Candid Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠢⣄⠀⠀⠀⢀⡤⠒⠒⠒⢤⠀⠀⠀⠀⣠⠔⠒⠒⠲⢄⠀⠀⠀⠀⡠⠒⠒⠒⢦⡀⠀⠀⠀⢠⠔⠒⠒⠲⢄
⠀⠀⠀⠀⠀⠀⢰⠁⠐⠆⠰⠀⠈⡆⠀⠀⡎⠀⠲⠀⠆⠀⢡⠀⠀⢰⠁⠐⠂⠰⠂⠈⡆⠀⠀⡜⠀⠒⠀⠒⠀⠱⠀⠀⢠⠃⠐⠀⠐⢑⠈⡇
⠀⠀⠀⠀⠀⠀⠸⡄⠢⢄⡠⠆⣠⠃⠀⠀⢣⠠⠖⠂⠲⢄⡜⠀⠀⠘⣄⠲⢄⡠⠖⢠⠃⠀⠀⢱⡀⠀⢀⠠⠂⡴⠀⠀⠈⢆⠹⣉⢉⡽⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠒⠚⠁⠀⠀⠀⠀⠙⠒⠢⠒⠋⠀⠀⠀⠀⠈⠑⠒⠒⠚⠁⠀⠀⠀⠀⠙⠒⠄⠒⠊⠀⠀⠀⠀⠈⠑⠂⠐⠒⠁


⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄⠀⠀⠀⠀⠀⢀⠤⠤⠤⡀⠀⠀⠀⠀⠀⡠⠤⠤⢄⠀⠀⠀⠀⠀⢀⠠⠤⠤⣀⠀⠀⠀⠀⠀⡀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢠⠪⡖⡦⢴⢢⠑⡀⠀⠀⡔⠁⠂⠀⠒⠈⢆⠀⠀⢀⠎⠰⡂⢈⡂⠑⡄⠀⠀⠰⠁⢤⠀⡤⠈⢣⠀⠀⢀⠎⢰⡂⢀⣲⠙⡄
⠀⠀⠀⠀⠀⠀⢸⡀⢩⡄⢨⠅⢀⠇⠀⠀⠇⠀⠉⢠⡍⡔⠞⡢⠀⠸⡀⠀⣁⣈⡀⠀⡇⠀⠀⢃⠀⠈⣀⠉⠷⢠⠀⠀⠘⡀⢈⠅⠂⢍⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠤⣁⣁⠤⠊⠀⠀⠀⠈⠢⢄⣈⡡⠜⠚⠁⠀⠀⠑⠤⣀⣀⠤⠊⠀⠀⠀⠈⠢⢄⣀⡨⠔⠃⠀⠀⠀⠙⠮⣒⣐⠮⠊


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⢈⠙⢦⠀⠀⠀⢠⠊⢍⠀⡨⠑⡄⠀⠀⠀⡔⠩⠄⠀⠍⢢⠀⠀⠀⢠⠞⠉⠀⠈⠑⢄⠿⠀⠀⡴⠊⡁⢀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⢸⠀⣈⠉⠈⡁⠀⡇⠀⠀⡇⠀⠒⣀⠂⠀⢸⠀⠀⢸⠀⠀⣃⣘⠁⠀⡇⠀⠀⠆⢀⠐⠀⠑⡁⢈⠀⠀⠰⠀⠀⣉⢈⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢦⡈⠻⠟⢀⠜⠁⠀⠀⠑⣄⠊⠀⠑⣠⠂⠀⠀⠀⠣⡐⠇⠘⢂⡴⠁⠀⠀⠘⢄⠑⠒⠊⣡⠎⠀⠀⠀⠣⡀⠚⠙⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠐⠢⡀⠀⠀⠀⢀⠤⠒⠀⠒⢤⠀⠀⠀⠀⣠⠔⠂⠐⠢⣄⠀⠀⠀⠀⡠⠒⠀⠒⠠⡀⠀⠀⠀⢀⠔⠂⠁⠒⢄
⠀⠀⠀⠀⠀⠀⢸⢇⣒⡲⢆⢒⡻⡄⠀⠀⡎⠀⠰⠀⠆⠀⢣⠀⠀⢰⠁⠀⠆⠰⠀⠈⡆⠀⠀⡜⠀⠭⠀⠬⠀⠱⠀⠀⢠⠁⠀⠦⠰⠆⠈⡇
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣀⠄⣰⠃⠀⠀⢣⠀⠒⠖⠢⢀⡌⠀⠀⠘⡄⠫⡉⢉⠝⢠⠃⠀⠀⢱⡀⠢⣄⠴⠀⡰⠀⠀⠈⢆⠀⠠⠔⠂⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠐⠊⠁⠀⠀⠀⠀⠙⠒⠀⠒⠉⠀⠀⠀⠀⠈⠑⠀⠀⠚⠁⠀⠀⠀⠀⠉⠒⠠⠐⠊⠀⠀⠀⠀⠀⠑⠒⠒⠒⠁

⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠙⠛⠛⠙⠛⠋⠉⠉⠛⠛⠛⠿⡿⠿⢿⠉⠛⠛⠛⠉⠉⠛⠛⠛⠻⠟⣛⡛⣟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢠⠀⠀⠀⣤
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣿
⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿
⣿⠇⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠠⠑⡆⠠⠌⠔⠣⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉
⣿⠇
⣏⠀⠀⠀⠀⠀⠀⠀⠄
⣿⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⣿⡀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⣿⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣶⣶⣶⣀⣴⣾⣿⣿

⠀⢀⡴⠚⠋⠉⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⢠⠴⠚⠉⠉⠙⠓⢦⡀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠙⠓⢦⡀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠙⠓⢦⡀
⢰⠋⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡰⠃⠀⣀⠀⠀⠀⣀⠀⠙
⡏⠀⠰⠋⠳⠀⠸⠉⠳⠀⢸⡆⠀⠀⢠⡇⠀⠐⠿⠂⠀⠸⠿⠀⠀⢸⠀⠀⠀⠀⡇⠀⠐⠿⠂⠀⠘⠿⠀⠀⢸⠀⠀⠀⢰⡇⠀⠈⠛⠁⠀⠘⠛
⢧⠀⠀⠦⣀⣀⣀⡴⠀⠀⣸⠁⠀⠀⠀⣇⠀⠀⢦⣀⣀⣀⠴⠀⠀⡼⠀⠀⠀⠀⣇⠀⠀⣠⠴⠶⠦⣄⠀⠀⡼⠀⠀⠀⠈⣇⠀⠀⡖⠒⠒⠒⣶
⠈⢣⣀⠀⠀⠉⠁⠀⢀⡴⠃⠀⠀⠀⠀⠘⢦⡀⠀⠈⠉⠀⠀⢀⡜⠁⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠘⢦⡀⠈⠲⠶⠚⠁⣀⠜
⠀⠀⠈⠓⠲⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠖⠚⠁

⠀⠀⢀⣠⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⡀
⢀⡴⠋⠀⠀⠀⠀⠀⠉⠳⡀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⠞⠉⢀⠀⠀⠀⡀⠙⢦⡀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⠙⢦
⡾⠀⠀⣤⡄⠀⢀⣤⡀⠀⢹⡀⠀⠀⠀⡏⠀⢤⣀⡄⠀⣄⢀⡄⠀⢳⠀⠀⠀⠀⡎⠀⠀⠼⠇⠀⠸⠧⠀⠀⢳⠀⠀⠀⢀⡏⠀⠐⢦⡄⠀⣠⡴⠂
⡇⠀⠀⣈⣀⣀⣀⣉⠀⠀⢸⠇⠀⠀⠘⡇⠀⠀⠉⣀⣀⡈⠉⠀⠀⢸⠀⠀⠀⠀⡇⢰⡓⠒⠒⠛⠛⠚⢻⠆⢸⠀⠀⠀⠸⡇⠀⠘⠉⣀⣀⡀⠉⠃
⠹⣄⠀⠈⠉⠉⠉⠉⠀⢀⡞⠀⠀⠀⠀⢳⡀⠀⠚⠁⠀⠉⠓⠀⢠⠏⠀⠀⠀⠀⠳⡀⠱⢄⡀⠀⣀⡠⠋⢠⠏⠀⠀⠀⠀⢳⡀⠀⠚⠁⠀⠉⠓⠀⣠
⠀⠈⠲⢤⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠀⠙⠢⣄⣈⣉⣁⡤⠖⠁⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⡤⠖⠁


⠀⢀⠴⠚⣉⠉⢉⠙⠢⣄⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠉⠓⠦⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠉⠓⠦⡀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⠦⡀
⣰⠋⠀⣚⠁⠀⠈⣙⠀⠈⢧⠀⠀⠀⠀⡴⠁⢀⡀⠀⠀⠀⣀⡀⠘⣆⠀⠀⠀⠀⡴⠁⢠⡀⡠⠀⣄⢀⡄⠘⣆⠀⠀⠀⠀⡼⠁⢀⡀⠀⠀⠀⠀⠀⠙
⡇⠀⠀⠛⠃⠀⠈⣿⠁⠀⢸⡆⠀⠀⠠⡇⠀⠠⠜⠃⠀⠚⠧⠄⠀⢸⠀⠀⠀⠀⡇⠀⠠⠛⠦⠀⠼⠻⠄⠀⢸⠀⠀⠀⢰⡇⠀⠠⠼⠓⠀⠀⠿⠃
⢷⠀⠀⣠⠶⠒⠲⢄⠀⠀⣸⠁⠀⠀⠀⣇⠀⠀⠦⣀⣀⣠⠔⠀⠀⡼⠀⠀⠀⠀⢧⠀⠀⢤⡤⡤⣤⡤⠀⠀⡼⠀⠀⠀⠈⣇⠀⠀⢦⣀⣀⣠⠔
⠈⠳⣄⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⢣⣀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁

⠀⠀⢀⡤⠴⠶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠦⢤⡀
⢀⡔⠉⢀⡤⠀⢠⣀⠈⠳⡄⠀⠀⠀⠀⢠⠞⠁⣀⡄⠀⢤⡀⠈⠳⡀⠀⠀⠀⠀⢠⠞⠁⣀⡀⠀⢀⡀⠈⢳⡀⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⡀⠉⢢
⡞⠀⢠⠀⣠⠀⢀⠈⣀⠀⢹⡀⠀⠀⠀⡏⠀⠀⣠⠖⠒⠦⡄⠀⠀⢳⠀⠀⠀⠀⡏⠀⢀⣤⡉⠀⢉⣄⠀⠀⢳⠀⠀⠀⢀⡏⠀⠀⣾⡇⠀⢻⣧
⣇⠀⠈⣉⠁⠀⠈⢉⠁⠀⢸⠇⠀⠀⠈⡇⠀⠠⡇⣀⣀⡀⢸⠀⠀⣸⠀⠀⠀⠀⡇⠀⠀⠙⣁⣀⣈⠋⠀⠀⢸⠀⠀⠀⠸⡇⠀⢀⣀⣀⣀⣀⣁
⠸⣄⠀⠈⠓⠒⠚⠉⠀⢠⠞⠀⠀⠀⠀⠹⣄⠀⠙⠧⣤⠼⠋⠀⣰⠃⠀⠀⠀⠀⠹⡄⠀⠊⠁⠀⠈⠓⠀⣠⠏⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⠳⢤⣄⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠀⠈⠳⠦⣤⣤⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣠⡤⠞⠁⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣠⠤⠞⠁

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⣶⠀⠀⠀⠀⠀⣴⣶⡤




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠶⠶⠀⢀⣤⠀⠀⣤⣤⠀⠀⠀⠀⢠⣶⠀⢰⣦⠀⢠⣾⡆⠀⣠⣶⡀⢀⣀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢀⡄⢰⣶⣦⡄⣶⣶⡀
⠀⠀⠀⠀⠐⠃⠀⠚⠛⠃⠀⠉⠁⠀⠀⠀⠀⠐⠒⠀⠀⠙⠗⠀⠉⠛⠀⠟⠛⠛⠘⠛⠛⠐⠟⠓⠀⠁⠐⠀⠘⠋⠀⠈⠙⠛⠀⠙⠛⠁
⠀⠀⠀⠀⠀⣠⠀⠀⣤⡀⠠⠂⠂⠀⡆⣠⠀⢀⣀⠀⠠⢎⡆⠀⣼⡿⠁⢰⣏⠀⢠⣾⣧⢀⣤⣄⠀⠠⣷⡆⠀⢶⠂⠠⠶⣶⡄⣀⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠉⠀⢀⣉⠉⠀⠈⢉⡀⠀⠈⣁⠀⠈⠁⠀⠈⠁⠀⢀⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉
⠀⠀⠀⠀⠲⠶⠄⠀⠶⠆⠈⢦⠀⠈⠋⣤⡄⢿⡿⠇⠘⣿⣿⠸⢿⣦⠀⠀⢼⡄⠙⣧⡄⠀⣤⣴⠀⢤⣤⠀⠰⣤⠀⠀⠀⡍⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⣠⣤⡀⣠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤⡀⠀⣤⠀⢀⡀⠀⠀⠀⠀⡀⠀⠄⠀⠀⢀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠘⠷⠄⠙⠛⠃⠘⠿⠟⠀⠿⠿⠃⠈⠻⠆⠺⠄⠶⠀⢿⠿⠀⠶⠿⠃⠘⠿⠆⠈⠛⠷⠀⠀⠾⠁⢴⡶⠦⠀⠾⠆⠀⠿⠼⠇
⠀⠀⠀⠀⣶⣶⠆⢀⡀⡀⠀⠘⠀⠀⠰⠗⠀⡀⠀⢀⠠⣦⡤⠀⢠⡄⠀⣴⣄⠀⠀⠀⠀⠀⢐⡄⠀⢰⣦⠀⠀⠀⠀⠀⣰⡄⠀⢰⣶
⠀⠀⠀⠀⠛⠛⠃⠘⠋⠁⢀⠺⠇⠀⠀⠆⠀⠛⠛⠋⠀⠈⠁⠀⠘⠃⠀⠈⠛⠓⠀⠀⠀⠀⠘⠃⠀⠈⠋⠀⠘⠀⠀⠘⠛⠛⠀⠘⠛
⠀⠀⠀⠀⢨⡄⠀⢰⣴⡆⠈⣿⠇⠀⠀⠀⠀⠐⠒⠀⠺⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⣤⡄⠀⢀⣀⠀⠀⣄⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⣤⠆⠀⣀⡀⠀⠀⠀⠀⠀⣤⠀⢀⡀⡀⠀⢠⡄
⠀⠀⠀⠀⠶⠆⠀⠀⠉⠀⠰⠆⠶⠀⠀⠀⠀⠉⠟⠁⠈⠉⠉⠀⠙⠋⠀⠊⠈⠀⠰⠀⠆⠀⠚⠃⠀⠀⠀⠀⠀⠿⠂⠀⠀⠓⠀⠐⠶
⠀⠀⠀⠀⢠⣤⠀⣄⢀⡄⠀⠶⠀⠀⠰⠀⠀⣶⢀⡄⣠⠶⣦⢠⡶⢶⡄⢰⣶⠄⠠⠶⠆⠀⡴⣦⠀⠰⠶⠀⣰⠶⣆⢰⡶⡶
⠀⠀⠀⠀⠈⠁⠀⠈⠛⠁⠘⠶⠛⠀⠐⠀⠀⠈⠐⠁⠻⠴⠿⠸⠧⠾⠃⠀⠟⠀⠰⠶⠆⠀⠟⠻⠀⠰⠶⠆⠰⠶⠂⠰⠷⠿⠂
⠀⠀⠀⠀⠚⠛⠀⠀⠖⠀⢰⣿⣧⠀⣿⣿⠀⣴⠴⡦⠠⣴⠟⢰⣦⣤⡄⢠⣦⠀⢰⣾⠗⠀⣀⡀⠀⢷⣶⡇⢀⣰⣿⠐⠷⣄⠠⣿⣿⡆
⠀⠀⠀⠀⣙⣙⠁⢀⣀⡀⠈⢘⠋⠀⠈⠁⠀⠀⣀⠀⠀⠉⠁⠀⠀⡉⠁⠀⠀⡀⢀⠉⠀⢀⣀⣀⠀⠈⠉⠁⠈⠋⠁⠀⠀⠉⠀⣉⣉
⠀⠀⠀⠰⣿⣿⠇⢸⣿⡿⠀⠉⠁⠠⣾⣷⠄⢀⣿⠀⣾⠿⣿⢠⣦⣴⡄⣠⠞⠁⢬⣤⣴⠈⠉⠉⠀⠳⡷⠀⣴⣶⠀⠀⣤⡆⠰⣿⣿⡇
⠀⠀⠀⢀⣬⣥⡄⢀⣭⡀⣠⣤⣤⠀⣤⣤⡀⠈⠉⠀⢀⣤⣄⢠⣭⣭⠀⣥⠀⠀⠈⣀⠁⠀⣠⣤⠀⠀⠀⠀⣨⠁⡀⠀⠀⣠⠀⠈⠁
⠀⠀⠀⠀⠻⠿⠃⠸⣿⠃⠙⠛⠛⠘⠿⠿⠓⠾⠿⠗⠛⠻⠟⠸⣿⡿⠆⠙⢦⡄⠐⠿⠧⠘⣿⡿⠀⣤⣶⠄⠹⠾⠇⠀⠚⠁
⠀⠀⠀⠐⣦⡀⠀⣶⣶⡦
⠀⠀⠀⠀⠀⠑⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⡆⠀⠀⠠⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠄⠀⠛⠓⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠂⠀⠀⠈⠀⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⢀⡀⢠⠀⠀⠀⡀⢀⠀⢠⠀⡀⠀⡀⠀⠀⠀⡀⠀⠀⢠⡀⢀⠀⡀⠀⡀⠀⠀⠀⡀⠀⠀⠀⡀⠀⡀⠀⠀⢀
⠀⠀⠀⠘⠷⠾⠃⠳⠶⠋⠸⠶⠞⠀⠷⠶⠃⠲⠶⠆⠙⠶⠟⠸⠗⠿⠀⠷⠲⠋⠸⠶⠖⠑⠶⠦⠘⠷⠶⠃⠲⠶⠞⠘⠶⠞⠃⠶⠶⠂
⠀⠀⠀⠀⢀⡀⠀⠀⣤⠀⠀⢤⠄⠀⢀⡀⠀⣴⣤⡄⠀⠀⠀⠀⢚⡧⠀⠘⠛⠀⢠⣒⡆⠀⣴⣦⠀⠚⠃⠀⠀⠛⠀⢠⡶⢶⠀⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠋⠙⠀⠀⠀⠀⠀⠉⠁⠀⠙⠉⠀⠀⠉⠀⠀⠋⠀⠈⠁

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠑⠀⡠⠒⠠⠀⠀⠀⠀⠀⠀⠀⢌⠀⠠⢀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠆⢀⡠⠄⠀⠐⠠⢀⠠⠉
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠠⠁⠒⠁⠔⠈⠉⠁⠢⠀⠄⠠⠈⢠⠋⠀⠀⠀⠀⠀⠀⠑
⠀⢀⠔⠐⠀⠐⠄⣀⡀⠄⠀⠀⠉⢀⣉⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠠⠀⠀⠀⠀⠀⠀⡇
⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠄⠄⡀⠀⢄⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⢄⠀⠀⠀⠀⡠⠀⢄⢧⠀⠀⠀⠀⠠⠀⠀⠠⠁
⠀⠐⠠⠤⠆⡡⠒⠉⠀⠀⠀⠀⠀⢀⠀⡀⠐⠀⠀⠁⠀⠀⠀⠀⠊⠉⠑⠐⠠⠀⠁⠄⠊⠸⡀⠀⡀⠀⠑⠄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢰⠀⠙⡡⠌⠀⡁⠀⠐⠀⠀⢀⠔⠀⢀⠠⠀⠀⠀⠀⠑⠂⠀⠀⢀⠀⢄⠀⠀⡠⠄⠀⠀⠀⢄⠀⠀⠀⠀⠀⠁⠀⠑⠄
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠔⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠌⠀⠁⠀⠀⠀⠀⠀⠀⠐⠄⢰⠀⠓⣀⡘⠀⡀
⠀⠠⣲⠔⠂⠒⢄⠀⢰⠥⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠌⠑⠡⠐
⠀⠀⡄⠀⠀⠀⢈⠀⠈⠐⠊⠈⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣀⡀⠀⠠⠀⠐⠀⠀⠐⠀⠀⢢
⠀⠀⠈⠢⠤⠜⠃⠀⠀⠂⠤⠄⠐⣁⢀⠡⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⢀⠢⡀⠀⠀⠀⠀⢀⠔⠁⠀⠄⠀⢀⢣⢄⣲⡠
⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠑⠄⠈⠀⠀⠡⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠊⠀⠀⠁⠀⣀⣩⠀⠀⠀⠤⠀⠤⢈⠉⠉⡩⢄
⠀⠀⠀⠊⠀⠀⠀⠀⠀⡠⠀⠈⠄⠀⠐⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⠉⠙⢄⠘⠮⠏⠀⠀⠀⠀⠀⠀⠀⠑⡄⠵⠾
⠀⠀⠀⠀⠀⠀⠄⠀⢈⢀⠀⠀⠂⠀⠈⠉⠀⠁⠢⡀⠀⠐⠊⠉⠁⠂⡀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠔⠚⠀⠀⠁⠌⠀⠀⢺⠆⠐⠿⠀⠂⠀⠀⠄⠀⠠⠂⢰⠀⢄⠀⠦⠂⠀⡐⢀⠠⢀⠠⠀⠀⠠⠀⠤⠂⠀⠌
⠀⠀⠀⠈⠂⠄⢀⠀⠀⠄⠊⠀⠀⠠⠀⠀⠁⡀⠔⠀⠀⠀⡀⠀⠈⡠⠂⠀⠀⠁⠐⠂⠈⠀⠡⠀⢀⠂⠑⠠⠀⠀⠀⠄⠊
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠟⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠙⣿⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⣴⡀⢀⣀⣀⣀⠀⠀⠀⠀⢰
⣶⣶⣶⣶⣶⣶⣦⣀⣀⣀⣀⣀⣀⡀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⣀⣸⣿⣇⡀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣧⣼⣿⣿⣿⣤⣤⣄⡀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡗⣻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧⡇⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠻⠛⠟⠻⠟⠻⠿⠿⠟⠻⣿⣿⣿⣿⣿⠋⠛⠋⠘⣿⣿⣷⣼⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠀⠀⢀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣿⠿⢿⣿⠇⠀⠀⠀⠈⣹⡟⣿⣧⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣟⣿⣿⠀⠀⣠⣄⣀⠀⠘⠿⠄⠀⠙⣆⠀⠀⠀⠀⢠⣤⣶⠀⠀⠉⠀⠀⠛⠀⠀⠀⠀⠈⠛⣇⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿
⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⠀⠀⣧⠀⠀⠀⠀⠀⢀⠀⣱⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⠷⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣯⡯⠀⠀⢠⡄⠀⠀⠀⠀⠘⣾⣿⢹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢈⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣇⡃⠀⠀⠀⠘⣿⢿⣸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⢿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⡇⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠇⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⡇⠀⠀⠀⠛⠿⠈⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣷⠀⠀⠀⡀⠀⠀⣹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⡇⣀⣴⠃⣤⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡁⠀⢨⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢻⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠀⠾⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠻⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⣠⡤⠶⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠐⠒⠂⠤⣀⠀⠀⠁⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢋⡉⠉⠛⠻⢶⣦⣀⠑⢄⠀⢀⠔⠂⠉⠉⠉⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣹⣿⣧⠀⠀⠀⠀⠙⠳⣬⡶⢁⣠⣤⡴⠶⠶⠶⠤⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⢛⣿⣿⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠖⠖⠶⠶⠦⢤⣄⣀⠀⡰⢋⠻⠟⠋⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠂⠒⠒⠚⠛⠛⠒⠲⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠈⠉⠙⠛⠛⠛⠛⠒⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠤




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠂⠀⠉⠉⠉⠉⠉⠁⠀⠐⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣄⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢀⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⣶⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡠⠄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠤⡀
⠀⠀⠀⠘⣄⠀⠈⠢⢆⠀⠀⡔⠉⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⡐⠉⠁⡄⠀⢀⠦⠊⠀⢀⡘
⠠⠄⠤⣀⠀⠑⢄⡀⠀⠑⠤⡘⢄⡀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢠⠃⠀⠀⢀⠠⠊⠀⡀⠜⡠⠔⠁⠀⣀⠔⠁⢀⡠⠄⠤
⠠⠀⠀⠀⠈⠑⠂⠌⣂⣄⠀⠈⠐⠬⠆⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⡰⠁⠀⢀⠔⠁⠀⠀⠮⠔⠊⠀⢀⣄⡪⠄⠒⠉⠀⠀⢀⡠
⡠⠀⠀⠀⢀⡤⣀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠰⡤⠄⢀⣀⡀⠠⠄⠂⠁⣀⠔⠊⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠠⣔⢂⠉⠤⠤
⠣⠀⣀⠀⠀⠀⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠒⠀⠂⠐⠀⠂⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⢀⣀⠤
⠀⠀⠀⡀⠌⠈⠁⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠩⠀⠀⡀
⠀⠀⠀⠂⠄⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠤⠀⣀⣀⠠⠔⠊⠉⠒⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠊⠉⠒⠤⢀⣀⡀⠠⠄⠐⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⣿⣿⣿⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠰⠀⠒⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢚⣛⣓⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢏⡉⠉⠉⠉⠉⡧⢤⡀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠤⠤⠀⢸⠀⠀⠀⠀⠀⠀⠐⡚⠐⢧⠀⡸⠉⡀
⠀⠀⠀⢸⣸⠀⠀⠀⠀⡇⢠⠇⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠓⡴⡏⠀⣇⣀⡇
⠀⠀⠀⠈⣇⢀⠀⠀⣠⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⠢⠤⣄⡸⠀⠀⠀⠀⠀⠀⠀⢐⠁⡇⠀⢹⡅⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠦⠇⠀⠘⠷⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⠰⠀⠀⠤⣤⠤⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣛⣛⣛⣛⡛⠋
⠀⠀⢠⠦⡤⠀⢸⠀⠀⣤⢾⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⡏⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢰⠶⡏⡄⠀⠀⠀⢹⠶⡄
⠀⠀⠞⠀⠣⠐⠋⠂⠠⠏⠈⠃⠀⠀⠀⠀⠀⠀⠀⠞⠚⢳⠐⠂⢸⠀⠀⠀⠀⠀⠀⠀⠈⠚⡇⠁⠀⠀⠀⢸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣉⣸⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⣜⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢠⠾⠤⠴⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⡇⠀⡆⡆⣢
⠀⠀⠀⠀⢸⣄⡠⠤⢺⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡤⠼⠷⠤⠤⠤⠤⠤⡄
⠀⠀⠀⠀⢸⣥⠤⠔⢺⠀⠀⠀⠀⠀⠀⠀⠈⡦⠤⠤⠤⠤⠒⡺⢅⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠸⣄⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠉⠓⠒⠒⠒⠋⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⣀⣀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⣀
⠀⠀⠀⠘⡯⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⢸⣅⣀⣀⣑⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣛⣛⡚⠛⡂⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠈⢇⣸
⠀⠀⠀⢸⡀⠀⠀⢠⣤⡽⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣓⣒⡋⠀⡇⠀⠀⠀⠀⠀⠀⠀⢨⠉⠑⠢⡀⠙⠈⠙⢦
⠀⠀⠀⠀⢧⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡩⠭⠅⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⢀⣴⡎
⠀⠀⠀⠀⠘⠤⠤⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠧⡀⣀⣀⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠤⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠿⠿⠛⠛⠛⠛⠿⠿⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄
⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠿⠿⠛⠛⠛⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⣀⣤⢤⡄⣠⣤⣤⠄⣀⠤⡤⠠⠤⠄⠠⠤⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⢤⢤⣤⠠⣤⣄⣤⣄⣄⡤⠀⣤⣤⣤⣤⡄
⠀⡿⢃⠐⠃⢸⣿⣿⠀⠁⠀⠰⡄⠴⠒⠀⠈⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠄⠀⠿⠀⠘⣿⣧⠀⢀⠈⣷⠀⢸⠟⠉⣻⣿⣷⣠⣬⡿⣿⣿⡇
⠀⠗⠈⠀⠀⠀⠉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠇⠀⣶⣠⣴⡞⠀⠀⠀⠀⠻⠟⠀⠉⠐⣿⠀⠀⠀⠀⢿⣿⣿⠛⡗⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⢘⣁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠆⠀⠠⠀⠈⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠓⠺⣄⠀⠀⠀⡀
⠀⡆⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⢰⡤⣿⡋⣴⡄⣜
⠀⡇⠀⠀⠀⠀⠀⠛⢨⣧⣘⠠⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠨⣹⠃⣿⣿⣿⣧⣤⣤⣄⣤⡆
⠀⢧⡤⠀⠀⠀⠀⡄⠀⠿⠿⠃⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⢀⠀⠀⠸⣧⡀⠀⠀⠀⠙⢀⣿⣝⣯⠼⣿⣿⣿⣿⡇
⠀⡏⠀⠀⡇⠀⠀⠀⢼⡃⠀⠀⠀⠀⠀⠀⡇⢀⣶⡌⠀⠁⠠⢠⣶⡟⠋⠙⠛⢧⠀⢰⡇⡀⢸⡆⠹⣷⠀⡄⠀⠀⣻⣿⣠⡄⣟⣿⣇⣛⠇
⠀⣿⣀⠀⠁⢀⡆⢀⡀⢁⠈⠓⠀⠀⠀⠀⠀⣼⣿⡃⢐⠊⢱⣷⡀⠉⠙⠢⠴⢲⡷⢰⡆⠀⢸⣧⠀⡿⢀⡤⡀⠀⣿⡿⠨⣿⢿⣬⣤⣙⠇
⠀⣿⠿⠤⠆⢀⡄⠈⡷⠋⣴⣾⠀⠀⠀⠀⣶⢻⣏⣡⠈⠉⣾⡄⠆⠀⢀⣀⠀⠸⠷⢺⣥⣄⢸⣿⣿⣿⣿⣇⡁⠀⣿⠀⢠⣿⣾⣿⣿⣿⡇
⠀⡧⠀⠀⠀⢸⡇⠀⡛⠠⡘⢨⠀⢀⣠⣴⣿⣷⡿⠥⠄⢰⣿⡗⢀⣁⣼⢿⠁⢸⣼⢯⡍⠀⠀⠀⠁⠾⡄⢹⡅⠀⢸⡄⠦⡿⠋⣻⣿⣿⡇
⠀⡇⠀⠀⠀⢸⡇⢈⣿⣶⣿⠗⣾⣛⠟⠟⢋⣹⣧⣤⡤⣺⣿⣿⣿⣷⣾⣿⠿⠛⣁⣼⠃⢀⠀⠀⠀⠀⠈⠛⢹⣷⣤⡙⠀⢼⣿⣿⣿⣻⡇
⠀⡇⠀⠀⠀⢸⡁⢀⠿⠟⠛⣸⣿⣯⠶⠛⣀⡀⠶⣞⣛⣯⣿⣿⣿⣿⡿⣿⣤⣬⣴⠆⠀⠈⠳⣄⠀⢀⣰⠆⠀⠙⢤⠀⠰⠳⣿⣿⣿⣿⡇
⠀⣷⠦⠐⠂⢸⡿⠛⡓⢂⣿⣿⣯⣵⡾⠿⢻⡿⠉⠉⠛⠋⢿⣿⣿⠟⢿⡌⣿⣿⢿⣄⠀⠢⡀⠈⠠⠘⠇⠀⠀⠀⠀⣠⣶⣾⣿⣻⣟⡻⡇
⠀⣸⣂⣠⣾⢈⡀⠀⢠⡟⣿⣭⢟⡻⣇⣀⣿⣠⣿⣶⣄⣈⠉⠻⠛⠛⡦⠙⣷⡄⠀⠉⠃⣄⠀⠀⠄⠈⠀⢈⣀⡄⠀⠉⠀⠀⢻⣟⣭⣽⡇
⠀⢿⣯⣥⣴⡈⡁⠀⣾⠃⣿⣇⠀⠀⠙⠿⠟⡀⢾⣿⣿⣿⣶⣦⣄⣀⡀⠀⣼⣿⢋⣶⡀⣈⣷⣄⡄⣀⠐⠞⣫⠀⠀⢠⣶⠞⠿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⠛⡇⣼⡏⠀⣿⣿⢳⣦⣤⠀⠘⢁⠈⣡⡀⠙⣿⡯⢿⣽⣿⣿⣿⢿⡆⢸⣿⢟⠉⠙⢿⡿⠓⣛⣡⠀⠀⠀⠀⡈⠀⣿⣟⣿⣇
⠀⢸⣿⣿⣿⠀⣵⣿⡇⠀⢻⣿⣿⣿⣿⣛⠛⠿⣿⡽⠛⠃⠹⠃⠈⠛⣿⣿⣯⣾⠁⠀⠇⣸⣷⣤⣈⡛⠋⡙⠧⡈⠀⠀⠢⣽⣽⣿⣿⣿⡇
⠀⢸⣿⣿⣿⢿⣿⣿⠇⠀⢫⡙⠁⠈⠀⢹⡟⢰⡶⠃⢀⡘⢾⣾⠀⣠⣿⣿⣿⠇⠀⠀⠀⣿⠋⠉⣩⡏⢢⠘⠦⠈⠀⠀⢹⣿⣷⣿⣿⣿⡇
⠀⢸⣷⣦⡷⠘⣿⣿⡄⠀⣿⣷⣦⣀⠀⣼⠇⠀⠀⣀⠈⠀⠸⣯⠀⣿⣿⣯⣷⣤⠤⣤⢀⣿⡟⠙⢹⡇⠀⠀⠐⢀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⡇⢸⣿⣿⡃⠀⣿⣿⣿⣿⣻⣿⣿⡟⠁⠀⠀⠀⢻⣟⠀⣻⣿⣿⣿⡟⣳⣾⠈⠛⣿⡆⢸⡇⠀⠀⠀⠘⡀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⡇⢻⣿⣿⣧⠀⣿⣿⣿⠸⢸⣿⢸⠁⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⣿⣧⡄⢿⡏⢸⠇⠀⢀⡄⠀⠀⠠⢘⣿⣿⣿⣿⣿⡇
⠀⢸⣿⠏⢱⠀⠙⣿⣿⠀⣿⣿⣿⣦⣼⣇⣸⣀⣤⠤⠤⢄⣸⣿⠀⣿⣿⣿⣿⣿⠻⡏⠟⢿⢾⠇⢸⠇⠀⠘⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣾⡇⠀⠀⣿⠙⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣴⠸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣿⣽⡿⢻⡇
⠀⢸⣿⣿⣧⣀⣦⠉⠛⠿⠏⠈⢻⣿⣿⣿⣿⣿⠀⠀⢰⠛⣿⠛⢿⠀⢿⣿⣿⡿⣿⣷⡄⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⡀⣾⡇
⠀⢸⣿⣿⣿⠿⣿⠗⠀⠀⣀⡄⢀⣀⠈⢿⣿⣿⣦⠀⠻⠀⣿⠀⡼⠁⠈⢿⣿⣿⡿⣿⣿⣆⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠿⡇
⠀⢸⣿⣿⣿⣾⣿⣴⡿⠛⠋⠉⠛⠉⠀⠈⣿⡿⣿⡆⠀⢹⣿⢤⣄⣀⡀⠘⣿⣿⣇⡬⢛⣿⠷⣄⠀⠀⢡⡀⠀⠘⠂⠀⠀⣆⠀⢯⠹⢛⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠁⢀⠀⠀⠀⠈⢿⣿⣿⣷⠀⠀⣿⣄⣴⢿⣏⠓⠸⣿⣹⡷⣾⡇⢀⠀⠠⢶⣤⡉⠀⡀⠀⠀⠀⢸⡀⢸⠆⠈⠁
⠀⢸⡿⣿⣿⡇⠈⠉⠀⢀⡂⠰⠆⠀⠉⠁⢹⣇⢸⣿⠀⠀⣼⣿⣿⠏⢉⣁⣠⣿⢻⣯⣿⠇⢸⡄⠀⠸⡿⣿⣷⣅⠀⠀⠀⢸⢁⡀⢂⡀
⠀⢸⣇⡤⣾⡈⠉⠁⠾⠿⢉⣠⣶⡯⣈⡄⠉⢿⠀⣿⠀⠀⣿⣬⣤⣷⠚⣿⣟⣛⣿⣿⣿⣤⣤⣧⣀⣤⡤⢽⣿⣿⠀⢶⠀⠈⠘⣿⣄⠳⡔
⠀⢸⣟⣷⣶⣄⣠⣤⠴⠶⣾⣟⣠⣶⣤⡤⢀⣿⣇⣼⡁⠰⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣏⣤⡴⣶⣾⣿⣿⡇⠀⠆⠀⢳⡈⢿⣦⡀
⠀⢸⡧⠀⠿⠟⠉⠁⣤⣴⣶⣾⣶⣿⣿⣿⣿⣿⣇⣻⡇⢾⣾⣾⣷⣿⣿⣦⣿⣾⣿⣿⡿⢋⣘⣯⣽⣶⣶⡾⠻⣿⣿⣄⠀⢠⣌⠻⡄⢙⡇
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⡉⣉⣬⣿⣿⠥⠾⡄⠀⠀⠀⢈⣭⣿⣏⣉⣁⣠⡬⠩⠇⠈⠙⠋⠒⠊⠀⠈⠉⠛⢛⠦⠶⠤⠭⠇
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠊⢁⡤⠖⠋
⠀⢸⡏⠀⠀⠀⠀⠀⠀⣰⠾⢿⡟⠤⠀⠀⠀⠙⢛⣃⡶⠖⠋⠁⠀⠊⠁⠀⠀⠀⠾⠛⠃⠀⠀⠀⠀⠀⠀⠠⠶⠿⠃⠀⠀⠛⠀⠀⠲⠶⡇
⠀⠘⠃⠙⠂⠀⠒⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⠒⠒⠂⠀⠀⠈
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⣶⣾⠿⠿⠟⠿⠷⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠾⠿⣶⣶⣤⣄
⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄
⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠉⠉⠓⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⣰⡟⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢀⣿⠀⣴⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⣿⣿⠀⠀⠘⠻⣿⣿⣿⣿⠿⠋⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⡄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⢠⡟⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠈⠻⣿⣿⣿⣿⡿⠟⠁⣼⠃⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣰⠏⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠀⠉⠙⠻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠛⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣰⣿⠃⠀⠀⠀⠀⡀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠏⠀⠀⢸⡇⢸⡇⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⡿⠿⠿⠿⢿⣷⣶⣄⠀⠉⠁⠀⠀⠀⣿⡇⢸⣷⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡄⠀⠀⠀⠀⠻⠇⢸⣿⠟⠁⠀⢀⣠⣤⣶⣶⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠛⣿⣆⠀⠀⠀⠀⠀⠀⠉⠀⣠⣶⠟⠋⠉⠀⠀⠀⣠⣤⣽⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠋⣿⡄⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣴⠌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⣿⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢰⡆⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⢶⣿⣦⣄⡀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠋⠀⠀⠀⠈⠙⠻⢶⣄⠀⠈⠳⠦⣄⣀⣀⣀⣀⡠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⡀⠀⠀⠀⣀⣀⣤⠞⣻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⣉⣹⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⢠⠴⠚⠉⠉⠁⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⡠⠚⠁⠀⠀⣀⣀⣀⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣿⡿⠿⠿⠛⠛⠛⠉⠉⠁


⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⢠⡟⠛⠛⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣦
⣿⠃⠀⠀⠀⠉⡀⠀⣀⣤⣤⣀⠀⠘⢿⣧⣤⣴⣶⣄⠀⠀⢴⣾⣿⣶⠟⢿⣿⠂⠀⠀⢸⡇⠀⠀⠀⢀⣤⡶⠿⢷⣾⡿⠀⢀⣀⣀⡀⠀⠀⢀⣀⣀⣰⣿
⣿⠀⠀⠀⠀⢸⣿⣾⠏⠁⠈⠙⣷⡀⢸⣿⠉⠀⠈⣿⡆⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⣸⣿⠀⠀⠀⠸⣧⣤⣤⣄⣿⠇⣰⡟⠉⠉⠻⣦⣠⡿⠋⠉⠙⣿⡀
⢻⣆⠀⠀⢀⣼⠟⣿⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣷⠀⠀⠈⣿⠀⠀⠘⣿⠀⠀⠀⣿⣿⠀⠀⠀⢸⣧⡀⠈⠉⣻⡷⣿⣷⠶⠶⠶⢿⣿⠃⠀⠀⠀⣿⡇
⠀⠙⠛⠛⠛⠁⠀⠙⢷⣦⣤⡾⠟⠐⠿⠿⠿⠀⢀⣼⠿⠷⢶⣴⣿⣦⣦⠀⠻⣧⣀⣴⠿⣿⣦⡄⠀⠾⠏⠙⠛⠛⠋⠀⠹⣧⣀⣀⣠⡄⢻⣦⣀⣀⣠⣾⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣭⣤⣤⣬⣽⣭⣥⣼⣿
⣿⣿⣿⣿⠛⠙⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣷⢿⣷⣿⡇⣷⡆⣤⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⡟⣿⣿⣷⣾⣿⢿⡇⣸⡇⣿⣗⣿⣾⡇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⢿⣷⠸⣷⢹⣿⣿⣿⣿⢻⡇⣿⠀⣿⣿⣿⣿⣟⣿⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡝⢿⣿⣿⣤⣿⡄⡧⣿⢸⣿⢸⡇⣿⣤⣿⣿⣿⡿⣿⣿⣿⣯⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡻⠎⠛⠙⠉⠈⠉⠡⣷⠸⣿⣹⣧⣿⣿⡟⠀⠀⠀⠙⠻⢿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢪⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠢⠀⣀⠀⠀⠀⠀⠀⢀⣤⣿⣇⠀⠀⠀⠀⣾⣿⣿⣷⣦⠀⠀⠀⠀⠊⡀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠀⠉⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡄⠀⠀⣸⡿⠟⠉⠙⡛⠿⣶⣦⠀⠀⠀⣽⡿⠄
⠀⠀⠀⠀⠀⠀⢠⠶⢦⡀⢻⣿⡇⠀⢀⣠⣴⣾⣿⣛⡏⠈⠁⠛⣤⠀⠀⢾⡟⠀⠀⡼⣡⠤⣌⠙⠿⢄⠜⣿⣇⣠⠦⡀
⠀⠀⠀⠀⠀⠀⣿⣴⣦⡈⢿⣿⡟⢳⠟⠁⡴⣯⠮⣿⡾⣿⡤⠈⠋⠀⠀⣶⠃⠀⣸⣿⣷⡛⠎⢷⡀⢠⢶⣿⡿⢁⣴⠑
⠀⠀⠀⠀⠀⠀⢙⠛⠽⢿⡆⢿⣗⠐⠦⣶⣷⡟⣷⠀⡥⠸⠷⠀⠀⠀⠀⢋⡀⠀⠋⠰⣏⣗⡀⠐⠷⠋⣼⣯⢀⡿⠟
⠀⠀⠀⠀⠀⠀⠸⡄⠐⣿⢃⣿⣿⣦⠉⠉⠘⡻⢷⣪⠴⢁⣤⡀⠾⠁⠀⠈⢻⠆⣄⡁⠻⠽⠆⠀⠀⠠⠿⡿⠘⢷⡄⡜
⠀⠀⠀⠀⠀⠀⠀⠘⢆⢻⡶⢩⠟⠋⠙⠁⠀⠉⠛⠁⣠⠟⢿⠉⠀⠀⠀⠀⠀⣀⡹⠳⢄⠀⠀⠀⠀⠀⠀⢲⢶⡾⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣿⠀⠀⠀⠀⠀⣠⠖⠈⠁⠀⠈⠉⠛⠷⣶⣶⠟⠉⠀⠀⠀⠈⠙⢢⡀⠀⠀⣼⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠜⢿⣄⣀⣤⠎⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣴⠟⠳⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢻⡇⠇⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠛⣿⢻⡆⠀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠇⠀⢸⣯⣿⣟⣒⣒⣲⣶⣶⣞⡛⠁⠀⠃⠀⠀⢀⣼⠁⠀⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣄⡄⠉⠉⠉⢉⣠⠀⠀⣀⣀⠈⠁⠒⠲⠴⠚⠋⠁⢰⠆⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢴⣆⠙⠳⠖⠛⠉⠀⠁⢰⣿⣿⣿⣶⣤⢀⣤⣤⣤⠶⠋⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢀⠀⠙⠉⠉⠉⠉⠀⠏⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣞⠀⠀⠀⠀⠀⠀⠀⣶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠦⠾⢤⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⢩⣟⣃⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⣠⠖⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠙⢓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠙⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⣀⣤⣤⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⠆⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠀⣠⠞⠁⠀⠙⢮⡁⠀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢱⠐⣆⡑⠒⠤⠄⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⢧⠀⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⣚⣷⠿⣋⣽⡶⠶⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠈⣏⣰⣾⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣶⢸⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⠛⢉⡴⠶⠶⠦⣄⠀⠀⠀⠀⢀⡴⠶⠶⣍⠛⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⣄⢸⢠⠏⠀⠀⠀⢀⠘⣧⠀⠀⠀⡟⠀⠀⣤⠈⢳⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣼⠁⣀⠀⠙⠾⡟⠀⠀⠀⠀⠙⠁⢸⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⢁⡼⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢿⠀⢨⢷⡀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣇⠀⠀⠀⠀⣸⠛⠏⣠⠴⠄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣇⢯⣈⠁⠀⠘⢦⣀⠀⠀⢀⣠⠏⣠⠶⣤⡈⠓⠦⠴⠞⠁⠀⡸⢹⠆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣭⣤⡇⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⠁⠜⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣿⠴⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⣤⣤⡤⠤⠶⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠲⠦⠾⡴⠻⠦⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⣀⡀⠠⠠⠤⠐⠺⢴⠒⢒⠉⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⡤⠔⠛⣋⢮⣉⠉⠁⠀⠀⠀⠀⠀⠀⢀⠤⠶⢄⡈⠆⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡁⢀⡏⠀⣤⠈⣧⠀⠀⠀⠀⠀⠀⢠⡇⠠⠦⠀⡧⠈⠦⢀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠔⢒⠷⠀⠱⢄⣈⡠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣚⣁⡀⢀⡠⠃⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠣⠅⡀⢠⠒⣐⠆⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠃⠐⢃⠎⠀⠀⠀⠀⠙⠛⢲⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣨⣦⡀⠀⠀⠀⠀⠀⠀⢀⠄⠤⠒⢄⠀⠀⠀⣀⡴⠧⣤⣤⣄⣀⡀⠀⠀⠀⠉⢧⢤⡀⠀⡀
⠀⠀⠀⢀⣠⠼⠃⠀⠀⠀⣀⡤⠞⠉⠁⠀⠉⠓⠦⢤⣀⣀⡀⢘⠀⠰⠂⣸⡤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⢹⡿⣏⣀
⢀⣀⣀⡞⠀⠀⠀⢀⣠⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡏⠂⠰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣀⣠⠟⡁⢸⡋
⠙⢯⠉⢳⣄⣠⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⣀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣴⠋⠙⠁
⠀⠿⠲⣆⣬⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣿⣥⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⣀⠄⠀⡀⠈⢣⠀⠀⠀⠀⠁⡰⠀⠀⠀⢀⠀⠀⠀⠀⣩⣯⠁⠀⠀⡀⠄⠀⠀⠄⣐⠘⢄
⠀⠀⠀⠀⠀⠂⢀⣍⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠊⠁⠈⢃⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡐⠀⠰⠀⠀⠀⠀⠀⠀⠀⠉⡀⠚⠂⢦⠀⠰⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠹⡉⣅⡁⠀⠀⠀⢛⠂⠀⠈⡄⠘⠀⠀⢦⠴⠒⠚⡆⠀⢄⡆⠀⠀⠀⠀⠀⠀⠁⠘⠚⠁⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠄⠀⠀⠉⠀⠀⠘
⠀⢠⠀⡇⠀⢸⠑⠉⡅⠀⠀⠐⠀⠀⠀⠡⣘⣄⠀⠀⠀⠀⠀⠀⡇⣬⠊⠀⠀⠀⠐⠂⠀⠀⢸⠙⠀⢢⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠐⠀⢀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣄⡀⠈⠸⠆⣖⣰⠶⠵⠞⠋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠡⣁⠀⠀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⢤⠀⠀⠀⠀⢀⠀⠀⠀⣠⠃⠈⠘⢀⠀⠀⠀⠀⠀⠀⠸⠩⢘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠰⠔⡖⠀⠀⠑⠀⠀⠀⣄⠌⠀⠀⠀⠤⡆⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠂⠘⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⢠
⠀⠀⠀⢺⡇⠀⠂⠁⢈⠅⠀⡠⠤⠀⠀⠀⠀⠀⠰⠉⣤⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢀⡀⠀⠀⠀⠀⠈⠀⠈⠐⢢⠀⠰⣐⡦⣂⠀⠀⠀⠀⠠⠃⠌⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⡆⠀⠀⠠⠄⢉⠙⣁⠆⠀⠀⠀⠀⠀⡰⠀⠠⠂⠀⠀⠁⠋⠃⠀⠀⠀⣆⠃⠀⠰⢀⡀⡀⠀⡠
⠀⢀⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠎⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠐⠁⠀⠀⡆⡄⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠈⢂⠀⡰⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣣⠀⠀⠀⠀⡠⠀⠗⡠⢆⢀⠒⠢⠀⠀⠀⠀⠀⡀⠀⠀⠙⢫⠆⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⠁⠀⠀⠀⠰⠖⠚⠺⠀⠀⠀⠀⠀⠤⠯⠤⠄⠀⢰⢡⠆⠀⠁⣨⣷⠦⢴⠂⠀⠀⠀⠌⡄⠀⠀⠀⠸⠀⠀⠀⡠⠊⡤⣠⠄⡀⠀⠀⡀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠺⠞⠃⡖⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⡀⠸⠋⠁⣫⣆⠀⠢⣀⠠⢂⠞⠀⠀⠀⠀⠊⠀⠀⢠⠀⠐⠀⠀⠀⠸⠀⠀⠀⠀⢙⠁
⠀⠀⠀⠀⢀⡀⡀⢻⠃⡄⠯⠈⠁⠀⠀⠀⡄⠀⠀⠴⠶⢦⠀⠑⢄⠂⠀⠈⡆⠀⠀⠀⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠂⠀⠀⠀⠀⢣⠠⠀⠀⠈
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣀⢿⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠀⢀⠄⠀⠈⠀⠀⠀⠀⠀⠢⠤⠀⠀⢸⡄⠀⠠⠁⠀⠀⡀⢀
⠀⠀⠀⠀⢠⡀⠀⠀⠀⣴⠟⠀⠈⠀⠀⠠⠀⠀⠀⠘⡷⢾⠀⠀⠀⠀⠀⡠⠄⠀⠀⢀⢤⡄⠀⠂⠐⠀⠀⠐⠀⠀⠀⢀⣀⠠⡀⠀⠀⠀⠀⠀⠀⣡⡇⠠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⠀⠀⠉⠀⠀⠀⠤⠀⠀⠳⠄⠀⠋⠀⠀⠀⠀⠀⠀⠘⠃⠂
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠠⠬⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠃⠀⡿⠀⠀⠠⡆⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⢠⡔⠙⢌⠗⡂⠀⠀⠀⠀⠀⠐⠄⠀⠀⠓⠒⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠃⠀⠀⡀⢁⠘⠛⠒⠈⠀⠀⡆⠀⠰⠀⠀⠀⠀⠶⠤⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⣀⠀⠐⠉⠈⠁⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠄⠀⠙⢓⠀⡴⣄⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠠⢀⡀⢠⣌⣸⠲⠄⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠯⣱⡈⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠃⢉⣹⡉⡍⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡀⠀⠀⠀⠈⢋⣕⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡴⠆⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠄⠀⠐⠻⡃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠰⡛⠲⡥⠈⠐
⠀⠀⠀⠀⢀⣈⡠⠁⠈⠋⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡄⣠⣄⡀⣀⠸⢲⠁⠌
⠀⠀⠀⡀⠞⠛⢳⡄⠁⠀⠀⠀⠊⠀⠁⠀⠄⢈⠀⠀⠀⠀⠀⠀⢀⡠⠶⣶⠗⠨⣙⣿⢷⡶⠷⡆
⠀⠀⠀⠀⠃⢰⡾⠺⠀⡀⠀⡄⡤⠂⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⣮⣀⡀⣽⠁⠀⠈⠙⠀⠁⡜⢁⡀
⠀⠀⠀⠀⠀⣐⠀⠀⠀⠀⠒⠒⠒⠀⠊⠠⣶⣿⣂⠀⠀⠀⠀⠀⢣⠀⠀⠈⠛⠗⠀⠀⠀⠈⠉⢀⠆⣰
⠀⠀⠠⠘⠀⠈⠄⠀⣸⠁⠀⠀⠀⠨⠄⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠑⠘⠣⣦⠄⡁⠀⠀⠀⠀⣸⠠⣶⣗⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠒⠓⠢⠈⠁⠈⠉

