Freshness Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣾⠿⠿⢿⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⡿⠿⠟⠛⠋⠉⠉⢿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⣀⣤⣶⣿⣿⣿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⠁⢀⣴⣿⠟⠋⠁⠀⠈⠻⢃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠀⠀⠀⢀⣴⡾⠿⠿⠿⢷⣦⡀⠀⠹⣿⡀
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠠⣿⠇⠀⠀⢠⣿⣧⡄⢠⣤⣤⣤⡄⢠⣤⣤⣤⠀⠀⣤⣤⣼⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⢿⣇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠁⠀⠉⠉⠉⠀⠈⠉⠉⢉⣀⠀⠈⠉⢹⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⣤⣤⣾⣿⣤⣤⣤⣤⣤⣿⣷⣤⣤⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⡄⠉⠁⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠰⣿⠟⠁⠀⠀⣠⣶⡿⡿⢿⠿⠿⠿⠿⠿⢿⡿⠿⣿⣿⣾⣿
⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⢀⣴⣿⠆⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡟⠉⠉⠉⠉⠉⠉⠉⣿⣏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⣶⣶⣾⠿⣷⣶⣶⣿⡧⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡏⠀⠁⠀⠀⠀⠉⠀⣽⣇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣧⣀⣀⣀⣀⣀⣀⣀⣿⡗⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣤⣤⣤⣤⣶⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣬⣉⠻⢿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡙⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠷⢶⣤⣉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⡀
⠀⠀⢀⣤⡾⠛⠉⠀⣀⣤⣴⠶⠶⠿⠛⠛⠻⠟⠲⢶⣄⣀⠀⠈⠙⠻⢦⡈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣧
⠀⣰⡟⠁⠀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠣⣄⠀⠀⠹⣆⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿
⣰⡟⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⡀⠀⠈⣷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿
⣿⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⣿⡄⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣄⠀⠈⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣼⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣶⣄⠀⠈⠛⠷⣦⣤⣀⣀⣀⡀⠀⣀⣀⣀⣠⣤⡶⠟⠋⠀⣀⣴⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠈⣿⣿⣿⣿⣿⣶⣤⣀⡀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉⢀⣀⣠⣴⣾⣿⣿⣿⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⠏
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⠏
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣾⣿⡿⠃
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠿⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡔⣤⠀⡤⣤⢄⠀⢖⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄
⠀⠀⠀⠀⡀⠘⢲⡏⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⡄⠠⠂⡌⠉⠂⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠁⠀⠰⢆⠀⠀⠈⠃⠂⢡⠀⠀⠀⠀⢀⠄⠀⠴⠚⠀⣆⠈⠈⡄⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠂⠀⡆⠀⠀⠀⢠⡸⡄⠀⠀⡔⠠⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠐⠰
⠀⠀⠚⠄⠰⡀⠀⠀⠱⢤⠀⢀⠀⠀⠀⠀⠀⡅⠸⠂⠤⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⡄⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢨⠀⠀⠀⠀⠀⠙⡀⠁⠀⠀⠀⠀⡠⠃⢠⡖⠀⠀⠀⠀⠀⠀⠄⠀⢣
⠀⠀⠈⠀⣀⠉⠂⡀⠀⣀⠄⠀⠀⡄⠀⠀⠀⠀⡇⡄⠀⠀⢀⠔⢀⡤⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠
⠀⠀⠀⠀⠈⠃⠀⠀⢀⡌⠀⠀⠄⠀⠀⠀⠀⠀⠁⠈⠑⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⠀⠊⠀⠀⠀⠀⠀⠀⠠⠄⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠏⠀⠐⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠠⡀⠀⠓⠈⡜⠌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇⠀⠀⠀⡠⠂⠁⡰⠃
⠀⠀⠀⠀⠀⠀⡃⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⣀⠀⠀⠀⠀⠀⠁⢂⠁⠌⣈⠭⠨⠔⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠘⠌⠐⠤⣀⣀⣠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⠀⠀⢀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣿⣲⡌⠈⠉⠉⠉⠉⠈⢉⣡⠤⣾⣏⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡂⢾⣼⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠸⢆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⠆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⣀⠀⠆⠰⣼⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠔⠀⠐⠂⢛⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠁⠀⠀⠀⠤⠒⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢆⣖⡈⠂⠈⠻⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠨⠗⠋⣁⣤⣶⣭⣀⠀⠀⠂⠂⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⠁⠀⠀⠖⠂⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⢉⣀⠴⠾⣯⣭⡵⠟⢩⠿⠰⠶⣶⣄⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠀⠈⠙⠀⠀⢺⣾⡏⠀⠀⠐⠀⠀⣀⡷⠒⠤⠐⢀⣠⠴⠚⢩⡷⠄⣘⡄⣁⢒⡞⢈⣇⢙⣳⡆⣈⠉⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡤⠀⠒⢠⣶⣾⠻⡇⢢⠲⣠⡀⠀⠈⠃⣠⡖⠉⢛⣳⠛⣟⢡⡴⠒⢻⡆⣠⠿⠏⠴⡖⢂⢰⡾⠘⣭⣦⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⣤⣶⣷⣺⣿⣦⣀⠁⠰⠷⣿⡵⣀⠀⢹⣿⢐⠿⣿⣿⠻⣿⣤⣤⣩⣾⠁⠚⠛⠀⠉⠈⠁⠀⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⢷⡶⠦⠉⠱⠯⠿⠛⠛⠨⠇⠏⠚⠟⠛⠛⠛⠚⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⣒⣉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣶⣿⣿⡿⠁⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠰⢻⣭⡤⢤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⡔⢡⡞⢠⣾⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣾⠁⣽⠀⣨⣿⠈⢷
⠀⠀⠀⠀⠀⠀⠀⡴⠁⢴⠃⢰⡇⠀⠼⣿⡄⠹⣆
⠀⠀⠀⠀⠀⢠⠞⠀⢀⡿⠀⢪⡇⠀⠘⣻⣷⡀⡉⠲⢄⡀
⠀⠀⠀⣠⠔⠁⠐⠄⡼⠁⠐⠚⣇⠀⠀⠟⠛⠳⡄⠀⢵⣪⡓⢄
⠀⢠⠞⠁⠀⠀⠀⡜⠀⠀⠉⠁⠈⢦⣦⡀⠐⢍⠪⡢⣈⠂⡙⢦⡱⡄
⢠⣇⠄⠀⠀⠀⡸⠀⠀⠀⠀⡀⠀⠀⢻⡟⡀⠑⢌⠊⠊⠳⡄⠀⠙⠈⢦⡀
⠜⡑⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⠀⠀⠀⠳⡍⢠⠀⠑⢄⠀⠘⢦⠀⠀⠀⠈⠒⢄⡀
⣯⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠎⡀⠀⠀⠙⣄⠱⣦⠀⠑⠄⠀⠑⢄⠀⠀⠀⠀⠈⠑⠒⠤⣀⡀
⣧⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠈⠐⣄⠀⠘⢆⢩⢳⡄⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠤⠤⠤⠤⠤⢄⣀⣀
⢷⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠏⣧⡀⠈⠳⣈⢸⢢⣀⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠦⣄
⢰⠆⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⢱⢄⠀⠙⢤⠀⠸⢳⢤⡀⠀⠀⠀⠀⠉⠺⠯⣷⣲⡢⡤⡄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠈⡏⠀⠀⠀⢠⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠓⡀⠀⠙⢦⡘⢰⢸⢓⠤⣀⠀⠀⠀⠀⠀⠈⠙⠓⠳⠴⣽⣝⡘⡝⠏⢳⠀⢇⠲⠐⡒⢖
⠀⢱⠀⠀⠀⠀⠇⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠙⢾⡀⠜⠀⡆⢉⢰⠢⡄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠚⠢⠷⠄⣠⣑⡬⠶⠋
⠀⠈⢆⠀⠀⠀⢈⣄⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠙⠣⣅⠡⠈⠀⡄⢡⠈⠏⢱⢲⢲⢤⢤⢤⢀⡀⣀⣀⡀⠀⠈⣷
⠀⠀⠘⣆⠀⠀⠈⠙⢦⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠙⠒⢤⣐⠀⠁⠘⡀⠀⠊⠌⠌⠌⠌⠂⠢⠑⡙⢷⣰⣻
⠀⠀⠀⠈⢆⠀⠀⠀⠈⠓⣄⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠈⠀⠂⢂⣀⡤⠼⠚⠋⠁
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠩⢣⡄⠀⠀⠀⠀⠑⢤⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⢄⡀⠀⠀⠀⠉⠓⠶⡒⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠊⡗⣄⡀⠀⠀⠀⠈⢗⠳⢵⣐⡀⡀⠀⡀⠀⠀⠀⠀⠀⠈⠱⢖⠤⣀⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠁⡇⡋⠖⣄⠀⢠⠼⡄⠀⠈⠙⠳⠷⣰⣤⣀⡐⡄⠀⠀⠄⢀⢀⡍⡍⠁⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠁⠀⠁⠂⢧⣰⠃⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠼⠾⠾⠮⠾⠿⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠓⠒⠒⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣖⠋⠉⢰⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⢤⣼⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠳⢤⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⣀⠀⠀⣠⠖⠒⡶⠛⠒⠒⠒⠒⠒⠒⠚⠓⠲⣄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡇⠀⠀⢀⣴⣿⣿⡿⠿⠛⠛⠛⠻⣿⡿⣣⠞⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⣸⠀⠀⢰⣿⣿⠟⣡⣶⣾⣿⣿⣿⣿⡿⢰⣧⣄⣠⠟⠀⠀⣀⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⠄⣩⠴⠚⣉⣁⣤⣀⡈⠙⢶⣄
⠀⢠⡏⠀⠀⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⠞⢁⡴⠛⠉⠀⠀⠀⠙⣆⠀⣿⣧⡀
⣠⡾⠳⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡼⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣷
⠈⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡾⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣆⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠻⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⢸⡀⠀⠀⠀⠀⢀⣠⠞⢁⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣦⣌⣻⣿⣿⣿⣿⣿⣿⡌⢧⡀⠙⠲⠶⠶⠒⢋⣡⣴⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠛⠋⠉⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⡿⠟⠀⠙⢶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣁⣀⣀⣀⣀⣀⣈⣛⣻⣿⣿⣿⣿⣿⣛⣋⣀⣀⣀⣀
⣿⣿⠋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣄⣀⡀⠀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠶⠿⠿⠿⠿⠷⣶⣼⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⣿⣄⡀⠀⠀⠀⠀⠘⠋⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠁⠀⠀⠈⠙⠛⢻⣧⠀⠀⠀⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣶⣦⠀⠈⢿⣆
⠀⠀⠀⠀⣠⡾⠋⠀⢀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⡄
⠀⠀⢠⣾⠏⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡏⠁
⠀⢠⣿⠃⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣤⣴⣶⣶⣶⣶⣶⣤⣤⣄⣀
⢀⣿⠃⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣶⣤⡀
⣼⡏⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⣶⣦⣤⣀⠀⠀⠉⠻⣦⡀
⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀⠀⠘⣿
⣿⡇⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢸
⢹⣧⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⢸
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠈⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⣠⣿
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠾⠟⠋⠁⠀⠀⣠⣶⠟⣹
⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠙⠻⢷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⠿⠋⠁⢠⣿
⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣆⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠟⠛⠛⠛⠉⠁⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠿⠛⠉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣦⣤⣀⠀⠀⣀⣤⣶⠾⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣡⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠤⠄⣼⣿⣀⣀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⡷⠀⣰⡆⠀⣾⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⣢⢸⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⣦⠀⢁⣄⠈⠁⣄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⠀⠙⠋⠀⠺⠟⠀⣸⡿⠀⡇⡀⠀⠀⠀⡔⠉⠉⢑⠎⠉⠉⡻⠉⠉⢑⠎⠉⠉⢢
⠀⠀⠀⢿⣦⡀⠀⠀⡰⠁⠀⢸⣿⠀⠀⢠⣶⠀⠀⡠⣇⠉⠲⣏⣏⠀⠀⠀⠣⣀⡔⠊⠐⢦⠒⠁⠂⡔⠊⠑⢤⢀⡸
⠀⠀⠀⢸⡫⡈⠑⠲⢍⡢⣄⡀⠀⠀⣀⠻⠿⠒⢂⢷⡆⠑⠤⣽⠻⡄⠀⠀⠀⠈⡄⠀⣀⠧⢄⣀⡸⠤⡀⠀⢠⠁
⠀⠀⠀⠈⡷⣌⠒⢄⡀⠀⠈⠉⠉⠁⠀⣀⠠⠒⣡⠮⠤⠤⢤⠟⠀⢳⡀⠀⠀⠀⠈⢹⠁⠀⠀⡏⠀⠀⠈⡏⠁
⠀⠀⠀⠀⠘⢌⠓⢄⣈⠉⠁⠀⠈⠉⣁⠠⠔⠚⢶⠆⠉⠒⠻⣞⠀⠔⠛⠒⠐⢄⠀⠈⣢⡔⠉⠉⠑⡤⠘
⠀⠀⠀⠀⠀⠈⠑⢄⡈⠙⠛⠛⠩⠤⠄⠐⠒⣊⡽⡀⠀⠀⢀⡜⣬⡀⠀⠀⠀⣸⠤⠊⠀⠣⡀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠈⠁⠀⠀⠉⠒⠒⠉⠀⠀⠈⠒⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⢠⠊⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢃⠀⠀⡔⠁⠀⡰⠁⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠘⢀⠎⠀⣀⠴⠒⠉⣁⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠸⢃⠴⠊⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡸⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠔⣊⠉⠈⢣⠸⡁⠀⠀⡠⠤⠘⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠔⠓⢄⡰⠁
⠀⠀⠀⠀⢀⠄⠊⠹⠙⠒⠤⠊⠉⠀⠈⠙⢄⠀⠀⠀⠀⠀⠣⠀⠀⠀⣠⠔⠊⠉⠙⠢⢵⡀⡌⠂⠑⣤⠤⠤⢀
⠀⠀⠀⢠⠃⠀⠀⠠⠀⣶⢠⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⢠⠞⠁⢤⣀⠀⠀⠀⠀⢡⣀⡤⠊⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠘⡀⠀⠀⡇⠀⢿⢸⡴⠩⠁⠀⠀⠀⡇⠀⠀⠀⠀⢸⠔⠁⠀⠀⠀⠉⠓⠤⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⢀⢱⠀⠀⡇⠀⢸⠋⠠⠁⠀⠀⠀⠀⡇⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠘⢌⠣⡀⡇⢠⠃⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠈⡆⠑⡇⡆⠀⠘⠇⠀⠀⠀⠀⠀⡇⠀⡰⠁⠀⠀⠀⠀⠀⠈⠲⢤⡀⠀⠀⡜⠈⢆⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠘⡀⠘⡇⠀⡇⠸⢀⠀⠀⠀⠀⠇⣐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠎⣀⡀⣈⣢⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠃⠀⢡⠀⡇⠀⢃⠆⠀⠀⠀⡟⠀⠙⠃⠀⠀⠀⠀⠀⢀⡠⠤⠒⠋⠉⠀⠀⠀⠀⠉⠐⠂⢝⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⣀⠔⠂⠹⠔⠙⠴⠓⠢⢸⡀⠀⠀⡜⠀⠀⠀⣀⡀⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠈⠢⡀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡘⠀⠀⡞⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠘⢄
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠧⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⠔⠒⢤⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠘
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⢿⡄⠀⠀⠀⠀⢀⠊⠀⠈⠃⠀⠀⠀⠀⠀⢠⠀⠀⢀⠂⠀⠀⡇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠣⠤⠤⠃⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⡇⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠢⡐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⡄⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠈⠑⠪⣁⠂⠀⠀⠀⣀⢤⠊⠁⠀⠀⠀⡠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⢹⠀⠀⠀⠀⡗⢄⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠑⢦⡰⠊⠀⠸⡀⢀⡠⠔⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠇⠀⠑⠢⠄⣀⠀⠀⠉⣑⡠⠄⠊
⣶⣶⡶⠶⢶⠶⣶⣾⣷⣶⣶⣶⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠈⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠘⠃⠀⢀⣀⣤⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠿⠟⠉⠁⠀⠀⠙⠛⠛⠛⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠈⠻⢷⣤⡀
⠀⠀⠀⣠⠀⠀⠀⠀⣠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠀⠀⠀⠙⢿⣆
⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠀⠀⠀⠈⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⢸⡆⠀⠰⠆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢰⠀⠀⢰⡆⠀⠈⠁
⣿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡄⠀⢸⠇
⣿⠀⠀⣿⠀⠻⣿⠗⠒⠲⣦⠀⠀⠀⣤⣤⠀⠐⠒⠲⡶⠿⠓⠀⢹⠁⠹⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢀⡜⠁
⢿⠀⠀⠻⣀⠀⢿⠀⠀⠀⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⢠⠀⠀⠀⠘⣦⠀⠀⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏
⠀⣿⡀⠀⠻⣤⡀⢀⠀⠀⣿⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢰⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠈⠿⣦⡀⠉⠿⣧⣄⠀⢻⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⡄⠲⠃⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠙⠿⣦⣄⠈⠉⠸⠿⠷⠂⢄⡀⠿⠇⠐⠖⠂⠉⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠈⠙⠿⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠶⠿⣷⠶⠶⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠶⠆⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⡁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠲⠶⠶⠶⠶⠄⣀⣀⣀⣀⣀⣀⣀⣀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⢠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⣴⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢸
⠀⠀⠀⣟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸
⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀
⠀⢀⣠⣿⣀⣿⣄⣀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣸
⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⠀⠀⠀⠀⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⠀⠀⠀⠀⣿
⠀⢻⣿⣿⣧⣿⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹
⠀⢸⣿⣿⡏⣿⣿⣿⠀⠀⠀⠀⠙
⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿
⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⣿⡇⣿⣿⣿⡆⠀⠀⠛
⠀⣿⣿⣿⣷⣿⣿⣿⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣾
⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠉⠁⠀⠉⠉⠀⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⣀⣀⣀⣀⣀
⠀⠀⠀⢠⣄⣀⣉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠉⠛⠛
⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⡿⠿⠋⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣾⣷⣿⣶⣿⣿⣿⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠄⠉⠄⠑⠆⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⢠⠁⠀⠀⠀⡀⠀⠀⠀⠀⡤⣀⠄⡠⢀⣠⡀⠀⡀⠀⡀⡀⡠⠀⠀⠀⢁⠀⠀⢀⠴⠚⠙⠉⠈⠀⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠁⠈⠁⠐⠈⠃⠁⠀⠃⠀⠈⠁⠈⠁⠀⠀⢘⠀⡴⠁⠀⢃⠴⢋⠇⡠⠶⠢⡀⠀⠙⣦
⠀⠀⠀⠘⠀⠀⣴⢤⠐⠀⠀⠀⠀⡶⠀⠆⠀⠰⣀⠐⠀⠀⢠⠐⡆⠐⠂⠀⠀⡘⠀⠓⠒⠎⠙⠀⠯⠎⠀⠀⠀⠈⢢⠈⠘⣇
⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣾⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⢇⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⢿⠏⠩⠛⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠁⠈⠀⠳⠆⠀⡟⠃⠀⠂⢟⡿⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⡴⡟⡀⠀⠀⢣⠐⠞⠀⢸⠃⠀⠀⠀⠀⢀⠜⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣀⣼⣋⣀⣸⣄⠀⢀⣠⠔⠉⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣧⣴⡿⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⣀⡤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣁⣠⣄⣀⣤⣤⣤⣴⣦⢾⠿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⠛⠟⡛⠉⠙⠉⠍⠙⠀⠀⢘⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣏⣥⣀⣠⣤⣤⠤⣶⣶⡒⢯⠿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡬⣥⣩⣤⣭⣤⢤⠤⣤⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣤⣤⡉⠉⠀⢀⣈⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣷⣶⣶⣾⠿⠿⠋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠾⠛⠉⠉⠉⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⣀⣠⡄⢶⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣻⣿⣿⠟⢻⣿⡏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⣿⠓⠺⣿⣷⣻⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⣿⠀⣠⡿⠿⠛⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣤⡋⢵⡖⣏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣿⡏⠀⠘⣷⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣭⣯⣼⡁⠀⠀⠈⣹⣌⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠿⠿⢿⠿⠷


