⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⠻⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠿⣅⠢⠔⠰⡸⡜⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣭⣾⣶⣾⣶⣥⣝⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡈⡿⣿⢟⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⠳⣿⣴⡿⡜⣿⡯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣯⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⣿⣿⡯⡷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡟⡵⢆⡈⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡟⠜⣁⢂⣀⢀⡀⢠⢙⡲⢤⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣿⣿⣿⣿⣿⣿⣿⣿⢟⡁⠂⠄⡤⣀⠦⡘⢢⠡⡘⠀⠈⠀⠀⢍⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡟⢧⣀⣴⣾⣿⣿⢿⡟⣟⢻⣷⣾⣳⣧⡿⣤⢇⣆⡳⠘⢃⢠⠀⡀⠀⡀⢦⠳⣸⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣯⡻⣥⢟⡿⣿⡳⣟⢿⡟⡽⡎⢸⣟⣯⡿⣮⣧⣷⠏⡎⡁⠘⠈⡀⣄⣡⢃⡝⣎⣳⣻⣿⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡳⣌⠵⣘⢮⣱⢋⠷⣙⠦⢳⢀⡁⣼⡟⣜⣣⢎⠳⡘⢂⠀⠁⡠⢄⣣⢜⣭⣾⢠⣋⢧⡓⣾⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣾⣿⣟⣮⣳⣛⢮⣙⠮⡅⢦⣐⣿⣟⣶⢣⢮⡁⠀⢄⡰⢊⡴⣣⣾⣿⣿⠟⠧⠓⢦⢹⡜⢷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣽⣮⣳⣜⣷⢾⠻⣿⣿⣯⣾⣭⣷⣽⣼⣯⣷⣿⣿⣿⣷⣌⡐⢀⠎⢧⠊⡒⢿⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⡻⣿⣶⣡⢩⣙⣻⣿⣿⣿⢿⡛⣟⢻⢻⣿⣷⣌⠢⡘⡘⠄⢹⣊⣾⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣾⣽⢻⡐⣚⣿⣿⣿⣿⣿⣿⣿⢿⡟⣿⣷⡱⣆⠡⢁⡈⣅⡉⢤⡰⣖⣟⡙⣎⠽⣿⣿⣿⣖⡡⠘⡎⠐⢪⢼⡄⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣷⢏⠳⡰⢌⣿⣿⣿⣿⣿⣿⡽⣞⡼⣜⣷⣿⣹⣝⣦⡹⣥⠟⣭⠷⢋⣴⣿⡹⣟⣿⡟⢻⣯⡽⣒⢬⠱⡠⠙⣿⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣎⢮⠱⣆⢯⣿⣿⡟⣿⣿⣿⣿⣿⣷⣯⣿⣿⣟⠻⣙⡹⠞⡉⢢⢳⠯⣞⣴⣿⣿⠋⠀⠀⠻⣿⣷⣺⢧⡥⡐⠘⣣⠀⠀
⠀⠀⠀⣰⠿⡈⡿⣷⣿⣼⣿⣿⠟⠀⢿⣿⣿⣿⡭⣭⢉⣿⣿⣜⠳⣜⠲⣝⠣⢍⠪⡕⣾⣿⡿⠁⠀⠀⠀⠀⢹⣷⡿⣿⡔⡇⢈⠸⡇⠀
⠀⠀⣰⡏⠡⢈⠒⠿⠟⣟⡿⠋⠀⠀⣿⣿⣿⣿⣿⣎⣻⣿⣿⢛⠛⠌⣓⠎⠢⠄⠓⣜⢲⣽⠃⠀⠀⠀⠀⠀⠀⢻⣛⣿⣎⡅⠀⡌⣽⠀
⠀⢠⡿⡌⠐⢁⠂⡖⢉⣿⠇⠀⠀⢰⡿⣧⣻⣿⣖⣖⡲⢿⡆⡯⢄⡍⡲⢬⠑⣈⡱⣌⣷⡗⠀⠀⠀⠀⠀⠀⠀⢸⣯⣟⠶⡌⢐⠒⡔⣇
⠀⣿⣳⠅⠌⢠⡜⠔⣞⣿⠀⠀⠀⢾⣿⣌⣿⣿⡷⣞⡽⣻⢸⡱⢊⡴⠡⣣⠿⢛⡓⣋⢧⢳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡵⢉⠆⡘⣔⢻
⢸⣗⡯⢌⡐⢋⢴⣂⣽⠏⠀⠀⠀⢨⣿⣷⢺⡿⣗⢯⡞⣽⠲⣡⢛⠰⣉⡅⠃⠀⠘⡶⣨⢭⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡘⠐⢌⣹
⣼⢧⡛⠠⠜⣊⢭⣿⠇⠀⠀⠀⢀⣾⡗⡽⣧⣿⣟⣮⣳⢽⡜⡥⢧⢯⠱⣘⢂⠀⢂⠰⠹⣮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣞⡳⡌⡔⢺
