Wording Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⡷⢀⢧⠀⣟⢾⢣⢳⣠⠃⠀⠀⠀⠀⠠⡀⠀⠠⡀⠄
⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠉⠀⡤⠈⠁⠀⠀⠀⠀⠀⡏⠉⣷⢸⠉⡆⡏⢸⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠋⠀⠘⠃⠸⠀⠚⠀⠀⠀⠀⢀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠖⣉⠲⡄
⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⡤⠀⠀⢠⡄⠀⡄⠀⠀⠀⠀⠀⡞⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⡇⢀⠄⢸⡇⠀⡇⢀⡤⠀⠀⠀⣇⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢠⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠀⠀⠀⠇⠀⠃⠘⠁⠘⠃⠀⠓⠘⠲⠊⠀⠀⠸⣄⠛⣠⠇
⠀⠁⠀⠀⠈⠀⢀⡀⠀⡐⢦⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠄⠀⠀⠀⠀⠀⠰⣍⡲⠀⠠⠐⠀⠀⠀⢀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠬⠩⠜⠐⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⡀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠀⠘⠂
⠀⠀⡤⠀⠀⠲⠀⠂⠀⠂⠀⡸⠂⠠⢄⢻⣀⣠⠤⠀⠒⠐⠂⠀⠀⠒⡦⢄⣠⠡⠐⠃
⠀⠀⠀⠀⠐⠀⠠⠤⠖⠀⠈⠑⣄⠀⠌⠄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠄⠀⠀⠀⡠⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣞⣶⣲⣴⠂⡶⠂⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠿⠂⢀⠔⠀⠀⠀⠀⠀⠸⠇⠿⢳⣯⠱⣏
⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⢄⠀⠀⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⡷⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠍⠀⠀⠀⠀⠀⠸⡆⡄
⠀⠀⠀⠈⠇⠻⠸⠇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠰⠠⠴⠀⠀⠖⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⡱⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠖⡣
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡠⣤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠩⢒
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠤⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡍
⠀⠒⠒⠀⠀⠋⠀⠀⠧⡄⢮⠂⣼⠇⠻⣌⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠘⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢐⡈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠠⠇⠀⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠀⣰⣷⢀⢤⠀⠀⠀⠀⠈⠀⠐⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠠⠀⢠⣿⣄⠂⠈⣦⡰⠙⢄⣾⣝⣻⠀⢸⠀⠀⠀⠐⠀⠂⢀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡝⣿⡿⡿⣄⣰⢿⡟⣤⣿⠟⠻⣿⣦⣼⣿⡿⠒⡇⠀⢀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣘⠉⠐⣿⣦⣿⣿⣿⣿⡿⠿⣿⣿⣷⠶⢴⣿⣿⣿⣿⣤⣶⣿⣿⡿⠠⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣭⣷⣦⣯⠷⢷⠋⠙⠋⠀⠀⠀⠙⠁⠀⠈⠉⠙⠉⠀⠙⠙⣿⡟⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢻⣛⣛⣿⠁⢀⣀⡀⠀⣠⣤⣤⣀⣴⣶⣤⣰⣶⣤⣶⣆⠀⢠⣟⣛⣟⣛⣲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠀⢾⣟⣿⣿⢸⣿⡿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⡄⠀⠙⣿⣿⢿⣩⣂
⠀⠀⠀⠀⠀⠀⠐⢎⣉⣿⣿⣿⣄⠀⢸⣿⠿⣿⣿⣿⣧⣿⣿⣷⣶⣿⣿⣿⠙⠛⣿⣷⢀⣰⣿⣿⠉⣻⣿⡿⠟⠂
⠀⠀⠀⠀⠠⢶⣿⣿⣿⣋⠉⣽⣽⠃⠈⣿⠷⠿⠟⠙⠛⠛⠁⠉⠉⠉⠉⢁⣀⡀⠉⠀⠀⣿⣿⣿⣺⣍⠙⣄
⠀⠀⠀⠀⠀⠀⠈⠙⡻⢿⣟⡿⣿⠀⠀⡀⠀⢀⡀⠀⠀⢰⣀⠀⣰⢦⣤⡼⣹⣷⣶⣶⣾⣻⡿⣿⡿⣟⣯⡁
⠀⠀⠀⠀⠀⠀⡤⠊⠠⢤⣼⣿⡿⣿⣿⣿⣤⣼⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⢁⣿⣿⣿⣿⡉⢻⠻⠷⡄
⠀⠀⢀⠄⠴⠀⠀⠀⣠⣿⠿⠿⠋⣽⣿⣿⣿⣿⣿⣿⠞⠉⢉⣷⣿⣿⣿⣿⣿⣾⠿⣝⡟⠳⢝⢻⣷⠒⠂
⠀⠀⠣⡀⢀⠀⠀⠄⠉⠀⠀⠓⠁⠹⠴⡟⣻⠟⢻⣝⣦⣤⣯⡟⠹⠿⢋⣿⣻⡿⠀⠉⠀⠁⠀⠙⠿⡇⠅⠀⢰
⠀⠀⢸⡀⠸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠟⠡⢀⠊⢿⣿⣿⠟⠐⠃⠀⠀⢹⡟⠀⢳⠀⠀⠀⠀⠀⠀⡠⠀⠠⢈⠆
⠀⠀⠀⠉⠳⣀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣷⠏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⠖⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠦⢤⣀⠀⠀⢠⠖⠲⢤⡀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠈⠓⠶⠋⠀⠀⠸⣿⡴⠚⠁⠀⣠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣼⡆⠀⢠⣀⠀⢠⣷⣄⡁⣠⡖⢀⣾⣿⠟⠁⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠦⣄⠀⡇⠀⠀⢀⣷⠀⠸⣿⡛⠿⡟⠈⠛⣿⢠⣿⡿⠁⢀⡴⠋⠁⠀⠀⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⣼⣷⠃⠀⠀⣸⣿⣤⣤⣭⡽⠛⠓⠲⣄⠙⢾⣿⠁⣰⠋⠀⠀⢀⣠⣾⡇⠀⠀⣿⡄
⠀⠀⠀⠀⢀⡴⠃⠀⠀⡀⠀⠀⢰⣿⠏⠀⠀⣴⣿⠟⠚⠉⣽⡇⣠⠀⠀⠸⠟⠛⠛⣶⠁⠀⠀⢠⣾⡿⢻⠇⠀⠐⣻⣷⡄⠀⠀⣀⡀
⠀⠀⠀⠀⡞⠀⠀⢀⣾⠀⠀⢠⡿⠃⠀⣠⡾⠿⠿⠀⠀⢘⣿⢿⡿⠀⠀⢀⡄⠀⣸⠃⠀⠀⢠⣿⣟⡁⠘⢦⣤⣾⣿⠟⠀⢀⡜⠁⣿⣦
⠀⠀⠀⢸⠁⠀⢀⣾⡏⠀⠀⠈⠁⠀⠈⣀⠀⠀⠀⠀⠀⠘⠿⢋⡄⠀⢀⣾⡇⠀⠋⠀⠀⠀⠀⠀⣉⣉⠉⠙⢿⡍⠁⣀⠴⠋⠀⣰⣿⠇
⠀⠀⠀⢸⡀⠀⣼⣿⡇⠀⠀⢀⣤⣶⣿⠋⠀⠀⣠⣷⣦⣴⣶⣿⣦⣤⣿⣿⣹⣤⡄⠀⠀⢰⣦⣀⠘⠻⠀⠀⠸⠛⠋⠁⠀⣠⣾⡿⠋
⠀⠀⠀⠀⢧⣼⣿⠇⡇⠀⠀⠈⣿⣏⡏⠀⠀⠀⢉⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢈⣿⣿⣻⠆⠀⢀⣴⣶⣶⣶⣾⡿⠟
⠀⠀⠀⠀⠀⠙⠏⠀⢷⠀⠀⠀⠸⣿⣇⣀⣠⣶⠏⠀⠀⣀⣀⣤⣴⡶⢶⣿⣿⣿⡿⡄⠀⠀⣿⠿⠃⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⢹⣏⠛⠻⣯⣴⣶⣿⡿⠛⣩⣴⣾⡿⠟⠋⠉⠀⠙⢦⣀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⣶⣾⡿⠂⠀⠈⠋⠉⠻⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢶⣾⣿⠏

⠀⣀⣀⣀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⡀⠀⠀⣀⣀
⠀⣿⣿⠛⢿⣷⡆⢸⣿⡀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⢠⣾⡿⠻⢿⣷⠀⢸⣿⡅⠀⣼⣿⠋
⠀⣿⣿⣀⣰⣿⡇⢸⣿⡀⠀⠀⠀⠀⠀⣾⡟⢿⣇⠀⢸⣿⡅⠀⠀⠀⠀⢸⣿⣄⣾⡿⠁
⠀⣿⣿⠛⢻⣿⡄⢸⣿⡇⠀⠀⠀⠀⢰⣿⣇⣸⣿⡄⢸⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀
⠀⣿⣿⣀⣰⣿⡏⢸⣿⣇⣀⣀⣀⠀⣿⣿⠛⠛⣿⣧⠸⣿⣦⣀⣴⣶⠄⢸⣿⡏⠀⢻⣷⡄
⠀⠛⠛⠛⠋⠉⠀⠘⠛⠛⠛⠛⠛⠀⠛⠃⠀⠀⠛⠛⠀⠉⠛⠛⠛⠁⠀⠘⠛⠃⠀⠈⠛⠛⠀⠀⠈⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀




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

⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢀⠀⠘⢠⡆⢸⠀⡘⡤⠂⢀⡀⢿⠀⠀⠀⢂⠄⠀⠀⢀⣧⠀⡄⢰⢀⣇⢤⠆⣶⣴⣐⡥⠸⠀⠀⣲⣺⠄⠀⠀⠀⠀⡀⠆⠄⢠⠆⣶⠐⠐⢤⠸
⠀⠀⢳⡿⡇⠗⠀⠋⠠⡄⢆⣷⢂⠈⣃⠀⠘⠜⠥⠀⠀⠏⠘⠰⠙⠃⠋⠙⠉⠀⠁⠈⠈⠀⠃⠀⠀⠀⠃⠀⠀⠤⠐⠜⠉⠋⠼⠈⠑⠉⠄⠀⠂⠐
⠀⠀⠀⢠⡄⠠⡄⣏⡑⠏⠺⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⡀⠀⣤⠴⢲⢤⢠⢤⣤⠀⡆⢸⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠢⡤⢀⢀
⠀⠀⠀⠀⢳⡴⠁⠈⠀⠀⡀⠀⡤⠀⢣⠀⠀⠀⢠⢋⢨⠗⡇⡾⡇⡟⠾⢡⢸⠁⠸⠣⠎⡿⢳⡇⣈⠀⠀⠀⡇⠀⣸⣶⠁⡆⡴⠂⣀⡀⠀⠀⠁⠈
⠀⢠⣀⡀⠈⠀⢀⢠⣶⢄⠹⡄⡒⣿⠈⣂⠀⠀⠘⠋⠀⠀⠁⠀⠀⠁⠀⢈⡈⠀⢀⡀⠀⠀⠀⠀⠁⣀⠀⠀⠙⠛⢇⠻⠜⢹⡉⢰⠗⢠⣿⢠⠂⡴
⠀⠘⡟⠃⠀⠢⣇⡀⠻⠂⠙⠷⠈⠈⠀⠀⠀⢸⢃⣤⠀⠀⠀⠠⣧⡀⢀⠟⡇⢠⢾⠁⠀⠀⠀⣀⠊⢤⡉⠀⠀⢀⠀⠀⡀⠀⠀⠉⠁⠋⢸⠃⡼⠁
⠀⠀⣀⣀⡀⠀⠁⠀⠁⠀⠀⢀⣀⡄⠀⠀⠀⠈⠂⡐⢇⠀⠀⠀⠰⣈⣋⣠⣬⣥⣼⠀⠀⠀⠀⣠⠟⠈⠀⠀⠀⢸⠀⠃⠃⡀⠀⠀⠀⠀⠀⠰
⠀⠀⠙⣄⠱⡄⠀⠀⠀⠀⠈⠝⠛⡋⢄⡤⢰⠀⠀⠀⠈⠁⠀⠀⠀⠋⠐⠂⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠀⡀⠀⠀⡄
⠀⠀⠈⠣⣴⡍⠀⠀⠀⡾⣅⠏⠀⡩⠆⠃⠈⠀⠀⠀⣀⢀⠀⠀⠀⡀⠀⡀⠀⠀⡄⣦⢀⡄⡤⠂⢸⠀⠀⢸⠆⠘⠃⠃⠸⠀⠂⠒⠘⠀⠙
⠀⠀⠀⡀⠀⠙⠆⠀⠀⠗⠁⠉⠀⠀⠀⠀⠁⠀⠀⢠⣻⠸⣰⣆⢻⠕⠀⠑⡄⣆⣼⠀⠏⡇⣯⠄⠀⠀⠀⠀⠀⠒⠊⠱⠏⠒⠈⠁⢀⣀⡤⡄⡆
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⢀⡦⡴⠀⠀⠀⠏⢹⠀⠃⠋⠼⠚⠁⠚⠁⠈⠈⠀⠀⠀⠁⠀⠋⠀⠀⠀⠀⢀⠤⢠⢰⢸⡄⡿⡼⣇⡟⠁⡇
⠀⠀⠠⣶⠞⠺⠏⣀⢄⡀⠀⠀⠀⠀⠹⠓⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⢲⢴⡞⠀⠀⠀⠀⣀⣰⡀⠀⠀⢸⣠⠸⠍⠏⠁⠃⠀⠻⠙⠀⠆
⠀⠀⢀⣀⠸⠲⠦⠉⠁⠀⠀⠀⠀⡀⢀⡠⠀⠀⡄⠀⠾⠇⡜⠀⡇⠀⠀⠀⠀⠀⠀⡠⡄⡆⠀⠀⢙⣯⣿⠋
⠀⠀⠘⠈⠀⠀⠀⡴⢛⡀⣯⠕⣸⣧⢨⠐⠯⡀⠇⠀⠀⠀⠉⠇⢹⣀⡀⠀⠘⣿⡹⢹⠀⡁⠀⠀⠈⠀⠈⠀⠀⣖⢠⠂⠀⠄⠆⠐⠸⡔⢺⠐⠁⠎
⠀⠀⠀⠐⢤⡆⠘⠧⢻⠀⠋⠳⢅⢸⠀⠀⠊⠁⠐⠀⠀⠀⢐⢃⠠⠀⠃⠀⠀⠁⠀⣀⠀⠀⠐⠩⡕⡄⠀⠀⠀⠀⠈⠁⠀⠁⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠉⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠒⠀⠀⠀⠀⢀⡤⢖⣄⠩⠤⠜⠁⠀⠀⠈⠲⠵⠁⠀⠀⡀⠀⠀⠀⠀⡀⠀⣀⣀⢀⠀⡀⢠⡄
⠀⠀⠀⡀⡄⣀⣀⣀⠀⠀⡀⣀⠀⣀⢠⠀⠐⠠⠂⣁⢀⡀⠀⢀⣀⠀⢀⣀⢀⢀⠀⢀⢠⡠⠀⢠⠀⠀⢻⢸⢹⢠⠖⠈⣏⡔⢲⣿⣜⠁⢳⠁⢸⠁
⠀⠀⢥⠛⢣⠏⠾⠹⠄⠃⠱⠁⠇⠧⢘⠀⠀⠀⠀⡧⠼⠈⡇⠺⠸⠶⠃⠀⠚⠾⠀⠈⠹⠓⠏⢘⠀⠀⠘⡏⠘⠘⠋⠀⠃⠛⠁⠃⠈⠃⠋⠀⢠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢦⡾⣧⣰⡇⢹⠏⠀⠀⠀⠂⡀⢒⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⠀⠀⠰⣰⠃⠀⠀⢀⠀⠀⡀
⠀⢠⠔⠊⠀⢀⠈⠑⠶⠀⠀⠀⠀⠀⢀⠄⠀⠈⠀⠈⠁⠁⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡁⠀⠁⠀⠋⠀⠈⠉⠰⠁⣠
⠀⠀⠀⠀⠋⠀⡄⠐⠀⠀⡀⠀⠀⡐⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣿⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠀⠀⠐⠁⠀⡀
⠀⠀⠁⠀⠀⡔⠀⠀⠂⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠸⠀⠀⠐⠋⣼⡿⠛⡾⠀⠀⢀⡔⠛⠁⠉⡂⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠠⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠀⠀⠀⠀⠙⠲⠧⣆⠔⠈⠀⠀⠀⢠⡗⠀⠀⣇⣠⢀⡞⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡎⠀⠀⠀⠋⠀⠟⢠⠋⡏⡴⢿⣆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠃⠀⠀⠆⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠑⠀⠘⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣁⡀⠀⠀⠐⠢⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠠⣦⠐⣷⣄⠀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠻⠟⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠆
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⣶⠘⣿⠀⠀⣻⡇⠈⠛⠿⣿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠈⢣⣰⠏
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠘⣿⠀⢹⣿⠟⠙⣿⡄⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡱⡄⠘⣷⠀⣀⠀⣿⠀⡆⠀⣼⠀⠀⢿
⠀⢀⢀⠀⠈⠻⣆⠀⢰⣷⣤⣿⠄⠘⣿⡀⠀⠹⠷⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⢹⣷⠟⢷⣿⠀⢹⠈⢻⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠘⢧⡼⠈⠻⠿⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠲⠶⠀⠘⡟⠀⠈⠛⠀⠈⠀⠈⠀⠀⠈
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢓
⠀⠠⣾⣟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠂⠆⡀
⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠰⠤⠄⠔⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠒⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⢀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣾⠃⠚
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠐⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⡀⠀⠘⢄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠰⠄⠀⠀⠉⠁⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣰⡶⢦⣄
⠀⠀⡀⢀⠚⡤⠆⢃⢈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⢠⡄⠀⡠⠀⠀⠀⠀⡀⠀⠀⠀⢠⠌⠘⠀⠀⠀⠻⣷⡀⣴⡄⢰⡿⣇⣼⢿⡇⠀⣿
⠀⠀⠐⠋⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠰⠂⡗⠡⠋⠁⠮⠧⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠹⣷⡟⢻⣿⠀⠯⠘⠇⠻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠠⠄⠀⠀⠠⠠⠠⠠⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠂⠆⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠃⠀⠀⠉⠑⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠖⠉⠁⠀⠀⠱⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠑⡄
⠀⠀⠀⠀⠀⣠⠴⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢣⠀⠀⠀⢠⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠦⢤⡀
⠀⠀⠀⢀⡞⠁⠀⠀⣀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⡆⢠⡤⢤⠘⡄⠀⠀⡌⢀⡤⢤⡀⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⣤⠀⠘⡄⢘⣯⣽⠇⠀⡇⣼⠀⢸⡆⡇⠀⠀⡇⣼⠀⠈⠁⡇⢀⡶⠒⢦⠐⡞⠀⠀⠀⠀⡇⠀⠀⠀⣇
⠀⠀⠀⠈⡀⣤⠀⣆⡁⢠⣏⣷⠀⡇⠀⣯⠙⣆⠀⣇⠼⣄⣼⠀⣇⠀⢀⡇⢿⣀⡤⢀⡇⣸⠃⠀⣿⠀⡇⢀⡟⢻⡇⠇⡀⠀⠀⡁
⠀⠀⠀⠀⠡⠸⡖⠻⣶⢸⠀⠀⠓⢿⡄⠁⠀⠀⣠⣿⣀⣀⣠⡾⢹⡀⣼⠹⣦⣀⣀⣸⣧⡘⠦⠼⠃⢼⠇⣼⢻⡟⢰⢠⡟⠳⠠⠁
⠀⠀⠀⠀⠀⠡⡛⠀⠉⢈⠀⠀⢀⣿⠷⠶⠶⡾⠋⠉⠁⠀⠀⠀⢘⡇⣏⠀⠀⠀⠀⠉⠉⠻⡶⠶⠶⢿⣏⠉⠸⢣⢣⡞⠙⡰⠁
⠀⠀⠀⠀⠀⠀⠈⠠⠠⠞⠉⠉⠉⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⣀⡼⠁⠹⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠛⠦⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠚⠓⠊⠉⠀⠀⠀⠈⠉⠓⠛⠒⠒⠊


⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡆⢠⡆⠀⠀⠀⠀⡠⠷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⢸⡇⢸⠃⠀⠀⠀⠀⠱⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠉⢝⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠞⠈⠀⠀⠀⠀⣿⠙⢿⣷⣤⣀⠸⠀⠀⣀⣼⡄⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠉⠛⠿⠿⠿⠿⠛⠻⢿⣶⣶⠿⢿⣿⠏⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣤⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⠀⠀⢿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠻⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⣠⠞⠋⠉⠉⠉⠙⢲⣾⡆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠞⠀⢀⣿⣆⣀⠀⠀⣤⢦⡄⠀⠀⠀⡇⠀⢹⣯⢂⣴⠾⢿⣶⣤⠿⢤⣧⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣦⣶⡶⠛⠉⠉⠉⠙⠲⣿⣿⠝⢦⡀⢸⠃⠀⠀⢿⣾⡁⢸⡍⠀⠀⠀⢸⣿⣀⣀⣷⠒
⠀⠀⢲⠒⠣⠀⠀⠀⠀⠀⢹⣿⡄⢀⣤⣶⣦⣄⠀⠈⢿⣁⠈⠳⣾⠀⢀⠀⠈⢿⡳⠬⠿⠟⠁⠀⡼⢹⡏⠀⢸⣦⣤⣤⣀
⠀⠀⠚⠒⠼⠀⠀⠀⠀⠀⣼⣿⣯⡟⠁⠀⠀⠹⡆⠀⠘⣗⣦⡀⠀⠀⢸⣧⠀⠘⣧⣀⣀⣀⣤⠞⣵⠞⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣷⡀⠀⠀⢠⠇⠀⢠⣿⣿⢷⣄⢀⡾⠸⣦⣴⠾⠛⠋⠉⠁⠘⣇⠀⠀⠀⠀⠠⡄⣀
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⡻⢿⣿⣷⡙⠓⠚⠉⠀⢀⣾⣿⣿⠆⠙⠛⠃⠀⠀⠀⠀⠐⣤⠶⡔⠶⠸⠦⠀⠀⠀⠘⢤⠼
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠞⠙⢿⣶⣤⣤⣤⡶⠿⣿⣅⠀⠀⠀⠀⢀⣀⣀⣀⠀⠈⠈⢣⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠴⠃⠀⠀⠘⠉⠈⠁⠀⠀⠀⠈⢻⡆⠀⢀⡴⠋⠀⣀⡈⠑⢦⠀⠀⠙⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⣷⢠⡞⠀⡴⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠍⠀⡑⠂⠀⠀⠀⠀⢹⡞⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠚⠲⠇⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡜⠀⢸⠀⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⡀⢸⢰⢠⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠈⠈⠈⠘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⢰⠀⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠁⢠⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠃⠘⢀⢄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢄⣀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⢰⠀⠈⠁⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠡⠐⠘⠘⠚⠀⠑⠒⠐⠺
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⡇⡑⢮⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⠃⠓⠀⠼
