Flu Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⠞⣀⢤⡴⠦⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠊⢠⡿⠊⠐⠋⠉⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⣐⣛⢪⡃⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠃⠀⠀⢀⣤⡔⡖⠒⠛⠆⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⣔⡊⠁⠸⣤⠤⢄⡀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡩⠋⠀⠀⠀⠀⠀⠀⠺⡆⠀⢿⠀⠀⠙⣦⠊⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⠄⠀⠀⠀⠀⠀⠀⢀⡱⠀⠘⢦⡴⠊⠉⣯⣭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⢰⣄⢀⠀⠀⠀⡰⣣⡇⠀⠀⠀⠉⣒⡶⠃⠠⣉⠉⠉⠉⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠛⠓⠋⠉⠓⠚⢱⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡐⠂⠤⠤⠶⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠕⣄⢠⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠹⡭⢴⠊⠉⠀⠙⢄⠀⠀⠐⠊⢁⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢣⠀⡇⢸⠀⠀⠀⠀⠀⠙⠒⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⠀⠀⠈⡇⢱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⡔⢡⠋⠀⠀⢸⠈⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠎⠀⢧⡀⢀⣀⡜⢱⠯⠹⡤⢄⡜⠱⡤⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠴⢇⡀⠀⠈⠉⡏⣞⠃⠉⠉⢉⢦⡋⢠⡤⠗⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⢀⣉⢵⡦⢤⢡⢹⣇⣀⡠⣜⠸⣙⣏⡀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⣢⡀⠇⢀⠬⠀⠁⡎⠉⠸⠬⠝⡼⡉⣷⣈⠗⠚⣥⡍⢦⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠔⢳⢸⠀⠋⡳⠖⠁⢀⡆⠀⢰⢡⢁⢩⠀⠀⠀⠉⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡛⢁⡴⠯⣿⠦⢄⡉⠉⠉⠀⠀⠀⠧⡼⢸⢸
⣤⢲⢷⡾⣣⠀⢀⣀⣀⠀⡀⢀⠔⠁⢀⡨⠋⠀⠀⠀⠀⠀⠈⠉⠒⣲⡶⠒⠒⠙⠋⠁
⠺⠤⠄⠠⠜⠀⣾⢄⠁⠼⢣⠘⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⣎⠤⠤⠄⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠁⠐⠉⠘⠛⠣⣄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠴⠆⠀⠀⠠⠦⠀⠀⠀⠀⢈⠆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⣀⡀⠤⠀⠒⠒⠒⠄⠀⠀⠀⠀⠀⠰⠀⢀⠀⣀⣠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠉⠀⠀⡀⠂⠀⠀⠰⢂⠀⠀⢀⠔⠁⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠀⠀⠉⠹⣀⣀⠀⠀⢀⣴⠣⠄⠳⢄⠁⠈⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡜⠖⠉⠈⠀⠀⠘⠜⠀⠀⠀⠒⠉⠊⠈⠢⢒⠒⠊⠉⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢬⠬⠀⠀⠀⠀⠀⢀⠃⠇⠀⢰⠀⠀⠀⠀⣠⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡒⠉⠁⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠵⠱⠤⠄⡠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠊⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⠘⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢄⡒⠲⢶⠤⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠁⠀⠀⠀⠈⠢⠀⠣⠈⠳⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠺⠏⠀⠸⠟⠉⠀⠀⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠐⡂⢂⣀⣑⠒⠂⠀⠀⣇⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡴⠍⡨⠁⠁⠀⠈⠉⠁⠀⠀⡜⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⡨⠔⠙⠦⣄⣀⣀⣀⣀⡠⠔⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠮⢋⠔⠈⠀⠀⠀⠀⠀⢈⠉⠁⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠀⠀⠀⠀⠀⠀⢀⠀⠄⠺⠢⣄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠠⡄⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⠀⡆⠀⠈⠑⢦⡀⠀⠀⠀⢸⠀⠀⢀⠀⠈⠀⠆⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠉⠢⣄⠀⢸⠀⠀⣀⢒⣀⣀⣰⢻⡄
⠀⠀⠀⠀⠀⣀⣀⣄⡈⠤⣤⣤⣀⣀⣀⣀⣂⢶⠒⠾⡀⣀⣀⣀⣀⣀⣈⡳⣼⢀⢌⡇⠉⠁⠀⠉⢨⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡹⠀⠀⣤⠤⠀⠀⠀⠀⠈⠢⢇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢗⣒⢀⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠤⠀⠠⢵⢖⣀⣐⡋⠐⠃⠀⠀⠀⣀⠤⣄⠀⠀⡠⢆⠒
⠀⠀⠀⠀⠀⡄⣀⡀⠈⡎⠀⠀⢡⠀⠀⠀⠀⠀⠋⠁⠁⠀⠀⠁⢀⢀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠏⠉⠁⠘⠀⢄⣒⠢⠀⠀⠀⠀⠀⠀⠐⠖⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣒⡂⠌⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡠⢲⣄⣴⣭⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡾⠋⠉⡜⠀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢸⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠘⠀⠀⣀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠐⠉⢁⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡈⠛⠁⠀⢀⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⠴⠒⠛⢦⣀⠤⠶⣤⡠⠤⠶⠦⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠴⠤⠀⡄⡰⢀⣀⣀⣠⣤⣤⣴⣏⢀⡠⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⢊⠀⡈⠀⠀⢀⠀⠀⠀⠀⠀⠈⣉⡹⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢠⡀⠀⠑⠦⡐⠘⢄⠈⠉⠁⠀⠀⣏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠙⡆⠢⡠⠣⠤⠂⠅⢀⣀⡠⠔⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⢹⡉⡔⠙⠻⠋⠸⠉⠉⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢰⢧⣤⣍⣐⢒⣂⣒⠦⢽⡆⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢴⡶⠒⠲⣄⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⠆⢀⡤⢊⣀⡠⠤⠔⠒⠚⣏⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡽⠲⠤⣄⡈⠳⡄⠀⠀⠀⠀⠀⡷⠶⠒⣧
⠈⢸⡶⠞⠓⠀⡡⠛⠒⠂⠳⡉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢻⠒⠚⠀⠀⠀⠀⠀⠀⠀⣧⢞⡻⠚⠋⠀⠉⠁⠀⠉⠛⠻⢦⠉⠉⠁⢸⠃⠀⠀⡟⠈⠉⠉
⠀⢸⣭⣩⡅⠀⣷⡶⠶⠖⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠤⠴⡇
⠀⢼⡀⣸⠀⢸⣏⣀⣀⡈⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣄⡀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⡀⠠⡄⠄⢀⡀⢀⠀⠀⠀⠈⠢⠤⠄⠃
⠀⠀⠉⠉⠀⠈⠓⠢⠬⠄⠂⠀⠀⣀⣤⣤⡆⠀⠙⠋⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡠⢄⠶⡖⠾⠛⠋⠉⠀⠀⠀⠦⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣶⣶⣶⢶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⢻⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⠇⠀⢘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠿⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠉⠀⠀⠞⠲⢤⣀⠀⠀⠀⣀⡠⠤⠒⡶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡴⠟⠋⠁⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠐⣶⣤⣀⠀⠈⠓⢶⣉⣥⣤⣄⣸⡷⠖⠛⠉⠉⠑⠲⢄⡀⣀⣠⡟⠀⢀⡴⠿⢿⣿⣄⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⣠⠞⠋⢀⡽⢳⣤⠶⠾⠅⠀⣴⣤⡞⠖⠀⠀⠀⠠⠞⠀⠀⠹⣍⠉⠛⠷⣏⠀⠀⠀⠈⠙⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢀⡴⢋⣴⠟⡁⠀⠀⠀⠀⠉⠀⡞⠉⠳⡄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢶⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⡾⢃⣼⠏⢀⡤⢤⣀⠀⠀⢧⠠⢀⡇⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⢈⡧⠀⠀⠀⠀⠀⠀⠙⢦⣰⠋⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡜⠁⠀⠘⠇⠀⢸⠀⢀⡈⣧⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠛⠛⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣰⠀⣏⠀⠀⠀⠀⠈⠳⣬⣁⡾⠀⠀⢀⠀⠀⠀⠀⢀⠀⠋⢸⡀⢀⡴⠋⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⡼⠁⠀⠀⠀⣠⠞⠁
⠀⣠⠶⢶⡄⠀⠀⠀⢠⡟⠀⣰⢿⠀⢈⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠷⠤⢊⡿⠋⠀⠀⠀⠀⠀⣠⡾⠻⢾⣷⡀⢠⡞⠁⠀⣾⣧⠀⣸⠇
⢸⡃⠀⠀⠙⠲⣤⠀⡾⢀⣀⣏⡤⠞⠉⠈⠁⠀⢰⡠⠀⠀⠀⠀⠀⠀⢀⠠⣼⡟⠓⠋⢀⡴⠋⠀⠀⠀⠀⠀⣠⡾⠋⠰⡆⠀⠈⠻⢏⠀⢀⠞⠁⢿⠙⠉
⠀⣻⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠚⠛⠛⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠈⣆⠁⠀⠀⠸⡇
⣸⠃⠀⠀⠀⣹⠓⣶⠶⢤⡴⠊⠁⠀⠈⠁⠀⠀⠀⠀⠘⣆⣠⠶⠶⠚⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⣾⠿⠾⠷⠶⠶⠶⠶⠶⣄⡀⠀⢹⣄⠀⠀⠀⣿
⠘⠲⠶⠶⠞⠋⠀⣿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⡾⠋⢀⣠⠤⠶⠀⠀⠠⣀⡀⠈⣷⠀⡿⠙⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⠶⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⣧⢀⡴⠋⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠁⠀⢀⣠⣄⣀⠀⠈⠁⣰⠏⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣴⠃⠀⢀⠀⠀⠀⠙⠁⠀⠀⠀⠃⠀⢀⡽⠋⠀⠀⠀⠀⠀⣠⡾⡿⠁⢀⣴⡶⠲⠶⢿⠝⠉⠉⠛⠶⠞⠁⠀⠀⠀⠀⣸⠀⠀⣿
⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⣠⢾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⣠⡾⠋⠀⡇⠰⣾⣹⡄⡏⢳⠃⠀⢰⠃⠀⠀⠀⠞⠀⠀⠀⢠⠿⣄⣰⡏
⠀⠀⠀⢀⡴⠋⠀⠀⢠⡞⠁⠈⢧⡀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⣀⡶⠋⠀⢠⣀⣿⠲⣧⠽⠓⠚⠁⠀⠀⢸⠀⢦⠀⠀⠀⠀⢀⣴⠋⠀⠈⠻⣧
⠀⠀⣠⣾⠀⠀⠀⢠⡟⣧⠀⠀⠀⠉⠳⠦⣤⣴⠟⠁⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠈⣯⢻⡿⠁⠀⠀⠀⠀⠀⠴⠚⢀⠇⠀⣠⣤⠶⠋⠈⠻⣦⡀⠀⠈⣧⠖⠛⣦
⢀⠞⠁⠀⠀⠀⠐⠚⠦⣜⣧⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⢀⡴⠋⠀⢧⡀⠀⢰⠗⠚⠁⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣸⢷⠀⠀⠀⠀⠺⢧
⢠⠆⠀⠀⠀⣀⣄⡀⠀⢸⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠠⠴⠚⠀⡰⠀⠀⢀⣠⠾⠁⠀⠀⠀⢀⣼⠋⣨⠟⠃⠀⠀⠀⣠
⢇⠀⠀⠀⠀⢁⣼⠛⠲⠋⠀⠈⠻⣄⠀⠀⠀⠀⢀⣤⢿⡀⠀⠀⠐⠾⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⣠⡞⠁⠰⣇⢀⣤⣤⡴⠛⠁
⠈⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⢠⠏⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢸⡇⠀⠀⢀⣤⠶⠚⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⢄⣨⠿⠋⠀⣠⠎⠉⠙⠢⠤⣀⣠⣤⠄⠀⠀⠈⡄⠘⣇⡤⠐⠋⠀⠀⠀⠀⠀⣀⣴⠞⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠈⠁⠀⣄⡴⠻⢦⣤⣀⠀⠀⠀⠀⠈⠙⠒⠒⠚⡇⠀⢹⣤⠶⠶⡄⣀⣤⡴⠞⢻⡄⢀⣤⠘⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⢻⡄⠀⠀⠈⠉⠛⠳⠶⢦⣤⣤⣤⣤⣤⣷⡶⠏⠀⠀⣴⠛⠉⠁⠀⠀⢀⣿⠊⠀⠀⠈⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢈⡗⠀⠀⠀⠀⡞⠀⢠⡄⠀⠀⠘⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠈⠛⢿⣀⣀⣠⠼⠃
⣶⣶⡶⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢩⢏⠴⣲⣞⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠰⠟⠁⠈⠁⠀⠀⣙⣒⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢤⠭⠥⢬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠠⢴⣮⢻⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣋⠀⠀⠀⠀⠀⠲⢁⠀⠈⢧⠖⠒⢄⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡵⠁⠀⠀⠀⠀⠀⠀⠐⢷⠀⠸⡄⠀⠀⣱⣎⢀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⠂⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠹⣤⠊⠀⢸⠭⣜⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠰⣤⢀⠀⠀⠀⡠⢫⣧⠀⠀⠀⠀⢙⡳⠟⠀⠠⣀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠛⠉⠉⠉⠑⠚⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠀⠒⠀⠀⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡋⡝⣂⡠⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠈⣏⠤⠒⠉⠀⠈⢢⡀⠀⠀⠂⠉⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⢣⠀⢁⠈⠀⠀⠀⠀⠀⠈⠒⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠇⢸⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⡔⢡⠋⠀⠀⠸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⠞⠀⣆⠀⠀⠀⡰⢣⣿⠙⣆⠀⣠⠒⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⢠⡋⠀⠀⠈⠉⡞⡵⠇⠘⠓⠒⢦⠘⠁⣀⣸⠈⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠈⠓⣢⣤⣀⠀⢻⡇⠀⠀⣨⠇⡏⣶⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠇⠀⣊⠉⠋⡜⠓⠈⣀⣿⢹⠚⢳⡈⡦⠹⣏⡓⢢⡘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡹⡄⠄⠸⠴⢄⡤⠃⠀⡆⠀⠀⠆⠇⠈⠉⠁⠀⠘⠋⠠⠗⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠗⠋⣠⣳⣦⣀⠈⠢⠄⠀⠒⠁⠀⢘⣠⢰⢆
⢀⢠⣄⣤⢄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠓⢲⠎⠀⠉⠉⠈⠙⠒⠤⠤⣀⣀⣀⣀⣰⡾⠸
⢱⠈⣙⡋⢱⠄⢠⡂⠒⢈⣆⡀⢏⡀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⡛⠁
⠈⠉⠀⠀⠁⠀⠚⠁⠢⠴⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣭⣤⡛⠉⠉⠁⠀⠒⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⡀⡴⠙⠁⠙⣨⠀⠀⣺⠒⠚⠛⠳⠶⣆⠀⢈⣒⣁⠈⠀⠐⠂⣡⣌⡁⢊⠀⠀⠦⢡⠢⡄⡂⠀⠀⠀⠀⣠⠠⠗⢆⠀⢄
⠀⠀⢆⡀⠀⠀⡬⠐⠀⢿⠀⠐⠑⠀⠀⡇⠀⠸⠭⠼⠀⠘⠀⠀⠣⡤⠃⠈⣀⠀⠘⡈⠩⠁⠄⠀⠀⠀⠀⡐⠀⢠⠏⡔⣀⠤
⠀⠀⠈⠋⠘⠓⠀⠀⠀⠸⣄⡈⠂⢀⡼⠃⡀⠀⠐⠂⡀⠀⠀⠁⠠⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⡁⠅⠀⠀⠈⠚⠈⠂
⠀⠠⠀⠀⠀⢐⡀⠈⠁⡀⠈⠛⠞⠋⠁⠀⠀⡤⠀⠄⠠⠀⠀⠀⣀⠀⡀⠐⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠠⢢⠀⠀⠀⠃⠘⠀⠀⠀⠲⠀⠀⠁⠰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠠⠄
⠀⠀⠀⠀⡠⠀⡄⠀⠘⠂⠄⢀⣀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠄⠀⠂⠈⠐⠁
⠀⠔⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⣠⣄⠁⡄⢠⣤⡀⠠⠀⠀⠘⠀⠀⢙⡀⠀⠀⠀⠃⠀⠀⠑⠀⠀⠈
⠀⣜⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣦⡀⠀⢹⣿⣄⠀⣸⣿⡀⠀⣰⣿⠆⠀⠀⠉⠓⠀⠀⠊⠀⠀⢰⣤⣤⣤⣰
⠀⢠⢬⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠐⢆⣹⡟⠓⢛⡟⢻⣿⡟⢻⡟⢿⣿⠁⠀⠃⠀⠀⠀⠈
⠀⠀⠀⠈⠁⢀⢐⡀⠀⠐⠀⢀⠔⢀⠂⠘⢛⠻⣿⡇⢛⣛⡇⣼⣿⡇⢸⡇⢸⣿⡿⠿⠿⠀⠀⠀⠀⠀⠿⢿⠀⠀⠀⠄
⠀⠀⠀⠀⠎⠁⠀⠀⠠⠀⠀⠑⠒⠁⢀⣤⣴⣾⣿⣇⣹⣿⣇⣉⣩⣧⣘⣁⣾⣿⣄⣠⡘⠁⠀⠀⠀⠌⠀⠀⡀
⠀⠀⠀⠀⠀⠤⠔⠢⠀⠠⠃⠀⡌⠀⠈⠩⠄⠀⣹⣿⢹⠛⡿⢹⣏⢹⣿⣿⡟⠉⢉⠙⠁
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠠⠀⠀⣶⡿⠛⢻⣿⡿⣿⣿⠟⢿⣿⠛⠿⣶⣦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢆⣀
⠀⠄⠀⠁⠀⠀⢨⠀⠀⡀⠀⠀⢀⠤⠒⠒⠈⠀⠀⠸⣿⠄⠘⠿⠠⠀⣿⣦⠀⠉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡄
⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠐⠒⠋⠲⠁⢀⣈⡉⠀⢡⣤⠸⡆⠀⠀⠈⣷
⠀⠂⠀⠀⡃⠀⠃⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⢲⠀⠀⠂⠀⠀⠉⠁⠙⠁⢀⠟⢟⠷⢷⣤⣁⠀⠛⠶⠖⠊⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⢀⠀⠣⡀⠀⠐⠀⢀⠀⠠⠀⠰⠉⡇⢀⠰⠀⠀⠀⡠⠐⠀⠀⠢⣀⠐⠠⠀⠈⠑⡀⠈⠁⠘
⠀⠰⠤⠤⠆⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⢀⡤⠳⠤⠤⠔⠈⠀⠂⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⠒⠀⠀⢰⡤⡄
⠀⠠⠀⠀⡀⡀⠀⠠⠀⠀⠠⠀⠀⠢⠄⠂⠀⠚⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠆⠀⠀⠀⠀⢀
⠀⣀⠀⡄⠀⢠⡀⠀⠀⡀⠀⠀⠀⡀⠀⠁⡇⠀⠀⠀⣰⠑⠈⠘⠂⡀⠀⢰⣦⣤⣤⢼⡂⠀⠀⠀⠀⠠⠁⡜⠀⡀
⠀⢀⠀⠄⠀⠀⠀⠂⠐⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠐⣄⠀⠀⠀⠀⡃⠀⠀⠀⠀⠠⠀⠀⠀⡀⠀⠠⠀⠌⠀⠒⠀⠀⠀⠀⠀⠠
⠀⠑⢕⠠⠠⡰⠀⠀⠀⠐⠀⠀⢤⡤⠦⠀⠀⠀⠀⠀⠐⠶⠠⠤⠘⠀⠀⢀⠠⡂⢟⠄⠀⠀⠙⠢⠤⠊⠀⡀⢀⠐⣅⡁⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠴⠀⠀⠀⠀⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠶⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠤⠴⠒⠋⠙⢢
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠲⣄⣀⡴⠋⠀⠀⠀⠀⠀⠀⠉⣿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠋⠀⠀⣀⣈⠁
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡷⢄⡀⠀⠀⢀⣀⡼⠀⠀⠀⠀⠀⠀⠉⠁⠀⠓⠲
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠙⠲⠄⢀⡀⠀⠃⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠓⠀⠀⠢⣄⠀⠀⢀⡉⠒⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢠⡀⠁⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠂⠀⠐⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠃⠀⡀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡰⠄⠘⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡴⢲⣌⣱⠒⣰⡇⠉⠠⠀⠄⠮⡔⠛⠳⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠔⠝⡧⠀⠀⠠⣀⠆⠀⠄⠈⠀⢈⡆⠀⠈⠢⡠⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡄⠀⠀⠀⠠⠀⡀⠀⢀⡀⣄⡴⠋⠀⣠⣄⠀⡘⢀⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠂⠤⠇⠬⠀⠀⠀⠀⠀⢀⣀⡖⢘⡴⠋⠀⢠⠞⠉⠙⢊⡠⠹⡞
⠀⠀⠀⠀⠀⠀⠀⠼⣀⡸⢲⠢⠀⠉⠀⠀⠐⢤⠚⢁⡴⠋⠀⣠⠖⠓⠒⠒⠢⡀⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡐⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⡼⢁⣀⡤⠤⢌⠔⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⡀⠲⡀⠀⢀⡴⠋⠀⢀⠜⢁⣦⡿⠤⠀⠀⠀⠈⢀⡔⠲⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠰⢳⡀⠈⠑⠋⠀⢀⠔⢃⢀⠜⠋⠀⠀⠀⠁⢀⡖⠁⢈⣆⠈⠡⡄
⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⠖⠀⠙⢄⠀⢀⠴⠁⠠⠀⠀⠀⢀⠀⠀⠀⠰⠋⠀⣠⠊⢎⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⢂⣄⠑⠀⠠⢄⣀⡀⡄⡈⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢃⠈⠙⠒⠲⠦⠤⡖⠁⢸⠋⠁⡸⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⠀⠀⠈⠢⠔⠋⠀⠀⠀⠣⠔⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⢤⠀⠀⠀⠈⠀⠀⢰⠀⠀⠀⠁⠀⠠⠆⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠐⠄⠀⠀⠀⢀⣨⡀⠀⠀⠀⠀⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠘⣗⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⢀⣄⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢀⣴⡄⠀⡠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠁⠙
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⠤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⡂⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠄⠀⠀⠠⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠀⠈⠀⢄⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠒⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠶⠖⠒⠒⠒⠛⠛⠛⠉⠉⠛⠛⠛⠓⠒⠒⠶⠤⢄⡀⢠⠞⠙⠲⣄
⠀⠀⠀⠀⠀⠀⢠⠏⠁⠉⢷⣠⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⣀⠤⠒⡶⠀⢀⣀⡤⠶⠒⠛⠉⠉⠉⠉⣑⣲⠦⠈⠉⠀⣀⣀⡸⠇
