Ferry Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠨⢩⠿⢿⡤⡤⡤⣤⣄⣸⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⣀⣀⣀⠀⠀⠀⠀⠤⠤⠠⠤⠀⣠⢀⣄⡄⣀⣠⣸⣠⣸⣇⣇⣇⣀⣇⣇⢀
⠀⠀⠀⠀⠀⠀⠀⡀⣤⡾⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠺⠳⠃⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠿⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢐⡆⡆⢰⠀⠂⡇⠒⢸⢹⢱⠀⠀⢀⠀⠀⡀
⠀⠀⡀⠤⠀⠀⠘⠷⠘⡃⠀⠀⠧⠧⠖⠒⠀⠐⠒⠒⠒⠀⠀⠠⠸⠐⠇⠃⠘⠀⠂⠃⠒⠚⠚⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠂⠀⠀⠒⠢⠤⠶⠋⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠾⠾⠄
⠀⠀⠀⣯⣭⣭⣭⡭⣭⡦
⠀⠀⠀⠀⠙⠋⣿⠛⠉
⠀⠀⠀⠀⠀⠰⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀
⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠀⠁⠀⠀⠀⠀⠁⠁⠈
⠀⠀⢀⠀⠀⠀⣛⡀⡇⠀⠀⣀⠆⠀⠀⢰⡄⠀⠀⢸⠇⠀⣠⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⢿⢇⣀⢀⣿⣉⡀⠀⠀⠈⠲⠤⣤⢸⠿⠂⠀⡨⠄⠂⠉⣉⠁⠉⠉⠉⠉⠉⡀⠀⠀⠀⠀⠄
⠁⠀⢈⣁⣼⣿⣿⣠⠤⠄⠀⠰⢚⣛⡆⠈⢠⠂⣭⠵⠤⣶⣰⣲⡶⠦⠦⠦⡤⡭⡭⣭⣹⠐⠃⠘⠆
⠀⠀⢬⡅⠈⠉⠁⣐⠙⠛⠃⠉⠉⠉⠁⣉⡀⣠⣅⠀⠀⠉⠉⠉⡄⣤⣤⣠⢠⣀⣀⠀⠐⠃⠀⠀⠀⠂⢀
⠀⡤⠼⠧⠀⢸⠁⠁⠄⠤⠄⠆⠒⠘⠃⣉⠁⣀⡀⠀⣤⣤⠀⠀⠀⠀⣤⣷⣤⣤⣤⡤⣀⣭⣭⡵⠀⠀⠈
⠐⠀⠲⠀⠂⠀⠀⠒⠒⠒⠘⠋⢃⢉⣀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢰⣿⣿⡏⡟
⠀⠀⠀⠀⠉⢿⡋⢫⣍⣩⣭⣭⣭⣭⠍⠁⠠⠤⠬⠭⠭⠭⠭⠭⠤⠬⠭⠭⢭⠿⠿⡼⢿⡿⠃⠃⢣⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠛⠻⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠠⣶⣤⣤⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⡄⠀⠀⣀⠀⠀⢀⠀⠀⠤⡄⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⣿⡆⣀⢿⡈⠀⠀⠀⠱⠄⡄⠸⠒⠀⡀⠁⠀⠉⠉⠉⠉⠉⠉⢋⡀
⠀⠀⠀⠀⠀⠤⠐⠿⠟⢐⠀⠀⠀⠭⠭⠇⠄⠀⠐⠂⠁⠛⠛⠛⠛⠓⠃⠒⠒⠒⠒⡆⠁⠈⠁
⠀⠀⠀⠀⠤⠽⠄⠀⠆⠈⠀⠠⠠⠐⢐⠂⣚⢈⣉⠀⣠⡄⠀⠈⠈⢩⣭⣭⣭⣥⢀⣠⣤⡤
⠀⠀⠀⠈⠀⠀⠀⢀⣀⡠⣤⣤⠥⠭⠬⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⡇⣾⣿⡋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠷⠶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠄⠀⡀⠀⡀⢀⠀⡀⡄⠀⡀⠀⠀⢠⢠⡄⡠⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠍⠗⠻⠌⠴⠽⠸⠀⠇⠃⠷⠌⠠⠀⠈⠎⠰⠇⠞⠚⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠛⠛⠛⠛⠛⠻⣿⡟⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⣴⣶⣶⣶⣴⣿⣷⣶⣶⣶⣶⣶⣦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠁⢀⣀⣀⣤⣤⣴⣤⣤⣀⣀⣀⠈⠉⠉⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣶⣦⣄⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣄⣀⣀⣀⡀⣀⣀⣀⣀⣤⣴⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣠⣥⣽⣶⣦⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⠿⠇⠿⠇⠿⠇⠿⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⣤⣤⣤⡤⣤⢤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣖⣚⣾⣿⣯⣷⣶⣾⡆⠘⠿⠾⠿⠻⠟⠟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⣉⣉⣉⣉⣉⣑⣦⠤⠤⠤⠶⢒⣒⣒⣖⣶⢻⣒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣤⣶⣶⡆⢰⣶⣶⡶⡇⣿⣄⣈⣿⣯⣿⣭⣩⣧⣧⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣉⣉⣙⣛⠒⡒⠒⠒⠒⠒⠚⡚⠛⢛⠛⢉⣉⣉⣹⣇⣤⣤⣴⣤⣤⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠦⣤⣀⣉⣉⣉⣉⣭⣭⣽⣿⣶⣤⣬⣉⣁⣀⣽⣃⡀⠀⢀⡀⢰⣏⣠⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠤⠖⠒⠶⠦⠭⠙⠛⠛⠛⠛⠁⠀⠉⠉⠉⠉⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠡⡀⢼⠄⠀⠀⠀⠀⡀⠃⠀⠀⢀⠊⢀⠎⠠
⠀⠀⠀⠀⢀⣠⡀⣶⠄⠠⠔⠀⠄⢀⠀⠀⠀⣠⣶⣶⣀⣬⣦⡀⢀⠈⡔⡥⠐⠀⢶⠃⠀⠀⠣⠬⠅
⠀⠀⢀⢠⠎⠻⠂⢩⠀⠔⠶⢆⠐⣌⠀⠀⠀⠀⠀⢙⠁⠀⠀⣬⠛⡢⢰⠠⠠⠬⠼⡆⢠⣶⡔⣦⢦⡀
⠀⠀⠴⢧⡾⢈⠰⢦⡤⠨⠲⠾⠭⠀⠀⣐⣂⡰⣈⢂⣀⠀⠀⢆⡀⠒⠚⠂⠒⢀⠒⠂⠛⠙⠳⡮⠍⠁
⠀⠀⢀⠚⠂⢄⠀⢘⡛⡦⠀⠀⠀⠐⠀⠄⠄⠤⠠⠤⠄⠀⠄⠍⠁⠁⢨⠀⠅⠀⠀⠁⠀⠀⠀⢐⡀
⠀⠀⠈⢨⠅⠨⠄⠨⠅⠀⠀⠀⠀⠈⠀⠉⠁⠉⠀⠀⠃⠐⠇⠁⠀⠀⠐⠂⠂⠀⠀⠦⢄⡄⣤⣤⡤⣛⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠆⠀⠀⢰⣇⡀⢦⣇⠀⠠⣶⠶⡿⠿⡯⠿⠿⠷⠻⠿⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⣹⠀⠀⣴⢤⣥⣴⣅⣤⣌⣱⣪⡉⢀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠧⠿⠦⠴⠻⠿⢿⡿⠷⠼⠞⠟⠓⠐⠈
⠀⠀⠀⠀⠀⠀⠀⣠⣤⢠⠈⠠⡄⠄⢠⣤⢤⣤⢠⠀⠀⠗⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠨⠓⠺⠒⠺⠗⠒⠿⠛⠛⠿⡛⠀⠀⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⢠⣴⡿⠠⢾⣤⣴⡧⣠⣄⣴⡇⠀⠀⠁⠌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠓⠀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⠦⠾⠭⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⡄⠀⠀⠀⠀⢤⣤⣿⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣯⣤⣤⣄⢠⣤⣤⣄⣀⡀⠀⠀⠀⣿⣿⡉⠉⠉⠀⠀⠀⠀⠀⣨⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⣿⣯⣭⣉⣬⣭⣭⠉⠙⠲⢶⣶⣾⣿⣯⣀⣸⣦⠈⣦⡀⣤⣬⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⠾⠿⠿⠿⠿⠿⠛⢛⣛⡛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠷⠾⠿⠿⠿⠿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⠐⣿⡷⢶⣶⡷⢰⣷⣶⡆⣿⡇⠀⠶⠶⣶⡆⠰⠶⠶⣶⡄⠰⢶⣶⠀⣤⣤⠀⣠⡄⢠⡀⣤⢀
⠀⠀⠀⠀⠀⠀⠰⢶⣶⣶⣶⣶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⢿⣿⠿⢛⣻⠟⠀⠀⠀⠀⠉⠉⠙⠇⠈⠉⠉⠻⠇⠀⠉⠛⠀⠘⠛⠃⠸⠇⠘⠇⠸⠀⣿
⠀⠀⠀⠀⠀⠀⠿⡿⠁⠰⠶⠿⠑⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠘⠿⣭⣭⣭⣭⣭⣭⣛⣛⣲⣶⣤⣤⣤⣤⣤⣀⣀⣘⣛⣛⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣻⣿⣿⣿⣯⣭⡵
⠀⠀⠠⣤⣤⣤⣤⠈⠋⠁⠀⠀⠀⠉⠙⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⡋⠁
⠀⠐⠛⠛⠻⠟⠁⠀⠀⢀⣀⣀⣀⣠⣤⣬⣿⣧⣤⣄⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⣀⣀⠍⠀⠀⠀⠀⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠖⢺⣯⣽⣿⠁⠿⠐⠂⠉⠀⣀⣀⡀⢀⡀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⣷⣦⣤⠴⠹⠟⠁⠀⠀⠀⠘⠛⠋⠩⠭⠛⡍⠁⢺⣿⣶⡆⠀⠀⠀⠀⡀⠀⣐⠀⠤⠀⢀
⠀⠀⠀⠀⡄⣶⣶⣴⡦⡔⠒⠈⠉⠁⠐⣤⢹⣟⣛⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⠀⠀⠀⢐⣈⠏⠉⠀⠰⣶⡯⣼⣚⣛⢦⣾⣦⣿⣿⣶⣦⣾⡆
⠀⠀⠀⠀⡍⣀⠀⠸⡗⡧⢄⣀⣤⣤⣴⣿⢀⠀⣿⠉⡖⢠⡤⢀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣤⣶⣖⣰⡅⣿⣿⡆⡄⣿⣿⣿⣿⢽⡏⣻⡇
⠀⣘⠲⣄⠀⠉⢹⠶⣦⣧⣽⣛⡟⢿⢻⣧⠬⢑⠛⠦⢇⣸⣇⢸⠀⣿⠋⢰⠚⣧⡭⣋⢻⠿⢿⠇⢸⣿⣿⣯⣿⡿⢛⣃⠨⠧⠻⣷⠉⠁
⠀⢘⡆⠀⠙⠛⢸⡇⠀⣿⠙⠛⣿⢷⣶⣥⣝⣐⠆⢷⠀⡆⢈⠙⠒⠻⠦⣼⣀⣿⠀⠀⠈⠁⠚⠆⠰⠇⠘⢷⣾⡏⡏⢹⠀⠀⠀⠻⣧⠀⠀⠀⠀⣀
⠀⠈⠁⢄⠀⠀⠀⠉⠑⠻⢤⣄⣿⠐⠀⣿⠙⠛⢷⣷⣤⣧⣘⡃⢺⠀⡖⠀⡌⠉⠓⠆⠀⠀⠀⠀⠀⠀⢠⣿⠋⡇⡟⣿⠀⠀⠀⡠⣿⣤⣄⣤⡤⠉⢀⣰⣦
⠀⠀⠀⠀⠀⠂⠄⠈⠓⠦⢄⣀⠉⠙⠒⠿⣄⡅⢸⡆⠉⠙⠃⠉⠛⠒⠧⣈⡇⠰⢇⠀⠀⠀⠀⠀⠈⡄⣿⠇⠀⡧⠷⠋⠀⣤⣤⡀⣽⣿⣿⣾⣳⡾⣿⠉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢄⡀⠈⠙⠓⠦⣨⠀⠀⠀⠀⠀⠀⠈⠉⠓⠄⢀⣀⠀⠀⠀⢸⡿⢄⣤⣀⣀⣸⣾⣶⣶⣿⡿⠛⢛⣉⡬⠵⠚⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠉⡃⠀⠀⣄⡀⠉⠑⠂⠄⣥⣀⡋⠀⠇⢸⡁⣠⡯⡤⠓⠚⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠅⠀⠀⠀⠀⠀⠈⠙⠉⠛⠉⢹⠹⡇⡇⠀⢀⠈⡗⠀⠀⠀⣤⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠶⠃⠀⠀⠀⠁⠀⢀⣤⠌⠻⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣠⠄⢲⣿⣿⣿⣀⠍⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⢈⠁⢸⠀⠀⢊⠁⢙⡂⠸⠛⠉⢀⡀⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠸⠀⠀⠒⠀⢁⡀⠀⠒⠿⣉⡃⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠀⠀⠠⣤⣤⡤⠶⠊⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠣⢄⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠹⠒⠉⠉⠑⠔⠒⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠚⠒⠒⠒⠲⢄⡀⠀⢀⡠⢘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡞⠉⠉⠉⠓⣆⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠼⡤⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢼⣤⢀⣀⠀⠀⣗⣒⡄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣛⣻⠷⣾⣛⣛⣭⣿⣷⡾⡧⣤⣴⢾⣛⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⢘⡇⠉⠉⠛⠻⠿⣿⠿⠛⠉⠁⣿⠻⠷⢿⠍⠛⠋⣉⣸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣶⣾⠿⠿⠛⢻⡇⠷⢀⡀⢠⠄⢸⠀⡀⠀⠀⣹⣤⣤⣼⣶⠟⣛⡥⠖⠛⡇⠀⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣽⣿⣷⢶⣿⣷⣾⣷⣷⣌⣁⣼⡦⠶⢚⣯⠿⠁⣁⡤⠞⠋⠁⢀⠰⠆⡿⢹⡽⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣮⣯⠐⡆⡭⢙⡛⣿⠉⣻⠾⠗⣏⣡⡷⠞⠋⠁⢀⠄⠠⠅⠈⣀⣴⠗⣫⢶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣧⢼⠿⡟⠷⣷⣬⡄⣿⣁⣴⣶⠞⠛⢉⠀⡆⣂⠀⠈⢁⣠⣶⡿⢋⡵⣊⠁⣘⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⠴⡞⠛⡇⣿⠴⠛⣧⣌⣠⠋⡛⡟⣿⠁⢸⠀⠘⠘⠀⢏⣸⠴⣚⡽⠛⣡⠞⡉⠂⢐⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⣉⠑⣢⠴⠛⢳⡿⠔⠊⣀⠤⠈⣙⠷⣾⢻⣄⣸⡇⢀⡤⠖⣫⠴⠚⣁⡴⠛⠑⡠⢊⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⡁⠂⣠⠴⠋⣁⠤⠂⢁⠠⠒⢉⠠⠐⡉⠄⣂⠽⣸⡏⠝⣛⡥⠖⠋⢀⡤⠚⡁⠀⠀⢊⡴⠋⡠⠃
⠀⠀⠀⠀⢤⢀⠀⣀⡴⠚⢁⠁⠂⣤⠴⠋⡡⠔⠈⣀⠄⠊⡡⠔⠊⠠⠀⢀⣔⣚⣧⠴⠟⠛⠉⠁⣀⡤⠞⢉⠀⢀⠔⢁⡴⠿⠖⠋
⠀⠀⠀⠀⠘⣧⣞⣁⣈⣒⣢⣔⣋⣀⣔⣈⣀⣤⣊⣠⣴⣥⡤⠴⠶⠖⠚⠋⠉⠁⠀⠀⢀⣠⠴⠚⠉⠰⠀⡠⢀⣤⠞⡛⠠
⠀⠀⠀⠀⠀⠙⢢⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠉⠀⠒⠀⠀⡠⢊⣠⢟⡹⠊
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⡠⠐⢁⡴⠿⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠈⠉⠉⠉⠀⠀⡔⠉⢻⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣠⣴⠏⢔⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠈⢧⣶⣾⠎⠀⠀⠀⠀⠀⠐⠈⠠⠔⣲⡞⠑⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡶⣼⣿⡷⠶⠬⣄⣀⡠⣴⣬⣴⢊⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣿⡿⠿⡒⠾⠛⠙⠋⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⡶⠾⣥⣤⣄⡀
⠀⠀⠀⠀⣇⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣳⡍⣯⢹⠉⣯⢩⡇⢻⢩⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⣿⠿⠿⠷⠿⠧⠤⣤⣤⣀⣀⡀⠀⠀⠀⠀⣰⣫⢩⠏⡭⣩⠏⡽⢱⠏⠈⠘⠿⠶⢦⣤⣄⣀⠀⠀⠀⡇⢠⣞⣿⢿⡀
⠀⠀⠀⠀⠹⣶⠶⠶⠶⢤⣤⣤⣀⣀⣉⠉⠙⠛⠓⠾⠷⢧⣬⣬⣥⠯⠬⣵⠋⠀⠛⠰⠆⠦⢠⢀⡈⡭⣉⣛⢶⣧⠏⡰⠧⢽⠃
⠀⠀⠀⠀⠀⠙⣟⠉⠙⠛⠓⠒⠶⠯⠭⣽⣟⣛⣲⡶⠶⠤⣤⣤⣍⣉⣙⠋⠀⠀⠰⠀⠤⠀⡀⢀⠁⠳⠶⠜⠁⢩⡍⠁⠀⠸⢦⣀
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠭⡉⠛⠛⠒⠶⠮⠭⣭⣿⣛⣒⣶⠦⠤⣤⣌⣀⣙⣐⠂⠀⠀⠀⠀⠀⢀⣀⠘⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⠀⠀⠀⠀⢠⠎⢾⣴⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠙⠛⠒⠒⠲⠦⠤⠅⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠉⠉⠉⠉⠉⠒⠒⠒⠓⠒⠶⠦⠤⠤⠤⠤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣭⣵⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠶⠖⠒⠒⠒⠛⠛⠛⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠅⢀⢤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣴⣟⣊⣀⣱⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠤⢴⣇⣉⣉⡹⢥⣠⠤⠜⠐⠣⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢹⢛⡛⠿⢛⡒⠾⠟⠉⢩⡝⠉⠃⣢⡎⠀⠈⠿⠈⠧⡸⣢⢄
