Poor Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣼⣿⣤⣉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⡿⢿⣞⡃
⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⣟⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⠋⠁⠀⠉⣉⠳⣄
⠀⠀⠀⠀⢠⢔⣿⣿⣟⡵⢸⢻⠹⠝⢿⣦⠀⠀⠀⠀⠀⠰⠟⣽⣿⣿⣿⠀⠶⠀⠀⠀⠩⢤⡽
⠀⠀⠀⠀⢦⣿⢻⠟⣉⠀⠌⠀⠀⠰⡆⢿⠀⠀⠀⠀⠀⢀⡼⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣈⢐⠃
⠀⠀⢀⠂⣾⣿⡟⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇⠀⠀⠀⠠⠎⢠⡿⣿⡟⢻⡝⠀⠀⠀⠀⠘⠃⡼
⠀⠀⠀⠀⣾⢻⡀⠀⠀⠀⠀⠀⡀⠀⠀⣸⡅⠀⠀⠀⠀⠀⠈⠧⣿⣿⣶⡾⢶⣦⣶⠖⠒⠛⠁
⠀⠀⠀⠀⢹⣶⣶⣦⣀⠀⠀⠀⢁⣠⣾⣯⠃⠀⠀⠀⠀⠀⠀⠜⠉⢿⡿⣇⠀⠸⣿⡀
⠀⠀⠀⠠⢻⡇⢿⣿⣿⡟⠒⠻⣿⣿⣿⣽⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⣣⣽⣆⣀⣿⠉⣦⡀
⠀⠀⠀⠠⠁⠀⣠⠿⢻⠀⣀⠀⡘⠈⢹⠘⠻⣆⠀⠀⠀⣠⠞⢡⠀⠀⡿⣿⢷⠤⣿⡆⢻⡟⠲⣄
⠀⠀⠀⠀⠀⠐⠃⣄⡞⠀⠉⡏⠀⡄⠈⡇⠀⢽⣷⣤⠞⠁⠀⣸⡠⠀⠉⢹⣌⣣⢿⢹⡘⣿⣄⠈⢧⡀
⠀⠀⠀⠀⠀⠃⢳⣹⣷⠀⠀⡇⠀⢃⠀⢸⡀⢸⠆⠘⢳⣴⣾⣿⣿⠂⠀⠘⣟⢢⣼⠉⢧⡹⠟⠳⡄⣷
⠀⠀⠀⠀⡾⠀⢸⡇⡅⠑⠀⠃⠀⢸⠀⠀⣷⣿⣶⣤⡂⠙⢿⣿⠏⠀⠈⠀⣿⡈⢰⡆⢸⣖⡥⢀⠘⠇
⠀⠀⠀⢰⠃⠀⣿⢀⡇⠀⠀⠇⠀⠈⢀⣠⣿⣿⣧⠛⠳⢤⣀⣸⠀⡄⠀⠀⠙⣿⢘⠁⠘⣿⣿⠓⠈
⠀⠀⠀⣌⠀⠠⡟⢺⣷⠷⠒⠚⡋⠉⠙⠁⣿⡿⣇⠀⠀⠀⠈⡟⠀⠀⠀⣷⣦⣽⣷⣦⣴⣿⣿
⠀⠀⠰⠓⠠⢸⡇⢸⣿⠀⠀⠀⢃⠀⠀⠄⢿⡇⢹⠀⠀⠀⢀⡇⣀⢠⡄⡿⢻⡏⠹⡏⠘⣿⣿⡆
⠀⠀⢸⡗⢧⣸⠁⣸⢸⠀⠀⠀⠘⠀⠄⢠⣾⣇⠘⡇⠀⠀⢸⠃⠉⢸⠀⡇⢸⠇⠈⡇⠀⣿⣿⣷
⠀⠀⣸⠃⠈⡟⠀⣿⡌⠀⠀⠀⠀⡀⠐⠸⣻⡇⠀⢿⠀⠀⣯⡀⠀⣸⣤⠀⣾⡀⠀⡇⢀⢸⡟⢿⡇
⠀⢰⡇⠀⣂⡇⠀⣿⡇⠀⡄⠀⠀⠁⠀⠀⣿⣷⠀⡘⡇⠀⠀⠘⢳⢹⢿⣦⢻⡄⠘⠇⠈⢸⣇⠈⡧⡀
⠀⠈⢧⣼⡟⠃⠀⣿⣱⠆⠸⠇⠀⠀⠀⠀⢙⢿⠀⡇⢻⠀⠀⣀⢸⣿⢸⡇⢸⡇⠀⢼⠂⠌⠛⣷⡷⠄
⠀⠀⠀⠙⠃⠀⠀⣿⣿⠀⣼⢠⠀⠀⠀⠀⡸⣿⡇⠁⠈⣧⠀⠙⣇⣉⣸⡷⢬⣇⠀⢸⣤⡄⡀⣿⠇
⠀⠀⠀⢸⢰⡇⢀⢻⠋⡀⢸⣸⠀⠀⠄⠠⡇⠉⢹⡠⠀⣿⠀⠀⠀⠉⣿⠀⠀⢹⡇⠀⠸⠀⠀⢸
⠀⠀⠀⠀⠘⡄⢘⢘⠀⠇⢸⣿⠀⠀⠇⠠⡇⠀⠀⡇⣤⣿⡇⠀⠀⠀⢻⠀⠀⣼⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠆⠀⠃⠘⣿⠀⢲⠀⢣⡇⠀⠀⠀⠁⠀⠸⣇⡜⠈⠁⠀⠀⠀⠈⢇⠀⣿⠀⠀⠀⠘⡆⠸⣇
⠀⠀⠀⠞⠀⠘⠀⢹⣀⢸⡟⠛⢫⣀⠀⡠⢦⣴⠖⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⠀⠀⠀⠀⠇⠀⠙⠲⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣯⠤⢇⠀⠀⠸⠎⣷⣀⢿⣀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⢷⣀⠀⠐⢧⠤⠤⢄⣈⣐⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣳⡟⠻⣦⠀⠀⢾⣽⣷⣾⣯⡽⢃⢰⡆⠀⠀⠀⠘⠦⣤⣀⢀⣬⠑
⠀⠀⠀⠀⢀⡠⣾⣋⡥⣤⡾⠛⠀⠀⠀⠉⠛⠻⠿⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠘⠷⣶⣦⣠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠁⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⢄⡀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⣈⡓⢄⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣨⡦⡀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡀⢦⠙⠿⠀⠃⠀⢳⠄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⡠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠲⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠈⠉⠀⠀⠀⠀⡇⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠀⠓⠀⢰⠁⠀⢀⠤⡁
⠀⠀⠀⠀⢠⡆⠰⠀⠀⢠⢾⣄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠙⡄⠠⠃⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠠⣋⡡⠄⠀⠴⠌⠍⢠⡭⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡀⠠⠂⡀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⠀⠀⠀⡄⠀⠀⢠⠀⠀⠑⡀⠈⢦
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠉⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠱⡀⠈⢧
⡀⠀⠀⠀⣈⢦⣀⠀⠀⣀⡠⠐⠀⠀⠀⠀⠀⠀⠀⢠⠒⠆⢀⠁⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠱⡀⠈⢇
⠁⠀⠀⡰⠁⠀⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢰⠘⠀⠀⠃⠠⠤⢠⡀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⢆
⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠉⠉⠉⠉⠉⠉⢠⠗⠒⠀⠀⠀⠀⢒⠶⠊⠀⣠⠾⢀⣀
⠐⠀⢸⠀⠀⡄⡀⠀⠀⠀⡀⠂⠢⡄⠠⠀⠀⠀⠀⠀⠑⠢⢄⣀⣀⣀⠤⠚⠁⠀⠀⠀⠀⣀⠔⠁⣠⠔⠋⠁
⠀⠀⢸⠀⠀⢠⠰⠔⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠈⠐⠠⠚⡁⠔⠊
⠀⠀⠀⢇⠀⠀⠑⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⠒⠁
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⣤⣿⣧⣵⣄⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⣿⣿⢿⣏⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣟⠓⠛⡁⠀⡌⠁⠀⠀⠀⢀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠈⠂⡄⠠⡔⠁⠀⠀⢀⠀⠤⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠄⡀⠒⠒⠓⠈⠐⠄⠀⠀⠘⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠸⠟⠀⠀⣀⡠⠐⠂⠄⠀⠄⠂⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣄⣰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠔⠈⠀⠑⡄⠀⡶⡀⠀⢀⠠⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⢸⠀⡇⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠈⠇⠠⠀⠂⠀⠀⢀⡄⠀⠃⠀⠀⠇⢀⣠⠤⠐⢌⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢼⣿⣻⣷⣯⣿⣿⣿⣿⣿⣧⣼⡀⠇⣿⣤⠴⡄⡀⠀⠀⠀⠀⢀⠠⠀⠀⢦⠀⠀⣀⠤⠐⠀⠀⠈⠙⢳⠁⠀⠂⢉⣭⣀⠤⠤⡴⠂⠰⠁⡇⢰
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⡿⠿⣛⣻⣿⣿⣿⣏⠶⡿⠟⠲⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⠁⡁⢄⢠⠀⣄⡠⠴⠒⠀⠀⠀⠈⡀⢈⠄⢊⠠⠄⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣯⣷⣟⢿⣜⣻⣿⣿⠿⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⡇⠈⠁⠘⠀⡀⠀⢀⠀⠀⠀⠀⠀⠁⠞⢡⠑⢠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠉⠁⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡀⠀⠄⠀⠀⠀⠀⠀⠠⠁⠀⣀⠀⡲⠄⠀⠀⠁⠌⠂⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢐⢀⡀⠁⠀⠠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢲⠗⢠⠦⠤⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⠠⠁⠀⠖⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠒⠂⠀⠐⠂⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠄⠀⠀⠀⠰⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡤⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⢠⡤⠀⣁⠉⠀⠴⢄⠀⢠⡢⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠀⠀⢩⠉⠁⠀⠈⠀⠁⠀⣀⠽⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠠⠀⠠⠀⠀⠀⠸⣄⡀⠀⠀⢀⣠⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠨⢀⡇⠨⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠾⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢛⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠈⠺⠗⠁⠶⢤⠞⠁
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⠀⣾⣇⠀⠀⠀⠀⠀⠀⢀⠀⠂⠐⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠦⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠑⡆⢹⡆⠀⠀⠀⠀⠀⠘⣀⣀⠠⣀⠀⠈⠀⠄⠀⣠⣶⣒⡒⢦⡀⠀⠀⠀⠀⠀⠁⠠⢤⡄⠀⠀⢺
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠪⣇⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠁⠂⢀⣼⠙⡏⠛⢧⡼⠆⡇⠀⠀⠀⠎⠀⠀⠀⠄⠘⢦⡀⢨⣤⠤⡀
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠦⣄⣀⢠⡾⠛⠄⡀⠀⠀⠀⠀⠙⢢⣷⠋⠈⢀⠈⣃⠀⠀⡐⠀⠀⠀⠰⠀⠀⢰⣇⢉⠔⠀⢑
⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠈⠐⣀⡀⠀⠀⠈⡛⢦⣦⡀⠈⠢⡀⠀⠀⠀⢰⡏⠠⠄⠀⠀⣿⠀⡐⠀⠀⠀⠀⡠⠀⢰⠎⣡⡮⢀⡤⠾
⠀⠀⠀⠀⢸⠈⠂⢄⡀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣸⢇⠀⠉⠛⠶⣄⠀⠀⠠⠤⠊⠁⢼⢶⣶⠠⡇⢰⠀⠀⠀⠀⠀⠀⠀⢨⢙⠻⣁⠻⣦⣤
⠀⠀⠀⠀⠘⠀⠀⠀⠉⠓⠂⠄⠀⠀⠀⠘⠋⠉⢙⣾⡆⠀⠀⠀⠀⠑⢦⣀⣄⣶⣆⣀⡤⠴⠈⠁⠀⠌⠒⠰⠦⡎⠀⠀⠀⠈⠁⠸⠋⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⢤⢣⢤⠀⠄⠀⠀⣄⠂⢲⣄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠂⠀⢹⡀⠀⠀⠀⠉⠀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⠀⠈⠑⠀⠀⠀⠀⠉⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⣉⠽⠛⠉⢱⣤⣤⡤⠶⠶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡴⠊⠀⣠⠴⠊⠉⠙⣿⠍⠉⠑⠒⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⣴⡼⠊⠁⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⡷⠋⠘⢆⠀⠀⡰⠦⢧⡠⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢚⢋⣄⣀⡀⣯⠀⢻⠀⠀⢀⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⣿⡀⠸⣀⡠⠎⠙⠒⠶⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢉⣍⡻⠇⠀⠀⠀⠀⢰⣄⠀⠀⠉⠑⠒⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⣉⡀⠀⠀⠀⠀⠘⡀⠳⡄⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡒⠒⠂⠒⠒⠓⠛⡇⠀⠀⠀⢷⣄⠀⠀⠉⠢⢄⡀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⡇⠀⢀⣤⡞⠀⠓⠄⠀⠀⠀⠉⠙⠁⠈⠉⠀⠀⠀⠀⡖⢲⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠀⠀⠀⠀⡇⢀⣥⣸⢃⣤⠔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢹⣠⢏⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⡟⡆⠈⠁⠈⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⡟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣼⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⢀⡀⠀⠀⠀⢹⠀⠀⠀⠀⡄⠢⡀⠀⠀⢀⣀⣠⠷⣏⠀⠀⠀⠀⠈⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⢸⠃⠀⠀⠀⠘⡆⠀⠀⠀⢸⡀⠑⠒⠺⣁⣸⣽⠄⢼⠀⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡈⠀⠀⡀⠀⠀⠀⡀⠈⠂⠀⠀⠀⠇⠀⠀⠀⠀⠙⡯⠄⡜⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠐⠳⠦⠦⢴⠿⠦⡄⠀⡠⠦⡘⣄⣀⣀⣀⡀⢱⣶⣇⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣧⡀⢀⢸⡴⠛⡎⢃⠘⢄⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⡿⢇⠈⢉⡇⢰⣃⡆⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠑⠒⠂⠤⠤⣜⡘⠉⡁⠉⠁⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢧⡀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢢⡀⠀⠀⠀⣄⠀⠀⢀⣀⣘⣄⣾⣷⣦⠴⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡶⠶⠶⠿⠿⢞⣛⣻⣿⣿⠿⠿⢇⣦⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡇⡀⠄⢠⠀⢸⠻⢙⡏⢢⣄⣸⡘⣼⣖⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⠤⠹⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠫⠵⠿⠭⠜⠲⠄

⠀⣠⣤⣦⠀⠀⠀⠀⠀⣀⡀⡴⢏⣡⣤⡄
⠸⣿⣯⣿⡀⠀⠀⠀⡠⠾⠟⠛⠉⠀⠉⠉⣙⡆⠀⣀⡤⠂⠀⠀⠀⠀⢀
⠀⠸⢿⣿⣷⡀⠀⢰⠗⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠻⠃⠀⠀⠀⠀⠀⢰⣿⣉⣉⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠙⢿⡄⠐⡧⠀⠀⣀⡼⡇⣶⣠⣄⠀⠀⢙⣆⠀⠀⠀⠀⠀⣠⠔⠚⣻⣿⡬⠖⢦⠀⠀⠀⠀⢀⡤⠖⠋⠈⠳⣄
⠀⠀⠀⠀⢀⣠⠶⢧⣀⣠⠟⠁⠙⠞⠉⠈⠙⢫⡝⠉⡇⠀⠀⠀⠀⠘⠒⠛⠋⠙⢿⣶⡧⡴⠖⠒⣫⣵⡎⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠐⠉⠙⠀⢸⡈⣅⠀⣠⠆⠀⠳⠔⢣⠀⢻⠞⠁⠀⠀⠀⠀⠀⡠⠔⢒⣦⣾⢷⠁⢦⣦⣜⠯⡷⣷⣢⠀⢀⣠⣴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠻⡄⠀⠀⢓⣀⠀⠐⠄⣸⠀⠀⠀⠀⠀⠀⠈⠳⠾⠟⠋⢸⣯⣷⠈⠛⢯⡉⣙⣿⠧⣒⠽⣊⡵⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⢦⣀⠋⠈⠃⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣉⣩⣭⣽⣿⣾⣿⣶⣽⡿
⠀⠀⠀⠀⠀⠀⡴⠉⠈⠉⠑⢢⣽⠟⠛⢿⣡⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⣿⡿⠿⠟⠛⠛⠯⡁
⠀⣀⠤⣀⠀⢸⡧⡄⢠⣴⡄⢸⡽⣄⣀⡼⠋⠀⢘⡆⠀⣀⠤⢄
⠐⡷⡂⠈⠱⡾⠉⣿⠘⠋⠃⠀⠉⠊⠉⠀⣦⡾⠟⣿⠊⠁⢀⣄⢧
⠀⠻⣷⣸⠣⣄⣠⠇⠀⠀⠀⠀⣄⡄⠀⢸⢻⡄⠀⣠⠞⠛⠿⠛⠚
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠛⠋⢀⡏⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠈⠉⠉⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⣀⣙⢦⡀⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡶⠀⠀⢈⠥⠠⢝⣄⠀⠠⡀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡼⠃⢀⠀⣁⣤⡖⢋⡏⠑⢤⣌⠄⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣥⣔⡘⢇⠈⠛⠃⠉⠀⠀⠀⡏⠢⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠖⠏⠉⠀⡜⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⡀⠀⠈⠳⠒⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠉⡁⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠸⡄⠀⢀⠤⠔⢀⣀⠀⠀⢸⠃⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠓⠒⢦⣦⡆⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠐⢄⠑⠶⣄⠀⣩⠁⠀⢀⡏⠀⠀⠀⡴⠦⣄
⠀⠀⠀⠀⠀⠀⢱⡄⢀⡆⠀⠀⠀⣠⢞⣁⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠱⣄⠈⠙⢆⠀⢀⡞⠀⠀⠀⢀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢹⠧⡜⠁⠀⠀⢰⠁⠸⠿⠆⠹⠿⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠈⢳⠀⡸⠁⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠸⡤⠖⠒⠀⠀⠈⠛⠖⠈⠈⡳⠞⡄⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠘⢠⠇⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⢄⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠬⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢂⢀⠇⢸⠀⠀⠀⣦⠀⠀⠀⠀⡇⠀⠀⠀⢳⠀⠀⠹⣄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡞⠓⢺⠀⡇⠀⠉⠀⠀⠀⢀⠁⠀⠀⠀⠀⢣⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠁⠀⠘⢀⠀⠀⢀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⢧⠀⠀⠸⡆
⡠⠒⣀⣀⡒⢄⡼⠳⢄⣀⣀⣀⣀⣤⡖⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡆⠀⡇⠀⠀⢇⠈⠀⠀⠸⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢹⡄
⡇⠀⠀⠀⢀⠟⠀⠀⠀⠈⢏⠉⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡱⡄⢸⠴⠦⠤⠤⠿⠤⠤⠤⢀⡉⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠻⡄
⢡⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠛⠚⠒⠒⠒⠒⠒⠒⠖⠒⠂⢸⡦⠤⠤⠤⠤⠤⠤⠤⠤⢤⣠⠖⠋⠀⠀⡿
⢸⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠠⠄⠒⠊⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⣠⡴⠛⠙⠓⠒⠢
⠀⡇⡀⢸⠀⠀⠀⡆⡆⠀⠀⠀⠀⣀⠐⠛⠉⠲⡦⠤⠒⠂⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⣀⣀⣀⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⣠⠴⠋⠁
⠀⠇⠀⢸⡀⠀⠀⢳⠸⣄⠔⠈⢁⠔⠃⠰⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡈⠙⠲⠤⣠⠞⠁⣈⠶⠊⠁
⠀⢠⠀⠈⣇⠀⠀⠈⢦⣠⠔⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠁⢰⠞⡠⠢⠤⣀⣠⠔⠋
⠀⢸⢰⠀⠸⡄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠜⠁
⠀⠈⠈⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠰⠠⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣟⢋⣯⣯⣯⣏⢛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠟⠛⠱⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⢏⣼⠞⠛⡄⠀⡒⡌⢿⣶⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⡱⣲⢦⠀⣾⡴⡜⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣾⠁⠸⡛⢃⢰⢛⡇⠠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⠀⢣⣾⡿⠸⣾⣧⠃⠟⠢⡀
⠀⠀⠀⠀⠀⠀⣰⢫⠛⠁⠀⠈⠋⠀⠉⠀⠀⠛⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣡⣈⠀⣀⠈⡆⠀⡌⢀⡀⡰⣏⠗
⠀⠀⠀⠀⠀⠀⠹⣬⣥⣇⠀⠀⠈⠒⠃⠀⣸⡤⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠥⢦⠹⡈⠙⠊⠉⡠⢱⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠉⠁⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢝⡒⠖⢊⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠙⡖⠊⡟⢙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣏⠉⡷⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠘⢄⠑⠒⢁⣸⢸⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣇⢈⣆⢀⢿⣿⣷⣄
⠀⠀⠀⡠⠔⣾⣿⣿⣿⣿⣄⠚⠃⠀⠈⣡⣿⣿⣿⣿⣿⣿⡖⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠿⣿⡏⢸⣿⣿⣿⡆
