Arms Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠛⣷⡟⢳⡶⠴⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢳⡟⢃⣿⠷⠀⣿⣷⡌⠉⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢀⣿⠁⢸⣿⠀⢸⡿⠉⠃⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣼⣿⠀⣿⣿⠀⢸⡇⠀⠀⢳⣤⣄⠀⠀⠀⠀⢻⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⣿⡀⣿⣿⠀⣸⣧⠀⠀⣾⣿⡉⠉⢠⣄⣤⠞⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣶⡔⠋⠁⢀⣀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⠏⠉⠙⠛⠛⠛⠛⠛⠛⠁⠈⠻⣆⠀⠀⠙⢿⣦⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠙⢿⣷⡀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⢸⣧⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣦⣤⣄⣸⣿⣧⡀⢻⣷⣄⠀⠀⠈⠛⢶⡄⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠁⠈⠉⠉⠛⠻⢿⣿⡿⣿⣷⣿⣿⣷⣤⡀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣬⡉⠙⠛⠻⠿⣷⣤⣄⣀⠀⠀⠀⢀⣀⡀⢘
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣾⡿⢣⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠊⠙⠿⣷⣤⡀⠀⠀⠈⠉⠛⠿⠷⠀⠀⠉⣴⡿
⠀⣀⣤⠶⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⣿⡿⢋⣴⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⣀⣀⣀⣀⣠⣼⠏
⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣠⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠂⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣴⣿⣿⡏⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⣫⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣼⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠁⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⡡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠃⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡄⢸⢱⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢤⠀⡀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⣴⣷⡀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠻⢿⡶⠀⣀⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂
⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠺⠀⢹⣿⠿⠑⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣱⣄
⠀⠀⠻⣿⣦⣄⣀⣀⡄⠀⠀⠀⠢⣄⣀⠀⡀⠀⠀⣀⣾⢻⡄⠓⠄⠀⠀⠀⠠⡔⠀⠀⠀⠀⠀⡉⠛⠳⠄
⠀⠀⠀⠈⣿⡏⠉⡉⠘⢧⣀⠀⠈⠁⢀⠀⠀⠀⢠⣿⡿⠈⣷⠀⠀⠀⠰⣤⣴⡇⠀⠀⠀⠀⣼⡷
⠀⠀⠀⠀⢹⣿⣴⣽⣤⡀⢿⣀⡀⠀⣼⠀⡔⢀⣾⣿⠇⠀⢻⡷⣄⠀⠀⠈⢻⡇⠀⣠⣴⣾⡿⠁⢀⡄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣌⣻⣧⣰⣿⣾⣾⣿⣿⠟⠀⠀⠘⣷⡽⣿⣗⣀⣠⠓⠛⠋⣭⠗⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠘⠻⢮⣍⠉⠁⠀⣴⡈⣡⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡿⠛⠉⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠡⠀⠀⠀⠀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢄⠀⠸⣆⠸⣆⢱⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠰⡶⠀⢞⡙⢀⡏⢠⡇⣰⠯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣤⡴⠦⣤⣈⣛⣫⣿⡿⠟⠃⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⡴⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⠞⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣀⣤⣴⠶⠶⠶⣦⣤⣀⠀⠀⢀⣠⡴⠶⠛⠉⠉⠉⠉⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡿⠟⠉⠐⠂⠙⠓⠒⠒⠉⠛⠶⣟⡵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⠃⡀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⢿⣆⠙⠂⠀⠀⠀⠠⣤⣴⡶⠟⠁⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢶⣦⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠦⠤⠀⠀⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⢯⣍⡉⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣭⣭⣭⣭⣭⣭⣴⣶⣾⠿⠿⠛⠛⠋⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⣿⣿⣧⡀
⠀⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄
⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦
⠀⢠⣿⣿⣿⣿⣿⣷⣤⣶⣶⣦⣴⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣤⣶⣶⣦⣸⣿⣿⣿⣿⣿⡆
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⣀⠒⠁⠀⢉⠑⠄
⠀⠀⠀⡠⠌⠀⠀⠀⠀⠀⠈⢢⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠶⠶⠤⠤⡀
⠀⢀⠊⠀⠀⠀⠀⠀⠐⡀⠀⠀⢦⠠⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁
⠀⠸⠀⠀⠀⠀⠢⡀⠀⠈⢆⠀⢼⠀⠆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⡇⠀⠀⢀⠀⠀⠈⢢⠀⣰⠠⠊⢁⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⢰⠀⠀⠀⠀⡕⠢⠀⠘⠊⠁⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⡇⠀⠀⠀⠀⠀⠀⡄⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⣀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⣀
⠐⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣀
⠀⠡⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢣⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡼
⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢠⠃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⣀⠀⠀⠀⠀⣀⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⣶⣠⣴⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣿⣿⡋⡀⠀⠀⠀⠀⠀⣾⢻⡏⠀⢠⣿⣿⣿⠿⣿⣿⡇⠀⠀⠂⠀⠀⢀⣠⠔⠀⡠⠬⠓⠓⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣇⣽⡽⣧⠀⠀⠀⠀⠀⡇⠀⡇⠠⢫⣿⣿⠦⠚⣻⡿⠃⠀⠀⠰⠉⠁⠀⠀⠀⠘⠁⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⣿⡧⢶⠟⠀⠀⠀⠀⠀⢻⠈⣧⡶⣿⣏⢀⡀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠀⠀⣿⠀⠁⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣿⢿⣾⡄⠀⠀⠀⠀⠀⠘⣇⠸⣀⣿⢿⡷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠀⢰⣇⢰⡄⠟⣠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣽⡾⠙⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣘⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢻⣤⣿⠠⠈⡇⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢷⡟⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⢿⣿⢻⣷⠀⠀⠀⠀⠀⠀⠀⠋⠀⣲⡆⠸⣿⣏⠀⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣄⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⢻⣏⠀⠈⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣸⣿⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠘⣿⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣯⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡹⡇⠀⣇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡜⢿⠁⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠻⣦⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣰⢢⣦⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠿⣿⢹⠋⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣠⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠂⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘

⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠒⡤⠰⡄⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠒⠂⠵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠄⠠⢤⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⠠⠀⣀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠂⠒⠒⠒⠂⠞⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠊⠁⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠶⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠚⢛⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⣠⣴⠇⠛⠱⢷⡦⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡄⠀⢀⡭⠷⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢰⠇⠀⠀⠀⢀⣀⣠⡤⠤⣤⣸⠁⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⢸⠀⢠⠔⠋⠉⠉⠁⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡤⠊⠉⢹⣴⠇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠃⠀⠀⠀⣀⣤⠤⠶⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠒⠻⢿⣯⡉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⣷⣦⣄⣀⣀⣠⡤⠖⠋⠀⠀⠀⣼⠲⢤⠒⠋⠁⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠀⠀⠀⠈⠐⠨⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠙⣟⢦⠀⠀⠀⣠⠇⠀⢈⡇⠀⢀⣤⣤⠤⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣳⠴⠚⠁⠀⣤⠊⣠⠊⡡⡊⣽⠃⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠋⠁⠀⠀⠀⠀⠀⢸⣀⡇⠐⣼⡿⣃⡴⠁
⠀⠀⡴⠒⠶⢶⣶⣤⠼⠄⠀⡀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠉⠉⠁⢰⡏
⠀⠀⡇⠀⠀⠀⢻⣏⡇⢀⡤⠛⠲⣤⣟⢦⡀⠀⠀⠀⠀⠀⣾⠛⠲⠶⠶⠖⠒⠚⠋⠉⠀⠙⠦⣄⣀⣤⢤⣀⣀⣠⡤⠊⠁⠀⠀⣠⣄⣘⣧
⠀⠀⠹⡄⠀⠀⣼⣿⣧⠎⠀⠀⠀⠀⢹⢿⢳⡒⠦⢄⣀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⣀⣲⠤⠖⠉⠀⠉⠁
⠀⠀⠀⠙⢦⣰⠃⡜⠁⠀⠀⠀⣀⣀⣜⡼⠀⣧⠀⠀⠈⠓⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠉⠁⢹⡟⠒⣶⠇⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⠄⢀⡆⢠⠀⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠤⣀⣴⠷⠴⣮⠴⣬⣬⡵⠋
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⣴⠋⠙⠓⠒⠳⠤⠶⠚⠁
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⡼⢹⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⡶⠿⠛⠛⠛⠛⠿⣶⣤⣀⣠⣴⠞⠃
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠁⣸⡷⠞⠋⠉⠉⠉⠛⠷⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡏
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⢦⡀
⠀⠸⣇⡀⠀⠀⠀⠀⢀⣀⡑⠲⢤⣤⣤⣤⣤⡶⣾⡋⠁⠀⠹⣆
⠀⠀⠉⠉⠛⠻⠿⣯⣍⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣷⣤⣀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠶⠶⠶⠾⠟⠛⢻⣏⠉⠙⠻⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢠⡀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠙⢦⣄⣷⡈⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠠⣍⣳⣄⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠠⠛⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⠟⠁⠑⡄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣷⣤⡀⠀⠙⠿⣿⡿⠋⠁⠀⠀⢀⠄⠒⠈⠉⠉⠀⠒⠤⡀⠀⠈⠙⠻⡿⠋⠀⣀⣴⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡴⠋⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠈⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⣴⣶⣤⡀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡣⠒⠄⠀⠀⠀⠀⠃⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⢸⠀⠀⠀⢀⠤⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢄⡠⠂⠀⠀⠀⠀⡀⠀⠀⠀⠙⠿⡿⠿⠃⠀⠀⠀⠈⠀⠀⠀⢠⡀⡨⠀⢀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⡆⠀⡠⢠⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢀⠀⠣⡸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠈⠙⣿⣿⣿⣿⣿⡏⠉⡡⢤⠑⠊⠀⠀⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⠀⠠⠜⠀⠒⠄⢸⣿⣿⣿⣿⡟⠁⠀⠈⣿⣿⣿
⣿⡿⠿⣿⡄⠀⠀⠘⣿⣿⣿⣿⡇⠀⡀⠊⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠁⢸⣿⣿⣿⣿⠀⠀⠀⣼⠿⠛⠻
⡟⠀⠀⠈⠙⢦⠀⠀⣿⣿⣿⡿⠗⠈⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢀⠁⠺⠿⣿⣿⡇⠀⢀⠞⠁
⣧⡀⠀⠀⠀⠀⠁⠀⠈⠉⠁⢀⡤⠊⠀⠄⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⢠⠀⠑⢢⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣠
⣿⠟⠓⠒⠒⠂⠀⠀⠀⣠⣴⣿⣷⠀⠰⠀⠄⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⢀⠀⢂⠀⣸⣿⣷⣄⡀⢀⣀⠐⠈⠉⠉⠙
⣇⠀⠀⢀⣠⡜⠀⠀⠈⠛⠿⣿⣿⣤⠁⠀⠂⠀⠀⠀⠀⠀⡌⠉⠒⢠⠄⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠈⡄⠈⢆⣿⣿⠿⠋⠁⠀⠀⢘⢦⣄⣀⣀
