀⣀⠀⣄⠀⠀⠀⠠
⠀⠀⡇⠀⠉⠀⠀⠀⠀⢀⠔⠁⠀⢀⠔⠉⠁⠀⠀⠀⢠⠊⢃⠿⠀⠀⠀⠀⠀⠀⠆⠉⠀⠀⠀⠀⠈⠀⠀⢀⡻⠋⢛⡶⠂⠀⡆⠀⢈
⠀⠀⠁⡀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⣾⠛⡆⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣶⠺⠧⠀⡘⠀⠀⣈
⠀⠀⠀⠩⡀⠀⠀⠠⠂⠀⠀⢠⢃⠤⠔⠒⠒⠁⠀⠀⠘⠿⠞⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣼
⠀⠀⠀⡇⠉⠁⠀⠀⠀⠀⣠⠋⣁⣴⣾⡟⠋⠲⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠠⠛
⠀⠀⠀⠙⣄⠀⠀⣀⣠⠞⠉⢀⠉⠁⠚⠒⠊⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠈⠑⠺⠇⠀⠠⢀⡈⣁⠀⠀⠀⠀⠀⠀⠀⠤⠔⠪⠁⠓⣒⠒⠢⡄⠀⠤⡀⠀⠀⠀⠀⢀⣀⠤⠐⠊⢁⣀⠀⣠⠀⡴⢩⠁⠀⢀⠀⢤
⠀⠀⠀⠀⠀⠀⠈⢄⡀⠆⠀⠀⠀⠉⠲⣤⣀⣀⡀⠀⣀⣀⡤⠂⠀⠀⠀⠀⠘⠀⠀⠘⢄⠔⠒⠽⠷⠃⠀⠀⠀⢠⠎⢂⣁⠴⠛⠁⠀⠀⡴⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠑⠤⣀⣀⠠⢤⠀⠐⠀⠀⠸⡛⢻⡋⠀⠐⠒⢊⣠⠅⠓⠘⠤⠤⠀⠀⠊⠴⢔⢋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⡀⠀⠀⡀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⢤⠤⠘⠀⠀⢃⣈⠭⠛⠁⠀⠀⠀⠄⠀⠀⠀⢀⢀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣄⣀⣘⠦⠄⠀⡇⠀⠀⠀⢀⠄⠂⠀⠀⠠⠀⠀⢸⠀⢧⠤⠤⡬⡥⠢⡀⠀⠀⠀⠀⡄⠀⠀⡨⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠱⡄⢲⡆⠀⢸⠀⠀⠀⡰⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⡴⠊⠉⠀⠀⠀⠀⠀⠹⣻⣇⠀⡜⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣠⣔⠰⢠⠏⠀⠀⠀⠙⡄⠀⠀⠀⠀⢻⠈⢦⢀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠪⡄⠉⣰⡟⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡹⠁⠀⢠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡚⠂⠀⡰⢡⣅⠀⠀⠑⠲⠤⣀⠀⠀⠀⠀⢸⣧⠎⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠢⣀⠁⡠⣉⠀⢢⠀⢄⠀⠀⠉⠒⠠⠀⡸⠁⠀⠀⠐⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠙⠲⡥⣀⠀⠑⠀⠑⠀⡁⠤⠄⠀⢁⡀⠀⠠⢚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠈⢳⣤⣤⣤⣤⣤⣄⣴⠾⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠀⠙⢿⣄⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠖⠂⠀⣠⣴⠶⠿⠿⠿⠿⠿⠿⢿⣷⣦⣄⣀⣼⠟⠋⠉⠀⠀⠈⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣧⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⠀⠉⢻⡟⢁⣤⠶⣦⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⣾⣡⣦⠘⡧⠀⠀⢀⡾⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⡼⠿⣿⣇⠀⠀⣸⣵⣶⡀⣸⠃⣀⣀⣀⣀⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⡶⠶⠶⠶⠶⣤⣿⠿⠛⠳⠶⠤⠀⠉⠙⠻⢏⣰⠿⠋⠉⠉⠀⠈⠉⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⢀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⡽⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡶⠋⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠶⠦⠤⠤⠤⣤⣤⡿⢱⣶⣶⡄⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⣆⠀⠀⠀⢀⣀⡤⠶⣻⡽⣿⣿⡇⣴⡶⠶⠿⠛⠋⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡬⠿⣷⣖⣋⣩⡴⠖⣋⣡⢟⡶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠒⠒⠒⠚⠛⠋⠉⠉⢿⣆⠀⣀⡬⠟⠛⠛⠟⣻⠟⢁⣞⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠿⣧⣤⣀⣀⢀⣼⣁⣀⡼⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠙⡻⠋⠛⠉⢹⠇⠀⠀⠘⢿⢿⣷⣶⣤⣤⣤⣶⠟⠋⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠈⢣⡈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠉⠉⠉⠩⠉⠀⠉⠉⠉⠉⠉⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⡇⠀⠐⠒⠀⠀⠠⠀⡁⠀⠀⠀⠀⠀⡀⢀⣀⣀⠀⠀⢀⣈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⡇⠀⢀⢀⠀⣀⣀⠀⡅⠀⠀⠀⠠⠎⢁⠀⠠⠀⠀⠀⠐⠐⠨⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⠇⢀⡀⢐⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⡇⠀⠀⠀⠀⠘⠂⠀⡆⠀⠀⠀⠀⢃⠈⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠤⠄⡇⠀⠠⠤⠀⣠⡄⠀⡁⠀⠀⠀⠐⠀⠓⢄⡀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⢀⠉⠉⠀⠀⠀⢀⣀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⡅⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠛⠒⠋⣀⠈⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠐⠀⠀⠇⠀⠐⠀⠀⠤⠆⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡰⠊⠁⠅⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⡀⠀⢀⣀⠀⢀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢻⠀⡆⠀⠣⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡃⠀⠀⠀⠀⠉⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢡⡁⠠⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠃⠀⠀⠀⠅⠀⠀⠀⠀⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠠⠤⠄⠇⠀⠠⠤⠀⣠⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠈⠉⠁⡁⠀⠈⠉⠀⠛⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⣃⡀⢀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣽⡇⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣿⡇⠀⠐⠀⠀⠒⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣷⣤⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⠿⠿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢘⣛⡛⠯⣪⢿⣿⣿⡿⡦⠹⠇⠺⣿⣐⣦⣼⡊⠞⡉⠊⡠⠐⠈⠈⡸⠃⠀⣶⠋⠀⠂⠀⠀⢈⠀⠲⠌⢀⢭⣴⣂⣿⠂⠻⢏⡼⣿⢿⣯⡯⣴⠟⣛⣛⡛
⠀⢸⡏⢩⡆⠰⣿⠿⠛⢻⡽⠎⠲⠶⣰⠈⡻⠒⡀⠁⠄⣠⠤⠴⠀⣊⣤⠴⡾⢬⣙⣒⣒⠀⠔⠤⣍⣑⠒⠶⣈⠓⠂⡒⢂⣒⣽⢿⡟⠛⠿⣿⢁⢮⣌⠈⡟
⠀⠘⠣⢙⠑⣡⡇⣬⡿⠋⢐⡿⡙⡢⢃⠔⠠⠂⣠⠖⢋⡤⣶⠛⣿⡩⡓⣏⠘⡾⡅⠿⣹⠏⣻⡶⢦⣬⣙⠲⣄⠑⢄⠑⠴⣥⡲⢦⢱⡿⣗⡨⣄⠀⠙⠐⠷
⠀⢸⠏⢸⣿⡿⠷⠭⣍⡛⡮⠍⠊⠔⣁⡴⢞⣷⢋⡾⠋⠻⣠⠂⣸⠆⡇⣘⣼⠶⠛⠲⠿⢾⣭⣧⡸⠰⠛⣳⣌⠁⠀⠀⠤⣤⣭⣭⢩⢉⠭⠶⠯⣽⣧⠙⡿
⠀⢸⢇⢘⠏⡶⣿⠗⠈⠃⡥⢋⠔⡁⢊⡴⢻⢛⡼⡈⡆⢆⢹⠰⡄⣱⠞⢋⠠⣤⠊⠉⡀⠀⣠⠌⠹⣼⢶⠋⡈⡷⣄⠀⠀⠈⠻⣿⣇⠘⠼⣛⠷⡌⡏⣰⣧
⠀⢸⡟⣽⠄⣒⠒⣪⢂⠞⠁⠀⠔⣠⣿⡘⣸⢻⡳⢁⠋⠜⠄⢎⣾⣇⠴⠃⣰⠃⡠⠊⡠⠃⠑⣔⢀⡟⡌⡎⣱⢁⠽⣶⡄⠀⠀⠈⢿⡌⢦⢛⢛⣀⣏⣿⣟
⠀⢸⣭⡟⠀⢤⠒⡴⠏⠠⠀⠌⣰⣿⣰⢶⣛⣿⣖⠵⠓⠋⣿⣛⣉⣁⣀⣠⡇⣊⣥⡬⠀⠰⠆⣌⣾⣃⡹⡸⠁⣁⣂⣯⡼⣄⠠⠀⠄⢻⡄⠲⡚⠉⠛⣽⡿
⠀⢸⢋⢔⣦⠆⠟⡐⠀⠁⢀⣴⡇⡿⣅⢀⣨⢣⣿⠀⢀⣔⡘⣋⣿⡍⠡⠈⢼⣿⠇⠀⠀⡐⣾⣹⣗⣿⡃⣷⣉⠀⣩⡎⣵⠜⣦⠀⡈⠀⢻⡄⠡⢷⢢⡜⢿
⠀⢠⠱⣷⣖⡡⠀⠀⠀⢀⣿⠃⡇⣹⡟⢫⣉⡎⢹⡀⢪⠳⠉⡜⢻⡘⠢⣄⠀⠉⠈⠀⠀⠁⠀⣰⢧⢡⢋⣩⠿⡜⢋⡜⣛⠁⠻⣧⠀⠀⠀⠛⢌⢲⡧⡸⣦
⠀⢐⠒⠆⡊⠀⠀⠐⠀⣎⠌⢢⡜⡇⣇⣼⠙⢘⣆⡌⡆⣆⡇⢳⠌⣷⣼⡈⠳⣤⢀⠐⢒⡠⣔⢧⣸⢞⡈⢋⢀⣿⠎⢞⢩⡄⢢⣽⢆⠀⠀⠀⠈⠔⢷⢬⡟
⠀⢸⣟⡄⠃⠀⢀⠁⣸⣯⠂⠈⣁⣿⢻⣇⣨⣣⣏⢌⣴⣭⣹⡌⢾⠟⢼⣛⣦⡈⢢⡼⠘⣧⡙⠘⠇⢜⣹⠀⢟⢀⣡⣽⡿⡠⠟⠍⣻⡄⠂⠀⠐⠘⣿⢸⣿
⠀⢰⠍⡔⣠⠁⠀⠀⡸⢻⣶⡷⠿⠿⠈⠻⣇⠝⢹⠯⠀⠀⣰⡿⢸⡞⢓⡏⠡⠀⣾⠇⠂⢀⠙⠲⣄⠠⢩⡼⠣⣡⣿⢿⡷⠿⣷⣻⠋⡇⠀⠀⠀⠃⢸⡷⣿
⠀⢸⠧⡔⠹⠀⠀⠀⠀⡾⣬⣝⣃⠀⠀⠀⢯⡞⠟⠛⣁⠛⢟⣀⣿⠁⣟⠀⠂⢸⡏⠀⢊⠁⡠⠂⠈⢷⣾⢂⢖⣿⠀⠀⠀⢛⣫⣥⣧⠀⠀⠄⠀⠀⠈⢣⠙
⠀⠀⠉⠁⠀⠄⠀⢀⢰⠟⡯⠥⢧⢤⡤⣤⣌⠳⣇⠨⠧⢊⠆⣿⠇⢸⠁⠈⢀⣿⡍⠀⠀⠀⢳⡄⠀⠀⠻⣆⣾⣁⣀⣠⣤⠬⠥⢊⢺⡄⣄⠂⡀⠀⠀⡈
⠀⢐⠒⠒⠒⠒⠒⠚⢸⡇⠉⠒⣸⢉⡅⠂⠙⢷⡌⠙⠲⢦⣔⡾⠀⠘⣇⠀⠈⣿⠇⠀⠀⠀⣼⢛⢦⡀⠀⠈⠻⡟⠉⣿⣫⠐⢧⣛⣿⡇⢙⢀⠈⠁⠉⠄⠤
⠀⢘⡀⢀⣀⣀⡀⢸⢸⣏⡁⢠⡟⢸⢾⠶⠲⠋⡽⢦⡀⠀⠈⠳⠀⠀⡼⣇⠀⢸⠀⠀⠀⢰⣏⡟⣸⠝⣦⡀⢸⠀⠀⣸⢟⠂⠀⡺⢏⡄⠀⢰⠀⠀⠀⠀⠥
⠀⢨⢶⠂⢕⡄⣩⢀⢸⡟⣡⠟⡘⡃⣎⠐⣡⠈⡷⣻⣼⠗⣦⣄⣡⡾⣡⢿⡄⠀⠀⠀⠀⣾⡟⣴⡅⣾⢹⣷⣬⣀⣴⢫⣎⢣⠱⠘⡈⡇⠀⢈⢩⡝⠉⣌⠛
⠀⢰⠂⢠⣸⢁⠛⢸⢸⡦⠳⠀⢇⠜⢈⡞⡠⡏⠁⡇⡝⠠⠷⣀⣭⢺⣼⠾⡇⢀⠀⠀⡀⢿⣇⢡⠋⡌⣧⣸⣮⢋⡸⢫⠟⡘⢸⢇⢣⡇⢠⢸⣆⢳⢀⠊⣶
⠀⠈⠀⣸⢃⠆⠀⢸⢸⡶⠖⠛⣌⣋⠒⠤⠤⣵⡳⠊⢲⠤⢲⣹⠾⣻⡇⢨⠀⠄⡠⠊⡠⠚⢿⠻⣤⡠⢷⡷⡥⠬⢴⡧⠾⠶⠓⠚⢢⡇⢰⢸⠈⠄⢿⠀⢹
⠀⢠⠀⣽⠘⡀⠀⠘⢸⡧⢀⣴⠏⣼⣅⡀⣀⣾⢡⢞⣦⣠⠞⠁⢰⠃⢡⢠⠰⢊⠔⠃⠀⠀⠘⡆⠈⠳⣼⢏⡀⢈⣍⣴⣯⡂⢀⠐⢋⡅⢸⠈⠀⡘⣸⠀⣨
⠀⠀⢠⠁⠱⠈⠳⠀⢸⢡⠡⢡⠶⠌⣸⢇⠹⣠⢏⣦⡾⠋⠀⢠⠏⠀⠈⡤⠞⠁⠀⠀⠀⠀⠀⢹⠀⠀⠈⢳⡟⡙⢇⡾⢫⠅⡰⠃⢣⡆⢸⠸⠃⠐⠁⢁⠈
⠀⠀⠃⡀⡱⡡⠀⢠⢸⡄⠇⠎⡤⣄⡣⢸⠗⣵⣿⠋⠀⠀⠀⠞⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠙⣷⡎⣷⣿⠀⡇⢃⠈⡃⢸⠀⠸⡕⠁⢈⠿
⠀⠀⠠⠋⠈⠁⠀⠈⠀⡵⢈⠂⠑⡄⠠⡟⣴⡟⠁⠀⠀⠀⠌⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢷⣷⠱⢠⢃⠼⡄⡆⠰⠀⠀⠁⠈⢣
⠀⠀⡆⢀⢂⠅⡆⠀⠀⣷⠘⢪⠖⢫⢋⣴⠏⠀⠀⠀⠀⠈⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⢻⡄⠱⡄⠒⣼⡆⢸⠀⡀⠀⣆⠀⡄
⠀⢰⢇⠸⣤⠀⠀⠀⠀⣿⡯⠴⢲⣿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢻⣇⠈⢩⡽⡇⢸⠀⠀⢀⡿⠰⢰
⠀⢠⡸⢆⠹⣆⡶⠀⠀⣿⢀⣎⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠈⡰⡇⢸⢠⡄⡾⢠⢧⠈
⠀⢠⡁⠘⣇⠹⣀⠀⠀⣿⣷⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠌⢷⣱⡇⢸⢈⡿⠰⠇⠀⡆
⠀⢨⠱⠾⠄⠲⢻⠀⢨⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⢸⡥⠁⠀⠎⣠
⠀⣠⣀⣀⣀⣀⡘⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⢀⡀⣀⣀⣁⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⣿⡿⠻⣿⣿⣿⣿⣿⡟⣙⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⣄⠀⠈⠙⠛⠿⣋⡉⢸⣿⣿⠀⠀⠈⣢⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠁⠈⠣⢄⣈⡉⠛⠳⣆⠈⠐⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⢿⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠈⡉⠀⠀⡄⠀⠀⠀⠒⠇⠀⠀⠀⠀⠀⠀⠀⡴⠀⠄⠂⠙⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⠇⡙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡸⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠨⡇⠀⠀⠀⠀⢀⡔⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⢠⣤⣀⣤⠀⠀⠀⠀⢀⠀⠀⠀⠠⡀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢰⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢤⣀⡀⠉⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⡀⠀⠀⠻⣄⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⢸⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢠⡿⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣴⣆⣀⡀⠀⠈⠂⠀⠼⠃⢀⣀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢰⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⠞⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠴⡺⣽⣿⣿⣿⣿⣿⣗⠒⠃⠈⠘⠋⢉⣼⣿⣿⣿⠀⠀⠀⠐⢠⠀⠀⠀⡀⠀⡄⠀⡼⠋⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⢠⢞⣡⠚⣠⠈⢿⣿⢿⣿⣿⣿⣿⣦⣄⣀⣴⣿⣿⡿⢧⣿⠀⠀⠀⠀⡾⠀⠅⠀⠈⢈⠄⠈⠀⠈⠿⠆⠀⠀⠀⠀⠀⣴⠀⡇⠀⠀⠀⠀⠸⢸⠀⡼

⠃⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠇⠀⠀⠀⡸⠃⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⢀⣴⣧⡀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠈⠀⠀⠀⢀⣴⣧⡀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠈
⠀⠀⢨⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⢻⡇⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢰⡁
⡀⠀⠀⠀⠳⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠳⠀⠀⠀⢰⣿⣿⡆
⣁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠂⠀⠀⠀⠀⠀⠀⠀⣸⣁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⣿⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⢀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⢀
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⣇⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⣇
⠃⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⡸⠃⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⢀⣴⣧⡀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠈
⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⢻⡇⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢰⡁
⠀⠀⠀⠀⠳⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠳⠀⠀⠀⢰⣿⣿⡆
⣁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⣺⣁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⣿⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⢀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⢀
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⡇⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⡇

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⠀⠀⢀⣀⠀⠀⢀⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⡀⠀⢀⣀⣀⠀⢀⣀
⠀⠀⣿⣿⡿⡻⠿⠉⡿⡿⣋⠛⠡⡂⠈⢀⣴⣾⣿⣿⡖⣤⡈⢩⣯⡿⣿⢟⠺⠟⠋⢁⣶⣶⡄⠀⣈⡀⠀⢉⠉⣿⠂⣀⠀⠀⢀⡀⠀⠇
⠀⠀⠏⠉⠐⠀⠀⠀⢈⣂⣤⣈⣁⠬⠤⠤⠴⣴⣒⠢⡄⣅⢩⡙⠇⠁⢘⡌⣧⢀⠀⠈⠀⠐⢨⠙⠿⢛⠚⢦⠁⣿⢛⡽⠖⠓⠋⢧⣤⠄
⠀⠀⣿⠄⠀⠠⠠⠀⠼⠿⡥⠤⠤⠐⠒⠒⠒⠈⠿⢤⠄⠙⢸⣿⣆⣤⡋⣰⣇⣈⣷⠘⣿⣷⡷⠶⡆⢺⣏⡋⣸⠷⠋⣠⢤⣈⣢⣡⣨⡇
⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣉⣛⣁⡥⠱⣶⣼⠚⠲⣤⣿⣿⡿⣿⣏⢹⡔⢻⣘⣥⠋⠑⠀⠀⡌⠉⢙⣿⢦⢃⡛⢹⠻⠧⠤⠶⠇
⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⢸⣿⠿⣶⣥⣿⣿⣿⡀⠘⢛⡿⣾⣿⠯⣬⡄⣈⣉⡛⠓⡶⣤⠏⣿⣴⣿⠟⠉⡻⣾⢷⡖⠂⡁⠘⠇
⠀⠀⢀⣀⠀⣧⣮⢴⡀⠀⠀⣈⣉⣩⣽⠀⠎⢸⢟⣿⢿⣗⠖⢤⣤⣤⣬⣷⡶⠖⢒⣀⣀⣀⡻⣈⡏⢀⡾⢩⡶⠿⢤⣧⢀⡿⠾⠿⠷⡆
⠀⠀⡤⠄⢠⡄⠀⠀⠀⠀⠀⠹⣯⡴⠰⣧⡤⣸⡾⠏⠺⠛⠛⠋⠉⢙⣛⣽⣿⣿⠯⠅⠈⡄⠐⢌⣠⠿⡿⠚⡄⠀⠀⣄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⠀⣀⣤⡀⠀⠀⠉⠙⠛⠋⠛⢛⣛⠛⠿⠻⠟⠛⡒⢾⢿⡿⠋⠙⠉⠒⠤⣤⢤⣲⡟⢉⢀⡀⠀⣥⡤⠶⢛⡦⠄⠀⢀⣄⡀
⠀⠀⢀⠀⠉⠉⠁⠁⠁⠈⠀⠀⠀⠐⠀⠀⢀⣁⣈⠉⠁⠔⠓⠐⠓⠀⠈⠁⣀⣀⣀⣀⣈⣙⠒⠒⠀⠹⠤⡤⠚⠓⢂⣈⠁⠴⣦⡾⣄⡇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⢟⠍⢀⣾⢏⣝⠀⠀⠀⠀⠀⠉⠙⢏⡁⣈⠩⠉⣀⡀⠁⠆
