⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢸⡆⠈⠳⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠙⢦⠀⠀⠀⠀⡟⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⢷⠀⠀⠈⠳⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢰⡇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡇⠀⠈⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣸⡇⠀⠙⣆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⠀⣄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠙⣆⠀⢰⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⠈⢣⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⠃⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⣿⠳⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠻⡝⠷⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠀⠀⠀⠘⣆⠈⠙⠓⠦⢤⣄⣀⣿⣀⣀⢀⡀⠀⠘⢦⡀⠙⠓⠲⢦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⢀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠈⢳⣦⣄⡀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⣤⣀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡇⢀⡀⠀⠀⠈⠷⣍⠙⠛⠛⠓⠶⠶⠖⠚⠛⠛⠉⠉⠀⠙⠾⣉⡉⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀
⡇⠀⣿⡛⠓⠒⠒⢾⣟⡢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣓⣶⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀
⡇⠀⠸⣿⣦⡀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⠋⢉⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠹⡄⠀⠀⠀⠀⠀⠀
⡇⠀⠀⠘⣿⣛⠦⣤⣀⠀⠀⠈⠙⠲⣤⣀⣠⠤⠴⠒⠛⠓⠚⣻⠿⠋⠁⠀⠀⣀⡤⠖⠃⠀⢀⡤⣾⡷⢋⡞⠀⠀⣰⠂⠀⣤⠀⠀⠀⠀⠀⠀⠀⠸⡟⢦⣹⡀⠀⠀⠀
⡇⠀⠀⠀⠸⣏⠷⢤⠤⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⢀⣀⠴⠛⣁⣀⠤⠶⢛⡽⠋⠀⣀⡤⠞⠋⠰⠋⠀⡼⠁⣠⠞⠁⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠙⢷⡀⠀⠀
⡇⠀⠀⠀⠀⠹⣆⠀⠀⢀⣤⠚⠁⠀⠀⠀⣀⣤⣶⠯⠖⠚⠋⠉⠀⣠⡴⣋⣤⢶⠟⠉⠀⠀⠀⠀⠀⣼⣃⣼⠋⠀⢀⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀
⡇⠀⠀⠀⠀⠀⢈⡿⠖⠋⠀⠀⠀⠀⠀⠈⢀⣠⠖⢫⠀⠀⠀⣠⡾⢛⣻⢟⣟⡏⠀⢀⠀⠀⠀⠀⣾⡿⡿⢣⠀⣠⠞⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠋⠀⢀⡀⠀⠀⢀⣠⡴⠚⠋⠁⠀⡏⠀⠀⣠⢏⡤⠚⠁⣸⣿⠁⢠⡾⠀⠀⢠⡾⠋⡾⠁⣎⡴⠃⠀⠀⠹⣆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀
⣦⠴⠾⠿⠶⢞⣛⣿⣯⡤⠾⠛⢉⠀⠀⠀⠀⠀⣼⠀⢀⣼⡽⠋⠀⠀⢀⣿⠃⢀⡎⠀⠀⢠⡏⢀⡾⠁⣼⣏⣁⣀⣀⣀⣀⣹⡄⢸⡄⠀⠀ ⡆
⡇⠀⠀⠀⠀⠈⡿⢠⡾⠂⠀⠀⢸⠀⠀⠀⠀⢰⠏⢀⣾⠟⠓⠛⠛⠛⢻⡏⣠⡟⠁⠀⣰⣟⢀⡾⠛⠛⠋⠉⠁⠀⠀⠀⠉⠉⢻⡌⣷⠀⠀⢸⠀⠀⠀⢰⡄⠈⠀⠀⠀
⡇⠀⠀⠀⠀⣼⡷⡿⠁⠀⠀⠀⢸⠀⠀⠀⢀⡏⣠⠟⠁⠀⠀⠀⠀⠀⢸⣱⢋⡇⠀⣠⢯⣧⡟⠀⠀⠀⠀⠀⢀⣀⠠⠤⠀⠀⠀⢳⣹⡀⠀⢸⡄⠀⠀⠸⡇⠀⣿⡟⠀⠀⠀⠀
⠂⠀⠀⠀⠸⠋⣴⠃⣠⡾⠀⠀⢸⠀⠀⠀⣼⡿⠋⢀⣴⣾⣿⣿⣶⣀⡟⠁⣼⠃⡼⠃⣾⠟⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⢦⣄⠘⣿⣧⠀⢸⡇⠀⠀⠀⣧⠀⡿⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣰⣯⣾⢿⡇⠀⢠⣿⡆⠀⣸⠏⢀⡴⢛⣿⣿⣿⣿⣿⡿⢧⡀⣏⡼⠁⣠⠟⠀⠀⠀⠀⠀⡞⣹⣿⣿⣿⣿⣿⣷⠙⢧⣹⣽⠀⢸⠻⠘⣆⠀⣿⣤⡇⠀⠀⠀
⠀⠀⠀⠀⢸⡟⠁⡿⢸⠀⣠⠟⠀⡇⠀⡏⠀⡾⠁⢸⣿⣿⣿⣿⣿⣷⠈⢳⡟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⠀⠘⠋⢻⠀⢸⠀⢠⢻⡄⡏⣿⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠈⠁⣾⡿⢿⣠⣿⠀⢂⢹⣸⡁⠀⠁⠀⠸⣿⡿⢉⠉⢹⡟⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⡿⠋⢹⣿⡿⠀⠀⠀⠈⣇⡇⢀⡾⣼⣷⣧⡿⠀⠀⠀⠀⠀
⡆⠀⠀⠀⠀⢀⡿⠁⣾⢿⠈⣧⣸⡜⣷⡇⠀⠀⠀⠀⠙⠓⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠋⠀⠀⠀⠀⠀⢻⡇⣸⠓⢡⣿⢹⡇⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠈⠀⢰⣿⠞⢷⡈⢻⣇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠇⣠⠏⢣⣘⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣤⣶⣫⣧⣤⣤⣝⣦⣙⢦⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠞⣉⣦⣤⣽⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣒⣒⣉⠉⣿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠃⠳⣶⣒⠒⠉⢻⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⡟⠀⢀⡽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠀⢄⣀⣀⣙⣓⣦⣼⡆⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⢀⣤⣎⡴⠿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣃⣘⣷⣌⣙⢿⣉⠉⠁⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢣⣔⣯⣭⡵⠒⠐⣫⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠀⠀⣿⠛⠒⠛⠓⠚⠓⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠩⣭⣶⡯⠶⠒⢻⢿⣾⠏⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⢀⣠⣾⣋⣙⣲⣜⣦⡈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣟⣥⡶⣼⠛⡏⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠈⠻⣿⠉⠙⠛⠛⠮⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠉⣼⠋⢁⣴⣇⠀⠀⡏⠑⠲⣄⡀⠀⠀⠀⣀⣤⡶⣿⡿⣶⢹⣿⣰⣄⣈⠳⣄⠀⠀⠀⠀⠁⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠞⠋⠀⣿⣄⡀⢰⠀⠀⢀⡌⠛⢻⣿⢿⢻⣿⣅⣿⣇⡿⢿⠛⢿⣌⠙⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⡿⠈⠹⣮⡀⠀⠰⡇⢀⡆⣿⣾⠸⣿⠸⣡⠏⠀⢸⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈
⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⣽⣿⣷⣦⣤⣄⡀⠀⠀⠀⢠⠇⠀⠀⠀⠙⢦⣀⡇⣸⡇⠃⣿⠀⣿⡟⠁⠀⠀⠘⢧⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣤⡀⠈⠙⠢⣤⡤⣤⣀⣀⡁⠀⢀⣤⡶⠶⣒⣚⣷⣿⣿⡇⠀⠙⢿⣿⣿⣿⣿⣿⣷⣶⣶⠋⠀⠀⠀⠀⠀⠀⠈⠙⠶⠯⡤⠟⠀⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠋⠀⠀⢀⣀⣸⣿⣶⣯⣽⡇⢺⡏⠺⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣀⡈⠛⢿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⣿⣿⣿⣿⣿⣥⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠳⣾⣿⣉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡇⠀⠀⠀⠈⠻⣿⣿⣷⣬⣝⣻⣛⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⠉⠉⠛⠛⠦⣄⠀⠀⣠⠴⠛⠉⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⢿⡛⠛⣛⣋⣭⣴⣶⣶⣿⡇