⠀⠀⠀⠀⠀⠀⠀⣼⡿⡳⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⢣⢣⡃⠖⡬⡙⠻⠷⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⣋⢦⡙⡵⢒⡍⣣⠝⡢⢬⡙⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡝⡲⡍⡖⣥⢚⡱⢢⢍⡳⢎⡖⣯⣿⢿⡷⣶⣤⠶⡶⢶⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⡶⢶⢾⣛⢿⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣻⣿⡱⢧⡹⣔⡫⢖⣭⢮⡓⡏⡜⢲⡐⢎⡔⠢⡔⡩⠔⣃⠲⡐⠦⢌⠭⡙⠻⢿⣿⣿⡭⢍⡳⣍⠶⡩⢎⡳⣍⢏⡶⣹⣾⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣽⢿⣧⣛⢶⣙⣿⣾⠟⣬⠱⡘⢥⠚⡴⣈⠧⡔⣡⡙⢤⢣⡑⢎⡜⡰⣉⠕⡢⢌⡙⢿⣇⡳⣌⢳⡙⢮⡱⢎⣞⣴⡿⣿⠃⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⢾⣽⣾⣻⣽⢋⠞⡤⢓⠭⣒⣍⣲⣥⣎⣱⡡⢚⡤⢣⠜⡢⡜⡱⣈⠞⣑⠪⢔⠢⢚⢳⣭⢖⡹⢦⣹⣾⢿⣽⣳⣿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣿⣽⣿⣿⣻⣿⢾⣳⡿⢣⢍⡚⣴⡽⠚⠋⢉⣧⡄⠈⠉⠙⠷⣜⡥⢫⠴⣡⢓⣬⡶⠗⠻⢶⠿⣤⡣⠜⣿⣳⣧⣿⢯⣟⣾⣷⡇⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⣽⣿⡿⣑⣋⠮⣼⠋⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⣿⡼⢣⢏⡒⣯⡏⠀⠀⠀⢸⣷⡀⠉⠳⣝⣷⣿⣟⣿⣻⣿⣾⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣳⣿⢲⡱⢎⡽⣯⠀⠀⠀⠀⠀⣿⡟⠀⢀⣠⣼⢻⡜⢧⣎⠵⣎⠷⣄⠀⠀⢸⣿⡇⠀⠀⠹⡾⣿⣯⣿⣿⣽⠃⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⢣⣟⣼⣛⠾⣽⣦⣤⣤⣤⣤⣬⢶⡾⣿⣻⡜⣿⣿⣷⠿⣿⣾⣿⢩⠷⣦⣌⡛⠁⠀⢀⣼⢳⢽⣿⣽⣷⡏⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣳⢻⡼⣖⢯⣛⡶⣝⡮⣗⢯⡾⣭⠿⣵⣏⡷⣻⣵⣻⢾⣟⣿⣻⣵⣫⢟⡴⣫⢽⡻⣟⢯⡳⢭⣚⣿⣯⡟⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣱⢯⡳⣏⠿⣜⣳⡽⣺⣭⢷⣛⣞⢿⣱⣾⣽⣷⠾⠷⠛⠛⠉⠉⠙⠻⢿⣷⡽⣞⡽⣛⣞⣳⢏⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⢿⣿⣿⣿⣝⣞⣳⡽⣫⡽⣖⢯⣗⣾⣭⡿⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡻⣿⣽⣻⡼⣭⡟⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠰⣿⣟⣿⣿⣿⣿⣾⣽⣶⣿⢷⣿⣿⡿⠟⠛⠻⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠿⣷⣟⣧⣟⣳⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣯⣿⣾⡿⠛⠁⠀⠀⠀⠀⠐⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠈⢻⣿⣾⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢰⠃⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠁⠀⠔⠂⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠠⠤⠤⣤⡀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀
⡇⢸⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣦⠴⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠶⠛⠶⠶⠋⠀⠀⠀⠐⠠⠤⣀⡀⠀⣸⣿⣿⣿⡇⠀⠀⠀⢰⠙⢆⣀
⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⠻⣿⡜⣻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⣿⣿⣿⡿⠃⠀⠀⠀⠈⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣽⣿⣆⣏⡞⡿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡾⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⢳⣿⢟⡭⢻⠽⣟⡿⣿⢿⣿⣿⣿⣿⣷⣄⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣾⣷⣿⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣟⣷⡿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡿⣿⡛⢿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⣿⣏⢷⡳⣞⣾⣿⣿⣍⣻⣏⣬⣹⣷⣿⡙⢦⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⢮⣏⣷⣽⡿⣿⢫⡝⣹⡿⣩⢛⣿⣻⣙⣎⢞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣾⡽⣻⡿⣝⣳⢽⣺⡱⢎⣽⡷⣩⠳⢮⡝⣿⣏⣞⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⡿⣿⢿⡾⣷⢿⡾⣾⡷⡿⢿⡷⣿⡿⣻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⢿⣠⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠲⠚⠛⣿⡟⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⢿⡿⠿⠿⠿⠿⠛⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠉⠙⠒⠒⠶⠦⠶⠴⠦⠶⠶⠷⠶⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⡄⠀⠀⢠⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠊⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣥⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡒⠉⠉⠉⠐⠒⠊⠻⣦⣀⣿⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠹⣷⣴⠟⠒⢒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠲⠤⠤⠬⠭⠥⠤⡤⡄⠈⠄⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