Sweaty Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠉⠉⠉⠉⠙⠚⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⠠⠄⠈⠅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⣿⠀⠸⠀⢠⣆⣠⠀⢀⣤⣤⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢰⡀⠀⠀⡷⣀⠛⣄⡎⣶⠉⠞⠈⣤⢇⢱⡆⠘⠟⣧⠎⠘⡇⠻⢿
⠀⠀⠀⠀⠀⢠⠒⢀⡰⠶⠂⠀⠀⠀⠀⠘⠀⠀⠂⠄⠀⢀⠇⠙⠀⠘⠀⠀⠀⠀⡀⠁⡐⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⡜⠸⠀⠀⣠⠤⠤⣤⣜⠁⠀⡋⠀⠀⠙⣦⣀⣀⡀⡽⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⠳⢤⡀⠀⢰⢁⡓⣠⡮⠤⠒⠋⠀⠈⢳⠀⠓⠀⢠⠎⠁⠙⠶⣽⣆⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⠚⢢⢻⣤⡜⢀⠀⣏⠀⠀⠀⠀⠀⣿⠆⡇⠀⠀⡇⠀⠀⠀⣸⡇⢸⣽⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡧⢲⡆⠀⠉⠋⡘⠀⢹⡀⠀⠀⠀⠀⢙⡼⠀⠀⠀⢧⣀⠀⠀⠈⢁⠞⠳⣟⡮⠀⠀⠤⠴⢔⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢠⣻⡄⠀⠀⢠⢳⠀⠀⠙⠲⠦⠤⠴⠋⠀⠀⢠⠤⢶⠉⠙⠒⠊⠁⢸⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣤⠄⠀⠀⡆⡘⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡠⢠⠖⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠘⠁⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⣏⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠚⠀⢀⡤⠚⠙⠳⡀⠀⠀⠀⠀⠶⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⠀⠀⠀⠘⠞⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠉⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠘⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢠⠂⠀⠀⠀⠸⡀⠉⠉⠒⠒⠶⠦⣤⣤⣤⣤⡄⠤⠤⡖⠺⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣠⠇⠀⣀⠀⠀⠀⠙⢄⠀⢀⠔⠚⠳⣧⡴⠓⢦⡀⠀⣠⠃⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠐⠋⠀⢠⠋⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠙⠁⠀⠀⡧⠚⡅⠀⠀⠀⠈⡇⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⠓⠦⢤⣀⡜⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⢱⠀⠀⠀⢀⣸⡀⠀⠀⠀⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢨⠁⠀⠀⢠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠈⡗⠂⠉⠉⠈⣧⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡘⠀⠀⠀⠰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢣⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠶⠿⠿⠛⠻⠿⠿⠷⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣟⠋⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⢰⣿⡇⢀⣤⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⢿⡆
⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠟⠛⠉⠀⠀⠀⠀⠈⠛⠁⢼⣿⡇⠀⠀⠉⠻⢷⣦⣄⡀⠀⠀⠘⣿
⢸⣟⠀⠀⣀⣀⣀⣠⣤⣴⣶⠶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠶⣿
⣿⡟⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢀⡀⠀⠀⠀⢠⣴⡶⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠶⠶⠀⠀⠀⠀⠀⣴⡄⠀⠘
⣿⡄⠀⠀⣾⣷⡀⠀⠀⠀⢀⣤⣴⡶⠿⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠶⢶⣶⣦⡄⠀⠀⠀⠀⢰⡿⣿⠀⢐
⢿⡇⠀⠸⣿⣿⠇⠀⠀⠀⢸⣯⡀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⢀⣼⠇⠀⠀⠀⠀⠘⠿⠟⠀⢸
⢸⣧⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠻⠶⠿⠛⠁⠀⠀⢠⡄⠀⠀⢠⡀⠀⠀⠈⠛⠷⠶⠟⠋⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾
⠀⣿⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⢰⡿
⠀⠘⣷⡀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠀⢀⣿⠃
⠀⠀⠹⣷⡀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⣄⠀⢀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠏⠻⠿⠋⠙⠿⠿⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⡾⠋⠁⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⣿⣷⡄⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⢋⣤⠀⠈⠔⢐⣤⠒⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢚⣿⡇⢈⠉⡸⠊⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠁⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢐⠀⠀⠀⠀⠠⡀⠀⠀⠈⠀⠄⠀⠀⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⢀⠀⠀⠁⠢⠄⣀⢀⣀⣀⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡌⠀⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠠⡀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⣀⠠⢖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⡍⠀⠀⠀⠀⠀⠀⠀⠲⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⡐⢇
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡴⢃⢤⡀⠎⢳⠀⠀⣤⡀⠀⢀⠐⠅⠒⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡉⢀⠁⢸⠈⢀⠀⡀⠀⠀⠁⢤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠈⠳⣗⡴⠀⢸⣀⡤⡠⣦⣤⡀⠉⡁
⠀⠀⠀⠀⠀⠀⠀⢘⠸⠁⠠⢴⣾⣷⣿⣩⣧⣿⣿⣷⣦⣦⣤⣼⣿⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠶⠛⠛⣷⣦⣶⣿⣷⣶⣶⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⢠⡶⣿⣁⠀⢻⣿⡿⠿⠿⢿⣿⣿⣿⣞⣶⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⢀⢠⣿⣿⣯⠁⣠⣿⠛⢿⣶⣶⣦⠌⠙⢦⠉⠘⠿⣧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠛⠁⠀⣠⡴⠃⠈⠳⠤⢈⣴⣞⡛⣷⠀⠀⣬⣿⣿⠏⠐⠄⠀⠀⠂⢿⡆⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⣁⣀⣠⣴⡶⠛⠉⠀⠀⣸⠃⠀⡟⠋⢛⣱⡏⠀⠀⠘⣿⠛⡳⠀⠀⡗⢦⠀⠁⠘⣦⣤⡹⣮⡧
⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⢺⡟⠛⠉⠀⣼⣀⣀⣠⡾⢶⣿⠲⠴⢿⣿⣿⣿⠃⠀⠀⠀⢸⣿⡓⠀⠀⠘⠛⠀⠀⢾⣏⢻⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣦⠴⠞⢳⡀⠀⠈⢻⣝⡛⠁⠀⠀⠉⠀⠀⠈⠁⣿⡟⠀⠀⠀⠀⢸⡇⠙⠢⠀⠀⠀⠀⠀⠀⠉⠹⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⣳⡄⠀⠀⣿⠁⠀⣴⣻⠄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⠛⠛⢧⡀⣾⣟⠀⠀⠙⠉⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠙⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣦⠉⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠁⣠⠞⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠘⢧⠀⠀⠈⠙⠷⢦⣄⠀⠀⣠⣤⣤⠶⠶⠛⠛⠋⠁⠀⣠⡞⠉⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣠⢾⡉⢧⣰⠏⠹⣆⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⡀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣸⠛⢷⣿⣮⣿⣤⠖⣛⣷⡄⢀⣴⠟⠁⠀⠀⠀⠀⠀⢿⡀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢸⠁⣸⠟⠋⠁⠘⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠙⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡞⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣛⣭⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣇⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠾⠟⣋⢹⣧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⢹⡇⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠁⣷⠀⠀⢻⡄⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣦⣤⣶⠾⣿⠀⠀⠸⡇⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣦⣄⣀⠀⠀⠀⣿⣵⠏⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⢉⣿⣋⡾⠀⢹⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⡿⠛⢹⡇⠀⢸⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⢈⣷⡀⣸⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡿⢿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠘⠉⠉⡏⠀⠀⣀⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠖⠋⠉⠐⠛⠋⠀⠙⠛⠛⠻⢶⣤⣄

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀
⠀⠀⠀⠀⢰⠋⢳⡞⢀⠀⢧⠀⠀⠀⢀⠀⠀⢀⡤⠖⠒⠒⢦⡀
⣠⢤⣠⢤⠏⠀⢼⡀⢬⠦⠘⡆⠀⣠⠋⠀⡰⠋⠀⠀⠀⠀⠀⢹⡀
⠛⢋⡱⢦⠀⠀⠀⠈⢳⡀⣟⣿⡀⡴⠒⠲⣇⣄⠀⠀⠀⠀⠀⢸⠃
⠀⠈⠁⠘⣆⣠⠞⠳⢶⣛⣆⠀⢿⠀⠀⠀⡟⠛⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠈⠀⠀⣀⣤⢽⣾⠀⠈⠓⠶⢺⠟⠲⠤⠤⠤⠴⠯⠤⠤⠶⠖⠒⠒⠒⠲⠶⣤⡀⠀⠀⠀⢰⡋⠓⢦⢠⠰
⠀⠀⠀⠀⠀⠀⣾⠠⠾⠟⠮⠁⢠⠀⠀⠁⠀⠀⠠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣧⣤⡏⠛⢻
⠀⠀⠀⠀⠀⠀⠘⠒⠶⠖⢲⡀⠒⠋⢠⣻⠄⠠⡀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢈⠧⢾⡏⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣤⠀⠉⠀⠐⠚⠀⠀⠀⠈⠓⠦⢬⣀⣊⣢⣐⣄⣤⠾⠃⠀⠰⣖⠯⢤⣬⢿⠡⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠓⠃⢞⡧⣠⢦⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⠉⠙⠞⠀⠀⠀⠀⠀⢠⠖⢷⡚⢹⠆
⠀⠀⠀⠀⠀⠀⠀⢰⡚⢩⡍⢹⡆⠀⠀⠀⠀⠈⠋⢀⣀⠤⡤⢶⠒⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠂⣖⣋⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠈⠙⡓⠤⢄⣀⣀⡴⠚⠉⢳⡀⣁⣀⠗⠀⠀⠸⡗⣤⠤⠤⠤⡔⠒⠉⢁⣠⠴⠯⠭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡗⠦⢤⣀⡀⠈⠉⠉⠐⠒⠛⠛⠒⠒⠒⠢⣀⣿⠉⠁⠀⠀⢳⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠙⠒⠦⠤⣀⣀⠀⠀⢸⠀⠀⠙⠻⠀⠘⢦⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣏⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢤⠴⠒⠋⠉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠾⣆⠀⢿⠒⠒⠒⠒⠛⠋⠉⠉⠁⡰⢁⡎⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠚⠄⠘⡄⠀⠀⠀⠀⠀⠀⠀⡴⢁⡞⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢮⣩⠄⣠⠏⠀⠀⠀⠉⠙⠛⣻⣛⣛⣋⣉⡤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡤⠴⠾⠾⠟⣟⠉⠑⠒⠒⠊⣉⠭⠛⠯⢄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡝⠛⠻⢭⡳⣄⣀⣨⣧⠀⠀⠀⣼⣀⠀⣀⡤⢖⣻⠿⢟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣕⣒⣛⣊⣩⠭⠭⠒⠉⠀⠀⠀⠙⠲⠤⣉⣉⡛⠒⠦⠤⣾

⠀⠀⠀⠀⢠⡖⠒⠒⠢⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⣀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠈⠐⠲⢤⣠⡀⣀⠔⠊⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠈⣀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠈⡝⠁⠀⠀⢀⣀⣀⠀⠀⠀⠀⠱⡀⠔⠀⠈⣀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣠⣴⡾⠟⡋⠛⠇⠀⠀⠀⠀⣧⠀⡶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠋⣥⠀⠀⠂⠀⠀⣠⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠀⠀⠀⠀⠀⠈⠈⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⢊⡡⠔⠋⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠏⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⡀⡤⣖⠋⠈⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠈⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣦⡀⠀⠀⠀⠺⣿⣿⣿⣶⣶⣾⣿⣿⡷⠀⠀⠀⠀⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡆⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣤⠂⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠆⠀⠀⠀⡿⠿⠿⠿⠿⠛⠿⠿⠛⢀⣼⣿⣿⡆⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⡀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠈⣳⣿⣿⣿⠟⠃⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⠿⠟⢋⠑⠊⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡠⡉⢉⠉⡉⠀⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠐⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠂

⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣤⣄⠀⣠⣤⡀⠀⠀⣀⣤
⠀⠀⠀⣀⣿⣿⡧⢤⣀⠀⠀⠀⠀⠀⣰⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣷⣿⣦
⠀⡴⢋⡥⢿⣿⠧⢤⡈⠳⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⢸⢣⠏⠀⠀⠀⠀⠀⠙⡆⢿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⢸⡸⡄⢀⣀⣀⣀⣀⣤⡇⣸⠀⠀⢸⣿⠛⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠋⢉⣉⠀⠀⢻⣿⣿⠀⠀⠀⣠⡤⣤⡀
⠈⠳⣼⡯⠿⠽⢧⠿⢾⡴⠃⠀⠀⣾⡏⠘⠿⠀⠀⠀⢷⠆⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⢿⣿⡇⠀⠀⠈⠓⢤⡽
⠀⠀⢈⣹⡖⠶⢯⠙⠋⣷⠀⠀⠀⣿⠇⠀⠀⠦⣄⡀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⢀⢀⠸⣿⡇
⠀⠀⠻⠃⠙⢦⡘⠃⢀⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠃⠙⠃⠈⠉⠀⠀⠀⠀⠀⠘⠋⠀⣿⠟⢦⡀
⠀⠀⠀⠀⠀⡟⠙⠋⠉⢹⡄⡞⠉⢹⠀⠀⠀⠰⢤⣄⡀⠀⠀⠀⠤⣶⣖⣃⠀⠀⠀⠀⠀⢻⠀⠀⡇
⠀⠀⠀⠀⠀⢳⡤⠤⠶⠚⣇⠱⣄⣸⠀⣄⡄⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⡶⠚⠁⠀⢶⣤⡀
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠸⡄⠀⢹⠀⠉⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⡇⠀⠀⠀⠈⠳⠿
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠹⣄⢸⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⡇
⠀⠀⠀⣠⢤⠖⠀⠘⢦⠀⠀⢀⠈⠙⢇⠈⠃⠀⠰⠲⠤⠶⠤⠖⠲⠞⠓⠚⠓⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠷⠋⠀⠀⠀⠈⠳⣄⠘⠞⠃⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⢀⡴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠈⠙⠲⠤⣄⣘⡛⠀⠀⠀⣀⣀⡤⠴⠚⠉⠀⠈⢳⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠉⠉⠓⢦⡀⠀⠀⣯⠼⣿⣿⣭⣉⡇⠀⠀⠀⠀⢦⡤⠀⢧⠀⠀⠹⣍⢳
⠀⠀⠀⠀⠀⠀⠀⣠⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣿⣿⡀⠈⠀⠀⠀⠀⢧⠀⠀⠀⠘⡇⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⣿⣿⡇⠀⠣⠶⣄⡄⠈⣇⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣸⣿⣿⣿⠀⠀⠀⠈⠀⠀⣿⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⣿⣿⡿⠁⠀⠀⠀⠀⠀⡟⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣝⣋⣀⣀⣀⣀⣀⣠⣤⡇⣿⠤⠶⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⣀⣀⣿⣿⣿⣿⣿⣿⡇⣸⠶⠶⠒⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣧⣶⠀⣀⣠⢿⣷⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣉⣭⣵⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⡟⢀⣿⣿⣿⣿⣿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠋⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠙⠻⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠒⡉⠉⣀⡀⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠛⠫⡒⣼⡇⠠⠊⢠⠾⠊⠁⠀⠀⣀⠠⠬⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠚⠳⠦⡈⠉⠚⠆⠘⠀⠀⠋⠀⠀⠀⠴⠚⠓⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡥⠤⠤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠄⠀⠰⡏⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⠖⠉⠀⣀⣀⠀⢀⡠⠔⠊⢁⡠⠤⠐⠒⠲⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠤⡲⢛⡉⢀⠔⠉⢀⡴⠊⠁⢀⡤⠢⠄⢸⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢸⡗⠁⠀⠈⣦⡖⠁⡬⠀⠀⡎⠀⣠⠜⠓⡤⣀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠎⣁⣬⡙⠁⠀⠀⢇⠁⠀⢠⢦⣄⢤⠤⠧⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠒⢀⠀⠀⠀⠀⠀⠑⡲⠋⠀⠈⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⠌⠁⢸⠀⠀⠀⠀⣀⢀⡄⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⢄⡀⢸⠀⠀⢀⡾⠚⠛⠃⢀⠎⠀⠀⠉⠁⠒⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣨⣿⡄⠀⣞⠷⠒⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠖⡱⠚⢉⡿⠋⠑⣄⠀⠀⣠⠞⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⢤⣤⠰⠖⠉⠁⠀⠀⢀⠃⢀⡟⠀⢀⡀⠀⠉⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡁⠀⠀⠀⠀⠀⡴⠂⣠⠆⠀⠀⠓⢹⠀⠀⠘⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠹⣧⡀⢀⠔⠒⠉⢀⠔⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠨⢿⣅⣀⠀⣠⠃⠀⠀⠤⠤⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⣈⡙⢾⠿⣁⡀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠐⡝⠉⣶⠤⣳⠐⠉⢦⣀⣀⢉⣁⣠⢾⣉⠤