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠄⠀⠀⠠⠀⡀
⠀⠀⠀⡞⠈⠀⠈⢐⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⢀⡀⠘⠌⠃⠀⠁
⠀⠀⠀⠘⠒⠐⠂⠇⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠈⠀⠀⠀⠀⠀⠀⠐⠼
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⡻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠤⢄⠀⡄⢀⡀⢠⡀⢠⢠⠀⡀
⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠀⣹⠠⡇⣈⣳⢸⠃⢸⢸⣞⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠘⠀⠛⠀⠃⠀⠈⠛⠂⠙⠘⠈⠂⠀⠀⠀⠀⠀⣀⡈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠃⡎⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢿⠁
⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣾⣤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠢⠄⠤⣰⠀⢀⣴⠆⣾⣃⡚⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠈⠙⠷⠀⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣴⣶⣶⣄⠀⣸⣿⣦⣤⣤⣀⣿⣿⣶⣄⠀⠀⠀⣠⣤⣤⣄⡀⢠⣶⡄
⠀⠀⠸⣿⣇⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⠟⠉⢹⣿⣿⠿⣿⡟⠛⠛⣿⣿⣿⡿⢿⣿⣦⣰⣿⣿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⡆⠀⢰⣿⣷⡀⠀⢠⣿⣿⡏⠀⠀⢀⣿⣇⣰⣿⣧⣴⣿⡿⢿⣿⣇⠀⠈⠻⣿⣿⣿⣷⣶⣬⣿⣿⣧
⠀⠀⠀⠀⢿⣿⣀⣿⡿⢻⣿⡄⢸⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⠛⠉⠀⠈⣿⣿⠀⠀⠀⠘⣿⣿⣭⡛⠻⠿⣿⣿⣄
⠀⠀⠀⠀⠈⣿⣿⣿⠃⠀⠹⣿⣿⣿⢿⣷⣤⣼⣿⡏⣸⣿⡿⣿⣷⣦⣤⣤⣿⣿⡄⠀⣀⣠⣿⣿⣯⣀⠀⠀⢸⢙⣿⣧
⠀⠀⠀⠀⠀⠘⠿⠏⠀⠀⠀⠘⢿⡟⠈⠛⠿⠟⠋⠀⠛⠋⠀⠀⠉⠛⠛⠛⠻⣿⠿⠿⠿⠟⠋⠉⠛⠿⠿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠘⠀⠀⠈⠀⠠⡀⠄⠀⠀⢀⠀⠀⠦⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⡂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢠⠤⠤⢄⣀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠀⠀⢛⠗⠉⠑⠀⠀⠉⠐⠀⠈⠑⢤
⠀⠀⠀⠀⠀⠀⢹⠀⠀⣀⣠⠴⢦⣴⣶⣶⣿⡄⠈⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣀⡰⠁⠀⠀⠀⣠⠤⠤⢤⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠴⠛⠛⠋⠉⡹⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⠀⠀⠀⢸⠈⢩⠉⠁⠀⢰⠎⠀⠀⣀⣀⢈⡉⠁
⠀⠀⠀⠀⠀⠀⣈⡁⠀⠀⣀⣀⣠⣤⣤⣾⣧⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠀⠀⠀⣸⣀⡆⠀⠀⠀⡞⠀⠀⡘⠀⠀⡘
⠀⠀⠀⠀⠀⠎⠈⣿⣿⣿⣿⣿⣿⣟⠛⠛⠉⠹⣆⠀⠀⠀⠻⠀⠀⠀⠀⠂⠀⠀⢀⡟⠫⠇⠀⠀⠰⡇⠀⠠⣅⣀⣤
⠀⠀⠀⠀⠀⢠⠂⡏⠀⠀⠀⣇⣀⣬⣥⣤⣶⠾⠟⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡾⠀⠀⡀⠀⠀⠀⢧⡀⠀⠀⠠⡀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠈⠀⢳⣶⣿⣿⣿⠿⠿⠟⠛⣣⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣠⢞⠉⠀⢠⣼⡄⠀⠀⡐⠀⠀⠀⠀⠈⠂⢄⡜⠿
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠀⣀⣠⠴⠋⠁⠀⠈⠀⠎⠳⠤⠤⠤⠤⠖⠋⠁⠀⠀⠀⠀⠉⠈⢳⣤⣄⣀⣀⣀⣤⠴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠶⠒⠛⠉⠁⢀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠄⠀⠀⣨⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢤⡀⠀⠀⠀⠀⠠⠀⠈⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠈⠀⠀⠀⠈⠳⡔⠓⠒⠤⢤⠀⠀⠀⠀⠀⡀⠀⢀⣸⣗⠒⠓⠄⡀⠈⠀⠐⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣠⠴⠒⠒⢄⠀⠀⠀⣿⣶⣶⣾⡿⠀⠀⠀⠀⢀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⠞⠁⠀⠀⠀⢸⠀⠀⠀⠏⠉⠉⠉⡇⠀⠀⠀⠀⠸⠀⠀⠀⣸⠀⠒⠚⢹⠄⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡎⠀⠀⠀⠀⠀⡸⠀⠀⠀⣶⣶⣶⣾⡇⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⣶⠟⠀⠀⠀⣼⢦
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢀⠔⠁⠀⠀⢠⡟⠛⠉⠉⠇⠠⡀⠀⠘⠉⠀⠀⣸⠃⠀⠀⠀⡏⠀⠐⠒⡇⠈
⠀⠀⠀⠀⠀⠀⡖⢣⠀⠀⠀⠙⠂⠀⠒⠁⠀⠀⠀⠒⠋⢴⣶⣶⣆⠀⠀⣤⣀⣠⣿⣿⣿⠋⠀⠀⠰⠟⠁⠀⠀⢠⢷⡄
⠀⠀⠀⠀⠀⠀⠁⡞⠳⣄⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⢿⡛⠛⠃⠉⠉⠉⠉⣩⣽⠉⠆⠈⠀⢀⣬⠉⠉⠀⢸⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠙⠒⠶⠶⠖⢺⠏⡼⠀⠀⠀⠀⠃⢹⠒⠶⠶⠶⠖⠋⠁⠀⠁⠀⠀⠀⠛⠙⠒⠶⠦⠞⠿
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠈⠍⠂⠀⠀⠀⠀⠀⢀⠤⠂⠉⠉⠉⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠤⠒⠒⠂⠒⠒⠢⢄
⠀⢠⠄⢀⡂⠬⠐⠀⠉⠀⠀⡀⠈⠠⠀⠀⠀⠀⠠⠁⡠⠰⠎⠉⣉⢦⣠⠌⡄⠀⠀⠀⡀⠀⢀⡀⠠⠄⠐⠀⠉⠑⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠂⠁⠀⠀⠀⠀⢉⣿⣾⣥⡄⠄⠆⠀⠀⠀⠀⠀⡀⢀⠜⠀⠱⠃⠋⡁⢰⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠉⠟⢻⡀⠀⠰⠘⡀⠀⠀⠆⡀⠰⣁⠀⠀⠀⡱⡄⠇⠄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⠀⢤⠚⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠇⠃⠀⠀⢀⡀⠤⠽⠳⠀⠀⢃⢁⠀⠀⢸⣡⢆⠀⠉⠀⠈⠀⢠⠘⠀⠀⠀⠀⠃⢀⡀⠀⠤⠉⠀⠀⠀⠈⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠸⠈⡄⠊⠁⢀⡀⠠⠀⠐⠂⣉⠰⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡇⠑⠒⠉⠀⠀⠠⠔⠂⠉⠁⠀⢀⠤⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠲⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠞
⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⡔⠒⠈⠉⠁⠀⠀⠀⠀⠀⡨⠂⠀⠀⢠⠖⢉⠇⠀⠀⠀⠀⣰⠖⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀⣠⡤⠤⠄⠒⠊⠁⠀⠀⢰⠃⠀⠈⠲⡄⠀⠀⠘⣉⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠭⠀⠛⠘⠓⠈⠉⢡⠀⠀⠀⠀⠟⡄⠀⠀⠀⣇⠀⠀⠐⢓⣙⢒⡷⠀⢺⠀⠀⡔⠤⠄⠀⠀⠒⠒⢐
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠈⠑⢄⠀⠰⡁⠀⠀⠀⢀⣀⣠⠤⠰⠀⠀⠀⠀⠂⣹⢾⣭⣭⡁⠀⠀⠘⣇⠀⠀⠀⡠⢿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠈⡀⠀⠀⠁⠈⠉⡋⠀⠄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⠀⠀⠀⠀⠑⣆⡀⢰⢡⠂⠀⠀⣇⣀⡀⠀⠀⠀⢀⣸⡀
⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠘⡀⠀⢀⣀⠀⠀⣠⠗⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠘⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠓⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⡠⠖⠛⢉⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⢡⢀⡤⠶⠂⡡⣀⡴⡊⠁⠀⠐⡀⠀⠀⠀⡠⠶⠀⠐⠃⠄⢀⣰⠀⠀⠀⠀⠀⠀⡠⣀⠤⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⢸⠻⣓⠴⢾⣿⣿⣿⣦⠀⠀⠀⢣⠀⠐⠊⠂⠈⠀⠰⠦⢀⣄⠀⠉⠀⠒⠂⠀⠠⠖⠁⡠⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢏⠀⠀⠐⣝⣛⡿⠟⢀⡠⠐⠁⠀⠐⠢⠤⣀⠀⠔⠒⠒⠠⡀⠛⠀⠓⠀⠀⠀⣰⡉⡄⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠈⢢⡔⠔⠊⡁⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⠀⢀⠧⠀⠐⠐⠒⠒⠉⠁⠈⠁⠀⠀⠀⠀⠀⠈⠂⠤⠤⠤⠤⠤⠤⠤⠤⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⢐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠤⠤⠤⠤⠬⢤⠄⡁
⠀⠀⠀⠀⠀⠀⠀⣇⠤⠔⣀⠔⠐⠀⢲⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⡀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠠⠤⢀⡀⠀⠀⠑⢄⠀⠀⢀⠔⠋⠊
⠀⠀⢠⠀⠀⡠⠁⠉⠢⣀⠋⠀⠀⠀⠈⠀⠀⠀⣠⡄⠼⠏⠁⠀⠚⠁⠉⠉⠉⢆⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠑⠐⠁
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠴⠦⠵⠶
⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⡀⠔⠁⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠈⠐⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠀⠠⠤⠄⠐⠒⠀⠁⠁⠀⠀⠀⠀⠙⠛⠳⠶⣶⣄⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⢰⣿⣿⣧⡀⠀⠀⢀⣄⣠⣬⣿⣇⠰⠂⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⢀⡀⠙⠛⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣦⣤⣾⣷⣾⣷⣶⣶⣤⡀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣀⠀⠙⢿⣿⡏⠈⠐⠞⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⢿⡿⠿⠿⣿⣿⣿⣿⡆⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣾⣿⣿⣁⡀⠀⠀⢸⣿⠏⢻⣿⣿⣿⣿⣿⣿⡇⠀⠈⣴⣶⡄⢀⣀⡀⢹⡏⣿⣿⣿⡟⣁⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⠿⠿⠿⠿⣦⠀⠈⠁⠀⠈⠻⣿⣿⣿⣿⣿⣷⡀⢢⣙⣋⣁⠹⠿⠟⢘⠀⢿⡿⠉⣿⣈⡠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⢏⣀⣀⠀⣴⣶⣿⣶⣾⣿⣿⡿⠂⠉⠀⠀⠉⠉⠙⠃⠘⣿⣯⣽⣷⣶⣾⣿⣿⣿⣷⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⢐⣨⣿⣿⡛⠛⠋⣤⣈⠁⢈⡿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢽⣿⡟⠉⠻⢿⣿⣏⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣿⣿⡿⠿⢻⠁⢶⣾⣭⣽⡿⠁⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⠀⢀⣀⣶⣤⣀⣠⣤⣬⠉⠐⠂⠀⠦⣰⣿⡿⠿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣁⣿⣷⢆⣼⣶⣾⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡶⢦⣿⣿⡿⠟⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡟⠛⠛⠋⠉⢻⣿⡏⠉⢁⣀⡀⠀⣿⣯⠁⠘⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣿⣏⣉⠀⠀⠀⠙⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢠⡄⠘⣿⣷⣿⣿⣿⣿⣷⣾⠿
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣟⠀⠀⠀⣠⣶⣶⣶⣤⠀⠀⠘⠃⠀⠀⠀⠀⠋⡟⠀⠙⠛⠛⢹⣿⣦⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⢻⣿⠁⠀⣤⣶⣾⣿⣿⡿⢇⡀⢰⣿⠟⠉⠉⠙⢶⣶⡄⣀⣤⣴⣤⣤⣀⡇⠒⠀⠀⠖⢸⠃⠙⠷⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⠀⢘⣿⣿⣿⣿⠁⠀⢨⣛⣛⡇⡀⠀⠀⡀⠀⣿⡿⠿⠿⣿⣿⣿⣿⣿⣶⣟⣷⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠘⠻⣿⣿⣿⡏⠈⠁⢀⣍⣉⣁⣉⣤⣀⣁⣠⣽⡇⠀⠀⢸⣿⣿⣿⣿⠟⠻⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⣌⣉⣉⣹⣿⣿⣿⣿⢟⣿⣿⣀⠀⠀⢸⣿⠟⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠛⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢀⣉⣉⣉⣉⡍⠉⠉⠁⢿⣿⣿⣿⣷⡆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡅⠀⠀⠀⠘⣿⠀⠀⢀⣠⣼⣿⣷⣶⣶⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⢰⣿⡄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⡀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡿⠘⠀⠀⢻⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣔⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠐⡆
⠀⠀⣠⡠⠀⠠⢤⠴⠄⠀⠀⢀⣀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄
⡀⠈⠛⠙⠚⣰⢊⡋⡛⠟⠧⠿⠎⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⠤⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠈⠁⠐⠒⠒⠚⠃⠀⠉⠈⠀⠰⠷⠿⢕⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢄⠀⠀⠀⠸⣉⡀⠂⠄⢠⣢⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠤⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠂⠀⠀⢸⡹⣆⡄⠄⢀⠀⠀⠀⠀⡀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠳⠈⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠘⡩⡺⢠⠀⠀⠀⠀⡀⠀⠐⢀⣁⠀⠀⠸
⠀⠄⠀⠠⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠒⠀⠂⠀⠁⣀⡀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠑⠁⠀⠀⠀⡀⠀⠠⠀⠀⠉⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠠⠤⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠁⠈⠀⡖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠇⠄⠀⠀⠀⠀⠀⢦⡰⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠂⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⡀⣀⣀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⣰⣀⣠⠤⠤⠠⠯⠿⠯⠿⠗⠀⠀⠀⡇⠀⠉⠉⠉⠛⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢁⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢤⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠒⠀⠄⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀⡐⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣀⡀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢻⡟⠉⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⣀⠀⠀⣿⢿⡇⠀⠀⠠⠤⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⣯⣤⣿⡄⠀⠀⠀⠀⣾⣿⣿⣇⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢳⣶⣦⣀⠀⠀⣀⣤⡀⠀⣰⡿⠛⡙⠛⣷⡄⢀⣴⣾⣿⣿⣿⡋⣴⣿⠋⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠘⣿⣿⣿⡶⣟⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⣿⣅⣶⣿⣿⡇⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⣸⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⢀⡀⡄⣆⢳⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⢀⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⢺⡜⡎⣧⢳⢹⡜⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠧⠼⠘⠈⠁⠀⠀⣶⡄
⠀⠀⠀⠀⢀⣤⣤⣴⡖⠒⣶⣶⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣤⣤⡀⠀⠀⠀⣀⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠷⠤⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⢽⣽⣧⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣾⣿⣿⠀⠀⢸⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡷⠿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠾⠚⠋⠁⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄⠀⢱⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⠴⠦⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢳⣀⣀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠛⠙⠯⣭⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠋⠍⠅⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⠿⡀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⣴⣿⡿⡟⠻⠆⣽⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⡿⠋⡀⠀⢠⣼⡿⠃⢸⣿⣿⣿⣿⡏⠉⢻⡟⠀⢹⣿⡄⠁⠉⠻⠿⡉⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠙⠀⣸⡿⠁⠀⢸⣿⣿⣼⣯⡇⠀⠘⡁⠀⢀⣽⣿⠖⠒⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⢸⣿⠏⢹⣿⠇⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠈⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⠤⣀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠢⡄⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡄⠀⠀⠑⠤⣔⠀⠀⠀⠀⠀⢀⡤⢤⡤⠤⢤⣼⡇⠀⠀⠀⠀⢀⡏⢠⠄⠀⢤⠈⣣⡈⠀⠀⠀⠀⠀⠀⠖⠋⠁⠈⠙⠲⣶
