Pickles Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠛⠉⠉⠀⠀⠀⠉⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣦⠀⠸⣿⣷⡀
⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⠀⠀⢹⣿⣧
⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢿⣿⣆
⢀⣤⣶⣿⣿⡇⠀⠀⢀⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⣿⣿⡟⠋⠁⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦
⠘⢿⣿⣿⣿⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏
⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡁
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣄⢻⣿⣇
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠈⣿⣿⡄
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠸⣿⣷
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠹⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⣴⣶⣶⡄⠀⠀⠀⠀⠈⢿⣿⣦⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⣿⣿⠀⠀⠀⠀⠀⠛⢻⣿⡟⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠃⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠈⠻⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠿⠟⠻⣿⣿⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⣠⣄⣴⣿⣇⠀⠀⠀⠀⠀⠸⢿⠏⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⢰⣿⣿⣃⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠈⠙⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡈⠛⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣷⡄⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣤⡀⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣛⣿⣿⣷⣤⣀⠀⠙⠛⠋⠀⠀⠀⠻⣿⣷⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠟⠛⠿⢿⣿⣶⣦⣄⣀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣠⣤⣴⣶⣾⣿⡿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣛⣻⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣁⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢭⠙⠛⢛⣛⠛⡛⠛⠛⢛⠿⠿⣛⢛⠛⠛⢛⠛⢉⡉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⡠⠊⠁⡠⠓⠨⣞⠉⠃⠀⠀⠈⢾⠒⠲⡚⡏⠣⢣⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⠐⡞⠀⠀⢰⠀⠀⠀⠈⢃⠃⠀⠀⠀⠀⢇⠀⠑⠀⠘⠣⡅⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠱⠀⡾⢐⡅⠀⡰⢆⠀⠀⠀⠀⠻⡄⠀⠀⠐⠘⡄⠀⠠⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠁⠀⠃⡘⠀⠈⢂⠀⠀⠀⠀⠱⡀⠀⠀⠀⢻⡉⠀⡄⠢⠀⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠀⠀⠀⠀⠀⢀⣨⠦⣀⠀⠀⠀⢑⠀⠀⠀⠀⢧⠀⡇⠐⡀⢰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⣴⠋⠀⠀⠀⢣⡀⡠⠘⠢⢄⡀⠀⠺⡀⡇⠀⡇⢼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠱⡀⠀⡳⣇⠀⠁⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢇⠀⢀⠀⠸⡁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠐⠀⢱⡴⠃⣿⠀⡄⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄⠐⠸⢰⡆⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠈⣧⠀⣿⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡖⠉⢹⠀⠸⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠀⠸⡀⣿⠀⢸⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⢸⡀⠀⣇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢳⣿⠀⠙⢸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⢰⠈⠢⣟⠀⠀⠀⠀⠀⠀⣻⢀⠀⠀⠀⠀⠀⢸⡏⠀⠸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡇⠀⢸⠀⢐⢻⣄⠀⠀⠀⠀⠀⢹⠃⡀⠀⠀⠀⠀⠈⡇⠀⡖⠀⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢣⠀⢸⡁⠠⠉⡏⡀⠀⠀⠀⠀⢸⠀⢯⡀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡘⢆⠈⡇⢰⡀⢿⡀⠀⠀⠀⠘⢸⠇⠈⢧⠀⠀⠀⠀⣿⠀⠜⠀⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠠⣀⠇⠄⠑⠤⠷⣄⣀⢀⣠⠾⢀⣀⡤⠓⠀⠴⠊⠟⣂⠄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢭⣑⣒⠒⠤⠤⠬⠭⠤⠤⠤⠤⠤⠤⠒⣒⣊⡭⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠒⠒⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⠶⠶⠺⠿⡿⠟⠳⠒⠖⢲⡶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡛⡛⠟⢻⡟⢻⠛⣿⠟⠛⢟⣿⣛⣏⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡠⣲⠦⢴⡋⢙⠖⠉⣳⠔⠋⠲⠀⢲⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠁⡇⠀⠀⠹⡜⠀⡰⠃⠀⠀⢰⠁⠐⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢾⠀⢣⠀⠀⠀⠙⡴⠁⠀⠀⢠⠿⢄⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⡆⠀⠀⠰⠁⠀⠀⢠⠃⠀⠀⠆⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠒⠉⢺⠄⢀⣄⡰⠂⠒⡇⠀⠠⠴⢄⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡨⠀⠀⠈⡞⠁⢀⡇⠀⠀⠸⢠⠁⠀⠀⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⠀⠀⠀⠅⠀⠀⡇⠀⠀⠰⡚⠀⠀⠀⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⢰⠀⠀⠀⣷⠀⠀⠀⡇⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⢄⣀⠼⠀⠀⢀⠹⡄⠀⠀⣇⠀⠀⡘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠈⠊⠓⠐⠪⠤⠛⠂⠚⠐⠓⠈⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⠴⠦⠤⠴⠶⠖⠒⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣛⣙⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⣀⠀⠁⠀⠀⠀⡄
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠳⠤⡤⠄⢀⡀⣉⣀⠤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴
⠀⠀⢈⠃⠂⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠤⠆⠀⠀⠀⠀⢀⠀⡤⢀⠠⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠆
⠀⢀⠅⠀⠀⠀⡀⠀⢀⠄⠀⠱⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠄⠀⠀⠀⢐
⠀⢰⡀⠀⠄⠀⠀⠠⠂⠀⠀⠊⠀⠀⠀⡅⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⢀⠀⠉
⠀⠈⠀⠀⠀⠄⠀⠁⠄⠀⠀⣡⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠐⠀⠸⠀⠀⠀⢨⠀⠀⠁⠀⠀⠐⠀⠈⠀⠠
⠀⠀⠆⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐
⠀⠀⠢⠈⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠨⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⡙
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠓⠀⠀⠂⠀⠒⠈
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀
⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠠⠀⠀⠄⠠⠠⠠⠀⢰
⠀⠀⡖⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠄⡄⠠⠠⠠⠀⠘
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢒⠁⠄
⠀⠀⣄⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠐⠅⠐⠀⠂⢀
⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⡄⠠⠀⡠⠤⠄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠂⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⠤⠤⠀⠄⠀⠀⠀⠁⠀⠀⠁⠂⠀⠦⠤⠀⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠙⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⠤⠤⠤⠄⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⡒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠁⠒⠂⠠⠀⠒⠒⠒⠒⠠⠄⣀⣀⠀⠄⠒⠒⠒⠊⠁⠈⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡰⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⢣⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢘⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⢀⠤⠤⠀⡛⠀⣀⡀⣀⣀⢀⠀⡠⡄⠀⢀⠀⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢡⠘⠂⡿⠁⡌⢔⣁⡅⡀⠎⡀⠁⢄⡀⠀⣻⠈⠒⡏⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⢒⡉⠀⠒⠈⠂⠚⠃⠋⠒⠁⠢⠤⠄⠂⠀⠘⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠀⠀⠀⣀⠠⠤⠀⠐⠒⠒⠐⠒⠒⠒⠒⠒⠒⠒⠒⠠⠀⠀⠀⠀⠀⠀⢀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠞⠛⠋⠉

