Cherry and star Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣤⣤⣤⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠛⠛⠋⠙⠛⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠉⠢⡕⢤⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢸⣿⣷⡥⢁⡌⠄⣠⠞⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠙⣿⣿⣿⣯⣞⣟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡠⢚⡿⡿⣿⣿⠿⠧⠤⣀⣀⡀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⣀⡤⢊⣴⠟⠁⠀⠈⠙⣆⠀⠀⠀⠉⠉⠛⠛⣿⠋
⠀⠀⠀⠀⠀⢀⠤⠤⠄⠒⠢⣀⠀⡐⠈⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠋⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⢀⠊⠁
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢹⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠉⢄⣀⠠⠀⢀⠀⠀⠀⠃⠑⠤⠀⣠⠟
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⣘⣧⠄⠀⠀⠀⠀⢀⣀⣀⣠⣴⣿⡿⠟⠋⠁⠀⢸⡇
⠀⠀⡠⠀⠀⠈⠙⠛⢷⣶⣦⣹⣿⣟⣦⣠⣶⣿⣿⣿⠿⠟⣉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠐⠠⡀
⠀⠀⢆⠀⠀⠀⠀⠀⠠⠽⣿⣿⣿⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠁⠀⠀⠀⠀⢅⡀
⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⠋⣿⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢸⠃⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢀⣠⣾⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠇⠀⠀⠀⠀⠀⠈⢻⠶⢤⣄⠴⠁⠀⠀⠀⠀⢠⣿⢿⣿⡟⠋⠛⠿⣷⠦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠠⣀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠈⠃⠀⠀⠀⠙⡄⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠐⠂⠒⠁⠀⠀⠀⠀⠀⠀⡤⠊⠙⣿⠀⠀⠀⠀⠀⠀⠀⢠⡃⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠔⠁⠀⠠⠤⠔⠓⠄⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠉⠛⠻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⣿⣷⣆⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠛⢿⡄⢀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡠⠤⡤⠤⡤⠄⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠒⢄⠀⣠⣮⠤⠴⠂⡎⠀⠀⠀⠀⠀⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠉⠠⣀⠐⠐⠈⠀⠀⠀⠀⠀⠳⣆⠴⠇⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣆⢀⠀⠙⠂⣀⠀⠐⠀⠀⠀⢠⡟⠀⠀⠀⡄⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⣠⠆⠀⡜⢈⠷⡦⣾⡀⠀⠀⠀⠀⣵⣿⣷⣄⠀⣸⣧⣶⡆⠀⠀⠀⠀⣄⡴⠖⠀⠀⠀⠀⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣺⠖⠋⠓⢀⠐⢋⠾⠀⡇⡌⠳⢀⡀⢀⡠⠿⠿⠿⠁⠈⠥⣿⣿⣿⠄⠀⠀⠠⠁⠀⣀⡤⠶⠓⠒⠚⠋⢉⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠃⢠⡾⢠⡀⢧⡇⡘⣮⠑⠂⠀⠀⠀⢰⡀⢰⠃⠈⠛⠁⠀⠀⢀⣤⡴⠛⠃⠀⠀⠀⠀⣠⡾⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠟⠀⣸⠁⡸⢱⠀⠉⣧⢇⠀⡀⠀⠈⠁⢂⣤⣠⢎⡀⠀⠀⠜⠀⠐⠲⠶⣴⢾⣻⡽⠷⠶⠚⡁⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⣰⢏⡼⢱⢰⠀⠀⠈⣧⠉⠙⡆⠀⠀⢸⠗⡄⠢⠉⢉⠙⠻⠶⣦⣴⡂⠉⠃⠀⠠⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠋⢀⡴⣣⠞⠀⠈⠈⣦⠀⠀⠈⢣⠀⣽⠀⠀⢸⠀⡁⠀⢱⠀⢣⠀⠀⠀⠀⠻⠻⢖⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣮⡄⢨⠟⠛⢾⡿⠂⠀⠀⢰⠈⣷⣤⠀⠀⠂⢸⠀⠀⠇⠀⡇⠀⠀⢧⠈⡆⠀⠀⠀⠀⡧⡐⠠⢍
⠀⠀⠀⠀⠀⡠⠀⠘⠻⣳⡏⠀⠀⠀⣄⢰⡀⠇⢨⠀⢹⡍⠙⢶⣄⠘⠀⠰⠀⠀⡇⠀⠀⠘⠄⢸⠀⠀⠀⠀⡄⠹⢦⡀⠓⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠘⡟⡃⠀⡀⠀⣠⡟⣿⠧⣄⢳⡀⠑⠀⠘⡏⠓⠀⠂⠀⢠⡇⠀⢠⠶⢿⠼⠓⠄⢦⣄⠑⣀⠈⠳⠀⣗
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠐⠇⢣⣿⠛⠁⠀⠀⠀⢨⠀⠘⠣⣄⠀⠇⠀⠀⠀⠀⠠⠁⢀⡁⠀⠀⠀⠀⠀⠘⢿⠀⠙⢦⡀⠀⢯
⠀⠀⠀⠀⠈⠂⠀⣠⠤⣀⡤⠛⣛⡇⠀⠀⠀⡴⡞⠀⠀⢀⡀⠀⢃⠀⠀⠀⠀⢰⠀⠈⢧⣀⣀⣄⣀⣄⣀⡾⠀⠀⠀⠉⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠚⠽⠚⠀⠔⠁⠀⠀⠰⣿⣤⡀⠂⠺⣿⠶⢀⣾⠁⠀⠀⠈⠀⢮⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢘⡗⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢀⠀⠠⠀⣤⡗⠀⠀⠀⠀⣀⢈⣇⠀⠀⣀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠶⠦⠒⠉⠑⠄⠀⣠⠶⠶⠉⠉⠙⠒⠈⠁




⡿⢿⡿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠋⠉⠙⠓⠦⣄⢀⣴⡧⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⢀⣀⣠⣤⠤⠤⠤⣤⣼⣟⣁⣀⣀⡀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠛⠲⢤⣀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠒⠲⠶⠶⠶⠶⣶⠖⠛⠁⠀⢸⡷⢤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢸⡇⠀⠀⠉⠙⠛⠓⠒⠒⠲⠶⠶⠤⣤⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣠⠴⠖⠒⠒⠶⢤⡀
⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⣤⣀⢳⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢀⡴⠚⠉⠀⠀⠀⠀⠈⠙⠲⣄⡟⢀⣤⠴⠒⠒⠒⠲⢤⣤⠞⠉⠀⠀⠀⠀⠀⢈⣙⣿⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣟⣉⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⠀⠀⠀⠀⠀⠀⢀⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠰⠉⠈⠃⠀⠀⠀⠀⢰
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠘⠁⠀⠁⠐⠶⠃⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠶⠶⠒⠚⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣲⣶⣦⣤⣤⣴⣶⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣤⣀⢠⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⢠⣤⣀⡀
⠀⠀⠀⢉⣽⣷⣦⡀⠀⠀⠀⢠⣾⣿⣿⣿⡄⠀⢘⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣶⣶⣶⣤⣤
⠀⠀⠀⣾⣿⣿⣿⣿⡀⠀⠀⠸⣿⣿⣿⣿⠏⠀⠈⠉⢿⡋⠑⡼⢠⣤⣀⠀⠀⠙⠻⠿⠿⠿⠿⠟⠋
⠀⠀⣰⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⢿⡿⠏⠀⠀⠀⢰⡉⠀⡼⡅⣿⣿⣿⢿⣧⣤⠀⠀⠀⠀⠀⢀
⠀⠀⢻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠻⡿⠛⠛⠃⠁⠀⠀⢀⢠⣶⡿
⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⣠⣶⣖⣇⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⣤⣬⣯⣭⣥⣀
⠀⠀⠀⢀⣴⣶⢀⡴⠀⠀⠀⢉⣿⣿⣿⣾⣷⡄⠀⠀⠀⣴⣾⣿⣿⣶⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⡿⠃⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠟⠁⠀⠀⠀⠘⢿⣿⣿⠏⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⢿⣿⣿⡿⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡀⠀⠀⠉⠁⠀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠉⠉⠸⣿⣿⣿⡇⠀⠀⠀⢀⣠⣾⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣠⢶⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣯⠞⠁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠂⠀⠣⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠤⢄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣷⡴⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⠟⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠰⡆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⢋⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⢄⡤⠔⠀⡀⠀⠀⠀⠀⠾⠶⣀⠀⠀⠉⠀⠀⠀⠁⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⣸⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣿⠀⠀⣠⠔⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢶⡀⠀⢀⣤⠤⠀⠀⢄⠀⠀⠀⠀⠀⠁⣠⢠⠖⠀⣤⡄
⠀⠀⠀⠀⠀⠀⣿⣦⣤⠀⢠⠈⡃⠀⠚⠁⢸⡇⠀⡀⠉⠀⠀⠀⠀⠀⠉⡄⠀⣀⠉
⠀⠀⠀⠀⠀⠀⣈⠯⠉⠐⢋⢠⠀⠘⠿⣆⡾⠃⠀⠁
⠀⠀⠀⠀⠀⢀⠐⠊⠀⠈⠉⠀⢀⡴⠖⣸⠓⠾⠛⠉⠁
⠀⠀⠀⠀⠘⠓⠙⠀⢀⠀⢸⠀⠊⠀⠀⠙⢧⠀⠈
⠀⠀⠀⠀⢀⣀⠀⠀⠉⣭⠂⠀⠀⠀⠀⠀⠀⠁⠀⢀⡤⡬⠓
⠀⠀⠀⡀⠚⠋⠠⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠐⠶⠉⡡⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠂⠈⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠛⠛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⣠⠞⢁⡤⠞⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⢤⣰⡧⠞⠉⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⡙⢿⣷⣦⡟⠘⢦⠉⠻⣍⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠛⡇⠀⠈⣧⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⣆⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⡀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠴⠚⠋⠉⠉⡟⠉⠉⠛⠲⢤⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠲⠤⠧⠶⠂⣠⠴⠖⠓⠚⣷⠒⠒⠒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⣀⡀⣿⠀⣀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣼⣿⣀⡀⠀⠀⠀⠀⢀⡏⠀⣠⢶⡆⠈⠉⠉⠉⠁⠀⠀⠀⢀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠁⠀⠀⠀⢠⠏⠀⠈⠉⠓⢦⡀⠀⢸⡇⠀⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸
⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⡀⠐⠶⠴⠿⠖⠀⠀⠀⠀⠀⢳⣀⣀⣧⠀⠹⢼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡾
⠀⠀⠀⠀⠀⠀⡇⠀⣼⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⣇⠀⢻⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⠏⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣤⠤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠰⡄

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠀⠈⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⣷⣄⣙⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡟⢧⡈⣿⡄⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠻⣿⠁⠈⠻⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣿⣼⠿⣤⠀⣻⠿⢿⣍⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⢠⣌⠳⢿⡀⠀⠈⠳⣦⡙⠻⡅⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣣⡟⠉⢷⣀⡼⠀⠂⢸⡃⠙⢦⡘⢧⣀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⢲⡄⢻⡄⠀⠀⢸⡁⠀⣀⣷⣄⠻⣿⠶⠾⣷⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣇⠀⣡⠀⠀⣞⣠⠞⠁⣀⣿⡄⢻⣷⣄⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠄⠀⠀⢀⠇⠈⣿⣿⣿⠄⢸⣿⡥⠖⠊⠉⠀⢳⡀⣿⣿⣦⡤⠼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣴⢹⣿⣿⠂⢸⠁⠀⠀⠀⠀⣠⠟⣧⢹⣿⠇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⠏⢸⡇⢸⠀⢸⡄⠀⠀⣠⠞⣡⠞⢿⣸⣧⣤⣤⠤⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡞⠀⢸⡇⢸⠀⠀⣷⣴⣾⣵⠞⠁⠀⠈⣿⡋⠛⠛⠛⠒⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡾⠀⠀⠘⡇⠈⡄⠀⠘⣿⣿⣷⡀⠀⠀⣠⡾⢷⣀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡜⠁⠀⠀⠀⢿⠀⢿⠀⠀⠈⢿⡿⠷⠶⠾⣃⣀⢀⡟⠛⠶⣤⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣴⠁⠀⠀⠀⠀⠸⡇⠸⡇⠀⠀⠀⠙⠦⣄⠀⠀⠉⠛⠶⢤⡄⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣿⣿⠃⣀⠀⠀⠀⠀⠀⠸⣄⢹⣄⠀⠀⠃⠀⠀⠙⠢⢄⣀⠀⠈⠙⣻⣦⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⠃⠀⠀⠀⠀⢠⠆⠀⠀⠀⠙⢿⣍⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠻⣶⡖⠂⠀⠀⠒⠢⢄⡀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠞⠻⣦⠈⠳⣄⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠘⣷⡀⠹⣶⣤⡄⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢹⡦⠒⠂⠤⢤⣀⡀⠀⠀⠀⡼⠁⠀⠀⠘⠿⠿⠷⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⣷⠈⣧⠀⠀⠀⠀⠈⠙⠲⣄⢸⡇
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠰⣴⣿⣆⣹⣿⠞⠀⠀⠀⠀⠀⠈⢻⡇⢠⣄
⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠸⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠹⣿⣿⣿⣷⣆⣠⣄⣀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⢸⠆⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡈⠻⢿⣿⣿⣿⣿⣥⣀⠀⠀⠀⠀⡠⠈
⠀⠀⠀⠀⠘⡀⢸⣿⣷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠦⢀⡈⠉⠉⠋⠉⠀⠀⡀⠤⠊
⠀⠀⠀⠀⠀⢻⡀⢻⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠑⣄⠙⢿⣿⣿⣿⣾⣿⣿⣿⣶⠖⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡈⠙⠛⠿⠛⠛⠉⠁⠀⡀⠖⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠒⠀⠈⠁⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⢢⠐⣴⠀⡆⢰⠀⠰⠀⠶⠀⠀⠰⠀⠄⠀⢀⠀⠆⠐⠄⡆⢰⡄⠠⡄⣀⠰⣤⠐⣆⠀⣖⠰⣦⠀⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⢻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⣠⠿⢋⣵⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣟⠲⣄⡱⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣭⣑⡀⢸⠀⠘⣧⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠘⣧⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⡆⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⣀⣠⣤⣤⣷⣤⣤⣀⣀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣴⣿⣿⡿⠿⣿⣿⡿⢿⣿⣿⡿⠦⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣶⣶⣶⡿⣫⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣟⠻⢿⠿⢛⣻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⣁⡙⠿⢿⣿⣿⣿⣿⣿⢰⣿⡿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⡻⣿⣿⣿⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⣿⣍⣛⣿⣋⣩⣿⣿⣿⣿⣿⣷⡘⣿⣿⡎⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢘⡖⠁⢢⣄⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠉⠱⡌⠀⠀⠀⢀⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠲⠠⠞⣇⠀⠀⠀⠀⠀⢀⡠⠞⠋⠘⠉⠛⡟⠀⣠⣤⣞⠹⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠤⣄⠎⠁⠀⠀⣠⠔⠋⠀⣀⣴⣖⣀⣤⠷⢚⣗⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠈⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠙⠀⣠⠴⠋⠁⢀⣤⣾⠿⣛⣭⣶⠿⣟⢩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢸⣶⣤⣄⡀⠀⠀⠀⠉⠛⠲⠤⣄⡴⠚⠁⠀⣠⠖⡫⢛⡵⣾⢿⣿⠁⠀⡇⡸⣿⣆⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢿⡿⠋⠉⠓⠲⢤⣄⡀⠀⠀⠀⠀⢀⡴⠚⣡⠞⡴⢋⡼⡵⠋⡿⠀⠀⡇⣷⢿⡻⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠘⣇⠀⠀⣄⠀⠀⠀⠉⠓⠦⣤⠖⠋⡀⣰⢃⡜⢁⢎⡞⠀⠀⡇⠀⢸⡇⠸⡎⡵⣌⠻⡀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡞⢀⡎⡞⠀⠃⠀⡇⠀⢸⡇⠀⢹⢷⣟⣷⣝⢶⣿⣏⢢
