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