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠀⢀⢀⢀⣚⣁⡚⡀⠸⠵⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠿⠞⠋⠚⠶⡆
⠀⠀⣠⣤⠤⠀⠀⠀⠐⠚⡲⠆⠠⠄⠚⣀⣀⠀⠀⠀⠀⠀⠁⠀⡠⢰⡦⠀⠉⢹⡶⠂⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠀⠀⠤⠂⠀⠀⠈⠀⠐⠀⠀⠀⠤⠄⠀⠐⠻⣖⢧⣤⠤⢏⣿⣌⠠⠐⡆⠀⠒⠂⠐⠀⢀⣀⣀⣤⣴⠾⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣒⡆⢒⣠⣲⣶⣤⣽⣦⣤⣦⣷⣿⣵⣛⣳⣶⣾⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⠆
⠀⠀⠀⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠈⠙⠋⠉⠉⠉⠉⠙⠛⠻⣿⠋⠟⢛⠉⠡⠀⠀⠠⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⡄⠼⡯⡆⠀⡀⣀⠀⠠⠄⠀⠀⠀⠉⢀⠀⠀⠈⠉⣀⡀⠀⠀⠠⠤⠤⠈⠛⠛⢀⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢉⡁⠈⠉⣠⠤⠖⢛⠋⠉⠐⠂⠀⠖⠀⠀⢀⡀⢒⡀⠀⠂⠂⠀⠙⠀⡀⠰⠂⠠⢤⡁
⠀⠀⠆⠸⠟⠛⠁⠤⠀⠈⠈⠇⠐⠁⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠠⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣷⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠉⠀⠙⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠞⣶⡞⠛⠛⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠠⣄⣀⣉⠒⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠈⠓⠉⣙⣍⡉⠧⠄
⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⣠⠇⠀⠀⠈⠑⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⣶⢠⡄⢰⡌⠳⢟⡀⠀⠀⠀⠐⠒⠙⠀⠐⠖⠀⠈⠲⢄⡀
⠀⠀⢠⠀⠀⠀⢘⣁⡤⢄⡐⠚⢁⠀⠀⠠⠀⠀⠀⠘⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠩⡥⠾⠳⣦⠙⢛⡋⢁⣠⣀⠀⠀⣠⡤⠤⠤⠄⠀⠡⣀⠀⠙⣦
⠀⠀⠿⠀⠀⣀⠀⡡⠐⠀⠀⠀⠀⠈⠢⡀⠀⢆⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⠀⣀⠴⣤⠶⠶⠀⠀⠀⠙⠃⣼⡙⣯⡁⠂⠀⠀⠀⠀⠀⠰⠄⣀⣀⠀⠋
⠀⣤⡆⠈⠁⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⠀⡠⠁⠀⠰⠃⠀⠀⠀⠀⠄⣹⡿⣹⢇⣠⠀⠀⠀⠰⠿⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘
⢠⣿⡒⣤⣤⠂⠀⠀⠂⠈⠉⠀⠀⠈⠐⠄⠀⣼⡂⠁⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⣨⣯⣤⠰⠷⣰⠶⣟⠂
⠃⢻⣝⠄⠀⠀⠁⠀⢀⠄⣀⠤⠀⠀⠉⢺⣦⣿⠗⠂⠀⠘⢻⢧⡴⠀⠀⠀⠀⠀⠀⠀⢴⢶⣼⠇⠀⠹⠿⡏⢋⠀⠀⠀⣄⡀⣠⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⠊⢃⣹⠀⠀⠀⠀⠀⠀⠀⠰⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠟⠙⢷⣶⠄⠀⡀⠀⠀⣙
⡄⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠠⡞⠁⠀⠀⠀⠀⠀⠀⠠⢀⣴⠿⡇⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣧⡤⢾⡆⠀⢠⠁⠀⠀⠙
⠘⣶⠒⠈⠁⠀⣠⠖⠀⠀⠀⠀⠀⢀⣀⣠⠴⠎⠀⠀⠀⡾⣿⡶⠋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⡌⠀⠀⠀⣿
⠀⠉⠤⠠⠄⠂⠁⠀⠀⠀⣼⠶⠋⢉⣠⣦⡤⣄⠀⠀⠈⡁⠋⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⡟
⢸⡆⠀⠀⠀⠀⠀⡐⢀⡾⠁⢤⣾⠟⣿⣟⣀⠈⠃⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠹⣦⣀⣀⣠⣈⡴⠛⡠⢣⠀⠀⠘⠉⠋⠀⠀⠀⠀⠀⠠⠀⢈⡁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠘⠀⠀⠀⢀⣰⠄
⠀⠀⠀⠉⠺⠷⠀⠀⢀⣀⠀⢉⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦⡻⠀⡰⢦⣌⠑⠶⣄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠉⠁⢀⣀⠀⣠⠆⢠⠋⡉⠀⠀⢀
⠀⠀⠀⠀⠀⣄⠐⢠⠄⠀⠋⠁⠘⠂⢤⡀⠀⠀⠀⠀⠀⠀⣀⢀⠔⠁⠀⠀⠀⠀⠈⡀⠀⠀⠳⣄⡤⠲⠲⣿⠷⠁⠀⠀⠀⠄⣨⡍⣶⠋⣠⠿⠊⠁⠀⠐⣼
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⡄⠀⠀⡀⠀⠉⣈⠉⠁⠀⠀⠉⠉⠣⣄⠀⠀⠀⠀⣀⡀⠠⢄⠀⠀⢹⡦⣶⡷⠂⠀⠤⠤⠴⠄⠚⠩⡆⠻⡉⢀⣀⠀⠠⠄⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⣧⠀⠀⠀⠇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠒⠉⠀⠉⢄⡀⠀⠀⠀⣸⠇⠈⢧⠀⠤⢤⣴⠶⠛⠁⠀⢸⠀⠀⠁⠀⠀⠀⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡆⠀⠀⠠⡄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠘⣟⠙⡀⠀⠀⠈⣶⢖⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣐⣦⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⣤⡘⠒⠒⠂⠳⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⡟⠐⢷⡒⠟⠶⠾⡌⠐⢄⠀⠀⠀⠀⢀⠖⠀⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡠⠤⠃⠀⠀⠙⣧⢸⣷⠁⠀⢸⡀⠀⠀⠀⠌⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠊⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⡀⠀⡜⠀⠀⢀⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣰⡔⣺⠋⠀⠃⢠⡘⠇⠀⠀⠀⠀⠀⢹⡇⠻⡄⠁⠀⠠⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣴⠛⠿⠉⣴⡇⠀⠀⠀⠁⠀⠻⠀⠀⠀⠀⠀⠈⣷⠀⠻⢞⠄⠁⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⢊⡄⠙⢃⠀⡐⢻⠍⠑⠢⣄⡀⠀⠀⠂⠀⠀⠀⠀⠀⣿⠀⡠⠊⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠀⠀⡔⠀⢷⡀⢀⠀⠀⠉⠐⠢⠆⣀⠀⠀⠀⠀⣿⠞⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠢⢼⠀⠰⣾⠀⠈⢢⡀⢤⡀⠀⠀⠈⠑⠂⠤⢰⠏⠀⠀⠀⠈⣁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠘⠳⣼⣄⠀⠀⠱⠀⠘⠂⠀⣁⡠⠤⠀⠃⠀⡀⠀⠠⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠳⡉⠑⠲⠦⣠⣄⣁⡀⠠⠴⠖⠛⠁⠀⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⣼⠉⠁⠒⠒⠠⠤⣤⣥⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠲⣿⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡌⠀⢰⡇⠀⢀⠀⠀⢸⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠙⠿⣀⣠⣿⡇⠀⣸⠀⠀⣻⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢈⠉⠉⠉⠉⠈⠓⠂⢿⣇⡀
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇
⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧

⠀⠀⠀⠀⠀⢀⢀⡆⣼⡀⠘⡄⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡇
⠀⠀⠀⡀⣤⣾⢸⡇⣿⣇⠀⠘⠄⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⢃⡀⠀⠐⢆
⠀⠀⠐⢁⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⡟⠋⠁
⠀⠀⠀⢸⠟⡿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠘⠀⠀⠹⠿⣿⣿⣷⡻⣇⠛⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠚⠿⣿⣿⣿⣷⣤⣄
⠀⠀⣠⣾⣿⣯⣤⣴⣒⣦⣿⣿⣿⣽⡿⠿⡋⠉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡰
⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣒⠈⣧⠄⠀⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣼⡿⠻⠀⠀⠀⢸⠈⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⢋⠁⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠏⠙⣦⡀⠀⠐⠉⠉⠻⣷⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⢿⡄⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⠀⠀⠀⢀⣾⣿⣷⣤⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣄⠀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣄⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠐⠒⠁⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠈⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⠀⠀⠀⠀⢈⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⢀⡠⠰⢀⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⣠⠂⠁⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢌⠀⢠⠃⢜⡂⠀⠀⠠⠀⣠⠖⠊⠀⠂⠀⡀⠈⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠉⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐
⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢧⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣾⡷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣿⣿⡞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣿⣿⡾⡇⢿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⢿⣿⣻⡏⣿⡇⡷⢸⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣿⣿⣿⡇⣿⠇⠁⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣼⡋⣿⣿⡇⣿⠀⠁⠀⢳⣿⣇⣀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢠⣿⣿⡇⣿⣀⣀⣀⣈⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢸⡇⠀⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣸⠁⠀⣿⣿⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⡄⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠿⠃⠀⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠤⠖⠀⠦⠰⠴⠖⠶⠶⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⣡⣼⠿⠿⣿⣿⣿⡟⣿⣿⠇⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠉⠀⠀⠀⠉⠀⠀⠀⠈⠉⠻⠏⠙⠁⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⢰⣿⣷⡄⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠉⠙⢿
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠛⢻⠶⣶⣄⡀⠀⠁⠀⠀⠀⣀⣠⠴⠷⠒⡓⠀⠀⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡐⠃⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠁⠠
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⡄⠀⠀⢠⠄⠀⠀⣆⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠘⠉
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠂⢾⣿⣄⠀⠀⠀⠐⠛⠋⠉⢿⣩⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠚⢹⣽⣿⠉
⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇⠀⠀⢀⡄⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠄⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣇⠀⠀⠀⠁⠀⠸⡇
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡀⠀⠀⢠⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⡄⢀⡄
⠶⣶⣶⣦⡴⠶⠴⠶⠖⠉⠁⣴⠀⠈⢰⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⣠⠀⠀⠀⠁⠀⠈⠉⠁⠀⢀
⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢷⡶⠿⠿⠿⠿⠀⠀⠈
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠟⠓⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⡙
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣠⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⠶⠖⠀⠀⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⠀⠰⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⣿⣿⣿⣿⣿⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⢀⣿⣿⣿⣿⣿⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠂⠀⠈⢻⣿⣿⣿⣿⡀⣀⣸⣇
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇
⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⢀⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⡀⠀⠰⣶⢮⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⠐⠠⡄⠀⠀⢐⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠓⠛⠾⠷⠾⠷⠿⠟⠛⠺⠿⠿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⢠
⠀⠀⠠⣌⠀⣀⣀⠀⠀⠲⣤⡀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠒⠒⠒⠲⡖⠒⠂⠈⠑⢂⣠⣤
⠀⢐⣈⣩⣦⡀⠀⠀⠀⠤⠩⣽⣶⣿⡏⠉⠉⢠⣿⣦⣤⣤⡄⠀⠀⠘⠀⠀⠀⠀⠀⢱⣶⣄⠀⠀⠀⠀⠈
⠀⠛⠀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢉⣽⣿⣿⣿⣯⡤⠀⠀⠀⢀⣤⣴⠂⢸⣿⣿⣷⡀⣤⣀⣀
⠀⢀⣼⠟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣤⣭⣉⠉⠉⠉⠉⠉⠉⢉⣉⣉⡍⠛
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠓⠒⠀⠐⠂⠈⠛⢲
⠰⣶⣶⣶⣴⣶⣾⣿⣿⣿⡆⠀⡀⠀⠀⠀⠀⠌⠽⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠇⠠⠴⡖⠀⠀⠀⠘⠛⠻
⢠⣿⣿⣿⣿⣿⣦⣿⣿⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠉⠛⠛⠻⠷⠶⠲⠶⠦⠤⣀⣀⢠⣾⠋
⢘⣻⣿⣟⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⠉⣙⣿⣿
⢀⡿⠿⠿⠿⠵⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠉⠉⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⣀⠤⠈⠀⠀⠑⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢿⣀⣀⣀⠀⠄⠈⠁⣠⠤⠀⢠⡞
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠘⠀⠀⣀⠀⠁⢐⣄⢘⣿⣿⣇⣀⠄⢀⡀⠀⢀⣰⠎⠀⢄
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⠃⠀⢐⠉⠉⠉⠛⠻⢿⣿⣀⣈⣀⣠⣿⡿⠎⠀⢸⠀⠀⠀⠀⠀⠀⣀⡀⢀
⢀⡀⠒⠀⠀⠀⠀⠀⠒⠀⠌⠀⠀⠀⠠⢕⡄⠀⠀⠐⠀⠈⠉⠙⠛⠉⠅⠀⠁⠠⠊⠄⡀⠀⠀⠤⠟⠛⠁⠀⡄
⠀⠳⡀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣀⠘⠐⢿⡦⠀⠈⠢⡀⠀⠀⠀⠀⠃⢰
