Saber Emojis & Symbols

⠀⢀⡀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⢠⣤⠠⠄⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤
⠈⠉⢧⠉⠙⢉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⣀⠏⠛⠀⠒⠒⠒⠒⠒⠀⠀⠀⠒
⠀⠀⠀⠉⠉⠁
⠠⢦⣤⣤⣤⣤⢡⡄⠀⠀⠀⠀⠤⠤⠄⠀⠀⠂⠒⢈⠝⠁⠀⠀⠀⠀⢰⡰⠶⠶⡆⣃⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠔⠚
⠈⠚⡟⠉⠙⢁⠋⠁⠁⠀⠛⠋⠉⠉⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠧⠤⠤⠔
⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢰⣶⣤⣤⣤⣴⣷⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⠿⠛⠀⠀⠀⠀⠀⢸⠿⠿⠿⢿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠉
⠈⠉⠳⣶⣶⡿⠃⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠶⠒⠒⠶⠀⠧⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⠤⠋⠀⠀⠀⠀⠀⠐⠐⠊⠓⠒⠀⠒
⠀⠀⠢⠤⠶⠊
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠘⢯⣛⣛⣿⣿⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠠⠤⠒⠒⠒⠾⠗
⠀⠈⠛⠛⠛⠁
⠀⣀⣀⣀⣀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠈⢿⠉⠉⠉⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠈⠛⠛⠋⠸⠙⢸⡋⠀⠀⠀⠈⠁⠆⠀⠀⠈⠁⠀⠀⠙⠿⠃
⠀⠀⣀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢦⡀⠀⠃⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠛⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠙⢿⣶⣤⣀⡀⠈⠁⣭⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠉⠀⠐⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣤⠀⠉⠁⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢯⡂⡀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⡀⠀⠲⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠛⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠀⠙⣿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠰⢿⡆⠂⠀⠀⠀⣶⡆
⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡦⣴⣶⣤⡀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠈⣸⡯⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁⠀⠙⠻⢿⣷⣦⣄⡀⢠⠀⢀⠀⢱⣴⠿⢇⠀⠀⠀⠉
⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡴⣦⠦⣼⠀⠸⣯⠀⠨⢦⠄
⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠄⠀⠀⠀⠀⠀⢲⡽⢢⢲⣿⠀⢈⠂⠀⠠⠄
⠀⠀⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣰⣴⣿⡲⣬⣅⡀
⣴⣦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⣸⣿⠻⣿⣿⣿⣿⣿⣿⣦
⠙⢋⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌⠀⠛⣰⠟⣿⠷⢿⣿⣿⣿⢗⠂⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣤⣤⣶⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠁⠈⠛⠉⠁⠀⠀⠀⠈⠹⣿⣿⠿⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⢠⣤⣶⣾⠿⠟⠋⠀⠀⠀⠈⠀⢈⣿⣿⠇⠀⠀⣀⣀⣤⣴⠟⠁⠀⠀⠀⠀⠀⠛⠿⣿⣶⣤⡀
⠀⠀⠀⢸⣿⠁⠀⣀⡀⣰⣿⠀⠀⠀⠀⣼⣿⠏⠀⣴⡿⢟⣿⣿⣿⡄⠀⠀⠀⠐⢦⣄⠀⠀⠀⠹⣿⣿⣦
⠀⠀⠀⢸⣿⠀⠀⣿⢷⣿⡏⠀⠀⠀⠀⣿⡏⠀⢸⣿⠁⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⢀⣿⡏⣿
⠀⠀⢀⣿⡏⠀⢸⣿⠀⢸⠇⣀⠀⠀⠘⣿⠀⠀⢸⣿⣴⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣧⠀⠀⣸⣿⠀⣿
⠀⢀⣾⠏⠀⠠⠿⠛⠛⠻⠀⣿⠀⠀⠀⢻⣧⠀⢠⣿⠛⠛⠛⠋⠁⠀⣠⠇⠀⠀⠀⢸⣿⣷⣾⡿⠃⢠⣿
⣰⣿⠁⣀⣀⣄⣠⣤⣴⣶⣼⣧⣄⡀⠀⠀⣿⠀⠘⠁⠀⢀⣤⣴⠶⠿⠋⠀⠀⠀⢀⣼⡟⠈⠻⣦⣀⣾⠏
⣿⣿⡿⠛⡏⢹⡏⣸⡏⢀⣿⠋⠛⢿⣆⢰⣿⠀⠀⠀⠀⠾⣿⠁⠀⠀⠀⠀⣤⣶⣿⠋⠀⠀⠀⠈⢿⣏
⣿⡿⣷⡀⣧⢸⣇⣿⣇⣼⡏⠀⠀⠀⣿⣾⣿⡶⠿⣦⡴⢶⣴⢦⣄⣤⣀⠀⣿⣿⠛⠿⣦⣄⠀⠀⠀⣿⡆
⣿⡇⠙⢿⣾⣿⡿⠿⠿⣿⡇⠀⠀⠀⣿⡟⢿⡄⠀⣿⡀⢠⣷⠀⣿⠁⢘⣿⣿⣧⠀⠀⢸⡟⠀⢀⣴⡿⠁
⣿⡇⠀⠀⢹⡏⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠈⢳⣾⣿⣷⣿⢿⣾⠋⢧⠞⠁⣿⣿⡀⠀⢸⣷⣶⡿⠛⠁
⢻⣧⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⢀⣿⡟⠀⠀⠀⢿⣿⣿⣿⣆⢹⡆⠀⠀⠀⠘⣿⣷⡀⠀⠙⢿⣶⡀
⠸⣿⡀⠀⣿⡇⠀⠀⠀⢿⡇⠀⢸⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿⠀⣿⠀⠀⡰⣾⣿⣿⡧⠀⠀⠀⢻⣷
⠀⢿⣧⠀⢸⡇⠀⠀⠀⢸⡇⠀⣾⡟⠀⠀⠀⠀⣴⣿⣿⣿⠟⣠⡿⢠⡞⣿⣯⣸⡿⠃⠀⠀⣠⣿⡿
⠀⠈⣿⣆⢸⡇⠀⠀⠀⠘⣷⢀⣿⠇⠀⠀⠀⣸⣿⣿⣿⠏⣴⠟⣴⣻⢿⣅⣼⠟⠀⠀⣠⣾⡿⠋
⠀⠀⠘⢿⣾⣷⠀⠀⠀⠀⢹⣾⣿⠀⠀⠀⠀⣿⣿⣿⢃⣾⠋⠀⢹⣷⣼⠟⠁⢀⣴⣿⠟⠁
⠀⠀⠀⠈⠻⣿⠀⢸⣧⡀⠀⢻⡇⠀⠀⣤⣠⣿⣿⣯⣿⠃⢀⣴⡿⠋⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠈⠀⠸⣿⢷⣀⣄⢁⡀⡀⣿⡟⢿⣿⣿⣇⣴⡿⠋⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢻⡿⣿⣿⣿⢿⣿⡀⠈⠻⣿⡋⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣇⢸⣇⠹⣆⠙⣷⣄⣠⣿⠃⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠛⠙⠛⠛⠻⠛⢿⣭⣀⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣤⣤⣤⣴⣶⣿⣿⣿⡿⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⢀⢼⡯⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢒⣤
⠀⠀⠀⠀⢠⠏⠑⢿⣭⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣤⡶⠟⠋
⠀⠀⠀⠀⢨⠀⠀⠈⢻⣶⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⣣⣾⠏
⠀⠀⠀⠀⠐⣄⠀⠀⠀⠻⣧⣜⣶⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠈⠐⠦⢶⣠⠛⢋⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠘⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⢈⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠠⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠀⠁⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠊⣻⡷⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠓⢄
⠀⠀⢀⡀⠀⠀⠀⢠⠤⠤⠖⣚⡉⠠⠄⠒⠉⠁⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣶⣦⣤⣾⣿⣭⣵⡆
⠀⠀⠀⠸⣷⠋⠉⡟⠙⢿⢷⣭⣐⡀⠄⡀
⠀⠀⠀⠀⠙⠤⢴⢃⠤⠏⠀⠀⠉⠙⠛⠶⢶⣥⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣀⣀⣀⣤⣔⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠿⠶⠶⠾⠿⠯⠭⠭⠤⠴⠞⠛⠉
⠀⠀⠀⢻⣟⡟⠛⣿⢿⣿⣷⣤⣀⡀
⠀⠀⠀⠈⢿⡀⢐⡇⢀⡜⠉⠛⠻⠿⣿⣶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣷⣶⣦⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠰⣷⣦⣶⣶⣭⣓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣽⣶⣦⡶⡷
⠀⠀⠀⠀⣧⠇⠈⡽⠛⢿⣟⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣚⣿⠿⠛⣍⠀⢣⡇
⠀⠀⠀⠀⠙⠦⣼⣃⣠⠎⠈⠛⠷⣮⣔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣬⡶⠟⠋⠈⢦⣀⣹⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣴⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣦⣤⣤⣶⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡿⠿⠛⠻⠿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⣔⣲⣦⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠿⣧⣴⣒⡤⠤⠤⣀⣀⡀
⠀⠀⠀⠈⠛⠛⠓⠒⠒⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠒⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠰⡖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣄⠃⢀⠀⡰⣾⡎⠢⠄
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡶⠾⠂⠈⠃⠸⡿⢿⢿⣿⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⡠⢄⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠀⠠⠟⢹⠀⠐⣆⢀⠀⢀⡠⡄⠙⠛⠞⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⣤⣄⣀⠰⢄⠏⢠⠔⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡄⢀⠀⠀⣄⢄⡏⠙⠓⠒⠲⡴⠚⠉⣿⣿⠳⣄⢢⣀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡃⢠⢸⡎⠀⠀⠀⣠⠜⡙⠒⢦⣌⢿⡆⠈⠃⠀⠀⠀⠀⠀⠀⠹⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣤⠀⣠⠡⢿⣽⣇⠀⢠⠞⠁⠀⠃⡗⠀⠉⠻⣿⣦⡔⣤⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢡⣤⣸⣾⣤⡝⣁⠶⣏⠈⣄⢠⡀⢇⡠⠠⠂⡼⡿⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⣫⣽⡿⢷⣿⢌⢉⣟⣠⠿⣿⣟⡿⠷⡠⢾⢰⣿⠀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠄⢜⡀⣁⣸⣇⡾⠞⢬⣳⣟⣁⣌⣼⣿⣟⣿⠀⢶⣤⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠰⠌⣯⣥⣶⡟⢿⣤⡜⣿⢿⠷⡆⡏⠻⣿⣿⠻⣧⣻⠌⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣤⢂⠄⠠⠀⠠⡈⠀⡆⢘⣾⠄⠹⣿⣿⣸⡃⣸⠃⠀⠈⢿⣿⣻⣿⡀⢈⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠻⣸⣿⠶⣧⣺⣿⣴⣾⣯⣴⢾⣿⣿⣿⣿⣷⡆⣄⠈⣿⣿⣏⣱⣞⠈⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡠⠀⢠⡵⣿⣫⣿⣿⣿⣿⣿⣿⣗⣻⣿⡽⣿⣿⣿⠄⠉⣇⠈⠙⠻⢿⡌⠁⣠⠘⠳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠈⢫⣿⣿⣿⡾⣿⣿⠯⠟⣫⣤⠴⠿⣿⣿⣷⠸⡏⢳⣄⠀⠈⠛⢷⡞⢦⣄⠈⢢
⠀⠐⡲⡤⠂⠀⠉⠙⣿⣯⠟⠀⠀⠀⠀⠀⣿⣿⣿⠋⢰⡇⠀⠀⠉⠀⠀⠀⠀⣿⣿⣿⣷⡌⠺⢿⣷⣀⠀⠀⠙⢿⣿⣧⣤
⢠⣞⠃⠀⠀⢀⣀⢲⣶⣏⠀⠀⠀⠀⢠⣾⣿⣿⡇⢀⣿⣿⡂⡀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⡄⠀⢨⠛⢧⡀⠀⠀⠙⣿⡆
⣿⣿⢀⠀⠞⢢⠝⠋⠀⢸⣶⡀⡄⢀⣿⣿⣟⠏⢃⡞⠘⣷⣠⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⣿⣦⣶⣳⣾⣿⡆⠀⠀⠈⠉
⣿⣷⣤⣃⠔⠁⠀⢠⣶⣿⣿⣿⣿⣾⢿⣿⡟⣠⣾⠃⠀⢹⣿⣷⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣧⣀

⠱⢦⡀
⠀⠀⠉⠶⣄
⠀⠀⠀⠀⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢧⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣴⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣼⠧⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣈⠉⡙⠛⢻⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡖⢶⣤⣈⡘⢲⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣧⡜⢛⣉⣁⠙⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣡⠿⠉⠱⠆⢠⠉⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠙⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⠳⣤⣰⣤⣄⠀⢙⡗⣦⣀⠀⠀⠸⡶⠠⠀⢀⡙⡀⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠙⠻⣾⣷⣆⠀⠜⠀⠉⡓⠲⠀⢶⣶⣿⡾⠟⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣶⣄⣈⡄⡐⣨⠏⠁⠀⢈⡤⠤⠬⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⡟⠐⠁⠀⠀⢠⡏⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠈⣴⠛⠃⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⣿⣾⡿⠗⠁⠀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡀⠀⠀⠀⠀⠀⡎⢹⣿⠟⠻⠿⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣰⡥⠚⠁⠠⠴⠂⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⢀⣏⠀⢀⠄⠀⠂⢀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⡌⠻⣆⠀⣀⡼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠁⠀⢸⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠀⠀⠀⠀⠐⠋⣴⡞⢻⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠣⠤⠀⠠⢤⣤⢶⣻⠀⠤⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⠒⠶⢶⣦⠀⢾⣿⡶⣶⡚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢹⡀⠈⢻⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⠀⠇⠀⣿⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⢣⠀⠀⢉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠀⠿⣦⡜⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⢣⡀⢻⡄⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠹⣾⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠸⣇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡄⢻⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡓⠨⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢰⡄⢸⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠷⠦⢸⠀⢸⠀⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣭⣿⣄⠀⠌⠀⠆⢠⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣯⣥⣾⣿⡶⢶⣤⠾
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣧⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣼⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠔⠉⠻⣿⣷⡙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⣾⣿⠟⠉⠢⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡰⠋⠉⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁⠙⢆⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠞⠁⠙⢿⡗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⡿⠋⠈⠳⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⢀⠀⠀⠀⠀⠀⠠⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠁⠀⠀⠀⠊⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠕⠀⠀⠈⠢⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡪⡂⠁⠀⠀⠀⠀⠀⠈⢐⢕⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣔⣵⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣮⣂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣔⣭⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣿⣶⣄
⠀⠀⠀⢀⡀⠀⠀⣀⣠⠴⣶⣴⣀⠀⠀⠀⠀⣀⣤⠼⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠿⣿⢶⣾⣿⣿⣷⣦⣽⣏⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⢸⣇⡾⠀⠀⡾⠋⠻⣿⣿⣿⣶⠢⠄⢀⡀