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⡇⠀⠀⢧⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⣰⢃⠞⠀⡼⢰⠁⠀⠀⠀⡇⠀⠸⣇⠐⠀⠹⢽⣧⣽⢳⣮⡑⠀⢳⡀
⠀⠀⠀⠀⠀⡶⠤⢾⠀⠀⠀⢹⡀⠀⠘⣆⣠⠤⠤⠤⠤⣀⠀⢀⡴⣣⠏⠀⢰⡇⡇⠀⠃⠀⠀⢧⡀⠀⠈⠙⢦⣀⠀⠉⠻⢿⣿⣏⢰⡀⣑⣄
⠀⠀⠀⠀⢠⠇⡀⡜⠂⠀⠀⢀⡇⢀⡞⠉⠀⠀⠀⠀⠂⣀⣱⣮⣾⠁⠸⠗⢸⢸⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡴⠋⢠⠏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣋⡬⠷⣤⣀⣾⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠓⢦⣄⠀⠈⠓⢦⣀⠐
⠀⠀⠀⠀⠀⠈⠁⠀⢠⠞⠁⠀⡼⠀⠀⡀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⣈⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠓⠀⢉⣷⣦⣀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣰⡇⢠⣶⣿⣀⡀⢀⢀⡎⠀⠀⠀⠀⠀⠀⠘⠻⢿⣦⠉⢢⡀⠀⠙⠀⠀⣀⠀⠀⠀⣸⣿⡿⠿⠛⠐⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⣠⠞⠀⠀⢠⠞⠁⢱⢸⣿⣿⣿⣷⣾⣼⠁⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣀⣠⡭⠶⠚⠋⠉⠀⠀⠀⣀⣠⡼⠿⠛⠛⠉
⠀⠀⠀⢀⡴⠃⠀⢀⣴⣿⣈⠃⠈⢆⠻⣿⣿⣿⣿⣿⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠉⠀⢀⣀⣠⡤⠴⠖⠚⠉⠉
⠀⠀⣠⡞⠄⠀⠰⠾⠿⠿⠷⠶⠶⠚⠓⢌⣙⠛⢛⣹⡇⢻⣿⣿⣷⣦⣤⣤⠀⠀⠀⠀⢀⡞⠀⣠⠖⠛⠉⠁
⢀⣴⣿⣿⣤⣦⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⡁⠘⢆⠛⢿⣿⣿⣿⣿⣯⠗⢀⣴⠋⠀⢰⠇⠀⠀⣠⡀
⠘⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀⠑⠢⣬⣉⣉⣩⠤⠖⠋⡿⠀⢀⡟⠀⢲⠒⠙⢦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⡠⢴⠂⠀⠀⠀⠳⡄⠀⠈⢧⡀⢀⡀⠀⢸⠃⠀⣸⠁⠀⠾⠲⣜⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡜⡀⢀⢿⠀⠀⠀⠀⠀⠹⣄⠀⠀⢳⣄⣀⣰⡟⠀⢀⡇⠀⠀⣤⠀⠁⠰⡶⠖⣿⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣀⢱⡎⠉⠁⠀⠘⠓⠀⠀⠘⢦⠀⠀⠹⣿⣿⠃⠀⡼⠀⠀⠀⠀⠀⠀⠶⠥⣀⠞⠃
⠀⢀⡄⠀⠀⠄⠀⠀⠀⢠⣜⡈⠩⡇⠀⠀⡖⠖⡻⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⠟⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠺⠿⠇⠀⠀⠀⠀⠀⠀⠀⢹⠞⠛⠀⠚⠤⣠⠟⠆⠀⠀⠀⠀⠀⠀⠙⣦⣤⣴⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⢠⢠⠄⠄⢠⠀⠄⠄⡠⢀⠀⢠⠀⡄⢄⠀⠠⠀⣀⢠⠀⡠⣀⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠛⠿⠶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠒⢒⣶⣶⣶⣦⣤⣤⣤⢤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠛⠛⠛⢻⡟⣿⠙⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⠀⠀⠀⢀⣠⠄⣿⢁⣿⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠁⠀⠀⠀⠀⣠⣴⠟⠁⢸⡿⣸⡟⣿⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣼⡇⣿⠁⢿⣮⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠁⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⢀⣿⢿⣿⠀⢸⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⣀⣀⣀⣀⣴⠟⣻⠃⠀⠀⠀⠀⢀⣾⣿⣼⡇⠀⠘⣿⢸⣷
⠀⠀⠀⠀⠀⢀⣾⡿⣃⡤⠖⠛⠉⠉⣠⠟⠁⢠⡏⠀⠀⠀⠀⢀⣾⢻⡿⣿⡇⠀⠀⣿⣿⣿
⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⣠⡞⠁⠀⠀⣾⠀⠀⠀⠀⢀⣾⠃⣸⡇⣿⠀⠀⠀⢻⡏⣿
⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⣠⢾⡿⠀⠀⠀⢰⡇⠀⠀⠀⢀⣾⠃⠀⣿⢡⣿⠀⠀⠀⢸⡇⢿⡄
⠀⠀⢀⣾⡟⠀⠀⠀⢀⡼⠋⣼⠁⠀⠀⠀⣼⠁⠀⠀⣠⣿⠃⠀⠀⣿⢸⡇⠀⠀⠀⠘⣷⢸⡇
⠀⠀⣸⡿⠀⠀⠀⡠⠋⠀⢰⡇⠀⠀⠀⠀⠉⠀⢀⣴⡿⠃⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⢸⡇
⠀⢠⣿⠃⠀⢀⠌⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⢸⡇⣸⠃⠀⠀⠀⠀⣿⠘⣇
⠀⢸⡟⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⠀⢸⡆⣿
⠀⣿⠇⠀⠁⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿⠀⠀⠀⠀⠀⢸⡇⢻
⠀⣿⡀⣀⣀⣤⣤⣴⣶⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⠀⠀⠀⠀⠀⠈⣧⢸⡇
⠀⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠃⠀⠀⠀⠀⠀⠀⣿⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢿⡀
⠀⠀⠀⠀⣠⣴⠾⠛⠋⠉⠉⠙⠛⠻⢶⣤⣄⡀⠀⠀⣰⠏⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣧⠀⠀⠀⠀⢀⣠⣴⡶⠿⠟⠛⠛⠷⢶⣤⣀
⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⠏⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠸⣦⣴⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⣰⠏⣰⠟⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⠿⣧⠹⣇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⠏⣼⠋⠀⠀⠀⠈⠙⠻⣦⡀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠘⣧⠘⣧⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡼⢃⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⢷⡘⢷⡀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠶⠾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⣦⣤⣤⣤⣶⠶⠿⠛⠋⠁

⠖⡠⠀⢀⣴⣿⠁⠀⠀⠙⢻⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠉⢠⠄⢄⡀
⠀⢠⡖⠉⠈⠛⠂⢤⠀⣀⣀⡄⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠋⠉⣀⣀⣤⠈⠻⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⡀⠀⠀⠯
⠂⠠⠁⠀⠀⢠⣴⠞⡉⠉⠉⠛⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠈⠃⠀⠐⠆⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠑⡀⠁⢀⡀⠓
⠀⠀⠃⠤⠀⢈⠉⠀⠙⠀⠀⠀⢦⠐⠀⠀⠠⠄⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⠑⠲⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠐⠂⠀⠈
⠀⠸⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠐⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠀⠀⠐⠂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠒⠒⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⣀⠗⠂⠀⠀⠄⠀⢀⠜⠁⠀⠀⠀⢀⡄⢀⣤⣴⣦⠶⠀⠀⠀⠀⠀⠀⠀⠠⡀⡈
⢠⣶⠀⠤⠄⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⠻⣿⡧⡀⠀⢀⣤⠚⠋⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢤⣍
⠀⠀⠀⠤⠴⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⣠⣀⠘⢦⣀⣲⣶⣤⡤⠆⠈⠈⠻⠁⠉⠀⠋⡉⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻
⠀⣀⠀⠀⠀⠀⡁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠲⣛⠿⠭⠏⣹⠀⠀⠀⠀⠠⢆⠀⠀⠀⠀⢦⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠙⠛⠀⢠⣀⡈⠁⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⢴⠷⠀⠀⠀⢲⣄⠀⠠⠐⢱⣄⠀⠀⡀⢀⠘⠀⠄⠀⠈⠆⠀⠀⢀⠀⠁⠀⠄
⠐⠂⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢀⡃⠀⠀⢀⠀⠼⠟⠛⢦⡈⠁⠙⢖⠋⠀⠉⠀⢠⠀⠀⠀⠈⠀⠀⠀⢠⠀⠠
⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠤⠼⠟⠛⠒⢾⣄⣂⠀⠀⠀⢫⠂⢤⠸⠄⠀⡐⠂⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⡄⠀⠀⠐⠈⠛⠓⠀⢀⠀⠆⠀⠈⠾⠀⠩⠄⠘⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠤⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠠⠋⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢀⠀⠀⠀⠀⠀⢀⠒⠄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡠⠠⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠤
⣦⣀⡀⠁⠀⠀⠉⠢⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠿⠶⢦⣄⣦
⣿⣟⡤⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⢤⠀⠀⠀⠄⠊⣉⣳⣤⡀⢀
⠈⠀⠀⠈⠓⣶⣾⣿⣥⣔⡁⢤⣶⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣙⡆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠠⠀⠀⢀⠔⠋⠁
⢃⡀⠀⠀⠀⠸⠀⠓⠛⠫⡍⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⡤⠀⠉⠀⢾⠗⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⣌⠛⢦⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣘⣧⡙⢧⡀⠹⣾⣿⡄
⠀⠀⠀⠀⠀⠰⣶⣶⣶⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢡⠛⢷⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⢠⠃⠀⠀⡿⣷⣭⡓⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣸⠁⠀⠙⠿⢷⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣼⣴⣶⣶⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢻⣿⣜⣻⣿⣿⣿⣿⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⣿⣿⣿⡏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣌⢻⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⡛⢛⠲⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⠢⢖⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⠀⠀⣿⡻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡒⢺⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⢿⣿⣿⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢿⣧⣬⣼⣇⣸⠉⠛⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⣷⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢼⣿⣿⣿⣿⣿⣿⣿⡿⡼⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⡏⠀⠈⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡇⠀⡇⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠈

⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣿⣿⢿⣿⣿⣿⡿⠃⢀⣾⣿⣿⣿⣿⠇⠀⢶⣿⣿⣿⣶⣄⣼⠃⠸⡇⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣿⣿⡀⢻⣿⣿⠳⠶⣾⣿⣿⣿⠿⠿⠀⠀⠀⠙⠻⠿⢻⡿⣿⣦⣠⡄⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣿⣿⡗⣿⣿⣿⢀⣴⣿⣿⡝⣿⣿⣷⣤⣤⣤⣴⣿⣿⡟⠀⣿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠿⢻⡃⠹⢿⣿⣿⣿⣿⣿⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⡇⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣆⢀⣿⣿⣿⠿⠟⠁⠀⢀⣼⣿⡿⠋⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣶⣬⣿⣿⣾⣭⣭⡀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣿⣿⣿⣿⡟⠛⣿⣗⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⢿⣿⣾⠏⠙⢿⣿⣿⣿⣄⠀⠀⠈⠛⢿⣿⣿⣿⣯⡽⣿⡛⠋⠉⣿⡅⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⢀⡼⠻⣿⣿⣮⠉⠻⠏⣩⣶⣿⣿⣿⣷⠾⢿⣿⡁⢠⣿⣿⣆⣸⣿⡇⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣿⣷⣶⣬⠉⠁⠀⠀⠀⠉⠛⠛⢛⣿⣧⣤⣿⣿⣷⣾⣿⣿⣻⣿⣿⡇⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣿⣿⠟⠟⢹⣿⣿⣤⣶⡄⠀⣸⢿⣯⣽⣿⣿⡿⠹⣿⣿⡿⢛⣿⡟⠃⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠿⠋⢀⣴⣿⣿⣉⣿⣿⠃⢰⣿⣿⡛⠙⠋⠀⠀⢀⣽⣿⣧⣿⣿⠇⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣀⣰⣿⣿⡿⠻⣿⣿⠇⠀⠘⣿⣿⡇⠀⢀⣀⣰⣿⣿⣿⣿⣿⣭⣤⡆⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣥⣴⣿⣧⠀⢀⣶⣿⣿⢇⣾⣿⣿⣿⣿⣿⠟⢻⣿⣿⡿⠇⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⢻⣿⣿⣿⢯⣶⣶⣿⣯⣼⣾⣿⠿⠻⢿⣿⣿⣤⣼⣿⣿⣷⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣀⣉⣹⣿⣿⣿⣯⣠⣽⡿⠿⠿⢛⣿⣶⣄⣾⣿⣿⣿⣿⡿⢿⡿⠟⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⣿⠿⢫⣿⣿⣿⢿⣿⣿⣿⣴⣶⣾⣿⣿⣿⣿⠟⢉⣩⣍⠀⣤⣤⡀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⡁⠈⠉⠉⠉⠉⠉⠉⡉⠉⡉⢀⠀⠈⡉⠉⠀⡀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠘⡄⠀⠇⠀⢡⠀⠀⢠⠀⠀⠈⠀⡇⠀⡇⢨⠀⠀⡇⠀⠁⡅⡅⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠱⠀⠄⠀⠈⠄⠀⠘⠢⠀⠀⠀⠇⠀⠇⠨⠀⠀⠇⠰⠄⠅⠇⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣉⣉⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡘⣄⢀⠀⠀⠀⣿⣿⣾⢿⠿⡌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠢⠁⠀⠛⠀⣷⣾⣿⣿⣵⣓⣿⠷⡟
⠀⠀⠀⠀⠀⠀⠀⣀⠒⠤⠖⠁⠀⠀⠀⠀⠀⢹⣿⠟⠋⣩⣿⠿⠶⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⣴⡟⢁⠴⠋⢡⠄⢄⡀⠳⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⣸⡟⢀⡎⠁⠔⣩⠃⠄⢋⣀⣄⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢠⣿⠃⠈⠘⣿⣿⣿⡐⠾⣿⣿⣿⡷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡀⠀⠀⠀⣾⡏⢀⢠⣿⣿⠿⣛⢁⣤⡬⣅⣼⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⢶⣄⣰⣿⠃⣆⡘⣿⣥⣴⣌⣡⣴⣾⣿⣿⣿⣿⠟⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠐⢛⣿⣿⢧⡞⢙⣷⣿⣿⣿⡟⠡⠷⢶⣿⡟⣻⠟⣛⣭⣭⣭⡛⠛⠷⠶⢶⣄
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣴⣿⠟⢠⣿⣾⣿⣿⣍⠉⠙⠛⠶⣤⡀⠀⠁⠾⣿⣏⣿⡋⠁⠀⠐⠀⣠⢼⣷⠶⣦
⢠⡦⢤⣀⡀⣾⣿⣀⣀⣤⣾⣿⠏⢠⡿⠔⠉⠉⠉⠉⠀⠀⣠⣤⣤⣿⡄⠀⠀⠈⠻⣼⠃⠀⢠⢰⣤⣥⣾⣿⡀⠸⣧⠀⠀⣀⣀
⠈⣧⠀⠉⠙⢻⣟⣻⣻⣿⡟⠁⣰⣿⡖⣦⠀⠰⠷⠶⠶⣾⣿⣿⣿⡿⣿⡄⠀⠀⢀⡙⣷⣤⣴⣿⣿⢿⡟⠻⣇⠀⠈⢧⣼⡏⠈⠳⣆
⠀⠘⣇⠀⠀⠘⠿⣏⣽⠋⢠⣞⣡⡟⠟⠀⠀⡀⠀⠀⠀⣼⣿⣻⣭⣬⡉⣇⢰⡏⣉⢉⣿⣿⡅⠈⣻⣿⠛⢧⣹⣀⣤⣀⠉⠻⣦⣤⡿
