⢀⣴⠖⣾⠟⣲⠿⣤⣞⢉⡷⠯⣄⡬⠷⣏⢙⣦⣼⢷⡚⢿⡖⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡶⣿⢏⡿⣻⠞⢳⣾⡁⣈⡿⣄⣠⠼⢦⣀⡼⢯⣀⣹⣷⡛⢺⡟⣏⢻⡻⣦⡀⠀⠀ ⠀
⣤⣿⡟⣽⢋⡽⠷⢴⡏⠀⢹⣇⢀⣈⣧⣀⣀⣿⣀⣀⣿⠃⠀⢹⡶⠻⣍⠻⣏⢻⣯⣆⠀⠀⠀⠀⠀
⣼⣧⡞⣽⠛⠺⣇⣠⡼⠿⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠳⠿⣦⣄⣼⠛⠻⣍⣷⢽⡇⠀
⣴⢿⣠⠷⢷⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⠿⠿⣬⣿
⠀⠀⠀⠀⣿⠻⣇⢀⣾⠋⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⢠⡴⣤⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠹⣧⢀⣿⡻⣧
⠀⠀ ⣿⡞⠉⣿⠃⣠⡾⠋⠁⠀⠀⠀⠉⠻⢶⠀⠘⠷⠟⠀⢠⡶⠛⠁⠀⠀⠀⠈⠙⢷⣄⢹⣏⠉⢻⣯
⠀⠀⢸⣏⣳⣴⡟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡶⣟⣽ ⠀⠀
⣿⣇⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣇⣸⣯⠀⠀⠀
⢿⡟⢿⡇⠀⠀⠀⠈⠛⠷⠶⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠖⠚⠁⠀⠀⠀⠀⢸⡟⢉⣿⠀⠀⠀
⠘⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠇⠀⠀⠀
⢹⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡏⠀⠀⠀⠀
⢸⡇⢹⣆⠀⠀⠀⠀⠀⠀⣠⣴⢦⣤⣤⣤⣤⣤⢤⢤⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⣸⠏⢸⡇⠀⠀⠀⠀
⢸⣇⠀⢻⣦⡀⠀⠀⠀⢠⡟⠀⠀⠀⠙⠻⠶⠶⠶⠶⣶⠟⠁⠀⠹⣷⡶⠶⠛⠻⣿⠀⢠⡗⠀⠀⠀⠀
⠻⢷⡴⠿⠷⣦⣀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⣟⣷⣿⣦⣀⠀⠀⢸⡷⠟⠁⠀⠀⠀⠀
⣤⣤⡶⣦⡶⠛⣟⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣨⣿⡃⠀⠀⣹⣟⠙⠻⣧⠀⠀⠀⠀⠀⠀
⢠⡿⣿⠀⣿⠘⣷⠀⣿⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⢶⡶⢿⠿⠀⣨⣿⡀⠀⠀⠀⠀⠀
⠀⣿⣇⢹⡆⠘⠇⡘⣷⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡾⢿⡘⣷⠀⠀⠀⠀⠀
⣿⢹⡄⠋⠀⠀⢈⣽⠛⣏⠉⠁⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⢹⡇⠀⠀⠀⠀
