Classroom Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⣿⡟⠻⠛⢉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠗⠀⠀⠘⠿⠿⢿⣿⣿⠿⠿⠿⠿⠡⠀⢊⡇
⠠⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⡢⠕⠒⠲⠀⠉⠉⠉⢉⣼⣿⡯⣠⠎⢁⣼⣥⢈⠀⠀⠀⠘⠀⠠⠼
⣀⠀⢠⣾⣶⣶⠀⠀⠀⠀⠀⠂⠀⣟⠤⠄⠠⠐⠄⠢⠄⠀⠈⠉⠙⣧⡃⠀⢤⠠⠞⠙
⢹⠃⢸⣿⣽⣿⡆⠀⠁⠀⠀⠀⠀⢀⢀⢁⠀⠉⠁⠀⠀⠀⠈⡀⠀⠃⠀⠀⠀⡀
⠄⠠⠉⠿⠛⠟⠀⠀⠀⠀⠀⠀⠀⠁⠈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤
⠀⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠰⠀⠈⠀⡴⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠓⠄⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠁⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡄⠀⠀⠀⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠀⠀⠉⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢙⡃⠈⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠁⠊⠉⠁⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠰⠖⠲⠾⠀⠀⠀⢠⠀⠆
⠀⠀⠀⠀⢈⠁⠀⢀⣀⣠⠀⠐⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠐⠂⠤⠀⠀⠀⠀⠀⠰⠖⡆⠀⢢⣀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠚⠀⠄⠀⠀⠀⠀⠶⠚⠓⠀⠀⠀⠀⠰⠺⠆⢠⠀⠀⠀⠸⡈⠓⠂⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⠁⠀⠀⠈
⠀⠀⠀⢶⣴⠀⠘⠀⠀⠀⠾⠁⠐⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⠤⠄⠈⠁⠂⠀⠀⠀⠄⢀⣤⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠏⠀⠀⠠⠰⠀⢀⣀⣠⡀⡀
⠀⣄⢀⣸⡷⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠒⠀⢈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠔⠀⠀⠼⠏
⠀⠀⠁⠀⠀⠀⠀⠳⠖⠺⢃⡀⣀⡀⠀⠀⠀⠀⣠⡄⢰⣆⣀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠒⢲⣶⠃⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⠀⠀⠀⠈⠀⠀⠀⡿⢦
⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠆⠁⠈⠁⠀⠀⠀⠈⠀⠀⠀⠈⠉⡀⠀⠀⠓⠀⠀⠀⠐⠉⠀⣾⠈⠀⢠⡀⠀⠒⠀⠀⠀⠀⠃⠀⠀⠸⠇⠀⠀⠠⠀⠰⠆⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠑⠆
⠀⠀⠀⢀⣠⡤⠄⠠⠐⢦⠀⠀⠀⠀⠀⠀⠀⣤⠲⡄⣀⠀⠀⠀⠠⠖⠀⠀⠀⢠⠀⠰⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⡏⠀⠀⠀⠤⣤⣀⡀⠀⠀⠀⠀⡀⠀⣀⠜⠏⠸⢠⡄⠈⠀⠀⠀⠀⠀⣀⣀⡄⢲⠤⠔⢨⠃⠽⠂⠀⠀⠀⠀⠰⡙⠶⡂⠾⠤
⠀⠀⠃⣨⡿⠦⣆⠀⠀⠀⠉⣁⠘⠀⠀⠀⠱⡆⠙⠀⠠⠊⢠⠗⠠⡦⢈⡀⠀⣤⣠⢤⡄⠀⠀⠀⠠⡷⠀⠀⠆⠀⠀⢠⢀⢠⣄⡁⠀⠀⠀⠀⠂⠀⢀
⣉⡀⠀⠉⠀⠀⠼⠀⠸⡇⠀⡍⠀⠀⠀⠀⠀⠀⠐⢀⠀⢉⣹⠀⠀⠸⠟⠀⠀⠀⠀⠀⠁⠂⣀⡀⠀⠃⠸⠃⠀⠀⠀⠘⠇⢀⡝⠇⠀⠀⢀⣸⢟⠙⣿⣄
⢀⠈⠀⢨⠀⠰⣀⠀⣀⠀⠀⠄⠀⠀⠀⠀⠀⣀⡀⢠⣀⣤⡝⠀⠀⠀⠠⠷⠀⠀⣀⠀⠰⣤⡤⢼⠆⠀⠀⠀⠀⠀⠀⠐⠀⢾⣹⡆⠀⣀⢀⣾⢻⠀⠀⢹⣆
⠈⠁⠐⠛⠀⡀⠙⠐⠃⠒⠀⠾⠀⠀⠀⠀⠠⠏⠙⠀⠓⠒⠂⠀⠰⠷⠀⠓⠀⠘⠋⠒⠎⠠⠀⠴⠂⠀⠀⠀⠀⠀⠀⢠⡦⡿⢙⣄⣉⠍⠘⠏⠘⣀⠀⠈⡏⠁
⣀⣀⣀⣀⣀⣁⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣇⣀⣁⣀⣀⣀⣀⣀⣀⣘⣋⣀⣀⣀⣀⣰⣄⣀⣀⣐⣁⣀⣀
⣿⡿⣿⡿⣿⠿⢿⠿⠻⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⡿⠟⣻⡿⢿⣿⣿⠿⠿⠿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣤⣀⣾⣿⣿⣷⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⣤⡀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣠⣤⡀⠀⢸⠀⢀⣤⣄⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢀⣤⣼⡇
⠀⠀⣿⣿⣷⠀⠀⠀⣾⣿⣿⠀⠀⠀⢰⣿⣿⡇⠀⢸⡀⢸⣿⣿⡆⠀⠀⢰⣿⣿⣗⠀⠀⠀⣾⣿⣿⡇
⢸⣿⣿⣿⣿⡍⣧⣿⣿⣿⣿⣏⢹⣾⣻⣿⣿⣿⡙⣏⣿⣿⣿⣿⣯⢹⣿⣻⣿⣿⣿⡝⣯⣟⣿⣿⣿⣯⢻
⣾⣿⠋⠈⢻⣿⣿⣿⡟⠁⠙⣿⣿⣿⣿⠋⠈⢻⣿⣿⣿⡟⠁⠙⣿⣿⣿⣿⠋⠈⢻⣿⣿⣿⡟⠁⠙⣿⣿
⣿⣿⡄⠀⣼⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣧⠀⢠⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣧⠀⢠⣿⣿
⣿⠃⠀⠀⠀⠹⣿⡟⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠹⣿⡟⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠹⣿⡟⠀⠀⠀⠈⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⡇⢹⡃⣸⢿⡇⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣹⡇⠉⠁⣿⣼⣧⠐⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⠿⠗⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠶⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⢸⣿⣛⠃⢦⡆⣾⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⡛⢹⡇⠞⠇⣿⢻⣧⢨⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠘⠿⠾⠃⠀⠀⠻⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡗⠀⠀⣴⢶⡄⢀⡀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣬⡅⣸⡇⣿⣧⣭⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠛⠛⠛⢻⣿⣿⣿⠀⠀⠀⣿⣼⡇⠀⠀⢿⣧⣿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣺⣿⣿⡏⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡀⠘⠓⠒⠚⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠺⠿⠿⠷⠶⠾⠿⠿⠷⠾⠃

⠀⢀⠔⠀⠐⡄⠀⠀⠀⠀⠀⢀⠔⢉⢉⠑⡄
⢠⡡⠊⠉⡆⡀⠀⠀⡎⠃⠀⠈⠒⠁⢠⢃⠇⢀⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡖⠛⡄
⠀⠀⠀⡜⡰⠁⣖⣂⠃⢘⣒⡆⠀⡠⢃⠎⠀⢈⣉⣁⣈⡩⠖⢊⡩⠟⠋⠉⠉⠉⠉⠳⣄⠀⠀⠀⠸⡀⢱
⠀⣠⠊⡴⠁⠀⠀⠀⡁⡏⠉⠁⡰⢁⠎⠀⠀⠘⠒⠒⢺⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣠⢷⡈⡟⠓⠲⡄
⢴⣁⠚⠀⠤⢤⠀⠀⠉⠁⠀⠴⣀⣉⣐⠒⠀⡄⠀⠀⢸⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠱⠇⠀⠀⠉⠳⡆
⠀⠀⠈⠉⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠤⠀⣠⡼⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠶⠚⠛⠋⠉⠙⠛⠛⠿⢤⣀⠀⢰⠇⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣅⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢲⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⡴⠚⠁⣰⠉⠉⠋⠉⠉⠉⠉⠋⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠠⡀⢀⣀⠠⠤⢤⣀⡴⠀⠀⠀⠀⠸⡏⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⠀⢠⡇
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⣇⠀⠀⠲⣀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠽⣦⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⠀⠀⠀⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇⠀⡏⢀⡠⠄⠀⠀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⡼⠉⠀⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⣷⠃⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢿⠀⠀⠀⠀⠀⠀⡠⠊⠙⠒⠒⠒⠉⠁⠈⠂⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠛⠶⣦⣀⡀⠀⠀⣀⣀⣤⠶⠋⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⢹⠋⠉⠉⠉⡇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠤⣴⣄⣀⣴⣤⠼⠇⠀⠀⠸⠧⢤⣦⣤⣤⠼⠦⠼⠃

⠀⢀⣀⣀⣠⣀⠀⠀⣀⡀⢀⡀⠀⠠⢄⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠈⠉⠉⠉⠃⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠓⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠇⠀⠒⢀⠀⠀⠀⠠
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⢠⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠉⠉⠉⠀⠀⠀⠀⠀⢀⣄
⠀⠠⡀⠀⠀⠘⠛⠻⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠒⠀⠀⠀⠁⠙⠘⢁⠐⠀⠐⠀⠀⠓⠃⠺⠿⠛⠋
⠀⠀⠹⢒⠤⠄⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠀⠀⡀⣈⠁⠃⡀⢀⢀⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠁⠀⠈⠀⡢⠀⠆⠐⠐⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⠀⠈⠀⠈⢡⠀⠀⢠⠀⠀⠀⠀⠀⠂
⠀⠀⢀⣠⡀⠀⠙⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠄⠀⠐⠂⠰⠀⠀⠀⠈⣀⠀⠀⠀⠈⡊⠉⠉
⠀⠙⠋⠉⠁⠀⠀⠀⠀⠀⢠⣷⣄⠀⠀⠀⣠⣄⠀⠀⠀⠐⠘⠂⠀⣾⣻⡷⠀⠀⠀⠐⠆⠀⠀⠂⠀⠀⣤
⢀⠐⠈⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠐⠀⠊⠀⠀⠀⠈⠉⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠈⠐⢢⠀⢀⠀⠀⠤⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠈⠘⠀⠀⠀⠀⠀⠀⠅⠀⠠
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠈⠛⠛⢁⠀⠀⠀⠀⠀⠀⢠⠲⡄⠀⠐⠀⠀⠀⠀⠀⢀⠒⢲
⠀⠀⠐⠀⠐⠈⡈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⣉⣈⠉⠀⣀⣴⣆⣀⠀⠀⠉⠀⠀⠂⠐⠂⣀⣤⡀⠀⢀
⠀⠀⢀⢀⢀⡀⠀⠀⢀⠠⠸⠐⠐⠀⠀⠀⠀⠀⢸⠯⡟⡧⠀⣿⣿⣿⣿⣿⣶⣶⠀⠁⠀⠀⢸⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀
⠀⠀⠀⠀⠀⡴⠊⠉⣉⠉⠑⢄⠀⠀⠀⠀⠀⠀⢠⡴⠊⢉⣉⡉⠛⢧⠀⠀⠀⠀⠀⠀⠀⠐⠀⣉⡉⠑⠆
⠀⠀⠀⠀⡼⠀⠀⠀⣿⠀⠀⠀⠃⠀⠀⠀⠀⢰⡿⠀⠀⠈⢀⡇⠀⠈⣧⠄⠀⠀⠀⠠⠁⠀⠀⢁⡷
⠀⠀⠀⠀⢳⠀⠀⠀⣿⠀⠀⠀⡞⠀⠀⠀⠀⠈⢧⠀⠀⢸⠋⠀⠀⠀⡿⠀⠀⠀⠀⠐⠀⠀⢀⡈⡷⠀⠀⠘
⠀⠀⠀⠀⠀⠳⣄⣀⠉⣀⡠⠞⠀⠀⠀⠀⠀⠀⠘⣷⣄⣘⠛⣃⣠⡞⠁⠀⠀⠀⠀⠀⠔⢄⣀⠉⢁⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⢀⠴⠚⠉⠓⠲⣄⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠋⠛⠳⢤⡀
⠀⠀⠀⠀⣰⠃⠀⢸⠀⡇⠀⠈⢳⠀⠀⠀⠀⢠⡟⠀⠀⣾⠉⠀⠀⠹⡆⠀⠀⠀⠀⣰⠃⠀⢰⡏⠃⠀⠀⢳
⠀⠀⠀⠀⢿⠀⠀⠸⠶⡇⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠙⢹⠀⠀⠀⡇⠀⠀⠀⠀⢻⠀⠀⢸⡟⡆⠀⠀⢸⠃
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠇⢀⡠⠋⠀⠀⠀⠀⠀⠳⣄⠀⠙⠞⠀⣀⡜⠁⠀⠀⠀⠀⠈⢧⣀⠘⠷⠃⢀⣠⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠋
⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠲⢦⣦
⠀⠀⠀⠀⣰⠃⠀⠐⢲⠂⠀⠘⡆⠀⠀⠀⠀⠀⡜⠀⠀⡖⣦⠀⠀⢣⠀⠀⠀⠀⠀⣼⠏⠀⠀⡞⣦⠀⠈⢦
⠀⠀⠀⠀⣯⠀⠀⠀⡾⠀⠀⠀⣹⠀⠀⠀⠀⠸⡅⠀⠀⡷⣿⠀⠀⢀⡆⠀⠀⠀⠀⢻⠀⠀⠀⠧⣿⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠀⠰⠇⠀⠀⡠⠃⠀⠀⠀⠀⠀⢣⡀⠀⠧⠟⠀⢀⡜⠀⠀⠀⠀⠀⠘⣧⡀⠀⠦⠟⠀⣠⠏
⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠏⠉⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠒⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⢀⣤⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⠀⡇⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⡇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣈⠉⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠒⠚⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠈⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⣿
⠀⠀⣤⡤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠴⠧⠤⠶⠀⠀⠀⠀⠻⠤⠤⠤⠴⠶⠶⣦
⣠⠞⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢠⠞⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⢿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡆⠀⠀⠀⣏⠀⠸⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇
⢸⡆⠀⠧⠤⠤⠤⠴⠖⠒⠒⠒⠒⠒⠚⠛⠛⣿⠃⠀⠀⠀⢹⡀⠀⠉⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡏⠉⠉⠉⠉⣩⠟⠁
⠀⢷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠈⠧⠤⠤⠤⠤⠴⢦⠀⠀⢸⠀⠀⠀⠀⢷⠖⠒⠒⠛⠁
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠓⠚⠛⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠲⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠳⣤⡀⠀⠀⠀⠀⢀⡞⠁⠀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠈⢙⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠙⠲⣄⠀⣠⠏⠀⠀⠀⣠⢾⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣙⣶⣾⣯⣤⡤⠤⠴⠶⠶⠶⠶⠶⣆⠀⠀⠀⠀⠀⠀⡼⠉⠓⠦⣀⠀⠀⠀⠈⠙⠃⠀⠀⢀⡜⠁⣼
⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠈⠓⢤⠀⠀⠀⠀⠀⢰⣋⡠⠞⠁
⠀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡴⠃⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀
⢸⡁⠀⠹⠤⠤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠒⠚⠋⠉⠉⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢠⠞⠲⢤⣀⠐⠀⠀⠀⠉⢹
⠈⣧⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠴⠶⠶⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⡰⠃⣀⡀⠀⠈⠑⠦⣀⠀⡠⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣤⠞⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠢⡀⢀⡜⠁⡼⠋⠙⠲⣤⡀⠀⠈⣹⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠢⢤⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠈⠋⢀⡞⠁⠀⠀⠀⠀⠉⠓⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢀⡞

⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⡠⠖⠀⠀⠀⢠
⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠃⠀⠀⠀⠋⠁⠀⠈⠀⠀⠀⡰⠃⢀⣈⠀⢀
⠀⠁⢀⢀⡀⢠⡶⣤⢹⡆⠰⠒⢨⢀⢁⢡⡂⠦⣤⣤⢀⡂⠢⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⡼⠀⣾
⠀⠀⠎⠀⠱⡀⠙⠋⢸⡇⣀⡁⠀⠀⠀⠀⠀⠀⠉⠈⠈⠁⠀⠀⠄⠀⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡣⢴⠇
⠀⡆⠀⠀⠀⢇⡠⠂⢸⡇⢿⣿⠀⡀⠈⠀⠀⠀⢀⠀⢰⡁⠀⢠⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠸⣁⡜
⠐⡃⠀⠀⠀⠞⠄⠀⠈⠃⠀⠠⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⡈⠤⠀⠀⠠
⠀⠆⠀⠀⠘⠀⠑⠘⠀⠰⠂⠰⠀⠰⠀⠀⠀⠀⠀⢀⣄⣀⣀⠀⠠⠀⠀⠀⠀⠈⠀⠐⢠⠐⠃⠰⢍⣁⠀⠒⠈
⠀⠄⠀⠂⡀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠐⠀⠀⠀⠀⠙⠛⠟⠛⠛⢁⠀⠀⠠⡄⠆⠀⠀⣀⠒⠀⢌⢀⠒⠠⠀⠁
⠀⠀⠀⠀⠣⠄⠀⠀⠄⠉⠀⠀⣀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⣠⠀⠠⠄⡁⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠄⠀⠀⠀⠠⠀⠀⢀⠀⢰⠋⡆⡿⠀⣼⠉⠧⡏⠙⢲⣶⡄⢸⢻⠀⣸⢻⡆⣀⠄⠠⠂⠀⠐⠃
⢀⡈⠀⠀⠀⠀⠀⠀⢠⡏⠀⣿⣼⡇⡇⠀⡇⢲⡆⡧⠄⢨⡿⣅⢸⣏⠀⡿⠶⡇⠠⠆⠀⠴⠀⠐⡁⠄
⠘⠁⠀⠐⠇⠐⠀⠀⢀⠁⢸⡇⢸⠇⠧⠤⢻⡼⢹⣇⣀⢸⡇⣸⢸⠈⣧⡇⠀⠷⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠
⠀⣐⣀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠠⠀⠀⠀⡀⠀⠀⠀⢉⣀⣉⣅⠈⠀⠀⠀⠀⠀⠀⠂⠀⠃⠂⠀⠀⠁⠀⣤⠤
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡏⠘⠁⠀⠀⠀⡀⠒⠀⠀⠀⢠⠀⠇⠈⠀⠀⣈⡉
⠀⠀⠀⠀⠀⠞⠆⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠒⠂⣄⣰⣇⡠⠀⡀⠐⣀⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣈⡉
⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⡀⣈⡁
⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠃⠀⣠⠖⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠖⡳⢶⣤⡤⠁⠀⠀⠀⠀⣄⣀
⠀⠑⠄⠀⠀⠀⡄⣀⠄⡈⠁⠀⠈⠀⠀⢠⠀⠁⠈⠀⠀⢀⠀⠀⢀⡀⠀⠀⢀⡼⠁⡎⡀⠀⢡⠀⠀⠸⠀⠤⣅
⠀⡄⠀⠀⡀⠀⠘⠄⠚⠀⢀⠀⠀⠀⠀⠁⠈⠀⢀⠤⠤⣬⠀⠀⠎⠀⠔⠈⠉⠀⠘⠛⠁⠀⠀⢆⢀⠂⠀⠢⢄
⠀⠀⠀⠰⠀⠀⠆⠈⢄⠀⡈⠀⠈⠀⠀⠀⠀⠄⠀⣠⠞⠁⠀⠀⠀⠀⠐⠃⠐⢀⠔⠂⠀⠀⠀⠀⠁⠀⠀⠠
⣀⣀⣈⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣂⣀⣀⣈⣀⣊⣙⣲⣛⣀⣀⣃⣀⣐⣃⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡉⡤⢤⠀⠀⠀⠀⠀⡠⢔⣬⢖⡐⡀⠀⢨⢙⡟⠂⡇
⠀⠀⠀⠀⢀⣠⣄⣤⣖⡔⡔⠀⠀⡠⠨⡠⡞⠁⠀⠈⠙⠾⢶⠀⠀⠀⠀⢠⠇⠀⡥⣕⣋⠉⣷⠃⢁⠈⢹
⠀⠀⢀⡴⠋⢀⡤⠤⡑⠗⢣⢀⡬⠣⢽⡁⠘⠚⠉⢹⠀⠀⢸⠀⠀⠀⠀⣼⠀⠘⠧⣀⠀⡹⠁⢠⢺⠀⢸⡄
⠀⠀⡞⠀⣰⠃⠀⠀⠘⣆⠼⠋⠀⢀⠼⡇⠀⣀⠴⠋⠀⣠⢏⠀⢀⡀⠀⠉⠉⢢⠀⠈⢻⠁⠐⠓⠚⠀⠘⠓⣦
⠀⢸⠃⠀⣇⠀⠀⠀⢰⠛⢦⡖⢻⠀⠀⡗⠉⠁⠀⣤⡚⢁⣮⣴⣥⣵⠖⠲⠤⠞⠀⠀⢸⠦⠤⠤⢤⠀⢠⡖⠟
⠀⠘⣇⠀⠘⠦⠤⠖⠃⠀⡼⠁⢸⠀⠀⣧⣄⣀⠀⠀⡸⠉⢈⠥⢍⠉⢳⣄⣀⣀⣠⠴⠃⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠈⠲⣄⣀⣀⣀⣠⠞⠁⠀⢸⠀⠀⢿⠀⠈⠉⠓⢧⡀⠘⠲⠚⢀⡼⠀⠀⠀⠀⡀⢦⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠘⢳⡖⠚⠀⠀⠀⢠⠖⠉⢀⣀⣀⠉⠲⣄⡠⠶⠶⣤⡸⣤⡀⢠⠚⢄
⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠲⠴⢞⠥⢹⣀⠀⠀⠀⣏⠀⠐⣇⣀⣸⠆⡰⠋⠀⣠⠴⠦⠤⣧⢆⡺⠫⠎⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠾⣤⣀⣀⡀⠈⠉⠁⣸⠤⢤⢡⡛⣐⣇⠈⠉⠁⣰⠁⠀⠀⠉⠉⠓⢤⡇⠀⡔⠉⠙⡆⡸⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣠⠔⠿⠤⣀⡉⠑⠚⡩⠋⠉⠉⣿⠀⢠⠋⠉⢱⠀⠘⣇⠀⠙⠶⠞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡴⠋⠀⠀⢠⠔⠊⠀⣠⠊⠀⠀⠀⠀⢹⡀⠈⠳⠴⠊⠀⣸⠉⠲⠤⡤⠴⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡞⠁⠀⠀⠀⠘⠶⢤⡀⠈⠱⡄⠀⠀⠀⠀⠙⠲⠤⠤⠴⠚⠁⠀⠀⣀⡤⠚⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠻⠤⢤⡞⠀⠀⠀⠀⡴⠤⣤⠼⠃⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢥⣀⣀⣀⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠁⠉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⡠⠒⣶⠀⠘⡄⠀⠀⢀⠄⠊⠢⡀⣠⡶⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠉⢉⢤⠾⠋⠄⡰⠁⠀⠀⢿⣦⡀⠀⢈⠁⠀⠀⡠⠂
⠀⠀⠀⠀⠀⢠⡤⠤⠛⠃⠀⠀⠀⠈⠈⠩⠍⡇⠀⠀⠀⢺⡇⠀⠀⢂⣀⠀⠤⠄⠠⠇⠀⠀⠀⠀⣰⣕⠁⢀⡴⠊⠀⠀⠀⠀⠀⣨⡟⢀⠀⠀⠀⢪⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠐⠀⠀⠀⠀⠂⠀⢀⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢙⣣⠤⠌⠀⠀⠀⠀⠀⠠⢾⡁⠀⢀⠤⣦⡀⠀⠑⡄
⠀⠀⠀⠀⠀⠹⣧⡠⠤⣶⡆⠀⠁⢸⠉⠉⠁⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⡀⠈⠀⠜⠀⠀⠀⠀⠻⣁⡂⠆⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠈⠛⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣴⠖⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠈⢣
⠀⠀⠀⠀⠀⢰⡟⠹⢑⡄⣴⣻⠋⠐⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣸⡇⠀⣶⠶⠒⡄⠀⠀⠀⠀⣿⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠑⠒⢋⣴⡷⠁⠰⢀⠜⠀⠀⠀⠀⠀⠀⢀⣄⡤⢤⠴⠶⠒⠚⠛⠛⠛⠓⡄⠀⠸⢇⣀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⢸⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⢎⠞⠀⠀⢠⠊⢀⡀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣼⠟⠀⣠⡐⠑⡄⠀⠀⠀⠀⠀⠈⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡴⡳⠋⠀⠁⡔⠁⠠⡟⠭⠳⡄⠀⠀⠀⠀⠙⠒⠒⠒⠂⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠲⠊⠈⠻⠒⠁⠀⠀⠀⠀⠀⠀⠙⣒⣒⠁
⠀⠀⠀⠀⠀⠀⠻⡁⠀⢤⠎⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁
⠀⠀⠀⢀⣠⣤⠤⢀⠀⠀⠀⠀⠀⠀⡴⡒⠀⠀⢠⣴⡂⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠙⢆⠀⠀⠀⣾⡏⠁⠂⠤⢀
⠀⠀⢠⢛⠋⠀⣠⡀⠙⡆⠀⠀⠀⢳⡇⠀⡁⠀⠀⠉⢹⡆⢸⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠄⣠⠖⠁⠀⠀⠀⠈⠁⠒⠤⢔⡀⠀⠑⢐⠄
⠀⠀⣇⡎⠀⠌⣸⣿⠀⢸⠀⠀⢀⢾⠋⢐⠁⠀⠀⠀⢅⡄⠸⢄⠀⠀⠀⠀⢠⣾⠟⠁⠀⢠⣋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡮⠟⢁⡴⠋
⠀⠀⣿⡇⠸⣴⡿⠃⢀⡎⠀⠀⠀⢀⣣⠘⡆⠀⠀⠀⠜⠟⠀⡎⠀⠀⠀⠀⠀⠉⠓⠦⢅⣀⠈⢉⠓⠶⢤⠀⠠⣴⡿⠛⠙⢀⡴⠋
⠀⠀⠈⠳⣄⣀⣀⡤⠊⠀⠀⠀⠀⢸⡟⠰⠇⠀⠀⣤⣤⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣤⠞⠀⠀⠘⢆⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⠤⠃⠀⠙⠃⠥⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⡇
⠀⠀⠀⠀⣠⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄⠀⠀⠀⢀⡴⠖⠒⠒⠛⠒⠃⠀⠀⠀⠳⠒⠒⠒⠒⢲⡄
⠀⠀⠀⡞⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⢳⠀⢇⣀⣀⣀⡀⠤⠤⠤⠤⠤⠤⢤⡇⠀⠀⠘⡆⠀⠓⠂⠉⠉⠉⠙⠀⠀⠀⢠⠖⠒⠒⠒⣲⠏
⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠎⠀⠀⠀⠀⠳⠤⠤⠤⠤⢤⠀⠀⡇⠀⠀⠈⡶⠶⠒⠚⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢃⠀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣈⣉⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢇⡀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠙⠲⣄⠀⠀⠀⢠⠞⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣀⠽⠋⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡧⣀⠀⠀⠈⠙⠦⣴⠋⠀⠀⡠⢺⠁
⠀⠀⠀⠀⢠⡴⠖⠒⠒⠒⠛⠛⠛⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⠀⠀⢼⡀⠀⠑⠢⣄⠀⠀⠀⠀⢀⠞⣡⠞
⠀⠀⠀⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠉⠲⣄⣠⠎⠀⠀⠀⠀⠈⠛⠧⣄⡀
⠀⠀⠀⢾⠀⠘⠦⠤⠤⠤⠒⠒⠒⠒⠒⠒⠚⠉⢉⡽⠋⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⡰⠒⠢⢄⡀⠀⠀⠉⣹
⠀⠀⠀⠘⣦⠤⠤⠤⠤⠤⠤⠶⠖⢒⣶⡚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⢀⠞⣠⢤⣀⠀⠉⠲⢤⣔⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢏⣀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠙⠦⠊⡰⠃⠀⠈⠓⠦⣄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⣀⢀⠀⣄⡐⠀⢀⢀⢀⠢⡐⠂⠀⠙⠙⠉⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠚⠀⠁⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠠⡀⠀⠐⠁⠢⠀⠀⠀⠁⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠀⠠⠀⠈⢠⡄
⠀⠀⠀⠀⠀⠀⢈⠂⢀⠀⠀⠂⠐⠈⠀⠀⠀⠀⡀⠀⠀⡆⢀⠢⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⢸⡇⡀⠀⠈⣤
⠀⠀⠀⠀⡀⠸⠮⠀⠀⠁⠢⢄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⠠⠊⡀⠀⠀⠘⠆⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠄⢸⡇⢰⣶⠀⣿
⠀⠁⠀⠀⠁⠃⠀⠀⣀⣤⠄⠀⠑⢄⡴⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠈⠉⠉⠈⠉⡉⠉⢩⠉⢩⡍⠉⡏⠉⠃⠐
⠀⢠⣂⡀⠀⠀⠀⠀⠛⠁⠀⠀⡰⠋⡠⠒⡀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⠠⣉⠕⠊⠝⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠊⠀⢈⠁⠥⠐⠂⠃⠀⠀⢀⡡⠤⠔⠒⠓⠋⠀⣀⠀⠀⢀⠀⠰⠢⠀⠒⠀⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢁⢊⡀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠰⠉⠀⠀⡀⠀⠐⠀⠆⠀⠀⠠⠀⡀
⠀⠀⡠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠘⠀⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⠀⠀⠀⠦⣀⠘⠀⠀⠀⠀⠀⠀⠀⢠⢹⣿⣷⡄
⠀⠀⠃⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⠀⠁⠈⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠈⠻⣿⠄
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠻⠶⣲⠴⠟⠁⠀⢀⠀⠀⠀⠀⠀⠀⢀⠆⠑⠒⣀⣤⣧⡄
⠀⠀⠀⡀⠀⠀⠂⣀⠀⠃⠂⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠐⠫⡨⠆⠀⠀⠀⠀⠀⠸⠄⠨⠜⠉⠛⠛⠁
⠀⠀⠰⠀⠐⠚⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠈⠃⠀⠀⠀⠀⠀⠰⠱⠀⠀⠀⠀⢀⡀⠀⢠⠀⣀⠀⠀⡀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠀⠀⠄⡒⠇⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠘⠀⠙⠃⠐⠂⠇⠘⠘⠃
⠀⠀⢠⠂⠀⠀⠀⠀⢸⠀⠀⠃⡀⠀⡜⢀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡰⢢⠕⠂⠀⠀⠀⠀⠀⠀⢀⡈⠀⡈⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠈⠒⠀⠀⠀⠀⠈⢸⠀⣦⠆⠀⡏⠀⠑⠀⣴⣁⣀⣩⠆⠀⡠⢊⠔⠁⠀⢀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⡂⠀⠀⠀⠐⠀⢀⣠⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠛⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⣀⠀⡰⠁⠀⠀⢠⢠⠀⠀⠀⠀⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠘⡀⡄⠀⠀⠀⠀⠀⣮⣤⣼⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⢀⠀⡠⡀⢠⡄⡄⠀⡄⢠⡄⠀⠀⠀⠀⠈⠱⠃⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⢀⡡⠶⠂⡀⢀⠀⠀⠀⠀⠀⠀⡀⡀⠤⢀⢄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠀⡠⠔⠁⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢀⠠⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠸
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠊⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⠀⠐⠀⠀⠀⠈⢀⠈⡄⠀⠀⠀⠀⠀⠠⠈⠀⡀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⡀⠈⠁⣀⠊⠀⠀⠀⡠⠈⡱⠈⠠⡀⠈⠀⡀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⢉⠦⡁⠀⠀⢀⠀⡑⢂⠐⠁
⠁⢠⠊⠀⠀⢀⠠⠚⡄⠊⠀⠀⠀⠈⠢⠄⠁⠀⠀⠀⠡⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⢆⠀⠀⡀⠈⠀⠄⠀⠔⠁⠀⠈⢢⡔⡤⢊⠔⠁⠀⢠
⠊⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠈⠁⠈⠠⡀⠀⠂⠀⠀⠀⠄⠀⠔⠁⠀⠀⠀⠀⠄⠁⠀⠀⠊⠀⠀⠀⠀⠁⠀⠀⢀⠄⡡⠊⠒⠁⠀⡀
⠀⠀⢠⠂⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢀⠔⠀⠀⠀⠄⠁⢀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡠⠊⠀⠀⠀⠀⢀⡐⣢⠂
⡀⠀⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠄⠀⠀⠀⡀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⡠⠈⠢⡀
⡑⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠉⢠⠔⠁⠀⠀⠀⠒⠈⠂⡀⠀⠆⠀⠀⠰⢆⠀⠀⢀⠐⠀⠀⠀⢀⠀⢤⠊⠁⢀⠀⠀⠀⠀⢡⠊⠀⠀⠀⡀
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠠⡔⠁⠀⠔⠁⠘⠀⡀⠀⠀⠀⠀⠠⠌⠁⡀⠀⠂⠀⠀⠉⠄⠁⠀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠂⡠
⠀⠀⠈⠆⠁⠀⠀⠀⠐⠁⠀⠒⠁⠀⠀⠀⠀⠁⠠⠀⢀⠀⠀⠀⠀⠈⠂⡄⠀⠀⠁⠀⠐⠀⠀⡠⠊⠁⠀⠀⠀⢀⠐⠀⠀⠁⠀⠀⠀⠊⡱⠂
⠀⠈⠁⠀⠈⠠⠐⠁⠀⠔⠀⠀⠀⠠⠌⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠔⠣⡀⠀⠀⠔⡠⠊⠀⠀⠀⠀⠀⠤⠀⠐⠀⠀⠀⠀⠐⠄⠊⠀⠐⠁
⠁⠀⠀⠀⠠⠀⢀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢀⠐⠁⠀⠀⣈⠲⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠠
⠀⠀⠡⠆⠀⠀⠁⠀⠀⠠⡐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠜⠂⡀⠀⠀⠐⡠⢂⠌⠀⠀⠀⠀⠀⡠⠒⢀⠀⠀⡠⠑⡄⠀⠀⠀⠀⠀⠁⠆
⡀⠀⠀⠀⠀⠌⢀⠄⠂⠀⠀⠢⡀⠆⠀⠀⠀⠀⢀⡰⣀⠄⠀⠀⠀⢀⠖⠡⢊⠔⠁⠀⠠⢄⠲⡀⠂⠀⠀⠀⡰⢎⡠⠊⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠁⢀⠀⠀⠠⠄⠁⠂⠀⠀⠀⠀⢀⠐⠈⠂⠀⢀⡄⠀⠀⠔⠁⠀⣀⠀⠀⠨⠒⠈⠢⠀⠀⠂⡠⠊⠁⠄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠠⠆⠀⠀⠠⠀⠀⠤⡉⠄⠀⠀⠀⠀⢜⠁⢀⠄⠁⠀⠰⠄⠀⡡⠀⠀⠀⠀⠠⢌⡠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠱⠆⠀⠀⠀⠀⠉⠀⡀⢀⠀⡡⢊⠔⠁⠀⠐⠎⠀⠀⠓⠀⠀⠀⣀⠈⡠⠊⠀⠈⠢⡀⠀⠆⡀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠄⠁⠀⡠⠀⠀⠀⠃⠔⠁⠀⠀⢀⠘⠣⡊⠔⠁⠀⠀⠘⠄⡠⠙⠀⠀⠀⠀⢌⠠⠋⠀⠀⠀⠀⠠⠌⠃⡀⠀⠀⡀⠈⠁⠄
⠀⠀⠡⠄⠑⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠐⡠⠊⠀⠀⠀⠀⢆⡀⡠⠂⠁⠈⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠁⠔⠁
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠘⠢⠘⠀⠀⠀⡠⠈⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠐⠀⠀⠀⠐⠁⠀⠐
⠁⢄⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡰⠎⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⡀⢂⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠀⠀⠀⠀⣀⠠⡀⠈⠀⡀
⠠⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⠀⠁⠄⠀⠀⠀⠘⠀⠀⠀⠠⠂⠀⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠘⠠⠀⠀⠀⠀⠀⠀⠈⠠⠄⠁
⠀⠀⢀⠀⠁⠀⠀⠀⠀⡐⢎⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠱⢎⡠⠊⠀⡀⠈⠢⡀⠆⠀⠀⠀⢀⠜⠃⡀⠐⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⠁⠈⠠
⠢⡄⠁⠀⠀⢀⠐⡠⠊⠀⠀⠑⠄⠀⠐⠀⠀⠀⠀⠁⢄⠊⡠⠊⠁⠄⠀⠀⠠⠄⠉⠠⠀⢀⠔⠁⠀⠀⠁⠂⡀⠀⠈⠀⠀⠀⠀⠐
⠁⠈⠢⣀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠈⠂⠀⠈⠁⢀⠄⠠⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⡌⢪⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠐⢄⡀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⢊⠐
⠄⠠⢀⠄⠁⠀⠀⢀⠀⡠⠊⠀⠀⠀⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⢀⠰⠁⠀⠠⠈⠱⣨⠢⡀⠀⠀⡠⠊⡱⠂⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢑⠄⠁⠀⠀⠀⠀⠆⡁⠀⠀⠀⠀⠐⢪⡲⡁⠀⠀⢀⠐
⠃⠄⠀⠂⠔⡀⠊⠀⠀⠈⠠⠈⠠⠊⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠁⠄⠀⠌⠁⠀⠈⠠⠀⠤⠀⠄⠁⠀⠀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⢚⠷⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⡀⠀⣠⠴⠦⡶⣥⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠉⠓⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠰⣧⣛⣇⡞⠁⠀⣀⣀⠀⠁⠸⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⡻⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢹⡉⠲⠋⠁⢸⡆⠀⢠⡇⠀⠀⠀⣀⠊⣰⢊⡙⢆⠀⠀⠀⢀⠔⠁⠀⠐⠣⠤⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⡖⠁⢀⡄⠘⠉⣇⠀⠀⢀⠞⠀⢀⡾⠀⣾⠉⠉⠉⠑⠉⠮⠴⠋⢹⠀⠀⠈⠛⠲⠦⢤⣀⠀⠀⠀⠉⡶