⠀⠀⣼⣤⣴⣿⣿⣿⣿⣿⣿⣿⣶⣾⢋⡙⣿⣿⣿⣿⣿⣿⣷⣤⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⣰⣿⡆⣾⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡿⠀⡇⣿⣿⣿⡇⠘⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣟⠁⢰⢇⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠻⢿⠿⠛⠣⡀⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡇⡠⠛⠿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⣈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣀⡈⡇
⠀⠀⠀⡏⠀⢾⠓⠃⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠓⠓⢻⠆⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣟⣠⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣈⣿⣷⠃
⠀⠀⠀⠙⠦⠴⠓⠢⣝⣠⣿⣿⣿⣿⣿⣿⣿⣧⣚⡥⠚⠳⠤⠚⠀⠀⠀⠀⠀⠀⠀⡤⣤⣾⣿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣦⡤⢄
⠀⠀⠀⠀⠀⠀⡠⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠩⡖⡄⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⠉⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⢙⣀⣸
⠀⠀⠀⠀⠀⠰⠶⠆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡎⠹⡇
⠀⠀⠀⠀⠀⠰⠄⠄⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢡⣄⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣧⣤⠉⢲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣞⢙⣧⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡿⣿⣿⣿⣷⣿⠋⣶⡾
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⡇⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣻⣿⠿⠛⣉⣤⣿⣿⣿⣿⢸⣿⣿⣿⣧⣬⡙⠻⢿⣿⡏⡽⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠋⠉⠙⠁⠉⠉⠉⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠉⠉⠉⠉⠉⠉⠈⠋⠉⠉⠉⠉⠉⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠒⠒⠒⠀⠶⠖⠳⡄⠈⠀⠈⠉⠋
⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⣸⣿⣿⠁
⠀⠀⣤⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣠⣤⣤⣤⣤⣤⣶⡶⢶⣶⣶⣶⣿⣿⣿⣿⣽⣷⡟⠁⢀⣿⣿⣿⠟⠚⠁
⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⣿⣿⣿⣿⠿⠿⠛⣻⣿⡇⠀⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⢄⣠⣿⣿⡇⠀⠸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣠⢇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣍⣙⡀⠀⠙⠿⣿⠇
⠀⠀⠀⠀⠉⠙⠉⠉⠉⠉⠁⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⡆⠀⠀⣸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⡄⠁⠀⠀⠀⢻⣿⠋⠹⠀⠀⡀⠀⣤⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⣀⢀⡃⠀⠀⢸⠟⠙⠀⠀⡔⣣⣿⣡⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡃⡇⡄⠀⠀⠈⠀⠀⠀⠀⢈⡛⣉⣉⣻⡏⠀⠀⠁⡜⠀⠸⡏⣠⠀⣿⠸⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⢤⠆⠀⢀⣀⠀⠀⠀⠀⠀⣷⣷⣇⠃⣇⠀⡀⠀⠀⠀⠀⠀⢸⣯⣯⣯⣿⠁⠁⠀⠈⠀⠀⠰⣿⠁⢰⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠠⠤⠄⠀⣳⣟⡻⠟⠋⠀⠀⠀⠀⠀⠀⠀⣸⣽⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠓⠂⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡏⠻⠋⠇⠀⠀⠀⠀⠀⠒⠀⠒⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠠⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⡾⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠁⠀⠀⠀⠀⠀⠈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠑⢄⡠⠂⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⣀⠀⠀⢘⡄⢀⡀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⡄⠀⠀⣿⠀⠀⠨⡇⠸⠇⠀⠀⢀⡏⠉⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡸⠣⡀⠀⠀⣀⠎⠀⠀⠀⠠⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢶⠀⠀⠀⠈⠒⠠⠐⠊⠀⠀⠀⠉⠁⠀⠀⠀⠀⡰⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⠁⢲⡤⠤⢀⡀⠀⢀⠤⠀⢲⠈⠁⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠆⠚⠀⠀⠐⠀⠠⠴⠤⠤⠠⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠠⠄⠛⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠂⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠙⠄
⠀⠀⠀⠀⠀⠀⣸⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣀
⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⢰⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⢲⡖⠒⠒⠒⠒⠲⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡶⠒⠒⠒⠒⠒⠒⠒⢲⡖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡟⠉⠁⠈
⠀⠀⠀⠀⠀⣠⣴⢿⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠷⠶⢶⣶⣤⣤⣤⣤⣤⣴⣶⠀⠀⡀⢀⠉⠒⠒⡞
⠀⠀⢀⣴⡾⠋⠁⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠓⠐⠚
⠀⣴⡟⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠹⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠘⣿⣄⣀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣜⠩⠉⠛⢾⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡅
⣀⣼⡹⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇
⢇⠈⢡⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠈⠓⠌⣀⢀⣀⣤⢾⡇⢀⣤⠶⠋⠁⠀⠙⢦⡀⠀⠀⠀⣀⡴⠊⠁⠀⠙⢦⡀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠉⠀⠀⠘⢿⡉⠀⠀⠀⠀⠀⠀⠀⠉⠲⡴⠚⠁⠀⠀⠀⠀⠀⠀⠙⢦⡤⠖⢉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣶⢶⣶⣶⣶⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠚⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⢾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠚⠛⠻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠈⠉⢃
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⣠⡤⡼⠈⠳⣄
⢠⡙⠦⠈⠉⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠇⣠⠞⣍⡺⠿⠃⠀⠈⢳⡀
⠀⠙⠂⠀⠄⡀⠘⠙⣦⣀⠀⠀⠀⠀⠀⠀⢠⡟⣿⣾⡽⠛⠀⠀⠀⠙⠄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⢤⡀⠀⠀⠀⣾⡿⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠙⠦⣤⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀⠀⠀⠀⠀⡆⢰⠐⠀⠀⠀⠀⠀⢀⣠⠐⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣀⣠⠤⣀⠨⠟⡆⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠤⢴⣏⣛⠻⢆⡻⡟⣹⠷⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠿⢲⢿⠖⢿⡧⠿⡆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣧⢲⣄⣠⣷⣄⢽⣄⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣄⢈⣭⡙⣻⡃⠀⣩⡆⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⣏⡙⣿⢟⣻⣿⣇⠫⠛⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡰⠂⠃⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠜⠀⡮⠀⠂⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠇⢸⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⣀⣤⠴⢶⣿⢻⣽⣽⣿⠻⣿⣋⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡤⣟⠈⢉⣰⣶⡏⡉⢋⠀⢩⣄⢛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡠⠀⢀⡞⡁⠀⠁⠈⡆⢠⣶⡄⢿⣟⣿⣯⣾⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠜⠁⠴⠋⠀⠀⠀⠀⠀⣾⠉⠋⠀⠙⣿⣿⣿⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠁⢸⠃⠀⠀⠀⠀⣿⣷⡟⠀⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣤⠴⢤⣀⠀⠀⠀⠀⠀⠀⠄⠊⠂⠋⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠛⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠖⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⡠⠒⣀⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣠⠊⣠⡾⢷⠆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣾⣵⠛⣱⣧⣌⡠⢒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⣞⣧⡜⣤⣧⢀⠏⠁⠘⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣁⣀⣀⣟⡦⠻⣿⡿⢶⣤⣤⡀⠀⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡵⢋⠔⠁⠀⠩⣙⠧⠾⣷⣼⢻⣟⣯⡓⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣋⠟⣴⠃⣶⠀⠀⠀⢹⣶⡀⢻⣟⣿⣿⣷⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡾⠃⢠⠇⠀⠹⣆⠀⠀⠘⣇⢧⣼⣿⣿⠏⣜⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⡴⢣⡀⠀⠀⠈⠓⠺⠷⢾⣾⣿⢱⣿⣼⠿⠾⠟⢶⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣠⣾⢳⣌⣇⠀⠀⠀⠀⠀⠀⠀⣿⣇⣾⣟⣡⠤⠖⣛⣿⠧⠾⠛⠒⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⣿⡟⣷⣤⣿⠀⠀⠀⠀⠀⠉⠃⠀⠻⣿⣤⣠⡴⠞⠉⠀⠀⠀⠀⡀⠀⠀⠙⣦⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⣿⠀⣿⢃⣻⠀⠀⠀⠀⠀⣠⣶⡯⠤⠚⠛⠃⢦⣄⠀⠀⠀⠀⢰⡇⢀⡀⠀⣽⠁⣿⠙⣿⣷⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡿⢸⣷⠉⠼⣇⠀⠀⠆⣸⡿⠁⠀⠀⠀⠀⢀⣠⠽⣿⢷⡀⢀⣿⠀⢸⡇⠀⢹⡇⣿⡾⠋⠉⠉⣼⡋⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠁⠀⠀⣸⢷⡀⢸⢣⡇⠀⢠⡆⢀⣴⠟⠁⣸⠷⣶⢃⣾⠇⠀⢸⡇⢰⣿⣇⣿⣿⣷⡀⠀⠀⣸⣯⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣠⡏⠀⢧⡌⠸⡇⠀⠸⡇⣿⠁⠀⠀⣿⣄⠀⠉⢉⣷⣤⣸⠇⣼⡇⣿⣿⢿⡟⠈⠙⠓⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣦⣉⠃⠀⠘⢧⡀⠻⣦⣀⣿⢋⣇⠀⢸⣿⣿⣷⣀⣤⣘⣿⣿⣴⡏⠀⣿⣏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣭⣳⣄⠀⠀⠹⣦⡈⢻⣿⡞⣙⣷⡿⠻⣿⢿⣿⣝⠛⢻⡟⢹⠇⢘⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⣠⣟⠛⠳⣄⡀⢀⡉⠛⠛⠛⠛⠉⠀⢡⡘⣧⣻⣿⣧⣸⡇⣾⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⣷⡄⠀⠸⣟⣿⣿⢷⣄⠀⠀⠀⠀⠀⢳⠘⣿⢿⡟⣿⡇⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⡀⢹⡿⣿⠀⠀⢹⣽⣻⡄⠙⢷⣿⣦⡀⠀⠀⠀⢘⣿⣷⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⢶⣀⠀⠈⠙⠛⠃⠻⣦⣰⣏⣹⣿⡇⠀⠈⠻⣏⠻⣆⠀⠀⠀⣿⣿⣿⣟⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠲⠶⠾⠿⠛⠿⠿⠿⠿⣧⣄⣀⠈⠁⠉⠀⢀⡴⠿⠿⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢠⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠧⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⢉⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠄⠀⠀⠀⢀⠈⠛⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠀⠀⠀⡄⠳⣤⡀
⠖⢦⡀⠀⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⣠⣾⣛⣻⣠⠀⠈⠻⣆
⠸⣆⠙⠓⠀⠀⠈⠙⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢛⠃⣠⣞⡱⣦⡉⠿⢋⠀⠀⠀⠈⢧
⠀⠈⠗⠀⠀⠄⠀⠀⠛⠉⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⣿⣾⣡⠟⠉⠀⠀⠀⠀⠱⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠊⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⣸⣛⡿⠿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⣿⠟⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠈⠛⠒⢿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⡇⠸⡀⠃⠀⠀⠀⠀⠀⠀⢀⣤⠐⠁⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠤⡂⣈⠠⡻⢧⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⢀⣠⣴⠺⠿⢿⣦⠾⢯⡶⠲⣿⡀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢻⣷⣷⡉⣴⣝⣨⣧⡙⢤⣅⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⢓⡠⠋⡟⠉⠿⣿⠙⡿⡄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣟⣷⢜⣷⣈⣠⣿⣦⡤⣽⣄⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡁⠀⢉⡁⡝⢻⡟⠠⠈⢉⣆⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣿⠲⣿⣷⡶⢿⣿⣶⠠⢟⡽⠄⠂⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣷⠤⠬⣧⡴⠖⠛⠛⠈⠁⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠃⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡴⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⠞⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⠀⢸⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡌⠀⢸⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣄⣤⣶⣾⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⣿⠉⢯⣹⣽⣿⡟⢻⡿⢏⣧⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢉⠛⠃⣰⣤⡝⠛⠫⡏⠈⠙⣃⠀⠿⣿⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣼⠉⠙⠂⠀⢻⡉⠛⣃⠐⢠⣾⣤⣴⡿⠇⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣇⠀⠀⠀⠀⠀⡰⠁⠀⣠⠏⠸⠀⠀⠀⠀⠸⣆⣩⣿⡿⢋⣿⣥⣾⣷⡾⢿⣿⣽⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠜⠁⠀⠞⠁⠀⠀⠀⠀⠄⠀⢸⠿⠁⠉⠀⠀⠙⣿⣿⢿⡇⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣼⠇⠀⠀⢻⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⣀⣀⣈⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡠⠒⠤⠼⠃⠀⠀⠀⠀⠀⠾⠟⠛⠁⠀⠀⠀⠸⡿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣶⣲⡀⠀⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢶⣟⠊⠁⠈⠿⡧⠀⠄⠂⢠⣄⣀⠀⠀⢀⣦⣷⡌⣿⣷⣄⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠇⠀⠸⣿⡄⠀⠀⠀⠀⠀⣀⡶⣿⣷⣿⡆⠀⠙⢼⣿⣦⢶⣀⣹⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⣼⣷⣴⣾⣷⡀⣿⣿⢠⠐⢈⠟⡃⠀⠼⠠⡌⢤⣒⠻⢿⠋⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣢⢤⣾⣆⣿⠛⢿⣿⣿⡗⠌⢻⣿⠟⠫⠀⠀⠀⢈⠂⣠⠞⠻⠷⣄⠲⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠿⢿⣧⡄⠙⠛⠛⠀⠀⠈⠹⢃⡀⠘⠆⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣀⢈⠑⣿⣿⢫⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠈⠻⣷⠀⠀⠀⣦⢠⣄⣋⣙⣓⣶⣤⠀⠀⠀⡀⠸⣿⣿⣶⣿⠿⣫⠜⠛⢈⣀⠿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⡀⠀⠀⠀⠸⣧⠀⠘⠡⣿⣿⠍⠉⢿⣯⣨⡗⣶⢠⡇⠀⣹⠿⠛⠓⢈⠅⡤⠞⠋⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢸⠄⢀⠰⣿⣿⣔⣫⣾⢿⠁⠃⢿⣾⡇⢰⣿⠣⣀⣀⢾⡌⣀⠀⠀⠀⠀⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠛⠈⢑⡀⠘⠙⠛⠛⠛⠛⢩⣾⡄⣿⣿⣤⡟⣡⣿⡿⠁⠠⣼⣍⣥⣤⡴⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣄⣀⣀⠉⠓⠌⠳⢮⣿⢺⣎⢿⣇⣿⠟⢹⣣⣿⢫⠖⢁⣾⣇⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⣻⢿⣿⣶⣌⢤⣌⢿⢘⣿⡞⣿⠟⠀⣿⡿⣷⣷⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣮⣝⣿⣧⢹⡆⢛⣿⣷⣿⠀⠈⢻⣿⡿⢋⡽⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣭⣟⠻⣷⣧⣿⡟⣿⡇⠀⢠⣾⣯⠀⠀⣀⣶⣶⡿⠷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣝⡿⣿⣿⣿⣿⣰⣿⣿⣷⡿⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣧⣿⣿⣿⣿⣿⣿⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠙⠿⣿⣀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡦⣄⡒⣠⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠺⣿⣿⡦⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⠀⠆⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⠶⠚⠆⠁
⠀⠀⠀⠀⠀⠀⢠⠄⢦⣴⣿⣿⣿⠿⠿⢿⠏⡿⣿⣿⠀⠀⠀⡀⠀⣀⣴⡿⡿⣹⣿⢿⣶⢚⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡀⣠⢀⣴⣤⣿⣿⠀⢸⣋⣤⡴⠌⢁⣾⢏⣇⣿⣼⣿⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣀⣿⣿⣿⣾⣿⣿⣿⣿⠀⠘⣿⠛⠒⠒⠂⠊⠘⠋⠹⠋⠋