⠀⠀⠀⠈⠻⣧⣀⣸⠇⢀⣰⠇⠀⠉⠙⠛⠷⢶⣦⣥⣀⣐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠉⠛⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠠⠀⠀⠀⠁⠀⢐⣂⠠⠤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠐⠒⣂⣭⣥⣭⣤⣒⣒⣂⣤⠤⠤⠀⣀⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⣠⣴⠿⠛⠛⠉⠛⠛⠿⣶⡶⠶⠶⣶⣶⡿⠟⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⡉⠉⠛⠛⠿⡟⠁⠠⠴⠶⣿⣻⠖⠂⠈⠀⠀⠀⠀⠉⢷⡦⢠⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠙⠳⣦⣀⠀⠀⠀⠀⠀⣀⣤⡤⠶⣶⡄⠀⠀⠠⠀⠀⠈⢷⣼⡴⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡶⠛⠋⠀⠀⠀⠀⢈⣡⠴⠒⣋⡿⣷⣄⣀⣀⣀⣀⣤⣾⣅⡒⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⢀⡄⠀⠀⠀⠉⢀⡴⠋⠁⢀⣾⣿⡟⠉⢉⡉⣉⣉⡉⠙⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⠀⠸⣿⣿⣿⣧⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣤⡤⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣼⠟⠿⣟⠻⣿⡀⠀⣿⡿⢻⣿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⣸⣯⣄⡀⠘⡆⢿⡇⠀⣿⠀⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠃⢸⡇⠀⣿⠀⢸⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣶⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄⠀⠀⠈⢿⣿⣿⠀⢠⠸⣿⢸⣿⠀⢸⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣴⠂⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠈⢿⣿⣦⠟⣷⣿⣾⢿⡇⢸⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢻⣿⠀⣠⣴⡏⠀⠀⠀⣀⣀⣈⡙⣷⡄⠈⣯⡋⠀⠿⠿⠃⣼⡇⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠋⣿⣥⣶⡾⠿⠛⠛⠛⠛⢿⣿⡀⠘⠛⠛⠛⠛⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡈⠉⠉⠁⠀⠀⠀⠀⣠⣄⣀⡄
⠀⠀⠀⠀⠀⠀⡀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠛⠛⢛⠀⠀⠀⢀⣤⡶⢾⢋⠍⠁⠀⠀⠀⠈⠀⣀⡀
⠀⠀⠀⠀⠀⢸⠃⠀⣹⠀⠀⠐⠚⠙⠿⠛⠀⠀⠈⠀⢦⣀⠀⠀⣴⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡈⣛⠿
⠀⠀⠀⠀⠀⣸⡱⡄⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢤⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⡿
⢀⡀⠀⠀⠀⠨⢹⡟⠓⢂⠀⠀⠀⠀⠐⠒⠀⠾⠿⢿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡿⣫⣿⠁
⠁⠀⠀⢂⢸⠛⠙⠛⠛⠀⠀⠀⠀⠒⠐⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇⠀⣪
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣶⡶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢉⣉⡉⠋⠁⠐⠿
⠀⠀⠀⠀⠉⢧⠀⠀⠀⠀⢸⣿⠀⠀⠉⠙⠛⠿⣿⣷⣶⠀⠀⡀⠀⡀⠀⠀⠀⠀⠐⠦⢤⣌⣉⠉⢹⡆⣴⡒
⠀⠀⠀⠀⠀⠘⠗⠀⠀⢾⠹⣿⡆⠀⠀⠀⠀⠀⠀⠉⡋⠀⣠⢇⣸⡿⣿⣶⠂⠀⠀⠀⢀⠀⠉⠛⠋⠀⠻⢭
⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢨⣻⣿⣄⡀⠀⠀⠀⢀⣀⣈⣏⣠⣾⠟⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣶⣤⡄⣀⣀

⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠤⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⢀⡔⢂⡠⠤⣶⡇⠀⢀⠀⠀⠀⠀⠻⢡⠆⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⡔⠁⠴⢹⡖⠆⠘⠀⠀⠘⠢⠀⠀⠀⠀⠁⢀⠀⠀⠀⣀⠀⢀⠀⠀⣿⣿⣿⡟⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠁⠀⢐⣻⣷⣌⢻⣶⣼⣿⣿⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⣤⢪⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⢦⠀⠀⠀⠀⠙⠿⣿⣿⣾⣷⣽⢿⡏⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢱⠀⣴⣦⢤⣄⣈⣙⣻⣏⠩⢯⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡳⢤⣀⣿⣿⡿⣷⣌⠻⢿⣿⠿⡷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠉⠒⠀⡄⠐⣾⣿⠿⣆⠀⠙⠿⣝⠈⠙⠿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠁⠀⠘⣡⡾⠛⢧⡀⠀⠈⠢⡀⠀⠈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠠⠀⠐⢄⠀⠀⠴⣄⠀⠀⠀⠀⠀⠀⠉⠙⠃⣼⡁⠀⠀⠀⢙⣆⠀⠀⠑⣄⣀⡀⠈⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢉⢀⡔⣀⣀⣤⣴⣶⣿⣿⣿⢃⣴⡅⠀⣠⣤⣾⣿⣿⡇⠀⢠⣿⣿⣿⣷⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢁⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠉⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠴⠖⠚⠋⠉⠀⢀⣀⡤⠴⠚⠉
⣠⠖⢛⣛⣉⣉⣛⣛⣛⠋⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁⢀⣀⣀⣀⣤⠤⠴⠖⠛⠉⠁
⣇⠀⣿⠉⠉⠉⠉⠉⠈⢹⣶⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⠁
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⢻⣶⡄
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠞⠛⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢀⠀⣠⣄⢡⣄⡀⢀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠚⢳⠀⠀⠀⢹⣿⢸⡟⠀⠀⠀⢰⠟⠐⠶⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠉⠀⠀⠀⢀⣀⣼⣧⣶⡾⠎⠁⠈⠉⠷⣶⣦⣿⡴⣀⠀⠀⠀⠈⢉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢰⣿⠛⠉⠉⡀⢀⡀⠃⠈⢃⡀⢀⡉⠉⠘⢻⣧⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⣷⠀⠀⠀⠙⠓⠖⠢⡔⠲⠶⠟⠀⠀⠀⢸⣿⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣿⠀⠀⠴⡄⡀⠀⠸⠇⠀⠀⠀⡄⡀⠀⢸⣿⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⣿⠀⠀⠀⠻⣄⡀⠀⠀⠀⣡⠞⠀⠀⠄⣾⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠑⠀⠄⠀⠈⡈⠳⡁⡱⠞⠉⠢⠀⠀⠀⠙⠇⠀⢸⠁⠀⠀⠀⠀⠀⠀⡠⣴⣷⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡴⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⢀⣀⣠⣄⣤⡄⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠀⠀⢀⡀⠀⠀⣿⡇⠀⠀⠘⠛⠿⣿⣿⣷⠀⣿⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⡀⠈⠀⢠⡾⠃⠀⠀⠿⠃⠀⠀⠀⠠⠀⣸⣿⣿⣀⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣿⣶⡶⢉⡤⠦⣄⠀⠀⠀⢠⠔⠒⠀⠸⣿⣿⣏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣠⣤⣤⣤⣤⣤⣤⡀⣬⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠙⠛⠛⠛⠛⠛⠋⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠁⠀⠀⣠⡆⠀⠀⠰⢦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠙⠓⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠉⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⢀⣼⣿⠟⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄⢠⣿⣿⠃⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠃⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣄⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢀⣠⣴⣾⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣏⢻⣿⣷⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣿⣿⠟⠉⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠙⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠃⠸⣿⣄⡀⠀⠀⠀⠀⠀⣸⣿⡿⠋⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⣿⣿⢿⣷⣦⣤⡀⠀⠀⠀⠸⣿⣷⣄⣀⣀⣼⣿⠇⠀⠀⠀⣀⣴⣿⣿⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢀⣿⡟⠀⠉⣿⣿⠁⠀⠀⠀⠀⠘⠻⣿⣿⣿⠿⠋⠀⠀⢠⣿⡟⠋⢻⣿⡀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠺⣿⣷⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣼⣿⣷⡄⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠭⠉⠉⠉⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⣤⣬⣭⣽⣿⣿⣿⣶⣾⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣍⠶⢆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡿⡟⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠠⠀⣼⣿⡇⠠⠤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⢻⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠛⠛⠁⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡣
⠀⠀⠀⠀⠀⠔⡀⠀⠀⠀⠀⠀⡀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠀⣦⣤⡔⢀⢴⠦⢀⡀
⠀⠀⠀⡠⠆⢀⠀⡶⠀⡄⠀⠄⠀⠄⠄⠲⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⡇⠈⠁⢠⣳⣏⠞⠉⠙⠂
⠀⠀⠀⣥⣘⡀⠠⠀⢁⠀⠂⡀⠀⠠⠀⢐⡠⠀⣀⡰⠀⠀⠠⢄⠀⠹⠇⠀⢠⢳⢧⣟⠀⠀⠀⠀⠂
⠀⠀⢰⣿⣿⣿⣿⣿⣶⣶⠖⡂⠀⠁⠹⣿⣿⣤⡤⠫⠁⠰⡦⡀⠑⢌⡄⢀⡏⣿⣼⡫⠄⠀⠀⠀⠰⡀
⠀⠀⣼⣿⣿⣿⣿⣿⡿⣿⠀⠁⠀⠀⢣⠘⣿⣿⣶⠶⠆⠀⠇⠈⠦⣾⣿⣼⣟⣽⣿⣯⠅⠐⠶⣤⣬⡅
⠀⠀⢿⣿⣿⠁⢰⠁⠀⡾⠘⠿⠿⠿⠿⣧⠈⣰⣤⠘⠀⠀⠠⠀⣾⣿⣏⣹⣾⣿⡿⠋⠀⣀⠔⠉⠁
⠀⠀⠘⣿⣇⠀⢾⢈⣿⡇⠀⠀⠀⠀⠀⢻⡇⠹⡇⠂⠀⠀⢠⠀⢻⣿⣿⣿⡿⠟⠁⣠⣾⣿⣆
⠀⠀⢸⣿⣿⣄⣀⣼⡿⠁⠀⠀⠀⠐⠀⠘⣿⠈⠉⠀⠀⢀⣶⡆⢠⣿⣿⡟⠁⢀⣺⣿⣿⣿⣿⡆
⠀⠀⠿⠿⠿⠿⠿⠿⠁⠦⠤⠤⠠⠤⠤⠴⠿⠇⠀⠀⠀⠼⠿⠿⠿⠿⠿⠷⠶⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⣠⣶⠃⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⢰⣿⡇⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⣿⠀⢠⣿⠀⠀⠈⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣾⡏⠀⠀⢸⣿⣇⢸⣿⡄⠀⠀⠘⣧⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢸⣿⡇⠀⠀⢸⣿⣿⡆⠻⣿⣦⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣦⣤⣾⡿⣿⡇⠀⢹⣿⣧⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠸⣿⣷⡙⠛⠛⣵⡿⠁⠀⠀⣿⣿⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⢿⣿⢀⣾⡟⠁⠀⠀⢀⣿⣿⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣾⣿⠀⠀⠀⠀⣼⡿⢹⣿⠏⠀⠀⢰⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣦⣴⣏⠀⠀⢀⣿⣿⣿⣤⣀⣤⠞⡉⢀⣿⡏⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠾⣿⣿⣦⣤⣾⣿⢻⣿⣿⣿⣿⠀⡇⢸⣿⡿⣶⣾⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡛⢿⣿⠀⠻⣿⣿⣿⠀⡷⠄⢀⣴⡿⠉⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⣧⣀⠁⠰⣆⠙⠛⠋⢰⠁⣴⣿⣿⡀⢀⣠⡿⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⣠⢀⣿⣿⣷⣦⡈⠻⣶⣶⣿⠀⠻⣿⠟⣿⡿⠋⠀⠀⠀⣬⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⢹⣿⣧⠀⢹⣿⣿⠷⠖⠀⣰⣿⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡙⠛⠋⠁⢸⣿⡏⣴⢿⡟⠁⢀⣴⣾⣿⣿⣀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣄⠀⠘⠋⢸⣿⠘⠀⠀⣿⣿⣿⠟⠛⠛⠋⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠘⡆⠀⣿⡀⠀⠀⠈⠉⣡⠖⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣰⣿⠇⢀⣵⠀⣽⠹⣦⣤⣤⣾⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡟⠀⣾⣿⢾⣿⡀⢻⣿⠏⣿⡿⢸⣿⣧⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠄⠘⠿⠄⢉⣤⠆⠁⣰⠟⠡⠾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢋⡏⡄⠀⠔⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡰⠁⠀⠐⠀⢠⠇⠀⡡⢢⡄⢀⡴⢊⠭⢍⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⡴⢡⠀⡇⡆⡆⡿⠀⠸⣇⠀⠀⠘⠀⣇⣶⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣇⠰⡀⢧⢸⠸⣿⡀⢆⠉⠳⣄⠐⡄⠻⣝⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠁⠈⠢⣳⡈⢆⢣⠻⣿⣌⢧⢀⠘⣷⠸⡄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠪⡅⠀⠀⠉⡡⡄⢱⣌⢿⣾⣷⣧⠹⣿⠘⣾⣦⣿⠇⡇⣈⠀⢸⡆⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣺⢩⣛⢦⣿⣿⣿⣶⣿⢀⣿⡿⣫⠞⠁⡡⢀⣼⠃⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠠⡸⠖⠀⢸⠂⠹⣿⣿⣿⣇⣾⣯⡟⢁⣴⣿⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⢀⠀⠀⢤⠀⢿⡄⢆⠹⣯⣿⣿⣿⣿⣷⣿⣴⣆⡀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣋⠍⢹⠒⡟⣰⠺⢿⣶⣷⣿⣿⠟⠛⠋⠉⠉⠁⠱⠉⠃⠙⠒⣶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠿⠃⣼⠀⠀⠁⠐⣠⠼⠛⠁⠁⠀⠠⠀⠀⠀⠰⢆⠀⡀⠀⣼⢡⢺⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠷⠶⠶⣧⠤⢤⣞⣭⣀⡀⠀⠀⠀⢀⡴⣋⠀⢤⠶⢦⠶⢾⣧⣷⡘⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡗⠀⣤⡤⡀⢀⢸⡀⢀⢮⣹⡆⠀⣨⣾⠛⢿⣃⡏⠠⠄⠀⣼⣿⡍⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢉⡇⢀⡛⡇⠀⣀⣸⣧⣐⣊⣿⢷⣿⡁⠠⠀⠘⣿⡷⢦⠤⠼⠋⢙⡃⢁⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠲⠶⢒⣻⠿⢛⠅⠈⢛⢽⠞⠻⢿⣴⣶⠟⠃⠐⠀⠁⠀⠃⠀⠀⠴⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⣇⢠⡿⣦⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⣧⠀⢰⠆
⠀⠀⠀⠀⠀⢸⠚⣿⠉⣛⡏⢻⡟⠞⠀⠀⠀⠀⠀⠀⠀⣆⢸⣇⢰⣇⢰⡇⣰⠀⠀⠀⠀⠀⠀⠘⢧⣼⣷⠟⡙⢷⣾⣤⠟
⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣷⣾⣷⡇⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⡟⢻⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣶⣷⣶⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠇⠠⠿⠾⠸⠣⠷⠈⠳⠀⠀⠀⠀⠀⢤⣼⣿⣿⣉⣿⣽⣽⣿⣹⣧⡤⠀⠀⠀⠀⠐⠌⠸⠦⠆⠟⠞⠆⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠆⠄⠀⠐⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠀⢀⡀
⠀⠀⠀⠀⠀⢶⡄⢀⣹⢤⢾⡁⠀⣴⠂⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣷⡠⠀⠀⠀⠀⠀⠀⠀⠰⡀⣸⣄⢼⢄⣼⡀⡰
⠀⠀⠀⠀⠀⢘⣾⣁⣹⣿⣿⣁⣹⡟⠀⠀⠀⠀⠀⠀⠀⣴⣩⣍⣯⣼⣭⣭⣥⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣤⠶⣢⣬⣽⡇
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣉⣉⣉⣁
⠀⠀⠀⠀⠠⡆⢸⠿⠾⠸⠩⠿⠈⢳⡄⠀⠀⠀⠀⠀⠐⠏⠹⠷⠆⠞⠾⠌⠳⠄⠀⠀⠀⠀⠀⢶⡿⠽⠯⠽⠿⠯⠽⠯⠿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠀⠀⠀⠘⠀⠀⠀⠘⠉⠃

⠀⠀⠀⠀⠀⠀⢀⣀⡀⢼⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢠⣿⣿⢿⣿⡿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣠⣿⣧⣼⡄⡀
⠀⠀⠀⠀⠀⠀⢻⣇⣀⣿⣀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⢹⣿⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣽⣅⣿⣇⣽⣭⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠛⢛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣾⣯⣷⡿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢛⠛⠛⠛
⠀⠀⠀⠀⢀⡄⢠⣤⣤⢠⢄⣄⠤⣄⠄⠀⠀⠀⠀⢀⣰⣶⣶⣒⣶⣲⣲⣶⢲⣆⣀⠀⠀⠀⠀⠀⡠⢠⣠⣠⢠⣤⣤⠠⣄
⠀⠀⠀⠀⠀⠁⠈⠉⣉⠈⢈⠈⠀⠈⠁⠀⠀⠀⠀⠼⠿⠍⠉⢉⠉⡍⡉⠉⠩⠿⠧⠀⠀⠀⠀⠀⠓⠈⠉⠉⢈⠈⡉⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁

⣿⠿⢿⢷⢿⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⡿⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⢴⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡶⣠⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠬⢇⡠⠁⢦⣀⠘⣦⡀⠀⣰⣧⠀⠀⣠⠖⠀⣀⡌⢠⡸⡯⠘
⠀⠀⠀⠀⠀⠀⠀⢰⡤⠶⣟⢃⡄⠀⠙⣷⣿⣿⣶⣿⣿⣷⣿⣿⣴⠟⠁⢠⡘⣛⠷⠦⣖
⠀⠀⠀⠀⠀⢀⡀⠋⠰⣦⣉⠌⣿⡶⣦⠈⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣶⣿⠃⣌⣱⠆⠸⢀⡀
⠀⠀⠀⢀⡀⠬⣻⠎⠲⠀⣼⣏⡺⣿⣿⣇⠘⠛⠉⠉⠉⠉⠛⠃⢨⣿⣻⡷⣙⣯⡀⠔⠱⢟⠥⠀⡀
⠀⠀⢴⡻⣿⣾⣇⡚⢷⣌⣀⡝⠁⣿⣏⣙⠦⣄⡀⠀⠀⢀⣀⡴⢋⣙⣿⠀⢻⣄⣈⡴⢛⣸⣷⣿⠟⡷
⠀⠀⠀⠈⠋⢿⡨⠅⠈⢿⢌⣁⣽⣠⡽⡉⠿⢁⡉⠀⠀⢈⣁⢿⠉⢹⣥⣻⣈⡡⣾⠃⠀⣍⡾⠏⠃