⠀⠀⠀⠀⠀⠀⠸⠤⣀⠀⠈⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⡴⠋⠁⠀⣼⠤⠚⠉⠀⠀⠀⠀⠀⢀⣤⣖⡋⠭⠄⣀⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⡀⠀⠀⠀⠀⢠⠋⠰⢃⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢀⠏⡘⠁⡇⠀⠀⡀⠀⡆⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠄⠀⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡈⠰⠿⠿⢧⠀⠀⡁⠠⠁⠀⠀⠀⠀⠀⡠⠴⠒⠋⠉⠉⠉⠉⠁⠊⠁⠀⠀⠢⡀⠀⠁⠄⠀⠁⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣄⣤⣤⣼⠀⠀⣘⡄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡠⠔⠒⠒⢮⡓⠀⠰⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⠙⢻⡆⡞⢁⢤⠀⠀⠀⠀⠠⡓⠀⠀⠀⠀⢠⠞⠁⠀⢀⠎⠀⠀⠀⠀⠀⠹⡄⠀⠱⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡀⣿⣶⣾⣿⣿⡇⠙⠁⡏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⠂⠀⠢⡎⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠇⣅⣠⣤⣤⣼⡇⠀⠘⠷⠒⠒⣚⢳⡀⢀⡇⠀⠀⠀⠀⠀⠀⣤⠚⡖⢤⣤⣀⡀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⡇⠀⢠⠖⠉⣉⣄⣈⡉⠉⠀⠀⠀⠀⠅⠀⠀⠈⣰⡇⠉⠁⠀⠈⢉⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⣦⣤⣶⣶⣶⡇⠀⣿⠀⠈⠙⠒⠚⠉⢀⠀⠀⠀⠀⠀⠀⠀⡾⠏⠙⠿⠶⠶⠚⣋⣀⡀⠀⠀⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡄⡏⠉⠉⠉⢉⡇⠀⠘⠳⠦⠤⠶⠒⠛⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⡀⠉⠀⠀⣹⠀⢀⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡁⣿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠉⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢳⣡⣤⣤⣤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠘⢀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠂⠦⠄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠸⣸⡏⢉⣉⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⠲⢄⡀⣀⣀⣤⡄⠀⠀⠒⠈⠁⠈⠒⠁⠀⠀⠀⠀⠐⠿⣿⣷⡖⠢⡄
⠀⠀⠀⠀⠀⠀⢣⢻⡿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠳⣄⠔⠀⢻⡠⠊⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠋⢳⣶⣿⠁⠀⠀⠀⠀⢀⡴⢢⠞⡿⢲⡆⣀⠀⠀⠘⠛⠀⡄⢣⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⣀⡤⠂⠉⠉⠈⠉⠉⢭⠉⠉⠓⠘⠢⠓⠊⠀⠁⠉⠉⠉⠉⠙⠛⠛⢆⣸⠀⡏⢳⣄⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⡀⠔⠋⠁⠀⠰⣖⠒⠀⡔⠁⢸⣤⠄⠤⠐⣦⡤⠤⠤⠤⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢆
⠀⠐⠉⠀⠀⠀⠰⠚⠉⠉⠣⠀⡀⠀⠰⠊⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠑⠒⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡉⡆⠀⠀⠀⠀⠀⠚⠃⢂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢴⡥⠤⢀⣀⠉⠀⠒⠒⠚⠋⡠⠒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⡁⠒⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠉⠁⠉⢐⠉⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⣤⣀⣈⠤⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠛⠛⠋⠉⠉⠛⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠱⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣠⠴⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠻⠒⠉⠷⣤⣀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⢀⡞⠁⠒⢤⣀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠒⠋⠉⠀⠀⠀⣰⡷⠀⠀⠀⡎⠀⠀⠀⠀⠈⢳⡀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢄⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⣠⡆⢸⠁⠀⠀⠀⠀⠀⠀⠹⡄⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡤⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠇⠀⠀⠀⠀⠀⠰⠋⢀⡜⠁⠈⠻⡀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢱⢀⣚⣷⠀⠀⠀⠀⢸⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⣾⡀⠛⠋⠀⠀⠀⢀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⢄⣀⣀⣀⣀⣀⠀⣿⠀⠀⠀⠀⠀⠘⢆⡀⠀⢀⣴⡟⠙⠶⠶⠶⠶⠖⠋⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⣀⣀⣀⡉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⡩⠟⠀⠀⠀⠀⡠⠚⠉⠙⣦⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠔⠛⠶⠶⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠃⠀⠀⡀⣀⡴⠃⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣀⡠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⢀⡀⠤⠤⠒⠒⠉⠉⠻⣿⣭⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠀⠀⡄⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠸⣴⡞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠱⢉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣰⣂⠈⠈⣲⣀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠲⠮⠖⣿⡇⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠈⠉⢫⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⡤⠤⣤⣾⠶⠛⢿⣉⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠽⠿⠿⠿⠿⠿⠓⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⡠⣄⠀⠀⢠⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣶⣿⣿⡄⠀⠀⣠⡀⠀⢻⠋⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠈⠹⣿⣧⠀⢠⠟⣃⠀⣸⠀⢻⢧⣽⠇⠀⠀⠀⠀⠀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣓⣻⠆⠀⠙⠛⠓⢷⣿⠟⠐⢛⡂⣾⠛⠛⠀⠀⠀⣴⡶⠾⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣆⠀⠀⠀⠀⠙⢡⡀⠄⠀⢈⠀⢀⣴⣦⣤⣘⣿⣿⢇⡀⠀⢀⣀⣽⠟⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⢿⣦⡀⢀⠀⠀⢉⣿⣾⠏⣤⣿⡿⣿⣿⢿⣿⣿⣶⢾⣽⣆⣸⠿⠷⠀⠀⠀⢀⣾⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠙⢻⠋⠀⢀⣤⣭⣿⣿⣿⣿⣦⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠰⣯⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣠⡟⠲⣤⣠⣶⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣻⣿⣿⣿⣿⣏⣯⣽⣷⣿⡈⠉⣀⠀⠀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠙⡿⢣⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣆⣿⣿⣿⣿⢻⣿⠿⣿⣿⢿⡟⣇⣠⢹⣶⣾⠟⣉⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢣⣤⣿⣿⣿⣿⣿⢾⡟⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣴⣦⣀⣀⣀⣼⣆⢠⣾⣿⣿⢿⣧⣾⣿⣿⣯⡛⠛⢿⣿⡿⠯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⣠⣴⣆⣠⣤⡀
⠀⠀⠀⠀⠀⠐⠹⠿⠋⠉⠉⠉⠇⠸⣹⣿⣿⣮⣭⣿⣿⣿⣿⣿⣠⣠⣿⣻⣔⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⢂⠀⠈⠛⠋⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠈⣽⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡀⠛⢚⣻⢿⣿⡃⡾⣷⣦⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣚⠴⢿⢇⣻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠋⣿⣿⣿⣿⣿⣿⣆⣿⣧⣮⣿⠀⠸⠿⠛⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢿⣿⠏⠉⢽⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣾⣿⣿⣿⣿⣿⣿⣃⣽⡿⣿⣤⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣤⣦⠀⠼⣿⣿⣹⣿⢿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠰⢧⣾⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠋⠀⠀⠀⠘⠛⡧⣽⣷⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⢋⣷⡿⠀⠀⠀⠀⠻⠟⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠠⣴⡆⣠⡔⠌⠉⠋⢉⠷⣄⠽⣿⢟⣿⣿⣤⠿⠃⠈⢉⣽⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣼⣿⡇⠀⢠⣶⠞⠛⠚⠀⢀⣌⠙⣿⣁⣤⡀⠀⠈⠉⠀⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⢴⣻⡿⠀⠀⠿⠇⠀⣿⠀⣨⡿⢿⣅⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⠀⠀⠀⠀⣯⠈⠃⠊⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠒⠂⢀⠈⠀⠰⠆⠀⠀⠀⠀⢀⡰⠉⠉⠀⠁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡀⢠⣄⠀⠈⠑⢦⡀⠀⢀⣤⠶⠶⡁⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⣀⡀⠀⠀⠉⠳⢦⡀⢡⡖⠉⠀⠀⠀⠇⡀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠔⠊⠉⠁⠒⢵⣄⣀⠀⠈⢾⡀⠀⠀⠀⠀⠀⠉⠠⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢤⡀⠀⠀⠙⠧⢂⠀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢸⡇⠀⠀⠁⠀⠹⠀⠀⠀⠀⠈⢙⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠴⠶⠤⡄⠀⠀⠀⠀⠈⠉⠁⡄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠔⠒⠂⠀⠀⠀⠀⠀⠀⢰⣦⡴⠏
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠂⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣴⣠⣾⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⢾⡋⠀⢷⠀⠀⠙⢫⣴⡄⠀⠀⠀⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⡇⠀⠀⠀⢀⡐⠂⠀⣀⡇⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⢶⡄⠈⣧⠐⠲⣆⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢿⣦⡼⠃⠀⢸⡏⠀⠉⠑⢦⡀
⠀⠀⣠⠤⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⠘⣿⠄⠀⠀⠀⠙⣆
⢀⡞⠁⠀⡄⠀⡨⠔⠢⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⢀⠈⣧
⣸⠇⠀⠀⢣⡞⢰⡟⠛⠋⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠒⢛⠛⠛⠛⠛⢁⡼⠀⡄⠀⠈⡄⠈⡆
⢷⠀⠀⠀⡿⠁⠈⠳⠾⡃⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⣇⡴⠋⠀⠀⠀⣠⠞⠁⢠⡆⠀⠀⠇⠀⠸⡆
⢹⡆⠀⢀⡇⠀⠀⠀⠀⢟⠀⠀⣹⠆⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠟⠀⠀⠀⢠⡞⠁⠀⠀⢸⡇⠀⠀⢰⠀⠀⢻
⠀⠻⣄⣿⠀⠀⠀⠀⠀⠀⠀⣰⠋⠰⣤⢛⣤⣀⣀⠴⠋⠀⠀⢸⠀⠀⠀⠀⡈⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⠀⠐⡇
⠀⠀⢠⡍⠀⠀⠀⠀⠀⠰⢦⠏⠑⢶⠀⡎⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⣀⣤⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⢱
⠤⠈⠉⠀⠀⠀⡇⠀⡄⢘⡄⠀⠀⠈⠲⢆⠀⠀⠀⠀⠀⣠⠞⠉⣷⡶⠟⠻⠌⠀⠀⠀⠀⣿⠀⠀⠀⠀⣇⠀⠀⢸⡆
⠀⠉⢢⡀⠀⢀⠃⠀⡇⠸⡇⠀⠀⠀⠀⠈⠳⠶⠶⠶⠋⠁⠸⠇⢸⠇⠈⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣿⠀⠀⢸⠇
⠀⠀⠀⠙⡄⢸⣀⢀⡇⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢻⡄⠀⠸
⠀⠀⠀⠀⠈⠿⠈⢹⠇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⡇⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣄⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣄⡀⣠⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣟⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣁⣙⡒⠀⠀⠀⠄⣀⡚⢿⡀⠀⠀⢀⡾⠋⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠙⠓⠚⢻⡅⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⣤⡦⢄⣀⣀⡠⣤⠄⠀⠀⠘⠋⠻⠞⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣴⠋⠀⠀⠀⠀⢠⠋⠀⠀⢀⠼⡳⠶⠋⠀⠐⠀⠀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⡾⠁⠀⠀⠀⠀⢀⡞⠀⠀⣰⠃⠀⠐⡀⠀⠀⠉⠰⢆⣠⠘⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠶⣋⠀⠀⠀⠀⠀⢠⠟⠀⠀⢀⠇⠀⠀⠀⢳⡄⠀⠀⠀⠈⠷⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢹⡄⠀⠀⠀⢀⡇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠹⢦⣄⣀⠀⠀⠀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⠀⣣⠀⠀⠀⠈⢧⡀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡿⠀⠀⠀⠀⠀⠙⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠀⠀⢸⡇⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠸⠦⠤⠾⠇⢱⡄⠀⠀⠀⠀⠀⢈⡇⠀⠀⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡜⠘⠷⠤⠶⠷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠃⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣀⡴⠶⠛⠃⠀⠀⠀⠀⢀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠴⠶⠶⠶⠶⠶⠶⠶⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⢷⢀⣤⣤⣤⡴⠶⠶⠾⠍⠛⠛⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠄⢀⣀⣀⠤⠤⠤⠐⠒⠒⠒⠀⠉⠉⠉⠉⠉⠉⠉⠑⠒⡶⠤⠤⣄⡤⠎⠙⠦⡄
⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⢳⠴⠒⠉⠉⠀⠀⠀⠀⠀⢀⡠⠖⢊⠟⢀⡠⠴⠒⠋⠉⠉⠁⢀⣩⠥⠖⠀⠀⢠⡤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠓⡲⠀⠀⠀⠀⠀⠀⣠⢶⡆⣠⠔⠋⠀⠀⠚⠊⠁⠀⠀⠀⠀⠀⠰⠞⠋⠒⣒⠶⠄⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⣰⢁⠞⡆⠀⠀⠀⢰⠃⢸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⢰⠄⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢠⢇⣾⣶⣇⠀⢐⢤⡟⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠒⠂⠤⠖⠊⠉⣍⠉⠙⣄⠨⠒⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⡸⣸⣀⣀⣽⠀⠈⣦⠁⠀⠀⢀⠐⠈⠁⠀⠀⠀⠀⣠⠀⠀⢀⡤⠴⠮⣕⠀⠈⡄⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⣧⠟⠛⠛⢻⠀⡏⣠⡄⠀⠀⠀⢼⠂⠀⠀⠀⣠⠞⠁⠀⡰⠉⠀⠀⠀⠈⢣⠀⠘⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⢿⡀⠋⣸⠀⠀⠀⠀⠀⢳⠀⠀⠈⣡⠊⠙⠲⡁⠀⠀⠀⠀⠀⠘⡄⠀⢱⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣾⡇⠀⠘⣛⣭⠭⢽⣄⡸⠀⠀⠀⣧⠤⠲⠆⢻⢲⡶⠤⢄⣀⣸⠁⠀⢈⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⣯⣉⣉⣉⣸⡇⢀⡞⠁⢴⣒⣢⠄⠀⠀⠀⠀⠳⣄⣀⣠⡷⣄⣀⣀⣀⡼⠃⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⣿⠿⠿⠟⠻⡇⠈⢧⣀⣀⣀⣠⠤⢲⡀⠀⠀⠀⠈⠉⠉⢀⣨⠉⠁⠐⠒⠒⡄⠀⡸⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⢸⣻⣶⣶⣶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠐⠆⠀⠒⠀⠐⠚⠁⢠⠃⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠈⣼⣥⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠲⣒⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣽⣤⣄⣀⡀⠀⡸
⠀⠀⠀⠀⠀⠀⢳⣟⢋⣉⣽⡇⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠲⢄⡀⣀⣠⣤⠤⠖⠒⠋⠉⠑⠛⠉⠉⠉⠛⠿⢿⣿⣶⠒⡄
⠀⠀⠀⠀⠀⠀⠈⣞⡿⠟⣻⠁⠀⠀⠀⠀⠀⠀⢀⡎⠈⠢⠔⠀⢱⡔⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣝⣿⡿⢤⠤⠤⠤⢤⡞⣹⢩⠋⡿⠤⠤⠤⢌⣉⣀⣸⡖⢦⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⣀⠴⠒⢉⡀⣀⡀⡠⠊⡇⠀⠀⣀⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃⠃⠛⠒⠂⠤⢄⣀⠀⡜
⠀⠀⠀⠔⠋⠁⢀⡠⠴⠗⠄⢞⠁⠀⡿⠁⠋⠀⣏⡉⠉⠉⠙⠛⠒⠒⠒⠒⠦⠤⠤⣀⡀⠀⠀⠀⠀⠈⠉⠓⠢⠤⣄
⠀⠀⠀⠀⠀⠀⠁⠀⣀⠔⠚⡄⠀⠀⠀⢀⠤⠊⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⡀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠠⡎⠅⣀⡁⠓⠒⠒⠒⠚⡢⠔⠁⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄⠁⠂⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠈⠉⠉⢐⠊⠀⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠤⠬⠔⠋⡀⡀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣦⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⣤⠶⠶⠶⠖⠛⠛⠛⠉⠉⠁⠀⠙⣧⡀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠶⠶⠛⠛⠃⠀⠀⠀⠛⠛⠳⠶⠄⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠠⣆⠀⠀⣤⣤⡴⠆⠀⠀⠀⠶⣤⣤⠄⠀⢰⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡄⠀⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⢻⣄⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢻⡄⠀⣿⠷⣤⡈⠉⠙⠻⣿⣷⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠟⠁⠀⣿⠀⠈⠛⢶⣤⣤⡾⠛⢿⡍⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠛⠛⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⢦⣤⡿⠛⠶⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⣀⣠⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠶⢶⣦⣶⣶⡿⠿⠿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⣈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⢔⣪⣿⣿⣿⣿⡄
⣶⠶⠶⠶⠶⠶⠶⠶⢶⣾⣿⣫⡴⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠠⣴⣿⣿⣿⣿⣿⣿⣿⡶⠶⢶⣶⠶⠶⣶⡶⠶⢶⣶⡶⠶⠶⣶⣶⣶⠶⠶⠶⠶⣶
⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⡏⠀⢰⡿⠀⠀⣾⠟⠀⢀⣴⣿⠟⠁⠀⠀⣠⣾⣿
⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢟⣫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⢷⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠇⠀⣾⠃⠀⣾⠏⠀⣠⣾⠟⠁⠀⢀⣴⣿⣿⠿⠋
⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⠶⠊⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡄⠘⠀⢠⠋⠀⣼⠃⢀⣴⠟⠁⠀⢀⣴⣿⠿⠋
⣿⣿⣄⠀⠀⠀⠀⢸⣿⣿⣿⡷⢋⡁⢀⣀⣤⣤⣴⣶⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣇⠀⠀⠀⠀⠰⠁⣠⠞⠁⠀⣠⣴⠟⠋⠀⠀⢀⣠⣶⣿
⣿⠻⣿⣧⡀⠀⠀⠈⣿⣿⣿⠞⣵⣾⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⣧⠀⠀⠀⠀⠀⠁⠀⣠⠞⠋⠀⠀⣀⣤⣾⠿⠟⠋⠉
⣿⠀⠈⠻⣷⡄⠀⠀⢹⣿⣿⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡿⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⠴⠟⠋⠉⠀⢀⣀⣤⣴
⣿⣷⣄⠀⠈⠻⣆⠀⠈⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠘⠇⠰⡏⢀⡴⠀⠀⣠⡞⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⣤⣴⠶⠿⠟⠛⠛⠛
⣿⠈⠻⢷⣄⠀⠙⢧⡀⠘⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣠⣀⣉⠀⠀⠀⠀⢈⣀⣴⠞⠋⣠⠔⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⣿⣦⣀⠀⠙⠳⣄⠀⠁⠀⣿⠀⠻⣿⣇⠀⠀⠀⠶⠦⣤⣤⣤⣤⣴⣦⡀⠀⠈⠿⣦⣶⠟⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠻
⣿⠉⠛⠷⢦⣄⠀⠁⠀⠀⠈⢷⡀⠘⠿⡆⠀⠀⠀⢀⣤⣤⣤⣤⣾⣿⣷⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⢤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠉⠑⠶⠶⣶⣤⣤⣤⣠
⣿⠶⣦⣤⣀⠈⠉⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠘⣿⠟⠉⠉⠛⣶⢦⣤⣀⡀⠀⣀⣠⣧⡴⢶⡄⠀⠀⠀⠀⠉⠉⠙⠛⠳⠶⠶⣤⣤⣉⡙⠻
⣿⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⢳⣦⣤⡀⢀⣀⠀⠀⠀⠀⠀⠀⣰⣏⡶⣆⠀⠀⢹⡶⠢⣍⡛⠿⣯⣍⣀⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠉⠙⠻
⣿⠛⠛⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡜⢿⣷⣶⣄⣀⣀⣼⣷⡟⢠⡿⢷⡄⠀⡟⢤⣀⠙⠓⢦⣌⣉⠛⠻⠿⢶⣄⡀⠀⠀⠀⠀⣴⣋⡈⢙⣦
⣿⣀⣀⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣌⠻⣿⣿⡿⠛⢉⡟⢀⡿⠁⣾⡄⠀⢣⡀⠉⠛⠦⢤⣀⠉⠙⠒⠀⠀⢸⣿⣦⡀⠀⠀⠈⢿⣿⡟⠹⡆
⣿⠉⠉⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⡽⠁⢀⣴⣿⠁⣼⠃⣸⠏⣿⠀⠘⢿⣶⡤⢤⣀⡉⠙⠓⠤⠀⠀⠈⣿⠈⠻⣦⠀⠀⠈⣹⡿⠚⠁
⣿⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠃⠀⣺⣿⠃⢰⡟⢀⡟⠀⣿⠀⡄⠀⢹⣦⡀⠉⠛⠳⢤⡀⠀⠀⠰⣿⣄⡀⠈⣷⡴⢶⣹⠶⣤⡀
⣿⠀⢀⣠⡴⠊⠀⡀⠀⠀⢀⣴⡿⠿⠛⠿⢿⣿⣿⠇⠀⠀⠛⠁⢰⠏⢀⡼⠁⢠⡏⢀⡇⠀⠀⠹⣿⣶⣤⡀⠀⠉⠃⠀⠀⢼⣅⠀⢸⣿⣄⣈⠛⣀⣼⡟
⣿⣾⠟⠉⢀⣴⠞⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣼⠁⠀⡇⠀⠀⠀⠈⠙⢿⣿⣶⣄⠀⠀⠀⠀⠈⠙⢲⡿⢹⠛⣿⢻⠩⣏
⣿⠀⢀⣴⡿⠁⠀⣠⡾⠃⡠⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣤⣄⠀⠀⠀⠀⠀⢰⡏⠀⠀⣧⣀⣤⣤⣤⣤⣄⠈⠻⣿⣷⡀⠀⠀⠀⠀⣸⣷⣸⠂⣿⢸⣢⣧⡀
⣿⣴⣿⠋⠀⢠⡾⠋⣠⠎⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⣧⠀⠀⢻⡅⠀⢰⣧⠴⠋⠀⢀⡙⣿⣿⣤⡀⠀⣼⠟⠙⠛⢿⣿⣿⣿⡛⢻⣦⡾
⣿⠟⠁⠀⣴⠟⣡⡾⠁⢀⠄⠀⠀⠀⠀⠀⢀⣾⠉⣿⣷⠹⣿⠈⣿⡏⠀⠀⠀⣿⠀⠀⣸⠃⠀⠘⠷⠴⠛⠛⠋⠙⠛⠛⢿⡿⢿⣿⠀⠈⠁⣾⣿⣿⣿⡇⠀⣿
⣿⠀⢠⡾⢃⣴⠏⢀⡴⠃⠀⡠⠀⠀⠀⠀⢸⡇⠀⢹⣿⣇⠹⠀⢿⠃⠀⠀⠀⢸⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣾⢿⣿⣿⠀⣿
⣿⣠⡟⣡⡾⢁⣴⠏⢀⡴⠊⠀⠀⠀⠀⠀⠛⠛⠛⠻⣿⣿⡄⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣄⣀⣀⣸⣿⣿⣷⡾⣿
⣿⠏⣴⡟⣠⡾⠁⣴⠟⠀⣠⠂⠀⢠⣦⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⢹⣷⢀⣭⣉⣛⣛⣋⣉⣤⡀⣿⡄
⣿⣼⠏⣴⠟⣠⡾⠁⣠⠞⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠟⠋⠁⠀⠀⢸⣿⠀⢿⣿⣿⣿⣿⣿⣿⡇⣿⡇