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠰⠁⠣⢿⢦⠀⠈⢳⠁⠸⣆⠦⢴⠏
⠀⠀⠀⠀⠀⠀⣔⡷⣤⣧⣀⡴⠊⠉⠳⠀⠀⡃⠀⠈⠃⠁
⠀⠀⠀⠀⠀⢨⠞⠂⠑⡽⣙⢦⣤⣤⣤⣀⣼⡀
⠀⠀⠀⠀⠈⢛⣲⣔⣼⢧⣄⣬⣌⣁⣲⠾⢶⠉⠒⢤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣾⣦⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣀⣤⣤⣶⠶⠾⠿⠿⠿⠿⠷⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣴⠾⠛⠋⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣴⡿⠉⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠈⢻⣦⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠞⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠛⢁⡤⠀⣠⠆⠀⠐⣦⡀⠀⠙⠗⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⡞⢻⠀⠀⠀⡟⢻⣆⠰⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⣠⠀⠀⠀⣾⣴⣿⡇⠘⡇⠀⠀⡇⢸⣿⣷⣧⠀⠀⠀⠀⠘⣧⠀⢀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠙⠂⠀⢠⣿⠀⠀⢀⣾⢿⡿⠃⢸⡇⠀⠀⡇⠈⢿⡿⢿⣦⠀⠀⠀⠀⢿⣶⠋⠙⣆
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⡾⣿⠀⠀⠉⢷⡀⠀⠀⣼⠁⠀⢶⢱⡀⠀⠀⣸⠁⠀⠀⢸⠀⢸⣇⣼⠃⡿
⠀⠀⠀⠀⣠⡶⠛⠻⣧⠀⠰⣧⠏⠀⠀⢰⡌⠳⠤⠜⠃⠀⠀⠘⢧⡙⠦⠞⢃⡖⠀⠀⢹⡆⠘⣧⠻⢶⠇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣿⠀⡘⢦⡈⠓⠀⠀⠀⣤⠀⠀⣙⡓⠒⠀⢠⠤⠀⠀⠈⠻⣦⡈⠥⠆⠀⠀⢸⡇⠀⢻⣧⣿⠀⠀⠀⠰⠊⢁⡤⠤⠤⠤⠄
⠀⠀⠀⠀⠹⣧⠿⢸⣷⡄⠀⠀⢀⣿⡄⠀⠈⠁⠀⡀⢠⣤⣤⣄⣀⠀⠈⠛⢦⣀⠀⠀⠈⠃⠀⠸⣿⡃⠀⠀⠀⠀⠀⣀⡤⠔⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠘⢷⡄⠙⠿⠀⠀⢘⢻⡇⠀⠀⠀⣼⠃⠀⠀⠀⠈⠉⠛⠲⠤⢀⡸⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣴⠀⡏⣸⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⢸⣿⠁⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠙⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣾⡟⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⣿⢀⣠⣤⠴⠶⠶⠶⠒⠀⢤⣄⣀⠀⣿⠀⢀⣾⡿⠁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢿⣧⡀⠀⠀⣤⠻⠛⠁⠀⠀⠐⠛⠿⠗⠀⠀⠀⠙⠇⡿⣠⣿⠟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠙⢿⣦⡀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⢀⣉⡛⠷⠶⣬⣷⣦⣄⣀⣀⣀⣀⣀⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠈⢛⡖⣣⡀⠀⠀⢀⣠⠞⠋⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⢻⡏⢹⣄⣤⠶⠋⢙⣿⠛⠻⠛⣿⠟⢿⣁⠀⠀⠀⠀⠀⣠⠖⢢⣤⣾⡜⢻⠟⠛⠛⠉⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠴⠋⠁⢠⡤⠞⣿⢻⣧⠶⢾⡷⢀⡟⢉⢻⡄⠀⠀⣰⠻⣦⣀⡈⠻⣄⣿⠀⠀⠀⠒⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠾⣿⡄⠀⠀⠀⠈⣷⢚⣿⠻⡟⠓⠦⠐⠛⠀⠈⣇⡻⣄⠀⢹⠦⢀⡿⠀⠀⠈⠻⣧⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠶⠖⠚⠋⠀⠀⠀⠀⠀⠀⣨⠟⢠⣧⠀⠀⠀⠀⠀⢤⣿⢇⣼⠗⠺⡗⠸⠃⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢠⡤⣤⠤⠶⣿⠁⠀⣿⣿⡄⠀⠀⠀⠀⠈⢳⡾⠁⢠⠖⠻⣆⡀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⣠⡾⠉⠀⠀⠀⠀⣿⣴⣾⣿⣿⣯⣥⣤⣤⣈⣁⣘⣿⠀⣯⠀⠀⠀⠡⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⢀⣤⣤⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣹⣦⣤⣴⣾⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⠉⢹⡿⠉⠉⠉⠉⣿⡏⠀⢿⣿⣿⡿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠾⣧⣬⠿⣯⠀⢸⡇⠀⠀⠀⠀⣿⡇⢠⡾⣅⣨⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣍⣉⣃⣀⣤⠶⠶⠶⠿⠁⠀⠀⠀⠀⠀⠀⠉⠀⠸⣦⣤⣄⣀⠈⠳⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⠲⠶⠶⠾⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠛⠛⠛⠛⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢿⣤⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠈⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣦⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⢠⠞⠁⠈⠉⠙⠛⠛⠲⠶⠶⠬⠭⢭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⠴⡟⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢠⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⠁⠀⠀⢤⣴⣶⣦⣤⣙⠚⠀⠘⠀⠻⠽⠛⢃⣀⣀⣀⣀⡀⠀⠀⣸⠃⢻⡘⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⢠⣿⠁⣀⣿⠀⠀⠀⠒⠒⠶⠚⠛⠛⠁⠀⠀⠀⠀⠛⠻⠿⢿⣿⣛⠓⠒⠀⠀⡟⠀⠀⡇⠀⠀⣠⣿⣅
⠀⠀⠀⠀⠀⠈⠿⣤⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣧⠀⢠⣇⡠⣫⠄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⣌⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣳⣶⣤⡆⠀⠀⠀⢀⣦⣤⣤⣤⣶⡶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⡤⠶⠞⠛⠉⠁⠈⠛⠿⣿⣶⣿⣥⣤⠞⠁⠈⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⡄
⠀⣀⣀⡟⢳⣤⡴⠚⠉⢿⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⡀
⢺⣥⡉⠁⠀⠁⢀⣠⡴⠾⣷⡀⢀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣼⣁⠀⠙⠳⠴⠋⣧⣀
⠻⢥⡴⠂⡴⠞⠋⠁⠀⠀⠘⠛⠛⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢦⣀⡼⠃⠉⠳⢦⡀⠀⠀⠁⣤
⠀⠈⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⢤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⣄⣀⣈⣉⣉⣉⣉⡉⠉⠙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⠀⠉⠉⠉⠉⠉⠉⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⡁⠈⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠋⠉⣁⣹⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣖⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠃⢠⠈⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡧⡀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣾⠀⠀⠰⠀⠒⡎⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠡⠉⠀⠀⠀⠒⠀⠁⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠁⡀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠐⠒⠆⠀⠊⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡅⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠠⢘⠂⡀⠀⠀⠈⠲⠤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠇⠉⠀⠀⠀⠈⠑⠀⠀⠀⠀⠠⠦⢤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠠⣠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠈⠉⠀⡰⠃⠐⠊
⠀⠀⢀⣀⣀⣀⠀⠀⢀⢀⡀⠀⠈⠠⡀⠀⠀⠈⠀⠀⠀⡀⣠⠄⠀⠀⠀⠀⠀⠉⠀⠀⠒⠿⠃⠀⢋
⠀⡎⠁⠀⠁⢠⡬⡄⢾⠚⡄⠉⠁⠀⢈⢀⠀⠀⠀⠀⡇⢺⠗⡊⠀⠀⡀
⠀⠀⠀⣀⣀⣘⣿⣷⣿⣄⠈⠀⠀⠀⠀⣯⣠⡀⠀⣀⣸⣯⡤⠿⠟⠟⢿⠵⠷
⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⡟⠓⢛⣋⡀⢱⣀⣾⣶⣶⣤⣤⣤⣤⣤⣿⡶
⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⠶⠛⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⢿⡿⢛⠓⢃⠴⣲⣾⠛⣻⣽⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣡⠶⡾⠟⡁⢈⣑⠚⡛⢡⣾⣿⣏⣶⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣷⣾⣿⣯⡰⠁⣆⡟⠛⠛⠁⠈⠛⠻⠶⠞⣿⡶
⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⣿⣿⣶⠈⠄⢂⡠⠤⠀⠐⠈⢭⣭⣿⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠠⢛⣲⢿⣿⣿⣿⠿⢋⣠⣂⡽⢄⣠⡴⠦⠀⠀⢿⣿⡭⢱⠀⠀⠀⠀⠰
⠀⠀⠚⠀⣠⠚⢡⠒⠒⢻⣿⣿⣭⣤⣿⠃⣿⡻⣯⡶⠖⠀⠐⠈⡁⠳⠀⠀⠀⠃
⠀⠀⠀⠀⢉⠑⡇⠀⣀⣼⠈⢿⡦⠶⣿⣷⠈⠁⠻⠦⠀⠀⢀⡆⠐⣦⠞
⠀⠀⠀⠀⠈⢸⣤⡞⣠⡾⠀⣈⣧⠀⠀⣿⡇⣀⢠⣄⡀⠀⠘⠇⠀⠉⢿⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡏⠴⢇⡀⠈⠻⣦⡀⣾⣧⣤⠏⠀⠉⠲⠀⠀⡼⠀⢸⠙⣎⠢⣄⡀
⠀⠀⠀⠀⠀⠠⣞⠹⣶⡀⠑⣄⠀⠙⢿⣿⢻⠥⢚⣻⠋⠁⠀⡰⠁⡀⡇⠀⢹⡄⠰⡍⠑⠢⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡗⠸⣿⡀⠈⠦⠤⢤⣻⣦⠀⠚⠋⠀⠀⡴⠁⠀⣿⠀⠀⢸⡗⢄⡁⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⢠⣿⡇⠀⢹⣷⠀⠀⠸⢸⣿⠎⢷⣦⣤⠖⠋⠀⠀⣸⢃⠀⠀⢸⣷⡸⠈⠑⠦⢄⡀⢀⣠⠽⢦⡀
⠀⠀⠀⠀⠀⣸⣿⡇⠀⣼⠇⠀⠀⢱⠋⣧⡀⠈⢿⣿⣦⣄⠀⣰⣃⣌⠀⠀⢸⠟⠁⠀⡠⠀⣠⠟⢁⠀⡀⠀⠹⡄
⠀⠀⠀⠀⢠⣿⡏⢷⣼⠏⠀⠀⠀⡞⢀⠙⣷⡄⠀⢻⣿⣿⣰⠇⡉⣿⣧⡀⢸⠀⠀⣰⠁⡰⢡⠞⠁⡴⠁⠀⠀⡘⡄
⠀⠀⠀⢠⡾⠀⢇⣼⡟⠀⠀⠀⢠⠃⠘⡄⠈⢿⣶⡘⣿⣿⡿⠀⢧⣿⣿⣧⣼⠀⣰⠃⣰⠳⠃⠀⡼⠁⠀⠀⠀⣷⢹⡄
⠀⠀⢀⣟⠀⢀⣾⡟⠀⠀⠀⠀⡞⠸⡄⠹⡄⠀⠻⣇⣿⣿⡇⣸⣾⣿⡿⠉⠁⢰⡏⢀⡇⠀⢠⠞⠁⠀⠀⠀⢰⡏⠀⣧
⠀⠀⣾⠛⢹⣿⡿⠁⠀⠀⠀⢰⠃⡀⢷⠀⢻⡄⠀⢸⣿⣿⢠⣇⠈⣿⠃⠀⡄⣼⠀⡸⢀⡴⢃⠔⠀⠀⠀⠠⠿⠁⣤⢸⡄
⠀⣰⣿⣶⣿⣿⠃⠀⠀⠀⢀⡏⠀⣧⠈⣧⠀⠹⣦⡀⢸⡇⣼⡏⠠⠏⠀⢰⡇⣿⠀⣼⠋⣰⠋⠀⠀⠀⠀⠀⠀⢀⠟⡀⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠩⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠐⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠄⠀⣀⠠⠀⠀⢠⠦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣴⠀⠀⠀⡤⣈⠤⣤⠂⠉⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⢀⠀⠀⢅⠀⡀⠼⢆⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠈⠀⠀⠐⠉⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡄⢄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠶⠉⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠀⡀⢀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠰⣤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡰⣀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣦⣦⠰⢤⠓⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⡦⡀⠀⠀⠀⠉⢻⠀⠀⢀⢣⠀⠀⠀⠀⠀⠀⢀⢀⠄⠒⠺⢥⠀⠀⢀⢐⣇⣄⠀⢠⡌
⠀⠀⢀⡠⠀⠀⠀⠄⠀⠀⠀⠀⠁⣤⣀⢰⢀⠁⠱⣄⠀⠀⠀⢰⠀⠀⢠⠀⢆⢰⡖⠀⠉⠁⠀⠐⠀⠀⠀⠀⠈⠀⠉⠉⠛⠀⠑⠁⠃
⠀⢀⡏⠁⠀⠀⢇⣰⣶⡤⡎⢨⡟⢻⠀⠀⠉⠀⠀⢈⠀⠀⢀⠆⠀⠀⠈⠀⠘⡷⠐⠃
⠀⠀⠣⡀⠀⠀⠄⢰⣾⢷⣡⡸⣆⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⠀⠀⠀⠀⠐⣜⣃⢀⣷⡆⣿⠏⡐⢢⣖⠆
⠀⣀⣤⣤⣤⣾⣿⣶⣶⣾⣿⣿⣿⣷⣄⣀⠀⠀⢀⣀⣼⣿⣿⣆⠠⠒⣒⡾⠛⠭⠀⠠⣑⣄⣀⠀⠉⠁⡑⠀⠀⠀⢀⣄⣀
⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣬⣤⣴⠀⣷⣤⣿⣿⣿⣿⣿⣶⣶⣾⣿⣶⡾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠲⠀⠀⠐⠰⠐⠐⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠩⣙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠈⠁⠹⡄
⠀⢀⣠⣤⣀⡀⠀⠀⠀⠙⠓⠦⣄⠀⢻
⡴⠉⠔⠢⠌⠉⠳⢄⡀⠀⠀⠀⠈⢳⣸⡇
⢧⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⣀⠀⢀⣀⣀
⠈⠑⠒⠒⠒⠛⠛⠒⢦⣈⢧⠀⠀⠀⠀⠀⠀⢠⡴⠋⠉⠉⠉⠉⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢳⡄
⠀⠀⢀⡤⠴⠦⠤⣄⣀⠀⠀⠁⠀⠀⠀⠀⣿⠀⢀⣤⠖⠛⠉⠳⣤⡀⠀⠀⠀⣷
⠀⠀⡏⠘⠉⠀⠀⣀⣉⣓⣦⡀⠀⠀⠀⠀⡽⠶⠏⠀⠤⣤⠴⠀⠀⠙⠳⠦⣤⡟
⠀⠀⠻⢤⡤⠖⠋⠁⠀⠉⠙⠻⠀⠀⠀⠸⣇⠀⠀⠀⠐⠦⠒⠀⠀⢤⠲⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⡀⠀⠀⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣄⠀⠀⠈⠉⠉⠃⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠾⠧⠉⠳⠦⣤⣤⣤⣤⣴⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣠⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⣭⡙⠛⠓⠶⠦⣤⣄⣀⡤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠸⣇⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⣼⠋⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠹⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣤⠟⣀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠿⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣞⣡⠾⠋⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣠⡄⠀⠀⠀⠘⢧⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⣰⠦⣤⣤⣩⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠳⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠚⠋⠉⠉⠉⠛⠛⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠉⠛⠲⠦⠤⠤⠤⠴⠺⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⢷⢀⡤⠄⠀⠀⠀⠀⠀⠀⠤⢤⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⢁⡘⠛⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⣠⠖⠛⢲⡘⣿⡷⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⠁⠀⠀⠀⣷⡇⠀⣤⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠘⢦⣀⣀⠀⠉⠀⠐⢯⠇⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⣧⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⡟⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠦⣤⣀⣀⣀⣀⣀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠦⣄⠀⠈⠹⡯⢭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⢤⣀⠀⠈⠙⠲⣄⣻⣼⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⢻⡓⠶⢦⣠⡽⢿⡀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢠⢳⡀⠀⠀⠀⢸⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⢹⠀⠀⠀⠀⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠤⠤⠤⠼⠤⠤⠾⠤⠤⠤⠴⠧⠤⠼⠿⠆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠻⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠦⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠉⠉⣠⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⠀⠠⣤⣜⡓⠘⠰⣶⣂⣀⣀⠀⢠⠋⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⠃⠀⠈⠉⠉⠀⠀⠀⠉⠉⠛⠋⠁⢸⠀⣇⣠⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⣀⠀⠀⢀⠀⢀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⠒⠋⠉⠒⠶⠮⠜⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠢⠤⠚⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡦⡠⠖⠋⠙⠶⠊⡇⠀⠀⠀⠀⠀⠀⠀⡶⢄⠜⠲⢄⡉⠙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠒⠂⠤⠤⠤⠤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡼⠉⠉⠉⢹⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢤⡇⠀⠀⠀⠀⣇⡸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠓⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠰⠆⢠⠄⠠⠄⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡒⠀⠀⢇⠀⠄⡌⡀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢁⠉⠀⢀⠏⢀⡀⠊⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢨⡤⠀⠸⠀⠀⢱⠰⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠤⠄⠤⠈⠀⠀⠘⢄⠀⢻⠃⠒⠠⠤⠤⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⡖⠀⠨⠀⠀⠉⠛⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⠀⢀⠂⠀⠀⠀⠀⠀⡨⠒⠤⢀⠀⠀⠀⢀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡦⠀⠀⢀⠤⠖⠀⠰⠀⠠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⠀⠀⠀⡀⠈⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠘⡄⣤⣴⠂⡞⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠜⡴⠁⢀⢿⡟⠈⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡜⠈⢈⠱⠀⠐⡖⠀⣀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠰⠂⠀⣌⠅⠀⠀⠀⠀⠁⠓⡈⠀⠸⠁⠉⣈⠠⡃
⠀⠀⠀⠐⢚⠁⠈⠃⠀⠠⠆⠀⢠⠀⠸⠀⠀⠀⠂⠉⠁⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⢠⠀⢀⣐⠆⠀⡆⠀⠀⠘⢠
⠀⠀⠀⠀⠀⠀⢰⠀⡎⠀⠈⠉⢢⠤⠦⡜⠀⡧⠆
⠀⠀⠀⠀⠀⠁⠏⠆⠀⠀⠁⠀⠘⠀⠀⠉⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⠋⠙⠁⠀⢀⡤⠁⠈⠙⠿⠛⡲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣁⡤⠖⠂⠀⠀⠙⠒⠀⠀⠀⠰⠬⠭⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣻⣿⣁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⣠⣹⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⢡⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣾⣼⣷⣿⣿⣿⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡧⠋⠀⠀⠚⠀⠀⠀⠀⠀⠘⠀⠀⣴⢠⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢋⣾⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⣋⣽⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⠿⠟⣻⠟⢫⣽⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢠⡏⢽⡿⡏⠉⢀⡖⢼⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⠑⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⡀⢰⣿⣤⣿⢰⠁⠀⠘⠃⣿⣿⣸⣿⣿⣿⣿⣿⠴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣹⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⢺⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⣿⣼⣿⣿⣸⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣤⢻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣇⣀⡀⠀⠀⠀⠀⣼⡿⢟⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠐⣹⣿⣿⣿⣷⡟⣿⣿⣿⣶⣾⣿⡷⠿⠟⣋⣁⣀⠀⠀⢠⣭⣽⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠂⢉⢿⣿⣿⣿⣷⣿⡿⣿⣿⣿⠛⠔⠖⢀⣘⠡⠏⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⡂⠐⠒⡀⠙⢧⡤⠨⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠐⠐⢺⣿⣿⣿⣿⡽⣿⣿⣇⣻⣿⣯⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠻⠟⠛⡛⠛⠿⠿⠿⠿⠶⠮⠉⠉⡁⠡⢬⣅⣤⣾⣿⣿⡿⣯⢿⢽⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣻⣾⣿⣿⣿⣿⠦⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠛⠢⡀