⠀⠀⠀⠀⠀⠉⠓⠀⠐⠚⠘⠂⢀⢮⡄⢹⡦⣾⣣⡄⢠⣌⣳⢴⡞⠀⣭⠄⠀⠂⠙⠂⠀⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠿⠏⠙⠃⠀⠉⠁⠈⠉⠁⠈⠋⠹⠿⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣦⣤⣖⢻⡟⣲⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⡀⠀⠀⠀⠀⣷⣿⣿⣯⣻⣿⣿⣿⣟⣟⣼⣿⣗⡿⡀⠀⠀⠀⢀⡀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣈⠀⢠⡀⠀⢿⣿⣿⣦⣶⣶⣶⣾⣷⣶⣶⣶⣼⣿⣿⡿⠀⢀⡄⠀⣀⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡉⣛⠓⢢⢹⡆⠀⢻⣟⣿⣭⣽⣧⣼⣧⣼⣯⣽⣿⣻⡟⠀⢰⡏⠰⠚⣛⡉⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣟⠁⠠⡿⢿⣃⡼⢁⣤⣻⠿⠿⣿⠋⢻⠙⠁⡋⠙⣿⡿⠿⣟⣤⡘⢇⣛⡿⢿⠄⠈⣛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡳⣶⣞⣼⣿⣷⣿⣛⠅⣐⣰⣤⣤⣰⣴⣷⣦⣤⣤⣆⣂⠘⡋⣿⣮⣿⣧⣳⣶⢞⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣶⢂⣾⣿⡍⡿⣯⣿⣿⣿⣿⣳⣾⣿⣿⠋⠉⠁⣙⣋⠀⠉⠙⢿⣿⣿⣗⣿⣿⣿⣿⣽⢿⠩⣿⣷⡐⡦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣽⣸⡁⠋⡶⢱⣿⣿⣿⠟⠋⣍⣉⣾⣧⣀⣠⣀⠈⠁⢀⣄⣀⣼⣷⣉⣭⠹⠻⣿⣿⣿⠞⢿⠙⢘⣇⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣯⣷⠀⠀⢸⣿⣿⣟⣰⡿⢉⣿⣷⡄⠘⡛⡇⠀⠀⢸⢛⠋⢠⣾⣿⡙⢿⣦⣻⣿⢻⡇⠀⠀⣾⣽⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣧⣿⡀⣤⣼⣧⣿⣟⣏⣁⣩⣴⣿⣽⣾⠃⣀⠀⢀⣀⠘⣵⣯⣿⣮⣉⣉⣹⣻⣿⣿⣧⣦⢀⣿⣼⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣬⡁⠸⣇⣿⣿⡝⢻⣿⠀⢸⣿⣿⣿⠀⠙⠳⠞⠋⠀⣿⣿⣿⡏⠀⣿⠿⢩⣿⣿⣸⠇⣨⣥⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠈⢲⣤⣹⠿⡄⠙⣧⣿⣿⡴⢲⣦⡼⡟⢻⣿⣶⠤⢾⣷⠦⣴⣿⡟⠻⠧⣴⡶⢦⣾⣿⣼⠋⢠⠟⣏⣤⡞⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⣙⣷⣦⣿⣀⣿⣿⣻⣧⢺⠋⠀⢁⡾⡿⣿⣦⣤⣠⣴⣿⠿⢳⡀⠀⠹⡇⣸⣟⣿⣿⢀⣿⣵⣿⣋⣡⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⡼⠟⢻⣿⣿⣿⣿⣽⡌⣿⣏⠘⣧⡀⢻⡿⢿⡿⢿⡟⢀⣾⠃⣹⣿⢡⣯⣿⣿⣿⣿⠟⠻⢧⣤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠉⠻⠿⡿⣿⣿⣿⣿⣿⣿⣬⣄⣢⡹⠟⢳⣤⣴⠞⠹⢏⡜⣰⣥⣿⣿⣿⣿⣿⣿⢿⠿⠟⠉⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⣧⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣾⣅⡈⢁⣨⣷⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⣦⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠘⣿⣿⣿⣟⣿⣿⢹⡿⣿⣿⣿⣿⣾⣿⣿⣿⢿⡏⣻⣿⣻⣿⣿⡿⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢴⣤⣸⣿⣿⣿⣯⣻⣿⣷⣿⣿⣿⡇⢸⣿⣿⣿⣾⣿⣏⣿⣿⣿⣿⣃⣤⡦⠀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣃⣶⢿⣟⣿⣿⣿⣶⣿⣾⣿⣟⠻⠃⡘⠿⣻⣿⣷⣿⣾⣿⣿⣿⣻⠿⣶⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣿⣾⣿⣿⣿⣿⣯⡄⢉⡏⢀⣿⣿⣿⣿⣿⡿⣿⣤⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠋⠁⠈⠻⢿⣿⣷⣼⣦⣴⣧⣾⣿⡿⠟⠁⠈⠙⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠟⠛⠶⣄⢳⣸⢃⡴⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡏⠀⢀⣠⠼⠎⠧⠯⢤⣀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⣿⡟⠛⣉⣀⠀⠾⠿⠆⢀⣉⠛⠻⣿⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢀⡟⢀⣶⣿⣿⡿⢣⣆⢿⣿⣿⣵⡄⢻⡄⠀⠈⡿
⠀⠀⠀⢀⣠⣶⡾⠶⠶⣤⣈⠳⢄⣾⢀⡀⢻⡿⢀⣀⣿⣿⣆⠀⢿⣿⢁⡀⢻⣠⠞⣁⣤⡶⠶⢶⣶⣤⡀
⠀⠀⠀⢿⣿⣯⡀⠀⠀⣀⣨⠽⢗⣂⢸⡿⢀⡺⢿⣿⣿⣿⣿⣿⠷⣃⢺⡗⢈⡲⠯⣍⣀⠀⠀⢀⣸⣿⡿⠂
⠀⠀⠀⠀⠉⠛⠻⠛⠛⠉⢠⡞⠁⢿⡀⢀⣾⣿⣀⡀⣿⡿⢀⣀⣾⣿⡄⠀⡼⠉⠳⡄⠉⠛⠻⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣄⠀⠈⣷⡀⠛⣿⣿⠿⠞⠳⠿⣿⣿⠛⠃⣼⠃⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣿⡷⠶⣤⡀⠀⠻⠟⠂⣀⣤⠶⢶⣿⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠉⣹⢳⡞⢏⠉⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠛⢡⡏⢹⡌⠛⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠀⠄⠀⠀⠘⢌⠢⠀⠀⠰⠀⠠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠊⠀⠘⡠⠀⠀⠀⠠⠑⠀⠀⠀⠐⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠂⠀⠀⠀⠀⠀⣄⠀⠀⡰⠁⠀⢹⠀⠀⠀⣀⠀⠈⠀⠀⠀⠄⠁⠄⠘⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⢀⠐⢀⠁⠳⠉⢦⣠⠁⠀⠀⢸⠀⠲⢀⡶⢱⠀⢀⠀⠸⠎⠀⠀⡄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⡉⠀⠀⠹⡀⠀⠀⠸⠀⠀⠰⠁⠀⡇⠈⠴⠃⠐⡆⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢾⠣⣄⣠⠀⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠀⡇⠂⠀⠀⢠⠃⠀⠀⣀⢀⡀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠈⠣⡄⠀⠀⢇⠀⠀⢠⢃⣄⣀⣤⡔⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡞⢈⠼⠊⢰⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠙⠀⠀⠀⠀⠀⢄⠀⠠⠀⠀⠀⠀⠀⢀⠔⢠⠀⠀⠱⠁⠀⠰⠁⡾⠁⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣄⣅⣀⠀⠀⠀⠀⠀⠀⠃⠄⠂⠖⠄⠀⡀⠈⠠⠄⠂⠀⠀⠠⠴⠋⠉
⠀⠀⠀⠀⠀⠀⢐⡖⣻⣉⣉⠉⣩⣽⠶⣀⠀⠀⣀⣠⠤⢤⣀⡄⡦⠀⡠⢦⠐⠠⣤⢄⠂⠠⠾⠿⢷⣿⡂⡀⢰
⠀⠀⠀⠀⠀⠀⠹⠷⠼⠷⠶⠚⢧⡇⢠⣾⣶⣛⡻⣄⣵⢠⣼⠃⣓⢲⠀⢘⣧⡂⠚⢺⣇⣀⡀⡶⡞⣮⣭⠴⣾⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠙⠛⠛⠁⠀⠙⠶⠖⠛⠉⠙⠳⠚⠋⠀⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣰⡿⠋⠛⣧⠀⠀⠀⠀⣿⠋⠀⠀⠙⣷⠀⠀⣾⡏⠉⠙⣷⠀⠀⢰⡿⠋⠉⢿⡄⠀⠀⣴⡿⠛⠻⣦⠀⠀⠀⠀⠀⣠⣴⠶⠶⣶⡄
⠀⠀⠀⠀⠀⢿⣇⠀⢀⣿⠀⠀⠀⠀⠻⣧⣤⣤⣼⠟⠀⠀⢿⣧⣠⣴⡿⠀⠀⢸⣧⡀⠀⣸⡇⠀⠀⢿⣄⡀⢀⣿⠀⠀⠀⠀⠀⣿⡁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⠻⢿⣿⡁⠀⠀⠀⠀⠀⢸⡿⢿⣄⠀⠀⠀⢠⡿⠛⣿⡀⠀⠀⠀⢹⣿⣿⣏⠀⠀⠀⢈⣿⠿⣿⠁⠀⠀⠀⠀⢀⣽⣿⣶⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣦⡀⠀⠀⢀⣿⠁⠈⢻⣦⠀⢠⡿⠁⠀⠘⣷⡄⠀⣠⡿⠁⠈⠻⣧⠀⢠⡿⠃⠀⠹⣷⡀⠀⠀⣴⡟⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢿⣦⢀⣾⠃⠀⠀⠀⠹⣷⡿⠁⠀⠀⠀⠘⣿⣴⡟⠀⠀⠀⠀⢹⣷⣿⠃⠀⠀⠀⠹⣷⣤⡾⠃⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢿⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠉⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⣷⠀⠀⠀⠀⠀⠀⣰⠞⢋⡿⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣷⠶⣿⣇
⠀⠀⠀⠀⠀⣸⡏⠀⠰⣄⣿⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠟⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⡟⠀⠈⠁⠐⠃⢹⡇
⠀⠀⠀⠀⠀⢻⣧⣀⣳⣼⠏⠀⠀⠀⠀⠀⠘⣷⡀⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣿⡴⢀⡄⠀⣼⠃⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠙⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠷⠛⠃⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⢿⣇
⠀⠀⠀⠀⠀⢠⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣦
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠷⠶⠶⠶⠶⣶⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣴⠶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠴
⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⣀⢺⠨⡊⠀⢤⠀⠠⣣⠈⢠⠀⠀⡣⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢄⡄⠐⠀⠀⠂⡤⠀⠀⠲⠃⠈⠄⠀⠘⠃⠁⠀⠀⡀⠁⠀⢀⠀⠀⠐⢄⡀⠉⠀⠀⠀⣠⠄⠲⡀
⠀⠀⠄⡀⡀⠘⠀⠀⠀⢀⡄⠡⠀⠇⠀⠀⠀⠀⣁⠀⠢⡐⠁⣀⠀⠀⠀⠱⠀⢀⠀⠆⠀⠀⠀⠈⠂⠀⠀⠐
⠀⠘⠈⠀⠀⠉⠀⠀⠀⠠⠈⢀⠀⡀⠀⠀⠄⠀⠞⠀⠄⠁⠀⠳⠀⠀⠀⠗⠋⠉⠉⠺⠀⠀⠀⠰⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠁⡀⠀⠀⠁⠀⠀⠀⠀⡰⢆⠀⠀⠀⠀⠀⠀⢀⠦⠀⠀⠀⠀⠀⠀⠠⠠⠢
⠀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢤⢀⣐⠨⠅⢠⣀⣀⠀⡀⠠⠄⠀⠠⢀⣀⠀⢠⠆⠀⠀⠀⠀⠴
⠀⠀⠂⠀⠀⠠⡈⠀⠀⠄⠀⡀⠀⠀⠂⠤⠀⠀⠙⣍⣇⢜⣉⠕⠁⠈⠙⢠⠒⡷⠇⠨⠈⠁⠀⠁⠢⠀⠀⠀⡊
⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⢰⡻⠖⠚⢟⡄⠀⠀⠀⣦⠪⠐⠠⢸⣄⠀⠀⠠⠄⠂⠈⠒⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠤⠤⢇⠁⠲⠀⠀⠀⠀⠀⠀⣠⠃⢀⡀⢀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣀⣀⠰⠈⢦⣀
⠨⠍⣦⡑⠂⠀⠀⠄⠀⠑⢠⡁⠀⠀⡤⠊⠀⠉⠱⡂⢆⡠⢀⠠⠃⠀⠑⢢⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠁⠂
⠈⠂⢄⡅⠀⠀⢀⠀⠀⠀⠆⡈⠖⢅⠈⡀⠀⠀⡐⡐⠄⠑⠈⡀⠀⠀⢠⡭⠴⠐⢶⣱⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢀⡰⡑⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠆⠀⠀⠀⢄⠀⠀⠀⠨⠀⠰⠀⠲⡲⠬⠉⠘⢓⣀⠤⠈⠣⡹⡚⠉⠠⢇⡐⠃⠈⠀⠀⠀⠀⠢⠄
⠀⠀⠀⠑⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⢁⠀⠀⠀⢀⠯⣡⣭⢧⠀⠀⠀⠀⡗⠖⣊⡾⠙⠀⠀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠂⠀⠐⠀⠀⠀⠒⠀⠀⠀⠐⠀⠀⠀⠘⠈⠀⠀⠁⠃⠀⠀⠀⠒⠁⠈⠀⠐⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢶⡀
⠀⠲⡂⡊⠂⣢⢤⠀⠀⢠⣤⣷⢷⣆⣰⡄⠀⠀⠀⡀⠌⠠⠀⠀⠀⠐⢆⢤⠎⣤⢳⠠⢄⡀
⠀⢁⠺⢶⡴⢏⠁⠀⠀⠀⠘⢷⣤⡻⡎⠀⠀⠀⠑⠆⢀⠀⠄⠁⠀⠀⠀⢙⢤⣿⠀⢖⠁⠀⠀⠀⠀⠁⠀⡀
⠀⠠⣈⠔⠁⠈⠀⠀⠀⠀⠐⠧⠊⠙⢷⠀⠀⠀⠐⠂⠁⠑⠆⠀⠀⠀⠀⠮⠚⠉⠑⠺⠆⠀⠀⠀⠀⠀⠁⠀⠈
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⣼⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣿⣿⡿⠋⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣠⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠻⣿⡿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡿⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣷⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠻⣿⣧⡀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣼⠀⠀⠀⠀⢻⣿⡇⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣿⡆⠀⠀⠀⢸⣿⠇⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣄⣀⣠⣿⡿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢹⣿⣿⣿⣿⡿⠁⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠀⠈⣿⣿⣿⣿⠃⠀⢀⣾⣿⠏⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⢹⣿⣿⣿⣷⣶⣿⣿⡟⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⢸⣿⣿⡻⣿⣿⣿⣿⡇⠀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⢀⣼⣿⣿⡇⠸⣿⣿⣿⣿⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠘⠛⠁⢰⣿⣿⣿⣿⣿⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⡄⠀⢰⣿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⣿⣿⠃⠀⠈⠛⠀⢻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠈⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⣀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠻⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠕⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠰⠆⣤⢠⡄⣠⣶⣯⣤⢀⡄⢠⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣒⣲⣇⡀
⠀⠀⣀⠠⠤⠠⡀⣀⡤⣾⠃⡀⠀⠠⣤⠤⣄⠀⠀⠤⠁⠊⠉⣐⣀⣀⡂⡇⣻⣹⠓⣒⣒⣒⠐⠢⢳⡦⠀⠀⠀⣀⣴⠾⠛⠉⡏⣿⡋⠛⠷⣦⡀
⢨⠃⠀⠀⠀⠀⠀⠒⢹⣿⡜⠑⠂⣁⡀⠉⠢⢂⠀⢒⢀⠀⠀⠛⣿⠁⠀⠁⢐⠀⢰⣿⠟⠻⣶⠀⢸⠓⠀⢠⡾⠛⠁⠀⠀⠀⡇⡗⡇⠀⠀⠈⠻⣆⡀
⠺⢠⠚⣏⢹⣤⣻⣿⣟⣫⠾⡅⡜⠿⢻⣦⣔⡼⠂⠘⢈⢇⠀⢸⣟⣦⣂⢧⢨⣶⣴⢠⠈⣿⡿⠇⠚⠁⠀⡼⠡⠀⢰⠄⣄⠀⡀⣰⠋⠄⣀⢰⡀⠝⢿
⠀⢣⣽⣾⢼⣿⣿⣷⣶⣿⣶⣿⣿⣧⣼⣿⣵⠁⠀⠀⢿⣴⣿⣦⣿⣿⣿⣶⣿⣦⣷⣿⣺⣯⣭⣽⡧⠀⠀⢷⡀⠋⣽⡏⠃⠞⣺⣷⠀⠖⢩⡼⠁⠈⡿
⠀⠸⢟⠛⢻⣿⣉⠛⡉⣿⡛⣿⣿⣿⣿⢿⠇⠀⠀⠀⢼⣫⣭⣭⣭⣥⠽⠿⡿⠿⢶⣿⣿⣯⣭⣿⡇⠀⠀⢸⠽⢼⣽⣯⣤⣼⢫⣨⣯⡐⢹⡽⣅⢌⡥
⠀⠨⣿⣭⣭⣽⣬⣦⣾⠿⡷⠿⠿⢿⣿⣿⠁⠀⠀⠀⢤⠶⡥⠤⠶⠸⠄⢤⠴⠶⠸⢿⣤⣿⣿⣿⡇⠀⠀⡼⠁⠉⠛⢈⡀⠃⢘⠋⠘⣃⠹⢿⡿⢿⡧
⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⣤⣥⣤⣬⣤⣤⣥⣤⣬⣶⡥⠼⠶⠼⢭⡷⠀⠀⠉⠛⠐⠒⠒⠐⠀⠀⠀⠐⠒⠐⠒⠑⠊
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⠙⠷⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⠻⣦
⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠈⢳⣄
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⣦
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠛⠋⠙⠛⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⣧
⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣇
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢹
⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⢀⣠⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠻⣆⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣟⣿⣿⡿⠛⠿⠟⠉⠻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠙⢧⡀⠀⠀⢻⣿⣿⣽⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠈⢿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⢤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⠉⢁⣼⣿⣄⢀⣤⣤⣄⠀⣿⡇⢸⣿⠀⣤⣤⣄⠀⢠⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡆⢸⣿⠀⣾⡷⠶⠿⠆⣿⡇⢸⣿⢀⣴⠶⣿⡇⢸⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠟⠁⠘⠿⠖⠘⠿⠶⠖⠀⠿⠇⠸⠟⠘⠿⠖⠿⠇⠸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⠀⠠⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠉⠈⠻⣄⣀⣀⣨⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⣻⣶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠶⢖⣚⣛⣭⣭⣴⣶⣾⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⢦⣀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠷⠦⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⠿⠿⠟⠛⣋⣉⣥⠤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⢿⠿⣿⣿⠿⡿⠿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⠭⠥⠤⠶⠶⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⠒⠂⠉⠀⠀⠈⠑⢒⣲⠴
⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣴⡤⠤⠤⠤⠤⠤⠤⠤⠀⠐⠒⠒⠒⠂⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⡀⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠘⠤⢤⡤⡤⠤⠤⠤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠐⠒⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠽⠶⠴⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣤⣄⣀⣤⣬⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠛⠛⠛⠛⠋⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣀⣤⣤⣀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣾⡿⠛⠛⢿⣷⡀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠸⣿⣇⠀⠀⢸⣿⠇⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⡏⠀⠀⠙⣿⣷⣾⣿⠋⠀⠀⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⣿⡇⠀⠀⠀⣿⡏⢸⣿⠀⠀⠀⢸⣿⠈⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣿⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⣾⡟⠀⣀⠀⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣿⠀⠀⣀⠀⢻⣷
⠀⠀⠀⠀⠀⢸⣿⠃⢸⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⡆⠘⣿⡇
⠀⠀⠀⠀⠀⣾⡿⠀⢸⣿⡇⠀⣿⡇⠀⠀⠀⣿⣇⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⡃⠀⢻⣿
