Penguin Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠛⠿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠆⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⢀⡀⠤⠤⡤⠠⠤⠤⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠺⡍⢀⡀⠄⠀⠀⠀⡠⠀⣀⠐⡸⠀⠀⠀⠀⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠈⠢⡔⠂⠀⠀⠀⠒⠈⡤⠜⠁⠀⠀⠀⠀⠈⢻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠻⠿⠿⠿⠿⠿⠟⠛⠋⠉⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏⠉⠛⠛⠿⠿⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡚⠝⡻⠿⣿⣿⣿⣿⣿⣿⢿⠛⠉⠁⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠿⡛⠅⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠌⠁⠀⢠⡁⠂⠘⣉⣭⣂⣒⠈⡄⠀⠀⠀⢠⡉⠀⠂⠀⣬⠍⢀⡐⠊⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠐⠦⠐⠠⠔⠉⠐⠤⠦⠃⠀⠀⠀⠈⠳⠶⠊⠉⠲⢤⠐⠤⠂⠁⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣰⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠛⠛⠋⠣⡀⠑⠩⡙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠊⠀⠀⠀⠀⠐⠄⠀⠈⠢⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⣤⠴⠒⠋⠉⠳⡄⠀⠀⠀⠀⣠⠴⠖⠋⠉⠉⠉⠛⢲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣸⣀⣠⣴⠤⠴⠖⠛⠉⠁⠀⠀⠀⠀⣀⡤⢿⡀⠀⣠⠞⠁⠀⠇⠀⠀⠀⡠⠊⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠛⠉⠁⠀⠀⠙⣶⢇⠀⠀⠀⠰⡀⠀⠐⠀⠀⢀⣀⠀⢀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠦⠴⠶⠶⠶⠒⠛⠋⠉⠃⢄⠀⠀⠀⠀⢀⠔⠒⠒⢠⡏⠀⠑⠠⣀⠀⠑⢄⣀⠔⠊⠁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣉⠝⡛⢍⠁⠐⠠⣀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠘⠀⠀⠀⠸⠗⠀⠀⠀⢸⢀⠀⢀⡃⠀⠸⡇⠀⢻⡂⠀⠀⠒⠈⠀⢀⠃⠀⠡⠀⠀⠀⠣⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣧⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢎⠁⠀⠀⢩⠂⠀⠀⠀⠀⠷⣄⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠙⠳⠾⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⢀⡟⠳⢤⣊⠀⠀⠀⠀⢀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠛⠀⠐⠀⠀⠀⠉⠙⠓⠲⠦⣤⣄⣀⣀⣀⣀⠀⣀⣀⣀⣀⣌⣼⣥⠀⠀⣼⠉⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⣀⠉⠀⠘⠃⡀⠁⢀⣀⠀⢤⠀⠀⠀⢈⠉⠉⠉⣉⠉⠉⠀⠀⠔⢸⠇⢀⡏⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⡾⠉⠙⠲⠦⣤⣀⡀⠈⠁⠠⠄⠀⠛⠀⡈⠀⠾⠀⣀⠀⠘⠋⠀⠤⢸⡀⢸⠁⡿
⠀⠀⠀⠀⠀⠀⢠⠖⢋⡠⠄⠐⢺⠃⠀⠀⠀⡘⠀⠈⠉⠛⠓⠲⠦⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠤⢶⡻⣄⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠻⣼⣁⣀⠀⢀⡟⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡗⢸⡧⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡇⡾⣆⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⢻⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡟⠘⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⡔⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠙⢦⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠁⠒⠠⠄⠄⠊⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠧⠴⣄⣼⠃⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡷⠒⠒⠚⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠓⠚⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⢿⣿⣿⣿⣿⣿⠟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⢤⣀⠀⠀⣿⣿⡇⠀⢀⣤⢀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠸⣾⡿⠀⢀⣿⣿⣇⠀⠸⣿⡾⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠏⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠶⢿⣿⣿⡿⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠰⠁⠀⠀⢻⡟⠀⠀⠈⡆⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠚⠻⣿⣿⠛⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠰⠿⠀⣼⣧⠀⠿⠆⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣦⡄⣠⡄⢸⡇⢠⡄⢡⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠙⠋⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢻⣿⣿⡷⠈⠀⠻⠇⠈⠁⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠁
⠀⠀⠀⠀⠈⠻⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣄⡀⠀⠀⠀⣀⣤⣤⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⢆⠀⣀⣀⠀⠀⣀⡀⠀⣼⣿⣿⣿⣿⠷⣶⣶⡶⢿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠥⠼⠿⠟⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠸⢿⣿⡿⠦⠴⢿⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠾⠿⠿⠶⣷⣴⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣤⣶⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⠟⠋⠉⠁⠠⠌⠙⠻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠚⢤⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠐⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣼⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣵⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢿⣧⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⡀⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣽⣇⠀⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠈⢿⣧⣠⣄⠀⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⣸⣿⣿⢿⡿⠦⠶⠀⣮⠁⠈⠀⢿⠂⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢁⣀⣄⣼⣿⣿⣧⣄⡀⠀⠸⡟⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠹⣯⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣼⣿⣿⣿⣿⣿⠿⠟⠃⠀⠘⠶⠆⠰⢆⣤⣶⣿⣷⣤⡄⠴⣶⠿⢿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠁⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⢀⣄⠀⠀⣠⡀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠈⠛⠀⠀⠙⠃⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠇⠀⠀⠀⠀⠠⣶⣶⣶⣦⠀⠀⠀⠀⠸⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⣿⣿⣿⣿⣿⠏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠏⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⠏⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢻⣿⣿⣿
⠀⠀⠀⢿⣿⠋⠀⠀⠸⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠻⣿⡟
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠙⠛⠻⠿⠷⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⠶⠟⠛⠋⠁⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⠃⠀⠀⠈⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⡿⠁⠀⠀⠀⠀⠘⠿⠛⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠖⢦⡄⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⡞⠛⢲⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣷⣤⣼⣿⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣶⣿⡇⠀⠀⠘⣿⡿⠛⠋⠉⠙⠛⣿⠀⠀⠙⠿⠿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠈⠙⠓⠲⠶⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠷⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⡷⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣷⣄⣈⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣠⣴⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⣸⣿⣿⣿⣿⠁⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢰⡗⢠⣿⡿⠻⣿⣟⠀⡄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⣿⣦⣠⣿⣿⠈⠋⠀⠸⡏
⠀⠀⠀⠀⢰⣃⠈⠀⠀⠐⠢⣄⡀⠀⠀⣠⣾⣄⠀⠀⠀⠙⠿⢿⡿⠿⠋⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠉⠳⡤⠄⠀⠀⠀⠈⣷⡶⠏⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠶⠂⠂⠀⠀⠘⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣠⡠⣄⣴⣹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢻⣿⣧⣀⣀⡠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣯⣁⢈⡀⠀⠀⠀⠀⠀⠀⢄⠆⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠸⣿⣿⣗⣺⠄⡔⠀⠀⢀⣠⡾⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⣿⣿⣿⣿⣴⣶⣷⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢹⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣃
⠀⠀⠀⠀⠀⠀⢠⡏⠹⣿⡉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣦
⠀⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠿⡀
⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠐⣣⠐⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠿⣟⡂
⠀⠀⠀⠀⢸⠦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⢠⠆⠘⠁⠀⠀⡴⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢴⣿⣿