⠀⠀⠀⠀⢘⠰⠆⠀⠰⠆⢰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡤⠃⠀⠀⠀⠀⠀⢠⡄⠤⠤⠄⢤⠁⣿
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡸⠀⠰⠀⠀⠈⠘⢧⡀⠀⠀⠀⢀⠡⠲⠦⠤⠶⣔⠇⠀⠀⠀⠀⡰⠁⣤⡀⠀⢠⡄⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠰⡄⣁⡀⣼⠁⡔⠀⠀⠀⠀⠀⠀⠃⢤⣠⠰⣀⠓⠰⠄⠰⠶⠟⡀⢨⠙⠒⢹⡱⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠐⠎⠁⠀⠀⠈⠙⢢⠀⢤⠀⠀⠶⠄⠠⢠⠀⡔⠉⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠊⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢱⠠⠤⣴⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠇⠈⠁⠀⠀⠀⠠⣄⡁⠠⠂⠀⠀⠀⠀⠀⠀⢀⡑⠤⠤⠤⣂⠀⠀⠙⠠⠤⠤⠅⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⢄⣀⠰⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⢂⡀⠀⠀⠀⠀⠐⣉⢈⠀⠠⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠒⠤⠶⠤⠊⠀⠀⠀⡀⠀⡶⠀⠈⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠁⠄⠂⠈⠁⠑⠀⢠⠀⠀⠀⠀⢼⡀⠉⠀⠈⠁⡁⡆⠀⠀⣠⠀⣤⠀⠀⣤⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠂⠐⠲⠽⣢⡄⠀⠀⠠⠀⢀⠀⠜⢀⠦⠤⡀⠐⡄⠀⠀⠐⡞⢛⠀⠈⠀⣘⠛⡃⠀⠘⢿⣆⣀⣠⣀⣀⡴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠇⠒⠂⠀⠲⠂⠫⢀⠀⠀⠁⠀⡄⠸⠂⠀⠰⠆⢸⠀⠁⠀⠐⡀⢼⣩⡃⣤⣬⣍⣉⣉⠀⠀⠈⡁⠀⠿⠆⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⡔⣠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠊⣅⡈⠠⠀⣤⠛⠣⠀⠀⠀⠑⣄⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢦⡀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠈⠉⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠈⠙⠁⠃⠄⠀⠀⠀⢀⠀⠠⡨⠈⠒⠂⠱⢠⢤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣠⡄⠀⢠⣄⠀⠀⢠⣄⠀⡀⠀⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⠟⢻⣿⡿⢻⣿⣿⡟⢻⣿⣿⣿⠟⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⢿⠇⠀⠸⡿⠀⠀⠈⢿⠄⠀⠀⢿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⣀⣸⣿⣇⣸⣿⣿⣿⣀⣿⣿⣿⣂⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄⠀⢠⣤⣤⣤⡄⠀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣏⣿⣿⡇⠈⢹⣿⣿⠁⣾⣿⡟⢛⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⢿⣿⣷⠀⢸⣿⣿⠀⣿⣿⡇⠻⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⢱⣿⣿⣿⡇⠙⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⣻⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡴⠘⣦⢀⡸⡄⣀⣠⡆⠘⢶⡄⠀⣿⢤⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢋⣿⠙⡿⣇⣿⣿⡟⣹⣧⡛⢻⣿⠀⡛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⡇⢰⡟⢸⡇⠀⢸⡏⠓⠂⣸⡇⢠⣿⢸⡁⣶⠆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣸⣿⢠⡇⢸⣿⡇⢰⣮⣽⡇⣼⡟⢰⡇⣾⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠂⠀⠈⠃⠀⠀⠘⠂⠈⠋⠘⠃⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣿⣿⣷⣼⣿⣿⣧⣿⣷⣼⣧⣽⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢠⣿⡇⣿⡿⢷⣄⢸⡿⢷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⠀⣿⣿⡿⠠⢸⠁⢀⡉⢻⡇⢀⡈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢀⣿⣿⡇⣿⡇⢈⣿⢾⡇⠈⣿⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⠀⣿⡿⠁⠀⢸⠀⢸⣷⠀⡇⢸⣷⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣿⠀⣾⣧⣿⡇⣿⣧⣾⠋⢸⡇⢀⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠘⠋⡀⣿⠣⠘⢀⢸⢇⠘⠁⣴⡇⢸⡿⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠏⠈⠿⠼⠏⠀⠿⠇⠿⠏⠻⠦⠸⠿⠟⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⣸⣷⣀⣃⣰⣿⣀⣸⣀⣸⣄⣹⣇⣊⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡶⠁⢀⣀⠀⠈⢻⣿⣿⡏⠐⠠⠤⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣸⣿⣿⣿⣿⣿⣿⣿⣷
⡇⠀⠀⠉⠠⣶⣿⣿⡟⠁⠄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠈⡟⢉⡈⠉⠛⠛⠛⢿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠩⠿
⠀⠀⠀⢀⣿⣿⣿⣯⣤⣤⡄⠀⠀⠀⠿⠋⠐⢿⣿⣿⣿⣿⣿⣿⠀⠀⢱⣿⣿⠇⣰⣶⣦⢸⡟⢹⣿⡿⠻⣇⠛⠀⠈⠛
⣠⣤⡀⣰⡿⠟⠛⠋⠙⠛⢷⡀⠀⠀⣀⣀⣀⣀⠹⠿⠿⣿⣿⣿⣇⠈⣦⣬⣤⣤⡈⠛⠛⣨⣴⣾⣿⣿⣿⣿⣿⣿⡽
⣿⣿⣷⣟⣠⣤⡄⠀⣾⣿⣿⣷⣾⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠀⠹⣿⣷⣾⣿⡿⠿⣿⣿⣿⠛⠿⢿⣿⣿⣧⣶⣿⣿
⣴⣭⣿⣏⠛⠛⢁⣤⣄⠀⢀⠾⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠸⠿⠋⣤⠀⢠⡄⠀⢹⣿⠋⠰
⡿⠿⠏⡇⠰⣶⣿⣭⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣀⡀⠀⣀⣸⣦⣄⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣶⣿
⣷⡦⢀⣧⣴⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢁⣤⣤⣄⡀⠸⡿⠡
⡍⣼⣿⣿⡿⠿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⢠⣿⠟⠿⠟⠛⠛⣿⣿⡿⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏⠢
⣾⣿⠛⠉⠀⠀⠈⠿⢿⣿⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⣧⠀⠀⠀⢀⠀⠸⣿⣧⣶⣿⣿⣷⣾⣿⡁⠀⠉⠀⢘⣿⣿⣿
⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢀⣾⣿⣿⣿⣿⡆⠀⠉⠴⠀⠀⠉⠀⠸⡿⣿⠋⠻⢿⣿⠿⣿⡇⠀⠐⢺⣿⣿⣿⣿
⢿⣿⡟⠀⠀⠂⠀⢀⣼⣿⣄⡈⠀⠀⢸⡿⠋⠀⠀⠀⠙⣿⣷⡄⠀⠀⠀⠀⠀⣄⡇⣀⠀⠀⢀⠀⣿⠁⠀⠀⠈⠛⠁
⠘⢿⡇⠀⣠⣾⣿⣿⣿⡿⠿⢳⣦⣤⣼⢇⣀⠀⠀⣀⠀⢹⣿⣁⣀⣀⣀⣀⣀⣴⣇⢀⣠⣄⠈⣠⡏⠀⠀⠀⠀⢠⡀⠀⠛
⠀⠀⠁⣠⣿⣿⣿⣿⡏⠀⠀⣠⣤⣤⣤⣄⣁⣤⣄⡈⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡟⠀⠀⠀⠀⠀⠈⠆⠀⡀
⢀⣴⣶⣬⣿⣿⣿⣿⠁⠉⠀⠀⣦⣤⣤⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉
⠈⣿⠟⠛⠛⠛⠿⣿⣇⠀⢀⣾⣿⣿⣿⣿⣿⡟⠛⠉⢉⣿⣿⣿⠻⠿⠿⣿⣿⣿⣿⡄
⠀⢁⡀⠀⠀⠀⠀⢻⡿⠃⢰⡿⣫⣍⣩⣹⣿⡷⣶⠂⠀⠉⠉⠙⠀⠀⠀⣿⢿⣿⣿⠂⢀⣀⣠⣤⣤⣤⣀
⡀⢀⠁⢀⡀⠉⠀⠾⣷⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢿⣿⠟⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⣤
⣿⣞⢿⣯⣽⣦⣴⣾⣿⣿⣿⣿⣆⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣀⠀⠀⠈⠋⠀⠀⡿⠋⠉⠀⠀⠉⢻⠃⢠⣾⣿⣿⣿
⠈⣿⠈⠉⠛⠛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢹⣴⣾⣿⣿⣿⣦⣤⣤⣶⣤⣁⠰⠂⠀⠀⠒⢀⢶⣿⣿⠃
⠖⠀⠀⠀⠀⠀⠉⠉⠙⠻⣿⣿⣿⣿⣇⠁⠀⠁⠀⠘⢡⣿⣿⠟⠉⠀⠀⣀⢹⣿⣿⣿⣿⣦⣶⣛⣶⣶⣟⣻⣿⣿⠣⡄
⠀⠀⠀⢀⣴⣿⣷⣆⠀⣼⡿⣿⣿⣿⣿⣀⣀⣀⣤⣴⣿⣿⣿⡀⡤⠀⠀⢀⣀⣿⣿⣿⣿⣿⡿⠿⠿⠋⠉⢹⣿⣆
⠀⠀⠀⠀⢿⣿⣿⣿⣦⣙⠀⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⢠⣾⣫⣿⣿⣿⣟⡛⠛⠃⢤⡤⣠⣴⣶⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣄⠀⢀⡀⠀⠀⠀⠀⠀⠤⡄
⠀⠀⠀⡔⣓⢒⢒⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠉⠀⠀⠈⣐⠀⠀⡂⠀⠈
⠀⠀⡀⠷⠟⡐⠾⠇⠀⢠⡄⠈⠀⢀⡀⠘⠀⠀⠀⠠⠨⠀⠀⠈⢀⠀⡈⠀⠀⠀⠀⠃⠐⠀⠐⠂⠂
⠀⠀⢠⣤⠄⠀⣤⡀⠀⠈⠁⠀⠉⠉⠀⠀⠤⡀⠀⠀⠀⠐⠖⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⠀⠄⠄⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠁⢀⡀⢠⠠⠀⠠⠈⠀⠈⠈⢀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣄⡀⠀⠁
⠀⠀⠁⠀⠂⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢠⠀⠣⡀⠀⠀⠀⠉⢳⢂⡀⠘⠛
⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠈⢰⡇
⠀⠀⢠⡤⢤⠄⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠇
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠘⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢀⡀⠀⣀⠀⠉⠈⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⡀⠀⣠⠤⠀⢀⠀⡠⣶⣠⠾⡄⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⣅⡀⢀⡀⠀⠀⠀⠀⡃⠀⠀⠀⣀⠋⠀⠣⣽⣿⣧⠇⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠛⠃⠘⠓⠀⠀⠐⠀⠀⠐⠀⡌⠉⠀⢀⠀⠁⠀⠁⡤
⠀⠀⠀⠀⠀⠀⠠⠮⠄⠈⠦⠀⢀⡀⠐⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠂
⠀⢠⣤⣀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡁⠀⠀⠀⠀⠈⠀⠴⠆⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣴⣶⠠
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡟⣱⡿⠈⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⡂⣲⠿⣿⣿⣿⣿⣶⣤⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣴⣾⣦⣤⡀⠀⠀⠀⣤⣄⠀⢻⡇⣷⢆⢠⣶⡆
⠀⠀⠀⠀⣇⣿⣸⣋⡿⣽⣻⠟⣏⠁⠀⠀⣯⠹⢇⡏⠶⣿⠫⠜⡏⠸⠎⣷⠀⠀⠀⠈⠟⢷⡾⣿⣿⣴⣟⣩⡄
⠀⠀⠀⠈⠙⠿⣿⣿⣿⣷⣾⡼⠏⠂⠀⠀⢻⣇⣿⣇⣛⣳⣰⣇⣇⣺⣇⡿⠀⢠⣤⡄⠛⣻⣜⣪⣀⣹⣺⣧⣤⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠰⡿⠟⢩⡟⢻⡛⣯⠓⠈⠛⠁
⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠀⠀⠈⢷⣦⡀⢀⡀⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠿⠇⠈
⠀⠀⠠⣤⡼⢿⣿⡿⣦⢔⣄⠀⠀⢈⠿⢿⡟⢻⣏⢿⡉⣧⠄⠀⣠⣤⣤⣴⣿⣿⠷⣄⠀⠀⠀⣂⣀
⠀⠀⠀⢿⣮⣧⣧⣿⣾⣼⠝⠁⠀⢨⡇⢈⣥⠘⣷⠘⣟⣳⠀⠀⣏⡉⢴⠋⠩⡄⢂⡜⠀⢠⣾⣿⣿⣷⣾⣷⢲⡄
⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⠀⠶⢰⣿⣶⢿⡟⠿⠉⠀⠀⠹⢧⣼⣿⣿⠟⠋⠀⠀⡞⢱⣾⢫⡝⠱⠎⡘⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠻⣬⣩⡶⠴⣷⡾⢿⡁
⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⣀⣰⣶⣶⣶⡄⠀⠀⠀⢀⡀⠂⢰⡤⡀
⠀⠀⣾⢻⣟⢟⣛⢛⢻⣦⠀⢀⡞⠟⢟⡛⢛⠻⢻⣹⡆⠀⢤⣍⣹⣶⣿⣶⣏⣶⡉⠀⢀⠴⣾⢿⡿⢿⠿⢷⠶⣦
⠀⠐⠿⣬⣴⣯⣵⣿⠿⠟⠀⠈⠳⢿⢷⣾⣷⠞⣛⣏⠀⠀⢺⣦⠷⣴⣦⣶⢿⣭⣄⠀⢿⣖⣯⣴⣇⣸⡆⣿⣀⣿
⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⠆⠀⠀⠘⠐⠛⡖⠃⠁⠁⠀⠀⠘⠿⠶⣿⡧⠾⠧⠿⠴⠋
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣾⣾⠿⡿⢿⣷⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣼⠆
⠀⠀⠀⠀⠀⠀⢸⣱⡎⠣⡦⠁⢫⡆⠽⢿⠀⠀⠀⣠⣴⠶⠞⡿⠻⠛⠛⡉⣽⣿⡶⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠈⢧⣶⣷⣶⣿⣿⣿⣿⠏⠀⠀⠘⠿⣿⣴⣦⣶⠶⠧⠞⠛⠛⠉⠀⠀⠀⢀⣀⣀⣀⣀⣠⣀⣀
⠀⠀⠀⢰⣦⣀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⣏⣉⣋⣉⣉⣉⣩⣭⣿⡿⠃
⠀⠀⣠⣾⣿⣿⡿⣿⣦⡀⠀⠀⠀⢀⡀⣄⢂⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣀⡀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⢰⡟⢿⢟⣭⣧⣹⣽⡏⠀⠔⣠⣻⡏⣟⢻⡚⡟⠤⡔⠄⢼⣍⢫⡟⡟⡟⠋⠋⢣⢿⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⡄
⠀⢀⢻⣬⣿⣶⣿⣿⠟⠁⠀⠀⣙⡫⣧⣽⣼⣷⠿⠟⠀⠀⢰⡿⠼⠷⠶⠶⠾⠟⠛⠁⠀⠀⠺⣿⣤⣴⣤⣦⣤⣷⡇
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣧⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠄⠀⠀⠀⠀⠀⠠⠡⠊⠀⠄⠀⠑⠀⠀⠀⠀⠀⢀⠤⠐⡶⢐⡟⠉⠓⣄
⠀⠀⠀⠁⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠈⠢⣌⠥⣝⡭⢯⠓⢺
⠀⠀⠅⠌⣉⠫⠀⠀⠀⠀⠀⠀⠀⠀⠠⡩⢀⠘⣦⠈⡱⠀⠀⠀⠀⠀⠈⢌⠠⠀⠄⠠⢈⠔⠀⠀⠀⠀⠀⠀⠀⠓⢥⠂⠋⠁⠁
⠀⠀⠐⠒⠛⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢰⣶⠠⠐⣀⠐⢀⠠⢢⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠢⡄⠀⠀⠀⠀⠀⢀⣤⣄⠁⠀⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡃
⠀⡎⢀⡀⠈⡕⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⢔⡛⣆⠀⠀⠀⠀⠀⠀⠀⠿⠝⠠⡚⠙⢲⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠅⣈⠀⠅
⠐⡆⠢⡙⠚⠟⠡⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⣠⠺⣠⣀⠇⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶


⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠠⠐⠤⠖⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠁⠀⠁⠒⠫⡭⠂⠁⠀⠀⠀⠀⠀⠠⢀⠀⠄⠀⢁⣔⡤⠀⠀⠀⠀⠀⡁⠀⡀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⣲⣄⠀⠶⠶⠖⠇
⢀⠀⠀⠀⠀⠀⡏⠀⡀⠀⠀⠀⠀⢐⣂⠀⡄⠀⠰⠍⡀⠀⠀⠀⠀⠀⠀⡈⠀⠓⠀⡀⠈⠀⠀⠀⠀⠀⠀⢀⢀⡁⠀⢀⣀⠀⠂
⠀⠀⠐⠐⠦⠄⠀⠀⠀⠀⠀⠀⠀⠻⠩⠈⠒⠒⠒⠒⠈⠀⠀⠀⠀⠀⠀⠙⠠⠀⠤⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢀
⠀⣠⠄⣀⣐⠠⡌⠀⢢⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠶⣶⠀⠀⠀⠀⠀⢀⠴⠶⠲⡊⠨⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡇⠀⠀⠀⣠⠔⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⠈⠁⠃⡀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠀⠓⠲⠒⢄⠀⠀⠀⠀⠀⠀⠈⠀⠛⠁⠈⠆⡄
⠁⠀⠀⠐⠁⠚⠁⡀⡀⠀⠀⠀⠀⠀⠸⣟⢲⡶⠦⢬⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠠⢲⣀⠀⢀⡐⠋
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⠛⠟⠲⠺⠟⠅⠀⠀⠀⠀⠀⠀⠀⣄⣣⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⠋⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡬⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⣻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡾⠀⠀⠀⠲⠦⠤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠀⠀⠀⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠠⠛⠀⠀⠀⠀⠀⠉⠁⠐⡂⠀⠉⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⠀⠀⠀⠀⠀⠀⢿⡇⠒⠛⠃
⠀⣠⠴⠿⠄⠀⠀⠀⠀⠀⠀⠀⠿⢸⣿⠀⠀⠀⠀⠀⠀⡀⠘⠀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⡅⠁⠀⠒⠀⠀⠀⠀⠀⠀⠄⠑⢊⣁⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢠⣠
⠀⠐⡄⠠⡴⡀⠀⡀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠀⢸⣿
⠀⠁⢨⠄⠈⣝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⡄⠀⠀⠀⠘⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⡇
⠀⠇⠂⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿
⠀⠐⢦⣄⣀⣸⣧⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠉⠙⠛⠿⣿⠟⠝⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠛⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⣿⠟⠛⠻⠟⠛⠻⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⣿⢽⡏⠉⢙
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠛⠛⠓⠛⠛
⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠂⢀
⠀⢸⣿⡉⠙⠏⢉⣿⡇
⠀⢸⣿⣷⠀⢰⣿⣿⡇⠀⠟⣧⠀⡿⠁⣏⡷⡿⠇
⠀⢸⣿⣿⣤⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠰⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⡀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠐⠁⠀⠀⠀⠀⠀⠀⣠⠍⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠠⣶⢀⡈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⡉⣉⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⣀⣀⡀⠀⠐⠁⠀⠀⠀⠀⠴
⠀⢀
⠀⠀⠈⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣭⣭⣿⣥⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣶⣶⣾⣿⣽⣿⣶⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠻⣿⠀⠀⠻⠀⠀⢸⢄⠰⠈⠉⠈⣧⣾⣿⣿⣿⣿⠋⠉⠉⠙⣾⣿⣿⣿⠧⠀⠀⠀⠀⢈⡀⣡⣤⣌⡻⣿⡇⠀⠀⠀⡿⠛⠀⠀⢸⣿⣿⣿⣧⠀⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢠⣧⠀⠀⢰⡿⠁⠈⠙⢿⡟⠰⠀⠐⠀⠋⠙⠻⠋⠀⢠⣶⣦⣤⡜⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠂⠉⠑⢾⠋⠉⠋⡿⠀⠀⠙⠉⠁
⣶⠶⠶⣦⠀⠀⢰⣿⣿⣿⣿⡏⢠⣶⣾⣷⣄⠀⠀⠈⢰⣤⡎⣱⣾⡇⠀⣀⣀⠀⠟⠉⠁⠙⣿⠈⡅⠀⠀⢀⣧⡔⣶⠶⣄⢤⣄⣢⣤⠈⠃⠀⠀⠁