⠀⠀⠀⠀⠀⠀⠀⢀⢰⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠒⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠙⠉⠛⠛⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣰⣤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡠⠚⡀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠺⠃⠀⢀⡠⠖⠉⢇⣐⡁⢠⡁⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠘⢧⡀⠀⢠⠋⠀⠨⢷⠋⠀⠀⢠⠇⠀⠀⣀⣀⣠⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⢄⡀⠀⠀⠙⣄⢸⡀⠀⠀⢀⡦⠤⠴⠟⠒⠛⠉⠉⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⠀⠘⣆⠑⠦⠔⠋⠁⠀⠀⠒⠦⠤⠤⠴⢲⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣀⣼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡞⠵⠖⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⠀⠀⠙⠂⠐⠊⠁⠐⡀⠀⠀⠀⠠⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣈⠤⠂⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣸⠃⠀⠈⠂⢤⣀⠀⣠⣾⣽⡈⠁⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠓⠛⠀⠈⠉⠁⡗⠑⢇⢠⡎⠀⠀⢀⡤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠤⠈⡟⢾⠀⣎⡿⠐⡟⣉⢗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⡅⠈⡃⡘⢣⠀⠭⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⢺⢁⣾⡎⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢱⣠⠺⢿⠸⠁⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⠬⡏⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣈⠀⠀⠒⠞⠉⠑⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢽⣋⠭⢁⣀⣒⣲⡒⠒⢛⣯⣉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠵⠛⠙⡙⠲⢀⡖⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡴⠾⠤⣤⣀⠀⠀⠉⣠⣧⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠮⢤⣄⣀⠀⠀⠀⠈⠉⠒⢺⣿⣿⣿⡿⢡⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢠⣀⠐⠈⠀⠘⢄⠀⠈⣉⡓⡶⣦⠀⠀⣿⣿⡿⢋⣤⣶⣇
⠀⠀⠀⠀⢰⣀⠡⠖⡁⠩⠀⠀⠀⠀⠀⠈⢉⣠⣮⣿⣿⠀⣾⡾⠏⢤⣿⣿⣟⡝
⠀⠀⠀⠀⢹⠫⢭⣑⠺⢵⣢⠄⢒⡶⢆⣦⣻⣿⣿⣿⣿⡴⠛⠁⢀⡄⣿⣿⡞⠢⢄⣀
⠀⠀⠀⠀⠪⣀⡈⠅⣛⠡⠛⢿⢷⣾⣿⣿⣿⣿⠛⠛⠁⣰⣶⣶⣯⢦⣿⣯⠽⢇⣀⠁⠬⣱⡆
⠀⠀⠀⠀⠀⠀⠀⢉⣰⠭⣕⠒⠴⣸⣿⠟⡛⠀⠀⠈⠡⣶⣿⡇⢃⡾⠁⢀⠄⠈⠈⠓⠾⣿⣿⡀
⠀⠀⠀⠀⠀⣤⡂⠉⠋⣄⠐⠨⣉⠉⠀⠀⡀⣀⠀⣸⣿⣿⣿⡧⠏⠀⠐⠁⠀⠀⠠⠈⠀⠘⠀⠈⢁⣲
⠀⠀⠀⠀⠀⣗⠀⠀⠀⡀⠀⠈⠚⠓⠒⠀⠴⢥⣾⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⢔⠦⣒⢝⣪⡇
⠀⠀⠀⠀⠀⠻⣠⢈⢀⠤⠘⠀⠀⠶⣛⣈⣛⣶⣿⣿⡿⠳⢶⣀⠲⢔⠢⢤⣀⣀⣤⣦⣼⣶⣿⠿⠟⡙
⠀⠀⠀⠀⠀⠠⡯⠊⠝⢍⠙⠙⠍⠙⠃⠋⠋⡃⠛⢳⣦⠀⠀⠙⢮⠗⣿⣯⣿⣿⡿⢟⠋⠉⣰⡬⠖⠃
⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠈⠚⣦⡀⢂⡲⠆⣤⣾⣾⣿⣷⡄⠀⠀⠀⣿⣯⣩⡵⠾⠇⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠂⢀⠀⠀⠘⣙⠓⢶⣿⣿⣿⠿⠋⠁⠀⠀⠀⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢈⣀⡾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⡠⠂⠠⠀⠂⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⣿⣿⣷⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢠⢮⡾⠛⠉⢟⡻⠉⠛⢷⡷⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣦⣤⡶⠶⠚⠻⠿⣧⣠⢔⡾⢷⡢⣄⣼⠿⠟⠒⠶⢦⣤⣴⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣶⣄⠁⠀⠀⡸⡳⠁⣨⣅⠈⢞⣇⠀⠀⠈⣠⡲⡟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠄⠙⡻⢎⣤⡶⢽⡷⢋⣉⣉⡙⢾⡯⠶⢤⣱⢟⠋⠠⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣿⡆⠀⠀⣠⡯⡮⣿⡟⡔⣵⢚⡓⣮⢧⢻⣿⣽⢿⡄⠀⠀⢰⣿⣲⢄
⠀⠀⠀⠀⠀⠀⢀⣞⣾⠋⠉⣻⢶⡪⢷⡷⢛⡭⡁⠀⣑⡃⢘⣊⠁⢈⠭⡛⢾⡾⢗⡶⣟⠉⠙⣷⣳⡀
⠀⠀⠀⣀⣀⣀⣾⣾⣧⢄⣐⡵⡁⣠⡏⢰⡱⢛⠵⠼⢿⣷⣾⡿⠇⠮⡛⣎⡆⢹⣄⢈⢮⣂⡠⣼⣷⣷⣀⣀⣀
⠀⠀⠀⠉⠉⠉⢿⢿⡟⠊⠹⡳⡁⠙⣇⠸⡹⣬⡲⢰⣾⡿⢿⣷⡆⢖⣥⢏⠇⣸⠋⢈⢾⠍⠑⢻⣿⡿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⢯⢿⣄⣀⣽⠾⢵⡾⡷⣭⣲⡁⢀⡩⡅⢨⣍⡀⢈⣖⣭⣾⠷⡮⠷⣯⣀⣠⡿⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠯⣿⠇⠀⠀⠘⣇⣗⣿⣧⠧⡻⢬⡥⢟⡜⣼⣾⣻⣺⠃⠀⠀⠸⣿⠽⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠂⣠⣵⢎⠛⠷⣺⡷⣌⣉⣉⣡⢾⡟⠶⠚⡳⣮⣄⠐⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⠾⠋⡀⠀⠀⢱⡵⡀⢙⡋⢀⢮⡎⠀⠀⢀⠙⠷⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠟⠛⠷⠶⢤⣴⣶⡟⠙⠪⢳⡾⠗⠋⢻⣶⣦⠤⠶⠾⠛⠻⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠘⢞⢷⣤⣀⣮⣵⣀⣤⡾⡳⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢽⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠁⠀⠀⠀⠠⢤⣶⣿⣿⡉⠙⠛⠻⢿⡀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠉⠀⠐⠀⠀⢸⡿⢿⡏⠁⠀⠀⠀⢸⣧⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠀⠀⢀⣴⣾⣿⣷⣀⣴⡷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠠⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡟⣿⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠙⣻⠿⠿⣦⣄⠀⠉⡀⠀⠀⠀⠀⡬⠿⠛⢿⠿⠿⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⣥⣄⡛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠷⠾⠷⠀⠀⠀⠀⠀⢀⡀⠀⢠⡴⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⣿⡿⣯⡤⠆⠀⠀⠀⠀⠐⠃⢀⠀⠀⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡼⣿⣽⣿⢻⣶⣄⣄⣀⠀⡀⠀⠀⢨⡄⠀⢴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠟⢃⠀⠙⠿⣿⣿⡏⠛⠿⣶⣥⣤⣠⣀⣤⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⢙⠛⠉⠰⢀⡀⡀⢠⠀⠵⠞⠀⠁⠀⠀⡆⠈⠙⠻⣿⣶⣄⣈⣿⣿⣿⠿⣿⣧
⠀⠀⠀⠀⣠⣶⣿⠃⢠⣤⣤⣤⣦⣆⣀⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠛⠻⠿⣿⣶⠏⠁
⠀⠀⠀⢀⣿⣇⢴⣲⡿⠛⠋⠉⠛⠣⢌⡡⡙⠿⠀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⢾⣧
⠀⠀⠀⠸⣿⣇⣸⠟⠀⠀⠀⠀⠀⠀⢈⣷⠉⢤⡸⡇⠀⠀⠀⠀⠀⠠⣄⠐⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠙⠛⠀⢀⣀⣀⣀⠀⠀⠀⣿⣦⠺⣇⠀⠀⠀⠙⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣶⣷⣾⣿
⠀⠀⠀⠀⠀⣀⡖⠈⠃⠀⠀⠈⠉⠁⠀⠈⠙⠶⠳⠦⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⣟⣈⣟⢹⣿
⠀⠀⠀⣤⡾⠉⠀⠀⣀⣤⣤⣤⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣢⠦⠛⠛⢁⡟⠓⠋⠸⢻
⠀⢠⣾⣯⣀⠀⣠⠞⠁⠀⠀⠀⠸⠶⠤⢤⣀⠐⠂⠠⠆⠀⠀⠀⠘⠁⣤⢤⡄⢠⠄⠋⠀⠀⠀⠀⠘⡇
⠀⢺⣿⣿⠏⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣦⣀⣀⣀⡤⣬⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠻⡆
⠀⠀⠻⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⠛⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡶⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⣧⡄⠀⠒⢲⡆⣠⡤⢴⡆⢀⠻⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣽⣦⣅⣶⠆⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠗⣸⣇⢄⡀⠀⢀⡴⠀⠀⠀⢀⣀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⣼⢿⣿⣙⣿⣿⣿⣿⣿⣌⣧⣀⠀⠀⠀⠀⠀⠀⠉⠲⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠰⢗⣿⣿⣿⣿⣻⣿⡿⠿⣯⣠⣄⣀⣀⠔⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡷⢞⣽⣟⣽⣿⣿⣿⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣡⣾⢋⣿⣿⣿⣿⣯⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠐⢤⠀⠀⠈⣷
⠀⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠒⠒⠲⠒⠂⠰⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⣆⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡋⠀⠀⣠⣄⣀⡀⣀⢀⠀⡰⠔⠃⠀⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣯⡛⢿⣿⠖⢶⣞⣻⣿⣻⠆⠀⠀⠀⠘⠉⠀⠀⠐⠉⠀⠐⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢁⣤⣼⣷⣾⣿⣭⣾⣟⣿⣶⣾⣷⣤⣠⣄⣀⡀⠠⢀⣀⣀⣴⡞⠙⡛⣦⣄⣀⣀⣀⣀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠟⠛⢉⣀⡀⢈⣉⣉⢉⣻⣿⣿⡿⢿⣿⣹⡛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣹⣟⣋⣉⣉⣭⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣡⡾⠂⠀⣀⡀⢀⠀⠀⠉⠀⠈⠛⠉⠛⠁⠈⣁⣹⣿⣿⠿⡿⢿⡟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣴⣿⣿⣶⡶⢴⠶⠶⠶⠖⠚⠓⠛⠛⠋⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿
⢀⣴⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣦⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣸⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡠⣄⠠⣄⠄⠒⠠⠔⠈⠩⠭⠀⠒⢄
⠀⠀⠀⠀⠀⡠⠊⡉⠀⠈⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⡐⡡⠋⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⢀⢠⠁
⠀⠀⠀⢠⠁⠁⢀⠔⠋⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢀⣀⡠⠔⡢⡢⠃
⠀⠀⠀⢸⢆⡀⠀⠀⠀⣀⠠⠔⠒⠂⠀⠀⠀⣀⠀⣀⣀⡠⠼⣿⡇
⠀⠀⠀⢸⢸⠁⡙⠶⠒⠂⠒⠒⠈⠉⠀⠈⠁⠀⠀⠀⠹⣁⠀⡿⡆
⠀⠀⠀⠘⡈⣴⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢁⠁
⠀⠀⢠⠐⡇⢃⠃⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⣸⠢⡄
⠀⠀⡇⠊⢡⠜⠀⠀⠈⠀⠉⠀⠀⠀⡄⠀⠀⠉⠀⠁⠀⠀⠈⠋⠙⢸
⠀⠀⠙⠦⢴⠀⠀⠀⠀⠀⣴⠋⡹⡿⠛⠉⠳⢦⠀⠀⠀⠀⠀⡶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠦⢄⡀
⠀⠀⠀⠀⠈⠆⠀⠘⣷⠾⠇⠠⠇⡠⢄⠈⠢⠀⢑⣸⠇⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠐⠂⠀⠛⠁⠈⠳⡀⠀⠘
⠀⠀⠀⠀⠀⠈⢦⡀⠀⠑⠒⠓⠉⠁⠐⠋⠙⠛⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠔⠳⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣤⠒⠀⠀⠀⣀⠈⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣧⡘⠁⠐⠄⢀⠀⠀⠖⠄⠸⣼⢻
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⢚⠀⢠⠀⠀⡁⠈⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡁⠀⠀⣠⠤⠤⢄⠀⠀⠀⡝⠋
⠀⠀⠀⠀⠀⣠⠊⠁⢤⠀⠘⡀⣠⠗⠺⣄⠀⠆⠀⠀⢳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠤⣀⠀⠀⠀⣀⠔⠒⠦⠀⠀⠀⠑⢄⡀⠈⠂⠐⠁⠀⣠⠞
⠀⠀⠀⠀⢠⠁⠀⠀⠀⠱⡄⠈⢰⠧⣼⣾⠉⠀⠀⠀⠘⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⡟⠉⢹⠀⠀⠀⢤⣀⡀⠀⠀⠀⠉⠢⡀⠀⡐⠊⠑⡆⠀⢳
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⢦⡸⠁⢀⣀⠀⠀⠀⠀⡄⠀⠀⠈⠑⠤⣀⠀⠀⡶⠭⢀⡇⠀⠀⣀⢧⠤⣼⢀⡀⠀⢰⡦⠚⢧⣀⡤⠤⡶⠁⠀⢳⡆⠠⢜⠂⠊
⠀⠀⠀⠀⢰⠀⠂⡀⠀⠀⠀⠀⠻⣎⠥⠺⡇⠀⠀⠀⠗⠤⢀⠀⠀⠀⠀⠈⠑⢣⠄⣀⣑⠈⡿⠁⡼⠠⠼⡆⠹⣍⣫⠤⠰⠏⠀⠀⠀⠷⡤⠶⠋⠔⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠙⢤⡀⠀⠀⠀⠈⢦⡠⣆⠀⠀⠀⡇⠀⠀⠉⠐⠢⠄⣀⠀⢰⠀⠀⠉⠙⣦⡼⠤⣦⠤⠒⣤⠾⠏⠀⠀⡦⣤⠀⠀⠀⢀⡰⢇⡀⢀⠀⡁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠻⣄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠹⠀⠀⠂⠀⡏⠀⠀⡇⠀⠀⠀⠀⢢⡠⢤⡄⣐⡀⠤⠒⠁⢀⡠⠛⠉
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢘⡷⢤⣀⠀⠀⠀⠉⠛⠳⠤⠄⣀⢀⡰⠉⠆⠒⡮⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠸⢅⠀⡁⠀⠠⠤⠐⠊⠉
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⣅⠔⡩⠛⡶⠤⣄⣀⠀⠀⠀⡄⠈⡇⠀⠀⠘⡷⠢⠤⣄⡀⡆⠀⠀⡇⠀⠀⢀⡀⠤⠀⠈⠀⣠⠀⠠⠤⠤⠤⠤⠤⠔⠺
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⠜⠀⠀⠀⠀⠛⠉⠘⠃⠒⠛⢦⣘⠋⠀⠀⠀⠀⠀⠙⠐⠂⠧⠐⠊⠉⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⢀⡰⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢱
⠀⠀⠀⢸⠀⠀⠀⣀⡠⠴⠚⠉⠀⠀⠉⠉⠀⠐⠛⠣⢄⡀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣅⣠⡇⠀⢀⣨⡄⣀⠀⣀⣤⡄⣠
⠀⠀⠀⠈⢲⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡉⠀⠃⠀⢹⠀⠀⢠
⣠⠶⠞⠛⠛⡳⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⠀⠀⡘
⠑⠤⠤⠶⠄⠧⡀⠀⠉⠐⠒⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠀⠀⢸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠀⣀⠀⠀⠈⠑⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⢸⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠠⠤⠄⢀⡉⠢⣄⡀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠋⣉⠃⠀⠎⠉⠑⠢⡄
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣷⣶
⣿⣿⣟⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣾⣿⣷⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣻⣿⠛⠛⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⣠⣿⣿⣿⣀⣀⣀⡀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⣼⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢾⣿⣷⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣭⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣼⣿⣧⣀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⡿⠟⠿⠿⢿⣶⣦⣽⣽⣿⣆⠸⣿⣶⣿⣿⣿⣦⡀⠀⠛⣶⣤⣀⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠙⠁⡉⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣶⣄⣀⣰⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠁⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋⠙⡟⠛⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠃⠀⣸⣷⣤⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⣹⣿⣿⣿⣿⣿⠏⠀⠀⢀⣶⡆⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⣤⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⣠⣿⣿⣿⣿⡟⠁⠀⠀⣠⣿⣿⣷⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⢰⡆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⡿⠀⠀⠀⡀⠀⠈⣿⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣄⣤⣿⣿⣿⡿⣿⣆⣀⣿⣿⣿⣿⣿⠀⠀⠀⠟⠁⠀⠀⠀⢻⣄⣳⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣇⠸⠿⠿⠉⠉⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢃⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢣⠀⠀⠀⠀⠀⡀⢀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⡀⠀⠀⠀⠈⢷⣘⣿⣿⣿⣿⣿⣿⠛⠉⠉⠻⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣰⣿⣿⣿⠻⠇⠀⠀⠀⠀⠀⢾⣿⣿⣿⠿⠃⠈⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠛⢿⣿⣿⣿⡆⢠⢠⠀⠀⠀⢤⡀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠘⢿⡌⣿⣿⣿⣷⠈⠀⠀⠀⠀⠈⢳⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⢰⡀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⢤⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⣸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠹⣦⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠘⣦⠈⢿⣿⣿⡆⠀⠀⠀⠀⠀⠠⠆⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⣾⡿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⠀⢹⣧⠀⢻⣿⣿⠶⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⠁⠀⠀⠀⠀⣾⡿⠁⢀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⢹⣿⣿⣿⣷⡄⠀⠀⢰⣄⠙⢷⣾⣿⣿⡄⠀⠀⠀⠀⣤⣹⣿⣿⣿⣿⣿⠀⢠⣶⣦⡾⠋⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⢈⣿⣿⣿⣿⣿⡄⠀⠀⢻⣷⡀⠙⣿⣿⡇⠀⠀⣀⢶⡈⢿⣿⣿⣿⣿⠃⢀⣼⣿⡏⠀⣠⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠸⣿⣿⣿⣿⣿⣷⡀⠲⣄⠙⠻⣶⣾⣿⣿⡀⠀⢻⣿⣿⣙⠻⣿⣿⣿⣾⣿⠟⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣷⡀⢹⣷⣤⣼⣿⣿⣿⣷⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣷⡄⣬⡍⠙⣿⣿⣿⣿⡿⠋⠁
⢀⣼⢿⣿⠏⡉⠛⠻⠿⣷⣶⣿⣿⠿⠋⣹⣿⣿⣿⣿⠀⣴⣦⠀⠀⠀⠀⠀⣀⠉⠉⠉⣡⣤⣤⣴⣾⣿⣿⣿⣿
⣿⣷⣾⣿⣿⣿⣷⣶⣦⣤⣤⣍⣁⣴⣿⣿⣿⣿⣿⣿⣼⣯⣴⣿⣿⣤⣴⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⡀⠀⠀⡀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⢴⣾⣿⣿⡿⠿⠂⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠢⠤⣾⡧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢇⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⠦⢄⣀⣀⣀⣈⣐⣒⡶⠤⠤⠤⠤⠄

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢀⣠⣤⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣾⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣿⣿⡿⠿⣿⣇⡿⣷⣯⣿⣿⣿⣿⣿⣿⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣠⣿⣿⣿⣤⣾⣿⣤⣄⣼⠋⠉⣿⠿⢿⡿⢿⣿⠟⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠟⠈⢻⣿⣿⣿⣾⣦⣥⣄⠀⠸⣿⣿⣼⣟⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡟⠀⠀⠠⣌⣽⠿⠿⣿⣿⢿⣿⣿⣾⣋⣹⣿⣿⣛⡉⠙⠛
