Evil smile Emojis & Symbols

⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⢠⡀⠀⢀⣄⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠘⠋⢠⠈⠋⠀⠀⠀⢘⠁⠀⠈⡃⠀⠀⣌⡉⠠⠉⣩⠀⠀⠀⢈⠋⠠⠘⣃⠀⠀⠀⠀⠈⠁⠄⠉⠁⠀⠀⠀⣦⠉⡀⠈⠉⣴⠀⠀⡘⠃⠀⠘⢋
⠀⠀⠀⠈⠒⠶⠖⠊⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠈⠛⠓⠛⠉⠀⠀⠀⠀⠘⠋⠉⠛⠁⠀⠀⠀⠉⠻⠟⠾⠛⠁⠀⠀⠈⠳⠶⠟⠋
⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⣤⠀⠀⠀⢀
⠀⠀⠀⠈⠙⠠⠐⠋⠀⠀⠀⠙⠃⡄⠙⠁⠀⠀⢘⠟⠀⠹⡃⠀⠀⣀⢘⠓⠀⠘⠛⢀⠀⠀⣆⠙⠓⠀⠛⣃⣴⠀⠀⣀⣛⣀⣈⣛⣀⠀⠀⡉⠃⣄⠐⠛⠃
⠀⠀⠀⠘⠳⠶⠶⠞⠁⠀⠀⠑⠶⠶⠖⠋⠀⠀⠈⠻⠿⠟⠁⠀⠀⠘⠻⠾⠶⠶⠿⠋⠀⠀⠘⠛⠿⠿⠾⠟⠁⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⠛⠶⠶⠶⠚

⠀⠀⠀⠰⠦⠀⠠⠾⠀⠀⠀⢾⡇⡀⢾⡇⠀⠀⠺⠧⢀⠰⠟⠀⠀⠀⠐⠷⠀⠰⠇⠀⠀⠀⠠⣾⡷⠀⢿⣷⠀⠀⢀⡘⠓⢀⠀⠚⢂⠀⠀⠻⡖⠀⠰⠾⠁
⠀⠀⠀⠀⠤⣬⣴⣦⠀⠀⠀⠠⣤⣭⡤⠀⠀⠀⠰⢤⣤⠴⠖⠀⠀⠀⠀⠻⣶⡾⠁⠀⠀⠀⠐⠶⣤⣤⡤⠶⠂⠀⠘⠿⢶⣶⣶⠶⠟⠁⠀⠻⣴⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢀⣆⠀⢀⣆⠀⠀⠀⢶⠄⠀⢤⠆⠀⠀⣶⡄⠀⢀⣠⡀⠀⠀⠀⣦⠀⢠⡄⠀⠀⠀⠀⣤⡀⠀⣀⡤⠀⠀⠀⠀⣴⠀⠀⠤⠀⠀⠀⣀⡀⠀⣠⡤
⠀⠀⠀⢠⣉⡀⣀⣩⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⣰⣶⡌⠉⠁⠀⠀⣀⣀⣐⣀⣀⠀⠀⠀⢺⣤⣀⣀⣠⣠⠀⠀⠠⢶⣤⣀⣄⣤⣶⠄⠀⣈⠁⣐⣀⣴
⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠈⠀⠉⠁⠉⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠈⠙⠉⠉⠁
⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠀⢀⣄⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠛⣀⡘⠋⠀⠀⠀⠘⠋⣄⠛⠁⠀⠀⠘⠛⠀⠙⠃⠀⠀⠀⡘⠛⠀⠘⠛⡀⠀⠀⣄⠈⠃⠀⠛⣁⡖⠀⠐⣄⠛⠀⠈⢛⣠⠆⠀⣈⡁⠀⠈⣁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⠶⠶⠖⠁⠀⠀⠈⠳⠶⠖⠁⠀⠀⠀⠙⠛⠷⠛⠋⠀⠀⠀⠈⠙⠾⠻⠟⠋⠀⠀⠀⠈⠛⠿⠿⠛⠀⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠰⠿⢀⠰⠷⠀⠀⠀⠘⠙⡀⠓⠀⠀⠀⠙⠇⢀⠘⠟⠀⠀⠀⣙⠛⠀⠘⢛⡀⠀⠀⢄⢺⡿⠀⣿⠇⡀⠀⠀⠀⠛⠀⡀⠞⠂⠀⠀⠙⠶⠀⠲⠟
⠀⠀⠀⠙⢶⣶⣶⠞⠁⠀⠀⠀⠤⠤⠾⠃⠀⠀⠢⠶⣤⠶⠴⠀⠀⠀⠿⠛⠛⠛⠿⠇⠀⠀⠈⠲⠤⠤⠤⠚⠀⠀⠀⠐⢶⣤⣤⡴⠂⠀⠀⠙⢷⡶⡶⠛

⠀⠀⠀⢠⣤⠀⢠⣤⠀⠀⠀⣴⡄⠀⣠⡄⠀⠀⣤⡄⠀⢠⡤⠀⠀⠀⡠⡄⠀⢠⣤⠀⠀⠀⠀⣠⡀⠀⣠⡄⠀⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⣠⣄⠀⢠⢤
⠀⠀⠀⢈⣛⣀⣘⣋⠀⠀⠀⠈⠁⣀⣉⡁⠀⠀⣨⣀⣂⣈⣀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠰⣤⣤⣤⣤⣴⠀⠀⠀⠀⢀⠘⣣⡄⠀⠀⠀⢠⣄⣤⣤⡤
⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠈⠛⠉⠋⠁⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠙⠛⠋⠁

⠀⠀⠀⠀⣀⠂⠐⣀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠐⡠⠀⠀⠀⠀⠀⠀⠀⢲⣄⠀⢢⠔⠀⠀⠀⠀⠀⠀⠀⠢⠔
⠀⠀⠀⠀⠛⠁⠈⠓⠀⠀⠀⠀⠀⢰⣤⣠⣁⣈⣄⣤⡆⠀⠀⠀⠀⡀⠈⠁⠀⠉⠁⢀⠀⠀⠀⠠⠀⠰⠆⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠛⠛⠛⠻⢿⡇⠀⠀⠀⠀⢧⣴⣦⣤⣴⣦⡜⠀⠀⠀⠰⠂⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⢢⢄⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⡀
⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣄⡀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠀⢤⡄⠀⠀⠀⠀⠐⢲⠦⠤⢺⡅⠁
⠀⠀⠀⢰⣶⡶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠉⠉⠛⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠈⢑⣄⠛⣠⠊⠀⠀⠀⠀⠀⠠⢄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⡤⠄⠀⠀⠀⠀⠀⠼⠧⠃⠾⠄
⠀⠀⠀⣀⠈⠁⠀⠈⢁⡀⠀⠀⠀⠀⠀⠘⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⣤⣤⣤⣴⣤⣤⣤
⠀⠀⠀⠘⠛⠋⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⠑⠀⠊⠄⠂⠀⠀⠀⠀⠀⣀⠰⡔⢀⣀⡀⠀⠀⠀⠀⠀⠀⣬⡓⢚⣥⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠐
⠀⠀⠀⠰⣤⣀⣀⣠⡴⠀⠀⠀⠀⠀⠀⡀⣛⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⠃
⠀⠀⠀⠀⠹⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⡯⠟⠛⢻⣦⡀⠀⠀⠀⢠⠞⣹⣿⠟⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠛⠁⠀⠀⢸⣿⣷⠀⠀⠀⣿⣾⡿⠁⠀⠀⠈⣿⡽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⢿⣿⡇⠀⠀⠀⢰⣿⡇⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢦⣀⣴⡿⣿⣿⠟⠀⠀⠀⠘⣿⡿⣄⣠⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠻⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠙⠺⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣶⣿⠻⢶⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣾⣿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⣹⡟⠀⢀⣼⣿⣿⠿⠿⣿⣿⠟⠛⣻⡿⣯⡀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇⣴⠟⠛⣷⣿⠀⣼⡟⠻⣷⢠⣿⠀⢈⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣦⣄⡀⢻⣿⣼⠏⠀⠀⠘⢿⣇⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠶⠶⠶⠾⠛⠛⠉

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


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

⢤⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⣷⣼⣦⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣴⣧
⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿
⠀⠈⣿⣿⠁⢸⠏⠹⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠋⢹⡇⢸⣿⣿
⠀⠀⢹⣿⡆⢸⡀⠀⣿⠀⢻⡟⠛⠛⢿⣿⠶⢶⣶⣶⣦⣤⣀⣀⢀⣀⣀⣤⣤⣴⣶⣶⣶⣶⡾⠟⠛⢿⡟⠉⣿⠀⢸⡇⢸⣿⡟
⠀⠀⠈⣿⣿⣼⡇⠀⢿⣇⠈⣷⠀⠀⠸⣧⠀⠀⢹⡏⠉⠉⣿⠿⠛⠛⣿⠉⠉⢹⡏⠀⠀⣿⠁⠀⢀⣿⠁⣸⡏⠀⢸⣧⣿⣿⠃
⠀⠀⠀⢹⣿⣿⣿⣄⢸⣿⡄⢹⣧⡀⠀⣿⣆⠀⠸⣿⡀⠀⣿⡄⠀⢠⣿⠀⢀⣿⡇⠀⣰⣿⠀⢀⣼⡇⢠⣿⡇⢠⣿⣿⣿⡏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣄⢻⣿⣦⠀⣿⣷⡀⣿⣧⠀⣾⡇⢀⣾⣿⠀⣰⣿⡇⣠⣾⣿⣴⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢸⣿⣿⡿⠋⢸⣿⡿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣾⣿⣾⣿⣷⣾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠙⢿⣿⣿⠃
⠀⠀⠀⠀⠀⢿⡟⠀⠀⣼⣿⠁⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⠈⣿⣷⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠘⢇⠀⠀⣿⡇⠀⢹⣿⣿⠀⢹⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⡇⠀⣿⣿⡇⠀⢸⣿⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠻⡇⠀⠈⣿⡏⠀⠀⣿⣿⠀⢿⣿⡇⣿⣿⣿⢹⣿⡏⠀⣿⡿⠀⠀⢹⣿⠀⠀⢸⠿⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣠⠿⣷⠀⠀⢸⣿⠀⠈⣿⡇⢹⣿⡇⢸⣿⠁⠀⣿⠇⠀⠀⣾⢧⣀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠤⠼⠿⡄⠀⢸⡇⢸⣿⠁⢸⡇⠀⢠⡿⠦⣤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊⠙⠶⠻⠦⠛⠓⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣶⠶⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠶⣶⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⡏⠉⢿⣿⡟⠛⠓⠶⢶⠶⠚⢻⠛⠙⢿⣿⠁⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⡇⠀⣶⣿⡇⠀⢀⡤⠞⠀⠀⢸⠀⠀⣾⣿⠇⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠘⢧⣤⡀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢸⣿⣏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⡇⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⢯⡉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⢉⡽⠋⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⣀⣠⡤⠖⠋⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠤⣤⣤⣀⣀⣀⣀⣤⡤⠤⠶⠚⠉⠁




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

⠀⠀⠀⠐⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡆⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠻⣿⣿⣿⠶⠄⠀⠀⠀⠶⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣄⡀⠀⠀⠀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⣦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠛⠉⠀⠀⠉⠛⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣠⣄⠀⣀⡀
⠀⠀⠀⠀⠀⣠⣦⣾⣿⣷⣿⣿⣶⣿⣷⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡤⠶⣆⣀⣴⣄⣠⣦⣀⣠⠶⢤⠞
⠀⠀⠀⢠⣾⠟⠿⠿⠘⠿⠏⠸⠿⠃⠿⠿⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠙⠋⠀⠙⠋




⠀⠀⠀⢀⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⠀⠀⠀⣰⣷⡄
⠀⠀⠀⢸⣿⣿⣿⣷⣤⠀⠀⠀⢀⣤⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⠀⠀⠀⠀⣴⣿⣿⣿⣄
⠀⠀⠀⠈⠛⠋⠉⠉⠀⠀⢠⡄⠀⠀⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⢠⡀⠀⣀⠀⠀⠀⠛⠋⠀⠀⠀⣀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢻⣿⣿⣶⣾⣦⣴⣦⣴⣷⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⣼⣶⣄⣰⣶⣦⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠉⠋⠙⠛⠋⠛⠛⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠹⢿⡿⠏⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⡀⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠘⠷⠷⠖⠀⠀⠀⠀⠀⠐⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠂⠀⠀⠀⠀⠀⠐⠶⠞⠃⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⡠⠶⠒⠒⠶⢤⡈⠀⠀⠀⠀⠀⠀⢁⡤⠶⠒⠒⠶⢤⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠠⠋⠀⠀⠀⣀⠀⠀⠙⠆⠀⠀⠀⠀⡰⠋⠀⠀⣀⠀⠀⠀⠙⠄⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠁⠀⡀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠠⡄⠀⠀⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⢠⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢄⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣠⠃⠃
⠀⠀⠀⠀⠀⡆⠀⢀⣤⠀⠀⠀⠀⠙⢖⣲⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣶⠶⠋⠁⠀⠀⠀⢲⣄⠀⢰
⠀⠀⠀⠀⠀⢂⠀⠺⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⡀
⠀⠀⠀⠀⠀⠘⡀⠀⣿⡇⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⣾⣿⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢣⠀⠘⣿⢶⠀⢀⠈⠉⢓⠶⠤⠤⣀⣀⣀⡀⠀⣀⣀⣀⠠⠤⠔⠒⠉⠁⠀⢰⣴⢿⠇⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠈⢯⣣⢼⣄⠀⣼⣤⣀⣀⣠⣇⣀⠀⢀⣠⣆⣀⣀⣠⣧⣠⣴⣧⣴⡟⣻⠋⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠙⠂⢹⠛⠻⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⡟⠀⢹⠉⢀⠞⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠸⠀⠀⠈⠻⠀⠀⠀⠀⠛⠉⠀⢸⠀⠀⡨⠐⠁⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠽⠯⠿⠿⠿⠯⠽⢿⠽⠽⠿⠿⠽⡿⡵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⢁⡀⠀⠜⡉⢀⣊⣔⣫⢀⣶⣡⣠⣤⠀⢸⠒⣿⣗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⢞⡁⢀⣿⣆⣾⡿⣷⡿⠻⠟⣿⣿⣿⣶⣿⣴⣷⡿⣧⡋⠉⢻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠾⢃⣩⡖⡿⢿⡄⠿⣧⠋⠀⠀⠀⢸⢿⢻⠟⠛⠋⢙⣾⣿⣣⠄⠈⣷⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⡿⣶⣿⠋⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣧⡏⠀⠟⠳⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠰⡌⣿⠉⠀⡀⣦⣤⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠈⠁⢰⡞⣡⠰⠃⠈⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⣇⡝⠀⠀⢹⣿⣿⡛⠿⢿⣷⣤⣄⠀⠀⣴⣶⣿⡶⢿⣷⣦⡤⠀⠀⠀⣿⠠⠆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⢟⣿⡄⠀⢾⣿⠂⠀⠀⠀⠉⠻⢿⣷⣾⠟⠉⠀⠀⠈⠋⢿⣿⣦⠀⢀⡥⢰⠃⠀⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠀⢀⡘⣿⣇⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡧⢀⣾⣥⢋⡀⠀⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠀⠘⠓⠚⡮⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⠟⠟⠘⠃⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠈⢁⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⠇⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠘⣄⡀⠠⠴⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠤⠀⡀⢁⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⡀⠀⠀⠀⠁⡀⠀⠀⠂⠑⠦⣄⡀⠀⠀⣀⡤⠖⠁⠂⠀⠀⡀⠉⠉⠀⠀⢀⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⢸⣷⠤⠓⠂⠀⠐⠉⣀⠀⠀⡌⠁⠀⠀⠒⠣⢤⡏⠀⠀⠀⠀⠈⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⡳⢄⠀⠠⠀⠀⠸⣏⠀⠀⠀⠀⢀⣴⣿⠀⢠⣿⣄⠀⠀⠀⠀⢈⡇⠀⠀⠀⠄⠀⡠⢞⢕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢵⣌⡂⢙⢦⡀⠀⠀⠹⡄⣷⢿⣤⡀⣦⣿⡆⠈⢠⡆⣠⣼⢿⡆⡸⠁⠀⠀⢀⡴⡋⠐⣡⡮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡕⠍⠢⡄⠀⠻⣾⠄⢿⣿⡮⣿⣷⣶⡯⣾⣿⠏⢸⡜⠤⠀⢠⠔⠩⢪⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⣶⣌⠂⡶⡄⠈⢇⠈⢿⣿⣷⣾⣶⣿⣿⠏⠀⠉⠀⢠⢶⠐⣡⣶⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣼⣿⡇⠀⠀⠂⢀⠹⣍⣛⣋⡭⢂⠀⠀⠀⠀⢸⣿⡯⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣯⣇⠀⠀⠀⠰⣅⣀⣨⣁⣀⠁⠀⠀⠀⠀⣸⣽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⠙⠙⠋⠉⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣷⣶⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⡿⠟⠛⠉⠛⠛⠛⠛⠛⠿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡿⠛⣩⣶⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟⠋⠀⣸⠏⠀⣠⡶⠛⢻⡆⠀⠀⠀⠀⠀⠀⢀⣤⠾⠿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⠀⠀⠀⡟⠀⣰⠏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⠛⠲⠶⣼⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠁⠀⠀⠀⠀⠀⢰⡏⠀⢀⣀⠀⣿⠀⠀⢀⠀⠀⢰⡏⠀⠀⠀⠈⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⣽⠆⢿⣀⣠⣿⠀⠀⣼⣣⢶⡄⠀⠀⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⢤⣄⠻⣦⡀⠀⢀⣼⠋⠙⠿⠶⢿⣅⠘⠛⠃⠀⣠⡟⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⠉⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠙⠳⠶⠶⠟⡋⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⣿⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣺⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣾⠋⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡾⠋⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠶⠶⠾⠛⠋⠁⠀⠀⠀⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠃⠈⠙⠛⠿⢷⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣾⡿⠟⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠋⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⣠⣾⣿⣿⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠋⢠⣾⣿⣿⣿⡟⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⣹⣿⣿⣿⣿⣦⡀⠀⠀⠀⣼⣶⣿⣿⣿⣿⣿⡟⣶⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡴⢯⣹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⢻⣿⣿⣿⣿⣿⣯⣀⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣺⣿⣿⣿⣿⣿⣿⣿⠖⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢁⡼⠁
⠀⠀⠀⠀⠘⣯⡝⣷⡄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⣇⡾⠁
⠀⠀⠀⠀⠀⠈⢷⡈⠿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠛⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠤⠀⠀⠠⠤⠶⠀⠀⠀⠀⢀⣠⣴⣾⢿⡏⢣⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣦⢹⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⠹⠻⠈⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠀⡿⢹⣿⠻⣿⣿⣶⣶⣶⣤⣤⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⡏⠗⠀⠹⡏⠀⢹⡀⢠⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⠈⠁⠀⣿⠀⢻⠻⠃⠈⠹⣿⠿⣿⡿⠋⢻⠟⢻⢻⣟⡏⠙⣯⠀⠹⡏⠉⢹⡄⠀⠀⠀⣷⠀⢸⠁⠈⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠃⠀⢏⠀⢸⡀⠃⠀⠀⡟⠀⠀⡇⠀⢈⠀⠀⠀⠿⠀⠀⢹⠀⠀⠇⠀⠈⣧⠀⠀⠀⡟⠀⠺⣠⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢆⠸⡄⠈⣇⠀⠀⢰⡇⠀⠀⡆⠀⠀⡆⠀⠀⡇⠀⠀⣾⠀⠀⠸⠀⠀⠛⠆⠀⢰⠇⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣷⠀⠀⠸⡄⠀⠞⠱⠀⠀⠁⠀⠀⠁⡀⠀⠙⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⡇⠰⡀⠀⠀⣠⠄⠀⠀⠀⠀⢧⣀⠀⢱⣴⠁⠀⣀⠀⠀⢿⠀⠀⣶⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠀⢹⡆⠀⢹⡀⠀⠀⡇⠀⢸⡇⠀⠀⣿⡆⠀⢿⠀⡄⢸⠀⣠⣿⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣬⣿⣧⣸⣧⡀⢀⣷⠀⢸⣷⡀⢀⣿⡇⢠⣼⣦⣧⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁




⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣵⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⡴⢢
⠀⠀⠀⠀⠀⠀⢸⣷⠈⠢⣀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣟⣿⣯⢉⡛⣈⣯⡛⢉⣉⣵⠄⢉⣿⣿⣿⣿⣿⣿⣄⠀⢀⣠⡖⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠱⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡟⣛⠿⠋⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠠⣼⠗⠀⣸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⠻⣿⣿⣷⡈⠀⠀⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⢀⣠⣼⣿⣿⣿⠋⠀⠀⠉⠛⠊⠙⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠘⣿⣿⣷⣤⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣅⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⡀⢸⣿⣿⣿⣿⣼⣿⡷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧⡴⣦⡀
⠀⠀⠀⠀⠀⠀⢿⣽⣛⢿⣿⣿⣿⣫⡜⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢀⣀⡀⠀⠈⢻⣿⣿⣿⣿⠏⡀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣆⢸⣿⣿⣿⣪⣴⣾⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣠⣾⠛⠛⠛⠳⢦⡀⢻⣿⣿⡏⣸⠇⡼⠁
⠀⠀⠀⠀⠀⠀⠸⣿⡿⣾⣿⣿⣾⣿⠁⠈⢛⣿⡿⠿⣿⣷⣤⣄⠀⠀⠀⠀⠈⢀⣶⣿⠿⠿⠧⠤⠀⠀⠀⠁⠀⢿⣿⣥⡟⢴⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠘⣹⣿⣿⣿⣿⣷⣾⣿⠷⠶⠶⢾⣿⣿⣿⡄⠀⠀⢀⣴⣿⢏⡴⠚⣛⠛⢶⣄⠄⡀⢠⡄⢸⣿⡝⠃⢼
⠀⠀⠀⠀⠀⠀⠀⣿⣆⣹⣿⣿⣿⣿⣿⣿⡅⢰⣿⣼⣆⣹⣟⣿⣷⠀⠄⠺⣿⠿⣿⠀⣿⣾⣿⡄⠙⣿⡖⠂⠀⢸⣿⡸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⡌⠻⣿⢻⣿⣿⣿⣿⣿⣯⡉⠙⠋⠉⢹⣿⣿⠀⠀⠀⡄⠰⡿⠻⡥⢤⠄⠀⠀⠈⠁⠀⠔⣾⣿⠃⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣿⣿⣿⣿⡃⠟⠉⠛⠃⠀⠀⢠⣾⢿⣿⠀⠀⠀⢳⠀⠀⠐⠤⠄⠀⠀⠀⠀⠀⠀⠐⣿⣷⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠰⣟⣻⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣤⢀⣄⣀⡀⠀⣠⣿⣿⠀⠀⠀⣧⠦⢄⡀⠀⣀⣀⡀⠀⠀⠀⠁⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠽⣵⣿⣿⠿⢿⣿⣏⣤⡙⡄⠀⠀⠀⢀⡀⠿⠿⠋⠉⠉⣶⠀⢀⣴⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⣿⡇⣿⣦⣀⣉⣿⣿⣿⡇⠀⠀⣴⣷⣾⣤⣀⣀⣴⠟⣼⢀⠈⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡗⢿⣤⣜⠿⣿⣿⣿⣿⣿⣿⣦⣾⠙⠻⠿⠿⠿⠟⣋⡄⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡜⣿⠛⢿⣖⠛⠛⠻⠋⠋⠛⠛⢻⠛⠛⢒⣶⡟⠉⠀⠀⠘⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⣿⣿⣿⣦⣼⣦⣤⣴⣶⣾⣶⣶⡿⢋⠀⠀⠆⠠⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⣿⣿⡿⠚⣿⣿⠸⢿⣿⣿⣿⠋⠐⠉⣿⣶⣿⠂⠘⣴⠅⢪⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠙⣿⣷⣿⣿⣿⡆⠘⠀⠻⡏⠀⠀⢰⣿⣿⣿⣿⣷⣽⣇⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡄⠀⣻⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⢧⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢋⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡇

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢰⠖⡆⠀⡉⣄⣀⠘⣿⠿⢿⣿⣿⣿⡟⠛⣛⠋⢠⣽⣛⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⢠⣀⠀⣼⠀⣿⠚⠁⠀⠀⠀⠈⠀⣾⣿⣿⡿⠷⣿⣿⣷⣸⣿⣿⣼⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⣂⣠⣄⠀⠀⠀⠀⠀⠑⠆⠀⠐⠃⠀⠈⠀⠁⠀⢀⣄⡤⠀⠀⢸⡿⠛⠉⠀⠀⠈⠉⢙⡻⣿⣽⣿⣿⣿⣿
⡧⡄⣀⢀⣤⠤⠄⠀⠀⠀⠀⠂⠈⠁⠀⠀⡉⠉⡙⡦⠀⠾⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣤⠄⡀⠀⠈⠃⠀⠀⢀⣤⣤⣤⣿⠟⠛⠛⣿⣿⣿⣿
⢳⣌⠇⠈⠽⠀⠂⠀⠀⠀⢐⠀⣀⠀⠀⠔⣡⢄⠀⠀⠀⠀⠀⠀⠀⢀⢀⣰⡀⠰⠀⠀⢜⡙⢣⣾⠾⢤⣀⠠⠤⠀⣀⡀⢨⣿⣿⣿⠉⠀⠁⠙⠁
⠀⠻⠈⠀⠀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠚⢍⣐⠞⣉⠱⠤⢀⡂⠀⠁⠈⠁⠀⠀⢿⣿⠇
⡘⠣⢈⠛⠢⣗⠎⠠⠀⠀⠁⠻⢗⠀⠀⠀⠀⣿⡄⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⢊⡀⠀⠀⡀⣈⠲⣾⠀⠀⠀⠀⠈⠁⠸⠗⢶⠇⠀⠂
⠦⠖⠼⠲⠶⡷⠛⡤⠃⠀⣢⡄⠺⡀⠀⢀⣀⣈⣀⠀⡼⠀⠀⠀⣀⣐⣀⣠⠶⠀⣀⣀⣠⢇⡈⡻⠀⢨⢀⣉⠇⢘⠀⠀⠀⠀⠀⠦⠀⠄
⡘⠁⠐⠀⣠⠃⠀⠀⠀⠀⠃⠓⢻⠗⠈⠀⠀⠀⢀⣰⡃⠀⠀⢨⣿⠉⢉⡏⠀⠀⢿⡉⢹⠐⣶⣶⣶⣶⣶⣶⡄⢠⠁⠀⠀⠀⠀⡀⠛⠋⠀⠀⠐⠂
⠈⣀⣤⣾⡅⠀⠀⠀⠀⠀⢻⠀⠈⠇⠬⢉⡗⠀⢛⢉⠀⠀⠀⠀⠻⠿⢋⣤⣴⣧⣬⡛⠋⠡⡙⢿⡟⢉⣉⠟⡠⠃⠀⠰⠍⠓⠤⣀⡀⠤⠷⡖⠒⢤
⣾⣿⣿⣿⣇⣠⠀⠀⠀⣤⢰⣦⠇⣾⣷⣆⣔⣄⣘⣮⢦⠀⠶⠂⠉⠛⠻⠿⣿⡶⠬⠛⠛⡲⡮⠱⠲⠶⠲⣿⡀⠀⠀⠀⢀⠠⢀⣀⣀⣤⣤⢼⠔⡏
⣿⣿⣿⣟⣿⠉⣿⡧⠀⠈⡦⠆⠈⠀⠀⠈⠉⠩⠥⠖⢊⣱⣄⡀⠀⠀⠀⠀⠈⠂⠀⠀⢀⠄⠀⠀⠀⠀⠁⠢⠁⠀⠀⠀⠀⣤⡤⠤⠤⠤⠤⠤⠤⠤⠄
⣿⣿⣿⣿⣿⠶⠚⡄⠀⠀⢀⠀⢘⡓⠈⠁⢭⠀⠄⢘⠉⠉⠙⠻⣶⠢⢤⣤⠀⠀⠠⠚⠁⠐⣄⠀⠀⠀⠀⠀⠀⠀⠄⣴⡤⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⠶⠟⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⢠⣿⣿⣿⣿⡧⠴⠀⢰⠀⢘⣨⠟⠻⡖⢂⡀⠀⠀⠀⠉⠓⡷⠀⠀⠀⠀⠀⠻
⠈⢙⣿⣿⣿⣦⣴⣦⠆⠀⣀⡀⠀⠐⠀⠀⠀⠀⡄⢀⡀⠀⠀⠘⣻⡏⡙⠀⠘⠋⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠘⠀⠛⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣄⠀⠱⡄⠀⠀⠀⠀⡴⠁⢀⣴⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⡈⠀⠀⠀⠈⣀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠈⠙⠛⠓⠶⠶⠤⠤⣤⣤⣤⠤⠤⠴⠶⠒⠛⠋⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡓⠐⠒⢺⠤⠤⠤⣼⠤⠤⠤⡷⠒⠂⠸⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡀⠸⠀⠀⠀⢘⠀⠀⠀⠇⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢤⣤⣤⣤⡤⠴⠖⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒⠲⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⠤⠤⠶⠲⠦⢄⣀⡤⠴⢒⣒⣒⡒⢤⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣠⠖⠛⢶⣊⠭⠽⠧⢬⣏⢹⣎⠭⠽⠒⠚⢃⣛⣩⣧⠟⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠷⠶⠤⠀⡧⣤⣤⡤⠬⠽⠛⠢⠤⠤⠤⠤⠴⠒⠛⠁⣶⠂⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠸⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠞⠋⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠿⡤⢶⠼⡇⠀⣼⡥⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢀⡠⣄⡀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠘⠂⠀⠒⠉⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠗⠦⢤⣀⡀⠀⠀⠀⠀⢀⣀⣠⠴⠊⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄

⠀⠀⠀⠀⠀⠀⠀⠈⠚⡡⠀⠔⣚⣔⠀⣤⠀⣈⡓⠠⠴⠀⠰⠦⠈⣃⣀⢠⡄⠀⣖⣲⡦⣀⠵⠁
⠀⠀⠀⠀⣠⣦⢶⡆⠀⢀⣀⠀⠈⠀⠀⢨⣕⠘⣋⡄⠀⡀⠀⢀⣎⠙⢕⣮⡅⠀⠉⠀⢀⣀⡀⠀⢴⣆⣤⣄
⠀⠀⠀⢈⣡⠞⠀⠀⡠⢋⣹⣿⠄⢠⢃⠄⠛⡇⠋⢀⠔⠳⠴⡤⠁⠁⡟⠙⢠⡃⡀⢰⣿⣏⡙⡄⠀⠐⠳⣌
⠀⠀⠀⠈⢀⠀⢀⠔⢁⣸⡿⠃⠈⣣⣻⣦⣄⣀⣠⠶⢶⣤⠴⠶⣄⣀⣀⣤⣿⣓⣀⠀⡛⣿⣃⡈⠢⠀⠀⠨⡥
⠀⠀⠀⠈⢀⡴⣶⠾⢿⣯⣀⣁⢻⠿⢋⡡⠤⠮⣹⠳⣉⣿⣉⡞⡫⠭⠦⠤⠈⠛⠻⢃⣉⣈⣯⠿⢦⡲⣤⡀
⠀⠀⠀⠀⠈⠰⠿⠹⣧⣿⠉⣵⣾⣻⣥⣤⣤⡀⠀⡇⡇⣿⢸⠸⠀⢀⣤⣤⣤⣽⡻⣶⢭⢹⣧⡻⠈⢿⠈⠁⠁
⠀⠀⠀⠀⢹⢸⡗⢠⣍⢷⣷⡿⣋⣁⡈⠉⠛⠻⣦⣤⢃⣿⣘⣤⣴⠟⠛⠉⢁⣀⣙⠻⣷⣾⢯⣥⠐⡾⠁⣀⣤
⠀⠀⠀⢸⣷⡄⢻⣀⢀⣾⣿⣟⣩⣤⣤⣤⣤⣴⣼⣿⠉⠀⠉⣿⣧⣶⣶⣤⣤⣤⣬⣙⣿⡟⡄⢀⡘⢃⣼⣿⡿
⠀⠀⠀⢰⣦⣭⡀⣙⣻⢹⡿⣉⣿⣿⠿⣬⣋⣿⣿⣿⠆⠀⢸⣿⢿⣿⡁⣤⠿⣿⣿⣉⣿⣿⢻⢛⠅⣨⣥⣴⣦
⠀⠀⠀⢈⣉⠙⢧⠘⡏⣼⡇⢲⡿⢭⣤⣤⣭⡿⠇⡟⡀⠀⢠⢿⠸⣿⣥⣤⣴⡿⢿⡖⢸⣿⢸⠟⡰⠋⣉⣉⡋
⠀⠀⠀⢸⣿⣿⣆⢱⡆⢸⣄⠀⡄⠛⠙⠋⢉⣡⢞⠀⠁⠀⠈⠨⠂⣈⡙⠉⠋⠛⢀⠀⣉⡟⢰⡞⣰⡿⢿⣿⣿
⠀⠀⠀⢨⣤⣴⣶⣼⣇⠀⣿⡿⠿⠶⠶⠿⠛⠉⠲⠀⠀⠀⠀⢠⠖⢉⠛⠻⠷⠶⠾⢿⣿⠀⣸⣷⣶⣶⣶⣤⣁
⠀⠀⠀⢸⡿⠟⠛⢛⣿⡠⣿⠀⠐⠒⠆⠀⠀⠲⣤⣴⣶⣶⣿⣧⡴⠆⠀⠀⠒⠒⢂⠀⣿⢣⣿⡋⠙⠛⠿⣿⣿
⠀⠀⠀⠈⠀⠠⠞⠉⣽⣧⢻⡄⢷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⢀⣠⣴⠇⢠⡟⣾⣯⠉⠢⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠻⣧⠈⣿⡍⠑⠦⠤⣤⠤⣀⣀⠀⡤⠀⢲⠀⠉⣼⡟⢀⣿⠞⠛⢿⣷⣤⡀
⠀⠀⠀⠀⣠⠎⣿⣿⠁⠀⠀⠘⣇⠘⣷⣄⠀⠀⠀⠀⠈⡇⠀⠀⠀⡀⣀⣼⣿⠁⣼⠃⠀⠀⠈⣿⣿⠉⠢⣀
⠀⠀⠀⠺⠁⠀⣿⡿⠀⠀⠀⠀⢿⡄⠘⣿⣷⣃⣐⢆⡀⢧⣀⣢⣼⣿⣿⣿⠃⢠⡟⠀⠀⠀⠀⢿⣿⠀⠀⢹⣿
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⣸⣷⠀⠘⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠃⠀⣼⡇⠀⠀⠀⠀⠸⣿⠀⠀⠀⢿
⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⢀⣿⠻⣷⡀⠈⠛⠛⢋⣉⣍⣉⡙⠛⠛⠁⢀⣼⢿⣷⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⣠⣾⠃⠀⢹⣿⣆⠀⠈⠉⠉⠉⠉⠉⠉⠀⣠⣾⠃⠀⣿⣷⡀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⣾⠏⠀⠀⢀⣠⣴⣿⠏⠀⠀⣰⡏⠙⢷⣤⣀⣀⣀⣀⣀⣠⡾⢻⣧⠀⠀⠘⣿⣿⣶⡄⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⢀⣴⣿⠇⠀⢸⡏⠉⢹⡟⠉⢹⣿⠀⠘⣿⣧⡀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣉⣭⣾⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢛⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠙⠿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣴⣾⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⢰⡿⠛⣻⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⠻⣿⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⣸⢃⣾⢿⣿⠟⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠃⠸⣇⠸⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢹⡏⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢿⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢋⡴⠟⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡤⠄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡠⣪⠞⠛⠛⢛⣛⡛⠛⡛⠛⣛⠛⠛⡛⠛⠛⠻⠛⣿⢧⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⣪⠞⠁⣀⠀⠠⡊⣁⢀⣨⣴⣯⣴⣠⣾⡦⢄⣤⠴⠀⡿⠛⣿⣯⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⣪⠞⣁⠀⣸⣷⢠⣾⣿⣿⡿⠛⠿⢻⣿⣿⣿⣶⣿⣧⣾⣿⣿⣷⡋⠋⠝⣿⡃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⣪⣾⠗⣸⣵⢾⣿⣿⠘⣿⣯⠚⠀⠀⠀⠀⢹⣿⠟⡿⠟⠛⠛⢫⣾⣿⣆⡄⠈⠳⣎⠢⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⢾⣏⣴⣾⡟⠿⠈⠀⠉⠁⠀⠙⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⢺⣿⣿⣿⣖⣼⠀⢹⡟⢮⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡞⠁⢨⢸⣿⠛⠁⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠛⢋⣩⡶⡄⢠⠀⠙⡌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⢸⠈⢿⠀⠀⢰⣼⣿⣿⣶⣼⣦⣄⠀⠀⠀⠀⠀⢀⡀⣠⢂⣀⣤⠀⠀⠀⠀⠈⠋⢰⡇⢛⡀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠸⣷⣇⠀⠀⠸⣿⣿⠿⠈⠙⠻⢿⣷⣷⣄⣀⣠⣾⡿⠿⠟⠛⣿⣿⣿⣟⠀⠀⠀⠸⠃⢘⡁⠀⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠘⣿⣿⣆⠀⠸⣿⣯⠀⠀⠀⠀⠀⠈⠛⠿⣿⠿⠋⠀⠀⠀⠀⠀⠈⢻⣿⣷⠀⠀⣤⠃⣻⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡁⠀⢀⣘⣿⣧⠀⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⠀⣾⣧⡌⣁⠀⠀⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠻⠓⠻⡏⣽⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⡿⢿⠃⠻⠃⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠘⠃⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢾⡷⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠘⡉⠈⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠟⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠛⠲⠀⠁⠀⢩⡒⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢚⠉⠀⠈⠀⠶⠶⠃⠀⠀⠀⠀⠀⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢴⠄⠀⠀⠀⠀⣤⠀⠠⣀⠀⠀⠉⡳⠖⠀⠀⠀⠲⣞⠋⠀⠀⣀⠀⠀⡆⠀⠀⠀⠀⠀⢶⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠈⠀⠀⠀⠀⣰⡄⠀⢠⡄⠀⠀⠀⠀⠉⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣕⢌⡳⣄⠀⠶⠀⠀⠀⢻⡄⠀⠀⡀⠀⠀⡴⢿⡇⠀⢸⡿⢦⠀⠀⢀⠀⠀⢸⡇⠀⠀⠀⠰⠀⣀⡴⡋⠔⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣜⠪⣙⢦⡀⠀⠀⠈⢧⡀⣿⢿⣦⣀⣱⣼⣿⠀⠉⣠⣆⣠⣴⡿⣷⢀⡾⠀⠀⠀⠀⣠⢞⠡⢊⣰⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⡑⢝⠦⣀⠀⠈⢷⣽⠈⣿⣿⣿⣽⣿⣦⣴⣯⣿⣿⡿⠡⣏⢎⠤⠀⢀⡤⢚⠕⡁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡇⢤⢑⠪⢰⣄⠀⠹⡄⠈⢿⣿⣾⡥⣭⣬⣷⣿⡿⠁⠐⠋⠀⢀⣤⡩⢚⢡⡆⣏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⢺⢲⣼⣼⡇⠀⠘⡀⠈⢻⡛⢛⣻⡟⢻⡛⠀⠀⠀⠀⠀⣿⣾⣖⢸⡰⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⢿⡇⠀⠀⠀⠈⠢⠙⠲⠶⠒⠋⠔⠁⠀⠀⠀⠀⣿⣿⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣧⠀⠀⠀⠈⠳⢤⣴⣶⣦⡤⠂⠀⠀⠀⠀⣠⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣞⣛⣛⡋⠙⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠛⠛⠛⠛⠛⠛⠻⠿⠏⠉⠓⢶⣄⡀⠹⣦⣄
⠀⠀⠀⠀⠀⢠⣶⢰⣶⡾⢻⣿⢲⣿⣶⣾⣰⡞⣷⠀⣾⡗⣾⣷⢰⡟⣿⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠈⣿⣼⣿⡁⢸⣿⣹⣿⠉⣿⠽⣧⣶⠀⣿⡏⣿⢿⡼⣧⡶⢸⣯⠈⢻⣿⣦⡀
⠀⠀⠀⠀⢀⣴⡟⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⢠⡿⠋⠀⣀⡴⠚⠋⠉⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⣦⠀⠾⢿⣾⡄
⠀⠀⣿⠋⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠈⢹⣆⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠉⠹⣆⠀⠀⠙⢿⡄
⠀⣼⣧⠀⠀⣸⣷⣶⣶⣶⡶⠶⠶⠶⠶⠶⣿⠀⠀⠀⢀⣾⣿⣤⣤⡤⠶⠶⠶⠶⠾⣧⠀⠀⠘⣷⣄
⡇⢹⡏⠀⠀⢹⡟⠛⠛⠋⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠉⠉⠁⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢹⣿
⡇⡟⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⢻⣴⡄
⣿⡇⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠷⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢸⡜⡆⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣇⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠘⣿⣧⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⣤⠤⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢠⡿
⠀⠀⠈⣿⣟⠓⣦⢨⠽⡇⣿⢙⡏⢻⢸⡏⢹⠄⡇⣾⠽⠆⠀⠀⢸⠀⣿⠽⠈⣳⡏⢸⡏⠀⠀⠀⢀⣾⠃
⠀⠀⠀⢿⠙⣿⡁⠈⠉⠁⠉⠀⠁⠈⠸⠏⠉⠀⠁⠈⠉⠀⠀⠀⠈⠀⠉⠉⠈⠁⠉⠈⠉⠀⠀⠀⣾⠏
⠀⠀⠀⠘⣧⠈⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠄⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠢⡀⠂⠂⡄⠀⠑⢄⠀⠀⠀⡠⠒⠈⠀⢠⠐⠒⢂⠔⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠄⠀⠀⠀⡀⠄⠀⠠⡀⠀⢀⡠⠀⠀⠀⠐⢄⡀⠀⢀⠆⠀⠨⠀⠀⠀⠀⢠⠂⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⢀⠔⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠢⡀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡆⠀⠀⠘⠀⠀⠀⠊⠀⢠⣄⡀⠀⠉⠀⠢⠔⠈⠉⠀⢀⣠⡄⠀⠑⠀⠀⠀⠇⠀⠀⢰⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠐⠤⠤⠚⢀⠀⠠⠃⠀⣇⠒⠨⢑⠂⡤⠊⠑⢤⠔⡊⠅⠐⣸⠀⠘⠄⠀⡀⠓⠤⠤⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⡸⠀⠀⢀⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠡⡀⠀⠀⢇⠀⠀⠀⠃
⠀⠀⠀⠀⢀⠀⠐⠀⠒⠁⠀⠀⠀⠀⠀⠐⠙⠒⡀⣀⣀⣀⡤⢤⠲⢖⡲⠖⠤⢤⣀⣀⣀⢠⠒⠋⠀⠀⠀⠀⠀⠀⠈⠒⠀⠂
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢛⠈⠛⠀⣸⠀⢨⡁⢀⡀⢈⡅⠀⣃⠀⠛⠁⡛⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠘⣆⢟⠛⠉⠁⠁⠀⠀⠀⠀⠈⠈⠉⠛⡻⣰⠇⠀⠀⠀⠞⢂⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⢨⠈⠁⠀⠀⢀⣀⠢⠔⣀⡀⠃⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⢢
⠀⠀⠀⠀⢀⠎⠀⠀⢠⠊⠀⠘⡀⠀⠀⠀⠀⠀⠀⠂⠂⠃⠐⠀⢀⠑⠋⡀⠀⠂⠘⠐⠰⠀⠀⠀⠀⢀⠀⢀⠃⠀⠑⡄⠀⠀⠱⡀
⠀⠀⠀⠀⡌⠀⠀⠀⣇⠠⠄⠠⠃⢀⠂⠀⠀⠀⠀⠀⡼⡺⠀⠀⠀⠀⠀⠀⠀⠀⡗⢧⠀⠀⠀⠀⠀⠀⡀⠘⠄⠠⠄⡸⠀⠀⠀⢡
⠀⠀⠀⠀⢡⠀⠀⠀⠀⢀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠘⣬⠒⢀⠄⠉⠉⢠⡀⠒⣡⠃⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⡀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠑⢄⣀⠀⠁⣀⠁⠀⡀⠠⠆⠂⠀⠀⠀⠀⠘⢄⡨⣆⢀⡄⣰⢇⡠⠃⠀⠀⠀⠀⠐⢰⠄⢀⠀⠘⣀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠀⠀⠁⠄⠀⣖⠀⠀⠀⠀⠀⠀⠉⠂⠠⠄⠐⠉⠀⠀⠀⠀⠀⠀⣲⠀⠠⠈⠀⠀⠒⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠒⠀⠀⠀⠈⠁⢦⢼⣀⠀⠀⠀⠠⠄⠠⠄⠐⠀⠀⣀⡧⡴⠈⠁⠀⠀⠀⠒⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠐⠀⠀⠀⠀⡆⣉⡒⠢⢄⡀⠀⠂⠀⡠⠔⢚⣉⢠⠁⠀⠀⠀⠂⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⡄⡀⠀⠀⠀⠀⡸⠀⠀⠀⠙⠢⣍⣒⣒⣩⠔⠉⠀⠀⠀⢧⠀⠀⠀⠀⢀⢠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠉⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠈⠉⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠒⠂⠀⠐⠒⠂⠀⠤⢀⣀
⠀⠀⢠⠀⡆⠀⠤⡐⢄⢀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡠⢪⠖⠀⠈⠀⡠
⠀⠀⠈⠢⠄⠂⠀⡷⠊⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠔⠁
⠀⠀⠀⠀⠀⢀⠜⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢢⡀
⠀⠀⠀⠀⢀⠎⠀⠀⠀⢤⣽⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⡀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠁⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢰⡇⢀⠀⠀⠀⠀⡀⠻⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠇⠀⣀⠀⠀⠀⠠⢄⣸⡆
⠀⠀⠀⢸⡗⠁⠀⠀⠀⠀⠈⠙⠻⣯⣉⠡⠀⠉⠉⠀⠀⠉⠉⠉⠙⣛⡵⠖⠋⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⢸⠁⠀⠴⢶⡖⠢⠄⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢲⣶⠲⠄⠀⡇
⠀⠀⠀⠘⡆⠀⠀⠈⡏⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⠇⠀⠀⢰⠇
⠀⠀⠀⠀⢱⡀⠀⠀⢳⣀⡈⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⣀⡞⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠱⣅⠂⡄⠀⡉⠙⠛⠻⠳⠶⠶⠶⠿⠛⠛⡉⠁⣠⠄⡪⠋⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠑⢦⣀⡁⠀⠀⠦⠤⢤⠤⠤⠴⠆⠒⣋⣡⠴⠋⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠈⠉⠓⠲⠶⠤⠴⠤⠴⠶⠖⠛⠉⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣶⣤⣤⣶⠶⠶⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⢀⠠⠀⠂⠀⡀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⠀⠂⢀⡤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠁⢄⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣿⠷⢶⣤⡀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⣶⣤⠀⠀⠀⠀⠘⢷⡄
⠀⠀⣠⡾⠛⠁⢀⣠⣤⣀⡀⠛⠀⠛⠻⠟⠛⣻⡛⠛⠷⣦⣬⡿⠀⠀⠀⠰⣶⣿⠿⠿⠟⠛⠛⠛⠁⠐⠐⠐⠂⠨⠻⣷⡀
⠀⣾⠋⠠⠀⣴⠟⠉⢩⡯⠛⠷⠶⣦⣤⡶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢠⣄⠀⣠⡶⠿⢶⣦⠡⠈⠘⣷
⠀⣿⠀⠀⢸⡏⣠⣤⣾⠻⣦⣄⡀⠀⠀⠀⠀⡀⠀⣀⢠⣶⠶⠶⠀⠀⠀⠀⠀⠛⢷⣦⡀⠈⠙⠛⠋⢰⣶⠀⠁⡰⠀⣸⡏
⠀⢿⣄⠀⠘⣷⠀⠀⢻⣦⣤⣉⠛⢷⣦⣄⣀⠀⠀⠀⠸⣧⡀⠶⠿⠶⠆⢀⡀⣠⣾⠛⠋⠁⠠⠀⣠⣾⢿⡆⠀⠐⣱⡟
⠀⠀⠙⢷⣄⡀⠀⠀⠀⢻⣿⣿⣷⣾⣧⣉⡙⠛⠷⢶⣤⣄⣀⣀⠀⠀⠀⠈⠛⠋⠁⠀⣀⣠⣴⠾⣿⢻⣾⣧⠀⢠⡟
⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠹⣧⡈⢹⡟⠿⢿⣿⣶⣾⣏⣉⠉⠛⠛⢻⡟⠛⠛⢿⡟⠛⠋⢹⣇⢀⣿⣴⣿⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠈⠻⣿⡁⠀⠀⠉⢛⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠈⠛⢶⣄⡀⣼⠇⠀⠀⠀⠉⣿⠛⠻⠿⢿⣿⣿⣿⣿⡿⢿⡿⢻⣿⡿⠁⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣮⣂⠀⠈⠀⠂⠍⠛⠻⠶⣶⣤⣤⣤⣿⣤⣤⣀⣸⣏⣀⣴⣿⣠⣿⣥⡿⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣅⠀⠠⢈⠐⠀⠀⣀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠄⠀⠠⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣅⣀⠀⠄⠀⡀⠐⠂⠀⠀⠀⠤⠤⠀⠀⠐⠈⠀⠠⠐⠁⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⣤⣀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⢶⣤⣤⣤⣤⣤⣴⣶⠶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠉⢉⣩⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣣⠤⠖⢲⣫⠞⠛⢮⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢿⢿⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠈⢮⣳⣶⣪⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⠦⢤⣀⣀⡬⠽⢫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠕⢻⡄⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠋⢳⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠀⣏⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠟⠛⠛⠻⠶⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠯⠥⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠉⠓⣆⠀⠀⠿⢷⡄
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⢳⣀⠘⢆⢻⡀
⠀⠀⠀⡆⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡀⠤⣶⣦⣤⡈⢧⡈⢻⣇
⠀⠀⢀⡟⠁⠀⠀⠀⢀⡤⠖⠒⠒⠒⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣾⣧⣀⣼⣿⣿⣿⣆⢳⠀⠹⣆
⢀⠀⣾⡇⠀⠀⠀⡴⠋⠀⠀⠀⣤⣤⣀⠈⣿⠲⡄⠀⠀⠀⠀⠀⠳⡄⠀⢈⡁⣸⣿⣿⣿⣿⣿⣿⠈⣷⠀⢻⣧⡀
⢸⠀⣻⠃⠀⠀⢰⠇⠀⢰⣧⣠⣿⣿⣿⣿⣎⠁⡇⠀⠀⠀⠀⠀⠀⠘⢆⡀⢻⣿⣿⣿⣿⣿⣿⠏⢀⡿⠀⠈⡧⢧
⢸⣠⠇⠀⠀⠀⠸⡤⠀⢀⣼⣿⣿⣿⣿⣿⣿⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠬⣿⣛⣛⣋⣁⠴⠊⠀⠀⠀⠹⡌⣇
⢸⣿⠀⠀⠀⠀⠀⣧⠀⠈⠻⢿⣿⣿⡿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣽
⢸⣿⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⠖⠒⠚⠛⢓⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣾⡉⣡⣄⣠⠴⠾⠳⠤⠴⠋⠉⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⣠⡼⠁⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢹⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢴⡾⠉⠉⠙⠯⠀⠀⠀⠈⣧⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⢆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣴⠢⣄⣀⣀⡴⠒⡶⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠈⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠦⠼⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⢷⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠘⡆⠀⠉⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠻⣄⠀⣀⣤⣤⣤⣤⣤⣤⣤⣠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣀⣠⡇⠀⠀⠈⠻⠯⣥⣀⡀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠻⣷⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⠀⠀⠀⣠⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⠆⠀⠀⠀⣰⠿⣆⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠬⠿⠖⠋⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢀⡀⠀⠀⠀⢰⡏⠀⠙⢷⣄⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣀⣠⡏⣿⠀⠀⠀⣾⠁⠀⢀⡴⠛⠉⠉⠉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⡇⠀⢀⡏⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⢀⣤⡴⠶⠾⢷⣤⣀⣀⣤⡾⢛⡇⠈⠑⣄
⠀⠀⠀⢀⡴⠒⠒⠒⠤⠀⣿⣄⠻⣦⣾⡇⠀⠀⠠⢴⣦⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠈⢻⡍⠁⠀⢸⡇⠀⠀⠓⠲⡄
⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⡿⠈⠳⣈⢻⡇⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢀⣀⠘⡆
⢀⡾⠁⠀⠀⠀⠀⢀⠀⢸⡇⠀⢸⠏⢷⣷⠀⠀⣇⠀⠠⢶⣿⣿⣄⠀⠀⠀⡷⢠⡏⠀⠀⠀⠀⢀⣠⣴⣶⣦⡀⠀⠀⠀⡿⠀⠀⢠⡞⢹⠇⢸
⣼⣁⡀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⣿⠁⠀⠈⠐⡎⢩⡾⣿⣿⣾⣷⡿⠃⢸⣧⣀⡀⣠⣶⣿⡟⠁⠀⠀⠀⠀⠀⣼⣧⣀⡴⠋⣠⡟⠀⢸
⠉⠉⠙⠲⣄⠀⠀⠀⣿⣿⣿⡀⠀⢀⣴⡟⡀⠀⠀⠀⠐⠮⣙⣋⡙⠛⡿⠃⠀⢼⣿⣿⣿⣿⢿⡳⢍⠂⢀⡼⠀⠀⣴⠟⢋⣩⡴⣾⠋⣀⡠⠚
⠀⠀⠀⠀⠈⠳⣄⣠⣿⣿⣿⣿⡄⡟⢸⣑⠁⠀⠈⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⢰⠏⠫⢄⣛⣚⣁⠜⠁⠉⠀⢠⣾⠷⢛⡏⠀⢀⣟⡉⠁⠀⠰⡆
⠀⠀⣠⡴⠶⠒⠚⠛⠿⣿⣿⣿⣿⢡⣿⣾⠉⠑⠀⠀⠈⠉⣉⣴⡀⣸⠀⠀⠀⡿⠒⠀⡀⠀⠀⠀⢀⠀⠀⠀⣾⠏⠀⠈⠀⠀⣼⠋⠛⠶⠤⠤⠾
⠀⢸⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⣾⡇⠘⢦⣤⣤⠤⢔⠚⣿⠀⢁⠃⠀⠀⠀⡇⠴⢧⠈⠒⠀⠀⠁⠀⠀⡆⡿⣧⠀⠀⢀⡼⠃
⠀⠀⢳⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣹⣇⠀⢸⠃⠀⣰⣏⣤⣿⡘⡇⠀⠀⠀⠀⣡⠀⣾⠳⣄⠀⠀⠀⠓⢦⣿⢇⡟⣠⣶⣿⣷⣤⣀⣀⣀⣀⡴⠊
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⢷⣾⣷⢾⢿⣟⠻⣿⣿⣯⠀⠀⠀⣼⣥⣾⣥⣸⡎⠓⢦⣤⡴⠋⣿⢆⣿⣿⣿⠟⠋⢷⡈⠉⠉
⠀⠀⠀⠀⠱⣤⠶⠦⣤⡀⠈⢿⣿⣿⣏⠀⠀⠀⠀⠈⣯⡓⠬⣿⣿⣷⣶⣾⣿⠿⠛⢛⣿⣧⡀⣸⠃⠀⣠⡿⢸⣿⣿⠃⠀⠀⠀⠻⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣾⣿⣿⣿⠀⠀⠀⠀⡆⢳⡁⠤⣜⠈⡙⠉⡙⠛⢛⣯⡟⠹⠙⢾⣷⣶⣾⣿⣤⣾⣿⣿⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⠿⣿⣿⣇⠀⠀⡄⠘⣌⠳⣤⣀⢉⠉⠒⠋⡓⣨⠞⡔⠁⠀⠀⠀⣸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣀⢀⡤⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠹⣇⠀⠓⠀⠈⠙⠲⠤⣭⣍⣘⣓⣋⡥⠊⡀⠀⠀⠀⣰⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠘⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠐⠿⣅⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠒⠒⠦⢤⡀⠀⠀⠀⡰⠁⢀⣠⣾⣿⡿⠛⣿⠳⣄⣀⡀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠙⠋⠀⠀⠸⡆⠀⠘⢧⣀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠉⢿⣄⣀⣴⣫⣷⡲⣄⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⣏⢻⣿⣿⣿⣿⣿⣤⣴⡟⠁⠘⢧⡀⠀⠀⠀⠀⠀⣀⣠⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⡿⠋⠀⠀⢸⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣦⣤⣤⣴⣚⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣄⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠙⠳⠀⢸⣆⠀⠀⠊⠀⣠⠄
⠀⠀⠀⠀⠀⠀⡏⠻⠶⠒⡟⠁⠀⠠⠤⠶⠶⠀⢀⡤⢰⣂⡏⡬⢁⠲⢄
⠀⣀⡀⠀⠀⣠⠁⠀⠀⠈⣳⡦⠤⠀⢀⡠⠴⠉⡄⠖⠈⠁⠀⠀⠀⡖⢈⡇
⠉⠉⢉⣠⣾⠥⣤⡀⣰⢿⢉⡒⣒⣸⠅⠧⠚⠁⠀⠀⠀⠀⠀⠀⠀⣏⣀⡄
⠀⠀⠉⠁⠀⣀⠀⣿⠇⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣿⣶⣶⣤⣤⣤⣄⣀
⢀⡴⠊⠀⣴⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠁
⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⢀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⠶⠤⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠠⠚⠛⠉⡟⠛⠻⠿⠿⣿⣿⣿⣷⣶⣦⣤⣤⣸⠀⠀⣧⣤⣶⣿⡿⠟⠛⠋⠉⠀⢸⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠙⠼⡿⠟⠉⠉⠙⠛⣻⠀⠀⢻⠻⣿⠧⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⣼⡇⠀⠀⠀⣀⣀⣀⡀⠀⠈⠛⠦⣤⣀⣀⣀⣀⣤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠸⣟⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⡏⠉⠉⠛⠓⣶⠒⠒⠒⠒⢲⡟⠋⠉⠉⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⢠⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣷⠞⢧⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠧⣄⣀⠀⠀⠀⢻⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠌
⠀⠀⠀⠀⠄⠀⡤⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠈⠀⠰⠀⡳⡀⠈⠁⠀⠔⠁⠀⠀⠀⡀
⠀⢀⠀⠀⣀⢀⣀⣁⣈⣶⣔⣁⡀⣄⡀⣀⡠⠃
⠀⠀⠑⠀⠈⠂⠈⡄⡰⠉⠁⠄⡰⠁⡔⡝
⠀⠀⠀⠠⠀⠈⣀⠋⠑⢢⠖⠚⠃⣸
⠀⠀⠀⠀⣀⠀⠀⠀⠀⢲⠖⠉⠉⢉⡦⠓⠂⠀⠀⠤⠤⠤⠤⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⢐⡴⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠀⠂⠀⠀⠀⠀⠤⠤⠤⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡐⠁⠀⢀⣴⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⢇⠂⠂⡰⠒⢒⠓⠒⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢂⠀⠀⠙⢿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠁⣼⠀⡰⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠋⠀⠉⢢⡇⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠖⠁⠀⠀⠑⢄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠐⡠⠐⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣶⣾⠿⠷⠶⠦⢤⣶⣴⣶⣾⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⡿⠛⠻⣷⡄⠀⢀⣬⣿⡿⠿⣿⣿⡆⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⣿⡽⠁⠀⠀⢹⡯⠀⠘⠛⠏⠀⠀⠀⠻⣿⡀⢻⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⣾⠟⢋⣁⣀⡀⠀⠈⠉⠀⠀⠀⠀⣀⣤⢤⣤⣽⣷⣌⢳⡜⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢷⠈⣡⠞⠋⢹⣿⣿⡄⠀⠀⠀⠀⢠⠞⠁⠀⠘⠿⠿⢳⡌⠀⢠⡟⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠄⠉⣼⣃⣤⡤⢤⣭⣉⡇⠀⠀⠀⢠⣯⡤⠶⢶⡶⠦⢤⣀⣳⠀⠘⠃⢿⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⡿⠀⠰⠟⠉⠶⠒⠒⠀⢈⠁⠀⠀⠀⠈⠁⠀⠈⠉⠉⠛⠂⠈⠙⠁⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣇⠀⠐⠶⠤⣤⡤⠴⠚⠉⠀⠀⠀⠀⠀⠛⠦⣤⣤⣤⣤⠴⠞⠃⠀⠀⢀⡾⡀⢿
⠀⠀⠀⠀⠀⠀⠀⣿⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⡏⣧⠘⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠿⢦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣇⢿⠀⠰⠚⠛⣶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⢴⣖⠚⢉⡿⠙⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢸⡄⠀⠀⠀⠹⡾⠃⢈⣧⣤⡿⣤⡤⣾⠀⠀⠀⠀⠈⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡜⡇⠀⠀⠀⠈⠁⠀⢼⠁⠀⢰⠏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠓⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠶⠚⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠃

⠀⠀⢸⣿⡿⠿⣻⣿⣆⠹⠿⠿⠟⣿⡄⢻⣸⣿⣿⡿⠗⣉⣠⣤⣴⣦⣤⣌⣙⢋⣡⣤⣶⣤⣤⣌⡉⠾⠿⣿⣿⢣⡟⣨⣟⠿⠿⠿⠏⣰⣿⣟⠿⣿⣿⡇
⠀⠀⢸⣿⣭⣳⣤⣼⣿⣆⠀⠀⢠⣿⣿⣄⠻⠉⣣⠄⠈⠁⠀⠛⠻⠏⠻⣿⣿⣿⣿⠉⡿⠟⠃⠀⠉⠀⢦⡉⠩⢋⣴⣿⣿⠀⠀⠀⣸⣿⣧⣤⣔⣬⣷⡇
⠀⠀⢰⣶⣬⡻⢿⣿⡿⠿⠗⢀⣙⡉⠉⢁⣤⣾⡿⠀⠀⠀⣰⣤⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⣤⣂⣀⠀⠈⣿⣯⣄⡉⠉⣩⣁⡀⠺⠿⢿⣯⡾⢏⣥⣴⡆
⠀⠀⢸⣿⣿⣿⣾⣿⣿⠆⠀⠉⢁⣠⣾⡟⠉⣙⣷⣀⢠⣼⢿⡿⠀⢀⣿⣿⣿⣿⣿⣧⠀⢰⣿⢿⣤⣀⣰⣟⣉⠉⢻⣶⣄⠉⠁⠀⢰⣿⣿⣷⣿⣿⣿⡇
⠀⠀⢸⣟⢛⣉⡛⠻⠿⠦⠀⠀⢨⠽⠛⠁⠈⣩⣯⣻⣿⣿⣧⣿⣦⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⣿⣿⣯⣟⣉⠁⠚⠻⢯⠁⠀⠀⠶⢿⠟⣋⣉⡙⣻⡇
⠀⠀⢸⡷⠟⠛⠻⠷⠀⣄⠀⡀⠐⠀⢀⣴⣷⣾⣿⣿⡿⠛⠛⠋⠛⠛⠿⠿⠿⠿⠿⠿⠛⠋⠙⠛⠻⣿⣿⣿⣶⣷⣦⠀⠀⠂⠀⢀⣠⠀⠾⠛⠛⠻⢮⡃
⠀⠀⢠⣶⣿⣿⣿⣦⣶⡌⠁⣀⡴⢿⣾⠟⣩⣽⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣯⡙⢻⣷⡾⣦⡀⠘⣡⣶⣴⣿⣿⣷⣦⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢠⡋⠀⢸⠏⢀⣠⣏⠃⠀⣠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠈⣯⣅⠈⠻⡖⠈⢻⡐⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⡿⣛⢛⣛⠛⣿⡇⠉⠁⠀⣀⣄⣺⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣇⣦⠀⠀⠀⠁⢻⣟⢛⣛⣛⣛⣿⡇
⠀⠀⢸⣿⣿⠿⠿⠓⣈⣤⡶⠀⣰⣿⣿⡉⢹⡏⠀⣠⣾⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣶⣄⠀⣿⣏⣙⣿⣿⡄⠂⣶⢨⣀⠛⠿⠿⣿⡿⡇
⠀⠀⢨⣿⠃⠀⣠⣾⣿⡿⣧⠈⢙⣛⢈⠘⢻⣇⠘⠡⠤⡀⠀⠤⣱⡀⠀⠀⠀⠀⠀⠀⣠⣩⠤⢀⡠⠤⠉⢂⣿⡯⣨⣙⣿⠃⠀⣾⢸⣿⣷⣄⠀⠘⣷⡅
⠀⠀⢸⡏⠀⣼⡿⠛⠉⠥⠹⢇⣠⡿⣟⢻⣿⣿⠀⠀⣤⣭⣆⠐⢾⣷⡄⠂⠀⠀⠂⣰⣿⢷⠒⣮⢤⣄⠀⢸⣿⣿⡏⣿⣿⣄⠻⠏⠌⠉⠻⢿⣆⠀⢻⡇
⠀⠀⢸⠃⣼⡿⢰⣿⣿⡟⠀⠨⡟⣿⣝⣻⣿⡏⢣⣿⣡⠟⣿⣿⣶⣿⡗⠀⠀⠀⠘⡿⢷⣾⡿⠟⣆⣽⣦⠋⣻⣿⣿⣿⡟⢩⠁⡀⣿⣿⣷⡌⢿⣆⢸⡇
⠀⠀⠈⣶⣿⠁⢺⣿⣿⣷⣸⠷⣿⣷⣿⣿⡿⠃⣠⣻⠦⡤⣤⣾⡟⠋⠀⣠⣤⣤⣄⠈⠻⣷⣦⡤⠤⢾⣣⡄⠻⣿⣿⣿⣿⠷⠖⡧⢿⣿⣿⡇⠘⣿⣼⠃
⠀⠀⠀⣿⠇⢠⣿⣿⠙⣯⡻⡒⣾⣿⣿⣿⠇⠀⠉⣩⡴⠚⠋⠁⢀⣤⣾⣿⣿⣿⣿⣦⣤⠈⠉⠛⢲⣬⡉⠀⠀⣿⣿⣿⣿⣖⣲⣿⡿⠉⣿⣯⡀⠻⣿
⠀⠀⠀⠃⢠⡿⣿⢿⡄⠘⣷⣻⣿⣿⣿⣿⠀⠀⡼⢳⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⢷⢷⡀⠀⢹⣿⣿⣿⣿⣣⡿⠁⢰⣿⣿⣷⠀⠸
⠀⠀⠀⠀⣴⡷⠾⡎⢳⣄⡈⣩⣾⣟⣿⣿⡇⢸⢳⡇⣀⡤⠂⠀⠀⠙⠛⠿⣿⣿⠿⠛⠁⠀⠈⠢⣄⡘⣎⣧⠀⣾⣿⢿⣿⣭⡝⣥⣀⡟⢨⠿⣿⡆
⠀⠀⠀⣶⣟⣠⣾⠇⢀⣿⣿⣿⣿⣿⣿⣿⡇⢹⡾⢋⠁⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡉⠻⢹⠀⣿⣿⣿⣿⣿⣯⣿⣿⡀⠸⣷⣄⣰⣦
⠀⠀⢀⢻⡿⣙⡇⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⠘⠀⠘⠿⣷⣶⣶⠺⠛⠛⠛⠛⠛⠛⠛⠓⢶⣶⣶⢿⠃⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡠⠹⣋⣹⠏
⠀⠀⢸⡇⠁⣡⣴⣿⣿⣿⣿⡿⣾⣻⣿⣿⣿⣇⠀⠀⠀⠈⢿⡻⣦⣀⡄⣦⢠⡀⣦⣔⣶⣿⣿⠇⠀⠀⠀⢠⣿⣿⣿⣯⣩⣿⢿⣿⣿⣿⣿⣦⢅⠈⢰⡇
⠀⠀⠈⢠⢠⣭⣟⣟⡻⣿⡿⠺⠶⡿⠿⣿⣿⣿⣦⡀⠀⠀⠀⠉⠻⠯⣭⣭⣭⣥⣭⡽⠗⠋⠁⠀⠀⠀⣠⣿⣿⣿⣿⠿⢿⠓⠏⢹⣿⢿⣿⣿⢦⣠⡄⠁
⠀⠀⠀⠸⣿⣻⣿⣿⣿⠘⣷⣷⣾⣿⣶⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣶⣾⣷⣾⣿⡧⣿⣿⣿⣿⣿⠇
⠀⠀⢸⣆⢙⢧⡝⠻⣯⣼⣿⡏⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣹⣿⣧⠻⠟⣫⠜⠋⣰⡇
⠀⠀⠸⣿⣿⣧⣿⣷⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢠⣿⣿⠇
⠀⠀⢀⣬⣽⣿⣿⢟⣿⠿⢋⣼⣿⣿⣿⣿⢾⠿⣿⣿⣿⣿⡇⠈⣿⡓⢦⣤⣤⣤⣤⠖⣫⡟⠀⣸⣿⣿⣿⣿⢟⡾⠿⣿⣿⣿⣧⣙⠻⣿⡿⣧⣿⣭⣤⡄
⠀⠀⢸⡿⠛⠍⠁⣴⢵⣟⣻⣿⣿⡟⣰⣿⣷⣷⡀⢹⣿⣿⣷⠀⠘⠇⠀⠈⠙⠋⠀⠀⠙⠀⠠⣿⣿⣿⠉⢀⣽⣿⣷⡜⢿⣿⣿⣟⣻⠮⣤⠈⠙⠛⣿⡇
⠀⠀⢸⣄⣀⠀⢰⣿⣟⠻⠿⠿⠋⣤⢿⣿⡿⠋⣿⣄⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣴⡏⠻⣿⣿⣿⡀⠛⠿⠿⠟⠻⣿⠄⠀⣀⣀⡄
⠀⠀⢘⣉⠀⠀⠈⡻⣷⠟⣓⢚⣥⣿⣿⣿⡗⣧⣹⣇⣾⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⣿⣑⡾⢿⣿⣿⣷⣬⣓⠛⣷⣿⡛⠀⠀⠀⠉⡃
⠀⠀⠸⠶⠖⠛⠊⡙⣿⣟⡉⠉⠿⢿⣿⡟⠹⡿⡿⢿⢧⣄⡘⣆⠀⠀⠀⠠⣀⣀⠀⡀⠀⢀⡴⢀⣤⣾⠷⢿⢿⠏⢹⣿⣿⠏⠉⠉⣻⣷⡘⠒⠚⠓⠶⠆

⠀⠀⠀⠀⣰⣿⣶⣷⣾⣶⣷⣾⣷⣿⣾⣷⠋⠘⢷⣿⣶⣷⣾⣶⣷⣾⣷⣿⣾⣷⣿⣾⣷⣾⣶⣷⣾⣶⠷⠊⠙⣿⣦
⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⠀⠀⠀⣿⣧
⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠂⠄⡀⠀⡀⠤⠐⠂⠉⠀⠀⠀⠀⠀⠀⡐⣿⡷
⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣖⣿⡿⠛⠒⠄⣆⠐⠚⠛⢿⣟⣦⡀⠀⠀⠀⠐⠀⣿⡟
⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠸⠻⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⣿⣟
⠀⠀⠀⠀⣽⣿⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⡶⠖⠁⠀⠘⠲⠶⠿⠿⠿⠛⠀⠀⠀⠀⠀⣿⣯
⠀⠀⠀⠀⢼⣿⠀⡠⠊⠁⠑⢤⠀⠈⢑⢦⠀⠀⠀⠀⢀⠔⠲⠉⠀⢠⢉⡒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⢾⣿⡜⢄⣸⢀⣴⣶⣦⣄⠰⠄⠡⠀⠀⠠⠃⠀⢛⣠⣤⣮⡁⠀⠇⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷
⠀⠀⠀⠀⢻⣿⠐⠁⠈⣿⣿⣟⣿⣿⠀⠀⡀⠂⠀⠁⢄⠀⣾⣿⡿⣿⣿⡎⠓⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⣹⣿⡄⢀⡀⡝⠿⣿⡿⠟⠐⠓⠀⠀⠀⠄⢲⠁⠹⣿⣿⣿⡿⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟
⠀⠀⠀⠀⣽⣿⠈⢄⠀⠘⡄⠀⠘⠒⢒⠄⠁⢀⠀⠈⢜⠀⢘⠖⠉⠡⠊⠹⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯
⠀⠀⠀⠀⢼⣿⠀⠀⠈⠐⠓⠀⠒⠂⠁⠀⠀⡏⠑⢄⣀⠁⠊⠤⠤⠧⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⡇⣿⣧
⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠓⠢⣄⠀⠀⢀⡠⠔⠒⠒⠒⠛⠉⠁⠀⢀⠇⣿⡷
⠀⠀⢠⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⢀⡤⣼⡆⢀⣾⡤⣀⠀⠀⠀⠀⠀⠀⣠⠎⠀⣿⣟
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⡿⠦⣈⡑⠢⠤⣄⣿⡟⠉⠃⠘⠉⢻⣿⣆⣀⠤⠤⠒⠉⠀⠀⠀⣿⣟
⠀⠀⠛⢿⣿⣿⠀⠀⠀⢀⡠⠴⠚⠉⠁⠀⠀⣷⠀⠀⠉⠓⠢⢤⣈⠁⠐⠃⠘⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯
⠀⠀⠀⠘⠿⡟⠀⢀⠔⠉⠀⠀⠀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷
⠠⠀⠠⠐⠢⠖⠂⠏⠀⠀⠀⢠⣿⣼⣿⡆⢰⠸⠀⣰⣿⣭⣧⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷
⢰⣤⠀⠁⠀⠀⢸⡆⠀⠀⠀⠈⠻⡿⠿⠃⡼⠀⡇⠹⢿⣿⠟⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟
⠀⠙⢷⣼⣆⡀⢸⣏⣢⣄⣀⠀⠀⠀⢀⣰⣃⣀⣸⣄⠀⠀⠀⠀⢀⣠⣞⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣏
⠀⠀⠀⠿⠿⠿⠿⠟⠿⠛⠿⠻⠟⠿⠻⠟⠿⠻⠟⠿⠻⠖⠾⠛⠿⠻⠟⠿⠻⠟⠿⠻⠟⠿⠻⠟⠿⠟⠿⠻⠟⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠒⠒⠒⠒⠒⠒⠒⠒⠶⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠐⠒⠀⠒⢀⣀⣀⣀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠒⠊⠁⠀⠀⠀⠈⡁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠉⠉⠉⠙⢂⡒⠒⠂⠉⠉⠀⠉⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠤⠤⢤⡀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠁⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⣀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣘⣿⣿⡿⣿⣟⣉⣙⡶⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣭⡭⠤⢹⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠭⠭⠭⠉⠓⠦⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣀⣴⠒⠃⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠁⠙⢦⣤⠀⠰⠖⠀⠀⣤⣰⠏⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠋⢰⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠦⠤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⢻⡟⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⠋⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠈⢧⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠓⠶⠦
⠘⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣀⠀⠀⠀⣀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡖⠀⠰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠈⠉⠙⠻⢿⣿⣿⡇⠀⠀⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠈⠉⠉⠛⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣼⣿⡿⠛⠁⢀⣠⣤⡶⠶⠶⠶⠶⠶⠶⢶⣤⣄⠉⠻⠇⠀⠀⠿⠋⢀⣤⣴⠶⠶⠶⠶⠶⠶⠶⢶⣤⣄⡀⠙⠻⣿⣿⡄
⠀⠀⠀⠀⣼⡿⠋⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⠀⠈⠻⣿⡄
⠀⠀⠀⠀⠈⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⠛⠷⣾⣧⠀⢠⣿⡶⠟⠃⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⣼⣯⡶⠞⢁⣠⣤⣶⣶⠿⠿⠛⠛⠛⢻⣿⣿⣿⣷⣦⣹⣷⣿⣡⣶⠿⠟⠛⠛⠛⠛⠛⠻⠿⢷⣶⣦⣤⣀⠛⠷⣦⣿⡄
⠀⠀⠀⢀⣴⠟⢉⣴⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⠙⣿⣿⣿⡟⠁⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣤⡙⢷⣄
⠀⠀⠀⣾⠃⢰⣿⠋⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣿⣿⣿⣁⢀⣀⣿⣿⣿⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠈⢻⣷⠀⢻⡆
⠀⠀⠀⢻⣦⣸⣿⣄⣀⣀⣀⣤⣶⠿⢛⣋⣭⣭⡭⠉⠉⠙⠛⡛⠛⢛⣿⣟⠛⠛⡛⠛⠉⠉⠉⢭⣭⣭⣙⡻⠿⣶⣄⣀⣀⣠⣼⣿⣤⣾⠃
⠀⠀⠀⠀⠈⣿⣍⠙⠛⠛⠛⣩⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⣿⠁⢹⡟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣤⡉⠛⠛⠛⢉⣽⠏
⠀⠀⠀⠀⠀⠈⠛⠿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠃⠀⠀⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣤⣤⣴⡶⠾⠛⣩⠾⠃⠀⠀⠀⠀⠙⢮⡙⠻⠶⣶⣤⣤⣤⣤⣴⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⢀⣰⣦⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣤⡀⠀⠀⢠⣄⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣿⣿⣷⣿⣿⣿⣿⣷⣄⣀⣴⣿⣿⣶⣶⣾⣿⣿⣦⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠻⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡋⠙⠻⣿⣧⣦⣄⢀⣴⠶⣦⡀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡈⠁⠀⠹⡿⠋⠀⠘⢿⠿⠛⠻⠿⠋⠁⣀⣼⡿⠋
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣶⣦⣤⣤⣄⣀⣀⣀⣠⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉


⠀⢀
⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠁⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⣀⣤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠲⢤⡀⠀⠀⠀⡼⠁⠀⠀⠀⢀⠀⠀⣠⠴⠚⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢻⡶⠖⠋⠁⠀⠀⠀⠀⠀⠓⡾⠁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⣤⣄⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢠⣤⣄⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢿⣿⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢻⣿⡿⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢳⡀⣠⠶⠒⠛⠓⠲⢴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡴⠖⠛⠓⠶⣄⡀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⣶⣦⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⣉⠉⠉⠉⠉⠉⠉⠉⣡⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣤⡏⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡀⠀⠴⣖⠢⢄⠀⠈⢣⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⣀⡠⠄⣀⢑⣃⠤⠶⠚⠛⣏⡠⠼⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠖⠀⠀⣜⠓⠄⡀⠀⢫⠀⠀⠀⡀⠤⠄⣀⠀⠀⠀⠙⠢⣠⠴⠚⠉⠉⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠸⣇⠀⠀⠿⣢⣚⣂⡴⠉⠂⠀⠉⠂⡑⡄⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠐⠶⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡞⡠⠗⠋⠉⠀⠀⠀⠈⡇⠉⠻⠃⠀⠀⠲⢱⠀⢀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠣⡡⢀⠀⠀⡠⡢⠃⠀⠰⠚⣓⠺⡄⠀⠀⠀⠀⠀⠠⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠂⠒⠊⠁⠀⠀⡴⢄⡀⢀⡼⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠁⢸⣀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⠈
⢠⡋⠳⡀⡎⠑⡄⠀⠀⠀⠀⠉⡇⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⢀⠜⠈⢮⡁⠀⠆⠀⠀⢸⡀⠀⡀⠀⠀⣠⠀⠀⠀⣸⠓⠚
⣇⡈⢦⠧⠞⢠⡃⠀⠀⠀⠀⢀⡗⢤⢤⠄⠀⠈⠁⠒⠀⠒⠂⠁⠀⠀⠀⡧⠀⠀⠀⠀⠀⠉⣹⠳⠤⠔⠋⠢⢤⠔⠃
⢏⠈⣏⠀⠈⢻⡄⠀⠀⠀⡰⢻⠇⡄⠹⡧⣤⠴⡄⠀⢀⣤⡀⢀⣼⣒⣊⣀⣤⡤⠄⠒⠊⠉⢳⠀⠀⠀⠀⡔⠉⡹⢠⠖⢢
⠈⠳⠤⠤⠲⡁⠘⢆⢀⡜⢁⣼⠀⠉⢢⠈⠢⣠⠊⠹⠉⢀⣉⠩⠄⠒⢈⠁⠀⠀⠀⠀⠀⢰⠈⢇⠀⠀⠀⡗⠈⠧⢏⠔⠙
⠀⠀⠀⠀⠀⠙⣄⠈⠋⢀⠞⢸⡀⠀⠀⠁⠀⠈⠒⠊⢉⣁⡴⠒⠤⠔⠁⠀⠤⡇⠀⠀⠀⣸⡇⠘⡄⢀⡠⣏⠁⠀⡸⠊⢻
⠀⠀⠀⠀⠀⠀⠈⠳⠤⠋⠀⠀⠉⢹⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣧⣴⡖⠋⠁⢸⡀⠑⠉⢀⡼⠓⠤⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠣⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠴⠒⠒⠒⠒⠦⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣻⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡿⢡⣶⣶⡌⢿⣿⣿⣿⣿⣿⣿⡗⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡆⠀⠀⠉⠉⠉⠀⠈⠉⢠⣾⣿⣿⣷⡄⠉⠀⠀⠉⠉⠉⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠋⠙⢿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠹⣯⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⣽⠏⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢻⡀⢸⠀⠀⠀⠀⣀⡀⢀⣀⠀⡀⠀⠀⡇⢠⡟⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣇⠈⣷⠑⡏⢻⠀⠸⠇⠀⡟⢹⠊⣾⠁⣸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡄⠈⠒⠃⠴⠤⠤⠤⠤⠦⠘⠂⠁⣰⡟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠟⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡀⠱⣫⠽⠀⠀⠀⠀⠀⠀⠀⠀⠯⣝⡆⢀⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⠤⡎⣢⢤⢀⡀⡠⣔⢱⠤⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠈⠙⠾⡰⢄⠷⠋⠁⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡰⡺⢝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠇⢎⢇⡍⡲⢝⡢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡜⠨⠃⢀⠀⠀⠀⠉⠂⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠁⢀⡤⠚⠉⠉⠉⠛⠻⣦⡀⠀⣍⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢰⡞⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⡀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇⡄⠈⠢⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠆⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⢣⠇⡄⠀⠘⢦⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢦⡐⠹⣄⠀⠀⠀⠀⠀⠀⣠⠏⠂⣟⡞⣜⡴⡀⠀⠀⢳⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⠀⠢⣝⢦⣌⠙⠲⠶⠶⠶⢟⣁⣀⡀⠀⠈⠙⠿⠀⢀⠀⠀⠱⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠐⠬⣑⠮⠗⠀⠀⣴⡮⡷⠊⡛⣟⠓⡶⡤⠀⠀⠈⡗⢄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠀⠀⠙⢶⣏⠀⠀⡾⢻⡸⡀⠀⡸⢸⠀⡹⡁⠀⠀⠀⢸⠈⢧⠀⠆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠁⠀⠾⢵⡀⠑⠬⠩⠔⢃⠔⠑⠁⠀⠀⣠⠇⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⡄⠀⢸⢇⠀⠀⠀⠀⠊⠑⠒⠶⠟⠓⢁⡓⢖⡒⠋⠁⠀⣠⠋⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⠈⠢⣀⠀⠀⢀⡴⡫⣰⣠⠤⠒⢉⡀⠉⠉⠉⠉⠀⠀⢠⢏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢄⠈⢧⡀⠈⠉⠉⠁⡰⠃⢀⡴⢊⣩⣤⣬⣍⣀⣀⠤⠤⣒⡥⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠃⠀⠙⠲⠤⠖⠊⠀⠀⢋⡴⠋⠀⠈⠀⣀⡀⠉⠉⠉⠉⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⢤⣀⠀⠀⢀⡴⡠⠏⠀⣠⠖⠋⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⠭⣭⡵⠚⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣷⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⠈⠀⠛⢿⣿⡟⢿⣿⣿⣿⣿⣿⣿⠿⣿⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣲⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣀⠘⠛⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡏⠀⠈⣉⠍⠙⠻⡄⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⢠⠀⠈⣉⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⠿⠧⠈⠀⠀⠀⠀⠀⠀⠉⠭⣙⠻⣷⡀⠀⠀⠀⠸⢠⣀⣻⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⡿⠀⠁⠈⠃⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣼⢟⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣤⣭⣀⣐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢟⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣴⣿⣿⠿⠿⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡷⠛⠛⢿⡟⠁
⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠉⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡔
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠜
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣶⡀⠀⠀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢀⣾⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠒⠀⠀⠀⠀⢶⣶⣶⣶⣝⣿⣯⢳⡄⠀⠀⠀⠀⠀⣹⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣾⣿⡄⠀⠀⢠⣾⣿⣿⣿⣷⣶⣤⣤⣤⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⢛⣉⣭⣭⡍⠉⠁⠉⠉⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢠⠾⢛⣩⣴⣾⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡿⠟⣋⣩⣥⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⢿

⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀
⢸⡏⠉⠉⠉⠛⣶⣶⣤⣤⣤⣴⣶⣶⠛⠉⠉⠉⢹⠀⠀⠀⣠⠾⣋⡉⠀⠈⠉⠛⢦⡀⠀⠀⣀⣀⣀⡀
⠘⣇⠀⠀⠀⠀⢻⣿⣿⠟⢿⣿⣿⠏⠀⠀⠀⠀⡼⠀⠀⣰⠏⣾⠿⣿⡇⠀⠀⠀⠈⢿⣀⡾⠛⢉⣙⡻⣦
⠀⠘⢷⣤⣀⣀⣠⡾⠋⠀⠀⠻⣦⣀⣀⣀⣤⡾⠁⠀⠀⣿⠀⠙⠿⠟⠁⠀⠀⠀⠀⢸⡏⠀⠀⣿⣿⣼⠎
⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⢀⡾⢧⠀⠀⠉⠛⠋⢠
⠀⠀⣰⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⣀⣀⣠⣴⡟⠁⠈⠓⠦⢤⠤⠖⠋
⠀⠼⢋⣈⡻⣷⣄⠀⠀⠀⠀⠀⢀⡤⢂⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠉
⠀⣰⠟⠉⠙⢮⡻⣷⡄⠀⣴⣾⢋⡴⠋⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀
⢰⡏⠀⠀⠀⠀⠹⣮⡻⠀⢟⣵⠟⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢀⠶⠛⠉⠛⢷⡠⠛⠋⠛⠳⣄
⣸⠁⠀⠀⣠⣶⣶⣌⢿⣤⣾⣡⣶⣶⣄⠀⠀⠀⣷⠀⠀⢠⣳⣾⠃⠀⠀⢀⣴⣶⣥⣶⣄⠀⠀⠈⢻⣶⣂
⢸⡄⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇⠀⢀⡏⠀⠀⢤⣾⡇⠀⠀⠀⠘⢿⣿⢿⣿⡿⠀⠀⠀⠀⣿⣤
⠈⢧⡀⠈⠻⣿⣿⣿⡟⠀⢻⣿⣿⣿⠟⠁⠀⣼⠃⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⡰⠣⡀⠀⠀⠀⠀⢀⠇
⠀⠈⠳⣤⣀⣠⡴⠋⠀⠀⠀⠙⢦⣄⣀⣤⠞⠁⠀⠀⠀⠀⠀⠈⠢⠄⠠⠴⠊⠀⠀⠙⠢⠄⣀⠤⠊
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣁⣀⡀
⠀⣠⠞⠁⠀⠀⠈⠓⢄⠀⢀⠔⠋⠀⠀⠀⠉⢦⡀⠀⠀⠀⢀⣔⣦⣦⣤⣀⠀⠀⠀⠀⣀⢠⣠⣤⣄⡄
⢠⡏⠀⠀⠀⠀⠀⠀⠈⢳⠋⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣴⣚⠛⠛⠛⠿⢿⡿⠤⠀⣴⣾⣿⠿⠟⠛⢒⣦
⢸⣧⠀⠀⠀⠀⢠⣾⣿⣶⣿⣿⣦⠀⠀⠀⠀⢀⣿⠀⠀⢧⠉⢿⣷⣶⣦⣤⣽⣦⢈⡟⣉⣠⣤⣶⣾⠋⣸
⠈⢿⣆⠀⠀⠀⠘⢿⣿⠿⢿⡿⠏⠀⠀⠀⢀⣼⠇⠀⠀⠈⠳⣄⣉⣉⣉⣠⠴⠋⠘⢯⣉⠙⠛⠛⣁⡴⠃
⠀⠀⠙⢦⣀⠀⠀⠀⢀⡴⢄⡀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉
⠀⠀⠀⠀⠈⠉⠙⠉⠁⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⣀⣀⣤⣾⠟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣟⡋⠁⠀⠀⠹⣷⡄
⢠⣤⣤⣤⣀⣠⣤⣤⣴⣶⡿⠿⠛⣉⡤⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠻⣯⠉⣙⣉⣉⣉⣥⣤⣴⣶⣿⣯⣤⣤⣶⣶⣿⣿⣿⣟⡻⢿⣿⣿⣿⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠹⣷⡝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢋⣛⢷⣬⡛⢙⡛⢿⣿⣛⠻⠷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠈⢻⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣧⡸⣶⣿⣿⡇⢿⣟⡻⣿⣿⣿⣿⡇⠙⢿⣿⣶⣴⣿⣿⣿⣦⣀⣀⣀⠀⠀⢀⣠⡾⠟
⠀⠀⠀⠀⠀⠉⠻⣷⣭⣻⣿⣿⣿⣿⣿⠿⠶⣿⣿⣿⡿⠮⠽⠛⠛⢛⣯⢭⡓⠀⠸⣿⣿⣿⣿⢻⣿⣿⣿⠿⡋⠤⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⡷⢚⣶⠈⢿⣿⣿⠛⠇⠀⠀⣉⣁⣛⡀⢀⣠⡿⠏⣉⣵⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣿⣿⣾⣧⠛⣼⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⢰⣿⣿⣾⣿⣿⣿⣴⡄⠉⣿⣿⣭⣁⣈⣉⣿⡿⣿⣿⣿⣟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣌⣫⣵⣿⡿⣿⣿⣿⣷⡄⠘⣿⣿⣏⣙⣩⣥⣴⠿⠿⠿⣷⣬⣄⣀⡄⠀⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣻⣿⣿⣿⣿⠹⣿⣿⣿⣿⣆⡘⠿⠟⠛⠋⠁⠰⣶⣶⣾⣽⣻⣿⣿⣿⣶⣶⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠈⢿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣷⣼⣿⣿⣮⡙⠿⣿⣿⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠙⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣷⣾⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡀⠀⠀⠀⠘⠃⠀⢻⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⣽⣿⠿⠦⠍⠛⠿⠿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠀⠀⠳⣄⠀⠀⠀⡀⠀⢸⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣄⠀⠈⠢⣄⣀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢦⣄⣀⠉⠁⠀⠙⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠐⠂⠀⠈⠙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠀⠀⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠁⠀⠈⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢠⣿⣧⠀⠀⢻⠿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣘⣽⣿⣶⣶⣿⣦⣤⣤⣴⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⠉⠁⣀⣀⣀⣀⣉⠉⠉⠈⢹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⡄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣦⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄⠾⠟⠛⠛⠻⢿⠿⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣤⣤⣄⣀⣄⣠⡾⠋⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣯⠉⠋⠛⠙⠋⠉⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⠄⠀⠀⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣠⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⠀⠀⡇⠳⢴⡮⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⣷⡴⠋⡤⠀⢀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⣌⠑⢤⡇⠀⠀⣷⣄⡈⠿⠛⢲⣦⠤⢀⣀⡤⢤⣶⠚⠛⠿⣁⣴⡏⠀⠀⢥⠔⠉⣠⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠏⢙⣧⡀⠀⢀⡞⠋⠈⠻⣿⠿⠛⠁⠀⠈⠉⠀⠀⠈⠻⢿⡿⠛⢁⠛⢦⠀⠀⣠⣞⠉⢳⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⠀⡞⠉⢻⣆⡎⠀⠀⠢⠀⠤⠔⠀⡀⠄⠀⠀⠀⠀⣀⠐⠠⠄⠠⠌⠀⠈⣇⣼⠛⠙⣧⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢧⡀⠁⠀⠙⢦⡀⠀⣠⠴⠊⠁⠀⠘⠂⠀⠐⠂⠀⠉⠒⢤⡀⠀⣠⠜⠁⠀⠁⣠⠟⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣹⡆⠀⠀⠀⠉⠀⠁⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠤⣤⡄⠀⠰⠁⠀⠀⠀⢼⡃⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⣀⠔⡄⠀⠀⠀⠀⡇⠀⠖⠒⢢⠤⡤⢤⠖⠒⠂⠀⡇⠀⠀⠀⠀⡤⢄⡀⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠸⠀⠀⠀⠀⣿⣶⣶⣦⣼⣀⣇⣠⣦⣴⣶⣶⡇⠀⠀⠀⠐⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠀⠈⡏⠙⠿⠿⠟⠋⠟⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣄⠠⣇⠀⠀⠀⠀⢀⣷⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠂⠈⠑⠂⠤⠢⠤⠶⠌⠒⠉⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠈⠉⣟⠉⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⠀⠀⠀⠿⠀⠀⢀⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⡀⠀⣀⠴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢻⣿⡟⠟⢲⣟⣶⢦⣄⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⣹⣇⣀⣨⣿⣿⣷⣼⣿⣦⣀⣬⣿⣿⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⠀⡶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢶⣼⣤⣙⣿⡿⠿⠋⠉⠋⠉⠉⠉⠙⢫⡍⠛⢷⣿⣿⣁⡤⠀⠀⠀⠀⠀⣰⠋⣉⣤⣄⣀⠀⠀⠿⣦
⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⣀⠳⣾⣿⡿⠉⢠⠇⠀⠀⢠⣶⣦⡀⠀⠀⣳⠀⠀⠈⢻⣿⣦⠀⠀⠀⠀⢰⢇⡾⣉⣵⣤⣈⡢⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣯⠀⠀⠸⡆⠀⠀⠘⠿⠟⠁⠀⠀⡿⠀⠀⠀⠀⣿⣦⣄⠀⠀⠀⠘⡸⣴⣿⣿⣿⡋⣺⠀⠀⢸⠇
⠀⠀⠀⠀⠈⠙⠛⠿⢿⡿⠿⣿⣿⡟⠀⠀⠀⠙⣆⡀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣠⣿⡿⣿⠀⠀⠀⠀⠁⢻⠾⣍⣁⡽⠋⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢹⣿⣧⣤⣄⣀⣀⠀⠉⠛⠛⠒⠋⠁⠀⠀⣠⣶⣿⡿⠟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡁⠻⣿⣿⣿⢿⣿⣷⣶⣾⣶⣦⣴⣿⠟⠛⢛⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢺⣿⠐⡟⠀⠉⠿⠏⠉⣿⡏⠉⢁⣠⡔⢺⣟⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠛⠓⠒⠖⠀⠘⣿⡇⠀⠉⠁⠀⠀⠀⢹⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⠕⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠲⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣯⣶⣤⣤⣶⣶⣤⣤⣀⠁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠠⣄⡀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣏⡿⣿⡟⠛⣿⣿⡿⢿⣦⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣋⣤⣤⣤⣤⣀⣀⠙⠳⠦⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠁⢸⠟⠁⣿⡇⠀⠈⠋⣿⠀⠈⢿⡿⠂
⠀⠀⠀⠀⠀⢀⣤⣞⣥⠾⠉⠀⣿⣿⠁⠉⠛⢷⣯⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠘⢿⡆⢿⠀⣄⢀⣼⠇⠀⢀⣿⠁
⠀⠀⠀⠀⠀⣤⣿⣿⣿⣀⠀⠀⢹⡏⠀⠀⠀⢈⣽⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡈⠛⠺⠴⠶⠊⠁⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠸⣛⣿⠻⠿⠶⣦⣤⣡⣤⣤⣴⡾⢻⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠄⠀⠀⣠⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠶⢦⣀⣈⣉⣀⣤⠴⠒⠈⠀⠀⠀⢀⣤⠴⠖⠚⠓⠲⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⢠⠴⠒⠒⠂⢄⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢰⠏⠀⠀⣠⡄⠈⢳⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡥⠀⠀⠀⠸⣆⠀⠀⠙⠁⠀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡇⠀⠀⠀⠀⠙⠢⣄⣀⣠⠔⠃⣤⡇⠀⠀⠀⠀⠀⠐⣂⣨⣭⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⣤⠞⠉⢿⣿⠆⠙⡗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⠓⠛⠋⠀⠀⠀⠀⠈⠑⠲⠅⠀⣀⣠⡤⠶⠛⠁⠀⠀⠀⢠⢂⡞⠉⢦⡀⠀⠀⠀⡠⠁⠀⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠉⠁⠉⠁⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣀⠀⠀⠀⠀⠀⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣉⠛⠲⠲⠚⠛⢉⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢀⡠⠄⠶⠶⠒⠒⠒⠒⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠴⠚⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⢀⠤⠐⠒⠀⠉⠉⠉⠁⠒⠠⡀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⡏⢀⠔⠂⠐⠢⡀⡠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⡆⠀⠀⠆⠀⠀⠀⠘⢶⣀⡀
⠀⠀⠀⠀⠀⠀⣇⡃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢠⣾⣷⠀⠇⠀⠀⢸⠀⠀⠀⠀⠈⠙⠤⣄⡀
⠀⠀⠀⠀⡤⠀⠁⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡡⠔⠁⠀⠘⠃⠀⠐⣒⢆⠠⠀⡰⠊⠉⠉⠉⠻⡅
⠀⠀⠀⠀⠱⣀⠄⡀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣨⠒⢻⠀⠀⠀⠀⠀⠀⠀⠜⠰⠈⠀⠀⡇⠀⠀⢀⡀⠀⢳
⠀⠀⠀⠀⠀⠉⠀⠸⠈⢷⣒⣶⠆⠀⠖⢧⣿⠿⠀⠈⠀⠀⠀⠀⠀⡠⠊⠀⡄⠀⠀⡜⠃⠀⠀⠟⠉⠀⠞
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠁⠃⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢠⠀⠀⢐⠇⠀⠀⠀⡴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⡤⠄⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠈⢲⡧⡀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠑⠵⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠊⠀⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠁⢰⠂⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠔⢺⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢤⣀⣿⣿⣿⣏⣀⣀⣀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣞⣛⣛⠋⠉⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠛⠛⠛⠛⠛⠛⠻⠿⠏⠉⠓⢦⣄⡀⠹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⢠⡾⠋⠀⣀⡴⠚⠋⠉⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⣦⠀⠾⢻⣾⡄
⠀⠀⣿⠋⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠈⢹⣆⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠉⠹⣆⠀⠀⠙⢿⡄
⠀⣼⣧⠀⠀⣸⣷⣶⣶⣶⡶⠶⠶⠶⠶⠶⣿⠀⠀⠀⢀⣼⣯⣤⣤⡤⠶⠶⠶⠶⠾⣧⠀⠀⠘⣷⣄
⡇⢹⡏⠀⠀⢹⡟⠛⠛⠋⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠉⠉⠁⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢹⣿
⡇⡟⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢻⣴⡄
⣿⡇⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠷⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⢸⡘⡆⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⠷⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠘⣇⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⢿⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠘⣇⠈⠙⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠛⠋⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠙⠳⣦⡀⠀⣠⠖⠋⠉⠉⠉⠉⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠉⠉⢿⣿⡿⠉⠉⠉⠉⠉⢹⣿⠀⠀⠻⠿⠋⠉⠉⠉⠉⠀⢒⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣄⣀⣀⣀⣤⡾⠋⠀⠀⠙⠻⣦⣤⣀⣀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⣧⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠶⠉⠉⡍⠀⠈⢾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣀⡤⠄⠀⢀⠞⠀⠀⠐⠒⠦⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠊⢁⣤⣶⣷⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣶⠟⠋⠉⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡈⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⢁⣴⣿⣾⣆
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢀⠴⠟⠛⠟⠛⠻⢦
⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠐⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⢤
⠀⠀⠀⢹⠇⠀⠀⠀⠀⠙⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢸⠋⠉⣿⣦⡄⠀⣀⣠⣴⣶⡀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠂⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣠⣾⣿⣿⣿⠖⠤⠊⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢰⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⢩⡿⠉⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠙⢦⢀⠀⠀⠈⠀⠱⣄⠳⢽⣿⣧⡿⠟⠉⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢀⠀⠀⠀⠀⢀⠏⠀⠠⠄⠀⢀⡀⠈⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠁⠠⠄⠀⠈⠀⠀⠀⠸⠔⠁⢀⠤⠀⠀⠀⢀⡴⠧⠶⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣍⣀⣤⠤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⠛⠛⠛⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⢤⣀⠀⢀⡴⠚⠉⠛⢷⡦⣄
⠀⠀⠀⣠⠏⣠⠴⠒⠒⠒⠲⢤⣀⠀⠀⠀⠀⠀⠘⠈⠉⠉⠀⠀⠀⠀⠀⡳⡌⢿⠲⡟⠛⠛⠲⣄⣀
⠀⠀⣰⣣⠎⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡌⢇⠘⠄⠀⠀⢸⠉⠙⢦⡀
⠀⢀⣿⠋⠉⣆⣀⣤⣤⡀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠸⠀⠀⠀⠀⠸⠀⠀⠀⠙⢦
⠀⠸⢇⠀⢀⣿⣿⣿⣿⣿⣆⢀⠠⣤⢀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠁⢿⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢳⣄
⠀⡇⠀⠉⣹⣿⣿⣿⣿⣿⣿⠀⠀⠈⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠈⠀⡞⠀⠀⢸⠇⢠⠃⢀⠀⠀⡰⠀⠀⠀⠡⠀⡌⢧
⠀⡇⠀⠀⢿⣿⣿⣿⣿⠿⣿⠀⠀⠀⠀⡀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⣾⢀⣮⡤⣤⡤⠚⠙⠢⡤⠀⠀⠀⠃⠸⡆
⠀⢳⠀⠀⠘⣿⣿⣿⣷⣶⢃⣠⣄⠀⡄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣧⠞⠁⠀⠀⠀⠀⠀⠀⢳⠀⢸⡆⠀⢀⡇
⠀⠘⣧⠀⠀⠈⠉⠛⠉⠀⠀⠟⢛⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠃⡠⣸⠁
⠀⠀⠘⢗⠮⣀⡀⠀⢀⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⢡⠏
⠀⠀⠀⠈⠳⣄⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡴⠃⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⡠⠁⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠂⠀⠀⠀⠀⠀⡠⠦⠤⢤⡀⢀⣠⠶⠋⣠⠊⠉⠀⠉⠙⠒⠶⠶⡚⠩⠁⠀⢔⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣄⣀⣀⣀⣤⡤⠶⠟⠋⠁⠀⢰⠁⣠⠾⠶⠤⣄⣂⣀⠤⠤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠭⠷⠞⠁

⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⣤⣀⣠⠤⠀⠀⠒⠒⠀⠐⠒⠂⠨⠍⠉⠩⠭⠍⠉⠙⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡘⠃⠀⠀⠀⠓⠀⠛⠛⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⢠⣶⣶⣶⡿⠛⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⣿⣿⣟⡛⠛⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⠀⠀⠙⣿⣿⡿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣆⡀⠀⢀⣾⣿⣿⣿⣿⡀⠀⠀⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⠃⠀⠀⠘⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⢰⣿⣿⡿⠀⠀⠀⠀⢠⣤⡀⠒⠄⠀⠀⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠂⠀⠀⠈⢻⣿⣤⠀⠃⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣀⣀⣀⣀⣀⠀⢀⣀⣀⣹⣿⡆⢰⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⢿⢇⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢤⣀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠐⠒⠀⠀⠤⠐⠂⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠈⣇⠀⠀⠀⠀⠀⠀⡟⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⢻⠀⠀⠀⠀⠀⢸⠃⠀⠀⣸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⢀⣶⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⣄⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⢀⣠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣿⡇⢠⣿⡆⠀⣴⣶⣶⣆⠀⣰⣿⡀⣾⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣼⣿⣷⢰⣿⣿⣿⣿⣀⣿⣿⣧⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠿⢿⠿⠿⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠤⣄⣀⣀⣀⣀⣠⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠶⠶⢶⣦⣄⠈⠻⣿⣶⠶⠀⠀⠀⠀⣀⣴⡿⠛⣉⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠘⡏⠁⠀⠀⠀⠀⠀⣾⣿⣿⣷⡀⠀⠀⠀⠀⠰⢶⣾⡿⠋⣠⣾⣿⣿⡀⠀⠀⠀⠈⠉⡱
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢦⡀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠈⣿⣿⣿⣿⣧⣠⠏⠀⠀⠀⠀⠸⣧⣠⣿⣿⣿⣿⡇⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠈⠻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠏⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⢄⠠⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⡤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣄⠀⠉⠓⠲⠤⢤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠚⠉⠀⢠⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠉⠙⠒⠒⠒⠒⠒⠒⠂⠉⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⡀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⠿⠿⠿⠿⠿⠷⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⢤⡀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣼⣿⠃
⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⣠⣾⣿⠇
⠀⢹⣿⣿⣦⡀⠀⠀⢠⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠠⣾⣿⣿⣿⡏
⠀⠀⢹⣿⣿⣿⣧⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠉⠉⠀⠀⢀⣿⡏
⠀⠀⠀⢹⣯⠙⠋⠉⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢨⣿⣧
⠀⠀⠀⠀⠹⣷⡀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠠⣤⡼⠏⠉⣿⡇
⠀⠀⠀⠀⠀⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⢠⡄⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣰⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣿⣇⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⢰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣄⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣠
⠀⠀⠀⠀⣠⣤⣴⠿⠃⠀⢿⣿⡇⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠟⠋⠀⢀⣀⣀⣶⣤⣿⣧⣤⡀⠀⠀⠀⠀⠘⣿⠆⠀⠀⢰⣿⡇⠀⠀⢀⣿
⠀⠀⠀⠘⣿⣄⣀⣄⠀⠀⠈⢁⣿⡆⠀⠀⢀⣴⡿⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣿⣿⣡⡿⣿⣿⣧⡀⠀⢀⣸⡏⠀⠀⠀⢸⣿⡇⢀⣴⡟⠋
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡷⢿⣁⠈⢀⣴⣿⠋⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠿⠛⠻⣿⣿⣶⠿⣿⠀⢸⣿⡇⠀⠀⠀⣸⣿⣷⡿⠋
⠀⠀⠀⠀⠀⡸⠟⠉⠛⠛⢿⣿⣻⣷⡿⠟⠁⠀⠀⠀⠀⠀⢀⣸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⢿⣶⠿⡦⠈⠛⠁⠀⢠⡾⠛⠉⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣤⣇⠀⠸⠿⠀⠀⣼⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⠀⠀⠀⣶⣷⡄⠀⠀⠀⠻⣶⣶⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠹⣿⣷⣶⣤⣶⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠈⠉⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⢠⣶⠿⣿⡆⠀⠀⢸
⠀⠀⠀⠀⠀⡾⠋⣹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣶⣤⣄⣀⡀⠀⣀⣤⣾⠿⣇⠀⠀⠀⠀⠀⢠⣿⢇⠐⣿⣷⠀⠀⠈
⠀⠀⠀⠀⠀⢰⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⠀⠻⢿⡟
⠀⠀⠀⢀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠛⠿⠛⠛⠋⠉⠀⠀⢀⡀⠀⠀⢠⣄⠘⠟⢠⣤⡿⠃⠀⠀⣶
⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⣶⡾⣿⣿⣿⠆⠀⠈⢻⣆⠀⠀⠀⠀⠀⢀⣾⠏
⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡿⠏⠁⣶⠛⠿⠀⠀⠀⠀⠈⢿⣆⡀⠀⣀⣠⡿⠃
⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠉⠛⠁
⠿⣷⣶⣶⣤⣶⣦⣤⣶⣶⣶⣶⡶⠶⠶⠶⠟⠛⠛⠛⠛⠿⢷⣦⣀⣠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷
⠀⠀⠀⠉⠀⠀⢸⣿⠁⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⢹⣿⡆⠀⠀⠀⠀⣤⣤⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡿⠛⢁⣀⠀⢸⣿⠁⠀⠀⠀⣾⡿⠉⢻⣿
⠀⠀⠀⠀⠀⢠⣼⠛⢷⡾⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⣁⣤⣶⣿⡏⠀⣼⣿⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣶⣾⠟⠛⠁⢸⡇⢰⡇⠀⣤⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠘⢿⣦⣀⡀⠴⣶⣾⡿⠟⠛⢛⠛⠉⠁⠀⠀⢰⣦⢠⣿⠁⢸⡇⠀⣿⡄⠀⠀⣿⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⣴⡄⢸⣷⡄⠀⣿⣷⠀⣾⣷⡀⠘⣿⣿⡟⠀⣿⡇⠀⢿⠁⠀⠀⣿⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣤⣿⣷⣾⡿⣧⠀⠹⠋⣴⣿⢸⣿⣾⡟⢹⡇⠀⣿⡇⠀⢸⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣷⠙⣷⠀⠀⣿⣿⢸⣿⣿⣷⢸⡇⠀⣿⠀⠀⠈⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠉⢻⣦⠛⠀⠀⣿⣿⠀⣿⣿⣿⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠁⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣿⡄⠀⠀⢿⣿⣤⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⣼⢿⣿⣿⣿⣿⣿⡇⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⠀⣾⠉⢀⣸⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⣶⠀⣰⠟⠁⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⡿⠀⢹⣆⠀⠀⢿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠋⠀⣾⠀⠀⠸⣿⠀⠀⢸⣇⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⣤⠋⢀⣤⠀⢸⡇⠀⠸⣿⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣠⣴⣯⠀⣸⠛⣧⣸⡇⠀⠀⢻⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡁⣹⠋⣶⡏⢰⣿⡇⢠⣾⢷⣈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⣻⠟⣿⣿⠀⠙⠀⠀⠉⠀⠸⠟⢸⣿⢿⡇⣿⡇⣰⡀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⣿⣇⢈⣿⡶⠛⠃⠙⠛⠛⠠⣤⠈⠁⢸⡇⣿⡇⢿⡇⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠉⠉⠛⠛⠁⣀⡀⢠⣤⣶⣶⣤⣀⣈⡁⠿⠇⢸⣇⣼⠃⠀⠀⠀⠀⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⡶⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠈⠛⠻⣶⣄⠈⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣿⣧⡀⠀⠀⠀⠀⠀⣸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⣤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⣷⣤⣴⢿⡇⠀⠀⣀⣀⣤⣤⣴⠖⠒⠚⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣾⣿⣶⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣀⣿⣿⣋⣀⣤⣄⣤⣄⣠⣤⣤⣤⣤⣄⣠⣤⣀⣀⣀⣤⣤⣀⣠⣤⣀⣤⣀⣤⣀⣤⣄⣀⣀⣤⣀⣀⣤⣤⣤⣤⣀⣀⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣏⣉⣙⣛⣋⣙⡉⣉⣏⣉⣡⣆⣈⣹⣉⣿⠋⣉⠉⡉⢙⣿⣤⣿⣿⣿⣤⣤⣿⣤⣤⣤⣤⣿⣿⣿⣤⣧⣤⣤⣼⣧⣤⣤⣤⣤⣤⣭⣽⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠾⠛⢻⡟⠛⠛⠛⠲⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠙⠳⢦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣀⣠⣤⣤⣤⣤⣀⣀⠀⢠⡏⠉⠛⢦⣀
⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⠿⣧⡀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠈⠛⣦⡀⠀⠀⠙⣦⡀
⠀⠀⠀⣼⠃⠰⠶⣄⠀⠀⢀⣀⣤⣄⣀⣾⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣆⠀⠀⠈⢻⣤⣤⠾⠛⢧⡀
⠀⢠⡟⠻⣦⢀⣀⣸⣧⣴⠋⠉⣠⡟⢹⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⡀⠀⠀⠈⣷
⠀⣾⠁⠀⣿⠈⠉⠉⠉⠀⠀⠛⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠿⠿⣿⣿⠃⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⣿⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠀⠈⣿⠀⠀⠀⠀⣼⠃
⠀⢿⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⢀⣹⣆⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⢀⣠⡟⠀⠀⠀⣰⡟
⠀⢸⡇⠀⢸⡿⢶⣤⣄⣀⣴⠟⠛⠋⢹⡏⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⢀⣴⢿⡄
⠀⠈⣷⠀⠈⣷⣿⠀⣿⠟⠁⠀⠀⠀⠼⠃⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁⢸⡿⢦⣄⣀⠀⣸
⠀⠀⣿⠀⠀⢹⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⣶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠘⢷⡀⠈⠙⢳⡟
⠀⠀⢻⡆⠀⠀⠀⠀⣧⠀⠀⣠⠞⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣷⠀⣠⡟
⠀⠀⢸⡇⠀⠀⠀⠀⠙⠳⠟⠁⠀⣰⠏⠀⠀⣀⡀⠀⣀⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢘⣷⠏
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⣤⠞⠋⢻⡾⠋⠀⠀⠀⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢠⡟⠀⠀⠀⠀⠀⢿⣄⣉⣽⣷⡦⣤⣀⣠⡴⠟⠁
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠛⠛⠉⠀⠀⠀⢸⡟⠉
⢠⡟⠀⠀⠀⠀⣀⣤⣤⣤⠶⠶⠦⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠟
⣸⠇⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉
⣿⠀⠀⣿⠁
⣿⠀⠀⣿
⢻⡄⠀⢻⡄
⠘⣧⠀⠈⢷⡀
⠀⠹⣧⠀⠈⢷⡀
⠀⠀⠙⣧⠀⠸⡇
⣀⣀⣀⣈⢳⣼⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤⠤⠤⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠈⠁⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠀⠀⠈⠑⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠁⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠄⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⡀⠠⠀⠀⠀⡀⠈⠠⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠌⠁⠠⠀⡀⠂⠀⠐⠢⡀⠂⠄⠈⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠠⠀⡈⠠⣼⣿⣾⡄⠈⡀⠀⠀⠀⠂⠀⢓
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⣀⠀⠀⢀⠀⠨⠀⠂⠀⢿⣿⣿⠇⠀⠂⠀⠀⠀⡁⠀⠀⢁
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⠀⢀⠀⠀⠀⠀⠈⣳⣤⣈⠀⠀⠠⡈⠂⢄⣀⢈⡤⠊⡀⠌⠀⠠⠀⠀⠁⠈⡆
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢧⣀⠀⠀⠀⠀⠈⠈⠀⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢽
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠣⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡝⠦⢶⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠰
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠥⠆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡷⠯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁

⣀⣀⣀⠀⡀⣀⣀⣀⠀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⠶⠶⠶⠾⠿⠿⡛⢛⠛⣿⠛⠛⢛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⡉⠉⠉⠉⢀⣈⣉⠉⠀⣀⡀⠀⠀⠠⠤⠤⠄⠠⠄⠀⢈⡑⠒⠤⣀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⣴⠟⠋⠀⠀⠀⢀⡀⠤⠤⠄⠠⢀⡀⠀⠀⠀⠀⠀⠀⢰⠒⠀⠀⠒⠂⠀⠬⢁⡒⠄⡁⠂⢀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⢸⡟⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠢⠀⠁⠀⠀⠀⠹⣷⡀
⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠟⢻⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⢠⣾⠟⣡⠔⠒⠒⠒⠂⠘⠿⠿⠿⠿⣿⣿⣿⣦⣤⡄⠀⠀⠀⢾⣟⣁⣤⣶⠾⠿⠟⠿⠿⠷⠿⠗⢀⠐⠂⠀⣀⣀⠐⠋⠛⢟⢷⣄
⣿⢣⠊⡴⢊⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠙⣿⠁⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠻⢷⣤⣀⣀⣠⣤⣶⠾⠛⡛⠛⠛⢷⣄⠀⠱⠹⣷
⣿⢀⠀⠃⠉⠉⠀⡄⠹⠿⠶⠿⠃⠀⠀⣠⣼⠟⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⢀⣠⣴⣿⣄⠀⠀⢻⡆⠀⠀⢸
⢿⣿⣄⠑⠀⠀⣼⣿⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⡉⠛⣷⠉⠂⠒⠒⠂⠀⣀⣠⣴⡾⠟⠋⠁⣼⡿⠛⠂⣸⡇⢀⠀⣸
⠈⢿⣆⠉⠂⢰⣿⣿⣷⣄⡀⠉⠀⠀⠉⠙⢷⣄⣠⡄⠀⠉⠉⠙⠃⠺⢋⣀⣤⣴⡶⠾⠟⠛⣻⡏⠀⣀⣤⣾⡿⠀⠀⠀⠛⣠⠎⣰⡿
⠀⠈⣿⠀⠀⢸⣿⡟⢸⡟⠻⣷⣶⣤⣤⣤⣤⣭⣭⣤⣤⣴⣶⠶⠾⠟⠛⣿⡉⠀⠀⢀⣀⣴⣿⣷⠿⢛⣿⠟⠀⠀⠀⠀⠀⣴⣾⠏
⠀⠀⣿⡇⠀⢸⣿⣷⣿⣇⣀⣿⡃⠀⠀⠉⣿⠋⠁⠀⢈⣿⠀⠀⠀⣀⣀⣽⣷⣶⣿⡿⠿⠛⣿⡇⣠⡾⠋⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣿⠉⠁⠀⠀⠀⣸⡿⠟⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⣿⡇⠀⠈⢿⡽⣿⠻⣿⠿⢿⣿⠿⠿⢿⣿⠛⠛⠛⠻⣿⠉⠀⠀⠀⠀⢻⣧⠀⣠⣴⠿⠋⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⣿⠀⠀⠀⠈⠻⣯⣤⣹⣧⠀⢿⣆⠀⠈⣿⠀⠀⠀⢀⣿⠀⠀⣀⣀⣤⣴⡿⠟⠋⠁⣀⠄⠀⣀⠤⢀⣠⡾⠋
⠀⠀⣿⠀⠀⠀⠀⢀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⢀⡠⠄⠒⣉⠤⠒⣉⣤⡶⠟⠋
⠀⠀⣿⠀⠀⢠⡀⠀⠙⠢⠄⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠠⠠⠐⣒⠉⠠⠐⠈⣁⣤⣶⠿⠛⠉
⠀⠐⣿⡀⠀⠀⠉⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠂⠈⠉⠀⢀⣠⣴⡶⠟⠋⠁
⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠾⠟⠋⠁
⠀⠀⠀⠀⠙⠻⢶⣦⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠿⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢄⣄⠀⠀⠀⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠻⡖⠤⣄⣀⠰⡜⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣒⡒⠛⠛⠛⠒⠧⠀⠀⠀⠀⠈⠙⠻⣌⠑⢄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⡫⢟⣛⣉⣩⠭⠠⠀⠀⠀⠀⠳⠤⠀⠀⠀⠀⠀⠀⠑⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⡿⢚⣉⠁⠀⢀⠤⢖⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⣫⠟⠉⢉⣩⣏⣥⣾⡿⣻⠋⠀⢀⠀⠀⠀⡆⠀⢰⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠩⢤⡶⠚⡟⠣⢬⠟⠋⢰⣁⠴⠛⠓⠦⣽⢰⠇⠀⠈⣇⠀⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣶⢚⠀⠀⠀⠈⠁⠞⠁⠀⠀⠀⠀⢨⡞⠀⡀⠀⢿⡆⣼⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠀⠀⣼⣤⠇⠀⢿⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣋⠥⠐⣃⣽⣿⣶⣄⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠀⡇⢘⠀⠀⣸⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠿⠿⠿⠿⠷⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⢠⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢀⡤⠒⡆⠀⠀⠀⠀⢳⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃⠐⡁⠀⠀⠀⠀⠀⠀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣳⠶⠦⠤⢄⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠉⠑⠒⠶⠒⠈⠀⠀⠀⣴⠀⡠⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠙⠂⠀⠀⠀⠀⠀⠀⡠⠄⠀⢀⡇⢠⠇⡘⠀⠀⠀⠀⠀⠀⡆⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠄⠀⣠⡴⠋⢀⠀⠀⢸⡇⢸⡆⠀⡀⠀⡆⠀⢀⢎⣇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠖⠁⠀⠀⠉⠀⠀⠹⠀⠀⠘⡷⠟⠀⠀⢹⡀⠃⠀⣬⠞⠹⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⢀⣤⣾⠿⠛⠉⠉⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡟⠓⠦
⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠉⢿⣿⣿⣿⡿⡇
⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣯⣄⠀⠀⠀⠀⠀⢻⣿⡿⠱
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⢿⣄⠀⠀⠀⣼⠟
⢿⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⣴⣿⠀⣀⡾⡋⠀⠀⠀⢀⣀⣀⣀⡀
⠘⠆⠀⣿⠁⠈⠙⢷⣦⣀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⣀⣴⡾⠋⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⢞⡩⢊⣠⡴⠶⠛⠛⠛⠛⠛⠿⣶⡀
⠀⠀⠀⠻⠄⡀⠀⣎⣹⣿⣿⣶⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⠿⢿⣥⢄⣀⣀⣀⣙⣿⣿⡿⠿⠛⠉⠒⠁⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠉⠉⢉⡩⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠤⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠻⣿⣿⣿⣿⣿⣿⡝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⢹⣿⣿⣿⣿⡏⢿⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡤⠶⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⣿⣿⣿⣿⣿⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠈⠉⠉⠉⠉⠉⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁⠀⠀⠈⢻⣿⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⢀⣠⡤⠴⠧⢤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠒⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠁
⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠶⠶⢶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠷⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣀⣀⣤⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣷⠀⠀⢠⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⢶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣴⣹⣧⣶⣿⠖⣫⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⣠⣤⣴⣞⡿⣿⣻⣦⢔⣀⣬⣽⡛⠳⢬⣉⣓⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡈⡻⣾⣿⣿⣿⠿⠟⣛⣛⡛⠿⣆⠹⡄⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣌⢮⣻⣟⣡⠜⠛⠳⢀⠏⣳⡌⠣⣷
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠹⣿⠇⠀⠀⠰⢋⡾⢿⡗⡄⡇
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⠀⠀⣀⡴⠋⢁⣿⡽⡿⠁
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠛⠉⠁⠀⣠⡾⢋⡼⠁
⠀⠀⠀⠀⠀⣸⠗⠒⠄⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠿⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠐⠋⠁⠐⠋
⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠁⣀⣀⠀⠀⠈⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠟⠿⣿⣆⠀⠀⠀⠇⠀⠀⠀⠀⠰⡄⢰⣿⡟⣠⠶⠋⠉⠉⠙⠳⢦⡀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣀⣀⢋⣔⠊⠙⠻⣆⠀⠀⠘⡄⠀⠀⠀⠀⢻⣿⣿⡼⣣⣖⣀⣈⣙⣒⣲⣤⣿⡄⠀⠈⠃⠀⠀⠀⠀⠀⠀⣿⣠⣤⣤⣤⣤⣄⡀
⠀⢰⠏⠉⠉⢿⣯⡉⢲⣶⣿⡷⠀⠀⠈⡆⠀⠀⠀⠈⣙⣽⡿⣿⣿⡷⠀⠀⠈⠉⢉⡩⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠉⠙⢷⡄
⠀⢻⡐⣏⠿⢮⠏⠛⠶⠤⢤⢗⡴⠒⠒⠦⡀⠀⠀⠀⠿⠿⣤⣤⣵⡒⠶⢒⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⠳⣌⠀⢸⡀⠀⠀⠀⠀⠘⣇⡀⢀⣠⠤⣀⠇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⠀⣿⣠⡶⢶⣦⣤⡀⠀⠀⡸⠁
⠀⠀⠀⠈⠛⠶⢧⠀⢀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⠿⢿⣿⠁⠀⠀⠀⠀⢠⡟⠁⠀⠞⠀⡸⠃⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⠻⢷⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⢸⣿⠀⠀⢠⠀⠀⢸⠇⠀⠀⠀⢀⣤⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⠻⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⡀⠀⠀⢸⡏⠀⠀⡏⠀⢀⡿⠶⢶⠾⢿⡟⠋⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠙⢦⣀⡀⠀⣸⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡇⠀⠀⢸⠇⠀⢸⠁⢠⡞⠀⠀⠘⠓⠛⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠈⢯⠉⠉⠙⣏⠉⠉⠉⠀⡽⠀⠀⠀⠀⢠⠁⠀⢀⡞⠀⠀⡾⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢀⡬⠶⠶⠲⠾⠶⠤⠤⠬⣅⣀⠀⠀⢠⠏⠀⢀⠎⠀⠀⣼⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠖⠋⠀⢀⣼⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠉⢲⡖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣼⣷⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠶⠶⠒⠛⠋⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢀⣠⣤⣤⡤⢤⣀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⣀⣤⣤⣐⠀⠛⠛⢛⡟⠓⠶⣬⠗⠀⠀⢶⡾⠿⠛⠛⠛⠀⠀⠀⡨⠻⣆
⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼⠁⣠⣦⣈⠙⠛⠛⠉⠀⠀⠀⣀⡀⠀⠀⠀⢧⣀⠀⠰⠶⠞⡉⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣇⢀⠻⠈⠙⣧⡈⠙⣷⠦⣤⣀⡀⠸⡍⠶⠶⠀⣀⣠⡿⠓⠀⠀⣠⣿⡄⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠙⢿⠓⣿⣦⣄⣈⢹⡟⠓⠲⢶⡤⠭⣭⡤⠴⣶⠛⣯⣻⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠳⣾⡀⠉⠛⣿⠷⣶⣶⣾⣷⣶⣾⣶⣶⣾⣶⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠈⠙⠦⣴⣇⠀⠀⠈⡏⠉⠉⡿⠛⣿⢋⡟⣹⡿⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⠀⠉⠉⠛⠚⠓⠒⠚⠓⠚⠛⠛⠋⢉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣤⣄⣀⡀⠀⠀⠀⠀⣀⣠⠞⠁

⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠈⠙⢿⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣻⠟⠉
⠀⠀⠈⢧⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡼⠃
⠀⠀⠀⠈⢧⠠⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⢀⡞
⠀⠀⠀⠀⠈⢧⠹⣦⡀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⡞⢀⡞
⠀⠀⠀⠀⠀⠈⣇⠹⣿⣷⣤⡉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢉⡴⠞⡽⢁⡾
⠀⠀⠀⠀⠀⠀⠘⣆⠹⣿⣿⣿⠷⣦⣌⡙⠓⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠴⠖⠚⠋⣁⣤⣾⣻⠀⡼⠁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⡆⠻⣿⣿⡄⣿⠈⡟⠓⠲⣦⢤⣤⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣤⣤⣤⠤⠶⡾⢩⡇⢨⠏⣼⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠹⣿⣧⢹⠀⣇⠀⠀⡇⠀⠀⡏⠀⠀⠉⠉⠙⡟⠉⠁⠀⠀⠈⡏⠀⠀⡇⠀⢠⡇⢸⢣⠟⣸⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢻⣿⣿⣇⢻⡀⠀⣷⡀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⣠⡇⢀⢸⣇⣿⠟⣰⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠻⣿⢩⡿⣷⢦⣽⣧⠀⢻⠀⠀⠀⠀⡄⡇⢦⣀⣀⠀⢸⠇⢠⣾⣧⠼⡟⢻⠸⣶⠇⢰⠏
⠀⠀⠒⠖⠰⠐⠰⠒⠂⠠⠘⢷⠈⢿⣾⠧⡟⢸⠁⠹⡿⠛⣷⠞⠿⣿⠷⠻⠿⡿⠶⠷⡿⠛⢿⠹⢘⠸⢷⠸⣷⠟⢰⠟⠐⠀⠄⠂⠦⠀⠂⠈⠰⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⣿⣼⡇⢸⠁⢸⠃⠀⡟⠀⠀⣿⠀⠀⠀⡇⠀⠀⡇⠀⢹⠀⢸⡇⢸⣾⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣏⣷⣸⠀⢸⠀⠁⡇⠀⠀⢿⠀⠀⢀⡇⠀⠀⡇⠀⢸⡇⢸⣇⣾⡾⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⣎⠻⣄⣸⡀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⢠⡇⠀⢸⣃⡼⢉⡾⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠈⢧⣈⠀⠙⠶⠟⢦⣀⡼⢧⣀⡾⢦⣀⡼⠧⠴⠋⠁⣠⠞⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠙⠒⠶⠤⣤⣬⣀⣀⣉⣀⣀⣀⣀⣤⡤⠶⠛⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣿⣽⣿⣿⣿⣛⠻⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣞⣭⣴⣿⡿⠿⠟⠻⣿⣿⣷⣜⣳⠈⠀⠁⠙⠉⠋⠙⠧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⠭⠿⠛⠋⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢼⣼⣧⣶⣰⣦⣠⣿⡻⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣫⣾⠏⠀⠀⠀⠀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠑⠢⡙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢃⣿⠁⠀⢀⠔⠤⠍⠡⠀⠀⠀⠀⢈⠙⢂⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⢠⣿⡗⠀⠠⣂⠄⠀⠀⠀⠀⠀⠀⠀⡔⢂⣌⢁⠙⠢⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⣰⡿⠀⡼⣿⡶⢃⠐⣮⠂⠀⠀⠀⠀⣤⠀⠀⠁⢈⡀⠑⠶⢉⡳⠀⠱⠆⡀⢀⡐⠆⠐⠢⡀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⣿⣵⡀⠰⢻⣿⡁⠀⣿⡀⠀⠀⠀⠄⢘⣓⣂⠀⠠⣍⡱⠀⠀⢈⡗⠂⠈⡉⢀⠉⠤⠈⠒⠁⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⣿⣟⡅⠀⣼⣿⣯⠀⣸⡧⡠⠄⡠⢰⣿⣿⣟⣁⠒⠀⣃⠱⢡⡀⣗⣙⡀⣄⣠⠖⣤⡞⠁⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣽⣿⠟⢀⣾⠇⣽⡦⠑⣺⣿⠿⣛⠟⠃⠛⢉⡸⠠⣤⣩⠁⠉⡄⡀⣸⠟⠁⠀⠀⠀⠀⠀⢰⡿⡇
⠀⠀⠀⠀⠀⠀⣿⣷⠂⠀⢺⣋⠀⣼⢳⣆⣿⣰⣾⣿⣿⣿⣿⣿⣿⣶⣌⣡⠰⢧⣿⠁⢠⡇⡍⠀⠀⠀⠀⠐⠀⠀⠀⣼⣧⠃
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⡇⠘⣊⢼⠇⣍⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣦⣭⡡⠀⡼⠀⣼⠀⢀⢠⣤⣴⣶⣾⣿⣿⢿⡀
⠀⠀⠀⠀⠀⢀⣿⣿⡷⢸⣿⡶⢋⡀⠃⠈⢬⣿⢟⣫⡵⠶⠖⣞⡷⠶⣮⣽⣿⣿⣶⣾⣛⣄⠘⠃⣉⣬⣉⣭⠭⣤⣄⠁⢸
⠀⠀⠀⠀⢰⣿⣾⢮⣡⡐⢟⣑⠒⢒⣬⣝⣿⣵⡿⡟⢳⣦⣦⣭⣿⣤⣿⣿⣿⣿⣿⣿⣿⠘⠄⢾⣿⣿⣥⣅⣯⣾⡹⢇⢹
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠿⡇⠜⢣⣉⣻⣷⣿⣿⣧⣭⣉⣭⣩⣥⣴⣿⣿⣿⣿⣿⣿⣿⡖⡀⠈⠳⣅⠘⠉⠁⠀⠀⠈⣼
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣼⣇⡘⢈⡈⢿⡿⠛⢿⠫⠛⠛⠛⠛⠿⣿⢿⢿⡿⢿⡿⣿⣿⣇⠆⠀⠀⠈⠁⠀⠀⢈⠻⠌⠻
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⡟⡰⠈⡃⠾⠁⢸⠆⠄⠀⣠⠀⠠⠰⣿⣿⣿⣷⣾⣶⣿⣿⣿⠘⡀⠀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠹⣧⢿⣾⣛⣿⣧⠘⢃⠶⡰⠖⠋⠀⠀⠀⠋⠀⡣⢉⣿⣿⣿⠁⢹⣿⣿⣿⣿⡅⢿⣄⣈⣷⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠙⢮⣿⣿⣿⣿⠀⡘⢄⢎⣩⢡⡑⢎⣱⠊⣪⡴⢋⣵⣿⣿⣿⣾⡟⣛⠿⠛⠟⠋⠉⠉⠀⠹⣦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡗⢣⠈⣾⣿⣼⣿⠨⢠⣿⣯⣵⣿⣿⣿⣿⣿⡏⠸⠁⠀⢠⡄⠀⠀⠀⠀⠀⠘⣷⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢠⠀⣿⣿⣿⡏⡅⣿⡟⣿⣿⣿⣿⣿⣿⣿⣷⡶⠦⠤⠬⠔⣒⣀⣀⡀⠀⠀⣿⡇⢀⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠐⡾⣿⣿⡇⡄⢻⡇⣱⣶⣿⣙⡛⡛⢻⠛⠻⠛⠛⠛⠉⢋⣉⣿⣿⠖⢃⣿⢁⡿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢃⠀⠛⢿⣇⢡⠈⣧⣿⣿⣿⣿⣿⣿⡩⣻⣿⣿⣾⣿⡛⣿⣿⣿⡿⠀⢸⠇⣼⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡹⣧⠠⠏⡈⡻⣧⡠⠉⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⣿⣯⡿⣿⣿⠃⠀⠸⢠⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣧⠺⡄⢃⢿⣿⣄⢽⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠩⠃⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⣿⣷⢌⠱⡜⣿⣿⣮⡿⣿⣿⣿⣿⣿⣿⣿⠟⡭⡩⠉⠀⠀⠀⠰⠃⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⣷⠸⠀⡰⠹⣿⣿⣷⡌⢝⠿⣿⣿⣿⡇⠂⡀⣇⠀⠀⠀⠀⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⢀⣷⣽⢥⠹⣿⣿⣿⣦⣑⠸⡻⣿⠛⣀⡏⠀⠀⠀⠐⠂⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣼⣻⣿⣿⡆⣙⢿⣿⣿⣿⣷⣤⣄⠒⡹⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣄⢠⡉⠟⡿⢿⣿⣿⣿⣇⣀⣀⣊⠔⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣶⣸⠞⠻⠷⠿⠏⢛⣫⠟⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣅⣀⣐⣤⣽⡯⠖⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣶⣶⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠛⠛⠛⠉⠀⠀⠐⠒⠒⠒⠒⠈⠉⠉⠉⠉⠉⠉⠉⠓⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠠⠖⠒⠒⠒⠢⠤⠤⠤⠤⠤⠤⠄⠒⠒⠉⠉⠉⠉⠉⠉⠓⠒⠢⢤⣈⠙⠢⣄⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⢈⠍⠉⠉⠉⠉⠑⠒⠦⣌⡑⠦⡀⠑⠦⡀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡈⠣⡀⠈⠀⠀⠀⠀⢻⣧
⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣤⣀⠀⠙⠀⠈⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠉⣿⣿⣿⣿⣿⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣄
⢠⣾⠟⢁⡴⠂⠒⠒⠚⠃⠙⠿⠿⠿⢿⣿⣿⣿⣿⣦⣤⡄⠀⠀⠀⢸⣿⣁⣤⣶⣿⠿⠿⠛⠻⠿⠿⢿⠿⠀⡈⠛⠀⠀⢀⠀⠀⠛⠛⢟⢿⣦⡀
⣾⡏⡔⢉⡴⠂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠉⠀⠀⠀⠀⠀⠙⠿⠛⠁⠀⠀⠘⢿⣦⣀⠀⠀⠀⢀⣠⣴⣾⠿⠿⠿⢿⣦⡀⠀⠳⡹⣷
⣿⣷⠀⠘⠰⡿⠟⠛⠻⣿⣦⣀⣤⡄⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠉⠀⣰⣆⠀⠀⠙⣿⡄⠀⠃⠸
⣿⣿⠀⣆⠀⠀⠀⢰⡆⠘⠛⠛⠋⠁⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⣤⣤⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣄⡀⠀⢸⣷
⢻⣧⣆⠘⠆⠀⢠⣾⣇⠀⠀⠀⠀⢀⣰⣿⣿⡁⠀⠀⠀⠀⠀⢀⣀⣀⡉⠙⣿⡇⠙⠛⠒⠒⠂⠀⢀⣠⣴⣾⠿⠋⠁⢀⣿⠿⠿⠓⣸⡟⠀⡆⢀
⠈⣿⡎⠓⠦⠀⣼⣿⣿⣄⠀⠠⠚⠉⠈⠉⠻⣷⣄⠀⢀⠀⠀⠛⠛⠛⠟⢰⣿⠃⠀⢀⣠⣤⣶⣿⠿⢿⣿⠁⠀⠀⣠⣾⣿⠀⠀⠀⡿⠁⣠⠃⣼
⠀⠘⣿⡀⠀⠀⣿⣿⡿⣿⣷⣦⣄⡀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠛⠋⠁⠀⢀⣼⣇⣠⣴⣿⣿⡿⠃⠀⠀⠀⠴⠚⢁⣾⡟
⠀⠀⢻⡇⠀⠀⣿⣿⠃⣿⡏⠙⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠟⠛⠋⠉⢹⣷⠀⠀⢀⣠⣤⣾⣿⣿⠟⠉⣴⡿⠁⠀⠀⠀⠀⠀⣺⣿⠋
⠀⠀⢸⡇⠀⠀⣿⣿⣶⣿⣇⠀⣿⡇⠀⠀⠀⢹⣿⠁⠀⠀⢸⣿⠀⠀⠀⠀⢀⣨⣿⣶⣾⣿⣿⠿⠛⢻⡇⢀⣼⡟⠁⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣶⣶⣶⣾⣿⣶⣶⣾⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠘⣿⣿⠏⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⢸⣿⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⢸⡇⠀⠀⠸⣿⣹⣿⠹⣿⡛⠻⣿⡟⠛⠻⣿⡟⠉⠉⠉⠙⣿⠀⠀⠀⠀⠀⠀⢿⣇⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⢸⡇⠀⠀⠀⠙⣿⣯⣀⣹⣷⠀⢻⣧⠀⠀⢻⡇⠀⠀⠀⠀⣿⠀⠀⠀⢀⣀⣤⣾⣿⠿⠋⠀⢀⡠⠔⠀⣀⠔⢀⣴⡿⠋
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠙⠛⠿⠿⣷⣾⣿⣷⣤⣾⣿⣦⣤⣤⣾⣿⣷⡾⠿⠿⠛⠋⠉⠀⣀⡤⠖⠉⣠⠔⠋⣁⣴⡿⠛
⠀⠀⣿⡇⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⡤⠖⠉⣀⣴⣾⠟⠋
⠀⠀⣿⠃⠀⠐⣆⠀⠀⠈⠓⠦⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠐⣒⣉⡥⠔⠒⠉⢀⣤⣶⡿⠟⠋
⠀⠀⣿⡄⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠤⠒⠒⠋⠉⠁⠀⠀⣀⣴⣾⠿⠛⠁
⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠟⠋
⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣤⣤⣤⣤⣴⣶⣶⣾⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⢄⡀⠀⠑⡄⠀⠀⠀⠀⢠⠊⠀⠀⢀⢠⢀⣀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠉⠉⣽⣷⡄⠀⢸⠀⠀⠀⢀⠇⠀⣴⣾⠛⠉⠹⠷⣷⡀⠈⡆
⠒⠒⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⣸⣿⠇⠀⢸⠀⠀⠀⢸⠀⠀⣟⣇⠀⠀⠀⢠⣟⡇⠀⡐⠀⠀⠀⢀⡠⠤⠤⠤⠤⢄
⠀⠠⣞⠋⠀⠈⢳⣦⣀⣀⣀⠀⠀⢴⠊⠙⠻⣿⣳⣿⠿⠋⠀⢠⠃⠀⠀⠀⠀⢇⠀⠙⢟⣷⣦⣖⣿⠟⠁⢠⠃⠀⠀⡔⠁⠀⡀⣄⢠⢀
⠀⢨⣿⣄⡀⢀⣬⣽⠇⠀⢸⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠈⠳⣄⡀⠈⠉⠁⠀⣀⠔⠁⠀⠀⡜⠀⢠⢦⡎⠈⠛⢿⣿
⠀⠀⠙⠿⠿⠿⠟⠁⠀⡠⠁⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠒⠈⠉⠀⠀⠀⠀⠀⡇⠀⢾⢺⠀⠀⠀⠀⣷
⠑⠦⣀⣀⠀⢀⣀⠤⠊⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⢞⣷⡤⣤⣺⡿
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⡄⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡀⣤⠾⡿⢿⡍⠁⠐⡆⢸⠉⠀⣺⡛⠛⠛⢯⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢺⡇⠀⠀⠸⣿⠀⠰⠀⠈⢆⠀⣽⣇⠀⠀⣸⡻⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣦
⠀⠀⢠⢦⣤⣀⡀⠀⠀⠀⠀⠀⠘⢄⠹⢿⠦⣤⣮⣏⠔⠀⠀⠀⠀⠑⢌⣾⣿⣿⣿⠵⠊⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⢸⡆
⠀⠀⣾⠀⠀⠈⠙⠓⠶⢤⣄⣀⠀⠀⠈⠘⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡖⠛⠉⠀⠙⠄⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠀⠀⠌⠀⠀⠀⠉⢹⠓⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⡇⢈⠉⠢⠀⠀⠀⢸⠀⠀⠀⢠⡇
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠃⢸⠀⠈⠉⠛⠲⣦⣤⣀⠀⠀⠀⢀⣠⠴⢾⠉⠁⠀⠀⠀⡇⠀⠃⠔⡦⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠈⢷⠀⠀⠀⡆⠀⠀⠠⠃⠸⡀⢸⠀⠀⠀⠀⠀⠃⠀⣸⠇⠀⠀⠈⢧⡀⠈⡄⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⠀⢀⡼⠃
⠀⠀⠀⠈⢳⡄⠀⠘⠄⠀⠀⠀⠀⢃⡄⠀⠀⠀⢀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠳⣄⡈⠢⡀⠀⠀⠈⢿⠀⠀⠀⠀⠀⢀⡴⠟
⠀⠀⠀⠀⠀⠙⠲⣄⡈⠒⢄⡀⠀⠘⠃⠀⠀⠐⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣕⣒⣂⣀⣂⣀⣡⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⢥⣤⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣄⠀⠀⠀⠀⠀⡞⠀⠀⠈⠑⢦⣄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣐⣈⣀⠀⠀⠀⠱⡄⠀⠀⠀⣇⠀⠀⠀⡠⣨⠿⠿⡶⣦⣤⠤⠒⢹⠇⠀⡶⡄
⠂⠀⠈⢉⢉⣶⢶⣖⠢⢄⠀⠀⠈⢼⡟⠻⠷⢯⣷⡄⠀⠀⢸⠀⠀⠀⠸⡄⠀⢰⣺⡇⠀⠀⠂⣼⣿⠀⠀⡜⠀⢸⠃⠈⢲⣤⣄⣀⣀⣤⠤
⠀⠀⠠⢽⡿⠻⠉⢿⣳⡸⠀⠀⣼⢜⣏⠀⠀⠀⣿⡧⠀⠀⢀⠀⠀⠀⠀⠘⢦⡘⢿⣷⣄⣀⣴⣿⢟⡠⠊⠀⠀⠸⡀⢀⣦⣿⡉⠛⠻⣯⢣
⠀⠀⢸⣿⡆⠀⠀⣸⣽⢿⠀⠀⠹⡿⣝⡳⡶⣿⠿⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⢳⣸⣿⣇⠀⠀⠀⡼⣽
⡀⠀⠀⠳⢟⣿⣞⡷⢋⠇⠀⠀⠀⠈⠢⢍⡉⠁⠀⣀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣶⣾⠿⠃
⠑⢤⣀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⢀⣤⡤⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⠁⠐⠇⠀⢠⠃⠈⠻⣦⡀⠀⣠⡴⠟⠙⠀⠀⠘⠀⡠⠐⠀⠉⠁⠈⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢹⠁⠂⠀⠐⠀⠘⣇⠀⠀⣨⠇⠀⢋⣄⣀⡼⠀⢀⡥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⢀⣠⡄⠀⠀⠈⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⠄⢰⠋⠀⠀⠀⠀⠀⠁⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡄⠀⠀⠀⠀⠀⠈⠓⠠⡀
⠀⠀⠀⡰⠁⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠐⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡎⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⡇⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⠀⢠⣶⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢠⠔⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠔⠰⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡠⠃
⠀⠘⢤⡀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠨⢤⡀
⠀⠀⠀⢨⠋⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠂⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⡠⠺⠀⠀⠀⠀⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡠⠀⠀⠀⠀⡤⠄⠊
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡨⠋⠑⠠⠤⠔⠊
⠀⠀⠈⠣⣄⢀⠀⣀⠔⠓⠲⡖⠂⠀⠀⠀⠀⢀⠀⠀⠀⣠⠃⠐⠁⠀⢀⠐⢦⡀⢀⡠⠃⡀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠔⠒⠁⠉⠐⠈⠁⠀⠐⠀⠐⠈⠀⠀⠀⠈⠀⠀⠈⠐⠠⠤⠄⠒⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠙⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣶⣦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠚⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠈⠻⣿⣷⠄⠀⢸⠀⠀⠀⣠⣶⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠙⠒⣦⣤⣀⣀⡀⠀⠺⠀⠀⢸⣿⣿⣿⠇⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⡿⠋⠀⡇⠀⢸⡀⠀⠀⠻⠿⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠀⠀⠀⣀⠜⠀⠀⠀⠳⣄⡀⠀⣠⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣤⣄⣤⣤⡤⠴⠾⠷⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣺⣿⣿⣿⣿⣶⣦⡱⡄
⠀⠀⠀⠀⣀⣤⡴⣶⣶⣶⣶⣶⣦⣽⣦⡀⣀⣴⣴⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠉⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣀⣀⣀⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠚⢉⣴⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣋⠁
⠀⠀⠀⠀⣠⠶⠛⢛⣭⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠿⠃⠈⢿⣿⣿⣿⣿⣿⡿⠿⢷⣦⡀
⠀⠀⠀⠀⣀⣴⠿⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⡿⣿⠏⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣦⣄⠈⠙⣄
⠀⠀⠀⠐⠁⠀⠐⢒⣶⣿⣿⣿⠀⠀⣀⣀⠀⢀⠀⠀⠀⠀⢀⠜⣸⠏⣠⠤⢄⠀⠀⢸⣿⣿⣿⣿⣯⡉⠉⠛⠆⠈
⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⡄⠀⢡⣄⠑⠀⠉⠒⠀⠀⠀⠰⠃⣼⠁⣰⣄⠀⠀⣼⣿⣿⣿⣯⡉⠙⠂
⠀⠀⠀⠀⠀⠐⣻⠿⠿⢿⣿⣿⡷⠰⠋⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣳⡟⠉⠂⠀⣿⣿⣿⣿⣿⣿⡦⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢶⡙⢿⣷⠐⢒⣶⠾⣿⣄⠀⠀⠀⠀⠀⠀⣼⣿⢶⣶⠂⢠⣿⠟⣩⡆⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⣾⠉⠀⠈⢿⣆⣭⣼⣷⣶⠆⠸⣶⣿⣧⣯⣴⡯⠁⠀⢹⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⡽⠉⠀⠉⠁⢠⣄⠉⠋⠉⠻⠉⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⣶⣶⡴⠒⠠⡰⣶⣿⣿⣶⢆⠄⠒⣴⣶⡆⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡘⠿⣿⡷⢤⣈⡉⠻⢿⣋⣤⡶⣿⣿⠛⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡀⠹⣷⣌⠉⡏⢻⠋⠿⢈⣰⣿⠃⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⡀⠹⡏⠁⠒⠲⠂⠂⢉⣼⠛⣶⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣿⣿⣷⡀⠹⣧⣦⣠⡀⣧⣾⡇⣰⣿⣿⣿⣿⣿⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⣿⣷⡀⠈⢙⣛⣛⠋⠘⣵⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⣿⣷⡀⠀⠉⠁⠀⣼⣿⡿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠙⠻⣷⣶⣶⠿⠛⠁⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠷⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣀⡽⢶⣄
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣸⣧⡴⠶⠶⠶⠶⠶⣦⣄⡀⠀⣠⣾⣏⣠⡶⢿⣯⣉⣀⣀⠙⢷⣄
⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣏⠀⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠹⣦⡀
⠀⠀⠀⣰⣿⣀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⣰⠏⠈⠉⠛⣿⠏⠀⠀⢀⣴⣾⣿⣿⣿⣷⡶⠶⠶⢦⣄⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡀
⠀⢰⡟⠀⠀⠀⣸⠏⠀⠀⢠⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢻⡆⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠙⣷
⠀⣿⠀⠀⠀⠀⡿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣼⠇⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣀⣽
⢸⡇⠀⠀⠀⢰⡇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⡶⠶⠟⠁⠀⠀⠀⠀⢀⣿⣤⣤⣤⣤⣤⣤⣾⣀⣤⠾⣏⠈
⣿⠁⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠙⠿⢿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⣿⠉⠉⠉⠁⠀⠙⢷
⣿⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣿
⣿⠶⠶⢶⣦⣄⣨⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⢠
⢿⡀⢀⡿⠀⠉⠁⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⣼
⠸⣷⠟⠁⠀⠀⠀⠀⠀⠉⠳⣦⣄⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⠛⠻⠶⠟⠛⠋⠉⠀⢸⡟⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠈⠛⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⣠⡾⠛⣿⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠹⣆⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠐⠟⠀⠀⠿⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠉⠀⣠⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣠⣤⠶⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣻⣷⣶⣶⣶⣾⣟⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠈⠻⣦⡀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⣀⡴⠋⠘⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠙⠳⠶⠴⣿⣥⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣽⡷⠶⠶⠛⠁⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⣤⣄⠀⠀⠀⢸⣿⣷⣤⣀⡀⢀⣀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⣀⠀⣀⣠⣴⣿⣿⠀⠀⠀⢀⣤⡄
⠀⠀⠀⣼⠃⢹⡇⠀⠀⢸⡇⠈⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠀⣿⠀⠀⢀⣿⠁⢻⡄
⠀⠀⠀⣿⡀⠀⢿⡀⠀⣿⡇⠀⢀⣠⣶⠶⠶⠶⠄⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⢀⠴⠶⠶⢶⣦⣀⠀⠀⣿⡆⠀⣸⠇⠀⣸⡇
⠀⠀⠀⢹⣷⢠⠈⢷⡀⢻⡇⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣀⣠⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⣿⠃⣰⠏⢠⢰⡿⠁
⠀⠀⠀⢀⡿⠈⢷⣌⠻⣾⡇⢸⣧⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠉⠉⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢀⣿⡾⢋⣴⠏⠸⣇
⠀⠀⠀⣼⠃⠀⠀⢹⣷⣼⣿⢛⣡⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣙⢻⣿⣴⣿⠁⠀⠀⢻⡄
⠀⠀⢰⡇⠀⠀⢀⡿⢡⡿⠁⠉⠉⠉⠙⠻⣿⣧⣄⡀⠀⣠⣤⣠⣄⠀⣸⡇⣿⡀⢀⣤⣠⣤⡀⠀⡀⣤⣿⡿⠛⠉⠉⠉⠁⠹⣇⠹⣇⠀⠀⠀⣿
⠀⠀⠸⣇⠀⠀⠸⠃⢸⡇⠀⠀⠀⠀⠀⠀⣈⣿⣿⣷⣶⣿⣿⣿⣿⣴⠟⠀⠘⢷⣼⣿⣿⣿⣷⣶⣽⣿⣋⡀⠀⠀⠀⠀⠀⠀⣿⠀⠹⠀⠀⢀⡟
⠀⠀⠀⠻⣦⡀⠀⠀⠀⣿⠄⠀⠀⠀⢰⣞⠁⠀⣾⠛⣿⡿⠿⣿⣿⠁⠀⠀⠀⠀⢹⣿⡿⠿⣿⡟⣻⡆⠀⢙⣶⠀⠀⠀⠀⢼⡇⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠙⢿⣄⣀⡾⠁⠀⠀⠀⠀⠘⠏⠛⠦⠼⠿⠯⠤⠴⠋⢿⡆⠀⠀⠀⠀⣾⠏⠳⠤⠬⠿⠿⠤⠞⠋⠝⠀⠀⠀⠀⠀⠹⣆⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡀⢸⡇⠀⠀⠀⠀⣿⠀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣇⣠⡜⠋⠉⠉⠑⠀⠀⠀⠀⣀⣤⣤⣈⠹⣿⡇⠀⠀⠀⠀⣿⡿⢉⣠⣤⣄⡀⠀⠀⠀⠐⠉⠉⠙⠛⣤⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡀⠀⠀⠀⠀⠀⢀⣴⠟⠋⣽⠋⠉⠀⢸⣿⠀⠀⠀⢸⣿⠀⠈⠉⢻⡍⠛⢷⣄⠀⠀⠀⠀⠀⠀⣸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢻⣦⡄⠀⠀⠀⠊⢀⡾⠀⠋⠀⣶⣤⣸⣿⠀⠀⠀⢸⣿⣀⣴⡆⠘⠃⠰⣆⠉⠂⠀⠀⠀⣤⣾⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡈⣿⡏⠆⠀⣀⣠⢼⠓⣒⣚⣚⣳⣬⣿⣿⠀⠀⠀⢸⣿⣯⣴⣓⣚⣒⡒⣻⢤⣀⡀⠀⠎⣿⡏⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣾⣷⣿⠿⣿⢿⣿⣏⡛⢿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠟⣉⣿⣿⢿⡿⢿⣿⣿⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⠁⠀⠀⠘⡆⠹⣿⡿⢷⣿⣿⣿⣄⣀⣀⣼⣿⣿⣷⠿⣿⡿⠁⡾⠀⠀⠀⠉⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⢧⠘⣿⣇⢸⡁⠉⠉⠛⠋⠉⠃⢹⢀⣿⡟⣠⠃⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠈⢷⡘⢿⣌⠀⠘⠀⠀⠀⠘⠀⢈⣼⠏⣴⠇⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⣶⠀⠈⠻⣦⡙⠷⣶⣤⣤⣤⣴⡶⠟⣡⡾⠃⠀⢸⡆⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠹⡄⠀⠀⠈⠻⢶⣤⣬⣉⣩⣤⣴⠾⠋⠀⠀⠀⡼⢁⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠘⠀⠀⠀⠀⠀⠀⢀⣭⣅⠀⠀⠀⠀⠀⠀⠈⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⢀⣾⠁⣾⣿⣿⡆⣹⣆⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣸⣧⣧⣿⣿⣿⣿⣿⣿⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣸⣽⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠻⠿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⠉⠀⠀⠀⢀⣠⣴⠞⠀⠀⠀⢀⠲⢾⡅⠀⠀⠀⢰⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⣀⣠⣤⣶⣿⠿⠟⠀⠀⠀⠀⡾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠘⠛⠻⠿⠛⠻⠿⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠊⠂⠀⠀⠠⠀⢻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡿⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠋⠀⢀⣠⣾⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠉⠙⠀⡀⠀⠀⠀⠀⠀⠀⢳⣦⣀⠀⠀⣀⣤⡴⠒⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠠⠤⡼⠁⠀⢀⣤⠀⠀⠀⠀⠉⢻⣿⣿⠟⠉⠃⠀⠈⢻⣶⣶⣾⣿⡿⠂
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠊⠈⠁⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠀⠀⠀⠈⠛⢻⠛
⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠚⣉⣀⣬⣿⣦⣤⣤⣶⣶⣦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤⠄
⠠⠶⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⢀⣀⣀⣤⡶⠟⠋⠉⠀⠀⠀⠀⢠⠂⠀⠀⠀⠈⠉⠙⠛⠷⢶⣇⡀⠁⠀⠀⠀⠙⠁⠀⠀⠶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⢁⣀⣠⣤⣤⣤⣤⣤⣴⣇⣀⣀⠀⠀⣀⡴⠁⠀⣠⠏⠈⠻⢦⣀⠀⠀⠀⠀⠀⢀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣃⠀⢀⣠⠔⠋⡶⢀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣵⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⣾⣿⣿⣿⡿⣟⡛⢿⡍⠪⡳⠀⢿⣿⣿⣿⣿⠿⠛⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⢷⡄⠀⠀⠀⠉⠛⠋⠁⠀⠀⣠⠟⠉⢓⣠⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⣤⣴⡿⠿⠿⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⣀⣀⣀⣠⡴⢞⣥⣴⣾⣿⣿⡿⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣉⣩⣭⣷⣾⣿⣿⡿⢟⠉⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⣟⠿⣍⠀⠀⠈⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠑⠈⠀⠀⠀⠑⠀⠑

⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣀⣤⣬⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⣠⣴⢂⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇
⠀⠀⠀⠀⣾⣿⣿⡀⡀⠀⠀⣤⣤⣴⣆⡀⣿⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⠀⣠⣶⣤⣤⡀⠆⠀⢀⣿⣿⡟⠀⢄
⠀⣰⡇⠀⣿⣿⣿⣿⣧⡀⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣷⣀⣼⣿⣿⣿⡇⠀⢸⣇
⠀⣿⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿
⣄⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⢈⡡⠂⠀⣽⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠐⠢⢌⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⠋⠀⠀⢀⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠑⢄⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣠⣄⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⣀⣀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⢿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⡿⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣆⡹⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣠⠇⡐⣦⣷⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⢋⣴⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⠀⢻⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣹⣿⣿⠇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠸⣿⣿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⢀⣿⡿⣿⣿⡿⠋⠁⠀⠀⢠⣴⠾⠟⠶⣦⡀⠀⠈⠙⠻⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡄⠀⠸⠏⠀⢹⣿⠁⠀⠀⣀⣴⣿⣿⣦⣀⣠⣿⣿⣦⠀⠀⠀⠹⣿⠀⠹⠇⠀⢠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣦⣄⡀⠀⢸⡇⠀⣠⡾⠃⢹⣿⣿⣿⣿⣿⣿⣿⡇⠷⣄⠀⠀⣿⠀⠀⣀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣦⣈⣧⡾⠋⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠳⣄⢸⣠⣿⣿⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣿⣿⡿⢿⠟⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠿⠿⣿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⣿⣿⣷⣦⣤⣀⣀⣀⣀⣤⣴⣽⣿⣿⣯⣤⣤⣀⣀⣀⣠⣤⣴⣾⣿⣿⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠹⣿⣿⣟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⡏⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣽⣿⣿⡆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣿⣿⣿⡿⠁⢠⣿⣿⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠉⠛⢻⣿⣿⣿⠛⠋⠁⠀⠀⣸⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⢿⣿⠋⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⣴⣄⠀⠀⠀⠁⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠿⠛⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠉⠛⠛⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠈⠙⢿⡿⡟⠉⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡿⠿⠿⣿⣿⣿⡿⡟⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⡇⠀⠀⣿⡯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠟⣵⣿⣿⣿⠟⠉⢉⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡏⣾⣿⣿⣿⠡⣚⣭⣭⣽⣿⣶⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⠛⠛⠯⠹⣿⣿⣿⣿⡆⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠰⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡀⣿⣿⣿⣿⠇⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡸⣄⡀⠙⢶⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣯⣤⡴⣡⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣿⡿⠟⠋⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠰⢬⣛⣻⣿⣿⣿⣿⣿⡅⠀⠀⠀⠉⠛⢿⣿⣿⡿⠋⠀⠀⠀⠀⢀⣿⣿⣿⣿⣶⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣯⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⡇⠀⠀⠀⢡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⡄⠀⠀⣈⣙⣷⡄⠀⠀⠀⠀⣠⣾⣋⣀⠀⠀⢸⣿⣟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⡛⢿⠇⠐⣾⡛⣿⠟⣿⣶⡂⢰⣾⡿⢻⣿⢛⡷⠀⠸⡿⢛⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⠿⠂⠀⠀⠈⠈⠩⣿⠟⠉⠀⢈⡏⠻⣯⠍⠁⠁⠀⠀⠐⠟⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢦⠀⠀⡀⠀⠀⠀⢀⢀⣤⣬⡁⡄⠀⠀⠀⣀⠀⠀⡔⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢣⠸⡟⣶⣦⣄⣀⠳⣿⣿⠞⣀⣠⣼⡟⢛⠃⠚⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠑⠘⢷⣬⡉⠻⠻⠟⠟⣋⣥⣾⢁⢉⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠈⣿⢿⣾⣴⣤⣷⠿⣿⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡄⠘⢷⣦⣅⣨⣴⡾⠃⢠⡎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⡘⣿⣆⠀⠙⠛⠛⠋⠀⣴⣿⢠⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣤⣤⣍⣛⠦⣀⣀⣀⣀⠼⣛⣩⣤⣬⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠻⠿⠿⠏⠻⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣄⡀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣠⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠙⢷⣤⡤⠀⠀⢤⣤⡾⠋⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⣤⣶⡀⠙⠁⠀⠀⠈⠋⢠⣶⣤⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠟⠿⠿⠛⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠋⢀⣀⣀⣀⣀⣀⠈⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣴⣦⠸⣆⣷⣄⠹⣧⢻⡶⠚⢻⣄⠀⡀⣠⠊⠉⠝⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⢸⠸⡗⠚⠆⠿⠀⠻⠀⠙⠀⠃⡀
⢃⣿⣤⡜⣿⣿⠷⠿⠈⠃⠀⠀⡠⠊⣰⠁⠀⠀⠀⠀⠀⠀⡼⠀⣸⡓⡄⠀⠀⠀⠀⠂⠀⢀⡀⠠⡄⢰⠀⢸⣷⡀⣷
⣿⠋⣸⡇⠈⠋⠀⠀⠀⣄⠔⡟⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⡏⠉⠁⠀⠐⣄⠀⠢⠀⠸⠀⢸⢷⡄⣷⠾⢧⢸⠟⠛⠙
⠙⡷⠏⠀⠀⠀⢀⠐⣶⠋⠘⠀⠀⠀⡄⠀⡆⢸⡄⠀⠘⣆⡇⠀⠀⠀⢀⠬⡟⡶⠰⣿⠀⢸⠛⠻⡜
⠀⠀⠀⠀⢀⡀⠸⣇⡏⠀⠀⠀⠀⡄⣇⠀⡷⣸⠘⢆⠀⠋⢳⠀⢀⡴⠓⠂⢣⢹⠛⠙⠇⠈⠀⠀⠀⠀⢀⠐⣆⠀⣷
⠞⠛⡿⣦⠘⣷⣄⢻⠀⠀⠀⠀⠀⢱⣸⣴⡗⣻⠧⡶⠷⢼⡀⠀⠉⠁⠀⠀⠈⡎⠀⠀⢀⡀⢶⠀⢻⣆⠈⣷⡿⡆⠿
⣇⣰⣧⣌⢷⣿⡉⠛⠇⠀⠀⠀⠀⢺⠛⠌⠛⠀⠐⠉⠀⠀⠀⢀⣀⣀⣀⣀⡀⢣⢰⢷⣀⣳⡼⣷⠸⡟⠛⠈⠁⠀⢠
⠸⡇⠀⢸⡇⠻⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⣒⣒⣲⣭⣭⣭⣥⣤⣶⣶⣾⣿⡈⣿⠛⠛⠆⠛⠉⣀⠀⠀⣤⢳⡀⢸
⠀⠳⣴⠟⠀⠀⢠⡀⢸⡇⠀⢀⠙⠢⣣⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢸⠀⠀⠀⢰⡀⢿⣳⣄⣸⠿⢿⢾
⠀⠀⣀⠀⡀⢀⠈⣷⣸⠇⠀⢀⠀⠐⠈⠙⠀⠘⣏⠉⠉⠉⠉⠛⠿⣿⣿⠏⠀⣰⣯⣰⡀⣤⠘⣧⣸⠯⢭⡻⡆⢸⠆
⠠⣞⢹⠇⠹⣿⣧⡘⣷⠦⠀⣀⠑⠢⠤⠾⣄⠀⠈⠓⠤⣤⣀⣀⡤⠬⠟⣠⣾⣁⣼⡁⢳⣽⣷⣽⠗⢦⠀⢧⣿⠈⡆
⠀⢸⣿⠛⣷⠙⠿⠙⠛⠀⠐⠀⠆⠠⠄⡀⠈⠑⢲⠒⠒⠒⠒⢲⠒⠛⠉⠵⠆⢻⡉⠙⣧⠙⣷⠚⠙⣜⡆⠈⠛⠀⠙
⠀⠀⣸⡾⢏⠀⠀⠀⢀⠀⢸⣿⡈⣰⠾⠛⠛⠙⢙⣷⠀⠀⠀⠘⣏⠉⠉⠙⠓⢦⡷⠞⠁⠀⠈⠙⢦⠈⠁
⠀⠀⠀⣠⠾⣟⢦⢳⣬⣧⠸⠉⢻⡁⠀⠀⠀⣾⣿⣇⢢⡀⠀⠰⢿⣷⡄⠀⠀⠀⢻⣄⠀⣄⢸⣧⣼
⠀⠀⠈⡵⠜⣻⠚⢻⠓⠀⢀⡴⠋⠙⠛⠂⠀⠙⣿⡿⠀⠙⠀⠀⠸⣿⡇⠀⢀⡴⠛⣼⣧⣼⣾⡟⢛⣆
⠀⠀⠀⢠⠊⠀⣤⡀⢸⣷⡞⣀⣀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣹⠁⣰⠋⠀⠼⢿⠇⠀⠛⠁⣸⠄⠀⠉⠉⠒⢤
⠀⠀⣀⡆⢸⡄⣿⠷⣾⠟⠉⠉⠉⠙⠳⢦⣀⠀⠀⠀⠙⢯⡭⠉⢩⡏⠀⠁⠀⠀⠴⠋⠙⢢⣾⣶⠃⠀⠀⠀⠀⠀⢠
⠉⠉⡏⡟⠉⠷⠁⢰⠃⠈⠑⠢⣄⠀⠀⠀⠈⢳⡀⠀⣄⠀⠙⠶⢋⡄⠀⠀⠀⠀⠀⣠⠖⠁⣏⡏⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠙⣆⣄⠀⠀⡞⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣯⣷⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢧⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⠀⣼⣹
⠀⠀⠀⠀⠈⠏⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠛⠀⠀⠀⠀⡴⠁⠏⠀⠀⠀⠀⠀⠀⠀⡼⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡑⢦⠀⠄⠐⣘⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⢠⣾⠷⣶⣌⠙⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⣿⡇⠀⠀⠙⢷⣜⢎⡳⡊⠆⠐⠆⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⠁⠀⠀⠀⠀⠻⢧⣴⣾⣿⣷⡀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢸⣿⠀⠀⣀⣀⣀⣤⣴⣿⠟⢹⣿⣶⣷⣤⣤⣄⣉⠠⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⠟⠛⠛⠉⠉⠉⠁⠀⢀⣾⠟⠉⠙⠛⠛⠿⢿⣿⣷⣦⣄⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠰⢤⣾⠋⠁⠀⠀⠀⣀⣠⣤⡤⠶⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣤⠂⣌⣀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡔⡋⠛⢦⣄⣴⠞⠋⠉⠁⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣮⡌⠲⠌⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠰⡁⣤⣿⣿⣧⣄⠀⠀⢿⡆⢠⣿⣿⠃⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣽⣿⠛⠛⠻⢿⣷⡀⠈⢳⢸⣿⢏⣴⠟⢁⣠⣴⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⠃⠀⠀⠀⠀⠙⣿⡄⠈⣿⠯⠞⣡⣴⡿⠟⠋⠉⠉⠛⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⠐⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡀⠟⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣀⠀⢸⣇⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⣤⣀⠀⢻⣆⠀⣿⡿⠁⠀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⣽⣿⣆⠾⣿⠾⠿⢷⣤⠛⠀⠛⠁⠀⡼⢋⣤⡶⣦⣄⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣁⢁⣴⡿⠋⢁⣴⡞⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⠛⠛⢿⡿⢷⣦⣄⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⣼⣿⡟⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⣾⡟⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠰⠄⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⠀⡆⠈⠀⠀⣀⣀⣀⣠⢾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⣷⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣿⣧⠹⣄⡀⣸⣿⣿⣿⣿⣿⣿⣿⣮⣿⣶⣦⡀⠀⠀⠀⠿⠇⠀⠀⠀⢿⣿⣿⣧⢹⡆⠀⠀⠀⣠⡶⠻⣾⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣿⣹⣷⣌⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⢀⡇⠀⠀⢰⡏⠀⢀⣿⡟⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢻⣏⢻⣍⣷⢤⡀⣀⣬⡛⠻⢿⡟⠙⠛⠛⠉⠙⠳⠶⠶⠶⠶⠛⣁⡴⣾⣿⣿⡏⣼⠃⢀⣴⣿⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠈⣿⡄⢻⣧⣸⠋⢛⡿⢷⣴⣿⣿⣄⣀⣀⣀⣀⣀⣤⣤⠶⡾⢻⣩⣶⠟⣽⠏⠰⠃⠀⣿⠿⠟⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⢻⣿⠀⢻⣿⣷⣾⣇⣀⡇⠀⠉⡏⠉⢹⡉⢉⣏⣀⣭⣤⣷⣾⡿⢁⣼⠏⠀⠀⠀⢾⠏⠀⣠⣾⠟⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⡾⠁⠀⠀⣀⠀⠀⣠⣾⠛⠁⢄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠾⠋⠀⠀⠀⠀⣿⣤⣴⡿⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣿⡛⢿⣻⣿⠿⢿⡿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢩⡔⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠘⣿⣍⣋⣙⣛⡶⡿⠷⢾⣤⣭⢥⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠣⡂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡄⠈⠛⠛⠛⠉⠛⠿⣷⣶⣴⣶⠞⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⢡⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏⡑⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⢋⡄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⢿⣷⣤⣀⣀⣀⣠⣤⣶⡿⠟⠋⠒⠐⠘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡶⠶⣆⡩⡛⠿⠟⡛⠛⠉⠁⠂⠐⢃⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠋⠐⠃⠘⠂⠉⠀⠀⠈⠑⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⠋⣠⣴⣶⣾⣷⣦⣤⣤⡄⠀⠀⠀⢀⣠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⢀⡞⠀⢀⣀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⢸⡿⠋⠀⡀⠀⠙⣻⠟⠁⠀⢀⣾⣿⣿⣿⣿⡿⣿⣿⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⣼⢀⣾⠛⢿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⢼⠃⠀⠘⠃⠀⣠⠟⠀⠀⠀⢸⣿⠟⣿⣿⣿⣷⢿⣿⠋⠁⣀⣤⣤⣦⣦⣤⣤⣀⡀⠀⠀⠀⠀⠸⣬⣿
⠀⠈⠿⠀⠀⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠶⠶⠞⠁⠀⠀⠀⠀⠀⠏⣀⠙⢉⣉⡵⠟⠁⠀⣼⠛⠉⠀⠈⠉⠛⠿⣿⣿⣷⠆⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⢿⣿⡛⣿⠈⠈⢻⣿⠿⠂⠀⠀⠀⠀⢠⠀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠸⡿⢸⣿⣾⣿⣿⡏⠀⠀⣸⠃⠀⠀⠀⠀⠀⣤⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⣿⣯⡿⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⢀⣀⣀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⠶⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢠⡃⠀⠈⡷⠀⠀⠀⠀⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⡇⢨⢿⠆⠀⢨⡹⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡀⠈⠑⠀⠬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡻⢀⠠⠤⠐⢒⣠⠤⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠹⣷⣼⠉⠁⠀⠀⣠⣼⡧⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠟⠛⠛⠻⢦⡄⠁⠀⠀⠈⠙⠻⠶⢶⠶⠟⠁⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⢡⣄⠀⠀⠤⠂⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣾⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠈⠙⠛⠉⢿⣧⠀⢰⣤⣥⡴⠃⠈⠙⠂⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⠀⣀⣤⡴⠾⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏⢿⠃⠀⠀⠀⣠⣶⠟⡉⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠻⠿⠉⠿⠛⠛⠁⠁⠀⠀⠀⣸⢟⣵⣾⣿⣿⡿⢿⣿⣿⣷⣶⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⢽⡿⣷⡆⠀⠹⡟⢻⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⢶⣶⣶⣶⣴⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⣄⣇⣤⡿⣣⠔⢡⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣾⠻⡏⢻⡏⢋⣿⠟⠉⠀⠀⠀⠀⢀⣠⣤⠴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣍⠛⠧⣟⠋⡤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣠⡤⠞⠋⠀⠀⠀⠀⢀⣠⢶⣫⣡⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣛⣻⣿⡿⣿⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⣿⡿⡿⠟⠃⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠰⢶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠉⠐⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡪⠀⠳⣄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⣁⠞⡠⠀⠐⠉⠳⢤⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠀⠀⠀⠀⠀⠀⠔⢂⠡⣀⠤⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢦⡀⡙⢢⣤⣤⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣤⣤⣀⣀⠀⠈⢿⡀⠱⡐⠄⠀⢸⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠚⣧⠸⠀⠀⠀⢸⠒⠉⠁⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⢀⣀⡤⠤⠄⠒⠀⠀⠙⠛⠷⠷⠶⠶⠛⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣄⡀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠈⢳⡄
⠀⠀⠀⠀⣠⡞⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢻⡄
⠀⠀⠀⣰⠏⠀⠀⠀⢠⠎⠀⠀⠀⠀⠐⢮⣵⣺⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣴⣂⡀⠀⠀⠀⠱⡀⠀⠀⢻⡄
⠀⠀⣼⠏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⡇⠈⠉⢻⠿⢿⣷⣤⣀⠀⠀⠈⣆⡜⠀⠀⣀⣤⣴⣾⢿⡟⠛⠋⢹⡇⠀⠀⠀⠀⠀⢱⠀⠀⠀⢿⡀
⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢈⠀⠀⣿⡏⠛⢻⠶⣤⣽⣤⠶⡿⠛⢻⣿⡀⠀⡇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⢠⡿⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠉⠓⠆⠀⡈⢧⡀⠈⢄⠀⠀⠀⡠⠎⠀⠠⠉⠄⠀⠣⡀⠀⠁⠀⡠⠃⠀⡴⠃⠀⠀⠐⠒⠛⠒⢤⠀⠀⠀⠀⣿
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⡖⠒⠀⠐⠢⠝⣦⣤⣈⣀⣈⣀⠤⠊⢀⣤⡈⠒⠤⣀⡉⠈⣉⣠⡴⢚⡡⠂⠀⠀⠒⣦⣄⠀⠀⠀⠀⠀⠀⢹
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢸⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣝⠦⡀⠀⠈⠉⠁⠉⠀⠀⠀⠀⢀⣤⣾⡿⠈⠃⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠉⠲⢄⡀⠀⠀⠀⠰⠾⠿⠛⠛⠛⠛⠓⠚⠓⠀⠀⠀⠀⢀⣠⠴⢊⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡆⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⢉⣷⡶⢤⣄⣀⣀⠀⠀⠀⠀⣀⣀⣀⣤⢤⣶⠛⠉⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⣧⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣀⣴⣿⣿⣿⣄⠀⠈⠉⢉⣽⣿⣍⠉⠁⠀⢀⣾⣿⣷⣄⢀⣾⣿⡿⠁⠀⠀⠀⢀⠂⠀⠀⠀⢰⡟
⠀⢿⣆⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⠛⣿⣿⣆⢀⣴⣿⣿⣿⣿⣷⣄⢀⣾⡿⠛⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⡜⠀⠀⠀⢠⡿⠁
⠀⠈⢿⣆⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣅⠀⠀⠹⣿⣿⣿⣿⡿⠃⠘⣿⣿⣿⣿⡿⠁⠀⠀⣹⡟⠁⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⡿⠁
⠀⠀⠀⠻⣧⡀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⢹⣿⡿⠋⠀⠀⠀⠈⢿⣿⡟⠁⣀⣴⠞⠁⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⣴⠟
⠀⠀⠀⠀⠘⢷⣄⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢦⣤⣤⣤⣤⣤⣤⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠳⣄⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣌⡳⢄⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣀⣀⣀⣀⣀⡤⠴⠊⠁⠀⠀⢀⣠⠴⠋⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣿⣦⣀⠀⠀⠀⠈⠙⠓⠒⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠤⢒⣋⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⡶⠾⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