⣿⢋⣾⠋⣴⠏⣠⡾⠁⣠⠖⠁⣠⣼⣿⣤⡀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣀⣠⣴⠒⠉⣁⣤⣴⡷⠾⠿⠿⠿⣿⣦⣄⢻⣿⣿⣿⣿⣿⡇⣿⡇
⣿⣿⢣⡾⢃⣴⠏⢀⡼⠃⠀⠠⣿⣬⣭⣴⡏⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠛⠁⠀⣀⡀⠚⠛⠉⠀⠀⠀⠻⢋⡀⠀⠀⢀⣀⣀⣀⣽⣿⡇⠻⣿⣿⣿⣿⡇⣿⡇
⣿⣷⡿⢡⡾⢃⣴⠏⢀⡴⠃⢀⣿⣤⠀⢸⣇⠀⠀⣼⣿⣿⣷⠀⠀⠐⠀⢀⣿⡶⠿⠛⠁⠒⠀⠀⠀⠀⠐⣿⡏⢿⡟⣿⡟⣿⠟⣿⡏⣿⡇⠀⠙⢿⣿⣿⠇⣿⠗
⣿⣿⣰⡟⢡⡾⠁⣰⠟⠀⣠⡟⠉⢛⣿⣿⡟⣷⠀⣿⣿⣿⣿⣆⢀⣤⠶⢻⠥⣤⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠷⠾⣧⣾⣧⣿⣦⣿⣷⣿⠷⣦⣄⠀⠙⠿⢀⣿
⣿⣿⡏⣴⠟⣠⡾⠁⣠⠞⢹⡇⠀⠈⣿⣿⡇⣸⢰⣿⣿⣿⡿⠟⠋⠀⢠⡟⠛⠙⢦⡀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠠⢤⣤⣽⣿⣿⣿⣿⣿⡄⠈⢿⣧⣤⣶⠟⣿
⣿⣿⣿⠏⣴⠟⣠⡾⠃⢠⣾⡷⢦⣤⣥⣿⠿⢿⣾⠿⠋⠁⠀⠀⠀⠀⣿⣦⣤⣤⢾⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⠻⠟⣻⣿⣿⣿⡇⠀⣸⣿⠁⠀⣰⡿
⣿⣿⣿⣿⠋⣴⠟⢀⡾⠋⢸⡇⢻⣶⣶⣶⡇⣿⠁⠀⠀⠀⠀⠀⠀⣠⣿⣤⣀⣡⣼⣿⣦⡀⠀⢸⣿⠷⢶⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣷⠾⢿⣿⠷⠟⠋
⣿⣿⣿⣿⣿⣋⣴⠟⢀⡴⣻⡇⠈⢻⣿⣿⡇⣿⠀⠀⠀⠀⠀⢀⣾⡁⢌⣻⡿⢛⣛⣉⢹⡇⠀⢸⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣶⡿⠟⢻⡇⠀⠀⠻⣿⡇⣿⠀⠀⠀⠀⠀⢸⣿⣿⣮⠁⠐⢿⣿⣿⡟⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⢸⣇⠀⠀⠀⠈⢋⣿⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠹⣿⡗⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⢸⡏⠛⠻⠾⢻⣏⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠻⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠾⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢙⡇⠀⢸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⢀⣠⡴⠞⠁⠀⢸⣿⠛⠿⠶⢶⣶⣦⣤⣤⣤⣤⣶⣶⣶⣿⠟⢻⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠞⠋⠉⠀⠀⠀⠀⠀⠛⢷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣶⠿⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠒⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡆⠀⠁⠈⠀⣀⠄⣀⠀⣠⣀⡋⡐⠀⢠⢊⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠈⢰⠤⠐⠉⠉⠈⠁⠀⠀⠐⠢⠄⠁⠈⠓⠲⢂⡂⢊⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⠈⠐⢄⠁⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠐⠂⠀⡴⠉⠀⠁⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢠⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⡐⠎⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠠⠤⠀⠀⠀⠈⠨⠀⠀⠠⠄⠀⠹⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢡⡀⠀⡀
⠀⠀⠀⠐⢀⣀⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⣱⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⡆⠀⠠
⠀⠀⠀⠈⠀⢸⠀⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⣃
⠀⠀⠀⠀⠰⣽⠀⠀⢠⣙⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠂⠀⠀⠀⠀⡍⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⠐⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠇⢀⡆⠂⠈
⠀⠀⠀⢤⠄⠀⢃⠀⠀⠀⠀⠀⠒⣀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠈
⠀⠀⠀⠀⠀⠀⢘⣆⠀⣁⠀⠀⠑⠃⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣸⡇⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⣶⠋⢀
⠀⠀⠀⠀⠀⢀⡀⡠⠊⠻⣦⣀⡀⢲⣀⠀⠀⠀⠀⠀⢂⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣤⡞⠇⠀⠦
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢈⡾⢷⣄⠀⠀⢀⠀⣄⠂⡀⠀⠀⣤⡠⠀⠀⠀⠀⠀⠀⣈⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠻⠶⢻⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠛⠁⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⡔⠋⠁⡽⠙⠛⠷⠒⠲⠖⠦⠨⠛⣏⠉⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⠀⠘⠒⠂⠸⠀⠀⠀⠀⢀⠴⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠓⢦⡀⠀⢠⠖⠃⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠈⣳⠀⠀⠀⠀⠀⠀⢸⠃⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣨⠇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣇⠀⢀⠀⠛⢦⣀⣠⠴⠖⠋⣿⠀⠀⠀⢿⠓⠲⢤⡀⠀⣠⠞⢁⡴⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⡄⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠋⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠆⠀⠀⢀⡴⠚⠉⠙⣦⠀⠀⠀⠀⠀⠀⠠⠞⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠸⣄⣀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⠀⠀⠀⠀⣾⣠⠴⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢴⠋⠙⠳⣄⠀⠀⠈⠛⠓⠲⠶⠤⠴⠋⠉⢳⡀
⠀⢀⡞⠉⠙⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠞⠀⠀⣀⣤⣤⣤⣤⣤⡄⠀⠀⢸⠇
⠀⠘⡇⠀⠀⢠⠤⠖⢶⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠙⠦⠴⠋
⠀⠀⠙⠦⠤⠞⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣞⠉⠉⢙⡆⠀⠀⠀⠀⠀⢠⡀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠹⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⣄⠀⠈⠛⠉⠙⡆
⠀⠀⠀⠀⠀⢀⣀⣀⡞⠁⢀⡴⠋⠉⠓⠦⣤⣦⠀⠀⣇⣠⡤⠶⠛⠁⠈⡗⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡆⠀⣧⡀⠀⠀⠀⠀⠀⠳⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠙⢦⣀⣀⡿⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢠⠶⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⣟⠀⣀⣨⠓⣅⣀⢀⣿⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀
⠀⠀⠀⠀⠀⣠⢤⠀⠀⡠⠋⠉⡤⢤⠀⠀⠀⠉⠸⢄⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⢹⢴⡃⠀⠀⢀⡤⢄
⠀⠀⠀⠀⠀⠑⠺⢕⠊⢀⡀⠀⠁⠉⠀⠀⠯⡵⠀⠀⠛⡫⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢎⠣⠴⢒⣋⡀⠉⠓⠦⢜⡴⠚
⠀⠀⠀⠀⠀⠀⠀⡜⠀⢯⠟⠀⠀⢯⣱⠀⠀⢠⠴⡄⠀⢧⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠘⠒⠁⢠⢤⡄⠘⢄
⠀⠀⠀⠀⣏⡱⢮⡁⠀⠀⠀⠀⠀⣀⣤⡀⠀⠈⠋⠀⠀⢈⠴⣍⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣙⣒⠚⠀⡤⡄⠀⠀⠀⠈⠉⠀⠀⠈⢒⣒⣉⡳
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢞⣳⠀⠀⠑⠚⠀⠀⣴⣲⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇⠀⠓⠁⠀⠯⡷⠀⠀⣞⠝⠀⡏⠀⠈
⠀⠀⠀⠀⠀⡴⢲⡯⠤⡀⠉⠀⠀⠰⣋⡗⠀⠈⠁⣠⠖⢽⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀⠀⢀⣄⡀⠀⠀⣀⣀⠀⠰⣇⣀⡀
⠀⠀⠀⠀⠀⠙⠊⠀⠀⢈⣆⣤⣀⣀⠀⢀⣀⣤⢺⠁⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡰⠋⠙⠦⣈⠓⠃⠀⠈⠓⣁⠴⠋⠲⣄⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡸⠀⠀⢈⣶⠉⠀⠀⣷⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡔⠒⠒⠒⢆⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⠳⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⣔⠋⢳⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⡀⡇⠀⠀⡀⣸⠃⣇⠀⠀⠀⢸⢆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠑⠒⠋⢉⠁⠀⠈⠉⠉⠒⠋⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢦⡀⢀⠞⠁⠀⠀⢼⣁⡽⠀⠀⢀⢤⣀⠀⠈⠳⡄⣀⡴⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠓⣌⠁⢀⣠⡀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠂⠀⠀⠈⣡⠖⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣀⠇⠀⠀⠐⣍⢱⡄⠀⠀⠀⡠⢤⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠦⠤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠧⠜⠀⠀⠀⠧⠤⠖⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠊⠉⢳⡀⠀⠀⣀⠀⠀⠀⢀⡖⢛⡆⠀⠀⠀⢀⠀⠀⠀⠀⡎⠉⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠸⣉⣱⠀⠀⠀⠉⠉⠀⠀⠀⢴⢉⡵⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⣃⢤⡀⠈⠁⠀⠀⠀⢰⠚⣳⠀⠀⠈⠉⠀⢀⠤⢌⡳⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠏⠀⠀⠑⢦⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡔⠋⠀⠀⠹⢤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣠⠒⠲⠤⢤⠀⡤⠤⠔⠒⢄⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡜⠀⠀⠀⢀⡇⡇⠀⠀⠀⢸⡃⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣀⡹⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢀⢠⣹⣴⣤⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠈⣽⣿⣿⣿⣿⣾⢃⡀⡀⡆⡀⢀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣜⢹⢿⣿⣿⢿⡳⢮⣵⣿⣿⣷⣯⠄⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⠈⠙⠂⠘⢫⣿⣿⣿⣿⡿⠝⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠐⠋⠟⣟⠻⣱⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠘⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⢀⡀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠒⣼⣶⣶⣧⣖⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠰⢽⣿⣿⣿⣿⣧⠶⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⣀⣩⠿⡿⡿⡟⣭⢠⠄⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⢢⣷⣾⣾⣼⡓⠃⣛⣽⣾⣾⣴⣃⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠮⣿⣿⣿⣿⣿⡥⠖⣻⣿⣿⣿⣿⣗⠆⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⡄⡈⣽⣻⢿⢿⣻⡅⡀⣽⢻⣿⣿⣿⣴⢢⣷⣿⣿⣼⣺⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⣀⣤⣤⡤⠶⠖⠛⢻⣄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣼⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁⢠⣤⡤⠶⠆⠀⠀⠰⠶⢤⣤⡄⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⣠⠀⠀⣀⣀⣠⠄⠀⠀⢠⣄⣀⣀⠀⢠⡍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣸⡀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠀⠠⢦⠶⠀⠀⠀⠀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠹⣦⡀⠀⢀⣀⣀⣀⡀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣧⠀⢸⡟⢦⣈⠉⠙⠻⣿⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⣸⠃⠀⠙⠳⠶⠶⠛⠙⣯⡉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠉⠳⣆⣀
⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠛⠉⠉⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⠶⠋⠉⠙⠛⠓⢶⣄
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⡆⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⣀⣤⣿⡿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣴⡏⠀⠀⠀⠀⠀⠀⢀⡴⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣧⣀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠉⢻⣦⣀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋⠀⢀⣿⣿⣿⣿⣿⣿⣦⣤⣤⣾⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣤⣤⡤⠴⠿⡧
⠀⠀⠀⠀⠀⢸⣦⣄⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠞⠋⠉⠉⠉⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⣀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠛⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⡿⠉⠉⠉⠉⠉⠉⠙⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⠖⠚⠛⠛⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣹⣿⣿⣿⠀⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣭⣭⡍⠁⠀⠀⠈⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠉⠻⣿⣿⣿⣿⣿⣟⡁⢉⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⠞⠁⠀⠀⠀⠀⠀⠀⣸⠟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠸⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢰⡾⢿⡆⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⣽⠆⠀⠀⠈⢻⡟⠁⠀⠀⠰⣿⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠈⠉⢿⡄⢀⣿⣀⣸⣇⣀⣿⡀⢀⡿⠉⠁⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣿⡀⠀⢷⣤⡾⠟⠛⠉⠉⢉⡉⠉⠙⠛⠿⢷⣤⡿⠀⢀⣿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠟⠁⠀⣤⣶⣄⠀⠈⠁⠀⣠⡶⣦⡀⠉⠻⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢷⡆⠀⠛⣷⡟⠁⢠⡄⠈⢿⣴⠟⠀⠀⠀⠀⠹⢷⠾⠃⢀⡀⠈⠻⣾⠋⠀⢰⡾⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠛⠷⣾⠏⠀⢀⣀⡀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⣁⡀⠀⢹⣷⠾⠛⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡿⠀⠰⣿⣉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⣿⠄⠀⢿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣦⣄⣀⣸⡇⠀⢀⠉⠉⠁⠀⠛⠀⠀⢰⡟⢻⡆⠀⠰⠆⠀⠈⠛⠋⠀⠀⢸⣇⣀⣀⣴⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠟⠉⠉⢹⡇⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⡤⠀⢸⡏⠉⠉⠿⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣧⠀⠀⣼⡟⣻⡆⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⡶⣦⡀⠀⠀⠀⣼⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⡶⢿⣆⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡾⠃⠀⠀⣰⡿⢶⣤⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡾⠃⠀⣠⡿⣧⡀⠀⢠⡄⠀⠀⢰⡾⢶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣤⠀⠸⣷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⣀⠀⠀⠀⠘⠷⠟⠃⠀⠀⠖⠀⠀⣀⣴⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣿⠁⠀⣾⠛⢷⣦⣤⣀⣀⣀⣀⣀⣠⣤⣴⡾⠛⢷⠀⠈⣿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⢀⣀⣾⠁⠈⣿⠉⢹⡏⠉⣿⠁⠘⣷⣀⡀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣿⠆⠀⠀⢠⣼⣧⡀⠀⠀⢰⣿⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠸⣷⣾⠇⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢰⣶⣆⡀⠀⠀⠀⠀⠈⠛⠋⠀⢀⣀⣠⣼⣿⣿⣤⣀⣀⠀⠈⠛⠋⠀⠀⠀⠀⠀⣠⣶⠆⢀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄⠀⠀⠀⣀⣴⣶⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣷⣦⣄⡀⠀⠀⢠⣼⠟⢁⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣴⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣼⠟⢁⣤⣼⣿⠇
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⢁⣴⠟⠛⠛⠋
⠀⠘⢿⣿⣿⣶⣤⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣿⣷⣶⣆
⠀⠀⠀⠀⠉⠛⠋⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⡟⠁⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⠿⠿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣿⣷⣶⣶⣤⠀⠀⢀
⢾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣟⣁⣠⣽⣿⠟⢋⣠⣾⣿⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣿⣿⠻⢿⣿⣿⣷⣶⣿⣿⣋⡀⢀⣴⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⣿⠁⣹⣿⣦⣄⣿⣿⡍⢻⣿⡿⠿⢿⣿⣿⣿⣦⣤⣶⣿⣿⣦⡀
⠀⠀⠀⢀⣀⣤⣤⠀⢻⣿⣧⠀⠀⠀⠀⣠⠞⢁⣴⡟⣿⣿⣶⣿⣿⣿⠿⢿⣿⣷⠀⢿⣿⡀⠀⢿⣿⡍⠛⢿⣿⣅⣽⣿⣷
⠀⢠⣾⣿⡿⠿⠛⠀⠀⢻⣿⣧⡀⣠⠞⢁⣴⢿⣿⡇⢸⣿⡿⠃⣼⣿⣷⣤⣌⡁⠀⠈⠻⠃⠀⠈⠿⠇⠀⠈⠿⠿⠛⠛⠉
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢹⣿⠟⢁⣴⣿⠁⠘⣿⣧⠀⠉⣠⣾⡿⣿⣿⠛⠿⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⣴⠟⢻⣿⡀⠀⠙⠋⠀⠰⣿⠟⠁⢸⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⣠⣿⣇⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠘⢋⣴⣿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣉⣤⠒⠒⢲⠒⠠⢌⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠐⣎⢦⠀⠘⡀⠀⣠⠜⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠃⣼⣁⠀⡀⠈⣆⠻⠀⡦⢊⠴⠊⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠂⠠⢤⣄⡉⠀⡀⠀⡋⠒⠒⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⣠⣞⡵⠞⣳⡌⢫⠉⠁⠀⢐⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠻⠙⢫⠋⠁⠈⠀⡇⢿⠉⢦⡱⡀⢀⠎⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠉⠀⡀⠀⢀⡸⠀⢀⡱⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠈⠈⣿⡉⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣀⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁⠀⠉⠉⠉⠁⠀⡀⠄⠂⠙⡗⠲⣤⣀⣀⣀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠙⢆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⢀⠔⠈⠁⠀⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⡈⠀⠠⠂⠁⠑⢄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡔⢃⠀⠀⣀⣀⣀⣀⠀⢦⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠈⢠⡡⣀⠤⠒⠉⠀⡀⠀⠀⡏⠉⠉⠉⠀⠀⢱⠀⠀⠀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⢀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⠓⠴⢦⡂⠀⠀⠀⠀⠀⡕⠸⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⡰⠁⠀⢠⠧⢄⡀⠀⢩⠁⠒⠒⠂⠐⠈⠁⠀⠀⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣁⠀⠰⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡀⠀⠀⠀⠈⠢⢀⡀⠀⢀⠄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⠙⠶⢤⣀⣀⣀⣀⡀⢀⣀⣀⡠⠴⠚⠋⣠⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⠲⠤⣤⣀⣀⣈⣉⣀⣀⣤⠤⠖⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠋⠩⠉⠉⠙⠻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⡀⢸⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣙⠛⠶⠶⠶⠶⠶⠛⣩⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⠀⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⠀⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣸⡇⢸⣿⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⡾⠉⣠⣼⣿⣧⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⠀⣼⣿⣿⣿⣿⣿⣇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢿⠀⢻⣿⣿⣿⣿⣿⡏⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢷⣀⠙⠛⠿⠛⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠉⠓⠲⠶⠖⠛⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣿⣻⣽⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣭⣭⣭⣭⣿⣯⣭⣯⣽⣍⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠈⠉⠉⠁⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠏⠢⠄⣀⣀⣀⢤⢤⣳⣄⠀⠀⠀⠀⠀⠀⠀⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⡀⡇⠀⠀⠌⠀⢙⠒⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠤⠀⣬⣯⡍⣉⠉⢢⢸⠄⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠻⠊⠉⠙⠒⠒⠚⠁⠸⡁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢎⡔⡦⡙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡰⠋⠁⠀⢰⢈⣆⡠⠀⠀⠀⠀⠀⡘⠒⠑⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠀⠀⠀⢻⠈⡄⠀⠀⠀⠀⢠⠗⡂⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠀⢀⠀⠀⠈⠀⠱⡀⠀⢀⣰⣏⡰⠙⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢣⠈⢆⠀⠀⠀⢀⡱⠒⡿⢻⠏⢀⠜⢄⠀⠀⠀⠑⠤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣼⠀⣾⡀⠀⠑⢄⠀⣍⡠⣬⠞⠁⡠⠂⠀⢸⠄⠒⠄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠜⠈⠑⠣⣠⠼⠆⠸⡇⠁⡠⠊⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁⠀⠀⡾⡖⠁⠀⡼⡄⡙⠀⠈⠀⠇⠀⠀⡆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠀⣀⠎⠀⢁⣎⢰⠀⡸⠁⠀⠰⢿⡀⠀⠀⡇⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔⢁⠌⠸⠈⠀⠀⢀⠃⠀⢻⠄⢰⢀⠠⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠓⣤⡠⠊⢣⡀⠀⠀⣸⠅⠊⠀⠉⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⠬⢽⠉⡏⠬⠤⠝⠦⠤⠃⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠌⡝⠃⠀⠀⡀⠠⠀⠤⠄⠂⢀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⡇⡇⠀⠀⠃⠀⠀⠀⠀⠀⡄⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠁⠃⠀⠀⠑⠠⠤⠤⠐⠊⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠙⡃⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠚⢳⣤⠤⢤⣤⣤⠤⠤⠤⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⠀⠒⠒⠂⢠⠨⠛⣆⠀⠹⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣀⣀⡤⠒⠂⠉⠉⠁⠉⠉⠻⣆⠉⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠠⠘⠒⠒⠒⠚⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠛⢛⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣬⣉⠙⢀⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⠥⡬⠀⠥⠬⠼⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠊⠀⠰⠀⠀⠀⠈⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢰⠞⠉⠉⠉⢳⡀⠀⠀⢀⡞⣁⣀⣀⡠⠤⠤⠖⠒⠀⠒⠢⠐⠄⠳⢤⣙⣧
