Morality Emojis & Symbols

⠀⠀⠀⠀⢰⣴⢷⣴⠀⠀⠀⠀⠀⠀⣀⣀⣐⢟⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⡀⠀⠀⡰⢇⠀⠀⠀⠀⠀⠀⡀⣀⣤⣤⣤⣀
⠀⢀⣟⡞⢿⣇⣄⣽⠗⠀⠀⠀⠀⠈⠉⠉⢹⣿⠉⢉⠁⡀⠀⠀⠀⠀⠀⠻⣯⣿⢤⣼⣿⣿⡇⠀⠀⠀⠀⠐⠟⠿⢛⣭⣤⡈
⠀⣾⡃⢰⣿⠙⠟⠙⠁⠀⠀⠀⠀⠉⠉⠀⢸⣿⠀⠈⠉⠁⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠇
⣰⠁⠿⡼⠩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⢤⣠⣴⣾⠿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⢠⡄⠀⣤⡀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⡀
⢠⣮⣵⡶⣮⡖⣶⣽⡆⠀⠀⠀⠀⢀⣤⣤⣸⡿⣷⣤⣤⡄⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠙⢦⣛⣴⠋
⠸⣿⣿⠇⣿⡇⢿⣿⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⣀⠸⣿⡇⣀
⠀⣿⣿⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠠⣾⣿⣭⣻⣫⣿⣿⡆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢠⣽⣤⣿⣤⣾⣦
⠀⠿⠿⠀⠿⠇⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⠆
⠀⠀⠀⠀⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⠙⠉⠋⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⢀⣤⣄⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣾⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣿⣿⣷
⡀⢠⣽⡁⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠇⣿⣿⣷⣤⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣤⣤⣨⣯⣠⣴⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⣴⣿⣯⣿⣷⡄
⣠⡬⣝⣽⣿⣝⣯⣤⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⡅⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⡇⠀⠀⠀⠀⠀⠀⢘⣷⣷⣜⣾⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⠁
⠛⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⢻⢿⡃⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠉⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠁⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣌⠛⠶⣤⣤⣤⣴⠾⢋⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠘⢻⣿⠛⠛⠛⠛⠀⢿⣄⣼⠇⠘⠛⠛⠛⢻⣿⠛⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣾⠙⣧⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⢠⡟⢹⣆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠃⠀⠹⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡿⠁⠀⢻⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⣇⣀⣀⣀⣹⣆⠀⠀⠀⢸⠀⠀⠀⢀⣾⣁⣀⣀⣀⣿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡉⠉⠉⠉⢉⡿⠀⠀⠀⢸⠀⠀⠀⠸⣏⠉⠉⠉⠉⣹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠻⠶⠶⠶⠛⠁⠀⠀⠀⢸⠀⠀⠀⠀⠙⠳⠶⠶⠾⠋⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠐⠛⠛⠛⠛⠛⠛⠛⠓⠀⠛⠂




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠤⣤⠃⣤⠤⡄⠀⠀⠀⠀⠀⢀⣾⠟⠹⣿⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣸⣇⡆⠀⠀⠀⠀⠀⢠⣄⣀⣀⣠⣀⣀⣠⡄
⠀⠀⠀⠀⠀⢀⠀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣸⣿⣄⠀⢹⡇⠀⠀⠀⠀⠀⠰⣆⣻⣿⣿⣾⠃⠀⠀⠀⠀⠀⠿⣿⣯⣻⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠘⠛⠃⠀⡇⠀⠙⠛⠃⠀⠀⠀⠀⠸⣷⣾⠿⣿⠻⣷⣾⠃⠀⠀⠀⠀⠀⠀⠹⣿⣇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡯⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⠿⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠆


⠀⠀⠀⠀⠀⢰⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⡆⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⢤⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣤⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⣾⣷⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣿⣶⡆⠀⠀⠀⠀⢸⡿⠟⢠⣤⡀⠿⣿⡆⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⢻⣞⠉⠋⢻
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣷⠀⢾⣿⡇⠀⡿⠀⠀⠀⠀⠀⢰⣿⣿⣿⠽⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⡟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠔⡟⠛⠉⠈⢢⠀⠀⠀⠀⠀⠀⠀⠶⣦⠀⣾⡿⠋⠁⠀⠀⠀⠀⠀⠘⠉⠉⣤⠉⠉⠁⠀⠀⠀⠀⠀⠀⣰⡟⣭⡍⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠸⡀⢰⣦⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠖⠀⣿⠀⠲⡶⠆⠀⠀⠀⠀⠀⢿⣷⣬⣥⣾⣿⣇
⠀⠀⠀⠀⠀⢀⣠⣵⣼⣿⣅⣠⡇⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⠶⠀⠀⠀⠀⠀⠀⠀⣀⣦⣿⠶⣀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠻⠛⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠏⠉⠁


⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡤⣀⡀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣤⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠈⣀⣁⠀⢰⠁⣀⣀⠀⠀⠀⠀⠀⠀⣂⣾⣿⡇⠀⠈⠒⠀⠀⠀⠀⠀⣾⣿⢋⠀⠀⠀⠙⣷⣷⠀⠀⠀⠀⢀⡞⣵⣾⠟⣿⣮⢳⡀
⠀⠀⠀⠀⠀⢠⠈⠋⢀⣸⣐⠉⠋⠀⠀⠀⠀⠀⠀⢃⠉⠉⢱⣶⡶⡀⠀⠀⠀⠀⠀⣨⣿⠰⠆⠅⠠⠆⣽⣅⠀⠀⠀⠀⠸⡇⣿⡄⠀⢠⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠳⢄⠸⡫⠊⠀⠀⠀⠀⠀⠀⠛⠛⣷⣌⣉⣡⣾⠛⠛⠀⠀⠀⠀⠀⠻⣮⣛⣿⣛⣡⠞
⠀⠀⠀⠀⠀⠀⠁⠘⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣶⣾⣿⣿⠛⠛⠛⠛⠛⠇⠀⠉⠁⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⣾⣷⣶⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠀⢀⣠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⢠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠉
⠀⠀⠺⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠤⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣩⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⢠⡦⠀⠀⢠⣿⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⡟⢹⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠈⠁⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡟
⠀⣼⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⢿⣿⣿⣿⣷⣰⣿⣿⣿⣿⣿⣿⡷
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟
⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⢉⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⢣⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⢿⣿⢇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣶⣿⡦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣻⣿⣿⠀⠀⠀⠀⠀⢀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⠉
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⣩⣤⣤⣤⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡆⠐⠀⠉⣀⣀
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠀⠀⠐⠛⠛⠁
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠂
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣾⡤⠴⠂
⠀⠀⠈⠻⠟⢿⣿⣿⠛⠋⠀⠸⠏⠈⢻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠈⠁
⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠋⠻⠿⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⡛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⡇⣿⢸⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡟⢧⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⢹⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⡇⠀⠹⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⢰⣿⢸⣿⣇⠀⠀⠀⠀⠀⠀⣠⠟⠀⢸⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣀⣀⣇⣀⣀⣘⣦⠀⠀⠀⠀⠀⢸⣿⣿⣦⣩⣾⣿⣿⠀⠀⠀⠀⠀⣴⣋⣀⣀⣸⣀⣀⣀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉


⠀⣀⣀⠀⡀⢀⠀⠀⠀⠀⢀
⣿⣯⢹⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣿⣄⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⠓⠒⠛⠋⠉⠉⠙⣿⠟⠉⠉⠉⠛⠓⠚⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠢⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠰⠋⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⢀⣴⠖⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⡀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⣥⣤⣬⣷⣤⣄⣀⡀⠀⣿⡇⠀⣀⣠⣤⣤⣽⣧⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣿⡇⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢁⣤⣤⣶⣶⣿⣷⣶⣦⣤⡀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿

⠀⠀⠀⢰⠛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠆
⠀⠀⠀⢨⠿⢧⠀⠀⠀⠀⠀⠀⠀⢴⡞⠀⠀⠀⣀⣩
⠀⠀⠀⠈⣶⣎⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣾⡀⣛⡿⠻⠷⠶
⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⠻⠛⠛⢻⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠆
⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⡀⢠⣤⡆
⠀⠀⠀⠀⠀⠀⠀⣦⣸⣧⡴⠀⠀⠀⠀⠀⣿⣿⣿⡧⠀⠀⠀⠦⢴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣀⣀⡀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣀⣰⣿⣆⣍⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠀⠀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠉⠛⠻⠿⠛⠉⠀⠀⠘⢧⣴⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢳⡄
⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⣆⠀⠴⠡⢀⡈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠳⣤⠀⠁⠀⠀⠀⠀⠀⠧⠼⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢤⡀⠀⠀⣀⠊⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢠⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢰⣷⠀⠀⠿⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠖⠀⠀⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⡀⠀⠀⠀⠀⠀⠳⠀⠀⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⣀⡤⠶⠻⠆⠀⠀⡠⠁⠀⠀⠀⠀⢰⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠉⠛⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⢻⡓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠠⣿⣿⠆⣄⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠈⠉⠁⠒⠤⡀⠀⠀⠀⢀⠠⠐⠈⠀⠀⠀⠐⢹⡇⠀⠀⠙⠢⣄⡀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢈⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⡄⢸⣷⠀⠀⠀⠀⠀⠉⠓⠤⠂
⠀⠀⠀⠀⠀⠀⢠⠀⢢⠄⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⠇⠀⠀⠀⠀⠀⠠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⠊⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠁⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⢢⠀⠀⢻⣿⡿⠟⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣾⡆⠀⠀⠀⠀⠑⢄⡈⠁⠀⣸⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣭⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⢁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣽⣧⡀⣀⡀⣠⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡿⠡⠁⠀⠀⠈⠂⠈⠁⠐⣠⠄⠀⠀⠸
⠀⠀⠀⠤⢤⡔⠶⠒⠈⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⡐⠁⠀⠀⠀⠀⠂⠰⠋⠀⠠⠠⠄⠀⠼⣄
⠀⠀⠀⠀⠀⠀⢈⣉⣦⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⢐⢀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠏⠀⣌⠀⠘⣦
⠀⠀⠀⠀⠀⠀⡠⠄⢻⠀⢿⣿⣿⣿⣿⠏⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣀⠀⠈⠉⠠⠄⢀⡀
⠀⠀⠠⠄⠀⣁⠀⡀⠀⠀⠀⠹⣿⣿⡟⠀⠰⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠛⠆
⠀⠉⠁⠉⠀⠀⠀⠙⠀⠐⠀⠀⣴⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠸⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠛⠈⠉⠙⠒⠂⠈⠈⠛⢿⡿⠓⠒⠀⠈⠉⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠆⠶⢶⠶⠆⣶⠀⣴⡄⢶⡶⢀⣴⠶⠦⢠⡶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⡁⠀⣿⡀⠀⣿⠉⣿⠀⣸⣇⠘⣧⣀⣀⢈⣛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠁⠀⠈⠀⠉⠉⠀⠀⠉⠁⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⡴⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢀⠀⠀⠀⠀⠀⡁⢈⡄⡈⠀⠀⠀⠀⠀⠰⠒⠂⠖⠒⠂⠀⠀⠀⠀⡈⠲⠞⠁
⠀⠀⠀⠀⠀⢰⠈⠈⠁⠄⠀⠀⠀⠀⠀⠀⠋⠬⠠⡈⠁⠀⠀⠀⠀⠀⠀⠃⠘⢠⡔⠀⠀⠀⠀⠀⡀⠐⢀⡠⠦⠀⠀⢠⠐⡄⠀⢀
⠀⠀⠀⠀⠀⠈⠤⡀⠌⠁⠀⠀⠀⠀⠀⠀⠙⢶⣤⠔⠉⠀⠀⠀⠀⠀⠀⢀⠺⠋⠁⠀⠀⠀⠐⠒⢠⠀⠈⠀⠀⠀⠀⠘⢆⣏⡀⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠔⢀⡀⠀⠀⠀⠀⠀⠀⠀⢴⢒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⠀⠀⠠⣶⠜⠙⢴⠇⠀⠀⠀⠀⠐⣚⠸⢀⡚⠀⠀⠀⠀⠀⠀⠠⠀⣛⡉⠀⠀⠀⠀⠀⠀⠄⠐⣀
⠀⠀⠀⠀⠀⠀⠀⢄⡐⠀⡄⠀⠀⠀⠀⠀⣾⢤⣠⢼⡖⠊⠀⠀⠀⢀⠈⠑⢊⡡⠲⠀⠀⠀⠀⠬⠄⠀⠈⠀⠀⠀⠀⠀⠀⣀⣀⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠦⠌⠉⠀⠃⠀⠀⠀⠀⠉⠠⠌⠈⠀⠀⠀⠀⠀⠂⠀⠰⠀⠐⡄⠀⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢼⠉⠁⡡⠉⠉⠀⠀⠀⠀⠀⠤⠊⢃⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠆⢀
⠀⠀⠀⠀⠀⠀⡀⠠⣠⠄⠀⠀⠀⠀⠀⣹⠀⠖⢉⠲⠇⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠈⠉⠉⡉⢉⡉⠀⠀⠀⢀⠘⢢⠊⠌⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⣠⠀⠀⠀⠀⠀⣺⠀⠀⠚⠀⠀⠀⠀⠀⠀⠒⠂⠀⠂⠒⠒⠀⠀⠀⠀⠤⠓⠄⠐⠂⢠⠀⠀⠸⡄⡸⠀⠀⠸⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠏⠧⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠥⠇⠀⠀⠀⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⢠⡖⢰⣴⠐⣤⠀⠀⠀⠀⢠⠐⣀⡀⢰⠀⠀⠀⠀⠀⣀⠀⣤⠀⣀⡀⠀⠀⠀⠤⠤⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⣷⠀⢥⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠀⢈⠉⠀⠀⠀⠉⡁⠀⠀⠀⠀⠒⠐⠆⢀⠀⠂
⠀⠀⠀⠀⠀⠄⢀⡚⣤⡗⡄⠀⠀⠀⠈⠃⢸⡇⡔⣁⡨⠇⠀⠀⢰⠀⡂⠀⠀⡒⠠⠀⠀⠀⠙⠁⠐⠦⠆⠈⠛⠀⠀⠀⣀⠉⠉⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠛⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠛⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⠀⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠹⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠏⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠙⣧⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣼⠃⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⣾⠷⠶⠶⠶⠶⠶⠾⣷⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠷⠶⠶⠶⠶⠶⠾⣷
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⣰⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣆⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠛⠉⠀⠀⠀⣿⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⣿⠀⠀⠀⠉⠛⠶⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⡴⠏⠻⢦⣤⣤⠾⠃

