Dear Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣦⣼⣿⣿⣷⣄⣤⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠇⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠀⠀⠀⣶⡄⠸⣾⣠⣿⣿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣀⣠⣾⣇⣤⣿⣿⡟⠓⡸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⣿⣿⠋⣿⣿⣿⣿⣿⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣴⠀⠀⠀⠀⢻⣿⡿⣿⣿⣿⡿⡿⢿⠗⠀⠠⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿⣷⣤⣤⣴⣿⣯⣾⣿⣿⣿⡤⠀⠰⠣⠀⣴⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢶⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡄⠀⠀⠀⠀⠠⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢺⣧⡀⢘⣠⡄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡏⠉⠙⠓⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣂⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⢀⢀⣠⣶⣦⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⢀⣠⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠿⣿⣿⣿⣿⣿⣧⡟⠉⢉⣷⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⠀⠐⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡛⢿⡟⢹⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣥⠀⢰⣾⣿⣿⣿⡏⣿⠏⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣾⠿⠟⣝⠃⠠⣼⣾⣿⠇⠈⠛⣿⣿⠉⠛⠿⠿⠇⢀⡈⠙⢿⣿⣿⣿⣿⡃⢸⣭⣿⣾⣿⡇
⠀⠀⠀⠀⠀⣠⠾⠆⠄⠀⠈⠁⢀⠈⠙⠿⠆⣴⠟⠁⠛⠀⠀⠀⢰⠀⠀⠁⢐⠠⣼⣿⣟⣿⠁⢈⢸⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠄⠀⠀⠀⠀⢠⠂⢤⠀⠀⠀⠘⠋⠁⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⡗

⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡏⠀⠀⠀⠠⠂⠀⠀⠀⠀⡎⠀⡸⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⣇⣠⠄⠀⠀⠀⣱⠛⠁⠐⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠞⠉⠀⠀⠀⠀⠀⠃⣀⣴⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡙⢶⣄⠀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢻⠸⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠈⠁⢠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⡆⠙⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠿⣛⣵⡿⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣼⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠈⢱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⣿⣿⡿⣿⣿⣿⣦⣤⠀⠀⠀⠀⠀⣶⣺⣿⣿⣧⠀⠀⢸⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡿⡏⠙⠟⠟⣿⣿⣂⠀⣰⣀⣠⣿⣿⣿⣿⣿⡆⠀⠀⢸⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣄⠀⠂⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣗⡀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⣼⣿⣿⡿⠟⠋⠁⠻⣿⣿⣿⡟⢿⣷⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢸⠀⡨⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡈⢻⣧⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄⠙⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠈⠀⠀⠀⠙⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠟⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠁⠀⠀⠀⠀⠀⠀⠰⠛⠃

⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢄
⠀⢀⠃⠀⠀⢠⣆⣀⣤⡴⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⣤⣄⣀⣆⡀⠀⠀⢃
⢠⡾⠀⠀⢀⣼⠟⠋⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣄⠀⠀⠸⣦
⣿⣇⢀⣴⠟⠁⢀⣠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣀⠀⠙⢷⣄⢀⣿
⢻⣿⣿⠃⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⠀⠀⢻⣿⣿
⢸⣿⠇⠀⠀⢸⣇⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⢠⣿⠀⠀⠀⢿⣿
⢸⣿⠀⣠⣴⡾⣿⡇⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣾⡿⣶⣤⡀⢸⣿
⠘⣿⣾⡟⠉⠀⠘⣿⣶⣾⠟⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠈⢿⣶⣾⡟⠀⠀⠙⣿⣾⡟
⠀⢹⣿⡀⠀⠀⠀⠸⣿⡄⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⣼⡿⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠻⢷⣶⣦⣤⣶⣿⣿⣷⣾⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣶⣿⣿⣷⣦⣤⣤⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣉⠉⠛⠻⣿⡶⣄⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠋⢉⣡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⣿⠙⢲⡄⠹⣿⣷⣤⣤⣤⣴⣿⡟⠁⣴⠞⠛⢟⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⡁⠈⢿⡾⢿⣿⣿⠿⢿⣿⣿⡟⣾⡟⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢾⠃⢿⣿⣿⣶⣾⣿⣿⠗⢻⠗⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⢻⣿⣿⣿⣿⠃⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣷⣼⢿⣿⣿⣿⣶⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠸⣿⡿⡀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠀⢷⣶⣾⠗⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣷⣎⣍⣭⣶⣿⣟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡼⠛⣿⣿⣿⢛⣻⣤⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣷⠀⠘⠻⣿⣿⣿⣿⡏⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⣿⣷⣄⠀⠛⠿⡏⢈⣧⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠛⢽⢿⣷⣤⣀⣿⣿⣿⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡆⠀⠛⢿⣿⣿⣿⠟⢳⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⢚⠉⠙⣀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⡈⣠⣴⠟⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⢀⣀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⡄⣄⠀⠀⢠⡀⠀⣀⡀⣀⣤⡘⠧⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢰⣦⠀⣻⣮⣹⣻⣒⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣿⢯⡉⠉⡹⡿⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠈⣏⣻⣦⢉⣷
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠞⠛⠙⡻⣿⣿⡾⠃⣸⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⠋⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠹⢿⣿⣿⠏⢀⢴⠊⠀⠉⠉⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢻⣿⡇⠡⣿⣀⡀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠁⢹⡦⣿⣿⣿⣷⣾⣧⣀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⠹⣿⠿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣻⣿⠇⠁
⠀⠀⠀⠀⠀⠀⢺⣿⣿⣯⠠⠁⠀⠀⠈⢹⣿⣿⣿⠛⠛⠋⠉⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿

⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣷⠀⠀⠀⠔⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⢢⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠸⡞⢆⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡰⢳⠇
⠀⠀⠀⠀⠀⠹⣎⠳⢄⠀⡁⠀⠀⡡⠟⠛⠒⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠒⠛⠻⢄⠀⠀⢈⠀⡠⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠈⡳⡄⠀⠀⠀⠀⢠⢾⠁⠀⠀⠀⠀⠀⠀⡇⠀⠌⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡈⠳⢤⣀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠙⡌⠀⠀⠀⠀⣀⡤⠞⢁⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡉⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⢉⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠉⠓⠀⠀⠀⠀⢀⡄⢠⡀⠀⠀⠀⠀⠚⠉⠁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠉⠛⠒⠀⠀⠀⠒⠉⣿⣿⠉⠒⠀⠀⠠⠒⠋⠉⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣀⠰⡇⢸⠆⣀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠢⠈⠳⣦⢱⡀⢀⡞⣴⠞⠁⠔⠲⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣈⣰⠇⠸⣆⣁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠂⠐⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠟


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢰⠀⢀⡴⠂
⡏⣰⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⢿⡟⠀⠀⠀⠀⠀⡼⠁⠰⡀⠀⠀⠀⡼⠀⠀⠀⠀⡆
⢸⣇⠀⠀⠀⠀⣼⠃⠀⠀⠹⣄⣀⡼⠁⠀⠀⠀⠀⡇
⠀⢻⡄⠀⠀⢰⡇⠀⣀⣠⠆⠈⣿⠁⠀⠀⠀⠀⣸⠃⠀⢀
⠀⠀⠹⣦⣤⡿⠟⠛⠉⠀⠀⠀⣿⡀⠀⠀⠀⣰⠏⢀⣠⠋
⠀⠀⣤⣌⡙⢷⣦⣀⡀⠀⠀⢀⢻⣧⣀⣤⣾⠿⠶⠋⠁
⠀⠀⢸⣇⠙⢦⡈⠻⣧⣄⡀⢾⣟⣩⡭⢤⡄
⠀⠀⠀⠻⢦⡚⠃⠞⠋⠉⠀⡀⡘⢁⣴⠟
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠟⠁⠘⣮⡁
⠀⠀⠀⠀⣼⢸⡆⠀⠀⠀⠀⠀⠀⠀⢉⣴⡆
⠀⠀⠀⢰⣿⣦⡻⢦⣤⠖⣀⣤⡤⠤⢿⣿⠖⠁
⠀⠀⠀⣾⡏⠛⠿⠶⠖⠀⠀⣾
⠀⠀⢠⣿⡇⠀⠀⠀⠀⡀⠀⠙⢶⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢸⣿⣇⠀⠀⢀⣼⠃⠀⠀⠘⠃⠈⠉⠉⠉⠉⠉⠙⠛⠻⢿⣿⣿⡟⠉⠉⠉⠉⠙⠓⠶⣄⡀
⠀⠀⠈⣿⣿⣦⣶⣿⡏⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢺⣷⣄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⢀⡘⣿⣿⣿⢸⢇⣾⢹⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⢿⣿⡆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢸⣷⠘⣿⡟⢸⣿⣿⡈⢿⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣾⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠈⢛⣃⣽⣿⣿⣿⣿⣇⠀⢠⡄⠀⠀⢰⡟⢣⣧⠀⠀⠀⠀⣼⡟⢀⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡇⢸⣷⠀⠀⢸⣷⡾⠙⠀⠀⠀⡴⠋⣠⣾⣿⢿⣿⣿⣆⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣼⣿⠀⠀⣺⢛⣁⣀⣀⣤⣴⣶⡿⢟⣯⣥⣬⡻⣿⣿⣦⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠘⣯⢿⣿⡟⢿⣿⡿⠀⡰⠿⡿⠿⠿⠟⠛⠉⠁⠺⣿⣿⣿⣿⣿⣮⡛⢿⣿⣦⡀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠹⣇⠈⣿⡇⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⢷⡀⠈⠻⢿⣷⣤⣈⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢻⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣆⠙⣶⡄⠀⠙⠻⢡⣼⣏⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣼⠀⠿⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣿⠃⠀⠀⠀⠀⠙⣿⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⣉⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⢰⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⢴⠛⠀⣠⡞⠁⣷⠀⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣧⠟⣘⡲⠞⠉⠀⢠⣿⣄⠞⠷⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣴⠋⣡⡆⢰⣏⣴⣫⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠙⢮⡛⠙⣇⣸⡉⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡓⠲⣭⡷⠆⠙⠳⠝⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠏⠀⠰⠷⠀⢦⣙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠈⢃⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⡆⠘⠲⣶⠖⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⠃⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⠶⠋⠀⠀⠀⣿⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠘⡇⢠⡿
⠀⠀⠀⠀⠀⠀⣸⠇⣰⠋⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢸⢃⡾⠁
⠀⠀⠀⠀⠀⠀⣿⢠⡇⠀⢸⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⠟⠁
⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⢸⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⣿⠘⣇⠀⡜⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⢿⡀⠘⠃⠳⣼⣦⠀⠀⠀⠀⠀⠀⢀⣴⠟⠒⠦⠄⣀⣀⣀⣀⣀⣠⣼⠃⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡼⠁⣼⠛⣷⠀⢀⡤⠆⠀⣸⠏⠉⠑⠒⠲⠦⠤⠤⢤⡄⠐⣾⡀⠀⠀⢛⡿
⠀⠀⠀⠀⠀⠀⢸⡇⡞⠀⡼⠁⠀⣿⢀⡞⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣼⡹⠀⡼⠁⠀⢠⠏⡜⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⣿⠁
⠀⠀⠀⠀⠀⡼⢱⣁⡾⠁⠀⣰⠏⢸⡤⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡏⣧⠀⢰⡏
⠀⠀⠀⠀⢈⡇⠀⣿⠀⠀⠀⢻⡄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣼⠃⢳⠀⣾⠁
⠀⠀⠀⠀⢸⡇⢰⠇⠀⠀⠀⠀⢻⡄⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠀⢸⡇⣿
⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢷⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⡇⢹
⠀⠀⠀⠀⠀⣿⣾⠀⠀⠀⠀⠀⠀⠈⢷⡸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠀⠀⣷⢸
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣧⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⠀⠀⢻⢸⡆
⠀⠀⠀⠀⢠⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⠘⣧
⠀⠀⠀⠀⣼⣷⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢻⡄⠀⢾⣿⠹⣆
⠀⠀⠀⠀⠀⢿⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠆⠀⢾⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⣀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣀⣼⡟⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡿⡯⣍⣽⣶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⡘⣱⡙⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡤⢠⡤⠄⠀⠀⠠⡀⠀⢀⠀⣀⣀⣤⢀⣤⣤⣬⡍⠙⠂⠻⣿⣿
⠀⠀⠀⠀⠀⡀⠲⢴⢻⠄⠉⠀⢀⡖⠲⡈⠈⣁⠠⣿⠂⢱⠀⣿⣺⣇⣼⣿⣿⣤⣤⣤⣿⣿
⠀⠀⠀⠀⣾⢙⡁⠈⠀⠀⢀⠀⢆⠁⠠⡀⠀⠰⠀⢙⡯⢼⣛⢉⣗⡑⠈⠻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣯⠘⣷⠀⠀⢀⠀⠀⠈⠃⠐⠃⠘⠃⢀⠁⢀⡉⡀⠀⠀⠃⢷⡀⠈⢻⣿⣿⡟
⠀⠀⠀⢸⡇⢠⠈⠷⡄⠀⠀⠈⠂⣸⣟⠂⢐⣯⣤⣍⣻⣧⣽⣶⣼⡷⡴⣿⡖⣻⣿⠟⠁
⠀⠀⠀⠘⠓⠁⠀⠁⠈⠂⠀⢠⡿⠛⣿⣿⣾⣛⣷⡞⠒⠽⢿⣿⣷⠙⢃⣏⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠼⢃⠀⢀⠀⠎⠀⠀⠉⠉⠙⠛⠓⠒⠚⠚⠋⠉⠙⣦⠻⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⢂⡀⣠⠀⠐⢩⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠿⡆⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⣠⡔⠁⠀⠀⠀⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣧⣱⠀⠀⠀⠠⠒⠀⡀⠀⡰⠗
⠀⠀⠀⠐⢢⠃⠀⠀⠀⠀⠑⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⠈⠁⠀⠉⠀⠀⠀⠀⣇⡀
⠀⠀⠀⠘⠎⠀⠀⠀⠀⠀⠀⠀⢻⡦⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡖⠂⠘⠋⠀⠁
⠀⠀⠠⠗⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠾⠄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠃⠀⠜⠁⠀⠀⠀⠀⠀⠈⠒⠓⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⡜⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⢹⠆
⠀⠀⠀⠀⠀⠀⡸⢰⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⠶⢶⠙⡣⠃
⠀⠀⠀⠀⠀⠀⠘⡄⢑⠊⠋⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⣄⣌⠜
⠀⠀⠀⠀⠀⠀⠀⠈⢎⢂⡀⠋⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣰⠂⠀⢸⠉⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣉⠱⡘⡦⡀⠀⢸⡀⠀⠀⠀⠀⠀⢰⡇⠀⡀⠿⣒⠪⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢉⣈⠑⠶⠸⠀⠀⠀⠀⠀⢘⠓⠋⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡖⢂⡄⠀⠑⠄⡸⠛⠉⠉⠟⠙⢅⠔⠉⠀⠒⠒⣺⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠁⠙⠦⠜⠀⢀⡀⠀⡤⠀⠈⠒⠱⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⢦⡄⣦⡀⢉⠀⠁⢐⣤⢤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠋⠃⡄⠇⢰⠚⠃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠄⠸⠁⠀⠈⠀⣄⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⡀⣻⣿⣿⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡛⠕⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣴⠚⠁⠀⠀⠀⠀⠈⠳⣄⠸⡆⠀⠀⠀⠀⠀⣰⠁⡴⠚⠁⠀⠀⠀⠀⠈⠲⣄⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣷⠀⠀⠀⠀⠀⣿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⡀⠀⢀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠹⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⠇⣰⠇⠀⢸⣄⢻⣆⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⣼⠇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣄⡀⠀⡇⠀⠀⣾⠀⢀⣠⣿⠞⠋⠀⠀⠀⠙⠿⣿⣆⠀⠀⡇⠀⠀⣷⠀⢀⣼⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠻⣶⣿⠀⠀⣿⣶⠟⡋⡀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠿⣶⣧⠀⢀⣿⡶⠟⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣍⢺⣫⠿⢿⣯⣕⢹⡵⠃⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⣸⣇⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢹⣟⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣿⣜⠛⠰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡜⢿⣆⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠙⢿⣿⣮⡹⠆⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⠻⣿⣦⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣈⠻⣇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⡿⢱⣷⣾⣭⠉⢿⣿⣿⡟⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣼⣿⡟⣱⣿⣿⡿⠃⠀⠘⣿⡿⠁⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⢹⣿⡿⠀⠈⢻⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢸⣿⠏⠀⢿⣿⡟⠀⠀⠀⠰⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⠀⢹⣿⣿⠁⠙⣿⣧
⠀⠀⠀⠀⢸⡏⠀⠀⠈⣿⡇⠀⠀⠀⠀⣿⠇⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⠀⠀⠀⠀⢸⣿⠃⠀⠀⢹⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣧⠀⠀⠀⠀⣿⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⣿⠀⠀⠀⠀⣼⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡆⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⢰⡿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢸⣿⡄⠀⠀⣿⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⢠⣿⡇⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⢨⣿⡀⠀⠀⠀⠀⠹⠷⠀⠠⣿⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠂⠀⣿⠀⠠⠿⠏⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠈⣆⠀⠎⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⢊⢻⡄⢻⠀⠀⠀⡄⠀⠀⠀⢼⠀⠀⠀⡆⠀⠀⠀⠀⢈⣇⡀⢀⠜
⠀⠀⠀⠀⠈⠲⣅⣀⣳⣤⣄⢻⡄⠀⠀⢸⡄⢀⣴⠃⠀⢀⣠⣖⣫⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣝⣿⣦⣴⣿⠷⠿⢿⣿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠫⠽⣿⣿⣷⣾⣰⡿⠃⠀⠈⢦⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⣤⠀⡀⠀⠐⠀⠀⢀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢸⣾⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢠⡤⠆⠂⢀⡠⠆⣰⡇⢀⣠⡠⠂⣀⠄⠀⡜⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠐⠀⠀⠀⡤⡞⠋⠈⠓⢲⠀⠠⠋⣹⣿⣷⣿⡟⠊⠀⡀⡤⠀⢠⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣻⣿⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⠁⠠⣤⠐⢻⣅⡀⠀⠀⠾⣿⣭⣿⣿⣆⠀⠁⡇⢠⠖⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢦⡈⠉⡀⠀⢀⢾⣿⣿⣿⣽⣿⠆⠐⡠⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠑⡀⠺⣷⣿⣾⣿⣿⣿⡿⣾⣿⣷⠚⢡⠐⠀⠀⠐⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠘⠂⡘⠻⣿⡶⢿⣿⡟⠮⣊⢃⣿⡇⠀⠂⢄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠷⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢣⣶⣿⣿⣿⠔⢂⣿⣿⣿⣿⠄⣠⡿
⠀⠀⠀⠀⠀⣠⠀⣀⡄⠂⠀⠹⣥⣇⡌⠁⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⡇⠀⠀⠹⠿⢋⣹⣏⡮⣼⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠐⠂⠆⠀⠀⠠⠀⠹⣿⣷⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠶⣋⢠⣮⡿⣸⣻⣿⡿⣟⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠁⢀⠰⢀⡀⠀⠀⢹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠃⢁⠗⢴⠞⢃⣠⣿⢹⣹⣗⣧⣭⠿⠁
⠀⠀⠀⠈⠂⠀⠀⠀⢘⠜⡡⢀⡀⠀⢿⣿⡘⣿⡟⠀⠉⠉⢻⣿⣿⠉⠈⠛⢿⣿⣿⣄⠀⠘⠎⡼⢻⣾⣷⣿⣵⣿⢿⡞⢁⡠⠄
⠀⠀⠀⠀⠀⠓⠀⠂⠀⣀⡪⣠⣀⡀⠸⣿⡯⣿⠃⠀⠀⠀⠀⠙⣿⣧⡀⠀⣀⠙⠻⢿⣶⣤⡌⢁⣼⣿⣿⣿⢿⣗⣉⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢄⣀⠈⠉⠉⠈⠈⠗⠀⢻⣷⣿⠀⢰⣬⡟⣀⠀⣼⡿⠀⢀⣴⣴⣶⣄⣜⢿⣷⢾⣿⣿⣿⣿⣿⣟⣉⡁⠉⠉⠁
⠀⠀⠀⠀⠀⠄⠀⢉⠀⣀⡀⢀⠰⠿⠷⣼⡟⣿⡀⠀⠘⡝⠟⢻⡿⢦⣶⣾⣿⣿⣿⣿⣿⣾⢿⡾⣿⣿⣿⣿⢻⣛⠻⣯⡑⠦
⠀⠀⠀⠀⠀⠀⠀⠉⠂⢈⠿⠦⠕⠄⠀⣸⡇⢾⡇⢀⡖⢾⡷⠁⠙⠛⠛⠻⠟⠛⢿⠻⠛⣀⣸⣿⡟⠛⠛⠛⠟⠣⠀⠀⠉⢀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠓⠂⣰⡀⠤⠛⠁⠈⡻⠛⠁⢿⣿⣴⣴⣸⡖⢃⣶⡈⢂⠴⠛⣉⢀⡡⠄⠈⠉⠀⢠⠈⡁⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣛⠆⢀⠀⠣⣅⣐⡀⠐⠛⠿⢿⣿⣿⣿⣿⣿⠟⢡⢀⣾⠟⠁⠀⢴⠀⡄⠀⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⠀⠀⣿⠽⡖⠀⢧⠸⠂⠠⠨⠛⠋⠓⣿⡁⠈⠉⠀⠴⠠⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⢿⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⣼⠏⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢿⣿⡆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠄⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⣿⣿⡋⠉⠉⠛⠉
⠀⠀⠀⢠⣀⣀⡴⠶⠶⢌⣿⣿⡆⠀⢀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣥⣾⣿⡟⠉⠙⠃
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠘⣿⣿⣿⣶⡿⠋⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⢨⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠓⣷⠀⠀⠀⠀⠀⠀⣠⣶⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡛⠳⢄⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠁⣿⠀⠀⠀⢀⣀⣨⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⠞⣡⣾⠟⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠤⠄⠈⠂⡀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢃⣼⣏⡤⢀⣤⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠚⢲⣴⠂⠻⣦⠀⢼⣇⠀⢠⣤⣤⣤⣄⣀⣼⣿⣿⣿⣿⡏⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠻⣿⡦⠀⠿⢷⣬⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣼⣿⣿⣄⠀⠙⠀⣀⠉⣟⣠⣿⣡⡴⠟⢿⣏⠀⠉⠩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⣤⡄⢻⣿⣾⣿⣿⣴⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡀⣿⣿⣿⣿⣿⡉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢰⣃⠾⣿⡿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡟⢠⣴⣿⣿⣦⣿⣿⣿⣿⡄⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⡇⠸⢻⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠁⠀⠀⠁⣽⣟⣵⡿⣏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢀⢈⠁⣴⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣰⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠙⡏⠉⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠿⢿⣿⣿⣿⣿⣿⣀⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣜⣿⣿⡟⣭⠿⠛⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠑⣄⡁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠋⠁⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣧⡬⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠈⡁⠀⠀⠠⠖⠹⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⠋⠀⠀⠊⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣤⠆⠀⠀⠀⠠⣬⠛⢉⣉⣽⣿⡷⠒⠀⠈⠛⠋⠀⠀⠀⠠⠀⠀⠘⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠋⠉⣴⣿⣿⣌⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⣿⡟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⣠⣤⣄⠀⠀⠀⣠⠄⢠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⣤⡄⠀⣤⣤⡄⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣥⠆⠀⣭⠁⡚⠀⠀⠀⡿⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠹⠀⠀⣿⣿⠃⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠁⢸⡿⠃⠃⠀⠀⢸⠇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣗⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⣠⣠⡶⢀⡟⠀⠀⠀⢸⡗⠀⠂⠀⠀⣿⡔⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠒⢸⣿⣿⣿⠀⡆⠀⠀⣿⢻⠀⡆⠀⠀⡿
⠀⠀⠀⠀⠀⣉⡹⠢⣼⠇⠀⠀⠀⢸⠈⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⠄⠀⠀⢰⠠⠀⠜⢺⣿⣿⣿⠠⠂⠑⠁⣿⣾⠀⡇⡈⠐⡇⢀
⠀⠀⠀⠀⠀⠈⣷⢒⡯⢀⡄⢀⡀⢼⡄⠉⡀⠀⣸⠇⠀⠀⣠⣶⣾⣿⣷⣶⣶⡿⠁⠀⠀⣐⡉⠈⠋⠁⣾⣿⣇⢈⠸⣆⠀⠀⢹⣿⠀⡇⠡⠄⢷⢨⠃
⠀⠀⠀⠀⠀⠘⠈⣾⡃⢈⠀⣬⣭⣿⡄⠀⣷⣐⡏⠀⠀⠀⣻⣿⣿⠛⠛⣿⣿⡃⠀⠀⢂⡛⠀⠠⠈⢸⣿⣿⣿⠀⠃⣈⠐⣀⣸⣿⡆⢘⠲⠲⢾⠈⣰
⠀⠀⠀⠀⠀⠀⢀⣿⢠⠍⠐⠁⠛⣿⡇⠀⠻⣿⣷⣷⡀⠀⡏⠀⢧⠀⠀⠻⠠⠃⢀⣠⣀⣤⡤⣶⣆⣾⣿⣿⡷⠀⣇⠀⠊⠱⣸⣿⡿⢰⡆⣷⣾⠿⠁
⠀⠀⠀⠀⠀⣇⣸⣧⢀⠎⠀⠀⠀⣿⠳⠄⠀⠀⠀⠀⠙⠻⠷⠖⠾⠶⣶⣾⢤⣬⣉⡁⠉⠉⠀⠉⢻⣿⣿⣏⢇⣦⠓⠂⠒⠨⣟⣿⣶⣦⣿⡏⢋⡴⠃
⠀⠀⠀⠀⠀⣿⣿⡇⠸⠀⠀⠀⠄⢿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣦⣄⣾⣿⣿⣧⢯⡏⣢⡅⠁⢑⣿⡿⠿⠇⠿⣧⣽⣶⣤⡄
⠀⠀⠀⠀⠀⡏⠘⠁⠸⠀⠀⢹⣶⣾⣿⡇⢠⣄⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠄⠐⠌⠈⡀⠀⢠⣿⣿⣿⡿⣿⠋⢩⣐⣶⠾⢿⣷⣿⣾⣴⠞⣘⢽⠟⠇
⠀⠀⠀⠀⠰⡿⣦⣤⣀⢀⠀⠈⠉⠛⠉⠁⠈⠛⠀⠀⠀⠀⠀⠈⠉⡉⠀⠀⠀⠀⠀⠘⣆⣵⡇⣾⣿⣿⣋⣶⣏⣰⡀⠠⠇⠒⣾⣏⡁⣶⡆⣼⣯⣶⠶⠆
⠀⠀⠀⠀⠘⣦⠀⠀⠈⠉⠂⢤⡄⠰⠆⡀⠀⠀⠀⠀⢙⠉⠀⠀⠈⠁⠀⡀⠀⢀⢀⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠞⠖⣶⡷⣤⣿⡋⠽⢻⢢⣡⣶⡆
⠀⠀⠀⠀⢀⣿⡯⠉⠀⠀⠀⠀⠁⢤⣀⡹⠛⠖⠠⠀⠀⠀⠀⠀⠀⠈⣶⣿⣤⢶⣿⣿⡟⢿⢿⠹⠿⠋⠙⠧⣿⡿⠿⣷⣬⡯⢙⡛⠛⠻⣾⣟⣿⣰⡿⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠀⠉⠙⠁⠀⠀⠀⠀⠃⠒⠒⠛⠃⠀⠀⠉⠛⠂⠑⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠠⡀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠠⠂⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠄⠀⠀⠀⠠⢤⣧
⠀⠀⠀⣰⡟⡰⠀⠀⠀⢠⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡄⠀⠀⠀⢆⢻⣶
⠀⠀⢢⠛⢸⠃⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠈⡇⠛⡔
⠀⢰⡆⣠⠄⠀⠀⢀⠂⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠠⣄⢰⡆
⠀⡎⠀⡟⠀⠀⠀⠌⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠡⠀⠀⠀⢻⠀⠱
⠰⠀⢸⡇⠀⠀⡘⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢃⠀⠀⢸⡇
⣄⠀⢸⡇⠀⢠⠁⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⡄⠀⢸⡇
⣿⠀⢸⡇⠀⠀⠀⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⣿⠀⠀⠀⢸⡇
⣿⢰⡆⢷⠀⡇⠀⡉⠀⠀⢾⠃⠀⠀⠀⠀⠀⠀⡐⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⢂⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⢀⠀⢸⠀⡾⢠⣆
⢻⣿⣷⠀⠣⢇⣴⡇⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⢸⣦⡸⠌⠀⣾⣿
⠈⢿⣯⠃⠀⠘⢿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡿⠃⠀⠘⣿⡿
⠀⠀⠙⢷⣄⠀⠀⠑⣄⣿⣦⠀⠀⠀⠀⠀⠀⡀⡠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⢀⠀⠀⠀⠀⠀⠀⣼⣿⣠⠊⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠙⢿⣦⣤⡈⠻⣿⡀⠀⠀⠀⠀⠀⢷⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡾⠀⠀⠀⠀⠀⢀⣿⠟⢁⣤⣶⡿⠋
⠀⠀⠀⠀⣀⣀⣈⠻⣿⣶⣬⡑⠠⠀⣀⠀⡰⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢆⠀⣀⡀⠄⢊⣥⣶⣿⠛⣁⣀⣀
⠀⠀⠀⢸⣿⠛⠛⠛⠛⠻⠿⣿⣷⣤⣀⣴⣶⡾⣷⣧⣄⣠⣤⣤⣤⣄⣄⣠⣼⣾⢷⣶⣤⣀⣤⣾⣿⠿⠟⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⢻⣄⠀⠀⠀⠘⢦⡀⠙⢻⣿⣿⣿⣟⣩⣯⣬⣭⣭⡁⢈⣭⣭⣭⣽⣍⣻⣿⣿⣿⡟⠋⢁⡴⠃⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢸⣯⡀⠘⠛⣿⣿⠋⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⣿⣿⠛⠃⣀⣽⡧⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠙⠧⣄⡘⠋⠙⠟⠀⢸⣿⢹⣿⢳⡄⣼⣿⣿⣿⣿⣿⣿⣧⢠⣾⣿⡏⣿⡇⠀⠻⠋⠘⢃⣠⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣤⣤⣿⣿⣾⠋⠉⠁⣿⡿⠟⠉⠉⠻⢿⣿⠈⠉⠛⣷⣿⣷⣤⣤⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣆⣴⡟⢁⣴⡿⢿⣦⡈⢻⣦⣠⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠋⠉⣿⢿⣿⣦⣴⣿⡟⣿⠉⠛⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣶⣿⡄⣿⣭⣿⣿⣿⣿⣭⣿⢠⣿⣷⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣷⣼⣯⡀⠀⠀⢀⣽⣃⣾⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⣿⠃⠀⢹⣿⣿⠟⠻⣿⣿⡏⠀⠘⣿⣿⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣄⡀⢀⣠⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⣄⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠘⢧⣀⠀⠈⣸⣷⣄⡀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⣈⣉⣁⣤⡬⠬⠯⢍⣙⡒⠂⠀⠀⢸⣄⡀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣤⠤⠀⣠⢤⣤⣤⣴⠄⠀⠀⠀⠸⣿⣯⣤⠄⠀⠀⢠⡎⠉⠛⠒⠲⠤⠤
⠀⠀⠀⠀⠀⠠⣴⠀⠀⠀⠀⠀⠀⠀⡜⠻⠿⢷⡄⢀⣿⠋⠀⠀⣤⣾⠿
⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⢠⣿⣿⣿⡟⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢾⡀⠀⠀⠸⣿⠀⠘⣿⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡀⠀⠀⠀⠸⣷⠀⠀⠀⢣⠀⠀⠸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣀⠀⠀⠁⠨⠿⢿⣦⣜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡄⠙⢿⡻⣷⣄⡀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠈⡙⢿⣦⣷⠀⠀⠀⠀⠀⠀⢀⠀⢻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⣿⣧⣤⡀⡄⠐⠀⣾⣦⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠹⠳⠛⠀⠀⠁⠀⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡙⢦⠐⠀⠰⡄⠀⠀⠀⠀⠀⢀⡆⡠⠔⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⢒⣦⣤⠀⣀⣀⡴⣂⣈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢁⣁⠁⠈⣋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⢺⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⣾⠀⠀⠀⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⡿⠂⠀⠀⣿⣤⣾⣶⣿⣿⣿⣷⠀⠀⠀⠀⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⢸⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠆⢀⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣴⣿⡟⠻⠛⠛⠛⠛⢻⣿⣷⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⡴⢚⣿⡿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠉⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠸⣆⠀⠀⠀⠀⣀⣀⠀⢀⣛⣀⢸⣇⣀⣀⣀⣀⣀
⠀⠀⠠⠶⠒⠒⠉⠐⠒⠻⠷⠾⠶⣶⠶⠶⠟⠿⠾⠟⠛⠛⠛⠋⠛⠻⠛⠛⠛⠛⠛⠉⠉⠋⠉⠉⠉⠉⠋⠑⠒⠂⠀⠐