⠀⡴⠖⠛⠂⠀⠀⠀⣸⡃⠀⣴⠛⠏⠠⠀⠀⡆⠀⠈⠀⠀⢘⠀⢰⠀⢠⠀⠘⡀⠉⣧
⢸⡇⠀⠀⠀⠈⢢⠀⠈⢿⡀⡏⢠⠀⢈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⡄⠀⠀⠀⡇⠘⣸
⠀⣿⠂⠀⠀⡀⠈⠀⠀⣸⡿⠛⠀⢤⣚⣩⣭⣩⣒⡉⠉⠀⠴⠖⠒⢆⣀⡐⣖⣳⠂⠉⢳⡄
⢀⣸⡇⠀⠀⠁⠀⣤⠞⢹⠃⠀⠀⢰⠀⢀⠊⣴⡿⢷⠀⠀⠀⡴⢷⣦⠑⡄⠈⡇⠀⠀⠈⣷
⡟⠀⠀⠀⣆⠀⠀⡹⠶⠛⠠⡀⢀⡸⢠⣬⣄⣛⣿⠞⣠⣤⣄⢻⣿⠟⡠⣁⢀⡃⠀⠀⠀⣿
⠻⢤⣠⣤⠁⣀⢀⣠⡶⠓⠲⣦⣠⣾⣻⠟⠟⠩⢋⠕⠛⣏⠙⢌⠫⠑⠟⢿⢙⣷⣤⡤⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠁⢀⣴⠞⠋⠉⠙⡇⠀⠀⠀⠀⠀⠚⡍⠂⠀⠀⠀⠀⠀⡋⠉⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠔⠊⠘⠀⠀⠀⠀⠀⠀⠜⠙⢆⠀⠀⠀⠀⠀⡈⠉⠑⠤⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣀⣀⡀⢳⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠄⠊⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⠒⠂⢲⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠤⢽⡇⠀⠐⠒⠀⠤⢼⠀⠀⠀⠀⠀⠀⡸⠄⠒⠊⣿⣤⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠁⠀⠐⠒⠂⢎⣰⣣⣆⣴⣰⣠⠗⠒⠈⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣛⣦⣤⣀⣀⣀⣤⣐⣿⣾⣓⣠⣄⣀⣀⣠⣴⣞⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠀⠈⠀⣼⣿⡀⠈⠀⠀⠀⠈⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⢷⡳⢄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡤⢤⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢄⠀⠀⠀⠀⠈⡇⢸⠁⠀⠀⠀⠀⡠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡞⠁⠀⣀⡴⠀⠀⠀⠀⢠⠇⠸⡄⠀⠀⠀⠀⢦⣀⠀⠈⢳⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢳⠀⠀⠀⠈⠳⠴⠯⣉⣧⠤⠖⠛⠋⠉⠓⠚⠋⠙⠛⠲⠤⣼⣀⠽⠦⠞⠁⠀⠀⠀⡞⠉⠉⠳⣄
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⡇⠀⠀⢀⣴⡋⠳⢤⣀⣀⠞⠁⠀⠀⡴⠋⠉⠙⡆⠀⠀⠀⠀⢰⠋⠉⠙⢦⠀⠀⠈⠳⣀⣀⡤⠞⣙⣦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠙⠦⠖⠋⠉⠉⠙⢦⣀⡼⠀⠀⠀⠀⠱⢤⣤⣾⡁⠀⠀⠀⠀⢈⣷⣤⡤⠞⠀⠀⠀⠀⢧⣀⡴⠋⠉⠉⠙⠲⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠷⠟⠀⠀⠀⠀⠻⠾⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⡾⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢿⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⡷⢤⡇⠀⢠⠏⠉⠙⣄⡤⣄⠀⠀⠀⡞⠉⠉⢳⠀⠀⠀⣠⢤⣠⠏⠉⠹⡄⠀⢸⡤⠾⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢧⠀⢈⡷⠚⡇⠀⠘⢦⣀⡴⠋⠙⠃⠀⠀⠀⢦⣀⣀⡴⠀⠀⠀⠘⠋⠙⣆⣀⡴⠃⠀⢸⠓⢾⡁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⢷⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡾⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⣖⡦⠀⠀⠀⠀⣴⣲⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃
⠀⠀⠀⠀⠀⡰⠒⠲⢄⣀⣀⣤⠞⠁⣱⠀⠀⠀⠀⡴⠋⠙⢻⡀⠀⠀⠀⠀⢀⡟⠋⠙⢦⠀⠀⠀⠀⡏⠈⠳⢤⣀⣀⡠⠖⠒⢆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⣄⡴⠚⠉⠉⢣⡀⠀⠀⠱⣄⣀⣰⠃⠀⠀⠀⠀⠸⣄⣀⣠⠎⠀⠀⢀⠜⠉⠉⠓⢦⣠⠟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⡟⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡼⠀⠀⠀⢀⡶⠒⢞⣁⡟⠓⠦⢤⣤⣠⠤⠤⣄⣤⡤⠴⠚⢹⣉⡷⠒⢶⡀⠀⠀⠀⢧⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠉⢑⠀⠀⠀⠀⠘⡆⢰⠃⠀⠀⠀⠀⡎⠉⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠀⠀⠀⠀⢀⡇⢸⡀⠀⠀⠀⠀⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠖⠒⠋⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡆⠀⠀⠒⠀⠈⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠂⠤⠐⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠠⠒⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠚⢀⠀⠁⠀⠈⡃⠒⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⢀⠓⠄⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⢒⣄⠀⠀⠀⠀⠀⠐⠒⠢⠤⠔⠀⠀⠑⠰⠄⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢨⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡌⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⡟⠀⠀⠀⠀⠀⠀⢀⢀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠰⠁⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂
⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠔⠀⠀⠀⠀⠀⢉⡀⠂⠠⠤⠤⠤⠀⠲⠂⠉⠀⠀⠀⠒⠒⠲⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠠⠄⠤⠄⠀⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠠⠤⠤⠀⠀⠀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠁⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣟⣛⣻⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣯⣭⣿⣿⣽⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠒⠀⠀⠔⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠰⠁⠀⠀⠀⠀⠀⠀⠈⠰⠒⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠤⠀⠀⠒⠀⠀⠐⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⠆⠀⠀⠀⣠⣾⠇⢰⣴⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡤⡀⠀⠸⠀⠀⠀⠚⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡌⠆⢀⠁⠀⠀⠀⢀⣠⠀⢰⠴
⠀⠀⠀⣀⡤⠤⠤⠤⢀⡀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠝⠋⠀⠀⠢⡀⠀⠀⠀⠀⠀⡀⠤⠐⣠⡙⠆
⡞⠙⠛⠁⠀⠀⠀⠀⠀⠈⢩⠒⠂⠀⠀⠀⠀⠀⠀⠸⣇⡴⢷⡴⠃⠀⢠⢐⣠⠥⠔⠂⠈⠭⠐⠁⠀⠀⠈
⠘⠦⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢼⣶⠾⠏⠃⠁⠈⠁
⠺⡄⠀⠀⠀⠀⠀⠀⢐⣺⣿⡇⠀⢀⣀⣀⠀⠀⠀⠋⠁⠰⠆⢠⠊
⠼⠁⠀⠀⠀⠀⠀⠀⠶⠟⠏⠀⠀⠀⠀⠈⠑⢦⡀⠀⢀⣀⡤⠾⡀
⢠⠀⠀⠄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢆⢀⡤⠼⠖⠒⠶⠤⠤⣀⡀
⢸⠆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢢⡀⠀⠀⠀⠀⠳⣴⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⡴⠚⠂⠀⢀⡀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀
⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⢤⣄⣀
⠠⡄⠙⠀⠀⠀⠀⠀⠀⠴⢺⣇⠀⠀⢀⡼⠃⠈⠓⠢⠤⢤⣤⡤⠶⠛⠉⠒⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⣄⡀
⠀⢳⠈⠆⠀⠀⠀⠀⢀⣴⣿⣿⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦
⠀⠀⠒⠒⠒⠂⠈⠉⠀⠀⡀⠝⠋⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠶⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣶⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣀⣀⣀⣐⣒⣒⣒⣶⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣒⣚⣉⣁⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⡊⠉⡄⠈⠒⢁⡀⠀⡄⢈⠆⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢢⠀⢠⠊⠙⠀⠀⠀⢀⣈⣍⡠⠤⠤⠤⠤⣀⡈⠡⡦⠘⠤⠃⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠀⠀⠓⢊⣀⡴⠊⠁⠀⡏⠀⠀⠀⠀⠀⠀⠈⢂⠀⢤⠆⠀⡠⢄⠘⠂⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠠⠀⡀⠁⠋⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠣⠀⠀⠀⠀⢸⠀⡠⠤⡄⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠂⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠊⡸⠂⠰⡀⡀⠀⠀⠀⡄⡜⠀⠀⠨⠀⠈⠠⠀⠀⠠⡀⡸⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡱⠠⡄⠀⠱⡭⠀⠀⠘⡷⠃⠀⠀⡴⠀⡀⠼⠤⠤⠤⠤⠤⠤⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡴⢄⠀⡰⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⡄⠁⠉⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⣶⠋⢠⡿⢄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰⠉⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡔⠒⡄⠈⠧⢀⣿⣀⣨⣷⣵⡒⠤⠤⠤⢒⠀⠀⠀⢈⡐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠊⢀⡤⠲⣏⠉⢻⣄⠀⠘⣿⡙⠒⠦⠁⡶⠂⠲⣎⠠⡇⠈⠑⢦⣄⣀⣀⡀⢀⣠⠔⠏⢠⡄⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢳⡀⠘⣷⠀⢻⡦⠔⠋⢉⠝⠃⠀⡇⠀⠀⣿⠀⠃⠀⠀⠀⠎⠁⠉⡉⠀⡍⣤⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢹⠗⠋⠀⠐⠀⠀⠀⠀⠈⠃⠀⠐⢻⠂⠀⠀⠀⢀⠀⠐⠀⠁⠀⠀⠙⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⡌⠀⠀⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠨⠀⡔⠢⠰⠆⣰⠃⠀⠀⢨⠀⠀⠀⠀⠀⡆⠀⠀⠘⡌⢆⠘⠄⠀⣠⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠘⡄⢳⠈⢆⠈⠀⢄⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠢⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠄⢣⠀⠁⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠚⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢰⡀⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⠀⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⠑⠢⠤⠤⠤⠤⠤⠔⠊⠀⠀⠀⠀⠀⢀⡇⠈⣧⡀⠤⠟⠉⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠲⠤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⡶⠿⠋⠀⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⡉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠂⠈⠉⠉⠉⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠒⠀⠐⠀⠂⠆⠂⠂⠀⠀⠂⠀⠀⠐⠐⠐⠂⠀⠀⠀⠀⠐⠀⠃⠃⠀⠀⠁⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⠶⢶⣄⣀⠀⠀⠀⠀⣀⡶⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠉⠙⠷⠲⢦⡼⢧⡀⠦⢄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠔⠲⡤⢄⢀⠀⠀⢀⣈⢥⠀⢈⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⢀⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠒⠃⠀⠀⢠⠀⠀⠀⠀⢀⣄⣼⠃⠰⠚⠁⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⣀⡴⢿⣰⠀⢀⣘⣿⣄⣠⣄⠀⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⠗⠚⡅⠐⠂⢹⡆⣸⠋⠁⠀⠀⣬⡟⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠓⠋⠀⠀⢠⠄⠀⠀⠀⢻⣧⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠫⠇⠀⠀⠀⠀⠀⠀⡀⢺⡧⠤⢤⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡆⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣿⡗⠒⠚⢁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠒⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡦⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣑⣑⣚⣘⣃⣄⣤⣤⣤⠤⠤⠬⠶⠮⢕⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠆⠀⠀⣀⣀⠀⢀⣜⣫⣿⡂⢤⡀⠈⡩⠖⢶⡋⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⣟⣩⣿⣽⣷⣶⠖⠁⠳⣵⣾⡶⢠⢦⡘⡇⠀⠀⠄⠀⢠⠶⢦⡄⠀⢀⣴⡏⢱⣦⣀
⠀⠀⠀⠁⡀⠀⢀⣀⣠⣀⠀⠀⠀⠀⣠⢠⠀⠀⠀⠀⠈⠻⠏⠛⠛⠃⠀⠀⢀⣨⣭⣽⡟⡾⠛⠉⢂⠀⢸⡀⠀⠀⠀⠀⠀⢸⣷⣿⣍⡿⢇⡳
⠀⠀⠀⠀⠀⠈⢣⣾⣿⣿⣶⣤⡀⠀⠈⠎⢇⠀⠀⢀⠀⢀⣤⣄⠀⠀⠀⠀⠇⢹⣿⡽⣷⣟⡱⢀⠀⢣⠘⣷⠄⠀⠀⠀⠁⣾⠿⣟⣷⣿⣯
⠀⠀⠀⠀⠀⠀⠀⢙⠛⢛⡻⣾⠁⠀⣀⠸⡌⢣⠀⠀⠀⡿⣿⣿⣿⣷⣶⡀⠘⠶⣑⣰⠆⠀⠀⠘⠀⣠⢓⡸⢂⠐⡲⢄⢰⣿⣷⡏⡫⢩⡃⢀⡀
⠀⠀⠰⠒⡂⢤⠤⣬⢭⠤⡁⠁⣄⡾⣽⡆⠹⢄⣱⡀⠀⠹⠿⣿⣷⢿⣿⣷⡄⠀⢉⠅⢐⢀⡀⡀⠀⠈⠛⠃⢸⡀⠀⠼⢼⠿⡻⠟⠛⢿⠃⠀⠁
⠀⠀⠀⢡⡁⣈⠾⣤⢃⢳⣾⠶⡿⡿⣷⣿⣯⠤⠀⢀⠶⠀⠀⠘⠢⠿⠻⠃⠀⠃⠂⠠⢘⣖⠇⢀⡤⠠⠤⡈⣧⠿⠶⠃⢀⡠⠀⠀⢀⢀
⠀⠀⠀⠀⢻⠑⢉⣭⠩⠋⢸⣴⣱⢻⣷⣿⡏⣰⡟⣿⣶⣤⡀⠀⢀⠀⠀⠠⡀⠉⠃⠀⠉⠁⠀⠘⣶⣖⠒⠃⡿⡇⡄⠀⠈⠁⠂⠈⠁⣢⠂
⠀⠀⠀⠆⣘⠚⠋⠁⠀⠀⢀⣻⢽⣷⣽⠋⠀⣡⣾⣿⣿⣿⡿⠀⣈⡁⠒⡲⢅⠈⢀⢈⡇⠀⣼⢹⡸⠏⠓⠀⠹⠉⠆⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠃⢨⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⣯⣿⡿⣿⣧⡄⠻⠿⡤⠒⠌⠀⠓⠉⢼⡈⣣⢴⣷⠂⠀⢧⠄⠀⠀⠘⠓⡈⠓⠀⠀⠀⡴⠀⠛
⢀⠀⠐⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⠻⡿⠻⠷⠂⠴⠤⠆⠀⠀⠈⣀⡄⠉⠉⢄⠏⠀⠀⠀⠀⠀⢀⡄⠋⠉⠉⠛⣂⠀⠁⠄
⡘⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⡷⠶⠀⠐⠀⠀⠀⠀⠀⢠⡌⠚⠠⣀⡀⠃⡄⢨⠉⢰⡿
⣤⣬⣃⠨⢧⡄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⠲⠁⣀⣀⠀⠀⠀⠀⠀⠺⠆⠀⠠⠌⢿⣿⣀⣭⡆⢈⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠩⠤⠰⢒⣲⣲⣟⣉⣡⣆⣖⣶⡶⡤⢄⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⣿⡷⠴⠛⠻⠿⠿⢿⢟⢻⢛⡛⠛⠃⢠⣤⣾⠃⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠂⠈⠑⢾⣿⢿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡤⠊⠀⡠⣿⡟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⠕⠶⣤⣀⡀⠙⠻⣿⣿⣿⣹⣿⣿⣿⣿⠿⠋⠀⣀⣬⣤⢟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡿⣿⣶⣤⣀⡉⠛⢷⣆⡈⠛⢿⣿⣽⡿⠛⢡⠴⠶⠛⢉⣩⣴⠞⠋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣴⣬⡿⠀⠀⠉⠛⠛⠿⢷⣬⣥⡶⢤⣽⢫⣴⢶⣀⣤⡴⠶⠛⠉⠀⠀⠈⠁⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣱⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⠊⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠠⠟⠻⢏⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢎⢽⡄
⠀⠀⠀⠀⠀⠀⣗⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠻⡀
⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡰⠀⠀⠈⠥⠀⣧
⠀⠀⠀⠀⠀⣿⠘⠄⠄⠈⠳⢘⠲⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠋⢻⡤⠂⠐⠠⢰⣽
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠐⢮⢀⣦⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⠞⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡿⠃⠁⠀⠀⠀⠀⣴⣆⣿⡜⠇⢠⡐⡠⣠⠀⠘⢁⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢻⣾⡿⠄⡖⡶⠆⠀⠀⠐⣿
⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠚⠟⢻⡿⣟⣿⣒⣛⣛⣈⣀⢀⠏⢐⢄⡟⠁⡽⠋⠉⠱⣖⠶⠺⡗⡻⢿⡇⡇⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢇⡗⠀⠐⠠⠶⢻⣿⡝⠫⠍⠀⢹⠟⠛⠇⡮⠀⠴⠮⠒⠀⠀⠀⠀⠈⠁⠀⠀⠫⠞⢹⡗⠦⠄⠀⠀⠀⠀⢰⡾
⠀⠀⠀⠀⠀⢸⠜⠉⠀⠀⠀⠀⠸⠋⠙⠀⠁⠀⠀⠀⠀⠀⢳⣦⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣽⢿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢺
⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡑⢀⡻⣿⠛⣛⠩⣶⢤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡄⡀⠀⠀⡀⠄⠀⢨⡴⣟⣫⣻⡿⡖⠱⠘⠈⠁⠹⠃⢽⡠⣢⣠⣔⠀⢀⡀⠀⠀⠀⠀⣧⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⡲⡶⣿⡳⠛⠟⡿⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⡙⣲⠤⢆⣠⣤⣶⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢩⠏⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉⠉⢡⢁⣞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠋⠄⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠈⠀⠚⠛⠛⠒⠒⠒⠠⠒⠚⠟⠉⠉⠛⠊⠉⠁⠀⠀⠀⠀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠄⠄⠖⢦⠐⢶⡶⠶⣶⣠⢶⣶⣶⡶⠲⠹⠟⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠉⠀⠘⡁⠀⠠⢀⠠⠀⢀⡃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⢀⠠⠀⠀⠑⠘⠀⠀⠀⠀⠀⠂⠀⠀⠴⣆⠀⠀⠂
⠀⢁⠀⠄⠴⠤⣒⡄⣔⡢⠼⠀⠀⡘⠀⠀⠐⢶⢦⣧⡤⣐⣀⡀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡴⠋⡅⡭⠘⠄⠁
⠀⠈⠀⠀⠐⠂⠁⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠰⠜⠒⢄⡈⠉⠁⠢⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡁⠀⣀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⢤⠉⠇
⠀⠀⠀⠐⠂⠀⢀⠀⢀⢀⢀⠃⠀⠀⠀⠀⠀⠀⠘⠀⡀⠉⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠂⢩⢄⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠠⠁
⠀⠀⠀⠀⠀⠪⠉⠙⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⣠⠃⠀⠰⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠁⢀⣀⣀⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠐⠚⠫⢤⡈⠙⠫⡙⠲⣤⡀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠞⠁⠀⠀⠀⠀⠀⠈⢦⠀⠘⡄⠀⠙⢦⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣀⣠⠄⠀⢠⣤⣄⣀⠀⠀⠀⣠⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠁⠻⠟⠀⠀⠸⠟⠁⠀⠁⠀⢀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠶⠄⠔⠂⠀⠓⠒⠒⠂⠀⠀⢸⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣠⠤⠒⣸⠭⠍⠛⠛⠓⠲⠤⠀⠀⠀⠈⣳⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣯⠟⣋⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣮⠛⢃⠴⠊⠙⠲⢤⣀⣀⠀⠀⠀⢀⣀⡤⠖⠉⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣊⠷⢈⡤⠚⠁⠀⠀⠀⠀⠀⠀⠉⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠑⠈⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠳⠈⠳⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⢹⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠘⢦⡀⢆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⣾⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠠⠁⠈⡆⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠀⠀⠀⢠⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡇⠀⠀⣴⠧⠤⠶⠒⠶⠾⠥⢾