⠀⠀⠀⠀⠀⠀⢀⡆⠀⣨⣏⣀⣠⣴⣳⣤⠤⠤⣮⣥⡾⢷⡷⠶⠾⠛⠋⢉⡉⣈⢀⠡⠊⠑⢄⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⡖⠙⢿⡟⠛⠃⡂⠈⠥⠉⠭⠁⠄⠒⠒⢐⠐⠈⠉⠉⠇⡇⠀⢈⠘⡀⣀⠤⠂⠠⠤⠤⠤⠤⣶⣶⡇
⠀⠀⢠⣴⣖⢾⣴⠦⠼⢋⠀⣀⡀⣇⣀⣀⡠⠄⠧⠤⠶⠚⠐⠒⠒⠈⠁⠉⠉⢉⡀⢀⡀⠀⠀⠀⠀⠐⠋⠍⢁⡟
⠀⠀⠘⢏⠚⣊⣉⣉⠡⡤⣤⠤⠤⠀⠤⠄⠀⠐⠒⠒⠐⠒⠀⠀⠉⠉⠀⠀⣀⠀⠀⣀⣀⣄⣀⣠⣤⣤⡤⠖⣊⠝
⠀⠀⠀⠀⠙⢄⣀⡀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣦⠤⢶⡶⢶⣶⣶⣾⢿⡛⠋⠛⠋⠑⠒⠊⠉⠉⠁⠡⡄
⠀⠀⠀⠐⠄⠀⠈⠙⠉⠫⠭⡭⠅⠤⠤⠀⠀⠉⠁⠈⠉⠁⠐⠒⠒⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⢻⣿⣿⣿⠛⠛⢻⠛⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢶⣶⣾⣿⣿⣿⣾⣿⣿⣿⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣀⣀⣌⣉⣏⣉⠉⢹⡏⢹⡏⠉⠉⢹⠉⠉⠩⠉⠉⠉⡏⠉⢹⣉⣉⣉⣓⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠄⠀⠘⠀⢀⣼⠞⠉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠶⠀⠀⠐⢋⠞⠃⠀⡄⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⢹⣇⠀⠀⢀⠁⠆⢀⠀⣐⡀⠠⠧⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠂⠁⠀⢨⢄⠀⠀⠈⢸⠀⠀⢨⠔⡈⠔⢈⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠐⢎⠀⠀⠀⠘⠃⣀⠔⠉⠔⠈⠀⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠤⠀⠠⠀⠂⠘⠄⠀⢀⠤⠋⠀⠀⢀⠀⠄⢐⡈⠤⠀⠂⢠⡤⠷⠀⠐⠂⠉⠓⠒⢀
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠔⠁⠀⠀⠀⠀⠈⠀⢒⡀⠤⠐⠂⠉⠀⠀⠀⠀⠀⣀⠀⠀⢡⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡇⠀⠀⠀⠀⠀⢐⡈⠄⢸⡀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡨⠀⠀⠐⠆⠼⡆
⠀⠀⠀⠀⠀⠀⠀⡰⡄⢲⣇⣠⡤⠖⢃⢸⠽⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠂⡗⢀⠁⠈⠃⠀⡇
⠀⠀⠀⠀⢠⠜⠋⠁⣁⠞⠋⡁⠄⠒⠉⠀⠀⠀⠀⠠⠀⠀⡀⠠⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⠑⠶⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠠⠐⡈⠄⠂⠁⠀⡀⠀⠀⡂⠨⠀⠐⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⡜
⠀⠀⠀⢈⡁⠐⠀⣀⠄⠠⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡱
⠀⠀⠰⠀⠒⠉⢈⠀⣠⠀⢐⠄⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠟⠁
⠀⠀⠀⠱⠒⠶⠶⠤⠤⠤⠤⢤⡤⠤⢤⣄⣤⣠⣀⣠⣄⣀⣀⣀⣀⣀⣀⣄⣀⣀⣂⡤⡤⠴⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀
⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢰⣧⣤⣀⡀⠀⠀⠈⠀⡀⡀⠀⠈⠓⠀⠀⠀⠀⠀⣀⣀⣠⡀
⠀⠀⠀⠉⠙⠒⠳⢮⣭⣔⡲⠧⢘⣛⠖⠿⣿⣿⣿⡿⠿⢿⡟⠓⡀⢤⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠒⠠⠀⠀⠀⠀⠉⠉⠛⠃⠉⠉⠛⠒⠶⠞⠛⠓⠀⠙⠛⠉⠠⠀⠀⠀⠈⠃⡒⠛⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠴⢛⣒⣡⡂⠀⣀⣤⠀⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠭⢿⡭⢖⡿⢿⡿⡆⣸⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠉⠑⠢⠔⠲⠤⣄⡀⠀⠀⠀⡈⣽⣚⣩⡼⠉⠀⠀⠈⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠉⠓⠲⢤⣌⡙⠋⠙⠃⠤⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⣈⠔⢁⣴⣶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⢷⣦⣄⡀⠀⠀⠀⠀⠀⠐⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣦⣤⣤⡾⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⢠⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡚⠓⠚⢒⡒⠒⠒⣒⠚⠓⠛⣒⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠫⠜⠀⠀⠧⠟⠀⠘⠧⠇⠀⠘⠭⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡩⡍⠉⠉⣩⣍⠉⠉⣩⢍⠉⠉⢉⢭⠉⠉⢉⣭⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣞⣀⣓⣋⣀⣀⣑⣊⣀⣀⣙⣚⣀⣀⣘⣚⣁⣀⣈⣒⣁⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠚⠉⠉⠓⠶⠖⠋⠉⠒⠶⠶⠊⠉⠙⠒⠶⠚⠉⠉⠓⠦⠖⠋⠉⠙⠲⠼⠧⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠄⠀⠀⠀⠀⠀⣴⣆⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣠⠀⠀⠀⠀⢰⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣲⣿⣖⠀⠀⠀⠀⣿⣿⣿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡏⠃⠀⢀⣹⣿⣿⣿⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠓⣶⢠⢙⣶⣶⣶⡞⠛⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠠⠿⠶⠦⠤⠼⠿⠿⠿⠿⠿⠯⠭⠽⠿⢿⣟⣻⠿⠿⠿⠿⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠓⠒⢶⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣼⣿⣿⣿⣿⣁⣇
⠀⢸⡏⢉⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣓⣒⠒⢲⣶⣶⣶⣶⣯⣭⣭⣭⣽⣄
⠀⠀⠷⢶⠆⠀⠀⠠⠤⠤⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠭⠭⠭⠭⠭⠍⢉⣉⣠⣿⣛⣛⠛⠻⠿⠿⠿⠿⠿⠽
⠀⠀⠀⡎⠀⠀⢰⣿⢻⣿⣾⣿⡿⣿⣿⣧⣿⣿⣇⣿⣿⣧⣿⣇⣿⡇⠀⠀⠀⢰⠃⠀⢀⣀⣀⣀⣠⣴⣋⣹⣶⠶⠶
⠀⠀⢸⠁⠀⠀⣰⣦⣶⣦⣶⣶⣤⣶⣶⢠⣶⣦⢤⣭⣭⢠⣭⢠⣭⠀⠀⠀⠀⢸⠀⠀⢸⠉⠉⠉⠀⣿⣿⣿⣿
⠀⣤⠗⠤⠤⠤⠿⠧⠿⠭⠿⠿⠬⠿⠯⠽⠿⠿⠽⠿⢯⣝⣛⣙⣛⣀⣀⣀⣀⡼⠀⠀⢸⣦⣶⣶⣿⣿⣿⣿⣿⡤⣶
⡜⠓⠶⠶⢶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠞⢿⠀⠀⠀⢸⣟⣿⣿⠿⣛⣫⣭⣶⣿⣿
⠙⢿⣶⣶⣶⣤⣤⣤⣤⣤⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣶⣶⣿⣟⣩⣵⣶⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡤⠀⠠⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠈⠀⠀⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠤⠄⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢡⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣠⠄⠀⠀⣀⣠⣠⣤⡶⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣴⣤⣤⣄⣤⣶⣶⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠉⠉⠭⠭⠩⠭⠍⠍⣉⠉⣉⠐⣯⣮⣉⣙⣿⣾⠆⢉⠻⣎⠓⡻⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠊⠉⣉⣉⣉⣩⣥⣶⣤⠌⠙⠿⠛⠛⠿⠒⠈⠀⠈⠉⠉
⠀⠀⠀⠑⠒⠒⠒⠀⠀⠀⠸⠍⢀⣀⣈⡉⠉⠉⠁⠉⠙⠉⠀⠉⠀⠀⠐⠒⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢠⠃⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣀⡀⠠⡯⠣⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⠀⠀⠀⢸⠀⠀⠠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣼⣏⣩⣖⢦⣤⣄⣀⣀⠀⢈⡇⠀⠀⠁⡀⢀⣆⣩⣤⡶⠃
⠀⠀⠀⢀⣀⣊⣛⣿⣷⣴⣶⣶⣦⣼⣿⣿⣿⣿⣓⡻⠶⠾⠶⠾⢒⡚⢳⣿⣿⣭⣉⣙⠝⢁
⠀⠒⠀⠀⠿⡉⠹⠿⠶⠟⡉⠽⢯⣭⠋⡐⠿⠿⠧⠽⠿⠟⢀⣽⣿⡋⠙⣛⡂⠘⠛⠯⠐⠛⠀⠀⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠀⠖⠐⠛⠛⠧⠄⠀⠛⠂⠰⠦⠉⠉⠡⠄⠀⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣈⣶⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠇⠀⢏⡉⠛⠂⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠑⢄⠀⠀⣸⠀⠀⠈⢳⣄⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡳⠀⠀⠀⠑⢀⡟⠀⠀⠀⣸⠛⠣⡀⠀⡿⠀⠒⡶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⡀⠀⣴⠛⠡⢤⠐⡄⠀⠀⠈⠓⡀⢀⡴⠣⠀⠀⠈⠲⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣽⠁⢄⠀⠈⠙⢃⢠⣤⣴⣟⠋⠶⠖⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣾⠀⠀⠙⠢⡀⠀⢩⣯⠻⣿⣿⣶⡆⠠⢑⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣿⢠⠀⠀⠀⠈⠱⣾⢀⠷⠈⠻⣿⡧⠀⠀⢹⡦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠐⢤⢆⡜⢤⡀⠀⠀⠀⠀⣬⣿⢸⡆⠀⠀⠑⢄⠀⣾⠁⠀⠐⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣀⣀⠀⠂⣿⠃⣼⡁⡀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠛⢿⠁⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠢⢄⡀⢰⡿⠀⠓⢈⣡⣗⣤⣿⢏⢬⡀⠀⢀⡀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⢤⣀⣤⠀⢭⡼⠾⠑⠍⢉⢜⣷⠒⡄⣠⡄⠀⠀⠀⠀⠉⠢⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣄⡙⢷⣤⣿⡿⣟⡑⢶⡶⢾⣭⣀⠁⢡⠇⢢⡄⣰⠄⠀⡄⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢡⠀⠈⠙⠿⢾⣿⣄⠀⠉⠳⠶⣄⣻⡟⠈⠉⢢⡄⠁⠉⣿⠀⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⠀⠀⠀⠀⠆⠙⣯⠀⠀⠀⠀⠀⠉⠳⠢⢄⣿⠀⠈⠐⣜⠃⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠀⣼⣷⣠⠀⠀⠀⠈⡎⣿⢀⡖⠁⠀⠀⠀⠀⠀⠀⠉⢲⢶⣀⠘⢗⠢⢽⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡘⠸⠎⣰⣻⣿⠹⢆⡆⢠⡞⢰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣿⣿⣁⠀⠈⠒⠠⣀⠀⢀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠔⠀⠨⢻⣿⣽⣏⡋⣶⢿⡟⡡⣿⣿⡇⠀⠘⡆⠈⡀⠀⠀⢀⠀⠠⣴⢀⡝⣿⢿⣆⡄⣳⡀⠀⠉⠛⢅⡀
⠀⠀⠀⠀⠀⠃⠐⠒⠒⠻⡿⣿⡿⢝⣿⡿⠁⣸⣿⣿⣇⣿⡀⣇⢸⣿⢠⠆⡈⠀⠀⣿⣸⠀⣾⣿⡟⠃⡧⢠⢀⣦⠀⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⢀⢀⣀⢳⡾⢀⣓⣈⣭⠥⣽⣯⣟⣻⢻⡏⠉⢱⠈⠋⠈⣀⣋⣠⣼⣽⣧⣼⣯⣭⣡⣤⡅⠐⣻⢱⠀⡸⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡿⢿⢿⣿⣿⣽⣿⣄⣀⣠⣌⡛⡈⣝⡤⢶⣶⣿⠿⢛⣩⣭⣤⡭⠽⢿⡿⠿⣿⣿⣿⣿⡀⠀⠀⠀⠢⠰⣴⡿⠁
⠀⠀⠀⠀⠀⣸⠇⠰⣚⠞⠈⠙⢹⣿⣿⣿⣯⡤⡾⢿⡖⣋⠣⠞⠛⠙⠁⠠⠤⠤⠤⠤⣤⣷⣿⣾⠇⣿⣿⡄⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⢿⠀⢒⠃⠈⢥⣀⠒⠠⠽⠛⣽⣯⣿⣿⢿⠿⠂⠀⠀⠀⠀⠒⣀⣀⣀⣀⣒⣻⣿⢏⣾⡉⡽⠓⠀⠀⠂
⠀⠀⠀⠀⠀⢸⡄⠀⠉⣀⠒⠢⠄⠤⠄⢨⢏⣉⢺⠙⣦⣀⠈⠀⠀⠀⠀⠁⠤⠤⣤⣭⣿⡿⣵⡟⢣⠋
⠀⠀⠀⠀⠀⡸⣿⡍⢐⡀⠤⢍⣁⠐⠂⣚⢀⠈⣩⡾⢷⡤⠄⠒⣂⠀⠤⠐⢒⣠⣶⣾⢟⣾⢇⡘⠁