⠀⠀⠀⠐⠂⠈⠁⠀⠀⠀⢠⡀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢧⡔⠀⠀⠈⡄⠀⠓⣻⡀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠊⡴⠁⠀⠀⡯⠀⠀⠀⠠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠂⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠢⠖⠉⠙⢦
⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣤⠤⠤⠶⠾⠦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢉⡒⠃⠒⢤⠀⠀⠈⠑⣦
⠀⠀⠀⢀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠇⠀⠀⠀⠸⡃⢠⡀
⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⢀⣀⡴⠂⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡍⠁⢀⠀⠀⠙⢦⣀
⠀⠀⠀⢡⠔⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⠵⢦⡀⠀⠈⠳⣤⡀
⠀⠀⠀⢦⠀⠀⢀⡤⢊⠀⠀⠀⠹⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠙⢦⣄⠀⠀⠙
⠀⠀⠀⠀⠙⠛⠁⠀⠸⡀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠓⠦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠙⣆⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠘⣆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢦⡄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⡤⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠈⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣄⣀⣀⣀⣀⣠⣤⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡀⠀⠀⣼⠀⠀⠀⣀⡠⠒⠉⠀⠉⢳⡀⡀⢀⡀⠤⢄
⠀⠀⣀⠠⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠠⠠⠄⠀⠀⠀⠉⠀⠀⢀⡨⠀⢠⠊⠀⡡⠤⠤⡀⠀⠈⠁⠈⡇⠀⠀⠀⠗⠢⡀
⠀⠉⠉⠁⠀⠀⠂⠀⠌⡑⠂⠤⢀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠪⣀⠠⠔⠀⠀⢸⠀⠀⠓⠀⠀⠹⠀⠀⠀⠴⠃⢀⣀⠀⠀⠀⠚⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⢀⠈⠒⠠⠄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣣⣀⠠⠄⠐⠁⠀⠀⠀⢀⠀⠈⡸⠀⠀⠐⠀⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣐⠀⠀⠄⢀⡀⠉⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠈⠉⠉⠀⣠⣤⡤⠶⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠒⠤⠤⢌⣁⡀⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣒⡒⠒⠬⣔⠄⡀⠀⠀⢠⠀⠀⠀⠈⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠢⡀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⡶⠶⠦⠤⣄⡀⢀⣸⡄⠀⠈⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠐⢤⡀⢀⠤⠶⠦⣄⡀⠀⠀⢱⠄⠀⠖⠀⡹⠁⠈⠳⠤⠖⠁⠠⣄⠀⠀⠈⡧⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠤⠉⠒⠀⠀⠒⢿⡀⠀⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⢀⡴⠄⠊⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠴⠦⠀⠤⠞⠉⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠐⠂⠐⠀⠂⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠰⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⡀⠀⣄⡠⠀⢤⢖⠀⢄⠀⠀⢰⣷⣀⣰⣸⡍⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠉⠚⢩⡅⠀⣤⠀⠑⠈⢛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⡴⠋⠉⠒⠛⠛⠃⠀⠀⢨⡿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣄⠈⢂⠀⡀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢁⠀⠒⠂⠐⠂⠒⠒⠢⡈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢉⡷⠲⢀⠿⡿⢀⣶⣐⡖⠲⢄⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱⠸⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢈⢠⡾⠆⣀⣀⠀⡇⠀⠀⣣⣠⡝⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⣘⡇⠀⠓⠈⠉⠉⠙⡀⠀⠀⠀⠂⢀⠀⣀⠀⠉⠢⡀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⡀⢀⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠉⠰⡮⠀⠀⠀⠀⠀⠉⠑⠀⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠈⠁⠋⠁⠀⠀⡌⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠠⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠈⠀⠀⣠⠎⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠶⠦⠤⠐⠒⠚⠉⠉⠀⠻⣄⡀⠀⠀⠠⡄⠀⣠⣤⠞⠋⠀⠄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⢳⠠⠉⠀⠀⠀⠁⢠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠴⠒⢦⠀⣧⠰⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠸⡆⢿⣆⠀⠘⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣀⣀⣹⡀⢻⣆⣀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠉⠉⠀⠀⠬⠣⢀⡿⠙⢧⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⠉⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⠒⠈⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠓⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠃⠉⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠀⠐⠘⠂⠀⠀⠀⠀⠆⠀⠘⠓⠀⠐⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢎
⠁⠀⠀⠀⠀⠈⠓⠋⠉⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⠋⠁⠈⠀⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢤⡤⢀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣤⡤⠄⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠙⣷⣿⣂⣀⣀⠹⡆⠋⠀⠁⠀⠉⠁⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢹⡄⠀⠀⠀⠀⠀⠂⠈⠁⠘⠉⠓⠐⠂⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠁⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠀⠀⠀⠀⠀⡀

⡀
⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡜⡰⠃⡠⠊⠁⠀⠀⠀⣈⠩⠝⠉⢆⠀⠙⢎⠳⣥⠀⠀⢀⣠⢏⢦⠀⢠⡔⣞⣠⡤⡄⡠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡜⡰⢁⡜⣠⠔⠊⠉⠉⠁⠀⠀⠀⠀⠈⢆⠀⠀⢣⠸⡆⠀⠀⠀⠈⠈⠀⠀⠀⠉⠈⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⢠⠃⠎⡴⠷⠦⣤⡀⠀⠀⠀⠀⣠⠤⠶⠬⣄⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠇⡇⣰⢮⣴⣶⣶⣤⡉⠀⠀⠀⠀⣡⣴⢶⣦⣬⢆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢸⢠⡑⡄⠈⣄⣻⣿⠛⠀⠀⠀⢀⠻⣿⢂⡽⠁⠠⢣⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡌⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡆⡸⠹⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⠈⢧⠀⠀⠠⣄⡀⠑⠀⠘⠀⣠⠄⠀⠀⡸⢠⠃⢸⠃
⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡂⠀⡗⢄⠀⠈⠣⣁⠀⢂⠜⠁⠀⢀⠜⠗⠁⢀⠎
⠶⠈⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢣⠀⠇⠀⣑⢤⡀⠀⠉⠀⢀⡠⢺⠁⠀⢀⠴⠃
⣲⠒⠃⠀⠸⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠚⠣⣴⠊⠀⣰⠊⠉⠀⠈⠉⡅⠸⠒⠂⠁
⠹⠞⢲⠊⠁⠀⠈⠷⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⣇⠀⠀⡥⠔⣯⣃⠀⠀⠀⠀⠀⠈⡧⠤⠄⣀⣀
⠀⠀⠀⡇⠀⠀⠀⠈⡄⢑⡤⠔⠖⠈⠉⠀⠀⠀⡰⠉⢦⠎⠀⢀⠎⠀⠑⠂⠤⠄⠒⠉⠀⠀⠀⠀⠀⠈⠉⠒⠠⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⡧⠔⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⢆⠂⠂⢧⡀⠀⠀⠠⠛⠢⡘⠀⢀⠔⠁⡀⠀⠠⠜⣡⠆⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡹
⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⡟⠢⣀⠃⠀⠀⠙⣴⠃⠀⠠⠙⣦⠀⠘⡏⠀⠠⠍⠄⠀⠀⠀⠀⠀⢀⠞⠁⠀⠙⢦⡀
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠣⠤⠌⣹⠦⠀⢰⢃⠀⣀⡢⣄⠀⠓⡤⠦⡞⠀⢀⣀⡄⠀⠀⠀⢀⠏⠳⡄⠀⠀⠀⠙⢦⡀
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢃⢨⠀⠈⠝⠋⠉⠩⠇⠐⢟⠉⠉⠂⠉⠉⠉⠁⡟⠀⠀⠈⢦⡀⠀⠀⠀⠙⢄
⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⡀⠈⠾⡀⠀⠠⠶⠾⠁⢀⣅⠀⢻⡶⡆⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⠢⣀⢀⣀⠀⢧
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢈⠚⠀⡱⠀⠀⠀⠀⠀⠇⡹⠄⠀⠉⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⡸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠠⣀⡔⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠔⠁
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣻⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠀⣀⣴⠃⠀⠀⢀⡠⠒⠁
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⠁⣤⠏⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠩⡁⣀⠞⠁⢀⡱⠔⠊⠁
⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⡄⠘⡏⠹⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⡔⠃⠀⢀⠂
⠀⠀⠀⠀⠈⠑⠒⠂⠀⠐⠒⠉⠁⠀⠀⠀⠙⠶⠓⠁⠀⣠⠮⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠫⢄⣂⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠈⠑⠦⡀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢛⣿⠿⠛⠛⠛⠛⠿⢿⣛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣠⣤⠀⠀⢀⣤⡄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠉⢀⣶⣌⡙⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠈⠛⠛⠿⠷⠄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠉⠐⠃⠘⠀⠁⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣬⣷⡄⠀⠀⠀⠀⠀⠀⠀⣠⠿⠂⠀⠀⠀⠀⠀⠀⠐⠻⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣤⡴⠆
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠛⠳⠶⣤⣀⡀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢀⣀⣠⣤⡴⠶⠚⠛⠻⢧⣤⣀⡀
⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠷⢤⣄⣀⠀⠀⣠⣶⣶⡄⠀⠀⢠⠶⢿⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠉⠀⠀⠻⣿⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡶⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠿⣿⡿⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣤⣤⣤⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠰⣾⠃⠀⣰⡀⠀⣠⠀⠀⡀
⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷⡄⢠⣤⣜⣿⣡⡘⠿⢠⠟⠁⠞⢠⣂⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣏⣃⣹⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠈⠉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠃⢀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⠂⠐⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⡜⡀⢸⣀⠜⣣⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⣭⡙⠿⣿⡟⣿⡙⡏
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠘⣡⠞⡡⡜⠃⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⡿⠛⠃⠀⠘⢷⡶⢏⣣⣴⣿⣿
⠀⠀⠀⢸⡏⠛⠿⠿⠻⡿⠟⠻⠿⠀⠀⠱⢎⣱⠄⠀⢼⠻⠛⠿⠛⠛⠛⠛⠛⠟⠻⠿⡇⣿⠟⢿⠟⠟⠃⠀⢠⠀⠀⢸⣗⠀⠀⠀⢰⡀⠳⠈⠿⠿⠛⣿
⠀⠀⠀⢸⡁⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠤⠤⠄⠠⠭⠤⠀⠼⠿⠄⠀⢤⠤⡥⠤⠀⠤⠤⠤⠯
⠀⠀⠀⠈⠁⠘⠀⠀⠂⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠉⠀⠁⠈⠉⠈⠉⠉⠉⠉⠀⠀⠀⠁⢰⠛⠛⠛⠓⠒⠒⠒⠂⠀⠐⣶⠂⠀⠀⠀⠀⠐⠻⠓⠒⠒⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢺⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡈⠉⢉⣉⠙⠛⠛⠀⠀⣀⣀⠀⡀⢸⣦⣤⣤⣤⣤⡀⠀⢠⣤⣤⣿⣇⣀⣤⠀⠀⢴⣶⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢟⣿⠉⣹⣿⣿⣿⣽⣿⠓⠿⠛⢿⣿⣿⣿⣿⡿⠏⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡋⠉⠀⠉⢉⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⣩⣧⣄⣀⣀⡀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠉⠀⠠⡄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠚⠋⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡧⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⢶⣶⡔⠲⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣿⣶⣶⣿
⠀⠀⠀⢸⣿⣶⣄⣰⣶⣶⣶⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠈⠈⠉⠈⠉⠻⣍⣁⠀⠈⠉⢹⣿⣿⠋
⠀⠀⠀⢸⣿⣿⣿⣻⣉⣉⣛⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠂
⠀⠀⠀⢀⣉⣉⣉⣉⣩⣭⣭⡅⠀⠀⠀⠀⢸⣧⠀⠀⠀⣀⡙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤
⠀⠀⠀⠸⣿⣿⣿⣿⣟⢠⣴⣆⠀⠀⠀⠀⢸⡿⠤⠀⢋⣁⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛
⠀⠀⠀⢸⣿⡏⠛⣿⣟⣽⣿⣿⡄⠀⠂⢉⣠⣤⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠈⠈⣷⣢⡿⠟⠛⣉⣡⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠐⠒⣉⣠⣤⣖⣚⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⣀⠀⢀⠀⠀⠀⠀⠀⣀⠀⡾⣯⣥⣤⣤⣤⣤⣤⢽⡇⠀⠘⠀⠈⢈⡉⠁⢄⣀⣀⣀⣀