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣏⡉⠀⠈⠛⠛⠻⠾⣮⣋⢝⡻⢛⡷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠇⡰⢠⠙⣧⠀⠀⠀⠀⠀⠀⠈⠻⢧⣶⡭⠶⠛⠋⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⢰⠀⡘⢠⢁⠇⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⡏⢠⠃⠆⡰⢡⢃⠎⡐⢰⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡽⣷⡄
⠀⠀⠀⠀⠀⠀⣼⡿⠡⢁⠇⡌⡰⢁⢂⠞⣄⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣄⣸⣯⣴⡾⢿⣆
⠀⠀⠀⠀⠀⣼⣿⢡⠃⠎⡌⡰⠁⢂⠎⢘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢟⠝⡩⣻⣯⢒⢝⣿⡆
⠀⠀⠀⠀⢰⡿⢸⠆⡜⡜⢰⠃⢀⠎⡰⢡⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣕⡡⣊⡼⠋⠙⠓⠛⠛⣿
⠀⠀⠀⠀⢸⡷⠿⣤⣵⢠⠃⢄⠎⡰⢡⣿⣵⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣀⣴⢿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠹⣇⣸⡎⡰⢱⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⢏⠕⡡⢻⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠉⠙⠓⠦⣭⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢕⣕⠡⠊⡠⣺⡇
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡽⢷⠶⢲⠖⡶⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣣⡿⠃⡠⣮⠎⣽⠇
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠎⢡⠃⠠⢁⠇⡈⢸⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠟⡠⢊⠔⢡⢾⡿
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠹⣶⣰⠆⡰⠃⡌⢠⢃⠇⢌⡼⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷⢊⠔⡡⠚⣡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠈⢻⡔⠁⡜⢀⢆⠎⢜⣞⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢊⢔⣩⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠹⡎⢀⢂⢏⢎⢎⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣳⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢹⣣⠏⢎⢎⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⣟⡘⠎⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⣿⡌⡜⡿⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠾⠷⠶⠶⠶⠾⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠲⠲⠲⠲⠲⠲⠞⠗⠖⠟⠗⠛⠃⠛⠛⠙⠋⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠀⠒⠀⠀⠐⠂⠒⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠰⠒⠂⠀⠀⠀⢐⣒⠂⠔⠒⠐⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠠⠀⠍⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⢬⢤⢥⠬⠤⡥⡤⠤⢤⣤⣤⠤⠦⠇
⠀⠀⠀⠀⠀⣀⠔⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠂⠐⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠆⠰⠄⠐⠦⡐⠠⠀⠀⠔⠊⠧⠐⠞⠃
⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠂⠃⠀⠀⢀⠃⠂⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠉⠀⠒⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠐⠊⠑⠁⠀⠠⠠⠠⠤⠀⠐⠂⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣤⣤⣶⠶⠶⢶⡖⢶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠹⠋⠩⢡⣢⡤⠄⠒⠛⢶⣤⣶⣾⣿⣿⣯⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠆⠀⠀⠀⠀⠀⠀⣠⠹⣿⠀⠀⠀⢠⣿⣯⣼⣿⡿⣷⡝⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⢠⣴⡾⢻⡁⣰⡿⠀⢀⣠⠾⣣⠞⠟⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⣀⠀⠈⠿⠁⠘⢻⣿⣿⠞⢿⣿⡟⠿⢴⣶⡁⡀⢠⣆⠀⠀⠀⠀⠀⠀⠸⣤
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠼⠁⠀⠀⠀⠀⠻⡆⠀⠀⠀⣤⠴⢿⣏⣤⡄⢀⣀⣀⡎⡙⣿⠇⠛⠃⠀⡄⠀⡀⠀⠻⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠎⣠⠀⠀⠀⠀⠀⠀⠈⠙⣁⣴⣤⣥⠶⠟⠗⠀⠁⠈⢻⣿⣶⣗⠂⠀⠀⠀⢈⢀⠸⡿⠗⠀⠘⠂
⠀⠀⠀⠀⠀⡄⠀⢰⣿⠀⠈⠀⠀⠀⠀⢀⣼⣿⣿⠉⠀⠀⠀⠀⠀⢤⠀⠀⢹⣿⠿⠇⣠⣶⣾⡆⣄⢠⣄⣤⡀⠀⠠⣤⣾⡄
⠀⠀⠀⠀⠀⠀⢠⡿⣿⡇⢠⣿⣟⢳⡀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣤⡤⠝⠛⠋⢻⣿⡿⠟⠛⢣⡀⠀⠈⢸⣧
⠀⠀⠀⠀⠀⠀⢸⣗⣿⣆⠸⠋⣽⣭⡻⠿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠛⠁⣾⡿⠏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⢆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠘⠛⢿⢿⣿⡄⢹⡟⣿⣤⣌⣻⣶⣧⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢻⡇
⠀⠀⠀⠀⠀⠀⡄⠀⠈⠈⢿⣷⣌⠛⠛⠛⢿⣿⣿⣿⡗⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⣸⠃⠰
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠛⢠⣹⣿⠃⠀⡤⣄⢨⠙⠻⣧⡀⠀⠀⠀⢀⡿⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⡿⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠸⣦⠦⡀⠀⢻⡁⠞⠉⠀⠀⠀⠀⠀⠈⠧⣀⠀⠀⠀⠀⠈⠿⢇⣀⣀⣠⣤⣶⣤⡆⣿⠿⠏⠀⠖⢀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⢄⠢⠈⠢⢀⣵⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠑⠢⣿⡷⠀⠀⠈⠉⠉⠿⢿⣿⣿⣇⢀⡖⠀⠀⡀⣠⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢷⣄⠀⣆⣄⡀⠀⠁⠀⠀⠀⠀⣠⣿⡾⠆⠀⣀⠀⠀⠔⠙⣿⣿⠘⢶⡀⢀⣼⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠱⣬⡀⠀⠀⠹⡇⡿⠉⠿⣠⡄⠀⠀⠀⢾⣿⡟⠀⠀⠈⠁⠀⠀⠀⣧⣾⡇⠀⠀⢠⡞⠝⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠳⣧⢃⠀⠀⠉⠀⣰⣿⣿⠋⠋⠀⠀⠀⠀⣀⡀⠴⠾⠿⣿⡇⢀⠔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣞⠻⣯⡀⠀⣠⣾⣿⣿⣋⣠⠤⠂⠀⢚⠁⠄⠀⢀⣀⣼⠿⢿⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠙⠻⠿⣦⣿⣿⣿⣿⣿⠿⠟⠛⠟⠛⠛⠛⠶⢿⠿⠟⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠐⠚⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠦⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⣄⠀⠀⠀⠀⠀⣀⣼⡟⠳⡶⠦⣄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠘⣾⠘⠀⢰⣸⣦⢷⣦⠵⣬⢕⡒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠁⠀⠉⠢⣀⠈⠳⣦⣞⡉⠙⠒⢄⣑⣦⡱⢬⣦⣄⠀⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⣎⣵⣄⣈⡏⠉⠑⣦⣞⡹⣄⣿⣾⣷⣿⣦⠒⠒⠘⢻⡏
⠀⠀⠀⠀⠀⠀⣀⣠⠤⠗⠚⠋⠛⠒⠒⠒⠒⠒⠢⠤⠼⠿⣟⣋⣱⣶⢇⣾⣾⣿⣿⣿⣿⣿⣿⣤⠤⠒⠁⠱⡄
⢀⣀⡠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂⠶⡒⠲⣶⠞⠋⠁⠀⢸⢲⣿⣿⣿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⢸
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢀⣿⣿⣿⡤⠤⠴⠒⠋⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⡆⢀⠜⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢠⢖⣂⣐⣼⣲⠝⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠴⠚⠝⠀⣀⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢏⣽⢿⣾⣮⣿⣟⠶⠃⠀⠀⣸
⠀⠀⠈⠉⠉⠉⠁⠀⠹⣿⣿⣶⣖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣢⠴⣵⣿⢿⠽⠿⠟⠩⡗⠆⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠟⠁⣄⢶⠬⣯⠿⠀⠀⠀⣀⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⣀⣠⠴⠾⠛⢟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⠷⠶⢶⣶⠶⠶⠖⠚⠋⠉⠉⢀⣀⡤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⡀⠀⠀⢀⡄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⣀⣀⣤⣠⡀⠀⠀⠀⠀⠀⠀⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢷⠀⠀⠀⣰⠏⠀⠀⠄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠁⢳⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢈⡇⠀⢰⠗⠀⠀⠀⠀⠀⢀⠀⠀⡽⡆⠀⠬⢻⣿⣿⠀⠀⠀⠀⠀⢀⠀⣧⠘⡏⠫⣌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄⠘⢣⣀⡏⠐⠀⠀⢰⠁⠤⣿⠀⣾⡇⠍⠀⠃⠸⢟⠃⢀⣄⢐⠀⠀⠀⠉⢸⣦⠇⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⢁⠀⣓⠘⣻⣦⣾⣿⣿⣿⡄⠲⠇⠀⢸⣿⣿⣾⣿⣄⣄⡢⠀⢀⠜⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠣⣸⢷⣿⣿⣿⣿⣿⣿⣿⣶⠖⠃⢸⣿⣿⣿⣿⣿⣿⠶⠿⠂⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠶⠖⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠱⣄⡐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣤⣾⣯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢻⣿⣿⣶⣤⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⢩⠑⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣻⣿⣿⣿⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣿⡇⡸⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣡⠈⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⢧⡇⡟⠅⠳⠛⣹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣯⢠⠌⠀⠀⢷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣹⠁⡿⠀⢄⠀⠩⡸⢿⢻⡇⠠⠀⡤⠄⠀⠀⠀⢦⠲⠂⣻⠙⡷⠄⡀⠀⢀⡔⠀⢸⠀⣡⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⡇⠀⠀⠣⢀⠀⠐⠊⠙⠠⠂⡄⡐⢰⡀⠂⠰⠂⣈⠉⠀⠐⡀⠀⡀⡀⠈⠀⢸⡆⠇⢊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⠿⡇⠆⢀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢂⠉⠀⠰⠀⠀⠀⠀⢈⡗⡾⣭⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡶⠚⠛⢲⣈⣁⡤⠖⠲⣄⣀⣀⣀⣠⠤⣤⠴⢦⡐⠀⢈⣈⡤⢄⡀⣀⡠⢤⣀⣠⠼⠷⠦⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠄⠲⠦⠭⠀⢀⡒⢒⣮⠭⠥⠀⠀⠀⠀⠀⠒⣭⡭⠽⢃⠬⢀⣈⡁⠀⠀⠀⠒⠩⠭⢝⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣧⠀⠀⠀⢀⡀⢰⠇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠰⡆⢽⢿⢀⣀⡀⢸⠅⡿⠀⡿⠀⠀⠀⢀⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣆⢠⣀⡄⢻⢸⣿⣸⣿⣯⣾⢰⠃⢰⣇⣤⣦⢄⣾⡟⣠
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡙⣟⣎⣻⣾⣿⣤⡏⣿⣿⣿⢷⡎⣶⣼⣿⣿⣯⢞⢟⡼⠁⠀⠀⠀⢀⡄
⠀⠀⢀⣶⣄⡀⢀⣤⢆⣷⣜⣎⢻⣿⣿⡜⡇⢸⣿⡟⡿⢠⡇⣸⣿⣿⠏⣼⣾⣶⣷⡶⣠⣴⣯⠞
⠀⠀⠀⡙⠻⣝⡮⣝⣿⣿⣿⡻⣇⠹⣿⠇⢣⠸⠉⠀⠃⠸⠁⠜⣿⢃⡞⣽⣿⣿⣯⢞⡵⢋⣡⠆
⠀⠀⠀⠉⠓⢦⣙⠮⣙⠿⣿⣿⡊⢦⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠎⠸⣿⠟⠫⢖⣥⣶⣩⣀
⠀⠀⠀⢀⣼⣶⣿⣿⣟⡦⣌⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣤⣾⣿⣿⡿⢛⣀⣤⣴⣶
⠶⢤⣶⣶⣬⣽⣿⣿⣿⣿⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠿⠿⣛⣻⠭⠿⠚⣋⣉
⠀⠰⠦⠤⢭⣭⣉⡛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣏⡍⠁
⠀⠀⠀⢰⣾⣿⣿⣿⣿⣶⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣿⣿⣿⣿⠿⠿
⣀⣀⣠⣬⠭⢽⣿⣿⣿⣿⡻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⣶⠦⢶⡗⠞⠯⠽⢿⣿⣿
⠹⠶⠛⣛⣋⡭⣵⣿⣽⣶⣷⣦⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢿⣿⣿⣿⣿⣏⡛⠒⠲
⠀⠀⠈⠉⠀⢲⣿⣿⣿⡿⣋⣠⢖⣡⣀⢄⠀⠀⠀⠀⠀⠀⠀⡄⠠⣄⣬⣝⢦⣬⣛⠿⣛⠿⠇
⠀⠀⠀⠀⠀⣈⡵⣾⠵⣪⣿⣵⣿⣿⢫⢎⣰⣴⣀⡄⡇⢲⣠⣜⢦⠘⢿⣿⣷⣿⣦⣝⠻⣽⣲⣄⣀
⠀⠀⠀⠴⣿⠷⢋⣥⠞⣹⣻⣿⣿⢷⣯⣾⣿⣼⣿⣧⢷⠘⣿⣿⣯⣿⢮⢿⡿⣿⡷⠈⠳⠆⠉⠛⠉
⠀⠀⠀⠀⠁⠀⠉⠀⠀⠘⠟⣴⢋⡾⣿⣿⢇⣿⣿⣿⣿⢦⢻⣿⣿⡿⣦⠳⣵⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢡⡟⠀⠻⡿⢸⢼⣿⡟⢻⠸⡿⡜⡿⠛⠈⢧⡙⢮⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠀⠀⠀⠀⣿⠈⠁⠀⢸⡄⢷⣷⠀⠀⠀⠀⠀⠈⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠛⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⠀⠆⠘⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡴⠃⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⣄⠀⢀⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⣤⣤⡀⠀⠘⢷⣾⣿⣿⣿⡟⣿⣿⣿⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠛⠀⠀⠈⠟⠛⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠿⠛⠋⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠻⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⢀⡠⢤⡀⢄⡀⢧⠀⣠⢠⡀⢀⠀⠀⠀⢠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡁⢠⠕⠂⠉⠻⣄⢹⡌⡆⣧⢰⠤⡈⠘⢻⠟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⠛⠂⠈⠈⠃⡿⠛⡙⢉⣉⣣⠜⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣉⣙⣿⠄⢚⢀⠚⠛⢡⠄⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠇⠀⠈⢂⣠⣤⣋⣬⡛⣿⠤⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠸⣿⣉⠗⠘⣻⠇⠈⣣⠈⠲⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠈⠻⢧⣭⣰⣚⠟⠉⡤⡓⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢽⣶⢖⠢⣐⠠⡦⣤⢄⠀⢨⣿⠦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠼⢠⠇⡜⢠⠀⣈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠑⠢⠄⠦⣌⠀⠐⣀⢀⡐⠀⠀⢀⡴⢂⡀⠀⠀⠀⢐⠀⡄⠀⠠⠛⣼⢃⢿⡀⡹⢄⠁
⠀⠀⠀⠀⠀⠀⠐⡀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠆⠒⠄⠀⠄⠄⠌⠻⣦⡘⠃⡄⠻⣀⣴⠟⠂⠀⢀⡀⢀⡀⠉⠎⠀⠀⠂⡀⠁⣈⢦⣹⡂⠂
⠀⠀⠀⠀⠀⠄⠀⢋⠃⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣤⣰⣦⣄⣬⡇⢾⣿⡇⢸⣧⢀⡃⣠⣠⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠇⢠⣉⠄⡁
⠀⠀⠀⠀⠀⠀⠀⢬⡛⢮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠉⠡⠘⣀⠉⣶⡻⠮⠭⠴⠿⣯⠍⢩⠰⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢦⡊⢱⡆⢸⠃⡶⠀⠀⠀⠀⠀⠀⠈⠁⠠⠒⠈⡠⢺⣟⠓⠶⠶⠶⠛⣻⣷⡀⠁⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠰⣄⠛⠀⠃⡞⢸⢁⡟⣀⠀⠀⣧⡄⠀⠀⠀⠀⠘⣡⠏⣍⡛⠛⠛⠛⣋⣹⠙⢬⡀
⠀⠀⠀⠀⠀⠦⠤⣿⣇⠀⡁⠛⠛⣋⣈⣑⠚⠙⠻⠦⡀⠀⠀⠀⠁⠀⠘⢋⣛⣛⣛⣩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡀⠠⢬⡐⢃⣠⡦⠎⡩⠍⠿⣦⣄⠀⠐⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⢀⠴⡁⠘⣡⠔⡿⣶⣤⡼⣼⣭⠀⢠⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠙⠪⡿⠶⠥⡞⢁⠀⠐⠂⣠⣾⠖⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⣉⡱⠶⠎⢀⢋⠀⠉⢉⣉⠃⡀⠀⠐⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠀⡌⠩⠤⡤⡖⢲⢹⡌⣇⠲⢆⠉⠋⢠⡓⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣦⠤⠈⠁⠰⠃⢷⢘⠆⠰⡏⠐⠄⡓⠂⢉⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠂⠀⠁⠐⠉⠀⠜⠈⠂⠄


⣴⣶⣤⣦⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤
⣿⣽⣭⣭⣽⣷⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣶⣿⣿⣿⣷⣶⣾⣶

⠀⠀⠀⠀⢀⣤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣆⠂⡀⠄
⠀⡀⠠⠶⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⣿⠼⡄
⠛⠱⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣷⠕⠿⣆⣦⣤⣄
⠀⠒⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⡰⣦⣿⡟⣟⠋
⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡶⢷⠉⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡎⣀⡏⣿⣿⣾⣾⣷⣾⣷⡴⠂⠴⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⠀⠶⠮⠁⠀⠀⢻⣿⣿⣿⠛⣻⣿⣿⠃⠀⠀⡉⠉⣴⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠚⠏⠁⣿⣇⣶⣿⢷⣿⠝⠀⠀⠀⠁⠀⠀⠈⠉⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⢈⣻⡆⠀⣀⠉⠀⠀⠉⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠐⠀⠀⠀⠒⠊⠥⣦⣾⠅⠀⠀⠀⠠⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣆
⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⡀⠙⣹⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⠁


⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢡⠈⣽⣩⠈⠍⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠄⠄⠤⠿⡄
⠀⠀⠀⠀⠀⢨⠀⠄⠠⣄⣄⠀⠀⠀⠀⠀⢸⠂⣿⠀⣧⢸⠇⡾⠄⢀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠛⠂⠣⠀⠐
⠀⠀⠀⠀⠀⠜⠀⠈⠀⢉⡹⠀⠀⠀⠀⠀⢀⡂⠀⢄⢰⠀⣀⡠⠨⣄⠀⠀⠀⠀⠀⠀⠈⠾⣧⢠⠐⣤⢶⡀
⠀⠀⠀⠀⠀⡆⡄⣤⡄⡖⠦⡄⠀⠀⠀⠀⠐⢱⣈⠊⠈⠙⠈⠀⠀⠛⠀⠀⠀⠀⠐⠀⠰⠀⠦⠄⠀⠦⡌⠁
⠀⠀⠀⠀⠀⠀⢃⡅⠀⠀⠐⠂⠀⠀⠀⠀⠘⠘⠉⠛⠲⠘⢻⠐⢺⣐⢷⠀⠀⠀⠀⠀⠘⣄⣾⣀⣠⠄⢰
⠀⠀⠀⠀⠀⠀⠀⢹⠊⠗⠊⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⠊⠀⠀⠀⠀⠀⠀⡁⠈⠁⠄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠄⠂⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡇⠆⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠄⢅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣤⣤
⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡓⠀⡀⣄⡄⢹⠀⡇⢒⡀⠀⠀⠀⡄⡤⡤⠀⠤⠬⠤⠴⠴
⠀⠀⠀⠀⠀⠜⠶⠆⠰⠳⠇⠶⠧⠂⠀⠀⠐⠂⢀⠀⠛⠆⠂⠃⠀⠐⠂⠀⠀⠀⠀⠁⠤⠄⠤⠰⠦⢦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⢨⠠⠄⢔⡠⠦⠤⠀⠀⠀⠀⠀⢤⡤⠤⠤⠰⠄⠀⢇⡤⠤⠰⠼
⠀⠀⠀⠀⠀⢇⠦⠂⠢⠲⠘⠓⡖⠀⠀⠀⠤⣤⢾⡤⠆⢴⣴⢠⡠⠄⡄⠀⠀⠀⠐⠀⠀⠠⠄⠁⢄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⣴⣧⣴⡄⠀⢀⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣾⡿⢿⣿⣧⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣍⠛⠛⠻⢿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠹⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⣿⣿⣿⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣴⣿⣷⣤⣄⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠻⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠉⠉⠁⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⠴⠖⢦⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡏⠀⠀⠀⠀⠙⡆⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣠⡀⡄⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠳⣄⠀⢀⠚⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣦⢾⡇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠶⠶⠶⠾⠛⠛⠛⠛⠶⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠲⠶⣤⡀⠠⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡴⡞⠛⢲⣶⣌⠙⢦⡀⠀⠀⢀⡞⣉⠀⠀⣶⣶⣦⠀⠹⣦⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣾⢼⣿⣿⣿⣿⡿⠀⠀⣿⠀⠀⣿⢰⣿⣷⣾⣿⣿⣿⠀⠀⣿⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⠈⠷⠿⠿⠛⠁⠀⠀⣾⠀⠀⣧⠈⠻⢿⣭⡭⠟⠁⠀⠀⣾⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⠘⢧⡀⠀⠀⠀⠀⣀⣼⡏⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⢋⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣨⠁⠀⠀⠉⠛⠶⠶⠟⠋⢠⡇⠀⠀⠀⠈⠻⠦⣤⣤⣤⡴⠞⠁⠀⠀⠀⠀⠀⣉⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⣠⠶⡿⠀⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠰⣷⡆⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⢻⠓⠓⣠⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠟⢉⣧
⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠂⠀⠉⠉⠉⠀⢀⠔⠋⠹⣆
⠀⠀⠈⢻⡈⣷⠀⠀⠀⠀⠀⠀⢠⡶⠛⢦⣈⡙⠛⠒⠲⠦⠶⠖⠊⠉⠙⢲⣦⣤⡴⠚⠁⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠘⡅⢘⠀⠀⠀⠀⠀⢀⡼⠄⠀⠀⠈⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⢸⠇⠘⣦⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢷⠀⡇⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣄⣀⣾⡁⠀⢹⣆⣀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠸⡆⣧⠀⣴⠋⠉⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣷⢸⣴⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠸⡀⠛⠀⢀⣤⡶⢿⠀⠀⠀⠀⠒⠛⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⡇⢠⡴⠋⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠳⠋⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠂⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠅⠀⠀⡸
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣴⣿⣾⣶⣶⣴⣷⣶⣶
⣿⣛⣻⣛⣛⣛⣛⣃⣙⣛⣿⣛⣛⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⣭⣭⣭⣭⣽⣿⣯⣭⣭⣽⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⣀⣀⣀⣠⡠⠻⡘⠇⠀⢰⠀⠀⡜⢰⣤⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠉⣻⣷⣾⡿⠿⠟⠻⣿⠇⢲⠃⠀⠀⡌⡀⠀⠁⠆⡾⢸⣤⣀⡀⠀⠀⣀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠨⢦⣬⣀⡠⣍⡇⡰⣾⠛⠉⠀⠀⠀⠀⠀⡘⠀⠀⡆⠸⠰⢀⠀⠘⠘⠰⢠⡎⠉⠉⠻⠿⣖⠻⣷⡞⢁⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠳⢖⣓⡹⣈⣁⣿⡄⣉⢐⠷⣏⡄⠀⠀⠀⠀⠀⢰⢁⠂⠰⢀⠀⠀⢸⠠⡇⠃⡇⢻⠃⠀⠀⠀⢀⡜⡫⠞⡠⢿⣋⣽⡷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡙⡿⠿⢩⡧⢷⡴⣦⠛⠖⠁⠀⠀⠀⠀⠀⢀⡏⠇⠀⠞⠀⠀⠀⠀⠀⡘⣸⡌⡿⠀⠀⠀⠀⠀⠉⠱⠯⠙⠫⠞⡿⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠀⠛⠉⠃⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⣾⠀⢰⡳⠰⠀⠁⢰⠙⢸⠇⢻⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣄⠀⠀⠀⢀⠇⠀⠆⠃⠀⠀⠀⣈⠆⣿⠀⣦⢸⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠛⢻⣿⠇⠀⠀⣼⠀⠈⠂⠌⠶⠀⠀⠘⡆⢠⠐⢫⡏⠀⣼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢧⡄⢰⡿⠛⢷⣤⡀⠸⠀⠌⠀⠀⠀⠂⡀⠰⠁⡎⡄⢸⡧⢶⠿⣿⣿⣿⣿⣿⡿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣠⠟⠁⠀⠀⠈⡹⢷⣿⣄⠀⠄⠀⢠⠀⠐⢰⢁⠃⣿⠀⠀⠀⠘⢿⣻⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠙⠷⣦⣤⣎⡀⠀⢸⠹⠀⡟⠀⠀⠀⢘⣀⡥⠬⢩⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠃⠐⠀⠀⠀⠈⡛⠏⢹⣆⢨⡇⠄⠇⠀⢀⡶⡋⠁⢀⡈⠉⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠣⢤⡀⠹⠳⣄⣸⠀⠀⠸⠊⠠⣄⠀⠀⠀⢉⣾⣿⡟⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⡄⠀⠀⣤⠬⢒⠘⠌⠁⡀⠒⠶⠀⠆⢿⡁⠀⠐⢉⣜⣟⢐⣝⡇
