Happiness Emojis & Symbols

⠀⢠⠤⡀⢠⠈⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⣼⡀⠀⣼⠀⢰⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢹⠀⡠⠚⣆
⡇⠸⠇⠀⢿⡄⠘⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⡘⠆⠁⣠⠋
⠃⠀⠈⠀⢀⣀⣀⣀⠛⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣴⠀⠘⠛⠀⣼⠁⠀⢀
⠀⢃⠀⠀⠀⠀⠀⠈⠛⢀⣤⢠⡇⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠉⠘⠂⠀⠀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠘⠧⣄⠀⠹⠀⠀⠀⠀⠀⠠⠄
⠀⠈⢦⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠐⠶⠀⠀⠛⠀⠉⢈⡵⠈⠹⡆⠀⠀⠀⠀⠀⠀⠈⡇⠀⠳⣤⡀⠀⢀⡤⣤⠁
⠀⠀⠀⠉⠲⣄⠀⢾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⣀⣀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡼⠃⠀⠈
⠀⠀⠀⠀⠀⠈⡀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠉⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠛⠀⠀⠀⢀⡛⠀⢠⡤⠚⠁
⠀⠀⠀⠀⠀⠀⣗⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⣹⠶⠒⠋⠉⠉⠉⠉⠁⠒⠲⣤⡀⠀⠀⠀⠀⠲⡅⠀⠀⠀⠀⣴⠋⠀⢀⡎
⠀⠀⠀⠀⠀⠀⢹⡄⠀⢻⡀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠹⠆⠀⢀⡴⠁⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠈⢃⠀⠘⢣⠀⢠⠞⠁⠀⠀⠀⠛⠃⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠹⡀⠀⠀⡀⠀⢀⡞⠁⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢀⣼⠦⠶⠛⢻⠇⠀⠀⠀⠀⠀⠀⡠⠒⠲⡄⠀⠀⠀⢀⣠⠾⠀⠀⠀⢫⡀⠀⢹⣦⠛⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠋⢁⣼⠀⠀⠀⠘⣇⡀⢸⡟⠀⠀⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢰⠀⠀⠀⠀⢸⣿⡗⠙⠛⠛⠛⠉⠀⢀⣀⣴⣿⡟⠀⠀⠀⠀⠀⠙⡿⢦⣀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⠀⠸⣿⣷⣤⣤⣤⣤⡶⠛⠋⠉⢹⡟⠁⠀⠀⠀⢷⡤⠛⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣷⠀⠀⠀⠀⠹⣿⣿⣿⠿⠟⠐⠒⠒⣴⠟⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⣆⠀⠀⠀⠀⠈⠻⠇⣀⣀⡠⠶⠞⠁⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠙⠶⢦⣤⣤⣤⣤⣤⣤⡴⠶⠛⠉⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠳⠾⠙⠛⠛⠛⠋⠁⠰⠶⠶⠦⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠈⠙⠛⠛⠛⠲⠦⢤⣤⡤⠤⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠐⠛⠋⣉⣁⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠳⣤⡤⠔⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡤⠴⠶⢶⣶⠶⠖⠛⣿⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠈⠉⠉⣿⠉⠙⠛⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠧⠤⣼⡁⠀⠀⠀⠀⠀⠀⡰⠓⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢘⡇⠀⠀⠀⢀⡤⠚⠀⠀⠀⢠⡇
⠀⠀⣀⠀⡀⣀⣠⡴⠖⠉⠀⠀⠀⠀⡀⠘⣧⣀⠰⠊⠉⠀⠀⠀⠀⠠⠺⡇
⠀⢘⠋⠤⠀⣀⣈⣡⡀⣀⡀⠀⢀⡠⠚⠛⠁⠉⢠⡀⠀⠀⠀⠀⡜⠁⣀⠃
⠀⠈⠑⠂⢤⣤⣤⣤⣤⣤⠀⠠⣏⠑⠂⠠⠀⠀⠀⣴⠄⢤⣤⠘⠓⢠⠌⠁
⣤⣤⣀⣤⣤⣤⣤⣤⣤⣠⣤⣤⣄⣉⣤⣤⣶⣶⣶⣤⣶⣤⣤⣌⣁⣠⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣠⣤⣤⣤⣄⣠
⣿⣿⣿⣿⠿⠛⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣀⣤⣌⣀⡀⠀⠐⠒⢭⡓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢛⣫⣭⣭⣭⣭⣇⣀⣀⣀⣐⣿⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⡵⡺⠟⠟⠛⠉⠀⠉⠁⠀⠀⠀⠀⠉⠹⣟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠏⠀⠀⠀⠀⠀⠈⠉⠙⠂⠀⠀⠀⠀⠀⠀⢿⡆⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⢹⠆⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠀⠀⠀⢸⠱⠀⠈⠔⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠙⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠃⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡠⠖⢣⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⢦⢀⣀⢄⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡆⠀⠀⠀⠀⠀⢴⠒⠚⠉⢀⡠⡾⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⢀⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⢶⠒⠚⣉⢴⠇⠀⠀⠀⠀⢀⡾⠁⠀⠀⣀⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠙⠲⠒⠁⠀⠀⠀⢀⣴⠏⠀⢀⡤⣄⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⣀⠀⠀⢀⣀⣠⠴⡟⠴⠒⠊⣁⡠⠟
⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠈⢉⡩⠝⠋⢅⣀⡠⠃⡖⠂⠉⠁
⠀⠀⠀⠀⠠⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠤⡏⠀⠉⢉⣠⢴⠆⠀⠈⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠞⠁⠒⠒⠒⠒⠉⠑⠒⠉⠉⢀⡞⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠤⠬⣉⠑⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⠀⠀⠉⠓⠢⠤⣈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠺⣆⠀⠀⠀⠀⠁⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣯⠙⡖⠒⠒⠃⠤⠤⠔
⠀⠠⢄⠀⠀⠀⠀⠀⢀⡔⢉⡴⠊⠉⠹⠦⠖⠒⠒⠓⠓⠂⢄⡈⠉⠉⠛⠒⠒⠒⣃⣀⠀⠀⠀⢀⠏⣸⠁
⠠⡑⢄⠀⠒⠤⣀⢠⠎⢨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢇⠀⠀⠀⢀⡏⣀⠠⠔⣚⠖⠁
⠀⠘⢄⠳⣀⠀⠀⠉⠲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⡼⠒⠒⠒⠉⠁
⠀⠀⠀⠙⠢⡑⢤⡀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣤⣼⣿⣿⣿⣷⣾⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠟⣁⠀⠀⠀⠀⠀⠀⣀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠁⠚⠉⠀⠀⠀⠀⠀⠀⠉⠃⠀⠸⣿⣿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⠋⠀⢠⣤⡀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠙⠁⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣸⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡀⠀⠀⠀⠔⠀⠀⠀⠀⠐⠀⠀⢀⢆⠠⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠐⡶⠖⠻⠁⢀⡀⠔⠈⠀⠀⡠⠂⠀⠈⢞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣠⠄⢴⡈⠠⠠⠠⣦⠒⠁⣀⠄⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠁⢿⣧⣾⡇⠀⠀⠀⢹⡖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⠄⠺⣇⠀⠀⠀⠀⠋⠈⠀⠀⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠃⠈⠁⣀⠠⠄⠛⠒⢺⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡏⠢⠐⠉⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣤⣤⣤⣤⣤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⠛⠈⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⢰⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣆⣀⡇⠀⠀⠀⡀⣠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢯⣷⣿⠟⠀⠀⠀⠀⢿⣿⣴⣛⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⡨⠊⠀⠀⠀⠀⠀⠈⢛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠁
⣿⣿⣻⣻⣛⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣻⣿⣿⣻⣛⣿⣟⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⢀⢄⡸⠀⠐⢅⡤⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠄⠀⠀⠀⠀⠄⡻⠦⠙⠁⡠⠚⠒⡄⠈⠃⢖⢠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠑⠀⠀⣔⠦⠊⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠪⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡮⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢕⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠑⠉⡄⠀⢀⠀⠀⠂⠀⢢⠀⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠆⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡂⢠⡇⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠙⠛⠶⣄⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠂⠀⠀⠀⠀⠠⠁⢈⠀⠀⠀⠀⠛⢈⠇⠀⠀⢄⠄⢀⠏⠀⠀⠀⠀⠄⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⡀⡀⠁⠐⠚⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⡰⠂⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡄⢴⠀⠀⠀⠀⠀⠀⡄⠀⠘⠀⠈⠁⠄⠠⠀⠠⠀⠠⠤⠡⠈⢢⠈⠢⠀⠀⠄⠁⠀⠀⠊⠀⠁⠌⠑⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠤⠀⠀⠢⡀⠀⠀⠀⠔⠁⡀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠈⠲⠤⡤⠤⠴⡊⠀⠐⠂⢬⢶⢰⠎⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡠⠁⠀⠀⠀⠀⠙⠒⡒⠂⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠄⠀⠀⠤⠀⠈⠧⠀⠚⠏⢈⣟⠚⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠀⠀⠀⠸⠁⠀⠁⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠄⣠⠐⠓⡂⠀⠀⢐⠀⢀⠂⠀⠀⢄⠀⠙⠃⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠃⠀⠘⠲⠄⠀⠀⠀⢀⠀⠀⢀⡤⠊⠁⠀⢀⠂⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠨⠀⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠐⠀⠃⠀⠀⠀⠀⠀⣆⠀⢀⡆⠀⠀⠀⠀⠀⠄⠀⠈⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠰⣀⠃⠀⠀⠀⠀⠀⡂⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠈⠳⠀⠀⠀⠀⠀⠶⠂⠀⠰⠤⠀⠀⠀⠀⠠⠬⠉⠀⠉⢥⠄

⠀⠀⣠⠖⠒⠒⢦⣀⣀⠀⠀⠀⢀⣠⣀
⠀⢰⠁⠀⠀⡞⠁⠀⠈⠱⡄⡴⠁⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⣇⠀⠀⠱⡀⠀⠀⢀⡿⡇⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠉⠲⡄
⠀⠀⠈⠲⣄⣠⠟⠢⣤⠞⢀⣧⣀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⡆⢸⡄
⠀⠀⠀⠀⠀⠻⡇⠘⡟⠀⡽⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡴⠉⠀⠀⠀⠀⠀⠉⠲⡄⠀⡇⠀⠀⠀⠀⠀⢠⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣼⣣⠞⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢜⣾⠁⠀⠀⣀⠔⢁⣠⠔⠀⣹⠀⠸⡄⠀⠀⠀⠀⠆⢀⡞
⠀⠀⠀⠀⠀⡔⠲⡿⠋⢠⠔⠋⠉⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⢴⣮⠭⣿⡖⣊⣉⡀⣈⡩⠤⠔⠊⠟⣆⠀⠘⢄⡀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢇⢀⣽⠀⡏⣠⠴⠖⠒⠒⠒⠒⠦⡀⢷⠀⠀⠀⢀⠀⠀⠀⠀⢯⡇⠠⠔⠲⠄⠀⠘⠉⠉⢸⠃⠀⠀⠀⢹⣻
⠀⠀⠀⣠⠔⣿⣿⠀⠀⣷⣟⠒⠒⠒⠒⠒⠒⠒⣇⣾⠀⠀⠀⠻⣉⡗⠀⠀⠀⠙⢄⠀⠀⠢⠤⠔⠂⣠⠏⠀⠀⠀⠀⢀⠧⣀
⠀⠀⠀⠁⠚⢹⢻⠀⠀⣿⢉⡭⡛⠒⠒⠚⠻⠥⠈⣽⡄⠀⠀⠀⠈⠳⢄⠀⠀⠀⠈⢑⡄⢤⠄⠐⠞⢹⣦⣀⣀⡠⠤⠺⣯⡼⠃
⠀⠀⠀⠀⠀⠛⠈⢧⠠⡻⡄⠀⢀⣀⠁⠀⣠⠀⣰⠃⠙⡄⠀⣠⡤⢤⠀⠙⠲⠤⣤⠚⠙⠪⡖⠐⡖⠁⠀⠙⣄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣇⠈⠒⣤⣬⣭⠥⣴⣊⣁⣀⣤⡿⠔⠓⠤⠼⠃⠀⢀⠔⢧⠀⢀⠀⡇⠀⠱⣀⡤⠴⠉⣆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢪⡀⠉⠉⡇⠈⠉⠀⠈⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡉⠉⢹⠧⠤⠔⢻⣀⡴⠚⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢹⠇⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡲⠊⠀⡀⠀⠀⢻⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣴⠛⣆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⢧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⠖⠋⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡦⠇⠘⠦⡤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⢀⣀⣀⣀⣀⣤⣴⠾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣇⠀⠉⠉⠉⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣵⡇⠀⠀⠀⠾⣩⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡀⠀⠀⠀⠀⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣀⢀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠔⠆
⢀⡀⠀⠀⢠⠆⠀⠀⠀⢀⠀⠤⠀⠖⠂⠀⠀⠈⠉⠁⠒⠒⠊⠋⠁⠀⠀⠁⢸⠠⠤⠐⠒⠈⠁⠀⠁⠈
⠆⠁⠔⠀⠘⠀⢀⠠⠀⠤⠤⠤⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠴
⠀⠐⠈⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⡀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⡔⡈⠰⠄⠀⠁⠌⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠤⠤⠤⠐⠒⢒⡔⠃⠀⠈⠐⢲⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠇⢸⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢈⠀⠸⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢈⡀⠀⡆⠀⢀⣀⣀⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⠉⡋⠉⠀⠀⠈⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⠀⠀⠀⠀⣃⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠹⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⢀⠒⠈⠀⢘⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠀⠒⠒⠊⠁⠀⠀⠀⠀⠉⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠄⡀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡠⠂⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠖⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠐⠀⠠⠁⠀⠀⠀⢠⠦⡀⠀⠐⠠⠀⡀⠀⠈⠉⠁⠀⠀⡩⠀⠀⠀⠀⠀⠰⠢⠔⠘⠐⢃⡮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⠑⠂⢄⣄⣀⣀⢀⣀⣠⡔⢺⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣠⣶⠤⢤⢀⠀⣀⣠⡛⢡⡌⢊⡷⠊⠀⣋⠏⠀⠀⠀⠀⠀⢀⠎⠀⠈⠁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢒⢢⠤⡴⣽⠀⠶⠀⡩⠗⢀⠉⠶⡤⠤⣢⠃⠀⠀⡇⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠆⠀⠈⢓⡒⠚⠋⠀⠠⠴⠀⠀⠀⠮⠹⢀⡜⠀⠀⣀⠤⠒⠁⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⡀⠧⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠪⠗⠲⡍⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣠⡜⠢⠤⣄⣀⣀⣀⡀⠤⠴⠊⢡⠃⠀⢀⡁⠘⡀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⠸⢀⡀⠀⠀⠑⢄⠈⢢⠄⠲⠀⠃⡤⠔⠉⢡⠀⢃⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠆⠀⠈⡗⠂⠐⠁⠁⠺⡀⠀⡏⢠⠂⠀⠀⠀⡀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠸⠀⠀⠀⠀⠀⠀⠙⠢⡝⠞⡐⠁⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠄⠀⡆⠀⠀⠀⠀⠀⠈⠄⡇⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡰⡇⠀⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡌⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠰⠇⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⣮⢀⠀⠠⠤⠤⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠆⠰⡘⠾⠁⠀⠀⠀⠉⠀⣀⣀⡀⠷⠠⠤⠤⠀⠒⠠⠜⠒⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠄⠀⣸⠀⠀⠈⢻⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢢⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠐⠞⣰⢠⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⢄⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢱⡋⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢀⣠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠈⠉⠉⠉⠀⠈⠴⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠀⣤⣀⠐⠆⢄⠂⠀⠀⠀⢀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⠉⠁⠀⠑⡐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠂⠀⠈⢦⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢰⣧⠀⠀⠀⢰⣦⡄⠀⠀⠀⠑⡦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡗⠀⠀⠀⡀⠀⠀⢀⠀⢨⠀⠁⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣦⡀⠀⠀⠉⠁⢠⡄⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠙⢥⠀⠺⣶⡶⠟⠁⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⣀⠈⠉⠁⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠘⠡⠀⠤⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⣀⠤⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠒⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠐⠄⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⡀