⠀⠀⠀⢰⡿⠛⠛⠛⠛⠛⢿⣿⣿⠟⡿⠟⣿⢿⣿⣿⣶⣾⣶⣾⠿⠟⠻⠿⠿⢶⣷⣿⣿⣿⣳⣋⣿⣉⣿⢉⡏⡟⠈⠁⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠉⣿
⠀⠀⠀⢘⣛⣛⣛⣋⣿⣿⣿⡿⠃⣀⠝⠀⣷⣼⡟⠛⢛⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠻⠟⠛⠋⠁⠉⠁⠈⠋⠓⢠⣤⠀⢀⡄⠀⢠⠟⣤⡄⢀⣀⣀⣿
⠀⠀⠀⢸⣿⡟⠛⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣨⣇⢀⡶⠛⣴⠀⠀⠙⣿⣿
⠀⠀⠀⢸⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣾⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⢎⠷⢋⣶⠀⠀⢨⣍⣿
⠀⠀⠀⠸⠿⠿⡇⠉⠁⠀⠠⣤⠀⠠⠼⠯⠠⠄⠤⠤⠄⠬⠭⠭⠭⠭⠭⠭⠭⠭⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠚⠓⠛⠓⠒⠒⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣴⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣀⣀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⢸⣥⢄⣠⣤⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⢶⡇⠀⠀⠀⠀⠀⠘⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣰⠃⠘⠠⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠠⠀⠀⠀⠀⢀⡠⠄⠘⠻⢿⣷⡄⠀⣄⡀⢠⠀⢀⡀⠀⠀⠀⠀⣠⡁⣀⣀⣀⣼⠟⣿⣿⣿⣿⣯⣭⣬⣏
⠀⠀⠀⠀⠀⠀⠠⠈⡀⢀⣦⠀⠀⠀⠀⣿⣿⣧⣀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠋⠀⠀⣞⣱⣞⡶⢛⠃⠀⠀⠀⣰⡗⣤⣤⣀⣀⣼⠂⠉⠙⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠋⠁⠈⠛⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡱⢎⡵⣛⠀⠀⠀⣸⣯⣥⣬⣉⣉⣉⣉⠀⠀⠀⠈⢉⣩⡉⠉⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⢿⡟⠛⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡱⢎⡰⣍⠀⠀⢨⡿⠿⡿⣿⣿⠟⠉⠁⠀⢰⠀⠀⠈⠩⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠚⣆⢐⣀⠺⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠑⠋⠄⢠⣟⠀⠀⢀⣤⣤⠀⠀⠀⠀⡶⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠛⠃⠿⠁⠀⠀⠀⠘⠆⣠⠀⠚⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⣀⠀⠀⠀⠀⠈⠁⣽⣇⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠃⠀⠀⠀⠛⠓⠀⠀⠀⠀⠀⠿⠿
⠀⠀⠀⠰⠦⠤⠾⠷⠿⠦⠿⠦⠤⠤⠤⠤⠤⠿⠷⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⠄⠤⠀⠠⠤⠤⠀⠄⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣼⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠻⣅⠀⠀⠈⠁⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣠⠴⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⣿⠀⠀⠀⠉⠛⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠘⢧⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠲⠶⠶⠶⠶⠶⠖⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠚⠛⠉⠉
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠈⠀⠀⠀⠀⠀⣰⠲⢆⣴⡏⠉⣋⣿⣄⣰⡞⠃⠀⢀⡀⢸⡏⠸⠁⠿⠜⠛⠓⠒⠰⡶⠒⠁⠀⠻⡄⠀⠀⡤⠶⣆⡇⠀⠀⠀⠀⠀⠀⣠⣌⣋⢻⣶⠶⠒⢿⡀
⠀⠀⠀⠀⣀⡴⠋⢙⠟⠀⠑⠢⠋⠀⢿⠏⢡⡿⢶⡋⠙⡄⠙⣦⠀⠀⠀⡀⠀⠀⠀⢴⣀⡤⠄⠷⠿⣄⠘⣶⠀⣨⣟⢶⣦⠀⢀⣤⢚⠁⠉⠉⠈⠱⢖⠈⠉⠘⠃
⠀⠉⠐⠶⣭⣔⠒⠂⠶⠂⠰⠤⠛⠄⠀⠛⠿⠁⢸⠁⠀⡇⠀⠘⢷⠀⠞⠱⣶⡶⢦⠀⠟⠀⠀⠀⠘⠻⡛⢿⠰⠛⠉⡙⢟⠀⠀⠁⠈⠑⠰⠦⣾⡶⠾⠟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠒⠲⠖⠂⠈⠀⢠⡶⠀⡄⢰⠀⠀⠐⢿⠀⠀⢉⠀⢸⢇⡀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⢈⠂⡤⠀⠀⢀⡄⠎⠀⠓⠒⠂⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠙⡆⠀⠀⠀⠀⠀⢏⣄⠀⢧⡄⢧⣀⡴⠿⠀⠀⠈⢧⡈⠘⠁⠀⠀⠀⠰⠮⠡⠴⠦⠀⠠⠔⠊⠈⢰⡞⠉⠁⠀⠀⠀⠀⠀⠀⢀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢘⡧⡴⠀⠀⠀⠀⢸⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠚⠛⢢⠀⢀⠤⠆⠈
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁⢧⡀⠀⠀⠀⠀⠈⠻⣤⠸⠋⠀⣇⠀⠀⠀⢀⡜⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣠⠄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠴⣦⡴⠬⠓⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢠⠌⠑⡆
⠀⡶⠛⣋⠠⠴⣷⠀⠀⠀⠀⠰⡀⠀⢱⠦⠤⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣧⡐⠀⠶⢳⡆⠀⢀⣀⠀⠀⣀⣀⠀⠄
⠈⠀⢈⣠⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠉⠉⠉⠁⠐⠛⠡⠶⣤⡶⠉⢠⡶⠔⠒⠒⠒⠶⢦⡴⠏⠉⠉⠐⢶⣿⣠⡞⠁⢀⣿⠉
⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠙⣤⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠛⢄⠀⠀⠀⠀⢠⠑⠆⡑⢦⣠⠿⣹⡟⠷⠚⠉⠀⠀⠀⡀⠀⠀⠀⠤⠀⠤⠀⠒
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⢳⣄⠐⠶⢶⣴⣞⠛⠈⠁⠀⠈⠙⠻⠦⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅⠀⣠⣄⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⢁⠀⠀⠈⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠛⠂⢤⣀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠙⠒⠢⠤⠤⠤⠤⠤⠤⠔⠒⠉⠘⠀⠀⠀⣠⡔⢾⠁⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⢰⠷
⠉⠀⠀⠉⠉⠉⠐⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠚⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠐⠀⠀⠂⣠⠟⠛⠁⠈⢦⣧⠀⡄
⠀⢀⡄⠀⠀⠀⠀⠀⠀⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠒⠀⠀⣠⡊⡤⠖⠈⠁⠘⠛⠒⠶⢤⡄⠀⠀⠀⠀⠀⢂⠠⠤⣄⠀⣠⠞⠃⠁⠀⠀⠀⠀⠞⣹⢞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠀⠑⢆⠀⠀⠀⠀⠀⠀⣰⠟⠃⠀⠀⠀⣀⣀⢀⡀⠀⠀⣷⠤⣀⠀⣠⠶⠣⠶⠦⠸⠶⠷⠒⠛⠲⡀⠀⠀⠀⠐⠋⡘
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⢁⡄⠀⠻⣄⠀⠀⠀⢠⡍⠀⠀⠀⠐⠋⡤⠀⠀⠈⠁⠀⢻⡀⠄⠉⠀⠀⠐⠀⠀⠀⠀⠐⠠⠀⢠⣧⡀⢶⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢀⣀⣌⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⢴⡆⡀⠀⠀⠀⠀⣠⠅⣄⠸⡀⠀⢀⠴⡄⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⢢⡄⡼⠟⠉⣤⠀⠀⠀⠀⠀⢠⣀⡀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢷⡌⢀⠀⠋⠀⠀⣠⠏⠀⠀⠘⠃⢰⠃⣀⠜
⠉⠀⠀⠀⢠⡞⠀⠀⠈⠁⢹⡄⠀⠀⠈⠁⢀⡟⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠛⠋⡰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠊⠁⠀⠀⠀⢀⡴⠋⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠹⢆⣀⣀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⣠⠔⠂⠾⠀⢠⠇
⠀⠠⠄⠠⠰⠆⠀⠀⠀⠀⠰⠃⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣬⣁⠤⠔⠛⠚⠉
⠀⠀⠁⠀⠀⠀⣀⣤⣤⣀⠀⢁⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⢰⠎⠁⠀⠀⠀⠀⣀⣄⡀⠀⠀⣀⣀⢀
⠀⠀⠘⠁⠀⠂⣁⠤⠀⣈⡑⠋⡀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⣶⡾⣄⣀⡀⠀⠀⢀⠀⠻⠀⠀⠀⠀⠀⠀⠛⢻⣗⣤⠚⠉⠀⠈
⠀⠠⠄⣄⢀⣴⣿⣇⠀⠉⠀⠠⠈⠀⠀⠈⣤⠀⡀⠀⠰⠄⠀⠀⠀⠦⠀⡀⢧⠀⠀⠀⠀⠀⠀⠀⢇⡼⠉⠙⢦⣠⠊⠀⠘⡂⠀⠀⠀⠀⠀⣠⡜⢛⣌⣦
⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⠁⠀⠋⢠⡿⠀⠀⠀⠀⠎⠉⠉⠡⠀⠀⢀⣼⠃⠀⠀⠀⠀⣦⢤⣼⣇⠀⠀⠀⠁⠀⠀⡐⠀⠀⠀⠀⠀⠰⠋⠀⠈⠻⣮⠱⡀
⢠⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠐⠶⠚⡈⠢⠽⣄⢀⠠⠄⠀⡀⠠⠀⠀⡌⠁⡄⠀⠠⠄⠀⠐⢢⢀⡟⠀⠀⠀⢷⠒⠛⠕⠢⣄⠀⠀⠀⡆⠀⠀⠀⠀⠈⢧⡹⡄
⠀⠁⠰⠀⢿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⢤⠤⠾⠁⠀⠀⠀⠀⠁⠀⠀⡏⠀⠀⡇⠀⠀⠀⡆⣠⣸⡜⠀⠄⠀⠀⠈⢆⠀⡀⡀⣨⣄⠀⣶⠀⠐⠀⠀⠀⠀⠀⢱⣻
⠀⢤⡀⠀⠸⡟⠋⠉⠀⠈⠑⠶⢤⣤⣤⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠂⠀⠉⠁⢡⡞⠀⠒⠀⠂⠀⠀⠈⢿⠉⠉⠀⠀⠈⠳⡄⢈⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠙⣆⣀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡴⠋⠅⠀⠀⢀⠀⠀⠀⡄⠀⡆⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠠⠀⣀⣀⠀⡀⠰⢇⠀⠀⠀⠀⣴⢋⣀⣤⠀⠀⠀⠘⡄⠀⠀⢱⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠶⠂⠀⠰⡏⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢧⡀⠀⣸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠉⠀⠄⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢆⠀⠀⠈⠙⠲⡄⠀⠀⠀⠀⠀⠰⡀⠀⠑⢄⡀⠀⢀⠏⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠈⣻⠖⠶⠶⠞⠃⠀⠀⠀⠀⠀⢠⣵⣆⡀⠀⢀⡴⠋⠀⠀⠀⣠⠴⠊⠉⣁⠾⠷⡄⠀⠀⠈⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣁⡙⢣⡞⠁⠀⠀⡠⠊⠀⠔⠀⠈⣏⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠾⠁⠀⠀⠄⢁⠂⠀⠀⠀⠀⠈⠛⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⢀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡤⢀⡄⠠⡔⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠓⠒⠒⠒⠂⠸⢄⢸⣄⠀⣸⣤⡡⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⣃⠀⠀⠀⠀⢀⣀⣮⣀⡉⠛⠁⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠈⠉⠉⠉⠁⠁⣀⡬⠶⠈⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢐⢲⣾⣿⣿⣥⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠘⣄⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠐⣄⠐⠃⣠⠀⠀⠀⠄⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢹⠀⠉⠐⠛⡄⠠⠟⢲⣂⠀⠤⢄⠉⠉⣠⠔⠊⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠦⣄⠀⠚⠇⠀⠘⣧⣀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⣠⠴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⣧⠀⠀⣀⡴⣇⢈⡗⢂⠔⠠⠄⢤⠐⠢⠀⠊⠁⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⣷⢀⠾⠉⠀⠙⠢⠁⡾⠁⠀⠀⠈⠀⢠⡀⠴⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⢠⠁⠀⠨⠀⠀⡀⠄⢧⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢦⣀⠄⠀⠀⠀⠀⡼⠀⢸⠆⠈⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣧⡀⣞⠀⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣿⠁⡿⠀⠐⠰⠚⠀⠀⢹⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢹⣠⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣨⠄⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡼⢹⠇⠀⠀⠀⠀⠠⠚⢘⡇⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠉⠉⡀⠚⠃⠸⢀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠸⠀⡷⠀⠀⡀⠠⢤⡀⠐⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿⡀⠀⣣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢻⠀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠐⠎⠡⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣤⡴⠞⠁⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠰⢦⣤⣀⣀⣤⠴⠂⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠸⡇⡘⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢱⡇⠀⠀⠀⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⡄⠀⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠻⣦⡀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠈⠙⠓⠶⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⣧⡀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⢀⣼⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣏⠙⣄⠀⠀⠀⠀⠀⢟⣆⠀⠀⠹⣾⣷⠏⠀⠀⣰⡿⠀⠀⠀⠀⠀⣠⠋⣹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣦⠤⡀⠀⠀⠘⡜⣆⠀⠀⢸⡇⠀⠀⣰⢣⠃⠀⠀⢀⠤⣴⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡻⡮⣝⢒⡳⢷⡘⣕⢦⣸⣇⡤⣪⢇⡾⢖⡒⣪⢵⢟⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢳⢙⠮⣣⣀⠤⢧⢹⠓⢼⡧⠚⡏⡼⠤⣀⣜⠵⡋⡎⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢷⣒⠦⢼⣎⠀⡜⠛⢄⠀⢣⣧⣸⣇⣼⡼⠀⡠⠛⢣⠀⣱⡧⠤⣒⡾⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⢬⣍⣓⠤⣄⠻⡦⣿⡜⢣⣿⢴⠟⣠⠤⣚⣩⡵⡲⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣳⣦⣀⣀⣈⣳⣙⣄⣉⣻⣛⠿⢮⡷⢾⡵⠿⣛⣟⣉⣠⣋⣞⣁⣀⣀⣴⣞⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡽⠟⠉⠉⢉⡽⣩⠋⣉⣽⣭⣷⢞⡷⢾⡳⣾⣭⣯⣉⠙⢍⢯⡉⠉⠉⠻⢯⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠮⢞⣋⡭⠒⠋⣴⠗⣿⢣⡞⣿⠺⣦⠙⠒⢭⣙⡳⠵⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡾⠭⠒⣺⢏⠀⢣⣤⠊⠀⡜⡟⢹⡏⢻⢳⠀⠑⣤⡜⠀⡹⣗⠒⠭⢷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡼⢨⢖⡝⠉⠒⡞⣸⡤⢺⡗⢤⣃⢳⠒⠉⢫⡲⣅⢧⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣵⢗⣫⠬⡵⡾⢡⡩⠞⢹⡏⠓⢝⡌⢷⠮⠥⣝⡺⣮⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠟⠒⠁⠀⠀⢰⢣⠏⠀⠀⢸⡇⠀⠀⠹⡜⡄⠀⠀⠈⠒⠻⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣏⣠⠋⠀⠀⠀⠀⠀⣾⠏⠀⠀⣰⢿⡿⣆⠀⠀⠹⣷⠀⠀⠀⠀⠀⠙⣄⣹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⡿⠁⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠀⠈⢿⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⢿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠁⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠙⢿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⠀⠐⢀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣷⡴⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣿⠟⠋⢰⠏⡸⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠋⠀⠀⠀⠀⡴⣥⣾⣦⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡄⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣻⣿⣿⣷⣄⠀⠀⠀⢠⣿⣿⣿⣧⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠂⠀⠘⠻⣿⣿⣿⣿⢃⠀⠀⢸⣿⣿⣿⣿⡧⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⠉⠀⠁⠈⠀⠀⠀⣿⣿⣟⣷⣤⣿⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠏⠠⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣯⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡄⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⠋⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⡄⡀⣾⣧⣿⣿⡇⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣟⠻⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⠛⠻⠿⠟⡷⢶⣤⣀⣀⠀⡈⠙⠛⠿⠿⠏⢉⣤⣿⡿⠿⣿⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⡈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⣻⣟⣋⣹⣿⣷⣶⣶⣆⡀⠈⠉⠙⠋⠁⢰⣿⠿⡿⠿⠿⠿⠛⠿⣿⣿⣿⣿⣿⣿⡉⢲⣊⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣿⣿⣿⣿⣿⣦⠤