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⡿⠿⠟⠛⠋⠉⠁⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠙⢿⣷⣶⣶⡿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⡟⠁⠀⠀⣀⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⠀⠀⠀⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⣿⠿⠟⠛⠛⠁⠀⠀⢾⣿⣿⠆⠀⠀⣠⣤⣴⣶⣾⡿⠿⠛⠛⠉⣽⡿⠀⠀⠀⢻⣷⡀
⠀⠀⠀⣀⣀⣤⣤⣴⣶⣿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⠁⠀⠀⣠⣿⠏⠉⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⢻⣷⡀
⠀⣼⣿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⡿⠿⠿⣿⣦⡀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⣴⡀⠀⠀⢻⣷⡀
⠀⢹⣿⠀⠀⠀⣀⡀⠀⠀⢰⣶⡿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⣼⣿⣷⡀⠀⠀⢻⣷⡀
⠀⠈⢿⣷⡿⢿⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⣼⣿⣿⣿⣷⡀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠿⠃⠀⠀⠻⠿⠿⣿⡆⠀⠀⠀⣼⡿⠁⠀⠀⠼⠿⠿⠿⠿⠿⠷⠀⠀⠀⢻⣷
⠀⠀⠀⢀⣾⡟⠀⠀⢀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣾⡟⠀⠀⢀⣾⣧⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⢿⣷⣶⣶⡆⠀⠀⣶⣶⣶⣿⠇⠀⠀⠈⣿⣇⠀⠀⠐⣶⣶⣶⣶⣶⣶⣶⠆⠀⠀⢰⣿
⠀⢀⣾⡟⠀⠀⢀⣾⣿⣿⣧⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠈⠛⠻⠿⠿⠛⠁⠀⠀⣠⣿⠏
⢀⣾⡟⠀⠀⢀⣾⣿⣿⣿⣿⣧⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣤⡀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⣾⡟⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣷⣶⣾⠿⠿⠛⠉
⣿⣇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⡿⠀⠀⠀⣾⡿⠿⠿⠃⠀⠀⠻⠿⠿⣿⡆
⠸⣿⡄⠀⠀⠙⢿⣿⣿⣿⣿⠿⠃⠀⠀⣰⣿⠃⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠙⢿⣦⡀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⣿⡇⠀⠀⣶⣶⣶⡆⠀⠀⣿⡇
⠀⠀⠈⠙⢿⣶⣤⣤⣄⣠⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇⠀⠀⣿⣿⣿⡇⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇
⣶⣴⣶⣶⣦⣦⣶⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣴⣶⣶⣶⣶⣤⣶⣴⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢀⣾⣿⡟⠻⣿⣿⡄⠀⠀⠀⠀⢀⣀⣀⣸⣉⣉⣹⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⠛⢻⣿⣶⡀
⠀⠀⠀⢸⡿⠟⠛⠚⠛⢿⡇⠀⠀⠀⢸⣿⣿⣿⡿⠛⡛⠻⣿⣿⣿⣿⠀⠀⠀⠀⣀⣀⣸⣿⣇⣀⣀⠀⠀⠀⢰⣿⠿⠿⠇⠀⠸⠿⠿⣿⡄
⠀⠀⠀⠈⠀⠀⣀⣀⡀⠀⠁⠀⠀⠀⢸⣿⣿⣿⢠⣴⣿⣤⢸⣿⣿⣿⠀⠀⠀⠀⠿⠿⢿⣿⡿⠿⠿⠀⠀⠀⢸⣿⣀⣀⡀⠀⢀⣀⣀⣿⡇
⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⢛⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠈⠻⣿⣿⠿⠃⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⡿⠿⠟⠋


⠀⢰⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣤⣾⣿⣷⣆
⠀⢸⡇⢰⣦⠀⢀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⡿
⠀⢸⡇⠸⡿⠀⠸⡯⠀⠸⡧⠀⣿⠀⠀⠀⠀⠀⠀⣹⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣿⣿⣷⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠁
⠀⠸⠗⠚⠛⠒⠚⠛⠒⠚⠓⠚⠿⠀⠀⠀⠀⠀⠀⢾⣻⠇⠀⠀⠀⠀⠀⠀⠀⠠⣟⣿⣿⣤⣿⣿⣷⠀⠀⠀⠀⠀⠀⠉⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⢠⣾⣿⣿⣷⣶⢰⣶⣿⣿⣿⣧⠀⠀⠀⠀⣴⣶⡟⢛⠒⣦⠀⠀⠀⠀⠀⠀⠎⠉⣠⡴⣭⡑
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃⠀⠀⠀⠻⠿⣿⣿⡏⠀⣿⣿⡿⠿⠋⠀⠀⠀⠀⣿⣿⣌⠻⠋⣼⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡼⠃
⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢅⡀⠀⠀⢀⡠
⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠠⠤⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣴⠟⢷⡜⠀⠀⠀⠀⠀⠀⣿⣿⠋⣴⡌⢻
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⢱⠀⠀⠀⠀⠀⠀⣿⣿⣧⣍⣥⣿
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⠻⠿⢿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠙⢻⡿⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⣾⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣉⠉⣉⣉⣉⣉⣉⣉⡉⠉⠀⠈⠉⣉⣉⣉⣉⣉⣉⠉⣉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢻⡄⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⢠⡟⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠈⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠃⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡏⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇⠀⣤⡄⠀⣿⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⠃⠀⠀⢸⣇⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⢠⡿⠀⠀⠀⠛⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠘⠃⠀⠀⠀⢻⡆⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⣾⠃⠀⠀⠀⠛⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠘⠃⠀⠀⠀⠘⠃⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠛⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⠛⠛⠛⠛⠛⠛⠛⠛⣻⡇⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⣿⡛⠛⠛⠛⠛⠛⠛⠛⢛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⣤⣴⠾⠋⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠈⠻⢶⣤⣄⣀⣠⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠾⠿⠶⠿⠷⠾⠿⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠰⠆⠰⠆⠰⠶⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠀⠶⠀⠶⠀⠶



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠉⠉⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡷⠀⠀⠠⣴⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣤⣀⣤⣾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠿⢿⣿⣿⣿⣿⣿⣦⣠⠤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠛⢿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⢤⣀⠀⠀⠉⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⣿⠀⢶⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠠⣴⣿⣾⡂⠀⠀⠁⠀⠀⠀⠀⢸⡇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠏⢀⣬⡄⠹⢿⡟⠁⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢶⣄⣹⡏⠀⠈⠛⠛⢀⣤⡄⢇⣀⠀⠀⠀⠀⢴⣿⣿⣿⣦⠀⠀⠀⠀⡄⠈⠀⠀⠀⠈⠀⡿⢃⣤⡀⣀⠀⠀⠀⡴⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠚⣿⣭⣻⡷⠖⣰⢃⣼⣶⣷⡄⠀⢀⠀⢒⠂⢻⣿⣿⣿⡿⠃⠀⠀⠈⠀⢀⣄⠈⠶⠄⠈⠁⣘⠛⠙⠿⠁⠞⣿⠁⠀⢠⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠸⠈⣿⠿⠏⠁⢀⡉⢸⣿⣿⣿⡿⢀⡃⠀⣀⠀⠀⠙⠙⢿⡠⡆⠀⠐⠛⠐⣤⡈⠑⠀⢀⡢⢠⡟⢰⣇⣠⠄⠀⢤⣀⡙⢾⠀⠀⢙⡃
⠀⠀⠀⠀⣠⠟⢠⣶⣷⠀⠐⣶⡿⣧⡤⠉⠛⠉⣠⠘⢧⣀⠁⠰⠐⠆⠀⠸⠗⢀⣠⡄⠀⠀⠀⠉⠀⣀⠀⠀⠛⠠⢾⣯⣿⣾⣿⣷⡄⠀⣠⣦⡀⠀⣛⠄
⠀⠀⠀⠀⠿⠀⣿⣿⢏⡀⢿⣿⣿⣿⡄⠂⠀⢾⠃⠀⢀⠙⠀⣠⣤⡀⢀⣀⠙⠉⠉⠀⠀⠀⠀⠀⠀⠙⠿⠲⠄⣰⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⠏⠀⠀⠃
⠀⠀⠀⠀⠀⢸⣿⠏⠸⡇⢸⣿⣿⠋⠀⢄⠀⠀⠀⣀⠸⣇⠀⢨⣽⣷⣿⣿⣷⡀⠀⡙⠁⠀⠀⠀⠄⢠⢀⠀⢀⡘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡆⠄
⠀⠀⠀⢀⡀⠘⣃⠀⣤⣆⠀⠈⡀⠀⣀⢚⡄⠧⠻⠌⠀⠉⠀⠸⣿⣿⣿⣿⣿⠇⠀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠘⠁⢈⠻⣿⣿⣿⣿⠿⠃⢰⠀⠈⠀⠀⠹
⠀⠀⠀⠈⣇⠈⣿⣷⣈⢻⣿⣠⣤⡀⠈⣉⣁⡀⢀⠘⠓⠚⠛⠀⠙⠻⠛⠛⠁⢰⣤⣴⡆⠀⣬⡃⠀⠀⠀⠀⣰⣿⣿⠄⠈⠛⠋⢉⡴⠖⠘⢄⠀⠄⠀⠈⣤
⠀⠀⠀⠀⠙⠁⠛⠛⠛⠷⣿⣏⡉⢀⡾⢹⡉⢻⡝⠷⠀⢀⢈⠁⢀⣠⣬⠑⠒⢈⡀⠈⣠⣄⠈⠛⠶⠀⢀⡀⠈⠙⠏⠀⣰⣷⠀⠙⢁⣤⣤⣠⣤⡄⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⣷⠀⣾⣿⣿⡿⠘⢧⣈⢁⣼⣯⠑⣀⣤⣾⣄⠘⣿⣿⠃⠀⠘⠿⢾⣿⠏⠀⠞⠀⢠⠞⣿⣄⠀⣠⣶⣿⣿⠀⠀⢻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠙⢻⠃⠀⢀⣉⣉⠀⠉⢸⡏⢸⡄⢻⡄⣈⠁⣠⣾⣿⣿⣿⣷⡆⢸⠀⠣⡈⠛⠛⠁⠀⠻⣿⣿⠟⠀⠀⠄⠙⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⣄⠀⠘⣿⣿⡟⠀⠀⠈⠳⢦⡤⠟⢠⡟⣼⣿⣿⣿⣿⣿⣿⠃⠰⡄⠀⠉⢠⣼⣿⡆⠀⠀⣠⠀⠞⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣬⠓⣄⡀⠀⠁⠘⠀⠀⠐⠀⢠⠀⠋⠀⠈⠻⣿⣿⡿⠿⠋⠀⣠⣴⣾⣆⠙⠿⠟⠁⠚⠋⢩⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢯⣽⡶⠖⠀⠀⠀⡀⣼⡼⠆⠀⠀⠀⠀⣠⣶⣾⡂⠈⢳⣿⣿⠏⠰⣾⣿⣿⠀⣾⣿⡄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢶⡌⢳⠀⠀⢰⣶⣶⡆⠀⢉⠛⠋⠃⠀⠀⢈⡀⠠⣄⠈⠋⠋⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠈⡄⠀⠀⠀⠀⠻⠏⠀⠀⠁⠈⢰⢄⠴⠀⢸⠀⠀⠀⢰⠇⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢠⣶⣥⣤⡀⠀⠀⢌⠡⠇⠀⠀⠀⠀⠈⠳⠄⠀⢠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢿⣿⣿⣶⣤⡀⠀⠀⣴⣿⣿⣿⣷⡆⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⣤⣤⡈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠷



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣵⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠈⠁⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⡜⢩⣦⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠰⢧⣤⣿⣇⠀⡎⢀⠜⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⠀⠀⢠⠃⠈⢿⣿⣿⠀⠳⠧⠤⠊⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⡎⠀⡠⠤⣉⠀⠀⠀⠀⢀⣴⠋⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡸⠀⢘⢀⠔⠉⡇⠀⠀⣼⣿⣿⣷⠯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣰⠃⠀⠈⠓⠒⠊⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠈⠑⠒⠁⠀⣸⠀⠈⠑⠒⠉⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠉⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠒⠒⠒⠒⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣤⣤⣤⣶⣾⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒⠚⠒⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣧⡤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⡹⠀⠈⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⣪⣻⡎⠀⠀⢀⡴⠋⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣒⠭⠗⠋⠁⠀⢳⢳⡀⢠⠋⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢠⡖⡪⠕⠚⠉⡔⣩⠉⠢⠀⠀⠀⢣⢣⣿⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢣⢣⠀⠀⠀⢧⡈⢟⡰⢀⡠⠴⣚⡵⢻⠆⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢯⢧⠀⠀⣀⡬⢖⣪⡵⠚⠉⠀⠉⠻⡄⠀⠈⢧⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⣠⠞⠉⢎⣓⠭⣖⣋⣉⣉⡉⠉⠉⠉⠑⣄⢹⠀⠀⠘⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢀⣼⣁⣀⣀⣈⣀⣀⣉⣉⣉⣉⣀⣀⣀⣀⣀⣈⣿⡧⡀⠀⠀⠀⣉⠢⡀⠀⠀⠀⠀⠀⢀⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠳⣄⠈⢿⣦⣝⣆⣀⣀⣀⣴⣿⡷⠋⠀⠀⠀⠀⣀⡤⠖⠒⢲⠃
⠀⠀⢸⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⡇⠀⠀⠸⡄⠀⠈⠙⣻⠛⠛⠋⠉⠘⢦⠀⠀⠀⠀⡴⠁⠀⠀⣠⠃
⠀⠀⢸⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⡇⠀⠀⢠⠃⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠁⠀⢀⡴⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⣴⠟⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠊⠀⢀⡴⠋
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡗⢤⣀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠤⠤⠖⠒⠋⠁