⣿⣿⣿⣿⣧⡐⣤⣀⡀⠀⠀⠈⠛⠁⠀⠘⠀⠀⠀⠀⠀⠀⠂⠤⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠋⠀⠀⣀⣠⣴⣊⣼⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠋⠉⠉⠉⠙⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠉⠉⠉⠉⠙⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠤⠀⡀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⡀⠐⢄⡀⠀⠀⠀⢀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣤⣶⠞⠁⠀⢠⠇⠀⢹⣿⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠀⢣⠀⠀⠉⢳⣶⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣠⣿⣷⣴⣾⣿⠃⠀⢱⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠊⢿⣿⣿⣶⣴⣿⣷⣀⠀⣠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡆⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠤⠂⠀⠈⠛⠇⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⡾⠛⠃⠐⠂⢌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣠⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣠⣤⣀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠻⢶⣤⣠⣤⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠻⠟⠁⡀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠙⢿⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠲⣾⠀⠀⠀⠀⠀⠙⣦⠎⠀⠀⠀⠀⢙⡛⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡰⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠃⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣷⠀⠀⠀⠀⠀⠀⢳⣀⣤⠄⠀⠀⣸⡄⠀⠀⠀⠀⢸⡆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢹⡆⣠⠆⠀⠀⠀⠸⣿⠁⠀⠀⠀⢸⡟⠃⠀⠀⠰⣾⡗⠀⠠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣻⣯⠄⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠃⠀⢠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡀⠀⣻⣀⣀⣀⣀⠀⢸⡇⠀⠀⠀⢸⠃⠀⠀⠀⢰⡏⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢽⠟⠋⠉⠀⠀⢈⡭⠉⠉⠛⢳⡄⠀⣿⠦⠤⠀⣠⣯⣉⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⠀⠀⠀⠈⣷⠶⠋⠙⠒⠚⠉⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⣼⠇⡀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠘⢦⣐⣤⣄⣠⣤⣤⡤⠶⠛⣡⣾⣥⣄⣤⡴⠶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠿⢿⣿⣥⣄⣀⣤⣤⠾⠛⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠁⠀⠀⠀⠻⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣠⣶⠞⠛⠉⠉⠉⠉⠉⠙⠛⠷⢶⣤⣀⡤⠶⠶⠛⠿⠶⠶⣤⣄⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠙⠢⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠈⣻⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠶⠶⠶⠶⣶⣶⣶⣶⣶⠶⣿⣶⣤⣶⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⡀⠀⠀⠀⠀⠠⢤⣤⣤⣤⣤⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣙⠿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⣿⣿⣿⣿⣷⣦⣍⠛⠿⢿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣿⣷⠈⠽⣿⣿⣿⣿⣿⣿⣶⣦⣬⡙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣬⣍⣛⠻⣿⣿⣿⣿⠘⠋⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣭⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣿⠏⣠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⣽⣶⣶⣶⣤⣤⣙⡻⠿⢿⣿⣿⣿⣿⠋⣵⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠛⠃⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⢹⣿⣿⣿⣿⣿⣿⣿⣟⣛⠀⣶⣴⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⠃⣴⡆⢲⡀
⠀⠀⠀⠀⠀⠀⣰⣷⢸⣿⣿⣿⣿⣿⠃⣼⣿⠁⣿⡇
⠀⠀⠀⠀⠀⢰⣿⣿⢸⣿⣿⣿⣿⠏⣼⣿⣿⠀⣿⡇
⠀⠀⠀⠀⢀⣿⣿⣿⣾⣿⣿⣿⡟⣸⣿⣿⡻⠂⢿⣷⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣆⠈⣿⣿⣆⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣤⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣇⢸⠏⣿⣇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢰⣿⣿⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⠿⠟⣋⣀⣀⡀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣴⣾⣿⣿⣿⣿⣿⣷
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⠟⣱⣿⣿⣿⣿⡏⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣠⣾⣿⣿⣿⣿⣿⠿⠿⠿⠛
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣾⣿⣿⣿⣿⡟⠀⣾⣷⣦⣄⣈⠉⠛⠛⠛⠛⠛⣉⣁⣤⣶⣿⣿⣿⣷⣦⣤⣤⣤⡤⠀⣠⣶
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⣿⠋⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣁⣤⣾⣿⣿
⠀⠠⣄⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⡿⠛⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⢹⣷⣦⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡛⠿⠿⢿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⢉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢉⠉⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢠⣠⡄⠈⡄⠀⡁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠁⠀⡎⢀⣤⣠⠀⠑⢄
⠀⠀⠀⠀⠀⢀⡔⠁⠀⢠⡾⠁⠙⠒⠃⠓⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠓⠚⠁⠹⣦⠀⠀⠱⣄
⠀⠀⠀⢀⡴⠋⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⠳⣄
⠀⠀⢠⠎⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⢦
⠀⢠⡏⠀⠀⠀⢀⣿⣿⣥⠤⠦⠠⠤⣤⣤⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣦⣤⡤⠤⠀⠦⢤⣹⣿⣆⠀⠀⠀⠈⣧
⠀⡾⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⢀⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣅⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠀⢸⡆
⠀⣇⠀⠀⠐⠙⠛⠁⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠙⠋⠐⠀⠀⢠⡇
⠀⠈⠳⣦⣄⣀⠀⠀⠀⠛⠛⣛⣛⣻⣿⣶⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣾⣟⣛⣛⡛⠛⠃⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⠃⠀⠃⡀⠀⠀⠀⠀⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⣿⣀⣜⢁⡼⠁⣴⠃⢀⡀⣀⡀⠀⠀⠀⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣼⡃⢀⣾⣿⣿⣧⣠⠄⢀⣈⡁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⣿⣿⣿⣥⣼⣿⣿⣿⣿⣿⣷⡤⠂⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠛⠛⠉⢻⣿⣿⣗⠃⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡹⡇⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡇⣟⠀⠀⠀⠀⢠⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠁⠀⠀⠀⠀⠈⠀⠈⢧
⠀⠀⠀⢀⣠⣴⣾⠿⠟⠓⠂⠀⠀⠀⠀⠉⠳⣦⣄⠀⠀⢀⠐⠄⠓⣀⣀⣀⣀⣀⡀⠀⠀⠐⠌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⣧
⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⢿⣷⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⣿⣿⣦⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢰⣾⣿⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣴⣿⣿⣿⣿⣿⣿⡿⢿⣷⡤⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢐⣿⣿⣷⣲⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⡐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢤⣿⣿⣿⣿⡿⣿⣿⠿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠲⠾⠛⣫⡿⠟⠛⠋⠀⠀⡸⣿⣿⣿⣿⣞⣿⣿⣷⣰⡟
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⠉⢁⣿⣿⣿⣿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⢀⢀⣠⣿⣷⣤⣼⣿⣛⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠪⢹⣿⣿⠃⠀⠀⠀⠈⠀⠀⠉⠛⠿⣿⣿⣿⣷⣶⣤⣀⣠⣤⣾⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠉⠋⢰⠟⠟⠁⠀⠀⠀⠀⠠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠛⠛⠿⠿⠿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣼⢷⠀⠀⠀⠀⠀⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⢤⣀⡀⢘⢿⠿⡟⠀⣀⡤⣤⢤⣀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⣀⣨⣶⣴⡚⣻⡆⠀⠀⠀⢠⣖⣷⡿⠿⢿⣷⣯⣿⣶⣾⣿⣽⣾⣿⠿⣿⣿⣷⡄⠀⠹⣷⣄
⠀⠀⠀⠀⣾⣯⡽⣿⢛⡏⠯⣅⠀⠀⠀⠸⣽⣿⠀⠀⣀⣈⣉⣻⣿⣻⣿⣉⣁⣀⡀⠀⢹⣿⡽⠀⢀⣹⣿⡶⢴⣤⣴⡀
⠀⠀⠀⠀⠈⢫⣓⣯⣥⡾⠛⡻⡿⣄⡀⠘⣿⣿⣾⣿⡏⢻⣿⣿⡟⢚⣿⣿⣿⠙⣿⣷⣿⣷⠃⢻⣿⣿⢿⠶⠿⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⣿⣛⠡⣶⠀⢙⢁⡿⣿⣇⡀⠈⢳⠟⢛⣭⣨⣮⣏⣭⢨⣅⣧⣬⣸⣽⠻⣾⠃⠀⢁⡴⠣⠟⠃⠀⣦⡀⠹⣿⣿⣷
⠀⠀⠀⠀⠠⣿⢿⣳⢴⣽⣿⣾⠳⡹⣿⣦⠀⠈⣟⠿⠿⣊⡻⠟⢣⣬⠻⠟⣋⡻⠿⢋⡏⠀⢰⣏⣿⣦⣤⡾⣻⠟⠀⠀⣿⣿⣿⡂
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣘⢷⡼⢇⢻⣿⣿⡄⠀⣿⠇⣤⣉⣁⡲⣀⢑⣶⣆⣉⣡⡤⢿⠃⠀⠈⢛⣿⣿⢛⣼⠣⣄⣄⣸⣿⣿⡿⠁
⠀⠠⢤⣷⣶⠘⣿⣿⣿⣿⣷⣦⣸⣷⣿⡹⣧⢴⣿⠟⡟⠳⡜⣇⣸⢸⡇⢀⢿⢉⣻⠿⣶⢤⣀⣾⡿⣳⠾⠭⠼⣿⡾⡹⢿⣋⣴⣶⠆
⠀⠀⡾⠽⣼⢷⠹⣿⡿⠛⠻⠿⣿⣿⣽⣶⣿⡿⣿⢒⢓⡚⣿⣛⡛⣿⡓⠛⣾⠻⣻⣶⡟⢳⣼⣿⣽⣡⠀⠀⢀⠀⢹⠁⢰⡟⡿⣿⣏
⠀⠀⠻⣔⠟⡾⠀⣿⣷⡀⠀⠤⢬⣿⣿⡿⠳⢤⣼⣒⢛⠋⣿⡉⠓⢸⢀⣧⣼⠷⣼⣻⡴⠛⢋⣉⣀⣀⣀⡤⠎⣄⢸⡀⢸⣘⣿⣿⡏
⠀⠀⠠⣽⣿⠃⠀⡿⣿⣷⣤⣀⣄⣈⠙⠻⠭⣷⢿⣬⣟⢛⣟⠛⠖⢻⠃⣀⡾⢷⡙⣿⢰⣋⣽⢿⣿⡅⠀⠀⠀⠸⡈⣿⣆⠙⣼⣡⣿
⠀⢤⣷⣿⡝⠀⢰⠇⠋⢙⡟⢋⣽⣿⣷⡲⣤⣺⢬⣿⣿⠛⡿⠟⠷⡾⠶⠵⢖⠉⠺⣿⢘⣏⣈⣹⡇⠹⣄⠀⠀⠀⡇⣸⣿⠀⠻⣿⣿
⢘⡾⡫⠋⠀⣠⠟⠀⠀⡈⢠⣿⠟⠁⣏⠙⢁⣹⣿⠃⣹⡶⡖⠞⡆⠨⣷⠷⣾⡷⢦⢼⡴⣏⣙⣻⡇⡤⠞⢷⣤⠄⠉⣹⡟⢧⠀⠙⢿
⢸⢻⡁⠀⡰⣋⠀⠐⡀⢹⡿⢍⢢⡀⢹⣏⡩⡭⢿⡄⢺⡇⢁⢰⡇⢀⣲⡧⢾⡶⣦⣸⡴⢧⣃⣻⢿⠱⠁⠀⣿⠀⣸⡿⢷⣿⣧⣀⣼
⠈⠷⣝⣫⣵⡇⠀⠀⠁⣼⡅⠀⠀⣇⠀⢿⣰⢗⣩⣳⣌⠟⡦⣼⡇⡑⢴⣦⣯⣔⡰⣏⣙⠫⣽⠏⣸⡄⠀⣾⢿⣶⡿⠁⠘⣿⠙⠛⠋
⠀⠀⠀⠀⢸⣧⠀⠀⣴⠻⣜⠀⢠⡏⠉⠲⣽⣼⡠⠜⡼⢳⠶⣭⣥⣥⡯⣼⠟⠛⢻⢮⣙⣿⡡⠊⠁⢹⡄⣷⣠⣿⢻⣦⡰⢻⣄⡀
⠀⠀⠀⠀⡼⡸⣢⡾⣧⣠⣾⡀⢸⣀⡀⠀⠀⠙⢷⣾⣀⢏⣿⡿⠀⠉⢉⣽⡓⠒⠒⣛⡿⠋⠀⠀⣠⡞⢠⣾⣿⣿⣞⡸⣿⣆⢿⠇
⠀⠀⠀⡜⡍⣴⣿⠁⣬⡻⣿⣝⠈⠻⠟⠦⣄⠳⣦⣌⣛⣿⣿⣶⠦⢤⢬⣭⣤⣴⣾⣭⡴⢂⣠⡾⢋⢀⣿⣺⣿⣻⣎⠹⣿⡟⢸⡀
⠀⠀⠀⢤⠀⢻⣿⡾⣅⡼⣨⢛⣷⢦⣬⣾⣾⣷⣶⣿⣇⠠⢢⢸⠀⡀⢸⣇⢲⢏⣿⣿⣾⣿⣏⣡⣽⠿⢿⣯⣼⣈⢏⣽⡟⣡⣬⠇
⠀⠀⠀⠈⢆⡄⢥⣛⣷⣔⡃⣞⠂⣾⢨⡏⠉⣭⡉⠳⡞⠂⢸⣾⡶⣿⢾⣷⡸⡧⣾⡼⢻⣽⣭⡆⠸⣏⡿⢿⣹⣴⣿⣏⣤⡿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠿⠶⠽⢾⣇⣀⣾⣷⡠⠿⠶⠚⠻⢿⣻⠯⠛⠓⠶⠿⠽⠎⠼⠮⠥⠴⠟⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡴⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠯⠀⢠⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢢⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠶⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⡀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠄⠀⢀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⣠⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠈⠉⠀⠈⠁⣀⠀⠀⠀⠰⠀⠀⠀⢀⣰⡄⡀⠀⠀⠀⡀⠀⠀⠀⢈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⡀⠠⠾⠆⠀⠀⠀⠸⣷⡝⢲⠖⠛⠁⠀⠂⠒⠛⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣀⠀⠀⠠⣌⠙⠛⠀⠂⠀⠀⠀⠀⣤⣴⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣼⣤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡉⠙⠛⡀⠠⠀⠀⠘⠛⠻⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⢄⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣾⣶⣷⣄⠀⠀⠀⠀⢉⣴⠶⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⣤⣷⣶⣶⣾⡿⣫⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠟⠛⠓⠞⠛⠉⠐⠛⠉⠥⠶⠌⠛⠛⠛⠿⠶⢿⡛⠟⢫⣽⣾⡿⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡛⣙⢟⣉⠛⣻⣻⣹⡟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠉⠉⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⣀⠰⡄⢳⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠐⠞⠐⢭⣠⣇⣼⡴⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣀⠼⠋⠙⠓⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⣠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢀⡜⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠠⠋⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣠⣤⠶⠶⢤⣤⣀⠀⢀⣠⡴⠖⠚⠉⠉⠒⠂
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠟⠉⠀⠉⠉⠉⠀⠉⠛⠯⠟⠉
⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠔⠚⡉⠁
⠀⠀⠀⠀⠀⢺⣇⠣⠀⠀⠀⢀⣠⣤⠞⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⢶⣶⣶⣦⠤⠀⠀⠀⠀⠈⠉⠙⠛⠓⠂⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⣤⣒⡲⠤⣤⣤⣀⣤⣤⣶⡶⠶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢻⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠸⣿⣿⣿⣿