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠲⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡇⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠙⢷⣤⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠆⣼⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣰⣿⡟⠁⢠⡎
⠀⠀⠀⠀⠘⣧⠀⠘⣿⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢠⡿⠋⠀⢀⡿
⠀⠀⠀⠀⠀⢹⡀⠀⠸⣷⡀⠷⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠟⠁⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠈⢷⠁⠀⠙⣷⣄⠠⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⠆⠀⠀⣠⡾⠏⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠈⠻⣶⣤⣀⠀⠀⠀⠘⢷⣀⠀⡀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⢀⣴⠏⠀⣠⣾⡟⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣈⣉⠉⠁⡀⠀⠀⠀⠈⠻⢦⣄⣡⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡆⣠⣦⣶⠾⠋⢁⣴⣾⣿⡿⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠻⠿⠿⠿⢿⣶⣤⣍⡛⢿⣿⣶⣄⠀⠀⠀⠀⢰⣿⣿⡿⢋⣥⣾⡿⠟⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠉⠙⠛⠷⣿⣿⠿⠇⠀⠀⠀⠘⠛⢿⣷⣿⣯⠤⠶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡙⠛⠻⠿⠶⣾⡋⠀⠀⠀⠀⠐⠀⠀⢸⡿⠛⠁⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣹⠆⠀⠀⠀⠀⠀⠀⠘⠆⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢸⡇⠀⠀⣾⡇⣰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⢀⡙⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⢠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⣶⢿⣝⣷⡎⣿⣿⣿⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣀⣿⣿⣿⣿⣃⣾⣿⣿⡷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣽⣿⣯⣥⣭⣵⣿⣿⣿⣿⣿⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣨⣿⣿⠯⡻⢿⣿⡿⠛⠀⠘⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠙⠄⠁⢈⠶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣶⣆⠀⠘⠛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⢰⠆⠐⢆⠀⠀⠀⠠⠀⠰⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢃⠘⣾⡄⠀⠈⢱⣦⣀⠀⣧⠀⢳⠶⠄⠀⡇⢰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠓⠛⠿⠿⣷⣀⠉⠛⢯⠉⠙⠛⢄⠚⡇⠀⣣⣇⣼⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣷⣤⣀⣠⣾⣷⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⡏⣹⣿⠶⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⢿⣶⣤⣛⣛⣯⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣶⢿⣻⣿⡭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣵⣾⣴⣖⡀⢀⡦⠄⣿⠁⡀⠀⠸⣽⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣹⣿⣦⣐⣧⠦⠧⣀⠀⣽⣿⣿⣿⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣶⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣤⣶⣦⣿⣿⡋⣉⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠟⢻⣿⣿⡟⠉⠙⠛⠿⣿⣿⣿⡿⢿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⣾⣿⡟⠋⣠⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⣿⠋⠀⢰⣿⡿⠃⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⢻⡇⠀⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣼⡿
⠀⠀⠀⠀⠀⢸⣇⠀⣀⠀⢸⣷⢀⣁⣀⣀⣀⢀⠀⣿⡀⠀⠀⢀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣽⣿⣿⣿⣿⣷⣶⢾⡖⠀⠒⡤
⠀⠀⠀⠀⠀⠈⠋⠩⠉⠝⠛⠋⠏⠟⠛⠛⠛⠉⠙⠙⠛⠛⠃⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⣆⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣳⣼⣀⠈⠳⠀⠀⠀⠀⠀⠀⡐⣣⠔⠭⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡽⢿⠆⣠⠤⢴⡋⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡄⠀⣨⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠐⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢠⣶⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿⣇⠀⠀⠀⢻⠀⣀⣄⣠⣤⣶⣦⣆⡀⠀⠀⠂⠈⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢻⠃⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⡇⠋⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠁⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠁⢀⣴⣾⣿⡿⠉⠉⠉⠉⠉⠉⢹⣷⣿⡂⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⠄⣠⠞⣹⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠊⠁⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⡇
⠀⠀⠀⣠⣤⠤⠴⠒⢠⡤⢴⣦⣴⣧⣴⣦⣤⣴⡶⣿⣴⡶⠶⠾⠿⠿⠻⠾⣿⡿⢿⠾⢿⠿⠟⠛⠻⠛⠛⠒⠒⠒⠶⠲⠤⠤⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣩⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣇⠀⢰⡀⠄⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⠇⠘⠧⠀⠀⠸⠶⠷⢤⣤⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠖⠛⠊⠙⠓⢯⣥⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡖⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣗⣙⣛⣿⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⣿⣿⣿⣿⠻⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣭⣉⣓⡒⠂⠤⠄⢀⣠⣴⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⣿⣿⡫⠉⢩⣿⣿⣿⠋⠀⠀⠀⠀⠈⠉⠷⢃⣿⠙⠛⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠀⢻⢯⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⢿⣿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⡋⠀⠀⠀⠈⣦⠀⠀⠨⣿⣿⣿⣯⣀⣀⣀⣀⠀⣈⣿⡇⠀⠀⠀⣬⢹⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣆⡨⢏⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢁⠀⠀⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣻⣵⡿⠃⠀⠈⠘⠻⠿⠿⠿⠿⠟⣿⣻⣦⠂⡼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⠿⢛⣷⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠺⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣶⠄⠰⣶⠦⣶⣶⡶⠆⠲⠄⠠⠤⠀⠤⡄⠤⠤⣦⣶⣶⣴⣤⣄⠱⣀⣠⣤⣤⣄⡄⣄⣤⠤⠦⠶⠶⠠⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣯⡽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠢⢀⠀⢀⠴⢒⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢚⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠈⢁⡈⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡉⣩⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⣦⡶⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣷⣾⣿⣿⣏⣹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢚⠀⠀⠀⢲⢄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡹⢿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⡀⠘⠀⠛⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠓⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡄⠀⠄⠀⠀⠀⠀⠖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⢰⠁⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠁⠀⠀⠠⠈⢁⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⡐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣈⢀⠳
⠀⠀⠀⠀⠀⠈⠁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠈⢁
⠀⠀⠀⠀⠀⠐⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠀⠄⠀⠐⠈
⠀⠀⠀⠀⠀⠌⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠂⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠂⠀⣤⢀
⠀⠀⠀⠀⠀⠀⣦⠠⢽⣀⢀⣤⠠⠀⠀⠀⠀⠀⠀⢀⣨⡄⠐⠃⠐⠶⠦⠰⠎⠁⠀⠀⠀⠀⠀⠀⠠⡰⢻⣹⠖⠒⠇⢱⣿⠋
⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠈⠙⡟⠄⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⢿⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⠂⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡄⠀⠀⠊⠀⠀⠀⠀⠀⠠⡄⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡔⠸⡡⡄
⠀⢠⡀⢇⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢰⠐⣇⢂⠑⡇
⠀⠈⣰⡬⠁⠀⠲⠀⢠⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣴⠀⠀⣴⡆⠸⠀⠀⠀⠀⠀⠀⠈⡘⣄⠀⣰⣤⠍⠳⠏
⠀⠄⠉⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣞⠛⠉⠉⠀⠈
⠀⠁⠀⠘⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢰⠆⠄
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⢠⠀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⡆⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡴⠎
⠀⠀⠀⠀⠀⠀⠸⣮⡄⣆⠀⡀⠀⠀⢀⠄⢰⠀⠀⣘⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣶⣷⡄⠀⣸⣤⡾⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⡟⢸⡯⡃⢯⠾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⢉⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣮⣟⣿⣇⣤⣤⣴⣔⡲⣶⠀⡀⠀⠀⢀⣴⢐⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⣿⣿⣯⣿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠉⣾⠁⢰⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⢀⣹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⡟⠀⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⣿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⣽⣇⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⣳⣦⣤⣠⣴⣯⣀⡀⣀⣀⣴⣾⣿⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⣼⣿⠉⣹⠿⠿⠿⠿⣿⣿⣿⠀⠙⣿⡇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣰⣿⠀⠀⠀⠀⠈⢻⡿⢧⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠐⣿⠃⠀⠀⠀⠀⠀⠀⣧⡟⠀⠀⠈⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣿⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⡀⠀⠀⠀⠀⣰⡏⠀⣀⣀⡀⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡿⣀⢀⢸⡇⠤⢤⠤⢾⡿⠶⠞⠛⢛⠛⣡⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⢋⣼⠷⣖⣒⡐⠊⠈⠛⠁⠶⠾⢏⠉⠙

⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠡⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⢦⡀⢰⡄⠀⠀⠀⠀⣄⣼⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⡆⢀⣧⡆⢀⣤⠁
⠀⠠⠷⠾⡟⣿⣷⣄⠰⢰⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⣶⡇⣿⣿⣁⣨⠤⠄
⠀⠰⢶⣤⡺⢿⣿⣿⡆⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢩⡏⣿⣿⣿⡤⠄
⠀⠀⠀⠀⠁⠈⠛⣿⡇⢼⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢸⢡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢸⡀⣷⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣹⣿⣥⡿⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡘⣿⣷⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣿⢋⣿⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣈⠛⢧⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⢨⣵⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣾⣿⣿⣷⡄⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⠀⠀⠀⠀⠈⣡⣧⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣇⡀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣭⡼⠃⣀⣀⣬⣷⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠛⢿⣶⣤⣷⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣉⢀⣽⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠈⠙⠶⢭⣿⣿⣿⣿⣿⣄⡀⠀⠀⢀⣀⢀⣴⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⣿⣿⡿⠟⡁⠴⠀⠀⡄⠉⠻⣿⣿⠋⠁⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠲⠀⢠⣿⠿⣭⠀⠀⠀⡀⠀⠀⠀⢹⣿⡷⠾⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢻⣿⣶⣼⣧⣤⡀⠀⠐⣦⡞⣠⣽⣿⠀⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠻⠈⠙⠻⣿⣧⠀⣼⠗⠘⠛⣿⡿⠓⠻⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⣤⣷⣶⣾⠿⠋⠀⠉⢁⣤⣾⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠈⣿⠙⣿⡇⢶⡄⡒⢠⡄⣿⣿⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠰⠃⠀⠘⠁⣤⣾⣿⣧⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢀⡄⠀⠀⢀⠀⠀⢀⣉⣉⡁⢀⣀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣸⡆⠀⠀⢌⣸⠆⣼⣿⣿⣁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠁⠀⠀⡾⢻⣿⣾⣿⣿⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣴⡆⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⣠⡍⢹⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡄
⠀⠀⠀⠀⠀⠀⠀⣶⠀⢀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢰
⠀⠀⠀⠀⠀⠀⠀⢻⡄⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡽⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣅⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣄⡀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡊⢻⣿⣦⣤⣤⣤⣼⣿⠯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢾⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⢿⡿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇⣾⣷⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠸⣿⣿⡿⠁⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⣼⣿⡎⠁⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⢻⣿⣷⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢻⡟⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡇⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⣼⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠸⣹⡿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠦⣄⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠹⣷⠀⠀⠀⠀⠀⣸⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣼⣦⣶⣶⣶⣤⣿⡨⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠋⠉⠟⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⣾⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢅⢰⠷⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠁⢠⣤⣴⣾⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⠀⠀⠀⢀⣴⣿⣿⡏⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⠛⠉⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢻⠀⣿⣿⣿⣿⣿⡿⢿⣿⡇⣾⣽⣿⣿⣿⡿⡟⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⡠⣿⣦⣀⠉⠛⠿⡿⡿⠿⠿⠿⠿⢻⣿⣿⣿⠃⠀⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡼⢋⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣷⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⡜⣠⣿⣟⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠈⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡸⠀⠺⣯⣽⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡇⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢿⣿⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠈⢿⣾⡄⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⢸⡜⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣷⡻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠹⣷⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡈⠀⠀⠈⠓⣄⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⢇⠀⠀⠀⣰⡿⠛⠛⠳⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣀⣿⢦⣤⡖⠀⠀⠀⡄
⠀⠀⠈⢳⣦⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠻⣧⠀⠀⢀⣹⡀
⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⣸⡟⠁
⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⡾⠋
⠀⠀⠀⢰⠀⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⡆⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠈⡆
⠀⠀⠐⣷⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢸
⠀⠀⠀⠻⣷⣦⣤⣙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠉⠙⣿⠿⣿⣷⣤⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠤⣧⣿⣷⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣉⣛⣿⣷⣤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣐⣤⠾⢟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠛⠛⠿⣿⡷⣦⡀⠀⠀⠀⠀⠀⢀⣴⣆⣺⡿⠿⠷⠶⢟⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣈⠐⢦⣄⢿⣿⣿⣿⣦⣤⣀⣤⣴⣾⣝⣩⡿⢀⡤⠒⠚⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⣼⣿⣾⣿⣷⣴⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣶⣿⣧⣴⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢏⣿⣿⣿⣿⡿⠻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡯⠉⠁⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣄⢠⡠⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⣿⠉⣿⣻⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⣿⣿⣩⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⣌⣣⣿⢤⠀⢴⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡃
⠀⠀⠀⠀⠀⠀⠞⣮⣿⣷⣟⣿⠁⠀⠛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣼⣿⣿⣶⣾⣿⣟⢷⡶⢺⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠨⣹⣿⣿⣿⣿⣿⣿⣉⣽⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢹⡖⡃⣁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠠⠦⠵⠾⠿⠿⣿⢿⡿⡿⣿⠿⣷⠶⢻⠓⠇⠙⠈⠿⠿⠿⢿⣿⣿⢿⣿⠿⢿⡿⢿⣿⠿⠿⣷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢠⡅
⠀⠀⠀⠀⡈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⢰⡖
⠀⠀⠀⠀⡇⠀⢠⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠈⠓
⠀⠀⠀⠀⠈⠁⠀⠀⣀⡤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠱⢤⣀⠀⠀⠃
⠀⠀⠀⡀⣀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠙⣿⢷⣶⡄
⠀⠀⢰⣿⠏⢙⠷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⢀⠀⠀⢉⣴⣿⠟
⠀⠀⠙⠻⠶⣤⣄⣀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣠⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠙⠛⣷⣶⣶⣦⣧⣐⣀⣠⡆⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡶⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⡉⢤⣌⠙⣿⣯⣍⡰⠻⢶⣤⣀⠀⠀⠀⠀⣀⣤⡛⠺⠟⢒⣶⣶⣶⠶⠞⣛⠛⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠷⠀⠈⣯⣟⣷⣦⣾⣿⣿⣷⣾⣦⣞⣿⣿⣤⢁⣴⡟⣩⠋⠁⠀⡈⠉⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠀⠾⠊⣽⣿⣿⢏⢩⣌⡛⠿⠿⠛⣹⣿⣿⡟⠛⠇⢠⠴⠞⠁⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠛⢷⣾⣿⡦⠶⢖⠰⣾⣿⣿⡿⠿⢀⣴⠃⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣠⣷⣤⣶⣿⣻⡿⣿⣏⣴⣷⠈⠉⠠⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣏⠉⠀⠰⢿⣿⡭⠀⢱⠛⢿⡄⠉⠹⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣯⠞⣿⠋⣀⣐⣛⣷⣄⣻⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡟⣹⣿⣿⣦⡏⣼⡟⣻⣏⣿⡿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢧⠚⠿⣿⣦⣤⣿⠥⠛⠛⠿⢏⣼⠟⣿⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡟⠀⠀⠙⠻⠟⢿⣶⣶⣦⣤⡾⠃⠀⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠐⠀⠈⠂⠀⢬⡴⠋⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⢸⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⣀⡙⠁⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢀⠞⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣸⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣐⡜⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⡿⣠⣤⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠛⠉⠉⠉⠉⠑⠌⡻⣿⣿⣿⣿⡟⣧⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⣻⡄⠀⠀⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⢸⡆⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠈⣷⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡎⣼⣿⣿⣿⣿⠁
⠀⠀⠠⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣧⢸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠜⣰⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⡒⠲⢦⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣔⣠⣾⣿⣿⣿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠤⣀⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣴⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣦⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣠⡾⡫⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⢷⣄⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⣴⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣶⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣶⠀⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⣶
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⢸⡇⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡇⡈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠘⡜⣇⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⢣⠇⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⠷⠘⢄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⡠⠃⠾⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣄⣱⣿⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣿⣏⣠⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡹⣔⠤⣀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠤⣢⢏⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣛⢒⠉⠒⠤⢠⡟⡀⠀⠀⠀⠀⢀⣏⡀⠄⠒⠉⢒⣛⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⠉⠙⠓⠿⣾⣷⣿⣄⠀⠀⢠⣼⣾⣶⠗⠒⠋⠉⣉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡌⠻⣿⡛⠉⢢⣌⣉⡽⠟⠛⠛⠹⢯⣉⣩⡔⠉⢛⣿⠟⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢧⣙⣿⣤⣸⣯⠀⢀⡀⠀⠀⢀⡀⠀⣽⣇⣤⣿⢏⡴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⠛⠛⢻⣟⡻⣾⠁⠀⠀⠈⣷⢟⣻⡟⠛⠛⠭⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠈⠀⠀⠀⠀⠂⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢠⣥⣄⣠⣤⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣿⢿⠀⠀⠀⠀⣾⣿⠏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⣤⠹⠄⠁⠀⠀⠀⠡⠏⣤⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢧⡀⠀⠀⠀⠀⢀⡼⢃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⠀⠀⠀⠀⠋⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⠀⢠⡀⠀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢠⡷⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠄⠀⠀⠀⠀⢼⡄
⠀⠀⠀⡿⢡⠀⠀⠀⠠⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠄⠀⠀⠀⡌⢿
⠀⢀⡈⠀⠃⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠘⠀⢃⡀
⠀⡘⢡⡞⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⢳⡄⠃
⢠⠀⢸⠁⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⡇⠈
⡀⠀⢸⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⡇
⣷⠀⢸⡀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⡅⠀⠀⢀⡇
⣧⣰⠘⡇⠀⠀⠀⠀⠀⢼⠆⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠠⡣⠀⠀⠀⠀⠀⠀⠸⡧⠀⠀⠁⠀⠀⢸⢃⣆
⢸⣿⣧⠀⠆⣠⣿⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣆⠰⠀⣸⣿
⠀⠻⣯⠂⠀⠹⣿⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⣿⠏⠀⠀⣹⠟
⠀⠀⠈⠳⣄⡀⠈⠠⣿⣷⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠁⢀⣠⠞⠉
⠀⠀⠀⠀⠈⠻⣶⣦⣈⠻⣆⠀⠀⠀⠀⠸⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⠇⠀⠀⠀⠀⣰⠟⢁⣤⣶⠟⠉
⠀⠀⠀⢠⣶⣶⣤⣽⣿⣷⣦⣁⠀⠒⢐⣀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⣀⠀⠂⠀⣀⣴⣾⣿⣯⣤⣶⣶⡄
⠀⠀⠀⠸⣇⠀⠀⠀⠨⣉⠛⢿⣿⣶⣿⣿⡿⠿⠟⠞⠻⠟⠻⠟⠶⠻⠿⢷⣿⣿⣶⣿⡿⠛⣉⠁⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠹⣆⠀⠀⠀⢈⣷⠀⠸⢿⣿⣿⠞⠿⣿⣿⣿⣷⣾⣿⣿⣿⠿⠳⣿⣿⡿⠇⠀⣾⡁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠘⢧⣀⠐⠎⠻⠗⠀⢸⡟⣵⣶⣄⢉⣿⣿⣿⣿⣿⣿⡉⣠⣶⣮⢿⡇⠀⠺⠟⠱⠂⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣠⣿⣯⡿⠛⠉⢹⣿⠿⠛⠛⠿⣿⡇⠉⠛⢿⣿⣿⣄⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣶⣠⣾⠁⣠⡾⢷⣄⠈⣷⣄⣴⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠙⠉⣹⡟⣻⣶⣶⣿⢻⡏⠉⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣴⣿⣧⢻⣧⣿⣿⣿⣿⣼⡟⣼⣷⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣿⣿⣿⣦⣿⣤⣀⣀⣤⣿⣾⣿⣿⣿⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⡟⠙⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⣿⣿⡏⠀⠀⠛⠛⠁⠉⠛⠛⠀⠀⢸⣿⣿⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣆⡀⢀⣠⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣠⣄⣬⣁⣤⣠⣄⣀⣤⣤⣀⣤⣤⣠⣄⣠⣤⣠⣤⣤⣤⣤⣤⣤
⣿⣛⣟⣻⣛⣻⣿⣛⣛⣿⡛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣻⣿⣿⣟⣛⣛⣛⣛⣻