⠀⠀⠀⠀⠀⠀⠀⠹⠶⠋⢹⡀⠀⢸⣄⠴⠃⠀⣠⣾⣤⣀⣿⠤⠤⠤⣤⠄⠀⠐⢁⡼⠀⠀⢀⠀⡠⠀⠀⡾⠀⢠⡟⠒⠃
⠀⠀⠀⠀⠀⢀⠄⣀⠴⡆⠈⣇⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠃⠀⢀⡖⠉⣠⣅⣶⢇⠷⣸⡀⠘⠧⣤⡼
⠀⠀⠀⠀⠀⣢⣤⢥⣄⡰⣤⢹⡄⠀⢸⡗⠒⠒⠒⠒⠒⠒⠃⢀⡞⠀⠀⣰⠋⢠⠞⠁⢀⡤⠤⣍⠀⠙⢦
⠀⠀⠀⣰⠋⠀⠀⣀⣡⣀⡏⠀⠓⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⠇⢀⡏⠀⢰⠃⠀⠀⠀⢳⠀⠘⡇
⠀⠀⣼⠁⠀⠠⠯⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡟⠀⢸⠃⡀⣿⡐⠆⠀⠀⢸⠀⠀⣿
⠀⠀⡇⠀⠀⢀⣠⣄⠀⠈⢳⠀⢠⠀⢀⣰⠄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⡼⣇⢤⡋⣌⡦⢤⣠⠞⠀⠀⡟⠀⠀⠀⢀
⠀⠘⡇⠀⠠⡏⠀⠀⣷⢀⡬⠧⡄⢮⡡⢮⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢁⣭⣬⣀⣸⠁⠀⢀⡼⣃⡤⠶⢶⠡⣰⠐⢶
⠀⠀⢷⡀⠀⠓⠦⠖⠃⢸⣇⣀⣈⡁⠈⠉⡴⠁⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⡇⠀⢸⣳⣦⣤⣤⠤⠶⠋⡼⠁⠀⣀⢬⣉⠫⠥⣎
⠀⠀⠀⠳⢤⣀⣀⣀⣤⠖⠃⡠⠊⠁⢀⡞⠀⠀⠀⣰⣤⣭⣤⣵⠀⠀⠀⠀⠧⠤⣄⡀⠈⠑⣆⠀⠀⢸⡃⠀⢸⡁⠀⢈⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢺⣀⣀⡀⠀⠉⠳⣤⠊⠁⣉⠭⡉⠉⢳⡀⠀⠀⠀⠀⠀⡷⠀⠀⢸⠂⠀⠈⢧⡀⠀⠙⠒⠋⠠⠔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣿⠀⠀⠧⡤⠟⠀⣸⠃⠀⡸⠉⠙⠉⠀⠀⢀⡼⢠⠶⠤⣤⣩⠷⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠛⠁⠀⠀⣠⣯⠷⠀⠀⠀⠀⠾⢅⠀⠀⠙⠶⠤⠤⠤⠖⠋⠀⢿⣀⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢤⣤⠤⠴⠚⣽⠁⠀⣰⠚⠛⢦⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠹⠤⠤⠞⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⡤⠞⠁

⣿⣿⠟⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⠤⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⣀⠀⣀⣂⣄⠀⡈⠀⣀⠀⠀⠀⣄⠀⠀⠀⠀⠁⡘⠃⠀⢀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠠⠉⠏⠥⠈⠉⠀⠀⠀⡃⠀⠉⠠⠄⠁⠀⠀⠀⠀⠀⠠⡏⠠⠔⠁
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠕⢂⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠈⠀⡀⠂⠀⠄⠠⠀⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠘⠓⢾⡇⠀⠄⠀⠰⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠡⠀⠂⠀⠀⠀⠐⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠠⠐⠀⢀⠀⠀⢠⡥⠀⢀
⠃⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠈⠀⡡⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇⠐⠂⠄
⠀⠀⠂⠀⠀⡠⠊⠀⠀⠀⠀⠘⢘⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⣀⠀⠀⠀⣀⠀⠀⠠⠒⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠊⠀⠐⠀⠀⠀⠀⠁
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠸⠀⠅⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠴⠶⠶⠄⡂⠀⠂⠶⠀⠀⠈⠁⠙⠂
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠃⠴⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠠⠄⠙⠢⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡉⠀⢀⣀
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠠⠀⡐⠚⡃⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠻
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠠⠉⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃
⠀⣀⡀⠀⠀⠀⠀⡀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⠀⢀⠂⡀
⠀⠙⠟⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠴⠠⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠇⠘⠡⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠀⠀⠜⠁⠀⠠⠀⠐⠀⠀⠀⠂⠀⡠⠠⠀⠁⠻⠀⠑⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰
⠀⠀⡀⠀⣀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁⠀⠀⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣋⣁⣆⡀⣀
⠀⢳⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⣏⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣾⣿⣿⣼⣯⣟⣿⣽⣿⣿⣿⣿⢿⢫⣿⣷⣾⣷⣶⣶⠄
⠢⡈⠒⠀⡀⠀⠀⠀⠀⠠⣀⡎⠀⠀⠈⠁⠂⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣙⣙⢳⣇⢹⣿⣿⣿⣿⠟⠃
⢀⡀⠄⠋⠃⠐⠃⠠⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡉⠀⠛⠛⠫⠙⠉⠻⠛⠿⠛⠋⠛⠟⢉⡏⠉
⠈⠟⠀⠀⠰⡀⠁⠠⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠒⢀⣀⠄⠓⠀⠀⡀⠀⠀⢠
⠀⣷⠂⠀⠀⡇⠀⠀⢀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠃⠀⠀⠉⠓⠂⠀⠠⠄⠀⠀⠐⠂
⠀⠉⠁⠀⠀⠀⠀⠀⠂⠐⠀⠂⠉⢀⠐⠒⠀⠀⠀⣀⡀⢀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⢠⡂⢲⡀⣴⣶⣦⡀⣶⣒⠂⢰⠶⢶⡖⠀⠀⣰⠐⢲⡴⠶⣶⡖⢠⠰⣆⠆⡐⣄⠀⣴⠖⠒⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀⠈⢁⣼⠹⡿⠀⣽⡇⢬⣉⡗⢰⠀⣾⠁⠀⠀⣿⢠⣬⠀⠾⢿⣄⢐⢸⣿⠇⠒⢸⣧⣹⡗⠒⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⣿⣿⢀⡀⢹⡄⢀⣾⠟⠀⣠⣷⣦⡀⢸⣿⠀⠀⠰⣿⠀⢶⠐⢶⡾⠇⢨⢘⠿⡀⠍⣸⣿⠘⢓⣒⣚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⢸⣿⠈⠃⠚⠰⠿⠿⠿⠇⣿⡇⡘⠃⢨⠭⡄⠀⠀⢸⠀⣿⢸⡇⢸⡇⠑⢾⣶⣷⣾⣿⠏⠸⠿⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⠟⢰⣋⣵⠐⠒⠂⠈⠀⣿⡟⢻⡇⢸⡀⡿⠀⠠⢖⡦⡾⡌⢛⢻⡗⠷⣶⢪⣍⢹⠓⠀⠐⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠭⡀⣰⠞⢻⠀⡆⠀⠀⠰⠂⣿⣇⣼⡇⢸⠛⣧⠀⢀⡘⣧⠙⠁⡀⡨⡁⢰⡟⠈⠛⣈⠙⡆⢸⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡽⠙⣒⡋⢀⣥⣝⡀⣾⢸⡎⣉⣋⣀⢸⡀⣿⠀⠀⢩⠱⣢⢀⡇⢀⡇⣿⠃⣾⠃⠤⠴⠃⣐⣦⢠⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⢸⣉⡁⣿⠋⣹⣷⠈⠼⠃⢉⣿⠃⢈⣛⡁⠀⠀⠸⠴⠥⠨⡌⠉⠧⣖⠂⣿⠻⣦⡆⣷⠀⣿⣸⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣸⠇⣿⠿⢻⣿⢸⢿⡂⠈⢹⡇⠸⢤⡄⠀⠀⣺⣼⠃⡜⢱⡾⠀⣿⠃⣿⠀⣿⠤⠟⠉⠁⠘⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠅⠀⠻⠿⠿⠏⠸⠼⠑⠦⠾⠃⠯⠼⠃⠀⠐⠧⠼⠁⠐⠸⠯⠘⠤⠏⠻⠾⠟⠰⠂⠀⠀⠄⠅⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⢀⣀⠀⣀⠀⣀⣀⢀⣀⠀⣀⣀⢀⡀⠀⠀⢀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡤⠘⢧⣜⠸⣿⡌⣑⡆⡏⢹⢺⣿⢨⡕⣺⣭⠃⠀⢰⡋⠀⢹⠤⣿⡇⠈⠀⠐⠛⠸⠟⠛⣿⡆⣭⣍⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠣⢰⡟⢹⣧⠈⣴⠒⠀⣥⠾⢀⠋⠰⠲⠶⣽⠧⠀⠀⡑⣲⠋⠀⢺⢶⡶⠶⢸⣻⣴⡟⠻⠿⠳⠆⠹⠁⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠄⠠⣼⡟⢰⣿⢹⡆⣧⣴⠸⠠⠇⢁⠎⣿⣷⠀⢀⡴⣧⠀⣦⡉⠸⠟⣷⠰⡿⠻⠿⡿⠿⠇⢈⡟⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣦⣄⢸⣿⠀⠻⠞⢡⣶⠀⡄⢦⢀⣿⢀⣭⡍⠀⠘⠧⣿⠸⠷⡷⠰⣦⣿⠸⣿⢰⣿⠅⢀⠘⠋⢸⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠈⠻⠿⠋⢸⠀⡇⢻⣿⣦⣅⡜⠀⠋⠞⣶⡄⠀⠀⡉⠵⢚⣶⢴⠶⠶⠶⣶⣴⣆⠀⠀⠔⠀⠤⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣶⠑⢹⡴⣿⢈⡐⡇⣴⠙⠻⢿⣿⣆⠺⠴⠟⠀⠀⢀⡅⢘⣿⢅⡆⠈⣱⢞⣁⣿⣟⣩⣭⠙⢬⠋⣈⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡶⡾⢿⡃⣿⢈⣌⡈⢿⡐⠇⠀⣿⠇⠀⡖⠒⣇⠀⠈⠲⠛⠋⢈⠋⢺⡇⢸⢲⣬⠁⢋⣿⠀⢢⢸⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠇⠿⣿⣸⡇⠀⢁⣀⠈⣥⡄⠀⣇⣀⡄⠀⠐⠟⣿⠈⢸⠰⣿⣿⡏⣭⣐⢠⣆⢻⡗⡃⠸⢷⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⢠⡞⠙⡆⣿⢻⡇⠀⠅⠄⠄⠰⡇⠀⡇⠀⡇⠀⢠⡾⢋⠀⢸⢼⣏⢹⡧⡾⢠⡏⡛⡋⢸⢾⠰⣦⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⠛⠁⠈⠛⠁⠀⠂⠈⠀⠛⠃⠘⠓⠚⠃⠀⠈⠛⠛⠀⠛⠈⠛⠛⠁⠃⠘⠁⠁⠈⠈⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠛⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡜⡇
⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⣶⠚⠛⠛⠛⠃⠀⠀⠀⠛⠛⠛⠛⢳⣄
⠀⠀⠀⣯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢸⡇⠀⠀⠀⠀⠀⢸⣤⠤⠶⠶⡆⠀⠀⠀⡖⢒⣒⣲⣧⡿
⠀⠀⠀⠸⣗⡚⢛⣛⣋⣉⣉⣉⣩⣭⣥⣼⠃⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⣧⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠶⢶⣃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⠴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⠦⣄⡼⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⠀⠀⠀⢀⠞⣻
⠀⠀⠀⠀⠀⠀⢀⣀⣈⣛⣛⣋⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⠻⣾⡁
⠀⠀⠀⠀⣾⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⣿⣀⣀⣀⣤⠤⠤⠤⠤⠤⠤⠾⡿⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⡴⣣⣝⡲⢄⡀⣠⣾⠇
⠀⠀⠀⠀⠈⠿⠗⠒⠒⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢻⣦⣄⣠⢞⡴⠋⠈⠙⢦⣍⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠉⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣭⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⢀⣀⣤⣶⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⡿⠛⠛⠛⠛⠉⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⢹⣿⣶⣤⡀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡏⠙⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢿⣿⣿⣿⣿⡆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠹⠿⣿⣿⣿⡿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠈⠻⠟⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣠⠔⠀⠉⡁⢶⠆⠉⠉⠀⢤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡎⠀⣠⠂⡈⢠⠄⠉⠠⢀⣰⠈⠉⢤⠈⡀⢠⡀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⢀⠀⣀⡰⣀⢀⡟⠉⡱⠖⠯⣀⡨⠖⠶⡁⠉⣇⢀⡰⡀⣀⢀⠀⠑⠄
⠀⠀⠀⠀⢀⠄⠾⠇⢡⡁⠀⣟⢀⡬⢣⣵⣭⠥⠿⠿⠖⠶⠿⠧⢭⣵⣥⠫⣀⢘⡇⠀⣁⠀⠿⠆⢄
⠀⠀⠀⢀⠁⢀⠄⠰⡊⠉⡙⣬⣦⠖⠋⠙⡄⠀⠀⡇⠀⠀⡇⠀⠀⡜⠉⠓⢦⣦⠜⡉⠙⡒⠂⢄⠀⢁
⠀⠀⡠⠾⠀⢠⣄⢤⠓⣌⡴⠋⠉⢦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠎⠉⠣⣌⡔⢣⢄⣴⠀⠸⠦⡀
⠀⡀⠀⢠⠐⢪⡀⡰⢨⠟⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⢯⠰⡀⣨⠒⢠
⠀⣠⡀⠠⣀⠼⢍⣶⣏⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⡀⠀⠐⠀⠀⠄⠀⠀⠀⠀⢈⣷⣎⠽⠄⡠⠀⣠⡀
⠐⠑⠀⠶⣇⠠⠄⡆⠀⠉⠀⠀⠀⠀⠐⠀⠀⢄⢢⣸⢸⢸⣠⢂⠄⠀⠐⠀⠀⠀⠀⠊⠁⠀⡄⠠⢀⡷⠆⠀⠑
⠇⠠⢃⡀⡸⡅⣿⡇⠤⠤⠀⠠⠀⠀⠀⠠⣈⡲⣵⣷⣿⣷⣷⡵⣊⡡⠀⠀⠀⠀⠀⠀⠤⠄⣿⡇⡽⡀⣀⠣
⠸⠂⢀⣱⡀⡸⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⣒⣚⣿⣿⣿⣿⣛⣟⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⣽⠸⡀⣰⣁⠁⠾
⡇⠀⢆⠀⢹⡅⢿⡇⠉⠉⠀⠀⠀⠀⠀⠈⠤⢚⢝⣿⣟⡿⣟⢝⠢⠈⠀⠀⠀⠀⠀⠈⠉⠁⣿⠇⣽⠁⢀⡆
⠀⣄⡀⠓⡇⠈⢇⣷⣀⡠⠄⠀⠀⠀⠀⠀⠀⠁⠎⡸⢸⠸⡈⠆⠁⠀⠀⠀⠀⠀⠀⠤⣀⣰⡇⠏⠀⡗⠃⣀⡔
⠀⠊⠠⣈⠉⢹⡥⡛⢧⠀⠀⢀⠀⠀⠀⠄⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⢀⡀⠀⢠⠟⠣⣽⠉⢉⡠⠈
⠀⠀⡀⠈⠰⢺⡀⢙⡨⣷⡎⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣠⡯⣘⠁⣨⠲⠈⠀⡀
⠀⠀⠈⢻⠀⢘⠉⠉⡖⠩⡙⢦⣠⠋⠀⠀⡀⠀⠀⡀⠀⠀⡀⠀⠀⡄⠀⠈⢣⣠⠞⡩⠑⡎⠉⢙⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠁⠒⢣⠤⢴⣉⢋⠓⢦⣼⣁⠀⠀⠇⠀⠀⠇⠀⠀⣱⣤⠖⢋⢋⡱⠤⢠⠓⠂⠁⢀
⠀⠀⠀⠀⠀⠁⠺⠇⠈⢄⢀⣵⡀⣉⣌⡙⠭⡙⡾⡷⡒⡲⡿⠞⡩⠝⣉⣌⡁⣀⣅⢀⡌⠀⠿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⣀⡀⠘⣀⠀⡣⢄⠼⠓⢧⠉⢡⠖⠻⢄⠤⡃⢀⡘⠀⢀⡀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠂⣠⠈⠁⠤⠘⠉⠙⠠⠄⠉⢠⡀⠈⠀⡠⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣈⣉⣃⣄⣀⣀⣚⣂⣀⣀⣄⣋⣉⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⢫⠏⠀⠀⠀⠀⠀⠀⠀⠠⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠃⠀⠀⠀⠂⠀⠈⠉⠁⠀⠀⠀⡞⡸⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠠⠁⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⢢⠀⡀⠀⠀⠀⠀⠀⣴⡇⠀⠀⡆⣥⣀⡀
⠀⠀⠀⠀⠁⠒⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠐⠁⠀⠀⠀⠤⠄⢠⡤⡀⠀⠀⠀⠀⢱⢙⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⡀⠧⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⢀⣴⣡⠊⠀⢀⡀⠀⠀⠀⠀⠀⠢⠷⠄⠀⠀⠀⡖⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣮⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠉⠉⠈⠈⠉⠁⠀⠁⠀⠉⠉⠀⠀⠀⠀⠉⠐⠐⠚⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠊⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠖⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢒⠆⠀⠀⠀⠀⠀⠀⠒⢀⡤⡀
⠀⠀⠀⠀⠀⢀⡤⣠⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡌⠀⠀⠀⠀⡠⡴⠀⠀⠀⢱⢹
⠀⢀⠄⠀⠀⠀⠀⠀⠉⠙⢢⠢⡀⠀⠀⠀⠀⠀⠀⢀⣎⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠜⠀⠀⠀⠀⠀⠢⠧⠄⠀⠀⡨⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠱⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⡖⡶⠀⠀⠀⠀⠱⡈⡄⠀⠀⠀⠀⠀⠀⡜⠐⠁⠀⠀⠀⠀⠀⢀⠔⡶⠀⠀⠀⠈⢢⠱
⠀⠀⠀⠠⠤⣤⣀⠀⠀⠀⠀⠎⡐⠀⠀⠀⠀⠀⡇⢇⠀⢀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡸⠠⠁⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⠀⠀⢨⠆⠃
⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⢜⡴⠁⠀⠀⠀⠀⠀⠉⠐⠓⠁⠀⠀⣰⢁⠆⠀⠀⠀⠀⣰⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⠚⠂⠀⠀⠀⢀⠔⡸
⠀⠀⠀⠀⠀⠀⠴⠶⠾⠷⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⣠⡤⠾⠵⠁⠀⠠⠤⠤⠤⠯⠾⠀⠀⠀⠀⠀⠀⠀⠒⠢⠤⠤⠤⠤⠴⠶⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢢⡠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠑⢬⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⠀⠀⠀⠀⠀⠇⠇⠀⠀⠀⠀⠀⡠⡺⠋⠀⠀⠀⠀⠀⠀⠀⠱⡘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣱⣄⠀⠀⠀⢀⢇⡇⠀⠀⠀⠀⡘⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠊⡘⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠎⠀⠀⠀⠀⠀⠀⢡⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⠤⠤⡄⠀⠀⠀⠀⢀⡌⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⢀⣠⣣⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⢄