⠀⠀⠀⠠⣾⣯⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡉⢁⡀⠸⣷⠈⠁⠀⠀⠀⠀⢀⣙⣿⣿⣏⡉⠛⢻⣿⡿⠿⢿
⠀⢀⣴⣶⣌⣹⡏⠙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢛⠛⠛⣿⡆⠀⠀⠰⠾⠟⢿⣆⠘⠿⢿⣿⡙⠻⠿⢿⣿⣇⣴⡾
⢰⣦⡙⢧⡙⠛⠃⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⣷⠀⠈⠀⠰⠶⢶⣶⣤⡀⢻⡆⠀⢠⣿⣿⣧⢀⣤⣉⣿⣿
⣾⠋⠙⠻⠇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣭⡴⠃⠀⠀⠀⣀⣀⡀⠀⢹⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠁
⢻⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢠⣤⡀⢀⡀⢰⣿⡿⢻⡟⠷⠈⠛⠀⠀⢀⣼⡟⣛⣿⣿⣿⡇⣠⡄
⠸⣏⣹⣆⠀⠀⠀⠀⢀⣶⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⢠⡞⠋⠉⠻⣿⡁⠸⣏⣁⣼⠻⣆⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠈⠙⢿⣦⣀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⣿⣿⣷⢀⡟⠀⠀⠈⢠⡀⠰⣄⠙⠓⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⡟⠀⠀⠀⢸⣷⠀⠀⠰⣿⣿⠟⠛⣿⣻⣿⣿⣿⣉⣥⡄
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⢀⣀⣸⣇⠈⣻⣯⡿⢿⡟⠀⠀⠀⠚⣛⣠⣾⣦⣭⣿⣶⡿⢛⣿⣯⣽⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣦⣬⣁⣤⣾⣁⣀⡀⣀⣼⣿⡟⠉⠉⠉⠁⠀⠀⠈⠻⣿⡿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠃⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢧⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢱⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣼⢿⣤⣼⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⠋⢸⣿⠿⢿⣿⣶⣿⣷⣶⣦⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⣾⠃⠀⠀⠀⠈⠉⠙⠛⠻⣿⡿⠶⠾⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣍⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠉⠙⢿⣿⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣠⣿⠏⠀⠀⣀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠠⣿⣿⣿⡀⠀⠀⣰⣿⠋⠀⠀⣴⣿⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⠀⢀⣤⣾⣿⣿⡛⢿⠄⠀⠀⠀⠀⠀⣹⣿⣿⡇⠀⢰⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⣠⣶⠟⠋⢀⠙⣿⣿⣿⣦⠀⠀⠀⠀⠀⢸⣿⣿⡇⢠⣿⠏⠀⠀⢀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠸⣿⣿⡇⣀⣴⠞⠋⠀⠀⢠⣾⢦⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢿⣿⣷⣿⡏⠀⠀⠀⣿⣿⣿⣿⡏⢸⣿⠈⢿⣿⣿⣿⠁
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣾⣿⣿⡿⠋⠀⠀⠀⣰⠎⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣾⣿⣿⣏⠀⠀⠀⢰⣿⣿⣿⣿⡄⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢸⡟⠀⠀⠀⠀⢂⣾⣿⡿⠀⠀⠀⠀⣼⣧⡼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⣾⠁⠀⠀⠀⠀⣸⣿⡙⠇⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣆⠀⠀⠀⠀⢻⣿⣿⣿⡀⣼⣿⡿⢋⣿⣟⣿⣟⣿⣿⣿⡟
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠈⠀⠀⢀⣤⣾⣿⣶⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠘⣿⡀⠀⠀⠀⠈⢿⣿⡏⢳⣿⣿⠃⠈⠻⠿⣷⣿⣿⣿⣿
⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⢠⣴⡿⠿⠛⠉⠀⠀⢈⣿⣿⣿⣿⣭⣿⠋⠀⠀⠘⣷⡀⠀⠀⠀⢸⡿⣧⣿⣿⣇⣀⣤⣤⣴⣾⣿⣿⣿⡃
⠀⠀⠀⠀⢻⣆⠀⠀⠀⢀⣾⣿⠋⠠⣀⣠⣤⣀⣤⣾⣿⣽⣿⣿⣿⣿⣾⣿⣿⣿⣻⣿⣄⠀⠀⠈⡹⣯⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠙⢷⣶⡶⠿⠛⠟⠋⠐⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠛⠛⢿⣷⣶⣾⣿⣿⠿⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣻⣷⣂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠩⢉⠉⠡⠈⣿⣇
⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠉⠛⠛⡿⠾⣳⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢸⠀⠀⠘⠀⠀⠀⢀⠽
⠀⣠⣾⣿⣿⣿⣿⣭⣦⣷⠀⢀⣤⣤⡀⢼⣥⠊⠙⢋⡉⠁⠠⠄⡟⠀⠀⠀⠀⠀⠀⢠⠞⠛⠋⠁⠀⣄⠀⢢⠀⠌⠤⠀⡴⠁
⣼⣿⠇⠀⢐⡋⠉⠉⠭⢸⣾⣿⠿⠿⠿⠻⠻⢿⠖⠀⠀⠔⢂⠴⠀⠀⠀⠀⠀⠀⠀⣞⡳⣄⢩⠄⠼⠛⡀⢀⡵⣤⣤⣎
⠻⣿⣆⡀⠀⠀⢤⠴⢀⣿⢿⣧⠀⠤⠄⠀⠐⠚⡏⢁⠦⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⡁⠀⠀⡈⠠⡎⠀⢻⠾⠄⠷⡀
⠀⠈⠉⠉⠁⢢⣀⣰⠉⠙⠾⣿⡀⠀⠐⠤⠃⡠⢡⡿⠴⢻⡿⠲⡀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢠⡏⠰⠀⢰⡋⡆⠘⠀⠈⢢
⠀⠀⠀⣰⣿⠕⣦⡀⢈⡗⠀⠉⠁⣱⡀⣔⠁⢀⠉⡍⡰⠈⣠⣶⠐⡀⠀⢆⠈⠀⠐⢀⠈⠉⢉⣆⠀⠈⡄⠀⠘⣏⠁⠈⠄⠀⢸
⠀⠀⣼⠋⠛⠠⡀⣿⠘⡟⠈⣦⣄⣉⣉⡑⢤⠈⠀⢀⠁⡠⢿⡏⣃⠱⠄⢢⣿⠟⠅⢋⡀⠈⢻⠃⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠸
⠀⢰⡇⠀⠀⠀⣴⢡⠂⠀⣿⠀⠀⠀⠀⠀⢀⢃⠀⢸⢸⠀⢼⠉⠉⠁⢰⠋⠒⠒⠀⠀⠇⠀⢘⠆⠀⠀⠀⠀⢤⡃⢴⠀⠐⠀⡇
⠀⣾⠀⣀⡤⡪⠊⢋⡄⢰⡇⠀⠀⠀⠀⠀⠀⠁⠂⡨⠙⢀⠼⠀⠀⠀⢸⠀⠀⠀⠀⠀⣁⡀⠺⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢰
⠀⣿⢀⡧⠈⠑⠢⣉⡌⣟⠀⡔⠀⠀⠀⠀⡰⠣⡀⠉⠀⠷⡀⢲⡶⠖⡞⠢⠀⠀⠆⠘⡀⢈⠃⠀⠰⠀⠀⠀⠀⠈⢢⡄⠀⢾
⠀⠙⠂⠀⢀⣤⡶⠿⠟⢿⡔⠉⢦⡀⠀⠰⠁⡐⠉⠢⠴⠀⠘⡀⢻⣃⠠⠤⡁⠒⠤⠀⠀⡏⠀⢀⠃⠀⠀⠀⠀⠀⠀⠳⠄⢸
⠀⠀⠀⣠⣾⠓⠠⠄⣀⣸⠣⡀⢀⣭⣿⣧⢠⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⣸⠁⣎⡀⣀⡀⠀⢨⠀⣱⡿⣿⣿⢿⣿⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠈⠑⠒⠐⠀⢸⠖⠀⠀⡠⠔⠈⠂⠀⠘⡄
⠀⠀⠛⠦⣄⣈⠛⠑⢢⠎⢰⣿⣿⣿⠹⢿⡿⣧⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⡄⠈⠄⠀⠀⠀⠁⠀⠈
⠀⠀⠀⢠⠇⠀⠈⠉⠁⠀⣿⣿⣏⡿⠃⠏⣸⢿⣇⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠐⡀⠈⢄⣀⠠⠀⢀⠠
⠀⠀⠀⣾⠀⠀⠀⠀⠀⣸⢭⣿⡇⣷⢠⠐⢉⡾⢿⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⢄⣀⠠⠔⠈
⠀⠀⢀⡟⠀⠀⠀⠀⠀⣿⡎⡻⢻⠏⡌⢠⠞⢡⠈⢻⡀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⢸⡇⠀⠀⠀⠀⢸⣯⠃⢿⠸⠀⣁⡞⠀⠈⡆⠈⢿⡀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⢸⠁⠀⠀⠀⠀⣿⢹⡀⢸⠇⠀⠏⠀⠄⠠⠙⠀⠘⣷⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡇
⠀⠀⣼⠀⠀⠀⠀⢈⠶⡆⢀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⡸⣧⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇
⠀⠀⡟⠀⠀⠀⠀⡿⠆⠃⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⠀⠀⠛⡆⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⢸
⠀⢐⡇⠀⠀⠀⢰⣧⠀⠘⠀⡀⢀⠀⠀⡐⠀⠃⠀⠀⠀⢀⠜⣡⠆⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⢸
⠀⠈⠋⠉⠧⣤⣾⠈⣦⠀⠄⡅⣸⠀⠰⠀⡞⠐⠀⠂⡀⠈⡘⠁⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⡆⠙⠀⠀⠁⡉⠀⠂⡘⠀⠆⠀⠐⠀⡀⠀⠀⠀⠂⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣏⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠁⡆⠀⡆⠀⡀⡀⠀⠀⠰⠀⡇⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⣻⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⡇⢠⠃⠀⠰⠡⠀⠀⠀⢃⠀⠀⠀⠀⢰⡆⡢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠅⣣⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡜⢃⠀⠀⠀⠀⠃⡘⠀⠀⠐⠔⠛⠶⠦⠈⣠⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣨⠀⠀⠀⡇
⠀⠀⠀⣠⡾⠓⠊⠀⠀⠀⠀⣄⠆⠀⠀⠀⠀⠀⠀⠀⢀⠏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠏⡀⠀⠀⠀⠀⠀⠀⠀⠘⡦⡘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠄⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡬⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢨⠿⠛⣋⡉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣄⣂⣉⣉⣁⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠄⠀⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠤⡄⠀⠀⠀⠀⠀⠀⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⢠⢀⡄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⣨⠕⠸⣿⡣⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠝⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠁⠈⠀⡄⠀⢲⡟⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⢰⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⠉⠰⠀⠳⡀⠀⠀⠀⠀⠀⢠⠀⢑⠋⠑⠂⠀⠀⠄⠀⣀⡀⠀⠀⠀⠀⠀⠸⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠐⠀⠀⠀⠚⠁⠀⠀⠈⠲⢐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡌⠀⢈⣁⣄⠀⠀⠀⠀⠀⠀⠀⠀⡏⢂⠀⠀⠸⣌⠉⠀⠀⠒⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠠⠀⠋⠜⠃⠀⠐⠉⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⢲⢂⠖⠁
⠀⠀⠀⠀⠀⠐⠹⠀⡀⢀⠀⠀⠀⠈⠀⠀⠀⣀⡰⠀⠦⡤⢐⠊⠁⠐⢄⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠁⡠⠈⠁⠈⠁⠀⢄⠀⠀⠀⠀⢈⡀⢀⣁⠔⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣄⠘⠃⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠜
⠠⠀⠤⠀⠐⠠⠢⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠈⠀⠰⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⠀⠀⢀⠀⠀⠈⠀⠀⠀⠠⠴⠂⡀⢀⠀⠈⣀⣃⣀⠁⠀⠀⠘⠆⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣁⣈⠁⠃⠀⠀⠈⡄⠀⠀⠀⠀⠀⠐⠂⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠀⠐⡊⣀⠀⠈⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠚⠛⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡉⠀⠀⢀⡀⡀⠀⠀⠀⢠⣤⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡏⠰⠚⣡⡏⢠⠆⠀⠀⠛⠁⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⠁⠰⣤⠇⡾⣰⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠉⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⡤⠶⢄⣠⠴⢞⣻⣶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⣴⠋⣁⡀⣀⣀⣤⣬⣉⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠗⠛⠉⠁⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⢰⡇⣠⠿⡝⢿⠋⠛⠛⠃⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠈⣧⢸⣦⠀⠀⢠⡄⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠘⣧
⠀⠀⠀⠀⠀⢸⣟⢃⠀⠀⠀⠀⣀⣉⡀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠉⠻⣦⢦⣀⠀⠉⢉⡼⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢉⣻⣉⡥⠤⡄⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⢻⣆
⠀⠀⠀⠀⣀⣤⣶⣛⣉⣷⣤⣿⡁⠀⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢠⡇⠹⣦⠀⠀⠀⢿⡄
⠀⠀⢠⣞⣛⡂⠀⣤⡭⢭⣤⣼⣇⠰⢦⣀⢻⣶⡄⠀⠀⠀⣸⠇⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠘⣧⠀⠀⠸⣧
⠀⠀⣼⣧⠬⡍⢠⡏⠓⠊⠉⣿⢿⣧⠀⠉⠙⠀⠙⠦⣤⣀⡿⠀⠀⠀⢠⡏⠙⠳⣄⠀⠀⠀⠀⠀⣸⠃⠀⣾⠀⠀⠀⠸⣇⠀⠀⣿
⠀⢰⠏⠉⠉⣠⠞⠁⠀⠀⢠⡷⠾⣏⠳⣤⣀⣀⣀⠀⣿⠁⠀⠀⠀⠀⡾⠁⠀⠀⠙⣧⣀⣀⣀⣀⣀⣠⣴⣛⢤⡀⠀⠀⢻⣶⣤⣿⡇
⠀⡿⢠⠴⠋⠁⠀⣴⠖⠒⠋⠛⠛⢛⣦⣤⡼⢻⡏⠻⣿⣆⡀⠀⠀⣼⠁⠀⠀⢀⡼⠋⣹⣿⠛⠛⠻⣟⣧⠀⠈⢻⣄⠀⠘⣧⠀⢹⡇
⢰⡇⠀⠀⠀⢀⣾⢋⠉⠉⠉⠉⠉⣉⣉⣸⡇⢸⡇⠀⠀⠉⠛⠻⠞⠁⠀⠀⢀⡾⠁⢰⣏⡇⠀⠀⠀⢹⡌⢇⠀⠀⢹⣆⠀⢻⡀⠘⣿
⠈⢷⣀⣠⠶⠋⣽⣭⣍⣩⣭⣭⣭⣭⣭⣽⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⣾⢸⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⡆⠘⡇⠀⢹
⠀⠀⠀⠀⠀⠀⢸⣓⣲⣤⣤⠴⠖⢒⣒⣻⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣇⢀⢸
⠀⠀⠀⠀⠀⠀⣸⣷⣮⣭⣭⠭⣽⣾⣛⣽⡇⠀⢸⡄⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠹⣿⢸
⠀⠀⠀⠀⠀⢰⡟⠺⠿⠯⠭⠭⠭⠖⠛⠚⣷⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⠟
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠


⠀⠀⠀⠀⢠⣠⣼⣁⠄⠀⠀⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣤
⠀⠀⠀⣐⣶⡉⠀⠈⢻⣁⠀⠀⠀⠀⠀⠀⢩⡍⠀⣀⠀⠀⠀⠀⢠⣤⢿⢿⣿⣿⡀⠀⠀⠀⠀⠀⣸⣟⣿⣿⣿⡁
⠀⠀⠠⠔⢫⡘⠉⢙⡼⠄⠀⠀⠀⠀⠹⣷⡄⠋⢴⡿⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠗⠀⠀⠀⠀⢠⣯⣿⣿⡿⡯⣿⡄
⠀⠀⠀⠀⠹⠋⢹⠙⠈⠃⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣽⣧⡀⠀⠀⠀⠀⠀⠀⢰⠟⠛⢳⢠
⠀⠀⠀⠀⠙⠛⠦⠶⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠗⠀⠀⠀⠀⠀⠠⣿⡏⢿⣿⠀⠀⠀⠀⠀⠀⢸⣶⣶⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢰⣶⢿⣾⡿⢶⡆⠀⠀⠀⠀⠀⠿⠷⠟⣩⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⡆⢠
⠀⠀⠀⣰⢶⣿⠿⣿⠿⡆⠀⠀⠀⠀⠀⣤⣠⡄⢠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⢤⠞⣩⡍⠳⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⡌⣿⣅⠹⠿⠿⠃⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠬⠽⡀⠨⠇⢠⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠓⠒⠒⠀⠀⠀⠀⠀⠘⠝⠑⡚⠍⠓

⠀⠀⠀⢀⣴⣰⣶⣶⣶⡆⠀⠀⠀⠀⠀⣀⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢻⣷⡄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣿⠉⢉⣬⡉⠀⠀⠀⠀⠰⠙⡟⡿⢻⢻⠋⠀⠀⠀⠀⠀⠹⣤⠽⡿⣴⡿⠃⠀⠀⠀⠀⣰⣿⡿⢿⣷⡄
⠀⠀⠀⠀⣿⠀⠘⠛⠁⠀⠀⠀⠀⠀⣀⣧⣭⣼⣼⣀⣄⠀⠀⠀⠀⠀⠀⠀⠙⠉⢹⣤⠀⠀⠀⠀⢿⣿⡢⣹⣿⡇
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠛⠿⠟⠋

⠀⢀⣠⡤⠶⠶⠶⠶⠤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠉⠉⠛⠛⠲⠶⣤⣀
⢰⡟⠁⠀⠀⠀⠀⢀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⣀⠀⠀⠀⠀⠈⠳⣄
⠸⣦⣠⣤⣤⣶⠞⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠉⢳⣤⣀⠀⠀⢀⡿
⠀⠀⠉⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠸⣧⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢻⡄⠀⠀⣼⠟⢁⣀⣀⣀⣀⡈⠛⢻⣦⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢻⣄⣸⣿⠞⠋⠉⠀⠀⠉⠙⠓⠦⣿⡇⣰⠇⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠦⣤⣽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⣖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡙⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣠⣾⣯⠙⠳⡄⠀⠀⠀⣴⣿⡍⠛⢳⡄⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⢿⣿⣿⣤⣴⡟⠀⠀⠀⣿⣿⣷⣶⣾⠇⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠀⠀⠀⠀⠈⠛⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⢀⣼⢋⣤⠀⠀⠀⣀⣠⣤⣤⠶⠶⠶⠶⠚⠛⠛⠛⠛⠛⠲⠶⠶⢦⣤⡀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⣼⠇⢼⡇⠀⢰⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢸⡿⠀⠸⣧⠀⠈⢳⣄⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣿⡇⠀⠀⠹⣧⠀⠀⠙⢷⣤⡀⠀⢸⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⢿⡇⠀⠀⢀⡼⢷⣄⠀⠀⠈⠛⠷⣿⣀⡀⠀⠀⠀⠸⡇⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⣿⡀⢠⡟⠀⡴⠛⠷⣤⡀⠀⠀⠀⠉⠛⠛⠷⠶⠾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠙⢿⣿⠀⠘⠁⠀⠀⣼⠟⠓⠶⠦⣤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠿⠛⠁
⠀⠀⠀⠀⠀⢘⣷⣄⣀⣀⣴⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡶⠶⠿⠛⠻⣿⣄
⠀⠀⠀⠀⢀⣾⠟⠉⠛⠛⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⢠⣿⠏⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⡀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⠶⠦⣤⠘⢿⡄
⠀⠀⢀⣾⠏⠀⠀⠤⠠⠀⣴⡶⠟⠛⠋⠉⠉⠁⠀⠀⠹⣆⠀⣼⠋⠁⠀⠀⠀⠀⢀⡀⠀⢸⡇⠈⢿⡆
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⠀⠀⠀⠸⣆⠀⢹⣾⡏⠀⢀⡟⠀⠀⠀⠘⣇⠀⠀⣷⠀⠈⣿⡄
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢹⣦⡾⠿⢦⣤⣼⡇⠀⠀⠀⠀⣿⣤⣴⠟⠀⠀⠸⣧
⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠿⢦⣤⠴⠾⣧⣤⣴⠶⠟⠃⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠟⠁⠀⠀⠀⠀⠀⢿
⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣸⡇
⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠛⠻⠷⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣾
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠷⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠾⠿⠿⠿⠛⠛⠛⠉⠁

⡋⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠹⠈⠉
⡇⠀⠀⠀⠀⠀⠀⢠⡼⢰⠀⡀⠀⠀⠀⠐⠒⠈⡇⡀
⡇⠀⠀⠀⢠⠀⠀⢸⠁⣸⠛⢁⠀⠀⠀⠀⢩⣿⠀⢨
⡇⠀⠀⠀⠈⠄⠀⠏⡰⠃⡞⢩⣶⡄⠀⠀⠘⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⡍⠑⡀
⡇⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠛⠋⠀⠀⡀⠀⠀⠐⠃⠸⠀⡀⠀⢀⣀⠆⠒⠔
⡇⠀⠀⠀⠠⡀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡀⢁⡀⠀⠐⠙⠻⣩⠟
