Cleanliness Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⣀⠘⠈⠁⠀⡀⠀⠀⢸⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡠⠤⠴⠧⠄⣀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠞⡡⠔⠀⠘⣟⠀⢢⣍⠺⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣠⠔⢀⣁⢀⣉⣛⡑⠠⢀⠀⠐⡑⡼⠁⡷⢌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠒⠲⠀⠀⠐⠡⡊⠀⠀⠀⡄⡁⠉⠑⠢⣄⠀⢬⡃⢠⡧⠜⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄⠒⠀⠀⠃⠀⠀⠀⠀⠀⢡⠡⠀⠀⠀⠀⠙⣦⡑⢏⠈⠈⠀⠀⠖⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢃⢣⡀⠄⠂⡜⠁⡟⢢⠑⡀⢀⡙⢢⠔⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣓⣋⣀⣐⡂⠤⣄⡀⠑⡑⢄⠜⢀⡜⠀⠀⢱⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠊⠉⠈⢆⠀⠉⠁⠂⢍⡓⠾⡛⢤⡾⣀⠀⠀⠀⡇⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠈⢌⣦⣠⡴⠒⠚⠳⢶⡈⠺⡏⠢⠍⣁⡉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⡄⠀⢀⡼⠓⠢⢍⡓⢤⡀⠀⢹⡑⢌⠢⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣘⡾⠖⠋⠀⠀⠀⠀⠈⠲⣝⢦⣸⠇⠀⠣⡘⡄⢀⡆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡋⠁⣀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢦⢹⡀⠀⠀⠐⠘⡄⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠙⢾⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠀⡀⠐⡡⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠙⢦⡀⠀⣠⠞⠁⠀⠀⡰⠃⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠙⠺⣯⢋⠄⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣳⠴⠋⠀⠐⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠌⠉⠉⠉⠪⢢⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡱⠌⠀⠀⠀⠀⠀⠡⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠸⢁⠀⡠⡄⠀⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠂⣠⠀⢤⠲⡏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠂⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠸⠖⠒⠒⠒⠒⠒⠒⠶⠞⠀⠀⠀⠈⣥⡛⡟⡛
⠀⠀⠀⠀⠀⠀⠰⡟⢢⠤⢄⠀⠛⢤⡀⣀⣀⣀⣀⡠⠂⠀⢸⠁⠀⠀⠈⡇⡗⠲⠶⢶⢠⣤⣤⣄⣀⣀⣤⣶⠘⠓⠒⠒⠒⣤⠾⠿⣏⡛
⠀⠀⠀⠀⠀⠀⠀⠘⢇⣀⠜⠁⠂⣤⡀⢁⡅⠀⢠⡀⠀⢀⠸⠶⠶⠶⠶⠇⡇⠀⠀⣾⡌⣿⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⢠⠊⠀⠀⠀⠐⣴⠁⠀⢀⣀⣀⡀⠀⠘⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⣈⠉⠈⣅
⠀⠀⠀⢠⠀⠀⠚⠁⠀⠀⠈⠉⠈⠙⠉⠂⠐⠂⠂⠀⠓⠐⠓⠀⠀⠀⠀⠘⠃⠛⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⡀⢸⡇⠀⠀⢹⠄
⠀⠀⠀⠀⢹⠟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⢻⡟⠛⡟⠛⣟⠛⣿⠛⣻⠛⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡄⢀⡖⠓⠒⠒⠚⢲
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⢸⠀⡇⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⢁⠀⡀⠀⠀⠈⢀⠀⣲
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⡇⠀⢸⠀⡇⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⡀⢸⠉⢰⡖⠘⣿⠒⣶⠂⡒⡎⢳⠀⢸⡇⠀⣧⠀⣸⣠⣧⣤⣷⣄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠄⠀⠠⠦⠀⠀⠠⠄⠀⠶⠇
⠀⠀⠀⠀⠀⡃⠨⠄⠸⡇⠀⠷⠀⢿⠀⠅⠇⠸⠀⠸⢧⡤⠿⠠⠼⡿⠓⠂⠛⢻⡿⠶⠤⣤⠠⠤⠤⠤⠤⠠⠄⠐⠒
⠀⠀⠀⠀⠀⠁⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⡃⠀⠀⠀⠀⠀⠀⠀⣺⢴⣶⣶⠿⢶⣿⣷⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠐⠦⠾⠤⠤⠴⠴⣏⠀⠀⠉⠉⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠴⠤⠶⠖⠚⠋⠉

⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣴⠿⠛⠙⠳⣤⣤⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣠⠎⠀⠀⠀⣠⢤⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⠖⠂⠀⢀⡜⠀⢀⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠒⡆⠀⠘⢧⡶⠟⠉⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠘⠗⠁⠀⠀⠀⠀⠀⣤⠀⠊⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡿⠀⠀⠀⠀⠀⠀⢊⡨⠐⠒⢦⠀⠀⠀⠀⠀⠀⡰⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠴⢒⢿⣄⠘⡄⠀⣠⣾⣾⣿⣷⣤⢔⢾⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⣉⡉⠀⠀⣼⣿⣿⣿⣟⢯⠎⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣿⣿⣷⣶⣿⣿⣿⣿⣿⠟⡧⠰⠯⢄⡀⠀⢀⣠⠤⢠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡽⢣⠞⠑⣧⠀⠀⠙⢾⡋⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣿⣿⣿⣿⣿⣿⠟⣵⣋⡤⠶⠛⠳⣤⣀⣠⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⢢⡼⠋⠉⠉⡹⢫⣞⡠⢿⣶⡒⠒⠒⠀⠩⢩⡤⢀⣀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠞⠀⠀⢀⡞⡵⠁⠙⡄⠀⠉⠻⣷⣄⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣧⠀⠹⣆⡠⣴⣫⠞⠀⠀⠀⡇⠀⢀⣤⣾⣿⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⢀⣠⠴⠛⠉⠁⠀⠀⠀⠀⠈⠳⣤⠏⡐⢸⡃⠀⠀⢀⡞⠀⠀⢻⣿⣿⠟⠙⠋