⠀⠀⠀⠀⠀⢄⠈⠀⠀⠀⣶⣄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⡏
⠀⠀⠀⠀⠀⠐⣠⠂⠀⠀⠈⢣⡞⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⡶
⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠈⢻⣿⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⣄⢦⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡷⠄⠀⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣣⣵⠄⠀⠀⠀⣈⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣰⣿⣿⣿⣿⣿⣧⣆⣠⠀⠠⡀⠀⠀⢀⠀⢀⡄⢀⣤⣟⡅⡀⢀⣤⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣦⣴⣮⣤⣼⣶⣿⣿⣹⣷⣿⣮⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⣻⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠻⠛⠉⠿⠛⠁⠀⠀⠈⠀⠀⠀⠀⠘⠻⠶⠛⠷⠿⠛⠋⠁

⣿⠿⡿⢿⠿⠿⠟⠿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⠿⢿⣿⡿⠿⠿⣿⣿⡿⠿
⣿⣿⣿⣶⣿⣿⣿⣿⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⣈⠉⡁⠢
⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⠈⠹⠂⢡⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠅⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠄⠀⠐⣁⠈
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡄⠀⠀⠀⠀⠠⠐⡒⠠
⠀⠀⠀⠀⠀⢄⠢⣀⠄⠀⠠⢀⠌⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠙⢀⠤⡊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢔⠀⠈⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠑⠄⠁⠀⢀⡈⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⢰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠄⠢⠐⢂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠀⠀⢤⣼⣶⣤⣄⡐⠦⣤⣶⣴
⠀⠀⠀⠀⢠⣀⡨⠖⠛⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⢄⠰⢁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⢀⣸⣻⣿⣿⣿⠉⠉⠛⠓⠏
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠚⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⣀⣉⣉⣉⣉⣹⠇
⠀⠀⠀⠀⢀⡀⠀⣁⠀⡀⡠⠌⢂⣤⣴⣈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣀⠀⠀⠀⠀⣀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⡅⠈⠉⠋⡩⠜⢀
⠀⠀⠀⠀⠨⠁⠀⠀⠀⠘⢾⣃⣈⡛⡛⣲⠄⡀⠀⠀⠀⠀⠀⠀⢹⣛⣛⢓⣲⠿⠋⠉⠉⠩⢁⠀⠀⠀⠀⠀⠀⠀⣔⠫⠛⠛⠀⡤⠵⠖⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠚⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣭⣽⣷⣶⣦⣤⠤⠴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣫⢏⣻⣿⣿⣿⣷⣮⡭⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠛⠛⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⢀⣤⣴⣶⠿⠟⠛⠛⠋⠙⠛⠛⠋⠛⠻⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣤⣤⣤⣀⣀
⢠⣾⡿⠛⠉⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠉⠙⠛⠷⢦⣄⡀
⠻⣿⣧⣤⠬⠭⠵⠒⠒⠋⠙⠛⠷⣬⣳⣆⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣤⣤⠴⠒⠲⢦⡀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠦⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠟⢷⣀⣴⣄⠀⠀⢀⣠⠖⠀⠈⠳⠋⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣂⡀⡛⠋⠙⠷⠞⠋⠁⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠷⣿⢷⣶⣦⣤⣾⡴⠦⢾⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⢥⣾⡧⠠⣭⣷⡄⠈⠁⠀⢠⣶⣋⣬⣍⣧⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣽⣥⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢀⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣀⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣼⠉⣿⡄⠀⢿⣿⣿⣿⣿⣿⣿⠁⠀⣾⡏⢹⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣿⣿⣿⠇⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⠛⠋⠀⣠⡿⠟⠉⠀⠉⠻⢿⣧⡀⠈⠙⠋⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣾⣧⣄⣀⣀⣀⣀⣀⣀⣤⡿⡆⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣽⣦⠀⠀⠀⠀⠀⠈⠳⢭⣉⣉⣉⣭⠶⠋⠀⠀⠀⠀⠀⣠⣾⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠳⠤⣀⡀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠬⠿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⠟⠓⠒⠢⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣥⣄⡀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⠷⠶⠶⠶⠶⠶⠶⠶⠾⠿⢿⣿⡛⠛⠋⠉⢀⠀⠀⢀⣀⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣶⣶⣶⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⡿⠿⠿⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣦⠀⠀⢀⠀⠐⡀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣤⡀
⠀⠀⢀⡾⠀⡀⠀⠀⠐⢷⣄⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠚⢉⠁⠀⠠⠃⠳⡀
⠀⠀⢾⠃⠀⠱⠀⠐⠋⠀⢩⠙⠒⠀⢿⢷⠀⠀⠀⢀⠀⠀⠨⠀⢀⠚⡂⠀⣼⣉⡤⠀⠀⠀⣇⠀⢠⣇
⠀⠀⠈⠀⣀⠀⠀⠀⠀⡄⣘⡀⠀⠀⢰⠀⠀⠁⡀⠈⠀⠀⠀⠀⢿⢟⠃⠀⠀⠈⠂⠀⠀⠐⠘⠀⠘⢹
⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠓⠀⠛⠛⠀⠀⠀⠈⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠲⠎
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⣨⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿⣿⣷⣴⣶⣌⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⣾⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡄⠹⡄
⠀⠀⠀⠀⢀⣴⠞⢛⠃⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠙⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠉⠉⠀⠀⣇
⠀⠀⠀⠀⣾⠇⠀⠉⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢷⣦⣤⣴⣿⡄
⠀⠀⠀⢰⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⠀⠉⠛⠿⣿⣿⣦
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⡇⠀⠀⠀⠀⠸⣿⣿⣧
⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠇⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠈⡆⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣧⣤⠿⠋⠀⠀⠀⠀⠀⢠⠇⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⢠⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢱⠀⠀⢸⣿⣿⣿⡟⠿⠿⠛⠋⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠼⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⢸⣿⣿⠿⠿⠿⠇⠀⠀⠀⠀⢸⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠱⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄
⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣦⠄⣀⢀⣀⠰⣿⣿⣷
⠀⠀⠀⠻⠀⠀⣀⣤⣄⣀⣴⣥⡄⠀⠀⠀⠀⠈⢻⣷⣄⣤⣤⣤⡀⣀⣠⣮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⠁⢀⠈⣿⡏⠀⠈⠙⠋⠓
⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠼⠟⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠰⠉⡙⠟⡉⢱⠀⠀⠀⢠⡀⠀⡎⠉⠻⠋⠉⠆⠀⣠⠀⠀⠀⡜⢉⠛⠋⠉⢣
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠁⠀⡸⠀⠀⠀⠘⢿⡄⢇⠀⠈⠀⢠⠇⣼⠏⠀⠀⠀⢡⡀⠈⠁⢀⡜
⠀⠀⠀⠀⠀⢠⡞⠈⠉⠈⠉⠘⣦⠀⠀⠀⠈⣻⠃⠉⠁⠉⠁⢻⡋⠀⠀⠀⢠⣏⠈⠉⠉⠁⣹⡄
⠀⠀⠀⠀⣰⢿⡇⠀⠀⠀⠀⠀⣿⢷⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢺⣿⡆⠀⠀⢰⣿⡷
⠀⠀⠀⠈⠁⠀⢻⣀⡀⠀⢀⣰⠋⠀⠁⠀⠀⠘⣆⣀⠀⠀⣀⡞⠁⠀⠀⠀⠘⢿⣃⠀⠀⣛⡽⠃
⠀⠀⠀⠀⠀⠀⠰⠥⠬⠾⠥⠼⠂⠀⠀⠀⠀⠀⠮⠤⠵⠯⠤⠗⠀⠀⠀⠀⠀⠸⠤⠽⠮⠤⠵
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠉⠙⡟⠉⢹⠀⠀⠀⠀⠀⠀⡎⠉⢛⠋⠉⠆⠀⢀⡀⠀⠀⡎⠠⢝⡫⠄⢱
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠁⢀⣜⠀⠀⠀⠀⠀⠀⢣⡀⠈⠀⣠⠃⣴⡟⠁⠀⠀⣱⢄⠀⠀⡠⣎
⠀⠀⠀⠀⣀⣴⡞⠀⠉⠉⠁⠈⣶⣄⠀⠀⠀⣴⠃⠈⠉⠉⠀⢹⡋⠀⠀⠀⣴⠇⠀⠈⠁⠀⠸⣦
⠀⠀⠀⡾⠟⢹⡇⠀⠀⠀⠀⠀⣿⠙⠷⡄⣼⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⡄⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⣀⣼⠃⠀⠀⠀⠛⠙⢧⣀⠀⢀⣠⡞⠀⠀⠀⠈⠈⢳⡄⣀⢀⡠⡞⠁⠁
⠀⠀⠀⠀⠀⠀⠰⠧⠼⠾⠤⠼⠂⠀⠀⠀⠀⠀⠺⠤⠷⠧⠤⠗⠀⠀⠀⠀⠀⠘⠒⠚⠓⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡽⢤⡀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣏⠀⢸⢰⣧⣷⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⢉⣀⡬⣧⠈⠉⠁⠈⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠿⠿⠭⠥⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡤⠖⠋⠉⠉⠉⠉⠉⠛⠲⣄⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢠⠇⠀⠀⣠⣾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠳⡀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡟⠀⠀⡴⠃⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⠀⠹⡄⠀⠀⢹
⠀⠀⠀⠀⢸⡇⠀⡾⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠙⢆⠀⢸⡆
⠀⠀⠀⠀⢸⣄⡼⠁⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠇⠀⠀⠀⠈⠳⣼⠇