⠀⢸⣇⠁⠀⠀⠀⡾⠑⠒⣿⡖⠀⠀⠈⠛⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⣾⣿⠀⠀⣀⣉⣹⡋⠉⣿⠀⠀⠀⠀
⠀⢻⣦⡀⠀⠀⠀⢀⣴⣿⠿⣖⡂⠀⠀⠀⠀⠀⠀⠳⠴⠃⠀⠀⠀⠀⠁⠀⢾⠉⠙⠻⣿⣿⣿⡀⠀⠀⠀
⣸⡏⣻⢳⡖⠛⣿⠃⣿⢧⡉⠻⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⠇⠀⠀⠘⣿⡟⣷⠀⠀⠀
⠀⣰⣟⡴⣣⣼⣃⣀⣻⣤⣿⡼⠿⣟⣋⣿⢻⣝⣛⣷⡶⠦⢤⣤⡾⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸⣷⡙⢷⡄⠀
⣴⣿⡿⢿⡏⠉⠙⢿⡍⢹⣯⡶⠖⠿⣿⣅⣀⣬⣽⣥⣿⠛⠛⣿⣷⡶⠶⠒⢚⣁⠀⠀⢀⣠⣿⡿⢿⣿⣿⡀
⠀⠀⣸⣿⡏⠀⢸⡇⠀⠀⠈⢿⣾⠃⠀⠀⠀⠈⣿⠟⠁⠀⠀⠙⢷⡄⠘⣯⠛⣿⣯⣭⣴⡶⠞⠛⠉⣿⠃⠀⣿⣿⣷
⠀⠀⣿⡏⣿⠀⠈⣿⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⣷⣤⡿⠟⠋⣿⣴⠟⠀⠀⠀⢠⡟⠀⠀⣿⢹⡿
⠀ ⠀⣿⠀⢿⡆⠀⢹⣇⠀⠀⠀⢿⡄⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⣏⠀⠀⠀⢀⣾⠃⠀⢸⡟⢸⣷
⠀⢻⡇⠈⣷⡀⠀⠻⣧⡀⠀⠈⠻⣦⣄⣀⣤⠿⢷⣄⠀⠀⣀⣼⠿⠶⠾⠛⠛⠛⠉⠀⠀⢠⣾⠇⠀⢠⡿⠀⣸⡇
⠀⠀⠈⣿⣄⣘⣷⣄⣀⣈⣻⣶⣤⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣤⡶⣟⣁⣀⣠⣿⣁⣰⡟⠁
⠀⠀⠐⠛⠿⠿⢿⠯⠛⢛⠛⠛⠛⠙⠛⠛⠋⠉⠛⠛⠙⠛⠛⠋⠛⠛⠛⠙⠋⠛⠛⢛⣛⠛⢛⠛⠛⠛⠛⠛⠉⠀⠀
⠀⠀⠀⠀⠀⠀⣠⡶⣿⢏⡿⣻⠞⢳⣾⡁⣈⡿⣄⣠⠼⢦⣀⡼⢯⣀⣹⣷⡛⢺⡟⣏⢻⡻⣦⡀⠀⠀ ⠀
⣤⣿⡟⣽⢋⡽⠷⢴⡏⠀⢹⣇⢀⣈⣧⣀⣀⣿⣀⣀⣿⠃⠀⢹⡶⠻⣍⠻⣏⢻⣯⣆⠀⠀⠀⠀⠀
⣼⣧⡞⣽⠛⠺⣇⣠⡼⠿⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠳⠿⣦⣄⣼⠛⠻⣍⣷⢽⡇⠀
⣴⢿⣠⠷⢷⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⠿⠿⣬⣿
⠀⠀⠀⠀⣿⠻⣇⢀⣾⠋⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⢠⡴⣤⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠹⣧⢀⣿⡻⣧
⠀⠀ ⣿⡞⠉⣿⠃⣠⡾⠋⠁⠀⠀⠀⠉⠻⢶⠀⠘⠷⠟⠀⢠⡶⠛⠁⠀⠀⠀⠈⠙⢷⣄⢹⣏⠉⢻⣯
⠀⠀⢸⣏⣳⣴⡟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡶⣟⣽ ⠀⠀
⣿⣇⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⣇⣸⣯⠀⠀⠀
⢿⡟⢿⡇⠀⠀⠀⠈⠛⠷⠶⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠖⠚⠁⠀⠀⠀⠀⢸⡟⢉⣿⠀⠀⠀