⠀⠙⠋⠂⠀⠀⠀⠀⠈⠰⢄⠀⠐⠄⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠲⣤⣐⣀⣀⣀⠀⠀⠀⠀⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠠⠸⠀⠀⠀⠀⠀⠀⠄
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠢⡊⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠒⠈⡁⠀⠀⠈
⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⢄⠀⠀⠀⣀⡀⡀⠀⠠⠤⠠⠤
⠀⠈⠀⠈⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠂⠤⠄⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⡀
⠀⠐⠀⠒⠃⠈⠁⠀⡀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠉⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠂⠈⠓⠒⠉⠉⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠲⣤⠀⢠⡄⠈⠢
⠂⠀⠀⡀⢀⡀⠀⠀⠀⠀⠈⠁⠁⠈⠑⠋⠂⠊⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠳⠀⠀⠜⡀⢀⠳⢀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠐⠲⠦⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⠀⠂⠀⠀⠀⠀⡂⠘⠀⠀⠀⠀⠈
⠕⠀⠀⠄⠁⠀⠀⠀⡀⠀⠀⠀⠀⠢⠀⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠠
⠀⠐⢆⠀⠀⠀⠀⠀⠐⠂⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢊
⠀⠀⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠄⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⡠⠃⠀⠀⠀⡐⠀⠀⢀⡀⠀⠉⠉⠙⠀⠀⠘⠀⠈⠆
⠀⠄⠀⠀⢀⠀⠀⢠⢀⡀⠀⠀⠈⠰⢤⡀⠈⠀⠀⠐⠀⠀⠀⢀⠉⠈⠓⠒⠛⠢⠤⢌⠀⠀⠀⠁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠂⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠱⠁⠀⠈⠙⠦⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣦⡀⠀⣠⠀⠙⢿⣿⣷⣄⠀⣠⡄⠘⢿⣿⣿⣦⡀⢀⣤⠈⠻⣿⣿⣦⡀⠀⣠⠀⠙⢿⣿⣷⣄⠀⣠⡄⠙⢿⣿⣿⣦⡀⢀⣤⠈⠻⣿
⠟⣡⣾⣿⠀⠀⠀⠙⠿⣫⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⣿⠀⠀⠈⠻⠟⣡⣾⣿⠀⠀⠀⠙⠿⣫⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⣿⠀⠀⠈
⠈⠻⣿⣿⣧⡀⠀⣠⡄⠙⢿⣿⣷⣄⠀⢀⡄⠙⢿⣿⣿⣦⡀⠀⣠⠈⠻⣿⣿⣧⡀⠀⣠⡄⠻⢿⣿⣷⣄⠀⢀⡄⠙⢿⣿⣿⣦⡀
⠀⠀⠈⠻⠿⣫⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⡇⠀⠀⠙⢿⠟⣡⣾⣿⠀⠀⠈⠻⡿⣫⣾⣿⡇⠀⠈⠹⡿⢋⣴⣿⡇⠀⠀⠙⢿⠟⣡⣾
⣦⡀⠀⣠⠈⠻⣿⣿⣧⡀⠀⢀⡄⠙⢿⣿⣷⣄⠀⠀⣠⠘⢿⣿⣿⣦⡀⠀⣀⠈⠻⣿⣿⣧⡀⠀⢀⡄⠻⣿⣿⣷⣄⠀⠀⣀⠘⢿⣿
⡿⣣⣾⣿⠀⠀⠈⠻⡿⢋⣴⣿⡇⠀⠀⠙⢿⠟⣡⣾⣿⠀⠀⠙⢿⡿⣃⣾⣿⠀⠀⠈⠻⡿⢋⣴⣿⡇⠀⠈⠻⣿⠟⣡⣾⣿⠀⠀⠙
⠘⢿⣿⣿⣦⡀⠀⢀⡀⠻⣿⣿⣧⡀⠀⠀⡀⠸⣿⣿⣿⣄⠀⠀⣀⠘⢿⣿⣿⣦⡀⠀⢀⡀⠻⣿⣿⣧⡀⠀⠀⡀⠸⣿⣿⣿⣄
⠀⠀⠙⢿⡿⢋⣴⣿⡇⠀⠈⠻⣿⠟⣠⣾⡇⠀⠀⠹⣿⡿⢃⣴⣿⠀⠀⠙⢿⡿⢋⣴⣿⡇⠀⠈⠻⣿⠟⣠⣾⡇⠀⠈⠙⢿⡿⣣⣾
⣄⠀⠀⢀⠐⢿⣿⣿⣧⡀⠀⠀⡀⠺⣿⣿⣷⡀⠀⠀⣀⠸⢿⣿⣿⣄⠀⠀⢀⠀⢿⣿⣿⣧⡀⠀⠀⡀⠺⣿⣿⣷⡀⠀⠀⣀⠘⢿⣿
⡿⢃⣴⣿⠀⠀⠙⢿⣿⠟⣠⣾⡇⠀⠈⠻⣿⡿⢂⣴⣿⠀⠀⠙⣿⡿⢃⣴⣿⠀⠀⠙⢿⣿⠟⣠⣾⡇⠀⠈⠻⣿⡿⢂⣴⣿⠀⠀⠙
⠐⢿⣿⣿⣄⠀⠀⠀⠁⢾⣿⣿⣧⡀⠀⠀⡈⠰⣿⣿⣿⡀⠀⠀⢀⠰⣿⣿⣿⣄⠀⠀⠀⡁⢾⣿⣿⣧⡀⠀⠀⠈⠰⣿⣿⣿⡀
⠀⠀⠙⢿⣿⠗⣠⣾⡇⠀⠙⢿⣿⡿⢠⣾⡇⠀⠈⠻⣿⡿⢂⣴⣿⠀⠀⠛⣿⣿⠗⣠⣾⡇⠀⠙⢿⣿⡿⢀⣴⡇⠀⠈⠻⣿⡿⢆⣴
⡀⠀⠀⠀⠁⢾⣿⣿⣇⠀⠀⠀⠉⢶⣿⣿⣧⡀⠀⠀⠈⠰⣿⣿⣿⡀⠀⠀⠈⠁⢾⣿⣿⣇⠀⠀⠀⠉⢴⣿⣿⣧⡀⠀⠀⠈⠱⣿⣿
⣿⠆⣠⣾⠀⠀⠙⢿⣿⡷⢀⣼⡇⠀⠙⢿⣿⡿⢂⣴⣿⠀⠈⠻⣿⣿⠆⣠⣾⠀⠈⠙⢿⣿⡷⢀⣴⡇⠀⠙⢿⣿⡿⢂⣴⣿⠀⠈⠻
⠡⣾⣿⣿⡄⠀⠀⠀⠉⢶⣿⣿⣇⠀⠀⠀⠉⢰⣿⣿⣿⡀⠀⠀⠈⠡⣾⣿⣿⡄⠀⠀⠀⠉⣴⣿⣿⣇⠀⠀⠀⠉⢰⣿⣿⣿⡀
⠀⠈⠻⣿⣿⡦⣀⣼⡇⠀⠙⢿⣿⡷⢀⣴⡇⠀⠙⢿⣿⣿⠆⣠⣾⠀⠈⠻⣿⣿⡦⢀⣼⡇⠈⠻⣿⣿⣷⢀⣴⡇⠀⠙⢿⣿⣿⠆⣠
⡀⠀⠀⠈⠋⣴⣿⣿⡇⠀⠀⠀⠉⢴⣿⣿⣇⠀⠀⠀⠙⢡⣾⣿⣿⡀⠀⠀⠈⠋⣴⣿⣿⡇⠀⠀⠈⠛⣵⣿⣿⣇⠀⠀⠀⠙⢡⣾⣿
⣿⡆⢀⣴⠀⠈⠻⣿⣿⡦⢀⣴⡇⠀⠹⢿⣿⣷⠄⣠⣾⠀⠙⢿⣿⣿⡆⢀⣴⠀⠈⠻⣿⣿⡦⢀⣴⡇⠀⠹⣿⣿⣷⠄⣠⣶⠀⠙⢿
⢫⣴⣿⣿⡀⠀⠀⠈⠋⣴⣿⣿⡇⠀⠀⠀⠙⢡⣾⣿⣿⠀⠀⠀⠙⢋⣴⣿⣿⡀⠀⠀⠈⠋⣴⣿⣿⡇⠀⠀⠈⠛⢡⣾⣿⣿
⠀⠙⢿⣿⣿⡦⢀⣴⡆⠈⠻⣿⣿⣦⠀⣠⡆⠈⠛⣿⣿⣷⡄⢀⣴⠀⠙⢿⣿⣿⣦⢀⣴⡆⠈⠻⣿⣿⣦⠀⣠⡖⠈⠻⣿⣿⣷⡄⢀
⠀⠀⠀⠙⠋⣴⣿⣿⡇⠀⠀⠈⠛⣡⣾⣿⣇⠀⠀⠈⠛⢋⣴⣿⣿⠀⠀⠀⠙⠛⣴⣿⣿⡇⠀⠀⠈⠛⣡⣾⣿⣇⠀⠀⠈⠛⢋⣴⣿
⣷⡄⢀⣴⡂⠙⢿⣿⣿⣦⠀⣠⡆⠈⠻⣿⣿⣦⡀⢀⣴⠈⠻⣿⣿⣷⡄⢀⣴⠀⠙⢿⣿⣿⣦⠀⣠⡆⠈⠻⣿⣿⣦⡀⢀⣴⠈⠙⢿
⢋⣴⣿⣿⠀⠀⠀⠙⠟⣡⣾⣿⡇⠀⠀⠈⠻⢋⣴⣿⣿⠀⠀⠈⠻⢋⣵⣿⣿⠀⠀⠀⠙⠟⣡⣾⣿⡇⠀⠀⠈⠻⢋⣴⣿⣿
⠈⠻⢿⣿⣷⣄⠀⣠⡆⠙⢿⣿⣿⣦⠀⢀⡄⠈⠻⣿⣿⣦⡀⢀⣴⠈⠻⣿⣿⣷⣄⠀⣠⡆⠙⢿⣿⣿⣦⠀⢀⡤⠈⠻⣿⣿⣦⡀⢀
⠀⠀⠈⠛⠟⣡⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⡇⠀⠀⠈⠻⢟⣵⣿⣿⠀⠀⠈⠻⠟⣡⣾⣿⡇⠀⠀⠙⠿⢋⣴⣿⡇⠀⠀⠈⠻⢟⣵⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠔⠒⠒⠊⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠐⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠀⠀⠂⠀⠀⠀⣿⢢⣄⣀⡀
⠀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣿⢸⣼⣿⡇
⠀⠀⠀⠀⠀⢸⠃⡟⠀⠀⠰⠄⠀⠀⠀⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠃⠀⠀⠀⣿⠸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠐⠀⠀⠇⠰⠂⠀⠁⠀⠀⡀⠀⠀⡄⠀⠀⠀⠂⠀⠀⠀⣿⢀⣼⣿⡇
⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠛⠀⠀⠀⢀⠀⡀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠀⣿⡇
⠀⠀⠀⠀⠀⡟⠸⠇⠀⠠⠆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⡠⢀⠀⠀⠀⠀⠀⣿⢀⠀⣿⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠀⠠⠂⡐⢀⠂⠀⣄⢀⠀⢠⡟⢸⢠⣿⡇
⠀⠀⠀⠀⠀⡇⢀⠀⠀⣠⡤⠀⠀⠀⢰⡿⠀⡖⢰⠀⠀⠀⢠⠄⠀⠈⠁⠀⠘⠁⠀⠀⢸⡇⣼⢘⣿⡇
⠀⠀⠀⠀⠀⡇⢸⠀⠈⠁⠂⠀⠀⢠⡿⠀⠀⢷⠈⣹⠀⠀⠨⠀⠀⠈⠐⠠⠀⠀⠀⠀⢸⡇⡋⣼⣿
⠀⠀⠀⠀⠀⡇⢸⠀⡀⠀⣄⠀⠀⣼⠁⠀⠀⠸⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠆⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⠀⡇⣾⠀⠁⣀⠂⠐⠻⠃⠀⣀⠀⠀⢰⠋⠉⡆⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⠀⡇⠛⠀⠀⠀⠀⣸⠏⠀⠿⠷⠀⠀⠈⠂⠀⠁⠇⠀⠀⢀⠔⠩⡇⡠⣁⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⡎⠀⠀⢀⢀⣤⡆⣴⢠⢤⡰⠤⣀⠰⠁⠀⠀⠔⠁⠾⠀⢸⡇⡇⣿⣿
⠀⠀⠀⠀⢸⠇⢠⠀⠀⢘⡟⠀⣴⣷⢻⣾⣿⡇⡏⣿⢸⢃⡠⠊⠀⠀⠀⣀⣀⡠⠂⠀⢸⡇⡇⢸⣿
⠀⠀⠀⠀⢸⠀⣾⠀⢤⡾⠁⡸⠟⣛⣸⣿⣿⠇⢿⣿⢸⣨⣶⠀⠀⠐⢻⠀⢸⠀⠀⠀⣸⡇⡇⢸⡇
⠀⠀⠀⠀⢸⠀⡏⠀⢀⠤⡀⠃⠀⠀⣠⢄⣹⣿⢘⡩⠮⣬⠒⡆⠀⠀⡾⠀⢬⠀⠀⠀⣿⡇⣇⣼⡇
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⡄⠀⠐⢸⣿⣾⢡⣿⠜⡬⠿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⡇
⠀⠀⠀⠀⢸⠀⡇⠀⠀⠰⠐⠁⠀⠀⠈⠉⠁⠉⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⠶⢿⠁⣿⣿⡇
⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠟⠻⡿⠿⡟⣞⠿⠯⠽⣦⣾⣶⡘⢦⣿⣿⡇
⠀⠀⠀⠀⠘⣴⣖⡺⢿⣿⣿⣻⣻⣝⣯⡧⠽⠦⠤⣄⣻⣿⠶⠤⠤⠐⠒⣊⣉⣩⡿⠥⠴⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠪⡻⣾⡉⠡⠤⠄⠒⢒⣂⣉⣉⣁⣭⣄⣀⣀⣤⣤⣼⣷⣾⠷⠶⠟⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣿⡿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢰⠂⠉⠱⠢⣤⣤⡐⣤⣤⡄⠂⠉⠀⠖⣈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⣀⠀⠀⠟⠓⠛⡆⠀⢠⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⢀⣐⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠊

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠹⡆⠘⢦⡴⠃⢰⠏⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠘⠶⠶⠶⠶⠃⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⡀⢈⡷⡶⠖⠛⢷⣄⠀⠀⡀⣀⡀⠀⣠⡾⠙⠲⢶⠾⡁⢀⣀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡨⠟⠁⠀⠀⠀⢀⣀⣨⣟⣉⠉⠉⣉⣻⣅⣀⡀⠀⠀⠀⠈⠻⢄⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⣠⠞⢳⡶⠋⣀⣀⣰⢯⡛⢞⡽⣆⣀⡀⠙⢶⡞⠳⣄⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡚⠙⣧⠞⠁⠀⣾⠀⣰⣏⠙⠛⣷⢄⡠⣾⠛⠋⢹⣆⠀⣷⠀⠈⠳⣼⠋⢓⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⢹⡇⠀⠀⠸⡌⠁⠉⢛⡻⠅⠋⠙⠈⢟⡛⠉⠈⢡⠇⠀⠀⢸⡏⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⢤⡤⠤⣽⠘⢚⡿⠃⢠⡆⢰⡄⠘⢯⡓⠃⣯⠤⢤⡤⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡿⠰⠟⠠⣄⡈⠛⠛⢁⣠⠄⠻⠆⢿⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢀⠀⠀⠙⢛⡛⠋⠀⢀⡀⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⡌⢛⣒⣚⣏⣹⣒⣒⡋⢡⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡹⣌⠀⠘⠶⠶⠃⠀⣡⢏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣌⠉⠛⠛⠛⠛⠉⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣆⣰⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠋⠙⠋⠉⣁⣀⡒⣦⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⢀⣂⢈⡐⣈⣀⣨⣄⣤⢰⣂⣀⣠⠀⠘⣉⣀⣀⠀⠀⠲⠐⠒⠠⠀⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣀⣤⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣴⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⢬⣩⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠷⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠩⠉⠉⠁⠁⠉⠈⠈⠙⠻⢿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣀⠀⣀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠙⢿⣿⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦⣄⠀⠹⣿⣿⣷⣔⡀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⠃⠻⠿⢏⣾⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢛⣇⠈⠙⢷⣆⡈⠻⣿⣿⣿⣥⣦⣴⣾⣶⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⣶⣶⣶⣾⣿⣧⣷⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠹⠛⠶⣬⣙⠛⠻⠻⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⠀⠐⢿⣷⠛⠛⠿⣿⣾⣁⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣶⣿⣿⣿⣛⠿⢷⣤⣀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⣈⣉⠙⠓⠒⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠂⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠻⢟⡭⠈⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⡆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⢔⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡂⠀⠀⠀⠀⣜⡁⠔⠻⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⡦⠀⠀⠀⠀⠀⠀⠀⠱⢶⡏⢣
⠀⠀⠀⠀⣠⠦⠿⠯⣛⡸⢄⠀⠀⠀⠀⠀⢠⣐⢇⣚⡇⠀⣠⠖⠋⠛⠛⠾⠟⣆⠀⠀⠀⣤⠤⠴⣰⣎⠶⡼⢤⠀⠀⠀⠀⢠⠞⠒⠓⠲⡎
⠀⠀⢀⡞⠁⢠⠴⠒⠦⡀⠀⢳⡀⠀⣀⡴⠋⠀⠀⢸⡇⠰⢇⡀⣠⠶⢤⠀⠀⢸⡆⠀⠀⢧⣀⣀⣈⠀⠉⣰⠏⠀⠀⠀⢠⠏⠀⣰⠂⠀⡇
⠀⠀⡾⠀⢰⠇⠀⠀⠀⢸⡄⠀⣧⠀⣇⣀⣤⡄⠐⢹⡇⠀⠀⠉⠁⠀⣸⠀⠀⣸⠁⠀⠀⠀⣠⠔⠁⠀⣴⠃⠀⠀⠀⣰⠃⢀⡜⢸⠀⠀⡇
⠀⠀⡇⠀⢸⡀⠀⠀⠀⢸⠇⠀⣹⠀⠉⠁⢸⡇⠀⢸⡇⠀⠀⠀⢀⡴⠁⠀⣰⠃⠀⠀⠀⠈⣧⣠⣀⠀⠀⠙⢦⠀⣰⠃⠀⠈⠉⠉⠀⠀⠙⢻⡆
⠀⠀⢷⠀⠀⠳⣤⣠⡴⠋⠀⢀⡟⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⡔⠋⠀⠀⠚⠓⠒⢲⡆⠀⠀⠀⠀⢈⡇⠀⠀⢸⡖⠳⠲⠶⠶⠶⢶⠀⠀⡖⠚⠃
⠀⠀⠈⢧⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢸⡇⠀⢸⡇⠀⠐⣇⣀⣀⣀⣀⣀⣀⣸⡇⢀⡏⠙⠛⠉⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠈⠙⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⠒⡤⠒⠄⠀⠀⠀⠀⠀⠀⠠⠁⠐⠈⢣⡀⠀⠀⣠⠤⠤⠴⣐⣾⠱⠄⡧⣤⠀⠀⠀⢠⠀⡠⢦⠀⠀⠀⠀⠀⠀⡐⢁⣶⡉⢃
⠀⠀⠀⠀⡏⠑⠙⠣⠭⠏⠉⡇⠀⠀⠀⣠⠞⠛⠛⠓⠊⣹⠀⠀⣿⠀⠀⠀⠀⠀⠙⠋⠅⢸⠀⠀⣠⣧⣼⣶⣴⣄⠀⠀⠀⡴⠋⠙⢒⣙⠒⠋⠳⡀
⠀⠀⠀⠀⡇⠀⢰⠋⠉⠉⠉⠁⠀⠀⡼⠁⠀⣠⣒⡛⠙⠃⠀⠀⠛⠒⠚⢛⡷⠃⠀⢀⡴⠛⠀⡞⠀⢠⠴⢬⡀⠘⣇⠀⢸⠃⠀⣸⠉⠈⢱⠀⢰⢷
⠀⠀⠀⠀⡇⠀⠈⠑⠒⢦⡀⠀⠀⢸⠁⠀⠀⠉⠉⠉⠓⢆⠀⠀⠀⠀⣠⠏⠀⢀⡴⠋⠀⠀⠘⣇⠀⠙⠦⠞⠁⢠⠏⠀⠸⡄⠀⠘⠲⠶⠋⠀⠉⢸
⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⠹⡄⠀⣿⠀⠀⡴⠉⠉⢣⠀⠘⡇⠀⠀⣰⠏⠀⢀⡾⠁⠀⠀⠀⣠⠞⠃⠀⠀⠀⠘⠳⢄⠀⠀⠙⠦⢤⣤⣄⠀⠀⠀⡟
⠀⠀⠀⢠⠤⠤⠤⠞⠀⠀⢠⠇⠀⢸⡀⠀⠳⣄⣠⠞⠀⢠⡇⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢰⡇⠀⢠⡎⠉⠙⡆⠀⠈⡇⠀⣠⣤⠤⠖⠁⠀⢀⡼⠁
⠀⠀⠀⢾⣀⡀⠀⠀⣀⡤⠋⠀⠀⠀⠳⣄⣀⠀⠀⣀⡤⠏⠀⠀⠸⣥⣤⣤⠇⠀⠀⠀⠀⠘⣇⠀⠀⠓⠒⠚⠁⠀⣠⠇⠀⣷⠀⢀⣀⣠⠴⠋
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠴⠚⠁⠀⠀⠈⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠖⣂⣀⣀⣀⣀⣀⣐⠢⠀⠀⠐⠒⠖⠂⠀⠀⠀⠀⠄⢀⡂⠠⠄⠀⠀⢰⢈⠉⠭⠭⢉⠒⠊⡉⠭⠭⠍⢰⠀⠀⠀⠀⠙⢮⢄⠀⠀⣀⠀⠀⣀⠀⡇
⠀⠀⡀⣇⣀⣀⣂⣀⣀⡸⠘⠀⠀⠘⡉⠉⠃⠀⠀⠀⠁⠜⣁⣀⠑⡌⠀⠀⢐⠐⢒⣀⡒⠒⢨⠀⠒⠒⠀⠒⢸⠀⠀⠀⠀⡠⣄⡙⢦⡄⠈⠲⡀⢸⠀⡇
⠀⠀⠇⠖⣤⠒⣤⠢⡔⢢⠘⠀⠀⠃⡇⠘⠀⠀⠀⠈⠀⢆⠃⠀⠠⠃⠁⠀⠠⢈⠥⠤⠤⢌⠸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠑⢢⠇⠀⠙⢖⡀⠈⠋⠀⡇
⠀⠀⡆⠟⣿⠛⣽⠛⡝⢫⢘⠀⠀⢂⡇⢸⡀⠀⠀⠀⠁⠀⡶⣖⡇⠈⠀⠀⠈⠐⠂⠐⠒⠒⠘⠀⠒⠀⠐⠒⢸⠀⠀⣴⠏⣴⠏⠀⠀⠀⠀⠙⢵⡀⠀⡇
⠀⠀⠇⡛⣿⠛⣿⢛⡇⢸⠠⠀⠀⠘⢭⡭⠃⠀⠀⠀⠀⠀⠛⠒⠁⠀⠀⠤⠘⠮⠍⠭⠭⠉⠾⠏⠭⠩⠭⠭⠜⠀⠸⣟⣾⠏⢀⢠⠤⠤⠠⣄⠀⠙⢇⡇
⠀⠀⠣⠭⠤⠭⠤⠥⠬⠥⠞⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠜⣌⡣⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠉⠀⡔⠃⢀⠤⠤⢤⡀⠙⢆⠀⠁
⠀⢀⣴⠖⡖⠸⠰⢒⢲⣦⠀⠀⠀⡠⠒⠉⣻⡯⢋⢓⣄⠀⠀⠀⠀⠀⡧⣭⢴⠀⠀⠀⠀⠀⠀⡜⢉⢣⠀⠀⠀⠀⠀⠀⣨⠀⢰⠁⠀⠀⠀⢙⡄⠘⡆
⠀⢸⠉⠉⠉⠁⠉⠉⠉⠩⠀⠀⡜⠀⠀⠀⡿⣗⣕⣕⣝⡆⠀⠀⠀⡸⣠⠍⣆⢧⠀⠀⠀⠀⡸⠀⢨⠀⢣⠀⠀⠀⠀⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠀⠀⠀⠙⠉⠉⠉⢙⠀⠀⠀⢠⢡⠃⠀⠘⡌⠆⠀⠀⡰⠁⠀⠘⠀⠀⢣⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣄⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠖⠋⠀⠀⠀⠈⠺⠀⠐⠻⢿⣯⣏⣿⡽⠾⠂⠀⠀⠀⠀⣤⠄⢠⠀⢶⠌⢠⢠
⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠑⠒⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠘⠧⠠⠄⡀⠀⣤⠤⠜⠀⠀⠀⠀⠀⠀⠉⠉⡉⣱⠀⠀⠘⠃⠈⠃⠸⠋⠀⠁⠃⠀⠀⠀⠀⠀⠁⠖⠒⠒⡒⠒⠒⢢⢹⠀⠀⠀⣀⠤⠦⣼⢒⢶⣄
⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⠁⠀⠇⠾⠀⠀⠀⠀⢀⣤⠤⡤⢄⡀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠂⠚⢸⠀⢀⠞⠀⠀⠀⣿⢕⢕⠝⣧
⠀⠈⡎⢁⠤⣁⠤⣉⠤⡌⠁⡇⠀⠀⠀⢰⠀⢠⠉⡇⠀⠀⢀⠚⠁⣀⠤⠤⣀⠈⠳⡄⠀⠀⠀⠀⠐⣁⢿⣨⢯⣹⢏⡹⢸⠀⢸⠀⠀⠀⠀⠿⠷⠿⠿⡞
⠀⠀⡏⠻⡠⠻⠄⠟⠄⠏⠙⡇⠀⠀⠀⢈⠀⢸⠠⡇⠀⠀⣌⠀⡜⠀⠀⠀⠀⢱⠀⢙⠀⠀⠀⠀⠸⣥⠿⣬⢯⣼⡇⢹⢸⠀⠘⡄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⣇⣰⠓⣦⠚⣦⠚⣆⣨⡇⠀⠀⠀⠋⠀⠘⢰⠃⠀⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠈⠉⠀⠀⠀⡘⠤⠿⠬⠧⠼⠧⠜⣸⠀⠀⠈⠳⠤⠀⠀⠤⠚⠁
⠀⢀⠣⡘⢒⡉⢒⣉⠒⣁⠜⡇⠀⠀⠀⡆⠀⡆⣹⠀⠀⠉⠉⠁⠉⠉⠉⠉⠈⠉⠁⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢁⣀⣀⣀⡀
⠀⠀⢩⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⠾⣥⣁⣇⡏⡯⠽⡆⠀⠀⣋⣙⣋⡄
⠀⠀⠀⠘⡦⡀⠀⠀⠀⠀⠀⢀⠀⡠⠬⠄⡀⡀⠀⠀⠒⣉⠭⠤⠍⡁⡒⢈⠉⠤⠭⢉⡂⠀⠀⢀⠞⣘⢢⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢡⠀⡌⡃
