Woof Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣤⡀⢀⣠⣀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⢀⣾⡿⣿⣆⠀⣴⣿⢿⣷⡀⠀⣾⣿⢿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⢀⣤⡄⢸⣿⡇⣼⣿⠁⢸⣿⢰⣿⡇⠀⣿⡇⢀⣿⣟⣈⡙⠘⣿⡿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢿⣿⣾⣿⣷⣿⣿⠇⢹⣿⣄⣾⡟⠈⣿⣧⣸⣿⠃⢿⣿⡿⠿⠟⠀⣿⡃⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠈⠛⠛⠀⠻⠿⠟⠀⠀⠙⠻⠟⠁⠀⠈⠻⠟⠃⠀⠀⣿⠇⠀⠀⠸⠿⠟⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠳⠶⠶⢶⣤⣤⣤⡄⠀⠀⠀⠀⣠⣤⣤⣴⠶⠶⠶⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡉⠉⠉⠉⠀⠀⠈⢈⣙⣆⣀⣀⣀⢴⣿⣿⡏⠉⠁⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⢶⢞⣭⠉⠁⣠⣬⠉⠋⠀⠉⠉⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠤⠀⠀⠸⠾⠷⠀⠐⠿⠷⠂⡀⠀⢱⠀⠀⠀⣶⣷⡏
⠀⠀⠀⠀⠀⠀⠀⣇⠀⣶⣬⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⣷⡄⠀⠈⡻⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡇⠀⠸⣿⣷⠤⠖⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⠷⡶⣦⣤⣤⣭⣤⣴⣶⣿⠿⢃⡀⠀⢈⠉⢻⡽⠁⠀⠀⠀⠀⣠⢄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⠟⠛⠿⠧⠤⠤⠤⣴⣶⣶⣷⣶⡾⣿⣶⣾⡞⣦⡀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣀⣀⠔⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⡉⠉⠛⠛⢋⣉⠉⠙⢿⡶⠆⠀⠀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⣞⡆⠀⠀⠿⣃⣨⣢⠆⠀⢀⣾⣿⢿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡶⢺⠄⠀⢀⡾⢻⣶⣶⣾⢿⡿⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⣾⣿⣷⣇⠀⠀⣸⣇⣎⠉⢛⣿⠉⠀⡀⣼⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢸⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢠⠃⠀⠀⣰⣿⣷⢸⣿⣿⢠⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢴⣿⠀⠀⠀⠀⠀⠀⢠⡶⢦⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⢠⣶⠒⣄⠀⠀⣸⣿⡵⠃⠀⠀⠀⠻⣿⠟⢀⣉⡁⢻⣿⠟⣀
⠀⠀⠀⠀⣀⠤⠊⠁⣾⡇⠀⠀⢀⣤⠀⠀⢸⣧⢸⠁⢀⣾⡿⣷⡈⡆⠀⣠⣿⡿⣿⣄⣸⢀⡠⣿⣿⠀⠀⠀⡀⠀⣀⣤⣶⣿⣿⣿⡆⢠⣾⣿⡇
⠀⠀⠀⠞⠁⠀⠀⠀⣿⡇⠀⠀⣸⣿⠀⠀⠀⢙⡟⠒⠁⠻⣇⠈⣻⠓⠋⢹⣿⠁⠈⠉⢹⠉⠀⢻⣿⣄⡤⠚⠁⠀⣿⣿⣿⣿⣿⣿⣧⠘⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⡠⠻⣿⠀⠀⢀⡞⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢿⣆⠀⡠⠃⠀⠀⠸⣿⡄⢣⠀⠀⠀⠈⠉⠉⠙⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠻⢦⠤⠊⠀⠀⠀⢀⣄⣠⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢻⣷⡼⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⢰⣶⡄⣿⣿⣄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣘⣫⣝⢵⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⡶⡄
⠀⠀⠀⣤⣬⡻⢟⣨⣍⡘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⡏
⠀⠀⠀⠹⢿⠇⣼⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⡜
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠟⠀⠀⠀⢀⡴⣿⡏⠀⠀⠀⠀⠀⠀⣴⡟⢦⠀⠀⢀⣠⡤⣄⠀⠀⠀⣀⣴⡏⠑⡄⠀⠀⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⡤⠚⠁⠀⣿⠀⠀⠀⢰⣦⠀⠀⢻⣇⡜⢀⡰⣿⠻⣧⣘⠀⢀⣼⣿⠻⣿⣄⣸⣠⠞⣿⡇⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠘⣿⡀⠀⠀⣾⡇⠀⠀⠀⣹⠓⠉⠀⠻⣆⣈⡝⠉⠁⢻⣿⠀⠀⠀⡏⠀⠀⣿⣷⢤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⠜⢹⣧⠀⠀⣰⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠻⣄⣠⠜⠀⠀⠀⢸⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⣤⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠿⠇⣉⣅⣻⠗⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡽⣿⣆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠸⡷⢸⣿⣿⣿⠇⠀⠀⠀⠀⢀⣤⣄⠀⢻⣿⡈⣋⡘⣿⢆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⢀⣤⣄⠀⣿⡏⠻⣷⣨⣿⣿⡟⠃⠰⣿⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⣿⡟⠻⣷⡹⣿⣄⣹⣧⠉⢻⡷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣄⠀⠀⠈⢿⣷⡹⣿⣄⣹⣷⠈⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠘⣿⣧⢠⣶⣜⣿⣷⠙⠻⠿⠋⠀⠀⠀⠀⣤⡀⢿⡷⢠⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠙⣿⣷⣿⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⣀⡛⢃⣴⣶⣮⣍⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⢾⣿⣿⡿⠟⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣦⣤⣤⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠘⣿⡛⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⢸⠀⠀⠀⣀⡀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⡟⣠⠃⠀⠀⣾⣿⣷⢿⣿⢇⣾⣿⡆
⠀⠀⠀⠀⠀⢀⡠⢚⣿⠁⠀⠀⠀⠀⠀⣼⡏⢢⠀⠀⣤⣴⠢⡀⠀⠀⣠⣼⣏⠙⡆⠀⢰⣿⡟⠁⠀⠀⠀⠙⠟⢃⣤⣤⣌⠻⢟⣥⡄
⠀⠀⠀⡠⠖⠉⠀⢸⡏⠀⠀⢠⣶⠀⠀⠹⣧⣆⣠⠾⣿⠹⣦⣇⣠⢼⣿⠛⠿⢦⣥⠔⢻⣿⡇⠀⢀⠔⠀⣴⣿⣿⣿⣿⣿⠀⣿⣿⡟
⠀⠀⠈⠁⠀⠀⠀⠸⣧⠀⠀⣼⣿⠀⠀⠀⡜⠁⠀⠀⠻⠤⠔⠁⠀⠸⣿⠀⠀⢀⠇⠀⠀⣿⡗⢎⠁⠀⠀⠻⠿⠿⣿⣿⣿⣦⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠊⠈⢿⡀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠔⠋⠀⠀⠀⢹⣷⢸⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋
⠀⠀⠀⠀⣠⣄⢠⣶⡄⢀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡿⠺⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣀⡹⣿⠎⡻⠃⣿⣿⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣎⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢹⠂
⠀⠀⠀⢿⣿⢀⣾⣿⣷⣬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢁⠞
⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⡿⠀⠀⠀⢀⡠⣿⠃⠀⠀⠀⠀⠀⣠⡖⢄⠀⠀⢀⣠⣄⠀⠀⠀⢀⢠⡶⠲⡄⠀⠀⣿⡧⠊
⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⡠⠖⠉⢰⡿⠀⠀⠀⣤⠀⠀⢻⣇⡸⠀⣰⣿⢻⣆⠣⠀⣠⣿⡟⣿⣄⣸⣠⠾⣿⡇⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⣧⠀⠀⢸⣿⠀⠀⠀⢹⠃⠉⠀⢿⣄⣩⠏⠉⠸⣿⠀⠀⠉⡍⠀⠀⣿⡧⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⠴⠋⣿⡀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⡠⠜⠀⠀⠀⢻⣧⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣿⡿⠿⣿⣿⣿⣿⣾⣿⣶⣶⣶⡶⣦⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⢀⣾⠟⠉⠀⠈⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠉⠉⠉
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⢹⣿⠈⢻⡿⢸⠏⢀⡌⣿⠁⣠⢸⡟⠁⣤⠴⠾⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠟⢰⠈⠇⣾⠀⣿⢁⣿⢸⡿⢸⡇⢠⣴⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣠⣿⣧⣀⣿⣦⣤⣾⣿⣦⣴⣿⣷⣼⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠟⠉⠉⠉⠙⠛⠛⠿⣿⣿⣿⣏⣉⣛⣛⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠛⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠷⠆⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠹⣷⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠚⠛⠉⣽⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣄⣿⣧⣤⣶⣾⠿⠟⠛⠉⠀⠀⠀⠀⢠⣿⠀⢹⡇
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⠉⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣼⠇
⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣸⡟
⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⣀⡀⠀⠀⣿⣟⣠⣤⡄⠀⠀⢀⣤⡶⠄⠀⠀⣿⣿⠟
⠀⠀⠀⠀⠻⢺⣿⣇⠀⠀⣿⡇⠀⢸⣿⣿⣿⠁⠘⣦⢠⣿⡟⣿⣻⣦⣴⣿⣷⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠈⣿⣿⠀⢠⣿⡇⠀⣼⡿⣿⣿⠀⠀⢹⣿⣿⠀⠈⢹⡏⢸⣿⢿⣆⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⢻⣿⡀⣼⣿⣧⢠⣿⠁⣿⣿⠀⢀⣿⠹⣿⡆⠀⣸⡇⢸⣿⢸⣿⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠏⠘⠿⠟⠃⠀⠘⠿⠧⠾⠋⠀⠻⣷⣴⡿⠁⢸⣿⢀⣿⡇⢀⣬⣡⡫⣆⠠
⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⣸⣿⢸⣿⠇⠈⠻⣿⡧⠱
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⢽⡟⣾⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⢋⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠀⣰⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠿⠒⣴⠞⣴⠛⣱⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢹⣿⡏⣀⠁⡴⢃⡶⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠔⠚⠘⣿⠻⢆⣿⣴⢳⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣈⣴⣿⣶⢀⣘⡙⠿⣟⠻⠈⣾⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣤⣤⢾⢶⣯⣍⡻⡎⣶⡕⢸⡿⢼⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠋⣾⣏⠙⢓⣣⣿⣷⢨⣿⠂⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠘⢿⣦⣭⣭⣿⣧⣤⡐⣾⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣶⣽⣿⡟⠁⠈⣷⠘⢺⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⠛⣣⡾⣮⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠈⠙⣿⣿⣿⣿⠟⠳⣮⣳⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡀⠀⠘⣿⣿⡇⠀⣄⠘⣷⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣧⣀⣰⣿⣿⣿⣶⣿⣆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⢻⣟⠛⠙⣿⡏⠀⠙⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠻⣦⡀⣿⡇⠀⠀⡿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠙⢿⣾⣷⡀⠐⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠙⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣶⣶⣿⣿⡿⢃⡀⢩⣿⣿⠉⠉⢉⣀⣀⣈⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠛⣋⡁⠈⠉⠙⠻⢿⣿⢀⡾⠁⣼⡟⠀⢀⢖⡛⠋⠉⣀⣤⣼⣿⠻⢷⡄
⠀⣿⣿⡷⢦⣤⣀⣀⣠⣴⡾⠋⠁⢀⣤⢰⠟⠁⠀⠀⠀⠀⠀⠷⣼⠇⢸⡿⣠⠀⣺⠸⣿⣭⡉⠉⠁⡀⠀⢀⠐⣽⣦⡀
⠀⠈⣧⠀⠀⠉⠙⠉⠉⣽⡆⠀⢠⣿⡇⠹⠀⠀⠀⠀⠀⢀⠀⠀⣿⠅⠈⠃⣿⠀⣽⡀⢻⣆⣽⠇⠶⠃⠀⢸⠇⠘⣿⣿⡿⣶⡖
⠀⠀⠘⠿⣶⣦⣤⣴⣿⣿⣿⠂⠸⣿⡄⠀⡄⠀⠀⠀⠀⣼⡂⠀⣿⡆⠀⠀⢿⣇⢹⣷⡞⠉⠀⡀⣆⠀⡤⠀⡀⠀⣿⣯⠼⠋
⠀⠀⠀⠀⠀⠉⠙⠛⠋⠩⣿⡀⢀⠻⡿⣄⠻⣆⠀⠀⠀⠘⣧⢠⢻⣿⠀⠀⠈⢻⣿⣟⠀⢶⠀⢷⡘⠇⠀⢄⡵⢶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠳⣄⠹⣧⠀⠀⠀⠀⠀⠈⢸⢀⣤⣶⣄⠀⢹⣿⣿⣦⣈⠳⡌⠀⢀⡘⠀⣰⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢰⠀⠀⢠⣿⣷⣶⣤⣄⣀⣠⡏⠈⠉⢹⣿⣴⣿⣏⠙⣛⣛⢧⡰⡆⢴⣽⣄⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⣴⡿⠻⣿⣿⣿⣿⣿⠋⣿⠀⠀⠘⣿⠛⠛⠻⣿⣽⣛⣿⣷⢽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣴⣿⠋⠀⣠⣿⣿⣿⠟⠁⠀⢻⡄⠀⠀⢸⠀⠀⠀⠀⠈⠛⠻⢯⢳⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠻⠷⡄⢰⣿⣿⣿⠃⠀⠀⠀⠀⢳⣃⠀⢸⠀⠀⠀⠀⠀⠀⣰⠋⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢌⢿⣿⣿⠀⠀⠀⠀⠀⠀⡇⠀⠈⡆⠀⠀⠀⠀⡼⠋⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢊⢋⠉⢣⡀⠀⠀⠀⠀⢡⠀⠀⢰⠀⠀⠀⢰⡇⣨⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⣀⡸⠀⢂⠀⠑⠒⢤⠀⠀⠸⣀⠀⠘⡄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠑⠒⠓⠛⠃⠀⠀⠙⢆⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢠⠀⠝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⠯⠴⠒⠛⣇
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠒⠒⠲⠼⢦⠀⡄⠀⠀⢻⡀
⠀⠀⠀⠀⠀⣰⡿⣋⡀⠀⠀⠂⠐⠂⠀⠈⢧⠀⠀⠀⠈⢿⠒⠶⠶⠦⠤⠤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⣼⠏⠤⠀⠠⠀⢀⣀⠀⣀⠀⢀⡞⠀⠀⢠⠀⠈⢫⡐⡂⠐⠀⠤⢀⣀⡈⠉⠉⠙⠛⠓⠲⠶⠶⠦⠤⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⠀⠐⢂⡘⠁⠀⠀⠜⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠒⠀⠀⠀⠀⠈⠁⠐⠒⢠⡴⠋⠁⠈⣿⠛⢦⡀
⠀⠀⠀⠀⢿⡆⠀⠒⠂⠠⠈⠉⠉⠐⠒⠒⠋⠥⠤⠶⢤⣤⣒⡒⠒⠒⠂⠠⠀⠐⠒⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡞⢠⠔⣊⡁⠟⠀⠀⢻⡄
⠀⠀⠀⠀⠸⣷⡒⠤⠤⠤⠀⣀⠀⣀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⡟⢱⢃⠞⢡⠄⡀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢻⣿⠿⠟⣓⡒⠢⠤⢀⣀⣀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠐⠒⠠⠥⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣦⡏⠀⡆⠰⠁⡔⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠹⢦⣤⣄⣀⡁⠀⠀⠀⠈⠭⠉⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠚⠋⠉⢹⡇⢰⠁⡇⢮⠆⡇⢸⠀⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢦⣤⣀⣀⡀⠀⠉⠑⠒⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣀⣀⡀⠀⢸⡇⡸⢀⢟⢸⠀⢡⢸⠀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠶⢶⣦⣤⣀⠈⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠀⠤⣄⣈⠉⢹⡇⠁⠌⠸⡘⠃⡼⢸⡆⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠾⢶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⡇⠠⠐⡄⠓⠚⢁⢼⡟⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠦⢤⣄⣀⡀⠀⠀⠈⢿⡄⠑⠄⠒⢒⠬⠚⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⠤⣿⣷⣦⣤⣤⡶⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣸⣿⢃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣹⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇⢱⠀⠀⣠⣤⣾⠲⡀⠀⠀⢠⣤⣿⡇⠈⢆⠀⢀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⣿⡇⠀⠀⠀⣾⡆⠀⠀⠻⣧⡎⣀⠔⣿⡏⢻⣆⣇⣀⠴⣿⡟⠙⠿⣦⣽⠴⠋⣿⣿⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢿⡇⠀⠀⢰⣿⡇⠀⠀⠀⡼⠉⠀⠀⠙⢧⣀⡼⠁⠀⠀⣿⡇⠀⠀⢀⠇⠀⠀⢻⣿⠶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⠴⠃⣿⡇⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣄⡠⠊⠀⠀⠀⠸⣿⡆⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⣿⠀⠀⠀⠀⠀⠀⠀⣶⡟⢦⠀⠀⢀⣀⣤⢄⠀⠀⠀⢀⣀⣾⡏⠳⡄⠀⠀⣾⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⣿⡇⠀⠀⠀⣤⡄⠀⠈⣿⡇⡘⠀⣠⣾⡟⣷⡀⡇⠀⣠⣾⣿⢻⣷⣄⣽⢀⠴⣿⣿⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⡇⠀⠀⠈⢻⠓⠚⠁⢻⣇⠈⢻⠟⠋⠁⣿⡇⠀⠈⠉⡟⠁⠀⣿⣿⣄⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣠⠞⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠹⣷⡀⢀⡴⠁⠀⠀⢸⣿⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⣧⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢿⣧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣿⣷⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠙⢋⣈⡛⢃⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣴⣿⣿⣿⣌⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄⣿⡇⣠⣄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡟⢃⣬⣅⡻⠿⠀⠀⠀⠈⠋⠉⠀⠉⠛⠁⠀⠀⢰⡶⠟⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⣼⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡖⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠿⠛⠛⠿⢿⣿⣦⣄⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠋⠉⠛⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠛⠛⠿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⢸⣿⡿⠿⠃⢸⣿⠆⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⢸⣿⠋⠀⠈⣿⣷⢸⣿⡿⠿⠂⣾⡿⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⠛⠙⢻⣷⡘⣿⣧⣤⣤⣿⠟⢸⣿⡇⠀⢠⣬⠁⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠁⠀⠀⠀⢿⣷⡀⢿⣷⡀⢀⣸⣿⠇⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⣸⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⣤⣬⣿⣷⡈⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⢿⣷⣦⣿⣿⡿⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣤⣤⣴⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠉⠛⠿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⠈⣿⣏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠻⢿⣶⣤⣬⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣶⣶⣶⣿⡿⣿⣷⣶⣶⣾⡿⠟⠋⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉






⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣤⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⠋⢉⣉⣿⠉⣻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⠁⣴⣶⡆⠸⠀⢈⣉⣏⢠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⠛⣿⠁⣴⣶⡆⠸⣄⠙⠛⢁⣼⣥⣼⣿⣅⣽⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠿⠀⠘⣄⠙⠛⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⡈⠙⠀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⣇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠈⠙⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠⡖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣠⠀⢰⠇⣠⠖⢆⢠⡖⢦⠀⢸⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠉⢇⡟⠀⣿⠀⠸⣿⠀⢸⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠙⠛⠁⠈⠛⠁⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠟⠛⢻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⢿⣿⠋⢻⡏⠉⠉⢿⡏⢀⡄⠙⠀⠐⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠘⠻⠄⠘⡃⢸⡇⢸⡇⠘⠿⠀⣇⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⢠⡀⠀⣧⡀⠁⣸⣿⣦⣤⣾⣿⣶⡿⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⣼⣿⡶⠿⣿⠟⠛⠻⣿⠏⠉⠙⢇⠀⠤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⢿⣧⠀⢹⠀⣶⡀⢻⠀⢸⡆⢸⡀⢠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠈⠀⠸⡄⠙⠃⣼⣦⣀⣀⣼⣧⣤⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⠀⣿⣦⣤⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠻⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣄⡙⢧⡄⠀⢀⣠⣤⠴⠶⠶⠶⠶⢦⣤⣀⣀⣾⣿⠛⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢀⡟⠀⠑⢦⡽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣏⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠚⠁⠀⢀⠎⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣤⠤⣤⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⡞⠀⠈⠛⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠋⠀⠀⠁⠀⣠⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡔⠀⠄⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⣈⡴⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠙⠒⠆⠠⠤⠤⠤⠴⠇⠖⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠓⠒⠠⠭⢉⣉⣛⡒⠒⠒⠒⡖⠒⠒⠒⠒⢒⣒⣚⣋⣉⠭⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡽⢿⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⠟⠉⠀⠀⠉⠛⠛⠛⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⠟⠉⠀⠀⠉⠛⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠀⠀⠀⠀⠀⣠⣄⡀⣶⠆⣴⠀⠈⣷⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⢀⣠⣄⢰⣦⡖⠀⡶⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⣀⢠⠖⢲⡼⣇⣨⠟⠏⠁⠏⢀⣰⠏⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⢴⣤⡈⣷⢿⡌⠟⠳⠠⠇⢀⣼⠇
⠀⠀⠀⠀⠀⢸⡏⠀⣀⣠⣿⣞⠷⠞⠁⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣼⠇⠀⣴⣷⣤⣸⠟⠙⠈⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⢸⣇⠀⠉⠛⠟⠁⠀⠀⢀⣀⠀⠺⣗⠛⠋⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠈⠳⠞⠀⠀⠀⢀⣀⠀⢻⡗⠛⠉
⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⠶⠦⡤⠶⠛⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣤⠶⠦⡤⠶⠋⠉⠛⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠶⣦⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⡀⠈⠻⡆⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢀⡄⢶⣚⡀⣯⡭⢸⡇⠀⢀⡿⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢀⣤⡄⢴⣞⡃⣿⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⢠⡾⠋⢀⣠⣤⢻⣄⢹⡜⠯⠀⠛⠀⠘⠀⣠⠞⠁⠀⠀⠀⠀⢠⡞⠋⠀⠀⣶⣄⠹⣶⢿⡸⠏⠀⠺⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠈⢳⣞⠷⠉⠉⠀⠀⠀⣠⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⣿⠛⠓⠉⠀⠀⠀⣤⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⣉⡀⠀⢀⣴⠶⣤⣬⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⣀⡀⠀⣀⣴⠶⣤⣼⡧
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢚⣈⡩⠭⠭⣭⣍⣓⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⡖⣥⢀⣀⢐⡒⠉⠹⠀⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠸⣇⡙⠦⠭⠬⠥⠴⠖⣀⣴⢫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠋⡹⢺⡞⣿⠋⢸⡏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣩⣤⣤⣤⢤⣈⠙⡆⠀⠀⠀⠀⠀⠀⢸⠁⠄⢸⣿⡴⡳⣿⢧⣿⠀⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⣧⠈⣿⣯⡄⣀⡽⠀⢹⠀⠀⠀⠀⠀⠀⢠⢸⠀⢸⣿⠀⣇⢿⢸⣿⠀⣼⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢛⣿⢿⠛⠛⣿⡀⠀⢸⠀⠀⠀⠀⠀⠀⣾⢸⠀⢸⡇⢀⠁⢸⣼⡇⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡟⠸⣆⠀⣿⡇⠀⡏⠀⠀⠀⠀⠀⠀⣿⢸⠀⢸⡇⠈⠀⢸⣿⡇⠀⡟⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⢋⡏⠀⠙⣦⣿⡇⢀⠇⠀⠀⠀⠀⠀⠀⣿⢸⠀⠈⠁⠀⠀⢸⢻⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⢸⡟⠉⢧⠘⣿⠀⢸⠀⠀⠀⢀⣤⣄⠀⡏⢸⠀⠀⠀⠀⠀⡜⢸⠇⢠⠂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⢧⠀⠸⢰⣿⠀⡜⠀⠀⠀⢸⣻⠛⣴⡇⢸⠀⠀⠀⡇⠀⠟⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠏⢸⠀⠀⣿⣿⢠⡇⠀⠀⠀⠘⣿⣁⢸⡇⢸⠀⠀⠀⡇⠀⠀⣿⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡆⠀⠈⣤⠀⢻⡿⢸⠃⠀⠀⠀⠀⠘⣿⣾⡇⡘⠀⠀⠰⡇⢠⠀⣿⢠⠃⡷
⠀⠀⠀⠀⠀⠀⣠⣖⠂⣤⡀⠀⠀⣷⢰⡇⠀⡆⣿⢸⠈⡇⢸⠀⠀⠀⠀⠀⠀⠘⣿⡇⡇⠀⠀⢰⠁⣸⠀⠁⢸⢀⠇⠀⠀⠀⠀⣾⢲⣷⡄
⠀⠀⠀⠀⠀⠀⣿⠿⠛⣿⡁⠀⠀⣿⠸⡇⠀⡇⣿⠈⠀⠃⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢸⠀⡟⣀⠀⡇⢸⠀⠀⠀⠀⡸⣻⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠸⣖⡜⢾⣷⡀⠀⣿⠀⠀⡇⡇⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⣿⠀⠇⠛⠘⠇⢸⠀⠀⠀⢠⠟⢡⡄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣞⠶⢿⡷⡀⣿⠀⠀⣇⣿⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡄⢸⣶⠀⢀⠀⠀⣾⠀⠀⠀⡌⠰⡟⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢧⡳⣿⠀⠀⡏⡿⢘⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢇⢸⡿⠀⠀⠀⠀⢸⠀⠀⠰⢁⡿⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠳⣅⠀⠀⢃⠻⠋⢀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄⡇⡄⠇⠀⡀⢸⣆⡴⢃⠎⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠋⢠⡆⢨⠀⢀⠎⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠸⡇⠉⠸⠀⠈⡇⠈⠛⣠⠎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠘⠶⠋⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⠀⡆⠀⢰⣿⠀⠀⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⢀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡇⠀⠀⢠⠀⢠⡟⡛⡄⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡾⣆⢸⠀⡆⠀⠸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠈⠀⡾⢠⡇⠇⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣿⠸⠀⠇⠀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⠀⠀⢸⠀⠈⡇⢸⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⠀⡄⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠁⠀⠀⠀⡞⠀⢠⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢱⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠰⠁⠀⣸⠀⢸⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠈⣆⢠⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠠⡇⣰⣆⢻⠀⡘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢰⡀⢸⠀⠇⠟⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢻⡀⢸⠀⠇⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⢸⠀⢸⢀⢠⠀⠀⢠⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠃⠀⡇⠈⠀⢠⠛⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⡆⠸⠀⣾⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⡇⢠⢀⡏⢰⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⡇⠀⠀⣿⠀⣄⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⡇⠀⢸⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⢿⠀⠀⣿⠀⠛⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⡆⡇⠀⣼⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⠘⣇⠀⢸⡀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡇⠃⠀⠟⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠹⡀⠈⠀⣰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃⢸⠀⢰⡄⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⢇⠀⠀⡿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⠀⢸⢣⡿⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠃⠀⠀⢸⠀⠀⠁⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⣠⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠈⠓⠒⠒⠒⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⣀⣤⢤⣤⣀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⢀⣴⣿⣿⣿⠷⠿⠟⣿⣦⣦⣄⠀⡀⠄⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢰⣿⣒⣶⣿⡿⠛⠃⠀⠀⠀⠀⠀⢤⣭⣻⡧⠀⠐⠂⡀
⠀⠀⠀⠀⢀⣀⡀⠀⣠⠤⠒⠠⢌⢀⣄⣿⣿⣿⣿⣯⡇⠀⠀⡄⠑⡄⠀⠀⠈⢹⣯⡧⠔⠚⠉
⠀⠀⠐⠋⠉⠉⠿⣿⣶⣤⡀⠀⠀⠙⣿⣿⣿⣿⡿⠁⠀⠀⠀⠉⠀⣀⠠⢄⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠛⣶⣥⣀⡈⠹⢿⣿⣦⡀⢀⣇⠈⠁⠀⢸⡀⠀⢠⣄⠀⡄⠐⠆⡬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠈⠙⢿⣷⣤⠄⠙⠯⣿⣧⡏⠀⡅⠀⠀⠱⡀⠀⠙⣃⢤⡲⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠈⠻⣶⣤⢂⠈⠛⢷⣤⠀⡘⠛⠿⣦⡁⢠⣴⡾⠟⠉⠁⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠈⠙⣷⣆⠀⠀⠉⠛⠢⢤⣄⣀⠉⠒⠥⣀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠡⢆⠀⠀⠀⠈⠻⣷⣤⡄⡀⠉⣦⣄⠀⠲⢤⣀⠀⠀⠀⠀⣠⣾⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢶⣀⠀⠙⠻⣷⣄⠠⠌⠻⣷⡄⠀⠈⠑⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠻⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠈⠉⠉⠉⠉⠉⠁⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡖⠐⠒⠢⠤⠄⠀⣀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢠⠖⠲⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡋⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠢⢄⣀⣀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣶⠀⠀⠀⠀⠀⠀⠻⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⡟⠉⠋⠀⠈⠑⠢⣄⠀⠀⠀⣠⠜⢻⠰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠉⠁⢸⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠇⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⡀⠠⡄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢠⡀⢠⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡌⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⡇⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⡎⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⠀⠀⠀⡀⢾⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢐⣇⢀⡤⠀⠒⠂⢉⡧⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠉⡶⠶⠁⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠆⠀⢀⡤⠚⠉⠀⠀⠀⣀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢀⡸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⣄⠀⢀⣀⠠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⡜⠁⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⠁⢀⠾⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢰⡆⣀⡜⠀⠀⠀⠀⠀⠀⠸⡆⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣤⠋⠀⣉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠓⠠⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠸⠂⢀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠸⣄⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠎⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⢀⣠⠜⢧⡀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀
⠀⠠⠔⠒⠈⠣⣄⡠⠷⣀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠫⡄⠀⠀⠀⠠⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀
⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢺
⠀⠀⠈⢦⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣀⠠⠟
⠀⠀⠀⠀⠱⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠂⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣀⡀⠤⠤⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣒⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⢀⡀⢀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣦⣤⣠⣿⣷⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢿⡇⠀⢠⡀⠀⠀⠀⢀⣠⠀⠀⣄⡀⠀⠀⠀⢀⡄⠀⢸⣿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠘⣧⠀⠀⠙⠻⠶⠾⠛⠁⠀⠀⠈⠛⠷⠶⠟⠋⠀⠀⣼⠏⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣽⡀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⢀⣯⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠈⢉⡁⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠿⠟⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣬⣿⠦⢄⣀⡀⠀⣀⡠⠴⣿⣥⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⢉⠂⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡄⠀⠀⠀⠀⠀⠀⢀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠶⠿

