⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡸⠋⢀⣾⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⣴⠞⠁⣰⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡀⣤⡋⠁⡀⠜⠋⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠰⣖⠒⠒⠲⠶⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⢠⡾⢣⠚⠉⣡⣷⠊⠁⣀⠔⠋⣱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣬⣷⣤⣀⠀⠠⠄⡀⠉⠛⠢⣤⣄⣄⠀⠀⠀⢀⣤⣴⣤⣰⣶⣸⣿⣾⣿⣿⣼⠀⡆⠒⠛⢉⡜⠚⠋⢁⣠⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠈⠳⣄⠀⠀⠉⠙⠛⠛⠦⠤⠶⠤⡀⠈⠳⣄⠀⢾⣾⣿⢿⣿⣿⣿⡿⠈⡿⢿⠇⢠⢱⡓⠚⢉⣻⠛⠉⠉⣁⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠠⣴⠜⠛⠒⠶⠶⠶⠤⠵⣠⠤⠤⠀⠩⡄⠈⠳⢼⡟⠻⠷⠼⠛⠳⠖⣫⣿⢺⢷⠞⠦⢄⣉⡿⠛⠺⠿⠿⢭⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠓⣦⣀⣀⣀⣀⣀⠿⢤⣒⣒⠰⢒⣈⠦⣀⣀⣹⡄⢀⠀⡀⡦⢇⢓⢹⢨⣿⣬⠲⣄⣉⠙⢯⠒⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠠⣚⣉⡀⠀⠀⣀⣀⡤⠔⣎⣀⣠⠞⢀⡜⢎⡍⣹⣧⠘⣶⣷⡆⣜⣿⣿⠟⢮⡶⠬⠯⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⢉⣽⠟⠋⠉⠀⢀⣡⠔⠛⢃⣶⠿⣤⣾⣼⣾⣟⣷⣻⣿⣿⣽⣿⣯⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠈⠛⠓⠛⣻⠿⠛⠁⣠⣾⢿⡇⣴⣿⡇⡸⠻⡟⠉⠙⠻⠿⢽⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠓⠒⠚⠛⠉⠀⠈⠋⠉⠀⠉⠁⠀⠀⠀⢀⠦⡀⢸⠘⡄⢠⠒⠀⠀⠀⢀⡤⢚⠵⠋⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢡⡙⣦⠆⣇⣼⢸⠀⠀⠀⡿⣸⠏⠀⠀⠀⠀⠈⠉⢲⣾⢯⢽⣗⡲⢶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠱⣌⣓⠧⠻⠖⠛⠁⡞⠀⠀⠀⣿⣿⣀⡴⠪⢷⢄⡀⠀⠀⠉⠉⠛⠛⠙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡄⠀⠀⢹⠀⠀⠀⡠⠐⠓⢋⣭⣏⣿⠟⠛⠂⠠⠄⢁⢨⡳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⠀⡴⠙⣆⠀⠀⠀⢀⡔⢧⣈⣹⢸⣯⡀⠀⡀⠀⢀⢑⠙⢖⢤⣽⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⡍⡿⠀⠀⢷⡀⠙⠿⣿⠿⠛⠁⣴⡟⠉⡆⡄⠱⣄⢳⡄⠀⠡⡈⢢⣀⠉⢈⣑⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡯⢿⣠⡷⠃⠀⠀⢸⣟⠤⢂⣀⣀⠴⣫⣿⠶⠾⠘⣇⠰⠿⠷⣿⣦⡀⠀⠒⠭⠛⠻⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣇⣠⠾⠋⠁⠀⠀⠀⢸⡤⣒⠄⠀⡀⢴⣿⡿⣶⣿⡇⠘⢀⣾⣾⣾⡍⣿⣷⣦⣴⣦⣤⣀⣬⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢸⡏⡳⠭⠉⢐⣵⢹⠷⠛⢂⡧⠄⠈⠈⠩⣾⣧⣸⣷⣍⡉⠉⠛⠛⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣎⡀⠀⣛⠜⢈⣆⠀⠀⡩⡀⠀⠀⠀⠀⠀⢸⢿⣯⣿⣷⣦⣄⠀⠙⢝⢦⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣀⣀⣀⢮⢹⣷⣄⠈⠁⠀⠀⢐⡤⠒⣽⡟⢻⣽⣿⣿⣿⣷⣄⠈⢎⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠻⡌⣀⢊⡎⣧⡘⣷⠶⠶⠾⢏⣀⠜⠉⠀⢸⡇⢻⡉⣻⠿⣿⣆⠊⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣦⡀⣽⣾⡆⣠⣿⠀⢹⡌⠢⢀⠀⠈⢳⡀⠀⣹⣻⡄⢳⠀⣳⢘⢻⡲⣾⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢛⣿⣾⣿⣿⣿⣿⣴⣿⣎⣿⣏⣿⠀⠈⣷⣀⡠⠀⢀⣸⣧⡼⠃⣻⣇⠈⣗⣿⣾⣷⢫⣿⡿⣄⡴⣲⣖⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢋⣴⣿⣿⣿⣿⣿⡟⠀⢈⣽⣿⣏⣽⣿⡷⠀⠀⡇⠀⠀⠉⠙⢻⡏⠀⠀⢹⣿⣤⣿⣾⣯⣿⣿⣿⣿⣿⠀⠀⠶⣮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣾⣿⣿⣈⣓⣚⣿⡄⠰⣰⢍⡓⠢⠀⠀⠀⣧⠀⢐⢼⣤⣾⣿⣿⣿⠟⠀⠈⠻⡀⣖⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣍⣙⣿⡿⣿⠊⠁⠀⠈⠀⠀⠀⡀⡏⠩⣱⡿⢿⣿⣿⡿⠋⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⢒⢢⣸⣽⣿⣷⡈⠒⢔⡄⠀⠀⠀⠈⢹⡃⡘⣿⣷⣼⣿⣭⣗⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⣼⣭⣿⣷⣿⣿⣿⣶⣤⡈⠉⠖⢒⡶⡾⠛⢡⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⢠⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⣯⣿⠟⣻⣿⣿⣿⣿⣯⡙⠉⣽⣿⣿⣿⣧⣉⠛⠿⠟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⣾⠏⠀⠀⣿⣿⣿⡿⠋⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠂⠉⠹⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⣿⠇⠀⠀⠀⠸⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⢻⡟⠁⠀⠀⠀⠈⠹⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⡏⠀⠀⠀⠀⠀⠈⠳⣄⢀⣴⣿⣿⣿⣿⢟⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