⠀⠀⠀⠀⠀⢰⠃⠈⠑⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢒⡒⢦⡀⠀⢸⡄
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⠛⠿⣶⣶⣮⠿⠋⠁⠀⠘⠛⣪⢭⡤⣡⡾⣀
⠀⠀⠀⠀⠀⢠⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠚⠃⣪⠃⠤⠄⢤⡀
⠀⠀⠀⠀⡰⠇⠀⢀⡀⠉⠑⠒⠲⠤⠤⠒⠒⠋⢁⡀⡀⠀⢠⡄⣀⠁⠠⠿⢨⡿⣿⣲⣤⣀
⠀⠀⠀⡘⠀⠀⠀⠀⠈⠁⠒⠠⢀⠀⠀⢀⠔⡙⡀⢃⠑⡈⠂⠐⠠⢘⠰⢤⡈⠉⠽⣶⠶⠿⠱⠂⠀⣀⠀⡤⠄⠤⢠⡄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠶⡬⢐⡈⠄⠀⠂⠁⠨⡒⠠⡈⢀⠐⠱⠂⠀⢆⠉⠘⡁⢎⠡⣂⠔⠄⠆⠊⠁⠘⢆
⠀⠀⣠⠒⠒⠢⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣈⠉⠙⠓⠒⠒⠀⠃⠀⠀⠀⠀⠀⠤⢈⣀⡁⠀⠈⠀⠀⢀⣀⠠⡴⠖⡏⠸
⠀⠀⡇⢀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠉⠉⠹⣟⣛⠒⠛⠮⠭⠭⠄⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⣀⡀⠤⠤⠄⠒⠂⠁⢀⣀⠔
⠀⠀⠈⠢⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠉⠒⠄⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠲⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⣀⡴⠛⠉⡠⠒⢁⡞⣠⡴⠶⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢧⠞⠁⠀⣠⠊⠀⣀⠾⠋⠁⢀⣀⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⡼⢁⣀⣴⣯⣤⡴⠛⠉⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡄⠀⣀⡤⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢠⡎⠈⠑⠦⡀⠀⠀⢰⣦⣟⠀⢸⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⡠⠚⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠛⠀⢀⡾⠁⢈⡷⠋⠀⠀⠀⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠛⠓⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⣤⠒⠒⣲⠄⠀⠀⣼⡀⢀⡜⠁⠀⠀⠀⠀⠻⢋⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⠀⢻⠀⠀⠀⠠⣔⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠹⠴⠛⠁⠀⠀⠀⡿⠌⢸⡇⠀⠀⠀⠀⣠⠞⠋⠉⠉⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡁⠀⠘⣇⠀⠀⠀⠈⠛⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠞⠹⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⡆⠀⠀⣴⠉⠀⠙⣆⠀⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡤⢿⠀⠀⠀⢿⣤⡤⠶⠶⠶⠟⢋⡇⠀⠀⠀⡄⠀⠀⢹⡙⠛⢻⠏⠀⢸⡍⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⢇⠀⠀⠘⢧⡀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢳⡄⠀⠀⠉⠈⠁⠀⠀⢸⣇⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠷⣤⣘⡄⠀⠀⠀⠙⠛⠚⠛⠁⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠶⠺⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣴⠾⠋⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣰⣿⣿⣷⣭⣿⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢠⣇⡘⠻⣿⣿⡏⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢰⠋⢙⡷⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠁⠉⠙⠻⠿⠀⠀⠀⢠⡾⠋⠀⠀⠀⢠⡞⠙⠓⢾⠁⢀⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣛⡭⠿⠿⢷⣶⡦⠤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢸⡆⠂⢴⡿⠤⡎⣀⡾⠁
⠀⠀⠀⣠⠶⠚⠛⠛⠉⢁⡴⠚⠁⠀⠀⠀⠀⠸⣤⣤⣬⠿⣤⣀⠀⠀⠀⠀⢀⣠⡴⣏⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⢀⡀⠠⠘⠁
⠀⠀⣾⠁⡀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠈⡼⠉⠛⠛⠛⠛⠋⣹⣦⢘⣷⠀⠀⠀⠀⢀⣤⠞⣡⡴⠋
⠀⠀⠉⢶⠁⢀⡀⢀⣀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡇⠀⢀⠀⠀⠀⠀⠌⢿⠟⠙⠻⢶⣶⣾⡯⠶⠚⠁
⠀⠀⠀⠈⠛⣿⡀⣸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⠀⠀⣹⠗⢐⣶⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠈⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣘⠀⠙⠃⠀⠾⠏⢀⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣦⠀⠀⠀⠀⠀⢀⣿⣤⣀⣀⠈⠛⠛⠶⠶⠶⠶⠖⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣼⠏⡰⠃⢸⣇⠀⠀⠀⠀⢸⡗⠀⠉⠉⠙⠛⠓⠶⠤⣤⣄⣀⣀⣼⠁
⠀⠀⠀⠀⣠⡿⢡⠞⣀⣴⣏⣹⡄⠀⢀⣴⠞⢻⣇⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⣹⠏
⠀⠀⢀⣾⠋⡴⠃⠰⣿⡟⢿⡀⢻⣄⣼⣿⣄⠀⢙⡷⣴⣆⡀⠀⠀⠈⠙⠛⣺⡧
⠀⣰⠟⣡⠞⢦⢀⣴⠋⠀⠈⢷⣀⣩⡼⠋⠙⠛⠉⠀⠀⠈⠙⠳⠶⣤⡶⠿⣿⣟⡳⠶⣤⡀
⣼⣋⣴⣁⣠⡾⠟⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠆⢀⡿
⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣥⣼⣿⣶⣶⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠧⣄⣀⣀⣙⠉⠁⠀⠀⡔⠋⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⣶⣦⣭⣤⣭⣉⣉⣛⡓⠶⠶⡤⢿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣙⣛⣛⣿⣷⣿⣇⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⡿⠟⠿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡸⣿⣿⣿⣿⢱⣿⣿⡞⣡⢨⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠌⡛⠿⣿⣿⣿⣿⣿⡿⠟⣫⣴⣿⡇⣿⣿⣿⣿⢸⣿⣿⢻⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⣉⢸⣿⣷⣶⣶⣤⣤⡰⢶⣿⣿⣿⡿⢰⣿⣿⣿⣿⢸⣿⣿⢸⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⣶⢰⣿⡆⣿⣿⣿⣿⣿⣿⣿⣶⡆⢠⣬⣄⢻⣿⣿⣿⣿⢸⣿⣿⠸⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⣿⣮⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⢸⣿⣿⠟⣵⣾⡿⢿⣷⣌⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠈⠉⢡⣿⣿⣿⢸⣿⠣⣶⣶⢹⣿⡆⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡘⢿⣷⣭⣥⣾⡿⢡⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢸⣿⣿⣿⣿⣦⣝⣛⣛⣩⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣦⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⢶⣿⣿⣿⣿⣿⣿⣷⡆⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⠟⣫⣾⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⢠⣾⣿⠋⠀⣀⣀⣀⣀⣠⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⣦⣤⣀⣀⣀⣀⣀⠀⠙⣿⣷⡄
⠀⠀⠀⣰⣿⡟⠁⠀⠘⠛⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠃⠀⠈⢻⣿⣆
⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⣰⣿⡟⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⢻⣿⣆
⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣛⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠰⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⡆⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠛⠛⠉⠉⠉⠉⠛⠛⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣴⣿⡟
⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠋⣁⣀⣤⣤⣀⣈⠙⠻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⣶⡿⠿⠛⠛⠛⠛⠿⣿⣶⣼⣿⣿⣿⣿⣿⠃⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠘⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⡿⠟⠁⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠠⣉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠴⠀⠀⠹⣦⡀⠠⠤⠞⠁⠀⠀⠀⠈⠂⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠉⣀⠔⠁⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠻⠆⠁⠀⣠⠤⠤⣀⡀⠀⠘⠱⠀⠀⠀⠀⠀⠀⣰⠃⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⠀⠚⢛⠿⠿⠦⠀⠀⠀⠈⠉⠒⠦⠤⣄⣠⣤⠴⠚⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⣆⣘⢛⠷⠷⠦⠀⠀⠀⠀⠀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢺⣄⢀⣙⠛⠛⠑⠀⢀⣠⠔⠒⠀⠉⠉⠁⠤⠤⠤⠤⠤⠤⠤⣴⠞⠁⣠⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢻⡄⢀⣩⠌⠙⠁⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⣁⡤⠚⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠻⣦⡁⠉⠊⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⡡⠴⠊⠁⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⢦⠀⢀⠈⠪⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⠵⠚⠉⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠱⡀⠀⠀⠀⠁⢸⠀⠀⠀⢠⠏⠀⠐⠦⣄⠀⠀⠀⠀⢀⡤⣾⠗⠋⠀⠀⠀⣠⠴⠚⠉
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠉⠀⠀⠀⡾⠀⠀⠀⠀⠈⠙⠢⢄⣚⠭⠊⠁⠀⢀⡤⠖⠉
⠀⠀⠀⠀⡀⠐⠦⣄⣀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢀⠴⠚⠁
⠀⠀⠀⢀⠱⣄⡀⠀⠈⠉⠉⠓⠚⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋⠀⢀⡤⠖⠁
⠀⠀⠀⠘⢦⡀⠉⠓⠶⠤⠤⣄⣀⣀⣀⡀⠀⣀⣠⡤⠤⠖⠒⠉⠁⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡔⢆
⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⡇⣿⠇⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠾⢿⣿⣿⣿⣿⣿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣿⣿⣿⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠛⠈⠻⣷⣔⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⣤⠋⢀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠦⣄⠀⠀⠀⠀⣿⣿⣿⣿⡇⠈⠁⠐⠋⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⡂⢰⣤⣠⣿⣿⣿⣿⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣹⣿⣿⣿⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠈⠙⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡿⣋⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠛⠛⠛⠛⠛⠋⢉⣼⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠘⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⢀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⡀⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⣰⣿⣿⡿⠋⠀⠀⠈⠻⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⠟⠁⠀⠀⠙⢿⣿⣿⣤⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⠃⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠘⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⡟⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⢻⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠁⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⢁⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⣸⣿⣿⢁⣾⣿⡟⠁⠈⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷
⠀⠀⠀⣠⣿⣿⡇⣾⣿⡿⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄
⠀⢀⣾⣿⡿⠋⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⢀⣿⣿⡟⠀⠀⠀⢹⣿⣿⣄⣀⣀⣠⣿⣿⠏⠀⠰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣼⣿⡿⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠻⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠟
⠀⠀⠀⠈⠛⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣠⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣤⣤⣤⣤⣶⣾⣿⣿⡿⠟⠉
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⠛⠿⢿⣿⣿⡿⠿⠛⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⣀⣤⣶⡿⠿⠟⠛⠁⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⣠⣿⠏⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⠆⠀⣾⣿⣥⣤⡄⠀⠐⢒⡶⠂⠀⠀⠀⠀⠀⠘⢿⣧⡀
⠀⠀⣰⣿⠋⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⡄⠀⠀⠀⠀⠀⠀⠀⠰⢿⢤⠄⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⢠⣿⠇⠀⠀⠀⢼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠿⣷⣶⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣤⣤⣤⣄⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⠋⠉⠉⢛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⣾⣻⣭⣥⠴⠶⣦⣹⣦⣄
⠀⠀⠀⠀⠀⢰⡿⢃⡀⠠⠤⠶⠾⣿⠇⣹⡀
⠀⠀⠀⠀⠀⣸⠃⢏⣀⠀⠤⠤⣤⣾⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠶⠒⠒⠒⠒⠀⠀⣀⡤
⠀⠀⠀⠀⠀⣿⠀⠈⠁⠐⣦⣤⡤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠐⠦⣤⣤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠙⣧⣴⢟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⣷⠀⣶⠀⠀⠀⢸⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣿⠀⠛⠶⠶⠶⢋⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡴⢒⡦⠖⣻⠓⠆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠻⡻⣌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣿⣤⣘⣵⣋⡉⠉
⠀⠀⠀⠀⠀⠀⢻⡀⢢⡀⠀⠀⠙⣦⠙⣧⡀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⢿⣟⢷⣄⠐⢶⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⡀⠀⠀⠈⠃⠈⢷⣄⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣆⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⠃⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣧⣄⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠛⠦⣌⠙⠻⠷⢶⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢠⡿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢸⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠸⣄⠀⣿⣿⢧⣀⣀⡀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸⡇⣸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡎⠒⠀⢹⡄⠑⠚⠒⠊⠀⠀⠀⠀⠀⣿⣿⢯⠐⠦⣤⣤⣤⣴⠾⠋⠀⡠⣞⠄⠀⢸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⣤⠾⠃⢹⡄⠀⠀⠀⠀⠀⠀⢄⠀⡼⡼⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⠷⠶⢶⣤⡤⠤⠄⠀⠀⠸⠃⢰⡄⠀⠀⠀⠉⠢⣭⣥⣷⠄⠀⢀⣾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⠋⠀⠙⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠙⠦⣤⣤⠤⠀⠈⠁⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠉⠛⠛⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⣀⣠⣴⠾⠛⠛⠋⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⣠⣤⡴⠶⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠞⠋⠉⠀⢀⣠⡄⠀⠀⠀⠠⣌⡙⠷⣄
⠀⠀⠀⠀⠀⠀⠠⢾⣿⠁⠀⠀⠀⠀⠿⢋⠀⡠⠀⢠⡀⠘⠷⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⢰⢃⣞⢣⠀⢰⢻⣦⣂⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠆⠀⢰⠄⠀⣸⣿⡏⢸⠀⢸⠘⣿⣿⣄⠀⠀⠸⣦⠔⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢠⢿⠀⠈⢳⡀⠀⡞⠀⢾⣆⠀⢰⠋⠀⢰⠀⣿⣞⣽