⠀⠀⠙⠷⠷⣄⣀⠀⠉⢀⡉⣉⡀⠀⠈⠯⠁⠀⠀⠀⠀⠈⠙⣿⣦⠈⠻⡇⣼⢠⣿⣿⡏⢿⣿⡀⠿⣧⡄⠈⣿⣿⣿⣿⡆⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠈⢩⡿⢠⣿⠏⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⡇⢰⢧⡉⡈⣻⠋⢉⡼⠋⣷⠀⣿⣇⣤⣿⣿⣁⣿⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⣾⢡⣿⡟⢸⣿⣿⠀⢀⣾⠀⠀⠁⠀⢸⣦⣼⣧⣿⠀⡜⣦⣌⡩⠭⠿⠟⠓⠒⠒⠲⠿⠿⠿⣿⣿⣟⣁⠀⠀⠼⣇⡀
⠀⠀⠀⠀⠀⢀⣿⡿⢹⡇⣸⡏⠻⣆⠀⠿⠀⠀⠀⠀⠁⠀⣩⡿⢁⡾⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⡈⠙⠳⢶⣄⡀
⠀⠀⠀⠀⠀⢸⣿⠇⠸⣷⢸⡇⠀⢈⣷⠂⠴⠋⠋⣀⠠⠾⠋⢠⡾⢹⠃⣀⢀⣤⠆⠒⠐⠂⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠈⠛⢦⡀⡀
⠀⠀⠀⠀⠀⢸⡟⠀⠀⢻⣞⣷⣶⣿⠟⠀⠀⠀⠘⢁⠀⠀⢀⣿⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠙⠙⣧
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠙⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⣐⣤⣄⣀⣀⣤⠀⠀⣿⣿⠀⠀⠹⣦⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠈⠁⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠛⣿⠀⠀⢿⣸⡄⠀⠀⠈⠙⠳⠞⠋⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢻⣦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡖⢿⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⢀⠃⠀⠀⠀⠘⣿⣁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣩⢼⡇⠀⠈⡉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⣄⣢⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⠁⡞⢳⠆⡤⠙⠀⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠠⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣁⠁⢀⠀⢂⠀⠺⡗⢸⣦⠀⠀⠀⠂⠀⠀⠀⢘⢘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠉⢰⣄⠀⠈⢈⣄⠀⢺⡟⠦⣤⡀⠀⠀⠀⠰⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⣴⠋⠠⠀⢀⣄⣾⢀⡇⠀⠈⠙⠛⠲⠶⠶⠾⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠤⣤⠴⣸⣏⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣫⡿⣦⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡋⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡴⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿
⣿
⣿
⣿
⣿
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣭⣿⣶⣾⣿⣿⣿⣿⣯⡀⠹⢶⣶⣤⣀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠿⢿⠿⠿⠿⣿⣷⡆⠀⢈⠷⣿⣷⣦⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢤⣤⣾⣿⣷⣿⣿⣿⣿⡟⠿⢿⡟⣀⠈⠓⠂⠀⠀⠀⠴⠂⠀⠀⠀⠈⠙⠛⠦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢿⣿⣟⠃⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡿⣷⡄
⡇⠀⠀⠀⠀⢰⣶⣶⣿⢟⣿⣿⣿⠛⡾⠇⠁⠀⠀⠀⢁⣶⣦⠀⣠⣤⡀⠀⠀⢀⣀⣤⡄⢠⣾⣿⣿⣿⠉⣷
⣇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠘⠃⠀⠀⠀⠀⢀⣾⡿⠿⣾⣿⣿⡇⠀⠀⠘⠛⣽⠓⠟⢉⣿⣿⣷⣦⣿⡇
⡇⠀⠀⠀⢸⣿⡿⣷⣿⣿⣿⣿⣇⢉⡃⠀⠀⠀⣰⣿⣿⠀⠀⠙⠃⠙⠇⠀⠀⠀⢠⠈⠂⠀⠋⣹⣿⣿⣿⣿⣷
⡇⠀⠀⠚⠛⣣⣾⣿⣟⣿⣿⣿⣿⣿⢠⣶⣶⡾⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣴⣼⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⣿⡀
⡇⠀⠀⠀⢸⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠺⣄⣄⠀⠀⠀⠀⢶⣦⡀⢼⣿⣿⣿⣿⣿⡇⢸⣇
⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢳⣼⠇⠛⠷⢂⡴⠻⢿⣿⣿⣿⣿⢿⡟⠆⠈⣿
⡇⠀⠀⠀⠀⠘⢷⣿⣆⢻⣧⡀⠀⣤⣤⣤⣶⣤⡤⠀⠀⠀⠀⠀⢸⠿⡀⠚⢠⡾⢃⣶⢶⠙⣿⣟⡵⢋⠈⠀⠀⢹⡄
⡇⠀⠀⠀⠀⠀⠀⣿⣿⣀⣽⣷⣼⣿⣿⣯⠀⣠⣴⠄⠀⠀⠀⠀⠀⡸⠁⡶⣿⠃⢸⡟⠀⢠⣿⣿⡷⠀⠀⠀⢾⣿⡇
⡇⠀⠀⠀⠀⠀⢠⣿⣿⠛⢹⡏⠀⠨⢭⡽⠏⠉⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣿⠀⣰⠇⣠⣿⣿⣿⣿⠀⠀⠀⠻⢹⣧
⡇⠀⠀⠀⠀⠀⢸⣿⠏⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠀⣿⢀⣉⠘⢡⡿⣿⠟⣿⣷⠀⠀⠀⠀⢿⡄
⡇⠀⠀⠀⠀⢠⣿⠇⣤⠸⣧⣤⣤⡴⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⣿⣯⠀⣾⣿⡏⠀⠛⢻⡆⠀⠀⠁⡼⣿⣄
⡇⠀⠀⠀⠀⣼⡟⠻⠄⠛⣤⣿⣿⣇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠘⣿⠀⡼⢡⣿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢿⣦⡀
⡇⠀⠀⠀⠀⣿⣸⡆⠀⠀⠙⢿⡏⠙⣿⣉⡀⠉⠁⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠉⢀⠃⣼⡿⠀⠀⠀⣴⣿⠀⠀⠀⠈⣀⠀⠛⣷⡀
⡇⠀⠀⠀⠀⢻⡿⣧⠠⠃⠀⠀⠙⢄⠈⠻⣏⠀⠀⠀⢀⣀⣤⣄⠀⠀⠈⠀⠳⣄⣿⡀⣿⠁⠀⠀⠀⠹⣽⢦⡀⠀⠐⠈⠀⣠⠈⢷⡄
⡇⠀⠀⠀⠀⠈⠻⢽⣷⣄⠀⠀⠀⠀⠰⢄⡙⠳⠶⠞⠛⠋⠉⠛⣷⠀⠀⠀⠀⠈⢿⣧⣿⠄⠀⠂⠀⠀⢻⣿⣷⢂⡰⠠⣾⡴⠀⠈⣷
⡇⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⣤⡁⠀⠀⢸⣿⡄⣾⠃⣠⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⢿⡿⠐⠀⠀⠀⠀⠈⢿⣿⣿⡴⣷⢋⣴⣶⠶⢿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⢷⣀⣄⢸⣿⣿⣿⢐⡆⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⣸⡷⠀⠐⢀⠈⠀⠀⠈⠿⣿⣷⣾⣿⣷⠿⠃⣾
⡇⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠁⢸⣿⢽⣿⡿⢿⣏⣿⠃⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡟⠀⢠⠄⠻⣷⣶⡄⢁⡆⠸⣿⣿⣿⣫⣶⠀⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣋⣼⡟⠁⠈⣿⣧⣤⣤⣴⠆⣾⠁⠀⣀⡀⠀⠶⠒⠀⠀⠀⠀⠣⢿⣼⣉⡉⠀⣨⡾⣻⣿⠟⣰⠏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠈⠻⠇⢘⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣯⢞⣽⣿⡟⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⢿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣴⡿⢷⡔⠢⣀⢀⡀⠀⢀⡀⠤⡤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣈⣿⣌⡓⣸⢇⡸⢿⡄⣹⣶⣯⠙⠖⣴⣆⣡⣤⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋
⡇
⡇
⣿
⣿
⣿
⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⢸⡟⢹⣿⣿⣄⠀⠀⢀⣼⣿⣿⣷⡙⣧⣀⣀⣀⣀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡓⢤⣄⣀⣴⣿⡿⠾⠿⠿⠏⠙⠚⠁⠿⠿⣿⣿⣟⢻⣷⣭⣿⣯⣳⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⣿⣦⣤⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⣯⣝⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠹⣷⣄⣉⣽⣿⠇⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣄⠀⠀⠸⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠈⠛⠿⣿⠀⠀⠀⠀⠀⢔⣢⡀⠀⠀⠀⠀⣿⠀⠀⠠⠷⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢤⣤⡤⠟⠁⠲⢶⣾⡿⠛⣧⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⣽⠋⠆⠀⠀⡀⠀⢀⠀⣸⡇⠀⢻⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣾⠁⠀⠈⠉⠀⠈⠀⠀⢠⣿⡇⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡰⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⢧⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣰⠏⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠘⣷⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⠃⠠⠃⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠈⣿⣷⠀⠸⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠃⠀⠤⡀⠀⣀⡀⠀⠀⠀⠀⠀⣀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣦⠀⢀⠔⠈⠀⠈⣿⣿⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⡄⠀⠀⠀⠀⠀⢹⣿⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠖⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⢸⣀⠆⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡒⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠁⠂⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢸⠇

⠀⠀⠀⠀⠀⠀⣤⠤⠤⠴⠶⠖⠒⠒⠒⠒⠒⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣠⣀⣀⣀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⢀⣀⣀⣀⣀⣉⣉⠙⠲⢤⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣤⠒⠊⠉⠉⠁⠀⠀⠀⠀⠉⠉⠛⢾⠙⢦⡀⠀⣠⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠒⠒⠒⠒⠒⠢⠤⣧⡀⣷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⠀⠀⣀⣀⠤⢚⣭⡿⢋⢮⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⣠⡾⠋⢀⢎⠃⣿⣇⠀⣠⡤⠖⠒⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⡜⠎⠀⠙⡟⣿⠧⢤⡄⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⣸⠱⠀⠀⠠⢓⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⡿⠁⠀⠀⢠⡇⠁⠀⠀⠀⠚⡄⠀⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⢠⣿⣿⠛⢦⣄⠀⣸⣿⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⠙⠋⠀⠀⠘⣷⣿⣿⠀⠀⠀⠀⠀⢻⣣⡴⠶⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠹⡏⠉⠛⠶⣤⡴⠋⠹⣧⡀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠴⣷⡄⠀⠀⠈⢳⡀⠀⠙⠟⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋⠉⠓⠒⠒⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⠤⠤⠖⠚⠉


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡴⢦⣠⠞⢆⣠⠶⡄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡏⠲⡎⠀⠙⢧⡰⠃⠀⠈⠳⣠⠞⠁⠈⡦⠚⢳⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⢤⣼⠈⠉⠓⠼⣀⡤⠴⠖⠒⠚⠛⠛⠒⠒⠶⠤⣄⡹⠔⠚⠉⢸⣀⣠⡄
⠀⠀⠀⠀⠀⣤⠄⠸⣀⡀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⡁⣀⡇⢤⡄
⠀⠀⠀⢠⡀⢹⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⢰⡁⣀⡀
⠀⠀⠀⠈⣆⣀⠆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢄⢀⡜
⠀⠀⠫⠀⠀⠀⡔⠀⢀⣠⠤⠤⠶⠦⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠤⠤⢤⣀⠀⠘⡄⠀⠀⠨⠃
⠀⢶⠊⠑⡀⡘⠀⠚⠉⢀⣀⣤⠤⢤⣀⡀⠀⠉⢤⠀⠀⠀⢀⠌⠁⠀⣀⣠⠤⠤⣄⣀⠈⠙⠂⠘⡀⡐⠁⢒⡆
⢀⡤⠓⠈⢠⠃⠀⠀⣚⡭⢤⢶⡖⢶⢦⢭⣁⠀⠈⠆⠀⠀⠆⠀⢀⣩⢥⢶⠖⣶⢦⠬⣝⡂⠀⠀⢡⠈⠐⠪⣄
⠈⢲⣄⠀⢸⠀⠀⠈⠩⣒⠬⢧⣭⣥⡮⠤⠚⠑⠀⠸⠀⠸⠀⠐⠛⠢⢬⣦⣭⣥⠯⢔⡪⠉⠀⠀⢸⠀⢀⣴⠋
⠺⣅⣈⡆⢀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠋⠁⠀⠀⢨⠀⢰⠀⠀⠀⠉⠓⠒⠒⠒⠚⠉⠀⠀⠀⠀⢀⠐⢧⣀⡽
⢠⠞⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠈⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢦
⠈⢙⠶⠤⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠑⠒⠛⠦⠞⠓⠒⠁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡠⠴⣎⠁
⠀⠓⠢⡼⠁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣤⣖⡺⢦⠾⣒⣦⣀⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠹⣤⠜⠃
⠀⠀⢾⣀⣀⣀⠱⡀⠀⠀⠀⠀⠀⠠⠞⠁⢻⡉⠀⣉⣓⣋⡁⠈⣹⠃⠙⠂⠀⠀⠀⠀⠀⠀⡰⢃⣀⣀⣨⠄
⠀⠀⠀⢰⣃⣈⡇⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢉⠀⢃
⠀⠀⠀⠀⠀⠸⠀⢀⣀⡑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡁⠀⠀⢸⡍⠈
⠀⠀⠀⠀⠀⠉⠉⡏⣀⣼⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⢸⡀⢸⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⣀⢤⠖⠻⡈⠉⠛⠒⡶⠶⠦⠶⠶⡒⠚⠋⠉⡸⠓⣦⢄⣸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠶⠚⢧⢀⣴⡊⢙⣆⠀⢀⡞⠉⡷⣄⢠⠟⠦⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠳⠊⠈⠳⠏⠈⠾⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠷⠶⠶⠶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣏⠉⠁⣽⣿⣿⣦
⠀⠀⠀⠀⣶⡶⠿⠛⢻⣇⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣤⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠹⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣴⣾⣯⠈⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣋⣀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣅⡀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢀⣾⣿⠿⠿⠅⠿⠿⣿⣦⡀⠀⠀⠉⠉⠉
⠀⠀⠀⢸⣿⣿⣷⣦⣤⣤⣾⣿⣿⣷
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠾⠛⠛⠿⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⢀⡄⠀⠀⠀⢰⠃⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠘⣁⣀⣤⣤⡤⠤⠤⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⣠⣶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⢻⣿⣿⠷⣶⣄⡀⢶⡄⠀⠀⢶⠀⠀⠀⠀⠈⠙⠿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⢀⣤⠀⠘⡿⠁⠀⠈⠙⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⢀⣴⠟⣿⠀⢸⡇⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣤⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣠⣿⠃⠀⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣰⡿⠁⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⣰⡿⠁⠀⠀⠀⣿⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢠⡿⠁⠀⠀⠀⠀⢿⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣾⠃⠀⠀⠀⠀⠀⢸⡀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢰⡟⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣿⣄⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢹⣧⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢿⣄⠀⠘⢿⣦⣴⠿⠛⠛⠛⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⣀⣤⣶⠶⠶⠾⢿⠀⢸⣿⠀⠀⠀⠉⠙⠷⣄⠀⠀⠀⠀⢻⣆⠀⠀⠹⣧⣀⡴⠂⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⢀⣴⡾⠋⠁⢀⣀⠀⠀⠸⡆⠀⢿⣦⡤⠶⠄⠀⠀⠘⢷⡀⢠⡾⠛⠙⣧⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⣠⡿⠋⠀⠀⠀⠈⠉⠛⠛⠿⠿⣤⡾⠛⠀⠀⠀⠀⠀⠀⠈⣷⡏⠀⠀⠀⠼⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠙⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠋
⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠟⠛⠋⠁