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⡄⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣷⣤⣼⣿⣧⣤⣤⣤⣾⣿⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣼⣿⣧⣤⣾⣿
⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⢉⣿⡟⠉⠉⣁⠀⠀⢀⡀⠀⠀⣈⠉⠉⠙⣿⣏⣹⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠁⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⡇⠀⠀⠙⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿




⠀⠀⠀⢠⣄⠀⠀⢀⠀⠀⣴⡄⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⢀
⢾⣯⡃⢸⡟⠁⣾⣏⣿⡆⣻⡇⣿⡇⣘⣭⣿⡀⣿⠟⠁⠀⠀⣿⠀⣿⠀⣿⠀⣿⠛⣿⠛⢻⡆⣼⣏⣻⡆⢸⡟⢻⡇⠸⣯
⣰⣬⣋⣘⣿⣆⣙⣶⣶⣀⣿⣅⣽⣇⣻⣦⣿⣅⣿⣀⣀⣀⣀⣿⣀⣻⣶⣿⣀⣿⣀⣿⣀⣸⣇⣙⣷⣶⣀⣸⣇⣸⣇⣰⣤
⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⡟⡆⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣦⢸⢸⠿⡆⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢰⢿⠇⡇⣴⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣿⠀⡼⠹⠘⢸⠀⣷⠀⠀⠀⠀⣶⡰⣭⡆⠀⠀⠸⠈⡇⣇⠏⢣⠀⣿⡝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⣡⣾⢸⣧⣙⣡⣧⠸⣤⣿⠀⠀⠀⣰⣿⣷⣜⠃⠀⠀⣷⡆⡇⣼⣌⣋⣼⠉⣧⣜⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣼⡟⣷⢧⡄⡇⢹⣥⣨⢣⣿⣾⠀⢀⢴⣿⡏⣿⢻⡦⡀⠀⣷⣿⡜⣅⣬⠋⢸⢰⡹⡿⢿⣧⢿⡇
⠀⠀⠀⠀⢠⡄⠀⢸⣦⣹⣷⠏⣿⠀⣿⣷⢻⡏⣿⠇⡿⢀⠎⣼⢹⡇⣿⢼⢃⣵⡀⢿⠸⣳⢹⡏⣾⣿⠘⣷⠻⣾⣿⢬⡇⠀⢠⡄
⠀⠀⠀⢠⡿⠀⠀⢸⣟⣹⡏⢆⡏⢸⡏⡏⡌⠑⡎⡀⡇⢸⡟⣿⠘⣇⢹⣧⣸⣿⡇⢸⢀⣿⠸⢡⣿⢻⡆⢣⢁⢹⣏⣾⣇⠀⠀⢿⡄
⠀⠀⢠⠏⣧⠀⣠⠇⡻⣿⣱⡟⣾⢸⡇⢣⢻⣄⠁⡇⢿⡜⣧⡇⣶⢻⡀⣿⣿⣼⣣⡟⣼⠈⣰⡟⡏⢸⡇⣧⢿⡆⣿⣿⠹⣄⠀⡸⢹⡄
⠀⠀⢸⢠⡘⣿⣡⣞⢠⡿⠿⢹⢿⢸⡇⣼⣆⠛⠆⢿⢸⣙⣿⢡⢿⣾⡇⢹⡇⣿⠉⡇⡿⠸⠋⣰⢧⢸⡇⣿⠘⠃⢿⡈⣣⣎⣿⢃⡌⡇
⠀⢠⣼⢼⡇⣹⡶⡽⣾⣷⣇⢸⡿⢸⡇⡟⡿⡜⡟⡌⣼⣿⢃⡿⢸⢸⣇⣼⡇⣏⠿⣏⢃⢿⢱⢿⢸⢸⡆⢿⠀⣼⣾⣇⢯⢶⣏⣼⢱⣧⡄
⢀⡾⢻⡏⣱⠏⣀⡰⣻⢿⡟⣧⢿⣸⠃⠀⠁⡇⡷⣱⢋⣿⣏⣷⡼⢸⡿⣿⢱⢟⣄⠹⣎⢾⢸⠈⢀⠘⡇⠿⣾⢹⡟⣿⣆⣀⠻⣟⢸⡟⣿
⣼⡷⣾⡿⣱⢇⣿⠾⣇⣾⠣⢻⣄⡿⢸⣧⢠⠃⣼⠃⢸⣿⡜⠘⢁⡾⢁⡏⡿⣾⠙⣦⡹⣦⢸⣠⣼⡆⣿⢰⡟⠸⣧⣼⠧⣿⡼⣞⢿⣗⢾
⡏⢻⣿⢣⢻⡘⣯⠘⡿⢣⡶⣸⣿⠇⣦⣿⢸⡾⡏⣶⠸⢀⡖⣸⢹⣧⣾⡗⣧⢣⡆⣼⢣⣿⢿⡇⣿⣰⠸⣿⣇⢶⡘⢯⠃⣽⢣⡟⡜⣿⡞
⢷⡟⢸⡸⠞⠄⣩⢞⣡⡞⣰⠿⡏⢸⣇⢻⠘⠀⡇⠙⠘⠏⡟⢉⠘⠛⡏⢣⠉⠘⠃⠛⠈⣿⠀⠁⡇⢸⡇⢻⠻⣦⠷⣌⠳⡍⢠⠳⢷⡇⣷
⠈⢿⡈⢧⠻⣷⣯⣞⣠⡾⢃⢸⠁⢸⡘⡏⣴⡄⣿⠀⡆⠀⢰⠞⠀⠀⠀⠈⠟⣧⠐⠑⠀⣿⠀⣆⢱⢙⡇⠘⡇⡘⣷⣌⢳⣽⣦⢟⡞⢀⡿
⠀⢬⣳⡌⢳⣿⢋⡞⣿⢣⣿⢸⣸⣇⠻⢠⡏⠀⡿⡄⠀⢸⡄⢶⠀⠀⠀⠳⠀⢹⡄⠎⣸⢿⠀⢹⡌⠟⣼⣇⡇⣿⡘⣟⢿⡙⣷⡞⢠⣞⡤
⠀⢸⣿⠇⢾⡻⣟⠠⣿⢸⣻⠸⣿⠘⡆⢸⢰⢰⡇⢰⣸⡿⡇⣾⢷⡆⢾⡆⢸⠂⣷⢰⡇⠸⠀⣌⡇⢸⠁⣿⢣⣟⡇⣿⢤⣿⢿⡁⣸⣿⡇
⠀⢸⡏⢧⡾⡁⣧⠃⣿⣸⢻⡇⢡⡄⣷⢸⢃⡟⢰⢸⡿⣦⢃⡟⠸⣳⢢⣿⠸⠁⡘⣿⢳⡄⢷⡘⡇⣿⢰⡅⢸⣿⡇⣿⠘⣾⢈⢧⡽⢹⡇
⠀⢸⣷⡜⢱⢃⣿⢷⡟⣿⡟⡴⢛⠃⣿⣴⡟⣠⢿⢸⡇⡿⣸⣇⢠⣏⡎⣿⣴⢸⡇⣿⡞⢳⣌⢻⣦⣿⠸⡛⢆⢻⡇⢿⡾⣿⡸⡎⣡⣾⡇
⠀⢸⡟⣸⠀⣼⣿⢀⡁⣿⢸⢱⡏⢸⢫⢏⣾⡟⡘⣼⠉⣵⡿⡏⣾⢏⡗⢹⡅⣼⠇⡿⢧⠎⣿⣦⡹⡝⡆⢹⡞⡇⣿⢈⡀⡿⣧⠀⣇⢻⡇
⠀⠀⢧⠻⢸⠇⣿⢘⢣⠈⣤⣾⠃⡾⠃⣎⢸⢱⢣⡇⣼⠹⡇⢳⡿⢠⢗⡟⡼⠁⣸⣧⠸⡜⡜⢂⢳⠘⣷⠘⣧⣥⢁⡞⡃⣿⢸⡆⢏⡾
⠀⠀⠈⢷⣼⡆⣿⠸⡾⣇⢿⡘⣸⢃⠸⡹⠘⣿⢸⡿⣏⣆⢳⡀⣷⠺⣬⢡⣽⣼⠋⣿⢿⡇⣿⠸⢾⠀⡘⣇⢇⡟⣸⣣⢇⣿⢸⣧⡾⠁
⠀⠀⠀⠀⠉⠃⠈⢧⡹⣿⣮⢡⡏⣷⡀⣧⡄⢻⠘⠁⣿⢙⣦⠳⣌⠷⠏⢣⡿⣱⡏⣿⠘⢠⡇⢠⡼⡄⣧⢻⡈⡿⣿⢋⡾⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠐⣦⡻⣍⣮⣾⡇⢿⣿⠘⠇⢸⠀⣷⠸⡎⣿⡆⠹⣷⣴⠏⢐⣯⣹⠇⣧⢸⡇⡞⠃⣿⡹⢸⣧⣽⣱⠟
⠀⠀⠀⠀⠀⠀⠀⢫⣳⡹⡏⢿⠀⡟⢸⣠⢰⣾⢰⡏⡄⣿⠘⣷⠀⢻⡏⢺⣘⠃⣿⢰⢹⠸⣇⠇⣄⠃⣷⢸⡇⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣥⢻⣾⡄⡇⣾⡿⡄⢿⡘⣧⡇⣿⠃⡁⣾⢸⣷⠸⡏⢸⣿⢾⣸⢰⡟⢸⣿⢰⣶⢸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣏⣣⢷⠉⡇⢷⢰⣄⠻⠇⣿⢠⣿⡏⣾⠛⣧⢻⢹⣿⠸⠏⣨⡄⣷⢸⢀⡏⣜⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣎⠳⣏⢸⠸⣘⢧⠀⣿⠸⢹⣧⡟⢀⢹⣦⡟⣿⠀⠾⢹⠀⡇⣸⠟⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⡻⠄⢿⡀⡆⣿⡼⣼⢹⡇⡿⢸⣿⡕⡿⢸⣠⡟⠰⢛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣍⣃⠘⣷⡉⢾⡕⣣⡟⣩⣼⢃⣉⣭⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣾⣷⡟⢞⣵⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⠏⠀⣼⡇⠈⢻⠈⢹⡆
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠲⠤⢤⣀⡤⠂⠀⣀⣀⣤⡀⠀⠘⢣⣞⣦⣥⣿⣧⣧⣞⣦⠚⠀⠀⣠⣄⣀⡀⠀⠢⣄⣠⠤⠴⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠲⣄⠀⡎⠉⢳⠮⠤⠠⠏⠉⠛⠋⠉⣿⣿⣻⣻⣛⣻⣛⡏⠉⠛⠋⠉⠧⠤⠬⢶⠊⠉⡇⢀⡴⠒⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢧⠀⠈⢄⠀⠀⣎⡗⠀⠀⣀⣤⡤⠴⠦⠶⠤⣤⣄⣀⠀⠀⢏⡆⠀⢀⠈⠀⢠⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠲⠤⣀⡀⠠⠔⠃⠀⠈⠓⠦⣤⣤⣬⣧⡶⠚⠋⠉⠀⠀⠀⢐⠀⠀⠀⠈⠉⠛⠲⣦⣯⣤⣥⡤⠖⠋⠀⠀⠑⠤⣀⠀⡀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠈⠉⠉⠑⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⣴⡿⣷⡄⢀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⣀⣹⣿⣿⣁⣀⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠉⠉⠓⠲⠤⣄⠀⠀⠀⠀⠀⢩⢭⣭⣭⣯⣽⣽⣭⢯⠁⠀⠀⠀⠀⢀⡤⠴⠒⠋⠉⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠂⠂⠀⠐⠲⡄⠀⠀⠈⠳⣤⠖⠋⠉⠒⠒⠒⢒⡖⣖⠓⠒⠒⠊⠉⠓⢦⡴⠋⠀⠀⠀⡴⠒⠀⠀⠂⠒⠲⣄⡀
⠀⠀⠀⠀⠀⢠⠎⠀⠘⢦⣐⢠⡀⢀⡴⠃⠀⠀⠀⣼⠉⠳⣀⠀⠀⠀⠐⠈⠁⠉⠐⠀⠀⠀⢀⡰⠋⢹⡀⠀⠀⠀⠳⣄⡀⣠⢐⣠⠞⠀⠈⢦
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⢉⡏⠁⠀⠀⠀⠀⠀⠸⣄⠀⠈⠓⠤⣆⡀⣦⠀⢠⡆⣀⡤⠔⠋⠀⢀⡼⠁⠀⠀⠀⠀⠈⠉⣏⠉⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠉⠒⠦⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⣀⠠⠶⠃⠀⠀⠳⠦⢀⡀⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠀⠸⠇⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⠀⠀⠀⠀⡇⣸⠃⢀⠀⢠⡈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠁⠀⠀⣰⠃⠀⠀⡞⠀⢈⡇⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠸⠁⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠚⢸⡆⠀⢰⡆⠀⠀⣠⠀⢀⡀⠀⠈⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣞⠀⠀⠀⢻⠀⠀⢸⠆⠈⠁⠀⠀⣇⡔⢰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠋⢠⠀⡄⠀⠀⠀⣀⠀⠀⠀⣼⠀⢉⠀⢸⣧⡇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠇⣼⠃⠀⡠⠀⢀⡀⠆⠀⢀⢉⠀⠀⣧⠁⠀⠈⠀⠏⠏⠀⢰⠀⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣇⠐⡇⠀⠀⠁⠀⢸⣷⠀⠃⣾⠀⠀⠂⠈⠀⠀⠁⠆⠀⠀⠀⢸⢃⡁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠁⣸⡆⣦⢠⠀⠀⢠⠀⠁⠀⠀⠛⠀⢀⢰⠀⡄⠀⣆⠀⠁⠀⠀⠀⠸⠃⢸⠆⠋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⡀⣠⠠⠈⡃⠞⠀⠀⠁⡀⠀⠀⢀⠀⢀⠸⠆⣆⠁⡞⠑⠀⡀⣤⠀⠀⣠⠀⠊⠀⣠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⢳⣉⠀⠁⠑⠰⠓⣠⣷⠇⠀⠰⡌⠂⠰⠀⡄⠉⠀⠇⠐⠾⠁⣸⠀⣶⠀⣶⠄⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠋⡤⠛⠐⣷⡀⠜⠿⣿⡏⣻⣧⡐⠁⢀⠀⢸⡇⠀⠀⠀⠀⢠⣘⡧⣶⠛⣷⣤⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡀⠘⠄⢀⡀⠀⣄⣀⠈⡄⠙⠉⣿⡄⠉⠀⡠⢁⡯⠀⢠⠀⣸⡟⠁⠇⠞⠟⢁⣀⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡄⠙⣻⡆⢨⢉⣠⣿⣾⣏⢀⣔⢃⠈⣁⠀⡀⠇⠋⠛⠀⠀⠀⠋⠅⢡⣴⡀⢸⡇⢦⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠓⠉⠙⠛⢾⡌⠙⣿⠇⠁⠈⠛⠙⢷⢈⣤⣅⠀⣄⢂⣼⣷⡀⢰⣾⠆⡿⠃⣾⣿⠆⠘⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⢠⣿⡿⠂⠉⠁⠒⢶⡄⠀⣤⣼⡿⠋⠁⠀⠈⠙⢿⣿⡆⠀⠀⠆⠉⠘⠳⣦⣤⡠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣗⡒⣶⠞⣻⣟⡀⢀⠀⣀⣼⣮⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⣸⣆⠀⠀⢀⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠯⢉⣛⣛⡽⣿⣦⢠⠞⣿⣿⠦⡀⢀⡀⣰⡿⠿⣿⣿⣯⣾⣽⣷⣀⣫⣾⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠦⠤⠔⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣤⣀⠀⠀⠉⠁⡴⠋⠁⠲⡄
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⣠⣤⣤⣴⡿⣠⣿⣿⣷⠀⠀⢈⠀⠀⠀⠀⢹⡄
⠀⠀⢀⡄⠸⢄⠞⣸⠗⠘⠉⢭⣟⣛⠿⣿⣿⣿⣿⣿⡆⢀⢸⠀⠀⠀⠀⢸⣧
⠀⠀⠺⠷⠖⣰⣶⡄⠀⠀⠀⠀⠀⠉⠻⢮⣝⢿⣿⣿⣤⣿⠀⠀⠀⢸⢠⣿⣿
⠀⠀⠀⠀⢸⣿⡿⠃⠀⠀⢀⠀⠀⠀⠀⠀⠉⠳⣝⠻⠟⠃⠀⠀⢠⠇⠀⠉⢻⣆
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⡌⣇⠀⠀⠀⠀⠀⠀⠈⢀⣀⣤⣿⠟⠋⠀⠀⠀⣄⠉⣳⡀
⠀⠀⠀⠀⣻⢻⣧⣀⣠⠞⠀⣿⠀⠀⠀⠀⣀⡀⠀⢈⠟⠉⠀⠀⠤⢤⠴⠾⠟⣀⣼⡇
⠀⠀⠀⠀⣿⠀⣹⠋⢁⣀⡠⣿⡶⠒⠚⠉⠀⣀⡔⠁⠀⠀⠀⠤⠞⣉⡀⠀⢻⡏⠛⠃⠀⠀⠐⠒⠲⠶⢤⣀
⠀⠀⠀⠐⢿⣿⠏⠀⠀⠀⠀⠸⣇⠀⠀⠀⢠⡇⣧⣶⡆⠀⠀⠀⣰⠛⠉⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠈⢷⣜⣿⣿⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠙⠛⣿⠀⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢀⣀⣴⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⣿⣿⠟⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⠀⠀⠉⠉⠉⣩⣭⣤⣄⣀⣀⣀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⡰⠀⠀⣾⢻⡏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢀⡄⠀⠀⢠⠊⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣠⡶⠟⠋⠀⠀⠀⣴⠃⠀⠀⠈⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡴⠟⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⡏⠀⠀⣧⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⣿⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣷⠀⠀⠘⣧⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⣿⡀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣇⠀⠀⠈⢷⡄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣽⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠀⠀⠀⠘⣿⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠋⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣷⣾⣷⡄⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡽⠿⠿⣅⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⣦⣴⢞⡿⢋⡙⠛⢧⣼⡀⠻⠃⣸⣤⠟⠛⣉⠻⣟⢶⣤⡶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠁⢀⣯⣬⡿⠛⠋⠉⠉⠀⣿⡃⣿⡇⣿⡇⠈⠉⠙⠛⠻⣯⣬⣇⠀⢹⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢸⣤⣥⡇⠀⠀⠀⠀⠀⡟⢳⣶⣶⠞⡇⠀⠀⠀⠀⠀⣧⣥⣼⠀⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⡀⢸⣬⣅⡇⠀⠀⠀⠀⠀⢧⠈⣩⡉⢰⠃⠀⠀⠀⠀⠀⣇⣭⣼⠀⢸⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠃⠈⢉⣅⠡⠀⣰⣷⣄⠀⣠⡜⢿⠿⣤⡀⢀⣴⡷⡀⠀⢀⣍⠈⠀⢛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣾⡿⡷⢾⡷⡿⢿⠟⢿⣟⢯⢟⣿⠟⢿⢻⡷⣿⠶⡿⣯⣧⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣬⣤⣵⣯⣤⣤⡷⢿⠶⢶⣾⢿⣾⣶⢿⣶⠶⢾⠷⣽⣦⣬⣧⣧⣬⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠟⡹⢂⣿⣐⣷⣾⣠⣷⣜⣡⣙⣴⣧⣸⣶⣗⣸⣇⠺⡙⠇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠟⠛⠣





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠋⠉⠍⠁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⡠⠀⡀⠀⠄⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡖⢦⣀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠿⠄⠀⠀⠀⠀⠀⠀⠉⠉⠲⢄⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⣄⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠈⢳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⠀⠀⢤⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠏⠀⠀⠀⠀⠀⠀⢩⠉⠢⣘⣦⡀⢸⠀⢹⡞⠳⡄⢙⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⠃⣀⡈⠃⠱⠎⠑⣼⡇⠁⢸⠘⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠄⠀⠀⠀⠀⠀⡇⠘⢿⣆⠀⡄⠀⠉⠀⡔⠋⠀⡇⠀⠀⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠧⠒⡧⠀⠀⠀⠀⡆⢰⠁⠀⠈⠉⠀⠀⠀⠀⠀⢰⢀⡎⠃⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⡆⠀⢀⠿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⠉⢻⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠎⠱⣄⠁⠀⠀⠀⣀⡤⣶⠫⠎⠀⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣉⣿⣿⡖⢒⣿⢯⠞⢡⠂⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢯⣵⠷⠒⠟⡛⠋⡽⣸⡴⠋⠀⠀⢠⣆⠄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠡⠞⠙⠳⢦⣔⣀⣼⣋⣁⡀⠀⢀⡤⠛⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⣇⠔⠋⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⠈⠉⠻⣤⣄⣒⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣷⠤⠞⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢰⡀⠀⠈⠁⠀⣹⠋⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡴⠊⢹⠀⠀⠀⠀⠀⠈⢇⠀⠀⣀⠔⠓⠄⠀⠀⠈⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢇⠀⠀⢀⡠⠴⢚⡗⠫⠥⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⣶⠖⣋⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⢀⡴⠛⢋⡡⠤⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢤⣈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠙⠛⠬⣀⡀⠀⣰⠋⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠶⠤⠤⠽⠖⠁⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣿⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡂⠁⢉⠉⢙⣥⡤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⡆⣷⣿⠋⠀⠀⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⣿⡉⡷⣇⣀⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⢙⠷⡀⡷⣬⣼⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡶⢚⣾⠃⢳⠶⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⡼⡠⣶⣿⣰⡿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣤⣶⡶⢶⣾⠟⣻⠯⠈⢡⠷⣿⡅⠉⠲⠿⠻⣿⣶⣶⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠛⣋⡵⠤⠒⢞⣻⠶⢚⣩⠐⠊⠀⠀⠀⠃⢀⡏⢻⠀⠀⠀⠀⠈⠙⠻⢭⢻⣯⠻⠹⢿⣿⢿⣿⣰⣤⣀
⠀⠀⠀⠀⠀⠀⠦⠞⠃⠀⠚⠁⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⠀⠀⡀⠀⣊⣧⠘⡇⠀⠀⠀⠀⠀⠀⠀⠙⢪⡽⠀⠀⠀⠁⠈⠛⢿⣿⣷⣄⡀
⠀⠀⠀⠀⡶⠎⠁⠐⠀⠀⠀⠀⠀⢀⣰⠏⠰⣰⣶⣶⣶⣿⣿⠟⢁⠃⠀⣀⣽⡆⢻⠽⠻⡟⠓⣶⣶⣄⠀⠀⠛⣉⡇⡀⠀⠀⠀⠀⠙⢿⣿⢧⡄
⠀⠀⠀⠶⠃⠀⠀⠀⠀⠀⢀⣤⠀⢤⡁⢠⢁⣿⣿⣿⠟⠉⠀⠀⠈⠀⠀⡀⣈⡁⠸⡆⢀⣠⣶⣯⣏⣹⡇⢀⠀⡁⢾⡤⣄⠀⠀⠀⠀⠀⢻⣟⣷⡄
⠀⢀⠐⠂⢠⠀⠀⠀⣠⣶⠟⠀⠀⢼⡆⠈⢸⣿⣿⠃⠀⠀⠀⠀⠀⢀⡜⠁⠘⢷⠀⣼⣾⣿⣿⣿⣿⣿⣿⢠⠄⠀⢼⡇⣿⣷⣦⡀⠀⠀⠈⣿⣽⣦
⠀⣘⢺⢀⠈⠀⢀⣜⣿⠂⠀⠐⠢⢶⡇⠀⢸⣿⡧⠀⠆⢀⡀⠀⢨⠃⠀⠀⠀⠀⠰⣿⣿⣿⡿⠏⠉⣄⢭⢸⡃⡠⠶⠗⢻⣿⣿⣷⣄⠀⠠⣽⣿⣷
⠀⢈⡻⢈⠈⠀⣾⣉⠋⠀⠀⠀⠀⠀⠀⢰⠸⣿⣦⣀⠆⠈⢡⠔⠘⠄⣀⠀⠀⢄⡴⠻⢿⣿⣃⢹⣯⡿⡖⣸⠃⠀⠀⢠⡸⣿⣿⣿⣿⡆⠀⣼⣾⣿⠄
⠀⣈⣽⣽⡄⣾⣷⡟⠂⠀⠀⠀⠀⠀⠀⡦⠃⡠⠀⠑⣧⣀⡎⢠⡴⠀⠈⢀⠉⠂⣴⠒⠀⢸⣭⡥⢋⡤⠄⠙⣴⣶⣠⡟⣳⣿⣿⣿⣿⡇⠤⣿⣿⡷
⠀⠉⠉⠛⠻⡹⠷⠿⠦⢤⠀⡤⠀⠠⠘⠃⢸⢘⣦⠀⣾⠗⠁⢻⡀⢻⠀⠠⠀⣸⣯⡷⢀⡬⠻⡇⢺⣶⣦⠀⡿⣳⢏⣵⣦⢸⣿⡿⠿⠫⣼⠟⠛⢃⡼
⠀⠀⠀⢤⡀⡁⣰⢲⠀⡺⢌⠧⣴⡆⡄⠠⠈⠇⢻⣶⡟⠒⠀⣾⣿⣉⣁⢀⣀⣈⢹⣿⣿⠃⠐⢿⣾⠇⢻⣀⠁⢁⣾⡏⣸⣿⡿⢡⣶⣇⣿⣾⡿⠟
⠀⠀⠀⠀⠉⢁⡘⡩⠣⡧⢰⠛⣾⣧⡤⠖⠖⠻⠞⣿⣿⠏⠻⠿⡿⠇⣡⡈⣧⡈⠻⠿⠿⠋⠻⣿⣏⠰⡿⢻⠷⣾⣿⣟⣿⠠⣧⣸⡾⢳⠟⠁