⠀⠀⠀⠀⠀⠀⡞⠡⡙⠦⠘⠋⠀⠈⠣⣭⠉⢀⡀⠈⢧⡉⠵⡋⠀⢸⠃⢹⣍⡇⠀⠀⡤⠖⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⡿⢹⣆⠀⠀⣿⠀⠉⠁⡀⢨⣤⣄⣀⠙⠲⣄⠀⠘⠃⠘⣿⠁⠀⠀⢀⡩⠴⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣀⢰⣿⠀⠀⣰⠃⠀⠀⠀⠈⠙⠒⢚⠀⠀⠀⠀⣿⠀⢀⡴⠋⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠘⠛⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣸⡏⣰⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⣧⡴⠖⢒⣒⣀⠒⠲⢤⣸⠀⣴⡟⢰⠃⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠹⣧⡀⠘⢁⠀⠀⠀⠀⠀⠀⠀⢀⣬⠾⠋⠀⢸⡀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣠⣝⡓⠲⣿⣦⣤⣤⣤⣤⡶⠋⠀⠀⠀⢀⡀⢉⢾⣤⣀⣀⠴⠋⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡜⡴⠓⣋⡤⣾⣧⢭⣽⣍⡗⢦⠀⠀⡰⣇⢙⢿⣁⡏⠀⠠⠔⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣿⠆⠀⠀⠸⢞⡟⣿⠓⠄⠋⠀⣯⡳⣄⣷⢈⠏⠀⠈⢳⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠉⠁⢀⣀⣀⣠⡞⢁⣿⠀⠀⠀⠐⢿⡵⢉⡹⣌⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠠⣇⡀⣀⡴⠋⠀⠀⢸⣤⣾⣿⣧⣥⣤⣄⣸⡇⢾⠀⠈⢡⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⣤⡶⠶⠶⣶⡌⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢿⣙⢿⡃⢻⡇⠀⠀⣿⠁⣼⣋⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠭⠤⠖⠒⠚⠃⠀⠀⠀⠀⠈⠈⠷⠤⠤⣈⡓⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠰⠂⠀⠀⠀⠀⠀⠀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣶⢰⠀⠀⠀⢰⠀⠀⢠⠐⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⢀⠐⣀⡸⠁⠀⠀⡘⠃⠀⠀⠀⣀⠀⠀⠈⠷⠀⠀⢠⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡉⠀⠆⠀⠀⢸⣀⢁⠁⠀⠀⠀⠁⠄⠀⠀⠀⡽⡇⠀⠀⢠⠀⠀⠀⠸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⢆⡇⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⢎⡵⠋⠀⠀⠀⠀⢀⠞⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠃⢈⠀⠀⠀⠸⠀⡄⠀⠀⠀⠀⠀⠀⠀⡸⠔⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠄⠀⠀⠀⢄⠻⠀⠀⠀⠀⠴⠀⠀⠀⢆⡀⠀⠀⠀⠀⣀⠀⢠⡄⠀⠀⠀⠀⠀⠇⡇⠀⠀⠀⡇⠀⣷⡀
⠀⠀⠰⠘⡀⠀⠀⡸⠅⠘⠀⠀⠀⠈⣧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢀⣴⡧⠧⡖⠴⠆⠀⠀⢰⡇⠀⠀⠀⢤⠁⣸⣅⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠤⣀⡀⣠⠀⢀⡄⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡬⣿⠀⠀⠀⢸⡉⠋
⠀⠀⠀⠀⡱⢿⣿⠷⣴⠮⡇⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡝⣿⡗⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⡉⠲⣈⣱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⡇⠀⠀⠀⡇⠀⠀⠀⢀⣀⠤
⠀⠀⠀⠀⠀⠀⢈⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠁⠄⠀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠔⢩⠘⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢱⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢧⠀⠀⠀⡐⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡽⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⣇⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⡁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠤⠟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠣⠌⠀⢤⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⣉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠠⠰⠗⣼⣧⡀⠊⠀⢀⡒⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣇⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠙⢫⡋⠛⠃⠀⡤⠈⠁⡗⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠓⢀⠀⠀⠀⠷⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠤⠤⠈⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⡄⠠⢦⠀⠀⠈⠈⢹⡄⠀⠀⠀⠀⠀⠀⠀⣴⣄⢆⠀⠀⠀⠢⠀⠰⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢣⠐⡇⠀⠀⣀⡀⠀⢀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⢢⢰⠀⠖⠀⢸⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⡨⠐⠂⠀⠀⠩⢣⣄⠈⣷⡘⢧⠀⠀⠀⠀⠀⠘⢧⡤⠌⠰⢷⣄⠠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡌⠀⠀⠀⠄⠀⠀⠈⠉⠷⣬⣷⣈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⡂⢀⠄⠀⢿⣳⣴⠀⠀⠈⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠠⢊⠤⠀⠻⢿⠞⠄⠀⠀⠒⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⠴⠊⠉⠉⠙⠲⣄⠀⡠⠖⠋⠉⠓⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡀⠚⠁⠀⠀⠀⠀⢀⣀⣈⣁⡀⠀⠀⠀⠀⠀⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠑⠮⡳⡍⠙⢒⣒⠂⠭⡀⠰⠤⢍⡓⠦⣄⣀⣠⢇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠡⣄⣀⡀⠩⢭⣉⡒⢬⠉⠙⠚⠀⠀⠀⠈⠑⠦⣄⡐⣡⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠉⠈⠳⠝⠆⠈⠓⠮⠷⢑⢦⣉⣉⣠⠴⠒⠒⠲⢤⣀⠀⠀⢀⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣐⡺⠦⣙⡌⠙⠿⠭⠷⢀⣠⠖⠉⠀⠀⠁⠀⠀⣠⠴⠒⣁⡬⠁⡴⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠨⢤⣀⡀⠉⠙⠒⠿⠆⣀⣠⠔⢋⠀⠀⠀⠀⠀⣀⡴⠊⠁⣠⣾⠁⠀⡾⠀⠀⢀⡤⠂⢰
⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉⠉⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠋⠁⠀⢄⠀⠉⠓⠦⠤⠮⠥⠴⠖⠛⠉⠀⠀⣼⣡⡤⠖⠋⠀⠀⢸⡇
⠀⠀⠀⠀⡴⠋⠀⠤⢤⣤⣤⠤⠤⠤⠖⠒⠒⠲⠤⣄⣀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢈⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠇⠀⠀⠀⠀⣀⡤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠁
⠀⠀⠀⠸⣄⠀⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠈⠓⠲⠌⠳⢭⣉⠛⠓⠒⠒⠒⠒⠖⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⠀⠀⣀⣠⡤⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣤⠤⠶⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡖⠛⠉⢿⣦⣶⣿⣿⣿⣿⣿⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⣖⣾⣧⠀⢸⣿⣟⡿⠟⠻⣿⣿⢿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⢀⠀⡀⢾⣿⣿⠄⣠⣾⠏⠻⣷⣷⣶⣂⠙⠲⠄⠘⠻⣗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠁⢀⣠⡴⠋⠀⣘⠲⢀⣴⣾⡋⢽⠂⠀⢰⣿⣿⡇⠑⠂⠀⠀⠁⡻⣆⠀⠀⣄⢄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣁⣤⣤⠴⠾⠛⠁⠀⠀⢰⡏⠀⢸⢉⣉⣵⣿⠀⠀⠈⢿⣉⡝⠀⠀⢇⢲⠀⠀⢘⣶⣦⡙⠾⠇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢻⡏⠁⠀⢰⣧⣤⣤⠾⠛⢿⡏⠓⠻⣿⢿⣿⠇⠀⠀⠀⢸⡟⣍⠁⠀⠈⠉⠀⠀⠿⣧⣻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣗⠛⠉⢷⡀⠀⠈⢳⣭⠅⠀⢀⡀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣧⠈⠑⠀⠀⠀⠀⠀⠀⣄⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣠⣷⠄⠀⣸⡇⠀⢰⣯⠗⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⠉⠈⢷⣬⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠙⢿⣟⠳⣤⡀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⣠⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠻⣆⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠞⠋⠁⣠⡞⠁⠀⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠉⢙⡷⢄⠀⣼⢿⡛⠋⠉⠁⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⣴⣏⣦⡘⣾⡁⢈⣿⣄⠀⠀⢀⣴⠞⠉⠀⠀⠀⢠⣆⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠃⠈⠛⡿⢺⣿⡭⠔⠻⣧⣰⠟⠁⠀⠀⠀⠀⠀⠀⢹⡄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣾⡏⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠴⡾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡞⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⠞⢿⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡇⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠶⠻⡏⠁⠐⣦⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣿⣇⡏⠀⠀⠀⠀⠀⠀⢠⣾⣿⣅⣀⣤⣴⡇⠀⠀⢹⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣦⣄⣀⠀⠀⠀⠀⢸⣇⡿⠙⠛⠃⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⠟⡏⠀⣿⠀⠀⢸⡅⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⣤⣬⣭⣁⣠⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⠿⢻⠇⠀⢿⠀⠀⠘⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠘⣦⠀⣾⠀⠀⠀⡇⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⢿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠘⠛⠓⠏⠀⠀⠀⡿⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠈⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠒⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⡒⠩⠅⠒⠛⠁⠘⠛⠛⠻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠙⢿⣶⣄⣀⣀⠀⣠⣾⡿⠋⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠟⠁⠀⠀⠀⢀⣼⣿⣆
⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠋⠹⣿⡄
⠀⠀⠀⠀⢀⣾⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⡀
⠀⠀⠀⠀⣾⡟⠈⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠹⣿⣄
⠀⠀⠀⠀⠹⣷⣄⠀⠀⠈⠉⠛⠻⠿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⢿⣧⡀
⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣶⣦⣤⣀⡀⠀⠙⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣶⣤⣀⠙⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⡉⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣬⣙⠻⢿⣶⣤⣤⣀⣀⠀⠀⠀⢀⣀⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣶⣭⣽⣻⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢂⣠⣴⣾⣷⣾⣷⣿⣿⣿⣿⣿⣿⠿⢷⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠⣶⣿⣟⣯⣿⣿⠟⣿⣿⣯⣀⠘⠻⠿⢦⣌⡙⠿⣦⡈⠻⣿⡟⠻⣶⣦⣄⡀
⠀⠀⠀⠀⠁⠀⠐⠁⢠⣾⡿⣫⣾⣿⢿⣿⣧⣤⣤⣭⣭⣭⣭⣝⣒⣒⠮⢭⣓⠦⣍⡲⢌⠻⢦⠙⢿⡏⢻
⠀⠀⠀⠀⠀⠈⠀⣰⠟⣩⣾⣿⣿⡃⢀⣉⣭⠤⠶⡒⠒⠒⠀⠀⢒⣈⡉⠑⠚⠭⣲⢝⡲⣕⢤⡑⢌⠻⡄
⠀⠀⠀⠀⠀⠀⠐⠁⣴⣿⣿⠻⠿⠟⣋⡥⣖⠯⣓⠠⠨⠅⣀⡭⠤⣤⣊⠉⠒⠤⡈⢑⢯⡪⣷⣝⢮⠳⡘
⠀⠀⠀⠀⠀⠀⠀⣸⢹⣿⣿⣤⡴⢞⡥⣾⢗⠍⠀⣐⠶⠋⠉⠉⠛⠛⠳⢖⡤⡀⠀⠀⠑⢌⢪⡪⡳⡱⡝
⠀⠀⠀⠀⡄⠀⢠⡇⡜⡏⠛⠋⣠⢞⢞⢝⢕⠕⡩⡢⢕⣠⠭⠭⠥⠴⢦⣔⣤⣄⠀⠀⠀⠀⠑⡑⡙⡝⡜
⠀⠀⠀⠀⡇⠀⢸⠀⠀⣷⣤⡾⢡⢎⣾⢣⢫⢊⢎⡔⢩⡶⠋⣩⡬⠭⢵⣾⣝⢿⣷⡄⠀⠀⠀⠘⡄⠘⡜
⠀⠀⠀⠀⡇⠀⢸⠀⠰⡘⠋⢀⡞⣸⣹⡎⡎⣾⢸⣇⠘⢷⡀⢻⡇⢠⣶⠮⣟⣷⣝⣿⡆⠀⠀⠀⠘⡀⠱
⠀⠠⠀⠀⢃⠀⠸⡇⢢⡻⣦⣾⠃⣿⢸⢷⢳⡹⣦⣉⡛⠦⡙⣄⢷⢸⡇⢀⣈⡻⣿⠻⡇⠀⠀⠀⠀⠁
⡇⠀⡀⠀⠘⡀⠀⣷⣾⣷⣌⡁⢠⣿⠘⣆⠳⡙⠦⣭⡉⢳⡽⡸⣾⢾⡇⣾⠟⠻⡜⠄⣷⠀⠀⠀⠀⠐
⡰⡀⠰⠀⠀⠐⠈⢿⣯⡻⣿⣟⡛⠛⠀⣿⡆⠹⣦⠈⣿⢸⢃⢧⢿⢸⢠⠇⣠⣤⡙⠀⢸⠀⠀⠀⢀
⢣⢣⠠⠣⠀⠀⠀⠀⠙⢿⣮⣝⡿⢟⣚⣋⣡⠴⢋⡴⡣⢫⢎⢎⠎⢃⠏⡼⠛⠙⡇⡀⣸⠀⠀⡀⠐
⢣⣳⡳⡱⡑⠀⠀⠀⢄⠀⢉⡛⠙⠻⠒⠒⠒⠩⠕⠪⣒⠕⠃⠀⡠⣫⠞⣠⣴⣆⣧⢆⡏⠀⠀⠁
⢳⡳⡝⣝⢎⢦⡢⠀⢐⠵⡢⢌⣑⠒⠤⠤⣤⣤⠶⣙⠢⣀⢤⣪⢞⡥⠞⠉⢹⣿⣿⡼⠁⠀⡌
⢧⡙⢌⢮⡳⣕⢯⡢⣄⠉⠚⠥⣊⡉⠛⠒⠒⠘⣐⠦⣽⡾⠿⣚⣩⣴⣶⣷⣾⣿⡿⢁⡔⡘
⡄⠳⣌⠳⣍⠢⣑⠮⣓⠭⣒⡤⢄⣂⣉⠀⠄⠤⠤⠴⠒⣚⣋⣉⣀⢀⣿⣿⣿⢏⣤⢞⠝
⢿⣤⣿⣧⡈⢳⣌⠓⢬⣙⠲⢭⣑⡒⠮⠭⢭⣍⣉⣉⣉⠉⢙⣿⣿⣿⡿⣻⣵⣿⡿⠃⠀⠀⢀⠄
⠀⠉⠛⠻⠷⣤⣿⣿⣆⠙⣿⣦⣌⠙⢻⣶⣦⠀⣩⣿⣿⣷⣿⣿⣿⣽⣿⣿⠟⣪⠂⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⠾⠿⠿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠛⠩⠠⠕⠉⠀⠠⠞⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣾⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠇⢀⣬⣙⣿⣇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⢋⣶⠘⢿⣿⣿⣿⢰⡏⠀⠈⠙⠏⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣿⡏⣠⡀⣼⣿⡇⠀⠙⢿⡇⠸⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⢰⣿⡇⣈⡉⠀⠀⠀⠀⠛⠦⣌⣳⣦⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠉⠁⠀⠀⣤⠀⠀⠠⡦⠉⠁⣿⣆⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⢣⡿⡇⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⣰⠿⠛⢀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⣄⠀⠲⠿⠿⣷⣦⠀⠀⢀⣴⡿⣶⣿⡟⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣿⣷⣤⣄⣀⣀⣀⣤⣶⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⣯⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠷⠶⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⣿⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠢⠤⡀
⠀⠀⠀⠀⡠⠐⠂⠉⠀⢀⠤⢀⠁⠢⡀⠈⠉⠓⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⡌⠀⠉⠭⠁⠤⠐⡊⡡⢔⠱⡈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⡀⠀⠀⣀⣠⠤⠒⢤⡀⢳⠊⢲⡀
⠀⠀⠠⢀⠒⠒⠒⠒⠀⠈⠇⠀⠄⢆⠑⢬⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡏⠈⡉⠉⠤⠄⠒⠀⠆⡿⢉⡆⠈⠉⡆
⠀⠀⠀⠦⠀⠂⠀⠀⠀⠒⠀⠄⠁⠀⠩⣬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⡎⠀⠀⠀⣀⡤⠤⠀⡌⢰⠋⠀⣏⠁
⠀⠀⠀⠸⠙⠛⠀⠀⠘⠛⠚⠀⡄⢠⠒⡣⠰⢑⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⡞⠇⢀⠀⢁⠤⣀⠁⡜⠉⠉⠁
⠀⠀⠀⠈⠢⡀⢀⡠⠀⠀⠀⠀⡠⠃⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠑⣄⡀⡔⣁⡠⠜⡋⠒⠤⡀
⠀⠀⠀⠀⠀⠈⠁⠀⠒⢰⣛⠅⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠈⡱⢿⠒⠊⠁⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠁⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⠁⡰⠁⠸⡆⠀⢸⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠘⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢲⠀⢹⡄⢸⠀⠀⠀⠀⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠻⡌⢲⠒⡦⠰⠃⢧⡀
⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⢾⠀⡀⠀⡀⠼⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠨⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⠀⢿⣁⡠⠆⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠄⠂⠁

⠀⠀⠀⢰⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱
⠀⢠⠒⢺⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡗⠒
⠀⢼⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⢀⣠⢤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⢤⢄⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡶
⣤⢸⡀⠀⡇⠀⠸⣒⡒⢒⣒⣒⣇⢠⡠⠀⢗⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⠂⡤⢠⡁⠀⠀⠀⢈⡇⠀⢸⡇⠀⡇
⠛⠀⡇⠀⡇⠀⠀⠁⠀⠀⠀⠀⠙⣆⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣴⠋⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡇
⠀⠒⡇⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⣀⡤⢤⣤⡄⠀⢀⢤⡖⠲⠤⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠉
⠀⠀⠣⠤⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠘⠉⠀⠈⢑⠈⠉⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⠒
⠀⠀⠀⠀⠣⠄⠤⠃⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⣀⠀⠎⣤⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠓⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠀⠀⠀⡄⠘⠛⠀⡆⠉⠁⢀⠃⠀⠀⠈⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⠀⡠⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⡀⡀⠐⠀⠀⠀⠀⠀⡀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠀⠀⠀⠀⠀⠀⠖⠂⠀⠃⠠⠃⠀⠔⠀⠀⠀⠀⠀⠀⠼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⣀⡉⠦⠀⠀⠀⠐⠤⠀⠀⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣀⢀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡤⡀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠑⠒⠒⠒⠉⣿⠀⢹⡏⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢂⠀⢀⡹⠂⢎⡀⠈⡐⠉⠈⠲⡀

⠀⠀⠀⠀⢈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁
⠀⠀⠀⢤⣌⣠⣤⣉⣳⢤⣀⡀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠁⠀⠁⣀⣀⣀
⠄⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣠⣤⣴⣿⠿⠟⠀⠀⠚⠻⢷⣦⣄⡀⣀⠀⠱⣦⣤⣄⣀⠀⣀⣠⣴⡿⢻⣿⠿⣿⢿⣿⣿⣿
⣤⣀⠀⠒⠂⠀⠀⠀⠐⠶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠁⠀⠐⢒⣢⣤⣦⣤⣤⣭⣽⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣾⣿⣵⣿⠿⢻⣌⡱⣶
⣿⣿⣿⣷⣶⣶⣶⣤⣤⣴⣶⣿⣿⣿⣿⡿⠿⠛⠛⠁⠀⠀⠀⣀⣀⣹⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣥⡼⠻⣧⣉⢳⣤
⠈⠉⠉⠛⠛⠻⢿⣿⣷⣤⣤⣭⣥⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠁⠄⠛⠛⠉⠉⠻⠛⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣝⣷⣌⠻⣧⣄
⠀⠀⠀⠀⠀⠂⣀⣀⣈⡉⠛⠛⠿⠿⢿⣿⣷⣶⣶⣦⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠋⢉⣉⠁⠤⠤
⣦⡔⠛⠓⣦⣤⣯⣭⣭⣤⡤⢶⣶⣆⣀⣭⣭⣥⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⡀⠀⠠⢤⠄⠒⠒⠢⢤⣭⡉⠒⠲
⠛⠛⠒⠛⠿⠣⠤⠤⠬⣴⣶⣮⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⠉⠉⢁⣀⡀⠀⠐⠒⠒⠀⠉⢉⣁⣀⠐⠲⢤⣈⣉⠒⠶⢶⣬⣿⣷