⠀⠀⠀⢤⡀⠀⢀⠠⠘⠈⠀⠀⠐⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⠀⠀⠀⠃
⠀⠂⠐⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠂
⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠲⡄⠀⠀⢨⠁⠀⡅⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠰⠶⠇⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠇⠀⠀⠀⡇⠘⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⢀⣠⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⡅⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⢲⠂⠋⠀⠀⠈⠳⠒⠲⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠘⠈⠁⡇⠀⠀⠀⠀⠉⠉⢋⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠎⠀⠀⢀⠀⠀⠀⣰⠀⠀⠀⡠⠀⠀⠀⠀⠈⢑⡦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⣀⣀⣀⡀⢀⡀⣀⣠⠇⠀⢰⡋⠀⠀⠀⠀⠉⠀⣀⡠⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣴⣾⣿⣯⣶⣶⣶⣶⣷⣷⣷⣶⣿⣿⣶⣧⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⠉⢉⡉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢀⠄⢸⡀⡀⢀⢀⠄⡀⠀⢀⡏⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢀⠀⣀⢄⢠⢰⣉⠀⡇⢁⢘⠀⣙⢈⣢⢸⠁⢸⢸⢸⠒⠁⠀⢸⠃⠀⡇⡈⢁⢀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⢸⠀⠘⠌⠠⠀⠁⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠊⠈⠤⠌⠘⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⡤⠤⡤⠜⠙⠉⠓⢲⣀⠀⠀⠀⠀⠰⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀⢀⠔⠃⠀⠀⠀⠀⠀⠀⡤⣀⡀⠀⠀⠋⠙⠊⠳⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣃⠔⠋⠁⠀⠀⠀⠀⢨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠠⡈⢩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⢐⠉⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠠⢌⣒⡲⠄⢺⠁⠀⠙⠀⠀⠀⣠⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣄⣀⠦⠬⠭⠹⣇⠀⠀⠀⢀⡠⠟⠁⠘⠤⡄⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣸⢸⠀⡝⣿⢿⠀⠀⠀⠸⢤⣀⡶⠞⠀⠀⠀⠀⠀⠈⠉⠓⠋⠒⠩⡃⠀⠀⠀⢦⣀⠀⠀⢀⡹
⠀⠀⠀⠀⠀⠀⠈⡄⠀⢀⡏⡼⢠⠃⠉⠀⠀⠀⠀⠀⢸⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢅⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢰⠃⡾⠀⠀⠀⠀⠀⠀⠤⡄⡄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡆⠀⠀⠀⠀⠀⠀⢤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⠜⠀⠀⢴⠆⠀⠀⠀⠀⢁⠡⡀⠀⠀⠀⠠⣞⡀⠀⠀⠀⠀⣤⠀⠀⠈⢆⣀⠀⡀⢀⡖⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣠⢷⢤⡀⠀⠀⠠⣀⠄⠀⠀⠀⠀⠀⠀⢀⠴⠩⢋⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢷⣦⠤⠤⠤⢤⢴⣖⠉⠁⠴⡄⠸⣗⡲⠤⠄⢤⡤⠤⠤⠤⢶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠡⠀⢻⡇⠀⠀⡾⠞⠉⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠧⠴⠟⠁⠀⠀⠟⠃⠰⢷⠄⢀⡀⠀⢤⠄⢢⢄⡘⠚⠷⡀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠦⠀⣀⡀⠈⠙⠀⢸⠀⠀⠉⠉⠂⠼⠟⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠐⣠⠄⠀⠉⠀⡒⠦⢼⣀⢀⠀⡀⢀⠀⡀⠠⣐⠢⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠩⠧⠤⠾⠦⡤⠴⠾⠳⠈⠀⠑⠚⠂⠈⠁⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⢠⠀⠀⠀⢨⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢀⠀⠈⡀⠀⠀⠈⢂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡄⠌⠸⡄⡤⠂⠀⠀⠐⠤⢠⠃⠈⠢⡄⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⢄⠀⢠⣀⠧⡀⠀⠀⢠⠲⠴⠀⠀⡠⠆⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠲⠂⠈⠉⠉⠉⠉⠉⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡁⠀⠁⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠸⠇⢀⠀⢀⡠⠄⢲⡾⠉⠉⠉⠉⠉⠉⠉⠀⠒⠒⠂⠠⠤⢈⣳⠀⠀⠀⠀⢄⠀⠀⠃
⠀⠀⠀⠀⠆⠀⣾⠞⠁⠀⠀⢈⡴⢂⣈⠁⠢⡀⠀⠀⠀⠀⡠⢊⣡⡌⠑⢤⡈⠑⠤⣀⠀⠀⠀⢸
⠀⢀⡤⠞⠂⢴⡇⠀⠀⠀⠰⠋⣰⣿⣯⣴⣦⣸⡆⠀⠀⢠⣧⠼⠿⠷⠷⠤⣵⠀⠀⠘⡷⠀⠀⢸
⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠙⡗⠒⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⠸
⠀⠀⠀⢇⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠀⠀⠀⠀⠀⢀⡏⠀⠉⢿
⠀⠀⠀⠘⢧⡀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠘⣏⠃⠠⠴⠦⠖⠘⢋⡼⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠈⠳⢤⣀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠑⠦⣤⣤⣤⠴⠋⠁⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣐⠂⠲⡄⠀⢀⠔⠁⣠⠞⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠳⢦⣀⡀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢈⣤⡁⠀⠘⠣⠚⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⡶⠶⠶⣶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠓⠀⠀⠰⣉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⠇⠀⠀⢷⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠈⠯⣀⠀⠀⠀⢀⣨⠋⠈⠁⠀⠙⠛⠒⢶⡆⠠⠤⠴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢤⣤⡤⣤⠞⠓⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠤⠤⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣸⡧⣤⣤⠤⠤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⠘⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠘⢄⢸⣧⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⢶⡿
⠀⠀⠀⠀⣌⠁⠀⠀⠀⠀⠈⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⢃⠀⠀⠀⠀⠀⢀⡴⣦⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢤⣀⡘⠀⣴⠀⠘⣧⣈⡄⠀⠀⠀⢀⡏⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠉⠛⠦⠻⡤⠚⠀⠈⠀⠀⠀⠀⣘⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠶⠒⠀⠀⠀⡟⠀⠀⠀⢸⡄⠀⠀⠠⠴⠶⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣀⣀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣰⠇⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⢀⠴⠚⠓⠠⣄⠀⠀⠀⠀⠀⠀⠀⡌⠉⢻⡄⠀⠀⠀⠀⠀⢸⠇⠀⣠⣤⠤⠴⠦⠄⢈⣦
⡰⠃⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠙⢒⠦⣷⠀⠀⠀⠀⠀⢸⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣧⣀⣀⣀⣀⣀⣀⣀⣘⣁⢀⠀⠀⠀⢘⠋⠀⣼⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣽⣿⣷⣶⣶⣶⣶⣶⣾⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⣉⣉⣭⢵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣄⣀⣀⣀⣀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠭⣦⣤⠤⠒⢂⣈⠴⢦⡖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⣁⡤⠤⠶⠦⠶⠤⠤⣀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠱⡀⠣⣀⠿⠗
⠀⠀⠀⠀⠀⠀⠀⢀⣯⠖⣩⠴⠒⠉⠉⣀⣀⣉⣉⠒⢬⣑⣺⠀⠀⠀⠀⠀⠀⢀⡀⠆⠃⠈⠀⡇⠀⠀⠀⠀⠀⢀⡀⠠⢄⣀⡀⠀⠀⠈⠙⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣡⠊⢁⡤⣖⣋⡩⠤⠴⠦⣀⠽⠲⡞⣇⠀⠀⠀⠀⠀⡀⠂⠁⠀⠀⠀⡠⠁⠀⣠⠴⠒⠉⠀⠀⠀⡀⠀⠈⠙⢦⠀⠀⠀⣰⢤
⠀⠀⠀⠀⠀⠀⠀⠈⣇⡾⠋⣠⠤⠄⠀⡀⠀⠚⠉⠂⠀⣷⠏⠀⣀⣀⡀⠀⠀⠀⠐⠂⣴⠞⡧⠀⢞⠁⠀⠀⠀⠀⢀⡦⠵⢄⡒⠰⣬⡇⠀⠀⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢏⢳⡀⠁⠀⠀⢀⠀⢠⠄⠀⠀⣸⠇⠀⣸⣡⡤⠟⠀⠀⠀⠀⠀⠈⠱⡇⠀⣾⠒⡲⠖⠈⠉⠀⠀⠀⠁⠈⠁⢸⣹⢤⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢄⡀⠀⠀⠉⠁⠀⣠⠴⠃⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⢿⡀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⢸⠁⠀⣸⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠙⢒⠒⡟⣯⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠢⡑⢤⡀⠀⠀⠈⠲⠁⠀⣀⡴⢷⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠒⠼⣾⠤⠼⣦⠠⢾⣉⣉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠛⠯⣉⣱⠒⠿⠋⢯⡥⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣊⠀⠀⠀⠀⠀⢘⣦⡤⠼⠓⠶⠤⠤⢤⡄⠀⠀⠀⣠⣞⣉⡽⠇⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡆
⠀⠀⢰⠒⢦⡀⠀⠀⠀⠈⠉⡳⠒⠂⢹⠧⣄⡀⠀⠀⠀⠀⠀⠀⡟⠒⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⡇
⠀⠀⠀⠑⠦⣼⠂⠀⠀⠀⡴⠁⠀⢠⡏⠀⠀⠈⠉⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠻⡖⠛⠁
⠀⠀⠀⠀⠀⠈⠳⡀⠠⡞⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠜⢦⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣷⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣀⣴⣶⣾⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⡿⠋⠉⠀⠉⠀⠀⠀⠀⠃⠀⢰⣿⡟⠉⠉⠛⠻⠛⠛⣿⠀⠀⠀⠀⠀⢻⡿⠋⠉⠉⠀⠉⠙⢻⣿⣿⠃
⠀⠀⠀⣀⠀⠈⡝⠀⠀⠀⠀⣀⣀⠀⠀⢀⠀⣸⣿⡆⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣿⣏
⠀⠀⠘⠋⠉⠠⣿⣄⠀⠀⠀⠉⠉⠀⠀⡈⠀⣏⡃⠀⠀⠐⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣠⡀⠁⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠪⠙⠻⠿⣶⠀⡶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠐⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠁⠀⠡⠘⠁⠀⠀⠀⠀⠀⠀⠀⡄⢀⠂⠊⠀⠐⠀⠀⠀⠀⠈⠀⠀⣠⠃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡤⡄⠀⠀⠀⠀⠀⠄⠀⡈⠀⠀⠀⠀⢢⢀
⠀⠀⠀⠀⢠⢨⡇⠀⠀⠀⣀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⢇⠀⠀⠀⡀⠀⠀⠔⠳⠀⠀⠀⠀⠊⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⡂⢐⣐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⠀⠀⠀⠀⠠⠔⠃⠃⠀⠀⠀⠀⠀⠈⠙⠒⠂⠈⠛⠍

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠁⠀⠉⠉⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢤⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⢀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⡄⠀⠀⠀⠀⠀⠠⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢴⣶⣾⣿⡿⠶⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢻⡛⠶⣤⣀⡀⠀⠀⠀⢀⣀⣤⣾⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠻⢦⣀⠉⠉⠙⠛⠛⣉⣽⠟⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠉⠛⠲⠖⠚⠛⠉⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⣀⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠤⠄⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠂⠀⠈⠉⠉⠒⠂⠄⡀⠀⠀⠀⡀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠑⠶⠚⠁⢀⣄⠐⠁⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠐⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⢰⠁⡌⠀⠀⠆⠀⠈⠉⠉⠉⠀⠉⢉⠇⠀⠐⡏⠀⠀⠀⠀⠙⠢⠤⠤⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⡤⠼⢧⡇⠀⠀⣇⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠘⠦⣀⣉⣁⡤⠚⠁⠀⠀⠀⠀⢈⡴
⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠶⠉
⠐⠀⡱⠐⠒⠈⠀⠀⠀⠁⠒⠠⠤⣩⣐⠠⢄⣀⡙⠦⣄⣀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⢀⠠⢐⡠⠆⠑⢄⡀⠀⡀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠥⣁⣉⡹⢛⣛⡻⠛⢛⡋⠀⠤⠤⠄⠒⠂⠉⠁⠀⠀⠀⠀⠉⠈⠃⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⢠⡄⢀⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⠶⠤⠤⠴⠖⠋⠡⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠠⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣠⣤⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠀⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠠⠄⠀⠄⣀⠀⠀⠀⢀⠔⠀⠉⣠⣀⣀⠤⠤⠤⣀⠀⠀⢸⣄⡀
⠀⠀⠀⠀⢀⠤⠒⠉⠉⠉⠉⠀⠀⠉⠑⠢⣀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⠁⣠⠖⠊⠁⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠙⡆
⠀⠀⠀⠔⠁⢀⠶⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⢂⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡌
⠀⠀⠄⠠⠚⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⠋⢠⡠⠤⠴⠒⠒⠒⠦⠤⣸⠀⠀⠀⠘⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢣
⠀⠀⡚⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠤⣀⠀⠀⠀⡇⡌⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⡀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡗⣦⠂
⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠙⠢⣤⠃⡇⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⣰⢓⡕
⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠀⡅⠀⠸⡄⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠦⡷⣄⠀⠀⠠⠄⠀⡊⠀⠀⢀⠔⠁
⠰⠦⣀⠀⠑⠤⡀⠈⠂⠤⠤⠔⠁⢀⡠⠞⠁⣠⣴⠐⡄⢀⣘⣄⠈⠀⠐⠁⠺⢅⣀⠀⠀⠀⠀⢀⡜⠀⠈⠑⠲⠤⣤⠭⣤⡔⠚⠁
⠀⠀⠈⠙⡊⠒⡠⠋⢒⠒⠒⡖⠺⡥⠔⢲⠋⠀⢀⣀⣀⡀⡀⠀⢹⠒⠲⠶⡶⣶⠋⣉⠉⢉⡉⣩⡆⢀⣀⠂⢒⠎⢣⡀⡀⠈⢇⠒⣆⣀⣀⡀⡀
⠀⠀⠀⠀⠉⡝⠁⠀⠀⠀⠀⠀⠀⠹⡂⠊⠀⠀⠚⠃⠈⠉⠣⣀⠃⠀⠀⠈⠀⢸⣀⡠⠋⠉⢱⡄⠊⠉⠈⢆⠂⠀⠀⠀⠀⠀⠀⠳⡃⠀⠀⠉⠋
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⡠⠴⠦⢈⡁⠒⠒⣦⠤⠤⠤⠀⠒⢞⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠸⣀⠀⠀⠠⠤⠤⠤⠤⠤⠤⢾⣀⡀
⠀⠀⠙⣆⠀⠀⠀⠉⠚⠁⠓⢠⠄⠐⠒⠚⠀⠀⠀⠀⠪⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⠀⠀⠀⣆⣀⣀⣀⡀⢢⡠⠤⠔⠚⠁⡼⠁
⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢶⠤⠤⠖⠒⠊⠱⣄⠀⣠⠄⠀⠀⢨⠃⠀⠀⠀⠈⠀⠀⠀⠀⠸⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠈⠳⠃⠀⠓⢦⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠻⡼⠀⠀⠀⠀⣸⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢰⠛⢦⠀⣠⠴⠚⠉⠉⠁⠀⠀⠀⠀⠉⠉⠓⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⡏⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠻⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠋⠉⠲⢄⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⣿⠀⠀⠀⠀⠀⡴⠋⡇
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⣄⠀⠀⣠⢦⡀⠀⢠⡄⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠞⠁⠀⢹
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⠤⠴⠁⣀⣁⠈⠁⠀⠉⠒⣁⠈⠂⢤⡤⠴⡞⠁⠀⠀⢀⡴⠃⣀⣀⣠⡼
⠀⠀⠀⠀⠀⢀⡇⠀⡄⢠⠀⠀⠀⠀⠀⠀⣀⠀⡸⠀⢠⠊⠁⡈⠑⡄⠀⠀⠀⠸⣴⠋⣠⡈⢣⣧⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⢈⠉⠀⠀⢸⠀⠺⠿⠂⡄⠀⠀⠀⠀⢷⠈⠛⠋⢸⣧
⠀⠀⠀⠀⠀⠀⣇⠀⢀⠀⠀⠔⠉⠁⠙⠦⠜⠀⠀⠀⠀⠑⠤⠤⠖⠁⠀⠀⠀⠀⢸⠉⠒⠀⠁⢸⠀⠀⠀⠀⣶⠒⠒⠒⠤⢤
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠨⠀⠀⠴⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢸⡇⠀⠀⠀⠈⢳⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠳⢄⣀⡀⠀⠀⠀⠀⡋⠳⣄⠀⠀⠀⠀⠀⠀⠀⠤⠤⠇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠹⡄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⢸⠇⠀⠀⠀⠀⢠⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡞⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⠑⠲⠤⠤⣀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣀⡤⠒⠁⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣳⠃⠀⠀⠀⢀⠀⠀⠀⠀⠈⠢⣀⠀⢠⡏⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠇⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⠓⠋⣸⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠸⡄⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠉⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠑⡄⠀⠀⠀⠀⣰⠛⠦⣄⡀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠢⣄⣀⠞⢧⠀⠀⠀⠉⠒⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⡔⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣠⠞⠉⠉⠓⠦⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠆⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⠀⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠆⠀⠀⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠡⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠠⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠉⠒⠢⣄⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣀⠖⠒⠉⠉⠉⠁⠒⠢⢌⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠔⠁⠈⢀⡀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣽⠢⠀⠑⢄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡇⠀⠀⡤⠀⠀⠤⠄⢀⣠⣾⡇⠀⠀⠀⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠻⣿⣶⡶⠿⠛⢻⡟⠀⠀⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠉⠻⠄⠀⠔⠋⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⠤⠤⠤⡤⣤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⢠⠃
⡄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠠⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⡎
⠱⡀⠀⠀⠢⡀⠀⠀⠀⠀⠑⠀⠀⢄⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠘⢎⠀⠀⠈⠂⡤⢆⠀⠀⣀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠑⠦⡀⠄⠀⠀⡀⠀⠀⠁⠂⠤⣀⠈⠢⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠦⠤⡄⠀⠀⠀⠀⠀⠀⠈⠶⠥⣄⠀⠀⠀⠢⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄⠀⠀⠈⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡸⠁⣀⠀⠀⣀⠜⠀⠀⠈⠢⣀⠌⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣦⢸⣧⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠿⠛⠉⢈⠶⠒⡕⠊⢁⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⠁⠀⡤⠊⣠⢺⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⠀⢞⣽⠟⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⣨⠀⡼⠟⠋⠉⢈⣀⠉⠉⣲⣶⠀⠀⠀⠀⠀⠉⡢⠛⢄⠈⠻⣮⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⡞⣁⡄⠀⠀⠟⠉⠀⠀⡌⢠⡑⢄⠀⠀⠀⠀⢹⡄⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⢻⠃⠀⠀⠀⠻⠂⠀⣈⠙⡦⠈⣧⠀⠀⣰⣷⡑⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⡀⣠⠔⠂⠀⠉⠉⣀⠀⢸⠀⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣄⠀⠃⠃⢲⣒⣀⣭⣭⡾⠀⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⡀⢀⠀⠻⠿⠿⠟⠁⣰⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡑⠤⢀⠠⢔⣵⣿⣿⣿⣿⣟⣡⣴⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⡟⠣⢅⡻⡿⢿⠿⢿⣿⣽⣿⣷⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠓⠤⣦⠀⠈⢦⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡙⠀⢀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠻⣿⢻⢻⠋⠟⢳⢦⣄⡀⠀⠀⠀⠈⡅⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣄⠂⠄⠀⡜⡼⠸⠻⣦⣀⣴⠞⠓⢦⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣾⠀⢠⢡⢡⠀⠃⢡⠹⠁⠘⢘⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢳⠆⠇⡎⠂⠀⣆⡾⠋⢂⣤⣴⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⡁⠘⠸⡸⠱⣆⡴⠋⣠⡴⡞⠟⢆⠇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠂⠃⢠⠀⢡⡥⣫⠴⠻⣯⠀⠀⠀⠘⠘⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣴⡜⠀⣞⣴⠞⠋⠀⠀⠀⠘⢿⣄⢧⠁⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣷⡋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣴⣮⣼⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⣀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⢦⣾⡄⢠⠾⢁⡐
⠀⠀⠀⠀⢰⡄⠈⠆⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢈⣄⠀⠀⠀⠁⢠⠄⠁
⠀⢀⠀⠀⠀⠙⣄⠈⢀⠀⡀⢰⠀⠀⠀⡀⠀⢀⠀⠀⠀⠛⢥⡄⠈⠉⠓⠒⠲⡄
⠀⠈⢢⡀⠑⢆⡀⠆⠈⠴⠗⠈⢀⠀⠘⠀⡤⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠈⢢⣄⡙⠞⠁⢀⡠⠤⠛⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠀⡀⠀⠃
⠀⠀⢠⡄⠀⠉⢁⡀⠀⠘⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢂⠀⠂
⠀⠀⠀⠀⠀⠀⠈⢣⣀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⢤⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠚⠁⣼⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⣀⣈⣧⠀⢡⡀⠀⠀⣀⣀⣀⣀⡠⠴⠶⠚⠋⠁⠀⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠃⡀⠀⠀⠀⠀⠀⠀⣀⠞⠉⠙⠛⠀⠉⠉⠉⠉⠙⠛⠿⣷⡀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢡⠀⠀⠀⣠⠶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠲⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣃⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠁⣠⣿⡆⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢠⠀⠀⢷⡖⠒⠀⠶⠶⠶⠊⠉⠁⠀⠀⣠⣴⣿⣿⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸⠄⠀⠀⢲⣄⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣦⠄⠀⠀⠀⠀⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠁⢆⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁⠉⠁⠈⣿⡿⣿⠇⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠙⠻⢇⣀⠄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠈⡍⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠉⣠⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣠⣾⡋⠀⠀⠀⠀⠀⠀⢀⣰⠞⠃
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠉⡛⢿⠓⠒⠒⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠶⢶⣶⠶⠖⠋⢻⠉⠲⣤⠀⣃⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⢨⠷⡀⠀⣾⠀⠀⠀⠈⠋⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⢤⣤⡀⣀⣀⣀⣀⠀⣤⡴⠛⠁⠀⠀⠈⠀⢃⣴⠁⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠴⣦⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠈⢃⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⠃⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠤⠖⠛⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⢀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢇⠀⠀⠀⠀⣸⠘⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣠⠾⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⠳⣄⣀⡴⠉⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⣉⣠⠶⠛⢳⣄⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠉⢀⣴⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⡀⣀⣠⠤⠚⠛⠉⠁⠀⠀⠀⠹⣦⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡏⠻⡀⠘⠿⠒⠲⠶⠒⠒⠒⠘⠋⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠘⠦⣀⠀⠀⠀⠀⢀⡟⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠈⠓⠶⠶⠶⠋⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠴⠤⠴⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠣⠴⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣄⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠲⣄⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡀⠀⢨⡛⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⢄⠀⣧⡉⣷⣾⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠀⠘⢳⡆⠀⣤⣿⡇⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢆⢸⣸⣧⡰⡶⣏⣳⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⣄⣦⢀⡈⢦⣀⣨⣯⣿⣯⣿⣿⣟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣻⡄⡤⣀⣻⢿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⡟⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢈⣷⣠⣿⣴⣤⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣀⠀⣤⣦⣤⣠⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡟⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⣀⣀⡾⠀⠟⣴⣿⣯⠤⢤⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⢻⣄⠀⢉⣡⠀⢀⡶⢸⠏⢀⡶⢀⡟⠻⣦
⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⣶⣿⠇⣼⠟⠛⠛⡇⠀⠚⠁⡄⠀⢨⣴⠿⢃⣴⣿⠁
⠀⠀⠀⠀⠀⢸⣇⢠⣾⣟⣡⡏⣸⠏⢀⣴⣿⣷⣤⣴⣾⣿⣤⣤⣴⣾⣿⣟⣡⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣻⠟⠋⠉⠉⣀⠉⣀⠛⢁⣴⠀⣸⠃⢈⡤⢀⡉⠀⣡⠄⢈⠁⢀⡟⠀⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣤⣴⠞⠀⣿⠀⡟⠀⠰⠃⢰⡏⢠⡿⠃⢨⠁⣼⠟⠀⡍⠀⠞⠀⠘⣉⠀⠉⣿⡄
⠀⠀⠀⠀⣿⠙⠶⠟⠋⠀⣸⡇⠀⢿⣤⣴⣦⡔⠁⣦⣤⣶⠇⢰⣤⣴⣾⣧⣴⠎⣰⣿⠟⠻⢷⣿⠇
⠀⠀⠀⠀⠹⣷⣄⣀⣠⣾⣿⣧⣀⣀⣿⡍⡿⠀⢸⣿⢉⡏⠀⣿⠃⠘⠷⠟⠃⣰⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠙⠛⠛⠋⠘⣷⣤⣿⠇⣿⣄⣰⣿⣆⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠋⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⠬⠶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢀⡀⢀⡀⠀⢀⣿⠁
⠀⠠⠖⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠸⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢳⠀⢀⣠⡀⠀⠠⡀⠢⡀
⠀⠁⠀⠀⠸⡉⢢⡀⢀⠴⠲⡆⠀⠀⠀⠀⠀⠀⣠⡴⠶⠿⠆⠀⠠⠶⠿⣥⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡇⢸⣴⠋⣠⠃⠀⠀⢸⡄⠹
⠀⠀⠺⣉⠳⣝⡦⠱⣼⢠⠎⠀⠀⠀⠀⠀⠀⡾⢡⡖⠀⠀⠀⠀⠀⠀⠀⣤⡙⢷⡀⠀⠀⠀⢸⣡⠤⡝⠳⣄⠀⠸⠵⠒⡦⠀⠀⡇
⠀⠀⢀⠨⠑⡎⢀⡞⠁⠈⡆⠀⠀⠀⠀⠀⢸⡇⢠⠻⣄⠀⠀⠀⠀⠀⠀⣠⣟⠈⢷⠀⠀⠀⠀⠀⠀⡆⠀⠈⠇⠀⠶⣉⠁⠀⢠⠇
⠀⠀⠙⠒⢲⠆⠈⠀⠀⠰⣇⠀⠀⠀⠀⠀⣼⠀⢸⡇⠈⠓⠶⠤⠤⠴⠞⠁⢸⠀⢸⣇⠀⠀⠀⠀⢀⡽⢀⣄⣤⠞⠒⠾⠃⠀⠈