⠀⠀⠀⢀⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⣿⣿⣷⣦⣀⣀⢠⣾⣿⣿⣿⣿⣷⣄⣀⣀⣴⣶⣿⣿⣷⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣷⣝⠻⠿⣿⣿⣉⢿⣿⣿⣿⣿⣿⣿⣿⣏⣩⣿⣿⣿⣿⣿⣿⣿⡿⣉⣿⣿⠿⠟⣫⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣿⣿⣿⣶⣦⣤⣤⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣤⣤⣴⣶⣿⣿⣿⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⢿⣿⣿⣿⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣿⣿⣿⡿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣬⣻⣿⣿⣿⣧⣼⣿⣿⣿⣟⣥⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⣿⣿⣷⣼⣿⣿⣍⣩⣿⣿⣧⣾⣿⣿⠈⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡌⣿⣿⠟⠻⣿⣿⢡⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⣿⣿⣿⣿⣿⣿⢟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⢿⡇⢸⣿⣧⣼⣿⡇⢸⡿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠿⠾⠿⠿⠿⠿⠷⠿⠁⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣠⠊⠒⢤⣔⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⡤⠔⠊⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠯⣻⣶⣤⡈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⣠⣴⢾⡫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠫⢟⡷⣦⣄⡉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⣁⣤⡶⣟⠯⠺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣦⣍⠓⠯⣻⢷⣤⣈⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⢉⣠⣴⢿⡫⠗⢋⣤⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣬⣙⠺⢽⡻⢶⣤⡉⠓⢦⣄⡀⠀⠀⠀⠀⢀⡤⠖⠋⣁⣴⡾⣻⡽⢚⣩⣴⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣍⡓⠯⣟⡷⣦⣄⠙⠳⢤⣴⠛⣁⣤⡶⣟⡯⠗⣋⣥⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣬⣙⠳⢽⣻⢶⣤⣈⠙⠻⢷⣛⣡⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠞⠛⣿⣶⣬⡙⠪⣝⡻⣦⣄⡈⠙⠢⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⠋⠁⠉⢲⣶⣿⡫⠟⣋⣿⣷⣦⣍⡓⠯⣟⣷⣦⠋⠉⠉⢓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣊⣥⢶⣫⠷⣀⣠⣴⣿⣡⣴⠾⠛⠉⠀⠈⠙⠻⢷⣤⣹⣷⣤⣀⡰⢯⣓⢦⣍⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣩⡴⠾⠿⣖⣯⣶⣿⠿⣟⣯⡶⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⣮⣿⡿⢿⣷⣯⣗⡾⠿⠶⣬⣑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣺⣥⣸⣏⢁⣠⣴⣾⣿⠟⣋⡥⣖⣽⣿⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢹⣿⣕⡦⣍⠛⢿⣿⣷⣤⣀⢉⣿⣠⣽⡂
⢀⡠⢒⣒⡒⠐⢺⢡⠃⠙⣿⣿⠿⠛⠁⢈⣿⣿⣿⣿⣿⡿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠿⣿⣿⣿⣿⣿⣍⠀⠙⠻⢿⣿⡟⠁⢣⢸⠒⠐⣒⣒⠢⡀
⠺⠟⠋⠉⠙⢧⣸⣾⣇⣀⠘⣿⣧⣴⣾⣿⣿⣿⠿⠛⢁⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠙⠻⢿⣿⣿⣿⣶⣤⣿⡟⢀⣀⣿⣼⣠⠟⠉⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠙⠻⢿⡿⣿⣿⡿⣿⡿⠛⡉⢀⣤⠾⠿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠿⢦⣄⠈⡙⠻⣿⡿⣿⣿⡿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⠏⠁⠀⠀⢀⣴⣿⠁⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢹⣷⣇⠀⠀⠀⠉⠷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⣴⡄⠀⠀⢠⣿⠁⠺⣧⣄⡀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣀⣤⡿⠂⢻⣧⠀⠀⠀⣴⣄⢦
⠀⠀⠀⠀⠀⠀⠀⡔⣱⣿⣿⣿⡄⢠⠟⠛⠷⣦⣤⣥⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣥⣤⡶⠟⠛⢧⠀⣼⣿⣿⣷⡑⡄
⠀⠀⠀⠀⠀⠀⡜⡼⣽⣿⣿⣿⣿⡆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⡜⡄
⠀⠀⠀⠀⠀⢰⢸⢳⣿⣽⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣽⣿⢻⢰
⠀⠀⠀⠀⠀⠈⣿⢸⣿⣿⣿⣿⣿⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⣿⣿⣿⣿⢸⡟
⠀⠀⠀⠀⠀⢰⢿⢸⣿⣿⣿⣿⣿⣧⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⣿⣿⣿⣿⣿⣿⢸⣻
⠀⠀⠀⠀⠀⠈⡾⣼⣿⣿⣿⣿⣿⣿⣞⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣾⣿⣿⣿⣿⣿⣿⡼⡌
⠀⠀⠀⠀⠀⠀⢱⣻⣿⣿⣿⡿⣿⣿⣿⣎⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⣾⣿⣿⡿⣿⣿⣿⣿⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿⣟⠵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢝⡿⣿⣿⣿⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠘⡹⣿⣿⡻⠝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⣻⣿⡿⡝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠛⢿⣿⡟⠋⠙⠛⠇⠀⠀⠀⠀⢀⡄
⠀⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⡁⢀⣶⡟⠑⠾⠋⠿⢇⣰⣄⣤⣴⣿⠇
⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣻⠠⣀⣈⢀⣙⠛⢛⡋⠉⠉
⠀⢿⣿⣿⣿⣿⣿⣆⢠⠀⠀⠀⠀⠀⠀⠀⣠⡜⠟⢉⠴⠖⠀⠁⢻⡧⠀⠉⠛⠻⠀⠀⠀⠀⠠⠶
⠀⠘⣿⣿⣿⣿⣿⣿⡄⣆⠀⢀⣄⡀⢠⣿⣿⢸⣞⣿⡈⠀⠀⠀⣻⡇⠀⠀⠀⢠⣰⡶⠶⠶⣷⣦⠀⡀
⠀⠀⠹⣿⣿⣿⣿⣿⡇⣿⣷⣾⣿⣿⣿⣿⣟⣄⣽⣿⣷⣀⡀⣠⢿⣤⠀⠀⠀⢠⡇⣀⡀⣀⡉⠛⢀⡀
⠀⠀⠀⠙⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠟⠉⠋⠉⠉⣿⢉⡿⠤⣦⠉⠛⢷⣿⣿⡿⠟⠟⠉⠀⢀⣼⢿⡤
⠀⠀⠀⠀⠈⠻⢿⡿⠣⣿⣿⡿⠛⠋⠀⠀⠀⠀⢀⣤⣿⣺⠿⠰⣟⡓⣚⠹⠃⠁⡀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣏⣧⠀⠆⠀⠈⢁⣴⡢⠞⠁⠠⠞⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠛⠛⠿⣿⣿⣶⣶⣶⣶⡿⠯⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡏⠀⠀⠀⠙⣿⣿⣿⡋⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢹⡇⠀⠀⠀⠀⢸⢿⣿⣿⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⡶⠀⠀⢷⠟⠸⡏⠻⣧⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⣿⡇⠀⠀⣸⡄⠀⣇⠀⠙⣧⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠹⠿⡦⢶⠟⠀⠀⠹⣧⡀⠈⣿⡀⢀⡴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢢⣄⣤⠄⢦⡀⠀⠀⠀⠈⠻⠾⢿⡟⠋⠀⠁⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⡿⣷⡆⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⣶⡆⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⢸⣿⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⡈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣽⣟⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣅⢽⣧⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣞⡉⠅⢿⣿⣿⣿⣞⢳⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⢀⣀⠀⢠⢿⢿⣷⠆⡄
⠀⠀⠀⠀⢀⣴⠉⠀⠀⠀⠀⠐⠶⣾⣿⣿⢟⣍⠐⠖⠉⠉⢀⣠⠤⠖⠒⠒⢲⣎⣅⡀⢀⣤⣾⡿⠾⠿⠷⠶⠮⣾⣯⠴⠁
⠀⠀⠀⢀⣾⡇⠀⠀⢀⣴⠀⠀⠀⠀⢻⣣⣾⣿⣿⢂⠈⣿⣧⣀⠀⠀⣀⣴⡾⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠈⣿⡇⠀⢠⡟⠻⣦⣀⠀⡰⣯⣻⣽⣿⡇⢋⣴⠿⠛⠛⠉⠛⠛⠉
⠀⠀⠀⠀⣿⡇⠀⠈⠀⠀⣿⣿⠖⠉⣻⠟⠋⠉⢹⣿⠃
⠀⠀⠀⠀⣿⣯⡀⠀⠀⠔⠉⠀⠀⣼⣿⣴⣤⡴⠟⠁
⠀⠀⠀⠀⢻⣿⣿⣦⠀⠀⠀⠀⣾⠏⠀⣽⠏⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣧⣶⣤⠾⠛⠋⠀⣿⠞
⠀⠀⠀⠀⠀⣿⣿⣿⠟⠋⠀⣚⠋⢁⣼⠇
⠀⠀⠀⠀⢠⠿⢿⣿⣧⣤⣞⣉⡹⣿⠞
⠀⠀⠀⠰⠋⠋⠛⠺⠺⠯⠷⠶⠶⠛

⣶⣶⡶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠾⠛⠷⣤⡐⠿⠀⠺⠇⠐⠿⠀⣴⡆⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠉⠁⢀⣠⡶⢿⣤⣾⠃⢠⣶⡿⢿⣦⣤⣤⣤⣀⠈⠉⠀⠿⠃⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⢠⣤⣶⡛⠁⠀⠀⠉⠁⣠⣿⡿⣱⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⠁⢼⡷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠻⣧⡀⠀⠐⢮⣝⢿⣦⣀⠀⠶⣾⣿⠿⠁⣿⣿⡿⢿⣿⠿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⠋⣀⡀
⠀⠀⠀⠀⠀⢀⣄⠘⠋⠀⠹⣷⡀⠀⠀⠉⠀⠉⠙⢿⣦⡀⠀⢀⣴⡿⠿⠁⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⣤⡀
⠀⠀⠀⠀⣠⣌⠛⠀⠀⠀⠀⠘⢿⣾⡆⣀⠀⠀⠀⠀⣻⣷⣾⠛⠋⢰⣄⣀⠀⢀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣡⣄
⠀⠀⠀⣀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣜⢷⣄⠀⣰⣿⣿⣿⣦⣄⣀⣻⣏⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠙⢉⣄
⠀⠀⣀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠶⠶⣦⣀⠀⠀⠀⠀⠀⢺⣏⠁⠙⣿⡄⠀⠀⠀⠈⠋⣀
⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠉⠛⠓⢦⡹⣧⠀⠀⠀⠀⠀⢙⣿⠆⣸⡇⠀⠀⠀⠀⠘⠛
⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠃⢹⣧⣄⡀⠀⣴⠟⠁⢠⡿⠀⠀⠀⠀⠀⠀⠸⠟
⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡛⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠘⠁⠉⠻⣿⠃⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣶
⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣿⠀⠈⠙⠻⣿⣿⡿⠟⠷⣦⣙⠶⣤⣀⡀⠈⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣤⣀⡀⠀⣾⠇⠀⠀⠀⠈⠙⠻⠶⣯⣭⣅⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠋⠁⠀⢸⣿⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠁⢀⣴⠃⠀⢸⣿⢷⣄⡀⠀⠀⠀⠀⠀⠉⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢰⡿⠁⠀⠀⠸⣿⠀⠉⠛⢃⣴⠆⠀⢀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⣿⠿⠿⢿⣿⡟⠀⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠉⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠘⣷⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠉
⠀⠀⠀⠙⣁⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣼⣿⡀⢻⡆⠀⠀⣼⡏⠀⠀⠘⣿⣧⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⠉
⠀⠀⠀⠀⢹⣿⠿⣷⣶⣶⠶⠾⠿⠛⠋⠉⠉⠉⠉⠛⠻⠆⠁⠀⢠⣿⠁⠀⠀⠀⢹⡿⣆⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡙⠋
⠀⠀⠀⠀⢹⡇⠀⢠⣤⣀⣀⣀⣀⣀⠉⠛⠛⠛⠛⢛⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⢸⡇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠙⠁
⠀⠀⠀⠀⢸⣷⠀⣿⠋⣩⣯⠉⠛⠛⠻⠿⠷⣶⣦⣾⣷⣤⣤⣾⠟⠀⠀⠀⠀⠀⢸⣿⠀⣼⡇⠀⠀⠀⠀⠀⠀⢴⣦⠉⠁
⠀⠀⠀⠀⠀⣿⡀⢻⣆⠈⠁⢾⡗⢀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣾⡏⠀⠘⠻⠷⣦⣄⡀⢺⡷⠀⠁
⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠘⠋⢠⣶⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣶⣶⣶⣦⣤⣈⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠸⠟⠀⣶⠆⢰⡦⠀⣶⠆⠻⠇⠈⠛⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣾⠃⣀⠀⢀⡂⢐⣄⣀⣀⣤⡀⠀⠀⠀⣾⡇
⠀⠀⢀⡀⠠⠄⠤⠄⠀⠀⠀⠄⠒⣾⣯⣿⣯⣯⣽⣿⣿⣿⣷⣶⣶⣶⣿⣿⣤⣤⣴⣤⣶⣤⣴⡬⠿⠿⠿⠿⠿⠷⠶⠾⠯⠟
⠀⠀⡃⢀⣀⣀⣠⣤⣶⣶⣷⡿⠟⠛⠛⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⡍⠀⠈⠉⠉
⠀⠀⢻⣿⣿⣯⢿⡿⠟⠋⠁⠀⠀⠀⣸⣿⣿⠁⠀⠉⠀⠀⢹⣿⣿⣿⡄
⠀⠀⠐⠭⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠁⠈⠉⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⡎⢀⡌⠀⠄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠳⠼⣤⣌⢠⣆⡀⢀⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⢿⡇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣿⠀⠀⡴⠀⠀⠀⣿⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠀⣄⠀⠘⠆⠀⠙⠀⠸⠁⠀⡤⠀⣿⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⣤⣈⡙⠻⣷⣦⡈⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⠀⠀⠐⡆⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠈⢻⣿⣶⠾⠟⠛⠛⠛⠻⠶⣤⡀⢸⡇⠀⠀⠀⠀⠀⠘⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣇⠀⠀⠀⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⡀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠙⠛⠻⠿⠟⠛⠋⣁⣤⣤⣤⣤⡤⠶⠖⠃⠀⠀⠀⢈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠋⡿⠛⠷⣦⣤⣄⣀⣀⣀⣀⣀⣤⣤⣶⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉

⠙⢿⣿⣿⣿⣿⡻⢦⡀
⠀⠀⠉⠻⢿⣿⣿⣶⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠉⠻⣿⣽⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣧⡀
⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣯⣶⣿⣿⣿⣿⣶⣦⣤⡄
⣿⣿⣿⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣷⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣶⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢠⣠⣠⡄⢠⣠⡄⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⢠⡀⠐⡛⣻⡙⠛⣃⠘⠐⣛⠀⣀⡀⠀⣩⡉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣤⣼⣧⣼⣿⠏⣿⣼⠋⣷⡀⣿⣼⢿⣧⢸⡏⢿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠛⠀⣼⡇⢸⣿⢰⡟⢻⡇⢸⡇⣿⡟⢸⣿⣾⠃⠸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⡄⠀⣿⠇⢸⣧⡿⠁⣾⣷⣼⠇⣿⡇⢸⡏⢿⡤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢲⣼⡇⠀⢸⣄⢸⣿⠦⠄⠸⣧⡿⠃⣿⠀⠸⡇⠸⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠸⠦⠤⠭⠄⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠲⢾⡉⠉⣩⠝⠛⠋⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠈⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄
⠀⣀⡀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠳⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡷⣄⣠⠞⠁⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀
⠀⠈⠉⠛⠲⠇⣼⠳⢤⡀⠀⠀⡰⠋⡇⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢰⠇⠈⢧⠀⠀⠀⣀⡤⠞⢧⠸⠒⠋⠉
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠉⠳⡶⠁⠀⢳⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡞⠀⠀⠈⢧⡴⠚⠉⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢰⠃⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⡠⠧⣀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢹⡀
⣀⣀⣀⡀⠐⡇⠀⣴⠤⣤⡎⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠉⠀⠀⠈⠉⠓⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⣇⣠⡤⠴⢦⠀⢸⡇⣀⣀⣀
⠀⠀⠉⠉⠉⠁⢰⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸⡆⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠀⠀⠀⠀⠀⠀⠀⢈⡟⠉⠀⠀⠀⠀⠀⠀⡼⢀⣀⣀⡀⠀⣧
⠀⠀⠀⠀⢀⣀⣸⡀⣾⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠉⠁⣿⠀⣏⣀⣀
⠀⠀⠈⠉⠉⠉⠀⠀⣿⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⣸⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣄
⠀⠀⠀⠀⠀⡞⠀⡰⠐⠂⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠂⠒⠰⡀⠘⡆
⠀⠀⠀⠀⡜⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠸⡄
⠀⠀⠀⢰⠃⠀⠀⢹⣠⠤⠤⠤⣤⡴⠚⠋⠛⠚⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠚⠋⠛⠲⣤⡠⠤⠤⢤⣸⠁⠀⠀⢳
⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⢄⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇
⠀⠀⠀⠿⣅⣀⣀⣀⠀⠈⠉⠁⠀⣠⡀⠈⠳⣄⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⡴⠋⠀⣠⡀⠀⠉⠉⠀⢀⣀⣀⣀⣽⠇
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠖⠒⠋⠁⠙⠶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠞⠁⠉⠓⠒⠶⠒⠚⠋⠉

⠀⢀⣀⣀⡀
⠀⡞⠤⠤⠬⠭⠭⠭⠷⠶⠶⠶⡖⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠱⣤⣤⠶⠦
⠀⡧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠁⠀⠀⢰⡏⠉⠩⠭⣭⣉⡉⠉⠉⢹⠛⢢⡀⠘⢇⣀⡀
⠀⢱⠉⠉⢰⠤⠉⠉⠉⠁⠠⠰⣦⣶⣶⣾⡇⠀⠠⠤⠤⠔⠋⠀⠀⢸⠀⠀⡇⠀⠀⠙⠻⠤⣄⣀
⠀⠈⠒⠀⠋⠀⠀⠀⠀⠀⠀⠀⢣⠉⠉⢈⡇⠠⠤⠤⢤⣤⡰⣶⣶⣼⠀⢠⡇⢀⣀⣀⡀⠀⠀⠰⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠸⣇⣀⣀⣀⣐⣒⣳⣿⣿⣻⣶⠟⠁⠣⠼⣂⡙⡦⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣈⣵⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣠⠊⠁⠀⠀⠈⢻⡏⠉⠱⡄⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣸⠃⠀⠀⣇⣤⣶⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠵⣦⣤⣈⣁⣀⣠⣴⠟⠉⠉⠉⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠠⠍⠘⡄⡀⡍⠀⠀⠠⠍⡄⡅⠀⢼⠀⠀⠐⠌⢠⠀⠀⠅⠀⠎⠠⠍⡐⡅⠀⠀⢹⣿⣷⣻⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠠⠤⠤⠤⠬⠤⠤⠥⠤⠤⠬⠤⠤⠤⠤⠤⠬⠤⠤⠤⠥⠤⠤⠥⠤⠤⠬⠄⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⡀⠠⠀⠀⠐⠂⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡄⠀⠀⠀⡀⢀⠔
⠀⠀⡁⠁⢨⡀⠒⠒⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉⠀⠀⠐⢒⢂⣬⣤⠀⠉⡀⠘⠄
⠀⠀⠠⠈⠢⡀⠑⠢⣄⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠠⠎⢉⡀⠄⠀⠀⢀⠲⢄⡀
⠀⠀⠀⠠⠀⠈⠄⠀⠀⠐⠒⠀⠘⠛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠀⢇⠀⠀⡐⠁
⠀⠀⠀⠀⠂⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠆⠤⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⡀
⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⢢⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠀⠀⠤⠀⣴⣦⡐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠀⢨⡼⢤⠈⠁⠀⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠐⣀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠦
⠀⠀⠀⠀⠈⠆⠀⠀⠈⢂⡄⠀⠠⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⠷⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡀⠀⠀⠂⢀⡀⠸⡆
⠀⠀⠀⠀⠀⠠⠀⠂⠀⢀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠂⠀⠜⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣾⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢻⣿⣿⢹⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⣿⣿⠸⣿⣿⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣽⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⡿⠛⠛⠛⠛⠛⠻⠟⠋⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⢻⣿⣿⣿⣿⣿⣟⣽⣾⣿⣿⣿⣿⣿⣷⣬⣙⡛⠿⠿⠿⠿⢿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣛⣿⣿⣟⣻⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣭⣿⣿⣭⣿⣿

⠀⠀⢀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⣀⡀⡠⠀⠀⠀⢀⠔⠁⢠⠊⠀⠰
⠀⠀⠀⠀⠀⢰⢀⣀⣀⣀⠤⣄⣀⠀⠀⠀⠰⠀⠐⣄⠤⠐⠂⠀⠈⠁⠀⠀⠰⡋⠀⠀⠒⠒⠲⠖⠒⠊
⠀⠀⢀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠋⠀⠀⠄⣤⡀⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠉⠒⠒⠤⠤⠴⠢⡄⠀⠀⠀⠀⠀⢇⠀⣀⣀⣀⠤⠚⡄⠡⡀⠀⠀⠀⠀⠀⠀⠉⠒⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠄⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀
⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣰⣃⣀⡀⣀⠠⠄⠀⠀⠀⠁⠀⠀⠀⣀⡀
⠀⠀⠀⡆⠀⠐⠋⠉⠉⠙⠁⠉⠁⠀⠀⡼⠁⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠙⠉⠉⠃
⠀⠀⠀⠈⠒⠤⣀⡀⠀⣨⠓⡄⠀⠀⠀⠈⠒⠤⢀⣀⣀⡰⡟⠄⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⢚⠀⠀⠀⠀⠀⠀⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠉⠘⠒⡄
⠀⡘⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⣀⡀⠤⠤⠤⠤⠀⠀⠀⠒⠒⠂⠠⠤⢀⣴⠋⠈⢣⠔⢢⠖⡄⠀⠀⡜⠀⠀⠐⠀⠀⠀⠀⠀⠀⡏
⠀⠇⠀⠀⠀⠀⠐⠀⠀⢳⠀⢠⢀⠈⠀⡅⢰⠁⠸⠃⠁⠀⢀⣀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⢄⡄⠬⣆⠀⢚⡀⠃⠼⠃⠒⣶⠁⠀⠀⠀⠀⠀⡀⠀⠀⢸⠃
⠀⠐⠀⠀⠀⠀⠀⠆⠀⢸⠊⠙⣄⠢⠄⠁⠘⠄⢼⠶⠚⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡈⠁⠉⠀⠁⠶⠠⢤⡏⠀⠀⡐⠀⠀⢠⠁⠀⢀⣏⣀
⡠⠒⡂⠀⠀⠀⠀⠈⠀⠘⡖⠉⠀⠀⢰⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣤⠊⠀⠀⢸⠁⠀⢠⠁⠀⠀⠂⠀⠀⡼⠀⠈
⠁⠀⢳⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⡥⡴⠶⠏⠁⠀⠀⠀⠀⢀⠤⠒⠒⠒⠀⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⢀⠇⠀⠀⠀⠀⠀⠈⠀⠀⢀⠟⠲⠖
⠀⠀⠘⡀⠀⢐⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⣠⡴⠂⠀⠐⠀⢄⠑⡀⠀⠀⠀⣸⠀⠀⠀⠘⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⢇⠀⠀⠀⠀⢀⣠⠴⠃⠀⠀⠀⠀⠀⠀⢸⠀⢀⡠⠴⠁⠈⢠⠊⠉⢱⠀⠀⠀⠓⢄⠀⠀⠃⠀⠀⠀⡄⠀⠀⠀⠈⠓⠲⠤⠤⢄⣀⣀⡴⠃
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢳⡄⣀⠀⠀⣀⣴⠶⡴⢋⠖⢢⣄⢨⡟⠀⠸⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⢂⡏⠀⠚⠋⠉⠀⠙⠻⢴⣋⠜⠀⡗⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⢃⠄⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢃⠝⠛⠁⠀⠀⠀⠈⠛⠄⠖⠉⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⣔⡠⠋⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠈⡘⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⠣⡀⠀⠀⠀⠀⠀⢠⠃⢠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢧⡀⣀⣀⡤⠂⠀⠙⠲⠦⠤⠄⠊⠁⠀⠀⠑⠄⡀⠀⣤⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡓⠒⣶⠋⠑⣶⠚⠛⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠉⠙⠦⠐⠋⠉⠉⠀⠈⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣀⣀⠤⠴⠄⣀⡀⠀⠀⠀⠀⠈⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡰⠋⠁⠀⠀⠀⠀⠀⠀⠈⣉⠇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠙⠢⢄⡀⠀⠀⠀⢠⣴⣋⣁⣀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠢⠶⠶⠾⠉⠀⣄⢴⠁⠞⠙⠟⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠤⠤⠠⣤⠀⠴⠊⣀⣀⡀⠀⣀⣀⡤⠼⠙⠄⠀⢀⡤⠤⠤⠾⠀⣠⣤⠀⣀⣀⣀⠀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣤⣾⣭⣭⣥⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣤⣤⡀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⣴⣶⣶⣶⣶⣾⣉⣀⣽⣿⣵⣴⣦⡀⠀⢈⠿⣏⠀⠀⠀⠀⠀⡘⠉⢛⣛⣯⣿⣶⣶⣶⣶⣤⣤⣤⣼⣽⣿⣿
⠀⢸⣿⣿⣿⣛⣿⣶⣶⣶⣿⣿⣿⣷⣾⣶⣶⣶⣾⡿⠃⠀⠰⣄⣁⡀⣀⣼⣿⣿⣗⣒⣒⣒⣲⣷⣲⣶⣖⣿⣿⣿
⠀⠀⣿⡿⠿⣿⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠺⠟⠹⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠸⣿⠁⠀⣿⡇⠀⠀⠀⠀⢰⣧⠄⠒⠲⡗⠰⣦⣤⣤⡖⢤⡤⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣸⣿⠋⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠈⣿⣛⣛⣿⣟⣺⣿⣶⣾⡒⣈⠤⢀⠀⠀⠀⠀⠈⠛⠻⠷⠻⣿⣿⣿⡄⠀⠀⡴⠋⠹⣇⣿
⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠈⠉⠁⢘⣧⠿⡿⠶⠶⠎⠁⠀⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠣⣀⣰⠿⣻⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡖⢺⡏⣿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⠀⠀⡠⠋⡫⢸⣿⣿⣦
⠀⠀⠀⢠⣴⣯⣺⣿⡛⣻⣿⣊⣷⣤⠀⠀⠀⠘⣇⢸⣷⠋⢾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠚⠻⠛⠛⠛⠋⠀⠀⡀⢳⠗⢺⡟⣛⡟⣧
⠀⠀⠀⢾⣶⢶⡏⢉⣥⣍⠉⣳⢶⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢰⡷⢾⡶⢾⣷⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⢉⡙⠛⠁⠀⠀⠀⠀⣀⣀⣀⣤⣤⡀⠀⡁⢿⣿⣾⣿⠇⠀⠲⠶⠶⢶⠶⠀⠀⠀⠀⠸⠷⠼⠧⠽⢿⣿⣿⣿⠃
⠀⠀⠀⢀⣀⣀⣀⣄⣸⣷⣤⣤⣤⣤⣤⣼⣧⣴⣶⣶⣿⣷⡀⠀⠁⠉⠉⠀⠀⠀⠈⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠣⠗⢸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠉⠙⠛⠛⠛⠋⣿⣿⣿⢿⣿⡿⠛⠛⠻⠿⣿⣿⣿⠀⠀⢘⡟⣿⣿⡈⡏⠙⠛⡂⠀⠀⠀⠘⠛⠛⠁
⠀⠀⠀⠀⠀⢸⣳⣏⣯⣵⣺⠀⠀⠀⠀⠀⣿⣿⡟⠚⠛⣿⣄⠀⠀⠀⠀⠈⠉⠀⠐⠋⠀⢹⣟⡃⠙⠛⠚⠓⠶⠶⠶⠖⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠘⠧⠤⠜⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⠟⠂⠀⠀⠀⠀⠀⠀⠀⠚⠦⣾⠿⠧⠤⠤⠤⠤⠤⠤⠤⠤⠭⢭⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⣴⡗⠀⠘⢁⣿⡆
⠀⠀⠀⠀⠀⡠⢿⣿⠀⠐⣿⣿⡁
⠀⠀⠀⠀⣸⣧⣼⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⡘⢧⡀⠀⠀⠀⠀⠀⣀⣤⣄⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢳⡀⠀⠀⠀⣰⡿⠁⠙⠛⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⢧⣀⠀⠀⡯⠀⠶⠂⣤⣄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣁⠀⡄⠈⠈⠳⡄⠉⣠⣤⣤⡀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⢿⣷⠀⠀⠀⠹⣶⡄⠁⢈⣡⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⠀⠀⠀⠀⠙⠿⣷⣿⣿⣛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠋⠙⠉⠉⠀⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢿⣿⣦⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⢻⠿⢷⣤⣤⠞⣀⠀⠀⠀⠀⢀⣉⡄⠀⠀⠼⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⢿⠇⠘⢿⣷⣦⣤⣤⣴⣾⣿⣦⡀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢀⣠⡤⣤⡀⠈⠛⢻⣿⣿⣿⠏⠈⠻⣦⣄⡀⠀⠘⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠁⠈⠀⠉⠛⢷⣶⣾⡿⣿⠏⠀⠀⠀⠈⠙⠻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠂⠀⠀⠚⢳⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠚⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠉⠈⠉⠑⠋⠉⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡼⣤⣄⣠⣄⣤⡴⣶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠸⣿⣿⣿⣟⣡⡘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆⢸⣿⣿⣿⣼⣷⣽⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠹⣿⣿⣿⣟⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠻⣿⣿⣿⠇⠀⠀⠀⠀⠈⢻⣿⡏⠋⠻⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢀⡟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⡀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢡⠀⠀⠀⠀⠀⠀⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣆⠀⠀⠀⠀⠀⠀⡼⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢇⠀⠀⠀⠀⢸⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡹⡄⠀⠀⠀⠀⢸⣇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⡄⠀⠀⠀⡾⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢹⡀⠀⠀⠀⢻⡟⢿⡀⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠀⠀⣼⠁⡜⠁⠀⢀⡼⣷⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣶⢄⡀⠈⢳⣄⠻⣄⠀⢠⡿⠛⠉⠉⠈⠻⣆⠀⣰⢣⠞⠀⢀⣴⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣕⠠⡙⣦⠹⣤⡿⠁⠀⢀⣀⣀⣀⣘⣶⡣⠋⣀⣴⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣝⣷⣼⣷⠾⠛⠛⠉⠉⠉⠉⠉⠻⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣽⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣶⣤⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⡤⠊⢰⠏⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠐⡄⠙⠷⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡽⠀⠀⡟⠰⠁⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠘⣆⠘⣄⠀⠘⣧⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠃⠀⢸⣷⡇⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⣿⢛⡇⠀⢸⣆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡀⠀⢸⡆⢧⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣧⠀⡇⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⢿⣈⣇⠀⠀⠀⠘⢿⣷⡀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿⣠⣧⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⢻⣍⢧⡀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⢠⡏⣼⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠳⡀⠀⠐⠋⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡎⢱⠀⠀⢠⠛