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⡆⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠦⠤⠤⠤⠤⣤⢤⡤⠤⠤⠶⠚⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⡤⣄⣀⣠⣀⣀⡾⠸⣅⣠⣤⣤⠶⠶⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⠳⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠓⢮⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⣴⣾⣭⣦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡞⠀⠀⠀⠈⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠸⢿⣷⡿⠀⠀⠀⠀⣠⢒⡁⠭⠟⠃⠀⠇⠀⠀⠀⠀⢸⡏⢉⣛⡻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣟⢻⡇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⢀⣃⣻⣿⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⠗⠉⢀⣦⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⡆⠀⠀⠀⠀⠀⠀⠠⡱⠦⢤⠪⠟⠇⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⢠⣴⠄⠀⣀⣇⠈⠀⠀⠀⠀⢰⠻⡇⠂⢿⠖
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠘⢿⣿⡄⠀⠀⠀⠀⡀⠀⠀⢈⠇⠀⢽⡀⠀⠀⠀⠘⠛⠀⠀⠸⠟⠀⠀⠀⠀⣴⢋⠀⠀⢀⢾
⠀⠀⠀⠀⠀⠀⡯⣷⡇⠀⠀⢻⡇⠀⠀⢀⣾⠃⠀⠀⠀⢀⡤⢿⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣣⣾⡞⠉⣾⡁
⠀⠀⠀⠀⠀⠀⢷⣿⡆⠀⠀⠀⠃⠀⠀⢸⠋⠀⠠⠀⣼⠋⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣿⡿⣋⣻⡿⣧
⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠞⠀⡃⠀⣄⣿⠀⠀⠀⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⢿⡿⣧⠉⠃⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⣴⢎⠀⡔⠀⠀⠸⣤⣼⡅⠀⢠⠄⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡄⠻⠐⢄⣜⣯⡛
⠀⠀⠀⠀⠀⠚⠛⠛⣡⣤⣥⣤⣈⣿⣤⣤⣀⣀⣿⣿⣳⣶⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣅⣳⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣷⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠛⠉⠛⣿⣿⣿⡇⠀⣠⣤⣶⣶⣿⣿⣿⣿⠿⠿⠿⠟⠛⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠀⣠⣿⣿⣿⠃⠀⠈⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⣠⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣯⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠁⣀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣄⡀⠻⠿⠿⢿⣿⣿⡿⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠃⠀⠀⣰⣿⣿⡿⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⢀⠀⢀
⠀⠀⠀⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⢀⠊⠀⠊⠂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣦⣀⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⠀⢀⠀⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⠋⢹⢏⠻⠿⠛⠉⠙⣿⣿⣿⣿⣿⣷⡀⣘⣀⣀
⠀⠀⢠⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣿⣭⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣤⣤⡤⡤⢀
⠀⠀⠀⠈⠻⠿⠏⠝⠋⠩⢛⠋⠫⠍⢍⣵⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣏⠉⠉⠉⠋⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⣤⠀⣠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠸⣽⣇
⠀⠀⠀⠠⠤⣴⠤⠀⠁⠀⠈⠉⣰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⡞⣿⣿⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⠄⢰⣿⣿⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢟⣿⣿⡻⣿⣷⡀⠀⠐⢣⣽⣿⣧⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠁⡉⠃⢠⣿⣿⣿⣿⠿⢽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣇⠹⣎⢷⡀⠀⠀⠉⠉⢾⡛⣛
⠀⠀⠀⠀⠰⠆⠀⠐⠂⠀⣼⣿⣿⣿⣧⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠹⣜⢷⡀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢿⡟⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠰⠨⡋⣇⠠⠇
⠀⠀⠀⡀⢀⣀⠀⠀⠀⠈⣿⣿⡿⠁⣻⣿⣿⣿⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣨⡁⢽⣿⡌⡇⢀⣴⠶⠂
⠀⠀⠈⠁⠀⠒⠒⠒⠃⠀⠻⠟⢉⢹⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢙⡷⡄⣿⣷⣥⡀⣷⣆⣀⡀
⠀⠀⠀⠀⠐⠲⢀⠀⠴⣄⠐⣶⣾⣌⢉⣷⣾⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⢺⣿⣿⣾⣿⣏⣰⣿⠁⠃
⠀⠀⠀⠠⢀⡄⠠⢀⠀⠈⠛⠁⠿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣾⣿⣿⠁⣸⣿⣿⣿⣿⣿⣿⡿⠷⠠
⠀⠀⠀⠀⠠⠤⢤⣾⣿⣥⣈⠋⠽⠿⢿⣷⣾⣿⡇⠀⡀⢘⣔⣶⢂⢶⣤⣴⠶⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠂⡀⠦⢭⡽⣿⣤⣦⠷⢦⣭⢍⠻⣿⣦⣤⣦⣿⣿⣾⣽⣐⣠⣤⣠⣤⣿⣿⣿⣿⣿⣿⣿⣏⠛⠳⠶⠦⠀⠠⠄
⠀⠀⠉⠀⢤⠶⠄⠀⠚⠛⠻⠟⠻⢿⣶⣾⣧⣽⣯⣽⣟⣼⣿⣿⣿⠿⣿⠿⠟⣻⣿⣿⣿⢟⣿⣿⣿⣿⡿⣧⣄⣈⡃⠀⠐⠤⠄
⠀⠀⠀⢀⠂⠒⠀⠖⠰⣺⣧⣽⣡⢊⡛⠛⣽⡿⢻⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣾⡿⢿⣾⢿⣋⣩⣿⣿⡁⠈⠿⠉⠤
⠀⠀⠂⠼⢿⡯⠤⣶⣾⡟⠃⡤⠉⣁⣥⠞⠉⣹⣿⠿⠺⠟⠀⠀⠐⠢⠶⠀⠂⠘⠀⠁⠘⠁⠀⠁⢉⡀
⠀⠀⠀⠀⠀⠠⠌⠛⠀⠀⠐⠒⠒⠀⡀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆
⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⢸⣏⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀
⢀⡴⠖⠛⢉⢉⣀⡀⠀⠀⠀⠀⠉⠙⠻⢶⣄⠀⠀⠀⢻⣼⣧⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁⠀⣀⣄⣠⡤⢤⡽
⠙⠋⢄⡉⠉⠉⠉⠙⠛⠓⠲⠦⣴⣀⠀⠀⡙⢧⡀⠀⠀⢻⣿⡀⠀⠀⠀⢀⣴⠞⠋⠀⣀⣠⣴⣖⠛⠉⠉⢀⡴⠋
⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⣉⣳⣦⣄⠈⢻⡄⠀⠀⣿⡇⠀⠀⣰⠟⠁⠀⣠⡾⠛⠉⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣌⣷⠀⠀⢸⡇⠀⣼⠋⢀⣠⣾⣫⡤⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣀⣀⣀⣀⣀⣀⣈⣿⣆⠀⢸⡇⣸⠇⣠⡾⠋⠉⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⠹⣦⢸⡇⢿⣴⢏⣀⣤⣤⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠶⠶⠶⠞⠛⠛⠛⠛⠛⠋⠉⢻⣿⡏⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⠶⢦⣤⣄⣀
⠀⠀⣀⣴⠶⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢶⡄
⠀⠀⣿⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⣧
⠀⠀⣿⠈⠉⠛⠳⠷⠶⢶⣤⣶⣶⣿⣭⣭⣉⣉⣉⣉⣉⣉⣹⣽⣏⣉⣉⣉⣩⣭⣭⣽⣷⣶⡶⠶⠶⠞⠛⠉⠀⠀⣿
⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠉⠙⠛⡶⠶⢤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠖⣻⠉
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⣶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠴⠶⠶⣖⢶⣿⣿⠟⣿⡟⢿⣛⠿⣿⣿⡟⢦⠀⠀⣰⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡧⠃⢻⠀⢿⣿⣿⣿⣷⣸⠀⠀⣿⡻⠿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠳⣄⢰⠞⢧⣸⢿⠿⡿⠏⣿⣆⢀⠃⠀⠀⠛⢌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈⣷⣰⠏⠀⠀⠡⣌⡉⠙⢯⠀⠀⠀⠀⠸⣆⢱⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠘⣷⣤⡈⢧⠀⠀⠀⠀⢿⡆⣷
⠀⠀⠀⠀⠀⠀⠀⣼⢳⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⢻⣇⠈⡆⠀⠀⠀⠀⢻⡾⡆
⠀⠀⠀⠀⠀⠀⢀⡏⢘⡄⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄⢱⠀⠀⠀⠀⠈⣷⣧
⠀⠀⠀⠀⠀⠀⢸⣫⠟⢇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣇⢸⠀⠀⠀⠀⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⡄⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣄⣀⣠⣬⣷⡄⠀⠀⠀⠀⠀⣰⠉⣃⣶⣤⣼⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠹⢿⣿⣿⠛⣄⣠⣆⣀⣰⡿⣠⣿⣿⠀⣹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉⠉⠀⠀⠁⠀⠈⠀⠘⠻⠿⠿⠟⠀⠁⠀⠉⠈⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣠⣶⡄⠀⠀⠈⠛⠛⠿⣿⣿⠷⠄⠀⠙⢻⣧⠀⠀⠀⠺⢿⣆⠀⠀⠨⢷⠀⠀⠀⣾⡶⠇⠀⠰⠿⣷⡄
⠀⢀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⠀⠀⢸⣿⣷⠀⠀⠀⢸⣿⡄⢠⠀⠀⢦⠀⢰⠇⠀⠀⠀⠀⠀⠘⠟
⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⠇⠀⢙⣿⣿⠇⠀⠀⠘⢹⡇⠈⠀⠀⠈⠀⠘⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⢩⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⣄⣤⣠⣾⠃⠀⠀⠀⣀⢀⡀⠀⢠⡿⣿⡀
⠀⠀⢠⣶⡀⠀⣄⣀⣀⡰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⡄⠀⢀⠛⠛⢃⠀⠘⠁⢹⣇
⠀⠀⡀⠢⣿⡀⠈⠁⠙⠿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⢛⣀⣈⠇⠀⠀⠀⠐⠒⠀⠀⠀⡀⡀⠀⠀⣀⣄⠀⡘⠀⠀⢣⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠑
⠀⠀⠀⠀⢀⣀⣠⣴⣶⣷⣦⡀⠀⠀⠀⢀⡤⢼⠛⠋⠀⠁⠀⠀⠈⠀⣠⡞⠈⠃⠀⣀⡀
⠀⠀⠀⠀⠚⠿⠛⠛⠿⣿⠝⠛⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣽⣿⡄⠀⣀⣀⣀⡀
⠀⠀⠀⢠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠘⢻⡄⠈⠀⠀⠀⠀⣠⣴⣦⣤⡀⠁⠀⠀⠀⠘⠿⠿⠃⣼⣭⠋⢻⣿⣆
⠀⠀⢠⣿⣿⠋⠀⠀⠐⠱⣄⡤⠀⠀⠘⡇⠀⠀⢀⡤⠞⢿⣿⣽⡿⠗⠦⡀⠀⠀⠀⠀⠀⠀⠻⠋⠀⢸⣿⣿⡄
⠀⠀⣼⣿⣿⣷⡀⠀⠀⠀⢰⣆⠀⠀⠀⣤⣤⡄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⣀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⣿⣿⣿⠈⠃⠀⠀⠀⠀⠿⠄⠀⠀⠘⠛⠼⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⣽⣿⣿⡋⠀⠀⠀⠀⠀⠛⣿⣿
⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠚⠉⠁⠉⠛⠒⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠶⠖⠛⠻⠛⠧⣐⠒⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠉⣀⣀⣀⡀⠀⠐⠢⣄⠀⠀⠀⠀⠑⢄⠀⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡾⠋⠁⠀⠀⠀⢉⡀⠀⠈⠳⡄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣿⣧⢀⡄⣴⣶⣦⣄⡈⠑⠀⠀⠙⢦⡀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣸⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⠁⠈⠉⠉⠉⠉⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠲⢬⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣷⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡻⣮⡙⠛⠻⠿⢿⣿⡻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣎⠻⣿⣶⣤⣄⣀⣙⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⣿⣿⡷⢶⣦⠀⠀⠀⠀⢠⣤⣶⣾⣶⣷⡒⠀⠀⠹⣆⠈⠻⣿⣏⣛⠙⠛⠛⠻⠯⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⣴⣿⣿⣿⣿⣿⣿⣶⡀⠀⣀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣤⣤⣿⡄⠀⠀⠉⠛⠛⠛⠛⠛⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠁⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠏⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢶⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡙⠿⠛⠉⢱⠀⠀⠀⠀⣀⠀⠀⢰⠋⢙⠻⠟⡉⠙⡆⠀⢀⡀⠀⠀⠀⢰⠋⢉⠻⠟⡉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠁⠒⠈⠀⢸⠀⠀⠀⠀⠹⣷⡄⢸⠀⠀⠐⠂⠀⢀⡇⣠⣿⠃⠀⠀⠀⢸⡀⠈⠐⠂⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠳⢤⣀⣠⠔⢣⡀⠀⠀⠀⠀⠙⣿⣤⠓⠤⣀⣠⠤⠺⣴⡿⠃⠀⠀⠀⠀⣠⠓⠤⣄⣠⠤⠚⣄
⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⣰⡿⣿⡀⠀⠀⠀⠀⠀⠀⣿⢿⣦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠈⠁⠀⠹⣧⠀⠀⠀⠀⠀⣰⠟⠀⠈⠁⠀⠀⠈⢿⡀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠈⢿⡙⠇⠀⠀⠸⢋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣑⣦⣖⣉⣳⡄⠀⠀⠀⠀⠀⠀⢠⣛⣉⣢⣴⣋⣙⡆⠀⠀⠀⠀⠀⠀⢠⣟⣉⢲⣴⣊⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⣁⣀⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⣿⣿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⣛⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢙⡋⠄⠈⡇⠀⠀⣠⠀⠀⠀⢸⠀⠰⢈⡁⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠉⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠁⠀⣠⠃⣠⣾⠏⠀⠀⠀⠘⣆⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢀⣠⡞⠙⠒⠤⠒⠊⠹⣄⡀⠀⠀⠀⠀⢀⡴⠉⠒⠢⠔⠚⠉⣶⠟⠁⠀⠀⠀⢠⡾⠀⠉⠒⠒⠉⠀⢷⡄
⠀⠀⠀⠀⣠⣶⡿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣶⣄⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠰⠛⠉⠀⣿⡄⠀⠀⠀⠀⠀⢀⣿⠀⠉⠻⠆⣼⣿⣇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣾⠿⣧⠀⠀⠀⠀⠀⠀⣼⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⠀⢀⣀⡼⠃⠀⠀⠀⠀⠉⠀⠻⣆⣀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠈⠀⠙⣧⠤⣀⢀⡤⢼⡋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠥⠬⠷⠯⠤⠼⠆⠀⠀⠀⠀⠀⠀⠰⠥⠤⠽⠮⠤⠬⠇⠀⠀⠀⠀⠀⠀⠘⠓⠒⠚⠛⠒⠒⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠉⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠋⣈⡇⠀⠀⣀⡀⠀⢰⢋⠱⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠸⣎⡨⣣⠚⠉⠀⠈⠲⣼⠁⣹⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⣅⡀⠀⠀⠀⢀⣼⠏⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⢌⡉⠀⢉⡡⠊⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠙⣦⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⢀⡜⠀⡜⠉⠉⠉⠉⠁⠀⠀⠉⠑⠒⠒⠒⢪⠉⢣⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⣠⠔⡟⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢻⡢⢄⡀⠀⠑⣄
⠀⠀⠀⢀⠜⠁⠀⡠⠔⠉⢀⡜⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢣⠀⠉⠢⡀⠈⠣⡀
⠀⢀⡔⠁⢀⠴⠊⠀⡠⠖⡹⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧⡀⠀⠘⢆⠀⠙⡄
⢀⠎⢀⠔⠁⠀⡠⠊⠀⢀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠉⠢⡀⠈⢣⠀⠘
⡎⡰⠃⠀⢠⠎⠀⠀⠀⡸⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠙⡄⠀⢇
⠻⢀⡠⠔⠁⠀⠀⠀⠀⡇⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠸⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⠀⠑⠤⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⣀⡠⠖⠁⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠟⠒⠆⠤⠤⢤⣀⡀⠀⠉⠉⠉⠉⠉⣉⣁⡤⠤⠤⠤⠤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠀⠀⠀⠀⠀⠀⠈⠑⡦⢤⡤⠒⠉⠀⠀⠀⠀⠀⣀⣀⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣴⣾⣿⣶⣦⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⡿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣿⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⣿⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣸⠇⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡟⢿⣷⣄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠻⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡏
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠘⢶⣴⣦⣴⣶⣤⣤⣤⡴⠞⠋⠉⠉⠹⣿⣿⣿⣿⣿⠇
⠀⠀⢀⣴⣿⡿⢿⣿⣿⠿⠛⠛⠉⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠙⠸⣿⣿⣿
⠀⠀⠙⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⣀⣀⣤⣄⣀⡀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⠉⠉⠙⠛⠛⠛⠻⠿⠛⢿⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠻⢿⣿⡛⢿⣯⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠳⡄⠀⠀⠀⠀⢀⣆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠹⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡗⠀⠘⠄
⠀⠀⠀⠀⠀⠀⢰⢦⣄⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⡄
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣰⡯⠛⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡟⢢⣦⣰⣦
⠀⠀⠀⢠⠒⠁⣿⣽⣨⣷⣾⡄⠸⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⣿⣿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⡿⠛⣿⠛⠻⠀⠀⠇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣄⢹⣿
⠀⠀⠀⠀⠴⢾⡷⠤⠄⠈⠀⠀⠀⠈⢆⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡆⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣿⣆⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠗⠀⠀⠸⣶⡆⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠉⡁⠀⠶⠖⠟⠻⠛⠛⠦⣤⣲⣶⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢰⡶⠾⠷⣿⡷⠾⣯⣁⣿⣿⣉⣭⣭⣽⣥⡆⠀⡄⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤
⠀⠀⠀⠀⠘⠋⠙⠒⠛⠛⠻⢿⣿⣿⣭⣿⣿⣮⣿⢿⡷⢶⣇⡈⠸⠷⣤⣴⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠋⠀⠉⠛⠑⣚⣁⣈⣻⣷⣆⡀⠻⣿⣷⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡠⠐⠚⠿⠿⠿⣿⣿⣿⣿⣷⣤⣼⣿⣷⣄⣤⣶⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡀⠀⠀⠀⠀⠨⠿⠿⠟⠋⠉⠉⠉⠉⠉⠉⠹⢿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⡟⠙⠛⠀⠁⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠈⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⢦⡀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠙⣄⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉⠉⠉⠓⢤⡙⢿⣿⡄⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣽⣿⣿⣿⣿⢿⠾⠃
⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⣿⣿⡄⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⣱⠏
⠀⠈⠉⠛⠛⠻⠿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣹⣿⡆⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⣉⣉⣩⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣏⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣼⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣹⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠴⠛⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⡿⣻⠿⡿⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⢿⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⣷⣶⣶⣷⣾⣿⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠈⠺⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠅⠀⠀⢀⣀⣀⣀⣠⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⣽⣿⣿⣿⣶⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⣉⡥⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠙⠻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤

⣀⣀⣀⢀⣀⣀⣀⣀⠀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡟⠁⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠀⠀⢀⡄⠀⠀⠀⠀⣎⡑⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠰⢃⣷⣆⣡⣤⡀⠀⠀⠀⠀⠀⢀⣤⢠⣶⡆
⠀⠀⢀⠀⡀⠀⠀⠻⠇⠀⠀⠀⠚⠛⠛⠉⠀⢤⣌⣀⠀⠀⠀⠀⠸⢿⡏⢸⡿⠃⠀⠀⠀⠀⣴⣿⣿⡏⠿⠛⠀⠀⢠⣤⡦⠐⠿⠛⠀⠙⢿⣿⡆
⠀⣠⡺⠆⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⣀⣀⠈⠛⠁⠀⢢⡤⣄⣰⠀⠀⠀⠀⣼⠀⠀⢰⡈⠉⠉⠀⠐⠿⠂⠀⠘⠟⡡⣶⠀⠀⠀⢀⣀⡀
⠀⠙⣿⣿⠀⣤⠀⠀⠀⠀⠀⠾⠆⠛⠀⠀⠛⠋⠰⠎⠁⠀⠀⠴⣜⣿⠀⡀⢰⡾⠇⠀⠀⠸⣧⡀⠀⢠⣶⠂⠀⠀⠠⣧⡄⠉⠀⠀⠀⠿⡋
⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠈⠁⠀⠀⠀⠀⠈⠀⠀⠈⠁
⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⣰⣿⡏⣁⡄⠀⣤⣀⠀⠀⡀⠛⠁⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡰⣶⡶⢀⠀⠀⢀⠀⠉⢰⡆⠀⢀⠀⠀⠀⢤⠄
⠀⣿⣿⡁⣿⡄⠅⢉⠉⠀⠰⣿⠆⠀⠀⠀⠀⠀⠀⡀⠀⡀⢂⣸⠃⠀⠀⠀⢠⣴⡶⠈⠛⠛⠃⠀⠻⣿⣷⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⢈⣴⢾⡿
⠀⠹⣿⣷⣮⣀⣀⣀⡤⠀⠀⠀⠀⠐⠀⠰⣿⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⢸⠟⢁⣾⠌⠀⠀⠀⣀⡂⠀⠀⠀⠘⠻⠟⠹⠿⣶⠇⠀⠈⠉⠉
⠀⠀⠙⠳⠾⠿⠟⠋⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸⠷⠼⠃⠀⠀⠀⠰⣶⡄⠉⠀⠰⠀⠺⠟⠁⠀⠀⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠻⠏
⠀⠀⠀⡠⠶⠤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⣡⣿⣷⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣠⡷⠀⠀⢀⣾⡟⢁⣤⠀⣠⣀⠀⠀⢀⠀⣤⣶⡜⢿⣧⣄⠀⠀⠀⠀⢀
⠀⠀⠸⣿⣿⠇⢙⣅⡀⠀⠠⢀⡄⠈⠀⠀⠀⠐⠀⠀⠀⢀⣶⣭⠀⠀⠀⢸⣿⡇⢻⡅⠀⠈⠉⠀⣾⡿⢁⡌⠉⠀⠀⠙⠛⠃⠀⠀⠀⣄⡝⠁
⠀⢸⡆⠀⠀⠀⢀⠉⠀⠀⠄⢘⡗⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉⠉⠀⠀⠀⠈⢿⣿⢶⣤⣤⣤⡤⠀⢨⡀⠺⠃⠀⠀⣴⡖⠀⠀⠀⠀⢀⡾⣿⡀
⠀⠈⠻⠂⠀⠰⠟⠁⠀⠀⠂⠈⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠈⠋⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⢀⣤⡶⠂⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⠀⠀⠀⢀⠀⠉⣰⡄⠀⢀⠀⠀⠀⣄⡀
⠀⣰⣿⡇⣠⡶⠀⣴⣤⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⡄⠀⠀⠃⢀⣀⣄⠰⣾⠿⠆⠛⣷⣶⣄⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⢀⣠⣶⣷
⠀⢿⣿⣇⢷⠆⠂⠀⠀⠀⠀⠉⠀⠀⠀⢰⡷⠀⠀⠀⠐⠀⢹⣶⡀⠀⠀⠀⢸⣿⠇⣤⠀⠀⠀⠀⠈⠉⠉⠀⠀⠘⠿⠇⠿⢿⣴⠃⠀⠙⠛⠚
⠀⠘⢿⣿⢳⣤⣤⣤⠖⠀⠀⠀⢂⠀⠀⠐⠁⠀⠀⠀⠀⢠⣟⣻⡇⠀⠀⠀⢰⣄⠈⠿⠀⢠⠀⣴⡶⠂⠀⠀⠀⠀⠀⠀⣰⣟⠋
⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠻⠏
⠀⠀⠀⠀⠠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠂⠀⠀⣰⡆⠀⠀⠄⠀⡞⠑⣀⠈⠉⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠰⢢⣶⣎⣠⣤⡀⠀⠀⠀⠀⠀⣀⣌⢴⣶⠆
⠀⠀⠀⣀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠺⠛⠛⠋⠀⢤⣘⠆⠀⠀⠀⠀⠸⣿⡇⢹⡿⠃⠀⠀⠀⠀⣴⣿⣿⠏⠿⠋⠀⠀⢠⣴⡆⠘⠟⠛⠀⠙⢿⣿
⠀⠀⣦⣝⢃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣁⡀⠙⠛⠁⠀⢦⡤⣀⠸⠀⠀⠀⢀⣼⠀⠀⣴⠈⠋⠉⠀⠜⠿⠀⠀⠘⠟⢡⡷⠀⠀⠀⠀⣀
⠀⠀⠘⠿⣿⠀⡶⠀⠀⠀⠀⠀⠶⠄⠗⠀⠀⠛⠃⠰⠏⠀⠀⠀⠷⣝⡿⢀⡀⣰⡾⠃⠀⠀⠸⣦⠀⠀⢠⡶⠂⠀⠀⠰⣶⠀⠃⠀⠀⠀⠿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠀⡠⠠⢄⡀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠘⣸⣿⣧⣴⣦⠀⠀⠀⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡆⠀⠀⢠⣾⡏⢁⡀⢀⣀⡀⠀⠀⠀⢀⣤⣦⠺⣿⣀⡀
⠀⠀⣾⣿⣿⠈⢛⣁⠀⠀⠄⢀⡀⠀⠀⢀⠀⠆⠀⠀⠀⢠⣬⣍⠀⠀⠀⣾⣿⡁⣯⡁⠈⠙⠉⠀⣾⡿⣀⠉⠉⠀⠈⠿⠟⠀⠀⠀⠠⣀⡒
⠀⣾⠀⠁⠀⠀⠈⠋⠀⠀⠀⣸⡇⠀⠀⠀⠀⢀⠀⠀⠀⢙⠉⠛⠀⠀⠀⠹⣿⣷⣯⣀⣀⣀⡄⠀⣉⠐⢿⠁⠀⢀⣤⠄⠀⠀⠀⠀⢠⣿⣷
⣄⠘⢷⠒⠀⢾⠟⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠈⠓⠿⠿⠛⠉⠀⠀⠙⠃⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠙⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣌⣻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣆⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠛⠿⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⣀⠀⠠⡄⢛⠀⠈⢻⣿⣾⣷⠆⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣧
⠀⠀⠆⠀⠀⠻⠈⠃⢠⡀⠉⣿⣿⣿⠆⡀⠙⢻⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⠇⢿⠯⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⣠⠀⠀⠀⠋⣙⡃⠄⢠⣿⠇⠐⣠⡴⢿⠇⠀⠀⡞⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠻⢐⡿⠇⣼⣧⠋⣼⠿⠿⢶⣴⣾⣣⣤⣄⣠⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⣾⡀⢠⡋⠀⢸⡇⢀⡶⠺⠗⠀⡽⠋⣹⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⣯⡛⠀⠀⠀⠀⠀⣞⠁⣀⠀⡀⢀⡀
⠀⠀⠹⣅⠀⢸⠇⠀⠾⠁⡏⠉⠀⠀⢸⡁⢀⡏⢠⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⡆⠀⠀⠀⢺⠋⠀⠉⣼⠃⡎⢡
⠀⠀⠀⠘⠀⣸⣦⡤⠀⠀⠁⠀⠀⠀⠈⠁⠀⢁⣿⣿⣿⣿⣿⣿⣿⣎⠻⣿⣿⣿⣿⡄⠀⠀⠈⣀⣀⣠⣿⣤⣧⣼⣶⣿⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡯⠀⠈⢿⣿⣿⣿⠀⠀⠀⠙⠉⠉⠛⣋⣀⣈⣉⣉⣉⣁⣀⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠⣴⣶⣶⣦⣤⡤⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇⡲⢧⡀⢿⣿⣿⡆⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠷⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠇
⠀⠀⠤⣼⣶⣿⣿⣿⣿⣿⣛⣽⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿
⠀⠀⠀⢈⣩⣭⡿⠿⠾⠿⠖⠲⣾⣿⣧⡕⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠘⠀⠀⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣧⣾⣿⣿⣾⡗
⠀⠀⠀⢀⣤⣤⣤⣤⣤⡞⠳⣶⣿⣤⣊⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠦⠀⣿⠟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⢿⣿⡿⠿⠛⠿⠿⠿⠷⠚
⠀⠀⢀⣀⢀⣉⣩⣭⣤⣾⣿⣛⡙⣛⣣⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠆
⠀⠀⠀⠁⠀⠙⠻⠿⠿⠿⠛⠻⠟⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢂⠀⣀⠞⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠛⢻⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠘⡿⠈⠀⠀⡀⠉⠉⠉⠙⠿⠿⠿⠿⠏⡹⠿⠟⠛⢋⠙⠓⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠿⡅⠀⠀⠂⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣼⣿⡝⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣉⣁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⡁⠀⠀⢞⣷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠋⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⢠⣶⣾⣟⡛⢻⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠛⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢭⠉⠉⠉⠉⠉⠉⠉⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⡴⠛⢎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡡⠋⠀⠀⠀⠱⣌⢣⡤⠤⠤⠤⠤⠤⠤⠤⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢡⠞⠁⠀⠀⠀⠀⠀⠈⠣⡙⢆⣀⣀⣀⣀⣀⣀⣀⣈⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠮⡴⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠽⡦⠷⠤⠤⠤⠤⠤⠤⠤⢤⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡤⠤⠤⠤⠤⡄⠀⠀⡇⠀⢀⣤⢤⡤⡤⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠐⠀⠀⠀⠂⡇⠀⠀⡇⠀⢸⠀⠀⡀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠁⠈⠁⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠚⠀⡇⠀⠀⡇⠀⠘⠒⠚⠓⠒⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠚⠒⠒⠒⠒⠒⠓⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⡀⠠⣤⡀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠠⠄⠠⠿⠃⣀⠀⠀⠀⠀⣴⡋⠀⠀⠀⠀⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠾⠿⠀⠀⣴⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢋⠁⠀⠀⠁⠠⣾⣿⡆⠀⠀⢀⣸⠇⠀⠀⠀⢀⡀⠀⠈⢱⣿⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⢠⣯⠁⠙⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⠃⠰⠆⠘⠹⣌⣄⠀⠲⠚⠛⠁⠀⠀⠀⠘⠛⠋⠁⠈
⠀⠀⠀⠀⠀⠀⠄⠆⠙⢻⣦⠀⠀⠀⣴⡟⠋⠐⡄⠀⠀⠀⠀⢀⠀⠀⢀⡴⠌⠃⠘⢿⣇⣠⠄⢢⣤⣶⠃
⠀⠀⠀⠀⠀⠀⠘⠎⠀⠀⠘⡒⠒⠚⡉⠀⠀⠉⠎⠀⠀⠀⠀⠡⡄⠉⠈⠀⠀⠤⠀⣸⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠠⠒⠀⠀⡀⠀⡀⠀⠙⠀⠊⠀⠀⠀⢀⠀⢠⣄⡀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠉⠙⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡄
⠀⠀⠀⠀⠛⠛⠻⠀⠠⠡⠀⢠⣉⡠⣤⠤⠀⠻⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠌⠑⠀⠀⡀⠉⡀⠀⢀⣀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠄⠀⠀⠀⠋⠈⠂⠀⣶⣾⣿⣦⠬⡟⢿⣿⡇⠀⡀⠀⠀⠸⣤⣄⢀⡀⠃⠀⠀⠀⠀⠀⠀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠂⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠖⠉⠈⣿⡗⠶⡆⢘⡿⢁⣬⡿⠀⠀⠐⠢⡖⠈⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⣤⡀⠀⠀⠀⣶⣶⡷⠀⣠⠹⢦⠼⠃⠸⣇⢠⣿⠁⠶⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠴⠂⠐⠋⠙⠀⠀⡴⠀⠻⠿⢇⠀⠁⢀⣀⣠⣴⠆⣀⣈⠻⢚⣥⣦⠀⠀⠀⠀⠀⠀⠀⡀⢸⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⢀⣤⢠⡾⠶⠀⠀⢷⣶⣼⡇⢈⣱⠄⢙⣿⣶⣗⣼⢏⢻⡄⢹⣏⣿⣷⣀⣀⡀⠀⠀⡄⢽⠿⣧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡃⠘⡷⠀⠀⠀⠀⠈⣿⣿⡟⢣⣴⣾⣿⠟⠸⠿⠾⠚⢷⣿⣿⡿⠉⠉⠉⠀⠀⠀⢁⡸⠒⢃⡿
⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠻⠟⠋⠀⠀⠀⠀⠀⠀⣿⣿⣷⣌⢫⣍⣨⣅⣠⣦⢀⡈⢳⡉⠤⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠟⠁⠀⣤⠚
⠀⠀⠀⠀⠀⠘⠟⢋⠈⠃⠀⠀⠀⢀⣄⠀⠠⠺⡆⠀⠀⠹⣿⣿⣿⣟⢿⠟⠁⠉⠿⠆⠰⠿⡴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠘⠻⠇⠀⠀⣧⣄⠀⠀⠙⠿⢃⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⣷⠀⠀⠰⠮⣥⠘⣧⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠙⡆⠀⢠⡾⠋⠉⠙⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠀⠐⣠⣶⠾⢿⡿⣄⠂⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⢁⡆⠟⠀⠀⠀⢀⡼⠃
⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣷⣾⢿⡿⣿⣿⣶⡀⠀⠀⢀⡀⠀⠀⠀⢀⣠⠾⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠐⠋⠀⠀⠒⠀⢠⣟⡆⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⡜⢿⣿⣷⠜⠿⣿⡿⠿⣥⠀⠠⠿⠛⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠈⠁⠐⠷