⠀⠀⢠⢶⣒⣒⣚⣛⣛⣫⣭⣭⠭⠭⠭⠭⡽⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⡄
⠀⠀⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠭⣟⠏⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠜⠛⠛⠉⠛⠉⠛⠉⠛⠉⠉⠉⠉⠉⠉⠙⢄⠀⠀⠀⠀⠀⠀⠛⠂⠂⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠉⠉⠱⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⢲⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣤⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠠⣧⣭⣿⡦⠄⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣣⣿⣧⡄⠀⡇⠀⢸⠇
⡇⠀⣿⣿⣿⣦⣀⣀⣤⢤⢤⢶⣤⣴⣿⣿⣿⣿⣿⠀⡇⠀⢨
⡇⠀⠈⢿⣿⣿⣿⣷⣮⣷⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⡇⠀⢸
⡇⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡁⠀⠀⡇⠀⠈
⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢯⡳⠍⣷⡄⠀⠇
⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣷⡒⠋⠀⡆
⡇⠀⢾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢶⢤⡈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⡎
⠈⠳⠤⠤⡀⢀⣀⣉⣉⣉⣉⣉⣉⣉⡠⠤⠤⠤⠴⠚⠁⠀⠀⠐⠤⠄⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠤⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡤⡤⡤⣤⣶⣤⣤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⣿⣯⣿⣾⣾⣿⣧⣿⣿⣿⣿⣯⣿⣿⣿⣞⢹⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⡆⠉⣽⣛⣟⣻⢚⣿⣻⢻⣿⣛⢘⡛⣯⣭⡦⡖⡞
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠳⢦⣬⣙⣛⣟⣻⣸⣿⣿⣿⣿⣟⣚⣛⣛⣭⣵⣶⠋⠓⠦⡀
⠀⠀⠀⠀⠀⠀⢰⣏⠉⠀⠀⣀⡹⠗⠛⠫⣉⠛⠿⢛⡻⠿⠋⢉⡋⢉⠁⠀⠈⠲⡄⡠⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠁⠐⠾⠭⢀⣀⣀⡔⠁⠀⠀⠈⢳⡀⠀⠀⣈⢂⠠⠤⠀⠒⡏⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢀⠈⠀⠀⠄⢀⠀⢈⠉⠀⠂⠀⠀⠀⠀⠀⠀⡧⠆⣇
⠀⠀⠀⠀⠀⠀⠘⣿⢰⣬⣍⣘⣉⡒⡐⣂⢒⡀⣈⡀⢐⡚⡀⠁⣘⣃⣢⣬⣥⣤⣶⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⣭⣭⣽⣗⣋⣟⣛⣋⣻⣛⣙⣃⣏⣟⣯⣯⣯⣽⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣾⣏⣿⣿⣿⣿⡏⠍⠻⠿⢿⠙⢿⠹⠟⡟⢻⣿⣿⣿⣿⣅⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠻⠿⠿⣿⣧⣾⣧⣤⣽⣤⣼⣴⣦⣿⡶⠿⠿⢿⣛⣿⣿⡟⣇
⠀⠀⠀⠀⠀⠀⠘⡇⠉⠿⠻⣿⣿⣿⣷⣶⣶⣶⣶⣾⣲⣶⣶⣿⣿⣿⡿⡿⠳⠛⡩⠁⣿
⠀⠀⠀⠀⠀⠀⠨⡇⢂⠈⠙⢰⠀⠂⣡⠁⠩⠁⡍⠭⠩⢥⠉⣉⠂⠀⠒⠀⡃⠃⡆⡃⣿
⠀⠀⠀⠀⠀⠀⠘⡧⢁⠀⠰⢰⠠⠡⢸⠀⠰⠠⠀⠢⠀⢸⠘⠀⢐⠀⠀⠀⡁⠀⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠘⡇⠂⠀⠸⢸⠀⠄⠘⠀⠀⠐⠀⠁⠀⠠⠈⡁⠂⡀⢀⠀⠁⠀⢫⠀⣹
⠀⠀⠀⠀⠀⠀⠀⣇⠀⡀⠀⠀⠀⠑⠲⡀⢸⠀⠆⠀⠀⢰⠀⠀⠂⠀⠀⠀⠀⠀⢰⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⢣⠀⠀⠀⠆⠈⣆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠈⠃⢘⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣦⠷⢜⡀⠈⢢⡈⠢⠀⠀⣦⣡⡀⠀⠀⢸⡾⣦⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠴⣦⣀⣝⣉⠐⠶⠾⢛⣯⣉⣛⡾⠯⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣀⣀⡸⣆⣤⠤⠤⠤⠤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢒⣿⣭⠖⠚⠋⠉⠙⢂⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⣬⣿⣓⠤⠄⠀⠀⠀⢈⠀⠀⠀⠀⢀⣀⣠⡤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠖⠲⠶⠤⡀⢸⣟⡿⠓⡯⢤⠍⠛⠁⠀⠉⠉⠉⠉⠉⠉⠁⠀⢀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡤⠒⢋⣩⣿⣿⣭⡟⣛⠲⠂⠤⠤⠤⠤⠠⠄⠒⠒⣒⣒⣚⣉⠭⠴⠿⠶⢦⡤⠤⠤⠤⠤⠤⠤⠶⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠋⠑⢲⣤⣄⠁⣰⡞⠛⡌⡹⠟⠉⠉⠉⠉⠙⡟⠉⠉⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⢀⣤⡤⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⢉⣀⣬⠷⢜⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⣀⡠⠴⢾⣅⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣠⡾⠋⠁⠀⢰⠈⠉⠓⣶⠒⠒⠒⠐⠊⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⡀⠙⢷⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠟⠀⠀⠀⠀⠸⢨⠉⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠘⠇⠀⠈⣷⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸⠀⠀⠀⠀⠀⡀⠲⠀⠀⠄⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠠⠄⠀⠈⠄⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⢀⠰⣄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡸⠄⠂⠀⠀⠈⠘⣏⠀⠀⠀⠀⠀⣀⠀⠁⠶⠁⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⡆⠀⠀⠀⠠⠀⠀⠐⠆⠀⠠⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠞⠂⠀⠈⠀⠀⠀⠀⠛⠀⠘⣀⣤⠀⠈⠃⠀⠀⠀⠀⡄⠀⠀⠀⡐⠁⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠃⠀⠀⠀⣠⠀⠀⠀⠀⠘⠀⠀⠀⠰⠀⠀⠈⠁⠀⠛⠒⡀⠟⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠠⠄⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⣀⠐⡦⠊⠀⠀⠀⢀⡀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⠀⠀⠀⠆⠀⠀⣠⠀⠀⠀⠀⠺⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⣬⠀⠠⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⠀⠀⠀⠃⠀⠀⠀⠶⠀⠀⠀⢄⠀⠀⠰⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣤⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠐⠀⠀⠀⡠⠊⠀⠀⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠛⠈⠉⠀⠀⠀⠀⠀⠈⣀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠉⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠈⠂⠀⠀⠘⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⣀⠀⠠⡀⠘⠁⠀⠐⠃⠀⠀⠉⡉⠀⠀⠀⠘⠂⠀⠘⠂⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠠⠆⠀⠀⠈⠀⠀⣈⠠⡀⠀⠀⡄⢀⣀⠀⠁⠀⠀⢀⠀⠔⠻⠒⠀⠀⠀⢰⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡀⠀⠀⠐⢀⠀⠀⠀⠀⠖⠀⠈⠁⠨⠔⠀⠠⠎⠉⠀⢐⠒⢀⠀⠀⠊⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢳⡤⠔⠂⠉⠢⡈⠀⠀⠀⢀⡀⠀⢀⠀⢈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠳⢄⠀⠀⠀⠀⠐⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⢹⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢄⠉⠒⠠⠤⣄⡀⠈⠉⠉⠉⠀⠂⠀⠀⠐⠂⠒⠀⠈⠀⠀⠀⠀⣀⣠⠤⠒⠁⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⡈⠉⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣁⡀⠤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⡀⠉⣉⠀⠀⠘⠛⠂⠀⠀⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠆⠀⠀⠀⠀⠎⠀⠀⠀⠣⡠⠒⠐⠢⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠀⠀⠀⠀⢸⠀⠀⠀⣀⢠⡇⣀⠀⠀⡃⠀⠀⠀⠀⢸⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢧⡀⠀⣀⠜⢣⡀⠀⣠⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⠠⡀⠀⠈⠉⠁⠀⠀⠈⠉⠁⠀⠀⠁⠀⠀⢸⠀⠀⠀⠄
⠀⠀⣀⢀⠔⢦⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠖⢄⢀⡀
⢀⣫⡀⠉⣄⠀⠣⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡠⠃⢀⡎⠁⣨⣂
⢏⡀⠈⡃⠠⠀⠀⠙⠒⠲⠤⢄⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⠤⠴⠒⠚⠁⠀⠠⠀⡉⠀⣈
⢸⠁⢀⣁⣀⣠⠖⠒⠒⠢⠤⠤⢤⡇⢠⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠤⠤⠒⠒⠒⢦⣀⣀⣁⠀⢹
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⠛⡟⡟⢻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠛⠉⠛⠒⠶⢼⠀⡇⡇⢸⠴⠒⠚⠛⠙⠓⠲⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣯⣤⣤⣤⣤⣠⣤⣤⣤⣤⣗⣧⣤⣤⣤⣤⣠⣤⣤⣤⣬⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠟⣳⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠛⠓⢲⣶⣶⣶⣦⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠖⠛⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⣀⣀⡤⠤⠀⠀⣀⣀⣀⣀⣠⣖⠚⠛⠛⢦⢀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣠⠔⠊⠉⠁⠀⣒⣤⢶⡞⠛⠉⠭⠉⠉⠉⠉⡷⠈⠀⣼⣉⡙⢦⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠣⣀⣠⣴⠾⣋⡩⠔⠊⠁⠀⠀⠀⠀⢆⣤⠞⣁⡤⠾⠛⠋⠉⠻⡄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣤⠞⣫⠔⠊⠁⠀⠀⠀⠀⠈⢀⣤⠖⣋⡵⠛⠁⠀⢀⡄⠀⠀⢠⡇⠀⢠⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠞⠁⠠⠀⠀⣀⠀⠀⠐⣈⡴⠞⠉⣠⡾⠋⢒⠐⠀⠀⠀⠀⠀⢀⡟⠈⡀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⡀⠀⠀⣀⣤⠶⠟⠥⠀⢀⡴⠋⢀⠜⠁⠀⠀⠀⠀⠠⢠⠞⠁⢀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡀⠘⠲⢦⠶⠾⠿⠿⠃⠀⠀⠀⣠⠏⠀⠔⠁⠀⠀⠀⡄⠀⢀⣴⣯⠖⠒⢾⡄⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⣠⠼⠤⣄⠁⠀⠀⢢⣤⡾⠁⠀⠈⠉⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠁⠀⠀⠈⢻⡂⠈⠚⡟⠀⠀⡀⠀⠀⠀⠀⠀⣠⠞⢹⣿⠁⠀⠀⠀⠂⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⠠⠀⠀⡈⢧⠀⢰⠁⠀⠈⠁⠀⠀⢈⣠⠞⢁⠀⢸⡇⠈⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⡇⠀⠀⠀⠀⠈⢧⡀⠳⣄⣀⣀⡤⠖⢋⠀⠀⠈⠀⣼⠀⠀⠀⠈⢣⢠⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⠀⠐⡄⠀⠀⠀⠹⣞⡉⢠⠀⠀⠁⢸⡀⠒⠀⢰⣇⠀⠀⢀⢀⡇⣸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣧⠐⠈⠀⠀⠀⠀⠈⠻⣄⠀⡀⠀⣠⣷⡀⢠⠏⠀⠀⠀⠈⡼⢀⡟⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢀⠼⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡾⠟⠙⣿⠏⠀⠀⠀⡀⠰⠃⣼⢳⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡎⠰⠈⣷⣄⠀⠀⠈⠀⠀⠀⠀⠘⢧⠒⢠⠏⠀⠀⠀⠀⠁⠁⣼⠃⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⠢⣄⣠⣽⣷⣄⡀⢀⠀⠀⠃⠀⠈⡇⡾⠀⠂⠀⠀⠔⢀⣼⣁⣀⠜⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠈⠙⠛⠻⠶⢤⣄⣀⣀⣴⠃⣳⣄⣀⣀⣤⠶⠛⠋⢉⡠⢊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢭⣭⣀⣒⣒⣂⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣲⣶⡯⠵⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠶⠶⠶⠟⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠲⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠉⠚⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣭⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠒⠲⠤⠤⣄⣀⠀⠤⠤⠶⠖⠒⠒⠦⠤⣀⡀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠒⠒⠲⠖⠚⠉⠀⠀⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣀
⠀⠀⢀⣴⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⣸⠃⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⣇
⡇⠀⠀⠀⠀⡸⠃⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠠⠞⠁⢀⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠈⡀⠀⠀⢣⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠓⠀⠀⠀⡇⠀⠀⠀⠀⠀⣃
⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻
⡇⠀⠀⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸
⢹⡄⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣶⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡴⠛⡉⠙⠢⠀⢤⡁⢀⡠⠄⢦⠀⡤⠲⣆⠴⣳⠀⠀⡞⢻⡆⠀⠀⡤⢖⡛⡆⢠⠔⠛⣛⡦⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠇⠸⣿⠀⠀⡷⠈⠁⡌⠁⡶⠛⠀⡇⠀⠀⣴⠁⠀⠀⠃⣌⠀⠀⠀⡇⠻⢴⠀⠙⠦⢬⡛⠳⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸
⢸⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠸⠀⣠⡤⠞⠁⡄⢸⢣⡄⠓⠒⣆⡇⠀⡶⡀⠙⡦⢠⠀⠉⠉⢱⠀⡇⠸⠿⡀⣠⠐⠶⠃⣠⠇⠀⠀⠀⠀⠀⡇
⢸⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢇⠸⡀⠀⠀⠛⠚⠀⠉⠉⠉⠁⠓⠶⠃⠙⠚⠁⠘⠖⠒⠢⠼⠀⠓⠒⠒⠃⠈⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⡇
⠀⣇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢹⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢘⡇⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠈⠣⠐⠀⠠⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠴⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠂⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠶⠛⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠳⠶⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠹⢦⣀⡀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠐⠒⠚⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠉⠙⠻⣧⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⢶⠶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠛⠁
⣿⣿⣿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣀⣤⣤⣶⡆⣴⡦⠾⣴⣿⣿⣿⡧⢾⣷⣿⣷⡶⡆⢴⣶⣶⣶⣶⣤⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠖⣴⣿⣿⠟⡿⢋⣼⣏⢀⣽⡅⣸⣶⠀⣾⣧⣾⣾⣷⣶⣶⣶⣶⣶⣰⣦⣿⣬⣭⣩⣿⣝⡟⠻⢿⣷⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠠⡾⢻⣿⢯⣾⡟⣹⠦⣼⣧⣿⣿⡟⣼⣿⢻⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⡿⢷⠞⢱⡿⢟⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠄⢀⠛⠹⢿⡟⣸⠏⣼⢧⣿⣯⡿⣹⣿⣟⣙⣿⠷⡾⠿⣿⠿⠿⠿⠿⠿⣿⠿⣛⣻⡘⣛⣛⣻⣧⣭⡷⣿⣶⣶⠘⣷⠿⠿⠁⠉⣴
⠀⠀⠀⠀⠀⠀⢰⣽⣷⢰⣦⢈⡁⠘⠁⠿⠻⠿⢹⣿⣿⣿⣿⣿⢸⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⢻⡇⢸⡿⣿⡿⢿⠜⠛⠛⣋⣅⢠⣶⡶⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠘⠙⣿⢀⣿⡀⠃⢸⡇⣿⣶⡆⣴⣶⣤⣄⣬⡅⢀⡈⣩⣉⣍⢈⣍⢉⣍⣩⣭⣥⣀⣥⠀⣶⣤⡖⣾⡗⣿⣿⣿⣿⢸⣿⡇⢹⡇⠈
⠀⠀⠀⢀⣀⣤⣤⣾⣯⡀⠛⠁⠁⢸⡇⠿⣿⣿⣿⣿⡿⢳⣿⡇⢸⣷⣿⣽⣿⢹⣿⣼⣿⣿⣿⣿⡿⣿⠀⣿⣿⣧⣿⣿⣿⠿⠿⠟⠈⣉⣠⣿⣷⣶⣤⣄⡀
⠀⣠⡶⢟⡫⠝⠛⠁⠙⠻⢿⣷⡶⣶⣤⣤⣉⣉⡙⠛⠛⠈⠻⠇⠸⠿⠿⠻⠿⠺⠿⠿⠿⠿⠿⠿⠋⠛⠀⠛⢉⣉⣈⣥⣤⣶⠶⠾⣻⣿⣟⣉⡀⠀⣀⣉⡻⢷⣄
⣰⣟⣀⣼⣧⣤⡤⠴⠶⠶⠶⠾⢻⣷⡶⢶⣭⣭⣙⣛⣛⣛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⣛⣛⣋⣩⣭⣥⡶⠶⠛⠋⠉⠻⣯⣀⠉⠁⠀⠀⠀⠈⢻
⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣭⠿⠿⠒⠿⢯⣭⣉⠉⠛⠛⠛⢛⣛⡛⠓⠒⠛⣻⡿⣿⣿⣍⣉⠉⠉⢹⠁⠀⢿⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⠀⠀⢀⣼
⣿⠙⠳⣄⠀⠀⠀⢀⣀⣤⣶⢟⡭⠔⠒⠒⠀⠀⠀⠀⠈⢉⣿⠟⠛⠉⠉⠉⠙⢷⣶⠏⠁⠀⠀⠀⠀⠉⠙⠻⢿⣀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⣡⠴⠋⢹
⣿⠀⠘⠧⠀⠀⠶⣾⣉⣻⠁⠋⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⡁⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠘⣆⠀⠀⣀⣀⣤⠴⠖⢻⡟⡧⠀⠛⢸
⣿⠀⠀⢀⡐⠷⠀⢩⠈⣿⠛⠛⠶⠶⠶⢦⣤⣤⣾⣯⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡋⣂⣀⣀⣀⣤⣤⣤⣤⠤⠶⠟⡆⢹⡟⠋⣩⡀⣤⣀⠀⢸⡇⣧⠘⣫⢸
⣿⡀⠀⠈⠓⣀⠀⢸⠀⣿⠀⢰⡇⠰⠷⠀⠀⠀⢻⠁⠀⠀⣿⠀⠀⠀⠀⢈⣍⠉⠉⠙⡏⢻⠋⠀⠀⢠⣄⠀⠀⠘⠂⠀⡇⠸⡇⠀⠀⣠⠀⠀⠀⢠⡇⣿⢀⣀⣸
⣿⡷⣤⣀⠀⠉⠀⡾⠀⣿⠀⢈⠓⠷⠀⣤⣀⠀⢸⡆⠀⠀⡿⠀⠻⠉⠀⣠⠉⠉⠀⠀⣧⢸⡆⠀⠀⠀⠉⠉⢰⣆⡤⠀⡇⢰⡇⠀⠀⢉⠉⠲⠄⢸⣧⡿⠞⠋⢸
⣿⣧⣤⢉⠙⠓⠸⠷⢤⣿⣀⣸⡧⣶⡀⢀⠁⠀⢸⡇⠀⢸⡇⠀⠀⠀⠈⠷⠖⠀⠀⠀⣿⢸⡇⠀⠀⠰⠶⠤⠀⢤⣤⢀⣇⣸⣧⣀⠴⠿⠖⠚⠛⠉⡁⡄⣰⣿⣸
⣿⣿⣿⢈⡛⣿⣿⡆⣤⣤⢀⣈⣉⢉⣛⠛⠛⠓⠚⠳⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠾⠷⠶⠒⠒⠚⠛⠛⠛⣉⠉⢉⣩⣄⣤⣴⢠⣶⣷⣷⠸⣿⣿⣽⣿⢸
⣿⣿⣿⢸⣧⣿⣿⡇⣿⣿⣼⣿⣿⣹⣿⡇⢸⣿⢰⣿⣶⡇⣾⡶⣶⣶⢰⣶⣦⣶⡴⣶⢶⡟⠰⣶⡦⢦⣸⣴⣶⢾⣿⣿⣿⣿⣧⣻⣿⣾⡿⠿⣿⠇⣿⣿⣿⣟⢾
⣿⣿⣿⠠⣿⣿⣿⡇⣭⣬⣉⣛⣛⠛⠿⠿⠸⡿⠸⠿⡧⠟⢻⡇⣿⠘⣾⣿⣿⡿⠇⣏⡸⢧⠀⣿⠇⠼⠿⠿⠟⠘⠛⣉⣛⣁⣩⣬⣤⣶⣶⡆⢿⠆⣿⢿⣿⣿⣾
⣿⣿⣿⠰⣿⣳⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣶⣶⣶⣥⣤⣤⣤⣦⣴⣶⣷⣷⣶⣶⣶⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⡆⣿⢺⣿⣅⢿
⣿⣯⣿⢰⣿⣿⡇⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠻⢿⣿⠛⣿⣿⣿⣿⡿⠛⣿⣿⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢙⣿⢀⣿⠛⣿⣿⣸
⣿⡿⣿⢸⣿⣿⣧⢳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠈⣿⠋⣿⠟⡛⢻⡇⠀⠟⢉⣿⠀⢸⠟⠉⠙⣏⠁⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⢸⣿⣿⣻
⣿⣷⣿⢸⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣤⣾⣿⠀⣿⠀⠿⠾⡇⠀⠀⠘⣿⠀⢸⡄⠐⠚⣛⠳⠂⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⢯⣿⠸⣿⢷⣿
⣿⣿⣿⢸⣿⣿⣿⣷⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣶⣾⣷⣶⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⣛⢉⡴⣿⠈⣷⡇⣿⢸⣿
⣿⡏⢻⢨⣟⢻⣿⣿⢸⡷⢰⣶⣦⣤⣍⣭⣉⣉⠙⡛⢛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡛⢛⡛⣛⣋⢉⣉⣉⣍⣭⣍⣴⡤⣶⢶⣶⣾⣿⣿⢸⣧⣿⢠⣿⣷⡿⠀⣸
⢿⡗⢌⡘⠋⢼⣿⣿⢸⣿⣎⢹⣿⣿⣷⢹⣿⣿⡄⣿⢹⣿⣿⣍⣿⣿⣿⣿⣸⣿⣿⢃⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣧⣇⣸⣿⣿⣿⠛⣾⡷⡟⠘⠃⣁⡤⠞⢹
⣿⡇⢰⡍⠑⠒⢤⣤⣈⡙⠻⠾⠟⠿⠿⣿⣿⣿⡇⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⡿⢿⠏⠿⠙⠛⠛⣉⣀⣤⡤⠔⢲⡟⡇⠀⠘⢸
⣿⡇⢠⣤⠀⠻⠄⢻⠉⣿⠁⣶⠒⠒⢤⣤⡥⢌⣤⣉⣁⣉⣉⣙⣋⣘⣛⢛⣙⠙⢛⢛⣛⣛⣈⣁⣩⣬⣥⣥⣴⣤⠶⠶⢶⠶⣾⠋⢿⣧⡬⢷⣦⢸⣇⣿⢿⣞⣸
⣿⡇⠀⠀⣠⠀⠀⢸⠀⣿⠀⠻⠀⠶⢆⠀⢤⣀⣿⠀⠀⠈⠉⡏⠉⢩⣧⣀⡀⠀⠀⠀⣯⢩⣏⠀⡅⢠⣤⡆⢀⡛⠛⠃⢸⡆⣿⠀⢺⡟⠻⢦⡉⢸⣿⣿⣤⣻⣿
⣾⡇⠀⢶⡈⠁⠀⢸⠀⣿⠀⣰⠀⠀⠈⠳⠆⠀⣿⠀⠀⠀⢀⡇⠀⢸⣯⠈⣁⣀⠀⠀⣏⠈⡋⠈⣇⣈⡉⠙⢋⣁⠀⠀⢸⠇⣿⠀⢸⡿⣶⡛⠟⢺⣿⣿⠈⠛⢹
⣻⡇⣴⡀⢠⡄⠀⣼⠀⣿⠀⣿⠀⢰⡆⣰⡦⠀⣿⠀⠀⠀⢰⡇⠻⠿⠖⣀⠉⣿⠄⠀⣿⠘⣿⠀⢛⠁⠩⣥⠈⠉⠉⠁⢻⠁⢻⠀⠘⡇⠙⠋⡀⢸⡟⡟⠘⠛⠸
⣿⡇⠈⠁⢬⡁⠀⣸⠀⣿⠀⣿⠀⠀⠀⠀⣄⣀⣽⠀⠀⠀⢠⡇⠀⡀⠄⠉⢉⣄⠀⠀⣿⠀⣿⠀⢸⣶⣀⠀⠀⠛⠒⠀⢸⠀⢸⠀⠀⡿⠤⠀⠉⠈⣷⡏⠛⠻⢤
⢿⡇⠀⠻⠄⠀⠀⢸⠀⡿⠀⣿⠀⠀⢶⣀⡈⠉⢹⡄⠀⠀⢸⠃⠀⠛⠒⢀⠈⠁⠀⠀⣿⠀⣿⠀⠀⠁⠀⠸⠧⠤⠄⠀⢸⠀⣸⠀⠀⡇⣤⣀⠛⠀⣿⡇⠰⠆
⢸⡇⠀⠀⠠⣦⠀⢸⠀⣿⠀⠹⠀⢰⡆⣈⠃⠀⢸⡇⠀⠀⢸⡄⠾⠂⠀⠈⠉⡁⠀⠀⣿⠀⢹⠀⠀⡷⠶⣁⡀⠀⠳⠀⢸⠀⣿⠀⠀⠄⠦⢤⣄⠀⠙⡇⠀⡘⢀
⠸⣧⠀⠘⠃⠀⠀⢸⡀⣿⠀⠘⡄⠀⠀⠙⢶⡆⣸⡇⠀⠀⠸⣧⠀⢀⡿⠖⠂⠉⠁⠀⣿⠀⢸⡇⠀⡇⠲⣅⡶⢦⣤⠀⢸⡄⢻⠀⠀⢀⣠⣤⠀⠀⠻⡇⠀⢋⣹
⠀⣿⡄⠀⠀⠺⠆⠈⣧⠸⣇⠀⢣⠀⠛⠂⣀⠈⢹⡇⠀⠀⠀⣿⠀⡈⠓⠀⠰⠦⠀⠀⣿⠀⢸⡇⠀⡧⠰⣦⣤⡄⠀⠀⠈⠀⢸⡀⠀⠸⣄⠀⠶⠀⢸⡇⠀⠈⣿
⠀⠸⣿⡄⠀⠛⠀⠀⢹⡀⢻⡄⠘⡄⠀⠀⣉⠓⠈⣷⠀⠀⠀⢻⡄⠙⢀⠘⠓⣀⡀⠀⣹⠀⠸⡇⠀⢀⠀⣀⠀⠘⠓⠂⢸⠀⠘⡇⠀⠀⠹⣟⣃⣀⠸⣷⡄⣰⡿
⠀⠀⠹⣿⣦⠀⢠⣄⠘⣇⠈⢷⡀⠹⣆⠀⠉⠋⠀⠹⣆⠀⠀⠸⣇⠀⠘⠃⠀⠀⠀⠀⡝⠀⠐⣧⠀⠘⣇⠙⠒⢳⣄⣀⠘⡄⠀⢿⡀⠀⠀⠙⢧⡀⠀⢹⣿⡿⠁
⠀⠀⠀⠹⣿⡳⣤⡀⠀⠹⣄⠀⢳⣄⠙⣆⠰⣆⣀⠀⠙⣦⠀⠀⠹⣷⡂⠀⠰⢄⠀⠀⢿⣀⠀⢻⡄⠀⠸⡄⢠⣄⡀⠀⠀⢷⣄⠘⣧⡀⠀⠀⠀⢉⣹⣿⡿⠁
⠀⠀⠀⠀⠘⢿⣾⣽⣶⣤⣼⣶⣾⣿⢷⣌⢳⡀⣠⡛⠂⠘⢷⣄⠀⠈⠻⣷⣀⠀⠀⠀⢸⣿⣦⡀⠹⣄⠀⢣⡀⢀⡘⠓⠂⠘⣿⣷⣾⣿⣶⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣯⣭⣈⡁⠀⠉⠛⢷⣉⣁⡀⣀⣀⣿⡷⢤⣀⠀⠉⠓⠶⣤⣀⣹⡿⣿⣦⣽⣧⣄⣻⣄⣀⣀⣤⣤⣟⣡⠤⢟⣫⣵⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢾⣯⣟⣛⠲⠶⠬⣭⣍⣉⣉⣙⣁⣀⠉⢙⣿⣷⣶⣶⣿⣿⣷⣨⣯⣉⣥⡭⠽⠿⠿⠛⣋⣩⣥⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⠶⣦⣤⣤⣤⣬⣭⣉⣉⣉⣉⣉⣉⣉⣭⣥⣤⣤⣤⣤⡴⠶⠾⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡼⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢰⠃⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠆⡿⢸⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡇⢰⠀⠇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣿⠘⠀⡇⢸⠀⠠⠀⠈⣷
⠈⢢⡀⡀⢀⣠⠴⠶⣤⣭⣁⣿⣷⣦⣿⢸⡆⢴⠶⠀⢹⡆
⠀⠀⠻⣮⣽⠅⣙⡓⠒⠶⣬⣙⠛⠻⣿⣎⠇⠀⠚⠋⠀⢿⡀
⠀⠀⠀⢺⣿⡠⣄⡙⠲⠤⡀⠈⠙⠦⢬⡛⠻⠶⣤⣉⠛⠘⣧
⠀⠀⠀⠘⣿⣿⣦⡝⢦⣤⡀⢀⡀⣶⡀⠈⠁⠀⠈⠙⠛⠳⣾⣆
⠀⠀⠀⠀⠸⣯⣿⣿⣎⡛⢷⣤⣤⣤⡙⠢⠄⢠⡄⢀⠙⠂⠈⠙⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠻⣿⣍⢿⣷⣍⠻⣿⣿⣷⡄⠀⠹⣿⣄⠀⢀⡀⠀⠀⢉⠙⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣿⣿⡷⣶⣝⠻⣿⣶⣶⣄⡉⠓⠀⠀⠠⣄⠀⠙⢶⣤⡉⠛⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣏⠻⣆⣈⡛⠿⣿⣿⣦⣀⠘⠶⢿⣦⣀⠀⠈⠙⠿⣶⣤⣙⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣀⣨⡙⢿⣷⣮⡛⢿⣿⣷⣦⣤⣉⠛⠿⣦⠀⢀⣤⠉⢿⣧⣀⣉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢻⣿⢿⣶⣝⠳⣭⡀⢝⣻⠿⣿⣿⣷⣄⣈⡀⠀⠁⠀⡀⠙⢿⣿⣷⣤⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢻⣿⣯⡻⣷⠘⢿⣶⣿⣿⣦⡻⢿⣿⣿⣿⣦⡀⠚⠟⠀⠀⠈⠛⠿⣿⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢻⣿⣿⣮⣷⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⠿⣿⣦⠀⢠⣾⠦⣲⣦⣌⠙⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢿⣿⣿⣎⢻⢿⣿⣿⣿⣯⣽⣿⣧⣙⠻⣦⣼⣷⣤⣍⡛⣿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣦⡈⠙⣿⣯⢷⠘⠷⡿⣿⣿⣿⣿⣿⣿⣍⡙⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠘⢷⡀⢈⣿⣮⡣⡀⠓⠊⢿⡿⣿⣛⣿⣿⣿⣶⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠁⠀⠀⠀⠹⣆⣀⠙⢿⣾⡀⠠⠄⢷⠈⠉⠙⠓⠛⠻⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣭⣻⣦⣄⡘⣇⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠿⠿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⢀⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠘⠿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣶⣿⡿⠟⠋⠀⣾⡆
⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⠷⢶⣶⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢸⡿
⠀⢀⣠⣴⣶⠶⠀⢿⣿⡀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⢰⣿⠃⠀⠻⠃⠀⠰⣶⣤⣀
⣴⡿⠋⠁⠀⣀⡀⠈⠻⣿⣦⣤⣀⡀⠘⢿⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⢈⣁⣤⣶⣿⡇⠀⠀⠀⠉⠛⢿⣦
⠻⣷⠀⣠⣾⠟⠁⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣦⣤⣤⣄⣀⣈⣁⣀⣀⣀⣀⣀⣀⣤⣬⣥⣤⣶⣶⣾⣿⠿⠟⠛⠉⠀⠀⠀⠀⣴⡀⠀⠀⣿
⠀⠀⣼⣿⠁⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣿⣷⠐⠻⠟
⠀⠀⢿⣷⣄⣀⣠⣤⣶⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠻⢿⣦⣀⠀⠀⣠⣿⣿
⠀⠀⠀⠙⠛⠛⠛⠛⠉⠀⢻⣿⣦⡀⠀⠀⠀⣰⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡀⠀⠀⠀⢠⣿⡏⠀⠀⠙⠻⢿⣿⣿⠟⠃
⠀⠀⠀⢀⣶⣦⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣿⠿⠋⠀⠙⢿⣷⣄⣀⣀⣀⣠⣾⡿⠋⢿⣿⣦⣀⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⡄⠀⠀⠙⠻⠿⣿⡿⠿⠋⠁⠀⠀⠙⠛⠿⠟⠋⣀⣠⣤⣄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠉⢻⣇⠀⠀⠀⢸⣿⣷
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢀⠀⢸⣿⠀⠀⠀⠸⣿⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢠⣿⠏⠀⠀⣀⠀⠀⣼⡟⠀⠀⢠⣾⠟⠉⠛⣿⡀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠉⠀⢸⣿⠀⠀⠀⠀⣿⣿
⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣾⡟⠀⠀⠀⠉⠀⠀⣿⠇⠀⢠⣿⠏⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿⡇⢀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠘⣿⣿⡇⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⣸⣿⠀⢀⣿⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠉⠀⠀⠀⠀⠈⠁⣀⣀⣠⠀⣿⣿⡇
⠀⠀⠀⣿⣿⡇⠙⠛⠿⠶⣶⣤⣤⣤⣀⣀⣀⣿⣏⠀⣸⣟⠀⠘⠃⠀⣀⣈⣁⣀⣀⣀⣿⣯⣠⣤⣤⣴⣶⡶⠿⠿⠛⠋⠁⠀⣿⣿⡇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⣾⠆⠀⠈⠉⠉⠉⢹⣿⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡏⠉⠉⠁⠀⠀⠀⣶⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⠀⠀⠰⠂⠀⠀⢸⣿⠀⢠⣿⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠈⠁⠀⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣿⣿⡇⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣾⡏⠀⢸⣿⠀⠀⢰⠄⠀⠀⢸⣷⠀⠀⢸⣿⠀⠘⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⣿⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠘⠂⣿⡇⠀⢸⣿⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢸⣿⠀⠀⠀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⣿⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣿⡇⠀⠀⠀⣀⡀⠀⢿⡇⠀⢸⣿⠀⠀⠀⠉⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢻⣿⡇⠀⠀⣿⡇⠀⠀⠐⣶⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠈⠀⠀⠸⣿⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢸⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢹⣷⠀⠀⠀⣀⠀⠀⠀⢿⡇⠀⢿⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⢸⣿⣧⠀⠀⣿⡆⠀⠲⠂⠀⠀⠀⣿⡟⠀⠀⠈⣿⣇⠀⠀⠛⠀⠀⠀⠈⣿⡀⠘⣿⣇⠀⠀⣀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⣿⣿⠀⠀⢻⣷⡀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⢹⣇⠀⠸⣿⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠘⣿⣿⡆⠀⠈⢿⣷⣄⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠘⣿⣦⡀⠀⠈⠁⠀⢸⣿⠀⠀⠙⣿⣆⠀⠀⠀⢸⡏⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣄⠀⠀⠙⠿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⢀⣼⡟⠀⠀⠀⠈⠻⣷⣦⣤⡿⠁⠀⠀⠀⣰⣿⣿⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠁⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢨⠁⡤⠀⠀⠄⠒⡲⠀⠐⠀⠰⠄⠀⡰⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠂⠀⠐⠀⠒⠔⠢⠂⠄⠄⠠⠅⠂⠁⠂⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⠄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢦⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣟⣛⣟⣛⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠈⡁⠰⠤⠤⠤⠤⣄⣀⣄⠁⠀⠀⠀⠀⠀⢉⠀⠂⠤⠤⠤⠤⢀⣠⣌⠀⠀⠀⠀⠀⠀⡁⠠⠄⠤⠤⠤⡄⣀⣤⡁
⠀⠀⠀⠀⡐⠀⢉⠓⠲⠶⠶⠾⠿⠛⠋⡀⢂⠀⠀⢀⠀⠈⡙⠒⠶⠶⠶⠿⠿⠛⢁⠐⡀⠀⠀⠐⠀⢉⠓⠲⠶⠶⠾⠿⠟⠛⣀⢂
⠀⠀⠀⠀⡞⢰⡷⠤⢤⡀⣀⢀⡴⢶⡴⢿⠅⠀⠀⠰⠇⣺⡂⠀⢄⣀⣀⡀⠷⣦⠺⣯⠄⠀⠀⠘⢶⡷⡤⣤⣀⣀⣀⣤⣶⣴⣿⢹
⠀⠀⠀⠀⡇⠎⠀⢢⠊⠀⡷⢦⡀⠸⠄⠘⢱⠀⠀⢰⠈⠉⠉⣲⣴⡿⠏⠀⢀⣷⣷⡟⡆⠀⠀⡦⡿⣷⢹⡿⡟⢻⢹⢨⣷⡯⣿⢸
⠀⠀⠀⠀⡇⠀⣦⠉⣦⠏⠀⣰⠄⣠⡇⢒⢸⠀⠀⢸⠀⢀⣀⣩⠷⠿⢶⣴⡿⢿⣿⣷⡇⠀⠀⡇⢦⡇⣾⢳⢲⢸⠊⣾⣿⣇⣯⢸
⠀⠀⠀⠀⡇⣄⡛⠀⣧⣠⢾⠇⠀⣿⣿⣿⢸⠀⠀⢸⢸⠒⠻⠁⠀⣠⠀⢹⣄⣀⣿⡏⡇⠀⠀⡇⠀⡇⢹⢸⣿⣾⢰⡿⣿⣧⣿⢹
⠀⠀⠀⠀⡇⣟⣇⣴⠟⠃⢀⡆⠀⣿⠀⡿⣼⠀⠀⢸⣸⣄⣀⡐⣀⠀⠀⠞⠻⣿⣿⡇⡇⠀⠀⡇⣷⣧⢠⢬⡋⣿⢸⢧⣻⣿⣯⢸
⠀⠀⠀⠀⡇⣧⣿⠀⢀⣠⠾⠀⢰⣯⠠⣅⢹⠀⠀⢸⠸⠛⣿⢉⣀⣱⣾⡲⣾⣿⣿⡇⡇⠀⠀⡇⡟⡏⡘⡀⠁⣷⢿⣼⣹⣿⣿⢺
⠀⠀⠀⠀⠇⢿⠟⠠⣾⣧⡀⣀⣽⡇⣸⡷⡈⠀⠀⠸⡐⢄⡀⢼⣛⢿⡯⢙⣻⣾⣧⡄⠇⠀⠀⢃⣃⡇⡀⣧⠀⡏⡍⣿⠻⣿⣿⡸
⠀⠀⠀⠀⠈⠂⠠⣄⡑⠒⠉⠉⠉⠉⠉⠐⠁⠀⠀⠀⠑⠀⢩⣈⠂⠓⠶⠿⠿⠛⠁⠊⠀⠀⠀⠈⠃⠙⢕⡛⠚⠉⠴⣿⣿⠛⠕⠁


⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢀⠘⢄⣈⣁⣀⣀⣰⣦⣴⡧⠀⠀⠀⠀⠀⠀⠠⣄⣉⣀⣀⣀⣤⣤⣶⠇⠀⠀⠀⠀⠀⠰⢤⣈⣁⣀⣀⣠⣤⣴⡶
⠀⠀⠀⠀⢪⣀⣀⠈⠉⠉⠉⠉⠉⣁⢐⣤⠄⠀⠀⢠⣀⣠⠊⠉⠉⠉⠉⠉⣉⠀⣰⣬⠂⠀⠀⣆⣀⡔⠉⠉⠉⠉⠉⢉⡁⢐⡤⡅
⠀⠀⠀⠀⡋⣸⡍⡏⠃⣶⡶⣶⡾⠻⡎⣼⡀⠀⠀⢈⠀⠛⢹⡗⡶⣶⣶⠒⣿⣿⣟⡇⡀⠀⠀⡳⠗⠿⣷⢤⣀⠖⠰⢿⣯⣭⣿⠁
⠀⠀⠀⠀⡇⣿⢧⡘⢩⡿⡇⠙⢙⣷⣷⡟⢸⠀⠀⢸⠀⡇⢸⠈⠇⡇⢽⡶⠛⢻⣏⠉⡇⠀⠀⡇⠄⣀⡇⠈⣫⣇⣠⣿⣷⡤⣿⢸
⠀⠀⠀⠀⡇⣿⡈⢹⢴⢰⣧⣼⣸⣿⣿⡇⢸⠀⠀⢸⠸⠁⢸⡄⠀⡇⢸⡇⠀⣿⢹⠘⡇⠀⠀⡇⠌⠉⠙⠴⢮⠀⠈⣯⡝⠀⣀⠘
⠀⠀⠀⠀⡇⡿⡇⣦⣼⣾⠉⣠⢿⡣⣏⡽⢸⠀⠀⢸⢀⠀⢨⡷⠀⢳⠘⣿⠀⡗⢸⡗⡇⠀⠀⡇⠉⢳⣽⠆⠖⠷⢺⠿⣿⣿⢥
⠀⠀⠀⠀⡇⢻⠳⡜⣿⡷⠚⢧⣌⡴⢿⡇⢼⠀⠀⢸⣺⠄⢺⡧⠀⢸⠀⣿⣼⡇⢺⡇⡇⠀⠀⣇⣀⡿⣏⠉⢐⣤⡀⣿⡿⠛⠛⡀
⠀⠀⠀⠀⡇⣏⢰⠁⣿⢙⣠⠞⣼⠙⣹⢧⠸⠀⠀⠸⢻⠾⣿⡄⠀⣸⣾⣿⣿⣿⣿⡇⠇⠀⠀⡟⣷⠀⢈⣻⡀⣀⣸⣿⣿⣷⡆⡇
⠀⠀⠀⠀⠡⡙⠺⢶⣿⣾⠁⡼⢿⣤⡏⢞⠆⠀⠀⠘⡌⠺⢽⢠⣾⡟⣁⠖⣿⣿⠟⡰⠀⠀⠀⢃⠀⣛⠛⠛⠹⠄⢋⣽⣯⡟⢃⠃
⠀⠀⠀⠀⠀⠀⠐⠒⠤⠘⠉⠩⠟⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠯⠄⠉⠩⠥⠉⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠤⠁⠀⠬⠍⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠠⠄⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠠⠀⠀⠐⠈⠁
⠀⠀⠀⠠⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠈
⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠀⠈⢁⠀⠤⠀⠀⠐⠂⠀⠀⠀⠒⠀⠀⠀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠠⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠑⢔⠀⠈⠙⠄⠐⠒⠂⠀⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠇
⠀⠀⠀⠀⡀⠄⠀⠀⠒⠂⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠌⢀⠃⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⢀⡠⠊⡠⠂⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠈⡁⠀⠒⠂⠒⠀⠈⠁⢀⠀⠔⠉⠒⠂⠡⠀⠘⠄⣀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠈
⣤⣤⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣄⠀⠘⠲⠶⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠁⠀⣸⠇⠀⠀⢠⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠴⠚⠁⠀⠀⠀⣿⠀⠠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢰⡇⠀⠀⠀⠀⠀⡞⠁⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠈⠳⣄⣀⣷⠀⠀⠀⡀⠀⠙⠶⠊⠙⠋⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⢢⣼⣠⠃⠀⠀⠀⠀⠈⠀⠀⠀⣠⠏⠳⡄⠀⠀⠀⠀⠀⣮⠽⠃⢿⣳⢰⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢯⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠿⡗⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣅⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠒⠂⠀⠀⠀⣿⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⣠⠖⠚⠢⠤⠤⠤⠤⠤⠤⠤⠔⠒⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢀⡖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠖⠒⠒⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⠀⠀⠸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠄⠒⠒⠒⠚⠛⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⣤⠤⠴⠚⠋⠷⣤⣤⠴⠓⠦⢤⡤⠴⠛⠓⠒⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣏⠉⣿⠀⠀⠀⠀⠀⢠⠞⠛⢣
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠘⠦⡤⠞⠀⢀⠀⠀⠀⠀⢀⡞⢳⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠸⠤⣿⡀⣧⠞⠀⠀⠀⠀⠀⠀⠀⢰⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡿⠀⠀⠀⠈⠳⣬⢶⣯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡄
⠀⠀⢀⣀⠤⠴⠒⠒⠲⢤⣀⢠⡴⠶⠶⠒⠒⠒⠲⡆⠀⢸⡇⠀⠀⢹⠀⠀⣠⠴⠚⠋⠉⠛⢲⡴⠏⠉⠉⢳⢘⡋⠉⠉⠉⢹⠀⢠⡟⠃⠀⠀⠸⠃
⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠘⣾⡄⠀⠀⠀⠀⠀⠀⢳⠀⡸⠀⠀⠀⡸⢀⣾⠃⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠸⡇⢸⡇⠀⠀⠀⣤
⢸⡆⠀⠀⠀⠀⣤⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣤⡴⠞⠀⡧⠀⠀⠀⣧⡞⠡⠀⠀⠀⠀⠀⡞⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⡄⠀⢀⡇
⡎⢃⠀⠀⠀⠀⠀⠀⠀⣤⣾⡁⣶⠀⠀⠀⠀⢸⣇⣀⣸⠃⠀⠀⠀⣿⠀⢀⡄⠀⠀⠀⠀⠙⠒⠙⡇⠀⠀⠀⣠⢤⡀⠀⠀⠀⣧⠸⣧⣗⣲⡾⠁
⣇⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠈⠉⠉⠏⠀⠀⠀⠀⠉⠁⠈⠉⡂⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡆⡼⡏⠉⣳
⠸⡄⠻⠀⠀⠀⠘⠛⠀⠀⠀⢀⡿⠀⣀⣀⣀⣀⣀⣀⣰⠀⠀⠀⠀⠀⠀⠀⢈⠙⠢⢄⣀⣀⣀⡠⡴⠦⠀⠐⣧⠀⠀⠒⠈⠉⡟⠙⠧⠼⠛⠁
⠀⣧⠰⢇⠀⠀⠀⠀⠀⢀⠰⣻⠁⠉⠉⣉⣉⣉⣠⣿⠏⠁⠈⠉⠉⠉⠉⣩⡟⠶⣤⣀⣀⣀⣀⡴⠶⠶⠶⠶⠋⠳⠶⠚⠛⠛⠁
⠀⠸⣆⡀⠉⠉⠀⠉⣉⣠⠞⠛⠛⠛⠋⠉⠉⠉⠉⠘⠓⠛⠛⠛⠉⠉⠙⠉⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⣤⡴⠶⠶⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡖⠳⢶⠋⠉⠀⠀⣠⠞⠋⣠⣤⠴⠦⣤⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠙⣆⠀⠀⠀⠀⠘⠁⢠⠞⠁⠀⠀⠀⠀⠉⢣⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡦⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣶⡆⠀⠀⠀⡇⠈⡷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠉⠁⠀⠀⢀⠇⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠤⠤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠐⠀⠀⠀⠀⠤⠀⠴⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⣠⡴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⣀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡄⢠⡾⠉⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⠚⠁⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢸⠀⠀⢀⡔⠛⠒⠦⣄⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠑⠀⠀⢀⡀⠀⢰⡂⠀⠀⠀⠀⠀⣼⠀⢠⠏⠀⠀⠀⠀⠀⠙⢦⡀⠀⢠⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣄⠀⠀⠀⢰⡏⢹⠀⠀⠁⠀⣀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡘⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡿⢹⡆⠀⠀⣿⠀⢸⠀⠀⠀⣼⠉⢳⡀⠀⡿⠀⢸⠀⠀⠀⠀⣀⣀⣀⣠⣤⣼⣿⠀⠀⠀⠘⠛⠛⠓⠶⢦⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⡇⢸⡇⠀⠀⡟⠀⢸⠀⠀⢠⡏⡠⠚⠧⡼⠃⠀⢸⢉⡏⠉⠉⠉⠉⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠠⣤⡤⠞⠁
⠀⠀⠀⠀⠀⢾⣋⠉⠀⠀⠀⠀⠀⠀⠈⠁⠟⢸⡇⠀⢰⡇⠀⠈⢇⣀⡼⠀⠓⠙⠳⠖⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣤⣤⠟⠳⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠉⠛⢶⡄⠀⠀⠀⣠⠤⠤⠦⠀⣇⠀⢸⠓⠢⡀⠈⠉⠀⡤⠤⠀⢀⡴⠀⠀⠸⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⣀⣠⠤⠴⠏⠀⠀⠀⠙⠲⠂⢤⠀⢿⣤⠞⠓⠋⠀⠀⠀⠀⠓⠒⠒⠒⠲⡆⠀⡖⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣄
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣄⠀⠀⠀⠛⢶⡄⠀⠈
⠀⠀⠀⠀⠀⠛⠦⠤⠤⠤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠾⠤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠶⠶⠦⠤⣤⣤⣀⣀⣀⠀⠀⠀⠀⢀⣠⡿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢀⣀⣀⣀⡈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠴⠦⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⣀⣤⣶⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢀⣴⣿⠟⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢠⣿⡟⠀⠀⠀⠀⢈⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⢍⣵⣿⠟⠁⠀⠈⢻⣷⡄⠐⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣿⢟⡁⠀⢀⣄⠀⠀⢿⣷⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢟⣵⣿⠟⣥⣾⡇⠀⠸⡏⠀⠀⠘⣿⡇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢟⣵⣿⠟⣥⣿⣿⣿⡇⠀⠀⠃⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⣵⣿⠟⣡⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⣿⡆⠀⠀⠀⠀⠀⠸⢟⣵⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⡇⠀⢀⡄⠀⠀⢸⣿⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢻⣿⡀⠀⠀⠀⢀⣴⣿⠟⣑⣛⣛⣛⣛⣛⣛⣛⣛⣛⠃⠀⠀⠀⠀⢠⣿⡟⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢻⣿⣄⢀⣴⣿⠟⣡⣾⣿⣿⣿⡱⢎⣿⣿⣿⣿⣿⠇⠀⠀⠀⣠⣿⠟⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⢿⣿⣟⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⣼⡿⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡿⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣶⣶⣶⣶⣶⣿⠿⠟⠋⠁⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠰⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣶⡰⣄⣆⡖⢦⣤⣶⢢⡀⣶⢲⣰⢰⣂⢰⡖⣶⡦⢰⣆⣰⠒⠰⡖⣦⡴⢲⣰⣆⡆⣖
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠛⠈⠓⠋⠋⠛⠊⠀⠛⠊⠘⠐⠊⠈⠃⠋⠙⠋⠙⠘⠒⠈⠃⠋⠙⠚⠘⠈⠃⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⢀⢤⡀⠀⠀⠀⠀⢀⣀⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠎⠀⠀⠀⠀⣇⣈⡇⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠰⠭⣆⢼⠝⠀⠀⠀⠀⠀⣔⡦⠀⠀⠀⠀⠀⠀⠁⠀⠾⠆⠀⠀⠸⢤⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡤⠤⢤⠀⠀⠀⣀⣀⣀⡀⣀⣤⣤⠀⡤⠤⠤⡄⠀⣠⠖⠂⢰
⢀⡤⠒⠉⠁⠀⠉⠳⣼⠉⠉⠁⠀⠈⡅⢰⠁⠀⢸⢀⡴⠋⠀⠀⠀⠛⠁⠀⠈⠶⠂⠀⠀⢸⠀⡏⠀⠀⠈
⢸⠀⠀⠀⣤⠀⠀⠀⣿⠀⠀⠀⢠⡤⠇⣸⠀⠀⢸⡞⠁⠀⠀⠀⡔⠋⡀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡆
⡆⠆⠀⠀⠀⠀⠐⠺⢤⡆⠀⠀⠠⠧⢤⡇⠀⠀⢸⡄⠤⡀⠀⠀⠈⠉⠙⠀⠀⠰⠲⠀⠀⠀⡆⢳⡧⢞⠁
⢣⠸⠀⠀⠀⠦⠀⠀⢈⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠧⢀⠀⠀⠀⣠⡄⠀⣤⠀⠡⠠⢐⠇⣏⡤⠞
⠘⡄⢆⣀⠀⢀⡀⠄⣾⣉⣉⣉⣉⣩⣿⠓⠒⠒⣂⣀⡸⠳⢤⣀⣈⡠⠶⠶⠶⠊⠲⠒⠒⠊
⠀⠑⠦⠤⣤⠤⠴⠊⠉⠉⠁⠀⠉⠀⠈⠉⠉⠁⠀⠉⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢤⡴⠚⠉⢉⡽⠛⣉⣁⣉⡉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠙⡆⠀⠀⠀⠋⡰⠋⠁⠀⠀⠉⢂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⠃⠀⠇⠀⠀⠀⠀⡅⠀⠐⠿⠀⠀⢸⠀⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⠄⠠⠄⠊⠀⠀⠀⠀⠀⠙⢤⣀⠀⢀⡠⠊⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠈⢳⡀⠀⢀⡴⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠔⠒⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡴⠁⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡦⠂⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠁⠀⠀⠀⠆⠀⠀⡀⠀⠀⢀⡁⠀⠀⣈⣀⠀⠀⠀⠀⠀⣰⠏⠀⡀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠆⠀⠀⠀⢀⠄⠀⠀⠀⢨⡀⣠⠌⠀⠉⠁⢠⣀⠀⣠⠁⠀⠃⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⠀⠀⢀⠞⡇⠀⠁⢀⡀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠈⢆⡏⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣸⠁⡇⠀⢸⠀⡷⠀⠀⠏⢹⡄⡸⠀⣧⣤⣤⡤⠤⠤⠤⠼⠷⠀⠀⠉⠉⠙⢓⡆
⠀⠀⠀⢠⡶⠒⠒⠒⠒⠒⠒⠛⡆⡇⠀⣿⠀⢡⠀⡸⠰⠡⢍⡇⠀⠰⠊⠀⠀⠀⠀⠀⠀⠣⣀⣀⡦⣌⡽⠋
⠀⠀⠀⠀⠉⠓⢦⠀⠀⢀⠤⠤⠁⢇⠀⡗⢢⠀⠉⠁⠤⠤⠤⠆⠀⠰⠀⡄⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⢠⠔⠒⠊⠀⠀⠈⠁⠐⡆⠸⠖⠉⠁⠀⠀⠀⠉⠉⠉⠳⠀⠋⠉⠁⠀⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⢠⣖⠒⠲
⠀⠀⠀⠸⠤⠀⣀⣀⣀⣀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⣼⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠚⠁

⠀⠀⠠⠒⠉⠉⠒⢒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⣆⡀⠀⠀⡀⠀⢀⠦⠛⠓⠤⡀⠀⠀⠀⠀⠀⠀⠤⠖⡀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠸⣤⣄⣀⠀⢰⠂⢀⠀⠀⠀⠘⡆⠀⢀⡎⠄⠀⠂⠄⡈⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠔⢁⠃
⠀⠀⠀⠙⢝⣪⣁⡆⠀⠄⠀⠀⠁⠀⠇⢀⠎⠀⠀⠀⠀⠀⠈⣦⡤⠘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⣀⣄⣘⠐⠁
⠀⠀⠀⠀⠀⠙⠾⣇⠀⠀⠀⠀⢀⠜⠀⡄⠀⠀⠀⠀⠀⠀⠀⡛⠉⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⢠⠆⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠐⠚⠁⠀⠀⠷⢀⠀⠀⠀⠀⣠⠔⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⣠⠔⢫⠿⠋⠁
⠀⠀⠀⠒⠒⠒⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠋⠁⠀⠀⠀⣀⣀⣀⢠⣤⡄⢀⡀⠂⠃⠁⠀⡀⠀⠤⠀⠤⢀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠀⠠⠤⠀⠀⠀⣀⡀⠀⠀⠈⠙⠀⠀⠀⠀⠁⠀⢀⠔⠊⠉⠉⠂⢄⠀⠀⠀⡁
⠀⠀⠘⢄⣔⡆⣀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠤⠄⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⡐⣤⠃
⠀⠀⠀⠀⠉⠂⠙⢂⡖⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡀⣀⡰⠀⠀⠇⠀⠀⠀⠠⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠪⠤⣬⣷⣶⡦⢤⣬⣤⡤⣤⣦⣴⡶⢶⡾⠿⠛⠉⠀⠀⠀⠈⠤⠀⠀⣀⣀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠈⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣸⡏⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠶⠒⠲⢤⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠲⢤⣀
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⢀⣀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⣼⠁⢠⠖⠛⠋⠉⠉⠛⠓⢲⡄⠘⣆⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢘⡷⣄⠸⡇
⠀⠀⠀⢀⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⣆⣀⣤⠶⠴⠶⠞⠁⠘⣦⡟⠀⠀⠀⠀⠀⣀⠀⠐
⠀⠀⠀⢸⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⣀⠉⠀⠀⠀⠀⠀⠀⢀⡟⠀⠠⠀⠐⠃
⠀⠀⠀⠀⣇⠀⠀⢳⡀⠀⠀⠀⠀⢠⡟⠀⠀⣸⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⠀⢀⡀⠀⣀⠀⢀
⠀⠀⠀⠀⢻⡀⠀⠀⣿⢦⣄⣠⠴⡏⠀⠀⢠⡏⠀⠀⠀⡀⠀⠀⠀⣀⣤⣤⣤⣼⠷⣄⡀⢀⣠⢾⣥⣀⣀
⠀⠀⠀⢀⡤⠷⠤⣶⠿⠀⠀⠀⠀⠷⣶⠤⠿⣤⡀⠀⠀⡇⠀⢠⡞⠁⠀⠀⠀⠹⡄⠈⠉⠉⣿⠀⠀⠀⠉⢳⡀⠀⠁⠀⢤
⠀⠀⠀⡏⠀⠀⠀⠹⣄⠀⠀⠀⠀⣠⠇⠀⠀⠀⢹⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠙⠦⣤⡴⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐
⠀⠀⠐⡇⠀⠀⠀⠀⠈⠓⠶⠶⠚⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⠇
⠀⠀⠀⡇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢸⠀⠀⡇⠀⢻⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣼
⠀⠀⠀⡇⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⡇⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠃⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁⠠⠤⠀⠀⠶⠶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠑⡀⠀⠠⠫⠹⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡎⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⡙⢿⣿⠀⠀⠀⢀⣀⠰⠄⠚⠻⡃⠀⠈⠁⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣀⠀⠀⡀⢀⠄⠀⢾⡿⠢⡟⠀⠐⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠚⠀⠀⠈⠀⢰⡇⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠣⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣀⠀⢀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠁⠀⠀⠀⠀⠀⠐⠀⠀⢠⠀⢨⠓⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠄⠁⠀⠀⠱⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⡠⢤⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠔⠀⠀⠀⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡧

⠀⣼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠄⠀⢀⡀⢀⠀⡀⠀⠐⠂⢀⢀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠃⠐⠀⠀⠀⠀⠀⠙⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡀⠀⠀⠂⠘⠂⠒⠒⠊⠓⠘⠀⠃⠚⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣦⠀⠠⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⡃⠀⠀⠀⠀⠀⠿⠇⠠⠭⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⢶⢤⣤⣿⣼⠃⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⢀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠠⣦⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⠆⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢶⠖⠒⠒⠂⠒⠆⠀⡀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠒⠠⠋⢻⠀⠀⠀⠀⠀⡒⠺⠋⠂⠰⣶⢤⣀⣶⣶⢚⣶⣛⣦⣤⣄⡄⠀⠙⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠺⡖⣤⣤⡄⠈⣣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢙⡄⣈⣼⣷⣼⣺⣇⠝⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢘⣷⣿⣿⣿⣿⣿⣿⣧⡀⠀⠸⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣶⣿⠿⣿⠏⠁⣽⠻⣿⣿⣿⣷⡀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠉⠀⠀⠀⠠⠀⠀⠀⠩⠛⠋⠙⠃⠀⠸⣷⠀⠀⠀⠀⡐⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣠⠊⣽⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠌⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠉⢀⢠⠆⢈⣀⡀⢈⠛⠚⣡⠞⢹⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⢀⠀⠊⢀⠞⠀⣀⠾⠚⠛⠿⡖⢆⡀⢀⡀⣠⣶⣿⣿⣿⣷⣿⣧⣤⠏⠀⠈⢧⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠛⠀⢰⠇⠀⠀⣀⠀⢻⣤⣧⣤⣿⠟⠉⠋⠛⠿⣿⣿⡏⠀⠀⠀⠀⠀⣿⡼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡆⢀⣀⠀⠀⣴⢺⣿⣿⠋⠡⠔⠁⣀⠀⢉⣿⡇⠀⠀⠀⠀⠈⣿⠃⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢡⡈⠉⠀⢀⣿⡿⡟⣧⣜⠲⣤⡄⢨⣤⢞⣿⡇⠀⠀⠀⠀⠀⢷⣄⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣠⡴⠋⠑⠒⠊⠘⢷⣿⠟⠻⠦⢠⡶⠛⡫⠀⣿⣄⠀⠀⠀⠀⠀⡞⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⡷⢶⡆⠙⠻⡆⠀⠀⠀⣠⣿⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠁⠈⠀⠀⠀⣦⠐⠆⠀⠀⠀⠀⠠⠤⡄⢀⠠⠁⠀⢹⡆⠘⣿⠀⠀⢰⢿⡏⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⢄⠀⠀⠀⢸⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡆⠀⠈⣿⡖⣿⡶⠏⢁⣿⣷⣦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣳⣿⣡⣠⠚⠃⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⣼⡿⠁⠀⠀⠒⠀⠀⠀⠀⠀⢸⣿⣏⡀⣒⣟⡥⣮⡼⠁⠀⠀⢿⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣼⠀⠀⠈⠁⢀⡤⠊⠀⠀⠀⠀⢠⠄⡈⣙⣿⡇⠉⢩⡄⣾⠂⠀⠀⠀⠀⠹⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠰⣯⠾⢻⡀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠐⠋⠃⣾⣷⣥⣠⣀⠀⠀⠀⣼⠏⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡂⢠⣏⣽⡆⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⣲⣶⣿⣿⣿⣷⣬⣷⣄⡀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢠⢲⡏⠁⠀⠈⠙⠛⠳⢾⡧⣤⣀⣄⠀⠀⢘⣷⣤⣷⣶⣿⡿⠿⠛⠉⠉⠋⣿⣿⣧⣟⣻⡿⣿⣫⣾⡇
⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠈⣶⣽⢦⣤⣀⣀⣄⠀⠀⠀⠐⠭⡛⢟⣋⣩⣥⠭⠉⠋⠁⢠⣤⣤⣾⣔⣤⡿⣿⣷⣿⣿⣷⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠚⠒⡅⠀⠛⠈⠉⠀⠀⠀⠈⠀⠀⠈⠙⠦⣤⣄⣀⠈⠛⠿⣿⡶⠶⠖⠛⠛⠛⠉⠙⠉⠉⠉⠀⠐⠛⠛⠛⠛⣏⣉⣉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡞⠁⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣌⠻⣦⡀⠀⠀⠀⠀⢀⡴⢋⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣌⠻⣦⡀⢀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣌⠛⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⠘⡇⢀⡴⢋⡴⣌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢋⡴⠋⠀⠈⠳⣌⠻⢦⡀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡴⠻⣄⠀⠀⠀⠀⠈⠓⢌⠛⡄⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⣠⡤⢼⠀⢸⡀⠀⠀⠀⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⡤⢾⡇⠀⠸⠀⠘⢧⡀⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⢀⡞⢹⠀⢸⡇⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⢸⡂⠀⠀⠀⠀⠀⠀⠀⢄⣀⣄⣗⣵
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢲⣄⠀⠀⠀⠀⢀⠀⠀⢸⢳⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣭⣿⠇⠀⢐⠂⠀⠐⣦⡀⠀⠑⠒⠾⠏
⠀⠀⠀⠀⠀⠀⠲⣤⣴⣼⣾⠀⠀⠀⠀⠈⠻⢯⣯⣿⠆⠀⠀⠀⠀⢠⠐⠒⠐⢄⠀⠀⠀⠀⠸⡀⣀⣄⡇⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢱⢦⡀⠀⠀⢀⠀⠄⠀⠠⠄⢀⠀⠈⣄⡄⠤⣏⣗⠀⠀⠀⠀⠈⠲⠶⠽⠿⠀⠀⠀⠀⢀⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⢼⣺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠑⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠑⠁⠀⠀⢀⡠⣤⣤⣄⡀⠀⠀⠀⠑⡄⠀⠀⠈⠀⠀⢸⠦⡄⠀⠀⠀⠀⣠⠤⣐⣶⠀⠀⡀⡄⢒⡩⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣴⡵⠛⡉⠁⢉⠙⠒⢄⠀⠀⠈⠢⡀⠰⢤⠴⡤⣔⡿⠁⠀⠀⠎⡵⠚⠉⠘⢦⣩⡵⠞⠋
⠀⠀⠀⠀⣀⣀⣀⠀⠀⡀⡀⠀⣀⣀⣠⣤⠷⠛⠀⠀⠀⠀⢀⠟⣆⠈⠢⡀⠀⠀⠈⠢⢀⣉⠹⠋⠀⠀⠠⣪⡞
⠀⠀⠀⠀⠉⠘⠛⠓⠛⠒⠚⠚⠛⠋⠋⠁⠀⠀⠀⠀⢶⡲⠖⢷⡳⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀⢄⠵⠋⠒⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠉⢆⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠈⠉⠛⠃⠀⢀⠀⣀⡈⠒⢤⣀⣀⡤⣚⠴⠚⠀⠀⣀⣀⣀⣡⣷
⠀⠀⠀⠀⠀⠐⢦⠴⠰⣟⠇⠀⠀⠀⠀⠀⠀⢘⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⢦⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠻⠼⠿⠯
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠰⢤⠴⠶⣽⠆⠀⠀⠀⠀⠀⠀⠰⣤⠴⠼⣾⠅⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢐⢄⠀⠀⠀⠋⠉⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⢀⢈⠙⡄⠀⠁⠀⠀⠀⢽⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⠤⢼⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡳⡄⠀⠀⠠⠄⠀⠠⡄⠀⠀⠙⠭⣁⣈⣿⠇⠀⠐⢦⠠⠦⢶⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠘⠲⣒⣒⣧⡿⠀⠀⠀⠀⠀⡀⡏⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠙⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠙⢭⣝⣷⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛
⣤⠤⠦⠤⢤⡀⠸⡧⣾⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⢸⡇⠀⠸⡳⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢹⡄
⣿⠀⠀⠀⠀⠙⣦⡟⢳⣠⠞⠁⣤⡏⢀⡔⠋⠁⢸⣷⠶⣦⣰⠲⣆⣞⡉⢩⡿⢾⠈⣛⣷⣰⠋⢻⣤⠖⢶⠖⠒⢦⠀⣠⠴⠟⠉⢻⠀⡇⢰
⣿⠀⠸⠟⠀⢠⠙⠃⢸⠛⠦⠤⡄⠈⣾⡀⠻⠃⠀⣿⡀⠉⠁⠀⡏⡴⠚⠀⢙⠸⡄⠳⠶⣤⡀⠈⡇⡄⢰⢲⡀⠸⡏⢧⠀⠛⠀⢸⣠⣟⣿
⢿⣀⣀⣀⡤⢋⠀⠧⠼⢃⡏⠉⠁⣀⢇⠙⠚⡆⠀⡇⠙⠦⡤⠞⠀⠓⠒⠒⢋⡀⠙⠒⠚⠃⠙⠉⣀⠉⢉⣈⠙⢛⣁⣌⠉⡇⠀⠘⡏⢣⡼
⡼⠮⠭⠤⠔⠋⠲⠶⠞⠦⠭⠭⠭⠕⠋⠙⠒⣇⠀⢻⠓⠦⠤⠴⠋⠑⠒⠒⠋⠙⠒⠖⠊⠉⠙⠋⠉⠋⠋⠈⠉⠉⠀⢸⠀⡇⠀⠀⡟⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠤⣼⣁⡇⠀⢸⠃
⣷⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⠶⠋⠀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡾⣏⠀⠈⠉⠀⢀⣾⡀
⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠁⢠⠟⣗⠦⠤⠤⢞⣿⡀⣠⢄⠀⠈⢠⣄⠀⢀⠀⠀⢀⢦⣀⠀⡄⠉⢀⣀⣈⣃⣈⣉⣃⣒⣾⠿⢹⡇
⢸⡀⡠⣄⠀⢸⠁⡇⠀⢰⠻⡄⢸⠀⠀⠉⡟⠋⢉⡇⠓⠃⠸⡆⠄⡏⠈⠓⠋⠹⡆⢸⠀⠀⠀⣿⠀⢸⠉⠈⠉⢸⡇⠀⣿⢹⡀⢨⠇
⠀⠉⠀⢸⡀⣸⠀⢹⣀⡟⠀⠳⠎⠀⠀⠀⢳⠀⢰⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⡀⡸⠀⠀⠀⠘⣇⠀⡏⠀⠉⠉
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠎⠀⠀⠀⠀⠳⡴⠁⠀⠀⠀⠀⠧⠏⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣤⠶⠖⠲⢶⡤⠶⠚⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠿⡉⠙⠏⠀⠀⢀⡴⠋⣠⠴⠒⠛⠓⠦⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣀⠀⠀⢹⡄⠀⠀⠀⠈⠀⡜⠁⠀⠀⡀⠀⠀⠈⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠿⠂⠀⢰⠃⠀⠀⠀⠀⠀⡇⠀⠀⠸⠿⠂⠀⠀⣸⠀⠟⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠁⠂⠀⢀⠀⢀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⢻⡄⠀⣠⠞⠋⠁⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢿⠀⢀⠔⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡴⠃⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠈⠳⣬⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⣤⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡆⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⡀⠀⠀⠈⠀⠀⠀⠳⠀⠀⠀⠀⠃⠀⠀⡠⠛⠲⢤⣀⠀⠀⠀⠀⢀⡞⠁⢀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠀⠀⡀⠀⢰⠂⠀⠀⠀⠀⢀⡇⠀⡞⠁⠀⠀⠀⠈⠓⢄⠀⠀⡾⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣤⡀⠀⠀⢠⡞⢹⡀⠀⠁⢀⣀⠀⠀⢸⡇⢠⠁⠀⠀⠀⠀⠀⠀⠀⠳⣠⡇⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⠇⣷⠀⠀⣸⠀⢸⠇⠀⠀⡞⠉⣧⠀⢸⠃⢸⡀⢀⣀⣀⣀⣀⣀⣠⣤⡿⡇⠀⠀⠈⠙⠛⠓⠶⢦⡀
⠀⣀⣤⣤⣤⣤⣤⣤⣤⣭⣿⡄⣿⠀⠀⣿⠀⢸⡆⠀⢰⢃⠖⢙⠶⡏⠀⠸⣉⡏⠉⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠀⠰⢦⡶⠚
⠘⢏⣀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢿⠀⠀⡏⠀⠀⢧⣠⠞⠈⠉⠙⠒⡇⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡴⠞⠙⠒⠛⠁
⠀⠀⠈⠑⣦⠀⠀⠀⡴⠒⠒⠃⢸⡀⢀⣏⣙⠄⠀⠀⢠⡖⠲⠤⠴⠇⠀⠀⠓⠒⣦⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⣠⠤⠖⠚⠋⠀⠀⠀⠙⠛⠓⡆⠘⠷⠞⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⣳⠀⢟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⣴⠒⠚⠲⢤
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⣳⠀⢀⣸
⠈⠉⠉⠙⠛⠓⠒⠲⠶⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠟⠀⠀⠈⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣛⣓⣲⣶⣶⣤⣤⣤⣶⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣋⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡶⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⢀⣀⣑⡂⢀⣐⡀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣉⡋⣠⠄⠤⢄⡙⠳⠚⠋⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠈⣓⡉⣦⠀⠐⠠⠛⡦⢤⣇⡄⠐
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡴⠉⠀⠀⢹⡔⠈⠉⠐⢧⠟⠉⠑⠀⠄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠶⡀⠀⠀⣸⡇⠀⠀⠀⡘⢦⣀
⠀⠀⠀⢠⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡶⢬⠂⠙⣧⠻⠆⠴⠚⠋⣤⠜⠙⠛⢦
⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠒⠄⠸⣧⠀⠐⠀⠓⠁⠳⢀⡠⠄⠢
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢉⣉⣏⣓⢶⡆⡤⢤⠐⠆⠙⠋⠓⢠⠌⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠃⠈⡀⢨⢛⠉⠀⠈⠑⣤⡀⠰⡏⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡋⠙⢤⠋⢡⡼⠛⠳⣴⡉⢠⡽⠋⢻⡄⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠟⠛⢫⢀⠀⠀⠡⠓⣼⣀⣠⣬⡅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠁⠀⠀⠈⣠⠠⢦⠤⠐⠫⣙⡄⢋⡡⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠈⡆⠚⠉⠒⠉⡡⢼⡦⠚⡠⠾⠋⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⣿⣘⣉⠁⢊⡉⣍⠑⢖⣢⣴⣲⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⣀⡴⠁⠀⠈⣃⠈⠑⠋⣀⡏⢴⠋⣷⣲⣃⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡁⠀⠀⠀⠀⠉⢿⢠⠌⢉⣭⠛⠟⠙⠋⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠓⢨⠤⡴⠊⠉⡷⠒⠁⠛⣷⡒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡞⠃⠀⠀⠀⠠⣟⠂⠔⣀⡴⠋⠉⠈⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠈⡗⠤⠀⠀⠀⡀⣼⡠⢤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⣷⣮⠓⠺⣁⠇⠀⠀⡛⡾⠷⢦⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣡⣤⣼⠿⠛⢩⡇⠀⠐⢞⠁⠠⠾⠉⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⡷⠄⣭⠁⣀⣴⡋⢈⡷⠤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠺⠄⠀⢠⡷⡤⠒⠋⠡⠄⠉⣿⠀⠀⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠤⢰⣌⠀⠀⠀⢰⡎⣙⠁⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠉⠓⠋⠀⢹⡴⠂⠏⠁⡀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠤⡠⡜⠁⠀⠀⠀⠀⣥⠄⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠶⣄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟

⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⢤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⣀⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡉⠹⢿⣿⣿⢿⣭⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⢿⣿⣿⠉⠙⠛⠻⡿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠐⠛⠛⠛⠉⡛⠛⠛
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⣶⣤⣤⡀⠀⠀⠀⢼⣿⣿⣟⠛⠛⠻⣿⣿⣿⡁⠈⠻⣿⡷⢬⡛
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠃⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⡗⠁⠤⣨⣿⣿⣿⠄⠀⢩⣝⠿⣶
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣴⠿⠛⣻⠿⠿⠿⠿⢿⢹⣿⣿⣿⣿⣯⠀⠀⣸⣿⣿⣿⡃⠀⠀⣉⣧⣴
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣠⣴⣤⣴⣿⣦⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⡅⠀⠀⠀⣩⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣶⣿⣿⣿⣿⡿⢻⣿⠏⠙⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⢽⣿⣿⣿⡇⠀⠀⠀⢀⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢠⣿⣿⠛⠿⠿⠋⣠⣾⠋⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠘⣿⣿⣿⡇⠀⠀⠀⢠⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣶⣾⠿⠛⠁⢀⣴⣿⣿⣿⣿⣿⡿⢊⣿⣿⣿⠶⠦⢻⣿⣿⡇⠀⠀⠀⢸⣿⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣿⣿⣿⡟⠉⠉⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡛⠛⢤⣿⣿⡟⠀⠀⢿⣿⣿⠅⠀⠀⠀⢸⣿⠄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣻⣹⣰⣿⡇⠀⣹⠆⠀⣸⣿⡟⠀⠀⠀⠀⠐⣿⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠈⠠⠀⠀⣿⣿⠃⠀⠀⠀⠀⠨⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⣐⣿⠄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢲⣶⣾⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠛⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⠀⠀⠀⠀⠀⠀⢿⢿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣦⡑⠀⠀⠀⠀⠀⣀⣴⡾⠟⢻⣿⣿⣿⣀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡀⠀⠀⢰⡾⠋⠀⠀⠀⠘⢻⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠿⣶⣄⠘⠇⠀⠀⠀⠀⠀⢀⣿⣿⡿⠇⠀⠀⠀⠀⠀⢀⣄⣴
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠈⠋⡀⠀⠀⠀⢀⢠⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⢀⣤⣶⡿⡛⢛⣩⡄⠀⠀⣀⡀⠀⠀⣾⣿⡿⢟⠻
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣆⠀⠀⠻⠋⢩⣽⡟⣱⣿⣿⠇⠀⢠⣻⣿⣤⣾⣿⡿⠃
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢿⡽⢿⣿⣿⠿⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢀⣤⣿⣛⣫⠬⠉⠉⢛⠀⠙⠻⠛⠛⠋⠁⠙
⠀⠀⢸⣍⣁⣾⣿⣿⣿⣞⣣⣴⣾⡿⠋⠀⠐⢿⣿⣿⣿⣿⠟⠛⠁⠉⠀⠀⠒⠈⠝⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⢸⣿⣿⡿⣫⣿⣿⣿⡿⠟⠋⠀⠀⠀⢰⣾⣿⠟⠉⠃⠀⠀⣀⡤⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡉
⠀⠀⣸⣿⣿⣼⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠈⠷⠄⠀⣠⣤⣶⠾⠋⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛
⠀⠀⢿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠐⠟⠛⠋⠉⠉⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⡘⠻⡆⠀⠀⠀⢥⡀
⠀⠀⠻⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⢻
⣀⣠⣤⣤⣀⡀⠀⢸⣇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⠀⠀⢸⠧⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡆⠀⢸
⢿⠀⠀⠀⠀⠉⢳⣈⣷⣾⡀⣠⠞⠋⠙⡗⠀⢀⠴⠞⠛⢳⣀⣤⣄⢠⣤⣤⢀⡖⠛⠛⣶⣸⠇⠻⢤⡄⢸⠷⢿⡀⣠⣤⣄⡤⢤⣀⠀⠀⢀⣠⡖⠚⣷⠈⡇⠀⣿
⢸⠀⠀⣖⡄⠀⠈⠿⡇⠈⡷⣇⠀⠘⠛⠳⣤⡏⢠⣤⠀⢸⣿⠈⠻⠼⠀⢸⠞⣓⣢⠈⠳⡸⡇⢴⣒⡓⠛⠀⠈⡏⡇⠈⢁⡀⠀⢹⡶⢾⠉⠡⡆⠀⢻⠀⣧⣤⡏
⢸⠀⠀⠉⠀⠀⡼⠀⡇⠀⡇⢈⣍⣩⠆⠀⡟⢧⣀⣁⠀⢸⡹⣆⠀⠀⠀⡜⢸⣁⡀⣀⡠⠃⢣⡀⠉⣹⠿⣄⣀⠇⠧⢤⠇⢹⣀⣈⡇⠘⠦⣤⠀⠀⢸⣶⡧⢾⡅
⢸⡤⣤⠤⠴⠋⣠⠀⠙⠛⡁⢾⣀⣀⣀⡴⢣⣄⠀⢹⠀⠈⣇⡈⠛⠛⠋⢀⣀⡈⠉⠉⣠⣄⠀⠉⠉⣩⣀⣀⣀⣤⣀⣀⡰⢦⣀⣠⠤⠶⡄⢸⠀⠀⢸⣇⣨⠟⠁
⡞⠓⠲⠶⠒⠋⠈⠓⠚⠛⠓⠶⠶⠶⠖⠚⠁⠈⠙⢻⡆⠀⣿⠙⠲⠶⠖⠋⠀⠉⠉⠉⠁⠈⠙⠓⠋⠁⠈⠉⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡏⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⡇⡼⠀⠀⢸⠁
⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠶⠾⠁⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⢠⢞⡏⠀⠀⠉⠀⠀⢀⣿
⢻⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⢠⠿⡷⢤⣀⣀⣠⢞⣏⠀⢀⣄⠀⠀⠃⢀⡀⠀⠀⠁⠀⠀⣠⡄⢠⢤⡀⠐⠀⡀⢀⡘⢦⣙⠒⠦⠤⠤⣒⡿⢹⡄
⢸⠀⠀⡀⠀⠀⢰⠛⡆⠀⠀⡴⣄⠀⢸⠀⠉⠒⣦⠥⠶⢻⠉⢧⡞⠈⡆⡀⢰⠏⠙⣤⡼⠶⡆⠀⡟⠈⠁⠀⢷⠀⠀⡟⠚⠙⠚⢹⡏⠛⢻⠻⡇⠀⢸⡇
⠘⠧⠞⠉⡆⠀⣟⠀⢧⠀⢸⠀⢸⣀⡿⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇⠀⢸⠃⠙⠦⠼⠁
⠀⠀⠀⠀⢳⣀⡇⠀⠘⠳⠞⠀⠀⠉⠁⠀⠀⠀⠘⣇⠀⡾⠀⠀⠀⠀⢻⠀⢸⠀⠀⠀⠀⠀⣟⢀⡇⠀⠀⠀⠘⣦⡼⠁⠀⠀⠀⠀⠳⣤⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠉⠉⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣀⠤⠤⠤⣀⣀⣀⠀⠀⢠⢤
⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠰⠛⠂⠀⠰⠦⠀⠀⠒⠁⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠚⠒⠂⠤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠙⡤⠀⠀⠀⢠⠊⠀⠈⠢⠀⠐⠂⠄⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⡇⠄⠀⠈⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠘⢤⣀⣀⠜⠣⣀⣀⠜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠀⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⠀⠀⠀⢹⠉⠉⠁⠒⠒⠂⢤⢤⡤⣀⣀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡜⠸⠀⠀⠀⠀⠉⠉⠑⠒⠊
⠀⠀⠀⠀⣀⠤⢼⠀⡠⠤⠸⣿⣿⣿⠟⠻⠛⢿⣿⣿⡿⠀⠀⠀⠸⡆⠀⠀⠀⡠⠤⣇⡇
⠀⠀⢠⠊⢡⠔⣿⠘⣄⠀⠀⠉⠛⠛⠀⠀⠀⠘⠛⠋⠁⠀⢤⡀⠀⡏⠢⠤⠼⠀⣄⣡⢈⡆
⠀⢀⠃⢠⠃⠀⢿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⡏⠒⠀⠀⡈⣣⠀⢠⠇
⢀⡼⠀⠈⢳⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠃⠀⠀⠀⠉⡗⡖⠚⠁
⡌⠀⣴⠦⣈⡱⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⢸⢰⠁
⠣⠴⠴⠁⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⢤⡤⡤⡤⣤⠤⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⡇⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠒⠲⢼⠇⡇⡇⢿⠴⠒⠒⠒⠒⠦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠉⠉⠀⠀⠀⠀⠉⠉⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⠤⢄⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣴⢟⡡⠲⡀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⠎⠀⠀⡸⠒⢒⡪⠋⠁⠀⠀⠉⢑⣶⠟⠓⢇⠀⠀⠱⡄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⠃⠀⠀⡜⠁⠰⡉⣀⡤⠂⠤⣀⣴⠟⠁⠀⠀⠀⢣⠀⠀⠘⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢼⠀⠣⢄⣀⡜⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠢⢄⡀⠀⠀⠀⢳⣀⡤⠚⠀⡷
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠱⡠⠋⢳⠐⢢⣴⠟⠁⠀⠀⠀⢀⠈⠑⠤⡔⠁⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠐⣅⣠⢃⣴⠟⠉⣆⡀⠀⠀⢄⠀⠑⢦⠴⠃⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢀⣼⠟⢇⣀⡎⠀⡜⠐⢤⣀⡹⠒⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣴⠟⠁⠀⠀⠀⠉⠈⠑⠒⠊⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠉⠉

⠀⠀⠀⠀⣠⣤⣀⣀⣀⣴⣶⣦⡀
⠀⠀⢀⣾⠋⠙⠛⠛⠋⠉⠈⠙⠿⢶⣶⣶⣦⡀
⠀⠀⣾⡏⠀⠤⣄⠀⠀⠀⠀⣠⡀⠀⠀⠈⢻⣷⣄
⠀⢀⣿⡀⠀⠀⠈⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠙⠿⣷⣦⣄
⢠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀
⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠉⠻⣆⠀⠚⠳⠀⠀⠰⠲⠄⠀⠀⠀⠀⠀⠀⠰⠛⠳⠀⠀⠀⠈⠻⠷⣶⣤⣤⣤⣄⠀⠀⠀⠀⣠⣤⣴⣤⣄⣠⣤⣤⣿⠏⠙⠿⠶⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠶⠿⠛⠁⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠖⠲⠄⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠙⠂⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠈⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠲⠀⣠⣤⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣄⡀⠀⠀⠊⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠱⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣦⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⠿⠿⠿⢿⣿⣿⣿⣷⣿⣿⡿⠿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⠴⠶⠒⢶⣤⠶⠚⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠓⠚⢯⡉⠛⠀⠀⢀⡴⠋⣠⠴⠚⠛⠓⠲⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⡀⠀⠀⣧⠀⠀⠀⠀⠀⡜⠁⠀⢀⣀⠀⠀⠈⡆⠸⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠋⠀⢀⡏⠀⠀⠀⠀⠀⣇⠀⠀⠘⠟⠁⠀⠀⡇⠀⠋⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣳⣄⣀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣀⠴⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠀⠀⠄⠀⠄⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣠⡤⠤⢤⣄
⠀⠀⠀⠀⠀⡟⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⢷⠀⣠⠞⠁⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⢷⡀⢴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣸⠞⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⠀⠀⠀⠀⠸⠁⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⣜⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⡇⠀⣠⠊⠀⠉⠓⠦⣀⠀⠀⠀⡞⠀⠀⠂⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⢶⡀⠈⠇⠀⠀⠀⠀⢀⡇⠀⠃⠀⠀⠀⠀⠀⠈⠳⡀⢸⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⢳⡀⠀⢀⡏⠀⡇⠀⠀⣰⠶⣄⠀⢸⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡀⠀⠀⠶⣯⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⢸⡇⠀⢸⠇⠀⡇⠀⠀⡏⣠⢿⣄⡜⠀⢸⠶⡖⠒⠂⠊⠛⠋⠉⠋⣹⠁⠀⠀⠀⢀⣀⣀⣽⠇
⢰⡞⠋⠉⠉⠉⠉⠉⠉⠉⢻⢸⡇⠀⢸⠀⠀⢧⠀⡸⠁⠓⠲⢬⡇⠀⠘⡛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣤⣀⣨⠥⣤⣼⠟
⠀⠙⠓⢦⡀⠀⠀⢀⣠⠤⠼⠀⡇⠀⣸⠒⡄⠈⠉⠁⡤⢤⣀⣀⡇⠀⠈⠧⠤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⢀⣠⡤⠴⠃⠀⠀⠈⠒⠶⢤⠀⢷⣤⠟⠛⠁⠀⠀⠀⠓⠒⠒⠒⢢⠀⡶⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⣠⠤⠴⢤⣄
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⠀⠀⠙⢲⠄⠀⢨
⠈⠓⠒⠒⠲⠶⠦⠤⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡼⠂⠀⠀⠛⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉

⠀⠀⠀⠀⣀⣤⣶⣶⣿⡿⣿⣿⣷⣶⣤⣀
⠀⢀⣴⡾⠛⢩⡿⠋⠀⠀⠀⠈⠻⣷⡈⠛⢷⣄
⣰⡿⠋⠀⠀⣿⠃⠀⠀⣤⣄⠀⠀⢹⣇⠀⠀⠹⣷⡀
⢻⣧⠀⠀⠀⣿⡀⠀⠈⠿⠟⠀⠀⢸⡿⠀⠀⢠⣾⠃
⠀⠙⢷⣄⡀⠹⣧⡀⠀⠀⠀⠀⣠⣿⠃⣠⣴⠟⠁
⠀⠀⠀⠙⠻⢷⣿⣿⣶⣶⣶⣾⣿⣷⠟⠛⠁⣠⣾⠿⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠁⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⣤⣤⣤⣄⣠⣴⣶⣶⣤⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟⠁⠀⠀⠙⠏⠀⠀⠀⠙⠟⠁⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠛⠛⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⣰⡿⠁⢀⣀⠀⠀⣀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⠻⣷⣾⠟⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⣤⣤⣴⣶⡶⣿⡄⠀⠠⣾⠟⠻⣷⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⠀⠀⢀⣴⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣻⣷⣶⣶⣾⣛⣋⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠰⠶⠤⢀⣀⡀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠐
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀⠀⠀⡠⠶⠖⠀⠄⠀⠀⠀⠀⠀⢨⣥⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⢀⡶⠋⠉⠉⠻⡄⠘⠟⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠀⠀⠀⠀⠀⠈⠇⠈⠉⢿⣿⣿⠃
⠀⠐⠚⠿⢛⠻⣤⣀⠀⠀⠀⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣿⠢⡀⠀⠀⠀⠀⠀⠈⠀⠀⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣾⡇⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠰⣿⡄⠘⣆⠀⠀⠀⢀⠄⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢛⣿⣏⠄⠘⡄⠀⠀⠈⢄⠀⢾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣤⣠⣴⣴⡶⠿⠟⠓⠤⠀⠀⡇⢠⣭⣿⡷⠂⠀⡇⠀⠀⠀⠀⠀⢈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⠿⠥⠒⠁⠀⠀⠀⠀⠀⠀⠀⠠⠠⠬⣿⣟⣁⢀⠇⢀⠀⠀⢠⣆⣀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡯⢀⠎⠀⠈⠻⣶⡋⠀⠀⣽⣿⣿⠋⡨
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠈⠻⣶⣾⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⣒⣢⠭⠟⠛⠃⠀⠀⠀⠀⠉⠉⠉⠒⠒⢦⡄⠈⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠻⠿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⡇⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠺⠭⠤⡤⠒⠒⠒⠒⠒⠒⠒⠒⠉⠉⠀⢰⣤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢒⡂⣴⣿⣍⣒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠀⠖⢒⢒⣾⠿⢿⡷⢤⣤⠤⠤⠤⠤⠤⠤⠤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠒⣤⡀⠀⠔⠚⠉⢈⠿⠃⠀⠒⠀⠘⠿⠉⠉⠁⠈⠋⠳⠀⠀⠀⠀⠀⢀⣠⡤⠤⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⣁⣠⡼⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡐⠀⠀⢀⣀⡠⠤⠶⣥⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣠⠞⠋⠀⠀⠀⠀⠉⠐⠒⠒⠒⠂⠈⠁⠀⠀⠀⠀⠀⡡⠁⠈⠀⠀⠀⠀⠀⠈⢻⡆⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⠏⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢨⠀⠀⠀⢀⡻⠔⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠀⠀⢀⠌⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡜⠀⠀⠀⠀⠛⠀⠀⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⠀⠀⠳⡀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠈⢆⣠⠀⠀⠠⡀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠉⢽⠒⠤⡀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠶⢄⠈⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠰⡲⠀⠀⠀⠀⠶⠀⠀⣸⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⠤⠤⠤⠴⠒⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢳⡤⠒⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⢸⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡀⠑⢄⡀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢨⠟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢈⠑⠒⠢⠤⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⡀⠤⠔⠊⠁⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⢈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠏⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⣠⡀⠀⠀⠀⠀⠀⣠⣤⡀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠟⠀⠀⠀⠀⠀⠷⠞⢁⣌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣲⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣤⠴⣄⠀⠀⢺⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡝⣉⡀⠀⠀⠀⠀⠘⣧⣼⠇⠀⠀⠀⢸⡇⠀⢸⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠈⢯⡽⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠙⠛⠋⠀⠀⠸⡽⠇⠀⠀⠀⢠⡖⢦
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢿⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣤⣀⠙⠋⠀⠀⠀⠀⣀
⠀⠀⣠⠶⠋⠉⠉⠉⠉⠉⠉⠛⢷⡀⠀⡴⠒⠛⢻⡴⠛⠉⠉⠙⢦⡀⠀⣠⠶⠛⠉⠉⠛⠶⣄⠀⢠⡞⠁⠀⠀⠀⠀⠉⢳⡀⢀⡴⠚⠉⠉⠙⠛⠲⢦
⠀⡴⠃⠀⠴⠀⠀⣀⣀⡀⠀⠀⢸⡇⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⢹⣴⠋⢠⠄⠀⠀⠀⠀⠈⢳⡟⠀⠀⠖⠀⠀⠀⠀⠀⣷⡏⠀⠀⠀⠀⢀⣠⣤⡾
⣸⠁⠀⠀⠀⣴⠛⠉⠉⢻⣴⣆⣼⠁⠀⡇⠠⡄⠀⢠⣤⠀⠀⢀⣤⢿⡇⠀⠈⠀⣀⣀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣠⠤⠴⠒⠙⣆⣀⣀⣀⡀⠈⠛⠓⠦⣄⠀⣴⢲
⡏⠀⠀⠀⢸⠃⠀⠀⠀⠈⠁⠈⠁⠀⠀⣷⠀⠀⠀⢸⣸⠀⠀⠀⢷⣸⠀⠀⠀⢰⠁⢸⠀⠀⠀⠀⣿⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⢳⡀⠀⠀⠈⣧⠈⠋
⣧⠀⠀⠀⢸⡀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠉⠀⠀⠀⠀⢻⡄⠀⠀⠈⠓⠋⠀⠀⠀⢀⣧⣤⣤⣀⠀⠈⠛⠛⢶⣠⡶⠶⢤⣄⣤⠞⠐⣦⠀⠀⣻
⢹⡀⠀⠀⠈⢳⡀⠀⠛⢶⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠶⠤⠷⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡿
⠈⢳⡀⠀⠀⠀⠉⠛⠒⠋⠀⠀⠀⠀⣿⢻⠉⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡏⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁
⠀⠀⠙⠳⣦⠀⠀⠀⠀⢀⣤⡄⠀⠀⣿⢸⣀⣀⠀⠀⣿⠙⣦⠀⠀⠀⢀⣸⠟⣆⢠⣶⠀⣤⡇⠀⣇⡾⣇⠀⠀⠀⢠⡶⢾⠁⠀⠀⠀⠀⠀⠀⣤⡀⡿
⠀⠀⠀⠀⢸⠀⣀⠀⡀⢸⠀⣷⡆⣴⣿⠈⠉⢹⡇⠀⣿⠀⢹⠀⣤⡄⣿⠉⠀⣿⢸⡏⠷⠿⠇⠀⠀⠀⢻⢰⢷⡀⢸⠁⠘⣇⢸⢳⡄⣾⢧⠀⡏⠛⠁
⠀⠀⠀⠀⢸⢸⡟⠞⢻⢸⡆⠀⣿⢿⠀⠀⠀⠘⣿⣧⣿⠀⢿⠀⡇⠙⠋⠀⠀⢹⢸⠇⠀⠀⠀⠀⠀⠀⠈⠛⠘⡇⣸⠀⠀⠹⣾⠈⠷⠏⠸⣦⡿
⠀⠀⠀⠀⠘⠿⠀⠀⠘⠿⠀⠀⠹⣾⠀⠀⠀⠀⠀⢸⣿⠀⢸⣄⡇⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⡿⢿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⣰⣿⠇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⢿⣷⡀
⢠⣿⡏⠀⠀⠀⠀⠀⠀⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠃⠀⠀⠀⠀⠀⠈⣿⣧
⣸⣿⠁⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⢹⣿
⣿⣿⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠙⠛⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⣄⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⣶⣿⠟⢿⣷⣾⡿⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁
⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⡿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠉⠛⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠣⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⠂⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⣠⠔⠞⠹⡟⠀⣶⠿⠂⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⢰⠁⠀⠀⠀⠘⣤⠊⠉⠉⠢⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⣏⢀⡀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⣠⢤⡔⠒⡴⠒⢦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣄⠀⠀⢀⡼⠻⡈⠀⠀⣠⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⡴⠒⢶⠒⢦⡤
⣇⠀⢃⠀⢱⠀⠼⣅⠀⠀⣿⠀⠀⠀⡜⠉⢆⠈⠙⠛⠉⠀⠀⠈⠛⠋⠁⡔⠉⣧⠀⠀⠀⣇⠀⢀⣸⠦⠀⡆⠀⡘
⠈⢶⠉⠁⠛⠑⠀⣸⠃⠀⣿⠀⠀⠀⠇⠤⠘⠓⢄⡀⠀⠀⠀⠀⢀⠠⠚⠣⠄⠜⠀⠀⠀⣿⠀⠸⣆⠀⠊⠻⠈⠉⡶
⠀⠈⠳⡆⠀⢰⠋⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢙⡆⠀⢰⠎⠁
⠀⠀⠀⢳⡀⠈⢧⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡼⠁⢀⡟
⠀⠀⠀⠀⠙⢄⡀⠙⠲⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠖⠋⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠉⠒⠲⢶⡇⢠⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢤⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣤⣤⣄⣤⣤⣤⣤⡤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠀⡇⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣾⠀⡇⣿⠀⡇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠙⠲⠿⠀⡇⣿⠀⠷⠚⠋⠉⠉⠉⠉⠙⠲⣄
⣀⢀⣀⣀⡀⠀⢀⡀⣀⡐⣧⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⡗⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⢀⣀⠀⠀⣀⢀⣀⢀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⢀⣀⣠⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢋⣭⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣌⣛⣴⣿⣿⣿⣿⣿⣿⣿⡟⣉⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⡋⠬⢹⣿⣿⣿⣷⣥⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢻⣿⣿⣿⢡⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣙⣛⣡⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠘⣛⣛⣛⣻⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠈⠉⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣀⡀⠀⠀⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡟⣋⠻⣿⣿⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⡇⢹⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡃⢾⣿⣷⣌⠻⠇⠀⣠⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣷⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠎⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⡿⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡅⢾⣿⡿⠋⠀⠀⠈⠹⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠒⠂⠀⠀⠐⠒⠒⠒⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⣾
⠘⣇⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⢠⡟
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠀⣸⠃
⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠶⢶⣤⣤⣀⣀⣀⣀⣀⣤⡴⠶⠶⠖⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣧⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣸⠇⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⣠⣴⠶⠶⠚⠛⠛⠛⠛⠛⠛⠶⢶⣤⡀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣾⣋⣁⣀⣀⣤⣤⣴⠶⠶⠶⣤⣤⣀⣀⡀⠙⣷⣄⣤⡾⠋⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣓⣒⣦⣔⣶⣤⣴⣶⣞⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠎⠘⠀⠀⠀⠀⢀⠀⠀⡠⠒⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⡤⠁⠘⢄⠀⢇⠀⢀⡠⠤⠀⠺⣄⢀⡜
⠀⠀⠀⠀⠀⠊⠁⠀⠈⠉⠁⠉⠑⠊⢢⡀⢄⢡⠀⠑⢟⣀⣈⣙⡷⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢂⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠈⠒⠁⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠉⠀⠀⠀⢀⠴⠊⠁⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡶⠀⠁
⣠⣤⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢋⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣿⠗
⠉⠁⠀⠀⢠⠏⠀⠀⣀⠀⡀⠀⠀⠀⠈⡄⠣⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡸⠱⡀⠀⠀⢀⢼⠔⢒⢼⡀
⠀⠀⠀⠀⣼⠀⠀⠈⠢⠥⠃⠀⠀⠀⠀⢱⠀⠑⣄⠀⠀⠀⠀⠀⠈⡡⠊⠀⠀⠀⠈⠒⠀⠆⢠⢇⣸⣭⠕
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠈⠀⠄⣀⡠⠤⠚⠀⠀⠀⠀⠀⠀⣀⡤⢦⣳⠄⠠⠛⠈
⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢠⠦⠔⠃⠀⠀⠤⠴⠒⠒⠂⠀⠐⠲⢄⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣶⡶⣷⠄
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡞⠚⠫⢉⡉⠉⠁⠀⠀⠀⠀⠒⠂⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠀⠀⠖⠐⢂⡨⠟⠁
⠈⠑⠂⠠⢐⢉⣽⢮⠂⠀⢸⠁⠀⢈⡩⠟⠁⠒⡲⠛⢻⡟⣁
⠀⠀⠀⠀⠀⠀⠫⠭⠤⣠⡈⡗⠊⠁⠀⠀⠀⠘⠒⠚⠙⠛⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣠⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢰⠟⣧⣰⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠘⠃⠀⠀⡴⠠⣄⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⡀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⡀⠠⠞⠻⣅
⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣍⡳⠀⢠⣾⣿⣿⣿⣿⣿⣿⣧⣄⠀⢠⣬⠶⠖
⢀⣀⠀⣠⣶⣿⣿⠛⣽⢧⠀⠀⠀⠀⢀⣠⣌⡀⠾⠍⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣉⡧
⢻⣿⣿⣿⣿⣿⣿⣿⣿⡝⠀⠀⠀⠀⣾⠁⠈⢷⡀⠠⣟⢾⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇⠉⠉⠀⠀⠀⠀⣴⣟⣿⣿⣷⣤⡄
⠀⠉⠛⠿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⣿⠶⠦⣼⠇⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠈⠉⠙⠟⠻⢷⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠉⡄⠉⠀⠀⠈⢿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⢀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⠻⣷⠶⠄⢀⣴⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣽⣷⢦⣤⣀⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠋⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣤⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣣⢿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢏⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣷⣿⡿⠿⢿⣟⣻⣿⣿⣿⣿⣯⣽⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡺⢯⣤⠧⣼⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⠶⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⢿⣿⣿⡿⠷⠟⠋⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠁⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠂⠀⡄⢄⠀⢠⠀⠀⠀⡤⢀⢠⢀⡇⢨⡖⢠
⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⢠⠞⠉⠢⡀⠀⠂⠀⠀⠀⠄⢀⠀⠠⠄⠀⠀⠀⠠⠚⠐⠘⠚⠂⠀⠁⠈⠉⠀⠁⠉⠑⠘
⠀⠀⠀⠀⠀⠁⣈⡁⠀⠀⠸⡆⠀⠀⢱⠀⠀⠂⠀⠀⠈⠤⠀⠀⠀⡔⠉⠑⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠁⢈⡁⠀⠀⠀⠘⠲⠶⠊⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠀⠑⠀⠀⠀⠀⡠⠊⠀⠀⡷
⠀⠀⠀⠀⠀⡉⣈⡁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡂⠀⢀⡀⠀⠀⠀⠁⠄⠤⠐⠁⠐
⠀⠀⠀⠀⠀⣉⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠓⠉⠳⡀⠈⠒⠋⠀⠉⢧⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⣉⣀⡄⠀⠀⢀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⡂⠀⠀⢀⠄⠀⠀⠀⣰⡀⠀⡃⠀⠀⠀⣠⠤⢄
⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠠⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⡀⢀⠀⠉⠑⠀⠡⠀⠀⡸⠁⠀⠀⠆
⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠑⠁⠀⠀⠀⠀⠒⢀⡸⠀⠂⢀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⡈
⠀⠀⠀⠀⠀⣠⣠⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠀⠄⠀⠀⠑⠒⠊
⠀⠀⠀⠀⠀⠠⢤⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠠⢤⡄⠀⠀⠄⠀⠀⠀⠠⠤⠠⠄⠀⠀⠀⠤⠂⠀⠁⠀⠀⠡⠄⠠⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠀⠐⠂⠀⢀⡤⠴⠒⠒⠒⠒⠒⠒⠒⠲⣦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠤⢤⡤⠀⠀⠀⠀⣀⠀⠀⠀⣀⠤⠐⠀⠀⠒⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾⣭
⠀⠀⠀⠀⠀⠤⢤⡤⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠤⡤⠤⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠠⣴⡦⡄⠀⠠⠀⠀⢰⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠤⠤⠦⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠁⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠤⠤⠦⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠴⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⡔⠁⠈⢢⠀⠀⠀⠀⠀⠀⠰⠄⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠶⠤⠆⠀⠀⠀⠀⢱⠀⠀⠀⠀⠂⠀⠀⠀⠘⠀⠀⠀⢇⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢒⠰⠆⠀⠀⠀⡄⠀⠹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢃⠤⠆⠀⠀⠘⠷⢄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠃⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠢
⠀⠀⣀⠠⠤⢄⡀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⡅
⢠⠊⠁⠀⠀⠀⠈⠢⡀⠀⠇⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠙⣄⠸⡀⠀⠀⠀⠀⢀⠁
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠱⡀⠀⡀⠀⡘
⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠈⣆⠙⢦⠀⣠⠃
⠀⠀⠈⠓⠤⣄⣀⠀⠀⠀⣀⡸⠤⠇⠉⠁⠑⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠙⢂⠶⠴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠸⠁⠋⠈⣋⣈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠐⢣⠴⢄⣀⣀⢸⠃
⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠚⠀⠒⠁⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⢀⡔⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠦⣤⢤⢀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠛⢦⠀⠀⢱
⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⣀⡄⢶⠒⡎⠒⠤⡖⠲⠤⠞⠉⠁⠀⠀⠀⡱⠀⠀⢉⠆
⠀⠀⠀⠀⠀⠀⠐⢄⣀⣸⡌⢳⡙⣦⡓⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⡏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡹⡿⣌⢲⠦⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⠈⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠃⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡤⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠥⢦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠈⣷⠈⠳⡀⢠⠖⣉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⡇⠸⡷⡄⠈⠛⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠃⠀⣷⠙⢄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⠀⠀⠑⣦⣄⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠔⠊⠉⠙⠓⢄⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠻⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠉⠲⢤⣤⡴⠋⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⢠⠗⢄
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠈⠓⠶⠤⠶⠋⠀⢈⡇
⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⢀⡼⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠘⢧⡀⢀⡴⠋⠀⠀⠀⠀⠀⢱⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⡛⠀⠀⠀⠀⠀⢺⣄⡈⠻⠿⣟⠉⠈⢷⢦⣀⣀⡤⠚⢁⡴⡃⠢⣄⠀⠀⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⢧⠀⠙⠻⢧⣛⡜⠦⣤⠏⢦⣭⢥⣖⡾⠿⠋⠀⠀⠈⠙⠒⢋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢤⣬⠿⠶⣄⠀⠈⣙⢷⡀⡤⣶⣟⣉⣠⣴⣶⣦⣤⣀⣀⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⡷⡀⣿⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠚⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡗⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⢙⡌⠈⣢⡈⣛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠉⠹⠤⠏⠉⢯⠟⠙⠪⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣀⡀⢤⠀⠀⣿⠰⠷⠀⠾⠆⠸⠷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠋⠉⣹⠊⠇⠀⠿⢤⣤⣤⡤⣤⡤⣤⢼⠇⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⣠⠖⠃⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠸⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠉⠉⠉⣿⠀⠀⢰⡇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠽⠚⠉⢳⡀⠀⠀⠀⠀⠀⡟⠀⠀⣀⡀⠀⠀⣠⠀⡇⣀⠀⠀⠳⢤⣤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⢻⣀⣾⡥⠭⠿⢯⣹⣿⢁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⠴⣚⡷⠀⣀⡤⠖⠋⠉⠀⣧⣀⣀⣀⣨⡇⠀⠀⠀⠀⠀⠀⣠⠖⢛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⣡⠴⠋⠁⠀⠀⠀⣠⣾⡉⠉⠉⠉⣸⣃⠀⠀⠀⠀⠀⣼⠁⡼⢉⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣠⠴⠛⠹⡄⠙⠦⣤⠴⠃⡽⠉⠳⢄⠀⠀⢿⠸⡇⣾⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠴⠖⠋⠁⠀⠀⠀⠘⢦⡀⠀⢀⡼⠁⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⠀⠀⠀⠀⠀⠀⠙⠦⠎⠀⠀⠀⠀⢲⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠠⠄⡄⡤⠠⢠⢤⠤⡤⢠⠀⠤⡀⠆⣤⢤⠠⡤⠤⠄⠠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣤⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣰⠋⣧⠎⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠁⣀⣀⣀⡀⠋⢈⣷⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣍⡳⠤⠀⣰⠊⠁⠀⠀⠉⠃⡀⠀⣙⣠⡤
⢀⡀⢀⡤⡎⣟⠉⠳⡄⠀⠀⠀⢀⣔⡀⠽⢃⡴⠇⠀⠀⠀⠀⠀⠀⢱⠀⢉⣣
⠣⡈⡏⢄⡳⠼⠃⡤⠁⠀⠀⢰⡏⠀⢻⡀⢸⠀⠀⣄⢠⡄⠀⠀⠀⢸⡆⠉⠁⠀⠀⢠⡞⠉⣻⢳⢤⡀⢀
⠀⠈⠉⠺⢤⣤⠘⣅⠀⠀⠀⠘⠛⠓⠚⠃⢸⠃⠀⠋⠘⠁⠀⠀⠀⢸⡇⠠⠄⠀⠀⠈⢧⠙⠧⢾⡰⢻⣁
⠀⠀⠀⠀⠀⠀⠡⣌⠑⢄⡀⠀⠀⠀⠀⣴⡏⠀⠀⠐⠚⠁⠀⠀⠀⠉⣧⢠⣿⣀⠀⠀⣨⠃⣤⡤⠗⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠉⠲⠤⣀⠀⡿⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠋⠀⠸⢈⡡⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⣽⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⣼⣀⡤⠖⢉⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣀⣀⣀⠤⠖⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣤⣤⣴⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠃⠀⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣯⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣾⣿⢿⡿⠀⣴⣿⣿⣷⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⣾⣿⣿⣾⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⢿⣿⣿⣿⣧⠘⣿⣿⣿⢿⣿⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣄⠈⠛⡟⠛⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠛⠀⠼⠿⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⣀⣈⡻⣿⣿⣿⣿⣷⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣼⣿⣿⣧⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠤⠌⠛⠾⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢀⣴⣶⣶⣄⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣿⣿⣶⣾⣿⣿⡟⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⢿⢿⣷⣿⣿⣽⡿⡗⠠⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠄⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣤⣿⣯⣾⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠁⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠘⢿⣿⣿⡿⠁⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢫⣾⣿⣿⣿⣶⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣀⣠⣿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣹⣿⣿⣾⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠈⠉⠁

⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠈⢀⠠⡄⠴⠠⠀⠀⠀⢀⠤⣀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⡄⠉⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠈⠈⠀⡀⠀⣀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠄⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠄
⠀⠀⠀⠀⠀⠀⡶⠛⢯⠛⣢⠀⠀⠠⠁⠀⠀⠀⣀⢂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠁⠈⠀⠀⠀⠀⡐⢀⠀⠀⠄⠀⠀⠣⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣇⣀⠀⠀⠠⣾⠇⠀⠈⠈⠉⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⠀⠀⠀⠀⠀⠀⠛⠉⠑⠀⠀⠈
⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠥⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠴⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠋
⠀⠀⠀⠀⠁⣄⠀⠀⠁⠀⡆⠀⠀⠀⠀⠸⠀⢀⠀⠀⠦⠄⠀⠀⠀⠢⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⡀⠀⠈⠋
⠀⠀⠀⠀⡰⠉⠉⠀⠀⠀⠁⢄⣀⣤⠰⠊⠀⠄⠀⠸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢀⠀⠀⠰⠶⠁⠀⠀⠀⠀⠀⠉⠀⠛⣉⠄
⠀⠀⠀⠀⠀⠐⠄⠀⢖⡀⠀⠀⠀⠀⠀⠤⠈⠀⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⢸⠃⠀⠒⠀⠀⠠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢠⠀⠆⠀⠀⠢⠀⠠⠀⠀⠀⠀⠀⠀⡀⠁⠀⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⣇⡼⠀⠀⠀⢀⢄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠘⠁⠃⠀⠀⠀⠀⠀⢠⡄⠀⢀⡠⠤⠐⠒⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠉⠀⠀⠀⠆⠘⠈⠴⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⡂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠛⠀⠄⠀⢀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⡴⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠈⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠠⢀⡀
⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⢚⣄⠀⢨⡆⠘⠊⠀⠠⠁⠀⢀⢠⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢤⡄⠀⠀⠀⠁⠵⡋⠀⠀⠀⠳⢄⠀⠀⠀⢀⡠⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⣾⡯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⣀⡁⠀⠀⠀⠂⠐⠀⠀⠂⡀⠀⠀⠀⠁⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣀
⠀⠀⠀⠀⠀⠌⢄⠀⠀⠀⠀⢩⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠠⠀⠃⠀⠀⠁⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠂⢀⠂⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠈⠀⠑⠴⠃⢛⠆⠤⠌⡀⡀⡀
⠀⠀⢀⢄⣀⠀⠀⡀⢀⠤⠀⢤⠄⡄⠀⠀⠠⣤⡊⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⢀⠀⠀⠂
⠀⠀⠐⡐⡉⣆⣀⠀⠀⠀⠋⠀⠄⠀⠠⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢜⢀⠁⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠐⠁
⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀
⠀⠀⠐⠀⠀⠀⢀⡀⠆⠀⠠⠘⠃⢀⠀⠀⢀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠂⢠⠀⠀⠠⠀⠉⠀⠰⠀⡄
⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⠀⠂⠠⠄⠐⠀⠀⠀⠈⠀⢰⠀⠠⠈⢀⠀⠁⠀⠀⠂⠐⠄⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠣⠀⠀⢀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢭⡙⠲⣾⠀⠀⢸⡀⣧⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡘⠓⠶⣧⣦⠙⠆⢦⠈⢷⣌⠳⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⣁⠀⠀⠀⠀⠛⠂⠈⠳⢤⣈⡉⠀⠀⠙⠷⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⢀⡴⠋⠁⠠⠒⠉⠉⠉⠁⠀⣀⡤⠖⢒⣠⡤⠤⠄⠀⠀⠉⢙⡛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⣰⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠙⠛⠓⠂⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⡰⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠲⣄
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣆
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡀
⠀⠀⠀⠀⠀⠀⡖⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇
⠀⠀⠀⠀⠀⣸⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⡇
⠀⠀⠀⠀⠀⣿⡄⠀⢸⠁⠀⠀⠀⠂⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⠀⢀⡼⣡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢢⡀⣀⢸⡇
⠀⠀⠀⠀⠀⣿⠃⠀⢸⡆⠀⠀⠀⢀⣤⣤⣤⣤⣤⠴⠖⠒⢾⡆⠉⠛⠋⠛⠁⠀⠀⣄⣀⣠⡴⢻⣿⣷⣦⣄⣀⠀⢹⡏⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠠⠟⠀⠀⠠⠤⠤⠔⡂⠔⠈⠁⠀⠀⡀⢠⡀⠀⠀⠈⣹⠟⠓⠾⠿⣿⣧⣄⣉⣷⠘⡇⢰⠇
⠀⣠⡴⠶⠶⣿⣤⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠀⢠⠊⠀⠲⢤⣤⣤⣤⡄⠀⠀⠀⢸⣇⡾
⡾⠁⣀⣀⣀⠀⠈⣿⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠂⠀⠀⠠⠋⠀⠀⠀⠸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⠛⠛⠳⢦
⠇⠚⠉⠉⠙⢷⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⡶⠶⣤
⠀⡀⠀⠀⠘⠻⣿⢀⠀⠀⢀⠄⠀⠀⠀⠀⣰⠟⠁⠸⣆⠘⠿⢷⡄⠀⠀⠀⠀⠀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⠂⠀⠀⢀
⠀⠹⣆⢀⠀⠀⢿⢸⡀⢠⡏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢀⣀⠚⠋⠉⠉⠘⣿⠀⠀⠀⠀⠀⢸⠀⢀⡼⠀⠀⣀⠔⠁
⠀⠀⠈⠻⣆⠀⠼⢸⡇⣾⠀⠀⠀⠐⠁⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⡇⠀⠋⠀⠀⠉
⠀⠀⠀⠀⢹⡄⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⡴⠟⠛⠹⣄⣴⠉⠳⣄⡙⠒⠤⠤⡀⠀⠀⠀⠀⡟⠀⠀⠀⢸⣾⠃⢀⡤
⠀⠀⠀⠀⠀⠙⠶⠶⢻⡇⠀⠀⠀⣠⠞⠁⠀⠀⣀⣤⡴⠿⣄⠀⠸⡟⠒⠤⣤⣤⣀⠀⠀⢸⠇⠀⠀⠀⣾⡟⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⡞⠁⠀⠀⡴⠛⠉⠀⠀⠀⠀⠉⠀⠃⠀⠞⠙⢿⣿⡆⢀⡟⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡿⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢴⣻⠖⡼⠁⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠈⠁⢀⡼⠁⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠞⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣤⣤⣤⠶⠛⠉
⣿⣿⣿⣿⣷⡾⢿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣷⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⡿⡿⡿⠿⠿⣿⣷⠿⠿⣿⠿⢿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠒⠒⠒⠒⠢⣴⡁⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⢀⠀⠀⠀⠀⠀⣀⠈⠻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡄⠤⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⣀⡤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠤⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠙⠋⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢄⠀⢊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣠⣶⣿⣿
⠀⠀⠀⠀⠄⠀⡀⠀⢠⠠⠈⢠⡷⠐⠱⠀⠀⡀⠀⠀⠀⠀⣠⣤⣤⣯⣾⣿⣿⣿⣿⣿⣿⡆⠀⢀⡀
⠀⠀⠀⠀⠈⠢⣀⠑⢜⡡⠐⠊⡄⠂⠀⠒⠁⠐⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠁
⠀⠀⠀⠀⠡⡀⠈⢢⢎⢀⠀⠀⠘⠉⠀⣰⠄⠂⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣉⢂⡀⠀⠺⠁⢳⡷⠒⠲⠃⠠⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠡⡈⣡⡐⢄⣀⠈⠐⡘⢠⡔⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⢆⠑⢄⠁⢰⣜⡁⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣷⣧⣄⣷⣯⣼⣝⣁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⡳⢻⠛⠟⡻⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⡀⣘⠀⣁⢁⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⢻⣿⣿⠀⠀⠀⠀⠉⠙⣿⣿⣧⠀⢰⣿⣿⠉⠉⠛⠛⠀⠘⠿⠿⠿⠟⠁⠀⣨⣿⣿⠀⠀⣿⣿⠁⢸⣿⣿⡏
⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⣼⣿⡿⠀⢹⣿⣿⣄⠀⠀⠀⠀⣠⣴⣶⣤⡀⠐⠿⠿⠟⠁⠀⠀⣿⣿⡇⠘⢿⣿⠇
⡀⠘⣿⣿⣧⣄⠀⠀⠀⢰⣿⡿⠃⠀⠈⠻⢿⣿⣷⣦⡀⠰⠿⠿⢿⣿⣿⡄⢀⣄⣀⠀⢀⣠⣿⣿⠀⠀⠀⣠⣤
⣷⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠁⠀⢀⡄⠀⠀⠈⠛⣿⣿⡆⠀⠀⢸⣿⣿⠁⠈⠻⠿⠿⠿⠿⠛⠁⠀⣰⣿⡿⣿
⠏⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢿⣷⣀⠀⠀⣠⣾⣿⡏⠀⠀⢿⣿⣿⣄⣀⣤⣄⡀⠀⠀⠀⢀⠀⠙⠿⣴⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠘⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣦⣀⣤⣿⣇⠀⠘⣿⣿
⣿⣿⣷⣆⠀⣼⣿⣿⠟⠛⢿⣿⣷⡀⠀⠀⠉⣁⣀⠀⠀⢀⣠⣤⣄⡀⠉⠉⠁⠀⠙⢿⣿⡿⠛⠻⠋⠀⠀⠸⢿
⠀⣸⣿⡟⠀⣿⣿⣿⣀⢀⣼⣿⣿⠇⠀⢰⣿⣿⣿⣷⠀⠘⠛⢿⣿⣿⡄⠀⠀⠀⠀⢀⣀
⣰⣿⣿⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣠⣿⣿⡟⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣄⠀⢰⣿⣿⣷⣆
⠈⠻⣿⣷⡄⠀⠀⠈⠉⠉⠉⠀⠀⠀⢠⣾⣿⣿⠏⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠘⣿⠈⠻⣿⣿⣿⡏⠉⢿⣿
⠀⠀⠀⣿⡇⠀⢀⣤⡄⠀⢰⣦⣤⡀⢸⣿⣿⣷⣶⣄⠀⠀⠘⣿⣿⣿⣀⡀⠀⠀⠐⠺⠇⠀⢉⠻⠿⠃⠀⢸⣿
⠀⠀⠀⢸⣿⣾⣿⡿⠃⠀⠀⣿⡿⠃⠀⠉⢻⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣦⠀⠀⣾⣿⣿⣷⣦⣴⣿⡟
⣀⡀⠀⠀⠉⠉⢀⣤⣶⣶⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⠉⠉⠛⠻⣿⣿⡇⠀⠿⠏⠙⠛⠛⠛⠋⠁
⣿⣿⣷⣤⡀⠀⣾⣿⣿⠿⠛⠛⠁⣠⣴⣦⣄⠘⠻⣿⣿⣷⣶⣿⣿⠇⠀⠀⢀⣿⣿⠀⠀⠀⠀⢀⣤⣄⡀⢀⣾
⠋⠛⢿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣾⣿⠿⢿⣿⣷⡀⠀⠉⠛⠛⠛⠉⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⣸⣿⣿⡷⠸⣿
⢀⣤⡀⠉⠛⠛⠛⠉⠀⠀⠀⣿⣿⡇⠀⠀⠁⢹⣷⠀⠀⠀⠀⢀⣠⣤⣀⠀⠘⢻⣿⣿⣷⣶⣾⣿⣿⠟⠁⠀⠉
⣹⣿⣿⠀⢠⣶⣿⣿⣦⣄⠀⢸⣿⣷⣄⠀⠀⠈⢿⣧⠀⠀⣼⣿⡿⠿⣿⣷⠀⠀⠈⠙⠿⠿⠟⠛⠁⢠⣴⣤⣄
⣿⣿⠏⠀⣿⣿⡏⠀⣸⣿⠀⠀⠙⣿⣿⣧⠀⠀⠻⣿⣷⣾⣿⡿⠀⠀⠘⠛⠀⠀⢀⣀⠀⠀⠀⠀⠀⢻⣿⠛⠿
⠀⠀⠀⠀⠸⢿⣿⣟⠋⠁⠀⠀⠀⣸⣿⣿⠀⠀⠀⠈⠛⠛⠋⣠⣤⣶⣄⠀⠀⢰⣿⠿⢿⣷⣤⣤⣴⣿⡿
⣶⣤⣀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠘⢿⡫⠃⠀⠀⠀⠀⠀⠀⣼⡿⠛⠻⣿⣷⠀⢿⣿⡀⠀⠀⠈⠙⠛⠉⠀⣠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠒⠒⠒⠒⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠐⠚⠋⠁⠀⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣤⠠⠄⢀⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣰⠃⡖⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢱⢀⠞⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠱⠒⠉⢱⡨⠀⠀⠀⠀⣠⡀⠀⠘⡆⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠰⠄⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠄⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠠⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠂⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⠀⢀⠀⠂⠀⠤⢀⣀⡄⢄⣸⠛⣯⡲⢂
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠔⡁⠀⠉⠱⠇⠀⢀⣀⠈⠀⠀⠈⠓⠈⠁⠁⡀⠈⣀⠀⠈⡀⠠⠠⠀⡄
⠀⠀⠀⠀⠀⢸⠀⢠⡰⠀⠀⠀⠂⠄⠀⠀⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠘⠀⠙⢠⡄⠁⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠱⡌⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠔⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠂⢀⠒⠁⠀⠀⠀⠀⢠⠀⠀⢀⠀⢤⢀⢲⠀⠠⠀⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠰⢸⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⣀⡀⢙⢚⢀⣀⢘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢙⢛⡟⠛⠛⠿⣛⠛⠻⢿⣉⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠋⢉⣉⣉⣉⣉⡉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⢁⣠⠤⠖⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣀⡴⠚⠉⠀⢀⣀⡤⠴⠶⠒⠒⠒⠒⠶⠦⢤⣄⡀⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⠞⠁⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⢀⡼⠃⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢠⡞⠁⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋⠀⢀⡼⢳⡀⠀⠹⣄⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢰⠏⠀⢠⠏⠀⠀⠀⠀⢸⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡽⠋⠀⣀⡴⠋⠀⠀⠹⡄⠀⠹⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⡏⠀⢠⡏⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⡴⠲⣄⠀⢹⡄⠀⢻⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠀⠀⡾⠀⠀⠀⠀⠀⣅⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠈⡇⠀⣧⠀⡟⠀⠀⣧⠀⠈⣇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⡇⠀⢹⢸⡇⠀⠀⢸⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠸⡇⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⡇⠀⠘⣿⠀⠀⠀⢸⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠟⠀⠀⠀⣼⠀⠀⡿⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡄⠀⢹⡀⠀⠀⠀⠀⠀⢸⣀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣴⠓⡆⠀⢀⡏⠀⢰⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢷⠀⠈⢧⠀⠀⠀⠀⢀⡼⠋⠀⢀⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡇⠀⠈⠛⠁⠀⡾⠀⠀⡞⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⢧⠀⠈⢷⡀⢀⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡾⠁⠀⡼⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠈⢷⡀⠀⠙⠋⠀⢀⡴⣏⣀⣀⣀⣀⣀⣀⣙⣋⣀⣀⣀⣀⣀⣀⣴⠇⠀⠀⣰⠏⠀⢀⡞⠁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠙⣦⡀⠀⠰⢯⡀⠀⠀⠁⠉⠀⠁⠈⠉⠁⠉⠀⠀⠀⠀⠁⠀⢀⡴⠞⠁⢀⡴⠋⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⢦⡀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉⠀⣀⡴⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⢀⣠⠴⠚⢉⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⠤⠤⠤⠤⠴⠶⠒⠋⢉⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⣶⣶⣶⣶⣶⡷⢾⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣷⢾⠶⣶⡶⠶⠶⣶⡾⠷⡾⠶⠶⠶⢿

⠀⠀⠀⣠⣤⣒⣒⣒⣒⣛⣛⣛⣛⣋⣉⣉⣉⣉⣙⣩⢽⢳⢦⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒⠂⠀⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⢢
⠀⠀⢰⢷⣶⣒⣒⣒⣒⣒⣶⣶⣶⣤⡶⠶⠶⠶⣶⣖⣓⣶⣸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠘⣭⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⠿⠯⠭⠭⠭⠭⠽⢶⡁⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣀⡀⣀⡀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠤⠤⢶⠁
⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣐⣐⣘⡄⠀⠀⢸⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠹⢉⢹⠀⠀⠀⢸⠀⠀⠀⡇
⡿⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣶⣾⢾⣷⣧⡀⠀⡇⠀⠀⡇
⡇⠀⠘⣿⣿⣿⣿⣦⡠⠤⠦⠖⠒⢈⣃⣶⣾⣿⣿⣿⣿⣿⣻⠆⠀⡇⠀⠀⡇
⡇⠀⠀⠈⢿⣿⣿⣿⣿⣦⣬⣴⣾⣿⣷⣿⣿⣿⣿⣿⡾⣿⠋⠀⠀⡇⠀⠀⡇
⡇⠀⣠⣴⣾⣿⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣽⢾⠁⠀⠀⠀⡇⠀⠀⡇
⡇⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⡡⠞⣦⡀⠀⠀⡇⠀⠀⡇
⡇⠀⠘⠳⣽⣯⣿⣿⣯⣿⣿⣿⣽⣗⣛⣉⣀⡈⠳⢍⡀⠒⠱⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⢀⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡉⠁⠀⠀⡇⠀⠀⡇
⠇⠀⠀⣏⢿⣿⣝⣿⢲⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠃⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢰⠀⡀⠙⠮⣽⣷⣻⣿⢿⢿⣿⣿⣿⣿⣿⣿⢿⣯⠞⠁⠀⠀⠀⠀⡇
⠘⢄⣈⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠓⠛⠋⠉⠀⠀⠀⠀⣀⣀⠜⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣓⢯⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠓⠘⠀⣦⡀
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠰⠏⢩⣿⡿⣭⠁⠀⢀⣴⣦⡤⣀⠀⠀⠀⣿⣿⣦⣤⣶⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⡷⢶⢀⠙⠁⠀⠄⣤⡄⠀⠀⠙⠞⠉⠁⠀⢰⣯⢞⣻⠿⠞⠁⠀⣀⣿⣿⣿⣿⣿⣿⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣦⣸⣿⠦
⠈⠀⠀⠶⠀⠀⡀⠀⠀⠄⣴⣦⣦⠀⠀⠀⠀⠈⢁⣼⣷⠀⠀⠀⠀⠙⣏⠙⠛⠛⠛⠋⠙⢢⣶⡔⠁⢀⡔⠆⠀⠀⠀⠘⠛⠓⠂
⠀⠀⠀⠀⠀⢸⡿⠀⠲⠖⠀⠈⠛⠀⠀⠀⠀⢠⣟⡵⢫⣦⣵⠂⠀⠐⠁⠀⣤⠀⢶⡀⠰⢟⡟⠻⢄⢾⡷⠤⡀
⠀⠀⠀⠀⠀⠠⢆⡆⠀⠀⣠⣶⡆⠀⠿⠿⠆⣼⠅⠀⢸⡿⡟⠀⠀⠀⢀⢤⣽⣶⠀⡿⠠⠈⣟⠀⣿⣎⠀⠀⣻⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⢀⣀⣤⣾⣿⣟⣡⠀⠑⠃⠒⡋⡅⡠⢋⢁⠛⣤⠀⠘⡄⡙⠂⣹⠯⠀⠀⠈⠀⣤⣄
⠀⠀⠀⠀⠋⠿⠀⠀⠀⢸⣦⡀⠀⠈⠻⣿⣿⣿⡿⠋⠀⢀⠤⣨⡖⢫⢣⣡⠀⢇⡅⢣⣴⠇⢿⡾⠛⢷⠀⠀⠺⣄⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠉⠁⢰⣦⣤⠈⢯⣟⡛⠀⠀⠈⠚⠋⠪⣵⣦⣬⡁⠀⠡⠵⢰⠀⠀⠰⣦⠟⠀⠀⢠⡄⡦⠀⠉⡉⠀⠘
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⣲⣂⣔⡏⠛⢁⠂⣹⣿⠀⢀⡄⢠⡆⢺⣿⣿⣻⡟⣆⡀⠀⣀⠀⠀⠀⣎⢝⡉⢢⣤⠈⡡⢾⠛⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢸⠢⡤⠜⠪⡿⠏⠁⠀⣈⣄⣻⣿⠵⠛⣇⢸⡇⣸⠉⢉⠟⠉⢹⠙⠯⣾⣷⡶⠀⠉⠀⠉⣹⠲⠤⢥⡤⣀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠘⢧⡝⣈⢄⠐⠀⢁⣴⣿⣿⡜⣱⢦⡄⢸⡸⡇⡇⡴⢁⡤⢞⣵⣿⣿⣍⠛⠉⠀⠀⠀⠀⢸⠂⠀⠈⢽⣭⣷⠄⣸
⠀⠀⠀⠀⠀⠀⠀⡈⣧⣬⠀⢀⠿⢿⢉⣿⣿⣿⣻⠀⣌⢢⢟⡉⡉⠳⡫⠴⢾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⡀⠀⣰⣾⡿⣷⡰⠁
⠀⠀⠀⠀⡈⠂⠀⠀⠀⠊⠀⠸⡆⢸⠎⡼⠁⠐⢙⡀⠀⠉⠈⣶⠈⠂⠟⠿⡍⡫⢯⠉⠀⠀⣰⣶⣤⡀⠀⠀⠁⠀⠉⠺⣿⠛⠝⠟
⠀⠐⢄⠀⠑⠀⠀⠀⠀⠀⠀⠀⣏⢨⡴⡇⠀⣰⠟⣿⣇⠀⠀⠋⠀⠀⠀⠁⠊⣵⢋⣀⡀⢰⡿⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡾⠃⠁⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢷⣄⠀⠀⣠⡖⢀⡁⢲⣄⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢀⠙⢷⣄⠙⠳⠌⠡⠞⠋⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⣧⠀⠙⢷⣄⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣼⣿⣿⣄⠀⠀⠙⢳⣄⠀⠺⣯⣉⣛⣭⡿⠂⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠺⠿⣿⡿⠿⠃⠀⢸⡏⢹⣿⣄⠈⠉⠉⠉⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣀⡀⠀⢠⣿⢠⡿⣡⡿⢷⣄⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢹⣧⢠⣿⣿⣿⣼⢏⣴⠃⠙⢷⣄⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣀⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣡⣤⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠋⠉⠢⡀⠀⠀⠀⠀⢀⡴⠚⠉⠓⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡌⠀⡀⠀⠀⠹⡄⠀⠀⣠⠋⠀⠀⠀⠀⢸⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⡇⠀⠀⠀⢀⠀⢳⠀⢰⠃⠀⡀⠀⠂⠀⡸⠀⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱⠀⠀⠐⠀⠀⢸⡄⡇⠀⠂⠀⠀⠀⢠⠃⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠀⠀⣿⠀⠀⢀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡤⡌⢣⠀⠀⠀⠀⡇⠀⠀⠀⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⢦⠀⠀⢸⢁⠇⠈⢢⠉⠀⢸⡇⠐⠈⠀⢀⡞⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠃⠓⠒⠛⠘⠛⠉⠉⠑⠒⠚⣄⠀⠀⢀⠌⠉⠙⠋⠉⠉⠳⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢀⡴⠉⠀⠀⢀⣀⠠⠀⠀⠀⡀⠀⡠⠄⠒⠢⢄⠀⠈⢦⡴⠃⠀⠀⠀⠐⡀⠀⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠈⠙⡄
⠀⠀⠀⡾⠀⢠⠦⡤⠤⡤⠵⡀⠀⠀⠁⠞⠒⠲⠰⠒⠘⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠁⠀⠀⠀⠀⠀⠓⠈⠒⠲
⠀⠀⠀⣧⠀⢇⣀⣐⣚⣁⣀⡇⠀⠀⠀⢦⠐⠒⠒⠒⢒⠇⠀⠀⠀⡀⠀⠂⢀⡎⠀⠂⠀⠀⣀⣀⣸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠤
⠀⠀⠀⢧⡀⠈⠢⣤⣠⡤⠊⠀⠀⠀⠀⠈⠒⠤⠤⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⢹⡀⠀⢀⡠⠄⠀⠀⠀⠀⠀⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠉⠁⢀⡞⠉⠙⠓⠲⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⢀⡤⠛⠋⠉⢀⣀⣤⣀⣄⠼⠉⠛⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⡤
⠀⠀⠀⡼⠀⢀⡔⠋⠉⠳⠤⢤⡄⢀⠀⠀⡤⡶⢋⠀⠀⡏⢹⠀⡆⣷⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢠⠇⠀⡼⠀⠀⠀⠀⠀⠈⠳⠋⠋⠛⢡⠃⡎⣼⠀⡇⢸⠀⡇⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠋⠁
⠀⢀⡞⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇⣿⡀⡇⢸⣠⢧⡎⢣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⡤⢤⠀⠀⠀⠀⢀⡖
⣞⠉⠄⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⠖⠋

⡇⡤⠤⠀⠀⠀⠒⠒⠂⠀⠠⠄⣀⡀
⠘⢄⡒⠤⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠀⠤⠄⢀⣀⠀⠀⠀⠠⡀
⠀⠈⠙⢇⠤⣀⡠⠇⠈⠉⠁⠹⠁⠀⠀⠀⡙⠉⠁⠙⠒⠒⠂⢠⠭⠅⢒⣤⣄⠀⠀⠠⠄⠀⠀⠈⠻⢓⣈⡦⣶⠒⠀⠀⠣
⠀⠀⠀⠀⠈⠐⠩⠍⣒⡒⠠⠬⠄⠤⠤⠀⢣⣀⣀⣠⣄⠀⡀⣠⣀⣀⡤⢽⣽⠛⠋⠈⠀⠀⠀⡠⠒⢣⠀⡰⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠐⠒⠐⠒⠀⠀⠀⠐⠂⠂⠀⠉⠁⡠⠀⠀⠀⢀⡄⠉⠙⡆⡾⡪⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠳⠤⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠛⢢⠠⡴⡪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠻⣀⢀⣡⡲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠻⡀⠀⢀⡧⡨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡠⠠⢔⠒⠉⠻⡀⠀⢀⣋⢝⠫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⡀⠀⠈⠄⠀⣀⡉⢔⡫⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠚⠓⠀⠁


⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⠀⠉⠉⠉⠀⠀⠀⠐⠒⠒⠂⠀⠢⠤⠄⠀⠀⠠⠤⠤⠤⢆⠀⠀⢀⡀⠀⠀⠀⡰⠄
⠀⠀⠠⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⠺⡷⠛⠋⠉
⠀⠀⠀⠀⠉⠒⠢⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠤⠄⠀⠀⠤⠤⠤⠤⠤⠀⠐⠒⠈⡿⠔⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⣶⣿⣷⣦⡀⠀⠀⠀⢠⣶⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⡓⢦⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡏⠈⠳⣤⡞⠁⠈⣧⠀⠹⢿⣿⣿⠿⠃⠀⠀⠀⠙⢿⣿⣿⡿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠴⠋⠙⠆⢠⢇⣤⣶⣶⣶⣶⣶⣦⣤⣤⣠⣴⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠖⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⡟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢻⣿⣿⡇⠀⢸⣿⣿⡏⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⣿⣿⣿⠁⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⣿⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⢹⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⡇⠀⢸⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⠃⠘⢿⠿⠃⠀⠘⠿⠿⠁⠘⠿⠿⠃