⠀⠀⠀⠀⠀⠷⠞⡛⠘⣿⣷⣒⣲⣒⣀⠩⣇⣀⡴⠾⠛⠓⠛⢟⣣⣴⣶⣭⠴⢶⣿⣟⣾⢁⣮⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠈⠉⠉⠙⠁⠈⠉⠀⠀⠀⠉⠉⠉⠉⠛⠛⠓⠂⠀⠙⠻⠿⠷⠾⠖⠋⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⣿⠈⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢒⣛⡓⣒⠒⠒⠛⠒⠒⠶⠶⠾⠶⠶⠶⠶⠶⡀⠔⠊⠀⣷⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣙⣿⣿⣿⣿⣿⣿⣿⡟⠿⡷⠶⣶⣾⣷⣶⣶⣶⣶⠿⠶⠶⠶⠼⠶⠶⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠿⠿⠿⡿⠿⠿⣿⡿⢋⣴⣿⣿⣿⡿⠿⠿⠛⢻⡿⠿⠿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠒⣒⣚⣛⣛⣒⣒⣒⣒⣒⣻⣟⠒⠒⢲⣶⠿⠛⠛⠛⠒⠒⠒⠒⠀⣼⡇⠀⠀⠀⠈⠙⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⢛⣛⣓⣒⣒⣒⣒⣒⣒⣲⠈⠓⢲⡶⡶⠶⠶⠶⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠛⢛⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⠶⣛⢅⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡏⠀⠀⠀⣠⠾⣫⣵⡿⢣⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣻⡇⠀⠀⠈⠛⢛⣻⣿⣿⣿⣿⣿⡇⣿⣿⣿⡟⠀⢀⣴⢟⣵⣾⣿⡟⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠯⢿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⠟⢀⡴⢋⣵⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⢋⡴⣋⣴⠿⠿⠿⠿⠋⢰⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢳⣶⠀⠀⢀⣛⣻⣿⣿⣿⣿⣿⣿⡿⢡⣟⡵⢋⣡⠥⢶⣒⡚⣛⠃⠀⠀⣹⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⣽⡇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⠿⢛⣫⣵⣶⡿⠟⣋⣱⠃⢠⣆⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣛⣿⣥⠾⠟⠋⠉⠁⢸⡟⢀⡤⢖⣫⣵⣾⡿⣟⣫⣵⣾⣿⢋⡟⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣾⡟⠀⠀⠀⠀⠀⢀⡿⣛⣥⣾⣿⣿⣿⣷⣿⣿⣿⣿⣶⡟⣾⠁⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣭⠿⠧⠤⠤⠤⠤⠤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⠇⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣭⣴⣶⣿⠛⣛⡛⢛⣟⣳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠏⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⣿⣧⣼⢟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⠏⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣳⠏⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡽⠁⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓⠋⠠⠴⠿⠿⠟⠛⠛⠛⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠽⠿⠿⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣾⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠄⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⢠⣶⣤⣿⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢤⠈⠹⠿⣧⣀
⠀⠀⢘⣻⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢰⣬⣾⣦⣄⣨⣿⣟
⠀⠀⠨⣿⣿⡇⢀⣾⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡗⠋⠙⣿⡈⠁⠚⠻
⠀⠀⢸⡇⣿⣿⢴⣿⣍⡇⠀⠀⠀⠀⠀⠀⣀⣤⡤⠐⠀⠐⠒⠒⠀⠀⢠⠤⠌⠐⢀⢐⣿⡆⣐⣿⣶⣷⠎⣳⣴
⠀⢸⣿⣯⣿⣿⣿⣿⢻⡇⠀⠀⣠⣴⡶⢛⡉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡽⣾⣿⣿⣿⣅⣝⣫⡤⠿⡟
⠐⠛⣿⡗⢿⣿⡿⢿⣿⣧⣶⡿⠟⠋⢉⡉⢺⣷⠤⢄⣠⣤⣄⣀⣀⣀⣀⡀⠀⣾⠱⣿⣿⡛⢿⢿⣭⡼⢿⣾⣷
⢰⢏⣽⣷⣼⣿⣯⣿⣾⡏⣃⣀⣠⠤⢺⣇⣀⣯⣤⣤⣤⣤⣤⣤⣤⣬⣸⣉⡸⢿⣯⣉⠹⣷⣾⣿⣿⠛⢻⣏⢳
⢠⣾⣿⣦⣿⣿⡡⣽⣿⡷⠛⣉⣤⡶⠿⣛⣩⣭⣤⣤⡶⠿⣿⣯⣽⣟⡛⠿⢿⣿⣧⡙⠶⣼⡟⠻⣿⣯⡉⢹⡄
⠸⢿⣽⣥⣽⣿⢓⢉⣠⣿⠾⠋⣡⣴⣿⣿⣿⠏⠒⠀⠀⠀⠀⠀⠈⠉⠛⠿⣦⣟⣿⡛⢷⣿⡙⠲⠜⣿⣿⠟⢀
⠀⠛⢹⠛⢿⣿⢹⣿⠋⠀⢀⣴⣿⡽⣿⣷⣿⣶⣤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣰⡏⠛⣷⣯⣽⣷⣶⣿
⣿⣶⣼⣄⣸⡏⢹⣧⣆⣴⣿⣿⡿⠆⠠⠀⠀⠀⠄⠉⢛⣿⣶⣦⣤⣤⣄⣀⡀⢈⣿⣿⣇⠀⠉⢿⣯⣭⣏⡬⢿
⢬⣽⣽⣿⣿⣿⣾⣿⢿⣿⣿⣿⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⠿⣿⣿⣿⣦⣀⣿⣿⣙⣃⣀⣀
⣀⣠⣽⣹⣿⠿⠛⣿⣀⢀⡘⣿⡏⠿⠿⠛⢻⣿⣷⠿⣿⠿⢟⣻⣿⣿⣷⣾⠾⣿⣿⣿⣻⣍⢉⣿⡿⡛⠿⠿⡿
⢹⣿⣿⠛⣾⣤⣤⢻⣏⡻⢿⣝⣿⣾⣟⡛⣿⣿⣛⣛⣿⣿⡾⠿⠛⠋⠁⠀⢠⣿⣿⣿⣩⣿⣿⡟⢈⣿⢠⣴⡦
⠰⢿⢻⡐⣽⣯⣿⠀⣯⡟⢿⣿⣿⣿⣿⣽⣿⡿⠷⢚⠩⠀⠀⠀⡀⣀⣤⠶⢛⣡⣿⣿⣿⣿⣿⣄⠀⠀⠀⠙
⢰⣿⢸⣷⣿⣿⢹⡆⣿⠀⠲⠀⢉⣻⢿⣿⣿⣿⣿⡷⠶⠿⠿⠿⣻⣿⣱⣾⣿⣿⣿⡏⠟⢿⣻⣿⠷⣤⣆
⢠⡷⢺⣯⣿⣿⣹⠁⣿⢖⠀⠀⠀⣩⠴⠇⠉⠉⠉⠛⠛⠛⢛⡛⠋⠉⠉⠍⠃⠸⣿⠃⠀⠘⣿⣿⣆⣸⣿⣇
⠘⣗⣻⣧⡟⣿⡟⠀⣿⠈⠃⢒⣋⣠⠐⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⣷⡄⣠⠸⢻⡏⡿⣿⠿⣶
⠀⣹⣮⣿⡀⢿⣿⠀⢹⣃⠈⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡸⢆⢈⡁⠳⠉⡆
⠀⠀⣾⣿⠀⠘⡏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⡇⠀⠺⣷⡄⢀⠉
⠀⠈⢯⣿⢀⡀⠹⠄⠀⠀⠀⠐⠂⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⣄⠿⡙⠛⠁
⠀⠀⢸⡏⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣧⠉⠻⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⢻⣛⢿⡀⠘⣿⣾
⠀⢀⣤⡧⠀⠀⠀⠀⠀⣸⣿⣤⣸⣿⣯⡀⠀⠙⢷⡀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣽⣾⣟⣿⣿⡿
⣤⣤⣬⣤⣶⣤⣀⢀⣾⣿⣧⣁⣈⣿⣿⣷⣀⣀⣈⣻⣧⣀⣠⣤⣤⠖⢲⣾⡿⢿⣯⣩⣿⣿⢻⣿⣿⣏⣿⣿⣿
⠈⠈⢻⡿⠘⠻⠿⠿⠿⢿⣿⣟⣛⣛⠻⠿⠿⠛⢛⣻⣯⣭⣥⣤⣽⣿⣽⠿⠷⢼⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⣴⡖⢤⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣦⡀⠀⠀⠀⢸⡆⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡷⠘⣏⠑⢄⡀⢸⡇⠀⠀⠀⣿⡿⠁⠢⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠘⠀⠘⡀⠀⠙⢾⣧⠀⠀⠀⡏⠁⠀⠀⠈⢂⣄⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡏⠀⠀⢀⠄⠀⠀⠀⠀⠀⢉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡆⠀⢸⠀⠀⠀⠀⠀⠀⠋⢹⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢼⡇⢀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢹⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣾⢷⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⡿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠃⠀⠀⠀⠀⠀⠁⢈⣷⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢻⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⡛⠠⠁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⢿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣼⠀⢣⠀⠀⠀⠀⠀⠀⠈⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⠀⠀⠀⠒⠻⣿⠟⣋⠴⣶⠶⠦⠄⠀⠀⠀⠀⠀⠾⢶⡁⠀⠀⠀⠐⠂⢀⣠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⣟⣀⠠⠘⣁⡀⣀⣀⣀⠤⣈⣿⣿⡿⢿⣿⠀⠒⠒⠛⠀⠀⠀⠀⠿⠿⠿⠶⢀⣠⣴⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠐⠛⠐⠊⠤⠤⠤⠄⠀⠀⠀⠀⠉⣻⣻⡿⣾⣿⡆⠀⠀⠀⠀⠀⢠⡤⣟⣭⣴⣾⣿⣿⣿⣿⣿⣿⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡃⠀⣸⣿⠏⢀⣀⣤⣶⣾⣿⡿⢿⣿⠿⠛⠃⠀⠈⠙⠃⡠⠎
⣀⣀⣀⣀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣬⣭⣶⡿⠛⠀⣴⡆⠀⣺⣿⠂⢀⣄
⣿⣿⣿⣿⣿⣖⠀⠑⠪⠄⡀⢀⣤⣀⠀⠛⢿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⣟⣻⣿⣭⠉⠁⠉⠉⣴⣶⣾⣿⣿⡇⢸⣿⡿⠗⢸⣿⡷⣄⣀⡀⠀⢀⡠⠀⡀
⠛⠛⠛⠳⠆⡀⠀⠀⠀⢰⣬⣀⠉⠙⠛⢶⣶⣭⣭⣭⣭⣭⣿⣿⣭⣤⣼⣿⣿⣿⣿⣷⣦⣿⢿⣿⣿⣿⣿⣿⡟⠘⢿⠀⠀⢾⣿⣿⣏⠻⢿⣿⣷⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣄⠈⠉⠻⠿⠟⠻⣯⣝⣛⡛⠛⠋⠉⠀⠀⠈⠉⠙⠷⠀⠉⠉⠉⠀⠀⠀⠀⠀⠆⠸⣿⣿⣛⠂⠀⢨⣝⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⣄⡀⠀⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⢾⠗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠿⣿⢤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡖⢻⡆⠠⠈
⠀⠀⠀⠀⠀⠀⠙⢗⠒⠒⠶⠼⣧⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠁⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠉⠁⣿⡿
⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣿⣴⣿⣛⣻⣿⡿⣿⡾⣦⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡵⢌⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠲⠤⠶⣼⠇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡀
⠀⠀⠀⠀⠀⠰⠧⠐⠒⠲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⢐⣃⣹⣿⣠⡤⠬⣽⡿⠶⠖⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⢀⣿⣄⣠⣴⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣻⡷⣧⣙⠤⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠾⢿⣛⣿⣿⣿⣿⣍⡉⣴⣜⣯⡾⠞⠓⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⠀⢻⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⠛⠻⣿⣍⢙⡿⢿⣿⡇⢠⣿⡿⣼⢀
⠀⠀⠀⠀⠀⠀⣨⠿⣿⣿⣿⣯⣮⡀⠛⢛⣤⢀⠈⠛⠲⢿⡟⠓⢿⣇⠳⣄⡀
⠀⠀⠀⠀⢠⣶⣿⠆⠠⠀⠘⠉⠉⠉⠒⠀⠀⠈⠠⠂⠀⠠⠄⠀⢘⠻⠿⢮⣉⠁⢤⣴⣶⣤⡤
⠀⠀⠀⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣤⣽⣄⣈⠙⠳⠦⠀⠀⢀⣠⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠋⠉⠐⠀⠈⠙⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣧⣸⣯⣝⣿⣭⣉⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠀⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣿⣶⣶⣤⣀⣀⠀⠀⠀⡀⢢⣀⠀⡀⠀⠀⠐⣆⡀⠄⠀⠀⢀⠦⠀⡷
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⢀⣀⡠⣞⣠⡀⠀⣻⣿⣿⣿⣿⣿⣷⣶⣟⣡⣬⣿⣿⣅⣀⡀⢀⣈⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⢀⣀⠁⢪⢁⣴⣦⣴⣼⣿⣾⣿⣿⣷⣶⣿⣿⠿⠿⢿⣿⣿⣿⣿⠟⠛⠛⠋⠛⠻⠲⢶⣶⣄⣀⣄⠽⡿⢆⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⢀⠉⢻⡿⢿⣿⡿⠛⠉⣁⣉⣀⣿⣻⠛⠻⣿⡿⣿⣟⠋⠛⠐⢻⣷⡜⠀⡹⠀⢉⠙⢿⣿⢷⣆⣠⠀⠉⡙⠟⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⢿⣛⣛⢈⣹⠛⠛⠛⡧⡙⢀⣾⣷⣾⣷⣦⣤⡴⠼⠿⢿⣖⠓⠟⢺⡚⠿⡯⣤⠬⢨⡆⠼⡯⣤⣁⣀⣀⣁⡀
⠀⠀⠀⡠⠤⠀⠀⡄⢠⣤⡤⠈⠀⠀⠀⢀⡤⡀⣄⠡⠫⠿⠽⠟⠛⠛⢛⡛⠋⠩⠄⣈⣿⣾⣄⣤⣈⠣⡄⡀⢠⣩⣽⣾⣿⣿⡿⠯⠄
⠀⠀⠠⠤⠤⠤⠤⠄⠤⠷⣈⡅⢾⠐⠶⠖⠶⠷⠷⠖⠒⠒⠒⠒⠚⠒⠚⠉⠛⠋⠉⠉⠙⠛⢿⡿⣦⣄⣭⢉⣐⣈⣀⣀⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠻⣇⢻⡷⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢁⣠⡄⠀⠀⠀⠒⠷⠈⡛⠉⠐⠓⢚⠛⠛⠿⠥⠐⠶⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⢂⣀⣹⣐⣏⣸⣧⡄⢈⠅⡀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠽⠛⠁⠈⠁⠸⢣⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣆⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠛⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣿⣶⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢠⣿⠹⡜⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡴⠁⣿⡿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⣠⠎⢸⠀⣿⠀⢸⣿⠀⠇
⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⣠⠞⠁⠀⠀⢸⣿⡿⢿⣿⡄⠸⡀⠀⢀⡀
⣤⣤⣼⣦⣶⣅⣀⣀⣀⢳⡞⠁⠀⠀⠀⠐⢸⣿⠀⢸⣿⢹⡀⢣⠀⢐⣂
⠀⠂⡀⠂⡀⠈⠉⠉⡙⢸⣫⣙⣩⣿⣿⣿⣿⣿⣶⣾⡿⣶⣿⣬⣦⣸⢿
⠀⠂⠓⠂⠁⠈⠀⠀⠉⢻⡏⠀⠀⠀⠀⠀⠀⣿⣜⣯⡏⠀⠀⠉⠉⢹⡿
⠀⢀⠔⢚⣫⣐⠢⡀⠀⢸⡇⠀⠀⠀⠀⠀⠐⣿⡳⣏⡷⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⡴⣇⠆⠁⠀⠀⠁⠜⡀⣼⡇⠀⠀⠀⠀⠀⢠⣿⣉⣹⣿
⣠⠃⠀⠀⠀⠀⠀⠈⣘⣾⡇⠀⠀⠀⠈⠉⠉⠉⠉⠙⠋⠁⠈⠁
⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡟⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠉⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠆⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠚⠈⠀⠈⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣀⣀⣄⠀⠀⢄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡴⣿⢻⣏⢙⣁⠀⠈⠈⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠵⠓⠛⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡠⠀⣸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠈⣤⡀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣀⣴⣺⡥⢿⣤⣉⣰⣄⣠⣨⣦⣼⣅⣀⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⡴⠞⣛⣩⡤⠷⠞⢻⠚⠛⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⢿⠚⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣃⣢⠘⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡄⣘⡿⣿⣿⣿⣿⣯⣝⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⠞⠁⣹⣁⡟⣻⣔⣻⣿⣿⣿⣷⣮⣟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⣦⣤⢿⣻⣮⣷⣿⣿⣿⣿⣿⣿⣿⡏⠈
