⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣟⣿⣽⣿⣆⣀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠒⠦⣀⠈⢦⠀⠀⠀⠀⠀⠀⣼⣿⡟⠉⠉⢉⣹⣿⣿⡿⠿⣟⣻⣿⣿⡿⢿⣿⣿⣷⣍⠛⠉⠹⣿⣿⢻⣿⣿⢏⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣾⣷⣦⣤⣦⣶⣶⣿⡟⠀⣠⣾⠿⣻⣽⣿⣶⣶⠿⣿⠶⠭⣙⢧⠀⠉⡙⢿⣷⡄⠚⢿⣿⣼⣿⡏⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠈⠁⠀⣸⣿⡟⠀⣰⣿⠋⣾⣿⡏⠀⣅⠸⡈⠇⠀⠐⠾⢃⠔⢀⠃⠀⣿⠇⠀⣿⣿⠏⢿⡇⠀⣿⣿⣷⣤⣀⠀⢀⠼⠀⢀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡆⠠⣿⡏⠀⣰⣿⡟⠀⠻⣿⣄⠀⠨⣷⣭⢯⣜⣉⠭⢂⡠⠊⢀⣾⠋⢀⣼⡿⡽⠀⢸⣿⣶⢿⣿⣿⡿⢿⣻⣯⣷⡄⠈⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠹⡆⠀⠘⠿⣅⠀⠀⠈⠻⢷⣦⣈⠙⠛⢲⣒⠚⠁⣠⣴⠟⡡⠶⠛⢫⠜⠀⠀⢸⣿⠃⠀⠟⠁⡈⣴⣿⣿⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣇⠀⠐⡄⠀⢠⠈⠓⠦⣾⣿⣶⣬⣉⠛⠒⣢⣤⠶⠛⠋⢀⣨⣴⠶⠚⠹⠄⠀⠀⠐⠃⠀⠌⠀⡞⣱⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⣿⣿⣆⠀⠈⡀⢸⡇⠸⣶⣤⡙⠻⠿⠿⣾⣏⣭⣤⣤⠶⠞⠛⠉⡡⠀⠀⢀⣃⣤⣶⣿⣿⡓⠦⢸⢇⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠹⣿⣿⣦⣄⣠⣸⠿⠀⠹⣿⣿⣧⣄⠠⠀⢠⣌⣉⣉⠉⠛⢛⣠⣤⣶⣾⡿⣟⠿⠛⠋⠀⡠⠁⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⠍⠉⠐⠀⠈⢿⣿⣿⣿⣿⣿⣷⣶⣬⣿⣿⣻⣀⣽⣤⣽⣿⣿⣶⣿⠿⣟⠻⠙⠈⠁⢀⣠⣤⣴⣾⣧⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⣯⢅⠈⠐⢀⠀⠀⠀⢣⠑⠢⠍⠙⡛⠿⠿⣿⣿⠿⢿⠟⠛⠿⠩⠛⠓⠈⠉⠀⠀⣀⣴⣾⣿⣿⣿⣿⠿⣿⡄⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣞⣆⠁⠠⣆⡀⠀⠀⣎⡇⠀⠀⠀⠉⠐⠢⢌⡁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡣⠔⠛⠑⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣧⣤⣄⡀⠘⠘⠄⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⡄⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣤⣀⠀⡀⠀⠀⠀⠀⢀⣨⣴⣿⣿⣿⣿⣿⣿⣿⡿⠟⣁⣰⡷⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⠿⢛⠽⠑⡋⢡⣴⣿⠟⣿⠇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⡤⠜⠛⢟⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⠉⠈⢀⣴⣿⣿⢿⣿⡏⣸⡟⢠⠟⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⡭⠄⠀⡀⠀⣈⠄⠀⠉⠉⠉⠉⠉⠉⠁⠁⠈⡄⠀⠙⠻⢿⣟⣿⠟⣰⡟⡠⠋⠀⠀⠀⠀⣀⣤⣿⣿⣿⣯⡶⢞⠳⣄⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣷⠊⠜⢠⠔⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⠜⣚⣡⠾⠍⠊⠀⠀⠀⢢⡒⣤⣿⣿⣿⠟⠋⠉⠁⠈⠛⠄⠂⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⢪⣶⣿⣾⣿⣷⣖⡀⢀⠄⡂⠀⠀⠀⠀⠀⡶⠊⠉⠈⠀⠀⠀⠀⠀⠠⠬⢥⣶⣿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⠿⠟⠛⠻⠟⠻⣳⣷⣿⢟⣡⢆⣀⠀⢸⡀⠀⠀⠠⠀⢀⠀⠀⣷⣽⣿⣿⣿⣿⡿⠿⠧⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠁⠀⠀⠀⠀⠀⠀⣿⡿⣡⣿⣾⣿⣿⠂⡸⠁⢀⠔⠁⠀⠀⠁⠐⠛⢿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣤⣀⡀⡀⠀⠀⠀⠀⣿⣝⣽⣿⣿⣿⠥⢺⣆⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡿⣿⣿⣦⣤⡀⢸⡿⠋⠁⠀⠘⣆⢁⠹⡇⠀⠀⠀⣀⣤⣄⡀⠀⢀⣀⣤⣶⣦⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⠟⠛⠋⠑⠀⠀⠢⠀⠈⠋⠙⢆⠁⠀⠀⠀⠀⣿⡈⠀⡇⠀⠀⢰⠋⠀⠀⡰⣴⣿⣿⣿⣿⣿⣿⣿⡶⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⢁⠈⠂⢀⡠⠄⠒⡂⠉⠀⠁⠒⠬⢀⠀⠀⠀⠀⢸⡇⠁⡇⠀⠀⠘⣆⠤⠤⡤⢄⣀⣁⠔⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⢿⣀⣤⡒⣉⣤⣴⣴⡆⠃⠀⠀⢀⡣⠃⠀⠈⠢⡀⠀⣸⠇⠀⡇⠀⠀⢀⠄⠋⠀⠸⠛⠠⠡⠈⠑⠢⣄⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⠠⠈⠀⠀⠀⠀⠀⠈⢢⡿⢘⡔⠇⢀⠎⠁⠀⠀⠀⢆⠀⠀⣠⣰⣾⣷⣷⣿⣶⣤⣀⠀⠈⢦⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢠⡾⠟⠛⠛⡻⠿⣿⠿⣿⡿⠟⠯⠁⠀⠂⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠷⠁⠰⢠⠃⠀⠀⠀⠀⠀⠀⠁⠸⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⢤⡑⣄⠀⠀⠀⠀
⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⢉⣀⡤⠄⡀⢀⡏⢀⠂⡏⢀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠋⠙⠛⠊⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠃⠁⠀⠀⠀⢼⠳⠃⡜⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠒⠒⠒⢤⡘⢄⢸⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢾⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢿⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