⡇⠀⠀⠀⠀⠀⠡⣀⠤⠀⣀⠒⢶⣀⢀⣨⠇⠛⣤⢀⡛⡀⣀⢀
⡇⠀⠀⠀⠀⠀⢦⡡⣠⡏⠉⣡⡀⠈⠉⠁⠀⠀⢀⡻⢗⢁⠉⠒⠉⠐⠂⠀⠀⠀⠀⠀⣀⠀⢤⡀
⡇⠀⠀⠀⠀⠀⣈⡞⢤⠘⠃⣀⠀⠀⠀⠠⠀⠀⠀⠈⠛⠂⠉⠀⣀⠀⠀⠈⠀⠀⠀⠈⠉
⡇⠀⠀⢀⡴⠿⠃⠁⠦⠰⠂⣩⡀⠀⠀⠐⡀⠋⡒⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠚⠛⠃⠊⢙⠉⠁⠄
⡇⠀⠀⠀⠙⠂⠀⠀⢀⠀⠉⠉⠁⠂⠐⠀⠑⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠁⠁⢸⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠨⢒⢀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⡂⠔⠹⢐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠁⠀⠀⠀⠀⠉⠢⡀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⢡⣴⣆⣴⡀⠀⢁
⡇⠀⠀⠀⠀⠀⠀⠀⡴⠋⣰⡗⣄⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢐⡀⠀⠈⠀⠒⠓⠶⠷⠀⠀⠀⠐⠂
⡇⠀⠀⠀⠀⠀⠀⢸⠃⠺⢻⡇⠈⢦⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢘⣒⣒⣒⣒⠛⠀⠸⠿⠒⠶⠶⠇
⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠓⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠁
⡇
⡃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⢀⣀⡀⠀⣀⢀⣀⢀⣀⣀⣀⣀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀
⠀⠀⠳⠆⠀⠀⣧⠄⠃⠠⠇⠠⠝⠉⣻⣿⡛⣿⣿⣿⣿⣟⠛⠛⣟⠛⠛⣻⠇⠀⠈⠂⠀⠀⠀⠀⢀⣀⡤⠴⠚⠫⡁⠀⠀⠑⢤⡀
⠀⠀⠐⣤⠰⠂⠀⠀⠀⠀⠀⠷⣦⡈⠁⠘⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⣀⣤⠴⠶⣟⠉⠁⠐⠀⠀⠀⠈⠳⣀⡀⠀⠙⠶⢠⣄
⠀⠀⠀⠋⠀⠀⠀⠀⠀⡀⠀⢸⣿⣷⡄⠀⠘⣿⣛⣿⣟⣻⣿⣿⠟⠽⢾⠿⡦⠐⠉⠙⠳⣄⠀⠈⠓⣄⢀⠀⠀⠦⠀⢀⣈⣡⣤⠶⠚⠛⠃
⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣣⣈⡈⣽⣿⠀⠀⢹⣿⢻⣿⣿⣽⣿⣿⣸⣿⣾⡇⠀⠀⠀⠀⠈⠳⠄⠀⣀⣉⡤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣿⣿⣾⣿⡟⠂⠀⠘⣿⣟⢻⣿⣿⣉⢉⢙⡛⢹⡆⠀⠀⠀⠀⠀⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣯⣝⡻⡙⡧⠀⠀⠀⣿⠗⠁⠀⡀⠀⢉⣈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⢀⣤⡍⠓⡴⢶⣦⡄⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⢿⡟⠀⢣⡀⠀⠀⠀⠁⠀⠀⠀⢠⣦⠀⢿⣷⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⡄⠀⠀⠀⠀⣸⡖⠁⢠⣯⣿⠟⠻⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠷⠀⠀⠀⠀⠀⠀⣀⡀⣿⣿⡸⣿⣿⣶⣿⣷⣦⣞⣿⠃⡠⢸⣿⡇⠀⠀⠀⠘⠻⢯⡧⣰⠻⣿⣆⣀⠄⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠁⣾⡿⣷⣿⣿⣿⣿⣿⣿⣯⣅⢸⠃⣾⣿⣷⠀⠀⠀⠀⠀⠀⠁⠁⠀⠉⡿⠿⡾⢻⣷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⠟⣿⡿⣿⠛⠛⠙⠉⢹⠈⢀⠛⣿⣽⡇⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣈⣄⣰⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡆⠀⠀⠀⠀⠀⢿⣦⢖⠀⠀⠀⠀⠙⠛⠿⠖⠗⢈⡅⠠⠆⠨⠀⣿⣃⣗⣀⣀⣀⢛⣻⡥⣶⡀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⢰⣦⣤⡀⠛⠀⠀⠀⠀⠐⠀⢰⠐⣋⢠⡧⠀⡀⠀⠀⠀⡾⠁⠀⠀⠼⠿⠛⠿⣾⣷⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠀⠀⠈⢉⡁⠀⠀⠶⠆⠀⠀⠀⣼⣶⣿⣾⣿⣶⡇⠐⡀⢸⠇⠀⠀⠀⠀⠀⠀⠂⠈⢻⣿⣿⣿⣿⢿⢽⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣤⣘⣅⠈⠒⠀⠀⠀⠀⢰⣿⣿⣿⣿⠏⠘⣿⣿⡏⠀⠀⢠⠀⣀⣀⠀⠀⠸⣆⣠⣿⡇⠈⣬⡯⢺⣶⡇
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡟⠛⠻⢦⠀⠀⠀⠀⠸⠿⣿⣿⡿⠠⠀⣋⡙⠀⠀⠀⢸⣿⣿⣿⠧⣄⠸⡟⠛⡻⣝⡀⠘⣨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠈⣿⣶⢲⠞⢎⣳⣥⣴⠶⣧⡼⠋⠀⠀⠐⢽⠿⠀⠀⠠⣿⣿⣿⣟⣀⣈⣷⣝⢼⣿⣯⣷⣽⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⡀⠀⠀⡇⠈⣿⠸⠗⣠⣿⣧⣼⣿⠁⠀⠀⠀⠀⠀⠀⠁⠈⠓⠢⠬⠙⠿⠛⠿⠿⠿⠿⠦⠙⠟⠛⠛⠂⠉⠛⠛
⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⣰⣿⡿⣡⠔⠃⠀⣀⠗⢴⣿⢐⠓⣿⠧⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣀⢀⣘⣛⣿⣶⣴⣾⣿⣟⣉⣹⣦⠀⠀⣿⣰⠈⣿⡞⡷⡻⢓⡻⣇⣠⣾⢇⣴⡶⣠⡄⢀⣴⡄⠠⣀⠀⢀⡄⣠⣤⡄⠀⠀⡀⠀⠀⣀⡀⣀⣤⡰⠃
⠀⠀⣿⡿⢹⡏⠉⢹⠟⠙⡷⠃⣾⣿⣿⠟⠛⠻⣟⢀⣿⣭⣭⣷⣶⣼⣿⣿⣷⣿⣟⠿⢻⣿⣾⡉⣠⣾⡟⢫⠿⡿⢹⣿⣿⣷⣿⣷⠆⣿⣿⣿⢗⣿⣿⡇
⠀⠀⣏⠁⠘⠃⢤⣾⣤⣤⣼⣦⣼⣿⡇⣠⡄⠐⠿⠾⠟⠋⣉⣍⠀⢨⡙⠻⡏⠿⣡⣾⠋⡿⠃⡴⠋⢻⠇⠠⠊⣥⠀⣿⣿⣿⣿⣿⣿⡿⢻⡟⠘⣿⣿⡅
⠀⠀⠓⠃⠐⠐⠂⠀⠀⠀⠀⠀⠀⠚⠷⢿⣧⣤⠄⠀⠀⠀⠹⡿⠿⣭⠤⠀⠁⠠⣉⣴⠟⠓⠂⠀⠀⢦⣤⣀⣀⠁⠀⣿⣿⣿⣽⣿⣿⣾⣿⣯⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣷⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢶⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢳⣤⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⢣⠞⠃⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣦⢸⢀⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣠⠤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠒⠂⠤⠄
⣶⣶⣶⣶⣶⡶⠀⠈⠀⠀⠀⠀⠰⠆⠀⣀⠀⣀⠀⠀⠠⠠⡄⠒⢈⡧⣴⣾⠛⠛⢯⣀⡀
⠀⢁⡈⠉⢉⠃⡀⠀⠀⠀⠀⠀⡊⠃⠤⠄⣠⠄⠀⠀⠀⠀⠀⠉⠉⣠⠟⣠⠀⠀⠀⠈⣁⠄⠠⡤⠄⣀
⣿⣿⣿⣿⣿⢠⡃⢀⠀⠀⢢⡄⠉⠒⠐⡲⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⣴⡿⠆⠀⠊⠉⠁⠀⠀⢸⡀
⣿⣿⣿⣿⣿⡸⠀⠀⠀⠈⠉⠉⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣥⠄
⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠿⠛⠛⠛⠿⠷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⢠⡿⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣤⡿⠁⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⣤⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄⡀
⠀⠀⠀⠀⠀⣀⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢻⡇⠀⠀⣰⠟⠀⠀⢠⣾⠟⠉⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⣠⡾⠛⢁⡿⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠸⣧⠀⣈⣉⠀⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⣠⡾⠋⠀⠀⠈⠀⠀⠀⣠⣤⡶⠟⠉⠀⠀⠀⠀⠀⢀⣿⠟⠛⠙⠻⢶⣿⠛⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿
⣰⡿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⣆⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢰⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⡶⠶⠶⢶
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣼⡇⠀⠀⠀⠀⠈⠉⠀⣿⡆⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣀⣀⣤⣴⡶⠾⠟⠛⠛⠉⠉⠀⠀⠀⠀⢀⣼
⠈⠙⠛⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣀⣤⣴⣾⣿⡿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠈⢿⡄⠀⠀⠀⠀⠀⣠⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠘⣷⡄⠈⠻⣶⣤⣤⣴⣾⠟⢩⡟⠀⠀⠀⠀⠀⠶⢿⣯⣤⣤⣤⣤⣤⣤⣤⣴⡶⢾⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⠉⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣴⡟⠉⣀⣤⣴⣶⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠸⠟⠿⣤⡾⠃⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⢀⣀⣠⣴⠿⠋⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠦⠀⠀⠀⢰⡿⠁
⠀⠀⠀⠀⢸⣿⠁⠀⢰⡆⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⣴⢿⡏⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⢸⣿⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⢀⣾⣇⠀⣿⣀⡀⠀⣰⡟⠁
⠀⠀⠀⠀⠸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⣾⣁⣿⣶⠟⠋⠁⣼⠟
⠀⠀⠀⠀⠀⢿⡆⠀⢸⣿⣇⠀⠀⠀⠈⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣆⠀⠀⣿⠟⠉⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠈⢿⣄⢸⡏⢿⣄⠀⠀⠀⠘⣿⠿⢶⣄⡀⠀⠀⠀⠀⢀⣴⣟⣡⣀⣿⡄⠀⠁⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠀⠻⣷⣄⣀⠀⠸⣧⠀⠉⠛⠻⢶⣶⡾⠟⠁⣿⠉⣍⠹⡇⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠳⡌⠻⣇⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠘⣷⣠⣾⡙⢷⡄⠀⠀⢀⣤⣠⡾⠋⢀⣤⣤⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣤⣤⠄⠈⣿⡟⠻⣿⡿⠀⢰⣟⠉⣿⣷⣴⢟⡴⠟⠉⠀⣸⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠉⠀⠀⠀⢹⡇⠀⢻⣧⠀⠈⣿⣤⣿⠀⠀⠸⣧⣤⢴⡾⠧⢤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠰⣦⡘⠃⠀⠘⣻⢷⡀⠙⢿⣤⡄⠀⠀⣈⣡⣼⠷⠤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⠀⠀⠈⢻⣄⠀⠀⣻⡄⠻⣦⡀⢈⡏⠀⢰⡟⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⣧⠀⠀⠀⠙⢷⣿⠟⠃⢀⣿⣇⣾⠁⠀⣾⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡆⠀⠀⠀⠀⣹⡷⢀⣾⣿⣿⡏⠀⢠⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⡆⠀⠀⣰⣿⣷⣾⣇⠙⣿⠇⠀⣼⠇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢿⣤⣾⠏⠙⠿⣿⡉⠛⠏⠀⢠⣿⣦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠈⢿⣷⣦⣤⣾⠇⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡶⢿⡿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡟⠛⠛⠛⠛⠛⠋⠉⠉⢀⣠⣴⣾⠿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀⠀⠀⠀⠀⠀⠔⠛⠋⣠⠞⠁⠀⠀⠀⠉⠛⠳⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠟⣿⣷⠶⣶⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣬⣽⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⣴⡟⠉⢀⣠⣄⣀⣈⡉⠉⠛⠛⠛⠿⠿⠟⠛⠛⠉⠉⠙⢻⣅⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣩⣷⠾⠋⠁⠈⠉⠉⠙⠛⠛⠶⠶⠶⠶⠶⣶⣤⣤⣴⠶⠾⠛⠓⠶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣤⠶⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⣹⣿
⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠋⠀⣠⣾⠿⠿⠿⠿⠿⠿⣷⡄
⠀⠀⠀⠀⢀⣀⣸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡿⠋⠀⣠⣾⣟⣁⣀⡀⠀⠀⠀⠀⣿⡇
⠀⠀⢠⣾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⡿⠋⠀⣠⣾⠟⠟⠛⠛⠛⠿⣷⡄⠀⠀⣿⡇
⠀⠀⢸⣿⠀⢀⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⢀⣴⡿⠋⠀⣠⣾⣿⣥⣤⣤⣤⣤⡀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠈⣿⡏⠉⠉⠉⠉⠉⠀⠀⠀⢀⣴⡿⠟⠛⠉⠉⢛⣿⡿⠋⠀⣠⣾⠟⠉⠉⠉⠉⠉⢹⣿⠁⠀⣿⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⢀⣴⡿⠋⠀⣠⣾⣟⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⢀⣴⡿⠋⠀⣠⣾⠟⠹⣿⡄⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⢀⣴⡿⠋⠀⣠⣾⠟⠁⠀⢀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⣿⣷⡿⠋⠀⣠⣾⠟⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⣾⡟⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣠⣾⠟⠁⠀⠀⢀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⣿⡇⠀⢀⣿⡇
⠀⠀⢸⣿⠀⠰⣿⣷⣶⣶⣶⣶⣶⡿⠋⠀⣠⣾⠟⠿⣷⣶⣶⣾⠿⠛⠁⠀⠀⠀⠀⢶⣶⣶⣶⣶⣾⣿⠆⠀⣿⡿⠿⠿⠟⠁
⠀⠀⢸⣿⡀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠛⠿⠿⠿⢿⣿⡿⠋⠀⣠⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⢸⣿⣀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠼⠷⠶⠾⠿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⣒⣋⣭⣤⣤⣈⠙⠢⢤⣀⣶⠤⠀⠀⠀⠀⠠⣴⣂⣠⠜⠋⣉⣤⣤⣭⣍⣒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⣿⣿⣿⣿⣿⠓⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠚⣿⣿⣿⣿⣿⡿⠿⣾
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡅⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⣽⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣏⠁⠀⠀⠀⣠⣤⣿⡄⠀⠀⠀⠀⢸⣷⠾⠄⠀⠀⠀⠈⣙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣸⣷⠀⠀⣠⣶⣦⣍⠓⠀⠀⠀⠀⠏⣤⣶⢶⣦⠀⠀⢾⣏⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣠⣿⠀⠔⢙⣷⡾⢻⠇⠀⠀⠀⠀⠘⠟⢿⣾⠋⠓⠀⠾⢿⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⡴⠶⠷⠶⣦⠀⠀⠀⣾⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠢⡦⣀⢶⠌⠀⠀⣸⠃⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠈⠷⣄⣀⣀⣤⣭⣄⣀⣠⡼⠃⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠳⣄⠀⠀⠀⠉⠻⠄⠠⠼⠋⠁⠀⠀⢠⠖⡡⠺⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠘⣧⣾⣆⣀⠀⣀⠀⠀⠀⢀⣴⣴⠋⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⡀⢈⠟⠙⣾⠿⣶⠾⣶⠛⠉⠻⣠⣄⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⢋⣉⣉⣉⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠤⣊⣭⣶⣶⣶⣶⣮⣍⡢⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡰⣣⡾⠋⠁⣀⣤⣼⣿⣿⣿⣿⣮⢷⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣰⢳⠏⠀⢠⠞⠉⠀⠀⠀⠈⠳⡄⠙⢿⣷⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⣾⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢱⠀⠘⣿⡆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢼⢹⣿⣆⣷⠀⠀⠀⠀⠀⠀⠀⣼⠂⢠⣿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⢇⢻⣿⠻⣷⣤⣀⣀⣀⣤⣾⠟⢀⣾⠇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠳⣿⣷⣌⡛⠛⠛⠛⠋⣁⣴⠿⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣙⡛⠿⠿⠿⠿⠟⣋⣡⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣗⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⣿⣿⣿⣏⠙⠻⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⣦⣛⣽⣿⣿⡏⠀⢀⣀⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡿⢿⢻⣾⡿⠃⠀⢀⡀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠚⠀⠉⠀⠀⠈⠉⠃⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠂⠀⠀⠀⠀⠀⠀⢀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⢀⣤⠚⢿⠁⠀⠀⠀⠀⠀⣤⣀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣦⣶⣄⡀
⠀⠀⠀⢀⣴⣾⣿⡅⡡⡈⢧⠀⢀⠀⠀⠀⢻⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢯⠙⠉⢻⣿⣿⣿⡿⣷
⠀⢀⡴⠛⣿⣿⣿⣿⣆⠛⠆⢳⣾⡷⠶⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣶⡞⠀⣀⠐⠟⢻⣿⣷⣼
⢀⡏⠀⠀⠘⣿⣿⣿⣿⣦⣀⡠⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣙⣿⡿⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣏⠀⠙⠓⠀⠀⠀⣻⣿⣿
⣾⠀⠁⠀⠀⢸⣿⣿⣿⣿⣿⣧⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⠷⣽⣭⣙⡿⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣬⣄⣀⠀⠀⠀⢀⡾⢻⣷
⣿⡀⠈⠀⠀⣸⣿⣿⣿⣿⣿⣳⣩⡿⣿⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠄⢀⣠⡄⠀⠀⣬⣿⣀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠛⠋⠁⠀⠀⣰⣦⣴⠿⠟
⣿⠀⠀⠁⢨⣿⣿⣿⣿⣿⣿⠟⠁⠚⠁⠠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⢠⣶⣿⡿⠙⠿⣍⣐⠒⠤⠤⣄⣀⠀⠙⢳⡶⠛⠀⡴⠋⠁
⣷⡄⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⣾⡿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢸⡅⠘⣿⣿⠀⣀⣀⣀⡈⠉⠉⠉⠛⠛⣿⠛⠛⠛⠛⠛⢻⣄
⣿⢇⡴⢲⣿⣿⠿⠟⢉⡈⠳⣤⣤⣾⢋⣰⠟⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠦⠤⠾⠿⠿⠿⠃⠈⠉⠓⠦⣤⣀⣰⡟⢀⡄⣀⣀⣴⠟⢹⡀
⣿⡾⢀⡞⠀⠀⠀⠀⠈⣷⣾⣿⣿⣿⣏⠁⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠟⠛⠛⠋⠉⠀⠀⢘⣇
⣿⣷⣾⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣦⣤⣼⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣓⣒⣒⣒⣒⣒⣒⣛⣀⣀⣀
⣿⣿⡿⡻⠿⢛⠻⠿⡿⢟⣿⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠄⠂⣀⣀⣤⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡶⠶⠛⠯⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠃⢭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡌⠰⢤⣼⠎⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢯⡀⠐⠚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠙⠀⠐⢘⣯⣤⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣿⡿⠀⠀⠘⡖⠀⢀⠐⠰⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠳⣽⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠢⣀⠈⠀⣠⣼⣿⡥⢤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⠀⠡⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢀⠸⠀⠉⠀⠀⠁⠀⣨⠀⠀⠀⠀⠀⠀⢐⣂⡤⢀⣟⣯⡂⠀⠄⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⢀⢔⠁⠀⠈⢐⠀⠀⠀⠀⠃⠘⣽⣃⣀⠀⠀⠀⠀⠀⠛⠁⡸⢸⠙⠇⠀⡈⠁⠑