⠀⠀⠄⢄⠀⡀⢀⠀⡀⠀⡀⢄⡀⠀⠀⠀⢄⡀⡀⡀⢄⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⡀

⠀⠀⠀⠀⣴⡾⠟⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠻⢷⣦
⠀⠀⠀⠘⠛⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⡾⠟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⣴⣶⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣟⣀⣀⣠⣿⣇⣀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠸⣷⡀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠉⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠙⠻⠷⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢀⠀⢡⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⣄⠻⠀⠘⡄⢀⣐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⢶⣷⢤⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣋⣗⣀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⢿⣉⢱⡿⢟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢷⣦⣿⣷⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⠀⠈⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣶⣶⣶⡾⠋⠁⠠⠂⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⣿⣿⠿⠟⠟⠋⠛⠟⢛⠋⠉⠁⠀⠀⠀⠀⢧⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠛⢛⣯⡍⠋⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⣦⣀⠺⣷⣦⣀⣄⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣨⣿⣀⠀⢹⣿⣷⣶⣶⣿⣶⡌⠀⠤⠀⠬⠆⠈⠿⣷⣮⠉⠀⠿⠶⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣷⣤⣿⠟⠉⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⣀⣀⡈⢻⡀⣤⣶⠟⠛⣻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠮⣀⣿⣿⣿⣿⣧⣠⣴⠿⠿⠿⠿⣿⣤⣤⣤⣀⣠⣶⣿⣿⡬⣿⣿⡿⢿⣿⣿⣿⣿⣟⢆
⠀⠀⠀⠀⠀⠀⡾⠿⠟⠛⣠⣾⣿⣿⣉⣵⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣎⢿⣶⣄
⠀⠀⠀⠀⠀⣸⢣⡶⠚⠋⠉⠉⠁⠀⡿⢻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡟⠛⠷⣦⡄⡀
⠀⠀⠀⠀⢠⣯⠏⠀⠀⠀⠀⠀⢀⡾⢡⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠲⢍⣷⣤⣤
⠀⠀⠀⠀⣞⡟⠀⠀⠀⠀⠀⠀⣿⢁⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⡀⢀⡼⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⢠⣼⡶⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⢀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⣼⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠈⢿⣆⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⠀⡀⠁
⠀⠀⠀⠀⢶⣀⠀⠙⢿⠿⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠤⢀⡜⠁⠀⣴
⠀⠀⠀⠀⠀⠙⢷⣦⣤⣹⣤⡀⣬⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⡀⢠⠋⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢋⣴⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡯⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣯⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠙⢿⣯⣀⡀⠀⠀⠉⠉⠉⠑⠀⠉⠈⠿⠏⠀⢀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢾⠿⡿⣶⡆⠈⣿⣿⣿⣷⡀⠀⠀⠀⣴⣶⣶⣦⡾⢦⣼⣿⠋⠍⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠾⢻⣿⠿⣾⣿⡋⢹⣿⠿⠇⠘⠛⠛⠉⢀⠀⢿⡟⠍⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠨⣭⣵⣿⣿⡇⠀⠀⢀⣶⣶⣾⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠁⠀⠀⠀⠿⣿⣿⣇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣅⠀⠀⠀⠀⢸⣿⣿⣿⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣆⠀⣠⣤⣾⣿⣿⣿⠄⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡛⠛⠉⠘⣿⣿⣿⣯⣗⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣧⣝⣷⡄⠀⠑⢤⣀⣀⣀⣠⣤⣶⣾⣿⣿⡭⠀⠀⠀⠀⠀⠉⠁⣲⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠉⠉⠛⢻⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣤⣴⣿⣿⣿⣿⢿⣿⣮⡫⠝⠄⠀⠀⠀⠀⠀⢉⣭⣿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⠙⢧⠝⠠⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢿⣯⠻⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡜⠄⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣄⣀⣠⣴⣾⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠯⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣟⠷⣝⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣟⡷

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡆⣰⠏⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⠎⠰⡀⠀⠀⡼⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠙⣤⡼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⢀⣼⣤⠴⠚⠀⢸⡇⠀⠀⠀⡼⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣈⠻⢧⡀⢀⠀⠀⠸⣧⣀⣠⣾⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡙⣦⠙⣷⣄⠀⠿⣩⠭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠏⠘⠁⢠⣤⠠⡡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⠀⠀⠀⠁⠀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣌⢷⣤⠄⣀⣠⠤⢾⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠿⠖⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⡄⠀⠙⡗⠦⠤⣤⣤⣤⣤⣤⣤⣤⣦⠶⠶⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣤⣾⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢠⣤⡀⠀⠀⠈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣈⢿⣿⡏⡇⣼⢷⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⠀⢻⣿⡀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⣿⣇⣿⣿⡘⠷⠀⠀⠀⢸⡄⡀⠀⠀⠀⢸⡿⢠⣿⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣧⠀⣷⠀⠀⣿⣹⢷⠀⠀⢀⡾⢁⣾⣿⣿⣇⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⣿⠀⠀⠿⠋⠀⣀⣠⣭⣶⢿⣟⣉⢿⣿⣦⡀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡻⣿⡟⣿⣿⢀⠼⠿⠿⠿⠛⠋⠉⠺⣿⣿⣿⣿⣿⠻⣿⣦⡀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⡇⢸⡏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⢦⡀⠙⠻⣷⡆⢳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠡⢸⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣹⠆⠀⠈⠙⢿⣦⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢰⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠏⠀⠀⠀⠀⠀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢨⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠳⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⡀⣼⣿⣠⣴⠟⠁⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠙⢷⣄⣹⣿⣀⡀
⠀⠀⠀⠀⣼⡟⣾⣿⡿⠁⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⢻⣿⣿⠻⣷⡀
⠀⠀⠀⠀⣿⡇⣿⣿⠇⠀⠀⠀⡎⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⡄⠀⠀⠀⣿⣿⡇⣏⡅
⠀⠀⠀⠀⣿⣇⡵⣿⡀⠀⠀⡜⡼⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠻⡰⡀⠀⠀⣿⡧⣷⣿⡇
⠀⠶⣒⣚⠸⣿⡇⣿⡇⠀⢰⢧⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⢧⠀⢀⣿⡇⣿⡿⢘⣒⡶⠆
⠀⠀⠀⠙⣷⣾⣧⣿⣷⣠⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⢠⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣸⣿⣷⣿⣴⡟⠁
⠀⠀⠀⠀⢡⢿⣿⣿⢷⣿⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⢻⣷⢿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⠟⣡⣿⣧⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣰⣿⣦⡙⢿⠃
⠀⠀⠀⠀⠀⡎⣼⣿⣽⣿⣀⠙⠦⢄⣀⣀⣀⡀⠀⠀⢸⣼⡟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣸⠀⠀⢀⣀⣀⣀⣀⠤⠚⢁⣽⣿⣿⣿⠈⡂
⠀⠀⠀⠀⠀⠹⣿⣿⣷⣭⣍⣙⣛⣳⣶⣶⣦⣭⢽⣲⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣶⣺⢭⣤⣶⣶⣖⣛⣛⣉⣭⣽⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠦⣨⡿⣧⣸⣄⠀⠀⠀⠀⠀⢀⣾⣸⡿⣧⡠⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⡿⡨⠓⠒⢲⠒⠒⠫⡺⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⠀⠀⠀⢸⠀⠀⠀⢼⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠄⠀⢸⠀⠀⠄⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡓⣄⠀⠀⣸⡀⠀⣀⠜⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠗⢬⣷⠀⠘⠀⣴⡏⠚⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⢸⢻⠀⢸⠀⢛⡛⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⠈⠇⢸⢠⠃⣰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸⢀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⣏⣸⣘⣷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⠛⠋⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢼⠄⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠂⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠑⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⣿⠆⠁⠀⠀⠀⠀⡀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢂⠀⠀⠀⠀⠘⠰⣷⡈
⠀⠀⠀⠈⣴⠋⡌⠀⠀⠀⠀⠀⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠄⠀⠀⠀⠀⢣⠹⣧
⠀⠀⠐⠀⠋⢸⠁⠀⠀⠀⠠⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠄⠀⠀⠀⠘⡇⠙⠠⠂
⠀⠀⣧⠀⢀⠀⠀⠀⠀⠀⠃⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠐⠀⠀⠀⠀⠁⡀⠀⣴
⠀⢠⠃⢠⡟⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⡀⠀⠀⠀⢻⡄⠙⡀
⠀⠈⠀⢸⠁⠀⠀⠀⡄⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢁⠀⠀⠀⠘⡇⠀⢁
⠈⠁⠀⣿⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⣿⠀⠘
⣆⠀⠀⣿⠀⠀⢰⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⡆⠀⠀⣿
⣿⠂⠀⣿⠀⠀⠀⠀⢨⡏⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢹⠃⠀⠀⠀⠀⣿⠀⠸
⣿⠀⣦⠘⡆⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠆⠀⠀⠀⠀⠀⠀⠀⠀⢼⡤⠀⠀⠀⠀⠀⢰⠀⢸⠃⣴
⢿⣿⣿⡄⠱⠀⡇⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣦⠀⠈⠀⠏⢠⣿⣿
⠀⢿⣿⣷⡀⠈⢱⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⣷⡄⠀⠀⣾⣿⡿
⠀⠘⢿⣆⠁⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠈⣴⡿⠃
⠀⠀⠀⠙⢷⣄⠀⠀⠈⠡⣄⣿⣶⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠀⡄⠀⠀⠀⠀⠀⠀⠀⣶⣿⣠⠌⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⢿⣶⣦⣀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⠸⢾⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡷⠇⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⣀⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⡈⠛⠄⣀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⠀⠀⠀⣀⠠⠛⢁⣴⣾⣿⠟⠁
⠀⠀⠀⠀⢰⣿⣿⣷⣶⣶⣿⣿⣿⣿⣦⣀⠀⠉⠀⣉⡄⣸⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⣇⣠⡁⠀⠉⠁⣀⣴⣿⣿⣿⣿⣶⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢌⡉⠻⢿⣿⣷⣶⣿⣿⣿⣬⣿⣶⠷⠶⠶⢾⠿⠿⡷⠶⠶⠾⣶⣿⣡⣿⣿⣾⣶⣾⣿⡿⠛⢉⡡⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠰⣦⡀⠈⢻⣏⣻⣿⣿⣡⣶⣶⣶⣷⣶⣶⣄⣠⣶⣶⣾⣶⣶⣶⣬⣿⣿⣟⣹⡟⠁⢀⣴⠌⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⢰⣿⣄⡀⠘⠛⣿⣿⡟⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⠛⠁⢀⣬⣿⡄⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠘⠇⠘⠿⠏⠀⢰⣿⠋⣿⣿⢶⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⡶⣶⣿⠙⣿⡆⠀⠹⠿⠃⠼⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⢀⣀⣾⣿⣇⡿⠿⠚⠋⢸⣿⣿⡿⠛⠛⢿⣿⣿⡇⠘⠓⠿⢿⣴⣿⣷⡀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⣿⣿⣧⣤⠀⠀⣼⠟⠁⢀⣤⣤⡀⠈⠻⣧⠀⠀⣤⣼⣿⣿⣿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣶⣾⣏⣄⣴⡟⠛⠛⢻⣦⣠⣿⣷⣶⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠈⣁⠀⣿⡝⣯⣿⣿⣿⣿⣽⢻⣿⠀⣈⠁⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣰⣾⣿⡄⢿⣇⣽⣿⣿⣿⣿⣧⣸⡿⢠⣿⣷⣆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⣌⣿⡉⠀⠀⠀⠀⢉⣿⣡⣿⣿⣿⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⣿⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢻⣿⣿⣿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⣿⣿⣿⠁⠀⠈⣿⣿⠿⠛⠛⠿⣿⣿⠁⠀⠈⣿⣿⣿⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⣦⡿⣹⡟⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⣦⠔⠛⠿⠿⠾⠋⠀⠀⠀⠀⠺⢧⣀⡀⢀⡾⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⣠⣾⡷⢷⣄⠀⠀⠀⠀⢀⣤⣠⡤⠞⠉⢀⠁⠱⠀⠒⠂⣴⡄⠀⠀⠀⠀⠀⢈⠿⠋⠀⠰⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⢀⡼⠋⠁⠀⠀⠙⢻⣦⠀⠦⣉⢱⡏⣀⠄⠀⠋⠐⣠⡄⠀⢀⣿⣷⠀⠀⠀⢀⣴⣯⠖⠀⠀⠀⢀⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⣀⡀⠂⢀⡀⠀⣻⣿⠀⣼⣿⣿⡀⡤⠚⠥⠋⠀⠘⢀⣼⠠⣻⣿⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠋⣠⠆⠠⠀⠀⠀⠀⠀⠀⢦⣼⣿⠞⠁⠀⠔⠉⠀⠠⣾⣿⣷⣿⣿⣿⣇⠀⠀⠀⠀⡠⠌⢉⣠⡾⠿⠋⠁
⠀⠀⠀⠀⠠⣿⣧⣤⣀⡐⢡⡔⡁⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠃⠀⠀⠀⡀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠀⡀⢀⠴⠶⠟⢁⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣮⣼⣿⣦⣤⣀⠀⠀⠀⠀⣤⣿⡷⠀⠀⣀⠀⠂⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡷⢾⣷⢀⣴⣦⣀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣌⠙⠛⠁⠀⠀⠹⣷⣄⣤⣿⡿⠋⠀⠀⢰⣿⣆⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣿⣿⠿⠛⠋⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠢⣤⡀⠀⠀⠈⠉⠻⠿⠦⢶⣤⣶⣾⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠐⠠⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⣀⣠⣄⠖⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠈⠻⠟⠑⣷⣄⣀⠀⠀⠈⠓⢶⣤⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠐⠋⠟⠋⠀⠀⠀⠀⠀⣀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠛⢦⣦⢀⣄⠙⠷⣦⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠻⣿⡷⢿⣿⣿⣿⣿⣿⣿⠛⠋⠋⠁⠙⠁⠀⠀⠀⢀⣴⣧⣠⡴⠂⠀⠀⣠⠞⠋⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢁⣾⣿⣿⠿⠛⠛⠛⠀⠀⠀⠀⠀⠔⠀⢀⠾⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡠⠀⠀⠀⠀⢤⡀⢣⠀⠀⠀⠀⢀⠂⣠⠄⠀⠀⠀⠠⣀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡾⠀⠀⠀⠀⠀⠀⠸⣜⡆⠀⠀⠀⣼⣼⠁⠀⠀⠀⠀⠀⠘⣆⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹⡇⡄⠀⡄⢿⠁⠀⠀⠀⠀⠀⠀⠀⢘⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡆⠀⢀⡄⠀⡄⠀⠀⣾⡴⠃⠀⠻⣼⣧⠀⠀⡄⠀⢠⠀⢀⣾⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠳⣼⡇⠀⣧⡴⠛⡉⠀⠀⠀⠀⠀⣙⡳⣦⣇⠀⣼⡴⢟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⣻⡽⢻⡿⡆⠗⠁⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⠛⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡹⣧⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣾⣆⠛⢿⣮⡁⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⢿⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣹⣿⣿⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⡿⣱⣿⡿⠋⠈⢿⡟⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢿⣿⠀⠙⣿⣿⣿⣿⣷
⠀⠀⠀⣾⡟⠀⢻⣿⠁⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢸⡟⠀⠀⠈⣿⡟⠈⢿⡇
⠀⠀⠀⢸⠀⠀⠈⣿⠀⠀⠀⢸⠁⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⢀⡿⠀⠀⢸⡇
⠀⠀⠀⢸⡀⠀⠀⢸⡆⠀⠀⣸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣼⠃⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⢿⡄⠀⣿⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣸⡟⠀⠀⠀⣸
⠀⠀⠀⠘⠃⠀⠀⠀⠈⠉⠀⣿⠀⠹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢸⡇⠀⠛⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣰⡏⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⠃⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⠀⠀⣿⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⠶⠛⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⣦⣄⣿⠁⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢻⣿⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠸⣷⣤⣄⣀⡀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢀⣤⣾⣿⣥⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠋⠛⠻⠿⢧⣷⣄⠀⠀⠀⠀⠀⠀⢸⣇⣶⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⣿⣦⠀⠀⠀⠀⢀⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⠀⣷⡛⠙⣧⠀⠠⠈⠻⣿⢁⡠⠴⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣽⢟⠁⠀⠉⠀⠀⢰⣿⠏⣤⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠈⣿⠉⠀⠀⠀⣤⣤⣤⡀⠈⠁⣸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠞⠛⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢉⣉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣦⣠⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠷⠀⠀⠀⠀⠀⠀⢀⡀⢀⣿⣴⣤⣿⣿⣿⣿⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣿⣧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠀⢻⣿⣷⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢻⣿⡏⠀⢈⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡇⠈⣿⡂⠀⠀⣿⣿⡿⠛⢿⣟⢰⣿⣯⢿⣿⡿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡁⠀⣙⠃⠀⠀⢻⣿⣾⣯⣽⣿⣾⣿⣿⣿⣾⣷⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢨⣥⡀⣀⣈⣽⣿⣿⡟⢩⣿⣿⣿⡿⠹⣿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣶⢤⣶⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡇⠀⠚⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢹⣿⣿⡇⣿⡟⠋⠃⠋⠹⢻⣿⠃⠨⠆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢼⣿⣿⡄⣽⣅⣀⠀⠀⠀⣈⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣷⣶⣤⣬⣽⣟⣿⣷⣮⣝⣿⣤⣤⣴⣂⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢻⣿⣿⣿⣿⣿⣻⠿⠿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣮⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡿⠃

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠔⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠑⢄⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⣼⡆⠐⠀⠀⠀⠀⠀⠂⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠂⠀⠀⠀⠀⠐⠂⣾⡄
⠀⠀⠀⠈⣰⠟⣀⠃⠀⠀⠀⢀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⢀⠀⠀⠀⠀⢣⡘⣿⡀
⠀⠀⠀⠄⠟⢰⠇⠀⠀⠀⠀⠄⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⡄⠀⠀⠀⠈⢧⠘⠇⡄
⠀⠀⣦⠀⠀⠈⠀⠀⠀⠀⠄⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠄⠀⠀⠀⠘⢀⠀⢠⡀
⠀⢀⠟⢀⣾⠀⠀⠀⠀⠐⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠰⠀⠀⠀⠀⢹⣆⠘⠇
⠀⠘⠀⢸⠇⠀⠀⠀⠀⠁⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠁⠀⠀⠀⠀⣿⠀⠘
⠀⠇⠀⣸⠇⠀⠀⢀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⠀⠀⠀⢿⡀
⡄⠀⠀⣹⠀⠀⠀⡄⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠈⡄⠀⠀⢼⡇
⣿⡄⠀⢹⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢸⡟⠀⠀⠀⠀⣼⡇
⣿⠀⣄⠸⡇⠀⡄⠀⠀⠃⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠈⠁⠀⠀⡀⢀⡿⢁⡀
⣿⣤⣿⡀⢻⡀⠇⠀⢀⡀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣀⠀⠀⠃⣸⠃⣸⣧
⠈⣿⣿⣇⠀⠡⢇⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠄⠀⢠⣿⣿
⠀⠹⣿⡍⠂⠀⠈⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠀⠀⠚⣿⡿
⠀⠀⠙⢿⣦⡀⠀⠀⠙⠇⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄⢠⠟⠁⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠙⠿⣦⣀⡀⠀⠓⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣇⣴⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⣄⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡖⠃⠀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣤⡈⠻⣿⡀⠀⠀⠀⠀⠀⠀⠉⣿⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡏⠁⠀⠀⠀⠀⠀⠀⣸⠟⠉⣠⣴⣿⠿⠋
⠀⠀⠀⠀⢀⣤⣤⣤⣀⣙⣿⣿⣶⣤⣉⠐⠂⠀⠄⠠⠜⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠄⠀⠀⠀⠐⢊⣡⣴⣾⣿⣛⣁⣠⣤⣤⡀
⠀⠀⠀⠀⢸⣿⠟⠛⠛⠛⠛⠿⠿⣿⣿⣷⣦⣀⣠⣠⣾⡔⢿⣾⣠⣀⣀⣀⣠⣤⣤⣀⣀⣀⣠⣹⣾⠷⣶⣦⣠⣀⣠⣴⣿⣿⡿⠿⠟⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠑⠢⠀⠙⠻⣿⣿⣿⣿⣿⡿⠞⢛⣋⣉⣉⣉⠛⠉⠙⢋⣙⣉⣙⣛⠛⠾⣿⣿⣿⣿⣿⡿⠋⠁⠰⠆⠁⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠈⣻⡶⠀⠀⣿⣶⣿⣿⡷⠾⢿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠟⠷⣿⣿⣷⣾⡇⠀⠰⣾⠃⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⣐⠿⣷⣦⠀⠀⣸⣿⣿⣁⣀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢀⣀⣹⣿⣿⡀⠀⢠⣶⡿⢗⡀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠙⠀⠙⠋⠀⠀⣿⣏⢸⣿⣟⣿⡄⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⣴⣏⣿⣿⢈⣿⣇⠀⠈⠛⠁⠘⢁⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⣤⣠⣤⣾⣿⣿⣾⠉⠉⠀⠀⣿⣿⠿⠋⠁⠈⠻⢿⣿⡇⠀⠈⠉⢻⣾⣿⣿⣤⣤⣤⡴⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠹⢿⣿⣿⣶⡆⠀⣰⡿⠁⢀⣴⣾⣶⣄⠀⠹⣧⡀⠀⣶⣾⣿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⠿⠿⣷⣦⣿⣍⣉⣉⣽⣧⣶⡿⠿⢿⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣠⡄⢰⣿⠈⣾⣿⣿⣿⣿⡞⣼⣧⠀⣤⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣿⣿⣿⡈⣿⣦⣿⡿⠿⠿⣿⣧⣾⡟⣸⣿⣿⣧⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⣿⣿⣿⣷⣸⣿⣀⠀⠀⠀⢀⣸⣿⣰⣿⣿⣿⣿⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣾⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢻⣿⣿⣿⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⡏⠀⠀⠸⠿⠿⠟⠋⠛⠿⠿⠿⠀⠀⠈⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣀⣼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡠⠄⠀⠀⠀⠠⣄⢂⠀⠀⠀⠀⡐⢀⠄⠀⠀⠀⠠⢄⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠁⠀⠀⠀⠀⠀⠸⣼⡄⠀⠀⢀⣧⠇⠀⠀⠀⠀⠀⠈⣧⡇
⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⢃⡆⢰⡈⣇⠀⠀⠀⠀⠀⠀⠀⢨⡇⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⣀⠀⡆⠀⣶⠀⣀⡿⠞⠀⠀⠳⢾⣆⠀⢰⠀⢰⠀⢀⣾⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⣳⣧⣀⣿⠟⡉⠄⠀⠀⠀⠀⠠⣬⣻⣿⣄⣼⣞⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⢮⣹⡟⠇⠋⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣿⢛⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣌⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⢿⣷⣝⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⣀⣀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣍⠻⣧⢘⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⣿⡿⣹⣿⡿⠀⢿⣿⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⢻⣿⠏⠻⣿⣿⣿⣿⡇
⠀⠀⠀⣾⡟⠘⣿⡟⠀⠀⢸⡧⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠹⣿⡏⠙⣿⡀
⠀⠀⠀⣿⠀⠀⢹⡇⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⡇⠀⠀⢀⡟⠀⠀⢹
⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⢸⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⣼⠁⠀⠀⣼
⠀⠀⠀⢸⠀⠀⠀⠸⣧⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⣰⡟⠀⠀⠀⣿
⠀⠀⠀⠘⠃⠀⠀⠀⠉⠁⠸⠇⠸⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠇⠠⠧⠀⠉⠀⠀⠀⠐⠛