⠀⠀⠀⠀⠀⠹⢢⡀⠀⠀⠀⡀⢸⢠⠒⢦⢹⢀⠀⠀⠀⠐⠂⠀⠒⠂⡁⠐⠒⠀⠀⠒⠆⠀⠀⠈⡮⣊⢴⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⡇⡁
⠀⠀⠀⠀⡷⢄⠀⠩⢢⡀⠀⠠⠌⢆⠀⢠⠋⠄⠀⠀⠄⡉⠤⠤⠭⠁⠃⠐⠀⠐⠀⠂⠀⠀⠀⣸⣠⣹⣄⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⣀⣇⠃
⠀⠀⠀⠀⣇⣀⣑⡄⠀⠊⢦⡀⠀⢸⡻⢹⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⡃⠐⠒⠒⠒⠂⠆⠀⢰⢡⠃⠀⠰⡘⡄⠀⠘⢇⣀⣉⡀⢈⣁⡠⠇⠀⠀⠑⠶⠊
⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠬⠼⠦⠀⠉⠁⠀⠀⠀⠀⠻⠭⠭⠉⠩⠅⠿⠨⠍⠉⠭⠡⠟⠀⠗⠃⠀⠀⠀⠑⠧⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⢀⠤⠐⢶⣟⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡔⠒⠒⢌⠄⠀⠀⠀⡜⢸⠈⡄
⠀⠀⢠⠃⠀⠀⣿⣷⣿⣛⣧⠀⠀⠀⠛⠀⠉⠀⠛⠁⠐⠘⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⣀⢀⡀⣀⡀⠀⠤⢸⠠⠋⠱⢨⠠⠀⠀⡜⠀⢸⠀⠘⡄
⠀⠀⢸⠀⠀⠀⠉⠉⠉⢙⠇⠀⠀⠀⣠⢲⠆⠤⠤⠤⠄⠒⠐⡄⠀⠀⠀⡇⠥⠤⠤⠥⠤⠤⠡⠄⠀⠀⠐⡇⠀⠐⠂⢱⣀⣠⠁⠂⠀⣼⣀⣀⣸⢀⣀⣘⡄
⠀⠀⠀⠳⣄⠀⠀⢀⡠⠊⠀⠀⠀⠀⠙⢼⢂⠂⢒⡒⡂⣠⡀⠇⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠐⠃⠀⠀⠀⠸⢝⡻⠀⠀⠀⠙⠛⠓⠷⠖⠒⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠁

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

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠋⠡⡀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⢀⣤⣶⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢠⣾⣿⡇⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⣻⣼⣿⣿⣿⠛⠋⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣬⣿⢿⡿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠻⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣴⣴⣿⣿⡿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⡿⣿⠁⠈⠁⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⢿⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⠀⠴⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⢀⠠⢀⣀⣀⣀⣀⣤⣀⣀⣠⣤⡤⢤⣤⣤⡤⠀⠀⢀⣀⡀⡽⠿⠛⡇⢀⣉⣀⣀⡀⡇⠀⠀⢀⡾⠃
⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡡⠔⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⡵⢀⠞⠉⠀⠀⣷⣤⡀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠖⠃⢀
⠂⠀⠀⠀⣴⣷⣶⣶⡄⠀⠂⠀⠀⠀⠀⠈⠀⠀⢨⠂⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠉⠉⠉⠎⣠⠜⠀⠀⢀⡀⠆⡿⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⢚⣤⠀⢸⣿⡏⣿⣿⣷⠀⠁⠤⠀⠀⠄⠀⠀⠀⠈⠁⢀⠤⠤⠄⠤⠤⠄⠀⡀⠀⠀⠐⠤⠀⠀⠐⡕⠙⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠠⠗⠀⣼⣿⣿⣿⣿⡿⠀⡇⠀⠀⠀⠀⢀⠀⠀⢀⡔⠀⣄⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠲⠐⠀⠂⠀⠀⠀⠀⠀⢠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠤⠀⢰⢁⡽⠟⠻⠿⡉⠉⠱⠄⠀⠀⠠⠀⠀⠄⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠁⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣈⡀
⣿⣿⡿⠿⣿⠿⢿⢿⡿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣶⣶⣷⣶⣷⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆

⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣽⡆⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⣰⠶⡄⠀⠀⠀⠀⣤⡙⢤⠈⠉⠐⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠠⣙⣿⡥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⡀⠀⠀⠈⠉⠁⠐⠂⠀⡠
⠀⠀⡤⠀⢤⡞⣶⣤⣀⠀⠀⠈⠻⣆⣰⡦⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠺⠫⠟⠠⣿⡇⠉⠻⢷⢤⡀⠀⠘⣷⡀⠀⠀⠀⢀⠴⢮⡀⠀⠐⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣀⣀⣘⡂⠀⠀⠀⠀⠐⢾⠉
⠀⠀⠀⣠⠀⣸⡇⠀⠀⠀⠘⢻⣷⣤⣿⢴⣀⣢⡠⠋⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠀⠜⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣶⣿⣿⣷⠃⢀⢠⠀⠈⠓⣄⢀⣀⠈⠃⠀⠁⠀⠓⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣬⣾⣷⢄⠤⠄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢈⠀⠀⠀⢸⡷⠀⣹⣾⡍⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⣿⣾⣿⣿⣮⠳⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣰⣟⠨⠇⠠⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⢈⣿⡟⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡴⠀⣺⣿⠠⣨⡄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣧⣼⣁⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⣤⠀⢠⡀⢀⠎⡀⠀⠀⠥⣼⡙⢷⣀⡘⠃⠘
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⡋⠒⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠞⢃⡌⠀⠀⠀⢸⠃⡈⢇⠈⠹⠿⣀
⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠓⠉⠉⠙⠚⠋⢮⢴⡜⠁⠈⠀⣠⣤⢀⡀
⠀⠀⠀⠴⠀⠿⠇⠐⠀⠀⠀⠀⠀⠻⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡏⠁⠀⠀⠀⠀⠀⠀⣨⠀⡃⠐⡀⠙⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⣄⠀⠀⠀⠀⠀⢀⠏⣀⢀⣽⡆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠈⠳⠻⣏⣄⣣⡐⠂⠠




⣀⣀⣄⣀⣀⣀⣀⣄⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡛⣙⣿⣉⡛⣻⡟⣻⡟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⠲⡀⢄⠂⠀⢠⢲⣤⡟⢇
⠀⠀⠀⠀⠀⡀⠀⣀⢀⢄⠀⠀⠀⠠⠀⣠⠴⠒⠺⢶⣜⡍⡇⠀⠀⠀⠀⢠⠏⠑⠛⢔⡦⠧⣄⡜⠲⠓⠤⡎
⠀⠀⣠⠴⠚⠺⠛⣵⣆⡇⢠⢠⡎⢬⡎⠣⣀⡤⢤⡀⠉⠙⡆⠀⠀⠀⠀⡼⠀⢰⠛⠒⠲⢤⡟⠀⣰⡆⠀⡇
⢀⡜⠁⢠⠔⠚⠓⢎⠀⢹⡤⠋⠉⠹⡇⠀⠀⠀⣠⠃⠀⢠⠇⠀⠀⠀⠀⠧⣄⡈⠑⠦⣀⠞⠀⡼⢡⡇⠀⡇
⣼⠀⢠⠇⠀⠀⠀⢸⢶⠉⢀⣀⠠⠂⣇⣀⠤⠚⠁⢀⡴⢋⠤⣀⠶⡀⣀⠀⠀⣱⠀⠀⢹⠀⠀⠉⠉⠀⠀⠉⢹
⡇⠀⠸⡄⠀⠀⢀⡼⠘⢻⠏⢹⠀⠀⣏⠀⠀⠀⠲⠯⣄⠼⡶⢷⣶⢿⡉⠉⠛⠁⠀⠀⡼⠛⠛⠛⢻⠀⠀⡏⠉
⠹⣄⠀⠈⠓⠒⠉⠀⢀⡞⠀⢸⠀⠀⣿⠲⠤⣄⣀⢰⠁⠀⡖⠒⡄⠀⢳⠦⠤⠤⠴⠊⠀⠀⠀⠀⢸⣄⣠⡇
⠀⠈⠳⠤⣤⣤⣤⠴⠋⠀⠀⢸⠀⠀⢸⠀⠀⠀⠈⠙⢆⠀⠙⠋⢁⣠⠋⠀⠀⡔⠠⠄⢷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣻⣏⠉⠀⠀⠀⢀⠖⠁⢀⡤⠤⡀⠈⠳⣤⠔⠛⠓⠶⠜⣦⡤⢠⣊⢣⣀
⠀⠀⠀⠀⠀⠀⣼⠉⠓⠚⠼⢏⠶⢨⣄⡀⠀⢀⢸⠀⢀⢻⣀⣀⡿⢀⠞⠁⢀⣤⡒⠒⠶⣣⠦⠽⢛⡊⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠧⢤⣤⣀⡀⠀⠉⠑⣸⠦⠤⡠⣜⢩⣻⡀⠀⠀⣀⡞⠀⠀⢀⣀⠉⠑⢤⡇⠀⡜⠉⠉⡇⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢀⣠⠖⠻⠤⢄⣉⠉⠒⣩⠟⠉⠉⢩⡇⠀⢰⠉⠈⢳⠀⢸⣇⠀⠙⠲⠖⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡴⠋⠀⠀⢀⠴⠊⠁⢀⡔⠁⠀⠀⠀⠀⢧⠀⠘⠢⠔⠃⠀⣸⠉⠲⢤⣤⠤⡦
⠀⠀⠀⠀⠀⣰⠃⠀⢀⡞⠁⠀⠀⠀⠸⠤⢤⡀⠀⠙⣆⠀⠀⠀⠀⠈⠓⠦⣤⣤⠤⠞⠁⠀⠀⢀⡠⠔⠁⠀⣠
⠀⠀⠀⠀⠰⠧⣤⣤⡞⠀⠀⠀⠀⡤⣤⣀⣠⠇⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣤⠠⠀⠠⢂⢀⡔⠂⣀⢀⠞⠁⠀⠀⠃⣠⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠠⠐⠂⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠋⠀⠔⢠⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⢀⣀⠞⡐⠀⠀⠀⠀⠀⠀⠂⢀⠀⡀⢠⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠁⠉⠀⠀⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢀⡀⣀⠀⣀⠀⣀⠀⣀⢀⣀⢀⡀⢀⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠋⠙⠛⠛⠙⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠶⠶⠿⠶⠶⠿⠶⠶⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⣔⣴⣶⣢⣢⣶⡴⠴⠶⠦⠆⠲⠴⠴⠦⢢⣶⣴⣔⣴⣶⠂⠀⠀⢠⠄⡔⢠⠢⡔⡤⠢⠤⠄⠀⠢⠄⠤⠤⢀⡔⢤⢢⡄⡤
⠀⠀⠀⣻⣞⡿⠓⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⢞⢾⣿⡁⠀⠀⢡⡘⣜⠃⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢫⡜⣤⠃
⠀⠀⠀⣽⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⢿⡅⠀⠀⢡⡜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠃
⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡳⠀⠀⠀⠉⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃
⠀⠀⠀⢫⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⡄⠀⠀⠉⠜⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠡⠃
⠀⠀⠀⣷⣻⣿⣟⣾⣷⣠⣤⣤⣄⣀⣤⣤⣤⣤⣖⣾⣷⣳⣻⣟⠆⠀⠀⠈⠤⠡⠉⠌⠱⠠⠄⠤⠠⢄⠀⠤⠤⠄⠴⠡⠉⠌⠤⠡⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⣀⣀⣄⣀⣤⣀⣠⣄⣠⣄⣀⣤⣀⣠⣀⣀⣄⣀⡄
⠀⠀⠀⠀⠒⠀⠒⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠒⠀⠂⠀⠀⠀⢻⣿⠿⣿⠿⠟⠋⠉⠉⠉⠁⠈⠉⠉⠉⠙⠿⢿⣿⢻⣿⠇
⠀⠀⠀⠙⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠃⠀⠀⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⡇
⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⢿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁⠀⠀⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠤⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠀⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡇
⠀⠀⠀⠀⠤⠀⠴⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠤⠀⠄⠀⠀⠀⣸⣿⣭⣿⣦⣤⣀⠀⠀⣀⠀⠀⡀⠀⠀⣀⣤⣶⣿⣼⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠋⠛⠛⠙⠛⠋⠛⠃⠛⠛⠙⠛⠉⠛⠋⠙⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠠⠀⠈⠠⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠐⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠏⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠈⠢⣀⡤⠓⠀⢠⠂⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢠⡤⠤
⠀⠀⢠⠂⠈⠉⠑⠿⠒⢤⠴⠃⠀⠀⡰⠋⠈⠑⠣⣀⠀⡀⢀⠀⠀⠋⠂⢀⠀⢀⣁⠀⠿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢⡄⠘⠁⠀⠀⠀⠀⠜⠀⠀⠀⠓⠀⠀⠀⠌⠀⡔⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡀⠀⢀⣀⣀⣀⡀⠀⠀⢠⢣⡶⢤⡶⢤⣄⠰⢊⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠁⢨⡇⠀⢸⠀⠀⠈⢿⡿⠋⠛⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠆⢀⠤⠶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣄⠀⠀⠀⢷⠄⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡷⢀⣀⣿⣳⣦⣤⣀⣾⣽⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢠⢘
⠤⠤⠤⠤⠤⠤⠤⠼⠃⠈⠛⢿⣙⣟⣍⣝⣾⣿⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⢸⠀⠀⠀⠀⢀
⠀⠀⠀⢠⠀⠀⡀⠀⢀⣀⠤⠚⠛⠛⠯⣽⣿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⠉⡏⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⡀⠀⢸⢇⣠⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠀⠀⠀⠀⠀⠀⢠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠤⢄⣀⠀⠀⠀⠀⡟⣸⠀⠀⠀⣠⠝⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠓⣄⠀⢈⡷⢨⡦⢠⠞⠁⠀⡠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⢷⣠⣧⡌⢿⡟⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⢃⠈⣻⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⢳⣤⣴⠃⠀⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⡿⠁⠀⠀⡀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣟⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠀⠀⠆⠀⠀⠈
⠀⠀⠀⡀⠀⠀⠐⡄⠀⠀⠀⠹⣟⡀⠀⠀⠀⠀⠀⣸⡇⣼⠀⠀⠀⡄⣰⠃
⠀⠀⠀⠙⣆⡀⠀⡇⠀⠀⠀⠀⠘⠷⠶⠶⠦⠤⠴⢿⣄⠎⠁⠀⠘⣷⠃⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠈⠛⠀⠛⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠰⠆⠀⠀⠃⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠦⢤⣄⣀⣤⣼⣯⢠⠀⠀⠀⢸⣥⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣾⣠⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢯⡋⠁⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣯⣿⡧⠉⣀⠀⡖⢜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣴⢀⣠⣿⣿⣿⣿⣷⣿⣭⣷⡽⠂⠑⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⡲⢼⡗⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠓

⠀⠀⠀⡇⡠⠒⠁⠀⠉⠐⠂⠤⣀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣇⠤⠄⣀⡀⠀⢸⠀⢀⣠⡼⠂
⠀⠀⠀⠙⠢⠤⣀⠀⠀⠀⠀⢀⠤⠋⠀⠀⠀⠀⠀⠈⠣⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠉⢹⠋⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠧⠤⠤⠤⠤⠤⠤⠤⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⢀⡧⠊
⢀⡠⠤⠠⣤⢀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠁⠀⠀⠀⠈⡆⠙⢆⠀⠀⠀⡞⠥⣀⠀⠀⠀⠀⠀⠈⣑⢦⠀⠀⠀⠀⡠⠒⠉⠈⠉⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⡴⠣⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⠀⠘⡆⠀⠀⡇⠀⠀⠑⠢⠴⠒⠊⠁⠀⢸⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠜⠀⠀⠱⡀⠀⠀⠀⠀⠀⡜
⣀⠀⠀⠀⠀⢸⣀⢴⠇⠀⠀⡇⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⠊⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢳
⡀⠉⠉⠉⢉⠏⣠⠎⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⢀⡀⠼⠀⠀⠀⢇⠉⠐⠒⠒⠒⠒⠈⣡⠃⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈
⠈⠐⠒⠒⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⠤⠒⠁⠀⠀⠀⠣⢄⣀⠀⠀⠀⠀⣀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠤⢀⡀⠀⠀⠀⠀⠀⠀⢀⡜⠁⡏⢆⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢩⠉⠑
⠀⠀⠈⢳⠀⠀⠀⠀⢀⠎⠀⠀⠧⠀⠣⡀⠀⠀⠀⠀⢸⠑⢄⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢀⡴⠊⡇⠑⠢⣀⠀⠀⠀⠀⠋⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠀⢠⠃⠀⠀⠀⢸⠀⠀⠑⡄⠀⠀⠀⢸⠀⠀⡏⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠋⠀⠀⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡨⢅⡀