⠀⠀⠀⠀⣠⣴⣦⣤⣤⣤⣤⣾⣻⣶⣤⣤⣤⣤⣴⣦⡄
⠀⠀⠀⠘⣿⡀⠀⣀⣀⣉⣩⣭⣭⣩⣉⣉⣀⡀⠀⢀⣿
⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⢹⣿⠉⣿⠉⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠾⠿⠿⠾⠿⠿⠷⢶⣤⡀
⠀⠀⣠⣤⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⣀
⠀⢸⣿⠉⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠛⠷⣦⣀
⠀⢸⣿⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠈⠛⠛⠛⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠷⢶⣄⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⠶⠶⠾⠟⠋⠀⠀⠀⠀⠙⣷⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⠿⠿⠿⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⢶⣶⣶⣶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠹⠘⠃⠛⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢙⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⢛⠚⡛⢛⢛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠹⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠛⠙⠙⠋⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠻⣧⠀⠀⠀⢀⣠⣤⣄⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣴⠟⠀⢀⣰⠟⠋⠈⢻⣷⠟⠋⠻⣧⠀⠀⠀⣴⠟⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣄⠀⠀⠀⠀⢀⣴⢟⣡⣤⣤⣴⣿⡁⠀⢀⣴⣯⡀⠀⢸⣏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣿⡄⠀⣠⡾⠛⠻⠟⣉⣀⡀⠀⠉⠻⠿⠛⠿⣯⣿⡆⠀⠻⣦⣄⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⢠⣴⢿⣤⢶⣤⣿⠋⠉⠻⣦⣀⣴⠶⢶⣄⠹⠿⢷⣤⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣸⣿⣿⠁⣾⡇⠀⠈⠻⣦⡀⠀⠈⠻⣧⡀⠀⠙⢷⣄⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢿⣅⣿⠆⢈⣿⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣄⠉⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠈⣿⠁⢰⣿⠁⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢷⣤⣿
⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⢀⣿⡀⠀⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠺⠋⠻⢶⣤⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠁⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⠻⣦⡀⠀⠈⠻⣦⡀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⡈⠻⣦⡀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣋⣿⣦⣌⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⠟⠋⣾⠋⠉⠉⠉⠛⠻⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠙⠻⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠈⠻⣦⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠟⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⣈⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⣀⠘⠈⠁⠀⡀⠀⠀⢸⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡠⠤⠴⠧⠄⣀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠞⡡⠔⠀⠘⣟⠀⢢⣍⠺⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣠⠔⢀⣁⢀⣉⣛⡑⠠⢀⠀⠐⡑⡼⠁⡷⢌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠒⠲⠀⠀⠐⠡⡊⠀⠀⠀⡄⡁⠉⠑⠢⣄⠀⢬⡃⢠⡧⠜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠒⠀⠀⠃⠀⠀⠀⠀⠀⢡⠡⠀⠀⠀⠀⠙⣦⡑⢏⠈⠈⠀⠀⠖⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢃⢣⡀⠄⠂⡜⠁⡟⢢⠑⡀⢀⡙⢢⠔⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣓⣋⣀⣐⡂⠤⣄⡀⠑⡑⢄⠜⢀⡜⠀⠀⢱⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠊⠉⠈⢆⠀⠉⠁⠂⢍⡓⠾⡛⢤⡾⣀⠀⠀⠀⡇⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠈⢌⣦⣠⡴⠒⠚⠳⢶⡈⠺⡏⠢⠍⣁⡉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⡄⠀⢀⡼⠓⠢⢍⡓⢤⡀⠀⢹⡑⢌⠢⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣘⡾⠖⠋⠀⠀⠀⠀⠈⠲⣝⢦⣸⠇⠀⠣⡘⡄⢀⡆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡋⠁⣀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢦⢹⡀⠀⠀⠐⠘⡄⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⢾⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠀⡀⠐⡡⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠙⢦⡀⠀⣠⠞⠁⠀⠀⡰⠃⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠙⠺⣯⢋⠄⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣳⠴⠋⠀⠐⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⢸⣿⣏⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⢠⣿⣟⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⢼⣿⣿⣤⡀⠀⠀⠀⢀⡾⣿⣿⠶⠂⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⢠⢿⣿⡇
⠀⠀⠀⠀⡸⣿⣿⡆⠉⠃⠀⢀⠞⠀⣿⣿⠃⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣼⣿⣏⠙⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⢠⠃⢼⣿⡛⣄
⠀⠀⠀⠀⢀⣿⠻⣷⠀⠀⠀⠀⠀⢀⣿⢿⡆⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⢸⡿⣿⡀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⣾⠿⣧
⠀⠀⠀⢠⡞⠁⠀⣿⠀⠀⠀⠀⢀⡾⠁⠀⣷⠀⠀⠀⢀⣴⠟⠀⣿⠀⠀⠀⢠⡟⠁⢸⣇⠀⠀⠀⢀⣤⠾⣿⠀⠀⠀⢠⡾⠃⠀⢻⡄
⠀⠀⢠⣟⠀⠀⠀⠸⣧⣀⠀⠠⣾⠁⠀⠀⠸⣦⡀⠠⣿⡁⠀⠀⢹⣆⡀⠀⣾⠀⠀⠀⢹⣄⠀⠘⡟⠁⠀⣯⡀⠀⢠⣞⠀⠀⠀⠈⢳⣄⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣷⣶⣤⣤⣤⡤⠆⠘⢆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⠀⠀⠄
⠀⠀⠀⠙⣾⣷⣿⣽⣯⣤⣶⠶⠋⠐⠀⠀⠀⠀⠀⠀⢨⣿⣿⡅⢀⣀⣠⣤⣤⣄⠀⠀⠀⠀⣼⢟⣽⣿⣿⣭⣥⡴⠆
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⡶⠦⡄⠢⠀⠀⠀⢸⣿⣿⡿⠛⠋⠉⠉⠹⠿⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⡿⢯⣝⡓⠆⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣟⠲⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠙⠳⠈⠉⡠⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠌⠙⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⣶⠶⢶⣤⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠒⣀⣀⣀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⢸⡟⠀⠀⢀⣹⣷⡀⠀⠀⠀⠀⠀⠀⡀⣠⠊⠁⠀⠀⠀⠀⠀⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣾⡇⡀⠶⠿⠿⠿⠿⠷⠄⠀⠀⠀⠴⠁⡇⠀⠁⠀⠀⠀⠀⠀⢰⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⣤⢠⡄⣤⢠⡄⡌⠀⠀⠀⠄⠀⢣⡀⠂⠀⢀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⠇⠀⠉⠈⠁⠿⠈⠁⠀⠀⠀⠀⠀⠀⠀⢙⠒⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⢀⣴⣤⣤⣤⣴⠶⠖⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣂⣀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⣤⣤⣤⣤⠆
⠀⠀⠀⠾⣳⣿⣷⣶⣶⣿⣛⠁⠀⠀⠄⠀⠈⠀⠀⠀⠀⢠⣿⠟⠛⠛⢿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣟⣧⣤⣄⣀
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣟⡓⠶⠄⢀⠀⠀⠀⠀⠀⢸⡿⢠⣴⣤⠘⣿⡇⠀⠀⠀⠀⠀⠀⠙⣻⡻⣿⣿⣿⡿⢯⣍⡛⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡻⢮⣍⡓⠂⠀⠀⠀⠀⠀⠀⣼⡇⠀⠘⠁⠀⣿⡇⠀⠀⠀⠀⣤⣶⠟⠋⠀⠘⠻⢯⡙⠷⣌⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⠁⠀⠁⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣶⠦⠤⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠿⠛⠛⠛⠿⠿⠿⠟⠋⠁⠀⠀⠀⡀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠙⢄⢢⠠⠀⠈⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠡⡆⢺⠆⢈⣿⣷⡦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣷⣿⠀⢸⣿⣿⣿⢿⣿⣕⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠤⢆⣹⡏⡀⠀⠙⢿⠟⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠂⠀⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠉⠺⣁⢤⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣀⡤⠚⢍⠉⠉⠁⢶⠢⠤⣄⡀⠀⠈⠢⠍⠾⢤
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⠿⡿⠏⢹⠀⠀⠀⠱⡀⠀⢸⠀⠀⠀⠈⠑⠒⠤⣀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⡆⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠣⡀⠀⠀⢸⡄⠨⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡇⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣠⠾⠗⣸⠋⠁⢀⢫⡟
⠀⠀⠀⠀⠀⠀⢃⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⢒⡪⠥⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠮⡑⠒⠤⠤⠤⠄⠒⠛⢛⣢⣴⠞⢉⠽⡓⡂⣁⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠈⠑⠒⠒⠒⠒⠲⢾⠿⠟⣁⢴⠁⠀⠀⠀⢀⡀⠀⢀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢠⣀⣀⠀⣀⣀⡠⠤⠒⠉⠀⠀⠢⠤⠒⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠒⠒⠒⡖⡖⠒⠒⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠶⠒⠒⠆
⠀⠀⠀⠀⡟⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠘⢲⠚⠒⠚⠛⠛⠛⠛⠓⣧⢰⠀⠀⠀⠐⠂⠐⠒⠐⢰⠄⡘
⠀⠀⠀⠀⢁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⢸⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠀⠀⠆⠀⠀⠀⠀⠸⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢐⡇⢸⠀⠀⢠⠄⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⡈⠀⠀⠀⠡⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸
⠀⠀⠀⠀⠀⠥⠀⠀⠀⣃⡄⠀⠑⠢⠄⠀⠆⠀⠀⠼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⢰⠀⣦
⠀⠀⠀⠀⠀⢠⡀⠀⡠⠃⠀⠀⠀⠉⠉⠀⠈⠂⠀⡘⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢀⣼⠤⢦⡀⠀⠀⢀⡀⠀⠀⠈⠀⡋
⠀⠀⠀⠀⠀⠀⣤⠀⡀⠤⠤⠀⠀⠐⠒⠂⠂⠀⠉⠁⠀⠀⠀⡆⠀⠀⡇⢸⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠐⢥⣴⠖⡇⠀⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠐⠢⠀⠀⢀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⡀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⢸⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠁⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣧⣤⣄⣤⢸⠀⠀⠀⢸⠀⢼⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⣠⠖⢊⣡⡼⠋⠻⠥⣀⠀⠀⠉⠚⢦⣤⠤⢼⣀⣾⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⢰⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡐⢂⠜⠒⠂⢤⣠⠞⠀⢀⠰⠋⠀⠀⠀⠀⠀⠈⠁⠐⠲⠤⢌⢢⣀⠄⠘⠓⢦⣄⣀⣀⣀⣤⡇⢸⠀⠀⠀⠀⠀⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣇⠃⠀⡠⠴⢾⢃⡤⠊⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⣿⠒⠤⣀⠀⢩⡉⠉⠁⠀⠁⠈⠀⠒⠒⠒⠒⠀⠃⠀⡌
⠀⠀⠀⠀⠀⠀⠀⢿⡤⠏⠀⠀⠈⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠢⣼⠉⠉⢉⡩⠭⠭⠌⣉⠉⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢀⡤⢾⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⣄⠀⠀⠁⠀⠀⠏⠐⠠⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡅⠸⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡸⠀⢸⠀⠀⡘⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠈⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢡⠀⢸⠀⡤⡟⠒⠠⠤⠤⠤⠴⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⢸⠀⢸⢐⡇⠀⢱⣦⠶⢶⡖⠈⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡼⠀⠀⠀⠀⠀⢠⠝⠶⠐⠒⠚⠃⠐⠒⠶⠤⢶⣉⣀⠀⠀⠀⠀⠀⢣⡈⢸⠱⠦⠴⠶⠿⠿⠦⠤⠦⠴⠶⠦
⠀⠀⢀⣀⡀⡀⢀⣀⢈⣷⠒⠀⣀⠞⠉⠀⢀⣄⣀⡀⠠⠄⣀⣀⠀⠀⢀⡀⠉⠫⣄⡈⠉⠈⣯⣵⢼⣄⣀⣀⣀⡀⣀⠁⡀⢀⠀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣀⣰⠁⠀⠀⠰⡿⠏⣀⡀⠤⠄⢀⣀⠉⠑⡾⢬⡇⠀⠘⡇⠀⠀⠈⢁⠛⠋⠘⠛⠙⠛⠛⠟⠋⠙⢻⢾⡃⠉⠀⠉⠈⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢁⣀⣈⣦⣠⠶⠅⡀⣠⠊⢠⠀⢀⠀⠀⢈⠙⠀⢀⠨⠶⣶⣦⢤⣤⣤⠈⣶⡦⠀⣀⣀⣀⠄⠀⣀⣀⢸⡟⠋⠛⠛⢷⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢨⠛⠀⠈⠁⠀⠃⢀⡴⠉⠀⠀⢀⡂⣈⡁⠨⣄⠐⠀⠁⠀⠀⡴⠦⣀⡤⣄⠀⠉⣴⣄⣀⣄⣀⣀⣀⣏⣻⣿⣃⣤⣤⣤⣼⣧⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡃⡆⢀⡔⠀⠄⠀⣭⣠⢀⡀⠀⠀⠀⠀⠀⠁⡌⠀⣤⢤⣰⡄⠈⠆⢸⡡⠖⠀⠀⠈⠅⠀⠀⠀⠀⣀⣀⠠⠀⣀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⡾⠁⠰⢀⣐⣀⠠⡎⠀⠀⡾⠶⠸⢤⣀⡅⣤⣧⡄⠠⠘⡀⠀⠈⠦⠠⢀⠀⠀⠃⠐⠂⠚⡄⠀⠠⢎⠀⠀⡀⣀⠤⠃⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡸⠓⠛⠂⠉⠉⢀⢀⣣⣤⣌⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠸⠦⠴⠆⣀⣀⣀⣈⣉⣉⠈⠀⠙⣄⠀⢀⡄⠛⠛⠁⠒⠓⢠⠀⠀⠀⠈⢆
⠀⠀⠀⢠⠒⠉⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠐⠊⠉⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠀⠑⠒⠐⠒⠒⠂⠁⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠈⡄⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠦⡠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠶
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠉⠉⠉⠉⠙⠉⠉⠀⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠌⠉⠉⠉⠪⢢⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠌⠀⠀⠀⠀⠀⠡⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠸⢁⠀⡠⡄⠀⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠂⣠⠀⢤⠲⡏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠂⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠸⠖⠒⠒⠒⠒⠒⠒⠶⠞⠀⠀⠀⠈⣥⡛⡟⡛
⠀⠀⠀⠀⠀⠀⠰⡟⢢⠤⢄⠀⠛⢤⡀⣀⣀⣀⣀⡠⠂⠀⢸⠁⠀⠀⠈⡇⡗⠲⠶⢶⢠⣤⣤⣄⣀⣀⣤⣶⠘⠓⠒⠒⠒⣤⠾⠿⣏⡛
⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⠜⠁⠂⣤⡀⢁⡅⠀⢠⡀⠀⢀⠸⠶⠶⠶⠶⠇⡇⠀⠀⣾⡌⣿⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢠⠊⠀⠀⠀⠐⣴⠁⠀⢀⣀⣀⡀⠀⠘⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⣈⠉⠈⣅
⠀⠀⠀⢠⠀⠀⠚⠁⠀⠀⠈⠉⠈⠙⠉⠂⠐⠂⠂⠀⠓⠐⠓⠀⠀⠀⠀⠘⠃⠛⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⡀⢸⡇⠀⠀⢹⠄
⠀⠀⠀⠀⢹⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⢻⡟⠛⡟⠛⣟⠛⣿⠛⣻⠛⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡄⢀⡖⠓⠒⠒⠚⢲
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⢸⠀⡇⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⢁⠀⡀⠀⠀⠈⢀⠀⣲