⠛⠓⠲⠶⠶⠦⠤⢄⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⡀⠀⠠⠤⠶⠶⠶⠒⠀⠉⠉⠉⢁⣀⣀⠠⠤⠶⠂⣈⣉⡀⠒⠆⣈⣉⣱⣶⣶⡿⣿⣷⣾⣿⣿⣿
⠤⠶⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠤⠤⠤⠤⠤⠀⠒⠛⠋⣉⣀⡀⠤⠤⠄⣘⡃⠤⢤⣄⣿⣷⣾⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣀⣀⣀⣤⠤⠤⠖⠒⢛⣉⣠⡤⠤⢒⡒⠢⣬⣽⣶⣿⣿⣿⣿⣭⣒⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉⠉⣉⣉⣡⣤⣤⠖⠒⠒⠋⣉⣡⣤⡔⣲⣮⣽⣿⣿⣿⣿⣿⣿⣟⣻⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠶⠶⠛⠉⠉⠉⠀⢀⣤⣤⠶⠖⢊⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣴⠶⠶⠛⠛⣉⣠⢴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠛⠛⠉⠁⣀⣠⡤⠖⣛⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠻⠿⣿⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣴⠶⠾⠟⠛⠋⠉⠁⢀⣀⣤⣤⠞⠛⢉⣡⣴⣾⣿⣿⣿⣟⠿⣭⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⣠⣤⡴⠶⠏⠉⠁⣠⣤⣶⣿⣿⣿⣿⣿⣿⣷⠻⢧⡝⢷⣹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⠶⠶⠾⠟⠛⠉⠀⠀⢀⣤⠶⣾⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣀⣤⠶⢛⣫⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⡴⠾⠛⢛⣩⡴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢛⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣤⣴⠶⠛⣫⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣶⡘⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣩⣤⣴⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠛⠛⠛⠛⠛⠛⠛⣤⡿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⡿⠿⣿⣿⣿⡿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣛⠛⠛⠛⠿⠿⠿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⡀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠷⠶⠿⠿⠿⣿⡿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣾⡛⢻⣿⣷⣦⣀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⢿⣭⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣥⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠿⢿⠟⣉⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢡⡦⠀⢉⣁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠖⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⠛⢳⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠶⠃⠀⠘⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡆⠀⠀⣇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠐⣦⡀⣳⢄⣀⠼⠉⣲⣤⣤⣴⣋⡷⠤⠤⠾⢽⣶⣴⠋⠒⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣀⣀⠀⠀⠀⢸⠉⠉⢰⡤⠒⠉⢠⡄⠀⠀⠀⡂⠀⠀⣤⠀⠀⠈⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠸⣁⠺⢧⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠠⠃⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠉⠉⢹⠒⠋⠀⠀⠀⠢⠐⠤⠀⠀⠤⠒⠀⠀⠒⠠⠤⠄⢂⡠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⠄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠸⠥⣤⠽⠶⠃⠀⠀⠀⠀⠀⣠⠞⠁⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠁⣦⣀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣀⣠⡾⠋⣇⠀⠀⢀⣴⣧⣀⡤⠖⢉⣀⡀⢈⣉⣩⢭⣭⣍⡙⢅⣠⣀⣀⣠⣤⠞⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠛⠉⠁⠀⡾⢡⡞⣉⠔⠋⠉⠀⠀⣀⡬⠗⠛⠂⢹⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠉⠀⣀⣠⠴⠒⠉⠙⠢⣤⣀⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠃⠀⠀⠀⠀⠀⠀⢿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣶⠶⢶⡶⠶⠶⣶⠶⠶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⣾⠁⠀⠀⠘⣶⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇⠀⠀⠀⠀⠘⣇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣡⡼⠀⠀⠀⠀⠀⠀⠸⣄⣹⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣵⣯⡜⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣄⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡿⠀⠃⣹⣶⣦⢀⡀⠀⠄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⣡⣿⡿⠁⠀⠀⡆⣾⣿⣿⡄⠙⢷⣶⣦⣌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣂⣩⢤⣴⣾⣿⠟⠁⠀⠀⡰⢰⣿⣿⣿⠁⠀⢸⣿⡿⠿⠖⠓⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⡤⢖⡟⠉⠉⠀⠀⠀⠀⠰⣁⣿⣿⣿⠃⠀⠀⣼⣏⠀⠀⠀⠀⠀⠠⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡵⣸⢡⣺⠃⠀⠀⠀⠀⠀⢰⣱⣿⣿⡟⠁⠀⢀⣼⢟⡼⠄⡒⢖⣺⣻⡃⠒⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢛⣷⢨⢬⡕⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠀⢀⣤⣿⠟⠘⡶⠜⡵⢃⢋⡹⣷⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⢠⠇⠨⣽⡁⠀⠀⠀⠀⠀⠀⢠⠟⠀⢀⣴⡟⠋⡏⡆⠀⢈⡼⡇⢸⠤⠶⢿⣧⠰⠦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡧⢋⣴⠥⠶⡆⠀⠀⠀⠀⠀⢠⡟⠀⣴⠟⢿⣘⢦⣙⠿⠋⠈⡿⢡⠷⠒⠀⣛⣿⣷⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣴⢱⠌⣴⡟⠁⠀⠀⠀⠀⠀⡿⠀⣼⠏⠀⠈⠻⣦⣈⠔⠒⡁⠐⣿⣾⣋⣁⡈⣽⣿⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠃⡞⢼⣄⣠⣼⡏⠀⠀⠀⠀⠀⣼⠃⣸⡇⠀⠀⠀⠀⣨⢵⡖⣿⠀⠀⠘⣻⣯⢡⠀⣰⡾⠁⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣠⡤⣥⢾⠏⠀⠀⠀⠀⡠⣰⠏⡴⠋⠀⠀⠀⠀⠀⠃⢠⣿⠸⡆⠀⠀⠙⠦⢬⠭⠿⠃⠀⠸⢿⣷⡀
⠀⠀⠀⠀⠀⣠⡾⠋⠘⠋⠙⠛⡁⠄⣠⣴⣀⣴⣵⢋⡼⠃⠀⠀⠀⠀⠀⠐⢸⣿⣿⠀⣷⣶⣄⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⡠⢊⣴⣾⣿⣿⣿⢟⣵⠟⣀⠤⠒⠀⠀⠀⠀⠀⠉⢻⣿⡆⣿⣿⣿⣷⣕⢄⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⢰⣯⠣⠀⠠⠄⣪⣾⣿⣿⣿⡿⣋⡵⠟⢛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡘⣿⣿⣿⣿⣷⣥⣆⡄⠀⣀⣴⣿⣿⠆
⠀⠀⠀⠈⢿⣿⣶⣾⣿⣿⣿⣿⣫⣷⣏⣽⣶⠎⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠉⠹⢻⣶⣿⣿⣷⠾⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣭⣛⠛⣻⣟⣿⣿⣯⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⢽⣟⡛⣛⣛⡻⣻⣛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣹⠳⡀
⠀⠀⠀⠀⠀⢀⡞⠁⡟⠀⢠⠃⠀⡇
⠀⠀⠀⠀⠀⡼⠀⢸⠃⠀⡎⠀⠀⠃
⠀⠀⠀⠀⢰⠇⠀⡟⠀⢰⠃⢀⢰
⠀⠀⠀⠀⣸⠀⢰⡇⠀⡾⠀⣾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠤⣀
⠀⠘⠀⠠⡿⠀⣸⠀⢀⡇⢠⡿⠧⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡸⠀⢸
⠀⠀⠀⠀⡇⠀⡿⠀⢸⠁⢸⣿⠃⠐⠂⠀⠨⢍⠋⣰⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⡇⠀⣸⠑⣆
⠀⠀⠀⠀⡇⠀⡟⠀⢸⠀⢸⢋⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠐⠚⠀⠬⢐⡂⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠜⡤⠀⡼⠀⢠⡇⠀⡏
⠀⠀⠀⠀⡇⠀⡇⠀⣾⠀⠀⠆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠤⣤⡉⣿⠆⠀⠀⠀⠁⠒⠇⡀⡐⠀⠄⠀⠀⡜⣼⠇⢰⠃⠀⡾⠀⢰⠃
⠀⠀⠀⠀⢃⠀⡇⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢄⡉⠈⠛⠑⠀⠈⠑⡄⠀⠀⢀⡴⠀⠀⠈⠑⠂⠨⣙⡟⢀⠏⠀⣸⠃⢀⡟
⠀⠀⠀⠀⠈⠉⢧⠀⢧⢠⠃⠀⠀⠀⠀⠀⠠⡀⠀⠀⠙⢄⠀⠀⠀⠀⠠⣼⡤⠔⠋⠀⠀⠀⠀⠀⢀⢱⡿⢀⡞⠀⢰⠏⠀⡼⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⢶⠢⢲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠞⠁⡜⠀⢠⠏⠀⡼⠁⠂⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠉⠉⣽⣦⣄⣀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡼⠁⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠻⠿⣿⣿⠿⡇⠑⣄⠀⠀⡀⠈⠀⡎⠀⡜⠀⣠⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⣿⣦⣤⣀⣀⣴⡇⠀⠈⡠⠊⠀⠀⠀⣇⠎⢀⡴⢇⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡈⠙⠻⠿⢻⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠷⠶⠞⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠸⠐⢠⢆⡠
⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⡠⠤⠦⢄⡀⠉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⡜⠀⠀⠀⠀⢱⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠈⢁⠀⠱⡀⠀⠀⢀⡜⠀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⢡⠔⢂⡈⠉⠉⠉⠀⢄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⠃⠸⠄⠈⠀⠀⠀⠀⢠⠞⠉⠉⢢⠀⠀⠀⠠⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢦⠀⠀⠀⢃⡜⠛⠲⣄⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣲⡿⣗⡇⠀⢀⠞⠍⠳⢤⠈⢹⡆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠣⡀⠉⠉⣏⢰⠄⠩⣈⡇⣸⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠤⠬⣉⠁⠈⠓⠒⠒⢟⠲⠋⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⡁⠈⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠈⠉⠉⠉⠉⠉⢉⣉⡏⠉⠙⢄⡀⠐⢀⡜⢱⠄⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠲⠤⠝⠒⠒⠊⠉⠉⡁⠀⠈⢢⠀⠀⠉⢲⠎⠀⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⠤⢄⣀⣀⣀⣁⣀⣀⣀⣀⠤⠧⠤⠔⠋⠀⠀⠀⠱⢌⣻⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⣠⠴⠒⠲⣄⠙⡄⠀⠀⣠⠖⠛⠋⢉⣉⣍⣈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⣠⠶⠶⢶⣇⡜⠀⢰⠇⠀⢀⣼⠫⠤⣤⡈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠑⠒⠠⠔⠛⠁⠀⣾⠀⠰⣏⠙⠦⠄⠚⣁⣤⠴⠚⠊⠐⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢒⣦⡤⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢹⡃⠀⠀⣠⠞⠋⠉⠉⠛⠷⣦⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠈⠙⠓⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠹⡀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣾⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣠⠶⠶⠤⢧⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⣠⣸⠇⣿⣠⠶⠶⣾⡉⣧⠀⠀⣀⣤⣿⣄⣀⠀⠀⠀⢸⠇⢸⡇⢠⠔⣒⡶⠶⢶⣖⠒⠲⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢰⠉⠟⠀⣿⠈⠓⠒⠋⠃⡿⠀⢰⡏⠀⠹⣷⣌⣷⣀⣰⠟⠀⢸⡇⢸⡼⡁⠀⢀⡤⠈⠑⢦⠀⠀⠀⢉⣷⡤⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⠀⡰⠇⠀⣾⠀⠀⠀⠀⢹⡟⠋⠁⠀⠀⡼⠀⠈⢷⣿⡾⠟⠀⠀⠀⠈⡇⠀⡴⠋⣽⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢷⣿⠀⠀⣷⠀⠀⠀⢷⣄⣴⠏⠀⠀⠀⠀⢸⡇⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠘⢷⠾⠃⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⡄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⠃⠀⠀⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢀⡀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢾⠋⠓⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠘⣄⡀⢀⣀⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣍⣉⠭⠟⠉⠉⠉⠓⢢⣄⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡿⠀⠀⢀⡴⠏⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⣤⣤⣀⡀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢸⣯⠴⠚⠁⠀⢠⡾⠋⠉⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠈⠉⠓⣾⣏⠀⠀⠀⠀⠀⠀⠀⠘⢤⡤⠶⠚⠉⣿⠀⠀⠀⢠⠟⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣨⠷⣶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡞⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡞⠁⣀⣈⡇⠈⡇⠀⠀⢀⡴⠶⣤⡀⠀⢸⡆⠀⠀⡇⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢀⡾⠃⢸⠀⢸⡏⠙⢶⡄⢩⠀⢀⠟⣠⣤⣌⣷⠀⢸⡇⠀⠀⡇⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢠⡾⠁⠀⢸⠀⢸⠛⠛⠛⠁⡆⠀⢸⠛⠧⠍⠿⠛⠻⢾⠷⠶⠾⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠟⠀⠀⠀⢸⠀⢸⡆⠀⠀⢠⠃⠀⣿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣧⠀⠀⠸⢦⣤⡇⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠈⣧⠀⠸⣆⣀⣀⣀⣀⣤⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠈⢻⡄⠀⠀⠸⡆⠀⠈⠉⠉⡩⠟⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠹⣆⡠⠖⠉⠀⠀⠀⣠⠟⠁⢀⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠀⠀⠀⣀⣀⣽⢦⡖⠋⠉⠳⢦⣤⣤⣤⠞⠁⣠⠔⠋⠉⠛⢶⣄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠁⠀⠀⠙⠲⣄⡀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠛⠋

⠀⠀⠀⠀⣠⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⣴⣴⠀⠀⠀⠀⢠⣤⣄
⠀⠀⣠⣿⣿⣄⣄⣼⣿⣧⡄⠀⠀⠀⣰⠛⢻⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠿⣿⠁⠀⢸⡿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣶⣄⣀⣀⣰⣶⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⡇⠀⠀⠀⠀⠀⠀⢠⣥⡄⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⣤⡄⠿⠏⢉⣯⡹⠿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢀⣹⣏⣀⣀⣀⣀⣀⣀
⠀⠀⣿⡟⠛⠛⠻⣿⠅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡏⠉⠉⠉⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠘⢷⠂⠀⠀⢠⡾⠏⠀⠀⠀⠀⣀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣀⠄⢀⠤⣀⢰⣄
⠀⠀⣤⣾⣶⣶⣶⠾⢧⡀⠀⠀⠀⠈⣿⠀⠉⠉⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⢿⣯⣼⣿⣿⣄⣠⠇⠀⠀⠀⠸⠿⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⣤⣤⣤⣤⣿⣇⠀⠀⠀⠀⣿⣿⣤⣄⣾⣿⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣄⠀⠀⠀⠀⠙⢻⠉⢹⣿⡍⠉⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣄⣅⣸⣿⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⠀⡇
⠀⠀⢶⣶⣶⡶⡶⣶⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣍⠀⠀⠀⠀⠀⠼⠄⠨⠿⠄⠤⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠎⠀⠀⠉⠁⠈⠓⠒⠓⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⢲⡀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣵⡾⣿⣶⣄⢷⠀⣰⠇⣠⠞⠙⢦⡲⣄⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠫⢎⡥⠖⢺⡽⡏⠁⢧⠀⠀⠀⠀⠀⢩⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠈⢠⣞⣉⡤⣇⠀⠀⣶⠀⢤⣀⡀⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢀⡟⠉⠈⢉⡁⠀⠀⠀⠀⣀⠈⠀⠀⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⡴⢿⠉⠀⠀⠀⠘⠋⠀⢀⡀⠈⠙⠀⠀⠀⣠⡴⠃
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡞⠁⠈⠙⢲⡀⣦⠀⢀⡤⣤⡤⢤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⠁⠀⠀⠀⠈⢳⡀⠀⠈⠓⠚⠓⠚⠁⠸⢆⡞⠀⠀⠀⡐⡆