⠀⠀⠀⠀⠀⠀⢠⣤⣀⣤⢎⡡⠤⠤⣴⣶⣦⣭⣭⣭⣭⣭⣭⡙⡏⠙⠻⢦⣠⣤⣤⣤⣤⣤⣤⣤⣤⡝⢦⡇⣠⢂⣼⡄⠀⠀⠀⠀⠀⠀⢺⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⣧⠀⠀⢀⡀⣤⠀⠀⠀⠀⠀⠀⠘⢦⣸⡇⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠰⣬⣀⡀⠩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠳⠖⠄⠲⡇⠀⠀⠀⠀⢠⡀⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠼⡾⣒⠒⣒⣚⠁⠈⠍⠀⠀⠀⠀⠀⡀⢤⢤⡀⠀⠀⢀⠴⢎⡸⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣥⠋⠉⠁⠘⢆⠀⠀⠀⠀⠀⠀⠸⣛⡓⠞⠁⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠤⠰⠂
⠀⠀⠀⠀⠀⠀⠸⣍⣒⣒⣊⠠⢹⠤⠤⠤⠒⢻⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⢇⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠸⠀⠀⠀⠀⢘⠀⠑⢤⣏⡉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⣒⣒⣉⠍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⢀⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠘⠁⠀⠀⠀⣈⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣋⣡⣴⡾⠇⠀⢠⣌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣤⣶⣶⣿⣿⣿⣿⠋⣴⣶⣦⡀⠹⣆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⠇⠀⠁⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⣛⠛⠻⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⢠⣿⣿⣿⣦⠘⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⠊⡉⠽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠀⠃⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠷⢤⣤⣀⠀⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠻⠶⢦⣄⡀⠉⢻⣌⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⡴⠞⠂⠀⠀⠀⠀⠈⠙⠳⢦⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠻⣌⣿⠓⠶⣤⣀⡀⠀⢀⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠉⣙⢷⣄⠀⠀⠀⠀⠈⢷⡀⠀⠀⣸⠏⠀⠀⣤⠟⠙⢳⣤⠹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⢸⣿⢛⣿⡦⠶⠶⠒⠲⠶⠦⣄⣰⠏⠀⠀⣾⣄⣠⡾⠛⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣄⣀⣀⢀⣹⣿⣥⡀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⣸⠓⣾⠏
⠀⣤⠄⣀⡄⠀⠀⢀⣀⣤⣤⡴⠶⠶⠶⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⢿⣗⠀⠀⣀⣀⣤⠶⠟⠁⠀⠀⢰⢏⣽⠷⠒⠚⠓⠳⣦⡀⠀⣠⠀⢀⡀
⡼⠁⣾⠋⢀⣤⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⣴⠄⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣾⠋⠁⣠⢴⣶⠦⡤⢸⡇⠀⢸⠀⢸⠃
⠃⢸⡇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠾⠓⠋⠀⠀⢠⡾⠃⢀⡾⢀⡟
⠀⠘⣇⠀⢻⣆⠀⠀⣤⠶⢂⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠈⠀⠚
⠀⠀⠈⠀⠀⠙⠳⢶⣤⣀⣉⣁⣀⣉⣁⣀⣠⣤⡶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡬⠉⡉⢹⡿⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⡇⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠾⠀⠀⠀⢀⣀⡤⠖⠛⠙⣿⣓⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⠀⠀⠀⠀⠀⠶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⢻⡭⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣹⠟⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣄⡤⠚⣁⡠⠤⠤⠤⠤⠤⠤⣄⣸⣿⣛⣂⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣉⠀⢤⣿⣀⣀⠀⠀⠀⠒⠻⢤⣄⣿⠀⢠⡟⠙⢷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣼⣧⠀⣿⠟⠒⠒⠒⠒⠶⢤⣤⣀⠀⠈⠉⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠑⠀⠉⠉⠉⠛⠛⠒⠒⠲⠞⠛⠖⠾⠷⠶⠶⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣩⣥⣤⣤⣾⠞⠉⠉⠑⠲⢤⣀⠀⠀⠀⠈⢙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠁⠀⠀⠈⣿⠀⠀⢷⠀⣄⠀⠙⢷⡀⠀⣠⣟⠻⣟⡳⠦
⠀⣠⠆⢀⣴⠀⢀⡴⠋⠀⢸⣏⠉⣉⣹⣟⣛⠛⠛⠛⠛⠛⠓⠒⠒⠶⠶⠶⠶⠖⣲⡶⠛⠁⠀⠀⠀⠀⠀⢀⣼⣯⠀⢀⡿⠀⣿⠀⠀⠈⢷⣾⡟⢌⡳⡄⠙⠓
⣼⠁⣴⠟⢁⣴⠋⣀⡤⠶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠒⠶⠦⣤⣀⣴⠞⠉⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢻⡄⠈⠀⠀⠁⠀⠀⠀⠀⠳⡸⣆⠙⠂
⠇⢸⡏⠀⣿⠁⠞⠉⣴⠃⠀⠀⠀⢀⣀⣀⣤⣤⡴⠶⠶⠖⠛⠛⢛⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈
⠀⠀⠙⠂⠉⠛⠲⠾⠷⢶⡶⠿⠛⢻⡏⠉⠛⢶⣄⡀⠀⠀⠀⣰⠏⣽⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⣷⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⠁⠀⠀⢸⡇⠀⠀⠀⠈⠛⠷⣤⣀⡟⠀⢹⡄⠀⠀⣀⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣿⠀⢠⡀
⠀⠀⠀⠀⢠⡗⠀⣿⢰⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡷⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⢸⡁
⠀⠀⠀⠀⣾⠁⣸⠇⣼⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⠸⡇
⠀⠀⠀⢸⣇⠀⢻⡄⠛⠶⣤⣄⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡏⠀⢸⣧⠀⣿
⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠉⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⡶⠖⠛⠛⠛⠉⠉⠁⠀⠶⠚⠉⠁⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠉⠉⢙⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⢉⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠶⠶⢾⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣤⣤⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠳⢶⡀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣀⣠⡼⠷⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡞⡓⠦⠟⠑⠛⠀⠀⢹⡆⠀⠀⠀⠀⢀⣼⢯⣍⣙⡶⠶⠟⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠛⠲⠶⠤⠤⠤⢤⣤⣤⣼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡤⢀⠴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠉⠠⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠳⣤⣤⣤⡤⠴⠚⢁
⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢀⣀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠛⠀⠑⠒⠒⠒⠛⠋⠉⡠⣄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠒⠚⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠶⠶⠖⠒⠒⠒⠒⠶⠤⢬⣽⠓⠦⢤⣤⣀⣀⣀⣀⣀⣤⠴⠶⠛⡏
⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠀⣤⠀⠹⠓⠒⠶⠦⠤⠤⠤⠤⠶⠒⠚⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢠⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠶⠀⠀⠀⠀⠿⠀⠀⠀⠘⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠰⠆⠀⠠⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠤⣀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡨⡳⡀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠷⢵⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠞⡴⢡⠀⠀⢀⣀⡠⠜⠣⢊⣥⡶⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⠀⠀⠀⠀⢹⢼⢣⡇⣤⢞⠥⡀⠀⠀⠀⠞⣡⠤⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠎⠀⠀⠀⠀⠀⣀⡀⠤⠤⢼⡀⠙⠛⠁⡆⡆⠀⠀⠀⠀⠔⠒⠀⠀⠀⡸⠲⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒
⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠈⠓⢄⡀⠑⠆⠀⠀⠀⠀⠀⠀⠀⡔⡏⠀⠀⠷⢦⡈⠁⠒⠀⠤⠤⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣀⠔⠂⠉⠉⠉⠑⠢⣄⠘⢦⡀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⡆⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠑⢄⠀⠀⠀⣀⡴⠛⢃⠀⠀⠠⠳⡄⠀⠀⠀⠀⠈⠉⠒⠂⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠲⠶⢟⠉⠀⠀⠸⢄⣀⢴⡀⠘⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠈⢢⠀⠀⢀⣀⠀⠘⣷⡀⠀⠙⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢣⠀⠈⠁⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⢇⡇⠀⠀⠀⠀⠀⢹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⠁⠀⠀⠀⠀⠀⠀⠆⠻⣷⣄⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠙⠷⣤⣀⣴⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠟⠛⠛⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣤⣤⣤⣤⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁


⣶⣶⣶⣶⣾⡿⢷⣶⣶⡶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣶⣿⢷⡶⠷⠷⠾⢷⣾⡶⠾⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣶⣶⣤⣄⠀⠀⠀⢹⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣿⣾⣿⣿⣿⣿⣿⣆⠀⠀⢳⡀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡇⠀⠀⠀⠀⢠⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⣵⣭⣛⡇⣩⣭⣭⣭⡹⡇⠀⡞⠁⠀⠀⠀⠀⢸⡇⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡅⠘⠇⠀⠤⠤⠤⠤⣼⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣸⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣭⣝⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢺⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢼⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⢸⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⠀⠀⢀⣴⣿⠇⣟⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⢸⣿⡇⣿⣿⡧⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⠘⢿⣷⣬⣭⣵⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⡀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠸⠇⠹⠇⠿⠸⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣉⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢉⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⣡⣤⣤⣄⠙⢿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣧⡈⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⠘⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣄⠙⠻⢶⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⢋⣴⣿⣿⣿⣿⣿⣿⡿⠏⠐⠶⣦⡙⢷⣄⠹⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⣿⣿⣿⣿⣿⣿⣿⠋⣤⣶⣦⡄⠈⠿⠀⢻⡄⢹⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⣤⠈⢉⡉⠐⣿⣿⣿⡿⢀⣶⣶⣤⣤⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠹⣧⡈⢷⣄⡈⠉⣉⣠⣾⡿⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣦⡘⢷⣤⣉⠛⠠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⣌⠙⠻⠂⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠛⠛⠃

⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠰⠄⡈⠗⠈⠁⠐⣁⡠⠀⠀⢴⠋⠀⠀⠈⠑⠢⢄⡀⢀⡀⠆⠒⠒⠒⠶⣀⠀⠀⠀⣠⠺⠚⠰⠷⢄⠄⠀⠀⠘⡀⠢⠵⠀⠆⣀⠔⠀⠀⠀⡄⡈⢁⡱⠀⢠
⠀⡤⠤⠁⠀⠀⠀⠀⠈⠤⠆⠀⠘⠖⠖⠂⠦⣀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⡸⠀⠐⢸⡍⠀⠀⠂⠀⠻⠄⠀⠀⡐⠃⠀⠀⠀⠀⠀⡣⠀⠀⠀⢀⢗⠈⠃⡇
⠀⢀⡰⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠈⠉⠀⠀⠉⠉⠁⠁⠀⠀⠈⢫⣄⢀⣀⡰⠛⠈⠀⠀⢑⠅⠀⠀⠀⠀⠀⡤⠈⠀⠀⠈⠀⢤⡄⠀⠰⠆
⠀⠀⠀⠞⠀⣀⠀⠢⠄⠀⠀⢠⠀⠒⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠎⠈⠱⠂⠀⠀⠀⢀⠀⠀⠍⠈⠐⢀⠀⠀⠀⠄⠸⠰⢀⡄⠦⠉⡁⠀⠀⡀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠘⠠⡈⠂⠀⠀⠀⠀⢰⠄⠀⠂⠈⠒⠀⠀⠀⢀⠀⠀⣐⠆⠞⠠⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠟
⠀⠀⠀⡔⠒⢢⠀⠀⠀⠀⠀⠀⠢⠀⠈⡀⠀⡈⠀⠀⠠⠐⠄⠠⠂⠀⠆⠀⠁⠄⠀⠀⠉⠀⠀⣩⠄⣀⣀⠓⠠⣠⠀⠀⣀⣀⡀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠠⠐
⠀⠀⠰⠀⠠⠀⠀⠀⠴⠀⠠⠁⢰⢐⠈⠡⠀⠀⠀⠈⠈⠀⠈⠀⠀⠠⡄⠄⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠀⠀⠱⢵⣄⡈⠉⠛⠲⣤
⠀⠀⠂⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⣀⣀⡀⠀⠤⠤⠐⠉⠉⠢⡄⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠑⡥⠀⠀⠀⠂⡀⠀⠔⠂
⠀⢘⠀⠀⢠⠃⠀⠰⠉⠈⠱⠀⠀⠀⠀⠀⠒⠀⠀⡄⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⡸⠀⠀⣠⠮⠃⠘⣵⠀⠈⡗⠀⠀⠀⠀⠀⠀⠁⠀⢁⠂
⠀⠈⢲⠐⠁⠀⠀⡄⠀⠀⠌⠀⠀⠀⢀⠀⣀⠀⢂⠈⠀⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡀⠠⠄⠖⠁⠀⢬⠋⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠈⠈⠄⣅⢆⠂
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⢀⠀⠈⢀⠀⠀⠀⠆⢠⠀⠈⠑⠒⠰⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠰⠜⠛⠋⠀⠰⠀⠀⢸⠀⠂⠀⠘⡄
⠀⠀⠂⠀⠀⠀⠀⡄⠀⡸⠀⠀⠀⠈⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⡀⢀⢀⣀⣀⣤⣴⣶⣾⣶⣶⣤⡀⠀⠀⠀⠋⠀⠀⠀⠰⠓⠁⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣤⠀⠀⡗⠀⡀⠀⠀⠀⠀⠀⣴⣾⡿⠟⠟⣛⠻⠻⡟⠟⠉⠹⠿⣿⣏⠉⡹⢿⣿⣿⡆⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡄⣀⠀⠢⣀⣄⡀⢁⠠⠆⠀⠠⣼⡏⢸⢰⡄⠀⠈⠰⢧⠙⡆⠹⡿⠷⠂⢹⣯⣥⣾⡟⠻⣿⠀⠀⠀⠀⠀⠀⠀⡀⢡⠀⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡁⠀⣋⠀⢀⡀⠀⢰⣿⡇⠀⣸⣇⣸⣤⣷⣴⣷⡴⢾⡿⠶⢾⡟⢛⠛⠙⡆⢁⢻⡇⠀⠀⢀⠄⠡⠂⠀⠄⠐⠌⠁⢄⡰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠋⠁⢀⡀⠀⠀⠐⣿⠟⡛⢿⠉⡝⢿⠀⣦⣶⣶⠸⣇⠘⠛⣣⡈⠠⣆⠁⢸⣾⡷⠀⠀⠸⡀⠀⢀⠈⡀⠉⡡⢀⠀⢀⡀
⠀⠀⡴⠒⠒⠲⣄⠀⠀⠀⠀⠀⠈⠁⠀⠴⠀⠀⠀⠀⠀⠀⡹⣧⠡⣜⠃⠰⣄⠡⡙⠋⢹⣧⢹⡄⠙⠛⢣⢠⣈⣀⣼⣿⠃⠀⠀⠀⠀⠀⠀⢄⠀⢠⠈⠢⠬⠤⠃
⠀⢼⠀⠀⠀⠀⢈⡄⠀⢀⠀⠴⠀⠀⢤⠀⢠⠀⢀⡀⠀⠀⠀⠨⠛⠶⢾⠶⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠋⠁⠀⣀⠀⡔⠆⠀⠀⠀⠑⠂
⠀⠘⢦⣀⣀⣀⠜⠀⠀⠀⠀⠀⠀⠓⢤⠦⢤⡔⢁⡀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡬⣃⡟⢠⣌⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⠀⠀⠀⣀⠤⠤⢄⡀⠘⠀⠀⠹⢄⣠⠏⠀⣀⠀⢀⡄⠀⡀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⢖⣄⣸⠶⠛⠉⠉⠀⠀⠀⠠⡀⠐
⠀⢘⠁⢸⠆⡼⠁⠀⠀⠀⠘⡆⠀⠒⠀⡐⠀⠦⠀⠈⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁⡰⠋⠙⡆⠀⠀⠀⠐⠢⠼⠛⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠀⠀⠁⢀
⠀⠈⠉⠉⠀⢇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⡎⠀⠈⢡⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⣠⢠⣤⣤⣤⠲⣾⣩⠻⡍⢣⠀⠉⠀⠀⡀⠀⠀⡠⠀⢀
⠀⠀⠄⠀⠀⠈⠓⠤⡤⠤⠊⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠀⠑⠄⠔⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠐⣩⡆⣿⡞⠷⠍⠛⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠘⠀⢀⡀⠀⡀⠀⠀⠤⠀⠀⡀⠀⠀⢀⡀⢀⡀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠄⠀⠈⠀⢈⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⢀⠂⢒⡠⡀
⠀⠀⣐⠂⢘⠀⡀⡎⢃⢩⡩⠤⠼⢄⡠⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠄⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⢀⡀⢀⠄⠲⠒⠠⡀⢀⠀⢀⠎⠂⠱⠀⠀⠼⢈⠆
⠀⠆⢨⠖⠉⠉⠈⠁⠈⠀⠀⠈⠉⠀⠓⢼⠀⠀⠀⣠⠴⠟⠉⠋⢹⡆⠀⠀⠀⣠⠖⠑⢠⠅⠂⠀⠀⠀⠁⠀⠀⢨⡠⠀⠀⠀⠐⣩⠀⠀⢨⢰⠀⡄⠀⡄⠀⠰⠈
⠀⡌⢸⡀⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀⢫⡉⠀⢠⠞⠁⣠⠤⠤⠖⠛⠀⣀⡠⠔⠁⠀⢀⡜⠀⠀⠀⠒⠀⠀⠀⠐⠨⢴⡄⠀⠀⢄⠈⠀⠀⠀⠢⡑⠄⢀⠀⢔⠡⠂
⠀⠀⣃⠜⡘⠤⠤⠄⢂⠠⡌⣤⠗⠈⠆⠀⠀⢯⡼⠊⠁⠀⠀⠀⢰⡋⠉⣀⣀⠤⠒⠉⠀⠀⠀⠠⠆⣀⠀⠀⠀⠀⢠⠂⠋⠣⠁⠀⠀⠀⢀⡀⠀⠁⠀⠈⠁
⠀⠀⠀⠈⠀⠆⠖⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠖⣀⡤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣰⡧⠊⠵⠚⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁⠀⠀⡞⠀⠀⠉⠀⠀⠀⠀⠀⠠⠤⢭⡵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡅⠀⠀⠀⠀⠀⢀⣀⣐⢶⣒⠒⠺⠆⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠶⡾⠍⣯⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⣉⠀⠀⠀⠀⠀⠰⠭⡀⠀⠘⣦⠒⠒⢤⡀⠀⣈⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢩⠞⠁⠀⠀⠀⠀⠀⠀⠐⢾⡆⠀⢻⠀⠀⠀⣹⣞⠁⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⢧⡠⠚⠉⠸⣍⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡇⣤⡀⠀⠀⠀⠀⢀⡴⣸⡁⠀⠀⠀⠙⠲⢤⡼⠁⢈⡉⠉⠉⠒⠒⠒⠒⠒⠒⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠇⠹⣾⠦⠶⠒⠻⡤⠝⣇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⢶⠢⠄⣀⣀⣀⣀⡠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡀⠀⢀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⡯⠜⢒⡹⠿⠛⢦⡀⠀⠀⠀⢀⡠⠤⠄⣲⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡄⠀⠷⠚⡇⠀⠀⠀⠀⠑⢤⡀⠀⠁⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⢱⡀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠉⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡄⡀⠀⠀⢣⠘⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢠⠊⡸⠋⠀⠀⠘⡀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⡰⠃⢠⡃⠀⠀⠀⣠⠓⣼⡛⣲⣀⡀⡰⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣀⡜⠁⠀⠈⠛⠒⡞⡩⠏⠘⠓⠢⠼⡁⠞⠁⡀⢹⠈⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠙⠲⢤⣀⡀⢠⠁⠗⡆⠀⠀⠀⣾⢸⢲⣴⠋⠁⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠤⠶⢚⡿⠍⢺⢨⣼⢳⣤⣦⡙⢧⡘⣸⠧⣤⡀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡩⢦⠀⡇⢠⢚⡀⠀⢸⠀⠀⡘⠒⠚⡜⡈⡄⠷⠼⠉⠉⣗⠇⢸⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠋⣗⣰⠀⠁⡟⠞⠁⠀⣠⠇⠀⢰⠁⡇⡇⡅⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⣠⠞⢼⣿⠦⣄⡈⠉⠉⠁⠀⠀⠀⠦⠴⢀⡇⡇
⣠⠴⣖⣴⡖⣆⠀⠀⠀⠀⢀⠀⠀⢀⠔⠁⠀⢉⠟⠁⠀⠀⠀⠀⠀⠉⠑⠒⠶⣤⡤⠤⠤⠬⠟⠘⠁
⣸⣈⣀⣋⣈⠷⠀⣞⡉⠋⢡⡯⡀⠫⠤⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠒⠢⠴⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣤⣜⣉⠁⠈⠉⠉⠙⠒⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢂⣉⣉⣉⣉⡐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⠴⡞⢇⠀⢀⡜⡇⠈⠑⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡴⠃⠀⠘⢎⠳⣘⠁⠇⡠⠖⡡⠛⢆⠱
⠀⠀⠀⠀⣀⣀⡀⠤⠤⠤⠤⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠠⡆⠰⠇⠤⠤⠤⠬⠧⠙⠀⢋⣀⠊⠀⠀⠈⡄
⣠⡖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠉⠙⠐⠒⣶⠄⣠⡄⢠⣤⡍⡉⢉⢐⡃
⡇⠉⠛⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⣘⣄⢀⣤⠾⠷⠟⠃⣿⢄⠳⡀⠀⠀⢠⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢍⠉⢊⡁⠊⠀⢠⡄⣿⠈⠳⡙⢄⡠⠋⡰
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠑⠤⣀⡀⣇⣿⠀⣠⠼⠋⣡⠞
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠀⢈⣿⠀⠤⠒⠋
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⢸⡀⢀⣄⣀⠀⠀⠀⣀⣀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢄⣀⠀⠀⢸⠇
⠀⡇⡏⠀⠈⠉⠉⠉⠀⠀⣀⠤⠒⠛⣗⠢⢤⣀⣀⣀⣠⠤⠒⠋⠁⠀⠀⠀⠀⠈⠳⡄⣸
⠀⡇⢹⠀⠀⠀⠀⢠⠖⠉⠀⠀⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿
⠀⢳⢸⠀⠀⠀⠀⡆⠀⠤⠒⠉⠙⡄⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡇
⠀⢸⠈⡇⠀⠀⠰⡀⠀⠀⠀⠀⠀⠈⠂⢠⠸⣀⣀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣸⢸⡇
⠀⠘⡆⡇⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣠⢳⠁⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⡿⣸
⠀⠀⡇⢻⠀⠀⠀⠀⠱⡌⢦⢀⣠⠴⠚⠁⢸⠀⠀⢸⠃⠉⠁⠀⠀⢹⠀⠀⠀⢠⡇⣿
⠀⠀⢣⢸⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢣⡇
⠀⠀⢸⠈⡇⠀⠀⠀⠀⠀⠀⢠⠞⠉⠒⠤⣈⣆⠆⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⣾⢸⠃
⠀⠀⠈⡆⣇⠀⠀⠀⠀⠀⢠⠏⠀⠀⣆⠀⠀⠉⢳⠦⠀⠀⠀⠀⠨⠜⠀⠀⠀⡇⣾
⠀⠀⠀⣇⢸⠀⠀⠀⠀⠰⠃⠀⢀⠎⠀⠉⠒⠄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡏
⠀⠀⠀⢸⠘⡆⠀⠀⠀⢠⡗⠀⠈⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⠃
⠀⠀⠀⠘⡄⣇⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾
⠀⠀⠀⠀⣇⢸⡀⠀⠀⠀⠀⠈⠑⠢⢄⡰⢊⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡇
⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠁
⠀⠀⠀⠀⠸⡆⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡟
⠀⠀⠀⠀⠀⣷⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⡇
⠀⠀⠀⠀⠀⢸⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼
⠀⠀⠀⠀⠀⠀⢷⠘⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠘⡷⢦⣈⠛⠲⠶⠤⣤⣤⣤⣤⡤⠴⠶⠚⠋⣁⡤⢞⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠉⠛⠲⠦⠤⢤⣤⣤⣤⠤⠴⠶⠚⠋⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣶⣶⣶⣶⣶⣶⣾⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⢀⠀⠀⠀⠀⠺⣷⣶⣶⣤⣤⣀⠙⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⢘⣉⡀⠀⣒⠀⠄⢹⣿⡉⠛⠻⢿⣷⠘⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠤⠒⠂⠀⠉⠀⠀⢸⣿⣿⣿⣷⣦⠹⠇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣧⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁




⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⢠⠀⠄⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢠⣆⠂⠤⠀⠠⠒⣸⠀⡠⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠎⠁⠉⠀⠶⠀⠀⠀⢉⠆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⢀⠃⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠃⠂⠠⠀⠂⠃⠀⠀⠀⡅⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠔⡊⡀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⢐⣉⣁⠀⠈⣉⡇⠀⠀⠁⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠈⡄⡠⠄
⠀⢀⠤⠒⠈⢁⠊⠈⠀⠁⠰⠀⠉⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠄⠈





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⢈⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠇
⠀⠀⠀⡀⠀⠁⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⣉⠔⠁⢘⠀⠀⣀
⠀⢰⠉⠁⠀⠀⠀⠀⠈⠉⡁⠀⠀⠀⠨⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡠⠔⠊⠀⠀⠀⠘⡀⢌⠁⢉⢂
⠀⢠⠀⡃⠁⠀⠀⢸⠘⠀⡇⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⡨⠈⠈⠁⢸⠗
⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⢆⠁⡆⣒⡂⠘⢢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⠀⢀⠀⡠⠀⠀⠈
⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃⠀⠀⡈⠀⠀⠀⠀⠜⠈⠑⡀
⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣬⣧⣯⣤⣤
⣿⣿⣁⣍⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⡻⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣄⠀⠀⠀⠀⠀⣀⣀⠀⣠⠀⠀⠀⢀⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠁⠀⠀⠀⠀⠈⢉⡹⠿⠟⠀⠀⠀⠀⠘⠏⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡆⠀⠀⢴⣦⠀⠀⠀⣀⣰⠈⠀⠀⠀⢀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢊⠉⣀⣤⡄⠀⢀⡀⠀⠚⡛⠁⠀⠈⢻⣾⣧⠀⠀⠰⢻⠃⠀⠀⠀⠀⠀⣶⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣢⣤⣽⣿⣶⣿⣠⠱⣦⡆⠀⢿⣿⠀⠀⢿⡏⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣦⣾⣿⣿⣿⣿⣿⣷⣷⣎⡟⠻⢶⣿⣿⣶⡶⢠⡄⣠⠀⢿⠇⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠰⠳⢦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⡛⣾⣏⡓⡸⣦⡍⢀⣀⡀⠀⠰⣄⣠⠀⠦⠾⠷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠛⠀⠀⠰⢿⠇⠀⠉⠛⠃⠀⢀⠀⠀⠀⠀⠀⠀⢒⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠸⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡴⠀⠀⠀⠀⠘⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠞⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠠⡞⠋⠀⠈⣻⠿⣿⣿⣿⣿⣿⣿⡿⠿⣏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢸⡏⠉⠉⢹⣄⠀⢙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⣠⠞⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠈⣀⡴⠚⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⡏⣴⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⢀⡴⠒⠦⣄⡤⠦⣄
⠀⠀⠀⠀⠀⢰⠀⠹⣜⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠃⠀⠀⢀⡏⠀⠀⠀⠈⠀⠀⣸⠧⠤⣀
⠀⠀⠀⠀⠀⠀⠆⠀⡏⠳⣤⠖⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠀⠀⠒⠋⠁⢸⠆⠀⠀⠀⠀⠘⠁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡦⠀⠀⠙⠛⠛⢯⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠉⠙⠲⠬⣽⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣷⠀⣷⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠞⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣀⣀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠸⠁⠀⠀⠀⠀⢸




⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠇⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⢄⢠⠠⠤⠀⣄⠀⡤⡄⠀⠀⡤⠄⠀⡠⠀⢠⠄⠠⢤⢀⠄⢄⢠⢠⢠⠤
⠀⠀⠀⠀⠀⠀⢀⣉⣀⣀⡠⢹⠀⣀⣸⠘⣆⣏⡇⣀⣀⣏⣁⠀⣈⡶⢸⡀⣈⢱⡸⣀⡘⣸⢱⣘⣉⡀
⣿⣿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠲⠤⢄⡤⠴⠒⠒⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡆⠀⠀⠀⠀⠀⣠⢋⡜⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⡴⠒⢄⢀⣠⠗⠉⢀⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠑⠒⣺⢡⠃⠀⢀⡎⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠊⠑⠤⠖⠁⠘⠦⠴⠋⠳⠔⠃⠘⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡟⠀⡠⡄⢠⣤⠀⠀⣠⡉⠀⠀⣠⡦⢦⠀⠈⣿⡔⠒⠆⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠁⠈⠀⠈⡏⠀⠀⢀⡷⠁⠀⠀⡸⠀⡼⠀⢀⠇⠘⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠣⠔⠂⠀⠣⠤⠴⠊⠳⠴⠚⠐⠃⠀⠘⠒⠊⠀⠀⠉⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠦⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣭⣀⣀⣸⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢋⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠇⠀⠀⠀⠀⠀⠀⣼⡏⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⡁⣷⠀⠀⠀⠀⢀⣠⣤⣼⣿⢤⣤⣀⡀⠀⠀⠀⣿⠳⣜⠦⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢱⠟⢠⣿⡄⣠⡴⠞⠋⠉⠀⠀⠀⠘⠃⠈⠉⠓⢦⣀⡇⣀⣬⡷⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣾⣯⣤⣻⡇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⢲⡄⠀⠹⠿⣏⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣷⡙⠇⣿⠀⠀⠀⠘⢷⣀⣠⠾⣧
⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠏⠀⠀⠀⠀⢸⡟⠛⠾⠋
⠀⠀⠀⠀⠀⠀⢀⡈⣷⣤⣀⢰⣷⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⣿⢸⠀⣉⣭⣿⠗⠀⢀⡾⢋⣠⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠯⠤⠴⣶⡟⢻
⠀⠀⠀⠀⠀⠀⠹⣧⡾⠛⠉⠀⠘⢧⡀⠸⣧⣈⣩⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⢶⣀⣟⡶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠈⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠈⢿⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣲⡶⠋⣷⣤⡀⠀⠀⢰⣦⡴⠚⠁⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣄⣀⣿⠈⠉⠉⠉⢹⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡴⠭⠙⢻⡆⠀⠀⠀⢸⣗⣠⠿⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⢸⣯⣵⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⠷⠶⠀⠀⠀⠀⠙⠛⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠛⠻⠟⠋⡡⠤⠀⠉⠀⣴⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢄⠀⠈⣧
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠁⠀⠠⠀⣼⣿⣶⠀⠀⠀⢀⠀⠀⠀⠀⡂⠀⠸⡆
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣄⠀⡠⠊⠀⠀⠀⠐⠀⠀⠀⢷
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣇⠀⠀⠀⣀⣀⡀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣷⣄⠈⠉⠉⠁⣀⣴⣿⡿⢿⣿⣿⠓⣇⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠀⠀⠀⠉⠋⢉⡀⠄⠁⠀⠀⠹⣆⢀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠆⠀⠀⠀⠀⠁⠉⡽⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⢉⢡⠐⠀⠀⠀⠀⡄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡁⠈⠙⠛⠁⠀⠀⠀⠁⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⢠⠤⠤⢠⡀⢀⡄⣤⠤⠄⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⢸⠓⠒⠈⢧⡼⠀⣿⠒⠂⣾⠲⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠀⠈⠁⠀⠉⠉⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⠠⠐⠒⠒⢦⠖⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠄⠚⠠⢀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠁⡔⠁⣐⡎⠀⠀⢩⣒⠙⢄⠀⠄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⣦⣃⠠⠀⡖⠉⠉⢣⠄⠠⣘⣴⠀⣀⡀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⠂⠁⠀⠀⢀⣾⠃⠔⡟⣰⢻⠘⠆⢛⠀⠘⣧⠀⠀⠈⠐⡀
⠀⠀⠀⠀⠀⠀⢠⠒⠀⠐⠁⠀⠀⢀⣠⣎⠉⢠⠎⢠⠇⠈⠀⠘⡀⢦⠂⠙⣧⡀⠀⠀⠈⢂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⡴⠋⠁⠈⣱⠏⠀⡞⠀⠀⠀⠀⢃⠈⢖⠈⠉⠻⣦⡀⠀⠀⠡⡀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⣠⠎⠀⠀⠀⣰⠃⠀⢰⠁⠀⠀⠀⠀⠘⠀⠘⡄⠀⠀⠈⠻⣆⠀⠀⠐⡄⠀⠐⡄
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠐⠏⠀⠀⠀⠀⢧⣤⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠉⠻⣦⣀⣤⠞⠋⠉⠁⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠤⠀⠊




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠐⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣁⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠯⢥⣉⠓⣄⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠒⠒⠀⠀⢄⠀⠀⢎⢡⢀⡀⠀⠀⠙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠉⢉⣳⣄⠀⠀⠀⠢⠀⠀⠏⠀⠀⠃⣢⠄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⠀⠘⢯⡟⠀⠁⠈⠁⠐⠎⠀⣰⣠⠎⠁
⠀⠀⠀⠀⠀⡠⢀⠄⠀⣀⡤⠴⠶⠒⠒⠚⠛⠛⠁⠈⠉⠉⠉⠙⠇⠀⠀⠤⠐⠈⠀⢠⣯⠟⠒⠒⠲⡄⢠
⠀⠀⠀⠀⠀⢀⠁⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡶⠟⠉⢠⠇⢠⢁⠂
⠀⠀⠀⠀⠀⠀⠄⠸⣄⠀⠖⠢⠄⡀⠀⠀⠀⣀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠓⡒⣶⠒⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡼⢸⠀⠀⠀⠀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡇⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠈⣀⠤⠒⠉⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠲⡇⠀⠀⠀⢸⠀⢸⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠴⣮⠄⠒⠒⠂⠀⠀⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠚⠀⠃⠀⠳⠶⠶⠒⠒⠒⢂⡀⠈⣩⠜⠀⠀⠀⡀⠀⠀⢂⡀
⠀⠀⠀⠀⠀⢀⡀⢀⠀⣀⠜⢲⠆⠤⠤⠄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⠔⠊⠀⠀⠀⡜⠀⠐⠀⠀⠀⢄⣠⡪⡙⠭
⠀⠀⠀⠀⢠⢋⡴⢋⠞⣁⠤⠞⠓⠒⠛⠒⠒⠒⠒⠂⠠⢤⣀⠴⠋⠀⠀⠀⠀⣠⠞⢹⡀⠃⠘⠀⠀⠈⢟⢍⠂
⠀⠀⠀⠀⠈⠈⠁⠫⢈⣡⣃⣠⡤⢤⠶⢶⡚⠋⠉⠉⡹⡟⠁⠀⠀⠀⢀⡴⠞⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⠁⠀⢸⠀⠀⠉⠒⠤⣸⠁⠃⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⣇⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⡏⡌⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣇⠸
⠀⠀⠀⠀⠀⠀⠀⣄⠘⢅⠓⠤⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠼⢀⣹⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠻⢻⡟⠛⠛⠛⠛⠛⣿⠿⠿⢿⠟⠉⠁⠀⠀⠀⠈⠁⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠓⠒⢿⠀⠀⠀⠀⠀⠀⡷⠤⠤⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣻⠎⠟⠀⢯⠀⠀⠀⢀⣶⡺⣭⡷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠁⠀⠀⠛⠒⠒⠒⠀⠤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠐⠄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⡤⠶⠛⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠏⢀⣀⣀⣀⡼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠘⠋⠩⠔⣚⣿⣿⣿⣿⣿⠟⠋⢻⣿⣿⣿⣿⣿⣧⠀⠐⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠡⢖⣻⠿⠛⠉⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⣀⡤⠴⠚⠉⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⣰⠁⠀⠀⠴⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠤⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢨⣿⡆⠀⠀⢠⠤⠀⠀⡆⠀⠉⠁⠀⡀⡠⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣾⣿⣧⠀⢀⢀⠀⠀⠈⠃⠀⠀⠀⠈⠐⠁⣹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠘⢿⣿⣿⡗⠁⠂⠀⠀⢔⣉⠭⠵⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠉⠻⣿⢿⣿⢷⡀⠀⠀⢸⠛⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⢶⠲⠔⠃⠀⠀⠀⠙⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠙⢶⣶⣶⡶⠖⠋⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠋⠉⠉⠉⠛⠓⢲⢤⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⢠⣦⠰⣿⠇⠀⣭⡓⠦⣄⡀⠀⠀⠀⠉⢳⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⠼⠃⠀⢸⣻⣮⠟⠛⠙⠲⢼⡟⠁⠀⡀⠉⠓⢦⡀⢀⡟⣦⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⡄⢀⣤⢀⣸⠃⠀⠀⠀⠀⠀⠓⢤⡿⠋⠀⠀⠀⣙⡾⠀⣇⣀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠖⠃⠀⢹⡏⠀⠽⠂⠀⠀⠠⢶⡆⠈⢣⡴⠶⠂⠈⠉⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣯⡿⠁⣠⣄⣠⣄⣀⣠⡀⢀⡀⢸⠀⠀⠀⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠳⡀⠀⠀⠁⠀⠉⠀⠈⠉⠀⢈⡷⠤⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠤⢤⣤⣄⣠⡤⣶⡿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠋⠙⠲⢶⣶⠶⢾⣋⣉⠀⢸⣧⠳⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣨⣧⣄⣠⣼⣧⣀⣀⣈⣉⣠⣼⡇⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⣿⢻⠀⢠⡄⠀⣄⠀⠀⠸⠀⠀⠀⣴⢃⡜⠉⠛⠒⣶⠶⢭⡉⠁⠀⢸⡇⠀⢳⡀⢘⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⡇⠘⡆⣸⢧⢠⢿⡀⠀⠀⠀⣠⠂⣿⢿⡁⠀⠀⢸⠁⠀⠀⠙⠒⠦⣼⡇⠀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣳⠈⢠⡄⠙⠃⠘⠋⣸⠁⠀⠀⠐⠇⠀⢁⣸⠧⠤⠤⣼⡀⠀⠀⢰⣧⡖⠃⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠀⡼⠙⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⠒⣾⣲⣶⣏⣙⣛⣻⣛⣲⠭⣟⡷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⠤⣾⠿⠭⢞⡿⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣧⣀⣀⣠⣧⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢡⢾⠉⡏⢇⠙⣍⠪⣝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣴⠃⣰⠖⢾⡘⡶⠋⣳⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠉⠒⠋⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⡀⢀⡀⣀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⣾⣬⣷⣦⣟⣧⣤⣾⣿⣿⣿⣾⣿⣶⣶⣷⣶⣿⣬⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣯⣿⣿⣿⣜⣿⣿⣯⣴⡻⣿⣿⣿⣻⣿⣿⣛⣯⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⠹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⡄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢾⡿⠓⣍⢻⣿⣿⣿⣿⡿⢿⡇⠀⠁⠀⠀⠠⣦⠀⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣤⣤⠃⠸⣿⣿⣿⠃⠀⢻⡁⠀⠀⣼⠈⠀⠀⢾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠿⠏⠀⠈⠺⣿⣿⠷⠀⠈⠥⠀⠀⠁⠀⢀⠀⠀⢻⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⠀⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⠄⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢿⣭⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠉⠛⠛⢶⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣿⣿⣏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⡆⠀⠀⠀⠀⠀⠛⢦⣄⠀⠀⠀⠀⠈⢻⣿⣿⠟⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⢟⢧⣿⣿⣿⣿⣿⣿⣯⣠⡂⠀⠋⡀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠐⣾⣿⣿⡄⠀⠀⢈⣿⣷⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⢰⡄⠀⠀⠸⣿⡀⢀⣠⡤⠄⠉⣿⣧⣤⣤⣌⣿⣿⣿⣿
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠁⠀⠀⠀⠀⠙⣿⠀⠀⢀⡶⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⡀⣶⣿⠀⠈⣿⠁⠀⠀⠀⠐⠀⠂⠒⠀⠒
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣽⣿⣉⢤⣤⣽⡆⠀⠀⠀⢀
⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣾⡿⠟⠋⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠊⠈⠁⠉⠈⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣨⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡘⢸⠁⠀⡀⢠⠠⠤⠤⠤⠤⡤⣀⠀⢸⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡼⠂⠁⠀⢀⡠⠀⠀⠀⠀⢀⣀⠉⠺⣯⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠔⠻⢤⡀⠀⠀⠉⠀⠀⠂⠄⠀⠀⠠⠑⠀⢈⣟
⠀⠀⠀⠀⠀⠀⡠⠔⠀⠁⠒⢤⡀⠀⢘⠀⠀⠀⠀⠃⠀⠁⠁⠀⡶⠀⢀⢰⡆⠀⠁⠛⠋⠈⡆
⢿⠒⠠⠤⠔⠊⠀⠀⠀⠀⠀⠀⠙⣤⠼⣄⠀⠀⣠⡣⠄⠀⠀⠐⠀⠀⠀⠀⠑⡈⠀⣀⡀⠀⡄
⢸⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⢦⡆⠀⠈⣉⣹⠃⠁⠀⠈⠁⠒⠶⣦⣄⣀⠔⠋⠉⠉⢳⠔
⢸⡇⠀⣀⣤⣾⠟⢉⠠⠀⠀⠀⠀⣨⠖⠊⠀⣸⡆⠀⢀⠀⠀⠁⠐⠂⠀⠀⠀⢀⠀⠀⢈⠷⣄⡀
⠀⠉⠁⠀⠈⣇⡰⠁⠀⠀⢀⡴⠊⠀⠀⢀⡴⠁⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡤⢾⡄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠎⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⢻⡀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣹⢆⡀
