⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠶⠶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⣋⣀⢀⣀⣤⣤⡾⠿⠿⠿⣿⣷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠛⢻⡿⣿⣿⣦⡀⠀⢠⡾⡿⠯⠽⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⣿⣿⣷⢴⡏⠉⣧⠀⠀⢹⠒⢻⣀⣘⡇⠀⢸⠷⠶⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠉⠁⠀⠀⠀⠘⣧⠀⠀⠀⢀⡼⠀⠉⠁⠀⠉⢱⡏⠀⠀⠰⢄⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⠇⠀⠀⠀⠀⠀⠀⣈⡙⠒⠒⠉⠀⢠⣄⣀⣀⣀⡼⠃⠀⠀⠀⠈⢣⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⡾⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⣀⠀⠈⠉⢉⣁⣐⡶⠀⠀⠀⠀⠀⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠉⢲⡄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋⠀⠀⠀⠀⠀⢈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣏⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠐⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣗⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣀⣀⣀⡴⠻⣄⣂⣀⡴⠟⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⣯⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠉⣀⣀⣈⣳⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⡟⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⣠⠖⠋⠁⠀⠀⠉⠻⣆⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⢀⡞⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠙⠶⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡠⠊⠁⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⡴⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠈⢳⠲⣄⠀
⠀⠀⠀⣠⡾⠥⠤⠤⠬⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠉⠓⠒⠤⢤⣀⡀⠀⣠⣞⣉⠀⠀⢠⠞⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⢄
⠀⡴⢋⣁⣠⠤⠄⠒⠠⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⢀⡴⠚⠛⢷⡏⠀⠈⢷⠔⠁⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⣸⠁⡸⠀
⢀⡿⠋⠉⢀⣀⣀⡤⠤⢤⣀⣀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢾⠀⠀⢠⣷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⣰⠇⠐⠁⢀
⡞⢀⡴⠚⠉⣠⠞⠁⠀⠀⠀⠈⠙⢦⡀⠀⠀⢳⡀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡃⢀⣤⠞⠁⠀⠀⣠⠏
⠙⣿⠀⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⢩⢦⠀⣤⣧⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣍⣥⢔⣠⡴⠛⠁⢠
⠀⢹⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢧⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠉⠁⠀⠀⢰⣜
⠀⠘⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⢿⠟⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠀⠀⠀⣌⠇
⠀⠀⠹⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡜⠀
⠀⠀⠀⢻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣰⠃⠀
⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡀⠀
⠀⠀⠀⠀⠀⣸⣷⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀
⠀⠀⠀⢠⣴⡿⠞⠿⣄⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⢣⡀
⠀⠀⠀⠸⣼⣦⣠⣶⠿⢦⣤⣾⣷⡀⠀⠀⠀⠀⡅⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠇⢰⠃
⠀⠀⠀⠀⠀⠉⢿⡇⠀⢰⡏⠀⣻⡟⠛⢦⠀⢀⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⠇⠀⠀⠀⠀⠀⠀⢠⡆⢀⣴⣃⢀⣠⠏⠀

⠀⡄⣀⠠⠀⠄⡀⠄⠠⢀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣄⣀⣀⠀⡀⠠⢀⠠⢀⠀⠀⠀⠀⢠⢲⢧⠀
⠑⠠⠙⣶⢆⠦⣄⠀⠁⠂⠀⠀⠐⠀⠁⢈⣀⢀⠀⠀⣿⣿⣿⠟⠀⢿⣿⣿⣿⠂⠀⠀⠀⠸⢿⣟⡻⣶⣤⣆⠀⠄⠂⠀⠀⢀⡠⣿⠏⠀
⢦⣔⡠⢹⡿⣷⢦⡙⣦⣄⠀⢀⠂⢈⣴⡟⠏⠁⢀⡞⠛⠉⠀⢀⠀⠤⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠏⠉⠓⠌⠳⢀⠡⠈⡀⠀⡟⠁⡀⠂
⠋⠚⡙⠳⡳⣬⢄⡉⢲⣟⣦⠀⠂⣿⠁⠀⣠⣶⣯⣀⣀⠀⠀⠉⠀⠀⠀⠈⠲⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠱⠀⡐⠀⡁
⠀⢂⠀⠡⠐⢭⢞⡵⣳⣬⢻⣧⠸⠃⠀⢸⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣶⣶⣿⣿⣿⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⣀
⠀⢂⠈⠄⡘⠈⢎⡽⣳⢯⣟⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⡤⠀⡁⢾⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠌⡀⢄⡑⠠⠌⡱⣟⡾⣝⣿⣧⠀⠀⠈⣿⣿⣿⣿⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠇⠠⠁⡈⣿⣆⡴⠾⣦⡀⠀⠀⠀⠀⠀⠀⢀
⣴⡶⣄⠎⠰⢁⠂⡱⢯⡿⡹⠞⢿⡀⠀⠀⢻⣿⣿⣷⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⡇⠀⠌⣡⡄⠂⢈⠻⣷⡀⠀⠀⠀⠀⠀
⠷⠛⠌⢀⠂⠌⡐⠐⠉⢷⡀⣠⣿⠄⠀⠀⢸⣿⣿⣿⣯⣿⣿⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢼⡇⢰⡾⠉⠙⠛⠶⣄⠸⡗⡀⠀⠀⠀⠀
⠀⠌⢀⠂⡀⠂⠄⣄⣰⡿⢿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣴⣶⡿⠛⠒⣾⣶⠀⠈⣠⣿⣦⡀⠀⠀⠀
⠀⠂⠄⡐⠀⠡⣰⡞⠉⠀⣾⣿⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⡟⣿⢇⡤⢴⣏⠁⣠⡟⠉⠉⠉⠀⠀⠀⠀
⠀⠡⢀⠐⠈⣴⡟⠀⠀⢰⣿⠃⠀⠀⠀⠀⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣴⠛⠀⢠⡾⠃⠀⡈⣩⣿⡾⠟⠻⠿⢤⠀⡀⠀
⠀⠡⠀⠂⢡⣿⠄⠀⣠⣿⠇⠀⠈⠀⠀⢀⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣾⣿⡇⣠⠚⣧⡀⠀⣼⣿⡽⠄⠀⣀⠄⠘⣇⠠⠁
⠀⡁⣂⣵⠿⠿⠶⠿⣿⣏⠀⠀⠀⠀⠀⢸⠃⣿⠟⡛⠉⠋⠙⠛⠛⢿⣿⣿⣇⠀⢀⣿⠛⠉⠉⠀⠀⠙⢿⠉⠈⢳⡌⠀⠛⠀⢀⡏⠀⠄
⠀⣠⡟⠁⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⢠⡟⠰⣿⣄⢳⡄⠀⠀⠀⠀⠈⢿⣿⣿⠀⢸⣿⠄⠀⠀⣶⣄⠀⣼⡇⠀⠀⠳⣴⣦⠀⠀⠉⠢⠄
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢀⡾⠁⢸⡿⢬⣹⢷⡀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠋⠀⢀⣰⣿⣿⣿⣿⣤⣄⣀⡸⣿⠁⠀⢀⠀⠀⠀
⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⢸⣿⣦⢻⢿⡇⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⢀⠀⣿⣿⣿⣿⣿⣟⠉⠉⠁⢈⠀⠀⠀⠃⠀⠀
⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠸⣯⣿⣿⣾⣿⠀⠀⠀⠀⠀⠈⠉⢿⣧⠀⢀⡴⠛⠹⠋⠝⠉⠉⠑⣶⣄⣸⠀⢠⡄⠀⢀⣄
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠂⠄⣿⠛⠿⢿⣿⣄⠀⠀⠀⠀⠀⠀⡈⢿⡆⢸⣁⡀⠀⠀⠀⣤⣶⣶⣿⣿⣹⣠⣾⢳⡀⠾⠛
⠀⢀⡠⣦⡰⣧⢠⠟⠓⠚⠏⠀⠀⠀⠈⠀⠌⢿⣄⠂⠀⠉⠹⡆⠀⠀⠀⠀⠀⠀⠀⣿⡈⢣⣷⡶⣶⣾⣿⣿⣿⣿⣿⠿⣋⢅⡀⡿⣤⣀
⢸⣷⣻⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡬⠐⠿⣷⣾⢶⡶⠃⠀⠀⠀⠀⠀⠀⢀⣼⢷⣄⡙⣦⡉⠘⠻⢿⣿⣿⡯⠓⡜⣎⢖⡹⢆⠇
⡷⣞⡽⠎⠁⠀⠀⠀⠀⠀⡄⠀⠀⠠⠟⠉⠀⠀⠀⠉⡝⠛⠋⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⣨⠟⠁⠙⠶⣶⣟⠹⠋⠀⣠⣴⢼⡾⡁⢂⣀
⠽⠺⡜⠀⠀⠀⠀⠀⠀⡒⢻⠖⡀⠀⠀⠀⠀⠀⠀⠨⡐⢃⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⣥⢨⣷⠽⣿⣿
⢀⠁⠀⠀⠀⠀⠀⠀⡒⣭⠳⡬⠵⠆⣀⣀⢀⠀⠀⢠⢁⡋⠔⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠙⡻⣿⣿⡟⢏⡶⣏⡌⠙