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⢲⡤⣤⡄⠀⠀⠤⢶⠋⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠙⡾⠲⢤⣀⡤⢼⡁⠀⠀⠀⠀⠀⢈⡧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⡼⠁⠀⠀⢻⣀⣠⡽⣆⠀⠀⢀⡴⣫⣀⣀⡟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡞⠁⠀⠀⠀⠈⠁⠀⠓⠮⣷⣤⡮⠴⠏⠀⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠁⠀⣇⣀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⠟⠻⠧⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠛⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⢉⢀⠀⠐⠒⣘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠞⡵⣇⣴⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠆⠀⠀⢀⣀⠀⢲⢋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣽⣇⡆⠀⠀⠀⢣⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣆⠀⢠⠀⠉⠉⡤⢾⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣭⣿⣿⡃⠀⠀⠀⠀⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢫⣿⡅⠈⠀⠀⠀⡁⠦⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠃⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⠤⠠⠄⠀⠀⠀⡀⠀⠠⣼⡏⠸⠙⢫⠀⠀⠀⠀⠔⠁⢀⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⠀⠀⠀⠀⡤⠤⢒⡉⠀⠈⠉⠛⠘⣀⠠⢤⡀⠁⠂⠄⠈⠙⣇⢀⠈⠀⠀⠀⠰⢸⡄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠀⠀⠀⠀⠊⢀⠀⢊⠐⠶⢀⣾⠧⠚⠉⠀⠂⠙⢀⡁⠠⠀⠐⠀⠰⠀⠀⢀⠀⠂⠀⠈⡓⠴⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠢⢀⡸⠣⠀⣩⡗⣶⢴⠀⢀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⣠⣴⣿⣶⣤⠈⣀⣫⣿⣶⠆⢈⣼⣿⣮⣽⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠀⠀⢀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣿⣿⠿⡛⣋⣽⣿⣿⣐⡀⠻⣿⣿⣿⠛⣿⠿⣿⣷⣄⣆⠀⠀⠀⠀⠈⢳⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⡿⢋⣠⣿⡿⣿⣿⣿⣿⣿⣿⣶⣌⣹⣿⣿⣿⣷⡿⣸⣿⡏⠀⠀⠀⠀⠀⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣟⣼⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣾⠟⡋⠀⠀⠀⢀⠀⠰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣯⣴⠃⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⢰⠃⢈⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⢃⣿⣿⣿⠵⡀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⡿⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣻⡿⢷⣾⠃⠊⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡇⢶⠀⠘⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠟⣲⣾⣿⡿⢀⣿⠇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠁⠀⠀⠓⠀⣤⠀⠀⢽⣛⢿⣶⣿⣿⣟⠛⣋⠀⠈⠀⠈⠁⠉⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠢⠀⠀⠈⢀⣴⠋⠀⠀⠀⠙⢆⣙⣿⣿⣽⠶⠇⠀⠀⠀⠀⠀⠀⠀⢴⡦⠀⠰⡆⠢⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣤⡵⠇⠀⠀⠀⠀⠘⣿⣿⠟⢁⣷⠜⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⢬⣿⡄⠐⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⠆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⡃⠉⠀⠁⠀⠀⠀⠀⣠⠄⠆⠀⠠⠠⠌⢸⣿⠟⠈⠀⢨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⢨⣦⡄⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⣿⡧⠄⠀⣠⢦⣤⢶⠘⠈⢲⡄⠀⠀⠒⠃⠈⠛⠀⢀⢀⣬⡿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢱⡀⠀⠀⠀⠀⠀⠀⢀⣔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⣀⣀⠀⢀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠔⡿⠋⠐⠁⠖⠂⠒⠀⠀⠀⠴⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰⠋⠀⡠⠖⡛⣤⠊⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠀⠀⠀⣀⡽⠻⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢠⠀⠀⠀⠀⠀⢶⡤⠤⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⠤⠶⣄⠀⠀⠀⠀⠀⠀⢀⣀⡸⠋⠀⣤⣖⣺⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⠛⠋⠉⠁⠀⣠⣀⠈⠻⠶⢤⣤⡔⠛⣯⠙⣧⣤⡟⢯⡩⣄⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣻⠖⠂⠳⣤⣄⡴⠋⠙⠙⢦⡀⠀⠀⢹⣧⠀⣹⢾⣟⡟⢳⣼⢧⠀⠉⢳⡀⠀⠀⡷⡀⠀⠀⠀⢀⡴⣤⠴⣄
⠀⠀⠀⠀⠀⠀⠘⢷⣟⣤⡞⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⣾⢻⠀⠛⢠⡷⡇⠈⣿⡼⣶⣄⠀⠳⡀⠀⠑⠃⢀⣠⣶⠏⢰⣱⣋⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣼⠃⢠⢀⣦⡆⢸⠇⢰⡏⣧⢸⡿⣦⡚⠛⠓⠒⠒⠋⢹⣙⣷⢀⣝⣓⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣤⠾⢯⠻⡅⠀⣾⠟⣿⣧⠎⠀⣼⠃⣿⠛⡇⠘⣧⡀⠀⢀⣀⣤⠼⠋
⠀⠀⢀⡀⠀⢀⣀⣠⡤⠞⠉⣿⠀⠀⠻⣏⠀⠀⠀⠑⠈⣰⢿⣀⡼⠋⢀⣼⣡⡾⠃⠀⡇⠀⠈⠉⠋⠉
⠀⠀⠸⣍⠛⢻⡍⢻⣿⣀⡰⠛⣧⠀⠀⠙⣦⡀⠀⣠⡾⣧⢶⣶⣶⣛⣿⣿⡿⢶⣤⣶⣛⣀
⠀⠀⠀⠈⠙⠻⠟⠋⢉⡟⠀⠀⠘⣆⠀⠀⠈⣛⢾⡯⠚⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⡿⠘⠞⠉⠁⠀⠀⢀⠴⠿⠿⠚⠛⠛⠛⠛⠛⢲⣾⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣀⣀⣸⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠋⠉⢨⢟⢟⠟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣀⣠⡭⠭⠴⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠁⠐⠂⠤⠀⠤⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠐⠒⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡈⠠⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠄⠀⠐⠃⠀⠀⠀⠀⠀⠀⣠⠞⢁⠴⠁⠀⠀⠈⠂⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⡀⡄⠀⠠⠁⠀⠀⠀⠀⠀⢀⠜⡁⠀⠁⡄⠀⠁⠀⢁⠀⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠠⢅⠀⠠⠠⠀⠥⠀⠀⠀⠀⠀⡴⠁⢀⠀⠀⠀⠠⡀⠀⢀⠈⢀⠢⠤⡀
⠀⠀⠀⠀⠀⠀⡀⠆⢈⣠⠀⠀⡹⠀⠀⠀⠀⢠⠎⢀⠔⠁⠀⠀⠀⠀⠰⠄⠀⠄⠈⣀⠲⠦
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⡰⠀⠀⠀⢀⡔⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠲⠆
⠀⠀⠀⠐⠚⠀⠀⠒⠠⢐⡀⠀⠀⡠⠔⠉⠀⡼⠁⠀⣄⠁⠢⡀⠀⠀⠀⠚⠂⠰⣢⡶⠾⠶⠾⣟⢦⡀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠵⠒⠉⠀⠀⢀⠜⠀⠀⠀⠈⠉⠐⠈⢢⡀⠀⠀⠓⢨⡥⠴⠶⠶⠶⠶⠿⠷⢦⠄
⠀⠈⠒⠄⠠⠄⠒⠊⠉⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠒⠠⠴⠁⠀⠀⠠⠤⠤⠀⠒⠈⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠂⠒⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠂⠁

⠀⠀⠀⠀⠀⣀⣠⢀⣆⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⣰⣀⣀⣀⣀⣀⣶⡀
⠀⠀⠀⠀⠀⠙⠶⣾⣷⡶⠞⠁⠀⠀⠀⠀⠀⢰⣿⠻⠦⠄⣴⠂⠀⠀⠀⠀⠀⠠⠞⣿⢦⣠⢀⡿⠀⠀⠀⠀⠘⠐⣆⡶⣠⠆⠛
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⠀⢶⣭⣷⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣸⠟⢶⡀⣸⠇⠀⠀⠀⠀⠀⠀⢘⣿⡏
⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠶⠿⠻⣗⠀⠀⠀⠀⠀⠀⢠⣭⣥⣴⣾⣶⡿⠀⠀⠀⠀⠀⠀⠀⣼⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢉⡳⢄⠀⠀⠀⠀⠀⠀⠀⠐⢲⡂⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣃⣀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡄⠀⠀⠀⠀⠀⠀⢈⣭⣀⣠⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣥⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠟⠂⠀⠀⠀⠀⠀⠀⠻⣿⣥⣄⠈⣻⡇⠀⠀⠀⠀⠀⣠⠿⣿⣿⢷⣄
⠀⠀⠀⠀⠀⣤⣬⣿⣏⣿⣤⣄⠀⠀⠀⠀⠀⠀⣾⠁⢻⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡟⣷⣿⡇⠀⠀⠀⠀⠀⠘⣿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢠⡴⣿⣿⠦⡄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣷⣆⠀⠀⠀⠀⠀⠀⠠⡾⣿⢧⡀⢀⡾⠃⠀⠀⠀⠀⢶⠎⣷⣦⣤⣄
⠀⠀⠀⠀⠀⢰⠀⣿⣿⠀⡰⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣮⡅⠀⠀⠀⠀⠀⠀⢹⡟⢳⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⡙⣷
⠀⠀⠀⠀⠀⠈⠢⡌⣿⠴⠁⠀⠀⠀⠀⠀⠠⠴⠚⠉⠀⠸⠁⠀⠀⠀⠀⠀⠀⢸⠃⠼⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠹⣄⡀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⡇⠀⠀⠀⠀⠀⠀⠰⡶⣻⠄⢀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣷⡘⠟
⠀⠀⠀⠀⠀⠀⠸⢿⣟⢻⠆⠀⠀⠀⠀⠀⠀⠀⠸⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠐⠿⣷⣾⡿⢻⡄⠀⠀⠀⠀⠠⣔⣛⡯⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠊⠉⠉⠉⠉⠉⠓⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⠠⠄⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⣾⠀⠸⠀⢀⢆⣠⠀⢀⣤⣠⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠰⡀⠀⠀⡷⢀⠛⣄⡌⣶⠁⠾⠀⣄⢆⢠⡆⠈⠟⣧⠎⠘⡃⠱⢟
⠀⠀⠀⠀⠀⢠⠐⢀⡠⠶⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠘⠀⠘⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢰
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⡘⠸⠀⠀⣠⠤⠤⣤⣔⠁⠀⠛⠀⠀⠙⣆⣀⣀⡐⣼⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠛⠳⢤⡀⠀⢠⢁⠐⣠⡮⠤⠖⠊⠀⠈⢳⠀⠓⠁⢠⠞⠁⠙⠦⣽⣆⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⠒⢢⠹⣤⡸⢀⠀⣏⠀⠀⠀⠀⠀⣾⠆⡇⠀⠀⡇⠀⠀⠀⢸⡇⢸⣸⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡧⠒⡆⠀⠉⠛⡘⠀⠹⡀⠀⠀⠀⠀⠙⡱⠀⠀⠀⢣⣀⠀⠀⠈⢁⠜⠳⣟⡮⠀⠀⠤⠤⢄⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢀⣺⡄⠀⠀⢀⠷⠀⠀⠙⠲⠤⠤⠴⠊⠀⠀⢠⠤⢶⠈⠛⠒⠚⠁⢠⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠄⠀⠀⡎⠐⠀⠀⠀⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡀⢄⠖⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠁⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣎⠔⠂⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠺⠀⢀⡠⠒⠉⠲⡀⠀⠀⠀⠀⠶⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠘⠖⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠋⠀⠀⠀⠀⠀⠘⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⠀⠀⠀⠀⠸⡀⠉⠉⠒⠒⠶⠦⣤⣤⣀⣤⣄⠠⠤⡔⠲⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠆⠀⢀⠀⠀⠀⠙⢄⠀⢀⠔⠚⠳⣧⡰⠗⢦⡀⠀⣠⠃⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠋⠀⢀⠃⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠸⠁⠀⠈⡧⠚⡅⠀⠀⠀⠈⣇⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⠦⢄⣀⡘⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢱⠀⠀⠀⠀⣸⡀⠀⠀⠀⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢈⠃⠀⠀⢠⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠀⠀⡗⠒⠈⠉⠁⢇⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢱⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠭⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠈⠈⢳⡀
⠀⣀⡤⠤⣤⣀⠀⠀⠀⠉⠓⠦⣄⠈⣇
⡞⠁⠒⠒⠂⠉⠓⢦⡀⠀⠀⠀⠈⢧⣿
⢧⡀⠀⠀⠀⣀⡀⠀⠙⣆⠀⠀⠀⠸⡏⠀⠀⠀⢀⣠⣀⣠⣤⣤⣀⡀
⠀⠉⠙⠋⠉⠉⠉⠙⠲⣌⣧⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⢠⡏⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠙⣆
⠀⠀⣠⢚⣛⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⢸⡇⣠⠴⠛⠉⠀⠈⠳⣤⣀⠀⠀⣹
⠀⠘⣇⠁⠀⢀⣤⠴⠦⢭⣷⡄⠀⠀⠀⣼⠛⠃⠀⠈⠓⠚⡁⠀⠀⢉⠛⢶⠏
⠀⠀⠈⠓⠛⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⣦⠀⠀⠀⠉⠉⠀⠀⠘⠬⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠶⠦⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⠻⢦⣀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠉⠁⠀⠉⠙⠛⠛⠛⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⡛⠶⠦⢤⣤⣀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠉⣻⠋⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⣧⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⡀⣠⡏⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢸⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡿⣋⡴⠳⠦⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⣀⠀⠀⠀⠈⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⢀⣤⣄⣉⣙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢲⡏⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠓⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢰⣴⣀⣀⣀⣐⣀⢐⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢺⣟⣭⣿⣿⣿⣿⣿⣽⣿⣿⣗⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠀⢱⣼⣤⣶⣾⣿⣿⡿⢿⣿⣟⣛⣻⣿⣟⣛⣿⣿⢿⣿⣿⣿⣿⣶⡠
⠀⠀⠀⠀⠀⠀⠀⢀⠰⡜⢄⡳⣦⣷⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠈⠉⠻⡿⠿⣷⣿⣿⣿⣿⣿⣦⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣲⣤⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠠⡀⢠⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠊⢀⣴⣿⣷⣾⣿⣿⣿⣿⣧⠙⢦⡀
⠀⠀⠀⠀⠀⠛⢚⣶⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⡀⠓⠤⠤⠤⠤⠔⠒⠋⠁⣀⣠⠔⢹⣿⣿⣿⣿⠋⠁⠀⠀⢙⡷⠀⠙⣄
⠀⠀⠀⠀⠀⠉⠈⠟⠋⠀⢀⠀⠈⣿⣿⣿⣷⣿⣿⣦⡤⠤⠤⠤⠔⠒⠚⠉⠁⠀⢀⣼⣿⣿⣿⡇⠀⢀⣠⣴⠾⢃⠄⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⠿⠟⡛⠫⠁⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠟⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠉⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠢⠀⠐⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡤⢄⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠛⡈⠀⠀⠀⠃⡄⢈⠙⠋⡄⠀⠀⠀⠀⠡⣀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠁⠀⠀⠂⣈⠉⠦⠠⢤⡀⠀⠀⠀⠀⠀⠀⡉⠉⠉⡁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⡱⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠈⠀⠀⠠⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⡆⠈⠂
⠀⡀⠀⢀⠐⢄⡀⠲⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢣⢆
⠀⠀⠈⡹⠁⠀⠀⠐⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀⠁⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠙⠋⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠅⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡤⠆
⠀⠀⠠⠀⠀⠀⠀⠀⡠⢆⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⠶⡵⠠⠀⠈⠂⠀⠀⡀⠀⢀⠀⢀
⠀⠀⠀⢠⠀⡀⠂⠀⠁⡨⢄⠀⠀⠀⠰⠐⠂⠀⠀⠀⠀⠀⠀⠚⠈⠀⠀⠒⠎⠂⠀⠂⠀⠀⠀⠈⠁⠘⠁
⠀⠀⠀⠀⠐⠀⢠⡜⠂⠡⣌⡀⠀⡀⠢⡀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒
⠀⠀⠀⠠⠀⠀⠛⠁⠰⠄⠃⠀⠀⡽⠀⠀⠀⠀⠘⢀⠄⠀⠀⡉⠁⠀⡀⠀⠀⠂⠁
⠀⠀⠀⠐⠂⠀⡀⠀⠀⠁⠀⡰⠊⢀⠀⠁⠀⠀⠘⠡⠤⠀⠀⠠⠀⠀⠘⠂⡀⠀⠀⠛⠡⠀⠡⣀⠒⠄
⠀⠀⠀⠀⠀⢉⠇⠲⣆⡶⠚⠁⠖⠁⠀⠀⠀⠀⠀⠛⠂⠀⠀⠰⠡⡄⠀⡀⠐⡀⠀⠀⠀⢈⠀⠀⠡⠄⠨
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⠀⠀⣠⡦⠀⠀⠒⠂⠀⠀⠐⣀⡬⠀⠆⠀⠀⠀⠁
⠀⠀⠀⠀⠂⠤⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠅⠤⠷⡶⡂⡂⡐⡋⠉
⠀⠀⠀⠀⠀⠂⠂⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⡀⠀⠄⣀⡀⠀⠀⠀⠐⠠⣰⣤⠄⠑
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⠀⡤⣠⣠⣿⣿⣷⣄⠀⠀⠀⠙⠋⣴⣃⠀⣤⡤⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠔⣋⣼⢀⣀⣶⣷⣶⡟⠿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⡇⢠⡴⠇⠀⠀⠉⠳⡀
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣾⣿⣿⣿⣿⣧⣄⠈⠿⣻⣿⣷⠀⠀⢠⣿⣿⣴⠟⢀⣀⣀⣤⣄⣀⡀
⠀⠀⡔⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠽⠿⢿⡄⠰⠿⣿⠿⠶⠿⣯⣥⣤⡄⠀⣈⣉⣻⡄
⠀⠀⠁⢰⣿⣿⣿⡃⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠀⠉⣍⡙⠄
⠀⠀⠀⢸⣿⣿⣏⠀⢴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡇
⠀⠀⠀⢺⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃
⠀⠀⠀⠸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⢴⠞⢻⣿⢂⣴⣶⣧⣤⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⣤⣴⣶⣶⣤⣄⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⣸⣿⣿⡿⡿⣿⣿⣿⣿⣿⣦⡐⡄⠀⠀⠀⠀⣤⣽⣿⣿⡛⠛⠳⠤⢉⣿⠳⠀⠀⢸⣿⠋⢹
⠀⠀⢸⡇⠀⣿⣿⠇⠓⣻⣿⣟⡛⠿⢿⣿⣿⠗⠒⠒⠈⠁⣿⣿⣭⣭⣿⣿⠦⡀⠉⢂⣠⡴⢻⣿⡄⣾
⠀⠀⢸⡧⣀⢸⡄⠀⢀⡙⠿⠟⠁⠀⣼⣿⡟⠀⠀⠀⠀⠀⠛⠋⠙⠿⠿⠉⠁⣿⣾⣬⠄⢀⣸⡿⠁⠏
⠀⠀⠈⢧⣸⣿⡇⠀⠈⠙⠓⠂⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⠉⠀⠀⣼⣿⠴⢃⠄
⠀⠀⠀⠀⢩⣿⡟⣿⡤⣤⠀⠀⠀⠀⠀⣿⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⢾⣿⡷⠖⠋
⠀⠀⠀⠀⠀⠹⣇⢸⡇⢱⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠸⢻⡇
⠀⠀⠀⠀⠀⠀⣿⢻⣧⢸⠀⠀⠀⢰⣾⣇⣀⡀⠲⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢹⠈⠿⢺⡄⢀⠀⢸⡇⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⡄⠈⢸⡇⣼⠀⠸⣷⣿⣧⣤⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠
⠀⠀⠀⠀⠀⠀⠀⡿⣦⠘⠇⣿⣧⣴⡿⠿⠿⠿⠿⠿⠿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⣿⣼⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣷⠀⣿⣻⣇⣷⣦⣴⣶⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⢹⢻⣿⢾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣌⡟⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⣷⣶⣶⣷⣶⣿⣷⣿⡿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠋⠙⠛⠛⠿⠿⠶

⣿⣿⡿⠻⠿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠒⢦⠀⣠⢫⠖⢆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⣳⣧⣤⣜⣠⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠷⠶⢶⣦⢾⠓⠛⠛⠛⠋⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠿⠿⢖⡚⡏⠀⠀⢀⣄⠀⠀⢀⡜⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠷⢶⣻⠁⠀⡴⠁⡨⢃⡴⢋⠜⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⡛⠛⠿⣞⡿⠀⡜⢀⠎⡴⢉⠔⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣛⣛⡛⢷⣻⠃⡸⢁⠎⡜⡰⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠒⢫⠟⠁⢰⠁⡜⡸⣰⡟⠀⣀⠄⢀⡤⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⣠⠃⠀⠀⠀⠙⡰⣱⢳⠑⣊⠵⢊⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠕⠀⠀⢀⡔⠁⠀⠀⠀⢀⠔⡱⢃⠇⣫⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⢀⠎⠀⠀⣀⡤⢒⡥⠊⡠⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡭⣑⣒⣚⣒⡈⠭⠔⢊⡡⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠛⢦⡀⠀⠀⠀⠹⡄⢲⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣙⢶⣄⠀⠀⠹⡌⢧⠀⠘⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣭⡳⡄⠀⢁⣤⣴⣶⣾⡿⠻⣦⣦⣀
