⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣴⣿⣿⠿⣟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣏⡏⠀⠀⠀⢣⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⣟⠧⠤⠤⠔⠋⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⣿⡀⢀⣶⠤⠒⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠈⢿⣆⣠⣤⣤⣤⣤⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⣾⢿⢿⠀⠀⠀⢀⣀⣀⠘⣿⠋⠁⠀⠙⢇⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀
⠀⠀⠀⢀⣾⢇⡞⠘⣧⠀⢖⡭⠞⢛⡄⠘⣆⠀⠀⠀⠈⢧⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀
⠀⠀⣠⣿⣛⣥⠤⠤⢿⡄⠀⠀⠈⠉⠀⠀⠹⡄⠀⠀⠀⠈⢧⠀⠀⠀⠈⠻⣦⠀⠀⠀
⠀⣼⡟⡱⠛⠙⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠹⣧⡀⠀
⢸⡏⢠⠃⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠸⣷⡀
⠸⣧⠘⡇⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿⠇
⠀⣿⡄⢳⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠁⠀⠀⠀⠀⣼⡟⠀
⠀⢹⡇⠘⣇⠀⠀⠀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣼⡟⠀⠀
⠀⢸⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢳⣼⠟⠀⠀⠀
⠀⠸⣧⣀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⣴⡿⠁⠀⠀⠀⠀
⠀⠀⠈⠙⢷⣄⢳⡀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣠⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⣀⣀⣠⣤⠾⠷⣦⣤⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

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