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠀⠛⠉⠉⠙⢿⡿⠋⠄
⠀⡂⠀⠀⠀⠀⢤⠤⣤⣤⠤⠶⠦⢤⣤⣬⣿⢿⠿⢁⣤⡀⠀⢀⢢⣼⡇⠀⡄⠖⠠⣤⠄⠀⠀⠀⠠⠶⠦⠠
⠀⠂⠀⠀⠀⠀⠀⠀⠡⢶⣾⢂⣶⢶⣦⣤⡤⠀⠐⢺⣽⡿⠈⣽⣾⣿⣇⣀⠻⢿⣿⣃⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢶⣶⡬⢙⣠⣴⠚⠿⠃⠀⠈⣛⣯⡌⠻⣿⣦⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣶⣶⣶⣿⣿⡿⢯⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠛⠲⠶⡤⣈⡙⠛⣿⣿⡿⣿⣿⣟⠋⠛⠙⠛⠛⠛⠻⠃⠀⠈⠉⢉⠉⠙⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢂⠀⢆⣧⣤⣈⣓⠀⠸⣧⣀⣀⣀⣀⠀⣠⣤⡀⠀⠀⣰⣦⣤⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠒⠂⠈⡄⣷⡂⠀⠈⢹⡆⠀⢀⣀⣈⡉⠉⠀⠙⠛⠓⠂⠘⠓⠂⠀⠀⠋⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠘⡀⣄⠀⠀⠠⣙⡀⠀⠓⠚⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠃⠀⠐⠂⠚⠁⠙⠀⠀⠀⠈⢷⡄⢰⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢓⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉



⣶⣶⣶⢶⠶⢶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⣿⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⢸⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠀⣿⢸⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⡇⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡇⢠⣿⣘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⡟⣿⡏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⣿⡷⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠏⠀⣿⡷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⠀⡿⣟⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⣀⠰⡿⠿⠿⠿⡟⠽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⡗⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⠀⠅⠀⠈⠁⠀⠀⠀⣿⣴⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣾⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⡀⠸⠽⠜⡀⣀⠀⠀⠘⡆⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⢠⠀⠀⡁⠀⠀⠀⠀⠿⢻⣻⣿⢐⢸
⠀⠀⣠⣀⣀⡀⠁⠀⣠⢀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣅⣒⠭⢝⡀⠀⠀⠃⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠸⠁⠀⠀⠀⠀⠀⢘⣀⣸⣿⣿⢐⢼
⠭⠽⠿⠯⠿⠿⠾⠽⠿⠈⠏⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡟⠉⠉⠉⠉⠙⠋⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠛⠛⠙⢻⣿⢻⣿⣿⢸⢸
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣴⢀⠾⣦⠸⣿⣿⣟⣿⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠹⠛⠁⠀⠀⠀⠀⠀⢠⣰⠀⢰⣶⠀⢰⡇⠀⠀⠀⠀⢸⣿⣿⣷⣿⣷⣿⣿⡯⣿⢈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠲⠞⠫⠍⠥⠶⣺⠍⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⡘⠛⠂⣸⣿⡷⠤⠤⠤⢤⣿⣿⣿⣿⣿⣿⣿⡇⣿⠨⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⢉⣉⣥⣶⣿⡼⣥⠭⠙⠟⢸⡇⢴⣿⣼⡿⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⡇⣿⡄⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡈⠉⠀⠀⠀⣀⣠⣴⡶⠟⣻⠍⢽⠛⠈⣉⣥⣾⣿⣟⣫⣴⡿⠿⠏⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣤⣶⠾⠋⣉⡠⢾⠒⠉⠐⣂⣤⣶⠿⡛⣉⠝⣛⡟⠙⠉⠀⠁⠀⠀⠈⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⣿⣷⣿
⠀⠀⠀⠀⣀⣤⡴⠞⠛⣿⣿⣿⠀⣷⣾⠋⠉⠳⠬⠤⠶⠿⠏⡭⠥⢿⣿⣴⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣻⣿
⣤⠶⠟⠛⠉⢀⣧⡤⠔⢿⣿⡏⣰⡿⢿⣿⣷⣷⣶⣶⣶⣞⣢⠖⢒⣿⣿⣿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠶⢾⡷⣿⢿⣿
⠀⣀⡤⠔⠚⠉⣇⠀⣠⣿⣿⡗⠞⢛⣩⡙⠛⠓⣒⣧⣬⣭⣥⣿⡿⢿⠿⠁⠀⠀⢀⣀⣐⡒⠐⠒⠒⠂⠀⠒⠒⢒⡒⠚⠛⠛⠛⠚⣗⣿⣿⣿⣿⣿⣿
⠭⠤⡤⣤⣤⣘⣿⣟⣩⠼⣿⢀⣤⠾⠛⣁⣴⣾⣿⣿⣿⣿⠟⠋⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⣭⣭⣭⣭⣭⣭⣭⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣴⡆
⠀⠀⠀⢸⣿⣿⣶⣤⣀⠈⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣄⡀⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⠉
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠉⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠇⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠈⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣴⠟⠉⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣷⣦⣀⣀⣉⣛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠋⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡏⠉⠉⣿⠉⠛⠿⣿⣿⣯⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢹⣿⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⠀⠀⠀⠙⢿⣿⢦⣉⣷⣤⠀⠀⠀⠀⠐⣿⣄⡇⣘⣿⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠻⠀⠀⠀⠀⠀⠀⠆⠀⠀⠙⠙⠀⠀⠀⠀⢀⣃⣸⣇⣿⣿⣐⡂⣀⠀⡄⣸⣧⣸⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⠛⠛⠛⠛⠻⠷⠶⠶⢦⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⣾⠈⢸⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠶⠶⠶⠦⠬⠭⠭⠭⠙⠛⣓⣒⣶⣾⠤⢼⣤⣀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢠⠘⡅⠀⠀⠈⠁⠒⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⢤⣤⣤⣤⣶⣖⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠖⠛⠉⠉⣀⣴⡾⢛⣿⣿⡟⠀⢹⡷⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⠄⣀⠀⠀⠀⠀⠀⢀⣀⠤⡤⣀⡀
⠀⠀⠀⠀⠀⢀⣤⣾⣿⠋⣰⣿⣿⣿⠃⠀⠀⣿⠸⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠚⠁⠀⠀⠀⠈⠓⢦⣤⣠⠖⠋⠀⠀⠂⠀⠙⠲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⣴⠆⠰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⣾⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢲⡯⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠋⠉⠀⠂⡀⠉⠀⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠠⠄⢀⣔⡏⠀⢰⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢿⣇⣀⢸⡇⢰⡶⣄⣀⠐⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠈⠙⡛⣸⠃⠀⠉⠙⠻⡾⠧⣤⣛⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⣤⠄⠀⠀⢨⣵⡏⠀⠀⠀⠀⠀⢹⡄⠀⠈⢿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠚⠂⠲⠀⡜⣨⡿⠀⣀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠛⠛⠿⠿⣿⣿⣾⣶⣿⣥⣶⣿⣿⡛⠛⠻⠿⣿⣆⡉⢰⡦⢿⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢩⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠯⠿⣿⠃⠀⠀⠀⠀⠀⠉⠙⠳⢾⡧⣤⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠚⣇⠀⠀⠀⠀⠀⠀⡠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣠⣾⠟⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠈⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣰⣃⢋⢌⣼⣦⣗⣴⣷⣧⣾⣿⠋⠀⠐⠇⠴⠃⠀⠀⠀⠀⠀⢰⣽⠃⠀⣀⣀⠐⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣤⣬⣅⣟⣻⣟⣿⡿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣶⣤⡀⠀⡀⣀⢀⣼⡿⢀⣴⠟⠉⠓⢢⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣉⡉⠛⢻⡿⣿⣿⣷⣮⣿⠋⠰⢟⡿⠿⡿⣿⡿⣿⣷⣦⣤⣤⣽⣿⣿⣷⣿⣧⣦⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⠷⢤⢄⣉⠛⠃⠀⠀⢈⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⠆
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣠⣞⣛⢻⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠏⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⣀⣸⠃⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⡅⡟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣭⡤⣄⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⢂⠀⠀⠀⢴⢞⣛⣩⣥⣶⣤⣿⣿⣿⣿⣿⡿⣿⢟⣩⣾⣏⠀⠀⠉⠿⢿⢿⡿⠿⠿⢿⡅⠾⢿⠀⠀⠂⠀⢀⣤⡞
⠀⠀⠀⠀⠀⠈⠳⣤⣀⣐⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣟⣛⠛⠿⣿⡅⠀⠀⠀⠀⢠⠷⣍⠢⣶⣿⣧⣀⢤⣀⣀⣰⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠐⠓⠶⢚⣋⣩⣭⣭⣭⣏⣉⡛⠿⣦⣝⠂⠀⢐⣕⣾⣶⣮⣷⣮⣽⣿⣿⣿⣿⣿⣟⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⣀⣁⣐⣨⣽⣿⣿⣿⣿⣿⣿⣿⣄⠈⠛⢷⣄⠀⠉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣭⣭⣤⣶⣾⣿⣿⣿⣿⢿⡏⠁⢾⣆⠙⠶⣤⣈⣛⠿⠿⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢈⠙⢿⣿⣧⡍⠙⠀⠶⢿⣷⣴⣤⣉⣉⣛⡛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠻⠶⠦⠉⠀⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣼⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣤⣤⣄⠸⢆⡀⢠⠀⡄⠀⢠⠀⠀⢀⠀⠸⢾⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠦⠴⠂⢻⣿⣿⣿⣿⣷⣮⣑⠈⠐⠗⠒⠻⠒⠛⠒⠂⢰⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢻⢿⣿⣿⣿⡟⠟⠁⠀⠈⡇⣤⢨⡀⠀⢰⣿⣿⣿⣿⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠁⠈⣃⣠⣴⡶⣖⣶⣯⣶⠞⠋⠀⠈⠛⠛⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣷⢴⣺⣽⢋⣡⡴⣟⣻⡟⠋⠁⠀⠀⠀⠈⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠅⢺⡏⢿⠷⠾⣶⣦⣤⣿⠛⠒⠒⠒⠒⠒⠒⠒⣲⣶⣲⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣯⡽⠇⣚⣵⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣯⣵⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣶⣿⣿⣿⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡇⠀⠀⠀⠀⠐⠀⢸⡇⠈⠀⠀⠀⠁⠀⠐⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡗⡀⣀⠀⠠⢄⣀⡈⢠⣶⠀⢀⡀⠀⠀⠀⠀⠀⠁
