⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣤⠘⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣖⣟⣿⡏⢸⣁⡥⢸⣾⣶⣒⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀
⠀⠀⣴⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡿⠵⣲⣴⠿⠋⠉⣸⡏⠈⣿⣆⠙⢠⣿⠋⠸⣷⡈⠓⠯⣣⣄⣠⣦⣶⣿⣱⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⡿⢻⣳⠀
⠀⣸⡏⠸⣝⣄⠀⢠⢴⢷⡄⠀⠀⠀⠀⠀⣹⣇⢰⠒⠍⣆⣴⣾⣿⠁⠀⠀⢻⣶⡿⠃⠀⠀⣿⣷⣤⡀⢈⠟⠉⠍⢀⣼⡏⠀⢀⡀⠀⣠⣿⠿⣣⠀⢀⣼⠟⠀⢠⣿⠀
⠀⣿⡇⠀⠘⢞⣷⣵⡇⠸⣵⡀⠀⠀⠀⠀⢸⡿⠌⣀⡒⢸⡿⣋⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⠻⣿⣿⡎⠘⠀⣂⣾⠯⠤⠤⠓⠀⣠⣿⠃⠀⣿⣶⡿⠃⠀⠀⢸⡹⠀
⠀⢸⡇⠀⠀⠘⢧⢿⡇⠀⢻⣣⠀⠀⠀⢀⣿⠃⠀⣼⡿⡟⠘⣿⠃⠀⠀⠀⣀⣤⣤⣄⣀⡘⠻⣿⡼⠛⣤⣿⠀⠛⣿⡟⠛⠃⠀⣸⡿⠃⠀⢀⣿⠟⠀⠀⠀⠀⣿⡇⠀
⠀⢸⣿⡀⠀⠀⠀⠹⢇⠀⠀⠹⣵⢄⣼⣿⡏⡴⢺⣿⣟⣼⡿⣿⣶⡖⢶⠞⠉⠀⠀⠀⠈⠉⠛⠺⢿⣎⣿⣿⡇⠀⢻⣇⠀⢀⣾⠟⠁⠀⠀⠞⠁⠀⠀⠀⠀⣸⣿⣾⡀
⣸⡿⣽⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⡷⡇⠀⠙⢯⡁⠀⠀⢀⡉⠋⠀⣀⠤⢤⣀⠀⠀⠀⢀⡴⠏⠁⢸⣿⠀⢸⣻⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠽⢻⡻
⢻⣇⠈⠻⢷⡀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢋⣇⣧⠀⠀⠀⠙⢦⡖⠗⠓⠀⠈⠉⠉⠛⠚⠳⣦⠶⠋⠀⠀⢀⡟⠙⠳⣼⣿⠍⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⢀⣾⠇
⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⣴⠋⢘⣦⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⢀⡾⠁⠀⠀⠈⢳⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣾⠋⠀
⠀⠈⡛⣦⡀⠀⠀⠀⠀⠀⢠⣿⠤⣠⡞⠁⠀⣏⠈⠑⠦⣀⠀⠀⠀⠀⠙⠲⣾⡋⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢻⡦⠶⡞⣳⠀⠀⠀⠀⠀⠀⣀⣴⢿⢃⠀⠀
⠀⠀⣿⡗⠻⠷⠄⠀⠀⠀⣾⢿⣴⠏⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⠀⢀⡾⠰⡀⠀⠀⠀⠀⠀⢻⡄⠙⣿⠆⠀⠀⠀⠀⠊⠁⣠⡾⠋⠀⠀
⠀⠀⠹⢽⣦⣄⡀⠀⠀⠀⢿⢿⠏⠀⠀⠀⠀⠀⢘⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠞⠁⠀⠙⡄⠀⠀⠀⠀⠀⠻⣴⣿⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀
⠀⠀⠀⠀⢸⡿⠟⠛⠀⠀⣸⢏⡀⠀⠀⠀⠀⡀⢸⡍⠙⠓⠢⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢆⠀⠸⡄⠀⠀⠀⠀⡆⠈⠳⣄⡀⠀⠰⠾⢿⣵⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠘⢿⣓⣶⣤⠞⢁⡞⠀⠀⠀⠀⠀⣇⡼⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⠳⢄⣷⠀⠀⠀⠀⢹⡀⢀⣀⡽⠃⣀⣤⣺⡻⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠐⠾⣥⣤⡾⠀⢰⠂⠀⠀⢠⡟⠁⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠙⠲⢤⣀⣀⠈⢿⠿⠷⠾⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣈⣹⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠤⡤⠤⣖⠋⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⢻⣠⠞⠁⠀⠀⠀⠀⢀⠀⠀⠿⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣞⡷⣄⣀⣀⣳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣿⣏⠁⢐⣤⢤⣀⣀⠀⠀⠙⠻⣝⡉⠉⠉⠛⣻⠿⠟⠋⠉⠀⠀⠀⢀⠀⠀⠀⠀⣠⠖⢹⣷⡗⠉⠙⢯⡫⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠙⠮⢟⣿⣏⢠⡀⠚⢯⡉⠙⠛⠋⠁⠀⠀⠀⠀⠈⠓⠲⠶⠶⢶⡶⠛⣉⣀⡤⠶⣻⢃⣤⠞⠉⠀⠈⠀⠀⠙⠪⣳⢄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣀⣴⡾⠋⠀⣠⠔⠀⠀⠀⠀⡙⠻⣿⠳⢶⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀ ⠈⠉⣹⠉⢁⣤⢾⣿⣍⣠⡀⠱⡀⠀⠀⠀⠀⠀⠈⠑⠮⣶⣤⣄⠀⠀⠀⠀
⠀⠀⠀⡟⣥⣴⣶⠋⠀⠀⠀⠀⠀⢰⠃⢠⣾⢰⣷⣿⠛⣶⣤⣤⡄⢠⠀⠀⠀⠀⢠⣤⣴⡖⣿⣷⣾⡏⢸⡇⣿⣿⡏⣿⣶⣷⠀⠀⢠⠀⠀⠐⣦⣤⣤⣽⣿⠃⠀⠀⠀
⠀⠀⢀⣭⣾⠟⠁⠀⠀⢀⡤⠀⣠⣿⣶⣿⠃⣼⣿⠃⠘⣆⠙⣿⠋⠙⢿⣻⠿⣿⢽⣷⣆⡽⠋⢡⢤⡏⣿⣤⣘⣿⣿⢯⣯⠽⣷⣄⠘⣷⣄⠀⠀⠀⠙⠪⣵⣤⠀⠀⠀
⠀⠀⠸⢿⣤⣤⣤⣶⠾⣿⣵⣾⡿⠿⠩⣿⣭⡿⢃⣀⣴⣿⣿⣿⣷⣌⣉⣀⣠⣿⣿⢾⡿⢶⣾⣧⡾⠔⠛⢋⣩⣱⠶⣠⡿⡆⠀⠙⠻⠾⠝⠻⠶⣶⣶⣶⣾⠿⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠁⢀⡤⢄⡀⢍⠷⣄⡓⢀⡉⠛⠋⢉⣬⣽⣍⢻⣿⣿⣍⠉⠠⡝⣿⠯⢽⠉⠉⢐⡪⠥⣄⣥⠶⣶⠮⢿⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣙⢚⢁⠀⠀⠀⠈⠑⠢⣤⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⢅⠒⢫⡈⣯⢀⠀⠠⣒⣲⣎⡲⣾⠇⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢫⠩⣍⣍⣩⣍⣙⣛⡉⣋⣁⠀⠘⠿⣿⠟⢙⣿⣿⣿⠇⠀⢹⣿⣿⡇⢀⣧⡿⠋⢋⣓⣍⡭⠉⣭⣝⣰⣃⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢺⠒⢶⠶⠓⢫⣀⡤⠒⠉⠒⠖⠒⠒⠚⠉⠒⢄⡩⣄⡀⠠⠓⠒⠯⠵⠖⢪⣯⢍⡰⠖⠓⠉⠀⠀⠀