⢂⠡⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢦⣄⠀⠀⠀⠀⣀⣠⣤⡶⠶⠶⠷⣚⣯⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢟⠻⢋⡝⠻⠷⣤⣀⠀⣠⡾⠛⡍⢌⠱⡙⢻⡶⢟⠛⠭⡉⢔⣠⣧⣾⣿⣿⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣋⠱⡰⢌⠲⡡⢌⡱⠘⡌⡙⡛⢍⡰⠣⠜⢢⠓⡌⠦⡐⢎⣜⣤⣽⣿⣟⣿⣿⣟⠿⣜⢯⣿⣿⡄⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡫⢄⠣⡘⢢⢑⠊⡥⢑⠊⡔⢩⠔⡡⠜⡰⣈⠇⣙⢢⢩⠐⢣⠘⡘⢿⣿⡿⣭⢟⡻⣝⣻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀
⠀⠀⠀⣀⣀⣤⣤⣤⣴⡶⠶⠾⢛⡛⠻⢛⠫⠍⠭⡑⢌⠣⡜⢡⠎⡱⡘⠬⡱⢌⠣⡜⠰⡩⡐⢅⠪⡐⢡⠂⣍⣂⣧⠘⠤⡙⢿⣾⡿⣿⣿⣿⡿⣟⢯⣿⣿⡇⠀⠀
⠀⠀⣼⠉⡍⢩⢑⣌⣴⣬⣥⣧⣶⣬⣵⣬⣦⣙⣦⣵⢈⠱⡘⠴⡨⢔⡡⢃⡴⣈⢒⣌⣱⣠⣱⣬⣦⣵⠷⠟⢋⣩⡴⢋⢒⢡⠂⡝⢿⣵⣛⡿⣷⡽⣎⣿⣿⡇⠀⠀
⠀⠀⢿⣷⣾⣿⢿⣻⣿⡿⣏⣯⣝⣻⣿⢻⡿⣿⡿⠣⢌⡒⢥⠢⣅⣂⣡⣉⣀⣁⣉⣈⣉⣉⣁⣤⣤⣤⠶⡞⡛⠥⡘⠤⣉⠆⢣⠘⡌⢻⣿⣿⣽⣿⣽⡿⣿⠁⠀⠀
⠀⠀⠸⣿⣿⣹⣾⣿⣿⣿⣿⣿⡿⣟⣼⣣⣿⡿⢁⠣⢆⡉⢆⡱⡀⢆⡰⢠⠉⡍⢡⡉⢍⠩⡉⢅⠒⡄⢣⡔⢡⠒⣉⠖⡡⢊⡅⢣⠘⠤⢻⣿⣟⣿⡿⣻⡏⠀⠀⠀
⠀⠀⠀⢿⣿⣷⣿⢻⣝⣻⣟⣳⣽⣾⡟⣽⡟⢠⢃⠚⠤⡑⠪⠔⡡⢎⠰⢡⠃⡜⠢⠜⡊⢖⠡⢊⣶⠈⣽⣿⣀⢳⣶⣌⠰⣁⢾⡄⣉⠆⡅⢋⠽⣿⣼⠏⠀⠀⠀⠀⠀
⠀⠀⠀⠘⣿⣿⣞⣳⣾⣿⣽⣿⣿⡿⣼⡟⡡⢎⡌⢣⡑⢌⠣⣸⡐⢌⣂⠣⡘⢄⢣⡎⣼⣯⢂⣽⣿⢸⡟⢻⣷⣌⣿⣿⠷⠾⠾⣿⣶⣼⣀⢃⠦⡙⣷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣿⢿⣷⣿⣿⡻⣭⣿⣟⣿⠑⡰⢃⠜⣠⠘⣄⢳⣿⠡⢢⣿⡐⢡⢎⣿⣷⣿⣿⣸⣿⣿⣾⠇⣼⢿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠙⠿⣆⠱⡈⢷⣆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠙⢿⡹⣿⣾⣱⢯⣿⣿⡇⢣⢙⠢⡑⢢⠑⣼⣿⢇⢃⣿⣏⣰⣱⣿⣿⣿⣹⣿⣿⣿⣿⢿⡟⠁⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙⣷⢧⣂⡝⢷⡀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠻⣮⡹⢷⣿⣿⣿⡘⠤⢊⡱⠘⣄⣻⣿⣿⣞⣿⣟⠋⠉⠉⠁⠈⠉⠉⠛⠿⣿⢯⡿⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠎⣻⣿⣮⣿⣄⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⢌⡛⢻⣡⠚⡄⢂⠧⣼⣿⣿⠟⢹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣇⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⣴⣿⣧⡐⢿⡍⠻⢿⣧⡀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⠂⣆⣼⣿⠇⡜⢠⢃⢼⣿⣿⠃⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠘⢻⣿⡧⠀⠀⠀⠀⣀⣾⣿⣿⣿⣷⣈⢻⡆⠀⠈⠙⠀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⣀⣿⣾⣿⣿⠈⡔⢣⢨⣿⣿⡇⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠙⠷⣦⣄⣀⣀⣻⣃⣀⣤⣴⠟⠋⣿⣿⣿⣿⣿⣿⣬⢿⡄⠀⠀
⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⠟⣿⣿⡧⢃⢎⣰⣿⣿⣿⡇⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⣠⡼⠏⢀⠠⢀⠀⠌⠉⢙⣉⠉⠉⠁⡀⠠⢰⣿⣿⣿⣿⣿⣿⣿⣯⣿⠀⠀