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀
⠀⠀⠀⡄⠀⠀⣴⡃⣀⣴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣀⢘⣆⠀⠀⢠
⠀⠀⣸⡇⠀⠀⣹⡿⠋⠁⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠙⢿⣏⠀⠀⢸⣇
⠀⢰⣿⠁⢀⣼⠏⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠹⣧⡀⠈⣿⡆
⠀⠘⣿⣦⣾⠃⠀⢰⡿⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢿⡆⠀⠘⣷⣴⣿⠃
⠀⠀⢹⣿⡟⠀⠀⢸⣇⣴⠏⠀⢰⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡆⠀⠹⣦⣸⡇⠀⠀⢹⣿⡏
⠀⠀⠀⣿⡇⠀⣠⣴⠿⣧⡀⠀⣸⡇⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⢸⣇⠀⢀⣼⠿⣦⣄⠀⢸⣿
⠀⠀⠀⢻⣷⣼⡿⠋⠀⠹⣿⣿⠟⠀⠐⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠂⠀⠻⣿⣿⠏⠀⠘⢿⣧⣾⡟
⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠘⣿⣦⣀⣠⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣀⣀⣴⣿⠃⠀⠀⠀⢨⣿⡟
⠀⠀⠀⠀⠀⠹⢿⣶⣶⣶⣾⠿⠿⠿⠛⠛⠛⠛⠛⠻⢿⣷⣦⡀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⢀⣴⣾⠿⠟⠛⠛⠛⠛⠛⠿⠿⠿⣶⣶⣶⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣾⣿⣿⣿⣿⣿⣿⣷⣾⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣿⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢀⡶⠁
⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⡰⠃⣄⠀⠀⠀⡎⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣰⠃⠀⠘⢦⣀⡾⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⣏⣀⡤⠆⠀⣿⠀⠀⠀⠀⣼⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣿⡉⠁⠀⠀⠀⢹⡄⠀⢀⣼⣡⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠓⢮⡛⢶⣆⠀⢰⡾⢛⣛⣋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣼⠗⠶⠛⠃⡀⢙⢉⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠋⠘⢮⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢧⡀⠀⠀⠀⠀⠀⣼⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣮⣝⡋⠑⢺⡏⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠘⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢀⡾⠀⠀⠘⠃⠉⠙⠛⠛⠛⠛⠛⢿⣿⣿⠋⠉⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣿⠇⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⣿⣦⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡜⣿⣿⢸⣼⡏⣷⡀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢘⣿⠃⡸⣿⡇⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⢹⣿⣼⣿⣷⠘⢃⡀⠀⠀⣹⣆⡄⠀⠀⠀⢸⡟⢠⣷⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧⢸⣧⠀⠀⣿⡼⠣⠀⠀⣠⠟⣠⣾⡿⣿⣿⡄⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣾⣿⠀⢘⣛⣁⣠⣤⣴⣶⢿⣻⣿⣤⣝⢿⣿⣄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⣿⠙⣿⡇⢠⠛⠛⠛⠛⠉⠉⠀⠙⢿⣿⣿⣿⣷⡙⠻⣷⣤⡀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢹⡀⣿⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡻⣤⡈⠙⢿⣷⢈⡷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡆⢿⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣼⡟⠀⠀⠀⠹⣿⣮⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⣨⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⣠⠾⠁⠀⣰⠆⣴⡄⠀⣹⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⡞⠺⣄⣴⠞⠁⠀⣸⣇⢀⣼⣏⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣫⡴⢚⡅⢀⣴⠀⣰⢏⣙⣋⡤⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣜⢿⣤⢾⠇⢸⠷⢚⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠈⠷⣤⣈⠳⣼⣆⢺⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⣻⣶⡏⠁⠀⠉⠃⠛⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠘⠛⠃⠲⢤⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠈⣁⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⡆⠘⠶⣶⡶⠚⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢸⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡾⠋⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠖⠓⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡆⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⡦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢸⠇⣸⡏
⠀⠀⠀⠀⠀⠀⢠⣿⢠⡞⠁⢠⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⡜⣰⠟
⠀⠀⠀⠀⠀⠀⢸⡇⢸⠁⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⢸⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⡞⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠙⠃⠹⣜⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠓⠲⢤⣄⣀⣀⣀⣀⣀⣠⣼⡇⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣸⠃⣸⠟⣿⡄⠀⢀⡤⠀⠀⢸⡏⠉⠓⠲⠶⠤⣤⣭⣭⢭⡥⠔⢶⡇⠀⠀⠰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⢇⡾⠁⣰⠏⠀⢹⡇⢠⠏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⣿⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢀⡿⡾⠀⣼⠃⠀⠀⣾⢁⡟⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡅⠀⣛⢛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⡼⣹⠁⣰⠃⠀⠀⡴⢃⡞⡀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⠀⣛⢙⠀⠀⣸⡃
⠀⠀⠀⠀⠀⣼⠃⠙⣾⠃⠀⠀⣾⠁⠈⠛⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢐⡏⢹⠁⢀⡿⠁
⠀⠀⠀⠀⠀⣾⠀⢸⡇⠀⠀⠀⠘⣷⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠁⠘⡇⢸⡇
⠀⠀⠀⠀⠀⢿⠀⣾⠁⠀⠀⠀⠀⠘⣧⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠘⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⢹⠸⡇
⠀⠀⠀⠀⠀⢸⣧⡇⠀⠀⠀⠀⠀⠀⠀⠹⣆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⠀⠀⢸⡄⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣧⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⣼⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⠀⠀⢸⡇⢻⡄
⠀⠀⠀⠀⢠⣿⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⢿⡄⠀⢸⣷⡎⣧⡀
⠀⠀⠀⠀⠈⢹⣧⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣧⣬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢾⣿⡆⠀⢩⡿⣾⣷⡄
⠀⠀⠀⠀⠀⠈⠋⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⣄⣾⡇⠀⠀⠀⠀⠠⣄⢺⢀⣼⡀⢰⡀
⠀⠀⠀⠀⡀⠼⠛⠀⠉⠳⣄⠀⢠⠄⠀⣹⠌⠘⠉⠉⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡀⠀⢻⠀⢠⡇⢠⣶⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣼⣤⠿⣴⣿⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⢩⣌⣫⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡄⠀⠀⠨⢃⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣰⡾⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⠀⠈⢸⡇⠙⢳⣶⠠⠄⠒⠒⠒⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⢠⠃⠀⠈⠁⠐⡄⠀⠀⠀⣰⠆⠻⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⡰⢃⡜⠀⠀⠀⢀⣠⡀⠄⠀⠋⢂⠀⠙⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠣⣿⠁⠀⠀⡀⣿⠉⠒⠀⠀⠀⢸⣤⡟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣇⢠⡆⠘⣻⣁⣀⠀⠀⢀⣠⣿⣿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡟⢻⢀⡞⠛⠛⠛⠛⠻⣿⠛⡟⣿⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⡄⠘⣸⡇⠀⠀⠀⠀⠀⠙⢧⣷⡌⢿⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⢰⣿⡆⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⡄⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠈⠛⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡙⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⣉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣌⠙⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡙⢷⣄
⠀⠀⠀⠀⠀⠀⣴⠟⣡⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣽⣦⠹⣧
⠀⠀⠀⠀⢀⣼⠃⣼⣿⣿⣿⣿⣧⡀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢼⣻⣿⣿⣷⡘⣷⡀
⠀⠀⠀⠀⣾⢃⣾⣿⣿⣿⣿⣿⣿⣷⡃⠀⠀⣠⡴⠚⣁⠤⠄⠀⠀⠐⠒⠠⠤⣈⡙⠦⣄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠸⣧
⠀⠀⠀⣸⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡾⢋⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⢳⣆⣀⢠⠿⠿⠿⢿⣿⣿⣧⡿⠂
⠀⠀⢠⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⠃⣀⣤⣴⠶⠟⠋⠉
⠀⠀⣸⡏⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠉⠁
⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⠇
⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⠤⣄⡀
⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠃⡀⢟⣡⣶⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠂⢤⣙⠷⣦⡀
⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣽⣿⣮⡻⣦⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣻⣦⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡾⢿⣄
⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡥⠶⠀⠀⠀⠀⠀⠀⠶⢦⣝⠳⣦⣄⣹⣿⣿⣿⣿⣿⣿⣷⣾⣿⣆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢰⣿⣿⣙⣋⣩⣄⡈⢹⣿⠃
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣾⣄⣀⠀⠀⠀⠀⠀⠀⢀⣲⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡂⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣍⡛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣫⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣦⣬⣍⣉⣉⣛⠛⠛⠋⠉⠉⠉⣉⣉⣤⣤⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠁⠀⠀⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠹⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀
⣸⠃⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢳
⣿⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣦
⠹⣷⣤⣾⢏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢿⣠
⠀⢻⣿⡇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⣏
⠀⠸⣿⠀⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⢻
⠀⠀⠻⣇⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢸⡇⠀⢸⣿⣿⢀
⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣾⠿⣣⡼
⠀⠀⠀⠀⠈⠻⣿⣶⣄⡀⢸⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣟⣠⣤⡶⠞⠋
⠀⠀⠀⠀⠀⠀⣀⡙⠻⢿⣷⣿⣷⣤⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⡿⠟⣋⣥⡄
⠀⠀⠀⠀⠀⢸⡿⠟⠳⢄⠈⠙⠻⣿⣯⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢉⡴⠋⠁⠀⠃
⠀⠀⠀⠀⠀⠸⣷⠀⠀⣀⠉⠂⢄⠀⠙⣿⣏⣀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢿⡏⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⠿⠀⠈⠇⠀⠛⠉⠀⠴⠀⠲⠶⠀⢱⡆⠀⠀⠀⠀⠇⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⢉⣁⡀⠀⣠⡖⢠⣤⠄⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠄⠀⠀⢠⠀⠀⠀⠉⠀⢹⣿⣿⠋⠙⠿⣿⣿⠀⣠⡅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡻⣿⡿⠄⠀⢸⣿⠇⠀⠀⠀⠙⢿⡄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⠁⠀⢀⣀⣀⣀⣀⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣤⣤⡄⢀⠀⠀⠈⣻⣿⣿⣿⡋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠛⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠘⡧⠀⠰⡞⠀⠀⠁⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡀⣴⣍⠀⠀⠀⣠⣴⣦⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⠿⢫⠟⠁

⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⡀⠀⠀⢠⢚⠏⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠈⢟⢢⠀⠀⠀⡀
⠀⣼⠀⠀⡴⢡⠃⠀⡰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⡀⠀⢣⠱⡄⠀⢹⡄
⡘⠙⠀⡜⢀⠏⠀⡴⢱⠁⠀⠀⠀⠀⠀⠀⠀⡠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⢱⠱⡄⠀⡇⠘⡄⠘⠙
⡇⡄⢰⡅⡘⠀⡸⢀⠇⠀⠀⠀⠀⠀⠀⡠⢊⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠢⡀⠀⠀⠀⠀⠀⠀⢃⠸⡀⠘⡀⣿⠀⡆
⡇⢠⣻⠇⡇⢠⠃⠼⠀⠀⠀⠀⢀⡴⢊⡴⠋⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠳⣌⠲⣄⠀⠀⠀⠀⠸⠀⢣⠀⡃⢿⣧⠁
⡇⠸⣿⢻⠃⢾⠀⡇⠀⢀⣤⠖⣁⠔⠋⠀⠀⣜⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⠈⠓⢄⡑⢦⣄⠀⠀⡇⢸⡆⡋⢻⡿⢀
⢰⠀⣙⢦⣆⣿⠧⣷⠾⢋⡤⠚⠁⠀⠀⠀⢰⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢱⠀⠀⠀⠀⠉⠢⣌⠻⢖⡧⢿⣆⣧⢞⡁⢸
⠈⡆⠈⠿⠿⢒⣫⡵⠊⠁⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠉⠲⣭⣓⠺⠿⠏⠀⡎
⠀⢱⠀⢠⣾⣿⡏⢡⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⣿⣿⣦⠀⢰
⠀⠀⠳⣿⣿⣧⡀⠈⠑⠂⠤⠤⠤⠄⠀⡀⠜⠈⢧⡀⠀⠀⠀⣀⣤⢶⣦⡶⣤⣄⠀⠀⠀⠀⣠⠏⠘⢄⣀⠀⠤⠤⠤⠄⠒⠉⢀⣠⣿⣿⡷⠃
⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠳⠶⠶⣶⣤⣤⣠⡌⠛⠦⢤⣼⠍⠀⠀⠙⠀⠀⠉⣷⣤⠤⠞⠋⣠⣠⣤⣴⡶⠶⠶⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣄⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠱⡄⠀⢸⠀⢀⠄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠘⠄⠘⠀⠊⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠐⠀⠀⠀⠆⣴⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡆⠀⢠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢷⠀⣼⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣗⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣼⡴⠃
⣿⡿⠿⠿⡿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⢿⣾⠃
⠀⠀⠀⠀⠀⠈⠀⠀⣰⣿⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢿⣧⠤⠤⠤⠤
⠀⠀⠀⠀⠀⣠⣤⣄⣿⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⣷⣠⣿⡷⠒⠦
⠀⠀⠀⠉⠉⠀⠀⠈⣿⣿⣶⣾⠟⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⡠⣮⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡉⠂⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠒⢄⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢸⣿⡟⣱⠟⠁⢀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠱⢦⣤⠄⢦⡀⢸⣇⠀⣀⣀⣄⣀⣀⣾⣿⣿⣿⢏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⣿⠆⠈⠷⣮⠿⢿⣿⡿⣿⡿⢿⣿⣿⠿⣃⣚⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠿⠷⠄⠈⠀⠄⣸⣠⣿⣴⢞⣙⣿⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⠆⢻⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠰⢸⣿⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡏⣠⣾⣷⣼⣿⣿⣿⡄⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠇⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠰⠿⡞⣋⡎⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣀⡀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⢾⡷⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⢿⢏⡽⠛⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣈⠢⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠦⠄⠀⠀⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⣠⠀⠀⠀⠈⡻⠟⠛⣻⣿⡯⠍⠉⠿⠟⠁⠀⢀⠀⠈⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⠂⢩⣿⣯⣰⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠻⣿⠋⠉⠁

⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡇⠀⠀⠀⢤⡀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢀
⠀⠀⢰⠀⠀⠀⠀⠀⢨⣶⣾⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⡀
⠀⠀⠘⣆⠀⠀⠀⢠⣾⡟⠉⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣠⣼⣟⢷⣶⣾⠁⠀⠀⠀⢰
⠀⠀⠀⠘⣷⣦⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠙⢿⣄⠀⠀⢠⣸⣇
⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⢦⡄⢸⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣴⡿⠋
⠀⠀⠀⠀⢠⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠸⠀⢠⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⢏
⠀⠀⠀⢠⡇⠀⠀⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠘⡆
⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⢹⡀
⠀⠀⠀⠸⣿⣦⣄⣀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠈⠛⠿⢿⡿⣷⣿⣿⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣤⢾⣿⣟⣀⣀⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠙⠻⣿⣿⣷⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣆⣿⣿⡿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⣉⣻⣿⣿⣷⣦⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣰⣭⣶⠿⢟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠛⠛⠻⠿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⢸⣿⣿⣿⠷⠶⠾⣋⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠓⠤⣄⡀⢾⣿⣿⣿⣿⣦⣀⠀⠀⢀⣠⣤⣟⠻⠟⣿⠃⢀⣠⡤⠴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠘⢷⣤⣨⣿⣮⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣴⣿⡁⢀⣤⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢭⣿⣿⣿⣿⣿⡿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⠉⢈⢈⠛⡻⢻⣿⣿⣟⡿⢟⣻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣶⣶⣴⢦⣤⣿⣿⣿⣿⣿⣷⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣿⣅⣈⣀⣀⣹⣧⣬⣍⣁⣹⣍⣉⣹⣇⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣿⣿⣿⣻⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣴⢦⢛⣹⣷⣤⠀⣠⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⣦⣾⣿⣧⣽⡿⠿⠋⠈⠻⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣷⣼⣿⣻⣿⣿⣿⣷⣌⣁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣩⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⢾⣿⣿⣿⣿⣿⣿⣿⣷⣔⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣼⣧⢟⠘⡀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠄⠾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣇⣴⣿⡿⣿⠈⡗⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠁⠀⠀⠀⠀⠁⠈⠉⠉⠉⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣹⣿⡟⣻⣿⡿⢿⡆⠀⠀⠀⠀⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠂⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⣿⣿⣿⣿⣿⣧⣼⡇⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⢀⣤⡾⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⢀⣠⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸
⠀⢰⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⣿⣿⣿⣿⣿⠟⣵⠟
⠀⢸⠢⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⢫⡾⠃
⠀⢸⡤⣿⣿⣿⣿⠟⠁⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⣰⠏
⠀⢸⣿⣿⣿⠟⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠏⣴⠃
⠀⠘⠛⠋⠀⠀⠀⠀⠀⠘⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠛⠋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣿⣿⣿⣿⣿⣿⡗⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣼⣿⣿⣿⣿⣿⣿⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣽⣿⣿⣿⣿⣿⣿⣿⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣽⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣿⣿⣿⣿⣿⠿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣷⣶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣼⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣻⣿⣟⣿⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠲⠶⠶⠾⠿⠿⠿⠿⠿⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢬⠍⠉⢩⣍⡭⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⡠⠂⠂⠀⠀⠀⠀⠊
⠀⠀⠀⡄⣢⡏⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠄⡀
⠀⣰⣢⢿⠃⠈⢀⡐⠊⠀⠀⠀⠀⠀⠀⠀⢐⠀⠃
⡆⠈⠛⢸⡴⡫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⡇⠀⠘⠿⠟⠠⠐⣀⣀⣀⣀⣀⣀⣴⣶⣿⠃
⡇⡡⠈⠀⠀⠀⠀⠈⢫⣿⣿⣿⣝⡡⠚⠁
⠃⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⠙⡾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⢹⣿⣿⣧⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⢶⣶⣶⣶⣾⣿⣿⠶⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠱⠇⠈⠀⠹⣿⡟⢿⡏⠃⠠⠿⠉⢩⣿⣯⡶⠀⠀⢸⣷⣏⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠃⢻⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠻⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⢿⠑⠂⠀⠀⠀⠀⠀⣷⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡣⡄⠀⠀⠀⢠⠀⠀⣀⣂⠀⠉⣀⡀⣀⣿⣷⠄⡀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡂⠀⣿⣿⣿⣏⣄⣬⣷⠿⠿⠿⠿⣟⢁⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⢸⡭⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⢘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠁⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⣾⠃⠀⠀⢳⣿⡟⠀⣦⠀⠀⠂⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠈⠀⠀⡀⣠⣞⣧⡄⣴⡟⠀⠀⠀⠸⡹⣷⣄⣷⠃⠀⠀⠸⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠠⡀⠙⠛⠾⣿⣿⣏⠀⠀⢀⠀⠀⠰⠉⢻⣿⠒⠀⠀⠀⢳⡄
⠀⠀⠀⠀⢀⣤⡶⠊⠈⠛⠷⠀⠀⠀⠀⠀⠀⠀⠙⢯⣻⡀⠀⠀⢀⠀⠀⠾⢿⣦⡀⠀⠀⠈⠻⠄⠀⠀⠀⠀⣀⡀⡤⠀⠀⣴
⠀⠀⠠⢶⣯⣿⡇⠀⠀⠀⠀⠈⠛⠷⣽⡀⡀⠀⠀⠀⠈⠙⠀⠀⠈⠀⠀⢠⡿⢹⣱⣄⡀⠀⠂⠀⠀⠀⣦⡀⠈⠋⠀⢀⣾⡟
⠀⠀⠀⠀⠀⢉⠻⣄⡀⠀⠀⠀⠀⠀⠈⠛⠿⣴⣄⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⣽⢟⡟⣦⠀⠀⠀⠀⠉⠻⠛⠒⠀⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢯⣷⣄⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠈⠁⢈⣿⠃⢎⡆⠀⠠⢄⠀⠀⡀⠴⠄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠁⠀⠀⠀⠀⣾⠀⠈⠁⠴⠿⠃⠀⠀⠀⠀⢀⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠀⠌⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⣌⡀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢼⣿⡇⠀⠀⠀⣤⠀⠀⠀⢀⣆⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⢾⠏⠀⣘⣴⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢠⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠈⠛⠲⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⣁⠔⠁⠐⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠲⢶⣼⣿⣷⠀⠀⣹⣿⡇⠀⢴⣧⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⣋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⣇⣿⣿⡇⠀⣿⣷⢠⣿⣇
⠀⠀⠀⠀⠤⢀⣀⣀⡈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢥⣸⣿⣿⢺⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢻⣇⣦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠊⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⣿⠿⠷⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢫⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣷⣮⣝⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⢟⣿⣿⣗⢦⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣄⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠀⣿⣿⠉⠁⢃⣸⠟⢿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠦⣾⣧⡀⠀⠀⠃⠀⠀⠀⢸⡇⡠⠌⣿⡟⠀⣦⣠⡿⣜⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣈⢽⣦⢤⣿⠀⠀⠀⣸⡿⣁⣴⢿⣧⣴⢿⣯⡾⢿⣏⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢹⣷⣝⢷⣼⣦⠀⣰⣿⢟⣥⢟⣾⡷⣿⢻⣷⣿⠰⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠻⣿⠇⣿⡿⠿⠿⡻⢿⣯⣿⣹⡿⣱⡿⣿⡟⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢡⣭⣦⢿⣷⣀⣤⠀⠌⢻⡿⠻⣕⣻⣾⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⣿⣿⡏⠂⣴⡿⣿⣶⣿⣿⡏⠙⣿⠀⠀⠀⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠻⢦⡀⢀⣽⠋⢹⣿⣷⡄⢻⡄⠀⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠀⢀⣾⣿⡏⠀⠀⢸⡇⠘⠀⣸⣿⡅⠀⣿⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⣾⣿⠟⠁⠀⠀⠘⠁⠀⠀⠁⢸⠇⠀⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡿⢿⣿⣇⠀⠀⠀⢤⣄⣴⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣿⣿⣧⠀⠀⢸⣿⣿⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢿⣧⡙⠄⠀⢸⡟⠁⡐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠐⢠⡿⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⣿⠀⠀⡴⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⣿⠀⢰⡿⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⠀⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡟⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠳⡀
⠀⠀⠀⠀⠀⠀⢀⡿⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢿⡀
⠀⠀⢀⡠⠔⡲⣾⡁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢨⣷⢖⠢⢄⡀
⠀⡰⠋⡰⠋⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⠀⠙⢆⠙⢄
⢸⣇⡸⠁⠀⠀⣿⡇⢇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡘⣸⡿⠀⠀⠈⡇⡸
⣿⡏⡇⠀⠀⠀⢸⣿⡘⣄⠀⠀⠀⠀⠀⢀⢚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡀⠀⠀⠀⠀⠀⣰⢃⣿⡃⠀⠀⠀⢸⢻
⢿⣧⢱⠀⠀⠀⣠⡿⠃⣚⠳⣄⡀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⡇⠀⠀⢀⡠⢞⡁⠘⢿⣄⠀⠀⠀⡆⣾
⠘⣿⡌⠳⠒⣊⠠⠀⣠⠈⠙⣂⠈⣉⠒⠾⣧⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⣾⠷⠒⣉⣁⢘⠛⠅⡀⠀⠄⣑⠲⠞⢱⣿
⠀⠈⣿⡆⢊⠴⠒⠉⠉⠉⠉⠉⠛⠺⢿⣷⣜⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡿⣣⣾⡯⠟⠛⠉⠉⠉⠉⠉⠒⢦⡙⢴⣿⠁
⠀⠀⢿⣧⠸⣄⡀⠀⠀⣀⠴⣪⠝⠁⠀⠙⢿⣷⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢨⣾⡿⠉⠀⠈⠫⣗⠤⣀⠀⠀⢀⡠⠇⣼⡿
⠀⠀⠈⠻⢷⣦⣬⣭⣥⠴⠚⠁⠀⠀⠀⠀⠀⢿⣁⠀⠛⢶⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠂⣨⡿⠀⠀⠀⠀⠀⠈⠓⢤⣬⣭⣥⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣬⣴⠛⠻⠀⣠⠤⣤⣤⠤⣄⢀⠟⠛⣦⣥⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠙⠓⠀⠸⠇⠀⠾⠋⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢘⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣆⠀⠱⣄⠈⠀⣠⠎⠀⣰⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡎⠀⠁⠀⠈⠁⡈⠁⠀⠈⠀⢱⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣼⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⠿⠣⠀⡀⠀⠠⠀⠀⢀⡀⠘⠿⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠸⣟⠿⣀⠀⠀⣀⠿⡿⠇⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⡄⢙⠀⠀⡏⢠⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⣸⠀⠀⣆⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠉⠀⠀⠉⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⡆⢰⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇⢸⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡇⢸⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢵⡼⠖⠁

⠀⠀⠀⠀⠀⣢⡤⢔⣠⠆⠲⢄⡠⡀⠀⠀⠀⠀⢈⠀⣀⠉⠆⠀⠅⣠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠂⡀⠀⠒⡤⢀⠀⠀⠀⠀⡤⠀⣀⠀⢀⡀⠃⠐⢀⡀
⠀⠀⠀⣠⠟⣁⠀⠘⣿⡃⠀⠠⡉⠻⢆⠀⠤⠞⡀⠈⢡⡘⣧⠀⣈⡒⠄⠑⡀⠀⠀⠀⢀⠄⡠⠦⡤⠤⠀⢄⢟⠉⣠⣝⡄⢀⠊⠐⠑⠚⠀⢩⣴⠌⡀⠀⠹⡄
⠀⠀⢸⠇⠘⠋⢹⠏⠉⢳⡶⡀⡁⡀⠘⡆⠠⢂⣄⣢⢰⢻⡳⡄⠀⢉⠀⠄⠃⠀⠀⠀⠈⢀⡀⠀⠇⠀⠀⠀⠜⢆⠃⠚⠘⡌⠀⠂⣀⢀⡀⠒⢂⡠⠰⠲⡄⢳
⠀⠀⠀⣐⡰⢀⡹⠶⠴⠒⠻⣧⡞⠃⠀⡇⠠⠋⣼⢱⠊⠷⠝⠂⠘⡀⠃⠀⠀⠀⠀⠀⠀⢾⠇⢨⠓⠋⣉⠡⡈⣿⣦⠀⢰⠃⠀⠘⢡⢂⡴⠀⠈⠛⠋⠳⠄⢸
⠀⠀⠰⡀⠀⢺⢳⠀⢻⠉⢷⠹⡿⣦⢠⠃⡀⠘⣁⠘⠠⢈⡠⡌⠧⡆⠀⠀⠀⠀⠀⠀⠐⣄⠀⣾⠀⢃⣹⣀⢸⡄⠛⢃⠆⠀⢆⠀⠀⠊⢁⠜⢈⢀⠀⡀⢥⡎
⠀⠀⠀⠑⢌⠛⠸⡏⠉⠉⣹⠀⢀⡤⠃⠀⠈⠢⠕⠈⠢⡌⠷⠇⠂⣈⠆⠀⠀⠀⠀⠀⠀⠈⠛⠡⠠⡼⡄⢉⠀⠤⠒⠋⠀⠀⠀⠁⠬⠭⠏⡅⠈⠋⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠷⠮⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠟⣿⠟⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠳⣿⡞⠁⠀⠐⡇⠀⢠⠆⠀⠀⠀⠀⢀⣴⣷⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⣿⣷⣄⠀⠀⣿⣤⣾⣤⣤⣴⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣮⡙⢻⣿⡾⠻⣿⣉⢕⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⣥⣤⣍⣻⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡷⡄⠴⠒⠀⠀⠒⠋⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠹⠥⢽⣿⣶⣯⣶⣦⣤⣿⣇⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢙⣿⢿⣿⣿⣿⣿⣿⣟⣧⡀⠉⠰⠂⠶⠄⠀⠠⡀⠀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠂⠳⠀⠀⢰⡇⢰⣶⣿⣿⣏
⠀⠀⠀⠀⢄⣀⠀⠀⠂⠀⣄⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠈⠀⠀⠈⠘⢻⣿⣿⠋
⠀⠀⢀⣬⣶⣷⣾⡒⣃⣸⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠈⢀⠀⠀⠀⠀⠚⢿⣿⡧
⠀⠀⢸⢿⣿⣿⣿⣷⣾⣿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⢀⡄⢸⠅⠀⠀⠀⣧⣼⡾⠇
⠀⠀⠀⠈⢿⣿⣿⣿⣽⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣆⣸⣶⣂⣠⣤⣿⣿
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠚⠁⣘⠻⣿⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡿⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠖⠊⠙⠂⠙⠿⠿⠿⠟⠙⠉⠉⠉⠻⠟

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⠀⣀⠀⠀⠀⠀⢀⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⣼⣵⠋⣭⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⡿⠿⢛⣻⣉⣛⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⣼⠳⣟⢛⣾⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣇⣈⣡⡭⡝⢿⢼⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⢤⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣉⣉⡛⠻⠎⠺⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣰⣿⡋⠓⠆⠈⠃⣀⣉⣀⠠⠤⠾⠭⠀⣽⣗⡚⢛⡊⢹⣯⣼⣿⠙⣻⣿⣿⣇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣰⢮⡤⠋⣇⠘⠃⠀⠐⠀⠀⢯⠀⠙⠦⠀⠐⣒⠚⠿⣭⣁⣸⣇⣉⣿⡶⣿⠜⣿⣿⢿⣿⣿⣿⣾⣟⣿⣿⠇
⠀⠀⠀⠀⠀⣼⡿⣈⡁⠀⠈⠁⠀⠉⡀⠨⢓⡈⠣⠤⣀⠀⡄⡛⠀⢈⣩⡥⠶⣏⡛⢘⣏⣡⠇⠸⠛⢾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢠⣿⡇⠹⣷⠀⠃⠀⢀⠀⠁⠀⠈⠷⡄⣶⡗⠀⠛⠂⠂⠘⢢⡘⠚⣉⠉⠉⠀⠟⢹⣷⠀⠃⠹⣿⣿⣻⣿⡟⠁
⠀⠀⠀⠀⢸⣿⠇⠀⠙⣷⣄⠈⠘⠁⠉⠡⡄⠀⣴⣤⡄⠀⣠⡓⠈⠶⠤⣿⣭⢙⣦⣤⢰⣦⣀⢹⣷⡀⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⢀⡾⠀⢪⠻⣇⠀⠀⠘⢀⣈⣥⣿⣷⣍⡠⣽⣿⣿⠿⠶⠼⣿⣿⣽⣷⡾⢿⣿⡟⣿⣿⣡⡿⣿⡟⠁
⠀⠀⠀⠀⠀⠓⠊⠀⠀⠈⠀⠀⠚⣤⡄⣴⡿⠛⢫⣿⣿⣷⣾⣛⣿⣿⡙⠉⠾⢿⣿⣇⣿⡌⠉⢑⣿⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡛⢁⠀⠈⠠⢂⠎⠀⠀⠈⠉⠛⠛⠻⠶⠶⠶⠶⠶⠶⠚⠛⠓⠛⣧⡘⢿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⢀⠂⠉⠉⢄⠁⢀⡾⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⢘⡤⢄⡴⠃⠀⠀⠸⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢧⡝⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉
⠀⠀⠀⠀⠀⢀⣤⡟⠁⠀⠀⠀⠀⠘⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⢷⡘⡇⠀⠀⠀⠀⡰⠊⠉⠀⡆⠀⢠⠾⠛⠁
⠀⠀⠀⠀⠀⠯⣼⠃⠀⠀⠀⠀⠀⠑⠺⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠈⠋⠉⠈⠉⠁⠀⠀⠀⠀⢰⡁⠐⠁
⠀⠀⠀⠀⠐⣄⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡤⠜⠉⠳⠤
⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠿⠒⠂⠚⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠿⠁
⠀⠀⠀⠠⠟⠉⠀⠀⠀⠀⠀⠔⠒⠀⠀⠈⠀⠀⢀⠖⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⠾⠿⠄⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⢺⡀⠀⣠⠟⠃⠀⠀⠀⠀⠀⠀⠀⠐⠦⠴⠞⠋⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡴⢾⣏⣀⣄⣤⣶⡾⢋⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣛⣻⣶⠋⠀⣠⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⡟⠛⠋⢨⣿⣿⡿⠛⢿⠙⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿⠿⢿⣿⣿⠁⠀⠀⢠⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⢹⢿⣿⣿⣿⣿
⠀⢀⣤⣤⣟⣡⣶⣷⣴⣿⠏⠐⠀⠀⠀⠀⢁⣽⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⡀⠀⠸⡅⠀⠀⠀⠛⢿⣿⣿⡿⠿⠛⢿⣿⣿⡟⠀⣠⣦⣿⣿⣿⣿
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣠⣬⣍⠉⣙⣿⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⣀⢀⣾⣿⣿⣿⣷⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣶⣤⡄⠘⠟⣻⣿⣿⣿⡿⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢰⣿⣓⣛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⢳⡀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣆⣠⣮⣊⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣷⣿⣾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠉⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀⢀⣈⡙⠛⠛⠿⠿⠿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠿⢿⣿⣿⣿⣿⣶⣶⣶⣦⣤⡤⠀⠀⣿⣿⣿⣷⣼⣿⣿⣿⣦⣄⣠⣶⣄⠀⠩⠁⣠⣄⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠙⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⢀⣤⣾⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣄⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢡⣠⣤⡞⣿⣿⣿⠀⠀⠀⠀⠀⢨⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣃⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣡⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣅
⣿⣿⣿⣿⣿⣿⣿⢿⡁⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣟⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⢿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⣿⣿⣿⣿⣿⢿⣿⣨⡇⠀⠀⠀⠀⠀⢂⣴⢾⣿⣿⡿⢫⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠈⠙⢿⣿⣿⠇⠀⠀⠈⠉⠉⠋
⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃⡄⠀⣶⠀⣾⣿⠙⡻⣭⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢺⣿⣿⡗⠀⠠⣤⣾⣿⡇⣰⡀
⣿⣿⣿⣿⣿⣿⣞⣿⣿⣷⡽⡀⣸⠀⣿⠏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣧⣠⣄⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣺⡇⠀⠀⠀⠀⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⣇⠀⠀⠀⢰⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣉⣸⣿⣿⣿⣿⣿⣿⣿⣘⡟⠁⠀⠀⢀⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠘⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠛⠉⠈⠉⠉⠉⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢠⣿⠷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠮⣿⡄⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⣿⠃⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠘⣿⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⢸⣿⠀⢀⡴⠛⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠙⢦⡀⠀⣿⡇⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⣾⠋⠀⢀⡾⠁⠀⠀⢰⠀⠀⠀⠀⡆⠀⠀⠈⢷⡀⠀⠙⣳⣿⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣷⣦⣼⣇⠀⠀⢀⡾⠀⠀⠀⠀⣿⡀⠀⠀⣸⣧⣴⣾⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠉⠛⠿⣿⣶⣿⡇⠀⠀⠀⠀⢸⣿⣶⣿⠿⠛⠉⠁⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣼⣗⣐⣴⣦⡈⢻⣿⣷⣤⣤⣤⣴⣾⣿⡟⢁⣴⣮⣁⣾⣷⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣘⣿⣿⣿⣿⣿⣿⠛⠿⢿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⣿⣥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⢃⡀⠀⠀⠹⡟⢁⠀⠀⠘⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⠛⣄⡀⠀⠀⠀⣠⠚⢛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡰⢄⠳⠀⢰⠀⠸⢆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣏⠇⠾⣿⣿⠖⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣧⠀⠛⠋⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⡿⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⢿⣿⣿⣿⣷⣽⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⢠⡄⠀⢀⣤⠖
⣾⠀⣰⠟⠁
⣿⣼⡏⠀⠀⠀⠀⠀⠀⢠⡞⠀⡀⠀⠀⠀⠀⠀⣰
⢹⣿⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢻⡄⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⣿
⠘⣿⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠻⣄⣀⣰⠟⠀⠀⠀⠀⠀⢀⣿
⠀⠹⣧⠀⠀⠀⠀⣼⠃⠀⠀⢀⣠⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠹⣧⡀⠀⣰⣿⣶⡶⠞⠋⠁⠀⠀⢿⡇⠀⠀⠀⠀⠀⣰⡟⠀⠀⣠⠇
⠀⠀⠀⠈⠻⣿⣿⡉⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢀⣴⣟⣀⣤⠾⠃
⠀⠀⠀⣷⣦⣌⠙⠿⣦⣤⡄⠀⠀⠀⢠⣸⣿⣦⣤⣶⠿⠿⠛⠋⠁
⠀⠀⠀⢿⣄⠈⢻⣆⠈⠻⣷⣤⣀⠀⣾⡟⣉⣥⠴⢴⣦
⠀⠀⠀⠘⢿⣦⡰⠿⠐⡶⠛⠉⠉⠀⠀⠈⠟⢀⣤⡾⠁
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣶⣶⠧⠸⣆⠻⠋
⠀⠀⠀⠀⠀⢀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡝⠄⡀
⠀⠀⠀⠀⠀⣾⡇⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⡿⢀
⠀⠀⠀⠀⢰⣿⣷⣌⠻⣦⣤⡴⠂⣀⣤⣤⠶⠼⣯⣿⡗⠋
⠀⠀⠀⠀⣿⡏⠻⢿⣷⣦⣤⠄⠈⠁⣼⠇
⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⣰⠀⠀⠀⠙⣿⠓⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⠶⠶⠤⣤⣤⣀⡀
⠀⠀⠀⢸⣿⣿⡀⠀⠀⣠⣾⠇⠀⡀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠙⠛⢦⣄
⠀⠀⠀⠸⣿⣿⣷⣶⣾⣿⡿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢀⠹⣿⣿⣿⣿⢸⡇⢰⡟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡷⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⢸⣆⠹⣿⣿⡇⢸⣧⣿⣧⢸⣷⡀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡌⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡎
⠀⠀⠀⢸⣿⡄⢻⣿⣇⣼⣿⣿⣿⠀⠻⠇⠀⠀⠀⠀⠀⢸⣷⠀⣀⠀⠀⠀⠀⠀⠀⣾⣿⠃⢸⣇⣸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⣉⣁⣼⣿⣿⣿⣿⣿⣿⣇⠀⠰⣶⡀⠀⠀⠀⣿⠘⢰⣿⠀⠀⠀⠀⠀⢰⣿⠋⢀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⡇⠀⠀⢀⣿⣶⡾⠉⠦⠀⠀⠀⣠⠟⠁⣠⣾⣿⡗⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣠⣿⡇⠀⠀⢸⡿⠟⠁⠀⠀⠀⣀⣨⣥⣶⣿⡿⢟⣛⣁⠻⣿⣿⣿⣧⡀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣶⣶⣶⣶⣾⣿⣿⣿⠿⠟⢫⣵⣾⣿⣿⣿⣿⣮⡛⢿⣿⣿⣦⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡘⢿⣿⡄⠹⣿⣿⡇⢀⡞⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣇⠉⠻⣿⣿⣦⣄⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⢻⡇⠀⣿⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⠻⣦⡀⠀⠙⠻⣿⣿⣶⣄⠹⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣷⠀⢿⡇⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⡈⢿⣦⡀⠀⠈⠙⠿⢉⣤⣿⡌⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⢿⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣤⣿⡏⠀⠀⠀⠀⠀⠻⣿⣿⣄⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⣬⣰⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⡀⣨⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡻⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣍⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⣯⣭⣵⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢘⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣄⡀⠀⠗⡀⠀⠀⠀⠠⡌⢀⣾⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣷⣝⢢⠈⢿⣇⠀⠀⣦⠁⣼⠹⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⣿⣶⡶⣬⣷⣄⣸⣿⣤⠀⣱⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠉⣾⣿⣿⠀⠀⠱⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⢸⠏⠀⠀⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠰⠃⠀⠀⠙⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⠀⠀⠀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣤⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⡦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⢿⣿⠃⢚⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡠⠔⠒⠒⠛⠛⠛⠛⠛⢛⣿⡟⣻⣿⡏⣿⣿⣿⣿⣏⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢾⠀⣿⣿⡷⢿⣿⣿⣿⣤⡜⠀⠀⠠⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⢠⣾⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠸⣿⣿⣿⣿⣿⡧⢣⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣿⠀⠀⠰⣿⡿⠟⠀⠀⠀⠀⠀⠀⢀⠐⠀⢸⣿⡆⠙⠟⠹⣿⣿⠁⢺⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠸⣿⠅⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣻⣿⣷⣄⠈⠾⣿⣁⣆⠀⠚⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡄⠀⢀⣿⣆⠀⠀⠀⠀⠀⠀⣛⣿⠀⢖⣧⡌⢿⣷⠀⠘⠛⠃⠀⠐⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢡⠴⣿⣿⣯⣷⣄⡀⠀⠀⢰⣿⡟⠀⡇⢹⣧⣔⠙⠃⠀⠀⠀⠀⡆⠰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣰⣶⣾⣿⡏⠈⣩⣿⣷⣤⣗⣶⣿⣿⣆⣹⣿⣿⣿⣧⠀⢀⡆⢀⣲⣧⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣠⣿⣿⢿⣿⡵⠾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⠼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠑⠹⠟⢡⣾⡟⠁⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠈⢻⣿⢿⠁⡛⠋⠀⠀⠉⠙⢻⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⠀⣠⡿⠋⠀⠀⠀⠀⣰⣿⣿⡿⢋⡽⠁⠀⠀⠀⠈⣿⡾⠀⡏⠀⠀⠀⠀⠀⠈⡿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢰⣿⡟⠋⠀⠀⠀⠀⠀⣰⣿⣿⡟⢡⠎⠀⠀⠀⠀⠀⠀⠸⣿⠀⢇⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣴⣿⠟⠁⠀⠀⠀⠀⠀⢀⣾⡟⣿⠟⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⢠⠀⠀⠀⠀⠀⢼⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠂⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣶⠀⠀⠀⣼⠿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⢆⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠀⠀⣾⣧⣸⠆
⠀⠠⠀⠀⠀⠀⠀⢰⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯
⠀⢠⠀⠀⠀⢀⣿⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣹⡇⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠰⠀⠀⠀⠈⠛⠛⠻⠀⠙⠀⠃⠀⠀⠀⠀⠀⡀⠠⡀⠧⠐⠆⠰⢿⠿⠿⠿⠃⠴⠠⢴⠀⠀⢀⣄⣸⣿⡗⡆⣀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠁

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡀
⠀⠀⠀⠀⠀⠀⡄⢸⣷⠎⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠹⣼⡇⢠
⠀⠀⠀⠀⠀⠀⣇⠸⣿⠀⠀⠀⢠⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣄⠀⠀⠀⣿⡇⢸⠄
⠀⠀⠀⠀⠀⢀⣿⢸⣿⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⣿⡇⣼⡀
⠀⠀⠀⠀⠈⠻⣿⠈⣿⡄⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢀⣿⠃⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢹⣧⣿⣷⣟⣿⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿⣻⣞⣿⣸⡟
⠀⠀⠀⠀⠀⠀⠈⢛⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣧⡙⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⣤⣤⣀⣀⡀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢀⣀⣀⣤⣤⣴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠻⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠟⠛⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠗⠀⠀⠀⠀⠺⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡦⠐⡄⠀⠀⢠⡆⢰⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠰⠗⠹



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⡉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⡴⡋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⣞⡵⣄⠀⠀⠀⣠⡴⠋⣁⠄⠀⠀⠐⠒⠦⢤⣀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠘⣆⠻⣆
⠀⠀⠀⠀⠀⠀⠀⣾⢁⣿⣿⣷⣯⣄⢀⡾⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡙⢶⡀⠀⠀⠀⠀⠘⣿⣇⢹⡆
⠀⠀⠀⠀⠀⠀⢰⡏⢸⣿⣿⣿⣿⣷⣿⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢰⡄⠀⠀⠀⣼⣿⣿⡄⢿⡀
⠀⠀⠀⠀⠀⠀⣼⠃⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣿⠀⠀⢰⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡆⠀⣸⣿⣿⣿⣧⢸⡇
⠀⠀⠀⠀⠀⠀⢹⠀⢿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣿⠀⢠⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣦⠸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⡏⠀⣼⣿⣿⣿⣿⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠹⣿⣿⣿⣿⣿⣿⣾⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣰⠏⠀⢸⣿⣿⣿⣿⠟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠹⣿⣿⣿⣿⣿⣿⣿⠿⢷⣄⡀⠀⠀⠀⠀⠀⠀⠠⢀⡾⠋⠀⠀⣼⣿⣿⣿⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⢿⣿⣉⣿⣿⡆⠀⠉⠻⢦⣄⡀⠀⠀⢂⣴⠟⠁⠀⠀⢰⣿⣿⣿⠃⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣿⣿⣿⣿⣏⠀⠀⠀⠀⠈⠙⠓⠒⠋⠀⠀⠀⠀⠀⣈⡿⠟⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣾⣠⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⣻⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⠁⠀⠀⠈⠙⠳⢦⣌⡳⢦⣄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣍⣻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣘⣃⣙⣀⡈⢹⣿⠁
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣮⣙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣻⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⣶⣤⣭⣉⣉⣉⠛⠛⠛⠉⠉⠉⣉⣉⣤⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠉⠀⠀⠀⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠹⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢢⣹⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣶⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⢀⢠⠀⠀⠀⢀⡄⡀⠜⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡽⡛⣿⣤⣄⣀⣼⣯⢟⣅⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⣾⡿⠛⠻⣿⢻⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⡡⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⠹⠃⢰⣿⡦⠤⠀⠠⠤⠄⠀⣀⣀⣀⣀⣀⡀⢀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡤⢴⣋⣟⡴⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⢰⣏⠀⠀⠁⠈⠉⠚⣿⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⠼⢫⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣄⠀⣾⠟⠀⠀⠀⠀⠀⣰⠏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠙⣡⡆⠘⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⢹⣼⣿⠀⠀⠀⠈⠀⠀⣿⡀⣸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣏⢡⡀⢀⠔⠹⣿⡄⡸⢦⡀⢸⣷⠄⢀⡀⠀⠀⠀⣈⣹⣿⣷⡙⠀⠀⠀⣼⡿⢐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⡥⠀⠜⠁⡀⠀⠻⣿⡵⠦⠁⢸⣿⣾⣿⣷⣶⣿⣿⣥⣽⡿⠿⣇⢠⡀⢀⡟⠀⡞⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⣉⠛⡞⠁⠀⠀⢸⣿⠂⡼⠉⣿⠏⢹⠀⠀⣀⠀⠉⠀⢀⡀⠼⢷⣍⢘⡏⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡝⣡⣈⠹⣏⠀⠀⠀⣸⡟⢰⣧⣤⣽⠀⢸⠋⠉⠀⢀⠒⡩⠟⠀⠀⠀⢹⡆⢻⠀⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠛⣋⡉⠐⢨⣶⠶⠻⢿⡇⡞⠀⠠⠜⣇⢸⠶⠾⠚⠋⠲⠽⠿⡷⠚⠀⢸⣇⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣩⣶⣿⠯⡴⠮⠂⠀⡀⣠⢸⣓⡂⠀⢀⢸⡘⡄⠄⣰⠀⠀⠀⠀⢀⡀⠀⣼⣼⠏⣿⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⠠⣀⣈⡁⣾⣾⠥⣠⡄⣶⣀⣇⡿⣥⣿⣧⢀⠀⠇⢼⣡⣼⢓⣎⣠⡏⣏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣦⠴⠷⠻⠟⠋⠋⠙⠂⠛⠱⠌⠉⠛⠛⠛⠛⠛⠻⠛⠛⠃⠸⠟⠆⠻⠟⢻⣍⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⡀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⣹⣥⣴⣶⣶⣶⣶⣶⣭⣄⡲⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⣾⣿⠟⣋⣉⣛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣷⡜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣰⣿⡿⢡⣾⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⡿⢁⣾⣿⣿⣎⢃
⠀⠀⠀⠀⠀⠀⠀⠰⢆⣿⣿⢁⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⠟⣰⣿⣿⣿⣿⣿⡜⡁
⠀⠀⠀⠀⠀⠀⠀⠺⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡿⢃⣤⣶⣶⣬⡙⢿⣿⡇⡃
⠀⠀⠀⠀⠀⠀⠀⠲⢸⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⠁⣿⣿⣿⣿⣿⣿⠈⣿⡇⡋
⠀⠀⠀⠀⠀⠀⠀⠸⠊⣿⣿⣧⠹⣿⣿⣿⣿⣿⢃⣾⣿⣿⡄⢿⣿⣿⣿⣿⡿⢠⣿⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠹⣿⣿⣷⣌⣛⢛⣋⣥⣾⣿⣟⣿⣿⣦⣙⣛⣛⣋⣴⣿⢏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⣿⡿⢫⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡝⠿⣿⣿⣿⣿⣿⣿⡆⣾⣿⣿⣿⣿⠿⠋⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⣽⠛⡻⠿⠿⠿⠿⠿⣛⠫⠅⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠋⠈⠋⠈⠁

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠀⣀
⠀⢠⡀⠀⠀⡔⡝⠀⡠⠜⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠣⢄⠀⢫⢢⠀⠀⢀⡄
⠀⡈⠁⠀⢸⠀⠓⢉⠴⠊⢠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠄⠑⠦⡉⠚⠀⡇⠀⠀⢁
⡸⠀⡆⢀⠜⢁⠞⠁⡠⠖⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠲⢄⠈⠳⡈⠣⡀⠰
⡇⠀⢦⠊⡰⠃⡰⠁⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠉⢆⠘⢆⠑⡴
⠱⡀⠀⢰⠃⢰⡇⢸⡠⢻⠀⡰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢆⠀⡟⢄⡇⢸⡆⠘⡄⠀⢀
⠀⢣⠀⠘⠀⢀⡧⠊⣠⠃⠀⡇⢇⠀⡕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⠀⡘⢸⠀⠘⣄⠁⢼⡀⠀⠃⠀⡜
⠀⢸⠀⢰⡔⠁⡠⠪⡈⠢⠚⢀⡎⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⢹⡀⠓⠔⢁⡕⢄⠈⢢⡇⠀⡇
⠀⠀⢣⠀⠀⡜⠀⠀⠑⡄⠀⢎⠀⢸⠀⠣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠜⠀⡃⠀⡱⠀⢠⠊⠀⠀⢣⠀⠀⡜
⠀⠀⠈⢧⠀⠳⢄⣀⡠⠼⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠈⠀⠀⠀⠀⠈⠈⠀⠀⠧⢄⣀⡠⠞⠀⡜⠁
⠀⠀⠀⠀⠑⠦⠤⠠⠤⠄⠒⠒⠚⠉⠉⠉⠉⠉⠐⠲⢄⠀⡹⠢⠴⠚⠉⠉⠉⠉⠓⠢⠖⢏⠀⡠⠖⠂⠉⠉⠉⠉⠁⠓⠒⠒⠠⠤⠤⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠁⠁⠀⠀⠀⠀⠀⠀⠈⠈⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⡀⠀⠀⠠⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠁⠀⠀⠀⠀⠀⠀⠈⢈⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠀⠀⠀⠀⠁⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⡆⠀⠀⢰⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠿⣷⣾⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡇⢸⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⣰⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡏⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣠⠿⠀⠀⠀⠀⣀⣀⣀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣐⣥⣤⠼⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⢇⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠋⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢼⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣧⣄⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⣋⣠⣼⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠷⢶⣤⣤⡘⣀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⢀⣠⣴⡾⠟⣋⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣑⢶⣶⣤⠈⢿⣿⣿⡂⣠⣄⠀⠀⠀⠠⣾⣷⣤⣾⣟⡭⢁⣠⡼⠷⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢏⡙⢿⣦⡼⢷⣿⣿⣿⣽⠒⢠⣗⠀⡼⢻⣿⣿⠿⢁⣼⣟⡻⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠞⢾⣻⣿⢋⣁⣀⠙⠂⣤⣄⢴⠄⣈⣻⠖⠉⣽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠻⢿⡿⣇⣼⣿⣿⡏⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣀⣠⣄⣹⣿⣿⡿⠧⠈⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣿⣿⣿⣿⣿⣿⠅⢀⣴⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣟⣟⣻⣿⣿⣑⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⡟⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⢩⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⢁⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣧⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