⠀⠀⠀⠀⠀⢀⠴⠁⠠⠱⢄⠀⢾⡀⠀⠰⣆⠀⠀⠀⠀⡂⠀⠀⢀⡠⠀⠀⣱⡇⠀⠀⠀⠀⠀⡄⢦
⠀⠀⠀⣀⣴⠋⠀⢀⡠⠔⠊⠁⠈⣌⠃⠀⠉⠀⠀⠀⢀⠁⠒⠋⠙⡗⠀⠀⢁⡇⠰⡀⠀⠀⠀⣁⡠⠳⣄
⠀⡴⣞⣿⣿⠔⠊⠁⠀⠀⠀⠀⠀⠜⠆⠀⠀⠀⠀⠀⠈⠀⠈⠑⠀⠁⠀⠀⠈⢳⠀⠁⠀⠀⠀⢠⡀⠀⠈⠂⠀⠀⠀⠀⢀
⣮⠄⢙⢻⢇⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠄⠀⠀⠥⡀⠈⠯⣀⠈⠙⠠⡔⢔⠀⠘⢤⡀⠀⠀⢀⠀⠙⠲⠖⠀⠀⢀⠧⠀⡀
⠻⠟⡧⠈⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⣟⠙⢢⠀⠀⠀⠈⠈⠀⠀⠉
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠈⠁⡄⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣥⣶⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⣂⣭⣭⣍⡲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⣼⣿⣽⣿⡟⢿⡿⠀⠀⠀⢀⣤⠀⠀⣠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢻⣿⠿⡿⠀⠘⢆⠀⠀⠀⠀⣄⠣⠜⠉⣼⢷⠄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⡟⢴⡂⠀⠀⠈⡆⠀⠀⠀⢹⠀⠀⢀⡴⠃⠀⠀⠀⠀⢀⡾⣇⣀⣤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⣀⡶⠀⢀⣤⠟⢀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡇⠉⠁⠙⢻⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⢰⠚⠉⢀⡴⠞⠁⢠⠞⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡶⠀⢱⠀⣀⡸⠖⢳⣏⠁⠀⠀⠋⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠁⠀⢸⠛⠁⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠒⢦⡴⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢣⠀⣀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⣤⠴⠒⠋⠀⢈⡿⢦⣄⣀⣀⣀⣀⣠⠴⠊⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣄⣄⣀⡀⠀⢀⡄⢀⡽⠛⢤⣉⠀⠀⣀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠈⠛⠯⣅⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠋⠉⠛⠉⠉⠙⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠯⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⠐⣿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠂⠀⠀⣄⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠗⠃⠀⠰⣿⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⢀⣤⡚⠳⠞⠃⠀⠀⠀⠀⠀⡇⠀⠁⢀⣀⡤⣤⠖⠒⠒⠶⣦⣄⠀⠀⠀⠀⢸
⠀⠘⣿⡟⣻⣦⣀⡀⠀⠀⡀⢀⣉⣀⣈⡡⠰⠞⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⣸
⠀⣸⠏⣽⣡⡟⡏⢧⡐⠤⢈⠉⢻⣿⠛⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⣰⠁⢰⠿⠿⣗⠳⣤⡘⢻⣾⣛⣿⠫⠀⣠⣾⣇⠀⠀⠀⠀⢀⣤⣤⡴⠞⠋
⣏⣠⠋⠀⠀⠙⢦⣀⣂⠈⠞⢻⢧⡥⠊⠁⠉⠙⠛⠛⠿⠶⠾⠛⠁
⠈⠁⠀⠀⠀⠀⠀⠙⢿⣷⣤⣾⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⣿⣿⠛⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠷⠋
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⡿⠿⠋⠁

⠀⠀⠀⠀⠀⡠⠀⠠⡀⠄⠀⢀⠀⠀⢠⡾⠁⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠄⠀⢺⣦⡀⠀⠇⠀⢠⡇⠀⠠⠆
⠀⠀⠀⠀⠀⣴⠂⠀⠀⠠⠴⣿⣿⡟⠀⠀⠀⠘⢁⠖⢀
⠀⠀⠀⠀⠀⠂⠀⠀⢀⣰⠾⠛⣩⡀⣠⣇⣤⠖⠀⠀⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡮⠿⣷⣾⣿⣭⣉⣀⣀⡀⠀⠉⠁⠰⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠜⣡⣄
⠀⠀⠀⡗⠀⠀⠀⠁⠀⠀⡼⢹⡿⣷⠄⠈⠉⠁⠘⠉⣿⣦⣰⡌⠛⠛⣛⣛⣭⣭⣭⡭⠶⠖⢒⣀⣀⡄⣤⣴⣶⡌⠉
⠀⠀⠀⠈⠢⣤⣤⣤⡤⠠⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⡆⠧⠞⠀⠀⡚⢋⣭⣭⡤⣶⡀⢸⣿⢹⡟⠉⠁⣿⠇⣿⡇
⠀⠀⠀⠀⠀⠈⠹⠄⠀⠀⣠⣴⠄⠀⠀⠀⠀⠀⣀⣠⠇⢰⣿⣧⢸⣿⢸⡏⢀⠀⢻⡇⣼⡏⢸⡿⠿⠀⣿⡿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣿⠶⠶⠶⠖⠒⠉⠁⠀⠀⢸⣿⢻⣾⣿⢸⡿⠛⠃⠘⣿⣿⠁⢻⣷⣶⠆⠿⣇⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⣀⣀⣀⣠⢴⡅⢸⣿⠈⣿⡿⢸⡿⠿⣃⣀⣿⣽⡆⢨⣿⠀⣹⣿⣿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠉⠉⠁⠀⠀⠀⠀⢽⡀⠈⣽⢶⣦⣼⡾⣿⠀⣿⢹⣯⣀⠀⠀⣿⠀⣿⣿⣿⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢺⣇⢸⡏⢠⣤⣾⡇⢿⣶⡏⢸⣿⣉⡀⠀⢿⣦⠿⠹⢿⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠤⠤⠤⠤⠴⠟⢻⡇⢸⣧⣠⣿⢿⡇⠸⠿⠃⢘⣋⠉⠁⠠⠐⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⡀⣠⢿⣀⣉⣩⣥⡤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠉⠋⠙⠉⠉⠁⠀⠘⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠉⠉⠉⠉⠉⠉⣩⣿⡿⣿⣯⣍⠉⠉⠉⠉⠉⠹⣿⣿⣿⡏⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠀⠀⠀⠀⠀⠀⣼⣿⠋⣁⣈⣹⣿⣆⠀⠀⠀⠀⠀⠈⠙⣿⡇⠀⣿⣿⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⠻⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣶⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⡀⠀⠀⠀⠀⠀⢻⣧⣌⡉⢁⣤⣿⠇⠀⠀⠀⠀⠀⢀⣤⣿⡇⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣙⣿⣷⣿⣟⣁⣀⣀⣀⣀⣀⣰⣿⣿⣿⡇⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠔⠊⠉⠀⠀⠀⠀⠀⠀⠂⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢁⠀⠀⣀⣀⡀⠉⠀⠉⢁⠀⡔⢹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡉⠉⠀⠀⠉⠉⠀⠀⠀⠀⠃⠨⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠋⠅⠒⠈⢁⠢⠠⠂⠌⠐⠒⠉⢢⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⠀⡱⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⠳⠠⠀⠀⠈⠀⡘⠛⠀⠈⠀⠀⠀⡸⠃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠄⡠⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠀⠀⠐⠀⠀⠂⠐⠊⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⢀⡌⢀⡠⡠⡢⡄⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠈⠉⠀⠀⠀⠺⢦⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠠⠀⠈⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢄⠘⠉⠛⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⡁⢂⣀⠀⠱⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠑⠤⠤⠔⠁⢈⡀⠀⠀⠀⠈⠂⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⡀⠡⢮⢹⡣⠆⠀⠀⠃⢄⠀⠀⠀⠀⡰⠈⡍⠀⣀⢡⠠⡄⣀⠀⡧⠐⣂⠠⠔⠊⠀⠈⡂⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡚⠉⣁⠀⠀⢰⠀⠀⠑⠄⡀⠀⠐⠂⢸⢈⠋⡘⡆⢣⠈⣬⠂⠘⢇⢀⠀⠈⠁⢌⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠱⡄⢐⠀⠀⢸⠀⠀⢀⠀⠀⠁⠁⠀⠀⠈⠏⠈⡏⠀⢙⠈⡤⢬⣉⡨⠄⠂⠁⡠⠛⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⢄⠀⠀⠈⠺⣄⠀⢸⠀⠀⠀⠈⠐⠂⠆⠀⠀⠀⡇⠀⡇⠀⠀⠀⠣⡠⠄⠤⠀⠂⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⣄⡀⠀⠉⠚⠤⠀⠀⡀⠒⠄⠌⠁⠚⠠⠇⢀⣇⠀⠴⠐⠉⠀⠀⡄⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠌⡉⠓⠂⠤⠀⢰⠀⣄⠀⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠤⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⢀⡀⢀⡠⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠤⠄⡀⢃⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠇⠸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠓⠐⠂⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠠⠀⠀⠁⠂⢀⡀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠒⠆⠰⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢞⣭⡽⣭⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣱⣿⡏⢤⣬⣿⣷⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⡿⠶⠈⣿⣿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣻⢷⣦⡾⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣒⣒⣲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣾⠏⣬⣿⣷⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⡷⠆⣹⣿⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢝⣻⣾⣛⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀
⠀⠀⠀⠀⠲⣶⣶⣦⣤⣤⣶⠿⠋⣉⡄⠀⠀⠀⠀⠀⣠⢔⣒⣖⡢⣄⠀⠀⠀⠀⣀⣄⣉⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣿⣿⣅⡀⠀⠀⠀⢠⢷⣿⡁⠾⣿⣾⡄⠀⠀⠈⢿⣿⣿⣷⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⢏⡿⣍⣡⠿⡽⠁⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣛⠋⢼⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣀⣼⣿⣯⣴⣶⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⣄⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⡟⢻⣷⡄
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠸⣦⣴⠟⣿
⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣀⠔⠻⠾⠄⠀⢻⠉⠈⠁⠸⠾⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠙⢻⡏⢯⢿⡍⠈⣭⣍⣩⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣿⣿⣿⣿⣿⣿
⣿⣿⣄⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿⡿⣿⣿⣿
⣿⣿⣿⣧⡀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⠉⠙⠟⠉⠛⠉⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⠿⣿⣿⡿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⡁⣀⣀⣁⣤⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⣩⣿⣿⢿⣿⣿⣿⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠏⠷⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠉⣧⠈⠉⠽⠃⠀⢻⣟⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣧⣀⠀⣀⣐⣻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠧⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢠⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⡀⠀⠀⢿⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⡆⠀⠀⠋⠋⣭⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠺⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣤⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⢈⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣃⠀⡀⠀⠀⠀⠁⠀⢀⣀⡠⠈⣷⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣯⣉⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⢀⠀⣴⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠀⣃⠀⠀⠀⠀⢒⣾⣀⣶⣟⣿⡟⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⣯⣠⣴⣷⣶⣿⣿⣿⣿⣿⣿⣯⣿⡖⣿⡇⢂⡀⢘⣛⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣤⣤⠀⠀⠀⣀⣀⢠⣤⡀⣆⠀⣠⡇⣀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆
⠀⠀⠀⠀⣼⣿⣿⡟⠁⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠻⠿⠿⠿⢶⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧
⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⢀⣠⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠹⣿⣿⣧
⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡴⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣦⡀⠀⠀⠀⠹⣿⣿⣧
⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⡀⠀⠀⢻⣿⣿⡆
⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣠⣼⣿⡿⣟⣯⠅⠀⠀⠈⣿⣿⣷
⠀⣿⣿⣿⠀⠀⠀⠰⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣴⣾⡿⣟⣯⡷⣛⣭⢾⡃⠀⠀⠀⣿⣿⣿
⢀⣿⣿⣿⠀⠀⠀⢘⢯⡻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣫⡵⣞⣯⠷⣛⣽⢾⣻⡅⠀⠀⠀⢹⣿⣿
⠀⣿⣿⣿⠀⠀⠀⠈⠳⣝⢮⡻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣻⡽⢞⣫⡷⣟⣯⢶⣻⣽⢞⣫⡵⠆⠀⠀⠀⣼⣿⣿
⠀⣿⣿⣿⠀⠀⠀⠰⣆⡈⠳⣝⢮⡻⣿⣷⣄⣠⣴⣶⣤⣄⣀⣤⣶⣿⡿⣿⣿⣿⣿⣿⣞⣯⡷⣛⣭⢶⣻⡽⠞⠋⠁⠀⠀⠀⠀⠀⣿⣿⣿
⠀⢹⣿⣿⡆⠀⠀⠘⠻⣿⢦⡈⠻⢿⣮⣿⣿⣿⣿⣿⣿⣿⡿⣟⣯⠷⣻⣽⠟⠻⣿⣿⣿⣷⣛⣭⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏
⠀⠈⣿⣿⣷⠀⠀⠀⠀⠈⠳⣝⢦⡀⠙⢮⡻⣝⢿⣟⣯⡷⣛⣭⠶⠛⠉⢀⣤⡶⣟⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁
⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠈⠳⣝⢦⡀⠙⢮⣛⣯⠷⠛⠉⢀⣠⡴⣞⣯⡷⠟⠉⠀⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏
⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠳⣝⢦⡀⠉⢀⣠⡴⢞⣫⡵⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⠀⠀⠀⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠙⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠳⣝⢶⣻⡽⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⢠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣾⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠠⡀⠀⠀⠀⣠⠄⠀⠲⠦⢄⠀⡈⠉⠛⠛⠛⠉⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠉⠀⠀⠀⠀⡼⠷⠀⠀⠀⠀⠈⠲⣆⠀⠺⠎⣶⠀⠀⠚⠏⠙⠁
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠒⠲⢤⣴⣿⣿⠑⠀⠀⠀⣀⠠⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢯⢠⡳⡏⠀⣐⢼⣶⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠤⠐⠊⠉⠁⣚⡂⠽⠟⣿⠀⠀⡤⠂⠈⠑⢷⡄⠀⠀⠰⠟⠴⠁⠀⠀⠀⢸⣷⣧⠀⢰⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠇⠀⠀⣼⡇⠀⠀⠀⢐⣿⠟⠀⣠⡎⠹⣆⠀⣠⣤⠄⠀⠈⢷⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⣿⠟⣤⣾⣧⡵⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⣋⡀⠀⠴⠿⠀⠀⢻⡼⠟⠁⠀⠀⠀⡘⡃⠀⠘⠿⠈⠀⠀⠀⢀⡾⠋⠁⠼⠟⢋⠀⡏⠀⠀⠀⣀⣀⡠⠤
⠀⠀⡀⠀⠉⠘⠀⠶⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⣸⡇⠀⠉⠉⠁
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⠆⠀⠻⢀⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣠⣤⣄⣀⣤⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠙⢿⠉⠛⠛⠛⠛⠻⣷⠀⢠⠤⡀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠟⠛⠛⠋⢹⣿⣿⣇
⠀⠀⠀⠀⢸⣿⠿⠿⠁⠀⢠⣤⡀⠀⠀⣴⡆⠀⠈⠁⠈⠀⢠⣦⣤⣦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟
⠀⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⠠⠤⣤⠄⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⣶⣶⣶⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠶⠶⠦⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣄⣀⣀⣀⣠⣴⣄⡀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣄⠀⠀⠀⠀⣠⣶⣾⣿⠟
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⡇⢸⣾⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢉⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣴⣶⣄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⢸⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇⢸⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠂⠀⢹⣏⡀⠀⠀⠀⣠⠖⠿⠿⠶⣦
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡛⣅⣿⠀⢠⣼⠏⠁⠀⠀⢠⣇⠀⠀⡀⡄⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠛⠛⢿⣍⠈⢀⡂⣁⢀⢸⣿⢰⡆⢀⢀⠞⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⡀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⣝⢃⣠⠊⢹⡾⣿⣯⠸⠡⠚⢸⢣⠀⢰⡟⡿⠤⠀⠀⠀⠀⠀⣵⣟⢸⡀
⠀⠀⠀⠀⠀⢰⡟⠟⣷⣿⡗⣀⡤⠊⠀⠀⠀⠀⠈⣿⠃⠘⠋⠓⢠⠀⠀⠀⠀⠀⣿⣿⢠⠃⠀⣠⠖⠻⣶⣄
⠀⠀⠀⠀⠀⣼⣧⣤⡾⣿⣶⡟⠀⡠⣶⡾⠑⣶⡆⠀⢈⠀⣿⣿⣾⣼⠀⠀⠀⠀⣿⣿⡃⡀⢀⠃⢠⠀⠸⣿⠏⠃⠂⠢⠄⠀⠤⡀
⠀⠀⠀⠰⣆⣿⠇⣸⣿⠧⠻⡀⢙⡤⢯⢻⡄⠮⠃⠀⢹⠀⠛⠛⠁⣹⡆⠒⠉⠁⣿⣿⠁⠀⠠⢇⣠⢸⡃⣻⠀⠀⠀⠒
⠀⠀⠀⠀⡭⣴⠞⠻⣍⣀⣤⢹⣿⡿⣯⠻⠟⠶⣄⡤⣴⠶⠒⠸⣿⡽⠃⠀⡀⠀⣿⣿⡄⠀⠀⡘⠛⠈⠁⠻⣄⡀⢠⣤
⠀⠀⠀⠀⠀⣇⠀⠀⡀⡿⡙⠦⠀⢀⠈⢳⣦⠜⠀⠀⠈⠀⠀⠀⢻⠗⢢⣤⡹⢋⢛⣋⠉⠉⢉⡝⠷⠤⠴⡀⡏⢿⣜⣻⣿⣶⣶⠶⠶⠶
⠀⠀⠀⠀⠀⢠⣏⠀⠈⠁⠈⠉⣭⣽⢶⠚⠛⠲⣾⡄⠐⠂⠀⠀⠘⡇⠀⠉⣢⣈⣁⣤⡤⠀⢾⣅⣱⠀⠆⠐⠇⡼⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢸⠇⠀⢀⡀⠀⢀⣀⣉⠀⠀⡄⡤⢌⠀⠀⡖⠸⠦⠤⡇⡀⠉⠉⠀⣠⠄⠤⠀⡀⣿⣭⣕⣒⡶⢀⡇⠀⠀⠠⠶⣿⠿⠂
⠀⠀⠀⠀⠀⡏⠀⠀⢸⣧⣾⣿⣿⣿⣿⣷⠃⠀⢦⣐⢼⣇⡀⠀⢠⠃⠀⠀⠀⠺⠻⠐⣦⠀⢀⣼⣬⣍⠉⠄⣮
⠀⠀⠀⠀⢀⡇⠀⣀⣸⣿⣿⣿⣿⡿⠟⢣⢀⡽⠒⢧⣀⠁⠘⢆⢸⠆⡦⠀⠁⠀⠻⠄⠁⠂⠈⡀⠄⣲⢠⣼⣿⠀⣤⣷⣚⣻⠶⠲⡄
⠀⠀⠀⠀⢸⣠⢼⡏⠾⠉⠛⠉⠁⢀⣀⡜⠀⡀⡀⢠⣿⣴⡐⠊⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⢿⣕⠄⢸⠀⣿⣿⣏⡛⠁⢀⡸⠃
⠀⠀⠀⣴⠟⠥⠀⠀⠀⠀⠎⠣⢻⣥⠀⣦⢸⠁⠅⣻⣿⣿⡟⢂⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡂⢙⣭⡆⢸⠀⠁⠘⠻⠁⠬⠋