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⡇⠀⢸⠀⡇⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⡀⢸⠉⢰⡖⠘⣿⠒⣶⠂⡒⡎⢳⠀⢸⡇⠀⣧⠀⣸⣠⣧⣤⣷⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠄⠀⠠⠦⠀⠀⠠⠄⠀⠶⠇
⠀⠀⠀⠀⠀⡃⠨⠄⠸⡇⠀⠷⠀⢿⠀⠅⠇⠸⠀⠸⢧⡤⠿⠠⠼⡿⠓⠂⠛⢻⡿⠶⠤⣤⠠⠤⠤⠤⠤⠠⠄⠐⠒
⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡃⠀⠀⠀⠀⠀⠀⠀⣺⢴⣶⣶⠿⢶⣿⣷⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠐⠦⠾⠤⠤⠴⠴⣏⠀⠀⠉⠉⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠴⠤⠶⠖⠚⠋⠉

⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣤⣤⣤⣤⣄⣀⣀⣰⣾⠛⣷⣀⣀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⢸⡟⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠈⣿
⠀⠀⠈⢿⣤⣤⣤⡶⠶⠶⠾⢿⡟⠛⢿⡿⠶⠶⠶⣦⣤⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠷⣦⡀
⢀⣀⣀⣀⣀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⣿⠋⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠛⠻⠶⢶⣤⣀
⣿⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠳⣦
⣿⡀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠛⠛⠛⠛⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠻⠶⣦⡀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠙⠳⢶⣦⣤⣤⣤⣤⣤⣤⣤⣴⠶⠟⠁⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⣀⣀⣀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠉⠉⠉⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡶⢶⡶⢶⠶⣶⠶⡶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠟⠷⣦⡘⠟⠘⠇⠸⠃⠻⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠘⣿⡶⠰⡆⢸⠆⢶⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⣀⣴⢿⣤⢠⡄⢠⡄⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠁⠀⡁⢈⡁⢈⠁⣉⠀⣉⠀⠀⠀⣠⡶⠛⠷⣦
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣷⣾⣧⣜⡁⠛⠀⠛⠀⠀⠀⣿⡀⠀⠀⣿⠇
⠀⠀⠀⠀⣼⠏⠁⠀⠀⠈⠹⣧⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠷⣶⡾⠋⠀⠀⠀⠀⠉⠻⣿⠿⠿⢶⣄⠀⠈⠻⠶⠟⠋
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣀⣀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⣠⡿⠀⠀⢠⡾⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠙⠻⠷⠾⠟⠋⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢿⣄⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠸⣯⡀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⣿⣥⣀⡀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⣶⣾⠋⠉⠙⢷⣄⠀⠀⠀⠀⠈⠻⠶⢶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣄⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣄⠀⠀⠀⣠⡶⠶⠶⠟⠋⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢷⣶⠟⠋⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠙⢶⣤⣤⣤⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠳⠀⠠⠄⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠒⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠄⠤⠄⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⣀⡴⠚⠉⠉⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠡⠀⠀⠠⠤⠁⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⢠⢤⣀⠀⠓⠈⠙⠛⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⣇⠀⡀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠀⠀⡠⣊⡂⠰⠀⠀⡀⠄⠀⢀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠠⠄⠀⠉⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠈⠀⢄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠑⠤⣀⡀⠤⠀⠀⠀⢀⣀⡀⠀⠀⠀⠐⠒⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠠⠊⣰
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠦⠤⠤⠤⠒⠒⠒⠀⠀⠀⠀⠀⠈⠒⠢⠤⠤⠤⠀⠠⠤⠤⠤⠴⠖⠿⠦⠤⠤⢤⣤⣤⣬⣅⣤⣤⣤⠤⠤⠤⠤⠴⠖⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⢷⣦⠀⣿⡿⠿⠿⠀⢸⡿⠿⢿⣦⠀⣴⡾⠛⠻⠇⠀⣴⡾⠛⢿⣦⡀⢰⣿⣧⡀⢸⣷⠀⠀⣴⣿⣧⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣾⠟⠀⣿⡿⠿⠗⠀⢹⣿⣶⣾⠋⠀⠙⠿⢷⣶⣤⢸⣿⠀⠀⠀⣿⡇⢸⣿⠹⣷⣼⣿⠀⢰⣿⣉⣿⣆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿⣷⣴⡦⠀⢺⡿⠀⠻⣷⡄⢶⣦⣤⡾⠟⠀⠻⢷⣤⣾⠟⠁⠸⣿⠀⠘⢿⡿⠀⣿⠏⠉⠹⣿⠄⢿⣿⣦⣶⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⣀⡀⠀⢀⣀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⢀⣀⣀⡀⠀⣀⡀⠀⠀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣿⣏⠀⠻⣷⣠⣿⠋⢀⣾⡟⠉⠉⠻⠀⣿⣟⣉⣉⠁⠈⢻⣿⠁⠀⣿⣿⣦⠘⣿⠀⢸⣿⣋⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠙⣿⡿⠀⠀⢹⣿⠃⠀⠘⣿⡄⠘⢿⣿⠀⣿⡟⠛⠋⠀⠀⢸⣿⠀⠀⣿⡇⢻⣿⣿⠀⢸⣿⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠋⠀⠀⠘⠛⠀⠀⠀⠙⠻⠿⠿⠛⠀⠛⠛⠛⠛⠂⠘⠟⠻⠓⠀⠛⠃⠀⠙⠛⠀⠘⠻⠛⠛⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣷⣤⣤⣶⡶⢶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⣿⠀⠈⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⣤⣾⠋⠀⣾⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢹⣿⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠐⢿⡅⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠀⣿⠀⣿⠀⠀⠀⠀⢸⣿⠀⢀⡿⠐⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⣿⡶⠟⠁⠀⠀⠈⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠀⣀⠀⠀⠀⠀⣿⠀⣿⠛⠛⠛⠛⠿⠟⠀⠀⠀⠀⠀⢰⡄⠀⠈⠙⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁⠀⣠⡾⣿⡟⠀⠀⠀⠀⢿⣄⠙⢷⡀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢸⡇⠀⠀⢀⣴⠄⠉⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠁⠀⠀⠀⣰⠟⢠⡟⠀⠀⠀⠀⠀⠀⠹⣧⠀⠻⣦⠀⠀⠀⠀⠙⠷⠄⣀⣬⣄⡀⠰⠟⠁⠀⠀⠀⠀⠈⠙⠷⣦⣀
⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⡆⢀⣀⣀⣀⠀⣼⠏⠉⠉⠻⣆⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡀
⠀⠀⠀⠀⢾⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⡀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⡇⠈⠛⠛⠋⠀⢿⣄⠀⠀⣰⡟⠀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⢷⣄⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⢀⣴⠄⠙⠛⠛⠋⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⣀⣤⡶⠟⠛⠛⠛⠛⠛⠛⢻⣦⠀⠀⠀⠀⠀⠈⠛⠷⣦⣽⣷⣶⡄⠀⣼⡇⠀⠀⣿⡇⠀⠀⠰⠟⠁⠀⠀⢸⡇⠀⠀⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠘⢿⣥⣤⡾⠛⠻⣿⠛⠻⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠈⠁⠀⠀⢀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠖⠛⠷⢦⣄
⠀⠀⠀⣨⣿⣀⣀⣀⣀⣀⣀⣀⣠⣿⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠙⣷
⠀⢀⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣦⡀⠀⠀⠀⠀⠀⠈⣿⡟⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣤⣄⠀⠀⢹⡇
⠀⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⢸⣷⠀⠀⣀⣤⣤⡾⠛⠛⠛⠛⠳⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⢸⡏⠀⢹⡇⠀⢸⣇
⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⣼⡏⠁⠀⠀⢀⣤⣤⣄⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⡇⠀⢸⡇⠀⢸⡇⠀⢸⡟⠻⣄
⠀⣿⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⣶⡶⢿⡅⢀⣿⠶⣶⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡾⠛⢷⡄⠀⣿⠀⢸⡇⢠⡿⠁⠀⠘⣷⠀⢸⡇⢠⣿
⠀⣿⠀⠀⠀⠀⢰⡟⠉⠙⢷⡄⠀⠀⠀⢸⣿⠀⣿⡇⠀⣿⡄⠈⠙⠛⠁⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣇⣀⣼⡇⠀⣿⠀⢸⡇⢸⡇⠀⠀⠀⣹⡇⢸⡇⢸⣿
⠀⣿⠀⠀⠀⣤⣼⣧⣤⣤⣾⣥⡄⠀⠀⢸⣿⠀⣿⡇⠀⣿⡄⠛⠛⠛⠛⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡟⠋⣹⡇⠀⣿⠀⢸⡇⠘⣷⡀⠀⢠⡿⠁⢸⡇⠘⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⣿⡁⠈⠛⠛⠋⠀⣿⠁⠀⣿⠀⠀⠀⠀⠀⣀⢸⡿⠟⢹⡇⠀⠻⣦⣼⡇⠀⢸⡇⠀⢸⡇⠀⢸⣧⣼⠟
⠀⣿⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⠀⣿⡇⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⠀⠀⠀⢀⡾⠋⢸⣷⠶⢻⡇⠀⠀⠈⢹⡇⠀⢸⣇⠀⣸⡇⠀⢸⡏
⠀⣿⠀⠀⠀⠀⠛⠛⠁⠘⠛⠛⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠟⠁⠀⠸⠿⣶⠾⠇⠀⠀⠀⢸⡇⠀⠈⠛⠛⠋⠀⠀⣸⡇
⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⣿⡇⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠾⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣠⡿
⠀⢻⣦⣤⣤⣤⣶⣤⣤⣶⣤⣤⣤⣤⣤⣾⡏⠀⠘⢷⣦⣦⣤⣤⣤⣤⣤⣤⣤⣤⣾⠋⠀⠀⠀⠀⠀⠀⣤⣾⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣦⣶⠞⠋
⣿⣿⣿⡿⠿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⣴⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠄⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣴⣷⣶⣾⣿⣶⢠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠠⠀⠀⠀⠆⠀⣿⣄⠀⠘⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⢸⡏⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⢿⡏⠟⠀⢿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣙⣬⡋⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡄⢀⠀⠀⢘⢠⠀⠀⠀⣻⢁⠁⠀⠘⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⢹⣷⢿⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣷⢀⢳⢸⠀⠀⠈⢳⣇⠀⠀⣿⢸⣷⢀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣾⣿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠄⣿⠘⡀⠛⠄⠀⣬⠀⣿⠀⠀⣹⣧⣟⡌⠿⣂⠙⡛⠿⢿⣿⢿⡟⠟⣿⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⡖⠀⣿⡐⣧⠐⢦⡠⠿⠇⠸⢣⡸⣻⡞⢻⣧⣧⠈⠃⠁⡜⡛⠻⠛⠀⠶⠤⢠⣭⣉⣛⡛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡆⣿⣿⣽⣦⠋⠀⠀⠀⠈⠛⣷⣿⣡⠐⠸⡃⠀⡇⠸⢷⣿⣶⣦⡄⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠟⠛⣻⣀⣠⣠⡀⠀⠀⠀⠬⢻⠉⠀⠀⠁⠀⠇⠀⠘⣻⡿⡿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠻⡛⠛⣂⣴⣿⣿⡟⣿⣿⣿⣦⡀⠀⠀⠈⠀⠃⠀⡆⠀⠁⠀⠰⠃⠘⡇⡀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠰⡤⢼⣿⡿⠉⠉⠛⠿⠛⠈⣿⣷⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡞⢀⠀⠃⠁
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠉⠻⠿⠁⠀⡀⠀⠀⠀⠘⠙⢿⣇⠀⠀⠀⠀⠀⡀⠘⠀⢢⠁⠸⠆
⠀⠀⠀⠂⡔⠁⠀⠀⠀⠀⠀⠀⠑⡀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠚⠀⠈⠀⠀⠀⠰⠐
⠀⠀⠀⠀⠋⠓⠒⣬⠁⠀⠀⠀⠈⣖⢏⡀⠠⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡜⠀⣄⢀⠀⠀⠀⠀⢀⣉⠂⠘⣷⡶⠀⠀⠨⣀⢀⠀⠀⠀⠀⠀⠀⡀⠚⠲⠂
⠀⠀⠊⡅⠀⡆⠀⡤⠀⠀⠀⣤⠙⠇⠀⠀⣄⠀⠀⠀⠉⠉⠀⠀⠀⢀⡠⠄⠐⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⢻⠐⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⠀⠁⠀⠀⡁⠀⠀⠇⠀⠠⠱⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠇⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⡀⠀⠀⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢀⣀⠈⠑⡤⢌⠁⠀⠀⠀⠀⣀⠤⠔⠒⠶⣶⠦⠤⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠣⡀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡘⠛⠕⣢⠀⠀⠉⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠐⢲⠀⠈⠁⠀⠀⠀⠀⡈⠀⠀⠀⠘⡤⢸⠤⢀⠀⠀⠀⠀⢠⠁⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠈⠑⠢
⠀⠀⠀⠀⠀⠀⡠⠔⢊⠅⠒⠀⠀⠒⠢⠉⠁⠒⠤⢌⡀⡄⠈⠀⠀⠃⠀⣳⠀⠀⠀⣉⠢⣀⣀⢸⡀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⡠⠊⠀⡰⢋⣰⠆⣠⡄⢴⣈⡙⠄⠀⠀⠀⠀⠁⠂⠤⢀⣠⣶⠟⠀⣐⠁⠀⠀⠀⠀⠈⠉⠓⠤⡀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⢰⠁⠀⠀⠳⣻⠟⠏⠙⢉⠻⢿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢩⡤⠄⠉⠒⠠⠀⠠⠤⠀⠀⠘⠓⠈⠉⠁⠀⠀⠀⢡
⠀⠀⠰⠁⠠⠤⠤⣴⠋⠀⠐⢄⠆⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⢯⢊⠅⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔
⠀⠀⠀⠀⠒⠒⠒⢺⣅⠀⠀⠈⠀⠀⠀⣸⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠛⠉⠈⠛⠢⡀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡄⢀⠀⣠⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠉⠉⠉⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⢀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣴⠿⠛⠙⠳⣤⣤⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣠⠎⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⠖⠂⠀⢀⡜⠀⢀⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠒⡆⠀⠘⢧⡶⠟⠉⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠘⠗⠁⠀⠀⠀⠀⠀⣤⠀⠊⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡿⠀⠀⠀⠀⠀⠀⢊⡨⠐⠒⢦⠀⠀⠀⠀⠀⠀⡰⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠴⢒⢿⣄⠘⡄⠀⣠⣾⣾⣿⣷⣤⢔⢾⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣉⡉⠀⠀⣼⣿⣿⣿⣟⢯⠎⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⠟⡧⠰⠯⢄⡀⠀⢀⣠⠤⢠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡽⢣⠞⠑⣧⠀⠀⠙⢾⡋⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⣿⣿⣿⣿⣿⠟⣵⣋⡤⠶⠛⠳⣤⣀⣠⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⢢⡼⠋⠉⠉⡹⢫⣞⡠⢿⣶⡒⠒⠒⠀⠩⢩⡤⢀⣀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠞⠀⠀⢀⡞⡵⠁⠙⡄⠀⠉⠻⣷⣄⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣧⠀⠹⣆⡠⣴⣫⠞⠀⠀⠀⡇⠀⢀⣤⣾⣿⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⢀⣠⠴⠛⠉⠁⠀⠀⠀⠀⠈⠳⣤⠏⡐⢸⡃⠀⠀⢀⡞⠀⠀⢻⣿⣿⠟⠙⠋
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⢔⣵⠏⠁⠀⢠⠏⠀⢠⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡱⠋⠁⠀⠀⡴⠃⠀⠀⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⠞⠁⠀⠀⢀⣼⠁⠀⠀⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡵⠃⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡤⠤⢤⣤⣏⠟⠀⠀⠀⠀⠰⠿⠛⠛⡟⠉⠉⣿⡏⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠇⠀⠀⠀⢀⠔⡁⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢻⠰⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠈⠢⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⠀⢷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⡠⠊⡀⠀⠈⠢⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡿⠀⠸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠴⠚⢁⡠⠀⣈⠔⠊⠀⡤⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢒⣉⡥⢔⡉⠄⣀⠔⣉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠤⣚⣩⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠄⠀⠀⠘⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣠⣶⣿⣿⣿⣷⣦⣴⣾⣶⣴⣶⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⠀⠀⣀⠤⢄⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⠃⠀⠀⢳⠀⠀⠀⢰⠇⠀⠈⢣⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠸⣌⠁⠀⡾⠀⠀⠀⠸⡎⠁⢀⡜⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠑⠈⠀⠀⠈⠉⢦⡈⠒⠋⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⠄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣾⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠉⠉⢈⡻⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠈⠛⠿⢶⣶⣤⣤⣤⣠⣤⣤⣤⡶⠞⢿⣇⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠋⠉⢹⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣄⠀⣹⠆
⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⡴⠚⠋⠛⠓⣶⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣞⠉⢳⡀
⠀⠀⠀⠀⠀⣰⠟⠀⣴⠏⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢳⡀⠙⠳⢤⣀
⠀⠀⠀⣠⠴⠋⢠⡞⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠈⠳⠦⣄⠉⠛⠶⣄⣀⣀
⠀⣰⠟⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠉⠳⠦⡀⠀⠈⠙⢦⣄
⣶⡋⢀⣴⣄⢰⡏⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣤⣄⣈
⠈⠛⠋⠁⠉⠋⠁⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠃⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⠶⣦⣤⣄⠀⠀⠀⠀⠀⠹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠁⠠⠀⡈⢿⣆⠀⠀⡀⠄⢀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠻⠷⣦⣄⣸⣿⣶⣀⣠⡶⠟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠀⠀⣠⡾⠉⠛⠁⠻⠟⠻⣦⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠛⠉⠁⣀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣯⣀⣀⣤⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠶⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⣾⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⡿⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠙⠻⠿⠿⠿⠟⠉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀⠀⣿⡇⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠆⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠻⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄⡀⢀⣀⣀⠠⣶⡿⠉⠉⠀⢠⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠻⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣴⣾⠛⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⠿⠿⠂⢀⣀⣀⣀⡀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠛⠋⠈⠙⠻⣦⣤⠀⠀⠀⣤⣤⡴⠛⠋⠀⠀⠀⠀⠈⢻⣿⠿⢿⣶⣦⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠓⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠃⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡋⠉⠉⠉⠉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⣿⠇⠀⣿⣤⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⢴⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⣿⡇⠀⣿⠟⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠉⠀⠀⠈⡇⠀⢠⣿
⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠛⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠸⣿
⠀⠀⠠⣾⣿⣿⣇⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⣷⡆
⠀⠀⠀⠉⠁⠙⡿⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⠀⠀⠀⠀⠙⢇
⠀⠀⠀⠀⠀⢰⣿⠀⠀⣠⣤⡾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⢀⣤⣤⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⡖⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠙⣥⡀⠀⠀⠀⢀⣶⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠘⠟⠁⠀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣀⣸⡿⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣿⠀⠙⠿⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠛⣄⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢈⡿⠀⠀⠀⠀⠀⠘⠿⣆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀⠀⠀⠀⠀⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠋⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠿⠋⠀⠀⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠉⠁
⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠛⠋
⣀⣀⣉⣉⣉⣈⣹⣿⣿⣿⣿⣋⣁⣠⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢠⣄⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⣠⣶⣿⣧⣀⣼⣿⣿⣿⣿⠋⠐⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣴⣷⣶⣿⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡁⢠⣾⣿⠉⢹⠏⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡏⢈⣿⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⠿⣿⠿⢷⠀⣶⣤⣤⣤⣤⣀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⣷⣶⣶⣶⣆⣠⣄⣈⣁⡘⢳⣝⠿⣿⣿⡿⠟⠀⠀⣠⣴⣿⢿⣿⣿⣿⣷⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣿⣿⣿⣿⠋⠉⣹⣿⣿⣿⣬⣇⠀⠶⠶⣤⠴⣶⣾⣿⣿⣶⡿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⠛⣿⣿⣿⣿⣿⣉⣿⣿⣇⣄⣤⣿⣿⡿⢿⣿⣿⣿⣶⣦⠀⠀⠻⠿⢿⣿⣷⣤⡾⠛⠉
⠀⠀⠀⠀⢀⣤⣶⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣹⣿⣿⣿⣷⡘⠀⢩⣿⣿⣿⣆⣀⡀⠀⠀⠀⠀⠈⠃
⠀⢠⣄⣴⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣤⣿⣿⣿⣿⣿⡿⢿⣿⣿⢿⣷⣠⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠈⢻⣧⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣴⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⢿⣿⣿⣿⣿⣀⡽⣻⣿⣿⣿⣿⢻⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣼⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣮⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠐⠋⠁⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣠⣄⠀⠀⠀⢈⣍⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠻⠟⡀⠀⠀⢸⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠘⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠘⠓⠛⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢶⣶⠒⢚⡿⢿⡛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢯⣹⢟⡭⠖⠛⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⠀⣾⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⡇⠀⡏⠀⠀⠀⠀⢤⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠁⢸⡅⠀⡇⠀⠀⠀⠀⢸⠙⢦⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠛⠁⢀⡤⢾⡇⠠⡇⠀⠀⠀⠀⢸⡆⠈⢧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢈⡽⠶⠚⠉⠀⢸⡇⠀⡇⠀⠀⠀⠀⢸⡇⠀⠘⢧⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⢀⣸⡇⠀⠀⢠⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠉⠉⠉⠉⠉⢹⠇⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡶⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡞⠁⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⡼⠙⢦⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡀⣰⠃⠀⠀⠳⢴⡎⠙⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡟⠉⠀⠀⠀⠀⠀⣹⢶⠞⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣺⣁⣼⠁⠀⠀⠀⠀⠀⠀⡿⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⣀⣉⣉⣹⠀⠀⠀⠀⠀⠀⠀⢧⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⠀⠀⠀⠀⢀⣠⣿⣤⡀
⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣻⣿⣿⡿⠀⠀⠀⠀⠀⢠⣿⣿⣿⠍⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣆⠀⠀⠀⠀⡾⣾⣿⣿⣯
⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠺⢸⡿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠃⢻⣿⣿⠛
⠀⠀⠀⠀⠀⢸⣏⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢸⡇⢿⡇⠀⠀⠀⠀⠀⢸⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡃
⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⡀⠀⠀⠀⠀⠀⣠⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⣿⠙⠲⠄⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠘⢿⣿⡿⠟⠷
⠀⠀⠀⠀⣼⠟⠈⢿⡄⠀⠀⠀⠀⢠⡿⠁⠙⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣴⣿⠟⣿⠀⠀⠀⠀⠀⠀⣿⡿⣷⡀
⠀⠀⠀⣸⠏⠀⠀⠈⠻⣤⣤⠀⢠⡿⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⢠⡟⠁⠀⢻⡀⠀⠀⠀⠀⣰⡿⠀⠙⣷⡀
⠀⠀⠀⠛⠓⠀⠀⠀⠀⠉⠀⠀⠘⠳⠂⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠛⠓⠀⠀⠈⠉⠁⠀⠀⠐⠛⠁⠀⠈⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠠⢤⣴⣶⣿⣿⣿⣷⣦⡤⠤⠄⠒⠒⡄⠀⠀⠀⠀⠀⡀⠠⠤⠄⠠⠔⠁⡥⠀⠀⡀
⠀⣀⠔⠊⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠱⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠸⡥⡤⠔
⢀⠁⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠈⠷⡤⠀⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⣉⠁⠀⢀⡀⠀⠐
⢸⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠻⠀⠀⠁⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⡠
⠈⠀⠀⠉⣿⣿⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⣴⣾
⠀⠣⠀⠀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢢⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢔⣡⣾⠟⠁
⠀⠀⠑⢄⠀⠀⠙⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣴⡯⠛⠁
⠀⠀⠀⠀⡶⠠⠄⠀⠉⢉⡋⠗⡄⡠⣤⠒⢒⠚⠐⠀⢤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠄⢸⠀⠀⠈⠘⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠐⠦⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⡄⢠⣦⡾⡄⠀⠀⢳⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠊⣀⢴⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⠙⠄⠀⠊⣛⡿⣿⣿⣿⣿⣯⣿⡇⠀⣿⣿⠟⢣⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠁⠀⠀⠉⠉⠛⠛⠇⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣿⣻⣿⣟⣻

⣿⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡿⠻⢳⣤⡿⠟⠟⠻⠛⠛⠛⠛⠻⠛⠛⠻⠛⠛⠛⠛⠛⠻⠻⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠨⠀⣤⠭⠉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠂⠀⠀⠀⠀⠀⠀⢄
⠀⢠⠄⠀⠀⠀⠤⠊⠉⠑⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡁⠀⠐⠠⠤⠐⠀⣀⠤⠐⠀⠈
⠈⠘⠀⠀⠀⠀⠢⣀⣀⡠⠓⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⠐⢖⢒⠒⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠀⠀⠀⠀⠀⠀⠀⡀⠂⠄⠀⠀⠀⠰⡇⠀⠀⠀⠀⠘⡇⠀⠀⢀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠄⠀⠀⠀⠀⠙⠲⠤⣤⣀⣸⣠⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⠄⠀⢀⠄⠀⠐⠀⠈⠓⠎⣀⡀⠀⠈⠀⣠⣈⠀⠀⡀
⠀⠀⠀⢀⠀⠆⠀⠐⠂⠀⠀⠀⠈⠀⠀⠉⠀⡀⠂⠀⠀⠘⠁⢄⣰⠁⠀⠉⠒⠤⡁⠂⠤⣀⡀
⠀⠀⠀⠀⠀⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠤⠂⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄⡀⠉⠒⠂⠍⣙⠒⠲⠤⠤⣄
⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠆⠀⢀⡀⠤⠀⠒⠂⠁⠀⠀⡀⠈⠁⠒⠤⡀⠀⠈⠉⠐⠂⠊
⠀⠀⠀⠀⠀⠘⠊⠁⡀⢀⠀⢀⡠⠄⠠⠤⠄⠁⠀⠀⡅⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⡆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⢢⠒⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⢸⢀⠔⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠑⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡔⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠉⡀
⠀⡤⠄⠤⠒⢪⠡⠉⡠⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁
⠀⠀⠂⣀⡹⠁⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡰⠀⠂⠰⠀⠀⠰⠀⠚⠀⠂⠰⠀⠆⠰⢀⣆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⣧⣶⣴⣶⣾⣴⣧⣼⣶⣷⣴⢳⣿⣶⣧⣾⢿⣦⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡄⢄⡀⠀⠐⢄⠀⢆⠀⠀⠀⠀⠀⠀⡄⠀⢦⠰⢄⠰⠀⡆⠆⡀⠄⠠⡀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣄⣀⡀
⠀⠀⡠⣾⣷⡀⠀⠀⢠⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠐⢋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡿⠓⣀⠀⠀⣬⣇
⠀⠀⢈⣼⣷⣄⣿⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⣽⡇⠀⣼⣿⠀⡀
⠀⠀⣼⣿⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⡇⠉⠉⠉⠉⠉⠉⠁
⠀⠸⢧⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠖⣩⣿⣿⠀⠀⠀⠀⠀⠀⠐⢣⣿⣿⣿⡄
⠀⠠⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏⣿⡿⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠿
⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠻⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⡀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡀⢠⣾⣦⣄⣀⡠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⣀⣀⡀⠀⠀⠀⠀⡐
⡇⢸⣿⣇⣉⡉⠉⠉⣿⡋⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⡍⠻⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⠐⠚⠛⠓⢺⡟⠛⠓⠒
⠥⠬⢿⠿⠿⣿⣆⢰⠃⢣⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠹⠆⠇⣠⣶⣇⠐⣶⣶⣶⡶⠀⠀⠀⢸⣀⣻⣿⣿⣿⣆⠀⡏⢱
⠀⢀⠎⢂⠀⠈⢿⡏⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢸⠘⣧⠀⢸⣍⠛⠛⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠠⢣⠀⠈⠻⣿⠁⠀⡇
⠀⠀⠀⠈⠀⠀⠼⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠂⠘⠿⠗⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠁⠀⠂⠀⢠⠇⠀⠀⢸⡄
⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠐⠀⠂⠀⠀⠀⠐⠐⠀⠀⠀⠀⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⡴⠚⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⡰⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣠⠢⠀⢠⢷⣬⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠁⠈⠀⣸⢿⠛⠀⠤⠐⢤⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠀⠀⢀⠀⠠⠔⠒⠀⠐⠲⠦⡈⠀⠑⠂⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠁⠀⠀⢀⡤⠆⣩⠥⢬⢸⢷⣠⠠⡄⢀⠭⠐⠂⠛⠀⠠⠔⡒⠦⢍⣱⡤⠔⠂⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠎⠀⠀⠀⠞⢸⠃⡄⠘⠶⠁⠚⢿⠼⢵⣱⢀⠀⠀⠀⠀⠀⠐⠀⠙⣇⠀⠘⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⢀⠂⢰⠀⠟⠀⠀⡄⠀⠄⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⢦⣲⡀