⠀⣤⡀⠂⠀⠀⠀⣶⣿⡟⠳⠾⠛⠿⡾⠿⠿⠶⠦⢬⣤⣄⣘⠛⠛⠛⠛⠛⠀⢦⠀⠃⠀⢸⡟⢰⡄⠀⠀⠀⠀⠃⠓
⢸⣿⣿⠀⠀⠐⠀⢸⣿⣷⡞⠒⠂⠂⠰⠐⡀⠀⠰⠀⠘⣹⣿⡷⠀⠀⠐⠀⣀⣸⡇⣥⠤⣾⡇⢸⡇⠀⠀⠀⠀⠠⠤⠀⠀⠀⠄
⢈⣙⣙⣀⣄⡀⠀⠊⠛⠛⠃⠐⣀⣀⣈⣀⡠⢬⣀⣀⣈⣙⣛⣻⣁⠤⠤⠐⠤⠤⠤⢬⠄⣿⠇⢾⡇⠀⠴⠄⠀⠀⠆⠄⠊⠠
⠈⠉⠉⠁⠀⠈⠉⠉⣍⣉⣩⣧⡄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣹⠋⢉⠁⢚⡁⠀⠀⠈⢀⣿⠀⢺⡅⠀⠀⠀⠈⠉⠉⠉⠛⠟⠉
⠀⠀⠁⠈⠈⠉⠉⠀⠈⢛⢛⣉⣀⡀⣀⣀⡀⠥⠤⠤⣄⣄⢂⣻⣛⣛⣩⣉⣯⣭⣭⠙⣹⣿⠀⢻⡇⠉⠉⣉⢢⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠞⠛⠛⠓⠶⠷⠶⠶⠾⠿⠿⠿⠛⠛⠛⠛⠋⠉⠴⣾⡏⢠⣼⠇⣠⣀⣀⣠⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣶⣿⣿⣿⠿⠷⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣄⠁⢼⡇⠠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡇⢡⠉⢠⠞⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⡇⠀⡠⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⢧⡴⠿⠲⠒⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⠀⠀⠀⢠⡎⠘⠀⠉⠈⠇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⡔⠃⡇⢸⠀⠀⢀⡀⠀⠀⢄
⠀⠀⠀⠀⠀⢀⡄⠃⠀⠀⠀⠀⡠⠊⠀⠀⡇⠐⠿⠛⠂⡇⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡇⢸⠉⠉⠉⡇⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡅⠸⡄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⣤⣤⠄⡇⠀⠱⡀⠀⠈⡄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠡⠤⠤⡇⠀⠀⢱⡀⠀⠈⡄⠀⠀⠀⠀⢰⠂⢲
⠀⠀⡠⠒⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⠀⠀⡇⠀⠀⠀⠡⡀⠀⠐⡀⠀⠀⠀⠀⡖⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠨⠀⠀⠀⠁⠐⠂⠀⠂⠓⣄⠀⠑⡀⠀⡀⠀⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡉⡉⠉⠉⠉⠍⠉⠁⢡⡴⠒⠒⠒⠤⠄⢦⠀⠈⣄⢀⠐⠃⠰⠀⠀⠐⠢⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢤⣤⠶⣶⡶⢲⠀⠀⠀⠀⠀⠀⠒⠒⠒⠲⠤⢬⣄⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡋⠳⣼⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢐⠹⢏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⢿⠀⠀⠠⠀⠀⠀⠈⡀⠀⠀⠀⢀⣐⣃⡨⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠄⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡶⢲⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⠉⠁⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠴⠂⢠⡤⢐⡺⡭⣭⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠸⠓⠲⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢠⣤⣤⢶⣶⢄⠀⠀⠀⣤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⡾⠛⣿⣏⠳⣾⡥⠤⠾⠛⠓⣿⡶⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⠃⢀⣠⠿⠒⠚⠉⠉⢻⠏⠀⠀⠀⠀⠀⠀⠀⣰⣦⡤⢶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡉⠀⡀⠛⡾⠋⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⣀⠀⠀⠀⣿⣿⠁⠀⢸⡇
⠀⠀⠀⡀⠀⠀⠀⠀⣾⣏⡉⣉⣤⡟⠀⢠⣦⠴⠶⢳⣶⣶⣜⣩⣭⣭⣿⣿⣶⣤⣹⣿⡄⠀⣼⡀
⠀⠀⠀⠈⠲⢤⡀⠀⢿⣿⣾⣿⠿⢇⣴⠯⠤⠴⠾⠛⠛⢿⣾⣿⠿⠟⠛⠛⠻⠿⣿⣿⡇⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠙⠢⣌⠻⡏⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⢧⣠⣦⣶⣦⣤⣀⡀⠀⣿⣷⣶⣦⣄⠀⠀⠁⣿⣿⣿⣶⣿⣿⣷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣽⣁⣀⣄⡀⣿⡿⣦⡉⠀⠈⢙⣿⠷⣦⡀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⢿⣿⣿⣿⣿⣧⣬⡙⠦⡄⠀⣿⠀⠹⣿⣷⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣯⢷⡿⢟⡛⠙⣿⣷⣯⣟⣿⢻⠶⠿⡷⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⣍⣻⣶⣿⣿⣿⣌⡿⢉⣿⡟⣷⣶⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢬⡘⢪⣿⡏⢉⠛⠻⡟⢻⣿⡿⢿⣿⣿⡿⢻⣿⠿⠟⢻⢿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠈⠉⠁⠀⠀⠐⠃⠀⠉⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⠠⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠠⠒⠈⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣀⢴⡤⠴⠓⠀⠀⠀⠀⢸⠀⠀⣀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⠐⠛⠉⠀⠀⠀⢀⡀⡀⠀⣈⣺⠼⠓⠚⠉⠙⣦⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠀⠀⠀⠀⠀⠀⢀⡰⠍⠋⠁⠀⢈⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⠉⠀⠀⠀⠀⠀⠀⢺⡇⠀⢈⣤⣤⣶⠶⠓⠒⠚⡂⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠿⠑⠛⠉⠁⠀⠀⠀⠀⠠⡄⣀⣠⣿⠇⠀⠋⠁⠀⠀⠀⠀⠀⠀⢰⣄⠀⠛⢧⣀⣀⣀⡀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣇⡯⠴⠾⣟⡋⠉⠩⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠈⠉⠋⠉⢁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡬⠉⢒⠈⠁⣥⠟⠷⠛⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⠾⡾⠿⢿⣀⣀⣀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠲⠶⠖⢒⡫⠈⠀⠀⠀⢸⡆⠀⡀⡴⢀⡄⠴⠒⠋⢹⡅⠹⣿⣤⣶⣾⣿⣿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠻⠈⠀⠀⢤⠀⠘⠀⠀⠀⠀⠀⠀⣷⠀⠀⡴⠛⠀⠀⠀⠀⣠⣷⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣫⣠⢴⣶⣶⣿⣿⣿⡿⢻⠿⠿⣿⣿⣿⣵⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣯⣼⣯⣤⣥⣾⣷⣖⡚⠻⢉⣑⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣭⣤⣤⣬⡁⠈⣿⣿⣿⣿⣦⡄
⠀⠀⠀⣶⣶⣤⣀⡀⠀⠀⠙⣶⣶⠒⢀⡤⣤⣶⣶⣶⣿⣿⣟⣛⣛⣛⠻⠿⠿⠟⠻⣂⣽⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣏
⠀⠀⠀⠈⠽⠟⠛⠿⢷⣶⣤⣬⣭⣶⠿⠿⠖⢸⡋⠋⠀⡉⣈⡉⠉⢉⣩⣄⣀⣴⣿⣿⣿⡿⣿⣿⣿⣿⠃⠟⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠁⠀⠀⠀⠓⠂⠈⠉⠁⠀⠈⠉⠉⠉⠉⠉⠈⠻⠛⠿⠛⢫⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢛⣛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣖⣯⠭⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⢲⡄⣿⠁⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣖⣾⣿⣧⡿⠀⠀⠀⢀⣀⣸⣧⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⢀⣀⣿⡷⠶⢚⣛⣉⣉⣀⣠⣤⣤⣤⣤⣤⣤⣤⣽⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣰⡏⠉⣟⣋⣉⣭⣭⣥⢤⢤⢶⢶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣬⣭⣉⠙⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⠶⠖⠚⡛⠉⠩⢹⡇⠀⡿⠷⠷⠓⣛⣛⣛⣉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣉⣛⣛⣤⠇
⠀⠀⠀⠀⠀⠀⢸⣐⣀⣁⣩⣬⠤⠴⢖⣒⣿⢀⠉⡟⢛⣛⣉⣉⣩⣭⣤⢤⡤⣤⢤⡤⡤⡤⣤⢤⢤⣤⣤⣬⣽⡇
⠀⠀⠀⠀⠀⣯⣭⣵⣶⠶⠶⠶⠿⢟⣛⣛⡿⠀⠤⣿⢹⡇⡇⢸⣀⣇⣸⣸⣇⣿⣼⣥⣧⣧⣿⣸⣼⣿⣷⠿⠿⠗⠚⠲⣦⡀
⠀⢀⣤⣴⠚⡓⢒⠒⡉⣉⢉⡉⣉⠁⠄⠠⠀⠆⢀⡟⠛⠛⠋⠉⢉⣁⣀⣤⣤⠤⠶⠶⠚⠛⠋⠉⠉⠀⠀⠀⢀⣀⣀⣀⣟⣿⠆
⠀⠘⠦⣤⠤⠥⠬⠤⠤⠤⠶⠶⠶⠖⠒⠒⠛⠉⠉⠙⠒⠚⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠒⢚⠋⠉⠉⣁⣸⡿⠃
⠀⠀⠀⣇⣐⣀⠀⠀⠀⠀⠰⠐⠐⠂⠂⠂⢀⣀⣁⣈⣀⣠⡤⠤⠤⠶⠖⠒⢛⣉⣉⣡⣤⠤⠴⠶⠒⢾⣿⡟⠉⠉⠉⡿⠁
⠀⣀⣀⢸⡟⠛⠛⠿⠿⠿⠿⠭⠭⠭⠭⠭⠭⠭⠴⠶⠖⠒⠒⠒⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠳⣌⣯⣤⡄⠀⢸⠇
⠈⠉⠉⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡿
⠚⣉⣱⣢⣿⣽⠷⠲⢶⣦⠤⢤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡷⢤⣤⠖⠲
⠚⠁⠀⠈⠑⠦⣀⣀⡞⠙⠻⢷⡲⠶⠋⢈⡽⠿⠭⣭⣽⠛⠻⢷⣒⠒⢒⡖⠶⣖⠒⠲⢲⡶⢶⡒⠒⢒⣿⡿⠭⣿⠛⠳⣤⣀⣀
⠀⠀⣠⢤⣀⡀⠀⠀⠀⠀⠀⠀⣈⡛⠒⠋⠀⠀⠀⠀⢈⣉⡓⠒⠚⠛⠋⢀⣀⠈⠙⠲⠶⠃⠀⠉⠳⢮⣍⡽⠂⠈⡷⠦⢤⡉⠁
⠉⠉⠁⠀⠀⢈⣉⣉⣉⡒⠒⠚⠉⠉⠙⠒⠦⠤⣄⡤⠞⠉⠉⠛⠲⠦⠤⠜⠉⠙⠒⠦⣄⣀⣠⡟⠙⠲⠤⣄⣠⠴⠃⠀⠀⠙⢦
⠀⣀⣠⡴⠞⠋⠁⠀⠀⠉⠙⠒⠤⣄⣀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠉⠒⠒⠤⢤⣀⡀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠉⠓⠒⠤⢄⣀

⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⣼⡏⠀⣠⣄⣠⣿⣄
⠀⠼⠮⣿⣬⣷⣿⡿⣿⡈⠀⠀⠐⢀⠀⡀⠀⡀⠀⠀⠀⢀⣤
⠘⠀⠀⠀⠀⠀⢈⣱⣿⣿⡃⠠⡀⠙⡀⣀⠆⡠⢄⣀⠀⣾⣿⣿⣧⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠛⠀⠀⢐⠂⢸⣿⣿⣯⣽⣷⠛⢒⡠⣣⡿⠟⠛⠉⠛⠶⢄⡈⠙⠐⢿⣄⢀⠀⠀⠐⢶⣮⣷⣿⣴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⠛⠿⠓⠛⠛⠷⠮⣿⣯⣤⡐⠒⠒⠒⠒⣚⣛⠗⡆⣀⢈⠹⠆⠐⠶⠾⡷⣽⣿⣿⣶⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠐⠒⢤⣿⡇⠀⠁⠉⠈⢠⡤⠄⠄⠀⠤⠄⡽⠍⡘⠋
⠈⠁⠆⠀⣿⡇⠈⢿⠿⠻⣿⠿⠗⠿⡿⠶⠛⠟⣷⣻⣿⣿⠻⠃⠀⡄⡤⠠⠘⠁⡆⢠⠠⠄⣸⣧⡍⠉⠁⠁⠚⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡄⠀⠏⠈⣇⠸⠀⠖⠋⠛⠛⠃⠀⠀⠀⢀⡁⢀⡄⠀⠁⠀⠀⠀⢡⡄⠀⠀⠀⠐⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⡁⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠓⠶⡶⢆⣼⡷⡲⠒⣠⣤⣬⡝⠀⠐⠒⠀⠀⠀⠀⠒⠀⠘⠉⡭⠑
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⣠⡭⢤⢠⡀⠮⢾⡉⠐⠀⠈⠍⢐⡍⠀⠿⠹⠀⠀⠋
⠀⠀⠀⡀⠀⠀⠀⡀⣀⢤⠤⢰⠆⠀⢠⠀⡀⠁⡤⢴⠆⢹⣭⣽⠀⠃⠃⠘⣿⠐⠀⠛⠐⠂⠒⠒⠛⠿⠆
⠀⠀⠀⠠⠄⠘⠋⣤⣤⣶⣦⠬⢤⣀⡘⠀⠀⣌⠁⠉⠁⠈⣍⠉⣁⠀⠀⣀⡛⠦⢀⡝
⠀⠀⠀⠀⠀⠈⠀⠚⠿⡏⠯⡄⠀⠘⠛⠛⠛⠛⠓⠀⠙⠚⠻⠒⠀⠙⠛⠋⠋⠀⢀⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣾⣭⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠝⠛⠛⠛⠛⠀⣹⡟⠀⠀⠀⠀⠀⠂
⣤⣤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣂⣯⣭⣭⣭⣭⣭⣬⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉
⣏⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠻⠟⠿⢻⣿⡿⠛⠛⠛⠛⠛⠛⢛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⢿⣿⣿⣤⣶⣤⣦⣤⣤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠁⠀⠰⣷⠀⠀⠀⠀⠀⢐⡈⠙⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢰⣶⣿⣦⣀⠀⠀⠉⣉⣋⣀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⠛⢚⣳⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⣀⠴⠒⠉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢾⠘⣿⠳⣄⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠠⠀⠃⠹⣧⠈⢃⠄⠀⠀⠁⠀⠀⠀⠀⠀⢠⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣼⡏⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠁⠐⠀⠀⢻⣆⠀⠀⣱⣇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣶⣶⣦⣰⣤⡠⢤⣤⣌⣿⣿⣷⡄
⠀⠀⠀⠀⡘⠀⠘⡄⢀⣀⣻⣄⣴⠟⣨⠓⢄⠀⠀⠀⠿⠿⢻⣟⣿⣿⢹⣿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣠⣇⠀⠰⢃⣼⣿⣿⣿⣿⣧⢹⣧⣌⠳⣄⠈⠼⡀⣾⣿⣿⣿⣸⣿⡇⠀⠈⠙⠻⠝⡻⢿⣿⣿⣿⣿⣧⠀⠀⠄
⠀⢀⣴⡿⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⣳⣆⣥⣿⣿⣿⣿⣿⣿⣷⣶⣴⣦⢤⣀⣉⡳⠿⠿⣿⣿⣿⣧⠀⠰⡄
⠀⣸⡟⠀⠀⠀⣾⠟⢩⣿⠟⠉⠹⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⢸⣿⠀⠈⠉⠙⠛⠿⢾⣿⣿⣿⣧⢰⣿⣦