⠀⠀⠀⠀⡠⠊⠁⠈⠛⠿⢷⣦⣤⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⣴⠟⠁⠀⠈⠓⢄
⠀⠀⠀⡜⠀⠀⠀⢀⡤⢶⠀⠀⠘⣾⡍⠛⠶⠤⣄⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠴⠛⣧⡏⠉⠀⣤⣄⡀
⠀⠀⠀⠙⠒⠖⠚⠉⠀⠸⣄⣀⡴⠃⠙⡶⢄⣀⠀⠈⠉⠛⠓⠒⠒⠶⠒⠒⠖⠒⠋⠉⠁⢀⡰⠋⣇⠀⢀⡇⠀⠉⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠙⠒⠦⢤⣤⣀⣀⣀⣀⣀⣠⡀⠤⠐⠊⠉⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠦⠀⠀⠀⠀⠀⠀⢀⡸⣄⠀⠀⣀⡀⠠⢎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠈⠀⠁⠀⢸⠃⠀⠀⠀⠐⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠄⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣏⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠾⠧⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠋⠍⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠁⠈⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠚⠓⠒⠒⠒⠀⠀⠤⠄⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢉⣭⣭⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢠⠀⢠⠤⡀⠀⠀⢈⠀⠉⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢄⠀⢸⠀⠀⠸⠀⠀⠀⠸⠀⠎⠀⠐⢢⠀⠀⡔⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣢⠘⠖⠂⠉⠉⠉⠁⠃⠘⠒⠰⢥⡁⢀⠔⠁⣀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢢⠀⡠⠖⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠢⠈⠁⠰⡉⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⡀⠛⠀⡀⢀⠈⠀⢱⠀⠀⠀⠀⠀⠀⡨⠢⣤⠄⠀⠀⠀⠈⠱⡨⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠄⡔⠁⠁⠀⠀⠈⠂⠴⠋⠑⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣈⡠⠄⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠡⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠑⣏⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠅⠄⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡰⠒⡄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡌⠀⠠⠒⠢⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠘⢃⣉⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠉⡇⠀⢓⠀⢠⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠒⠐⠊⠉⡄⠈⡇⢀⡀
⠀⠀⠀⠀⠀⡏⠁⠒⢂⣁⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⡆⠀⠀⠆⠀⠀⠐⠒⠦⣠⠃⢰⠃⠀⠍
⠀⠀⠀⠀⠀⠑⡠⠂⠁⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⢐⠈⠑⡄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠂⠀⠀⡼⠉⠑⢦⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣤⠀⠀⠀⠀⠉⠲⠁⠀⠀⠀⠀⠀⠀⠀⢰⠔⠛⠀⠀⠀⠀⢀⠼⡣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠒⢄⠀⠂⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠔⠀⠀⢶⠃⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠄⠀⡠⡱⠢⢄⢀⡀⠀⠀⠀⠀⠀⠠⠀⡀⠀⡀⡤⠚⠑⡈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠁⢀⠆⢨⠉⠒⠒⣶⠒⠒⠒⠆⠹⡉⠀⡨⠂⠀⢐⢀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⡣⠀⠠⠠⠆⠀⠸⡀⣀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⠀⠀⡎⠀⠁⢘⠀⠀⠀⠡⠀⠔⠁⡤⢄⠀⠀⡔⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⠀⠱⡀⢣⠤⠞⠚⠒⠒⠲⠃⠸⠤⢤⣣⠊⠀⡰⠁⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀⠉⣪⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢌⠑⠊⣠⠚⠉⡡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠱⠞⠁⠀⠠⠂⠐⠄⠀⠀⠀⠀⠀⠀⢰⢄⢀⠸⠀⠀⠈⠑⢮⢐⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢉⡝⠀⠀⠉⠀⠡⠀⡠⠛⠄⠀⠀⠀⠀⠀⠁⠠⠋⠀⠀⠀⠀⢀⡈⠁⠈⢣
⠴⠂⣀⣀⠀⡀⠀⠀⠀⠠⠂⠼⣐⡎⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠄⠤⠠⡼⣀⡀⠀⠀⠀⠀⠀⣠⠴⠖
⢹⠁⠀⣿⠍⠐⠶⣤⠀⡇⢀⣀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⢥⡼⣀⠤⣀⠔⠋⢀⣴⡍
⢨⠀⠀⠀⠀⠘⠀⠈⡳⣎⡁⣠⠇⠀⢀⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠿⠗⠁⠀⢹⠀⡴⠁⢸
⠀⡀⠀⣦⠀⠀⣮⠟⠁⢣⠤⣽⠀⠰⠁⠀⢹⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠤⠠⡀⠠⡶⡺⠋⠙⠺⢿⡅⠀⢨
⢀⠀⠀⡿⣀⡴⠋⣰⠲⠧⠤⠾⠂⠀⠑⠀⠋⠁⠂⠀⠨⠀⠀⠈⠀⠀⠀⠀⢀⠖⠁⠁⡄⠀⠀⣀⣀⣀⠀⠇⢀⡟⡓⢲⠰⢅⠸⡣⡄⢸
⢀⣄⣠⡿⠋⣀⠔⠻⡀⢀⠔⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⠀⠀⠀⠢⠀⠔⠀⠀⠀⠀⠀⠈⠈⠀⢼⣀⠔⣁⣀⠈⠑⢀⠈⢺⣄
⠒⠚⠓⠒⠊⠁⠀⠀⠙⠁⣀⠠⠜⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠤⢬⡉⠀⢸⠂⠀⠀⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠊⠀⠘⣄⠀⠀⠀⠀⠰⠧⣀⠄⠀⠀⠀⠀⠀⠀⠀⢄⢀⣦⠀⠠⠀⠀⠀⢠⣏⠀⠀⠉⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⢃⡀⠀⢁⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡰⠀⠀⢀⡴⠉⠪⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠇⠈⢢⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⣠⡀⠻⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠜⠀⡔⢱⠋⠉⠇⢰⢤⣀⣀⢀⡀⠀⣀⠀⣤⠴⠖⠫⡀⠈⠆⠀⠈⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠜⠀⠸⡀⠀⠈⢩⠉⠉⡸⠀⠘⢆⠐⠚⠁⠀⠰⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠉⠙⠛⠻⠶⢶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢹⣇⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣴⠟⠀⠀⠀⢸⣇
⠀⠀⠀⣠⡾⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⢰⡶⣦⣤⣤⣄⡀⠀⢿⣦⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢰⡏⠀⠀⠀⠀⠈⣿⠀⠀⠀⢤⣤⣴⣾⠇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠦⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⣀⣤⣄⠀⠀⠀⠸⣧
⠀⠀⢸⡇⢀⣶⣶⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⣀⣤⣴⢾⣿⣿⣿⠟⠀⠀⠀⠀⢿⡄
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⢀⠀⠠⠶⠟⠋⠁⠀⢀⣴⠾⠋⠀⠀⠀⠻⠛⠋⠁⠀⣸⡿⠟⠁⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠂⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⢸⡇⢸⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⢀⣼⠇⢸⣿⣿⣿⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣾⠃⠀⣼⣿⣿⣿⡄⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⣿⣿⣿⣿⡗⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠹⣧⡀⠈⠛⠛⠋⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠷⢶⣤⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠻⠯⠤⠤⠤⠤⠭⠭⠭⠽⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣭⣙⣒⣒⣒⣒⣒⣒⣒⣒⣚⣩⣬⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠁⡏⢹⠉⡏⢹⠉⡏⢹⠉⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠾⣦⣧⣼⣤⣧⣼⣤⣧⣼⣤⠷⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⠶⢶⡶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠻⠶⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⢀⡴⡖⢶⡄
⠀⠀⠀⠀⠀⢀⡴⠶⣤⣤⣄⡀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⣿⡀⠀⠀⣴⠟⢮⣁⣼⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣸⠐⠚⣻⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠆⠘⣧⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠋⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣉⣸⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠛⠷⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠿⠛⠿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠈⠹⣧
⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⣠⡶⠶⣶⣾⠟⠛⣷⣄⣠⣤⣤⣴⣶⣶⡶⢶⣶⢶⣶⣤⣤⣀⣠⣾⠛⠻⣶⣶⠶⣶⣄⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢀⣸⡏⠀⠀⣈⣥⡶⠾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢶⣤⣀⠀⠀⢻⣆⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣤⡾⠛⠉⣁⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣈⠉⠛⢷⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⢀⣀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢷⡄⠀⠘⠷⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣀⡀
⠀⠀⢻⡆⠀⢰⣿⠀⠀⠀⠀⠀⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⢠⡄⠀⠀⣄⠀⠀⣄⠈⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⣄⠀⠀⠀⠀⠀⣿⡆⠀⢰⡟⠁
⠀⣠⣾⠇⠀⢸⣯⠀⠀⠀⠀⣰⡏⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠹⣆⠀⠀⠀⠀⣿⡅⠀⠸⣧⣄
⢸⣿⡁⠀⠀⢸⣏⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡇⠀⠀⡿⠀⠀⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣿⡁⠀⠀⢀⣿
⠀⠉⣿⡆⠀⠘⣿⡀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢸⣇⠀⠀⣿⠀⢀⡟⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⠃⠀⢰⣿⠁
⢀⣴⠟⠀⠀⣠⣿⠷⢶⣦⣿⠀⠀⠀⠀⠀⠀⠐⠓⠛⠋⠁⠀⠀⠘⣿⠀⠀⣟⠀⢸⡇⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⣿⣤⡶⠾⣿⣄⠀⠀⠻⣦
⢿⣇⠀⠀⠀⣿⠁⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠋⠀⠀⠁⠀⠈⠃⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠈⣿⠀⠀⠀⣸
⠀⢹⡇⠀⠀⣿⡀⠀⠀⠘⣿⠀⠀⢀⣤⣴⣤⣤⠀⣀⣀⣙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⣁⣀⠀⣤⣴⣦⣤⡀⠀⠀⣾⠃⠀⠀⢠⣿⠀⠀⢸⡏
⠀⣾⠃⠀⠀⠸⣷⡀⠀⠀⢹⡆⣴⠏⠁⠀⣾⠋⠀⠉⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠉⠀⠘⣷⠀⠈⠹⣆⢸⣿⠀⠀⢀⣾⠃⠀⠀⠘⣷
⠀⠻⣶⣄⡀⠀⠘⠻⣦⣤⣼⡇⠹⢧⣄⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⡼⠏⢸⣯⣤⣴⠿⠁⠀⢀⣠⣴⠟
⠀⠀⠀⠉⢿⡄⠀⠀⠀⠈⢹⣧⠀⠀⠉⢁⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡈⠉⠀⠀⣸⡏⠉⠀⠀⠀⢠⣿⠉
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠘⣿⡀⠀⢰⡟⢹⡇⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢸⡏⢻⡆⠀⢀⣿⠃⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣦⣈⠛⠛⠁⠀⠘⠋⠉⠻⠶⠛⠛⠷⣶⡶⠾⠛⠛⠷⠟⠉⠙⠃⠀⠈⠛⠛⣁⣴⣿⡁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠈⠙⠻⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟⠋⠁⠈⣿⡆
⠀⠀⠀⠀⠘⠷⣦⡄⠀⠀⢸⣿⠀⠀⠀⢠⣾⡿⢿⣿⠟⠿⣿⠶⢶⠶⣶⣤⣤⣴⣶⠶⠶⠾⣿⠟⠛⠉⠿⣷⡄⠀⠀⠀⣿⡁⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣼⡇⠀⠀⢰⣿⡛⠛⠀⠀⢀⣤⣿⣶⣶⣶⣾⣟⣉⠀⠀⠀⠀⢰⡿⠀⠀⠀⢰⣿⠁⠀⠀⠀⢸⡇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠘⢿⣆⠀⢸⣟⠃⠀⣠⣶⠟⠁⠀⠀⠀⠀⠈⠉⠛⠻⠷⣶⣦⣾⣧⣀⠀⠀⣴⡿⠃⠀⠀⣰⡿⠁⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⢸⣿⠀⠈⣿⣃⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠷⠿⠃⠀⠀⠀⣿⠀⠀⠀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠈⢿⡄⠀⠉⠛⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢸⣯
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⢻⣆⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⣦⣄⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠉⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣴⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠉⠛⠿⢶⣦⣤⣤⣶⡶⠟⠋⠙⠛⠷⠶⣦⣤⣤⣄⣀⣀⣀⣠⡿⠋⠀⠀⢰⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣄⣀
⠀⢸⡿⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢛⣿⣿⡇
⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⡂⠐⠞⠻⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠆⠂⠐⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠃⠀⠀⡀⠄⠀⠀⠀⠉⠀⠀⠄⠀⠀⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⢠⡄⠀⢐⠂⠐⢢⠀⠀⠀⠀⢠⡀⣠⠤⠀⢨⣉⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠆⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢴⡶⢉⠩⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡟⠟⢳⠀⠀⠀⠀⠉⠉⠉⠉⠛⠒⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠀⣸⠓⣬⠓⡀
⠀⠀⠀⠀⠀⠈⠩⠷⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠭⠿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠉⠉⠒⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡜
⠀⠀⠀⠘⠀⢀⠖⠶⡿⠀⠀⠀⠀⠀⢢⠀⢠⣂
⠀⠀⠀⠀⠀⡅⠹⣶⡆⠀⠀⠀⠀⠀⠀⢀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠰⢿⣇⢸⡇⣸⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢐⣷⣤⣽⣿⣿⣧⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⡀⠻⣿⣿⣿⣿⡿⠃⣀⠀⠀⢡⣤⡞⢻⡟⢷⣤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣰⣿⣿⣦⣌⣉⣉⣉⣤⣾⣿⣿⣦⠀⠙⠁⠾⠷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⣿⣿⣿⠻⣿⣿⣿⣿⣿⡋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣿⣿⣿⠀⣿⡿⠿⣿⣿⣿⣿⣿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣿⣿⣿⣶⣿⣿⢠⣄⡉⠛⠿⠿⠟⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠿⠿⠿⠿⠿⢸⣿⣿⣷⣦⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⣶⠆⣠⡀⠀⠀⠀⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠛⠻⣦⣿⣤⡿⠛⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⣿⡁
⠀⠀⠀⠀⠀⠀⢸⠿⠟⣿⣿⣿⡿⠻⡯⠀⠀⣼⣿⣿⣿⠿⣿⣿⣿⡆⠀⠐⣿⡅
⠀⠀⠀⠀⠀⠀⠀⢶⣾⠋⣿⠉⣿⡶⠀⠀⢀⣿⣿⣿⣿⠀⢿⣿⣿⣷⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠃⠈⠀⠀⠀⢸⣿⣿⣿⡇⠀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣲⣤⢄⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠼⠴⣧⡼⣮⣉⣆⣹⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠈⠀⠣⠚⢹⢓⡧⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡜⣩⣻⣦⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠈⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠤⠭⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⡠⠒⠲⢄⠀⠀⠀⠀⠈⢢⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⡟⠟⠓⠋⡡⠤⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠲⡃⠀⠀⠠⠤⠴⠊⠀⠀⠀⠀⠀⢒⣚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠃⣰⠃⣀⠤⠒⣉⡦⠤⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠑⢄⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠊⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢷⠿⠓⢻⠉⠇⠇⠃⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣀⣀⠵⠴⠚⠁⠀⠀⡘⠀⠀⣀⠤⠚⠁⠘⣄⠀⠀⠀⠀⠀⠀⢀⡴⢎⡑⣠⠓⠓⠒⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣮⠤⢄⠀⠀⠀⠀⠀⠀⢀⡤⠒⠃⠀⣰⠧⣄⣀⣀⠀⠈⢢⣀⣀⣤⢴⡚⢧⡑⣢⢝⠁⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢄⡤⠊⢀⡠⠴⠒⠁⠀⠀⠀⠀⠈⠉⠁⠈⠑⠾⠦⠭⢶⠞⠋⠁⠀⢀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⡤⠖⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠊⠓⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⠖⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠦⢎⡀⣀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⡦⢋⡀⢀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠪⣁⡝⠀⠰⡉⢱⠀⠀⢰⠒⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠣⣶⣇⡠⠤⠆⢙⡏⠊⢉⠋⣧⣉⠑⠖⣶⢉⡀⠰⡒⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢄⣧⠀⣠⠔⠚⢏⠓⠀⠀⠀⠀⠀⠀⣸⠯⢭⡀⢀⡽⢅⠈⢉⣴⣉⡥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠍⣡⢄⠀⠈⠀⢀⡤⠤⣄⠀⠀⠙⡆⡷⠂⢠⢇⡶⠃⠀⠤⢫⣴⡜⠀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡆⣠⠞⠁⠉⠮⢤⣸⣀⠀⠀⠛⠠⡈⣎⠀⠀⠈⠵⠽⢁⡤⠚⢯⠂⠀⡠⠛⡇⡙⠐⣣⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠒⣌⠣⠄⠀⠀⠀⠀⣙⣀⠀⠀⠀⠀⠩⠂⡡⠀⠠⢄⠀⠈⡇⢰⠤⠃⢊⢊⠎⠔⢌⢌⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⠃⠀⠀⢀⠤⡄⠀⠉⠀⠀⠀⠀⠁⠀⠓⣤⠊⣞⠀⢀⡬⣿⢃⠔⣱⢥⡪⢀⠒⡊⢤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⡀⠁⠀⠀⢌⢀⣑⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠒⠃⠠⠜⢀⢨⠄⠠⠂⠀⡃⢑⡌⡰⠠⣷⡠⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣩⠩⣴⠤⠀⠀⠈⠁⠨⠇⠀⠀⠀⡅⠈⡅⠀⡔⢈⠙⢢⠤⡨⠈⠭⠪⡫⣈⠗⠑⠡⡪⡪⠠⠖⠢⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡂⠽⢰⠒⠦⠄⠀⠀⠁⠀⠀⠀⠙⠾⠽⠀⣻⡶⠗⠃⠺⣂⣉⢊⡉⣎⠕⠀⡬⡙⢘⣚⠔⠧⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⣴⣤⠀⠘⠦⠛⣟⠗⠀⢠⠦⠔⣢⡄⠀⠀⠐⢙⡠⢐⠀⢓⢪⢄⡧⣰⠕⡡⠪⣐⡗⡩⢕⡱⠒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠹⡄⢀⠀⠀⠀⠁⠀⠈⠦⢦⡿⢖⡫⡢⠆⠯⡸⢼⢒⡋⢑⠑⡪⢔⡩⢜⢕⠈⢉⡗⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⠛⣡⢴⢆⠀⢀⣠⠀⠒⢭⡲⣵⠞⡳⢖⡳⢖⣡⡺⢙⠄⣉⠗⠡⢎⢕⣌⡗⠙⡤⣕⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠮⠛⢮⡳⣖⠒⢡⠊⠥⢒⡋⣠⠉⢒⣩⢄⢹⠠⢚⠭⢋⢔⡠⣈⣦⣀⣵⣊⢤⣄⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⡠⣽⡓⡂⠂⢼⣉⡀⡗⠾⣋⢥⡎⢈⣺⡃⢠⠸⣸⣂⣤⣤⣩⣧⠏⠩⣦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡸⠀⡠⠿⢡⡶⣂⠫⢉⡠⠕⡪⠕⡩⣋⠥⢒⢝⣀⡸⡧⠤⡔⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠜⠈⠒⢎⠱⣂⠴⡀⡵⠢⢔⡪⠔⢪⣔⠋⡞⠋⠉⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠘⠅⢩⠃⠳⡚⠛⠃⠈⠁⠀⡵⡈⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠃⠈⠉⠋⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⣀⠀⢀⠀⢀⡠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠔⠒⠭⢒⠢⠔⡒⠤⣒⠎⢒⡪⠥⢒⡩⠔⠢⠅⠒⠬⠓⠪⠕⠂⠩⠒⠉⠔⠊⠁⠒⠉⠒⠁

⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠒⢤⡀
⠀⠀⡠⠐⠁⠀⠀⠀⠀⠈⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠁⣄⡆⠧⠀⠄⠀⠐⢆
⠀⠀⡀⠀⠀⠀⢂⣀⢀⠀⠀⠀⠶⢖⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢊⣁⣀⠀⠀⠀⠀⠀⠈⢀⣠⣤⣄
⠀⠀⠠⢰⠒⠈⢁⢸⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⡠⠐⠀⠉⠁⠂⢀⠀⠀⠘⡄⠀⡔⢱
⠀⠀⠆⢸⢠⣆⠏⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠱⠄⠀⣫⣼⡡⠎
⠀⠀⠰⡌⠁⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠄
⠀⠀⠀⠣⡀⠀⠀⠀⠠⢄⡄⠂⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⢸⡶⣴⣶⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡾
⠀⠀⠀⠀⠈⠀⠀⢀⣀⢀⡀⣀⠀⢈⡴⠊⠁⠀⠀⠀⢉⠳⣄⠀⠀⠀⠐⠋⠉⢉⢀⢀⡀⡀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠃⠀⠁⣈⡟⠠⡀⢀⣀⣠⣰⣀⡀⠘⡆⠀⢀⣼⠿⠿⠯⠹⠿⠟⠳⠦⠄⠱⠀⢰⢦⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⢸⠉⠀⣧⡀⠀⠀⠐⡀⠀⠀⢃⠀⡈⣀⠀⡀⠠⡄⢠⠀⣄⣠⡜⠀⠀⠀⠀⠂