⠀⠀⠀⠀⠀⠀⢸⠀⢃⠀⠀⠀⢸⠀⢸⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠘⡆⢠⡀⢸⠀⠀⠇⢰
⠀⠀⠀⢀⣀⠀⠈⡇⠘⡀⠀⠀⠘⠀⡎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠈⠀⢸⠀⢰⠀⡸⠀⢀⣀
⠀⠀⠀⠸⡈⡆⠀⢃⠀⡇⠀⡀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢰⠀⢸⠀⡎⠀⠇⠀⡜⢸
⠀⠀⠀⠀⡇⠘⡀⠸⠀⢸⠀⠇⠀⠀⡇⠀⡇⠀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⠀⠀⢸⠀⠇⢸⠀⢸⢰⠇⢰⠀⢰⠀⡎
⠀⠀⠀⠀⠸⡀⢣⠀⡇⠘⡆⠀⠀⣀⠃⠀⡇⠀⠀⢀⠎⢠⠁⠀⢸⠀⢧⠀⠀⠀⡟⠀⢸⡸⠀⢸⠸⠀⠸⢀⠇⢰⠁
⠀⠀⠀⠀⠀⡇⠈⢧⠇⠀⠉⠀⠀⠉⠀⠀⡇⠀⠀⡼⠀⡜⠀⠀⠀⡆⠈⡆⠀⠀⡇⠀⠀⠁⠀⠈⠁⠀⢧⠜⠀⡞
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⠇⠀⡇⠀⠀⠀⢡⠀⢣⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡼⠀⢰⠀⠀⠀⠀⢸⠀⠘⢦⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠉⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡅⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣦⠘⣿⢿⡇⠀⢀⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢈⡏⠀⠀⠀⠀⠀⢈⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⡷⠿⢿⣿⠁⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠭⠽⠉⠉⣄⣴⡏⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣠⣀⣠⣼⣿⡛⠀⠈⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⢣⣿⠁⠀⠀⣀⣤⠜⡑⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢸⣏⣹⣿⣿⣿⣺⣿⠶⠤⠾⢻⠇⢰⠃⠀⠀⠙⠓⢤⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠚⠋⠁⠀⠁⡇⠈⡇⠀⠀⠉⢻⠿⠁⠀⠀⠀⡜⠀⠇⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⡸⠁⢀⠀⣤⡄⠀⠃⠀⠁⠀⠀⠀⣹⠀⠀⠀⠀⢠⠁⢸⠀⣺⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⣇⠀⠘⠀⣿⣇⠀⠆⠀⢄⠀⠀⠀⢸⡁⠀⠀⠀⡀⠀⢸⢠⣿⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⡇⠀⠀⠐⣾⣿⣾⠀⠀⠈⠣⣀⢀⣿⡇⠀⣀⠞⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⣿⣿⣧⡀⣀⣀⣬⡿⠛⠛⠛⠁⣀⣠⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⢀⣿⠁⠀⠀⢀⣴⣿⣿⡿⠿⠋⠉⠉⠙⠤⣄⣀⣰⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⡏⠀⠀⠾⠛⠛⠛⠻⠾⣶⣶⣶⡶⡂⠀⠀⠉⠙⠛⠻⠿⠿⠿⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣽⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⡠⠃
⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⡷⠁
⠀⠀⠀⣧⡠⣴⣦⡀⠀⠀⠀⣀⣀⣤⣵⣾⣿⣿⣿⣿⣷⣤⣄⣀⣀⣈⣐⣲⣚⣛⣴⠃
⠀⠀⠀⠉⠉⠉⠉⠙⠋⠉⠉⠁⢸⣿⣿⠟⡋⠉⡀⠈⡛⠛⠻⠿⠿⣿⡿⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠁⠒⠂⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠒⠀⠒⠚⠈⠀⢨⣿⣿⣿⣷⣎⠉⠉⠒⠂⠈⠁⠀⠀⠉⠉⠐⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢿⡻⣷⡀⠀⠀⠀⠀⠀⠈⠳⡶⢃⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣐⠲⣷⣤⠀⠠⠴⠘⡓⠾⢿⡎⣉⠈⣁⡉⠻⢒⣃⠀⢀⣀⣤⠾⡁⠈⠹⠁⠀⢠⣸⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣸⣷⣄⣹⣶⠀⠀⠀⣡⠍⠑⠀⠳⣄⣀⣼⠁⠂⠉⠁⡀⠀⣀⣀⣼⣷⣶⣇⠀⠀⣼⣷⣷
⠀⠀⠀⠀⠀⠀⠀⣇⢿⡾⠟⠉⢿⣿⠛⠻⣿⣿⠋⠀⠀⠀⠀⠙⢿⡃⠀⠀⠀⠀⠹⣿⣿⡿⠉⠹⣿⣿⣶⠀⠁⡟⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⣤⣴⠿⠃⠀⠀⠙⣟⠶⣤⣀⣀⣀⣠⡼⢦⣀⣀⣀⣠⣾⣯⡟⠁⠀⠀⠙⢿⣿⠀⡄⠙⣇
⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠙⠤⡀⠀⠀⠀⠀⠈⠑⢺⣿⢹⣯⠀⠰⡆⠀⢀⣾⣾⡟⠉⠀⠀⠀⠀⠀⣀⠟⡀⠇⡄⣹
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⢀⣿⣽⣿⡀⠀⡇⠀⣨⣯⣿⣧⠀⠀⠀⠀⠀⢸⠄⠀⠙⠃⠀⣹
⠀⠀⠀⠀⠀⠀⢘⡗⠀⠀⠛⢳⣾⣷⡆⠀⠀⠀⡠⠎⠈⣻⣿⣷⣶⣷⣶⣿⠋⠉⠛⢇⠀⠀⠀⣶⣿⣶⠲⠖⠀⠀⣿
⠀⠀⠀⢰⡟⣋⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⠊⠀⠀⠰⣾⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠣⡀⢰⣿⣿⣿⣷⣶⣶⣾⣿⣛⠛⣶
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡇⠀⠀⢀⡞⣻⢛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣇⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⡿⠛⣿⠋⢹⡍⠀⣷⠀⢸⣇⣿⣛⣛⣛⣉⣉⣭⣭⣭⣭⣭⣭⣭⣍⣛⣛⣿⣄⡿⠀⢸⠇⠉⣿⠙⢻⡟⠿⣿⠁
⠀⠀⢈⢡⣬⣇⠀⢹⡀⣈⣧⣠⣼⣤⣾⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣦⣿⣤⣸⣇⡀⣿⠀⢰⣯⣍⢱
⠀⠀⠘⠿⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠿⠿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠒⡟⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠳⡈⠀⠀⠐⠒⠈⠀⠀⠀⣸⠁⠀⠀⠀⢀⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠳⠲⠤⢤⣤⣤⠤⠒⠋⡏⠀⠀⠀⣠⠏⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠐⠀⠀⠀⡠⠊⠀⠀⢁⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡰⠋⠁⠀⠀⣀⡴⠋⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡙⠢⣤⠀⠒⠀⠀⠀⠀⠀⡠⠓⠀⠀⠀⠀⠂⠗⠀⢀⡜⠁⠀⠀⢀⠔⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡿⠉⠢⢄⣀⡠⠀⠈⠀⠈⠀⠀⠀⠀⠠⠀⢀⠎⠁⠀⣠⡞⠁⠀⠀⠀⡀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡇⠀⠀⠀⡏⠢⣀⠀⠀⠀⠘⠀⠀⠀⠃⢠⠊⠀⡠⠊⠀⠀⠀⠀⠀⢀⣈⡵⡶⠒⣺⠏⠁
⠀⠀⠀⠀⠀⠀⠀⡌⠀⡸⠁⠀⠀⢀⠁⠀⠀⠛⢀⠀⠀⠴⠀⠰⢠⢀⡠⠚⠀⠀⣀⡠⢶⠖⠋⠁⠀⡸⠁⢰⠏
⠀⠀⠀⠀⠀⢀⡜⠀⣰⠃⠀⠀⢀⠌⠀⠀⠀⠀⠀⠙⢀⡀⠀⠂⠄⠈⢠⠀⠘⠋⠀⠠⠁⠀⠀⠀⠀⠃⠀⡟
⠀⠀⠀⠀⢠⠞⠀⡴⠁⠀⠀⣠⠊⠀⢀⣠⡎⠀⣤⠀⠤⠀⠰⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠶⢿⡒⠛⠻⢯⡉⠉⠉⠛⢥⡀⠀⠁⠀⠀⠱⠀⠒⠤⠐⠉⠀⠦⠀⣄⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠙⠢⡀⠀⠈⠑⢄⠀⠀⠙⣆⠀⠀⠀⠀⠃⠀⠤⠀⠘⠀⠀⠀⠁⠁⠐⣺⡦⢄⣀⠀⠀⣇⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⢢⠀⠀⢸⡀⠀⡸⠂⠀⠀⣆⠄⠐⠐⢆⠀⠀⠀⠈⠀⠀⠀⠀⣉⡱⠾⠦⢤⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠈⠆⠀⠀⡇⡔⠀⠀⠀⠀⡇⠀⠀⠀⠈⢂⠐⠀⠀⠀⢀⠀⠊⠀⠀⢀⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢸⠀⢀⠟⠀⠀⠀⠠⣰⣇⠄⠀⠀⠉⠘⢣⡀⠰⢡⠃⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠸⣶⣃⣀⡀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠱⡆⠇⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡞⠁⠀⠀⠀⠁⠢⡀⡧⢀⡀⠄⠀⠀⠀⠒⠒⠿⡄⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠒⠒⠂⠒⠤⣀⠀⢹⡟⢁⣀⠤⠀⠀⠐⠒⠲⠦⢽⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀
⠀⠀⣠⣼⣿⣯⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣛⠿⢛⣛⢛⣛⡋⣛⣛⠛⠛⣛⠛⢓⣒⣒⣒⣀⡀
⣾⣤⣬⣤⣷⣤⣤⣤⣾⣿⣿⣿⣷⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣾⣷⣾⣿⣾⣷⣦⣶⣷⣖⣿⣷⣦⣭⣄⢀⣀⣀⡀⠀⠀⢀⣀⡀
⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⡿⡿⢍⣉⣛⣿⣬⣴⣿⣿⣷⣤⣤⣭⣉⣉⣉⢉⣉⣻⡟⣿⣿⣿⣿⣯⠛⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⠉⠉⣹⠟⣿⣽⣿⣿⣿⠟⡛⠻⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣿⡗⣲⣿⣿⣿⣿⣿⡇⠽⣯⠷⣶⣶⣾⣿⣿⣿⣿⣿⡞⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣶⣒⣋⣉⣩⣥⡤⠶⠶⢯⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠛⣛⣫⣩⣅⣠⣤⣤⡽⣿⣿⣿⣷⠛⠾⠿⠿⢿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⣠⣭⠭⠽⠭⠻⠿⣿⣿⣭⣽⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣾⣿⣾⡟⠉⠉⠹⠁⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻
⢀⣀⣈⣉⣤⡤⠤⠖⠞⠻⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠛⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧
⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠋⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋⠀⢰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣴⣮⣷⡀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⣾⣧⣤⣭⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⠉⠋⠉⠉⠁⠀⠀⠀⠀⢸⣿⣷⣧⣷⣾⣿⡇⠀⠀⠀⠀⠈⠉⠉⠙⠋⠙⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⡟⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⢹⡄
⠀⠀⠀⠀⠀⢸⣠⣿⣿⣿⣿⣾⠟⠓⢦⣤⡶⠶⢤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣶⣶⣤⡴⠞⠛⣷⣿⣿⣿⣿⣄⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣷⣄⠀⠀⠈⠉⢿⣿⣿⣿⣿⠟⠋⠉⠁⠀⣠⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣾⣟⣿⣿⣻⣶⣦⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠿⠋⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣬⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⣿⣿⡻⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣿⣿⣶⣬⣙⠻⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣽⣿⣄⠸⠿⠿⣿⣿⣿⣿⣶⣤⡝⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣍⡻⣿⣿⡇⢉⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣒⣂⠛⠛⠿⣿⣿⣿⣿⣿⣿⡯⢟⣰⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣍⡛⢻⠟⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣟⡀⢤⣭⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⣿⣿⣿⣿⣿⢃⣾⠁⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⢸⣿⣿⣿⣿⢃⣾⣿⢠⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣸⣿⣿⣿⠇⣾⣟⠿⠘⣿⡄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣷⡄⠹⣿⣦⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣦⣄⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡄⠿⢹⣷⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠃
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⠟⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣨⣵⣶⣾⣿⣿⣷⡄
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⠟⣹⣿⣿⣿⡟⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⣿⣿⣿⡿⠿⠿⠃
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣤⣾⣿⣿⣿⡿⢁⣾⣷⣦⣄⣉⡉⠛⠛⠛⣉⣩⣤⣶⣿⣿⣿⣶⣦⣤⣤⠄⢀⣴⣾⠆
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⠟⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣉⣥⣶⣿⣿⠟
⠀⠀⠀⠀⣦⣙⠻⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣁⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⣻⡟⢳⡖⠲⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢱⡟⢃⣿⠗⢀⣿⣶⠈⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣿⠇⢸⣿⠀⢸⡏⠈⢁⠀⠀⠀⠀⠀⠈⣓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠀⣸⣿⠀⢸⡇⠀⠀⣷⣶⣄⡄⠀⠀⢸⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣿⣿⡀⣼⣧⣠⣴⣿⣧⣀⠀⣲⠶⠋⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣻⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⠟⢿⡄⠀⠐⢶⣤⡀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⢿⣦⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠙⢿⣦⡀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠻⣄⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢻⣆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣤⣸⣿⣦⡘⢿⣧⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠉⠁⠀⠀⠀⠈⠉⠛⠿⣿⡿⠿⣿⣿⣿⣷⣄⡀⠀⠙⠆⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⢷⣦⣈⠉⠙⠛⠻⠷⣦⣤⣀⡀⠀⢠⡤⢀
⠀⠀⠀⣀⣠⣤⣴⣶⡶⠶⠶⠶⢶⣶⣾⡿⢋⣼⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠉⠻⠿⣦⡀⠀⠀⠀⠉⠛⠋⠀⠀⣰⡿
⣠⡴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠋⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣶⣶⣾⠟
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⣻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⣿⣿⣿⣧⣤⣤⣤⡴⠶⠾⠛⠉⠁⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⡿⠟⣩⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⢁⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠰⢿⡟⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣤⣤⣀⣀⣀⣠⣤⣤⣴⣶⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⡀⡰⠢⣄⡀
⠀⠀⠀⠀⠀⢠⠛⠺⠷⣮⣧⡀
⠀⠀⠀⠀⠀⣯⣀⡀⠀⠀⠉⠛⠻⢶⣦⠤⣀
⠀⠀⠀⠀⢠⠃⠀⠉⠓⣆⢀⠀⠀⠀⠈⠙⠒⠭⣕⣆
⠀⠀⠀⠀⠘⠦⣄⡀⢠⠏⠉⠛⠳⢆⣀⠀⠀⠀⠀⠉⠙⠲⠦⣴⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣂⠤⣀⠀⠀⠀⠉⠒⢠⢀⡤⢀⡀⠀⠀⠉⠒⠮⣕⣢⠤⣀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠭⡒⢦⠀⠀⠻⣄⣼⡳⠺⢵⣆⠀⠀⠀⠂⡈⡝⡶⠽⢶⣥⠮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠓⡤⣄⣀⠈⠉⢓⠋⢸⠉⠙⣬⣠⠧⢇⡇⡇⡆⠈⠉⢻⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢰⠃⠀⠀⠉⢙⠞⢤⡀⠉⡢⠕⠈⢦⣙⠓⣧⣧⣷⠀⡼⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠀⠀⢀⣤⣯⡤⠶⡇⠀⡇⠀⠀⠀⠀⠉⠀⠈⢛⡍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⣷⣄⣀⠀⠀⠀⠀⣿⢠⠋⢰⡾⣳⠀⠀⣠⢖⣙⣃⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣦⠴⠏⡯⠗⠀⠉⠁⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠳⡄⠀⠀⠀⠀⢸⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡾⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢠⡇⠀⣔⡻⣦⠀⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢭⣝⣳⢤⣈⣛⡁⠀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠶⠧⠞

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣀⣤⣤⣄⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠈⠁⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠐⠶⣤⠶⠇⠀⣿⠀⠀⠳⠶⠶⠃⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣀⠀⠀⣀⡄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠉⠙⠉⠉⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠓⠦⢤⣤⣤⣤⠴⠚⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢦⡈⠳⢄⡈⠻⡌⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣤⠶⠋⣹⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠉⠲⡄⠁⠀⠈⠀⢹⡄⠀⠀⠀⠀⠀⣠⠟⠠⠞⢁⡴⠚⢁⡤⢾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣌⠙⠆⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣴⠃⠀⠀⠀⠀⠠⠚⢉⣠⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠹⣆⢀⡼⠁⠀⠀⠀⠀⠀⠀⠐⢋⣡⠞⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⢿⡁⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠙⢦⣠⡞⠁⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⣠⡼⠳⣦⡀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠶⠞⠋⠀⠀⠈⠙⠛⠲⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⣸⣤⡄⣀⣀⠀⠀⠀⠀⠀⠉⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⣭⣾⣿⡿⣟⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠋⢡⠇⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄⠀⠀⠀⠇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣼⠅⢁⣰⣇⡃⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣠⣴⣦⣄⠀⢀⠈⢻⣷⣦⣤⣤⣤⣀⠀⠀⣯⣦⠍⢧⡘⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣿⣿⣿⣷⠊⠁⠰⢿⣿⣿⣿⣿⣿⣧⡀⢿⣿⣿⣿⡿⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⣿⣿⣿⣥⡀⠀⠠⣮⣿⣿⣿⣿⣿⡿⠀⠀⢿⣿⣿⠇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⣹⣿⣿⡄⠀⠈⠛⠻⠿⠿⠛⠀⠚⠃⠈⠉⠁⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⣀⣠⣾⣽⣿⣿⡇⠀⠀⠀⠐⢦⣄⠁⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠚⠀⠀⠌⠀⠀⠉⠁⠀⣴⣾⣷⣦⣽⣦⣄⣠⡴⢾⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠰⠄⢠⢠⠠⡙⠿⠛⢿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠤⠠⠤⠄⠊



⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣶⠀⠀⠀⢀⣀⣤⡄⢠⣤⣀⡀⠀⠀⠀⣶⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⢀⣄⢾⣿⣿⣿⡇⢸⣿⣿⣿⡷⣠⡀⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣴⣿⣿⡌⣿⣿⣿⡇⢸⣿⣿⣿⢡⣿⣿⣦⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣎⠻⣿⡇⢸⣿⠟⣱⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⣅⢨⣴⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶⣿⣷⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢰⡶⣠⣶⣶⣶⡄⢶⡖⣴⣶⣶⣶⡰⡶⢠⣶⣶⣶⣦⡰⠆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⡰⣿⣿⣿⣿⣿⠎⢼⣿⣿⣿⣿⡿⡀⢿⣿⣿⣿⣿⢇⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡜⢿⣿⣿⢫⣾⣧⠻⣿⣿⠟⣼⣿⣎⢻⣿⡿⢣⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣎⠿⣡⣿⣿⣿⣷⡹⢏⣾⣿⣿⣿⣧⠛⣱⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⣤⢻⣿⣿⣿⡿⢣⡜⢿⣿⣿⣿⠟⣴⡙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⣿⣷⡹⣿⡿⣱⣿⣿⣎⢿⣿⢏⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠌⣼⣿⣿⣿⣿⣦⢡⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏⠀⠈⢿⣿⣿⡿⠁⠀⠹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠘⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⣤⠀⠀⣀⡀⡀
⠀⠀⠀⠀⡇⠒⠀⠀⠀⠙⠀⠀⠀⠀⠋⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠈⠃⣼⣿⣇⠘⠁⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠐⠒⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿⣿⣿⣿⣿⡇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠸⠿⠿⠿⠿⠿⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠘⠃⠀⣿⣿⣿⣿⣿⡇⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣤⣤⡀⠀⣤⠀⢠⣤⣤⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡉⠉⢉⠉⠉⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠛⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠀⠸⠇⠀⠺⠿⠇



⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠴⠶⠒⠛⠋⠙⠛⠒⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠚⠛⠙⠛⠓⠲⠶⠄
⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⣿⡀⠀⠀⠀⠀⠀⢀⣰⣦⡄⠀⠀⠰⣄⠀⣆⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⠀⠀⠀⠠⠄⢿⣿⡆⠀⢀⡀⠙⠛⢛⠟⡇⠀⠐⠂⠈⡻⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⣀⣉⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⢈⣀⣤⣄⠀⣼⢂⠁⠀⠀⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢻⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠿⠇⠈⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⣁⡄⠀⣼⠂⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⢀⡜⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠰⣿⠓⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠻⢿⣿⠙⠷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠈⢿⡀⠀⠀⠀⣠⡆⠀⠈⠛⠻⡦⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣷⠀⠀⢴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⢿⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡾⠟⡧⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠇⠉⣭⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣀⠀⠀⣠⣾⠇⠀⠀⠀⠚⣶⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⢿⣅⠀⠀⠛⣿⣦⣄⡀⠀⣰⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠨⠀⠉⠙⠻⡿⣿⠟⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣸⣤⡆⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⢤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠻⡷⣦⢄⣀⠀⠀⣀⣀⣽⣷⡧⠈⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⡴⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢷⣹⢛⢇⡈⠭⠭⣽⣿⣽⡟⡅⢀⠀⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣟⡝⢀⣅⣿⣆⢿⣟⣿⣿⠠⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠠⠀⠀⢀⠰⣀⣠⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣋⣾⣻⠟⣿⣿⣀⠒⡇⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣗⠀⠀⠀⠀⣠⣿⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢙⠛⠛⠋⠈⠙⣿⣿⣿⣯⠄⢐⠃⠀⠀⣰⠿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⣴⣿⣿⣿⣿⣿⡿⣏⡓
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣤⡀⡀⠀⠀⠀⠀⠖⠀⠀⠻⣿⣿⣧⣮⣌⠀⠀⠀⣷⣤⣾⣿⣿⣿⣿⡏⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠁⠉⠀⠀⠒⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠸⠘⠉⣿⣿⣿⣿⣿⣿⣿⡃⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⣄⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣏⣹⣿⣟⣹⣿⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⢀⣤⣄⠀⠻⡿⠇⠘⣿⣿⣿⣿⣿⣿⣯⠀⣴⣿⣿⣿⣷⣿⣿⣿⡷⢤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡤⢆⠀⢰⡶⢶⡦⠘⣄⠀⠀⠉⠛⠿⣿⡋⠉⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢰⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⠉⣽⣿⡿⣧⣣⣮⢰⡐⣆⢘⣶⣶⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠃⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠋⢙⣾⠣⣿⣿⣸⢇⣿⡄⣝⢿⣟⣻⣿⡿⠁⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⡶⠂⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠟⠛⠋⠈⠉⢿⣿⣿⣭⣭⠟⠁⢰⣿⣿⣿⣿⣿⣿⣇⠀⠻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣞⣠⡎⡠⠂⡔⠀⡀⢀⠀⠀⠀⠑⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣾⡁⠰⢿⣿⣤⣄⣀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣷⣶⣿⡿⢿⣿⣷⣂⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢿⣿⣷⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡎⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡄⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⣀⣤⡶⠾⠏⠀⠀⠈⠉⠑⠦⣄⡀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⠘⡄
⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠶⢄⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⣧⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣗⣤⣀⣿⡀⠀⠀⠀⢀⣼⣿⣷⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣾⣿⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣶⡹⣿⣦⡀⠀⣸⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⣲⣾⣶⣿⣿⣿⣿⣿⣿⡆⣹⣷⣦⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡹⠿⠿⣿⣿⣿⣿⣿⡟⠁⣬⣿⣿⣿⢻⡿⠛⢀⡇
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠉⠉⠑⠋⠁⠁⣠⡀⠻⣿⣿⣿⣵⣿⣷⠞
⠀⠀⠀⠀⣠⣿⣿⣿⣿⠏⠀⠀⢻⣿⡿⠁⠀⠀⠈⠙⠻⢿⣿⣿⣷⣄⣀⠀⠀⣀⣤⣾⣿⣿⣿⡶⠶⠿⠿⠟⠁
⠀⠀⠀⠀⠋⠋⠼⠿⠁⠀⠀⠀⢘⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⡀⠀⠀⠀⠀⢀⠴⡶⢄⠀⠀⠀⠀⠀⡰⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣇⠀⠀⠀⢠⠋⢰⣷⠈⢣⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⣶⡄⠘⡿⠀⣴⠆⠀⠀⣸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢙⡙⢦⠀⠸⣷⢸⡟⢰⡿⠀⣠⠞⣉⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡳⣄⢹⣆⢃⣾⢁⡴⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⠿⠿⢿⣿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡶⠲⣦⢶⢺⣿⣤⠆⠐⢦⣽⣿⣶⢶⡶⠲⣶⣶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠿⠈⠃⠈⠁⠀⠀⠀⠀⣠⢺⣿⣵⠏⠈⢷⣽⣿⢢⡀⠀⠀⠀⠀⠉⠀⠊⠸⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⣽⡛⢿⣡⣇⢀⣧⣹⡟⣻⣝⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⡋⠀⣰⠋⡔⢠⣿⣿⣾⣿⢧⡐⡌⢷⡀⠈⡛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠷⠊⠀⢰⡋⡸⠀⢿⠀⠙⠏⠁⢹⠇⠸⡈⣳⠀⠈⠲⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠀⠀⠀⠀⣼⡿⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠹⣿⡆⠀⠀⠀⠨⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠍⠁⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⢀⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣁⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇




⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠙⠻⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠖⠀⠀⠉⠉⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡼⣿⠀⢿⢶⣤⡄⢠⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢡⠀⠀⠀⠀⢀⠀⢰⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⠀⠀⠀⠠⡤⠟⠟⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠃⠘⣷⣄⡀⠀⠲⠶⠮⢰⣿⣿⢷⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡋⠉⠙⢷⣄⠀⠉⠻⣤⣤⣤⡤⠞⠈⣿⠀⠻⣿⡉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠀⠉⢻⣆⣠⣬⣿⣷⣤⣄⡈⠻⣷⣀⣤⠴⠿⢤⣄⣻⣧⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠙⢷⣼⡿⠁⠀⠀⠀⠈⠙⣿⡀⠀⠀⠘⣿⡀
⠀⢠⣶⣿⣿⣶⣶⣶⣶⣿⠷⠟⠛⠶⣴⣿⠃⠀⣀⣀⣀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣏⡀⠀⠻⢿⣦
⢀⣿⠻⢻⡟⡿⣿⠏⢹⣿⣄⣠⣤⣄⣼⣯⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⢻⡇⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⣿⣷⣄
⣼⡏⠀⣼⠏⠀⣿⢷⡟⢀⡼⠁⣸⠋⢈⣿⣶⡟⠹⣿⣿⠟⣿⠏⠹⣇⣿⣿⣄⣀⣀⣠⣴⣾⣿⣿⣦⡀⢀⣤⣾⡏⠈⢻⣆
⣿⡇⠀⣿⠀⢸⣿⡞⠀⡞⠀⢠⠃⠀⣼⣇⣿⠁⠀⣾⠀⢸⡟⢠⠀⡿⠛⠋⠙⠛⠛⠛⣫⣿⣿⡿⣿⣿⣿⣿⣿⡆⠀⠈⣿
⣿⣇⡀⣿⣀⢸⣿⣄⠀⡀⠀⠀⠀⠀⣿⣿⣿⠀⢀⡇⠀⣾⡇⣾⢰⡇⠀⠠⣤⣤⣶⣿⣿⣿⡿⠁⠸⣿⢹⠁⠀⣁⠀⠀⣿
⢹⣿⣷⣿⣿⣾⣧⠙⣷⢿⣄⡀⢣⣀⠉⠉⣿⠀⣸⡇⡀⣿⡇⠛⣸⠇⢸⡇⣀⣠⣴⣿⣿⠋⠀⠀⠀⠻⣮⡂⠀⠉⠀⣠⡿
⠀⠛⠿⠿⣿⣿⣿⣿⠏⠀⠈⠙⠛⠛⠛⠛⢿⣿⣿⣷⣿⣿⣇⣠⡟⢀⣼⣷⣈⣩⣽⣿⠇⠀⠀⠀⠀⠀⠈⠛⠷⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⠟⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⣀⠘⡄⠀⡆⢀
⠀⠀⠀⣤⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠘⣶⡊⠀⠀⠀⠈⠓⣷⣾⠕⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⣤⠔⠁⠀⠀⠀⢦⣤⠖
⠀⠀⠀⠻⠇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠊⣿⠓⠄⠀⠀⠀⠱⣿⣿⢧⠁⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠺⣿⠦⠀⠀⠀⠀⠺⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⣀⣈⢑⣄⠀⣰⣽⡩⠭⣍⡐⣄⠀⠀⠀⠀⠀⠀⠐⢤⣀⣇⣼⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⣶⠏⠓⠐
⠀⠀⠀⠐⠉⠁⣼⡿⠟⠻⠿⠋⠑⡄⠀⠀⠈⠂⠀⠀⠀⢀⠀⠚⠛⠛⠟⠻⡒⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⣿⣿⡿⢶⡤⢄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠠⠔⠒⠛⠩⠏⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠧⠴⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢸⢡⡖⠀⡄⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⢾⣦⡄⠀⡀⢀⣄⠐⢠⣿⣄⢠
⠀⠀⠀⠀⠀⠀⢀⣾⣾⡛⢾⣿⣾⡟⠁⠈⠉⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣰⣿⣿⠟⢡⣼⣿⣿⣃⢀⡀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣷⡀
⠀⠀⠀⣠⣾⡿⠋⠀⠀⢀⣼⣍⡉⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣿⣿⣿⣿⠁
⠀⣤⣺⡿⠋⠀⠀⠀⣶⣇⣤⠞⠷⠈⠀⠠⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⠻⣿⠇
⢾⣿⠟⠀⠀⠀⢀⣼⣿⢏⣠⣶⢠⣴⠂⢤⡆⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠂⠀⢀⡀⣶⡄
⠋⠸⠓⠀⠀⠀⣾⣿⣿⣿⡾⢻⣿⢁⡄⠀⡇⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢤⠆⠀⠀⠀⠀⠀⣰⠏⠡⢢⣿⠀⢸⡇⢺⡆⣿⠆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣉⣀⢸⣿⡏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⡼⢃⣴⢀⣿⠇⠀⠄⠁⢸⡇⣿⣶⡀
⠀⠀⠀⠀⠀⠀⣿⣿⡟⠹⠃⠘⣿⡷⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢤⠀⠹⡅⣾⡧⣼⡟⠀⠀⠀⣄⠘⠃⣿⣌⣁⠸
⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣤⣷⣀⣾⣿⠁⢸⡇⠀⣷⢀⡿⢸⠟⠇⢀⠈⣷
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠃⠀⠸⠀⣧⣽⣾⣷⣴⣷⣄⢉⣼⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣷⠀⠀⡇⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⢼⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⢹⣿⠋⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠄⠀⣜⠑⣤⣶⢞⣳⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢻⣸⡿⠀⠘⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿⣦⣿⣿⣼⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡾⣻⡇⠀⠀⠈⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⣹⡇⠀⠀⠀⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣷⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠠⠠⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢼⡇⠀⠀⠀⠙⠒⠲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠑⢄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢀⣴⢃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⡬⠛⠙⢸⡆⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⣠⣴⣿⡐⢤⣀⠀⡇⠀⠀⠈⠿⣶⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⣿⠛⠹⣄⢨⣟⠂⡇⠀⠀⠀⣦⣘⢷⣍⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⣋⠞⢿⠀⡀⠈⠙⠛⠃⣇⠀⠀⠀⢹⡝⣧⡉⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡏⠀⠈⣷⣝⣄⢤⣀⣠⡿⡄⠀⠀⠀⣿⡆⢿⠉⠋⠈⠣⡀
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠸⡇⠀⠀⢸⣿⣿⣷⡹⣿⣇⡇⠀⠀⢠⠧⠁⢸⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠘⡄⠀⢸⡏⣿⣿⣷⢻⣿⣷⡀⠀⢸⠀⠀⡘⠀⠀⠀⢠
⠀⠀⠀⠀⠀⣼⠇⠀⢀⡄⠀⠁⠀⠈⣧⣿⣿⣿⣿⣿⣿⡻⠦⠬⢧⡀⠁⠀⠀⢠⠃⠐
⠀⠀⠀⢀⡾⠛⠀⠀⢼⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠹
⠀⠀⢠⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣾⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢹⡟⣿⣿⣿⣿⢣⠨⡓⠦⣀⠀⠀⠀⠀⠀⠠
⠀⣸⠇⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣠⠟⢉⡿⡟⣿⢠⠃⠑⠄⠀⠀⠀⠀⠀⢀⣂⣀⡀
⢠⡟⠀⠀⠀⠀⠀⡜⠀⣀⡀⠤⠒⠚⠃⠀⠘⢤⡇⠟⠀⠀⠀⠈⠲⡄⢀⣀⣠⣟⣿⠿⠓⠀⠀⠀⣀
⣸⠃⠀⠀⠀⠀⡰⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣢⣽⣿⠿⠴⠞⠂⠀⠈⠈⠁
⣿⠃⠀⣠⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣏⣉⠑
⢷⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠛⡏⠀⠉⠉⠁⠀⠈⢣⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠊
⠀⠙⠲⠶⠦⠤⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠀⠀⠀⠀⠀⢿⠂⢠⠀⠤⠀⠀⠌⠀⠀⠀⠂⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⠙⢒⢂⡀⢀⣀⣀⣂⣠⣤⣀⢠⣀⡀⠀⠀⠐⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⢸⠤⠈⠉⠀⠉⢻⣦⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⢾⡄⠰⠏⠋⠙⢿⣿⣷⣸⣿⣥⣤⣬⣭⣉⣉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⡀⠀⠀⠘⠀⠀⠀⡰⠀⢻⠶⠄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⢷⡼⠀⠀⠈⡀⠰⠞⠋⠈⠙⢢⣀⠜⠂
⠀⠀⠀⠀⠀⢠⡊⠀⠀⠀⠀⠀⠐⠁⠀⣇⠀⠠⡀⠀⢀⡴⠋
⠀⠀⠀⢀⡜⠉⡀⠀⠀⠀⡰⠊⠀⠁⠈⠉⠑⠒⠒⠚⠋
⠀⠀⠀⣾⢀⡦⢏⣤⠄⠠⠀⠀⠘
⠀⠀⠀⣿⣭⣿⡿⢛⠀⠀⠀⠀⢠
⠀⠀⠀⠹⠟⠉⣎⢀⡸
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠠⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⠰⠄⡀
⠀⠀⠀⠀⠀⠀⢠⣴⠟⣀⣴⣷⢤⠀⠂⠂⠂⠂⠀⠀⠀⢠⣿⣿⠛⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣷⣄⡘⢷⣦⡀
⠀⠀⠀⢀⡴⠞⠋⠀⡞⣱⡟⠁⠈⠛⠶⣶⠆⠀⠀⠀⠀⠘⠹⣭⣭⣭⡽⠙⡀⠀⠀⠀⠀⢶⡶⠞⠋⠀⠙⣷⡙⣆⠈⠛⠶⣄
⠀⠀⣼⡟⠀⠀⠀⢺⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢦⢀⠠⣁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⠂⠀⠀⠈⣿⡄
⠀⠀⠻⣧⣀⠰⢦⣴⠟⠋⠀⠀⣰⡴⠀⠀⠉⢲⣄⠀⠀⣿⣦⠈⣒⡊⢠⣾⡇⠀⠀⣤⠂⠁⠀⠀⣴⣀⠀⠈⠛⢷⣤⠶⠀⣠⡿⠃
⠀⠀⠛⢧⡀⠀⢐⣺⡦⠀⠀⠀⠘⣿⣷⣄⡀⠈⠉⠚⠓⠇⣭⣿⣿⣿⣯⡅⠗⠛⠊⠉⠀⣀⣴⣿⡟⠀⠀⠀⠠⣾⣓⠂⠀⣀⠿⠃
⠀⠀⠀⠀⠙⣿⠶⠍⠻⣷⣤⣤⣴⡿⠿⣿⡆⠀⠀⠀⠀⠀⠈⣻⣿⣿⡋⠀⠀⠀⠀⠀⠀⣺⣿⠿⣷⣤⣤⣴⡿⠋⠵⢾⡿⠁
⠀⠀⠀⠀⠀⠈⠳⣄⣀⣈⣽⣿⣿⡿⠓⠻⣿⣶⣤⣀⡀⠀⣐⣾⣯⣿⣖⡀⠀⣀⣀⣴⣾⣿⠓⠺⢿⣿⣿⣍⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⡏⢠⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣧⠈⡟⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣷⢠⠞⣺⠛⣿⠟⠉⠁⠈⣿⡏⠀⠉⠉⢿⡟⠻⡞⢧⢰⣿⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⢻⡇⢠⣿⠶⠛⠛⠺⣿⡿⠚⠛⠳⠾⣧⠀⢿⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣻⡇⢸⡇⠀⢀⣀⣠⡟⣧⣀⣀⠀⠀⣿⠀⣼⡳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠲⣿⡿⠛⠉⠹⣿⡿⠋⠙⠛⢿⡷⠂⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣸⠀⠀⠀⠀⣿⣇⠀⠀⠀⢸⡁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠹⣷⠤⠶⠾⢿⠿⠶⠦⢴⣿⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣇⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣀⣀⣠⣀⣀⣠⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⠀⠀⠀⣄⣀⣠⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡾⠋⠀⠦⠴⢂⣀⣉⣭⣭⣴⣶⣿⣻⣷⣴⣶⣯⣿⠂⠉⢻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠿⠿⠿⠿⠃⠀⠛⠻⠛⠁
⠀⠀⠀⠀⣠⣤⣴⣶⡦⠦⣀⣒⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣾⣿⢹⡇
⠀⠀⠀⣰⣭⣭⣶⣶⣿⣿⣿⣿⣿⣿⠛⢿⡿⢿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡿⠾⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠝⠋⠉⠉
⠀⠀⠀⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⣿⣿⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠈⠛⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣤⣤⣾⣙⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⠙⠛⠋⠉⡉⠀⣀⡀⣠⣤⣤⣤⣶⣾⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣴⣶⣿⣶⣼⣿⢰⣿⢿⣿⠋⣿⡇⢸⣿⠇⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣿⣇⣙⠋⣿⣷⣼⣿⢸⣿⠀⣿⡇⢸⣿⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⢈⣛⠛⣿⡇⣿⡏⣹⡿⣾⣯⣸⣿⠀⣼⡿⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣶⡿⠻⠿⠁⠿⠇⠙⣛⠛⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡅⢀⣤⠀⠴⠆⠸⠯⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠛⠛⠛⢛⣛⣙⠇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠈⢁⠬⢷⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡰⠁⢀⡀⠀⠀⢀⣘⣢⡀
⠀⠀⠀⠐⡔⠃⠐⣠⠃⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠄⠁⠀⠀⢉⠀⠀⡉
⠀⠀⠀⠀⠀⠒⡄⢂⠀⠀⠀⠀⠀⠀⠡⠀⢂⠀⠀⠀⠀⠀⠀⢆⣠⠀⡞⠀⠀⠀⠀⡈⢆⡀⠈
⠀⠀⠀⠀⠀⠀⠘⡌⢆⠀⠀⠀⠀⠀⠀⢡⠠⠦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⡙⠛⢠
⠀⠀⠀⠀⠀⠀⠀⠘⢌⢂⠀⠀⠀⠀⠀⢈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡉⠒⠒
⠀⠀⠀⡀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢨⢘⢙⠠⡁⠀⠀⠀⣀⣼⢉⣄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠐⠁
⠀⠀⠸⣈⠊⢁⡆⠀⠀⠀⢹⠈⠎⡀⠑⠀⠀⠀⠀⢀⠇⠐⠒⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢉⠁⡇
⠀⠀⠀⠫⡎⢲⠃⠀⠀⠀⠘⣆⠀⡰⠁⠀⠀⠀⠀⠀⢆⠀⣀⡄⠀⠀⢀⡼⠂⠀⠀⠀⠀⠀⠀⠂⡈⢠⠃
⠀⠀⠀⠀⢠⠈⡄⠀⠀⠀⠀⡈⠀⡆⠀⠀⠀⠀⠀⠀⠀⢨⠀⡆⠀⠀⢰⢀⢢⠀⠀⠀⠀⠀⠀⢠⢁⠆
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠠⣀⠀⣁⠜⠀⠀⠀⠀⠑⠘⠀⠀⠀⠀⠀⣀⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣄⢄
⠀⠀⠀⠀⠀⠀⢀⠊⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⡲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⢹⣍⡔
⠀⠀⠀⠀⡠⠶⢦⡦⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠺⢀⣿⠃
⠀⠀⠀⠸⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣾⣿⣍⡆⠀⠀⠀⠀⠀⠀⣆⣀⣴⠟⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠢⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠘⣄⠀⠀⠀⠀⠀⠠⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠔⠁⢀⡠⠤⠒⠢⢄
