⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣦⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣟⣀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⢻⣿⡈⢲⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⣿⣠⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⣦⠙⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢸⣿⣷⣀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢛⡐⠀⣺⣿⡟⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⣿⣯⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢫⣿⣿⣷⣤⢷⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠻⠃⠞⠁⣾⠫⣯⠆⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠻⣏⡀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣨⣤⠿⡿⠋⢀⣊⠹⢇⡀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠅⢴⡾⣃⢀⣾⣟⣿⡅⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠀⠀⢻⣿⣖⢀⠨⣱⡄⠀⢀⣠⠖⠊⠉⠁⠈⢻⣿⡼⠁⠶⣿⣞⠀⠘⣧⡀⢀⣠⡶⢏⣿⠟⠁⣀⣠⢿⣼⣣⣿⡿⣎⠟⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢠⡼⠻⠏⠳⣷⣮⢎⣷⠋⠁⠀⠀⠀⠀⠀⠀⣿⡪⡀⠀⢉⠁⠀⢠⣽⣷⡿⠋⢀⡟⢁⣠⡾⢋⣱⢟⣽⣿⣻⢿⠋⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠛⢃⣤⣀⠘⠓⡏⢻⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣧⠀⡌⣗⢦⣸⡞⠁⠀⠀⣠⣾⢿⡟⣡⣾⣵⣿⣿⠛⠋⡹⠱⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⠓⠞⠛⠀⠄⣱⢈⡏⠁⠀⠀⠀⠀⠀⢠⣽⣿⡿⠟⡷⠓⠚⣉⠕⠊⡱⣣⣾⣋⢻⣧⣿⣿⣿⡿⠷⠋⢀⡶⠁⠀⢱⣀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣇⠒⡄⠀⢣⢘⢣⣾⣥⣤⣤⣤⣤⡤⠖⠛⠉⠀⠀⠀⠀⡐⠋⠂⠀⣠⣿⣷⣚⣷⣾⣿⣿⣿⡿⢊⣀⢴⡟⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀
⠐⣻⣏⠒⣒⠠⢤⣄⠀⠀⠀⠀⠠⠤⠥⠥⠯⠬⠥⡿⠛⠋⠉⠉⠁⠀⠀⣠⢀⣠⣔⣖⣧⣶⢟⣿⣿⣼⣿⣿⣿⣿⣿⣛⣿⣿⡿⢟⣺⠋⠀⠈⠀⠀⣠⡆⣸⠁⠀⠀⠀
⠀⠘⠷⣕⡉⣶⣨⣤⣄⣁⡐⠊⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⣫⣾⣿⣿⡿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⢿⣑⣋⣉⡀⠀⠀⠀⠀⡜⡿⠀⣿⠀⠀⠀⠀⠀⠀
⠀⠀⠉⠙⠓⠶⣄⠛⣉⣩⣟⣵⠶⣶⡀⣤⠤⠤⢤⣄⡒⢀⡀⣀⣤⣿⣿⣿⢿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣟⣗⣒⣒⣦⣾⠟⠁⠀⣰⣴⣠⠘⣸⢹⣰⣸⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢷⣾⣿⡿⣟⣿⣧⣽⣛⣹⣷⣏⣳⣿⣿⣿⣽⣾⣿⣿⣿⣿⣠⢋⣼⣿⣷⣿⣍⣻⣐⣶⣾⢿⣿⡏⠀⢠⣾⠋⣼⠁⠀⠘⡀⡇⣻⠀⠀⢰⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠛⠿⠹⢿⣿⣿⣿⣻⣿⣿⣿⡟⠙⣿⣿⣿⠿⣿⣹⢾⢿⣿⣿⣏⢿⢹⣟⡿⣿⣿⣿⣿⣦⣼⣡⠞⠁⠀⣠⡿⢧⢱⣯⢳⢰⡇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⠿⠟⣻⢻⣿⣷⢿⣿⣿⣽⣶⣈⡇⠀⠙⠛