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡽⠂⠀⠸⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⢻⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢻⣿⣿⣿⣿⣿⣿⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡈⡇⠀⠀⠀⠀⠀⢸⣸⠃⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠙⠁⠀⠀⠀⠀⠀⠘⠃⠀⣠⠃
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣤⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⡀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠍⠉⠁⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠈⠙⠿⢿⣿⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⠐⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠁⠈⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠻⢿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠻⢦⣄⣀⣀⣠⡴⠟⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢀⡀⠀⠈⠉⠉⠁⠀⢀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐⠈⠀⠉⠀⠀⠀⠀⠉⠀⠁⠂⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠎⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⣀⣠⠴⣞⣯⣭⢧⠀⠈⠀⢄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠔⠃⢀⡤⢖⣻⣽⣶⢿⣿⣿⣿⣞⣧⠀⠀⢘⠂⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡆⠘⣆⢿⣿⣯⣿⡎⣿⡿⣟⣻⠼⠂⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⣞⢿⢿⣛⣯⡖⠛⠉⠀⡀⠀⠀⢸⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⠊⠀⠹⠚⠻⠷⠶⠶⠀⠀⠀⠀⠠⡀⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⢨⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠸⡀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⠎⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⠢⢀⠈⠋⠀⠀⠀⠀⠈⠋⠀⠀⡀⠔⠁⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠁⡀⠛⠶⠴⠶⠋⠀⢀⠊⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠁

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣤
⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠀⠰⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⢸⠀⠀⠓⠉⠃⠃⠛⠈⠙⠀⠀⠀⠉⠉⠀⠀⠀⣠⣄⠀⢠⣿⡀
⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠈⠓⠛⠙⠛⡉⠋⠋⠉⡉⠉⠉⠀⠰⠆⠀⠀⠘⠿⠏⠀⠘⠻⠃
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠊⠀⠀⣤⡀⠀⠁⠀⠀⠀⠀⠀⠀⠁⣴⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⡇⠀⠓⠃⠓⠛⠙⠓⠑⠈⠒⠀⠁⠃⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠇⠈⠀⠀⠄⠀⠀⠀⠀⠀⠙⡿⠃⠀⠀⠀⣀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠩⣭⣩⡍⠉⠉⡍⠉⢀⡄⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣤⡠⣠⣀⢀⣄⠠⣴⡾⠏
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢈⡉⠉⠀⠀⠀⠀⠀⢸⠀⢀⣄⣠⣄⣠⣄⣠⡄⣀⢀⣀⡀⢀⢀⡀⣀⣠⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠈⢦⡘⠃⢀⠀⠈⠁⢀⢠⠳⠛⠀⠀⢸⠀⠿⠼⢧⠼⠗⠫⢿⠄⠘⠾⠼⠳⠼⠬⢧⠟⣿⠈⡱⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠐⠚⠛⠛⠛⠂
⠀⠀⠀⠀⡄⠀⠉⠉⠉⠀⠀⠀⠈⢆⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠖⠈⠦⠴⠊⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠁⢰⣀⣀⣀⣀⣀⣀⣀⡀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠉⠀⠀⠀⠠⢴⡂⡦⣠⠔⡄⠒⡄⠀⠤⡄⠀⠆⣤⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠓⠒⠒⠚⠂
⠀⠀⠀⠀⠀⠈⠈⠛⠂⠒⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡡⠬⠔⠲⠊⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⢀⣀⣀⣁⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠄⠐⠂⠚⠉⠀⠀⠀⠀⠀⠀⢀⣿⡧⠉⢠⣿⡀
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠿⠋⠁⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⢠⠤⢄⢀⡀⠀⠀⠀⠀⡀⢠⡶⠀⠀⠀⡄⢠⡀⣠⡀⠠⢄⡄⠀⠀⠀⠀⠙⣿⣧⣀⣴⣿⠛
⠀⠀⠀⠐⠲⠆⠒⠔⠀⠀⠀⠀⠀⠘⣿⠟⠀⠳⠤⠜⠀⠀⠸⠤⠜⠙⠻⠶⣶⡶⠾⠃⠸⣷⣤⣤⣤⣥⣭⣬⣤⣭⣤⣤⣤⣤⣤⡄⠀⠀⠛⠙⠿⠋⠛

⠀⠀⠀⠀⠀⠄⠀⠒⠄⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠢
⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠁⠈⠂⠀⠀⠀⠀⠀⠀⠄⠰⠂
⠀⠀⠀⢀⠀⠁⠐⠀⠑⠄⠀⠀⠀⠀⠦⠀⠀⠐⠂⡀⠀⠀⠀⠀⠐⠀⠅⠀⠣⠀⠀⠀⠀⠀⢂⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠠
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈⠀⢠⢤⡄⣤⣈⠀⠀⠀⡀⢀⠀⠀⠀⢀⠁⠀⠀⠠⠤⣀⠀⠉⢀⣀⣀⠀⠀⠀⢀⠜⠰⠖⠀⠣
⠀⠀⡂⠀⠰⠀⢀⢴⡲⡯⣯⠛⠿⣉⡆⡿⠘⠃⢯⠿⠡⠀⠠⠀⢸⠁⠉⠀⣇⠀⠀⠀⠘⡃⠀⠈⠈⠀⠀⠀⠀⠀⠠⠂⠆⠀⠀⠀⣀⢀⠑⢀
⠀⠀⠇⠀⠀⠀⢸⠛⣿⡗⡧⣍⡒⣀⡏⠁⠀⠀⠀⠂⠔⠀⠀⠀⡼⢤⠀⢀⡀⠂⠠⠀⡀⠀⠁⠀⠀⠀⠐⠘⠁⠀⣀⡇⠀⠀⠀⠀⠀⢀⡍
⠀⠀⠀⠁⡀⠐⠉⡄⠈⠥⠝⠀⠀⢀⢀⡀⠀⠠⠤⠠⡄⠀⢀⠀⢀⠀⠠⠄⠄⠤⠠⠀⠁⢀⡀⠀⠀⠀⠀⡀⡀⠀⠁⠁⣰⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠑⠀⠀⠂⢐⠁⠀⠀⠀⠀⠃⠈⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⡀⠀⠀⠀⢀⡆⢠⣤⡀⠠⠄⠀⠀⠀⡤⠤⠴⠶⠆⠓⠦⠀⠀⣠⠴⣤⣠⡇⢘⠀⠀⠀⠀⢀⣀⢀⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠈⠻⢿⣦⡀⠀⠀⠀⢠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⡀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣤⣾⣿⣿⠇
⠀⠀⣇⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⡇⠀⡎⠀⠈⠀⠸⠄⣈⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋
⠀⠀⠧⠤⠤⠀⠀⠇⠀⠀⠇⠀⠸⠀⠅⠀⠣⠤⠔⠀⠰⠤⠬⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠦⠀⠀⢀⠀⠀⠀⠀⠀⠀⢾⠄⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠈⠀⠰⠉⠀⠆⠁⣸⠆⠀⠀⠀⠰⢶⣶⣶⣦⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠁⢀⠀⣴
⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⡄⠀⠈⢫⢿⣿
⠀⠀⠀⠀⠀⣶⣾⣿⣶⡆⠀⢀⠀⠀⠀⠀⢙⣿⣿⣿⠋⠀⢀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⢸⡀⠀⠀⠀⠀⢸⣾⠃⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⢀⡞⠀⠀⠀⠀⠻⠿⠛⠿⠋⠀⡼⠀⠀⠀⠀⠸⠛⠁⠀⠀⢀⣾⠃⠀⠀⠀⠀⡾⠙⢷⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠤⠒⠈⠉⠉⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣄⣦⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣄⣀⠀⠀⠀⠀⠀⠀⢠⠴⠶⠦⡀⠀⠠⠀⣠⠏⠀⠀⣀⣤⣤⣤⡀⢹⣿⣿
⠀⠀⠀⠀⠀⢠⡞⢻⣏⠀⠀⢰⠀⠀⠀⢸⣿⣇⠆⣽⣿⠀⢰⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⢠⣧⢴⠀⢀⣶⣾⣿⣬⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠟⠃⠀⢀⡞⠀⠀⠀⠀⠉⠉⠉⠉⠉⣠⡿⠀⠀⠀⠀⠐⠤⠤⠤⠂⢀⣾⡟⣿⣆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠤⠴⠞⠋⠀⠀⠀⠀⠐⠀⢀⣠⣤⡶⢿⡉⢸⣿⣿⣷⣦⣤⣤⣤⣶⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠚⣴⣾⣿⣿⠟⠉⢀⣀⣀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠀⣰⣾⢿⣷⡆⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⠀⠀⠀⠠⡇⠀⠀⣶⣶⣶⡄⠀⠈⣿⣿⡏⠀⠀⢸⣿⡿⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠺⣷⣴⣿⠋⠀⢰⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⢰⡇⠀⡄⠀⠀⢹⡟⣿⠀⠀⢀⣿⣿⡇⠀⠀⣶⣿⣷⡄⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢆⡘⡔⠀⠀⠈⠀⠉⠀⣠⣾⣿⣿⣷⢄⠀⠿⠿⠿⠃⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠀⠏⠁⣤⠾⣷⢶⠶⡶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣡⣤⡄⡀⢠⡤⢄⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠃⠀⠐⠀⢀⠀⠀⠀⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇⠈⠉⠀⠸⠀⠈⠉⠀⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⢢⣀⣀⣀⣤⠚⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠙⢦⠀⢠⠞⠁⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠘⠲⢄⣳⣃⠴⠚⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢰⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣰⡇⠀⠀⡀⠀⠀⢸⠀⠀⠀⣀⠀⠀⣷⡀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⣰⢿⢀⡤⠊⠁⠉⠉⠹⠉⠉⠁⠉⠢⣀⢸⢷⡀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⢄⡀⠀⠀⠀⣰⣃⣾⣼⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣸⣇⣳⡀⠀⠀⠀⣀⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠥⣀⠀⠙⣄⡠⠚⠉⠁⠀⠀⠈⠑⠒⠦⠤⠼⠤⠤⠖⠒⠉⠀⠀⠀⠉⠙⠦⣀⡜⠁⢀⡠⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠱⡔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡑⡴⠁⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣜⢰⢁⡶⢠⠇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠢⣤⠒⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢧⠰⡄⢱⠘⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⣠⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠶⠔⠂⠁⠲⢄⠀⠀⠀⠀⠀⠈⠒⢡⡀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡆⠀⠀⠀⠀⠀⠙⠒⠤⣀⠀⠀⠀⠀⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠷⠤⠤⠴⠶⠶⠶⠦⠤⠴⠷⠶⠖⠒⠉