⣀⣀⣀⣀⣀⣈⣙⣛⣿⣿⣷⣶⣶⣾⣿⣟⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡰⠢⣀⠜⢣⣀⡔⢦⠀⣠⡄⠀⠀⠀⠄⠀⠀⠠⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⠛⢢⡎⠀⠙⢧⣠⠏⠀⠀⠹⣄⡼⠋⠀⢹⡴⠚⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⡏⠉⠛⠦⠼⢁⣠⡤⠴⠷⠒⠒⠒⠒⠾⠦⢤⣄⡈⠧⠴⠛⠉⢹⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣇⠀⡇⠀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⠀⢸⠉⢠⣇⣀⡀
⠀⠀⠀⠀⠀⣀⣘⡇⠀⠉⢉⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡉⠉⠀⢰⠃
⠀⠀⠀⠀⠀⢯⠉⢱⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡎⠉⣹⠃
⠀⠀⠀⢲⡚⠋⠉⢉⡞⠁⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠈⢳⡉⠉⠙⢓⡖
⠀⠀⣠⠤⠷⡄⢀⡞⠠⠴⠚⠉⠁⢀⣀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⣀⠖⠋⠉⠀⢀⣀⠀⠈⠉⠓⠦⠄⢳⡀⢠⠾⢤⣀
⠀⠀⠈⣣⡤⠟⣼⠁⠀⠀⠴⢚⣉⣉⣉⣉⣙⠲⠄⠀⠘⣆⠀⠀⣰⠃⠀⠠⢖⣋⣉⣉⣉⣉⡓⠦⠀⠀⠀⣷⠻⠤⣴⠋
⠀⠐⢏⠁⠀⢠⡇⠀⠀⠖⢯⣉⢹⡘⠦⢏⡼⢉⣳⢦⡀⢸⡀⢀⡇⠀⡴⣞⡉⢯⡹⠴⢃⡏⣉⡽⠲⠀⠀⢸⡀⠀⢀⡽⠃
⠀⣠⡔⠛⢦⣸⡇⠀⠀⠀⠙⠲⢭⣙⣛⣛⣩⠭⠄⠀⠀⠈⡇⢸⠁⠀⠀⠰⠭⣍⣛⣛⣋⡭⠔⠋⠀⠀⠀⢸⣇⡴⠛⠦⣄
⠀⠈⢙⡶⠋⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢶⡊⠁
⠀⠰⢯⣀⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢾⣑⣣⡀⢀⣜⣊⡷⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣠⠽⠂
⠀⠀⣰⡋⠉⡷⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢁⣀⡍⢩⣄⡈⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢾⠉⠱⣄
⠀⠀⠀⢉⡟⠁⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣦⡾⠭⢭⣛⣋⡭⠭⢷⣴⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⢻⡉⠁
⠀⠀⠀⠚⠒⣦⠤⢤⠳⡄⠀⠀⠀⠀⠀⠈⠁⠀⠸⢤⣠⠤⠶⠶⠤⣄⣤⠇⠀⠈⠁⠀⠀⠀⠀⠀⢠⠞⡤⠤⡴⠒⠓
⠀⠀⠀⠀⠰⠧⠤⡾⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢧⣀⣽
⠀⠀⠀⠀⠀⠀⢰⣃⣀⡤⠤⡝⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢫⠤⣄⣀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡤⡇⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠋⠁⠀⢸⣄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⢶⠋⢹⡄⠈⠉⢙⣟⠒⠒⠒⠒⣻⡋⠉⠁⢠⡏⠙⡶⢤⣼⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠋⢧⣀⠴⡏⢈⣧⡀⠀⣼⠁⣹⠦⣀⡜⠑⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⣄⠀⠀⠀⢀⡤⠤⠀⠢⢄⡀⠀⠀⠀⠹⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⠀⠀⠀⠀⠀⡿⡄⠀⠈⠛⠦⡀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢸⣿⣿⣆⠀⠈⠢⠀⠀⠀⢀⢴⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠙⢦⡀⠀⠀⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⢀⡿⢿⣿⠀⠀⠨⠀⠐⠊⠵⢶⣎⣀⠀⡠
⠀⠀⠀⠀⠀⠤⠔⠂⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠉⠑⠒⠦⣄⠀⠄⠔⠊⠉⠀⠀⠀⡰⢐⡿⠃⠀⠙⠀⠔⠊⠁⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢧⠀⠘⢦⡀⠀⠀⠀⠈⢳⣄⡀⠀⠀⢀⣠⠜⠁⠉⣠⠴⠒⢢
⠀⠀⠀⢸⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠽⠖⠀⢸⣿⠟⡋⣿⠏⠉⠀⢀⠔⠋⠀⠀⠀⠸⣦⣄
⠀⠀⠀⠀⠓⣄⠀⡆⠀⠀⡆⠀⠀⠀⢸⣷⣤⣀⠀⠀⠀⢀⣀⣑⢹⡇⠘⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠒⠂⢀⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠉⠁⢀⠀⠘⢄⠀⠀⠸⣿⣿⠉⠛⠳⠊⠁⠀⠈⠉⠻⡄⢠⡟⣴⣶⡄⠉⠑⠢⢄⣠⠘⢿⣧⠀⠀⠀⠈⠧⢀⠄
⠀⠀⠀⠀⠒⢀⡠⠖⠃⠀⠀⠀⠀⠀⠲⢄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠻⣿⣿⡿⠛⠶⠋⠁⠀⠀⠉⠑⠀⠀⠸⠀⠁
⠀⠀⠀⢧⡀⢎⠀⠀⠀⠀⠀⠀⡠⠂⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡠⠤⠤⠤⣄
⠀⠀⠀⠀⠉⠛⠛⠒⠀⠀⢀⡬⠖⠋⠉⠉⠀⢧⠀⠀⠀⠀⠀⠀⣀⠐⠊⠁⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⡀⠀⠠⠋⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠁⠀⢠⣾⢆⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⣦⣶⣷⣷⣯⠍⠀⠤⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⣀⠛⢀⣨⢿⣿⡿⡏⠀⠀⠀⠢⠤⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠀⢠⠟⠁⠴⠆⡀⠀⠐⠷⠀⢘⠂⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⡎⠀⠀⠁⠘⢰⠃⠡⠈⠁⠀⠀⠈⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠳⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠵⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⢛⣁⣀⣤⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣥⣶⣶⡾⠿⠟⢻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠉⠀⠀⠀⡠⢾⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡇⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡼⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢠⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⡸⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⣠⣤⣿⠤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣼⣿⠻⣿⣿⡋⣿⡷⣤⡀⠀⠀⠀⠀⠀⢰⠇⣠⣾⣿⣿⣛⠟⣻⣾⠟⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢮⣿⣿⣿⣀⡙⢿⣿⣶⣿⣿⣿⣆⠀⠀⠀⠀⢸⠐⣼⣿⡈⡛⡿⣿⣿⣿⣿⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⣤⣤⣄⡾⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢫⣿⡿⣿⣿⣭⣭⣭⣭⣿⣾⡝⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⡸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣤⣤⣼⣿⣿⣿⣷⣿⣿⣷⠘⣿⣿⣿⣋⣩⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⡟⠻⣿⣿⣿⡿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢛⢿⣿⣿⣿⣿⣿⣿⡿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢻⣟⡿⣻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⠀⠐⠂⠄⡀⠀⠀⠐⠂⠄
⡿⠓⢠⡀⠀⠰⢀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡊
⢩⠀⠀⢷⣄⡀⠈⠁⣠⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⡀
⠀⢈⣠⠌⠙⠛⠻⠦⠄⡀⠚⠀⠀⠀⣤⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠉⠁
⠆⠠⠀⠀⠀⠀⠁⠀⣀⣅⠀⠈⠃⠀⠀⡤⠀⠙⠲⠄⠀⢀⠀⠀⠀⠀⠀⠄⠀⡀
⠀⠀⠁⢰⡆⠀⠀⠰⠏⢻⠿⠄⠀⠺⣄⠀⠀⠀⡀⠘⠃⠛⠶⠆⠀⠰⠎⠛⠂⠘⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢃⠀⠛⢀⣨⠡⠄⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠐⠀⠈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡄⠀⠀⣀⡤⠉⠀⠋⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠒⠛⠁⠀⠀⠀⠹⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⢀⡇⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⡄⠀⠀⠀⠀⠀⠈⠉⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠠⡀⠀⢀⠀⠢⡀⠀⠀⠙⣦⡀⠀⣠⡾⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠀⠀⠂⠉⠀⠉⠀⠀⠀⠔⠀⠠⠀⠁⢈⣟⡻⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢞⣿⣿⢧⣀⣴⣏⡷⠶⠶⠶⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠦⠴⠶⠶⠚⠛⢉⣴⢟⠝⠁⡼⣹⠃⢹⡇⠀⠠⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡕⠁⠀⢠⢇⡟⠀⠸⡇⠀⠀⠀⠠⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡵⠋⠀⠀⠀⣼⢸⠇⠀⠀⣿⠀⠀⠀⠀⠐⣀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢣⠞⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣱⠋⠀⠀⠀⠀⠀⢸⢣⡏⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡰⠃⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠸⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣸⠁⠀⠀⠀⠀⠀⠀⠀⡼⢸⡇⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⠷⠶⠶⠶⠤⣄⠀⠀⠀⠀⠀⠘⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⢻⠁⣸⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⣧⠘⣇⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠉⠀⠀⠾⠤⠛⠀⠀⠐⠓⠀⠀⢀⡼⠖⠚⠛⠛⢻⡀⢻⠾⠋⠉⠈⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢽⡷⠋⠀⠀⠀⠀⠀⠀⣧⣨⡦⠂⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣽⡆⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⢠⣲⡄⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠀⠀⠉⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢤⣤⣤⣤⣤⠶⠞⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⠀⠀⢀⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⡖⠁⠙⢦⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡜⠃⠀⣰⡀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡴⠏⠱⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⣨⣵⢦⣀⢠⣿⠂⠸⡇⣸⡄⢸⠀⠸⣷⡄⢠⢶⡍⡔⠀⠀⠰⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣁⣴⢦⡀⠀⠹⠦⣄⢹⡎⠀⣴⠼⡇⠹⠉⠺⠵⣃⠈⢷⡇⡠⠾⠃⠀⢀⠴⣄⣈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣈⣁⡤⢾⡆⠒⠚⠻⢿⡧⢜⡙⠢⣰⡶⢶⡁⠘⢛⡤⢼⠾⠛⠒⠒⢰⡧⡄⣨⢁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣭⣥⢿⠀⢻⡄⠀⣄⠑⢮⣓⢀⡿⢳⡞⢇⢀⣫⡴⠃⣰⠀⣴⠏⠀⣷⣬⣭⣯⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡒⠦⠝⠿⢦⣼⡟⠏⢸⠀⠈⠀⢀⠉⣻⣴⣿⣷⣼⡟⠉⡐⠈⠆⠰⠀⡕⢻⣦⠴⡟⠫⠶⡲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣰⠀⢤⣦⣄⡈⠻⢷⣼⠆⠀⣴⠖⠹⠅⣿⡿⢻⡟⣿⣟⠸⠝⠛⠆⠀⢱⣧⠿⠅⢁⣤⣴⡄⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠐⠳⣁⠔⢐⠏⢳⠀⠠⢻⠦⠔⠺⢂⣀⡼⢿⣟⣈⣠⣻⢿⢆⣀⠲⠛⠤⢴⡏⠀⢠⡏⢸⡈⠀⣪⠟⠂⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠂⠀⡿⠛⠣⡀⣸⠂⣾⠉⠀⣤⣄⠀⠋⠀⠠⠈⢿⡟⠁⠔⠀⠃⠀⣧⡄⠀⠉⣆⠂⡅⢠⠟⠛⣏⠀⢲⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⡈⠛⢿⣧⡘⢖⢡⣛⣸⠉⠀⠠⣿⣿⣦⡴⣿⠷⣦⣜⣠⡴⠿⡷⢤⣴⣻⣿⠄⠀⢙⣠⣳⡈⡾⢁⣞⠷⠋⢀⣼⡀
⠀⠀⠀⠀⠀⠀⢀⣬⡀⠙⢧⣦⡀⠈⠻⣾⡏⣼⡟⡃⠀⠀⢻⣿⣿⠐⡯⠀⠀⠻⠏⠀⠀⣿⢀⣿⣾⡟⠀⠀⢹⣻⡆⢿⣶⠟⠁⣀⣴⡿⠋⣨⣅
⠀⠀⠀⠀⠀⠀⣨⣀⠉⢲⠜⠋⠉⠢⡀⢻⠇⠸⣧⠙⡆⠀⠀⠉⠹⢆⣗⠂⢰⡏⢻⡄⢀⣿⡸⠋⠉⠀⠀⢺⢃⡾⠃⢸⡏⣠⠞⠉⠙⢦⠚⠁⣀⣇
⠀⠀⠀⠀⠀⢀⣀⡈⢙⡎⠰⣣⠀⠀⠙⣾⠇⠀⠙⣷⡟⠓⢦⣴⡤⠞⢻⣦⠌⢳⡟⢀⣾⠟⠣⢤⣤⡴⠚⣿⡟⠁⠀⢸⣷⠁⠀⢀⡛⡀⣷⠋⣁⣀⡁
⠀⠀⠀⠀⠀⢁⣀⣉⣙⠁⢸⢹⣗⡠⡀⢹⡅⠀⠀⠈⣧⠀⠂⢷⣄⣠⣾⡟⢷⡈⢀⡾⣻⣷⣀⣴⠟⠐⢠⡟⠀⠀⠀⣹⠃⠠⣐⣽⡏⡇⢈⣋⣁⣀⡈
⠀⠀⠀⠀⠀⠀⠬⣥⣤⠤⠸⡘⣯⢻⠿⣻⣧⣦⡄⠐⠘⣦⡀⠈⠙⣿⡃⠀⠊⢻⠟⠀⠀⣛⡟⠏⠀⣀⡾⠁⠀⠀⢀⣽⣿⡿⡻⣿⣰⢁⣿⣉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡴⠦⡘⣝⢮⠛⣿⣿⠀⣿⠀⠀⠀⠹⢦⣔⣧⣿⡈⠂⡠⡀⠲⣀⣧⣿⣠⠶⠋⠀⠀⠀⢠⠞⠐⠛⣟⡵⡳⢡⠞⠦⣄⠀⢙
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠸⡷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⣶⢀⠰⣶⡄⣿⣿⣏⣴⣶⠀⣤⠖⢲⣶⡆⢰⣶⡠⢶⡶⢺⣿⡗⢲⣶⡄⠐⣶⣦⠀⣠⡶⠒⣶⣦⢰⣶⡀⠉⠀⢠⣴
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⠃⠀⣿⡇⣿⣿⢷⣌⠉⢸⣿⣀⣸⣿⡇⢸⣿⠁⠀⠀⢸⣿⡇⢸⣿⡟⠀⣿⣿⢸⣿⣀⣠⣿⣿⢸⣿⡇⢀⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠛⠀⠀⠙⠃⠻⣿⠀⠙⢷⡜⠻⠟⠁⠙⠃⠈⠛⠀⠀⠀⢸⣿⡇⠈⠛⠃⠀⠘⠛⠈⠻⠟⠁⠘⠛⠈⠻⠿⠋⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⣠⡀⢀⣀⡀⣄⠀⣀⣀⣀⣀⣄⣄⠀⡀⢀⠀⣤⡄⠈⠙⠃⣀⣄⣄⣀⣀⣄⣀⠀⡄⢀⢠⡀⢠⡄⢀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠳⣙⣿⢶⠞⢡⠇⠀⣿⠿⠛⡷⠟⠻⣷⢆⢠⠀⠘⠃⠀⠄⣱⣻⠞⡳⣿⠛⠟⡧⠀⢳⡘⢶⢷⡿⡉⢊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠷⠞⠁⠀⠀⠈⠓⠚⠡⣸⡇⠈⣿⣽⡐⢠⠠⣣⣿⡾⠀⢸⢁⠜⠓⠚⠁⠀⠀⠙⠲⠷⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢋⡼⠿⣴⡷⢿⣦⠙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠅⢾⠙⠀⢹⡇⠀⢟⡆⢩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠄⢾⡅⠀⠳⠞⠀⢬⡧⠨⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣲⡈⢳⡥⠸⢂⢬⡟⢡⡚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣭⡄⠛⣾⣿⠋⣠⣽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠶⠾⣧⣼⢵⣶⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠬⠀⠁⠈⣰⣆⠀⠀⠀⠀⣀⢡⡀⠀⠀⢠⣆⡿⣮⣉⣁⠀⠸⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⣶⠤⠶⠈⠛⠛⢃⣀⠀⠀⠁⠀⠈⠢⡀⠈⠀⠀⠛⠤⠉⢛⠀⣴⡦⠀⠀⠀⡀⠀⠀⢀⣶⠌⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⡾⢻⠏⣉⡉⠲⢢⣤⣤⢦⣄⣀⠤⠤⣤⢾⣶⣤⣤⣶⣦⣄⣹⣄⣀⡈⠁⠀⢈⣶
⠀⠀⠀⠀⠀⠀⠀⢀⡈⠭⠙⠃⠉⠉⢉⡝⠋⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⣀⠰⠤⠌⠿⣿⡄⠛⠛⠲⢯⡁
⠀⠀⠄⠒⠒⠚⠉⠁⠀⠀⠠⠀⣀⣀⠈⠆⠀⠀⢺⣴⠀⠀⡤⡄⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠉⠑⣤⣤⣈⣷⠤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣥⠀⠹⡿⠷⠀⠀⠄⠀⢁⣀⡠⠴⢁⣤⡀⡀⠸⣿⣦⠡⠤⢐⡶⠖⠚⠉⠉⡉⣉⡥⠲⠦⣄⡉⠢⠀⠀⢠⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠆⣧⠀⠀⠀⠀⠿⠂⠀⣠⣄⠀⠠⠯⢁⠘⠆⢀⠠⠖⡋⢀⡀⠄⠒⠾⠉⠀⠀⠀⠀⠀⠉⣳⣦⣀⠰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⡀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠈⠁⡠⠘⠛⠋⠀⢠⣜⠂⠀⠀⠀⠀⣠⡴⠛⠉⠀⠉⠓⠦⣄⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠤⠀⢠⣦⠦⢀⡀⠀⠠⠀⠀⣀⠈⡿⣿⠀⠀⠀⠀⠀⠀⣰⡌⠉⣡⣔⡄⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠀⢀⢴⡤⢉⡀⢨⠿⢧⠀⠀⠀⠙⠃⠀⠀⢀⠀⢀⣠⣴⠞⣉⢣⡀⠛⠋⠂⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠖⡀⠂⠸⡎⠀⠀⠤⠘⠂⠀⠀⠀⠀⠉⠀⠀⡠⢼⣯⡵⠋⠀⠐⠛⢀⡇⣾⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠙⠀⠀⣀⠀⠀⡄⠊⠉⠀⠜⠫⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡾⠆⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⠀⠂⠀⠰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠐⡿⠀⠀⠀⠀⠈⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷


⠀⡀
⠀⠆
⠀⠁

⠀⠄
⠀⠄
⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⣄⡀⠀⠀⠀⣤⠾⣫⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⡨⠵⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠥⠀⡇⠘⣆⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⡆⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⣿⡿⣿⣿⡿⣿⣷⡤⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⡿⣭⣶⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢿⣿⣿⣿⣿⣿⣼⡿⠛⣻⣷⣽⣯⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣶⣭⡻⣿⣇⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⡿⣼⡇
⠀⠀⠀⠀⠀⠀⢠⣿⡟⢛⣿⣮⣿⣴⣿⣿⣿⣿⣧⠛⠛⠸⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⢿⣧⣤⣿⣿⣿⣿⣿⣿⡟⣿⠇⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣯⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⣶⣦⣾⣿⡿⠋⢻⣝⣿⣿⡿⢳⣿⣿⣿⣿⣿⡿⠟⠉⠀⠈⠀⠀⠘⠂⠙⠋⣠⣿⣿⣶⣶⡴⠠⠸⠃⠹⠠⠌⢀⠀⠀⠀⠀⠀⠀⠁
⣀⡁⠙⠛⡗⣼⠷⣿⡏⠀⢀⢸⣿⣿⣟⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠿⠘⠇⠀⢸⠀⡀⠈⠀⠀⠀⠔⢤⡈⣢⠀⡄
⠀⠀⠀⠁⡆⠙⠛⠛⢱⠄⠀⢀⠙⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠉⠉⠀⠀⠀⠘⠂⠀⠐⠀⠀⠀⠀⠀⢀⠘⢋⠁⠀⠉⠙⢳
⣤⣀⠤⠄⡧⠀⠀⠀⢀⠀⠤⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠎⢠⠄
⠿⠧⠤⡃⠂⠈⠀⠐⠂⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢘⠠⡭⠄
⠀⠚⠒⠀⠃⠀⠀⢠⠴⠶⢈⠩⠀⠀⡈⠀⠀⢠⠔⠐⣀⣠⣘⢁⡄⠀⠀⠀⠓⠉⠀⠀⠀⠀⠀⠀⠐⠂⠘⠁⠀⠀⣾⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠳⣤⣿⠋⠉⢹⡄⠂⠀⢀⡠⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡸⡆⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠐⢄⠀⠀⠀⠠⠄⠁⠀⠀⠚⠃⢈⣥⣤⠖⡀⠀⡐⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⡇
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢂⣀⠀⠀⠉⠘⠂⠈⠠⣠⠛⠁⠀⠡⠀⠁⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⡠⠞⠉⠀⠀⠀⠀⠡⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⠄⡀⣠⠀⠀⠈⢍⠁⠠⠀⢢⠄⢀⡈⢄⠀⠈⠀⠀⠀⣠⠊⠀⠀⠀⠀⠈⠂⡀⢰⡭⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⠃⠀⢩⢆⠠⠀⠀⣄⠐⠂⠀⠀⠤⠉⠀⡀⠀⣼⠁⠀⠀⠀⠀⢠⡞⠒⢷⡨⣯⣒⣒⠢⠌⠁
⠄⢀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⡂⢨⣡⠴⠥⢢⢞⣡⣚⣛⡃⡛⡀⠀⠹⣄⠀⢠⣤⣴⣿⣷⡄⠠⠈⠒⠠⠤⠐
⠓⠋⠉⡒⠦⣄⠀⠀⠀⠀⠉⠁⠀⠀⠈⠓⢄⡀⡖⠂⠘⠁⣀⣠⣦⣤⣾⣿⣿⣿⠁⢁⡀⠀⠙⠶⣿⣿⣿⡿⣝⣀⠈⠀⠀⠀⠀⠓
⠀⠔⠈⠀⠆⡈⠳⡀⠊⠀⣀⣤⡴⠤⢀⡀⠀⠙⠢⣾⣾⣿⣿⣿⣿⡿⠿⡿⡟⠹⠐⠈⠟⠀⠀⠀⠀⠈⠀⠀⠚⠠
⠀⠀⠀⠄⠈⠠⠀⢸⣦⣶⡟⠁⠀⠀⠀⠑⡄⠀⠱⡙⡛⠻⠟⠙⠠⡆⡸⠚⠀⠀⠀⠀⠀⢀⣀⣈⠑⢒⣠⠶⠋⠀⣄
⠀⠀⠀⠀⠀⠀⠐⠀⡇⠈⠁⠀⠀⠀⠀⠀⢳⠀⠀⠃⠈⠀⠀⠀⠀⠃⠀⡀⠃⠀⠀⠀⠀⠤⣄⣠⣶⣿⣿⣦⠲⠄⠠⠄
⠀⠐⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡀⠀⣠⡿⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠁⠐⡎⠛⠋
⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠊⢘⣵⣾⡿⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⣡⣤⣤⣄⣀⡤⠀⠀⠠⠇⠘⠿⠃
⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⢢⡀⢿⠛⠻⠕⠀⡴⢀⣁⢠⡴⢾⣶⣶⠒⣉⣁⣀⠀⣀⣙⣛⡻⣿⡿⠃⠀⠴⣰⠀⢆⡦⠠⣴⡆⠀⢆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⡀⠱⣞⣀⣀⣴⢺⣷⡌⣿⡿⠃⣸⣿⣿⠀⣿⡿⣿⡇⣿⡟⣛⢣⣯⣀⠀⠀⠀⣠⢤⠀⡄⣤⡀
⢀⠞⠀⠀⠀⠀⠀⠀⠀⢈⠄⠂⡌⠀⢸⣿⣈⣉⡁⣿⣷⡄⣿⣿⣿⡄⣿⣷⣿⠇⣿⡿⣿⣧⠉⠉⠱⡀⠀⠁⠉⠈⠈⠉⠀⠀⡀
⠃⠀⠀⠀⠀⠠⣀⠀⠀⠄⠚⡁⠀⠀⢸⣿⢻⣿⠇⢿⢏⠁⣿⣧⠿⠇⢿⡏⢿⠇⣤⡄⣿⣿⠀⠀⠀⠤⠀⠒⠤⡤⠤⡀⠀⠀⠈
⣤⣤⣤⣔⣶⡯⠬⣵⠁⠉⠁⠀⠀⠀⠸⣿⣾⣿⡆⣿⣵⣶⣿⡏⣵⣧⣿⡇⣿⡅⢽⣧⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⡄

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀
⠀⠀⠀⡔⠉⠀⠀⢡⠈⠉⠑⢤⠀⠀⠀⠀⠀⠠⠊⠀⠑⢤⠂⠀⠈⡆⠀⠀⠀⠀⠀⠀⣀⢀⠁⠀⠀⠈⠄⠂⠀⢄
⠀⠀⢀⠕⠄⡀⠀⣀⠀⠀⣀⡔⠁⠀⠀⠀⠀⠘⢄⠀⠀⢀⠀⢀⠀⠂⠀⢄⠀⠀⠀⡐⠀⠀⢀⠀⠀⡀⠀⠀⠀⢈⠆
⠀⠀⡇⠀⠀⠀⠨⣀⡗⠀⠀⠀⠳⠀⠀⠀⢀⠖⠉⠁⠀⢾⣿⡅⠀⠀⠀⠸⠀⠀⠀⠐⠤⣀⠀⣑⣾⣷⡤⠠⠔⠛⢄
⠀⠀⠑⠤⠔⠊⠀⠀⠈⢄⢀⠠⠂⠀⠀⠀⢰⠀⠀⠀⢰⠁⠀⠉⠲⠀⠐⠊⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠱⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢺⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⡀⠀⠈⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉
⠀⠀⠀⠀⠀⠀⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠢⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠈⢆⣀⡠⠤⡀⠀⠀⠀⡠⠄⠀⢆⠀⠀⠀⠈⢀⡀⣀⠀⠀⠀⠀⠀⡠⠂⠂⠒⠤⡖⠀⠀⠀⡃⣀⡀
⠀⡠⠄⠀⠀⢇⠀⠀⢨⠊⠀⠀⠀⡇⠀⠀⣾⠀⠀⠀⠂⣰⣀⣠⠈⠀⠀⠘⡄⠀⠀⠀⠀⢐⠀⣂⣀⠀⢃⣀⡠⠎⠁⠀⡜
⠀⠳⡄⠀⠀⠂⣲⣮⣾⣀⠀⣂⠔⠁⠀⠀⠘⠄⣀⡀⠰⠾⣿⠻⠄⣀⣀⡴⠁⠀⠀⠀⡔⠋⠁⠡⠈⢡⠿⠻⠄⠀⠤⠚
⠀⠀⠈⡳⠂⠐⠛⠻⢋⠀⠀⠈⢆⠀⠀⠀⠀⢰⠁⠀⠂⠀⡋⠄⠀⠀⠃⠀⠀⠀⠀⠀⠙⠂⠄⢰⠋⠀⣀⠔
⠀⠀⢰⡀⠀⠀⠀⡼⠤⣄⣀⣀⡼⠀⠀⠀⠀⠈⠣⣀⡀⡠⢇⣀⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁

⠀⠀⠀⣠⠤⢤⠖⠓⢦⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠓⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠙⠋⠲
⠀⠀⡠⠧⣀⢸⠀⠀⠸⢀⡠⠧⡀⠀⠀⠀⣀⠠⠄⣼⠀⠀⠀⢸⣀⣀⣀⡀⠀⠀⠀⢀⡴⠒⠢⣌⡄⠀⠀⢰⡤⠒⠒⢦
⠀⠺⡀⠀⡐⠉⣢⣤⡏⠐⡀⠀⡸⠀⠀⢸⠀⠀⠀⠸⣄⣉⣤⠛⠠⠀⠀⠘⡄⠀⠀⢸⠁⠀⠀⠂⢳⣤⣤⡏⠂⠄⠀⢈⡇
⠀⢰⠉⠑⢂⠖⠻⣿⠓⢆⡒⠉⢱⠀⠀⠈⠳⢄⡰⠶⢿⣿⡿⠒⠥⣀⡠⠊⠀⠀⠀⠀⠑⠒⠐⡶⢾⣿⣿⠗⠶⢗⡒⠉
⠀⠈⠓⡴⠁⠁⢠⢳⠄⠁⠱⡄⠊⠀⠀⠀⠀⡼⠀⠁⠂⣤⠂⠢⠁⠘⡄⠀⠀⠀⠀⠀⠀⡴⠁⠈⠀⢌⢳⠀⠄⠀⢹⡄
⠀⠀⠀⠳⢤⠴⡃⠀⡳⢤⡴⠃⠀⠀⠀⠀⠀⢧⡀⢀⣠⠞⠢⣄⣀⣠⠇⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠎⠈⠣⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⣤⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⡴⠃⠘⠋⠉⠉⠉⣡⡴⢦⡄⢰⡄⠀⢠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠀⠰⡿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⠀⣼⣥⣀⣀⣠⣤⡾⢷⣾⣿⣥⣾⣷⡤⠀⣠⠃⢠⡄⠀⣠⡞⠁⠀⠀⠀⠀⣀⡀⠛⠁⠀⠀⠀⠀⢸⣿
⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣁⣀⣬⠿⢻⣯⣉⣠⡴⢿⣿⠉⣿⣷⣿⢷⣆⡤⢀⡞⢁⣴⠋⢀⡶⠀⠀⣀⣴⠏⠁⠀⠀⠀⠀⠀⠂⢻⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⡈⠛⢀⣾⡿⠟⠁⠀⢈⣽⣿⣿⣶⣶⣿⣿⣿⣿⣿⣄⣩⣤⣬⣅⡘⠃⠀⠈⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⠇⠀⠀⠀⠀⠀⣀⠈⠿⠁⢠⣾⠟⠶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡇⣸⡷⠸⢃⣤⠞⠋⠀⣴⠞⠀⠀⢀⣴⠦⠀⠉⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠈⠀⢀⡿⠇⠀⣴⡿⠁⠀⠀⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⢀⡴⠟⣡⡶⠃⢀⣀⡶⠞⠁⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣿⠀⠀⠀⠀⣀⣿⠇⠀⢰⡿⠀⠀⠀⣾⣿⣿⠥⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠋⠠⠛⣉⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿
⣿⣿⣿⠀⠀⠀⢀⣬⠙⢂⣾⣿⣵⣶⣿⣿⣿⣿⣿⣶⣀⣶⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢁⠖⣋⣩⠄⠀⠀⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠃⠀⠀⠀⣸⣿⠇⣿⡟⠻⣯⣿⡿⠿⢿⣿⣿⣿⠿⠀⣙⠀⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣾⠷⠂⢠⣍⡤⠶⠶⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⣷⡀⠀⠀⠙⠏⢸⡿⠀⠈⣹⣿⣿⡿⢿⣿⣿⣿⣦⣿⣏⣀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠏⠀⠴⠶⠶⠞⠲⠟⡉⠛⠉⠙⠛⠁⠀⠀⠀⢸⣿
⣿⣿⡟⠁⠀⠀⠀⠀⢸⣇⣾⡟⠛⠈⠛⠛⢺⣿⣿⣿⣿⣿⣿⠛⣠⡼⠛⣿⣿⣿⣿⣿⢿⣯⣽⣶⣄⡙⠛⢃⣀⡛⠃⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⢀⡹⣿⣇⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣷⣾⡏⠙⠉⢋⣁⣤⣿⢉⣿⡋⠻⣧⣀⣉⢙⠛⣾⣿⠤⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⡆⠀⠀⠀⠀⠀⠉⢁⡘⢿⡆⠀⠀⠀⠀⠀⠹⣿⣾⣿⣿⡏⠉⠉⠉⠁⠀⠒⠈⠁⢺⡇⣾⡇⠹⣦⢙⣍⣿⣌⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⡇⠀⠀⠀⢀⡀⠶⠛⠁⠈⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⣤⣈⣁⣀⣀⡀⠀⣰⡏⠀⣿⣇⠀⢿⣼⡇⢘⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⣿⣿⣇⡀⠀⠀⠀⢀⣀⡴⠾⠋⠘⢻⣆⠀⠀⠀⠀⠀⠀⠙⠿⣿⣍⡉⠙⠛⠛⠿⠟⠋⠀⢠⣿⣿⣇⢸⣿⣧⣸⣟⡁⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⣿⣿⣿⠁⠀⠀⠘⠷⠀⠀⣀⣴⠞⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⢶⣤⣤⣤⣤⣤⣾⣿⠿⠿⣿⣷⣿⣿⡿⠛⠃⠻⣷⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⣿⠀⠀⠀⠀⣀⣴⠟⠉⠀⢀⡴⠃⢀⡙⠿⣦⣄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠉⠉⠉⠀⠀⣀⣾⣿⣷⣮⣝⣿⠀⢀⣧⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣤⣤⣴⡾⠋⠁⠀⢀⡴⠛⠀⣠⠟⠁⣠⡈⠙⠛⢿⣶⣤⣄⣘⣿⣷⠶⣤⣀⣠⣤⣶⠿⠋⠁⠀⠀⠘⣿⣿⣷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⣯⡀⠈⠁⠀⠀⢠⡄⠛⠁⣠⡾⠃⠀⣠⠉⠀⢀⡗⠀⣸⠹⢿⣍⡋⢉⠀⠈⠛⠛⢁⣤⣤⣤⣶⣶⣶⣶⣿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠊⠁⢀⣴⠋⠀⢀⣼⠋⠀⢰⡟⠀⢰⡏⠀⢸⠉⠛⠛⠛⠲⠶⠟⠋⠉⠁⠀⠘⣿⣿⣿⣿⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠛⠁⠀⣠⡟⠀⠀⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣇⠀⠀⠀⢰⡾⠃⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⣿⣿⣿⣿⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣹⣿
⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣄⡀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣀⣤⣤⣤⣤⣤⣭⣷⣦⣴⣿⡿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠠⢤⣤⡴⠶⠛⠋⢹⠋⠀⠀⠀⠀⠀⠀⢉⣿⡿⠋⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⢀⠠⠤⣤⣤⠴⠚⠋⣻⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢀⣠⠔⠊⢹⠃⠀⠀⠀⡇⠀⠀⠠⠋⠀⠀⠀⠀⣠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⢀⡞⠁⠀⢉⡤⢟⡏⠀⠀⢀⠏⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣠⣴⠟⢋⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⣠⠏⠀⣠⠞⠉⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠾⠛⠉⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⣰⠋⠀⠈⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠸⠷⠶⠖⠒⠒⠚⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠻⠿⢿⣿⣿⣿⣿⣷⣤⣿⣿⣠⣤⣤⣄⣀⠀⠀⠀⣠⣴⡶⠿⠿⣷⣶⣦⣰⣿⡃⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⠟⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠃⠀⠀⠀⠐⢤⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⠀⢀⣄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠘⢳⣄⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠈⠁⠀⠀⠂⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⠂⠐⠀⠀⣠⣷⠈⢿⣦⡀⠀⠀⠀⢠⣄⣤⣽⣬⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡀⠀⠀⠀⠙⣿⠀⠛⠛⠃⠀⠀⠀⠀⢷⠻⠛⠉⠉⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠿⣦⣷⡀⠀⠱⣄⠀⠀⠸⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀⠘⣦⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⢀⣀⣀⣀⠉⠻⣦⣘⣧⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢪⣿⣿⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠶⡾⠿⠿⢀⡀⠀⠀⠈⠉⠉⠉⠛⠛⠿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⡐⠀⠶⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠙⠻⢧⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡀⠁⠄⣠⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⣋⠀⠀⠀⠀⠙⠏⠀⠀⠠⣶⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢬⣍⡄⠀⠀⠀⠀⠰⣦⣾⣧⡀⣀⠀⠀⡀⠀⠁⣾⣿⠟⠁⣰⣄⠀⠀⠀⢀⠂⢀⡀⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠁⠀⠀⠘⠻⠟⠃⠀⠈⠻⠞⠁⠀⠀⠿⠉⣄⡈⠪⣻⣷⣄⠀⠸⡀⢸⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣄⡀⠀⠀⢀⡀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠉⠳⢽⣿⣿⣷⡀⢹⡀⣧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣦⢬⣷⣄⠀⠨⣧⠀⢶⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣄⣇⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⡇⠀⠀⠙⢷⣄⢹⡄⠸⠎⠉⠉⠀⠀⠀⠠⣾⣿⠏⠀⢀⠤⣄⣈⣛⣿⣿⣿⡆⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠉⠒⠤⣭⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠀⠀⠀⠀⠉⠙⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢠⡴⠶⠟⠛⠧⠀⣀⣴⣶⠄⠀⠀⠈⠉⠃⠀⠀⠐⠀⠀⠶⢦⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠘⣆⠀⠀⠀⢠⣼⡿⠞⠃⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣶⣾⣦⡀⠀⢠⠀⠰⡀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠻⢿⣿⣦⡈⣇⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⡀⠀⢠⠟⠀⠀⠒⠒⠤⠬⣿⣿⣿⣦⣈⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠯⠁⠀⠀⠀⣄⡀⠛⠛⠛⠉⢉⣛⣿⠿⠿⠿⣷⣤⣄⣀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠶⠾⠿


⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿⣿⢿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢡⣽⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⡙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⢿⣿⣿⣿⠿⢿⣿⣿⡿⢿⣿⢿⡿⢿⡟⠻⠉⠉⠃⠀⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠀⠿⡿⢿⣿⢃⡎⠁⠈⠀⠁⠀⠈⠀⠁⠀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡠⠤⠔⠊
⠀⠀⠀⠀⠀⠈⠹⣄⣀⣆⣀⡀⠀⢠⠴⠁⠸⠤⣆⣠⣀⡰⠤⠶⣤⠴⠦⠞⠢⠤⠴⠒⠛⠒⠚⠒⠚⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠁


⣿⢿⠿⠿⡿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢻⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⡛⠙⠻⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⢉⣿⢿⣯⠉⢻⣿⣿⣟⠉⠉⠛⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⠀⠀⣼⡇⠈⢿⣇⢸⣿⠉⠛⣿⣷⣿⡿⠟⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⢰⣿⠁⠀⠈⢻⣿⡏⠀⠀⣿⡟⠿⣷⣄⣀⣘⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠻⣿⣄⠀⣿⡇⠀⠘⣿⣿⡟⠛⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⡇⠀⠀⣿⡟⠻⣷⣄⡈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣿⣧⡀⠈⣙⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣠⣴⡾⠿⠿⠛⠻⠷⣶⣿⣿⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⢀⣾⠟⠁⠀⠀⠀⠀⣀⣀⠈⣿⠋⠉⠙⠛⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⣾⠏⢀⣴⣶⣦⠀⠀⠈⠛⠿⠿⣷⠶⠦⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⣀⣀⣀⣀⣀⡀