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

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠘⠛⢛⣛⠻⠆⠀⠀⠀⠟⢛⣛⠛⠳⠀⠀⠀⠴⠛⠛⠻⠷⠆⠀⠀⠰⠾⠿⠛⠛⠧⠀⠀⠐⠛⠛⢻⡷⢶⠀⠀⠀⢠⡶⣾⠟⠛⠛
⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠒⠘⠂⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠙⠋⠁

⠀⠀⠰⠛⠛⠻⠷⠆⠀⠀⠠⠶⠿⠛⠛⠷⠀⠀⠀⣰⠶⠿⠿⠿⠆⠀⠀⠰⠿⠿⠿⠶⣆⠀⠀⠀⠾⠷⠶⣦⣤⡄⠀⠀⢠⣤⣴⠶⠟⠶
⠀⠀⠀⠘⠿⠉⠁⠀⠀⠀⠀⠀⠻⠋⠁⠀⠀⠀⠀⠁⠠⣶⡶⠀⠀⠀⠀⠀⠀⢶⣶⠄⠈⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠀⠀⠀⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢠⠶⠶⠶⣶⡄⠀⠀⢀⣴⠶⠶⠶⢦⠀⠀⠀⣠⠶⠶⠿⠷⠄⠀⠀⠠⠶⠷⠶⠶⣄⠀⠀⠀⠴⠶⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠐⠚⠷⠆⠀⠀⠀⠀⠀⠚⢿⠆⠀⠀⠀⠀⠁⠐⢶⡖⠀⠀⠀⠀⠀⠀⢲⣷⠂⠈⠀⠀⠀⠀⢠⣤⣍⠛⠂⠀⠀⠘⠛⣿⣭⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢠⣶⣶⣦⣤⡀⠀⠀⠀⣤⣤⣶⣶⣄⠀⠀⠀⢠⣴⣤⣤⣤⣄⠀⠀⣠⣤⣤⣤⣦⡄⠀⠀⠀⠴⢶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠤
⠀⠀⠀⠀⣤⣤⠀⠁⠀⠀⠀⠁⢠⣤⠀⠀⠀⠀⠀⠀⠠⣤⡬⠁⠀⠀⠀⠀⠁⢥⣤⠄⠀⠀⠀⠀⠀⠠⠾⠿⠛⠃⠀⠀⠘⠻⠿⠧

⠀⠀⢀⣤⣤⣤⣄⡀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⢀⣀⣀⣠⡄⠀⠀⢠⣄⣀⣀⡀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⠠⢶⡮⠍⠁⠀⠀⠀⠉⠽⣶⠤⠉⠀⠀⠀⠘⢛⣉⣉⠉⠀⠀⠀⠀⠉⣉⣉⡛⠃⠀⠀⠀⠀⠀⠉⠛⠿⠄⠀⠀⠰⠶⠞⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠐⠙⠿⠂⠀⠀⠀⠀⠰⠟⠁⠂
⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠘⠉⠉⠉⠙⠃⠀⠀⠐⠛⠉⠉⠉⠙⠀⠀⠀⠘⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠃⠀⠀⠀⠛⠛⢷⣦⣤⡀⠀⠀⢠⣤⣴⠞⠛⠃
⠀⠀⠀⠐⠶⠒⠀⠀⠀⠀⠀⠀⠐⠶⠒⠀⠀⠀⠀⠀⠐⠶⠖⠀⠀⠀⠀⠀⠀⠲⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠘⠋⣙⣛⠛⠃⠀⠀⠐⠛⣛⣋⡉⠛⠀⠀⠀⠘⢛⣛⣛⠛⠉⠀⠀⠉⠛⣛⣛⡛⠃⠀⠀⠀⠚⠉⢛⡛⠛⠂⠀⠀⠘⠛⠛⡛⠙⠓
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠰⠟⠻⠿⠶⠆⠀⠀⠠⠶⠾⠿⠛⠧⠀⠀⠀⠴⠛⠿⠿⠶⠄⠀⠀⠠⠶⠿⠟⠛⠦⠀⠀⠀⠞⠛⠉⠉⠙⠀⠀⠀⠘⠉⠉⠙⠛⠧
⠀⠀⠀⠐⠊⠒⠀⠀⠀⠀⠀⠀⠐⠚⠂⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠠⢤⠤⠀⠀⠀⠀⠀⠀⢤⡤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠰⠶⣤⣤⠀⢀⣠⣤⣤⠄⠀⠀⠀⠀⠀⠋⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠃⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣉⠁⠘⠋⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡍⠀⠀⠀⠀⣰⡶
⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠘⢿⠇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⠀⣶⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠈⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠴⣾⣿⣟⡁⠀⠀⢀⣻⣿⣟⠦⠀⠀⠀⠀⠀⣀⣤⠴⢶⡄⠀⠸⠋⠉⠛⠒⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢀⣾⠟⠉⣀⣄⠀⠀⣀⣄⠈⠻⣷⡄⠀⠀⠀⠀⠀⢠⡤⠀⠁⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡴⠛⠋⠉⠀⠀⠀⠀⢀⡈⠙⠛⠶⡄
⠀⢸⣿⠀⠀⣤⣿⣇⠀⣾⣷⣆⠀⢹⣷⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⢠⣴⣶⣿⠆⠀⠀⢦⣴⣾⣿
⠀⠸⣿⡀⠀⢿⣿⠏⠀⣿⣿⠇⠀⣼⡟⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠀⠀⠀⠈⠛⠟⠁
⠀⠀⠹⢷⣤⣠⡤⠀⠀⢤⣤⣤⣾⠟

⠀⠀⠀⠀⠒⠶⢦⣄⠀⢀⣤⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣤⡶⢖⠂⠀⠀⠀⠀⠀⢀⣠⣤⢶⡿⠂⠀⠀⠰⠶⢤⣄⣀
⠀⠠⣄⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠙⠓⠂⠄⠀⠁⠀⠁⢀⡀
⠀⠀⠈⠙⠛⣻⣿⠂⠀⠀⠈⠙⢻⣿⡋⠀⠀⠀⠀⣨⣷⠀⠀⠀⠀⠀⣭⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠸⠿⠃
⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠛⠫

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢦⣤⡀⠀⠀⠀⠀⢤⣤⣄⡀⠀⠀⢀⣠⣤⠤⠀⠀⠀⠀⠀⠀⠒⠶⢶⣄⠀⠀⣠⡶⠖⠒
⠀⠀⠠⠖⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡆⠀⠀⠀⠀⠀⢮⣿⠉⢻⣷⠀⠀⣾⡟⠈⣿⡴⠀⠀⠀⠀⠀⢠⣿⣧⡄⠀⠀⠀⠀⣾⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠹⠿⠃⠀⠀⠀⠀⠀⠘⣿⠠⠜⣿⠀⠀⣿⠢⠄⣿⠃⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠈⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠒⠲⠶⣤⠀⠀⣠⠶⠞⠓⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠠⠾⠛⠉⠁⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⡀⣀⢀⠀⠀⠀⠁⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⠀⠀⠀⠀⠤⣤⡀
⠀⠀⠀⠀⢰⣿⣶⠀⠀⠀⠀⠰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⣶⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠀⠀⠀⠈⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢻⡟⢻⣛⣿⣋⣛⣻⣟⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣛⣛⣿⣿⣟⣛⣛⣻⣛⣛⣛⣻⣛⣛⣛⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔
⠀⠀⠀⠀⠘⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁
⣄⣀⠀⢀⡀⣿⢰⠀⠀⣇⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⣇⠠⠴⠦⢄
⢷⠉⠉⠁⢰⣿⠜⣆⠀⢹⡄⠀⠈⠃⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢐⡍⠓⠦⡄
⠸⡆⠀⠀⢸⡏⡀⠹⡆⠀⣿⠀⠀⠀⡇⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⡇⠀⢸⡇⠀⣠⠇
⠀⢱⠀⠀⣸⠁⢱⡄⢷⡀⠹⡆⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇⠀⠸⡇⠀⢿⠀⣠⣀⣀
⠀⠘⡆⠀⣿⠀⠀⣷⠈⢳⡄⢻⡄⢀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢰⡇⠀⠀⠇⠀⠀⡓⣡⠬⣟⡉⠁
⠀⠀⠻⠶⠋⠀⠀⣸⠀⢸⠀⢸⡀⠈⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⡇⠀⠀⣴⠀⠀⣣⠾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠘⣇⠀⣇⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠸⡇⠀⠀⠟⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⣁⠀⠈⠀⠈⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠒⠉⠙⠀⠐⠒⢄⡀⢈⢢
⠀⠀⠀⡀⠒⠋⢹⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠠⢄⣀⡀⠀⠀⠀⠉⠈⣸
⠀⠀⢰⠇⠀⠀⠁⠀⠳⢤⣷⠤⠶⠤⠄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢙⡷⠒⠛⠋⠁⠹⡄
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠐⠉⠉⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⡀
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠛⠛⠛⠿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠶⠶⠶⠶⠶⣶⣦⣤⣄⣀⡀⠀⠀⠀⣀⣴⡟⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠟⠛⠉⠀⠀⠀⠀⢀⣤⣶⠾⠿⠛⠛⠿⠿⣶⣤⡀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠘⣿⡄
⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠉⠉⠁⠀⠈⠉⠉⠙⠛⠿⣶⣄⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⠈⠻⣷⡀⣼⡟⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠉⢻⣿⡆⠀⠹⣿⣿⠁⠀⢀⣼⣿⣿⠉⢻⣷⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣿⣿⡗⠀⠀⣿⡏⠀⠀⢸⣿⣿⣿⣶⣾⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠁⠀⠀⣿⣧⠀⠀⠈⠻⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⣾⠆
⠀⠀⠀⠀⠀⠀⣀⠘⣿⣄⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣦⣤⣤⣶⡶⠟⠋⢀⣼⡟⣠⡄
⠀⠀⠀⠀⠀⠀⠻⣷⡈⠻⢷⣦⣄⡀⠈⠙⠛⠿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⡾⢋⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣤⣀⡀
⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡛⠛⠉⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠹⠛⣿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣶⣦⡀⠀⠀⠀⢿⡄⠀⠀⠀⣾⠁⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⡟
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠋⠀⠀⢀⣿⠀⠀⠀⠀⢹⡆⠀⠀⠈⠿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠶⠶⠶⠟⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠟⠛⠉⠉⠀⠀⠀⠀⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣤⣤⣤⣴⣶⣾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉


⣿⣿⡿⠿⣿⡿⢿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠠⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠓
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠈⠀⡐⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠒⢲⠒⢂⠖⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⢠⠀⠙⡆⠠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢀⠫⡄⠸⠀⠂⡀⠀⡞⠀⠀⠀⠈⠀⢸⠀⢰⠆⠁⣤⠀⠀⠀⢘⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠁⠀⠈⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠋⠉⠎⠀⠀⠀⠀⠠⠀⠐⠈⠸⠁⠈⠿⠀⠀⠀⠀⣀⡠⠆
⢠⢠⡀⠀⠀⠀⠀⠀⠠⠆⠀⢁⡀⠀⠀⠂⠀⠀⡄⢠⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⣀⠀⡀⠀⠰⣤⣴⢠⠀⣿⠀⠀⢀⠔⠀⠀⢀⡀⠀⠀⠆
⠀⠀⠤⡀⠈⠀⠄⠈⠉⠑⠊⠁⠙⠳⡀⠄⢀⠀⠶⠾⠀⠀⠀⠶⠀⡀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠋⠉⠈⠉⠹⠀⡠⠋⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠸⠂⠀⠘⠄⢀⠀⠀⠋⠀⠀⠀⡘⠢⠀⠂⠚⠐⠶⠶⠄⠀⠀⡀⣸⠇⠀⠀⠀⠀⢀⡦⠀⠓⢒⣒⣶⠾⠋⠀⠀⢀⡅⠀⠄⢘⠄⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠁⠀⠀⠴⠩⢐⠁⠀⠀⠠⠤⠀⠂⠐⠀⠀⡍⠀⠀⠀⡠⠚⠋⠁⠉⠙⠛⠠⠤⠒⠀⠉⠉⠁⠀⠀⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠘⢮⡀⠀⠀⠀⠀⠀⠀⠀⠠⡘⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠀⡉⠲⣄⠀⠀⠀⠀⠀⢰⣧⠞⠉⠀⣤⣤⣄⣤⣀⡀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣸⢲⣏⢳⡤⣶⡄⠄⣼⠁⠰⡛⠀⣇⣟⣃⣟⣘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠐⠀⢚⠸⠀⣷⠉⠉⠃⠻⣬⡇⡼⢹⡆⠀⠁⠀⡇⣿⠏⣿⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠄⠀⠀⠸⡀⠀⣡⠀⠘⠀⠀⠊⣿⣇⢸⡇⠀⠀⠀⡟⢿⠇⡟⠲⠶⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢁⠀⣿⠀⠀⠀⢀⠂⣯⠻⣾⠁⠀⠀⠀⣧⢸⡅⡷⣿⣿⠀⠀⠀⠀⠀⠒⠖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⢲⠈⠀⠉⢸⠀⣿⠀⣿⠀⠀⠀⠀⠛⠜⠁⠓⡋⡻⠀⠀⠀⠀⠀⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠨⠀⢠⣀⣴⢸⢸⠀⣟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠉⠀⠆⢀⡀⢶⠿⢾⣿⣼⣿⣷⣜⣸⡇⠀⡉⠀⠀⠀⠠⠂⠀⠀⠀⠄⠀⠀⠠⠀⢀⡉⢻⢁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠐⠓⠂⠀⠮⠤⠄⠁⠀⠜⠛⠻⠶⠶⠶⠶⠤⠀⠂⠐⠂⠀⠀⠀⠐⠚⠓⠘⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠠

⠀⠀⢐⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠰⠁⠀⠐⠖⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⡤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠤⡀⢀⣀⡀⠄⠀⠀⠀⠀⠀⠀⡰⠂⠐⢄⡀⢄
⠀⢁⠀⣴⡴⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣤⠄⣶⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠘⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⡈⢀⡀
⠀⢀⡈⠉⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠔⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⣻⠶⡀⢶⢆⠀⠀⠀⠀⠀⠀⠀⠓⠐⠚⢇⠙⢃
⠀⠨⣄⣀⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢀⡀
⠀⠀⠙⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⡤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢖⠀⠀⢤
⠀⠀⠀⡠⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠉⠱⡔⠉⠐⡁⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⡀⠖⠉⠉⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⣀⡀
⠀⢰⠚⠁⣨⡁⠈⣁⣦⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⢄⣈⠹⡀⠶⠇⡀⠀⠀⠀⠀⠀⠀⠀⠂⢈⠁⠉⠀⠉⠱
⠀⠘⢄⠀⡙⠃⢀⣈⠃⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠐⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⣘⠿⠳⡀⢻⠟
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠤⠤⢉⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠴⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⣄⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⣀⣀⢀⡴⡁
⠀⢠⠋⣄⠈⠎⢀⡈⠆⠀⠀⠀⠀⠀⠀⢐⢲⣶⠒⠗⣶⡆⠆⠀⠀⠀⠀⠀⠀⡃⢠⠄⡈⠀⣄⢙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡍⡁⢡⡄⢰
⠀⠸⡀⠛⢡⠆⠙⠁⡄⠀⠀⠀⠀⠀⠀⠈⡀⠁⠰⢆⠈⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠄⠙⢀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠜⠑⢀⡀⠂
⠀⠀⠈⠀⠁⢀⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⠒⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠠⠄
⠀⠀⠐⠒⢂⠀⠾⢂⠀⠀⠀⠀⠀⠀⡖⠂⠀⣤⠐⠚⢇⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁⢢⡔⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⢀⠀⣀⣀⡀
⠀⠀⠀⠀⠈⠒⠒⠉⠀⠀⠀⠀⠀⢀⡀⢠⣤⡀⠀⣤⣌⠀⠀⠀⠀⠀⠀⠀⢈⠸⡷⢀⡀⢾⠇⡆⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⢤⢤⣤⠤⡀
⠀⠀⢰⡀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠂⠙⠃⠀⠀⠀⠀⠀⠀⠀⣈⠁⠐⠊⠑⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⣈⡁⠜⢄⣉⠠⠁
⠀⠀⠸⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠡⠈⠐⠂⠀⠐⠂⡙⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈


⠀⡄⡀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠒⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⡀⢀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠠⠶⡀
⠐⠆⣸⡿⡎⢩⡧⢰⠀⠀⠀⠀⠀⠩⡀⠛⢩⡉⢿⠏⡅⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠘⠁⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠠⠞⠓⠀⠃
⢀⠄⠀⠈⠑⠄⠠⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠑⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⠠⠛⣴⡆⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⢀⡄
⠿⠶⠴⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠒⠒⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⠿⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣐⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣂⣀⣈⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⣄⠀⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠎⠈⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠻⢄⠉⠑⣦⠀⣄⠜⣀⡀⠀⠀⠦⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠒⠃⠀⠀⠀⠈⢁⠔⠒⠠⠋⠀⠈⡆⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡠⠂⠠⡀⠀⠀⠀⠀⠀⠈⢢⠀⡔⠀⢢⠈⠉⢢⠀⠀⠀⠀⡴⡀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡄⡦⢀⣄⠷⠀⣄⠀⠀⠀⢰⠁⠀⢑⠒⠊⠀⡖⠁⠀⠀⠀⢰⠀⠰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢇⠀⠀⣣⡤⢃⡀⡜⠀⠀⠀⠀⠑⠐⠈⠓⠄⠔⠃⠀⠀⠀⢀⠘⠤⠃⠀⢠⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠈⣁⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠀⠡⠆⣀⣀⠀⠈⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠠⠊⠁⢀⡀⠁⠀⠀⣴⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⡎⠀⠀⠱⠤⠤⢄⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠤⠊⠀⠈⠙⡄⠀⠀⠉⠁⠀⠀⠸⡀⠀⢠⠀⠀⠀⠉⠁⠀⠀⢀⠵⠀⡔⠐⠄⠀⢈⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⣑⠒⠁⠺⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠁⠉⠉⠦⠀⠔⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠎⠀⠑⠤⠒⠉⢢⠀⢀⠔⢲⠀⠀⠀⠀⠀⠀⡋⠢⡀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢜
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡠⠆⢀⠔⠂⠄⠀⡰⠀⠋⡠⠆⠀⠀⠀⠀⠀⠀⠈⠢⠇⠘⢤⠇⠀⠀⠀⠀⠀⠀⡔⠁⡍⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢸⠀⠀⠠⡀⠀⠨⠀⠉⠆⠉⢤⠂⠠⣀⠀⠀⠀⠀⠤⠀⠤⡀⠀⠐⠁⠉⠢⠀⠄⠈⡠⠚⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠑⠒⠀⠀⠀⠤⣀⡠⠆⠀⠀⠑⠠⠄⠇⠀⠈⠐⠂⠐⠀⠠⠒⠛⡠⠀⢄⠀⡀⢀⡔⠒⠒⢢⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣀⡀⠢⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡀⠣⠀⠜⠀⠈⡀⠀⠁⠈⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠊⢠⠄⢰⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⠀⠀⠀⠀⠇⠀⢀⠖⠒⠊⠁⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠔⠁⠀⠨⡀⠁⠀⠀⠀⢰⠓⠄⠀⠀⠀⠀⠀⢘⡠⠂⠀⠀⠀⠀⠈⠉⠁⡘⢣⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠈⢄⠈⠀⢠⠖⠒⠢⣨⠄⠀⢄⠀⠀⠀⠀⠀⠀⠠⣠⠀⠀⡄⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡙⡁⠡⡀⢠⠄⠥⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⣴⠁⣀⠴⠃⡔⠁⠐⠄⠰⠀⠫⡀⢏⠑⠢⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⡀⠀⠀⠀⠀⠀⠱⢄⡀⠖⢄⢀⡴⠁⠀⠑⠦⠵⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠢⢤⣀⠀⠀⡄⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢤⣤⠀⢠⠄⠰⠊⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣇⡀⠀⠀⢀⠔⠁⠀⠑⢄⠀⢀⠷⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠈⠑⠲⠋⠀⠀⠀⠀⠀⠑⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢄⣀⣀⣀⣀⣀⣀⣀⠠⠔⠊

⠀⠀⠀⠀⠀⡀⠠⠤⠀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠤⢀⡀⠀⠀⠀⠀⠀⢀⠀⠤⠤⢀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⣀
⠀⠀⠀⠠⠊⢀⠀⠀⢀⣈⠠⡀⠀⠀⠐⠁⠀⠀⠀⠀⠈⢢⠀⠀⠀⡐⠁⠀⠀⠀⠀⠈⢆⠀⠀⢀⠈⡀⣀⠀⢀⠀⠑⠄⠀⠀⠠⠊⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⡁⢠⡴⢐⣤⠀⠆⢁⠀⠀⠀⠀⣀⡁⢈⣀⠀⠀⡇⠀⢠⠀⢠⡄⠀⠰⠛⠁⠘⡇⠀⡈⠀⠀⣤⠂⣤⠄⡀⢸⠀⠀⡆⠀⠉⠳⠀⠞⠋⠀⢸
⠀⠀⠀⢂⠀⠀⠁⠀⠁⢀⢀⡆⠀⠀⠀⠁⠉⠤⠤⢋⠀⢠⠃⠀⠠⡀⠠⢀⣀⣠⣤⡖⣸⠃⠀⠰⠀⠠⠃⢠⠠⢩⠀⡰⠀⠀⢁⠀⠀⣀⡀⣀⠀⠀⡜
⠀⠀⠀⠈⠂⢌⣀⣂⣈⠥⠊⠀⠀⠀⠈⠢⢀⡀⢀⡀⠔⠃⠀⠀⠀⠈⠪⢅⣉⣴⡫⠗⠁⠀⠀⠀⠑⠤⣉⣉⣉⡡⠞⠁⠀⠀⠀⠑⠤⠀⠀⢀⠤⠊

⠀⠀⠀⢀⠴⣅⠀⠀⢀⡵⢆⠀⠀⠀⠠⠴⠂⠀⡀⠘⠳⢀⠀⠀⠀⡠⠈⠄⡀⠀⠤⠑⣄⠀⠀⠀⣴⣦⠀⠀⠀⣬⣦⡀⠀⠀⠀⠔⢁⠀⠀⠀⡁⠢
⠀⠀⠀⠂⠁⠙⢃⢀⣚⠁⠙⠆⠀⠰⠄⠀⠀⡘⠐⢀⠀⠐⡆⠀⠀⢈⠀⢠⡼⣤⡄⠀⠘⡄⠀⠸⠋⠀⣈⡄⣈⡀⠀⠑⠀⠀⡌⠞⠋⠐⠀⠂⠙⠒⢳
⠀⠀⠀⡄⠁⠈⠛⠈⢫⡀⠂⡌⠀⠰⠀⠄⢭⣤⣤⡤⢤⠀⠃⠀⠈⠐⢁⠈⠀⠈⢀⠁⢠⠇⠀⢠⠐⠠⠛⠀⠙⠁⠈⢠⠀⠀⠇⠁⠸⠗⠀⠻⠃⠈⡸
⠀⠀⠀⠘⢤⣿⠀⠀⠈⣿⠜⠀⠀⠀⠑⠌⢀⣠⡶⠾⡣⠎⠀⠀⠀⠑⢄⡈⠐⠈⣁⡠⠋⠀⠀⠀⠣⣄⠀⠐⠀⣀⡴⠁⠀⠀⠈⠦⡀⠘⠛⠃⣀⡴⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⠄⠀⡀⠀⠀⠄⠀⠀⠀⠀⣠⠔⢀⠈⠀⠂⢄⠀⠀⠀⠀⣀⠤⠀⠀⠀⠤⡀
⠀⠀⠀⠌⠉⠉⠃⠀⠚⠓⠀⠄⠀⢀⠺⠀⠴⠀⠄⠶⠀⠹⡀⠀⢠⢃⢠⣶⡄⠀⣀⡀⠀⠀⠀⠰⠃⠰⡶⠀⢰⡷⠈⠃⠀⠀⡐⠁⢠⡌⠀⢡⡄⢹⢂
⠀⠀⠀⣰⠈⠛⠶⠶⠶⠂⠠⡀⠀⢠⠀⠀⠀⣀⣄⡀⠀⠀⡇⠀⢸⠀⠀⠉⠀⠀⠿⠃⠀⠀⠀⢁⠀⠀⣠⣤⣤⡐⠀⢰⠀⠀⡇⠀⢀⠘⠀⢁⡀⠂⢸
⠀⠀⠀⠡⣀⢀⣴⣾⣿⡷⡣⠃⠀⠀⠢⡀⣾⠛⠛⢻⢄⡴⠀⠀⠈⢦⡀⠀⠶⠶⠆⢀⠌⠀⠀⠈⢦⡘⠏⠀⠈⠓⣠⠊⠀⠀⠐⣄⠀⠉⣻⣥⠄⣠⠂
⠀⠀⠀⠀⠈⠑⠻⠿⠓⠊⠀⠀⠀⠀⠀⠈⠙⠛⠋⠛⠉⠀⠀⠀⠀⠀⠉⠒⠂⠒⠊⠁⠀⠀⠀⠀⠀⠉⠓⠒⠒⠋⠀⠀⠀⠀⠀⠈⠑⠺⠿⠓⠊⠁
⠀⠀⠀⠀⢀⡀⠄⠀⠄⣀⠀⠀⠀⠀⠀⣀⠄⠐⠀⠠⣀⠀⠀⠀⠀⠀⢢⠠⠀⠠⠄⡀⠀⠀⠀⠀⢀⡄⠒⠀⠐⠠⣀⠀⠀⠀⠀⢀⠄⠀⠂⠂⠠⡀
⠀⠀⠀⡰⠏⠀⢰⠀⠀⡈⢳⠀⠀⠀⢼⠁⠀⠁⡄⠁⠈⠳⠀⠀⢀⡄⣀⠀⠀⠀⢀⠈⢂⠀⠀⢠⠿⠋⠠⠀⠀⠙⠛⠆⠀⠀⠐⠁⠀⠀⠀⠄⠀⠈⢂
⠀⠀⠀⠀⠢⠈⠟⠀⠘⠣⠂⠀⠀⠈⠀⠢⠽⢃⣘⠻⠄⠀⡇⠀⠤⠈⠁⣤⠀⢀⡄⠈⠀⠀⠀⠀⠀⠀⠶⠁⠶⠆⠠⠀⠀⠀⡇⠁⠰⠦⠀⠶⠆⠀⢨
⠀⠀⠀⢂⠀⠠⡞⠛⣶⠀⢀⠃⠀⠈⢄⠀⢰⡏⢙⡇⠀⠠⠁⠀⠐⡄⢀⠉⠀⠐⠁⠀⡸⠁⠀⠘⡀⠀⣠⡶⣶⡏⠀⠎⠀⠀⠐⡀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠑⠤⢀⣀⡠⠔⠁⠀⠀⠀⠀⠑⠠⠤⠠⠤⠊⠀⠀⠀⠀⠈⠢⢄⣀⣀⠤⠚⠁⠀⠀⠀⠈⠑⠠⠤⠄⠒⠉⠀⠀⠀⠀⠈⠒⠩⠭⠍⠚⠁
⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀⠀⠀⠀⠀⠀⢀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⡾⠁⠄⠠⠂⠓⠀⠀⠀⠀⡰⠉⠀⠀⠀⠀⠈⢢⠀⠀⠀⡔⠁⠀⠀⠀⠀⠉⢆⠀⠀⢀⡜⢉⣁⢄⢈⡉⠙⡆⠀⠀⢀⡾⠋⠢⠀⠜⠛⢧⡀
⠀⠀⢀⠻⡀⢶⡆⠁⠘⠗⠀⠀⠀⠰⢀⠀⣀⡀⢀⣀⠀⠂⡇⠀⠈⠀⠀⣀⠀⢀⣀⠀⠈⡀⠀⠈⢹⣤⠕⠁⠨⢥⡼⠐⠄⠀⡎⢀⠘⡡⠁⢜⠃⡘⢱
⠀⠀⠀⡆⠈⠈⠀⠀⡆⠊⠑⢢⠀⠈⡄⠐⡘⠥⠤⢏⠀⢀⠃⠀⠀⡤⢂⡙⠤⠨⠃⠀⢰⠀⠀⠰⣾⠃⣴⠶⢶⡀⢻⡔⠀⠀⢃⠀⠀⣰⣶⣆⠀⠀⡸
⠀⠀⠀⠘⠦⣀⠀⠀⢊⡰⠖⠁⠀⠀⠈⠢⠀⣀⢀⡀⠐⠀⠀⠀⠀⠀⠨⣀⣀⣀⡀⠐⠁⠀⠀⠀⠉⠢⠍⠀⠈⠤⠊⠀⠀⠀⠈⠢⣄⡙⠩⢋⣠⠞
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢀⠀⠀⡀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⠀⠉⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢼⡀⠀⠀⠀⠀⠀⠀⢀⡀⠈⡉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠁⠀⠀⠀⠈⢢⠀⠀⠀⠀⢠⠊⠀⠀⠈⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠈⡆⠀⡇⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢰⣿⣿⣶⠆⠀⡅⠀⡀⠀⣿⣿⡶⠆⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣀⣀⣻⣿⣿⣤⡼⠁⠀⠱⠤⠽⠿⠧⣤⣀⣀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠛⠉⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠂⠀⠠⠀⠀⣀⠀⠠⠀⠐⠒⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠶⠖⠒⠚⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣧⠀⠀⣰⡷⠟⠁⠀⣸⣷⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠾⣷⣿⡟⠀⠐⠋⠀⠀⠀⡴⠉⣿⡧⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠙⢂⠙⠀⢸⡿⠀⠀⠀⠛⠀⠀⢠⣶⣷⣿⠇⠀⠀⡴
⠀⠀⠀⠀⠀⢀⣀⠰⠈⠀⠠⠀⠀⠀⢀⣴⡆⣠⣿⣧⠀⠀⠈⢀⢤⡀⠉⢫⢻⣿⣆⠀⠀⠀⠁
⠀⠀⠀⠀⢸⣆⣈⡀⠀⠀⠀⠀⣠⠲⣬⡜⣿⠉⣿⣿⡄⠀⠀⠀⠀⣅⣠⠃⠀⠈⣉⣀⣀⡀
⠀⠀⠀⠀⢀⣸⣿⣷⣼⣷⣆⠘⠫⡀⠀⠙⢿⣏⠻⣿⣿⣤⡖⠀⠀⢸⠁⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀
⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣟⠀⠀⢩⡆⠘⢨⡻⠀⠘⣿⣿⣧⡀⠀⡎⠀⠀⠀⢠⠂⠀⠀⢆⣀⢀⣴⠒⠈⠣⣄⠠⣬⣿⡷⢆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢐⣇⢀⣯⠀⠌⠀⣸⣿⣿⣧⡠⡄⠀⠀⠀⣸⠆⠀⠀⢸⣿⣿⠁⠀⡀⢱⠌⢢⢀⣙⠻⣶⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢨⠿⣟⠁⢰⣾⣿⣿⣿⠿⠟⠻⣶⣶⣶⠛⠻⠷⠄⠀⣸⣿⠷⠒⠐⠒⠋⠀⢿⣶⣭⣹⣿⡇
⠀⠀⠀⠀⢸⣿⡿⠿⢿⣿⣿⠀⠀⢚⠆⠹⣀⠸⣿⣿⡿⠟⠳⣦⣤⣀⡉⠀⠀⢀⠀⠂⠐⣿⡿⠀⠀⠀⠀⠁⠀⠘⠛⢻⣿⣿⢆
⠀⠀⠀⠀⢸⠟⠀⡀⠀⠀⠉⢦⣴⣾⠀⢀⣷⡶⠛⠍⣿⠀⠀⠉⠛⣿⣿⣿⣾⣧⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣤⣾⣿⠇⠀⠈⠋⠷⠒⠈⠙⢿⣿⣦⠀⠀⠁⢠⡆⠀⣷⠂⠈⠀⠀⠀⢄⡀⡀⠈⠁
⠀⠀⠀⠀⠠⡀⡀⠀⠁⠀⢀⠀⠀⠈⢻⣏⡟⠀⠀⠀⠀⠀⠚⢇⡈⢻⠿⣿⣦⢀⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠈⠙⢶
⠀⠀⠀⠀⠀⢻⣿⣤⠄⠊⠀⠀⠀⡤⣻⠟⠀⠀⠀⠀⠀⠀⠀⠎⠙⠊⢇⠸⠿⣶⣄⢄⠈⣀⠀⣠⣴⡟⠖⠀⣷⠀⣰⡖⠄
⠀⠀⠀⠀⠀⢨⢿⡁⠀⠀⢀⣁⣴⢗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡙⡿⠹⣶⢾⣶⢟⣅⡼⠀⠀⢸⣷⠏⣀
⠀⠀⠀⠀⠀⢀⣾⢣⡇⠀⠈⠁⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠄⠀⠐⠁⠾⠦⣄⣼⣾⣡⣼⢏⡑⠀⡀⠈⢋⢲⠁
⠀⠀⠀⠀⠀⠀⠉⠼⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠲⠾⠛⠻⠛⠛⠋⠛⠃⠀⠐⠈