⠀⠀⠒⠶⠤⣤⣄⣀⡀⠀⠀⠶⠤⣤⣀⣀⠀⠀⠁⠀⣴⣿⣿⣿⢏⡿⠿⠶⠿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠶⠤⣬⣍⣙⡛⠂⢸⣿⡿⠛⠛⢿⣇⡀⠀⠀⠀⠙⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⣿⡧⡔⠉⠩⠂⠒⠒⠒⠂⠀⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⣷⣶⢶⡜⠀⢺⣟⣛⠂⠈⡿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢫⠾⢤⡀⠀⢠⢟⣻⠿⢸⠘⠢⠉⠙⠁⠀⣵⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣠⣞⡒⢌⣂⠸⣄⠀⠀⣼⡀⢀⣄⠀⠀⡴⡆⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⣉⠦⢤⢹⡀⠀⠀⢹⡄⠚⠁⣁⣀⣈⠁⣇⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢸⡄⠈⠓⠅⡖⠀⠀⣿⡀⠉⣓⣂⣉⠀⡟⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣀⡴⡏⢷⠀⠀⣼⢻⡆⠀⢻⣅⠀⠈⠋⠀⢀⡾⠁⡏⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⣷⠖⡇⠘⠆⠘⠁⠘⣷⠀⠸⡿⡷⣶⠶⠶⠋⢀⡼⣥⡈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢷⡶⠦⢤⡀⠀⠈⢧⠀⢳⣷⠈⢷⡄⣰⠋⢀⣾⠟⢮⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠛⠲⠶⠶⠶⠿⢷⣄⣿⠁⢠⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⣾⣿⣫⡾⠃⠀⠈⠙⣦⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⠙⠋⣠⣦⣤⡀⠀⠀⠀⠀⢀⠀⣀⡴⠚⠙⠐⡄⠈⠁⣀⠀⠀⠀⠀⠈⠙⢶⠜⠉⣠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⢠⡾⠛⠁⠀⠻⢶⡄⠠⣀⠋⡽⢁⡀⠠⠃⠀⣀⠀⢀⣿⣧⠀⠀⠀⣀⣶⡤⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠈⠙⢦⡈⠁⣉⡁⠈⢀⠀⢸⣿⡆⢸⣿⣿⣀⡤⢊⠝⠉⠐⢀⣴⠰⣿⣏
⠀⠀⠀⠀⢀⡀⠀⠀⠀⡴⢁⡴⠀⠀⠀⠀⠀⠀⠀⣤⣿⡷⠉⠀⠔⠉⠀⢰⣿⣷⣿⣿⣿⣇⠀⠀⠀⠀⠔⢉⣴⠾⠟⠉
⠀⠀⠀⠀⠺⢷⣦⣤⡈⣠⢋⡀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠄⠀⣠⣿⣿⣿⣿⣿⣿⣿⠃⣀⠀⠔⠚⠛⣡⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣾⠿⠶⢤⣀⠀⠀⣠⣾⡿⠃⠀⣠⡀⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇⣾⣷⣤⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠑⢦⣌⠁⠀⠀⠈⠹⣶⣷⣿⠋⡀⢀⣸⣿⣷⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⠋⢉⠀⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠙⠓⣤⡄⠀⠀⠈⠉⠉⠙⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢶⡿⠧⣄⠀⠀⠈⠓⠤⣄⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠠⢶⡿⠛⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠓⢆⣠⡀⣈⠻⣤⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠻⣿⡶⣷⣿⣿⣿⣿⣿⠿⠛⠛⠉⠋⠉⠁⠀⢀⣠⣆⣠⡴⠀⠀⢀⠴⠋⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢡⣾⣿⡿⠟⠛⠛⠁⠀⠀⠀⠀⠂⠀⠴⠛⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣾⣶⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣈⠻⣦⣄⣀⣀⣀⣠⠴⣫⣾⣿⡿⣫⣿⣿⣇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡿⢛⣿⢋⣽⣿⣿⣿⣿⡉⠛⠷⣮⣷⣯⣭⣭⣴⣾⣿⡿⢋⡼⢵⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⢋⣴⣿⣷⣯⡟⣿⡿⣿⣿⡇⣷⣦⣄⣉⠛⠛⠛⠛⢛⣉⡴⠋⠐⠙⢛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⠟⢻⡿⢀⡴⣿⣇⠿⣿⡷⠁⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠈⢽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢮⠇⣠⡾⠋⠀⢿⣏⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠙⠛⢁⡴⠊⠙⢳⣦⡀⠀⠙⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⢸⣿⣿⣿
⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠝⣿⣆⣀⣀⣀⣀⣹⣿⣦⠀⠀⠀⣰⢏⣽⣿⣿⡿⢻⣿⣟⣿⣿⣿
⢀⢀⣴⣾⣿⣼⣀⣀⣀⣤⣤⣤⠤⠤⠶⠒⠒⣛⣭⣭⣭⣁⣤⣤⣞⠳⣬⣙⣷⡤⢞⣵⣿⣿⣿⣿⠇⡿⣿⣿⣿⠋
⠘⣾⣿⣟⣟⣭⠭⠽⠀⠀⠀⠀⠀⠀⠀⣠⣺⠟⠛⢉⣿⣛⡿⣿⣿⠷⣌⡻⣷⣾⣿⣿⣿⣿⣿⡿⣸⢃⣼⣿⠉
⢰⣿⣿⣷⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⢰⣷⣏⣀⣴⣯⣿⠟⣷⣏⢿⡷⡄⠉⠻⣿⣿⣿⣿⡿⠟⣱⠇⢸⣿⣿⡇
⠈⠛⠿⢿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⢸⣿⣟⣛⣯⢻⠻⠋⠀⠀⢻⣿⢸⠿⠶⠮⣭⡤⠴⠞⠁⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠛⠛⠻⣽⣾⣯⢽⣿⣿⡟⣀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣾⣿⣾⣿⠇⠉⠛⠶⣤⣀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣉⣉⡉⢻⡅⣴⡇⠀⠀⠀⠀⠈⠙⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠻⠿⣿⣿⠛⢱⡿⣄⡀⠀⠀⠀⠀⠀⣿⣿⢿⣦⡀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠓⠶⣤⣈⣉⣀⡾⠁⠈⠻⣦⡀⠀⠀⠀⣸⣿⣧⢽⣿⣦⣮⣴⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⣠⡾⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢸⣿⡦⠶⠛⠉⢻⣿⢦⢈⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⣠⣾⢫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⢹⣯⡻⣿⢟⠝⠁
⠀⢀⣾⣿⣿⣌⣂⠀⠀⠀⠀⢀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣟⢿⣷⣄
⠀⣸⣿⡿⣿⣿⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠟⠋⠀⠀⠁⠙⣯⣧
⠰⣿⣿⣿⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠈⠛⠿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣷⣦⣀⡀
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣞⣛⣿⣿⣿⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣹⠉⡁⠀⣿⣿⣤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣿⣿⠏⣿⣿⣿⢽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢹⢿⣿⣿⣓⣿⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢸⠟⣿⡏⠁⠉⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠹⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣀⠄⡇⡀⠀⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣶⣾⣷⣶⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠒
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⣸⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠛⠿⠿⠟⡩⣿⣿⣿⣿⣷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⢀⣿⡟⣿⣿⣷⡀⣿⣿⣿⡛⠀⠻⣿⣿⠟⠁⠈⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠏⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⡀
⢸⣿⠁⣿⣿⣿⣷⣼⣿⣿⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣧⠄
⢸⡇⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢸⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿
⢸⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⣿⣿⣿⣿⡟⣿⣿⠟⠁
⠘⣷⣄⡘⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣰⣿⣿⡿⠟⢋⣼⠿⠃
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⣿⠃⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠁⠈⠁
⠀⠀⠀⠉⠛⠛⠛⠿⠿⠛⠉⠉⠉⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣤⣀
⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣄⣀⣀⡀
⠐⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣾⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⣷⡀⣸⣿⡿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣀⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠏⠀⠀⠀⠀⠈⣿⠇⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡄⢠⣿⣿⣿⣿⣿⡀⢸⣿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢋⣤⠀⡟⠀⠀⠀⠀⠀⠀⢻⠆⣤⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⠞⣿⠁⣿⣿⣿⣿⣿⣿⣧⢸⡟⢶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⡄⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣆⢹⣿⣿⣿⣿⣿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⠿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠛⠀⠀⠉⠉⠀⠀⠛⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠿⢛⠉⢡⣿⣿⣿⣶⣿⣿⣧⡍⠙⡛⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡞⠉⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠉⢳⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢋⣴⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣷⣎⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠁⠀⠀⢸⣿⣿⣦⣀⣀⣀⣀⣤⣿⣿⡇⠀⠀⠀⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢸⣿⣿⣿⠀⠀⠙⠿⠿⠿⠿⠟⠁⠀⢸⣿⣿⣿⡆⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣀⡠⠔⠒⠿⠟⠣⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠠⣄⠀⣀⢀⡒⠢⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠃⠀⢨⡍⠉⢉⠀⡼⠀⠀⠉⠉⡁⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢀⣀⠀⣣⠀⠒⢰⢡⠆⠀⠀⠀⠉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡇⠁⢠⡄⠸⠀⢄⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠐⠀⠀⠀⠀⠑⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⢀⣙⣓⠢⢟⠃⠀⠀⠀⠀⠀⠤⡤⢤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⡉⠑⠚⠙⠀⠀⠀⠄⠀⠠⢤⠴⡃⠀⢀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠐⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢀⠟⠒⠤⣀⡐⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢄⢀⠞⠁⠀⠀⢀⡀⢀⠀⣀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⢸⡼⠃⠀⠀⠠⠂⠈⠀⠈⠈⠁⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠘⠀⠀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣀⡐⠀⠘⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡀⠀⠀⣃⣤⠃⡀⠀⠈⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠚⠡⠮⡀⠀⠈⣉⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠈⠁⠀⠀⡏⢰⠰⡶⠒⠶⡤⢀⡤⢄⠀⢀
⠀⠀⠀⠀⡎⡇⠀⣰⡑⢩⠁⠀⠀⠀⢠⠁⡌⢰⠀⠀⠰⢁⢄⢀⣴⠀⢎
⠀⠀⣖⡂⠘⠀⠀⠡⠀⡘⢀⠔⣄⠀⢸⠀⠁⠃⠀⠀⠀⡈⠈⠀⠢⠁⠉
⠀⠀⠀⠠⣂⠄⠀⠀⠀⢶⠁⠀⡟⣤⡇⠀⠒⠀⢀⠀⢠⠃⡆
⠀⠀⠀⠀⠀⠀⠀⡄⢀⡞⠀⠰⠶⢶⡀⠀⢰⠀⡜⠀⡌⢰
⠀⠀⠀⠀⠀⡄⢰⠇⠘⠇⠀⢠⠂⠀⢸⠓⠋⠐⣇⣀⡏⠁
⠀⠀⠀⠀⠀⠀⢸⠺⡤⢤⡔⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠂⠇⠀⠙⠓⠢⠼⢃⣐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠲⣼⠃⠀⠿⠞⠛⠛⠉⠉⠙⠛⠛⠛⠳⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⢸⡀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄⠀⠀⢸⠧⣄⠀⠀⢀⡿⢄⡀⠀⢠⡷⣄⣀⣸⢧⣠⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⣟⠦⠟⢑⠒⡩⠤⠚⠛⠋⠉⠓⠎⠉⠉⠉⠉⠙⠛⢧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡵⠒⢦⡀⠀⠀⢸⠀⠀⠘⠋⠀⠀⠐⠗⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢈⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠙⠦⠭⠷⣄⣀⡼⠀⠀⢀⡠⠀⠀⠀⠀⠀⢀⠀⠣⣀⠀⠀⠀⠀⠠⢂⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠍⠊⠑⠂⠐⠂⠁⠀⠀⠀⠉⠁⠈⠠⠖⠫⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢯⡛⣦⠀⠀⠀⠀⠀⠀⠀⣴⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠛⠁⠀⠀⠀⠀⣀⡴⠚⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣾⠁⠀⠀⠀⢀⡀⣰⠿⣄⣀⠀⠠⠤⠤⣄⣀⣀⣤⣤⡤⠤⠶⠖⠋⠁⠀⣸⢦⣤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⠞⠋⠸⣦⣀⣀⣠⠞⡾⠁⠀⠀⠉⠙⠓⠒⠲⠶⢾⣄⣀⣀⣀⣤⠦⢤⣤⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⠃⠀⠀⠀⢀⣀⡤⠤⡄⢀⣼⡇⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠈⢸⡋⠀⠀⠀⢣⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⡾⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⣷⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠸⡄⠘⠧⢤⡄⠀⠀⣀⡀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣀⠠⠼⢷⣀⠐⠛⢲⡖⠒⠛⠶⣾⢻⡟⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⣠⠖⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠃⢸⣁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠓⠒⠲⠶⠶⠶⠶⠶⠶⠦⠤⣤⠾⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠟⠻⣿⠛⠻⣿⡟⠛⣿⠟⣿⠛⢻⡟⠛⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢻⡷⣾⣷⣷⣾⣿⢻⣞⣷⣶⣶⡟⢿⡞⢻⣦⣿⣟⣶⡿⣿⣟⢳⣟⢻⡟⢣⡼⢶⡾⠃⠛⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣧⣾⣤⣿⣤⣿⣴⣯⣼⣧⣼⣧⡿⢇⣿⣦⣧⡜⢿⣧⣼⣯⡜⢫⣼⣿⣼⣧⣾⠷⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠀⠄⠘⠀⠈⠀⡀⠀⢠⣤⢀⣤⡄⠈⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿
⣿⣿⣿⣿⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⣸⡛⣻⡏⣹⡟⣻⡛⣿⣌⣿⣙⣿⡽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠛⠛⠉⠛⠋⠛⠉⠙⠋⠈⠋⣉⣤⣤⣤⣤⡀
⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣤⣀⠀⠀⠀⠀⠀⣴⡿⠋⢩⣽⣟⠉⠙⠃⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⣦⡌⠙⣧⣀⣤⣤⣼⡟⠁⠀⠛⠉⠉⠛⢧⣀⡀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠉⠁⠀⠈⠉⠀⠛⠋⠁⠉⠉⠛⢦⣀⠀⠀⠀⠀⠀⠉⠉⠓⠊⠉⢁⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠘⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣤⣤⡀⢀⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⢨⡑⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡇⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠃⢀⣾⢿⣼⣄⠀⠀⠀⠀⣇⢀⣵⣤⠀⠀⢄⠀⠀⠀⠀⠀⠀⡘⠃⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⣿⠜⣋⣀⣀⣉⣿⣷⡀⠀⠀⠹⣏⣉⡙⣧⣄⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠲⣤⠞⠋⣠⣤⣾⠆⠀⠀⣹⠾⠋⠉⠉⠉⠻⠟⠛⠀⠀⠾⠟⠉⠉⠙⠻⡿⡇⠀⠀⢴⣤⣔⣠⡞⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡙⢃⣻⠆⣼⣧⠈⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠟⣀⠀⠆⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟⠉⠀⢿⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠆⢻⡧⠀⢀⡾⠋⢻⡆⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠻⢆⡠⣾⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣮⡀⢀⡜⠁⠀⣸⣇⠋⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣼⠋⣠⣶⡄⠀⠘⢷⡀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢀⣼⣿⠁⡈⠃⠀⠀⠛⠁⠀⢀⣼⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠏⠉⠉⡀⣤⠖⣲⠄⠛⢦⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠸⣷⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣍⡰⠶⠞⠋⡠⢶⡄⠉⠳⣶⣡⣄⣀⡀⠀⠀⠀⠁⠀⠀⢸⡿⣄⠀⠀⠁⠠⣄⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⡄⠀⠀⢸⡥⢾⡁⠀⠰⠇⢛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣃⢾⣄⡘⠃⢀⣠⠾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠻⣦⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢋⣽⠶⠋⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠀⠈⢷⡄⠀⠀⠀⠀⠀⠉⠀⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠉⠳⠶⠶⠶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡍⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⢦⣀⠀⠀⠀⠀⠀⣠
⣄⡀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠛⢷⣶⣤⣤⣶⡿
⣿⣧⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣠⡿⠉⠉⠁
⣿⣿⡆⠀⠀⠀⠘⢰⣿⠃⠀⠀⠀⠹⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣰⡟⠁
⣿⣿⣿⣶⣦⢢⠄⣼⣿⠀⠀⠀⠀⠀⠹⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡟⠃⡀