⡄⠀⠀⢠⠀⠀⠀⠀⠀⠘⠢⢌⣉⡙⠒⠣⣄⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⢘⣷⡂⠀⢻⠀⣇⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⡇⠀⡇⠀⠀⠠⡒⡉
⠀⠀⢀⠎⠀⠀⠀⠀⠀⣤⣀⡀⠠⢈⠑⠀⠀⠀⢀⠘⠀⠀⠃⠄⠀⣠⣑⣒⡮⣧⠠⠀⠀⠏⣸⠀⠸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠸⣄⠀⠀⠄⣂⡱⠭⠖
⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠁⠀⢄⠀⠀⠈⢀⠀⢉⠉⠿⣷⣐⠀⠐⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠑⠂⠠⠔⠒⣂⢀
⠉⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠈⠈⠰⢀⠄⠀⠀⠀⠀⠀⠀⠡⠈⠀⡀⠀⠀⠤⡀⠣⠴⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠑⠐⣆⠐⠈⠉⠈
⠄⠀⠀⠄⣀⠤⠴⠶⠌⠀⡢⢂⡤⢖⣠⠴⠤⠀⠑⠀⠠⠀⠀⠨⡐⠀⠀⠀⠀⠄⠄⠔⠉⣤⠶⠖⠛⠐⢤⡤⢊⠔⣝⡲⢤⣄⡀⠀⠀⠀⠐⠀⢀⠀⠈⢦⠀⢀
⠁⠀⢀⡀⠀⠀⠀⠀⠀⢀⡰⠋⡴⠫⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢸⠀⠸⠀⢀⠴⠶⡋⠡⣴⡁⡢⣄⣙⠛⠲⣍⠳⠄⠀⠀⠀⠈⢣⡀⠀⠀⠀⠑
⠁⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠜⡠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⢀⠈⠀⠀⠀⠁⠀⠈⠘⠉⠈⡁⠀⡈⠆⠆⠀⠂⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠘⠀⠀⠰⠀⠀⠠⠀⢁⡘⢦⠢⡠⡀⢦⣄⡤⠴⠶⡷⢶⡳⣶⣴⠖⠂⡀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⠈⠀⠠⠂⠀⣀⠤⠶⣤⡀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠈⢿⠀⠂⠙⢦⠤⠠⠤⠀⡊⠠⢫⡾⠃⡀⠐⠄⡀⠀⠀⠀⡀⠀⠀⠀⠠⢀
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠀⠀⠔⠀⡠⠋⠀⠀⣀⡈⠙⠧⠊⡑⢀⠈⠀⠀⠀⠀⠂⠲⣆⠀⠀⠀⠀⠀⠀⠀⠠⠐⠹⠉⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⡌⢰⢃⠀⠀⠀⠀⠙⠲⢄⠀⢀⢘⡤⠠⠀⠀⠀⠀⠀⢀⡉⠳⠲⠄⠀⠠⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠤⠀⠀⠀⠀⢀⡀⠀⡀⠠⠤⠤⡊⠀⠈⣸⢡⠐⠄⠀⠀⠀⠀⠔⠡⢂⢚⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠈⠀⠀⠑⡀⠀⡀⠀⠀⠀⠀⣤⠔⠊⠑
⣀⣒⠒⠢⣄⡈⠂⠁⠀⠀⠀⣿⠀⠀⠀⡧⠈⠸⣆⠀⠀⠀⣟⣠⡇⢸⣸⡅⠀⠈⠉⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠁⠀⠀⠁⠀⢠⠔⠁⠊⠀⣀
⡁⠅⠀⠀⠀⠑⣄⠀⠎⠀⠁⣈⢧⠀⠀⠈⠀⠀⠌⠙⢦⡄⠈⠸⠳⠈⢳⢳⡒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂⠈⠀⠂⠤⠀⠀⠀⠐⠊⡀
⠀⠀⢀⣸⠀⢠⡏⡀⠀⢄⠈⠁⠀⠣⡀⠀⠀⠀⠀⢀⣒⣛⡷⡀⠄⠀⢸⢀⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠄⠀⠊⠀⠀⠀⢠⠀⠐⡀⡀⢰⠏
⠀⢌⢢⠇⠀⠞⠀⢁⠔⠀⠀⠀⠀⠀⠈⠀⠄⠠⢒⣀⠦⢤⠮⢯⣅⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢠⠆⢀⡎⣐⠋⠂⠀⢀⣀⠈⣆⢣⠀⢠⡞
⠫⠔⠇⣂⠊⡁⠑⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠜⠁⠀⠀⠀⠀⠋⠉⢀⠔⠈⠀⠀⠀⠀⠆⠀⠠⠁⠀⠀⠂⠀⢠⠏⠀⡚⢀⠁⠠⠀⠀⠀⠍⠈⢿⡌⣦⠟⢁⣰
⡥⠬⡂⡈⠺⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠄⠀⢀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⡟⠃⠀⠠⠞
⠖⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⢠⠀⢀⣚⡀⠀⡃⠰⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠤⠤⢖
⠉⠀⠀⡀⠀⠀⣆⠀⠀⠀⠀⠐⠀⠀⣠⡤⢤⢀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠋⠉⠙⠳⣥⣾⡀⠀⠀⠀⡄⠀⠀⠈⠁⠀⠰⠚⠉
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢀⣠⣾⠗⠋⠱⣸⡇⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠐⡇⣾⡠⠐⢉⣛⢶⣤⢫⡟⣦⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣶⣯⢟⣭⣾⠧⣴⣿⣗⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⡞⠠⢫⢷⣎⠂⠀⠛⢿⣻⡝⡞⡅⠀⠁⠀⠰⠄⠀⡀
⠠⠄⠀⠀⠀⠀⠀⠀⠀⣻⠿⣽⣿⠿⠋⢰⡿⡜⢞⡷⠂⠀⠠⠀⠀⠀⠀⠀⠀⢈⠀⠐⠀⡀⠀⢹⡀⠃⠸⣎⡘⠦⣄⡀⠸⠿⣧⢳⣷⡀⠀⠄⡀
⠀⠀⠀⠀⠁⠐⠠⢄⡀⣿⠁⣿⠉⠀⠀⣾⠀⢃⡘⠇⡁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠈⢱⣄⠀⠈⠈⠂⠈⠻⣦⡀⠙⠀⡽⢻⡀⣀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠀⢀⠱⣏⠀⠫⣀⢚⣿⣃⡼⠡⠑⡌⠈⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠁⡀⠒⢀⣂⠐⣛⣷⡾⢿⣦⡈⢀⠂⠀⠀⠀⠁⠀⠀⢁⣀⣀
⣲⠀⠄⡀⠀⠀⠄⠀⠁⠈⢓⠶⠤⠽⠷⢁⠐⢀⠾⣍⠀⠄⠀⠀⢀⡠⠖⠐⠚⢶⣄⠄⠈⠀⠀⠀⠈⠀⠀⠈⠀⠈⠀⠁⠠⠈⠀⠉⠁⠀⠀⠀⠀⠀⢀⡜⣩⠄⠂
⣿⡄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠋⠉⠙⠉⠛⠉⠀⠋⠱⢀⡜⢡⢟⠀⠀⠄⢤⡁⠙⢷⡐⢂⠀⠰⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣪⠞⢀⡀
⢣⢳⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⠞⠀⡔⠋⠀⢁⡶⣀⠀⠀⠀⠙⠢⠀⠕⡡⢀⠂⠀⠀⠀⠀⠀⠀⠌⠀⢀⠀⠀⠀⠀⠀⠀⠒⡩⠚⡁⣴⣏⠀⢐
⠀⠃⠋⠱⠀⠀⠀⠓⠆⠀⠀⠀⠀⢀⠀⠀⠀⠉⠀⠰⡇⠀⠀⢞⠐⠈⡆⠀⠀⢠⡄⢡⡎⠀⡟⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠈⠀⢠⠴⡑⢌⠠⡁⡈⠉⠀⣼
⠘⠘⠀⠀⡇⠀⠀⢀⠀⡈⠢⣀⠀⠀⠁⠀⠀⠀⢱⠀⠡⠀⠀⠈⠂⠈⠳⣄⠀⢨⡅⣧⠁⡆⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢠⡎⠀⡇⠘⠀⠀⠀⣾

⣴⣦⣀⡟⠀⢸⣿⣧⡀⠀⠀⣠⣶⣷⡀⠘⣿⣷⣶⣤⡀⠙⣿⣿⣶⣤⣄⠀⢻⣿⡄⢰⡿⠛⠻⠿⠏⠀⠿⠿⠿⣦
⡟⠉⠉⠳⠶⠟⠉⢹⣟⣿⣿⣿⠁⠉⣿⠏⠁⠀⠀⠙⣿⣿⠟⠉⠙⣿⡟⠀⠀⠙⣷⣾⡄⠀⠀⠀⢀⣤⣄⡀⢀⣏⣸
⠿⣦⣶⣶⡆⢠⡎⠙⣟⣿⣬⣿⠀⣴⡏⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⢸⣧⠀⠀⢀⣿⡿⠛⠒⠲⢾⣿⠀⢈⣛⠉⠉⠙
⣠⣿⠟⣨⣤⣴⣿⠀⣿⠉⣠⠟⠀⢿⣿⣦⣀⠀⠀⠀⣿⣿⣦⣄⣀⣀⡛⠿⠿⢛⣡⣀⠀⠀⠀⠀⢻⣿⡿⠛⠛⠶⣾
⢿⣧⣴⡟⠉⣿⠏⠀⠈⣿⠏⠀⠀⠀⠉⢻⣿⣷⠀⣠⡀⠈⠉⠉⠙⢿⣿⡿⠂⣿⠟⠛⢧⠀⠀⠀⢠⡟⠀⠀⢠⣄⠙
⠀⠹⣿⡇⣸⣿⣄⣀⣠⣿⡀⠀⠀⠀⢀⣴⡿⠁⠀⣿⣿⣄⡀⠀⢀⣾⡏⠀⠀⢸⣷⣤⠈⣷⣦⡶⠟⠁⢀⣴⣿⣿⣿
⣀⣴⣿⠇⠈⠉⠉⠉⠉⠙⣧⠀⠀⢰⣿⣿⣧⣀⣠⠟⠉⣹⣿⠟⢋⣿⣷⣤⣴⣿⣁⣿⣷⠋⠀⡾⣿⠇⠘⠋⠀⠈⣿
⠛⠉⠁⢠⡾⠷⣄⠀⠀⠀⣿⠳⣤⣀⠈⠉⣻⡟⣡⣶⣾⠟⠛⠒⣿⡅⣸⡟⠁⠀⠀⠙⣧⡀⣠⣧⡼⣷⣴⣶⣄⢸⣿
⡀⠀⣠⣿⡇⢀⣽⣶⣤⣾⠿⠟⠳⠿⢿⣟⣉⣿⡿⠟⠁⠀⠀⠀⢸⣧⣿⡅⠀⠀⠀⠀⠘⢿⣿⣏⣀⣠⣌⡉⠛⣾⣿
⠿⠛⢿⣿⠋⢸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣠⣿⣿⡝⣿⡿⠛⠛⢿⡄⠀⠈⠉⠉⠛⣿⡉⢻⣻⡿
⠀⠀⠈⣿⣦⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣾⡿⠋⠉⠉⠙⢿⣷⠠⣤⣾⡇⠀⠀⠀⠀⠀⣼⣧⣾⣿
⠀⠀⠀⣿⣀⣻⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⠈⠻⣿⣄⠀⠀⠀⣠⣿⠃⠈⠙⢷⣀⠀⠀⢀⣴⣿⠁⠀⢻⣿
⣀⣤⣤⣀⠉⣻⣿⣦⣄⣀⣀⣀⣠⠴⣿⣿⡇⠀⠀⠀⠀⢠⠛⠉⢀⣴⣾⡿⣇⠀⠀⠀⣸⣿⣋⣻⠿⠿⣿⢧⣴⠋
⠻⣿⡟⠉⠙⣿⠉⣿⡏⠉⠉⣫⣥⣴⣾⣿⣿⣤⣀⠀⠀⣿⣦⣤⠞⠛⠛⢦⣀⠙⣶⣿⡟⠙⢿⡁⠀⣠⣿⡾⠋
⣾⡿⠃⠀⣼⣿⡇⠈⠑⢲⣾⡿⣿⠋⠀⠀⠀⠙⢻⣿⣦⡍⠋⠁⠀⠀⠀⠀⠹⣿⣮⠻⠿⠿⣿⣿⡄⠻⣿⠀⠀⣶⣶
⠋⠤⠄⠀⠘⠛⠓⠀⠀⠀⠛⠀⠘⠀⠀⠒⠒⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠀⠛⠁⠀⠙⠂⠀⠘⠂
⠀⠢⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⡀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠶⠶⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠋⠁
⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⠤⣄⣀⠀⠀⠹⡄⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢈⡙⠿⣯⣉⠉⠛⢉⡉⠟⠛⠻⠿⣿⣶⣄⣽⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠖⠉⠉⠉⠁⠐
⠀⠀⠈⠻⡄⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⣈⣹⣿⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣾⣁⡀⢤⠤⣤⣤⣀⣀⢀
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠙⠛⠿⠿⠟⠛⠉⠁⣴⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢙⡛⠻⠶⣶⣟⢿⡛⢿⣿⣷⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡈⣧⠀⠀⠀⠙⠷⣤⣉⠀⠀⠀⠀⣠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣀⣀⠀⣀⣠⣴⣾⠿⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢷⣄⠀⠀⠀⠀⠉⠛⠶⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠫⠾⠿⡿⠿⠋⠁⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠻⣿⣿⣷⣶⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠙⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⡶⠟⠛⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢠⣾⠉⠀⠀⠀⠀⢸⡟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇⠀⠀⠀⠀⠀⠈⣧⠈⠙⠳⢦⣄⡀⠀⠀⠀⣰⡿⣤⣉⣁⣀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⣋⠙⣿⡷⠶⠟⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣠⡶⠏⣡⠀⠀⢹⣇⠀⠀⠀⢹⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠁⠁⣊⡡⠀⠀⠀⠻⣄⠀⠀⣘⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠛⠉⠳⠞⠋⠙⠰⣶⣶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠙⢷⡀⠀⠀⣠⡶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⢀⢀⣀⠀⠀⢹⡇⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⢸⡇⢠⡾⠋⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠐⠁⠀⠀⠙⠃⠼⢿⣡⣿⢃⣀⠤⠤⠄⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⣿⢀⡤⠴⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⣿⡀⠀⢠⡔⠒⠐⠒⠀⢀⠀⠉⠉⠉⡠⠆⠒⠂⢠⠀⠀⠀⠟⠋⠉⠀⠀⠀⠁⠀⠙⠋⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⡒⠂⠁⠹⣧⠀⢨⠀⠀⠀⠀⠀⠀⠀⠢⡄⠀⢃⠀⠀⠀⠀⠀⢱⡀⠀⢰⡏⠀⠀⠀⢠⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠉⠛⠦⢤⣿⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⣻⠀⠸⡄⠀⠀⠀⠀⢨⡅⠀⠈⢃⠀⠀⠀⠠⠆⠀⢠⡎⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣄⠀⠀⠀⣀⠔⠁⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠘⢇⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⠿⠿⠿⠻⡿⠟⣽⠛⢿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣻⣟⣛⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁⠀⢹⣿⠟⠁⠀⢹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠤⡄⠀⠀⠀⢸⣿⣿⣿⣇⠀⣴⣄⡏⣠⣄⠀⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⡟⠀⠀⠀⢸⣿⣿⣿⣿⣦⣽⣿⣇⠙⢃⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⢤⣽⡄⠀⡏⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠙⡎⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠙⠲⠤⠤⠤⢺⡁⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠳⣄⣀⣀⣀⣴⡿⣧⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣈⣉⡽⠋⣠⡏⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠉⠻⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣿⣷⣶⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⡘⢿⡿⠀⡟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡗⠦⠔⠋⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⡇⢀⣀⠀⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⠒⠻⣄⣠⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⣶⣿⣿⣷⣤⠀⢀⣴⣿⣿⣷⣶⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠉⠉⠻⢿⣿⣿⣿⣿⣿⠃⢿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⢀⣀⣀⣀⠀⢻⣿⡿⠋⠁⠀⠀⠙⠻⣿⡿⠃⣀⡀⠀⢀⡠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛
⣀⣤⡤⢤⣄⡀⠀⢻⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⠧⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻
⣿⠀⠀⠀⠀⠉⢧⣨⣶⣇⠀⡴⠋⠉⣹⠂⢀⡤⠞⠋⠹⣆⣤⣤⠀⣤⣤⡄⡾⠛⠛⣷⣼⡇⠻⠶⡄⢸⠟⢿⣀⣤⣤⣤⠤⢤⡀⠀⢀⣀⡴⠚⢻⡇⢸⠀⢸
⣿⠀⢠⣶⠀⠀⢸⢿⠀⢸⠾⣁⣀⡉⠙⢦⣼⠀⣴⡄⠀⣿⡇⠘⠶⠃⢨⡟⣙⣲⠀⠳⡄⡇⢰⣚⣛⢻⠀⠘⡏⡇⠈⣡⡀⠀⢹⠖⡟⠁⠴⠄⠀⡇⢸⠦⣼
⣿⠀⠀⠀⠀⢀⠎⢸⠀⢸⡄⡨⠤⠽⠀⢸⠛⢤⣠⡀⠀⡏⢷⣀⠀⠀⡸⠀⣇⣀⣀⡴⠃⠹⣄⣈⣹⠟⢤⡤⠇⠷⠤⠇⠹⣤⡼⠀⠙⠤⡆⠀⠀⣷⢾⡖⣧
⢹⠦⠤⠔⠚⢁⣤⠈⠉⢉⠐⠧⣄⣀⡤⢊⡤⣄⣀⡇⠀⢳⣀⠉⠛⠋⢀⣤⣀⣀⣀⣠⢦⣀⠀⢉⡠⣤⣀⣀⢶⣀⣀⡖⢦⣠⠴⠶⢲⠀⣧⠀⠀⢹⣀⡽⠃
⠀⠙⠓⠚⠋⠁⠈⠉⠉⠉⠙⠒⠒⠒⠊⠁⠀⠈⠉⣷⠀⢸⡇⠙⠒⠒⠉⠀⠀⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⢾⣣⠇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠒⠃⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⠤⢤⡤⢞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣌⣙⡒⢒⣒⡽⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣾⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⣴⣦⢀⣶⡆⢠⣶⡄⢰⣦
⠀⠀⣴⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣾⣷⣶⣿⣷⣶⣿⣶⣿⣿⣷
⠀⠀⣿⣿⣿⣿⡇⠀⠶⠤⠤⠤⠄⠰⠤⠶⠴⠆⠶⠶⠶⠤⠤⠤⠄⠤⠤⠤⠴⠆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠈⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⣿⣿⣻⣿⣿⣿⣿⡇⢿⣿
⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠸⠿⠿⠈⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠠⠀⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿
⠀⣿⣿⣿⡿⠻⠟⠛⠃⡚⣻⣿⣿⣧⣴⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣩⣽⣻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠋⠉⠉⠉⠉⠉
⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⢁⠀⠠⠠⠐⠐⠐⠀⠀⠀⠀⠄⠀⡉⠙⠻⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⡠⠔⠈⠀⣤⡰⠒⣄⡎⢹⠄⣤⠀⠀⠀⠀⠀⠁⠢⢀⠈⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⢀⠀⠁⠀⠀⠀⢠⠇⣇⣰⠛⠦⠞⠀⠉⠁⠀⠀⠀⢀⣾⡇⠀⠈⠀⡀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠄⠀⠀⠀⠀⡀⠀⠈⠀⢠⡄⠀⠀⠀⣤⣠⠀⣠⠶⡶⢸⠟⠀⠀⠀⠀⠀⠠⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠃⢀⠈⠀⠀⣼⡇⢸⠇⡤⣴⠌⡟⢡⡇⣸⠃⣯⡟⢰⡏⣼⣃⠿⠤⠃⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠃⢀⠂⠀⠀⢰⡏⣷⡟⣾⢁⡟⣸⣇⠼⠵⠻⠞⠀⠑⢉⣉⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠸⠀⠙⠃⠙⠋⣉⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⣹⡏⠀⣠⡄⢘⣿⡇
⠀⠀⠀⠀⠠⠿⠀⠐⣀⣀⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣟⣉⠉⠈⢣⡄⠀⣾⣿⡄⠘⣿⡇⠀⣿⣿⠋⢻⣷⡀
⢠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠟⠉⣉⠉⢻⣿⣿⠀⢠⠀⢻⣿⣿⡀⢰⡄⠙⠀⢻⣿⠇⠀⠛⢻⣄⣈⣁⣠⣾⣿⣿⡄
⠀⠻⣿⣿⣿⡿⠛⠛⠻⣿⣉⡁⢠⣤⠀⢸⣿⠀⢸⡟⠒⠚⢻⡟⠀⣠⡤⠀⠻⠟⠃⢈⣙⣦⣠⣼⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠹⣿⡟⠀⣼⣿⡄⠈⣿⡇⠀⣠⡀⠉⢿⡄⠈⠛⠃⠀⣿⠁⢠⣤⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⣭
⠀⠀⠀⢹⣿⡀⠸⠿⠇⢀⣿⠋⣀⣈⣵⣤⣤⣿⣷⣶⣿⣿⣿⡆⣶⣬⡛⢿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⠁⢰⡿
⠀⠀⠀⣼⣿⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠁⣾⣿⣿⣦⡀⠀⠀⢀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠂⢀⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⣿⢟⣩⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⠂⢀⣾⠏
⠀⠀⠀⠉⠉⠉⠀⠐⣶⡄⠀⠄⠀⠀⠀⠀⠀⠀⠤⣶⣶⣿⣿⣿⣿⣿⣶⣌⡻⣿⡿⣿⣿⣿⠀⠀⠀⠀⠠⠂⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠐⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣻⡿⠿⠿⠿⣿⡔⣿⣿⣿⠇⠀⠀⠀⠂⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠁⠢⠄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣅⡛⠛⠁⠀⠐⠈⢀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⡈⠐⠠⠠⢀⢀⢉⠉⠉⠉⠉⠀⠄⠙⠒⠁⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⢶⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣶⠾⠟⠋⠁
⣶⣶⡶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠒⣶
⠀⢾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣰⣿⣷⠄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣴⣿⡟⠁
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣤⣤⣤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⢶⡶⣶⠒⠒⠚⠛⠛⠋⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⢿⣿⠟⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣴⣸⢇⣤⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡟⠁⠈⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣽⣿⣎⣋⣴⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠙⣧⡤⠂⠿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠒⠋⣹⣂⠀⣠⣧⣄⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣻⡏⠉⠁⠙⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⠛⢿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢈⠻⠷⣨⠓⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡜⠑⠈⠁⠀⠐⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠳⣔⣦⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⠿⠿⠛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣻⣿⠀⠀⠀⢸⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠾⣿⣽⣽⣿⣿⣷⡆⣴⣿⣿⣿⣿⣿⡿⠷⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⠀⠀⠀⠙⢿⣿⡿⠟⠁⠘⠻⣿⣿⡟⠁⠀⠀⠀⣨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠷⠾⠗⠚⠉⠀⠀⠀⠀⠀⠈⠙⠛⠶⠤⠤⠽⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⣶⣶⣾⣶⣶⣶⣶⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠀⠀⠉⠛⠿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠟⠉⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⣠⣾⣿⠿⠿⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣰⣿⣿⡟⠁⠀⣰⣾⣿⠋⠀⠀⠀⠀⣼⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀
⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⣼⣿⢿⣿⣦⣀⣀⣀⣴⣿⠏⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⢠⣿⣿⠏⠀⠀⢸⣿⠇⠀⠙⠛⠿⠿⠟⠛⠁⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⢀⣾⣿⡏⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⣼⣿⣿⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣷⣦⣤⣴⣶⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠸⣿⣿⠉⢻⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⣿⣿⡇⠀⠉⠛⠿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧
⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⢻⣿⡆
⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⣸⣿
⠀⠀⠀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠃⠀⢰⣿⡿
⠀⠀⠀⠀⠈⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠀⠀⠀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋⠁⠀⣀⡀⠀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣶⣾⠿⠟⠉⠁⠀⠀⠀⠀⠻⢇⣀⡀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⣤⣴⣶⣶⣶⣶⣄⡀⠈⠛⠁⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠻⠿⢿⣿⡿⠿⠛⠛⠛⠛⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡇⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣾⣿⣿⠿⠛⠉⠀⠀⠀⠀⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣾⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⡿⠿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣿⣿⡿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣤⡄⠀⠀⠀⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠸⠿⠇⠀⢿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣦⣤⣤⣤⣶⣶⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠟⠉⠉⠉⠉⠉⠉⠻⣿⣿⡀⠀⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⣠⣶⣿⠟⠁⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⢠⣶⣾⡶⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠻⣿⣿⣤⣄⣀⣠⣴⣾⣿⠿⠛⠁⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣶⣶⣦⣶⣶⣶⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⣿⣿⡿⠉⠀⠀⣹⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠁⠀⠀⠀⢿⣿⣿⣦⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⢠⣄⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠘⣿⣷⠀⠀⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⣿⣿⣧⠀⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⢰⣿⣿⣿⡄⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⣠⣿⡿⠻⣿⣿⣦⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣠⣼⣿⡿⠁⠀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡼⣍⠒⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡻⠶⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢠⠁
⠀⠀⢣⡅⣝⣶⣍⣇⠀⠀⠀⠀⠀⠀⣠⠎⠉⠁⠀⠈⠑⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⠈⠉⣶⠏
⠀⠀⠀⠉⠙⠛⠋⠛⠂⠀⠀⠀⠀⠠⢉⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠓⠀⠀⠀⠀⠀⣼⣟⡐⢒⡒⣚⡼⡏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣞⣉⣉⡭⠊⠁⡌⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣉⣭⠟⠋⢰⢇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢀⠿⠀⣀⣀⣴⣧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢟⣿⠏⣠⣃⣴⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡾⠿⠿⣿⠯⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⢿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠶⢶⣄⡀
⠀⠀⠀⠀⠀⡔⠣⡐⠒⠤⠤⠴⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣖⠶⠦⠤⢼⣿⣶⡄
⠀⠀⠀⠀⢈⠀⠀⢉⠢⠀⠀⠀⠀⡌⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢹⡗⢦⡀⢀⣀⣤⠿⣦⡀
⠀⠀⠀⠀⠘⢣⡀⣿⠀⠈⠳⢤⡀⠀⠠⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⡇⠀⡻⢦⣤⡠⣵⣽⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠙⠿⠂⣰⢤⠀⢹⡷⣻⣷⣿⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⢾⣿⣦⣼⣿⣿⣿⣿⣿⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢁⠔⠈⠉⠆⢥⠒⠊⠀⠀⠱⡇⠤⡺⠓⠝⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡁⠰⠃⠀⠰⠀⠀⠀⠡⣁⠀⠀⠀⠱⡀⠑⠀⠚⢠⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠃⢆⠀⠜⢧⠀⠀⠀⠘⡆⠀⠀⠀⢇⡀⢀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠘⡄⠀⠀⠘⡆⢐⠀⡀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⣠⠞⠉⠉⢢⠀⢀⠲⣀⠀⠀⢡⢰⠀⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢀⡏⠀⠀⠀⠈⢿⠃⠀⠀⠁⡀⠈⣶⡆⢈⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡃⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⠁⡇⢰⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⡄⡀⢀⡆⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⢧⠰⡇⢈⣸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡅⠀⡇⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⣴⡇⢸⢹⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡞⢄⣣⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢸⡃⢀⡖⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡇⠀⠁⢀⢿⡀⠀⠀⠀⠀⢻⢖⡀⠀⠀⠀⢸⡅⠰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⣅⠀⠙⡇⠀⠀⠀⠀⢸⠘⢅⠀⠀⠀⠘⡇⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⠦⢸⠐⡄⢻⡄⠀⠀⠈⢸⠀⠘⣄⠀⠀⣠⣇⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣔⡨⠄⣈⣐⠛⠢⠤⠖⠉⠂⠐⠁⣁⣈⡸⢅⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠬⠭⠭⠭⠭⠭⠭⠥⠖⠒⠋⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠋⠉⠉⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⢀⡠⠄⠒⠒⠒⠢⢤⡀⠀⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⠃⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⣨⠾⠛⠛⠋⠉⠉⠙⠓⠶⣤⡀⠀⠀⠀⡇⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢠⣇⢠⡄⣿⡴⢧⠹⡄
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣾⣡⡤⢟⣽⠞⠳⠏⠀⠈⢧⢹⡀
⠀⠀⠀⠀⠀⠀⢨⡅⠀⢰⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢸⡄⡆
⠀⠀⠀⠀⠀⠀⢘⡇⠀⣸⠀⠀⠐⠋⢉⣁⣉⣛⣦⡀⠀⠀⢀⣤⣤⣴⣟⣁⡀⠀⠀⠀⠀⠘⡇⣿
⠀⠀⠀⠀⠀⢀⡤⣧⠞⠁⠀⠀⢠⠞⠁⠀⣀⡈⠙⣟⠀⠀⠀⢀⠔⠉⣀⡀⠉⠢⡀⠀⠀⠀⠓⣿⡀
⠀⠀⠀⠀⢀⣸⡀⣇⠀⠀⠀⠀⡜⠀⠀⢸⣿⣷⡆⢸⠀⠀⠀⢸⠀⣾⣿⣷⠄⠀⣣⠀⠀⠀⠀⡿⢹
⢀⣀⠤⣰⠋⠉⠙⡿⠋⠉⢢⠀⠈⠀⣀⣈⣙⠋⠀⠄⠀⠀⠀⠘⣄⠈⣉⣉⣀⡠⠃⠀⠀⠀⢀⡥⠊
⡏⠀⠀⢿⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡼
⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⣇⠀⠀⢸⠀⠀⠀⣷⠀⠀⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢴⡇⠀⠀⢀⡞⠁
⣿⠀⠀⠸⠀⠀⠀⠛⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⠔⠉
⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠧⠀⢻⣓⣦⢤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⠶⢯⣁
⢿⠀⠀⠁⠀⠀⠀⢠⠋⠀⠀⠀⠀⢸⡇⠀⢠⠇⠉⠉⠻⠯⡭⠿⠋⠀⢻⡀⠀⠈⠳⣄
⢸⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⣇⡤⢾⠀⠀⠀⣠⣶⣶⣤⠀⠀⢸⠉⢧⡀⠀⠘⡆
⢸⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠸⠋⠀⣿⠀⠀⠀⠻⠻⠛⢿⠂⠀⠘⠆⢈⠇⠀⠀⡟
⠘⢧⣀⣀⣀⣀⣠⡽⠤⠤⠤⠤⠶⠃⠀⠀⡇⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⡗⠉⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡗⠒⠲⠶⠶⠤⠤⢤⣤⣤⣄⣧⠰⣞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⣀⠀⠀⡟⠛⠓⢲⡀⠀⠀⢈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠉⢙⡆⠀⢸⠛⠉⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠒⢲⡖⠚⠀⠀⠘⠒⢲⡒⠒⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠻⣄⠀⠀⠀⠀⠀⢈⡷⠚⠛⢦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠰⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠂⠀⠀⠀⠀⠀⠸⡀⠀⠀⠠⣀⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⠃⠀⠀⢀⡐⠁⣀⠤⠒⠘⡞⠁⠀⠀⣇⣀⠆⠀⠀⢀⡠⠒⠉⠀⠀⢱
⠀⠀⠀⠠⢦⠀⠀⠀⠀⢇⡀⣀⠔⢊⠩⠿⡏⠀⠠⢤⠄⠑⠦⡤⠊⡩⠋⣱⠄⠒⠉⠀⠀⠀⠀⣠⠃⡴⠞⠁⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠚⣉⠀⠀⠓⠾⣄⠀⠸⢤⣋⡠⠴⠳⣆⣀⣀⡀⢤⣐⡨⠔⠚⠣⡀⠀⠀⠀⠀⠘⠤⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠁⠀⠀⠀⠀⠙⢆⠀⢀⡁⢀⣴⡆⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠾⠂⢎⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠰⡰⠀⠀⠀⢀⠤⣄⢀
⠀⠀⠀⠀⣠⢬⡗⣎⣠⠤⠤⠊⠃⠀⠉⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣐⠀⠙⢶⣄⢀⠀⠀⠀⠀⢈⠂⠠⠄
⠀⠀⠀⠀⢯⣉⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⣀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⡋⠻⠷⠿⠫⢥⢾⣕⣒⣒⢒⠒⢒⢒⣒⣒⣾⠡⠮⠢
⠀⠀⣀⠀⢈⣿⡿⡟⡆⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⢸⠀⠀⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⡎⠘⢷⣺⡿⠁⡅⠈⠄⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠈⠁⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤
⠀⠀⠀⠀⠠⡇⠀⢀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠐⠚⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣽⠲⢋
⠀⠀⠀⠀⠀⢀⡀⠀⣆⠀⠈⠻⠮⣄⠀⠀⠀⠀⠀⠁⠢⠠⠄⠀⠄⠤⠊⠉⠒⠦⠤⣤⠤⣤⠀⡤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠁
⠀⠀⠀⠀⢀⣀⢴⠉⠁⠀⠀⢀⠇⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠁⠠⢀⠔⠁⢰⠾⡀⠀⠀⠀⠳⢤⣴⣀⡑⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠈⠒⠒⠁⠀⠑⠤⠀⠀⠠⠴⠶⠯⢝⡛⠲⠶⣤⣄⣈⠉⢒⠲⠦⠖⢒⡒⠢⢠⠴⠓⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣚⠛⠭⠿⠿⠶⢦⡤⠄⠄⠴⠁⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠒⠀⠐⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠐⠒⠒⠾⢗⣒⣒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣤⣤⣤⣤⣤⣶⣶⣿⣶⣿⡷⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣦⣾⡟⢻⣟⣿⣿⣿⣿⣿⣿⡉⠹⣿⣿⣟⠉⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⣿⣿⢿⣿⣧⠈⢻⡿⠛⠛⠛⠿⠿⠇⠀⢈⣿⣿⣆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠟⠋⠻⣿⠟⠻⣦⢻⣿⠀⠀⢀⣾⣦⠀⢠⡀⠀⠀⢸⣿⣿⣿⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣶⣤⣦⡀⠀⠈⣿⣶⣿⡇⠀⢸⣿⣿⡆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⡆⠀⣠⠀⠀⠀⠀⠀⠀⠉⠿⡇⠘⣟⣿⣿⣿⣿⣿⡄⠀⢿⣟⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠒⠛⠁⠉⠀⠀⠀⠀⠀⠀⣄⠀⠇⠀⣿⣿⣿⣿⣟⠃⠇⠀⢹⢿⣿⠃⠀⠀⠙⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡃⠺⠀⠐⠋⠛⠋⠀⠀⢰⣿⣵⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡈⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣟⠻⣿⣠⡤⠀⠙⢿⣿⣿⣿⣿
⣭⡿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠛⠻⠀⠀⠼⠿⠏⠘⢿
⣿⣿⣿⣶⣭⡓⠶⣤⣀⠀⠀⢄⢠⠥⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠿⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣿⣿⣿⣿⣿⣿⣶⣌⠙⢿⣧⣄⡄⠀⠀⠀⠤⠖⠁
⢿⣿⣿⣿⣿⡿⠛⠁⠀⣼⣿⣿⣏⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⠟⠀⠀⢀⣰⣿⣿⣟⣭⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⣿⣿⠀⠀⠀⣾⣿⣿⣧⣽⠖⣥⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣾⣧⣲⣦⣤⣄⡀
⣿⣿⣿⣿⡗⠀⠀⢸⣿⣿⣯⣷⢟⡹⡇⡩⢬⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠈⠀⠀⠈⠁⠔
⣿⣿⣿⣿⠂⠀⠀⠀⢙⣿⣯⣾⣊⣵⣆⢓⣫⢴⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠶⠂⠀⠀⠀⠀⢀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣫⣵⣦⣭⣭⣭⣭⣥⣝⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⢛⠻⠿⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⢀⣴⢟⣵⠿⡁⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢀⣴⢟⣵⡟⣩⡾⢋⣬⣥⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣡⡾⢋⣴⢆⣿⣿⣿⣧⣀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠟⠵⠿⠿⠿⠿⠿⠿⠏⣼⡟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣇⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⣿⣿⣿⡄