⠀⠀⡾⠛⢘⠏⠀⠀⡀⠐⠀⠀⢛⣿⠿⣟⠧⣠⠄⠟⠙⠉⣻⡿⠆⢰⡶⠂⠐⠦⡀⠀⠀⠀⠀⠋⠉⠛⠟⠃⡟⠀⢀⣀⣤⣶⣦⣴⣼⡷⣼⣦⠀⢀
⠀⣸⡏⢹⣦⣄⠐⢀⠀⢤⡷⢀⣿⣧⢤⣍⡻⣿⣄⢘⣒⡘⠛⠃⡴⠲⣷⡖⠀⢠⠃⠀⠀⠀⠀⠃⠀⡀⢠⠀⠋⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⢿⡿⣃⡀
⠾⣯⠓⣾⣤⣞⠻⢻⢀⣀⠀⢾⠿⠿⠷⠿⠷⡄⣨⠏⠉⠀⢸⡧⠛⠋⠀⡀⣀⢻⣄⣀⠀⠀⠀⠁⢠⠁⢸⠀⠀⠀⠈⠿⣦⣽⡛⠹⠿⠿⠟⠢⢩⠍
⠀⡏⠉⠛⠦⢈⠁⢈⢈⠁⠀⠸⡀⠀⠀⣀⠀⡼⠴⢖⠤⣴⡀⢀⣤⡄⠊⠉⠉⠉⠁⠠⠄⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠸⠛
⠀⠹⡀⠀⠀⠀⠀⠘⢷⣶⣶⣏⡜⢲⣿⡆⢘⣵⣾⣷⣶⣿⣿⡿⠛⢻⣥⠀⠀⠀⠀⣀⣀⣀⣀⢈⠇⠀⡐⠀⣤⡖
⠀⠀⠙⠲⣤⣀⣤⠄⢺⣾⣯⣿⣞⣷⠙⠦⠼⡿⢿⢿⣿⣿⠇⢇⡀⠰⠀⠀⠶⠶⠂⠉⠉⠁⠈⠛⢴⣎⠠⠂⠉
⠀⠀⠀⠀⣾⠑⠄⠠⠚⢛⣻⠟⠉⠙⢆⠀⠀⠀⠀⠒⠙⠻⢷
⠀⠀⠀⠀⠙⠛⠑⠦⣀⣀⡼⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⢱⠟⠒⢄⠙⢧⡀⢀⣀⠤⠤⠤⠤⠤⠤⠄⣀⣷⡀⣀⡽⣋⠁⣈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢠⣟⢆⡌⠠⠵⠦⠋⠊⢀⠄⠤⠒⠒⠢⢄⡀⠀⢻⠓⠍⢠⠴⣛⢡⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠴⠃⢨⢫⠊⠉⠁⠂⡄⠂⠅⠀⠀⡀⠠⠁⡀⠈⠁⠀⠻⠓⠁⠀⠈⠺⣲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢀⡠⠤⠤⡀⠈⠘⢠⠊⠀⢠⣦⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡔⠉⣴⡦⠀⠈⣄⠀⣃⠀⠀⠀⠁⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⠀⠀⠀⠀⡐⣨⣥⢤⣤⢉⠒⢄⡀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⠢⣤⠔⣋⠀⠉⠙⠀⠉⡠⣌⡐⢍⠛⠉⠁⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣰⢉⡿⣹⠆⣠⠶⣄⠈⠷⣟⠙⣴⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢣⠿⣽⠋⣸⢏⣀⡀⠹⣆⠙⢯⣌⠙⠄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠑⠒⠖⠓⠦⣭⣀⣂⣈⢳⣄⠈⠐⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠐⢆⠀⠀⡇⠀⢠⠉⠙⢍⠀⠈⢉⡉⠉⠀⢦⠀⠀⠀⠀⠄⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡄⠑⠀⠘⠦⡀⠑⢄⠀⠈⠑⠀⠉⠒⠠⠄⠑⠢⠄⠀⢀⡴⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠲⣄⡷⣶⡏⣡⣤⣿⣣⡾⠋⠉⢳⣼⣿⠟⠛⢶⣰⡍⠁⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡀⠀⠀⠀⣎⣩⠏⠄⡄⢀⠈⣤⠋⠀⠀⠀⠉⠀⠀⠂⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡇⢀⠔⠀⠀⠘⢬⣬⣥⠏⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⠴⠁⠈⠹⠰⠀⠀⠀⠀⠘⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠠⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠋⠠⠌⠀⠀⠠⠀⡀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡞⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠭⠒⠒⠒⠒⠶⡖⠂⠳⢄⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⢄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣠⠇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠙⢄⠢⠠⡀⡐⡀⠀⠘⠀⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠛⡳⢲⠄⠀⠉⠛⡦⣤⣄⣀⣐⣀⣃⣙⠦⢤⣀⣀⣀⡂⣠⣀⣀⣀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠷⠚⠒⠛⠋⠁⠀⠀⠀⠉⠉⠉⠁⠧⣀⣇⣠⢦⠤⠞⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠀⠠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⠔⠊⠉⠁⠉⠉⠑⠢⡙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠁⠉⠉⠀⠀⠒⠒⠂⠀⠠⠤⠷⠆⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠋⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡔⢡⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠊⠀⡌⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⠔⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠤⣉⢢⠤⢤⠀⠤⠐⣊⠤⠒⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⡠⠤⠄⠤⠭⢓⣋⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢄⡀⠀⠀⠀⠀⢸⠀⢠⢂⡢⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢸⠀⢧⠃⠸⢸
⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢠⡎⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢨⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠇⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠐⡌⢦⡑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⡇⢸⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣁⡒⠠⠍⣁⡒⠒⠠⠄⣀⣀⠀⣀⠎⢸⠀⢀⠸⠀⠀⠀⡜⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠉⠩⠤⠒⣀⣀⠤⠐⠁⠀⠸⡀⠳⠤⢊⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠱⠆⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣀⣤⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢺⣶⣿⣿⡟⣿⣹⣿⣿⣿⣿⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⣿⣿⣡⣽⣏⣉⡿⠛⣻⡿⣿⠿⣿⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠛⠙⣿⣿⣷⣶⣭⣄⠈⣾⣷⣾⣳⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠀⢀⡰⠼⠛⠛⠛⠛⢿⣿⡥⢼⣿⣭⣤⣉⡁
⠀⠀⠀⠀⠀⢀⣌⠻⡟⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠶⣦⡈⠃⠀⣤⢤⣄⠰⢾⣿⠳⢦⣬⣿⠏⣨⡿
⠀⠀⠀⠀⢠⣟⢯⠙⠛⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⡆⠈⠁⠠⠦⣤⣄⠸⡆⠀⣽⣿⡆⣠⣙⣿⡏
⠀⠀⠀⠀⣿⡉⠛⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠷⠚⠁⠀⣠⣤⣤⡀⢹⡆⠀⠀⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⢼⠿⣇⠀⠀⠀⢀⣠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⡴⠾⢦⡆⢸⡿⠃⣷⡁⠀⠀⢠⣾⣿⢶⣿⣿⣧⣾
⠀⠀⠀⠀⠈⠙⢿⣄⠀⠀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢺⣶⣶⣰⡇⠀⠉⢉⠠⣍⠒⢀⣿⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⡇⠀⠀⣸⠆⢀⢸⣿⠟⣹⣿⣿⣿⣏⣵
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣠⣴⣶⣧⡀⠟⠛⣹⠃⠀⠀⣹⣶⠿⠷⠾⠛⠉⢿⣷⣟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣙⣛⢛⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣣⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⢃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⡴⣾⣷⣄⣀⣀⣀⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⣿⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠐⠇⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣀⣤⣬⣽⣧⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠉⢻⣇⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⣠⡾⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⢠⣾⣿⡄⠀⠀⠀⠀⣿⣿⡇⠀⠀⣰⡿⠁⠀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⣠⣾⣿⡛⢧⠀⠀⠀⠀⢹⣿⣿⠀⢰⡿⠁⠀⣸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣰⣿⡏⠀⣠⡴⠛⠁⣀⢹⣿⣿⡆⠀⠀⠀⠸⣿⣿⣠⡿⠁⠀⢰⣿⣿⡿⢻⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣿⣿⡴⠚⠉⠀⡠⣸⣿⣿⣿⣿⣿⡀⠀⠀⠀⣹⣿⣿⠁⠀⠀⣿⣿⣿⡇⣼⣷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣸⣿⡿⠀⠀⠀⣴⣱⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢹⣿⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠿⠟⠁⠀⢠⣾⢿⣿⣿⣿⣿⣿⣿⣿⠻⡆⠀⠀⠀⢿⣿⣿⣀⣿⡟⢹⣿⣿⣟⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠛⠛⣿⣿⣿⢿⣿⠏⠀⢻⡀⠀⠀⠘⣻⣇⣿⡟⠀⠀⢙⣻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣠⣾⠛⠁⣀⣀⣀⣰⣿⣿⣿⣿⣷⣶⣶⣶⣿⣄⠀⠀⢫⢾⣿⣿⣿⣿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠛⠛⠉⠈⠉⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠛⠻⣷⣶⡿⠿⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⡤⠴⠶⠂⠐⢒⣐⠦⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠐⠒⠤⠤⠬⠤⢴⡃⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡟⠲⢔⣚⣒⣖⣛⣻⡄⠀⠀⠀⠀⠀⠀⠦⠄⣐⣠⣶⣤⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⣀⠀⠤⠄⢤⡛⠢⠤⢤⣤⠤⠴⠾⢷⡄⠀⠀⠀⠀⠠⡓⠂⠴⠶⠶⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⣠⢖⣒⣏⠁⠈⢻⢹⡗⠤⣤⣀⣆⣄⣤⣤⣿⡟⣞⡇⠀⠀⠀⣏⠒⠒⡞⠶⠟⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢟⣺⣿⣇⠀⠉⣿⠉⠷⠠⠤⢧⢭⢬⢬⢶⢺⡿⣿⢓⡖⠒⠒⢯⢅⠐⡓⡛⢛⣳⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡤⠒⠐⣿⣏⠉⠘⠛⠛⣟⠂⠤⠼⠼⠴⠴⢶⠿⣟⠓⠒⠒⠒⡶⠚⢸⣿⠋⠛⠛⣹⠀⠀⠀⠀⠀⠈⠐⢁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⢄⣨⣉⣉⣭⡉⢘⣟⣛⣿⠢⠤⢠⣰⣠⣸⡤⢾⡿⣷⠶⠤⢶⡿⠾⣻⡿⠶⠛⠿⠻⠀⡾⠭⠤⡤⢿⡆⠀⠠
⠀⠀⠀⠀⠀⠀⡶⠒⣒⣲⣶⣀⡀⡾⢽⣉⠁⠀⠘⢿⢹⠦⢤⣀⣤⣤⣬⣤⣾⡟⠷⠤⠤⡼⡶⠶⢛⣿⠓⠰⢛⡃⠀⣏⡒⠲⢒⢻⡇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠉⠒⠺⠶⢏⠙⢻⢷⠄⣀⣈⣙⣛⣛⣹⣆⡀⢨⠉⠩⠉⢉⣹⡟⠖⠤⠤⡦⣷⣶⣿⣿⠿⠛⠛⢛⣲⣿⣍⣸⣿⣿⣷⣶⣲⣦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣄⡀⢈⢉⡉⡉⣻⣌⣉⡉⣙⢛⢙⣉⣅⡟⠻⠒⠒⠶⣿⣇⣉⣁⡀⢉⣉⣉⣉⣹⣽⣬⣿⣿⣿⠿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠉⠙⠛⢻⠻⢻⣄⣈⠉⢹⢛⢙⠟⡋⣿⢳⠒⠒⠟⣿⣏⣯⣉⣉⣉⣉⣉⣉⣩⣋⣉⡉⣀⣀⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⢤⡤⠄⡁⠒⠲⠾⠶⢿⡑⠂⠘⢩⢭⢿⡿⠿⣿⣾⡦⡶⢷⣿⣿⢽⣯⡉⠉⠉⢉⢉⣉⡏⢻⣙⢳⠶⡶⣲⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠤⣄⢀⣀⣀⣸⣦⠀⠀⠀⠐⣉⠐⠒⡒⡶⡾⠷⣷⣷⣾⣤⣤⣼⣴⣶⠿⣭⣉⣩⣽⣿⣭⠙⠃⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠺⠶⠶⣷⠿⠏⠁⠀⠀⠀⠙⠒⠠⠧⣧⡷⣿⠟⣿⣉⢒⣷⡖⢒⣲⡞⠩⠖⠀⠀⠀⢨⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠸⣍⠐⠒⠒⠖⠚⣏⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉

⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀

⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠠⠴⣤⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⠉
⢀⣁⣀⠀⠀⠀⠀⠀⠐⢐⢂⣁⠤⠄⠠⠤⠤⠄⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢈⡓⢲⡤⠬⠿⠾⠧⠤⠀⠤⠄⠒⠒⠒⠒⠒⠊
⠀⡅⠀⠠⣦⢄⠀⠀⠀⣸⣙⠃⠀⠀⠀⠀⢰⠚⠇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠋⠉⠁
⠀⢠⣬⣤⣿⢶⡟⢀⠔⢉⠃⣠⣾⣽⡆⢠⣞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠈⠹⠅⢠⡷⠀⠀⠫⠘⠉⡄⢔⡷⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡐⠉⠁⠂⢸⠄
⠠⠄⠰⠁⠀⣴⡷⠰⠀⠀⢀⢮⡖⠯⢉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠠⡿⠁⠘⠊⠉⡩
⠀⠀⠃⠰⠀⡽⠁⢀⣧⣶⡾⣿⡉⠰⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠛⠀⣠⡊
⠀⠀⠸⠶⠍⠭⠉⠸⠛⠋⠹⠾⣷⠴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⡄⢀⡴⠃⠀⠀⠀⠀⠀⢀⠚⠛⢓⠮⠄
⠀⠀⠀⣶⠀⠀⠀⢀⠁⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠈⠩⠦⣀⠀⠀⠀⠀⢠⠃⠀⠀⠈
⠀⠈⠓⡾⡀⠀⢀⣀⠀⠀⠀⠰⣀⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠈⠳⠀⠀⢠⣃⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠱⢞⠉⠈⠉⠐⠤⣀⣘⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⣠⠈⢻⢏⠁⠀⠀⠀⠀⠐⠐⠂⠀⠐⠂
⠈⠲⢀⠀⠾⡛⠄⠀⠀⠀⠀⠘⠹⠟⠃⠲⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠃⠀⠉⠀⠀⠀⠐⠂⢰⡄
⠀⠀⠀⠁⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣤⣀⠠⢄⡠⠠⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡗⠂⠠⠤⠤⠤⠔⢰⠛⠀⠀⢀
⠀⠉⠀⣠⡀⠀⣀⣐⠦⣠⠤⣀⡠⠒⡤⠚⠢⡔⠉⢲⠃⠀⠉⠀⠀⠃⠀⡘⠀⢀⠂⠀⠀⠀⠀⠀⠀⢰⡖⠀⢀⠀⠇⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢸
⠈⠑⠞⠁⠀⠉⠁⠀⠈⠃⠀⠈⡁⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠕⠒⠊⠀⠀⠀⠀⠀⠀⣀⠾⠁⠀⠸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢋⣯⠔⠚⡁⠐⠒⠀⠈⠉⠁⠉⠀⡀⠀⠸⠀⢸⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⣸⣠⠤⠄
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠅⣀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠰⠦⢀⣀⠀⠘⠀⠀⠀⠀⢘⣇⡠⠤⠓⠁⠍⢀⡀
⠀⠀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠈⠀⡆⠀⠀⠀⣎⣉⠀⠄⠐⠈⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡆⠀⠀⢰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢰
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠤⠂⠂⠐⠂⠁⠀⠀⡀⢠⡀⠐⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠤⠐⠐⠒⠈⠁⢀⣀⡀⠤⠤⠀⠒⠀⢰⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣳⠞
⠀⠀⣁⣠⡠⠤⠤⠒⠒⠀⠉⠁⢀⣀⡀⠤⠤⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠐⠀⠐⠀⠄
⠀⠀⠀⠀⠀⣀⠀⠤⠄⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠶⠿⠇
⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⠀⢠⢄⣔⢀⡀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠉⠃⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡤⠤⠄⠀⢦⠠⠄⠠⠤⠀⣤⠤⠤⡤⠀⠦⠄⠤⠤⠄⠀⠶⠤⠄⠄⠠⠶⠠⠦⠤⠀⠤⠄⠰⠤⠴⠆⠀⠲⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⣀⡀⢀⣀⣀⠀⣀⣀⣀⣀⣀⢀⣀⡀⣀⠀⣀⠀⢠⣀⡀⡀⠀⣀⣀⢠⡀⢀⣀⡀⡠⠀⡀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠀⠀⠁⠈⠀⠀⠀⠉

⣤⣀⠀⢀⡀⠀⣤⡀⠈⠉⠛⠿⣶⣾⣿⣿⣿⣡⠼⠿⠋⢀⣤⣾⣿⣿⠿⠷⠶⠶⠆⠀⠠⠄⢘⣛⡛⠛⠛⣛⣛⣛⣃
⣿⣿⣿⣶⣿⣿⣿⣿⣿⣦⠀⠱⡌⠙⢿⡿⢿⡏⠀⣠⣴⣿⣿⣿⣿⡏⠛⡿⠲⠒⠖⠒⠶⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣧⡀⠡⠀⠈⠃⣘⣥⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀
⣿⣿⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣤
⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡏⣿⣿⣿⣿⣿⠛⢻⣿⣿⣿⡿⢷⡾⢿⣿⣿⣿⣿⣿⣿⡿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⣿⢻⣿⣶⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣏⣥⣼⡛⠀⡀⢤⡄⣤⣀⣄⣀⡄⣀⣿⣟⢛⣛⠉⣉⣛⣿⡿⣿⢣⣄⡀⠀⠀⣰⣾⣦⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣿⢼⣿⣟⢿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡏⠬⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠋⠛⠙⠉⠁⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿
⣿⣿⢿⣿⣿⢸⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠿
⣿⣿⣿⣿⡏⢺⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⢙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁
⣿⣧⣿⣿⣷⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠁⣉⣿
⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠀⠿⠿
⣿⣿⣿⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠁
⡿⢿⡟⢸⣏⣿⣿⣿⣿⣿⣿⣧⣵⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⣶⠤
⠋⠉⠛⣸⠛⣿⠏⣿⣿⣿⣿⡿⢿⣿⣫⡿⢇⣠⣤⢀⡴⠂⠘⠁
⣶⣶⣶⣾⣤⣿⡄⣿⣿⡿⣿⠿⣿⡿⢿⣷⣾⣿⡟⠉⠀⠀⢰⠠
⣿⣿⣿⣿⣿⣿⣧⣿⣿⣧⣿⡀⠀⠀⢸⠆⣿⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣯⣿⣿⡟⣿⣿⠃⢻⡄⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣟⣩⣿⣿⣇⣿⣿⣇⣻⣇⣀⣷⣘⣿⣿⣿⣿⡆⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⣀⢀⣀
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⢀⠀⢀⣀⡀⠀⠔⠠⠴⠦⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⢀⢨⡄⠁⠀⠀⠀⠀⠀⠤⠤⠶⠂⢀⣍⣍⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠄⢠⠀⠀⠀⠀⠀⠀⠴⢃⡀⠀⣄⣠⢕⡢⠤⠤⣄⣤⡀⡀⠀⠀⠀⢂⡠⠄⠒⠃
⠀⠀⠀⡀⠀⠀⣐⡠⣀⠂⠤⠤⠤⠔⠇⠉⠀⢺⠙⠀⠈⠉⠁⠀⠩⠌⠀⠁⠐⠋⢉⢁⠄⢀⡆⠠
⠀⠀⠀⠀⠁⢘⠁⠀⠁⡁⠀⠀⠀⢀⡿⠄⠤⢼⡉⡌⠁⠂⠛⠓⠷⠶⠄⠐⠪⠭⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠁⠀⠀⠀⣇⠀⠀⠒⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠒⠒⠹⠗⠋⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣀⠤⠒⠒⠈⠉⠁⢂⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⠀⣀⡀⠀⠀⠉⠳⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠊⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣴⠞⢠⠊⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣥⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠛⢻⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⠛⠁⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⣀⣀⣀⣠⣴⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠉⢩⠌⣹⢍⡍⢹⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡇⠊⠀⡇⠁⠘⠀⢈⠗⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠄⢹⠈⡇⡰⠃⠀⠀⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢁⣁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢊⣼⠌⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠐⠀⢁⠀⢀⠋⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠄⠰⡆⠉⠢⡀⠀⡏⠀⠀⠀⠀⡸⠁⡸
