⠀⣐⡝⡟⣞⡣⡗⡯⢲⢭⢻⡝⡺⣵⢫⣯⠿⣽⣳⣯⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⡖⡵⣚⢵⠳⡜⡧⣎⢳⡸⡕⢮⢳⣎⡿⣱⣟⡾⣽⣯⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡝⢬⡘⣥⢩⢎⡵⣩⢳⡩⠖⡥⣋⢭⢣⠞⣜⣳⢾⣹⢷⣻⢾⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⡘⠦⡙⢢⢋⡞⣔⢣⠧⣓⣛⠲⣉⢖⡹⢎⢧⢫⡗⣯⢏⣿⣫⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⡝⠞⡅⠧⣉⠖⣜⡢⢏⡳⢼⡢⢏⠲⡌⢷⣋⣞⢯⣞⡷⣯⢷⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢠⢣⣸⡘⡴⢣⠶⣙⢎⡳⢭⣾⣭⡖⣭⢷⣏⣾⣷⢯⣷⣻⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣬⢃⠖⡀⢎⢣⢣⢚⡥⣛⢥⣫⡿⠻⢷⣯⣾⢾⣻⣯⣿⢷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢣⣽⠂⠌⠐⡭⡖⣱⢚⡼⢩⣿⡟⠀⠀⢸⢯⣟⣻⢹⣻⣯⢿⣿⣿⣿⣻⢿⣿⠧⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡞⡏⣧⠨⢚⣴⠻⡴⢿⢕⣳⢸⠃⠀⠀⡼⣟⢾⠇⣸⢿⡿⠋⣿⣿⡿⡿⢘⠗⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠁⣿⡇⢸⣇⣾⡻⣷⣟⣺⣼⣘⣀⡀⡐⠁⢉⡞⢀⡏⠞⠁⠘⣊⣥⣴⣶⣾⡤⣄⣰⡿⠿⣿⣿⣿⡿⠻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢟⠮⣼⣯⢟⣵⡇⠹⠈⠁⠁⠀⠀⠀⡸⠁⠀⠀⠀⠀⢈⡎⢿⠽⠿⣿⠇⠈⠍⠁⠁⣧⠈⢻⠋⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⠸⣿⣿⡟⣿⣿⡉⠀⣠⠖⣶⣶⣤⡒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⡉⠴⠋⠀⠀⠀⢿⠿⠃⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⣻⣯⡷⣿⢿⠘⠃⠀⢷⡛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡀⠀⠀⠀⠈⠀⠀⠐⣡⣴⣿⣦⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⣿⣿⣯⢟⣆⠀⠒⠄⢉⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢻⣿⢻⣿⣿⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⢿⡹⡝⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠔⣹⣿⣿⣿⣇⣀
⣶⠒⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠳⣬⠀⢉⡢⢄⣂⠀⠀⠒⠶⠆⠀⠤⠤⠄⠀⠒⠂⠀⠉⠉⠁⠁⠀⠈⠉⠙⣻
⡟⠀⠀⠀⠀⠘⠡⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠒⠉⠀⠀⠀⠀⠉⠉⢿⡁⠉⠉⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠱⡀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠐⠄⡀⢆⠀⠀⠀⠀⠀⠀⠑⣌⠀⠀⠀⠈⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠒⠦⢤⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠦⡆⠀⠀⠀⠀⠀⠀⠘⠳⣆⠀⠀⠀⠈⠁⠮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠙⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠦⣬⢄⠀⠀⠀⠀⠀⠀⠘⢷⢄⣀⠀⠀⠀⠀⠀⠈⠛⣷⡄⠀⠀⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠣⡼⠓⡦⡀⠀⠀⠀⠀⠀⠙⢣⡀⠀⠀⢿⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠰⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠋⢆⡀⠙⢷⡄⡀⠀⠀⠀⠀⠙⣇⢀⠸⣇⠀⠄⠀⣀⠤⢀⠂⢙⠀⣄⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⢀⣴⠷⠒⢄⠀⠀⠀⠀⠀⠁⣄⠀⠈⠳⡄⠀⠀⠀⠀⠘⢷⡘⢣⣧⡊⠄⢠⠀⡉⢆⠣⠴⣐⠩⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠐⠂⠀⠐⠊⠁⠀⠀⠀⠀⠷⡄⠀⠀⠀⠀⣀⢹⣿⣿⣿⣆⠀⠀⠀⠀⠹⣶⢌⣻⣿⣷⣧⣎⣒⢤⠉⡞⣤⢓⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣵⣄⠀⡀⠀⠈⢪⣿⣿⣿⣧⡀⠀⠀⠀⠘⢷⣿⣿⣿⣿⣿⣿⣿⣾⣶⣬⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣆⡣⡀⢀⠠⠚⣿⣿⣿⣿⣆⠀⠀⠀⡝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⣿⣿⣿⣿⣿⣿⣿⣶⣥⣴⣾⣿⣿⣿⣿⣿⣇⠀⠀⣗⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠶⠶⠚⠑⠈⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠃⣯⡻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