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⠀⠀⠀⠀⠀⠀⠛⠀⢰⠇⠀⠀⠀⠀⠀⠐⠟⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠐⣆⠀⠀⠀⠑⠘⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣀⣤⣤⣤⠶⠛
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⣻⡆⠀⠈⠀⠀⠀⠀⠀⣀⣤⣶⣴⣤⠶⣶⣶⣤⡄⠀⠀⣀⡀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⠀⠀⠀⠀⠀⣠⣶⣛⠛⣡⣤⣤⣶⣤⡿⠞⠙⢻⡆⠈⠛⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⣼⢟⣿⠟⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠋⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⢾⡷⠀⢸⡇⢾⡟⠀⠀⠀⠀⠒⠀⠁⠀⠀⠈⠀⠀⢿⠀⠀⠀⠀⢀⡤⠖⠋⠁
⠀⢠⣄⠀⠀⠀⠙⠦⡘⠂⠀⠀⠀⠀⢸⣇⢘⣿⡀⠀⠀⢀⣀⣀⣤⠀⠃⠶⢶⣛⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠉⠻⢦⣀⠀⠀⠈⠢⣀⠀⠀⠀⢸⣽⣽⣿⠗⠀⠰⢋⣽⣿⠅⡀⠀⡔⢟⣋⠘⡇⠀⠀⠀⠀⠀⣀⣀⠤⠤⠶⠒⠒⠋⠉⠉⠀⢀
⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠈⠃⠀⠀⠈⣿⢿⣿⡆⠀⠀⠀⠑⠒⠀⠀⡀⠙⣆⠀⠀⢷⠀⠀⠀⠀⠉⠀⠀⠀⢤⣤⣒⡒⠒⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⣿⠛⡆⠠⠀⠀⠀⠀⠀⢀⠠⠴⠰⠋⠠⡀⢸⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠳⠀⠀⠀⠀⠀⢀⠎⣀⠤⠤⠶⡀⠏⢸⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠀⠀⠀⠀⠈⠙⢶⣀⠀⠀⢦⡈⠀⠀⢠⠞⠉⡑⠢⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠀⠀⠀⠀⠀⣠⣾⢜⢷⣄⡀⠃⠀⣰⠋⠖⠊⠁⣁⢧⡀⠀⠀⠀⠐⠦⠠⠀⢀⣀⣄⣀⡀⠀⠠⠤⢤
⠀⠀⠀⠀⠀⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⠈⠐⢍⡛⠿⣶⡇⠀⠀⢉⠉⢀⣠⣇⠀⠀⠀⠀⠦⠤⢄⣀⣀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠘⣇⠀⠀⠈⠓⢼⠇⠀⠀⠉⠀⢉⠀⣸⣇⡀⠀⠀⠈⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠇⢹⡄⠀⠀⢀⣾⠀⠀⠀⠀⠰⠈⡼⢱⠟⠛⠿⢶⣦⣤⣀⡀
⠀⠀⠀⠀⣀⣤⡶⠟⠋⠉⠀⠈⠉⠉⢨⠉⠁⠀⠀⠀⢻⣆⢀⣾⡟⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⣶⣄
⠀⠀⢀⣾⠿⣅⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⣯⡿⠛⢤⡀⠀⠀⢀⡴⠃⢹⡆⠀⠀⣼⠀⠀⠀⠀⠀⠀⢹⡎⣦
⠀⠀⣼⡟⠀⠹⣆⠀⠀⠀⡆⠀⠀⠀⠈⠻⣦⠀⠀⣠⣴⡋⠀⠀⠀⠈⢦⢠⣾⣿⡄⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣿⠘⣧
⠀⢸⡿⠀⠀⠀⠹⣇⠀⢠⡇⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠉⠳⣄⠀⠀⠀⢿⣿⣿⣷⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢹⣆
⠀⣿⠇⠀⠀⠀⠀⠹⡆⢸⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠈⢦⢀⡰⢈⣿⣿⣿⣇⣿⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⢠⣿⠀⠀⠀⠀⠀⠀⠀⠛⢀⣠⣶⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⢴⣿⣿⣿⣿⣿⣟⠀⠀⠀⡼⠁⠀⢸⡆⠀⠀⠀⠀⠀⠸⣷
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣹⠿⢿⠻⠿⢿⣿⣿
⠀⠀⠻⠛⢛⣉⣀⣠⣴⠟⣦⣠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⣅
⠀⠀⠈⣠⣿⡟⢻⣿⣏⡠⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠛⢿⣯⣴⣄⡀⠀⠀⡀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠛⠛⣏⣹⣿⣿⣧⣾⡿⣹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣽⣿⡑⠾⣿⢾⣷⣿⡷⢤⣠⠄⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣏⡅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠒⣩⣿⣿⣶⣶⣿⣿⣿⣡⢨⣿⣶⣤⡀⣰⣋⠀⢤⡴⠆
⠀⠀⠀⠀⣠⣀⣾⣿⣏⣿⣿⣿⣿⢿⣿⠃⠀⠀⠀⢴⠟⠙⢦⠀⠀⠀⠠⣀⠀⠀⢰⠞⠀⢦⠀⢀⠟⣠⣿⣿⡿⢿⣿⢿⣟⢻⡎⣉⡟⢉⡿⠞⠁
⠀⠀⠠⣾⣿⣿⢻⣿⣿⣿⣿⣾⣿⣻⡟⠷⠀⠀⠀⠈⠃⢠⠀⠂⢘⣮⣵⣿⣧⠀⠀⠀⠀⠘⠐⢠⠀⠙⢿⣿⢣⣼⣿⣏⣙⠟⢿⣻⡿⠉⠁⠀⠉
⠀⠀⠀⡉⠀⣿⣧⣿⣇⣹⡿⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⢦⣴⣿⡿⣿⣿⠏⡐⢁⣀⣤⠾⠷⠯⠀⠀⢈⡹⣾⣿⠿⠉⠙⢛⡔⠋⠁⡀⠀⣠⢠
⠀⠀⠀⠙⢉⣻⣟⣿⣧⠴⠛⢫⣾⡟⠻⣿⠀⠀⠀⢀⠄⠀⠈⠛⠿⣿⠋⠁⠀⠰⠛⣿⡋⠀⢀⣌⣤⡀⣮⡵⠆⢋⡴⠀⢀⡇⢀⣴⠀⢸⣷⡶⢣⡐
⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⢙⠸⢷⣴⢤⡀⠀⠀⠀⣴⡶⣾⣿⣦⣀⠄⡤⢄⣴⠞⠁⠙⢦⡸⣿⠟⠻⠉⠀⠐⡨⣿⡿⢫⠵⡯⣽⠷⡙⠛⠿⢌⣿
⠀⠀⠀⠀⠀⠀⠠⢶⣦⣤⠀⠀⠄⠀⠙⢦⣇⠀⠀⠶⠿⣿⣿⠉⣿⠉⠲⠿⣿⡏⠀⠀⣀⢠⡇⢰⣤⣴⡿⣹⣿⣷⠟⡴⣿⠆⡴⢂⡴⣈⣁⡻⡧⢴⣆
⠀⠀⠀⠀⠀⠀⠑⠾⣿⡇⠀⠆⠀⢦⠀⡀⠹⢷⣆⠘⠰⢦⠠⢎⡇⠀⠀⠐⢹⠷⠶⢄⢁⣼⡃⢸⠟⣿⠁⠽⣿⣷⣤⠟⣡⠚⠁⠀⠠⣮⣤⡀⣷⡾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠶⠐⠃⠴⠂⢉⡃⠀⠀⠙⡇⠀⠀⠑⠟⠁⠀⠀⠀⡏⠁⠒⠊⣉⣀⣽⣄⠀⢿⡖⠀⠙⠿⠾⢠⣭⡴⢃⣶⡿⠛⣿⠖⢻⡏⡇
⠀⠀⠀⠀⠀⠀⠀⣰⡀⣀⣶⣄⠀⣶⠄⣀⣤⠀⡀⠀⠓⠀⣀⣘⣛⣧⣴⢟⣰⠂⢀⡸⠿⠛⠉⢉⣻⠶⢦⡀⣄⡀⣀⢀⣽⣿⠟⢋⢁⠞⠉⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⢸⣾⠫⣿⣯⣼⡿⡷⢤⣤⣤⣶⠞⠋⠻⣿⣿⣷⣿⣿⠆⠸⣿⣤⣴⣶⠿⠋⠀⠸⣿⡟⠀⠉⠀⠈⠁⠀⠀⠃⠀⠊⠐⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠃⣀⠈⠀⡉⣚⠛⠻⡀⠀⠉⠀⢏⡀⠀⣶⣽⠏⡹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣟⡀⡈⠻⠀⠀⣄⠀⠦⠀⠄⠀⠒⠀⢀⠈⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⢺⣿⣿⣿⣶⠄⢁⣀⣤⣤⣟⣧⠘⢨⡇⠀⠃⠘⠛⠁⠀⣠⣤⣄⣰⣯⣄⣹⡏⢴⣄⠀⠤⠤⢓⣠⡾⠄⠂⢠⡄⠈⠐⠋
⠀⠀⠀⠀⠀⢀⣤⠶⢾⣎⠡⣿⣿⣡⡭⠀⠿⣋⡀⢛⣯⠏⠠⣾⡇⣠⠀⠀⠶⠀⣼⣿⠻⢿⣿⡿⡏⠛⠡⠀⠈⢀⡀⠰⠿⣧⣰⣄⡌⣠⣥⣴⡄
⠀⠀⠀⠀⠀⠀⢡⡶⠈⣹⡶⠈⢪⡿⠗⢠⣾⣿⣿⣇⠈⢨⡵⠙⣹⡇⠀⠀⠈⣶⠈⠢⠶⠏⣡⣤⠊⠡⠶⠤⠁⠤⠶⢤⣤⠈⢿⣿⣾⣿⢏⠁⠒⠀⣀⠀⠆
⠀⠀⠻⠆⠀⠀⣀⡑⠄⣿⣷⣶⣶⣦⣻⡿⢿⣿⣿⡌⡃⢀⣠⡴⠛⠀⠀⢀⣼⢇⣀⢶⣄⠀⠁⠀⣰⡟⢀⠀⢀⠀⠀⢲⢴⡧⢿⣿⣿⣶⣄⡰⠛⡟⣿
⠀⠀⠀⠀⠀⠸⠁⡌⢀⠛⠻⠟⣿⡿⠋⢀⡜⣿⡿⠛⠁⢸⡏⢠⢀⠀⠀⣾⣧⣈⣽⢇⡤⠀⠘⠄⣸⣓⠀⠰⠀⢀⣠⡞⠈⠱⢏⣴⣿⠟⠁⠄⢰⡀
⠀⠀⠀⠀⠀⡄⠛⠑⠀⣈⣥⣤⡀⠀⢴⠿⣿⠛⠃⠀⠀⢻⣇⠈⢡⡆⢠⣿⣿⠟⣡⢀⡞⠡⢠⣾⣿⣿⡷⠞⠿⠛⣁⡀⠀⠀⠸⠄⠉⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠁⡰⠀⠀⠀⢠⣶⣦⣾⠗⡈⠁⣑⠆⢤⣼⡿⠃⠠⡟⢠⠛⢉⡴⢀⣾⠅⣠⣉⠿⠿⣿⢿⣌⣰⠾⡿⠄⠤⣀⣴⣶⢹⣤⣿⣖⣦⣄
⠀⠀⠀⠀⠀⢹⡀⠀⣀⠺⣶⣶⣏⣈⣥⢙⣛⠋⢈⣁⠀⣀⠀⠀⠀⠀⠀⠰⠋⠁⣿⣿⠶⠀⢩⡞⠛⠇⠘⡿⠋⠂⠈⢺⣶⣿⣿⣧⢺⣿⣿⣏⣽⣻⡆
⠀⠀⠀⠀⠀⠈⠁⣠⠀⠐⣀⣀⣀⠉⠳⣾⡛⠓⠛⣿⣦⡸⣦⣀⠀⠀⠀⠀⡀⣸⣿⣿⣿⡆⣀⠑⢶⣂⠀⣦⡔⠀⠺⠟⠛⠉⠻⠷⠿⡏⠁⠋⠩⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠄⠚⠸⣿⣿⡏⠁⠀⣟⣀⠀⢀⡼⣟⠿⣦⣻⢆⣀⠀⢸⠀⠙⠛⠛⠛⠳⠌⢲⠞⠙⢣⠟⣉⣀⠀⠛⠀⠀⠐⠈⠀⣄
⠀⠀⠀⠀⢀⣲⡤⠴⠖⠀⠉⠘⠃⠤⠀⠉⠙⠣⢾⣒⡍⣠⠙⢻⠆⠛⠋⡉⡗⢀⣤⣄⡀⠀⣀⣴⠈⠑⠒⢤⢩⣵⠀⡴⠂⠀⠐⠀⠠⠛⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡈⠻⣦⣰⠒⠀⠹⣦⡟⠀⠋⢀⠾⠇⠐⢶⣼⠁⡚⢁⡌⢹⣆⣿⣻⣶⣤⡠⣶⠷⣆⠰⠴⡇⠈⢶⠺⠀⢋⠉⠘⠁
⠀⠀⠀⠀⠀⣠⣴⣾⣤⣶⡟⠿⢢⣾⣏⣺⡄⠀⠻⣥⣴⣌⣯⡕⠀⠀⠀⢻⢠⣿⢿⣷⡾⣸⠟⣭⠛⢿⣷⣾⣷⣏⡹⠶⠿⠶⣿⢧⡖⠈⠆⠀⢄
⠀⠀⠀⠀⣿⣿⡿⠋⢻⣿⣿⣎⠛⣓⢉⡟⠃⣤⣦⢿⣿⣿⣯⡅⣠⡌⠿⣿⠘⣭⣧⠙⣳⣿⡄⠹⣦⣈⡙⢿⣿⣿⣀⠀⠀⠀⢸⡗⠓⡀⠀⠀⠘⠃
⠀⠀⠀⣽⠀⠈⠓⠀⠀⠙⢿⠏⣾⣿⣿⣿⣔⠻⠆⠈⠙⣿⣇⡜⢁⠀⠀⠈⠿⠟⢛⣿⣋⣛⠁⠀⡹⠋⢁⣼⣿⣿⣿⣦⠀⣀⣿⡇⠀⠈⠁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣟⣾⣿⣿⣿⠉⠀⠀⡀⠈⠀⠹⠳⠞⠓⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠙⠀⠿⠋⠻⣿⣿⣟⣷⢿⣯⡷⢆⠀⠀⠀⠰
⠀⠀⠀⠀⢰⠀⠀⠁⠀⣤⠀⠙⣾⣿⣿⠿⠞⠃⠀⢶⣿⠀⢠⣾⣿⣿⣧⢰⣦⡽⢿⣿⢉⣤⠀⠀⠀⣰⣏⣈⣿⣯⢡⡾⢿⠄⣉⣁⡼⠃⠀⠐⣆
⠀⠀⠀⠀⠀⠘⠀⢀⠘⠉⢺⣿⣟⣩⠏⢀⡀⠀⠘⠛⠀⠐⢻⣿⣿⡿⣻⡿⢿⣗⣚⡋⠚⠿⡷⠐⠶⢿⠾⣿⣿⣿⣿⣇⡿⣑⣿⡿⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠐⣁⣸⣿⡿⠋⠁⠔⠂⠀⣀⣴⡖⠂⠀⣀⣔⣾⣟⣉⣀⠀⣄⠈⢉⣻⣶⣄⠀⣀⡀⠀⠀⠉⠛⣿⢿⣿⣿⣿⠃⡀⠈⡑⠂⣤⣶
⠀⠀⠀⠀⠀⠘⠃⠈⣻⣶⣿⣧⣀⠀⠰⢶⠟⠁⠀⠀⣀⣠⣿⢿⣿⡟⣻⠉⠈⡃⠀⠂⠁⠀⢂⡀⠍⠛⢒⣶⠀⠀⠸⢠⢿⠁⠘⠳⡀⠤⠁⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⠲⠶⠀⠀⣀⣤⣾⣿⡿⠷⢾⣿⣷⣶⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⢀⡄⡀⢀⠘⠀⠀⠂⠀⠀⠷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠿⠹⠇⠀⠀⠀⠘⠛⠉⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠧⠈⠁⠀⠀⠘⠦⢀⣶⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠿⠛⠛⠿⢿⣿⣶⣄
⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⣀⣠⣤⣄⣈⠛⢿⣧⡀
⠀⠀⢀⣾⣿⣿⣿⠃⢠⡴⠛⠉⠀⠀⠀⠉⠙⢦⡙⢷⡀
⠀⠀⣼⣿⣿⣿⡏⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣎⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣾⡇
⠀⢰⣿⣿⣿⡟⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠁⠀⠀⠀⣀⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⠉⠉⠉⢁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢻⡀⠀⠀⢰⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⢸⡇⠀⠀⢸⡇⠀⢀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⢸⠇⠀⠀⢸⡇⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⣿⠀⠀⠀⠘⣧⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⣤⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶⠶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠿⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷



⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⣀⣀⣀⣀⣀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢹⡉⠁⠀⠀⠀⢨⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢧⣤⣤⣤⣀⣼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡶⠒⠒⠒⠛⠛⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠙⢦⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⣼⠀⢀⣄⠀⠀⠀⢀⣠⡤⠶⠖⠒⠛⠉⠉⣿⠀⠀⠀⠀⠀⠑⢬⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⣿⠀⡜⢠⠇⢀⠞⠉⠀⠀⠀⣀⣠⡤⠤⠤⠿⠶⠶⠤⢤⣀⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣧⠏⠀⢸⣦⡀⠀⠀⠀⢿⠦⣀⣀⣀⣀⣀⣠⡀⠀⠈⠙⢦⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠈⡇⠉⠳⢤⣀⢸⠀⠀⠈⠉⠁⠀⢸⡇⠀⣀⡤⢾⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠚⠛⠉⠁⠀⢸⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⣀⣠⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠉⠁⠀⠘⢧⣀⡀⠀⠀⠀⠀⢀⣀⠼⠉⠙⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠾⠿⠟⠛⠻⠿⠿⠛⠻⠿⠷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠛⢷⣄
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⢠⡾⠛⠉⠙⠻⣦⠀⠀⠀⠀⢀⣾⠋⢀⣴⠆⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⣴⠟⠋⠉⠛⢷
⣿⠃⠀⠀⠀⠀⣸⡇⠀⠀⠀⣼⠇⢠⣾⠃⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⢿⡆⠀⠀⠀⢠⡇⠀⠀⠀⠀⢈
⠉⠀⠀⠀⠀⣴⡿⠀⠀⠀⢀⣿⠀⣾⠇⠀⠀⠀⢀⣀⣤⣼⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢸⣷⠀⠀⠀⠈⠁⠀⠀⠀⢀⣾
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢸⣿⠀⠿⠀⠀⢀⣴⠟⠋⠉⠁⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⡀⠀⠀⠀⠀⣿⡇⠀⢸⡿⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢀⡟⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢉⣙⣻⣿⠿⠶⣶⠿⠀⢀⣾⠃⠀⠀⠀⠀⠀⢿⡇
⠀⠀⣈⣀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣴⠿⠃⠀⠈⠉⠉⠉⠉⠛⣷⡄⠀⠀⠀⠛⠋⠉⠛⠃⠀⠻⢷⣤⣾⠋⠀⠀⠀⠀⠀⠀⢀⣁⡀
⠀⣾⡏⣹⡧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⣠⡶⠶⢶⣦⠀⢸⡇⠀⠀⠀⣴⡶⠿⢶⣄⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⢰⣿⢉⣿⠄
⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⠿⠶⠶⡶⠶⠿⠛⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠉⠉⠉⠉⠁⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠈⠉⠉⠉⠉⠛⠛⠻⠷⣦⣄⡀
⠀⠀⠀⠀⢀⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⡀
⠀⠀⠀⢠⡿⠁⢀⣤⡾⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⣾⠃⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⢠⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠿⣶⣶⣤⣤⣄⣀⣀⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣀⣠⣤⣤⣶⣶⠿
⠀⠀⠀⣿⡇⠀⠈⠉⠉⠙⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠉⠉⠉⠀⠀⠈⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢸⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣼⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⣧⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⣀⣀⣹⣷⣄⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀⣀⣀⣠⣾⣏⣀⣀
⠀⢀⣴⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣦⡀
⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣯⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠉⣉⣤⣶⢟⣽⣿⣿⡿⠏⠀⠀⢈⣻⠟⠛⠓⠆⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡟⣩⣷⡿⠋⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠛⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⡿⠛⠁⠀⠀⢈⠀⠀⣠⣤⣶⡿⠷⠿⠛⠛⠛⢷⣶⣶⣀⡀⠀⠀⠀⠈⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⣀⠀⠀⣤⣼⡷⠟⠋⠁⠀⠀⢀⣀⠀⠀⠀⠀⠈⠉⠛⠻⡆⠀⠀⠀⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠉⠀⣰⣿⠋⠠⣀⣽⣦⡂⡠⢂⡀⢀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⣼⡟⠁⢀⣼⣿⣿⣿⣿⡷⣯⡀⠀⠈⠉⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢀⣿⣿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠀⣼⡏⣠⠀⠀⢹⣿⣿⣿⡏⠀⢼⢳⡸⠷⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⢸⣿⠟⠁⠀⠀⠈⠈⠙⠋⠙⢶⣿⣿⠇⠀⠀⠀⣀⣠⡾⠋⠀⠀⠀⢀⣠⣿⣿⡟⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡆⠀⣿⡟⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡶⠶⠶⠟⠛⠉⠁⠀⠀⠀⣠⣿⣿⣿⣿⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⣿⣧⠀⠀⢀⠄⠀⠀⠀⠀⠀⠘⣿⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⠟⢻⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠘⣿⣷⣠⡆⠀⠀⠀⠀⠀⠀⢰⡿⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⠀⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠀⠀⠹⣿⣿⣿⣆⡐⠀⠀⠀⣠⡿⠇⢂⠗⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⣷⣤⡀⠈⠛⢿⣿⣿⣦⣀⣀⡘⠀⣄⠸⠀⠀⠀⢐⣨⣿⠟⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣦⣄⡈⠛⠻⢿⣿⣿⡇⣴⣿⣿⡼⣿⣿⣿⣋⣡⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣠⡉⠙⠃⢸⠿⣯⣄⡿⠛⢟⠛⠁⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡛⢿⣶⡆⢸⡀⠀⠘⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⢸⣿⣷⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⣿⣧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠹⣎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⢀⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢓⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠚⠀⠀⠈⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⣿⣿⣿⣿⣿⣿⣿⡿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠃⠀⠈⢿⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠉⠉⠁⠠⡤⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣴⣿⣿⣿⣿⠟⣵⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⠇⣿⣿⣿⡛⠟⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣇⠸⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣦⣈⠛⠿⠛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⡿⠟⠋⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣶⣶⣶⣶⣶⣶⣶⠿⠿⠛⠉⠁

⠀⠀⠀⠀⠀⠒⠄⢰⠐⠂⣦⠀⢰⢰⢢⢰⠀⠐⠂⣤⢰⡂⢰⡆⡆⡆⠀⡆⠀⣶⢶⡆⡆⢰⡔⠀⡆⢰⣶
⠀⠀⠀⠀⠀⠁⠁⠈⠈⠁⠁⠀⠀⠈⠉⠈⠀⠀⠀⠀⠈⠀⠀⠀⠉⠈⠀⠁⠀⠁⠈⠀⠁⠈⠃⠀⠁⠈⠉

⠀⠀⠀⠀⡠⠒⠉⠉⠓⢄⠀⠀⠀⠀⡠⠚⠉⠉⠓⢄⠀⠀⠀⠀⡠⠚⠉⠉⠓⢄⠀⠀⠀⢀⡴⠚⠉⠙⠲⣄
⠀⠀⠀⢰⠁⠘⠃⠘⠃⠀⡇⠀⠀⢸⠁⠘⠃⠘⠃⠈⡇⠀⠀⠸⠀⠘⠃⠘⠃⠈⡆⠀⠀⡌⠀⠐⠀⠺⠃⠘⠄
⠀⠀⠀⠈⢆⠰⣀⣀⠔⣠⠃⠀⠀⠘⢄⠠⠄⠀⠄⡠⠃⠀⠀⠘⣄⠢⠀⢠⠖⡰⠁⠀⠀⠰⡄⢦⣀⡠⠆⡴
⠀⠀⠀⠀⠀⠙⠒⠒⠊⠁⠀⠀⠀⠀⠀⠑⠒⠒⠊⠀⠀⠀⠀⠀⠈⠑⠒⠒⠋⠀⠀⠀⠀⠀⠈⠑⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡠⠚⠉⠉⠑⢦⠀⠀⠀⠀⡠⡞⠉⠉⠷⢄⠀⠀⠀⠀⡴⠋⠉⠉⠓⢄⠀⠀⠀⢀⠖⠉⠁⠉⠙⢆
⠀⠀⠀⢰⠁⠘⠃⠘⠃⠀⡇⠀⠀⢸⠀⠘⠃⠘⠃⠀⡇⠀⠀⢘⠀⠘⠃⠘⠃⠈⡆⠀⠀⠎⠀⠘⠀⠘⠀⠈⠆
⠀⠀⠀⠈⢆⡠⠂⠀⠂⣠⠃⠀⠀⠘⢄⠀⠰⠆⠀⡰⠃⠀⠀⠘⣄⠻⣶⡾⢇⡰⠁⠀⠀⠘⢄⢢⣤⣤⢃⠜
⠀⠀⠀⠀⠀⠉⠒⠒⠈⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⠀⠀⠀⠀⠀⠀⠉⠚⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡠⠊⡉⠉⡙⢢⡀⠀⠀⠀⡴⢊⡁⢉⡑⢦⠀⠀⠀⢀⡔⢋⡁⢉⡑⢄⠀⠀⠀⡠⠊⠉⠈⠉⢢⡀
⠀⠀⠀⢰⠀⠘⢃⡘⠃⠀⡇⠀⠀⢸⠀⠘⠃⠘⠃⠀⡇⠀⠀⢘⠀⠘⠃⠙⠃⠀⡆⠀⢰⡃⠘⠃⠈⠓⠀⡇
⠀⠀⠀⠈⢦⡴⠿⠻⠷⡠⠃⠀⠀⠈⢆⡰⠿⠿⠆⡰⠁⠀⠀⠘⢄⠰⠚⠲⢂⡴⠁⠀⠈⢧⡘⢿⡿⠟⡠⠃
⠀⠀⠀⠀⠀⠉⠐⠒⠉⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⠀⠀⠀⠀⠀⠉⠒⠂⠉⠀⠀⠀⠀⠀⠉⠐⠒⠉
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡔⠀⠈⠈⠑⢦⡀⠀⠀⢠⠎⢀⠀⠀⠉⠢⡀⠀⠀⢀⠔⠊⠁⠈⠑⢦⡀⠀⠀⢠⠄⠀⠀⠀⠈⢢⡀
⠀⠀⠀⢰⡇⠘⠀⠘⠀⠀⡇⠀⠀⣏⠀⡘⠀⠘⢁⡀⣿⠀⠀⡏⠀⠈⠀⠈⠂⠀⣷⠀⠀⡇⠀⠘⠀⠘⠀⠀⡇
⠀⠀⠀⠈⢧⡐⠦⠔⠃⡠⠃⠀⠀⠹⣄⠉⢶⠒⡻⣠⠏⠀⠀⠳⣄⠻⣁⠀⠟⡠⠃⠀⠀⠳⣄⠐⠀⠈⠂⡔⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠋⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡔⢉⣀⢀⣙⣦⡀⠀⠀⢀⠔⠕⠀⠀⠝⢢⡀⠀⠀⢀⠔⢀⠀⠀⡑⢦⡀⠀⠀⢀⡄⠈⠀⠈⠑⢆
⠀⠀⠀⢸⠀⠣⠜⠈⠓⠁⣷⠀⠀⣿⠀⠈⠁⠈⠀⠀⣷⠀⠀⣾⠀⠘⠃⠘⠁⠀⡇⠀⠀⢼⠀⠙⠀⠘⠀⢸⠆
⠀⠀⠀⠘⢦⡀⠐⠐⠀⡰⠃⠀⠀⠙⢆⡐⠉⠉⠂⠠⠃⠀⠀⠙⢆⡈⠁⠁⢀⠜⠁⠀⠀⠘⢦⡀⠀⠷⠠⠊
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⢀⡠⠄
⠀⠐⠊⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣀⣤⠤⠀⠀⠴⠶⠦⣄⠀⠀⠀⠀⠴⠋⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠠⠋⠀⢀
⠀⠀⠀⣜⣻⣷⢀⣾⣹⡆⠀⠀⠀⠀⢸⠋⠀⢀⣀⠀⢀⣀⣤⡌⠃⠀⠀⠀⠀⠀⣠⣄⠀⠀⣼⣿⡆⠀⠀⠀⠀⢀⣴⣶⡄⠀⣠⣶⣦
⠀⠀⠀⢯⣉⡽⠾⢁⣩⡇⠀⠀⠀⠀⢨⠀⠻⣿⠏⡆⠈⠛⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠻⠿⠁⠀⠀⠀⠀⢸⣿⠓⣿⢹⣿⡿⢺⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠈⠻⣿⡾⠁