⠀⠀⠀⠀⠀⠀⠩⣘⣤⣵⡾⢉⡛⢙⡁⠖⠒⠋⠒⠠⠰⠊⠙⠲⠆⠈⠐⠸⠇⠈⠄⠐⠆⠿⠶⠤⡄⣨⣷⣼⣷⣿⡟⣛⢹⣿⠯⣤⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡵⠃⣀⢀⠤⠤⠀⠄⠒⠂⠂⠀⠀⠀⠀⠀⠒⠒⠖⢲⡲⢚⢒⣢⣤⣤⣴⢖⠰⣟⣿⣿⣿⣼⣿⣯⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠞⠉⣀⣀⠀⠀⢪⣭⢽⡆⠘⢣⣶⣜⠀⠠⢠⡴⠾⠒⣿⣰⢮⡵⣷⣿⠛⢠⡾⠉⣿⣟⠿⣿⢿⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠈⢿⡁⡂⠀⠾⠷⠶⠃⠀⠚⠁⠘⠀⠀⢽⢭⠛⠫⠓⠸⠾⠶⣦⢿⣀⣾⣤⣠⣿⣟⣾⡽⣿⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⢁⣀⢀⣀⠤⠄⠄⢀⡄⠠⠄⠀⠀⠐⠀⠂⠀⠐⠀⠄⠀⠀⠂⠐⠧⠤⠠⠭⣿⣯⣿⣿⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⣿⣁⠀⢨⡦⠄⢠⣨⢇⡰⠀⠐⠢⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠈⠉⠊⠛⠛⠛⠛⠛⠙⠂⠐⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⣿⡇⣠⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠛⣿⣿⣿⡟⢛⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠀⠁⠀⣸⣿⣿⡆⠀⢀⠤⣒⢦⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⢹⣿⣷⣶⣤⣿⣿⣿⣷⣾⣵⣿⣿⡏⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠹⣶⣄⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⢟⣿⣿⣿⣿⡇⢻⣿⣿⣷⡟⡟⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⢀⣴⣾⠋
⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⢀⣎⢮⣹⣿⣿⡀⣠⣿⡿⣩⡔⣠⣀⣠⣿⣿⡿⠀⣀⡀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⢠⠊⡷⠀⣿⡟⢻⣟⢿⣿⣯⢟⢿⣿⣿⣿⢟⡞⣝⣼⣿⣯⡿⠋⣿⡇⠘⠃⢸⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣀⠓⣤⣔⠻⣧⡀⣿⣼⣿⣿⣷⡣⣯⢿⣏⣎⢮⣾⣿⣿⣿⣀⣼⠏⣽⣶⣮⢁⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⣯⣾⣟⢿⣿⣹⣿⣿⣿⣿⣟⠎⡸⣳⣿⣿⣿⣦⡽⣟⣻⣷⢽⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣭⡽⣿⣞⣻⣿⣿⣭⣭⣭⠎⠰⡹⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢿⣛⣫⣭⣵⠶⣶⣿⣿⣛⡧⣴⣹⣟⣛⣶⠾⣯⣽⣟⡷⢿⣽⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣟⣿⣽⠿⠞⠛⠛⠉⠉⠉⠉⢞⡟⠉⠉⠉⠉⠛⠛⠶⢿⣽⣻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠟⠃⠀⠀⠀⠀⠀⣠⡾⣛⣿⣿⣿⣝⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣁⡀⠀⠀⠀⠀⢀⣾⢯⣾⣿⣿⣿⣿⣿⣍⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⠀⠀⠀⢐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣾⣿⣽⣿⣿⣿⣿⣟⢇⠠⢛⠻⠷⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⣿⣏⣿⣿⣿⣿⣿⣿⣾⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⡾⢟⢿⣟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠙⢝⣯⣤⣼⣿⣦⣌⡀⠉⠛⠷⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⡏⠀⠀⠀⠙⢿⣧⠙⢿⣿⣿⣿⣷⣶⣤⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⣀⣀⣻⣿⡟⠲⢦⣤⡀⠀⢹⡇⠈⣿⣿⣿⣿⣿⣿⣼⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⣿⣆⠈⣿⣧⣄⣾⡇⢀⣿⣿⣿⣿⡿⣿⣥⣴⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣄⢿⣿⣽⡿⣷⣾⣿⣿⣿⣿⣥⣯⡾⠟⠉⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⠁⢻⣿⣿⣿⣾⣿⣦⣷⣿⣿⣿⡿⠿⣿⣿⣝⣄⠀⠀⠀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠁⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⢰⣿⠻⣿⣿⣿⣿⣿⢿⣿⠛⠉⠀⠀⠀⠉⢻⣿⡀⠀⣴⣾⡟⠁
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣆⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⣸⣿⠟⠛⠛⠿⠿⠛⠊⠁⣤⣶⣦⣠⣤⣶⠿⠟⠁⠀⠙⠉
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⡟⢿⣿⣿⣿⡄⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉
⠀⠀⠀⣀⣴⣿⣿⣿⣿⡟⣿⠀⠀⠠⣾⣿⣿⣾⣿⣿⡿⠟⠁
⠀⠀⠀⢉⣵⣿⣿⣿⣿⠃⠏⠀⣠⣴⣿⣿⣿⣿⣯⡁
⠀⢀⣘⣿⣿⣿⢿⡿⠃⠀⠀⠀⢻⣿⡿⠉⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⢀⣬⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠛⠛⠋⠸⠟⠃⣟⠁⠀⠀⠀⠘⠠⠄⠀⠀⠈⠁⠀⠀⠙⠿⠇
⠀⠀⣀⠀⠀⠀⠀⠈⠛⡙⠉⠀⠀⠀⠠⣄⡀⠀⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆
⠀⠀⠉⠀⠠⣀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠉⠃⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠛⢀⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠙⠂⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣤⡠⠐⠛⠁⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡧⠀⡀⠰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣄⠀⠀⢤⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⠎⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠙⠃⠀⠈⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⢀⠀⠀⠀⢠⣤⡄
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣵⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠰⢿⡄⠀⠀⠄⢸⡟⠁⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠙⠿⣿⣷⣤⣀⠀⠀⠀⠀⠘⠓⠢⣀⢰⣾⠁⠀⠒⠀⠿
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡖⣀⢃⠀⡖⠀⣿⡏⠉⡳⡀
⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠛⠙⠊⡹⣇⢀⣹⡇⠀⢍⠛
⠈⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠑⠀⠀⠀⠀⠀⠈⠟⠁⠇⢹⣧⡀⠲⠀⠀⠈⠁
⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠐⣦⣿⣿⣷⣭⣾⣶⣤⣄
⣿⣿⠀⠰⠀⠀⠀⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⣫⡄⢰⣿⣏⢹⣿⣿⣿⣿⡟⠛⠟
⠀⠠⡄⠀⠀⢀⣼⣿⣿⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠈⢡⠏⠙⡟⠛⣿⣿⣿⣟⠣⠁⢺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣾⣿⣆⡀⣀⣀⠀⣠⣽⣿⣿⣿⣷⣤⣄⣀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠙⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠛⠋⠁⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⠀⢠⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⠟⠿⠁⠀⠀⠀⠀⠀⠀⠘⠟⠻⠿⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄
⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣠⣶⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣦⣄⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣠⣴⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⡉⠙⠛⠻⠿⠦⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⠴⠾⠟⠛⠋⢉⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣶⣶⣤⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢤⣶⣶⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⢰⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⡆⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟⠋⠀⠀⠀⠀⠉⠻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈






⣿⣿⠿⠿⡿⢻⠿⡿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⢀⡠⣿⣏⣹⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⣠⣿⣯⢿⣿⣶⡤⢤⣄⠐⣿⡿⢽⣧⣿⡭⢿⣿⠒⣤⡤⢠⣶⣿⡿⣴⣿⣄⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠙⣷⣶⢤⡄⣤⡎⢿⡏⢹⣿⣏⠙⠿⣾⣿⣶⡝⠆⢨⣿⣿⡁⠸⢯⣶⣿⢧⢿⠋⣽⣿⡟⢹⡿⣰⣆⢤⣤⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣞⣿⣿⣍⣴⣿⣿⣷⣶⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣾⣿⣿⣿⣶⣾⣿⣿⣦⣬⣿⣿⣳⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣟⣳⡿⣟⢻⢹⣟⣛⣻⡽⠯⣿⡿⢭⢩⣿⣛⣻⡟⡝⣻⢿⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣴⣾⠷⠖⠛⠓⠚⣚⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣓⣒⠚⠛⠾⠿⢷⣶⣿⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠟⣋⣭⣥⣶⣶⠟⠛⠋⠉⠉⠉⠉⠉⠁⠀⠀⠈⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠳⣶⣶⣬⣭⣛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠿⣿⣿⣿⣿⠶⠶⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⣿⣿⣿⣿⠿⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⡄⠀⠀⠀⣠⣿⣶⣤⣤⣤⣾⣿⢿⣷⣤⣤⣤⣴⣿⣄⠀⠀⠀⢀
⠀⠈⢀⣤⡀⠀⠈⠃⠈⠀⠠⠀⠈⠁⠀⠄⠀⠀⠘⠁⠀⠀⣠⣄⠁⠀⠀⠘⠷⠲⣾⣿⣿⣽⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣯⣿⣿⣷⠖⢾⠃
⠀⠀⢨⡿⢉⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⡉⠻⡇⠀⠀⠀⠀⠘⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣄⠃
⠀⠀⠐⠋⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣭⡝⠃⠀⠀⠀⠀⢠⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄⣱⣿⢻⣧⠀⠀⠀⠀⠀⠀⢀⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿
⠀⠀⠀⠀⢸⣿⠘⣿⡀⠀⠀⠀⠀⠀⠐⢠⠘⣏⢻⢿⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⢁⣾⡏⠀⠀⠀⠀⠀⠰⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣦
⠀⠀⠀⢰⣼⠀⣤⠛⡇⠀⠀⠀⠀⠀⢰⢿⣿⢃⡀⢸⣶⠂⠀⠀⠀⠀⠀⢸⡇⠋⢀⢸⣯⣔⡇⠀⠀⠀⠀⣾⠉⡀⢸⡇⠀⠀⠀⠠⢠⣠⡾⣿⢠⡌⠹⣷⡀
⠀⠀⠀⢈⣏⠃⡝⣆⣷⠀⠀⠀⠀⠸⣿⠆⢁⡞⣇⠀⢛⡟⠀⠀⠀⠀⠀⠘⢿⣄⢼⣦⣿⡟⠀⠀⠀⠀⠀⣾⠄⣧⣈⡇⠀⠀⢀⣴⠛⢛⣄⣁⣼⣧⡆⡹⠁
⠀⠀⠀⠀⠉⠛⠓⠋⠁⠀⠀⠀⠀⠀⠈⠑⠂⠉⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠚⠃⠀⠀⠀⠀⠀⠈⠑⠛⠉⠀⠀⠀⠘⠛⠀⠛⠈⠉⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠰⢃⣴⡎⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠴⠀⠠⠀⠀⠀⠀⣶⣄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠄⢀
⠀⠀⠀⠀⢾⡟⢿⣷⡈⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠀⣠⣴⣶⡿⠀⠀⠀⠀⠀⠀⢿⣿⠿⣷⣆⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣴⠁⠀⢀
⠀⠀⠀⠀⣸⡇⠀⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣆⡿⠋⣜⣥⣠⡌⠀⠀⠀⠀⠀⠻⣆⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⢃⡴⣿⠠⣶⡿⢻⣿⢀⡴⣯
⠀⠀⠀⠀⠿⣟⢸⡄⡿⠀⠀⠀⠀⢀⣠⣰⣶⠏⣸⠟⢡⠜⠙⠻⢛⡵⠆⠀⠀⠀⠰⡶⣿⢠⡄⢸⣦⠀⠀⠀⣠⣴⣤⡴⠋⣸⡿⢚⡉⣠⢿⡿⠋⣴⠏⡀
⠀⠀⠀⠀⠀⠈⢿⠟⠁⠀⠀⠀⠠⣿⢋⣛⣡⢦⣠⠶⠻⠴⣦⣤⣀⡶⣧⡀⠀⠀⠀⢹⡦⣾⡁⡸⠃⠀⠀⠞⡋⡉⣩⣞⣀⡀⢴⣋⣈⣥⣠⡄⠚⠟⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉
⠀⠀⠀⢀⣅⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⣟⣻⣦⢀⠀⠀⠀⠀⠀⣀⠂⢀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀
⠀⠀⠀⠃⠘⣿⡏⢻⣆⠀⠀⠀⢸⣿⣷⣄⡠⢀⣁⣀⣒⣠⣤⣌⡁⠀⠀⠀⠀⠀⠀⣿⡿⣦⣤⣇⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣧⠈⡀
⠀⠀⠀⣠⣦⡿⡁⠘⠉⡇⠀⠀⠀⠻⢿⣟⠛⠟⠛⠻⠿⣿⣿⠛⣿⣤⡀⠀⠀⠀⠀⠹⣧⠈⠙⠻⡿⣦⣷⣦⠀⣂⠀⠀⠀⠀⠀⠀⠀⣿⠻⣷⣿⣦⣤
⠀⠀⢼⡏⠋⣼⣧⠄⠸⣿⠀⠀⠀⠀⠐⠉⢻⣷⣶⡄⢲⣤⣈⡆⠙⢿⠇⠀⠀⠀⢀⢳⣾⢷⡄⠰⣤⣤⣍⣉⠻⡟⢷⠀⠀⠀⠀⢀⣗⠾⣆⢉⡛⠈⠻⠿⣆
⠀⠀⠈⠻⠦⠿⠸⠧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠸⠿⠧⠼⠷⠽⠷⠦⠟⠀⠀⠀⠰⠾⠿⠋⠁⠠⠶⠾⠧⠉⠉⠳⠤⠌⠦⠀⠀⠀⠸⠿⠆⠠⠾⠟⠓⠛⠲⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⢀⠀⡀⣀⡀⠀⣀⠀⢀
⠀⠀⢀⡗⠀⠀⠀⠀⢀⣴⣿⣾⡆⠀⣼⣿⣷⠀⠀⠀⢀⡀⡀⠀⠀⠐⣾⣿⣷⣿⣿⣿⣾⣿⣿⡿⢀⣉⡇⣨⡽⢃⡀
⠀⢠⣾⣿⣴⡄⠀⠀⣸⣿⣿⣿⣿⡀⠁⠹⠏⠀⡐⡜⣨⡵⣿⣷⣤⡠⠙⠛⠛⡛⠛⣛⣛⣛⣋⠀⡸⣿⣜⡿⢳⡞⣖
⢀⣿⡟⠉⢿⣷⠀⠀⠻⣿⠿⢿⣿⠇⠀⠀⣠⣾⣷⣿⣿⣿⣿⣯⣿⣿⣦⣄⢼⣿⣙⢺⣟⡋⠛⠁⠘⠿⠻⣷⣾⠟⠋
⠀⠛⠄⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⣼⣿⣿⠋⠀⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⢰⣠⢼⡰⢠⢠⣀⠀⠀⠁⠊⠘⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠀⠢⠀⠀⠰⡄⣠
⠀⠀⠐⡄⠀⢰⣺⣽⣿⣿⣾⣽⣼⣿⣿⠀⠀⠀⠀⠀⣠⣦⣤⣄⠀⠀⠀⠀⠐⠀⠈⠼⠷⠁⣨⢷⣿⢕⠗⠀⢀⣘⡆
⠀⡄⣾⢁⠀⠈⠙⠛⡉⠉⠉⠙⠛⠛⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⠀⠀⣠⢞⢢⡇⠀⠀⢀⣈⣿⣿⣏⣀⠰⣞⣮⣷
⠈⢷⣙⣾⣦⠀⠀⣴⣻⣶⡄⠀⢀⡺⣿⣎⠠⣄⠈⠻⣿⣿⣿⣿⡿⢠⣶⣿⡏⣿⣷⡇⠀⠉⠉⠉⠉⠉⠉⠀⣤⣿⠇
⢀⣾⣼⣿⡏⠀⣼⡏⠹⢿⣿⠀⢢⣹⣯⣟⣵⡝⡄⠠⣿⡿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣗⠀⠀⢀⣠⡄⠀⠀⠈⠻⣧⣄
⠀⠛⠿⠟⠃⠀⠈⠓⠀⠘⠃⠀⠘⠛⠛⠛⠛⠛⠋⠀⠘⠇⠸⠋⠀⠰⠿⠿⠿⠿⠿⠟⠀⠶⠿⠿⠿⠷⠶⠀⠶⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣦⣀⡀
⠀⠀⣠⡠⡖⠻⠛⠉⠙⠓⠲⢤⡀⢀⣤⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⢹⣗⣿⢲⣲⣦⡀⠀⡠⠒⠁⡚⢉⣀⠀⠀⠁⠀⠀⠈
⠀⠀⣟⣻⣅⣆⣩⠿⢿⡏⠲⣤⢴⠋⠈⣿⠇⠀⢀⣀⣠⣤
⠀⠀⣿⡿⠛⠉⠀⠀⠘⣷⣆⡴⠁⠀⢰⣿⣦⢠⣿⠉⠛⢃⡄⠀⠀⠤⠤⣀⡀⠀⠀⣀⣴⣶⣶⣄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡏⣥⣴⠊⠘⣿⣿⡿⣿⣧⠀⠀⠀⠠⣄⣄⠀⠀⠉⠳⣾⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣶⣶⡤⠘⣛⢯⠁⠀⣴⡿⠀⠀⠘⣿⡇⠀⠀⠀⢹⡎⠁⠀⠀⠀⢸⡆⣸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣼⠋⣨⣽⣷⡶⢦⡰⠻⠿⣧⣄⡰⠆⠿⠿⢀⣤⣶⣿⣿⣦⠀⠀⠀⢸⢳⡟⣿⣿⣿
⠀⠀⠀⠀⠰⣿⠀⠈⠛⠷⣤⣄⣠⣤⡀⢠⣬⣍⣽⣶⣾⣿⣩⣭⣹⣧⣼⣧⣤⣤⣿⣤⣷⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⢻⡷⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠧⣅

⣏⣹⢿⣋⣉⢹⣉⡍⣟⣉⣹⣏⣉⢹⣉⡉⣿⣉⡉⣏⣉⢹⣉⣉⣿⣉⡉⣏⣩⢹⣏⣉⢻⣉⡉⣏⣉⣹⣏⣉⢹⣉⡉⣿⣉⡹⣏
⠛⠛⠒⠛⠓⠚⠛⠓⠚⠛⠒⠛⠛⠚⠛⠓⠚⠛⠒⠛⠛⠒⠛⠓⠚⠛⠲⠛⠛⠒⠛⠓⠚⠛⠓⠛⠛⠒⠛⠓⠚⠛⠓⠚⠛⠲⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣯⣷⣿⣽⣭⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡿⡇
⠀⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣠⣴⡶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⣿⠟⠀⢹⣿⣿⣾⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠿⣿⣿⠿⠿⠛⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣟⣫⣷⣿⣿⣿⣿⣿⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠘⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿
⣀⣀⣤⣀⣠⣄⣀⣠⣄⣄⣤⣀⣄⣄⣀⣠⣄⣀⣤⣤⣀⣄⣠⣠⣄⣠⣤⣠⣀⣤⣀⣠⣄⣀⣠⣄⣠⣤⣀⣤⣄⣀⣠⣤⣀⣤⣀
⡟⢻⡔⠛⣷⠘⠛⡇⠚⢻⡖⠛⢻⠘⠛⣇⠚⢻⡆⠛⢻⠐⠛⣷⠚⢛⡆⠛⢻⡐⠛⣷⠘⠛⡇⠛⢻⡖⠛⣳⠸⠛⡇⠚⢛⣦⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠻⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢴⠊⢿⣿⣿⣿⣧⡈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⢁⠍⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢾⣅⡐⣿⣿⣿⣿⣯⡐⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡨⢉⣤⣿⠏⠙⣿⣿⣿⣿⣿⣇⠀⠀⠰⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⡾⢃⣾⣿⠤⠤⣾⣇⠉⠻⢿⣿⣿⣷⡀⠀⠐⠆⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠸⠇⠸⣿⣠⠀⠠⠋⠙⠃⠀⠀⠉⠻⣿⣷⣶⣾⣶⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⣬⣿⣄⣀⣤⡾⡿⢿⣾⠃⣀⣉⠐⢆⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⢻⣦⢉⣿⣿⠀⠀⠰⢾⡏⠀⠛⠉⠀⠀⠀⠀⣀⣄⠘⣿⣿⣿⣿⣷⣾⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣿⡷⣾⠋⠉⠶⠄⠀⡸⠰⡟⠀⠀⠀⢀⣦⣄⣤⠀⣄⠀⠟⠻⣿⣿⣿⣆⠀⣉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠘⡟⢀⣴⣤⣷⡴⣿⠈⢻⣶⣶⣄⣠⣿⣿⡶⠀⢸⡇⠀⠀⠉⢹⣿⣿⣿⡟⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠘⢷⣏⣙⣿⠋⠛⠷⣤⣤⣄⢿⣿⠛⠛⡿⢿⢷⡎⠀⠈⠰⣤⣷⣶⢿⣿⣿⣄⢀⠰⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠛⠻⠿⣤⣶⣤⠀⠈⠟⣠⣿⠇⣠⣿⠒⡿⠃⣀⣛⣿⠿⠿⠏⠀⢹⣿⡿⠿⢷⣴⠖⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠆⠀⠀⢀⡀⢀⣄⠀⣀⣤⠀⢺⡿⢿⣿⣿⣿⣧⡀⢰⣿⡟⡟⠃⠀⠐⠛⠿⣿⡇⢀⣾⡅⢀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣤⠀⣴⠞⣿⢧⣤⣶⣿⣏⡉⠻⣧⠽⣿⣿⣿⣿⣿⣿⣿⣽⢻⣤⢛⣼⣿⣷⠋⣠⣾⣿⣷⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣤⣿⡄⠉⠐⠉⢿⣷⡿⠋⣀⣹⣶⣾⣿⣿⣽⠟⣭⣿⠏⠸⣿⣿⡝⠋⣹⣤⣽⡛⠛⣯⡀⠀⢈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⣿⣿⣷⢆⣤⣥⣿⣿⣿⡀⢉⡀⢺⣿⣟⣩⣿⣿⣿⣿⣶⣿⣿⣶⠷⢎⡁⠀⠚⠉⠙⠛⢿⡿⠄⢁⣤⠺⣿⠆
⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠈⣿⣿⣿⣤⣤⣈⣀⣨⡉⢿⡿⠉⢻⡿⢿⣿⠟⠛⠿⢟⢻⡇⠻⠛⢠⣀⣿⠀⠀⠀⢺⡆⢼⣧⣄⡈⢉⣥⡀
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣯⣥⠘⣷⣀⣼⠓⢂⣠⣶⠶⡶⠋⠈⠿⠀⠀⣟⠻⢻⣤⠄⠀⠀⠛⣮⣉⣾⣧⣤⣽⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢀⠘⠙⢿⣿⣿⣿⣿⣿⢿⣿⣿⣾⣿⣿⣿⡷⢏⣹⣿⣿⣧⣼⣇⠀⠀⠾⣧⣴⣿⡁⠀⠀⠀⠀⠉⠁⢩⣙⣿⠟⠉⠀⣡
⠀⠀⠀⠀⡀⢀⡇⣀⣼⣧⣷⣤⣿⣿⡿⠛⣿⣿⠉⣿⣿⡿⠛⢿⣤⣾⣿⣿⣿⣿⣿⡻⣿⣶⣾⡏⢁⣄⠙⢓⣤⣤⠄⣤⣴⠛⠛⠻⣶⣾⣷⡿
⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⡙⣻⣿⢁⣀⠁⠙⡟⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⠟⢻⣧⣤⣶⣤⣉⣋⣛⣯⣤⣾⡇⢠⣿⣿⣿⣇
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣉⣿⣿⠖⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢙⣷⠂⣹⣿⢉⠽⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣶⣷⣄
⠈⠿⠿⢻⣿⣿⣿⠛⡜⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡿⠋⠙⣿⣶⣿⡁⢙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟
⠀⠀⠀⠀⠉⠙⠃⠰⠿⠿⢿⣿⣯⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⡿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⣾⠿⠿⣿⡟⣛⣀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠈⠛⠿⠟⠿⠿⠛⠛⠛⠻⣿⣿⠷⠿⣿⣟⣿⠖⠋⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⠏⠙⢻⡏⠉⣙⠛⢛⣏⠛⢹⣽⠟⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠊⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠐
⠀⠀⠀⠀⠂⣠⠈⠛⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡝⠀⣄
⠀⠀⠀⠀⠙⢟⢁⠀⢳⠑⢄⠘⢉⡆⠀⠀⠀⢸⣿⣿⣷⡄⠀⠀⠀⡐⠢⠀⠀⠀⣠⣾⣿⣿⡀⠀⠀⠀⢐⣈⠃⡠⢾⣿⠀⠀⡛⠋
⠀⠀⠀⠀⠀⠀⢣⣣⠸⡇⠈⣦⣻⣧⡀⠀⢠⣿⣿⣿⣿⣿⢦⡀⠀⢛⡆⠀⠀⠈⠛⠻⢿⣿⢷⡀⠀⢀⣼⣯⠜⠀⣼⣇⣀⠌
⠀⠀⠀⠀⠀⠀⠀⢻⣿⢧⠀⣿⣿⣿⣿⣴⢟⣼⣿⣿⣿⣿⣷⣝⡷⡉⠉⠁⠀⠀⠀⠀⢸⣿⣯⠻⣴⣘⣭⡆⠀⣴⠘⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡘⠀⢹⣿⣿⠿⠺⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣾⣽⡞⠀⢸⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢼⣟⣛⡤⢤⣌⣩⣭⣿⣭⣭⣭⣭⣍⣁⣀⣀⣀⣀⠉⠉⢉⣙⡛⠿⣿⠿⠃⢠⣾⠇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡙⢩⣽⣿⠉⠉⠀⢾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣾⣿⣽⠛⠒⣚⡛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣸⣿⣷⣄⡀⣠⣾⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠻⢿⠿⠿⢿⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠙⠲⠾⣿⣿⣿⣿⣿⣿⣛⣛⣛⣋⣓⠂⠀⠀⣴⣤⣤⣤⣤⣤⣬⣟⣩⣅⣉⣉⣉⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠻⣿⣟⣛⠛⠛⠿⠏⠉⠉⠉⠉⠉⠀⠠⠿⠿⠿⠿⢿⣿⣿⣟⣿⣿⣯⣭⡷⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠓⠒⠚⠚⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉








⣿⢿⢿⠿⢿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡻⢶⣶⡲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⣶⣶⠾⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢀⡴⠛⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⡟⠳⡄⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠤⠴⠋⢻⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡿⠋⠲⠤⠼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⠀⠀⠀⠀⠈⠂⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣢⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⢧⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣷⣦⣀
⠀⠀⢀⣤⣀⢀⣤⣴⠾⢶⣦⠀⠀⠀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄
⠀⠀⠈⣿⢻⠟⠻⣿⣿⣷⣷⣤⣀
⠀⠀⠀⢿⣞⠀⢸⠃⠀⡹⠙⠻⠷⣬⣤⣀⡀
⠀⠀⠀⠀⠙⠛⠿⠶⠛⠁⠀⠀⠀⠀⠈⠉⠙⠛⠳⠦⠀⠀⠀⠂⠀⠠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠂⠡⣭⣤⣖⣂⣂⣀⣠⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⢈⣭⡄⠤⠬⠴⠦⢠⣤⡉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢈⣤⠚⠋⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠙⠓⣤⡐⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢣⡶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡄⠀⠀⠀⠀⠉⢶⡅⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠄⣴⠂⣀⣀⣄⣤⣠⠄⠀⠀⠀⢀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠐⣦⠛⠄
⠀⠀⠀⠀⠀⠀⢠⠌⡶⠀⠀⠉⠈⠀⠀⠀⢀⡀⠐⢧⡜⠁⣠⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⢶⠸⠂
⠀⠀⠀⠀⠀⠀⡀⣴⠀⣀⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣾⡿⣯⣿⣿⣝⣿⣿⡇⠀⠀⠀⠀⣦⢀
⠀⠀⠀⠀⠀⢠⠅⡍⠸⠿⠆⠀⠀⠀⣀⣀⣤⣤⠀⣴⣿⣿⡿⣿⣿⣿⣿⣽⣿⠇⠀⣀⣀⠀⢩⣌⡀
⠀⠀⠀⠀⠀⢸⣈⡃⠈⠙⠀⠀⠀⠛⠛⠿⡿⣫⣾⣿⣿⣿⣿⣮⣿⣯⣿⡿⠃⠰⠦⣾⡷⠀⢘⡁⠂
⠀⠀⠀⠀⠀⠐⠊⠇⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠒⠂⠀⠸⢀⡅
⠀⠀⠀⠀⠀⠀⢤⠻⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⣛⡁⠀⠀⠀⠀⠀⠘⠷⠂⠀⠟⠼⠂
⠀⠀⠀⠀⠀⠀⠘⠄⢷⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠙⠿⠫⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡾⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠰⣆⢳⡄⠘⠇⢀⡀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢠⡦⣤⣠⣤⡤⠀⢠⡞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠦⢀⠈⠁⠀⠀⠀⢼⣧⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀⠴⢃⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠙⠓⢤⣄⠀⠀⢻⠿⠀⠀⠀⠀⠀⠀⣠⡤⠚⠋⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⠀⠀⠉⢹⣶⣶⠀⠐⣛⢣⠡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⣼⡟⠁⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁⠀⠀⢀⣼⠀⠀⠀⠀⢀⣼⣿⣿⡟⠀⠀⠀⠀⣼⣿⠁⣸⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⠀⣠⣾⠇⠀⠀⠀⣠⣿⣿⣿⡟⠀⠀⢀⠃⢰⣿⣿⢀⡇⢸⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⢀⣼⣿⡏⠀⠀⣠⣾⣿⣿⣿⡟⠀⠀⠀⣿⠀⢸⣿⡟⠸⡇⠋
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠀⣰⣿⣿⡏⠀⣠⣾⣿⣿⣿⣿⠋⢠⠂⠀⢰⡿⠀⢸⡿⢁⡆⠁
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⡇⣸⢿⣿⡟⢀⣾⣿⣿⣿⣿⠟⠁⡰⢃⠆⢠⣿⠇⠀⡞⢁⡾⢱⠆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⠟⠁⢸⢻⡿⢰⡏⣸⣿⣰⣿⡿⢋⣾⡿⠋⠄⡼⣡⠟⣴⡿⠃⠀⠀⣰⣟⣷⠏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⣿⠏⠀⠀⣼⠈⡇⣿⠀⣿⣿⣿⡟⢀⣾⣿⢃⡏⣼⣷⣯⣾⡿⠁⢀⣴⣾⣿⠟⠁⠀⠀⣀
⠀⠀⠀⣶⠀⣼⢿⡀⠀⠀⢸⣿⠀⢠⠀⡟⠀⠃⠟⢸⣿⣿⠛⠀⣾⣿⠇⣾⢡⣿⣿⣿⡟⠀⣰⣿⣿⠟⠁⠀⣠⣴⣿⠇
⠀⠀⠀⣿⠀⠸⣟⣧⠀⠀⢸⡇⠀⣾⣴⡇⠀⠀⠀⢸⣿⠇⠀⢸⣿⡿⢸⣿⢸⣿⣿⣿⡇⣼⣿⣿⠃⢀⣴⣾⣿⡿⠃⢀⠆⢀⠄
⠀⠀⠀⣿⡄⠠⢹⣿⡆⠀⢸⠃⢠⣿⣿⠃⠀⠀⠀⢸⡏⠀⠀⣿⡿⡇⣿⣿⢸⣿⣿⣿⣷⣿⡿⠁⢀⠼⡫⠓⢉⣠⡶⣋⠔⠋
⠀⠀⠀⣿⣧⠀⣧⣻⣷⠀⢸⠀⣼⣿⡿⠀⠀⠀⠀⢸⠃⠀⢠⡿⢰⢱⣿⣿⠸⣿⣿⢹⣿⣿⡇⢀⡾⠋⣠⣶⣿⡿⠛⠁
⠀⠀⠀⢻⣿⣧⢸⣿⣿⠀⡀⢠⣿⣿⠃⠠⠀⠀⠀⣎⠀⠀⢸⢇⡟⣼⣿⣿⢸⣿⣿⠸⣿⣿⡇⢸⠇⣰⣿⣿⡩
⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⢃⣾⣿⠃⠀⢰⠂⠀⠀⠋⠀⢸⣿⡼⢡⣿⣿⢇⣾⣿⡟⣸⣿⣿⡇⠀⢰⣿⣿⠋
⠀⠀⠀⠀⠀⢻⣻⣿⡟⠀⣾⡿⠃⡀⠀⣸⠃⠀⠀⠀⢠⣿⣿⢃⠟⣸⢏⣾⣿⣟⣵⣿⣿⡟⠀⣴⣿⠟⠁⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠈⢿⣿⠃⣼⡿⠁⣼⠃⣰⣿⠀⠀⠀⠀⣾⣿⢃⢂⢠⣫⡾⣻⣿⣿⡿⠟⠋⡠⠞⠋⠀⠀⠀⠀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡟⣸⡟⠀⣼⠃⣰⣿⡏⠀⠀⢀⣼⡿⠁⢂⣾⣰⣟⣽⣿⠿⠋⠀⢀⣠⠄⠀⢀⣠⣴⠾⠟⢉⣴⠋
⠀⠀⠀⠀⠀⠀⢀⢱⡟⠀⣸⢃⣼⣿⡿⠁⠀⢀⣾⠟⠁⠀⣼⡇⣏⣿⡟⠁⠀⣠⡾⠋⠀⣠⡶⠟⠉⠀⣴⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢰⣿⣿⣿⡟⠁⠀⢀⣾⠋⠀⠀⣼⣿⣇⢻⡟⠀⢠⣾⡿⠀⢀⠾⠋⠀⢀⣴⣾⢿⣿⡏⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠁⠀⣾⣿⣿⠋⠀⠀⠀⡾⠁⠀⠀⣼⣿⡟⢻⢸⡇⣰⣿⣿⠃⠀⠁⠀⠀⣰⣿⣿⣳⣿⡟⠀⡜⣯
⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⣿⠃⠀⠀⠀⠀⠁⠀⢀⡾⢋⣿⡧⣀⣼⣠⣿⣿⠏⠀⠀⠀⢀⣼⣿⡿⣷⣿⠟⠀⡰⠁⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣰⠏⣴⣿⣿⡇⣿⣿⣿⣿⠋⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠀⡰⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡰⢡⣾⣿⣿⣿⣼⣿⠿⠋⠁⠀⠀⠀⣰⣿⣻⣿⡿⠋⠀⢀⡔⠁⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⡄⠀⠀⢀⣴⢟⣽⣿⡿⠛⠉⠀⠀⠀⠀⠀⣠⣾⣿⡷⠛⠁⠀⣠⣴⣿⣶⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⣼⠁⠀⢀⠞⣱⡿⠋⠁⠀⠀⣀⣤⣔⣀⡴⠞⠛⠉⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⣳⠏⠀⠀⢊⡞⠁⠀⢀⣤⡾⠿⠋⠁⠉⣁⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡘⣼⣿⠏⠀⠀⠀⢋⠀⢀⡴⠟⢁⡀⠀⠀⠔⢊⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⢀⡴⢃⠤⢊⣠⠞⠁⠀⠀⣠⣾⡿⠿⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⣼⡿⠋⠀⣠⣴⣾⠟⠀⢀⣴⡿⠁⠀⠀⣠⠾⠛⠁
⠀⠀⠀⠀⣼⠋⢀⣴⣿⡿⠟⠁⣠⡴⠿⠋⠀⠀⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠐⠀⠴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠚⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠛⠿⠿⢷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣶⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣾⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⣿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⣿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⡶⢒⠀⠈⢰⢠⣷⣄⠴⠖⣲⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⠫⣅⠢⣷⡓⠀⠀⠃⠠⣼⠟⢿⣿⣿⣿⠿⣍⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢝⣥⣏⡙⠛⣿⣧⣽⡧⢤⡤⢀⣽⣧⠀⣉⠁⡈⢰⣿⡿⡟⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣥⡾⣿⣦⣩⡿⢶⠏⣻⣷⣽⣿⣓⣚⠋⠻⣷⡟⢻⡇⠀⣗⠀⣼⣉⣳⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⠜⡋⣹⣾⡟⣃⢌⣹⣽⠟⣿⠂⠿⣿⡟⢿⣿⢿⣿⣹⣬⡻⣄⣷⣤⣉⡄⠙⣿⣏⢅⠙⢿⣶⡄
⠀⠀⠀⠰⠟⠩⣍⣩⡿⢧⡡⣿⡿⢿⣼⠃⠀⣀⣨⠀⣿⡜⣿⣷⡷⣾⣿⣿⠿⣿⣿⣿⣮⣿⡟⠻⠿⣶⣎⢻⣎⡙⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠛⠀⠀⠀⠀⠙⠛⠋⢗⢈⣇⠸⣿⣷⡹⢻⣿⣶⣮⣿⡿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣩⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡉⢻⡓⠉⠹⢿⠿⣿⣿⣞⣷⣶⣌⠻⠿⠿⣿⡿⠯⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠨⠀⠀⢀⠘⠀⠛⠩⠻⢿⣏⡿⢷⣿⠓⠪⠱⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉⠉


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣦⣷⣴⣤⣶⣤⣾⣵⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⢠⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢾⡿⠚⢠⠏⠀⠀⠘⢿⣿⣿⣿⣿⣯⡼⠃⡩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠠⣈⠓⠀⠀⠿⡀⠀⠀⢡⠈⢿⣿⣿⠿⢋⠀⠊⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⠆⠀⠀⠸⠾⠀⠀⠀⠁⠀⠀⠀⠀⠉⠡⣥⡐⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢢⠀⠄⠀⠵⠄⠰⡆⣀⣤⡔⡒⠒⠶⠤⣀⠀⡇
⠀⠀⠀⠀⠀⠄⠀⠀⢰⢠⡇⠁⣤⠘⢆⢠⢀⠈⠵⢳⡏⠉⠉⠀⠁⣾⢠⣀⠀⠙⢧
⠀⠀⢀⡀⠀⠀⠀⠀⣾⣼⡀⢰⡌⠂⢘⣸⡆⠀⠀⣠⠠⠠⠖⠆⢸⠟⢿⠉⢇⠀⠸⣄⣠⠀⠀⢀
⣠⣘⣦⡇⣀⣤⡄⠄⡑⣿⠃⠀⠉⡖⠁⠸⣧⡀⠀⣀⠤⣒⡒⠢⡀⠀⢼⣧⡀⠀⠀⠀⠁⠀⠀⠀⡀
⠘⠻⠟⠙⢿⣷⠶⠋⢸⡟⠀⠠⡇⠈⢀⠀⠸⠃⠴⣥⣥⣁⠀⠀⠉⠑⢢⣌⢷⣤⣀⣀⣀⣀⣀⣀⣈⠂
⠀⠀⠀⠀⠘⠈⠀⠀⠈⡷⠀⠀⠁⠀⠈⢁⡀⢀⣀⡈⢙⣇⠙⠪⣝⢢⡈⢹⣿⣿⣿⣿⡿⠦⡀⠙⡛⠿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠘⠆⠉⠄⠘⠂⣻⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣧⠀⡄⠀⠱⣈⣒⡒⠿⢿⣷⣤⡀⠀⠀⠀⣀⡄⣤⣤⣦⠴⣛⣽⣿⣿⣿⢻⣿⡆
⠀⠀⠄⢳⡈⠀⠻⠀⠴⣣⡟⠛⠻⣶⣤⣭⣉⣛⣛⣯⣽⠞⣉⣩⣵⣚⡙⣡⢾⣿⡟⣾⣿⣿⢀⢀⠤⡀
⠀⠀⠈⠈⢷⡀⣀⣀⣤⡽⠀⣀⣠⠟⠛⠛⠻⠿⠿⠷⢾⣿⣿⣿⣿⣿⠥⢰⣿⠟⣼⣿⣿⣇⣾⡼⢇⠀⡆
⠀⠀⠀⠀⠀⠉⠉⠊⠉⣇⠈⡿⠃⠀⠰⣄⠀⠀⠐⢤⣀⣹⣿⣿⠏⢻⣿⣿⣿⢸⣿⣿⠟⢧⠈⡛⠆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠀⡀⠀⠈⢷⡀⠀⠀⠈⢻⣿⡇⠐⢁⣿⣿⡇⣾⡿⠁⠴⠈⢣⣷⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢠⢠⠀⠀⢀⠀⢿⡄⠀⢀⠀⣿⠇⠐⡡⣸⣿⣾⡟⠀⣾⢀⠀⠀⠙⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢸⡆⠀⢸⡇⠘⣿⡄⠸⣧⣿⢀⣾⣾⡟⠋⢾⣀⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣇⠃⠀⣿⠃⢠⣿⣷⡀⣿⣷⣿⣿⡿⠡⠀⠘⣿⣧⠙⠓⠿⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠳⠴⠿⠶⢿⣿⣿⣷⡿⠿⣿⡿⠞⠁⠀⠀⠈⠙⠳
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⡋⠉⠉⠉⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡖⠂⠘⠉⠓⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⡀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠈⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢤⠀⠀⠀⠀⠀⡤⢄⠐⡄⠀⡆⢠⣀⡀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⠂⠑⠟⢦⡷⣄⡗⠈⠆⡞⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠀⠀⠀⠀⢸⠀⢿⣆⢀⡀⠘⠁⡤⠃⢰⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠒⢫⢀⠀⠀⣦⢸⠀⠀⠉⠀⠀⠀⠀⣱⡸⠉⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡄⢠⣟⠃⡀⠀⠄⠀⠀⣠⠾⠀⢳⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⡶⡷⣤⡤⣤⣖⡟⡇⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡿⠽⠯⠖⢹⢸⡴⠁⠀⢠⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢊⠔⠉⠳⠦⢄⣴⣯⢅⣀⣀⡴⠋⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⢻⠞⠁⠀⠀⠀⢰⠃⠀⠀⠀⠀⠉⠳⠖⠤⣬⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠯⠋⠁⠀⠀⠀⢀⡠⢦⠀⠀⠀⠀⢰⠀⠀⠀⡰⡁⠀⠈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠘⡄⠀⠀⣠⠤⣷⠖⢉⡀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⢹⡖⣫⠴⠚⠁⠀⠀⠀⡀⠀⠀⠀⠀⢄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⠀⠀⠀⡠⠊⡩⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠥⣀⣀⣉⣒⣤⠜⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢲⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢁⡀⢀⡀⢀⡀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠆⠰⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠛⠶⣤⣠⣀⢻⡖⠐⠂⢲⣖⣁⣠⣤⠖⠛⠉⠉⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣠⣤⣶⣦⣤⣤⣄⣀⣉⣹⡇⣿⣿⢸⣏⣉⣀⣤⣤⣤⣴⣦⣄⡀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣤⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡘⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢿⢇⠛⠛⠸⠿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⢇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢹⣿⡿⢶⣾⣿⣿⣿⡏⢿⣦⡀⠶⠶⢀⣴⠟⢹⣿⣿⣿⢱⣦⣿⣿⡟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢾⠟⠔⠜⣥⣄⢹⡇⢠⡌⠁⣴⣦⠉⢡⡄⢸⠏⣶⡌⠣⠦⢻⡿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⡀⢨⡮⢸⡇⢈⣡⡦⠙⠋⣴⣌⠀⢸⡆⣶⡄⣀⣀⡿⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣐⣋⣀⡀⠘⢧⣿⣷⠿⠋⠀⠛⠓⠈⠛⠷⣼⣷⡼⠁⢀⢀⣛⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠩⠿⢟⠛⠲⠶⠤⠤⠤⠤⢤⠤⠤⠤⠤⠤⠖⠖⠛⠻⠯⠍⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠐⠟⣢⠀⡌⢵⡌⡀⠀⢰⣿⣷⠀⠀⡀⣴⡍⠄⠀⠐⡻⠀⢸
