⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣲⢮⣿⣷⣶⣖⣿⡉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣽⣿⣿⣯⢿⣿⣿⣿⣿⣄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣻⣿⣿⣿⣋⣯⣖⣧⡀⢀⣤⣂⣠⣐⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⠿⣿⣇⠀⡾⣿⡍⢿⣿⣿⣽⠇⣿⣿⣿⣿⣿⣮⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡙⠃⠘⠛⠛⠀⣏⠉⠀⢘⣿⡏⢹⠶⣿⣙⠿⣿⣿⣿⣷⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢚⡃⠀⠀⠀⠰⠧⠼⠂⠀⠘⣿⠇⣨⣿⣿⣿⠛⠩⡟⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠈⠳⡄⠀⢴⣖⣺⣿⣇⣠⠃⠂⣿⣿⠟⠓⠦⣄⡈⢚⠻⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠷⢠⠒⠶⢤⣀⡀⠘⢦⡀⠛⠉⠉⢉⡟⢀⢰⣿⡏⡀⣠⢣⡀⠙⢍⠉⠳⡵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⣻⠋⢩⠏⢆⣀⣸⠄⠉⠉⠒⢝⠦⣀⣠⠞⠀⢸⡼⠋⢀⡜⠁⠀⠱⡄⠈⢆⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠃⠀⢸⠈⠉⠑⠲⠤⡖⠲⡯⠀⠀⠀⢸⢯⠀⢠⠋⠀⠀⠀⠀⠘⢆⠈⢇⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⡧⣄⢧⠀⠀⢀⣏⣼⡄⡈⠲⣄⠀⠀⠠⠊⠈⠣⠞⠠⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡴⠁⠂⠸⡄⠀⢸⡄⠀⠀⢀⣼⠁⠈⠻⡳⣄⡾⣿⣿⣿⠗⢦⣈⠟⠦⣄⡀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⢦⣠⣅⠀⢀⣸⣻⣷⣶⣶⣿⠉⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢠⢾⠃⠀⠀⠀⠈⣿⢛⠛⠛⢯⣀⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡽⡙⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠑⢄⡰⠤⠛⠋⠀⠀⠀⠀⠀⡝⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⣾⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡏⣷⣤⣄⣀⡀⠀⠀⠀⣀⣀⣀⣠⣾⠿⠛⠛⠉⠉⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⣅⣀⠩⠛⠛⠿⣧⠉⠉⠉⠉⠁⢹⣿⣟⣵⠶⠶⠖⠶⣮⣭⣄⣀⠀⡉⠢⢄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⢁⣀⣤⣤⣤⣤⣤⣥⣀⠀⠈⠓⠒⠒⠒⠒⣚⣿⣿⣯⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡱⣄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⠔⢋⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡤⠀⠀⠀⣴⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢆⠀⠀⠀
⠀⠀⠀⠀⡠⢊⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⢸⠀⠀⠀
⠀⠀⢠⣊⣴⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠻⠿⠟⠯⠽⠯⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀
⠀⠀⢸⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡯⠝⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣹⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀
⠀⠀⠀⠑⢻⣿⣿⣿⣿⣿⣿⣿⣷⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠀⠀⠀
⠀⠀⠀⢠⣸⣿⣿⣿⣭⣾⣿⣿⣿⣷⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣾⠀⠀⠀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠛⠛⠛⠛⡿⠽⠛⠀⠀⠀
⠀⠀⠀⠈⠿⠿⡟⠿⠟⠛⠋⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⢰⣸⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣤⣺⣿⣿⣦⣤⣼⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣇⠀⢀⣸⣯⡳⡀⠀⠀⠀⠀⠀
⢀⡠⢔⣊⣵⣿⣿⡳⡽⣿⣿⣿⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢟⣿⣮⣑⢤⡀⠀⠀
⣷⢿⣿⣿⣿⣿⣿⣟⣣⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⢣⢫⣾⣿⣿⣷⣬⣽⣒
⠀⠉⠑⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢛⢼⣿⣿⣿⡿⠿⠛⠛

⠀⠀⢀⣀⡔⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡴⢋⢄⠀⡔⡆⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠁⠸⣱⠃⣋⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣇⠀⣤⣤⣤⡄⠀⣸⠀⠀⢀⣀⣤⣀⠀⠀⠀
⠀⠈⡇⢻⣿⣿⠃⣄⠇⡰⠊⣭⠀⢆⡈⠙⢆⠀
⠀⠀⠘⢦⣉⣡⣾⠋⢰⢱⡌⣫⡋⢪⢧⡄⡄⣇
⠀⠀⠀⠈⡆⢀⢧⡆⡞⡘⠑⡂⡭⡫⢎⡂⣥⢸
⠀⠀⠀⠀⡿⣾⣿⡇⡇⣹⠕⡲⠲⢔⡈⠒⣽⢸
⠀⠀⠀⠀⣇⣛⣺⢇⡇⢻⡗⠯⣏⠆⡈⠣⣺⣸
⠀⠀⠀⠀⢸⠬⣷⡇⠙⠓⠒⢵⣥⠾⠐⢣⢕⡝
⠀⠀⠀⠀⣸⣷⠻⣯⠁⠉⠓⠲⣈⠙⣧⡷⡇⡇
⠀⠀⡠⠊⣸⣿⣿⣯⣟⣦⡀⠀⠀⠑⠲⣍⠉⠁
⠀⡔⠁⡞⡇⡟⡿⠟⠀⣨⠋⠀⠀⣀⡴⡏⠀⠀
⠀⢷⡀⢹⣀⣁⣁⠴⠊⠁⢀⣠⣖⣿⣷⠁⠀⠀
⠀⢸⠙⠓⡶⢢⡤⣤⠴⣺⣥⡷⡖⣿⣿⠀⠀⠀
⠀⠈⠳⠤⣕⣷⣷⡔⣵⠨⡞⣗⡏⣿⣻⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⡟⢭⠒⢾⢷⠛⢿⠀⠀⠀