⠀⠀⠀⠈⠀⡀⢹⣿⣿⡿⣿⣿⣾⣿⠟⢛⡿⠇⠀⠀⠀⠙⠿⡿⠟⠀⣾⣿⣿⣷⠀⠀⠀⢠⡿⠀⠋⠀⠀⢨⣧⠈⠀⢀⣨⣿⣿⣿⣿⣥⣶⣾⣷⣄⠀⠀⣀⣤⣤
⠀⣀⣀⣀⡀⠀⢠⡈⠀⠀⠀⠉⠉⠉⠀⠛⠓⠀⢠⣤⣤⠀⣤⣤⡄⠀⠟⠛⠛⠛⠃⠜⠋⠙⣦⠀⠀⠀⠀⢸⢏⣆⠰⣾⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⠀⠀⠏⠉⠉
⣼⣿⣿⣿⣿⣶⣿⣧⡀⠀⢀⣼⣿⣷⣶⣾⣿⣷⣾⣯⣍⠁⠉⢩⣤⣀⠾⠿⢷⣦⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⣈⡉⢡⡭⡾⠀⠾⠟⠻⠋⠉⠉⠛⠃⠈
⣿⡇⠀⠀⢈⣿⣭⣯⣽⣄⣿⣿⣿⣿⣿⠏⠀⠈⢹⣿⣿⡆⠚⢿⡟⠋⠀⠀⠀⠛⣃⣿⣿⣿⡟⢈⠛⠿⠀⠿⢿⣯⣤⣴⣤⣤⡀⠳⣦⠀⠀⠀⢰⣷⠀⢀⣀⣀⡀
⣿⣧⡄⢠⣼⣿⣿⣿⣿⣿⣇⠸⠿⠿⠛⠀⠀⠀⢸⣿⠟⣁⣤⠿⣷⣤⣾⣽⣶⠶⡿⠟⠋⡙⢧⠚⠉⠀⠀⣼⣿⡿⠛⠛⠻⣿⣿⣇⠻⢡⠀⣄⣻⡃⠰⣿⣿⣿⡷
⡏⠀⠀⠈⠙⠁⣿⠛⠉⠻⡀⢀⠄⠖⠰⢄⣠⣶⣿⣿⣿⡿⠿⠶⠶⣸⣿⣿⣿⣎⠃⠀⠀⠀⠸⠅⢆⡀⠐⢿⣿⡇⠀⠀⠀⣿⡿⠃⠀⢡⣾⣿⣿⣷⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⣎⠀⠀⠀⣿⠟⠛⠿⣿⣿⢀⡀⢠⠀⡿⠻⣿⣿⠉⠀⣀⣀⣀⠰⣷⣿⣿⣿⣦⡝⠃⠀⠀⠀⠉⠀⠰⢦⣸⡿⠛⠿⣿⣧⢀⡿⣿⣳
⣀⣤⣄⣀⠀⠀⠀⠀⠁⡀⠀⠿⠀⠀⠐⡏⠀⠀⠀⠸⢃⠀⡠⢤⣠⡄⠀⠀⠀⠀⣾⠟⠛⠿⣷⠘⡟⠛⠉⠉⡃⢀⣀⣀⡄⢀⠀⣀⠀⠜⠇⠀⠀⡞⠁⣤⠘⣤⠓
⠙⠉⠉⠹⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣿⠃⣰⣶⣶⣤⠉⠀⠀⠀⣿⠀⠷⠀⠀⠰⡟⠃⠙⠉⠹⣾⣿⣶⣿⠆⠀⠀⠀⠀⠀⣉⠶⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⣿⡆⠀⠀⠈⠉⠀⠀⠀⠀⢸⡏⡁⠀⠀⠀⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⡫⠾⠛⠟
⢠⣾⡿⠖⢶⣄⢀⣳⡄⠀⣀⣤⣤⣤⣤⣤⣀⠀⣤⣾⣿⣿⣷⣦⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢽⡂⣀⣾⣀⣀⡀⠀⠀⠀⠀⠀⠀⠘⢃
⣿⡏⠁⠀⠀⣿⡟⡟⠠⣶⣿⣿⡿⠋⠉⠉⠙⣿⣿⣿⡿⠿⠿⣿⣿⠆⠀⠀⠀⣠⣾⣿⣷⣦⡀⠀⠀⠀⠀⠘⠷⣿⣿⠛⠛⠹⣿⣿⣄⠀⠀⠀⣠⣄⠀⣫⣴⣤⣤
⣿⣇⠀⠀⣠⣿⡟⠁⠀⠙⠻⣿⣧⠀⠀⠀⠀⠉⠻⡿⠀⠀⠀⢸⣧⠀⠀⠀⣸⣿⣿⡿⣿⢿⣿⣄⣀⣠⣄⣀⡀⢸⣿⠀⠀⠀⢱⣾⣿⠄⠀⠀⣿⢿⣿⡉⠀⠈⢻
⣿⡿⠃⠸⣿⣿⠗⠛⠁⠲⠄⠀⠀⡴⠀⣠⡀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠚⢠⣿⣿⡇⠀⠀⠸⣿⡟⠛⠋⠉⠛⣷⡄⠈⢀⣠⡀⠘⠋⠁⢰⠏⠀⠀⢢⣿⡆⠀⠀⢠
⢻⣧⣶⣦⡝⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠠⣿⣶⣶⡤⢶⣿⢿⡇⣰⠟⠁⠙⣦⠀⠀⠀⠀⠀⠀⢸⣿⣷⡦⠰⣾
⣾⡏⠙⠻⡇⠀⠀⠀⠀⠀⠀⠀⣦⣾⣿⣿⣿⣿⣆⣠⠀⠳⣦⢤⠀⠀⠀⠀⠈⠀⣀⠀⠀⢀⣿⡍⢩⠟⢻⡀⢸⡇⣿⡀⠀⢀⣿⠀⠀⣦⠀⠀⠀⠎⠉⡿⣀⣀⣸
⡿⠓⠂⠒⠃⠀⢀⣀⠀⣤⣶⣄⣽⣿⣿⣿⣿⣿⣿⠧⠠⡤⠿⢾⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⢀⡀⠷⢾⠃⢀⣀⠙⠛⠁⠉⠉⠀⠈⢠⣦⣴⣤⡀⠀⢿⣿⠿⢻
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠘⡿⠻⠋⠉⠁⠀⠉⠛⠀⠂⠀⠂⢰⠃⢀⣀⣀⣀⣸⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⡏⠁⠾⠵⠐⠿⠖⠀⣀⣀⣈⡉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣶⣾⣿⣿⣿⣿⣿⣿⡟⠛⢿⣿⡄⢿⠋⠉⠉⠉⡇⠀⠀⠛⠙⠂⠀⣾⡿⠛⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣄⠀⠀⠀⠀⠀⡀⠀⠘⠿⠟⠛⠛⠛⢣⠃⠀⠀⠀⠈⡇⠀⠁⢀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⡈⠛⠀⠀⠀⠀⣿⠉⠀⠀⠈⡆⠀⠀⣾⢿⣿⣿⠿⢶⡀⠀⠀⠀⢼⡄⠀⠀⠀⠀⡿⠀⠙⠉⠁⢰⡟⠁⠈⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⣛⣛
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⣿⡀⠀⠀⣰⠟⠀⠀⠅⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣏⣡⣤⣼⡁⣆⣠⠀⠀⠀⣾⣧⠀⠀⠀⣿⣀⣀⣀⠀⡀⢄⣠⢠⠄⠈⠉⠉⠉
⢀⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠿⠗⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⠃⠀⠀⠀⠛⠿⠀⠀⠀⡿⠉⠉⠉⢻⡕⠉⣀⠀⠀⡄
⣾⠁⠈⠙⠁⠀⠀⠐⠂⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⢂⠀⡀⣀
⠓⠀⠀⠀⠃⠀⠖⠀⣴⠂⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣾⣷⣾⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢰⣷⠀⠘⠛⢶⣶⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠉⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠿⣿⠿⢿⠿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⣰⡖⢥⣃⡂⠄⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⡁⢀⣴⠾⣧⣼⢟⠘⢍⠀⠉⢳⢌⠢⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠊⠀⠈⠀⠉⠑⣊⡟⡄⣡⢀⠀⢀⡼⠀⠑⠌⢂⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠂⠡⠀⠀⠀⠀⠘⢄⡌⠙⢂⠀⠒⠁⠀⠀⠀⢈⡆⢃⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠈⡀⠸⠙⠂⠀⠀⠀⠀⠘⡷⢛⡀⡀⢀
⠀⠀⠀⠀⠀⠠⠂⣀⣀⣐⠂⠄⠀⠐⣄⠀⢀⠤⢤⠘⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠚⠇⠁
⠀⠀⠀⢀⠤⠤⠤⠤⠤⠬⠧⠦⢤⡀⠈⠑⠴⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠃⠆⢠
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠀⠀⠀⠀⠙⠉⠁⠁⠢⠄⡀⠀⠀⠀⠈⠀⡙⠠⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢁⢠⡄⠀⠀⠀⢠⠀⡄⠀⡀⠀⠀⠀⠈⠢⡀⠀⠀⠒⠚⠓⠒⠒⠒⠒⠗⠛⠉⠉⢝⡭⢭⣍⢂⠀⠀⠀⠀⡰
⠀⠀⠸⠀⠀⠸⠇⠀⠀⠀⢹⠋⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸⣤⠂⣿⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠄⡀⠀⠀⢃⡈⠀⢀⣘⠂⠄⠀⢀⡀⢸⠄⠄⠊⢂⠈⠉⠁⠰⠀⠀⠀⠀⠠⠂
⡔⠐⡇⠀⠊⠀⠀⠀⠀⠀⠀⠀⠱⠀⠘⠉⢳⠂⠀⠈⠉⠁⠀⠐⠊⠒⠀⠈⠁⠀⠒⡀⠀⠀⠀⠣⠀⠀⡄⠀⢀⡀⠤⠐⡞
⡯⢁⠀⢠⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠍⠁⠀⠒⠀⠀⠒⠀⠉⡀⠀⠀⠀⠀⠀⠛⠀⠀⢀⢔⣑⡀⠃⠀⠀⠀⠀⠀⠐
⡇⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠇⢈⢀⠀⣄⠀⠀⣠⢀⠈⢠⠀⠀⠀⠀⠀⠘⠈⡠⠢⠈⢹⡟⢻⠀⠀⠀⠀⠀⠰
⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠠⠀⠀⠀⠀⠀⠀⠃⡀⣄⣀⣀⣠⠀⠐⣋⢀⣀⣀⣤⠤⠼⠤⠤⠤⠤⠤⠴
⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣾⣿⠟⠻⢿⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣰⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠇⠄⠀⠀⠤⠘⡇⠀⠀⠀⠀⠀⠀⢸⣿⠟⠙⠛⠏⠙⢿⡿⠀⠀⠀⠀⠀⠀⢻⡇⠠⠀⠀⠉⠅⣻⡇
⠀⠀⠀⠀⠀⠀⢸⡀⠂⠀⠀⠂⢸⠃⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠘⣷⠐⣀⣀⣐⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠣⣀⣠⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣍⣿⣩⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣆⣀⣀⣴⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠈⠉⠉⣩⣭⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠋⠻⣿⣷⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠙⠛⡉⢿⡇⠀⠀⠀⠀⠀⠀⢸⡿⢻⠿⣿⠿⢿⣿⣇
⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⠈⠀⠀⠀⠁⢸⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⠀⢹⣿⡆
⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠄⠀⢀⣠⣿⣧
⠀⠀⠀⠀⠀⢿⣿⣿⣦⠀⢠⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⡄⠀⣴⣾⣿⣿⠛
⠀⠀⠀⠀⠀⠈⠻⠟⠋⠀⠈⠛⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣥⣤⣬⣿⣋⠉
⠀⠀⠀⠀⠀⠐⠛⠒⠒⠒⠒⠒⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⢠⣶⣿⣿⡿⠿⢿⣿⣄⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⢻⣿⠂⠀⠀⠀⠀⠀⢸⣿⠟⠉⠛⠋⠈⠻⣿⡇⠀⠀⠀⠀⠀⣿⣿⠁⠀⠙⠀⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠹⠀⠂⠀⠀⠂⠘⠃⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠙⠏⠐⠀⠀⠐⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣦⡀⠀⠀⢀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠀⠈⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠛⠃⠠⠦⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠃⠀⠀⡀⠀⡀⠀⠀⠁⠤⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠰⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠈⠁⠉⠀⠁⠀⠃⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠆⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠡⠀⠀⠲⡀⠀⠀⠀⠬⠀⠀⠐⠀⠈⢀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⠂⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰⠃⣀⢀⡀⠀⡄⠀⡰⠀⠐⠀⠐
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠰⠆⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⠁
⠀⠀⠀⠀⠈⠀⠀⠉⠁⡀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⢀⡀⠀⠄⠠⠀⠀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⣀⠀⠀⠀⠀⠀⠒⢀⡀⢀⠀⠊
⠀⠀⠀⠀⠀⠈⠀⠠⠞⠋⠉⠉⠛⠆⠁⠀⠄⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⠂⡀⣀⠀⠀⠀⠀⢠⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⡀⠀⠀⠰⠦⠀⠰⠄⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠈⠒⠀⠨⠁⣀⠈⠀⠀⠠⠀⠀⠄⠀⠀⡀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠀⠀⠀⠉⠀⠈⠡⠀⠘⢠⠰⢀⠱
⠀⠀⠀⢀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠐⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠉⠀⠒⠤⣀
⠀⠀⠀⠀⣠⠖⠒⠁⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠈⠁⡀⠀⠀⢀⡤⠤⠤⣄⣀⠤⠒⠀⣀
⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡔⠁⠀⠀⠀⠀⡇⠀⠀⠀⢈
⠀⠀⠀⠸⠀⠉⠗⠤⢠⣀⣀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⡸⠀⣧⠀⠀⠀⠀⠀⡘⠦⣄⢠⡎
⠀⠀⠀⠀⢀⢼⣀⠔⠁⠀⠴⢠⠃⠀⠀⠀⠀⢀⠤⠂⠈⠁⠀⠀⠑⢄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢰⠁⠀⢇⠙⠚⠓⠤⣀⡷⡀⠀⡟⠉
⠀⠀⠀⠘⡐⢼⠁⠢⡀⠀⠉⢸⠀⠀⠀⠀⡰⠁⠀⠠⠤⠤⠤⠤⠤⠄⠝⠶⠠⠤⢠⡀⠠⠤⠒⠢⣀⣀⡣⠀⠸⠈⠀⠀⠀⠈⢱⠆⡰
⠀⠀⠀⠀⠈⣦⡀⠀⠈⠻⣹⠁⠀⠀⠀⠀⡁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡸⠀⠀⢀⢾⠊⠁
⠀⠀⠀⠀⢠⠋⠛⠲⠦⡴⠊⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠖⠋⠁⣸⣀
⠀⠀⢀⠔⠁⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠉⠀⠈⠳⡀
⠀⠔⡕⠒⠢⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠦⠤⠤⠒⠒⠒⠀⠄⠀⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠠⢀⠀⠸⡀
⠘⠘⠀⠀⠀⠘⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡠⠁⠀⠀⠃⠀⢃
⠀⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠙⠞⠀⠀⢟⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢠⠁⠀⠀⠀⠀⠀⢸
⢠⢃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢀⡠⢲⣗⠥⠂⠀⠀⠀⠀⢱⠴⣗⠠⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸
⢸⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣠⠖⠁⠐⢋⡴⠃⠀⠀⠀⠀⠀⠀⠑⢌⡁⠀⠑⢄⠀⠀⠀⠀⠸⠀⠰⠀⠀⠀⠀⠀⠆⠀⢸
⠸⠀⢇⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⣀⠠⠔⠊⢁⡀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢄⡉⠀⠐⠠⢠⣄⠆⠀⠀⠀⠀⡌⠀⠀⢸
⠈⠀⠈⡄⠀⠀⠀⠀⢀⠧⠒⠉⠁⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠈⠢⠀⠀⠀⡰
⠀⠀⠀⠘⡄⠀⠀⠀⠎⠀⠀⢀⣠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠢⠄⣄⠀⡐⠁
⡄⠀⠀⠀⠘⢄⣀⣼⠠⠔⠊⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⠤⣀⣀⠤⠒⠒⠲⠤⢤
⠸⡆⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡄
⠀⢰⠁⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢡
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢬⠉⠙⠒⠀⠀⠐⡆⠀⠐⠊⠩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣼⠀⢀⣀⣀⠀⠀⢇⣀⣀⣀⡀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠒⠊⠉⢀⠚⠛⠉⠀⠀⠀⡈⠋⠋⠉⠉⠈⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⢄⣀⠀⠀⢠⡉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠨⢆⠈⠀⠀⡸⠐⠒⠒⠒⠒⠒⠲⡀⠄⠀⠀⠄⠀⠉⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢰⡐⠄⠀⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠒⠒⠈⣀⠠⠘⠋⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠔⠈⠀⠉⡠⠂⠀⠀⠀⠀⠀⠠⠐⠀⡡⠔⠋⠙⠉⠻⠲⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠔⠁⠀⡠⠊⠀⠀⠌⠀⠀⠀⠄⠈⢢⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⢰⠁⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣠⠀⣤⣤⣶⣆⣀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⡲⠄⢀⣀⡀⠉⠁⡒⠒⠒⠂⠤⠀⣈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡿⠧⡌⣯⠁⣹⠽⢿⣷⣧⠠⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠁⠀⠀⣕⠂⡤⢠⠄⠀⠤⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠴⠃⣿⣷⣸⡦⠀⣿⠁⠀⠀⠁⠐⠠⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⠀⢀⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⢈⡀⡀⠘⣆⣀⣀⣽⣿⣿⣷⣄⠀⢈⠂⠄⡀⠀⠀⠈⢄⢀⡴⠃⠀⠀⠀⠀⠀⢆⠠⠺⢉⡗⢒⠤⣠⢲
⠀⠀⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠠⠀⠀⠒⠁⠀⠀⠐⡄⠰⢊⡴⢁⡞⠀⣠⢆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠃⠈⠀⠀⠀⢠⣄⠀⢀⡜⠵⣋⣴⢏⣠⠞⠀⠏
⠀⠀⠀⠀⠀⠀⡘⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠄⠀⠀⠀⠀⠀⢈⣕⡶⢭⡋⢥⣠⠯⣣⡀
⠀⠀⠀⠀⠀⠀⠠⠰⠄⠤⠠⣤⠚⠓⢲⡴⠒⠒⢦⡀⣀⣀⣀⣀⣀⣇⠘⠁⠀⠀⠀⠀⠀⠀⠉⠐⠠⣈⡑⠪⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠙⠷⠔⠚⠛⠢⠴⠟⠉⠛⠃⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⠀⠀⠀⠀⠑⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠈⠉⠉⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠈⠂⠀⠀⠀⠀⠑⠢⠤⠄⠄⠀⠀⠀⠀⠀⢀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⣴⣉⠑⠀⢰⠁⠀⠀⢸⡇⠀⠀⠙⡆⠀⠀⠀⠀⡠⠖⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⣠⠀⢠⠈⠉⢧⠀⣿⡀⠀⢀⢾⣇⠀⢀⢾⡇⠀⠀⢠⠇⠀⠀⠀⢸⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠈⢻⣆⠀⠀⢸⡆⣼⡇⠀⢸⠀⠈⠛⠂⠁⠀⠉⠉⠁⠀⠻⠦⠤⠊⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⢹⡆⠀⠈⠛⠀⢿⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⣇⠀⠀⠀⢀⠌
⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡇⣠⠊⠹⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢺⣦⠁⣿⠀⠀⢀⠿⣦⡀⡠⠂
⣇⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣸⢡⡇⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢲⡆⢀⣿⠃⠸⠇⠀⡟⠀⠀⡜⠀⠈⢿⡄
⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠥⣼⠀⢰⣇⠈⣆⡀⢠⡎⠉⠇⢸⡁⢀⣾⠀⢈⡿⠀⠀⠀⠀⣧⣀⠎⢀⠄⠀⠈⣿
⠀⠀⠁⠢⠤⣄⣀⠤⠄⠂⣽⠁⠀⣿⡀⠀⠉⢹⢻⠃⠸⣧⣀⡠⠏⠛⠁⠘⠧⠊⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠼⠉⠳⠤⠔⠃⣾⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⠜
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⣿⣧⣀⣿⣿⠂⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⣀⣤⣤⣽⣿⣿⣟⣡⣴⣿⣶⣶⣦⣤⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⠛⠛⣿⣿⣿⢿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠸⣿⣿⡀⢀⣾⣿⣦⡀⣾⣿⣿⣿⣁⣠⣤⣾⣿⣿⣇⣸⣿⣿⣴⣾⣿⣿⣿⡇⠙⢿⣿⣷⡄⠀⠀⢸⣿⡿
⠀⠀⠀⠀⢻⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⠛⠉⣿⣿⣿⣿⣿⣿⡟⠛⠉⢸⣿⡇⠀⠈⢿⣿⣿⡄⠀⠘⠿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⠹⣿⣿⣿⣿⣷⣤⣼⣶⣿⣿⣿⣷⣿⡿⠛⠿⢷⣦⣾⣿⣧⣀⣀⣾⣿⣿⡇⣤⣶⣦⡄
⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠈⢿⡿⠋⠛⠛⢻⠋⠀⠈⠋⠈⠉⠁⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⠟⠀⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⡇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣭⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⠀⢀⠀⠀⠀⣀⣄⣠⣀⠀⠀⢀⣀⣀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢼⡁⠀⡇⠀⡇⠀⠙⠂⢸⠀⢸⠀⢠⡾⠋⠀⠀⠉⠛⠛⠋⠉⠉⠉⠁⠑⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⣀⠔⠆
⠀⠀⠀⡞⠁⠀⠷⢀⡇⠀⡆⠀⣾⠠⠘⠀⠉⡀⠐⠀⠀⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⢀⠀⠤⠀⠐⠒⠉⠁⠀⠀⠉⡖⠒⢲⡏
⠀⠀⠈⠃⠀⠀⠀⠚⠁⠀⠁⠐⠋⠐⠋⠀⠀⠐⠈⠈⠀⠀⠁⠀⠈⠐⠄⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⡠⠒⠐⠐⠒⠠⡄⠀⠀⢀⣧⣤⢼⡇⡄
⠀⠀⣀⠂⠀⢐⠂⠛⠘⡒⠀⠀⢰⠈⠉⡆⢀⡄⠀⠐⠆⠀⠀⠲⡂⠀⣀⠀⠀⣠⠄⣠⡤⢠⣄⢠⣶⠀⣤⠌⢡⡀⠠⠄⡄⣰⡀⠀⡄⠉⠀⠀⡐⡈⢁⡄
⠀⢘⡜⠀⣀⢠⢤⠀⠠⡄⣀⠀⠀⠓⠐⠀⡈⠁⠀⠀⠀⠀⡀⠐⠀⠀⠋⠁⠀⠙⠂⠉⠑⠉⠉⠋⠃⠀⢋⠀⠀⠈⠀⠂⠈⠐⠀⢀⡀⠀⠄⢀⠄
⠀⠀⠇⠀⠰⡀⡁⠑⡈⡀⠇⠐⠀⠅⠈⠀⠀⠄⠀⠁⠤⠀⠀⠠⠂⠰⠤⠀⠀⡁⠀⠐⠀⠀⠈⠀⠈⠀⠁⣤⠀⠈⠳⠲⠞⠀⣠⡄⠁⠀⠸⡏⠀⢆⠀⢰
⠀⠀⠄⠀⠀⠆⣙⢚⣋⠰⠀⠀⠀⠁⠠⠀⠤⠀⠀⠀⢰⡆⠀⠿⠀⠀⢀⡠⠂⢠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠉⠀⠅⠈⠠⢈⣠⡶⠀⠀⠇⠀⠈⠉⠙
⠀⠀⠀⠠⢀⠀⠹⠨⠇⠤⡈⠂⠠⠀⠀⠁⠀⠀⣀⠁⠀⠈⠁⠁⠉⠀⠉⠀⠀⣀⣀⣈⣀⣂⣀⣁⡀⠀⠀⢄⠀⠠⠄⠁⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠂⢴⠠⠠⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⣿⡁⢀⣁⡀⠀⠀⠀⠉⠉⣉⣉⣉⣉⢉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠡⠈⠀⠀⠀⣄
⠀⠀⠀⡀⠓⠬⠂⠀⠀⠓⠒⠁⠠⠀⠈⠀⠀⠀⢰⡏⠀⠉⠁⣿⠈⠉⠻⠋⠉⢳⡄⢿⡄⢠⣭⣉⢀⣥⡿⠀⢀⣀⠀⢀⠀⠀⠈⠀⠨⠁⠈⠀⠀⢀⡄⣷
⠀⠀⠠⠀⠀⢸⠀⠊⠀⠚⠀⠀⢀⡀⠈⠀⠀⣀⠘⡳⡄⠀⠀⣿⢸⣷⠀⢠⣿⠀⡇⠀⡇⢸⣿⣿⠸⡇⠀⠐⠛⠛⠒⠛⠛⠀⠀⠀⠀⠀⠂⠀⠁⠈⣤⠄
⠀⢀⡀⠀⠂⠈⢀⠀⢀⠈⠀⠀⠐⢲⠐⢤⠤⡖⠀⠈⢪⣂⠀⣿⢸⡿⠀⠹⠟⢰⡇⣀⡇⢸⣿⡿⠀⡇⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⢨⣧⠀⡀⠃⢚⠀⢐⠐⢂⡀⢀⣺⠀⣘⢀⡇⠀⠀⠀⣽⡀⢿⡤⠴⠳⣤⡴⠎⠘⡯⠤⠤⠽⣷⣴⡇⠀⠁⢈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⢀⣀⡁⣤⡁⠀⢀⠀⡈⠁⠀⠉⠉⠁⠀⠀⠀⣀⣀⣬⣥⣄⣀⣀⣀⣋⣀⣀⣀⡀⢀⣀⢀⣀⣛⣀⣀⡀⢈⣀⣈⣀⡀⠉⠈⢀⠀⠀⠀⠀⠀⠀⠅⢀⠬
⠀⠐⠈⢡⡤⣤⡄⢻⠁⠠⠤⠀⠀⡀⠠⠆⢻⡙⣿⠏⣩⠀⣤⡀⠉⢿⣿⠈⠀⣶⡌⠹⢋⣉⠙⢠⣤⣾⡇⣿⢸⡅⠶⢾⡇⠠⡀⠀⠀⠠⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⣶⡀⠁⠀⠀⠀⢷⡘⢠⡿⢸⣿⡇⠀⢸⣿⠀⠀⠛⣡⣶⡸⢿⣶⠘⢿⣿⡇⣿⢸⡇⡶⠺⡇⠀⠄⠀⠀⠄⠸⡤⠶⣾⣿⠄
⠀⠠⠰⠹⢻⠣⠄⠈⠈⠀⠀⠈⠁⠀⠀⠀⠀⢸⡇⢸⣿⠘⠿⠃⡄⢸⣿⠀⠀⣷⡌⣿⢻⡆⠙⠠⣿⣿⠀⣇⣸⡇⣷⠀⠠⢄⡤⠀⠀⠀⠀⢆⢹⣸⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⡀⢀⠀⠀⢰⣤⣤⡄⠀⠀⣧⣸⡟⠷⠴⠞⠳⣤⣥⣾⣤⣿⣧⣽⣦⣤⡾⡦⠤⠾⠷⠭⢽⣧⣿⠈⣿⢻⣹⡇⣰⣾⠀⠚⣿⠋⠉⡀
⠀⠀⠈⢶⠃⠀⠀⠀⠄⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⡉⣁⣠⡄⣀⠀⠉⠀⠈⠉⠙⠀⠀⢸⠇⠀⡴⠖⠶⠂⠀⠀⠛⠉⠀⠸⢿⡿⠐⠟⠁⠀⡼⡏⠐⣿⣿⠃
⠀⠀⢀⡀⠀⠀⠀⠈⠉⡉⠁⢀⣀⢀⣀⠀⠀⠀⠀⠉⠁⠠⠀⠁⠀⠠⡀⠀⠀⠂⠀⢀⠀⢀⠐⠷⣀⡰⠟⢀⣠⣄⣀⡀⠀⠛⠃⠘⣿⢿⣿⡏⠃⠀⢹⡏
⠀⠀⢈⡀⣥⠀⠠⠄⠀⠀⠀⠤⣬⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁⠆⠀⢠⣂⡞⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⠏⡁⠀⠀⠸⠇
⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⡀⠁⠇⢆⠀⠀⠈⠀⢀⣾⠀⠀⡀⠄⠀⠀⠈⠠⠤⠁⠀⠀⢀⠀⡀⠀⢀⡀⠈⠤⠁⠀⠀⢀⠄⠀⠒⠑⠂⢄⠃⠁⠀⠀⢤⡀
⠀⠀⠀⠀⠁⠀⡷⠂⠀⠐⠺⠇⠀⡇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠁⠀⠐⠀⠀⠀⠀⢀⢶⡤⣒⡆⠀⠀⠰⠀⠘⡆⠀⠊⠂⠍⠀⢀⡀⣀⡹⠀⠙⠈
⠀⠀⠀⠀⠄⠀⠀⠅⠀⠨⠄⠀⠁⡄⡁⠀⠈⠀⠀⠀⠀⠀⣀⠈⠀⡁⢀⠀⢀⠁⠂⠀⠀⠂⠁⠀⠀⠀⠂⠀⡀⠀⠨⠅⠀⢶⠂⠀⠀⠀⠈⠀⠀⠀⠀⢨
⠀⠀⢸⠀⠇⠀⠠⠀⠠⠀⡀⠀⡄⠃⣡⠀⠀⣨⠃⠀⠀⠂⠈⠐⠀⠄⠂⠐⠀⠀⠃⠀⡐⠂⢄⣐⠄⠀⠀⠀⠀⠀⠈⡥⠀⠙⠁⣠⣤⠀⠀⢠⣤⡀⠀⢻
⠀⠀⠘⠀⢂⡀⣀⠘⠓⠛⢀⣀⡅⠀⠃⠀⠀⣮⠀⠀⠀⠂⠀⠀⠂⠀⠀⠉⢠⢀⠀⠀⢀⠀⠀⡇⠀⣀⢀⡁⠀⣀⠀⡉⢀⣀⡈⡉⠉⠀⠀⠈⣉⣉⣀
⠀⠀⢈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠩⠐⡂⠀⠀⠃⠀⢠⠀⠈⠀⡘⠀⣀⠀⢀⡘⠌⠀⠰⠀⠀⠀⡇⢸⣏⡸⠇⠀⡷⠀⠿⠌⠉⠁⡇⠹⣾⠁⡏⠉⠉⠘⡇
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠀⠠⠤⢀⠀⠀⠀⢀⠤⠒⠀⠈⠉⠉⠁⠐⠢⢄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠐⠀⠀⠐⠆⠁⠉⢂
⠀⠀⠀⠀⠀⠈⡆⠠⡄⠄⠤⠞⠀⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣤⠀⠀⠀⠀⡦⢀⢠⡎
⠀⠀⠀⠀⠀⠐⠇⢰⠀⠈⠄⡆⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢀⠔⠉⠑⠄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠈⡀⠀⠑⠚⠂⢄⡧⠀⡛⠁
⠀⠀⠀⠀⠈⢬⠂⠀⠐⢶⢲⠂⠀⠎⠄⠠⠒⠂⠤⠤⠤⠄⠐⠊⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈⠀⠀⠉⠠⡀⡄⠀⠀⣟⠴⠁
⠀⠀⠀⠀⠀⠀⡑⠠⠄⢠⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢥⠄⠊⣃
⠀⠀⠀⠀⣠⣊⡉⠂⠴⠊⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠊⠀⠑⡄
⠀⠀⠀⡼⠁⠀⠈⡄⠀⠀⠀⠘⡡⡀⠀⠀⠀⠀⢀⡴⣟⣦⠞⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠑⠘⡄
⠀⠀⡜⠀⠀⠀⠀⠃⠀⠀⠀⠀⠇⠈⢆⠀⡠⠂⠁⠈⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠆⠀⠀⠀⡇⡆
⠀⡘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⢯⠂⢀⠀⠀⠀⠀⢰⠀⠐⠀⠀⠀⠀⠁
⠀⠂⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢀⠑⠀⠀⡀⠘⠀⠁⠀⠀⠀⡐
⢠⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡀⠀⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠐⡄⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠤⢰⡀⡠⠁⠀⠀⠈
⣆⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡠⠂⢤⡀⠀⠀⠀⠀⢀⡠⠔⠂⢉⠀⠀⠉⠑⢢⡀⠀⠀⠀⠀⠀⢀⠔⠊⠉⡁⠀⠡⡐⢀⠀⠀⢀⡀⠀⠀⢀⠔⠈⠣⠄⠠⠜⠁⠢⡀
⠀⠀⠣⡀⠀⠈⠐⠂⢀⠍⢠⠀⢁⠚⠥⠈⠀⠁⠈⠁⠀⠕⠀⠀⠀⢀⠃⠀⠀⠀⠁⠀⠀⠀⠑⢆⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⢀⠜⠂⣈⠕⠀⠈⠀⡡⠐⠒⠢⢄⠀⠀⠀⠀⡀⣀⠜⠁⠀⠀⢸⠠⠈⢈⡈⡀⠀⣠⠀⠔⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠘⠒⠫⠤⠄⠄⡤⠊⢀⠀⢀⡠⠚⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⢳⠀⠀⡆⠁⢀⠀⡘⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠮⠤⠴⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣢⡈⢲⡰⠁⠀⢘⠖⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⣀⡀⡠⠔⠢⢄⠀⠀⠀⠀⠁⠈⠀⠀⠈⠀⣀⠀⠀⠀⣀⠀⠀⠈⠒⣄⠀⠀⠀⠀⡤⠊
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠇⢯⡱⢸⠒⠀⠀⠆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⡤⠴⠎⠀⠀⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⢄⡠⠃⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⢀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠛⠁⠀⣠⠄⠒⠒⡤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠰⠆⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠠⠁⣠⡀⡄⠀⢡⣧⠀⠀⠀⢨⠉⠉⡇
⠀⠀⠀⡀⠀⠐⠚⠀⠀⠐⠻⣄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⢰⠃⠀⣙⡷⠃⠀⣿⠇⠀⠀⠀⠈⠳⠞
⠀⠀⢰⠁⠀⠀⠀⠀⠠⡄⢠⠀⠃⠀⠑⠐⠊⠉⠉⠐⠢⠤⠀⠤⠔⠊⠀⠀⢀⠀⠀⠀⠇⠀⠐⠇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠁⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⢸⠂⣠⣀⡀⠀⠀⣼⠇⠀⠀⠀⠈⠁⠀⣀
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠲⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡈⠈⠁⢐⡏⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⡧⠀⠐⠒⠒⠀⢤⢄⠈⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠂⠚⠉⠢⣄⠀⠀⠀⠀⠀⠙⠶⠶⢥⣤⣤⡾⠁⠀⠀⠀⠀⡚⠓
⠀⠀⣇⠀⠀⠈⠀⠂⠈⣾⠘⠃⡅⠀⠀⠀⠀⠔⠁⠀⠀⠀⢀⣄⣴⠟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⡀⣠⣖⠠⠀⠐⠀⣷
⠀⠀⠉⡄⠀⠀⠀⠀⠀⠙⠐⡀⡇⠀⠀⠀⠊⠀⠀⠀⠀⢰⠃⠙⠋⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⢠⡄⠀⠀⠀⠀⠄⠀⢀⢸⣿⣿⡇⠀⠀⢸⡅
⠀⠀⠀⠲⠶⢤⠤⠤⡾⠡⡾⠐⠁⠀⠀⠀⠀⠀⠀⣀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⠈⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⠛⠶⠶⠶⢴⣄⡺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⣉⠀⢀⡂⠀⠀⠀⠁
⠀⠀⡀⠀⣀⠠⠖⠒⠊⠱⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠁⠀⠠⠔⠒⠂⡄⠀⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⣤⣀⣀⣀⡀⢈⠀⠀⠀⠀⠀⠀⠀⢀⢄⠤⠄⠀⠀⠄⠀⠀⠰⠂
⠀⠀⢸⡀⠀⠀⠒⠡⡐⠀⠰⡀⠀⠀⠈⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⡄⠕⠁⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠈⡃⠀⠐⠂⠚⠁⠀⠀⡁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⡈⠀⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡀⡀⠀⠀⠀⡧⡀
⠀⠀⠀⢣⠀⣤⠴⠖⠚⠀⠀⡁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣄⢣⣤⠇⠀⠀⠀⠀⠀⡀⠀⠁⠀⠰⡀⠀⠀⠀⠀⠂⠀⠒⠊⠁⠈⠒⢄
⠀⠀⠀⠈⠆⡀⠀⠠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠙⠋⠘⠻⠂⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠀⠀⠉⠉⠐⠲⠀⡀⠤⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⠀⡄⠰⠀⡄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⣿⠛⡇⢸⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡔⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠻⣤⣤⠀⣿⡇⢠⣤⡴⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠅⣠⡝⠀⠹⠀⠘⣥⡀⠭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠈⠠⠠⠀⠀⠀⢠⠀⠈⠠⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠤⠤⠀⠈⠛⠀⠀⠀⠘⠃⠀⠠⠤⠴⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡀⠈⠀⠀⠁⢠⣤⣀⣠⣤⠂⠁⠀⠈⠀⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢃⠀⠈⢂⠀⠀⠙⢻⠛⠁⠀⢀⠊⠀⢀⠃⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠦⣀⠀⠐⣤⡴⢾⠶⣤⡔⠀⢀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢈⠒⠹⡄⠀⠀⡼⠑⢊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠓⠀⢰⣄⠀⠀⠀⢀⣴⠀⡐⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠸⡿⠟⠋⠛⠿⡿⠀⢀⣀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠙⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠕⢀⠀⠀⠄⢀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠞⠀⢀⠚⡁⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⡴
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠆⠇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡏
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⢠⠘⠁⠀⣀⠀⠀⠀⠀⢀⡴⠋⢃⠀⢱⠀⢀⡀
⠀⠀⠀⠀⠈⠤⠂⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡄⡘⠀⠀⠀⡆⠸⠀⣠⠋⠀⠀⠀⠔⠋⠀⠀⢸⠀⠈⡆⢾⢋⡴⠒
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢾⠔⠁⠙⢦⠀⠀⠀⠀⠀⢰⢡⠃⠀⠀⠸⢀⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢱⣬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⡃⠀⠀⢀⡦⠃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⡍⡅
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠃⢀⠔⢛⢳⠀⠀⠀⣾⠁⠀⡘⣃⣼⠋⠂⢸⣻⠉⣀⠰⠅⠈⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡆⠀⠀⠀⠇⠘⠀⢀⡼⠃⠀⠘⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠣⠀⠈⠂⠀⠠⠖⠉⠀⠀⠐⠁⢠⡄⠃⠀⠀⠀⠀⣀⠤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠶⣶⣄⠀⠀⠀⠀⠀⠀⠐⠿⠂⠀⣸⠓⠒⠒⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠇⢀⡴⠆⣠⡄⢠⡄⣴⢺⡇⠀⢠⠇⠀⠀⠀⢀⣔⡿⢄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠏⠀⢼⣏⠴⡽⡧⣾⡼⠻⠛⡃⣼⣿⠀⠀⠀⠀⢦⡜⠷⠊⠄
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢤⣾⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠈⠆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣴⠾⠁⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠇⠀⠀⠘⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣀⣀⡴⠤⠤⠤⣇⡀⠠⣾⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠙⠣⠔⠋⠀⠀⠀⢠⠀⠀⠀⠈⠛⠿⠟⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⢀⡠⢤⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⡊⠀⠀⠀⠠⠀⠘⠀⠸⣦⣤⣤⢠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣼⠀⢠⠀⣀⡄⠀⠀⢳⠖⠀⠀⠀⠀⠀⢸⣆⣀⠀⠀⠀⣈⣃⣀⣀⣀⣤⣤⣤⣤⣤⣀⣀⣀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡴⠉⠀⠁⠀⠁⠳⠒⠄⢠⡀⠀⡤⠀⠀⠀⣀⣉⠉⠙⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠶⣄⠀⠀⢵
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠑⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡖⠈⠀⠀⠀⠀⠉⠙⠒⠤⠤⢀⣀⡀⠀⢈⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⣄⠀⠀⠀⠀⣠⠞⠁⣰⠏
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⣶⡞⠀⢀⣤⠞⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣦⣤⣴⣾⣯⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠶⠶⠶⠒⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁

⣀⢀⣠⡀⠀⠀⣀⡀⠐⠂⠀⠄⠀⠚⠀⣀⡀⠀⠀⡈⠛⠙⠉⠃⠀⠀⠀⠀⠀⠁⠀⠀⠢⡀⠁⣠⣀⠀⠁⠘⢀⡀⢀⠑⠈⠉⠀⠀⠀⠀⠁⠀⠘⡟⠄
⠈⠉⠀⢀⣾⡁⣀⠀⠱⠀⠀⠀⡴⠋⠩⣽⣿⣿⣯⡽⣆⠀⠀⠀⢨⠿⠟⠷⢤⠀⠀⠀⠀⠀⢰⠃⠬⠽⣆⠀⠀⠘⠀⠀⡀⠁⠈⠲⠖⠀⠈⠀⠀⣀⠉⠀⡈⣷
⠀⠀⠀⠸⡄⠀⠈⠀⢸⠀⠀⠐⠃⢀⠀⠆⠀⠘⠉⠄⡋⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡞⠀⠀⠀⣼⠀⠀⢠⠀⠀⣧⠴⠄⠴⠤⣖⠒⠒⠒⢣⠂⠀⠀⠀⢷
⠀⠀⠀⠀⠁⠀⠥⠀⠀⠀⠀⢨⠀⠈⣀⠀⠦⠀⠀⢀⠁⠀⠀⠀⠁⠀⠤⠄⠀⠁⠀⠀⠀⠀⠈⠀⠤⠀⠁⠀⡄⣤⠄⠀⢻⡀⢠⠄⠐⣉⣉⣉⣉⣉⡀⠠⠤⠀⠊
⠀⠀⠀⠀⠀⣅⣀⣠⠀⠀⠀⢈⠀⠀⢨⠂⠀⢺⠀⠀⠀⡆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⡠⢄⠀⠙⠛⠋⠀⠀⠀⠈⠢⠇⡄⠀⠀⠈⣇⣤⠜⠁⢀⠀⠀⠒⠦⠒⠂⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⡤⣀⣀⣢⣄⠀⣀⡀⠐
⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⡙⠯⠯⡲⡄⠀⠀⠀⠀⠠⡀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠚⠝⢀⣶⠀⠀⠈⠀⠀⢰⠋⣡⠉⡆⠠⠉⠂⠋⢀⠶⠝⠧⡠⣵
⠀⠀⠐⢃⡘⠀⠀⠀⠀⣦⠠⡤⠤⢴⡅⠀⠀⠀⠸⡀⠀⠀⠀⣀⢀⠀⠀⠀⠀⡌⠀⠀⠀⠀⡅⠀⠀⠀⠀⡟⠀⠀⠀⢻⡆⠀⠀⠀⠘⡆⠀⠀⠀⠸⠀⠀⠀⠀⠐
⠀⠠⠀⠈⠀⠠⠀⠀⠠⠐⠄⠴⠄⠐⠀⠀⠀⠀⠐⠳⠀⢂⠀⠊⠀⠀⠀⠀⠀⠁⠠⠔⠤⠀⠃⠀⠀⠀⠀⡇⡀⠀⠀⣸
⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⡖⠀⠐⠀⠀⠀⠀⠀⠀⠀⡄⡀⢀⡄⠀⠀⠀⠀⠀⣇⡗⠀⠘⡯⠀⠀⠀⠀⠀⢰⠁⠀⠈⡀
⡀⠀⣀⣤⣀⣤⠀⣀⡤⠈⠂⠠⠀⠄⠀⢀⠀⠀⠀⠐⠀⠄⠀⠀⢀⠀⠀⡀⠀⠀⠒⠠⠄⠓⢀⡄⠀⢠⡀⠋⠀⢀⠠⠉⠂⠀⠀⠀⠄⠀⠠⠤⠄⠀⠠
⠉⡛⠉⠉⠇⠉⠁⠀⢰⠂⠀⠀⠀⠀⠀⢺⠁⠀⠀⡀⠀⠀⠀⠀⠨⠏⠭⠟⢦⠀⠀⠀⠀⠀⠿⠤⣀⠀⢙⡄⢀⠀⣀⠀⠠⠁⡏⠁⠀⠀⠀⠀⠀⢀⡛⠣
⠀⠁⠀⠀⠏⠉⠀⠋⠸⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠃⢀⠀⠠⠀⠀⠀⠀⢻⠀⠀⠀⠀⠐⡂⠀⡀⠀⠸⠁⠢⡐⠂⠀⠀⠄⠁⠠⡁⡞⢀⠀⠀⣾⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⡀⠋⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠘⣀⠸⡄⠀⠄⠀⡸⠈⠤⠀⠀⠈⠤⢾⠆⠠⠁
⠀⠀⠀⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰
⡤⣠⠀⠀⠀⠄⠒⠀⠀⠀⠢⢄⣀⠀⠀⠀⠀⠁⠀⣠⠀⠀⢄⠀⠁⠂⠀⠂⠁⠀⡄⡀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠘⣫⡄⠀⠀⠀⢀⡀⢾⡀⠀⠈⠀⠀⠀⠀⠰⠂⠓⠀⠀⠀⡐⠀⠀⠀⠀⢀⣀⠀⢸⡗⠀⠀⠀⠀⠀⢠⡶⠪⠳⣶⡉⠀⠀⠀⠀⢈⠃⠈⠉⣷⠀⠰⠋⠐⢴⡳
⠀⠀⢰⡆⠀⠀⠀⢠⡇⠀⠈⠀⢲⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⣺⡿⡄⢤⣿⣄⠀⠀⠀⠀⢿⢔⠀⠐⢿⡇⠀⠆⠆⠆⢰⠂⠀⠈⠀⣿⠀⠀⠀⠘⡇