⠀⣿⠁⢴⠀⣠⣸⣇⣛⣀⣀⣀⣀⣹⣆⣏⣛⣤⣿⣿⣿⣿⣯⣥⣼⣧⣤⣤⣄⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣈⣙⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠙⢉⣿⣿⣿⡿⠿⠻⠏⢿⣿⡏⠉⠉⠉⠉⡻⠏⠉⡟⠻⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⢷⡀
⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⡴⠁⠀⠺⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⡏⠋⠉⢉⣉
⣧⣴⣦⣤⣤⣤⣼⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⢦⡤⠤⠴⡦⠶⠤⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣌⣾⣇⠀⠀⠀⠁
⢥⠀⠀⠀⠀⠀⢰⡇⢶⡀⠀⠀⠀⠀⠀⠀⢿⣿⡄⢀⠶⣤⡄⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠈⣿⣿⡏
⠀⠃⠀⠀⠀⠀⠈⣿⣶⣾⢶⣤⠀⣠⣦⣤⣾⣻⣿⣟⣴⣯⣤⣤⣴⣤⣶⣶⣶⣶⣷⣶⣦⣤⣤⣀⣴⣷⣶⣤⡜⣧⣿⣿⣿⠁
⠀⠀⠀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠹⣿⣿⠿⠟⠉⣿⡟⠻⠿⣿⠏⢙⣿⣿⣿⠿⠻⢿⣿⣿⠿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⢹⠗⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣀⣿⣧⡄⠀⢀⡀⢸⣿⠘⡀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢀⣶⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣧⣾⣿⣤⣦⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡖⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣄⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠶⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡉⠉⠛⠛⠟⠛⠉⠛⠢⠦⣤⣀⡀
⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣤⣤⣄⣤⣄
⠀⠀⠀⠀⠈⠉⣛⠿⠛⠛⣉⣉⣀⣀⡈⠋⠙⠛⠛⠛⠛⠛⣻⣿⣿⠛⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠻⣿⣋⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠟⠋⠉⠛⠻⠷⢶⣦⣴⡶⠾⠿⠛⠿⠟⠛⠛⠿⠿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢀⣀⠀⠀⠀⠀⣀⣠⡀⠀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⢿⣼⣟⣻⣿⣿⣟⠃⣰⢚⣻⠿⢯⣤⣿⣿⣿⣾⣿⣷⠟⣿⣿⣿⣿⠿⣷⣶⣦⣤⡖⣶⣶⠤⠤⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢛⣋⣈⣉⣀⣚⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⣼⡿⠿⠿⠿⠓⠫⢽⡵⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣛⣻⣟⣉⡉⠋⠈⢩⠄⠉⠀⠠⠍⠁
⠀⠀⠚⠛⠛⠀⠀⠀⠀⠀⣀⣴⣿⣣⡤⠤⢾⣷⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⣾⣿⣿⠿⠷⠾⠿⠷⠤⠦⠶⠛⠿⠓⠂
⠀⠀⠀⠀⠀⠐⠂⠂⠐⠚⠉⠿⠟⠛⠉⣡⣼⣥⢐⣆⢻⣇⣽⡫⢽⡿⠟⠿⠿⣧⣭⢉⣥⠤⠄
⠀⠀⠀⠀⠀⠀⠀⡠⣤⣤⠴⠀⠀⠀⠀⢈⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉
⠀⠀⠈⠱⠀⠉⠀⠁⠉⠓⠂⠂⠀⠀⠀⠛⠛⠛
⠀⠠⠖⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣄⣠⣴⣶⣤⣶⣶⣦⣤⣶⣶⣦⣶⣶⣿⣶⣶⡶⠖⠂
⠀⠠⠄⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠀⠉⠉⠉⠉⠉⠸⠿⠿⠟⠛⠛⠻⠛⠛⠙⠛⠛⠿⢿⡟⠿⠿⠷⠂
⠀⠀⢰⡓⣉⡀⢀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣗⣺⣷
⠀⠀⠌⠉⠛⠹⢃⡠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣤
⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⣄⢸⣿⠀⣴⣦⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⣀⣾⣇⣀⣄⣀⣠⡶⢤⣤⣀⣀⣀⣴⣄⣀
⠀⣠⣦⣾⣴⣿⣿⣿⣶⣾⣿⣿⣿⣶⣿⣿⣦⣼⣠⣠⣤⣄⣧⣤⣦⢀⡀⣄⣸⣄⠀⣿⡀⢹⣿⣿⣿⣿⣿⣿⠛⢻⢿⠿⡟⠋⣿⣿⣟⣿⣾⣻⡟⢿⣿⣿⣿⣿⡇
⠀⢘⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡤⣶⣿⣿⣿⣶⣿⣧⡆⣿⣿⣿⣿⣽⣿⣤⣾⣼⣀⣷⣴⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠘⠛⠋⢻⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⠙⠛⠛⠛⠛⠛⠛⠿⠟⠉⠛⠿⡿⢿⡏
⠀⢠⣤⣄⣸⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣿⣿⣛⡿⢿⠛⣛⡛⡛⠉⠛⠉⠉⠁⠉⠁⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⣰⣟⣿⣿⣿⣿⣿⣿⣶⣦⣤⠬⣽⠀⠀⠀⠀⠰⣾⣷⣶⣶⣶⢶⣿⣿⣿⣿⣿⣯⡄⠀⠀⠰⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣤⣀⣀⡀
⠀⢺⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣻⣿⡿⡿⠿⠀⠀⠀⠀⡍⠁⠉⠈⠀⠚⣻⣿⠿⠿⠿⠟⠁⠀⠀⠀⠿⣿⣿⡿⣿⣿⠿⠯⠯⠽⣿⢹⣿⣿⡃
⠀⠸⠿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣄⣀⣠⣤⣂⣨⣭⣤⣽⣿⠄⠉⠸⣭⣥⣽⣿⡿⠇
⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⢿⡿⠿⢿⣿⡿⢿⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠁
⣶⣶⣶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣲⣶⣶⣦⣤⣽⣛⣄⠀⢀⡄⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠋⣀⣈⡉⢡⣤⣤⣤⠭⢿⣿⣤⡟⠀⡀⢺⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⣁⣤⣶⣶⣦⣤⣹⡇⣼⡇⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⠟⠉⠸⡟⢙⡿⣦⠟⠓⠡⣿
⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠋⣶⢠⣾⠁⣼⠿⣿⣞⡻⣶⠾
⠀⢰⣾⣏⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣳⣿⣿⢋⣰⠋⠀⣿⣇⠛⠉
⠀⠀⠀⠴⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⢻⣿⣛⢿⠉⠓⠀⣿⠇
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠏⠀⠀⠀⠀⠛⠁
⣀⣀⣀⣀⣨⣬⣥⣤⣤⣤⣤⣤⣶⣶⣶⣦⡆⠰⣶⡶⢶⣶⠶⠾⠟⠋⠉⠉⠉⠉⠯⠿⠛⠏⠐⠂⠀⠈⠋⠉⠉⠉⠉⠉⠉⠁⡟⠀⠉⠉⠉⢠⡏⠀⠐⠀⠉
⠙⠷⠮⣭⣭⠟⠟⠻⠛⠿⠿⠟⠋⠉⠉⣉⣁⢀⣉⢉⣁⣀⡀⠀⠀⣀⠀⠀⠰⠀⠠⠞⠠⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠛⠻⠿⠿⠻⠿⠛⠋⠁⠀⠀⠐⠂⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⢀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⣶⣶⣶⣶⡴⠒⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⣲⣿⣏⡭⠉⠀⠀⠀⢸⠄
⠀⠀⠀⠀⢠⣼⣿⣷⣤⣀⣈⣁⣀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠉⠋⠉⠛⣋⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠙⢛⣭⠭⠿⠻⢟⣳⠦⣤⣤⣤⣤⣴⣶⣇⣀⣠⢤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠩⠭⣍⣫⣑⣛⣻⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠈⠈⠉⠙⠛⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⠀⠀⠀⢀⣀
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡉⣿⣄⣁⣀⣀⣨⠈
⠀⠀⢂⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠀⠈⠉⠉⠁
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠋⠙⠋⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢴⣤⠤⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠰⠐⢀⠄⣄⠄⠀⠘⡂⢻⠐⠀⠀⢸⡟⡗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠒⢢⠇⢀⠡⠀⡖⠀⠀⠐⣦⠀⢧⠀⠀⠀⠀⢹⡄⠛⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡿⣾⣿⣆⢸⣿⡂⠀⠀⡞⠀⢸⡆⠀⠀⠀⠈⢻⡀⠈⢂⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⡇⠀⠀⠀⠀⠘⢧⠀⠑⣷⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠙⠛⠛⢛⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡩⠀⣶⡶⠆⠀⢸⡆⠀⠀⢻⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⠉⠉⠉⠉⠉⢼⡀⠀⠀⢿⡄⠀⠀⠈⢻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣻⡇⠀⠀⢀⠀⠀⡀⠀⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠘⠻⠄⣤⣤⡤⢅⣙⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⢀⣼⡿⠀⡀⢀⠀⠄⠠⠣⢐⣾⡻⡏⠀⢤⡀⠀⠀⠀⠀⠀⠻⡦⠤⢤⠾⠬⠭⣽⣿⣦⡀⣀⡭⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡡⠤⠞⣿⢃⣤⠴⣶⠖⣲⠀⠐⠚⠛⢿⠃⣈⠶⠑⣀⣀⣀⡤⡀⢶⡇⡰⠃⠀⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣄⣀⣀⣀⣿⡁⢀⠐⠼⠀⢸⠀⠀⠁⢀⡎⠠⠔⣞⡏⣿⠼⣻⣀⣀⣿⣾⣤⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄⠛⣛⠻⠛⠟⠻⠛⠳⠤⠾⠶⠶⠶⠿⠷⠶⠺⣛⣛⠛⠭⢽⡿⠿⠿⠿⢟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣒⣂⣈⠀⠉⠈⠁⠉⠈⠁⠉⠈⠁⠀⠈⠉⡁⡀⠠⣤⣴⣷⣾⣿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠈⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⡀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⠀⠀⠀⣀⣭⣇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⢠⣤⣌⣉⣉⡀⢹⡆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢈⣅⣀⠀⡀⠘⣿⣿⣿⣿⣿⣦⣄⠂⡀⠀⢸⣀⣀⣅⣀⣀⣀⣀⢀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠀⠀⠤⠤⣽⣿⣿⣿⣿⣿⣿⠟⠀⠀⢠⠀⠀⠀⠀⠀⢠⠰⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠰⠀⠀⠀⠀⠀⠛⠙⠛⠉⠛⠁⠀⠀⠀⣀⡸⢘⣃⣶⠄⠀⠼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠘⠀⠀⠀⢀⣆⠀⢀⡀⢤⠴⠚⠛⣋⣴⣷⣧⠿⠋⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢈⣀⣠⠤⢺⣿⢫⣥⢒⠄⠤⠖⠫⣧⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢩⠥⠐⣂⣸⡿⠚⣛⠛⣻⣟⣥⡾⠟⠉⠉⠉⠉⠉⠉⠉⠙⣛⣛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣴⢖⣾⣯⣮⠓⣪⣵⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⣈⣩⣭⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⡿⢟⣫⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠶⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⣆⠀⠁
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⡷⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡁
⢀⣤⣎⠀⠀⢸⡋⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠿⣆⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⡀⠀⠀⠴⡇⠀⠈⣿
⠀⠛⠛⠲⠄⠈⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢢⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⣠⣾⣤⡀⠈⠛⠄⡀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠁⠀⠀⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠦⠄⠆⣀⠀⠈⠛⠒⠒⠒⠒⠲⠀⠀⠠⠤⠤⠤⠠⠤⢤⡄⠤⠤⠤⠠⠤⠤⠤⠠⠤⠤⡄⢠⣤⣤⣄⢤⢠⡤⢠⠤⠄⠤⠄⠀⠠⡶⠶⠖⠒⢲⠒⠂⣀
⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢀⠀⠀⠦⠀⠀⢠⡄⡀⠀⠀⢸⣇⣰⡆⣤⣾⠁⠀⠀⠀⡀
⠀⠀⠀⡄⡄⠀⠀⠀⠀⣀⣤⣠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠉⠀⠀⣮⠀⠀⠀⠀⠀⣼⣷⣯⠀⠀⠸⠟⠛⠃⠙⠻⠌⠀⠢⠉
⠄⠀⠀⠹⠀⣀⠀⠀⠀⠿⠿⠛⠀⠀⠀⠸⠷⢇⠀⠀⢠⣼⣵⡇⣠⢀⠀⠀⠈⠛⠃⠀⠀⠰⠷⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠂⠀⠀⡄⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢀
⠅⠀⠀⠀⠀⠀⠀⠀⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠄⠀⠀⠂⠀⠡⣶⠂⠉⣁⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠡⢀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣄⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠻⢿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣷⢿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠽⠿⣿⣿⣷⣶⣦⣄⡀
⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣾⣾⣿⣦⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠀⠀⠀⠄⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠄⠂⡏⠁⢀⠘⠋⠉⣉⣠⣤⠤⠤⣄⡁⠠⡀
⠀⠀⠀⠀⠀⢀⠀⠤⠥⣂⠀⠀⠀⠀⢀⠀⢺⠀⠀⠀⡧⠈⠀⢤⣶⠉⢹⠘⢻⠀⣆⠈⡏⠰⣌⠂
⠀⠀⠀⠀⠀⠚⠁⠀⠐⡈⠂⠈⢀⡄⠁⠀⣸⠠⡐⠋⢀⢔⠃⢚⣿⢰⠛⠖⠚⠂⠛⠒⠓⠄⢸⣣⡐
⠀⠀⠀⠀⠀⠀⠐⠐⠚⠀⠁⡀⠈⣗⡐⠋⢀⠆⠀⠔⣿⠀⠩⠞⠉⠀⠤⣶⣶⣿⣽⢿⢷⣶⣄⡙⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠇⢠⡄⡀⢹⡦⠟⠀⡠⡁⡀⢒⠠⣿⠈⢁⡄⠀⠀⠀⣹⢿⡿⣿⣟⣟⣾⣿⣷⡘⡂⠁⠐⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡈⢄⣘⠓⣇⡺⠀⠀⠀⣿⡐⡇⢈⠙⢀⣴⠟⠷⠿⢿⢦⣿⣫⣀⡀⡲⢦⣌⡿⠚⡁⠀⠀⠀⠴⠧⠠⠰⣄
⠀⠀⠀⠀⠀⠐⠛⠙⠻⠠⠟⠲⠤⠥⠤⣀⣤⡧⠃⣠⣿⢈⣧⠐⠿⢆⠐⢩⣬⠀⠉⠉⠉⢈⠀⠉⠙⠛⠂⣖⣤⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈⠄⠀⠀⠀⣀⡠⠡⠼⠭⠿⠀⠛⠉⠀⠈⠀⠀⢬⠤⠀⠸⠄⠉⠀⠀⠢⠤⠀⠀⠀⠀⠭⠙⠁⣭⠙⡢⡄⠐⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣄⡀⠈⠃⣀⡀⠀⢉⠁⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢅⠀⠀⠀⠀⠄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠊⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⣄⡂⠀⠀⠀⠉⠙⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢼⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⠃⠀⢀⢀⠈⠀⣰⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠠⠀⠀⠀⣀⡄⠀⢸⠃⠤⠈⠧⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢠⠖⠀⠀⠀⠀⢀⠁⠀⠀⢀⢀⣀⣤⣴⠊⠇⠐⢚⡒⠀⠀⠸⠁⢤⠐⡦⢀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡀⠀⠀⠐⢢⠴⠐⠀⠀⢀⡀⢀⣁⢀⠐⠺⠻⠙⠋⠃⠉⠐⡇⠀⢀⡁⠺⠸⠇⠀⠉⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣠⠋⣁⠀⠀⠈⢩⠇⢤⠰⠒⣛⠋⠋⠉⢀⡀⢉⢀⢰⢺⢴⢼⣣⣟⡼⠀⠂⠁⠀⠀⠀⢀⡀⠀⡄⢨⡅⢰⡆⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠀⣠⣼⠇⢠⠋⠀⢀⡆⠋⠈⠉⡇⠈⠄⠀⠀⠀⠀⠀⠙⠈⠈⢹⣿⣭⣽⣿⣤⠰⡆⢘⣿⡀⠃⠈⠃⠀⠇⠀⠃⠘⠃⠉
⠀⣠⠖⠲⣶⠀⡒⢀⢐⡆⡇⠸⠄⢉⣁⢘⠐⠚⡏⠀⢸⡆⠀⠀⠀⡇⡈⠉⠀⠀⠀⠀⠖⠾⠾⠻⢼⣼⣿⢿⡗⣯⠀⠀⠀⢉⠀⡀⢠⠀⢀⣀⢠⡄⢠⡄⢠
⠍⡙⡧⡋⠙⡂⣀⣀⠸⠃⡇⢈⣆⣈⡀⣸⢀⡀⣇⣂⣸⣂⣀⢀⣀⡃⠁⣠⢀⣀⣀⣀⡤⣤⡠⠤⠾⠥⡬⠾⠁⣡⠸⢇⡇⠸⠄⠳⢰⠆⠈⠟⠸⠇⢸⠂⢹
⠶⠞⠱⡷⠶⡿⠷⢤⣾⣴⠇⡵⣽⣯⣄⡉⡬⣤⣧⣴⢾⣥⣶⣂⣠⠥⡄⠀⠀⣀⠀⠀⢀⡆⠠⠀⠀⠶⣴⣶⠦⢦⣠⣼⠀⠀⠰⣶⢶⠄⢰⣖⢠⡶⠶⡆⡠
⠀⠁⠀⠀⠛⠛⠋⣱⣉⠷⢷⣶⢟⠒⠛⣋⡗⣂⣚⣿⡈⣉⠙⣛⣱⣾⢻⡟⣗⣛⣓⡚⠇⠯⠿⠲⢮⣍⠋⣟⡋⠗⢯⠶⣄⡀⠀⠉⢈⣩⠉⢉⣤⡝⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠲⣿⠓⠋⠉⠉⢙⠒⠀⠉⠈⠉⠉⠀⠀⠐⠒⠲⠦⠬⡯⡝⡃⠀⠂⠾⠂⠁⠀⠢⠒⠦⠉⣛⠌⢏⡈⡉⠈⡁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⢭⡀⠀⠀⠃⢀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠙⠁⠀⠈⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡃⠀⠀⠀⠀⡄⠀⢲⡆⠈⢠⡉⠳⠷⢶⣤⣤⡀⡠⢏⡠⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢲⠀⠘⡆⠀⣹⠃⠀⠀⠘⠛⠛⠃⡀⠉⠀⢀⣣⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⢀⠇⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠓

⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿
⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣿⣿⡆
⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠉⠉⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⡿⣿⣄⣀⠀⠀⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣧⣤⣿⢠⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣼⣿⣿⣿⡟⠻⠿⣷⣿⣾⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠢⣿⠀⢣⡠⢻⣄⣾⡄⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣠⠔⢻⣷⣾⣷⡇⠀⠀⠀⢹⣛⣟⡃
⠀⠀⠀⣠⠀⠀⢀⡀⠀⣠⣤⣤⣤⣤⣿⣶⡀⠀⣿⠁⠈⠙⣿⣯⣼⠀⠀⠀⠠⠴⠶⠶⣶⣶⣶⣦⣤⣤⣙⣀⡻⣿⣇⠀⠀⠀⢘⣹⣯⣿
⠀⠀⠀⢹⡆⠀⠐⡇⠀⠀⠉⠉⠛⠛⣿⠿⣿⣿⣿⣦⣤⣀⡛⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⡀⠀⡷⢶⣤⣼⣽⠿⢇⡐⣄⡀
⠀⠀⠀⣸⠓⡀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠉⠉⢉⣿⠈⠉⠙⢻⠿⢾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⡄⣆⣼⣿⣿⣷⣬⢋⠛⠉
⠀⠀⢀⣙⣀⣀⣐⣃⠀⠀⠀⠀⠀⠀⠿⠉⠉⠉⠿⠀⠀⠀⠸⠒⠚
⠀⠀⠨⡏⣿⠿⡟⣿⣷⣦⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠄⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⡀⠀⠀⠐⠢⣤⣀⡀
⠀⠀⠀⡇⠀⠀⡧⢸⠋⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠀⣿⢦⣄⣀⡀⠀⡀⠀⠀⠀⠀⢠⣄⠻⠄⣈⣹⣗⣲⣶⣿⣥⣉⣳⡦⢤⣀
⠀⠀⠀⠃⠀⠀⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠆⣿⢶⣴⠿⢿⣷⣦⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠙⣻⣶⣧⣽⣗⣆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠁⠀⣿⡌⠀⢠⡀⠙⣿⠟⠛⡃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡴⠛⣩⣤⣶⣤⣙⢯⠙⢳
⠀⠀⠀⠀⠀⠀⣠⣶⠾⡿⣦⣾⢿⠶⣄⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⠁⠀⠀⡇⠀⠸⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡾⢡⠞⣿⣿⡿⠟⠙⣷⡣⣸
⠀⠀⠀⠀⢀⣼⡏⢸⡀⣼⠋⣿⢻⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡏⠀⠘⣯⠀⠀⠀⠘⣧⣿
⠀⠀⠀⠀⠾⠛⠇⠘⣼⣿⡀⣿⢸⢻⠀⢠⣷⠦⣠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁⠀⠀⠛⠒⠒⠒⠒⠛⠚
⠀⠀⠐⢲⡶⠶⠶⢦⡿⣭⣤⣉⣘⣘⠃⡿⢸⣆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠈⠉⢹⠋⠛⣻⠛⠖⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢦⡄⠀⠀⠀⢼⠷⣄
⠀⠀⠀⠙⠁⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠟⢦⣀⠀⣛⣉⢸⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢄⡄⠰⠀⡟⢶⣷⣶⡾⠤⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢫⡧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⣉⣀⣁⣁⣙⠻⢹⡿⣯⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⣠⣦⣤⣤⡔⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⢠⠀⡀⠀⠀⠈⢏⠉⠛⠛⠛⠛⠻⠿⣿⣯⣁⢿⢻⣿⣿
⠀⠀⠀⢐⣀⣀⣈⣁⡈⣀⣁⣐⠄⡀⠀⣻⠀⠈⡇⠀⠀⠀⠉⠙⠛⠻⠿⣶⣦⣥⣀⡀⠀⡇⠀⠀⠀⡀⠀⡁⠀⠀⠀⠀⢹⠛⠉⠉⠉⠛⡻⠿⣿⣿⣷⣮⣽⢿⡄
⠀⠀⠀⠀⢏⡙⠛⠛⠛⠛⠻⠿⣿⣯⣐⠿⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠛⠻⠇⣷⣦⣤⣀⡃⠀⢅⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠉⠉⠉⠻⠛⠃
⠀⠀⠀⠀⠀⢸⡟⡇⠀⠉⣿⠻⠿⣿⣿⣿⣦⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠄⢿⠀⠉⠙⡿⠻⢸⣤⣄⠀⠀⢸⣀⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿⠁⠀⠀⠉⢩⠉⢹⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⠀⠀⠀⠃⠀⠘
⠀⠀⠀⠀⢀⡼⠧⠇⠠⠀⠿⠶⡄⠀⠀⡸⠀⠸⡀
⣄⠀⠀⠀⠈⠷⠒⠒⠒⠒⠒⠒⠃
⣿⣷⡄

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠔⢒⡼⠛⢯⣉⣶⠖⠒⢲⢶⣶⣶⣒⣢⠤⠤⢄⣀⣀⡀
⣶⣶⣤⣤⣶⣾⣭⣭⣾⣞⣡⣔⣚⡉⠳⣼⡤⠚⠁⢸⠈⠳⣄⢹⠉⡳⠺⣿⣌⠉⡟⠒⠢⣤⣀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠷⣶⣾⣶⣤⣬⣿⣏⣀⣀⡇⠈⢳⣇⡠⠊⡝⢯⡑⣦⡀
⣶⣶⣶⣶⣶⣦⣤⣤⣄⡀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠻⠿⢿⣿⣷⣤⣧⣤⣱⣧⠞⢦⡀
⠉⢻⠟⠉⣠⠞⠁⠙⢯⡉⢢⡔⠋⣠⠞⠁⠀⠉⠹⣟⠶⣖⠒⠶⠦⣤⣤⣤⣀⠀⠀⠉⠉⠛⠛⠿⠿⣿⣷⣿⣶⣤⣤⣀⡀
⠳⣼⣠⠞⠁⠀⠀⠀⠀⠑⢬⣧⠞⠁⠀⠀⠀⠀⠀⠈⠣⡀⢱⡖⢉⡼⠋⠙⢯⡑⢤⠔⢲⠟⠒⠒⢶⣤⣤⣨⣉⣙⠛⠿⢿⣿⣶⣦⣤⣀⡀
⠀⢸⡋⠉⠉⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣷⣋⣀⣀⣀⣀⣑⣾⠞⠁⠀⠀⠀⠀⠳⡝⢦⠴⣻⠟⢿⡒⢤⠭⣽⢿⣿⣿⣿⣶⣦⣄⣀
⠀⠘⢧⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢾⣞⣁⣀⣀⣙⣾⠞⠁⠀⠙⣿⣍⣛⣿⣿⡿⢿
⠀⠀⠀⡏⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠈⢿⣞⣁⣈⣿⠟
⢀⣀⣠⡇⠀⠀⠀⠀⢸⣿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠄⠀⠀⠀⣿⣿⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣿⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠘⣤⠤⠤⡼
⠉⠀⠀⣇⠀⠀⠀⠀⢸⣿⠀⢴⡒⠒⠒⠤⠤⠤⠤⠤⠤⠤⣄⢐⡀⠀⠀⠀⣿⣿⢀⡤⠴⠄⠀⠐⠒⡶⠀⠀⣿⠀⠀⢈⣿⠉⠉⠉⠉⠉⠉⠉⣇⠀⠀⡗⠒
⣄⡀⠀⢀⣈⣈⣀⣈⣈⣩⣤⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣅⣠⡤⠤⠭⠥⠊⠀⠀⠀⠀⠀⠀⠑⠒⠒⠚⠒⠛⠒⠓⠒⣤⠀⠀⠀⠀⠀⠉⠀⠁⠁
⣠⠇⠀⠛⠒⠠⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠲⠦⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠐⠯⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⣀⠤⠤⠔⠒⠒⠒⠒⠒⠒⠒⠢⠤⠤⣀⠀⠀⠀⠀⡰⠋⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠈⠒⠲⠦⣄
⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⣀⣀⣠⡤⠴⠋
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⡀⠀⠀⠀⠀⠀⠀⠐⣧⣤⣀⣀⣀⠀⠀⠀⠀⣀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠠⡴⢤⠀⣴⡤⢶⢠⣇⣤⣄⢨⣉⣄⣠⣄⣸⡁⠀⠀⣤⣶⡀⠀⢀⡀⠀⡀⢰⡆⠀⠀⠀⠀⠀⢀⢠⠄⠀⠀⠀⠀⠀⢰
⠀⠒⠣⡾⠖⠾⢷⡶⢮⠿⣶⣦⣙⣏⣡⣬⣿⣟⣾⣶⣿⣿⣿⡷⠖⠛⠛⠓⠇⠉⠁⠀⠀⠀⢠⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣘⡁⠀⠀⠰⠀⠀⠀⠉⠀⠀⠃⠈⡉⠿⡧⠶⠖⠺⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣀⣀⣤⣼⣂⣀⣀⣤⣤⣤⣶⣶⠂⢸
⠀⢠⢶⡆⠀⠀⣀⡁⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠀⠀⢰⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠥⠀⠺
⠀⠒⠛⢿⣥⣀⠁⠀⠀⠀⠀⠀⠴⠶⠄⠶⠒⡀⠀⠰⠾⠦⢄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⣿⣀⣤⣤⣤⣤⣤⣶⣶⣶⣶
⠀⠀⠀⠹⡿⣄⣲⣶⣆⣤⠴⠶⠒⣳⡾⠿⠟⠛⠓⠂⢄⡀⠀⠀⠀⣰⣶⣤⡤⠀⠀⠀⠀⠀⣇⠈⠉⠛⠛⠋⠉⠉⠉⠉⠹
⠀⠀⠀⠀⠀⢈⣿⣤⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠰⢶⣦⣤⠴⠚⠋⠁⠀⠀⠀⢀⣾⣷⣶⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡴⠿⠉⠙⠻⣿⢿⡿⠷⣄
⠀⠀⠀⠀⢀⡀⠀⠤⠦⠭⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⣴⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⣀⡔⠀⠀⢀⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠁⣭⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⣴⡀⠀⠘⣿
⠈⠿⠻⢤⡀⠈⢣⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡨⡀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣴⡀⠀⠀⠭⡁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣄⡀⠘⠄⠀⢀⣠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠉
⡀⠀⡄⠀⠀⠀⠤⠤⠄⠤⠤⣠⡀⠀⠀⣀⣀⣀⡀⣀⡀⣀⠀⣀⢀⣀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⣀⢀⣀⠀⠀⣀⣀⣀⡄⡤⣤⠤⠄
⣁⠀⠀⠋⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⠀⠩⡌⠉⠈⠁⠉⠉⠈⠉⠉⠉⠈⣉⠉⠉⠈⢉⠉⠁⠉⣉⠈⠁⠀⠀⢸⡄⣠⠀⡀⣼⠀⠀⠁
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠈⠁⢐⠀⠀⠀⠁⠀⠀⣟⣰⠄⠀⠀⣿⣷⣿⡄⠿⢏⡡⠀⡀⠔⠂