⣿⣿⣿⣿⣿⣇⠰⣿⣿⣤⣄⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⠿⠿⠋⣁⠰⠂⠀⠀⣀⣴⣾
⣿⣿⣿⣿⣿⣟⣻⣍⣙⣻⠿⠿⠿⣷⣶⣶⣶⡿⣧⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⡶⠿⠛⠛⢋⣁⡀⠠⡤⠆⣀⠴⢶⣉⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣂⣈⣉⡉⣥⣈⣉⡉⠙⠿⠛⠛⠛⠛⠛⠛⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⣺⡀⢈⣉⣴⣶⣷⣶⣿⣶⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣍⠛⣛⣛⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣷⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡟⠉⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠈⠉⠀⠙⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⣽⣿⡄⢳⠜⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⠏⠀⠀⢰⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠄⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣿⣿⣿⣿⣤⡀⢀⢾⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣶⣿⣿⣿⣿⣿⣿⡅⢸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣄⡀
⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⡿⢏⣿⢀⢿⣿⣿⣿⣿⣿⡿⠛⠶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢉⠁⢸⡏⡸⠀⠙⣽⣿⣿⡿⣿⣿⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⣇⠔⣿⡿⣁⠀⢀⣿⣿⡿⠀⠀⠉
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡏⢉⣿⡁⡀⢹⠛⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢌⠋⡀⠜⠉⠁⣸⣧⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠁
⣿⣿⡿⢿⣿⠿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣷⣷⣾⣶⣾⣶⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⢤⣤⡤⣤⣤⣤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣺⣴⡉⠿⣬⠟⠛⠓⠋⠉⠀⢴⣾⡏⢳⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡟⠶⣋⢛⣽⠟⠉⠀⠀⠀⠀⠂⡠⠀⠐⠙⠙⠟⠗⢿⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡭⣝⠟⣷⡏⠔⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠈⠛⢢⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⣷⣦⡔⣀⠉⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢃⠈⠀⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣻⣏⢻⠙⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⣴⠉⣋⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣧⠀⠈⠉⠁⠉⢁⠀⠢⣀⠀⠀⡀⢀⠀⠀⠀⣴⠂⠀⠀⠐⠀⡀⢀⣹⣽⣃⣼⢿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠛⠛⣿⣿⡷⣟⣤⣐⣀⣐⣾⣛⡀⠀⠁⠀⣈⠳⡦⣮⣁⣀⣻⣆⡀⠙⣿⣿⢍⣬⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣟⣿⣿⣯⣿⡯⠦⣾⣟⣲⣶⣾⣭⡇⣽⣿⣿⣿⣿⣯⣥⣤⣼⣥⣶⣾⣯⢥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣼⣷⣉⣉⡈⠉⠛⠋⠉⠉⠹⠿⠿⠾⠛⢷⣭⠿⢿⣿⡛⠛⣿⣟⣛⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⡿⣿⢿⠿⡿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠉⠀⠀⠀⠀⠁⣼⡿⣟⣭⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡟⣻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⣀⣸⣟⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⠟⣗⣦⣠⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠉⡙⠛⠻⠒⠿⠗⠿⠿⠟⠛⠛⠻⠛⠛⠛⠋⢉⣁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠀⠄⠠⠀⠀⣀⡀⠀⠀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠤⣒⠼⠟⠛⠛⠻⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠈⠉⠛⠅⠢⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣀⣀⣠⣤⣤⣷⣵⣆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡾⠛⢿⣿⣿⣿⠇⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠸⣿⣯⠀⢟⠙⠿⠿⠟⠉⠛⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠱⣄⡀⠉⣀⣞⡤⣺⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡕⠾⣦⣤⣠⣤⣒⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠛⠛⠋⣁⢼⣿⠙⠉⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡶⣄⡈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⣴⠿⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀⠞⠟⢷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠩⠀⠘⡀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠀⢀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣀⣀⣀⣤⣤⣤⣤⣄⠤⠤⠀⠐⠀⠂⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠋⠉⠉⠉⠀⠀⠀⠀⠀⠒⠲⠬⣵⣒⡢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠒⠊⠉⠀⠀⠀⠀⢀⣀⡠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠮⠍⣀⠀⠀⠀⠈⠉⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠇⠀⠘⣄⠀⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠎⠀⠀⠀⢸⠠⠘⠈⢣⣄
⠀⠀⠀⠀⠀⠀⠀⡠⠊⣎⠀⠀⢀⡠⠎⠀⠀⠀⢸⡇⢰
⠀⠀⠀⠀⢀⡤⠀⠀⠀⠏⠀⠀⠰⠀⠀⠀⠀⠀⢸⠃⢸
⠀⠀⠀⠀⣸⡀⠉⠉⠽⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⣠⢞⣹⠛⣠⡶⠖⢀⠀⠀⠀⠀⠀⠀⢀⡎⣰⣿⣦⡀
⠀⢸⣃⠀⠘⢷⡄⠰⠂⠀⠀⠀⠀⠀⠀⠀⣞⣴⣋⣀⣀⡀⢀⣀⣨⠢⣀⣀
⢀⠈⠉⠀⠀⣾⠻⣶⣄⠀⠘⠀⠀⠀⠀⣸⡟⠉⠫⣀⠴⠄⢀⣤⢞⣯⣽⣏⠓⢄
⠇⠀⠀⠀⢠⠔⢂⣽⡈⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⡜⢀⡜⡹⠉⠓⡛⠲⡎⡄⠀⠃
⠀⠀⠀⠀⣠⡽⢈⡁⠙⢷⠀⠀⠀⠀⢀⣟⢧⡀⠄⠀⢀⠼⢆⠀⠂⠈⢳⠀⠶⠒
⠀⠀⠀⢀⣉⠀⣌⢁⠀⠘⠆⠀⠀⠀⠈⡿⢍⡻⢴⡚⠋⠀⠀⢣⠀⠉⠀⢨⠩⢿⣝⣦⣀
⠀⠀⢈⡿⠍⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠙⢦⡀⠀⠀⠑⡄⠉⠀⢋⡉⣙⠿⢿⣵⣦⣄
⠀⠀⣎⠀⠀⠀⠀⢃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⠀⠸⡄⠘⠁⠉⠀⠀⠀⠀⠀⠈⠑
⠀⠀⡇⠀⠀⠀⠀⠀⠳⣄⠀⠀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⡗⠀⠀⢀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢇⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣤⣀⣀⣀⣠⡤⠔⡪⠝
⠀⠀⠈⠛⣒⣒⠒⢉⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠐

⠀⠀⠀⠀⢀⢰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣆⣀⡆⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠐⢻⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢾⡷⡴⠂
⠀⠀⠀⠀⠀⣞⡷⠀⠀⠀⠀⠀⢀⣯⣍⠁⠀⣻⣇⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⢀⣀⡿⣇⢣
⠀⠀⠀⠀⡀⠘⠃⢀⠀⠀⠀⠀⠘⢿⡿⠃⠀⢏⠉⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⢠⡄⢠⣼⣼⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣻⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡟⠀⠀⠀⠀⠀⠀⡞⣿⡦⠓⠀⠀⠀⠀⠀⠀⣠⡾⣷⡄
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⣰⠿⣬⠁⠀⠀⠀⠀⠀⠀⠓⣿⡄⠀⠀⠀⠀⠀⢀⡼⢻⡄⠸⣿⡀
⠀⠀⠀⠀⠰⠁⢸⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠰⠁⡇⠀⠀⠀⠀⠀⠏⠀⠞⠀⠀⠻⠇

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢈⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⣤⣾⣿⡇⠀⠀⠀⠀⠀⣀⣤⠟⠻⢿⠀⠀⠀⠀⠀⠀⠼⣿⠄⠀⠀⠀⠀⠀⣶⣾⣷⣶⣶⣦
⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠾⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢶⠠⢠⡄⡄⡦⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠙⢾⡗⠋⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠁
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⡎⠋⡇
⠀⠀⠀⠀⠀⡃⠃⠀⠀⠀⠀⠀⠀⣸⠀⠀⠓⠂⡄⠀⠀⠀⠀⠀⡞⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢉⣉⡑⠢⡀⠀⠀⣀⣤⠴⠶⠖⠲⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⢁⣤⣬⣥⡰⠀⢰⠁⠀⣰⣚⣉⡙⠻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠙⠒⠀⠚⠁⠀⡇⠀⣎⡑⠀⢚⣃⡤⠖⠂⠐⠢⢄⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠷⠂⠀⠀⠀⠀⣇⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⠻⢤⣤⠤⣤⠀⠀⢀⡤⠶⠶⣤⣀⠀⠀⠀⠙⡄
⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠉⠲⠦⢤⣄⠀⠀⠀⠀⠀⠀⡾⢀⡴⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⢀⡞⠁⠈⢳⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⡇⡎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠊
⣿⠀⠀⠀⠀⠀⠀⢸⢡⡖⢁⠘⠧⡀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⡄⠀⢀⡀⠀⠀⢀
⣿⠀⠀⠀⠀⠀⠶⡞⢸⡗⢍⣙⠧⠸⠀⢠⠖⢻⣖⢢⡀⢀⡾⠀⡇⢰⣡⠚⠋⠙⠫⣍⠉⠙⠒⢄⡀
⣿⠀⠀⠀⠀⠸⠇⠀⠈⣇⠀⠀⢀⡖⠀⡾⠀⠀⠙⣿⠓⠋⠁⢀⠇⠈⢿⣦⡾⠃⠀⠈⠇⠀⡴⢚⠃⠑⡄
⣿⠀⠀⠀⠀⠀⠳⡇⠀⢹⠀⠀⠸⢤⣼⠃⠀⠀⠀⣿⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⠷⠏⠀⢠⠃
⣿⠀⠀⠀⠀⠀⠀⠸⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠓⣦⠀⠀⠀⠀⠀⠀⢠⠞⠀⢠⣄⢀⡰⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠳⡀⢱⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⣰⠏⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢝⡒⣚⠿⠛⠛⠢⢤⡀⠀⠀⠀⠘⡇⠀⠀⠀⢀⡟⠀⠀⣠⠞⠀⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⣀⡀⠀⣼⠃⠀⠀⠀⠀⢡⠀⠀⠀⢈⣧⠐⠋⠀⣠⠞⠉⠙⣧
⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⡽⠧⣀⡀⠀⠀⠀⠈⠓⠒⠊⠁⢧⠀⠀⢠⠃⠀⠀⠀⣸
⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡼⢡⠞⠙⢏⠳⡄⠀⠀⣀⣀⠀⠀⢸⠀⠀⡎⠀⠀⠀⢠⠏
⣿⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡾⠁⠃⢰⠶⠮⡀⠃⠀⡸⢁⣌⣳⡄⠘⠀⠀⡇⠀⣀⠴⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⡞⠀⢸⠀⢸⠛⠛⠁⡆⢀⡟⠫⠭⠟⠛⢾⠶⠖⡟⠋⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠋⠀⠀⠈⡄⠸⡀⠀⢠⠁⢸⠃⠀⠀⠀⠀⡜⠀⠀⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣇⠀⢧⠀⠈⠙⠋⠀⠀⠀⢀⠜⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⢷⡀⠀⠸⡀⠈⠛⠛⣻⠟⠁⠀⡴⠋⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⢳⡀⠀⢹⣤⠔⠋⠀⠀⣠⠞⢀⣠⣄⠀⠀⠀⢸⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣄⣀⣤⠤⠴⠛⢮⡁⠈⠛⠶⠶⠚⢁⡴⠋⠀⠈⠑⢦⣀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠈⠉
⣿
⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢆⠀⣁⣀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡘⠑⣤⠀⠉⠋⠀⠉⠓⢷⡦⣀⡀⠀⠀⢀⡠⣾⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢲⣆⠀⠸⣃⣠⠴⠒⠋⠁⢄⡀⢄⡀⠈⠒⠭⣍⣉⡭⠞⠁⣀⣠⠤⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⠒⠉⡀⠐⢆⣈⡥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠙⢦⡈⠓⠦⢄⠀⢀⠤⠒⠉⠁⠀⠀⠈⡆
⠀⠀⠀⢠⠴⠒⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠐⠒⠲⠦⠤⠴⠒⠒⠀⠒⠢⣜⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⣀⣀⣀⣀⣀⣠⠤⠤
⠀⠀⠈⣞⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⠠⠤⠤⠐⠒⠒⠚⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠘⠦⣤⡀⠉⠉⠓⠒⠂⠀⠀⠤⠤⠔⠒⠂⠉⠉⠀⠀⠀⠀⠀⢀⣀⡀⠤⠐⠒⠈⠉⠉⠑⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠉⠓⠖⠦⣄⣀⣤⣀⡀⠀⣠⡀⠀⠀⠀⢀⡀⠀⠐⠉⠁⢀⣀⠀⣀⣀⣀⣀⣀⢀⡀⢀⣀⣀⣀⣀⣀⡤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠁⠀⠉⠉⠉⠁⠀⠉⠉⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣀⠛⠿⢿⣷⣤⡀⠀⠀⣿⡏⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⣉⣉⠉⠉⠛⠛⠿⢿⣶⣤⡈⠻⣷⡄⠀⣿⣧⣼⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⠿⠿⠿⠿⠿⠿⠿⣷⣶⣦⣄⡉⠻⣿⣦⣼⣿⣿⡿⠟⠛⠛⠿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠉⠙⣿⣷⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣤⡀
⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⣿⡿⠛⠛⠻⢿⣷⣾⡿⠟⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⡀
⠀⠀⢰⣿⣋⣩⣽⣿⡿⠿⠃⠀⣠⣴⡿⠛⠉⢀⣴⣿⠟⠁⠀⠀⢀⣿⡿⠉⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⢀⣾⠟⠁⠀⠀⣠⣿⠟⠁⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⣤⡀⣤⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⣶⠶⠾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⢹⣿⠃⠀⠀⠲⣮⣿⣿⣿⣷⣄⠀⠀⠀⠀⠘⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⠛⠛⠋⠀⠀⠀⢀⣴⣾⣷⠘⣿⣆⠀⠀⠀⠀⠈⠀⠀⣶⣶⣌⠻⣯⡙⢿⣿⣷⣄⠀⠀⠀⠙⠿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⢰⣟⣡⣾⠇⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠙⢷⣄⠻⣿⣿⣷⣄⠀⠀⠀⠀⠉⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⣼⣿⠃⢀⣠⣄⡀⠀⠀⠘⠻⠿⠋⠀⠀⠀⠀⠻⣿⣦⣄⣀⠀⠀⠀⠹⣿⣿⣦⠈⠻⣧⠈⠋⢹⣿⡆⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⣠⣤⡶⢾⣿⣿⡿⠁⢠⣿⡿⠀⠉⠉⠙⠿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣶⣄⠘⣿⣿⠃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⣿⠃
⠀⠀⣾⣿⣁⣤⣾⡿⠋⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⣶⡀⠈⢹⣿⠀⠘⣿⡀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣸⣿⡏
⠀⠀⠻⠿⠿⠟⠋⠀⠀⠀⠀⣼⣿⠁⣀⣀⡀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⣿⠀⠀⢹⣷⠀⠀⠀⠀⢸⣿⣧⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠻⣿⡿⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⡀⠀⠈⣿⡄⠀⠀⠀⢸⣿⣿⣀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢸⣿⣇⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠉⢫⣥⣄⡀⣹⣯⡀⠀⠀⢸⣿⣿⣿⠟⠋
⠀⠀⠀⠀⣠⣴⣾⣿⡿⠀⠀⠀⣿⣿⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣇⠀⠀⠈⣿⣿⠁
⠀⠀⣰⣿⣋⣰⣿⠟⠀⠀⠀⠀⠘⣿⣧⠀⣰⣿⠏⠀⠀⣀⣤⣶⣾⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡹⣿⡄⠀⠀⢻⣿⡀
⠀⠀⠙⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣏⣠⣶⡿⠟⠛⠉⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣿⣿⣆⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠟⠋⠁⠀⢴⣤⣤⣀⡀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⡿⠛⣿⣦⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣄⠀⠀⠉⠙⠻⢷⣶⡄⠘⢿⠄⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠀⠀⠈⢿⣧⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣀⣀⠀⠈⠁⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣶⣶⣶⣿⣿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢨⡏⠀⢳⣄⡀⠀⠀⠀⠀⢀⣾⠃⢈⣧⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠷⢀⢜⡭⠚⢃⣺⡖⠻⠯⣻⣆⢸⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠰⠟⢂⣴⣶⣿⣿⣿⣷⣶⡌⠻⣿⡄⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠋⣠⡿⠅⠀⣹⢿⠟⠀⡨⢿⣷⡘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⣯⠴⣾⣶⣏⣠⢹⣾⣿⠦⣽⣿⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠙⠁⡶⠄⠁⠀⠀⠀⠙⢦⣶⣏⠹⡏⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣤⡄⠀⠀⠀⢰⣶⣿⣷⠀⠀⠀⣨⠄⣱⣼⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣧⠀⠁⡄⠀⣿⣿⣅⠀⢀⣾⠉⢰⣿⡟⢉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠁⠀⠱⣠⠙⣛⡹⣶⡿⠁⠘⠋⢩⡤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢻⣷⣤⠀⠀⠀⠀⠉⠛⡸⠀⠀⢀⡴⠋⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡟⠂⠘⣿⡏⠁⠀⠀⣤⢤⠞⠁⢐⡖⠋⠁⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣼⡇⣺⠀⠘⣿⣄⠀⠀⠀⠀⠈⠛⠊⠀⢀⣰⣿⡿⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⠁⣾⠂⠀⠈⢻⣷⣆⡀⠠⠀⠀⣀⣾⣿⡿⠋⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⣾⡂⠲⣄⠀⠈⠛⢻⣿⠿⡿⠿⠛⠋⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠸⠇⠀⠈⠐⡂⠀⠀⠁⠘⠁⢠⣠⡶⠛⢋⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡠⠀⠀⠀⠀⠀⠀⠀⣾⠇⣄⣶⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⣾⣯⣾⡿⠟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣿⣀⣾⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