Coughing Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣷⣄⢀⡏⡇⠏⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠈⢿⣿⡄
⠀⠀⠀⠀⢸⣿⡿⠿⠋⢉⡄⢙⠛⢧⡘⠀⡀⠀⡇⠀⠀⠀⠀⠀⢤⠄⠀⢻⠛⠆⡆⠺⠀⠀⠉⡇
⠀⠀⠀⠀⢸⣿⡃⠈⡦⢖⠛⠚⢤⠁⠑⣄⠥⠤⡇⠤⠤⠐⠒⠒⣒⡒⠀⠘⡄⠠⡒⢢⠀⢠⣿⣿⣂
⠀⠀⠀⠀⠀⢆⣀⣩⠈⣒⠨⢔⠜⠀⠀⢰⠀⠀⠁⠀⠀⠀⠈⠁⠄⠀⠀⡀⢾⠂⠙⡗⠊⡍⠁⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣉⠴⡖⠒⣩⣆⠀⠐⠚⠀⠀⢸⠀⠀⠀⣠⠀⢀⠴⡪⠂⡼⢄⣀⠏⠉⠀⠀⠀⠐⢄⠀⠘⡄
⠀⠀⠀⠀⠀⡰⠊⠀⠀⠈⢰⠛⠿⡀⠀⠀⠀⢀⡰⠀⠀⠀⣀⠔⠁⠈⠀⢰⠁⠀⡸⢆⠀⠀⠀⠀⣀⣼⡀⢀⠇
⠀⠀⠀⠀⢸⠀⠀⠔⠊⠉⢡⠀⢠⠃⠈⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⢸⠀⠀⠀⣎⣄⣀⡽⠋
⠀⠀⠀⠀⠸⡀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠖⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣦⠔⠋⠀⠉⠀⠀⠀⠀⠸⠧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⢀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠏⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠧⢤⠀⠀⢀⡤⢄⡀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⢦⢀⡏⣖⡀⣷⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠦⣀⠀⠀⠀⠳⡀⠀⢳⢠⡧⠆⠀⠀⠀⠘⠶⠃⠇⣤⡋⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡄⠀⠀⠀⣸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢤⡀⢀⣠⡀⠀⠀⠀⠀⢠⠏⠀⠧⠴⣲⠞⢋⣨⠷⢤⡀
⠀⠀⠀⠉⠓⠲⢤⠀⠀⠀⢀⡤⣀⠙⣩⠞⠁⠀⠀⠀⣰⠃⠀⣀⠴⠋⣡⠖⠉⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⡷⠋⠳⣌⠙⡆⠀⠀⡰⣧⠴⠊⢁⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡝⢾⣷⠞⠀⢸⡆⠑⠲⠊⠁⢸⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⠒⣿⡀⠀⢸⣇⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠁⠳⠀⡼⢹⡄⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠤⠞⠁⢠⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⡤⠊⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⣶⣤⣽⣿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣇⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠛⠁⠀⠘⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡂⠀⠀⠀⠘⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢛⠹⢷⣤⡀⠐⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠢⣀⢀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠰⡑⣄⠈⠇⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⢣⠀⠑⠀⡉⠂
⠀⠀⠀⠀⠀⢀⠀⡞⠀⠀⠀⠀⠀⡎⠀⠀⠀⣙⢰
⠀⠀⠀⠀⢀⡃⢸⡇⠀⠀⠀⠀⣸⣀⣀⡴⠀⠈⠇⡆
⠀⠀⠀⠀⢸⢠⣾⣿⠀⠀⠀⠠⣬⡿⠇⣊⢗⣠⡇⢇
⠀⠀⠀⠀⡟⠈⢸⣿⠀⠀⠀⠀⡟⠙⠿⣷⣟⠁⢻⡇
⠀⠀⠀⠀⢷⢷⠸⣿⡀⢀⣀⡐⢧⠀⢰⣿⠀⠀⢸⡉
⠀⠀⠀⠀⢘⣿⣷⣏⣴⡒⠤⣌⣾⠀⣞⢿⡗⠔⠛
⠀⠀⠀⠀⣼⠈⢻⣿⡗⡥⠀⠈⠃⠀⢹
⠀⠀⠀⠀⢦⣰⣄⣿⣷⣭⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⠀⣶⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡀⢹⣿⣇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡧⡄⠙⠿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⠂⠀⢠
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣄⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⢿⣿⣿⣦⣘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡮⠀⠉⠙⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡏⠑⠒⠤⢄⣀⣀⣲⡆⠴⠖⠒⡖⠒⠋⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⢙⣛⠛⠿⠿⠏⠙⠋⠉⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇
⠀⠀⠀⢡⠀⢠⠖⠒⠤⠔⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠈⠙⠋⠀⠀⠀⠀⠠⠶⠄⠀⠀⡀⢠⣤⠀⡇
⠀⠀⠀⠀⢇⠘⢄⠀⠀⠀⠀⠀⠀⠀⡠⠠⣆⠀⠠⠂⠈⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⠀⠀⠀⢀⡾⠛⢻⣤⣜⡁⠀⠀⢰⠃⣀
⠀⠀⠀⠀⠀⠳⢎⠀⠀⠀⠀⠀⠀⠐⣇⣀⣘⠈⠑⠒⠤⣀⠞⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠈⠿⣦⠾⣿⣿⣿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⠉⢈⡹⠂⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣲⣤⣤⢤⣤⣴⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⣸⡓⠩⣹⣏⢧⠀⢀⡠⠊⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣾⣿⣿⣧⣙⣺⣿⣿⣿⡿⢟⠁⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⢰⡡⠒⡀⢹⠈⠑⠉⠈⠆⠗⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡇⢸⠀⠀⠀⠀⠸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⠀⠐⢾⣀⡎⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣛⣩⣹⢛⣿⣿⡛⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢱⠢⢤⡋⣘⡅⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣃⣞⣛⣛⣛⣛⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡏⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠛⠻⠿⠿⠿⠛⣛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠂⠀⠀⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣯⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸⢧⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⣿⢙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠎⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⡵⠃⢸⣿⣝⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢓⡒⡴⠋⠀⢨⣑⣒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡟⠀⠀⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃⠀⠀⠈⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⢅⣀⠈⢿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠀⠀⠀⠀⣿⣿⣿⠟⢋⡁⠀⠀⠀⣀⠀⠀⢿⣿⡿⠀⠀⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠸⠛⣀⡇⠙⠻⠀⣿⠀⠉⣣
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠋⠉⠁⠀⠀⠀⠀⠑⣤⡀⠀⠀⠀⢓⣀⡀⠀⠈⠋⣤⡖⠁⠀⠀⠀⠀⠛⠛⠛⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢀⣀⣠⣾⣿⣷⣄⠀⢮⣁⣀⡹⠆⣀⣾⣿⣧⣀⡠⠤⢤⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠀⠀⡠⠏⠀⠘⠛⣻⣿⠿⢷⠶⠚⠉⢲⡾⠿⣟⡛⠓⡆⠀⠀⣨⠇⠀⠈⠙⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⣨⠉⡠⠖⢇⠀⠀⢀⡎⠀⠀⠀⠀⠙⢧⡠⢄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⢋⠴⠋⠀⠀⠈⡳⠖⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⣀⠔⠋⠀⡠⣄⣶⡒⢤⣤⡤⠴⠃⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⢻⠁⠀⠀⢰⣟⣒⠀⠀⢸⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⢟⣃⣠⡤⠼⠤⠤⠤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣤⡄⠀⠀⠀⣠⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠿⣿⡟⠀⠀⠀⣽⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣭⡄⠀⠀⠀⠰⣷⣬⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡀⠀⢀⠀⢀⠀⢀⡀⢀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢰⠆⠈⣇⠔⠹⠆⠈⠆⠁⠗⠉⢷⠈⢧⠤⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠃⠠⠈⠀⠀⠀⠢⠤⠀⠀⠀⢸⡠⠃⡮⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠐⠁⠀⠀⠀⠀⣠⣊⠤⢀⠀⠈⠁⠀⠁⠇⡸⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⣵⠶⢦⠀⢀⣴⠟⠒⠉⠉⠀⠀⠀⠀⠀⠖⠃⠀⣂⡀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠐⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠤⣄⠁⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠈⠉⠙⠫⡉⠉⠢⡄⠀⠀⠀⠀⢀⡠⠛⠂⠐⠳⡦⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡚⠟⠋⠻⣄⡀⠀⠀⠀⠖⠛⠀⠁⠉⠀⣀⣤⠰⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⡀⠀⣌⠀⠀⠀⠘⣭⡓⣤⣬⡤⠤⠤⢴⣂⡉⠔⡁⠷⣄⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠳⣤⠀⠀⠛⢄⣹⠀⠀⠀⠀⠀⠉⠛⠻⢄⣀⠀⠀⠀⡠⠊⠀⠀⠈⢳⡀⢀⡜⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⢤⠀⡠⠝⠁⠀⠀⢀⣀⠑⠲⠤⢤⣀⡀⠀⠀⠀⠀⠉⠑⠊⠀⠀⠀⢀⡴⠋⠉⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢄⡁⠀⢛⡃⠀⠀⠀⠀⠉⢳⠢⢄⣀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠵⠦⠀⠀⠀⠓⠂⠊⠀⠀⠀⠀⠀⠀⠘⡖⠲⠼⠙⠒⠒⠙⠉⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠈⠉⠉⠉⠉⠉⠉⠁⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠷⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠈⠉⠉⠀⠉⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣄⢠⠜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⠁⠀⠀⠱⡄⠀⠀⢀⡖⢉⡉⠱⡀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣤⣀⠀⠀⠀⠿⠞⠁⠀⠀⡀⠀⠀⠀⠙⠦⡠⠞⠠⡋⠉⠁⡇⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠉⢳⡀⠀⠀⠀⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⢀⡇⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠊⠁⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡁⣀⡼⢣⡀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢻⣯⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣼⠀⠀⠀⢀⡴⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡇⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⠦⠴⠖⡋⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠟⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⢰⣮⢭⣿⠖⢶⡄⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⠟⠀⠀⠀⠀⠀⠀⡿⢤⣀
⣀⣀⣀⣀⣀⠠⠤⠤⠤⠶⠤⠶⠒⠒⠒⠒⠒⢂⠈⠳⣄⠉⢻⠈⣠⡿⠁⠀⠀⠀⠀⠀⠀⣠⠞⣰⠃⠀⠀⠀⠀⠀⠀⢰⠇⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⣤⣀⠀⠀⠀⠖⠉⠁⠀⢀⠈⠳⣌⡛⠋⠀⠀⠀⠀⠀⠀⢀⡼⢁⠞⠁⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⣠⠖⠉⠠⣄⡈⠒⠤⣌⠙⢦⠀⠀⠀⠀⣠⠏⠀⠀⣀⣹⡦⣀⡀⠀⠀⣠⠔⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⣠⠞⠁⠠⢄⣀⠀⠉⠳⣄⠈⣧⠠⠁⠀⠀⡴⠃⢀⡤⠞⠉⡜⠀⠈⠉⠛⡿⡁⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⣿⠀⠀⠀⠀⠀⠙⢦⠀⢨⠄⠱⠀⡇⢀⡞⠀⣴⠋⠀⠀⢰⠃⠀⠀⠀⠀⢇⢻⡎⢀⣠⣄⣀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⢿⠀⠀⠀⠙⣷⡀⢸⠀⢸⣀⣸⡖⠡⠋⠀⡞⠁⠀⠀⠀⢸⢀⡴⠊⠉⠓⢾⣾⠉⠁⠀⠀⠉⠳⣄⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠈⡆⠀⠀⠀⢸⡗⠘⠓⠋⠉⣸⠇⠀⠀⡼⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⣆⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠸⣄⠀⠀⠸⠇⠀⠀⠀⢰⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠸⡄⠀⢠⠇⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⡰⢷⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⠞⠁⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⣴⠛⣀⡤⠞⠁⠀⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠿⠚⠋⠀⠀⠀⠀⣠⠴⠋⢧⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⠉⠉⠉⣿⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⣀⡴⠛⠁⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⣿⠀⠀⣿
⠀⠀⠀⠀⠘⠶⡶⠊⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢧⡀⠀⠀⠀⣴⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⠴⠋⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣀⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠠⠤⠤⠀⢀⣀⣀⠀⠀⠀⠀⠀⠸⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⣀⣠⡄⠀⠒⠚⠃⠀⠁⠀⢀⠀⠀⠸⣅⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠉⠉⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢀⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠈⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄⠀⠀⢤⠤⠠⣀⣀⣀⠤⠤⠤⠴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⢀⡴⠚⢉⡄⠤⠀⠀⠒⠒⠐⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⡴⠒⢄⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣅⣆⡉⠀⢣⣀⠤⠂⢁⠔⠊⣿⣿⣿⣿⣿⣿⣿⠋⠉⠛⢦⡀
⠀⠀⠀⢀⡞⠁⠀⠈⣣⠤⠀⠀⢻⣿⣿⣿⣿⡿⠡⠖⣛⣿⡅⠄⠬⠶⡊⣁⣔⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠈⠣⢤⡀⠀⡄⠀⠀⠀⠀⠙⢿⣿⣿⣷⡀⠈⠀⠚⣻⠉⠉⢻⠉⠀⠙⠛⠋⠉⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠉⠉⣀⣀⠀⣠⡄⠀⠀⠀⠉⠉⠙⡦⠀⠀⡇⠀⠀⠀⢣⣀⣀⣀⣀⡴⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠙⠻⢄⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⠀⠙⡿⠿⢟⠲⢄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢈⠆⠀⠀⠀⢀⣾⡇⣀⠴⢻⡇⠀⠀⠀⠀⡇⠀⠀⠀⠈⠓⠢⠤⢴⠒⠚⠓⠒⠲⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠉⠉⠀⠀⠀⠀⠀⡇⠉⠁⠀⠘⢷⠀⠀⠀⠀⡇⠀⠀⠀⣀⡀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠓⣇⣀⣀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢀⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣠⡴⠈⢉⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠚⠁⠀⢠⣿⠋⠀⣰⡧⠴⠛⠃⣙⣳⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢻⡶⠀⠀⠀⠀⠻⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⣼⠈⠁⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⠚⠓⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⠀⠀⠀⠀⢸⡀⠀⢀⡄⠀⠀⢶⠂⠀⠀⣠⡀⠀⠀⠀⢠⡴⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⣆⣼⠁⣶⠙⠿⠋⠀⣬⠖⠈⠁⠸⠷⠃⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠈⠢⣤⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢀⡄⠀⠀⠀⠰⠿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠈⠃⠀⠀⠁⠀⣀⣀⡀⠀⠀⠀⠼⠟⠁⠀⠀⠀⢾⣀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⣀⠀⠘⠋⠉⠛⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠙⠇⡀⢀⡤⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⠀⢁⠀⣠⣤⡤⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⢀⠂⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢿⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠀⠀⠲⠀⠈⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠖
⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⢾⡁⠆⠰⠛⢣⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠈⠛⠾⠟⠳⠦⣀⠀⠀⠀⠀⣀⡰⠂⠀⠋⠑⢆⠸⡄⠀⠀⠈⠱⡦⠀⠀⡹⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠈⣶⠙⢦⣀⡀⠀⠁⠀⠋⠀⠀⣀⡠⣤⠿⡧⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢤⠼⠒⠻⠀⠀⣈⠽⣷⣦⡀⠀⠤⠞⠉⠀⠈⠀⡇⠀⠀⠁⠒⠠⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠒⠂⠀⠀⠀⠘⢷⡠⠔⠚⠃⢀⣀⠦⢀⡜⠁⠀⢙⡌⢁⠀⠀⢀⣤⡤⠐⣠⠃⠀⠀⠀⠀⠀⠈⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣠⡄⠘⠋⠀⠎⠀⠀⠀⢾⡉⠚⠛⣥⣭⠀⢶⠊⠁⠀⢸⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠀⠀⠀⠀⠀⠶⣀⣤⣀⠐⢷⡏⠀⢰⣦⡁⠀⠙⠛⠂⠀⠉⠁⠘⡀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠿⠇⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠘⠀⠀⠈⣤⠐⠞⠛⠀⠀⠀⠀⠀⠀⡗⠈⠛⠓⢿⣄⠀⡀⠊⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⠀⠀⣟⠀⣿⣛⣃⡀⣀⣀⣀⣴⠟⠀⠄⠀⠀⠈⠛⠛⠓⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⣀⡴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠒⠋⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⣤⣄⣀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡛⠶⠦⣀⣀⣀⣀⡤⠴⠶⠚⠷⠂⠀⢉⣉⡤⠴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠛⠛⠉⠥⠤⠤⠤⠴⠶⠒⠚⠛⠋⠈⠏⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠠⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢢⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠈⡷⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠖⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠰⠄⡀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣤⣤⣠⠴⠞⣋⢿⣦⣄⣀⣀⣀⡤⠶⣾⠟⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⣩⣠⠄⠀⠀⠤⢀⠀⣿⠭⣉⠧⠀⠀⣡⠞⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣛⣫⢭⣶⠀⠁⠀⠈⢰⢉⣤⣙⡶⠌⠞⠱⣶⣧⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠈⠉⠁⠢⣍⠳⠚⠁⠀⠀⠀⠀⢈⣻⣿⠯⠙⠿⠻⠿⢿⣴⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣤⠀⠀⠀⠀⢙⡄⠉⣁⣤⡀⠒⢢⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⢿⣿⣿⡿⠶⠶⠶⠖⠈⠙⠉⠿⢄⡀⠀⣀⠐⠂⠀⣀⣀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠄⣀⣀⣀⣀⣀⣀⡴⠟⠁
⣶⣿⣷⣶⣶⣾⣶⣶⣶⣶⣶⣾⣷⣾⣿⣿⣶⣶⣾⣾⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣶
⣿⣿⣿⡟⠻⠉⠛⢿⠋⢩⡟⠛⢻⡏⠯⢹⣻⠿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣿⣷⣾⣷⣾⣷⣾⣿⣿⣶⣿⣿⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⣤⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⠟⠋⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣽⣷⡀⠀⠀⣠⡄⠀⠀⠀⠀⢹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠿⠿⠃⠀⠀⠀⠸⠟⠀⠻⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠠⣶⠀⢰⣶⠀⢶⡆⠀⠀⣀⣤⣀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣤⡄⠀⠀⠀⢠⣄⠀⣠⡄⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⡇⠀⠀⠀⠈⠉⠀⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠻⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣄⡀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠉⠻⠿⢿⡿⠿⠟⠛⠉⠀⠀⢀⣠⣴⣶⠀⠀⠀⢠⣾⣿⣿⣶⣬⣿⣿⣿⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠘⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣤⣤⣼⣿⣿⣿⣿⣿⠀⠀⢀⣾⣧⡀⠀⠀⢿⣿⣿⣿⣿⣿⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠻⠿⢿⠿⠟⠁⠉⣿⣿⡄⠀⣾⣿⣿⣿⣶⣶⣦⣽⣿⣿⡟⠛⠻⠿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣤⣤⠄⠀⢀⣿⣿⣿⣶⣼⣿⣿⣿⢿⣿⡿⠿⠟⠉⠀⠀⠀⠀⢤⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠃⢀⣴⣿⣿⣿⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⠀⠀⠀⠹⣿⣿⣉⠹⣿⣿⢟⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⡏⣴⣿⣿⣿⡿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣴⠞⠛⠋⠀⠀⠀⠀⢿⣿⡟⠀⣿⣿⢣⢫⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⢀⣿⣿⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢻⠀⠀⢰⢃⠞⡵⠋⠀⠀⠀⠀⠀⠀⣿⠏⢀⣴⣿⡟⣴⣙⣭⣿⢿⣷
⠀⠀⠀⠀⠀⠔⠂⠁⠀⠁⣸⣿⣿⣿⣿⣏⠺⣧⠀⠀⢀⣀⡀⠀⠈⡀⠀⠀⠀⠀⡠⢋⠂⠈⠀⠀⠀⠐⠢⡀⠀⠈⠁⠀⢸⣿⣿⣳⡎⡏⠀⠀⠀⠻⣇
⠀⠀⠀⠀⠀⠁⠀⠀⠙⠲⢿⣿⣗⡭⡭⠿⠏⣿⠀⠀⠤⠌⠙⠲⡞⠉⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⣼⠟⣥⣻⡵⠶⠛⠀⠀⠈⢹⡀
⠀⠀⠀⠀⠀⢧⡀⢘⡟⢳⣶⣤⣀⠻⠥⠭⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣴⢟⡡⣟⣀⣤⡴⢾⢿⡃⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⣿⣿⣮⡷⠶⠟⠛⠛⠒⠶⠦⢤⣤⣀⠀⢻⣤⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⣫⡥⢮⣭⣴⣻⢶⢷⡍⢺⡽⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣟⠉⠈⠀⠼⠁⠀⠀⢀⣠⡴⠛⠉⠟⠁⠀⠀⠀⠀⠉⠛⢿⡌⡳⢻⡆⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠓⠒⠒⠶⠾⣿⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣽⢵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡳⣤⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⣽⢳⣦⣬⣉⡙⢻⠋⠉⠙⢿⣦⣀⡀⠀⠀⣸⠀⢸⠀⡾⠏⠀⠀⠀⠘⡀⠀⠀⢀⣠⡾⠛⠉⠁⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣜⢸⣿⠿⠛⠛⠟⠀⣆⠀⠀⠘⢻⣄⣀⣠⣟⣠⠿⠚⠁⠀⠀⠀⠀⠀⢡⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡙⡶⣴⡾⠶⣛⢻⡆⠀⠀⠀⠀⢉⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠫⠄⠓⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣼⣾⣒⢻⣏⡾⢿⡄⠀⢠⠖⢻⡲⠤⠌⠳⠦⢄⣀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣾⣟⡻⢾⢸⡇⢰⠃⠀⣠⡤⠆⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣾⣁⠘⢆⡼⠁⠙⠂⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣇⠀⠀⣰⠁⠐⠀⣠⡆⠀⠀⢠⡦⠄⠠⠤⠆⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⣇⠀⠀⣠⡟⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⣄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢖⠀⠀⠀⠀⠒⠂⠤⠄⡀⢋⠒⢄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠑⠊⢁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠲⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⢆⠑⡀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠀⠀⠀⠈⡆⢁
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣀⠖⠉⠁⠢⡀⠀⠀⠀⠀⠈⢢⠀⠀⠀⡇⠸
⢀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⢋⡠⠊⠁⠸⡔⠁⠀⠀⠀⠀⠈⢂⣠⠀⠀⠀⠀⢡⠀⠀⠁
⡆⢰⠀⠀⢈⡴⠀⠀⠀⠀⠀⠀⠀⡄⢲⡠⠊⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠰⠀⠀⠠⡤⠆⠀⠀⠀⠀⡀⢀⠀⣀⢀
⡇⠸⠀⠀⠈⡄⠀⠀⠀⠀⢀⠻⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⢓⠀⠀⢠⠀⠀⠀⠀⠰⠀⠈⠁⠀⠀⡁
⢡⠀⢣⠀⠀⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠷⠂⢤⠀⠀⠀⣠⢂⡁⢀⠀⠀⠀⠉⣄
⠀⢢⡀⠁⢠⠀⢀⢤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⡃⠀⠈⠁⠀⠀⠈⣄⢀⠖⠐⠁
⠀⠀⠁⠀⠘⠀⠁⠀⠠⠀⠁⠀⠀⠀⠀⠀⠁⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢳⡀⠶⠀⠀⠀⠀⠀⠀⣀⣀⣈⣀⡀
⠀⠀⠀⠀⠀⠲⣦⡀⠀⢶⠀⠀⠀⠤⠠⡄⠀⠀⠀⠀⠀⠀⠀⢠⠶⠄⠀⠈⠲⡄⠀⠀⠀⠀⠓⠤⡎⢰⠞⠒⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠠⠄⠉⠉⢆⠀⠈⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⣠⠤⠠⠤⠄⠤⣸⠀⠀⠀⠠⠔⠈⠁⠙⢤⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠧⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠐⠠⠔⠁⠀⠀⠀⡃⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠂⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠓⢤⡀⠠⠀⠀⠉⠐⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢄⠀⣀⠀⠀⠀⠀⣀⠀⠀⠤⠐⢲⠋⠀⠀⠀⠈⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠈⠡⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠉⢣⡄⠄⠗⠀⠀⠀⠸⠀⠀⠰⡦⣄⠀⢀⢀⠀⠀⠉⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠉⠀⢀⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⠄⠁⠁⠀⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠃⡰⠁⣀⠀⢀⠀⠀⠀⡇⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢇⠀⡀⠀⢀⠀⡨
⠀⠀⠀⠀⠀⠀⠠⠤⠄⠐⠒⠈⢇⣀⣀⣀⣀⢠⣬⡎⠤⠊⠀⠀⠀⢡⠀⠀⡧⠤⠀⠞⠒⠒⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠈⠣⠠⠊⠉
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣀⢀⠠⠒⠁⢇⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠐⠈⠀⠀⠀⠃⠀⢠
⠀⠀⠀⠀⠀⠀⢖⠉⠀⠀⠀⠀⠈⡲⠀⠄⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡠⠔⢛⠁⠀⠀⠀⣀⣀⠀⠀⢤⡤⠄⠤⠄⠀⣀⣀⡄⠀⢃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⡀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠉⢆⠀⠈⢢⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢣⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⡄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡘⠦⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡕⠠⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠠⢈⠀⠂⠠⠄⠊⠀⠀⠀⠹⠀⠂⠀⣀⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠈⠁⠱⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⡁⠀⠀⠀⠀⠑⠤⢀⡀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡤⠴⠚⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⣀⠴⠋⠈⠋⠁⠀⠀⠀⠀⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠞⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠓⢻⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⢹⡀⠀⡴⣩⠙⣦⠀⠀⠀⠀⢰⡇
⠀⠀⠀⡀⠀⠀⠀⠈⢧⡀⠀⢹⡄⢠⠆⡀⠀⠀⠀⢹⣸⡇⡟⠂⣼⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠉⠳⢄⡀⠀⠀⠓⠀⠀⡇⠟⠋⠁⠀⠀⠀⠀⠉⠁⠁⢺⢧⡀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⣀⣀⣴⠷⣄
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠓⢲⣀⡴⢲⠄⠀⠀⠀⠀⡴⠃⠀⠙⢚⡵⠛⣡⠴⠚⠛⠲⣄
⠀⠀⠀⠀⠈⠉⠓⠀⠀⠀⣠⣴⠲⣄⡀⣾⣁⠀⠀⠀⠀⡼⠁⢀⡤⠚⢁⡴⠚⠁⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅⣸⠋⠀⠙⣄⢸⡄⠀⢀⡜⢷⠞⠁⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡙⢿⡾⠁⠀⢸⠀⠉⠛⠉⠀⢸⣧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠙⢻⣧⠀⠀⢸⡄⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡁⠈⠧⢀⡏⢹⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠴⠋⠀⢸⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⠴⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠩⠀⠐⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠉⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠄⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠠⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠉⠉⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⢀⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠲⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡗⠶⠾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠛⠛⠛⠲⠶⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣄⣠⣤⣦⣄⡀⠀⠀⠀⠀⠀⠹⠟⠋⠁⠀⠀⠙⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⡧⢶⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⣈⢤⠀⠀⢠⡆⠐⠐⠚⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⠐⢀⠸⠀⠶⠁⠀⠀⠀⠄⠀⡀⠀⠀⠀⢀⢀⠀⢀⠠⠀⠀⠘⠒⢰⠖⡞⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢂⠂⢀⠄⠀⠚⠖⠘⠉⢑⠂⠀⠀⠘⡀⢁⡈⢘⠀⠀⠀⠁⠀⠒⠂
⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⢪⣤⠀⠃⠀⠀⡀⠀⠀⠀⠀⠀⣆⠔⠀⠒⠲⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁
⠀⠀⠀⠀⢈⠰⣤⣃⠀⠀⠃⠀⠀⠀⠀⠀⠡⠠⢄⠠⠀⠀⠀⡠⠀⠤⠄⠀⠀⠀⠀⠄⠄⠀⠀⠠⠄⠀⠠⣀⠤⠀⠀⠀⡇⠀⠀⠀⠠⠄⠀⠒⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠄⣠⠀⠐⠀⠀⢱⣀⡀⣤⢠⡄⠀⠀⠀⡶⡂⢀⡤⠧⠀⠀⠉⡀⠉⠀⠁⠁⠀⠀⠀⡀⠀⠂⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠮⠘⠥⢾⠃⠀⠀⠄⠐⠀⠃⠀⠀⠙⠦⣃⣑⠚⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠁⠀⣻⡁⠀⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡶⠄⠀⠀⠀⠀⠀⠄⠌⠀⠀⠀⠀⠀⠭⠤⠊⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠊⣀⠄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⣷⠂⠋⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⡆⠤⠈⣄⡇⠀⠀⠀⢸⠈⢀⣊⠀⡆⠀⠀⠈⠂⠉⡀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠈⠇⠀⠀⠀⠀⠀⠁⣀⣀⠀⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⣀⣽⣿⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣛⣧⣄⠀⠀⠀⠀⣿⣿⡇⢸⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣷⣾⣿⡟⠀⠀⠀⠀⣾⡟⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢹⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⢷⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣾⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣿⡏⠀⠀⠀⠀⠀⢸⣿⡏⣿⡟⠀⠀⠀⠀⠀⠀⠀⠨⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⠃⠀⠀⠀⠀⠀⠀⢿⣧⣿⠇⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⡿⠇⢈⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⡇⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⡟⣿⣿⢷⡀⠀⠀⠀⠀⠀⢰⣿⡏⣹⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⡝⠛⣡⣤⠀⠀⠀⠀⠀⣾⣿⣿⠙⢱⣿⣿⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣷⣴⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡆⣾⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣾⣿⢾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠛⠁⠈⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⣿⣿⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⣿⡏⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠘⠻⣿⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⣿⡏
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠹⣿⣦
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⣿⣿
⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⠇⠀⠀⢻⣿⣇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⡄⠀⠀⠀⢿⣿⡄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠈⠛⠁⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣷⣦⡀
⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⣀⣀⣀⡀⠀⠀⣀⣀⠀⠈⠙⠻⢿⣷⣄⠀⠙⢿⣷⡄
⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⣿⣿⠀⠀⠿⠿⣿⣷⣦⣄⠀⠙⣿⣷⡀⠈⢿⣿⡄
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠘⣿⣷⠀⠘⣿⣷
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢹⣿⡆⠀⣿⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠛⠃⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢠⣤⡄⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⢸⣿⠂
⠀⠀⠀⠀⠸⠿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠸⠿⠇⠀⠸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣤⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⢿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢁⠀⠀⠈⣉⠹⡏⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠻⠿⡟⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⢈⠉⠉⠙⣛⢹⣿
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠄⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠠⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠶⢶⠶⢂⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣤⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠟⠉⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⠀⠉⠉⣀⢹⡟⠀⠀⠀⠀⠀⠀⠀⢸⡿⠹⠛⠿⡟⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⡟⠠⠀⠠⠀⠄⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⠄⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⠀⠀⠤⠀⠀⣀⣿⣿⣄
⠀⠀⠀⠀⠀⠸⣿⣿⣷⣤⠀⢠⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠻⣿⠿⠃⠀⠘⠻⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣃⣀⣘⣻⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠴⠶⠤⠤⠤⠤⠤⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⢷⣴⣶⣿⣶⣶⣶⡿⣿⡆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣿⣿⠋⠛⠉⠉⠙⣿⣷⡄⠀⠀⠀⠀⠀⠈⣿⣾⡿⠟⣿⠿⠛⣿⣿⣿⠁⠀⠀⠀⠀⢠⣿⣿⠋⠙⠿⠟⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢻⣿⢈⠀⠀⠀⡀⢸⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⡀⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⠈⢿⡿⢀⠀⠀⠀⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⣍⠀⠀⠀⠀⠀⠀⣩⣿⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⢀⣠⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣤
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⢊⢈⠀⠀⠀⠀⡁⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡝⠿⣿⣿⣿⣿⣿⣿⠟⣫⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢟⣱⣿⣶⣭⣭⣍⡱⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⣫⣭⣭⣿⠿⠿⣿⣭⣭⣝⠸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢇⡶⣼⢧⡝⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⡿⣸⢧⡟⣼⢣⡝⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢇⣿⣼⣣⡟⣼⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⣿⣿⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡼⣿⣿⣿⣿⣿⣿⢸⢟⣵⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡷⣹⣿⣿⣿⡿⣣⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠇⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠋⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣴⣶⣆⠀⠀⠀⢀⢀⣸⣿⠁⡄⠀⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⢀⠙⠛⢿⣿⣶⣿⣿⣿⣿⣿⣧⣄⣼⣯⣿⣦⣤⣤⡀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣻⣿⡇⠀⠒⠻⣿⣿⣿⣿⣿⣿⣿⡏⠛⢻⠟⠛⠿⠿⢿⣿⣄⣾⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⣤⡀⠀⠸⠟⠁⣿⣿⣿⣿⣿⣿⠀⠀⣾⣷⢄⣴⣄⣹⢻⣿⣿⣿⠤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣄⣸⠟⠋⠉⣉⣹⣆⣠⣤⣤⣄⠻⠛⠛⠿⠿⡇⠸⣦⡶⢦⣙⣩⣴⣬⣿⣁⣿⣶⡶⠂
⠀⠀⠀⠀⠀⠀⢺⣿⠟⠛⢛⣿⡷⠾⠗⣀⠛⠋⣻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢻⣿⡟⠻⢿⣿⡟⢋⣀⣀
⠀⠀⠀⠀⠀⠀⠘⣧⣴⣾⡿⠿⡏⢤⣼⣯⣷⡞⠁⠀⠉⠁⠀⣀⠀⠀⡀⠀⢀⣦⣄⣀⣠⣬⡋⠱⠀⠠⣨⣿⡿⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠹⣍⢿⡷⣡⣷⣶⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⣀⠀⠀⡈⠇
⠀⠀⠀⠀⠀⠀⠀⡀⣤⣿⣽⣿⠿⠋⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠁⠀⢻⣿⣀⣤⣤⣄⣸⣿⡀⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣷⡆⠀⠀⠀⣿⣿⠀⠀⠀⢀⣀⣀⡀⠀⠈⢤⠀⠐⠂⢸⢿⠟⠿⣿⡿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠉⠀⢀⡙⣿⡿⠁⢀⣤⣴⣿⣿⡦⠀⢰⣿⠿⠛⠻⣤⣄⠀⣀⣀⣀⡀⢸⠠⠀⠠⡄⡿⠻⣦⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠰⠂⠈⠀⠈⠃⠀⣿⣿⣿⡟⠁⢀⣛⣺⢃⠀⠀⡀⢸⣿⡿⠿⣿⣿⣿⣿⣦⣖⣦⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠀⠙⢿⣿⣿⡅⠉⢀⣬⣭⣌⣠⣤⣁⣠⣿⡆⠀⢸⢿⣿⣿⡟⠛⠛⣇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠶⠤⠤⠼⢿⣿⠿⣻⣿⣷⣄⡀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣶⠖⠒⠒⠀⣤⠄⠛⠛⢛⣿⣇⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⣿⡄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢲⠀⠀⠀⠀⣠⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣗⡣⠀⠀⠀⠈⠀⠉⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠈⢹⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⠂⠀⠀⢀⠐⠳⠦⣤⣤⠠⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣠⢠⡛⠡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠐⢄⣀⣠⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠜⠃⢀⠀⢀⢠⠊⠚⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠐⠄⢈⠄⢀⠂⡰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠉⠁⣃⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠁⠀⠁⠀⠀⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⢀⣀⠀⠀⢦⣦⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⠄⠐⠊⢉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⡀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⡶⠒⠿⠃⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⢓⢤⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢣⠀⣠⢌⡀⠌⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠖⡻⠛⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠙⢄⡨⢀⣠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠒⠂
⠀⢈⠈⡉⠀⠐⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄⡙⡺⢈⢉⠀⠀⠀⠀⠀⠀⢀⠂⢀⠨⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠆⠀⢸⢠
⠀⠎⢰⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢈⠀⠀⠀⠀⠀⠀⠀⠀⡈⠢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠄⠀⢸⣿
⠀⢄⢠⢏⡀⡹⢃⣳⣀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠌⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⣿
⠀⠂⠀⢡⠥⠐⠉⢡⡉⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣦⣤⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⣄⠀⢼⣀⠀⠔⢉⡽⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⣿⣿⡇
⠀⢿⡀⠀⡀⠄⠂⠀⢀⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿
⠀⠸⠈⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣿⣿
⠀⠀⠡⣦⣀⠀⠀⠀⣿⣅⡹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡟
⠀⠀⠀⠙⠻⠿⣷⣾⣿⣿⣿⣿⡿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡟⠉⠣⣀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡄⠋⠠⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡻⢁⡴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣷⠋⣼⣡⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣾⠁⡼⢫⠟
⠀⠀⠀⢀⠀⠀⠀⠀⣸⠃⠀⠁⢈⡴⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣰⠋⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⡇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢋⡤⠖⠒⠦⣄⠀⠀⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣏⣀⡀⠘⠤⢄⡀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⡾⠁⠀⠀⠀⠙⣶⠻⠅⠀⠈⢦
⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠐⠳⠤⣀⠀⠀⠀⠙⢦⡀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣄⣀⣀⡠⠋
⠀⠀⠀⠀⠀⢸⡇⠀⣠⢾⠁⣴⠀⠀⣠⣤⣟⣦⡀⠀⠀⠉⠳⢾⡇⠀⠀⠀⠀⠀⢰⡏⠉
⠀⠀⠀⠀⠀⠈⢷⣼⠃⣾⠀⡟⣆⡼⠁⠀⠉⠻⣿⣦⠀⠀⠀⠀⠻⣄⠀⠀⠶⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠈⠀⠀⡴⠒⠛⠛⠿⢧⠀⠀⠀⠀⠈⠙⢛⡿⠋⠛⠶⢤⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠴⠖⢲⢤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠲⣄⠀⠉⠙⠓⠶⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡗⢤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠉⠓⠦⣌⣀⡉⠂⠀⢂⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⣦⣄⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠄⠐⣯⠙⣧⠈⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡰⠀⠀⠀⠀⠀⠀⠈⣇⠈⠙⠾⣦⡀⠹⣮⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣤⠀⠀⣹⣦⣙⠀⠀⠀⠈⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠛⠦⣄⡀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢺⣁⣀⡽⠟⠢⠤⣄⣀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣲⠶⢤⣀⣈⡏⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠈⡷⣄⠀⠀⠀⠀⠳⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⠀⠀⠈⠉⠉⠒⠒⠤⠤⢤⣀⣀⠀⠀⠀⠀⠀⣷⢤⡿⠆⠀⢀⣀⣘⣦⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣄⡀⠈⠀⠂⠀⠀⠀⠀⠈⠉⢛⠶⣄⢠⠇⠀⠙⠚⠛⠉⠉⣿⣿⠱⡄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠁⠀⢉⡙⠢⣤⣀⠀⠀⠀⠀⣰⠏⠀⠘⠻⢦⡀⠀⠈⢦⡀⠀⣿⣿⣿⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡉⠀⠀⠀⢀⡴⠋⠁⠀⠀⠉⠛⠲⢦⣴⠏⠀⠀⠠⢴⣾⣷⠀⠀⠀⠙⢶⣿⡛⠛⠒⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣦⣿⡄⠀⢠⠞⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠷⠛⡄⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⣳⠞⠉⢳⡄⠀⣠⣿⣿⢿⣠⠏⠰⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣸⠁⢀⡖⠈⡇⣰⢻⡟⠉⠻⠏⠀⣠⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠖⠒⡙⠶⠞⠀⠀⣿⠃⡾⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⢶⠚⠉⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣰⠻⣄⣐⣦⠀⠀⠐⢇⡀⠀⠀⠀⠉⢻⠖⠲⠲⠶⠶⠶⠒⠒⠒⠛⠛⠉⠉⠁⠀⢸⠁⡼⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠁⠀⠈⠉⢈⡷⠒⠲⡬⠻⡄⠀⠀⢀⡟⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡇⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣀⡴⠇⠀⠀⠀⢙⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣼⣀⣤⡤⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠃⠄⠀⠀⠀⠀⠀⠀⠐⠚⠛⣛⠾⢻⠿⠍⡁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣑⣀⢤⡐⠄⠀⠀⠀⠐⠂⠉⠀⠀⢀⡏⠀⠀⠀⠀⠀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠉⣻⣿⠿⠛⠛⠲⠤⠴⠦⠤⠶⠶⠒⠚⠛⠓⠒⠒⠶⠞⣽⣩⠏⠉⠙⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⢀⣾⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠙⢿⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠈⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣠⣄⠀⠀⠀⠘⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡿⡿⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⣿⣦