⠀⠀⠀⠀⠀⢾⡇⣴⣶⠀⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠻⣿⠿⠷⣿⠛⠛⠧⠀⠀⠀⠀⢀⣀⣉⣀⣀⣀⡀⠀⠀⢀⣀⠛
⠀⠀⠀⠀⠀⠀⠛⠶⣴⣌⡻⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢻⣿⣿⣟⣻⣿⣿⣾⣿⣅
⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢿⣛⡛⢛⡉⠉⠉
⠀⠀⠀⠀⢅⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣀⣆⣀⡊⠉⠃⣄⠂⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠈⢋⠡⠝⠁
⠀⠀⠀⠀⠠⠀⠀⠀⢰⡀⠀⡗⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡾⣿⣿⡗⠉⠀⠀⠀⠂⠀⣾⠉⠀⠀⠀⠀⠀⠈⠐⠃
⠀⠀⠀⠀⠀⣀⠀⢠⠸⡗⠐⣿⣦⠉⠇⢠⡆⠀⠀⠀⠀⠀⠀⢿⣈⣳⠉⠉⠳⠄⠠⠀⠀⡽⠞⠁⠀⠀⠀⠠⢤⡄
⠀⠀⠀⠀⠀⡻⣄⠀⢀⡅⠁⠘⠃⠀⡾⠉⢳⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡶⣿⠷
⠀⠀⠀⠀⠀⠀⠘⠀⠋⠁⣀⣠⠠⣴⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠀⠀⠀⠀⢀⡤⠿⠖⠛⠧⠀⠀⡆
⠀⠀⠀⠀⠀⢤⡀⣀⠩⣯⡽⢻⡔⢻⡏⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠁⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⢀⣄⣴⡇⠀⣀⣀⡀
⠀⠀⠀⠀⣠⢞⡅⠩⠧⠉⠇⠘⣓⠂⣁⣼⠋⠀⠀⢠⣶⣾⣤⣤⣤⣤⣤⣀⠿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠁⠀⠈⠉⣡⡄⣰
⠀⠀⠀⠀⠀⠊⠀⠀⢀⣠⡄⠠⠀⡠⣉⠹⠀⠀⠀⢈⣉⣉⠉⠙⠛⠛⠿⢿⣷⣤⣄⣯⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣤⣀⠀⣨⣰⡿
⠀⠀⠀⠈⠠⢾⡇⠀⠀⣴⡶⣶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠐⠆⠀⠈⠉⠀⠀⠀⠀⠀⠐⠶⠄⠀⠘⠷⢦⡝⠛⠛⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣾⣿⡿⠟⠛⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣵⣿⣿⣿⣿⣷⣶⣦⣄⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⡿⠛⠉⢻⣿⣿⣿⣿⠟⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠈⠉⠉⠉⠉⢀⣠⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠞⠃⠀⠀⢲⣠⣤⣌⣳⠾⢿⠋⠙⣿⡝⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⠀⢀⠀⠀⠉⠀⠀⢀⡀⣠⠆⠈⣿⣷⡘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⢸⣦⣤⡄⠀⢠⣶⢿⣷⠀⡀⢹⣿⣷⡸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢻⣷⣾⣿⣿⣷⣷⣾⣿⣿⣿⣷⣧⣼⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⠁⠀⠈⠀⠀⠉⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠿⠟⠛⠿⣿⣿⣿⣿⠿⠛⠛⠿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠁⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢰⣷⠀⠀⣀⣀⠀⠀⢾⡷⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⠿⠛⠻⠿⣿⣿⣿⣿⡿⠿⠃

⠀⠀⠀⠀⣠⡴⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⣤⣾⣤⣤⣶⣆⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢸⣿
⠀⠀⠛⢿⠛⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣀⣽⣀⣀⣤⡄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿
⠀⠀⠯⣿⠿⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢾⠀⠀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿
⠀⠀⣛⣿⣟⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣽⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠶⣾⡶⣶⣶⣦⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⣿
⠀⠀⠛⢻⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣄⣼⣅⣤⣤⣄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⣿
⠀⠀⠶⢿⠷⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⣸⡀⢀⣀⡀⠀⠀⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⢸⣿⡇
⠀⠀⣭⣿⣯⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢼⠀⠀⠉⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⢸⣿⡇
⠀⠀⣒⣿⣗⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠉⣽⠉⠙⠛⠁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⢸⣿⡇
⠀⠀⣤⣿⣤⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠛⢿⠛⠻⠿⠟⠀⠀⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⣸⣿
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠀⠀⣿⣿
⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣾⣣⠏
⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⠭⠭⠭⠯⠭⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠊⠉⠠⡀⣀⢀⣤⡦⠛⢄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣦⠀⠸⠟⠀⢀⡀⠒⠙⣛⢻⠟⠁⢀⡀⢻⠾⠛⢱⣤⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣿⣶⣯⠛⠉⠃⠀⠔⠁⠱⣤⣶⣿⣿⣶⡴⠈⠉⢲⣧⡀⠘⠪⣽⣴⣶⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠓⠒⠒⠚⠛⠓⠒⠒⠛⠛⠛⠛⠛⢥⡀
⠀⠀⠀⠀⠀⠀⡸⠁⣤⣭⣭⣭⡍⢩⣽⣻⢛⠉⠉⠳⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠛⠛⠛⠛⣻⣭⣭⣭⣌⢇
⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡌⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣎⢂
⠀⠀⠀⠀⠀⢠⠄⠤⠭⠭⠭⠭⠥⠬⠍⠓⠒⠒⠐⠂⣿⣿⣿⣿⣿⣿⣿⣾⣿⡗⠒⠒⠒⠂⠋⠟⠛⠛⠛⠋⠀⠡
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣶⣶⣶⣶⣿⠈⠃⠀⠀⠀⣟⠛⠛⠻⠛⠛⠛⠛⢛⡇⠀⠀⠀⠋⢰⣿⣿⣿⣿⣿⣶⣾⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣯⢥⠉⢲⡄⠀⠀⠐⢍⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⡠⣿⣿⣿⣿⡿⡿⠿⠿⣿⣀⠀⠀⠀⠀⣇⠄⠚⠁⠀⠀⠀⠀⣸⡇⠀⠀⠀⣀⣸⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠢⠤⠈⠛⢻⣸⠀⠀⢀⣇⡿⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⣏⡍⠀⠀⣸⢸⠇⠠⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⣶⣭⠞⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠙⢮⣵⣮⡵⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡋⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⠀⢻⠋⣡⣽⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⢏⠁⠀⠀⠾⣯⣼⣥⣶⡾⣧⡀⠀⠀⠀⠀⠀⠉⣡⠒⣦⣀⣼⡀⣾⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⢀⣿⣿⣇⠈⠃⢀⠰⣄⡀⠉⠛⢃⠐⠒⠀⢀⠀⣀⠀⠀⠘⠃⣸⣿⣿⡿⢿⡧
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠹⠿⠁⠀⠀⠀⠁⠸⠿⡖⠀⠀⠀⠀⠀⢘⣿⠛⠦⣄⠀⠀⣈⠛⠉⣀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢈⡀⢀⣠⣠⣷⣼⣿⣷⡦⠀⣠⣄⡄⠁⢠⣿⣧⣾⣄⣠⠀⠀⣤⠀⢀
⠀⠀⠀⣴⣷⣼⣶⣿⣷⣴⣿⣤⣴⣤⣆⣤⣄⣼⣶⣿⣴⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣷⣾⣿⣼⣾⣷⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠙⠛⠉⠉⠙⠛⠛⠻⢛⣻⣭⣿⣿⣿⣿⠿⠿⠛⠛⠛⠋⠉⠉⠛⠻⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⣀⣀⡤⠤⠤⠶⠶⠒⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠠⠤⠄⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣴⡆⠀⠀⣴⣄⡀⠀⠀⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣾⠟⠉⢱⣶⣶⠉⠙⢿⣦⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣟⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⣘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣀⣀⣀⣤⣄⣀⣀⣀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠙⢿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣾⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣛⣿⣿⣟⣟⣻⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠻⠿⣿⢿⢿⣿⣟⠁⠀⠀⢀⣀⣀⣀⣀⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠿⠛⠿⠿⠇⠀⠀⠈⠛⠛⠛⠛⠛⠋⠿⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠈
⠀⠀⠀⠀⠀⠁
⠀⠀⠴⠀⠗
⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⢠⡞⠀⠁
⠀⠀⠀⢸⡅⢀
⠀⠀⠀⠀⡿⠭⠗
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢈⣷⣤⣿⣷⠞⠉⠉⢳⡀
⠀⠀⠀⣴⠋⠉⡀⠀⠁⢰⣦⠀⠀⣱
⠀⠀⠰⡇⠀⠀⠛⣂⢠⣤⣤⡿⠀⡟
⠀⠀⠀⢻⣄⠀⠙⠛⠟⢙⠉⣠⠞⣁⡀
⠀⢠⡞⠉⠉⠱⠶⣶⠷⠻⢿⡿⠶⣏⠁
⠀⡞⠀⠀⢀⣴⡄⠅⠀⠀⣀⠀⠀⠈⢷⡀
⠀⡇⠀⢠⣜⠛⡃⠀⠀⠰⠿⠆⠀⠀⢸⡇⠠
⠀⠹⡀⠘⠻⣾⣿⡿⣦⣿⡿⠂⠀⠀⣾⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⡙⢧⣠⣄⢈⡁⠉⢁⠀⣀⣤⣾⠛⠋⠉⠛⢦⡀⢀⡄⠀⠀⢠⡴⢄⣠⣤⠐⠶⣊⡁
⠀⠘⢣⡾⠛⠉⡿⠛⠛⠛⠛⢻⠁⢈⠀⠀⢄⠀⠈⢳⡜⠓⢀⡤⠦⠧⣾⣿⣿⠿⠿⢿⠛⠛⢶⣄
⠀⣼⡟⠁⠀⠀⢀⣄⠀⠁⠀⠈⠀⢸⣷⣄⠀⢀⡀⣾⣿⣤⡏⠀⠀⠀⣁⠈⠁⠀⠀⢀⡀⠀⠀⠙⣧
⠀⣿⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠂⠛⠛⣋⡴⠋⠉⠹⠿⡿⠿⢶⡶⢴⣿⡄⠀⠀⠀⠿⠷⠀⠀⠀⢸⠆
⠀⢿⡄⠁⠀⠀⢀⣤⣤⣤⣀⣶⣶⣿⣧⣿⠀⠀⢠⣶⠀⠁⠀⢀⡀⠀⠹⣧⣤⣦⣰⣶⡶⠂⠀⠀⣾
⠀⠘⢷⡄⠄⠀⠈⠙⠿⣿⣿⡟⢻⠿⠛⢻⠀⢰⣮⣩⣄⡀⢀⡛⠃⠀⠀⣿⠿⠉⠛⠋⠁⠀⣀⡼⠁⢀⠀⢰⠁⠉⠠
⠀⠀⠈⠻⢦⣀⡀⠀⠀⣄⠀⠀⣰⣀⣀⣼⣧⡀⠘⠻⠟⠻⠿⠟⠁⠀⣰⣧⣤⣦⡤⠶⠶⠞⠛⠐⠶⠌⠤⠜
⠀⠀⠀⠀⠀⠉⠛⠻⠶⠛⠛⠛⠛⠉⠉⠘⠁⠈⠓⠲⠶⠤⠦⢤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣠⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣷⣤⣴⣶⣿⣆⠀⠀⣼⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣯⣀
⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣷⣀⣻⣿⣿⣟⣛⣋⣀⣠⣷⣿⣿⣿⣿⣷⣀⣀⣀⠀⠀⠀⠀⣀⣉⣉⣙⣻⣿⣿⡟
⣤⣶⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠀⢀
⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢾⣿
⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣼⣿
⣤⣻⠟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⣀
⣿⣧⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠸⣿
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⢶