⠁⠀⠘⠀⠀⠀⠀⠀⢣⠀⠤⠀⣈⠂⠀⠀⠀⠀⢲⣀⠢⠄⣰⠃⠀⠀⠀⠀⡟⡀⠰⠀⢨⠅⠀⠀⠀⠀⣏⠁⠉⠉⠉⢃⠀⠀⠀⠀⠘⠆⢀⣢⠀⠟⠀⠀⠀⠀⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠈⢗⣀⣺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⡘⠀⠀⣧⠃⠀⠀⠀⠀⢻⠆⠘⠁⠀⠻
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⠓⠛⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⢀⡀⢀⠠⠤⠤⠤⠼⠠⠀⢀⡄⣀⠓⠃⠀⠀⢀⠋⠀⠀⠀⠀⠀⠚⠢⠔⠚⠀⢀⣀⣀⡤⡄
⡄⠀⠀⠀⠀⠄⢉⡁⠦⠀⠀⠀⠀⢠⠔⠇⠶⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡝⠊⠉⠚⡆⢀⡠⠀⠀⠠⠔⠋⠢⣤⡀⠀⠀⣀⡀⢎⡁⢩⡀
⠉⠀⠀⠀⢪⠀⠀⠀⠀⣤⠀⠀⢲⣎⡉⠀⠀⠁⢁⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡔⠐⠒⢂⠂⠂⡇⠀⠀⠀⡇⠈⡇⠀⠀⢰⣶⠀⠈⣧⡐⠒⠒⠺⣗⣅⠀⠈⠉⠉
⠀⠀⠀⠀⠸⡄⢀⡀⢠⡧⠀⠀⠀⠈⠆⢀⣦⡄⠠⠁⠀⠀⠀⠀⠈⠣⠼⠆⠜⠁⠀⠀⠀⠀⠀⢰⠀⠄⢰⠀⠀⠀⠀⠀⢿⠋⢃⠙⢙⠇⠀⠀⠀⠘⠻⡄⠨⠀⠸
⠀⠀⠀⠀⠠⠸⠆⠠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⢦⠀⠀⠀⠀⠀⠀⢈⠁⠀⢀⠀⠀⠀⠀⠀⡇⡄⠀⠀⡼⠐⠂⠒⠒⠒⠂⠈⠀⠈
⠀⠀⠀⠀⠁⡀⠠⡀⠁⡈⠠⠄⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠉⠒⠒⠈⠀⣠⠀⠀⣀⠐⠀⢀⣤⡀⠐⠀⣀⣀⠀⠉⠋⠀⠀⢛⠉⠀⠀⠁⠈⠉
⠀⠀⡌⠓⣄⠈⠒⣄⡐⠁⠀⠈⠑⣀⡄⠀⠀⠀⠀⠈⠀⠐⣶⣤⡀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠁⡐⠉⠄⠹⢻⠀⠘⠡⠈⠀⠀⢴⠀⠀⠀⠀⡀⣀
⠀⠀⠑⠄⢠⠈⠉⠉⠚⣆⠁⠀⠀⢀⠁⠀⠀⠀⢘⠀⠠⠀⠀⢹⡇⠀⠀⠀⠀⡆⠁⠀⠀⠀⡔⠀⠀⠀⠰⢖⠀⠀⠀⢘⡇⠀⠀⠀⠀⣠⠈⠀⠄⣠⠰⠀⠀⠀⡀
⠀⠀⠀⠰⠈⠀⠀⠀⠁⠛⠄⠲⠂⠋⠈⠈⠀⠀⠈⣤⠀⣤⠀⠸⠁⠀⠀⠀⠀⢠⡀⠂⠂⣐⠇⠀⠀⠀⠐⠘⠀⠠⠀⠠⠡⠀⠀⠀⠀⠇⠀⠄⠀⠘⠈⠈⠉⠀⠉
⠀⠚⡠⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⡄⠀⠀⡍⠀⢂⠀⠀⠀⠘⡄⠉⢰⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠲⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠉⠙⠳⣦⡀⠀⠀⠀⠀⢀⣠⣤⠶⠾⠿⠿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠘⣧⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⣨⣿⣷⠀⠀⠀⠀⣿⠀⣼⣅⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣽⣿⠟⠀⠀⠀⠀⣿⠀⠘⠧⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⢀⣀⠀⠈⢷⡀⠀⠀⠀⠤⠄⣄⣀⣀⣀⣀⠠⠤⠤⠤⠤⣰⡟⢻⣤⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⢀⣼⠋
⠀⣠⠊⠁⠀⡜⠁⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠙⠻⣦⣄⣀⡀⠀⠀⣀⣀⡄⠀⠈⠁⠀⠀⠀⢲⠂⠤⡀
⠀⠙⢤⣤⠞⠀⠀⠀⠀⠀⡀⠀⠀⠰⢦⣤⣀⣀⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠁⠀⠀⠂⠀⠀⠀⠀⢧⠀⣸⠆
⠀⠀⠀⠀⠀⢀⠀⡖⠀⠀⢇⠀⠀⠠⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢇⣠⠃⠀⠀⠈⠑⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠟⠛⠛⠛⠛⠛⠛⠳⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠠⠞⠛⠉⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣾⣷⣄⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣠⣄⠀⠈⢻⣿⣧⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⢹⣿⣧⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⢿⣿⡆⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⣿⡇⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢸⣿⡇⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⣿⣿⠃⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⣼⣿⡏⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⣾⣿⠏⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠀⠀⠀⢀⡤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⢦⡘⠁⠀⢧⡠⠒⠁⠀⣏⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠒⠦⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠩⠿⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠰⢤⠤⠼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⣀⣠⡌⠑⠚⠉⢉⠙⠲⢭⣑⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⣀⣀⣈⡉⢀⠀⠔⣛⡋⠁⠀⠀⠈⠳⣜⠉⠉⡹⠋⠀⠀⢀⠤⢄⠀⡀⠀⢙⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡖⠊⠁⠀⠀⠈⠲⠊⠉⠀⠀⠈⠑⢤⡀⠀⢈⢦⠀⠈⠓⣀⠄⠳⣄⠀⠙⠑⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡟⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⣼⠊⡇⢠⠊⠁⠠⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠁⠀⣨⠟⠉⠉⠀⠀⠀⠰⠶⣴⠄⠀⠀⠀⢀⠘⠀⢱⠀⢉⣦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡸⠒⣲⣾⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠈⠀⠀⠂⠀⠀⠀⠤⠀⠀⠀⡇⠀⠀⡰⢁⠞⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⡀⠀⠀⠀⠀⠀⠀⡔⠄⠀⠀⢀⡴⠃⠀⠀⣇⡼⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⢀⠊⠈⢇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡠⢈⡁⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠁⠀⠀⠈⠳⣄⠀⠀⠀⠳⢖⠶⠆⠀⢀⠀⢀⠔⠁⠀⠑⠶⠊⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠒⠒⡡⠔⠁⡀⠀⠙⠒⢦⠄⠀⢀⠔⠉⠀⠧⣎⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣀⠀⠀⣀⡤⠖⠒⠒⡛⠀⠀⢊⡀⠈⠀⠀⠀⢙⣦⡀⣀⡤⣪⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⡠⠺⣅⠀⠀⠀⠉⣿⠶⠖⢉⣉⡘⠀⣦⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠈⠑⠒⠐⠚⠷⠖⠀⠀⠀⠉⠉⢻⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢚⡃⠀⠀⠀⡀⠘⡇⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡍⡁⠀⠀⠘⣇⠀⢳⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⡉⠀⠀⠀⠸⡄⠎⠧⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⢠⣇⣀⣀⣀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⡟⠂⠀⠀⠀⠀⠀⠀⠘⢟⣀⠀⢰⠉⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⢀⠸⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⡂⢸⣹⡽⡸⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠋⠃⠀⠀⣈⠁⠈⠁⠀⠀⠀⠀⠄⠝⢻⢯⡀⠀⠀⠀⣞⠹⢉⡆⠀⠀⠀⢀⣴⣤⢶⣶⡶⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⢰⣧⣾⣶⡇
⠀⠀⠀⠀⠐⢠⠘⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠓⠊⠙⠒⠁⠀⠀⢾⡏⠀⠙⣟⠃⠀⠀⢸⣿⡿⠒⠟⠀⠀⠀⠀⠈⠙⡓⠛⣿⠆⠀⠀⠘⠛⠛⠉⠃
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠁⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣴
⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡍⣡⡐⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠦⢤⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠁⠉⠋⢘⠀⠀⠀⢘⡧⠰⣶⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠃⢠⣄⣉⠛⠛⠿⠆⠀⠀⠈⣯⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠈⠀⠀⠀⠈⠛⠋⠋⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠁⠀⠀⠄⠈⠉⠀⠀⠀⠉⠀⠈⠉⠉⠁⠉

⠀⠀⠀⣠⠀⠀⠀⢠⡄⠀⠀⠀⠀⠁⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⠀⠀⠀⢶⣤⡄
⠀⠀⠀⠫⡾⢂⠰⣶⠁⠀⠀⠀⠀⠊⠙⠒⠒⠀⠀⠀⠀⠀⠠⠀⠒⠠⠄⠀⠀⠀⠀⠐⢃⠀⠀⠀⢀⡐⠀⠀⠀⠻⠿⠃⠻⠿⠃⠀⠀⠸⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣄⣀⣀⣠⠀⠀⢀⣠⣤⣄⣀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠱⠀⠀⠀⣰⣶⣶⣶⡀
⠀⠀⠀⠘⠻⣿⡿⠟⠀⠀⠿⣿⣽⣽⣿⣧⠈⢣⣤⠀⣤⡈⠁⠀⠎⢥⡌⠳⠀⣠⡄⠰⣿⣿⣿⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠑⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠈⠁⠀⠈⠀⠀⠐⠃⠈⢵⠞⠀⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⠓⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠐⣖⠔⠶⢶⣖⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠖⠀⠀⠀⢀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⠆⠀⠀⢀⡀⠀⣣⡀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢸⡈⠋⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⠿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠲⢄⣠⠖⠤⣰⠃⠀⠀⠀⠀⠀⠀⢀⣀⡠⠃⠈⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠑⣄⣓⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⣀⠀⠀⠈⠛⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⡏⠁⠀⠀⠀⣀⣀⣀⠴⠂⠀⠀⢈⡥⠎⠀⠀⠐⠔⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢳⠀⠀⠀⡜⢀⢉⢪⠃⠀⠀⠀⠸⣀⣀⡀⠀⠀⢠⠟⠋⠁
⠀⠀⠀⠀⠀⢀⡏⠀⠀⢰⠀⠀⠀⢸⠀⠀⢠⠃⣸⠷⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⡀⠀⠀⢸⠀⢠⠯⣲⠃⠀⠀⠀⠀⠀⢀⣴⣶⢿⡻⣿⡟⣻⣷⡶⣦
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⡇⠀⠀⢸⢀⡏⠀⢸⠇⠀⠀⠀⠀⢀⠞⣗⡀⢉⣼⡿⣿⡤⡨⣼⠋⠲⣄
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡇⠀⠀⢸⡞⠀⢀⡟⠀⠀⠀⠀⠀⡞⠀⡌⠳⠋⢿⠃⠽⠌⠛⠃⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢷⠀⠀⠸⠁⠀⡾⠀⠀⠀⠀⠀⢸⠁⢸⡇⠀⣴⠂⢀⣠⡄⠀⠀⢸⠀⠀⣷
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣸⠃⠀⠀⠀⠀⢀⡏⠀⠀⢣⠎⠛⠙⣾⣿⣾⠅⠀⡎⠀⠀⣿
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢳⡀⠀⢠⡏⠀⠀⠀⠀⢠⠞⠀⢀⡴⢁⡠⠔⠒⠋⠉⢉⠆⡼⡇⠀⠀⣿
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡷⣤⠞⠀⠀⠀⠀⢠⡏⠀⣠⠊⡴⠉⢢⠘⡃⠀⠈⡟⡼⠁⢇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢈⠇⡖⢁⡜⠁⠀⠈⡇⠋⠀⢲⠂⡇⠀⢹⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠣⠜⠀⠀⠀⡼⢙⠋⠀⠙⡅⢳⠀⢻⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⢀⡜⠁⡈⠁⠀⠛⡇⠈⢧⠸⣃⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉⠀⠀⠈⠸⡄⠀⠃⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠋⠐⠀⠀⠀⠀⠀⠀⢂⠀⡀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠈⠈⡐⠀⠂⠀⠀⠠⢄⠊⠐⠄⠀⠄⠡⠜⠀⠀⠀⠁⡀⠐⡂⠢⠀⠀⠰⠀⡐⠁⠁⠀⠀⢠⠁⢄⡔
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠑⠀⠀⠀⠁⠀⠂⠀⠀⠀⠠⠀⠈⠑⠆⠀⠀⠢⠀⠀⠁⠂⠀⠀⠀⠂⠤⠀⠁⠁
⠀⠀⠀⠐⢀⠀⠠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⣀
⠀⠀⠀⠠⠮⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠄⡀⠀⠀⠈⡀⢀⢂⡂⠁⠀⢠⠔⠉⠀⠐⠀⠀⠐⢀⡄⢡⠀⠉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠄⡀⢁⠀⠀⠀⠀⠈⠂⡀⡀⣀⠀⠀⠀⠄⠀⠋⣈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠂⠈⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠔⠒⠒⠒⠒⠦⣄⠕
⠀⠀⠀⠀⠀⣀⣀⢠⠴⠶⣒⡶⠶⣬⣱⡿⠖⢂⣰⣿⣦⠀⢀⣀⣤⣤⣿⣤⣤⡴⠦⣀⣒⠶⠦⠤⠤⠴⠤⠄⢄
⠀⠀⠀⠀⠀⡁⠀⢀⡠⠎⠉⠀⠘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣼⠋⠀⢂⠀⠀⢀⠁⠆⡁⢸
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠐⠀⠐⠃⠀⠀⠈⢄⠉⠁⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⢛⠀⠀⠀⠑⢤⠚⠁⢰⡇⢸
⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠈⠉⠉⡇⠀⠀⠠⠀⠀⠠⠉⠀⠄⠀⠀⠀⠀⡈⠁⠀⣿⠀⠀⠀⠀⠀⡑⠄⠜⢣⠸
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢠⡀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠑⠤⠊⠀⠀⠀⠀⠠⢰⡌⠁⢰⢸⠀⡇
⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠇⠆⢸⠘⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⠙⠂⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⢀⠀⠀⠵⡠⠀⡉⠏⠀⡇⢸
⠀⠀⠀⠀⠀⠧⠀⠀⠀⠠⡀⠀⠀⡇⠀⠀⠀⠀⠒⠂⠀⠀⠀⠐⠠⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠁⠛⠀⠀⠁⢸
⠀⠀⠀⠀⠀⡄⢈⠢⠐⠈⠁⠀⠀⢡⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢂⢀⠀⠀⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠢⡇⢸
⠀⠀⠀⠀⠀⠧⠬⠤⣰⠤⠤⠄⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠀⠀⠀⡇⠀⢸⠂⠀⠀⠀⠀⠀⠀⢀⣧⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠓⠒⠒⠤⠤⢭⡤⠎⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⡟⢻⠀⠀⠀⠀⠀⠀⠀⠒⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⡭⡀⠀⠀⠀⠀⠀⠀⠀⡑⢙⠀⠀⡀
⠀⠀⠀⠀⠎⠉⣭⠆⠀⠀⠀⠀⣀⣀⠻⢡⠃⠀⠀⠀⠀⠀⢠⡶⡆⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⡅⠢⡀⠀⠀⠀⠀⢠⢜⠓⢄⡠⠟
⠀⠀⠀⠀⢀⡣⠛⡤⠀⠀⠀⠀⠀⢺⠄⡆⠀⠀⠀⠀⠀⠀⣄⠀⡵⠖⢄⠀⠀⠀⣀⠀⣪⠀⠀⡄⠐⣤⠄⠀⠀⠀⣒⡌⠀⠑
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⡀⣁⠀⠀⠀⠀⠀⠀⠹⣾⣲⢀⠀⠀⠀⠀⠈⠁⠀⡇⢈⠁⠀⠀⠀⠀⠀⠀⠀⠥⠠⠆
⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠰⢀⢆⠀⠀⠀⠀⠀⠀⠀⠸⢀⣨⠀⠀⠀⠀⠀⠀⠀⠀⡈⢢⠀⣄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⡆⠃⠀⠀⠀⠀⠀⠂⡘⢀⠂⠀⠀⠀⠀⠀⢀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠤⠀⠀⠱⢁⠀⠀⠀⢀⠂⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠈⠲⠒⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠈⠀⠀⠀⠈⠊⠀⠰⠤⠀⠀⠀⠀⠀⠀⠲⠚⠓⠒⠆⠀⠀⠀⠀⠀⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢚⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡇⠀⠀⠀⠀⠀⢸⢠⡆⠀⠀⠀⠀⠀⢈⢹⠟⠀⠀⠀⠀⠀⠀⢘⢉⡇
⠀⠀⢠⡀⠀⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⣿⠀⠀⠀⠀⠐⠉⣏⠉⠀⠀⠀⠀⠀⢠⠟⡄⠀⠀⠀⠀⣀⠀⢠⠁⠁
⠀⠀⠀⠋⢐⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⣇⡌⠉⠀⠀⠀⠀⠈⡇⢯⡚⠁⠀⠀⠀⠀⡽⠰⠳⠀⠀⠀⠀⠶⡂⡉⠀⠓⡃
⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠈⠄⠀⠀⠀⠀⢐⡥⠄⢬⠀⠀⠀⠀⠀⢇⠖⡒⠁⠀⠀⠀⠀⠈⠅⣤⡡⠁
⠀⠀⠀⠀⢀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠁⠹⢸⠀⠀⠀⠀⠀⠀⠰⡅⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⡄⢚⡤⠊⣠⠀⠀⠐⠤⢤⠔⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠴⠐⡀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⡌⠠⠐⡀⠀⠀⣭⢉⡉⣅⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠁⠑⣰⠄⠀⠀⠀⠘⢠⠣⠈⠄
⠀⠀⠀⠐⠒⠀⠀⠑⠁⠀⠀⠙⠀⠘⠈⠀⠀⠀⠘⠀⠀⠋⠀⠀⠀⠀⠓⠈⠚⠀⠀⠀⠀⠐⠚⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠓⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠠⢤⣤⣀⣀⣀⣤⣤⠤⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠂⠀⠀⠤⠤⠄⠄⠐⠊⠁⠀⠀⠀⠀⠀⠀⣀⠔⠒⠢⣀⠁⠢⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢠⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⢸⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⢇⠀⢀⢿⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⢀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠒⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡐⡇
⠀⠀⠀⠀⠀⢸⡜⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⢰⡇