⡿⠈⠄⡃⢜⣰⡾⡣⠀⠀⠀⠀⢸⣿⠖⣱⠙⣿⣾⣵⣻⣬⡛⡜⣻⢎⢳⡃⠎⡀⠀⢆⡙⣺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣓⡐⢜⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠟⠻⣿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠿⣅⠢⠔⠰⡸⡜⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣭⣾⣶⣾⣶⣥⣝⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡈⡿⣿⢟⡿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⠳⣿⣴⡿⡜⣿⡯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣯⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⣿⣿⡯⡷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡟⡵⢆⡈⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡟⠜⣁⢂⣀⢀⡀⢠⢙⡲⢤⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣿⣿⣿⣿⣿⣿⣿⣿⢟⡁⠂⠄⡤⣀⠦⡘⢢⠡⡘⠀⠈⠀⠀⢍⠲⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡟⢧⣀⣴⣾⣿⣿⢿⡟⣟⢻⣷⣾⣳⣧⡿⣤⢇⣆⡳⠘⢃⢠⠀⡀⠀⡀⢦⠳⣸⣆⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣯⡻⣥⢟⡿⣿⡳⣟⢿⡟⡽⡎⢸⣟⣯⡿⣮⣧⣷⠏⡎⡁⠘⠈⡀⣄⣡⢃⡝⣎⣳⣻⣿⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡳⣌⠵⣘⢮⣱⢋⠷⣙⠦⢳⢀⡁⣼⡟⣜⣣⢎⠳⡘⢂⠀⠁⡠⢄⣣⢜⣭⣾⢠⣋⢧⡓⣾⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣾⣿⣟⣮⣳⣛⢮⣙⠮⡅⢦⣐⣿⣟⣶⢣⢮⡁⠀⢄⡰⢊⡴⣣⣾⣿⣿⠟⠧⠓⢦⢹⡜⢷⡀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣽⣮⣳⣜⣷⢾⠻⣿⣿⣯⣾⣭⣷⣽⣼⣯⣷⣿⣿⣿⣷⣌⡐⢀⠎⢧⠊⡒⢿⡄⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⡻⣿⣶⣡⢩⣙⣻⣿⣿⣿⢿⡛⣟⢻⢻⣿⣷⣌⠢⡘⡘⠄⢹⣊⣾⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢀⣾⣽⢻⡐⣚⣿⣿⣿⣿⣿⣿⣿⢿⡟⣿⣷⡱⣆⠡⢁⡈⣅⡉⢤⡰⣖⣟⡙⣎⠽⣿⣿⣿⣖⡡⠘⡎⠐⢪⢼⡄⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣷⢏⠳⡰⢌⣿⣿⣿⣿⣿⣿⡽⣞⡼⣜⣷⣿⣹⣝⣦⡹⣥⠟⣭⠷⢋⣴⣿⡹⣟⣿⡟⢻⣯⡽⣒⢬⠱⡠⠙⣿⠀⠀⠀
⠀⠀⠀⠀⢠⣿⣎⢮⠱⣆⢯⣿⣿⡟⣿⣿⣿⣿⣿⣷⣯⣿⣿⣟⠻⣙⡹⠞⡉⢢⢳⠯⣞⣴⣿⣿⠋⠀⠀⠻⣿⣷⣺⢧⡥⡐⠘⣣⠀⠀
⠀⠀⠀⣰⠿⡈⡿⣷⣿⣼⣿⣿⠟⠀⢿⣿⣿⣿⡭⣭⢉⣿⣿⣜⠳⣜⠲⣝⠣⢍⠪⡕⣾⣿⡿⠁⠀⠀⠀⠀⢹⣷⡿⣿⡔⡇⢈⠸⡇⠀
⠀⠀⣰⡏⠡⢈⠒⠿⠟⣟⡿⠋⠀⠀⣿⣿⣿⣿⣿⣎⣻⣿⣿⢛⠛⠌⣓⠎⠢⠄⠓⣜⢲⣽⠃⠀⠀⠀⠀⠀⠀⢻⣛⣿⣎⡅⠀⡌⣽⠀
⠀⢠⡿⡌⠐⢁⠂⡖⢉⣿⠇⠀⠀⢰⡿⣧⣻⣿⣖⣖⡲⢿⡆⡯⢄⡍⡲⢬⠑⣈⡱⣌⣷⡗⠀⠀⠀⠀⠀⠀⠀⢸⣯⣟⠶⡌⢐⠒⡔⣇
⠀⣿⣳⠅⠌⢠⡜⠔⣞⣿⠀⠀⠀⢾⣿⣌⣿⣿⡷⣞⡽⣻⢸⡱⢊⡴⠡⣣⠿⢛⡓⣋⢧⢳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⡵⢉⠆⡘⣔⢻
⢸⣗⡯⢌⡐⢋⢴⣂⣽⠏⠀⠀⠀⢨⣿⣷⢺⡿⣗⢯⡞⣽⠲⣡⢛⠰⣉⡅⠃⠀⠘⡶⣨⢭⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡘⠐⢌⣹
⣼⢧⡛⠠⠜⣊⢭⣿⠇⠀⠀⠀⢀⣾⡗⡽⣧⣿⣟⣮⣳⢽⡜⡥⢧⢯⠱⣘⢂⠀⢂⠰⠹⣮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣞⡳⡌⡔⢺
⡿⠈⠄⡃⢜⣰⡾⡣⠀⠀⠀⠀⢸⣿⠖⣱⠙⣿⣾⣵⣻⣬⡛⡜⣻⢎⢳⡃⠎⡀⠀⢆⡙⣺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣓⡐⢜⡹