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⣸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢰⣶⣿⣿
⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⡄⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⣠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⣿⡿⣿⣿⣇⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢘
⠋⠀⠈⠻⠃⠀⠀⠀⠙⢿⡿⢿⣧⣴⣾⣿⣿⡆⠀⢀⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠀⠀⣤⣾⠟
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⠀⢻⣿⠿⠛⠉⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠟⢂⣠⡈
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣟⣛⣛⣛⣛⣁⣈⣉⣛⣋⣀⣀⣉⣉⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⡏⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣇⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⢁⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠇⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣠⣠⡀⠀⠀⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⠟⢳⠀⠀⠀⠘⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢾⣕⣾⠃⠀⠀⡀⢻⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⢀⡄⠁⠀⠀⠀⠈⢻⣿⡾⠦⠄⠃⣼⡟⢿⣿⣿⣽⣿⣿⣿⣿⣿⡟⣄⠀⣤⣶⣶⣾⣾⣯⣿⣴⣆⣀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠰⣚⠙⠫⠀⣭⣶⣄⡰⠆⠂⠐⠀⠿⣿⡿⣄⠂⣼⡅⠀⠉⠻⠟⢿⣿⣿⣿⣿⣇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⢀⠀⡂⠀⠤⠁⠀⠀⠀⠀⠨⡤⠴⠟⠹⠧⠈⠀⠀⢲⣾⣟⣿⣿⣿⣺⠆⠈⣧⠀⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋⠛⣻⣿⣿⡿⠿⠟⠛⣾⢼⣭⣌⠣⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣶⣶⣶⣿⣶⣇⣾⣷⣄⣷⣼⣿⣿⣿⣿⣿⣿⣿⠿⢿⣟⣛⠛⠛⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣋⣉⣀⣀⣀⣀⣀⣀⣈⣉⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⢠⣤⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡼⠇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢈⡁
⠀⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠉⠉⠐⠒⠠⢄⡀⠀⢰⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠙⢦⢸⠀⠀⣴⣷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠠⠀⠀⠰⠆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠘⠿⠿⠟⢀⠔⠈⣤⣀⣀⣀⠀⠀⠆⠀⠀⠀⡠⠂
⠀⠀⠀⢰⡄⠀⠀⠀⠀⠰⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⡴⠁⠀⠀⠀⠉⠈⠛⠀⢰⠶⠒⢒⣉⣀⣤⡀
⠀⠀⠀⢿⡇⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⠛⠂⠘⣧⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠶⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡎⠀⠀⠀⠀⠖⠀⠀⠀⠀⢸⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⣄⠀⠁⠐⠒⠒⠛⠛⠛⠛⠿⠓⠒⢒⡶⠶⠦⣤⣀⠀⠙⣧⠀⠀⠀⠉⠀⠀⠀⠀⠀⡇⢸⣧⠀⠀⢀⡴⠁⠀⢠⡄
⠀⠀⠘⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⡀⠉⠳⣦⡸⣧⡀⠀⠀⠀⠀⠀⠀⠀⣧⣈⣁⣤⡴⠋⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠻⣿⣷⠀⠀⢀⡀⠀⠀⢰⠏⠉⠁⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⠀⠰⠶⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠈⢿⣇⠀⢊⡁⠀⢠⡟⠀⢼⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠟⠇⠀⠀⠀⠀⠀⢉⣋⣀⣀⣀⡀⠀⠘⣿⡀⠉⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠉⠁⠀⠀⠉⠙⠳⣦⣿⣇⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠃


⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣷⣍⣶⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣁⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠁⠙⢾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⣴⣟⣥⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡟⠀⠀⠀⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠸⣿⣿⣎⡗⠀⠀⠀⠀⠀⡠⠒⠀⠈⠡⠟⠀⠀⠇⠀⠀⠀⠀⠀⢸⡏⠉⠙⠚⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠏⣿⠀⠀⠀⠀⠀⠀⠉⢫⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠉⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡤⢭⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠐⠂⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⣀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣫⠷⠤⡴⣹⠟⠃⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣻⣧⡀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠀⢠⠀⠀⠀⠀⠀⠀⣠⣤⡅⠀⠀⣁⡇⠒⠂⠀⠀⠀⠀⢠⠻⣿⠱⠘⢿⡦
⠀⠀⠀⠀⠀⠀⠀⢹⢹⣿⣯⠹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⣇⠀⠀⠀⠀⠀⠿⢿⠀⠀⠀⣯⠷⠀⠀⠀⠀⠀⣠⠞⠀⠈⠀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⢸⠇⣙⡇⠀⠈⢿⣿⠀⠀⠀⠀⣀⣴⠁⠀⠀⠀⠋⠀⢀⣿⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⣴⣷⡄⢀⢀⡀⠨⣹
⠀⠀⠀⠀⠀⠀⠀⢸⢀⣿⠀⠀⠀⠈⢿⠇⠀⠀⠀⣾⡇⠀⠀⠀⠀⣠⡴⠛⠃⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡿⢋⣶⣾⡟⣀⣘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣠⣺⡄⠀⠀⠀⠈⠀⠀⠀⢸⠇⢀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⣿⣿⡟⢡⣀⢹⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠀⠀⣄⣸⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠻⢿⡿⢡⡄⠉⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⢀⣦⡞⠀⡰⠀⠀⠀⢧⡀⣈⡏⠀⠀⣄⠀⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠐⢮⠀⠄⣀⢻⡘⠇
⠀⠀⠀⠀⠀⠀⡴⠿⣿⡿⠏⠀⠔⠉⠁⢱⣶⠁⠀⠀⠀⠈⣿⣿⣷⣀⣰⣿⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣖⠀⣤⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠷⠾⠟⠛⠻⠟⠖⠚⠛⠓⠖⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠛⠷⠶⢿⣿⣿⠇


⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⣀⣄⡀⠀⠀⠀⠈⠉⠀⠀⠀⢀⣀⡀⠙⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠈⣁⢈⡉⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⡉⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠞⠀⢀⡘⠄⠀⠀⠀⠀⠀⠀⠰⢁⡀⠈⢢⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠐⡀⢠⣼⡷⠆⠀⠀⠀⠀⠀⠀⢠⣿⣧⠄⢨⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠑⠤⠨⠜⢁⣀⠤⠤⠤⠤⠀⣘⠮⢁⡤⠋⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠰⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠈⠳⢍⡒⠂⠤⠐⣊⠥⠊⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣼⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢫⠟⠐⡶⠄⠠⣤⠤⠀⠀⠀⠀⠠⢤⡤⠠⣶⠒⠻⡝⢷⣄
⠀⠀⠀⠀⠀⠀⣠⣿⡿⠡⠃⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⣄⢻⣷⣄
⠀⠀⠀⠀⢀⣼⣿⡟⡰⠁⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠘⡄⢻⣿⣦
⠀⠀⠀⢀⣾⣿⡟⢠⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠘⡄⢻⣿⣧⡀
⠀⠀⢀⣾⣿⣿⢃⠇⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠸⡈⣿⣿⣷
⠀⠀⣼⣿⣿⡟⡜⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢃⠸⣿⣿⣧
⠀⢰⣿⣿⣿⢇⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠘⡀⢿⣿⣿⡆
⠀⣿⣿⣿⣿⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠸⣿⣿⣷
⢸⣿⣿⣿⡇⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⠀⣿⣿⣿
⣼⣿⣿⣿⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⣿⣿⣿
⣿⣿⣿⡟⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇⠀⢹⣿⣿
⣿⣿⣿⠇⠀⠀⡆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠁⠀⠘⣿⣿
⠹⣿⡟⠀⠀⢀⢷⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣾⡀⠀⠀⠹⣿
⠀⠈⠑⠒⠊⠁⠀⢣⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡸⠀⠉⠒⠂⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠉⠁⠒⠒⠒⠂⠉⠁⠀⠀⢀⡠⠊
⠀⠀⠀⢀⠤⠐⢒⣉⡉⠉⠁⠀⠀⠀⠈⠉⠻⣷⣶⡦⢴⣖⠂⠉⠉⠉⠉⠉⠉⠉⠉⢀⡐⠒⠤⢄⡀
⠀⠀⣰⠁⡔⠉⠁⠀⢀⡴⠒⠀⠀⠀⠀⠀⢠⣿⡟⠁⠈⢿⣆⠀⠀⠀⠀⠀⠐⠒⢄⠀⠀⠉⠑⢤⠈⠆
⠀⠀⢹⣿⣧⡀⠀⢀⡎⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⣠⣧⠀⠀⠀⠀⣷⡟
⠀⠀⠈⠛⠿⢿⣿⣿⣧⣀⣠⣤⡴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣦⣶⣾⣿⣿⣷⣶⣶⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⣴⣾⣿⣿⣦⣄⠀⠀⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⢀⡴⠾⣿⣿⣷⣤⠀⠀⠀⠀⣠⣶⣶⣿⣷⣤⡀
⠀⠀⠀⢰⠋⠀⠀⣿⠋⠀⠹⡆⠀⠀⣰⠟⠛⢿⣿⠿⢷⡄⠀⠀⡞⢀⣀⢸⡏⠀⠈⣧⠀⠀⡼⠁⠀⠘⡿⠉⠉⢳⡀
⠀⠀⠀⢸⡀⠻⠿⠿⠺⠗⠀⡇⠀⢰⠃⢀⣤⢸⣇⣀⠀⢻⠀⠀⠣⠈⠛⠛⠛⠟⠀⡼⠀⠀⡇⠐⠿⠾⠇⣶⠆⢸⡇
⠀⠀⠀⠠⣤⠀⠀⠀⠀⣄⠊⠀⠀⠘⢆⠈⠉⠋⠋⠛⠀⡼⠀⠀⠤⠤⠀⠀⠠⣴⣮⣀⠀⠀⠈⢄⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⣴⠀⠀⠀⢰⠈⠀⠀⠀⠀⢀⠀⠀⢠⣿⣿⠉⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢿⣦⠄⣠⣴⠃⠀⠀⠀⠀⠀⠛⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⠷⠶⣿⡿⠁⠀⠀⠀⠀⠘⢧⣄⢠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠀⢀⣤⣴⣶⣷⣤⡀⠀⠀⠀⠀⣠⣶⣾⣿⣷⣦⡀⠀⠀⠀⢀⣠⣤⣶⣶⣄
⠀⠀⢀⡞⠉⠙⣿⠋⠉⢳⡀⠀⠀⢰⠟⠉⠙⣿⡟⠛⠻⣆⠀⠀⡼⠁⠀⠘⣿⠋⠉⢻⡄⠀⣰⠟⠉⠙⣿⠟⠛⢷⡄
⠀⠀⢸⠀⢰⡶⣿⠶⡆⠀⡇⠀⠀⡏⠀⣶⣦⣿⢠⣤⠀⢹⠀⠀⡇⠐⠿⠶⠷⣶⡆⠀⡇⠀⡇⠀⣴⣄⣿⣀⣄⠀⣷