⠀⠀⠁⠀⠀⠘⠀⠀⠀⢸⡀⠄⢹⣠⣲⣦⡾⠂⠀⠀⠈⠀⠀⡀⠀⠀⡇⠀⣈⠣⡀⠴⠆⠀⠀⠈⢣⡧⢨⢻⠇⡄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢣⡀⢳⢘⡿⣿⠁⢀⠀⡁⠀⠚⠰⠗⠀⢀⣇⠂⣀⡰⠟⠉⠂⠀⠀⠀⢘⣇⡿⣿⣷⠂⢀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠁⠾⠏⠷⠞⠒⠮⣐⠃⠀⠀⢀⠤⠀⠚⠛⠖⠁⠀⠀⠀⠀⠀⢲⣄⡼⡟⠱⢇⠏⢀⠌
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠢⠁⣀⣀⡀⠀⠉⠉⠙⠛⠚⠦⡀⠀⠀⠀⠀⢀⣀⣤⡴⠻⠿⠛⣱⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⠀⠀⠀⠀⡖⠛⠻⠭⣽⠟⣽⡄⠄⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠞⠋⠑⠲⠄⢀⣠⠤⠞⠁⠀⠀⡼⠃⠀⠟⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⡜⠀⢠⣞⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠶⠶⠶⡶⠄⠀⠀⠸⠃⠀⣌⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣭⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣶⠲⠲⣂⡦⠽⠍⠚
⠀⠀⠀⠠⢀⠀⣀⡀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠇⡀
⠀⠀⠀⠘⠂⠚⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠈⠂⠈⠉⠉⠁⠉⠉⠁⠢
⠀⠀⠀⠀⢀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⣀⡀⠀⠤⠀⣀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡔⠈⠁⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢄
⠀⠀⠀⠀⡆⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⢀⠠⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠑⠐⠐
⠀⠀⠀⠀⡇⠠⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⣇⢂⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡖⢉⡁⠀⠃⠀⣲⡀⠀⠀⠀⠀⠀⠀⢄⡐⠠⠂⠀⠤
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠁⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⠇⠀⠀⠀⠀⠐⡀⢁⣰⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠂⠀⠀⠀⠀⡄⢸⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠄⠠⠀⠐⠈⡄⠀⠈
⠀⠀⠀⠀⢄⣀⠀⠈⣀⢖⠁⠀⠀⢀⠎⢁⢀⠁⣐⡂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡭⠀⠀⢈⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠘⠈⠙⠉⠀⠀⠀⠀⣠⠧⠆⠀⠀⠀⠀⠀⠀⠑⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⢀⠁⢀⠀⣀⢀⡀⣀⠠⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠈⠃⠁⠁⠀⠁⠈⠀⠀⡨⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠁⠁⠀⠀⠀⠀⠔⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠚⠌⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠒⠀⢁⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⡤⠤⠤⠤⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠻⠷⠶⠦⣤⡄⠀⠀⠀⠀⠀⠉⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣀⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⡏⠙⠛⢿⣿⠶⠶⠦⠀⠀⠀⠀⠙⣷⠒⠶⢠⣤⣀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣄⠁⠀⠀⠀⠀⠙⠛⠶⠤⢀⣀⣀⠀⠈⢳⡀⠀⠀⠈⠙⢷⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠚⠛⠀⠀⠀⠀⢀⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠘⠇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡄⠀⠀⡠⠀⠐⠒⠀⠀⢠⡄⠆⠀⠀⢠⡄⠀⠐⠒⠂⠀⠀⠀⠀⠀⠈⣿⣀
⠀⠀⠀⠀⠀⠀⢠⡞⠉⢹⡏⠀⠈⠃⠀⠰⢶⣶⠄⠈⠀⠀⠀⠀⠀⠀⠀⠒⣾⡶⠀⠀⠀⠀⠀⠈⣿⠉⠓⣄
⠀⠀⠀⠀⠀⠀⢾⠀⠀⠸⡇⠀⠀⠀⠀⠀⠴⠿⠁⠀⠀⠀⠂⠀⠀⠈⠁⠔⠻⠟⠁⠀⠀⠀⠀⠀⡇⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠈⠇⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠀⠀⠀⠀⠀⠀⠀⠔⠒⠂⠀⠀⠀⠀⠀⠀⢀⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢶⡿⣧⠉⠘⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢉⡿⠁⠉⠉⠁⠈⣰⡆⠋⠀⠀⠉⠁⣠⣾⢳⠏⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠈⢃⣾⠀⠀⠀⠚⠁⠙⠄⠀⠀⠀⠊⣿⣹⠇⠀⠀⠀⠀⠀⣰⠏⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⡟⠀⠀⠸⠃⣦⠀⠀⠀⡆⠀⠀⠀⠀⣠⢞⡷⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢰⡇⠀⠀⠀⠻⡈⢧⣀⡀⠁⠀⣀⣠⡞⣡⠞⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠘⠁⠀⠀⠀⠀⠙⢦⡉⠛⠒⠚⣉⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣄⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⣸⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠠⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⠑⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠠⠙⠷⠶⠀⠸⠁
⠀⠀⠀⠀⢠⣿⠦⠛⠋⢻⣦⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⢠⣞⣁⡶⢸⠀⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠉⠙⠿⣧⣼⣯⣸⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⢦⣤⣤⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⢿⠃⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠷⠶⣶⣄⡈⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣘⣻⣷⣦⣴⣟⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠿⠛⠇⠀⠀⠀⠀⠀⠀⠉⠛⠒⠷⠒⢊⣉⣉⡉⠉⢉⠙⣿⣿⣇⣶⣼⠟⠁⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⢠⣦⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠛⠛⠛⠛⠛⣛⣛⣉⣙⣿⣻⣿⣴⣶⢾⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠘⡏⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡾⢿⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⢸⠁⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠤⠤⠤⠤⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⠀⠀⠐⠒⠒⠂⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⣄⣠⣤⣤⣄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣧⡴⠆⣈⣋⣀⠴⣤⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⣉⣉⠉⣉⣉⣻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠗⠊⠉⠉⠀⠐⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⢴⣶⣶⣶⡤⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡲⠒⠤⣀⢤⡀⠀⠀⠑⠺⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡰⠃⠀⠈⣷⣀⠀⠀⠀⡏⠉⢩⠟⡖⠚⠒⠓⡞⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢅⠀⢀⠞⠀⠈⠑⠢⣤⣣⠔⠁⢀⣇⠀⠀⢀⣇⠀⠀⠀⠀⢀⣀⠿⠭⠭⠭⠭⠭⠽⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠈⡇⠀⡠⠊⠀⠉⠉⠁⠀⠳⡄⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⣀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣡⣤⣤⣀⣀⣠⣔⣹⣄⣸⣇⣀⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢰⡋⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡆
⠀⠀⠀⠀⠀⠹⣭⠭⠭⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣭⠏
⠀⠀⠀⠀⠀⠀⣷⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡞⢳⠀⢰⠋⡆⠀⡖⢳⠀⢠⠊⣦⠀⢰⠑⡄⠀⡎⢲⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⡇⠀⠇⢸⡀⢸⠀⣿⠀⠀⠀⡇⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⣿⠀⠀⠀⡇⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⣿⠀⠰⠀⡇⠀⡇⣸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢹⠋⢱⠒⡖⠒⣾⡻⡆⢲⡶⢶⠒⢶⠶⡖⠋⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⠀⠀⠀⠀⡃⠀⡇⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⢸⠀⡇⠀⣾⠁⡇⢸⣷⣾⠀⢸⠀⡇⠀⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠦⣿⠀⠰⠚⡇⠀⡇⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣧⠀⣇⢸⠀⠸⡖⡇⠀⢿⢀⠇⠸⣇⣿⠀⢸⢀⡇⠀⣇⣸⠀⢸⡀⡇⠀⣇⢸⠃⠸⡀⡿⠀⢸⢀⠇⠀⣇⣻⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢩⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⣠⠎⢁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢀⣠⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢻⣤⣿⠃⠀⠜⠁⠀⠘⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣴⣿⠋
⠀⠀⠀⢀⣄⠀⠀⠀⠛⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⢳⡶⣤⡀
⠀⠀⢸⡏⠀⢰⣤⣤⣤⣼⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠘⢷⣴⣅⠀⠉⢀⣿⡇⢸⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⢹⡉⠀⠀⢸⡿⠃⣼⡇⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⣠⠟⠀⠀⣠⣶⡿⠟⠋⠁
⠀⠀⢀⣤⣤⣴⡾⠃⣿⠃⢀⣿⣿⠀⠀⣿⡆⠀⠀⣷⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⣿⡇⠀⠀⣤⣄⠀⠀⢀⣼⡿⠋
⠀⠀⣾⠃⠀⠀⠀⠀⢻⡆⠘⣿⣿⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⣿⠀⡀⠀⢀⣿⠃⠀⣼⣿⠃
⠀⠀⠘⣷⡄⠀⠀⠀⠀⢿⡄⠹⣿⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⣼⡟⢡⡀⠀⠀⢰⣿⠀⠙⢿⠛⠁⠀⣰⣿⡏
⠀⠀⠀⢿⣄⣴⠟⠻⡆⠈⠻⣦⡘⣧⠀⢿⡄⠀⢰⣿⠀⠀⠀⢰⣿⡁⣿⡇⠀⠀⣾⡟⢠⣤⡿⠁⠀⣴⣿⠿⣶⡀
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣼⡟⠛⢻⡇⠸⣧⠀⢸⣿⠀⠀⠀⢿⡏⠀⢹⡇⠀⢸⣿⣧⣼⠏⠀⣠⣾⠿⠋⠀⣿⡇
⠀⠀⠀⠀⣿⣷⣶⡟⠀⣼⡟⠀⠀⡈⢿⣄⣿⡄⠈⣿⡆⠀⠀⠸⣧⠀⠈⢿⣄⣈⡿⠀⠀⣠⣾⠟⠁⠀⠀⠀⢻⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⢀⣾⠃⠀⠉⠻⣧⡀⢿⣷⠀⠀⠀⣿⡆⠀⠀⠉⠉⣡⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⢆⣀⠀⠀⣿⣇⣾⢃⣼⠃⠀⢀⣼⡿⢾⣿⣧⡀⠀⢻⣧⠀⠀⢀⣼⡿⠉⠀⠀⠀⠀⠀⠀⠀⣄⣠⡟
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⣿⠃⣾⠃⠀⢰⣿⡟⠀⠀⠀⠹⣷⡀⠘⣿⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⢀⠀⢰⡿⠋
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠈⠿⣾⡏⠀⣠⣿⠏⠀⠀⣴⠇⠀⢙⣷⣴⠟⠉⠀⠀⠀⠀⣤⠀⠀⠀⢀⣼⣷⠟⠁
⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠻⠿⠿⠟⠁⠀⠀⠸⣧⣤⣶⠿⢛⠁⠀⠀⠀⠲⣤⡴⠏⠀⠀⢀⣼⠟
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢀⣾⡇⠀⠒⠶⣿⠋
⠀⠀⠀⠀⠀⠛⣷⠀⠀⣶⠀⠀⠀⠀⠙⢷⣿⣀⣤⠀⣴⣶⡶⠏⠙⢷⣴⡾⠿⠋⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠻⣶⠾⠋⠀⠀⠀⠀⠀⢀⣈⣽⣿⣏⡉⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠞⠛⠋⣉⣉⡛⠿⣷⣤⡀⠀⠀⠈⠙⠳⠶⠖⠋⠁
⠀⠀⠀⠀⢀⣤⢶⣄⢠⣿⣏⠀⠀⠀⠀⢦⣀⣿⠀⣉⣀⣀⠉⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠙⠷⠾⠏⢸⣿⠛⢷⣤⡀⠀⠀⠉⠀⢾⣋⣉⡽⠇⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠻⣷⣤⡀⠀⠀⠉⠹⠶⠾⣿⣻⡿⠀⢀⠀⠙⠻⣿⣶⣤⡀⠀⠀⠀⠀⣧⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⠉⠻⢷⣤⣀⠀⠀⠀⠚⢡⡴⣶⣏⣿⠆⠀⠀⠉⠙⢿⣷⣦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡶⠛⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀⠀⠀⠀⠀⠉⠁⣀⣀⣠⣤⡶⠛⠉⢻⡆⠀⢀⣀⡀⠀⠀⠀⠀⢀⣴⠖⢶⡀
⠀⠀⠀⠀⢠⡾⠛⢶⣄⣠⡿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣤⣤⣶⡿⠿⠛⠉⠀⠀⠀⠀⢸⠃⡴⠉⠉⢻⣆⠀⠀⠀⠈⠃⠴⠾⠃
⠀⠀⣠⣤⣤⣿⣄⣀⣿⡷⠶⣦⡈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠴⢤⣾⡇⠀⢀⣀⣀⡀
⠀⠀⠘⢧⡽⠉⠉⠙⣿⠀⠀⢸⡿⠀⠈⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣴⠟⠉⠉⠙⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠛⠃⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠋⠁⠀⠀⠀⠀⠀⢿⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⣾⠀⠀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣿⡶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠤⠄⠄⣠⠖⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠁⠀⠀⠀⠉⠲⢂
⠀⠀⠀⠀⢀⣀⠴⠊⠀⠀⠀⠀⠀⣰⠁⠀⠈⢦⣀⠀⠀⠀⠀⠀⡠⠚⢅⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢱⠀⠀⢉⡆⠀⡤⠘⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡤⠼⣄⣀⣀⠀⠀⠀⠀⠀⢀⢠⠄⠀⢤⠔⠏⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⡀⢙⠦⣀⠀⠀⠀⠀⡇
⠀⢠⣞⠁⠀⡧⠀⠈⠁⠀⠉⡉⠉⠁⠜⠀⠀⠀⠀⡤⡀⠀⠀⠙⠛⠋⠒⠂⠉⠈⣍⠀⠀⠚⠳⣄⠀⢠⠃
⠀⢀⠃⣀⣀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠇⠀⢀⠀⡠⠁⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⡨⣧⠋⠢⡀
⠀⠀⠀⠸⠊⠣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡫⢛⠁⠀⠀⠀⣠⠀⠡⡀⠐⠆⠂⠀⠀⠀⠀⠀⣩⡕⠙⡄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠈⣓⢤⣤⣀⣠⡤⣤⠒⠙⠢⣼⠻⠀⡰⡴⠀⠀⠀⡠⠰⣶⣤⣤⡤⠐⡶⠮⣖⡓⡄⠘⠢⠔⠁
⠀⠀⠀⠀⠀⠀⡰⠁⠀⡷⠀⣠⠒⠦⡙⢍⠈⠚⣶⠏⠀⠀⠀⠀⠀⠁⠀⡿⠀⠀⠀⠀⡇⠀⠈⡂⣹⠆⢠⡄
⠀⠀⠀⠀⠀⠀⡇⡤⠚⠀⠀⡇⠀⠀⠠⡀⠙⡄⠈⢆⠀⠀⠀⠀⠀⡆⣞⠁⠀⠀⠀⠀⡘⠶⡈⠡⠸⠈⣤⠃
⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠀⠑⠤⡀⠀⠀⣠⡶⠐⠈⠄⠀⠀⠀⠀⠈⣻⠀⠀⠀⠀⠀⠀⡐⠀⠘⠈⠈⢹
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠉⠉⣠⠃⠀⢀⠃⠀⠀⠀⠀⠀⠳⡤⠤⢤⣤⣤⣀⠀⢀⠃⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠑⠂⠤⣤⣀⣀⣀⣀⡠⣶⡻⣑⠒⡁⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠀⠀⠸⡵⡋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⡟⢀⠇⢐⠉⠙⢮⡀⠀⠀⡀⠀⠀⠀⠀⡠⠉⠀⢹⡄⠊⡠⠉⠑⠠⡀
⠀⠀⠀⠀⠀⡠⡴⠣⣌⡢⠤⡼⠳⣮⠔⠉⡄⠀⠀⢱⠀⠁⠀⡰⠖⠆⠺⠱⢄⣀⡆⠘⡯⠀⠀⠀⠀⢸
⠀⠀⠀⠠⡎⠀⠑⠄⠈⠁⠀⡄⠀⠘⡓⠊⠀⠀⢀⡜⠀⠀⢘⠀⠀⠀⠀⠀⠀⢰⠀⠀⡆⠀⠁⠀⢠⠋⠀⠲⠄⠂
⠀⠀⠀⠀⢣⡀⢀⢀⠀⢀⡰⠀⠀⠀⠙⠢⠤⠒⠉⠀⠀⠀⠈⠢⢄⡀⠀⠀⢀⠎⠀⠀⠈⠢⠤⠔⠁⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣟⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠒⢢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄⠛⠀⠈⠁⢑⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠌⠀⠀⠀⠀⠀⠀⢊⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠀⠀⠀⠀⠀⠀⠀⠘⣰⠀⢀⡤⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡧⠒⢠⡌⢤⡒⠮⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠀⠀⠀⠀⠀⠐⢢⠀⠀⢀⠀⢀⢀⠀⠀⣤⡴⠶⠆⠗⠒⠒⠒⠒⠒⠒⠒⠶⠁⠀⠀⠀⢩⠖⡿⢛⠇
⠀⠀⠀⠀⠀⠀⠰⡟⢂⠤⢤⡀⠛⢆⡀⣀⣀⣀⣀⡠⠞⠀⢐⠁⠀⠀⠀⡇⢸⠺⠶⢶⠄⣦⣤⣤⣠⣤⣠⣶⡂⠛⠚⠒⠒⢢⠜⠿⣿⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢇⢀⠜⠁⠂⢼⡄⠈⣄⣀⣨⡄⠀⢀⡨⠶⠶⠶⠶⠧⣼⠀⠀⢼⣧⢻⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢨⠀⠎⠀⠀⠀⠈⢤⠋⠀⠀⠀⡀⠀⠀⠈⢂⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠀⡆⠇⠀⠀⠀⠀⠀⢸⠀⠎⠁⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣰⠉⠉⢹
⠀⠀⠀⠀⠂⠀⠚⠃⠀⠀⠈⠉⠉⠉⠉⠉⠐⡆⠀⠀⠚⠀⠓⠀⠀⠀⠀⠐⠃⠘⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠠⡿⠀⠀⠈⡇
⠀⠀⠀⠀⢹⠛⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⣟⠛⢛⠛⠛⠛⠛⠛⢛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⡔⠒⠂⠀⠀⠓⡄
⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⠀⠀⠁⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠀⠁⠈⠉⠈⠉⠁⢉⡆