⣿⠿⠿⠿⡿⣟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣿⣟⣻⣿⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠜⠟⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣶⣜⠟⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠄⠁⠐⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠤⡑⠚⡄⠠⣊⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⡙⠛⡄⣰⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣵⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣵⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠐⠀⠈⢆⠀⢀⠄⠂⠀⠐⠠⠀⠀⠀⠀⠐⢄⠘⢇⠀⢀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⢀⣤⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠙⢀⣧⠖⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⡉⠙⢠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⠈⡦⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢰⡤⠈⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⢿⣿⡤⠊⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡆⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⢠⣖⠉⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⢠⣶⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⢀⠀⢀
⠀⠀⠰⠒⠀⠀⠢⡀⠀⠀⠀⢀⣠⡐⠒⣲⠄⠀⠀⢃⠀⠘⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠭⢛⡄
⠀⠀⠀⠀⠠⠄⢀⣄⢠⠄⠀⣬⡟⠉⠉⠈⠳⠀⠀⠈⢆⣀⣤⠊⠀⠀⡠⠒⣆⡀⢀⡀⠀⢀⣔⠈⠑⠢⠀⠸⡉⡡⢔⠵⡎⡄
⠀⠰⠒⠋⠀⠀⠀⢘⠀⠀⠀⠙⡀⠤⣤⡤⠄⠂⠀⠀⡾⣩⣻⡧⠀⡘⠒⡗⠁⠀⠀⠐⠐⠁⠀⠉⢣⠀⠃⠀⢩⡜⡼⢋⣴⢿⡀
⠀⠀⠀⠐⢀⠀⢠⠏⠃⠀⢀⡤⢿⣿⣾⡞⢟⡇⠀⠀⠁⠻⠟⡻⠀⡇⠘⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠄⢰⠀⠀⠏⠀⠀⠑⡡⠁
⠀⠀⠠⠞⠁⠀⠘⠃⠈⠀⢸⠀⢀⣸⣿⣏⠁⡀⠀⠀⠀⠀⢀⠁⠀⠰⠅⢀⣰⠀⠀⠀⠀⠀⠀⢸⠀⢠⠰⠀⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢸⠀⢸⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡟⠀⠀⠀⠀⠀⠀⠘⠙⠉⠀⠀⠀⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⡆⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⢸⡆⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠁⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠈⠀⠀⢰⠀⠀⠈⡇⢸⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠒⢰⠆⠀⠀⠀⠀⠀⠇⢀⠀⠘⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠘⠛⠛⠛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⡛⠛⠃⠻⣿⡿⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢠⣧⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠛⣋⣛⠛⠛⠛⠛⠛⢛⣋⣡⣴⣿⣦⣉⣉⡛⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠛⠻⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠛⠋
⠀⠀⠀⠐⠲⢶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⣿⡿⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⢡⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⣌⠻⢋⣴⣾⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⢀⣤⣶⣿⣷⡌⢿⣿⣿⢋⣴⡌⢿⣿⣿⢋⣴⣿⣷⣦⣄⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⣴⣿⣿⣿⣿⣿⣿⣦⠙⣡⠘⠿⠟⣠⠙⢡⣾⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⠘⣿⣿⢸⣿⣿⣿⣿⡟⢸⣿⣷⠸⣿⣿⣿⣿⣿⢸⣿⡟⣰⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⣹⣿⢸⣿⣿⣿⣿⡇⢾⣿⣿⠄⣿⣿⣿⣿⣿⢸⣿⣠⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣦⡙⣡⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢤⡤⢠⡄⢠⡄⡄⢀⡤⠄⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⡀⢸⡇⢸⡟⢻⡇⡇⢿⣀⡀⣙⣳⠄
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⢀⣴⠀⠀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⡀⠀⢸⢹⣈⡃⣀⣷⠀⠨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣜⠷⣟⡬⣃⣼⣿⣿⣿⣻⣽⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡯⣭⢞⢽⢳⣷⣿⣿⣿⣿⣿⡿⣿⣽⡀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⢬⣾⣿⣿⠿⠿⠛⢿⣿⣿⣦⣿⣽⡄⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠉⠻⣿⣿⣿⠃⢀⠀⠀⠀⢈⠛⠿⣿⣿⣿⣦⣇⡐⣀⢡⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⢴⣂⡈⠁⠠⣿⣿⠁⠀⠈⠉⠀⠈⠁⠀⠀⠈⣿⣿⣿⣻⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿⠉⠉⠁⠀⢾⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⢉⣹⠈⢠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡧⠂⠾⠿⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣿⣿⡁⣨⡀⠘⠇⠀⠳⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠹⢟⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣠⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢈⣧⡄⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⢹⡏⠩⠙⢳⡄⡨⢁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠈⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠈⢌⠸⠋⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡛⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⡆⢸⠯⡀⠀⣙⡤⠀⠀⠀⠀⠠⣄⣀⣤⣀⣀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣇⣿⣿⣤⢸⣞⠆⠆⣹⡇⠀⠀⠀⠀⠀⠹⣟⡻⢾⣿⣷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⢀⣴⣌⣿⣟⣛⣩⠟⣾⡤⠖⠒⠓⠯⣌⠀⢀⠀⣼⣣⣿⣽⣿⣿⠄⠀⢐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣧⣿⣼⡿⢿⣿⡻⣶⣶⡗⠚⠂⡆⢀⠲⢶⣿⣿⣿⢿⣽⣿⣇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠀⣠⠤⣶⣿⣿⣿⣆⣀⣽⠛⠉⠉⢯⠹⠁⠓⡲⠖⠏⢴⠞⢃⡼⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡶⠟⠲⢭⡿⠋⣹⣿⣿⠂⠀⢠⣶⣾⣶⣦⣤⣄⣨⡤⠤⠴⣿⡀⣱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡞⣿⣔⢙⣀⣨⣼⠛⠋⣹⣷⠖⠉⠁⠈⠹⣏⠻⣿⡇⠀⠀⠀⣀⡼⣿⣿⠟⠂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣃⢹⣍⣿⣭⣤⠟⠢⢀⣿⡅⠀⢸⡇⠀⠀⢹⣆⣯⣤⡴⠶⢾⡁⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣯⢘⣿⡇⠤⠄⠀⣀⠞⢿⡧⠹⣿⡇⠀⠀⢸⠟⠉⠀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠃⠀⠀⠀⢀⣾⢇⠘⣏⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣴⡿⣸⠄⠙⣤⣤⠀⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢁⡾⠀⢷⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣶⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀
⠀⠀⠀⠐⠒⠒⠲⣉⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣾⣇⡷⠀⠀⠀⠀⠀⠀⠲⢦⠀⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠻⢦
⠀⠀⢠⣤⣤⡄⠀⣿⠀⠀⣤⣤⡄⠀⠀⠀⠀⠐⢦⣸⣿⠿⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠓⢰⣶⣤⠄⣸
⠀⠀⠀⠉⠉⠀⠀⣿⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⠺⣦⣤⣿⣿⣶⣷⣦⣤
⠀⠀⠀⠀⠀⠴⠿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠻⢿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡖⠒⠲⣆
⠀⠀⠀⣴⣶⣤⣴⣶⣦⣤⣴⣶⡀⠀⠀⠀⠀⣰⣿⣽⣿⢻⣿⣝⣧⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠈⣿
⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⣿⣿⣯⠁⠀⢩⣿⣿⡆⠀⠀⠀⠀⢸⠀⠀⠀⠛⠁⣠⡀⠛⠀⠀⠀⠀⠀⢹⡟⠛⠛⢣⣤⣤⣤⡏
⠀⠀⠀⠀⠈⠻⢿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⢻⣟⢿⣶⣶⣾⣿⡿⠁⠀⠀⠀⠀⢸⡄⠀⣤⣤⣬⣭⣥⣤⠀⠀⠀⠀⠀⠀⠳⣄⠀⢸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠷⠾⠻⠉⠀⠀⠀⠀⠀⠀⠀⠓⠶⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⠟⠁


⠀⠀⠀⠠⠀⢔⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣛⡻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣴⣶⡀
⠀⠀⠀⠶⠆⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣽⣅⣭⢻⣿⡄⠀⠀⠀⠀⠀⠀⣦⠀⣸⣿⣿⠧⢠⡆⠀⠀⠀⠀⠀⣠⣼⠟⠉⡀⡈⠻⣿⠆
⠀⠀⠀⠀⠠⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣾⣿⡷⠀⠀⠀⠀⠀⠀⣿⣦⠛⣿⡿⢀⣾⡇⠀⠀⠀⠀⠀⠛⣿⠰⠆⢀⠰⠆⣻⣦
⠀⠀⠘⠻⢷⣾⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠐⣿⢦⣘⣛⣃⣴⡏⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠿⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠉⠋⠁