⠀⠀⠀⡀⠄⠂⠈⡀⠁⠀⠨⢤⠈⣱⠈⠀⢠⡀⠀⢹⡄⠀⠀⠀⠀⠀⠨⠀⠀⠀⠰⠀⠚⠂⢇⣀⣀⡠⠔⠁⢠⠇
⠀⠠⠊⠀⠀⢰⢱⠀⢠⣯⣠⡎⠉⠁⠀⣶⠞⠁⠀⣸⠃⠀⠀⠀⠀⠠⠀⠄⠀⣴⠀⠀⠀⠀⠀⡇⠀⠀⣀⠴⢻
⠀⡇⠀⠀⠀⠀⠛⠤⣼⣁⠸⠧⠴⠒⠊⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⠀⠀⢰⠊⠙⡉⠉⠀⠀⣼
⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⢛⠽⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⢆⠀⠀⢹
⠀⠀⠈⠙⠛⠒⠲⡒⠛⠛⠉⠉⠁⠀⠀⠈⠀⣰⡇⠀⠀⠀⠀⢀⣀⠤⠲⠤⢤⣴⣀⣀⣀⣀⣟⠢⣀⠀⠳⣄⡜
⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠶⠒⠒⠛⠛⠛⠉⠀⠀⠀⠘⠻⠤⢤⡤⠖⠚⠉⠀⣀⣠⣄⣀⡥⠀⠀⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠋⠉⠉⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣎⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠠⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠠⠀⡀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣵⠆⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣖⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⣈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡴⠖⣋⣠⣴⠶⠛⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡶⠭⢃⡀⠀⠀⠀⠀⠀⠨⠽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⡛⠳⠄⠀⠀⠀⠀⠀⠀⠘⠋⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣐⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⠶⠌⠁⠀⠀⢀⠀⠀⠀⠛⠛⠓⠾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⢀⣄⡀⠀⠀⠀⣀⣠⣤⣤⣤⣴⢿⡛⡛⠛⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⡉⢳⣄
⠀⠀⠀⠀⠀⢀⡴⠊⠁⣀⠀⢀⡾⠟⢿⢶⣌⡉⢠⡆⢋⣽⠋⢸⡯⠥⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣻⣦⡀
⠀⠀⠀⢀⡴⠋⢀⡴⠋⠁⠀⠀⠀⠀⠀⣾⠁⠉⠻⣿⡉⠁⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⣫⠊⣿⣄
⠀⠀⠀⡼⠁⣰⠀⠀⠀⠀⠀⠀⠄⠀⢀⠏⠀⠀⠀⠀⠙⢶⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡴⢸⠍⡷⣄
⠀⠀⠀⡇⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⣷⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣀⠁⡿⢆
⠀⠀⠀⡇⣸⣿⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡴⠀⣿⢆
⠀⠀⢠⠇⣻⣿⣦⣀⣤⣶⣃⠔⠠⢂⡄⠀⠀⠀⠀⠀⠀⣦⠀⣿⡷⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⣾⡄
⠀⣴⠏⠰⠋⠈⠩⠝⠋⠉⠛⠛⠳⠮⣔⣁⡖⡠⢀⡐⣠⠿⣠⣿⠛⢦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣿⡀
⣼⠇⠀⠀⢀⡾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⢧⣶⣿⣿⡗⢦⣉⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⢿⡇
⣿⢠⡇⢰⣾⡃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢿⣷⣦⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡾⢻
⢻⣾⣇⢸⡿⢧⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣷⢸⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⡾⡃
⠈⣿⢿⣜⣧⠀⠀⢠⠇⢀⠀⣄⢂⠀⡀⠀⡀⢀⠀⠀⣿⣆⣿⢸⣿⣿⢬⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢃⡇⢠
⠀⣿⠀⠙⠿⡶⢦⠿⠷⢿⡿⠶⢶⠶⡷⢷⣼⣮⣼⣤⣧⣽⣷⣿⣿⠛⣦⣤⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢠⠏⣼⠃⣼
⠀⣿⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢷⠐⠀⠀⠈⠀⠀⠀⠁⠈⠝⣿⠣⣌⠳⣭⣙⠓⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢰⠏⢰⠇
⠀⣿⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣏⡓⢬⡁⠈⠻⣄⡑⠠⠤⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣠⡟⢀⡇
⠀⢻⡀⢀⢸⡇⢀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⡄⢻⣼⣿⡏⠲⣆⠀⠀⢀⣹⣿⡶⠶⠶⠶⠶⠆⠀⠀⠋⢁⣴⡟⠀⣾
⠀⠈⣧⢿⣤⡹⣌⢥⠀⡀⢀⡇⠀⣾⢰⠀⠀⡀⣀⣠⣦⣷⣾⣿⡟⠂⠉⠡⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁⡇⢸⡏
⠀⠀⢹⠀⠙⢛⣾⣿⣷⣼⣿⣷⣾⠷⠖⠚⠛⠋⠉⠁⠀⠈⣿⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢀⡿
⠀⠀⢸⡄⠀⠋⠈⠀⠻⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣘⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣼⠃
⠀⠀⠘⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⡿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇
⠀⠀⠀⢣⡀⡄⢆⠀⠘⢦⠀⣆⠀⠐⡇⠀⠀⡀⠀⢀⣾⣰⣿⡇⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠛⢿⣜⣧⡑⠘⠷⣬⣦⣄⣹⠘⣃⣧⣤⣾⣿⠏⡻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⢀⡖⢀⡴⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣿⣿⣷⣿⡿⢛⡝⣁⢆⡼⢁⡂⣠⢀⡄⣀⣶⠰⣠⣆⣄⣀⣀⣼⣴⣯⣴⣯⣴⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠿⠷⠿⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠴⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⢶⣒⣋⣁⣀⣈⠓⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠒⡶⠻⠉⢹⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣗⣒⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⢤⣘⣓⣒⡺⠤⠤⠤⠤⠤⢤⡤⢤⣤⢤⠤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡏⠉⣟⢒⡖⢺⠓⠒⠚⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠘⡟⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⡾⣏⡇⠀⠀⠀⠀⢸⣿⢿⣿⣷⢿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠉⠉⠉⠛⢇⣀⣀⣀⣀⡸⠛⠉⠉⠉⠙⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣓⣒⣒⣒⣒⣒⣚⡇⠀⠀⢸⣓⣒⣒⣒⣒⣒⣒⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄⢄⠀⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⣄⣤⠴⣖⣒⡞⡋⣀⣠⣼⢉⢉⣱⠷⣤⡴⠒⠒⣦⣤⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠈⠉⠉⠛⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠁⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠤⠤⠤⠤⣄⣀⣀⡠⠤⠤⠤⠤⠐⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢰⠄⠀⣄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣹⣿⣙⣿⠰⣆⠀⠀⠀⠀⢠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣄⠀⠐⠀⠉
⠀⠀⠀⠀⡤⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠉⢻⡏⢻⠺⣏⣀⠀⠀⠀⣿⣻⣿⢷⢟⣞⣹⣸⣹⣟⣿⠀⠀⠀⠀⢀⠴⠒⠀⢀⠀⠀⠄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠚⠓⠿⠺⠛⠛⠛⠂⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⢀⠐⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢴⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠄⠈⠈⠉
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣯⣻⣿⣝⠃⡳⣌⣿⣃⣘⣻⡇⠀⠀⠀⠸⠶⡄⠀⠀⠈⠛⠉⠅⠐⠀⠀⠀⣶⠰⠆⠀⠀⠀⠰⣴⣀⣀⣀⠤⠔
⠀⠀⠀⢀⢀⣀⣀⣤⣤⣀⡀⡄⢀⠛⠟⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠠⠈⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠠⢀⡀⠈⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠁⠈⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠹⣯⡏⠀⠀⠀⠀⣈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢿⣿⣿⣧⣀⣀⣼⣿⣿⠇⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠐⢀⣤⣴⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⠀⢊⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠐⠸⣿⣿⡟⣿⣿⣷⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠘⢰⣿⣿⣼⡻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢠⠀⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢰⢸⣿⣿⣯⣵⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⣀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢟⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⠏⢿⣜⢿⡟⣿⢻⣿⢧⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣱⣶⣶⣦⣶⣦⡄⠐⠀⠀⠀⠀⠀⠀⠻⣿⣞⣼⣾⣿⣿⣧⣷⣿⣾⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠠⠁
⠀⠀⠀⠀⠀⠠⠀⠉⠉⠁⠉⠉⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠓⠀⠘⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⡾⠦
⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠙⠀⠈⠌⠀⠀⢀⡤⠖⠚⡻⣛⠛⢦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⡀
⠀⠀⠀⢠⠀⠁⠀⠈⠀⠲⠀⠐⠀⠀⠀⢠⣀⠀⠉⠊⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡟
⠀⠀⠀⠈⠿⠿⠿⠿⠆⠀⠃⠐⠀⠀⠀⠀⠙⠛⠂⠘⠀⠀⠘⠈⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠚⠉⠉⠉⠲⡀
⠀⢀⠈⠀⢀⠴⠋⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⡇
⢀⠀⢢⡀⢨⠆⠀⠐⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⢠⠔⠂⠀⠐⠷⠄
⡇⠐⠤⠎⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣗⠀⠀⠀⠀⠀⠐
⠈⠀⡄⠀⠀⠀⠀⡠⠟⡙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠉⠉⠘⠀⠀⠀⠺⠃⢀⣐⠿
⠀⠀⠈⠒⠂⢻⣈⠔⠊⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠘⡀⠀⠙⠶⠀⠀⠀⠀⠈⠉⠀⠈⠉⠙⠒⠤⣄
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠈⠒⠖⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠑⢄⡠⡔⠁⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡅⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠈⠁⠒⠀⠘⡀⠀⠀⣩⠴⢖⠀⠀⠐⠂⠁⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠁⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⡀⠈⡃⠀⠀⠰⠀⡠⠞⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠘⠁⠀⡱⠀⢠⠀⠋⠀⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠌⢀⡴⠂⠀⠐⣄⣇⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠋⠀⠈⠉⡗⠚⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠶⠇⠤⠋⢰⣈⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⡆⠀⢠⠇⠀⠀⠀⠀⠸⠀⣌⠙⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡆⠀⠀⠀⡀⠜⢰⡇⠀⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠀⠒⠒⠂⠀⠠⠤⠤⠤⠤⠼⠀⠑⠒⠉⠀⠀⢐⡀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢀⠾⠤⠤⠀⠀⠄⠋⠔⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠁⠉⠉⠀⠀⠀⠀⠀⣀⣤⠾⠄⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠉⠀⠀⠀⠀⠁⠀⠈⠁⠀⠒⠒⠂⠤⠤⠄⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡤⢤⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡜⠉⠉⠙⠋⣀⣀⣙⠛⠁⠀⢶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⣈⡴⠖⠓⣀⠉⣉⣉⣉⡉⠓⠒⢤⣍⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠖⢋⡀⠊⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠈⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠞⠁⠄⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠈⠀⡀⢳⡄⢀⡁
⠀⠀⠀⠀⠀⠀⠀⢠⡔⠈⢠⡆⠠⠀⠀⠀⠀⣀⡤⠴⠾⠟⠛⠻⢿⣴⣀⠀⠀⠀⠘⣄⢰⣄⠙⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠐⠂⠀⠒⠀⠀⠚⠃⠀⠀⠒⠒⠀⠐⠘⠋⠀⠀⠀⠀⠈⠀⠉⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠆⠀⠆⠀⢠⠆⠀⠀⠀⠀⠠⠀⡀⠀⠰⠀⡆⠀⢀⠀⡄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⢀⡀⠀⢸⡄⠀⡆⢰⡆⢦⠀⠃⠀⠀⠀⠁⠀⡰⠀⣆⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢤⡍⠈⣠⠤⡄⠀⠀⠈⠀⢀⡨⠅⠀⠠⠤⠤⠄⢀⣤⠄⠀⠄⠀⠀⠀⡤⠄⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠸⢦⡀⠙⣆⠱⡄⠀⠀⠀⠉⠁⠀⠤⠄⠠⠤⠾⠛⠉⠀⠀⠀⢀⠆⢠⠃⢀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠰⣦⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡤⠄⢀⡤⠀⣠⡤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⢀⣙⠳⢦⠀⠀⣀⣀⣈⣉⣉⣤⠴⠞⣋⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢷⣄⢀⣠⣌⠉⠉⣩⠄⠀⠀⡸⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠘⠂⠀⠁⠀⠈⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⢿⣿⡿⠿⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⢄⠛⠿⡇⡸⠏⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢨⣿⣿⣿⣿⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢿⣿⣤⣀⣤⠏⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢠⢀⢀⠄⡈⠻⣿⣿⣿⣷⣿⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠳⣴⡿⢀⠀⠙⣿⠿⠿⢾⡿⣷⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⣿⣿⡈⢀⠃⢻⣇⡆⣸⠁⣿⡀⡄⢀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⢠⡀⣸⣿⣧⣼⣤⢸⣿⢥⡞⠀⣿⠇⢰⣾⣿⡛⣿⣿⣇⠀⣀⠀⠘⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣏⢀⠀⣸⠇⣿⣿⣾⡿⣿⢸⣿⡼⡿⢠⣿⣸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡟⢸⣾⣿⣿⣿⣯⣶⣿⣿⢿⣿⢻⣿⣼⡷⣿⣿⣿⣿⣿⣿⠋⢋⣯⣈⡛⢛⣽⣏
⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⢋⣼⡿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⡇⠈⠿⣿⣿⣿⣿⣷⣬⣭⣭⣙⣷⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣠⡬⢛⣻⣿⣿⣿⣿⣿⣷⣶⡿⣿⣿⣿⣿⣿⣿⣯⣵⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⢀⡀⠀⣄⠀⣀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠔
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡷⠀⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣔⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡿⠀⠀⢀⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⠶⢦⢀⣴⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣰⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠴⠶⠶⠶⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⣀⣤⠴⠖⠒⠓⠛⠒⠲⠦⣤⣀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠾⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠚⠛⠒⠒⢶⣄⠀⠀⠈⠳⣖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠒⠒⠒⠛⠓⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡞⠉⠉⠉⠙⠳⣄⠀⠀⠈⢻⣍⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⣄⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠉⠷⣄⠀⠀⠈⢻⣌⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡄⠀⠀⠈⠳⣀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⢲⣄⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣈⣳⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢧⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣭⣭⣭⣷⣄⠀⠀⠈⠳⣤⣠⣤⣠⡼⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⣶⠶⠶⠶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⣄⠀⠀⠈⠳⠶⠶⣶⠶⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠈⠙⠓⠶⠦⠤⠤⠴⠶⠚⠋⠁⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠛⠋⠉⠁







⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