⠀⠀⣀⣀⢀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⡴⠒⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⣠⣤⣀⡀
⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⠀⠈⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠈⠙⠆
⠀⠀⠀⠀⢗⢻⢧⡠⣄⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⢠⢠⣲⡄⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⣿⣷⠀⠀⠀⠀⠀⠀⣰⣶⡄⡄⣰⣶⡄
⠀⠀⠀⠀⠀⠁⠸⡟⣻⠀⠀⠀⠀⠀⠀⠀⠹⠩⠃⠘⠈⠩⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠙⠋⠀⠀⠀⠀⠀⠀⢿⣤⠇⠇⢻⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡄⣀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⡀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢺⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⠆⣸⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣿⣿⢷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠻⠛⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣦⣵⣦⣤⣧⣴⣯⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠤⣄⡀⠀⠀⠀⣀⠤
⠀⠀⠖⠒⠒⠂⠀⠀⠀⠒⠛⠓⠂⠀⠀⠀⠀⢀⣀⣤⡀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⢀⡴⠒⠀⠀⢰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢆⢠⠎⠁
⠀⠀⠀⢀⠶⣶⡄⢀⡶⢶⣄⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⠀⠀⣀⠙⠆⠀⠀⠀⠀⠀⠈⢀⠠⠀⠐⢠⣶⣦⠀⠀⠀⠀⠀⠀⢁⣤⣈⢂⠀⢊⣤⣄⠁
⠀⠀⠀⢸⠖⠿⡇⡞⠛⠛⣿⠀⠀⠀⠀⠀⠙⠀⣴⣶⣶⢢⠀⠺⣿⡿⠁⠀⠀⠀⠀⠈⠀⢠⣿⣿⡦⠀⢸⣿⡿⠁⠀⠀⠀⠀⢠⣿⣿⣽⡆⣦⣿⣿⣻⡇⠂
⠀⠀⠘⠈⠓⠞⠁⠃⠑⠞⠃⠀⠀⠀⠀⠀⠘⠀⠈⠙⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⢳⠀⠀⠉⠁⠀⠀⠀⠀⠀⠸⣿⣧⣼⠇⡟⣿⣧⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠙⠛⠋⠐⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁
⠀⠀⣠⠤⠤⠄⠀⠀⢀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⡀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⡄⢠⣴⣦⣤⣄
⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠸⡛⢻⢢⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣆⠁⡄⣰⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⡄⣴⣖⡄
⠀⠀⠀⠀⠀⠉⠁⢸⡗⢻⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⡟⠀⠇⢉⣩⠇⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠇⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⢸⣇⣸⠀⠄⢿⣀⡅
⠀⠀⠀⠀⡀⠀⠐⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⠀⠀⠀⠐⡚⠛⠒⠀⠀⠀⢀⡤⠆⠀⠀⠀⣠⠔⠂⠀⠀⠀⠀⠀⠀⡴⠛⠁⠀⠀⠀⠀⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⢢⣦⡄⠀⠀⠀⠀⠀⢀⣴⣿⣶⡀⡆⣴⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⡇⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⡇⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢉⠛⠋⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣮⣍⣤⣬⣭⣏⣵⣿⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⣷⠀⡄⠀⠑⡆⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⡶⢦⣤⣀⡤⠀⣽⡏⣇⠀⠀⣿⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠄⢿⣄⠘⠉⠉⠳⠄
⢻⡄⠀⠀⠀⢸⡏⡇⢻⡄⠀⠸⣇⠀⠀⠀⠣⡠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⠈⠛⠲⡄
⠘⣇⠀⠀⠀⢸⡇⠁⠀⢻⡄⠀⢻⡇⠀⠀⠀⡇⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢺⠀⠀⠀⣧⠀⠀⣠⠆
⠀⠙⡄⠀⠀⢸⡇⠘⣆⠘⣧⠀⠘⣧⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢰⡆⠀⠀⣿⠀⠀⣏
⠀⠀⣇⠀⠀⣿⠀⠀⢹⡆⠹⣧⡀⠹⣆⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠙⡆⠏⢙⢛⣖⠒⠒⠂
⠀⠀⢸⣆⢀⣿⠀⠀⠀⣿⠀⢸⡇⠀⣿⠃⠀⣿⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢨⠀⠀⠀⡇⢰⣫⡭⠴⠒
⠀⠀⠀⠙⠛⠁⠀⠀⠀⡿⠀⢸⡇⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⠂⠀⠀⢸⡆⠀⠀⣱⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡅⠀⠙⣆⠀⢿⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠀⢸⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⢸⠀⠀⠜⠀⠀⠃⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠐⠋⠉⠑⠀⠐⠚⢦⡀⠀⢸⢢⡀
⠀⠀⠀⠀⠤⠤⠴⠋⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠙⢲⡇
⠀⠀⠀⢸⠀⠀⠄⠴⠃⠒⣄⠀⢦⡀⢀⣀⡀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⠉⠙⠓⠦⢤⣤⣤⠤⠤⠄⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠛⠳⢭⡛⠉⠉⠉⠁⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣀⡤⡴⠊⠁⠀⠀⠀⠀⠀⠳⣄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣤⣄
⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉
⣿⣿⡟⠻⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⢦⠀⣠⠊⠁⠀⢹⠀⢀⡤⠒⠋⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣽⡴⡆⠀⠀⠀⠀⠀⢠⠋⠀⠀⣸⡜⠁⠀⠀⠀⣼⠖⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣉⡘⢓⡿⠂⠀⠀⢀⠏⠀⠀⠀⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⠁⠀⠀⠉⠙⠲⢄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠰⡦⢄⣀⡞⠁⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠱⣴⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠙⣦⡼⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⡠⠔⠚⠉⠁⠀⠀⢈⡧⠖⠚⠉⠉⠉⠙⡆
⠀⠀⠀⣾⠵⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠋⠉⠉⢢
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⠒⠶⠖⠒⠋⠁⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⣩⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⣁⣀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⠀⠀⠀⠀⠀⣀⣀⠤⠒⢫⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⣂⠔⡪⠄⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⢠⢋⣜⡵⢊⣠⠔⠊⠀⡀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⡀⢄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⣠⢮⠞⣰⢣⡿⢋⠔⢋⣠⢖⣋⠥⠔⠂⠀⠢⣌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠑⢦⡀⠙⢦⠈⠲⣌⠓⢄⡢⣄
⠀⠀⠀⢀⣀⣰⢣⡏⣰⢣⣿⡵⢁⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢀⠔⠀⠘⣆⠀⠀⠳⡀⠙⣄⠀⢳⡀⠈⢳⡀⠑⣌⠳⡄
⠀⠀⢰⡿⠫⣟⡿⡱⠁⣾⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⢠⣃⣀⠀⠀⠙⣦⠀⠀⢳⠀⠘⣆⠀⠙⣄⠀⠹⡄⠘⣆⢹⡄
⠀⠀⣾⣇⠀⣿⠳⠁⡼⠁⣼⠋⠀⠀⠠⠴⣶⠆⠀⠀⠀⢠⣮⣭⡀⠀⠀⠀⠀⠀⠎⡉⠀⠁⠀⠀⠈⢷⣄⠘⡆⠀⠘⣆⠀⠈⢣⡀⢱⡀⢸⠀⡇
⠀⠀⣿⢼⣺⣾⣡⠊⠤⠞⡏⠀⠀⠠⠶⢛⣿⡋⠀⠀⠀⠀⠘⠋⠁⣀⡀⠀⠀⠀⠐⢲⡾⠆⠀⠀⠀⠀⢻⣦⣷⠀⠀⠘⢦⡀⠀⠙⢄⢃⠀⡄⠇
⠀⠀⣾⢻⡋⢹⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⡇⠑⠀⠀⠄⠙⠦⣀⠈⠘⠀⠁
⠀⠀⠘⢾⡇⠈⢆⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠀⡆⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⢸⢀⠠⠬⠁⣦⠤⠶⡞⢶⡄
⠀⠀⠀⠀⠻⠀⠈⠣⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣀⣀⠀⢰⠁⡘⠀⠃⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⢻⣁⡁⠉⡰⠃⠀⠓⠬⠭⠔⠀⠀⠘⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠋⠭⢄⣭⠀⡘⠀⠁⠀⠀⠀⢨⠀⣠⠀⠀⠀⠀⠀⠀⠇⠈⠁⠀⠀⢀⠆⠀⡀⡀⠀⠀⡆⢸⡆⡔⡌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠋⠀⠁⠀⠀⠀⠀⠀⠈⣠⠁⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡞⠀⢀⣷⠀⠀⢠⠃⢰⣧⡇⢸⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡌⠀⠀⠖⠋⠀⠀⠀⠀⠀⠀⡇⠀⣸⡏⠀⢀⠏⢀⡞⣿⠇⢸⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢠⢿⠁⣠⠏⢀⡞⡸⡿⠀⡸⠀⡸
⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠞⠁⣠⢋⡼⠁⠃⠀⠃⢠⠃
⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⠜⢡⠎⠀⠰⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠈⠉⠀⠉⠉⠉⠁⠐⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣷⣶⣦⣄⠀⠀⠛⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⣸⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠉⠄⠀⠀⠀⠅
⠀⠀⣿⣿⣿⡿⠋⠀⠀⠘⠻⣿⣿⣿⣿⣿⣧
⠀⠀⣿⣿⣿⠴⠖⠀⠀⠘⢛⡀⠹⠿⣿⣿⣿⡀
⠀⠀⣿⣿⣿⡐⠿⠂⠀⠀⠙⠃⢠⣵⣿⣿⣿⣷⣄
⠀⢀⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡆
⢀⣾⣿⣿⣿⣿⣧⣀⠀⢄⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⡿⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠸⣿⣿⡿⠋⠀⠀⠐⠀⠀⠀⠒⠀⠙⠻⣿⣿⣿⠏
⠀⠹⠏⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠃

⠐⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠡⡀⢠⣦⣤⣤⣤⣤⣤⣤⣀⣀⣠⣤
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣀⣀⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠀⠉⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⣸⣭⠏⠉⠉⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠈⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢖⡰⠀⠰⣱⠄⠸⠀⠀⠀⠀⠀⠀⢰⠁⢰⣲⠄⠠⣞⡆⠈⡆⠀⠀⠀⠀⠀⠀⠁⠀⢆⠆⠀⢔⡂⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⣀⡀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠂⢠⡀⠒⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤⣄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠐⠛⠃⠀⢀⠌⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠘⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠡⡀⠙⠚⠙⠒⠃⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠂⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠐⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠉⠁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠈⠉⠑⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠢⠀⢀⢢⠀⠐⠀⠀⠀⠀⠀⠀⢐⠁⡖⠒⢒⡒⠒⢢⠈⠀⠀⠀⠀⠀⠀⢀⠂⠀⡖⡄⠀⢔⡂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠑⠒⠋⠑⠒⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠗⠒⠒⠅⠀⡰⠀⠀⠀⠀⠀⠀⠐⢄⠀⠢⠄⠀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠞⠉⠉⠙⠢⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠤⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠤⠄⠂⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢠⡠⠀⠠⣠⠀⢣⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠞⠀⣒⠀⠀⢰⡢⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠊⠙⠀⠘⠉⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠙⠊⠀⠀⠑⠋⡀⡇⠀⠀⠀⠀⠀⠈⠀⠈⠛⠁⠀⠙⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢠⡔⠒⡒⣄⠀⡰⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⣎⣱⠀⠂⢀⠇⠀⠀⠀⠀⠀⠈⢄⠀⠲⣶⣶⡶⠂⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⡀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⣀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀⣀⠠⠄⠊


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣠⣤⠄⣄⣤⡨⣆⠀⠀⠀⠀⠀⠀⢀⠎⢀⠤⠀⠀⠠⣀⠡⡀⠀⠀⠀⠀⠀⠀⡐⢁⡀⠀⠀⠀⣀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣦⣝⣀⣘⣤⣗⣨⡄⠀⠀⠀⠀⠀⠰⠀⠲⠲⠀⠀⠖⠶⠀⠆⠀⠀⠀⠀⠀⠀⡇⠫⣈⡇⠀⣏⡠⠇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢀⣄⣠⡀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢣⠀⢀⡀⠀⡀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠑⠤⣈⠀⠀⣁⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣙⢊⡡⠔⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣔⣾⠟⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡵⠋⠁⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠠⡏⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⠀⠀⢘⡆⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠗⠀⠀⠀⢳⡀⠀⠀⠰⡇⠀⠈⠓⠆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣼⣠⠦⣀⠹⡄⠀⠀⢧⠀⣀⣀⣀⣀⡤⠒⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠈⢳⣵⡀⢠⣞⣩⡥⠴⠆⢹⡗⢂⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠘⠦⠬⠤⢤⡾⠋⠈⠉⠑⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠿⠄⠀⠀⠈⠉⠀⠐⣄⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⢰⠃⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠰⠛⠓⡀⠀⠀⠀⠀⠀⠀⢙⠳⡀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢼⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡤⠞⡇⠀⠀⠀⠀⠀⠀⠀⡀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠙⠢⣄⣀⠦⠠⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⣹⠟⣻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢠⠃⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡎⢀⣠⣞⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⡒⢻⠀⠀⠀⠀⠠⠒⠀⠀⢸⠀⠀⠀⠀⡼⡰⠋⣀⣼⣏⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣏⠛⠦⣤⣀⠀⠀⢀⣀⠀⢸⣦⠶⡆⢰⣣⡇⠘⠿⠋⢸⣅⣀⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠒⠢⢬⣉⠑⠛⢌⠙⢫⢙⡀⣇⠈⠁⢸⣆⠀⠀⢸⡤⣤⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢀⡙⠯⣦⢬⣇⡜⠉⠻⡿⣀⠀⡎⠈⠲⢄⡸⠤⣀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠉⠳⣌⠑⠻⣝⠢⡈⡌⣹⠞⠣⡀⠭⣴⣵⣖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠑⢦⡀⠙⠞⠛⠁⢀⡔⠋⣶⢇⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠀⠉⢦⣄⣀⡴⠋⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣿⠇⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠈⠉⠙⢓⡶⠒⡤⠤⢤⡤⠞⠙⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⢀⡴⢋⡴⠋⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠤⠴⢶⠿⡶⠃⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣤⣤⢏⣀⣀⣀⣀⣀⣴⣃⡠⠤⣾⣥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠂⠒⠚⠚⠒⠉⠉⠉⠁⠀⠀⠉⠉⠘⠚⠢⠤⠤⠭⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⣷⡄⠀⠀⠀⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣼⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⣿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢠⣿⠃⠀⠀⣼⡟⠀⣀⣀⣤⣤⣶⣶⠾⠿⠟⠛⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⢿⣇⠀⣰⡿⠃⠀⠀⠰⢿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⣀⣠⣴⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠘⢿⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣀⡀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⡿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠿⠿⠋⠒⠣⢶⡬⠻⠿⠿⠟⣛⡿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣁⣚⠁⣴⠐⠚⠀⠤⠀⠂⠠⠼⢷⣮⣻⣶⡝⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢋⣢⡙⠟⢋⣤⠄⠐⠚⠻⢊⣅⢤⣦⠀⠈⠁⠉⣝⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠴⢪⡐⣢⡤⢀⡃⢂⢌⣿⢳⡀⡻⡔⡂⠐⠣⠆⠓⠘⠗⣈⡻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⢏⡶⠒⢃⡐⠁⠻⠟⢈⡁⠀⢙⠛⢋⣁⡉⠀⠁⢠⡁⠐⠂⣸⢗⣄⣛⢮⢻⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡸⢱⣿⢢⡤⢀⠒⢀⡅⣶⡌⠤⣴⡸⣿⡙⠠⠭⡄⠙⢀⢌⡛⠫⢆⡹⡞⡏⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣷⣬⡁⣉⣡⡀⠿⠸⠿⠙⠃⠨⠛⠴⢉⠁⠈⣋⠀⣰⣏⠾⢃⡠⣌⠃⢣⡇⣿⣿
⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣷⣝⡿⠟⠱⢆⣀⣠⣤⣴⢦⡶⢆⣤⣣⣅⣛⣈⢥⣔⠿⠗⢀⡐⢟⣽⣿⣿
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠻⠿⠿⠷⣨⣥⣵⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠟⠟⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣧⡰⠀⠀⠀⣀⡀⠀⠻⠟⠀⠀⢠⣤⣴⠄⠀⠀⠀⠀⣀⡴⢦⡀⠀⠀⣰⠛⢦⣀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⡆⡀
⠀⠀⠀⢠⡞⠉⠙⢦⡴⠛⢉⠷⡏⠀⠀⠀⠀⠀⠈⣁⡤⣤⣶⡒⠶⣴⠋⠟⠀⠀⠀⠀⠘⢇⡀⠌⢳⡀⡰⠉⠢⣠⠏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠘⣧⠀⠀⠈⠁⠐⠋⣸⠃⠀⠀⠀⠀⠀⢰⡏⠰⡇⠈⣷⠀⣹⠄⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣗⣧⡤⠞⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠈⠳⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠠⣄⣨⢷⣄⣹⣶⠿⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠟⢇⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢤⡤⠠⠐⠈⡇
⠀⠀⠀⠀⢰⣤⠋⠳⠞⠁⠀⠀⠀⠀⠀⠀⠐⠟⠧⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢄⡀
⠀⠀⠀⠀⠀⠀⠘⢛⡙⡹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⠶⣦⠀⠀⠀⠀⠀⠀⢀⣠⢞⣥⣴⡆⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⡴⠟⠛⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡿⢯⡉⠙⠛⢻⣍⣹⠿⡃⠀⠀⠀⠀⢀⣴⠞⠛⠳⠛⠛⢦⡀⢀
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢿⡾⠂⣿⠀⠀⠠⠄⣿⠀⠀⠀⠀⠀⠀⣿⠀⠙⣿⠖⢦⡊⠁⢸⡇⠀⠀⠀⠈⢻⡟⠒⠶⠶⠶⠶⢾⡿⠋⠁
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⢰⣾⣿⠀⠀⠀⠀⠀⠀⣿⣴⣛⣁⣀⣀⣙⢲⣼⡇⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠈⠛⠶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣤⣀
⠀⠀⠀⠀⠀⣤⠶⠾⠁⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⣀⢬⠑⡀
⠀⠀⠀⠀⠀⢷⣄⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⢰⡟⠛⢛⣻⣛⣟⡛⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠁⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠉⠙⠿⠋⠉⠉⡟⠀⠀⠀⠀⠀⠀⠀⡴⠋⠙⠛⢛⣉⡳⡄⠀⠀⠀⠀⠀⠀⢠⡼⠓⠶⠶⠛⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣠⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠿⠃⣿⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠘⠿⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡷⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢠⢖⣛⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⢠⣠
⠀⠀⠀⠀⠀⠀⢾⣏⢶⣙⣎⡇⠀⠀⠀⠀⠀⣤⡤⠤⠤⠤⠤⡤⡤⠤⢤⣤⠀⠀⠀⠀⠀⠀⣠⣴⣇⠀⢀⣿⣤⡀⠀⠀⠀⠀⠀⣟⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠈⠙⢻⠚⠋⠀⠀⠀⠀⠀⠀⡏⠳⣄⠀⢴⣤⡆⠀⣠⠞⢹⠀⠀⠀⠀⠀⣾⠁⠀⠘⢷⠟⠀⠀⢹⡆⠀⠀⠀⠀⣟⠉⡩⢿⢯⣵⠻⡍⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣤⠀⠀⠀⠀⡇⠀⠈⠳⣄⠉⣠⠞⠁⠀⢸⠀⠀⠀⠀⠀⢻⡄⠀⠀⢴⠄⠀⠀⡼⠃⠀⠀⠀⠀⣇⠀⢣⡘⠇⠺⣇⠜⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⢊⡥⠋⠀⠀⠀⠀⠓⠒⠒⠒⠚⠛⠳⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠙⢦⣄⠈⢀⣤⠞⠁⠀⠀⠀⠀⠀⣧⣀⣀⣉⣲⣞⣁⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋


⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣴⣿⣴⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⣀⣶⠂⠀⢠⡆⠀⠀⠀⣠⡾⠀⠀⠀⢀⣾⡀⣄⣀⣀⠀⠀⠀⣰⠃⠀⢀⣾⣷⡶⠀⣴⣴⢷⡆⢠⣴⠿⣤⠀⠁⠄⠆⠄⢤⡄⢻⣄⢀⣸⡶⣶
⠙⠋⠉⠃⠀⠙⠛⠲⠄⢸⣿⠁⠀⠀⠀⠿⠿⠀⠹⠛⠿⠀⠀⢛⣟⠁⠀⠈⢿⡿⠿⠁⠿⠷⠾⠃⠘⠖⢴⠆⠀⠈⠑⠀⠀⠸⠧⢶⠏⠀⠻⡿⠛
⣤⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⢀⡀⢀⡀⢀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣠⠀⠀⢀⣠⠀⠀⢀⣠⣄⠀⠀⣀⣄⡀⠀⠀⣀⠀⠀⠀⢀⣀⠀⠀⢠⣀⡀
⠻⣷⠿⠇⠀⣹⢷⣦⠀⣠⣶⣼⡄⢸⣷⣿⣟⠀⠀⢴⢏⠀⠀⠀⡿⠁⠀⠰⣿⠁⡿⠀⣿⣯⣿⠇⠸⣿⣿⡇⠀⢯⢋⢽⠂⢶⣾⣻⡷⠀⢶⣯⢽
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠀⠉⠉⠁⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠉⠀⠀⠉⠀⠀⠀⠈⠉⠋⠀⠈⠋⠙⠁⠀⠈⠉⠋
⠾⣿⣿⡆⠐⣿⣴⣿⠀⣷⣤⣴⡇⢰⣦⣠⣶⠀⣶⣄⣴⡆⢰⣦⣴⡾⠀⢀⢄⠭⡐⠄⣶⠞⢻⡄⠰⣶⠾⣷⠀⠘⢿⣾⠃⠀⢹⣼⠁⠀⠈⣷⡯
⠀⠀⠿⠇⠈⠉⠈⠉⠀⠻⠟⠻⠃⠀⠛⠙⠋⠀⠘⠟⠛⠁⠘⠋⠙⠃⠀⠀⠉⠻⠅⠀⠻⠶⠾⠁⠘⠟⠾⠏⠀⠈⠼⠿⠀⠀⠼⢿⠄⠀⠳⠿⠷
⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣄⠀⠀⠀⢠⠀⠀⠀⢀⡄⠀⠀⠀⢠⣄⡤⠀⠠⣤⣤⠄⠀⣄⣀⡄⠀⢠⣤⣤⠄⢠⣤⣤⡤
⠸⣿⣿⠷⠀⠰⠿⠀⠀⡀⠿⡃⠀⠐⢻⣿⠓⠀⠚⣻⡘⠂⠐⢙⣏⠃⠀⢠⢶⣿⡇⠀⣄⣿⣿⠀⢀⣸⣿⠀⠠⣠⣿⣿⠀⢤⣿⣿⡀⠘⠛⢻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠉⠉⠁⠀⠈⠉⠉⠀⠀⠉⠉⠀⠀⠈⠉⠉⠀⠈⠉⠉⠁⠀⠀⠈
⠀⣠⣦⡀⠀⣄⣴⣠⡄⢠⣴⣶⣤⠀⡴⠒⢦⡀⢠⡴⢦⣄⢀⡴⠶⣦⡀⣸⣴⡶⣶⢀⣀⣾⣿⠀⢸⣷⣲⢦⠰⣏⣁⡴⠀⠲⣦⣄⠀⠠⢶⣿⡶
⠀⠉⡏⠁⠀⠾⢿⠻⠄⠸⢿⣿⠿⠈⢧⣙⡼⠃⠸⢯⡽⠷⠘⢧⣯⡼⠂⠈⠛⠿⠏⠀⠙⠻⠛⠁⠈⠳⠽⠟⠀⠈⠙⠟⠀⠀⠘⠙⠓⠀⠘⠛⠓
⠀⡀⡀⡀⠀⣀⡀⣀⡀⢀⣀⢀⡀⠀⣀⠀⣀⠀⠀⡀⢀⡀⠀⣀⠀⣀⠀⢠⡀⠀⠀⠀⣤⣀⣤⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⣀⠀⢀⣀
⢰⣟⢝⣗⠈⢿⣿⣿⠇⠸⣿⣿⠟⠘⣏⠉⣹⠇⠃⠀⢀⡿⠸⠉⠉⢸⠇⢸⣿⣦⣄⠀⢾⣿⡏⠁⠾⢿⣶⠏⠐⢾⡷⠶⠆⠰⣿⣿⡏⢸⢿⣿⠛
⠀⠉⠋⠁⠀⠀⠙⠁⠀⠀⠈⠋⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠈⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⢰⣶⣴⣶⡀⣀⣴⣀⡀⢀⠠⠆⢀⠀⣠⣤⣄⠀⢀⡤⢤⡀⢀⣠⣤⣤⡀⢰⣶⡶⢆⠀⠀⢤⣤⡀⠘⢴⣿⣷⠀⣤⣤⣾⠁⢠⣶⣿⠃⠘⣷⣤⣤
⠈⠻⠿⠋⠀⢹⠿⢯⠀⠈⡄⢤⠁⠀⠸⠿⠟⠀⠘⠧⠼⠃⠘⠿⠿⠿⠃⠈⠈⠀⠀⠀⠀⠘⠉⠃⠀⠀⠁⠘⠀⠘⠀⠉⠀⠈⠈⠋⠀⠀⠈⠀⠉
⠀⠀⢀⠀⠀⠀⠀⢀
⢰⠉⠀⢑⠀⡙⠋⣈⠃
⣤⣬⣬⣤⣤⣭⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣮⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣽⣷⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⡠⠊⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⠊⠀⡔⠁⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡴⠁⢀⠌⠀⠀⢀⠄⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢃⡾⠁⢀⠎⠀⠀⠀⠈⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠈⠀⢀⠊⠀⠀⠔⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⡔⠁⠀⣠⠊⠀⢀⡞⠀⠀⠤⠖⠚⠛⠛⠳⢶⣤⡄⠀⠀⠀⠀⠀⣀⣀⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠋⠀⢀⡔⠁⠀⢀⡞⠀⠀⠀⠀⠀⣀⡤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡩⠥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜⠁⠀⠴⠋⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠏⣐⡀⠬⠄⠀⠀⠀⠀⠀⠀⢀⣢⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠿⢀⡄⢀⠀⠀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⢧⠀⠙⢦⠙⢦⠀⢸⡏⠉⣷⠀⠉
⠀⠀⢀⠀⠘⢦⡀⠀⠈⢧⠈⠳⣼⡇⠀⠈
⠀⠀⣀⠓⠀⠈⠛⢦⣀⠈⠳⡄⠘⣧⠀⠀⠀⢦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠤
⠀⠀⢀⠀⠰⣄⠀⠀⠙⢦⠀⠘⡀⠘⣦⠀⠀⠈⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠳⡀⠘⠲⣤⡀⠈⠃⠀⠁⠀⢯⡑⠤⠀⠀⢻⡀
⠀⠀⠀⠀⠈⢢⡀⠀⠙⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⢳⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠉⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠙⠦⢄⡀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠚⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠙⠳⢤⣄⣀⠀⠀⠀⠀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇
⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣠⣤⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠛⠉⠉⠛⠻⢿⣿⣷⡄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠁⠀⠀⠀⢀⣀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣿⣿⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠀⢀⣴⠾⠛⠛⠛⠛⠛⠶⣤⡈⠻⣧⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠈⢧⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠉⠉⠉⠉⠉⠉⠉⠙⢿⣏⠉⠛⠛⠁
⠀⠀⠀⢰⣿⣿⣿⣿⢣⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠈⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠘⠛⠛⠛⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇⣀⠀⠀⢹⡇⠀⠀⠀⢀⣿⠁⠀⢀⣠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠈⣿⠀⠀⠀⣸⡇⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢻⡇⠀⠀⣿⠁⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⢸⡇⠀⢠⣿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣐⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠞⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⣀⠤⠀⠠⢄⡀⠀⠀⠀⠀⢀⡠⠄⠀⠤⣀⠀⠀⠀⠀⠀⣀⠤⠀⠠⢄⡀⠀⠀⠀⠀⢀⡠⠄⠀⠤⣀⠀⠀⠀⠀⠀⣀⠤⠀⠠⢄⡀
⠀⠀⠀⡜⢁⡀⠀⠀⣀⠙⡄⠀⠀⢠⠋⣀⠀⠀⣀⡈⢣⠀⠀⠀⡜⢁⡀⠀⠀⣀⠙⡄⠀⠀⢠⠋⣀⠀⠀⢀⡈⢣⠀⠀⠀⡜⢁⡰⠀⢰⣀⠙⡄
⠀⠀⠀⡇⠀⠁⠀⠀⠁⠀⡇⠀⠀⢸⠀⠈⢀⣀⠈⠀⢸⠀⠀⠀⡇⠀⠁⠀⠀⡁⠀⡇⠀⠀⢸⠀⣈⠀⠀⢈⡀⢸⠀⠀⠀⡇⠀⢁⡀⣀⠁⠀⡇
⠀⠀⠀⠙⢄⠑⠒⠒⣁⠜⠀⠀⠀⠈⠢⣈⠁⠀⢙⡠⠋⠀⠀⠀⠑⢄⠒⠒⠊⣀⠜⠁⠀⠀⠈⠣⣙⠦⠤⢞⡡⠊⠀⠀⠀⠑⢄⡁⠀⠀⣁⠜⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⡰⠊⠀⠀⠀⠈⠢⡀⠀⠀⢀⠔⣡⠀⠀⢠⡑⢄⠀⠀⠀⡰⠋⠀⠀⠀⠈⠢⡀⠀⠀⢀⠔⠁⠀⠀⠀⠑⢄⠀⠀⠀⡠⢊⣀⠀⢀⣈⠢⡀
⠀⠀⠀⡇⠈⠃⠀⠀⠋⠀⡇⠀⠀⢸⠀⠘⠃⠀⠛⣴⢼⠀⠀⠀⡇⠈⠃⠀⠀⠉⠀⡇⠀⠀⢸⠀⠘⠀⠀⠘⠀⢸⠀⠀⠀⡇⠘⠊⠀⠈⠚⠀⡇
⠀⠀⠀⠣⡀⠐⠒⠒⠀⡠⠃⠀⠀⠘⢄⠐⠊⠉⠒⢀⠞⠀⠀⠀⠣⡀⠠⣀⠠⠊⡰⠃⠀⠀⠘⢄⠰⠯⠭⠗⢀⠜⠀⠀⠀⠣⡘⢏⣉⣉⠟⡠⠃
⠀⠀⠀⠀⠈⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠁⠀⠀⠀⠀⠀⠈⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠈⠓⠒⠒⠊