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⢔⣵⠏⠁⠀⢠⠏⠀⢠⣶⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡱⠋⠁⠀⠀⡴⠃⠀⠀⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⠞⠁⠀⠀⢀⣼⠁⠀⠀⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡵⠃⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡤⠤⢤⣤⣏⠟⠀⠀⠀⠀⠰⠿⠛⠛⡟⠉⠉⣿⡏⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠇⠀⠀⠀⢀⠔⡁⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢻⠰⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠈⠢⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⠀⢷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⡠⠊⡀⠀⠈⠢⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡿⠀⠸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠴⠚⢁⡠⠀⣈⠔⠊⠀⡤⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢒⣉⡥⢔⡉⠄⣀⠔⣉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢸⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠤⣚⣩⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠄⠀⠀⠘⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉

⠀⠀⠀⣠⡶⠶⢦⣤⣤⣾⣿⣶⣤⣤⠶⠶⢦⡄
⠀⠀⠀⠻⣤⣤⣤⣤⣴⣶⢶⣶⣤⣤⣤⣤⣼⠏
⠀⠀⠀⠀⠀⠁⢀⣠⣤⣿⣤⣿⣤⣀⡀
⠀⠀⣀⣀⣀⡾⠋⠁⠀⠀⠀⠀⠀⠈⠙⣦⡀
⠀⢸⡏⠉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠛⠳⠶⣤⡀
⠀⢸⣇⣀⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢻⡄
⠀⠈⠉⠉⠉⠷⣄⣀⣀⣀⣀⣀⣀⣀⣴⠟⠙⠛⢷⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣻⣄⣀⣄⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠹⠟⠟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢛⢙⡛⡛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣉⣉⣉⣉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⢨⢬⣥⡥⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢦⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡟⠀⢀⡴⠛⠙⢷⣴⠞⠳⣆⠀⠀⢠⡶⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⡄⠀⠉⠁⢀⡴⣋⣤⣤⣴⣏⠁⢀⣴⣯⡀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⣰⠞⠋⠛⣫⣤⣀⠀⠉⠛⠋⠛⣿⣿⠀⠘⠷⣤⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣸⣧⡾⢻⡾⠛⢾⣇⠀⠙⢷⣴⠛⠛⢦⡘⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣿⢻⣇⠘⣧⣀⠀⠙⢷⣄⠀⠙⢷⣄⠀⠙⢦⣄⠛⢶⡄
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢹⡿⠁⣼⠋⠛⢦⡀⠀⠙⢷⣄⠀⠙⢷⣄⠀⠙⢧⣈⣿
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢸⣇⠀⠹⣦⡀⠀⠙⢦⡀⠀⠙⢷⣄⠀⠙⢷⣄⠀⠙⢿⡄
⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠛⠉⠳⣶⠈⠻⣦⡀⠀⠙⢦⡄⠀⠙⢷⣄⠀⠉⠀⠀⠀⠙⢦⣄
⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠈⠻⣦⡀⠀⠙⢶⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡶⢶⣌⡻⣦⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⢾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡴⢋⡵⠛⠛⠛⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⡴⠟⠉⠉⠁⠀⠈⠻⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠙⢷⣄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠙⠷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡴⠋
⣶⣤⣶⣤⣦⣦⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣼⣿⣴⣦⣴⣦⣤
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⣤⣤⣤⣀⣀⣀⣀⣶⠟⢻⣆⣀⣀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⢰⡟⠁⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠙⣷
⠀⠀⠈⢿⣤⣤⣤⣤⣴⡶⠶⢶⣶⠿⢷⣶⠶⠶⣶⣤⣤⣤⣤⣴⠟