⠘⣧⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠇⠀⠀⠀
⢹⡟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡏⠀⠀⠀⠀
⢸⡇⢹⣆⠀⠀⠀⠀⠀⠀⣠⣴⢦⣤⣤⣤⣤⣤⢤⢤⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⣸⠏⢸⡇⠀⠀⠀⠀
⢸⣇⠀⢻⣦⡀⠀⠀⠀⢠⡟⠀⠀⠀⠙⠻⠶⠶⠶⠶⣶⠟⠁⠀⠹⣷⡶⠶⠛⠻⣿⠀⢠⡗⠀⠀⠀⠀
⠻⢷⡴⠿⠷⣦⣀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⣟⣷⣿⣦⣀⠀⠀⢸⡷⠟⠁⠀⠀⠀⠀
⣤⣤⡶⣦⡶⠛⣟⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⣨⣿⡃⠀⠀⣹⣟⠙⠻⣧⠀⠀⠀⠀⠀⠀
⢠⡿⣿⠀⣿⠘⣷⠀⣿⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⢶⡶⢿⠿⠀⣨⣿⡀⠀⠀⠀⠀⠀
⠀⣿⣇⢹⡆⠘⠇⡘⣷⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡾⢿⡘⣷⠀⠀⠀⠀⠀
⣿⢹⡄⠋⠀⠀⢈⣽⠛⣏⠉⠁⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⢹⡇⠀⠀⠀⠀
⠀⢸⣇⠁⠀⠀⠀⡾⠑⠒⣿⡖⠀⠀⠈⠛⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⣾⣿⠀⠀⣀⣉⣹⡋⠉⣿⠀⠀⠀⠀
⠀⢻⣦⡀⠀⠀⠀⢀⣴⣿⠿⣖⡂⠀⠀⠀⠀⠀⠀⠳⠴⠃⠀⠀⠀⠀⠁⠀⢾⠉⠙⠻⣿⣿⣿⡀⠀⠀⠀
⣸⡏⣻⢳⡖⠛⣿⠃⣿⢧⡉⠻⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⠇⠀⠀⠘⣿⡟⣷⠀⠀⠀
⠀⣰⣟⡴⣣⣼⣃⣀⣻⣤⣿⡼⠿⣟⣋⣿⢻⣝⣛⣷⡶⠦⢤⣤⡾⠛⠉⠉⠀⠀⠀⠀⠀⠀⢸⣷⡙⢷⡄⠀
⣴⣿⡿⢿⡏⠉⠙⢿⡍⢹⣯⡶⠖⠿⣿⣅⣀⣬⣽⣥⣿⠛⠛⣿⣷⡶⠶⠒⢚⣁⠀⠀⢀⣠⣿⡿⢿⣿⣿⡀
⠀⠀⣸⣿⡏⠀⢸⡇⠀⠀⠈⢿⣾⠃⠀⠀⠀⠈⣿⠟⠁⠀⠀⠙⢷⡄⠘⣯⠛⣿⣯⣭⣴⡶⠞⠛⠉⣿⠃⠀⣿⣿⣷
⠀⠀⣿⡏⣿⠀⠈⣿⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⣷⣤⡿⠟⠋⣿⣴⠟⠀⠀⠀⢠⡟⠀⠀⣿⢹⡿
⠀ ⠀⣿⠀⢿⡆⠀⢹⣇⠀⠀⠀⢿⡄⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢸⣏⠀⠀⠀⢀⣾⠃⠀⢸⡟⢸⣷
⠀⢻⡇⠈⣷⡀⠀⠻⣧⡀⠀⠈⠻⣦⣄⣀⣤⠿⢷⣄⠀⠀⣀⣼⠿⠶⠾⠛⠛⠛⠉⠀⠀⢠⣾⠇⠀⢠⡿⠀⣸⡇
⠀⠀⠈⣿⣄⣘⣷⣄⣀⣈⣻⣶⣤⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣤⡶⣟⣁⣀⣠⣿⣁⣰⡟⠁
⠀⠀⠐⠛⠿⠿⢿⠯⠛⢛⠛⠛⠛⠙⠛⠛⠋⠉⠛⠛⠙⠛⠛⠋⠛⠛⠛⠙⠋⠛⠛⢛⣛⠛⢛⠛⠛⠛⠛⠛⠉⠀⠀