⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠐⠀⠐⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠀⠀:¨ ·.· ¨:
⠀⠀⠀⠀⢰⡿⠉⠀⡜⣿⣿⣿⣿⣿⣿⣿⣿⠃.⠀⠀`· . ꔫ⠀
⠀⠀⠒⠒⠸⣿⣄⢀⣃⣿⣿⡟⠉⠉⠉⢹⣿⡇⠀⠀⠀⠀⠀
⠀⠀⠚⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠘⠠⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠁⠀⠒⠤⠀⠀⠀⠀
⠨⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀
⠁⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠳⢿⣦⡀⠀⠀⠀⠀⠀⠈⢖⠠⢀⡐⡄⣸⠑⣼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠳⣌⠊⢙⠓⡡⣺⢷⠓⡆⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠙⢿⣢⡀⡀⠀⠀⠀⠀⠘⢗⢠⠉⣤⡡⢒⢥⠄⠒⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⣷⣄⠁⢄⠀⡠⣢⡕⣧⢰⠏⠁⢥⣼⡤⣽⠼⡭⣔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠸⣾⣿⠿⣿⣗⣯⠋⣉⡽⡘⢿⡜⠻⠛⠛⠿⣹⡬⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

⠀⠀⠀⠀⠸⣾⣿⠿⣿⣗⣯⠋⣉⡽⡘⢿⡜⠻⠛⠛⠿⣹⡬⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⢿⣻⣽⣷⣿⣻⣶⢾⠫⣿⣿⢾⢔⢂⡀⠈⢧⡘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢾⣿⣷⣷⢯⣿⣿⠝⡧⣠⣿⣿⣾⡟⣷⣥⣾⣷⣾⣿⡴⣶⣴⣹⢂⣄⠀

⠀⠀⠀⢾⣿⣷⣷⢯⣿⣿⠝⡧⣠⣿⣿⣾⡟⣷⣥⣾⣷⣾⣿⡴⣶⣴⣹⢂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⣿⡏⣿⣿⣿⢑⣴⣿⡿⠟⠋⣽⢿⣿⣿⣾⣷⣿⣿⡷⡅⠩⣻⢻⣟⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⢿⡓⠻⣿⡯⠟⠛⣋⡀⣀⡰⡎⡟⣾⣿⡿⢿⡟⢿⣷⣾⣿⣿⣈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢹⡤⢵⠔⠯⠄⠀⠐⢛⣺⣦⣷⣧⣷⣿⣿⣷⣾⣿⠏⣿⣷⣿⣿⣿⡿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠻⢃⣐⠤⠔⠒⢖⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡂⠈⢻⣽⣾⣿⣿⣿⣿⣽⣿⣿⣿⣯⡇⠑⡅⣽⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⢠⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡇⠀⡆⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣻⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⡴⡁⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢗⣃⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⡿⢻⣿⢾⣿⣟⣷⣽⡟⢿⣿⣛⣴⡤⣯⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⡻⣾⣿⣾⣿⣟⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡿⣿⣿⣾⣿⣿⣿⣿⣿⡿⣿⣿⡀⠀⠝⣿⣏⣿⣿⣿⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢿⣍⡻⣿⣿⣿⣿⣿⣿⡿⡻⣿⣷⡇⠀⢠⢿⡯⣭⣿⣿⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡝⣿⣿⣷⣿⣻⢿⣿⣿⣿⣷⣌⣿⣇⠸⢯⣻⢍⣽⣟⣽⣿⣿⣯⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣝⠸⣿⣿⣿⣿⣎⣿⣿⣿⣿⣿⣿⣿⣧⣹⣿⣿⣿⣿⢿⣿⣯⢷⣻⢏⡗⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣻⣿⣿⣿⣷⣿⣿⣽⣿⣿⣿⣿⢿⣿⢿⣿⣿⠿⣻⠀⠹⣯⣟⢶⡫⣜⢭⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢩⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⢹⣏⣿⡟⠀⠈⢈⢟⣮⢳⠬⣆⢏⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⣿⣿⣿⣿⣿⣿⣿⣯⠙⣿⣿⣿⣿⣿⣿⡈⠙⢍⠀⠀⠀⠀⠁⡪⣓⠮⡱⢊⡅⠛⣄⠀