⠀⠀⠀⠀⠀⢸⡘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⢷⡀⠀⠀⢸⡏⢹⡄⠀⣰⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡖⠻⡄⠀⠈⠛⠉⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠸⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⢀⣀⣀⣴⠇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⠀⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⠙⠢⠤⠀⠠⠄⠋⠀⠀⡽⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠓⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠙⠶⠈⠲⠀⠀⠀⠀⠀⠑⠶⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠤⠤⠀⠐⣒⣒⡒⠒⣒⣒⡒⠀⠀⠀⠀⠀⠀⠉⢸⠉⠉⠉⠉⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠀⠀⠒⠒⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣄⣀⣀⣠⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⠿⢿⣿⣿⣿⠛⠿⠿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⢋⣩⣾⣿⣿⠟⠛⠉⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⢿⣿⣦⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⢀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦
⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⣼⣿⡿⠁⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⣀⡀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆
⠀⠀⣼⣿⡟⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⣰⠞⣩⣽⣿⣷⡄⠀⢸⣿⣿⠀⢀⡴⢋⣽⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⢠⣿⣿⠁⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⢸⡇⣼⣿⣟⠁⠈⣿⠀⣼⣿⡇⠀⣿⢠⣿⣿⡇⠀⢘⡟⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠸⣿⣿⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⣧⣿⣿⣿⣶⣶⡟⢠⣿⣿⠁⠀⠻⣜⣿⣿⣿⣶⣿⠇⠈⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⢻⣿⣷⣄⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠋⣠⣿⣿⠇⠀⠀⠀⠉⠛⠿⠟⠛⠁⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⢀⣠⣾⣿⠏
⠀⠀⠀⠙⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣀⣀⣀⣀⣤⣤⣴⣶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⢀⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣀⣀⣀⣤⣾⣿⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⡀⠀⠀⢸⣿⡿⠿⢿⣿⣿⢿⣿⣿⠀⠀⠀⢀⣠⣴⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣼⣿⣿⠀⠀⠀⠀⣸⣿⣿⣤⣶⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣦⡀⢀⣰⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠁

⠀⠀⠀⢠⣤⣤⠀⠀⠀⣠⣶⣦⠀⠀⠀⣰⢶⡄⠀⠀⠀⡴⡶⠆⠀⠀⠀⡠⠶⠆⠀⠀⠀⠠⠶⠆
⠀⠀⠀⢸⠀⡸⠀⠀⠀⣯⠉⣸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠰⠶⠶⠂⠀⠀⠻⠤⠟⠃⠀⠀⠲⠤⠖⠀⠀⠐⠢⠤⠖⠀⠀⠀⠢⠤⠔⠀⠀⠀⠲⠤⠔⠂
⠀⠀⠀⢠⢴⣤⠀⠀⠀⢴⣤⣴⠀⠀⠀⣤⣶⣄⠀⠀⠀⣠⠶⡄⠀⠀⠀⣠⣶⣄⠀⠀⠀⢠⣴⠄
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠈⠉⠉⠀⠀⢠⣇⢀⣹⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠠⠀⠠⠀⠀⠀⠴⠤⠴⠀⠀⠀⠏⠈⠿⠀⠀⠀⠦⠭⠴⠀⠀⠀⠦⠬⠴⠀⠀⠀⠦⠌⠴
⠀⠀⠀⢀⣤⣄⠀⠀⠀⣠⣤⣄⠀⠀⠀⢠⣤⡀⠀⠀⠀⢠⣤⡄⠀⠀⠀⣠⣤⣄⠀⠀⠀⢠⣤⣄
⠀⠀⠀⣿⠉⢹⡆⠀⠀⠛⠉⢹⠀⠀⠀⠈⠀⠁⠀⠀⠀⢀⣀⡀⠀⠀⠀⢉⣀⡙⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠿⠁⡻⠁⠀⠀⠤⠀⡠⠀⠀⠀⢤⣀⡤⠀⠀⠠⢤⣉⠤⠀⠀⠀⠤⣉⠤⠀⠀⠀⠤⣉⠡⠄

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⠀⠀⣠⣄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣤⠞⠀⠀⠀⢀⡀
⠀⠸⠛⢉⠀⠀⠀⠘⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓
⠀⠌⠀⠀⠀⢂⡀⠀⠀⠀⢀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡦⠤⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠈⡄⠀⠀⠌⠀⠀⠀⠀⢠⢠⡄
⢸⠀⠀⣤⠀⠀⡇⣠⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢶⡤⢷⠶⠾⠿⠏⠉⠀⡇⠀⠀⡀⠀⠀⢐⣆⠀⢸⡆⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠠
⠘⡄⠀⠛⠀⠀⡇⠉⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠢⢀⡀⢀⡀⠜⠀⠀⠀⢣⠀⠀⠈⠃⠶⠈⠀⠀⡰⠁⠀⠀⠠⠀⠀⢠⡦⠠⠀⠀⠀⠀⠃
⠀⠈⠢⢀⠠⠊⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠦⠄⠀⠀⠀⠀⠀⠀⠈⠙⠂⠈⠙⠒⠂⠀⠀⠀⢀⡤⠶⠀⠀⠀⠒⠢⡀
⠀⠀⠀⢀⣴⠿⡀⢀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⡀⠀⠀⠈⠄⠀⠀⠀⢀
⢀⡤⠞⠋⠁⠀⠈⠆⠀⠈⠙⠲⡄⠀⠀⠋⠙⠛⠳⣦⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢴
⠀⢲⠀⠐⠛⠀⠀⡇⠙⠃⠀⠀⡇⠀⠀⠀⠀⠀⢠⡈⠟⠸⠀⢸⡇⠀⠀⠀⠀⠀⡄⠀⠀⠘⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠻
⠀⠘⡀⠀⠀⠀⢀⠇⠀⠀⠀⡔⠀⠀⠀⠈⠉⠩⠥⠷⠄⠀⠢⡈⠁⢀⠔⠀⠀⠀⠀⠄⡀⣠⠖⣄⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠐⠂
⠀⠀⠈⠒⠀⠔⠁⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠤⣄⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠋⠀⠀⠀⠈⠛⠳⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠘⠛⠇⠀⠀⠀⠀⠀⠀⢀⣤⠀⠐⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢀⡀⠀⢠⠂⢀⣀⣀⠄⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠋⠀⣐⠂⡀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡁⠀⠀⠀⠀⡆⠸⣯⣟⠓⠀⠐⢯⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃
⠀⠀⡀⠀⠀⠀⢣⠀⠀⠀⠈⠇⠀⠀⠀⠀⠇⠚⢿⠿⠃⡸⡀⠞⠷⠟⠀⠀⠀⠀⠃⠀⠉⠁⠀⠇⠉⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠃⡀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠊⠀⠈⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠴⠶⢄⠀⠀⠀⠀⠸⠟⠋⠀⠀⠀⠘⠛⠧⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠉⠓
⠀⠸⠛⠉⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄
⡀⠀⠀⣀⣀⠀⠀⠀⢈⣁⡀⠀⠀⡀⠀⠀⡇⠀⢠⠀⠀⠘⡄⢀⠄⠀⠀⠀⠀⢰⠀⠀⠀⢸⡇⠃⠀⠀⠀⠟⡆⠀⠀⠈⠀⠀⠀⠀⠖⠆
⠡⠀⠀⠈⠁⠀⠐⠀⠀⠈⠁⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⢀⠁⠀⠀⠀⡐⠀⠀⠘⡄⠀⠀⠀⠀⡀⠀⠀⠀⢀⠃⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⢀⠔⠁⠂⠀⠐⠁⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢀⡀⢀⡈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠸⡝⣿⠃⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣶⠀⢸⡇⢰⠶⣶⠀⣶⡀⠀⢰⡆⢰⡟⠒⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣿⣄⣼⡇⣿⠀⣿⠀⣿⣧⢠⡿⡇⢸⣇⡀⠀⠀⠈⠳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢸⡇⣿⠀⣿⠀⣿⢸⡿⢸⡇⢸⡏⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⢸⡇⠙⢷⠟⠸⣯⠀⠀⠸⡇⢻⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⡀⠀⢀⣤⠀⠀⣰⠆⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠜⠁⠐⢞⡋⠺⠯⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢠⡆⣠⠀⡄⣰⡾⢂⣾⠗⣰⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣠⠇⠺⠀⣿⡰⠉⢿⡤⠸⣧⢜⡟⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠁⢀⡀⠀⣀⡀⠀⠀⠀⠀⠀⡀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣼⠋⣿⠀⣿⡆⠀⣸⡏⢻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⢾⡇⣿⠀⣿⠀⣿⢷⣰⢟⡇⢸⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣿⣄⣿⢀⡿⠘⠟⢸⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠸⠃⠈⠛⠋⠈⠗⠀⠀⠘⠇⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠸⠛⠏⠙⠇⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠘⠁⠈⠹⠁⠀⠀⠀⠀⠰⣿⣶⣶⡆⠀⠀⠀⠀⠙⠛⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⠉⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠉⠈⡇⠀⠀⠀⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡦⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠐⠒⠀⢸⣿⣿⠀⠀⡇⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠃⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣘⡿⠃
⠀⠀⠀⠀⠀⠀⡸⠃⠐⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣤⣤⣤⠀⠀⢸⣿⣿⣿⣿⡁⠀⠀⠀⢀⠀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⠆⠀⢠⣿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣷⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⠁⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡃⠀⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣿⣿
⠀⠀⠀⠀⣀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⢀⠀⠘⢿⣿⣿⠀⠀⠀⠀⢀⣠⣴⣤⡦⠀⠀⠀⠿⠿⠰⣿⣿
⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠃⠀⠘⠢⠤⠤⠔⠀⠀⠈⠿⠥⠤⠤⠅⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠉⠉⠛⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢻⣿⣿⣶⣄⠀⠀⠀⠀⣸⣛⢛⣓⣀⠀⠀⠀⠀⣴⣶⠶⣆⠀⠀⠀⢀⣶⣾⣶⣦⠀⠀⠀⠀⠠⢒⣆
⠀⠀⠀⠀⠀⢻⠿⠿⠿⣿⠀⠀⠀⠛⠋⠀⣀⠈⠛⠃⠀⠀⠀⡟⠀⠀⢻⠀⠀⠀⣾⡧⠀⠠⢿⡇⠀⠀⠀⡖⠋⠋⢲
⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄⠀⠀⠀⠀⢀⣠⣄⣀⣦⡄⠀⠀⠹⣷⡄⠀⣾⠃⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠐⠄⠀⠀⠀⠄⠀⠀⠂⠀⠀⢀⠀⠀⠈⠗⠻⡛⡗⠃⠀⠀⠀⣀⠁⠀⢁
⠀⠀⠀⠀⠀⡀⠀⡀⠁⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⣷⠹⠄⠀⠀⠀⠈⠂⠔⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠁⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣶⣶⣶⡄⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⡿⡿⠟⢿⠀⠀⠀⠀⣾⠋⠈⢿⠀⠀⠀⠀⣾⠛⠛⢻⠀⠀⠀⠀⠸⡇⠀⠉⡓⠀⠀⠀⠀⡿⠛⠋⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣄⠀⠀⠀⠰⣤⣤⡼⠁⠀⠀⠀⠀⠁⡀⢠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢿⡟⠀⠀⠀⢀⠉⠉⠁⡀⠀⠀⠀⠀⠴⢠⣀⠦⠀⠀⠀⠀⣄⠈⠀⠡⠆
⠀⠀⠀⠀⢠⣾⣿⣷⣷⣿⣷⡄⠀⠈⠀⠂⠀⠂⠈⠃⠀⠀⡙⠄⠀⠼⡃⠀⠀⠀⠀⠳⠀⠠⠇
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠠⠶⠀⠀⠰⠆⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⡄

⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⡟⠛⠛⢿⠀⠀⠀⢠⣿⡿⠋⣿⡆⠀⠀⠀⢐⣋⣉⣙⣓⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⣴⣿⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠌⠉⠉⠡⠀⠀⠀⢸⣇⠁⠀⢠⡇⠀⠀⠀⠘⠁⠀⠀⠋⠀⠀⠀⠐⡟⢛⠛⣿⠀⠀⠀⢸⣿⠃⠐⢉⣧
⠀⠀⠀⠀⠀⠀⢀⠂⠐⡀⠀⠀⠀⢸⣿⠇⠀⢿⡇⠀⠀⠀⠀⠴⢀⠴⠄⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠘⢿⡦⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠁⠐⠃⠁⠀⠀⠀⠀⠋⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢐⣤⡐⠀⠐⢁⣴⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⡮⢵⣿⣿⡇⠀⠀⠀⠘⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠛⠛⠃⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣠⣤⣤⣤⣀⣾⠷⠶⢶⡀⢠⣶⠶⠶⠞⠛⢻⣇⣴⠟⠋⠙⠛⢻⣶⡟⠛⠛⠛⣻⡷⠀⣠⣤⣤⣤⠀⣠⡾⠛⠛⠛⣷⠀⢀⣴⡟⠉⣩⣽
⠀⠀⣰⡏⠀⠀⠈⠻⠁⠀⠀⠘⣧⡟⣷⠀⠀⠰⣶⡞⣿⠻⣄⣀⠀⠛⢯⣯⠛⠒⣆⠀⠛⢷⡾⠏⢀⡀⠘⣿⣿⠀⢠⢞⡿⠿⠶⣾⠋⡇⠀⠙⢹
⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⠀⠀⢠⣾⣧⣿⣦⠶⠞⠃⠀⢠⣿⡗⠒⠋⠀⠀⡼⡿⠀⠈⠉⠀⢿⣿⡀⠈⠋⡁⠀⣀⣾⠀⡇⠀⠺⠿
⢠⣿⡏⠀⠀⢠⠶⣀⣰⢷⣄⣀⣀⣹⣾⣀⣀⣀⣀⣸⡇⢿⣀⣀⣀⣤⣾⠹⣇⣀⣀⣤⣾⠱⣧⣤⡾⠳⣄⣼⡇⠙⠲⠚⢷⣬⡏⣿⠀⣇⣀⣀⣀
⣼⠉⠉⠉⠉⠋⣀⠉⠁⣀⡉⠀⣀⣸⣏⣉⣉⣉⣹⠿⣧⣀⣉⣩⡼⠛⣧⣀⣀⣀⣤⠾⢿⣤⣤⡾⠳⠶⠶⠞⠛⠒⠻⢦⣴⠟⠀⠛⠶⠮⠥⠴⠟
⠙⠛⠲⠶⠶⠞⠉⠓⢺⡏⠙⠛⠉⠁⠉⠉⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⢀⣀⡀⠀⣨⣧⣀⣀⠀⢀⣀⣀⣤⣤⣤⢤⡄⢠⡴⠶⠶⠦⣤⣀⣤⠴⠶⠶⢶⡄⠀⠀⠀⠀⠀⠀⢀⣠⠶⠶⠿⣆⠀⠀⣠⡴⠖⠒⢶
⠀⠀⢠⡟⠋⠉⠻⣾⠋⠉⠉⢻⣆⡼⡏⠀⠀⢠⣤⣼⣷⣿⠀⠀⠀⣤⣾⡟⣧⣤⡀⠀⢾⣇⣴⡟⠛⠛⢻⣤⣿⠃⢀⡴⣤⣿⣀⣴⠿⠀⠐⠿⢿
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣧⠀⠀⢈⣹⡇⣿⣈⣛⣳⠄⠀⢹⣿⣤⣤⠗⠀⠀⣿⣹⠀⠸⠇⠀⣿⣯⠀⠸⠼⠃⠉⠉⣿⠀⡇⠀⣤⣾
⢀⣿⡟⠀⠀⢀⣄⠀⢠⣄⠀⠀⠈⣧⣿⠀⠀⠘⠛⢻⡏⣏⠀⠀⠀⢀⣼⢿⠀⠀⠀⢀⡴⢫⡇⠀⣴⣄⠀⢹⡝⢦⣄⡴⣆⢰⡿⣿⢠⠁⠀⠈⠉
⣸⠻⠷⠶⠦⠞⠉⠷⠋⠘⠓⠚⠛⢻⡟⠒⠒⠒⢶⣾⡇⠙⠒⠒⣺⢿⡏⠈⠛⠛⢛⣹⣇⠈⢉⣽⣥⣈⣉⣿⢧⣤⣤⣀⣉⣿⠁⢿⣜⣓⣒⣲⡾
⠿⣤⣤⣄⣀⣴⠿⣤⣴⣿⡶⠴⠶⠛⠳⠶⠒⠚⠛⠉⠛⠛⠛⠛⠁⠀⠙⠒⠛⠛⠋⠁⠙⠛⠋⠁⠀⠉⠉⠀⠀⠀⠀⠉⠉⣿⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣤⣤⠤⠶⠶⢶⣤⡀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠖⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣹⣧

⠀⠀⠀⠀⠀⠀⣤⠄⣀⣀⣀⣀⡀⣀⠀⣠⠄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠠⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠛⠀⠳⣤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⢸⠀⣿⢦⡾⣇⠀⣧⢀
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡆⢹⠀⠀⢻⡆⣿⠉
⠀⠀⠀⠀⠀⠐⠛⠗⠀⠀⠉⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠃⠘⠃⠀⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠕⠈⠉⠉⠓⢌⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡤⠤⡤⠤⡮⠀⠀⠀⠠⠀⠬⣮⣇⣤⡤⢄⡠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⠔⠁⢀⠇⡸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⢮⠥⡽⠀⠀⠀⠀⠀⠀⠀⠀⠮⡅⡽⠂⠈⠢⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡾⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⠀⠀⠀⢿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⣄⢾⣼⡷⠀⠀⠀⠀⠀⠀⠀⠰⣞⣐⡳⠀⠠⠔⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠈⡎⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⣸⠀⡰⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠛⠛⠒⠒⠒⠂⠀⠀⠀⠐⠓⠓⠒⠓⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⣿⠙⢱⡎⢹⡆⠀⠀⠀⠀⠐⠒⡶⠒⠂⢶⢲⠀⠀⢀⣶⡀⢰⡀⢰⡇⢺⠉⠀⣴
⠀⠀⠀⠀⠀⠀⢿⠀⠸⣇⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⣄⠀⢼⠗⢷⠀⢳⣼⠁⢸⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠃⠈⠀⠛⠀⠈⠀⠀⠋⠀⠘⠋⠁⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣄⢰⣂⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠚⠋⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⣶⣦⣌⠁⢏⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⠀⢻⡏⠟⠻⣷⣦⣀⠢⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠱⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢿⣄⠐⠻⣿⣿⣿⣷⣄⡀⢁⡶
