⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⣉⣉⡉⡐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⣉⣉⣁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⣠⣴⣒⢆⢌⣆⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡰⡎⣱⣒⢦⡁⡀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣊⡠⠀⣁⣽⣶⣺⣿⣧⡪⡎⠄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢠⢳⢗⣼⣿⣷⡚⠧⣉⠔⢄⣱⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣐⡿⢋⡸⣾⣿⣿⠟⠋⣩⣿⣿⠾⡈⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⡀⢠⣾⣿⣍⠙⠻⢷⣿⣷⣮⡋⢟⣂⡀⠀⠀⠀⠀
⠀⠀⡔⠉⢀⣴⣟⢾⡿⠟⢁⣤⡾⢿⣿⡟⡃⠠⢂⡸⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⠠⠔⣨⢿⣟⠚⣵⣠⡀⠹⢿⡿⡝⢧⡀⠉⢢⠀⠀⠀
⠀⠀⠀⢠⠟⠐⢡⣿⣥⣶⡿⠋⣤⣶⣿⣿⣌⣦⠀⠑⠳⡀⠀⣁⡂⣄⠀⡀⠀⠀⢀⠠⢀⢀⠄⣂⠀⢀⠜⠁⠀⢺⣫⣿⣗⢿⡤⠙⢿⣶⣬⣷⡄⠢⠹⢄⠀⠀⠀⠀
⠀⠀⢠⠃⠀⢠⣿⣯⣾⣿⢂⣠⣼⣿⣟⠏⣉⣹⣔⠀⠀⠈⠁⠀⠀⠀⢀⡔⠀⠀⢨⠀⠀⠀⠀⠀⠈⠁⠀⢀⣣⣍⣉⠽⣿⣧⣥⣀⠐⣿⢿⣿⣿⡄⠀⠉⠆⠀⠀⠀
⠀⢠⠏⠀⢠⣿⢿⡟⢩⢳⣿⣹⣿⣿⣟⣳⣿⣻⡇⣭⠥⢆⣀⠀⢀⡤⡞⠀⠀⠀⠀⢳⣕⠤⣀⣀⣀⡴⠺⣍⣙⣙⣟⣦⣿⣿⣿⣷⣿⡟⡎⢻⡽⣿⡀⠀⡘⡄⠀⠀
⠀⠎⠀⠀⣸⣏⢿⣁⣘⡍⢰⣿⠏⢸⣿⣿⡷⣿⣿⠕⢻⣭⠯⡏⣩⡴⡇⠀⠀⠀⠀⣸⢦⣍⣹⠭⣠⠛⢩⣿⣿⣻⣿⣿⡿⣇⠹⣿⣆⢩⡡⣘⡇⡽⣧⠀⠱⢹⠀⠀
⠸⠀⠀⠀⡟⡍⣿⣿⣏⡇⣿⣿⡤⣿⣯⣿⣿⡿⣥⣶⡿⠗⠘⢹⡪⡾⢻⠀⠀⠀⢠⢟⠶⣵⠙⡄⠸⣷⣆⣼⢿⣾⣿⣯⢽⣿⣆⣻⣿⢸⣷⢿⣧⠸⡭⠀⠀⠀⠇⠀
⡆⠀⠀⠀⣿⢱⣟⡏⠛⡘⡿⡂⢰⡟⠿⣿⣿⣿⣿⣿⣷⣾⣾⣼⣿⡷⡿⠀⠀⠀⠈⣾⣼⡟⣶⣾⣷⣾⣿⣽⣿⣿⣷⡿⠿⣻⡆⢤⣿⢃⠃⢸⣿⣣⣷⠀⠀⠀⢸⠀
⠃⠀⠀⢸⣾⣿⡟⢣⣠⣗⢳⠈⢻⠿⠄⡷⢈⣿⣼⣿⣼⡾⣽⢿⡇⡎⠁⠀⠀⠀⠀⠋⡉⣸⢿⣭⢾⣤⣶⣯⣧⣼⠁⢁⠸⠷⣿⠉⡝⣹⡆⡼⢿⣇⣜⠀⠀⠀⢸⡀
⠀⠀⠀⠀⢻⢻⡧⠀⠙⣧⠈⣷⡞⣧⡀⣿⣤⣿⣷⡙⣿⣷⡤⣼⣴⠃⠀⠀⠀⠀⠀⠀⠡⣶⠧⣤⣿⡟⠘⣻⣾⣟⣄⣾⠀⣼⢧⣼⠃⣼⠋⠀⣸⡟⡟⠀⠀⠀⢸⡇
⢀⠀⠀⠀⢾⣇⣳⡀⠠⣻⣧⢹⢷⣿⣟⠿⣽⡙⢻⣿⢏⠙⠻⠧⡉⡀⠀⠀⠀⠀⠀⠀⢀⢋⡿⠛⠉⢵⣾⡏⢁⣴⡾⠟⣻⣿⡾⡏⣰⣟⠄⠀⡿⣸⡷⠀⠀⠀⢸⡇
⣾⡄⠀⠀⢨⣿⣿⣣⠸⣿⣿⢟⠁⢻⣿⣤⣿⣯⣾⣟⡷⢯⣷⡀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡁⢣⡿⠥⣾⡏⣔⣽⣯⣶⣼⣾⡟⠀⣻⣿⣻⠃⣸⣿⢿⡇⠀⠀⢀⣸⠁
⡇⣧⠀⠀⠀⣷⣵⡌⠘⢷⣿⣿⣷⣔⣿⣿⣟⠿⣿⣿⣝⡂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢱⣡⣿⡿⠫⣾⣿⣷⣿⣤⣾⣿⣿⢻⠁⣈⣾⣾⠁⠀⠀⣸⢻⠀
⢰⢹⡇⠘⣷⢸⣿⣟⣄⠸⣻⣿⣶⣿⣿⣿⣟⢄⡀⠙⣯⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠟⡽⠋⢀⠀⣶⡿⣿⣿⣿⣿⣣⣿⠇⣮⣿⣿⡏⣰⠄⢀⠿⡄⠀
⠈⣎⣿⡄⠈⣧⣿⢟⡘⣷⣿⣵⠸⣿⣿⢡⣻⣿⡿⡷⢷⣵⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣮⡿⠶⣻⣿⢛⡆⣻⣿⡿⠀⣼⣿⣾⢃⢻⡿⣼⠃⠀⣾⢣⠁⠀
⠀⢰⣿⣻⠀⠘⣿⡏⢇⠈⡻⣦⡄⣘⣿⣷⢟⠉⠿⢿⣦⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣴⢟⠛⠕⢿⣿⣾⡿⢁⣢⣴⣟⠃⡸⣹⣷⠋⠀⣸⡏⡆⠀⠀
⠀⠀⢹⣿⣆⠀⢸⣿⣿⡵⡌⠹⣿⣾⣿⡫⣳⣕⣦⣄⡸⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⠋⡀⢠⣤⢾⣿⣿⣷⣵⣿⡟⠠⢾⣽⠟⡏⠀⣰⣿⡜⠀⠀⠀
⠀⠀⠀⢛⢻⢦⡈⣿⣿⣟⣆⣐⣿⣿⣝⢿⣏⠮⡛⣻⠿⢷⣭⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢫⣆⠥⣶⠛⠕⢻⡿⢫⣿⣿⢏⢀⣴⣫⣾⣽⠁⣠⣾⡿⠀⠀⠀⠀