⠀⠀⠀⠐⠧⠊⠙⠒⠚⠳⡈⠳⣄⠀⠀⢰⡋⠁⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⣸⠄⠘⠉⣹⠀⠀⣠⠎⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠣⡀⠈⢻⠁⠀⣿⣿⣷⣶⣤⣤⣤⣶⣾⣿⠀⠈⣿⠉⢀⡞⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠢⣼⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡴⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⠀⠀⠈⣿⠿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠘⢧⡀⠀⠉⢉⡽⠋⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣄⠀⠙⠒⠚⠋⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢤⣀⠈⠙⠶⢤⡤⠶⠋⠀⠀⣀⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠓⠲⠶⠶⠶⠶⠚⠋⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡠⠂⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢁⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡖⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠑⡄⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡇⢸⡄⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⠋⣻⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠳⣄⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠴⠾⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢦⡀⠀⢈⡟⣷⠶⢤⣤⣤⣤⣤⣤⣤⡶⠶⢾⠻⡁⢀⡴⠋⠁⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠙⠲⠞⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠋⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣠⡴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠐⢦⡈⡷⡄⣾⡰⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠋⠙⡷⣧⠘⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢤⡴⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢚⣛⢦⣠⣾⣮⣭⣭⣚⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⡾⠛⣹⡟⣿⠙⡟⢟⠻⢿⣶⣦⣍⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠴⠊⠀⢀⡠⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣴⣿⣏⣠⠞⢁⢣⠃⡧⠇⡈⠇⢀⣿⠻⣷⣍⠺⣾⡣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢯⡀⠀⠋⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⢾⡟⣿⡞⢏⡡⠞⠻⠽⠀⠈⠳⠃⠈⠋⣄⠈⠻⣳⣜⢷⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡏⣾⢡⣹⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠙⣿⠻⣧⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡸⢠⡏⢸⡇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⠁⠀⡴⣂⠀⢹⡀⢻⡄⡇⠀⠀⠀⠀⠀⠀⡴⠉⠙⣦⠤⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⢨⠱⣾⠇⢠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⢸⡇⢸⢷⢸⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⢸⠀⣿⡀⠸⣌⡸⠀⠀⠀⢄⠀⠀⠀⢀⡆⠀⠀⠀⣠⡇⢠⣸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠘⣠⡿⡏⠲⣀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⣠⠞⠁⡇⢸⣿⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣦⣿⠁⡇⠀⠀⠉⠒⠤⣤⣀⣀⣀⢠⡤⠲⣿⡁⠀⠀⠘⠮⣷⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡏⠀⠀⠀⢀⠴⠊⣿⠀⢀⡠⠴⠳⣄⣷⠙⠦⣀⠀⠀⠈⡶⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡇⣿⡀⠀⠈⠳⠦⣼⣿⡦⠋⠀⠀⠀⠀⢹⡏⠓⠚⠉⠀⠀⡇⢀⠱⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⡇⡇⠀⠀⠀⢀⠇⡇⢥⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⡦⠈⢆⡃⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢸⣵⠃⠇⠀⠀⢀⡌⢸⠀⠸⡄⠀⠀⠀⠀⡟⢨⢣⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡜⢀⠇⠀⠀⠹⣀⣤⣄⣼⠁⢸⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⡜⠀⠀⠀⠀⡟⠟⠻⠿⠆⢸⡀⠂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⡄⢰⢠⣏⠟⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠰⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠆⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠶⡄⣄⠇⠈⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠖⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⣠⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠥⠈⠤⠵⠮⢤⣶⠦⢤⣀⠀⠀⠀⠀⣀⣀⣀⣀⠾⠤⣀⣁⣠⡇⠀⠀⠀⠀⠀⠀⣀⣧⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠈⠉⠉⠉⠉⠀⠈⠙⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣤⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⡠⢀⣀⣀⣀⣀⣀⣀⠀⠀⣾⣿⡿⠛⠛⠿⢿⣿⡿⢿⣿⣯⠉⠀⠀⠀⠀⠀⢐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⢀⢈⣾⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠄⣿⣿⣿⠟⠙⠛⠻⠿⢿⣿⣿⡿⣿⠇⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠏⠋⠙⠁⠈⠁⠀⠈⠛⣿⣿⣿
⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠐⠃⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡟
⠀⠀⠀⠀⢸⢳⠀⠀⠀⠠⡄⠀⠀⠀⣤⠀⠳⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⡇⢄⣄⣠⠄⠀⡄⠀⠆⠀⠀⠀⠰⠀⠀⠀⠀⣏⠉
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢄⠀⡂⠀⠀⢘⠈⠀⠀⠀⢀⠀⢀⣀⢠⠔⠁⠀⠿⢉⡩⡀⠘⡀⠀⠀⠢⠀⠐⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢸⣿⣷⣤⣀⣀⠀⠀⠀⠀⣀⡠⠂⠀⠀⠀⢀⡜⠀⢢⣀⠂⠀⢀⠂⠀⠀⠈⠂⢁⠀⢤⡀⢀⡀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠤⢻⣿⣿⣿⠀⠀⠀⠀⠣⡈⡠⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠁⠘⠚⠐⠳⠀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠄⠀⠀⢻⣿⣿⡇⠀⠀⠂⠀⠀⠁⠨⡒⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠢⣜⠂
⠀⠀⠀⠀⠻⠿⠿⢿⢋⠋⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠼⠀⠀⠀⠀⠀⠀⠠⠂⠀⢠⠆⠀⠀⡠⢤⠡⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠏⠀⠀⠀⠀⠀⢠⢀⠀⠀⠂⠀⠀⣀⡀⢀⢠⠀⠀⠦⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠘⠚⠗⠀⠀⠀⠀⠀⠺⠓⠃⠠⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠡⠐⢋⡉⣂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡠⠂⠀⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠂⠀⠀⠀⠔⠁⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡂⠐⡂⡖⠀⠁⠚⠲⡆⠀⠀⠀⠀⠀⠑⢈⣱⠐⡆⠀⠀⠀⠀⠀⣀⣠⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡅⠀⢩⢅⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠞⠁⠀⠘⠚⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠒⢶⣎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⣸⠟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠉⠁⠀⠈⣿⠹⣆⡀⠀⠀⠀⣀⣀⣤⠞⠋⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⡞⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠈⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⠙⢦⣤⣄⣠⣤⡴⠶⠋⠁⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠂⠀⠀⠀⢸⣷⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⠠⣄⣀⡤⠶⠚⠛⠛⠋⠙⠛⠛⠻⠦⣤⣀⠀⣼⠋⢀⣠⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢠⣀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠻⣧⡀⠀⠈⢉⡴
⠀⠀⠀⠀⠀⠀⠀⣶⡄⠸⣿⡄⠀⠙⣦⠞⠁⣀⣠⣤⣤⣴⠆⠀⠀⠀⠀⠀⠉⠻⣟⠲⠄⠀⠈⠻⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠙⠁⠀⣴⣧⠶⠛⣩⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠸⣇
⠀⠀⢀⣠⡤⠶⠦⣤⡀⠉⠛⠶⣼⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡤⠤⠦⣤⣀
⠀⢠⡟⠁⠀⠀⣀⣀⣽⣦⣄⣀⡟⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣀⡀⠀⠀⠙⢷
⠀⣿⠀⣠⡴⠚⠉⠉⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠳⢦⡀⢸
⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⢀⣀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠙⢿
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈
⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⢦⣄⣀⣀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠶⣤⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⡄⠀⠈⠛⢯⣙⠛⠛⠛⠉⢉⣥⠞⠉⠀⠀⣼⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⣴⠏⠀⠀⢻⡄⠀⠀⠀⠙⠷⣄⣀⡴⠋⠁⠀⠀⢀⡾⠉⢳⡄⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡾⠃⠀⠀⠀⠀⠻⣄⠀⠀⠀⢀⡼⢿⡄⠀⠀⠀⢀⡾⠁⠀⠀⢻⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⣿⣁⣀⣽⣦⡀⢠⡞⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⠾⠁⠈⠉⠉⠀⠈⠳⠟⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⢀⣀⣀⣠⣤⣤⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣴⠞⠋⠉⠉⠀⠀⢀⣠⢴⡏⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⣷
⠻⣤⣀⣀⠀⠀⣰⠟⠁⢰⡇⠀⠀⠀⠀⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⢾⣏⣠⡴⠋
⠀⠀⠉⠉⠙⠛⠻⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠙⠶⣤⣀⣀⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠈⠉⠛⠒⠲⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠀⠀⠀⢱⣠⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⠙⡄⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣄⠐⢀⡜⠁⠄⠀⢀⠀⠀⡇⠀⠀⠀⠘⢆⠀⠀⠀⢸⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢿⠟⣻⠟⡝⡟⢷⣶⣤⣀⣀⠞⠁⠀⢀⠀⠀⠈⠧⠀⠀⢈⣀⡀⠈⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠋⢀⡬⠐⠋⠘⠁⠁⠀⠐⠦⠙⢷⣦⣀⠀⠈⠙⠂⠀⡠⠚⠉⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣡⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢿⣯⡣⡀⠒⠀⡼⠁⣰⠃⠀⠁⠀⠀⠀⢳⠐⠚
⠀⠀⠀⠀⠀⠀⠐⠟⡴⢡⣇⠛⠀⡀⠀⠀⠀⠐⠂⠀⠈⠀⢀⠀⠛⢸⣻⡷⣵⡀⠀⠱⡀⠹⣄⣀⣀⡤⠀⢠⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠹⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⠏⠀⠇⠻⠁⠀⠀⣹⢶⠤⢍⡭⠤⠖⠁⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⠦⢄⣀⣀⡠⠴⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⣀⠀⢰⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠦⠤⠤⠤⠤⠔⠚⢿⢄⡀⠀⠀⣀⢀⡠⠊⠁⠀⠀⠀⡇⠀⠀⢇
⠀⢠⠊⢉⣚⠉⢹⠄⠀⢀⣀⣠⠔⠋⡽⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠑⣾⡷⠏⠀⠀⠀⠀⠀⠀⣃⣀⠀⠈
⠀⢿⢠⠋⢀⢱⠈⠓⡄⠀⠞⠿⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠞⠁⠀⠀⠀⠀⠀⣀⣴⡉⠵⠮⣗⠒⣄
⠀⣰⠛⢻⠾⠊⡤⠞⠁⠀⠀⠀⠀⠘⠓⠋⢹⡏⠉⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣅⢎⣥⡼⢴⠃
⠀⠈⠋⢉⣓⣚⣆⡴⣦⣀⡠⠤⠤⠒⠒⢃⣸⠇⠉⠉⠁⣇⡀⠉⠙⠓⠒⠒⠦⠤⢄⣀⠉⢱⣌⡹⠯⣀⡼
⠀⠀⠀⠈⠓⣮⠿⠛⠉⠁⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⡀
⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢘⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠋⠔⠋⠄⡰⠀⠀⠀⠂⠠⠀⠀⠂⠱⠘⠀⠶⠂⠋⠇⠃⡇⠱⠀⠀⠰⠂⠁⠀⠀⠴⠀⠆⠀⠸⠉⠱⠋⠄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣴⣦⣦⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣪⣵⣾⣿⣟⣭⣯⡹⣿⣿⣷⣦⣕⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣾⣿⣿⣿⡿⢻⣿⣿⣷⣜⠻⣿⡇⢹⣿⣝⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣘⢛⣛⣻⣵⣾⣿⣿⣿⣿⣿⣿⣶⣥⣌⣩⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⣿⣿⠿⠋⠁⠀⠈⠙⠿⢿⣿⣿⣿⠟⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠿⠛⠛⠓⠂⠀⠀⠐⠚⠒⠒⠌⢻⣿⣦⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠿⣿⣿⡿⢠⡶⠶⠄⠹⠄⠀⠀⠴⠶⢦⡄⠀⢿⣿⡿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⣤⣿⣻⠇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⣿⣿⣹⡆⠀⠀⠀⠀⠛⠓⠈⠀⠠⡀⠀⠀⢰⣧⣿⣿⣏⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⣿⣿⣿⣿⣿⣧⠀⠀⢰⣾⣡⣄⣍⣝⣶⠽⠀⠀⣼⣹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⣠⠄⠉⠋⣽⣿⣿⡄⠀⠀⠻⠿⠻⡿⠟⠋⠀⠀⢀⣿⣿⣿⣍⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⣤⣤⣼⣿⣿⣿⣷⣄⠀⠀⠒⠖⠂⠀⠀⠀⣠⣾⣿⣿⣿⣿⣾⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣀⣤⡾⢻⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣛⣛⣙⢛⡛⣻⣿⣿⣿⢿⣿⠿⠛⠋⠉⠀⠘⣿⣿⣽⣷⣶⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⢫⡹⣿⣿⣿⠋⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡌⢿⣧⣼⣿⣿⠿⠟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢀⡀⣆⢧⢻⣿⣇⠀⠐⢝⢿⡄⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⢻⣿⣿⠇⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⣼⣿⣼⡞⡆⣿⣿⣿⣦⣤⡀⠙⢦⡀⠀⠀⠀⣰⡿⠃⠀⣀⣴⣿⡇⠁⠀⢀⣾⣇⡴⠉⢣
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡿⣿⣷⠹⡼⠻⣿⣿⢿⣿⣷⣦⣑⢄⢀⡾⣁⣤⣶⣿⠟⠋⠁⠀⢰⣀⡾⣿⣿⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠫⠀⠛⠛⠃⠁⢻⠻⣿⣷⡿⡟⠛⠛⠉⠀⠀⠀⠀⢠⣸⣽⣿⣿⠏⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠈⠃⠘⠃⠙⠛⠀⠀⠀⠀⠀⠚⠀⠀⠛⠻⠻⠋⠀⠛⠛⠁⠛

⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣾⣟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⢺⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣠⡤⣄⠀⠀⢀⣴⠄⢀⣴⠆⠀⢠⣾⡟⢀⣤⠀⠀⣠⡀⢀⣾⣏⡼⠁
⠀⢀⡴⠶⠶⣤⡀⠀⠀⢀⣿⡏⣴⣿⡁⠘⡇⢀⣾⡟⠀⣼⡏⠀⣰⣿⠏⢠⣿⠏⠀⣾⡟⠀⣼⡿⠋⠀⡀
⢰⡏⠀⠀⠀⠸⡷⠀⠀⣼⡿⠀⣿⡏⠛⣾⠗⠻⣿⣀⣴⣿⠣⢶⣿⠷⠖⢾⡏⣠⣼⣿⢀⣼⡿⠀⠀⡼⠁
⠸⣇⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠛⠚⠋⠀⠀⠉⠉⣼⡏⢠⣿⠋⣇⠀⠈⠛⠉⠀⠛⠋⠸⣧⣠⠞⠁
⠀⠈⠛⠶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⣴⠋⠉⢉⣿⠟⠲⢾⡟⠋⣿
⢰⣇⠀⡴⣿⡄⣰⢻⣷⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠸⣧⣠⡏⠀⠀⠀⠀⣼⣄⡀
⢸⣿⣸⠃⣿⣷⠇⢸⣿⠀⢀⣤⡄⣤⡄⠀⠀⢠⢤⠀⠀⠀⠀⣨⠉⠀⣤⠀⠀⢠⣿⣿⡟
⢸⣿⡏⠀⣿⡿⠀⢸⣿⢠⣿⣣⠇⠙⡿⢤⣤⠈⢻⠦⣤⡄⠀⣿⠀⢠⣿⡆⢠⠈⣿⣿⣿⠆
⢸⣿⡁⠀⣿⡇⠀⢸⣿⢸⣿⠁⠀⡰⠁⣼⡿⢀⠏⢠⣿⠇⣸⣿⣄⡞⣿⡷⠋⠘⠻⣿⡏⣤⣀⣤⣦⣄⣀
⠈⠛⠁⠀⠀⠀⠀⠀⠋⠀⠙⠛⠋⠀⠀⣿⣧⠞⠀⢸⣿⡴⠁⢙⣋⢤⣿⡇⠀⠰⣿⠿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣠⠖⠚⢻⣦⣀⣀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠀⢰⡋⢀⣼⡿⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⡉
⠀⠸⠥⠂⣠⡿⠋⠉⣸⡷⠀⣰⡿⢡⠇⠀⠀⠀⠀⣀⣀⠀⠀⢙⣛⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⢀⣀⣼⡟⠁⠶⣾⡋⠀⢰⣿⡵⢃⣴⡾⢛⡆⠸⣿⡀⠁⠀⢿⣄⠉⠀⢠⣴⠞⢳⠀⢀⣴⠶⣾⡏
⡼⠉⢸⣿⠙⣦⠀⠈⣿⢀⣿⠋⠀⣼⡟⠓⠋⢡⢞⣘⣿⣆⡠⣋⣻⣷⣠⣿⠟⠚⠋⣰⣿⠃⣸⡟⢀⡖
⢧⡀⠀⠓⠀⠟⠀⢠⣿⢹⡇⢀⡼⠙⠧⡤⠖⠋⠯⡥⠿⠃⠸⢭⠼⠟⠀⠻⢦⠤⠚⠙⠷⠖⠻⣧⠞
⠈⠛⢶⣤⣤⣤⣾⠟⠃⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣰⣦⡒⠶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣓⣤⣤⣀⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢭⣉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠒⠀⠈⢑⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡟⠋⠉⠙⢦⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡼⠀⠀⠀⠀⠈⣇⠀⢀⣈⣋⠉⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇⠀⠀⠀⢀⠀⢸⠔⠉⠀⠈⡇⠀⠀⠉⠐⢲⠖⠀⠀⠀⠀⠀⠀⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⡀⢿⠀⠀⢰⣏⣴⠃⠀⠀⠀⢀⠃⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢦⣝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢈⠘⢧⡀⠈⢁⣧⣦⠀⢀⠀⠚⠀⠀⢠⠉⠁⠿⠤⠤⠤⢄⡀⣀⣀⣈⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠀⠂⠍⠃⠉⢇⣠⠞⠁⠀⠀⠀⠀⠋⠀⠀⠒⢶⣓⣲⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⡠⠃⡆⠀⠀⠤⣀⡀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⠃⠀⠈⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠂⠁⡀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠠⣄⡀⠈⠁⠀⠀⢰⡀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠸⠈⢀⣀⣹⠒⠤⣀⠀⠘⠏⠁⠀⠀⠰⣶⠀⠀⠀⢢⢰⡢⣵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⡡⠆⡐⠀⠀⠛⣄⡠⢤⠑⠊⠀⠀⠐⣖⠒⠛⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠒⠤⡤⣤⠤⠤⠤⠤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢴⣤⣆⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣀⠀⡘⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣴⣉⣬⣟⣷⡖⠂⠤⠤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠛⠓⠓⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣴⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠈⠉⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⠤⠀⠀⣄⣀⠈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢰⠂⠐⠺⠗⠂⠈⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠸⠓⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠁⠀⠐⠀⠀⠀⣠⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⣤⣤⣶⠟⠁⠀⡟⠛⠛⠛⠛⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠝⠋⠁⠀⠀⠀⢧⣤⣤⣤⣤⣤⣤⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠁⠀⢀⠀⠤⠀⠂⠁⠙⢛⣿⣿⠿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣴⣾⣧⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠈⠉⠙⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣄⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡀⠀⠘⠺⢝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣭⣽⣶⡶⢾⡥⠤⠤⠤⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡹⠊⠛⠉⠉⠀⠤⣄⠀⠀⠀⠀⠀⠈⣧⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠀⠀⢠⢆⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⠂⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠞⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⠈⠀⢀⡠⠖⡆⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠄⢀⣠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢔⠀⠀⠀⠀⠰⡒⠊⠉⣀⠄⡇⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠹⣌⡉⢅⡾⠀⠀⠀⠀⣠⠏⠀⠀⠠⢐⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠉⠉⠀⠀⠀⣠⠔⢁⣀⠴⢳⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤⢴⠒⠋⡍⡉⠀⠤⠒⠊
⠀⠀⠀⠀⡀⠈⠤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠐⢊⣠⠀⠑⠚⠁⢡
⠀⠀⠀⠀⠠⠀⠀⢁⣀⡠⠖⠀⠤⠤⠔⠚⠤⠐⠒⠉⡜⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡴⠤⢍⡑⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⢄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣇⣀⠀⠈⢑⠖⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⡭⢷⡀⢀⣀⣀⣈⡑⠲⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⢸⢀⠇⠉⠉⠈⠁⠈
⠀⠈⠂⠀⡀⠀⠀⡴⢢⠜⠁⠀⠀⠉⠉⠀⠀⠀⠈⠐⠢⢤⣠⠀⠉⠉⠒⡖⠀⠀⡠⣇⠜
⠈⠢⠠⡄⠈⠑⠲⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣸⢅⠀⠥⠒⠉
⠀⠀⠀⠪⡲⢄⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣤⣤⣤⣤⣬⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡷⠦⠴⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠆⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣃⡟⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠤⠤⠤⣄⠀⠀⠀⠀⠀⢸⡦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣟⢹⡀⠀⠀⠀⠀⢰⠇⣿⠶⠚⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠹⣆⡳⢦⣤⣤⠴⢋⣴⠋⠀⠀⢀⣀⣠⠤⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠳⠶⠶⠦⠤⠤⠬⠽⠶⠶⠶⠶⠿⠗⠒⠛⠋⠉⠁⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠋⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠓⠒⣾⠶⠶⠶⠶⠶⠶⠶⠖⠒⠚⠛⠋⠉⠉⠀⢸⡀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⠴⠖⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠲⠶⠶⡷⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠒⠚⠛⠉⠉⠈⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⡤⠴⠒⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠲⠶⢾⡧⠤⠤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠖⠒⠛⠉⠉⡇⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⢀⣀⣀⣤⠶⢾⣿⠿⠿⣿⣶⣤⡀
⠀⢣⡀⣩⡧⣤⠎⠀⠀⠀⠀⡈⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣠⠞⢉⡳⢤⣀⣠⣤⣤⣤⠤⢤⣀⡤⠾⡉⠙⢦⣀
⠀⡿⣹⣿⢁⡎⠀⠀⣶⣀⣀⢹⡀⠹⣷⢹⠓⢦⡄⠀⢰⠉⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣸⣀⠀⠀⢱
⠀⡇⢹⠇⡜⠙⠉⠉⠀⠁⠉⠉⠱⡀⢹⡈⡇⡴⠁⢠⠊⠀⡤⠊⢿⠁⢀⠔⠉⠓⠒⠒⠒⠚⠲⡄⠀⡟⠻⣦⠀⠈⢢
⠀⣇⢼⡴⠁⠚⠙⠀⠀⠀⠚⠙⠂⠳⣤⣷⡏⠀⠀⢹⠄⢠⠏⠀⢸⣤⡎⠀⣀⡀⠀⠀⢀⣀⠀⠸⣤⣇⡴⠛⣦⠀⢞
⣠⠊⠀⣇⡀⠘⢗⡒⠾⠖⠒⡾⠀⣤⡏⠸⡄⠀⠀⠸⡄⢸⠀⠀⢧⡀⠀⠀⣁⣁⠠⠄⢈⣈⠀⠀⠀⡸⠀⠀⣳⠀⡞
⢧⠀⡴⣏⠓⠒⠬⡓⠦⠖⡞⠚⠉⣁⢧⠀⡽⠀⠀⠀⠳⠞⠀⠀⠀⣿⡷⣄⣘⢯⣉⣉⣡⣯⠤⢿⣿⠃⠀⠀⠙⠚⠁
⠀⠹⡀⠈⡷⠀⠀⢻⠛⣻⠀⠀⠸⡅⢀⣟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠳⡄⠈⢱⡬⢥⠏⠀⢰⠊⢹
⢀⡴⢳⡀⡇⠀⠀⠘⡦⡇⠀⠀⠀⣇⡞⠈⢧⠀⠀⠀⠀⠀⠀⢀⡜⢇⢠⠇⠀⠀⣇⣸⠀⠀⠸⣆⡎⢣
⠛⠠⠤⠽⣇⠀⠀⣸⣁⣱⡀⠀⣠⣏⣀⣀⣨⠇⠀⠀⠀⠀⠀⠛⠲⠼⢿⡀⠀⢀⣇⣹⡀⠀⣠⣏⣠⡤⠗
⣤⣤⣤⣤⣬⣵⣾⣥⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣮⣤⣤⣭⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⢀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠈⠁⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠤⣤⣀⡀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢹⠀⠀⠈⡇⠀⢰⠛⠢⣤⠚⢻⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢡⠀⠀⢸⠀⠀⢰⡇⠀⠈⢦⠀⠀⢀⠎⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⢣⠀⢸⠀⢀⠊⠉⠉⠉⣉⡧⠀⢘⣉⠉⠉⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡴⡆⢸⠀⡸⣄⠀⠀⠘⣴⠁⢠⠈⣮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠐⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⡱⠾⠼⠶⡷⣃⠀⠀⢠⡘⡦⠼⠴⡟⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠢⠀⠀⠀⠁⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠿⠶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣥⣴⣶⣶⣶⣶⣦⣤⣉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠙⠛⠛⠛⠛⠛⠟⠻⠿⣿⠂⠀⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⠀⠈⠉⠁⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢇
⠀⠀⠀⠀⠀⠀⠀⢀⠿⢿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⡞⢙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠊⣇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⡄⠀⠀⠀⠀⠀⠠⢬⣤⡄⠀⠀⠀⠀⠀⠀⣴⡆⢰⡤⢧⣦
⠀⠀⠀⣀⣢⡷⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠛⠤⢀⡠⣦⡶⡀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢀⣴⣿⠧⠘⠀⠐⠛
⠀⠀⠀⠐⠋⠀⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠌⣁⠠⡉⠁⢴⣤⣤⣤⡀⠀⣠⣤⣶⣾⠟⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠑⠀⡁⠂⢁⠂⠄⢀⠀⠀⠀⠀⢀⠀⠀⠠⡊⠄⢈⠀⠊⠀⠀⠀⠐⢀⡘⠙⠛⠣⠄⠻⠋⠁⢡⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠂⠀⢀⠤⣀⠨⣀⠄⠆⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠂⠀⠀⠀⠐⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠃⠀⠖⢂⠢
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⢀⠠⠐⠻⣀⠇
⠀⠀⠀⠀⠀⠀⡔⣀⣡⠉⠑⠐⠀⠀⠀⠀⠐⠒⠒⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠮⡄⢨
⠀⠀⠀⠀⠀⠀⠃⠘⠁⠑⢀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⢣⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠂⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠱⡀⣀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡄⠀⠆
⠀⠀⠀⠀⠙⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠎⠀⠀⠀⠀⠀⠑⠢⣶⣧⠀⠀⠀⠀⠀⠀⣢⣿⡤⡇⠀⠀⠀⠀⠀⢀⣿⠗⠊
⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⢼⣿⡀
⠀⠀⠀⠀⠸⠃⠻⠀⠀⠀⠀⠀⠐⢢⣼⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⠀⠀⠸⡜⡇
⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠄⠀⠀⠀⠀⠀⠰⠀⠡⠀⠀⠀⠀⠀⠀⠐⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠀⠀⣠⣀⠔⠁⠀⠀⠘⡀⠀⠀⢀⠄
⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠳⢄⣶⣀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠈⠙⢿⣶⠄⠀⠀⠺⣿⠁⠀⠀⠀⠀⠀⠳⣿⡴⠊
⠀⠀⠀⠓⢲⣾⠆⠒⠈⠀⠀⠀⠀⢿⡏⠑⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⣰⢿⡟⠀⠀⠀⠀⠀⠟⢻⠀⠀⠀⠀⠀⠀⠀⡟⢹⠀⠀⠀⠘⠃⣇⠀⠀⠀⠀⠀⢸⠋⣷
⠀⠀⠀⠀⠚⠟⠃⠀⠀⠀⠀⠀⠸⠈⠛⠐⠆⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠉⠉⠉⠉⠉⠰⠆⣀⠀⢀⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠰⠶⠀⠀⠀⠐⠶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠈⠃⠀⠀⠀⠀⠈⠁⠐⢶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠉⠁⠒⢀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠒⠛⡀⠀⠐⠤⡀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠘⢦⣀⠀⠈⠳⢄⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠤⠤⢱⣄⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⢡⡌⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠰⠀⠀⠀⠀⢰⡆⠀⢰⠀⡄
⠀⠈⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢰⣆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣠⠸⣏⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⢙⠀⠏⠢⡈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⢀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⠀⠐⠛⠋⠀⠀⠀⢠⡀⠀⠀⢀⡴⠋⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠘⠠⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠑⠂⠉⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠾⣠
⠀⠀⢃⠔⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠈⠑⠒⣤⠄⠤⠐⠒⠲⣋⢣⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠎⢠⡉
⠀⠀⠀⠠⢃⠀⠀⠀⠀⠀⠈⠓⠤⢀⡀⢠⠞⠁⠀⠀⠀⠀⠀⠐⠦⢄⡀⠀⡄⠀⠀⣩⡌⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⡄
⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄⠀⢠⣤⠀⠀⠀⡀⠀⠀⠉⠚⠛⣤⠊⣡⢂⠈⢣⡀⠀⠀⠀⠀⠀⣀⡤⠒⠁⠀⢰⠇⠁
⠀⠀⠀⠀⢀⡘⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠄⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣹⠀⠠⡀⠉⠒⠒⠒⠒⠺⡉⢀⣔⣀⠔⠀⠆⠙⢤⠆
⠀⠀⠀⠐⠈⠀⠀⠙⠦⣀⠀⠀⠀⢀⡠⠜⠃⡠⠀⡠⠒⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠱⡀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠈⡀⠔⢁⠄⠊⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣄
⢀⡄⠀⠀⠀⠀⠄⠀⢀⡔⢳⠤⠘⢋⡠⠚⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠱⡄⠙⠦⡀⠀⠀⠀⢀⡀⠀⠀⠠⠊⢀⠙⢄⡀
⠶⠀⠀⠠⡀⠀⠀⠘⠇⠀⠀⡶⠜⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈⠦⡀⠈⠃⠤⠀⡜⠳⡀⠠⠃⠀⠋⠁⠤⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠛⠓⢤⡅⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⠳⢄⡀⠀⡇⠀⡜⠁⠠⠂⠈⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠙⠃⡀⠶⠄⠔⠈⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⢬⡥⠤⠒⠂⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⠉⠀⠒⠒⠨⠤⠤⠄⢀⡀⠀⠀⣤⠤⠤⠤⠀⢀⣀⣀⠠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣷⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠺⠆⠀⠀⠀⠀⠀⢰⣄⠤⣤⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⠰⠖⠶⠊⣄⠀⠀⠀⡌⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⢣⡄⠀⠁⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠘⣦⠀⠀⠀⠀⠈⡄⢿⣄⠀⠀⢀⡜⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠳⣤⣤⣤⠞⠀⢸⠈⠑⠒⠉⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⣀⡿⣦⣤⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⡍⢿⡿⣻⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣿⣷⢿⣾⣷⡷⠷⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣌⣥⣾⠿⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣏⣤⣿⣿⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠙⠟⠙⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠶⠆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣤⣤⡀⢀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⢻⣿⣿⣿⣧⣀⠀⢀⣤⡀⠀⠀⣀⡀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉
⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⢀⣤⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠈⠉⢩⣿⣿⣿⠋⠛⠀⠀⠙⠛⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠉⠁⠈⠁⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⣄⠀⢀⣀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠊⠁⠀⣀⠈⠀⠀⠀⠑⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⡄⠀⠀⠐⠀⠐⡄⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠃⠀⢆⣀⠀⡀⢀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠚⠒⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡤⢃⡴⠂⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⠈⠀⠃⠠⠤⠟⠉⠁⠀⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⣐⡄⠀⠀⠐⡄⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⢤⠀⠱⠀⠀⠀⠐⣀⠀⠀⠱⣀⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠄⠈⠉⠀⠀⠀⠁⠢⢀⡀⠀⠀⠀⠄⠡⢨⡄⠤⠭⠌⡁⠀⠀⠀⠙⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠲⠠⠄⣐⣴⠇⠀⠀⠀⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠄⠀⣏⡀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⠌⠠⠡⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠛⠚⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⢾⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡶⠒⠒⠒⠒⠚⡇
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠁⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣉⣀⡐⠀⠐⠒⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢟⣛⣩⣭⣭⣛⣓⡀⠀⠀⠀⢮⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⠵⣾⢿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⢿⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡼⠁⠈⠀⠀⠀⠀⠐⠒⠲⠄⠀⠀⠀⠀⠀⠀⠸⣷⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠠⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠁⠀⠀⢀⠤⡀⠀⠔⡡⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡤⣄⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣧⠃⠀⠈⢀⠡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠁⠀⢰⠚⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⡇⠀⠀⠀⠀⣄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢙⠂⠤⠞⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣏⠀⠀⠀⠀⠀⠈⣏⠉⠁⠀⣠⠔⣻⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢠⢂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠘⢇⡉⣉⡠⣺⠏⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡠⠚⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⣌⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⡀⠀⠀⠀⣀⣠⡴⢾⠁⠒⠒⠉⣁⡠⠟
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢋⡀⢀⡞⢰⠒⠊⠉⠁
⠀⠀⠀⠀⠀⡠⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠒⠒⠒⠉⣀⣤⡆⠀⠉⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡴⠂⠤⠤⠤⠤⠖⠚⠦⠔⠒⠋⠁⡼⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⠈⠙⠲⠤⣀⡉⠑⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠲⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣦⠤⣄⢀⣀⣠⠋⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⡭⢷⡄⠀⢀⣀⣀⣀⣈⠑⠲⠦⢤⣤⣀⣀⡀⠀⣀⡁⠀⠀⠀⠀⠀⢸⠀⡟⠈⠀⠀⠈⠉⠉⠉
⠀⠀⠒⠄⠀⠀⠀⠀⠀⡰⢮⡠⠚⠁⠀⠀⠛⠉⠉⠉⠁⠈⠉⠁⠢⢤⣀⡀⢀⠉⠉⠉⠉⠴⡦⠀⠀⠀⢠⠋⡰⠃
⠘⢌⠢⡀⠈⠑⠢⢄⡘⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡀⠀⠀⠀⣸⠣⠤⠔⣊⡭⠛⠁
⠀⠈⠱⣈⠢⡀⠀⠀⠈⢳⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠏⠙⠋⠉⠁
⠀⠀⠀⠈⠑⢌⠳⢄⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⣀⡉⣒⡶⢤⣍⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⢦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⣿⣯⣉⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠠⡹⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣾⣿⣿⣿⣿⣿⣿⣧⣿⣭⢫⣞⣿⣿⠐⣷⠻⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠸⢸⡟⠛⠛⠛⠋⠙⠁⠉⠋⠿⢻⣾⣿⣿⣼⠘⢢⡟⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡆⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣯⣂⠛⢹⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢿⣴⣯⢟⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣿⣿⣯⣐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡇⠀⣷⣤⣤⠀⠀⠀⠰⠿⣿⣯⣭⣍⠑⠀⠘⣿⣿⡷⣭⣋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠹⠾⠦⠀⠀⠀⠀⠀⠁⠀⠈⠙⠃⠀⠀⠹⣿⣿⣦⠌⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡳⣄⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⣻⣿⣿⣮⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⣀⢀⣼⣿⣿⣿⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡖⠛⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⣿⣿⣿⡿⢹⢸⠹⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇⠰⢸⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⣿⣿⣿⣿⡃⠀⠈⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡲⢤⣀⣀⣀⠠⠒⠉⠀⠀⢠⣿⣿⣿⣿⡇⠀⠁⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡀⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⡃⠀⠣⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣧⡰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⢦⣤⡂
⠀⠀⠀⠀⠀⡀⠀⡀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⠶⢮⣭⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠊⠢⠙⢿⣷⡖
⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠛⢀⠀⠀⠹⣆⠀⠀⠀⠀⢠⣿⣿⡿⣿⣧⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠈⠘⡉⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠹⠀⠀⠀⢀⠃⣿⣿⠃⠘⢿⣷⠀⠀⣀⣀⢿⢿⡀
⠀⠀⠀⡀⠹⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⠀⠀⢀⠂⢠⣿⣡⢖⡒⢚⠻⡈⠋⠉⠀⠀⠂⠃
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢲⠏⠀⠀⠀⠀⢝⠉⠑⠋⠀⠀⠀⠈
⠜⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⢠⡄⠀⣼⠁⠀⠀⠀⢀⣀⡀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣿⣿⣿⡿⣿⡿⠿⣿⣿⣿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣾⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⡠⠖⢚⣉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⢠⡄⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠖⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆⠀⣀⣤⡄⠀⠈⢦⠈⢢⡀
⠀⠁⠀⠀⠀⢿⡉⢢⡀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠄⠀⠠⡶⠿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⢠⣇⡼⠋⣰⠇⠀⠀⠀⣧⠀⢳
⠀⠀⠰⣏⠓⢮⣳⣄⠱⣼⡇⣠⠏⠀⠀⠀⠀⠀⠀⢀⡾⣫⡴⠆⠀⠀⠀⠀⠀⠀⠀⣀⡉⠻⣆⠀⠀⠀⠀⠀⣞⢉⣈⡙⢿⣄⠀⠀⢾⡥⠶⣦⠀⠀⢸
⠀⠀⠀⠈⠳⢄⡙⠁⣰⠟⠁⢳⡀⠀⠀⠀⠀⠀⠀⣸⠇⢁⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡦⠹⣧⠀⠀⠀⠀⠉⠁⠀⡝⠀⠈⢷⠀⠀⡤⠖⠋⠀⠀⢸
⠀⠀⠰⣏⣉⣉⡁⢰⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠙⢦⣄⡀⠀⠀⢀⣠⡴⠛⡇⠀⢻⡆⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠀⣤⣉⣱⡄⠀⠀⠏
⠀⠀⠀⠀⢠⠞⣡⣄⣀⣀⣄⠙⢦⡀⠀⠀⠀⠀⣠⣿⡀⢸⣇⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⣿⠀⢸⡷⣤⡀⠀⠀⠀⣰⠋⢠⡷⠒⠋⠁⠈⠉
⠀⠀⠀⠀⠘⠋⠀⠈⠉⠉⠈⠳⣄⠙⢦⡀⠀⠀⣧⣂⡀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢀⣀⣸⠇⠀⢠⡞⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢦⠀⠈⣿⠀⠀⢿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠀⠀⣿⠁⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠳⣄⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣠⠞⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠈⢧⡈⠉⠙⠛⠛⢛⡿⠃⠀⠀⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠢⡀⠈⠳⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠙⢦⣀⡀⠀⠀⢀⣠⠞⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⣄⣀⠉⠛⠛⠛⠋⠁⠀⣀⣠⡴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⡆⢸⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣀⣼⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡷⠀⠀⢀⣠⠶⠛⠙⠃⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠙⢦⡀⠀⣠⠟⣽⠛⠳⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⢻⡙⢦⣀⡴⠋⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠲⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡉⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣤⣤⣤⣄⡈⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣤⣤⣤⣤⣀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⢸⣧⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠛⠟⠃⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⣠⡼⠏⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⢀⣠⡾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠈⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠶⠶⠶⠊




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⡟⣷⡴⠞⠉⠉⠀⠀⠀⠉⠉⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡈⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢤⡼⢆⠀⠀⡼⢧⠀⠀⣸⢣⡀⠀⠀⡷⣄⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⡄⠀⠀⠘⡄⠀⠑⠚⠁⠀⢑⠒⠁⠁⠑⢦⡚⠀⠈⠙⠒⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠉⢣⡀⠀⢀⠇⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣭⣝⠒⠋⠀⠀⠀⠀⠀⠀⠀⠐⣁⣀⣀⠂⠒⠁⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠠⠔⠦⠤⠤⢤⡤⠭⢦⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⡀⠀⠀⠢⠤⠤⠀⢀⣀⣘⣦⠞⠀⠀⣀⣠⠞⠁⢀⡴⠞⢛⣛⠓⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠙⠒⠒⠤⠤⠤⠤⠤⠶⣶⣖⠛⠋⠉⢀⣠⡴⢿⢀⣀⢻⡾⣹⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠾⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠈⣿⠙⠿⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣡⢀⡀⣸⣦⡴⠶⠶⠶⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣤⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠳⠃⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠈⠉⠙⠓⠒⠢⠤⠶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⣠⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⡇⠀⠀⠀⢀⣠⠞⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣧⣤⡴⠖⠋⠁⠀⡿⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣉⣷⠀⠀⠀⠀⠀⠛⣶⠒⠒⠚⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠁⠀⠱⠀⠀⠀⠀⠀⡼⣒⣂⣤⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣤⣄⠀⠀⠐⣀⣴⢖⡶⠆⠀⠀⠀⠀⢾⡛⠾⢿⠓⠄⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠸⠯⠐⡿⠀⠀⠀⠘⣯⡁⠀⢸⠢⢀⡀⠀⠀⡏⠉⣔⠀⠘⠀⠀⠀⢠⣴⡴⣶⡇
⠀⠀⠀⠐⣉⡷⠤⢦⡀⠠⠂⠲⠟⢻⣦⠀⠀⠀⠆⠇⠀⠀⢀⡴⠋⠀⠀⢸⠀⢸⠉⠂⠀⣟⣲⣟⣋⡿⡇⠀⠀⠿⠷⠾⠛⢳
⠀⠀⠀⠀⠙⢿⠛⢛⢡⠃⠀⠀⠀⠀⠘⡄⠀⠀⠠⠀⠀⠀⠀⠙⣀⣀⣀⡴⠀⢼⠤⣤⣀⠀⠈⠉⠁⡐⠀⠀⠀⠀⠠⠊⢰⡆
⠀⠀⠀⠀⠀⠨⡄⠸⠲⠘⠂⠀⠀⠀⠀⢺⠆⠀⢰⠆⠀⠀⠀⠈⠛⣻⠛⠀⠀⠀⠀⠀⠀⢩⠃⠀⡰⠁⠀⠀⠀⠀⠀⠁⡏
⠀⠀⠀⠀⠀⠀⢣⡆⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢸⠀⠀⠀⠀⡠⠖⠉⠃⠄⡀⠀⠀⢀⠔⠁⠠⡶⢡⣄⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡏⠂⠀⣀⣠⠔⠁⠀⠀⠀⢀⠼⠀⠐⠒⠈⠀⠀⠀⠀⠀⠈⠁⠀⠙⡄⠀⠀⠀⣀⡿⢧⡀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⣀⠀⠀⠀⢀⡀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⢉⡤⠤⢽⣄⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢼⡄⠀⠴⠏⠠





⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠈⠁⠒⢄⠀⠀⡠⠒⠀⠉⠀⠐⠒⢄⠀⣀⠀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠑⣄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠎⠀⠀⠀⠀⠀⠀⠀⠈⡞⠁⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣠⣤⢤⣀⠀⡀⢀⣀⣀⡀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠡⣾⣿⡆⢠⡀⠀⢠⢶⣄⠸⢯⡟⠀⢈⣿⠀⠀⠘⣿⢻⡿⢀⠀⠀⣀⣴⡖⠓⢾⡃⠀⠠⢦⢀⡤⢤⣤⠀⡇
⠀⠀⠀⠐⡝⢷⡀⢀⢣⠀⢷⠀⠐⠴⣮⡓⠴⢅⠀⠸⠈⡀⠀⠀⠸⡄⢠⠆⠀⡘⠻⠈⠛⠥⠂⣯⢠⠃⣸⣜⠷⠶⠛⠀⡇
⠀⠀⠀⠀⠘⠮⠏⠉⠀⠲⠾⢧⠀⣠⢯⣤⡀⠘⠄⣦⡄⣡⠀⠀⠀⠀⡼⣰⡔⠀⢇⠀⠀⠀⣠⠎⠛⣶⠇⢸⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠻⠧⠞⢡⠃⠀⠀⣠⣀⣍⠀⠀⠀⡰⠇⠈⣧⣤⡀⠉⣶⠚⠡⠾⣿⣿⡷⠄⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢀⡴⠓⢄⡀⠈⠃⠀⠘⠂⠀⡰⠁⠀⠘⠛⠶⠛⠋⠀⠑⢄⡀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠑⠂⠤⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠊⠉⠊⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠨⠅⠒⠠⢈⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣶⣴⣒⠒⠂⠀⠀⠀⠀⠀⠀⠉⢲⡔⠋⠃⠘⢑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠶⡛⠉⠀⠈⠁⠘⠛⡲⢦⣀⣀⠀⢰⡏⠆⠀⠀⠀⠀⣣⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡂⠀⠀⠀⠀⣀⡠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠲⠾⠿⢿⢻⠀⠀⠀⠀⠀⠀⢡⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢁⣠⣤⣼⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠈⣿⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣿⣿⣿⣿⡄⠘⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣿⠿⠟⠉⠀⠳⠀⠈⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢱⡀⢩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠁⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⠀⠀⠀⠀⠀⢴⣦⣤⡀⠀⠇⠀⠀⠀⠀⠀⢆⠘⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣴⡀⠀⠀⠀⠀⠈⠀⢃⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠂⢄⡀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢹⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠦⣀⡎⠉⢲⣡⠀⠀⠀⢣⠀⠀⠀⠀⢰⠂⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣾⠀⠀⠀⠀⠈⠳⠀⠈⠉⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣅⣿⣆⠀⠀⠀⠢⡀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣿⣶⣦⣤⠀⠀⠀⠀⠛⠢⠀⠀⠀⠸⠿⢿⣿⠿⠛⠛⣿⠛⠉⠁⠀⠀⠀⠀⣄⠀⠄⠈⢢⠀⠄⠁
⠀⠀⠀⠀⠀⠀⣀⡠⠈⠀⠀⠻⠛⢻⣿⣷⡄⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⢻⠀⠈⠀⠘⡄
⠀⠀⠀⠀⣠⣬⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⡀⠀⢠⣿⠇⠀⢀⡠⢤⣿⡗⠛⠛⠛⠿⠛⠛⠛⠓⠂⠄⠀⢸⡀⠀⠀⠀⠡⡀⢸
⠀⠀⠀⠐⠯⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⣿⣿⠁⠁⣀⣀⣼⣿⣶⣶⣶⣶⣦⣶⣶⣶⣤⣤⣄⡀⠈⠁⠀⠀⠀⠀⢡⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⠖⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠻⠋⠀⠀⡜⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⢒
⢰⠛⠉⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⡇⠀⠀⠖⢾⠽⠿⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤
⠘⠁⠀⠀⡍⠹⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠉⣭⠭⠥⠤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠋⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠙⣿⡇⠀⠀⣼⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣸⡿⠀⠀⣼⡟⠀⠀⡀⠀⠀⠀⢠⣾⠂⠀⠀⢀⣶⠆⠀⠁⢀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠷⠶⣾⡿⠀⣠⡾⠛⣿⠂⢠⣿⡗⢩⣿⠀⣾⡧⠊⣿⠂⢻⣿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿⠇⢰⡟⢁⣾⡇⢀⣾⣯⢤⣾⠏⣼⣿⢤⣼⠟⢀⣿⠃⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢿⣄⠈⠛⠁⠘⠛⣽⡏⠙⠋⠁⢰⡿⠙⠛⠉⠀⠀⠛⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠃⠀⠀⠀⣴⠞⣻⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣴⡆⢀⣴⡷⠀⠀⢰⣄⠀⠀⣶⠄⠀⠀⠀⠀⠀⠐⠛⠟⠁
⠀⠀⠀⠀⠀⢠⣶⠿⣿⡟⠉⠙⢦⠀⠁⢀⣤⠉⠉⠉⠛⠛⢻⣾⢿⣶⣿⣯⣤⣀⣠⣼⠟⠀⣼⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠹⠅⣰⣿⣀⣀⣤⠏⢀⣠⣌⠉⢰⣛⡆⠀⠀⣼⠏⢠⡿⢁⣭⣭⡍⢉⣡⣤⣴⡟⠀⠀⢠⣴⠶⣤⠀⢀⣴⠆⢠⣶
⠀⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⢹⡄⢹⡿⠁⢀⣼⢛⡿⠀⣸⡏⠀⣾⡷⢋⣿⠋⣴⠟⠉⢠⡟⠀⠀⣴⡿⢁⣼⠇⢀⣾⠏⣠⣿⠃
⠀⠀⠀⠀⠀⢠⣿⡃⢀⣀⣤⠟⠁⣿⣧⠶⠋⢰⣿⣥⢾⣿⠀⢸⡟⠁⢼⣇⡤⢿⣤⡶⠻⣧⣴⠞⢿⡥⠊⣿⡤⢺⣿⠔⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣴⡟⠀⠐⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠶⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣾⣯⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣬⣿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⡾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠒⠚⠛⠉⠉⠉⠉⠉⠛⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠳⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠛⢦⣄⡀⠀⠀⢀⡏⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⢿⡀⠀⠀⠀⠀⠈⠉⠛⠶⠼⠇⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢀⡞⠛⠾⣦⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣀⡼⠟⠛⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠙⠿⠀⠀⢠⡶⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢷⣄⣤⡄⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⢀⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡿
⠀⢠⣦⣤⣶⡀⠀⠀⠈⢧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⠃⠀⠀⢠⣶⣤⣶⡄
⢴⡾⣯⢻⡏⣷⢻⠄⠀⠈⣧⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⠏⠀⠠⡟⣾⣽⡿⣽⢷
⣴⢿⡍⠀⠀⠉⡏⠀⠀⠀⠈⢧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⠋⠀⠀⠀⢻⠉⠀⠀⠱⠿
⠈⠛⢦⣀⠀⠀⠛⢦⡀⠀⠀⠈⠳⣽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡼⠃⠀⠀⢀⡴⠟⠀⢀⣀⡴⠛
⠀⠀⠀⠈⠳⡀⠀⠀⠙⠳⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠉⠉⠉⠉⠉⠉⢩⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⠋⠀⠀⣠⠞⠋⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⣤⡀⠀⠉⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⣤⡤⠖⠋⠀⠀⠀⠀⠀⠀⢀⣤⣾⡷⠋⠀⢀⣤⠞⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠙⠳⢤⣀⠈⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠁⣀⡤⠞⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠈⠙⠛⠿⠶⢶⣶⠤⠤⠤⠤⣴⡶⠶⠿⠛⠋⠉⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠉⠳⡶⣄⡀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⢀⣠⣶⠞⠉⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⣰⠃⠀⠙⠳⠤⠞⠛⢦⡀⣀⣴⠛⠳⣤⠞⠋⠀⠘⣆⠀⠀⠀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣠⣤⡄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢉⣀⣀⡀⠂⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠂⠴⠒⠒⠈⢛⣶⣲⣤⣐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣦⡶⠶⠲⣶⠛⢿⣍⠻⣯⣍⠻⢿⣿⣿⣿⣿⣿⣮⡂
⠀⠀⠀⠀⢠⡴⢛⠏⠉⠀⠃⠂⠝⠷⣦⡙⢿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⡣⡀
⠀⠀⠀⠐⠋⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡻⣿⣿⣿⣿⣦⡻⣿⣿⣿⣻⣷⡑⢄
⠀⢀⠀⠀⡐⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⠿⠟⠉⠛⠿⣿⡿⢿⣿⣿⡉⠻⠷⣴⡄
⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣹⢦⣙⣏⠃⠀⢂⠐⡹⡀
⠀⢠⡆⠀⢶⣶⡲⢤⣠⣄⡀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠿⠿⠿⠿⠷⠀⣄⠀⢧⠀⢠
⠔⠈⠀⠀⠀⠉⠉⠀⠉⡛⠻⣾⣿⡄⠀⠈⠻⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⡦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⠉
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀⠀⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡐⢿⣿⣟⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢲⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣮⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠹⡌⣿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢿⣆⢹⠞⢀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠀⡁⠀⠀⠈⡿⠃⠀⠀⠀⠀⠀⠙⠲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⢀⡠⠚⠀⠀⠀⠀⠄⠀⠀⡀⠀⠁⠙⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⢷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⠀⡠⢻⡟⢻⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⢅⡔⠁⣔⠕⢋⣔⡤⠚⢛⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢋⣴⡟⢁⠊⠀⣀⣣⡴⠛⣁⡴⣾⠿⢫⠿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⢀⣴⣿⠏⠀⠀⢀⣴⠟⢁⡤⠚⡡⠞⣡⠞⢁⡴⢋⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣤⠙⠰⢩⣿⠇⠀⠀⣠⣾⣯⣶⢉⣤⣊⡤⠞⢁⠔⣡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⣾⡟⢀⠔⠋⠀⠴⠈⠁⡘⠈⠙⣇⠀⣡⠞⠁
⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣸⣿⣿⣧⣤⣀⣀⠀⠀⠀⠒⣉⡀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⢾⣿⣿⠻⣿⣿⣿⣿⣿⣼⣿⣟⣩⡟⠓⠀⠤⢠⣷⣀
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠾⠋⠁⠉⣽⣿⣧⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣦⣤⣠⣆⢚⣧
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠈⠀⢒⣶⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷⠆⠀⠀⠐⠠⠤
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡄⠀⠀⣤⠄⠀⢠⠸⣿⣿⣶⣶⣤⣮⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣷⠀⢰⣷⣤⣤⡦⠶⠒⣦
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⢠⡿⠀⢠⡾⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⡉⠉⠀⠀⣀⡜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⡙⠙⠓⠲⣤⡑⣶⠄
⠀⠀⠀⠀⠀⢱⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣠⣄⠉⠉
⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⠙⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⡿⣿⣿⣝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡙⢿⡻⣯⡛⣍⢉⣻⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠞⠟⠓⣄⣿⣏⣻⣿⣷⣿⣛⣙⠻⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⡿⣿⣿⣿⣿⣦⡀⠀⠀⠉⡉⠉⠰⠶⢶⣤⡎⢋⡁⠰⠚⠛⠻⢏⠉⠙⢇⠀⠙⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢿⡇⠀⠀⠀⠀⠀⠀⠈⠈⠃⠉⠁⠀⠀⠀⠀⠀⠱⢄⠀⠙⠲⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣽⣿⣿⣿⡿⣿⣿⣿⣿⣶⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣀⠀⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣦⣤⣠⣤⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠙⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⡙⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣼⣿⣷⣶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣾⣿⣿⣷⣿⣿⣌⢻⣿⣿⣿⣿⣿⣿⣿⣗⣱⣶⣤⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧⡙⢿⣿⣿⣿⣿⣟⠻⣿⣿⣿⠘⡷⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢻⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣮⣻⣿⠷⣧⣈⣁⣀⣀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠚⣿⣿⣿⣿⡿⠀⠙⠻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣄⡏⢈⢛⡻⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣤⡀⠈⠛⢿⡿⢦⣌⣻⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⣧⣍⠛⢷⣦⡀⠈⠀⠉⠉⠙⠛⠿⢿⣿⡿⠿⣿⣿⣿⣛⣳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣤⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⡀⠀⠀⢠⡇⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢰⡇⠀⣰⢸⡇⣰⣿⠟⠁⠉⢉⢏⡟⠹⠿⠟⣫⣶⣿⣿⣶⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢠⠸⡇⢰⣴⣤⣿⣿⣷⣾⣷⣿⣿⣾⣦⡀⢠⣾⣿⣿⣿⡿⠿⠟⠀⠀⠠⡉⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠘⣆⢻⡜⣿⣿⣿⣿⣿⣿⠿⣿⠿⣿⣿⣷⣿⣿⣿⣿⣯⣾⣿⣿⣿⣷⣦⡘⣦⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠘⢦⣻⣮⣿⣿⣿⡯⠶⣶⣶⣶⣶⡤⠉⠉⠙⠁⠁⠘⠛⠛⠛⠉⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⠃
⠙⠲⠩⠝⢻⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠃⢣
⠀⠀⠀⠀⠏⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠃⠀⣀⣸⡄
⠀⠀⠀⠀⠀⣿⣯⡀⠀⠀⠀⠘⢻⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠁⠀⣠⡾⠛⠻⡇
⠀⠀⠀⠀⠀⠸⣿⣷⠀⣴⠂⠀⠈⣴⣿⠆⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡄⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣆⠕⠀⠀⠘⣛⣵⡖⠀⠀⠀⠀⡴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣬⣔⠦⣤⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣟⠀⠀⣀⣾⣿⣿⡟⠀⠀⠀⠜⠁⠀⠀⣠⣔⡂⠀⠀⠀⠀⠀⠀⣻⣿⣦⢘⠃⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣸⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⣼⣿⣍⠉⢀⠀⡷⠀⣀⠀⠋⠙⠛⢁⠄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠰⢀⣉⣁⠉⠋⠙⠛⠋⠀⠀⠀⠀⠀⠀⠛⠿⠛⠀⠁⠀⠀⠀⢠⡄⠀⠀⢠⡀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡼⠃⠉⣳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⡄⠤⠄⢠⠾⠃⢶⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢣⢠⡀⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣀⣠⣤⣤⡄⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡳⠤⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⣊⡤⠤⠒⠊⠉⠁⣀⣤⠏⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠶⠾⣆⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠉⠙⠛⠿⠶⠿⡿⠿⠋⡐⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣤⡶⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣴⣚⣡⡤⠤⠤⠤⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⡿⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠈⠛⢷⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⡿⠿⠋⠀⠀⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⣠⣤⣀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣀⣤⣴⣾⣿⠿⠛⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⠿⠿⠛⠉⠁⣀⣤⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣉⣉⣀⣀⣀⣠⣤⣤⣶⣾⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠁⠈⡇

⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⢄⠀⠀⢀⠇
⠘⢻⣿⣾⠒⠒⠉⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⢠⣤⣤⣾⣷⣶⣦⡀⠀⠀⠀⠀⠀⡀⢠⡀⡀
⠀⣺⣿⣿⡇⠀⠀⠀⠀⠞⠛⠛⢿⣿⣧⡄⠀⣠⣼⣿⣿⣿⡿⠟⠛⢻⣷⣦⡄⠀⠠⣰⠿⠻⣿⣶⣦⣤⠀⢄⡀
⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇⠈⢿⣿⣿⠟⠃⠀⠀⠀⠀⣿⡟⠁⠀⠀⠁⠀⠀⠹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠋⠀⠀⠘⠿⢿⠧⢀⠑⠀⠊⠘⠋⠀⠀⠀⠀⠀⣄⣀⠀⢀⣽⠿⢋⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢃⠀⠀⠀⠀⠀⢀⣠⣾⣿⣧⣤⣦⡄⠀⠀⣴⡄⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣀⣼⠀⠀⠀⠀⠐⣿⣿⣿⣿⣽⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠻⠿⠀⠀⠀⠀⠀⠈⠛⠛⠿⠛⠛⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠲⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠛⠻⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠈⠁⢳⡦⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣹⡗⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⢀⣸⣿⠆⡀⣀⡀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⣉⠁⠐⠳⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣏⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠤⠤⠷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠙⠻⠿⠟

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠕⠉⠀⠊⠀⠀⠀⠉⠑⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠀⢀⣀⠄⢀⠤⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⣠⢚⠍⠐⠈⠁⠀⣘⡄⡠⢆⠄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠀⠀⠁⠀⣀⠀⠀⠀⠀⣀⠏⢁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⣢⠁⠀⠛⠀⠀⠀⠀⠛⠁⠠⡤⠤⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢈⣦⣤⣀⣀⢀⣀⠀⠄⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡩⠉⢇⡿⡋⠃⠀⣠⡘⠱⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠃⠀⡇⠀⠙⠀⠸⠙⠇⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⣠⠛⠆⢀⡇⣠⠀⠀⠀⠀⢠⣼⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⣠⠁⠀⣼⡿⣠⠃⠀⠀⠀⠀⠘⡻⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠃⠠⠪⢈⡔⠀⠀⠀⠀⠀⠀⠀⢃⠈⠒⠄⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⣁⠿⡦⢄⣀⠀⠀⠀⠀⢀⣸⠀⠀⠐⠁⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠁⠈⢀⡅⠀⠉⠉⠉⢉⣉⠁⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⡘⣏⠀⢀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⠹⠉⠀⠀⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣀⣘⠀⠀⠀⠀⠘⡃⢀⡭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⢦⣘⣆⠀⠀⠀⢰⠄⠁⠒⠝⠶⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠀⠁⠀⠑⡄⠀⠸⡄⠀⢄⣈⣀⣀⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠖⠁⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢠⠞⠁⠀⡴⠁⣰⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣠⠞⣛⣉⠉⠉⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⣰⠏⠀⠀⣸⠃⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⡍⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⢠⡏⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢻⣄⡀⠉⠒⢦⣄⣀⠀⠀⠀⠀⢸⡷⠒⣟⣠⡴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠘⠓⠚⠛⢿⠛⣷⣿⣶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣽⣶⡿⢿⣟⢦⠀⠀⠀⠉⠙⠛⠲⠶⢾⡷⢦⣸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⢿⣿⣷⢟⡿⠁⣿⠀⠀⠸⣿⣿⣿⣷⣶⡟⡸⠃⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⣤⣤⠴⠞⠁⠀⠜⢷⣄⠀⠙⠻⠿⠿⢛⣜⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡾⠋⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠠⡶⠂⠀⠀⠀⠝⠛⠶⣦⠶⠾⡋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢋⠀⠀⡆⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠳⢤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⡇⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡟⠀⣸⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣀⣤⣤⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⠃⢠⠇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠦⠤⠤⠝⠉⣏⡿⠉⠙⠓⠶⠶⠖⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡆⢀⠎⠀⡀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣀⣀⠀⠀⣠⡀⠀⡠⢤⡰⡛⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢰⠀⡞⠀⢠⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⠉⢀⡏⠀⠑⠊⠀⠈⠉⠀⢸⡀⠉⠳⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡘⡄⣇⠀⠸⡀⢻⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⠤⣊⠷⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣴⡞⣅⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⡄⠀⠱⢤⣑⣦⣀⣀
⣔⠂⡀⠙⠀⠈⡂⠬⠴⠒⠋⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⢮⣑⠦⣄⡀⠀⢀⡴⢂⠄⢀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠾⠿⠛⠋
⠠⢪⠄⡤⠀⠐⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠈⠙⠲⠭⢭⡁⠀⠂⠘⢃⠶
⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⡐⡂⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣱⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣦⢤⠶⡶⣾⠛⠲⠶⠤⣤⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠸⢤⠇⡿⠀⠊⢋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⢀⠋⠂⠑⡌⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢌⠀⠀⢠⡇⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢈⡑⠒⠲⠟⠶⣶⠁

⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢠⡈⠳⣄⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⠂⣠⡞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡞⠉
⠀⠀⠀⠀⠀⠈⢡⠊⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⣿⠷⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⢿⣦⠀⠀⠀⠀⠀⢸⠃⡀⠀⢀⡀⢿⡄⠀⠀⠀⠀⣠⡾⠋⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠹⣷⡀⠀⠀⠠⢾⠀⠁⠂⠈⠁⣸⢧⠀⠀⣴⡾⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠈⠻⢦⡀⠀⢻⡄⠺⣯⡗⢀⡿⠃⢀⣾⠟⠀⠀⣠⠞⣠⣴⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠙⢦⡀⢻⠦⣤⡴⣿⣠⣴⡿⠋⠀⣠⣾⢫⡞⡀⠀⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣦⠀⠀⠈⢷⡈⢰⣦⣴⠛⠛⠁⠀⢀⣾⠛⠁⢸⠀⣉⣁⡀⠀⢙⠆⠀⢤⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠁⠈⠀⠉⠀⠀⠀⢀⡾⠁⠀⠀⠸⣆⠻⠾⢁⢠⡋⠀⠀⡂⢿⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠈⢓⡖⠉⠸⣀⠀⠀⢹⡄⠀⡼⠁
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣆⣤⣶⣋⢳⣤⡾⠉⠳⣶⠟⠀⡸⠁
⠀⠘⡕⢦⣄⣏⢳⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣻⣿⢋⠀⠀⠀⠙⢿⣿⠀⠤⠤⠄⠒⠒⣲⡾⠁⠀⣼⠃
⠀⠀⠙⣦⡈⠁⠘⣇⡀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣼⣿⡇⢠⡄⡄⢠⡦⠸⣿⣿⡆⢒⣛⣛⣻⠏⠀⢀⡾⠃
⠀⠀⠀⠈⠙⠶⣤⡈⠛⠷⣤⣄⡀⠀⠀⢿⣇⠀⠀⠀⣿⣿⡇⠈⠁⠀⠀⠀⠀⢻⣿⡏⣉⣁⣼⠏⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⠷⣦⣸⣿⠲⢤⣀⣙⢿⣿⡀⢿⣥⣽⠃⣰⣿⠟⢀⣠⡿⠃⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠙⠿⣧⣤⣀⣀⠀⠉⠳⣄⣉⣡⢴⡏⠀⣴⣿⡋⠙⢦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⡿⠙⢻⣿⣷⣦⣼⠉⠁⢘⣧⣾⣿⣿⣿⣦⡾⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢧⣠⣿⣿⣿⣿⡏⠛⣶⠛⠀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠉⠻⣿⣿⣿⣷⣾⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⢠⡖⠉⠋⠙⠦⡀⠀⠀⠀⠀⢴⣤⠚⣁⠙⠒⠤⣀⢀⠴⠋⡠⠈⠉⢳⠄
⠀⠀⠀⠀⢠⢾⡄⠀⡤⠀⠀⠉⢦⡀⠀⠀⡜⠳⣄⡼⠃⠀⠀⡸⠋⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢀⡉⢣⡀⠀⠀⠒⠁⠀⢹⠄⢀⡃⠀⠠⠀⡀⠘⣰⠁⠠⠁⠈⠀⠁⣼⢃⣤⣤⠀⠀⢀
⠀⠀⢠⠎⠉⠙⢦⠱⢄⠀⣠⠶⠊⢉⡿⡆⢧⠀⣀⡤⠤⠶⠳⣶⠦⣤⡄⢠⣴⡭⠞⠛⠋⠃⠛⠮⣡⣄⡄
⠀⠀⢸⡀⠀⠂⣺⠴⠊⠉⠁⠹⡄⠉⠀⣿⡶⢋⣵⣶⣶⣄⠀⠈⠓⠒⢉⣻⣅⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟
⠀⠀⠀⠉⢓⡞⠁⠀⠀⠀⠀⢀⣯⣭⣰⠋⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⠙⠐⠠⠄⠀⢀⠀⢷
⠀⠀⠀⠀⢨⢧⡀⠀⠀⠀⢀⣸⠿⠋⣹⠒⢤⡈⠛⠛⠛⢡⠖⠉⠉⢲⡙⠻⠿⡿⢧⡞⣱⢀⢠⡞⠋⠑⠻
⠀⠀⠀⢠⡧⠾⠷⠖⢂⣈⢩⣤⡶⢛⠋⠳⣤⣷⠤⢤⣤⠼⠤⠤⣶⠞⢻⢻⡁⢧⠜⡟⣡⣿⠾⠥⢄⡀⢀
⠀⠀⠀⠈⢗⠂⣉⠭⠔⡻⠋⡽⢁⠸⠀⠀⢸⠇⢀⣀⣀⣀⣀⣀⣀⣀⢸⡀⣇⡏⣸⡟⡱⢧⠘⠂⢀⠌⣧
⠀⠀⠀⠀⠈⢷⠁⠀⡼⠁⠀⡇⠐⣠⢦⣤⠜⢻⡉⣀⢰⡀⣠⠀⡼⠁⠀⠈⠉⠉⠛⠴⣗⠋⠀⠀⠀⠀⣴
⠀⠀⠀⣴⣦⢀⡷⠖⠓⠦⣀⢱⣸⠡⡦⠙⣦⠼⠥⢿⡼⠥⠿⢴⡃⠄⠀⣠⣾⡷⠀⠀⠀⠉⠒⠤⢤⡴⠃
⠀⠀⠐⠾⢡⠏⡐⠀⠀⠀⠙⡎⠈⣧⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠉⠢⣠⣿⡿⡝⠀⠀⠀⣠⣶⣦⠀⠈⣢
⢠⡴⠒⢦⣾⠀⠀⡃⠀⠤⠀⢹⣰⠋⢷⠀⢻⡐⠠⠀⠀⠀⠂⠀⠀⠀⠹⡏⠊⠀⠀⠀⣴⣿⣿⡿⠀⢠⠏
⠀⠟⠟⢸⢻⡀⠈⠁⢀⠀⠀⠘⡇⠀⠈⡇⠈⡗⠀⠉⠡⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠹⠿⠛⠀⢠⠏
⠈⣇⠀⢸⠀⢳⡀⠀⠘⠇⠀⠀⠹⠤⠞⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢀⠄⠀⡄⠀⢠⠃
⠀⠸⡄⠈⣇⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⣶⠶⠷⠶⣦⡀⢀⡤⢲⣎⡩⢁⣶⢤⠀⠂⣠⠏
⠀⠀⢻⠀⠈⠦⠼⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠁⡐⢰⡿⠒⠉⣧⣾⣖⡼⠁⠉⠒⢼⡄⢇⡰⠋
⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠔⠒⠉⠉⠉⠉⠙⠛⠓⠺⢬⡁⠀⠈⠠⠀⠀⣿⢻
⠀⠀⠀⠀⠁⠀⠉⣤⣄⠀⠸⣷⠊⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠈⢻⡀⡀⡀⠄⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠐⣫⠆⢰⠃⠀⠀⠀⣤⡴⢿⠀⠀⡾⢷⣄⠀⠀⠀⠀⢷⠀⠀⢰⣃⠇
⠀⠀⠀⠀⠀⠀⣀⣠⠼⠟⠺⠀⠀⠀⠀⡿⣧⡾⠀⠸⣧⡼⠏⠀⠀⠀⠀⢸⠀⢀⡞⡿
⠀⠀⠀⠈⠧⠈⠻⣬⣉⠙⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠞⢀⢚⡜⠁
⠀⠀⠀⠀⠰⣦⢦⠐⠀⠉⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡿⠊⣴⡄
⠀⠀⠀⠀⠀⠐⠂⠀⠠⡄⢸⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠉⢡
⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠘⡀⠀⠠⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⢤⠀⠀⠀⣀⠀⢰⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠒⠀⠀⠀⠒⠀⠀⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣼
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⠶⠶⠶⣶⡶⢶⡶⢶⢶⣶⣶

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠈⠉⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣄⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⢁⠪⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠃⠀⠀⠀⠘⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣤⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⡀
⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀⣀⡀
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣤⡀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠋⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⣼⣿⣿⣿⣧⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣶⣦⣤⣭⣽⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣰⣁⣀⣀⣀⣀⣠⡬⠛
⠀⠀⠀⢀⣠⡴⠶⠛⠋⠉⣽⢏⣀⠈⢹⡄⠈⠉⠻⣿⣯⢿⡄⠀⠀⠀⢰⠋⢩⠟⠉⠉⠻⣌⣷⡶
⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠙⢦⣌⣭⣞⠁⠀⠀⠀⠈⢿⣿⠇⠀⠀⠀⣿⠀⢻⡒⣶⡞⢛⡟⠙⣇
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⡀⠀⠀⠀⢈⣷⣄⡀⠀⠀⣿⠀⠈⠛⠶⠶⠋⠀⠀⠙⡆
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⢙⡷⠀⠀⢻⡄⠀⠀⢀⡔⠢⢲⠀⡟⠁
⠈⠛⠶⢤⣤⣤⣤⣤⣤⣶⠶⠚⠃⠀⠀⣠⠃⠀⠀⠀⠘⢷⡞⠁⠀⠀⠈⢷⡀⠀⠈⢗⠒⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠤⣤⠤⠖⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠻⣄⡀⠈⠙⠉⠀⡇
⠀⠀⠀⠀⠀⠸⣿⣿⡟⢸⡇⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣬⠟⠂⠀⠀⠀⡟⣿⣿⣿⠃
⠀⠀⠀⠀⢀⣠⣼⠇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣶⣤⣴⡶⠶⠛⠉⠀⠀⠀⠀⠀⠀⡇⠀⢺⣥⣀
⠀⠀⠀⡴⠋⠁⢀⣠⣼⣧⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⣾⠁⠀⠀⣼⠷⠶⠶⠶⠤⠤⣄⡀⢤⣀⡟⠶⣄⡀⠈⠙
⠀⠀⠀⠛⠛⠛⠉⢡⡏⠡⡱⡴⠋⠙⣄⣠⠋⡔⠁⠀⠸⡇⠀⢸⡇⠀⠀⠀⠀⢠⡄⠀⠉⠛⠛⠀⢀⡿⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠱⡙⣆⣀⠞⡡⠊⠀⠀⡆⠀⢿⢀⣼⠗⠀⡦⠀⠀⠀⠙⠶⠶⠞⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⢌⡡⠊⠀⠀⠀⠀⣧⠀⢸⡟⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣃⣀⠀⠀⠀⠈⠀⠀⢀⣀⣀⣀⣿⣠⣤⣇⣀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⡿⠉⠉⠉⠉⠙⡟⠉⢿⡀⣰⣾⣿⡉⣻⠛⠛⢻⡟⠛⣿⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣤⢤⠤⣀⣧⠀⠀⠀⠀⢸⣇⠠⣬⣉⠉⠀⠈⠉⠁⠀⣀⣼⢇⢠⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠘⠳⠴⠤⠶⣌⠧⠀⠀⢠⣋⡶⠤⠤⠼⠃⠀⠀⠀⡔⣊⠭⠚⡼⠘⠼⠒⠤⠥⠖