⠗⠋⣸⣳⣿⡇⣿⣿⣿⣿⣿⠟⠛⠉⢀⣴⠋⠀⢨⠶⡾⣼⡼⠓⡇⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢠⣆⠀⠀⠀⠀⣠⠄⢀⣾⢀⣼⣧⣾⡻⣻⣮⣿⣛⡇⠀⣬⠉⠀⠀⠀⠀⠀⢅⣾⢻⢷⣿⣿⣿⣿⣿⣶⣶⣵⠟⠁⠀⠀⢏⠋⡴⣿⣺⠏⡇⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢆⢣⣬⡶⠙⠶⢶⣖⡿⣇⣸⣭⣾⣿⣿⣳⣣⣧⣵⣄⠁⠀⠀⠈⣠⡴⠒⢻⠀⠀⣨⠛⡟⣸⠑⣦⠈⡝⢿⣿⣿⠏⡀⠀⣀⣼⡞⣱⡁⠈⠃⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡾⠋⠀⠙⡦⣆⢹⡊⢿⣿⣏⢹⣿⢻⢿⣿⣿⢻⣟⠉⠀⠀⠙⠤⠤⠊⢀⡔⢻⠀⣷⡟⣸⢻⠇⠹⡌⢻⣿⣿⢔⡾⢃⠞⡜⠫⣏⠉⠃⢀⡇
⠀⣀⡀⠀⠀⠀⢀⣠⠠⣆⣡⠀⠀⠀⢸⡿⡄⣿⡞⠿⠻⠿⣿⣿⡿⠛⠉⣯⡝⣾⢹⡖⠲⡄⠀⠤⠊⠀⠘⠀⣏⣄⡿⡼⠀⠀⡇⡄⡿⣕⣻⠀⠀⢠⣧⠙⣙⢶⠒⠚⡇
⠀⠑⠮⣑⠢⢄⢻⢄⡑⢤⢩⣖⠀⠀⣼⣇⠃⢛⣿⣾⡀⣼⡿⠣⢤⠀⠘⡞⣷⢹⢿⣷⣀⠈⠀⠀⠀⠀⠀⠠⢿⣿⡜⡿⣦⣴⢱⢭⡮⠀⠙⢦⣴⢿⡥⢙⠮⠧⠤⠶⡏
⠀⠀⠀⠀⠙⠢⡽⢉⠝⡹⡿⣋⡑⡆⣏⢸⡼⣌⢻⡿⡿⢹⡧⠐⠁⠀⢀⠼⡻⣷⡛⣽⡌⠁⠒⠒⠤⣀⡠⠃⠀⠱⣿⣴⡏⠻⣯⡎⠀⠀⠀⠀⠈⠫⡀⡜⠀⠀⠀⠀⢰⠀
⠀⠀⠀⠀⠀⠀⠙⣍⠚⠀⢇⡈⣂⠃⠹⡀⣻⡮⢝⣒⢫⣿⠀⠀⠀⠒⠁⠀⣷⠝⠑⢜⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠷⠤⠘⢾⣶⢄⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢾⣷⡒⠋⠫⠙⣗⣔⡷⣿⣿⣖⣯⣽⠏⠐⢲⣝⢆⠰⣾⡏⠀⠀⠀⢿⣿⡢⡀⠀⠀⠀⢠⠀⠀⠀⡘⠀⠂⢉⣽⣼⡿⣷⡀⠀⠀⠀⢧⠀⠀⠀⠀⢸⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣶⣶⣿⣷⣿⢀⣾⡿⠛⡛⡏⠀⠀⠀⠙⣎⣶⠿⣧⠀⠀⠀⣼⣷⡉⡌⠒⢤⠤⠸⡤⠤⡚⠁⠀⠀⠈⢻⠃⢫⣾⢿⠳⢦⡀⠈⠀⠀⠀⢀⡆⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣿⠛⠉⠀⠙⢯⡇⠈⡘⠀⠀⠀⠀⠀⣼⠛⡘⣿⠀⠀⠰⠏⠁⢻⣌⠀⠈⠢⢤⡘⡄⢸⠀⠀⠀⠀⣿⠀⣰⠽⠏⡇⠀⠈⠑⠤⣄⢐⡾⠁⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠈⠺⡅⠀⠀⠀⢀⡜⣷⠀⢣⡇⠀⠀⣿⢳⠀⠀⠙⢷⣄⠀⠀⠙⡜⣺⢄⠀⠀⣸⣧⡾⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣬⢆⠀⠀⠀⠀⠀⠈⢂⠀⢀⠞⠘⢿⣷⣄⢹⢆⢸⠇⢳⢧⠀⠀⠀⠙⢷⣄⡀⡸⣧⡀⢣⣰⣏⣤⡴⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿⢦⠀⠀⠀⠀⠀⠈⢀⠎⠐⢆⠀⠹⣝⣷⣯⢻⠤⢮⠃⢿⡶⣶⣶⣶⣿⣿⣿⣷⣾⠯⠽⠝⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠚⡁⣧⣿⣷⣦⣀⡀⠀⢠⠎⣺⢴⣬⣣⣤⣸⣧⣉⣱⢅⣷⠤⢤⡷⠀⠤⠞⣋⣽⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⢻⣽⢁⠀⢧⠉⡟⠹⣏⠀⠙⠒⠋⠀⠀⠙⠮⣿⢾⣿⣨⣿⣶⡩⠞⠁⠒⠒⠒⠀⠉⠁⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⢸⠀⡈⡄⠗⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⢦⣆⠳⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠣⣿⣷⣿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⣜⣗⣦⣹⣾⡇⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠰⢹⣟⢦⡿⣫⣽⣿⣧⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡞⡼⢼⢥⢵⢿⠛⣿⣧⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⠦⣙⠾⠾⣤⡟⠈⠆⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠿⠛⠉⠀⠀⢸⡀⠀⠣⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠙⢇⠀⠀⠀