⠀⠀⠀⠀⠀⢸⣿⡏⠀⡇⠀⠀⠀⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡶⠀⠀⠀⠉⠉
⠀⠀⠀⠉⠀⣼⣿⠇⢀⡇⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⣿⠏⣠⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣤⣤⣴⡾⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⠀⣸⣷⣿⣿⠏⠀⢸⣿⣿⣿⣷⠀⠀⠀⣶⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⢀⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⠀⠀⠈⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠘⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣿
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⣀⣤⣠⠀⠀⠠⣿⡇⣿⣿
⠀⠀⠀⠉⠙⠛⠻⠟⠒⠀⠐⠤⠤⠭⠀⠀⠘⠦⠤⠭⠅⠀⠀⠀⠛⠛⠛⠁⠀⠁⠀⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⠲⣿⣿⣿⡗⢲⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠛⢛⣛⣛⠛⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣼⣿⣏⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢫⣿⣿⣿⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣿⣿⣿⣭⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣾⣿⣛⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣻⡿⣭⣹⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠟⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠀⠀⠈⠻⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟





⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠶⠶⠿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣄⠀⢀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣅⠀⠲⠦⠖⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠁⢿⣿⣷⣤⣀⣠⣿⡿⠙⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠰⠄⠀⠀⠀⠀⠈⠙⣿⡉⢹⣿⠀⠀⠀⢠⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⡆⠀⠀⡆⠀⠀⠀⣿⠀⡾⣿⠀⠀⠀⡆⠀⣴⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⡇⠀⠀⢰⠀⠀⠀⣿⠶⢷⣿⡆⠀⠀⡇⠀⣿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⠀⠸⣧⡀⠀⣿⢇⣾⢿⣇⠀⢀⠃⠀⢹⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⡇⠀⠀⠀⢿⠀⠀⣿⠚⢻⣻⡟⠀⢸⠀⠀⣾⡇⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⣀⠀⠸⡄⢠⡿⢻⣿⡏⣿⠀⠀⠀⠀⣿⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠶⣿⢃⠾⠉⢦⡀⠃⢸⡿⡟⢉⡏⣿⠀⠀⠸⣻⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⠿⠀⠀⠀⠳⡀⢸⡟⢻⡏⠀⣿⠀⠞⠀⢹⣿⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⢸⡟⣿⡅⠀⣼⡌⠀⠀⢸⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣷⣿⣶⣻⣿⣀⠀⠀⢸⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣿⣿⢻⣄⣀⣾⡀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠒⡆⢹⣁⣀⣉⠛⠛⠐⠿⣿⣿⣿⣿⣿⣆⣉⢀⡛⠋⣡⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠞⠓⠛⢿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡟⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣀⢻⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣯⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣮⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣶⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⢿⣿⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠋⠉⠉⠉⠉⠙⠓⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡀⠀⡠⡇⠀⢀⢴⣆⡴⢻⣀⡴⢺⣇⡤⢿⣤⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼⣇⠞⠀⣧⠔⠁⠈⠋⢀⠈⠋⠀⠈⠋⠀⠈⢹⠀⠈⡷⢚⡉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠁⠋⠀⡄⠀⠀⠀⠀⠀⠙⠦⡤⠴⠀⠀⠀⠀⢸⣄⠾⠁⣞⣤⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⠀⠘⠋⠀⠘⠋⡼⠁⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣄⡀⠀⠀⠀⠀⠀⠀⢠⣾⢔⣛⣩⠤⠄⠀⠀⠀⠀⠀⠀⠙⢂⠞⠀⠀⠀⣀⡠⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣰⠿⠿⢷⠆⠀⠀⣴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠉⢉⣉⡤⠊⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠰⣏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣀⣈⣉⣉⠭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣤⠤⢶⡖⠶⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣁⣤⣄⣤⣍⣓⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣘⣇⠀⠀⠀⠀⠀⢀⡀⠈⠳⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠉⢻⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣒⠛⠯⠓⠈⠻⣦⡀⠀⠒⠊⡡⠆⠴⠒⠛⠋⠉⠉⠉⠁⠀⢀⣀⡤⢲⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠀⠈⠛⠀⠀⠀⠀⠘⢿⠳⣶⣉⣀⠔⠀⠀⠀⠀⣀⣀⠤⠴⠚⢉⠟⠰⣇⡀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⢀⡠⠴⢤⠀⠀⠀⠀⠸⣄⠀⠸⡧⠀⠀⠀⠀⠀⠘⠻⣤⣬⣟⣿⠟⠉⠉⠉⠉⠙⠒⠒⠚⢁⡴⠊⠁⠙⣄⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⣠⣾⠀⠀⠛⣷⡦⠀⠀⠈⠙⢦⣤⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⣤⣀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠈⢳⡄⠀⣠⠞⢹
⠀⠀⠀⢰⡁⢀⡄⠀⠀⣘⡗⠉⠀⠀⠀⠀⠀⠀⠳⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠋⠀⠀⠀⠀⠀⢀⡴⠋⠙⠚⠁⠀⢸
⠀⠀⠀⠀⠉⠉⠳⠤⠚⠉⠀⠀⢀⡠⠶⠉⢩⣷⣤⠂⠀⠀⠈⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⠀⠸⢤⠄⠀⠉⣅⠜⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⠳⢦⣤⡀⠀⠀⠀⠀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⣠⠾⠀⠀⠷⢮⣅⠀⠀⠀⠈⠓⠳⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⠒⠶⢼⠍⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣇⣀⠀⠐⠊⣡⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠈⠙⢦⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣤⣴⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠛⠀⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠙⣿⣿⡿⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡄⠀⣀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠉⠛⠋⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠉⠉⠉⠉⠛⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⣀⡀⠀⡀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⢠⣶⣤⣀⣠⣤⣶⡿⠟⠈⠙⠻⠿⠿⣿⡆⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢸⣿⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⣿⡆⠀⠀⣀⣀⠀⠀⣀⣀⠀⠀⢰⣿⠇⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⢻⣷⣾⠿⠿⠿⢿⣿⠿⠿⠿⣷⣾⡟⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⡟⠁⠀⠀⠀⢸⣿⠃⠀⠀⠈⢻⣷⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣠⣼⣿⡁⠀⠀⠀⢀⣾⣿⡀⠀⠀⠀⢈⣿⣧⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⣿⠋⢿⣿⣷⣦⣴⣿⠟⠻⣿⣦⣴⣾⣿⡿⠙⣿⣦⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⠀⠉⣻⣿⢿⣷⣄⣠⣾⡿⣿⣟⠉⠀⠀⠈⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣾⡿⠋⠀⠈⠛⠛⠁⠀⠘⢿⣷⡀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⣠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣀⣠⣼⡿⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⢿⣧⣄⣀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⠶⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⡀
⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡷
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⢠⠄⢀⡤⠀⣠
⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠀⠀⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⠒⠒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠐⢤⡀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⡀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠉⠉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠦⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⡴⠶⠶⠟⠃
⣿⣷⠿⠿⠾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠂⠉⠉⠉⠉⠉⠉⠐⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠛⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⡤⠒⠋⠉⠉⠑⠒⠂⠐⠲⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣄⣠⠞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠉⠀⠀⠹⡄⠀⠀⡴⢋⣉⡙⡀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡶⣤⣄⠀⠘⠛⠁⠀⣀⡄⠀⠀⠀⠈⠢⠚⠁⣯⠀⠁⡇⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⡼⣆⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⢠⡇⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠋⠘⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢘⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⣄⡠⠴⠊⠀⠉⢲⡀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡎⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⢠⣞⣷⡶⢤⡄⠀⠀⠀⠀⠀⠀⠀⡸⢀⠞⠀⠀⠀⠀⠀⢱⢄⡀
⣀⣀⣀⣀⡀⠠⠤⠤⠤⠤⠄⠒⠒⠒⠒⢂⠈⠢⡈⠑⡇⢁⣾⠃⠀⠀⠀⠀⠀⢀⡴⣡⠋⠀⠀⠀⠀⠀⠀⡆⠀⠉⠓⠦⣀
⠀⠀⠀⠀⢀⡠⠔⠒⠒⠤⣀⠀⠀⠐⠉⠁⠀⣠⠈⠲⣍⠋⠁⠀⠀⠀⠀⢀⡰⢋⡜⠁⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⣠⠔⠉⠀⠦⣌⡙⠲⣄⠙⢆⠀⠀⠀⡰⠃⠀⣀⡤⡷⠦⣄⣀⣀⠴⠊⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⡾⠁⠀⠒⠦⢄⡀⠙⢦⠘⡇⢀⠀⢀⠞⠁⡠⠞⠁⢰⠁⠀⠀⠉⡟⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⡇⠀⠀⠠⣄⠀⢹⠀⢸⠀⣅⡈⢠⠏⢠⠞⠀⠀⠀⡏⠀⣀⠤⢄⣡⣹⠤⠒⠒⠢⢄⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⢣⠀⠀⠀⠸⣧⢸⣤⠼⠒⢻⠀⠁⢠⠃⠀⠀⠀⠀⣧⠎⠀⠀⠀⠙⢇⠀⠀⠀⠀⠀⠳⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⢆⠀⠀⠀⡇⠀⠀⠀⣴⠃⠀⢀⡎⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢱⠇
⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢹⠀⠀⡜⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⡰⢆⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⡇⠀⢀⡠⠊⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⢠⡏⣡⡴⠋⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⢸
⠀⠀⠀⢘⡋⠁⠀⠀⠀⣠⠔⠋⠘⣆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣨⡇⠀⢸
⠀⠀⠀⠘⣇⣀⣠⠴⠊⠁⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠈⢩⠁⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⢸⠃⠀⡀
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣇⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⣄⡤⠚⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣤⣴⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣞

⠀⠀⠀⠀⠀⠀⢴⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣿⢿⣿⣿⣷⠀⠀⠀⢀⣤⣼⠖⠒⠒⢶⣦⣄⠀⠀⠀⠀⣿⠟⠋⢻⡆⠀⠀⠀⢠⣿⠿⠛⠛⢿⣆⠀⠀⠀⠀⣘⣡⣚⣆⡀
⠀⠀⠀⠀⠀⠀⠹⡛⠟⠛⠛⡿⠀⠀⠀⠀⠉⠓⢀⣠⡀⠘⠉⠁⠀⠀⠀⠀⡃⠀⠀⠈⠇⠀⠀⠀⢸⣿⡓⠀⠐⢺⣿⠀⠀⠀⠀⡟⠁⠈⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⢀⣰⣄⣀⣴⣦⠀⠀⠀⠈⢿⣧⡀⠀⣾⠏⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⢤⣄⠀⠀⠀⢀⠀⠀⡈⠠⠄⠑⠀⢀⠀⠀⠀⠸⠥⠽⡗⢿⠾⠀⠀⠀⠀⠀⡈⠀⠀⢁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⢻⠘⣆⠀⠀⠀⠀⠈⢓⢄⡠⠎⠁⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠅⠅⠀⠜⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⠂⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠱⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⣴⣶⣤⠀⠀⠀⠀⠀⠀⣼⠿⣿⣿⣦⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⢿⡀⠀⠀⠀⠀⣾⠏⠀⠙⣧⠀⠀⠀⠀⠀⡟⡋⠙⢻⡇⠀⠀⠀⠀⠀⠿⠀⠀⠈⣛⠀⠀⠀⠀⠀⣿⠻⠛⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⣠⡀⠀⠀⠀⠈⢄⣀⣄⣠⠋⠀⠀⠀⠀⠀⠙⠀⠀⢀⠃⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⣿⡷⠀⠀⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⢀⡈⠀⠈⣀⠀⠀⠀⠀⠀⢀⠠⠂⠐⡀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣦⣠⣀⣴⣄⠀⠀⠀⢠⠐⡂⠀⢈⠛⢣⡤⠀⠀⠠⣆⠁⠀⠀⣶⠀⠀⠀⠀⠀⢉⡁⠚⠒⣍⠁⠀⠀⠀⠀⠛⠈⠀⠀⠈⠙