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⠀⠀⠀⠆⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡇⠈⢹⠙⠻⠉⡿⠉⣷⡆⢲⠂⠉⡇⠀⢻⠀⠸⠀⣀⣤⣦⣤⣼⣆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠰⠠⠀⠶⠀⠤⠠⠤⠀⠤⠜
⠀⠀⠀⠀⠀⡇⠀⡸⠘⡶⠀⠇⠀⢿⠇⢸⠀⢠⡇⠀⡾⡰⠴⠤⠮⣿⠧⠄⠘⠫⡿⠶⠶⢤⠄⠀⠀⠀⠀⠠⠤⠤⠀⠊
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠉⠉⠉⠉⠉⠉⢹⣥⣶⣾⣿⣶⣿⣷⣶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠜⠆⠾⡠⠆⠴⠤⢿⠀⠀⠉⠉⠉⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⡤⠴⠒⠒⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣁⣛⣻⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣯⣭⣽⣭⣿⣿⣿⣽⣍⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢿⣶⢸⣿⠛⠛⠀⣿⠟⢿⣦⢰⣿⡛⠷⠀⣴⡿⠛⢷⣆⢸⣿⣆⠠⣿⠀⢠⣿⣿⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠟⠋⢸⣿⠛⠛⠀⣿⠿⣿⡋⢈⡛⠛⣿⡜⣿⣄⠀⣸⡿⢸⣿⠹⣿⣿⢀⣾⣧⣼⣷⠀⣿⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠛⠛⠛⠀⠛⠀⠙⠃⠈⠛⠛⠋⠀⠈⠛⠛⠋⠀⠘⠋⠀⠘⠛⠈⠋⠀⠀⠛⠃⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣄⢠⣄⠀⢀⣄⠀⣀⣤⣀⠀⢠⡀⢀⣀⣤⣄⢀⣄⡀⠀⣠⠀⣠⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣼⣿⠈⢻⣶⡿⠁⣾⠋⢉⣙⡁⢸⡇⢸⣿⣭⣭⢸⣿⣿⣄⣿⠁⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣿⠀⠀⣿⡇⠀⠻⣧⣬⣿⠇⢸⡇⢸⣿⣤⣤⠸⣿⠈⢻⣿⠇⣿⣧⣤⡄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⢾⣏⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⡉⠈⠙⣿⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠁⠀⠀⠈⢻⣦⣴⣿⣦⣄⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣘⣿⣀⣀⣀⣀⣀⣈⡉⠀⠀⢸⡟⠁⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣦⣄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆⢀⣄⠀⠀⠻⣷⡀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⢰⡿⠁⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⡄⠀⠸⣷⠀⠹⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣤⣀⣼⣿⣶⠾⠟⠛⠃⣼⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⡇⢰⣿⠀⠶⣿⣧⣤⣤⡄
⠀⠀⠀⢠⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⢠⣿⡷⠟⠁⠀⠀⠀⠈⠉⠛⠿⣶⣶
⠀⢀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠛⢷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⠿⠋⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤
⠀⢨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⣷⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⠛⠛⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣿
⠀⠛⢿⡄⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣄⡀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡷⠀⠀⣸⣿
⠀⠀⢾⢿⣄⠀⠀⠀⠀⢸⣿⣀⣀⣽⡏⠙⢿⣆⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣼⡿⠛⠛⠛⠛⠛⢻⡟⢻⣿⠋⠀⠀⣴⡿⠛
⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠙⢷⣄⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣶⣶⣾⣷⣾⣿⣶⣶⡄⠉
⠀⠀⠀⠀⠀⠀⣹⡿⣶⠀⣾⠇⠘⣿⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣦⣀⣀⣀⣿⣀⣀⣀⣀⡀⢀⣿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⢿⡇⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⢨⣿⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣀⣀⣀⣾⣟⡉⠉⢉⣉⣩⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⣿⡇⠀⠀⠀⢀⣾⠿⣷⡀⠀⠀⠀⠀⣿⡇
⠀⠀⣰⡿⠛⠛⠛⠛⢛⣿⡀⠘⠛⠛⠛⠛⠛⠛⠙⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⣧⣀⣼⡇⠀⠀⠀⠀⣿⡇
⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣀⣉⣉⣁⣀⣀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠷⠿⠿⣷⡿⠶⠶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣶⣶⣶⣶⣶⣿⣧⣴⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⣿⡇⠀⠀⠀⠀⠾⠿⠿⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠻⢷⣤⣄⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣿⠉⠉⠉⠁⠀⣻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠙⠀⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⣿⡇
⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠉⠉⣿⣶⣄⣀⣀⣀⣀⣀⣤⣾⠟⠛⠃⠀⠀⢿⣷⣤⣴⡄⠠⣤⠀⢴⣴⣤⣴⣤⣤⣿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣉⣿⣛⣛⣿⣋⣉⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣷⣦⣴⣾⣿⣿⣿⣶⣴⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣤⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠻⣿⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⢸⣿⣿⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣴⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢷⡀⠀⣿⡏
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣼⡇⠀⢛⡛⣷⡀
⠀⠀⠀⢀⣼⠿⠿⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠙⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏⠀⠀⠈⠁⣸⡇
⠀⠀⢀⣿⠁⠶⠶⠆⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛
⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠘⠻⢦⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠃⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡿⠷⢶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣴⣶⠟⠋⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠉⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠁⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣠⡾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢻⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁⠀⠘⠻⣦⣀⠀⠀⣀⣀⣀⣤⣶⠶⢶
⠀⣀⣀⣀⣀⣀⣠⣴⠟⠁⠀⠀⠀⣠⣶⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠁⠀⠀⢈
⣾⠏⠉⠉⠉⠉⠉⠀⠀⠀⠀⣠⡾⠋⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⢿
⢻⣦⣀⣀⣀⣀⣀⡀⠀⠀⠰⣿⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠛⣿⠆⠀⠀⣾⣿⣯⣥⣤⡿
⠀⠸⣿⡉⢉⣩⡿⣿⡄⠀⠀⢻⡆⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣄⠀⢀⣿⠁⠀⠉⠁
⠀⠀⠈⠛⠛⠉⠀⠈⠻⣶⡶⠟⠁⠀⠀⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠇⠀⠀⠀⠀⠈⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠻⢶⣄⡈⠻⣦⡀⠀⠀⠀⣠⣴⠿⢷⡄⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠋⠁⠀⠀⠀⠉⠛⢷⣾⠿⣦⣴⠞⠋⠁⠀⠈⢿⣆⢀⣴⡞⣫⣝⣿⡿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠉⢻⣦⠀⠀⠀⠀⢀⣠⣶⠟⠁⠀⠈⢿⣆⠀⠀⠀⠀⠀⣿⣟⡵⣫⡿⠛⠁⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⢳⠀⠀⢹⠟⣹⠟⣺⣿⣉⠀⠀⠀⠀⠀⠀⠙⢷⡄⢀⣴⣾⡿⣫⡾⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢣⡀⠈⠛⠛⢶⡋⠈⠻⣧⠀⠀⠀⠀⠀⠀⠈⢻⡟⣡⢮⣾⠋⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣵⡟⠁⠀⢀⣤⡾⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⠉⣉⠉⡉⢉⡉⢉⣉⣉⣉⠀⠀⡀⣀⠀⣀⣀⣀⠈⣻⣷⣶⡿⠛⣁⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢤⢎⠒⢠⠀⠀⠀⠀⠀⢠⡀⠄⠀⠀⠀⠀⢁⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠐⠀⠀⠀⠐⠖⠀⠀⠀⠈⠃⠘⠤⠈⠀⠀⠀⠀⢈⡁⣀⣠⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⡈
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈

⠀⡀⠀⡠⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⢀
⠀⢀⠁⠈⠉⠑⠀⠀⠀⠀⠀⣦⠂⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠠⠄⡈⡀⠤⠀⠀⠀⠀⠂⠂
⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠽⠚⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠉⠑⠂⠀⠀⠀⠠⠀⠀⢀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⣀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠈⠀⠀⠄⠠⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠠⠉⠉⠁⠀⠀⠀⠐⡄⢠
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠢⡈⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠠⠈⠃
⣤⣤⢦⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣷⣤⣤⣤⣤⣤⣤⣶⣤⣤⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲⣴⣖⡂⣖⣶⢴⣶⢠⡖⢶⣰⣦⣰⢀⡶⡄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠘⠓⠂⠋⠓⠐⠞⠋⠓⠚⠙⠃⠛⠘⠛⠓⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⡖⢦⡖⣴⢒⡂⡆⣶⣒⢰⣆⣶⣶⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘⠀⠙⠚⠃⠃⠛⠒⠙⠈⠋⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⣏⢙⣦⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣈⣋⠉⠟⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣡⡤⠤⠤⠤⠤⢀⡉⠻⣆⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⢸⠃⢸⠃⠀⠀⠀⠀⠀⠀⢰⡆⢸⠆⡷⢠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠊⠁⠀⠀⠀⠘⣆⠘⢦⣀⣀⣀⣀⣀⣠⠼⠁⣼⠋⠁⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⣤⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣶⠛⠛⠛⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⢤⡀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀⠀⠹⣯⡽⠙⢦⡀⠙⢦⡀⠀⠀⠀⠀⠸⠷⠟⠛⢛⣻⣻⣟⠁⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠶⢠⡆⣶⠀⠀⠙⢦⡀⠙⠂⠀⠀⠀⢠⠾⣆⣀⣿⣉⣉⣉⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣤⣈⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣟⠀⣹⠄⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠿⣆⠰⠾⠷⠶⢶⡄⠀⠈⠁⠀⠀⠀⠈⠉⣿⠀⠀⣞⣹⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⢉⣭⣭⣭⣿⣭⣭⣽⠂⠀⠀⠀⠀⠀⠀⠀⣿⠀⠤⠤⠥⠤⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠒⠒⣶⡗⣶⠋⠀⠀⠀⠀⠀⠀⢀⡆⣿⠀⠀⠘⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⢤⣤⣬⣭⣭⡽⠏⠉⠁⠶⣄⠀⠀⠀⠀⣠⡞⠃⣿⣒⠒⠒⢒⣒⣒⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠛⠃⠀⠀⠈⠉⠁⠁⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢰⢶⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⡓⠀⠀⠀⠀⠀⠀⠀⠀⡐⡐⠆
⠀⠀⠀⠀⠠⠐⠐⢃⠀⠀⠀⠀⠀⠀⠀⢤⡟⠐⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠑⠆⡀⠀⠀⠀⠀⠀⠐⢈⣤⡆⠀⢀⡆
⠀⠀⠀⠀⠀⢠⢨⡟⠆⠀⠀⠀⠀⠒⣲⠁⣹⠁⠀⠀⠀⠀⠀⠀⠙⡉⣇⣀⠀⠀⠀⠀⠀⠀⢀⢾⠀⠈⠁⢌⠂⠀⠀⠀⠀⢈⠋⡆⢨⣐⠕
⠀⠀⠀⠀⠐⢲⠁⠘⠁⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠀⠀⠀⠀⠀⢦⣀⣋⠑⠑⠀⠀⠀⠠⠦⠕⢺⠀⣠⠀⠀⠚⠃⠀⠀⠀⠘⢾⠀⢠
⠀⠀⠀⠀⠀⢘⠐⠚⠀⠀⠀⠀⠀⠀⢀⠂⠸⠀⠀⠀⠀⠀⠀⠀⠘⢯⠝⠲⠄⠀⠀⠀⠀⠀⠀⠈⠇⣰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈
⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠙⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢤⠀⠀⠀⠀⠀⠀⠀⢰⢀⣇⣐⠂⠷⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⡌⠑⡈⢄⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⠀⠈⠫⡂⠀⠀⠀⡰⢁⠴⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠑⠃⠃⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠰⠤⠀⠀⠀⠀⠀⠀⠐⠞⠉⠁⠐⠂⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⠅⢵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣿⠀⠀⠀⠀⠀⠀⢸⡁⡍⠀⠀⠀⠀⠀⠀⡎⢻⠟⠀⠀⠀⠀⠀⠀⠀⠞⢻
⠀⠀⢀⡀⠀⠑⠜⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⣿⣧⠀⠀⠀⠀⠀⠚⢻⠓⠄⠀⠀⠀⠀⠀⢉⠿⡀⠀⠀⠀⠀⠀⠀⠀⡑⠙⠄
⠀⠀⠀⠻⡂⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣻⠂⠟⠃⠀⠀⠀⠀⠐⡆⣛⢤⠂⠀⠀⠀⠀⠐⡿⣠⢧⠀⠀⠀⠀⢉⣅⠘⠇⠐⠤
⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⠉⢂⠀⠀⠀⠀⠀⢠⡃⠘⢑⠂⠀⠀⠀⠀⠰⡅⡀⡀⠆⠀⠀⠀⠀⠁⢚⠀⡰
⠀⠀⠀⠀⠀⣹⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢂⠀⠀⠀⠀⠐⠇⢰⠈⠀⠀⠀⠀⠀⠀⡗⢀⠂⠀⠀⠀⠀⠀⠀⠈⠘⢳⠁
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣠⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⡌⢂⠀⠀⠀⠀⠀⠀⠸⡀⠘
⠀⠀⠀⠀⠘⠀⢆⠐⡀⠀⠀⠯⡤⠮⠤⡴⠀⠀⠀⠉⠈⠫⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠙⡌⢆⠀⠀⠀⠀⠀⠀⢡⠀⢃
⠀⠀⠀⠀⡀⠐⠀⠣⣐⠄⠀⠠⣶⠔⢆⠧⠄⠀⠀⠀⣆⠀⢡⡄⠀⠀⠀⠀⡄⢸⠀⡀⠀⠀⠀⠀⠈⡠⠀⠈⡼⠂⠀⠀⠀⣠⢀⠂⠠⠀⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⠀⠀⠀⠀⠂⠀⠀⢰⠀⠀⠀⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡐⢻⣿⣿⣆⠀⠀⠀⢸⠀⠀⠘⠀⠀⠀⢨⣿⡅⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣹⣿⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⢿⣿⣿⣿⣦⠀⠀⠀⢃⠀⠀⠆⠀⠀⢸⣿⣆⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣽⣿⡁
⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠇⠀⠀⠀⠔⠀⠀⣠⣾⣶⡆⢰⣶⣶⣶⣶⣷⡀⠀⠈⢆⠀⠀⠀⠀⠸⣟⡿⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠊⠀⢀⣴⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡟⢢⡀⠈⠔⠀⢠⣾⠟⠛⠻⣶⣶⣶⣶⣄⠀⠉⠛⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠆⠀⡜⠁⠀⠔⠁⢀⣴⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⢸⡿⠃⡀⢠⣿⡟⠃⠀⠀⠈⢻⣿⣿⣅⠀⠀⠀⠀⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡁⢠⡴⡿⠰⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠇⢸⠁⠈⠠⠘⠋⠀⠙⠋⠀⠀⠀⠉⡿⠟⠀⠀⠀⠀⠀⠈⣹⣿⣿⡇
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂⠒⢻⣶⣶⠀⣶⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶⣶⣶⠆⠀⠀⠄⠀⠀⠀⠁⠀⢀⠈⠓⣠⠀⢀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣞⣀⡤⠤⠤⠴⠤⠿⣿⣤⡟⠛⠛⠉⠉⠉⠉⠩⠍⡉⠛⣿⢿⣤⣀⠰⠀⠀⠈⠀⠀⠀⠀⡡⠄⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀⠀⠀⠀⠐⠂⡀⠈⢼⡏⠈⠀⢧⠀⠀⠀⢀⡴⠊⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⣁⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⠁⠀⠉⣧⡦⢶⡤⠷⣶⡀⠉⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠁⠈⢳⠚⣳⣞⣛⣭⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠙⢅⢀⣀⣀⡙⠃⠄⣀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠈⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢋⠄⠀⠀⠀⠀⠀⠀⠡⡀⠂⠖⠀⠀⡈⠉⠩⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠛⢥⠄⣀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠉⠀⣀⡂⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠨⣍⠉⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠈⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⢀⣿⠀⠀⠀⠀⠀⠀⡠⠔⠒⢆⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠕⠋⠉⠝⢆⠀⠀⠀⠀⠀⡼⢋⠻⡄⠀⣽⠀⠀⠀⠀⢠⣯⣄⠀⠀⠀⣭⠆
