⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢶⣳⣞⣶⡻⣟⣿⣻⢿⡿⣿⣿⣿⣿⣶⣶⣤⣤⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⢶⣶⢲⡖⣦⢖⡦⢶⣤
⠀⠉⠙⠛⠚⠙⠷⣎⢷⢫⡞⡵⢯⣟⣿⣻⢿⣿⡿⣿⡿⣟⣿⣿⣻⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⡿⣟⣿⡿⣿⣿⣷⣿⢿⣿⣿⢿⣻⢯⡟⡾⣙⠦⢧⡹⣎⠿⠘⠃⠉
⠀⠀⠀⠀⠀⠀⠀⡽⢎⡳⡜⡭⣓⠾⡼⣹⣛⣾⢻⣟⣿⣿⡿⣿⣿⣻⣿⣿⢿⣿⣻⣿⣾⡿⣷⣿⣿⡿⣿⣿⣯⣿⢿⣿⣻⣾⡻⣭⢳⡹⡱⢍⡞⣥⢳⡹⣂⠀⠀⠀⠀
⠀⠀⠀⠀⣰⢦⡻⣭⢳⡱⢃⠖⡡⢊⠱⢣⡝⣮⣛⡾⣽⣾⣿⣿⣻⣿⣷⡿⣿⣻⣿⣽⣷⣿⡿⣟⣿⣿⣿⣻⣽⣿⣿⢿⣽⣷⡻⣌⠣⠒⢥⢋⠴⢣⢇⡷⣻⢶⡄⠀
⠀⠀⠀⠀⠉⢟⣳⡝⡶⣙⢎⠬⡑⢌⡡⢧⡹⢶⣹⣿⣽⡿⣯⣿⣿⣽⣷⣿⢿⣿⣽⢿⣾⢿⣽⡿⣿⣾⣟⣿⣿⣯⣿⣿⢿⣾⣽⢧⣊⠍⣂⠎⡜⣣⢞⡵⢯⡳⠆⠀
⠀⠀⠀⠀⠀⠀⠈⢽⡚⡵⣊⢳⣘⢦⣛⣶⣻⡿⣿⣳⡿⣟⣿⣟⣾⡿⣽⣾⣿⣻⣾⣟⣯⣿⢯⣿⣟⣷⣿⣻⣾⢿⣽⣟⡿⣽⣯⣿⣳⣞⡴⣊⢕⡣⢞⡽⣣⠟⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣾⣱⢣⣝⣮⡽⣯⣟⣾⡷⣿⣻⢽⡻⣽⣻⣞⡷⣻⠽⣞⣳⢯⣷⣻⡽⣯⢿⣻⢾⣻⣞⡿⣽⢯⣟⢾⣹⢷⡿⣽⡳⣟⠾⣽⢦⡝⣎⠷⡏⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠘⠧⣟⣾⢷⡿⣯⣟⣷⡻⡵⢋⠎⡑⠄⠃⢎⠳⣭⢛⡼⢣⡟⣶⣹⢳⢯⡟⡽⣯⢷⣫⢟⡼⡳⢎⡷⢭⢋⠓⠡⠑⡌⢏⢷⣫⣟⢮⠋⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢯⡝⣧⣻⢼⡳⣍⠣⠌⠰⢈⠁⢂⠐⡀⢫⠜⡳⣜⢲⡭⣛⢮⣝⡻⣜⢧⡛⢮⡱⡝⢮⠜⢁⠂⠌⡂⠡⠈⡜⢮⣳⢎⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢮⡽⣞⡵⣏⠷⣍⠒⡈⠐⠠⠈⢀⠂⠄⡱⢊⡵⣊⠷⡼⣙⠾⣜⡳⣭⠞⣭⢣⢳⡩⢖⡈⠠⠈⠐⢀⠡⠐⣨⢳⣏⠿⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡳⣜⡼⣭⢻⡼⣡⠆⡁⠐⠀⠂⣀⠲⣁⠎⡴⣙⢮⡳⣭⢻⣜⠷⣭⢻⣜⡣⢧⡙⡆⢆⡡⢀⠡⠀⣄⠲⣥⢻⢬⡛⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡲⣝⡲⢧⡓⢧⡛⣜⢣⠞⡔⢣⢣⡜⡼⡸⣝⢮⡳⣭⢷⣫⣟⣭⡗⣮⠝⡶⡹⡜⢦⣡⢃⡞⣱⢎⡳⣎⢏⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⡴⢫⡖⣹⠣⡝⣬⢣⡛⡼⢣⡗⣾⣱⢻⣜⡧⢿⣼⣳⣻⣼⢮⡽⣎⢯⡵⣳⡽⣣⢏⡧⣝⡱⢎⡵⣊⢮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠜⣧⡙⢦⡛⡜⡴⢣⡝⡼⣣⢟⡶⣝⣯⢾⣽⣻⣞⣷⣻⣞⣯⢷⣛⣮⢗⣯⢷⡯⣟⡞⡶⣙⠮⢴⡩⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣲⢛⡴⣙⢦⡙⢦⡙⢦⡙⢶⡙⣮⢳⡻⣼⣻⢾⣵⣻⣞⡷⣯⣟⣯⣟⣼⢻⣞⢯⣳⡝⣞⡱⡍⠞⣦⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢳⡝⣮⡱⣍⢦⡙⢦⡙⢦⡙⢦⡝⣎⠷⣹⢣⢯⡟⡾⣵⢻⣼⢳⣻⠼⣞⡼⣫⢞⡯⢶⣙⢦⠳⡜⡩⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡧⣻⡔⣳⠜⣦⡙⢦⡙⢦⡙⢦⡹⢎⣝⡣⢟⢮⡹⣝⢮⡓⣎⠳⣍⠻⣌⠳⢥⡛⣜⢣⠞⣬⠓⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢭⡖⣧⣝⡣⡟⡴⣙⢦⡙⢦⡙⢦⡙⣎⠶⣙⢎⠶⡹⡜⢦⡱⣈⠱⣀⠣⡐⡌⢦⡑⢎⡱⢊⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⢣⢟⢦⢳⡹⣱⢣⡝⢦⡙⢦⡙⢦⡙⢤⢋⠼⣈⢇⠳⣌⢣⠒⡥⠒⡀⢆⠱⡘⢢⠉⡆⢅⡋⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢲⣭⣛⢮⢏⡶⣱⢃⠧⡚⢥⠛⢦⡙⠦⣙⠢⣍⠒⠥⣊⡑⢢⠃⣍⠰⢡⠘⡄⢣⠑⢢⠑⡌⠒⡌⢦⡙⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢠⡞⣵⣫⢶⡹⣎⢯⢖⣣⢏⡞⢭⢲⡙⢦⡙⢦⢡⠓⡤⢋⡒⡡⠜⢢⠑⡄⢃⠆⡱⢈⠆⡉⠆⡱⢈⠱⡘⠴⡘⢆⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⢷⣹⠶⣝⣮⢳⡝⣮⢏⡶⢭⣚⡥⢧⡙⢦⡹⢌⡣⡝⡰⢣⢌⠱⡘⢄⠣⡘⢄⠊⡔⠡⢂⠱⡈⠔⡡⢊⠜⣢⠙⣬⡑⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣯⢞⣧⢿⣝⡮⢷⡹⡞⣎⡳⡝⣦⢝⡲⣍⢧⢣⠏⡴⢣⡑⢣⢊⠖⣉⠦⡑⢌⠢⡑⢌⠱⡈⢆⠱⡘⢄⢣⠚⢤⢋⠴⣉⢇⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⢎⡿⣺⡽⣺⡝⣧⢻⡝⣎⢷⡹⣜⠮⣕⡺⣌⠳⣎⡱⢣⡙⢦⡉⠞⡤⢒⡡⢎⠰⡁⢎⠰⡁⠎⡔⡑⡊⠦⣙⠢⡍⢎⡕⡪⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢸⡹⣞⡵⣫⢷⡹⣎⢷⣹⢺⣬⠳⣭⠞⣥⢳⢬⠳⣌⢳⢣⢝⡢⡝⢪⠔⣣⠒⣌⢒⠡⢎⠰⣉⠜⡰⢡⢣⠱⢌⡱⢌⢣⢎⡱⢃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣘⢳⡭⣞⡵⣫⠷⣹⢎⣧⠳⢮⡝⣖⣫⡜⣣⢞⣣⡝⣎⠳⣎⠵⣊⠧⣙⠤⣋⠔⣊⠱⢊⡱⢄⢋⡔⢃⠦⣉⠦⡑⢎⡱⢊⡴⢋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