⠀⠀⠀⠠⠤⠤⣴⣶⡦⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣄⡀⠀⣀⣴⣶⡄⠀⠀⠀⠀⢰⣶⣶⣶⢶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠻⣿⣿⣯⣷⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠿⠿⠀⢸⣿⡇⠀⠿⠿⠃⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠙⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⣾⣿⣯
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣶⠙⣦⢤⡀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇⠀⡗⢢⡤⣤⠞⠉⠀⠀⠀⠶⠶⠞⠳⠶⠶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣼⠀⣿⠀⡇⠀⡇⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠈⠀⠁⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠀⢸⠶⠶⠤⣀⣀⠀⠀⠀⠀⠉⠉⢉⣻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⠉⢹⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣼⣤⣄⣀⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠦⠤⠤⠄⠀⠀⠀⠀⠀⠉⠓⠒⣿⠀⠀⠀⠀⠀⠀⢠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⡄⢠⡀⢸⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠶⠰⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⡇⢈⡀⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⣶⠒⠒⠒⠀⠀⣀⡴⠋⠉⠙⠒⣇⢀⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠣⠼⠓⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢸⠗⢦⡀⠀⠀⢤⣰⠈⣿⡿⣦⣀⢀⡀⢀⠁⠀⠴⠶⠶⠄⠐⠂
⠀⠀⠀⠀⠀⠀⣀⣤⡴⢶⠇⠈⠸⠦⡀⠈⠁⠈⠁⠀⢀⣿⣿⢂⡙⠻⠍⠁⠠⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⢠⠞⠉⠀⠤⠂⣨⣭⣍⣀⠀⠀⠀⠀⡀⠂⠹⠉⠀⢀⠀⠀⠀⠀⠠⣠⣶⣾⣿⢷⡷⠚⠿⣷⠀⢀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⢴⣶⠋⠀⠈⠟⠋⢹⣧⣶⣅⡀⠀⠀⢸⡀⠀⠸⠀⠀⠀⠀⠀⡽⠁⠀⠀⠘⠁⣀⣀⣨⣷⠿⠴⠚⠁
⠀⠀⠀⠀⠀⠘⠛⣟⣰⣆⠀⠀⠀⠀⢻⠏⣷⠀⠀⠚⠈⢷⡀⠀⠁⠀⠀⣀⠤⢀⢚⣽⣂⠖⢈⣥⠌⠋⠁⠀⠀⠀⢀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠤⣀⠀⠀⠀⣰⡏⠀⠀⣠⢤⡌⢻⡖⠹⠉⣭⠶⣾⠈⠿⢃⣴⣎⢋⠀⢂⠠⠐⠂⠀⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠁⣆⢠⣚⣛⠷⠒⣋⠷⡆⠠⡵⠖⠀⢮⡵⠚⠍⡝⠚⠿⠛⠃⠀⠐⠁⠀⠀⡶⠂
⠀⠀⠀⠢⣀⡀⢸⢀⣙⠁⠠⠾⣈⢀⠉⠚⠛⣋⣠⣶⡇⠄⠀⠒⢋⡘⢶⠉⠀⠌⠉⢀⠰⣿⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢰⢲⠒⠒⠒⡤⢰⡅⠀⣸⡟⠛⠋⠀⢀⣨⡴⢟⡫⠱⡉⠴⠏⠠⠔⠻⠰⣿⡇⠩⢦⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠆⢠⠀⠀⢠⠇⠀⢷⣿⢿⡴⠦⠤⠖⣋⣡⠃⠀⠀⠀⠀⠀⠀⡀⠒⠜⢷⡌⠀⣀⠘⢧
⠀⠀⠀⠀⠀⢘⣤⡦⢤⣶⣢⡄⠀⢀⣀⣼⠗⠒⠖⠄⠩⡖⠒⠀⢤⣤⠖⡰⠋⠀⠁⠉⠁⠈⠉⠉⠹⡆⠄
⠀⠀⠀⠀⠀⢀⣾⡷⠛⠯⠟⣒⣺⣏⠉⡏⢁⠀⠀⡨⢤⠃⠀⢀⣸⡉⣻⠔⠋⠉⠉⠉⠓⠥⠦⠤⠚⠁⢺⡄
⠀⠀⠀⠀⠀⠈⡼⠀⠘⠿⠿⠛⣉⣭⡾⠷⠾⠒⠊⠉⠁⠀⠉⢸⠇⢀⡠⠀⠂⠉⠀⠀⠈⢦⠀⡀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠠⢶⣦⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢧⠀⠀⠀⠀⠈⠀⠻⣧⣧⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡦⠌⢂⠀⠀⠀⠀⠀⠀⠘⢿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣮⠥⠆⠀⢳⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⢄⣀⠀⠀⣀⣤⣴⡶⠛⠋⠁⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠈⠉⠛⠟⠋⠈⠙⢦⣀⠀⠀⠀⠀⣀⠀⠘⢆⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⢦⠈⠉⠀⠹⣷⡄⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠓⠲⠀⢀⢀⣀⠠⢤⣤⠶⣛⣀⠀⡆⢤⡴⠉⠁⠀⠘⢷⣄⣠⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣌⣁⡘⠛⠋⠀⠀⠈⢢⡙⠛⢠⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠀⠀⠠⠼⣿⠟⠀⠀⠀⠀⠀⠀⢙⢀⣙⣉⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡛⠌⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠁⢠⡞⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⣀⣀⣤⡾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡡⠄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⡀⠠⠄⠀⢨⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⢀⣾⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⡏⠩⠿⠿⠿⠿⠄⠀⠀⠀⠠⠶⠒⠒⠶⠄⢸⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣄⠘⡀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢇⣠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣷⣄⠀⠀⠀⠉⠍⠍⠀⠀⠀⣠⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⠈⢉⠛⠛⠛⠷⣤⡀⠀⠀⠀⢀⣤⠞⠛⠛⠉⠁⠀⠀⠀⠀⠀⢀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⡌⠱⡏⢣⠋⢣⠀⠀⠀⠈⠙⠻⠶⠟⠋⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠉⠓⢄⠀⢨⠀⠸⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⢸⣿⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢓⡂⢀⣿⣻⣦⡗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠲⢦⣄⡈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣂⣉⣉⣁⠟⠫⣸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠙⢸⣈⠔⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⣤⣤⡌⠙⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢠⣀⣠⠀⡄⠀⣄⣀⡄⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠢⠤⠴⠂⠀⣠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣏⣀⣀⣀⣹⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡆⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢻⣸⠇⠀⠀⠀⠀⠀⠀⠀⢸⣠⠇⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠙⢦⣀⢀⣀⣀⣀⠀⢀⡼⠋⠀⠀⣠⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠙⢦⣀⠀⠀⢉⣙⠒⣄⠉⠛⠉⠀⠀⣀⡴⠃⠈⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠙⠓⠲⠦⣄⣈⠛⠒⠯⣌⣉⣁⣀⣠⡤⠶⢚⣁⣠⠤⠴⠖⠒⠂⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⡀⢀⣠⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠷⠶⠤⠤⠤⠶⠶⠖⠚⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⣀⠀⣀⣀⣀⣠⢶⣀⠤⠔⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⠉⠙⣹⣿⣿⢏⡿⡧⠿⠰⠈⣰⡦⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡿⣭⡫⠁⣥⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢻⡀⠀⠀⠈⢿⣹⣹⣹⢇⢘⣢⢾⣹⣿⣿⣷⣆
⠀⠀⠀⠀⠀⢠⣿⢿⣿⣗⣀⣿⡀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⡟⠁⢀⠀⢀⠆⠈⢯⠀⠐⠦⢤⣛⡛⢻⡿⠋⢷⣿⣿⢛⣽⣿⣿
⠀⠀⠀⠀⠀⠾⣏⠥⠡⠖⢻⣯⣦⡤⠤⠴⠞⠉⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⣴⠇⠀⡄⠈⢧⡀⠀⠀⠭⢃⡼⠤⠃⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⢶⣶⠖⠒⡟⠁⠀⠀⢴⠗⠀⠃⠀⠀⠀⠀⠀⡀⡟⠀⠀⡘⣀⢀⠀⢹⠀⠸⡍⢦⢤⠤⣼⡀⠀⠠⠚⢿⣋⠙⠓⠛⠟⠃
⠀⠀⠀⠀⠀⠀⣿⢿⣿⣟⣧⣔⠂⡴⠃⢀⡆⣦⠀⠀⠀⠀⠀⠐⠀⠀⢰⢁⢹⠈⠀⠘⠁⢰⡗⢄⣀⠀⢤⠑⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣋⣫⣹⣇⢀⡞⠀⢻⠀⠀⠀⠀⢨⠀⠀⢀⣾⣌⣈⠀⠀⢀⡔⢘⡇⠈⠻⣄⠀⠸⣀⣴⣤⣤⣀⡀⠑
⠀⠀⠀⠀⣀⠴⢿⣿⣿⣿⣿⣟⠟⡡⢂⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⣸⣿⣾⣿⣷⡄⢈⡈⠹⠁⠄⠀⢈⠑⣌⣟⢿⣿⣿⣦⣿⡆
⠀⠀⠀⠁⠀⠀⠀⠐⠻⣿⠞⣡⡾⣡⣾⠀⠀⢼⡖⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣋⠃⠀⠀⠀⠀⠈⢳⡈⠳⣿⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠞⠁⠚⠙⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠀⠙⣿⣿⣿⣿⣿⣿⣿⣦⡶⣶⣷⣾⡙⣦⣌⢻⡙⠉⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣛⣟⡿⡆⠐⢽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠿⢿⣿⣿⣿⣿⡿⢿⣟⡟⠁⢸⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠰⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢀⠀⣧⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⢀⣴⡏⠀⣯⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢀⣀⣤⠿⡟⠀⢷⠏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⢋⠵⠊⠀⠀⢸⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠻⠍⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⠿⠿⠟⠛⠛⠻⠿⠿⠿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠘⣿⣷⠀⠀⢀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣴⣿⡏⠀⠀⠀⠀⠀⠀⠹⣿⣧⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣷⣶⣶⣤⣤⣶⣶⣾⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃⠈⠉⠛⠛⠛⠛⠛⠛⠉⠁⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠰⣿⣷⣦⣄⡀⠀⠘⣿⣷⡀⠀⠀⢀⣀⣤⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣀⡀⠀⠀⠀⣼⣿⠏⠀⢀⣠⣤⣾⣿⠆
⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣾⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⣠⠀⠂⣈⢂⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⠀⠒⢀⠀⠒⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢢⡀⠠⠄⢀⡌⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡊⠀⠺⡀⢀⠗⠀⢑⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠐⢰⡦⠂⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⡜⣏⠄⠂⠀⠘⠃⠀⠐⢄⢹⢧⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠢⡀⢀⠼⠔⠓⠢⠄⡀⠀⠀⢀⠠⠔⠚⠢⠧⡀⠀⠐⠘⢧
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠑⠤⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢸⡇⠄⢆⠂⠀⠀⠀⠀⠀⠀⠀⠉⠐⡌⢁⢀⡀⡀⠀⠀⠀⠀⠐⣰⠠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠑⠠⢀⠀⠀⠀⢙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠐⠒⠒⠒⠒⠚⠒⠚⠂⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡹⢿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢧⣀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⣘⣿⠿⠶⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠸⠿⠀⢄⣉⣿⣿⡃⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⠀⠻⠿⠿⠿⠃⢀⠀⠀⠈⢛⣯⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⣾⣦⣤⣤⣤⣼⣿⣿⣿⣷⣦⠔⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠿⣿⣦⣴⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠙⠛⠁⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⠙⣿⣿⡀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣠⡀⠀⢿⣿⣿⣿⣿⣿⣷⡆⠀⠰⠂⠀⠀⠸⣿⣿⣿
⠀⠀⠀⣸⣿⡇⠀⠹⣿⣧⠀⠀⠀⢠⣴⣿⡿⠟⠿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⡟⠀⢰⣿⣷⠀⠘⣿⣿⣿⣿⠛⠁⢀⣠⣄⡀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠈⠋⠀⠀⠀⠉⠁⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣷⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⣠⣄⠀⣿⡀⠀⠀⠀⡠⢺⠆⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⡀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⢀⡀⠙⠋⢀⡘⠇⡴⠒⢲⠀⣿⣀⠤⢄⡀⠀⡠⠤⠤⡀⠰⡹⡄⡘⣷⡀⢠⣤⠀⢾⡷⢀⢃⢀⠅
⠀⠀⠀⠀⠀⠀⠈⠰⣔⠶⡁⠈⢲⠃⢀⡠⠾⠟⠀⠠⠀⢱⣼⠀⠀⠀⠘⠆⠼⢃⠀⣿⡇⣀⡄⠀⠀⠀⣸⣸⠏⠀⠄
⠀⠀⠀⠀⠘⠇⠀⠐⢸⠛⠛⠢⣬⣧⡎⠀⠀⡜⠂⠀⠀⣼⣿⣄⠀⠀⠚⡄⠀⠈⣇⣿⢏⡼⣥⠋⠉⣴⣵⠏⠤⠂
⠀⠀⠀⠀⠀⢀⠤⠊⠉⡕⠂⠉⡝⠁⡣⠀⢀⡅⠀⠠⠊⢸⣿⠈⠂⠀⠀⣄⠀⠐⠁⠙⣟⠀⠀⠀⠀⠋⠉⢱⠀⢠⣄
⠀⠀⠀⠀⠀⠈⠙⠻⢿⢷⡖⠈⠁⠀⠀⠀⠘⡀⠀⡀⠀⣿⣿⠇⠀⡀⢀⠄⠀⠀⠁⠐⠈⠑⣤⣄⠀⠀⠐⠣⡀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣊⣴⢿⠀⠀⠘⣤⣀⠀⢐⠁⠈⠁⠀⢸⣿⠀⠀⠀⠀⠱⠀⢀⠤⡈⠀⠀⡸⠋⠀⠠⣀⣠⣃⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠛⠉⠁⡜⠳⢄⣘⠀⢨⡀⠀⠐⠂⠀⠀⢾⣿⠆⠀⠄⢢⠀⠀⡇⠀⢘⡀⠔⠙⡄⢦⣀⣨⣭⠟⢩⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠄⡀⠘⠀⠐⠄⠀⢸⠀⠀⠀⣸⣿⡄⡀⠀⡼⠀⠀⠆⠀⠈⠀⡤⠐⡧⠚⠫⡒⠲⡀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⢻⣿⠋⠀⠀⠘⢄⠀⠀⠀⠁⠈⠀⠀⣅⡰⢦⠍⣉⣀⡀⣠⠄⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡆⠀⠀⢀⠀⢀⠎⠀⠒⠄⡴⣿⣿⡷⠤⠐⠂⠈⣆⠀⣀⡃⠀⠘⠑⠋⢷⣾⡿⠟⣻⣿⣋⣀⣀⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠐⠀⠉⠉⣄⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣀⡋⠉⠀⠝⠈⣤⣅⠤⠈⠁⠉⢫⡉⠉⠐⠶⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⠔⠉⠈⠑⢆⠀⠐⢻⣿⠓⠀⢠⠊⠀⠈⠱⢄⣀⡤⠻⠛⠢⢄⣸⠷⣦⣘⣄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠀⠁⠀⠀⠈⡄⠀⢸⣏⠀⠀⠆⠀⠀⠀⠀⢠⣉⠀⠀⣇⢀⣠⡦⣙⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣄⠀⠀⢀⡜⠘⣄⠀⠀⣠⠤⠄⠒⠁⠀⠉⠍⢩⡅⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠈⠉⢛⣀⠀⠀⠀⠀⠙⠓⠂⠠⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣶⣓⠉⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠻⡛⠛⣳⣤⠀⠀⠀⣠⡆⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⡤⠒⢤⡙⡄⠀⠈⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡉⢠⣇⠀⠀⣿⡾⣿⡿⠦⠀⠀⡘⠂⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⣀⣨⢿⠆⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⢸⣿⣄⣿⣃⣤⣌⡑⢬⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡿⠂⠉⢀⠊⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠛⠣⣤⠈⢙⣿⡟⠿⠻⣧⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠉⢡⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠆
⠀⠀⠀⠀⠀⠀⢀⡞⠁⡀⢠⡴⠀⠰⠾⠛⠃⠠⠠⡌⠢⠈⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⣄⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⢡⡀
⠀⠀⠀⠀⠀⢠⡝⠁⠀⠁⠈⠀⠀⣠⢤⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⢃⡴⣶⠆⠀⠘⡿⢀⠁⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⢀⣾⣛⣷⠄⠀⠀⠀⠀⠉⠀⠀⠀⣠⣾⣈⡗⠈⠿⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠃⠀⠀⠀⠉⠰⠀⢀⣤⣤⣄
⠀⠀⠀⢀⣿⡟⠉⠰⠄⠀⠀⠀⠀⠀⠐⠀⢳⣿⡤⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡜⠃⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⣶⠀⠀⠋⢿
⠀⠀⠀⢹⣯⡆⢀⣀⣤⣀⠀⡐⢋⣀⠀⣠⣾⣿⡷⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⢰⠆⠀⠀⠃
⠀⠀⠀⣿⣿⡷⢛⠋⠀⢀⣾⣿⣿⣿⣆⣿⣷⣿⣟⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠠⢘⡏⠹⣿⣿⣷⣶⣿⣿⣟⣻⣿⣿⣿⠿⣿⣯⣿⣿⠆⠀⢤⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⢸⣿⣆⢻⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠨⠟⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁
⠀⠀⠸⣿⣿⣇⠿⠛⢻⠉⠉⠉⠀⢀⠶⠄⠀⠀⠐⠆⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⢀⣾⡔⣧⡈⠀⠀⠀⠀⠀⡔⠉⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠇
⠀⠀⠀⢹⣿⣿⣿⣿⡏⠀⣿⣷⡘⠀⠀⠀⠀⠀⣾⣷⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠶⠉⠀⠸
⠀⠀⠀⠈⣿⣿⣿⣿⣷⠀⠿⠿⠛⠾⠁⠀⠀⠈⢻⣾⣯⣰⣀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠛⠋⠑⠶⡶⣶⠀⠀⠀⣠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⢀⠀⠀⣠⣿⣀⡄⠀⠰⢆⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣤⠈⠀⠀⠻⣿⣏⣠⡴⣆⣹⠦⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣀⠤⠀⠀⢸⣿⡿⢿⡋⢉⡲⣏⣥⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⡀⠀⡀⠀⠈⠻⠶⠤⠄⡴⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡈⠙⠿⣿⣿⣿⣷⣎⣥⠺⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠐⣚⢁⡄⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡟⢻⠏⣽⡏⣽⣿⣿⣶⣶⣿⣿⣿⣿⣟⣿⣿⣦⡆⠐⠄⠀⠀⠀⠀⠰⢦⡄⣽⣶⣿⠟⣶⣶⠆⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠀⢿⡇⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠂⠀⠀⢳⣾⣇⠙⢰⡿⠘⢣⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠈⠁⠛⢁⣿⣿⣿⣿⡿⠏⠁⠈⠉⠙⠛⢻⣿⣿⡧⣀⣔⢃⣼⣀⠿⡾⠀⠀⠀⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⢿⣿⣏⣀⣀⠀⠀⠒⠤⠤⠀⠈⠋⠁⠀⠛⠁⠀⢹⣿⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⠀⠈⠉⠙⠓⠒⠒⠶⠒⠲⠶⠂⠒⠒⠂⠀⠀⢹⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠑⢄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠃⠀⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣧⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠸⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠾⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠋⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠈⠙⢦⡀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠈⠻⠛⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⢀⣴⠄⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡶⠦⣤⣠⠏⠙⢦⡀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁⠀⠀⠀⣀⣴⡯⠇⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣚⡭⢶⡹⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠉⠉⠉⢲⠀⢀⣠⠴⣒⡭⠵⠚⠉⠀⠀⠀⢳⡹⡄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⢀⣠⣾⣿⣶⣶⣶⣶⣶⣶⢟⣪⠭⠖⠋⢁⠞⡥⠉⠳⡄⠀⠀⠀⢷⢱⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣇⠀⠀⠀⠸⡀⠉⣹⢀⡇⠀⢀⣠⠼⣃⡷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣀⣿⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⡍⠉⠉⠉⠉⠹⡜⣆⠀⠀⠀⠈⠓⣚⡭⢖⣚⠭⠖⠋⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠏⠀⠀⠀⠀⣠⢽⡜⣆⣠⠴⣚⣫⣵⢾⣭⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠻⠬⠖⢻⣷⣶⣶⣶⡆⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠹⠿⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠿⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⡡⠤⠁⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⢠⡃⠀⠐⢂⠂⠉⡂⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠂⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⡆⢁⠇⠈⡀⠀⠉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢠⠠⢂⠔⠁⠀⠀⠀⠀⠀⠉⠉⠛⠚⠂⠙⠒⢤⣤⣌⡦⠤⠤⠤⠄⠀⠀⣀⠀⠀⠀⠀⠑⠢⠄⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠒⠄⠂⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠖⠲⠲⡖⢤⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⠌⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⡌⠀⠰⠀⣈⣶⠛⣆
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡂⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠀⢸⠉⠉⢹⡉⠀⠸⡄⢸
⠀⠀⠀⠀⠀⠸⡧⡂⠀⠀⠀⠀⠀⠀⢀⡏⠠⠀⠀⠀⠀⠀⠀⠧⠈⠁⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠂⠁⢸⠁⠉⠒⣧⡸
⠀⠀⠀⠀⠀⠀⠱⣌⠢⠤⢀⡀⢤⣶⠭⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣆⠀⠈⡀⠀⣸⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠘⠢⠵⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡠⠄⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠒⠀⠀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠑⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠟⠋⠁⠀⠀⠀⠀⠀⠈⠙⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣑⣤⣤⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠀⡂⠘⠈⠀⠁⢛⠀⠀⠀⠀⠀⠀⣤⠤⠬⡆⠀⠀⠀⠀⡀⠀⠀⢠⠥⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⢈⣀⡀⠀⡰⠃⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⡀⣸⠀⣇⢀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⣣⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⢀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢛⣀⣀⣀⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡠⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣾⡿⠋⠉⠉⠙⠉⠉⠙⠻⢿⣧⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⠏⠀⠀⢠⣤⠀⣤⣄⣀⡀⠈⡏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⠀⠐⢢⣦⠀⠀⠘⢋⣯⣹⠆⡏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⢀⣤⣄⡀⠀⢠⡞⠗⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡤⠞⢩⣥⣆⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠛⠋⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⠛⢿⡻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠄⠀⠀⢤⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠐⠈⠚⢻⡗⣿
⢠⣄⠀⠀⠀⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠣⣼⢿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡴⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⢷⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⡄⢀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟
⠀⠀⠀⠃⠀⠀⠀⠀⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢀⠄⢾⣿⠀⠸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣀⠠⠤⠒⠋⠀⠳⡀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⠚⠙⠓⠂⠀⠀⠐⠒⠂⠘⡶⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⣽⠀⠈⠉⠁⣸⠀⠈⠉⠁⢠⣳⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⢧⠀⠀⠀⠁⠀⠀⠀⠀⡜⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣨⠷⣄⠀⠉⠉⢀⣠⠾⣄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⠋⠀⠐⠾⡉⠑⠂⢉⠗⠂⠀⠙⡆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠚⠂⠀⢄⡀⠘⣆⣰⠃⢀⡀⠀⠐⠓⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⠑⠾⠧⠊⠁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⡞⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢳⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⢀⡞⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢳⡀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠒⢤⡀⠀⠀⣠⣮⠤⠶⠓⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠶⠤⣵⣄⠀⠀⢀⡴⠒⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢠⠿⠲⢤⡀⠀⢹⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡏⠀⢀⡤⠖⠿⡄
⠀⠀⠀⠀⠀⣰⢃⠀⠀⠀⠘⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠃⠀⠀⠀⡙⣆
⠀⠀⠀⠀⠀⡏⡎⣸⢁⡶⢠⠇⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠲⢤⠖⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠸⡆⢶⡈⣇⢱⢹
⠀⠀⠀⠀⠀⢳⡄⡇⢸⠱⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠛⠓⡦⠴⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠳⠎⡇⢸⢠⡞
⠀⠀⠀⠀⠀⠀⠉⠳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⠀⠀⠈⠑⠢⣄⣀⠀⠀⠀⠀⠀⠈⢊⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣵⣤⣤⣤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠛⠙⠛⠛⠶⣤⣀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠛⠛⠛⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⢠⡶⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣆⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢠⡟⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠻⣏⠀⠀⠀⠹⣆⣀⣀⣀⣀⣀⠀⣿⢸⡇⠀⢀⣀⡼⠁⠀⠀⠀⠹⣆⡀⢀⣠⣤⣄⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠈⠉⠉⠉⠉⠙⣷⣿⢸⡇⣼⠟⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠉⠛⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠸⠿⢸⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣆⢢⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⡀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠈⢎⣮⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣥⡴⠆⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠀⡤⠀⠀⠀⢦⡈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⠀⠀⠀⠀⠠⣦⣤⣰⡀⢀⡀⠀⠀⠀⠀⠀⠐⣌⢻⣶⡿⡿
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣄⡀⠀⠀⠤⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣝⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢋⣠⣀⠀⠀⠀⠀⠀⠙⠟⣷⡤⠒⠀⠀⠀⠀⠀⠀⠤⣄⡙⠷⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠒⠀⠀⠀⠀⠀⠀⣰⡖⠀⠀⠀⠀⠀⠀⠀⣀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⢀⣀⣠⣞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⠠⣀⠀⠒⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡰⠀⠀⠀⠐⢓⣄⢬⡅⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⢞⡥⠂⠀⠀⡶⠀⢡⣦⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡏⠆⣹⠃⡀⠀⠀⢾⣶⠘⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠱⡴⢡⡾⠁⠀⠀⣬⢱⡄⢳⡄⠀⠀⠀⠀⠀⠁⠠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠣⡧⢿⠀⠀⠀⠀⠉⠈⠁⡀⠔⠀⠀⠀⠀⠄⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣃⡇⠀⠀⠀⠀⠀⠀⠈⡟⠃⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⣤⣤⣤⠤⠤⢤⠀⠀⠀⠀⠀⣄⣀⣀⣀⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⢰⠃⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⠤⠤⣄⠀⠀⠀⠀⠀⠀⡇⠀⢾⣉⣤⣀⣉⡷⠀⢸⠀⠀⠀⠀⢰⠃⠉⠉⠁⠀⠀⠉⠉⠈⢧
⠀⠀⠀⠀⣠⠤⠤⡀⢠⠇⢸⠀⠀⠀⠀⠀⢀⣀⣿⣀⡴⣠⣶⣠⣼⣤⡀⠀⠀⠀⠀⡇⠠⠒⠛⠛⠛⠉⠀⢤⢸⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠀⠀⠀⠀⡼
⠀⠀⠀⠋⡁⠀⢀⡙⠚⠁⢸⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠛⢈⡠⠋⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⢸⡗⠒⠒⠚⠒⠒⠒⢾⠁
⠀⠀⠀⢀⡉⠉⠉⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣄⣀⣸⠀⠀⠀⠀⠀⠀⣧⠀⢠⠏⠳⡄⠀⠼
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄
⠀⠀⢘⣙⣉⣋⣉⣋⣙⣋⡛⢳⠀⠀⠀⠀⣠⠤⠤⡄⠀⠀⠀⢠⡴⠲⣆⠀⠀⠀⠀⠀⠘⣶⢴⣦⣶⣴⣶⠟⠀⠀⠀⠀⠀⠀⢤⢸⡀⠀⠀⠀⢸⡡⡄
⠀⠀⢸⡏⡿⡏⡿⠿⢿⢹⢻⢸⠀⠀⠀⣼⢹⠀⠀⣷⠤⠤⠤⣼⠁⠀⡏⣷⡄⠀⠀⠀⠀⣷⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⣇
⠀⠀⢸⡯⡏⡇⡏⢹⢹⢸⢸⢸⠀⠀⠀⢹⣼⡄⠀⡟⠉⠉⠉⢹⡂⠀⣧⡏⠀⠀⠀⠀⠀⣿⠽⣏⢱⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡶⡀⠀⢸
⠀⠀⢸⣙⣓⣓⣓⣚⣒⣚⣚⣸⠀⠀⠀⠀⠈⠓⠚⠃⠀⠀⠀⠘⠓⠒⠋⠀⠀⠀⠀⠀⠀⢸⠖⠉⠉⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⣧⠀⢸
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠘⠀⠘⠀⠉