⠀⠀⠀⢳⠃⠀⠀⠀⠀⣰⣾⣶⡇⠀⠀⠀⣶⣠⠀⠀⠀⠀⣰⢀⡀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⣤⣿⣇⠀⠀⠀⠀⠘⠛⠋⠃
⡁⠀⠀⠀⠀⠒⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠂⠀⠀⣿⣿⣿⣥⣧⣖⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄
⠄⠆⠀⠃⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠖⢒⡒⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠁⡗⠀⠀⡤⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠓⠀⠀⠐⠂⠀⡠⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⡀
⠰⡇⠀⠐⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠘⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠻⠯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣟⣂⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣏⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣿⣿⣄⣀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣞⣛⡛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⡿⢷⢦⣀

⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠬⢹⠟⠁⠀⢘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⡖⢨⢀⣐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠀⠁⠘⠈⠉⠙⠛⠒⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡀⠤⠀⠀⣂⣶⣀⣀⣀⣤⣿⣤⣤
⠀⠀⠀⣀⣠⡄⠀⠀⠈⠀⠀⠀⠀⣉⣉⠉⣩⡉⣩⠍⢉⢓⠶⠀⢀⡆⠀⢠⡀⠄⣀⠀⠀⠀⠠⠐⢶⣾⣟⢑⡋⢹⠋⢻⣿⡿⠉⣽⠉⢻⡷
⠀⠀⡞⢻⠟⠿⠟⡀⢀⠀⠀⣄⠀⣾⠀⣸⣯⡧⣿⠁⠀⠈⢀⠀⠚⢷⠀⢸⡇⠀⣿⠘⣻⡛⠁⢹⣾⠺⠁⠀⠀⣸⠀⢸⣿⣳⢠⣸⡂⣨⣷⣀
⠀⠀⠁⢰⠀⠀⠀⠁⠘⢸⣷⣿⣿⡯⠀⡅⢿⣷⣼⡇⣶⢀⣸⣀⣴⡞⠀⣼⡇⢠⠇⠀⣽⡇⢠⣾⣻⢶⣇⡒⠀⢾⣦⡞⣽⣿⠋⢸⣏⡿⣿⣿⡄
⠀⠀⠀⠀⠀⢘⣒⣀⣠⣿⣿⣿⣿⣼⣷⣛⡿⣿⣼⣷⣯⣿⣷⣷⣬⣥⣷⣟⣹⣶⣷⣾⣶⣧⣾⣶⣶⣾⣿⣿⣷⣾⣿⣥⣤⣏⣔⣿⣿⣷⣷⣿⠁
⠀⠀⠲⠖⠪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣳⣿⣿⡍⢻⢋⠉⠹⠛⠈⠛⠉⠹⢉⣉⣸⠫⢹⡟⠋⠹⢿⣿⠟⠛⡿⢻⠿⣿⢿⣿⣿
⠀⠤⠤⠦⠤⠾⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠤⠶⠦⠆⣆⠀⠚⠈⢓⣦⢭⣿⠀⢸⡇⠀⠀⠘⠁⠁⢛⡁⠀⢸⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣟⣻⣿⣿⣹⠛⠋⠻⠏⠀⠀⠀⢀⣤⢹⡇⢈⣻⡄⣤⣤⣽⣧⣬⣧⠴⠀⣀⣤⣰⣾⣷⣤⣽⣃⣤⣦⣀
⠀⠀⠀⢀⡤⣤⠀⠀⠀⠀⠈⠙⠻⠏⠁⢿⣿⣿⣷⡄⢓⢲⣃⣀⣖⣳⠾⠿⠦⠴⢶⣟⣃⣒⣈⣯⣵⣼⣿⣯⣶⣶⣶⣾⣿⡾⠁⠈⠙⢟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⢿⠿⠿⠿⠿⠿⠿⠛⠛⣛⣿⣿⣿⣿⣿⡁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠤⠶⠋⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⡖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠏⠉⠉⠁⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣰⣿⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣼⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠃⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⢸⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢛⠛⠛⠃⠿⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⣠⣄⣀⡀⠀⢸⣷⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣿⣿⠇⣿⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄⡚⠃⣠⡤⠀⠀⠃⠀⢸⡄⠀⠀⠀⡏⠀⠀⢤⠀⠀⠀⠀⠀⠈⢹⣿⣿⡀⣿⡇
⠀⠀⠀⠠⢰⣷⢶⣆⣤⣠⡄⢤⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣁⠐⡧⠀⠀⠤⢸⠤⠤⠤⠤⡧⠤⠤⠤⠤⠶⠤⠦⠤⣶⣾⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡄⢿⣷⡟⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⣽⣿⣿⣿⣿⣿⣿⣿⡿⠹⠟⠁⠀⠀⠀⠀⠀⣹⠀⣴⡄⠀⣄⠀⠀⠀⠀⣿⣿⣴⣿⣾⣿⣟⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠋⠁⠲⠶⠅⠀⡀⠀⠀⠀⠀⠀⣀⠀⡾⢀⡙⠃⣸⣿⡷⠤⠤⠤⢿⣿⣿⣿⣿⣿⡇⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣾⡿⠖⠒⣂⣭⣥⠇⣸⣷⡿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⠀⣀⣠⡴⠒⢛⡋⠍⠚⠈⢃⣠⡶⠿⡿⢿⣷⠿⠋⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣏⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢾⣿⣿⡏⣯⣥⢖⢦⣘⣀⣠⣴⠞⠛⣱⣶⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣿⣿⡇
⠀⠀⠀⣠⡴⠶⠛⠉⣃⠠⢸⣿⣿⢠⡿⣷⣶⣶⣶⣶⣶⣖⡤⠄⣾⣿⣿⠿⠤⠤⠤⠤⢤⡤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣿⣿⣿⡇
⠀⠀⠀⢀⣠⠤⠒⠉⣏⢀⣰⣿⡷⠚⢋⣉⠉⠉⣉⣭⣭⣭⣵⡿⢿⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣐⣒⣒⣒⣒⣲⣾⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠃⠠⣠⣴⣮⣿⠉⡴⣿⣥⡴⠛⣋⣴⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠶⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⣛⣭⣿⣿⣿⣿⡿⠋⣠⣴⣾⣿⣿⣿⡿⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⡿⠟⣠⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⡿⠟⢋⣤⣶⣿⣿⣿⣿⣿⡿⠟⠡⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⡟⢋⡐⠋⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⣄
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣮⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⣠⡀⣠⣄⣀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⡈⣀⣀⣀⣀⣤⣄⣀⣀⣀⣈⣀⣄⣂⣘⡆⠀⡌⠠⣀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠒⠒⠀⣪⡤⠽⠓⠚⠛⠉⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠙⢉⠑⠈⠀⡙⢂⠳⠄⢄⣤⣤⡀⣄
⠀⠀⠀⠀⠀⠰⠀⠉⠁⠛⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠐⢀⠆⠁⣤⠿⠃⠀⢀⣤⠍
⠀⠀⠈⠄⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⣠⣾⣶⠟⠀⠀⠀⠀⠟⢿⣄⡈⠹⠿⣶⣄⠀⠀⠈⠀⠜⠀⠀⠀⠀⠐⣲⢤
⠀⠀⠀⡀⠀⠀⠀⣶⢀⠀⣀⣠⣼⣾⣿⣿⣿⠿⣤⣄⡀⠀⠀⢀⣴⣿⣦⣴⣮⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠁
⠀⠀⠀⠀⠉⠩⣽⣿⣟⣛⣿⣯⣽⣿⢿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⠿⣛⠿⠷⠶⠶⠶⣷⡶⣶⠶⠾⠓
⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠿⠿⠊⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠄⠸⠏⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡆⠈⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⣤⡤⠼⠗⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⠀⠀⠀⢀⠀⠀⣀⢀⣀⣼⠦⠻⠚⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠴⠖⠒⠛⡟⠁⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣼⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡍⣀⡀⠀⠀⠶⢺⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣀⣠⣤⠶⠖⣶⠏⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠉⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢹⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⢷⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠘⠀⠂⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠁⠀⢀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⡠⢀⣴⠿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠤⣂⣠⡾⠋⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡄⠀⠀⠀⠀⢀⣰⣤⡶⢞⡁⡀⢈⡟⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣃⣤⣤⠴⠆⢻⡏⠄⠋⢀⣀⣠⣄⣼⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢢⡤⠟⠞⢹⡃⠀⡠⠔⠈⣠⣴⣶⣿⣿⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⣀⠸⠗⢈⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣨⣥⣴⣾⣿⣿⣿⣿⡿⠟⠙⣿⢿⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡭⠛⠟⠛⠛⠛⢛⣛⣉⠻⠞⣻⣗⡜⢼⣟⣿⣍⠛⢿⣶⣾⣿⡛⠿⢿⣤⣴⣬⣶⣶⣋⠁
⠀⠀⠀⠀⠀⠺⠛⠀⠀⠀⠀⠶⠖⠋⢲⣶⠚⣳⣶⢿⣦⡾⢿⣿⣋⣀⣴⣿⡿⠿⣿⡿⠿⢛⣡⡘⠋⢘⠛⠶⠿⠿⢿⣿⣶⣿⣷⣬⣽⣿⣿⣿⡿⠟⠁⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠥⠤⠶⠾⠛⠋⠹⣿⣿⣿⠿⠂⠒⠻⣭⣿⠿⠿⠿⢓⡠⠀⢾⣿⣿⠇⠀⢨⡛⠻⢶⣶⠀⠀⠛⢛⣉⣩⣍⡙⠛⠛⠃⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⢀⣙⠫⠥⠤⠄⠈⠉⠋⢡⣤⣤⣤⣉⡁⠀⠐⢛⣫⡵⠀⠀⠈⠻⣷⣾⣯⡭⠉⠉⠛⠿⣿⡿⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⠖⠂

⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢒⣒⡻⠯⠭⠯⠽⠭⠽⢟⣒⡒⠤⢤⣀⡀
⠀⠀⠀⠀⠀⣶⣆⡤⠖⠊⣉⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣉⠑⠲⢤⣰⣶
⠀⠀⠛⠛⠛⣿⡟⢛⡿⠛⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢿⡛⢻⣿⠛⠛⠛
⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣄⣤
⡟⠉⢻⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⢸⠛⡟⢹⠋⡇⠀⠀⠀⢀⣀⠤⠄⠒⠚⠋⠉⠉⠉⠉⠙⠓⠒⠠⠤⣀⡀⠀⠀⠀⠸⠙⡏⢻⠛⡇
⠀⠀⢸⠀⡇⢸⠀⣇⣠⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⣸⠀⡇⢸⠀⡇
⠀⠀⠈⠀⠁⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠀⠁

⠀⠀⠀⠀⣶⣄⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣠⣶
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⡿⠋⠉⠙⠻⣿⣿⣿⠟⠉⠉⠙⢿⣿⣿⡿⠋⠉⠉⠻⣿⣿⣿⠟⠋⠉⠙⢿⣿⣿
⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⢹⣿

⠀⠀⠀⠰⡀
⠀⠀⢈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠠⣄⡀
⠀⠀⡉⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠛⠂⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠒
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠡⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠂⠈⠃⠀⠀⠀⠀⢀⣠⠴⠦⢄⡀⠀⠀⠀⠀⠠⢀⠨⠀⠀⠀⠀⠃⠂⠀⠀⠀⣤⡀⣤⣀⡀
⠀⠀⠐⠀⠀⡷⠐⠀⠀⠀⠉⠀⢀⠀⠀⠄⠀⡀⠀⠀⠀⠐⡂⠀⠀⠐⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢡⠦⡀⠀⠄⠀⠀⠀⢤⡈⠉⠠⠀⠀⡀⠀⠄⠀⠀⠁⠃⣴⠆⣀⠀⠀⢶⠀⠀⠀⠰⠄⠀⠀⢠
⠀⠀⠀⠀⢰⠠⠀⠀⠀⠀⠀⠙⠀⠀⠀⡄⠀⠀⠀⣀⠀⠀⢐⠀⠀⠁⢀⢀⣠⠀⠁⠀⠀⠀⠀⠀⡆
⠀⠀⡧⠠⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⡀⠀⠀⠄⠀⠀⠀⠃
⠀⠀⠑⠋⠀⠀⠂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠢⠄⠈⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⡸⣿⡅⠀⠐⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠴⠠⠂⠈⠉⢀⠀⠀⠚⠃⠐⠀⠀⠢⠒⠂⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡤⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⢨⠟⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣻⣤⣀⣀⣂⣀⣀⣀⣀
⣿⣿⠛⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