⠀⠀⢀⡀⠀⡴⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡈⠀⠐⠀⡁⠀⠀⣾⠦⡹⢎⠒⢧⣈⠃⡠⣨⣶⢿⠂⠀⢁⣼⣟⣿⣿⡟
⢠⣤⣀⣀⣾⡧⡾⡋⢩⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⢠⠄⣘⠀⠙⠄⠀⣯⠁⡷⠃⣵⠀⠈⠑⠿⠵⠛⣻⠀⢀⣨⣿⣟⣿⡏
⠿⠛⠉⢉⠔⠴⠀⠐⠸⡠⠋⡔⠶⣄⠀⠀⠀⠀⠀⠀⠀⣴⠘⠆⢄⠀⠀⡄⠀⠀⢹⢸⡋⣿⠍⠀⠀⠀⠀⣧⠛⠀⠀⠉⠏⠘⢻⡝⣧
⠇⠀⠀⢀⠔⡁⢀⠀⠀⠀⠉⡐⢀⡜⠑⢦⡀⠀⠀⠀⠀⠇⠘⠆⢸⠻⡄⠠⠐⠀⠘⢈⡈⡟⠃⠀⠀⠀⠀⢁⡀⡀⠀⠀⠀⠈⣫⣴⠿
⣷⣶⣷⣧⠔⢠⣠⣦⣠⣀⢎⢠⠎⢀⡄⡜⠙⠷⠀⠀⠀⣸⡀⠀⣀⢀⠀⠀⠀⠀⠘⠘⢇⣧⠆⠀⠀⠀⠀⣖⠉⠹⠛⠛⠛⠛⠋⠛⠆
⠀⠉⠁⠉⠀⠉⠉⠉⠉⠙⠛⠛⠓⠚⠐⠓⠛⠀⠐⢲⣾⡁⠙⠛⢋⠃⣠⠁⠖⡒⣂⢀⢸⢻⣿⡀⠐⠒⠀⠌⠁⢠⣤⣄⠀⣠⣤⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢉⢙⡂⠉⠉⠁⠄⢠⠀⠁⢪⠀⠈⣵⣗⠀⠀⠸⠃⠀⠴⣰⠂⢆⡴⢞⣴⡇⠀⠀⠀⠒⠂
⢰⡾⠀⠄⣀⡀⠀⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣈⠈⠁⠀⠇⠀⠈⠀⡄⢲⢈⣶⠇⢻⢦⣿⠇⡀⠀⠀⣀⠁⠀⢠⠀⠉⣹⠀⠐⠂
⢸⠦⢀⣦⣉⠀⣦⣄⣀⣈⡉⠛⣶⣤⣤⣄⣀⠀⡸⢇⠀⠀⠀⡄⢘⠀⠀⠰⠀⠀⠀⡄⢹⠚⠸⣬⡏⢰⠀⠀⣀⣻⠍⡅⠘⠋⠒⣿⠀⠀⣀⡀⣀⢀⣀
⣀⣀⣀⡀⢑⠌⠁⠈⢙⠯⠙⠛⣛⡛⣯⣿⣿⡿⠁⠁⣰⣿⢘⡄⢀⣆⠠⡀⠄⠀⠀⡧⣿⣖⢛⢸⣿⣈⡄⠀⠙⣿⢲⠗⢀⡤⠞⣧⣂⠛⣠⡠⢆⡽⣻
⣼⣀⣀⠽⣋⠈⣉⣉⣉⣿⣿⣿⣷⣾⠟⠛⣽⡶⣶⠂⢹⣟⢙⣟⢱⠸⠇⠢⠀⠈⠀⢡⢈⡧⠈⢿⣯⡹⣇⠀⠪⣟⣼⣻⣿⠁⠈⠐⣿⣾⢥⡖⠛⢰⡄
⣌⡀⠀⢳⣌⢋⣿⣸⣧⣀⣉⣿⣿⣷⡶⠾⣝⠃⣀⠾⠷⠽⠿⠞⢉⣟⠤⣴⡆⣒⣆⣠⣆⣠⡐⢾⡯⡷⣯⠀⠰⣆⣿⣿⡿⠇⠊⡀⡇⠀⡻⢁⣀⠌⠓
⣈⠹⣦⢠⣷⡞⡶⣿⡛⠍⣿⣿⣿⣿⣷⣾⡿⣿⠛⢦⣄⣠⣤⠶⣿⣾⣻⣉⣽⡭⠃⣨⡰⢛⠇⠈⢟⠛⣇⠀⢠⡟⠻⢽⣷⠀⢠⣬⣇⠘⠑⠃⡦⠠⢾
⠀⣤⠹⣇⣉⣽⣻⣿⣧⠀⢽⣿⣿⣿⣿⠋⡁⠀⠀⠤⠛⠛⠃⠈⣉⠀⠉⣿⠆⠀⡺⠛⠀⢙⡀⣮⡄⢆⡛⠀⠘⠁⢀⡘⠘⡀⠚⣡⢇⣆⣤⣠⢿⡰⣄
⣦⡉⠲⢷⣹⣦⣾⡇⢸⣶⣄⣿⣍⣹⣄⣀⣈⣁⠒⠛⠻⠲⠾⢿⣿⢛⡛⣿⠈⢻⡧⢓⠨⠘⠋⠁⠠⠋⣁⣈⠉⢠⡽⠿⠓⢲⡌⢠⠼⠋⡁⢈⣉⠡⡌
⠫⠓⠖⠺⠷⠄⣉⡳⠿⢧⠾⢿⣿⣟⠙⠿⠄⢴⣹⣖⡚⠛⠆⢘⡉⢉⡀⢻⣓⣾⣆⣉⣥⣤⣬⢴⣤⣤⠈⠉⣈⣁⡴⠶⣶⣶⣖⠒⠒⡉⠀⢉⣄⠒⢠

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠁
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠡⠀⠀⠀⠀⢹⡀⠀⠀⠐⠂⠀⠀⢀⡀⠀⠸⡋⠁⠋
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠠⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠉⠃⠀⠘⠠⠤⢀⠄
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⢀⡄⠀⠀⠀⢠⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⢠⡞⠃⠀⠀⠤
⠰⠒⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠮⠒⠃⠀⠀⠀⠁⠀⠀⠀⢰⣞⣵⣋⣒⠀⠈
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⡔⠊⠙⡐⠀⢰⡗⠞⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡤⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⣶⢖⡶⠶⠶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⣾⡖
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⢀⡤⡿⠉⢉⡛⠻⡿⠿
⠀⢀⡏⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣖⠎⠨⣴⠖⣠⣴⠀⢸⡀
⠀⡸⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡈⠀⠀⠀⢸⠀⡈⠓⠂⣈⠧⠀⢰⠗⠃⠀⠀⠀⠢⠄⢁
⢠⠇⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⡀⠈⢎⢉⠞⠀⢀⡜⠀⠀⠀⢀⣠⢦⠐⠂
⣜⣀⡠⢴⠀⡞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣔⡠⠴⠊⠀⡀⠀⠌⠐⠉
⠉⠁⠀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠂⠀⠀⠉
⠀⠀⠀⠘⡇⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⡄
⠀⠀⠀⠀⢳⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠
⠀⠀⠀⠀⠘⣇⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠤⠼⡀⠑⠊⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠙⠢⢌⡁⠒⠒⠒⢲⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢰⠃⠀⠀⠀⣀⢴⣁⡓
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠫⢙⣒⣒⡊⢕⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⢆⠀⠐⢈⡤⠚⠀⠈⢂⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠘⠤⠔⠊⠁⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣄⣀⡀⠀⠀⣠⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠉⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⣀⠠⠄⠋⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄⣰⡃⠀⢠⠾⠁⠀⠀⠀⠀⠀⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣿⢸⣄⣿⣤⠀⠀⠀⠀⠀⠘⣇⡿⠤⣄⡀⢀⣀⣤⡶⠚⠛⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⢂⠀⢠⣀⣿⣏⢿⡸⣾⠜⡀⠀⠀⢐⡷⣹⠇⣤⣼⡻⠉⠁⠉⠃⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠿⢶⠶⠿⣾⡛⢉⠁⠙⣦⣕⠉⠀⠹⠦⢞⣩⣶⠋⢰⠏⡩⢀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡠⡾⣆⠀⠀⠀⠪⣻⣯⡀⠘⠀⢸⠋⠛⠒⣿⢛⣿⡶⠷⠿⣧⣧⡊⠀⠀⠀⠀⢀⢸⡆⠀⠀⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠀⠳⡰⣄⢀⠘⣿⣶⠂⠀⢀⠀⢀⣠⡿⠋⠁⣠⠀⡄⠀⡹⠛⢶⣇⠀⠀⠈⠸⠇⠰⠆⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⢁⣟⠄⠈⢸⠀⢸⣏⠀⠀⠀⠜⣧⠀⠀⠀⠀⠀⢘⣿⢠⢸⠀⡔⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠆⢀⠀⠀⠀⠀⠀⡀⣿⠀⣼⡏⠀⣀⣿⡀⣾⡏⡆⠀⠀⠀⠙⠀⠀⢠⠀⠸⡇⣹⠋⢉⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠨⠣⣠⣠⡀⠠⠀⣩⠾⠷⣤⣈⣙⣋⣥⢾⡋⡇⠀⠀⠀⠀⢰⣦⣾⣇⡘⢠⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠖⠚⠋⠁⠀⠀⠀⠉⠉⠉⠀⠀⠳⣌⠂⠀⡀⡿⢸⢿⣿⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣡⣾⡤⠴⠞⠉







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠛⠉⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⢾⣭⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠉⠁⠀⠀⢀⣤⣤⣘⣿⣧
⠀⠀⠀⠀⠀⢀⣿⡿⠶⠶⠶⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⣤⣄⢸⡇⠀⠉⢻⣿⡆
⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠻⡾⠇⠙⡆⠀⠀⣿⣷
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠚⢶⡄⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣤⣤⣤⠾⠃⠀⠀⢹⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣠⠟⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⣀⣠⡼⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⣾⠉⠀⠀⠀⠀⠀⠀⠈⠷⠴⠶⣦⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⣠⠞⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⢹⣿⠏⠀⠀⢻⡄⣀⣤⣄⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⠛⠶⣤⡀⢠⡴⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣼⡿
⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠈⠛⠋⠀⣼⠃⠀⠀⠀⠀⠀⣯⡈⠙⠛⠻⣆⠀⠀⠀⢻⣾⡄⠀⢀⡴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⠈⠷⣤⣤⣤⠟⠀⠀⠀⣾⠀⠙⠓⠋⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢳⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⣀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⣠⣿⠀⣠⡴⠶⣄⠀⠀⠀⠀⠀⣰⠟⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠹⢦⡀⣹⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⡙⠋⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣶⣤⣄⣀⣀⣼⣇⣀⣀⣈⣀⣀⣀⣤⣤⣶⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡆⢰⢰⠖⠀⠀⢠⠤⡀⠀⢀⣆⠀⠀⠀⠀⠀⠀⢰⡋⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⢸⠀⠀⠀⢸⢨⠘⣖⠀⡇⠀⠉⠉⠙⠒⠢⠄⠉⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢺⢸⠖⠀⡴⣿⢸⠀⠘⡆⡇⢠⢶⡆⣇⣄⢀⡔⢀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠉⠀⠉⠙⠈⠁⠉⠁⠉⠈⠉⠁⠁⠈⠉⠉⠉⠁⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠈⡈⠁⣁⠉⣁⢈⡉⡀⠉⠁⣉⢁⡈⣉⠀⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⢸⠉⣦⡇⢸⠁⠀⡇⣿⠈⣃⡇⢸⣇⣿⠀⠀⠀⣀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠦⠤⢉⡉⠀⠀⢸⣞⠁⣇⢸⡄⠀⡇⣿⣀⡇⣇⢸⡘⣿⠀⠀⠨⠥⣄⠒⠒⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠋⢁⠓⠈⠓⠀⠃⠘⢙⠁⠓⡈⠃⠉⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⣈⡁⡁⠉⢈⣈⠀⣈⡁⣁⡁⣈⠈⡀⢁⠀⣀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠘⢿⣧⠃⠀⠈⡯⡆⣷⠋⡇⠀⡏⡇⡟⢻⢸⡈⠁⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡏⡠⣤⢤⢁⡄⡷⠀⡯⡇⡿⡀⣧⠀⣇⡇⡇⢸⠀⢳⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡗⠈⠁⠀⠉⠃⠁⠀⡛⠀⠃⠃⠑⠂⢉⡅⠃⢘⠀⠛⠁⠄⠀⡡⠐⠢⡄⣊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⠙⠁⢰⢢⢰⢰⢰⢢⠘⢤⠵⠂⠀⡀⠀⢸⡇⡀⢸⠁⠀⠀⢐⠀⢄⠂⠐⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠀⠤⣒⠀⠀⢸⣼⢠⢻⢸⡸⠀⣦⣄⣠⡀⡠⡤⢺⠁⡇⢸⠀⠶⡆⠀⠀⢀⠉⠃⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⠨⠥⣒⠀⠀⠠⡤⠀⠀⠄⠄⠀⠃⠘⠋⠚⠓⠳⠘⠞⠓⠘⠒⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠬⠭⣄⡀⠀⠀⠀⠀⠀⣶⢠⡆⠀⠀⠀⢠⠄⠀⣀⠀⣀⠀⣀⢓⠃⣀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⢠⠀⢾⠀⢱⡄⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⢸⠀⢸⡹⢠⠁⠇⣯⢸⢸⠝⡇⣷⣸
⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⡈⡇⢀⣧⠖⢠⢲⡆⡇⣸⠁⡴⠆⣰⢺⠀⢸⣃⠘⠄⡀⢹⢸⢸⡯⡇⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣈⣁⣉⣡⠞⠈⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠙⠀⣘⣉⣀⣓⣁⣘⣘⣀⣓⣁⣃⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠠⣤⡀⠀⣠⡄⠀⣤⡀⢠⣤⣄⠀⠀⠠⣦⢠⠆⣤⡤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⢰⣿⠁⠀⣿⣷⠀⠘⣿⣆⠀⠀⠀⠈⠀⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣼⡟⠸⣿⡆⠀⣿⡟⠀⠀⠘⣿⣧⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠈⠉⠈⠁⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠘⠆⢠⣾⠀⢸⣦⡀⣾⣷⠀⠀⠂⠘⢿⣧⠀⠀⠸⠃⣿⣷⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣯⠀⠈⣿⡇⣿⡿⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡀⢠⡆⠸⢿⠀⢸⠟⠁⠻⣿⡀⠀⠀⠀⠀⠈⢿⡇⠀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⣀⠀⣀⡀⠀⣀⡀⠀⣀⠀⢀⡀⠀⠀⣀⡀⠀⡀⠀⠀⠀⣀⠀⢀⣀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⣿⡇⠙⠀⣿⡇⠀⣿⡇⢸⣿⡇⡈⠁⠀⠀⣿⡇⠀⢻⣧⠀⠸⣿⡄⠈⢿⣧⠀⠀⣾⣄⣀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⣿⡏⠈⣿⡇⢸⣿⡏⠛⠀⠀⠀⣿⡇⠀⣸⣿⠠⠀⢿⣷⠀⢈⣿⡇⠀⡉⠛⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠐⠿⠃⠀⠻⠇⠘⠿⠃⠰⠟⠀⠀⠿⠇⠐⠛⠁⠂⠀⠘⠿⠇⠐⠿⠇⠀⠛⠂⠀⠋⠐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⣠⣤⠀⠀⢠⣄⠀⠀⢠⣤⠀⠀⠄⠀⣤⡄⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⣿⠀⠀⠸⣿⡆⠀⢸⣿⣴⣆⠀⠀⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢸⣿⠀⠀⠀⢻⣿⡀⢸⣿⠹⣿⣆⠀⣿⡇⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠈⠉⠀⠀⠀⠈⠉⠁⠉⠉⠀⠉⠉⠀⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠰⣷⡆⢲⠀⣶⡆⠀⣶⡆⢰⣶⡆⠐⠆⠀⠀⣶⡆⠐⣶⣄⠀⢠⣷⡀⠐⣶⣆⠀⠀⣴⡀⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣿⡇⠀⣿⡇⢨⣿⡇⠶⠀⠀⠀⣿⡇⠀⢸⣿⠀⠈⣿⣧⠀⠸⣿⡆⠀⠻⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⣿⡇⠀⣿⡇⠸⣿⡇⢠⡶⠀⠀⣿⡇⢀⡾⠋⠈⠀⠘⣿⡆⠀⣿⡇⠀⣦⡀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⡀⢀⡀⠀⢀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⣀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠙⢀⣾⡇⠀⢻⣧⢸⣿⡇⠀⠀⠀⠙⣿⣇⠀⠀⠋⢸⣿⡇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠸⣿⡇⠀⣸⣿⢸⣿⡇⠀⠀⠀⠀⠘⢿⣧⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠀⠾⠁⠙⠧⠀⠟⠁⠀⠛⠷⠀⠀⠀⠀⠀⠈⠿⠃⠀⠸⠿⠇⠐⠿







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⢺⡆⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠈⣿⠀⠀⣾⠇⠀⣠⣾⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⢻⣦⣠⣿⣤⣰⣿⠁⠀⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⡀⠀⢙⣷⣿⣿⣿⣿⣿⣿⣿⣷⣾⠋⠀⣠⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣿⣿⡿⠋⣠⣤⣿⣏⣿⣿⣿⣷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣾⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠾⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠻⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣤⡾⠋⣻⡿⣿⣿⣿⣿⡿⠋⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⡟⢁⡤⠀⣿⠁⠀⣿⡀⠘⢿⣄⠀⠘⠻⠶⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣤⣾⠁⣼⠏⢠⠄⢹⡇⢠⠀⣿⡆⢹⣶⣤⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣿⣀⠛⢁⣼⣦⣄⣤⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠉⠄⠀⠒⠈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⢀⢠⠄⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⡇⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠴⠤⢄⣀⠀⠀⠀⠀⢂⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠉⠁⠀⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀⡀⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠰⡆⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⢹⡼⡇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠒⠦⠤⠀⠄⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⣀⡀⢀⣀⣀⡤⠶⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠠⠀⠀⠀⠘⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣄⠀⢉⡤⠚⠁
⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠺⠿⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠛⠛⠛⠛⠛⠉⢩⡍⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⢱⡎⢡⣈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠀⠀⠀⠀⠘⠓⠢⠀⠃⠀⠀⠴⠃⢀⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠦⡄⡀⠐⠈⠀⠀⠀⠚⠀⠀⠇⡰⠂⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠦⡎⠀⠀⠀⠀⠀⡀⠀⠀⣘⠀⠁⢀
⠀⢀⡀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠀⠀⠈⠉⠉⠉⠃⠀⠈⠫⡄⠀⠠⠖⠋⠀⠀⠀⠉⠴⠶⠄⡡
⠀⣈⣏⠉⠉⠉⠉⠨⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠴⠚⠁
⢘⠤⠬⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⢧⡀⠠⠄⡀⢀⡀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⢤⣄⠀⠀⢸⠃
⠈⣠⠆⠁⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⡤⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⣼
⠐⠧⡐⠀⠀⠁⠤⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡗⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⠤⣄⡀⠀⢀⣸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠠⠤⠄⠒⠊⠉⣲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⡅⠀⠀⠈⠑⢚⠄⢡⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠠⣄⣀⣠⠴⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠸⢒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣰⢁⣨⠤⠤⠭⠭⠭⡀⠀⡧⠤⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠉⠛⠁⠀⠀⠀⠀⠀⠀⠑⠒⢏⣙⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢯⠈⢏⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠸⠄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢀⡶⣼⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠒⢺⡲⢄⣀⣀⣘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⠉⠐⠒⠒⠒⠲⠶⣛⣉⠙⠉⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠐⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⡀⠀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠗
⠀⠀⠀⣤⣀⠤⠤⠤⠴⡂⠀⣀⠤⠄⢓⠀⠂⠀⠀⠀⠀⠀⣎⠋⣻⣙⡶⠀⡉⠁⠀⠀⢀⣴⠛⢂⡴⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡤⢈⣀⣤⡆⢀⣤⠀⠀⠉⠿⣿⣿⣦⣼⠟⠀⠀⠀⠿⣿⠿⡿⠿⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⣴⣏⣉⣷⣯⣤⣄⠀⠖⠀⢺⡛⠋⠀⠀⠀⠀⠈⠸⢿⡖⠃⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠢⠀⠙⠛⠟⣏⢙⣿⢿⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣰⢀⣠⠞⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣈⢀⣀⣀⠤⠛⢣⡬⠀⠀⠀⠀⠀⡄⠂⠀⡀⠈⠀⡼⣧⣨⣽⣿⣟⡏⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠑⢠⣿⣷⠿⠉⠻⣶⣤⣢⣄⣠⠀⠀⡐⠖⠀⢀⠀⢀⠀⠀⠁⠺⣯⣌⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⢈⣿⣤⣾⣆⠀⠈⠻⢿⣿⣷⣌⠉⢙⣠⣌⣻⣇⠀⠆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠀⠁⠉⠈⠁⠀⣀⣿⡿⣿⠿⣿⠳⣭⣿⡉⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⣄⠀⠀⠙⢿⣤⡴⠏⠁⠀⠘⠿⡷⠊⣁⡄⠀⠀⠀⠀⠀⢐⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠠⠀⠀⠀⠀⠈⠞⠋⠓⠂⠐⠄⣤⠶⣴⣾⣧⣶⣄⢀⡀⠀⣲⠂⢸⡉⢀⡍⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⢃⠀⢀⣦⣶⡞⢛⠈⠀⢀⡰⠺⢟⡄⠹⠿⠋⢁⠘⠋⠓⠒⡄⠀⢿⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢳⣾⣿⢏⣍⡰⠰⣅⠀⠀⠈⢸⠶⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠈⠰⠌⠿⣿⡀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣶⣤⡀⠉⠁⠐⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡶⣨⡿⣆⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠒⠊⠤⠄⣤⣀⡠⠀⠀⠀⠀⠀⢹⣿⠃⢸⣷⢺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⣿⠀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣺⣿⣿⠀⠉⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡴⠶⢲⡀⡀⢿⣿⣿⡧⢄⢠⠐⢿⠓⠀⠀⠀⠀⢰⣷⣆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡾⠁⠀⠈⠈⠿⡧⠸⠈⠀⠀⠀⠀⠀⠀⣀⢽⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⢰⡷⠶⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⢸⠃⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⢲⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠑⠀⠀⠀⠀⠀⠁

⠂⠀⠀⠀⠀⠀⢄⠀⠀⢀⠀⢀⠀⠀⠀⠔⢦⠀⠀⠀⠈⠈⠄⠀⠀⠀⠀⡀⡀⠀⡤⠄⠀⢀⠄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠑⠲⠔⠊⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠑⠦⠄⢀⠤⠄⠈⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡄⠀⠀⠀⢮⠠⠊
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠄⠀⠀⣠⡆⠁⠀⠀⠀⠈⢣
⠀⠔⠋⠀⢀⠄⠀⠐⠂⢀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⠂⠄⠂⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⡀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠤⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠐⡄⠀⠀⠀⠀⠑⠀⠉⠀⠢⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⡰
⠐⠀⠀⠀⠱⠄⠀⠀⠀⡀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣀⠀⡀⠁⠈⠀⠀⡠⠀⠀⠰⠂⠀⠀⠈⠀⠀⠀⢰⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠳⢌⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠘⣀⡂⠐⠀⠠
⠀⠀⣀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠀⠀⠉⠰⡀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠐⠁
⠀⠟⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠠⠀⠉⠈⠁⠈⠁⠄⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢀⢀
⠀⠢⡀⠈⠁⠀⡠⠆⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⢄⢸⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠁⠂⠄⠀⠀⢀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠑⠀⠀⡄⠀⠈⠑⠄⠀⠠⢈⠜
⠀⠀⠀⠑⢆⠀⠀⠀⢠⣴⠀⠊⡁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎
⠀⠀⠀⠀⠀⡆⠀⢠⠉⠀⠀⠀⢰⠀⠠⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠑⢀⠀⠀⢀⣀⡥⠞⠀⠀⣀⠄⠒⠐⠀⠤⡀⠀⠀⢀⠂
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠈⡀⠀⠣⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠁⠄⠐⠂⠀⠀⠀⠀⠀⠀⠄
⠄⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠘⠢⠀⣀⠁⠬⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⠓⢄⡀⠀⠀⠊
⠀⠈⠂⠐⠢⡀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⡀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠤⢼⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠐⠃⠀⡀⠀⠀⠀⠀⢄⠀⠐⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢀⠴⡚⠓⠢⢄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⡠⠁⢠⡁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⡀⠀⢀⠀⠀⠀⠀⠠⠀⠁⠀⠉
⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠠⡄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠐
⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠁⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠐⠒⠒⠁⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠁⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠤
⣐⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣐⣀⣀⣀⣔⣀⣀⣀⣀⣀⣀⣀⣠⣘⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢤⣰⠤⠒⡲⢩⣉⠙⣯⡸⢇⣵⣶⣦⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠋⠓⠶⠾⠞⠛⠷⠖⠋⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⡤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡞⠁⠠⡀⠀⠭⠛⠦⡤⠒⠶⠦⢚⣩⠉⣩⠅⡠⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣇⠀⠀⡀⠄⠀⠀⠀⠁⠀⠘⣟⠰⣦⣙⠋⣄⣀⡡⣬⡶⡀
⠀⠀⠀⠀⠀⢀⣴⠿⣬⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠁⠀⠀⠀⡀⠀⣄⢀⡉⠀⠉⠉⡤⣄⣬⠨⠏⢀⢠⣦
⠀⠀⠀⠀⣀⠘⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⠇⠀⢄⡄⠀⠀⠑⢀⠛⠆⠀⠢⠸⣀⠀⠉⡀⠁⠘⠀⠠⠄⠈
⠀⠀⠀⢠⣤⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⠪⠚⣶⣁⠀⠁⠀⠀⠂⠈⠀⠉⠀⠀⠀⢀⠀⡄⠁⠀⠀⠀⠀⠠⠀⢨⢶⣸⠇
⠀⠀⢠⢄⣠⠀⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢀⣴⢏⠟⠇⢄⠀⠩⠀⠖⠁⠈⡁⠰⠆⢀⡀⠀⠀⠀⠀⠄⠀⠈⠐⠀⠀⢁⠀⠰⣬⠿⡴⠆
⠀⠀⣀⣴⢶⠀⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⣟⣸⠏⢂⠀⠲⣬⣀⠈⡀⢀⠑⠂⠀⠊⠀⠀⠅⡀⠀⠀⡀⠢⠠⠌⠀⠈⠀⠁⠄⠁⠉⠺⠳⠃
⠀⣰⠿⠶⠦⠊⠠⣢⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢣⡤⠀⠒⢊⠴⡖⣟⠐⠉⠢⠶⠀⠐⢒⢶⠌⢠⠀⣤⠀⣠⠀⠀⠈⠐⢠⠐⠀⢂⠀⠇⠀⠘
⠰⠃⣠⣤⢲⡖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⡯⢉⡭⠴⢴⡊⢉⣂⡐⢀⠓⠢⠴⠊⠀⠐⠎⠀⢧⠈⠁⡈⢀⡦⢀⠀⠈⠀⠀⠠⠀⡀
⠰⢃⠶⠢⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣃⠀⢰⡋⢀⣀⣤⡹⣞⠉⠳⢤⣀⠘⡲⠘⠆⡀⠰⡼⣠⠄⢘⡃⠀⡰⠁⠀⠛⠈⠉⢅⡐⡄⠀⢀
⠈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠿⠿⠟⠋⠝⢫⡋⠙⢿⣶⣶⣬⡓⢠⢄⠘⠀⡀⢿⡙⠶⠼⣧⠀⠏⠀⠀⠀⠰⡖⠄⣫⢍
⠂⢹⡆⣀⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠛⠅⠀⠀⠁⠀⠀⠌⠀⣀⡄⠀⡘⠀⢻⡙⣆⠀⠱⡶⠠⠹⣀⠀⠙⢾⡆⢀⣠⡇⢀⠴⢓⡯⢱⣤⢂
⠳⣏⣟⣩⣧⣠⣴⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡟⣇⡀⠀⢀⠗⣀⢀⠐⠐⠀⠠⠆⠈⢠⡀⠱⠜⣏⠳⣴⣷⡤⠖⠉⠀⠀⠀⠉⠻⣼⣽⠏⠀⢸⡔⣿⠁
⠀⠀⣼⢩⡧⡴⠎⡷⢶⠾⣇⠀⡀⠀⠀⠐⠀⢸⡇⠋⠀⠉⠙⠋⠉⠉⠀⠀⠀⠀⠀⢀⠀⠋⠁⠢⠉⠓⠶⠲⢟⡳⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠹⣿⡇
⠀⠀⠳⣽⢄⡘⠀⠏⡼⢼⣭⡟⡛⣦⠀⠀⠀⠀⠱⣦⣐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠉⠛
⠀⠀⠀⢸⡂⡨⠘⢃⡇⢖⣢⣞⣙⣼⠃⠀⠀⠀⠐⠀⠉⠛⠛⠛⠲⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢀⣳⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⡓⠂⠐⡋
⠀⠀⠀⣾⢃⡀⣴⠢⡔⡆⠫⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠐⠐⠦⣂⡼⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣨⠿⠀⠂
⠀⠀⠀⠹⣆⣤⣿⢠⢀⣏⢲⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠐⠲⢤⡂⠜⢻⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢥⡀⠂
⠀⠀⠀⠀⠹⡄⠙⠿⠾⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⡀⠑⠬⠯⠺⠟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⠀⠀⢻⠀⠐⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣏⠡⢉⡂⡂⢌⡿⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠐⢾⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡄⢤⣰⠟⠁⢾⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⠶⠲⢶⣿⠯⣿⣖⣲⣤⣠⣤⣤⡁⣀⠴⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠶⠦⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢾⣁⣀⡠⠤⠖⠒⠲⠦⠤⠴⠒⠛⠛⠓⠲⠤⠤⠤⠴⠒⠲⠼⠷⣄
⠀⠀⠀⠀⠀⣠⠿⠤⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄
⠀⠀⠀⢀⡼⠁⠀⠀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠓⠒⠦⠤⠤⠔⠚⠓⠒⠒⠋⠁⠈⢧⡀
⠀⠀⢀⡾⠁⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⡾⠁⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⣸⠇⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⣇
⠀⣿⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⡇⠀⠀⡞⠹⡄⢻
⢰⡏⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣀⠤⣄⠀⠀⠀⠀⠀⢠⠞⠀⠀⠳⣀⡏⠀⠀⡇⢸
⢸⡇⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠖⠁⠀⠈⣇⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⠁⠀⢠⠇⢸
⠀⣷⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣼
⠀⢹⡄⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡏
⠀⠀⢷⠀⠀⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⡼⠀⡾
⠀⠀⠈⢧⡀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⢀⡰⠃⡼⠁
⠀⠀⠀⠈⢳⡄⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠙⢦⡳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠤⠖⠋⠓⠶⠖⠒⢲⡄⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣴⣒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⠢⣄⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⠳⡄⠀⠀⢸⡟⣆⠀⠀⠀⡼⠁⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠙⢦⠀⢸⡇⠈⣦⠀⢸⠃⠀⠀⠀⠀⠀⣷⠀⠀⠀⣀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⢧⠀⢧⣀⡼⠀⠸⣄⠀⠀⠀⠀⠀⡿⠀⡤⠞⢡⠋⠀⠀⠀⣀⡤⠴⣲⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡒⠲⢦⡀⠳⣤⣤⢞⣀⣤⠤⠶⠶⠒⠚⠷⠶⢶⣶⡚⠁⣞⠀⢠⠏⠀⣠⠴⠋⠁⣠⠞⠁
⢀⣠⡤⠖⠒⠒⠒⠒⠒⠶⠤⢤⡀⠹⣄⣀⡟⢀⡤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣟⠋⢠⠞⠁⠀⢀⡴⠃
⠉⠉⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠙⣦⠈⢩⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣄⣀⡴⠋
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣾⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢰⠒⠚⠒⣶⣄
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⡼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢦⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡀⠉⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡴⠛⠉⠉⠉⠁⠒⠲⢤⣄⡀
⠀⠀⠀⠀⣠⣴⣚⣉⡉⠀⢙⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠉⠙
⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠙⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠶⠤⠶⠚⠉
⠀⢀⡴⢚⣉⣀⣀⣀⣀⣠⡤⠞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠞⠓⠲⣄⡀
⠴⠛⠋⠉⠉⠉⠉⠁⠀⢀⡴⠲⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠳⠦⠤⠤⣝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣁⣠⠼⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠛⠲⢤⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⢀⣠⡘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠘⢦⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢙⡆⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⡟⠉⢣⠀⠉⠑⠦⣤⣀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠛⠉⣠⠞⠛⢻⡓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⢋⣀⡀⠓⢦⡈⡇⠀⠀⠀⠀⠈⠙⠮⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡼⠀⡴⠛⢳⠉⣩⠿⠛⠛⠛⠻⢅⠀⢠⠞⠉⠉⠙⣦⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣠⠞⠁⢀⡇⢀⡾⠀⡇⠀⠀⠀⠀⠀⠘⡇⠈⢧⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠞⠁⠀⠀⣸⣡⠞⠀⠀⢻⡀⠀⠀⠀⠀⢀⡇⠀⠈⢳⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠷⠋⠁⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⢻⡀⠀⠀⠀⣸⠁⠀⠀⠀⢳⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢠⠏⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠃
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⠀⡀⢠⠖⢢⢄⣀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⣾⠀⡤⠈⣰⠼⠷⠒⠛⠉⠉⠉⠑⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠠⡎⢀⢆⡘⠖⠋⠁
⠀⠀⢀⡀⠰⢤⣶⡎⡀⣀⡂⠀⠂⡈⠀⠀⠀⠀⠀⢀⠈⢠⡆⢈⠀⢓⠞⠋
⠀⠀⠀⠀⠂⠄⠀⠐⣉⠙⠓⢦⣤⡇⠀⠀⠀⠑⢠⡀⠀⡄⠻⣼⠞⠁
⠀⠀⠀⠀⠀⠀⠐⠐⠂⠐⠀⠀⠘⠳⣄⠀⢰⠀⡀⠱⣄⣧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠁⠀⠁⠀⠈⢷⡄⠀⠹⡄⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠋⠈⢳⠐⠾⣟⠄⠒⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣰⠃⠀⠀⢠⣬⣩⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠉⡁⠁⠀⢀⡞⠁⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠁⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠹⣦⣤⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⢀⡤⠔⠒⠲⢤⣤⣄
⠀⢀⡞⡡⠊⠀⠀⠀⡨⢷⠞
⠀⢸⣧⡇⠀⠀⣠⠞⠀⢸
⠀⠀⠙⢷⣶⠞⠁⠀⢀⡜⠀⠀⠀⠀⣀⡤⠠⣤⠐⡄⠀⠀⠀⠀⠀⠀⣎⣌⡉⠒⠒⢤⡀
⠀⠀⠀⠈⠻⠶⠤⠴⠋⠀⠀⠀⣠⠎⠀⠀⠀⠘⠒⢣⡀⠀⠀⠀⢀⠞⠁⠘⠇⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⢰⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣠⠞⠀⠀⠀⠀⢰⠃⠀⠀⠀⠐⡄⠀⠀⠀⠀⢁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠴⣾⣶⣶⠿⠃⠀⠀⠀⠀⠀⠈⠑⠠⠴⠞⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠉⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⢟⣿⣁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡝⠁⢸⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⡞⠈⢁⡈⠟⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢻⣳⣄⣼⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⣋⡿⠟⠁⠘⣿⡿⠃⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⢤⣠⣔⠮⢛⡉⡠⠭⠴⣒⡉⠻⢷⣾⣷⣶⣦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡷⠖⠂⠀⢉⣽⡟⠇⠈⢳⣦⣴⣾⡯⠉⠁⠈⠙⠿⣿⣿⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⣾⣶⣶⠤⢚⣵⠋⠠⢶⣶⣷⡉⠉⠉⠉⢁⣀⠒⠂⠀⠃⠀⠸⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣷⢰⣿⠂⠀⢺⡦⠉⠛⠋⠀⠠⠀⠙⠁⡠⠀⠀⠀⠀⣰⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⢋⠀⣉⣠⡟⠀⠀⠿⣿⣶⠀⠀⠀⠀⠀⠐⠋⠀⣠⠆⠀⡐⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡀⠀⠀⠿⣷⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠶⠁⠀⠀⠁⢘⢹⠇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠳⣄⢡⡀⠘⠀⠀⣇⠀⠀⠀⢀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠹⠌⣿⣤⡀⠀⢿⡀⠀⠈⠛⠛⠂⠀⢀⣴⣶⠖⠀⠀⣄⡴⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⡘⢿⣧⡀⣾⣗⣀⠀⠀⠀⠀⠀⣄⠁⠀⠀⢖⡄⡈⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡝⢿⣶⡝⠿⠄⠉⠀⢀⡀⠀⢀⣀⣸⣿⣶⣶⠟⢋⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣽⣧⠴⣶⣤⡆⠉⣉⡭⢄⣭⣿⣏⣍⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⣿⣾⣿⡕⢶⡟⠁⢠⣾⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠓⢶⣋⣠⣸⡿⠟⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢈⠢⡀⠀⠀⡜⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⠀⣀⡉⠈⠂⠣⡘⢦⠀⠀⢈⠀⢷⡸⠄⠀⢀⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣍⡶⠝⠓⠚⡿⣻⡀⠘⠀⠸⠰⠀⢰⠂⣆⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣨⠄⠈⠋⠈⡹⢿⣾⡈⠀⠈⠀⢸⠅⡆⠀⠤⠚⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⠦⠐⠡⠊⣡⡀⠐⣫⠙⠢⡀⠁⠘⣠⠇⠎⠀⠀⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣌⣩⠤⠜⡡⠄⠐⠘⣻⣿⠁⢠⠀⡏⠀⡠⢁⠄⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⡤⠖⠃⠀⡠⠈⠳⣤⣄⣤⠔⢰⠃⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠤⠴⢒⡩⠒⠒⣉⡸⣿⢄⠀⡎⢠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠠⠐⡟⠉⣠⠂⠈⠳⣶⣿⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⡨⠋⢁⠀⠀⢺⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡜⠁⢡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⡀⠻⠀⠀⠐⠖⢡⡦⠀⠀⠒⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠃⣽⡥⠄⣀⠀⡇⠀⠀⠻⡝⣡⠄⠐⠂⠈⡄⠀⠙⣦
⠀⠀⢀⠤⠂⠀⡀⠀⠐⠀⠀⠀⠤⠤⣤⠀⠀⢀⠜⠒⢄⣴⠃⣰⣄⣀⠀⡴⠁⠀⢀⣾⠠⠤⢄⠀⠀⠀⠀⣀⠄⢀⡀
⠀⠀⣮⠁⠀⠀⢨⠀⠘⣢⣠⠀⠀⠀⠚⣇⣈⢙⠀⠀⠀⢹⠎⠀⠀⠙⡏⣹⣄⣤⡟⠠⠆⠉⠳⣁⠀⠀⣴⠁⠀⠀⠈⡆
⠀⠀⢹⣆⣀⣠⡷⠃⠈⠻⢡⡀⠀⢀⣸⣿⣿⣮⡁⠚⣁⡜⢄⣀⣀⣠⣺⠏⠀⠈⠳⣄⡀⠀⠀⣯⠀⠀⢫⡧⡀⠀⣰⣯
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠈⠉⠙⠙⠙⠛⠛⠛⠊⠁⠀⠀⢤⣀⠈⠙⠛⠛⠉⠀⠀⠈⠁⠚⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠢⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣦⢰⣿⣧⠀⣤⣤⣼⣿⣹⣿⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⠀⠀⠀⢿⣿⣿⣿⣿⡄⠈⢹⣿⠋⢹⣿⠀⣸⡿⠁⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣶⣶⡄⢸⣿⠙⠃⠻⠷⠀⠾⠟⠀⠸⣿⢠⣿⣇⣀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⣶⣤⣿⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⢼⣿⣁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣳⡶⢀⠉⠉⠉⠀⠀⢀⣶⡶⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠉⠛⡃⢶⣚⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣹⡟⠀⠿⠀⠀⣠⠖⠒⣾⡿⠧⠤⣤⠤⠴⠞⠋⠀⣤⣶⠄⠀⠀⠀⠀⠀⠿⠀⢻⡏⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣴⣿⠂⠀⣠⣾⣿⠃⠀⢀⣤⡶⣦⠀⠀⣼⣿⡧⠖⢲⣦⠀⠀⢀⣴⢶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠚⣿⣄⠤⠚⣹⣿⠃⠀⣰⣿⣏⠴⠋⢀⣼⣿⡟⠁⠀⣼⣿⣠⠴⠻⣿⡌⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡟⠁⢀⡠⠒⣿⣧⠀⢰⣿⠇⢀⣼⣿⠏⠀⢀⡴⢻⣿⣟⡤⢒⣿⡿⠁⠀⣀⣠⣿⣷⣀⣀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⣰⠋⠀⠀⣿⡟⠀⠸⠿⠶⠋⠘⠿⠦⠶⠋⢠⣿⡟⠻⠶⠛⠋⠀⢠⡟⠁⠀⣸⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣠⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠈⠷⣤⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⠀⢀⣀⣀⡀⢀⣀⣿⣋⣀⣀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠻⢿⡇⠀⢻⣿⡇⠀⢸⣿⡿⠁⠀⣿⣿⠛⠛⠿⠀⠈⣿⣿⠛⠻⣿⣷⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢀⡀⠁⠀⠸⣿⣷⠀⣼⣿⠇⠀⠀⣿⣿⣴⣶⠀⠀⠀⣿⣿⣀⣤⣿⣿⠁⢸⣿⣧⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣿⠇⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⡁⠀⠈⢿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⠀⠀⢸⣿⣿⣀⣤⣴⡆⠀⣿⣿⠀⢸⣿⣧⠀⠀⣀⣈⡉⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣀⣴⣿⠀⠀⢸⣿⣿⣿⠇⠀⠀⠘⢛⣛⣛⡛⠛⠃⠐⠿⠿⠀⠀⣿⣿⡀⠀⢹⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⠀⠉⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠉⠙⠃⠀⠸⣿⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠈⠙⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣠⠀⠀⢀⣄⡀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠀⢀⣤⣦⠀⠀⣄⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⣼⠿⣄⠸⠀⢀⣾⣿⣿⣷⣦⡄⠀⠀⣀⣀⣀⣀⣀⡀⢸⣿⣧⠀⠘⣿⣿⡄⠸⠇⣠⠿⣧⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠞⠋⠁⢀⣾⣿⡟⠈⢹⣿⣿⠀⠀⣿⣿⠿⣿⣿⣇⠀⢿⣿⣄⣀⣿⣿⣷⠀⠈⠛⠳⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣾⣿⡿⠀⠀⣼⣿⣿⠀⣸⣿⣟⣀⣸⣿⣿⠀⠈⠿⣿⣿⠟⣿⣿⡆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣧⣤⣼⣿⣿⠃⢰⣿⣿⠿⠿⠿⣿⣿⣇⠀⠀⣠⣤⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠁⢠⣿⣿⡏⠀⠀⠀⢸⣿⣿⣆⠀⢸⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠃⠀⠀⠀⠘⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢠⡆⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⣾⡇⠀⠀⣾⡇⠀⢠⣿⠀⠀⢠⣾⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠸⣿⡄⠀⢻⢿⡄⠀⣟⣧⠀⡼⣿⠀⢀⣿⡏⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⢢⣄⠀⠀⠹⣿⣄⠀⢹⣟⠢⡈⣯⠹⢦⡏⠈⠿⠡⣏⡴⠏⡿⠁⣠⣾⡿⠀⠀⣠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⠙⣿⠙⠢⣯⣠⣼⣷⣶⣿⣷⣷⣶⣶⣿⣤⣹⡷⠊⢡⡞⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠐⣦⣄⡀⠀⠈⠾⣯⠉⢻⣦⣾⣿⣿⡿⢿⡿⠟⠏⠿⠙⢻⠿⡿⣿⣿⣷⣾⡉⢠⡴⠋⠀⣀⣴⣾⠋
⠀⠀⠀⠀⠀⠈⠻⠿⣖⡶⠂⢹⣶⣿⡿⠟⠉⠁⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠈⠋⠻⣿⣿⣾⡓⠙⣋⡽⠛⠁⠀⣀⣀
⠀⠀⠈⠻⣷⣦⣤⣤⠌⢻⣤⣿⣿⠿⠀⠀⣴⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣆⠀⠀⠛⣿⣿⣾⠃⠴⠶⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠷⣆⣰⣿⡿⠟⠀⢠⠎⠁⣀⣀⣀⠀⠀⠠⠈⠀⠀⠀⣀⡀⠈⢱⡄⠀⠻⣿⣿⣦⡞⠋⠁⠀⠀⣀⣀⡀
⠀⠘⠻⢿⣶⣶⣖⠋⠉⣿⣿⠟⠀⠀⢈⣾⣟⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣭⣍⠛⢦⠁⠀⠀⠿⣿⣿⡀⣯⣽⡿⠿⠛⠁
⠀⠀⠀⠀⠀⢀⣉⠝⢾⣿⠏⠀⠀⠀⢸⣿⣿⣗⣚⣶⣿⣿⡇⢸⡿⣟⣶⣲⣻⡷⠀⠀⠀⠀⠀⢻⣿⡟⠓⢤⣄⣀⣀⣀⣀
⠐⠒⠿⢿⡿⠿⠯⣤⣸⣿⡇⠀⠀⠀⢨⣿⡋⠛⠛⠋⢹⣿⠇⢸⡇⠈⠁⠉⠉⠀⣀⡀⠂⠀⠀⢹⣿⡧⣾⠓⠛⠟⠛⠋⠉
⠀⠀⠀⠀⠀⣠⡴⢊⠹⣿⣷⡀⠀⢑⠐⢿⣷⡄⠀⢀⣼⠛⣄⠈⠛⡆⠀⠀⢀⣘⡿⠀⠈⠀⣠⣾⣿⣃⣀⡙⣦⣤⣤⣀⡀
⠀⠐⠚⠿⠿⠟⠛⢙⠟⢿⣿⣿⣦⠀⠀⠀⠀⠀⣴⠟⠉⠓⣿⠛⠒⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠉⢦⡈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣀⣠⣴⣧⡴⢬⣿⣿⣿⡄⠀⠀⠀⠰⣧⣶⡶⠿⠿⠿⠗⣖⣴⡄⠀⠀⠀⠀⢠⣿⣿⡿⠷⠖⠦⣿⣶⣤⣄⡀
⠀⠀⠐⠛⠛⠛⠉⠁⣠⣇⣈⣻⣿⣿⣆⠀⠀⠰⠀⠀⠿⠷⠖⠲⠾⠿⠀⠀⠆⠀⠀⣴⣿⣿⣿⣥⣄⣷⣄⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⢡⡋⢘⣿⣿⣧⡀⠀⠀⠀⣠⡤⡶⠲⠤⡀⠀⠀⠀⣀⣾⣿⣿⣅⡈⣧⠈⠛⠿⢷⣦⡀
⠀⠀⠀⠀⠐⠉⠉⠀⠀⣠⣾⡷⠋⡍⡈⣙⡿⣿⣦⣀⠀⠀⠁⠀⠀⢀⣀⣴⣿⠿⣿⣁⠘⡆⠳⣾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠀⢠⣗⡞⠉⣧⣄⡼⡏⢙⣿⠛⡛⡿⣏⠉⡿⣄⠄⡇⠙⣦⣧⡀⠀⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⣿⠋⠀⢀⣿⡟⠀⢷⣾⠁⢷⣹⡇⢹⣰⠇⠈⣷⣷⠀⠈⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢀⣾⠟⠀⠀⣸⣿⠀⢸⣿⡇⠸⣿⡄⠀⠈⠻⣧⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣿⠃⠀⠸⡿⠀⠀⠙⡇⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠑⠀⠀⠀⠘⠉⠃⠛⠘⠐⠃⠘⠚⠐⠂⠀⠁⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⡋⠉⠙⠛⠛⠒⠒⠒⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡀⠀⣤⣄⣀⡀⠀⠀⢴⣚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⠑⠀⠀⠀⠀⠀⣀⠢⡌⢹⠏⠃⠀⠀⣨⠗⠀⠐⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣨⡵⣄⣠⠴⠚⠉⠉⠙⡷⠋⠀⠀⠀⣼⣡⠖⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠛⣳⠟⠁⠀⠀⣀⠀⣸⠇⠀⠀⠀⣸⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣛⠿⠟⢿⠀⠀⠀⠘⢿⡴⠃⠀⠀⠀⢀⡯⠟⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠒⠛⠉⠁⣶⠂⠀⠈⣳⠤⠖⠚⠛⠃⠀⠀⠀⠀⠉⠰⠮⠭⠛⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠛⠉⠉⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⢐⣶⠤⣄⠀⠀⠀⢠⠖⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠥⣧⣬⣤⡷⠖⠋⠁⠀⠀⠀⠀⠀⢀⣀⡴⠚⢯⡈⢣⡞⠓⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠠⠴⣶⠒⠻⣍⠀⠙⢦⣀⡽⠋⠀⠀⠀⢸⠁
⠀⠀⡴⠋⠉⢉⡓⡎⢁⣠⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⣳⡤⠞⠉⠀⠀⠀⠀⠀⣿⣀⣀⣀
⠀⢰⠛⠶⠤⢶⠁⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⡏⣈⡴⣉⡽⠃
⠀⢸⠲⠦⠴⡞⠀⣀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⡋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠞⠛⣁⣨⠇
⠀⠈⠓⠒⠚⠙⠒⠉⠙⢮⡑⢤⡀⠀⢠⡤⠤⠖⠚⠉⣷⣥⠤⠤⠖⠒⠒⠖⠲⠒⠒⠒⠒⠒⠛⠉⠉⣉⣀⣤⠴⠖⢚⡟⢓⣻⣿⠒⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣍⠲⢬⣧⢀⡴⡠⢘⣃⠀⠀⠀⠀⠀⠈⢣⡀⢀⣀⣠⡤⠴⠒⠚⠉⠉⠀⠀⠀⡀⢸⢁⡟⠳⠛⠤⠶⡤⢦⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢖⡿⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠢⣴⠃⠀⢰⠇⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡴⠁⠀⠀⢸⢰⢀⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣼⡓⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠒⠚⠳⣌⡑⣤⠽⠖⠒⠒⠛⠛⠛⠛⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⣀⢬⠭⠿⠾⠶⠧⣭⣍⡑⠒⠒⠒⠒⠶⠶⡟⢹⠓⠒⢒⣶⠖⠉⣀⡤⠖⣒⣫⣭⣭⡍⢉⣛⣒⠦⣍⡛⢦⡀
⠀⠀⠀⠀⠀⠀⣇⠀⠀⢇⠀⠀⠀⣀⣀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣀⣀⡇⣟⠈⣩⠋⢀⡴⢋⡥⠲⣆⢻⠶⢭⣿⣳⠃⠀⣨⢇⣠⡙⢦⡙⣦
⠀⠀⠀⢀⣀⣀⣻⡀⠀⢸⠀⠀⠀⠀⠐⠛⠛⣿⠟⠿⠟⣻⠏⡠⠄⠸⠧⢭⡟⠁⡴⢋⡴⠯⠿⣿⣿⣧⣳⣩⢿⠇⡠⢊⠵⠋⠀⠙⣎⢳⠘⣧