⠀⠀⠀⠐⠢⢤⣀⣀⣀⠀⠀⢸⠀⢡⠘⡆⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⣀⠀⡠⠔⠂⠀⠀⠀⠀⠀⡐⠁⠀⢀⡔⠊⠉⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢸⠀⠸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠈⠀⣀⡀⠀⠐⠻⡄
⠀⠀⠀⠀⠀⠀⠂⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠰⡄⠀⠀⠀⠑⠦⠤⠤⠤⠤⠤⠤⠒⠒⢄⠈⠂
⠀⠀⠀⠀⠀⠐⠠⠤⠤⠤⠤⠤⠶⠤⠄⠒⠁⠀⠀⠀⠉⠒⠀⠤⠀⠀⠄⠐⠂⠐⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢨⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢠⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠜
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⡞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⡇
⠀⠀⡜⠀⠀⠀⢀⡠⠖⠒⠒⠂⠂⠉⠀⠀⠀⠉⠀⠀⢰⠀⠀⠹⡀⡠⠚⠉⠁⠀⠀⢠⡀⠀⠀⠀⢰⠇⠀⠀⡆⠀⣀⣀⣀⣠⠴⠒⠃
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⠀⠀⢀⠀⠠⠊⠳⣄⠀⠀⢸⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⢀⡀
⠀⠀⢀⠀⢀⣀⡀⠀⠀⠈⢉⣀⠀⠀⢀⡈⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⣀⡠⠂⠈⡆⠀⠀⠀⠘⡆⠀⠀⠀⠀⠠⠤⠤⠂⡠⢄⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠱⡀⠀⠀⠀⠸⣄⣀⠀⠤⠐⠚⠉⠁⠀⠀⠀⠘⠀⠀⠀⠀⠙⠊⠁⠒⠒⠒⠒⠒⠉⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠉⠉⠉⠹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡀⠀⠀⠀⠀⠀⣀⠈⠻⣿⢿⡏
⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⢺⡶⠀⠀⠀⢻⠿⠂⠀⣿⢸
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢺⣷⣄
⠀⠀⢀⣿⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣁⣀⡀⠀⠀⠀⠈⠀⠀⢻⣿⣷⣄
⠀⠀⣸⣿⡇⠠⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⠿⠁⠀⣠⠆⠀⠀⠈⣿⣿⣿⣷⡄
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⣀⣀⣾⣇⡀⠀⠒⠀⣹⣿⣿⣿⣿⡄
⠀⢸⣿⣿⣷⠀⠀⠀⠀⡀⠀⢀⡠⠂⠈⠃⠀⢂⠀⡄⠀⠀⠀⠈⠉⠁⠈⠉⠙⠓⠄⠀⠈⠻⣿⣿⣿⣿⡀
⠀⢸⣿⠻⣿⣆⠀⠀⠴⣋⠴⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣿⣿⣿⣿⡇
⠀⠘⣿⣄⠉⣟⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⢻⣿⣿⣿⣷
⠀⠀⢩⣛⠷⠿⠿⠩⠀⠀⠀⠀⢀⡒⢶⣤⣀⣀⣀⡀⠈⢻⣿⣿⣶⣶⣾⡶⠶⠟⠋⠁⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠈⠙⠻⠿⣿⣶⣶⣾⣿⣿⣿⣿⣶⣬⣭⣭⣭⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⠶⠖⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⣁⠀⠀⠀⢀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠛⠙⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⠟⠋⠘⠋⠁⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⣿⠃⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣦⣤⣤⣦⣶⡾⠃⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠻⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠛⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠁⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢹⡆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠻⣄⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠈⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⠃⠀⠀⠹⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠀⠀⢀⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠀⠀⠀⠀⠀⠀⡠⢚⡹⢯⠓⢄⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠌⢀⣿⡇⠘⣧⡀⠱⡀⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠈⠟⢄⠀⠀⢸⠀⢈⣿⠇⠨⢻⣇⠀⣧⠀⠀⠀⡐⡿⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠙⢀⢀⠀⠀⠀⠀⠈⠻⡁⠠⣼⠀⢈⣺⣿⠐⣞⡁⠀⣏⠠⢶⡦⠖⠁⠀⠀⠀⣠⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠙⢇⠠⠀⠀⠀⠀⠀⠐⠼⣧⡈⣩⣥⢀⣬⣥⣼⡧⠖⠉⠀⠀⠀⣀⢀⡦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣅⠠⣤⡄⠀⣶⣮⣿⡟⢻⡿⠛⢿⣡⠦⣯⣈⣿⣵⣫⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢁⡴⣺⢿⣧⣘⢧⣤⣿⡽⣖⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠼⢫⡞⣯⣸⣺⣸⡟⣯⠪⡓⢬⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⠛⢁⣣⠎⠐⡏⠙⠙⠁⠇⠈⠣⣴⢆⠈⠣⣕⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠞⠁⢠⢎⠏⠀⠀⠘⠀⠀⠸⠀⠀⠀⠈⢣⡠⡀⠈⠳⢯⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟⠁⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⢂⠀⠀⠉⠢⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡼⠁⠀⠀⠀⢺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⡇⠀⢻⢿⣿⡛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡟⣧⠀⠘⠀⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⢹⣷⣿⠀⠀⠀⠀⢻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡸⢿⣼⡀⠀⠇⠀⠈⢨⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⡼⠻⠇⠰⠀⠀⠀⠀⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⡀⡄⢀⢤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠀⠀⠀⡠⠖⠫⠀⡉⢈⡀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠄⡼⢀⠇⠐⡄⠁⢸⡷⢈⣿⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⢀⣴⠁⢸⢹⠀⠀⠀⢈⠔⠘⠟⠉⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⢀⡿⠏⠀⣾⠘⣴⠀⡤⣆⠀⣾⡇⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠅⠀⢰⡿⠃⠀⢀⡿⠀⢿⡀⢻⡟⠀⣿⣗⣠⡦⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢀⣴⣿⡗⠀⠀⢸⡇⠀⣾⣇⠘⡗⠀⣿⣿⣿⣧⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠚⠻⠟⠀⠀⣄⠈⡇⠀⣿⣿⠀⠏⠀⣿⣿⡿⢧⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⡏⠀⠀⠘⣿⠆⡇⠀⢿⣿⡆⠰⣀⢿⣿⠁⠀⠑⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡄⠀⠀⠼⢀⠉⠀⠀⡇⢸⡟⠁⠀⣇⠘⣿⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢠⢀⡾⠀⠀⠀⠇⢈⠃⠀⠀⠁⢰⡇⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠆⠀⣿⠋⠀⡶⠂⢸⣆⡜⢢⠀⠀⢄⡼⠀⠀⠀⠀⠀⠁⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⡏⠀⠀⠇⠀⢸⡟⠀⠀⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠃⠀⠸⡇⠀⠀⠀⠀⣽⠃⠀⠀⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡸⠀⠀⠐⠂⠀⡟⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣸⠁⠀⠀⢧⠀⠀⣸⠀⠀⠀⡄⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⡰⠀⠚⠀⠀⠀⠀⠂⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⣠⠀⠀⠀⠀⢲⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡇⠀⠀⠀⠀⢀⡠
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠃

⠀⠴⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡟⡦⣀
⣙⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠶⠶⢶⣶⣶⣶⠒⠲⣖⢲⡶⠒⠒⠒⠒⠒⢠⠄⢠⠀⢠⡝⠓⠾⠍⠉⠑
⠈⢶⣶⣶⣶⣶⣶⣶⣶⣶⣧⣤⣤⣼⣿⣿⣿⣦⡤⠭⢼⠦⣤⣤⣤⠤⠤⣬⡥⠬⠥⢬⡥⠤⠤⢼
⠀⠘⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⢿⣿⣿⣧⠀⠘⠀⢩⣿⣿⣶⣶⣿⣷⡜⠛⠛⠋⠀⣠⡞⠛⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⠟⠛⠛⠿⣿⣿⣿⣿⡝⠟⠛⠢⠄⠀⠠⠒⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣧⠏⠀⢹⣿⡌⠄⠀⠀⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⣘⣁⣀⡠⠞⠿⣧⠠⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢡⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠃⠀⠀⠀⠀⠠⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡈⡄⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢁⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⢀⣄⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣾⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠉⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠈⢃⡀⡤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣦⣤⣤⣶⣰
⠀⠀⠀⠤⣤⣴⠁⠀⠀⣀⠀⠀⠀⠀⣴⠁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡷⡄
⠀⢰⣤⣀⠀⠉⠀⠀⠀⠁⠀⠈⠀⢴⡼⠋⠁⠀⠀⠜⢒⠂⠀⠀⠀⢀⣴⠶⠒⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⠷⠁
⠀⠀⠙⢿⣦⣄⣀⣴⣦⡅⠀⠀⠀⢠⠀⠀⠀⠀⠀⡴⠿⣄⣀⣀⣀⡸⠃⠀⠀⠀⠀⠀⠉⠹⠿⢿⡿⠿⠋
⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⠀⠀⠀⠘⠷⠀⠀⠀⠸⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡈⠓⠁⠀⠀⠤⠄⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⡁⠀⠀⠀⠀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠔

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡿⢧⣾⣶⢋⣷⡞⠛⣶⣷⡖⢻⡟⣛⣿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠙⣿⠚⠉⠛⠚⠋⠘⠃⠘⠛⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠉⠀⠀⠀⠠⡀⠀⠉⠀⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠖⠀⠀⠀⠠⡀⣇⣼⡦⠄⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠰⠒⠲⣄⡀⠀⠀⠀⠀⣀⣄⠀⠀⠈⢻⡉⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠄⠂⠂⣀⣻⢶⡄⠀⠸⣿⣿⡟⠀⠀⣤⢻⣆⣀⡉⠑⠤
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢠⡶⠞⠛⠛⠛⠼⣧⠀⠄⠀⠉⠁⢀⣸⣿⣦⠙⠋
⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢰⣿⡡⠊⠀⠀⣴⣾⠟⠋⣦⣶⢁⢠⠞⠉⠉⣿⣶⣅⣠⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣿⣿⡀⠀⣀⣾⣿⡿⣇⣸⣿⣿⣷⣀⠀⠀⢠⡿⠙⠿⢷⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠛⠛⠛⠻⠧⣿⣿⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⢸⣿⣿⣿⣿⣿⣿⠿⠛⠉⢉⣽⣿⣿⣷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣼⣿⣿⣿⡿⠋⠁⠀⠀⣴⣿⣿⣿⣏⣭⣿⣯⣝⠻⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣉⠙⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠁⠀⠀⢀⣴⣾⣿⣿⡟⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣷⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠃⠀⠀⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠀⢠⣾⣿⣿⡿⠛⠉⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣾⣿⣿⡟⠁⠀⠀⠀⠈⠀⠉⠛⢶⡀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⢠⣤⠙⠋⠉⠀⠀⠀⠀⠀⢶⣶⡶⢦⡀⠀⠀⠀⡴⠶⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠁⠀⠀⢸⣷⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣿⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⢟⡼⣿⠇⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠏⢸⡇⣿⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⣴⠋⠀⠸⣅⡿⠀⠀⠀⠘⠳⣤⡀⠀⠀⠀⠀⠀⠀⠠⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠥⠀⠐⠛⠁⠀⢀⣠⢿⡇⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⣀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⠛⣋⣠⣧⡀⠀⠀⠀⠀⠀⠀⣸⠋⣿⣟⣹⣷⣶⡆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⣟⣻⣿⡿⢿⣏⠙⢷⣄⠀⠀⠀⠀⠀⢸⣟⣿⣿⣶⣿⣿⡿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⠿⣿⣷⣄⣿⣿⡿⢻⣧⢀⡀⠀⢀⡾⢿⣧⣿⣷⣾⡿⠆⢀⣆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⢨⡏⠁⠀⠄⡁⠀⢸⣄⣼⣷⣿⣏⠻⣧⣠⣿⣿⡟⢀⣾⣇⠘⣿⠉⣿⠏⢠⣠⡿⠃⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣾⣶⣿⠿⠿⣷⣾⣿⣶⣿⡿⠛⠙⣧⠙⣶⣿⣿⢁⣹⣧⣼⣧⣰⣦⣿⣴⡏⠀⠈⠉⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⢀⡿⠋⠁⠀⢘⣀⡀⠀⠉⠧⠀⢰⣗⠀⣠⣿⣿⡿⢿⣿⠟⠛⢿⡇⠘⣿⣻⣏⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀
⠀⠀⠀⠀⠀⣸⣧⣴⣿⣿⣿⠿⠿⣷⣾⣿⣷⣿⣿⡿⢿⠋⢹⡂⣀⣿⣦⣤⣼⣷⡶⣿⠿⡿⠻⡟⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟
⠀⠀⠀⠀⠀⣿⡟⠛⡉⢀⣸⣶⣤⣀⣤⣶⣆⣀⣽⣷⣾⣧⣼⣿⠟⠛⣫⠀⠈⣿⠀⢸⣤⣷⣾⠃⠀⠀⠀⠀⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠸⣷⣤⣿⣿⣿⡟⠉⠉⢉⣉⡉⢛⡉⠉⣂⣿⠉⣇⣠⣾⣿⣷⢾⣿⡷⣿⣏⠛⠁⠀⠀⠀⠀⠀⣠⡿⠉⣿⣄
⠀⠀⠀⠀⠀⠀⣿⢛⣿⠟⢁⣴⡾⠟⢻⢿⣿⣿⠿⠻⣿⣿⣶⣿⡟⠉⣿⣧⣤⣼⡀⠚⠃⠀⠀⠀⠀⠀⣠⣴⣿⣷⡾⣿⡟
⠀⠀⠀⠀⠀⠀⣿⡿⠃⣠⡾⠋⠀⣀⡼⠟⠋⠀⠀⠀⢸⣿⣄⣾⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⢀⣼⣏⣽⣤⣿⠀⢸⣧
⠀⠀⠀⠀⠀⢀⣿⠀⢰⡏⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⣿⣷⠿⠋⠀⠀⠀⠀⠀⠀⠀⣠⠟⢿⡿⠻⣿⣿⡿⢿⣿⠆
⠀⠀⠀⠀⠀⠸⣿⣷⡿⢡⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣿⣧⣀⣿⣿⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠸⣿⡇⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⣭⣀⣿⣟⣛⣿⠟⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠛⠻⠏⠙⠻⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢤⣀⠾⠃