⠀⠀⠀⠀⠀⠀⣤⣶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠙⢾⡱⣟⢿⣳⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡷⣭⣟⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠽⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⢸⡿⣷⣄⡤⠶⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣶⣾⣿⣯⣷⣾⣷⣷⣧⣝⣓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿⣿⣿⣿⣿⣽⣫⢿⣿⡿⣿⣿⣿⣷⣮⣼⣗⣲⣒⠲⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣏⣷⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣜⠢⡍⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣿⣿⡥⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣳⠛⢼⡻⣏⡯⣽⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠮⣵⠋⣴⢯⣳⣳⢟⡲⣝⠶⣹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣽⠃⣷⣛⢮⣵⣛⢮⣳⣽⣞⡵⣣⠟⣿⣿⢿⡻⣿⢿⣿⣏⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⢾⠃⣼⡳⣭⡟⡮⣧⣿⣿⣿⣿⣿⡵⢫⡝⣿⣯⠷⣭⢏⡟⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡼⡏⢰⣯⢟⢧⣻⣿⣿⣿⣿⣿⠿⢿⣿⡳⡜⡹⣿⣿⣷⡟⣾⡱⢿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⡽⣠⡿⢙⡾⢯⣿⣿⣿⣿⡿⠁⠀⠀⠙⣷⡹⠴⡹⣿⣟⣿⢶⣫⢟⣿⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀ ⠀ ⠀⠀⠀⠀⠀⠀⠀⢸⣺⢟⢵⡵⢹⣙⢯⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⢷⣩⢳⢹⣿⣻⣧⣛⢾⣿⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣑⣲⣤⡀⠀
⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣱⡟⢹⡝⣢⢫⢶⡹⢿⣿⣿⡀⠀⠀⠀⠀⠀⠈⢧⢎⡳⣜⢿⣽⡾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⣹⣥⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠥⢺⡟⣷⠸⡫⡜⣫⢮⡝⣯⢿⣿⢧⠀⠀⠀⠀⠀⠀⠀⠉⠓⣿⣮⠿⢽⣮⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣼⠐⢴⣹⢣⡗⢾⣩⢞⡹⢯⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡉⢉⣛⣻⣿⡛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⢇⠸⣧⡛⣼⢣⠟⣮⡝⣧⢛⡷⣄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣸⡆⠻⣝⢷⣋⣟⡲⣝⡎⡷⡹⢜⡳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡿⠁⠀⠉⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣻⣧⠀⠀⠀⠀⠀⠈⢧⣹⣶⠙⣾⠵⣌⠵⢮⣝⢲⡙⡎⣕⣣⢿⠳⡍⢯⣙⢲⠲⡤⢤⣀⣤⢤⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣱⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣿⣿⣧⣀⣀⣠⣴⣶⣾⣧⣻⢧⡈⠻⣭⡳⣆⡉⢒⡙⣴⡿⡙⢎⡱⢎⡱⣘⠦⢣⡙⠢⠍⢭⠳⣝⣖⣻⣆⡀⠀⠀⠀⠀⢠⡟⣧⢹⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣮⢟⣦⡌⠳⢷⡸⡠⢼⡟⡖⡹⣘⠴⢣⡱⢮⠼⡑⡎⠽⣘⠦⣡⣿⣿⢣⠝⣫⢷⣦⣄⣠⢯⡝⣮⢹⠀⠀
⠀⠀⠀⠀⠀⠀⢀⣼⣻⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣏⠉⢿⢯⣽⣿⣼⡹⢷⣄⡙⠣⣿⡹⢔⢣⣱⣮⣷⣾⣮⣶⣩⣼⣷⣾⣿⣿⠿⡿⣿⣾⣥⡚⢬⡙⠿⣎⡷⣃⣾⣥⡀
⠀⠀⠀⠀⣠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣦⠈⣝⢿⣿⣿⣿⣧⡯⡽⢶⣿⣷⣯⣼⡿⡹⢀⢆⢣⣛⣿⣿⡛⡽⡘⢦⡙⣔⠣⡭⣛⢿⣷⣯⣜⠢⢝⣻⣿⣿⣿