⣿⣿⢻⣻⡇
⣿⡯⡸⣿⡃
⣿⣿⣿⣿⡇

⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣀⣀⣀⡀⢀⣀⣀⠀⢀⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠸⣿⣿⣿⣿⣿⣿⣯⣿⠃⣿⣷⡀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣿⣿⣇⡇⠈⠙⠿⠿⠿⠟⠋⢰⣿⡷⡆⠀⢸⠀⠀⣼⣦⡀
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣏⣻⣿⢼⣧⠀⠀⠀⠀⠀⠀⠀⡀⣠⠷⣿⠀⢸⠀⢠⣿⣿⣿⣷⡄
⠐⡿⠶⢀⣀⠀⢀⣄⡆⠀⠀⠸⡏⢻⣿⠀⡿⠶⠶⠦⣤⡀⠀⠀⢐⠀⢴⣿⡀⢸⠠⣿⢿⣿⣿⡿⡀⠞
⠀⡇⠀⠈⠙⢿⡿⢻⠀⠀⠀⠰⡇⠀⠹⠀⡇⠀⠀⠀⠈⢿⡄⢘⣿⣦⣤⣿⡇⢸⠀⢀⢈⠥⠦⣰⠁⠀⢀⣤⣴
⠀⡇⠀⠀⠀⠈⠻⣮⠀⠀⠀⠐⠃⡀⠀⠀⡇⠀⠀⠀⠀⠘⣿⣾⣿⣿⡷⣂⠀⢸⠀⠘⠇⠀⠀⠛⠁⠈⠁⠘⢿
⠀⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠹⣇⠀⠀⡇⠀⠀⠀⠀⢀⣿⣿⣿⡿⠏⠛⠁⢸⣄⠀⠀⠀⠀⠀⢰⣤⣤⣦⢸
⠀⡇⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⣸
⠀⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣇⣰⣿
⠘⣷⣢⣤⣤⣾⣿⣿⣿⣷⣤⣶⣧⣀⡀⠀⠇⠀⠀⠠⠤⠤⠤⠀⠀⠀⠐⠒⠀⠈⠒⠀⠀⠀⠀⠀⠁⠁
⠀⠛⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛⠉⠉⠉⠙⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡜⡆⣠⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⠔⠓⠀⠀⠀⠀⠀⡴⠶⠀⠀⢠⣄⠀⠀⠠⠞⠁⢣⡇⠀⠀⠀⠈⠉⠙⠒⠒⠚⠼⠤⢤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⢀⠐⠒⠂⠄⠀⡀⢉⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠎⡀⠀⠐⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢇⠀⠀⠀⡸⢇⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⢀⠀⠙⠒⠊⠁⠈⠓⠒⠁⢀⠐⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿⣷⣦⣄⣀⣀⣀⣀⣤⣶⣿⡀⠀⠘⢲⡶⠤⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⡝⢹⠒⠒⠒⠦⠤⠤⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⡆
⠀⠀⠀⢀⡤⠒⢺⠃⡴⠊⠀⠹⣿⣿⡿⠉⠙⠉⢹⣿⣿⡿⠁⠀⠀⠀⣿⣄⠀⠀⣠⠊⠹⡴⢧⡀
⠀⠀⣰⠃⣠⠊⢹⠀⠱⢄⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠒⢦⡀⢸⡈⠑⠒⠃⣠⡦⠧⠰⡃
⠀⠀⡇⠀⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠃⡿⠈⠑⠒⠢⣀⡗⠀⢴⠃
⢀⠔⠀⠀⢹⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢨⢉⠂⠈
⡎⠀⡴⡶⠄⠨⠂⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣎⣸
⠳⠴⠷⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⢤⠤⡤⡤⣤⠤⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠒⠒⠲⠤⣼⠀⡇⡇⣿⡤⠴⠒⠒⠒⠦⣄
⣀⣀⣀⣀⣀⣀⣐⣧⣤⣤⣤⣤⣤⣤⣤⣤⡗⣧⣤⣤⣤⣤⣤⣤⣤⣬⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⡿⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣃⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⠿⠃⢀⣄⡸⠆⠀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠝⠟⣣⠀⢐⣿⣾⣿⣵⣐⣿⣷⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣶⡆⠀⢰⠟⠿⠏⢲⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⢠⠛⣿⠀⠋⠈⣴⣤⣾⣿⣦⣾⣿⣵⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⢀⠸⠁⠀⡀⣾⣿⣿⠟⢙⡻⣿⣿⠝⠿⢛⠛⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⡴⠉⠀⠿⢋⢛⣿⣿⠋⣰⣿⣦⣝⣿⣶⣾⣿⡖⠀⠰⠖⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠚⠿⣿⠿⢛⠀⠠⢿⣬⣻⣿⣡⣿⣿⣿⡿⣿⣿⣿⢋⡠⠤⠀⠀⠁⠀⠘⠛⠀⢀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠈⡻⣶⣄⡠⣄⡄⠀⣠⠔⠃⢁⡀⣹⣿⣶⣿⣇⠀⢉⡋⠙⠛⠛⠛⠚⠳⠍⠀⠀⠄⠀⠀⠀⠀⠀⠐⠊
⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢀⡔⣿⣿⣼⣿⡼⠿⠝⠉⠀⠀⢹⣿⣿⣿⣿⣿⢛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⡀⠀⣠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡄⢀⣀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⣾⣿⣿⣷⣴⣶⣷⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠊⠀⠉⠈⠀⠈⠻⣿⣿⣿⣿⣟⣻⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⠷
⠀⡔⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠁
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⡀⣀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢨⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⠿⠿⠟⠀⠀⠀⢠⣶⣤⠀⠀⠀⠐⠚⠋⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠦⣄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠄⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠈⢣⠀⡴⠊⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡀⠀⡟⠀⡀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠈⠁⢀⡇⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣠⠞⠹⣄⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⣄⠀⠀⠀⠙⠓⠒⠛⠁⠀⠀⠈⠛⠒⠛⠁⠀⠀⢀⠶⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠤⣄⣀⣤⡤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢶⡇⠀⢀⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⣤⣀
⠀⠀⠀⠀⠀⣠⠞⠋⠀⣀⣼⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠈⠓⢦⡀
⠀⠀⠀⠀⡴⠋⠀⣠⠞⠉⢸⡇⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠶⣤⠀⠀⣿⠈⠙⢦⡀⠀⠻⡄
⠀⠀⠀⣸⠁⠀⢰⠇⠀⠀⢸⡇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠆⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⣿⠀⠀⠀⢷⠀⠀⢻⡀
⠀⠀⠀⡏⠀⠀⢸⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⣿⠀⠀⠀⣸⠀⠀⠈⡇
⠀⢀⡴⠃⠀⠀⠈⢻⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢠⠉⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠉⠀⠀⠀⠳⣄
⢀⡟⠀⠀⣀⡀⠀⠘⠷⣤⠀⢹⡄⠀⠀⠀⠀⠀⠈⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⡴⠛⠀⠀⣀⡀⠀⠘
⡼⠀⠀⣼⢻⡝⠦⣄⣠⠾⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠸⢦⣠⡤⠞⣽⢻⡄
⢧⡀⣠⣏⣨⡇⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⣧⣈⣧
⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠤⣤⣤⣀⡀⣸⠀⢸⡇⣿⠀⣿⠀⣀⣤⣤⡤⠶⠶⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠀⢸⡇⣿⠀⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠲⠋⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⣶⠀⠀⠀⠀⢠⣖⡶⠰⠾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣀⠈⠉⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠿⠃⠀⠀⠀⠀⢰⣟⣷⠀⠀⠀⢰⡏⠉⢷⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠟⠸⣟⡆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠳⠶⠋⠀⠠⣏⡇⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⣯⠇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡈⠛⠃⠀⠀⠀⣀
⠀⠀⢀⡴⠛⠉⠉⠉⠉⠉⠙⢳⡄⢀⡴⠚⠛⣷⠞⠉⠉⠙⢦⡀⢀⡴⠛⠉⠉⠛⠶⣄⠀⡴⠋⠀⠀⠀⠀⠙⢦⠀⡠⠞⠉⠉⠉⠛⠲⡄
⠀⣰⠋⠀⠸⠁⣀⣤⣤⡀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠀⡖⠀⠀⠀⠀⠈⢿⠃⠀⠘⠁⠀⠀⠀⢈⡿⠁⠀⠀⠀⢠⣤⡴⠃
⢠⡏⠀⠀⢀⡾⠉⠀⠀⢷⠷⣼⠃⠀⡇⠐⠀⢀⠶⡄⠀⠠⡞⢹⠀⠀⠀⡠⢤⠀⠀⠀⠘⡆⠀⠀⠀⡔⠒⠋⠉⠱⠤⠤⠤⣤⡀⠉⠉⠲⣄⢸⣿
⢸⡇⠀⠀⢸⠁⠀⠀⣤⣀⣀⣀⣀⣀⣧⣀⠀⠀⠷⠃⠀⠀⠹⣿⠀⠀⠀⣇⡸⠀⠀⠀⢀⡇⠀⠀⠀⠧⣀⣀⡀⠀⡀⠀⠀⠀⣹⠀⠀⠀⢹⡄
⠘⡇⠀⠀⠘⣇⣀⣸⣧⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⠀⠀⠀⡼⠻⢿⣶⣄⠀⠀⠀⠹⡏⠉⠉⠛⠋⠁⠨⠃⠀⢸⣧
⠀⣿⡄⠀⠀⠈⠻⠭⡽⠇⠀⠀⠀⣴⢶⠚⠀⠀⢠⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⡞⢳⠀⠀⠉⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⢸⡄
⢰⠃⠙⠶⣄⠀⠀⠀⠀⢠⣄⠀⠀⣿⢸⠀⠀⠀⢸⡟⣆⠀⠀⠀⠀⣶⢦⠀⣤⠀⣸⡇⢸⣠⣦⠀⠀⠀⢀⣤⡔⠁⠀⠀⠀⠀⠀⣀⢸⡇⠀⢸⡇
⡿⠀⠀⠀⢸⠀⡀⠀⠀⣏⢹⣤⢠⣼⠈⠛⢻⠀⢸⠀⢸⠀⣠⡀⡟⠁⢸⢰⡟⠷⠿⠇⠀⠉⠸⣄⣶⡀⢸⠀⢻⡀⡶⡄⣴⢦⠐⡟⠛⠀⠀⠘⡇
⡇⠀⠀⠀⢸⣸⠟⠞⣇⣿⠀⢸⢸⠀⠀⠀⠸⢾⣼⠀⢸⠀⡏⠛⠁⠀⢸⣆⡇⠀⠀⠀⠀⠀⠀⠛⠁⣇⣼⠀⠀⢿⡇⠳⠟⠸⣦⠇⢀⠀⠀⠀⡇
⡇⠀⠀⠀⠈⠟⠀⠀⠙⠃⠀⠘⠿⠀⠀⠀⠀⠀⣿⠄⠸⣦⠇⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⣇
⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡀⠀⣿
⣿⠀⠀⠀⠂⠀⠀⠘⠆⠀⠀⠐⠄⠀⣰⣆⢀⡀⠰⠄⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻
⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⢸⠞⣧⠀⠀⠀⣴⠀⢠⣦⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⣿⣀⠞⣧⠀⠀⣤⠀⡄⠀⣴⠀⣀⢀⣄⠀⠀⢸
⢻⠀⠀⠀⠀⠀⡼⢻⠀⠀⢀⣄⠀⢰⠃⠀⠀⣿⠀⠀⢰⡏⡇⡾⠸⡄⠀⢸⠇⢳⡀⣷⣴⠀⢸⠉⠛⠀⢸⠀⠀⢸⠶⠻⠞⢹⡆⠉⢸⢹⡄⠀⢸
⢸⠀⢠⣤⠀⠀⡇⢸⠀⠀⣸⢹⡄⢸⠀⠀⠀⢸⠀⠀⢸⠀⠙⠁⠀⡇⠦⢸⠀⠈⠻⠁⢹⡆⢸⠀⠀⠀⢸⡆⠀⢸⠀⠀⠀⢸⡇⠀⢸⠈⣇⠀⢸
⠘⢷⠏⢸⠀⠀⡇⠈⡇⠀⡁⠀⡇⢸⠀⠀⠀⢸⡄⠀⢸⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠸⡇⠀⢸⠀⠹⣦⡞
⠀⠀⠀⠀⡆⠀⡇⠀⣧⠀⡇⠀⠹⠟⠀⠀⠀⠈⡇⠀⢸⠀⠀⠀⠀⣷⠀⣿⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⢷⣸⠁⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡾⠀⠀⠀⠀⣿⠀⡻⠀⠀⠀⠀⠸⡇⣸⠀⠀⠀⠀⢻⡼⠁⠀⠀⠀⠀⠘⢦⡟
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠃⠀⠀⠀⠀⠸⣆⡇⠀⠀⠀⠀⠀⣷⠏