⢸⡟⢠⣿⠋⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⠿⠟⠛⠛⠛⠛⠻⢷⣦⡀
⣿⡇⢸⣿⣀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣠⣾⠟⠋⠉⠀⣀⣈⣿⣄⣀⣤⡤⠀⠀⠀⠀⠙⣿⡄
⣿⡇⠀⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣵⡿⠃⠀⠀⠀⠀⠙⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⣿⡀
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢠⣶⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⠀⣿⡋⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⣿⡀⢹⣷⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⢻⣇⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠈⠛⢿⣶⣤⣤⣀⣠⣤⣤⣶⠿⠛⠁⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣶⣶⣶⠾⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣤⢀⡐⣚⡋⡙⠉⠛⠀⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡿⣿⢡⡆⣰⢂⡤⣩⠉⠙⠻⢶⣤⣀⠀⠉⠛⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡿⠋⣽⡿⠟⠈⠀⡏⡞⣵⢣⢞⡴⣡⢀⡌⠳⣦⠤⠀⠈⠡⡿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡵⠋⠀⢰⡿⠀⠀⠀⠠⠐⢣⢇⡟⡾⣵⢣⢏⡖⡤⠘⢿⣫⠀⠀⠈⢫⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⢠⣿⠁⠀⠀⠀⠀⠘⠸⢸⢹⢷⣯⡿⣾⣸⢣⢆⡌⢳⣗⣒⣂⠀⠱⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠰⠈⢸⢸⢻⣷⣏⣟⣾⣾⢰⠀⠹⣯⡭⠤⠄⠉⡭⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⢰⣿⠂⠀⠀⠀⠀⠀⠀⠀⠸⠘⢸⠸⡿⡿⡟⣿⣿⣾⢸⢁⡘⢿⣛⣛⣒⣚⣯⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⡄⠃⢧⢿⢿⣿⣿⣼⡜⡇⣷⢩⠻⣶⡶⠶⠽⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⣆⡈⠎⢏⢿⢿⣷⣹⡘⣎⢧⢰⡍⢛⠯⢥⣭⡽⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢮⣀⡁⡙⠓⠝⠌⠪⠳⠝⠦⠙⠦⠈⠳⠯⠉⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠊⠓⠓⠗⠈⠂⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣀⣤⣴⠶⠖⢒⣿⣧⠤⣄⡀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⢀⣴⣿⣱⣿⣵⠞⣻⣿⣿⣿⣷⢾⣻⣷⣄⠀⠀⠀⢸⣿
⠀⣰⣿⢿⣻⣼⣿⣿⣄⠈⠈⠋⢻⡿⢻⣿⣇⡟⣷⡄⠀⠀⣿⡄
⢸⣟⣼⣿⣿⣿⣿⣿⣿⣗⣲⣤⣤⣶⣿⣿⣿⣷⠏⣿⡄⠀⠸⣷⣤⠶⣶⡟⢷⣶⣶⣤⣄
⣿⣿⣿⣿⣿⣟⣿⠿⣿⣿⣯⣭⣿⣿⡿⣿⣿⣿⢟⣉⣥⠖⣋⣹⣿⣿⣿⣿⣿⣯⣿⢹⣾⢻⣦⡀
⣿⣿⢿⡻⢿⣽⣟⠻⣿⠶⠤⠤⠴⠾⢿⣟⢋⣴⣿⣹⣿⣿⠻⠾⠿⠿⠿⠟⣠⣿⣿⣿⣿⣼⢋⣿⡄
⢻⣿⣿⣝⡿⠯⣍⣿⡛⠲⠶⠶⠖⠚⠛⠁⣾⢻⣼⣯⣿⣿⡷⣶⣶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣾⣱⣿
⠈⢿⣿⡷⣭⣗⠲⢯⣍⣙⣛⣛⣛⣋⡉⢸⣧⣿⣿⣿⣿⣿⣟⣿⢶⠞⣛⣿⡿⢿⣯⣿⣿⣿⣿⣩⣿⡇
⠀⠀⠹⢿⡶⢯⣙⣛⣶⠤⠤⠭⠭⠥⠄⢸⣧⣿⣿⣿⣿⣿⣿⣛⣿⣉⣭⡤⠶⠟⣻⣷⠿⢋⡼⣯⣿⡇
⠀⠀⠀⠀⠙⠻⠷⣭⣿⣟⣓⣒⣒⣋⡥⠀⣿⡷⣿⣛⢶⣿⣭⣽⣯⣥⣤⠴⠖⠛⣉⡠⠖⣋⡴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘⣿⣿⡯⢽⣟⠒⠶⣾⠿⠖⠒⠚⠉⠀⠀⢊⠥⣮⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣽⡶⢿⣭⣉⣻⣗⣀⠀⠀⠀⠀⢀⠠⣾⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣟⣿⡦⠤⣿⣭⠤⠄⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠓⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠠⠄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢶⣾⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣯⣽⣿⣦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⡿⣳⣶⡾⣿⣶⣿⣾⣭⣭⣻⣿⣷⢾⣷⣷⣾⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⣿⣿⠿⣷⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⣨⣿⣟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡄⢸⣿⣿⡿⣶⣤⣙⠻⣿⣦⣀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣿⣿⣧⣄⡀
⠀⠀⠀⣠⣴⣤⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣧⣿⣿⣿⣿⣾⣝⡻⣷⣜⠻⣿⣶⡀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⡞⢰⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣼⣿⣿⣿⣼⣿⣿⣿⣿⣿⣞⣿⣧⠹⣿⣷⣄⠀⠴⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢳⣜⢿⣿⣿⣿⣿⣧⣤⣄⣉⡛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠐⣼⣿⣷⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡹⢋⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣿⣿⡾⣖⠀⣾⣿⣿⢹⣿⣷⠻⣴⣿⣿⣿⣿⣿⣿⣿⠋⡄⢿⣿⣿⣾⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠠⢻⣿⣿⣣⠈⣟⣿⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠛⣿⣯⣿⣆⠋⣻⣿⣿⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣷⣿⣦⡔⠻⢿⣟⣷⣾⣿⣿⣿⣋⣻⣿⣿⣿⢿⣿⡿⣹⣿⣿⣿⣿⣿⣿⡿⣿⢁⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠛⣷⣿⢟⣾⣀⣻⠟⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠟⢿⠟⠻⣿⣿⣿⣿⣿⣿⣶⣶⣿⠛⠾⣭⡻⣿⡿⠀⠀⠈⠙⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠑⢿⡟⣽⡟⣾⣿⣾⡏⢻⣿⣽⣿⣿⣿⣿⣿⣽⣿⣾⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⠈⠙⣮⢧⡀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠣⠿⢿⣿⣧⣌⣛⣹⣿⠾⣿⣯⣟⢻⣿⣙⣾⣿⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠹⠏⢿⣷⣼⢯⣧⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣶⣾⣛⣛⣛⣛⣫⣗⣿⣿⣿⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣽⢻⣷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣻⣿⣿⣿⣿⠸⣿⠉⢻⣿⣇⠀⠘⢿⣿⣿⣿⣿⣏⣉⣉⡿⢼⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⢿⣿⣿⣿⣿⡄⢿⡄⠘⣿⣷⡀⠀⠈⣿⠙⣿⣿⣿⡿⣧⡀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⣯⣿⣿⣟⣷⡈⠻⣦⣸⢿⣳⡀⢤⣿⠇⡽⢿⣿⣿⣟⣿⡀⢉⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠾⠿⣦⣈⠛⠿⠿⣿⣶⣩⡰⠃⠀⠙⠛⠿⢿⡇⠈⡾⣷⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠟⠑⠿⢳⣶⣤⡇⣠⢆⣠⠀⡀⣸⣿⣿⡿⠩⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠃⠻⠟⡿⠹⡿⢿⡿⣍⡁⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠇⠀⢿⣿⣦⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣴⡴⡄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣷⣿⣿⣶⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣥⣀⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣁⡤⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⡀⠀⢠⠞⣁⡴⠞⠁⠀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⢸⡟⣟⠲⣿⣞⣁⣠⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣐⠲⠯⣷⣾⠀⠘⣦⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⣿⠀⠀⠘⣇⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⡆⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⣤⠴⠶⠒⠒⣷⠲⠶⠶⢤⣄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⢀⣀⠀⡟⢀⣀⠀⠀⢈⣻⣄⣸⣧⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⠉⠉⠉⣠⠴⠋⠉⠁⠀⠀⢿⠀⠀⠀⠈⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠤⣄⣼⣄⡤⠄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⠤⠤⣼⠽⢦⣄⡀⠀⠀⠀⠀⠀⡏⠀⢠⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠉⠓⣦⠀⠀⠀⡇⠀⢸⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢀⡀⠀⠲⠦⠴⠷⠖⠀⠀⠀⠀⠀⠈⣧⡀⠀⣳⠀⠘⢧⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼
⠀⠀⠀⠀⠀⠀⢰⡇⠀⣰⠋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⡟⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠖⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡦⠱⡕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠒⠒⠂⢤⡾⠁⢑⢹⢊⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⣀⡤⠔⠒⢉⠏⠀⠀⠈⠘⠀⢆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⣀⡤⠒⠉⠁⠀⣀⠴⠋⠀⠀⠀⠀⢸⠀⠈⡌⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡴⠊⠁⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⡠⡸⠀⠀⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⡰⠋⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡰⡱⠁⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡜⠁⠀⠀⢀⢣⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡤⠊⠁⠀⠀⢀⣀⣤⣤⣄⣀⡀⢀⢪⢎⣀⣀⣀⣀⡜⡎⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠎⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠈⣹⣳⠿⣍⠀⠀⠀⠚⠒⠋⠉⠉⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢼⣁⡇⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠣⢤⣤⣤⣤⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠖⠛⠋⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⢀⠀⡀⠀⠀⠀⢀⡤⠊⠁⢰⡇⡎⠉⠀⠈⢢⡰⠂⠙⠒⢦⣀⠴⠚⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⠀⠀⠈⠣⡐⠀⠀⠀⠀⠱⡀⠁⠀⠀⠀⠀⣿⠀⠀⠀⠠⣼⣧⡀⠀⠀⢸⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⡀⣧⠀⠀⠀⠀⠀⡅⠀⠠⠀⠀⡰⠛⠉⠙⣻⣿⣿⣻⣷⣶⠖⠣⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⣄⣀⣠⣼⠿⣯⣤⣤⠶⠦⠶⡈⠓⠢⣀⠇⠀⠀⠀⠀⢩⠟⢿⣿⡄⠀⠀⠘⡄⠐⢇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠨⠟⠛⣿⣿⣿⣏⠁⠀⠀⠀⠀⠣⠀⡹⠓⠢⢄⡤⠶⠋⠀⠀⠈⠻⣄⠀⡰⠁⠀⠀⢱⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠤⠀⠀⠀⣸⠏⠙⠉⠙⣆⠀⠀⠀⢀⡔⠀⣇⠀⠀⠀⠀⢠⣄⠀⠀⢀⡰⠁⠉⢇⠀⠀⠀⢹⣾⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠰⡃⠀⠚⠑⠢⡜⠆⠀⠰⢮⣴⣶⣿⡂⢀⣠⠀⠀⠈⠣⠴⡋⠀⠀⢀⠈⣡⠀⢀⠞⠉⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠄⢆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⣺⣿⣧⣽⡟⠛⠺⡀⠀⠀⠀⢀⣷⣤⣷⠋⠁⠀⠈⢁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢰⢉⣉⣀⠤⡴⠊⠉⠉⠓⢦⣘⣆⣾⣥⡀⢀⣀⡠⠒⠤⡴⠞⢉⠟⢿⡏⠁⠀⠀⠀⠓⠒⠲⢾⣿⣯⣷⠄⠀⠀⠀⡌⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠋⠉⠀⢀⠇⠀⠀⠀⠀⠠⣿⣟⣉⣿⠟⠋⠉⠛⠂⢎⠀⠀⠀⠀⢈⠣⡀⠀⠈⡀⠀⠀⠄⢸⠏⠀⠈⠙⠂⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⣀⠠⠾⠋⠛⠻⡏⠁⠀⠀⠀⠀⡸⢆⠀⢀⣠⠂⠀⠈⠈⠛⢥⡠⠠⠴⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⡆⡀⠀⣀⡰⠃⠤⣉⡁⢘⡆⠀⠀⠀⠀⠀⠉⠓⠴⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⡀⠀⠃⠈⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⢀⣼⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠈⠀⠁⠀⠀⠀⠀⠀⠁⠈⣠⡾⠋⠀⠀⠀⠀⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⠛⠋⠣⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⣀⣠⣴⣎⣠⡤⠄⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⠖⡚⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠀⠀⠒⢠⠄⣀⣤⣞⡀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡠
⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⣼⣁⣀⣀⣀⡤⠴⠞⠋⠉⠀⠈⠉⠉⠀⠀⠀⠀⢠⠄⠀⠀⠀⡀⠀⠀⠀⠐⢀⠀⠀⣰⠏⠉
⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⡀⠘⠁⠀⠀⠀⢠⡏⣠⡾⠃
⠀⠀⠀⠀⠀⢸⣶⡶⠞⠋⠀⡂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠞⠋⠀⠀⠀⠀⠀⣠⡿⠟⠋
⠀⠀⠀⠀⢀⣾⣏⣠⣴⡾⠧⠤⠶⠖⠒⠒⠒⠒⠲⢶⡄⠀⠀⠀⠀⠀⠈⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⣰⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠠⠶⣶⡄
⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠙⠒⠒⠒⠒⠀⠀⠂
⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣴⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣛⣟⣙⣛⣟⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠾⠛⠛⠛⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠋⠁⠀⠀⠀⠀⢄⠀⠀⢀⠈⠛⢷⣄⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⢀⠀⢡⣄⠀⠱⡤⠠⠕⠦⠀⠑⢄⡀⠻⣧⣀⣀⣠⡾⠋⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠘⠀⠍⠁⠐⠀⢀⠎⠀⠀⠔⠀⠀⠄⠂⠁⠀⣹⣏⣹⠏⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢿⣿⢿⣇⠀⢀⣴⠛⣿⣦⣤⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣤⣀⣀⡀⠀⣀⣀⣠⣤⣴⠾⠛⢉⣴⢟⡵⠋⢹⠟⣿⠛⣿⡏⠁⠀⡀⠀⠀⠈⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠉⠁⠀⢀⣴⢟⡵⠋⠀⢀⡟⢸⡏⠀⢸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣱⠟⠀⠀⠀⢸⢃⣿⠁⠀⢸⣿⠀⠀⠀⠀⠐⢆⠠⠄⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣡⠞⠁⠀⠀⠀⠀⡟⢸⡏⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⡴⠋⠀⠀⠀⠀⠀⢸⠇⣿⠃⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠙⡉⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢡⡞⠁⠀⠀⠀⠀⠀⠀⣼⢠⣿⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⠀⢰⡄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⡄⠱⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⢀⠻⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣿⣀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠁⢸⠏⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠀⠀⠀⠀⢸⡄⢸⣧