⠀⠀⠀⢀⡴⠟⢦⣀⠀⠀⠀⠀⠘⢿⡀⠒⢒⡮⠗⠀⠀⠀⢸⠁⢻⠃⣷⠀⣿⠀⠀⠀⠀⠸⠿⣿⣿⣿⡿⠛
⠀⠀⠀⠙⠦⠤⠦⠋⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⢸⡒⠚⠒⣿⣠⣿⣄⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⠉⠉⠛
⠀⢠⠖⡩⠍⡑⢢⠤⠤⠄⠀⠀⠀⠀⢀⡤⠔⡢⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣖⣒⣺⢷⣶⣶
⠀⢸⠐⠀⠁⠀⢸⠀⢸⠀⠀⠀⢠⠊⠡⢢⡟⠟⠈⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣤⠠⠄⠄⡏
⠀⢸⠀⠀⠀⠀⢈⠀⢸⠀⠀⠀⢸⡀⠢⡥⢐⣨⠖⠁⠀⠀⠀⠀⣀⣠⣧⡤⠤⣄⠀⠀⠀⠀⢸⣿⠀⠀⢡⡇
⠀⠈⠲⠤⠤⠔⠋⠉⠉⠀⠀⠀⠀⠘⠓⠛⠃⠀⠀⠀⠀⠀⠀⣿⢄⣈⣩⣥⡴⠟⠀⠀⠀⠀⠘⠿⠤⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⡶⣏⢿⣿⣷⡀⠀⠀⠀⠀⠀⢀⡠⠔⠓⠦⣄⠀⠀⠀⠀⠀⢀⡠⣴⡶⠀⠀⠀⠀⠀⠀⠀⠰⡛⢻⠂
⠀⠀⣸⡜⠂⠀⠹⣧⠀⠀⠀⢠⡒⠁⠀⠀⠀⠀⡈⢽⡂⠀⠀⢸⢳⠟⠛⠉⢉⣷⡀⠀⠀⠀⣀⠔⠋⠙⠢⡀
⠀⠀⠓⡤⡀⠀⠀⠣⡀⠀⠀⢸⡀⠀⠂⠠⠀⠈⣠⡾⠁⠀⣸⣍⡚⢀⣠⢞⡿⠛⠁⠀⠀⢸⠁⠀⠀⠀⠀⢱
⠀⠀⢀⡇⠈⡄⣀⣠⠗⠀⠀⠈⠙⠷⣤⣤⣤⠟⠁⠀⠀⠀⠓⠿⣿⡯⠟⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠊⠉⠁
⠀⠀⠀⠀⡤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⢷⣴⡟⠅⠀⠀⠀⢾⠙⡦⠤⠤⢤⡤⣤⠚⡆⠀⠀⣴⣊⡉⠉⢉⣲⡄⠀⠀⠀⠀⡼⠉⠻⠯⠿⡆
⠀⠀⠀⣠⠞⠀⢆⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠘⡶⠃⠀⠀⢸⣌⡉⠉⣉⢼⠁⠀⠀⠀⠀⣧⠀⠀⠀⠀⣴
⠀⠀⠀⣏⠉⠀⢸⡆⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠚⠁⠀⠀⠀⠀⢧⣍⣉⣠⠎⠀⠀⠀⠀⠀⡯⢀⣀⣀⡀⢽
⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠀⠉

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢰⣶⣶⣶⠀⠀⠀⠀⣀⣀⠀⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠸⣿⡿⠀⢸⣿⣿⣿⠀⠀⠀⠐⣿⣿⠂⣿⣿⣿⣿⠀⠀⢠⣤⣄
⠀⠀⢠⡄⠰⣦⣀⣾⣷⡀⠀⠀⢠⡄⠀⠀⠀⣰⣷⡆⠀⠀⠀⣠⣶⣶⣶⣦⣄⣠⡦⠀⠀⠀⠀⢰⣶⣶⣦⣤⣤⡀⠀⠀⠀⠘⠿⣟⡀
⠀⣀⣼⣄⣀⣈⣉⢻⣿⡇⠀⢀⣸⣇⣤⣴⣾⣻⣿⣿⠀⠀⠀⣿⣿⣿⣿⡇⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠉⠉⠁⠀⠀⠀⠀⢀⣿⣿⡀
⠀⢸⣿⣿⣿⣿⠇⢸⣿⠃⠀⠀⣿⣿⣿⣿⡿⢸⣿⠏⠀⠀⠀⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣠⣴⣟⣿⣿⡇⠄⢠
⠀⠠⡄⠈⠁⠀⠀⢸⣿⠀⠀⠀⣤⠀⡍⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠀⠀⢸
⠀⠀⠑⠊⠀⠀⠀⢸⣿⠀⠀⠀⠈⠒⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⡆⡀⠀⠀⠓⣼
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠙⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠃⠀⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠾⠿⠷⠂⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⢀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣛⣋⡉⠉⠳⣄⠀⠀⠀⠀⣰⣶⣿⣯⣤⡶⣀
⠀⠀⠀⠹⣦⡀⢠⣤⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠉⠉⠁⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⢸⣿⡏⠉⢉⢉⣿⣦⡀⢿⠃⠀⠀⠀⣿⣿⡀⠀⠀⣠⠾
⠀⠀⠀⠀⠘⢷⣝⢿⠟⣙⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⠀⠸⣿⣇⠀⠠⡚⣿⣿⠇⢸⠀⠀⠀⠀⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠿⠶⠶⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢸⣿⠀⠰⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡟⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⡏⣿⠀⠀⠀⠀⠀⢸⣿⠀⢐⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⡇⠀⠘⠿⠂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣃⣀⠀⠀⠀⢸⣿⣿⡇⠿⠀⠀⠀⠀⠀⠻⠇⠀⠐⠶⠶⠶⠶⠾⠀⠀⠀⠀⠘⠿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⢀⣴⣶⣄⣠⣶⣶⣀⠀⠀⠀⠀⠀⣰⣶⣦⣠⣶⢶⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣙⣟⣽⣷⣛⣏⣹⣦⠁⠀⠀⢀⣈⣻⣋⣻⣟⣞⣙⣧
⠀⠀⠀⠀⠀⠀⣸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠻⠿⠿⠏⠫⠄⠀⢀⣿⣿⣿⣿⠿⠿⠿⠏⠋⠄
⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢳⣦⣔⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣉⣡⣀⣀⡀⠀⢸⣿⣿⣿⣿⣀⣈⣁⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣋⣉⠉⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⣿⠿⠿⠿⠿⠿⠇⠀⠀⠁⣿⣿⣿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⢀⣀⣙⡛⠛⢻⣷⠀⠀⠀⠀⠀⠀⢀⣙⣛⡛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⠀⠈⠻⠇⡀⠀⠀⠀⠀⣼⣿⣿⣧⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿

⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣝⣄⡠⣤⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⢸⣽⠀⠀⠀⠀⠀⣠⠔⠊⠉⠈⠳⢄
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢰⡉⠀⠀⠓⠵⣄⠀⠀⠀⠀⢀⡞⡁⢈⢳⠀⢸⢸⠀⠀⠀⠀⠀⣿⣶⣄⣀⣀⣀⣴
⠀⠀⣠⠞⠛⠳⣄⠀⠀⠀⠀⠀⠘⠺⣄⠛⢁⡵⠚⠁⠀⠀⠀⢸⠀⡑⠀⢸⡄⢸⡎⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⠟⠉
⠀⠀⠙⠶⠤⠖⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⡛⠒⠒⢺⣥⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠉⡙⠛⠛⠛
⣠⠖⢒⡒⠒⢤⣀⡠⡄⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠶⢦⠤⢤
⡧⠀⠀⠈⠁⢸⠀⠀⡇⠀⠀⠀⢀⠔⢊⣥⣢⢟⠎⢱⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⡉⢡⠈⡅⡏
⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢟⠠⡀⠘⢉⠄⢂⡽⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⢀⡇
⢧⡀⠀⠀⢀⡴⠤⠔⠃⠀⠀⠀⠸⢦⣄⣯⡴⠚⠁⠀⠀⠀⠀⠀⣤⠖⠪⠿⠿⠀⢙⡇⠀⠀⠀⠀⠘⡇⠀⠂⠈⢸⠇
⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠵⢤⠶⠖⠛⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠰⡻⣎⢻⠿⢿⣦⠀⠀⠀⠀⠀⠀⡠⠔⠚⠁⠑⠲⣀⡀⠀⠀⠀⠀⣠⢴⣺⠽⣃⣀⠀⠀⠀⠀⠀⠀⠀⢫⣉⡼
⢀⣵⡈⠀⠀⠸⣿⠄⠀⠀⠀⡶⢉⠀⠀⠀⠀⠀⡀⠔⢿⠀⠀⠀⢸⠙⡟⠋⠉⠀⣨⡷⡀⠀⠀⠀⢀⠔⠋⠉⠉⠢⡀
⠈⠲⡄⠀⠀⠀⠹⡄⠀⠀⠀⣯⡀⠈⠐⠠⠄⠈⢀⣴⠟⠀⠀⣸⣍⡙⠁⣠⡴⣻⠿⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱
⠀⢸⠀⠙⡀⠀⣀⡼⠀⠀⠀⠉⠻⢶⣤⢠⡠⣲⠞⠁⠀⠀⠀⠓⢿⣟⣟⡵⠚⠁⠀⠀⠀⠀⠀⠰⡅⠀⠀⠀⠀⠀⢠
⠀⠈⠛⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠋
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠐⣇⣀⣼⡏⠀⠀⠀⠀⡴⢢⣀⡀⠀⠀⠀⠀⢀⢤⠀⠀⠀⢀⡤⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⢈⡟⢻⣧⡶⣦⡀
⠀⠀⠀⣹⠛⡇⠁⠀⠀⠀⠀⢳⣴⠏⠉⠉⠉⠋⠉⣧⢸⠁⠀⠀⢻⠒⠶⠶⠶⠶⣾⠃⠀⠀⠀⠀⡸⠀⠀⠀⠉⠉⢇
⠀⢀⡎⠁⠀⢱⡄⠀⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠈⡗⠤⠤⠤⠚⡏⠀⠀⠀⠀⠀⡗⠉⠉⠉⠉⠉⣿
⠀⢸⣇⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠞⠁⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠐⣺

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠐⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣧
⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⣿⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣶⣶⡄
⠀⠀⠀⠀⠀⠉⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⠇
⠀⠉⠉⠓⠂⠤⢤⡈⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⣶⠀⠀⠀⠀⢠⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀
⠀⠀⢸⡟⠀⠀⠀⣠⣟⡼⠃⠀⠀⠀⠀⠀⠠⠐⣀⠠⡔⠁⠀⡰⢿⠀⠀⠀⠀⠀⠀⠀⣾⣛⣿⣿⣿⠟⠋⠁
⠀⠀⣾⠇⠀⠀⢀⣯⠟⠁⠀⠀⠀⠀⠀⠀⠁⠈⢁⠾⠀⢀⡜⠁⢸⠀⠀⠀⠀⠀⠀⠀⠉⠋⢻⣏
⠀⠀⣿⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠊⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠉⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⣾⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⡄⠀⠀⠀⣀
⠀⠀⠀⠀⢰⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡼⠁⠀⣠⡾⠁
⠀⠀⠀⠀⣼⣿⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣼⠃⢀⢼⠟
⠀⠀⠀⠐⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⣴⠃⠀⠘⠋
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠶⠖⠒⠂⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⢆⠈⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣰⡀⠹⡀
⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢼⠀⢆⠀⢰⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢣⠘⠀⡇
⠀⠀⠀⠀⠀⢀⠁⡀⢀⠔⢀⠠⡄⢀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠛⠈⠒⠂⠛⢠⠉⢆⠀⠀⠀⠀⠀⠀⠀⢀⠐⡄
⠀⠀⠀⠀⠀⠀⠈⡦⠀⠈⠀⠀⠈⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠰⠆⠀⠀⠁⠀⠠⡀⠀⢀⡄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡤⠀⠀⢠⡄⠀⠸⠆⠐⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠄⡀⠚⣧⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠊⠀⢰⡀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣀⡤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠦⡤⠄⠲⢧⡀⠀⠀⠀⢀⡙⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⡠⠤⠺⠛⠭⠥⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣃⣝⠀⡀⠀⠀⠠⡀⠀⠀⠓⠊⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠁⠀⠰⠀⠀⠐⣄⡃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣀⠜⠀⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢄⢰⡉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠁⠰⠠⣀⠀⠀⠀⠀⠀⠀⡀⡔⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⠀⠀⠈⠀⠀⠐⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡄⠰⠦⠄⠀⠀⠀⢠⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢠⡀⢄⡒
⠀⠀⠀⠀⠀⢠⠈⠉⠀⠀⠀⠀⠀⣇⠀⣀⠀⠀⠀⢤⠀⢀⠃⠀⡀⠀⠀⠀⠀⠀⠀⡐⠲⠰⠄⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠈⠂⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⢀⠎⠀⠀⠀⠟⢠⠂⠀⠀⠀⠰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢒⣒⡲⢤⠄⠀⢀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⠏⠀⡘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢇⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢐⠄⣸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣸⠀⠀⡷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⢀⠃⠡⠀⠀⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⡇⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠄⠀⠀⠡⡀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣟⠳⢄⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠌⠀⠀⠀⠀⠱⡀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿⠀⠀⠙⠢⣄⢸⠀⠀⢀
⠀⠀⣀⠠⠤⢄⡀⠀⢀⡴⠁⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⣎⠉⠀⠀⠈
⠀⠘⢃⠺⢦⣄⠙⠐⠧⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⠈⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⢸⠀⠀⠀⠀⡿⠀⠀⢀⠆
⠀⠀⠀⠑⠀⠑⠳⣄⠀⠀⠀⠑⢆⠈⠀⠀⠀⠀⠀⡠⠔⠚⠀⠀⠀⢠⣢⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠊⠀⠃⠉⡆⠀⠀⡼⠁⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠐⠀⡉⣂⡀⣠⠋⠀⠀⠀⠀⠀⠸⣠⣤⣂⣀⣡⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠙⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⠀⠀⠉⠐⠄⠀⠀⠀⠀⠀⠠⠮⠄⠀⠀⢀⡤⠔⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠤⠁⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠔⠤⡀⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡌⠀⠀⠀⠁⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⡀
⠀⠀⠈⠢⢅⡀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁⠀⠀⠀⠀⠀⠝⣠
⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠀⠀⠐⠒⠀⠀⠀⠀⠀⠈⠒⠠⠤⠤⠤⠤⠤⠤⠒⠘⠦⠤⠤⠤⠤⠭⠤⠤⠤⠤⠤⠤⠒⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣾⣿⣿⣿⣶⣶⣄⣤⡀
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢻⣿⣶⣾⣿⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠿⠛⠛⠛⠃⠀⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣀⡀⠀⠀⠀⢀⣀⣀⠀⠀⠘⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣄⠃⢴⣤⡤⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⢻⣿⡿⢟⡛⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⠋⠀⠀⠀⠀⡆⠀⠀⣀⠈⠙⠛⠉⠀⠀⠀⠁⢰⡏⣹⢆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣑⣡⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠙⠛⠒⠲⠒⠒⠚⠃⠀⠀⠀⢀⣠⡾⠉⠉⠉⠛⠛⠛⠻⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢦⣄⣀⣀⣀⣀⣠⣤⣴⠟⣋⡁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡍⢻⠉⠉⢀⡠⠛⣧⠍⣸⡛⠽⣟⢲⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣯⠀⠙⠒⠂⠉⣴⠊⣠⡖⠉⠛⣦⠈⣿⡚⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⣰⣇⣴⠿⣤⣷⣶⠁⣠⣟⡀⠀⠀⠸⡇⢸⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⣤⣀⣰⡟⢰⠇⠀⠀⠀⢏⣉⠉⠒⠋⠈⠹⠀⠀⠀⢱⡞⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠇⠙⠺⣿⣿⣄⠇⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣤⡀⠀⢸⣿⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠠⣦⣼⠗⢤⣤⣀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣀⠀⢈⣿⠀⢸⡿⢠⠏⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⠉⠹⣿⠀⠀⠀⠀⠀⠀⠀⠈⡃⠹⣸⠿⠋⢠⣆⠘⣅⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢻⣷⠀⠀⢸⡗⢻⡃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠸⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣿⠀⠀⠀⠁⣼⣡⠟
⠀⠀⠀⠀⠀⠀⠀⢿⣸⣇⣀⣀⠴⣦⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡸⣄⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛⢫⣤⠃⣽⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡟⠁⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣋⠛⠶⢦⣤⣤⣤⣤⣤⣤⣤⣴⠶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠙⠶⣾⣍⠀⢀⣈⣉⣁⣤⠛⢶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⣀⠁⠈⠉⠉⠷⠤⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡅⠈⠉⠁⠀⠀⣀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢠⡏⠀⠀⠀⠀⠀⠀⠉⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣸⠇⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⣠⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⠀⠀⠀⡰⠋⠈⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣦⠀⢀⡇⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠉⢀⠻⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢈⡇⠸⣇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠘⡀⣠⠟⠁⠀⠙⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣄⡸⠀⠀⠀⠁⠟⠀⠀⠀⠀⢧⡀⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣶⣦⣄⢸⠇⠀⠀⠀⠀⠘⢧⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠘⠻⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