⠀⠀⠀⢀⠔⠉⠉⠉⠓⢄⠀⠀⠀⠀⡠⠚⠉⠉⠙⠢⡀⠀⠀⠀⢀⠔⠉⠉⠉⠓⢄⠀⠀⠀⠀⡠⠊⠉⠉⠙⠢⡀⠀⠀⠀⢀⢔⠋⠉⠉⢑⢄
⠀⠀⠀⡇⠐⠆⠀⠀⠖⠀⡇⠀⠀⢸⠀⠲⠀⠀⠰⠂⢹⠀⠀⠀⣇⡔⠑⠀⠘⠑⣌⡇⠀⠀⢸⠀⠰⠀⠀⠰⠀⢸⠀⠀⠀⡇⠈⠶⠀⠰⠎⠀⡇
⠀⠀⠀⢇⠀⠀⣤⡀⠀⢀⠇⠀⠀⠸⡀⡤⢤⢤⠤⡄⡸⠀⠀⠈⢷⠡⡴⠶⠶⡤⢳⠇⠀⠀⠸⡀⣠⣄⢄⣤⡀⡸⠀⠀⠀⢇⠀⡖⠉⠑⡄⢀⠇
⠀⠀⠀⠈⠓⠤⠤⠤⠔⠁⠀⠀⠀⠀⠈⠢⠤⠤⠤⠊⠀⠀⠀⠀⠈⠑⠬⠭⠭⠔⠉⠀⠀⠀⠀⠈⠢⠤⠤⠤⠊⠀⠀⠀⠀⠀⠑⠬⠶⠮⠕⠁

⠀⠀⠀⠀⡠⠔⠒⠒⠤⡀⠀⠀⠀⠀⢀⠤⠒⠒⠢⢄⠀⠀⠀⠀⠀⡠⠖⠒⠒⠤⡀⠀⠀⠀⠀⢀⠤⠒⠒⠲⢄⠀⠀⠀⠀⠀⡠⠔⠒⠒⠤⡀
⠀⠀⠀⡞⢠⡄⠀⠀⣤⠈⡄⠀⠀⢰⢁⠰⠆⠀⠶⠀⢳⠀⠀⠀⡎⠠⠆⠀⠠⠢⠘⡆⠀⠀⢰⠃⢤⠀⠀⢠⠄⢣⠀⠀⠀⡎⢠⡔⠀⠐⣤⠘⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⢎⠇⢀⣀⠀⠯⢾⠀⠀⠀⡇⠀⠀⣖⢖⣲⠀⡇⠀⠀⢸⠀⠠⣀⣀⣠⠀⢸⠀⠀⠀⡇⠠⡤⠤⠤⡤⠀⡇
⠀⠀⠀⠘⠦⣈⠐⢈⡠⠞⠀⠀⠀⠀⠳⢄⡀⠀⣀⠴⠃⠀⠀⠀⠘⠦⣀⠀⢀⡠⠞⠀⠀⠀⠀⠳⢄⡘⠚⣀⠴⠃⠀⠀⠀⠘⠦⣈⠒⢈⡡⠞

⠀⠀⠀⣠⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⣀
⠀⠀⠈⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⢀⡾⢋⣉⣉⠙⢳⡄⠀⠀⠀⣠⠞⢋⣉⡉⠙⢦⡀⠀⠀⢀⡴⠚⠋⠉⠁⠀⠀⠈⠉⠑⠂⠀⠀⠀⠤⣀⣀⣀⣀⣤⡴⠶⠋⠁
⠀⠀⣾⠀⣯⣨⣿⣷⠀⣿⠀⠀⠀⡟⢰⣧⣼⣿⡆⠈⣧⠀⠀⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣭⣥⣤⣄⡀
⠀⠀⠹⣆⠙⠿⠟⠁⣠⡟⠀⠀⠀⢻⣌⠻⠿⠗⠁⣰⠇⠀⠀⠀⠀⣠⣶⡿⠟⠛⠛⠿⢷⣦⡀⠀⠀⣴⣾⠟⠋⠉⠉⠉⠛⠿⣷⣄
⠀⠀⠀⠈⠛⠶⠶⠚⠉⠀⠀⠀⠀⠀⠙⠓⠶⠶⠛⠁⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣦⣾⡟⢁⡴⢶⣶⣦⣄⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⢀⣤⣄⡈⣿⣿⢰⣿⡀⢀⣽⣿⣿⣆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢰⣧⣀⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⠿⡟⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠘⢿⣿⣿⣽⡿⣿⣆⠙⠿⣿⣿⠿⠚⠀⠀⠀⣰⡿⠁
⠀⠀⠀⣠⡴⠖⠲⠶⣄⠈⠂⠀⠀⢃⣤⠶⠒⠲⢦⣄⠀⠀⠀⠀⠈⠻⣷⣤⣄⣀⣀⣠⣴⡿⠋⠀⠈⠻⣷⣤⣀⡀⠀⢀⣀⣤⣾⠟⠁
⠀⠀⣸⠋⢀⣀⣶⡄⠈⣷⠀⠀⠀⡾⠁⣀⣰⣦⠀⢹⡆⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠋⠁
⠀⠀⢻⣀⣈⣛⣃⣀⣀⣿⠀⠀⠈⣧⣀⣈⣛⣁⣀⣰⡇
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠉⠻⣆
⠀⠀⠀⢄⣀⣀⣠⠤⠒⠀⠀⠀⠀⠀⠢⢤⣀⣀⣀⠄⠀⠀⠀⠀⠾⢧⣤⣤⣀⣀⣀⣀⣤⣿⠆⠀⠀⠀⠀⠺⢧⣤⣤⣄⣀⣀⣠⣤⡿⠂
⠀⠀⠀⢀⣤⠶⠶⣤⣀⠀⠀⠀⠀⢀⣤⠴⠶⢤⣄⠀⠀⠀⠀⠀⠀⠸⣇⠈⠙⠿⠋⠁⣰⡏⠀⠀⠀⠀⠀⠀⠸⣇⠈⠙⠿⠋⠀⣰⠇
⠀⠀⣰⠏⠀⢐⣦⠀⠙⣧⠀⠀⢠⡟⠁⠀⣶⠄⠈⢷⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⣤⠾⠋
⠀⠀⢿⣀⣠⣽⣯⣄⣀⣿⠀⠀⢸⣄⣀⣬⣯⣄⣀⣸⠂
⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠉⠉⠉⠁⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⣀⣀⠀⠀⠄⠀⠐⠐⠒⠂⠀⠀⠤⢍⡒⢤⡀
⠀⠀⠀⠀⠀⢀⣠⣦⣒⣊⣉⡀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⠤⠬⠷⢿⡄
⠀⠀⠀⣀⣴⠟⠛⠉⠁⠈⠍⠉⠉⠉⠉⠉⠉⡉⠩⠑⠢⢄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⣠⠆
⠀⠀⢀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠐⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠈⡇⣿⢀⡼⠁⢀⣀⠀⡀⡀⢀⢤⢀⡄
⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠰⠃⠈⠁⠀⠈⠙⠈⠡⠇⠈⢻⠙⠉
⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⠀⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢤⡄⠈⠑⠖⠒⠚⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢸⡦⡀⠀⢀⡠⠚⡇⠀⢀⡇⠀⠀⢸⠀⣀⠄⢄
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢈⡃⠈⠉⠁⠀⣀⠷⠚⠉⠈⠢⣄⠜⠂⠀⢢⠀⠱⡀
⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠡⢀⠄⠢⣈⠣⡀⠀⠀⠀⢀⡬⠧⠀⠀⠀⣇⣀⣵
⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⡁⠫⡀⢀⠄⠁⠀⠀⠈⠢⡈⠢⣠⠞⠁⠀⠇⣠⣔⣊⡭⢔⡁
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⡐⠁⠢⢄⠀⠉⠓⢬⡈⠲⣄⠀⠀⣀⠔⢻⠀⡇⣀⡤⠞⠛⠉⠉⠀⠀⠀⣨⠁
⠀⠀⠀⠀⠀⠀⠉⠲⣤⣰⢄⡀⢀⠞⠙⠢⠀⠀⠈⠳⢄⠑⠞⠁⠀⣸⡴⠛⠁⠀⠀⠀⠀⠀⣨⠶⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠛⢀⠔⠁⠀⠀⠢⣀⠀⠙⠦⣤⠚⠁⠀⠑⠠⢀⣀⡠⠔⠉⠀⠀⢇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠲⢄⡀⠀⢠⠞⢢⢀⠔⠁⢆⠀⠀⠀⢀⡠⠛⢧⠀⠀⠀⠀⢘⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠉⠲⠅⣀⡠⠋⠀⠀⠈⣧⡠⠔⠉⠀⠀⠘⣆⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠋⠑⠢⣠⠞⡅⠀⠀⠄⠊⠁⠙⢄⠀⠀⠀⠀⣈⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⡔⠁⠀⠈⠢⣄⠀⠀⠀⠀⣈⣡⠤⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠂⠠⡀⠀⠀⢀⠊⠡⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡀⠀⠢
⠀⠀⠀⠀⠀⠀⠌⠀⠀⡀⡀⠀⠐⠀⠀⠂⠀⠐⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢀⠀⠐⠀⠐⢀⡸
⠀⠀⠀⠀⠀⠀⡄⠀⠸⢀⠜⠀⠀⠃⠐⠀⠀⠀⠄⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠃⠑⠈⠦⢀⣀⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⢀⠠⠊⠀⠄⣆⠀⢀⠁⠌⠀⠀⠀⠀⠀⠀⢀⠔⠉⠂⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⡠⠄⠒⠂⠲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠐⠀⠙⢦⡚⠁⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠱⣏⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠄⠪⠊⠀⠀⠀⠀⠀⠀⠠⠊⠀⠄⠒⡖⠂⠄⡀⠢⡀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⢀⠔⠁⠐⠀⠀⠀⡇⠀⠀⠘⠄⠘⢄⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⡆⠀⠀⠀⠗⠀⠀⠀⠂⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠠⡀⠀⠀⡇⠀⠀⡠⠃⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠈⠐⠂⠧⠐⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⡤⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠐⠒⠂⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠔⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠀⠀⠸⠀⠀⢠⠀⠀⡀⠄⠀⠀⠀⠀⠐⠂⠀⢄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢈⡀⠀⠀⠀⠂⠀⠘⠂⠀⠐⠂⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠀⠀⠒⠀⠀⠐⠀⠀⠈⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠨⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠰⡂
⠀⠈⢄⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢰⢰⠀⠀⠀⠀⠀⠠⠒⠈⠁⢂⢸⡀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⡠⠀⡠⠂⠘⠢⣀⠄⠠⡀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢐⢘⠈⠀⡠⠀⠠⢀⠃⠀⠀⠀⠘⠁⠀⠈⠑⠔⠈⠑⡆⠈⠁
⠀⠀⠀⠀⠈⠢⠀⠀⢒⠎⠒⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢩⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁
⠀⠀⠀⠀⢀⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⠀⡇⠀⢘⠀⠀⢀⠀⠀⢸⢸⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠑⠀⠀⠀⠀⠢⠄⠀⠀⢠⠀⠀⠀⠀⠀⢊⠀⠀⠀⢀⠠
⠀⠀⠀⠰⠆⠀⠰⠂⠀⠀⠀⠀⢀⠈⠁⠀⠉⢀⠈⠀⠀⠀⠀⠡⠀⠀⠦⠀⠀⠀⠀⠀⡶⢶⣦⣤⣶⣦⠀⠀⠀⢀⠈⠁⠀⠀⠉⠀⠁⠀⠀⠀⠐⠖⠀⠀⠖
⠀⠀⢀⠰⡖⠖⢲⠂⡀⠀⠀⠐⠈⠢⡀⢀⡠⠞⠄⠀⠀⠀⠀⠲⠒⠖⠶⠂⠄⠀⠀⢠⡙⠞⠉⠉⠉⢁⡄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠄⠀⠶⠶⠶⠂⡀
⠀⠀⠀⠁⠣⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠐⠀⠠⠒⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⡴⠠⢤⣤⡤⠌⢦⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠑⠀⠀⠀⢠⠂⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⢸⣤⣴⡴⠛⢴⣦⣼⡇⠀⠀⠀⠈⠀⠁⠈⠂⠀⠀⠀⠀⠀⠠⠰⠂⠄⠀⠀⠀⠀⠀⠀⠂⠐⠂⠰⠀⠀⠀⠀⠀⠀⠰⠲⠀⠐⠆⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠠⠂⠄⠀⠀⠈⢄⠈⠙⠙⠓⠀⠌⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡠⠁⠀⠀⢀⡐⠶⠶⠶⢂
⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠂⠀⠂⠁⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠶⠶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠄⠰⠂⠀⡰⠂⠀⠀⢀⣤⠊⢠⠀⠀⣠⠈⢠⣄⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠰⠀⣀⠄⠆
⠀⠀⠀⠀⠋⠀⠁⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⣀⠒⠀⠀⠀⠀⠀⠜⠋⠀⠐⢃⠀⠀⠀⠀⠀⡬⠋⠘⠘⢡
⠀⠀⠀⠀⠲⣶⠆⢀⠄⠀⠀⠙⠄⠑⠄⠠⠔⠁⠜⠁⠀⠀⠀⠀⠚⠓⠀⡁⡃⠀⠀⠐⡀⠈⠉⠉⠀⠀⠂⠀⠀⠁⠀⠀⠈⠀⠀⠱⠀⠀⠀⠲⡁⠐⠒⠂⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢀⣤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⠀⢀⠀⠀⠘⠆⠀⠀⠐⠀⢀⠀⢀⠀⠈⠀⠀⠀⠀⣤⣤⡄⣤⣤⡐⠀⠀⠀⠀⣠⡀⠀⢀⣄
⠀⠀⠀⠈⢁⠀⠈⠁⠀⠀⠀⠀⠀⠘⠂⠀⠚⠀⢀⠀⠀⠒⠀⠀⠀⠀⠁⠂⠠⠀⠀⠀⠀⠈⢀⡈⣡⠄⠀⠀⠀⡀⠙⠛⣀⠘⠛⠁⠀⠀⠀⠀⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠈⠀⠀⠂⠀⠀⠀⠀⡀⠒⠒⠀⠀⠀⠀⠀⠈⠄⠘⠛⠛⠃⠀⠀⠀⠀⠀⠄⠈⠉⢙⠉⡐⠀⠀⠀⠀⠀⠑⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⢀⠠⠀⠒⠀⠄⠀⠀⠀⠀⠀⠀⠐⠒⠀⢄⠀⠀⠀⠀⢀⠄⠒⠒⠀⠀⠀⠀⠀⠀⡯⡅⠐⠂⣨⡑⠀⠀⠀⠀⣫⡄⠒⠂⣬⡐⠀⠀⠀⠃⠤⠐⠒⠂⠠⢘
⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠁⣠⠀⠀⣠⠀⠠⠀⠀⢰⣷⣶⣶⢴⣦⣶⣦⠀⠀⠐⠀⣀⠀⠀⣀⠀⠂⠀⠀⠂⠀⠁⠀⠈⠀⠀⡄⠀⠀⠉⠀⠀⠀⠀⠂⠈
⠀⠀⠀⢈⠁⠀⠈⠁⠀⠀⠀⢀⠀⠸⠃⠀⠹⠂⢀⠀⠀⠀⠙⠛⠁⣈⠙⠋⡈⠀⠀⠁⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠐⠂⢀⠀⠒⠀⡀⠀⠀⠀⣈⡃⠀⠘⣁
⠀⠀⠀⢈⠉⠋⠉⡁⠈⠀⠀⠀⠀⢀⠈⠃⢀⠠⠀⠀⠀⠀⠀⠀⠀⠓⢀⡄⠜⠀⠀⠈⠀⡈⠁⠈⢀⠀⠀⠀⠀⠐⠄⡀⠛⠁⠀⠈⠀⠀⠀⠀⢈⠛⠛⠛⠁⠂
⠀⠠⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠠⢶⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠄⠀⠀⠀⠀⣠⣶⣶⣶⣦⡀⠀⠀⠀⠀⡰⠀⠀⠀⠖
⠀⠀⢀⣘⠁⠀⡈⢡⡘⠃⠀⠀⠀⢀⡀⠀⠀⠐⠐⣉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⣡⡜⠂⠀⠀⠀⠀⠊⠉⡉⠻⠋⡉⠉⠀⠀⠐⠊⢡⡀⠤⢀⡄⠑⠂
⠀⠀⠸⡷⠶⠐⠶⢿⠇⠀⠀⠠⠀⠀⠀⠀⠰⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠉⠡⠈⠀⠀⠀⠀⣀⠘⣃⣤⣄⡛⢀⠄⠀⠀⣄⢀⡢⠀⢐⣀⢢
⠀⠀⠀⠘⢓⠂⠚⠁⠀⠀⠀⠀⠁⠀⡀⠀⡀⠄⠀⠀⠀⠀⠀⠄⡀⢀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢀⡀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠋⠀⠀⠀⠈⠢⣉⣈⣈⠤⠁
⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠠⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢀
⠀⠀⠀⠠⠈⠀⠀⠀⠀⠈⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⡄⢠⣀⠀⣠⠀⠀⢰⠀⠀⠠⠀⣀⠀⠀⣀⠀⠀⢡⠀⢠⠐⠀⣄⠀⠠⠞⠉⠀⠃⠀⢰⠀⠁⢠⡄⢨⡄⠀⠀⠇⠀⠘⠀⠈⠛⠄⠰⠛⠁⠀⡆
⠀⠀⠀⢂⠀⠀⠁⠀⠉⠀⠀⡌⠀⠀⠄⠁⠈⠁⠀⠉⠀⢁⡎⠀⠈⡈⠀⠀⢀⣀⣤⣴⢂⠆⠀⠐⡀⠀⠦⠀⠄⠄⠀⢀⠆⠀⠐⡀⠢⢀⠀⣀⠀⠐⢀⠃
⠀⠀⠀⠀⠢⢀⠂⠀⢂⡠⠊⠀⠀⠀⠀⠢⢀⠀⠀⢀⡠⠊⠀⠀⠀⠈⠢⣁⠉⠱⢟⠫⠊⠀⠀⠀⠘⠦⣉⠀⠀⣁⠴⠃⠀⠀⠀⠀⠤⡀⠈⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠴⡀⠀⠀⠈⡰⢄⠀⠀⠀⠀⠤⠊⠀⠀⠘⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⢠⣤⡁⠀⠀⢀⣤⡄⠀⠀⠀⠀⠐⠈⠀⠀⠁⠠⡀
⠀⠀⠀⠊⠀⠙⠂⠀⠘⠁⢠⢃⠀⠀⠄⠀⠀⠠⠀⠄⠀⡠⠇⠀⢀⠁⠀⢠⣴⢀⡄⠀⠀⠆⠀⢠⠛⠀⢀⠂⠐⡀⠀⠉⠄⠀⠰⠰⠛⠁⠄⠀⠘⠓⠈⡄
⠀⠀⠀⡀⠀⠘⠛⠀⢛⣀⠀⠸⠀⠀⠀⠀⢀⣀⣀⣠⡄⠀⡀⠀⠘⠀⠄⠀⠁⠈⠀⠀⠀⡄⠀⠸⠈⠀⠙⠃⠘⠋⠀⠀⠄⠀⢠⠐⠀⠷⠀⠐⠗⠠⠀⡇
⠀⠀⠀⠐⢄⢿⠀⠀⠈⣿⡤⠁⠀⠀⠑⢄⠀⢉⣭⡷⢂⠔⠁⠀⠀⠑⢄⡀⠀⠀⢃⢀⠜⠀⠀⠀⠱⣀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠢⡀⠀⠺⠗⠀⣀⠎
⠀⠀⠀⠀⠀⠁⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠁⠐⠒⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠉⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠄⠂⠀⠂⠄⡀⠀⠀⠀⠀⣀⠐⠀⠀⠐⣠⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⡠⢂⠀⠀⠀⣀⠀⠀⠀⠀⠀⡠⠐⠀⠐⠀⡀
⠀⠀⠀⡐⠑⠛⠆⠀⠴⠲⠀⠄⠀⠀⠜⠁⢠⠀⠀⢠⠀⠑⠄⠀⢀⠎⠀⣤⣄⠀⢀⡀⠀⠀⠀⢀⠊⠀⣴⠁⢈⣶⠀⠑⡀⠀⢀⢊⠁⣄⠀⠀⣄⠈⠡
⠀⠀⠀⠀⠈⠲⣶⣤⣤⠖⠂⢸⠀⠰⠀⠀⠀⢂⣀⠂⠀⠀⠰⠀⢸⠀⠀⠙⠁⠀⢾⠇⠀⠀⠀⢸⠀⠀⠀⣀⣈⡀⠀⠀⡀⠀⠀⠀⢀⡉⠂⠐⣉⡀⠀⡇
⠀⠀⠀⢣⡂⠀⣠⣶⣶⣶⢆⠆⠀⠀⠢⠀⣴⠿⠻⢿⡄⡠⠃⠀⠀⢂⠀⠀⢠⣤⡄⠀⡠⠁⠀⠈⢦⠀⠾⠉⠉⠿⠀⡰⠁⠀⠘⢄⠀⠘⢛⣛⣀⠀⡰
⠀⠀⠀⠀⠙⠪⠝⠻⠭⠐⠁⠀⠀⠀⠀⠁⠊⠥⠤⠍⠂⠁⠀⠀⠀⠀⠑⠂⠤⠤⠄⠊⠀⠀⠀⠀⠀⠙⠒⠤⠤⠒⠋⠀⠀⠀⠀⠈⠑⠤⠚⠫⠅⠊
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠠⠞⠀⠢⠀⠀⠘⠦⡀⠀⠀⡠⠊⠁⠂⠀⠀⠈⠦⡀⠀⠀⡈⠀⡀⠀⠀⠀⠀⢂⠀⠀⠀⣴⠟⠀⠀⠀⠐⠛⠦⠀⠀⢀⠀⠀⠀⡀⠀⠀⠈⠂
⠀⠀⠀⠁⢀⠐⠷⠀⢐⠃⠀⢠⠀⠠⠀⠀⠰⠆⠀⠶⡀⠀⠀⠀⢰⠠⠉⣁⠀⠀⣀⠀⠀⠃⠀⠘⠀⠀⢤⡄⢠⡤⠀⠁⠁⠀⠈⠀⠀⣤⠄⢠⣤⠀⠀⡂
⠀⠀⠀⠄⠀⢀⡴⠶⣤⠀⠀⡈⠀⠀⠄⠀⠀⡾⠻⣆⠀⠀⠄⠀⠘⡀⠀⠙⠁⠀⠋⠀⠁⠆⠀⠈⡄⠁⢀⣤⣤⡔⠂⢠⠃⠀⠘⡀⠀⠄⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠈⠢⣀⠁⠀⢈⡠⠊⠀⠀⠀⠀⠢⢌⣉⢀⣉⠄⠀⠀⠀⠀⠈⠦⣀⠀⠀⢀⡡⠊⠀⠀⠀⠈⠂⠌⣁⣈⠡⠒⠈⠀⠀⠀⠈⠂⠄⣀⣀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉
⠀⠀⠀⢀⡤⠊⠁⢀⠀⠄⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠑⢄⠀⠀⠀⢀⠔⠁⠀⠀⠀⠑⢄⠀⠀⠀⡠⠎⠒⠄⠠⠐⠒⢄⠀⠀⠀⢠⠶⠆⡀⠀⠶⢶⡀
⠀⠀⠀⠎⠀⣠⣠⠀⠰⣆⠀⢀⠀⠀⠂⠀⠀⠀⢀⠀⠀⢀⢁⠀⢀⢁⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⠸⣄⣛⠀⠠⡛⣠⠆⠂⠀⠠⢉⠀⠆⡀⢀⠶⠀⠛⡄
⠀⠀⠀⡀⠁⠈⠁⠀⠄⠀⠢⢈⠀⠀⡀⠠⠘⠂⠀⠻⠀⠀⡘⠀⠈⠀⠀⠻⠂⠀⠟⠠⠀⡄⠀⠘⣠⡟⢁⣤⣤⡀⢻⣆⠆⠀⢠⠀⠁⠈⣠⣄⠀⠀⠀⠇
⠀⠀⠀⠱⣄⠀⠀⠀⠕⠂⡄⠀⠀⠀⠐⢄⠀⠀⠀⠀⢀⠔⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⢇⡘⠂⠀⢋⡸⠋⠀⠀⠀⠢⡀⠸⠛⠛⠂⣀⠞
⠀⠀⠀⠀⠀⠁⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠁

⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⢶⠶⠶⣶⣶⠶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠤⠀⢠⣔⡀⠀⠀⠀⠠⡷⠀⠀⠀⠳⣄⡀⢠⣄⠈⠇⣠⡄⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠑⢦⡀⠀⠀⢾⠀⠀⠀⠀⠀⠀⣖⠆⣠⠖⣾⣿⣷⣮⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⠢⣅⣇⣸⣨⠔⡹⡃
⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⢸⠀⠀⠀⢱⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡾⢁⡀⣿⣿⢟⣿⣇⠷⠉⠀⠀⠀⠀⠀⣀⣀⣛⣨⡷⡉⠎⠹⣷⣾⣀⣛⣀⡀
⠀⠀⠀⠀⠀⠹⣇⣀⠀⠉⡹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣔⡆⣧⠈⠀⠿⣿⣿⣿⡟⡀⠀⠀⠀⠀⠈⠙⠛⠓⣒⠲⣦⡀⢘⣰⣿⡷⠐⣒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠸⠛⣢⠴⣁⡀⠀⠀⢰⠋⠓⠤⡀⠀⠀⠀⠀⠀⠀⣘⢦⣄⣀⣻⣿⠟⠱⠏⠂⠀⠀⠀⠀⠀⠀⠀⢉⠞⣡⠛⡟⢻⠃⢌⢦⡉
⠀⠀⠀⠀⠐⢖⡒⠉⠁⠀⠈⠉⢓⡶⢧⣶⠤⠴⠇⠀⠀⠀⠀⠀⠀⠏⠃⠈⠉⢭⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⠊⠘⠛⡇⡌⠺⠃⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠈⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⢶⢆⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠭⠊⠀⠀⠀⠀⢩⢢⠀⡀⠀⡔⡏⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⣤
⠀⠀⢰⣿⡆⠀⠀⠀⠛⢍⠉⠑⠢⢄⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⢄⠈⣺⣁⣁⣘⠴⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣩⠔⠊⢉⡴⠋⠀⠀⠀⠉
⠀⠀⠀⠉⠀⠀⠀⢢⡆⠂⢳⠀⠀⢀⠙⣄⠀⠀⠀⠀⠀⠑⠪⠵⢤⠊⡡⢤⣤⣌⠑⢄⣠⢒⣲⠖⠊⠀⠀⠀⠀⡜⢡⡟⠁⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢘⠆⠀⢈⣇⠘⡄⠀⠀⠀⠀⠀⠤⠀⡇⢸⠁⠸⣯⣿⡆⢸⠈⣁⠀⠀⠀⠀⠀⠀⢸⠉⣻⠀⠓⠊⠀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⡤⣉⡿⠀⡇⠀⠀⠀⣀⡄⣒⠴⠸⣌⠢⣤⡿⠟⢁⠮⡄⢄⡀⠀⠀⠀⠀⠀⢸⣀⠼⡀⠀⠀⠸⡁⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠤⢸⡿⠁⡼⠀⠀⠀⠉⠉⠀⠀⠤⠂⣨⡗⠒⠒⣚⡁⠠⡈⠑⠚⠲⠄⠀⠀⠀⠀⢣⡐⠓⣤⣀⠐⠛⢓⣒⡾⡻⠋
⠀⠀⠀⠀⠀⠛⢯⣉⠀⠐⠛⠋⣁⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡡⠃⠰⠀⢳⡱⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠙⠮⣄⣉⢉⣏⣛⡬⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⡄⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠃⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠉⠀⠀⠀⠉⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠤⢄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⣄⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⡄⠀⠀⠀⡆⠀⠀⠀⠀⣾⡄⠀⢰⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣡⠑⡀⠀⡇⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠧⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⢤⡄⠀⡠⠀⠀⠘⢦⡀⠀⠂⠠⠤⠤⠤⠀⠒⠈⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠱⠃⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠑⠢⠤⠤⠤⠤⠤⠔⠊⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠑⠢⠤⣀⣀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠔⠀⠠⠃⠀⠀⠀⠆⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣦⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣤⡠⠂⠁⠀⠀⠀⢠⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⢃
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇⠈⡄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⢡
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠀⠈⡄
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠀⠀⢡⡀
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⠀⠀⠀⢢
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢃
⠀⠀⠀⠀⡀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠹⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⡇⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⡆⠀⠀⠀⠀⢣
⠀⠀⠀⠀⡇⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣆⠀⠀⠀⠀⠀⠃⠀⢠⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠋⠀⠀⠀⠇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⢠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠂⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀
⣫⣥⣤⣬⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣒⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉
⣿⣿⣿⠟⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢸⡆⣀⣤⠴⠶⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣀⣀⣀⣻⡀⠀⠀⠼⠛⠁⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢀⠜⠀⠈⠓⠦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢀⠴⠋⢀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣉⣥⣄⣀⡈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇⢠⡼⠋⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣠⣄⠈⢷⡄⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⠀⢺⠀⠰⣷⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠋⠀⣸⠃⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⢰⣿
⣴⠚⠛⠓⠶⣼⡇⠀⠀⠀⠀⠀⠘⢆⣀⢀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠢⡀⠀⢸⡏
⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠁⢀⠇⠀⠸⠷⠚⠛⠛⠳
⠈⠓⣦⡤⠤⠤⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠙⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠙⠊⠁
⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⣠⣀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠠⠤⠄⠠⣤⡴⠛
⠘⣇⠀⠀⠀⠀⠀⣀⠀⠀⠀⢧⠀⠀⠀⠘⣄⠈⣡⡀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠈⠲⠦⠤⢖⡾⠃⠀⠀⠀⠈⣦⡀⠀⠀⢇⠀⠉⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠲⠖⢳⡀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⡴⠋⠙⢦⣄⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠘⣏⠳⢤⣄⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠙⠲⠶⠚⠋⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠴⠶⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⡴⠃⠀⠀⠀⠈⠁
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⢤⣀⣤⣤⣀⡄⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣿⣭⣿⡆⠀⠀⠀⠀⣾⣿⣿⣿⣽⣷⠀⠀⠀⠀⢰⣿⣏⣿⣏⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢿⣿⣿⠃⠀⠀⠀⠀⢿⣟⡛⢛⣻⡿⠀⠀⠀⠀⠘⣿⣟⠛⢙⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⡀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣏⣿⣿⣹⣿⡀⠀⠀⠀⢰⣿⣟⣿⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢿⣿⣿⣟⣿⡿⠀⠀⠀⠀⠸⣿⠟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣿⣛⣿⡆⠀⠀⠀⠀⣾⣟⣿⣿⣻⣷⠀⠀⠀⠀⢰⣿⣛⣿⣟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣻⣿⣿⠇⠀⠀⠀⠈⣿⣿⣿⣻⣿⣿⠀⠀⠀⠀⠸⣿⡟⣛⡛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠁⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⠀⠀⠀⠀⠈⠛⠻⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣿⣛⣿⡄⠀⠀⠀⠀⣾⠛⢛⡛⠛⣷⠀⠀⠀⠀⢠⣿⣛⣿⣏⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠸⣿⣿⣛⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠉⠉⠉⠉⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠐⠀⠀⠀⠀⠂⠤⣘⠀⠀⠀⠀⠠⢒⣀⣂⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣄⣠⣤⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠋⠀⠈⠉⠛⠛⠛⠛⠛⠻⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⣿⠟⠛⠉⠛⠛⠛⠷⠀⠀⠀⠀⠀⠀⠐⠿⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⣀⣴⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⡏⠀⠈⠳⠌⠿⠇⠀⠒⠀⡇⠀⠀⠀⠖⠀⠻⠟⠠⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⢠⢆⠀⠀⠀⠈⠄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡈⠀⠀⠀⡴⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡁⠳⡄⠀⠈⠢⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠀⠈⠁⠈⠠⡀⠀⠈⢂⠀⠀⠑⢠⠆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠆⠀⢠⠂⠀⠀⠀⣠⣤⣤⣦⣤⣄⡀⠀⠀⢢⠀⠀⠂⢸⠃⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢰⡏⠀⠔⢂⠸⣿⡿⠿⠿⠿⣿⣿⠓⠄⠘⣷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠈⠛⠟⠛⠛⠋⠁⠀⠀⠀⠀⡇⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠐⠀⠀⢀⠀⠀⠀⠀⠀⠃⣼⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⡆⠃⠀⠀⠀⢶⠀⠀⠠⠄⢀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠋⠀⠀⠀⠀⠀⠀⠉⠀⢠⠑⠠⠀⠀⠀⠀⢀⡀⠀⠰⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢹⡏⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⣿⠖⠂
⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣷⣦⣴⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠙⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠛⠛⠛⠛⠉⠛⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠟⣿⣀⣀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣏
⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠄
⣿⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠊⠀⠀⠀⠄
⠉⠁⣼⣧⣴⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡍⠲⠤⠶⠶⢴⠀⢠
⠀⠀⠀⠀⠻⣿⠋⠁⠀⠐⠒⠒⠂⠀⠀⠈⠉⠁⠀⠀⠀⠀⠠⡌⠊⣡⣄⠀⠀⡜⠀⠀⢆⡀
⠀⠀⠀⠀⠀⣿⡆⣀⣀⣠⠶⠶⠒⣠⢄⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠙⠋⣀⠞⠁⠀⠀⠀⠐⠃⣐⠂⡄⠀⠀⢠
⠀⠀⠀⠀⠀⢸⡇⡙⢧⠀⠀⠀⣠⣄⠈⢳⡄⠀⠀⠀⠀⠀⠻⠀⠤⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠘⣷⢃⠀⠳⣄⠀⠻⠟⠀⠀⢻⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢹⡞⢠⠀⠀⠀⠐⠂⠒⠒⠉⠐⠋⠀⠀⣀⣠⠔⣀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠉⠱⠆⠈⡆⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⢀⠤⠋⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⢀⡀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠘⠻⠶⢤⣤⠶⢶⣄⣀⣴⣤⣤⣤⠖⠊⠉⠑⠻⣇⢀⣶⣴⣶⣶⣤⣤⠶⣦⡀⣀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⣿⠀⠀⠈⠛⠙⢷⠞⠳⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠊⠀⠀⠀⠀⠀⠀⠡⡀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⡀⠀⠀⠀⠀⢀⡠⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡛⠒⠀⠠⠀⠒⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠈⠉⠒⠒⠒⠒⠒⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⠄⣤⣤⡄⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿⠈⣇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢹⡆⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠘⣧⢸⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣿⢸⡇⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⣿⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢻⣾⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢸⣿⡇⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠐⣿⠋⣿⢻⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠉⠀⣿⣶⠏⠉⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠶⠶⠖⠛⠋⠘⢷⣶⣦⣤⣤⣿⡆
⣿⣿⣿⡿⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣠⠖⠚⠉⠛⠲⣄⠀⠀⢀⠴⠚⠋⠙⠒⢤⡀⠀⠀⣠⠖⠋⠉⠙⠲⢄⠀⠀⠀⣠⠖⠚⠉⠛⠲⣄⠀⠀⢀⡴⠚⠋⠙⠒⢦⡀
⠀⣼⠁⢀⡀⠀⢀⡀⠈⢧⢰⠋⠀⣀⠀⠀⣀⠀⢱⡀⡼⠁⢀⡀⠀⢀⡀⠈⢧⠀⣸⠁⢀⡀⠀⢀⡀⠈⢧⢠⠏⠀⣀⠀⠀⣀⠀⠹⡄
⠀⣇⠀⠈⠁⠀⠈⠁⠀⢸⢺⠀⠀⠁⠀⠀⠉⠀⢸⡇⣇⠀⠈⠁⠀⠈⠁⠀⣸⠀⣧⠀⠈⠁⠀⠈⠁⠀⢸⢸⠀⠀⠉⠀⠀⠉⠀⢀⡇
⠀⠘⢆⡀⠓⠤⠚⠀⣠⠋⠈⢧⡀⠈⠒⠂⠁⣠⠞⠀⠘⢆⡀⠉⠉⠉⠀⣰⠋⠀⠘⢦⡀⠈⠀⠈⠀⣠⠏⠈⢣⡀⠘⠁⠈⠂⢀⠞
⠀⠀⠀⠙⠒⠒⠒⠋⠀⠀⠀⠀⠉⠓⠒⠒⠋⠁⠀⠀⠀⠀⠉⠓⠒⠒⠉⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋⠁⠀⠀⠀⠈⠓⠒⠒⠚⠁





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣝⣫⣍⣹⣝⣽⣛⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣠⡤⠤⠤⠤⡀
⠀⠀⠐⠊⠁⢀⡀⠀⠀⠀⠀⠀⠉⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⡞⠉⠉⠳⡄⠀⠀⠀⠀⡾⠁⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠛⠛⠓⠦⢄⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠶⠚⠋
⠀⠀⠀⣾⠀⠀⣲⣦⢹⡀⠀⠀⢸⡇⡀⣸⣷⡌⡇⠀⠀⠀⠀⠀⠀⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⣶⣤⡀
⠀⠀⠀⣯⠸⣾⣿⠝⢸⠃⠀⠀⢸⡇⠻⠿⠗⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠛⠳⢦⡄⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠹⣄⠀⠀⢠⡟⠀⠀⠀⠀⢳⣄⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⠀⠹⣦⠀⣰⡿⠁⠀⠀⠠⣄⠀⠀⠘⣿
⠀⠀⠀⠀⠈⠓⠚⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣻⣿⣆⢹⣶⡿⢁⡄⠀⠀⢀⣿⣿⡄⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⣦⣶⣿⣧⡟⢸⣿⡇⠸⣿⣶⣶⣿⠿⢿⡿⢠⣿
⠀⠀⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠈⠛⠛⠋⠀⢸⣿⠀⠀⠹⢿⣿⣿⣶⠞⠁⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⣠⡾⣿⡇⠀⠀⠀⠀⠁⠀⢀⣼⡟
⠀⠀⠀⠀⢀⡞⠋⠙⢳⡀⠈⠀⠀⠀⠀⢀⡶⠋⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠸⣿⣄⠀⠀⠀⢀⣠⣾⠋
⠀⠀⠀⠀⡾⠀⠀⣶⣄⢷⠀⠀⠀⠀⠀⣾⠀⠀⣶⣄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠾⠟⠋
⠀⠀⠀⠀⣇⠲⣾⣯⡝⣸⠀⠀⠀⠀⠀⣧⠰⣶⣯⡝⢸
⠀⠀⠀⠀⠹⠤⠤⠤⠤⠟⠀⠀⠀⠀⠀⠹⠦⠤⠤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠛⠷⣄⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢀⣸⡏⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠈⠓⠒⣛⡉⠁⠀⠀⠀⠀⠈⠉⠛⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⡿⣿⣾⡿⡟⠋⠁⠀⠀⠀⠀⠉⠛⢻⢿⠶⣶⣶⣶⡟⠋⠁
⠀⠀⠀⠀⣠⠞⠉⠉⢳⡀⠀⠀⠀⢠⡖⠋⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⠿⠗⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠿⠿⠞⢸⡇
⠀⠀⠀⢀⡏⢀⣨⡷⡄⣷⠀⠀⠀⡟⢀⢀⣶⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣠⡞
⠀⠀⠀⠘⡇⢈⣛⣋⡀⣿⠀⠀⠀⡇⠈⠻⠛⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉
⠀⠀⠀⠀⠙⠉⠁⠈⠉⠃⠀⠀⠀⠛⠋⠉⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡜⠁⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⠼⠁⠀⢀⡴⠚⠉⠃⠀⠀⠀⠀⠀⠹⠤⣀
⠀⠀⠀⠀⠀⠀⢀⡔⠉⠁⠈⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠈⡇
⠀⠀⠀⡰⠚⠉⠉⠀⠀⠀⠀⠀⠀⠷⢤⡀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠿⠀⠀⢠⡄⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⣠⠇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣀⡀⠀⠉⣀⠤⠤⠤⣀⠉⠀⢀⣠⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⠉⠛⢀⠊⠀⠀⠀⠀⠀⠑⡄⠙⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠤⠤⠀⠀⠀⠋⠃⠀⠘⠙⠀⠁⠀⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⡀⠐⡀⠀⠐⠤⠃⠀⢠⠇
⠀⠀⠀⠀⠀⢀⣠⣤⠎⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠚⠋⠀⠈⠒⠠⠤⠄⠒⠁⠀⠘⠳⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⢤⡀
⠀⠀⠀⠀⡜⠁⠀⠀⠀⡀⠀⠀⠀⡀⠉⠓⢦⠀⠀⠀⠀⢠⡟⠀⠀⣦⠀⠀⢻⠆⠀⠀⠀⠀⠀⣀⡤⠤⠏⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⡇⠀⠀⠀⠈⠈⢠⢤⠀⠈⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢
⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⣀⣀⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄⠀⡠⢢⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠋⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⠳⠤⢤⢤⠤⠤⠤⠤⠤⠤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠒⡡⢢⠑⠊⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠀⠀⡀⠀⣀⠀⠓⠲⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⣸⠄⠈⠒⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣀⣀⣀⣈⣁⣀⣈⡱⠟⠉⠉⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡔⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠸⠛⠛⠙⠓⠀⢀⣤⢀⡾⣁⣴⣤⣢⣦⣮⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠏
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⢀⡤⠬⢿⡙⠑⠖⠹⠜⢵⡄⢏⣚⣳⡼⠿⡏⢈⡀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠾⢝⣇⠴⣀⣴⣶⣿⣿⠿⣿⣿⣶⣿⣄⡻⢑⣩⣿⡅⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⣮⣯⢉⣼⣿⠷⠟⡝⠚⣤⡟⣹⢌⣩⡻⣿⣦⡙⠉⣹⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢃⣫⠆⣼⣻⣇⠘⢋⣘⣯⣧⣯⣾⡤⢛⣠⣽⣿⣷⠈⢋⣯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⢟⡿⢾⣿⣏⣘⣈⡎⢡⡈⣀⣁⣈⣁⣀⣰⣈⣿⣿⡉⢭⠌⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡛⠱⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢼⣿⣿⣿⣿⢿⣿⣹⡇⣿⢻⣿⠻⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⣿⣿⣿⣿⡏⣿⣿⡇⣿⢾⡟⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣇⣿⢹⡇⣿⠸⣏⢿⡿⣿⣿⣿⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿⣿⡷⣿⢸⡇⣿⠀⣿⣾⣿⣿⣿⣿⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠓⠀⠀⢸⣿⣿⢿⣿⣯⢻⡎⣷⣼⣷⣿⡟⣿⣿⣿⣿⡇⠀⠀⠀⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢸⠄⢸⠀⣿⠇⢸⡅⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⢸⠀⢿⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠂⠀⠀⠀⠀⠀⠁⠃⠘⠂⠘⠂⠘⠈⠀⠓⠀⠃⠀⠀⠀⢰⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⢰⡆⣶⠆⢰⢲⡀⣆⠀⡆⢰⠀⠀⠀⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⣸⡇⣿⠀⢸⠀⡇⡇⠀⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⠠⠸⠇⠹⠀⠸⠄⠁⠷⠂⠇⠓