⠀⠀⠀⠀⢀⣠⣤⡶⠾⠛⢉⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠈⣇⠀⣿⡀⢀⣠⣤⣶⣤⣤⣀
⠀⠀⣤⡾⠛⠉⠁⠀⠀⠀⠸⠧⠴⠛⠁⠀⠀⠘⠛⠃⠀⠀⢀⣨⠷⠚⠛⠉⠉⠛⢻⡀⢸⣷⠟⠉⠀⠀⠀⠈⠙⠻⣦⡀
⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣷⣀⠄⠀⠐⠀⠀⠂⠀⠙⣷⡀
⣸⡏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠂⠀⠈⠀⠘⣷
⣿⠀⠀⠀⠀⠐⠋⢁⣀⡀⠀⠀⠀⣀⠀⡠⠋⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⢿⡄⠀⠀⠀⠀⠀⠰⣾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⢸⡇
⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣦⠀⠀⠀⠀⢸⡇
⠀⠘⢷⣄⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⣼⡇
⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠘⠿⠋⠀⠀⣄⠀⠀⠀⠀⣀⠀⠀⠻⡾⠃⠀⠀⢠⣿
⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⠶⣶⣤⣤⣤⣤⣶⡶⠿⠛⠛⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣶⣦⣤⣤⣤⣶⡶⠾⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠌⢦⠂⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠄⣧⠠⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢸⠆⢸⠀⠀⣀⣀⣠⢤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⠼⢚⣩⣋⣤⣤⣄⣀⠠⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣄⠀⠁⠀⠀⠀⠈⠑⠄⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡀⢸⠉⠓⠶⠤⣤⣤⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡼⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡜⠀⡇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡼⠀⠀⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣸⠃⠀⠀⢹⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣰⠃⠀⠀⠀⠸⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⡏⠀⠀⠀⠀⠀⣧⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣼⠀⢸⣁⣀⣀⠀⠀⠀⠸⡄⠸⣄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠐⠻⣀⣸⠤⡀⠉⢳⣀⡠⠤⢧⠀⢻⣀⡤⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣯⠀⠤⠼⣆⡞⠁⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠒⠛⠋⠁⠈⠛⠲⢤⣤⣀⣀⣀⣤⡤⠖⠋


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⢰⣤⢶⠄
⠀⠀⠀⠀⠀⠀⣀⣠⡄⣰⣿⣿⠃⠀⠀⠙⠃⣠⡀
⠀⠀⠀⠀⠀⠀⣿⡷⡋⠻⢭⣶⢦⣤⣞⣳⣖⠻⠽⡇⠀⠀⠀⢀⡀⠀⠀⣠⣀⢠⣄
⠀⠀⠀⠀⠀⣿⣹⡇⢁⣀⣦⠌⠀⢹⡃⢉⠉⠀⠀⢀⣀⠀⣶⡎⣳⣲⣿⣩⣯⣧⢿
⠀⢠⣤⡄⠀⠈⢸⢇⣼⣿⠟⠛⢻⣏⣿⠚⠀⣄⠠⣾⡽⠉⠙⠋⠹⡯⠷⠄
⢰⡟⢻⣧⢛⣦⣸⣿⡏⣹⡗⡀⠀⠉⠙⢶⡞⢛⣷⠀⠁⠀⢰⣶⣀⠀⠀⡤⠤⢔
⠀⠙⢻⡟⠛⡻⠛⠻⠟⠿⠟⠀⠀⠀⠀⠸⠿⠛⠟⣉⠉⠉⠉⠉⡛⠁⢨⣄
⠀⠀⠀⠀⠰⠅⠀⠒⠠⣄⣤⡴⠆⠀⢀⠀⠀⠀⣺⣭⡇⡠⣔⣿⣹⠆⢾⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠐⢻⡗⠿⠿⠀⠠⠦⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡠⠟⠁⠀⠀⠀⢀⣽⣚⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠛⠉⠉⠁⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠻⢷⣄⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠷⣾⣿⡀⠀⠈⣿⣷⡿⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠙⢿⣶⣾⢿⣿⣄⣀⣠⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⠈⠙⢻⣿⡿⠿⣿⣤⣠⣿⠻⣿⡋⢩⣴⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⢺⣏⠀⣀⣴⣿⣿⠁⠀⣿⡇⠀⠉⠉⠛⠛⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⢻⣧⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣾⡿⢿⣷⣼⣿⣿⣿⡀⠀⠀⣴⣿⡿⣿⣦⠀⢀⣿⣿⡄⣸⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣾⣿⠁⢸⣿⣿⠏⢸⣿⣧⠀⢸⣿⠋⠀⣿⣿⠀⣼⡟⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⠏⠀⠘⣿⣿⠀⠸⣿⣿⠀⢸⣿⣆⣾⣿⠏⠠⣿⠁⠈⢻⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠋⠀⠀⠉⠛⠛⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣠⣤⡀⠀⢠⣦⠀⠀⢠⣶⠀⠀⠀⣀⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢠⣿⡿⠋⠁⠀⣸⣿⠀⠀⣾⣿⠀⢠⣿⠿⠿⠛⠀⢀⣤⣴⣤⣄⠀⢸⣿⠇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣦⣄⠀⠀⣾⣿⣫⣿⣿⠀⢸⣿⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠸⣿⣷⣦⡄⠀⣿⣿⠀⠀⣿⣿⡄⢸⣿⣏⣁⠀⢰⣿⡿⣿⣿⡁⠀⣾⣿⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠈⠉⠉⠀⠀⠙⠋⠀⠀⠙⠛⠁⠘⠻⠿⠛⠀⠸⠿⠀⠘⠿⠃⠀⠙⠋⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣶⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⣤⡄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠻⠦⠉⣤⣞⣹⣷⣄⣻⣷⣶⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠾⢿⣷⣿⠇⢻⢈⣿⣿⣿⣿⣁⢠⣴⣆⣴⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢛⢿⣿⣋⢋⡜⣏⡻⠻⡯⢽⣶⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠻⣶⣿⣿⡇⠀⢀⣠⠤⣾⠒⠋⠁⠀⠀⠀⣰⠆⠀⠀⣸⠁⡾⠢⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠢⣿⣿⣿⣷⡀⠈⢀⣀⠀⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⠹⣌⢁⣄⢟⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣧⣾⣏⣰⠓⣽⣿⣿⠆⠶⢀⡄⠀⠀⣹⠡⣄⣠⣀⠀⠉⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠷⡠⣶⣿⣼⣿⣷⣔⣿⣷⣶⠰⠏⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠩⣿⣿⡿⢻⡍⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢀⣀⡜⣿⡗⠞⡉⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣺⣩⣿⣿⣽⣿⣿⡿⢷⣤⣿⣿⡟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⢻⣷⣿⡿⢿⡍⣽⣿⣏⢨⠗⢾⣿⠿⠿⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣵⣿⣿⣿⠒⠊⠉⠉⠙⠋⠀⠐⠉⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠠⡔⢲⡿⢻⣿⠉⣿⢸⣿
⠀⠀⠠⣤⣤⣾⣧⠚⠋⠀⢈⣿⣉⣡⣼⣋⣤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⣶⣷⠄
⠀⠀⠀⠈⠈⠻⠿⢄⠀⠀⠀⠚⣿⠾⢿⣿⣿⡉⢋⣼⡟⣹⣿⣉⢋⡬⡟⠙⢻⣦⣢⣀⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠨⠣⢿⣿⣿⣿⣶⣿⣟⢛⣿⣿⣶⣿⣿⣧⠀⠙⠋⢽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠛⠺⡿⣿⠛⢹⣄⣿⠀⠀⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⡉⣃⡤⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣐⣤⣷⣾⠿⣿⢹⣻⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⢹⡿⡿⡏⢽⣿⣷⠰⠋⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⣡⣼⣋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠚⠿⠛⠛

⣿⠿⠿⠿⠿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠶⠒⠒⠒⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠐⠦⠀⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠀⠀⢸⣿⣅⠀⠀⠀⠀⣀⠀⢀⣬⣾⠇⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠂⢋⠀⠀⡶⠾⠛⢛⣴⡋⠉⠉⢭⡍⠘⠛⠡⣆⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⠀⠈⠀⠀⠀⠙⠀⠀⠀⠈⠁⠀⠀⢤⡼⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⢂⡀⠀⠠⠀⠀⠀⢀⣃⡉⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠃⡁⡄⡇⣇⠀⢏⠀⠀⠀⢨⣭⠀⠀⠀⢀⠐⣿⡿⠉⣤⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢰⠃⠃⠁⠙⠃⡌⢰⠀⠀⠈⠛⠋⠀⡠⢚⢆⡍⠛⠛⢬⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡈⢡⢸⢤⠀⠀⠠⠐⠋⠀⣸⠈⡏⢸⢦⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠘⢼⡀⠀⠀⠀⢀⡾⠁⠀⡀⢀⡄⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⢠⡀⣠⠏⡠⠈⡇⠀⠀⠀⠜⢠⠀⠸⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠘⠉⠘⠡⠂⠀⠀⠀⠀⠠⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠐⢤⡀⠀⠀⠀⣠⠕⠖⠀⠐⠦⣄⠀⠀⠀⠀⠹⣿⣷⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⡿⡄⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠈⢆⠀⠀⠀⢀⣤⡎⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⢦⡀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠊⢁⣿⢿⣿⡀⠀⠀⡠⠀⠀⠲⠥⣤⣬⡓⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠀⠀⠑⠀⠀⢈⠀⠀⠀⠀⠀⠈⠑⠒⠢⢤⣀⠀⠀⣀⡠⠔⠊⠉⠀⠀⠀⡠⠤⣺⠇⠀⠙⠇⢀⡤⠖⠉⠀⠀⠀⠀⠉⠙⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠉⢲⡀⠀⢣⡀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⢀⡴⠥⠚⠁⣀⡤⢤
⠀⠀⠀⠀⢰⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠁⠈⠓⢢⣤⡄⠀⠀⠚⠁⠂⣤⡤⠴⠚⠉⠀⣠⠔⠋⠁⠀⢠⣇⡀
⠀⠀⠀⠀⠀⢳⡀⠀⢀⡀⠀⠈⡀⠀⠀⠀⠀⣷⣀⠀⠀⠀⠈⠁⠀⠀⠠⣀⣴⡆⢻⡇⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠙⠦⢬⡀⠀⠀⢳⠀⠀⠀⠀⣿⣿⡷⢦⣤⣀⣀⡠⠤⠤⠤⣽⣇⠀⠀⢠⡞⡁⠀⠀⠀⣀⠀⠀⠀⠸⣿⣿⠀⠀⠨⠉⠉⡍
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠂⠀⠓⠄⠀⠀⠻⢿⣿⠀⠀⠙⠁⠀⠀⠀⠀⠀⠙⣆⢰⡟⢸⣿⣷⣄⣀⣈⠑⢂⡴⠄⠘⠿⣦⠀⠀⠀⢀⠘⢠⠄
⠀⠀⠀⠀⡀⠊⠁⣠⠔⠋⠁⠀⠀⠀⠀⠀⠘⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠘⠿⣿⠿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠉⠀⠀⠘⠃
⠀⠀⠀⠀⠳⡀⠸⣅⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡠⠤⠤⠦⠤⣀
⠀⠀⠀⠀⠀⠈⠑⠊⠑⠒⠀⠀⢀⣨⠵⠚⠉⠉⠉⠂⢧⠀⠀⠀⠀⠀⠀⠀⡀⠠⠊⠁⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⡸⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⠀⠀⢁⣴⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠉⢲⣿⣿⣿⣿⣾⠗⠀⢈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣿⣿⣿⣿⣿⡟⠉⠉⡀⠁⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣀⡀⠀⠀⠀⣶⠞⠃⡐⠚⠻⠿⣿⡟⠛⢀⡀⠀⠈⠘⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡜⠁⣠⠘⠋⣰⠀⡄⢘⠛⡀⢠⡁⠀⠀⠀⣤⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢰⠇⠀⠀⠀⠀⠁⡜⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠀⠉⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠉⠙⢦⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣀⣤⠴⠖⠒⠒⠛⢻⡅
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⣸⠃⣀⡤⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⠋⠁⢀⣀⠀⠀⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⢠⡟⠀⣀⣀⣀⣠⣤⣤⣤⠤⠶⠖⠋⠁⢠⡟⠛⠉⠉⠉⠉⠙⠛⠲⠦⣤⣀⠉⠳⣄
⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠛⠒⠶⣄⡇⣠⠞⠋⠀⠉⠉⠉⠉⠈⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⢬⣽⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡄⠀⢀⠀⠘⠛⠯⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠒⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⣃
⠀⠀⠀⠀⠀⠀⢀⣀⣉⡭⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣭⣥⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣛⣉⣉⣁⣀⡀
⠀⠀⠀⠀⠀⢀⣈⠉⠓⣒⡮⠋⠁⢀⡀⠀⠀⠀⠀⠀⠰⣿⣥⣤⣴⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⠽⠿⠿⠛⠋
⠀⠀⠀⢀⡀⠀⠙⠛⠓⠒⠲⢒⡺⠏⠀⠀⠀⠀⠀⠀⣴⠋⣁⣀⣠⣤⢤⣤⣄⠀⠀⠀⠀⠀⢀⣉⣛⣿⣿⣿⣿⣿⣿⠿⠿⠗
⠀⠀⠀⠈⠻⣕⡒⠠⠆⠒⠊⠉⣀⡀⠀⠀⠀⠀⠀⣠⠟⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⢙⣛⣛⣿⣿⣿⣿⣷⣶⣶⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠉⠑⠒⡒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣞⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣈⣙⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠭⢭⣁⣀
⠀⠀⠀⠀⠀⢰⣟⠉⢉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⢴⣄⡀⠀⠙⠿⣿⣭⣥⣤⣶⠆⠀⠀⠀⠀⠀⠀⠀⠴⣶⣶⣒⣚⣛⣋⡉
⠀⠀⠀⢀⡤⠖⠛⠛⠻⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⡶⣦⣤⣀⣀⣤⡶⠋⠀⠀⣀⣴⡆⠀⠀⠀⣀⣤⠤⠤⠤⠐⠒⠛⠃
⠀⠀⠀⣯⡀⢀⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠛⠶⣦⣭⣭⣤⣤⣤⠶⠟⢫⡟⠀⠀⢀⣀⣈⣩⣭⣭⣽⣿⡶⠶⠶
⠀⠀⠀⡼⠟⠻⠷⢤⣤⣤⣄⣈⣉⣹⡷⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣉⣉⣩⠵⠂
⠀⠀⢸⠁⠀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠶⠶⠶⠶⠟⠛⠉⠀⠀⢀⣠⡆⠀⠠⣤⣴⣶⣿⣉⣉⣀⣀⣀
⠀⠀⢨⠷⢿⣭⣭⣤⣤⣀⣀⣤⣬⣭⣭⠿⠀⠀⣠⣀⡀⠀⠀⠙⠻⠶⢶⣤⣤⣤⣶⣶⣿⠟⠋⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠝⠋
⠀⠀⠘⢦⣄⣀⣀⣀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⣶⡶⠶⠶⠶⠟⠛⠉⠀⠀⣠⣤⣴⣖⣚⣋⣉⣉⡀
⠀⠀⠀⠀⠉⠉⠛⠛⢻⡟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⣉⣩⡭⠽⠿⠿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⠀⠄⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣯⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⢃⠆⣼⣾⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⡀⢰⠂⡜⣰⢣⣯⡾⠾⠯⠔⠂⠀⠀⠀⠀⢀⣠⣀⠀⠀⢀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⡸⢡⠏⣼⣴⣿⣟⣋⣋⠩⠍⠁⠀⠀⠀⠀⠀⣘⣁⡽⠀⢰⠏⠚⣻⠆⠀⠀⣠⠟⠀⢀⡞⠱⢟⣷
⠀⠀⠀⠸⣷⣛⣛⣿⡶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣷⠶⠂⠘⠷⠞⠋⠀⠶⠿⣷⠶⠀⠈⠳⠶⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠑⣄⠀⢀⡠⢴⠀⠀⢈⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠳⢶⢼⡀⠈⢦⠈⡤⠋⣠⢠⠀⡰⠁⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠐⢷⡀⠀⠰⢵⠄⠉⠉⠀⠲⠁⢘⣦⣧⡀⠀⠀⣀⠜⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢀⣀⣸⣷⡤⠀⠀⠻⡀⠀⠀⠰⠀⠈⢻⠻⣿⡋⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⢀⣿⡟⡇⠀⠀⠀⣹⡖⢾⡆⠀⠀⠀⠞⠻⠝⠓⠶⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠔⠋⢉⣀⡭⣤⠖⠈⠉⢿⡟⠛⠲⣀⡒⠒⠠⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠂⠊⢀⢤⠖⠣⠀⠀⠀⠸⡀⠀⠀⠈⠟⠙⠠⢈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡹⠁⠉⠀⠀⣢⠄⠀⠀⠑⠀⠐⠀⠑⠀⠄⠤⠀⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⣿⣆⣰⣿⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠈⢿⡿⠁⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣤⣤⣄⣀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⢀⣴⡿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⢿⣦⡀
⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⣠⣤⣴⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣦⣤⣄
⠀⠀⢹⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡆⠀⠀⢸⡇⠀⠀⢰⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡏
⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⡀⢸⡇⢀⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣈⣻⣼⣧⣟⣁⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟
⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⢉⣽⢿⡿⣯⡉⠉⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣶⣤⣤⣤⡄⠀⠀⠀⠀⠀⣀⣴⠟⠁⢸⡇⠈⠻⣦⣀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠀⠀⠀⠀⠀⢾⣿⠇⠀⠀⢸⡇⠀⠀⠸⣿⡷⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⠿⣿⡇⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⢸⣿⠿⣶⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣿⣃⣀⣀⣤⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⣀⣀⣘⣿⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⣀⡉⠳⣤⠟⠛⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠤⠒⠋⠉⠉⠉⠉⢹⠋⠉⠛⠒⠢⢤⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣁⣀⣀⣀⠀⠀⠀⠀⢀⣀⠼⢧⡀⠀⠀⠀⠀⠈⠓⢄⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣹⠋⠀⠀⢸⠉⠓⠦⠤⢤⣤⣄⣀⣀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⣠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣜⣆⡴⠋⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠑⢾⠴⠚⠉⠉⠉⠓⢾⡁⠀⠀⠀⠀⠰⠿⠒⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠠⠒⠢⢀⣠⠈⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠢⠔⠀⠈⠋⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠙⠦⢄⣀⡀⠀⠀⠀⣀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⢤⣀⣀⣀⣀⡤⠴⠊⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠜⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣄⢈⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢢⢳⡝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠚⠛⠛⠓⠒⣒⡾⠃⠀⠆⢏⢎⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⢀⣀⠤⠖⠚⢉⡜⠀⠀⠀⢠⢸⠀⢇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⢸⠀⠘⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣠⠖⠋⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⡄⡜⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣠⠊⠁⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⠁⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠘⠁⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡴⠃⠀⠀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠞⠀⠀⠀⠀⢰⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡴⠋⠀⠀⠀⢀⣠⠴⠶⠶⠶⠶⢤⣄⡰⣳⣋⡤⠶⠶⠶⢤⠎⣇⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢀⠎⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡼⣱⠛⠧⣄⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡎⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⡇⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡃⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⣀⣀⣀⣤⣤⠶⠋⠀⠉⠙⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠂⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠂⡐⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡌⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⠔⠁⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⠄⠂⡁⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠂⠄⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠄⠊⢁⡠⠴⠒⠋⠉⠉⠉⠁⠉⠉⠉⠉⠓⠒⠦⠤⣀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡡⠖⠚⠛⠓⠂⠀⠀⠀⠀⠐⠉⢉⣉⡛⠒⠄⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⣙⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠨⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢠⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠀⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢐⡈⠤⠢⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠂⠀⠀⠀⠀⠈⡐⠐⠄
⠀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⢤⣀⣀⠀⠀⠀⠀⠑
⠀⠊⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠠
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠡⡁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠢⠠⠈⠀⠀⡠⠀⠀⠀⠀⢠⠀⠀⢃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡈⠀⠀⠈⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣿⣿⡦
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣾⣿⡄
⠀⠀⠀⠀⢶⣿⣿⣿⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣆
⠀⠀⣠⣤⣜⠻⠿⠛⣠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢻⡄
⣾⠿⠛⠛⠛⣿⡅⠀⠻⠿⡿⠛⢀⣦⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣶⣶⠀⠰⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀
⣤⠀⣴⣦⣿⣾⣷⡀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢟⡠⠀⠛⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣿⡅
⣿⡆⢻⡿⣿⣿⣎⠙⠲⠶⠶⠶⠶⢾⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣶⣿⣆⡀
⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣄⢻⣄⠀⠀⠀⠀⠀⠀⢀⣠⡾⢿⣿⣿⣿⣍⠉⠓⠦⣄⣀⡀⠀⢀⣀⣀⠀⢀⣤⣿⣿⣿⣟⣉⣡⣿⣿⣿⠃
⣿⣧⣴⣤⣴⣿⠉⠛⠛⣿⡿⠋⠉⠉⠈⠻⣷⣤⣤⣤⣴⣾⠿⣿⡄⠈⠻⠟⠻⠇⠀⠀⠀⠀⠈⠉⢻⣿⣿⣧⢤⡤⠿⠟⠋⠙⢿⣿⠧
⣿⣿⣿⣇⣀⠀⠀⠀⠐⣿⣿⡁⠀⢀⣼⣶⣄⢉⣛⠛⠭⢤⡀⠈⠻⢦⣀⠀⢠⣿⣦⣶⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣤⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⠟⠛⣶⣌⢦⡀⠀⠈⠉⠉⢻⣿⡿⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣠⣿⣿⣷⡝⣦⠀⠀⠀⠀⠉⠀⠀⠀⣤⣤⡀⢰⣷⣶⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠙⢿⣿⣿⣿⣎⠳⣄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣾⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⢺⣿⣿⣿⣿⣿⣷⣌⠳⣤⣀⣀⣀⣈⣭⢌⣩⣬⣿⡄⠀⠀⠀⠀⢀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠺⣦⠾⠋⠿⣿⣿⣿⣿⣿⣷⣌⣉⠉⠉⠽⣾⣿⡟⠁⠀⠁⣀⡀⣀⠀⠈⠙⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠙⢻⣧⢀⣄⢰⣿⠷⠻⣿⣷⡄⠀⠀⠀⠀⠀⢠⣶⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣠⡟⠓⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣼⠋⠙⢿⣿⣿⣷⣀⠀⢉⣁⣠⣤⣤⣶⡶⠾⢿⡋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠘⠋⢻⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣄⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣀⣀⡠⠤
⠀⠀⠀⠀⢀⠀⠀⠀⡀⢀⠦⡤⠠⢄⡀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠠⣄⠀⣠⣄⣀⡀
⠀⠀⠀⠀⢾⣧⠀⠀⡙⠀⠀⠀⠁⠉⠙⡄⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⠿⠍⠀⠀⠀⠀⢤⠌⠑⠀⠄⡉⠹⣦
⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠹⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠦⣄⠇⠍⠀⠀⠀⠀⠀⡦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢀⠤⠚⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠃⠀⠂⣵⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢄⡠⣇
⠀⠀⠀⠀⠀⢄⠀⣦⡄⠁⠒⠒⢆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⠚⠒⠚⠓⠋⠘⢄⣀⠠⠄⢰⠞⠋⠀⠀⠀⡘⠁⠈⢦
⠀⠀⠀⠀⠀⠈⠶⠾⠿⣾⣧⣠⠜⠀⠀⠀⠄⢀⡀⠀⠀⠀⣀⣩⡽⠗⠒⠒⠈⠁⠀⡀⠀⠒⠁⠀⠀⠀⡠⠊⠀⠄⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡀⠀⠀⠀⠐⡈⠉⠉⠉⢉⢁⣠⠶⠛⠐⣒⠴⠎⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠁⠀⠀⠀⠀⠈⠓⠢⠴⠾⡟⠁⢠⡴⠋⠁⠀⣶⡀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⢰⠋⠀⠀⡀⣸⣾⠇⠀⠀⠀⣠⣇⡀⠀⠀⠀⠀⠀⠀⠱⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⠃⠀⣦⣰⡷⠿⠋⠀⠀⠀⠀⠈⠉⢁⡦⠀⢀⣇⢷⣄⡀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⣦⡤⡀⣰⠀⣸⣧⣝⡀⠄⠀⠀⠀⠀⠀⠘⣴⣴⡿⠛⠉⠀⠀⠀⠀⢠⣤⣀⣀⣶⡞⠀⠀⠈⠉⠀⢸⣿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⣆⣀⣼⠏⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⢷⣄⣀⠀⢀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠚⢉⡞⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣾⠁⠀⠀⠀⣹⣦⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⢰⡇⠀⠀⠀⠀⡇⠙⢷⡶⣵⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠃⠀⠈⢿⣞⣿⣮⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡀⠀⠀⠻⣞⣿⣿⣔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠐⡀⠀⠀⢹⡞⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠇⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⢻⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢸⣏⣿⠳
⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⣣⡗⠂⠉⢈⣱⣦⡀⠀⠀⢊⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣿⢿⡀
⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⠉⠀⠀⣰⣿⣿⣿⠷⡀⠒⠐⣜⣦⠐⠂⠐⠢⢄⠀⠀⠀⠀⢻⡰
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣵⣿⠁⠀⠀⠈⠉⠀⠀⠀⠀⠈⠂⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠂⣄⠀⠀⠀⠀⢠⣿⣿⣿⡇⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌
⠀⠀⠀⠀⠀⠈⢝⢷⣦⣤⣴⣿⣿⣿⣿⣇⣿⣿⣿⣷⣆⣤⣠⣀⣤⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠑⠝⡻⢿⣿⣿⣿⣿⢿⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠈⢝⢿⣿⣿⣿⣿⣿⣿⣿⢟⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠦⠤⠤⣤⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠈⢳⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠉⠁⠂⠀⠀⠄⠠⠼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣹⣄⡀⠀⠀⠀⠀⡀⣠⣿⠇⠼⢦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⠋⠉⣛⣿⣿⣿⡿⣄⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⢿⣿⡿⠟⠃⠀⠈⠉⣩⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣥⣴⣾⣷⣤⣤⡄⠐⢻⠁⠀⠀⠀⠀⠈⡁⠴⠛⣶⡆⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⡁⣠⣾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠐⡔⠀⠀⠀⠙⠛⠈⠈⠁⠒⠒⠐⠒⢂⠉⠁⢀⡀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢸⠟⢻⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠅⠀⠀⠀⠂⠀⣁⣠⣶⣒⠒⠉⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠈⡀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠐⣄⠀⡀⠀⠀⠀⠀⡴⠆⠀⢂⡤⠀⠠⠔⠉⢉⠝⠁
⠀⠀⠀⠀⠀⠀⠹⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠉⢋⡑⠀⠀⠠⠽⠢⢡⡦⠛⠂⠀⠀⣀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡘⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠘⠀⠄⠁⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣧⣤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠮⠧⣦⣤⣴⣶⣾⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⢀⣀⡴⢞⣿⠧⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠛⠙⠛⠛⠛⠛⠛⠙⠛⠙⠛⠛⠋⠓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂⠂⠀⢀⠀⠀⠀⠀⠀⢀⡠⢀⠀⢀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠑⣦⠀⠀⠀⢀⠀⠀⠀⠀⡎⠀⠀⠀⢳⡇⠀⠀⠀⡆
⠀⢀⣀⠤⠤⢄⣀⠀⢠⠀⠀⠀⠀⣺⢶⣤⡶⠉⠈⠉⠐⠀⢆⣀⣀⣨⣾⢷⣆⣤⡴⢅⡀
⠀⠀⠉⠉⠉⠉⠛⠛⠷⣷⠶⠒⡿⢻⣿⡿⢧⡀⠀⠀⢀⡏⠁⠀⠈⠛⢿⢶⣿⡉⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠃⠁⠀⠈⠆⠐⠀⣇⠀⠀⢀⣠⠏⠀⠉⠻⣆⡀⣠⠃⠀⠐⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⡀⡀⠘⠀⠀⠀⠀⡸⠶⠤⣤⣩⣓⣉⢨⡀⠀⠀⢀⡜⢨⣾⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⢷⣶⣔⠯⠍⠉⢢⠤⠋⠉⠚⠻⡄⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⠛⠋⠑⠢⡀⠀⠂⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠨⠀⠀⠀⠀⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠱⠠⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