⠀⠀⠀⠀⠀⠀⠺⠛⠋⠀⠀⣿⣿⡗⡈⢆⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠹⣿⡿⠀⠀⣠⡴⠾⠋⢀⡐⠀⠄⣄⠈⡀⠉⡀⣿⠂⢁⠃⠄⣱⣾⣿⣿⣆⠙⠻⢿⣿⣿⣿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠾⠿⠻⠿⢿⣿⣿⣿⣿⣶⣤⣤⣴⡾⠿⠛⠛⠉⢻⣄⠀⠂⢈⠠⣨⡿⠻⠶⣶⡾⠋⢀⠂⠠⣘⣼⣿⣿⣿⣿⠀⠀⠀⠉⠻⢿⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢠⠀⡄⠢⠄⡀⠠⠉⢻⣿⣿⣿⣿⣤⡀⠀⠄⢒⠈⡀⠛⠷⣮⣤⡾⠏⢀⡐⢄⠀⠠⠐⢀⢂⣵⣿⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠁⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢀⠄⡉⠄⢂⠠⠁⢂⠁⠀⢀⠂⠹⣿⣿⣿⣿⣷⣮⣄⠠⠈⢀⠂⠄⠀⡀⠀⠄⠂⠠⢀⠈⢄⣶⣾⣿⣿⣿⠀⠀⠉⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⡿⢁⠠⠌⢂⠐⠈⠄⢂⠰⠀⠌⡈⠄⡐⠈⣼⣿⣿⡿⣿⣿⣿⣿⣶⣤⣀⡌⣀⠄⣁⣠⣁⣦⣶⣾⣿⣟⠀⠙⢿⣿⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣸⣗⡠⠐⡈⠆⡈⠐⡈⢀⠂⠌⡐⢀⠂⡐⠀⣹⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⢿⣟⠋⠉⠛⢿⡷⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢸⣿⡆⠐⢈⠐⡠⠁⠄⠂⠌⠠⠐⣀⠂⠁⢄⣾⡿⠿⣿⣾⣏⠁⠀⢳⡉⠛⠻⣧⣄⡀⠀⠀⢘⣻⡏⠀⠀⠰⣎⡇⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣮⡀⢀⠁⠌⠠⠡⠈⠄⠡⠀⠂⢡⣾⠏⢿⣆⠀⠙⢿⣆⠀⠈⢻⡄⠀⠀⠉⠛⠳⠶⠛⠉⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣦⣌⣀⣄⣁⣈⣄⣢⣵⣿⣷⣶⣾⡿⠀⠀⠀⠹⢧⡀⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣀⣀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣽⣾⣿⡏⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⠏⠁⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠘⢿⣯⣟⣿⡟⠀⠀⣀⣠⣤⣶⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣁⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣹⣿⣿⣿⠿⣿⣿⡹⢿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠹⠿⢷⣶⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠁⠀⠀⢻⣷⣻⢏⣿⣧⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠄⣀⠂⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡟⡾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠠⠄⡀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡽⣏⣷⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⡐⠄⡀⠡⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