⠀⠀⠀⢀⡔⠲⣖⣒⠒⡛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣤⣤⣀⣀
⠀⠀⢀⡞⣄⢘⠀⠉⠀⢀⠀⣳⠀⠀⠀⠀⠀⠀⣠⠴⠚⢋⣉⣥⠽⠷⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠶⠶⠶⠶⠶⣶⡞
⠀⠀⠀⢳⠮⠯⣥⠤⠭⡥⢾⣁⡀⠀⠀⠀⠀⠘⣧⠶⠯⣭⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣳⡟⢻⢿⣁⣄⡀⠀⠀⠀⠀⠀⣼⣡⣄⣀⣀⣀⠀⠀⢸⡆
⠀⠀⠀⢸⠀⡇⠀⡇⠀⡇⢸⠓⢸⠄⠀⠀⠀⠀⣉⣲⣦⡤⣤⣍⣙⡇⠀⠀⠀⠀⠀⣿⢹⡏⣿⣘⡿⣾⣿⡍⣿⡆⠀⠀⠀⠀⡇⢠⣤⡀⣀⡀⣀⣀⢸⡇
⠀⠀⠀⢸⠀⡇⠀⡇⠀⡇⢸⢀⢸⠀⠀⠀⠀⠀⠻⣤⡤⠶⠶⢾⣯⣅⠀⠀⠀⠀⠀⠻⠷⠶⠋⠈⠁⠘⠷⠶⠟⠁⠀⠀⠀⠀⣧⠈⠀⠉⠉⠁⠉⠉⢸⡇
⠀⠀⣴⠉⣷⣇⣀⣇⣀⣣⡿⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣤⣭⣍⣉⣉⣉⡽
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣏⣹⠆⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⣀⣀⣀⣀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢠⣠⣿⢷⠿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢨⢷⡀⠀⠀⠀⠀⠀⠀⠸⣟⡻⢡⢎⣕⣶⣖⡊⡳⡄⢸⢿⠀⠀⠀⠀⡤⠒⠋⣚⣗⡛⢛⠦⡀⠀⠀⠀⠀⠀⠀⡾⢯⠼⡇⢰⠯⡏⢷
⠀⠀⠀⠀⠀⠤⠿⣤⡿⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⣾⠙⣿⠍⣻⣿⣿⣾⢸⠀⠀⠀⢸⡁⠀⠀⠀⠉⠀⠈⠁⡇⠀⠀⠀⠀⠀⠀⡇⣟⡇⡇⢸⢸⣿⢸
⠀⠀⠀⠀⠀⠀⣞⠸⠦⠤⡄⠀⠀⠀⠀⠀⢸⣿⠳⡙⢙⢶⡋⡡⣪⠏⢹⣿⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣇⣿⣷⡇⢸⣼⣿⡼
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⠓⠚⠛⠋⠁⠀⠸⠿⠀⠀⠀⠀⠀⠙⠦⡤⠶⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃⠈⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠁⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠚⠙⠃⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⡟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⠀⠙⠚⠁⣸⠀⠙⠚⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣀⣀⠄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣏⠀⢀⡠⢟⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠈⠉⠙⠋⠉⠀⠀⠉⠻⢿⣿⣿⣾⣿⣦⣛⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠀⠀⢸⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠃⣇⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢧⡝⠀⠀⠀⠀⠀⠀⠀⢸⣴⠃⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠷⡄⠀⠀⠙⢦⣄⡤⠤⢤⣀⣠⠞⠁⠀⢀⡼⢧⣀⡀
⠀⠀⠀⢀⡴⠋⠉⣀⣀⠀⠙⠲⣄⣀⣠⠤⢍⠵⠀⠀⠀⠀⣀⡴⠋⠀⠀⢈⠙⠳⣄
⠀⠀⢠⠏⠀⠀⠀⠀⠉⠙⠛⠲⢤⣌⡉⠓⠶⠶⠶⠖⢚⣋⡥⠴⠒⠛⠋⠉⠁⠀⠈⣧
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⣀⠀⢀⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠉⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣁⠤⠤⠤⠤⠤⠤⠤⢄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠒⠆⠀⠀⠀⠀⠖⠆⠀⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠒⠒⠀⡆⠀⠐⠒⠂⠀⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣧⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠂⠂⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⣦⢄⣀⣀⢴⡊⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⠀⠈⠙⢄⠀⢀⠜⠉⠀⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠲⢤⡈⢦⠏⣠⠴⠂⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣴⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⡄⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠙⠢⡄⠀⠀⢀⣾⠥⠖⠓⠒⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠓⠶⢽⣆⠀⠀⠀⡤⠚⠉⠹⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠒⠤⣀⠀⢸⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣸⠀⢀⡤⠔⠚⢧⡀
⠀⠀⠀⠀⠀⠀⣼⢃⠀⠀⠀⠈⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠎⠀⠀⠀⢀⢹⡄
⠀⠀⠀⠀⠀⢰⡇⡏⣸⠁⣴⠂⡜⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠦⣤⠖⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠘⡄⢲⡄⢹⡈⡆⣷
⠀⠀⠀⠀⠀⠘⢷⡀⡇⢸⠙⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠶⢤⣄⡤⠴⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠱⠜⢹⠀⡇⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢻⡓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠁⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠈⠒⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⠀⠀⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣖⢪⠁⠀⠀⠀⠀⠑⣾⣦⣤⣴⣦⡴⠾⠿⠾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣥⣾⣿⣿⣿⣿⣿⡗⢳⢚⣿⠶⠉⢙⣻⣷⣶⣶⣶⣦⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣯⣖⣩⡖⡚⠷⢒⣿⢻⡁⢸⣜⣻⢸⣷⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢻⣅⠳⣬⣿⣽⣾⣿⠿⠛⡏⠉⠙⢦⠋⠶⣄⣀⣤⣦⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣷⣿⣛⠙⠛⢰⣫⣿⣁⠀⠀⠀⠀⠀⠀⠱⣀⣸⣻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣷⣦⠤⠤⡤⢤⣴⣿⡿⣯⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣤⡿⣿⣿⣿⣿⡶⠿⣶⣾⣷⠟⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠿⠋⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠷⠤⠖⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢦⣄⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠐⡂⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠒⢠⣤⣤⣴⠠⠀⠀⠀⠀⠀⢀⣀⣠⡶
⠀⠀⢸⣄⠟⠀⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⠂⠃⠀⠀⠀⠀⠀⠀⣴⣀⣮⠙⠙⠛⠈⠀⠀⠀⠀⠀⠻⠛⠁
⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢶⠿⠛⠛⠿⠒⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⠀⣤⠶⠶⠴⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⢚⠷⠚⠛⠀⠀⠀⠀⢸⢴⣶⣲⡶⢸⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⡶⢦⣵⣴⢴⡀
⠀⠀⣠⣤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠸⠤⢤⣤⠬⠼⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠈⢹⠀⠀⠀⠀⢠⣿⣬⣤⣣⣼⡄
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠓⠒⠚⠓⠚⠁

⠀⠀⠀⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⣤⣤⠼⠀⠀⠀⠀⠀⠀⠉⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠃
⠀⠀⠀⡼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢦⣴⡄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢀⣀⣰⠋⠭⠿⠛
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⢙⡇⠀⠀⠀⠀⠀⠀⢼⣿⢿⣭⣷⠀⠀⠀⠀⠈⠛⠛⠒⠚⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣾⣻⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡞⠛⠁⠀⠀⢠⣤⡄⠀⠀⢔⡢
⠀⠀⢸⡿⠿⠿⢿⠁⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⠿⠁⠀⠀⠀⠀⠀⠚⣿⠇⠀⠀⢸⡇
⠀⠀⠸⠷⠄⠠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠚⠁⠀⠀⠀⠀⠀⠀⠿⠅⠛⠛⠁⠃⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⣀⣤⣴⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢃⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢃⣾⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⣶⣶⣾⣿⣿⣿⣼⣿⣿⣿⢃⣴⣶⣶⣶⣶⣶⣾⡟
⠀⠀⠀⠀⣴⣎⠳⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢃⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⡴⢋⣶⡆
⠀⠀⠀⠀⠸⣿⣧⠹⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡼⢡⣿⡿
⠀⠀⠀⠀⠀⢹⡇⣶⣦⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣠⣶⡆⣿⠁
⠀⠀⠀⠀⠀⠀⢿⣌⢻⣿⣦⣀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢀⣠⣾⣿⢋⣼⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣧⡙⢿⣿⣿⣷⣶⣤⣀⠀⠀⠈⠙⠿⠟⠋⠀⠀⢀⣠⣴⣶⣿⣿⣿⠟⣡⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣎⠻⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⡿⢋⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣬⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠁⠀⠀⠙⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢶⣶⡶⣶⣶⣶⡶⡶⣶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣏⣿⣿⣿⣇⣷⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠄⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⡁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⢻⠋⢀⡀⠀⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣶⣿⣷⣤⣴⣾⣷⣶⣤⣶⣿⣷⣤⣴⣾⣷⣦⡀⠀⠀⠀⠈⠀⠀⠦⠤⠈⠀⠄⠀⠀⠶⠀⠀⡉⠀⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡀⠀⡀⠀⠚⠁⠀⢀⡠⠀⠙⠋⢀⠈
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠻⠛⠋⢋⣙⡛⠯⠤⠬⣄⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠐⠄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⢛⣛⣛⣉⣉⣉⣙⣓⣒⣐⣒⠲⠙⢂⠀⠀⠀⣿⢀⠀⠀⠆⠀⢀⡋⠀⡐⠂⠀⡃⠀⠀⡀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⢸⠀⠀⢠⠞⣿⠓⡄⠀⠀⢸⠀⠀⢸⠀⠀⠀⣿⠀⠐⠀⢀⠉⠉⠠⠈⠀⠀⠀⠙⠒⠂⠁⡀⠄
⠀⠀⠀⠀⠀⠢⡈⠛⠛⣼⠀⠀⠘⢄⣾⣢⠇⠀⠀⢸⠀⠀⠸⠰⠒⠒⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠓⠒⠒⠒⠚⠸⠉⠁





