⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢟⢻⢍⢕⢔⢔⢔⢮⢭⡛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣛⢍⢆⢇⢇⢝⢜⢜⢼⢸⢨⢹⢸⢨⢣⣪⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢫⢢⢣⢪⢪⢪⡺⣜⣜⢜⢜⢜⢜⢜⠜⢜⢜⢔⢜⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡱⣸⡸⡎⡎⡎⡎⡍⡯⡳⡜⢼⢼⢸⢘⣼⠀⣌⢳⡳⣕⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⡌⡎⡎⢯⢹⢸⢸⢸⢸⢸⢸⡸⡸⡸⡘⣾⠀⠀⠀⣆⢏⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠿⠀⠀⠀⠀⠀⠀⡿⡑⣕⡕⡝⡜⠼⡜⡌⣗⢧⣳⢳⢱⢑⢧⢳⠀⠀⠀⠀⠀⣌⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣧⢲⠰⠬⢭⢩⣙⢣⢳⢸⢱⢱⢱⠱⠵⡝⢮⡳⡝⠷⢵⢱⠩⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣧⡩⡹⣘⢲⢸⢸⢸⢸⢸⢸⢱⢩⢣⢹⢸⠣⠭⢽⢸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣮⡜⢜⢜⢜⢜⢜⢜⢜⢜⢜⢜⢜⢜⢜⢍⢎⢎⢎⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣯⡁⡇⡧⡕⣕⢱⠱⡱⡱⡱⡱⡱⡱⡱⡱⡱⡱⡱⡱⡨⡛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠿⠤⡅⣇⠣⢣⢣⢣⡓⡎⡎⡮⡸⡌⣎⢪⢊⣎⣪⣪⣪⣪⣢⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢸⢹⢸⢸⠭⡀⠈⠬⣪⣮⢊⢎⢎⢎⢮⢮⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣧⣣⢣⢣⢫⢪⢪⡚⡏⡇⡇⡇⡇⡧⣯⣳⢱⢩⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣷⣧⡇⡇⡇⡇⡇⡇⡇⣇⢷⡝⡞⡜⡜⡜⡌⠿⣟⡛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢣⢣⢣⢣⢣⢣⢣⠳⡣⡪⡪⡪⡪⡪⡪⡑⡗⡕⡕⣝⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⢸⢸⢸⢸⢸⠸⡸⢸⢸⢸⢸⢸⠨⢮⢮⣇⣳⢱⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡇⣵⠀⡲⣭⡳⡅⡇⡇⡇⡇⡇⡇⡎⡜⢜⢜⢸⢑⢇⡮⢾⠽⡎⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠮⢏⢆⢗⢜⢜⢜⢜⢜⢜⢜⢜⠴⣅⢧⢇⢽⢸⢱⢱⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⡱⡱⡱⡱⡱⡅⡇⡇⡇⡇⡏⡝⣜⡪⡫⡪⡧⣓⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢪⢪⢪⢪⢪⢪⢪⣪⠪⡪⡪⡪⡢⢧⡳⡽⣝⢮⠣⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⡜⡜⡜⣼⠀⠀⣎⢎⢪⢪⢪⢲⢵⢹⢮⠹⣝⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡣⡣⡣⡣⣹⠀⠀⠀⡸⡸⡸⡨⡹⡪⡺⡨⣹⢘⢎⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⡇⡗⠼⠀⠀⠀⡇⡇⡗⡜⢜⢜⢜⢜⢜⢜⢜⢢⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡱⡱⡱⡱⡍⡇⠀⠀⠀⢣⢣⢣⢫⢪⢪⢪⢪⢪⢪⢪⢪⢪⢙⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡱⡱⡱⡱⡱⡱⡱⣹⠀⡟⡜⡜⡜⡜⡜⣔⠱⡱⡱⡱⡱⡱⡱⡱⡑⡆⡍⡍⣍⣛⡛⡛⠿⢿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡱⡱⡱⡱⡱⡱⣱⣱⣼⠀⢱⢱⢱⢱⢱⢱⠸⣸⣮⡪⡪⡪⡪⡪⡪⡪⡪⡪⡪⡒⡆⠧⢭⢹⢒⢜⡙⢿⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣜⣜⣜⣜⣜⣼⠀⠀⠀⣷⣵⣵⣵⣵⣵⣵⠀⠀⠀⠀⣮⣊⡎⢎⠎⡎⢎⠎⣎⣎⣪⣙⣼⣬⣮⣦⣭⣆⣽