Multicellular Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣽⣯⢭⡿⣿⣯⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⣵⣾⣼⣿⣿⣿⣿⣦⣷⣧⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣟⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡞⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠒⠛⢻⡟⣿⢿⠷⡶⠶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢿⡋⠁⣀⠸⡄⢠⣦⠈⡇⠹⣿⢠⡇⢀⣤⢸⠇⠙⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠻⡀⢠⣄⢳⡈⠛⠀⢱⠈⠁⠀⡇⠀⠁⠨⡇⠘⠁⢸⠀⣴⡆⢀⠋⠉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢳⡀⢰⡆⠹⡌⢁⣠⣷⠶⢶⠞⠛⠛⠛⠛⠛⠖⠚⠳⠶⠦⣼⣀⡈⠀⣼⠀⢰⡿⠀⡼⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⡀⠰⣦⠹⣄⣠⡴⠻⡏⠁⠀⠀⠈⡄⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢠⠉⠙⠳⢧⣀⠀⢀⡞⢀⣠⠀⢙⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⢴⡌⠢⣀⣴⠛⠁⠀⠀⠘⡄⠀⣠⡀⢡⠀⢀⡀⠀⢸⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⡙⠳⢯⡀⠈⣫⠔⠋⠀⠻⣄
⠀⠀⠀⠀⠀⣴⠋⢀⣈⠀⣄⢀⣴⠏⠑⢄⠀⠀⢰⣦⠐⡄⠛⠁⠀⠆⢿⡇⠀⠀⠀⠿⠁⠀⠀⣠⡤⠀⠀⠘⠀⠀⠀⠙⢿⣅⠀⠲⠇⢀⠼⢦
⠀⠀⠀⢀⡼⠉⠲⢌⡙⠁⢠⡟⠁⠀⠀⠀⠑⢄⠀⠋⠀⠈⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠄⠀⠚⠁⠀⠀⢀⡀⠀⠀⠀⠔⠉⢷⡴⠊⢁⡀⠈⢳⡀
⠀⠀⢀⣾⣄⠐⠶⠀⠉⣶⠋⠈⠀⠀⠀⢠⣶⠀⠀⢀⠀⠀⠀⠀⠀⠀⠧⠀⡀⠆⠀⢀⡄⠀⠀⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠀⠀⠹⡄⠉⢁⡠⠜⢷⡀
⠀⠀⣼⠃⣀⠁⠂⢀⡼⠋⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⡄⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠚⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠻⡞⠁⣠⡄⠈⣷
⠀⣰⣇⠀⠉⠉⠀⡼⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠁⠀⢀⣀⠀⠀⠀⢹⡄⠈⡀⠔⠛⣇
⢀⣿⠀⠩⡁⠒⢾⠃⠀⠀⠀⠀⠿⠟⠀⠀⠀⠲⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⢻⠀⢠⠶⠀⣻⡀
⢸⠏⠵⠮⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡤⠀⠀⠀⠀⣠⢄⠀⠀⠰⣆⠀⠐⠀⠈⣇
⣼⠛⠒⠲⠶⣾⠇⠀⠀⠀⠐⠻⠿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠛⠛⠉⠀⠀⠀⣿⡀⡈⢛⠀⣻
⣿⠀⠛⠓⠀⣾⠂⠒⠀⠀⠀⠀⠤⠠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠁⠀⠀⠈⠁⢹⡇⠉⣈⠁⢹
⣿⠛⠛⠒⠚⢿⠀⠀⠀⠀⢴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⢯⣽⠆⠀⠀⠀⣸⡀⠈⠉⠀⢸
⢿⠀⠾⠗⠀⢸⡄⠀⠀⠀⠉⠉⢀⡀⠤⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠁⠒⠠⠤⣄⡀⢀⡟⠉⠉⠙⠒⢺
⠸⡷⠶⠒⠛⠹⣧⠀⠠⠀⠂⢉⣁⣄⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠢⡼⠆⠀⠀⠀⣹⡃⠀⠺⢶⡛⣸
⠀⢷⠠⣶⠶⠀⣿⠀⠀⠀⠀⠺⠿⠋⠀⣀⠔⠋⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⢰⠏⠈⠉⠒⠦⢌⡿
⠀⠘⣧⠶⠒⠉⠙⢷⡀⠀⠀⠀⡀⠔⠋⣀⣀⠀⠀⡴⠓⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⠄⠀⠁⢰⣯⡀⠀⠿⠇⠀⢠⡇
⠀⠀⢸⡀⠠⠶⠀⣈⢷⣤⠖⠉⠀⠀⠈⠉⠁⡠⠊⠀⡀⠀⠀⠞⠁⢑⡴⠂⠀⣀⠀⠀⣠⠀⠀⠀⠳⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⡑⠢⠄⢠⡎
⠀⠀⠀⢳⣤⠖⠊⢁⡀⠙⢦⡀⠀⠀⠀⢀⠜⠁⠠⣿⠃⠀⠀⠀⠀⢸⠁⠀⠀⠂⠀⠀⠘⡀⠀⣀⡀⠁⠀⠀⣿⣡⠀⠠⠀⠀⣰⠟⠢⣄⠈⠛⠀⣠⠟
⠀⠀⠀⠀⠻⣄⠀⠛⢃⡠⠞⠻⣦⣀⠔⠁⠀⠀⠀⠀⠀⠌⢠⣤⠀⢰⠀⣀⡀⢀⠀⣴⡄⢁⠀⢿⡵⠀⢠⠀⠈⠁⠀⠀⢀⣾⠋⠀⣴⣤⠑⢤⡴⠃
⠀⠀⠀⠀⠀⠙⢦⡶⠉⠀⣠⡄⠈⢻⣦⡀⠀⠀⠀⠀⠌⠀⠛⠟⠀⢸⠐⣽⠇⢸⠀⠳⠃⠀⡄⠀⠀⠀⠈⡆⠀⠀⠀⣠⠚⠁⠑⣄⠀⠁⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢄⡀⠈⠠⠒⢁⠀⠙⢿⣄⣀⡈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⢈⡀⠀⠀⠀⢠⣤⡴⠚⢣⠀⠸⢷⡈⣳⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡁⠀⠿⠋⢠⠂⠀⠈⢱⠶⢤⣄⣀⠀⣸⠀⠀⠀⠀⠇⠀⠀⣀⣀⣤⣤⠶⠚⠉⠹⡀⢶⡆⠱⣄⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⣴⠀⠀⠾⠃⡜⢠⣤⢀⡟⠙⠛⡷⠒⠒⢲⠛⠙⠛⡏⢉⣀⠸⡄⠰⠷⠀⢧⠀⣀⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣀⣼⠁⠘⠋⣸⠀⠾⠀⡇⢴⡆⢸⠀⠾⠂⢱⠈⠛⠀⢷⢀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⣧⣄⣀⣸⣇⡀⠀⣸⣀⣀⣀⣼⠤⠶⠚⠓⠉
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⣠⡄⣠⣆⣀⣤⣄⠀⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢂⡴⠶⠈⠉⠋⠀⠀⠻⠀⠀⠀⠙⠛⠆⠛⠳⢦⣄⡐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⠤⢴⣆⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⠨⣷⡦⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⣹⠋⠙⠂⠒⠚⠙⠒⠈⠈⠉⢿⠉⠁⠀⠘⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡇⠀⠀⣤⡌⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⡇⠀⡄⢀⡇⠘⠆⠈⠁⠀⠀⠀⠀⠀⢉⡀⡄⠀⠀⠀⠺⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⡗⠂⠁⣽⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⢿⡿⣇⠀⠀⢣⡀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢰⡷⠀⠛⣟⠀⠀⠀⠀⠘⠣⠄⠄⠀⠀⠈⢣⠸⠀⠀⠀⣰⡇⠀⢹⣇
⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⣨⠁⢠⡀⠁⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢰⠀⠀⠀⠀⠻⣇⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠈⠀⣤⠞⢋⠀⢬⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⠂⠀⠀⠀⠀⣁⠀⠑⠆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢼⡗⡮⠀⠀⠈⠓⠒⠀⢘⠁⠀⠂⠀⠀⠀⠀⣴⠎⡇⠀⠀⣿⠃⠀⠰⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⠓⠐⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠻⠀⠁⠀⠀⢸⠀⠘⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⡿⠀⣠⢸⡅⠀⢀⠀⠃⠀⣀⢠⣶⣧⣶⣶⣄⠀⠀⢸⠀⠈⠀⠃⠀⢸⠀⢠⡇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠰⣏⣀⣀⠏⠘⠋⠀⠉⠀⢠⣶⣿⣮⣿⣿⡿⢻⣯⣓⡀⢠⠀⠀⠀⢸⠀⢸⡆⠀⡇⠀⠀⠀⢹⡆
⠀⠀⠀⡄⠀⠀⠈⢉⣿⠀⢰⡄⠐⠂⠀⡿⠻⠏⣻⠟⣿⠇⢋⣻⣿⣿⣎⠀⠀⠀⠈⠀⢸⡇⢰⡇⠀⠀⠀⠀⢳
⠀⠀⢸⠃⠀⢠⡖⠋⢹⠀⣠⠀⠀⠀⢰⡿⠀⠂⠈⠑⣷⣴⠿⠏⠀⠀⠿⡄⠀⠀⠀⠀⢸⣄⠀⠀⠀⢰⡇⠀⠈⡇
⠀⠀⠟⠀⠀⠈⢙⣿⠸⠀⠀⠀⡀⠀⠘⢧⡀⠸⠛⢫⡉⢉⢠⡖⠰⣧⠀⢻⡆⠀⠀⢀⣼⣧⠀⠀⠀⢸⠃⠀⠀⠳
⠀⢸⠂⠀⢀⣴⠿⢁⠀⠀⠀⠀⠀⠀⠀⢸⠟⠦⣀⠀⣀⣉⠀⠀⠀⣼⠀⠀⠑⠶⠷⠟⠁⠟⡁⠀⣀⠇⠀⠀⠀⠀⡇
⠀⣿⠀⠀⣿⡋⠀⠀⠀⠀⠀⠀⠀⡆⠀⣾⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⠀⠀⣇⠀⠀⣀⣷⣿⣿⣾⣿⣷⣤⡀
⠀⣿⠀⠀⠻⢧⣈⠀⠀⠀⠀⠀⠀⡃⠀⣿⠃⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢻⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀
⣰⠁⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⢠⡇⠀⢿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻⣿⣆
⣿⠀⠀⡀⠁⢰⡟⠀⠀⠀⠀⠀⠆⠹⡆⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠾⢿⣾⢻⣿⣿⣿⡿⠿⡹⢿⢻⣿⡧
⡿⠀⠀⠙⠀⠈⠳⠒⢶⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠇⠀⠀⠀⠘⠀⣴⣠⡾⠿⣤⣿⣿⣿⣿⡿⢿⣿⣾⣿⡄
⣿⢀⠀⣧⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣼⣿⣿⣿⣻⣿⣿⣶⣤⡍⢻⡇
⣿⡞⠀⠃⠘⠀⠀⢠⡏⠀⠀⠀⠀⠀⣄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢀⣀⣹⣧⣻⣿⣿⣿⣿⡛⣿⣷
⠋⣠⡆⢀⡀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠈⢦⠀⠧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠯⣿⣿⠛⢻⣿⣿⣿⡟⢻⣿⠁
⢀⡏⠀⡸⠃⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⢲⠀⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡖⠻⢋⣷⣮⣿⠏⣿⣷⣬⡿
⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⡆⢰⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠈⣿⣿⡿⣷⣿⣿⣿⠇⠀⢀
⠀⠸⠃⠀⠀⠐⢆⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠛⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣾⠰⣿⣿⡄⣠⣿⣿⣿⠿⠀⠀⡸
⠀⠀⠾⠀⠀⠀⠀⠶⠀⠀⠸⢧⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣶⣿⣿⣹⣿⣧⣿⡟⢀⠀⢀⠃
⠀⠀⠀⢆⡀⠀⠀⠀⢠⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣽⣿⣿⣿⡿⢉⠀⠀⠀⠏
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠠⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠈⣦⡀⠀⠀⠀⠀⠰⡀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠛⠉⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠁⠂⣀⠀⠀⠀⠀⠀⢰⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⢼⠏
⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠉⠀⠆⠀⠀⠀⠀⠀⢸⠆⠐⠀⠶⠖⠀⠀⠠⠀⠀⠀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢠⣦⠀⢀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⡀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⠀⠐⡄⠀⠄⢠⡤⣮⡛⣃⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⣀⣀⠜⢻⡄⢠⡄⠰⢷⣿⠃⢶⣬⣀⣾⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢀⡾⠃⢸⡇⠸⡇⠈⢷⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣼⠃⠀⣧⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣿⠀⠀⣿⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠙⠛⠛⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣴⣶⣶⣶⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡠⠖⠒⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⢹⢿⠏⠀⠀⠀⠀⠀⠈⡷⠖⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠦⢄⣀⣀⠞⠀⠀⣴⠁⡞⠀⠀⠀⠀⢀⣤⡞⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢻⠀⠀⠀⡇⠀⢇⠀⠀⠀⠀⣾⡉⢇⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⢀⡎⠀⠀⠀⡇⠀⢸⣤⠤⠖⠚⠁⢣⠈⠓⢤⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⣀⣀⣀⣸⠁⠹⡄⠀⠀⢸⠀⠀⠀⡰⠃⠀⡜⠁⠀⠀⠀⠀⢸⠀⠀⢸⠃⠀⣠⣾⡀
⠀⠀⠀⢠⢾⠋⠀⠀⠙⡆⠀⢱⠀⠀⠘⢆⣠⠎⠀⠀⢸⠁⠀⠀⠀⠀⡠⠃⠀⠀⡏⢠⠎⠀⠀⠙⢦⡀
⠀⠀⠀⡞⢸⠀⠀⠀⠀⡇⠀⠈⣆⠀⠀⠈⡏⠀⠀⠀⠸⡀⠀⠀⠀⣞⡀⠀⠀⠀⢧⣸⠀⠀⠀⠀⠀⢳
⠀⠀⢸⠀⠸⣗⠦⡀⠀⠱⣄⠀⠈⠳⢤⣰⠃⠀⠀⠀⠀⣇⢀⣀⡴⢻⠉⠑⡄⠀⠀⠹⡇⠀⠀⠀⠀⣼
⠀⠀⠸⣄⠀⠈⢧⠹⡄⠀⢘⡆⠀⠀⠀⢿⠀⠀⠀⢀⠞⠋⠉⠁⠀⢸⡄⠀⠘⣄⣀⡼⠁⠀⢀⣤⢾⡏
⠀⠀⠀⠈⠣⡀⢸⡀⢳⡀⠘⢆⠀⠀⠀⢸⠀⠀⢀⡏⠀⠀⠀⣀⡤⠋⠘⡆⠀⣼⣫⠴⠒⠚⠻⠁⠘⣇
⠀⠀⠀⠀⠀⢱⠀⠙⢦⠑⢦⣈⡇⠀⠀⡞⠀⣀⡼⣧⣀⣠⠞⠉⢢⠀⠀⡇⠀⢻⡁⠀⠀⠀⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠀⠙⢇⠀⠸⣰⠋⠁⠀⡟⠁⠀⠀⠀⢠⠇⠀⢣⠀⢸⠇⠀⠀⢠⣇⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⠀⠈⠳⢤⡇⠀⠀⠀⡇⠀⠀⢀⡴⠃⠀⠀⠀⣱⠏⢀⣠⠴⠋⣽⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠹⡍⠑⢦⡀⠈⣇⠀⢀⡼⠁⠀⢠⣾⠀⠀⠀⠀⢰⣡⠞⠋⣯⠀⠀⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢳⡄⠱⣄⠹⣤⣞⠀⠀⣴⠁⠘⢦⣀⣀⣀⠞⢿⠀⠀⢸⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣷⡀⠈⠉⠉⠙⠢⡀⢸⠀⢀⡞⠁⠀⠀⣠⠇⠀⢀⡟⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠀⢹⡾⠀⢸⠁⠀⢀⣰⠃⠀⠀⣸⡧⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢸⠁⠀⣾⢀⡖⢉⡿⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⣧⠎⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠿⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢳⣀⠀⠤⠀⠀⠈⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣀⠴⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠴⠊⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡤⢄⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⠀⠘⠧⠞⠀⠀⠀⣸⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣰⠃⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⢠⡖⢢⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⢀⡴⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁⢀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡴⠃⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⡼⠀⠀⠀⣰⠏⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣤⠞⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⢳⠀⠀⠀⠀⠀⠀⡴⠫⣄
⠀⠀⠀⠀⠀⢠⣀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡀⠀⠀⠀⣠⠞⠁⢀⠏⠀⠀⠀⠀⣠⠞⠁⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠈⢩⠞⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠛⠁⠀⠀⣼⠋⠀⠀⡎⠀⠀⠀⣠⣾⡃
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢠⠾⡇⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠑⠀⣠⠞⠉⠀⠙⠦
⠀⠀⢠⣤⣀⣰⠏⠀⠀⠐⣄⠀⠀⠋⠀⠧⣤⣄⣀⡀⠀⢀⡾⠁⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⡦⠀⠠⠀⠀⣠⠾⠋⠁⢀⡾⠁⠀⠀⣀⡴⠛⠁⠈⢷⡄
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⣴⢁⣠⢤⣄⢸⡇⠀⠀⠀⡼⠁⢀⣠⢾⡋
⠀⠀⢀⣀⣸⠀⠀⠀⠀⠘⠛⠉⠀⠀⠈⠻⡇⠀⠀⢰⣧⡾⠋⠁⠀⠳⠄
⠀⠀⠉⠁⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣏⠁
⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣤⡴⠖⠋⠉⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⣠⠇⠈⠁⠈⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠤⠰⡎⠀⠀⢠⡄⠀⠀⠀⠀⢀⠀⠆⣠⣶⣦⠀⠀⠀⠀⠀⠀⢰⣷⠄⠀⣶⣦⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⡸⠂⠲⡆⠀⠀⠀⠀⠀⢀⣅⠀⠈⡛⠋⠀⠀⠀⠀⠿⢆⠀⢈⡆⣆⣉⠁⠀⠀⠀⠀⠀⠀⠃⠂⠀⣀
⠀⠀⠀⠀⠘⠁⠘⡀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠋⠐⠀⡁⠀⠀⠀⠀⠀⠀⠸⠠⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠒⠦⡀⠀⢀
⠀⠀⠀⠀⠐⢀⠐⠀⡿⡿⠇⠀⠀⠀⠀⠈⠘⠑⠠⠠⠀⠑⠁⠀⠀⠀⠸⠿⠇⢀⡘⠿⠿⠁⠀⠀⠀⢀⠀⠦⡠⠚⠢⡴⢠⡄
⠀⠀⠀⠀⠀⣠⡄⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣁⠌⠀⠀⠀⠀⠀⠀⣀⠨⠀⢅⣀⡀⠀⠀⠀⠀⠀⠀⠬⠡⣄⠤⠡
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢈⣀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠱⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⢠⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠰⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣐⠹⢀⣐⡀⠀⠀⠀⠀⡢⠀⠙⣼⣄⠁⠤⠀⠀⠀⠀⠀⠀⠻⠋⠐⢴⣶⡄
⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⠿⠂⠀⠀⠀⠀⠠⡛⠂⠤⠙⠁⠀⠀⠀⠀⠀⠠⠆⠹⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠔⠁⠉
⠀⠀⠀⠀⠀⠈⠁⠐⠠⠁⠀⠀⠀⠀⠀⠀⠈⠉⠰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠑⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢐⠆⠀⢙⡋⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⣀⡀⠀⠤⣉⡀⠀⠀⠀⠀⠀⠐⠈⠓⣤⠀⠈⠁
⠀⠀⠀⠀⢰⠀⠀⡹⢅⠀⠀⠀⠀⠀⠀⠎⠆⠀⣶⡎⠐⣀⢀⠀⠀⠀⠈⠛⠃⠒⠠⠦⣶⠄⠀⠀⠀⠀⠀⠰⠌⠐⠀⢆
⠀⠀⠀⠀⠘⠢⠈⣘⠀⠉⠾⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠏⠀⠀⠀⠀⠀⠀⠀⠢⠢⠈⠀⠀⠀⠀⠀⠀⢠⡁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠒⠒⠉⠉⠉⠉⢉⣉⣉⣙⣒⠒⠂⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢉⡡⣴⣶⣿⣇⣠⣾⣷⣶⣤⣤⡤⢄⣀⣀⠉⠉⠑⠒⠦⢉⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢉⣠⠶⡞⣡⣾⣿⠿⢿⣿⣿⣶⣿⣷⠈⣿⣿⣟⣻⡿⢻⣟⡲⣤⣀⡀⠀⠐⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣁⡤⠊⢡⠊⢀⣼⣿⣿⢱⣿⣿⠝⣿⣿⣿⠿⣀⣿⡿⣿⢿⣿⣿⣭⣿⣿⣽⣿⣓⣦⣤⡈⠑⠨⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡴⠊⠁⢠⣜⣛⣴⣿⣿⣿⣿⣿⣛⣫⣼⣿⣯⣶⣾⣧⡙⣻⣧⣿⣿⣿⣿⣶⢿⣤⠬⠷⣽⣿⠛⣗⣄⠈⠢⣈⠳⣄
⠀⠀⠀⠀⠀⠀⡴⢋⡴⠉⠂⣠⣾⣷⣾⣿⢿⣿⣿⠿⡇⠟⠟⠻⠛⠉⠛⠈⠁⠉⣼⡿⣷⣿⣿⣧⣿⢾⡏⢰⣿⣷⣆⠹⣿⣿⡍⣷⣄⠈⠃⡌⠱⣄
⠀⠀⠀⠀⣠⠌⣠⠎⢁⣠⣴⣿⣿⣿⣿⣿⣿⣿⡷⠄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢈⡉⢻⠾⣿⡿⢧⡘⠿⠿⠋⢈⣯⣭⡝⢿⣾⣳⡀⠀⠱⡌⠳⡀
⠀⠀⠀⡰⢋⡴⠃⣤⣼⣿⣿⡏⣠⣤⣬⠛⣿⣿⣿⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠺⠃⠻⡍⠛⠓⠆⠲⣗⡌⠻⢿⠇⢿⣿⣾⣷⣀⠠⠘⠆⠙⡀
⠀⠀⡘⢁⡞⠀⣾⣿⣿⣿⣿⣦⠿⢿⡿⢃⠟⠟⠋⠁⠀⠀⢈⠀⣀⣠⠽⠶⠶⢦⢼⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣾⡾⣿⣿⣿⣻⡯⢄⠀⠀⣆⠰⡀
⠀⠸⢁⡾⠀⣀⣼⡟⣿⣿⣥⢿⣿⡶⢖⡀⠀⢀⢄⠀⠠⢀⡴⠚⠉⠀⠀⠀⠁⠀⠀⠀⠀⢙⠻⣄⢀⠂⠁⠀⠀⠀⡴⠿⢿⣿⣷⣿⣷⣟⣯⣻⢿⣄⡿⠜⠀⢱
⢠⠃⣼⠠⡑⢿⣿⣹⣿⣾⢿⣹⣿⣿⠯⠀⠀⠀⠀⠀⢐⠮⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠈⢳⣄⡀⠈⠀⠀⠀⢀⣼⣟⣿⡾⣿⡚⠿⢿⣼⣿⡻⠀⠰⠀⡄
⢸⠀⠇⠀⡐⢘⣿⣿⣶⠷⠿⠟⠛⠉⠀⠀⠀⠀⠀⢠⡋⠀⠄⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠂⠐⠀⠀⢹⡂⠀⠀⠀⠈⢻⠉⣿⣿⣶⣼⣟⣳⣌⡿⢟⣳⣃⢀⡅⢰
⡘⢰⠀⠀⠀⠀⣿⣿⣇⣆⠈⠀⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⠀⣟⡊⠀⠀⠀⠀⠀⠈⠄⠹⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠘
⡇⢰⠀⠨⢡⣤⣿⣿⣻⢿⣷⡆⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⠿⠻⠿⣿⣿⣧⡀
⡇⢸⠀⠁⢠⣿⢱⣿⣿⣿⣿⢋⠀⠈⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠤⠀⠁⠀⠔⠀⠀⠀⠀⠀⠂⠀⠀⢀⢸⣯⡜⢀⡤⠀⠉⠛⣻⣿⡟⢩⣾⣿⡦⢿⣿⡛⡗⠀⠋
⡇⢸⠀⠛⢻⣿⣿⠛⣿⣵⣿⠄⠀⠀⠀⠀⠀⢀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢘⡇⡀⠉⠀⠀⠀⢰⠏⣴⣿⣾⣛⡛⣧⣿⡿⣿⣷⡀⡀
⡇⢸⣆⠀⣨⣿⣿⠿⣿⡏⣻⡇⠀⠾⠃⠀⠐⠀⠀⠄⠚⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠇⠄⠀⡀⢀⠀⠀⠤⠈⠛⠋⠟⣿⣟⣿⣷⣿⣏⠉⡥
⢹⠘⣏⠉⢉⠙⣿⡿⣿⣿⣏⣁⠀⣀⠁⣀⠀⠀⠀⠁⠈⠘⢦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⡞⠀⠀⣠⣷⣤⣦⣴⣧⣴⡆⣠⣶⣿⢏⣴⣶⣤⢿⠉⠁
⠘⣆⠙⡄⣼⢿⣿⣿⣿⣿⣾⣿⣿⣟⡋⠉⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡞⢁⣱⡾⠏⣉⣤⡤⠶⣦⣬⡙⢿⣟⣻⣟⢿⣿⣿⠊⡈⢀
⠀⠹⡄⢹⡀⢀⠑⣿⣿⣿⣿⠿⠛⠋⠉⢡⡁⠀⠠⢤⡁⠀⠁⠀⠚⠉⠲⢦⣤⡀⡀⠀⠀⠀⣀⡴⠋⣠⣿⠟⣵⠟⢫⣴⣾⣷⣄⠉⢻⡆⢹⣽⡿⣷⡶⠃⡀⠀⡌
⠀⠀⠡⡈⢇⠈⢰⣾⣿⡿⢁⣠⣴⡶⠶⠚⠷⢦⣄⠀⠘⣧⢀⣐⠀⠀⠀⠈⠈⠉⠉⠓⠒⠛⠁⠶⣾⣿⡟⢸⡇⢀⣿⣿⣿⣿⣿⡇⠈⣿⠈⣷⠅⡘⠁⠜⠀⠈
⠀⠀⠀⢣⠈⢏⠀⠘⣿⢠⣟⢹⣵⣿⣿⣿⣷⣤⠙⢷⡀⣷⣾⡋⠛⠀⠀⠀⠀⠠⠀⠀⠄⠀⠤⢾⣿⣿⣧⢸⣇⠘⢿⣿⣿⣿⣿⠗⣺⡟⠚⡛⠚⢲⠞⡴⠃
⠀⠀⠀⠀⢳⡈⢧⡀⠙⢸⡇⣾⣿⣿⣿⣿⣿⣿⠃⢸⣇⣿⣿⡃⣄⠀⣦⣀⠀⣴⣦⡃⠀⢀⣠⡶⣻⣟⣻⣆⠻⢦⣄⣉⣋⡉⢠⣤⠟⢁⠔⢁⠔⢁⠞
⠀⠀⠀⠀⠀⠙⢆⠑⢄⠈⠻⣬⡻⡿⣿⡿⠟⠁⣠⣾⣽⣿⣿⠁⣽⣶⣿⣿⣶⣿⣿⣿⣛⣯⣿⡰⣿⣿⡿⢭⣷⣦⣬⣉⣉⣉⣩⣤⡶⠁⡀⢊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠳⢦⡙⠦⣈⢛⣳⡶⠶⠶⢛⣩⢞⣽⣿⣏⣿⡏⣾⣿⣿⣿⣿⣿⡿⠷⠿⣿⣿⣾⣯⡟⣶⣿⣿⣿⢿⣿⡟⠡⢬⡠⢊⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⠐⢍⠻⠲⣿⣿⣿⡻⣿⣿⣿⣾⣏⣭⣽⣿⣿⣿⣷⣾⣿⣷⠈⣿⣿⣟⣇⣿⣿⣿⢟⡿⠋⢁⠰⢋⡴⠊⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢌⡓⠤⡉⠙⠻⢿⣿⣿⣷⢻⣿⣿⡿⣿⣿⣷⣿⣻⣯⡾⣿⡿⠛⡟⣉⡉⠛⢀⠀⠚⣁⠔⠋⠀⠀⠀⠀⠀⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡑⠦⢀⣸⡿⠈⠻⣎⠉⠻⠟⠛⠛⠛⠛⠉⠺⢿⡇⢀⢀⣀⠴⠊⢁⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠠⠬⣁⣊⠛⠓⠒⠒⠒⠒⠀⠀⠀⢘⣋⣉⠩⠤⠒⠚⠋⠁
⣶⣶⣶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛

⠀⠀⠀⠀⢰⣶⣴⣶⡀⠀⠀⢠⣴⣀⣶⡀⠀⠀⠀⣤⡰⡶⣦⠀⠀⠀⣴⣶⣶⡀⠀⠀⠀⢠⣖⣲
⠀⠀⠀⠀⢸⣿⣼⣿⠃⠀⠀⠠⣴⠻⣦⠆⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⢸⢿⣿⣷⠀⠀⠀⣼⣿⢷⣤
⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⢹⣇⡀⠀⠀⠀⠘⠿⣯⡴⠇⠀⠀⠀⣱⣼⠦⡀⠀⠀⠀⣀⣴⣾⣿⠀⠀⠀⣼⣿⣏⠁
⠀⠀⠀⠀⠸⠾⠚⠷⠂⠀⠀⠰⠾⠋⠒⠃⠀⠀⠀⠻⠟⠟⠀⠀⠀⠀⠿⠿⠟⠁⠀⠀⠀⠙⠁⠹⠷

⠀⠀⠀⠀⢀⣤⢀⣤⠀⠀⠀⠀⣄⠀⢀⡀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢀⣠⢄⠀⠀⠀⠀⣀⠐⡂⢀
⠀⠀⠀⠀⢈⣥⡞⢋⡀⠀⠀⢀⣿⡎⣼⡇⠀⠀⠀⠸⣉⢱⡄⠀⠀⠀⣹⣍⡟⣏⠀⠀⠀⡀⢯⡹⢁
⠀⠀⠀⠀⠈⠉⠘⠋⠁⠀⠀⠘⠛⠛⠛⠁⠀⠀⠀⠀⠘⠁⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠐⠂⠈

⠀⠀⠀⠀⢰⡟⣶⡀⠀⠀⠀⢠⣂⣰⣿⡇⠀⠀⠀⢖⠐⠄⡀⠀⠀⠀⠀⣼⣥⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠈⠙⠻⣟⠆⠀⠀⠀⣠⡥⣯⠀⠀⠀⠀⠀⢣⠀⠒⠀⠀⠀⠀⡜⡇⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠒⠛⠻⡟⣻⢿⡿⡶⠶⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢾⡋⠉⣀⠸⡄⢀⣤⠀⣇⠘⣿⠃⣯⠀⣠⡀⣽⠀⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢯⠁⣠⣈⢧⠘⠿⠀⢳⠈⠉⠀⢿⠀⠉⠀⣿⠀⠛⠁⡛⢀⣶⠀⢸⠏⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⣏⠀⣤⡌⢣⠈⠉⣈⣧⣤⣤⠾⠷⠶⠶⠿⠶⡶⠶⠿⠦⣤⣤⣇⠈⠋⠀⡞⠀⢰⣶⠀⣽⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡋⠀⣤⡘⢦⠈⣀⣤⢿⡟⠋⠁⠀⠈⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢙⠛⠳⢶⣇⠀⠈⠀⡼⠁⢀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⣍⠢⡈⢁⣴⠟⠋⠀⠀⢳⡀⠀⢀⠀⢡⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⢀⠃⠀⠀⠀⠉⡻⢦⣞⠀⠘⠟⣠⠞⠉⢳⣄
⠀⠀⠀⠀⠀⢀⡼⠋⠙⠂⡈⠉⠀⣼⠿⢇⠀⠀⠀⣠⡄⢡⠐⢿⠃⠈⡆⢾⣷⠀⠘⠀⢼⡟⠀⠀⠂⣀⡀⠀⠀⡰⠁⠀⠉⠳⣤⡋⠀⢠⡦⠀⢹⣦
⠀⠀⠀⠀⢠⠟⢤⡀⠻⠆⠈⢲⡾⠁⠀⠀⠑⢄⠀⠙⠿⠀⠣⠀⠀⠀⠰⠈⠃⠀⠀⠀⠀⠀⢀⠂⠸⠿⠃⠀⠈⠀⠀⠀⠀⠀⡨⠻⣦⣀⡤⠚⠁⠈⢧⡀
⠀⠀⠀⣰⠋⠀⣤⡈⠳⢄⣴⠟⠄⠀⠀⠀⢠⣄⠁⠀⠀⠀⠀⠁⠀⠀⠀⣇⠀⠀⣠⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠊⠀⠀⠈⢿⡀⠶⠟⠀⢈⣷⡀
⠀⠀⢰⡟⠛⠢⣌⠀⠀⡼⠃⠀⠀⠀⠢⠀⠙⠋⠀⠀⠀⠀⠀⠀⠰⠄⡀⠈⠠⠂⠈⠀⠀⠻⠀⠀⢀⡔⠀⠀⠈⠁⠀⠀⢀⣀⠀⠀⠀⠀⢷⣤⠴⠋⠁⠈⢷
⠀⢀⡿⠀⠶⠦⠀⢀⡾⠁⠀⠀⠀⣀⠳⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠞⠋⠀⠀⠈⠀⠀⢳⡀⠰⠟⠀⣸⣧
⠀⣼⠗⣢⠤⣀⣀⣾⠁⠀⠀⠀⢠⣤⡄⠀⠀⢀⠀⠀⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠀⢀⠀⠘⣧⠀⠂⢉⠀⢸⡆
⢠⡿⢀⣈⠆⠀⣹⠏⠀⠐⠂⠤⠈⠉⠁⠀⠀⠀⠙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⡆⠐⠛⠁⠀⢷
⢸⣇⣚⣛⠀⢀⡿⠀⠀⠀⠀⣠⣤⣄⠀⠈⠙⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⠀⠀⢠⣤⣶⡲⠀⠀⠀⢿⠖⠈⣠⡄⢸
⣿⠉⣉⣉⠉⢻⡇⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⡈⠁⠀⠀⠀⠀⣸⣧⣤⣬⣤⣤
⣿⣀⣉⣉⣀⣸⡇⠀⠁⠀⠀⠈⠉⠁⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢸⡇⠀⣤⡄
⣿⠉⢉⡉⠉⢹⡇⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡀⠀⡀⠀⠳⠿⠃⠀⠀⠀⢸⣇⣀⣀⡀
⢻⡀⠛⠛⢀⣸⡇⠀⠀⠀⠀⠀⣀⡠⠤⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠉⠁⠒⠲⠦⢄⣀⣿⠁⠀⠉⠉⠙
⠘⡷⠚⠛⠋⠉⣷⠄⠐⠒⠉⠉⣠⣤⣄⠀⠀⠀⢀⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠳⠾⠆⠀⠀⠀⣸⣇⣀⠘⠻⢟⣋
⠀⢻⠀⠾⠟⠀⣹⡆⠀⠀⠀⠘⠿⠛⠁⢀⡤⠚⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⢀⡑⠢⢄⡀⠀⢠⡟⠀⠉⠙⠒⠢⣭
⠀⠘⣷⠖⠚⠉⠉⢻⡄⠀⠀⠀⣀⠤⠚⢁⣀⡀⠀⢀⡼⠃⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢯⣿⠀⠀⠀⢠⣿⣀⠀⠘⠟⠀⠀⡾
⠀⠀⢸⡇⠀⠶⠆⢀⣽⣦⡴⠊⠁⠀⠀⠘⠋⢁⡴⠋⢀⡀⠀⢀⠞⠉⠙⣤⠒⠀⢀⡄⠀⢠⡆⠀⠀⠘⢧⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣉⠲⢄⠀⣼⠃
⠀⠀⠀⢻⣤⠴⠊⠉⡀⠈⢷⣄⠀⠀⠀⠀⡠⠊⠀⢠⣿⠁⠀⠀⠀⠀⢀⠇⠀⠀⠘⠀⠀⠈⢃⠀⠀⣀⠈⠃⠀⠸⣯⣡⠀⠢⡀⠀⣠⡟⠳⣄⠈⠛⠃⢀⡼⠃
⠀⠀⠀⠀⠻⣆⠀⠸⠛⣀⡴⠛⢦⡀⢀⠜⠁⠀⠀⠀⠀⠀⠊⢀⣤⠀⢸⠀⢀⡀⢠⠀⣠⣄⠈⡀⠸⣿⡆⠀⡄⠀⠉⠁⠀⠀⠈⣶⠏⠀⢠⣤⡑⢤⣠⠞⠁
⠀⠀⠀⠀⠀⠙⢧⣴⠞⠉⢀⣠⠀⠙⣷⡀⠀⠀⠀⠀⠀⠈⠀⠻⠿⠁⢸⠀⣿⡗⢸⠀⠹⠟⠀⢡⠀⠈⠀⠀⢱⠀⠀⠀⠀⣠⠞⠉⠳⣄⠈⠉⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠈⠉⣠⠞⠁⠙⠳⣦⡀⠀⡐⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠈⡆⠀⠀⠀⠀⢇⣀⣤⠾⢏⠀⢴⣦⡜⢷⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡞⠀⢰⣿⠂⣠⠋⠉⠛⢷⣤⣀⡀⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⠀⢘⣀⣠⡤⠶⠛⢫⠀⣴⡌⢢⡀⠈⣹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⡰⠁⢠⣶⠀⡼⢀⡈⠙⡿⠶⠾⣦⠤⠤⢤⡴⠶⠶⢾⠛⠛⠉⢿⠀⢰⣆⠘⡆⠉⠁⣠⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⣤⡀⠀⣰⠁⠺⠟⢰⠁⣴⠀⡟⢀⣤⢰⡇⢠⡆⠸⡀⢾⠇⠸⡆⠀⢁⣀⡽⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣤⣄⣀⡏⠀⠀⢀⡇⠘⠋⢸⡃⠈⠀⢈⣇⣀⣠⣤⠥⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠺⠷⠶⠶⠾⠗⠚⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⡁⠀⠘⠂⣉⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⣤⠛⢷⠂⠿⠂⠉⣴⠍⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢦⠄⠀⠒⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣱⠟⣡⡔⠂⡤⠀⠘⠂⠀⢴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⣤⠂⠈⢳
⠀⠀⠀⠀⠀⠀⠀⢀⠙⣌⣳⣘⣋⠸⠂⡰⠆⠀⠀⢰⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⢴⠂⠠⠀⠀⠐⠂⠀⠶⠈⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⡼⢏⠙⠋⠋⢩⠆⠀⠃⠀⠀⠀⠀⣻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⠀⠀⠀⠣⠀⠒⠃⠐⠒⠒⠆
⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠑⠊⣡⠚⠒⢠⡄⠀⢉⢀⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⢶⠀⠄⠀⠰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣼⢂⡀⢠⠒⠼⢏⠿⠀⠂⠀⠀⠀⣤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠈⠤⠀⠀⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣼⠇⠲⠋⢰⢆⡉⡀⠀⠀⠀⠀⠰⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠐⠃⠀⠂⠀⠀⢈⡇
⢀⡀⠀⠀⠀⠀⠻⢎⠀⠠⠀⣀⠀⠁⠀⠁⠀⡀⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠈⡀⢀⠆⠀⠀⢰
⠀⠀⠀⠀⢠⣤⠛⠀⠙⠂⠸⢃⡶⡃⢴⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠔⢀⡜⠀⠈⣀⠷⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣡⠶⠀⠀⠀⠀⠈⠁⢤⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠙⠀⠀⢶⠀⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠱⢆⣀⣀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢀⡀⠠⠀⠄⠈⣧⠀⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠗⠀⠙⢇⠀⠀⠱⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⡘⠀⠠⣀⣜⠂⡰⠂⢠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠁⠀⠀⠻⢉⡘⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠓⠈⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣄⠰⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣦⡄⠉⢠⡠⣞⡁⢨⡅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣠⠀⠀⠈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢃⡀⠈⠒⠋⠱⠟⠉⠊⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠁⠂⡄⠀⠀⡄⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠀⢸⡀⢻⠇⡀⠳⢰⠁⢶⠀⣾⡁⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⠀⠶⣄⠀⠀⠀⠐⠐⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠁⢀⡄⠁⠀⠀⠁⢀⣠⣌⠁⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⠆⠻⠂⡄⠀⠉⠀⠀⠀⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠊⠿⢆⠀⠂⠰⠘⠃⠉⢀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣧⠙⣶⠞⣶⡆⢰⡏⢸⠇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣬⠁⡼⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢤⡈⢉⡀⠈⠁⠀⠄⣀⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢈⠀⠀⢠⡄⠰⢾⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⣸⡇⣙⢹⡶⠀⠛⠘⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡟⠂⠀⠀⠀⠀⠋⠠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢷⡇⢠⡄⠠⠀⣀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠺⠇⠻⠁⠛⠡⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⣤⠘⠳⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⣠⡆⠀⣾⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⣤⠶⠀⢀⠀⠠⠉⠠⠄⠻⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣽⣵⢦⣴⣟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠻⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠿⠟⢡⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠰⠄⡀⠀⠀⠠⠀⠤⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠱⡄⠀⠀⠀⡀⠰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢂⡄⠀⡀⢀⠀⠀⢠⣤⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠠⡀⠀⠀⢠⡀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠸⠇⠀⠁⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠦⠠⠄⠀⢧⠀⠄⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠀⠀⠴⠄⠈⢁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣧⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣴
⣿⣿⣿⡿⠟⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣀⣀⣠⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠂⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠓⣄⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠊⠐⠀⠐⠄⠀⠀⠀⠀⠀⠂⠀⠠⡀⠀⠀⠀⡄⠀⠀⠀⠝⣶⣄⠀⠀⡠⠀⠀⣀⡀
⠀⠀⠠⠀⢁⠔⠂⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠂⠋⠀⠀⠂⠀⠱⡀
⠀⢠⠁⠀⠋⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⢱
⠀⠠⠊⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠠⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡧
⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠹
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⡁⠀⠠⠤⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀
⢀⡤⠤⠄⠨⡀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⣼
⡇⠀⠀⠀⢠⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣈⣸⠇
⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠄⠀⡀⠒⠀⠁⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡇
⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠂⠀⠀⠐⠁⠀⠀⠀⡂⠀⠀⠀⠀⠀⢀⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇
⠀⠈⢀⠈⠁⠀⠀⠀⡜⠁⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠎
⠀⠀⠀⠀⠀⠆⠈⠀⣤⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠠⠋
⠀⠀⠀⠀⠀⠀⠑⡎⠀⠀⠀⠀⠄⠁⠀⠂⠀⢀⢀⠀⠀⠀⠠⠈⠀⢀⠂⠀⠂⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⢠⠀⠀⠁⠀⢀⠔⠁⠀⠀⠀⠀⠒⠀⠀⢤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⢄⣀⣀⠄⠀⠀⠣⡀⣀⠀⠀⠠
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⣄⡄⠀⠀⣤⢠⣤⠀⠀⠀⢀⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠔⠎⡛⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠂⠘⠀⠃⠇⠘⠂⠉⠈⠀⠀⠀⠀⠀⠁⠈⠁⠀⢀⡀⢸⣳⡠⠔⠁⠀⠀⢀⡜⠓⢤⡀⢰⠋⠀⠀⠠⠀⠠⠀⡇⠀⣰⣦⡁
⠀⠀⠸⢇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⣼⣠⠏⠀⠀⠀⠐⠀⠈⠀⠀⠈⠀⠸⡆⢄⠀⢅⠀⣐⠼⠁⠀⡇⡜⠃⠀⠀⠀⣀
⠀⠀⠀⠘⡆⡜⠀⠀⠀⠀⣤⠀⡰⠂⠀⠀⠼⢾⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠒⠒⠟⣶⠛⠛⠉⠁⠀⠀⠘⡇⣇⠀⢀⡰⢶⣝
⠀⠀⠛⢄⢻⡇⠀⠀⣠⡞⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⢉⠦⠤⠤⠴⢠⡤⣧⡄⢼⠆⠀⠀⢰⠉⠂⠌⣐⠀⠁⠀⠈⢳⠀⠀⠀⠀⠀⠀⢳⣈⠉⢉⣀⠜⠋
⠀⠀⠀⠈⢢⡇⠀⠀⢈⣧⡀⠀⣸⡄⠀⠀⡠⡄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠘⣆⠌⠐⠀⢁⠂⠈⠂⡼⠀⣠⣤⣄⠀⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢳⠀⠀⠁⠘⠛⠙⡍⠀⠀⠀⠉⠁⠀⠀⠀⣄⠀⠀⠀⠀⠀⣴⡽⠀⣀⡠⢤⠤⡬⠷⣤⠤⠤⠤⠖⠋⢀⢮⠞⠁⠀⡴⢹⢀⣤
⠀⠀⠀⠀⠀⢹⢠⠶⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡿⠋⠀⢸⠁⠠⠀⠀⠀⠄⠈⢦⠀⠀⠀⠀⢸⠘⠒⣦⡸⠁⠓⠊⡼⠀⣾⣿⡆
⠀⠀⠀⠰⢆⢸⡏⠀⠀⡠⠤⠤⣀⣀⠴⠊⠳⡀⠀⠀⠀⠀⣀⡠⣤⠄⣀⣀⣀⠘⣦⠈⡀⠀⠀⠄⠀⣘⠇⠀⠀⠀⠈⠓⠊⠡⡷⠒⡆⣸⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠈⢾⡇⠀⠘⣅⠀⠀⡼⠉⠓⢦⣸⠁⠰⠀⠖⠉⠃⠀⠀⢀⠠⠐⠉⠉⠉⢷⠚⠛⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⠖⢲
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⢦⡞⠁⠺⠇⢠⡏⢢⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⣤⣤⠀⢠⣤⡄⠀⣠⣤⡀⠀⠀⣿⡇⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠠⠂⢼⠙⠲⢄⣠⠋⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠐⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣠⡇⠀⢸⢹⠀⠀⠀⡿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡴⠞⠁⠉⠉⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣾⠀⢸⣿⡇⠀⢸⢸⠀⠀⠀⣙⠇⢸
⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡀⠠⠍⡇⠀⠀⢀⢏⣌⢧⠀⠀⢸⣿⠀⢸⣿⡇⠀⢸⣿⠀⠀⠀⣧⠄⢾
⠀⠀⠀⠀⠠⠀⡷⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠛⠛⠀⡇⠀⠀⢸⢸⣸⢸⠀⠀⠘⠋⠀⠈⠛⠁⠀⠘⠛⠀⠀⠀⠿⠤⠼⠰⠆
⠀⠀⠀⠀⠀⠀⡧⡇⠀⠁⠀⠀⠀⠀⠀⠀⠐⡿⠷⢶⠂⠀⠀⢸⢰⡏⠙⢆⡇⠀⡆⠘⣆⣡⠞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⠀⠀⠀⠀⠀⠀⢀⠇⠂⠘⡆⠀⠀⠀⠶⠁⠀⠘⠃⠘⠃⠀⠀⠀⠀⠀⠀⢠⣾⡁⢀⣀⣠⠔⠛⠓⢤⣠⠎⠀⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⡴⣇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⠀⢹⡀⠀⠀⠀⢀⠀⢀⠀⠀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠈⠛⠿⡀⠀⠀⠀⠈⠋⠀⠤⠆⠀⠈⠁
⠀⠀⠀⠀⢀⡞⠤⣈⣆⠀⠀⠀⠀⠀⠀⣸⠃⠠⣕⠀⠀⢧⠀⠀⠀⠉⠀⣿⠀⠀⠀⢰⡆⠀⢇⣤⣤⣀⣀⣀⣀⡀⠘⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡞⠈⠘⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⠀⣤⡄⠀⢶⠀⠀⠀⠣⠤⠄⣀⣀⣀⣀⣤⠀⠀⢠⣤
⠀⠀⠀⠀⢣⣀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⠐⣿⠀⠀⠀⢸⢿⠀⣿⡇⠀⣿⠀⠀⡤⡄⠀⠀⠀⣶⠀⠀⡷⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣠⡀⠙⠿⢿⡿⢿⡿⠿⠿⠿⠿⠛⠀⠀⠀⢿⠀⠀⠀⢸⢸⠀⡇⡇⠀⣿⠀⠀⡇⡇⠀⠀⠀⢸⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢖⣢⣀⣀⣹⣷⣤⡤⠼⢷⣦⠤⠤⠄⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⢸⢸⠀⡇⡇⠀⡿⡆⠀⡇⠇⠀⠀⢸⢸⠀⠀⣷⡇⠀⠐⢶
⠀⠀⠀⢀⣤⣤⠔⠊⠉⡿⠃⠀⠐⠖⠁⠀⠺⠊⠀⠀⠀⠀⠀⢤⡄⠤⠤⢾⠤⠶⠶⠾⠾⠶⠷⠷⠶⠻⠗⠦⠗⡖⠒⠒⣼⠛⠒⠲⠟⠓⠿⠆⠙
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠘⠈⠀⠈⠉⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣠⣤⡶⠶⠿⠚⠓⠲⠦⠤⣼⣁⠀⠀⠀⠀⠀⣠⡾⠛⠃
⠀⠀⠀⠀⠘⠛⢷⣶⣆⡀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠦⣤⡞⠁⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⣀⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⡠⠂⡀⠀⠀⠀⠀⠀⠁⠤⡀⠀⠀⠀⠀⠈⢷⡀⠀⣀⣠⣾⠿⠋
⠀⢰⣶⣦⣠⣀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠀⠀⠀⢿⣼⠆⠀⠀⠀⠰⣺⣿⠣⡀⠀⠀⠀⠘⢛⡾⠛⠉
⠀⠀⠀⠉⠙⠛⠿⣷⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣼⣟⠉⠁⠀⠱⡀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠚⠛⣶⠀⠀⠙⣧⡀⠀⠀⣀⠀⠀⠀⠀⢿⠀⢀⣀⣤⣶⠿⠟⠁
⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⡶⣤⡀⢹⡇⠀⠀⢹⠀⠀⠀⠀⢸⡙⠛⠛⠉
⠘⠛⠛⠻⠟⠻⠿⠟⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣧⠀⠤⠾⠁⣾⠃⠀⠀⢨⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⠙⠁⠀⣀⣠⡾⠁⠀⠀⠀⢸⡄⠀⠀⠀⢸⡁⢀⣀⣀⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⡄⠀⠀⠉⠉⠉⠉⠁⠁⠀⠀⢞⣷⠘⡇⠀⠀⠀⠀⠛⠟⠛⠋⠉
⠶⠿⠶⠾⠿⢶⡶⠖⠲⠀⠀⠀⠀⠀⣿⠀⠀⠸⣏⢰⡀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠃⠀⣿⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠿⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠈⢷⡀⢀⣀⣤⣴⡶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡠⣶⣦⠀⠀⠰⣉⡿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠘⢷⡙⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣤⣦⣤⣶⣶⣶⣶⣾⡆⠀⠀⠀⠀⣿⠀⠀⠀⣤⡴⠞⠉⢀⣶⣰⡟⠀⠀⠀⠀⣀⡀⠀⠀⢻⣆⠀⠀⠀⠀⠙⢦⡀⠀⢀⣀⣠⣴⡶⢿⠟⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣐⣁⡾⠃⠀⠀⠀⢻⣦⡀⠀⠀⠀⠈⠻⢆⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠸⣇⠀⠀⠠⢴⣦⠀⢶⡠⡀⠀⠀⠀⠉⠉⠀⠀⢴⣤⣾⣿⠿⣳⣄⠀⠀⠀⠀⠈⠻⣦⡀
⠀⢀⣤⣄⣠⣀⣤⣤⣤⣦⠀⠀⠀⠀⠀⣏⠀⠙⠛⠛⠉⠀⠈⠳⣾⣧⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠙⣷⡀⠀⠀⢀⣀⣤⣴⣾⠷
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡴⠟⠋⣡⣄⡈⠉⠛⠦⣄⠀⠀⠈⢷⡄⠀⠀⠀⠸⢟⡾⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢀⣐⣪⣿⡇⠀⠀⠞⠀⠀⠀⢤⣼⠇⢠⣤⣄⠙⣷⠀⠀⠀⢻⡄⠀⠀⠀⠘⣷
⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠿⡇⠀⠀⠀⠀⠈⣧⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⠶⣄⠐⠛⣳⡦⠤⡿⠃⠸⣧⠀⠀⠀⣿⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢿⣤⣍⡛⠲⠛⣧⠀⢸⡟⠀⠀⠀⢸⡇⠀⠀⠀⠘⢧⣤⣤⣤⣦⣴⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠰⣻⡖⢀⠀⠀⠀⢀⣼⠃⠈⠛⠋⢠⡿⠀⣀⣤⡀⣸⡇⠀⠀⠀⢠⣗⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡿⢷⠀⠀⠀⠀⠀⠹⣤⡀⠘⠛⠋⠀⠀⠓⢄⣀⡀⠀⠀⢀⣠⣴⠟⠀⢠⡵⠟⢃⣿⠁⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠛⠋⠁⠀⠀⠳⡀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠈⠉⠛⠛⣋⠉⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⠀⠀⠀⠀⠀⠉⠣⢤⣄⡀⠀⠀⠀⠀⠀⠹⢶⡿⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⣸⠿⠿⢾⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠶⠶⠶⠶⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠙⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠙⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠈⠛⠶⠶⣤⣤⣀⣀⣀⣀⣀⣠⣄⣤⣤⣠⠶⡾⠟⠉⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣿⡉⠈⠉⠉⠉⠉⢻⣿⡃⠈⠉⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣀⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⡉⠁⢹⡙⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠞⠡⠀⣀⣽⣽⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣞⢡⣤⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣀⣁⣀⣀⡀
⠀⠀⠀⠀⢀⣠⣴⣿⣿⡶⣤⣴⣶⣷⣿⣿⣿⣶⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣴⣶⣿⣿⣿⣧⣤⣤⣿⣿⣷⡦⢀
⠀⠀⢀⣴⣽⣿⣿⣿⣿⣿⣞⡿⢿⣿⣿⣿⣿⣿⣿⣇⢆⡉⠛⠛⠛⠛⠋⣱⣮⣳⢾⣿⢿⣿⣿⣆⣠⣿⣿⣿⣿⣷⠶⡗
⠀⣰⣿⣿⣿⣏⠙⠛⠛⠫⣉⣧⣬⣿⡿⢟⣻⣿⣿⣇⣹⣷⡄⠀⠀⢀⢎⣩⣷⠝⠛⠉⠒⣉⣽⣏⣉⣿⡿⠋⠋⠻⣿⣿⣿⠷⡄
⢠⣿⣿⣿⣿⡿⣲⣶⣶⣿⣿⣿⣯⣿⣮⣵⣿⣿⣿⣿⣿⣿⣿⡄⠀⣧⣠⣿⣿⡶⣿⢧⣽⣿⣿⣻⣿⡟⢡⣄⠀⣴⠈⣻⣿⣇⣽
⠶⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⣗⣿⣗⢸⣿⣿⡿⠃⠀⠀⠈⣿⣿⡿⠛⠛⢷⣠⣽⣶⣿⣿⣿⣿⣯⣠
⢸⣯⣭⣭⣿⣿⣷⣺⣿⣿⣿⣿⣿⣿⣿⣿⢋⠉⢉⣽⣿⣿⣯⣿⢀⡉⣿⡇⠀⢀⠄⣴⣿⡏⠀⢀⣾⠟⠉⣿⣿⣿⣿⣿⣿⠋⢽
⠸⣿⣭⣽⣿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡿⠋⢤⣼⣿⣿⢋⣾⣿⠈⠻⣿⣿⣶⣷⣾⣿⣿⣿⣶⣾⣁⢠⡞⢁⡽⣿⣿⣿⣧⣤⣼
⠀⣹⣧⣼⣿⣞⣻⣿⠿⠻⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⡿⣿⠇⠀⠀⣿⣿⣿⠛⠉⢻⣿⣿⠚⢻⣿⣿⣧⣾⣡⡟⣿⣿⣿⣿⣿
⠀⠁⢻⣾⣿⣿⣿⣿⣛⣻⣾⣿⣿⣿⣿⣿⡿⡹⢏⣻⣿⢷⠟⠀⠀⠰⣿⣿⣿⢋⣟⣿⣿⣯⣀⣁⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠈⠿⠏⣙⢿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⢱⡟⠉⣻⣧⣴⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡏⠀⠀⢘⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠆⣉⣹⣿⣯⣿⢿⣿⣿⣿⢧⡽⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣤⣇⣆⣹⣿⣿⣷⣤⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠫⠿⠟⠛⠋⠉⠉⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠉⠛⠃⠸⠿⠿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿⣿⣿⣿⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣝⠓⣒⣀⡀
⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣲⣶⣿⣿⣿⡟⠓⠀⠀⠀⢀⣀⠠⠠⢦⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⡇⠀⠀⠀⠀⠀⠘⠁⠉⠟⠁⠀⠀⠀⠀⠀⠀⠻⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠶⠠⡄⠀⠀⠀⠘⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠘⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⣀⠅⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖
⠀⠀⠀⠰⠶⠈⠀⠀⠀⢀⡀⠀⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⠀⣨⢄⡀⠀⢨⣀⡌⣠⢠⠞⠉⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢸⠖⠠⣄⠼⣀⠀⢳⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠀⠀⠀⠀⠀⢀⣬⢷⣼⡅
⠀⠀⠀⠂⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⢧⡀⣰⡃⠀⠈⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠶⠆⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠐⠂⠐⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⣀⠀⠀⡥⠀⠀⠀⠀⠤⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⡂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢠⡎⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢃⠀⠀⠊
⠀⠀⠀⠀⡀⠀⢄⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠁⠠⠀⠄⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠐⠂⠄
⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⡃⠀⠀⠀⡤⣖⡧⠀⣂⣀⠔⢒⠄⢀⣠⠀⠀⠀⢠⠀⢀⢢⣄⣐⠄⠀⠀⡅⠀⠀⠀⠀⠀⠀⡐⠀⠀⠁
⠀⠀⡀⠀⠠⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠨⢰⡌⢇⠀⢀⣼⠀⢠⠔⠀⢠⡜⢸⢀⠇⢀⡠⠮⠂⠀⠀⡇⠀⠀⠙⠦⠀⠙⠁
⠀⠀⠃⠃⠉⠁⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠋⠁⠚⠁⠐⠚⠀⠙⠀⠋⠀⠀⠊⠁⠛⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⠆⠀⠀⠁⠶⠦⠠⠶⠦⠼⠿⠧⠷⠄⠴⠶⠴⠄⠠⠶⠶⠦⠿⠇⠀⠘⠀⠀⣖⠀⠤⠀⢐⡀⠀⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢀⣀⠀⠀⠈⢠⡄⠀⠀⠈
⠀⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⢠⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⠉⢀⣀⣀⡀⠊⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠢

⠀⠀⠀⠀⠀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⡤⡔⡶⣄⣀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⢸⣿⢩⣭⣽⣩⡇⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢾⣻⣿⡟⣿⣵⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠰⠟

⠀⠀⠀⠀⠀⢠⠀⠰⡄⠀⠀⠀⠀⠀⠀⢀⣶⠞⠳⣦⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣶⡄⠀⠀⠀⠀⢀⡀⠀⣶⠄
⠀⠀⠀⠀⠀⣼⠲⠖⣠⡄⠀⠀⠀⠀⠀⠈⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣉⣿⣿⣍⠁⠀⠀⠀⠀⣬⣿⣶⣿⡾⠟
⠀⠀⠀⠀⠀⠘⢒⡾⢯⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠻⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⢠⣾⢹⡟⢿
⠀⠀⠀⠀⠀⠀⠙⠑⠈⠀⠀⠀⠀⠀⠀⠈⠿⠤⣴⡟⠀⠀⠀⠀⠀⠀⠻⠀⡿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠹⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡭⢭⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠞⡡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣚⡼⠁⠀⠀⠀⠀⠀⠀⡄⠀⠜⠀⠀⠀⠀⠀⠀⠀⠈⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠐⠾⣧⡅⡸⣏⣀⠀⠀⠀⠀⢼⠉⠹⠏⢉⡆⠀⠀⠀⠀⠀⠈⡷⠒⣻⡆⠀⠀⠀⠀⠀⠸⣿⣿⠂
⠀⠀⠀⠀⢚⢻⠲⡅⢏⣿⠀⠀⠀⠀⠀⢠⠁⠀⣀⠑⡄⠀⠀⠀⠀⠀⠀⣿⠭⢿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⡀
⠀⠀⠀⠀⠈⠻⡶⠗⠊⠀⠀⠀⠀⠀⠀⠐⠤⠄⠡⠐⠋⠀⠀⠀⠀⠀⠸⣧⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃

⣀⡆⠦⢰⡂⡀⠀⠀⢨⡠⠒⠘⡄⠐⠜⠄⠦⢁⠪⢪⠧⢀⠀⠀⢺⣩⡗⡀⢤⣀⡴⢩⠞⣀⠌⠀⠠⣂⠎⢐⠘⠐⠄⠀⠀⡠⣪
⠚⣜⢌⢦⠈⠑⢞⢀⡈⠢⡤⠐⢢⣄⠰⡰⡀⠀⠈⠉⠣⣀⠈⠐⢔⠈⠀⠀⣮⡅⢀⣍⠄⠛⡆⡐⢡⠃⠀⠀⢁⣴⠚⢶⠉⣿⠋
⠈⢠⠷⡄⢁⣬⡣⢫⡀⠀⢠⡔⠁⠘⠀⢀⡰⠀⠥⠕⠀⠈⠐⠤⠄⠇⠀⠴⠋⠀⢉⡇⠀⠀⠘⢀⠁⠀⢀⢶⡗⠀⢀⡚⣠⠌⠑
⠄⡀⡰⠛⣕⡭⠐⡰⠤⠀⠀⠉⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⢸⠂⣀⠐⣎⠙⢄⠀⡠
⣞⠴⠑⠘⠁⠙⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠈⠀⠀⠀⠀⡁
⠀⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠓
⡷⠶⢧⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⣤⡀⣤⢠⢀⡤⠤⢠⠤⢤⡄⠀⡄⠀⣤⠀⣤⣤⠀⠀⣤⡀⢠⡤⣄⠀⠀⠀⠀⠀⠈⢉⡦⣀
⠀⠰⡞⠉⠀⠀⠀⠀⠀⠀⢧⣠⠇⡏⠳⣿⢸⠸⣄⣀⢸⣛⡸⣇⡀⣧⣀⢿⣀⡟⣿⣀⣰⠧⢷⢸⠗⣇⠀⠀⠀⠀⠀⠀⠀⠀⠛
⡲⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣷⢸⣹⠆⡞⢉⡁⣰⢻⡀⣿⣆⣿⢸⠰⣯⡁⡿⣄⣼⡇⢯⣁⠀⠀⠀⠀⠀⠀⠀⠐⠁⣦
⠞⠤⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠘⠈⠃⠙⠒⠃⠋⠉⠓⠛⠈⠛⠘⠐⠒⠃⠃⠉⠘⠃⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠪⠤
⡀⡀⠜⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢰⡦⡤⡤
⠀⠀⡀⢤⠄⠉⠉⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣢⣠⢀⠈⠉⢉
⠠⢊⡤⠂⠀⢀⢴⠇⢀⡠⡪⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡒⠄⠀⠒⡎⣦⣠⠑⡂⠀⠙
⢤⢛⣁⣀⣴⢁⡎⠀⠘⡵⠯⠙⢀⡴⢞⠝⡆⠀⠠⠐⡠⠀⡀⠐⠀⠀⠀⠀⠀⠀⠊⡹⢆⡀⢘⠫⠅⣹⠀⢁⠕⢥⢀⠈⠊⠂⢆
⠁⠊⢛⠋⢸⠞⠀⠀⠀⠄⠀⠠⢾⡏⢁⡼⠁⢠⢡⠁⠀⠠⡰⠤⢔⡠⠒⠀⠠⡀⠙⢤⠐⠣⡀⠑⢶⠃⢠⢀⣄⠀⠠⠇⢸⣾⠆
⠀⢀⡆⠀⠉⠀⢀⠔⢱⡤⠖⠒⠛⠙⠙⠀⠀⡄⡀⠀⠠⢸⡘⠟⢀⡯⠄⠀⠢⠟⠀⢰⠂⠘⡅⠀⢸⡀⠈⡼⠥⡄⠐⠉⡖⡄⠺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⢠⣄⠀⠀⠀⠀⠀⠀⠀⠙⠃
⡇⠊⠀⠀⠀⢸⣥⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠽⠗⠦⠾⠌⠛⠉⠉⠈⠉
⡀⠅⠁⠀⠀⠘⡿⠀⠀⠀⠘⠛
⡃⣗⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⡠⣤⣤⡀
⠀⠀⠂⠀⠀⢠⣿⡂⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠿⠄⠀⠿⠟⠘⠃⠁⠉⠁
⠀⠀⠀⠀⠀⠌⠿⠁⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠓⣤⠤⠤⠤⠤⠶⢍⡀⠀⠀⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠃⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠓⢄⠀⢞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⡤⠃⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⢚⠦⣀⢀⠓⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢸⠀⠀⠁⠀⢱⠀⠑⠂⠔⠂⠤⠤⠤⠴⢂⠀⡖⠊⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣂⣀⡀⢀⡤⠐⠒⢷⡦
⠀⠀⠀⠀⠀⠀⠘⠴⡀⢠⢧⡄⢆⠰⠤⣀⠀⡷⠶⡏⠁⢠⠀⣹⠹⣄⠀⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠳⡳⠈⠉⠀⠀⠘⠦⠤⠔⠣⢀⠌⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⡀⠀⠀⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠂⠽⣝⠉⣝⢉⡎⡃⠀⢤⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠛⢆⠈⣷⡱⣤⠜⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⢄⠀⠀⠀⠀⠉⠀⠀⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⡦⠤⣀⠀⠀⣀⣰⠖⠉⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠤⠇⠀⠀⠀⠀⠘⡇⠀⠀⠀⣸⡑
⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠜⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣠⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⣇⠀⠀⠀⠀⢸⠀⠀⣀⣀⠤⠄⠃⠀⠰⡤⠤⡴
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⡩⢷⠈⠉⠉⣹⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⠀⠀⢀⠔⠊⢆⡀⠑⠒⢶⡁⠉⣉⠽⣄⠀⠀⠀⢀⠄⠀⢇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠡⠀⠈⠀⠀⠀⢋⠉⠉⣷⠀⠙⠤⠤⠋⠳⡀⠐⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠆⢸⠀⠀⠀⠀⠑⠊⠁⢇⠀⠀⠀⠀⡰⠁⠃⠀⠀⠀⠀⢀⠇⠐
⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠤⠤⠊⠀⠑⢄⡀⠀⠀⠀⡠⠊⠒⠒⠒⠉⠀⠀⠈⠢⠄⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠈⠉⠁⠈⠈⠉⠉⠁⢀⡀
⠀⠀⠀⠀⠀⠁⠈⡓⠤⣀⠄⠊⠉⠀⠀⠀⡁⠀⠀⡀⠁⢐⠠⡀⠀⠈⠓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⢁⠈⣦⠀⠀⢀⠂⡄⠠⠀⠄⠀⠀⠁⢀⣀⠂⠀⠁⠄⠂⠄⠀⠈⢣
⠀⠀⠀⠀⠀⡠⠊⠀⣾⠃⠰⠊⠀⠀⣀⠀⣀⣿⡿⢈⣼⠟⣄⣈⠂⠁⠀⡀⠏⠀⠀⢣
⠀⠀⠀⠀⡀⠀⠀⣄⠁⢥⣄⡆⡀⢱⡟⠲⠋⠀⠀⣿⡿⠀⠙⠙⠚⠂⠄⠐⠴⠤⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠠⠂⢀⢠⡤⠘⠉⠀⠀⠀⢠⣿⠿⠈⠡⡻⠀⡿⠋⠈⠖⢀⠶⡆⠂⠶⣾⠎⢠⠄⢸⠁⡉⢀⡂⠀⠠⠀⠄⢀⠀⠀⣤⠠⢀⠄⡃⠀⠄
⢰⣁⡀⠀⠊⣌⡷⠈⡀⣀⠀⢰⠚⠃⡰⠰⣾⠃⢠⡇⢀⣈⣀⣢⣆⣥⠀⡀⠀⠀⠠⠅⠀⠧⠀⠂⠀⠀⢀⠀⠬⠁⠉⠀⠈⠈⠁⠀⠀⠀⠁
⠀⢀⠀⠃⠀⣈⠀⠀⠀⡀⢀⡀⠀⠤⢶⡟⢋⣨⡟⠋⣹⠳⠟⠹⠊⠂⠀⡀⢆⣠⠀⠀⠀⣠⡤⠀⠂⠈⠀⠀⢠⠆⠀⠀⠀⠀⢀⡀⢀⡄
⠎⠀⢀⠈⠐⠉⠀⠀⠈⠋⠂⠁⠀⠤⠂⢀⡬⡿⠛⣸⣷⢁⣴⠀⡲⠆⢀⣄⣼⠀⠂⠈⠁⢀⡘⠢⣀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠀⠈⠁
⠀⠀⠀⠃⠀⢀⠐⠀⠀⢄⠀⠀⢠⡬⠤⠨⠉⢳⠾⠋⢩⣿⡩⣸⠿⠝⠊⠀⠀⠂⠶⠀⠀⢈⠉⠀⢤⡉⠐⠀⠦⡀
⠀⠀⠀⠄⠀⠌⠀⠀⠀⠀⠀⠄⠡⠤⣤⣀⡤⢀⢖⠾⣿⣯⠰⣯⡤⠀⠀⣀⢀⡀⠀⢠⠜⠀⢈⠥⠀⣁⠈⢍⠉⠁
⠀⠀⡠⠃⠀⠈⠅⠀⠀⠀⠀⠀⠁⠀⠈⠋⠀⠻⠒⠉⠉⢿⠞⢹⡉⡇⠘⠋⠀⠁⢀⠁⠀⡠⠁⠀⠀⠀⠉⠢⠇
⠀⢅⠀⠀⠀⠌⠁⠃⠈⢀⠀⠀⠃⠀⠀⠀⠀⡀⢀⠀⠀⠀⡈⠀⠀⠀⠐⠀⠀⣤⡎⠀⡰⠁
⠀⠀⠀⠀⠀⠈⠐⠀⠠⠢⠀⠀⠂⠀⠀⠀⢠⠀⠛⠀⢀⡶⠀⣠⣮⠂⠀⠀⠀⠽⠀⡰
⠀⠀⠈⠀⠀⠤⠄⢀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠀⢀⠈⠁⠀⠈⠊⠀⠀⢀⡜⠀⠐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠄⣀⠀⠀⠰⠴⠐⠈⠐⢄⡂⠘⠀⢀⠀⠀⠀⢀⠴⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⡀⢀⠀⣇⠂⢈⡀⠤⠀⠄⠀⠱⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡠⠋⠀⠀⠀⣼⠀⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡔⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⣭⠙⠒⠒⠒⠒⢉⠙⠒⠒⠒⢻⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⠉⠉⠙⢢⣄⣀⣤⠬⣁⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⢃⡼⠁⡏⠀⠀⡀⠈⢻⢶⣧⠀⠀⢀⣠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠋⠩⡀⠀⣸⣤⣀⣉⡰⠃⠸⣄⡤⠖⣉⡽⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠀⠀⠀⠑⠦⣝⡋⠀⠀⠀⠀⠀⠀⢀⣀⣤⢚⡡⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣞⠣⣄⠀⠀⠀⠀⢀⡎⠉⣿⣒⢒⣺⠋⡹⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡆⠗⠀⠏⠑⠠⡔⠉⡇⢀⠂⠀⢀⠃⣞⠁⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠁⠳⠞⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⡘⢠⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⠒⠤⢀⡀⢴⢰⠃⠀⠀⠀⡇⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⡠⠴⠋⠀⠀⠀⠀⡆⡃⠀⠀⠀⠀⢠⠔⠋⠂⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡕⠀⠀⠀⣀⠴⠥⡀⠀⠀⠀⡠⠃⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠈⡶⢄⣈⡷⠀⠀⠈⣢⠤⠒⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⠀⠀⠈⢳⠧⠶⢊⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠃⠀⠀⠀⣴⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⡟⠀⢦⣀⣀⠀⠀⠀⣀⣀⠠⠜⢄⠀⠐⡠⠀⠀⠀⢢⠉⠉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⢧⣀⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⠀⢀⡇⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠈⠉⠒⠒⢲⠁⠀⠀⠀⠠⠀⢀⠀⠀⡇⠀⠀⢸⠀⠈⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡞⠒⠦⢄⣀⠀⠀⠀⠀⠀⢱⠀⡠⠊⠀⠀⠑⢄⠀⠀⠌⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢸⠀⢀⠂⠀⠐⠒⠩⡉⠉⠉⠉⢣⡀⠀⠀⠀⠈⠉⠉⠙⣳⠃⠰⡀⠀⠀⠀⠀⠀⡆⢠⠂⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠸⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠊⠉⠁⠀⠈⢆⠀⠀⠀⠀⡰⢁⠆⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⢨⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⣦⠀⠂⠊⠀⡈⠀⠀⠀⠀⠀⡠⠋⠀⠀⠁⠃⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣄⡀⠀⢀⡀⠀⠀⠀⢀⠀⠁⠔⢉⠂⡀⠠⠀⠈⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠤⠊⠀⠀⠁⠀⠀⠈⠁⠀⠈⠔⠁⠀⡇⠠⠀⢀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠠⡈⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⢠⠀⢀⡂⠀⢰
⠀⠀⠀⠀⠀⠀⠑⢀⠑⡰⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠄⢀⠠
⠀⠀⠀⠀⣰⠀⠁⠄⠁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⡀⡀
⠀⠀⠀⠀⠤⠀⠁⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠅⠈
⠀⠀⠀⠀⡂⠁⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡢⢄
⠀⠀⠀⢒⠐⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⣀⠀⣀⠀⢀⡀⢀⣀⢀⠀⣀⠀⠀⠀⢀⡀⠀⣀⠀⡀⡀⠀⠀⠀⠠⠈⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠈⠱⠀⢸⢳⢿⢸⠺⣅⢸⡷⡗⣇⣹⢸⣺⢼⠸⣁⡷⣿⡀⣇⣹⢺⣑⡆⢳⠁⠀⠂⠃⠀⢠⠀⠂⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠊⢀⢉⡀
⠀⠀⠀⠀⠀⠀⢨⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣀⠄⠁
⠀⠀⠀⠀⡈⠐⠈⠀⡰⠴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠐⠬⠂⠀⠡⠒⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠈⠂
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠀⠬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠁⣀⠰
⠀⠀⠀⠀⠀⠀⠠⠈⠠⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠴⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠈⠐⠀⠀⠀⠀⢀⡀⠀⠀⡄⠀⢠⠀⢠⢀⡈⣎⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠂⠑⢻⢦⠀⠐⠊⢀⡀⠐⠀⠂⢀⡀⠁⠈⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠐⣈⠄⢎⡃⠈⠀⠀⡀⠀⡬⠀⡀⠀⠠⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠉⠀⠀⠁⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡈⠻⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⣿⠈⠻⣦⣈⣻⣶⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⠀⣷⣄⠙⢷⠌⠉⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠻⣯⡙⠃⣿⡏⠻⠿⠛⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢠⣄⠰⣶⣾⡄⢻⣤⣤⣤⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠻⣦⡈⠻⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢶⣄⠙⢷⣌⠻⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡈⠻⣦⣨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣦⡙⢷⣄⠙⠷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣶⣤⣸⣿⢠⣌⣻⣦⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠛⠛⠛⣧⠘⡿⠿⠆⠙⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠿⣯⡉⠻⣦⡀⣿⡈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣀⡐⢷⣄⠙⢿⠀⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣦⡈⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡙⢷⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢶⣦⠀⠀⠀⠀⠠⣁⣹⠟⠀⠀⠀⡘⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⡿⠀⠀⠀⠀⠀⣨⡏⠁⠀⠀⠀⠹⡶⡟⠃⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡄⠀⠀⠀⠀⠐⣷⡀⠀⠀⠀⠀⣾⣇⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢦⣈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣽⡗⠀⠀⠀⠀⠀⢿⣷⡄⠀⠀⣀⣿⣿⣀⣀⣀⣿⣿⠀⠀⠀⢀⣴⡞⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣄⠀⠀⣀⣼⣿⠿⠟⠛⠉⠙⠉⠉⠛⠻⠿⣷⣤⣀⣴⣿⡟⠀⠀⠀⠀⠀⡄⠺⣷⡀
⠀⠀⠀⠀⠀⢠⣤⣶⡀⠀⠀⠀⠀⠀⠙⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣯⠀⠀⠀⣀⣤⡤⢿⠶⠿
⠀⠀⠀⠀⠀⠻⠃⣸⠯⣅⣀⠀⠀⠀⢠⡾⠁⠀⠀⢰⡶⠶⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠙⢷⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣾⠏⠀⠀⠀⠀⢠⡀⢀⣿⠇⠀⠀⠀⠀⠀⠛⠋⠛⣷⡀⠀⠀⠀⠻⣯⠀⠀⠀⠀⠀⠀⡤⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⠀⠀⠀⠳⢤⡴⡏⠀⠀⠀⠀⠀⢹⣧⣤⢶⣶⠶⠞⡷⣴⡿⠁
⠀⠀⠀⣴⡾⢷⣤⡀⠀⣀⣀⣀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠺⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠉
⠀⠀⠀⠻⠃⠼⠋⠛⠛⠿⠿⣿⠀⠀⠀⠀⠛⢻⣷⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣶⡄⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠚⠇⠀⠀⠀⠀⠀⠀⠰⠶⢷⣦⠀⠀⠀⠀⠐⠦⢾⡃⠀⠀⣻⣿⣶⣶⣤⣤⣠⡖⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠉⠉⠁⠈⠝⢷⡾⠟
⠀⠀⠀⢀⣾⠿⢾⡴⠶⠿⠿⠛⠛⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣼⣏
⠀⠀⠀⠀⠻⠆⠚⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠀⠀⠋⠉⢙⣷⠀⠀⠀⠀⠀⠀⠰⠾⡋⠀⠀⢀⣼⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⢦⡀⠀⠀⠀⠀⠑⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠉⢙⣶⡏⢠⡦
⠀⠀⠀⠀⠀⠀⠀⣶⠶⣷⠚⠛⠉⠀⠀⠀⣼⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢾⣿⡿⣄⠀⠀⠀⠀⠀⠈⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠈⢻⡦⠘⠀⠀⠀⠀⠀⣼⣷⠏⠁⠉⢲⢶⣤⣤⣤⣄⣠⣠⣤⣤⣴⣿⡟⠃⠁⠀⠙⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡼⠟⠁⠀⠀⠀⣿⣿⠁⠀⠈⣿⣿⠓⠀⠀⠙⢿⣷⠀⠀⠀⠀⠀⢬⣿⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⠳⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢹⡿⠀⠀⠀⠀⠈⢿⠄⠀⠀⠀⠀⠘⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢠⣼⠾⣆⠀⠀⠀⢀⣸⡋⠀⠀⠀⠀⠐⣾⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡌⠀⠀⠀⣴⣏⢉⠂⠀⠀⠀⠀⠻⠷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠺⡶⠿⠿⡷⠀⢠⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢠⠄⠤⣀⡤⠔⠢⡀⠀⠀⠘⡷⠤⠒⠉⠀⡀
⠀⠀⠀⠈⠁⡀⠀⡿⠀⠀⡇⢀⡀⠀⠀⠀⢀⡎⠉⠑⡍⠀⠀⠀⢀⣧⢠⡼⠵⢤⣀⣱⠀⠄⠊⠀⠀⢘⠉⢵⠈
⠀⠀⠀⠀⠀⠁⡰⠥⠀⠀⢣⠀⠁⡠⢤⣴⠟⠻⠀⢀⣼⠀⠀⠐⣅⠈⢲⡀⠀⣤⠑⣾⠉⠛⢆⠀⢠⢬⠉⢁⣤⡀
⠀⠀⠀⠀⠀⢠⠃⣠⣀⣠⣌⣆⠀⠣⡴⡅⠀⠀⠙⡿⠿⠀⠀⠀⢈⡉⢶⠬⠦⣄⣠⡏⣦⠤⠚⠀⡈⢊⡴⣋⠜⠁
⠀⠀⠀⠘⢁⣿⣿⣿⣿⣿⣿⣿⣆⠀⣷⠹⡄⠀⣐⣒⣒⡄⠀⠀⢀⣸⠈⢆⣀⡤⠛⠒⠊⠀⠀⠀⡤⣋⠜⢁⣀⡀
⠀⠀⠀⠀⢎⢻⣿⣿⣿⣿⣿⡿⢈⠀⢸⠖⠚⠚⠒⠒⠒⠚⣆⠀⠀⠙⠀⠀⠀⠀⣠⣴⠶⢒⡚⠿⣿⠁⠀⠀⢸⣏
⠀⠀⠀⠀⣈⠑⠒⠂⡒⠒⠒⠈⠁⠉⠘⠒⠦⠤⠤⠦⠆⠒⠃⠀⠀⣤⡆⠀⠀⡾⢋⠖⡒⠒⡌⢢⠙⡄⠀⠀⢸⣍⠁
⠀⠀⠀⠈⠚⠁⠀⠀⠻⠯⠀⠀⠀⠀⠀⠄⠀⠀⠀⠔⠒⠄⠴⠀⠀⠈⠁⠀⠀⡇⠸⣀⡛⠘⡜⡸⠀⡇⠀⡆⢸⠮
⠀⠀⠀⠀⢴⢀⡶⠚⢲⡀⠀⠓⠺⠰⠀⠀⠀⠀⠀⠊⠒⠤⠴⠂⠀⢠⣤⠀⣄⠹⣤⠈⢉⠉⠐⣡⠞⠀⠀⠧⠿⠁
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠀⠀⢠⠴⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⣠⠞⠛⠳⡄⠀⢰⡆⠉⠂⣴⠶⠶⠄⠈⢀⡀⠀⢀⡄⠀⣠⠤⢤⡒⠀⢸⣂⣀⣀
⠀⠀⠀⠀⠐⣆⠀⠀⢀⡇⢰⡇⠀⠀⠀⠉⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⣾⢧⠀⢸⡇⢰⡇⠀⠈⠁⠀⢸⠉
⠀⠀⠀⠀⠀⡌⠛⠒⠋⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣟⠛⠛⠂⠀⡟⠈⣇⢹⠃⢺⢁⠀⠀⢀⡀⠸⠶⠖⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢧⣀⣀⣠⠄⠀⣸⡇⠈⠁⣿⣤⣤⡄⠸⠃⠀⠘⠟⠀⠈⠦⠤⠖⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⢒⣤⣤⡄⢀⡴⣊⠥⢄⠉⢆⠲
⠀⠀⠀⠀⢠⣄⠈⠉⠀⠀⠠⡆⠀⠀⠘⠀⠈⢫⡂⢠⠛⠒⠠⢼⡀⠛⠑⠙⠀⠀⣀⠁⡀⡄⢠⣾⡔⠁⠀⢀⠃⣼⢠⣤
⠀⠀⠀⠀⠀⠁⢰⠭⡆⢰⠯⡆⠀⡟⡇⠀⠀⠀⡸⠡⠀⠀⠠⠀⠈⠟⢠⣄⣤⠀⠳⣄⢀⡗⠋⠀⠀⠀⢴⣧⡴⠃⢸⠏
⠀⠀⠀⠀⢶⣾⣯⡿⡽⠽⠯⡿⠿⠿⣿⠟⠀⠘⢇⠀⠀⠀⠀⠀⠀⡆⠈⠻⠃⠀⢂⡬⠛⠬⡀⠀⠀⠀⠔⠈⠀⢻⣮⡄
⠀⠀⠀⠀⢸⣿⠀⣶⡇⠀⣿⡇⢠⣿⣿⠀⠀⢀⡠⠣⡀⠀⠐⣂⣀⡇⠀⠀⠀⡰⠉⠀⠀⠀⠉⢦⠀⠀⠀⠀⣦⣠⠟⠁
⠀⠀⠀⠀⢸⣿⠀⣿⡇⠘⠿⠇⠀⠿⣿⡆⠀⡉⣀⢀⠙⠉⠀⠀⠀⠀⠐⠀⠙⠁⠀⠀⠀⠀⠀⠸⠟⠄⠀⠀⠸⠛⠋
⠀⠀⠀⠰⠿⠿⠿⠯⠿⠿⠯⠿⠽⠽⠽⠿⠃⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢦⠀⠀⠀⠑⢦⣁⠚⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢀⠀⠘⡐⠓⡀⠀⠀⠀⢻⠻⣄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠂⠀⠂⠀⠋⠀⠀⠀⠀⠀⢿⠏⠳⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡔⠈⠂⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠠⠀⠀⠀⠁⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⣄⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⣱⡀⠜⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠓⢀⠀⠀⠈⠉⠰⠆
⠀⠀⠀⠀⠀⠀⢉⠀⢄⠀⠀⠀⠀⠀⠀⠉⢀⡀⠄⠀⠘
⠀⠀⠀⠀⠀⠀⢸⡀⢈⡆⠀⠀⠀⠀⠠⡄⠀⢛⣆⠀⠸⣅⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠇⠈⠁⠀⠀⠀⠀⠈⢠⠄⠘⢀⠱⡀⠀⢰⠄⠀⠀⠀⠀⢐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠙⣷⠀⠀⠐⠁⠀⠁⠈⠀⠀⠄⠀⠀⠲⣄⠀⠀⢐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠳⣮⣾⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠙⣆⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣴⠏⠳⣄⠀⠀⠀⠀⠀⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣈⣁⣀⣀⣀⣀⣠⣚⣁⣀⣀⣌⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⠤⠤⢤⠒⠉⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢄⣀⠀⠀⢀⡠⡤⣄⠀⣀⣀⣀
⠀⠀⠀⡼⢡⡶⢢⠀⢧⠺⠂⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⠀⠈⠀⠀⠑⣤⠃⣤⠒⠀⠙⢁⠁⠈⠳
⠀⠀⢀⠃⠀⠉⠠⢀⡴⠚⠚⠋⢄⠀⢀⡀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢀⠺⠍⣀⡄⠀⠀⠀⠀⢀⠀⠀⠛⠁⠀⣠
⠀⢠⠋⠀⠀⠀⣀⢾⠂⠀⡖⠖⢸⡆⡁⠀⡀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢰⠉⢉⣨⠁⠾⡃⠠⣤⠀⠡⠆⠠⠤⢎⠐⠈⠉⢃
⠀⢿⠀⣔⠄⠀⠘⠈⠣⣄⣉⡨⠊⢣⠀⠙⠂⠀⠸⢄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠁⣠⠼⢥⣈⡁⣀⠀⠀⣤⡈⠀⠀⠀⠟⠸
⠀⠈⠣⣀⡴⠞⣶⣄⠀⢰⢀⡶⠖⠺⠦⡄⠀⠀⣀⠀⢡⠀⠀⠀⠀⠀⠀⠙⢶⡋⠀⢠⢶⡂⠁⡻⢆⣀⠃⠁⢈⡄⡀⠔⠁
⠀⠀⢰⠃⠀⠁⠈⢘⡆⢀⣿⠀⠰⡖⠀⢻⡀⠀⠒⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠬⠴⠊⠀⠀⠀⠉⠉⠉
⠀⠀⠰⡄⢰⣋⢇⡰⠉⠁⢱⡀⠀⠀⢂⡼⠁⠒⠒⠉
⠀⠀⠀⠈⠀⢲⠀⣤⡀⠀⠇⠙⠛⢒⠋⢹
⠀⠀⠀⠠⠊⠈⠀⠠⠀⠚⠣⣀⣈⣛⡡⠂⠀⠀⠀⢀⡠⠤⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢍⡀⠁⠀⢀⠀⠀⣀
⠀⠀⠀⠠⡀⠀⠶⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍⠁⢐⠅⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⣾⡑⢚⣁⠀⠀⠈⠂⠑⠈⡀
⠀⠀⠀⠀⠉⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⠉⠃⢄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣆⡀⡂⠀⢀⣬⠀⠀⠰⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠐⠷⠃⡠⠊⠀⠙⠀⠀⠤⡄⢀⠃⠀⠀⠀⠀⡴⣪⠁⢀⡉⢲⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⡄⢀⢄⠀⠄⠀⠀⠈⢀⠊⠀⠀⠀⠀⠐⡆⠈⠘⠮⠃⢸⡤⢂⠉⠉⠓⠆
⠀⠀⢠⠔⠖⠀⢄⡀⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⣤⢀⠁⠀⡂⣀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠱⢄⣐⢀⠠⡎⠀⠳⠋⠀⠀⡘
⠀⢠⠁⢐⡰⠀⠀⡉⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠦⠀⠤⠐⠋
⠀⢈⡀⡈⠀⠀⡰⠁⠀⠈⠤⠃⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⡐⠀⠀⠀⢤⠈⠗⠒⠲⠤⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠐⠢⡚⠉⠁⣉⠳⡄
⡇⠀⠀⠈⠋⠀⠃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠸⡿⠀⠀⢣⡤⠀⠙⠃⡿
⠘⢄⡀⠁⡠⠴⡀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠠⠄⢨⣆⣀⡀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠙⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⢠⣶⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠃⠀⠀⠈⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⠀⠈⢂⡠⠎
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣬⣭⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⣀⣀⠀⣀⡀⠀⣀⡀⢀⣀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢾⡇⣿⣒⢸⣇⣹⠆⣿⠰⣏⢀⢠⣯⣧⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠉⠀⠉⠁⠀⠉⠁⠈⠉⠈⠀⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⠆⣿⠭⠰⣯⡇⣾⠭⠀⣾⣧⢸⣏⡷⢰⡏⠙⢸⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠃⠛⠒⠘⠚⠃⠙⠓⠘⠉⠙⠈⠃⠙⠈⠛⠋⠘⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⡯⢽⡏⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡘⡇⠀⣿⢸⡇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠳⢒⠖⢳⡝⠶⣇⣸⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠁⢻⡒⠋⠓⡾⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⣀⡤⠤⢶⡻⣖⡾⡷⠤⢤⣀⠀⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⢸⠁⡀⠀⠀⠑⠦⠞⠁⠀⠀⡈⢷⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣠⠤⢶⠦⣄⠀⠀⠀⠀⠀⠐⣁⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡫⣶⠒⠒⢮⡳⡄⠀⢀⡽⠿⠿⠯⠀⢰⢡⠚⢢⠀⠈⣧⠀⠀⠀⣴⡛⢬⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠛⣯⠝⣳⡄⢹⢹⠠⠎⢀⣐⣒⣀⠀⠸⡜⠿⠃⠾⢂⡏⢀⣴⢊⠠⢙⣦⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⣄⠙⠺⢷⣗⡞⡜⣀⡀⠘⢻⣤⣟⠀⠀⠈⠓⠒⠒⠋⢠⣻⣌⠢⣴⣯⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⢽⣒⣺⠽⢾⡷⢤⡈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠯⠛⠁⠈⢦⣸⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡄⠀⠀⠀⠙⢦⡙⢦⠈⡇⢦⠀⠀⠀⢀⠀⣴⣒⣒⣒⣶⠀⢀⣀⣸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡙⢤⣀⠀⠀⠀⠙⢋⣴⠇⠀⠳⣄⡰⠋⠀⠀⠼⣧⡷⣒⠒⠛⢲⣄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣒⠮⠭⠤⠴⣒⡽⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⢴⣛⣒⣒⣒⣚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣍⠽⣧⠀⠀⠀⠀⠀⠀⢀⢪⠟⢉⢢⢄⠀⠀⠀⠀⠀⠀⠀⡴⠚⠛⣑⠢⣂⠀⠀⠀⠀⠀⠀⠐⡽⣢⡍⢁⢪⠗
⠀⠀⠀⠀⠀⠀⠀⢰⡯⠚⠁⠀⢨⡆⠀⠀⠀⠀⠀⠘⣟⢄⡀⠉⢫⣳⠀⠀⠀⠀⠀⠘⠣⣑⡋⢠⡆⢹⣂⠀⠀⠀⠀⠰⡷⠁⣍⠀⢙⠃⡷⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⠲⡟⠀⠀⠀⠀⠀⠀⠀⠈⠚⢈⣠⣜⠜⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⣀⣼⠃⠀⠀⠀⠀⠀⠠⣳⠭⣈⡫⢜⡤
⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠪⢕⡬⠦⣠⡫⠂⠀⠀⠀⠀⠀⠀⡔⣁⣈⢢⠀⠀⠀⠀⠀⠀⠀⠜⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠆⠃⠒⠛⡔⡄
⠀⠀⠀⠀⠀⠀⢐⡠⢾⠀⣤⢈⠦⢐⠆⠀⠀⠀⠀⡌⢨⠃⠐⣟⠳⠀⠀⠀⠀⠀⠀⠰⣀⠀⠉⠐⡤⢄⣀⠀⠀⠀⠀⠀⢨⠣⠪⡐⢄⢃⢇
⠀⠀⠀⠀⠀⠀⠀⢔⡺⠓⡖⠻⣒⠄⠀⠀⠀⠀⠀⢇⢾⠦⠴⠿⣙⠤⡀⠀⠀⠀⠀⠀⠈⠁⠒⠘⡌⠆⠀⠀⠀⠀⠀⠀⢄⠆⡀⠈⡉⡠⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠐⠊⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⠍⢑⡄⠀⡇⠀⠀⠀⠀⠐⡤⠋⢉⢍⠣⡄⠀⠀⠀⠀⠀⠰⠡⠆⢞⠖⠈⡄⠀⠀⠀⠀⠀⠀⠀⣾⢰⠋⣳⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠷⡁⠠⣨⠇⠀⡇⠀⠀⠀⠀⠲⡇⡔⢠⠡⡄⠺⠀⠀⠀⠀⠀⠰⡈⠋⠶⠝⣠⡃⠀⠀⠀⠀⠀⠐⡶⠎⠂⠉⠕⠃⠷⡆
⠀⠀⠀⠀⠀⠀⠠⠠⠀⡬⠉⡥⠠⠠⠃⠀⠀⠀⠀⠀⢑⠬⢁⡠⡔⠑⠀⠀⠀⠀⠀⠀⠈⠑⠒⠈⠳⡙⢄⠀⠀⠀⠀⠀⠘⡔⠁⠂⡄⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄
⠀⠀⠀⠀⠀⠀⠐⣐⣉⣄⠧⣬⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢴⡘⠷⠀⠀⠀⠀⠀⠀⠓⢃⣠⠖⢰⠏⠒⠀⠀⠀⠀⠀⠀⡠⢮⡹⢯⡄
⠀⠀⠀⠀⠀⠀⠀⠞⠈⡇⠀⠈⢸⠀⠀⠀⠀⠀⠀⣀⣀⡄⢿⡆⡃⠉⠀⠀⠀⠀⠀⠀⢀⡋⠀⣠⠖⣀⡀⠀⠀⠀⠀⠀⢾⠔⢁⡤⢤⡅
⠀⠀⠀⠀⠀⠀⠈⠊⢴⡹⠒⠒⠳⣦⡀⠀⠀⠀⠀⠐⢶⡽⠁⠂⠁⠀⠀⠀⠀⠀⠀⠐⠙⠉⢉⠥⠐⢅⠜⠀⠀⠀⠀⠀⣀⣀⡘⣦⣉⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠐⠂⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣤⢠⡤⠄⡤⢤⡀⣤⢀⡤⠤⢀⣤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠿⠸⠯⠄⠧⠼⠃⠿⠘⠧⠤⠾⠻⠆⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⢠⣤⢀⣠⡄⢠⣤⠀⣠⡀⢠⣤⡀⣠⣤⢀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⢯⢸⣯⠀⣝⡷⢸⣏⢰⠿⢷⢸⠻⡇⢿⣤⠸⡏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⣖⣿⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢸⠀⢸⡇⣿⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⠤⠬⢮⠳⣼⡃⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⢱⠖⠛⠒⡾⠈⠲⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⢀⣠⡤⢤⡷⣖⠾⡧⠤⣤⡀⠈⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⢀⡏⡀⠀⠀⠑⠶⠚⠀⠀⢀⠘⡆⠀⠀⠀⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠸⠇⠿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⣭⣀⠀⠀⠀⣠⢤⣤⣤⠀⠀⡴⠒⠒⠲⣄⠀⠀⠀⠀⣘⣥⡔⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣕⣯⣉⡉⢺⡳⡀⢀⡼⠛⣛⠛⠀⢼⢸⣉⡇⣀⠘⡆⠀⢀⡼⢏⠳⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣷⣫⢿⣀⡇⡇⠋⢀⣚⡛⣒⠀⠘⢮⣋⠈⣋⠜⢠⡶⣏⠈⣠⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡳⣌⣉⣽⢟⣽⡁⠶⢤⡘⠒⠚⠀⠀⠀⠈⠉⠁⠀⠻⣳⣬⡿⠳⡟⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠒⠚⠉⠹⣍⠢⡀⠙⣆⡀⠀⠀⠀⠀⠀⣀⣀⣀⣙⡁⠀⠀⢸⡞⣯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢇⠀⠀⠀⠈⠳⡼⢂⡇⠙⢦⠀⢀⠖⠘⠓⣷⣳⣎⣁⣴⣒⡚⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⢝⣒⣂⣀⣠⢴⡿⠃⠀⠀⠱⠋⠀⠀⠀⠉⡿⠼⠧⠤⠤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⣠⠄⠈⡁⠀⠘⠀⠾⠂⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣦⡠
⠀⠀⠀⠀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣷⡀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣝⠷⠾⢓⣆⠀⠂
⠀⠀⠀⢠⡤⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣦⠖⣾⣯⡉⠀⠀⡀⠀⣴⣷⣿⣼⣿⠗⠀⠀⡀⠀⠀⠀⠀⠀⠟⣷⢿⡏⠉⠂
⠀⠀⠀⢀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋⠐⣴⣷⣏⣩⡅⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⣄⠀⠀⠀⠀⠀⢤⠿⠙⠍⠋⠈⠀⠀⠀⠀⣠⣤⢰⣦⣄⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⡀⢰⡤⠀⡀⠀⠀⠀⠀⢺⡃⢰⡆⠀⠈⡀⠀⠀⠀⠻⠀⠀⡅⢻⠋
⠀⠀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠰⠆⠀⠀⠈⠁⠀⠙⣦⡀⣿⠀⠀⠀⠘⢶⡰⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠁⠀⢠⡄⠀⠀⠀⠀⠀⠿⠿⠋⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠚⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢠
⠀⠀⠀⠰⠂⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢶⣴⣇⣀⠀⠀⠀⡀
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⠀⠀⣾⣭⣻⢻⡷⡔⠚⠁
⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠈⠀⠀⠐⣟⣿⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⣼⣩⡄⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢿⡿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠄⠀⠀⠀⠁⣼
⠀⣀⠀⠀⠀⠰⣄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠐⢀⣠⣞⠁⠀⠀⠀⢀
⠀⠁⠀⠀⠀⠀⠘⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠿⠀⠀⠀⠀⠄⠉⠙⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢈⠀⠀⠀⠀⠰⠦⠠⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⠈⢘⠀⠀⠐⠀⠀⠀⠘
⠀⠀⠀⠀⠁⡄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠂⠐⣲⣾⣿⣞⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡚⢲⡄⠀⠀⠘⠟⢻⣿⠛⠃⠀⠀⠂⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠎⣰⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠉⠳⡖⠀⠀⢠⠆⠀⠈⠀⠂⠀⠀⠀⡀⠠⢥⣮⣿⣿⣷⡀⠀⠠⠄⠀⣇⠀⠉⢸⠀⠈
⠀⠀⠀⠀⠂⠀⠀⠀⢠⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠛⠻⠻⠟⠑⠦⠀⠀⠀⠉⠀⠀⡀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣴⠆⠁⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠒⠀⠀⠠⣤⠉⠀⠀⠄⠀⢨⠀⠀⡀⠀⢼⡴⠁⠀⣀⣀
⠀⠀⠀⠀⠁⠀⠀⡼⠀⠀⠀⠀⠀⣀⡴⠒⣫⡟⠁⠀⠀⠀⠀⢰⡀⣠⡔⠀⠀⠀⠛⠀⠀⠀⠀⠀⠗⠀⠞⠀⠀⠈⠁⠐⣴⠛⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⠞⣁⡴⠚⠁⠀⠀⠀⠀⠀⠐⠘⢿⣿⣆⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡾⠁⠀⠀⢄⣸⣿⣯⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⣸⣧⠖⠚⠙⠋⠉⠁⠀⠀⠀⠀⠀⠙⠲⠦⠤⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢴⣶⡄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⡉⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⢀⠰⠀⠒⠄⠀⠀⠀⠀⠀⠠⠰⠀⠀⠄⠆⢀
⠀⠀⠀⠀⠀⠀⠸⠤⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⢃⢀⠞⡍⠌⣂⠄⡀⠀⠀⠀⠀⠐⢯⢨⠭⠅⠭⠝⠹⡆
⠀⠀⠀⠀⠀⠓⠖⠀⢀⡤⠃⠀⠀⠀⠀⠀⠀⠀⡤⢁⡀⣤⠀⠀⡖⠒⠀⠀⠀⠀⠀⢀⠉⠲⡠⠔⣇⡄⠴⠀⠀⠀⠀⠀⠇⡀⠀⢆⠂⠀⠠⠇
⠀⠀⠀⠀⠀⠰⠉⠈⠉⠡⠀⠀⠀⠀⠀⠀⠀⠨⠀⠨⠤⠄⠅⠀⠑⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠈⠀⠀⠀⠱