⡶⣶⢶⠶⡶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⢶⣶⣶⣶⣶⡶⠶⢶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠓⠀⡀⠀⠀⠀⠀⠀⠈⠀⢤⡀⣈⡄⡀⣎⡄⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠤⠀⠀⢠⢃⠁⠀⠀⠀⠀⡜⠀⡀
⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣡⠀⠄⠀⠐⢀⠀⠀⠀⠊⠀⠠⠄⢀⠐⢘⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢁⡄⠀⠀⠀⠰⠀⠠⠃⠌⠀⠂⠀⠀⢀⡇⠀⠁⢸
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠠⠀⢂⠀⠀⠀⠀⠀⠂⠰⡀⠊⠀⠀⠀⠀⢀⠆⠀⠀⠀⠘⠀⠀⠀⢸⠃⠀⠀⢈
⠀⠰⠄⡀⠠⢄⣀⠀⠀⠠⢀⠈⣠⠈⠀⠀⠀⠀⠑⠀⢠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⢘⣀⠃
⠀⢀⡐⠁⠀⠀⠀⠉⡁⠐⠀⠓⡌⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢰⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀⡸⠁⠀⠰
⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⠀⠀⣆⠐⢀⣀⠤⠀⠄⠀⠀⠀⠏
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠈⠀⠋⠁⠀⠀⠀⠀⣴⣶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠁⠀⠉⠀⠀⠀⠰⠚⠁⠸⡾⠆
⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡀⠀⢰
⠹⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠠⠊⠀⠀⠀⠀⠀⠈⠇
⠀⡃⠤⠐⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⠈⠀⠀⠀⠄⠀⠀⠀⠀⣠
⣂⠁⢀⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖
⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠁⠐
⠀⢀⠁⢤⣀⠀⠀⠀⢀
⡈⠀⠀⠀⢠⡈⠐⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⡴⠂⠠
⠈⠉⢀⠁⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠘⡀⡀
⠀⠀⠀⠑⠀⠀⠀⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢱
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠑⠁⠄⠀⠁
⠀⠀⠠⠀⠀⠌⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠃⠀⣁⠀⠀⣠⠀⠀⠀⠠⠆⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⣺⣿⣤⠈⠁⠀⠀⣠⠔⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠑
⠘⢿⠛⠉⠀⠀⠀⠊⠡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⣴⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡀
⠂⠘⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠁⠄
⠸⠀⠐⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡎⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣰⡀
⠀⠀⢀⠌⡀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢀⠘⠘⠂⡀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠒⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄⢤
⣀⠐⢌⡄⠀⠊⠀⠀⠀⠀⠀⠀⠁⡙⠳⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⡐⠒⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠊⠠⠚⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠠
⠌⡁⠂⣡⠐⠂⠀⠀⠔⠒⠀⠉⢀⠀⠀⢁⠠⠀⠀⡀⠄⠐⠀⠀⠀⣀⡀⣀⡀⠠⠄⠒⠉⠰⠃⠐⢰⠳⠀⠄⠀⠀⠚⠀⢠⠄⠈⠂⠀⠀⠀⠀⠀⠤⠀⣀⡠⠤
⠈⠈⠐⠒⡈⠀⠀⠀⠀⢈⠉⠄⠃⠀⠀⠠⡀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠐⠄⠁⠀⠀⠀⡨⠒⠃⠀⠐⠑⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⢠⣽⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⠔⠒⠢⡀⠀⠀⠀⠾⠿⠿⠿⠿⠧
⠀⠀⠀⠘⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⣿⠁⠀⠀⢄⢀⣠⠃⠀⣀⠀⠀⢠⣴⣐⣂⣠⠞
⠀⢠⣴⡂⠐⡞⠀⣶⣤⠀⠀⠀⠀⠀⠀⠀⠛⢷⣿⢯⡀⠀⠀⢠⠁⠀⠀⠀⣿⣿⠿⠿⠟⠛⠉⠁
⠀⠀⠹⣧⡀⣇⢀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⣖⣗⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣦⣦⣶⢶⣶⣵⡄
⠀⠀⠀⠘⠺⡿⠟⠀⠀⠀⣀⡀⢀⡀⡀⣀⡀⢠⡄⣤⠀⡄⣤⠤⢀⡤⠀⣤⠄⠀⠛⠻⡿⣻⣿⣿⣿⣙⠋
⠀⢀⣄⣀⣴⣶⣭⠔⠁⠀⡷⣫⠘⡇⠇⠉⡶⠸⠇⠇⠱⠇⠼⠴⠤⠼⠇⠤⠗⠀⠀⠀⠀⠙⠛⠉⠉⡀
⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⢀⡄⣠⢠⡄⣠⢤⢠⡤
⠀⠀⠀⡀⠀⠀⡠⠄⠀⠀⠀⠀⢨⣇⡄⢠⡇⠀⠇⠻⠠⠇⠣⠴⠠⠼⠂⠀⠀⠀⠀⠀⠐⠇
⠀⢀⡀⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢸⣦
⠀⢸⣆⣦⡇⠀⣀⠀⢀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⡜⠩⠼⠟⡆⣀
⠀⡞⢻⢻⡇⠀⠀⠀⢀⡠⠄⠀⠀⡀⠀⢿⡿⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⣈⣁⠁⢠⣤⠄⣡⠉⣤
⠀⠁⣾⠙⣇⠀⠀⠀⢸⠀⠀⡠⠂⡂⠀⢸⠃⠀⠀⠀⠀⠀⠀⢾⠿⠀⠀⠀⢸⡏⢸⣧⠀⠁⢀⡟⣦⡟⠆
⠀⠀⣿⠀⢿⠀⠀⠀⠈⢀⠈⠀⡀⠂⢀⣸⣐⣀⠀⠀⠀⠀⢀⣠⣬⣤⠀⠀⡇⢐⣾⠉⡄⠀⠘⢰⣿⣷⠈
⠀⠀⠃⠀⠘⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠃⠀⠈⠉⠉⠀⠀⠀⠀⢹⢹⡄⠀⠀⠀⠀⡆⡦
⠀⠀⠀⠀⠀⠀⢀⣼⡧⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⡇⢸
⠀⠀⠀⢰⣿⠀⣿⣯⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠘⠛⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⢀⣠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠛⠒⠦⡀⠀⠀⡠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣠⠖⠉⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠳⣼⠁
⠀⠀⠀⠀⠀⢀⣀⠤⠔⠋⠁⠀⣀⣀⣀⣀⡤⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠙⠲⠤⠤⠤⠄⠀⠈⠑⠢⠤⣀
⠀⠀⠀⣀⣀⣨⠤⠴⠒⠂⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⠒⠶⠥⣀⣀
⠀⠀⢘⠤⠄⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⢠⠋⠀⠀⢀⠁⠀⠀⠀⠤⢤⡗
⠀⠀⠛⠤⡄⠒⠒⠀⢀⡁⠀⠀⠀⢣⠀⣀⡤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⡏⣀⣀⠂⠤⠀⠉⠑⠒⢠⠤⠛
⠀⠀⠀⠀⠉⠉⠓⠤⠖⠒⠊⠉⠉⠙⡅⠀⠀⠀⠀⠀⢀⡤⠤⠤⣀⠀⣀⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠈⠉⠉⠒⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⡀⠀⠀⢸⠀⠀⠀⠈⠉⠁⠀⠀⠀⡇⠀⣀⣀⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠷⣄⠀⠀⠀⠀⠀⣀⠜⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⣷⣶⣤⣄⡀⠀⠀⠀⣀⣠⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⡏⠙⢿⣆⢀⡴⠋⠁⠀⠈⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠛⠛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⠾⠋⠉⠉⠻⢦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄
⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠿⠶⣤⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣀⡀⠀⣀⣴⣿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⠦⣤⣄
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢠⣿⣧⣄
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠁⠀⡹
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢹⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢸
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⢰⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⣿
⢹⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣧⡈⠙⢿⣿⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⠿⠻⠿⠁⠀⠀⣿
⠈⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣍⣷⠶⠀⠀⠀⠀⢠⣿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⣿⡌⠻⣿⡙⠛⠛⠿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢿⡟⣿⡉⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠒⠓⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠓⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠏⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡉⠃⠀⠉⠿⠁⢀⡞⠉⠉⣹⡅⠛⢐⣧⡿⢋⡿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢠⣤⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⣆⠀⣴⣿
⠀⠘⠿⣿⠛⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⢸⡄⠀⢸⣷⣿⡟
⠀⠀⠀⣀⣤⡉⠉⠉⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢿⣿⡇⠈⠛⠋⠁⢀⠀⠀⠀⠱⣼⣷⠀⠈⢿⣿⠃⠀⠀⠀⣀⠄
⠀⠀⢼⣿⣿⣿⣦⣤⠀⠉⢀⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⠈⠛⢃⣤⣶⣿⣛⠋⠁⠀⠀⠀⠘⣿⣧⡀⠈⠃⠀⢀⣴⣾⠋
⠀⠀⠀⠉⠛⠿⠿⣿⣷⣶⣿⣿⣿⣧⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⠃⠀⠈⠛⠿⠿⠿⠷⠀⠀⠀⠀⠀⠙⢿⡗⠀⠀⣰⣿⠟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠈⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣶⣶⠄⠀⠀⠀⠀⠀⠀⣴⠀⠀⠃⠀⠰⠟⢁⣤⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠉⠙⠿⠃⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣦⠀⠀⢿⣿⣿⡿⠟⠁⠀⠀⠘⢶⣤⣄⣀⣿⡆⠀⠀⠀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⣠⣾⣿⡆⠀⠀⠀⣿⣿⣿⡍⠉⠀⠠⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠀⠀⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣶⣶⣄⠀⢸⣿⣿⡿⠁⠀⠀⠀⠹⣿⣿⠃⠀⣀⣀⣀⡉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠹⡿⠂⠀⢀⣠⣤⣶⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡦⢀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⣿⣿⣿⣿⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠘⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣿⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣾⣶⣷⣿⣿⣾⣶⣿⣿⣿⣾⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣨⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⠀⠀⠀⠀⠀⠀⠀⢼⣶⡆⠀⠀⠀⠀⠀⠀⢸⣯⡁⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⢠⣅⢀⣠⣮⢉⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣿⣦⡀⠀⠀⠀⢀⣴⣦⣤⣶⣆⠀⠀⠀⢀⣤⣴⣹⣷⡄⠀⠀⠀⠀⣀⡬⣿⠥⣤⡀
⠀⠀⠻⠿⠋⣿⣿⡟⣿⡄⠒⠀⠂⠀⠀⠀⣾⣿⣿⣿⢻⣧⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⢤⣨⣽⣿⣿⣿⠀⠀⠀⠀⣾⡟⠁⣾⠄⢙⣿⡆
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠷⠷⡄⠀⠀⠀⣿⣾⣿⣿⣿⣿⠀⠀⠀⣿⢿⣯⣿⣿⢹⡇⠀⠀⠀⠁⢸⣿⣿⡆⠀⠀⠀⠙⠿⢲⣿⣶⠙⠛
⠀⠀⠀⠀⠘⣿⣿⣿⠈⠁⣤⠠⡄⠀⠀⠀⣿⢸⣿⣿⡇⠈⠀⠀⠀⠷⢼⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⣿⡿⣿⡆
⠀⠀⠀⠀⠀⢻⣿⡇⠀⢈⣉⣁⡁⠀⠀⠀⠛⠈⣿⣿⠀⠀⠀⠀⠸⠿⢿⣿⢻⣿⡄⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⢸⣿⠃⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⡇⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⠶⠆⣒⣦⠠⠒⠒⠒⠒⡄
⠀⠀⢰⡞⠃⠀⠀⠀⠀⠀⣀⣘⣉⣃⡀⠀⠐⠺⢛⢻⡯⢓⣀⠀⡇
⠀⠐⣿⣿⣷⣶⣀⣠⠀⣾⡟⢿⣿⣿⣿⡆⠀⠀⠀⠈⠉⠀⠉⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⡀
⠀⠀⣿⣿⡿⠿⠿⢛⣄⡙⡷⠺⠟⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⡅⠀⠀⠀⠀⠀⠀⠈⡇⠙⠁⠀⠀⠀⠀⠀⣀⠀⠀⣶⡏⠉⠉⢉
⠀⢸⣿⣿⡇⠀⠀⣻⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣇⡀⠛⡏⠘⠃⠀⣤⠀⠀⠀⠀⠀⡇⠀⢀⣄⢀⣀⣀⣼⣿⣷⠀⣿⣧⣤⣤⣶⡀
⠀⠀⢿⣿⡇⠀⠀⠛⠛⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⢛⣿⡀⡇⠀⠀⠀⣿⠂⠈⠉⠉⠉⡏⢿⣶⠾⠿⣿⣿⣿⣋⣉⣹⡿⠀⠀⠀⠀⡇
⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡇⠀⠀⠀⣿⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⠛⠛⠛⠛⠛⠛⣟⠀⠀⠀⠀⡇
⠀⠀⠘⠿⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠈⠁⠀⠀⠀⠻⠀⠀⠀⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠇
⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶
⠀⠀⢀⣴⣶⣤⣶⣦⠀⠀⠀⢀⠀⠀⠀⠀⢀⣤⣽⣿⣤⡀⠀⠀⠀⠀⠀⠀⢠⣼⣯⠁⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣥⣄
⠀⠀⣾⡟⣿⣿⡏⣿⡇⠀⢴⣿⣷⠄⠀⠀⣿⣻⠋⠘⢛⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⣠⣽⡟⠁⠀⠀⠀⠀⠀⢀⣿⠟⢻⣿⢻⣧
⠀⠠⢿⡇⣿⣿⣇⢹⣧⣤⣾⣿⣿⣆⠀⠀⠈⢿⣀⣠⡮⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢿⣦⣤⣿⣿⣧⡀⠀⠀⠀⠀⠀⢸⣿⢰⣾⣿⠈⠁
⠀⠀⢸⢸⣿⢻⣿⡀⠀⠀⢸⣿⣿⠿⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠉⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⣾⡿⣿⡇
⠀⠀⠈⣿⡿⠈⣿⡇⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆⣿⡇⢰⠁⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⢠⣿⡇⢻⣿
⠀⠀⠆⠻⠇⠀⠿⠇⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣿⡇⠘⢄⡀⠀⣠⠞⣿⡿⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⡀⠀⣴⣶⣴⣶⣦⠀⣀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣶⣶⡶⠆⠀⠀⢀⣴⣶⣶⣄⣴⣶⣶⣄
⠀⠀⠸⣿⣆⠻⢿⣿⡿⢋⣾⡿⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣶⣾⡆⠀⠀⠀⠀⢸⣿⡿⠋⠀⠈⢻⣿⣿
⠀⠀⠀⠹⣿⣷⣄⢁⣴⣿⡿⠁⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⡿⠃⠀⠀⠀⣤⣾⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⠻⣿⡀⠀⠀⣸⡿⠋
⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠹⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⠉
⠀⠀⠀⠀⠀⠉⠛⠘⠋⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣶⡦⣄⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣤⡶
⠀⠀⠀⠀⠀⢿⣿⡿⠻⣠⣠⣀⠀⠀⠀⣿⣿⣿⠃⠛⠀⠿⢿⣿⡇⠀⠀⠀⠀⠀⢸⣥⣀⣽⣿⠿⢿⣷⠀⢠⣶⣄⢀⣴⡿⠿⠿⢃⣴⡆
⠀⠀⠀⢀⣠⣴⣯⣵⣿⣿⣿⠏⠀⠀⠀⠙⣿⢨⡄⣼⠀⣇⢸⠟⠁⠀⠀⣤⣴⣦⣤⣴⣷⡆⠀⠀⠀⠀⠀⠀⢻⣿⣮⣙⠃⢀⣠⣾⡿⠃
⠀⠀⠘⣿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⠿⠛⠁⠀⠀⠀⠀⠉⠉⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡆⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡀⣶⢰⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢸⣇⣿⣰⡇⣀
⠀⠀⠀⠀⠀⢱⣿⣿⡏⢀⣀⣄⡀⠀⠀⣶⣄⢴⠛⢿⡄⢠⢆⣴⡆⠀⠀⠀⣾⣿⣿⡿⠞⠛⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⢀⣠⣤⣤⣤⣾⣿⣿⡟⠀⠀⠀⢻⣿⣮⡑⠿⠟⣫⣾⣿⠃⠀⠀⠀⠻⣿⣿⠀⠀⢀⣿⡿⠃⠀⠀⠀⠘⢶⣄⡿⠛⠛⠻⡇
⠀⠀⠺⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠙⢿⣿⡆⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⠙⠷⡶⠟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿

⠀⠀⠀⢀⣤⣄⡀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢀⡴⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⡀
⠀⠀⢰⣿⣿⣿⠛⡟⢻⣿⣿⣷⠀⠀⠀⠀⠀⢸⣇⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⡶⠀⠀⠀⠀⠀⣾⣿⣿⣟⢳⣿⣿⣿⣿⡆
⠀⠀⠸⣿⣿⡼⠃⠁⠋⣼⣿⡟⠀⠀⠀⠀⠀⠈⢿⣦⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⢇⢀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠻⢿⣿⣿⠇
⠀⠀⠀⠈⠛⢷⣤⣶⡴⠟⠋⠀⠀⠀⠀⠀⢀⣤⣾⠟⢿⣦⣄⠀⠀⠀⠀⢴⣶⣾⣿⣷⣶⣿⣿⣿⠏⠀⠀⠀⠈⠻⢿⣿⣿⣇⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠙⠃⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋

⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡔⠒⡄⠉⠢⡀
⠀⠀⠀⠀⠀⠠⣤⠤⠤⠤⢤⣷⠤⠤⠤⢤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢱⣸⠀⠀⠀⢡⠀⠀⠀⠀⢰⠚⠳⡄⠀⠀⠀⢰⠚⠳⡄
⠀⠀⠀⠀⠀⠠⠁⡀⠀⠀⠀⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠔⠁⠀⠀⢀⣥⠀⠀⠀⢸⠀⠀⠀⠀⠈⠶⠞⢁⠀⠀⢀⡈⠳⠞⢁⡀
⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⣿⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⢢⠀⠀⠀⠈⠋⠀⠀⡠⠃⠀⠀⠀⢀⠀⠀⠀⡀⢂⠀⡘⣠⠀⠀⢀⡡
⠀⠀⠀⠀⡴⠀⠀⠀⢦⠀⠀⣿⠀⢠⡖⠀⠀⠐⡆⠀⠀⠀⠰⣄⢀⡀⠄⠀⠀⡏⠀⠀⠀⠀⢘⠘⠀⠀⠀⠛⢌⢠⡵⡻⢀⠀⢈⢳⡆
⠀⠀⠀⠀⠈⠲⠶⠒⠁⢀⣠⣿⣀⣀⡙⠲⠖⠊⠀⠀⠀⠀⢀⠀⠸⠆⠀⠀⠤⢇⠀⠀⠀⠀⠈⠀⡀⠸⠀⠀⠀⠀⠀⡇⠸⡇⢠
⠀⠀⠀⠀⠀⠀⢄⣐⣃⣠⣤⣤⣤⣤⣬⣦⣠⠀⠀⠀⠀⠐⠁⠶⠖⠦⠶⠖⠐⠲⠃⠀⠀⠀⠀⠀⡇⡄⢀⠀⠀⠀⠀⠃⠇⠀⢸
⠀⠀⠀⠀⠀⢀⢀⠀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠁⠈⠛⣀⡀⢀⣛⣀⠘⠛
⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠁⠁⠁⠀⠀⢤⣶⣶⡄⠀⠀⣀⣀⣠⣄⡀⢀⡀⣀⢀⣀⣄⣀⣠⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣹⡆⣿⣤⠟⣙⡷⢸⡇⣾⣧⡏⣿⣓⠈⣛⡆⢙⣳⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢰⣾⡛⢢⣭⡤⣬⠀⡅⢠⠀⣩⡭⣀⣭⠤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣥⠀⡞⠀⡿⢶⡇⢾⠸⣯⣄⢈⡽⠂
⠀⠀⠀⠀⠀⡀⡀⢀⡀⡀⡀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠈⠀⢀⣀⠀⢀⡉⠑⠢
⠀⠀⠀⠀⠀⠉⠁⠈⠁⢉⣉⣈⡈⠁⠁⠀⠀⠀⠠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⠀⠀⠀⠀⠄⠈⠈⠈⠉⠉⢀⡁⠠
⠀⠀⠀⠀⢀⡀⣀⣀⠔⣉⢁⡀⠹⠓⣄⣠⡄⠀⠀⠙⣆⠀⠀⠡⢀⠀⠀⢀⠼⠉⠀⢀⡜⠁⠀⠀⠀⣀⠤⠔⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠈⢡⠁⠆⠀⠄⠀⠄⠀⠆⢸⡇⢠⠀⠀⠀⠘⠦⠦⣀⠀⠁⠆⠀⢀⡠⠢⠊⠀⠀⠀⡠⢋⠤⠂⠉⠉⠉⠐⠢⡙⣆⠉
⠀⠀⠀⠀⠀⡎⠶⢾⣄⢠⠀⠆⠠⣆⣼⠗⠎⠄⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⣁⠀⠰⠁⠐⠤⠊⠐⡄⣅⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠈⣄⠉⠓⠒⠚⢁⡌⠀⠀⡰⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠒⣄⠈⢄⠀⠀⠀⡠⠁⠈⣒
⠀⠀⠀⠀⠀⠑⠀⠀⠈⡆⠀⠀⠀⡞⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⡀⠀⠀⠀⠀⠀⠈⡆⢨⠀⠈⠲⢀⠜⠀⠀⡐⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠳⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⢦⡑⠢⠄⠀⠠⠐⣈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⡄⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡠⠐⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠙⣯⣀⡄
⠁⠀⠀⠀⣾⣷⠄⠀⠀⠈⠀⠀⠀⢸⣿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠠⠊⠁⢿
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠻⢿⠿⠟⠀⠀⠠⠀⠀⠈⢉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠁
⠀⠀⠀⠀⣠⣤⡀
⠀⢀⣶⣤⣿⣿⣧⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠘⢻⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⠀⠀⠀⣰⣿⡿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⢸⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⣬⣩⡁⢸⣿⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠈⢣⣄
⠀⠀⠉⠀⠿⠿⠇⠈⠁⠀⠀⠀⠀⠀⣿⣿⡟⠸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣴⣶⣿⣿⣿⣿⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠀⠾⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⣖⡆⠀⠀⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠶⢒⠲⣶⣶⣶⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢀⡴⠋⡀⢀⠙⢦⡀
⠀⠀⠀⠀⢀⣰⣷⣄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠐⢺⠒⢸⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠙⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠷⠶⠶⠿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⢰⡆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣬⣥⣤⣤⣤
⣿⣿⠛⣝⢟⣉⡛⣻⢛⣽⡟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⠀⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧⣶⣠
⠀⠀⢘⣟⠐⣿⣿⠂⣻⡃⠀⠀⠀⠀⢀⣰⠁⢶⡶⠈⣆⡀⠀⠀⠀⠀⠈⠀⠘⣤⣀⢿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠐⠛⠛⣵⣶⣶⣮⠛⠛⠂⠀⠀⠀⠈⠹⡴⠾⠷⢦⠏⠁⠀⠀⠀⠈⠂⠄⣼⣿⡿⣸⡿⠛⠁⠀⠀⠀⠀⢷⣼⣿⠟⠛⢿⡄⠀⠀⠀⠀⠀⢷⣬⣿⠛⠛⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣗⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣾⠿


⠀⠀⠀⣠⣶⠟⢰⣶⣶⡆⠀⠀⠀⠀⠀⠀⣠⠴⠒⠦⠄⠀⠀⠀⠀⠀⢠⣴⢶⣤⣤⡶⣦⡄⠀⠀⠀⠀⠀⠀⣴⡿⢷⣦⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠻⣶⣄
⠀⠀⣼⣿⠟⠂⠀⢙⣫⣤⠀⠀⠀⢠⡖⠖⢢⠀⠀⡔⠶⢲⡄⠀⠀⠀⢿⡅⠀⠈⠁⠀⢨⡿⠀⠀⠀⠀⠀⠀⢻⢇⣼⡟⠀⠀⠀⠀⠀⠀⣼⠟⠛⠀⠀⠛⠻⣧
⠀⠀⢻⡘⢶⣤⡀⠘⢛⡟⠀⠀⠀⠀⠑⠶⠋⠀⠀⠙⠶⠊⠀⠀⠀⠀⠈⠻⣦⡀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⢟⣤⡀⠀⠀⠀⠀⠀⢻⣦⣤⠀⠀⣤⣴⡟
⠀⠀⠀⠙⠾⣿⣥⠴⠋⠀⠀⠀⠀⠀⠀⠀⠒⠦⠴⠚⠂⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠻⡿⠋⠘⢿⠟⠀⠀⠀⠀⠀⠀⠛⠿⣦⣴⠿⠋


⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠒⠤⣀⠀⠀⠀⠀⠀⣟⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢀⡤⡒⢒⢤⡀
⠀⠀⠀⣀⣀⣼⣿⣤⣤⠀⠀⠀⠀⠀⢀⣤⣤⠴⠚⠙⠛⠁⠀⠀⠀⣠⣾⣿⣿⣿⣶⡀⠀⠀⢘⠄⠀⠀⢀⡞⠁⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠲⢤⣠⣤⡀⠀⠀⠀⢿⣷⣿⣼⣷⣿⠇⣄⣴⠊⠀⠀⠀⠈⠃⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠟⠉⠛⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⣀⣄⣘⣛⣛⠁⠀⠀⠀⠀⠀⠘⠻⠿⠿⠟⠃


⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⢸⡏⢙⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⡟⡉⣽⡯⣽⣯⢉⢻⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⣿⠀⠀⠀⠀⡎⢉⣉⠉⣩⡭⡍⢻
⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣧⡁⢻⣟⣾⡟⢉⣼⠀⠀⠀⠀⣿⠠⠤⠤⠤⠈⡇⣿⠀⠀⠀⠀⣇⣐⣂⣒⣂⣒⣂⣼
⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⡀⠀⠀⣀⣤⣶⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣦⣿⣿⣿⣿⡄⣰⣿⣿⣿⣥⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡏⣹⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣀⠻⣿⣿⣿⣿⣷⡌⣿⣿⣿⣿⡇⠻⠿⠿⠿⠋⢴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⣾⣿⣧⡹⢿⣿⣿⣿⣧⠻⢿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⢏⣶⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣶⣶⣶⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⢿⣿⣿⣟⣛⣩⣴⣿⣿⣿⡇
⠀⠀⠀⠀⢹⣿⡿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⢩⣴⣿⣿⣿⣿⣎⡻⠿⠿⠿⠿⠇⣿⣿⣿⣿⣿⣿⣿⣿⡿⣩⣍⢋⣴⣷⣆⢀⣤⡄
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⢋⣶⣿⣿⣿⡆⣿⣿⢿⣿⣿⣿⠛⢿⢡⣿⡏⣾⣿⣿⣿⣇⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⣾⣿⣿⣿⣿⡇⣩⣵⣿⣿⣿⡇⣿⣦⡘⢿⡁⣿⢟⣉⠻⣿⢸⡿⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⠃⠹⣿⣿⣿⣿⡆⣿⣿⣿⣦⡀⢡⣾⣿⣷⡉⣠⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠈⠙⠛⢉⣁⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣌⡛⠿⣿⣿⣿⠿⣿⣿⡿⠿⣋⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠶⠶⠒⠀⠒⠶⠶⠿⠟⠋



⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣛⣛⣛⣿⣛