⠀⠀⠀⢀⣰⡑⡄⠀⠀⠀⠀⠀⣸⠀⢸⡇⠀⠀⠀⠤⡧⠀⠀⡆⠀⢸⣉⠈⠁⠁⠀⠀⠀⠀⠅⢧⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠈⢹⣿⣤⠀⠈⠀⠐⠂⣀⣀⣍⣍⢀⣀⣀⣙⡀⠐⡃⠀⡘⠁⢀⣀⡀⣀⠀⡀⠀⠨⠁⠃⠀⠀⠀⡀⣴⠁
⠀⠀⠀⠀⠀⢹⣿⠙⠃⠘⠀⠀⠋⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠃⠀⠁⠀⠀⠀⠀⠁⠐⢘⢇
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⠈
⠀⠀⠀⠀⠀⠀⣉⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠓⢀⠈⠀⠠⠀⠂⢀⠔⡁⠀⠠
⠀⠀⠀⠀⠀⠀⠐⠂⠤⠀⠀⠀⠐⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠀⠀⠒⠂⢂⣈⡀⢁
⠀⠀⠀⠀⠀⠨⠈⠀⡇⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢐⠂⢰⠀⠀⠀⠀⠀⠠⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠨⠅⠜⠀⠀⠀⠀⠀⠸⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⢦⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡸⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠌⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠤⣄⡀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣰⠉⠉⠓⠲⢤⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀⠀⠀⠀⢢⠀⠘⢶⡖⠢⢄⣀⠀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣰⢿⣦⣠⠤⠬⣃⣀⡜⠀⣔⣉⡴⠷⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣳⠦⠤⠬⣿⡆⠀⠀⠀⣀⠝⠀⢰⣷⣏⣀⣀
⠀⠀⠀⠀⡜⣱⠋⣠⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⠠⡤⠈⠓⠒⠒⠉⠁⠀⠀⡷⢋⣡⡤⡌⢳⠀⡇⢸
⠀⠀⠀⠀⠀⡇⢸⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⠋⠠⢁⠃
⠀⠀⠀⠀⠀⠉⠀⠑⠦⣥⣈⣁⣈⣀⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡆⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡆⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠁⠇⠀⣀⠤⠒⠊⣿⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⢹⠛⢲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠖⣉⠀⠄⢀⣀⣸⡯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⡰⠧⠤⢄⣀⣙⣢⣾⣠⣏⣙⣆⣀⣀⣠⣤⠤⠤⠤⠞⣆⠿⢋⡀⠒⠒⠢⢄⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⢙⡇⠠⡄⡈⠑⣄⢀⣴⠿⣦
⠀⠀⠀⠀⡰⢃⠖⣠⠞⣁⣸⠧⠤⠿⠭⠭⠭⠭⢉⣉⡉⢉⡩⠟⠉⠀⠀⠀⣠⠞⡇⠰⠁⠇⠀⠘⢟⠗⠕⠀⠁
⠀⠀⠀⠀⠃⣏⢰⡁⢊⡔⠀⣀⣀⣠⣤⠤⠶⠒⠒⣲⡞⠋⠀⠀⠀⠀⣠⠞⠁⠀⢿⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡏⠉⢹⠀⠈⠓⠦⣀⣰⠁⡇⠀⢀⣠⠴⠋⠁⠀⠀⠀⢸⡀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠆⡼⣸⠀⠀⢸⠀⠀⠀⠀⠈⠙⠲⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⣜⠀⢇⠻⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⢘⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡿⠿⢷⡶⠒⠒⠒⠒⠲⣶⡶⠶⣶⠟⠋⠉⠉⠁⠀⠈⠉⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠤⢾⠀⠀⠀⠀⠀⠀⣧⡤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⠭⠿⠀⢯⠀⠀⠀⢀⣔⠾⣕⡶⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠁⠀⠀⠛⠒⠩⠥⠤⠬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠁
⠀⢀⣤⠶⠶⠶⠶⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⢠⡟⠉⠢⡀⠈⠠⡀⠀⠈⠙⠶⣄⣿⠛⠒⠒⠒⠓⠛⠛⠋⠉⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⢧⠀⠀⠈⠳⣄⠈⢦⡀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠘⣆⠀⠀⠀⠈⠳⣄⠙⢦⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠸⣆⠈⢢⡀⠀⠄⣀⠈⠳⢄⠙⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠁⠀⢀⡾⠁
⠀⠘⢦⡀⠙⢦⡀⠀⠀⠀⠀⠑⢄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⢦⣀⠈⠓⠶⢤⣤⣀⣤⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣤⣤⠶⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⢉⣽⠟⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠾⠋⠁⠠⠄⠀⢀⡤⠞⠉⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⢀⣀⣠⠴⠛⠁⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⡠⠚⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⠦⠤⢤⣄⣀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⣠⠖⠁⠀⣠⠖⠉⠀⠐⠀⠀⣠⠎
⠀⠀⠀⠀⣠⣴⡛⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢴⣟⡁⢀⣠⠞⠁⠀⣀⣀⡤⠖⠋⢁⡼
⠀⠀⠀⣰⠏⠈⠙⠲⢦⣀⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠉⠉⠉⠉⠉⢉⣀⣠⡤⠶⠋
⠀⠀⠀⣿⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢤⣀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⢹⣏⠑⠦⣄⡀⠀⠂⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢦⣤⢿⡆
⠀⠀⠀⠀⠻⣆⠀⠀⠉⠓⠒⠤⠤⣀⣀⣀⣀⣬⠽⠶⠒⠉⠀⢀⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⠀⠀⣀⣀⣀⣤⡴⠞⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⡄⠁⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣭⢷⠞⠓⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡹⠁⠀⡞⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠤⠒⠂⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠿⠏⠀⠀⠛⢃⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠉⠛⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⡠⠔⠒⢢⣀⡠⠤⣄⣠⠤⠶⡄⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠖⠲⠄⠤⡠⢀⡠⣤⠤⠤⠤⢾⣁⣀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠄⡀⣌⠀⠀⢠⠀⠀⣀⠀⠀⢨⡬⢏⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⢄⡀⠈⠑⡍⠀⠪⡉⠀⠀⠀⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣹⡉⢐⣁⣀⢡⣢⠤⠴⠒⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⡠⠇⢺⣐⣒⣒⣙⡠⢄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⡏⠛⠓⠒⠂⠒⠚⠛⢹⣆⢀⣀⢀⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡧⠴⡇
⠀⠀⠀⠀⠀⠀⢀⠤⠤⣄⠀⢀⡤⠊⠁⣀⣀⣤⣴⠚⠳⣄⠀⠀⠀⠀⠀⠀⢸⡇⡝⠁⠀⢹⣄⣀⠀⠹⣄⠀⠀⠀⠀⠀⢨⣀⣀⣸
⠀⢠⣄⡴⠄⠀⡸⠯⠭⢼⡴⠓⠒⠚⠉⠁⠀⠀⠈⠳⣤⡸⠀⠀⠀⠀⠀⠀⢸⣇⡷⣤⠤⠚⠲⠎⠛⠒⠮⣷⡄⠖⠒⠆⣾⠈⠀⢸⠰⠲⠖
⠀⢸⣒⢲⠀⢸⣤⡤⠤⢄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣯⣀⡀⣼
⠀⣸⠉⣸⠀⣘⡋⠀⠀⠊⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⡄⠄⠀⡀⠀⠀⠀⠀⠠⠤⠤⠜
⠀⠈⠉⠉⠀⠙⠭⠭⠭⠄⠀⠀⣀⣀⣤⣄⠐⠚⠋⠁⠈⠉⠛⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⢀⠤⣤⠶⠺⠛⠋⠉⠁⠀⠰⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠈⠁⠐⠂⠀⠀⠀⠀⡠⠉⠉⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠠⠄⠠⢤⣀⠀⠢⡀⢀⡠⠖⠛⣇⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣀⡤⢄⣠⡀⠈⠑⢂⣸⠁⠀⠀⠀⠃⣤⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠉⠻⣷⠀⠀⠙⣦⠀⠀⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠿⠀⠀⠂⠉⢢⠀⠀⠈⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠠⠤⢤⠀⠀⠀⠀⠈⠋⠡⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠖⠒⠂⠀⠀⠀⠀⠀⢰⡦⠾⠁
⠀⠀⠀⠀⠀⠀⢰⠚⠉⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠘⣦⣀⣰⣧⠟⢄⠒⠠⣄⡀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⠘⡄⠀⠀⠘⠛⡀⠀⠀⡆⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠳⠒⠒⠒⠒⠒⠒⠺⣥⠀⢳⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣶⣈⡷⠀⢸⡇⠉⠙⠢⣄
⠀⢀⡤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠁⠀⠀⠘⣿⠄⠀⠀⠈⢢⡀
⢠⠛⠀⢰⠀⡤⢟⣳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣀⣀⡀⣿⢾⠀⠀⠀⠀⢳
⡸⠀⠀⢰⡟⢰⡏⢉⡁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⡹⠉⠉⠉⢡⠞⢰⠀⠀⢰⠀⢣
⢿⠀⠀⢸⠁⠀⠙⠋⡄⠸⢧⣀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣷⠋⠀⠀⢀⠔⠉⠀⢸⠀⠀⠘⠀⠈⣇
⠸⣇⢠⡞⠀⠀⠀⠀⠃⠀⣰⠏⣀⡠⠀⠀⠀⣠⠞⠁⢰⠁⠀⠀⠀⠏⠀⠀⠀⢸⠀⠀⠀⡇⠀⢸⡄
⠀⠈⣙⠇⠀⠀⠀⠀⢠⣠⠓⢤⡌⡘⠛⠛⠋⠀⠀⠀⢸⠀⠀⠀⢀⡀⠀⠀⠀⡾⠀⠀⠀⡇⠀⠈⣇
⡄⠐⠋⠀⠀⣆⠀⠁⢰⠀⠀⠀⠣⢆⠀⠀⠀⠀⢀⡴⠛⣧⡴⠚⣯⠣⠀⠀⠀⡇⠀⠀⠀⢇⠀⠀⣿
⠀⠑⢤⡀⠀⠇⠀⡇⢸⠀⠀⠀⠀⠈⠳⠶⠶⠞⠉⠀⠇⢸⠃⠈⠀⠀⠀⠀⠀⣷⠀⠀⠀⢸⠀⠀⢿
⠀⠀⠀⠳⣸⣄⣀⡇⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⡆⠀⡘
⠀⠀⠀⠀⠙⠁⢨⠁⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣟⠀⣫
⠀⠀⠀⠀⠀⠀⠘⠓⠒⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⢏⠣⠄⣦⡀⠀⠀⠀⢀⣸⡆⠀⠀⠀⢠⡿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠒⠒⠀⠀⠀⠈⠐⠂⠛⣦⢠⠴⡏⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠐⠶⢶⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⠀⠉⠉⢁⡟⠀⠀⣠⣞⠤⠒⠈⡀⠀⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡼⠁⠀⠀⠀⢀⡼⠁⠀⡰⠁⠈⢆⠀⠀⠉⠲⣄⡔⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠐⣫⠀⠀⠀⠀⢀⠞⠀⠀⢠⠃⠀⠀⠘⣆⠀⠀⠀⠘⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠸⡇⠀⠀⠀⢾⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⠦⣤⣄⣠⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠈⢧⡀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⠀⠀⠹⣄⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠷⣤⣼⡏⢱⠀⠀⠀⠀⠀⢈⡇⠀⢸⣥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡐⠙⠳⠶⠾⠆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉⠃⠀⠀⠀⠀⠀⣇⠀⠀⢀⣧⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠖⠛⠉⠁⠀⠀⠀⠀⠠⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡤⠄⣀⣀⣀⣀⣀⣀⣀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠸⣧⡀⠀⠀⢀⣀⣠⣤⡴⠶⠶⠟⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣾⠇⣸⡣⠀⢨⠆⠀⠀⠀⢰⣤⣴⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢛⢻⣿⠾⢹⣏⣄⣸⣇⢤⠀⠀⠀⢙⣿⣿⣿⣧⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⢿⣿⣿⣿⣷⣾⣯⣤⠤⠤⡍⠀⠤⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⡄⣶⣿⣿⣧⣨⠟⠛⠉⠏⠋⠒⠲⠒⠡⠞⣁⢾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢛⡙⠺⣫⣬⣿⣯⡀⠤⠾⣿⢷⣶⡦⠖⠒⠚⣿⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⢲⣖⣚⡯⡀⣼⠛⠁⡄⠀⠈⣎⣹⢃⣀⣠⡔⠋⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⢘⣿⠍⡉⢀⡰⢿⡇⢶⣿⠀⠀⡿⠛⠉⠁⠀⠱⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⣼⣇⢹⡟⠀⠀⣇⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣿⠞⡀⠰⠖⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠂⠀⠀⠀⢀⣀⣠⣤⣤⣤
⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣀⣀⣀⡀⠀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢉⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣦⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⢀⣴⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣷⣦⣍⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢉⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⠟⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠉⠉⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣀⣴⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠻⢿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠛⢿⣷⣤⣤⣤⣄⣠⣤⣾⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣶⣀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣟⣛⣋
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⢿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣷⣶⣤⣤⣀⡀
⠀⠀⠀⣀⣴⣾⣿⣿⠿⠛⠋⠉⠉⠉⠁⠀⠀⠙⠻⢿⣿⣿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠉⠛⠛⠛⠿⣿⣿⣿⣶⣦⣄
⢀⣤⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣦⡍⠛⠿⠿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣄
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⣈⠙⠛⠙⠻⠿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⡀⠀⠀⠀⠈⠻⣿⣷⣄
⣿⣿⡟⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⣠⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣿⣿⣿⣷⣶⣤⡀⠀⠘⢿⣿⣦
⣿⠏⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⣿⣿⣷⣤⠀⠹⣿
⠋⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⣿
⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷
⠀⠀⣰⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏
⠀⣰⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠁
⣰⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡏
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡏
⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⢀⡾⠃⠀⠀⢀⣀⣀⣸⣷⣤⡀⠀⣀⣀⣀⣀⣠⣀⣀⣀⠀⠀⣠⣾⣏⣀⣄⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢸⡇⠀⠀⣠⡟⠁⢉⣩⡴⠾⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⢦⣌⡉⠈⢻⣆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠻⣶⠟⢉⣥⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣬⡉⠛⣶⠟⠁
⠀⠀⠀⠀⠀⠀⢹⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡟
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢰⠇⠀⠀⠘⣇⠀⠀⣷⠀⠀⠀⠀⠀⢀⡇⠀⠀⠸⡆⠀⠀⠀⠀⠀⢻⡄
⠀⢶⡄⠀⣼⠇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠘⠷⠀⠈⠁⠀⠀⠀⠠⠞⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧⠀⢀⡶
⠀⣠⡟⠀⣿⠀⠀⠀⢠⡏⠀⠘⠂⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⣾⠀⠀⠀⠀⠀⠀⠇⠀⠹⡄⠀⠀⠀⣿⠀⢻⣄⡀
⠸⣏⠀⠀⣿⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⡀⠀⡇⠀⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⣸⠗
⠀⣸⠇⠀⢹⣇⣀⣰⡇⠀⠀⠀⠀⣀⣠⡴⠋⠀⢸⡇⠀⡇⠀⣿⠀⠙⠦⣤⣀⠀⠀⠀⠀⢸⡇⣀⣰⣏⠀⠸⣏
⣾⡋⠀⢰⡟⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠘⠃⠀⠃⠀⠛⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⢹⡆⠀⢈⣷
⠈⣿⠀⠸⣇⠀⠀⢻⡆⢀⣤⠶⢦⡄⢠⣈⣻⣦⠀⠀⠀⠀⠀⠀⣴⣟⣁⣄⢠⡴⠶⣤⡀⢠⡟⠀⠀⣸⠇⠀⣾⠁
⠰⣏⡀⠀⠹⣦⡀⠸⡇⢾⡁⢀⡟⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿⢸⡇⢀⣴⠏⠀⠀⣹⡇
⠀⠉⠛⣷⠀⠈⠙⠻⣇⠈⠙⠛⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⡛⠛⠁⢸⡟⠋⠁⠀⣴⠟⠉
⠀⠀⠀⡿⠀⠀⠀⠀⢿⡀⠀⣞⢹⡇⠀⣠⣄⠀⣀⣀⠀⠀⢀⣀⠀⣠⣄⡀⠀⣏⣻⠀⢀⡿⠀⠀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢶⣭⣉⠀⠈⠁⠈⠉⠉⠉⠛⠛⠉⠉⠙⠁⠈⠁⠀⢉⣭⡴⠿⣦
⠀⠀⠀⠰⣤⡄⠀⠀⣿⠀⠀⠈⣽⢿⣷⡶⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⡴⠶⢻⣿⡁⠀⠀⣾⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠀⣿⠀⢰⡏⠀⠀⣾⡛⠃⠀⣠⡼⠷⠶⢶⣿⣉⡀⠀⠀⣾⠁⠀⠀⢿⡁⠀⠀⣹⠆⠀⢿⡁
⠀⠀⠀⠀⢸⣇⠀⠀⠙⡇⠈⣯⡅⢠⡞⠉⠀⠀⠀⠀⠀⠉⠙⠛⠳⠿⣦⣤⣰⡟⠁⠀⢰⡋⠀⠀⣸⡗
⠀⠀⠀⠀⠀⠉⣿⠄⠘⣧⠀⠙⠶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⠁⠀⣾⠉
⠀⠀⠀⠀⠀⢰⣏⠀⠀⠘⣷⡀⠀⠈⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠈⠙⠛⢶⡀⠈⠛⢦⣄⣀⠀⠀⢉⣻⣶⢦⣤⣀⡀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⣰⣿⠟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣰⣿⠃⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⡇⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣿
⣤⣄⠀⠀⠠⣤⡄⠀⠀⢠⣤⡀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣤⣤⣾⡿
⢘⣿⡆⠀⠀⢹⣿⠀⠀⠈⣿⣷⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠉⠉⠁
⣾⡟⠁⠀⢠⣿⡏⠀⠀⢰⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣀⣀⡀
⢻⣿⡄⠀⠀⢿⣷⠀⠀⠘⣿⣆⠀⣿⡇⠀⠀⠀⠀⠀⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠷⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡄
⣼⣿⠃⠀⢀⣾⡿⠀⠀⢠⣿⠏⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣽⣿⡿⠗⠀⠀⠀⠐⠿⣿⣿⣭⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠉⠁⠀⠀⠀⠉⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⠀⠀⠀⠐⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠿⢿⣷⣶⣶⣶⣶⣾⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⣀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡉⠉⣉⣍⠛⠛⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢼⠇⢲⡏⣸⠇⠀⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡾⢠⡟⠀⠀⢸⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣠⡀⢸⠇⣼⠁⠀⠀⣿⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠻⠑⡿⢠⡟⠀⠀⢠⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢸⡇⣼⠁⠀⠀⣼⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⢠⡏⠀⠀⢠⡏⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⡟⢴⡇⣸⡅⠀⠀⣼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣼⢃⡏⠀⠀⠀⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⡿⣸⠃⠀⠀⢸⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⡿⠀⠀⠀⣾⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡗⢠⣾⢸⡇⠀⠀⢀⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢠⡇⣼⠁⠀⠀⣸⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢸⣧⡟⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣿⣿⠇⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢴⠆⢰⣿⣿⠀⠀⠀⡿⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⠁⣹⣿⡏⠁⠀⢸⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣿⣿⡇⠀⠀⣾⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⠀⠀⢰⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣤⣀⣸⣿⣟⠀⠀⣼⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠃⠈⣿⣿⡏⠀⢠⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⣿⠀⠀⣼⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣄⣸⣿⣟⣠⡾⢃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⣿⠛⠉⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡏⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⣿⡃⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣴⠟⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠿⠻⢿⠟⠛⠿⠿⡿⠟⢻⣿⢿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣾⣷⣾⣷⣾⣿⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