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣟⠀⢈⣿⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠶⣦⣄
⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⣾⠟⠛⠛⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠷⠶⠶⣦⣤⣀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠈⠉⠻⢶⣄
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠿⠷⠶⠶⠾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠶⣦⣤⣀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡾⠋⠀⠀⠀⠈⠙⣷⡄⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠛⠛⠛⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣀⣄⣀⣤⣀⣤⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⢩⣯⢹⡏⢹⡏⣽⡍⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠉⢻⣦⣁⢀⡀⢨⡀⣨⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢠⣿⠋⠘⠃⠘⠃⠙⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡶⠟⠹⠗⠸⠇⠸⠇⠿⠂⠿⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⢠⡆⢠⡄⣴⠄⣶⠀⠀⠀⣸⡟⠉⠉⢻⣆
⠀⠀⠀⠀⢀⣴⠾⠛⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⡶⠟⠛⠛⠻⢶⣤⣠⣀⣤⡀⠀⠀⠹⣧⣀⣀⣼⠏
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠙⠻⠋⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠙⢷⣄⠀⠈⠉⠋⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣠⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀
⠀⠀⠀⠀⠘⢷⣄⣀⢀⣀⣴⡟⠁⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⣠⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡟⠁⠈⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⣄⣀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠘⣷⣄⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣧⣤⣀⡀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢷⣶⡟⠉⠉⠙⢷⣄⠀⠀⠀⠀⠈⠛⠷⠶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠈⠻⣦⡀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣀⣠⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠈⠻⣦⡀⠀⠈⠻⢷⣄⠀⠀⠀⣠⣶⢶⠶⠟⠋⠁⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠻⣦⡀⠀⠈⠻⣷⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠻⣦⡀⠀⠈⠙⢶⣤⣤⣤⣤⣤⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⠻⠶⠶⠶⠾⠿⠿⠿⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣶⣶⣦⣶⣶⣦⣤⣤⣤⣤⣦⣶⣤⣶⣴⣤⣤⣤⣤⣴⣶⣶⣶⣤⣶⣴⣤⣤⣤⣤⣤⣤⣴
⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡄⢄⡀⠀⠐⢄⠀⢆⠀⠀⠀⠀⠀⠀⡄⠀⢦⠰⢄⠰⠀⡆⠆⡀⠄⠠⡀⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣄⣀⡀
⠀⠀⡠⣾⣷⡀⠀⠀⢠⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠐⢋⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡿⠓⣀⠀⠀⣬⣇
⠀⠀⢈⣼⣷⣄⣿⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣦⣽⡇⠀⣼⣿⠀⡀
⠀⠀⣼⣿⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⡇⠉⠉⠉⠉⠉⠉⠁
⠀⠸⢧⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠖⣩⣿⣿⠀⠀⠀⠀⠀⠀⠐⢣⣿⣿⣿⡄
⠀⠠⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠏⣿⡿⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⡿⠿
⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠻⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠘⠃⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⡀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡀⢠⣾⣦⣄⣀⡠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⣀⣀⡀⠀⠀⠀⠀⡐
⡇⢸⣿⣇⣉⡉⠉⠉⣿⡋⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⡍⠻⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⠐⠚⠛⠓⢺⡟⠛⠓⠒
⠥⠬⢿⠿⠿⣿⣆⢰⠃⢣⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠹⠆⠇⣠⣶⣇⠐⣶⣶⣶⡶⠀⠀⠀⢸⣀⣻⣿⣿⣿⣆⠀⡏⢱
⠀⢀⠎⢂⠀⠈⢿⡏⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢸⠘⣧⠀⢸⣍⠛⠛⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠠⢣⠀⠈⠻⣿⠁⠀⡇