⠒⠢⠄⣸⠀⠀⠰⣃⡀⠀⠀⠀⠀⡆⠀⢀⠜⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⠔⠊
⢀⡠⠖⠁⠀⠀⠀⠀⠈⠉⠐⠂⠤⠥⠚⠁⠀⠀⠀⠀⠈⠣⡀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣇⡀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⣀
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⣂⠔⠁⠀⠈⠒⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢠⠦⡐⠒⠀⠂⠐⠐⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⠤⣀⠔⠋⢸⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢢⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠱⡀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡶⡈⠉⠀⠀⠀⠀⠀⠀⣈⢶⠄⠀⠀⠀⠀⡔⠁
⡆⠀⠘⣄⠀⠀⠈⠲⣀⠀⠀⢸⠀⢀⠞⠀⠀⠀⠈⡇⠂⠤⠤⠤⠤⠤⠐⢊⠇⠀⠀⠀⡇⠈⠑⠤⡀⠤⠐⠊⠁⠀⠀⠀⠀⠀⢀⡞
⠃⢀⡠⠊⠀⠀⠀⠀⠀⠉⠢⢼⠔⠁⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⢀⠤⠋⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⠤⢀⡀
⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⡇⠀⠀⢀⡠⠤⠚⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡤⠚⡇⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠒⠗⠒⠉⠀⠀⠀⠀⣠⣦⡀
⠀⠀⠀⡖⠁⠀⠀⡇⠀⠀⠀⠈⠉⢲⠀⠀⠀⠀⠀⢸⠙⠢⡀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠸⡈⠢⡀
⠀⠀⠀⡇⠀⠀⢀⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠈⡇⠁⠀⠀⠁⠀⠀⠈⠀⢹⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⣇⠀⠈⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠿⠂⠀⠀⠀⠀⠀⢀⠀⣀⡉⠙⠀⠀⢀⣀⠠⢤⣦⠀⠀⠠⣤⣄⡙⠿⠃
⠀⠀⠀⠀⠀⠀⢀⣠⣟⠁⠀⠀⠀⠀⠀⠈⠑⢾⡿⠏⢠⣤⡀⠋⠀⠁⠀⢹⡀⠀⠀⠙⠿⠡⠒⣀⠀⢠⠒
⠀⠀⠀⢠⣤⣴⣿⢿⣿⣷⣷⠀⠀⠀⠀⣴⠄⠖⠃⠂⠘⣿⡇⠐⠒⢠⣾⣿⢙⣠⡀⠀⠈⢦⡹⣿⠀⠀⠠
⢠⣤⣤⡄⣇⣿⢷⣼⣿⣿⣿⣧⠀⠀⢠⡿⡆⠀⠀⢀⠘⣿⡇⠀⠀⢸⣿⣟⣺⡿⡅⠀⡆⢸⣦⣿⡆
⠸⣿⠿⠧⠿⢿⣿⣿⠿⠛⢻⣿⡃⣤⠸⡇⢡⠀⠀⠼⠤⡇⠆⠀⠀⢸⣿⡿⢿⡇⡇⠠⣿⢸⣿⣿⣿⠀⢀
⠀⠐⠒⣀⣿⣿⣭⡙⢿⣿⣿⣿⣿⣿⡃⡛⣯⢷⣤⢀⣴⣶⢄⠀⠀⠨⣿⣷⢰⡧⡇⣰⠟⣿⡿⡟⣽⣴⣿
⠀⠀⠀⡇⣿⣿⣿⣿⣿⢛⡇⣼⣿⠛⠺⠂⡝⡿⣷⢿⡿⢃⠈⢹⠉⣹⣹⣿⣿⡇⠨⠿⢿⣯⣰⡿⣿⠿⠿
⠀⠂⣀⡇⣿⣄⣿⣿⣿⣸⣿⡏⣿⠀⠀⠄⢀⡇⣿⢠⣇⣾⢰⢸⣴⣸⣹⣿⣹⣿⡶⠀⢺⣷⡁⣷⠲⠶⠖
⠀⠘⠛⡃⡿⠿⢿⣿⣿⡿⠿⠿⣿⠄⠰⠖⠋⠇⡟⠀⡏⠿⢪⡾⠋⣼⡿⣿⣿⣿⣇⠀⠀⣿⣵⣿⡀
⠀⠀⠁⠀⠁⠀⠈⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇⠀⠀⠀⠀⠀⠛⠉⠁⠰⡿⠀⢀⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠾⠿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠟
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣖⣲⣶⣶⣶⣶⡄⢀⣶⣶⣾⣯⣴⣶⣶⣶⣶⣶⣶⣶⣶⠆⡼⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⡿⢟⣵⣿⣿⣿⡿⢿⣏⣭⣭⣿⠟⢻⡿⠿⠛⠛⠀⠈⠁⠚⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡝⢻⣿⢿⣿⡇⠰⣿⣿⣿⣿⠋⠁⢀⣠⠄⢀⣀⠀⠀⠀⣠⣶⣶⣄⣠⣄⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢣⣴⢶⢶⣄⢷⣄⠙⠿⠃⠉⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠸⣿⣿⣿⠟⠀⣻⡷⠀⣸⣷⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⢀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢹⠏⠁⠀⠀⣛⣥⡶⢰⠂⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⢠⣶⣤⣾⣿⡿⠙⠘⢀⣼⣿⣿⣿⡟⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⠈⠻⣯⣿⣿⠿⠛⠛⠻⣶⠳⠶⢾⣿⣧⣀⠀⠀⠀⠀⠘⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡼⠋⣠⠀⣹⡟⠁⢂⣀⡀⠈⠉⠀⠀⠠⠉⠛⠿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⡀⢻⣾⠟⠁⢆⠀⠈⠁⠁⠀⠀⠀⠀⠁⠸⠗⡄⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣙⣡⡮⠨⠀⠈⢀⠀⠀⠀⠁⠀⠀⠀⠐⠀⣀⡐⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⣦⣤⠀⠀⠑⠂⢀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠠⢸⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣼⡇⠀⢿⣿⠓⢄⠀⠀⠊⠻⠷⣤⣤⡤⠠⣾⣷⣆⠀⠀⠙⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⡇⠀⢸⣿⡔⣆⢀⢀⣀⠀⠀⠀⠀⠀⣤⣼⣿⡘⠂⢠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣤⣠⣼⣿⣥⣬⣿⣿⡛⠱⣤⣤⣠⣶⣿⢩⣿⣧⠄⠁⣬⠸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡟⣿⣽⢻⣿⣿⣿⣿⣿⣥⣌⣉⣁⡿⠿⣿⣿⣿⢻⡄⣢⠽⠺⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠙⠛⠿⠒⠛⠁⣿⡟⣿⣟⡉⠀⢈⣿⣿⣿⣿⣿⢻⡇⠀⠂⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢸⣖⣲⡆⢀⣇⢀⡳⠄⠀⢀⣀⠈⠉⠉⠉⠋⠸⠇⡰⣤⣨⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠉⠉⢡⠊⠉⠁⠉⠀⠀⠉⠉⣻⠀⠀⠈⠀⠀⢈⡟⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣥⣤⣤⣄⡀⠀⠀⠀⢠⡀⢽⣷⣶⣶⣶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠉⠙⠛⠏⠉⠏⠱⠄

⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠏⠀⡐⡶⠀⠀⠀⠀⠀⠀⡀
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢟⡿⠄⠓
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡉⠙⠁⠀⠀⠑⠲⣶⣟⣤⣤⣄⣠⣶⣶
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣤⣤⣤⡀⠀⠈⣉⣿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣠⣶⡌⠁⠀⠈⠡⣶⠀⠀⠙⣿⣿⡇
⠀⠀⠀⢀⡀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⡇
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⣶⣤⣦⠀⠀⠀⠀⣼⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⣀⣤⣤⣄⡀⠀⠀⢸⣿⠇⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⡀⠉⠻⠿⠋⠁⢀⣴⣿⡿⢀⡟⡆
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣴⣿⣿⣿⠁⣸⠀⢁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣆⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢀
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⡀⡂⠀⠀⠀⠀⠀⡀⡀⠄⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠠⣴⠂⠀⠀⠀⠀⠀⠠
⠀⠀⡀⠔⠁⡃⠀⠀⠀⡴⠉⣉⠙⠱⠀⠀⠀⠀⠸⠀⡈⠉⡝⠀⠀⠀⠀⡐⢁⠁⡇⠀⠀⠀⠀⠈⢈⠽⠅⠅
⠀⠀⠦⢲⠀⠁⠀⠀⠀⠀⠉⢀⠃⢰⠁⠀⠀⠀⠀⣊⡀⠘⢄⠀⠀⠀⡜⠠⠨⠀⠇⡀⠀⠀⠀⢤⣈⠐⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠔⠁⠐⠃⢤⠀⠀⠀⢠⠤⠜⠀⢸⠀⠀⠘⠀⠀⢤⠀⡔⠃⠀⠀⠀⡤⠤⠃⠀⡇
⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠒⠒⠒⠒⠚⠀⠀⠀⠓⠂⠐⠒⠁⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠈⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡠⠤⡌⢄⠀⠀⠀⡤⠤⠠⠮⡡⡄⠀⠀⠀⣀⣠⣩⡀⠀⠀⠀⠀⡠⢠⠌⢄⠀⠀⠀⠀⢀⡤⢤⣬⡀
⠀⢀⠎⠀⠔⣀⠂⠀⠀⠀⠦⠤⡤⠀⣠⠇⠀⠀⢘⠀⣆⠄⢹⠀⠀⠀⡎⢠⠂⢲⢀⡇⠀⠀⡰⢁⠔⠒⢄⠑⡀
⠀⢸⠀⠠⠀⠄⠱⠀⠀⠀⢀⠞⢀⠞⠁⠀⠀⠀⠜⢃⠀⡐⠣⡀⠀⠀⠱⢄⣉⠁⠀⡆⠀⠀⡇⢸⠀⠀⢸⠀⡇
⠀⠘⣄⠐⠒⠃⡸⠀⠀⠀⡞⠀⡞⠀⠀⠀⠀⠘⡄⠘⠤⠂⢀⠇⠀⠀⢰⠒⠊⢀⠔⠁⠀⠀⠳⡀⠑⠒⠁⡠⠃
⠀⠀⠀⠉⠂⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠒⠊⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠊⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠹⡆⠀⠀⠈⠒⠸⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣇⠀⠀⠀⠹⢄⡀⠀⡠⠖⣡⠖⣄
⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠤⠤⠤⠀⠒⠒⠒⢂⡈⠉⠁⠉⠁⠀⠀⠀⠑⢶⠁⠀⠀⣀⣀⣹⠤⡄⠀⠐⣎⢴⣋⡴⠋⠀⠀⠘⡆⢀⡠⠤⠄⣀
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⢠⠟⣗⢒⣢⣉⠁⠀⠀⡇⠀⠀⣏⠇⠀⡀⠀⠀⠀⠀⣷⠊⠀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠀⠀⠉⠹⣄⣽⠧⠤⠤⠌⡤⣄⣗⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⣑
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡤⠜⠋⠤⢤⣤⡎⠀⠀⠀⠉⠉⠒⠲⠤⡟⠉⠉⢹⠃⡇⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠉⠙⠳⢤⣄⡀⠀⠀⠀⠈⠀⠀⡞⢘⡇⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠠⠀⠒⠒⠂⠈⠉⠉⠓⠲⢤⣤⠞⠁⢸⢁⠤⢄⡀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠈⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣾⠿⠿⠛⠛⠁⠀⢸⡇⠀⠀⠈⠑⡄⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⠶⠶⠒⠒⠒⠛⠻⣷⣤⣤⣤⣴⣶⣶⡿⠿⠿⠟⠛⠛⠛⠉⣉⣉⣉⣀⢠⡄⠈⠙⠲⣄⢀⣠⣤⣼⠀⠀⢄⡀⢀⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⢹⡥⣤⣤⡤⠴⠖⠒⠒⠋⠉⡭⠷⠟⠉⠀⢨⡇⠀⠀⠀⠈⠉⣠⠞⠁⠀⠀⠀⠉⢉⡜⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⢰⡎⠳⣄⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⢸⠻⠀⢀⣠⣴⣾⣷⣶⣖⣒⣒⣒⣒⠋⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠨⠀⢸⠁⠀⠈⠳⣄⠰⠁⠀⠀⠀⠀⠀⢀⣼⣾⠋⠁⠀⠉⢹⣿⠉⠉⠉⠉⠉⠉⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⠀⢸⠀⠀⠀⠀⠈⠙⢆⠀⠀⠀⠀⠀⠙⣿⣿⡗⠀⠀⢠⡻⢯⡄⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣾⠀⠀⠀⠀⠀⠀⠈⢣⡀⢀⣀⡤⢤⣿⡇⠙⣄⢠⣳⠃⠘⠃⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣇⣀⣉⣀⣠⣤⣀⣦⠤⠶⠾⠻⣟⠛⠉⢉⣿⠀⠀⢸⢣⠃⡤⠄⣀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠧⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⢦⡀⢸⣿⠀⣠⢣⡇⠀⣹⠀⡼⠇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⡷⢚⢫⣶⠿⡊⢡⡮⠈⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⢆⡾⠋⠀⡩⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⡜⠓⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠠⡄⠀⠀⠀⠰⢷⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⣆⢐⠀⠀⠐⠀⠀⢀⣦⠀⢈⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠣⠁⠀⡀⠈⠀⠀⠀⠀⢀⡲⠋⠀⠀⢀⠈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡰⠀⠀⠠⠀⠀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠃⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠞⠒⠂⠀⠀⠘⠀⠀⢰⠠⣧⠀⠀⡰⠦
⠀⠀⠀⢿⡾⢥⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠊⠀⠀⠀⠀⠀⠀⢬⣿⡆⠀⢸
⠀⠀⠀⠀⠌⣢⣽⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠰⠀⠀⠠⠐
⠀⠀⠀⠀⠈⢻⢻⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠀⠀⢂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈
⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠖⠀⡀⠀⠀⠀⠀⠀⠀⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚
⠀⠀⠀⠀⠀⠀⠸⠈⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⢛⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠋⠳⠄⠀⠀⠀⠀⠀⠀⣀⣠⡁⠀⠐⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠓⡛⠃⠀⠜⢢⠛⢄⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⡈⠃⠈⢷⠀⠀⠀⠀⠀⢀⢙⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⣄⠠⠰⠀⢀⡀⠀⡀⢃⠀⠀⠀⠀⡘⠋⠀⡀⠄⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠋⠐⠈⠀⠀⠙⠦⣙⣌⡆⠀⠀⠀⠙⠂⠈⠳⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⠀⠸⡀⠀⡀⠐⣷⡄⠈⠙⠻⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠇⠈⠊⠀⠙⠉⠀⠈⠀⢧⠀⠀⠀⠀⠀⠀⠀⠐⠄

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠛⠛⠃⠘⠛⠛⠛⠀⢀⡀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⢀⡀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡇⠀⠀⣺⡅⠀⠰⣿⠄⠀⢘⣧⠀⢰⡟⣧⠀⠈⣹⠃⠀⢸⣿⡅⠀⢻⣽⠀⢈⣿⣏⡇
⢀⣶⠶⢤⠀⡠⠶⣄⠀⠈⠁⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠌⠀⠀⡈⠁⠀⠐⠉⢁⠀⠀⠁⢐⠐⠂⢉⡒⢐⠨⠁⡀⢀⠡⢙⡀
⠈⠙⠶⠎⠉⠱⠶⠋⠸⠇⠺⠇⠻⠇⠿⠃⠿⠘⠿⠘⠿⠸⠗⠸⠟⠸⠇⠺⠇⠿⠇⠿⠘⠿⠸⠿⠸⠟⠸⠏⠹⠇⠸⠇⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⠤⠄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⢀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠒
⠀⠀⢐⠄⠂⠐⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠰⠻⠒⠚⠑⢀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠔⠁⠀⠒⠐⢀⠁
⠄⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣠⣤⣤⣀⡀⠋⠀⠀⡀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣠⣤⣤⣄⡀
⠠⠀⠀⠀⠀⠀⠰⠤⠶⠿⠾⠶⠦⠤⠤⠤⠝⠇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠶⠿⠾⠶⠦⠤⠤⠤⠘⠳
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠆⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠲⠦⠤⠆⠀⡆⠦⠤⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠔⡄⣤⣤⣤
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠠⠤⠀⠁⠀⠀⠤⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡛⠛⠛⠛⠛⠋⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣠⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⡤⠶⠾⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢀⡆⠀⠀⠀⠘⠁⡼⠀⠀⠀⠶⠒⠒⠀⠀⠀⠾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠰⢻⠁⠀⠀⠐⢻⡏⠉⠀⢀⣠⣞⡁⠀⠀⠘⠛⠉⠉⠀⣀⣀⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣄⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⡄⠀⠀⠀⣀⣀⡀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢋⣽⠀⠀⠀⡤⠴⠦⠀⠀⠀⢀⡼⠁⠀⠀⢀⣉⣠⡀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠋⢩⠇⠀⠀⠀⠀⠀⠀⣠⡶⠿⢤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠾⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⠀⢀⣇⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠒⠒⠒⠒⢒⣚⣛⣛⣛⣛⣋⣉⡉⠉⠙⠓⠒⠒⠒⠶⠶⠇⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⣴⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠤⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡷⠤⠤⠤⠤⠤⠤⠶⠶⠒⠚⣻⡆
⠀⠀⠀⣠⠶⠛⠉⢉⡭⠭⠭⠭⠭⠭⠭⠤⠤⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁
⠀⠀⠸⣧⣀⣀⣀⣈⣓⣒⣒⣒⣒⣒⣛⣛⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠒⠒⠒⠒⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠄⠀⠀⠀⠀⠁⠀⠀⠀⢠⠖⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡈⢤⣀⣀⢠⡄⡀⣀⡤⠃⡠⠃
⠀⠀⠀⣀⡐⠀⠀⠠⠠⠄⡨⠐⠂⠤⠄⠀⠀⠈⠲⣀⣤⣀⣅⣠⣀⡴⠊⠀⠀⠀⠤⠔⠒⢩⠀⠄⠄⠀⠀⢂⣄⡀
⠀⠀⠐⠩⠅⡀⢀⠀⡀⠈⠀⠀⠀⠉⠀⠒⠒⠠⢤⣴⣶⡽⠻⣷⣦⡤⠔⠐⠒⠀⠈⠀⠀⠀⠈⠀⡀⣀⠀⡈⠍⠁
⠀⠀⠀⠈⠸⡶⠅⠀⠠⠶⠂⣈⠄⠓⠀⠒⠈⢈⢑⣾⠇⠀⠂⠀⢷⣚⢉⠀⠰⠀⠐⠳⣅⡐⠲⠔⠀⠠⠵⡏
⠀⠀⠀⠀⠀⠀⠿⡶⢀⠀⠀⠁⠶⠚⢃⡀⠀⢝⠻⠛⠤⠀⠀⡰⢮⠟⡩⠆⢀⣐⠛⠲⣌⡁⠀⣀⢰⡞⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠖⠀⠀⢆⣀⣤⠆⢃⠘⢲⣾⡷⢰⢷⡶⠂⢸⠱⢦⣀⡀⠆⢸⡕⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡝⠁⠀⡜⠁⢿⡆⣨⢈⠀⡿⢎⠫⡄⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡡⠈⢤⡁⠌⣠⡈⢠⠀⢣⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠡⠁⠸⢺⠿⠾⢳⠆⠈⢀⠀⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠠⠀⠑⠚⠁⠄⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠼⠿⠄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠀⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄
⠀⠀⠀⠀⠀⠀⠰⠋⡀⠘⢶⠀⠀⢤⠶⠎⡐⢆⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⡠⠖⠒⠤⣀
⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠈⠀⠀⡆⠀⠀⠈⢻⡆⠀⠀⠀⠀⢀⠴⠲⢦⡀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⢡⠴⠒⠲⢦⡀⠀⣬⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⡇⠀⣴⠶⢄⣆⢸⣠⣶⣄⠀⢀⡇⠀⠀⠀⠀⡏⠀⠀⠀⢹⠀⠀⠠⠊⠁⠉⢢⠀⠀⢐⣴⡏⠀⠀⠀⠀⢳⣤⣾⠀⢠⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠱⣀⣹⣦⠞⠉⠉⢧⣼⠟⢀⡞⠀⢀⣠⠴⠚⠁⣠⣾⣦⠸⠦⡴⠁⠀⠀⠀⢀⡇⠀⠀⠈⢳⡀⠀⣼⡿⢶⠀⠀⣣⡬⠿⣤⠁⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢉⠉⢻⢀⡴⠋⠀⠀⠀⢣⣹⣇⡽⠃⠀⠀⠀⣾⠿⡇⡸⠀⠀⠀⠀⢠⠟⠢⢘⠿⠚⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢀⠤⠊⠀⠀⠙⣄⡀⠀⠀⣠⡏⠁⣼⠏⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠄⠿⠤⢿⡁⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘⠳⠤⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⣴⠒⠋⠁⠀⠀⠀⠀⠀⠘⡈⢉⢉⡜⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠀⠁⠀⠀⠀⠀⢳⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡴⠋⠉⠑⡄⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢸⡀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠒⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⠀⠀⠀⣀⣀⠽⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡠⠋
⢸⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠁⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⡀
⠸⣤⣤⡤⠶⠚⢉⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠉⠀⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣀⡤⠤⠿⣄⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠠⡾⠴⠒⠋⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⠦⣤⣤⡤⠊⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠛⠉⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠱⣄⠀⣀⣠⠤⠄⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣀⣠⡤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡄⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠆⠀⠀⠆

⠀⠀⢸⡠⠔⠉⠀⠈⠐⠂⠤⢀⡇⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠻⡀⠤⠀⣀⡀⢁⢀⠠⡼⠃
⠀⠀⠈⠁⠒⠤⢀⡀⠀⢀⠤⠊⠁⠀⢎⡹⠀⠈⠢⢄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡍⡠⠊
⠐⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡔⠂⠐⠄⠀⠀⠈⠢⡴⠊
⠔⠈⠉⠙⠖⠢⡀⠀⠀⠀⣠⠀⠒⠈⠁⠈⠐⠤⡀⠀⠀⠀⠀⠀⡀⠤⠤⠄⡀⠀⠀⢆⡀⢀⠌⠀⢀⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠘⠀⠘⡀⠀⠀⠀⠑⠢⢄⣀⡤⠤⠒⠈⡇⠀⠀⡠⠊⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⢀⠊⠈⢆⠀⠘⠤⠃⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠃⠀⠀⠀⢣⠀⠀⠀⠀⢸
⠉⠒⠒⠒⡺⢉⠔⠀⠀⠀⠣⢄⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⢏⠒⠂⠀⠀⠀⠒⢊⠇⠀⠀⡰⠁⠀⠀⠀⠀⠀⠡⡀⠀⠀⠈
⠉⠒⠒⠒⠓⠁⠀⠀⠀⠀⠀⠀⠈⠒⠒⠂⠉⠀⠀⠀⢀⡀⠀⠑⠤⣀⣀⡠⠔⠁⠀⠀⠀⠤⣀⠀⣀⠀⡀⢀⡠⠃
⠀⠀⠀⠀⢠⠂⠐⡄⠀⢀⣦⡀⠀⠀⡤⠤⢤⠀⠀⠀⠣⠜⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠒⠤⢀⡀⠈⠒⠒⠁⠠⠊⠸⠑⠄⠀⠘⢤⠃⠀⣤⠀⠀⠀⠀⠠⢄⠀⠀⡇⢀⠇⠀⠀⣠⣄⠀⠀⠸⣉⠆⠀⡠⠐⠁⡇⠈
⠀⠀⠀⠅⠀⠀⠀⡰⠁⠀⠀⡂⠈⢂⠀⠀⠀⠀⡇⠑⢤⠀⠀⠀⠀⠱⡄⠀⠀⠀⡠⠊⠀⠀⠈⠒⢄⠀⠀⢨⠀⠀⠀⡇
⢀⡀⠀⠂⠀⢀⡼⠀⠀⠀⠀⢃⠀⠀⢡⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⡀⠔⠁⠂
⠀⢈⠵⠃⠀⠈⠁⠒⠠⠄⢀⣘⡀⠔⠁⠀⠀⠀⠣⡀⢐⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠘⠮⢄⡀
⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠤⡀⠀⠈⠲⠤⠤⠤⠤⠤⠇⠀⠀⠀⠆⡠⠊⠁⠢⢄⠀⠇⠀⠀⠀⠀⠀⠉⠒
⠀⡐⠀⠂⠀⠀⣤⡄⠤⠤⠤⠤⢤⡄⠀⢇⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠛⠐⠀⠀⠂⠂⠙⠁⠀⡰⠓⡀
⠀⠈⠀⠁⠀⠀⡇⠈⠒⠄⡠⠊⠀⠀⠀⠀⠀⠀⡠⠀⠒⠒⠠⢄⠀⠀⣐⣁⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣀⣘⡄⠀⢀
⠢⡀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠢⢄⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠑⡄⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣨⠆⠀⠀⡜⠡⡀⠀⠀⠀⢀⡀⠤⠚⠀⠀⠀⠀⡰⠁
⡇⠂⢈⠆⠀⠀⠉⠢⢄⡀⡇⡠⠊⠀⠀⠀⠈⢄⠉⠁⠀⠀⠉⢁⠜⠀⠀⠀⡇⠀⠈⠑⡖⠈⠁⠀⠀⠀⠀⠀⠀⡜
⣴⠖⠁⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣀⣀⣀⠀⠀⠑⠂⠠⠀⠐⠉⢀⠐⡄⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠈⠁⠒
⠀⠀⠀⠀⠀⡠⠤⠄⣀⠀⠀⠀⠀⠱⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠈⠐⢤⡄⠤⠒⠈⠁⠀⢀⣀
⢩⠇⢠⠔⠁⠀⡃⠀⠀⠈⠐⠂⡄⠀⠁⠀⠀⠘⠫⡉⠉⠉⠉⠉⠉⠉⠓⢄⠀⠀⠀⢀⣀⠀⠀⠀⢀⠔⠋⢩⠰⣄⠀⠀⠘
⠋⠀⢸⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠂⠀⠀⠀⠙⡆⠀⡅⠀⡁⢀⠔⠁⠀⠀⠈⡄⠈⠳⡄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢔⡀⠀⠀⠀⠀⠀⠄⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠁⢙⠿⣁⢀⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢸⣷⡀⢄⣇⣿⣛⠶⠷⠼⠟⣛⠧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢣⡘⠄⠁⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⣾⣿⣷⢀⡀⣹⡅⣐⣦⡞⠋⠁⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢤⣭⡤⠀⠀⠀⠀⠀⠀⠀⠱⠶⣤⡐
⠀⠀⠀⠀⣿⣿⣿⢨⣿⣵⡆⢸⣯⣉⣵⠾⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡵⠛
⠀⠀⠀⠐⣿⡛⢹⣧⣿⣿⢿⣂⢛⣿⡙⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢄⠀⠀⠀⠀⠀⠄⠀⣟⠀⠀⠄⠈⠵⠁
⠀⠀⠀⠀⠘⣧⠀⠉⢩⣯⣦⠛⢽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠈⢂⠀⠀⠀⠀⠇⡙⢳⡌⠀⢖⢀
⠀⠀⠀⠀⣴⣿⣳⣦⣀⣙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣄⠀⠀⢀⣴⣾⣇⡨⡿⢀⣎⣺
⠀⠀⢠⣾⣿⣯⠙⠦⢝⣹⣏⢰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠒⠻⣿⣿⣿⡄⣡⠾⡣⠴⠊
⠀⠀⢸⣿⣿⣿⣷⡄⢸⣿⡿⢸⣿⣿⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢦⠶⠆⠂⠀⠈⠁⠀⣀⣩⠔⠀⠀⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣄⣿⣿⣌⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣴⣟⡛⠛⠿⣿⡿⣀⡂
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣠⣐⢀⡙⢿⣿⣿⡶⠦⠬⠄⠀⡀⠀⢳
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠀⠀⠐⢖⣂⠤⠀⠉⠥⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠐⠀⠀⠀⠀⠀⠀⠐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠄⠤⠤⢬⣤⣤⣔⣒⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣉⣉⣠⣴⡄⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠝⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠔⣒⠾⣻⠟⠩⠟⠫⠹⠅⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⣠⣈⠁⢈⣦⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠄⠠⠈⢀⣐⣠⠍⣠⢸⠀⠀⠀⠀⠀⢈⣿⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣿⠏⠀⠀⠂⣀⠵⠀⠀⠀⠀⠈⠉⠈⠉⠲⣤⡀⠀⠀⠀⠀⠀⠀⢸⣇⡐⣊⢔⡭⠖⠁⠂⡤⢚⠀⠀⠀⠀⠀⠘⣿⢧
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢘⠃⢠⣀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢦⡀⠀⠀⠀⠀⠸⣯⠄⢠⠂⠁⠉⠠⠈⠀⢺⠀⠀⠀⠀⠀⠀⢹⡧⣧
⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⣘⠡⡜⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⢣⠀⠀⠀⠀⢠⠦⠈⠀⡀⠀⠀⠀⠐⠂⢸⡀⠀⠀⠀⠀⠀⠰⣿⡝⡄
⠀⠀⠀⠒⠄⡀⠀⠀⠀⢠⢃⣸⣷⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣻⣧⠘⡅⠀⠀⠀⢀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⣿⣧⡸⡄
⠀⠀⠀⠀⠀⠀⠐⠀⡀⢸⣷⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⣿⣯⡍⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠄⠐⠘⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⡦⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⠠⠁⠉⠁⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠤⢠⣀⣀⡤⠦⣾⣿⣿⣿⣿⡏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠤⠄⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣖⣛⢯⣿⣷⠿⣿⣿⣉⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⣼⡧⠶⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠟⠁⠀⠀⠀⠀⠀⠱⡌⠛⣷⣤⣀
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡿⢰⡀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠈⠏⠟⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠨⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡇⠀⠃⡀⠀⠀⠀⠀⠀⠀⡘⠀⠔⠋⠉⠙⠻⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼
⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠉⠒⠤⠀⠀⢄⠀⠠⠀⣰⠑⠃⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡏⡇
⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣠⡶⠖⠛⠶⠷⢶⠶⠗⠚⠒⠲⢦⡀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⡇
⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⡇⢸⠀⠒⠛⠷⠆⠀⠀⠀⠀⠀⠶⠟⠓⠀⢧⢸⠀⣾⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣰⡷⠚⣫⣿⣿⡲⣄⠀⠀⢠⠶⣺⣿⣿⣛⢿⣦⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢹⡃⠈⢳⣾⣿⡿⢸⡶⠲⣟⠀⢷⣾⣿⡟⠘⣿⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣸⢧⡀⠠⠍⠉⢀⡼⠁⠀⢻⣄⠀⠉⠭⢀⣸⣇⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⡉⠛⠛⠛⠉⠰⣦⣤⡶⠉⠛⠓⠛⢉⣴⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⠟⢿⣶⣤⣄⣀⠀⠈⠋⠀⢀⣀⣤⣴⡟⠟⢧⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣇⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣦⣤⣮⣽⣿⣿⠛⠋⠁⠈⠙⠛⣿⣿⣧⣵⣤⣤⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⠀⠀⢣⡤⣤⠴⠒⠲⣄⣀⣀⠀⠀⠀⢀⣀⣀⡈⠙⣋⡽⢁⣤⠀⠺⣶⡂⢀⠘⣏⠛⢋⣉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠤⡤⣬⣄⣀⣀
⠀⡴⠋⠉⠱⣏⠀⠀⠜⠁⠀⣼⠋⠉⠉⣟⣁⣀⢹⣰⠏⢠⡞⡟⠀⠈⠛⠁⢸⣧⠘⢦⡸⣹⠋⠀⠈⡆⠀⡀⡀⢠⠞⠉⠉⢳⠀⠀⣸⡏⠀⠀⠙
⠀⠼⢀⠀⣰⡇⠑⢤⡜⢌⠀⠼⡄⠀⢠⠏⠉⣼⠛⠋⣠⡟⣴⠇⠀⠀⠀⠀⠠⣿⢣⣀⠛⠼⣦⠀⢠⡧⠐⠄⠀⠘⡆⠀⠀⡴⡠⠺⠅⢧⣠⢀⠼
⠑⠒⠂⠀⢠⠀⠀⠸⠍⡀⠃⢈⡞⢱⢀⠓⠂⠙⠳⢶⢧⣀⣯⡀⠀⢀⢂⠀⠀⣸⠀⠙⣦⣴⠟⠈⡉⠋⠙⠀⠀⠀⠺⡙⠘⡅⠈⠀⠀⠠⠙⠁⡖
⠉⠀⢠⣤⡌⠀⠀⠀⠀⠙⠈⠈⠃⠐⠀⢲⣀⠀⠀⠀⠈⠉⢻⣄⡁⠁⡆⠁⣁⡿⠈⠉⠉⠉⠈⢑⠠⠀⠀⠀⠤⠄⠠⠁⠀⠁⠀⡄⠀⢠⣀⣬⡇
⠀⠀⡾⠉⠃⠀⠀⠁⠀⠨⢶⡀⣴⢲⠈⢷⠃⠸⠓⡇⠀⠀⠀⣿⣿⡟⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠲⢰⢲⠤⠃⠀⠁⠀⠘⠋⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⣿⡇⠀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠾⣿⣷⣶⣿⡿⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠒⠒⠚⠛⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠶⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠋⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡠⠤⠤⢤⡀⠀⣖⣛⣛⡒⣆⠀⠀⠀⠈⣇
⠀⠀⠀⢠⠖⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⡇⠀⢰⢻⠋⠉⢳⡀⡇⠀⠈⠀⠀⢹⠈⡇⠀⠀⠀⢹⡄
⠀⠀⢀⡏⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠾⡇⡇⠀⠘⠛⠁⠀⢠⢧⡇⠀⠰⣶⣖⠋⣴⠃⠀⠀⠀⠀⢷
⠀⠀⣼⢀⠞⡄⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇⣧⠀⠀⢀⣠⡴⢫⡾⢤⠀⢠⣄⡤⠇⣸⠀⠀⠀⠀⠀⠘⡆
⠀⠰⡇⡎⠀⠘⠌⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠚⠃⠀⠘⠷⠶⠒⠒⠚⠀⠘⠓⠒⠚⠁⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⢷⠧⠔⠉⡤⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠘⡆⠀⡜⠘⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⢷⠘⠃⠀⠴⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠘⡆⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⢹⡀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀⣸⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢣⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠺⡿⡇⡏⡟⢰⣲⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠘⡆⠀⠀⠸⡅⢧⣠⠴⡄⠀⡀⠀⠀⠀⠀⠉⠉⢰⢣⠇⠹⠽⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢳⠀⢰⡖⢁⡀⢴⠒⠃⠀⡏⠙⠒⠒⢶⡀⠀⠙⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⢇⠀⠛⠉⢣⣘⡦⠀⠀⠉⠛⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠒⠒⠚⠋⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠘⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠔⠒⠒⠋⠉⠁⠀⠀⠀⠀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⢱⣀⡠⠤⠶⠒⠒⠒⠚⠉⠉⠉⠉⠁⠀⠀⣀⣀⠤⠤⠴⠄⠒⠒⠒⠀⠉⢉⣁⣀⠤⠤⠴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⡞⠁⣀⣀⡀⠤⠤⠤⠄⠒⠒⠒⠚⣉⣉⣁⡠⠤⠤⠐⠒⣒⣂⣚⣩⡭⠭⠥⠖⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⠀⠀⠁⠀⠀⠀⠒⠒⠂⢉⣉⣉⣉⣭⠤⠤⠶⠒⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠶⠶⠒⠒⠒⠚⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣾⡿⠁⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣣⣶⣿⠟⠁⠀⠀⡀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⠛⠛⠛⠛⠉⠁⣀⡀⠀⠀⢹⡄⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠛⠻⢿⡷⠀⠀⢾⡿⠛⠛⠉⠙⡿⢸⣿⡄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣿⣧⡀⠀⠀⠻⠛⠛⢿⣷⡀⠀⠀⠀⠀⣼⣿⣷⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⠻⠿⠶⠀⠀⡀⢀⠈⠛⠿⠶⠄⠀⠀⣿⣿⣿⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢀⣁⣀⡀⠀⠀⠀⠀⠀⣸⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠰⣿⣭⣭⣿⠄⠀⠀⠀⢢⣿⣿⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠉⠉⠀⠀⠀⢀⣞⣾⣿⡿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⣀⠀⠀⠀⠀⣀⣴⢋⣼⣿⣿⠁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣄⠀⠘⣿⣷⣶⣿⡿⠁⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢸⣿⣿⣿⠁⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣿⣿⡇⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⠁⠀⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⡇⡀⠀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣇⣷⠀⠸⣇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⣌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠠⢤⠂⠀⠀⠀⠀⠀⢄
⠀⠀⠠⠊⠁⡇⠀⠀⠠⡊⡈⡉⠉⡄⠀⠀⠀⠫⢤⠍⢩⠃⠀⠀⠀⠠⠋⡄⠃⠀⠀⠀⠀⡇⠩⠽⠨
⠀⠀⠒⠃⠀⡇⠀⠀⠀⠈⢀⠆⢠⠃⠀⠀⠀⠐⠅⡀⠑⠄⠀⠀⡰⠁⠦⠇⠠⡄⠀⠀⠀⠧⢌⠁⢢
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠔⠁⠐⠓⢲⠀⠀⠀⡴⠐⠊⢀⡔⠀⠀⠓⠒⠒⠂⠐⠁⠀⠀⢀⠂⠐⠀⡰
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠉⠁⠉



⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡰⠊⡉⠩⠃⠀⠀⢈⠉⠉⠉⠈⡇⠀⠀⠠⢚⠛⠛⡄⠀⠀⢀⠎⠁⢉⠐⡄⠀⠀⢀⠖⡉⢝⠓⡄
⠀⢸⠀⢈⢉⠐⡄⠀⠀⠀⢠⠋⢠⠎⠁⠀⠀⡰⠈⠀⠠⡃⠀⠀⠸⡀⠃⠐⠀⠹⠀⠀⡇⠘⠀⠀⠃⢸
⠀⢸⡀⠣⡠⠀⡀⠀⠀⢠⠃⢰⠃⠀⠀⠀⢰⠁⢨⣉⠆⢸⠀⠀⠀⡈⠽⠃⢠⠃⠀⠀⢇⠘⠦⠴⠃⡸
⠀⠀⠑⠂⠔⠚⠀⠀⠀⠘⠒⠊⠀⠀⠀⠀⠀⠑⠢⠤⠐⠁⠀⠀⠀⠂⠒⠊⠁⠀⠀⠀⠈⠑⠢⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⣿⡷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣤⢀⠀⠀⣄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⠢⠈⢸⣼⠒⣣⡇⠀⠐⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠾⠦⣀⡀⠄⠪⠐⢄⣡⠉⠑⠺⡖⢘⣮⡃⢸⠗⠊⠁⣌⡀⠓⠱⠀⢀⣀⠴⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢃⠠⠄⠀⠙⢋⣒⢶⠶⣾⣿⠉⣿⣵⠶⠶⣒⠛⠋⠀⠠⠤⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠔⢪⡿⣿⡤⡟⢿⡕⠂⠀⠀⠀⠀⡀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠂⡴⠯⠀⠀⠀⠸⠑⣸⣿⣇⠊⠇⠀⠀⠀⠽⢦⠐⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣀⠀⠀⠐⡆⠀⠀⠀⢫⠈⠗⠀⠀⠀⢰⠈⠀⠈⣀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡓⣿⠙⢵⡠⠄⠀⠀⠈⠓⣄⢄⣿⡠⣢⠚⠁⠀⠀⠄⢌⡮⠁⣿⣚⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠞⢺⡍⣌⠈⡤⠀⡰⠀⠄⣀⢀⠆⣛⠰⡀⣀⠠⠀⡆⡀⢠⠁⡡⢡⡟⠻
⠀⠀⠀⠀⠀⠀⠀⢠⡀⣄⡻⣏⢂⠚⠂⣱⣦⣴⣸⡼⠓⠙⠚⢧⢈⣆⣷⣟⠸⠓⡐⣹⢟⣤⢂⡄
⠀⠀⠀⠀⠀⢄⠀⠈⠈⠉⠙⠂⠛⠳⠦⣘⢮⢷⡠⢀⣠⠤⣄⡀⢄⡾⡵⣃⡴⠞⠟⠐⠋⠍⠁⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠁⠀⠀⠈⣻⣤⣴⣷⣿⣶⣿⣿⣦⣤⡟⠁⠀⠀⠈⠁⠁⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢛⣐⣂⣀⣠⣶⡿⠛⠛⢿⣿⣀⣿⣿⠛⠛⢿⣶⣂⣀⣐⣂⣛⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠉⠉⡩⠋⠷⢦⣀⣐⣹⣿⣏⣃⣠⣴⠞⡙⠍⠉⠉⠉⠉⠙⣷⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣤⣀⠴⠃⠄⣡⡞⢛⢿⣳⣾⣞⣿⡛⠳⣌⠂⠘⠦⡀⣤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⢀⠀⠀⠂⡾⣟⠀⠟⠀⡹⠿⢏⠀⢛⠀⡛⣷⠐⡀⠀⡀⠈⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢸⡇⢢⠄⠀⠀⠈⣀⠀⠀⠀⠠⡄⢸⡇⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢶⣬⣳⡄⠆⠟⠐⢠⣞⣭⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠈⠙⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⠛⠒⢶⠖⠀⣠⡶⠇⠀⣀⠀⠀⠀⢀⡀⣠⡟⠑⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⢟⠀⠀⡰⠀⠀⠀⢀⢎⠝⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠾⡊
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⣸⠀⠀⠀⡀⣖⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠳⠀⠀⢈⡄⠀⠐
⠀⠀⠀⠐⢄⢀⣀⠀⣀⣄⠉⠀⠀⠈⠚⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⡸⠐⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠈⠉⠉⠉⠀⠀⠠⡄⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠦⣤⣧⡇⠀⠀⠀⢀⣀⡀⠀⠹⡆⠀⠀⠙⠒⠲⠴⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠤⠆⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣦⠋⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⠀⠀⣯⠀⠀⠀⠚⠚⠙⠆⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠇⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⡤⢀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⣠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠋⠁⠀⢤⡀
⠀⠀⠀⠀⠀⠠⠤⣤⣤⡄⠀⠀⣤⣼⡇⠀⠀⠘⠀⠀⠀⠀⣀⣤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⡀⠀⠰⢤⣄⠶⠦⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠉⠛⠛⠛⡻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠉⠁⠀⠀⢄⠀⠀⠀⠀⠀⡶⠶⠤⡀⠈⣆⠀⠀⠀⠀⠀⠀⣐⡆⡀⠀⠐⣆⠀⠀⠀⠈⢦⣂⣀⣀⣠⣼⠋
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢰⠟⠒⠀⠀⡆⠀⠀⡀⠀⠑⠤⠀⠀⢀⡟⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⡠⡏⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⢀⣀⠀⠀⡇⠀⠀⠘⠲⠤⣤⣤⠦⠔⠁⠀⠀⠀⠀⠀⠀⢀⢀⡀⠈⠫⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠉⠉⠉⠀⢠⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢸⡕⠉⠉⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⣁⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠉⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠴⠔⠎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠂⠛⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣦⣤⣿⡀⠀⡴⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢀⣴⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⡖⠃⡮⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣸⣿⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢒⣧⠀⠀⠠⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⢿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠧⠦⠔⠂⠉⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⢸⣿⣿⡿⠿⠟⠛⠛⠉⠉⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⢀⠀⠀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⣀⣠⣤⣴⣶⣿⣿⣿⡿⢃⣷⢠⡠⠀⣿⠀⠃⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⣰⠀⠀⡀⢰⡇⠀⠘⠂⢠⡀⠀⣐⣒⣔⠒⢿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⡛⠀⣸⣿⡐⠠⢰⣿⡄⣈⡩⠉⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣆⣿⡤⠎⠉⢻⠥⡤⢀⠀⢞⠑⠚⢉⢡⠘⠛⣓⠿⠛⠫⠁⠡⠈⡀⠪⠩⣖⢄⢀⠈⠉⡃⠶⠸⠿⡧⠘⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠰⠋⢉⠀⣠⢶⡁⢀⠈⠁⠀⠈⠐⠀⠾⠖⠒⠉⠀⢀⡠⠄⠳⠀⢰⣇⠀⢄⠀⠉⠑⠙⢸⡧⣄⠀⠐⠈⡶⠚⠁
⠀⠀⠀⠀⠀⢀⠀⠘⠫⣀⠔⠂⠀⠀⠠⠀⡀⣿⢠⡆⠈⠒⠂⠀⠀⠀⢠⠀⠂⠁⢀⠀⢠⡇⣸⣿⠀⠀⠁⠀⠀⠒⠠⠒⠉⡆⠀⠀⡇
⠀⠀⢄⠀⡠⠀⠀⢀⠀⠀⠀⠀⠁⠈⣀⣀⣀⡻⠊⠁⢀⣄⠄⠀⠀⠀⣼⡆⠤⣎⣀⡠⠼⠷⠿⠛⠓⠉⢋⠄⠂⠁⠀⠀⠁⡇⠀⠀⠃
⠀⠀⠋⠠⠦⠁⡂⠠⠀⠈⠁⠂⠀⠻⠿⠿⠽⠿⠂⠀⠀⠀⠀⠈⠙⠀⠿⠷⠈⣀⡀⠪⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠶⠄⠘⠀⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁⠃⠀⠠⠦⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠋⠭⢜⡀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣷⣿⣿⣶⣶⣶⣤⣲⣞⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⡾⡻⣿⡁⢿⣿⣿⣿⠿⡌⠙⠉⣠⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣥⣤⣠⣏⣳⣾⣇⠈⠁⠱⢷⣶⠃⠀⢠⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣾⣿⣿⡇⠻⢿⣿⣿⣷⣵⣤⣀⣀⠀⣴⣿⣿⣿⣿⣿⡿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡅⠀⠻⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠢⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⡀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠶⣄⣠⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⣿⣿⣿⣿⠛⡿⣿⣿⣷⡄⠀⠀⠈⠛⠻⢿⠿⠿⣿⠘⣿⣷⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⠘⠀⠀⢸⣿⣿⣿⣶⣷⣶⣿⡆⣠⣶⣾⣿⠰⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠉⠁⠀⠛⠈⠉⣴⡿⢻⣿⣿⣿⣿⣿⣿⣿⣳⣷⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣷⠟⠷⣿⣿⣯⣽⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠉⠀⢸⡟⡿⠿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢀⠀⢘⡀⢠⣴⠟⠋⠀⠑⠆⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⣿⣷⣄⣈⣛⣿⣏⠆⠀⠀⠀⠀⠀⠀⠀⠒⠿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠐⠂⡀⠰⠀⠀⠀⠀⢰⠁⣴⣿⣿⣯⡙⠿⠿⣻⣤⣙⣲⣦⣄⡀⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠙⠂⠀⠀⢀⠀⠀⠀⠀⠀⡈⠀⢿⣿⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⡿⢛⣦⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⢷⢽⣿⢇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠷⠋⠀⠀⠹⠠⠄⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢘⢉⡉⣻⣟⠉⠁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣼⢿⡷⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⣟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣐⠛⠿⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⠂⠀⢀⣈⠁⠀⠀⠀⠂⠀⠀⠀⠄⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡷⠶⠿⠟⠁⠳⠶⢶⣀⡤⠀⠀⠛⢷⣦⣭⣼⣷⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠴⢧⠠⣤⠀⢸⣷⣽⣿⣿⣿⣷⡀⢃⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠻⠿⠷⠢⠀⠀⠠⠿⠶⠼⠿⠿⢿⣿⣿⣿⣷⡀⢷⡄⠀⠀⠀⠀⠀⠈⣥⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣦⣄⣀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣉⡀⠀⠀⠀⢠⣼⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡍⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡄⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡀
⠇⠀⠀⠀⠃⠀⡠⣀⡂⢀⣤⣀⣄⣤⣤⣴⣀⣀⠀⠀⠀⣀⣠⣀⣀⣤⣤⢤⡤⠀⢠⣤⠤⡤⠀⣀⣼⣶⣴⣶⣶⣖⠀⢀⠈⡄
⠀⠀⠀⠀⠀⠘⠃⠀⡀⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⢿⣷⢽⡿⣿⢿⣿⣿⣷⣶⣿⣽⣿⣿⣿⢏⠹⠙⠛⢹⠗⠀⠈⠃⢄⡀
⠀⠀⠀⠀⠋⠀⠀⠀⠁⢸⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⠿⠹⢿⡿⠿⠟⠻⢿⢿⣿⡼⠛⡛⣋⣀⡀⠀⠢⠁⠐⠢
⠀⠀⢀⣠⡏⠀⠀⠀⠀⢸⡿⢾⣿⠿⠿⣿⣿⠟⢿⠛⢻⡋⢻⣿⣻⢻⣿⣿⠿⠿⢛⣺⢷⢾⢚⢿⣻⠟⢋⣭⣭⠉⣁⡙⣦⡄
⠀⢀⣾⣿⣷⡆⠀⠀⢀⣨⡵⠥⠍⠁⠄⠙⠫⠤⠀⠿⡻⢡⠤⠏⠩⠮⡟⠛⠁⠀⣀⠍⠤⢠⠀⠘⡓⢂⠀⠭⠩⠀⠽⢆⣊⠑⣵⡀
⢶⣿⣿⣿⣿⠇⠀⠀⠡⠨⠂⠈⣶⣤⣄⠀⢠⣰⠊⡉⣰⠀⣀⣀⠀⢠⠍⣹⣀⠀⣇⡀⠘⣃⣀⢠⣉⡈⠢⣄⣀⡀⠀⢨⣤⠠⠘⠟⢤⡀
⡿⢿⣾⣿⣿⠔⢁⠒⠀⡀⠀⣀⠹⢛⠯⠿⠿⣻⣦⣅⠈⢺⣿⣿⣿⡖⢧⣈⣨⠀⢠⢄⣺⣿⣿⡓⢀⣠⢦⣒⢊⣴⣶⣾⠐⠒⠒⠂⠀⠙⠆
⠋⡴⠘⣽⣦⣤⡄⠈⢈⡔⠁⠀⠀⠀⠀⡄⢨⢀⠀⠀⠎⠉⠉⠉⠀⠋⠀⠀⠀⡀⠀⡦⠈⠋⠒⠒⠐⠀⠀⡍⣈⠨⠄⠤⠒⠘⠋⣀⣀⣀⡁⠑
⠷⣠⠎⣻⢻⠏⠳⠀⢾⣴⣶⣦⣤⣀⣶⣗⡀⡀⠀⢤⠄⣠⣀⣀⡀⡀⠁⠁⠁⠃⠀⣇⡀⠀⣀⣀⠀⣈⠉⠁⢊⠢⣀⠀⠀⣀⣀⡉⡀⢀⡀
⠒⠁⠈⠹⠉⢃⡄⠤⠤⠿⢿⣧⣤⢿⠈⠎⠀⠁⠀⡀⢮⢹⣿⣿⡇⠠⠀⠤⠄⠀⠀⠈⢠⣾⣿⣿⣗⠈⠋⠉⣾⣐⠺⡋⠘⠿⣿⣛⢀⣀⢀
⠄⠁⠂⠐⡠⠀⠀⠀⠀⠀⠀⠀⡌⠈⠈⠁⠁⠀⠈⠁⠀⠀⠁⠉⠀⠈⠀⣀⡀⠀⠀⢰⠐⠁⠒⠃⠀⠂⠛⠉⠱⠈⠙⠃⡉⠋⠙⠐⠂
⠒⠠⢀⠜⣦⠀⠀⠀⠀⠀⠠⣼⠇⡐⠂⠂⠠⣤⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⣠⣀⡤⠄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠃
⠘⠀⠀⠀⠈⣃⠂⡶⢀⣒⡞⡄⡤⠡⠄⠀⠀⠀⠤⢢⠤⠤⠤⠀⠤⣉⡈⠀⠉⠀⠀⠈⣉⣋⠀⠀⣀⠀⢀⠈⠀⠤⠀⠀⢘⠠⣹⣩⡁⠀⠀⢀⡌
⠘⠀⠀⠐⠀⢹⠓⡐⠲⠿⡿⠻⠇⠀⠀⠀⠀⠀⢰⢯⡤⣿⣿⣴⣶⡦⠷⠤⠀⠀⠀⠀⠀⣿⡴⣶⣶⢠⡶⠄⠀⠀⠀⠀⡜⠓⣼⣴⣶⣶⣦⣼⠁
⠈⡀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠁⠀⠀⠀⠘⠀⠐⠀⠙⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠛⠇⠝⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡅⠀⠀⠘⠀⡏
⠠⠤⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⡄⡄⠀⣀⠠⠀⠄⠀⠠⠀⠀⡄⠀⠀⠄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⡄⠀⠠