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢠⠍⠃⠐⠊⠱⡀⠀⠀⠀⠀⡌⠋⠈⠈⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠸⠿⠀⠀⠀⠿⠇⠀⠀⠀⠘⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡇⠀⠀⠀⠄⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⠀⢀⣾⣿⡿⢿⣷⡀⠀⠀⠀⠀⡠⠾⠟⠭⡦⡀⠀⠀⠀⠀⢠⣾⣿⣿⣦⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠟⠤⠤⠤⠽⠄⠀⠀⠀⢸⠟⠿⠀⠴⠻⡇⠀⠀⠀⠀⣧⠶⠓⠣⢷⠆⠀⠀⠀⠀⣿⣿⣟⣿⣿⡇⠀⠀⠀⢀⣿⣿⠋⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡐⠀⠀⠀⠀⢸⣷⡀⠀⠀⣼⡇⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⣍⣡⣉⣽⠁⠀⠀⠀⢸⣿⡋⠀⠈⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⡀⠀⠀⠀⠀⢸⣿⡟⠀⠘⣿⡇⠀⠀⠀⠀⠀⠚⢠⠐⠳⠀⠀⠀⠀⠀⠀⢻⣿⡿⠏⠀⠀⠀⠀⠈⠿⣷⠀⠰⡿⠟
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠘⠀⠁⠀⠀⠀⠀⠘⠋⠀⢀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣦⡀⠐⠀⠐⠃⣠⡆⠀⠀⠀⠀⠂⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡄⢰⣿⣿⣶⡔⣢⣾⣿⣿⠀⠰⠀⠀⠐⢳⡾⠒⠀⠐
⠀⠀⠀⠀⠀⠐⠐⠀⠈⠀⠀⠀⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠙⣿⣿⣿⠇⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣾⡟⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣫⣶⡿⣫⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣭⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢫⣶⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⢀⡀⠀⢁⡤⠞⢃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⡴⠋⠤⠞⠉⠀⣶⠩⢻⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⡜⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠉⠀⢀⣀⠀⠀⠀⠹⢿⣧⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⢿⣷⣹⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣮⢻⣷⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠲⣄⠀⣤⡀⠀⠐⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡝⣷⡻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠳⠄⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣎⢷⡻⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠐⣶⣠⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠈⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢱⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠊⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠈⢁⣈⠆⠀⠀⠀⠀⠀⠀⠀⠰⠁⠈⠀⠀⠈⢳⠀⠀⠀⢸⠀⣀⠀⠀⢀⡀⢈⠄
⠀⠀⠀⠀⠀⠀⠐⠮⡀⠒⢭⠁⠀⠀⠀⠠⠀⠉⢈⠀⠀⢂⡞⢸⠀⠀⠀⠀⠉⡄⠒⠐⠊⡌⠉
⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠚⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠠⡡⠠⡅⠀⠀⠀⠀⠀⠀⠘⢵⠀⠀⢺⣢⡠⡲⠀⠀⠀⠀⠐⡀⢇⣀⠧⡐
⠀⠀⠀⠀⠀⠀⠀⢇⣀⠌⠀⠀⠀⠀⠀⠀⢈⠗⠺⡎⠋⠳⠉⠀⠀⠀⠀⠀⢸⢀⠀⠀⢀⢸
⠀⠀⠀⠀⠀⡠⠂⠀⠉⠀⠒⡄⠀⠀⠀⠀⠘⡣⠎⡼⢬⡧⠋⠐⠀⠀⠀⠈⢸⢸⠀⠀⢸⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⡇⠀⠀⠀⢀⢄⠏⢜⠀⠀⠐⡤⡠⠀⠀⠀⠀⠘⢼⠀⡇⢸⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡤⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀
⠀⠀⠀⠀⠀⢗⡺⠀⢐⠀⡀⠀⡄⠀⠀⠀⠀⢀⠤⠦⡧⠴⠃⠀⠀⠀⠀⠀⠀⠀⢇⢀⡄
⠀⠀⠀⠀⠀⠐⣒⠒⢒⠒⣄⡤⡀⠀⠀⠀⢀⠎⠀⠀⠇⠀⢴⠆⠀⠀⠀⠀⡠⠒⡺⢍⡒⠒⡄
⠀⠀⠀⠀⠨⠭⠭⠭⠭⠤⠥⠥⡅⠀⠀⠀⡘⠀⠀⠧⠨⠀⠁⠀⠀⠀⠀⠀⡇⣦⢘⢤⠣⡆⠆
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠩⠉⠀⠀⠀⠀⠏⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡠⠃⠀⠑⢄⡄

⠀⠀⣠⠀⠀⠀⣀⡀⡀⠀⠀⢠⣤⠀⠀⠀⢠⡄⠀⡀⡀⠀⠀⣠⡀⠀⣤⠀⠀⣠⣄⠀⠀⢠⡄⠀⣀⣤⡀⢀⡀
⠀⢰⣿⡇⠀⣰⡿⠸⣧⡀⠀⣿⣿⣇⣀⢠⣿⡇⢸⣿⣿⡆⠰⣿⡇⢰⣿⡆⢠⣿⣿⣷⠀⣿⡷⠀⢹⣿⠃⣾⡧
⠀⠈⣿⡇⠀⣿⠁⠀⢸⡏⠀⣿⣿⡟⠋⢸⢻⡀⠘⣿⣿⠁⠀⣿⡟⠸⣿⠇⢸⣿⠹⣿⠀⢿⡿⠀⢿⡇⠀⣿⠇
⠀⠀⠃⠃⠀⢸⠂⠀⢺⠇⠀⠁⠺⠃⠀⢸⠀⡇⠀⠏⠸⠄⠀⠿⠗⠀⡽⠀⠀⣿⠘⡏⠀⢸⠀⠀⢸⠃⠀⠸
⠀⠀⣿⣧⠀⢀⣠⣀⠀⢀⣶⠀⢀⣾⡀⢀⣦⡀⢀⣾⡇⠀⠀⡄⠀⢀⣶⠀⢀⣶⡀⠀⣰⡆⠀⢠⣄⠀⢰⣆
⠀⠀⢸⣿⡆⠀⢹⡇⠀⣿⣿⡇⢸⣿⣿⢸⣿⠇⠀⣿⡟⠀⣾⣿⡆⢻⣿⡇⣾⣿⡇⢸⣿⡃⠀⣿⣿⡆⢸⣿
⠀⠀⢸⣿⠃⠀⢸⠁⠀⢹⣿⠁⠈⢿⡇⠸⣟⠀⠰⣿⠤⠀⢿⣿⠃⠘⡿⠁⢸⢿⠁⣼⡟⢇⠀⣿⣿⠀⢸⢿
⠀⠀⢸⠻⠅⠀⠘⠀⠀⠸⠛⠀⠀⠋⠇⠀⠛⠀⠀⠸⠀⠀⠸⣿⠀⠀⠇⠀⠸⠛⠀⠉⠃⠘⠀⠃⠘⠀⣈⠚
⠀⠀⣴⠀⠀⣶⠀⢐⣦⣼⣧⣶⡀⢰⣧⡀⢠⣿⣿⡄⢠⣦⡀⠀⠀⢠⣶⣀⠀⣴⡀⢀⣦⠀⣰⣦⠀⢰⣿⣦
⠀⠘⣿⡇⢸⣿⡇⠸⣿⣿⣿⣿⡇⢸⣿⡇⢸⣿⢿⣿⠘⢿⡿⠦⠤⢸⣿⣿⢠⣿⠉⢠⣿⠀⣿⣿⡂⠀⣿⣿
⠀⠀⣿⠀⠘⡟⡇⠀⢿⣿⠻⠻⠁⠈⣿⠃⢸⣿⠀⡷⡄⢰⡆⠀⠀⠘⡿⣿⠈⢿⠁⠐⡇⠀⣿⢻⡁⢠⡏⣇
⠀⢀⡝⠀⠀⢃⡁⠀⠘⠃⠉⠈⡀⠀⠉⣀⡈⠉⢀⠀⢀⣾⣿⢀⣶⡀⠃⢻⠀⠉⠀⠀⠁⠀⠈⠈⠀⠸⠀⠘⠆
⠀⣾⣷⠀⣴⣿⣿⡀⢠⣿⠄⢰⣿⡄⢸⣿⡏⠀⣾⣦⠀⢹⣿⠸⣿⠇⣾⣄⢰⣿⡄⣼⣀⣾⣾⣷⠀⣼⡄⣼⣧
⠀⢿⣿⡀⢻⣿⣿⠃⢻⡿⠀⢸⣿⠁⢸⣿⡇⠀⣿⡟⠀⢨⡿⠀⣿⠀⣿⣷⣸⣿⡆⣿⡟⣿⣽⡿⠀⣿⠃⢸⡇
⠀⠈⣿⠀⢸⣿⣿⠀⠸⡇⠀⠀⠟⡄⢘⣟⢣⠀⠇⢿⠠⡎⣇⠀⠀⠐⠯⠘⠽⠿⡇⠱⠁⠏⡿⠹⠀⠿⠀⠨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠒⠋⠉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢠⡄⠀⠀⣤⠀⡠⡆⢀⡠⡖⠀⣰⣀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡆⡰⠉⡇⡠⠋⢹⠞⠀⠻⠊⠀⠿⠋⠘⢿⠀⠘⢧⡠⠖⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠃⡿⠀⠀⠛⠀⠀⠀⠀⣆⠀⠀⡀⠀⠀⠀⠈⡇⢠⡏⣰⠃⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢈⠃⠀⠀⠀⠀⠀⠈⠙⢉⠁⠀⠀⠀⠀⣷⠋⠀⡷⢻⠇⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠒⠋⠀⠀⠀⠀⠀⠀⢀⡴⣉⣤⠀⠀⠀⠀⠀⠀⠀⠁⢾⡀⣸⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠶⣤⣀⣀⠀⠀⠀⢠⣾⣪⠥⠖⠒⠋⠀⠀⠀⠀⠀⠀⢀⣴⣃⣀⣀⠤⠔⡪⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠽⠛⠋⡹⠀⠀⣰⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢤⣴⣊⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠚⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢤⣤⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠶⠖⢶⣛⠓⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡥⠤⢤⣤⣤⠽⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣧⠀⠀⠀⠀⠀⣀⠀⠈⢳⡀⠀⠀⢹⣄⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⢻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⠛⠯⠑⠈⠙⣧⡀⠐⠒⢉⡠⠂⠖⠚⠋⠉⠉⠉⠁⠀⠀⣀⡤⢖⠟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠈⠙⠂⠀⠀⠀⠘⢿⠲⣴⣁⢀⠔⠀⠀⠀⠀⢀⣀⠤⠖⠋⡽⠁⣯⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⣠⠤⡄⠀⠀⠀⠈⣧⡀⠐⣷⠀⠀⠀⠀⠀⠘⢿⣅⣭⣿⣿⠟⠛⠛⠉⠙⠳⠴⠖⠋⣠⠔⠉⠑⣄⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⢀⡎⠀⠐⢓⣦⡀⠄⠀⠚⢧⣀⣸⡄⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣄⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⢳⡀⠀⢀⡞⢻
⠀⠀⠀⢰⠋⠁⠀⠀⠀⢾⠔⠂⠁⠀⠀⠉⠉⠳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⡴⠁⠀⠀⠀⠀⠀⣠⠞⠻⠴⠋⠀⢸
⠀⠀⠀⠈⠒⠺⣅⣀⠜⠋⠀⠀⠀⣀⠤⠒⣢⣀⠀⠉⠉⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣉⠀⠰⠟⢹⠀⠀⠀⠀⠀⠀⠀⠉⡷⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡔⠒⠒⣄⡤⠀⠀⠀⠈⢇⣤⣀⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣈⡙⣿⠦⣤⣤⡤⠴⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡞⠁⠀⢈⡭⠛⡇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠦⣶⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠦⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⣀⠂⠀⠀⢘⡿⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⡀⢐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣿⣿⣿⣿⣿⣿⣷⣀⡄⠀⠀⠀⠀⠀⠀⣶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⡇⢸⣿⣽⣿⣿⣿⣿⡙⠛⣤⣤⣀⣀⣤⠀⢾⣿⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⡏⠿⠛⠿⠿⠿⠿⠗⢠⣿⣿⣿⣿⣿⣴⣿⣿⣧⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠐⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⣿⣿⠈⠻⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣿⣿⠇⠀⠈⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣿⣿⠀⠀⠠⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣀⡀⣄⣀⠀⠰⠿⠃⠀⠀⠹⠿⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠺⣿⡆⠀⠠⠀⠀⠀⠀⠈⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠚⠀⢲⣿⣿⡧⠀⠁⠉⠛⠀⠀⠒⠀⠀⠀⠚
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠉⣾⣿⠇⠀⡀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⣀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣠⠋⠀⠀⠀⠀⠄⠀⠀⠀⠐⠻⠍⠀⠀⠀⠀⢀⠠⠤⣀⠀⠀⠁⠢⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣿⡿
⠄⠤⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠘⠁⠀⠀⠀⠑⡀⠀⠀⠀⠠⣀⣀⡈⠁⠀⠀⠀⠀⠀⠤⠀⣰⣦⣤⣷⣶⣤⣾⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⢳⠎⠈⠀⠁⠀⠀⠄⡀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣿⣿⣿⣶⣶
⠧⠀⠀⠀⠀⠁⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠙⠂⠀⠀⢀⠴⠀⠀⠐⢻⣿⠧⠉⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡔⠀⠀⠀⠀⠀⠀⠀⠀⠠⣘⣧⡄⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠒⣤⡀⠀⠀⠀⠸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠸⡌⠁⠉⠀⢀⠱⢤⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣼⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⡀⠀⠈⠉⠛⣿⣿⣷⢠⣿⣶⣾⣧⠆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⣿⣿⠿⠿⠯⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠐⢃⠀⢀⠆⢀⠠⡤
⠀⠄⠀⡀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣶⠿⣷⡆⠠⡀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀⡀⠀⠀⠀⠠⠀⠉
⠀⠀⠀⠻⣄⠀⠀⡀⠀⠘⠀⠂⠀⠀⠀⠀⣀⡈⠉⡤⠶⢋⠄⡀⠈⠳⡄⠀⠀⠀⠉⠉⠂⠀⠂⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠒⠃
⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠐⠁⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠒⠻⢷⣶⣆⣀⠀⠀⠀⠀⠀⠠
⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⠤⠴⣀⡀⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠀⠀⢀⠀⡀⠀⠀⣠
⠀⠀⠀⠰⠠⢄⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣷⡀⣸⣧⣥⣴⡿⠛
⠤⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢳⣿⡀⠀⠀⡰⠃⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠘⢿⣿⣆⠐⢀⠀⢀⣰⣿⡿⠋⠀⠀⠀⠀⢀⡀⠀⠾⠛⠝⢶⡠⠀⢀⣀
⠤⠀⠒⠂⠀⠂⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠉⠑⠢⡀⠀⠘⠿⣿⣿⠾⠀⣼⣿⣟⡁⠊⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠈⠓⠦⠤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠆⠀⢘⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣶⣦⣬⣋⡰⠤⢙⣿⣋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠐⠒⠉⠉⠩⠝⠛⠛⢻⡿⢿⠁⣰⡔⢩⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡀⠘⣿⡹⣧⠹⣿⡻⣿⣷⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣾⣷⠹⣧⠘⢷⡙⢷⡝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢹⡆⠈⣷⡈⢿⡄⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⡀⠸⣧⠈⢿⡄⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡇⠀⣿⠀⠸⣧⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠀⠀⠁⠀⠈⠁

⣷⣄⠀⠀⠈⠻⣿⣿⣿⠆
⣿⣿⠗⠀⠀⠀⠙⠛⠁
⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀
⠀⠀⢀⣴⡦⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⢀⣴⣌⢉⣴⣿⣦⡀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠙⢿⣿⣿⣿⣆⡀
⣿⣿⣿⣿⣿⣿⠟⠋⢿⣿⣦⠀⠀⠀⠙⢿⣿⡿⠃
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠉⠁⠀⠀⢀⣴⡿⠋⣠⣄
⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⢋⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡀⠀⠈⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡀⠀⠈⠋⠀⠀⠀⢀⣴⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣷⣄
⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⣴⣿⣿⠿⣿⣿⣦⡀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠙⢿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡤⠖⠚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡄⢀⡴⠛⠙⠞⠋⠁⠀⠀⠀⣼⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠏⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⡤⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⠋⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⢻⣠⣀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⡞⣡⠀⢻⡄⠀⠀⠀⠀⢸⡁
⠀⠀⠀⢀⠀⠀⠀⠀⠘⢧⡀⠀⠸⡆⢀⡖⣀⠀⠀⠀⠀⢹⣸⡇⣿⠃⣸⠇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠓⠀⠀⣿⠸⠋⠁⠀⠀⠀⠀⠀⠉⠀⠈⢸⡷⡄⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢻⠀⠀⠀⢀⣼⡷⣄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⡄⣀⡤⣤⠀⠀⠀⠀⠀⢠⠏⠀⠘⠶⢛⡭⠟⢁⡤⠾⠓⠦⣄
⠀⠀⠀⠈⠉⠓⠶⢤⠀⠀⠀⠀⣠⢤⣀⠉⢩⠖⠃⠀⠀⠀⠀⣰⠏⠀⢀⣠⠞⠋⣠⠞⠉⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢹⡴⠋⠳⣌⠉⣷⠀⠀⠀⣰⣯⡤⠖⠋⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⣸⣥⡶⠀⠘⡆⠙⠦⠤⠞⠁⢸⡆⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢦⣽⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡼⠻⡄⠀⢠⡿⡄⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠙⢀⡾⠀⣷⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠚⠋⠀⢀⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣠⠴⠚⠁

⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣄
⢀⣰⣿⣷⣷⣤⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⣠⡆⠀⣴⣶⣧⣤⣶⣾⣷⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣴⣶⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣧⣶⢀⣰⣤⣤⣾⣿⣷⣿⣴⣶⣶⣦⣷⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⣼⣿⡿⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣴⣿⠛⠛⠛⢿⣧⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⣤⡤⢤⣤
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠉⢹⣿⡇⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠂⣿⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⣻⡇⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⡁⠀⠀⡁⠀⠘⣻⣇
⠀⠀⠀⠀⢠⣿⣿⣦⡀⢀⣤⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠦⡄⠀⠤⣾⡿⠃
⠀⠀⠀⠀⠈⠿⡿⠛⠀⠀⠛⠿⡿⠋⠀⠀⠀⠀⠀⢀⣠⣤⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣄
⠀⠀⠀⠀⠀⢸⣿⠟⠙⠻⠿⢿⣯⠀⠀⠀⠀⠀⠀⣾⣿⡿⠋⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⠋⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⢸⡟⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠉⢻⡇⠀⠀⠀⠀⠀⠀⣟⠈⠀⠀⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠠⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣰⣦⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣈⣀⣀⣄⠀⠀⠀⠀⠀⠀⢀⡤⠖⠀⠀⠐⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠠⠖⢂⡛⠓⢦⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⣶⣶⣄⠀⠀⠀⠀⠀⢰⣿⣷⣶⣶⣶⣶⣾⣿⣦
⠀⠀⠀⠀⢀⡞⣴⡿⠛⠻⣿⣦⣱⡄⠀⠀⠀⠀⠀⢸⡿⠛⠛⠉⠉⢿⡇⠀⠀⠀⠀⠈⣿⣿⠿⠛⠋⠉⢿⣿⡋
⠀⠀⠀⠀⢸⣿⡏⢀⠀⠀⠀⢹⣿⡷⠀⠀⠀⠀⠀⠘⣷⣷⡆⢰⣷⣾⠃⠀⠀⠀⠀⠐⣿⡏⠀⠀⠀⢀⢘⣿⡇
⠀⠀⠀⠀⠈⢻⣅⠀⠀⠀⠀⣈⡿⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣤⣦⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣴⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣔⣤⢤⣞⣋⣭⠇⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⣀⣴⠀⣾⣿⣾⣿⣿⣛⢡⣾⣉⣴⣋⠉⠠⣀⣈⣥⣶⣶⣾⣤⣤⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⢁⣼⢿⣿⣿⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣦⣶⣶⣶⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢘⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣽⣯⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢲⢹⡰⣿⣿⣶⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⠟⡁⣠⣴⣦⣀⠙⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⡿⣼⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⡧⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣰⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⢠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⣯⣿⣽⡀⣨⠛⣿⣿⣿⣿⣿⣿⣿⣿⡀⠻⣿⣿⣽⡿⠃⣸⡛⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢺⣧⠻⣿⣿⠀⣿⡿⣿⣿⣿⣿⣿⣿⣷⣦⣈⣉⣉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⡀⠫⣿⣦⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣻⡀⠉⠛⢿⡮⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠝⡿⢿⡿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠿⠿⠛⠛⠿⣾⣿⣿⠿⣿⡿⢟⡫⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡏⠑⠒⠤⢄⣀⣀⣲⡆⠴⠖⠒⡖⠒⠋⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⢙⣻⠻⠿⠿⠏⠙⠋⠉⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇
⠀⠀⠀⢡⠀⢠⠖⠒⢤⠜⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡋⠈⠙⠋⠀⠀⠀⠀⠠⠶⠀⠀⠀⡀⢠⣄⠀⡇
⠀⠀⠀⠀⢆⠘⢄⠀⠀⠀⠀⠀⠀⠀⡠⠠⣆⠀⠠⠂⠈⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⠀⠀⠀⢀⡾⠛⢻⣤⣜⡁⠀⠀⢰⡃⢀
⠀⠀⠀⠀⠀⠳⢎⠀⠀⠀⠀⠀⠀⠐⣇⡀⣘⠉⠑⠒⠤⣀⠞⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠈⠿⣦⠾⣿⣿⣿⣷⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⠉⢈⡹⠂⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣲⣤⡤⢤⣤⣤⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⣸⡓⠫⣹⣏⢧⠀⢀⡤⠊⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣾⣿⣿⣧⣙⣺⣿⣿⣿⡿⢟⠁⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⢰⡡⠖⡀⢹⠈⠑⠉⠈⠆⠗⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡇⢸⠀⠀⠀⠀⠸⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢾⣀⡎⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠢⢤⡋⢐⡅⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿⣧⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡏⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠛⠻⠿⠿⠟⠛⣛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠂⠀⠀⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸⢣⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⢟⣿⢙⡟⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠎⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣫⣿⡷⠃⢸⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢓⡒⡴⠋⠀⢨⣑⣒⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡟⠀⠀⠸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠈⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡄⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⡟⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⣠⣿⣿⣿⣿⣿⠏
⠉⠻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⣰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣠
⠀⠀⠈⠛⢿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠘⣿⣿⣿⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⠘⣿⣿⣿⡄⠀⠀⢿⣿⣿⠀⠀⢠⣿⣿⡿⠀⠀⢀⣼⣿⣿⣿⠏⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿
⣦⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠘⢿⣿⣷⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠁⠀⢀⣾⣿⣿⠟⠁⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣶⣄⡀⠀⠀⠀⠈⠙⢿⣿⣿⣷⡀⠀⠈⢿⣿⣧⠀⠈⣿⣿⠀⠀⣿⣿⠇⠀⢠⣾⣿⡿⠋⠀⠀⢀⣴⣿⣿⣿⣿⡿⠟⠋
⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠙⠻⣿⣿⣦⡀⠈⢻⣿⡆⠀⢻⣿⠀⢠⣿⡟⠀⢠⣿⣿⠋⠀⠀⣠⣾⣿⣿⣿⠿⠛⠁
⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠻⢿⣷⣄⠀⠻⣿⡀⢸⣿⠀⣸⡿⠁⣰⣿⠟⠁⢀⣤⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⣀⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣷⣦⣀⠀⠙⢿⣷⡀⠹⣷⠀⣿⠀⣿⠃⣴⡿⠃⣀⣴⣿⠿⠛⠉⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿
⣶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣦⣄⡈⠻⣦⡙⣇⢻⢰⡏⣼⠋⣠⡾⠟⠋⢁⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣀⣀⣈⠙⠛⠷⢬⣑⢜⢼⢺⣞⠵⢛⣩⣤⣶⠾⠿⠿⠛⠛⠛⠉⠉⠉
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣒⣲⣗⢵⣷⢽⡯⠭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⡾⠿⠛⣋⣭⢖⡝⡧⡷⡕⢍⡓⢶⣤⣄⡉⠉⠉⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⢀⣠⣴⡿⠋⣠⡟⣰⠇⣧⢹⣌⠻⣦⡈⠙⠻⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛
⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⣀⣤⣶⣿⠟⠉⢠⣾⠟⢠⣿⠀⣿⠀⢿⣆⠈⢿⣷⣄⠀⠉⠛⢿⣿⣿⣷⣶⣤⣄⡀
⣿⠿⠟⠛⠉⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠛⠁⢀⣴⣿⠏⢀⣾⡇⠀⣿⡇⠈⣿⣦⠀⠙⢿⣷⣦⡀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⠿⠋⠀⠀⣠⣿⣿⠃⠀⣼⣿⠃⠀⣿⣧⠀⠸⣿⣧⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠟⠁⠀⠀⣠⣾⣿⡿⠃⠀⢰⣿⣿⠀⠀⣿⣿⡀⠀⢻⣿⣷⡀⠀⠈⢿⣿⣿⣷⣄⠀⠀⠀⠀⠈⠛⠿⣿⣿
⣤⣶⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⢀⣴⣿⣿⡿⠁⠀⢀⣿⣿⡇⠀⠀⣿⣿⡇⠀⠀⢿⣿⣷⡄⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⣾⣿⣿⠃⠀⠀⣿⣿⣷⠀⠀⠘⣿⣿⣿⡄⠀⠀⠈⠻⣿⣿⣿⣿⣶⣄
⣿⣿⡿⠟⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⣿⣿⣿⡄⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣤⡀
⡿⠋⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣆⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠋⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⢻⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⣿⣯⣶⣶⣮⣝⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠊⠉⠛⠿⣿⣻⣷⣮⣭⣑⣯⣿⣿⠿⣿⣿⣿⣾⣿⣿⣿⣯⣽⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣯⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠃⠊⠉⠉⠉⠙⠛⠛⠛⠓⠋⠉⠿⠽⠯⠷⢾⣿⠿⠿⠿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠖⠀⠀⣀⣉⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⣾⠿⢿⠿⠿⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣦⣤⣭⣍⣁⣀⣀⣚⣛⣛⣛⣚⠲⠦⠤⣤⡤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠠⠤⢤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⣤⣄⣀⠀⠁⠉⠉⠙⠋⠉
⣿⣿⣿⣿⣿⠿⢛⣻⣿⣭⣭⣭⣭⣽⣿⣷⣶⣤⣤⣴⣒⣒⣒⠲⠶⢶⣶⣶⣤⣤⣤⣤⣤⣤⣭⣭⣭⣍⣛⣛⡛⠛⠦⠤⠤⠿⠿⠿⢿⣿⢶⠶⠶⠶⠶

⠀⠀⠀⣠⡀⠀⠀⠀⣾⣷⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀⣤⡄⠀⠀⣠⣦⡀⠀⠀⠀⠀⣿⣷⠀⠀⠀⣠⣤⠀⠀⢀⣀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⣴⣤
⠀⠀⢰⣿⣷⠀⠀⣠⣿⣧⣄⠀⠀⢀⣤⣿⣤⡀⠀⠀⠀⢀⣿⣷⠀⠀⣸⣿⠃⠀⠀⠀⠀⣹⣿⡄⠀⣀⣿⣏⠀⠀⢾⣿⠇⠀⠀⠀⢀⣤⣿⣦⣄⠀⠀⢘⣿⣿⡀
⢀⣾⣿⣿⣿⡆⢸⣿⣿⣿⣿⡆⣰⣿⣿⣿⣿⣿⣄⠲⣴⣿⣿⣿⣿⠀⣿⣿⣆⠀⠀⠀⣰⣿⣿⡇⢸⣿⣿⣿⣇⢠⣾⣿⣶⣆⠀⢄⣸⣿⣿⣿⣿⡀⣰⢿⣿⣿⣿
⠻⣥⣿⣿⣿⡇⠸⣿⣿⣿⣿⠁⢻⣄⣿⣿⣿⣨⡟⠀⠛⣻⣿⣿⠞⠃⣿⣿⣿⠶⣤⠞⢹⣿⣿⡇⠈⣹⣿⣿⡋⢸⣿⣿⣿⡿⠀⠀⠛⣿⣿⣿⠛⠁⢧⣰⣿⣿⣿
⠀⠈⣿⣿⣿⣇⠀⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⡄⠐⣿⣿⡿⠀⠀⠀⠸⣿⣿⡇⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣿⣿⣿⠃⠀⣿⡟⣿⣇⠀⠀⢸⣿⢿⣿⠀⠀⠀⠐⣿⣿⣿⠇⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⡿⣿⣧⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣿⣿⣿⠀⠀⣿⠃⢸⣯⠀⠀⢸⡟⠘⣿⠀⠀⠀⠀⢸⡇⣿⠀⠀⢸⣿⠇⠀⠀⠀⠀⣿⣿⡇⠸⣿⠉⣿⠁⠀⣿⠇⠸⣿⡀⠀⠀⣿⡃⣿⠀⠀⠀⢹⡏⠙⣿
⠀⠀⣿⣿⡟⠀⢀⣿⠀⠈⣿⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⡃⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⡟⠀⠀⣿⡄⠀⣿⣇⠀⠀⣿⡇⣿⠀⠀⠀⠈⣏⠀⢻
⠀⠠⠟⠸⠇⠀⣼⡟⠀⠀⢿⡄⠀⣾⠇⠀⢿⡄⠀⠀⠀⢸⠇⡿⠀⠀⢹⣿⣤⠀⠀⠀⢀⣼⣿⠇⠤⠾⠾⠇⠀⠀⠿⠁⠀⠸⠿⡤⠠⠿⠁⠿⠄⠀⠀⠀⠿⠀⠸
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆⠀⢀⣴⣄⠀⠀⢠⣶⡄⠀⠀⠀⣶⣦⠀⠀⢠⣤⣴⡄⠀⢀⣤⡄⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⢰⣶⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠐⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⣸⣿⠃⠀⢀⣀⣿⣧⣀⠀⠀⣹⣧⣤⡀⠸⣿⡟⠃⠀⠸⣿⡇⠀⠀⠀⢀⣸⣿⣅⠀⠀⣀⣼⣿⣄⡀⠀⠀⣈⣿⣇⡀
⠀⠀⢀⣾⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⡷⠀⣼⣿⣿⣿⣿⡆⢸⣿⣿⣿⡇⢰⣿⣿⠀⢠⡟⡟⡛⣻⠀⠀⣿⣿⢿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿
⠀⢀⡾⣿⣿⣆⠀⠀⠀⠀⠀⠀⣿⣿⣧⠸⣿⣿⣇⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⠃⣸⣿⣿⠀⢸⣷⣛⣿⣋⡖⢠⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⠂⣿⣿⣿⣿⣿
⠀⡼⢸⣿⣿⠍⢳⡄⠀⠀⠀⠀⣿⣿⡿⢀⣿⣿⣿⠀⠻⣿⣿⣿⡿⠀⢿⣿⣿⡏⠀⣿⣿⣿⡄⢸⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⠃⠀⠹⣿⣿⣿⡿
⠈⠁⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⡿⠀⠀⣿⡟⣿⡇⠀⠸⣿⣿⣧⠀⠛⣿⣿⠃⠈⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⠀⠀⢸⣿⢿⣿⠀⠀⠀⣿⡟⣿⡇
⠀⠀⣼⠏⠈⢻⡇⠀⠀⠀⠀⠀⣿⣿⠁⠀⣿⣿⡇⠀⠀⣿⠁⢹⡇⠀⠀⢻⣿⣿⠀⠀⢹⣿⠀⠀⣿⠛⢿⡟⠀⠿⢿⣿⢹⣿⠀⠀⢸⡟⠸⣿⠀⠀⠀⣿⠁⢹⡇
⢀⡾⠃⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿⡿⠀⠀⣿⣿⠁⠀⢰⡿⠀⠘⣿⠀⠀⢸⣿⣿⠀⠀⠘⣿⠀⠘⡟⠀⣿⠃⠀⠀⢸⣿⢸⣿⠀⠀⢸⡇⠀⣿⠀⠀⢰⣿⠀⠸⣿
⠿⣅⠀⠀⠀⠀⠿⣦⠄⠀⠀⠀⣿⣧⣄⠀⣿⣿⠦⠀⣸⡇⠀⠀⣿⡄⠀⣼⣿⣿⠄⢀⣴⣿⠀⢠⡇⠀⣿⠀⠀⠀⣸⣿⠸⣿⡀⢀⣿⠇⠀⢿⡆⠀⢸⡇⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣰⣆⠀⢠⡀⠀⢀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⠀⠀⢠⠀⠀⢠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣸⣿⣿⣷⣼⣿⣸⣺⣾⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⢀⣼⣧⠀⣸⠀⠀⣾⣿⣿⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡺⣿⡇⠀⡏⠀⢰⣿⣿⣿⡟⣇⣇⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⡇⠀⠼⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢆⣿⠅⢀⠀⠀⣸⣿⣷⣿⡇⣿⣿⠇
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣼⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⡀
⠀⠀⠀⠀⢠⡿⠲⡄⣀⣀⠀⢀⣤⣀⣀⡀⠀⣀⠤⠓⠒⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⡀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢹⡇
⠀⠀⠀⠀⣽⣧⣤⡇⣧⣀⣾⣿⣿⣿⣿⣿⠀⡇⠀⠀⠀⠀⡀⣀⠀⠁⠀⠠⠄⣀⠀⠀⢘⣿⣿⣿⡇⠀⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇
⢀⣀⣤⡶⢿⣿⣿⣷⣿⣿⣿⡿⠛⠿⠿⣿⣧⣷⣤⡀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⠉⠒⢸⣿⣿⣿⠇⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠈⢱⡤⢤⣥⣭⣭⣽⡿⠉⢻⣷⣟⣦⣴⣹⣿⣿⣮⣳⣶⣶⡀⢁⠈⠙⢿⡧⡀⠀⠀⠀⠈⢿⣿⣿⣿⣾⠛⠋⠉⠁⠈⠈⢀⠈⠛⠛⢿⣿⡇
⠀⣼⡇⠀⡖⠒⢺⣿⣧⣯⡿⠛⠿⢿⣿⣿⣿⣿⣿⣿⡋⠛⣛⣛⣙⣷⣾⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⣟⠀⣠⣦⠼⠂⠙⢶⡷⢶⣒⣞⣉⣇
⢀⣿⡇⠀⣇⣀⣀⣉⣛⣉⣀⣀⣀⡀⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣄⠀⠀⠀⠈⣿⣿⣿⣿⣾⠟⠁⠀⠀⢠⣿⣿⣿⡿⡿⠿⣿
⢘⣿⡇⠀⠆⠀⠀⠐⠆⠀⠰⠀⠀⠰⣷⠖⠲⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⡄⣾⣿⣿⣳⣿⣿⣿⣦
⠀⣿⡇⠀⠀⠀⠰⣆⠂⠀⠀⠀⠀⠀⣯⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⢿⠂⠀⡇⠀⣿⣿⣷⣿⣿⣿⣿⡿
⣠⣿⡇⠀⡀⣀⢀⣀⣀⠀⠀⠀⠀⣘⣟⡀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⡄⠀⠀⢸⣿⣿⣿⣟⣸⡷⣾⡅⠃⣿⡟⣿⣿⣿⣿⣿⡇
⠟⣿⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡇⠀⠀⢀⣿⣿⣿⣿⣹⡷⣏⣷⣴⣿⣽⣿⣿⣿⣿⣿⡇
⠈⣿⠃⠀⠃⠀⢀⡤⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⡇⡷⠀⠀⠘⣿⣿⢻⣿⣿⣃⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⡅
⠰⣿⡆⠀⡄⠀⠀⠉⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⢰⣿⣿⣿⣷⣶⣶⣶⣦⣭⣽⢛⣛⣉⣽⣿⣷⡇
⠘⣿⡇⠀⠓⠐⠒⠒⠒⠒⠒⠂⠐⢲⣿⠒⠒⠆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢉⣀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡁
⢠⣿⡇⠀⣴⣶⣶⣦⣴⣤⣶⣶⣷⣿⣽⣷⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣼⣿⣿⡄
⠈⢿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⢿⠻⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣇⣿⣾⣿⡿⣿⣿
⠀⠀⠈⠛⠉⠫⠿⢟⠻⠿⠿⡿⠿⠦⣤⣤⣾⣿⣿⣿⣿⣿⣟⣋⣱⣦⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠿⢿⠿⢿⣶⣷⡀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣉⣈⣁⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣉⣉⣛⣛⣇⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢄⣀⠄⠀⣀⠠⢄⠈⠀⠀⠀⣀⣠⣄⣀⣀⡈⠉⠋⠃⢀⣀⠀⠀⠀⠀⠔⠄⠀⡠⣄⠀⠀⠁⡄⠀⠁⠈⠀⠀⠀⠈⡀⠀⠛
⠀⠀⠀⡜⠇⠤⠀⡂⠀⢀⡞⠁⢐⠛⢿⠷⢻⠆⠀⠀⡜⠋⠙⠓⡄⠀⠀⠀⢰⠃⠐⠙⡄⠀⠈⠀⢀⠂⠀⠘⢃⢀⣀⣀⡠⡁⠠⠘⢂
⠀⠀⠀⠃⠀⠀⠀⠃⠀⠈⠀⠀⠈⠀⠈⠀⠰⠀⠀⠀⠇⢀⡀⠀⠇⠀⠀⠀⠘⠀⠀⠀⠃⠀⠐⠀⢰⡏⠉⠈⢹⠦⠤⠤⠽⠀⠀⠀⠸
⠀⠀⠀⠀⣀⠁⡀⠀⠀⠘⠀⠀⡢⠈⢰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠀⠁⠈⠁⠀⠃⠀⠀⠂⠁⠈
⠀⠀⢤⡀⠛⠛⠃⠀⠀⠘⠦⢇⠁⠀⠈⣇⡄⠚⢀⠀⠀⠲⠔⠂⠀⠀⠠⠀⠄⠀⠀⠀⠀⢀⣀⡀⠀⢠⣀⣀⣤⡀⠀⣀
⠀⠘⠇⠀⠀⠀⠀⠀⠈⠫⠫⢦⠀⠀⠀⢀⠢⠠⠀⠀⠀⠀⠀⠀⢀⠊⠋⠰⢆⠀⠈⠀⢀⡋⠩⠀⡆⠀⠁⠉⢀⠟⠛⠦⢵
⠀⠀⠈⠄⠁⠀⠀⠰⠒⠒⠒⠺⠀⠀⠀⢺⡄⠀⠀⢠⠠⠀⠀⠀⠆⠀⡀⠀⠰⠀⠀⠀⢸⠁⠀⠀⡻⠀⠀⠀⠘⠀⠀⠀⠘
⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠀⠀⠀⠀⢸⢢⠀⢠⡇⠀⠀⠀⠀⠠⠀⠠
⠀⠀⢀⡀⣀⠀⠀⡀⠆⢀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠹⠀⡀⠀⣀⠸⠃⠀⠈⠳⠀⠀⠀⠀⠁⣀⠀⠂
⠛⡟⠛⠛⠙⠂⠈⡆⠀⠀⠀⠂⠨⡃⠀⢀⠀⠀⠀⠀⠈⠫⠹⢦⠀⠀⠀⠀⠸⠥⡀⠈⣂⢀⡀⣀⠀⠀⠸⠁⠀⠀⠀⠀⢀⡚⠠
⠀⠁⠀⠸⠉⡈⠁⠃⠀⠀⠀⡀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠋⠁⠄⠀⠀⠀⠀⠠⢁⡂⠀⠀⠼⠀⠀⠀⠠
⠀⠀⠀⠀⠄⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠠⠱⠀⠠⠀⠜⠀⠂⠀⠀⠐⠛
⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠩⠫⣂⠀⠀⠀⠀⢠⡆⠉⠂⠀⠀⠀⢠⠐⠏⠀⠈⠀⠀⠀⠀⠀⠐⠁⢲⠄⠀⠀⠂⠀⣠⡤⢦⣄⠂⠀⠀⠀⠰⠄⠒⢢⠀⠠⠂⢀⣴
⠀⠀⡽⠀⠀⠀⢸⠀⠁⠀⢶⠀⠀⠀⠀⡆⠀⠀⠀⡅⠀⠀⠀⢰⣿⡀⣨⣦⠀⠀⠀⠰⣯⡄⠀⢿⠆⠠⠤⠄⢠⠂⠀⠉⢲⠀⠀⠀⡏
⠡⠀⠃⠀⠀⠀⠈⡄⠠⠀⡌⠀⠀⠀⠀⠣⡀⠒⣠⠃⠀⠀⠀⠘⢀⠠⢀⠌⠀⠀⠀⢘⡍⠉⠉⠉⡁⠀⠀⠀⠘⠄⠴⠄⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠘⡎⠀⠀⠀⢈⢟⠀⠁⢈⡅⠀⠀⠀⠀⣀⠀⢀
⠀⠈⠁⠀⢀⠤⢀⡀⠀⠀⠀⠀⣤⢀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠁⠀⢰⡢⠆⡅⠀⠀⠐⠀⠀⡠⢀⠀⠀⠉⠀⠀⡰⠲⠖⠊
⠃⠀⠀⢠⠀⠀⠀⢁⠀⠀⢤⡞⠀⠉⠐⠋⠀⠀⠀⠀⢀⠀⠀⠀⡣⠠⠄⠀⠄⡇⠀⠀⢸⠈⡏⠀⠀⣭⡄⠐⡏⠍⠉⠹⡯⡈⠁⠓⠒
⠀⠀⠀⠘⡄⢀⡀⣸⠀⠀⠈⠙⠀⣠⡀⠈⠁⠀⠀⠀⠘⠀⠦⠠⠃⠀⠀⠀⠀⢰⠠⠄⡄⠀⠁⠀⠰⡟⠓⠐⢹⠂⠀⠀⠘⢷⡀⠆⠠
⠀⠀⠀⢀⠘⠂⢴⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠲⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣐⠀⠀⣞⣐⠀⢐⠂⣂
⠀⠀⣀⠀⠀⡀⠀⢠⠀⠂⢀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠉⠁⠀⡴⠀⠠⠄⠀⠐⠢⠀⢀⠠⣤⡀⠈⡄⠠⠉
⠀⠰⠀⠑⠀⠬⢖⢄⠄⠀⠀⠲⠀⠀⠀⠠⠀⠀⠛⢻⡄⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⣐⠀⠀⠈⢶⠀⠈⠀⠀⠀⠇⠀⠀⢀
⠀⡀⠀⡜⠁⠀⠐⠸⠆⢤⠄⠆⠂⠀⠀⠐⡀⢀⠀⢸⠃⠀⠀⠀⢁⠀⠠⠀⡏⠀⠀⠀⠁⠀⢀⠀⠘⡀⠀⠀⠀⠍⢀⠈⠸⠂⠐⠂⠒
⠀⠃⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⡁⠰⠀⠀⠀⢡⠈⠠⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠶⣤⣄⣀⣀⣀⣠⣤⠴⠞⠉⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠈⠉⠉⠳⣤⣀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠉⢻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠋⠙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⣸⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⠄⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣄⣼⠖⠋⠉⠉⢉⣀⢀⡤⣄⠀⠈⠉⠉⠉⠓⢾⣄⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⢀⡟⠉⡿⠀⣼⠶⣄⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸⠃⢸⠇⢠⡏⠀⣟⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⣦⠀⢀⡟⠀⡿⠀⣼⠁⢸⠏⣹⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡆⣸⠁⢸⠇⢠⡇⠀⡞⠀⣿⡴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣧⡏⠀⠀⠀⠈⠁⠸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⠟⠉⠉⠛⠓⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢈⡗⠀⠀⠀⠀⠀⠀⣀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧



⣇⢸⠆⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠂⠒⠒⠒⠒⠒⠒⠒⠒⠂⠂⠀⠐⠒⠒⠒⠒⠒⠒⠐⠂⢾⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⡇⠈⡆⡇⠈⠁⢠⢻⡀⡧⢼⢸⠀⠀⡇⠀⠀⠪⣄⠀⡇⠀⣅⡕⣸⠀⢱⢈⣖⠁⠬
⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦⣷⣶⣥⣧⣤⣤⣦⣤⣥⣷⣮⣼⣶⣤⣷⣦⣤⣴⣮⣤⣥⣤⣤⣼⣬⣶⣮⣤⣤⣥⣴⣦⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠈⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⡄⠀⢀⡄⢀⢴⠀⡠⣦⢀⣤⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⣗⡰⠉⣇⠔⠁⠷⠁⠘⠊⠀⠛⠋⠘⢻⠀⢳⡤⠒⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠎⠘⠀⡄⠉⠀⠀⠀⠸⢤⡤⠄⠀⠀⠀⢸⣠⠞⢰⣡⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢠⠴⠁⠀⠀⠀⠀⠀⢀⠔⢀⡀⠀⠀⠘⠃⠀⠋⢰⠁⢸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠰⣤⣀⣀⠀⠀⠀⣰⣟⡮⠥⠖⠒⠀⠀⠀⠀⠀⢈⣱⣃⣀⡀⠤⢔⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠘⠋⢉⡝⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠶⠦⠥⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⣈⣓⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠒⠶⣇⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡄⠀⠀⠀⠉⠀⠈⠳⣄⠀⠉⠳⣄⠀⠀⠀⠀⠀⢀⡠⠛⠉⠉⠉⠻⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡒⠿⠓⠉⠻⣄⠀⠀⠂⡩⠀⠴⠚⠒⠚⠋⠉⠀⠀⣀⡤⢲⠋⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣉⠂⠀⠀⠀⠙⣟⡶⢤⣠⣒⣀⣀⣀⣀⣤⠴⠒⣉⠜⢐⣧⡀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⢀⠔⠒⣆⡀⠀⠀⢘⣦⠀⢹⡀⠀⠀⠀⠀⠉⠳⠶⣿⣿⠉⠉⠀⠀⠉⠉⠉⣡⠔⠉⠀⠙⢦⠀⠀⠀⣠⡇
⠀⠀⠀⡰⠛⠀⠀⠐⣿⡁⠀⠀⠈⠓⠚⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠤⣄⣀⣀⠞⠁⠀⠀⠀⠀⣠⠷⡄⠔⠁⡇
⠀⠀⠀⠓⠖⢇⣀⠴⠚⠀⠀⢀⡠⠤⣤⣀⠉⠉⠛⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣥⠄⠀⢛⡩⠂⠀⠀⠀⠀⠀⠈⡙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢀⡴⠁⠈⠷⣦⡀⠀⠀⠘⠒⠦⠜⠁⠀⠀⠀⠀⠀⠀⠀⠓⡖⠲⠬⡏⠓⠒⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠸⣀⡄⠀⠉⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠑⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠟⣋⣤⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢁⣶⣶⣦⣡⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⢃⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣦⡙⠻⢿⣿⣿⣿⣿⣿⣿⡿⢡⣶⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠈⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⢋⣩⣤⣤⣤⣍⣁⠀⠀⠀⠀⠀⠀⠈⢿⣟⠻⣿⣿⣦⡈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠈⢿⣧⣙⣿⡿⢿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢰⣿⣿⣿⡿⢋⣥⣤⣌⠻⣿⣿⣧⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆⠹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⣾⣿⣿⣿⠆⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⣼⣿⣿⣿⡟⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣸⣿⣿⣿⡟⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢰⣿⣿⣿⣿⠁⢿⣿⣿⣿⣿⠃⠀⠀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣿⣿⣿⣿⣿⣿⣶⣤⣉⡛⢿⠀⠀⠘⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣌⡛⠻⢿⣿⣿⣿⣿⣿⣿⠄⠀⠀⣴⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣤⣉⡛⠿⠿⠋⡀⠀⢀⡉⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣬⣍⣙⡛⠻⠷⢦⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣦⣴⣶⣶⣶⣶⣶⣶⡶⠾⠛⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣿⣿⣿⡿⠟⢛⣉⣥⣶⠞⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣛⣩⣤⣶⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠛⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡤⠖⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣦⠴⠋⠀⠉⠁⠀⠀⠀⠸⠧⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠀⠀⣠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠏⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠧⢤⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠸⡄⠀⠀⠀⠀⠀⠘⢦⠠⡏⣖⡀⣧⠀⠀⠀⠀⠻⡀
⠀⠀⠀⢤⣀⠀⠀⠈⠳⡀⠀⢳⢠⡧⠄⠀⠀⠀⠘⠶⠇⠇⣠⡏⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⣸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡄⠀⠀⠀⣰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡀⢀⣀⡀⠀⠀⠀⠀⢠⠏⠀⢧⠴⣲⠾⢋⣩⠦⢤⡀
⠀⠀⠀⠙⠒⠦⣄⠀⠀⠀⢀⣤⡀⠛⣫⠞⠁⠀⠀⠀⣰⠋⠀⢀⡴⠚⣡⠔⠋⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠙⣧⠛⠳⣌⠙⡆⠀⠀⣠⣧⡴⠚⢁⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⢾⣧⡖⠀⢸⡆⠓⠶⠚⠁⢸⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠲⣿⠀⠀⢸⡇⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡞⠉⢳⠀⣸⢹⡄⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠤⠞⠁⢀⣧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⣿
⠙⢿⣿⣷⣄⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠙⢿⣿⣷⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠁⠀⠀⢀⣶⣦⡀⠀⡀⠋⠁
⠀⠀⠀⠈⠙⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⣀⣴⡿⠀⠀⠀⠈⠻⣿⣿⠟⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⡀⠀⠀⠐⠟⠁⠀⠀⣀⠀⠀⠀⠈⣡⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠀⠀⠀⢠⡾⠋⠀⢀⣠⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠁⠀⠀⡀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⣷⣄⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣄⠠⡀⠀⠀⠈⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⡿⠃⠀⠀⠀⢀⣦⡀⠀⠀⠙⢿⣿⣿⣿⣿⣷⡽⠂⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠉⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⠙⢿⣿⡿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡼⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⢀⡴⠋⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⢸⠁⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠛⠋⠉⣀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠋⠀⠀⢀⡴⠂⠀⣠⠴⠚⢁⣠⠞⠓⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢀⡴⠋⠀⠴⠋⠁⠀⠀⢸⠇⢾⡷⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠋⠀⠀⢀⣀⡀⠀⠀⠀⠘⣧⣀⠀⢛⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣖⠋⠁⠀⠀⠀⠀⠀⠘⢯⣀⡽⠀⠀⠀⠀⠈⡉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠚⠙⢲⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠳⣄⠀⠀⣀⠀⠀⠀⠘⣇⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠳⣄⠈⠙⠲⣄⡀⠉⠙⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢧⡀⠀⠀⠀⠀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠳⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣯⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠



⡏⠉⡦⠄⠀⠀⠀⠤⠀⠀⠄⠤⠄⠄⠤⠠⠠⠄⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠠⠄⠄⠀⠀⠀⠀⠄⠄⠠⠠⠄⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠄⠠⡎⠉
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠔⠒⢦⢰⠀⢲⠒⠀⢶⠀⢰⠒⡆⠠⠀⠀⠒⠂⠀⢠⠒⠀⠒⠖⢠⠒⢢⢠⠒⠒⡄⣄⡠⠂⡖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⠀⢀⠀⡼⢸⠀⠸⠀⠸⠈⠇⢸⣉⠇⢘⣀⠀⡍⡁⠀⠀⣉⠆⠀⠄⠈⠃⠃⠸⣄⣠⠇⡟⠱⡀⣏⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡖⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣇⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⢳⡈⠳⢤⣤⠴⠚⠡⢄⡀⠀⠀⠀⠀⡇⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⢧⠀⠈⢧⠀⠀⠀⠀⢳⡀⢦⠙⣄⠀⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠘⡆⠀⠘⣆⠀⠀⠀⠀⠃⠘⠂⠘⡆⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⡠⠋⠹⡄⠀⠉⠉⠉⠉⠉⠉⠉⢳⡄⠛⠀⠀⡇⠀⠀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠐⢦⡰⠊⠀⠀⠀⠙⠶⠤⠤⢤⡤⠤⠤⠤⠾⠤⠤⠤⠤⠇⠀⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠓⠲⠶⠆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣘⣆⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡖⠒⠒⠒⠒⠀⠀⠀⠒⠒⢺
⠀⠀⠀⠀⠀⢀⡞⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢧⣀⣀⣀⣀⣀⣀⣀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠆⠈⠁⠧⠘⠀⠀⠈⠀⠐⠀⠉⠸⠀⠀⠀⠦⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣦⣀⣀⣀⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡏⣿⣿⣿⣿⠛⠛⣡⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⣤⣤⣽⣾⣧⣈⠘⠿⠾⠂⢠⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣧⠉⢻⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⡈⠙⢿⣿⣿⣿⣿⣿⣿⣿⢷⣀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠿⣿⣷⡄⠀⠈⠛⠻⠿⠿⣿⢻⣷⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣀⠾⠿⣿⣿⣿⣿⣦⣾⣿⣿⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠃⢸⢿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⢄⣨⡄⠴⠟⠁⠈⠓⢄⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⣿⣿⡶⣾⣿⣷⠠⢄⣀⠀⠀⠀⠈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠰⠁⠒⠄⠀⠀⠀⠀⠀⠀⠈⠸⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣤⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢣⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡇⠀⡿⣿⣿⣿⡿⠻⠋⠉⠀⠀⡃⠁⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⣶⣧⡿⠁⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣟⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⣤⡶⠂⣤⣤⠀⡀⠀⠤⣤⣘⣻⣄⠀⡀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⣤⣀⡀⠀⠸⠏⢤⠀⢰⣯⣿⣿⣿⣆⢁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⣄⠋⠉⠉⠛⠛⠛⠛⠓⠀⠀⠘⠛⠚⠛⠛⠻⠿⢿⣆⠳⡄⠀⠀⠀⢀⣧⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠄⠠⠾⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣘⣛⡛⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⠙⣿⣿⣿⣿⠟⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠋⠐⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⡁⠀⠀⠀⠀⠐⡞⠛⠉⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⡁⡄⠀⢸⠆⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣆⠀⠀⢀⠈⠀⠀⠀⡾⠀⡄⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⡆⢾⡷⠀⠀⠼⠁⠀⡇⠀⣠⠖⠂⠺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡹⢻⢡⣿⠀⢀⠀⠀⣀⣸⢃⡞⠁⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⣿⡟⠠⠀⠀⢻⠁⠀⢿⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⣸⣿⠇⠀⢀⡤⠂⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⣺⣷⢿⡟⠀⡴⠋⠀⣄⣔⠈⣹⡄⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣤⣵⣿⣿⠊⠳⠘⠓⣦⣀⢻⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠶⣶⣾⣿⣿⡿⠟⠀⠀⠀⠀⠚⠛⠿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠀⢹⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⡶⣄⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣍⠀⠔⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡀⠀⠀⠀⢠⠱⣄⡀⠀⠀⢠⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣄⡑⠤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠴⠋

⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⠟⣰⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⣴⣿⣿⡟⣡⣾⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⣿⠟⣡⣾⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣥⣾⣿⣿⣿⣿⣿⣿⣧⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣤⡶⠛⠁
⣿⣿⣿⣿⣿⣿⢸⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠁⠀⠀⠀⢀⣴⠟⠀⣀⣤⠾⠛⠁⢀⣰⣶⣴⣶⡀
⣿⣿⣿⣿⣿⡇⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣴⠟⠁⣴⠾⠋⠁⠀⠀⠀⣿⠡⡄⢹⣿⣿
⣿⣿⣿⣿⣿⣇⢸⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣟⠠⣿
⣿⣿⣿⣿⣿⣿⡈⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⠀⠀⠀⠀⠀⠘⠛⢿⡿⠓⠛
⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣆⢻⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠙⢷⣤⠀⠀⣤⣀⠀⠀⠀⠀⢻⣿⣿⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⢻⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠙⠻⣦⣄⡀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠙⠻⢶⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⢿⣆⢻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⠉⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⡆⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⣿
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡃⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⢿⣿⣿⣿⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⣛⢻⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢙⠿⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⡰⠊⠀⢻⣿⣿⣿⡿⠃⠀⠀⠀⠐⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡀⠐⠀⠀⠈⠻⡿⠋⠀⠀⠀⠤⠶⠶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠸⡛⠛⠋⠋⠀⢳⠀⠈⠀⢰⣇⠀⠀⠀⠀⠀⠸⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠺⢷⣄⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠐⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⢰⠚⠁⠀⠀⠀⠋⠉⡀⠀⠐⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠸⡄⠀⠀⠀⠠⢴⠈⠉⠡⠀⠈⠀⠀⠀⡠⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡠⠀⠀⠀⡶⠀⠀⠀⠀⡠⠀⠀⠀⠀⠖⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡼⠒⠦⠮⢀⠤⠃⠀⠂⠀⠁⠰⣀⠀⡠⠀⢀⠌⠀⢀⠎⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡰⠈⠀⠀⠸⠂⠉⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⡰⠦⡰⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠐⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠿⣿⣶⣄⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⢹⠟⣡⡆⢻⣿⣿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢿⠿⠟⣛⣭⣶⣧⣶⣾⣿⣷⣶⣶⣶⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣉⠀⠀⠀⢾⣿⣿⢫⣝⣿⡿⢿⣿⣋⡍⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⢻⣿⣦⣿⣿⣿⡌⢿⣿⣵⣾⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⣛⠻⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢦⠀⠀⠀⠈⢿⣿⢋⢸⢿⠘⡘⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡜⣎⢧⣅⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡐⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣦⠻⣿⣌⡻⣿⡿⢟⣠⣴⣆⢤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⣿⠟⣤⣶⣿⣿⣿⣿⣌⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡁⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡇⣾⣇⣿⢸⠗⢯⣸⡯⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠈⠈⠉⠈⠉⠈⠉⠉⠀⠁

⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣧⡐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀⣀⡀⠈⠻⣿⣿⣦⡀⠈⠳⣦⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠻⣿⣶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠻⣿⣿⣿⣶⣤⣀
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⠻⣿⣿⣿⣿⣷⣦
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢿⣿⣿⣿⣿
⣦⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿⣿⣿
⣿⣿⣦⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠙⢿
⣿⣿⣿⣿⣦⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⣿⣿⣿⣿⣿⣷⣄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠉⠻⢿⣿⣿⣿⣷⣄⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⠀⠙⢿⣿⣿⣟⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠆


⣤⣤⣤⡤⢤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⡤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣧⡇⢸⠀⡄⢩⠀⡜⠁⠉⠋⣡⣭⡄⢸⠀⡏⢩⡉⡏⢩⠉⣿⣷⢸⠀⡇⠈⢩⡍⡟⠉⠉⣏⠀⢽⡁⠨⣿⣿⣿
⠿⠿⠿⠧⠼⠤⠣⠼⠤⠷⠤⠤⠷⠤⠴⠧⠼⠤⠷⠤⠴⠧⠾⠦⠿⠿⠼⠦⠧⠶⠼⠧⠷⠦⠴⠷⠤⠶⠤⠴⠿⠿⠿

⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠛⠛⠛⠛⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡷⠀⡀⢚⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⠠⠀⠲⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢈⡿⢧⡈⠻⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠂⠐⠄⢄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢺⢾⣽⠗⣾⣿⡟
⠀⠀⠀⠀⠸⣿⣿⣿⡿⠛⠋⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢀⣀⣼⣷⣖⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⣀⠀⣀⣀⣀⠀⣀⣀⣀⣀⠀⣀⠀⠀⣀⠀⢀⣠⣀⡀⠀⣀⣀⣀⠀⣀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡷⢸⣿⣀⡀⠀⣿⠁⣹⡇⠀⢹⣿⠈⠀⣿⠀⠀⣿⠀⣿⣄⣈⠀⢸⣧⣀⡁⠀⣿⠀⢺⣧⣀⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⠁⢸⣿⠉⠁⠀⣿⠙⣿⡀⠀⢸⣿⠀⠀⣿⡀⢠⣿⠀⡀⠈⣹⡷⢀⠈⢉⣿⠀⣿⠀⢀⠈⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠉⠀⠉⠀⠈⠉⠀⠈⠉⠀⠀⠈⠉⠉⠁⠀⠉⠉⠉⠀⠈⠉⠉⠉⠀⠉⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⣸⣷⠀⠀⢀⣿⣀⠀⢠⣶⣿⣿⣦⡀⠀⣀⠀⠀⠀⢤⡀⠰⡿⠀⣀⠄⠀⣠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣉⣉⣀⡀⠛⠛⠛⠀⣫⣶⣯⣭⣭⣶⠀⠛⠁⠀⠀⠀⠈⢷⣷⡎⠁⠀⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠾⣿⡇⠀⠀⠀⠀⣿⣷⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⣠⠚⠿⢣⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣷⡏⣷⠀⠀⠀⠀⠘⢿⣍⣹⠿⠁⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠸⣿⣿⡇⣿⠀⣠⣤⣤⡤⠷⠂⠐⠷⣤⣤⣤⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠿⠘⠇⠁⠀⣿⣿⣿⢳⣶⡆⢠⣿⣸⣿⣿⣿⠀⠠⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⣿⢟⣵⢟⣵⣾⣷⡝⣷⡹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠋⠚⠿⠿⠿⠿⠊⠛⠈⠉



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

⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢶⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇
⠀⠙⢦⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢹⠄
⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⣠⡴⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⡷⠟⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠹⣦⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠚⠛⠋⣉⡽⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⣴⠞⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿
⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⣀⣀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⣤⣾⣯⡤⠴⠶⠾⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠛⣿⠀⠀⠀⠀⠀⠀⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣾⠀⠀⣠⡶⠋
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠛⠛⠛⠲⠶⠶⠶⣿⣥⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣿⣴⠞⠉
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⠳⠶⠶⠤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠳⠲⠾⣧⠀⠀⠀⢀⡟
⠀⡇⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣠⡇
⠀⡇⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠿⠚⠛⢹⡇
⠈⡇⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⢈⡇⠀⠀⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠚⠛⢋⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⢈⡇⢀⣴⠋⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠛⠛⠋⠉⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢨⣧⡾⠃⠀⠀⠀⠀⠀⠈⠙⠳⢦⣶⠖⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠈⠉⠉⠙⠛⠓⠶⠶⢤⣤⣄⣀⣀⡀⠀⠀⠀⡟
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠙⠛⠛⠾⠷⢦⣤⣤⣀⣀⣀
⠛⠳⠶⠶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⣿⠉⠛⠛⠲⠶⠦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠲⠶⠦⣤⣤⣄⣀⣉⡛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⡿⠛⠻⠻⠷⠦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣈⣷⣄⣀⣀⣠
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣭⣯⣭⣿⣯⣿⣅⣽⣬⣭⣭⣭⣽⣯⣭⣭