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