⠀⠀⠀⠧⣤⣽⣽⡭⠿⠿⣷⠶⢤⣄⣀⣀⣀⣹⢀⡀⢤⡟⣰⡁⠀⠀⠀⣸⠁⡼⢁⣈⣙⡓⠲⠶⢶⠋⡖⠛⠒⣯⢖⣁⣀⡠⠤⠤⠾⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠿⣷⣒⠞⠋⠿⣽⠟⣵⡿⡿⢰⣧⡝⢦⠀⠀⡟⢰⡇⣾⠀⠀⠀⠀⠀⣠⣇⣱⠶⣶⡫⡞⠛⠛⠒⠒⢲⡆⣼⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⣯⢸⣛⡯⡓⡫⠢⡿⢸⠆⠀⣿⠈⣇⢻⡄⠀⣀⡴⣺⠟⠁⣸⣹⠁⠘⢎⠣⡀⠀⣠⠞⡼⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠛⠦⠾⡹⠵⣣⠞⠀⠀⠘⣦⠘⢦⡙⠛⠉⡔⠁⠀⢰⢇⡏⠀⠀⠈⣷⡘⠛⣡⠞⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⠀⠀⠀⠈⠓⢦⣈⡓⠦⢬⣉⣒⣋⣘⡛⣒⣛⣋⠭⠖⣋⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⠭⣭⡭⠥⠤⠶⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡤⢠⠀⣤⢤⡤⡀⠱⡂⢄⠀⠀⠀⠠⠀⠀⠆⠀⢢⠀⠔⠃
⠀⠀⠀⠀⣀⠀⢀⠸⢄⠀⠈⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠁⠁⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⢀⡄⠀⠮⠗⠀⠈
⠀⠀⠀⠀⠀⠐⠂⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠀⠀⢀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⡈⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⡀⠀⠀⢋⠄⠀⠀⠀⠀⠀⠈⠀⠘⠀⠠⠠⠄⠀⠀⢤⣀⠀⠀⠒⠀⢠⠄⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⢀⠍⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡄⢀
⠀⠘⠂⠔⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⠂⠀⠀⠠⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠄⠀⠒⠀⠈⠁⢏⠀⣀⠀⠀⢀⣀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠀⠄⠄
⠀⠀⠀⠀⠈⣠⠀⠀⠀⠁⠀⠀⡀⢁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠒⡄⠀⠀⢀
⠀⠀⠀⢠⠂⠨⠄⠀⠀⠀⡠⠈⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠘⠓⠀⠲⢌⠁⠀⠠⠄⠀⠀⠀⠄⢒⠀⢰⠄
⠀⠀⠀⠀⠁⠄⢀⣀⡀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠉⠓⠀⠤⡀⠀⠀⢀⠀⠀⠀⠒⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⢀⠈⠓⣀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠤⠀⠤⣇
⠀⠀⠀⠀⠀⠐⠂⠒⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢠⠀⠈⠁⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠁⠂⠀⡄
⠀⠀⠀⢐⡄⢀⠄⠠⠄⠊⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠔⠀⡸
⠀⠀⠀⠈⠀⠀⠐⠀⢀⡀⠀⡀⠄⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠐⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠀⠀⠒⠀⠀⠉⠀⠀⠀⠀⠌⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠖⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠲⢶⣚⢽⠛⠀⠀⠒⠒⢦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣛⡾⣿⣸⡣⢝⡙⡆⠀⠀⠀⣸⠀⣠⢝⡓⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⣾⡅⡪⠒⢫⣔⡭⢚⣿⢀⣴⡫⢔⡫⢖⡩⢚⡥⠂⣱⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠗⠁⢀⡤⠷⣺⢕⡫⢟⡭⢚⡡⢔⡡⠞⢉⠔⠛⣴⢊⠵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⢓⡩⢒⡥⢚⡥⢒⡭⠒⣉⣤⡚⣏⠀⠀⠓⠊⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡀⠀⠀⠀⠀⠀⠀⢀⡾⢔⡥⣚⠥⢚⡥⠚⣡⠔⣪⣿⣷⣹⠈⢖⣹⠃⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡩⢇⠀⠀⠀⡀⢠⣤⡘⢶⡥⢊⡥⠚⣡⠔⣫⠔⣫⡴⠟⣫⢔⡏⠁⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⣹⣄⡀⠸⣿⣝⡩⢘⡩⠞⣡⠔⣉⢔⡫⣰⠟⣋⢔⣪⠏⠉⡔⣰⢦⢫⢹⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣊⠔⣋⠝⣆⠀⠫⡖⣉⠔⢋⢔⡪⣶⠕⣊⠴⣪⠗⣡⢜⠶⣼⢟⡵⢋⡵⠝⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡥⢊⡥⠚⣩⣷⠀⠀⠀⠉⠙⢍⣋⡴⢊⠵⣋⠴⡪⠖⡡⢺⠉⠁⠉⠉⠀⠀⣀⠀⡰⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣩⠔⢋⡤⢿⡀⠀⠀⠀⠀⠀⠈⢘⡕⢊⡴⢊⠔⣩⢔⡇⠀⠀⠀⠀⠀⠀⠘⣷⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢊⡡⠒⡩⠟⣦⢤⡀⠀⠀⠀⣼⠔⡡⢚⡥⣪⢕⡽⢛⠶⠲⣄⠀⠀⠀⠀⣇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡴⢋⠴⢊⠤⢊⡽⡄⢀⣴⠗⣋⢔⡫⢚⠕⣡⠖⢁⠀⢋⠙⠃⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣖⣡⠖⣡⠔⣇⢸⡡⢚⡵⣋⠔⡡⠚⢁⠔⢫⡜⣉⢈⣹⣻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⠤⣚⣹⢺⡶⣫⢞⡵⠋⣠⠞⢁⠔⢁⠜⣡⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠓⠫⠤⠾⠥⠴⠇⠒⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡠⢀⡠⢀⡀⣀⢀⡀⢀⠀⣀⠀⡀⠀⡀⠀⡠⠀⣠⢀⡀⢀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠉⠀⠉⠈⠁⠉⠈⠁⠈⠁⠉⠀⠈⠀⠈⠀⠈⠁⠈⠀⠀⠀⠉

⣿⡆⠀⠀⠀⢻⣿⣿⣇⠀⠀⠘⣿⣿⣇⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⠁⢀⣿⣿⡟⠀⠀⠀⣰⣿⣿⠏⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿
⣿⣿⡀⠀⠀⠈⣿⣿⣿⡀⠀⠀⢻⣿⣿⠀⠀⠘⣿⣿⡇⠀⠀⣾⣿⡏⠀⣼⣿⡟⠀⠀⠀⣰⣿⡿⠃⠀⣠⣾⣿⡿⠋⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠿⠟⠛⠋
⣿⣿⣧⠀⠀⠀⠘⣿⣿⣧⠀⠀⠘⣿⣿⡆⠀⠀⣿⣿⡇⠀⠀⣿⣿⠇⢀⣿⣿⠁⠀⠀⣰⣿⡿⠁⢀⣾⣿⡿⠋⠀⠀⢀⣤⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⣀⣠
⢿⣿⣿⣧⠀⠀⠀⠸⣿⣿⣆⠀⠀⢹⣿⣷⠀⠀⢻⣿⡇⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⢰⣿⡟⠀⣠⣿⡿⠋⠀⠀⣠⣾⣿⡿⠟⠋⠁⠀⠀⣀⣤⣴⣾⣿⣿⣿
⠀⢻⣿⣿⣧⠀⠀⠀⠙⣿⣿⡆⠀⠀⢿⣿⡆⠀⠘⣿⣇⠀⠀⣿⣿⠀⣼⣿⠀⠀⢀⣿⡟⠀⣴⣿⠟⠀⠀⣠⣾⡿⠟⠉⠀⠀⣀⣤⣶⣿⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠹⣿⣿⣧⡀⠀⠀⠘⣿⣿⣆⠀⠈⢿⣿⡀⠀⢻⣿⠀⠀⣿⣿⠀⣿⡟⠀⠀⣼⡿⠁⣼⣿⠃⠀⣠⣾⡿⠋⠀⠀⣠⣴⣾⣿⠿⠛⠉⠁
⣄⠀⠀⠈⢿⣿⣷⡄⠀⠀⠈⢿⣿⣆⠀⠈⢿⣷⠀⠘⣿⡇⠀⢹⣿⠀⣿⡇⠀⢠⣿⠃⣸⡿⠁⢀⣾⡿⠋⠀⢀⣴⣾⡿⠟⠉⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿
⣿⣧⡀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠹⣿⣧⡀⠈⢿⣧⠀⠹⣿⡀⠸⣿⠀⣿⡇⠀⢸⡟⢠⡿⠁⣠⡿⠋⠀⣀⣴⡿⠛⠁⠀⠀⣀⣠⣴⣾⣿⣿⡿⠿⠿⠛⠛⠛
⢿⣿⣿⣦⡀⠀⠀⠙⢿⣷⣄⠀⠀⠈⢻⣷⣄⠀⢻⣧⡀⢻⣧⠀⢿⡆⢹⡇⠀⣿⠁⣿⠁⣴⠟⠁⢀⣼⠟⠉⠀⢀⣠⣶⣿⠿⠟⠛⠉⠁⠀⢀⣀⣀⣀⣀⣀
⣀⠉⠻⣿⣿⣦⡀⠀⠀⠙⢿⣷⣄⡀⠀⠙⢿⣦⡀⠙⣷⡄⢻⣆⠸⣇⢸⡇⠀⡟⣸⠇⣼⠏⢀⣴⠟⠁⢀⣤⡾⠟⠋⢉⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣤⡈⠛⢿⣿⣶⣄⠀⠀⠉⠻⢿⣦⣀⠀⠙⠻⣦⡈⠻⣆⠹⣆⢻⡀⣇⠀⡇⡟⣸⠃⣠⠟⢁⣠⠾⢛⣩⣴⡶⠿⠛⠛⠋⠉⠉
⠙⠿⣿⣿⣶⣤⡈⠛⠿⣿⣦⣄⡀⠀⠉⠻⢷⣤⣀⠈⠻⢦⡈⢷⡜⢦⢳⢹⠀⣿⢱⢃⡼⢁⡴⣋⡥⠾⢛⣩⣤⣤⣶⡶⠾⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠀⠉⠛⠿⣿⣷⣦⣄⣉⠛⠿⣷⣦⣄⣀⠈⠙⠳⢦⣄⡙⠲⢌⡪⡣⡷⡀⣻⠁⢊⣴⣿⣪⠵⠖⢛⣉⣉⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛
⣤⣄⡀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣶⣦⣤⣍⣙⠛⠶⠶⢤⣤⣉⣙⡒⠍⠊⠐⣥⣇⡴⠋⠊⢑⣒⣛⣉⣉⣉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣶⣶⣤⣄⣀
⠿⣿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠒⠒⠒⠛⢛⣛⠊⢉⠴⢛⠉⠒⠶⠶⣤⣭⣭⣉⣉⠙⠛⠛⠷⢶⣶⣤⣄⣀⠀⠀⠉⠙⠻⠿⣿⣿⣶
⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⠶⠶⠶⠶⠒⠛⣉⡥⣒⣾⠏⡔⠀⡇⣦⡢⣍⠲⢍⡙⠻⣯⣛⠷⢶⣤⣀⡀⠉⠛⠻⢿⣷⣦⣄⡀⠀⠀⠉⠻
⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠖⢛⡥⢞⡕⣡⢣⠃⡇⣿⠰⡹⣌⠳⣄⠙⠷⣄⠉⠻⢶⣬⡙⠻⣷⣦⣄⡀⠈⠙⠿⣿⣷⣦⣄
⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⠿⠿⠛⠛⢉⣡⣴⠞⣫⡴⢋⡴⢃⡟⢰⠇⣿⡆⢳⠘⣦⠘⢷⡀⠈⢳⣦⡀⠙⠻⣷⣄⠙⠻⣿⣶⣄⡀⠈⠙⠿⣿⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣤⣤⣶⠿⠛⣉⣴⠞⠋⣠⡟⢁⡾⠁⣾⠀⣿⣿⠘⣧⠘⣧⠀⢻⣦⠀⠙⢿⣆⠀⠈⠻⣿⣦⡈⠙⢿⣿⣦⣄⠀⠈⠻
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠿⠟⠛⢉⣠⣴⡿⠛⠁⣠⡾⠋⢠⡿⠁⢰⡏⠀⣿⣿⡀⢿⡄⠹⣧⠀⠻⣧⡀⠈⢻⣷⡄⠀⠈⢿⣿⣦⠀⠙⢿⣿⣷⣄
⠛⠛⠛⠛⠛⠛⠉⠉⠉⢁⣀⣠⣤⣶⣿⠿⠛⠁⠀⣠⣾⠟⠁⣰⡿⠁⢀⣿⠁⢰⡟⣼⡇⢸⣧⠀⢻⣧⠀⢻⣷⡀⠀⠹⣿⣆⠀⠀⠙⣿⣷⣄⠀⠙⢿⣿⣷
⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⡿⠟⠛⠉⠀⠀⣀⣴⣿⠟⠁⣠⣾⠟⠀⢀⣾⠇⠀⣾⡇⣿⡇⢸⣿⠀⠈⣿⡆⠀⢿⣷⡀⠀⠹⣿⣧⠀⠀⠈⢿⣿⣦⠀⠀⠻⣿
⣿⣿⡿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⣀⣴⣾⡿⠛⠁⢀⣴⡿⠋⠀⢀⣾⡏⠀⣰⣿⠀⣿⡇⠘⣿⡆⠀⢹⣿⡀⠘⣿⣧⠀⠀⠹⣿⣧⠀⠀⠈⢻⣿⣷⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⠟⠉⠀⣠⣶⣿⠟⠁⠀⣠⣿⡟⠀⢠⣿⡇⢸⣿⠇⢰⣿⡇⠀⠈⣿⣇⠀⢹⣿⡆⠀⠀⢻⣿⣧⠀⠀⠀⢻⣿⣿⡄
⣀⣀⣀⣤⣤⣴⣶⣿⣿⣿⡿⠛⠉⠀⢀⣴⣾⣿⠟⠁⠀⢀⣴⣿⠏⠀⢠⣿⡟⠀⣾⣿⠀⢸⣿⡇⠀⠀⣿⣿⠀⠸⣿⣷⠀⠀⠈⣿⣿⣇⠀⠀⠀⢿⣿⣿⡄
⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⣀⣴⣾⣿⡿⠛⠁⠀⠀⣠⣾⣿⠋⠀⢠⣾⣿⠁⢠⣿⡟⠀⣸⣿⡇⠀⠀⢻⣿⡇⠀⣿⣿⡇⠀⠀⢸⣿⣿⡄⠀⠀⠈⣿⣿⣿
⠛⠛⠉⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⠋⠀⠀⠀⣠⣾⣿⡟⠁⠀⣠⣿⣿⠃⠀⣾⣿⠃⠀⣿⣿⠃⠀⠀⢸⣿⡇⠀⣿⣿⣧⠀⠀⠀⣿⣿⣷⠀⠀⠀⠸⣿⣿
⠀⢀⣀⣤⣴⣾⣿⣿⣿⡿⠟⠋⠀⠀⠀⢀⣠⣾⣿⡿⠋⠀⢀⣴⣿⡿⠃⠀⣼⣿⡟⠀⢸⣿⣿⠀⠀⠀⣾⣿⡇⠀⢹⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⣠⣴⣿⣿⡿⠋⠀⠀⣠⣾⣿⡟⠁⠀⣰⣿⣿⠁⠀⣾⣿⡟⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿⡀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣋⣭⡵⠖⢀⣠⣴⠖⠀⢀⣙⣯⠍⠉⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠃⣠⠾⠛⠁⢀⡴⠟⠉⢀⣤⠾⠛⠉⢀⣤⣶⠞⠋⠁⠀⡱⡄
⠀⠀⠀⠀⢀⣀⣠⠤⠤⠴⠒⠒⡖⠁⠴⠶⠶⠶⠦⠤⠤⠤⠤⢤⣤⣉⣀⠀⠀⠖⠛⠉⠀⠀⠀⢀⡼⠁⡇
⠀⡴⡒⠋⢉⣀⣠⣤⣤⣴⣆⣀⣇⣀⣐⣲⣶⣶⣶⣶⠦⠤⠤⢤⣄⣀⡉⠉⠛⠒⠲⠦⠤⠴⠒⠋⢀⢠⠇
⠀⠻⠧⠼⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠑⠒⠺⠭⢝⣒⠦⢤⣀⣀⣀⡤⢖⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠭⠴⠚⠁

⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠤⢒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠉⠁⠀⠀⠀⠀⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠒⠒⠂⠘⠆⠀⠀⠀⠀⢀⡀⠔⠃⠀⠘⣊⡀⢢
⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠔⠋
⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠳⣄⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢠⡇
⠀⠀⠀⠀⠀⢈⢣⠀⠀⠀⠀⠀⠈⢹⠖⠂⠉⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣾⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠄⡀⢀⠼⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠂⢉⡱⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡂⠀⡆⡠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠛⠁⠀⠀⠀⠀⠐⠺⠅⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠆⠤⠤⢤⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠨⡳⢤⣄⣰⣄⡀⠀⠀⠀⢀⣀⣀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠂⠤⠌⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⠙⠛⠛⠛⠛⠛⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⣴⣤⣤⣤⣀⠀⠉⠛⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠈⠉⠙⠻⢿⣦⣄⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⣶⡿⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣼⡿⠁⠘⣿⣦⡀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠿⠿⢷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣠⣾⠿⠋⠀⠀⠀⠘⢿⣿⣷⣤⣀⣠⣾⠟⠋⠁⢀⣀⣠⣄⠀⠀⣈⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⠻⣿⣧⡀⠀⠀⠙⠛⠋⠉⠀⠘⠻⠿⣶⣦⣝⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣿
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣷⠘⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣰⣿⣧⣀⠀⠀⠀⢀⣀⣴⡿⠏⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠙⠛⠿⠿⠿⠿⠛⠉⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⠟⠿⣷⣤⣄⣀⡀⠀⠀⢀⣀⣠⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢷⣶⣶⡾⠿⠿⠟⠛⠉⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⢠⡄⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠁⢸⣧⠴⢦⡄⠀⣦⠀⣠⡶⠶⠀⣿⠀⣠⠦⠀⣦⡴⢦⡄⠀⣠⠶⢦⡀⠀⠴⠶⣦⠀⣠⡶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⠀⢸⡇⠀⣿⠀⣿⠀⠀⠀⣿⡾⣏⠀⠀⣿⠀⠈⣿⢰⣿⠶⠾⠇⢠⡴⠶⣿⠀⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠆⠸⠇⠀⠸⠇⠀⠿⠀⠙⠶⠶⠀⠿⠀⠙⠷⠀⣿⠶⠾⠋⠀⠻⠶⠶⠆⠘⠷⠶⠻⠀⠲⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡠⠠⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⢀⠂⢸⠁⡄⢉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢠⣄⠐⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⡠⠂⡔⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠈⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠒⠂⠀⠈⠙⢎⠁⢀⠁⠀⠒⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡑⠀⠐⠀⠢⣄⠀⢈⣠⠑⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣰⡏⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⢀⠀⠈⡗⢶⢄⡀⠀⠐⠛⠘⠉⠉⠀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣧⠀⠀⠀⡼⢹
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢚⠀⠀⠱⠈⠢⢀⠀⠀⢂⡤⠔⠀⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠈⢦⡑⠒⠋⠁⣘
⠀⠀⠀⠀⠀⡄⠀⠀⢷⡄⢀⡾⢀⠀⠀⠀⠢⠤⣁⡄⣮⡤⠴⠚⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⣂⢤⡤⠚⠁⠀⠀⠈⠁⠀⠁⠈
⠀⠀⠀⠀⠀⠈⠧⠀⠀⢹⣼⡅⠈⠠⡀⠀⠀⠀⠈⢰⠀⢀⠀⠘⠀⢀⡤⠤⠃⠄⠀⠀⠀⠀⠀⣾⠀⠁⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠤⠽⡃⠀⠀⠀⠂⠀⠀⠐⠘⠀⠂⠀⣀⢤⡏⠀⠀⢀⡎⠐⠀⠀⠉⢸⠋⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠱⠀⠌⠢⣤⠄⠤⢿⠸⠋⠉⠁⢠⣎⠣⡴⠒⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣷⠴⢚⠁⢈⣡⣰⢸⠀⠀⠀⠋⠈⠁⠙⠒⠒⠃⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠀⢴⣆⣠⠴⠊⠀⠸⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢽⡣⡀⠀⠀⡘⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡆⠀⠀⠀⢰⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠻⠕⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⡘⠠⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠈⠲⣄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠃⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠸⠋⠁⠀⠑⢆⠀⠀⠀⠀⠀⠣⠄⣀⡀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠂⠁⠘⢀⣀⣀⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢶⣄⠀⠀⠀⡀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⣯⣤⡀⢸⣿⠀⣼⡟⠘⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⢀⣼⠟⢸⣿⡄⣿⣧⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋⠁⠀⠘⠛⠃⠈⠛⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⣤⠐⢳⡆⢲⣦⢠⣶⣄⡖⢹⣷⡄⢀⣤⡶⣦⢀⣴⣦⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⡄⠀⠙⠻⣷⣦⣄⡤⣾⡟⢻⡄⣿⣧⣿⣆⣾⢡⣿⣿⠋⣿⣿⠃⢸⣿⣇⣼⣿⠁⣡⣾⣿⠇⣰⣿⡗⣰⣿⡽⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣿⣿⣇⢿⣧⣸⡇⠙⠋⠀⠉⠁⠸⠟⠁⠀⠈⠉⠀⠈⠛⠛⠀⠉⠉⠁⠈⠉⣉⣿⣿⢋⣽⣿⣷⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠹⣿⣿⠈⠙⠉⠀⠀⣴⠄⢦⡀⢠⣤⠀⢀⣤⡄⣠⣀⣄⠀⢠⣄⠀⢀⡴⠛⢹⣿⠇⠙⠢⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣿⡏⠀⠀⠀⠀⠀⣿⣧⢿⣀⢸⣿⡄⣿⣏⣁⢸⡟⣿⣦⣸⣿⠀⡟⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣿⠃⠀⠀⠀⠀⠀⣿⣇⢀⣿⠟⣿⡀⣿⣇⣹⢿⡇⣿⡟⢿⣿⠀⠙⠲⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣠⠋⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠛⠋⠈⠛⠉⠘⢃⠙⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⢀⠴⠲⣶⣆⠀⠀⠀⠀⢰⣦⡄⣷⡖⢺⡁⡦⢾⡀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢻⣿⠄⠀⠀⠀⠈⠟⠻⠏⠛⠘⠃⠋⠘⠃⠀⠀⠀⣰⣿⣣⣟⣴⡾⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀⠀⠀⠘⠛⠀⠀⡠⢲⣶⡀⣴⣦⣠⠖⣶⠀⣠⣤⣶⡄⢰⣿⡿⢫⣿⡟⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⣿⣷⣾⣄⠈⣿⣷⡏⣿⡇⢸⣿⣾⡟⣱⣿⡇⣿⡟⠀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⣿⠆⠻⠿⠀⠛⠃⠸⡿⠙⠛⠉⠻⠟⠻⣧⠴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣠⠟⣠⢤⣀⣀⣤⣤⡀⣠⣤⠀⢠⣤⡀⢀⡤⣤⠈⠛⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⡛⠀⣿⠀⠀⣿⠀⠀⢸⡇⣿⠸⣧⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠹⣷⡄⣿⠀⠀⣿⡟⠀⢸⡟⠃⢀⡈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⠿⠃⠿⠀⠀⠻⠷⠆⠸⠇⠀⠘⠷⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡆⢹⣿⣿⣿⠠⠏⢀⣼⣿⣡⠏⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣿⣿⣿⣿⠃⠀⣾⣿⠘⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢸⡇⢸⣽⣿⡏⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⡠⣾⡟⠀⠀⣿⣿⣿⠏⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣳⠀⡇⠈⣿⣿⠀⠀⢺⣿⡏⠀⠀⠀⠀⠀⠀⣠⠞⣱⡟⠀⠀⠀⣟⠻⠏⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⢇⢂⣿⣿⡄⢀⠀⢸⡆⠀⠀⠀⠀⠀⣰⡏⣾⣿⡅⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⡜⢿⣿⣄⢢⠀⠣⠀⠀⠀⠀⠀⢿⡅⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣻⣷⠘⢮⡻⣿⣦⡱⡄⠁⠀⠀⠀⠀⠸⣧⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠿⠟⠃⠀⠳⣌⠻⣿⣾⢆⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣂⣷⠀⠀⠀⠀⠀⠀⢹⡄⠈⢻⣯⢆⠀⠀⠀⠀⣸⣿⣿⣿⣯⢷⡕⢤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⣼⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡙⠷⣝⡢⡀⠀⠀⠀⠀⠙⢿⣷⠦⢄⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⢀⡜⠀⠀⠀⣿⣟⠀⠀⢀⣾⣿⣿⣿⡿⢻⣿⣿⣷⡈⠹⢮⣦⡀⠀⠀⠀⠀⠙⢄⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠞⠀⠀⠀⢠⣿⡆⠀⢠⣿⣿⣵⣿⣯⣤⣌⢿⣿⣿⣇⣀⣈⢿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⢠⣿⡟⠀⠀⣾⣯⣾⣿⣿⣷⣄⠙⣮⢿⣿⣿⣿⣿⣿⣷⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⠀⠀⠀⣰⢟⡿⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠘⡎⢿⣿⣿⣿⣿⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⠈⢿⣿⣿⡁⠀⠁⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣀⣤⣴⣶⡾⠿⠿⠛⠛⠋⠉⠀⠀⠀⣠⣾⠟⣿⠋⠀⠈⠉⠛⣿⣿⣿⠀⠀⠈⣿⣿⣿⢀⠀⠻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⣴⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⣋⣙⣻⣤⣶⣲⢶⣦⡀⠘⣿⣿⡇⠀⠀⠘⣿⡇⠀⠀⢠⠈⠉⠈⠙⡄
⠀⠀⠀⠀⠀⣰⠟⢁⣼⣿⣿⠟⠉⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢸⣿⡇⠀⠀⢠⣿⣶⣷⡀
⠀⢀⠀⠀⣰⡏⢠⣿⣿⠟⠁⠀⠀⠀⠀⣠⣶⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠇⠀⠀⠀⣿⣿⣿⣿⡆
⠀⡘⠀⢰⡟⢠⣿⣿⠃⠀⠀⠀⢀⣠⣾⢟⣩⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠙⣿⣿⣿⣿⡇⢠⡿⠀⠀⠀⠀⠙⠛⣻⠆⠛
⠀⡇⠀⣿⠃⣼⣿⠇⠀⠀⢀⣴⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃⡼⠃⠀⠀⠀⠀⠀⠀⠛
⠀⡇⠀⣿⠀⣿⣿⠀⠀⣰⣿⠟⢡⣾⣿⣿⣿⣿⣿⣿⠟⣁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡇⠀⣿⢸⣿⣿⠀⢸⣿⠏⢀⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⡴⠀⠀⣀⡀⠀⠀⠀⣀⠀⢸
⠀⡇⠀⡏⢸⣿⣿⠀⣼⡇⠀⣸⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⡟⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⢎⣤⠞⠋⠉⢻⣴⣶⣯⢽⣤⡼
⣴⠃⢰⠇⠀⣿⣿⡄⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠇⠀⣠⡾⠟⠛⢻⡄⠘⣿⣿⠏⠁⠀⠀⠀⠈⠉⠉⠀⠀⣿⠏
⠏⠀⠸⠀⠀⢹⣿⡇⣿⠀⣸⣿⣿⣿⣿⣿⣿⡿⠘⣿⣿⡿⠀⡾⠋⠀⠀⠀⢸⣿⣶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⠛⠃⣸⣇⠀⠀⠀⠀⠀⠉⠻⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⡀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⠀⣿⣻⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⣴⢻⣿⣧⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⢀⠈⣜⣾⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣀⣬⣤⣵⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠀⠉⠁⠁⠀⠀⠀⠐⠙⡟⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⡄⠀⠀⠀⠀⢠⡠⠀⠀⠀⣇⣠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠁⢙⠛⡡⠀⠀⠀⠐⠛⠡⠀⠀⠀⣷⡧⣾⡇
⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠡⣹⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠀⡇⠛⡟⠁
⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⠠⢀⠀⠈⠁⠀⠀⢀⠀⠀⠀⣯⢰⡇
⠀⠀⠀⠀⠀⡀⠴⣄⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣭⣥⠄⠀⠀⠒⠛⠊
⠀⠀⠀⠀⠀⠖⠐⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣗⣈⣿⣿⣀
⠀⠀⠀⠀⠀⠈⢤⣨⣇⡅⠀⠀⠀⠀⠀⠀⠀⡀⠔⣉⠂⡀⢀⢀⣠⣠⣼⣭⡤⣀⡢⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢫⠀⢳⡀⠀⠀⢀⡠⠂⠁⠠⠂⢱⠉⠀⠀⠀⡢⠂⠨⣿⠦⡽⣅⠀⠀⠀⠀⠀⠁⠐⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⢷⣀⠔⠁⢀⡠⠚⡃⠀⢾⠀⠀⠀⠊⠀⠀⠐⣿⠀⡧⠈⢹⠉⠙⠒⠲⠤⢤⡵⢧⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡜⠁⣐⠔⠉⠀⠀⠀⠐⣚⠠⢠⣴⣤⣤⣤⣬⣿⠆⡟⠀⢾⠀⠀⠀⠀⠀⢠⠃⠄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⢀⣇⣴⡇⠀⠀⠀⠀⢀⠃⢐⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠁⠀⠀⠀⠀⠙⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⣤⣤⡄⠀⡄⣀⣴⣿⠋⠁⠀⠀⠀⠀⡈⢀⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⢹⠀⠀⠘⠉⠉⡇⠀⠀⠀⢀⠤⠐⠃⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡄⠀⠠⡇⠀⠀⠀⠐⠄⠙⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢹⠀⠀⠀⠀⢜⡇⠀⠀⠀⠀⠀⠔⡰⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⠤⣬⡆⠀⢠⠀⡭⡇⠀⠀⠀⠀⠈⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣀⣀⣀⡄⢰⡁⠀⠈⢉⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠰⢧⣠⣠⣡⣤⡾⣇⣀⣉⣉⣉⣷⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠄⠤⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣤⠲⠶⢶⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣯⠿⣿⣿⣿⣿⣶⣴⣶⣤⣶⣍⠻⣿⣿⣻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢿⣾⣿⣶⣿⣿⣟⠉⠁⠈⠁⢠⣾⣿⡇⠻⣿⣤⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠶⢻⣬⣿⣿⣿⣿⣿⣷⡄⣰⣿⣷⣍⣉⠀⠀⠀⠙⠿⠋⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⡿⠉⠈⠁⢹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠻⣿⢿⣿⣯⡀⣠⡀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢟⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⣿⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠻⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡟⠁⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠁⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣤⣀⣀⢀⡀⠀⠀⠈⣡⣤⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠾⠷⠿⠶⠶⠶⠚⠛⠉