⠀⠀⠀⠈⡄⢃⠆⣿⣩⢿⣿⣿⣿⣿⣿⣷⣝⠇⢈⠲⡳⡄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⢚⠅⠀⠰⣫⣽⣿⣏⡾⣿⣿⣿⣿⣿⡞⠋⣿⢣⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠘⡐⡤⢽⣿⢿⣽⢿⢿⣿⣿⣿⡿⣷⣧⡒⢌⢯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣋⣐⢔⣾⡿⣿⣿⣻⣿⣯⡿⡿⣿⢿⢢⠀⣇⠂⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣡⣾⣿⣿⣾⡅⢿⣿⡿⣿⣻⡿⣿⡙⣿⢷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡬⡿⢫⡿⢿⣿⣿⢫⣿⠎⣼⣟⣿⣿⣿⣰⠆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣿⣧⡹⡏⢿⣿⣌⢣⢳⣼⣿⣿⣎⣧⠈⣆⠈⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠃⠁⡜⠀⢰⣵⣺⣿⢇⠎⡔⣰⣿⣯⢻⡿⢷⣻⣀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⡎⢷⠙⡘⡿⡏⠀⠋⢿⢿⣿⣞⣡⡀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡞⣻⡷⡷⡟⠀⠀⡿⣿⠃⠈⣏⢳⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢰⠘⣡⠐⠸⡙⡀⠀⠘⣇⠻⣾⣧⡑⠐⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠂⠈⣰⣟⡳⢰⡁⠀⢸⣱⡇⠀⣘⠆⡏⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠆⠀⢳⣽⢆⢠⢽⠻⠸⢟⣷⠀⢯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡼⡿⠃⠀⣏⡆⢴⡿⡟⠀⢠⡏⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⡿⠀⢘⡎⣟⣀⣽⡇⠃⠈⣫⣾⣲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢢⡾⣽⠁⠀⢸⢀⠀⡾⢹⡁⠀⢮⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢫⢧⠻⣿⡸⡂⠹⡧⢀⠄⢹⡈⢳⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡿⠂⡇⢀⡈⣼⠎⢠⢇⣿⠀⣴⡌⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡘⣟⣿⣳⢗⣸⣧⠙⡀⠀⣧⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣠⠁⠌⠇⣸⡅⣾⣾⣿⣸⠏⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡇⣿⢞⡾⡞⢛⢿⠀⠐⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠈⠀⡟⡇⣷⡞⣷⣿⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢇⢹⡾⡗⣷⣪⢿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣹⣇⠻⡘⣿⡟⢸⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⢸⡇⡇⢸⣟⣯⡆⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣙⡀⠀⣾⣡⡇⢰⢹⡇⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⡶⢹⡆⢳⢦⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢱⠹⢸⡆⠓⠈⣸⡇⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⢸⡇⠀⡀⣿⠀⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠃⠈⣴⡆⠘⢺⡇⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡘⢸⡇⠀⡷⡟⠠⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⡆⢿⢹⠀⢻⢇⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⢰⡇⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⠀⢸⣿⢰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠼⡃⠸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠃⢸⡏⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⢠⠀⢴⠃⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠸⣟⠀⡆⠈⡄⠀⠀⠀⠀⠀⠀