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠀⠀⠈⠀⠠⢀⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡉⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠛⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⢥⠄⠀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠊⠀⠀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠰⣤⠚⡖⣰⠒⠀⠀⠀⠀⠀⠀⢠⣾⠒⢠⡘⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⠀⠀⠀⡐⠁⠠⡢⠀⣠⠀⡔⠄⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠈⢬⡱⡀⠰⠹⠀⠀⠀⠀⡔⠁⠀⠈⣤⡀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⠉⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠄⣈⠉⠀⡀⠡⠂⠀⠀⠀⠀⡄⠀⣠⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⢀⠁⠐⢄⠀⡠⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢦⠀⢤⠂⠀⠀⠀⠀⠀⠀⢀⡔⠓⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠂⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠸⠇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣁⠀⠃⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠦⠀⠀⠀⠀⠰⠃⠀⢹⡀
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠛⢦⡀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⡤⠀⠰⠀⠀⡘
⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⡀⠀⠀⡰⠁
⠀⠀⠀⠀⢸⠇⠿⠀⣤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⡀⠄⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠒⠲
⠀⠀⠀⠀⢸⡇⡄⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠁⡀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡆⠀⠄⣴⡄⠀⠀⢰⠀⠐⠀⠷⠈⠁⡀⠀⠀⡰⠁⠉⠉⠀⡀⠈⢱⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⢥⠀⠀⠉⠁⠀⠀⠀⢆⣨⡇⠀⠀⢀⡁⠀⠞⠁⠀⣀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⡘⢀⠀⠀⢀⠀⠂⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠦⠀⠀⠀⠀⠀⢀⠀⠳⣬⢀⠀⠀⠁⠀⡀⠀⠀⠘⠆⡀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠰⠗⠀⣡⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢠⡈⠀⠀⠀⠠⣶⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⡦⠀⣠⡄⠀⠀⠀⠀⠁
⠀⠀⠀⣠⠶⠀⠠⣠⠀⢀⠈⠁⠺⠂⠀⠀⠀⠀⠀⢀⠀⠢⠤⠜⢀⣠⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⢀⡴⠁⠀⠀⠀⠈⠀⢨⡄⠀⠀⢀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠀⠀⠀⢳⠀⠀⡀⠀⠁⠀⢠⠊⠀⠈⢀⠀⠀⠀⠀⠀⠀⢀⡔⠉
⠀⠸⠃⠀⠶⠀⠀⠀⠀⠀⠀⠁⠀⠸⠁⠀⠀⢚⣻⡷⠀⠀⠀⠈⠁⡄⠰⠄⠀⠀⠀⡀⠈⠀⠀⠴⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡗⠀⠠⠀⠓⠄⠀⠀⣈⠡⡀⠠⠴⠀⢠⡄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⢀
⢰⠄⠀⠀⠀⠐⡷⠀⠀⠀⠆⠀⠠⣤⣶⠶⢦⠀⠤⢠⠀⠐⠗⠀⠀⠀⠀⠀⠔⠁⠀⠐⠋⠁⠀⢀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⣰⠆
⡌⠀⠀⠉⠒⠂⠀⢐⡀⠀⠀⠀⢸⡿⣼⣿⣶⠆⠀⠈⠴⠀⠀⠀⠀⠈⢀⠈⠀⠀⠀⠰⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠒⢀⠀⠈⠀⠀⠓⠄⠈⠻⠿⠟⠿⠀⠈⣠⠶⡀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠡⠄⣚⠀⢀
⠀⠀⠀⠐⠀⠙⡃⠀⠈⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠶⠀⠀⠉⣉⠀⠘⠃⠀⠙⣠⠋
⠀⠀⠀⠀⠀⠐⣛⠈⠁⠀⠀⠖⠀⠤⠀⠀⠘⠀⢀⡀⠀⠂⣴⡆⠀⡀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⣀⠀⠠⠗⠀⠉⣀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠲⠶⢤⣄⠀⠀⠀⠀⠋⢀⠀⠀⠐⡄⠀⡀⢀⡤⠎⠀⠀⣀⠀⠈⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠟⡥⠀⠠⠀⠉⠌⠳⡄⠀⠀⠀⢀⡐⠂⠠⠀⠈⠉⢥⠀⠲⠄⠸⡽⢃⡀⠀⠀⠠⠀⣠⠆⠀⢀⠔⠒⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠐⠃⢹⣇⠘⠂⣢⠄⠠⠘⢤⡙⢆⠁⠀⢉⡹⠀⠀⠀⠀⠈⠀⠀⠰⠆⢠⡆⠈⠁⠘⠂⠀⡘⠁⠀⠀⢨⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠳⣌⠁⠊⠀⠁⠀⠙⢺⡆⠀⢿⠇⠀⠴⠀⠈⠉⠙⡆⠙⠀⠐⠀⠒⠐⠀⠀⠀⠃⠀⠀⢀⠧⠀⠂⠂⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣆⠘⢧⠀⠀⠀⠀⠀⢼⡇⠰⠀⠒⠀⠀⠣⣀⢀⠀⠃⠀⠛⠀⠊⠀⠒⠲⢤⠄⠐⠒⠒⠁⢀⠀⠠⠦⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣣⡀⠁⠀⢂⡼⠀⠀⠀⠀⠀⠀⠃⠀⢠⣤⠀⠀⠐⠂⠀⠰⠦⠘⠀⠠⠀⠈⠃⠈⣀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠛⠋⢁⡀⠈⠀⣠⠖⠉⠀⠀⠀⠈⠙⠶⣆⢀⠆⠀⢰⡄⠐⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⣀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠀⠀⠠⡄⠀⠀⠻⠇⠐⢶⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠀⢀⡴⠂⠴⠀⠿⠀⡸⠃⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⣈⠁⠀⠀⠘⠃⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠉⠀⠀⠀⠀⠘⠀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠀⠀⠀⢀⣄⣻⠀⢀⡀⠠
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣀⡀⠀⠀⠘⠿⠃⠀⠪⠃⠀⠨⣀⡈⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⣴⡄
⠀⠀⠻⣿⡿⠂⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠼⠀⠀⠉⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢁⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠙⠀⠀⢀⠀⠀⠈⠙⠉⠀⠀⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡢
⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢄⡀⢀⠀⠀⣷⡀⠀⠈⣷⠈⠂⠀⠀⠀⢀⣀⠀⠀⠀⠡⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤⡀⠈⣷⠟⠁⠀⠈⠈⠛⠀⠁⢀⡴⠃⠘⠟⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠨⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⢦⣤⣬⣈⠟⠃⠀⠀⠀⠀⠆⠀⠀⠀⢀⢻⡇⠀⠀⢀⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠎
⠀⠀⠀⠠⠦⣀⠀⠀⠀⠀⠀⢀⡄⠂⠀⢢⣀⡈⠑⠀⠨⠃⢀⠀⠀⠀⠀⠀⣀⠀⠈⠳⣂⠾⣇⣾⠟⠉⡡⠀⠀⡈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠄⠀⠈⠁⠉⠀⠀⠀⠀⢉⡙⠻⣿⠁⠀⠀⠀⢀⡴⠚⠉⠃⠉⠙⢆⠘⢶⢌⡆⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠠⠋
⠀⠀⡄⢡⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠄⡃⠀⠀⡀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠳⣔⠲⢠⢀⡌⠋
⠀⢸⠃⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡶⠾⠇⠀⠐⠂⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠟⠹⡦⠀⡀⠀⠀⢐⡠⠀⠀⠈⠀⠀⠀⡀
⠀⣀⠘⠠⡤⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠐⠁⠀⠀⠁⣄⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠑⡦⠀⠐⣴⡶⠋
⠀⠛⠆⠀⠀⢈⠄⠀⠀⠀⢀⣀⡀⠀⠀⠒⠂⣀⣠⣾⠟⠣⢀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⠂⠀⠟⠁⠀⣂⡤⠀⠀⠀⠀⠀⢘⠀⡤⢤
⠀⠀⠈⠘⠃⠈⠂⢀⣤⠶⣿⣿⣿⡄⠀⠀⡞⠉⠁⠠⠂⣀⡼⠓⠄⠀⠀⠙⢃⣀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⣶⠾⠋⠀⠀⠀⠀⠀⠀⢸⠀⠙⠋
⠀⢠⡄⠀⠂⠀⢀⣿⣿⡶⢟⣿⣿⣿⠀⠒⠀⠆⠀⠀⠈⠀⠀⠀⠀⠈⠀⢄⠀⣈⠑⢦⣄⣀⣀⣠⠞⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠸
⠀⡠⣄⠀⠀⠀⠛⠿⣿⣶⣿⣿⣿⣿⣆⠘⠃⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⠼⢧⠼⣿⣠⠄⠈⠉⠀⠀⢀⡀⠀⠀⠀⢰⠶⢤⠀⠀⠀⢀⣀⣠⡄⠊⠀⠁
⠀⠓⢋⡄⠈⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣷⡀⠀⠀⠀⠴⠀⠀⠤⠀⠀⠀⠀⠀⢠⣼⢋⡴⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⡤⠄⠘⡏⡇⣇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡉⠉⠛⠿⢿⣿⡂⠀⠀⠀⠀⠆⠀⠀⠀⠀⠴⠞⠛⠁⠋⡘⠁⠈⠀⠀⠀⠐⠀⠀⠆⠀⠀⠀⠀⠀⣀⠀⡿⢻⢈⡇⣎⢹
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⣠⠴⢤⣄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣂⡤⠾⠃⢠⣄⠈⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⠙⢧⡉⠋
⠀⠀⠀⠄⠀⠠⠀⠈⠃⠀⠀⠀⢸⠁⠀⠀⣸⡆⠀⠐⠀⠘⠂⠀⠀⠀⠀⠀⠈⠀⢀⠄⠀⠀⠀⠁⠰⣿⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢾⡇⢠⡇
⠀⠐⠽⠀⠀⠐⠀⠀⢠⠀⠀⠀⠀⠓⠤⠴⠋⡠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡤⢲⡄⠁⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⣿⢹⡘⣆
⠀⠀⠀⠄⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⣀⠀⠀⠐⠁⠀⠀⢀⣷⠒⠧⣔⠗⠑⡡⢂⠀⠙⠚⠁⠀⠀⠀⡟⠀⢀⠀⠄⠀⠀⠀⠂⠁⠑⠀⠀⠹⡀⠃⣿
⠀⠀⠀⠀⠀⣍⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠯⢀⣄⠈⠙⠒⠥⣬⠀⠀⡀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠈⣸⠆⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⡀⠀⠀⢼⠀⠠⠊⠁⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠈⣡⠬⢄⠣⣀⠀⠀⠀⠈⠈⣶⢖⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠒⠀⠀⠂⠀⠀⠀⠀⠠⠼⠘⠉⠀⠒⠲⠐⠋⡀⠀⠘⠀⠘⠃⠀⡏⠙⣧⠀⠀⢀⡀⢀
⠀⠴⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠁⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠉⠛⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠙⠛⠸⠸⠕⠍⠊⠐⠗⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠸⠕⠍⠪⠰⠗⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠨⠝⠩⠋⠐⠗⠤
⠀⠠⢐⠆⠴⠴⠰⠆⠆⠆⠤⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠶⠰⠄⠆⠴⠴⠦⠆⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠰⠦⠄⢴⠰⠒⠀⠀⠆⠴⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⢠⣐⠒⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⡋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢻⣀⣉⡀
⠀⠀⠀⠀⠀⠄⠨⢀⣠⡀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣨⣀⣥⣍⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠄⠀⠀⠀⠀⢀
⠀⠀⠂⠀⠀⠀⠀⠘⠂⠄⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣁⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠒⠒⠒⠒⠒⠒⠃
⠀⠀⠀⢀⣀⠀⡄⠘⠀⢀⣃⣉⠀⣼⣽⠁⠀⠀⠀⠀⠀⢀⡞⣣⡌⢳⠀⢊⣁⠀⠼⠥⠤⢀⡀⠋⡽⠀⠀⠀⠀⠀⠀⠀⢸⠈⠄⠀⠁⠙⢀⠀⠀⠀⠀⠰⠆⠠
⠀⠀⠒⠄⠀⠎⠁⠀⠀⠀⠀⢀⣨⣡⢸⠂⠀⠀⠀⠀⠀⠀⠣⠌⠿⢯⠀⠈⠉⠀⠀⠀⠀⠹⠗⠚⠁⡀⠀⠀⠀⠀⠀⠀⢼⢸⡗⣿⡇⠀⢸⠀⡀⠀⠑⠊⠐⠂⠄
⠀⠀⠐⠓⠚⠁⠐⠄⠀⠐⡞⠁⠀⡀⣸⠁⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⢠⡕⠀⡀⠀⠻⣾⠂⠤⠈⠁⠀⠀⠀⠀⠀⠀⢺⠀⠥⠄⠀⢈⢸⠀⡀⠀⠀⢦⡀
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠓⠚⠒⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠉⠒⠓⠒⠋⠀⠀⠁⠀⠀⠛⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠘⠒⠐⠒⠒⠃⠘⠀⠁⠀⠀⠀⠙