⠠⢡⠃⠆⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠧⠊⡜⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴
⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉
⡌⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡂
⠀⠀⠠⠀⢄⠀⠀⠀⡜⠀⠀⢸⣿⣿⣿⣿⣿⢉⣩⣵⡦⣭⡙⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠑⠂⢀⠔⠁
⠀⠀⠡⣀⣀⣧⠤⠊⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣥⣪⣱⣢⣽⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣛⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣇⣾
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠉⠑
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣡
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⣿⡇⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⡣
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⣡⠖⣡⡝⢤⣼⢿⢿⡄⠀⠀⠀⠀⠀⠀⠌
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣠⠾⠿⠐⠙⣌⠳⡇⠘⣄⢿⠀⠀⠀⠈⠀⡘
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡠⢌⠛⠃⠀⠀⠀⠈⠓⢼⠀⢹⠐⠀⠀⠀⢠⠀⡇
⠀⠇⢀⣠⠤⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠀⠀⠀⢠⣷⠀⢀⣸⠀⢸⠀⠀⠀⠀⠀⠢⡱
⠔⠚⡅⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⡜⢹⠀⠘⡟⠀⠀⡆⠀⠀⠀⠀⠀⠑
⠀⠀⠈⠒⠤⠤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠁⣿⣷⠖⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢀⡱⠖⠉⠀⠀⢇⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠈⡘⠂⠀⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣄⢸⣆⢠⣾⠀⢀⣴⡏⠀⠀⣴⡇⠀⠀⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢙⠀⠁⣄⢀⣿⡌⢛⡁⢰⣷⡍⠴⠃⠀⠀⡀⠀⠉⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠀⢐⠀⠀⠀⠀⠀⠀⠀⢻⠁⣾⡇⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⢠⡀⠀⠀⠀⠄⣤⠰⠾⢤⠉⠀⠀⠀⠀⢀⠀⢀⠀⠀⣤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⢡⠈⠻⠟⢧⢸⣦⠓⢤⣻⣆⣴⠈⠀⠁⡀⠀⠂⠀⣴⢃⣤⣖⣒⣘⡛⣉⣀⡳⠆
⠀⠀⠀⠀⠑⠤⠀⠀⠻⣿⣷⡌⣆⢰⣷⠈⢻⣅⡐⢷⠙⢿⡅⠀⠀⠀⠁⠀⠁⣦⣹⡟⢁⣨⠹⢷⡄⠘⠛⠟⠂⢀⡀
⠀⠀⠀⠠⠔⠂⠄⠐⣲⠶⠦⢤⣬⣷⣧⣐⣛⠉⠻⣶⣄⠘⡇⡀⠀⡀⠀⠀⢀⣽⣿⣤⣼⣷⣶⠶⠖⢒⣛⠋⠙⢿⣿⠐
⠀⠀⠀⠀⠀⠀⠀⠚⠋⣀⣀⣀⠖⣋⣉⡙⢛⣿⣷⣾⣿⣿⡆⠂⠀⠀⠁⣴⣿⠿⢻⣍⠀⣴⠎⠳⢄⣘⠻⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠟⠁⠀⠉⣩⡵⠋⠀⠀⠈⠙⢿⠃⠀⠀⠀⠀⠟⠁⠀⠀⠉⠀⠉⠀⠀⠸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢐⣠⣾⡿⠿⣦⣐⠀⠀⠠⠄⠄
⠀⢀⣶⣦⣤⣄⢺⣶⣶⣄⡀⠀⠀⠀⠁⠀⠀⢐⡀⣤⢰⣿⣿⢸⡇⣶⣦⡌⣷⣦⢠⣀⠀⠀⠀⠀⠀⠀⠀⣂⣤⣶⡶⢂⣤⣤⣶⣆
⠀⠈⢿⣿⡿⣿⢥⣤⣙⠻⣷⠙⣿⠟⢡⡗⢸⣟⣣⣿⡀⣿⣿⢸⡇⢻⣿⣷⢸⡿⢸⠃⣿⣿⠸⡇⢸⣿⢸⣿⢟⣩⣤⢽⡿⣿⣿⠏
⠀⠀⠀⠈⠿⠙⠷⣽⣿⡷⡘⢷⡈⠳⣶⣭⡢⣝⠿⣿⡇⣿⣿⢸⡇⣸⣿⣿⣼⣧⢸⣧⡙⠻⢀⣟⣼⣿⠟⠡⠿⠿⣵⡿⣹⠏
⠀⠀⠀⠀⠀⠳⣤⣾⣯⡃⢸⣶⡄⡙⠾⠿⣿⠾⢕⣾⣇⣛⡭⠸⠇⣿⣿⣿⢿⡿⢮⣻⠇⠰⠿⢟⣋⢀⣴⣶⣢⣤⠈⠽⠋
⠀⠀⠀⠀⠀⠀⠈⢷⣿⠇⢘⣿⡇⣿⣷⠀⣤⣍⡁⠘⠿⣧⣾⣾⠷⠘⣿⡷⠀⠚⢃⣉⡀⢠⣶⣿⡁⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⢸⣿⡇⣿⣿⠀⢻⣿⡿⢀⣦⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⢸⣿⣿⣧⡈⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠿⠀⣿⣶⣶⣿⣿⣷⡄⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣶⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠐⢒⣒⣭⢭⣍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠈⠉⠴⣶⣶⣾⣿⣿⡿⣿⠋⢁⡶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢂⣩⠴⠺⠩⠝⠝⠋⠙⣻⡿⠓⠒⠈⡡⠊⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠾⡋⠉⠁⠘⠀⡀⠀⣳⠊⠁⠀⠀⢀⠌⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⡠⢡⠎⠁⠀⠀⠀⠀⠀⠀⠀⡽⠁⠀⠀⠀⢀⠊⠀⡴⠁
⠀⠀⠀⠀⠀⡐⢠⠃⢀⠂⠀⠀⠀⠀⠀⠀⢰⠃⠀⣀⡀⢀⠃⠀⡼⠁
⠀⠀⠀⠀⢀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡷⠁⡄⠀⡼⠁
⠀⠀⠀⠀⢀⠀⠡⡠⠀⠀⠀⠀⠀⠀⠀⠀⡸⢠⢿⠃⠠⠀⢠⠃
⠀⠀⠀⠀⠘⣄⠀⠑⢤⣀⠀⠀⠀⠀⢠⡺⣡⣟⡏⢀⠃⢀⣿⠶⠶⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠘⢗⠢⡀⠈⠓⠴⠤⠒⠉⠸⢿⡽⡣⠃⠀⣼⣿⠷⣖⣒⠢⠤⣀⣉⠙⠒⠶⠤⣤⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠳⣌⠐⠢⠀⠀⠤⠊⠉⠂⠊⠁⠀⣰⡿⠒⠠⠤⢬⣭⣙⡒⠻⠿⠿⠶⠦⠀⣰⣚⠛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣦⣀⣀⣤⡶⣄⣀⣠⣤⡶⠋⠀⠄⠀⠀⠂⣀⠦⠬⠍⠂⢶⣶⣲⡯⠿⢤⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡐⣡⠞⠋⠉⠀⠀⠊⠉⠉⠁⠀⠀⠀⠀⠁⢀⠔⠁⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠸⢰⠑⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡸⠁⠀⠀⠀⢀⠔⠊⠁⢀⡠⠶⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡠⠊⣠⣶⠆⡠⠒⠁⢀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣌⠲⣅⠀⢀⠀⠀⠀⢀⠀⠀⢀⡐⣡⢺⡟⡡⠊⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⢌⡉⠒⠤⢤⣀⢐⣔⡤⠊⠸⢕⡯⠊⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠘⠢⢤⣤⣬⣭⣤⡤⠴⠚⠉⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⣀⣀⠀⠀⠀⣀⣀⡠⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠿⢃⡄⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⣛⡴⠂⠀⠀⠀⠀⠲⣜⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡴⠊⠀⠀⠀⣈⡰⣌⣁⡀⠀⠀⠀⠀⠀⠀⣀⠸⠤⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠠⣄⠱⠶⠤⠀⠀⠀⠀⠄⣐⠀⠈⢀⠁⡄⡠⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⡄⢠⠀⠀⠀⠘⠁⠀⠀⠀⠠⠠⠴⠄⠀⠙⠐⠀⠤⠀⠰⠄⠙⠀⠄⠀⠀⠀⠀⠀⠀⣄⣇⡀
⠀⠀⠀⠀⠀⠈⢡⠟⣸⢂⡗⡼⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡯⣭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠊⠁⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠳⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⢷⣌⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣾⣶⣢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡌⢳⠀⠀⠀⠌⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠟⢿⢿⢹⡿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠩⠀⠃⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠈⠁⠀⠉⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡠
⠀⠀⠀⠀⠀⠐⣶⠧⣸⡀⡇⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡜⣠⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠞⠉⡴⠋⡼⠃⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⡼⠁⣠⠖⠚⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣴⢫⣖⣋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⠃⠘⢧⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠈⠶⠀⠀⠀⠀⠀⠀⠒⠶⢦⡴⢂⣴⠆⠄⠀⠀⢦⣐⡒⡆⢲⠀⢀⡀⠈
⠀⠀⠀⠀⠐⠛⠞⠿⠙⠃⠈⠀⡄⠀⠀⠀⠀⠈⠑⢛⠓⠄⡀⠈⢽⡗⠄⠀⣙⠲⡎⢟⠛⠀⠀⠀⠘⠀⢠⡟⡠⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠉⢉⣭⠓⠠⠀⠀⠀⠀⠀⠈⠁⠀⠣⠀⠀⣄⣹⣿⣯⠀⣸⢸⣱⡏⣴⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣨⣦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢚⣛⡣⠀⠀⠘⠁⠇⠈⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣶⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣜⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢤⣀⣀⠀⠀⢠⠎⠀⠀⠀⠀⠸⡄⠀⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠉⢹⣷⠲⡄⠀⠀⠀⠀⢳⣄⡀⠌⡨⠂
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⡠⣀⠀⢼⢿⣄⡽⠀⠀⠀⠀⡜⢀⡯⠥⠤⠤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠘⢢⣜⣑⠯⡠⡳⣄⠀⠀⢀⡜⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢈⠭⠷⣦⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠔⣽⢧⠔⣋⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⡡⣚⠁⡜⣧
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡍⠛⣄⡘⢨⡶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡜⡜⠁⡖⡇⡟
⠀⠀⠀⢣⡀⠀⡠⠖⠋⠉⠢⠀⠀⠀⠓⠋⠠⡞⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠸⡸⠀⡸⡸⣸⡇
⠀⠀⢀⡠⠟⠉⣀⠤⠔⠚⠓⠒⢒⣲⡶⠒⢶⠤⢤⡤⢤⣀⢀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢧⠔⡱⣱⡽
⠀⠐⠛⢲⡦⠊⠁⠀⠀⠀⠀⣀⠛⡵⣍⢉⠉⢠⣳⣶⡋⠁⣀⣻⠒⠤⣄⣀⠀⠀⢳⡙⠤⢞⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠄⠈⠃⡠⢀⡅⢰⡏⠯⠩⠜⠁⠲⠦⠈⠉⠙⠒⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠀⢸⣷⠟⢻⡄⢀⡾⠛⢻⡄⢸⣿⠟⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⢀⠀⢸⡇⠀⢸⡇⢸⣇⠀⢸⡇⠘⣷⠀⢠⡿
⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠿⠿⠿⠥⠼⠧⠤⠼⠷⠤⠽⠿⠿⠤⠴⡿⠿⠿⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⡀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠿⠿⣿⣿⣿⣦⡀⠀⠀⠀⠙⣿⣿⣿⣿⠗⠀⠢⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⢀⡤⠔⠚⠉⠀⠐⠀⠈⢿⣿⣿⣿⣦⡀⠀⠀⢈⠿⠋⠁⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣤⡀⠔⠊⠁⠀⠀⠀⠀⠀⠂⠀⠀⠘⣿⣿⣿⣿⣿⠖⠉⠀⠀⠀⠀⠀⠀⠀⢡⠀⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⠹⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⣿⣿⣿⡄⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠇⠘⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⡄⣸⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⢀⣴
⠈⠻⣿⣿⡄⠀⠐⠀⠁⠀⠀⠂⠉⠀⠠⠀⠀⠸⠀⠈⢻⡏⠀⠀⠀⠀⠀⠴⢁⠀⠀⣇⠈⠉⠙⠓⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣴⣿⣿
⠀⠀⠈⠻⣿⣄⠀⠀⢂⡀⠀⠈⠀⠀⠀⠀⠀⠀⠟⢰⡏⠠⠀⠀⠀⢀⡌⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠁⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠄⠀⠀⡝⠉⢲⡄⠀⣸⠏⠰⠁⠀⠀⢀⡎⠉⠉⢒⣄⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢨⡄⠀⠁⡀⠀⡻⠁⠀⠀⢱⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠙⠉⠈⠉⠉⠁⠉⠛⠲⢦⣀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⡴⠃⠘⡀⠀⠀⣰⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⢀⡼⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⠁⠀⠀⢡⠀⣰⠇⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⠟
⣿⠟⠋⠉⠉⠀⠀⠀⠈⣴⠏⢀⠂⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣤⡤⢀⡀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠉⠉⠉⠉⠒⠢⡀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠞⠁⠀⠀⠀⠙⢢⠀⠀⢠⠃⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⠂⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠳⠀⠀⠀⠀⠐⠀⠀⡇⠀⠿⢀
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠌⠀⠀⡐⠀⠀⠀⠀⠀⡠⠒⣀⠐⢦⡀⠀⠀⠀⠀⠸⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⡃⠀⣶⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⢀⡾⠀⠈⠀⠀⠀⠀⠀⡄⠈⠀⠪⢀⠻⣿⣿⣆⠹⠴⠀⣒⢒⣛⣠⣤⡀⣦⡀⠀⠰⠀⢀⠜⠁⠀⠉⠉⠓⠒⠤⠤⣀⣀⣀⣀⠤⠚⠁
⠀⠀⠀⠀⡾⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡸⠟⠁⠀⠀⠀⠈⠛⠉⠉⠛⠓⠈⢧⡁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⡆
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡐⠒⢄⠀⠙⠷⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢸⠄⠀⠀⠈⠀⠀⠀⡴⠁⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⠀⢡⠀⠀⠙⣆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡤⠶⠒⠒⠚⠳⠤⣀⠀⠀⠀⡾⢀⡄⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢝⠛⢻⠀⠀⡆⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠄⠀⠈⠑⣆⢠⡇⠚⠁⡿⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠉⠀⢀⠃⠀⠀⠀⠰⡄⠀⠀⢀⠀⠀⠀⠀⠂⠖⠀⠀⠀⡇
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡃⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⣷⣀⣠⠄⡁⢀
⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠸⠗⢀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠉⠤⠴⠉⠙⠀⠀⠁⠁⠀⠀⡀⢰⣶⣾
⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠸⠂⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠱⠤⣰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⡿⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣤⣤⠀⢸⣷⣿
⠀⠀⠀⠀⠤⠄⠀⠀⠀⣀⣠⣴⠾⣛⠭⠚⠡⠀⠙⣦⣄⠀⠀⠀⠀⠀⣤⣄⣀⡀⣀⠀⠀⢀⠤⠚⠍⠉⠉⠚⠂⠀⠀⡤⠄⠒⠊⠉⢉⣛⣿⣿⣿⣿⣿
⣄⣀⣤⣤⣤⣴⣶⣾⣿⡿⢋⡥⠞⠁⠀⠀⢀⣴⣾⣿⣿⣷⠖⢤⣤⠔⣸⣦⣄⠀⠀⠀⡠⠛⠀⠆⠐⠀⠀⠀⠲⠐⢶⡀⢀⡀⠊⠉⠀⠀⠈⠙⢦⡀
⢀⣠⣾⣿⣿⣿⣿⠟⠁⢠⠟⠀⠀⠀⠀⠀⠈⠁⠉⠙⠿⠁⠀⠀⢀⣾⣿⣿⣿⡇⠐⠀⠀⠀⣆⠀⠀⠀⢠⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠠⠀⠀⠹⣆
⣿⣿⣿⣿⣿⠟⠁⠀⢀⡏⠀⠀⠀⠁⠀⢠⡄⠀⠀⠀⠀⢡⠀⢠⣿⣿⣿⠟⠉⠀⠈⠑⢄⠠⠈⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⣿⣿⠟⠁⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⡸⠀⣾⣿⣿⡇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⣿⡿⠃⠀⠀⢀⣾⣿⣿⣷⡀⠀⠠⠀⠉⠉⠀⠀⠐⢀⡴⠃⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢿⣿⣷⣤⣀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇
⡿⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣠⣤⠶⠋⠁⠀⠀⣿⣿⣿⣿⣿⣷⣦⣤⣤⠴⠶⠛⠁⠀⠀⠙⢿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿
⣧⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣛⣛⣛⣟⣛⣛⣉⣛⣻⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣽⣍⣭⣭⣿⣿⣏⣽⣿⣯⣉⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣴⣶⠀⣴⣦⣴⡆⠀⠀⠀⠀⠀⢰⣶⡶⣶⣶⡆⠶⣦⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡞⣶⣶⢲⣷⢁⣤⣦⣴⢶⣶⢶⣾⡄⢰⣰⣶⣶⢶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⢸⣿⠀⣿⢿⡿⠿⢿⢿⢿⡇⢸⣿⣿⣿⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢻⣇⣿⡿⢸⡿⠀⣿⣯⣷⡴⠾⠾⠾⠇⣸⢿⣿⢿⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⢶⠶⣶⢶⡶⣶⢶⣶⣖⣶⠶⡶⣶⣶⠶⣶⢶⡶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣛⡛⣛⣛⣛⢛⡛⢛⡛⣛⣛⡛⢛⣛⣛⢛⣛⡛⣛⣛⣛⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣉⠁⠉⠈⡉⣈⠁⡨⠌⢡⣽⣡⣈⡅⣥⣌⡍⣤⢡⣭⣬⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢀⠀⠀⡇⣿⠀⢧⠀⣿⣀⣸⢾⢧⡷⡟⣿⣿⣼⣶⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠭⠬⠤⠤⠤⠄⣩⡍⡉⠨⢬⠥⠥⠭⠬⠭⠭⠭⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⣿⣷⣾⡷⡾⢿⡆⠀⢰⠀⢸⡏⢸⣿⡇⡏⡇⡇⢰⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠟⢻⡟⠛⢃⣃⣻⣃⠀⠘⣀⡸⠷⣘⠛⣃⠟⢃⡷⠆⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣾⡅⠚⡦⢹⣿⣧⠘⣳⢘⣿⡇⣿⡇⣿⣿⢸⣽⣻⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠳⠼⠿⠛⠖⠲⠿⠿⠟⠲⠷⠺⠿⠓⠖⠴⠿⠻⠸⠿⠛⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢨⣤⢤⣤⠀⣤⣤⣤⠀⢬⡀⠤⣦⣥⡴⠰⣦⣦⣤⣤⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⢘⡘⢛⡟⢘⡛⠛⡛⢀⡞⣘⡞⢛⣛⢛⡀⢛⣛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⣶⡶⠶⣶⠀⢰⡆⣰⢻⣷⡇⢢⣿⣥⢾⡭⢭⡩⣭⡯⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣆⣿⣀⣸⣇⣯⠀⣹⣿⠃⢿⣿⣽⣿⣏⣹⣿⣥⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⢲⠌⣀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠁⠀⠀⠀⠐⢀⠐⢤⡀⣀⡀⠀⢇⣠⡌⠁⢰⠄⡳⠂⠀⠀⠀⠀⠀⠀⢠⠂⢁⣧⣦⠀⢀⡗⣠⠠⠆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⡀⠸⠀⠀⠰⣀⠈⢲⠏⣠⠀⠣⠀⠀⠀⠑⠚⠉⠂⠁⠀⠀⠀⠀⠀⠀⠁⠍⠉⢧⠈⠂⢰⠿⠴⣥⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠪⠅⠈⠆⠀⠀⠀⠀⠀⠀⠀⡔⢀⡄⠀⠀⠀⢇⠀⠀⢘⣫⠃⡄⡇⠀⠂⠀⢸⡆⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢷⡇⢀⢀⠇⠀⠀⠂⠐⠂⠁⠀⠈⠱⠀⣸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠘⢀⣨⠀⠀⠀⠀⡐⠀⢤⠉⢦⡀⠀⠀⠈⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠦⡄⠃⠀⠀⠀⣠⠔⢓⠦⡜⠢⠈⠃⢱⡄⠀⠀⠀⠀⠀⠀⠀⠂⢈⠁⠒⡄⠀⠀⠀⠀⢀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⡆⠉⠁⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⡁⠀⠼⠀⢸⣏⢀⡄⠠⠭⡀⢈⠄⠀⠀⠀⠀⡀⠘⡦⢄⠉⠀⠀⠀⢰⠃⠀⢀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠉⡀⢀⠮⡤⠀⠀⠀⠀⠀⡖⠀⠀⢹⣄⡯⠕⠋⠛⠿⠀⠐⠀⠘⣯⣾⣢⡀⠀⠀⠈⠑⠋⠁⠀⠀⠀⠀⠘⣄⠀⠘⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢠⠈⠈⡛⣦⡴⢋⠉⠛⢶⠋⠉⠳⣠⠀⣐⡣⠃⠝⠁⠀⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣲⣄⣀⠙⣳⠀⠈⢢⡀⢨⣧⣄⠀⢸⡿⣤⡴⠛⠄⡈⣠⣠⡦⠛⠃⠢⠀⠀⠀⠀⠀⡀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠁⠆⠙⣆⠀⣨⠝⠲⢿⣶⣪⣴⠀⠀⠈⢓⢁⢠⡤⢬⣇⠀⣀⠽⢿⡇⠀⠀⢰⠀⠀⠀⠀⢸⠀⠈⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣶⠠⢿⣴⣇⡀⢠⠌⢉⡉⠙⠆⢀⡔⠺⠿⢏⠀⠀⠉⡟⠡⢄⡀⠸⠤⠀⠟⠁⠀⠀⠀⠈⠋⠴⠥⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠠⣇⠀⠀⠙⣦⢸⡁⠀⢠⠄⠀⡆⠆⢀⠦⡀⠁⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠀⠛⠀⠣⣄⣴⠀⠞⠉⠉⠀⠀⠈⠀⠁




⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⣵⣶⣦⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⣭⣥⣾⡿⠿⢿⣷⣤⣭⡙⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠸⣿⣿⡏⣴⣾⣶⠹⣿⣿⡇⣸⣿⣿⡿⠿⢃⣶⣷⣎⠻⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡌⣿⣷⣘⠛⣋⣼⣿⡏⣴⣿⣿⡏⣴⣿⡿⢋⣉⠻⣿⣷⡌⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⣿⣿⣿⠿⣿⣿⣿⠇⣿⣿⣿⣷⡙⢿⣇⠻⠿⢃⣿⠟⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣭⣴⡆⣦⣭⣥⣾⣿⣿⣿⣿⡇⢿⣿⣿⢾⣿⣿⠆⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠿⢋⣭⣥⡌⣿⣿⣿⣦⣤⣦⢰⣤⣥⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢉⣾⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⢛⣵⣿⣿⣿⡉⣭⣙⠿⢸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣤⣴⣶⣿⣿⣿⣿⣿⣧⡙⢿⣧⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⢛⣛⡛⢿⣿⣿⣶⣦⢸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣡⣾⣿⣿⣌⣛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢸⣿⡟⢡⣶⡌⢻⣿⡇⣸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣌⣛⣡⣾⡏⢴⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠿⠟⣛⠻⠿⢃⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢄⠀⠀⠀⢀⣴⠚⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠄⠀⠀⢠⡾⠁⠀⡠⣱⠋⠀⠀⢠⡖⠆⠀⢠⣦⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⣼⡇⡠⠂⠀⠀⢀⣿⠃⢀⠐⠀⢻⠀⠀⠀⣿⡇⠆⠀⣺⡇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢸⡀⢀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡯⠐⠁⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⠃⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⢻⡄⠀⡀⠹⢻⡆⠀⣀⠀⠙⠂⠀⠀⠀⣿⡇⢀⢤⡀⠀⢀⡤⡄⠀⠠⢻⣇⢰⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠙⣆⠀⠘⣿⡌⢻⡇⣼⡇⢰⡏⢸⡇⡠⢤⡀⠀⣿⡃⠀⢸⣷⢠⡿⠐⡠⠲⣦⠈⣿⢸⣿⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠢⠋⠀⠀⣿⡁⠸⠂⣿⡇⢸⣇⢻⣟⠀⠸⠇⠀⣿⡇⠀⢸⡿⢸⡇⢀⡄⠀⣿⠀⣿⠈⣿⢠⠁
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠉⠁⠀⠀⢿⡇⠀⠉⠘⣿⠀⣠⠄⠀⠙⠃⠀⡼⠃⣈⣁⠈⠢⠀⠀⠜⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢠⠀⣿⢰⡏⠀⠆⡤⠀⠀⢀⡾⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⢠⠄⢸⢠⠟⠘⢿⡔⠈⠧⠀⠀⣾⠁⠀⡁⢤⠀⠀⠀⣠⢀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣿⠘⢀⢀⠀⢄⡠⠉⣴⠆⠀⢸⣿⢀⠈⢀⡿⠀⠀⠀⠳⣠⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⠁⢿⡀⢸⣿⠀⢘⠀⢰⣿⠀⠀⣼⣧⢀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⣿⠀⢘⠃⠸⣇⠀⡘⠀⠸⣿⠀⢠⢸⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⠷⠜⠙⠷⠃⠀⠀⠈⠈⠀⠀⠀⠙⠤⠃⠈⠛⠀⠙⠦⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉






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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣷⣶⣾⢿⣿⣿⣷⣿⣿⣯⠓⠄⠈⠣⠄
⣿⣿⣿⣛⠀⠋⠍⠙⢿⣿⣿⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⣠⡝⠿⠟⠙⣿⡟⠀⠀⠀⢠⣤
⣿⣿⣿⣯⠀⠀⠘⠃⠀⠉⠛⢷⣎⠉⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠹⠹⢿⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⣰⠆⠻⠁⠀⢀⡼⠛⣻⡆
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠙⠻⠿⢿⣶⣤⣀⡠⢄⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠘⠇⢰⣾⠋⠀⠃⠀⠀⠿⠇⠈⠿⠁
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣤⠀⠀⠀⠙⠿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣾⣿⡷⠀⠀⠀⣠⡀
⣿⣿⣿⣿⣿⣇⢠⣶⠟⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉⠙⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄⠀⠀⠀⠀⢠⣿⣿⡃⠀⡀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣮⣻⣿⠟⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⣿⣿⠉⢻⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡷⣦⣤⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⢸⣿⣿⣷⠾⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣍⠻⣷⣌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣾⡟⠁
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⢿⣷⣶⣴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⢸⣿⣿⣿⣿⣿⣿⣿⡟⢛⣿⣿⣧⠀⠹⣿⣿⠿⠶⠤⣤⡀⠀⠀⠀⢠⡀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀
⠀⢿⣿⣿⣿⡿⣿⣿⣷⣿⣿⣿⣿⠀⠀⠘⢟⢷⣄⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠉⠙⠛⢷⡄⠉⠀⠀⢸⡿⣿⣷⣦⣄⠈⢆⠙⠳⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠀⠈⢿⣿⣿⣷⡄⠣⡀⠀⠈⠑⠉⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡴⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⣾⣵⣿⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⣿⣿⣿⣿⣿⣿⠛⢿⡄⠀⠀⠀⠀⠀⠀⠂⠀⣠⣾⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣿⣿⣿⢻⣿⣿⡇⠈⣿⠀⠀⠀⠀⠀⠀⠇⣴⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⣾⣿⣿⣟⣿⡇⣸⣿⣿⢃⣼⠋⠀⠀⠀⠀⠀⣸⣸⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⢀⣴⣄⣴⣿⣿⣿⣿⣿⣾⣿⣶⣿⣿⠷⠟⠓⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⡿⢿⣿⢃⣠⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣏⡡⠄⠀⠀⠀⠀⠀⠀⢤⣶⣶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠿⠖⠀⠀⠀⠀⠒⠀⠀⠹⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣞⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣠⣾⣅⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡇
⢀⣠⠴⠖⠛⠉⠉⠀⢸⣟⣠⠞⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⠟⣸⠁
⠉⠀⠀⠀⠀⠀⠀⠀⣸⠈⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⣠⠏⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⢃⡴⠋
⣄⡀⠀⣀⣤⣞⡵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣫⠴⠋
⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉





⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶
⠀⠀⠀⣀⡤⠶⢶⡋⣩⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣾⣿⣿⠟⠋⠁
⣠⠴⠚⠁⠀⠀⢀⣯⣠⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⢸⣯⣤⣤⣤⣤⣀⡀
⠁⠀⠀⠀⠀⣠⠞⣽⡤⠴⠶⠀⠀⢹⣉⣉⣛⣓⣒⡦⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⠀⠀⣰⣿⣧⡤⠶⠶⠀⠈⣿⣿
⠀⠀⠀⠀⢰⣣⠞⠁⠀⠀⣀⣤⢴⡏⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⣰⢣⡞⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⣀⣤⣾⠏
⠀⠀⠀⠀⠈⡏⠀⠀⢠⣾⡥⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⣿⠿⠟⠉
⠀⠀⠀⠀⢀⣿⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⢸⣇⠀⢠⠏
⢤⣤⣤⠶⢋⡾⢖⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⣤⣤⣶⣿⣷⣶⠟
⠒⠶⠶⠚⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠛⠛⠋⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠛⠛⠋⠉⠉⣉⡉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠚⠓⠀⠀⠀⠀⣤⠔⠀⢀⡤⢬⠀⠀⠀⠀⠀⠀⠈⠙⣿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⠾⠏⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠚⠋⠙⠻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣠⢴⠶⡄⠀⠀⠀⠀⠀⠀⠀⣰⣥⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠘⢱⣁⡴⠃⠀⠀⢀⣀⣀⣠⣾⣿⠛⢿⣤⣀⣀⣀⣀⣀⡀⠀⢀⣴⣾⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠙⡿⢿⣿⢿⡟⠉⠀⠀⠀⠀⢈⣥⣌⡙⢿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢳⡄⠀⢀⠖⠀⠸⠁⠀⢠⣠⠀⠰⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣿⣿⠇⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢇⠋⠀⠀⡀⠹⠀⠠⢱⣤⣺⣿⡿⣼⣿⡿⠋⠀⠀⡀
⣿⣿⣿⣿⣿⣿⡟⢀⣿⠉⢻⡇⠀⠀⠀⠠⡬⢀⣾⠿⠁⠀⠀⢁⣱⡦⣴⣞⡇⣿⣿⡿⣵⡿⢋⠀⠀⠴⠛⠛⠛⠳⣄
⣿⣿⣿⣿⣿⣿⡇⠘⣿⠀⠀⣧⠀⠀⠀⠐⠚⠁⠀⠀⠀⠶⢀⣼⣿⠗⢀⣾⣴⣿⣿⡟⠋⠀⠀⠀⠀⣀⣀⠀⠀⠀⢹
⣿⣿⣿⣿⣿⣿⡇⠀⢹⣶⣦⡝⠀⠀⠀⠀⠀⠀⠀⠀⢄⢲⡞⠋⢋⠀⢸⣏⣦⡙⠋⠀⠀⠀⢀⣴⠟⠋⠀⢱⡄
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣾⣧⣶⣾⡏⠋⠀⠀⠀⠀⢠⣿⣿⠄⠀⠀⢸⠇⠀⠠
⣿⣿⣿⣿⣿⣿⣼⡂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣯⡽⣿⠀⠀⣀⠀⠀⠀⠉⣡⣾⠓⣰⡏⠀⢠⡇
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⢸⣲⣟⢿⣿⣷⣿⢀⣴⡿⠛⣆⠀⠀⠻⠧⠞⠉⠀⢀⡞⠀⠀⣰
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣧⡛⣿⣿⡟⣿⣿⣿⡈⠧⠀⣸⠏⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡴⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠛⢿⡟⡁⢷⣿⣿⡗⢸⣿⣿⣇⣠⡾⠃⠀⠀⠀⠀⠀⢠⣴⠋⢀⠔⣩⢤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⠀⠈⠀⠀⠂⠀⠘⢿⠿⢇⠀⠿⠿⣻⡍⠀⠀⠀⠀⠀⣀⣴⠟⠀⣰⠇⡾⢁⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠙⠀⠀⠂⡠⠀⠀⢠⡤⠀⠀⠀⢸⣯⠀⠀⣀⡟⣿⡀⠀⢀⣴⡾⠛⠁⠀⣼⠃⠀⠷⠾⣼⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⠃⣴⡄⢿⡋⠀⠀⠀⣀⡀⠀⠈⠉⢿⢻⠀⠀⠘⠀⠘⡇⠘⠿⠋⠀⠀⠀⠀⢿⡀⢀⣠⣾⢿⠤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠢⠈⠶⡛⣆⠀⠻⠿⢿⣷⡦⠀⠈⣼⠀⠀⠀⠈⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠛⠁⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠢⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠤⢿⡀⠀⢠⡀⢰⣸⡀⡆⠀⠀⡀⠀⢠⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣔⠦⡀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠾⢷⣤⣼⣧⣼⣿⣷⣿⣤⣤⣿⣿⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣌⠀⡀⠁⠀⠈⠙⠛⠚⠛⠛⠛⠛⠛⠛⠛⣛⢻⣛⠛⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣿⣶⣶⣦⣀⣀⡀⣀⣀⣀⣠⣤⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⢤⣄⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⡀
⠀⠀⠀⠈⠙⠲⠦⣤⣀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠤⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢀⣀⣀⣀⡀
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡟⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢶⣤⣄⣀⣀⣀⣀⣀⡤⣺⠜⠀⠘⢿⣦⡀⠀⠀⠀⢀⣠⡾⠃⢱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⠀⠀⠉⠁⠚⠛⠓⠊⠉⠀⠀⠀⠀⠀⠉⠛⠼⠿⠯⠟⠈⠀⠀⠀⢿⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡣⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣍⡒⠠⡤⣤⣤⡶⠿⣽⠃
⠀⢦⡠⣤⣤⣤⣠⣴⣖⠜⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⢀⣎⡃
⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⣄
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣳⣄⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠺⣳⣦⣤
⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣫⠟⠉
⠀⠀⠀⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⢋⠃
⢀⣴⣞⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠄⠀⠔⠗⠠⠤⠚⠃⠤⠤⡄⠀⠀⠀⠀⠀⢀⠨⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⣟
⠋⠛⠓⠂⠊⠭⣓⠢⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠡⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡼⡄
⠀⠀⠀⠀⠀⠀⠀⢳⢱⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠡⠀⠀⢤⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢖⡽⠟⠋⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠠⠗⠀⠉⠰⠄⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡽⠁
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠄⠀⠀⠀⠠⠸⠀⠠⠀⡄⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡇
⠀⠀⠀⠀⠀⢐⣛⣧⠫⠮⣕⢦⡀⠀⠀⠀⠀⠑⠀⠅⠀⠀⠀⡥⠄⠩⠌⠄⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡠⣴⣤⣤⣤⣀⢻⣿⡄
⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠈⢣⢳⠀⠀⠀⠀⠀⡀⠁⠀⠀⠘⠃⠀⠰⠀⢰⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⣚⡛⠋⠉⠉⠉⠛⠛⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡇⠀⠀⣀⣀⠱⣄⠀⠀⠀⠇⠀⠀⠘⢨⠀⠀⢔⡐⠀⡀⠀⠀⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢗⡮⠕⠒⠐⠋⠈⣿⢲⠒⠆⠀⠐⠖⢊⣀⣸⡿⠀⠋⠓⠛⠵⣦⣀⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠉⠀⠀⠀⠀⠀⠀⡷⠖⠖⠂⠈⠉⠛⠁⠀⣉⡥⠀⠀⠀⠀⠀⠈⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⢶⢶⠆⠐⠂⠿⠁⠀⢋⣢⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⣤⣤⠀⠤⠶⠒⠈⢛⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⣀⣤⣀⠠⠤⠖⠆⠘⢟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣭⣤⣄⡀⠤⠶⠶⢚⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⣶⣦⣶⠞⠃⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⡀⢀⡀⣀⢀⡀⠀⠀⠀⠀⢀⠀⡀⢀⡀⡀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣷⣶⣤⣥⣌⣰⡀⠃⠒⡈⠀⠀⠁⠂⠒⠀⠎⠠⠁⠠⢀⣉⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⢿⠿⢿⢻⣷⣄⢀⣾⣶⣴⣶⣶⠶⠶⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣻⡿⢂⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣤⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⠟⠀⢉⣤⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡉⠀⠛⠓⠀⠻⢿⣿⣿⣿⣿⣿⣿⣧⡛⠂⠚⠛⠈⣁⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠈⠉⠘⠓⠀⢰⣶⣷⣿⣿⣿⡿⣿⣭⣻⣿⡗⠐⠛⠃⠉⠁⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⠀⢠⠀⠛⡁⠈⢻⣿⣿⣿⣦⣿⣘⣿⣿⡿⠁⡘⠋⢀⡀⢀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⣈⠆⢸⣿⡷⠀⠘⠿⢿⡿⢿⠿⠛⠋⠀⢾⣿⡆⢦⡀⠺⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⡀⡒⠈⠤⠄⢡⠀⠀⡌⠠⠤⠁⢂⢀⢀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠀⠀⡎⠉⠉⣴⠒⢠⠲⡄⢰⡆⢰⢲⡄⠉⠈⠴⠀⢈⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡐⠀⠀⠴⠦⠀⠈⢦⢸⠀⡇⣾⣷⢸⣾⠁⠀⠶⠆⠀⠀⡂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣌⠀⠀⡓⣤⠀⠘⠋⡘⠛⠁⠃⢘⠘⠉⠃⠀⡤⢃⠀⠀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠆⡥⠀⠀⠀⣀⣤⡐⠀⠂⡀⣁⢀⡴⢀⣦⡀⠀⠀⠀⢭⠰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠂⠀⠰⠂⣿⣬⡗⠀⠆⠠⢰⣾⠀⣻⣼⡯⠐⠄⠀⠒⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⡀⣤⡀⡀⠀⠁⠈⡅⠲⠑⢸⡗⠠⠀⠁⠀⢀⣀⡤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣤⣤⣶⣶⣦⣤⣄⣀⣀⣧⣴⣿⣇⣈⣤⣤⣴⣶⣶⣤⣬⣄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿⣼⣿⣿⣽⣿⢹⣿⣿⣿⣿⣿⣿⠿⠽⠿⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⢀⠠⠀⠈⠉⠙⠛⡻⢻⣏⢿⣿⡿⠛⠛⢫⠀⠐⡄⠀⢠⡀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢘⠁⣀⠰⠘⠀⠀⠺⢿⢸⡿⠃⠤⠀⠈⠠⠈⡰⠈⡂⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠋⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡋⣤⢤⣤⣠⠖⠚⠛⠛⠓⢦⡀⠀⠀⠀⣀⣤⡶⠛⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣿⣥⣬⣛⣃⣛⠲⡄⠀⠀⠀⠘⢧⠀⠀⠐⢿⣤⣤⣤⡀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠉⠉⢹⡇⠉⡇⠀⠀⣀⡾⠀⠀⣀⣀⣀⣠⡼⠃⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢧⣀⡾⠁⠀⠀⢺⡏⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠀⠀⠀⣀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢀⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⢸⡷⣄⠀⢠⡞⠋⠉⠛⠓⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠳⢦⣿⠷⢦⡄⠀⠀⠀⣠⣀⠀⠀⠀⠸⡇
⠀⣾⠀⢸⡇⢸⡇⠸⠿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠀⠀⠀⢀⣼⣇⣈⢻⣆⠀⢸⡇⠘⠳⣄⠀⠀⣿
⠀⣿⠀⠀⣻⠞⠒⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠳⣿⣿⣿⣥⠀⠐⣇⢠⡴⡿
⠀⣿⠀⢸⡏⠀⠀⠀⠀⠉⠙⢷⡀⠀⠀⠀⠀⠀⠈⠿⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⠋⢀⡇
⠀⢹⣄⡿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⣀⣀⣸⠇
⠀⠘⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢸⡏⠀⠀⣠⡞⠉⠁⣽
⠀⠀⠸⣆⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠘⣷⣶⡄⢿⠀⠀⣼⠏
⠀⠀⠀⠹⣆⠀⠀⠙⢷⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣰⣿⣿⠀⣿⣄⣼⠏
⠀⠀⠀⠀⠘⢧⡀⠀⢸⡇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⢀⡾⠁⠿⠃⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣈⠳⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡴⠟⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣬⣳⣟⣲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠄⠈⢁⡶⠛⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣿⣒⣒⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⣀⠀⠈⠀⠰⠂⠈⠂⠀⠀⠀⠀⠈⠒⡀⠂⠒⠒⠚⠀⠀⠀⠈⣉⡁⢀⣀⠀⠀⠤⢢
⠀⠀⠀⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡁⠀⠀⠀⠉⠉⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠱⣀⣨⠎⠁⠘⢄⠈⠩⢀⢂⡤⠐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢀⠈⠈⠀⠐⢀⡀⠐⠴⠉⠀⠐⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠦⠤⡤⣠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⢸⠰⠆⠁⠠⠖⠆⠈⠠⣐⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠖⠐⠒⠊⠃⠀⡀⠀⢠⡀⣠⢅⣤⣤⠤⠤⠄⠐⠂⠈⡒⢀⣤⠴⢒⣩⠷⠿⠉⠀⣀⡀⠀⠤⠄
⠀⠀⠀⢀⠀⠀⠀⠘⠂⠀⠈⠀⢀⠠⠀⠓⠦⢤⣤⣤⣤⡤⠴⢲⡞⠟⠉⣀⠈⠈⠉⠉⠈⠉⣹⠛⢛⠙⠗⣛⠿⠿⠿⢶⣶⣶⣿⣿⣿⣶⣶⠶⠖⠋⠁
⠀⠀⠀⠈⠑⠀⡀⠀⢀⠀⠀⢀⠀⠇⠠⠠⠤⠤⠤⠤⠶⠒⣻⠏⠀⠀⠈⠀⠀⠔⣊⠄⢒⡤⠾⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡀⠀⠁⠐⠀⠠⠤⠤⠖⠒⠛⢋⠁⡀⠀⠈⡀⠤⣒⡁⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠈⠀⠀⢉⠤⢒⣭⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⣄⠀⠀⢀⣨⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠴⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣭⣉⡀⢀⣀⣨⣵⣾⣿⠉⠉⠉⠛⠻⠶⢦⣤⣤⣤⣤⣤⡤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡔⠒⡌⣿⠋⠉⠁⠀⣿⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⢧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡄⠀⠀⢹⡆⠀⠀⠀⢸⣿⠀⣈⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⡤⠖⠚⠉⠁⠀⠀⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⠇⠀⠀⠀⠸⠿⢟⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠻⡖⠤⠂⣿⠀⠀⠀⠀⢰⠛⠉
⠀⠀⠀⠀⠀⢀⣴⠞⢉⡠⠤⠀⠒⠒⢿⡀⢸⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⡀
⠀⠀⠀⠀⢠⡟⢁⠔⢁⠔⠒⠉⣉⠭⢽⡇⣼⠁⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢀⣠⣶⠛⠫⡑⢦⡙⢦⡀
⠀⠀⠀⠀⣾⡁⢸⠀⢢⠀⠀⠘⣅⣤⣾⡇⣿⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣴⣾⣋⠕⢈⠇⢀⡇⠀⢱⠀⢻⡀
⠀⠀⠀⠀⣿⢧⡈⠢⣀⣱⣶⡿⢻⢹⣿⣧⡇⠀⠀⠀⠀⠀⠀⣀⣴⣞⠋⠑⠒⠒⠒⢉⡠⠊⠀⢀⡞⠀⣸⡇
⠀⠀⠀⠀⢿⠈⠛⢾⠻⠉⢡⢸⠈⡄⣿⣿⠃⠀⠀⠀⣠⡴⠾⢭⣀⠀⠀⠀⠉⠉⠉⠁⣀⡠⠔⠋⢀⡴⢿⠁
⠀⠀⠀⠀⢸⡄⠀⢸⡆⠂⡨⡄⡆⢣⢻⡟⠀⠀⠀⢰⡿⠶⢤⣄⣀⡀⠉⠁⠐⢂⣉⣉⣡⣤⠴⣞⣫⡆⡟
⠀⠀⠀⠀⠈⣇⠀⠀⣧⢀⠁⠃⢰⠘⣾⠇⠀⠀⠀⣸⠇⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠁⠀⠤⣾⣿⢰⡇
⠀⠀⠀⠀⠀⢻⡀⠀⢸⡈⡌⠘⠸⡀⣿⠀⢐⣊⣩⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡏⣼⠁
⠀⠀⠀⠀⠀⠘⡇⠀⠘⡇⠁⡄⠆⢇⢿⡾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢃⡏
⠀⠀⠀⠀⠀⠀⢻⠀⠀⢿⠀⢰⠀⠸⡘⣧⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣾⡿⢸⠇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠸⡇⡜⡄⠀⢧⢻⡀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣾⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢻⢠⢳⠀⢸⠸⡇⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⣿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣜⡇⡌⣄⣬⡷⠿⠚⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡻⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⠤⠤⠤⠤⠤⠶⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣀⣀⣄⡀⠀⠈⢤⣠⣌⠹⢷⠖⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠁⠉⠉⠛⣉⣱⡦⢀⡀⠉⠉⠛⣶⣶⣤⣦⡈⢯⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠶⣤⢧⠀⠀⠀⠹⣿⣷⣤⠀⠀⠀⠐⣶⢿⣟⣉⣷⣿⣿⣟⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⡠⠀⠀⢂⣴⣹⣆⠀⢀⣴⣿⣿⣿⣇⠀⠉⣹⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣀⠐⠁⠀⢀⣀⣥⣾⣿⣿⣶⣶⣿⣿⣿⣽⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠉⠉⣿⣷⡄
⠀⠀⠀⣾⣿⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⡀
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠿⠛⣟⣡⣼⣿⠁⢠⣴⣿⣿⣿⣿⣿⣿⡟⣛⣫⡿⠛⠟⠛⣋⠛⢿⣿⣿⣷
⠀⢠⣿⣿⣿⣿⣿⣼⣿⣿⣀⠀⠞⠉⠛⢻⣿⠀⢩⣱⢿⠟⢛⡛⢻⣿⣀⣳⣖⣩⣄⡈⢁⣿⣷⠀⠸⣿⣿⡆
⠀⠈⣾⣽⣾⢿⢿⣬⣿⡫⠒⠠⠮⢂⣅⣺⡟⢡⣤⣏⣹⣿⣿⣿⢷⡄⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡇
⠀⢀⣯⡿⣿⣷⣿⣿⣿⣿⣶⠠⣤⣾⡿⡟⢃⣹⣿⣿⣿⣿⣻⣿⣾⣿⣿⣿⣿⣾⣿⡭⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⡿⡿⣩⣏⢸⣿⣷⣾⠛⠋⠀⠩⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣟⣿⣜⢋⣿⣿⣿⣿⣿⡇
⠀⠀⢹⣿⣿⣿⣿⣋⣻⣿⢉⡙⢣⢸⡯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣻⣿⠻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⣿⣿⣿⡟⣿⢿⣿⣿⣿⣿⣝⠉⠳⠴⠚⣿⠯⠟⠾⠿⠟⣿⣿⣿⣿⣿⣿⣙⣿⠇⠹⢿⣿⣿⣿⡏
⠀⠀⠀⠈⣏⠦⠁⠸⣿⣿⣋⠀⠉⠈⠡⢿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡯⢴⣿⣿⣧⠀⠸⢹⣿⡟
⠀⠀⠀⠀⠈⠁⢤⠀⠸⣿⣿⣿⣭⣬⡓⢿⡿⣿⠟⣿⣶⡆⠀⠀⠀⠀⠀⠈⣰⡿⣃⠋⠋⠀⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠛⠷⠀⠙⢯⡝⡿⣿⣷⠄⠁⣿⡐⠛⠛⠁⣤⠀⠀⣄⠀⠀⠸⠟⠉⠀⠤⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢵⣵⠶⠌⠓⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⢉⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠹⠂⠄⠠⠊⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠒⠀⠀⠀⣴⠄⢤⠞⠳⠊⠈⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⣀⠠⢾⠀⠘⠀⣰⠄⠤⠓⢄⠀⡴⠓⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠚⠉⠉⠎⠁⠀⠀⡗⠐⠊⣇⠀⠀⡠⠺⡄⠠⠀⠔⠃⡈⠁⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠈⣦⡀⠤⢼⢶⠟⠘⠓⠠⡞⠋⠀⠀⠉⡆⢈⡠⠈⢌⠋⠱⣀⣠⡞⠂⠐⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡶⠚⢄⡈⠷⠄⠰⡋⠀⠀⡰⠿⡀⠀⡀⠐⠛⢤⡀⣀⡔⠁⠀⠈⢀⡀⡀⡴⠁⠐⡄⢱⣠⡰⠿⠭⣖⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠿⠀⠀⠀⣠⡅⠀⠀⢀⠗⠀⢎⠀⡆⠀⠛⠀⠀⠠⢀⠟⠛⠃⢀⡜⠤⡊⠀⡀⠴⢄⣠⠗⠛⠻⡄⠀⠀⣸⣧⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠱⠃⠀⢸⣾⠋⠉⢻⡶⠶⡇⠀⠀⢸⠗⠑⢰⠋⠐⢢⡶⠛⠐⠦⡀⡎⠀⠀⢠⠋⠀⠀⢰⡇⢠⢰⢀⣷⡤⠸⡟⢂⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣴⣾⠉⠀⢀⡌⡀⠀⢈⡖⠙⠃⡀⠤⢠⡆⢈⣠⣅⢀⣠⠀⡘⠓⠀⡴⡊⠁⠁⠄⠀⠙⢶⠋⠙⢧⡇⠀⣡⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣏⡛⠶⣿⣄⣀⣠⡎⠀⣔⠀⣴⠂⡈⠀⠉⠙⡇⠀⠀⢜⡄⡔⠔⠀⠈⢹⣤⡀⠀⢀⣼⣀⣠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠈⠉⠓⠓⠾⠷⣿⣹⣤⣄⣀⣠⠾⣅⣀⣠⡼⢳⣧⣄⣤⣴⠯⠿⠟⠛⡿⣫⣵⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠰⣸⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠄⢀⣶⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣦⣀⣀⠀⢄⠀⠀⢀⢀⢀⣰⣧⣾⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠾⠿⠿⠿⠿⠟⠛⠋⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⡂⠤⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠊⢀⣀⠀⠀⠑⡄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡅⠊⠁⠀⣀⠀⠀⠰⢿⣇⠇⠀⡇⠀⠁⠀⠙⡆⠀⡷⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⢸⡇⠀⣷⠀⠁⡁⠉⠀⡿⠶⣤⠀⠀⠉⠀⠀⠹⣄⠀⠀⣠⠇⢠⠇⠐⠁⠐⠚⠛⠃
⠀⠀⠀⠀⠀⠐⠛⡓⠲⢄⠳⢸⡗⠺⣿⢠⡿⠿⡶⠃⡿⠲⠚⢡⡿⣽⠇⢀⡀⠈⠉⠉⣁⠴⠃⠂⠉⠉⠉⠙⠂
⠀⠀⠀⠀⠀⠚⠉⠉⠉⠁⠀⠸⠇⠀⠟⠈⠷⠾⠃⠀⣧⠀⠀⠘⠷⠴⠖⠋⠁⠀⣀⣩⠤⠒⠒⢀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⢀⡀⠀⢀⣀⢀⠀⠀⣀⠀⣿⠨⠭⠭⠀⠀⠀⠒⠚⠉⠁⠀⠀⠀⠀⡏⠀⠀⠨⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠛⠀⠀⡏⢸⢸⣷⡇⠀⠛⠈⠉⠁⢰⠀⠀⠀⢰⠄⡄⠀⠀⠀⠀⠙⠦⠤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⡄⠀⠀⣷⢸⣨⣿⣇⠀⢀⣀⡀⠀⢸⠠⣤⠀⢸⠀⡇⠀⡄⢀⠀⠠⡆
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠇⠲⠏⠀⠀⠙⠸⠀⠏⠧⠄⡘⠋⡋⣑⡚⢂⢸⠀⠘⠂⠋⠑⠓⠘⠑⠂⡇
⠀⠀⠀⠀⠀⠸⡇⠸⢆⠤⠖⡒⠢⠀⡀⢀⣀⢀⡀⠀⣀⠀⡀⢀⠀⣈⠁⣀⢀⣀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠙⠂⠀⠰⣤⠝⡆⣏⠙⠈⡇⣧⠜⢸⠙⡆⣷⢸⢸⠉⢸⠁⢸⡬⠃⠀⡀⡠⠤⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⢰⠀⠁⠘⣆⠀⡇⣿⡆⢸⠀⡇⡟⣿⢸⣀⢸⣀⢸⣧⠀⢀⠀⠀⣀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠐⠞⠀⠇⠛⠓⠘⠖⠁⠇⠙⠈⠻⠘⠧⠘⠘⠂⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣄⣀⢠⢲⡆⣇⣄⠀⠀⢰⠖⠂⡶⠒⢠⡶⡄⢰⠒⢦⠀⢀⠊⡉⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⠈⠡⢈⠈⠉⡃⢁⡈⠋⡁⢸⠛⠁⡇⠀⢸⡄⣧⢸⡗⠋⠀⠣⠀⡁⡘⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠀⠀⠁⠀⠀⢸⠀⠀⣇⠀⢸⡇⣿⢸⣿⡀⠀⠤⠥⠂⢀⢂⠔
⠀⠀⠀⠀⠀⠀⠀⠸⣯⣾⠇⠈⠈⢉⠅⢐⢐⡂⣒⠀⢸⡀⠀⣏⣉⢸⡗⣿⢸⡏⣇⠀⠀⢐⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⠿⢿⡿⠟⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣧⣽⣽⣿⣿⣿⡿⣿⣾⣿⣛⣷⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⣿⣷⣾⣿⣿⣿⣿⣿⣷⣾⣿⣹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⡿⣻⡿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢤⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⢶⠉⣩⠿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠈⠙⢿⣿⣿⣿⣷⣤⣉⠛⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠇⣤⡀⠠⡐⢌⠢⠀⠀⠀⠙⠲⣄⡀⠀⠀⠙⠯⣿⣿⣿⣶⣿⣿⣿⣿⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣴⣿⣿⣆⠙⢆⠑⢄⠢⣀⠀⠀⠙⠻⣧⣶⣤⠂⣌⡙⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⡈⠳⠀⠱⣌⢷⣄⠀⠀⠈⠻⣿⣿⣿⣿⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣟⣄⠈⢦⡈⠳⣝⠓⣄⡀⠹⡞⢋⣳⣿⢿⣿⣿⣿⡿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠻⣦⡈⠳⣝⢿⣦⡈⠛⢻⣿⣾⣿⣟⣿⣻⡯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⠘⢿⠓⢝⢷⣝⢻⣶⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠻⣿⣷⡙⢷⣝⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⢽⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧⡀⠻⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠘⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠴⠶⠶⠶⣤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⣀⣀⠀⠀⠀⠘⣋⣴⡟⠛⢋⣭⣵⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⣠⣶⡀⢋⠉⡀⡀⠙⠦⠙⠁⠀⢿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁⣤⣤⣤⣀⣀⣤⣀⠰⣿⠿⠟⠈⡁⠠⢾⣿⣤⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣼⣷⣤⠀⠀⢹⣿⡀⠀⠀⢹⣿⣿⣿⣿⣿⡏⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠆⠀⠰⢿⡿⠋⠀⠀⣾⣿⣿⡿⠟⠓⠁⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢑⡿⠟⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣶⠀⣀⠀⠀⠀⠙⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⡛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢧⡀⠉⣽⡄
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⠋⠉⢻
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⠛⠛⢿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⣶⣾⡅
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢻⣿⣿⠏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣄⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⢀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠶⠮⠿⠶⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄
⠀⠀⠀⠀⠀⠀⠸⣅⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠄⠀⠀⠀⠀⠀⠀⠀⠘⠤⠔⠃⠀⠀⠀⠀⠀⠀⠠⠊⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⠤⢤⣤⣀⣀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⣾⠿⠓⠀⠀⠀⠀⠀⠘⠛⠿⠄⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠈⡡⠂⠀⠀⠢⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡠⠚⠉⢳
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡔⠁⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢐⣷⠀⠀⢰⠠⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣠⠾⣀⡠⠋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠛⠀⠀⡸⡀⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⡇⠀⣀⠱⡠⠖⠙
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⠄⠀⠀⠘⠦⢀⣀⣀⠔⠁⠈⠐⠀⠂⠁⠀⠀⠀⡐⠀⠀⠀⠀⢠⡟⠀⠀⠀⢹⣂⠀⣀⣱⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠈⡈⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠠⠀⠀⠀⠈⡛⠀⠀⢐⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣏⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢣⠦⠤⠤⠦⠤⠤⠼⢤⠄⠂⠀⠀⠀⠀⠀⠀⢀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠳⡀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢷⡀⠀⠈⠲⣄⣠⠂⠉⣃⡀⠷⠐⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⣻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⠾⣄
⠀⠀⠀⢀⠤⠐⠚⠹⡏⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⣀⣀⣤
⣠⠖⠚⠁⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⣀⣀⠆
⠓⣲⠃⠀⠀⢠⠔⠢⣀⠉⡷⠀⠀⠀⠀⠀⡷⠒⠖⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⡆
⠀⢧⡠⢦⠀⡸⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠣⠤⣤⠤⠤⠤⠤⠤⠒⠒⠒⠒⠒⠒⢒⡶⠋⠉
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⢀⣀⡤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠉⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠠⡏⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⡍
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠤⠺⠇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢰⠇⠐
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⠃⢰⠁⢸⠀⠂⠀⡌⠀⠀⠀⠀⠀⠐⠚⠟⠛⡲⣦⣸⣃⣾⠀⢃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢾⣽⣤⠎⠀⢀⠜⠀⠀⠀⠀⠀⠠⡶⠶⠤⣄⡀⢄⠙⢻⡽⠷⠚⠳⠶⠿⣶⡄⠀⠀⠀⠀⠀⡠⠀⠠⠖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⣄⠈⢒⢥⢆⣠⡐⠒⠂⡄⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠘⡍⣑⡀⠌⠉⢦⢈⠉⠉⠂⢀⠀⠉⠙⣆⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡳⣦⣤⣤⣄⣠⠘⣧⢁⠄⢠⠖⢤⠦⣤⣀⠡⠘⡾⠶⠞⠛⠻⢷⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣠⠤⠿⣿⠿⡷⣿⣿⣇⣘⠘⣎⠷⣄⡀⢂⡘⠀⢀⡀⠈⢁⣀⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠄⠈⠂⣸⡜⠇⠀⠈⡀⢡⢉⣛⣰⣧⡀⠈⠙⠻⢡⣤⣤⠶⠴⠿⠊⠀⢀⠀⠀⠀⣀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠉⠴⠚⠉⠀⠀⠘⢆⠀⡇⡸⠀⢈⡇⠸⡷⢔⠀⠈⡆⢀⠄⠀⠀⠀⠀⡴⠟⢂⠀⠈⢠⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠁⠲⠛⠣⣸⠃⠀⠰⡇⠕⠁⠎⠀⠀⣀⠠⠤⢀⢈⠰⡀⢈⡘⢀⡠⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠦⡸⠗⠳⢾⠙⠀⠀⣸⡙⢀⣳⣯⣔⠡⠄⠀⠒⠒⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⢳⣆⠠⢿⣏⡁⠈⡛⠏⠙⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠸⡆⠀⠀⠀⢜⣃⣁⠀⠀⠙⢷⠈⢮⣀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠐⢄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⣣⡀⡄⠀⠀⢀⠟⣛⠑⠮⡑⢤⣀⣲⡄⠈⠉⠙
⠀⠀⠀⠀⠀⠀⢀⢻⠀⠀⣠⠇⠉⠙⠯⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡇⠀⢨⠅⠈⠁⡖⠁⠀⠑⠀⠀⠀⠱⠄
⠀⠀⠀⠀⠀⠀⡌⡇⠀⠀⡿⣧⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣠⡐⡴⠦⣴⠀⢸
⠀⠀⠀⠀⠀⠀⠃⡆⠀⠴⠃⢋⠄⠀⠀⠀⠱⠀⠀⠀⠀⠀⣠⠁⠀⠀⠈⠣⠈⡀
⠀⠀⠀⠀⠀⠀⠸⣱⠀⠀⠀⠈⠚⢀⠀⡀⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠲⠄⢠⣖⠊⠁⠐⠤
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢱⡀⠀⠐⢇
⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣠⠤⠞⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠈⡥⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⣢⠜⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠉⠛⣩⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⢤⡀⠀⠀⠀⢠⠖⠛⠉⠉⣣
⠀⠀⡀⠀⠐⠠⢀⠀⠌⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠇⠀⠀⠀⢠⡟⢸⡆
⠀⠀⠡⡈⠳⣄⠈⠐⡁⡀⠰⡄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⣧⣤⣀⣀⣤⣾⠃⠀⠀⠀⣁⠀⢀⡠⠟⠁⣼⠇
⠀⠀⠀⠈⠁⠁⠒⠣⣤⣠⣀⠀⢠⠂⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠖⠚⠁⠀⠀⠀⠀⠠⠀⡬⠤⠶⠛⠁
⠀⠀⠀⠀⠀⠩⣍⠉⠩⠿⣁⠀⠆⢀⡇⠰
⠀⠀⠀⠀⠁⠢⠤⠀⠂⠀⣈⣶⣦⡈⡰⠀⠀⠄⡊⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⢁⣠⠶
⠀⠀⠀⠀⠀⢀⡠⠤⢤⠤⠀⠬⠻⡀⠀⠰⠀⢀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠉⠓⠡
⠀⠀⠀⠀⠀⠀⠈⠓⠲⠆⠱⠂⠀⠱⡀⠀⢠⠞⢰⠆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣟⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⢉⣉⡉⢐⠷⣤⠤⠔⠣⠆⠉⠅⠈⠂⡐⢂⠀⠀⠂⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⣀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⢀⡀⡈⠍⢣⠀⠠⢁⡔⢤⠛⡐⠠⠆⡄⠄⠀⣰⣤⠥⠶⠁⠄⠃⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠢⠤⢄⣧⣄⡊⢀⠀⢀⠈⠠⢴⣴⡶⠚⠤⢑⠄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠒⢢⠖⠂⠀⣿⠏⠀⠀⠣⡈⣦⣠⡞⠁⠡⣐⢂⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢂⠀⠄⠈⢸⠀⣌⠡⡔⢀⡽⡫⠀⢀⡈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠠⠤⡀⠈⡀⠙⢆⢡⡞⠁⠙⠲⠄⠍⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⠄⣤⣈⣠⡼⣿⢀⣈⠁⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⠒⢋⠀⠀⠘⣿⠇⠀⢉⠳⠒⢒⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠖⠊⢦⠏⠁⠀⣼⣠⠄⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢰⡏⠀⠀⠀⡿⡅⠀⠠⢤⠤⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠈⠀⠀⠂⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠂⢄⣐⣤⠏⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠛⠛⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣟⣻⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣻⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢳⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⣷⠀⢤⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⡆⠈⢷⣄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠈⢿⣧⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣿⣧⣤⣄⠀⢻⣿⡌⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⡿⣿⣿⣄⡈⣿⣿⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣽⣿⣿⠃⠹⣿⣿⡆⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠷⢉⣉⣁⣈⣑⠈⠿⣿⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⢇⣴⣿⣿⣿⣿⣿⣿⣷⣆⠹⣧⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣩⡴⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⢹⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣽⣿⣿⣫⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣾⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣻⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⣾⠋⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣝⣷⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⢃⡾⠋⠜⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⢰⠙⣿⣌⣛⣛⣻⣿⣿⣛⣿⣿⣷⣿⣿⠉⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢀⡁⣸⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⢀⠜⠁⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⣿⣿⣿⣿⣿⣿⣿⣿⣏⡁⠀⠀⠩⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠻⠿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⣶⣶⣄⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⣶⣶⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶
⠛⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻
⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣶⡀
⠁⠀⠀⠀⣀⡀⠀⠀⠈⢿⣿⣿⡿⠃⠀⠀⠀⣀⡀⠀⠀⠀⢿⣿⣿⡿⠃⠀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹⢿⡀⠀⠀⠀⠈⠉
⣀⣀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠝⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⢀⣀
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿
⠀⠙⠋⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣄⣀
⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠒⠁⠀⠀⠀⠐⢤⠡⠀⠀⠉⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⠁⠈⠁⠀⢀⡀⠀⠀⣧⢶⣗⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠘⠀⢀⡀⢚⡄⣴⠓⠁⠀⠀⠁⠀⠀⠀⠀⠀⢈⠆⠀⠀⠈⢓
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣀⡔⢰⣯⢚⢬⠚⠁⠠⠀⢰⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⢀⠈⢆⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⡾⠐⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡀⠀⠀⠀⠈⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢜⣤⣦⣀⡀⠀⢀⠴⡮⣖⠀⢐⠠⡁⠐⠁⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⢀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠗⠁⠀⠘⣷⢵⡶⡴⠒⠐⠐⠁⠀⢇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢁
⠀⠀⠀⠀⠀⡄⠀⠀⡔⠉⠉⠀⠽⡀⠀⠈⠁⠙⠛⢙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠢⠘
⠀⠀⠀⠀⠀⠁⢠⡈⠁⠀⠀⠀⠀⠉⠒⠉⠓⠀⢂⣣⠀⠀⠀⠰⠓⠦⠤⡠⠴⠒⢄⡀⡀⠀⠀⠀⠀⠈⢱⠰⠈⠂
⠀⠀⠀⠀⠀⡄⢨⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠱⣤⠀⠀⠐⠑⠊⢀⠆⠀⠀⠉⠳⣤⠐⠀⠀⠀⠀⠢⢰⡄
⠀⠀⠀⠀⠀⠁⢺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠙⢦⣀⠀⠀⡠⠌⠀⠀⠀⠀⠀⠈⠂⢠⠀⠀⠀⠶⣤⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠱⠌⣿⠶⢚⠁⠀⠀⠀⢐⠠⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠁⠀⠳⠗⢾⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠀⠀⠂⣀⠄⠀⠠⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠀⣧⡄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠴⠁⠀⠐⠀⢈⡠⠄⢀⠤⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠳⡀⠈⢿⣠⠂⠀⠀⠀⡐⠟⠀⢰⠂⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡂⠀⣻⣟⠀⠀⠀⠙⠀⣄⠙⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠘⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣌⡻⣔⣀⠀⠀⠀⢈⡴⠉⢐⣄⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣶⣶⣼⣁⣈⣀⣁⣀⠠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠐⠂⠐⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⣀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠘⠛⠛⠛⠛⠀⠈⠛⠛⠛⠛⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⣿⠇⠀⡀⠀⠀⢿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠄⠄⠀⢀⠁⠀⠀⠀⠀⠤⠄⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠐⠀⠤⡄⠀⠀⠀⠠⠤⠐⠊⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣀⣀⡇⠀⠀⠀⠀⠀⠀⡅⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠗⠐⠂⠈⠁⠀⠀⠡⡀⠀⠀⠀⡠⠈⠀⠀⢀⠀⠀⠀⠀⠉⠁⠂⠀⢀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣵⣶⣅⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡠⠻⣿⣿⠟⠠⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⣿⣿⡆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀⠤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠿⠿⠿⠿⠃⠀⠐⠂⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⢀
⠀⠀⠀⠀⢤⣤⣶⣽⣿⣾⡆⠁⠀⠀⠀⢀⠄⠀⠀⠀⣀⣤⡤⠶⠒
⠀⠀⠀⠈⡛⠻⣿⣿⣿⣿⣿⣆⣠⣧⢼⣧⣢⣴⣾⣿⣏⡁
⠀⠀⠀⠐⠣⣦⣿⣿⢿⣿⣯⡙⣭⣷⣾⣿⣿⣿⣿⣿⠛⠛⠉
⠀⠀⠀⣠⠖⠉⠉⠀⠀⠘⠻⠛⠋⠀⠀⠀⠉⢻⣿⣷⣦⣄
⠀⠀⠀⠾⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢻⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⣤⣽⣗⡁⠀⠀⡄⢀⠀⢠⣤⣥⡀⠨⢿⣿⣿⣿⣿⣦⡙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠠⢤⢤
⠀⠀⠀⠀⢺⣯⣍⡻⠅⠀⠀⠀⠀⠁⡿⠛⢫⡷⠂⠐⣿⣷⣿⣿⣿⣿⡀
⠀⠀⠀⢀⣤⡋⠁⣀⢠⠀⠀⠀⠀⢰⣆⡀⠀⠀⡸⢰⣿⢿⣿⣿⣏⠙⣷⡀
⠀⠀⠀⣽⡝⢿⣿⡿⠋⠀⠀⠀⠀⠘⠼⢿⡄⢦⣡⣼⣿⣾⣿⣿⡙⠆⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠘⠀⠃⠃⠘
⠀⠀⠀⣿⣿⡸⠟⢡⡀⢀⠀⠼⣷⣦⡄⠀⠁⣼⣿⣿⣿⣿⣿⣿⣷⣄⠀⡈⠃
⠀⠀⠀⢻⣿⡿⠀⠀⠉⠙⢧⣰⠋⠉⠁⠀⠈⣿⡿⣿⠧⢹⣿⣿⣟⣿⣷⡘⠀⠀⠀⠀⠀⠀⠄⡄⠀⡄⢠⡄⢀⠀⠀⢤
⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⢀⣿⢷⡘⣷⡄⢿⣟⣿⣯⡹⡿⠄
⠀⠀⠀⣾⣿⣿⣷⣤⠖⠒⠋⠉⠒⠲⠤⣶⣾⣿⣿⣿⣿⣧⣈⣻⣿⡟⠙
⠀⠀⠀⣿⣿⣿⣿⣿⣤⣀⡀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⡿⣿⣏⠛⠁⠀⠀⠀⠀⠘⠒⠚⠀⠘⠈⠘⠀⠛⠃⠃⠀⠃⠛⠚⠂⠘
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⡜⣿
⠀⠀⠀⢻⣿⣿⣿⣿⣼⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠀⡛
⠀⠀⠀⢸⣿⣿⣿⣿⣯⢹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣾⣿⡝⢸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠋⠛⠛⠛⠛⠛⠛⠁⠛⠛⠛⠛⠙⠛⠃⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⡠⠀⢉⡁⠉⠀⠈⢀⠉⠐⣢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡂⠁⡠⠔⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠘⣄⠀⡀⠀⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⠴⢣⡮⠄⠔⡠⢌⠀⠀⢀⡤⠂⠓⠐⣆⡔⠢⠇⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⢠⠎⡠⠠⠎⠀⠀⠀⠀⠀⠉⠁⠀⠀⢎⠀⣞⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠰⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢭⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠣⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠲⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⣠⠔⠒⠈⠢⡀⠀⣠⡩⠴⠀⠀⠘⢄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⢠⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⠫⠤⠀⠤⠀⠀⠹⠒⠁⠣⠤⢰⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣁
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠑⠂⢀⠖⠂⠠⠤⠘⢄⠀⠀⠀⢷⠦⢀⡤⠰⣄⡀⠀⠂⢹
⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⢄⠀⠀⠀⠓⠁⢡⠀⠀⠱⡀⠃⠘
⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⠢⡀⠀⠀⡠⠊⠀⠀⠀⠡⣌⢀
⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠨⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢜⠒⣉⢀⠀⠀⠀⠀⠀⠈⠸
⠀⠀⠰⠀⠈⢀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠎⠘⠀⠀⠀⠀⠰
⠀⠀⠀⢆⠀⢈⠂⠀⠀⠀⠀⠑⠊⠁⠀⠀⠁⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠈⢆⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⠘⢡⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠑⢀⡀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⣠⠊⠀⠨⠆⠁⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢠⠤⠂⠁⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠤⠀⣀⣀⣀⣀⣀⣀⡀⠤⠴⠒