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠶⠶⠤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠛⢋⠉⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠂⠄⠀⠀⢀⠀⣿⢀⢀⠀⠀⣀⣤⣶⣿⠿⠛⠁⠀⠀⠸⣿⡀⠀⠀⠀⠀⠄⠀⠀⣠⠿⡔⠠⢤⣶⣔
⠀⠐⠂⠀⠀⠠⠀⡀⢱⠇⠉⣠⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢻⣧⠀⠅⠀⠀⠀⣀⣴⠋⠀⠈⢷⠀⢹⠀⡀
⠐⠀⠀⠀⠁⢀⣼⠛⡄⠀⣾⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣇⠀⠁⠀⡠⢚⡁⠀⠀⠐⠀⠑⠪⠀⠠
⢀⠀⠀⠀⢰⡿⠃⠀⠘⠦⡘⢿⣿⣆⠀⠀⠀⠀⠀⢀⠀⠀⣐⣶⣶⣿⣿⣀⠂⠈⠉⠽⠧⣶⡆⣶⣚⠁⠑⠂
⠀⠀⠀⠠⠾⡶⠀⠀⠀⣤⠤⠂⠘⣿⡆⠀⠀⣀⣴⣼⣗⣴⣿⣿⡿⠿⠿⣛⣛⣛⣻⣿⣶⣤⡀⢝⠙⠁⠄⠀⠈
⠀⠄⠘⣷⣙⣁⣶⠀⡔⠫⠂⠀⠀⢸⣿⣠⣼⣿⡿⠿⣛⣩⣥⢶⣾⣛⣿⠽⠿⠿⠟⠛⢻⣿⣷⠘⠏⠀⡀⠂
⠐⠀⠀⠁⠀⣴⣝⡀⣯⠢⢀⣠⣴⣾⡿⢟⣫⣵⢶⣻⡯⠷⠛⠋⠉⠀⠀⣀⣀⣤⠤⠤⣾⣿⡏⠀⠀⠆⠱
⠘⠀⠀⠀⠀⢈⣿⠀⠃⣴⣿⠟⣫⣵⢾⣿⡿⠞⠉⠁⢀⣀⡤⠴⠒⠛⠉⢁⣀⣠⣤⣴⣿⡏⠀⠠⠼⠀⡤⠑
⠀⠀⠀⠛⠉⠁⠸⠀⠀⣿⣏⣼⢫⡿⠋⠁⢀⣠⠴⣚⡩⠅⠀⠀⠀⠀⠈⡉⠁⠀⠀⠀⠹⣿⡄⣀⠗⠀⣈⠳⠐
⠀⠀⠀⠀⠀⡢⠠⡀⣀⠈⠻⣿⣿⣷⣤⣞⣫⡴⠋⠁⣾⣿⠄⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⣿⡇⢐⣀⠠⣄⠋
⠀⠀⠠⣠⡼⠀⡐⣤⠏⢶⣂⠄⠙⠻⣿⡏⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⢀⡤⠀⠀⠀⣿⠇⣿⢿⡇⣡⣤⡀
⠈⠀⠀⢰⡿⣼⡅⠁⠀⠀⠋⢲⠀⠀⣿⡇⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⣼⡟⠀⣿⣿⡿⣋⣽⣿
⠀⠀⠐⡿⠁⠀⠭⡉⠁⣴⣿⠈⠀⠀⢸⣿⡤⣄⡀⠀⠀⠙⢿⣶⣶⣶⡾⢟⣁⡤⠤⠔⣿⣷⡶⠟⢻⣾⠟⡡⠈
⠀⠀⠈⣱⣇⠀⠈⠓⠂⣉⣋⣡⣶⣶⣿⣟⠙⢾⠉⠉⡷⠒⣶⠒⠒⠂⠘⡁⠀⣀⡤⠞⠛⠷⢭⣳⢜⣿⡆⠑
⠀⠀⠂⣾⣣⠀⣠⣾⢿⣻⠿⠟⠋⠉⠁⠈⢳⠞⠀⣸⠃⢀⡿⠤⠤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠙⢿⣿⠠⠄
⠀⠉⠃⠀⢃⣼⣿⣟⠉⠀⠀⠀⠄⠀⣠⠖⢁⡠⠶⠿⠴⠾⠦⠤⠤⢤⣤⣴⠶⠶⠶⠶⠚⠛⠛⠛⠉⢻⣷⠈⠄
⠀⠈⠀⣴⣿⡟⠉⠹⣧⡀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣧
⠀⠂⠀⢿⣯⡀⠀⠀⠈⢻⣶⣾⣥⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠠⠐⡌⠻⣿⣦⣤⣴⡿⠋⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⡇
⠐⡀⠀⣸⠃⠀⢍⠉⠁⠐⡀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠰⣾⠋⠀⠀⠀⠉⠒⣆⠙⡀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡄
⢠⢊⣉⣉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠉⠁⠈⢳⡄⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⢆
⠀⠀⠀⠀⠀⢻⡀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠙⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠉⠛⠒⠚⠋⠉⢣
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡠⠞⠀⣈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣇⣀⠀⣀⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡟⠒⠶⠒⡖⠋⠑⠢⠊⡀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡤⠦⠗⠀⡜⠁⠀⢽⠁⠆⠀⢱⠀⠀⢳
⠀⠀⠀⠀⠀⢳⠀⠀⠐⡀⠘⠀⠀⣇⠉⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡜⠀⠀⠀⠀⠱⢄⣀⠼⠳⠤⠔⣁⠴⠂⢸⠀⠁
⠀⠀⢀⠀⠀⠈⢧⠀⠰⡌⣂⣐⠊⠘⠉⣈⡸⠆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠰⡇⠀⠀⠀⢀⠀⢸⡉⠓⠉⠉⠉⠀⢴⠆⢸
⠀⠀⠀⠀⠀⠀⠀⢧⠀⣇⣀⠀⠉⠉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠳⡀⠀⠀⠀⠀⠀⢻⠋⠀⠈⠁⠀⠊⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡼⠳⢤⣀⣀⣀⣀⡗⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣏⠁⠀⠀⠃⠀⠹⠀⠀⡠⠊⠉⠙⢣⡠⠤⠠⢄⠀⠀⢹⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⡄⠀⠀⠀⠀⢸⠁⠀⢇⠀⠁⠀⠘⠀⠀⠂⠀⡆⠀⡼⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢇⠀⠀⠀⠀⣸⠐⠀⠈⠓⠰⡖⠙⢄⡀⣀⠴⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⡜⠇⠀⠀⠈⠁⢒⠖⠈⠀⠀⠀⠀⠀⠈⠁⠀⠐⠒⠂⡘⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⡿⢶⣿⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠞⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠘⣇⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠬⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠤⣀⡧⠤⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠉⠉⣐⡀⠀⠠⠠⠤⠤⣤⡄⠀⢀⠀⠀⠀⠐⠀⠠⠄⠠⠄⠄⠐⡢⢀⠀⠂⠐⠀⠀⣐⡀⢐⠂⠀⠀⠀⠐⠀⠤⢀⣀⣀⠀⠀⠀⢢⡠⠀⢠⠞
⠀⢀⣀⣭⣩⣭⣿⣄⣉⣕⡒⠂⠂⣤⣤⡈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠾⠗⠛⠉⠀⠀⠀⢀⣀⠀⠀⠀⠈⠉⠑⠚⠠⠀⠀⠀⠀⠐⠶
⠀⠾⢽⠟⠷⠿⠿⡿⣿⠿⡿⢤⠀⠀⠀⣵⣠⣄⢠⣤⡄⠚⠛⠂⠰⢦⣄⠀⣀⠀⠀⠀⠀⠘⠻⠛⠉⢠⡄⠀⠀⠉⠁
⠀⢠⣾⣶⣶⡀⠀⠀⠠⠀⠓⠀⣒⡤⡨⠿⣻⣞⣲⣦⣦⢠⣶⣶⣢⣤⠘⢗⠠⡐⢀⠀⠀⠀⣠⡤⣄⠬⣳⣿⡇⣀⢲⣤⣀⣶⣶⠂⠀⢀⣀
⠀⢺⢩⣿⣯⣿⠄⡄⠀⠀⠀⣀⡻⠃⢷⠀⠀⠈⠉⠻⠿⠞⠿⠿⠿⠷⠀⠈⠀⠀⠀⠈⠀⡟⢻⣴⣥⠰⠿⠿⠿⢯⣥⣝⢿⡀⠀⠀⠀⢹⣿⣆⢰⣶⣶⣶
⠀⠸⣾⣿⣿⠘⡋⢱⡀⠰⠿⠷⡦⠤⠤⠾⠖⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⢀⣾⣶⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠙⢷⣄⢰⣮⣿⣿⣏⠛⠻⠿⠤
⠀⠘⣿⣿⡇⠀⢠⡀⢧⣀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⠷⣾⡷⠂⠀⠘⣿⣷⣿⣿⣿⣿⣿⣄⢀⢀⡀⠀⠀⠀⠀⠙⠳⣿⣿⠛⠙⠂
⠀⢤⣿⣿⣤⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣶⣦⡶⠊⠀⣀⠈⣭⣉⣚⣫⣶⠁⠀⡠⠉⣿⣿⣿⣽⣿⢿⣿⣠⣿⠟⠂⠀⠀⠀⠀⠀⠈⢿⡷⢶⣤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠛⠉⠛⠋⠉⠉⣭⣡⡿⠛⢿⣿⣿⣿⡿⠘⠗⠚⠛⠓⠂⠀⠀⠀⠀⠀⠤⣸⡇⠨⡿⠙⠻⠷⣤⡀
⠀⠠⢤⠂⠀⢠⣠⣴⣶⣤⣄⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣤⣄⠀⠁⠒
⠀⢶⡷⠖⢺⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠤⠤⣶⣤⣷⣀⡀⠈⣿⠿⣿⠇⠀⠀⠀⠀⠀⠁⠶⠦⢾⡿⠿⠿⣿⣿⠿⠿⠿⠿⠷⠶⡆
⠀⠈⠁⠀⠿⠛⠛⠛⠉⠁⠉⠁⠀⠀⠀⠸⠷⠿⠟⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠖⠟⠓⠛⠂⡀⠀⠠⢄⠀⠈⠙⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀
⠀⣄⣀⣀⣀⣀⣀⣀⣀⡶⠴⠆⢀⡲⣆⣀⣀⠀⢀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢁⣀⢀⣀⡀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠉⠛⠒⠒⠀⠀⠀⠉⠋⠉⠁⠈⠹⠋⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠓⠦⠊⠉⠙⢦⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠂⠉⠆⠀⢧⠀⠀⠀⠀⠀⠀⣸⠀⠀⢏⠈⠋⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠀⠀⠈⠳⣄⠀⢀⡤⠞⠁⠀⠀⠀⠐⣀⣀⣠⣤⣤⣄⣀⣀
⠀⢀⡠⠤⠒⠒⠋⠉⠉⠉⠉⠉⠉⠙⠛⠒⠦⣄⠀⠈⠛⠉⠀⠀⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⡆⠀⠀⠀⠀⠀⠠⠐⠈⠉⠉⠀⠢⢄⡀⠀⠀⠀⢠⡇⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⣇⠀⠀⠀⠐⠀⠠⣠⠀⠀⠀⢀⠀⠂⠙⢆⠀⠀⣸⠁⡼⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠈⠉⠑⢦⡀⠀⠀⠀⠀⠀⠈
⠸⡄⠀⠀⠀⠀⢶⣀⠠⠤⠤⠄⠂⠉⠙⢮⣆⢀⡏⠀⡇⠀⠀⠀⠀⠀⠠⠃⢀⠠⢀⠀⠀⢀⠠⢀⣛⡠
⠀⠻⣄⠔⠀⠀⠸⡿⠀⠀⠁⠢⣹⡛⠀⡀⢾⡾⠀⠀⡇⠀⠀⠀⠀⡌⡄⠀⢺⣿⡆⠀⠉⠁⣾⣯⡃⢈⢾⡄
⠀⡀⢈⠈⠨⢀⠠⠔⡀⠀⠀⢀⡵⠭⠾⠓⢙⡀⠀⠀⢻⠀⠀⠀⠀⢇⠣⣄⣲⣿⡕⠀⠀⠒⢜⠶⣂⣠⢎⠇⠀⠀⢠
⠀⢆⠀⠡⠀⠀⡠⠊⣉⣉⣉⣉⣉⠢⡀⢀⣏⣠⡂⠀⠈⢧⡀⠀⠀⠸⠉⠒⠊⣁⢓⡶⠖⡚⢀⡉⢀⢘⠇⠀⢀⣴⠋
⠀⠈⠑⢺⢣⡌⠀⠀⠑⠂⠀⣢⠄⡇⢱⡎⠩⡠⠇⠀⠀⠀⠙⠦⣄⣀⣧⠈⠈⢃⠒⠪⠥⠒⡡⠁⠀⣼⣤⠴⠋⠁
⠀⠈⠉⠁⠀⢇⡀⠑⠒⠋⠉⠁⠊⢁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠉⠒⠚⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⡚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠒⠒⢺⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⡀
⠀⠀⠀⠀⠀⠀⢀⠴⢏⠱⢠⠁⢳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⢇⠀⡏⠉⠲⣄
⠀⠀⠀⠀⠀⡰⠁⠀⠑⠭⣳⠔⠋⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡔⠀⠀⠘⡼⠀⠀⠱⡌⢳⡀
⠀⠀⠀⠀⠰⠁⡰⡆⠀⠀⠙⠀⠀⢘⢣⡀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡘⠳⣄⣠⠄⡇⠤⣤⠴⠉⢆⢳⡀
⠀⠀⠀⠀⠃⢰⠁⢰⠀⠀⠀⡀⠀⢸⠀⢳⢈⡆⠀⠀⠀⠀⠀⠀⢰⠊⠫⢳⠀⠀⣧⢰⠀⠀⡏⠀⠀⡌⠙⠳⡄
⠀⠀⠀⠀⡍⠘⠢⢸⠀⠀⠀⠀⠀⢸⠀⠎⡘⢣⠀⠀⠀⠀⠀⠀⡜⠉⠉⠘⠢⣠⡃⢘⠀⢀⣹⣄⠔⡛⠉⠙⣇
⠀⠀⠀⠘⣄⢸⡑⠺⣀⣀⣰⡀⣀⡼⠀⠋⣀⡴⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⡆⡹⠉⠉⡭⡉⠉⣿⣼⠃⠀⠀⢸
⠀⠀⠀⠀⠈⠉⠁⠀⢻⠀⡄⢢⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⣰⠏⢳⠀⢸⠀⢳⠀⠃⠸⢄⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡇⠈⡆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡎⠀⢸⡀⢸⠀⣸⣄
⠀⠂⠀⠀⠀⠠⠆⠉⠙⠶⠇⠐⠷⠭⠍⠹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠤⠌⠙⢚⡇⢾⡙⠒⠒⠛⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣭⣿⡀
⠀⠀⠀⡘⠛⠻⠿⣿⣿⣤⣾⣽⣿⢏⣤⠖⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠂⠀⠀⠀⠨⣿⣿⣿⣆
⠀⠀⠀⠙⠻⠷⣦⣶⢟⣿⠿⠛⠋⠉⠀⠀⠀⠀⠠⢉⣍⣁⡒⠛⣀⣘⣂⡀⠀⠀⠀⠀⠀⠘⢼⡿⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠒⠄⠀⠐⠶⠤⠀⠀⣀⠒⠚⠂⠀⠬⠉⢛⣀⠀⠀⠀⢈⡖⣫⣎
⠀⠀⠀⢠⡞⠉⠀⠀⢽⣿⣿⣿⣷⡄⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢇
⠀⠀⠀⠘⡂⠀⠀⡀⠛⠋⠙⠋⣉⣇⠀⠀⠀⠀⠀⠀⢄⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢴⣾⢿⣆
⠀⠀⠀⠸⣿⣧⣴⣶⣷⣶⣿⣿⡿⠷⠀⢀⣀⣀⡀⢬⠐⢌⠀⢅⠠⣍⠑⢌⠁⣤⠳⠀⠀⣀⣀⠀⠀⠀⢹⠛
⠀⠀⠀⠀⠙⠯⣶⣦⣾⣿⣽⠿⠋⠀⠀⠀⠀⠀⠐⠂⠐⠄⠀⠀⠀⢀⡀⠀⠀⠈⠀⠬⠀⠋⠁⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢠⣆⠻⢷⣆⡈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠐⠒⠒⠦⠀⠀⠀⠀⠠⡴⠖⢒⡂⠤
⠀⠀⠀⠘⣿⣦⡈⠻⢿⣦⣄⡄⠉⠲⢤⡀⠀⠀⢀⣾⠟⠁⠠⣄⡠⠾⢷⣷⡄⠀⠀⠀⢸⡇⠀⠈⢻⣿⣶⠄
⠀⠀⠀⠀⠘⢿⣿⣦⣄⠉⠛⠿⣿⣷⣈⣻⠀⠀⠼⣿⣄⣀⡄⢈⣦⠀⠀⣽⡆⠀⠀⠀⠀⢻⡄⠀⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣶⣶⣬⣿⠛⠀⠀⠘⢮⣏⣩⠄⠸⢋⣠⡼⠋⠀⠀⠀⠀⠀⠐⠧⢈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠴⠶⠒⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⢳⢏⣠⡄⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠞⣉⠁⠀⢀⣤⣉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⢀⡎⠈⡇⢠⡏⠀⠘⡆⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣠⠖⠒⠚⢿⣆⡇⢸⣦⣄⡠⠟⠋⠉⠁⠀⠈⠉⠂⠀⠀⠀⠐⠋⠉⠀⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠉⠀⣇⠀⠀⠀⠀⠩⠤⢬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠘⢦⣀⡀⠠⢤⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠒⠀⠲⠤⠤⠀⠄⠀⡀⣾⣀⡀⠀⠀⣰⠛⠓⠚⠛⠃⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⣉⣡⡽⢀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠤⠤⢶⡀⠈⠁⠈⠳⠶⠛⠦⣤⣀⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⢤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠀⠀⠀⣀⡤⠟⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠶⠛⠁⠀⠈⠙⠲⠶⠒⠛⠉⠀⠀⠈⠓⠶⠤⠤⠶⠋⠁

⠀⢠⠊⡽
⠀⢸⠀⡇
⠀⠘⡆⢧
⠀⠀⠘⢎⠢⠄⢀⣀⣀⠤⠄⣒⣒⣒⡒⠒⠀⠤⢀⣀
⠀⠀⠀⠀⢱⠀⠉⠛⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠘⣆⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡒⠝⡻⡸
⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⠒⠋⠉⠁⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢭⡒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡯⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⣍⣒⡒⠤⠤⢤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⡤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉

⣿⣿⣿⣿⢿⣿⠗⠀⠔⢦⢱⣏⢀⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢧⣤⠓⠰⠋⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠟⠛⠙⠻⠉⣓⡋⠈⣓⠛⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠽⠀⠀⡀⠠⠀⠁⠀⠐⣺⢿⣿⣦⣮⣿⣿⣾⣏⣿⠛⠉⠩⠹⡷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟
⠀⠀⠈⠁⠐⠁⠀⠈⠉⠁⠀⠈⠙⠛⠙⠃⠋⠉⠉⠀⠀⠀⠀⢀⣶⡟⠻⢛⠙⢯⣿⡛⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠁⣤
⠀⠀⣀⠀⢴⡀⣀⡀⠀⠀⣆⠠⣤⠤⠜⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣞⣿⣏⣀⣶⣥⣼⣿⣿⣿⣿⣿⣀⣀⣀⡀⣠⡄⠀⠀⣿
⠒⠂⠒⢀⣾⣶⣤⣤⣼⡤⢼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠀⠀⢠⣿
⢾⠀⢚⣙⣛⣿⣿⣟⠛⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣿⣿⣿⣿⣿⣯⠙⠋⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣰⡛⣿
⣿⣶⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠟⢛⣿⡟⣿⣿⡿⣛⣿⠿⠻⢿⡿⠶⣶⣿⣿⠿⠓⠀⠀⠀⣠⣾⣿⣿⣿
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⠾⠿⠋⠍⠃⠀⡸⣿⢁⡐⠿⠋⠠⣬⠉⠘⠉⠱⣄⡀⢹⣿⡿⢶⡦⠾⠿⣿⣿⣿⠿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⢸⣿⢇⡀⠀⠁⠀⢸⣦⠀⢐⠀⠙⠂⢠⣄⡀⠄⢶⣦⡐⣺⣿⣿⠿⠿⠧⢤⢤⠴⠋⠉
⠀⠀⠀⣀⠠⣀⡀⠀⡀⠀⠀⠀⠉⠈⠀⠀⠶⠗⠚⠛⠀⠈⢳⣄⡉⣸⣿⣷⣦⣽⠿⠛⠛⢉⢠⣴⣤⣴⠦⢖⡀⠉
⣀⣀⣀⠀⣦⣤⡄⠀⠄⠀⠀⢀⣠⣾⣦⡄⠀⠀⠀⠀⠀⠀⢠⣯⣩⡍⠉⠀⠀⠀⠀⢀⡀⢀⡀⠀⠐⠶⠿⣿⣷⣶⣴⣶⣶⣄⣦
⣿⣿⣿⣦⠀⠒⠀⠀⠀⠀⠀⢈⣯⣍⡉⠉⠁⠀⠀⢀⠀⠀⠸⣏⣿⣇⡀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣉⡙
⣿⣿⣿⣿⣿⣷⣶⣶⡄⢀⣠⣤⣿⣿⣿⣶⣶⣷⣾⣷⣿⣿⣿⣿⣾⣿⣆⣀⣀⢀⠀⠀⠀⠀⠋⠋⢰⣦⣤⣀⠀⠀⠀⠀⠈⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣬⣿⣿⣿⣶⣤⣤⣬⣭⣭
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣶⣶⣶⣶⣤⣄⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠉⠁⠀⢀⣾⣿⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠴⢠⣼⣿⠿⢛⣽⣿⣿⠟⣿⣿⡿⣿⣿⡟⠛⠉⠉⠉⠉⠉⠉⠀⢈⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣿⡛⣿⣿⣷⣤⣁⠛⠿⠤
⠀⠀⠀⠀⣦⣴⡶⠒⠀⣸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⣿⣿⣼⣿⣯⣿⣿⣿⣿⣌⣛⠻⠟⣶⣶⣤
⠀⠀⠀⠀⣀⣠⡴⠋⣽⣿⣿⣿⣟⣻⣿⣿⣿⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣮⡛⠳⢶⣶⠖
⠀⠀⠀⠈⠉⣁⠀⢺⣿⣏⠻⣿⡙⢿⣿⣻⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣠⣤⣿⣿⣿⡿⣿⣿⣿⣏⣿⡿⢻⣿⣿⣿⣛⣿⣿⣿⣷⡦⣬⡤
⠀⠀⠀⠰⠟⢣⣶⡚⠿⣿⡆⠈⠓⠈⢻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣟⡿⠁⠀⣴⣯⡽⢿⣿⣯⣿⣿⣿⡿⣷⣗⡀
⠀⠀⠀⠀⢠⢠⣞⠻⢦⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣤⣀⣀⡀⠀⠀⠀⣀⣀⣿⣿⣿⣿⡟⠀⠀⠈⠋⢁⣴⣿⣿⣶⣿⣯⣿⣷⡌⠋⡈
⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠘⣉⣩⣾⢻⣿⣆⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣹⣾⡿⢉⣾⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⣀⠀⠉⠉⢁⡀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣿⣿⣿⣽⣿⣿⣿⣿⢳⡄⠙⠻⠛
⠀⠀⠀⠀⣠⠄⣩⢴⣽⣿⠿⢡⣿⢿⣿⣿⣿⡇⠁⠀⠀⠀⠉⢀⡀⠀⠘⠃⠀⠀⠈⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠱⣶⣠⣀
⠀⠀⠀⠀⣤⣾⡿⠟⠋⢡⡖⣿⣿⣿⣿⣿⣿⣧⢃⠀⠀⠖⡀⠻⠿⠦⠄⠰⠆⠀⠰⣸⣿⣿⣿⣽⣿⣿⣿⣭⣿⣿⣿⢿⣿⣿⣷⡈⠻⡶
⠀⠀⠀⠀⠉⢁⣴⣿⡾⢂⣺⣿⣿⣾⣿⣿⣷⣾⣧⣠⡀⠠⠇⡃⢲⢰⡎⠃⠀⢀⣵⣿⣟⣹⣿⣿⣿⢻⣿⣏⣿⣿⣿⣆⢭⣛⡛⠿⠦
⠀⠀⠀⠀⠚⠋⠛⢋⣴⣿⣿⣯⣿⣿⣽⣿⢿⣿⣿⣿⢻⣶⣤⣥⣍⣀⣤⣴⣾⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⣉⣉⠛⠗
⠀⠀⠀⠀⣠⢴⣎⣻⣿⣿⣿⠿⣿⣿⣿⣿⣾⣿⠟⠙⠿⢿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⢻⣿⣿⣿⠿⣿⣿⣿⢦⣿⣧⡈⣉⣡⡌⠛⠷⡄
⠀⠀⠀⠀⣺⣯⣿⠟⣋⣼⣿⣿⣿⣻⣿⣿⠋⠁⠀⠀⠀⠀⠈⣿⣽⣿⠁⠀⠀⠀⠀⠉⢿⣿⣷⢿⣿⣿⡯⣭⣽⣿⣿⣿⣦⣭⣈⣛⣢
⠀⠀⠀⠰⠊⢉⣀⣾⡿⣻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⠇⢁⠀⠸⣿⣿⣿⣝⠷⠷⣟
⠀⠀⠀⠈⣱⡾⠛⣩⣾⣿⣿⣿⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣡⣀⣸⣄⣀⣈⣿⢿⣿⣇⣠⣍⣀
⠀⠀⠀⢀⣥⣴⣾⡟⠉⠓⣩⣾⣿⣿⣿⠏⠳⠤⣀⡀⠀⠀⠀⠈⣿⣿⣿⡄⠀⢀⡠⠴⠋⢻⡭⠭⡇⠀⠀⢸⠁⠀⠀⣿⠆⠛⢿⣿⣻⡆
⠀⠀⠀⢠⣴⣶⢶⡶⠂⣶⣿⣼⣿⣿⣿⠀⠀⠀⠀⠉⠁⠀⠀⠀⢿⣽⡿⠃⠀⠀⠀⠀⠀⠀⣿⡀⡇⠀⠀⢸⠀⠀⠀⣿⣗⡾⢶⣬⣉⠃
⠀⠀⠀⠰⠟⠋⣀⡴⣿⣭⣉⡉⠛⠻⢿⡀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⡇⠀⠀⢸⠀⠀⠀⣿⣿⣿⣬⡙⠻⠿
⠀⠀⠀⠀⠶⣼⣿⣻⣿⠿⠿⠿⢿⣶⣼⠳⣄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⣶⡇⠀⠀⢸⠀⠀⠀⣿⣿⣿⡾⣿⣢⣄
⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⡀⢀⣼⣿⠁⠀⠀⠙⠲⠤⣄⣀⡀⠀⠀⠀⢀⣀⣠⠤⠖⠋⠁⠀⠹⣟⠃⠀⠀⠸⠦⠐⠐⠉⠉⠛⠛⠻⠶
⠀⠀⠀⠀⢴⠶⠖⠓⢂⣀⣠⣽⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣭⡀⠀⠀⠀⠤⢀⣀
⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣶⣶⣤⣤⣄⣉⣉⠐⠒⠒
⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣶⡶⣯⣽⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⢿⠿⠿⡿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡛⢋⢻⣛⡚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⡿⠉⠈⠢⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠒⢤⡀⢸⡁⠛⠀⢀⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠐⠉⠋⠀⠀⠀⠀⠀⠊⠂⠉⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣤⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠐⠄⠚⠒⠔⠰⣶⣶⣶⣶⣶⣾⣷⣦⡀⠀⠀⡆⣀⣸⣿⣿⣛⣏⣻⣿⣿⠆
⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠻⣿⣿⣿⣿⣿⣿⣿⠟⠂⠐⢿⠛⠻⢿⣷⣬⣦⡟⣿⠃
⠀⠀⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⢋⣠⠆⠀⠀⠀⢡⠀⠀⠈⠙⠛⠻⠿⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⢁⠂⢘⣃⠀⠀⢘⠀⠀⠀⠀⠤⣦⣷⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠉⠀⠀⠈⡩⠂⠈⢢⡈⠀⢿⢷⣶⣶⣤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠀⠒⠁⠀⠀⠀⠀⠑⣠⣿⣿⣿⣿⡿⠣⠀⢰⣤⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⢛⠟⡟⠁⠀⠀⠀⠙⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠘⡂⠀⠀⠀⠀⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⢀⡀⠅⠊⠉⠀⠰⠗⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣤⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠓⢤⡀⠀⠀⠹⡆
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠙⣄⠀⠀⢿
⠀⠀⠀⢰⡟⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⣀⣀⣤⡴⠶⠛⠻⠋⠀⠀⢀⠐⠀⠀⠀⢿⠀⠀⢸⡄
⠀⠀⠀⣿⠁⠀⠊⠀⠀⠀⠀⠐⢀⡤⠖⢋⡙⠉⠉⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⢀⣀⣀⣸⠀⠀⢸⡇
⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⣠⠥⡞⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢉⡁⠉⠛⠋⡄⠀⣼⠃
⠀⠀⠀⢻⣦⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠰⠞⠋⠁⢀⡄⠱⡄⠀⠀⠀⠀⠀⡀⠀⠰⠾⠇⠀⠰⠀⡇⠀⣿
⠀⠀⠀⠀⡿⠀⠂⠀⠀⠀⠀⢸⡄⠀⠀⠀⢤⣀⠈⠛⢃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⡇
⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⡀⠈⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠈⣿⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠑⢄⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢤⠂⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠘⢷⡀⢀⠆⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠉⠙⠓⠶⠤⣄⣀⣀⣀⣴⡟⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠐⠠⠀⠠⠀⠀⢤⠤⠀⣀⡀⠀⠉⠁⠀⡗⠠⢀⡘⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢶⡶⠁⠀⠀⠉⠀⠀⠀⡦⠀⠀⠈⠑⠢⡀⠀⢷⠀⠀⠅⠀⠀⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠈⠃⣜⠰⠀⠐⡀⠀⢈⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠂⠄⠀⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠃⠠⠀⠈⠆⡈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢀⠀⠀⠈⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠀⠙⢣⠀⠘⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠐⠀⠄⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠣⡀⠀⠑⠧⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⠠⢤⠄⠀⢸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢣⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠙⢀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠤⠀⠃⠈⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠑⡦⢀⠀⠠⣽⡄
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⡀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠇⠀⠉⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠈⠄⠀⠀⠀⠀⠀⢀⠈⡑⠊⠀⠀⠀⠀⠀⠀⠀⢈⠢⣖⠀⠈⡒⢄⡀⢀⣼⡂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠸⠉⠀⣷
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢰⠀⠀⠀⠀⠀⠢⠠⠆⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢤⠀⠀⠸⡀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⡸⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠔⠁⠀⠀⠀⢰⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣷⡄⢠⡟
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡇⢹⡞⠁
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⡆⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣧⠏⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣍⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠀⢨⣿⣿⡟⣵⢞⣤⡳⣮⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆⠀⠀⠀⢼⣿⣿⣧⣶⣿⣿⣿⣶⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠀⠀⢸⣿⣿⡟⣿⣿⣿⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠿⢿⠟⡌⢿⠿⠿⢠⠀⠐⢿⣿⣿⣮⣭⣭⣭⣵⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢈⣾⣿⣎⢸⣿⣿⡧⣀⣬⡻⣿⣿⣿⣿⣿⣿⣿⢿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢏⣾⣿⣿⣿⣎⢿⣿⢰⣿⣿⣿⣿⣯⣾⡩⢾⣵⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠋⠾⠿⠿⠿⠿⠿⠎⠻⢸⣿⡿⣿⣿⣿⣿⠔⣿⣿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠼⠿⠿⠿⠿⠸⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠖⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠤⠶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠛⠀⠉⠉⠁⡇⠸⠈⠉⠁⠀⠁⠛⠀⡇⠀⠀⠀⠀⠀⣠⠈⡤⠤⠠⠄⠀⠀⠤⠤⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⠀⠀⡇⠐⠀⠀⠄⠠⣤⣤⠀⡇⠀⠀⠀⠀⠘⠀⡀⠀⠄⠀⠒⠒⠒⠒⠒⠒⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠁⢠⡀⢰⣔⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠛⠚⠂⡇⠈⠘⠒⠃⠈⠛⠃⠀⡇⠀⠀⠀⢀⠀⡆⠈⠋⠈⠉⠀⠀⠀⠀⠀⠀⡘⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⡇⢀⠀⡀⡀⢀⣀⠀⠀⡇⠀⠀⠀⠀⠇⠈⠦⣀⠒⠘⠋⠁⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠁⠀⠀⠀⠀⡇⠈⠀⠀⠀⠈⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠀⠈⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠒⠒⠂⡇⠐⠐⠒⠂⠀⠺⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠤⢀⣠⣤⣶⢤⡗⠊⠛⠁⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⡵⡖⡄⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠈⠀⡇⠈⠀⠀⠀⠘⠋⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠌⢳⡏⠸⠈⠂⠇⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠀⠤⠤⠀⡇⠀⠠⠤⠄⠠⣤⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢷⠘⡀⠀⠘⢶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠀⢳⡇⠰⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠙⠛⠁⡇⠸⠘⠋⠃⠐⠓⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⣠⣠⠀⢀⠀⠀⡇⢠⠀⠀⠀⢠⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠶⠆⠀⠒⠒⠂⡇⠐⠐⠒⠒⠰⠶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢀⢀⠀⡇⢈⠀⣀⡀⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠶⠀⠠⠤⠀⣷⡆⠠⠤⠆⠀⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⢀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠉⠀⠀⠉⠀⣿⡇⠀⠀⠁⠀⠈⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⢠⡶⣿⣿⣶⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⢿⠿⠿⠿⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡠⠀⠀⠤⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠠⡄⠤⠀⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠄⠦⠐⠛⠉⠑⠦⡀⡀⠠⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⢡⠐⠀⠀⠀⠀⣖⣢⣥⡶⠒⠒⠀⠀⣠⠤⠄⠀⠀⠀⠠⢄⢹⡄⣾⡙⠠
⠀⠀⢀⠄⣀⠤⠲⠓⠉⠁⠀⠀⠀⠁⠓⠢⢴⢠⠏⣠⡦⠑⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣒⣾⣿⣯⠚⢡
⠀⠀⢠⠚⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⣠⣀⣴⠟⣡⡶⢋⣤⡼⠷⢟⣿⣿⣷⡀⠈⢂
⠀⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣰⣳⣭⣭⣥⣤⣤⣴⣶⣶⣾⡿⠃⣿⡘⡇⠀⠐⣆
⡞⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠵⢿⡟⣽⡛⠿⠿⠿⠿⠟⣛⣁⣠⣾⣷⣷⢀⠀⣿
⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣓⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷
⠊⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⠍⣿⣿⣿⣿⣿⣿⣏⠁⢸⣽⠋
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣏⠲⠌⠙⢲⢿⣿⣿⣿⣟⣀⠟⣆⣿⣯⡤
⢃⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠛⠙⠛⠁⠄⣺⣣⡏⠉⠉
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡋⠝⠋
⠀⠐⡡⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠌
⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠁⠄
⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⠿⠋⠘⠒⠢⠔⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠈⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠒
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⠛⠋⠁⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠀⠤⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢠⠤⠤⠤⠄⠰⠶⠂⠀⠀⠀⠀⠀⢀⣴⣷⣶⣄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣾⠶⠆⣠⣤⣄⣤⠠⢤⠴⠞⠁⣀⣴⣿⣿⣿⣿⡟⣠⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⣤⣶⠾⠿⡿⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⠏⠹⣿⠙⢿⣿⣿⣙⣇⢩⣿⣿⡦⢾⣿⣿⣿⡷⠙⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⣠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠘⠀⠙⢢⣿⣿⡟⣿⣿⣋⣿⣿⣧⣼⣿⢸⣿⣷⡀⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠶⢦⣤⠆⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠹⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣤⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡴⠀⠀⠀⠀⠀⠀⠐⡟⠀⠀⠀⠀⠀⠀⠙⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⢷⣄⣀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄⠀⠀⢀⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣭⣭⠀⣭⣽⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠠⢋⣉⠛⠁⠀⠀⠀⠀⣿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⢸⣿⣿⡈⣿
⠀⠀⠀⣿⣿⣦⣄⢨⣀⣠⣤⣴⣤⣮⡬⠷⣶⣴⠂⡀⣰⣿⠀⣿⠋⠛⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢃⣾⣿⣿⣿⡿⢸⣿⣿⣷⣻
⠀⠀⢸⣿⣿⣿⣿⡭⠓⡾⢛⡻⣽⣿⡐⠀⣀⠙⠂⠙⠿⡏⠸⡁⡀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣫⣾⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿
⠀⠀⣸⣿⣿⣿⣿⡿⣿⠃⠙⠛⢉⣬⠉⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣦⠘⠿⢿⣶⡦⠀⣠⣿⣿⣿⣿⣿⡿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⣤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠐⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿
⠀⣼⣿⣿⣿⡿⣿⣿⣍⡉⢁⡠⠄⠀⠀⠀⠀⠀⠀⢁⡠⠀⣠⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿
⢰⣿⣿⣿⣷⠀⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⢒⣩⣴⣾⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢿⣛⣀⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣾⣿⡿⠿⠿⠿⠿⠿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⣿⣿⣿⣿⣿⣿⡌⣿⡿⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢠⣿⣿⣿⣏⢻⣿⠗⠛⢀⡤⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⢸⣿⣷⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡾⠁⠀⠀⣀⡤⠂⠀⠀⠀⣠⣴⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡤⢤⡀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⠉⠉⠀⠀⠀⠀⣀⣹⡄⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⢴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠁⠀⠀⠀⠀⢀⣀⠀⠀⣀⡴⠛⠉⠁⠀⣠⣷⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⢰⢶⠀⠀⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣄⣀⣀⠀⣀⣀⡴⠋⠉⠙⠋⠉⠀⢀⣠⣴⣾⣿⠿⠓⠚⢉⣩⠭⠟⠁⠀⠀⠀⠀⣿⠀⠀⢸⢸⠀⠁⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣷⠈⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠉⠀⣠⠴⣚⠉⠀⠀⠀⠀⣰⡄⠀⠀⣿⠀⠀⢸⢸⠀⠂⠠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣆⣀⣀⣤⣤⣶⣶⣿⣿⣿⡿⠛⠁⢀⣤⠖⠋⠁⠸⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⠀⠄⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⣿⡿⠿⢛⡹⠛⠁⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⢀⡀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠀⠛⠁⢁⡤⢒⣉⡤⠖⠋⠁⠀⠀⠀⣶⠀⠀⠀⠀⠀⢠⡀⠈⣧⠀⠀⠀⠀⠀⣿⠀⠀⢸⢸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⣀⣀⣀⣴⠾⠟⠚⠉⠀⠀⠀⣠⠶⠶⣄⠀⢿⠇⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⠸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢀⣤⡿⠉⢉⡇⠀⠀⠀⣿⠀⠀⢸⢸⠀⠄⠀⡇⡇
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢧⠀⣀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠒⣟⠁⣀⣀⠀⠀⢹⠀⠀⢸⢸⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⡈⣳⡘⠊⠃⠀⠀⠀⠀⠛⢦⣄⣴⠇⠀⠀⠀⠀⣦⣠⠤⡴⠋⠀⠀⠸⡄⠀⢸⠀⠀⢸⢠⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣾⠉⣉⣙⣓⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⡞⡀⠀⠀⠀⠀⢬⠀⢸⠀⠀⢸⣤⠀⠁⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⣿⣿⣿⣿⣿⣷⣾⣝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠱⡄⠀⠀⢀⠜⠀⢸⠀⠀⢸⣿⠀⠂⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⡻⡄⠀⠀⠀⠰⣶⠞⣡⠟⠀⠀⠀⠙⢠⡘⡯⠀⠀⢸⠀⠀⣾⣿⣷⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡓⠀⠀⣀⡴⠛⠋⠁⠀⠀⠀⠀⠀⢸⠇⠁⠀⠀⢸⠀⠀⠙⠛⠁⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢾⠁⢤⡀⠀⠀⠀⠘⣆⠀⠀⠸⠄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈⢳⣠⠛⣆⠀⠀⠀⠙⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠠⣶⣄⣿
⠀⠀⠀⠀⣴⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡠⢻⡆⠀⠀⠀⠘⣦⠀⠀⠠⠶⠀⠀⠿⠾⣧⣦⣴⣿⠁⠉
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠻⣿⣿⣿⣿⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢿⠃⣾⠆
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠓⢦⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⡀⠀⠀⠈⠙⢦⠀⠀⢿⢳⣄⠀⠀⠀⠛⠋⠀⠀⠀⠀⢀⣿⠀⠀⣿⠇
⠀⠀⠀⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⠇⠀⢸⡆⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠉⠀⠀⠙⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠘⢳⡄⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⠆⡔⡠⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠾⢾⡴⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠙⠋⠉⠉⠁

⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡈⢛
⣻⣿⡭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠏⢤⣴
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠰⠂⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⡇⠀⠀⠙
⣿⣿⣟⣋⡀⠀⠀⠀⠀⠀⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡿⠤⠄⡀
⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠶⣶⣶
⣿⣿⣿⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠁⠀⠀⠀⢐⣛⣿⣿
⣿⣿⣿⣟⣋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢩⣽⣿⣿
⣿⣿⣿⠯⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣻⣿
⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠛
⣿⣿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢾
⣭⣇⣐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⠤⠤⣄⢀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠄⠀⠀⣿⠈⢧⡴⣃⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠀⠀⠀⣿⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⣿⠀⠀⠘⢧⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡟⢻⡆⠈⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⢸⡇⢾⠀⠀⠀⢠⣠⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡞⠀⠀⢳⡌⢳⡄⠲⣄⢹⡇⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡏⠀⠀⠀⠀⢹⠀⣷⠤⢬⣿⣿⠥⢴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢳⣄⠀⠀⣀⡞⢠⡏⣠⠞⢹⡏⠳⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⣉⣉⣡⡴⠋⠀⠁⠠⠿⠳⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠿⠀⠀⠇⠻⠼⠃⠇⠀⠇⠿⠸⠀⠿⣸⢯⡄⠀⠿⠀⠘⠧⠟⢸⢯⡍⠷⠼⠃
⢀⢀⣠⠫⢂⠢⠍⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⢁⠀⡼⠠⠐⠒⠻⠄⡀
⡴⠂⠀⠀⠁⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠂⠀⠀⢀⠀⠀⠀⠰⠶⡄⠀⠀⢬⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠁
⠀⠀⣀⣤⡄⠀⠀⢰⣿⣾⠁⠀⠈⠀⠀⠀⠀⢀⠀⡔⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠁⠀⠀⠴⠂⠀⠀⠀⠖⢖
⠀⠀⠘⠿⠃⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⠤⠎⠑⠙⣫⠀⠀⠠⡀⠠⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⢤⣀⣤⡎⠀⠐⠋⠁⠀⠀⠀⠀⡨⠆⠀⠀⠄⠃⠆⠣⠐⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠺⠃
⢀⡄⠀⠀⠀⢀⠀⠀⠄⠀⣄⠀⢡⠀⠀⢀⠜⠀⣠⠊⠀⠀⠈⠈⠀⣀⢀⡁⠀⠀⠀⠀⢁⠀⠀⠀⡀
⠀⠁⠀⠠⠒⠂⠀⠀⢀⠁⠅⠀⠆⠒⠀⠈⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠂⠀⠀⠈⠂⠤⠌⠉⠀⠐
⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠂⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠄⠠⠀⠇⠀⠁⠀⠀⠀⠀⠂⠠⡀⠀⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢄
⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠒⠈⠈⠀⢄⠰⠀⠀⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀
⠀⠀⠂⡀⠢⣀⠀⠀⠀⢀⠤⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢾⣧⣴⡶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡽⣿⣾⠛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣻⣷⠟⠁⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢁⣴⠟⠀⠀⢀⣴⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⣴⡟⠁⠀⠀⠀⢸⣿⣿⣿⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⠿⢱⡶⢄⠀⠀⠀⠘⢸⣿⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣟⣻⠃⢠⣿⣇⠘⠃⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⡿⣿⣿⣿⣇⠀⠀⠀⠀⠸⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣾⢏⠋⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣇⠀⠀⠀⠀⣼⡏⠘⠀⢰⣿⣿⣿⣿⢴⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢻⣿⣿⡇⠀⠀⣼⡟⠀⠃⠀⢸⣿⣿⣿⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣿⣿⣿⣇⠀⣾⡿⠀⢠⠀⠀⣿⣿⣿⡿⣆⡈⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⠿⠆⣿⠃⠀⢸⠀⢰⣿⣿⡿⠛⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢧⣾⡇⠀⠀⠁⠀⠀⡀⠀⣾⣿⣿⠁⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣽⣿⠀⠀⠀⠀⢰⡀⠀⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⠀⠀⠀⢸⡇⠀⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢣⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣿⡿⠁⠀⠀⠀⠀⠆⠀⣸⣿⣿⠃⠸⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣤⣿⠃⠁⠀⠀⠀⠀⢀⠀⢠⣿⣿⠏⠀⠀⠀⠀⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡟⠀⠀⠀⠀⠀⠀⡌⠀⣾⣿⠟⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⡼⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⢀⠎⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⡠⢎⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⣀⣴⣷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡿⣿⣿⣿⢿⣿⡿⠻⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣀⣀⣀⣀⡒⠲⠶⠚⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣚⣙⣉⣁⣀⣀⣈⣉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠲⠶⠄⠀⠀⠀⠀⠀⣍⠉⠛⠛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⣄⣀⣀⣀⡀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣽⠦⠯⠽⠫⣻⣿⣿⣿⣆⣰⠃⠀⠀⢀⣠⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠟⣛⣛⣙⡛⢿⣿⣿⣿⠏⠀⠀⣼⣯⠽⠛⠻⢯⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣇⠎⢰⣾⣿⠙⣿⣿⣿⣿⡀⠀⢾⣯⡕⢹⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠀⠀⠋⣿⢿⣿⡇⠀⢸⣿⠇⠈⠉⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⡽⣷⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⢶⣿⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣾⣿⣧⣄⣬⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣠⣬⣀⣤⣤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠄⠀⠠⣀⠀⢀⣀⣋⣀⠈⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠹⠿⠛⠛⠛⠋



⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣟⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠒⠒⠒⠂⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠉⠐⠒⠂⠀⢴⢶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠒⢛⠀⠈⠑⢦⣄⣀⠀⣤⠶⢿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⢦⡘⢧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠇⢀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡦⣌⡐⠈⠲⢦⣤⣤⣠⣤⣴⡫⠥⠒⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠈⠉⠁⠀⠒⠒⠒⠒⠒⠀⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⣾⡆⠀⢀⣴⣤⠀⢀⣶⡄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠘⣿⠟⠁⠈⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⢰⠇⢷⡀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠠⠔⠉⠐⠒⠚⡆⠘⡇⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠛⠒⣤⣀⡀⠀⠀⠀⠀⣳⣶⠗⠚⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡁⢯⠀⠰⢆⠀⢸⡈⠉⠘⡟⠉⠻⠀⢰⡀⠀⠀⣰⠷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⢆⡡⠀⠋⡀⠀⡗⢲⠒⣇⠀⠀⠀⠉⠻⠛⠈⠀⠀⠀⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠈⠳⠀⠈⠷⠀⠀⡇⠀⡇⠀⡄⢻⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄
⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⣦⡀⠀⠀⢸⡁⠀⡇⠀⠇⢿⣤⣤⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠁⠹⣄⣀⣼⡇⠠⡇⠀⠷⣤⣀⣉⡉⠙⠻⠋⠁⠀⠀⠀⡰
⠀⠎⠀⠀⠀⠀⠀⠠⠆⠀⢠⡶⠋⠁⠀⠀⠀⠀⢰⡄⠀⠈⠉⠉⠀⠘⠀⢀⣀⣀⣤⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡴⠞⠋⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣠⣴⠋
⠀⠀⠙⠶⣤⣄⣀⣀⣤⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⣠⠾⠿⠻⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠖⠛⠁⠀⠀⠀⠃⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠦⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⢤⡀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⣠⡴⠶⠛⠛⠛⠓⠲⢤⡀⠀⠀⠀⣠⡴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⠀⠀⠀⠀⠈⠀⣠⡼⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣿⣧⣤⡁⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⡏⠀⠤⠴⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⢻⣿⣥⣀⠀⠀⣊⡄⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠸⠷⠄⢼⣿⣿⡋⢹⣋⣹⠇⠀⠀⠀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣠⡤⠤⣤⣤⣀⠉⠓⠦⠉⠹⠆⠀⢠⠜⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢀⣼⣿⢷⡀⠁⠀⢈⡹⠶⠄⣠⠂⠀⠘⠻⡀⠾⣅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣿⣿⡁⣻⡄⠈⠁⠀⠀⣤⠑⢤⣠⣷⣤⡼⠦⣄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⢀⣤⣶⡼⠻⣦⡀⠻⣿⣿⣿⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧⣿⣿⣿⣧⣇⠈⠁⠀⡹⢿⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣽⣶⡷⠊⠀⠘⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡏⠘⢉⡉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡶⢿⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣄⠑⢄⡀⠀⠀⠀⠀⢠⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⣿⠿⠿⠿⢮⠳⣄⠉⠲⠄⠀⣰⠃⠀⢠⣷⣄
⠀⠀⠀⠀⢀⠀⣼⡇⢀⣾⣅⠀⠀⠀⠀⢻⣿⣷⣶⣶⣤⣤⣌⡓⠄⠀⠀⠁⠀⠀⠈⠡⠀⢆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠛⠒⠿⠳⠛⠉⠁⠀⠀⠀⢠⣼⣿⡿⠟⠋⢉⠁⣈⣹⣆⠑⣄⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⢴⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⣿⣿⡶⠋⠉⠀⠀⠀⠈⠀⠘⣦⠀⠀⠀⣸⠇⠀⠀⡀⢀⡀⣻⡟⢲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢤⣿⣷⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠋⣡⡔⠀⠀⠀⠀⠉⠈⠙⠛⠿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⣹⣾⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣿⡀⠀⠀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣄⡀⠀⠀⢀⡀⠀⠀⢠⣿⣥⣤⣦⡇⠠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⢶⣿⣿⣿⣮⣼⠿⠛⠛⠋⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡘⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⠀⠀⠀⠀⠀⠀⠘⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢧
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠱⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇
⠠⠴⠐⠐⠚⠁⠉⠉⠉⠁⠉⠉⠁⠀⠐⠒⠀⠀⠤⠤⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠠⠠
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠁⠈⠐⠀⠒⠒⠐⠂⠒⠒⠋⠉⢉⠅⠤
⠀⠀⠀⠀⠀⠀⢀⣀⠐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣸⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠟⣫⢎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⣡⡾⢱⣇⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⡿⠟⠋⣡⣾⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠛⠛⠉⠉⠀⣠⣴⣾⠟⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣉⣁⠂⢀⣠⡤⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠈⡧⢸⠁⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀⠀⠀⠓⠚⢁⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢁⡎⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣊⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠃⡘⠠⠊⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠁⠀⠉⠉⢹⡆⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢠⠄⠀⠀⠀⠀⠀⠀⢼⠁⠀⠲⠤⠀⠀⠀⢸⠷⠛⠛⠛⣦⠀⠈⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⠙⡆⠀⠀⢀⡖⠻⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠸⠶⣄
⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠈⠓⠋⢁⠆⠀⢸⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠠⠾⢤⡀⠂⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠘⣆⣰⢠⠀⠀⠀⠀⠀⠀⡰⠋⢀⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣀⢀⣴⣾⠶⠿⠓⠒⠦⠤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠉⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⢷⡄⠀⠀⠀⣸
⠀⠀⠀⠤⠤⠒⠚⠉⣀⠤⠒⠀⠀⠀⠈⠉⠑⠒⠲⠶⠶⠶⠖⠒⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠤⠐⠁⠀⠓⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⢀⡖⠀⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠐⠋⠒⠒⠶⠦⠤⠶⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⣀⠈⠀⣠⡟
⠀⠀⠀⠀⠀⡠⠔⠊⠁⢀⣀⠠⢤⣀⠀⠀⠀⠀⠒⠢⢄⣠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠺⡝⢠⣄⠈⢿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⡙⢦⠀⠀⠀⠀⠀⠈⠑⠄⠀⠐⠶⠖⠂⠀⠀⠀⠀⠀⢹⡆⢸⣧⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡔⠀⢹⠄⠃⠀⠀⠀⢸⡋⠉⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣘⠒⠊⠀⠀⠀⠀⠀⠀⠻⣈⡠⢆⠀⠀⣶⡄⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠃⠹⣄⣀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠁⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠿⠛⠛⠛⠉⠉⠉⠉⠙⠛⠛⠻⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠛⠛⠀⠀⠀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⢤⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠱⣀⠞⠁⠁⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡄⠀⡟⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⠀⠀⠙⢦⣄⣀⣠⡴⠃⠘⢧⣄⣀⣠⠞⠁⠀⠀⡀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠙⢆⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⠞⠁⠀⠀⣸⠇⠀⠀⠀⠀⢠⠶⢦⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⢀⣀⠠⠴⠚⠁⠀⠀⠀⢠⡟⠀⠀⣠⠶⢤⡏⠀⠀⡏⠀⠀⣽⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢣⡀⠀⠹⣄⡘⠄⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣷⠀⠀⠀⠱⠀⠈⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢈⡷⠀⠀⢀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⠞⠁⠀⣠⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠶⠶⠖⠛⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣀⣤⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠶⠤⠤⠶⠒⠋⠁
⠀⠀⠀⠀⢀⡼⠁⠀⣠⠞⠉⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡼⠁⠀⢸⠃⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣠⠖⠃⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⡞⠁⠀⣀⡀⠀⠠⣇⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⡞⠀⠀⡼⢻⠹⣄⡀⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⢀⣤⠟⠁
⠳⠤⢾⣅⡼⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠛⡟⢳⡟⢻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀⠈⡇⠀⡇⢸⡇⢸⠃⢀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠁⠀⠀⠀⠀⠀⠈⠉⠛⠇⠀⡇⢸⡇⠸⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⡗⢺⣤⣠⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⡿
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⠖⠁⠈⠈⠁⠀⠀⠀⠀⠉⠷⢄⡀⠀⠀⠀⠀⠀⠀⣀⡀⠿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣠⣤⣶⣶⣶⣦⠤⣀⠀⠈⢳⡄⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠻⡅
⠀⠀⠀⠀⠀⡌⠉⠀⠀⢀⣴⣿⣿⡻⣝⢿⡿⠉⠀⠈⠳⡀⠀⢷⠀⠀⠀⢀⠟⠈⠁⠒⠳⣀⠀⠀⠈⡆
⠀⠀⠀⠀⢰⠃⠀⠀⣰⣿⣿⣿⢻⡿⠘⠃⡴⠉⠀⠀⠀⣧⠀⢸⠀⣠⠖⠁⠀⠀⠀⠀⠀⠈⣄⠀⠀⡇
⠀⠀⠀⠀⢈⠀⠀⠀⡿⠟⠛⢁⣀⣀⠀⡠⠄⠀⣴⠀⠀⡟⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡤⠞⠁
⠀⠀⠀⠀⠸⡀⠀⠀⢧⠀⠀⠋⠀⠀⠀⢀⣠⠞⡙⠀⡰⢁⡘⠁⠀⠀⠀⠀⠀⠀⢀⠀⡰⠋
⠀⠀⠀⠀⠀⢣⡀⠀⠈⠣⣀⠀⠀⠀⠘⠛⠃⠁⣠⠞⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠘⠧⢤⣀⠈⠑⠒⠒⠀⠒⣂⣉⣔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⠋⡄⠒⢓⣶⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡰⠋⣠⠋⠀⢠⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢰⠁⠀⢹⠄⠀⠈⢿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⣀⠀⠀⠀⠉⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢦⠟⠋⠀⠀⣀⣤⡏
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠙⠢⢄⣀⣀⣠⠼⠋⠋⠶⠰⢦⢄⣤⢠⣔⡔⡶⠝⠙⠉⠁⠀⠀⢠⣶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠋⠁⠀⠹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣍⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣶⣴⣤⣤⣄⣠⣄⣠⡤⠤⠴⠛⠻⣿⣿⣿⣿⣿⣿⣿⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣾⣶⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠛⠛⠛⠛⠛⠛⠛⢛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠿⠿⠿⠿⠿⠛⠋⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠐⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⠶⠖⠒⠋⠁