⠀⠀⠀⠓⣀⠀⠀⠀⣠⣎⣀⠀⠀⠙⠄⠀⠀⠈⠈⠉⣠⠞⠀⠀⠬⣀⡀⠀⠀⠀⣀⡼⠁⠀⠳⠀⠈⠈⠉⠉⠁⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠀⠀⠀⠀⣾⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⠀⣾⣷⡀⠀⣴⣾
⠀⠀⠀⠀⠀⠐⣤⣴⣿⡿⠀⠀⠀⠀⠀⠹⣄⡀⣀⡼⠀⠀⠀⠀⠀⠘⠆⢀⣀⣠⠟⠀⠀⠀⠀⠘⠉⠁⠈⠉⠸
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⠯⠉⠈⠙⢿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⢀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠿⢿⣿⡿⠛⠀⢴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢃⣴⣿⠟⠀⠀⠀⠀⠙⢿⣿⣟⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠏⠈⣯⠀⠀⠀⠀⠀⠀⡒⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⣿⡇⣦⣦⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣸⣷⣼⣿⣿⣶⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡞⠿⢻⣿⣿⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡜⠰⣼⡿⠀⣿⣿⣺⠁⠀⠀⠰⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢉⠇⢹⡇⠀⣿⣟⡟⠀⠀⠀⣾⡿⣿⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠐⣿⠃⠀⣿⡿⠁⠀⠀⢠⣿⣯⡉⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡁⠀⠀⠀⠐⠛⠀⠀⠀⠀⠠⣾⡟⠀⢀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⡆⢸⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⢀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣯⣿⣷⡀⠀⠀⠀⠀⠀⣀⣠⣾⡟⠇⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⠶⠿⠿⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠐⠛⠛⠛⠛⠚⠛⠛⠉⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⣦⠀⠀⣴⠒⣦⠀⠀⣴⠒⣦⠀⠀⣴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠆⣿⠰⠶⠶⠆⣿⠰⠶⠶⠆⣿⠰⠶⠶⠦⣿⠰⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣴⡿⣦⠀⠀⣴⣿⣦⠀⠀⣴⢿⣦⠀⠀⣴⢿⣦⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢛⠃⠀⠀⠘⠛⠋⠀⠀⠘⠛⠋⠀⠀⠈⠛⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢫⣼⡯⢸⠽⢹⠁⡿⢸⣧⣟⡿⣇⡯⢸⣽⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣉⣈⣉⣈⣀⣈⣀⣉⣁⣈⣀⣉⣁⣉⣈⣈⣀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⢀⣴⡾⠿⣶⣆⠀⠀⠶⠶⠶⢶⣶⡆⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⢸⡿⠀⠀⢸⣿⠆⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⢀⣴⣾⣛⣁⣀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣀⢀⣀⣀⣀⠀⠀⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣴⣾⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡛⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠋⣾⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣏⢁⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣶⣦⡿⡁⠀⠀⢸⠉⡛⢻⡇⣉⣉⢨⣿⣦⣼⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣹⠬⠹⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣼⠿⣫⠿⠟⠋⠀⣴⣤⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠉⠛⠻⣿⠅⠀⠀⠀⠀⠀⠀⠹⠄⡛⢿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠘⠻⢿⣿⡏⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠔⢣⡖⣿⢉⣢⠀⠀⠀⠀⠀⠼⠛⣧⡀⠾⠃⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠚
⠀⠀⠀⠀⠀⠀⠰⢻⣇⠹⣦⡸⠁⠀⠀⠀⠀⠀⠐⠞⠛⣿⣶⢷⣄⡀⠀⠀⠀⠀⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠚⠂⠛⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡂⠀⠙⠻⣷⣶⣦⡤⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⠒⠄⣹⠇⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣤⡀⠀⠀⠀⠀⠀⠀⣠⠞⢻⡄⣰⡿⢿⡌⠓⠲⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠙⠲⣄⡀⠀⠀⣴⠏⠀⠈⠳⣿⡅⠀⠋⠀⠀⣾⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⡏⠀⠀⠀⠠⣮⢳⡀⠀⢀⠀⠙⠈⠁⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠘⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⠻⡄⠀⠛⠀⢣⡀⣸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣤⡀⠀⠀⠈⠙⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡠⢺⡀⠈⠀⠀⡀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣦⣄⠀⠀⢩⣾⠛⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠳⠆⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠀⠀⠀⠈⣹⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠻⢶⣄⡀⠀⠀⠀⠀⠀⣿⠆⢷⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠈⠉⠙⠲⢤⣀⣼⢿⣧⢸⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⣀⣀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⢸⡇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠉⠉⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢠⡀⢀⣿⣿⣿⣿⣿⠷⣶⣄⠀⠀⠀⢀⣾⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠛⢁⣾⣿⠋⠸⣿⣿⢠⣦⠹⣷⣤⣤⣿⢿⠂⠃
⠀⠀⠀⠀⢀⡀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⣷⣾⣿⣿⠀⠀⠀⠘⣿⡿⢿⡞⠁⠻⡄
⠀⠀⠀⠀⠺⣦⡀⠀⠀⠀⠙⠒⢤⣀⠀⣠⡾⠷⣤⡀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⢸⣟⠀⠸⣶⣤⣴⠇⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠈⠙⣷⠀⠀⠀⠀⠀⢹⡿⠛⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⡄⠀⠈⠉⠀⠀⣀⠤⠖⠒⠚⠉⠁⠁⠀⠀⠈⠉⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣤⣷⣼⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣿⣿⣤⣤⠴⡒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠛⠈⢿⣿⣟⣻⣖⡰⠆⠀⢀⣈⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣿⣿⣿⣿⣧⣷⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣻⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⢹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣃⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣭⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⣿⣾
⠀⠀⠀⠀⠀⠀⢀⡛⢻⣿⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠹⠁⢈⢛⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢋⣡⣾⠄
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦
⠀⠀⠀⠀⢐⡺⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿
⠀⠀⠀⠀⣬⡓⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⣱⣾⣆
⠀⠀⠀⠀⠛⠁⢀⡀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠻⣿⣆
⠀⠀⠀⠀⠙⠈⡁⠀⠀⠀⠿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣿⡏
⠀⠀⠀⠀⠐⠂⣀⡀⠀⠀⢾⣦⣄⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠇
⠀⠀⠀⠀⠀⠉⢿⡁⠀⡀⠈⣡⣟⡑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⡛
⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠸⣿⣷⣄⢴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⢰⣴⣶⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡂⠄⣹⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⣾⠆⠀⠀⢀⣠⣬⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣬⡿⣿⣿⣿⣿⣯⣴⣀⣄⡑⣆⡀⠀⢀⣠⠀⢀⡄⢀⣤⣿⣌⣀⠀⡴⣬⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣷⣼⣶⣇⣾⣾⣿⣿⣳⣾⣟⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣛⡿⣛⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⠿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠋⠹⠟⠋⠀⠀⠈⠉⠁⠀⠀⠀⠈⠻⠶⠿⠷⠾⠾⠛⠉⠁

⠀⠀⠀⠀⡠⢤⣶⣤⡀⠀⠀⠀⢀⠔⢶⡶⠦⡀⠀⠀⠀⠀⠀⠀⡔⠉⢻⡷⢄⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠐⠀⠀⠟⠠⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠘⠀⠀⠀⠀⢰⠛⠻⣿⣿⣿⡆
⠠⣤⣤⣤⠀⠀⠉⠀⢀⣀⡤⠀⣣⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀⡞⠁⠀⠀⢀⣄⠀⠀⠀⠂⡁⠀⣹⣿⣿⣿⣷⠶⠂
⠀⠈⢙⣿⠀⠀⠀⠀⣸⠛⢁⣾⡇⠀⠀⠀⠈⣿⡿⣆⠐⠲⢿⡇⠀⠀⠀⠰⣿⣦⡄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣦⣤
⠀⠀⢸⡇⠀⠀⠀⠀⢸⡆⠚⠉⡟⠀⠀⠀⠀⠈⣧⡈⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⡟
⠀⠀⠈⠃⡀⠀⠀⡀⠘⠀⠀⠀⢳⡀⠀⠀⠀⣠⡿⠁⠀⠀⠘⢦⣀⡀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠉⠉⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉
⠀⠀⠀⡠⠒⣶⠖⠢⠀⠀⠀⠀⡠⠒⢿⡟⠂⠀⠀⠀⠀⠀⠀⠀⢁⢘⡏⢉⢣
⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠇⠂⠘⡃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣦⠀⠀⠀⢰⡄⠀⠀⢀⣡⠀⠀⠀⠀⠰⣶⡿⠠⣤⣤⡄⠀⠀⠀⠀⢰⣤⡀⠀⠀⢦⠀⠀⠀⠀⠀⣸
⠀⠀⢸⣿⣄⠀⢠⣾⡟⠠⣶⣿⣿⡀⠀⠀⠀⠠⣏⠀⠀⠈⣽⠃⠀⠀⠀⠀⣿⡿⠿⠒⠀⢸⠀⠀⣠⡀⠀⣹
⠀⠀⢾⠀⠉⠀⠈⠉⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠆⠀⠀⣿⠀⠀⠀⠀⠀⣸⠇⠀⣤⣄⣼⣿⣿⣿⣿⣿⣿⣤⣤
⠀⠀⠘⠧⠀⠀⣠⠼⠃⠀⠀⠀⠹⢆⡀⠀⠀⡀⠚⠀⠀⠀⠈⠠⠄⢀⣠⠴⠋⠀⠀⠀⠈⠉⠻⠿⣿⣿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀
⠴⠶⠤⠶⠦⠤⠦⠤⠶⠤⠴⠦⠴⠶⠤⠶⠦⠴⠶⠤⢶⣀⢅⣩⡰⠶⠤⠶⠦⠴⠦⡤⠶⠤⠤⠦⠤⠶⠤⠶⠦⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣄⣀