Vomiting Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢦⣄⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣷⣌⠛⠷⣦⣄⡀⠈⠻⣿⣦⣄⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⠉⠉⠉⠉⠀⠀⠀⠉⠛⠷⣤⡙⣆⠙⢷⡘⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣁⣀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠈⢷⡷⢻⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⣉⣭⣿⠿⠟⠛⠉⠉⠀⢀⣠⣤⣤⣤⣀⡀⡀⠀⠀⠈⠇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣤⡾⠟⠉⣀⣀⣤⣄⣀⡀⣰⣿⠟⠉⠉⠉⠛⢿⣿⣦⠀⠀⠀⠚⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣶⡿⠿⢻⣿⣿⡉⠉⡉⠛⠁⣠⠶⣶⣤⡀⠀⣬⣿⣷⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠋⠁⠀⠀⢸⡿⠙⠛⢾⣧⠰⣶⣿⠟⠋⢉⠀⠀⢾⣿⣿⣇⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⢸⠧⣄⢀⡆⢨⡁⠙⢥⠀⢀⡞⠀⠀⢹⣽⣿⠂⡾⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⣿⠃⠈⡇⠀⠈⢧⡸⠁⢒⠉⠸⣿⡿⣼⠖⣫⡥⢤⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⡿⣧⣰⠃⠀⠀⣸⣷⡄⠀⠀⣸⣿⣿⠏⣼⣡⣄⡼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢇⠛⣁⡼⣡⠴⡦⠀⡿⠈⢿⡄⠈⣿⡟⣿⢸⠏⢡⠟⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣭⣍⣉⣁⣘⣥⠞⠁⠈⠒⠒⠀⢀⠋⠻⣷⡿⣌⣀⣡⠔⣁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣛⠛⠻⠿⢟⣫⣵⡶⡖⠒⠶⣄⠈⠁⣁⠀⣿⠁⠈⠉⣛⣩⠟⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣬⣙⡛⠛⠛⢻⡽⠟⡏⠉⣀⣴⠞⢳⡄⠹⣇⢻⣷⢦⣤⣉⣭⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠛⠛⠉⢀⠀⠀⠹⣆⢹⡆⠙⠦⣄⣽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣦⠀⠀⠀⠟⠀⡖⣳⠀⠀⠙⠃⠀⠀⢻⡌⣧⣤⣀⠀⣀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠟⠀⠀⠀⢈⡀⠈⠀⠀⠀⣀⣤⠶⠖⠾⣇⢹⡀⠿⣤⠟⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠋⡉⠁⠀⠀⠀⠀⠀⠀⢠⡟⢹⠦⠄⢠⣾⡏⡔⢺⢩⣛⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⣞⡩⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠀⠀⣾⡏⣇⢻⣄⣸⢉⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣀⣴⡿⠛⣻⡆⠀⣠⡶⣦⠀⠀⣾⠻⣇⢰⣿⡇⣿⢠⣉⠁⣼⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠻⠿⠟⠁⢀⣼⠟⠀⢰⡏⢀⣿⠀⠀⣿⡆⠙⠋⠈⣷⡸⣾⣸⡿⠋⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠒⢦⠀⠀⣴⡿⢋⢤⠀⣾⠀⢸⡿⠀⠀⣿⠁⠀⠁⠀⠘⢿⣶⣥⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠁⠀⣼⡟⡠⢌⠁⠀⣿⠀⢸⣷⠀⢠⡿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠁⢰⠃⢸⡟⠀⠈⠻⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡑⣁⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⡘⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⠇⠈⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⢿⣿⡏⠀⠀⠀⠀⠀⢰⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣾⣿⣧⠀⠀⠀⠀⠀⠘⠉⣿⡄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣿⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⡀⠀⣸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢨⣷⣿⣿⣿⣿⣿⠿⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠄⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠐⠀⣴⠂⠀⠀⠀⠈⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠡⠄⠈⠀⠀⠀⠈⠀⠀⣌⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠂⢄⣀⣨⣂⠤⠉⠛⢋⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠂⢄⡀⠔⠁⠠⢂⠡⠐⠊⠈⢂⠁⠓⢄⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠘⠀⢤⡊⠐⠁⠀⠸⢤⠀⢠⠀⠀⠀⠆⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠐⢤⡈⠓⠒⠠⠀⠀⠰⢢⠀⠀⠀⠈⡀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⡈⠢⠀⣀⢁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠐⠄⠓⠛⠅⣐⠂⠠⠄⠤⠄⠤⠊⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠜⠒⠠⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⣉⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡰⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡐⡐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠉⠉⠉⠉⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠆⣀⣀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠁⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡄⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣈⠖⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⡵⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡃⢠
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡔⠒⠦⠀⠀⡀⠀⠠⠤⠐⠈⠙⠒⠒⠒⠂⠈⠛⠤⢀⡀⠀⠀⠀⣀⠠⠔⠊⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢱⠈
⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠞
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⡇⠀⠀⠀⢠⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡓⢀⣈⠳⠀⠀⠀⠄
⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢀⠤⠄⣀⡀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢣⠀⡘
⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠀⢀⠎⠄⠁⠂
⠀⠀⠀⠈⠑⢤⠀⠀⠀⠀⠀⠀⠎⠀⢀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⢣⠀⠀⠀⠀⠈⡕⠂⠲⠎⣌⠀⡠⠂
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢀⡌⠀⠠⠊⣀⠘⠶⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⠰⠂⠈⠑⠢⡀⠀⠀⢀⠟⠁⢸
⠀⠀⠀⠀⢠⣜⣁⣀⠠⠔⠈⡇⠰⠀⠜⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⠀⠀⠀⠀⠐⣤⠂⠹⠀⠀⠸
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⣏⠒⣂⠀⠤⠀⢀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠘⡀⠀⢢⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠀⠀⠀⠀⠀⠰⠀⠑⠤⠉⠀⠒⠂⠐⢀⡽⠋⠑⣤⠀⠀⠀⠀⠀⠀⢃⠀⡰⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⡐⠂⠀⠤⢤⠌⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡎⠀⠀⢲⠀⣀⠀⠈⣴⠀⠠⠃
⠀⠀⠀⠀⠀⡖⠉⠀⠀⠀⠀⡰⢩⠀⠀⠀⠈⠉⢐⠉⠀⠀⠘⠀⢣⠀⠀⠀⠀⠀⡆⠀⢀⡰⠧⠠⠀⠘⠀⠿⡀⢀⠷⡀
⠀⢀⡴⠀⠀⢹⡄⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠥⡸⠋⠀⠀⠀⢀⠁⠀⠘⢃⠈⠀⠈⠷
⠀⠸⡄⠀⢰⡋⠀⠀⢀⠊⠀⠀⡇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⡤⠀⢀⡀⢀⠃⠀⢀⡄⠀⢀⠀⠀⠀⢘
⠀⠀⢸⠀⠀⠀⠀⢀⠂⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⢻⠀⠀⡌⢠⠀⠘⢄⠀⣠⠌⢘⢣
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⣄⠀⠀⠀⠀⡈⠀⠀⡄⠈⡆⠀⠀⠀⠀⠀⡌⢀
⠀⠀⠀⠀⠀⠀⠘⠓⠲⢄⡀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠈⠢⢄⡀⠀⠁⠀⠐⠚⡄⠀⠀⠑⠘⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠒⠐⠂⠈⠁⠀⠛⢦⣄⠀⠀⠀⠀⠀⠑⠆⢀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠀⠀⠀⠀⠈⠀⠚⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠰⠇⠀⠀⠀⠀⠀⠱⡄⠀⢀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢢⠴⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘⢆⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠢⢀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡌⠀⠀⣨⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⡀⠀⠈⠐⠒⠂⠠⠤⠴⠛⠂⠤⠄⢀⣀⣀⠐⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠁⠀⢀⡀⠀⠀⢠⠶⢀⡀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠏⠁⠀⠀⠀⠁⠐⢢⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢄⠀⠀⠀⠀⠀⢠⡾⡀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠈⠁⠐⠒⠈⠉⣀⠈⠒⠠⠤⠤⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⢀⣀⠉⠐⠢⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠉⠁⠑⢦⡈⠹⡕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠈⣿⡆⠀⢸⡏⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠐⠲⢤⠠⠲⢤⠠⠞⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣤⠤⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡠⠤⠀⠒⠀⠀⠀⠁⠒⠒⠒⠒⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡏⠒⠶⠤⠤⠤⠤⠤⠄⠀⠀⠀⠒⡈⣇⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠉⠑⠲⠶⠶⠶⠶⠂⠁⠀⡸⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⢳⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⠤⠒⠉⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣨⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢁⣄⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠢⣤⣀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠼⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡿⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⡠⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢐⠄⠀⠀⠀⠀⢸⠈⠳⢄⡀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢫⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⠀⢠⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠠⡇⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠷⠂⠀⠘⠀⠀⠀⠀⠀⠘⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢀⠃⠀⠀⢀⢤⡄⢤⡒⠚⠀⠈⡆⠈
⠀⠀⠀⠀⠀⡎⠐⠐⠒⢀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢇⠀⣸⠀⠀⠀⠀⠀⢧⡀⠀⡘⠀⠀⠀⢠⡀⠊⠄⣛⠡⠖⣂⢀⠀⠀⠢⡀
⠀⠀⠀⢠⠰⠀⠀⠀⠀⠀⠑⠦⢤⣀⡈⠀⠃⢠⡄⠘⢀⡀⠤⠤⣼⡔⠀⠀⠀⠀⠀⠀⠀⠙⡀⣧⠀⠐⠒⠢⠉⠒⠉⠀⠀⠠⡅⢂⡀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠈⠀⠀⠀⠀⣈⠁⠒⠂⠒⠒⢀⣤⣈⣀⣈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠚⠛⠋⠁⠀⢠⣴⣄⣍⣥⣤⠄⠦⠝⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡉⠛⠛⠛⠛⠛⠻⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠌⠛⠛⠋⠙⠛⠛⠳⢶⣦⡀⠀⠀⠀⠙⢿⣟⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠙⣷⡄⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠈⣿⡄⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣿⠃⠀⠀⠀⠀⠰⣶⣶⡶⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠙⠷⠶⠶⠶⠖⠀⠀⠀⠀⠀⠈⠋⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠰⠿⠿⠿⣷⣄⠀⠀⣠⡶⠿⣶⣤⡀⠀⣠⡾⣿⠋⠛⠀⠀⢀⣼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠈⠙⠿⠟⠉⠀⠀⠈⠙⠻⠟⠉⠀⠛⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠷⣶⣶⣶⣶⣦⣴⣶⣶⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡷⣦⣀
⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠋⠀⠀⠀⠈⠛⠶⣆⠀⠀⠀⠄⠀⠤⠤⠄⢤⣤⠤⠤⠀⠄⣤⡀
⠀⠀⠀⢸⡏⠀⠀⠀⣀⣀⣀⠀⠀⠒⠶⠿⠷⠾⠷⠶⠶⠶⠿⠷⠶⣶⡶⢾⡣⠤⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠹⠆
⠀⠀⣠⣤⣥⠶⠶⢆⡈⠉⠉⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⢾⠋⠉⠉⠉⠉⠐⠛⢶⡴⠿⣧⡀
⠀⠀⠉⠉⢻⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠀⠀⢀⣄⣴⡾⠁⠀⢸⡇
⠀⢺⣀⠀⠘⣏⠉⠻⢿⣿⠷⠂⢀⣤⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣀⣤⠄⠤⠐⠛⠛⠷⠀⠀⢈⡇⠀⠀⣸⠇
⠀⠈⠻⣦⣾⣿⣄⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠛⡄⠘⠇⢸⣿⣦⠞⠃
⠀⠀⠀⠀⢸⣿⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⣿⠃⠀⣤⡄
⠀⠀⠀⠀⠀⢹⡆⠀⠙⠻⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠉⢀⣷⠈⠁⣼⡇
⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠉⠻⢶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣠⠶⠊⠁⠀⡀⢸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢰⣿⢹⡆⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣶⣾⣿⣿⣷⣾⣿⣿⣉⣿⠟⠉⠁⠀⠀⠀⠀⣧⢠⡿⠀⣸⡿
⠀⠀⠀⠀⢸⣧⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠀⠀⢸⡧
⠀⠀⠀⠀⢸⣿⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠃⠀⠠⡾⠃
⣿⠟⢷⣦⣤⣉⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢠⣷⠀⠀⠀⠀⢀⣀⣤⣤⠴
⠘⣷⠀⠀⠙⠛⣷⡦⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⣿⠀⠀⠸⣏⣀⣴⣶⠞⠛⠋⠀⠀⣀
⠀⢻⣆⠀⠀⠸⣇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣇⣠⡶⢆⠙⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠸⣿⠀⠀⠸⠟⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⡁
⠀⠀⢻⣇⠀⢠⣶⠚⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠒⠲⣦⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠸⣿⣴⡿⠛⠶⠖⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⠶⠞⠛⢿⡆⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⢶⣤⠚⢢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣤⡸⠃⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠙⠻⢧⣤⣼⠟⠛⢂⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠊⠀⠀⠀⡄⢸⡇⠀⠀⠀⠀⢀⡟⠁
⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠯⠥⣤⠤⠤⠶⠶⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⢸⣿⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠟⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠈⣿⡄⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠈⠻⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠙⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⣈⣿⡀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣼⣃⣀⢠⣴⣶⣤⡶⠶⠶⠤⣶⣄
⣀⡴⠖⠛⠛⠉⠁⠈⠉⠙⣻⣿⡿⢶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⢿⣿⣉⣀⣀⣀⣀⣀⣀⣀⣤⡴⠋
⣹⣷⣶⣶⣶⣶⣶⣶⣦⣍⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠁⠈⠉⠈⠉⡈⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⢀⣴⡾⣿⡟⢿⣿⣿⣿⣶⣤⡀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⢻⣧⠘⣿⡄⠹⣷⡝⢿⣿⠿⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣿⣧⠘⣿⡄⠘⣿⣆⠻⣷⡌⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣇⠹⣿⡀⠘⣿⡆⠙⣿⣄⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⢻⣧⠀⠘⣿⡄⠘⣿⡄⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⡄⠀⢹⣧⠀⠸⣿⡄⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⢻⣇⠀⠈⣿⡄⠀⢻⣧⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠛⠀⠀⠛⠃⠀⠈⠛⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣖⡀⣀⣄⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢀⡀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣼⡿⢀⡇⠀⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣟⣿⡇⠈⠀⠀⣿⣿⣿⣻⣿⡆
⠀⠀⠀⠀⣸⡄⠀⠸⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⢈⣿⣧⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⡟⢢⢀⠀⢀⣠⣤⣠⣀⠀⡤⠘⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠄⢠⣿⣿⣿⣿⣿⣿⢿⡇
⠀⠀⢀⣸⣷⣾⣸⣦⣾⣿⣿⣿⣿⡄⡀⠀⠀⠀⢠⣄⠀⠀⠐⠂⠤⢀⢸⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠰⣚⣛⣙⣿⣻⣿⠿⢿⣥⣌⣉⠹⣿⣟⢲⣀⣀⠠⠙⠿⣦⡄⠀⠀⠀⠸⣿⣿⣧⣣⡿⠿⠛⠛⠛⠛⠿⣿⣿⣧
⠀⣿⠉⡭⠩⣿⣿⣴⣾⠿⢿⣿⣿⣿⣿⣿⠟⠻⢶⡗⢦⣼⣿⡆⠀⠀⠀⢹⣿⣿⣿⠀⣠⣤⠆⠲⣴⣤⣄⣘⣻
⢨⣿⠀⠃⠀⠈⠙⠋⢁⡀⠀⣸⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄⠀⠀⢸⣿⣿⣿⣾⠋⠀⠀⢰⣿⣿⢿⡿⠿
⢸⣿⠀⠒⠀⢀⡒⠀⠒⠀⠀⢻⠒⠒⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣇⠀⢀⠄⣾⣿⣿⣾⣿⣿
⢸⣿⠀⠀⠀⠘⠂⠀⠀⠀⠀⣾⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢸⣿⣿⣿⢻⣄⡌⠀⣿⣿⣿⣿⣿⣿
⡿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡂⠂⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⠀⠀⢸⣿⣿⣿⣿⣿⣳⣠⣿⣿⣿⣿⣿⣿
⠘⡿⠀⠁⠀⢠⠄⠀⠀⠀⠀⢼⠅⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠀⠀⠺⣿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿
⢸⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⢿⡃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠄⠀⠸⣿⣿⣿⣷⣶⣶⣶⣿⣩⣭⣽⣿⣿
⢸⣿⠀⢀⣀⣁⣀⣀⣀⣀⣠⣿⣍⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣾⣿⣿
⠸⣿⣀⣻⣿⣿⣿⣿⣿⣿⣟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣯⣿⣿⣟
⠀⠉⠛⠛⠻⠿⠿⠿⠿⣿⠯⢤⣤⣼⣿⣿⣿⣿⣿⣛⣯⣭⣿⣿⣿⣿⣶⣿⣿⡿⢿⣿⣿⣿⡿⢿⠿⣯⣿⡉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣬⣭⣭⣿⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⣤⣤⣴⣶⣆⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠉⢿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣄⠉⠉⠀⠀⠛⠛⠛⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇
⠀⠀⢀⠄⠀⠐⢸⣿⣿⣿⣿⡌⠀⠀⠀⠄⠀⠠⣀⠀⠀⢀⣀⣀⣤⣿⣿⣿⣿⣿⡆
⠄⠈⠙⡻⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣵⣿⠿⢛⠫⠁⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠋⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡿⠟⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠾⠛⠛⠛⠛⠻⠷⠶⣤⣄⡀
⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠉⠻⣦⡀
⠀⠀⢀⡾⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡇⠀⠀⠀⠀⠈⢿⡄
⠀⢰⡟⠀⠀⠈⠑⠃⠀⠀⠀⠀⠀⠀⠀⠙⠊⠠⡒⡂⠀⠀⠀⠈⢿⡄
⢀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠺⣧
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣇⣀⡠⠄⠀⢠⠊⠉⠐⡄⠀⠁⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣿⠄
⣿⠉⠀⠀⠀⠀⠘⢄⣀⡠⠃⠀⠀⠠⡂⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣀⣀⡀⠀⠀⠀⠀⠀⢠⠶⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⢀⣿⡀
⢹⡏⠉⠀⠀⠀⠀⠀⠀⠁⢀⡤⠒⠢⢄⠀⠀⠀⢀⠒⠦⠈⠁⠀⢸⡟⠙⠄
⢸⣇⠀⠀⠀⠀⠀⠀⠀⣰⢋⡀⠤⠀⠀⢧⠀⠀⠀⠲⠊⠀⠀⠀⣾⢇
⠀⣿⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣸⡟⢺
⠀⢹⡇⠀⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠃⢀⣸⣙⡏⠀⢀⡿⠀⢰⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⠖⠒⣄
⠀⠈⣿⠀⢹⡀⠀⠀⡀⢧⠀⠀⠀⠀⢶⡞⠀⠀⠁⠈⠁⠀⣼⣧⣄⠎⠀⠀⠀⠰⠰⠂⢰⠇⢹⡄⢀⣞⣡⡤⠖⠃
⠀⠀⢹⣇⣸⡧⠤⠀⠹⣌⠳⢤⣤⣴⠞⠀⠀⠀⠀⠀⠀⢀⣿⢼⠋⠀⠀⠀⠀⢀⣀⠴⠋⠀⣸⣁⡬⠥⠖⠒⠲⣄⠀⠰⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢸⣧⢤⡷⠒⠀⠀⠈⠛⠓⠛⣡⣴⠶⠿⠿⣿⠿⠿⠿⠿⠷⠶⠶⠶⠒⠚⠉⠀⠀⠀⠚⠋⠁⠀⠰⢀⣀⣤⣎⣠⣤⡤⠥⢤⡀⠀⠴⠁⠐⠁
⠀⠀⣸⡇⠰⠇⡎⢑⠀⠀⠀⠀⣼⣿⣿⣒⠒⠚⠴⠃⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠲⠔⠒⠋⠉⠉⠀⠀⣤⣤⣤⡖⠚⢁⣤⠴⢖⠲⢦
⠀⢀⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⡯⠭⠭⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⠆⠈⠉⠓⠲⠧⠴
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠭⠥⠤⢤⠀⠀⠀⠀⢄⠀⠘⠂⠀⠀⠀⠤⠤⠄⢀⣀⠀⠀⠀⢀⠀⠉⠉⠙⠛⠛⠷⠤⣤⣀
⢀⡟⠀⠀⠀⠀⣀⠤⢄⠀⠀⠀⢿⣷⣖⣒⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠠⣠⡄⠀⠀⠀⠈⠉⠐⠒⠠⣤⠤⢠⣤⣤⣀⣀⣤⣬⠗⠀⢪⡵
⣼⡇⠀⠀⠀⠰⡁⠀⠀⡇⢀⠤⣸⣿⣄⣤⣶⡾⠿⠛⠛⠻⠿⢿⣿⣍⠉⠙⠛⠓⠦⢤⣀⠀⠐⠲⣦⠤⠤⠼⠂⠀⠀⠀⠀⠀⠀⠀⠐
⣿⠀⠀⠀⠀⠀⠙⠂⠊⠀⠰⠤⠃⠻⠿⠟⠁⠀⠀⠀⠀⠐⠲⠦⢬⣻⡆⠀⠀⠀⠀⠀⠈⠙⠲⣄⣸⠇⠀⠀⠀⠙⠯⣉⠉⢣
⣿⠀⠀⠀⠀⠀⠀⠀⠔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⠀⠀⢸⣇⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⢻⡆⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠀⠀⠀⠀⢀⠖⠁⢦⠀⠁⠀⠀⠀⣿⠀⠀⠀⠈⠁
⠘⣷⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⡄⠀⠀⠀⠀⢰⡟
⠀⠙⣷⡤⠼⢷⡖⠒⠒⠂⠈⠀⠀⢢⡴⢮⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠈⠻⣦⣤⠛⠶⣆⡀⠀⠀⠀⠀⠟⠋⠑⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⠶⠟⠉
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢇⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠋⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢫⢿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠃⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠖⠒⠈⠉⠉⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⢲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠀⠰
⠀⠀⠀⠀⠀⠀⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠁⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢰⠔⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆⣀⣀⡀⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠸⡄⣤⠘⣷⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⠀⢀⠊⠉⠙⠒⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠱⣍⠀⠀⠈⠀⡀⠔⠊⠁⡠⠈⡠⠃⡐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⡀⠀⠀⠀⡘⠀⠈⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⢀⣈⠤⠊⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⡁⠒⢦⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇
⠀⠠⢐⡲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡇
⢈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠬⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣭⣭⣥⣶⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠓⠒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣤⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⢠⠃⠉⠙⠲⣆⢸⠀⢣⠀⠑⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⠘⢶⠋⠋⠀⠠⢴⠂⠑⠾⠀⠈⣧⠀⠸⣾⠀⠀⠀⣠⠂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⡄⠈⢧⠳⣄⡠⠴⠦⡐⢠⠀⠀⢸⡀⠀⢹⡇⢀⣼⠇⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣼⠀⠀⠂⠈⠙⠛⠋⠁⡏⠀⠀⢸⠆⠀⠀⣷⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢀⡀⠀⣰⢃⠀⠀⠈⠀⢀⠀⣹⣿⣿⣿⣥⣤⡤⠖⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠆⠀⠀⠉⠉⢁⠮⠒⡉⠀⠀⠒⡜⢷⣿⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡀⢀⣀⣀⡤⠄⠁⠀⢈⣠⠀⠀⠀⢰⣾⣿⣿⣉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢠⣐⠒⡆⠀⠀⠀⢿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠂⠄⣀⢸⣜⡄⠀⠀⠀⠸⣿⣿⣆⠀⠀⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠲⠲⠒⠚⠉⠁⠀⢀⡤⣤⡟⠀⠹⠀⠀⠀⠀⠀⢇⠉⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⡐⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠉⠈⠉⠁⠀⠀⢀⣠⠤⠀⢀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠁⠀⠀⠀⠈⢆⠀⠀⠀⡇⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡀⠠⢤⣤⡀⢀⡟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢋⠀⠀⠀⢸⢠⠫⢠⠆⠀⠀⠀⠀⠀⡆⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠄⠜⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⣾⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠒⠒⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠲⠴⠶⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢠⢿⡃⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢿⡠⠒⠋⣉⠍⠉⠈⢉⠓⢤⣀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢩⣄⠀⠀⠀⠈⠳⢤⡈⠓⢦⡀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⠲⡄⠀⠀⠀⠀⠀⡏⠀⠀⠉⣀⡀⠀⠀⠀⠙⠏⠀⠀⠀⠳⡄⢀⡾⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⠀⠀⠀⠀⠀⢧⠀⡰⠋⠈⠉⠉⠉⠉⠳⣄⠀⠀⠀⠀⠙⠍⡷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠈⢧⡁⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣀⣤⣠⡼⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠤⣼⠖⡆⠀⠀⠀⠀⠀⠉⠓⠦⡤⣤⣀⣀⣤⡥⠴⠚⠁⢀⣅⡀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⣘⡄⢧⠀⠀⠀⣠⢲⠎⠉⡟⠣⠤⠴⢳⠒⠦⣄⣀⠞⠁⢰⠓⠤⠦⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠢⣤⢯⠟⠈⢦⠀⡼⢁⡞⠀⠀⠓⠶⠦⠤⠾⠀⢠⢇⡏⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠓⠚⠈⠀⠀⢾⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠴⠲⣤⠞⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣷⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠒⠚⠛⠛⠛⠛⠛⠛⠓⠛⠧⠼⠓⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢒⣚⠚⢲⠖⠚⠒⠛⠓⣶⢲⣶⠒⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠎⠥⠬⠂⠸⠦⣤⣄⣠⡤⠟⠒⠚⠁⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠒⠚⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣠⣤⠤⠠⠄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠋⡍⠓⢦⡀⠀⠊⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠘⣆⠀⢀⠤⠀⠀⠘⠛⠲⢦⡄⠀⠀⠀⠀⠀⢿⡄⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠻⠀⠀⠀⣿⠴⠁⠀⠀⠀⠀⠘⠛⠻⠛⠀⠀⠀⠀⠀⠘⢷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢀⣀⠀⠀⠀⠹⢦⣠⡶⠤⢄⣸⣧⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢽⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣧⡀⠀⠉⠛⠛⠶⠾⠛⠁⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠠⣬⣁⣠⠄⠀⠀⠀⠀⠐⠀⠄⠀⠀⠇
⠀⠀⠀⠀⠀⣤⡶⠒⠛⠋⠉⠁⠈⢿⣆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⣧⡀⢀⣀⣄⣀⣠⣼⠟⠀⠀⠠⠰⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠞⠛⠉⠀⠉⠻⠶⣤⠶⠶⠚⠛⠁⣀⠀⠀⣩⣽⠟⠋⠉⠻⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⢀⣠⠔⠀⣀⣀⠀⢀⡀⠀⠀⠀⠀⢀⣠⡀⣀⣴⠞⢋⣠⣾⡿⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠈⠉⠛⣛⡿⠋⠀⠸⣿⣿⠶⣤⣤⣿⡿⣿⣭⡤⠖⠋⠁⠹⣧⠀⠀⠀⠀⠈⠈⠧⠀⣤⣀⠀⠀⠀⠸⠿⣧⡀
⠀⠀⠀⠀⠀⣠⡞⠁⣀⣤⠶⣻⠋⠣⣤⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣄⡙⠛⠒⠒⠀⠀⠀⣀⣠⣼⣿⣄⠐⠻⣦⣤⣀⡙⣧⡀
⠀⠀⠀⠀⣴⣿⠿⠛⠉⢀⣾⢁⣤⠞⠉⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠾⣿⠿⠛⢻⡍⠈⠙⠻⠶⠾⠟⠉⠉⠙⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠿⠛⠁⠀⠀⠸⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⠗⠀⠀⢙⡇⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⡿⠃⠀⠴⠟⠀⠹⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠰⣦⠀⣧⠀⡼⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡿⠀⢈⣤⣤⣄⠀⠰⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠀⢰⣏⣤⡴⢿⣷⠀⢠⡞⠳⡆
⠀⠀⠀⠀⠀⢠⡴⠿⠿⢦⡄⠀⠀⠀⠈⠉⠳⣤⣤⠟⠀⠈⠓⠾⠃
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢀⡾⢻⡆⠀⠀⠀⡴⠛⠛⢦⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢀⡾⢡⡟⠀⠀⠀⠈⢧⡴⠶⣆⠻⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⡾⢡⡟⠀⠀⠀⠀⣠⡞⠁⣠⠏⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢸⣇⣀⣀⣸⣿⣡⣟⣀⣀⣀⣀⡴⢋⣤⣞⠻⣦⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡯⠭⠭⠭⠽⠿⠯⠾⠿⢭⡽⠗⠈⢷⡤⠶⠛⢹⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⡿⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⣸⠃⠀⠀⣰⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡗⠒⢲⡶⠚⠛⠉⠀⠀⠀⠀⢠⡏⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠛⠛⠛⠒⠶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠾⠷⠶⠶⠾⠷⠆⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣀⣀⣀⣠⣤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠋⠘⣢⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣯⢀⠠⡞⠁⠀⠀⠀⢠⢿⠀⠀⠀⢰⡥⠲⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠋⠓⠿⠀⠀⠀⢻⢉⢀⡭⠶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠼⠊⠻⠃⠀⠐⠀⠰⠀⢀⡾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠧⣤⡄⠐⠂⠀⢀⡤⠶⣛⣉⠉⠙⠛⠓⣦⡀⠀⠘⢻⢉⡀⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠶⣾⠁⢀⠀⠀⢠⠾⢤⣀⡀⠈⣉⡷⠖⠋⢡⣽⣆⡀⠾⠋⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⢠⣧⣄⣀⣨⠟⠋⢁⣠⠴⠚⠉⠐⠄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣠⠞⢉⣀⣴⣚⣉⣀⣠⣤⣤⣤⣭⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡀⡄⠄⣂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢇⣂⣀⣀⣀⣀⣠⣤⡤⠤⠶⠒⠛⠉⢩⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢲⡄⠉⠀⣠⣤⠆⠀⠀⠀⠐⣶⢦⡀⠀⠀⣩⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠛⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠈⠆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢲⠀⠀⠀⠀⠀⠐⠋⠓⠀⠀⠀⣴⠅⢀⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⠉⠀⠀⠀⠈⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠗⣲⠦⡤⢤⡴⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣴⢦⡜⠃⠈⠡⠞⡡⢋⠜⣡⣟⢲⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠻⣀⠌⢧⡙⢦⣀⠀⠈⠑⠋⠘⣁⡵⢋⡴⣣⠟⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠈⠻⣤⡛⢲⠮⣭⣙⣋⣉⣩⠤⣶⣫⠟⠁⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠄⢀⠀⠀⠉⠙⠒⠓⠚⠷⠞⠓⠚⠉⠀⠀⠀⢀⠄⢠⠄⢀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣡⠔⢁⠔⢠⠆⢀⡤⢀⡔⢀⡴⠀⡠⠃⡴⠃⡴⠁⡴⠃⡴⢋⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⢁⡔⢁⠔⠁⡠⠋⠠⠋⡴⠋⣠⠎⣠⠞⢠⠞⢠⠞⢀⠞⣠⢋⡤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠈⠀⠁⠈⠐⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣁⠀⠀⠀⠀⠉⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠆⡠⢺⢃⠞⢧⠀⢠⢿⡀⠀⠘⣎⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣏⡞⠁⠈⠋⣀⠌⢧⡞⡀⠱⣜⣆⢹⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢱⠋⣡⠤⢤⡀⠀⢠⢚⣿⠀⠀⠐⢦⡀⠀⠀⠀⠈⠃⠈⠙⢿⠀⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠞⠁⠀⠀⡙⢆⠹⣌⡇⠀⠀⠘⠉⠁⡄⠀⠀⢾⠇⠀⠀⢸⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠓⢦⡀⢘⣇⣿⡷⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢀⢰⡘⣆⣇⢷⡎⠀⠉⠛⣛⣧⡀⠀⠀⠀⠋⠲⠀⠀⢀⣇⠀⣸⢋⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⠮⢷⣿⣭⡡⠏⠹⡄⣰⣻⡻⠙⠝⡗⠦⠀⠀⠀⠀⠀⠈⠋⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣑⠂⠀⠀⠀⠀⠀⠘⡟⠒⣍⠀⠀⢸⡀⠀⠀⠀⢯⣉⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠟⡄⠀⠀⢿⠒⠲⠒⢊⡏⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠈⢳⡀⠈⠓⠢⢄⣸⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠙⢦⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣖⣒⣲⠶⠴⠤⢤⣤⣤⣤⣤⣤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⠿⠬⠽⠭⣭⠯⣭⠽⠭⠭⠯⠭⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠚⠒⠚⠒⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠁⠀⠀⠀⠈⠉⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣀⠀⠀⣰⡆⢠⡄⢀⠀⢀⣶⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼⡇⣸⣇⡰⠻⣤⠞⠀⠻⠛⣧⠻⣤⠏⢸⢀⣴⠂⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⣰⠁⠻⠁⠈⠁⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠘⠏⢸⣴⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢸⠃⠀⠀⠀⠘⣇⣀⡤⠆⠀⠀⠀⠈⢰⠃⢀⡷⠒⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢿⠀⠀⠀⠲⠶⠋⠀⠀⠀⠀⠀⠈⠁⣠⣄⣠⡄⠀⠀⢸⡴⢟⣾⣏⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠞⠀⣰⣶⣦⣤⣀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠖⠒⠀⠀⠈⠀⠈⢱⡃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠚⠛⠛⠛⠛⣣⠄⠀⠀⠺⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠖⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡤⠶⠚⠛⠉⠉⠉⠉⠉⠙⠛⠶⣄⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣸⡀⢀⣤⠴⠚⣶⡤⠶⠲⠶⢦⣄⡀⠀⡼⠁⢀⣤⠞⠁⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⠋⠁⣠⠞⠉⠀⠀⣀⣤⣤⣬⣿⡾⠗⠚⢩⡟⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣠⠞⠁⠀⢀⣴⡿⢯⣿⣉⣿⠿⣆⠀⣠⣿⠃⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠈⠁⠀⢀⡴⣿⠛⠀⠀⣽⠋⠁⠀⠈⠟⢁⡏⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⢀⣠⣾⠁⠃⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⡦⠤⠤⠤⢤⡤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠙⠉⢠⡇⢸⡇⠀⠀⡎⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡶⠦⣤⣾⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡏⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣸
⠀⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠼⠧⠤⣤⣼⣧⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀⠒⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠭⣍⠁⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠉⣉⣠⠤⠴⠤⠴⠏⠀⢠⣭⡉⠓⠉⣽⡦⠴⠒⢶⣄⣀⣀⣉⣒⣃⣀
⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⠀⣀⡤⣄⠀⠀⠀⠀⣤⣴⠀⠀⠈⠉⢉⣤⡀⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠉⠉⠛⢦⡀⠀⠀⠙⠒⠃⠀⢠⣄⣀⠈⠉⠀⠀⠀⠀⠚⠒⠃⢠⣖⢋⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠶⠦⣄⡀⠀⠈⠉⢉⣀⣀⠀⢀⡴⠒⠒⠶⠴⠚⠉⠈⠹⢤⡬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠉⠉⠉⠉⢩⣗⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀⠀⡀⠀⢀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠯⠛⠭⠻⠽⠀⠛⠹⠼⠛⠽⠺⠼⠛⠻⠿⠛⠟⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡯⠝⠭⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡞⡎⠀⣞⠆⢸⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⣸⠽⡷⢧⠀⠁⢀⢞⢾⣢⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢰⣱⠋⠉⢳⣱⠀⡞⣎⢴⡽⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡴⠲⢜⢞⡢⠤⣊⠞⠺⠝⠉⠉⢀⠤⠤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⣀⣼⣀⠈⠉⠀⠀⠀⠀⠀⢠⢀⣀⠴⢤⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⡀⠀⢀⠱⡄⠀⠀⠀⠀⠀⡞⢸⠖⡖⢺⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢼⡀⢾⡹⠀⣸⠆⢹⠀⠀⠀⠀⢀⡽⢉⠾⠽⢎⠙⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣍⡆⠃⢞⣡⠒⢣⡀⠀⠀⢠⠊⣰⠋⠀⠀⠀⢳⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠿⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠁⢀⠀⠀⠙⠻⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠋⠛⠛⠛⠿⠿⠦⣠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⠶⠿⠷⠶⠶⠂⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣉⣽⣿⣿⠀⠀⠀⢴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⡿⠃⠀⢠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣿⣿⣿⣿⣶⣦⠀⢸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⠉⠈⠁⣠⣿⣿⣿⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣿⡿⢿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡇⠈⠁⠙⣿⣿⣿⡿⠛⠛⣿⣿⡿⣿⣿⣿⣿⣿⠿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠿⠿⣿⠛⠙⠿⣿⡿⣿⡗⠀⠀⠀⠘⣿⣿⡃⠀⢀⣿⡿⠀⢹⠊⣿⣿⣇⠀⠀⠙⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠸⣿⣷⣤⣾⣿⡇⠀⠸⡄⠸⣿⣿⡄⠀⠀⠘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠹⣿⠙⢿⣿⡇⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⢀⣴⣶⣶⣶⣾⣶⣿⣿⣿⣿⠋⠀⠀⠀⢰⣾⣿⣆⣀⡀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠸⢷⣈⣿⣧⡀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⣼⣿⣿⣿⡏⠉⢻⣿⣿⡟⠁⠀⠀⠀⢀⣿⣿⡿⢿⣿⣧⡀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠁⠈⢿⣿⣄⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠈⢿⣿⣿⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣿⣿⡇⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⢻⣿⣆⣀⠀⢀⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡟⢿⣿⠇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠉⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⣸⣿⣿⣿⡿⠿⣿⣿⡿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⠀⠘⠇⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⣾⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⣼⣿⣿⣿⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⡆⢸⣿⣿⣿⠈⣿⣿⠀⠀⠀⣴⣿⢿⣿⣿⣿⠿⣿⣿⡄⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠰⡀⠘⢿⣧⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⢀⣿⣿⣿⣿⣿⠀⣿⣿⣤⣤⣼⣿⣧⠀⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢿⣷⡀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⡇⠀⣹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠟⢹⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⡧⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠁⠈⢿⣿⣿⡟⠀⣼⣿⣿⣿⣿⣿⣿⣧⣀⣿⣿⣧⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⢸⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣶⠀⢀⣰⣿⢓⣦⡀⢀⣼⡅⠀⠀⣰⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣤⣤⣴⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⠿⠟⠛⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⠇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢠⣿⣿⡏⠀⢸⣿⣿⣿⠿⠋⣤⡈⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⠉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣸⣿⣿⠃⠀⣾⣿⣿⡇⠀⣾⡿⠁⠀⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣷⣤⣉⠉⠉⣠⡘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣷⣾⣿⣿⣿⠀⠀⣿⣿⣿⣄⣰⣿⡇⠀⣾⡏⢻⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡿⠀⣾⡟⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠋⠈⣿⣿⠀⠀⣿⣿⣿⣿⣿⠏⠀⠀⢿⣷⣾⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⠃⠀⠛⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⢠⣿⣿⠀⢈⣿⣿⣿⣿⠁⣠⠄⠀⢸⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡄⠀⣤⠀⠉⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⢠⣾⣿⡏⠀⢸⣿⣿⣿⣿⣏⠁⢀⡀⠸⢿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣧⡀⠙⢠⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⢸⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠆⠘⣷⣄⢸⣿⣿⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠛⠃⢸⣿⣿⠇⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⠇⢀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢻⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⣿⡏⠀⠀⢠⣿⣿⣿⣿⠏⢀⠛⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⡇⠀⠀⢸⣿⣿⣿⠃⣠⣿⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⡆⠀⢀⠀⠀⣀⣀⣤⣤⣀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣿⣿⡇⠀⠀⢸⣿⣿⣿⣾⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠈⠉⠉⠉⠉⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⢸⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣿⡿⠁⠀⣀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠠⠶⠶⢶⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣀⣿⠆⢰⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠠⣶⡦⠤⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠁⠀⠀⠀⣸⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢀⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡿⠉⠻⠂⠀⠀⢠⣴⣶⣦⣤⣤⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⢸⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⣿⣿⣧⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⡇⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠃⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣄⣤⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣧⡄⠀⠀⠐⠀⠀⠀⠀⠀⣠⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⣨⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠃⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⢾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠃⠠⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⠃⠘⠁⠀⠀⠀⠀⢀⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣼⣿⣿⠀⠀⠀⠀⠀⠀⠘⠛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡁⠀⠀⠀⠀⠀⢠⠄⠀⠘⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣇
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠛⠁⣀⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⣤⡀
⠀⢀⣴⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠟⠻⢿⣿⣿⣷⣄⡀
⠀⣾⣿⣿⣅⣠⣦⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣄
⢀⣿⣿⣿⣿⣿⠿⢿⣿⠿⢿⣷⣀⣀⣤⣤⣶⣤⣴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⢲⣶⣶⣶⣶⣾⣷⣾⣿⣶⣽⣿
⣿⣿⣯⣿⣄⣼⣶⣾⣷⣤⣤⣿⣿⡿⠿⠿⠟⠛⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣤⣀⣀⣀⣀⣴⣶⣦⣴⣾⣿⣧⣽⣭⣠⣼⡋⢙⣿⣿⣿
⠉⠙⠛⣛⠛⣿⣿⣟⠛⣻⣿⠋⢁⣠⣶⣦⣀⣀⠈⠉⠉⣀⣀⣀⡀⢀⣀⣀⡀⢀⣀⠀⣀⣀⣈⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⣿⡟⢿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⢿⣿⠛⢿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡆⠈⢁⣄⠀⠸⣿⡀⠀⣰⣯⠀⠀⢠⣶⡄⠙⠀⣴⣶⡿⠃⣴⣆⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⣿⣿⢿⢿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⣶⣤⣤⣄⣠⣤⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⣄⣠⣤⣤⣦⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣦⣤⣄⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠈⠛⠋⠉⠉⠉⠉⠉⠉⠀⢀⣠⣤⣴⣶⣶⣤⣄⡀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠛⠁⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟⠉⠉⠉⢉⡉⠛⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⡀⠀⠀⢀⣴⣶⣶⣾⣿⠀⢠⡄⠀⠈⠁⠀⣿⣿⣶⣶⣶⡄⠀⠀⠀⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣶⣾⣿⡁⠀⠉⠙⠀⠀⠀⠀⢰⡆⠀⠋⠉⠀⢀⣿⣿⣦⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⡉⠈⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠁⢉⣙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠿⠿⠿⠛⠁



⠀⠀⡀⠀⢀⠀⡀
⣿⣏⢩⣟⡛⣛⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠠⠚⣸⠂
⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⡀
⠀⠀⠀⠀⠀⢀⡆⣀⡀⠠⠀⣤⣤⡄⣠⢖⡙⣗⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡤⢀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⢀⠜⠱⠤⠃⠊⢁⡌⢩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⡟⠻⠗⠲⢊⡤⠚
⠀⠀⠀⠘⣦⣀⠎⠉⠒⠤⠎⠀⠀⠀⠀⢺⠉⠀⣙⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢰⠤⢧⠀⠀⠀⠨⡯⢒⠠⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠑⣄⡘⠶⠀⠀⠀⠀⢀⠄⠠⢄⠈⠳⠒⠉⠀⠀⠶⡩⣔⣂⠾⣀⡀⠉⠂⠭⣒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠳⠀⠀⣏⡙⡇⠈⠢⣀⢀⠀⠀⢀⠆⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠚⠥⠒⢶⡅⡉⠊⠷⠄⢀
⢇⡀⢀⣀⠀⠄⠐⠈⠀⠀⠀⠀⠑⣀⠀⢁⠇⠀⠀⢘⠚⢷⢞⠁⠀⠀⠀⠀⣀⠤⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂⠤⠁⡒⢴⢦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠒⣤⡤⠂⢀⠺⡆⠆⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠤⡀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⣖⠁⠀⢀⠈⡀⠀⠀⠀⠀⠈⠀⠁⠒⢪⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠑⠄⢈⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⡿⡻⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠀⠀⠀⠙⠢⡄⢵⣰⠗⠒⣲⠀⢀⡾⠋⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⣠⣴⠀⠀⠀⠀⠈⠛⢿⡴⡒⢰⡥⠃⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠀⢴⣉⡸⣇⠠⢇⠀⠀⠠⡀⠈⢠⡤⠛⠋⠀⡀⠀⠀⠀⣀⣀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠻⠔⠒⣣⠤⢶⠞⢻⡁⠀⠀⠀⠀⢀⣠⠖⠀⠀⠠⠀⠛⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠶⠆⠀⢰⠁⠀⠀⠀⠚⠙⢆⠀⠀⠀⠈⠹⣀⠀⠀⠀⣀⠎⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⣈⡆⠀⢰⡆⠘⡤⣀⠀⡤⢀⠀⠸⡄⠐⠀⠀⠠⠈⣱⣖⡊⠁⠂⠀⠀⠈⡀⠑⠄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠃⢀⡘⠂⡉⠁⠀⣙⠢⣀⠥⡐⢾⣄⣭⡴⣾⣽⣖⣤⢀⠀⠀⠀⣀⢴⠄⣳⠉⠐⠢⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠁⢧⠄⠀⢠⡄⠈⠐⠢⢄⡙⣶⠒⠂⠀⠒⣞⢑⣖⠂⠒⠒⠊⠉⡝⠣⢄⡐⠒⢄⠀⠀⠀⣈⡑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣴⠿⢧⠤⠴⠉⢏⠁⢾⠀⠀⠀⠀⠀⠀⠀⠢⠠⠃⠈⠂⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠺⢄⣀⣀⡠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢄⠈⠂⠴⠣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠘⢤⠋⠢⡀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⠀⠀⠸⠆⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢛⣥⣶⣾⣶⣦⣝⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣧⡹⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⢷⣦⣍⣛⡛⢿⡟⢛⣛⣛⣛⣛⡻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⢈⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⡆⠻⢿⣿⣿⢸⣇⢹⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣜⠿⢈⣛⣋⣉⣉⣙⣛⡛⠿⢋⣼⣿⣿⣿⡆⣻⣿⠘⣿⢸⣿⣿⣿⣿⡿⢟⣣⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠛⣡⣮⣭⣴⠟⣥⠾⠻⢮⣙⣛⣩⣴⠶⢛⣩⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⢸⡇⢿⣿⢈⡟⣩⣥⣶⣾⣿⣿⢰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠛⢿⣿⣿⣿⣿⡿⢛⢻⣿⡇⣿⣿⣿⣿⣧⡙⣶⠶⢛⣴⣿⣿⣿⣿⣿⢃⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢿⠷⢌⣿⣿⡋⠶⠿⢈⣿⢡⣿⣿⣿⣿⣿⢡⡿⢸⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⣿⠟⡹⣿⣿⣶⣿⣇⢾⣿⣿⣿⣿⠏⣾⢃⣿⣿⣿⣿⠿⢋⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⣿⣿⣿⣾⣷⣿⣿⣿⡿⠟⣘⢿⣿⠿⠟⣼⡏⠺⢛⣛⣩⣵⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⠟⠀⠀⣿⡇⢿⣿⠸⣿⠇⠀⠻⣿⣷⣴⣶⡆⢻⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠋⠁⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢉⡾⠒⠵⠋⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡋⠀⠀⠀⠀⠀⡀⠀⡶⠒⠢⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠹⠀⠱⡀⠀⠜⡧⠤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢀⠀⠀⠀⠀⡠⣚⠀⠀⠙⠢⢤⠏⢉⠒⠒⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠒⠈⠃⠛⡀⠀⠀⠀⠀⠀⠀⠙⢔⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⣕⠄⣶⡿⣄⠀⠀⠀⠀⠀⠀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠐⠊⠀⠀⠈⠢⣀⠀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡜⠀⣇⣀⡀⡜⢰⠆⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠚⢤⣀⠀⠈⠸⡅⠀⠀⡕⠦⡠⠖⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠶⠸⠟⠂⢠⠳⠴⠘⢄⠠⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⢀⠀⠤⡀⠃⠀⠀⠈⠀⠀⠘⠁⠀⠚⠀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠘⡦⢄⣈⠀⠂⠈⠀⠨⠀⡀
⠠⡐⠶⠶⡆⠀⠀⡀⢄⠀⡔⠁⠀⠌⠀⠀⠀⠀⠀⠉⠐⣶⠖⠒⠚⠃⠁
⠘⠃⠀⠐⠁⠸⠢⢀⠋⠆⠁⠁⠀⠀⠀⠀⠀⠀⢠⣔⣀⠖⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡖⠒⠲⠛⠳⠖⠲⢶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠘⢄⠘⡀⠀⠀⠀⠈⡄⠀⠉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠑⢄⡀⠀⠀⠳⣄⠀⢣⠀⠘⣤⡤⠛⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⢄⡀⠀⠙⠢⣀⡀⢈⡱⠦⠛⠉⠀⠀⣀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣄⣀⣈⠴⠦⠒⠊⠉⠀⠀⢀⡠⠤⠒⠉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣗⠒⠦⢤⣴⣦⣉⣁⣀⣀⣠⡴⠚⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣹⣿⣿⡟⣛⣭⣭⣍⠀⠀⢀⣤⣾⡿⠍⠉⢉⠿⢿⣿⣽⡿⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢟⡁⠀⠘⡆⡧⣶⠀⠉⣿⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠲⠶⠦⡄⠙⣿⡿⠉⠀⠸⣿⡇⠀⠀⢡⡘⠟⠀⠀⣸⠓⠲⠶⢶⣎⢧⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠐⢀⠘⣆⢹⡇⠀⠀⠀⠛⠛⠀⠀⠀⠉⠢⠤⡴⠃⠀⠀⠀⠀⢸⡎⠻⡏
⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⠾⠛⠛⠷⣿⣼⣷⣄⡀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠲⠤⣤⣤⣤⡾⢃⡀⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠐⠒⠒⢸⣿⣿⣿⣟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏⠀⠀⠀⢣⣀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢶⣴⡶⠟⠋⠉⠛⢯⡳⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠶⣤⠈⠓⠋⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣤⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠙⢦⣉⠻⣦⣄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠃⢀⣼⠋
⠀⠀⠀⠀⠀⣼⠁⢀⣿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠈⠻⣷⣄⠙⠿⡶⣄⡀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⢻⣤⣼⠃⠀⠀⠀⠀⠀⢀⣴⡿⠿⢷⣶⣦⣤⣝⢿⡕⢄⠘⠮⣙⠶⣤⣶⡿⠋⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢈⣿⡈⢻⣆⠙⢦⡈⠳⣌⠙⢷⣄⡀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⡟⣿⣤⣄⣀⣀⣀⣰⡿⠿⠶⣶⣶⣴⣶⡿⢛⣤⠦⣿⣦⠀⠙⠦⡈⠑⢦⡈⠛⢷⣖⠾⣿⣦
⠀⠀⠀⠀⠀⠀⡇⣿⠀⠈⠉⠉⠉⣿⠁⠀⠀⢰⡟⢀⣿⠃⢸⡇⠀⠘⣿⣷⡀⠀⠈⠢⡀⠙⠢⡀⠉⠳⢦⣉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⡇⢿⣄⣀⣀⣤⣴⣿⡀⠀⠀⣼⠇⢸⡟⠀⣸⣷⢴⠞⠛⢋⠻⣄⠀⢀⠘⠂⠀⠀⠀⠀⠀⠙⠷⣤⣼⠇
⠀⠀⠀⠀⠀⠀⣷⡾⠋⠀⠀⠈⠉⠉⠻⣧⢠⡿⠀⣾⠃⣼⠋⠢⠀⠀⠀⠈⠳⠹⣦⠀⠑⣄⠀⠀⠀⠀⠲⡀⠀⣈⣿
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣰⡦⣿⣾⠃⢸⡏⢸⡏⠀⠀⠀⡄⠀⠀⠀⠀⠈⢷⡀⠈⢣⠀⠀⠀⢀⣷⣾⠟⠉
⠀⠀⠀⠀⠀⠘⣧⣀⡀⠀⠀⠀⠀⠻⣤⣾⠇⢠⣿⣀⣸⣧⡀⠠⠀⠐⢀⠀⠰⠀⢁⣈⣷⡀⠀⣣⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣤⣤⣦⣤⣤⣿⠃⢠⡿⠛⠛⠛⣿⠻⣶⣦⣀⣽⣄⣴⣷⡟⠛⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠸⡏⢿⣄⣼⠋⢠⡿⠁⠀⠀⠀⢹⣆⣸⣏⣉⣽⠋⠁⠈⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢿⣈⡿⠃⣰⠟⠁⠀⠀⠀⢰⡿⠿⠿⢧⡀⠀⣠⡶⠛⣉⣉⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⣠⠾⣿⡀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢹⡿⠋⠉⠁⣀⠠⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡞⢁⡶⢾⣿⣦⡀⠀⠀⠀⠈⢿⡄⠀⢸⡇⢀⡴⠊⣡⠔⠀⠀⢿⣦⣤⠶⠛⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣼⠃⣼⠁⢸⣇⠀⠙⠃⠀⠀⢀⣼⠇⠀⠀⢻⣏⢀⠔⠀⠀⠀⠀⣠⠉⠁⠀⡀⠀⠡⠈⢷
⠀⠀⠀⠀⠀⠀⠀⣼⠏⢰⡏⠀⠻⣦⡀⢻⣦⣀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠻⣏⠀⢠⠃⠀⠀⢻⠀⠸⡀⠸⣄⠀⢱⣿
⠀⠀⠀⠀⠀⢀⣠⡟⠀⣾⠁⠀⠀⢸⡇⠀⠈⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣼⠀⢀⡀⠘⡀⠀⠳⡄⢈⣶⠟⠁
⠀⠀⠀⠀⠀⢸⡛⠻⢾⣧⠀⠀⠀⠘⣧⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣸⠀⠀⢳⡀⣠⣾⠟⠁
⠀⠀⠀⠀⢠⡞⠁⠀⢴⠟⠁⠀⠀⠀⠘⠷⣤⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠦⠶⠟⠋
⠀⠀⠀⠀⠈⠙⠳⠦⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢀⡤⢤⡀⣤⠤⣄⢠⠤⣄⠀⢠⡤⣤⢠⡤⢤⡀⡄⣤⠤⢀⣤⢤⡄⣤⡀⣤⢠⢠⣄⢠⡄⣤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠸⣧⣼⠟⢧⣤⠟⢸⣤⡾⠀⢸⡟⠋⠹⣤⡼⠣⡇⢭⣿⠞⢧⣤⠇⣿⠹⡿⢸⠸⡏⢿⡏⢧⣼⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣡⠶⠛⠛⠓⠦⣍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⡾⠁⠀⣤⠾⣄⠀⠘⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣿⢸⡇⠀⠀⢳⡴⠏⠀⠀⣸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣀⡄⡾⢡⣏⣹⡈⣇⠀⠀⠸⠁⠀⠀⣰⠏⣼⠷⣄⢠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠋⠉⠁⣠⠟⢋⣩⣭⣙⠻⣦⠀⠀⠀⠀⣴⢃⡾⠉⣷⢹⡎⠉⠛⠶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⣸⠃⣼⠋⠄⠠⠉⢷⡈⣷⠀⠀⣸⠇⡾⠁⣠⡟⣸⠇⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢿⠘⣇⠐⡀⢂⠐⢸⠇⣼⠀⢀⡿⢸⣟⣛⣩⡴⠋⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢀⣤⠶⠶⣤⡘⣧⡹⠶⣤⣤⡴⠟⣰⣟⢛⣋⡴⠏⠉⠉⠀⠀⠀⣠⣤⠤⢤⣄⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⠁⠀⠀⠈⣷⠈⠙⠶⠦⠤⠶⠛⠁⠀⠈⠁⠀⠀⠀⠀⠀⢀⠘⠁⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡄⠀⠀⣠⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⣀⣀⡴⠷⢤⣤⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠉⠉⠛⠛⠉⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢿⠉⣉⠀⣀⠀⣸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠁⢀⡀⠀⢰⣦⠀⠀⠀⢀⡀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠟⠉⢹⠏⠙⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣾⠁⢠⣼⠏⢷⡀⠀⠀⣷⡀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⠛⢦⣤⠟⣦⡤⠞⢷⣼
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠛⢿⡀⠀⠻⣄⣿⠉⠀⣀⡿⠋⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢠⡞⠉⠛⠋⠻⣄⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠉⠳⡄⠙⠿⢠⠞⠉⢀⣤⠞⣯⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⣴⠋⠀⠀⠀⠀⠀⠈⢷⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠃⠀⠀⠘⠶⠶⣿⠀⠀⠚⢳⣤⣀⡀⢀⣠⣄⣿⡀⣠⣾⡁⠀⠀⠀⠀⠀⠀⠀⢀⡿⣦⡀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠁⠁⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁



⣿⢿⠿⡿⠿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣛⣛⣻⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣔⣒⣀⠒⠶⠶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣺⣽⠟⢉⣩⣤⣤⣤⣤⣤⣬⣿⣒⠂⠀⠀⠀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠯⠛⠁⠢⢴⡿⠟⠙⣓⡚⢓⠛⡛⠳⠮⢤⣀⣄⡐⠻⡢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⡫⠂⣰⡆⢴⣆⣤⣤⣈⠛⠿⢛⠉⠁⢀⣅⢀⡀⠀⠈⠙⠻⣦⣌⠐⠕⢄
⠀⠀⠀⠀⠀⠀⣠⣿⢿⠏⠠⣺⠟⣡⠀⠸⠟⠉⣤⢶⣳⣬⡛⢿⠿⣶⣦⣬⣉⠒⢦⡀⠈⠻⣷⣄⠲⡡⢀
⠀⠀⠀⠀⠀⣰⣳⠋⠁⢀⡼⢁⠞⡱⠃⡨⢤⣼⡾⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣙⢾⡀⡈⢿⣷⣔⡹⣷⡀
⠀⠀⠀⠀⢰⢳⡏⢀⢠⡿⢠⢃⠠⠀⠐⣁⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣄⢩⣍⡈⠛⠻⠎⢻⢳⡀
⠀⠀⠀⠀⣿⢿⠇⠆⡟⠁⠋⢁⡄⡸⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⣆⢻⡟⣦⣶⠀⠈⣏⣧
⠀⠀⠀⠀⣿⡞⢠⠁⠀⠁⠠⣿⢣⠃⣿⡇⣀⡤⡄⠀⠀⠀⠀⡠⣄⣀⣀⣃⡀⠄⠀⠀⢸⡘⡈⠗⢇⡀⣴⢻⢸⣿⡀
⠀⠀⠀⣶⣿⡇⡀⢸⣠⡆⣄⣈⠘⢢⠟⠋⠁⢀⠇⠀⠀⠀⡜⠀⢿⣿⣶⠏⡘⠀⠀⠀⠈⠓⠐⣿⢹⠆⣿⠸⢸⣿⡇
⠀⠀⠀⣿⣿⡇⣷⠈⢿⡇⡇⢻⠞⠁⠀⠀⢀⠎⠀⠀⠀⠀⣇⠀⠀⠀⣠⠜⡤⠊⠙⠀⡘⢠⠰⣿⠈⡄⡿⢀⢹⣿⡇
⠀⠀⠀⢻⢿⠀⢸⣇⠸⣟⢰⣸⡄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⣜⣠⣄⡸⡰⢃⠃⣦⠁⠘⠄⠁⡈⣸⣻⠃
⠀⠀⠀⠸⣾⡇⠈⢏⡆⢿⣖⠗⣷⡀⠀⠀⠈⠷⢄⠀⠀⠶⠮⣥⣦⠀⠀⠈⠉⣩⡞⠁⢃⣼⣿⠃⠈⠀⢠⢡⠇⡿
⠀⠀⠀⠀⢿⣿⣆⠘⣺⡜⢿⢆⠻⡷⠀⠀⠀⠀⠀⠉⠒⠦⠤⢤⡤⠤⠤⢶⠻⠗⢀⡀⠾⠙⠁⠞⠀⠐⣿⠏⡼⠃
⠀⠀⠀⠀⠈⢻⣎⢆⠱⡹⣎⠻⣧⣿⠀⠀⠀⠀⠀⠀⢀⡠⠤⢶⡲⢤⣴⣋⣤⠴⣪⡵⡣⢦⠄⠀⠀⠠⢊⡾⠁
⠀⠀⠀⠀⠀⠀⠛⢯⣧⡙⢿⢧⡈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⣤⣤⠴⠾⣋⠤⠖⢁⠔⡹⢋⡴⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠪⡻⣟⠎⠳⣝⡻⡆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠷⠾⠋⣀⠤⣚⡵⢊⠕⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⣤⡀⠙⢳⠀⠀⠀⠀⢸⠍⢉⣳⣄⣀⣴⣛⡤⢄⣠⣶⠟⢉⣪⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡻⣶⣧⠀⠀⠀⠈⣗⣒⡲⠞⠒⠒⢒⣉⣉⣥⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⡀⠀⠀⢸⣽⡿⠿⠟⠛⠛⠛⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣵⣄⡀⠀⢀⣥⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⣏⣤⣄
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⢸⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⡆⠀⠀⠀⠾⠻⡿⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⡟⠋⠉⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠈⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⣠⠾⠋⠁⠈⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠛⠉⠀⠙⢷⣾⠃⠀⢀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⢀⣠⡀⠀⠀⠙⢷⣤⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⣀⣤⠾⠛⠉⠻⣦⡀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⢀⣼⠟⠀⠀⢀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⢀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⡟⠀⠀⢀⣤⣤⣤⡀⠀⠀⢰⡟⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣾⡁⠀⢘⡷⠀⠀⣸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⣧⠀⠀⠘⠻⠶⠟⠁⠀⢀⡿⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⢷⣄⡀⠀⠀⠀⢀⣠⠞⠁⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⠀⠀⠀⢀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⣇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⣾⣹⣿⣿⣧⠤⠆⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢹⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣂⣀⣴⣂⣾⣿⣿⣿⣿⣿⣿⣿⣶⣾⡁⣐⣦⣄⣀⣶⣠⣖⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣋⣴⣭⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⡤⣀⣈⠑⠀⠀⢠⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠷⡤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿⣿⣿⡿⠟⠉⠛⠛⠛⢿⣿⣿⣿⣿⣿⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⣿⣷⣾⣦⣄⣀⠀⢠⣀⣀⡀⢀
⠀⠀⠀⠀⠰⠦⠀⠀⢚⣿⠻⢻⡟⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆⠀⠁⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣥⣷⣾⣿⣿⣿⣟⣿⣷⣿⣦⣕⢦
⠀⠀⠀⠀⠀⠈⠁⠀⠈⢓⣤⣼⠄⠀⠀⢰⣶⣦⡀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣠⣄⣳⣄⣰⣶⣦⣴⣦⣤⣄⣀⣀⣠⣿⣳⣶⣾⣿⣿⣿⣯⣿⡿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⣦⡀⢨⣿⣿⣿⣦⡀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⠈⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠙
⠀⢀⣀⣀⣠⣤⣤⣴⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣦⣶⣤⣶⣿⣿⡿⠿⠿⠛⠋⢁⠀⠀⠁
⠀⠈⠉⠉⠉⠀⣿⣿⣿⣿⣿⣟⠈⡆⠐⡟⢹⣿⣿⣿⣿⣿⣿⣷⣄⠀⣙⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⢀⣀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠀⢳⣀⣾⢿⣿⡿⣿⡿⠟⠛⠻⣿⣷⣿⡞⠿⢷⡀⢀⡀⠀⠀⠀⠀⠀⠈⠈⣵⣿⣯⣤⣼⡟⠙⠓⠚⠋
⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⡿⢿⠉⠈⠛⠻⠟⣿⣷⣶⡄⠀⠀⠀⠈⠿⢷⣄⣉⣙⠻⠆⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠈⠉⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣭⣉⡉⠠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠛⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠛⠛⠛⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠐⠂⠀⠀⠀⠀⠀⠂⠂⠃⠂⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠂⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠓⠁⡔⣡⢶⠂⢀⣠⣴⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⢡⠃⣺⡿⠋⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠋⢠⠀⠈⢁⡠⡶⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⣽⣖⣎⠉⡰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠙⠛⠋⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣀⠀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⠒⠐⠃⠀⠁⢀⡠⠔⠒⠒⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⢀⡠⠊⢁⠆⠀⢰⠋⠀⢒⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠋⠀⠀⠈⠒⠉⢡⣾⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠉⢉⡩⠝⠋⢁⡀⠤⠃⢠⠊⠀⢠⠻⠏⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠥⠒⠊⠉⠁⠀⠀⠀⡇⠀⠀⠰⡄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠸⠤⠤⢤⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⡠⠔⢲⠀⠀⠀⠀⠶⠆⠀⠀⠙⢦⡀⠀⠀⠀⡀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠀⠈⠉⠁⠀⢀⡜⠀⠀⠀⢳⠀⠀⠀⣀⠤⣀⠙⡄⠀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠝⠁⠀⠀⠀⡔⠉⠀⡰⠁⠀⠀⢀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⢀⠴⠊⠁⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⠀⠀⠈⢢⡀⠀⢠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠠⠀⠀⠀⠀⠀⠈⡏⠀⠀⢀⡀⠀⠀⠀⠀⠱⡀⠀⠉⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠂⡟⠁⠀⢠⠃⠀⠀⠀⠀⠀⡠⡀⠀⠚⠛⠀⠀⡁⠈⢦⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢰⠁⠀⠀⡘⠀⠀⠀⠀⢠⠎⠀⠈⢆⠀⠀⠀⠀⢣⠀⠀⠑⠤⠔⠋⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⡠⠎⠀⠀⠀⢇⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⠀⠀⠀⠈⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⠃
⠀⠀⠀⠀⡠⢾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠋⠀⢀⡀⠀⠀⡇⠀⠀⠀⠀⢰⠁⠀⡗⢄⠀⠀⠀⠀⠀⢰⠃⢀⠃
⠀⠀⠀⢘⠀⢘⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡰⠉⠈⢳⠀⠈⠢⠤⠤⠔⠃⠀⠀⠧⠄⠵⠀⠀⠀⠀⡎⠀⡎
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠁

⠀⠀⢸⣿⣛⡛⣰⡾⠛⢷⡄⣰⡞⠛⢷⡄⣿⡟⠻⣶⡀⠀⢸⣿⠛⣷⢠⡾⠛⠻⣦⢸⡇⢰⣟⡛⠆⣰⡾⠛⢷⡄⢾⣷⡄⣿⠀⣿⠀⣿⣦⢸⣿⢠⡾⠛⠻⠂
⠀⠀⢸⣿⠉⠁⠻⣧⣀⣼⠇⢻⣦⣀⣼⠇⣿⣇⣠⡿⠃⠀⢸⣟⠛⠋⠸⣧⣀⣴⡿⢸⡇⢨⣝⣻⡷⢻⣦⣀⣼⠇⣿⡏⢻⣿⠀⣿⠀⣿⠙⢿⣿⠸⣧⣈⣿⡇
⠀⠀⠀⠁⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠁⠀⠈⠁⠀⠉⠉⠀⠀⠈⠉⠁⠀⠈⠀⠀⠉⠀⠉⠀⠉⠀⠈⠁⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⣉⣉⣉⣉⡉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣠⣶⠟⠋⠉⠉⠉⠛⠷⣦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣼⠟⠀⠀⠀⣠⣾⣆⠀⠀⠈⢻⣆⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡇⢸⡏⠀⠀⠀⣾⡏⠈⣻⡆⠀⠀⠀⣿⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣠⣿⡇⢸⡇⠀⠀⠀⠈⣿⡾⠛⠁⠀⠀⠀⣿⠃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢀⣾⠃⣴⠟⢹⣇⠸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠏⣠⡿⣦⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣴⠾⠟⠛⠋⢸⣯⣴⠿⠷⠾⠿⢶⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣰⡿⢷⡄⠻⣧⠻⠷⣶⣷⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⠋⠁⠀⠀⠀⠀⣰⡿⢋⣀⣤⣶⢷⣦⣄⣈⠻⣷⡀⠀⠀⠀⠀⢀⣾⠃⣼⠏⠀⠈⣿⠀⣿⠀⠀⠀⠈⠙⠻⣧⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⣸⡟⢀⣾⠋⢡⡄⠀⣄⠉⢻⣆⠈⢿⠀⠀⠀⠀⣾⠃⣼⠏⠀⠀⣰⡟⢀⣿⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠐⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠁⣼⡇⢠⡄⠠⡤⠀⣤⠀⣿⡄⢸⡇⠀⠀⢸⡟⢰⣿⡶⠶⠿⠋⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⣷⡀⠠⡆⠀⡦⠀⣰⡿⠀⣿⣇⣀⣠⡿⠁⣼⣧⣤⣶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢹⣇⢀⣴⠶⠿⠷⢶⣄⠈⢿⣄⠸⠿⣶⣤⣤⣤⡾⠿⢀⣼⣏⠉⢋⣉⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣄⠸⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠙⢡⣿⠁⠀⠀⠀⠀⠹⣷⠀⠙⠷⣦⣤⣀⣉⣀⣤⡶⠟⠁⠉⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣄⠀⠀⠀⢀⣴⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠆⢀⣀⡀⣀⣤⡾⠛⠷⢶⣶⡄⠸⣧
⠀⠀⠀⠀⢀⣴⠿⠛⠉⠉⠙⠻⠷⠾⠛⠋⠉⠙⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢸⡏⠉⠉⠁⠀⠀⠀⠀⣸⡇⠀⣿⡀
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣷⣤⡶⢶⣶⡶⢶⣤⣿⠁⠀⢹⡇
⠀⠀⣠⠈⠀⠀⣠⡄⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⣠⡀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡟⠁⠀⠀⣿⠃⠀⠈⢿⡄⠀⢸⡇
⢀⣤⠀⠀⠀⣰⣿⠁⠀⣀⣸⣏⠘⢷⡄⠀⠀⠀⢸⣷⡄⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⠿⢷⣤⣀⣼⠿⣦⣀⣤⡾⠿⣦⣼⡇
⣾⠃⠀⠀⢰⣿⣿⠀⠀⠹⣿⡁⠀⣾⠿⠆⠀⠀⢸⣿⣿⡄⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⢀⣼⠿⢷⣤⡿⢿⣧⠀⠀⠙⢿⣄
⠻⣦⣀⠀⠀⠙⠻⣦⣀⠀⠈⢿⣄⣿⠀⠀⠀⣠⡾⠟⠁⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠻⣦⡀
⠀⠈⠛⢷⣤⡀⠀⠈⠙⢷⣄⠀⠻⣿⠀⣴⠟⠋⠀⠀⣠⣴⠟⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠈⢿⣄
⠀⠀⠀⠀⠈⠻⣷⣤⣀⣸⡟⠀⠀⠀⠀⣿⣄⣀⣴⣿⠋⠁⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠿⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠸⠿⠀⠀⠀⠀⠀⠹⠿⢶⣤⣤⣤⣤⡶⢿⡆⢿⣦⣤⡾⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⣧⣤⣴
⣶⣶⣤⣶⣶⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣤⣶⣦⣴⣶⣶⣶⣿⣦⣼⣿⣦⣶⣴⣶⣤⣤⣴⣦⣤⣴⣶⣶⣶⣶⣶⣶⣦⣶⣶⣤⣤⣤⣴⣦⣦⣤⣴
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣤⠀⠀⠀⠀⣠⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣷⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⠿⠟⠁⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃⠀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⡷⠶⠛⣷⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢩⣡⣨⠛⣯⣵⡎⡟⣻⡇
⠀⠀⡤⠤⡴⠏⠙⢦⠴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠏⢸⣿⣿⣋⣛⣛⣋⡣⡻⣧
⢀⣠⠇⢠⣀⠀⠀⣀⣀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣎⠺⡮⠀⣿⢹⣧⠷⢣⢩⡾
⢿⡀⠀⠚⠛⢀⡀⠛⠇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠰⣦⣤⡶⠶⠿⠿⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣰⡂⢻⠝⠿⡿⣔⣣⣹⡇
⣼⠃⠀⠀⢰⡏⣻⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣤⡞⣳⣶⣎⣸⣟⣏⠷⡚⠽⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣌⡶⢷⣍⡾⠉⠉
⠙⠲⡄⠀⠈⠳⠋⠀⠀⢠⡖⠋⠀⠀⠀⢀⡾⣣⡴⣝⡺⣿⢟⡏⣿⣧⡕⣿⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠷⠞⠛⣦⡀⠀⠀⠀⠉
⠀⠀⠷⠴⠶⣄⣀⡴⠶⠾⠃⠀⠀⠀⠀⣼⣓⣬⣶⣟⣳⣵⠿⣧⣿⣿⡇⣿⢸⢸⣇⠀⠀⠀⠀⠀⢸⡏⠉⢁⠀⠀⣤⡤⠈⠉⣻
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠀⠉⠉⠀⠀⠈⣿⣿⣗⣿⣈⢿⡟⠀⠀⠀⠀⢀⡼⠃⠸⠿⠄⠀⠋⠉⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⠀⠀⠐⠂⠀⠀⠛⠈⠙⠋⢹⣿⠃⠀⠀⠀⠀⠈⢷⡄⠀⢠⠶⠛⠛⠲⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠠⠦⠀⠀⠀⠀⠀⢀⣠⡟⠁⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⢀⣀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠸⡴⠀⢀⣼⣿⡟⣷⣶⣤⣀⡀⠀⠀⠀⠉⠉⢧⣠⡤⣤⣀⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⡄⠀⠀⠀⢀⣴⢟⣿⣿⢋⣼⡿⣢⠝⣛⠷⣦⣀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠼⣧⠙⣿⡶⣿⣿⣁⡼⣹⡅⢹⡏⢩⡉⠋⢡⣄⣀⣩⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡫⣱⠀⠀⣿⣤⡜⣿⢿⣠⡏⢠⡏⣀⣸⡟⣈⠁⡀⠙⣾⣿⢡⡞⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣓⣿⣾⣤⡦⣼⡏⡼⣷⣟⢧⣼⠣⣱⡿⢣⣿⡿⠿⢋⣾⣏⣼⣟⣽⡙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢫⡝⢏⣾⣕⡹⣷⣕⠸⣿⣸⡟⣰⣿⠑⡎⢿⣿⣶⣌⢳⡱⣏⢿⣿⣧⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡜⣿⣎⣿⣿⣿⡊⠻⣧⡹⣿⢡⡿⠣⢛⠝⣿⣿⣿⡬⢸⣿⠸⣿⣿⣿⡾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡵⠳⣬⢙⣿⠉⢰⣗⢿⡿⣿⡿⢡⣷⣿⣤⣿⣿⣿⢸⡌⣿⡶⣆⡙⢿⠧⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⠃⣿⡆⢀⣄⣇⣿⣷⡬⣻⣇⠓⣿⣿⣿⣿⣿⣿⡼⣉⢹⣷⢹⣧⡀⢚⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⡇⣿⡇⣿⣿⣿⣿⣿⡆⢹⣿⠹⣇⡾⢳⢝⡊⡟⠗⡏⡌⣿⡇⡾⣹⡳⣌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢇⢻⣧⢦⢹⢿⣿⣿⣰⡋⣿⡰⡦⢞⣴⢿⠣⣜⣺⢣⡤⣾⡟⣰⣻⣖⣇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⢿⣸⣿⠲⣼⣯⣹⠵⢎⡇⣿⡇⣴⡿⣏⣵⣎⠹⣉⣭⣶⡿⣟⢻⡺⣿⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠿⠿⠟⠛⠻⣿⣿⣿⣶⣦⣹⣏⢧⠾⣛⣭⣾⠿⢻⣭⣶⣷⣿⣭⣷⣿⠗⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠛⢻⣌⣉⣉⠉⠛⠻⢶⣿⢛⡍⣦⣚⣿⣿⣿⣿⣿⣿⣧⡜⣲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⣾⣛⠲⠦⠽⠿⠟⠋⠀⠀⠀⠀⣿⡌⣿⣷⣿⣭⣙⠦⢛⣲⣭⣶⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⡓⠦⠤⠄⠀⠀⠀⠀⠀⠀⢹⣇⠿⢟⣻⣭⣵⡾⠟⢛⣥⣚⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠷⣶⣾⣛⣀⣠⣤⠴⠾⠛⠿⠞⠛⠛⠙⠈⠁⠉⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢢⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⣻⢦⡴⠭⠷⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⡿⠉⠁⡞⠀⠀⠀⠀⠀⠈⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⢰⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠦⠤⠀⠀⠀⠦⠒⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠋⠀⠀⠙⠒⠋⠉⠛⠚⠉⠉⠉⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠀⢩⠂⠐⠒⠚⠉⠉⠉⠉⠲⠰⡤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠁⠂⠧⣀⠤⠸⡠⠤⠀⠀⠀⢀⡗⠑⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⠲⣄⣀⠈⠇⣀⡈⡢⠠⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⢒⠞⢤⠢⠶⠦⢐⠖⠒⠋⠉⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⢀⣜⣁⡱⠭⠭⠭⠬⣄⣲⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⢸⠀⠉⠉⠉⠉⠉⠉⠉⠀⡧⣤⡤⠠⢤⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠨⠗⠙⠄
⠀⠀⠀⠀⠀⠀⠀⣔⣒⣢⡄⢀⡤⠊⣀⣀⡤⠤⠶⢞⠉⠓⣄⠀⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠈⡷⢤⣀⡈⠳⣄⠀⠀⠀⠀⠀⡧⣤⠤⢾
⠐⢰⠤⠼⠆⠒⡰⠿⠆⠒⢖⠛⠋⠉⠁⠀⠀⠒⠒⠊⢳⠦⠼⠀⠀⠀⠀⠀⠀⠀⣇⠜⡲⠖⠊⠐⠒⠊⠉⠓⠾⢷⡐⠒⠒⠂⡇⠀⠀⢸⠒⠒⠒
⠀⢸⣉⣹⠀⢸⣦⠤⠤⠤⣈⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢾
⠀⢸⣀⣸⠀⢸⣋⠀⠀⠚⢩⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣄⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⡀⢤⠠⠀⢀⠀⠀⠀⠀⠀⠣⠤⠤⠊
⠀⠈⠉⠉⠀⠘⠲⠬⠭⠔⠀⠀⢀⣠⣤⣤⡀⠐⠛⠉⠀⠀⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠠⣤⡶⠶⠟⠛⠉⠁⠀⠀⠲⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠

⠀⠀⠀⢀⣶⣾⢿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣴⣿⣶⡄
⠀⠀⣠⣾⠗⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠁⠀⣀⣽⣷
⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠉⠙⣧
⢀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣤⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⣿⣿⠆
⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠛⠛⠿⠿⠿⠿⢿⣶⣠⣦⣦⣬⣷⣶⣦⣶⣴⠾⠿⣝⣦
⣿⠆⠀⠀⠀⠀⠠⣀⣀⣼⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⢹⠇
⣿⣀⢀⣠⢀⣴⡿⠻⠍⢻⣿⡅⠀⣴⡶⠶⢶⣤⡄⠀⠀⠀⠉⢿⡟
⡿⣯⣷⣿⣿⡿⡱⠛⠓⠉⣿⣷⠈⢁⣤⣤⣦⣬⣇⠀⠀⡀⠴⠛⠁
⣿⣿⣿⣿⣿⡟⠀⡊⢠⡄⣿⣿⢀⣿⠟⠓⠒⣻⣿⡼⠁
⣿⣿⣿⣿⣿⠷⠀⡉⢀⠀⠀⠉⠈⠀⠒⢶⣜⣿⡯⡇
⣿⣿⣿⣿⣿⠂⢷⣀⢠⠇⠀⠀⠀⠀⠀⠀⠙⠻⠃⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⢜⡁⠈⡄⢀
⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠖⠋⠉⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠊⠁⠀⢀⢀⣀⡈⠁⠀⢤
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡇⠀⢀⠀⠀⣀⣀⣠⣾⡤⡜⠀⠀⠀⢀⢀⡀⠀⡈⠉⢀⡀⠀⠂⠊
⠁⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠡⠈⠂⠀⠈⠀⡀⠀⠈⠄⠠⡄⠀⠻⢹⡃⣀⠈⠖
⠀⠠⡾⠁⠀⢀⣾⣶⣀⡀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⡊⠀⠉⡍⡠⢀⣀⢀⠀⠀⠀⠀⠄⠈⠁⠀⠉
⣆⠀⠀⠀⠀⢸⡿⠁⠙⠿⣤⡀⠀⠀⢀⣼⠃⠀⠉⠛⠻⠿⢿⠖⠳⡟⠳⠀⠀⠀⢐⠀⣠⠤⠠⠀⠀⠀⡄⠐⢤⣴⠅
⣿⣦⠀⠀⠠⣿⠀⠀⠀⠀⠈⠛⢦⣴⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠢⢼⣧⠀⠂⠢⠀⠀⠈⡀⠀⠂⠀⠈⠃⠀⢀⠀⠑
⣿⣿⣷⣴⣾⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢄⡀⠂⠀⠀⠀⠚⠂⡐⠀⢐⡂⣴
⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠯⣀⡐⢲⠀⠀⠊⠉⠸⡞⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⣛⣙⠛⣩⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⣩⣶⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣌⡻⢿⣿⣿⣿⣿⡟⣡⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠷⠆⠀⠀⠀⠀⢿⣿⣿⣿⡻⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣵⣾⣿⣿⣿⣶⣄⠀⠀⠀⠈⣷⣝⣿⡿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⣾⣿⣿⢫⣴⣮⢻⣿⣧⠀⠀⠀⠈⢿⣿⣷⣝⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢃⣿⣿⡿⣸⣿⣿⠀⠀⠀⠀⠀⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢏⣾⣿⣿⢡⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣼⣿⣿⣧⣝⡛⠿⣿⠀⢠⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣜⡛⠿⣿⣿⣿⣷⠀⠀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣬⣍⣭⡄⢠⣌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠁⠈⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠶⢶⣚⣿⣿⣿⣿⣿⣿⣶⣤⣈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⡓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣭⣿⣿⣿⣿⠿⣿⣿⣿⣯⣭⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣿⣋⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣶⣶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣾⣿⣿⣷⣄⣤⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣷⣽⣿⣿⣯⣾⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠏⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠸⠟⠳⢖⣛⣉⣉⣭⣷⣮⣭⣭⣭⣭⡭⠝⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⣵⣣⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣧⣖⡝⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢛⣡⣶⠿⠟⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣮⡛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠫⣠⣾⣿⣿⣶⣶⡾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠈⢹⣧⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣘⣿⣿⣿⣛⣉⣉⣤⣤⣶⣿⣿⣿⡿⠿⣏⡳⢎⠅⠀⠀⠀⠀⠀⠀⠀⣀⠤⣾⣿⣿⡰⣏
⠀⠀⠀⠀⠀⠀⢀⡞⣅⣶⣿⣿⣿⣿⣿⠛⠋⠉⡉⢉⣭⠼⠟⠘⠛⡈⠀⠀⠀⠀⠀⡀⠀⠀⣠⠞⠁⠀⣿⠉⢿⣷⠙⡆
⠀⠀⠀⠀⠀⠀⢸⡇⣿⡏⠸⣟⢻⣿⣿⣇⠀⢀⡴⠋⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢿⣧⠘⢿⠀⣷
⠀⠀⠀⠀⠀⠀⣿⢸⣿⣇⢘⣿⣤⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠶⣶⣤⡀⠀⠀⢹⣿⣿⠀⣿⢀⡀
⠀⠀⠀⠀⠀⠀⣿⠘⣿⣿⣿⣿⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡐⠁⠀⣤⢤⡄⠉⠀⠀⠀⢹⣏⠸⣛⣫⣭⣙⡳⢦⣀
⠀⠀⠀⠀⠀⠀⢻⡄⢸⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣺⢀⠀⠀⠀⠀⠀⠀⠈⣿⠾⠛⢋⣉⠙⠻⣦⡹⡆
⠀⠀⠀⠀⠀⠀⢸⣧⢸⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⡁⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠉⣠⠚⠳⢄⠁⠀⢸⡇⣿
⠀⠀⠀⠀⢀⡾⣫⣴⠾⣧⠒⠶⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣹⠇⠀⢃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣀⠰⣇⠀⠀⠘⠀⠀⣸⠇⡟
⠀⠀⠀⢠⡟⣰⡟⣠⣤⡀⠙⢤⢀⣀⠀⢀⣀⣀⢤⠤⠒⢊⡡⡤⠚⠁⠀⠀⠸⠇⠀⠀⠀⠀⠀⣀⣤⠶⢫⡀⢀⣦⠈⣳⠆⠐⠀⣰⡟⣼⠃
⠀⠀⠀⢸⡇⣿⡇⢙⠀⢹⡀⠀⠈⠉⠀⠉⠉⠙⠓⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠰⣀⡀⠀⠀⢾⡁⠀⣐⠶⠖⠋⠁⠰⡃⠀⠀⣴⠟⣼⠋
⠀⠀⠀⠸⣇⢿⡇⠘⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠙⠋⠁⠀⠀⠀⠀⢀⣡⣴⡾⣋⡾⠁
⠀⠀⠀⠀⠻⣌⠻⣦⡀⠀⠀⢹⡆⢀⡤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⡵⠞⠋
⠀⠀⠀⠀⠀⠈⠳⣌⡻⢦⣤⣄⣁⣈⣡⢠⢿⡐⢦⡀⠀⣀⣤⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣠⡾⣫⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢮⣭⣭⡙⣿⡧⢠⠗⠀⠹⢶⠉⣠⠗⠀⠀⢀⣴⣬⣴⣾⡷⠄⠀⠀⠀⠀⠀⣠⡾⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠻⣤⡀⠀⠳⠴⠛⢧⠀⠀⠀⠈⠛⠛⠙⠟⠀⠀⠀⠀⠀⢀⣾⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣙⠷⣦⣴⡰⢮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣭⡛⢦⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡌⣷⣦⣄⠀⣀⣀⣠⣤⣴⡟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⡿⠐⢦⡀⠁⠀⡸⠉⢷⣜⡛⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⣇⠀⠀⣹⠛⠛⠛⣦⡈⣿⣿⣿⣦⣭⣛⡳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣾⣿⣷⠻⠀⣼⠻⣦⠀⠀⢹⡟⠿⣿⣟⣿⣿⣿⣿⣎⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⣵⡞⣿⣿⣿⣿⠀⣷⠁⠀⠹⣧⣀⣼⡇⠀⠸⣿⡽⣿⣿⣿⣿⣦⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣧⢹⣿⣿⣿⡇⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⢹⡟⣬⣿⣿⣿⣿⣇⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣾⣿⣿⣆⠸⢿⣿⣇⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠈⣿⡿⣿⡿⣿⣿⣾⢧⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⣿⣿⣿⣿⡄⢻⣦⣟⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⣿⢳⣿⡇⣿⣿⣷⡼⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡇⢸⣿⣿⣿⡄⢹⣿⡇⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⢹⣿⣿⣇⣿⣿⣿⣿⣷⡸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⢣⣿⣿⣿⣿⣿⡌⢻⣿⡀⠀⠀⠀⢸⣿⣿⣿⡄⠀⣾⣿⣿⣇⣿⡿⠟⠋⠙⣇⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣿⡄⢙⡇⠀⠀⠀⢸⣿⣿⣿⣇⣀⡽⠟⠛⠉⠁⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⠈⠉⠛⠛⠋⢻⣿⣿⣿⡌⢿⣦⡀⠀⣈⡿⠿⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡾⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⢳⣤⣀⠀⣀⣀⣨⣿⣿⣭⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣟⡋⠤⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠶⣛⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣯⣤⠒⡀⠈⣡⣀⣀⣀⣀⣤⣤⡤⠶⠶⢟⣛⣫⡭⠥⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣙⢿⣿⣿⠿⠟⣫⣝⡛⠿⢛⣛⣩⡭⠽⠶⠶⠖⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠛⠋⠁⠀⠉⠛⠉

⠀⠀⢸⣿⣛⡛⣰⡿⠛⢷⡄⣴⡟⠛⢷⡄⣿⡟⠻⣦⠀⠀⢸⡟⢻⣷⣰⡾⠛⢷⣆⢸⡇⣾⣟⡻⠂⣴⠟⠛⣷⡀⣿⣧⡀⣿⢸⣿⢸⣿⣄⢸⡇⣴⡟⠛⠷
⠀⠀⢸⣿⠉⠉⢻⣧⣀⣼⠏⢿⣦⣀⣼⠇⣿⣇⣠⡿⠁⠀⢸⡟⠛⠋⠹⣧⣀⣼⡟⢸⡇⣬⣙⣿⡗⢿⣄⣠⣾⠃⣿⠙⢿⣿⢸⣿⢸⡏⠹⣿⡇⢿⣦⣛⣿⠇
⠀⠀⠈⠁⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠁⠀⠉⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠁⠀⠈⠁⠈⠉⠉⠀⠀⠉⠉⠀⠀⠉⠀⠈⠉⠀⠁⠈⠁⠀⠈⠁⠀⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⣉⣉⣉⣉⡙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⣠⡶⠟⠋⠉⠉⠙⠛⢷⣦⡙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣾⠋⠀⠀⢀⣠⣿⡀⠀⠀⠙⢿⡄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡇⣸⡏⠀⠀⠀⣿⠋⠉⣿⡄⠀⠀⠘⣷⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⣠⣿⡆⢻⡇⠀⠀⠀⠙⣿⠾⠋⠀⠀⠀⢀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⢀⣿⠃⣼⠏⢹⡇⢸⣇⠀⠀⠀⢠⡿⠀⠀⠀⠀⢀⣾⠃⣼⡿⣦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣴⠾⠟⠛⠋⢸⣧⣼⠿⠾⠿⠿⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣼⠿⣦⡈⢿⡄⠿⢶⣶⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠁⠀⠀⠀⠀⣰⡟⢉⣠⣤⣶⣷⣦⣄⣉⠻⣦⡀⠀⠀⠀⠀⢠⡿⢁⣾⠋⠀⢸⡇⢸⣷⠀⠀⠀⠉⠛⢿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⣼⡏⢀⣿⠋⢡⡄⢠⡄⠙⢿⡄⠹⣷⠀⠀⠀⢠⣿⠁⣾⠃⠀⢀⣼⠇⢸⡟⠀⠀⠀⠀⠀⠀⠙⣿⣀
⠀⠀⠀⠀⠀⠀⠰⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡇⢠⡄⢠⡄⠠⡤⢸⣿⡄⣿⡆⠀⠀⣸⡇⢸⣿⣶⠶⠟⢋⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠸⣧⡀⢠⡆⠠⡆⠀⣸⡟⢀⣿⣁⣀⣴⡿⢀⣾⣧⣴⣶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠂
⠀⠀⠀⠀⠀⠀⠀⢻⣇⢀⣴⡶⠿⠷⣶⣄⠘⢿⣄⠻⠷⣦⣤⣤⣴⡾⠿⢀⣾⡋⠙⢋⣉⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣤⣀⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠛⢩⣿⠃⠀⠀⠀⠀⢹⣧⠈⠛⢷⣤⣄⣉⣋⣀⣤⡶⠟⠉⠉⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠉⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣄⠀⠀⠀⢀⣼⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⣀⣀⣀⣀⣴⠟⠻⠶⣶⣦⠀⢿⡆
⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠛⠿⠶⠾⠟⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣿⡛⠛⠋⠁⠀⠀⠀⠀⣿⠀⢸⣷
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⣧⣴⣶⣶⣴⣶⣦⣴⡿⠀⠀⣿
⠀⠀⣤⡈⠀⠀⣠⡀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⣠⡀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⡏⠀⠀⢸⣿⠀⠀⠹⣷⠀⠀⣿
⢀⣤⠀⠀⠀⣰⣿⠁⠀⣀⣸⡏⠙⣷⡀⠀⠀⠀⢹⣷⡀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣾⠿⣧⣄⣠⡾⢿⣄⣠⣴⠿⢷⣄⣿
⣾⠃⠀⠀⢰⣿⣿⠀⠀⠻⣿⡁⢀⣾⣿⠆⠀⠀⢸⣿⣷⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⢀⣿⠿⣦⣴⠿⣿⡄⠀⠈⠻⣯⡀
⠻⣦⡀⠀⠀⠙⠿⣦⡀⠀⠘⢷⣼⣿⠀⠀⠀⣠⡾⠟⠁⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣴⡟⠁⠀⠈⠁⠀⠈⠻⣦⡀⠀⠘⢿⣄
⠀⠈⠛⢷⣤⡀⠀⠈⠻⢷⡄⠈⢻⣿⠀⣴⠿⠋⠀⠀⣠⣴⠟⣫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠈⠻⣷⣄⣀⣸⡟⠀⠀⠀⠸⣿⣀⣀⣴⡿⠋⠁⠀⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⡀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⡿⠉⠙⠉⠀⠀⠀⠀⠀⠈⠙⠋⠹⠏⠀⠀⠀⠀⠀⢽⠿⣦⣤⣤⣤⣤⡶⣿⠘⢿⣤⣤⡾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢿⣤⣤⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⣶⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠉⠻⠻⠟⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡼⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⡀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿⣿⡖⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄
⢀⡀⠴⢷⣦⣶⣤⣿⣿⣿⣿⣿⣧⣤⣦⡆⣠⣶⣦⠶⢶⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠴⠶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠄⠁⠀⠀⠀⠀⠉⢙⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠄⠚⠿⠿⢿⠿⠿⠿⠿⠿⠿⠟⠿⠏⠂⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠉⠉⠉⠉⠁
⠀⢄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠛⠛⠛⠉⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠠⡤⠄⢰⣀⡀⠠⠤⠤⠤⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⣠⠔⠋⣱⡀⠄⠀⠀⠒⠐⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⠴⢦⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣏⣠⣒⠁⠈⢆⣀⠤⠒⢉⡠⠒⣽⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⢤⡀
⠀⠀⠀⠀⡠⠛⠀⠀⠹⣤⡤⠀⠀⢿⣿⣿⣿⣿⣿⠟⡥⠞⢻⣿⠉⠉⠹⢤⠖⢁⡠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠳⣀⡀⠀⢀⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄⠀⠊⠡⠽⡗⠒⠒⢞⠉⠉⠙⠟⠛⠛⠋⢩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢦⡀⠀⢸⠁⠀⠀⠘⣆⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠓⠦⣾⠁⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠀⠀⠈⢳⣿⣿⡿⢏⡀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣳⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀⣼⠀⠀⣸⣷⠀⠀⠀⠀⠀⡏⠉⠉⠁⠈⢦⣀⡀⠀⣀⣀⣸⣄⣀⣀⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⡤⠤⠴⠁⠀⠀⠀⠀⡿⠻⠴⠚⠁⢻⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠣⠤⠤⠤⠴⠦⠤⣄⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⠤⠤⣤⣤⣤⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⡴⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠁⠀⠈⣿⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⣄⢄⣠⣤⣤⣆⢤⣀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠤⢾⡏⠿⠿⢿⠿⢿⡇⣿⠦⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢁⣀⣤⠼⠷⠒⠷⣾⣶⡾⡷⠿⢦⣄⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢟⣿⣿⣿⠀⠀⣀⣀⣈⣙⢁⠀⠒⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢀⣼⢿⣿⡿⠀⢈⣡⣤⣬⣭⡝⢠⣽⣟⡲⠆⢻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣦⡻⣗⠘⠛⣚⣻⠝⠟⡀⢺⡟⣛⡿⠁⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡇⠀⠀⣠⠀⠀⣠⠞⡁⠈⢠⠀⠀⠀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡇⡄⢠⡇⠀⣼⠟⠻⢿⠟⢿⡇⠀⠀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⣳⢈⠁⠀⢫⣞⣛⣓⣻⣧⣇⡜⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣞⢂⡴⢚⣿⢿⣟⣿⡿⢿⢷⣏⣠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣡⠿⡻⠛⣱⢣⣟⡯⣽⡌⠃⠃⡉⢙⣍⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⠿⠂⢠⠀⡇⠎⣸⣇⣧⡝⠘⠰⢅⠨⠁⠸⣶⣴⠆
⠀⢠⡟⠓⠲⠞⠛⠲⣄⠀⣼⠛⠾⡷⠈⡐⢠⡾⢃⠀⣩⠛⠛⣧⡀⣾⠚⠓⠓⢈⡡⠼⢳⣤⡖⠛⢷⡀⣴⠛⢉⣧
⠀⢸⡆⠀⡟⣚⡆⠀⡏⢸⡇⢉⡀⢉⣷⠰⢠⡇⠀⣾⣛⣳⠄⣈⡷⢸⢾⢷⡆⠁⣴⠛⠾⢉⢻⡆⡈⠿⠁⠀⡿⠼
⠀⠈⡿⠀⣉⣉⣀⣴⢳⡏⠀⠼⠿⠀⠈⢳⠈⣿⠀⣉⣉⣉⠀⢿⣰⣪⠉⡞⡁⠠⣼⠛⠀⠉⠈⣻⣁⠀⢸⡷⠇
⠀⢸⡧⠀⣼⠏⠁⢀⡞⠁⢰⠶⠶⣦⠈⢻⡐⣯⠀⠸⡏⣯⠀⠸⡅⠁⠆⣹⠃⠀⢿⡀⠀⠀⠀⢈⣿⠀⢬⡇
⠀⠈⠛⠛⠋⠀⠀⠈⠛⠛⠛⠁⠀⠙⠛⠛⠉⠓⠚⠛⠃⠙⠚⠓⠃⠀⠀⠙⠒⠒⠛⠀⠀⠀⠀⠘⠓⠚⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠉⠙⠿⣶⣦⣤⣀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄⢶⣦⣀⠨⣿⡛⢿⣦⣄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⠻⣷⣌⠻⠆⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠈⢻⣧⡀⠈⢻⣎⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡼⣧⠙⠃⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⣀⠀⠀⠀⠀⢀⠀⠈⠢⠄⠀⣀⣀⡀⠜
⠀⠀⠀⠀⠀⡀⠂⠩⠓⠈⠥⢀⠇⠀⠀⢻⣿⠶⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠀⠈⠉⠂⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⠃
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⣀⣤⣤⣾⣷⣴⣦⣠⣦⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠙⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠄⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⠛⠋⠛⠀⠀⠀⠀⠀⢿⠗⠀⠀⠀⠈⠁⠂⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠘⠻⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠈⠀⢀⠁⠀⢠⣆⣀⣀⡀⢀⣀⡀⠀⡁⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⢂⠀⠀⢀⠀⠆⠀⠀⠀⠙⠋⠉⠛⠛⢩⠃⡀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠴⣄⠴⠶⠒⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠜⠉⠉⠙⠒⠒⠒⠒⠚⠉⠀⠀⠈⢢⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠉⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣀⣠⡤⢶⡋⠀⠙⠲⠤⠤⠷⢦⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⢉⡟⠀⠀⠀⢻⡀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡴⠋⠀⠀⠀⠀⠈⢷⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⠀⢀⣠⡼⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠎⠁⠀⠈⠉⠉⠉⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠉⠈⠉⠉⠁⠈⠱⠀⠀⠀⠀⠀⠠⣄⣠⣤⣤⠴⠶⠶⠶⠶⠞⠛⠛⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠶⢤⣄⡀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠈⣏⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠈⢈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠂⠉⠙⡛⠛⡉⣡⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠚⠒⠒⠚⠓⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣜⠃⠀⠀⠀⠀⢀⣊⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢤⣤
⠀⠀⠀⠰⡀⠀⢀⣀⣀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣌⣿⣧⣀⠀⠀⣄⢲⡀
⠀⠀⠀⠆⠁⢸⣿⣿⣧⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠔⠀⠀⠀⠀⠀⠀⢈⣿⠈⠿⠈⣻⠃⠈⡿⡞⠷⠄
⠀⠀⠀⠀⠀⠘⡿⠙⡏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠅⠀⠀⠀⠀⠀⠀⠀⠈⢙⣄⠀⠀⠙⢁⣀⠆⢠
⠀⠀⢀⡀⢴⡇⠀⠘⣅⣀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡦⠐⠁⠘⠢⡀
⠀⠀⠉⢳⡀⠧⡀⢀⡿⢉⡆⠀⠀⠀⠂⠀⠀⠠⠣⡄⠀⠀⠀⠀⣠⣖⣶⡶⣀⢀⣼⡁⠀⣿⣴⠃⠀⠀⠈⠣
⠀⠀⠀⠀⠁⠀⠀⠦⠹⠟⠀⠀⠐⠂⠀⠀⠀⠆⠀⠡⡀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠇⢠⡞⣍⠂⠀⠀⢀⠆⣄
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠚⠂⠀⠀⠀⠀⠀⠀⢐⠀⠙⣿⣿⣿⣿⣷⡻⣧⠛⠃⢀⡴⡇⢐⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣼⣿⣿⠟⣿⡷⠦⢠⡖⢹⡈
⠀⠀⠀⠀⠀⢠⠀⠀⡪⠀⠐⠾⠀⠀⠈⠤⠁⠈⠉⢡⣴⣶⠧⠀⠀⠀⢠⣿⣿⢿⢠⠏⠀⠀⠸⠃⠘⠇
⠀⠀⠠⠀⠀⠀⠉⠉⠁⠀⠀⠄⠀⠀⠀⠀⢀⡠⠔⠾⠛⣣⠀⠀⠀⠀⣼⣿⡏⠀⠀⣰⠀⠀⠀⠠⠀⠰
⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢄⠐⠊⠁⢀⠴⠒⢉⣿⠀⠀⠀⢠⣿⣿⡇⠀⢠⡏⠀⠀⠀⠀⠀⠀⢴⡀⢀⣀⡀
⠀⠀⠀⠀⢈⠀⠀⣀⡤⠖⠛⠁⣦⠀⠀⠀⠀⠈⠀⡆⠀⣿⠰⠀⠀⣾⣿⣿⣷⣰⣿⡇⠀⠀⠀⠀⠀⢀⡏⠀⢩⠉⠁
⠀⠀⠀⣁⡤⠖⠋⠁⢀⣡⡶⢿⣿⠀⠀⠀⠀⠀⠀⠿⠀⠙⠈⠁⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠀⠀⢸⠃
⠀⠀⠀⠀⢀⣤⠖⠛⠉⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠸⣷⡷⡤⠆⣼⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⣰⠀⠸⠂⠀⢸⡆
⠀⠀⠀⠀⡈⠀⠀⢀⣠⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠹⢦⠇⡆⠉⠀⣿⣿⣿⣿⣿⠁⠀⠀⢀⠁⠀⢠⡄⠂⠈⠁
⠀⠀⠀⠀⠀⠠⠞⠋⠁⠀⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⢸⡻⡏⠁⡀⠀⣿⣿⣿⣯⠉⣿⡏⠻⠸⠉⠒⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣷⢻⢊⠀⠀⢻⣿⣿⣿⡄⠘⣿⣆⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⠀⠀⠀⣿⣿⣿⡇⠀⢿⡌⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠘⣿⣿⣿⡄⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⠀⠀⠁⢀⢸⡞⠄⠀⠀⠀⣸⣿⣿⣿⡀⠻⡄⠁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠀⠀⢸⣇⣠⠴⠋⠀⠀⠀⠀⠈⠀⢷⠙⢶⣶⣾⠿⡿⣿⣟⣀⣀⣽⣦⣧⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠐


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢀⣶⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣼⡟⢱⣾⡟⢻⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠛⠷⣮⣭⣦⣸⡹⢟⣼⡷⣼⣿⣿⢟⢷⣶⣤⣄⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⢉⠙⠛⠿⠿⣷⣿⣼⣇⠻⣡⣿⡀⠿⣻⣿⠆
⠀⠀⠀⡀⠀⠀⠀⠀⠗⠹⡗⣰⣆⠀⠶⢈⣉⠛⠛⠿⢷⣿⣶⣿⡿
⠀⠀⠀⠁⠀⠀⠀⠀⣄⡀⠀⡀⠙⠘⠀⠺⣤⢀⣧⣴⠆⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢁⡎⡙⢀⠀⣤⢀⡤⣀⠀⠈⠀⠀⠀⢺⠉⣐⠂
⠀⠀⠁⠀⠀⠀⡤⠀⡀⢀⣉⡀⠘⠚⠁⠸⢶⡛⠀⢀⡀⠀⠀⠈⠑⠉⠀⣀⠠⢀
⠀⠐⠀⢀⡀⠐⠛⠻⠃⡾⠶⠀⢠⣦⠀⣀⡀⠁⠀⡆⢈⠂⠁⠀⠀⠀⠰⠂⡀⠴
⠁⠀⠀⠀⠀⠀⠉⠐⠂⠉⠉⠐⠋⠻⠄⣜⣹⡆⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠶⣌⠉⢀⠀⠀⠐⠆⠀⠀⠃⠀⠀⠀⠀⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡀⣤⠓⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠲⣄⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠁⣠⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠲⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠐⠂⢨⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠖⠧⠠⠀⠤⠘⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢀⣀⣀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣠⡴⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠛⠛⠛⠛⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⠶⣤⡀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠻⣆⠀⠙⢶⣤⣀⣀⣠⣴⠞⠋⠀⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⢠⡀⠀⠀⠀⢻⡄⠀⠀⠀⠉⠉⠁⠀⣤⡀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠘⣦⠀⠙⣧⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡇⠀⠘⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⠀⠀⢹⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢻⡄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣷⠀⠀⠛⠀⠀⠀⠀⣿⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠙⢳⣤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠒⠲⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠉⠛⠶⠶⠤⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠙⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠘⠛⠛⠛⠛
⢀⡴⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠉⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾


⠀⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⠀⠧⠼⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠱⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⢀⠀⣀⠀⢀⡀⠀⡀⡀⢀⠀⠀⡀⠀⠀⢀⠀⠀⣀⠀⢀⢀⠀⣀⢀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢘⢈⠀⠀⠀⡌⢡⠀⡇⢎⠘⠀⠀⠆⠀⠀⠐⢢⠀⢸⠀⠰⢸⠡⡁⠀⡇⡰⢆⠀⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣾⣷⣷⣾⣶⣶⣶⣶⣶⣶⣿⣷⣾⣿⣶⣿⣷⣶⣾⣿⣶⣾⣶⣶⣶⣶⣾⣾⣶⣶⣶⣷⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣂⡶⠲⣄⡶⠒⣆⣶⠒⣆⠀⣶⣲⣦⡶⠲⣄⣦⢴⣒⢠⡖⢲⡄⣶⣰⡆⡆⣶⣄⣆⣴⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠓⠞⠃⠛⠖⠃⠛⠒⠋⠀⠛⠁⠀⠓⠞⠃⠃⠒⠞⠉⠓⠚⠁⠃⠙⠃⠃⠛⠘⠃⠙⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⣭⠭⣭⣙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠞⠁⢀⣠⡀⠈⠳⡌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⡇⣿⠀⠀⢿⣀⡿⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡞⣱⠛⡇⢿⠀⠀⠰⠇⠀⠀⣠⠏⡼⢧⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠋⠉⣸⠗⢛⣩⣙⠻⢦⡀⠀⠀⠀⣰⢏⡾⠛⡆⢳⠙⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⣼⢁⡟⠩⠅⠍⠻⡆⢳⡀⠀⢰⠇⡾⠁⣠⠇⣼⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⣿⠸⡇⢂⡘⡀⢂⡿⢸⡇⢀⡿⢸⣟⣋⣥⠞⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡴⠖⠲⢦⡘⢦⡛⠶⣤⠴⠞⣡⢿⢉⣩⡴⠋⠁⠀⠀⠀⢀⡤⠶⠶⠦⣄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⣷⠀⠙⠓⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉⠀⠀⢀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣦⣀⣀⣴⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⣤⣤⡴⠛⠓⢶⡆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⡉⠉⠀⠀⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⣠⣤⣤⣤⣸⠃⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡆⠀⢸⡷⡄⠀⠀⢰⡄⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣟⠀⢠⣇⠀⢹⣆⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⣿⠄⠐⣿⠁⣽⠆⠀⢸⣿⡄⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⢈⡿⢯⡽⢿⡉⠈⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠳⢦⣀⠉⠳⣤⡈⢳⡇⢀⡤⠞⠉⣀⡴⠞⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁⠀⠀⠀⠙⣦⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⣠⡇⠀⠁⢺⣄⣤⡞⠉⠈⠷⣄⠀⠀⠀⠀⠀⣼⠃⢀⣼⠃⠀⠀⠀⠀⠀⠀⠈⣷⣄⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠋⠛⠖⠲⠛⠛⠙⠓⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠓⠋



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣀⣠⡤⢞⡿⠁⠀⠀⠹⣟⠦⣤⣀⡀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⢉⡋⠉⠀⠀⠀⠀⠀⠈⠙⣛⠉⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⣙⣷⠶⠀⠀⠀⠰⢾⣛⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣠⠤⢶⣖⣶⠦⢤⡀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠸⣇⣰⠋⡉⠉⢳⣀⡿⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣬⡿⠀⠀⢀⠸⣯⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠃⠀⠈⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⠤⠴⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣶⣬⡉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠀⣿⣿⣷⡀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠸⣿⡿⠛⠉⠁⠀⠀⠀⠀⢀⣤⣾⣿⠋⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠈⠻⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠙⣿⠿⠛⠉⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡄⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠣⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⠀⠀⠀⠉⠉⠉⠁⠒⠂⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠲⠶⣄⡀⠀⣶⣦⣤⣤⣀⣀⣀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡴⢬⣟⠀⠸⣿⡄⢀⣀⣈⠉⠛⠦⡀⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⠘⠋⠆⠀⢰⣾⠛⣿⣿⣿⡆⠀⢹⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⠃⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃⠀⢸⠀⣿⣿⣿⣿⣿⡧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⢸⠀⠀⡴⠺⣅⣀⣤⣤⡄⢢⡈⠀⠀⠀⠀⣸⠀⣿⣿⣿⣿⡟⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠸⡇⠀⠐⣦⣌⠉⠉⠉⠀⠀⠙⠲⠤⡤⠔⠃⠀⣿⣿⣿⠟⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢻⡀⠀⢻⣯⣚⣙⣶⣲⠦⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏⠀⠀⠀⠀⠀⠀⢀⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠰⢏⠤⠤⠴⠒⠊⠉⠉⢳⣳⡀⠀⢩⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢃⡤⠤⠤⠴⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢳⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢻⣲⣤⣀⣐⢊⣀⠀⢀⣀⠤⣺⠏⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠻⣷⡄⠀⠉⠀⠉⠁⢀⣾⠃⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⣝⠲⠤⠤⠤⠒⣹⠃⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠀⡰⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣼⠕⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣀⠀⠏⢀⠴⣂⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣉⣉⣹⣿⣿⣿⣷⣴⣿⣾⣿⣯⣭⣽⣂⡀⠁⠦⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠔⠉⢁⣠⣶⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣝⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣿⢿⣿⣿⡿⠋⣹⡆⡄⠀⠙⣿⣟⢿⣿⡉⢛⢿⣿⣿⣦⡙⠇
⠀⡄⠤⢀⣀⣀⠠⠄⠒⠉⠉⠀⣰⠟⣱⣿⣿⠟⢸⡞⣱⠀⠃⠀⠀⢻⣿⠈⣿⢇⠄⣻⡎⢻⣿⣿⣖⠲⠤⣀
⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢰⣿⣿⡏⢸⠀⠈⢁⣴⣾⠦⠀⠘⠃⠀⠛⢴⣶⣄⠐⠀⢻⣿⣿⣆⠀⠀⠉⠒⠦⣄⡀
⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠘⢛⡭⢆⢒⠲⡀⠀⠀⠀⠈⢛⢻⡇⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠉⠓⠆⠀⠀⠀⢀
⠀⠀⠀⠹⣶⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⣿⠇⠀⠀⠀⡾⠀⠈⠆⠁⠀⠀⠀⣀⠀⠈⡎⠀⠀⢀⣿⣿⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⣰⠃⠀⠀⠀⠀⡔⢡⢋⣠⣿⣿⡿⠀⠀⠀⠀⣧⠠⠐⠀⠛⠁⠀⠀⠉⠀⣠⠃⠀⠀⣼⣿⣿⠇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠘⣱⠏⠈⡐⠀⠉⠀⠀⠀⠐⢌⣓⣢⡶⠂⠀⢉⢲⢴⣞⡥⠂⠀⠰⣟⠿⠟⠲⡄⢳⡀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡄
⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⠉⣷⡖⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠖⠋⠀⠀⠀⠀⠀⠀⢸⣿⠜⡆⣷⠰⠃⠀⠀⠀⠀⢀⡏⠀⠀⢸⡇
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠄⠀⠀⢀⡾⠁⠐⣡⠏⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣴⡿⣄⠀⠀⠀⢀⣠⣶⠶⠲⠤⠞⠋⠀⢀⣀⣤⠶⣯⣤⣤⡶⠋⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣼⠃
⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢀⡠⠄⣚⠯⠝⣂⣜⣁⣀⡠⠀⠀⣤⣶⣾⣿⣷⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⠁⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢁⠤⠒⣯⠷⠚⣉⠤⡖⠛⣋⣉⠛⢿⠀⠀⡼⢃⣀⠈⢹⡿⠀⠀⠀⠈⠉⠲⢄⡀⠀⠀⠀⢠⢣⠇⠀⠀⠀⣸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⢜⠐⠛⣛⠥⠒⠉⠀⢈⡽⠃⢀⣾⣷⣌⠀⡞⢶⣿⡍⠉⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡞⠀⠀⠀⢠⡿
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⡴⢋⠜⢡⣿⣿⠁⣽⣾⡇⢸⣿⣧⠀⢦⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⣾⠃
⠀⠀⢿⡀⠀⠀⢀⠔⠲⡔⠉⠉⠀⠀⠎⢠⠏⢰⣿⣿⠇⠀⣿⣿⡇⠀⢿⣿⣆⠈⢣⡀⠈⢳⡄⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⢀⣾⠃
⠀⠀⠸⣧⠀⣀⡇⠀⠹⡇⠀⢰⠀⠀⢱⠾⠤⠿⢿⣿⣀⢸⣿⣿⣇⣀⣸⡿⠿⠤⠼⢧⣄⣀⣻⡀⠀⠀⠀⠀⠀⡇⠀⠀⢠⡿⠁
⠀⣠⠔⠛⠋⠉⢳⡀⠀⣣⡀⢈⣆⢀⡼⠀⠀⠀⠀⠀⠈⠉⠛⠻⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢄⡀⠀⠁⠀⣰⡟⠁
⠈⠀⠀⠀⠀⠀⠀⠉⠊⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣼⠟
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢠⡖⣦⣴⢲⣤⡖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠈⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠳⣏⡷⢸⡁⣻⡥⣏⣹⢸⢳⡇⣿⢸⠷⡇⣎⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠰⠀⠀⢐⢆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢃⡤⠶⣖⠢⡌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⣞⡓⠒⠢⣐⡵⢎⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠞⠁⠀⠪⠳⣄⠉⡲⣅⣽⠹⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⢼⣆⠀⠀⠀⠀⠈⠱⡄⡻⢨⠞⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⢀⠀⠈⠛⠻⠥⣀⣀⣀⣀⡧⠛⠁⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⣆⡯⣇⣰⠆⣠⡀⠀⠀⠀⠁⠀⠀⡇⢸⢀⣠⣬⣔⡐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣺⠛⠀⡥⡌⠘⢫⣞⣡⡀⠀⠀⠀⠀⠀⠹⣌⣍⠉⠉⠹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠼⠇⠀⠀⡩⡅⠀⠀⠸⠬⢤⠀⠀⠀⠀⠰⢆⢰⣶⢒⡲⠖⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡭⣇⠰⣽⠌⠱⡹⢂⡹⣭⠁⠀⠀⢠⠔⠢⢄⣺⢇⠈⣁⠀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣪⠶⣄⡄⣤⣴⠶⣓⠌⠰⡄⠀⡇⠐⠳⣴⢗⣋⣀⡩⣮⢎⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠪⠇⠘⠃⠀⠀⠀⠈⠃⠇⠂⠀⠀⠉⠚⠛⠉⠀⠘⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠙⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠈⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠦⡄⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⠀⠊⠋⠀⠀⠀⠈⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠈⠁⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡀⠀⢦⡤⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠀⠀⠀⠈⠐⠀⢳⡀⠀⠀⠈⢷⡠⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠉⠠⠀⠶⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠲⣄⣬⠿⠀⠀⠀⠀⡁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠰⠊⠀⠳⠤⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠐⠐⠒⠀⠀⠀⣠⡀⠔⠁⠀⠀⠀⠀⠗⠀⠀⠔⠀⠐⡈⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠴⣶⣦⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠂⢀⡄⠄⠐⠀⠀⢀⠄⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⠄⠄⠀⠀⣤⠴⠷⠖⠒⠒⠀⠈⡀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠢⡀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠔⠋⠁⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣌⣂⠀⠀⠀⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠥⠜⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⡄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡦⠤⠤⠤⠤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⣠⠔⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡜⠁⠑⠄⠀⠀⢀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠁⠀⠀⠈⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⠀⠀⠙⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠁⠀⢀⠀⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⣇⣠⠀⠂⠀⠀⠀⠀⢀⡠⠐⠀⠉⠁⠄⢠⡄⠀⠀⣀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠈⠁⠀⠀⠉⠀⠀⠀⠐⠒⠈⠁⠀⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣶⣶⣶⣶⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣶⣶⣶⣦⣥⣤⣤⣤
⣿⣹⣿⣻⣹⣟⣵⣍⣿⣋⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣯⣭⣭⣭⣿⣯⣭⣭⣿⣯⣭⣭⣭⣽⣭⣭⣭⣭⣭⣭⣿