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠓⠚⠿⠤⠤⣤⣜⣈⣀⣀⣘⣻⣛⣀⣀⣈⣐⣣⣤⡤⠴⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⡄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⠉⠉⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢉⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠻⣼⡋⠀⠀⠀⠀⢀⡴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣄⢨⡇⠀⠀⢀⡴⠋⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢹⠦⣭⣛⣛⣿⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⠓⠶⢴⣋⠀⢠⡼⠋⡇⠀⢠⡟⠶⣶⣄⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠴⠛⠉⠉⡇⠘⠳⣤⣀⠈⠓⢦⠀⠀⠻⣄⣸⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⢠⣤⠀⠀⠀⠀⠀⢿⡀⠀⠀⠈⣽⣆⣸⠃⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡏⣿⠀⠀⠀⠀⠀⠈⠙⠛⠛⢻⡇⢹⠇⠀⠀⠀⠀⠾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⢸⡆⠀⠀⠀⠀⣠⠦⣤⢀⡼⠁⣯⣤⣤⠀⠀⡶⢤⣄⡉⢳⠄
⠀⠀⠀⠀⠀⠀⢀⣟⣠⠟⢠⡞⠁⠀⢀⣤⠞⠁⠀⠀⠈⠓⢦⣽⠀⢸⡇⢰⡇⠀⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢸⡀⢰⠦⣄⡉⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠃⣼⠀⢀⡞⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠙⣇⣸⡇⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⠈⠳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢸⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢿⡀⣧
⠀⠀⠀⠀⠀⠀⠤⠀⠶⠶⠶⠾⠶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠾⠷⠶⠶⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣠⡴⠶⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠠⠟⠋⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠐⠁⠀⠀⠀⠀⠀⠐⡠⠀⠈⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢤⣤⣾⣿⣿⣽⠟⠡
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠋⢉⣆
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠉⠀⡠⠋⣸
⠀⠀⢠⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⢀⣠⣴⣾⣿⣿⡿⠟⠋⠀⢀⡠⠊⢀⠜⠉
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡉⠀⠀⠠⠐⠁⣠⠔⠁
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠈⠊⠠⠀⠒⠉
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠙⠻⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡴⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⠀⠀⠀⠀⣆⠀⢃⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⡏⠳⢄⠐⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡈⠀⠀⠀⠀⠹⣀⠄⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠌⡇⢀⡄⠀⠀⠀⠐⢤⠀⠀⠙⢆⠀⠀⠀⠀⣠⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⡏⢸⠱⡀⣀⠀⠀⠀⠑⣄⠀⠈⣇⠀⠀⠌⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠤⠐⡇⠀⠑⠹⡌⣏⢢⡀⠀⢹⢀⠈⢸⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠲⡀⠻⡄⠀⠀⢀⡇⠐⡧⢠⠏⠸⠀⢸⠀⢠⡾⡀
⠀⠀⠀⠀⠸⡗⢄⡀⠀⠀⢳⠀⠡⠈⠢⡈⢢⣀⠞⢠⠀⠿⠁⡐⢀⠀⡜⠀⢸⠁⢇
⠀⠀⠀⠀⠀⠑⢄⠈⢦⠀⠀⠱⣄⠡⠀⠈⢄⠛⡐⠁⠃⡠⠊⠀⠌⡸⠁⣠⠏⠀⢹⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠐⠲⡤⣍⠀⠀⠀⠘⡆⠑⠀⠈⣤⠁⠀⠞⠀⠀⠌⣰⠵⠚⢁⡄⠀⣼⠀⠀⡠⢊⡾
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠉⠢⡀⠀⢸⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡁⠔⠂⡑⠀⣰⠃⠀⠨⠐⣉⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠐⢆⠹⡄⡞⠠⠀⠀⠀⠀⠀⠀⠀⠰⠎⠀⠀⡎⢀⡔⠁⣠⢀⡴⢋⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠂⠛⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡞⣰⢾⢃⡎⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠘⡀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠀⡿⢨⢸⣼⠰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢁⠈⡄⠁⠁⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠒⠀⢃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠄⠀⠀⠀⣀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠨⠀⠄⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠠⠊⠁⠀⠈⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣅⡀⠀⠀⠀⠀⠀⠀⠨⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⡭⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢜⠉⢹⣿⡀⠹⣿⡘⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠐⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡁⣶⣶⣾⠀⢰⣿⡇⠀⢋⣛⡙⠿⣆⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⣶⠟⠫⠙⣾⣿⣷⡇⠀⠈⠹⡇⠀⢹⡟⣷⣶⣤⠘⡄⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⠋
⠀⢀⡀⠠⠶⣶⣶⡟⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠃⠋⣿⣿⣿⣿⡞⢉⡿⠇⣄⢲⢆⣶⣿⢀
⠀⣸⠣⠒⢊⣛⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠂⠀⠘⣿⡿⢿⣿⣷⡄⠆⢠⣾⣿⣟⣋⠀⠀⢀⡀
⡸⢇⠤⠉⣸⣿⠃⠀⠀⠀⣀⣮⠭⠝⠛⠀⣾⣞⠃⡠⠀⣀⠀⣀⣡⡾⣻⣿⣄⣠⣖⣚⣉⠁⠙⣿⣝⣇⠀⢸⣿⣷⡄
⠃⠌⠀⠸⠛⢁⠀⠊⣀⣸⠟⠁⠀⠀⢀⣰⣿⢟⠀⠀⠆⠠⠈⣩⣿⣿⡿⡻⠿⠗⠊⠉⠈⠃⠀⢿⣿⣿⣞⡉⠉⣿⣷
⠆⠀⠀⠀⠀⠉⠀⢈⡟⠋⠀⠀⠀⢠⣯⡿⠓⠐⠃⢋⠃⠀⢠⡽⠋⠙⠒⠀⣀⠀⠀⠀⠀⠀⠐⠚⠛⠻⣿⡇⠤⠾⣿
⠀⠀⠀⠀⠈⠀⠀⠊⠀⠀⠀⢀⣿⡿⠋⠀⠀⠀⡀⠚⡅⠀⠈⣀⠄⠀⣹⠋⢠⡝⠲⠀⠤⠤⢤⡀⠀⠐⠁⠙⠀⠀⠹
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠄⢀⣀⣤⡼⢷⡄⠃⢘⠀⢀⡏⠑⠤⠆⠂⠐⢉⡀⠑⠒⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠟⠉⠀⠀⢄⡀⠈⢹⣿⡀⠀⠉⠃⢀⠀⠛⢉⣛⣛⡟⠋⠠⢬⠀⠀⠀⠀⣿⣿⣷⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠋⠁⠀⠒⠀⠙⠆⠀⢉⠁⠀⢀⠄⠩⠷⠐⡄⢡⣴⠀⠀⠘⢲⣿⣷⣶⣶⡘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡐⠁⣀⣐⣂⡄⣻⢰⣿⡿⠀⠉⢻⣿⣿⣿⣿⣿⣦⡅⠚⠿⢿
⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⠉⠈⣧⠀⠀⠛⢐⣥⣾⣿⠄⣁⠀⠁⠈⠐⠣⣤⠀⠊⠾⠿⠿⠿⠛⠛⠛⠿⠿⣿
⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠱⠶⠛⠋⠉⢉⠀⣴⠁⣠⣔⣤⣤⡔⢈⣾⠋⠁⠀⠀⠀⢈⡆⠀⠀⠲⡄
⢲⣿⣿⡄⠀⠀⠀⠀⠀⢰⠀⡀⠀⠀⠀⠸⣶⣤⠚⠁⠒⠰⣼⣷⣯⣤⣋⣨⣿⠃⢿⡷⣰⠀⠀⢀⠀⢀⣿⠰⡄⠀⠒
⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⢴⢣⠀⠀⠀⡄⠀⠃⠠⡦⠙⠛⠉⠀⠘⠛⣿⣿⣃⣠⣿⣶⡯⠀⣠⠊⣠⣼⡇⡜⣷⣴⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠉⡾⡆⠀⠈⠀⢀⡄⠀⠀⠰⡄⠀⠀⣴⣿⣿⣿⡶⣿⡟⠀⠀⠁⣼⣿⣿⣧⣷⣿⣿⣷
⠀⠀⠀⠀⡀⠠⠄⠀⠀⣀⠔⠪⠀⣿⡆⠈⠙⠌⠠⣤⠁⡠⠖⣆⠀⣀⣠⣼⡿⣾⡯⠚⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏
⠀⠠⠄⠀⢀⣄⠤⠒⠛⠛⠆⣴⣿⣿⠸⣄⣿⢆⣦⣽⣎⣅⣠⢉⣉⣥⣿⣿⣷⡿⠁⠀⠀⠀⠀⠀⠈⠛⠿⠟⠛⠋
⠠⣶⣤⢀⣴⡿⣄⠀⢈⣠⡶⣿⣿⣿⣀⣼⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡆⣀
⠀⢸⣿⣾⣹⣕⣽⣿⡿⠟⠁⣽⣿⠟⠛⢁⡟⠟⡿⣟⢿⣷⡀⠀⠀⣸⣿⡟⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠂
⢶⣾⡿⠸⣷⣝⣇⣤⣴⣶⠾⢫⠏⠀⣀⡌⠀⠀⠇⠘⡆⢻⣷⡀⢰⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⢹⣿⡄
⣿⣿⠁⠀⣿⣿⣏⠐⠋⠁⠴⠭⠒⠛⠃⠀⠀⠀⠀⠀⠈⠀⣿⡇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⣿
⠉⠀⠀⠀⠈⠉⠛⠛⠛⠓⠒⠀⠀⠀⠀⠀⢀⡀⠀⠀⡠⠀⠉⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠤⢄⠺⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⡶⢖⠒⠘⠉⠉⠉⡉⠙⠑⠐⢒⠖⣦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣾⣿⣉⡏⠻⡐⢿⢻⠄⠀⠀⠀⠀⠃⠀⠀⠀⡘⢿⠟⣰⠋⡍⣩⣿⡶⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠫⢀⠀⠘⠟⡉⡇⠀⠙⣌⠧⠠⠀⠀⠀⢸⠀⠀⠀⢀⠠⢮⡾⠁⠀⡟⡌⠿⠀⢀⠨⠓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡉⠤⡀⢊⢫⣤⠀⠀⠁⡇⠀⠀⠈⢻⣖⣥⠀⠀⠈⡀⡀⢠⣷⣰⠊⠀⠀⠀⡇⠈⠀⢠⣾⢣⠐⡰⠎⠱⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣞⣤⣳⣪⡄⡄⠀⢣⡈⢷⡄⠀⡇⠀⠀⠀⠀⠙⣿⡇⣾⢸⡇⡐⣾⡿⠁⠀⠀⠀⠀⣇⠀⣰⠏⣠⠃⠀⡤⣆⡳⣧⣘⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡹⣯⠙⠻⢷⣷⣮⡐⠂⢱⡌⣿⣆⡇⠀⠀⠀⠀⠀⠈⠻⣧⣜⣣⣷⠏⠀⠀⠀⠀⠀⠀⣇⣾⡇⣴⢃⠖⢊⣴⣿⠿⠛⢩⡿⣱⣄
⠀⠀⠀⠀⢀⠾⠻⢻⣿⣷⡀⠀⠈⠙⠛⢿⣮⣻⣈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠁⠀⠀⠀⠀⠀⠀⠰⣿⣏⣺⣯⣶⠿⠛⠉⠀⠀⣰⣿⣿⠛⣻⢦
⠀⠀⠀⡠⠁⠀⠀⠂⠛⠿⣿⡄⠀⠀⠀⠀⠈⠉⠛⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠋⠉⠀⠀⠀⠀⠀⣴⡻⠟⠉⠀⠠⠈⠡
⠀⠀⠠⠃⠀⠀⠀⠐⠢⡄⣠⡽⣆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡾⣥⣀⡤⠔⠀⠀⠀⠀⠳
⠀⢠⣷⠶⢴⣤⣄⡀⣀⠈⠛⠯⣝⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡬⠟⠋⢀⢀⣀⣠⢔⠤⡶⣧
⠀⣾⣝⣷⣖⣐⢈⠩⣟⡻⢓⣦⣌⣙⣷⡀⠀⠀⠀⣠⣤⡴⠶⠦⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠶⠶⣦⣤⡀⠀⠀⠀⣰⣟⣉⣤⣖⠻⣛⡾⢉⢀⣐⣲⣽⣼⡆
⢠⣿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠁⠀⠀⠈⣁⠤⠴⣄⡄⠀⠈⠱⡄⠀⠀⠀⠀⡴⠉⠀⠀⣤⡴⠤⠄⡉⠀⠀⠀⠁⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⢿⣧
⢸⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡪⢶⣿⣯⣽⡟⠢⢀⡀⠰⠀⠀⠀⠰⠀⣀⠠⠚⣿⣯⣿⣷⠮⣒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿
⡼⢿⣿⣭⠛⠿⡿⢶⣦⣄⡀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠚⠉⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠟⢻⣽⣿⠿
⣆⣈⣭⣭⣩⣄⣈⣛⣹⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣝⣋⣀⣤⣭⣭⣅⢀
⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛
⢰⣢⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣤⣲
⠸⣯⣽⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠤⠄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿
⠀⢿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣀⠀⢀⣀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿⡇
⠀⠘⣿⣿⣿⠿⠿⠋⠛⡻⢟⣫⡿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢺⣏⠉⠓⠊⢉⣻⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣛⠯⡛⠉⠻⡻⢿⣿⣿⡟
⠀⠀⠹⡟⠃⠀⠀⢀⣠⡴⣨⠷⣛⣿⡿⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠉⠁⣤⡄⠉⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⣿⣟⡻⢭⣳⣦⣌⠀⠀⠀⠛⡿⠁
⠀⠀⠀⠙⡄⣤⣺⡟⡫⡓⣦⠶⢉⡾⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠘⣎⠑⢦⣗⡛⠻⣿⡒⡄⡜⠁
⠀⠀⠀⠀⠘⣍⢐⣸⠿⣛⠃⢀⡿⠁⠀⠀⠀⢀⣠⠶⠋⣟⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣟⣏⠳⢦⣀⠀⠀⠀⠀⠹⣇⠀⢝⡻⢿⣀⢈⡝
⠀⠀⠀⠀⠀⠀⠻⡩⢀⣜⢲⡾⠁⠀⣀⣴⣾⡿⠅⠀⢐⠛⢻⡄⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⣸⠛⢓⠀⠀⠯⣿⣶⣄⡀⠀⠹⣶⢺⣅⠰⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣷⣜⣡⣴⣾⣿⡿⠻⠀⠀⠀⠋⠀⠈⡇⠀⠀⠀⢀⣮⣿⣿⢧⠀⠀⠀⠀⡟⠀⠈⠃⠀⠀⠨⢻⣿⣿⣶⣤⣙⣾⡾⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣮⣿⠟⡟⠁⠀⢠⡆⢀⠀⢠⣷⠀⠀⠀⣾⢠⢿⣸⠘⡆⠀⠀⢀⡧⠀⠄⠀⣧⠀⠀⠘⡝⢿⣯⣬⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢄⡁⠀⠀⠘⡀⠀⠀⠄⢹⠀⠀⡜⢸⠟⢸⣿⡂⠘⣄⠀⢸⠁⡄⠀⠀⡈⠂⠀⠀⣉⡤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⣤⣀⡨⢀⢤⠘⡇⣸⠛⢸⡗⣼⡙⠀⠀⠘⡄⡞⠠⢠⠘⡀⣠⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠿⠥⠇⠿⣇⡂⡇⠧⠤⠤⠽⠗⠒⠛⠁⠉

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠰⣏⣝⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣍⣹
⠀⠈⠻⠻⣿⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⡿⠟⠟
⠀⠀⠀⠀⠀⠻⢯⡷⣄⡀⢀⢠⠠⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡠⠤⡄⡀⢀⣴⢮⡽⠋
⠀⠀⠀⠀⠀⠀⠈⠹⢿⣏⠟⣴⠟⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⠻⣙⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣣⣼⢟⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣻⣧⡜⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠋⠈⢻⣯⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠙⣌⠂
⠀⠀⠀⠀⠀⠀⠀⡺⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠋⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠑⢄⠀⠀⡠⠊⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢎⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⣧⣀⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⢻⣧⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠙⢿⣦⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣶⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⢠⣾⢞⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡳⣷⡄
⠀⠀⠀⣰⠟⣱⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣎⠻⢆
⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣴⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠐⠶⠈⠋⠀⠀⠀⠀⠈⠉⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠄⠀⣤⠤⠔⠖⢈⠋⠀⠄⠀⠼⠀⠀⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠉⠂⠀⠀⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠁⠀⠀⠀⠆⠀⢸⣗⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠉⠛⠋⢱⠋⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣠⢞⣿⡀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⣴⣆⣸⡴⠊⢁⠄⠘⠻⣦⡀⠀⠀⠄⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀
⠀⠀⠀⠀⠀⠀⠉⠀⢀⠠⠊⠀⠀⠀⣻⣿⡄⠀⠀⠀⢀⡼⠉⢁⡀⠀⠀⠀⢀⣤⣤⣶⣦⣤⡀⠀⠀⠀⠀⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⣡⣀⠴⠛⠁⢸⡄⠀⣀⠝⠀⠀⠀⠀⠀⠀⡜⠁⠀⣠⠼⠚⠉⠀⠙⢄⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⢠⠖⠉⠉⠀⠀⠀⠀⢸⠁⠀⡍⠀⠀⠀⠀⠀⠀⠸⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠣⣀⠀⠘⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠐⡀⠀⠀⠀⠀⢠⠃⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠘⣄⠀⠁⡂⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠀⢀⠀⠀⠸⡄⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠛⠉⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⡔⢦⣀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣷⣤⡀⠈⠉⠛⠛⠻⠿⠿⢿⡿⠟⠋⠁⣠⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠒⢛⣋⣥⣴⣮⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⣶⣶⣶⣶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⢠⣿⣿⡿
⠀⠀⢸⣿⣿⣿⣿⣷⣦⣄⣀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣷⣦⣀
⠀⠀⠀⠉⠛⠉⠈⠛⠿⣿⣿⣿⣿⣷⣶⣶⠶⡶⢒⡒⢚⣛⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠟⠿⠿⣿⣶⣚⡻⢇⡘⢟⣓⣲⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠉⠉⡹⢿⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣱⣎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⣿⣿⣿⣿⢸⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢿⡿⣿⣺⣿⣿⡿⣿⣿⡿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢻⣻⣷⣽⣿⣾⡟⡟⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣼⣧⣿⣶⣶⣿⣽⣧⡀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⡟⠁⠀⠀⢸⡇⠀⠀⠈⢻⣧⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡿⠀⠤⣤⣤⣿⣿⣤⣤⠤⠀⢿⡇⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣇⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⣸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠘⠉⠀⠀⠙⠃⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠘⠉

⠀⠀⠀⣠⣿⠀⠀⠀⠀⣸⣧⠀⠀⠀⢀⣰⣷⡄⠀⠀⢀⢰⣧⠀⠀⠀⠀⣾⣧⠀⠀⠀⢀⣼⣧⡄⠀⠀⠀⣾⣄
⠀⠀⢾⠿⡹⣿⠀⠀⢰⣿⣿⣿⠀⠀⢾⠟⠙⣿⠀⠀⢸⣾⣯⡤⠀⠀⢰⣿⣿⡆⠀⠀⣽⡟⠻⣷⠀⠀⣿⠏⢻⣶
⠀⠀⠈⠘⠃⠉⠀⠀⠈⠛⠛⠁⠀⠀⠈⠃⠐⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠛⠛⠁⠀⠀⠈⠐⠃⠉⠀⠀⠈⠀⠀⠁
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣄
⠀⠀⣰⣾⣿⣧⠀⠀⠀⣼⣿⠀⠀⠀⢀⣿⣷⡀⠀⠀⢀⣿⢷⣄⠀⠀⢠⣾⢿⡆⠀⠀⣠⣿⣿⡀⠀⠀⢠⣶⢿⡄
⠀⠀⠻⢿⡽⠟⠀⠀⠰⣿⣿⠟⠀⠀⠸⡏⠘⠟⠀⠀⠸⣧⣆⠟⠀⠀⢸⡇⠸⡇⠀⠀⢻⣿⣿⡟⠀⠀⢸⣣⡌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣰⣷⡄⠀⠀⠀⢰⣦⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⢰⣶⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣾⣆
⠀⠀⢴⠟⡈⣷⠀⠀⢨⡟⢹⣶⠀⠀⢰⡿⠹⣧⠀⠀⢀⣿⢿⣀⠀⠀⣰⣿⣿⡇⠀⠀⣤⣿⢿⣧⠀⠀⣤⡟⢹⣧
⠀⠀⠘⠚⠃⠋⠀⠀⠘⠃⠀⠋⠀⠀⠘⠃⠀⠃⠀⠀⠈⠃⠈⠃⠀⠀⠘⠻⠿⠋⠀⠀⠙⠋⠈⠋⠀⠀⠙⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⣶⣤⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴⣆⠀⡑⢄⣠⣤⠀⠀⠀⠀⠀⣠⣼⣿⠟⠛⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣦⡀⠀⣀⡀⠀⠀⣸⡿⠻⣇⣸⠈⠿⠟⠀⣀⠤⠒⠉⠙⠋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣿⡉⠗⠀⠀⠙⠻⣾⣏⣤⡶⠋⠁⢀⡼⢓⣾⣇⡀⠀⠀⠀⠀⠀⡄⢀⡝⠙
⠀⠀⠀⠀⢨⡀⢠⠀⢼⡀⠀⠄⠀⣿⠟⣠⢆⣀⢀⣀⠫⠉⢘⠳⢤⡀⠀⠀⢀⣷⣿⣧
⠀⠀⠀⠠⣧⣄⠀⠀⠀⠙⣆⢀⠈⠀⠀⢿⠮⠻⡆⢻⡷⠀⠈⣷⣤⣿⡆⠀⣸⣿⣥⣿⣶⠦
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣹⠎⠀⠀⠀⠀⠉⣀⢁⠀⠀⣠⣾⣫⣤⣤⣍⢋⡅⣨⣭⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⡀⠀⢀⣄⣠⣿⣿⡿⠟⠋⠀⠉⠉⠉⠁⠀⠁⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠔⠛⠛⠋⢹⡷⠶⠶⢶⣶⣶⣶⣶⣶⠠⡆⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢱⠇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀⡀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣤⠄⠀⣀⣠⠾⠛⣿⠄
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠙⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣡⣿⣿⣿⣴⣿⣿⣿⣷⢀⡛
⠀⠀⠀⠀⠀⠰⠾⠇⢀⣤⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠻⠴⠎⠅⠵⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⡿⠷⠶⠶⠶⠄⠄⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣱⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⢀⣠⠖⠛⣿⡆
⢀⡴⣾⣿⣷⣶⣾⣿⣦⡀
⣿⡟⠁⠀⠀⠘⣧⠤⣾⣿⡄
⢹⣦⠀⠀⠀⠀⠘⣷⠟⠋⣹⡄
⠀⢻⡀⠀⠀⠀⠀⠘⢷⣽⣿⣿⡄
⠀⠀⢻⣆⠀⠀⠀⠀⠈⢻⣦⡿⠟⣛⣙⣛⡓⢦⡀
⠀⠀⠀⠹⣕⡤⣤⣤⡴⢞⣩⣶⣿⣿⣧⡀⠉⣳⠇
⠀⠀⠀⠀⠈⠙⠛⠿⠚⠋⠁⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠻⣟⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡜⢦⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⠦⡀⠈⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠢⡀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠢⡀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢈⠒⢄⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡁⠢⣄⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣙⠢⣄⣙⠲⠦⣍⡒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣛⣶⣬⣟⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣿⣶⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠃⢒⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⣀⣷⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠖⢠⡀⢩⡽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡇⠀⠀⢨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⣷⡘⣿⡟⢦⣤⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣿⡇⢹⣇⣠⣿⣟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣿⠃⠸⠿⠿⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣀⣼⡿⣿⣇⠹⢿⣿⣦⡔⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠄⠸⡟⠀⠀⠀⠀⣼⡏⢹⣿⣿⡇⠀⢀⣤⣾⣟⣿⣷⠆⠘⣆⣀⣀⣉⡛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⢧⣀⠀⠀⣰⠏⠀⠈⣿⣿⡇⢀⣽⣿⣿⣿⣿⡏⠀⠀⡟⠉⠉⠉⠏⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠛⠦⣄⡀⠈⣹⠶⢯⣄⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣼⠁⠀⠀⡀⠀⡀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣽⣿⣿⣆⠀⠀⠙⣶⣿⣿⣿⣿⣿⣿⣿⡟⢻⣠⣿⠻⠀⠻⠆⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢘⠻⣿⣿⣷⣤⣾⣿⣿⣿⣿⡿⣿⣿⣿⠇⢠⣿⣇⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠷⢾⣟⠛⢛⡷⠶⠶⣾⡆⠀⠀⠀⢸⡟⠻⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣤⣿⣿⠻⣆⡸⡀⣀⣀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠟⣡⣴⣶⣶⣿⣿⣿⣅⠀⠀⢘⡏⠀⠀⢀⣾⠃⢠⡞⣱⣌⢿⣿⣿⣿⣿⣿⣿⣏⢸⣿⣷⣿⠤⣈⠛⣯⣉⣩⣬⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣡⣾⣿⣿⣿⣿⣿⠏⠀⠙⠳⢦⣼⡶⢶⣤⡾⠃⣰⢏⣼⣿⣿⣦⠻⣿⣿⣿⣿⣿⣿⢸⣿⠇⣿⠀⠈⠙⢿⡏⣑⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⣿⣿⣿⣿⣿⡿⠃⠀⡔⠀⠀⣸⢿⠆⣿⣿⢃⡞⣡⣾⣿⣿⣿⣿⣷⡝⢿⣿⣿⣿⡏⣼⡿⡰⢻⣆⠀⠀⢿⣀⣙⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⣿⣿⣿⣿⣿⣅⣐⠊⠀⠈⢀⣿⠈⢰⣏⡾⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣷⣿⡗⣡⠁⢹⠀⠀⠀⠉⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡧⠀⡾⣨⣴⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠛⢋⣴⡿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⡟⢿⣯⡻⣿⣿⣿⠇⠀⣠⠿⠿⣿⣿⣥⣿⣿⣿⣿⠏⣿⣿⣿⣯⢻⡿⢰⣿⣿⠃⣰⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠁⢸⣆⠳⣌⡙⣿⣠⡎⠁⣀⠴⣿⣿⣿⡿⣿⣿⠋⣼⣿⣿⣿⡿⣸⠇⢸⣿⢣⣾⠻⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⡿⠁⣠⣿⣿⣷⣌⠙⠿⣤⣾⣿⣧⠀⠹⣿⣿⠀⢿⣿⣿⣿⣿⣿⡿⢃⡿⢀⣾⣿⣿⡃⠀⠹⣷⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠁⣴⣿⣿⢻⣿⣿⣷⣦⣄⣈⡉⠉⠑⢾⣇⣾⡇⠈⢻⣿⣿⣯⠉⣀⣉⣴⣿⣿⣿⣬⣙⠳⠦⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠘⣿⡙⠿⠿⠋⣠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⠛⠻⢦⡀⠹⣿⣿⣷⡈⢿⠉⠉⠀⠀⠉⠛⠻⠶⣤⣤⡾⣿⣆
⠀⠀⠀⠀⢠⣧⣿⠳⣦⡴⠞⠁⢸⣿⣿⠿⣿⣿⣿⣛⣛⣻⡯⠷⠚⠉⠀⠀⠀⠀⠀⠙⠶⣌⠻⣿⣿⣌⢧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢿⡿
⠀⠀⠀⠀⠇⠀⣿⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣸⣏⡸⠁
⠀⠀⠀⡸⠀⣼⠇⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢱⠀⠀⠀⠀⠀⣰⠋⢙⡇
⠀⠀⣀⣧⣸⡟⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢋⠜⠀⠀⠀⠀⠰⠁⢀⡞
⣴⣞⠉⣽⠁⠀⠀⢰⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⡏⠀⠀⠀⠀⢀⣇⣀⡾
⠻⠮⠟⠁⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⢀⣤⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢀⡼⠀⠀⢿⣀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠺⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⠙
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⣛⣉⣙⣛⣓⣛⡛⠻⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡾⣿⣛⡛⣻⣿⣿⣿⣿⣿⠷⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⡿⢸⣿⠟⣋⡤⠄⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢟⣼⣿⣿⣿⣿⢸⡏⠉⠉⠉⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡞⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠞⠻⣿⣟⢢⡝⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣠⠖⠻⣿⣧⡧⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣤⡤⠿⣏⠀⠸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⢷⣄
⠀⠀⠀⠀⠀⢀⣔⣑⣐⣒⡒⠒⠠⢄⣀⠹⣧⠂⢿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿
⠀⠀⠀⠀⠀⠈⠿⣿⣯⡿⠿⠿⣶⣄⣛⣻⣿⣧⣈⣿⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⠛⣿⣅⢈⣿⢈⣿⣍⢿⣿⣿⣿⣿⣿⣿⣿⣷⢰⡇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣤⣄⡀⠀⠘⠻⣽⣿⢟⣾⡣⠝⢿⣿⣿⣿⣿⣿⡏⢸⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⠀⠯⣝⡛⠷⣦⡀⠈⠻⣎⡻⣿⣦⡈⠻⣿⣿⣿⣿⡇⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠠⣉⠓⡮⣝⠷⣄⠈⠹⣿⣿⣿⣦⡈⢿⣿⣿⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣝⣷⣝⢶⣝⡨⠳⣄⠙⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣝⢿⣦⡌⠳⡈⢻⣿⣝⢿⣿⣓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠻⣿⣧⡙⢿⣷⡌⠀⠹⣿⣦⡉⠻⣷⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠙⢿⣮⡻⣿⣦⡀⠈⠻⣿⣤⠈⠻⢿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⡙⢿⣦⡀⠙⢿⣕⣄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠛⠶⠄⠙⢿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡤⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⢀⡀⠄⠀⠀⠀⠁⠀⠀⠐⢦⡤⠾⠋⠀⠘⣁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠂⠀⠀⣼⠫⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⣠⠌⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠶⠤⠀⢄⣴⠋⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠈⢡⡄⠀⠀⠀⠀⠛⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠀⠄⣼⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠿⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠐⠀⠈⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠲⣌⠁⡰⠛⠋⠉⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣦⣀⠀⠈⠀⠀⣼⠀⢄⡀⠀⠀⣀⠀⠐⢦⣶⡾⠋⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣾⣧⡧⢾⣦⣼⣶⠿⠷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡆⢾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⠀⠀⠀⠀⢺⣿⠓⢶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠂⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⣿⠏⠁⣤⠆⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣛⣿⡿⠀⠀⣾⠠⣤⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣤⣀⠀⠒⠀⠀⠔⠀⠀⠀⠳⠒⠶⠟⠷⢄⡀⣹⠀⠉⠀⠰⠄⠀⢀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⣽⣽⣿⣷⣿⣤⣽⣤⣴⣀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠀⠇⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠁⠙⠛⠖⠺⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠒⡀⠐⠒⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣌⡙⠛⠛⠛⠛⠛⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠉⠉⠉⠉⠁
⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣶⣶⣦⣾⣦⣾⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣤⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣹⣿⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠇⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄⢀⣸⣿⣿⣿⣿⣿⣿⣿⡋⠁⠈⢠⡟⢸⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⢸⡏⢹⣿⣿⣿⣿⢿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⣿⣿⠁⢈⣿⣿⣴⣿⣿⣿⣿⡿⠿⠶⠂⢀⠀⣀⣀⣤⡶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣯⣽⣶⣶⣿⣿⣿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠹⠿⠿⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⡿⠿⠿⠿⠿⠻⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⣿⣿⣿⣿⣿⣿⡟⠛⠿⠿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