⠀⠀⠀⠀⠀⠀⠀⢐⣿⠉⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠦⢤⣭⣿⣿⣿⣿⣿⡉⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠿⠀⠀⢩⡏⠀⠸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢰⢿⣤⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⢷⣶⡎⠀⠀⠀⢏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢻⣷⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣼⣿⣾⣿⡁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢨⣉⣽⣿⣯⣿⣿⣔⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⡿⠊⠀⠀⣮⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠐⢼⣿⣿⡿⠟⠋⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢷⣿⣴⡗⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣾⢋⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡆⠉⠀⠘⠇⠀⠀⠈⢷⣹⡇⠀⠀⠀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠺⣄⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⢠⣤⡄⠀⠀⠀⠀⠈⣿⡧⠀⠐⠀⠛⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⣾⠘⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⠐⠄⠀⠀⠐⠁⠀⠀⠈⠃⡟⠂⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⣀⢀⣄⠀⡀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⡛⠈⠁⣀⡀⠀⠀⠀⠇⠳⣞⣇
⠀⠀⠀⠀⠀⠀⣧⡧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⣿⣿⣿⣿⡇⠀⠀⣇⠉
⠀⠀⠀⠀⠀⢀⡟⡇⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⣿⡿
⠀⠀⠀⠀⠀⢸⡇⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠁⢳⣷⣤⣄⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣼⣽⠿⢿⣟⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠀⠲⠀⠀⢰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⠈⠡⠀⡄⣤⠀⠀⠀⠋⢸⠤⠈⠀⠤⠸⠀⠀⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⡁⣿⣜⠆⣹⠃⠸⠀⠀⠀⠀⣻⠐⠐⡀⠆⠀⠠⠐⠀⠀⠀⠀⠰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠒⠂⠔⠲⠖⠷⠐⠚⠲⠶⠶⠶⡶⠦⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢄⣀⢀⣀⣀⣄⣠⣴⡄⡀⢀⡇⢛⡀⠡⢈⡧⡑⢀⡇⠀⠠⠀⠀⠘⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣬⣿⢿⣿⢿⣿⢿⠋⢁⠃⣠⡄⢸⣷⣧⣤⣤⣴⣤⣤⣤⠀⠰⠔⣀⠠⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠛⠛⠀⢉⠉⡑⢙⠀⠘⠀⠚⠉⡘⢛⠛⡛⢛⡛⡛⣻⣿⣀⣒⣒⢈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠎⠁⠀⠀⠨⠳⠝⠮⢻⠢⠉⠺⠀⣥⢈⡄⠇⠀⠅⡇⣿⣿⣤⣤⣤⡀⢔⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⠆⣰⣺⢣⠼⠴⢶⣏⠚⠒⣾⣿⣻⣿⣿⣟⠁⠲⣆⣨⣭⡅⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠉⡍⠁⠉⠉⡄⢁⢈⢸⡧⢐⣓⣿⣿⣿⣿⣿⣿⠀⠀⠤⠴⠅⠂⢸⣷⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠾⠶⢿⠖⠲⢶⣷⡭⠹⠸⡗⢀⠀⡿⠿⠭⠻⢿⠷⠄⠫⠙⠉⣥⡥⢸⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢈⠐⣻⣍⣀⣢⣗⢰⠀⠒⢐⡞⠁⠁⠀⠀⠇⡴⠀⠂⠐⠀⠀⠀⠀⢹⣿⣅⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠈⠿⠻⡿⠛⡛⠀⠒⠠⢤⠄⢔⢠⡄⠠⠀⣥⠀⠰⠴⠴⣄⠀⠄⢼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⠨⠇⠁⠀⠽⠀⡇⢋⡀⠀⠀⡀⠀⠁⠀⠀⠐⢠⡄⠤⢐⣿⠏⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢲⠔⠂⣷⣿⣾⡶⠿⡷⠾⡶⣿⠿⢿⢧⠰⡤⠡⠤⣄⠀⠀⠤⠄⠀⠀⠸⠿⠃⣈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⠀⣿⣿⣿⡇⡃⠆⠣⠧⠸⠀⠘⢠⡱⡇⢺⢸⠀⠀⣀⣐⣒⣀⡀⠀⡗⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠁⢡⣤⣧⣴⣤⣤⣷⣤⣥⣧⣼⣬⡄⠀⡉⠁⢬⣥⣀⡀⢁⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠖⠸⠀⠀⠀⠉⠠⠀⠈⠀⠸⠀⠀⠀⠈⠉⡟⠉⠏⡛⠻⠛⢿⡟⠛⠛⠃⠀⠈⡄⠘⠛⠃⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠕⠌⠘⠈⠃⠀⠘⠃⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡀⠀⠀⠀⢀⠀⢀⡀⢀⡀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡱
⠀⠀⠀⠀⠀⣟⣶⡓⢀⣤⠀⢀⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢥⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠸⣯⡙⢦⠉⠂⠀⠀⠀⣦⡘⡖⡇⠀⠀⠀⠀⠈⠙⠰⢦⠄
⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⡖⠙⠂⢳⣴⠈⠦⠀⠉⠀⠀⠀⠀⠀⢀⠐⢀
⠀⠀⠀⠀⠀⢿⣠⠇⣄⠑⠆⠁⠀⠀⠈⢉⣠⣤⡀⢠⣄⣀⠀⠀⠀⢾⡹⣇⠄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⡀⠰⠆⣀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⢀⠤⠂⠈⠉⠀⠀⠱⡄⠙⡌⠙⠲⣄⠀⠀⠘⠂⠘⠟⢠⣁⠂⠀⠀⠀⡴⠉⣀⣀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣄⠐⣤⠿⡠⠂⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⠀⠀⠈⢳⡄⠀⢰⣆⠀⠀⠁⠀⠀⠀⡜⢀⣤⡄⠈⠛⠁⠀⠈⢷⡀⠀⠀⠀⠀⠀⢻
⡀⢠⠀⢀⠀⠀⠻⠃⠀⣠⠃⠀⠀⠂⠈⠀⠀⠂⡂⠀⠁⠀⠀⠀⠀⠀⢹⡄⣾⠛⠦⠤⠾⠴⣄⢆⠁⠈⠛⠁⠀⠀⠀⠀⠀⠀⢣
⠱⣞⢳⠈⠛⠀⠀⠀⣆⡹⠀⠀⠀⠰⠿⠂⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⡇⠀⠀⠄⢀⣀⠘⠦⠄⢀⣀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠉⠀⣀⡔⢦⠀⠀⠈⡇⠀⠀⠈⠓⠒⠀⠀⠑⠂⠉⠀⠀⠀⠀⠀⠀⠈⠶⠲⡁⢠⡤⡄⠀⠀⢀⡀⠀⠀⡄⢉⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠰⢄⣟⠳⠄⠀⠘⢿⡀⠀⢠⡆⢚⠛⣿⠚⢻⡂⣻⣁⠗⠀⠀⠀⢰⠃⠀⢣⠘⣟⠳⣧⣀⠼⠷⠘⠛⢃⠂⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠙⢦⠀⣠⣅⠀⠳⣄⠀⠳⠚⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⣠⢇⡀⠀⠘⠆⠀⠀⠀⠀⠄⠀⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⢨⠇⣴⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢆⡀⠛⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⡹⠟⠢⠤⣄⡤⡀⠤⢼⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⣤⡂⢦⣼⡟⠀⠀⠀⠈⣯⡕⠲⠖⠒⠈⠉⡔⣤⠘⢁⣼⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢿⠞⠁⠀⠀⠀⢸⡀⠀⠀⢸⠀⠀⠀⡌
⠀⠆⢀⠰⢀⠀⢫⠁⠀⠀⠀⠀⠀⠀⣐⣳⠀⠸⠀⠀⠀⠙⠟⠀⠀⠈⠋⠙⠢⠾⠂⡿⣀⠆⠰⢶⡟⠂⠀⠀⠀⠀⠀⣸⡇⠀⠀⠸⡇⠀⠀⡇
⠀⠘⠄⠀⠉⠀⡎⠀⠀⠀⠀⠀⠀⣀⡙⠇⠐⠉⢟⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠙⣆⠀⢠⡄
⠀⠀⠀⠀⠀⣴⡀⠤⠄⠒⠂⠉⣡⣌⡒⠃⠀⠀⠸⡆⠀⠀⠈⠙⠒⠢⠤⢄⣀⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⢏⠀⠀⠀⠀⠈⠙⠮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡙⢯⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⡨⠄⠰⠀⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⡄⠀⣀⣴⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠰⢣⡞⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠴⡄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠊⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⠇⠀⠀⠀⠀⢸⣅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣚⣂⣐⢒⣉⣀⣀⣀⣀⣀⣈⢳⣆⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣺⣅⣙⣂⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠀⠀⠀⢈⣥⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⢠⠀⠠⣶⣄⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠠⠄⠀⠂⠈⠉⢀⠤⠈⢀⡴⣤⠀⠀⠀⠀⠈⠻⠧⠀⢴⠀⠀⠀⠀⢳⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⢘⠟⠀⠘⠃⠐⠀⡇⠀⠀⠀⠂⠁⠀⠀⠀⢀⣀⠀⠂⡀⠀⠑⣄⠀⠀⢸⡦⠃
⠀⠀⠸⠃⠠⡖⠀⠀⠀⠀⠀⡀⠉⠀⠀⣼⠀⠀⠂⠀⠀⠀⠀⠀⡆⣀⣰⠀⠀⠈⠉⠉⠀⠀⠈⠌⠠⢢⠎⠀⠀⠀⠀⡄
⠀⠀⠈⢆⠨⠛⠀⢰⡎⠂⠀⠈⢒⠥⠀⠀⠠⠀⠀⠠⡤⠤⠴⠊⠀⡄⠡⠄⠀⢀⣍⠀⠀⠀⠀⣠⠜⠁⠀⣖⠀⠀⠀⢥⡀
⠀⠀⠀⠀⠀⣠⠀⠀⢆⣀⣠⠴⠀⠃⠀⠀⡸⠀⠖⠳⠀⠀⠀⠀⠀⡁⠢⣷⣄⠈⠉⢸⠇⠰⣤⠃⠀⠀⠀⠛⠁⠀⢿⠇⠁
⠀⠀⠀⢀⣠⡀⠀⠂⠀⠀⠀⠤⠤⠤⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠈⠀⠰⡋⢑⠀⠀⠀⠰⠄⠈⠑⠀⠈⡤⠤⠞
⠀⠐⠴⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡀⠀⣙⠄⠀⠀⠀⠀⠀⠀⠀⠈⠔⢢⠀⠀⠙⠅⡀⡠⡀⠀⠠⡄⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⠀⠀⢀⠀⠙⠈⠀⠐⠛⢡⠀⠀⡠⡄⠀⠠⡀⢠⠀⠘⡆⠀⠀⢀⠉⠨⠟⠁⣛⠃⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⠀⠀⠀⠀⡜⠖⠠⡸⠖⠀⡸⣦⢸⠂⠐⠃⠀⢀⠀⠐⠒⠂⠀⢀⡴⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠀⠀⠀⠀⡇⣀⠀⠀⣤⣤⡄⠊⢸⠀⠀⠀⠀⠈⠁⠘⠁⠀⠠⠋⠈⠢⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠀⠀⠀⢰⠀⠀⠉⠃⡇⠀⠘⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠠⠴⠃⠀⠀⠠⠼⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠳⣆⢤⣠⠤⢔⢂⣰⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠠⠀⡀⢀⠠⣄⣠⣤⣿⡚⡟⢁⡰⡎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡡⠒⠉⢈⠹⠎⠉⠀⠀⠙⢾⣥⣹⡀
⠀⠀⠀⠀⡀⠠⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢢⣴⠒⠒⠿⠋⠑⠄⠂⠙⣿⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠤⠄⢢⡀⠇⠀⡰⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠉⡆
⠀⠀⠀⡀⢠⠔⠚⠒⠒⠒⠒⠻⠟⠀⠘⠷⠯⠤⢒⠷⢾⣶⣦⠢⣑⡀⠀⠴⠀⠐⠄⠀⠀⠈⣴⡠
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠄⠀⢈⠏⠀⠀⠀⠀⠀⠐⣄⠀⡔⠜⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢸⡄⠀⠀⠐⠒⠖⠙⠐⡥⠤⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢸⠄⠀⠀⠀⠀⠀⠀⢆⠸⡐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠸⠀⡀⢀⢀⠀⢀⠠⢸⢐⠌
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⣷⠶⠤⠤⠬⠠⠄⣊⡜⠋
⠀⠀⠀⠿⠦⢤⠤⣄⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⢀⢒⠂⢄⣔⣵⣷⣭⠨⠤⠴⠚⠉⠀⣀⣠⡠⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠙⠃⠒⠒⠻⠚⠛⠋⠋⠁⠀⠠⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠈⠈⠉

⠀⠀⠀⠀⠀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡤⠴⠖⠒⠦⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣁⣠⠤⠦⠤⣤⣀⣙⡆⠪⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⢔⢄⣠⣤⣴⡶⡟⣷⠾⣛⣽⡿⠶⢖⣿⣷⠲⠶⢶⣶⣾⣿⣬⡶⠮⣁⣾⣶⠶⠶⠴⠶⡶⢦⡀⢠
⠀⠀⠀⠀⠀⡁⣿⣀⠀⡠⠜⠉⠀⢸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢰⠏⠁⢂⠀⠈⠉⢐⣐⠀⣷⢀
⠀⠀⠀⠀⠀⠀⣗⢠⠋⠀⠀⢐⣀⣘⡃⠀⠀⠡⣈⠁⠀⠀⠀⠀⠀⠀⠀⣂⣀⣘⡇⠀⠀⠑⢄⠊⠈⣿⠐⡏⢸
⠀⠀⠀⠀⠀⠃⣼⠁⠀⠀⠀⠈⠀⢸⡇⠀⠀⠂⠀⠀⢊⠀⠀⠀⠀⠀⠠⠁⠀⡸⠄⠀⠀⠀⠀⣑⠰⠹⠀⣿⢸
⠀⠀⠀⠀⠀⠄⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢱⠈⠀⠀⠀⠀⠀⠼⠄⠀⠀⠀⣿⢨
⠀⠀⠀⠀⠀⡆⡇⠀⠈⠶⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⣿⢨
⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⢑⡀⢀⠀⢤⠂⠀⠀⣿⢸
⠀⠀⠀⠀⠀⡂⢻⠄⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠂⠀⠈⠑⠲⠆⠀⠀⢀⡇⢸
⠀⠀⠀⠀⠀⡃⣧⠀⠢⡀⠐⠃⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⢀⠀⠄⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⠈⣷⢸
⠀⠀⠀⠀⠀⠃⢿⣐⣡⣈⣦⣤⣄⣀⣧⣀⡀⠀⠀⠀⠀⠀⠚⠃⠈⠁⠀⢸⠀⠀⠆⠀⠀⠀⠀⢀⠀⣹⣤⡟⢸
⠀⠀⠀⠀⠀⠠⠲⠠⠄⠘⠐⠀⠉⠘⠅⠐⠉⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠺⠦⠴⠗⠒⠶⣤⣤⣥⡼⠋⠁⠖⢸
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠐⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⣄⣀⣀⣤⠂⠀⠀⠀⣆⡀⣀⠠⠀⢠⣤⠀⠀⠐
⠀⠀⠀⠀⠸⣧⣀⡘⢠⣤⢠⣤⢠⣤⣠⠄⣤⢀⣤⢠⣄⣰⣇⠀⠀⠀⡏⢸⠀⢻⠀⣀⢀⡀⢹⠁⣿⢀⣀⡀⡇
⡀⠀⠀⠀⠠⡈⠻⠻⣦⢻⠀⢻⠠⠁⢹⡀⠿⠀⣇⠠⠇⢸⡇⠀⠀⠘⠀⢸⠀⠈⠀⡅⣨⡇⢸⠀⣿⢠⠋⠀⡇
⠀⠀⠀⠀⠠⠷⡄⠠⠋⠸⠧⠸⠧⠀⠿⠅⠠⠐⠧⠀⠀⠸⠧⠀⠀⠆⠠⣾⣦⡀⠜⣇⡸⣧⣼⣄⣿⡀⢶⠰⡦⠀⠀⠀⢀
⠀⠡⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠺⡄⠀⠀⠀⢀⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂
⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠂⠄⠀⠀⠀⡀⢀⢀⠀⡀⢀⠠⠀⠄⠀⠐⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡋⢹⣿⡋⣻⣿⠉⣿⣿⣿⣿⠄⠀⠀⠈⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠈⠛⠶⢤⣤⣤⣤⣤⣤⣄⣺⡏⠁⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⢸⣿⣿⣿⣿⡟⠿⢿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⡿⠿⢻⣿⣿⣿⣿⡇⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠙⠿⠿⠷⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠾⠿⠿⠋⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣿⣿⣿⣧⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣀⣼⣿⣿⣿⣿⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⣿⣿⣿⣿⣿⡏⠉⠙⢿⣿⣷⡀⠀⠀⢸⡇⠀⠀⢀⣾⣿⡿⠋⠉⢹⣿⣿⣿⣿⣿⣿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣿⣿⣷⠀⠀⢸⡇⠀⠀⣾⣿⣿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⣿⣧⠀⢸⡇⠀⣼⣿⣿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⣿⡇⢸⡇⢸⣿⣿⠇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠹⣿⣿⣟⣛⣣⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠋⠉⠉⠉⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⠿⢶⡶⠒⣶⣶⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠉⠉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠇⠂⠀⠀⠀⠐⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡿⠂⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⠇⠐⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⠀⠀⠀⠀⠂⠸⢿⣧
⠀⠀⠀⠀⠀⠀⢸⣷⣤⠀⠀⠀⠀⠀⢀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡄⠀⠀⠂⠀⢀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡦⠀⠀⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠳⠄⠀⠔⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⢿⡿⠟⠃⠀⠀⠙⠻⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠿⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠛⠛⠿⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠙⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⠀⠀⠀⠈⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⠀⠀⠈⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠈⠀⠀⠀⠁⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠀⠀⠀⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠉⠉⠉⠉⠉⠘⠿⠆⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠐⠊⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠐⠋⣉⣤⣈⣉⠙⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣰⠇⣴⡾⠟⠻⠻⣿⣿⣄⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠋⠉⠉⠹⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠟⠛⠁⠹⣿⣿⡏⠁
⠀⠀⠀⠀⠀⢠⣿⣶⡟⠁⠀⠀⠀⠈⠙⢿⣼⣧⠀⠀⠀⠀⠀⠀⠀⢸⣇⣬⣤⡄⢠⣬⣤⣽⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠈⢿⡿⠃⠀⠀⠀⠀⠈⠀⠾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠃⠈⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⠀⠀⠀⠁⠘⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠻⠦⡀⠀⠀⠀⠀⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⠀⠀⠀⠀⢀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠷