⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢸⠀⠀⠀⣼⡀⠀⢀⠖⠉⠀⠉⠓⡄⠀⠀
⣸⠀⠀⠀⣿⡇⡰⠁⠀⠀⠀⠀⠀⢹⠀⠀
⣻⠈⠉⠉⡏⡏⠏⠁⠀⠀⠀⠀⠀⡋⠀⠀
⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀
⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⡥⠋⢙⡆
⡇⢠⠉⠙⢢⠀⢀⡇⠀⠀⢠⡏⠀⠠⠎⠀
⡇⠀⠀⠀⠀⣠⠏⠈⣆⠀⠀⠀⠀⠀⠀⠀
⡇⠐⠒⠚⠯⡁⠀⠀⠀⣉⠽⠟⠒⠉⠁⠀
⡇⠀⠀⠀⠀⡈⣦⢠⠞⠁⠀⠀⠀⠀⠀⠀
⣷⠀⡀⠀⢠⡇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀
⡟⢸⠀⠀⣾⠃⢰⡇⠀⠀⠀⣀⠀⠀⠀⠀
⣿⠘⠤⢴⠃⠀⢸⠁⠀⠀⠸⣌⡿⠀⠀⠀
⡇⠀⠀⡜⠀⠀⠘⠀⠀⠀⡀⠀⠀⠀⠀⠀
⡧⠀⢠⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀
⣿⠀⢸⠀⠀⠀⠀⠈⠙⣆⣧⣀⠀⠀⠀⠀
⣯⠀⠈⢧⠀⠀⠐⠚⢛⠏⡏⠉⠉⠁⠀⠀
⣗⠀⠀⠀⠙⠲⠤⠔⠋⠰⡇⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀
⠇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢤⣀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢋⣉⣀⣠⣄⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣴⣾⣯⠴⠚⠉⠉⠀⠀⠀⠀⣤⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⡿⡇⠁⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⣠⣴⡿⠿⢛⠁⠁⣸⠀⠀⠀⠀⠀⣤⣾⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⣠⠀⡇⢧⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠘⣿⠀⣀⡠⠞⠛⠁⠂⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣻⡦⣞⡿⣷⠸⣄⣠⢶⡟⠁⠀⠀⠀⣀⣴⠟⠋⠁⠀⠀⠀⠀⠐⠠⡤⣾⣙⣶⡶⠃⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡷⠰⣔⣾⣖⣾⡷⢿⣀⣀⣀⣤⢾⣋⠁⠀⠀⠀⣀⢀⣀⣀⣀⣀⠀⢀⢿⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠠⡦⠴⠴⠤⠦⠤⠤⠤⠤⠤⠴⠶⢾⣽⣙⠒⢺⣿⣿⣿⣿⢾⠶⣧⡼⢏⠑⠚⠋⠉⠉⡉⡉⠉⠉⠹⠈⠁⠉⠀⠨⢾⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠐⠀⠀⠀⠈⣇⡿⢯⢻⣟⣇⣷⣞⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣆⠀⠀⠀⠀⢠⡷⡛⣛⣼⣿⠟⢙⣧⠇⡄⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢶⠏⠉⠀⠀⠀⠀⠀⠿⢠⣴⡟⡗⡾⡒⡿⠯⢏⡅⠀⠀⠀⠀⣀⢀⣠⣧⣀⣀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣷⢿⣫⡽⡿⠿⠗⢷⠀⠀⠑⣗⡀⠀⠀⠀⠈⠙⣿⢭⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⡴⢏⡵⠛⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠀⠰⡟⣿⠀⡌⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠂⢿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⣼⠛⣲⡏⠁⠀⠀⠀⠀⠀⢀⣠⡾⠋⠉⠀⠀⠀⠀⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡴⠟⠀⢰⡯⠄⠀⠀⠀⠀⣠⢴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡾⠁⠁⠀⠘⠧⠤⢤⣤⠶⠏⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠘⣇⠂⢀⣀⣀⠤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀
⠀⠀⢀⡤⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣀⡤⠀⠀
⠀⢀⠏⠀⠀⠀⢀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠙⡇⠀⠀⠀
⠀⠸⡀⠀⠀⠒⠁⠀⠀⠀⠀⣠⢯⠄⠀⠀⠀⠀⠀⠀⠨⠀⠀
⠀⠀⢣⡀⠀⠀⠀⠀⠀⠄⡜⢽⣿⣾⡵⠤⠆⠐⢀⠀⠈⡆⠀
⠀⠀⠀⠀⠑⠒⠒⠒⠀⠀⠀⠀⢯⠆⠀⠀⠀⠀⠀⠀⠀⡃⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⢂⠀⡜⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣢⠶⣟⠓⠦⠂⠀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠘⠉⠉⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡎⠀⢠⠖⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⡄⠀⠈⠣⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠒⠢⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠐⣸⣤⢻⡘⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡷⢮⣔⡠⢀⠀⠀⠀⠀
⠀⠐⣴⡆⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢲⣌⠦⣽⣶⣿⣧⣥⣾⢿⡛⠛⠋⠉⠛⠛⠻⢿⣿⣿⣿⣿⣾⣥⢊⠀⠄⠀
⠀⠀⠠⠐⠧⠀⠍⠄⠆⠀⠀⠀⠀⠄⡐⢘⣈⢦⣱⣻⠭⠿⣿⡿⠿⠛⠛⢵⠗⠃⠆⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣹⠀⠄⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣤⢁⡾⣈⠦⡌⠀⠀⠀⣀⠰⠀⠀⠀⠀⠠⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣹⠀⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢤⣻⣼⣟⡽⠋⠄⣀⡁⠒⠚⠤⠠⢒⡒⡲⠡⠀⠀⢄⣀⣁⠼⠂⠀⣀⠀⠀⠀⠀⠀⠹⡿⢱⣁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠡⢎⢳⡛⣮⣜⣁⣂⣀⣀⣀⠀⠐⠂⠊⠀⠀⠀⠀⠀⠀⠈⠱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠱⠈⢆⡑⠆⢎⠹⡹⠿⢿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠉⠄⠐⠈⠠⠈⠍⠃⠺⢹⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢄⢋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠎⠈⠴⠙⡚⠝⡻⠿⡿⢿⣿⡄⠀⠂⠀⠄⠁⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡴⣆⢰⣦⣰⡆⠀⣶⣰⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠐⡈⠐⠈⠑⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠰⠁⠹⠼⠇⠹⠇⠀⠿⠸⢤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠐⣶⠒⡴⠒⣦⣰⠒⢢⠀⠀⣏⣰⡆⡶⣰⠒⠒⢰⣀⣰⠐⢲⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠛⠀⠓⠖⠋⠙⠲⠚⠀⠀⠃⠘⠃⠓⠙⠲⠛⠘⠀⠘⠀⠖⠀⠀⠀⠀⠀
▀█ ▀ ▄█ ▀█
▄▄█ ▀█ ▄▀▀▀▄ █▄▀▀▄ ▀▀▀▄ █ ▄▄█ ▄▀▀▀▄ █ █
█ █ █ █▀▀▀▀ █ █ ▄▀▀█ █ █ █ █▀▀▀▀ ▀▄ ▄▀
▀▀ ▀ ▀▀▀ ▀▀▀ ▀ ▀ ▀▀ ▀ ▀▀▀ ▀▀ ▀ ▀▀▀ ▀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡅⡿⡃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣺⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⠇⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣠⡟⠃⠀⠀⢿⢳⡄⠀⠀⡀⠀⠀⠀
⠶⠶⠛⠓⣯⣽⡄⠀⠀⠀⠀⢮⣩⣿⣤⡤⠷⠧⡀
⠀⠀⠀⠀⠀⠉⣯⢆⠀⠀⢄⢯⠟⠉⠙⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡖⢇⠀⣞⡏⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠩⡇⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡻⡅⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