⠀⠀⠀⠚⠛⠸⠘⠐⠂⠃⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠘⠘⠂⠂⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠘⠐⠐⠃⠐⠗⠂
⠀⠀⠀⠰⠐⠦⠐⠘⠒⠰⠶⠖⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠰⠄⠶⠐⠒⠆⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢶⠰⠂⠆⠂⠒⠲⠒

⠀⠀⠀⠀⠀⠀⠀⠘⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢠⡀⢂⡄⠀⣠⠙
⠀⠀⠀⠀⠀⠀⠀⣡⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠔⠂⡈⠒⢀⠘
⠀⠀⠀⠀⠀⢸⢴⠃⠁⠤⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣀⣀⠀⠀⣈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠑⠈⠀⠁⠀⠀⠰⠀⠀⢀
⠀⠀⠠⢤⠤⢀⠠⡄⠀⠀⠀⢀⡀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣤⢀⣀⠈⠉⠉⠉⠉⠀⣈⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠈⠂⣀⠀⠀⢀⡀
⠀⠂⠀⠈⢑⠂⠐⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⢹⠆⠀⠀⠀⠀⠀⠉⠀⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⠀⠀⠀⠀⠑⠉⠋⢇⠙⠙
⢀⠆⠔⢀⠒⠀⠀⡄⠀⢠⡤⠆⡰⢣⡌⠀⠀⠀⠀⠀⠀⠀⠀⠰⢴⠈⠶⠆⡀⠀⢠⣤⣀⠶⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⡶⣤⠶⠂⠀⠀⠐⣧⡶⣼
⠋⠀⠀⠀⠈⠀⡹⢃⠀⡀⠐⠂⠐⠂⠆⠀⠀⠀⠀⠀⠀⠀⡁⢠⠀⣥⠀⠀⠘⠀⠱⢶⠓⠒⠒⠓⠶⡎⠀⠀⠀⠀⠀⠀⠠⣉⣦⣎⣤⣝⣱⣀⠀⠂⠀⣧⣿⡞
⠁⠐⠒⠒⠊⠙⠁⠀⠀⠁⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠐⠈⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⠲⠶⠺⠰⠰⠆⠦⠀⠶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠶⠰⠐⠀⠆⠰⠶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢶⠐⠰⠰⠆⠰⠖⠤
⠀⠀⠀⠀⠀⠂⣔⡦⣠⢄⡠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⣦⢠⢀⣲⠤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⢶⡤⣰⢤⡆⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠛⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁
⠀⠀⠀⠀⠀⠀⣬⠺⡄⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠀⡉⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠰⠂⠶⣶⡞⠓
⠀⠀⠄⠂⠀⢠⠄⠀⣶⠀⣟⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⡐⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⣦⡼⠻⠛⠿⢹⠀⠢⢀
⠀⠀⠄⠀⠀⠸⠆⠀⠹⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠶⠦⠤⠄⠪⠰⠀⠀⠈
⠀⡀⠚⣘⡟⠈⠀⢀⠔⠒⠢⡠⠖⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠋⣀⠂⠀⠀⠀⠂⠀⠀⣂⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⡖⠂⠁⠀⠀⠀⠀⣳⢒⣀⣐⡒⣚⣌
⠀⠉⠈⢽⡇⠠⠀⢸⣄⣀⠤⠤⢄⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢘⠀⠰⠃⠀⢂⡐⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣠⡀⡀⠀⠀⣹⣭⣽⣭⣭⡍⣭
⠀⠲⡴⠈⡇⠀⠀⠤⢀⠤⣄⠐⣒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠨⠀⡼⢀⠀⢠⡏⠀⠀⠰⢠⡀⠀⠀⠀⠀⠀⢀⡞⠂⢸⡿⠃⠀⠡⠀⣠⢷⣠⢧⣤⢷⠞
⠀⠀⠇⠀⠃⠀⠀⠀⠈⠓⠁⠀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠡⠾⠧⠶⠤⠂⠠⠃⠀⠀⠀⠀⠀⠀⠣⢀⣀⠗⠐⢄⠒⠀⠁⠀⠈⠉⠀⠉⠁
⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢰⡖⠤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⡶⠋⠀⢀⢸⠀⡠⠛⠛⠛⠛⠛⠉⠉⠀⠀⢀⡴⠿⠛⠛⢛⠛⠛⠋⠈⡇⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠖⠉⢰⡴⠚⠁⠀⠀⠀⠘⠾⠊⠀⡠⣶⡄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠐⡀⣰⠁⠀⠀⡠⡀
⠀⠀⠀⠀⡀⠄⢠⣤⠔⠉⢀⡤⠞⠁⠀⠀⠀⠀⠀⣀⠀⢠⡀⠀⠳⠿⠥⠀⠀⠶⠃⠸⠀⠀⠈⠣⣀⠀⠀⢀⣨⠔⠁⠀⠀⠀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣼⠋⠀⠀⠀⠀⠶⠂⠀⢠⠏⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⡤⡸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠿⠀⠀⠀⠀⠸⢦⣾⣥⢤⣄⠀⠀⢆⠠⢴⢀⢀⠎⠀⠉⠒⠢⢄⣀⠀⢀⣀⠀⠀⠀⠀⠙⣿⡤⣤
⠀⠀⠀⠀⠀⠀⠋⠀⢰⣖⡒⢤⡀⠀⣄⢠⠀⠀⠀⣾⢁⣀⣠⣟⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⠟⠢⠀⣠⠀⡮⠖⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡮⠉⢹⠗⠋⡀⠐⠂⠂⠀⠀⠉⠛⢡⠏⢀⣽⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⡿
⠀⠀⠀⠀⢀⠤⠐⠋⠁⢀⠏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⡿⠋⠉⢳⡄⠀⠀⠈⠢⡐⠀⠀⠠⢀⡸⠀⠀⠀⠀⢀⠀⣹⠧⢒⠆
⠀⠀⠀⢰⣿⣶⣤⣄⡀⠘⢄⠀⠀⡆⢀⠀⠀⢠⡄⠀⠀⠀⠸⣅⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠒⠊⠉⠀⠀⠀⠀⠀⢺⡷⠛⠊⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣷⣦⡼⠀⠀⡏⠙⠀⠓⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡞⠙⡆⠀⠼⠁
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⢄⣀⠀⢠⣤⡌⢀⡄⠀⠀⠀⠸⡿⠀⠀⠀⢀⣼⠙⡣
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣠⠖⠛⠃⡀⣀⠀⠀⠀⣞⣽⣿⣿⣿⢶⠚⠁⠘⠀⠉⠀⠁⠀⠀⠀⠀⢿⠀⢀⣴⠟⠊⢹⡅
⠀⠀⠀⠀⠄⠸⣔⡽⠀⠀⠠⢀⣘⣧⣤⠤⠴⠞⠁⠀⠀⠀⠀⢹⢹⠁⣀⣌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠭⡟⠁⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢳⣏⠉⠐⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⣇⢸⡇⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠉⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⠹⣧⠀⠀⠂⢉⣻⣦⣤⡼⠀⠀⣀⡀⠀⠀⠀⠀⢃⢿⣿⠀⣿⠀⠀⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⠸⣍⣀⠈⡧⠀⠀⠘⠉⠀⠃⠀⠀⠄⠸⡸⣿⡇⢻⡇⠘⠋⠉⠻⡁⢘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠧⡡⠃
⠀⠀⠀⠀⣀⡀⠀⠀⢈⡴⠒⠛⠻⣏⠃⠀⠀⠀⣀⡠⢠⡀⠀⠠⠀⡇⢿⣿⣘⣿⠀⠀⠀⠀⢸⢎⠀⠀⠀⠀⠑⠁⠈
⠀⠀⠀⠀⠛⠁⠀⠀⠾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠘⢼⣟⡩⠋⠀⠀⠀⠰⠃⠀⠳⣄⠀⠀⣠⣐⣒⣒⢒⡄
⠀⠀⠀⠀⢞⣹⠀⢴⡦⠀⠀⠐⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢠⠆⠀⢀⡠⠖⠉⠀⠀⠈⡟⠛⠛⡟
⠀⠀⠀⠀⠀⣀⣶⠶⠦⢤⣤⡤⢤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠙⠙⠈⠂⠾⠃⠀⠀⢰⣹⠆⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⣀⣤⣤⣤⣤⣖⠚⠉⠉⠉⠉⠉⣷⠴⠶⠶⠶⠶⠚⠃⢀⠀⠀⢠⢊⣁⣀⣨⣌⣲
⠀⠀⠀⠀⡀⠸⣄⣀⣀⣠⠇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠸⢿⣿⣿⣿⣿⠟⠰⠄
⠀⠀⠀⠰⠀⠀⠀⠀⠘⠒⠒⠒⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢤⣀⠀⠀⢀⣠⠶⠛⠉⠉⠳⣦
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠓⠶⠋⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣿⣠⡤⠶⠒⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠙⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠈⠁⠈⠷⠄⠀⠘⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡴⠋⠉⠳⣆⠀⠀⢸⡏⠁⠀⢻⡀⠀⠀⠀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⠃⠀⠀⠀⠘⣆⠀⠀⢳⡀⠀⠈⣁⢠⠀⠈⠁⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⡟⠀⠀⠀⠀⠀⠹⣆⡀⣀⡿⠦⣄⣁⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⢧⢀⢷⢠⢠⢠⠄⢠⠠⣀⠤⡄⡄⣤⡤⠀⡄⡤⢤⢠⢠⠀⡤⡤⢤⢠
⠀⠀⠀⠀⠀⠀⠀⠁⠃⠘⠈⠈⠈⠒⠘⠚⠈⠒⠁⠁⠞⠲⠑⠃⠁⠈⠘⠐⠃⠃⠁⠈⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠛⣋⡉⠉⠉⠉⠙⠛⠒⠒⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⠻⣤⡀⠀⠀⠸⡉⢇⠀⠀⠀⠀⠀⠀⡠⠤⠄⡀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⡴⠞⠍⠻⣅⡴⠋⣹⢦⠀⡔⢻⡿⡆⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⣠⢞⠩⠀⠀⠀⠈⠻⣾⠁⢸⠀⠱⡝⣧⢘⠄⠀⠀⡎⠈⢣⡀⠀⢀⢠⢱⠀⠀⠀⠀⠀⠀⢨⡻⣷⣦⣀
⠀⠀⠀⣰⠯⠐⠀⠀⠀⠀⠀⠀⢈⣥⠀⢠⣞⠭⡏⠉⠀⠀⠀⠧⡀⣴⣧⣿⣼⣼⡾⠀⠀⠀⠀⠀⠀⢐⣾⣿⣿⣿⣷⣄⣀⡀
⠀⠀⢠⣟⠭⠀⠀⠀⠀⢀⠠⡄⢏⠇⢰⠉⠀⠀⡆⠀⠀⠀⣀⣤⢧⢀⠽⣿⡛⠋⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢿⣻⣯⠿⡻⣷
⠀⠀⢸⢒⠀⠀⠀⠀⢸⠹⠊⠁⠀⠀⢸⠀⠀⠀⡇⣀⣤⣾⠋⣼⢾⣡⢎⡔⠉⠓⣄⠀⠀⠀⢀⣠⢾⣻⠯⠑⠋⠉⠀⠀⠑⢍⡪
⠀⠀⢺⠀⠀⠀⠀⡠⠼⣇⠀⠀⠀⠀⢸⠀⠀⠀⣋⣥⣿⡿⢠⡟⣀⠕⢫⠒⠃⣪⣭⣦⣶⠿⠟⠛⠉⠀⠀⠀⣀⣀⡀⠀⠀⠀⠌
⠀⠀⢸⡄⠀⠀⠀⢯⣉⣛⣧⠀⠀⠀⠘⣄⣴⣾⣿⣿⣿⡇⢸⠚⢁⠴⠚⠀⠀⠉⢻⡿⣀⣀⣀⣀⣀⣀⣀⡼⣿⠛⢿⡷⡄
⠀⠀⣠⢛⣉⣽⣿⣾⣷⡤⠼⣇⠀⠀⠀⠙⠛⠛⠉⢹⣿⡟⡡⠊⠀⠀⠀⠀⠀⠀⠀⣧⡏⠉⠉⠃⠀⢸⢸⣣⠀⢤⡀⢙⡇
⠀⠀⡇⣿⣿⠟⠻⡟⢻⣻⠒⠛⠒⢒⣦⣤⠀⢀⡀⣼⡿⠈⠀⠀⠀⠀⢀⡜⣧⠀⢀⣿⣷⠀⠀⠀⢠⠸⠼⢏⣢⣤⠼⠃⢙⢦
⠀⢀⣽⣯⡅⠠⠀⢀⣿⠃⡇⠀⠀⠋⠁⢈⣷⣋⡽⠿⡇⠀⠀⠀⠀⣀⡼⠤⠇⠠⣽⣿⣸⢀⣀⣀⣸⡇⠀⡸⠃⢠⢶⠚⠛⠛⠓
⢠⣿⣿⣧⢼⡶⣴⣿⣯⢤⣿⠀⡀⠀⠀⢸⠁⠀⢀⠀⢹⠒⢒⢛⠁⢀⣠⣤⣴⢞⠯⣾⡯⢝⣂⣀⣉⣧⠜⢁⡼⡏⠀⢹⣶⣶⣶
⣿⣿⡿⠧⣤⣸⣿⣿⣾⠀⠈⠀⢸⡀⠀⠀⡇⠀⠘⢿⣷⣤⣾⣼⡀⣞⣯⡝⠊⠁⠀⣿⣷⢿⠁⠀⡿⠋⢠⣾⣹⠁⢀⣿⣏⠢⡀
⢻⣿⠀⢰⣿⡙⢿⣿⣿⠀⠀⠀⢠⣷⡤⠤⢳⣀⣀⣠⣤⣿⣿⣿⣿⣿⠟⡇⠀⠀⢀⣿⣿⡼⠀⠀⣇⣴⡿⢷⠇⠀⣸⣿⡪⠕⠘
⠈⣿⠀⢸⣿⣿⢈⣻⣿⡄⠀⣠⣾⡿⠗⠛⢹⠁⢸⣿⡿⠟⠛⠉⠀⠀⠀⢳⠀⢠⣿⣿⡿⢿⠀⠰⣏⠀⠀⡼⠀⢠⣿⣗⠯⠀⢠
⠀⢸⣄⣈⣿⣿⣼⣿⡿⠓⡏⠉⠀⣀⣠⣤⣾⢤⣸⡀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠛⢍⠀⠀⠈⠳⠤⣴⣿⡆⢨⡢⠾⢯⣓⠅⠀⣼
⠀⢸⣿⠋⠀⠀⣿⠉⠉⠓⠛⢻⡿⣿⣟⣫⣵⡿⣿⠉⠉⠉⠛⠓⠒⠛⣿⠋⠀⠀⠀⠑⢄⡀⠀⠀⠈⠉⠁⠈⣿⣿⣷⣯⣓⢶⣿
⣶⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣤⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣠⠴⠒⠒⠒⠒⠶⠬⣍⣓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠋⠀⠀⠀⢀⣤⡖⠦⢶⡄⠉⠓⠦⣝⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣇⠀⣿⠠⣞⡦⠀⣹⠆⡞⢛⡎⣻⠄⠀⠀⠈⠙⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢋⣉⣉⣉⡓⢦⣄⠀⠸⣆⠘⢦⣈⠀⠀⠙⣷⣩⣉⡴⠟⠀⢠⠖⠲⡄⠀⠳⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⠞⠉⠀⠀⠀⠉⠳⡝⢦⠀⠈⠛⠶⢬⣙⡒⠶⢤⣄⡉⠀⠀⠀⠘⠶⠶⠃⠀⠀⠹⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡞⠁⢠⡖⠞⠙⠳⢲⡄⠙⣎⢧⠀⠀⠀⠀⠈⠉⠙⠒⠦⣍⡳⢦⡀⠀⠀⢀⣤⡀⠀⢀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠁⠀⣞⠁⣞⠉⣳⢘⣳⠀⠘⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡳⣄⠀⠸⠤⠇⢀⡼⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠀⠀⠸⣇⣌⠛⣡⣸⡇⠀⠀⠈⢧⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠓⠦⠤⠶⠋⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡇⠀⠀⠈⠘⠛⠁⠉⠀⠀⢀⡴⠦⡝⠲⣝⡲⢤⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⡄⠀⠀⠀⠀⣠⣄⡀⠀⠈⠳⠴⠃⠀⠀⠙⠲⢭⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠹⣆⠀⠀⣾⠁⠀⢹⠀⠀⠀⠀⠀⢠⣤⡀⡀⠀⠈⠓⢮⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⢳⡀⠈⠓⠒⠋⠀⠀⠀⣰⡏⣉⣬⡉⢹⡂⠀⢀⣀⠙⣆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⣀⠀⠀⠀⠀⠘⢯⡀⢧⣀⡿⢨⠽⠀⢿⣬⠇⢸⠄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⠳⢤⡀⠀⠀⠘⠷⢦⣠⡶⠼⠃⠀⠀⠀⠀⡼⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣍⠳⠦⣄⣀⠀⠉⠀⠀⠀⠀⢀⣠⠞⣁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣍⣙⡛⠒⠒⠚⢛⣉⣥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠐⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡔⢂
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠁⢱⠐⠈⡃⡘⡆⢸⢰⠐⠀⠨⠀⠀⠢⡀⢸⠀⡆⡄⡆⠈⡆⣴⠁⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠀⠀⠁⠈⠈⠈⠈⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠈⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢶⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⢇⠈⢧⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣧⠈⢇⠈⢆⠈⢇⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠠⣦⠘⢇⠘⣇⠘⣇⠙⢧⠈⢆⠈⢆⠈⢆⠈⣦⠈⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣈⣿⣿⣿⣿⣿⣿⡙⡇⠘⣄⠈⣆⠘⣆⠈⣆⠸⡧⠸⠇⠸⠇⠸⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠛⠋⠈⠛⠛⠿⣿⣿⣿⣿⣧⠘⡄⠈⡄⠘⡄⠈⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢋⣬⠐⡄⠘⣄⠸⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡷⠞⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡠⠞⠋⠁⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⣀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⠀⢶⣦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠈⣿⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠟⠁




⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠤⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠂⠒⠒⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⢠⠠⡄⢀⠀⡀⠄⣄⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠘⠈⣘⠇⠈⠀⠀⠀⠁⠀⠃
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢠⡀⢀⣀⣀⢀⢀⢀⢀⢰⡄⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠃⠑⠐⠘⠈⠐⠔⠘⠄⠸⠤⠋⠠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⠤⠔⠂⠉⠉⠁⠀⠀⠉⠑⠒⠤⢄
⠀⠀⣤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠉⠉⠒⢄
⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⡀⠀⠠⠔⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠈⠑⠂⠤⡀⠀⠀⠀⠀⢀⡀⣩⠷⠦⠤⠤⣀⣀⣀⣀⡀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠗⠈⠇⠽⠸⠐⠅⡇⠇⠅⠵⠇⠓⠇⠀⠇⠆⠇⠇⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⢠⢀⠀⡀⣀⣠⣀⢀⡀⣀⠀⡄⡀⡀⡀⡀⡀⠀⣀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢆⡀⢆⣸⢸⣸⠨⣈⡷⣀⡇⡇⣇⡇⡏⡉⢱⠀⠀⠀⠀⠀⠀⠳⠖⠀⠘⠊⠃⠒⠃⠀⠘⣸⠈⠂⠊⠂⠊⠂⠃⠛⠀⠛⠀⠛⠀⠃
⣿⠿⠿⠿⣿⠿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣟⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠀⠀⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠂⠠⢄⣀⣀⠠⠤⠶⢀⠀⠤⡄⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠒⠻⢤⣀⠀⣀⣐⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠰⢠⠏⢿⠁⠀⠉⠙⣶⣦⡀⠀⠀⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⠊⢅⠀⢘⢦⣀⣁⡼⠁⢇⣠⠔⣉⢽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡰⠀⠀⠀⠈⠑⠬⣙⡁⠀⠀⠀⢀⣀⢠⠤⠎⠡⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡾⣝⠢⣀⠀⠀⢀⡼⠀⡝⠭⠭⡟⣁⠋⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⢈⢀⠃⠈⠐⠁⢰⠀⠃⠀⠰⢀⠇⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢱⡀⠀⠀⠀⡘⡠⠀⠀⢠⡆⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡜⠀⠈⠁⢉⠁⠁⠀⠀⠀⠃⡠⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢈⠄⠀⠀⢀⣠⡊⠀⠀⠀⢀⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⢄⡀⡯⠀⠀⠑⣄⠠⠊⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠏⠀⠈⢳⠤⠖⡊⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠣⠄⠀⠀⠀⢠⠄⠒⠁⠤⠀⠤⠀⠀⠀⠳⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠤⢀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⢸⢄⡀⠀⠀⠀⠀⠀⠀⢇⠀⢀⠀⠀⠀⡀⠀⠀⡈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠠⠀⠀⠉⠒⠒⠠⠐⢦⠀⠈⠉⠒⠒⠂⠤⢤⡄⠐⠁⠀⠀⠀⠈⠀⢀⠄⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠒⠉⠀⠨⡀⠀⠀⠀⢀⠂⠂⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠂⠁⢠⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣟⣟⣿⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣿⣟⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠄⠤⠀⠀⠀⠀⠀⠀⣠⣶⡆⠀⠀⠀⢀⣴⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⣐⣆⣠⠆⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⢀⣾⣿⣿⣷⡞⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣷⡃⠀⠀⠀⠌⣾⣿⣷⣉⡀
⠀⠀⠀⠘⠘⣿⣿⣿⣿⠗⠆⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠈⠙⠉⠉⠉⠛⠋⢁⣴⣶⡀⠀⠀⠠⠚⣿⣿⣿⣟⠉⠀⠀⠀⠂⣿⣿⣿⡇⠆
⠀⠀⠀⠀⠙⠔⠉⠭⡁⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⣾⣿⣿⠿⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⠀⠀⠈⡹⣿⣿⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠀⠀⠐⣿⣿⠃⠀⠀⠙⠉⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⡝⠩⠀⠁
⠀⠀⠀⣠⣶⣦⡆⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡆⠼⠁⠚⣒⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠴⠋⢠⠍⢥⡄⠀⠀⠠⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡥⠀⠀⠀⠀⠿⣈⠓⢂⡙⢰⢆⡅⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠹⣿⣿⣿⢀⠀⠀⠀⠀⠿⠈⣩⠄⣉⡄⠓⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠗
⠀⠀⢰⣿⣷⡄⠀⠀⠀⠀⢼⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⢀⣥⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠸⣿⣿⣧⡀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢟⠅⠀⠀⢀⣰⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠲⡄
⠀⠀⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠉⠙⡆⠀⠀⠀⠀⠀⠀⣀⠀⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠆⢸⣿⣧⣤⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⢟⠂⠀⠀⠀⠀⢠⠞⠋⠁⠘⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠐⢶⣄
⠀⠀⠀⢰⠃⠳⣄⠘⠛⠃⠀⠀⢙⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢈⣶⣾⣦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⣠⠸⡄⠰⣌⣉⡀⠀⠀⠀⠘⠛⠛⣿⣿⠋⢉⣁⣄⠀⠀⠀⢼⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣿⣿⡄⠰⣷⣤⡀
⠀⠰⠏⠀⣿⢀⡀⠀⠻⣄⠀⠀⠀⠀⠀⠈⠋⠀⠈⠉⠁⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠹⠿⠇
⠀⠀⠀⠀⠃⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠿⢟⠛⠋⠂⠀⠀⠀⠤⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡜⢹⢿⠣⡀⠀⠀⠀⢠⣤⡄⠈⣻⣿⡀
⠀⠀⠀⠀⠀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠀⠃⠀⠀⠀⠨⣿⣿⣯⠀⠘⠙
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⢻⣿⠁⠀⠀⠀⣀⣀⠰⠶⣖⣘⡛⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⢀
⠀⠀⠀⠀⠀⢾⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠙⠋⠱⣶⡯⣩⣭⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠰⠾⠯⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠁

⡿⢿⡿⢿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠒⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠖⠋⠉⠉⠉⡉⠒⠲⢭⣑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢰⠃⣀⡀⠀⣴⢚⣭⡛⢧⡀⠈⠙⠦⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⠦⢤⡀⠀⠘⣇⠸⣄⠓⠃⠀⢯⡘⠦⢏⣺⠁⢀⣤⣄⠈⠳⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣡⠖⠛⠉⠛⠲⢌⢣⡀⠈⠳⢬⣓⠦⢤⣌⣛⠙⠋⠁⠀⠸⣄⡼⠀⠀⠹⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡞⠁⣠⠴⠒⠦⢦⡈⢳⣳⡀⠀⠀⠈⠉⠓⠲⠬⣝⡲⣄⠀⠀⠀⢀⣀⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡇⣼⠀⢰⡋⢰⠋⢳⢘⣶⠀⢳⡱⡄⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⠀⠻⠼⠀⣰⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⠀⠀⢿⣬⡙⣥⣸⠅⠀⠀⠱⣜⠦⣀⠀⠀⠀⠀⠀⠀⠀⠙⢎⡓⠶⠶⠚⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⡄⠀⠀⠀⠉⠁⠀⠀⠀⣴⠒⣮⠑⠮⣓⠦⣄⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⡄⠀⠀⢠⠴⠶⡄⠀⠈⠛⠁⠀⠀⠈⠙⠲⢝⡲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⢆⠀⠸⣄⣀⠟⠀⠀⠀⡤⡴⠲⠦⢤⠀⠀⠉⠢⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠳⣄⠀⠀⠀⠀⠀⣴⠃⣴⠛⣦⢙⡆⢠⡖⣦⢹⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡓⢤⡀⠀⠀⠨⣇⣌⢛⣥⣻⠁⠀⠉⠁⢸⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣍⡓⢤⣀⡀⠈⠉⠁⠀⠀⠀⢀⡴⠋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣍⣙⡒⠒⠒⠒⣚⣩⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠅⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⣄⠀⠀⢀⠀⠀⡄⠀⡄⢀⠀⡀⠀⣀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠂⠁⠂⠐⠀⠀⠈⠀⠃⠈⠐⠑⠐⠘⠀⠐

⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⢰⢋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠠⠁⠠⠬⠐⢂⠀⠀⠀⠀⡦⠀⡐⠉⢂⢀⡀⠀⠀⠀⣀⣧⡵⠖
⠀⠀⠀⠀⠀⣥⠜⠁⢢⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠸⡀⠁⠐⠀⠂⢉⡜⠀⠀⠀⠈⢢⡠⣯⡟⣍⣈⠏⠀⠀⢼⠁⢈⡇
⠀⠀⢀⠀⡀⠀⠀⠰⢀⠀⠀⠘⡁⣤⠁⠠⡀⢀⠀⠀⠑⠤⡄⠀⡠⠎⠀⠀⠀⠀⠐⣁⣸⠨⢵⡋⠑⡀⢀⡴⠚⢯⠉
⠀⠀⠈⡉⠅⠀⠠⢂⠌⠀⠀⠀⠐⢄⠀⠀⠐⣄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⠈⠉⠀⠀⠈⠉⠚
⠀⠀⠀⢰⣶⣿⣷⣷⣀⠀⠀⠀⠀⠈⠰⢤⡾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠤⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠀⠑⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⡈⢀⠀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠆⠁⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠃⠀⠐⠃⠀⠀⠀⠠⠘⠀⢀⣀⣀⠀⡀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⡠⢊⠆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠁⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠉⡇⠠⠀⠀⠀⠀⠀⢻⣷⠄⠀⠠⠊⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡦⢄⠤⠖⠁⠀⠀⠀⠀⠀⠠⣠⣶⣶⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⠢⠄⠄⠊
⠀⠀⠀⠀⠀⠠⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡡⠀⠀⠀⠀⠈⢤⡐⢀⠆
⠀⠀⠀⠀⢠⠁⠏⠀⠀⠀⠄⢀⠴⠀⠀⠀⢠⠿⢿⠿⣿⣿⡿⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠁⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠂⠐⠔⠁⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠒⢀⠸⠈⠂⠀⠀⠀⠀⠀⠀⢂⡀⠀⡰⢀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣈⠁⢈⣤⣤⡗⠀⠀⠀⠀⠀⢠⠊⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡍⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂
⠀⠀⠀⠀⠀⡄⢠
⠀⠀⠀⠀⠀⡆⠀⠄⣀⠀⠀⠀⠀⠀⠀⡠⢐⣤⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠠⠐⠁⠀⠑⢩⠁⠀⠀⠀⠀⠀⠀⠈⢿⡇⣀⣀⠀⡄⠀⠀⠀⢰⡸⠏⠟⢀⢰⠀⠀⠀⠈⡀⣬⢢⠀⠀⡠⡂⠥⠀⠠⠁
⠀⠀⠀⡁⡄⠀⠐⠒⠄⠅⠀⠀⠀⠀⠀⣠⣴⡾⠛⠻⢿⠃⠀⠀⠀⠀⠀⠃⠈⠄⢉⡀⠂⠀⠀⠀⠐⠠⢙⢠⣌⠜⢁⠠
⠀⠀⠀⠑⠈⠀⠒⠨⠈⠀⠀⠀⠀⠀⠠⡈⠛⠃⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⣀⣰⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠓⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢇⢰⢲⢲⢠⠀⣤⠂⢰⠂⣤⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠈⠈⠀⠉⠀⠁⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⠠⠐⠂⠉⠁⠀⠀⠀⠀⠈⠉⠀⠢⠄⢀⡀⣀
⠀⠀⢖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠑⠤⢀⠁⠀⠀⠀⠀⠀⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠀⣀⢀⡠⠠⠤⠚⠓⠒⠂⠠⠤⠤⠤⠄⠐⠉

⠀⠀⠀⠀⡄⢀⡠⡄⢠⢀⢠⡀⢤⡤⢀⠤⢀⠀⢀⠀⢀⢠⢄⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢤⢠⡠⠠⢰⢠⠠⢠⠠⣄⠠
⠀⠀⠀⠀⠃⠀⠃⠂⠀⠘⠒⠂⠀⠀⠈⠒⠈⠒⠘⠐⠘⠘⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠈⠚⠈⢺⠈⠈⠂⠈⠀⠉⠒
⠀⠀⠀⠀⠀⠀⡠⠀⡠⡄⡄⢠⠠⢤⠀⢤⢰⢀⠀⢠⢤⢄
⣤⣤⣤⣤⣤⣤⣵⣤⣦⣦⣥⣾⣥⣼⣵⣼⣼⣤⣴⣼⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣗⣾⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣶⣶⢶⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣫⣵⣶⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣡⣿⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡜⣿⣯⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣭⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⡜⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠸⣿⣿⡿⠗⣽⣿⣿⣿⣿⣿⣿⣯⣶⣶⣶⢾⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠉⠑⢞⣣⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⣵⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣯⣭⣭⣭⣭⣭⣿⣯⣭⣭⣭⣭⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⢒⡒⢒⣒⣂⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⡀⠀⠀⣀⣠⣾⠿⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣶⣶⣶⣶⣶⣶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣿⣛⣿⣟⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠀⠀⠀⠘⠋⣡⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢈⣤⡀⠈⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡼⠟⠿⣿⡀⠈⠑
⠀⠀⠀⠀⠀⠀⡛⠁⠀⠀⢠⣤⣤⡌⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠁⠈⠀⠢⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣈⣥⡦
⠀⠀⠀⠀⢀⡀⡁⠀⠀⠠⠿⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣄⠋⠁⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡂⠀⠀⠀⠀⠀⠀⠃⠻⠀⠀⠀⠀⠀⢰⣶



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣙⣋⣍⣹⣟⣽⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢸⠇⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢆⣀⣤⠼⠄⣀⣾⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⢀⠔⠋⠀⠀⠀⠀⠀⠀⠑⢶⡋⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠋⠀⠀⠀⢠⡄⠀⠀⣠⡀⠀⠘⢦⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠀⠀⠀⠀⠀⠞⣋⡑⠺⡉⠀⠀⠀⢸⡀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢠⣌⢓⡦⡿⠀⠁⠀⠀⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠤⠴⠁⣀⢸⠀⠀⢥⣤⠴⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠖⠂⠀⠀⢀⠀⢳⡄⠀⠀⠀⠐⠞⠸⡀⠀⢱⡀⠀⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⠀⠀⣰⠄⠀⡿⠀⠀⠀⢣⡀⠀⠳⡛⠉⠉⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢸⠀⠈⠁⠀⠛⠀⢀⡼⠃⢀⣳⡄⠀⠘⢲⡚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠈⡁⠒⠛⠈⢷⠆⠀⣀⡠⢭⣉⠘⠢⡀⠀⠙⣆⣀⣠⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢡⠀⢀⣴⡦⠀⠚⠁⢸⡆⢤⡱⡄⠘⡄⠀⠸⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠉⠀⠀⠀⢧⠀⠀⠀⠀⠀⠽⣤⡟⢾⡁⡇⠀⢸⠀⠀⢷⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⠳⣀⠰⠋⠀⠀⣘⣁⣨⠜⢠⠶⣸⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⢰⣄⠀⠈⠓⠄⣀⠀⠀⠐⢦⠀⣀⡴⠁⠀⢠⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠑⢤⣀⠀⠀⠉⠉⠋⠉⠉⠁⠀⢀⡴⠃⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠉⢲⡖⠒⠶⢶⡖⠒⠛⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣾⠁⠀⠀⠸⡇⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣶⣾⣿⢿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⡀⠀⠀⣾⡿⠛⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⣾⣟⠉⢹⣿⠄⠘⣿⣇⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠙⢿⣷⡿⠟⠀⠀⠘⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠛⢿⣶⣤⣶⣶⣤⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⣴⣾⠟⠋⠉⠉⠹⣿⣤⣶⡿⠿⠟⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠐⠿⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⣿⠟⠉⠻⣿⡄⠀⠀⠀⠸⠿⠿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⡀⠀⠀⢸⣿⠀⠀⣴⡿⠟⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠸⣿⣄⣀⣰⣿⠇⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣭⣍⡀⠀⠀⠸⣿⣄⣸⡿⠁⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣯⣉⠛⠁⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠙⠛
⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⠟⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣦⣀⢀⣴⣿⠏⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⣼⡿⠋⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⢀⣿⡿⠿⠋⠀⠀⠀⠀⣴⣾⠟⠛⠛⠛⢿⣷⡄⠀⢻⣷⣄⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡃⠀⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢹⣿⡀⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢱⣿⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣀⣀⣀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣿⡏⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⢠⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠘⠿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⡀⠀⠀⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡇⡷⠒⠒⠒⢺⢈⡇⠈⠀⠙⠒⠊⠁⠈⠓⠚⠁⠈⠓⠚⠁⠀⠙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⢧⠹⡔⠚⢃⡞⣸⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡙⢄⠊⡴⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⢦⡑⢄⠀⠀⢀⣠⣤⣄⣀⡼⠿⣿⣿⣿⣿⠿⢤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⢁⠞⠀⠹⡌⢧⠀⣿⣿⣿⣿⣧⠀⠀⠈⢹⠉⠀⠀⠀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡇⣞⣈⣉⣁⣹⠸⡆⠹⢿⣿⠿⠃⠀⠀⠀⠸⠀⠀⠀⠀⠻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⡇⢯⢠⣤⡄⣸⢰⠃⠀⠀⠀⠀⠀⢴⠒⠒⠛⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⡌⢧⠀⡰⢃⠞⠀⠀⠀⠀⠀⠀⠀⢳⡀⣠⣴⣶⣿⣿⢗⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⢄⠋⠀⠀⠀⠹⣿⣿⣦⡄⣴⣿⣿⣿⠿⢿⢣⣿⣟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠙⢦⠱⡄⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣕⠿⣿⢸⣿⣿⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⡞⣰⠡⠤⠌⢧⢹⡀⠀⢴⣿⠿⠛⠁⠹⣿⣿⣿⣿⣶⣎⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡇⡷⠒⠒⠒⢺⢈⡇⠀⠀⠀⠀⠀⠀⢰⣃⣉⣛⣿⠿⠿⠇⠉⠁




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣾⣷⢸⣟⣛⢸⡟⠛⣦⢸⡇⣴⠛⠻⠂⣸⢿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠻⠋⠿⠸⡯⠥⠸⠧⠴⠟⠸⠇⠻⢦⡴⠢⠿⠚⠿⠼⢷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡀⢀⣀⡀⢀⣀⣀⠀⢀⣀⠀⢀⣀⣀⠀⢀⣀⣀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣽⠇⣿⢭⠅⠿⣯⣅⢸⣯⡭⢀⣿⢻⡆⢸⣯⣽⠇⣿⠉⠉⢰⣷⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠻⠂⠿⠶⠆⠻⠶⠛⠸⠷⠖⠺⠋⠙⠛⠸⠃⠻⠂⠙⠷⠟⠘⠇⠀⠻


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⣉⣥⣤⠦⠶⢦⣤⣍⡙⠓⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣴⠟⠉⠀⠀⢤⠄⠀⢀⠈⠙⢷⣄⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠠⠄⠀⠚⠂⠀⠀⢤⡀⠀⣾⢷⣄⠀⠙⢷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢰⡆⠀⠀⠀⠰⠆⢀⣠⣤⠶⣿⣦⣽⣷⠄⠘⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠁⢦⠀⢴⠄⢠⣿⡉⠻⣦⣼⡇⠀⠀⠰⡄⢹⡆⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⣤⣴⠷⠞⠃⢸⡇⢻⡄⠀⢀⣀⣀⣸⡏⠻⣦⣨⠟⠠⣄⠀⢠⡄⣼⠃⣿⠀⠛⠷⢶⣦⣀⡀
⠀⠀⠀⠀⢀⣶⠟⠉⠀⠀⠀⠀⠘⣧⠘⣧⠀⠙⢿⣍⢹⡟⠋⠉⠁⠀⢀⠀⠀⠀⢠⡟⢰⡏⠀⠀⠀⠀⠈⠙⢿⣄⡀
⠀⠀⠀⠰⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣧⡘⢷⡀⠀⠙⢿⠇⠀⠀⠰⠆⠙⠁⠀⣰⠟⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⢴⡏⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠳⣄⡙⢷⣤⣀⡀⠙⠂⠀⣀⣠⣴⠟⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟
⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣿⢹⣿⠀⠀⠀⠈⠻⢶⣬⣉⣙⡛⠛⣛⣉⣩⣴⠾⠋⠀⠀⠀⠀⠀⣠⡶⢶⣄⠀⠀⢸⡟⠁
⠀⠀⠀⠀⢸⣧⡀⠀⣀⣿⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠁⠀⠀⠀⠀⣠⣄⣠⡾⠋⠀⢀⣿⠀⣠⠟⠃
⠀⠀⠀⠀⢀⣠⡴⢾⣏⣹⡇⠘⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣅⠙⠿⣄⢀⡴⢟⣡⡾⠛⠃
⠀⠀⣠⡴⠟⢁⣠⡶⠛⣿⣀⣀⣸⡇⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⣄⠈⢿⡄⠋⠉⠉
⣾⠛⣿⠀⣰⠟⠁⠀⠀⢹⣏⣉⣿⠁⠀⣿⣉⣉⣩⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣴⠟⠷⠞⠁⠀⢀⣀
⢹⡟⠋⢰⡏⠀⠀⠀⠀⠀⣿⣼⡯⠀⠀⢹⠏⠉⢹⡇⠀⠀⠀⠀⠀⣠⡾⠛⠛⢛⣿⠟⠁⠀⢀⣴⠾⠛⠛⠉⠛⣿⢦⣄
⢸⡇⠀⠛⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣼⡇⠀⠀⠀⠀⣰⠏⠀⢀⣴⠟⢡⣶⡀⢠⡟⠁⠀⠀⣀⣤⣤⣿⣄⣹⣧
⠈⢷⡀⣼⠋⠉⠹⣆⠀⠀⣤⣤⣤⣤⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⢿⣤⠞⠛⠁⢀⣿⡉⠀⣿⠀⠀⢠⣟⠙⢷⣄⣹⠉⠉⢹
⠀⠈⠻⣿⣧⣤⣾⣧⣤⣼⣧⣤⣤⣤⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⣾⠋⠁⠀⣿⠀⠀⣿⠙⢷⣄⣸⡟⠀⠀⢸
⠀⠀⢠⣾⣧⣤⣤⣤⣤⣤⣤⣤⣴⣿⡄⠘⠷⠶⠾⠁⠀⠀⠀⠀⣰⠟⣧⢀⣾⠋⠁⠀⠀⠹⣿⠛⢻⡟⠛⠛⠉⠀⠀⢠⡿
⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇⠀⠀⠀⠐⢷⣄⡀⠀⠀⣿⣀⣼⠇⠀⠀⠀⠀⠀⠀⠈⠳⣾⣇⣀⣀⣀⠀⠆⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣿⣃⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣶⡖⣖⣢⣦⣖⢂⣶⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤⢠⡅⣬⠅⣄⢨⣤⣠⠄⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠛⠐⠛⠛⠛⠛⠛⠙⠙⠒⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⠭⢓⠚⠭⣇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠐⡀⢀⣀⣷⣌⢳⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢸⢸⠀⠐⢰⢯⣳⠏⠀⡂⡇⡧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠈⢏⢆⠙⢿⠋⢁⠠⠀⡼⡹⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⢠⣤⠈⠳⢕⡢⠬⠤⣔⡫⠞⠁⠀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⣴⣯⠹⡆⠀⠀⠈⠉⠉⠁⠀⢠⡲⣎⣠⣏⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⢋⠕⠋⣷⣲⠃⣖⣲⡄⠀⠀⠀⡠⠊⣹⢮⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⢾⣀⠀⠘⠋⠀⣯⣸⠀⠀⢠⠎⣩⢟⡁⡴⠋⣉⣹⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣯⣨⣇⣴⣒⡂⡇⢸⠀⠀⠙⡋⢁⡞⠸⣁⣸⢯⣳⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠬⠭⠭⠭⠗⠈⠋⣄⣀⢸⣱⠊⠀⠀⠹⢽⣉⡁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢠⢢⠀⡀⠌⠛⠀⠰⣦⡠⠀⠀⠀⠀⣀⣄⣀⡠⠒⠒⠦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣴⠾⠿⠋⠝⠻⠓⢦⣀
⠀⠀⢀⡸⠋⡀⠂⠉⠰⢄⠈⢀⡠⠀⡆⠐⡇⠐⠆⠀⠀⠉⢀⡿⠀⠀⠀⠀⣰⠾⠿⠙⠛⠿⣦⠀⠀⠀⢸⣧⠀⢀⣴⣲⠆⠈⠘⠿⣷
⠀⠌⢫⠁⡞⠁⠀⠀⠀⢘⠀⢸⡬⡇⠀⢾⡀⠰⠂⠠⠀⠀⠙⡀⠀⠀⠀⢰⣿⣇⣶⡟⠛⢲⣾⠂⠀⠀⣾⠏⣠⣄⠙⣄⣀⡆⠀⢀⣿
⠀⡀⠠⠀⠳⡄⠀⠀⠠⠎⡀⠁⠈⠀⠀⠀⠉⡆⠀⢀⡀⠀⡰⠃⠀⢀⠀⠈⢿⣿⣿⣠⣈⡼⠋⠀⠀⢠⣯⠀⣿⢐⡀⠈⣉⠀⢴⣿⡟
⠀⠘⠻⠓⠀⣀⢉⠁⠠⠎⣩⠀⠀⣀⢀⣤⡀⠀⢈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠈⢿⡀⢀⣉⠀⠀⠛⠂⣀⣿⠁
⠀⠀⠀⠀⢰⡡⠌⠁⠀⠀⠡⠶⡈⠉⢧⣠⠂⠀⠀⢀⣠⢦⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠄⡀⠙⠛⠿⠿⢿⣶⣴⡿⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠰⠛⢄⠀⡹⠶⠀⢉⣤⠟⣡⣶⣶⠀⠀⢠⣍⠛⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠚⢅⠀⡀⠛⠃⠀⠀⢀⡅⠸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠰⡄⠀⡌⡀⠀⠀⢀⣤⠀⠀⠀⠀⣇⠀⢰⠀⠠⢀⢀⡀
⠀⢀⠠⣦⠐⠛⠣⣀⡀⠛⠃⠀⠀⠀⠀⠘⠋⠀⣌⡛⠿⣿⣿⠿⢿⣿⣿⡆⢀⠀⢳⡀⠀⢰⣿⠟⠀⠀⠀⢀⠇⠀⠘⠈⠀⢀⣾⣿⡀
⠀⢼⡀⠈⢷⡤⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢤⡉⠀⢀⣿⡿⠃⡸⠃⠀⠣⡀⠀⠀⠀⠀⢀⡠⠊⠀⠀⢰⣶⣾⣿⣿⡿⠁
⠀⠈⠁⢀⣄⣀⣦⣶⣆⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠶⠖⠶⠊⠀⠀⠀⠀⠈⠑⠒⠒⠛⣉⠀⠀⠀⠀⠘⢿⠿⠟⠋
⠀⠀⢠⣷⣿⠟⠉⠉⠙⠻⣿⣧⠀⠀⠀⢀⡾⡯⣧⡵⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⣤⡴⣶⣶⡶⣦⡀⠀⠀⠀⠀⠐⠈⠂⠀⣠
⠀⢠⣰⡟⠁⡀⠈⠀⠀⠀⠈⢿⣇⠀⠀⠈⣗⣿⢷⢷⣟⡾⡧⣦⣄⡀⠀⠀⡀⠀⢱⢫⡟⣹⣷⣶⣿⡥⠿⣧⠀⠀⣰⣦⣤⣤⣶⣾⣿
⠀⠠⣿⠀⠀⠈⠠⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠫⣟⣿⢿⣹⣻⣻⡾⣻⠀⠀⠀⠁⢿⣛⣻⣿⣏⣥⣿⠳⠿⣿⠀⠀⠻⣿⣿⡿⠿⠟⠁
⠀⢹⣿⣇⠀⠀⠀⢀⠀⠂⣠⣿⠇⠀⠀⠀⢀⠀⠈⠙⠻⢽⣯⣯⠿⠋⠀⠐⠀⠠⠸⣿⣿⢸⣯⣍⣻⡿⢻⣿⠀⠀⠠⣩⡕⠠⠊
⠀⠀⠉⢿⣷⣤⣤⣤⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⣻⣯⣭⡿⠿⠛⠁⠀⢀⣀
⠀⠀⠀⠉⠋⠙⠋⠋⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⣇⡼⠈⠚⢹⠂⣤
⠀⠀⠀⣀⡠⠤⣒⣉⣉⡉⠙⢦⣄⠀⠀⠀⠀⠀⢄⠀⠸⠀⠀⠀⣂⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠉⠚⡰⠂⠀⠐⣾⣟⢟⣀⣤
⠀⢰⠃⣀⣴⣾⡛⢿⣿⣿⣦⡀⠈⢧⠀⢀⠀⠀⠀⠃⣠⣰⣿⣿⡟⢻⣿⡖⢤⡀⠀⠀⠀⠀⠀⠰⣗⠚⣡⠷⣾⠀⠀⠈⠁⠹⣤⠚⠉
⠀⡜⢀⡾⢛⣿⣷⣾⡫⣿⣿⣿⢠⠆⠀⠀⠀⠀⠄⢸⢿⡋⢽⣿⣿⣿⣿⣿⣿⣧⠈⠀⠂⠀⠀⢠⠬⠃⢹⠀⣿⠀⠀⠀⠀⢨⡇⣉⣢
⠀⡗⢸⣷⣿⣁⣾⣿⣿⣟⣿⣿⡇⢆⠀⠀⠠⠤⠠⠈⢿⣴⡌⠹⢻⡿⣿⣻⣿⠏⠀⠈⠐⠠⠄⠙⢊⣽⠹⡆⠙⣷⠄⢀⠔⡈⠸⠭⣍
⠀⡇⠈⠻⣿⡿⠟⢛⣿⠛⠛⣻⠃⢸⠂⠀⠀⠀⠀⠔⠒⠉⠓⠿⠦⠶⠟⠛⠉⠀⠦⡀⠀⠀⠀⠀⠿⠎⣱⠘⠒⠋⠤⠄⠈⠀⣏⠉⠋
⠀⠹⡂⠀⠈⠛⠻⠿⠿⠟⠋⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡚⠀⠀⠡⠀⠀⠙⠀⠀⠀⠀⠀⠀⠓⠊⢹⢠⠖⡆⡏⠳⠜
⠀⠀⠀⠀⠐⠒⠒⠂⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⣀⣀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢶⣶⡶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠛⠛⠛⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠿⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣤⣤⣬⣥⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣉⣉⣩⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣉⣩⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣉⣉⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠋⠛⠛⠋⣛⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⡀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⡇⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣤⣤⣤⣤⣤⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⣽⣿⡇⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠾⡟⠟⢟⠻⠛⠻⢿⠶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢿⣛⠉⣇⣼⠀⢦⣅⡏⡷⣼⣜⢰⡆⡈⠳⣴⣇⠈⡟⣛⣻⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠏⡁⣷⣨⢹⣇⣿⢩⠙⢷⡎⡟⢻⢦⣿⢶⣀⣧⠔⡏⢨⡟⠉⢀⡧⢊⠈⡙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢩⡗⢦⣷⡼⣏⡷⣿⢸⡌⣶⢋⡿⣄⢸⠀⠹⣾⡏⡇⠀⣧⠞⢳⠚⣿⣠⡿⠾⢥⢦⣞⣹⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⣏⠰⣿⡄⣿⣄⣿⡙⣿⠁⣿⢦⣿⡉⣿⡀⠙⣿⡴⣾⢋⠷⣇⢸⡇⠀⣨⣿⠋⣹⣷⣖⣏⣉⣠⣥⣪⡟⢷⣄
⠀⠀⠀⠀⢀⣾⠟⠠⣌⠳⣦⣇⢹⣿⢳⣧⣼⡀⣿⢸⠁⢳⣏⡽⣾⢻⣦⢏⣾⣼⠛⣿⣷⠛⣱⣷⣾⣏⣉⣹⣇⣠⡯⢖⣏⠴⢤⡿⣧⡀
⠀⠀⠀⣠⡿⠃⠀⠢⡄⠙⣧⠙⣿⡇⡞⠀⣙⣷⡇⢸⡀⣸⣿⠁⢸⣿⡿⢣⣿⣁⡼⣻⣿⣿⠿⠋⣉⣀⣤⡾⢚⡟⢀⣼⣥⣰⡷⢊⡝⣿⡄
⠀⠀⣰⡿⠉⠓⢷⢤⣦⠘⢺⣄⡈⣿⡇⣠⣟⠁⠹⣾⣷⠀⣿⡴⢻⣿⣁⣾⣽⡟⢡⢏⣿⣟⣡⠶⡟⠁⡾⢠⡿⠖⢻⡯⠻⣼⣧⡚⠭⣸⢿⡄
⠀⢠⡿⢁⠱⡤⢨⣧⣙⣧⡉⠹⡍⠹⣿⡟⠛⣟⠛⠻⢿⣷⣟⠁⡾⢸⣿⠟⢡⣿⠾⢋⣿⠟⠁⣸⢁⡾⣡⠏⠀⣰⣟⣠⣤⣿⡦⠄⣺⠕⣈⣿⡄
⠀⣿⠃⢰⠶⡳⢤⣈⣧⠈⠛⢷⣿⣄⣿⣇⠀⢸⡄⠀⣰⡿⢻⣾⡀⣸⡏⢰⠟⠁⠀⣼⣏⣀⣼⣷⣿⣾⠷⠒⢻⡯⠟⢁⡞⢸⡷⢋⡿⢚⠓⠺⣷
⢸⡿⠈⢢⣳⡈⠢⣀⠝⠳⡖⠛⡟⠛⢿⣿⣄⠀⣿⡿⢻⡇⢸⠏⢷⣿⣧⡟⢀⣠⣼⣿⠿⠛⠉⢁⣼⣃⣀⣀⡏⢀⣴⢋⡴⢋⡴⢋⣔⣵⡒⣉⢿
⣼⡇⣠⡶⡟⣻⣯⣌⣳⣦⣬⣷⣜⢦⡈⣻⣿⣿⣿⡇⢼⠀⣾⠀⠀⢻⣿⣿⣿⣿⣏⣡⣤⡴⠶⠿⣯⣉⣉⣿⣽⠿⣿⠟⢛⢛⡻⠋⣉⠉⠙⣢⢾
⣿⡝⢻⡉⣿⣫⣴⣿⠻⣦⣀⣀⣩⣿⣿⣿⠛⠻⣿⣧⣸⣠⣿⠀⠀⣠⣿⣿⡿⠟⠋⠉⠀⣠⣤⣴⠶⠟⠛⢧⣀⣴⠗⣟⣉⠽⠷⣏⢉⣽⠋⢁⣸
⣿⠂⠘⠷⣌⣳⣄⣹⣽⡿⠿⣉⠉⠀⠀⣿⠙⣄⠘⣿⣿⣿⣇⢀⣴⣿⡟⢻⣿⡀⠀⢀⣼⣿⣫⣤⣤⣤⣤⣼⣿⣍⡟⠉⠛⢦⣤⣞⡻⠯⣟⢋⢲
⣿⡄⣤⣞⣯⣉⡉⣹⠟⢧⡀⠈⠳⣦⣀⣿⠀⠈⢷⣌⢻⣿⣿⣿⣿⣿⠀⠈⣿⣧⠀⣼⡿⠟⠋⠁⠀⠀⢀⡤⠞⠋⣉⣛⣻⣿⡟⠉⠓⠶⣉⡱⢾
⢹⡇⠈⣻⢤⣤⣽⣿⣶⣶⣿⠷⣶⣤⣍⣻⣷⣦⣤⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⣶⣿⠁⠀⠀⠀⣀⣼⡷⠒⠛⠉⠉⢉⣹⣶⣿⠶⠶⢶⣏⣉⣼
⠘⣿⠀⠤⠴⣾⠃⢠⠞⠋⣠⡴⣾⡿⠿⠿⢯⣉⣉⣉⣀⡉⠻⣿⣿⣿⣷⣤⣾⣿⣿⢏⣠⣶⠾⠟⠛⠛⠻⢿⣿⠾⢿⣿⡁⠀⠙⢷⣒⣒⣊⣳⣿
⠀⢻⡆⢲⠞⡇⡠⢃⡴⢞⣥⣾⠋⠀⠀⠀⠀⠀⠈⠉⠉⠙⠳⣬⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆⠈⠙⢶⡖⠊⣉⣒⡖⣾⡟
⠀⠀⢻⡧⢾⣇⣒⣫⠔⠋⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣙⣋⠻⡓⢶⡟
⠀⠀⠈⢿⡎⠀⣸⠃⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⠽⣤⠄⣸⡿⠁
⠀⠀⠀⠈⢻⣆⢈⡶⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠳⣴⡟⠁
⠀⠀⠀⠀⠀⠹⣿⡄⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣾⠏
⠀⠀⠀⠀⠀⠀⠈⠻⣮⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⢀⣠⡴⢖⡻⠛⣻⣯⢽⠿⢋⣾⠵⡏⢘⣷⠟⣹⠏⣿⣍⣻⣝⣲⣤⣀⣀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣬⣛⡰⣋⣿⣴⣯⢀⣿⢀⡼⣧⣿⢽⢺⡇⢠⢿⣸⠦⣿⡒⠻⡽⣭⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣏⣑⠻⣯⠴⠺⠛⢾⣞⡹⠋⢂⣇⣠⣫⣷⣶⠿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⡻⢿⣿⣿⣶⣷⣿⣾⣷⣿⡿⢛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠈⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢜⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⡒⠶⢶⠤⢄⠀⠀⠑⠊⠀⠀⠀⠀⠀⢀⠔⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠃⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠠
⠠⢀⠀⡇⠀⠀⠀⡠⢤⣶⣆⢀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡾⠀⠀⠀⠀⠱⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡸
⠀⢀⠆⠀⠀⠀⠀⠀⠉⢄⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠙⠋⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠸⠀⠀⠠⠂⠀⠀⠀⠀⠑⠇⡀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠐⠄⠉⠉⠉⢠⠁⠀⠀⠀⠈⠳⣤⣀⣀⢴⢣⠀⠀⠀⠀⠀⠠⠎⠉⠑⡄⠀⠀⠀⠀⠸
⢀⠇⠀⠀⠐⠄⣀⡀⠀⠀⢀⡀⠈⡀⠤⣄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡼⠶⣄⠀⠰⠋⠙⠳⡄⠀⠀⠈⠧⣀⡴⠁⠀⠀⠀⠀⠠⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢺⡀⢀⡸⠀⠘⢤⣀⠴⠃⠀⠀⠀⠀⠀⡶⠖⠢⡀
⠀⡠⠶⢤⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡞
⠀⣇⠀⣠⡦⣀⣀⢄⡀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠒⠚⠁
⢀⣈⠉⠁⡸⠋⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠁⠈⢳⠄⣇⠀⠀⠀⢠⠆⠀⠀⠀⢀⠄⢤⡀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡠⠀⠈⠀⠀⠀⠘⠀⠙⢄⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⠀⠀⣀⠠
⢤⠴⠋⠀⠈⠲⠤⠖⠃⠀⠀⠀⠀⠹⢄⣀⠇⠀⠀⠀⠀⢀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢣⡀⠀⠀⠆⠻⠟⠀⠚⠂⣤⠄⠀⠀⠀⠀⠠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠁
⢀⣤⠆⢱⠀⠀⠀⠀⡀⠤⡀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⠠⠉⠐
⠉⠀⠀⠀⢢⠀⠀⠘⢅⣀⡼⠓⠢⢼⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠄⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⠾⢁⡤⠦⠠⣄⠀⠀⠀⡠⠄⢄⠀⠀⠀⡠⠲⠶⠟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⢄⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠈⡧⠀⠘⢅⣀⡼⠀⠀⠀⠈⡂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠙⠢⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠢⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞⠁⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠁
⠀⠀⠘⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢠⠋⠀⠀⠀⠉⢣⠀⠀⠀⠀⢠⠀⠲⡎⠉
⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⡄⠀⠀⠘⢄⡀⠀⠀⣠⡎⠀⠀⠀⠀⠸⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⠃⠀⠀⠀⣸⣙⠉⠉⢱⠼⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠲⢴⠘⠣⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⢱⠀⢣⡀⠀⣨⠃
⠀⠀⠀⠀⠀⢠⠒⠒⠄⠀⠀⠀⠀⠀⣇⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠡⠀⢀⠌⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠀⡄⠀⠀⠠⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠂⠀⢀⠀⠀⠀⠀⠀⠘⢣⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠉⠉⠀⠂⠀⠀⠠⢄⣀⠤⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⢤⣀⠡⠄⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠒⠛⠂⠐⢢⣀⢈⢁⡒⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠭⠀⠀⠀⠀⠄⠐⠀⠒⠐⠶⢶⢎⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡉⠉⠙⣫⣍⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡟⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⡦⠄⠠⠠⢠⣄⡀⠀⠀⠀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣔⣤⣀⣩⠥⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠦⠉⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠚⠋⠁⠐⠶⣀⠂⢀⢙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠥⠌⠀⠀⠀⠸⠀⠀⠈⠠⡥⡍⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡠⠟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⣃⣥⡠⠤⠤⠀⠀⢀⡤⠔⠋⢠⠤⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⣔⣓⡶⠒⠒⠀⠀⡄⠄⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⢀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠰⣯⡽⠇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠓⠒⠐⢊⠔⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⢣⡀⠀⠀⠀⠀⢀⠈⠒⠄⠠⠀⢑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠒⢆⠀⠀⠀⠀⠀⠀⠁⠀⠲⢜⡀⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣚⡂⣀⣀⢀⣀⡀⠀⠀⠀⢙⡓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠕⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄

⠀⠀⠀⠀⢀⠖⠦⠀⢀⠀⡀⠀⡐⠀⠀⡀⢀⢰
⠀⠀⠀⠀⠸⣄⡶⢛⠫⢼⡏⢸⡊⠺⡛⠥⠯⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡄
⠀⠀⠀⠀⠀⢰⡆⠀⣀⠘⠂⢀⣀⢠⠤⢄⡠⡂⣀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠚⣇⣠⠃
⠀⠀⠀⠀⠠⠁⠻⠚⠽⠳⠄⠻⠻⠓⠯⠟⠯⠁⠮⠗⠀⠀⠀⠀⠈⠀⡀⠄⢀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⢀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠈⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⢶⣤⣄
⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢰⣦⣌⡛⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢫⣥⣼⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢠⠄⠀⢈⣬⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣏⣽⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⣪⠟⠀⣰⡄⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡜⠤⠸⠳⠾⡞⢱⢇⡼⠁
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠐⠂⡞⠰⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡶⣶⡤⣛⣧⡀⠒
⠀⠀⠀⠀⠀⢸⠀⠠⠅⣠⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠞⠋⠛⠃⠠⠉⣀⠯⣵⡟⠾⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⠿⠿⠂⠿⠓⠶⡟⠻⠆⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠐⠴⠠⠤⠆⠀⠴⠦⠤⠀⠠⠠⠴⠰⠠⠔⠠
⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠂
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠺⠀⠀⠀⠬⠁⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠤⠀⠀⡁⠀⠀⢠⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠁⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠠⠤⠠⠀⠤⠤⠀⠤⠤⠄⠄⠀⠆⠀⠴⠦
⠀⠀⠠⠤⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠄⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣠⡀⠀⠀⢈⣫⠁⠀⠀⣄⣻⠂⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢘⡿⠁⠀⠀⡼⡞⠀⠀⢨⡎⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠁⠀⠀⠀⠀⠀⠉⡀⠀⣤⡀
⠀⠀⢘⠃⠀⠀⠼⣽⠃⠀⡀⡾⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠰⠃⢠⡖⠂⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠀⡀⠁⣤⡿⠁
⠀⠀⠀⣢⠀⣼⡞⠁⡀⠀⡼⠁⠀⠀⠀⠀⠁⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⡆⠀⡔⠁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢀⣐⠠⢄⠰⠀⣾⡾⣤⣀⠁
⠀⠀⠠⡇⠘⠹⢠⠊⠝⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠊⡝⠿⢶⣤⣿⠎⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠆⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣽⠃⠄⠈
⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠁⠊⠀⠀⠀⠀⠁⠈⠀⠓⠁⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣀⣿⡀⠀⠀⣾⠋⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣤⣀⠀⠀⠀⠀⣨⠕⠚⠋⠉⠁⠉⠉⠓⠳⢤⡀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠦⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢠⣄⡀⠀⠀⢈⣠⡀⠀⠀⠘⣧⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢦⡤⠄⠁⠀⠀⠀⠀⠀⣠⡽⠷⠶⢤⡞⠋⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢻⡆⠀⠙⣆⠀⢠⡀⠀⠀⣷⣠⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢶⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠶⣄⢉⣽⢂⡟⠀⠀⡃⠀⠀⢽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣃⣀⣤⠞⠁⢀⠀⡇⠀⠀⠨⣀⣀⣤⣴⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⡀⠀⠀⠀⢠⠀⠀⣦⡀⠀⠀⠀⠀⠀⢀⣸⠇⣇⠀⠀⠠⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠿⠀⠀⠀⢀⡀⠀⠁⠀⠸⡀⠀⠀⢻⡄⢀⣠⣤⠶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢸⡇⠀⠀⣐⡷⠀⠀⣹⠏⠀⠀⠀⠀⢳⡀⠀⠀⠻⣍⠁⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠛⠛⠁⠀⠀⢸⡇⠀⠉⠉⠀⠈⠛⠃⠀⠀⣘⡿⠀⠀⠻⣄⠀⠀⠈⠳⣤⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣀⣲⡆⢲⡠⡀⠀⠈⠁⠀⠰⠾⠿⠛⢦⡀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠿⠇⠀⠀⠀⢱⡀⠉⠀⠀⠀⠙⠛⠀⢀⡤⠖⡒⠒⠦⣄⠀⠉⢲⡀⠀⠀⢻⣤⡤⠶⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢇⠀⠀⣠⣴⡶⠀⠐⠋⠀⠈⣽⡃⡤⣌⢳⡀⠀⢹⡄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠶⠿⠛⠋⡀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠐⣻⡆⢩⣿⣴⡏⠈⡇⠀⠀⣷⠀⠀⠈⣇⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠘⣦⠀⠀⠀⣶⡆⠀⠀⠉⣹⠆⠲⠏⣸⠃⣀⡀⣿⠀⠀⠀⣿⠉⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⡄⠀⠀⠈⠳⣄⠙⠉⠀⠀⠂⠠⣄⣤⡤⠞⠃⠰⠟⢡⡏⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠈⢢⡀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠐⣦⡤⠀⢀⣴⠏⠀⠀⠀⣼⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠲⣄⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠖⠚⠋⠁⠀⠀⢀⡼⠃⠀⠈⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⢀⡿⠲⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣹⡏⠉⠙⠛⠻⣿⠉⠉⠉⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣾⣿⣧⣼⣿⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣿⣶⣿⣿⣿⣿⣿⣿⣶⣆⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⣀⣶⣿⣿⣿⡿⣿⣿⡛⢟⡙⢿⣿⣿⣿⣧⡌⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣹⣶⣿⣿⣿⣯⣿⣦⠙⠻⠁⢸⣿⠾⠛⢻⣿⣿⣿⣆⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠰⣶⣿⣿⣿⣿⠏⠉⠉⠉⠁⣀⠀⠐⠊⠁⠰⠆⢀⣌⣿⣿⠟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⣠⣿⣿⣿⠛⠁⣠⡀⠀⣠⡞⢹⣷⠀⠐⠶⠆⠀⠘⠛⢡⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⢿⡿⢿⣉⠻⠆⠀⠉⠁⠀⠉⠀⠿⠋⠀⡄⠀⢀⣄⡀⠀⠀⢀⣿⣿⣶⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣶⡿⠃⠀⠀⢀⣄⠀⠀⠿⠀⠠⡄⠀⠀⠀⠀⠰⣾⢻⡇⢠⡿⢿⡟⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⢠⣦⠀⢠⣄⠘⠛⠁⠀⠀⠀⠙⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣰⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣼⡁⠀⢠⣿⠆⠀⠀⣤⣄⠀⠾⣿⣿⣿⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢀⣤⠋⠁⡀⠀⠀⠀⠀⠀⠻⠋⠀⠀⣼⡟⣿⣧⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⡿⠁⠀⠀⠛⠃⠀⠀⠀⠀⠈⣷⡀⠛⠛⠰⠿⠿⣷⣰⡞⠁⠀⠀⠀⠀⠀⠸⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠿⠿⣷⠀⣈⡀⠀⠀⠀⠀⣴⠆⠀⠈⣿⣆⠀⠀⠀⢀⣼⡏⠀⠀⠀⢀⣤⠄⠀⡠⠠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⣻⣿⣷⣶⣿⡤⠛⠃⠀⠀⠀⠀⠀⠤⠀⣀⠛⢻⣷⣤⣴⣿⢿⣇⠀⠈⠁⠈⠃⠀⠰⡄⠠⣿⣿⣧⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⡿⢿⣦⠉⠰⠀⣾⡿⣷⡄⠀⠀⡀⠰⠿⠁⠘⣿⢿⣿⡷⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣿⣿⣷⣾⠟⢿⠆⠀⠻⢷⠿⠀⠀⠀⠁⠀⠀⢀⡀⠀⢼⡏⠀⠀⠀⠉⠻⠆⠀⠀⠞⠀⣠⡀⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢋⣿⣿⡿⢣⣶⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣿⣿⣿⢷⡌⠙⢳⣶⣦⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢻⣿⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄⠀⠀⠸⠿⠏⠀⠀⣠⣶⠶⡀⠀⣀⢰⣦⠈⣷⠀⠀⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠈⠀⠀⣾⣇⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢹⡟⢦⡀⠀⠀⠀⠀⠀⣿⣷⣾⠀⠀⠈⠘⠛⠿⠟⠀⠀⠀⣀⠀⠀⠀⠸⠟⠻⠆⠀⠀⠀⠀⢹⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⠿⠗⠀⠀⠀⠀⠀⢰⡾⠋⠉⠉⠐⠚⢳⣤⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⢻⣿⣼⠿⠋⢠⡀⠀⢠⣤⡀⠀⢸⣿⠆⣤⡄⠀⠀⠀⣿⠿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦⠀⢀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠃⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⢿⡇⠀⠀⠀⣄⠀⠀⠀⢠⣼⠿⣦⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⣿⣯⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⡧⠀⠈⢷⠆⢀⣤⣄⠀⠀⢠⡄⠈⢿⣿⠀⠀⠀⠚⠃⣀⠀⢡⡴⢿⣤⡀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠉⠉⣻⣿⡿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡄⠀⠀⠀⠀⠸⣿⡞⠀⢠⠀⠀⠀⠀⠀⣶⣆⠀⠰⣶⠋⠀⠀⣠⡄⠀⢿⣦⠀⠀⢿⡋⢹⠆⠀⡀⠀⠀⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠤⣴⣿⣿⣴⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⢀⣀⠄⠀⣴⡏⢀⣄⢸⣿⠀⠀⠈⠛⠃⠀⠈⠃⠀⠰⣿⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡏⢰⣾⣛⡇⠀⣴⠄⠀⠙⠃⠀⠀⠀⠀⠛⠀⠀⠈⠙⢿⣤⣄⣀⠁⣈⠋⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠃⠈⠉⠛⠁⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠛⠀⠀⠀⠀⠀⠰⠖⠀⠈⠁⠠⢾⣿⣿⠛⠛
⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⠁⠀⠀⠈⠁⢀⡀⠀⠀⢿⣀⡸⠂⠀⠀⠀⠀⣞⠁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡶⠄
⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⡆⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠁⠀⣀⣀⡀⠁⠀⠀⠀⠈⠿⠿⠀⠀⢠⣾⡧⠀⠀⠀⣴⣿⣿⡇⣀
⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⡿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⣿⣯⡁⠉
⠀⠀⠀⠀⠀⡀⠸⣿⣿⡏⠁⠰⠀⠀⢰⡟⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠉⠁⠀⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣶⣶⣏⡁
⠀⠀⠀⠀⣀⠙⠻⣿⣿⠁⠀⠀⠀⠀⠈⠑⠚⠁⠀⠀⠀⠀⣀⠀⠀⠈⠉⠀⠀⣿⡄⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⣰⠾⢦⡀⠀⠀⠸⣿⣿⡏⠁
⠀⠀⠀⠀⣀⡀⢲⣿⣧⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠻⣧⠀⢀⠀⠀⣤⠉⣷⠀⠀⠀⣠⡈⠉⠁⢀⡀⠀⢡⣤⠈⠁⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⡠⠾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠃⠀⠐⠂⣀⡀⠀⠐⠶⠿⠏⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⡰⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⢀⣤⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄⠘⠇⢀⣟⣾⣿⣯⣀
⠀⠀⠀⠀⠘⠋⢸⣿⠀⠀⠀⠀⠸⣿⠁⠘⠆⠀⠀⠉⠁⢀⣤⣤⣄⠀⠉⠀⠀⠀⠀⠘⢻⣤⡀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠐⢶⣾⣿⡄⠀⠀⢀⡄⠈⠛⠋⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⢠⣤⣤⣤⠀⠠⣤⣠⡿⠟⠀⣀⠀⠀⠀⠀⢰⡄⠀⠂⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢤⣴⣿⣷⠀⠀⠈⡁⠀⠀⠀⢴⠄⠀⠘⠀⠀⠀⠀⠀⣀⠠⠭⠏⠀⣿⠀⠀⠀⠀⠀⠘⠛⠀⠰⠿⠃⠀⠀⠀⢙⣻⣿⡗⠰
⠀⠀⠀⠀⠀⣤⣿⣿⣇⠘⠁⠈⠛⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣾⠟⠁⣀⠀⠘⢷⡄⠀⢠⣶⡄⠀⠀⠀⣀⠀⠘⣧⣤⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⣤⣿⣿⡟⠀⣴⡆⠀⠀⠀⢀⡞⠛⢻⠀⠀⠈⠀⠀⠀⠀⠀⠛⠀⠀⠈⠓⠀⠀⣶⠗⠀⠀⠈⢙⠃⠀⠈⠀⢀⣽⣿⡿
⠀⠀⠀⠀⠀⢉⣹⣿⣧⠀⠉⠀⢰⡆⠀⠀⠑⠆⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠠⡦⠀⠀⠀⠉⠀⡀⠀⠀⢘⣿⣿⣇
⠀⠀⠀⠀⠴⠈⣹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠙⠃⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠇⠀⠀⣸⣿⡿⠛
⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⡄⠀⢀⡀⠈⠀⡀⠀⠀⠀⠛⠛⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⢀⣄⠀⠀⠀⠀⠀⣠⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣧⠈⠀⠈⢁⡀⠈⠁⠀⠀⠀⠀⠀⠐⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣿⣧⡐⠃
⠀⠀⠀⠀⠀⠀⠂⢰⣿⣿⣷⡀⠀⠈⠁⠀⠀⢀⣠⡀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⢸⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⢾⣉⣿⣿⡀⠀⠶⢀⠀⠀⢸⡋⠹⡆⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠶⡀⠀⠸⠷⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⡇⠀⠀⠀⠀⠀⠈⢱⠞⠁⣷⠂⠀⠀⠀⠀⠀⠀⡶⠀⠸⠟⠀⠃⠐⠟⠀⠀⠀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣧⠀⠰⠿⠂⠀⠘⢦⣄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⣶⠀⢀⣼⣿⡏⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣶⠀⠀⠂⠀⠀⠀⠀⠀⢤⠀⠀⠀⠐⠀⠀⢀⣤⣄⠀⠀⠛⠇⢀⡀⢠⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⣿⣿⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣿⣈⡿⠃⠀⠂⠀⠈⢀⣜⢻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢻⣿⣆⣀⠀⠀⠀⢠⡟⣣⠀⠀⠀⠀⣤⠀⠀⠘⠛⠃⠀⠀⠀⠀⢀⣸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠹⢿⣿⣶⡄⢀⠘⠿⠟⠀⠀⠀⣴⣦⡄⣤⣴⠚⠀⠀⠀⢀⣀⣼⣿⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷⣾⣿⣿⡄⠀⠀⣀⠀⠀⠙⠛⠀⡈⠁⢶⠄⣀⣠⡿⢿⡟⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⠘⢋⣿⣿⣿⣆⣀⣿⡦⠘⠃⠀⠈⠛⠀⣀⣴⣿⣿⠶⠟⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠹⢿⠿⣿⣿⣿⣾⣿⣷⣾⣿⣿⡿⢻⡿⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠀⢸⠛⠛⠛⠙⣟⠉⠛⠋⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠁⠀⠚


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⢁⣶⠴⣶⣿⠉⠀⠉⠉⠉⠉⣉⠉⢁⡈⠉⠉⣀⡉⠉⠉⠉⠉⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡆⡆⡆⡆⡦⢴⠀⠀⠀⠀⠀⠀⠀⠠⢔⠄⠀⠀⠀⠀⠀⠲⠀⠀⡖⠂⠀⠀⠀⠀⡀⠀⡐⠒⡀⠠⡀
⠀⠀⠀⠀⢀⣇⣀⡇⣀⣇⣸⠀⠀⠀⠀⠀⢀⢔⡕⠈⡄⠀⠀⠀⠀⢴⣶⠄⢰⠷⡆⠀⠀⠀⠀⠱⣠⠟⡛⢇⡀
⠀⠀⠀⠀⠰⠄⠤⠄⠤⡤⢰⠀⠀⠀⠀⠀⠀⠁⡄⡔⠀⠀⠀⠀⠀⠀⢠⠀⢀⣭⣀⠀⠀⠀⠀⡔⢙⡣⢴⡏⠑⡀
⠀⠀⠀⠀⠀⠂⠀⠒⠂⠓⠊⠀⠀⠀⠀⠘⠀⠈⠁⠉⠃⠀⠀⠀⠀⠀⠒⠀⠂⠀⠑⠀⠀⠀⠀⠈⠁⠁⠀⠁⠉

⠀⠀⠀⠀⠀⢠⢢⠀⠀⡔⠢⠀⠀⠀⠀⠀⢀⠄⡀⡒⢄⠀⠀⠀⠀⠒⠒⠒⠒⣒⠲⠀⠀⠀⠀⡖⢲⠀⣖⠒⡆
⠀⠀⠀⠀⠀⠀⠉⡖⠐⡋⠈⠀⠀⠀⠀⠀⠀⠎⠀⠈⡆⠆⠀⠀⠀⠀⠡⡇⡸⠤⢣⠀⠀⠀⠀⡇⢸⠀⠤⠀⡇
⠀⠀⠀⠀⠈⠡⠀⠌⢊⣁⠀⠀⠀⠀⠀⡀⢾⠮⠐⠪⠊⠀⠀⠀⠀⠀⠀⠂⠊⠀⢐⠀⠀⠀⠀⠉⢹⢀⠋⠉⣡
⠀⠀⠀⠀⠠⠔⠒⠨⠍⠒⠈⠀⠀⠀⠐⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠫⠬⠄⠙⠚⠹⠀⠀⠀⠀⠢⠘⠢⠤⠤⠤⠆

⠀⠀⠀⠀⢀⡠⠔⠠⣄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣒⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠢
⠀⠀⠀⠀⠀⢈⠍⠩⠅⡇⠛⠀⠀⠀⠀⠀⢀⢔⢆⡓⠉⠀⠀⠀⠀⢀⡀⠸⡀⢸⠘⡆⠀⠀⠀⡔⠈⠘⠄⢀⣠⠃
⠀⠀⠀⠀⠀⠐⡂⠐⠂⡇⣀⠀⠀⠀⠀⣀⢼⠤⠕⠁⠀⠀⠀⠀⢀⠃⠀⠀⠀⠿⠐⡇⠀⠀⠀⠣⠀⠠⢢⠂⠰⡀
⠀⠀⠀⠀⠠⠬⠤⠤⠤⠇⠲⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠤⠬⠃⠤⠁⠀⠀⠀⠀⠀⠀⠈⠢⠐⠄

⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⣄⠀⠀⠀⠀⠀⠀⡠⠀⣤⠀⠀⠀⠀⢠⠀⠀⠀⠀⣠⡀⡄⠀⠀⠀⢀⠀⡀⠀⡄⠤
⠀⠀⠀⠀⠀⠀⠀⢸⢹⢀⡀⠀⠀⠀⢀⠔⠀⡖⢲⠀⠣⠀⠀⠀⢨⠘⠸⡤⢠⠤⠌⡆⠀⠀⠀⢸⡂⡅⠀⡇⡀
⠀⠀⠀⠀⠀⠐⠒⢨⠰⣀⠀⠀⠀⠀⠈⠠⠄⠧⢼⠠⠔⠁⠀⠀⢸⠀⠘⠄⠔⠰⠆⡇⠀⠀⠀⠯⠤⡭⠬⠥⠭⠆
⠀⠀⠀⠀⠸⠀⠀⠺⠂⠀⠸⠀⠀⠀⠀⠀⠀⠤⠜⠀⠀⠀⠀⠀⠀⠀⠨⠇⠨⠅⠀⠀⠀⠀⠀⠏⠀⠁⠈⠁⠉⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⣋⣭⣴⣶⣶⣶⣬⣭⣙⡲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣾⣿⣿⣿⡿⠛⣡⣌⣩⢙⣿⣷⣮⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣟⠠⢘⣿⠗⣸⠡⣶⢘⡖⣸⣿⣿⣿⣷⣌⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠶⠶⣤⡀⠀⠀⠀⠻⣧⡙⠿⣿⣿⣷⡈⠗⠲⠞⣁⣾⣿⢋⣭⡝⢿⣷⣌⢷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⣩⣶⣿⣿⣿⣷⣦⡙⢆⠀⠀⠀⠈⠙⠳⠦⣭⣙⡛⠿⢷⣿⣿⣿⣿⣌⣛⣡⣿⣿⣿⡆⢿⡄
⠀⠀⠀⠀⠀⠀⣾⢋⣾⣿⢋⣉⣴⣄⣉⠙⣿⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢬⡙⢿⣿⣿⣿⡿⠿⣿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⢸⡏⣼⣿⠋⣴⡏⣴⣬⢻⣬⢹⣿⣎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣿⣿⡘⢃⣼⣿⢃⣿⠇
⠀⠀⠀⠀⠀⢸⡇⣿⣿⣶⢨⠷⣬⣥⢾⡀⣾⣿⣿⣆⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣙⠻⠟⣋⣥⣿⠏
⠀⠀⠀⠀⠀⠘⣧⢹⣿⣿⣶⣶⣈⣡⣦⣾⣿⣿⡿⠛⠳⣌⡳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢹⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⠇⣼⣿⣶⣍⡓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⣆⢻⣿⣿⡟⢡⣶⣦⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡙⣿⣷⣘⠻⢛⣼⣿⣿⣿⡟⣉⣡⣦⣉⡍⢻⣿⣿⣷⣮⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⣿⡏⢡⣾⢡⣶⡙⣧⡍⣿⡏⣭⠙⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣮⡛⢿⣿⣿⣿⣷⠂⡽⢮⣭⠼⡇⢲⣿⣷⣤⣾⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣍⡛⠿⣿⣷⣶⣦⣁⣴⣶⣿⣿⣿⣿⡿⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣍⣛⠻⠿⠿⠿⠿⠿⢛⣩⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠶⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠈⡹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠸⣀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉
⡌⠛⠛⣱⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠄
⠀⠀⢀⠃⠀⠀⠀⢀⣀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⡸⠀⠀⠀⡎⠘⠛⠉⠐⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡇⠀⠀⠀⠡⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢠
⠀⢸⠀⠀⠀⡀⠀⠑⢄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡜
⠀⡇⠀⠀⠊⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⠈⠉⠁⡄⠀⠀⠀⠀⠱⢤⠤⠴⡟⡀⠀⠀⠀⠀⡜⠁⠈⡆⠀⠀⠀⢠⠃
⡜⠀⠀⠀⠒⠠⠤⠀⠀⡤⠠⠌⠒⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠚⠓⣄⠀⡏⠈⠙⡆⠀⠀⠘⠢⠾⠁⠀⠀⠀⠀⠢
⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠦⠤⠃⠀⠙⠒⠊⠀⠀⠀⠀⠀⡜⠉⠑⡄
⢰⠋⠈⣷⠀⠀⠀⠀⠈⠙⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠴⠃
⠀⠑⠒⢉⡞⠉⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠉⠙⣦⢸⠀⠀⠀⢠⠂⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠂⠈⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠤⠔⠉⠈⠳⠤⠖⠁⠀⠀⠀⠘⢄⣠⠃⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠘⠛⠃⠓⠀⡤⠀⠀⠀⠀⠠⣶
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⣤
⡤⠞⠁⢆⠀⠀⢠⡔⠒⣄⢀⣀⡴⠛⠲⡄⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠐⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠀⠀⠈⢂⠀⠈⠓⠒⠉⠀⠘⣄⠀⠀⣽⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⡀
⠀⠢⠀⠀⠀⠱⣀⡀⡀⠀⠀⠀⠈⠉⠉⠀⡞⠁⠀⠱⣄⡀⣞⠁⢘⠄⠀⠀⢢⠈⠀⢸
⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠙⠦⣄⠴⠁⠀⠀⠁⠁⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠁⠈⡄⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⡀
⠀⠀⡇⠀⡀⠀⠐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡞⠉⣢⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢰⠀⠀⠀⢀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠰⣄⡠⠇
⠀⠀⠰⡀⠁⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡎⠀⠀⠀⠈⡆⠀⠀⠀⠐⠒⠚⡄
⠀⠀⠀⠈⠂⠤⢀⣀⡀⠄⠊⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀⠀⠀⠑⢤⣤⠤⠞⣀⠀⠀⠀⠈⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡠⣧⡀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⢀⠔⠛⢄⠀⡎⠉⠉⡆
⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⡏⠀⠈⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠠⠞⠀⠘⠒⠚⠁
⠀⠀⠀⠀⠘⡀⢀⡅⠀⠀⠀⠀⠉⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠨⠅⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⢢⣔⢁⠀⠔⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠦⠤⢤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢐⠄⠈⠩⠩⠉⠉⠉⠀⠐⠀⠀⠈⠀⠰⠆⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠱⢚⠃
⠀⠀⠀⠂⠀⠀⠀⠀⠉⢸⣳⠀⢿⠁⣼⠉⢧⢰⡇⠀⢰⠏⢳⡄⡞⠉⡃⠈⢧⣰⠃⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠂⣼⡀⢿⣤⡞⠘⣧⣄⠸⣄⡼⠃⢧⣀⡟⠁⢨⠇⠀⠀⠈⠐⠀⠐⠂
⠀⠀⠀⠀⢠⠀⠠⠐⠀⡀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠄⠀⠀⠈⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠄⢠⠀⢀⠀⡀⠀⠀⠠⡀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠐⠈⠀⠀⠐⠄⠂⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠐⠀⠀⠐⠋⠀⠑⠂⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣠⢠⠆⣴⢣⠎⣠⠖⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⣼⣸⡥⠷⠚⠋⠉⠉⠉⠉⠙⡆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡆⢇⡵⠊⠁⠀⠀⣾⣷⠄⠀⣶⠀⢠⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣇⡷⠋⣠⠖⢒⢦⠀⠀⠁⠀⠀⠀⠀⡼⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢣⠎⢠⠎⣰⣾⡿⣻⠀⠀⣾⣿⠆⠀⣰⢃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⣷⠋⠀⠹⣤⣨⠭⠚⠁⢀⡀⠀⠀⠀⣰⣃⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⢀⡷⠁⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⡇⠀⡴⢡⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⠀⣃⠼⠋⣠⡄⠀⠀⢀⡠⠤⠤⠞⣠⠴⠚⣡⢞⣹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢇⡸⠟⣁⡀⠿⠋⠀⠀⠀⡏⠀⠀⣠⠞⢁⡠⠾⠥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣳⠎⠀⣼⣿⣿⠠⠶⠶⠂⠀⠙⠦⢔⡡⠖⠫⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠁⢠⡶⠈⡛⠁⣤⡀⠀⠀⢀⣠⣞⣙⡒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠛⠁⢸⡇⠀⠈⠛⢀⣴⣋⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢴⡷⠀⠘⠃⠀⠀⣠⢫⠤⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠈⠀⣀⡤⠊⣭⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠭⢍⡙⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠠⠐⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠐⠠⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠄⠒⠁⠀⠀⠀⠁⠀⠀⠀⠀⠐⠁⠀⠀⠀⠁⠀⣀⠀⠀⠤⣅⠠⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠀⡡⡀⠈⠀⠘⠁⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠜⠀⠈⠢⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⠁⠀⠐⠁⠀⠀⠀⠀⠆⠰⠃⠀⠀⠠⠀⠀⠀⢠⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠠⠀⠀⠀⠀⠀⠐⠢⠀⠀⠠⠒⢢⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠂⢀⠀⢀⠀⠀⢂⠀⠀⠄
⠀⠀⠀⠀⠀⠰⠀⠀⠄⠈⠠⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠁⠐⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠁⠀⠐⠠⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠐⠡⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠐⠥⠎⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠄⠀⠸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⡔⢦⠀⠀⠀⠀⠀⠀⠀⠐⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠪⠔⠈⠀⠈⠄⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⡂⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⡰⠒⠐⠄⠀⠀⠀⠀⠀⠐⠃⠀⠀⡀⠀⠁⠀⠘
⠀⠀⠀⠀⠀⢃⠀⠈⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢹⠀⠀⠀⠂⡄⠀⠀⠀⠀⡀⠀⢀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⠀⢱⠀⠨⠀⠀⠉⠁⠀⠰⠀⠀⠀⠐⠀⡈⠓⡀⠀⠁⠒⠂⠀⠒⠀⠤⠀⢀⠀⠀⡔⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠂⠈⠁⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⢀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠔⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠄⠂⠀⠀⠄⠀⠀⠀⠠⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠊⠆⠀⠀⡠⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠔⠂⡠⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠑⠠⠀⠀⠤⠀⡐⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠈⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠁⠂⠀⠀⡀⠀⢈⠀⠇⠃⠀⠀⠀⡀⠐⠀⠂⠀⢀⠠⠊⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⡀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈


⡿⠿⠿⠿⠿⠻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄⣤⡄⣤⢤⡀⣤⢀⡤⠄⢠⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠃⠿⠅⠻⠔⠃⠿⠘⠦⠆⠛⠛⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⢀⣤⡄⣴⡆⣤⣤⢀⣤⠀⣤⣤⢠⠴⠄⣄⣠⢠⣤⢠⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠊⠓⠂⠚⠛⠛⠒⠘⠙⠃⠋⠓⠘⠒⠂⠃⠙⠘⠓⠘⠋⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡁⠶⢀⣄⣹⢷⡄⣈⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⢌⣿⡤⣴⣵⢋⣁⣰⢟⣤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡘⠿⢉⣉⣀⡡⣼⣿⠋⢿⣌⢠⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠈⢗⢤⡸⠟⢹⡿⢠⡅⠀⢀⠴⡻⠁⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠑⠬⣑⣒⣒⣒⣒⣩⠕⠊⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣱⢄⣀⡀⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⣼⣯⡿⡼⣯⢯⡿⣽⢧⡖⡆⠀⠀⡇⣀⣼⡀⠀⢀⠞⡴⠂⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢹⡏⣽⢹⠋⡏⢹⠉⡏⡇⡇⠀⠀⢇⣓⣶⣃⠰⣁⣤⣟⠉⠉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡇⣹⢸⠍⡇⢸⠀⡇⡇⡇⠀⠀⠸⡘⠥⣤⡙⠲⣉⢀⣝⡩⠈⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⠉⣽⢸⠄⡇⢸⠒⡇⡇⡇⠀⠀⢀⢵⠴⠃⠙⣆⡈⢉⣇⡠⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢾⣼⣆⣿⣸⣆⣇⣾⣴⣧⡷⢧⣄⡰⠋⠀⢹⣭⠏⠀⠉⢹⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠈⠉⠀⠀⠀⠘⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⠄⠏⠋⠰⡨⠲⠈⠸⠉⠪⠨⠅


⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠒⠒⠂⠤⢄⡀
⠀⠀⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠐⠀⠐⠤⢀
⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠈⠐⠢⠄⣀⠀⠀⠀⠀⠀⠈⣑⣶⣄⣀⣀⠀⠀⠀⢀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⡀⡄⡀⢀⡀⡀⣀
⠀⠀⠀⠀⠃⠙⠨⠇⠅⠃⠭⠉⠉⠑⠇⠑⠅⠀⠈⠨⠸⠸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠣⠏⠠⡇⠃⠣⠁⠀⠛⠤
⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⣀⡀⣀⡀⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣘⣤⣘⣬⣜⣤⣏⣤⣟⣤⣇⣇⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣽⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢺⠓⡀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠊⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡉⢀⢬⠝⠀⠀⠀⠀⠀⠀⠤⠤⡀⣃⠀⢠⠔⢓⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠾⡄⠀⠀⠀⠀⠀⠀⠈⠢⢻⣟⡔⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡚⣇⠴⠃⠀⠀⠀⠀⠀⠀⢀⣧⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⢏⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣖⣲⠖⢒⣒⡒⠲⣲⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⠲⣆⠀⠘⢆⠹⣤⠤⠤⠤⢤⣤⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠿⠃⡸⢀⣠⡈⠓⢌⡻⣦⠼⢛⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡒⠛⢧⡏⡹⠁⠀⡴⢛⡴⢎⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠤⣿⠃⢠⠞⣴⡯⠠⢤⣵⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠⢼⣤⠋⣼⢉⠉⠉⢉⣉⣙⣆⢹⡀⠀⠀⢀⣨⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢨⠓⠛⠛⠒⠒⠒⠚⢻⡄⡗⢦⣀⠜⣡⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⡇⠤⠒⠨⠀⠀⠲⣾⠃⡇⠸⢡⣾⣩⣾⢄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠱⣤⠤⠤⠤⢤⣴⠇⡼⡡⠖⠣⡀⠳⠵⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⡙⢮⠽⢋⡤⠞⠊⢱⡂⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡥⢆⠛⢄⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣠⠿⠅⠨⠷⣌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣼⣓⣲⠖⠒⢲⡾⣆⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣸⠁⠀⠀⠀⠀⠀⠀⠘⡆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⢲⠾⡀⠀⠀⠀⠀⠀⠀⠀⠰⠟⡴⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣦⢧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠊⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠦⠦⡍⠃⠀⠀⠀⠀⠀⠀⠀⠘⢢⠤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣂⠓⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⠤⠤⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠤⣒⣶⠆⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⢀⣴⣿⣿⣛⣋⡉⠭⠭⠭⠉⠉⠙⠉⠉⠉⠉⠓⢦⡀⠀⠙⢦⡀
⠀⠀⠀⠀⣠⠟⠀⢀⡔⢁⣾⠟⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠙⢦
⠀⠀⠀⢠⠏⠀⢀⡎⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣧⠴⠶⠈⢧
⠀⠀⢀⡟⠀⠀⣼⠞⠋⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠤⠤⠤⠤⢤⣤⣤⣤⣿⣧⣀⣀⣀⣀⣤⠤⠖⠚⠉⠀⠈⢧⠀⠀⠈⣧
⠀⠀⣼⠁⠀⡸⠁⠀⠀⠀⠀⣴⣶⣬⣥⣤⠶⠶⠶⠶⠖⠟⠛⠛⠉⠀⣸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠸⡆
⠀⠀⡟⠀⢠⡇⠀⠀⠀⠀⠀⣨⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⣀⡤⠖⣊⣩⣤⣤⣼⡷⠀⠀⠀⣿
⠀⠀⡇⠀⠸⡇⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣹⣿⠟⠁⠀⠀⣀⠴⣊⣥⡶⠟⠋⠉⠀⣠⣤⣄⠀⠀⠀⢸
⠀⠀⣇⠀⠀⡇⠀⠀⠀⠠⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣵⠟⠋⠀⠀⠀⢀⣾⣷⠾⠛⠁⠀⠀⠀⣠⠟⠁⠀⢹⡇⠀⠀⢸⠂
⠀⠀⣿⠀⠀⣿⠀⠀⠀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠋⠀⠀⠀⠀⠀⣰⠟⠉⠁⠀⢠⢚⡇⠀⠚⠁⠀⠀⠀⢸⡇⠀⠀⣸
⠀⠀⠸⡆⠀⣿⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⡾⠁⠀⢠⡄⠠⠃⣼⠁⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⡟
⠀⠀⠀⢷⠀⢹⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢀⡞⢀⡆⣼⡏⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⣼⠁
⠀⠀⠀⠈⢇⠘⠂⠀⠀⢷⠏⠀⠀⠀⠀⢰⠇⠀⠀⠀⣀⣠⠴⠊⠀⠀⣠⠖⠁⠀⡜⢰⡿⠀⠀⠀⠀⠀⢀⣯⣿⠃⠀⣰⠃
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠘⡆⠀⠀⠀⢀⡿⠀⣠⠔⠊⠉⠀⠀⠀⢠⠞⠀⠀⠀⢠⢡⣿⠇⠀⠀⠀⠀⡤⣽⡿⠃⠀⡰⠃
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠨⣄⢀⡤⠞⠁⠀⠀⢀⣄⠀⠀⠀⣰⠃⠀⠀⠀⠀⢸⡿⠋⠀⠀⣀⠴⣪⣾⠟⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠨⠛⣄⠀⠀⠀⡴⣱⡟⠀⢰⢿⡗⠀⠀⠀⠀⢠⡏⠀⢠⣾⣯⡿⠟⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠈⠃⠀⣸⣷⠏⠀⣠⢯⣿⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⢹⡿⠤⢞⣵⠟⠁⠀⠀⠀⣀⠴⠃⠀⠀⣀⡠⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣏⣀⣀⣀⣀⣊⣡⣤⠤⠶⠒⠋⠁

⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣼⣧⣭⣿⣴⣵⣯⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⢉⣡⡤⠤⠤⠤⣭⣍⣛⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⡞⠉⠀⠀⠀⠀⢠⣤⣀⣉⠙⠳⢦⣍⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡏⠀⣠⡀⠀⠰⣟⢋⡬⣍⠻⣤⠀⠀⠈⠙⢦⣙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⢤⣄⡀⠀⠀⠸⡇⠀⢧⠘⠷⠟⠀⢸⣇⠸⣤⡼⠀⣟⠀⠀⢀⣀⠀⠙⢷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣠⡴⠶⠶⠦⣤⡙⢦⡀⠀⠙⢦⣈⠳⢦⣄⡀⠀⠘⠷⠶⠴⠛⠁⠀⢸⡋⠈⣷⠀⠀⠹⣆⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣴⠋⠁⠀⢀⣀⡀⠀⠙⢧⡹⣆⠀⠀⠉⠛⠶⢬⣍⣙⡓⠶⢤⣀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⢹⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡾⠁⠀⣾⠛⢋⣈⠛⢻⡆⠀⢳⡹⣆⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣌⡛⢦⡀⠀⠀⢀⡤⣄⠀⠀⢸⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠃⠀⢸⣋⢰⡏⠉⣷⠈⣻⡆⠀⢳⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡹⣆⠀⠘⠷⠞⠀⢠⡟⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡄⠀⠀⣿⣀⣙⠛⣁⣘⡇⠀⠀⠀⠹⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠓⠦⠤⠤⠖⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⣇⠀⠀⠈⠉⠙⠞⠋⠉⠀⠀⠀⣠⡤⣌⠳⢮⡛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⠤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡼⠃⠀⠉⠳⢮⣝⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⣆⠀⠀⠀⡼⠋⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣍⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡘⢧⡀⠀⠳⣄⣀⡼⠁⠀⠀⠀⣠⠦⠞⠻⠶⢶⡄⠀⠀⠈⠙⢦⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠹⣦⡀⠀⠀⠀⠀⠀⠀⣠⠿⢠⡖⠛⣦⠸⢷⡄⢠⡶⢦⡈⢷⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠛⢦⡀⠀⠀⠀⠀⠙⣷⠈⠳⠴⠋⢰⡟⠁⠈⠳⠚⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡙⠳⣄⡀⠀⠀⠙⠛⢷⣴⠟⠚⠁⠀⠀⠀⠀⢠⠟⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠⠶⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣌⣉⡙⠛⠛⠛⢋⣉⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢨⠨⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠇⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⠀⡀⠀⡀⠀⡀⡀⠀⠀⢀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⢀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠀⡸⢀⠀⡇⡰⠱⡀⡗⡃⢀⠀⠐⠀⠀⠘⢢⠀⡇⠀⠲⡁⣇⢀⠇⡗⢅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠿⢿⣷⣦⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡇⠀⠀⣴⣿⠿⣿⡎⠻⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠀⠀⠻⣿⣶⣿⠃⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠋⠉⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⢀⣤⣼⡿⠃⠀⠀⠀⠀⠈⢿⣷⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡟⠛⠿⣷⣄⠀⠀⠀⠀⠀⠘⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠈⣻⣿⠃⣠⣤⣤⣄⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣿⠋⢹⣿⡆⠀⠀⠘⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣸⣿⠀⠈⠿⣿⣿⠟⠁⠀⣠⣴⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠙⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣄⡀⣀⣼⡿⠃⣠⣴⣶⣦⡀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠘⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠻⠿⠿⠛⠁⢸⣿⡋⠉⢻⣿⠀⣿⣷⣤
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠈⠻⣷⣄⢀⡀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠿⣷⣶⣿⠏⢸⣿⠋⠁
⠀⠀⠀⠀⠀⢀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠙⢿⣿⣟⠀⠀⠀⢸⡿⠃⠀⢀⣴⣿⢿⣶⡄⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠉⢹⣿⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤⣀⠀⠀⣠⣿⡟⢿⣷⣄⣴⡄⠀⠀⠀⢸⣿⣅⣠⣿⡇⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⢀⣴⣶⣦⡄⠈⠛⠿⠿⠿⠿⠋⠀⠀⠙⠿⣿⣧⣀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⣼⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣧⠀⠀⢸⣿⣍⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣴⣶⠆⠀⠀⠀⠀⠀⠸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣌⠉⠛⠿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⣼⡿⠟⢿⣷⡄⠀⠀⠀⢠⣿⡟⠉⠙⣿⣧⠀⠀⠀⠈⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⣿⣧⣀⣠⣿⠇⠀⠀⠀⠈⢿⣧⣄⣤⣿⡟⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⡀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⣠⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⢿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠟⠛⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⣾⣿⡿⠟⠋⠉⠀⠀⠀⠈⠹⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡿⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣮⡽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡆⠀⠨⠋⠂⢀⠠⠀⠀⠀⠀⣠⣴⠾⠿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢧⡀⠀⢀⣀⣧⣥⣤⢴⣾⠏⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠙⠛⠛⠿⠛⠉⠈

⠀⠒⠀⠀⠀⠂⠐⠀⠀⠀⠒⠐⠀⠂⠂⠀⠀⠒⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠈⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⢀⢀⢀⣀⡀⣀⣀⡀⠀⢀⠀⣀⢀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⡠⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠒⠶⡖⠢⢤⣀
⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠀⠀⢀⡜⠒
⠀⠀⠀⠈⠒⠤⠄⣀⠀⠀⠀⠉⠉⠙⠙⠛⠛⠑⠋⠑⠂⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠤⠤⡤⠤⠀⠀⠀⢈⡁⠀⢈⡙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠰⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⢋⣭⡝⣿⣦⠀⠀⠀⠀⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⣿⢃⣾⡿⢡⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡅⢿⣷⣥⣾⡿⣡⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⡿⣃⣬⣍⠻⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣎⢻⣿⡿⣡⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⢰⣿⣿⣿⣷⠘⠻⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣭⣵⣿⣿⣿⠟⣡⣤⣀⠀⠀⣿⣿⣿⣦⡛⠿⠟⠁⠀⠀⠀⠈⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣙⣛⣋⣭⣴⣾⣿⣿⣿⣿⣶⣝⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⣫⣥⣤⣭⡻⣿⣿⣷⡙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣷⢸⣿⣿⣿⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⣡⣦⡻⣿⣿⡿⠿⠿⠟⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣡⣾⣿⣿⣿⡎⣡⣾⣿⡷⢦⡹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣭⣭⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⣿⣿⠡⣿⡿⢋⣴⣿⡇⣹⣿⣿⡇⣀⣀⣀⣀⣠⠀⣠⣾⣿⣿⢡⡎⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡙⢿⣿⣿⠟⣡⡹⢶⣿⣿⠟⣡⣿⣿⡿⠐⠛⠛⠛⠛⣡⣾⣿⣿⣿⣿⣷⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣶⣶⣿⣿⣿⣶⣶⣶⣾⣿⣿⠟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣠⠴⠒⠒⠒⠒⠶⠬⣍⣓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠋⠀⠀⠀⢀⣤⡖⠦⢶⡄⠉⠛⠦⣝⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣧⠀⣿⠠⣞⡦⠀⣹⠆⡞⢛⡎⣻⠄⠀⠀⠈⠓⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢋⣉⣉⣉⡓⢦⣀⠀⠹⣆⠘⢦⣈⠀⠀⠙⣷⣩⣉⡴⠟⠀⢠⠖⠲⡄⠀⠳⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⠞⠉⠀⠀⠀⠉⠳⡝⢦⠀⠈⠛⠶⢬⣙⡒⠶⢤⣄⡉⠀⠀⠀⠘⠶⠶⠃⠀⠀⠹⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡞⠁⢠⡖⠞⠙⠳⢲⡄⠘⣎⢧⠀⠀⠀⠀⠀⠉⠙⠒⠦⣍⡳⢦⡀⠀⠀⢀⣤⡀⠀⢀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠁⠀⣞⠁⣞⠉⣳⠘⣳⠀⠘⣎⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡳⣄⠀⠸⠤⠇⢀⡼⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠀⠀⠸⣇⣌⠛⣡⣸⡇⠀⠀⠈⢧⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠓⠦⠤⠶⠋⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡆⠀⠀⠈⠘⠛⠁⠉⠀⠀⢀⡴⠶⡝⠲⣝⡲⢤⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠹⡄⠀⠀⠀⢀⣠⣄⡀⠀⠈⠳⠴⠃⠀⠀⠙⠲⢭⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠹⣆⠀⠀⣾⠁⠀⢹⠀⠀⠀⠀⠀⢠⣤⡀⡀⠀⠈⠓⢮⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⢳⡀⠈⠓⠒⠋⠀⠀⠀⣸⡏⣩⣬⡉⢹⡂⠀⢀⣀⠙⣆⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⣀⠀⠀⠀⠀⠘⢯⡀⢧⣀⡿⢨⠽⠀⢿⣬⠇⢸⠄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣌⠳⢤⡀⠀⠀⠘⠷⢦⣠⡶⠼⠃⠀⠀⠀⠀⡼⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣉⠓⢦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣍⣙⡛⠒⠒⠚⢋⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠠⠒⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡔⠂
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢱⠐⠈⡇⡘⡆⢸⢰⠐⠀⠨⠀⠀⠢⡀⢸⠀⡆⡄⡆⠈⡆⣤⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠀⠀⠁⠈⠈⠈⠈⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠈⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠟⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠋⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⣠⣤⣶⠶⠶⠶⠶⠶⢶⣶⣤⣤⣀⣀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⢀⣀⣠⣤⣤⣶⡶⠶⠶⠶⠶⢶⣦⣄⡀
⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⡟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠉⠛⠻⢶⣦⣄⡀⣀⣤⣴⠾⠟⠋⠁⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣤⣿⠿⠿⣯⣄⡀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠙⠻⢷⣤⣀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣀⣴⠿⠋⠀⣿⠀⠀⠀⠀⠀⠀⣴⡾⠛⠛⠛⠿⣦⡀⠀⠀⠀⠀⠀⢿⡇⠈⠛⢷⣤⡀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⡾⠛⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⣦⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠻⣷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⡿⠻⣷⣄
⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠙⢷⣦⡀⠀⣿⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣤⡾⠃⠀⠀⠀⠀⠀⣼⡇⠀⣠⣴⠟⠉⠀⠀⠈⠻⣷⣄
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣷⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⢻⣶⡶⣦⣄
⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣿
⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠉⠛⢷⣦⣀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣾⣏
⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠈⠙⠻⣶⣤⣴⡾⠛⠉⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣀⣤⣴⠾⠟⠋⠙⠻⢶⣦⣄⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠹⣧⡀⠀⠀⠀⠀⢀⣤⣦⣤⡀⠀⠀⠀⣀⣀⣿⣷⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠈⠻⠷⣶⣤⣤⣿⠁⠀⠈⣿⠾⠿⠛⠛⠉⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠛⠛⠿⠶⢶⣶⣤⣤⣤⣤⣶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⡾⠋⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⣤⣴⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢤⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠠⡆⠀⠀⠀⠀⠠⡅⠀⠰⠂⡀⠀⣾⣟⣶⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⡿⠃⠀⢀⢀⡀⠀⠀⠀⠰⠆⠀⠀⠤
⠀⠀⠀⠀⠤⠀⠀⢠⡄⠀⢀⠀⠀⢠⣤⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠉⠋⠀⠸⡇⠀⠀⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⡁⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠅⠀⠀⠰⠀⠀⠰⠀⠀⠀⠀⠛⠑⠀⠀⠀⠈
⠀⠀⠀⠀⢤⡄⠀⢀⠶⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠰⣦⡄
⠀⠀⠀⠀⣲⠄⠀⠸⠠⠀⠀⠁⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢻⣄⣀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⢠⣞⣿⣷⠀⠀⢼⣞⠀⠀⠐⠘⠀⠀⠰⠀⠀⠘⠃⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠘⠃⠀⢴⣿⣿⠿⠂⠀⢜⡶⣘⡞⠀⠀⠈⠓⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠈