⠀⢀⡀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀
⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⠤⠐⠂⠈⠉⠉⠁⠉⠈⠀⠉⠈⠁⠉⠀⠉⠈⠉⠀⠒⠀⠤⠤⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀
⢸⠀⠀⠀⢶⣄⡠⠤⠤⠒⠒⠒⠂⠤⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⣀⣀⡀⡄⣀⣀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀
⠸⡆⠀⠀⠈⡇⠈⠁⠀⠀⠓⢦⣀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⣀⠤⠔⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠑⠒⠒⠤⠤⠐⠒⠊⠉⠀⠀⠀⠀⡤⣄⡀⣐⣴⡿⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠱⡀⠀⠀⠘⡄⠀⢠⡀⠄⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡛⢡⠐⣹⣏⠁⢀⡇⠀⠀⠀⡎⠀
⠀⠀⢳⠀⠀⠀⠘⣤⡈⢣⠘⣴⢻⣾⣤⣰⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣯⣶⣿⣿⢡⢆⡙⠄⠻⠶⠞⡇⠀⠀⢘⠀⠀
⠀⠀⠀⢣⠀⠀⠀⣿⠀⢂⠹⣾⣯⢿⣿⣿⣆⠀⣿⡏⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢿⣿⣿⣷⣯⣷⠌⢂⠀⠀⣰⠃⠀⠀⢸⠀⠀
⠀⠀⠀⠘⡄⠀⠀⠸⣇⠠⠘⢸⣿⣿⣿⠻⣯⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠻⢿⣿⣿⡏⡜⠀⠀⣼⠏⠀⠀⠀⡜⠀⠀
⠀⠀⠀⠀⢣⠀⠀⠀⢻⡄⣱⣾⠟⠋⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠉⠙⠿⣷⣌⣠⠏⠀⠀⠀⡸⠁⠀⠀⠀
⠀⠀⠀⠀⠀⡆⠀⠀⠈⣿⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣀⣤⣶⣦⣤⡀⠀⠉⢻⡟⠀⠀⠀⡰⠁⠀⠀
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠸⣄⠀⠀⢀⠀⠀⠀⢰⣿⣽⣿⣿⣿⣿⣿⣷⣤⠢⡐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⣲⢇⣶⣿⣿⣿⣿⣿⣾⣿⠀⢠⡾⠀⠀⠀⢰⠃⠀⠀
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⢻⣼⠟⠂⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠂⡜⢀⠂⠀⠀⠀⠀⠀⠀⠠⠈⣄⡞⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⠃⠀⠀⢠⠇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠹⣤⠀⠄⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡇⠐⢂⠈⠄⠀⠀⠀⠀⠈⠠⠑⢨⠑⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⠀⠀⠀⡸⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠘⣇⠀⠀⢀⠈⢻⣿⠿⣿⣿⣿⣿⣫⣿⣿⡷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠄⠂⣾⣿⣿⣿⣿⠿⣿⣻⠟⢁⣼⠏⠀⠀⠀⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⣻⠐⠈⠄⡨⠄⡌⢛⠶⠦⠷⠿⠛⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠡⠉⠀⠉⠛⠛⢛⣛⣡⡶⢯⡏⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⡇⠘⡈⠰⣀⢣⠘⡤⢢⠑⡰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠀⠀⠈⠀⠐⢈⠡⢩⣷⠟⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⡇⠰⢈⠡⢂⠦⢩⠔⡡⢊⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⠋⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡼⢃⠐⠌⢢⠍⣎⠱⣊⠱⡌⡘⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢫⣽⠃⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡼⠁⢈⠐⡈⢆⡉⢆⠳⡌⡓⢬⠑⡌⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⡤⢐⠈⣆⣿⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣸⢃⠐⠀⠂⠌⢠⠘⡌⡓⢬⡑⢎⡒⠄⢡⠠⢁⠢⠄⡀⢀⠠⣴⡲⢦⡲⢶⡼⣿⣶⢰⡟⢡⡌⢦⡹⣾⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⡴⢃⠄⠀⠌⠐⠈⠄⢊⡐⢉⠆⡙⠦⣉⠞⣠⠢⢡⢆⠱⣈⠦⣁⠛⣿⣳⣿⣯⣿⠟⣥⢧⣚⣥⢺⣽⠓⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⡰⠁⠀⠀⢀⡾⣑⠈⠀⢈⠠⢁⠢⠌⡄⢌⠢⢌⢡⠓⣬⢚⡤⣛⡴⢪⢵⣘⢦⡝⣺⢴⣻⣿⣿⢣⣟⡞⣧⠿⣜⣿⡎⢧⣹⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⡰⠁⠀⠀⢠⢟⠰⠁⠀⠀⠀⠐⡈⠰⠩⢜⢢⠳⡌⢦⡝⢦⣋⠶⣣⢻⡝⣮⡝⣶⣹⢷⣫⣟⡿⣞⡿⣾⡽⣯⢿⣽⢺⡝⢦⡹⡇⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢰⠁⠀⠀⢠⠏⠰⠁⠀⠀⠀⠀⠐⠀⠡⢁⠊⡄⢣⠘⣅⠚⡤⢋⢞⡱⢧⣛⢶⡻⣵⣻⣞⣷⣫⢿⣽⣻⢷⣻⣯⣟⡾⣻⢼⢣⡱⣧⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢀⠇⠀⠀⢀⡾⣸⠁⠀⠀⠀⠀⠀⠀⡀⠁⢂⠐⡈⢄⠣⢄⠋⡴⢉⠦⣙⠶⣹⢞⡽⣳⡽⣞⡾⣽⣻⢾⣽⣻⡽⣾⡽⣳⢏⣟⢦⡑⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀
⠀⣸⠀⠀⠀⣼⢣⡅⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠂⡐⠠⠒⣈⠒⡌⡱⢊⡥⢫⡕⣫⢞⡵⣻⣝⣻⢷⣯⣟⡾⣳⢟⣧⢿⡹⣞⢼⡣⢞⡸⡇⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡯⠽⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠂⠠⢀⠡⠒⣀⠣⢌⡑⠣⡜⡡⢞⡱⢎⡷⣣⢯⣛⡾⣳⠾⣽⡝⣾⢣⣏⢷⡹⣆⠟⡤⢓⣷⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀
⠰⠇⠀⠀⢸⡿⠆⠀⠀⠀⠀⠀⠀⠀⢈⠀⠠⠁⢂⠐⠠⢁⠂⢅⠢⠌⡱⢐⠱⢌⠲⡍⠶⣭⢳⣏⢷⣫⢟⡵⣻⣜⡳⣎⢷⠳⣭⢚⣥⢃⢿⡄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀
⠰⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠠⠐⠀⠠⠐⠠⠀⠌⢂⠡⠈⠄⢂⠂⠡⠈⠆⡌⡱⣘⠳⣌⠷⣪⠷⣹⢎⡷⣳⢎⡷⣹⢎⡟⡴⢫⡜⢪⡜⣷⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀
⠰⠀⠀⠀⢸⣿⡃⢀⠁⠠⠀⠁⠐⠀⠌⢀⠐⠠⠁⠌⡀⢂⠡⠈⢄⠈⠅⡘⠰⢠⠑⠬⡱⢌⡳⣕⣫⢧⡛⣼⢣⡟⣼⢣⡟⣼⡙⣧⡚⡵⣊⢿⡄⠀⠀⠸⡀⠀⠀⠀⠀
⠈⡇⠀⠀⢹⣿⡅⠂⠌⢀⠐⠈⠄⠡⢀⠂⠌⠠⢁⠂⠔⡀⢂⡁⠂⢌⡐⢨⠐⣡⢊⠵⣡⢏⡖⣳⢬⡳⣝⠶⣫⡜⣧⢻⡼⣣⡟⣶⡹⡵⣩⢞⣷⠀⠀⠀⢣⠀⠀⠀⠀
⠀⣇⠀⠀⠀⣿⣶⠉⡔⡈⠤⢁⠌⡰⠠⡈⠤⠑⡠⠌⡐⠤⠁⡄⠣⡐⢌⠢⡑⢦⢩⢞⡱⢎⡞⣥⢏⡷⣭⢻⡵⣻⡼⣏⡷⢯⡽⢶⣹⠵⣫⠎⡽⣆⠀⠀⠀⠳⡀⠀⠀
⠀⢸⠀⠀⠀⢻⣯⡑⢦⠱⢌⠦⡘⢤⠣⢜⡠⠓⡄⢣⠘⡄⠣⢌⠱⣈⢆⢣⡙⡆⣏⢮⣙⢧⡻⣜⢯⣞⣵⣻⣼⣳⢿⣹⡽⣯⣟⣯⣏⢿⢲⢻⠶⣿⣆⠀⠀⠀⠙⢄⠀
⠀⠀⢇⠀⠀⠈⣿⣘⢦⢫⡜⢦⣙⢦⡛⣤⢣⡙⡔⢣⠱⣌⠓⣌⠓⣌⢎⢦⡹⣜⠼⣎⡽⣎⢷⣹⢾⡽⣞⣷⣯⣿⣻⣯⣿⣳⣟⡾⣼⢛⡎⢧⢻⣦⣽⣶⡄⠀⠀⠑
⠀⠀⠘⡄⠀⠀⢸⣿⢺⣇⡻⣼⢸⢶⣙⢦⡳⣜⡜⣣⠳⣬⠹⣌⣛⡜⣮⢖⣳⣭⣟⢾⡵⣯⢿⡽⣯⣿⣟⣷⣿⣿⣿⢿⣿⣿⣞⡷⣍⠞⡬⢣⢃⠞⣷⣿⣿⣤⠈⠄
⠀⠀⢀⡇⠀⠀⢸⣿⣹⠲⣝⡲⢯⡾⣽⣺⡵⣏⡾⣵⣛⢶⡻⣜⢧⣻⡼⣯⢷⣻⡾⣯⣿⣽⣟⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⡻⣜⡣⢚⠴⣉⠞⣼⣹⣿⣿⣿⣿⣄