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