⠀⠀⠀⠈⠀⠀⠼⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠂⠘⠿⠗⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠐⠁⠀⠂⠀⢠⠇⠀⠀⢸⡄
⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠐⠀⠂⠀⠀⠀⠐⠐⠀⠀⠀⠀⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠤⠤⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡲⠒⠤⣀⢤⡀⠀⠀⠑⠺⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡰⠃⠀⠈⣷⣀⠀⠀⠀⡏⠉⢩⠟⡖⠚⠒⠓⡞⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢅⠀⢀⠞⠀⠈⠑⠢⣤⣣⠔⠁⢀⣇⠀⠀⢀⣇⠀⠀⠀⠀⢀⣀⠿⠭⠭⠭⠭⠭⠽⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠈⡇⠀⡠⠊⠀⠉⠉⠁⠀⠳⡄⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⣀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣡⣤⣤⣀⣀⣠⣔⣹⣄⣸⣇⣀⣇⣀⣀⣀⡀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢰⡋⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡆
⠀⠀⠀⠀⠀⠹⣭⠭⠭⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠬⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣭⠏
⠀⠀⠀⠀⠀⠀⣷⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡞⢳⠀⢰⠋⡆⠀⡖⢳⠀⢠⠊⣦⠀⢰⠑⡄⠀⡎⢲⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⡇⠀⠇⢸⡀⢸⠀⣿⠀⠀⠀⡇⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⣿⠀⠀⠀⡇⠀⡇⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⣿⠀⠰⠀⡇⠀⡇⣸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢹⠋⢱⠒⡖⠒⣾⡻⡆⢲⡶⢶⠒⢶⠶⡖⠋⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠀⠀⠀⠀⠀⡃⠀⡇⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⢸⠀⡇⠀⣾⠁⡇⢸⣷⣾⠀⢸⠀⡇⠀⡇⢸⠀⢸⠀⡇⠀⡇⢸⡇⢸⠦⣿⠀⠰⠚⡇⠀⡇⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣧⠀⣇⢸⠀⠸⡖⡇⠀⢿⢀⠇⠸⣇⣿⠀⢸⢀⡇⠀⣇⣸⠀⢸⡀⡇⠀⣇⢸⠃⠸⡀⡿⠀⢸⢀⠇⠀⣇⣻⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢩⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠒⢢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠄⠛⠀⠈⠁⢑⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠌⠀⠀⠀⠀⠀⠀⢊⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠀⠀⠀⠀⠀⠀⠀⠘⣰⠀⢀⡤⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡧⠒⢠⡌⢤⡒⠮⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠀⠀⠀⠀⠀⠐⢢⠀⠀⢀⠀⢀⢀⠀⠀⣤⡴⠶⠆⠗⠒⠒⠒⠒⠒⠒⠒⠶⠁⠀⠀⠀⢩⠖⡿⢛⠇
⠀⠀⠀⠀⠀⠀⠰⡟⢂⠤⢤⡀⠛⢆⡀⣀⣀⣀⣀⡠⠞⠀⢐⠁⠀⠀⠀⡇⢸⠺⠶⢶⠄⣦⣤⣤⣠⣤⣠⣶⡂⠛⠚⠒⠒⢢⠜⠿⣿⡉⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢇⢀⠜⠁⠂⢼⡄⠈⣄⣀⣨⡄⠀⢀⡨⠶⠶⠶⠶⠧⣼⠀⠀⢼⣧⢻⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢨⠀⠎⠀⠀⠀⠈⢤⠋⠀⠀⠀⡀⠀⠀⠈⢂⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣀⠀⡆⠇⠀⠀⠀⠀⠀⢸⠀⠎⠁⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣰⠉⠉⢹
⠀⠀⠀⠀⠂⠀⠚⠃⠀⠀⠈⠉⠉⠉⠉⠉⠐⡆⠀⠀⠚⠀⠓⠀⠀⠀⠀⠐⠃⠘⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠠⡿⠀⠀⠈⡇
⠀⠀⠀⠀⢹⠛⢛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⣟⠛⢛⠛⠛⠛⠛⠛⢛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠀⡔⠒⠂⠀⠀⠓⡄
⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⠀⠀⠁⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠀⠁⠈⠉⠈⠉⠁⢉⡆
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⠀⠀⠀⠆⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡇⠈⢹⠙⠻⠉⡿⠉⣷⡆⢲⠂⠉⡇⠀⢻⠀⠸⠀⣀⣤⣦⣤⣼⣆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠰⠠⠀⠶⠀⠤⠠⠤⠀⠤⠜
⠀⠀⠀⠀⠀⡇⠀⡸⠘⡶⠀⠇⠀⢿⠇⢸⠀⢠⡇⠀⡾⡰⠴⠤⠮⣿⠧⠄⠘⠫⡿⠶⠶⢤⠄⠀⠀⠀⠀⠠⠤⠤⠀⠊
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠉⠉⠉⠉⠉⠉⢹⣥⣶⣾⣿⣶⣿⣷⣶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠜⠆⠾⡠⠆⠴⠤⢿⠀⠀⠉⠉⠉⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⡤⠴⠒⠒⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣁⣛⣻⣟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣯⣭⣽⣭⣿⣿⣿⣽⣍⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢤⢎⠒⢠⠀⠀⠀⠀⠀⢠⡀⠄⠀⠀⠀⠀⢁⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠐⠀⠀⠀⠐⠖⠀⠀⠀⠈⠃⠘⠤⠈⠀⠀⠀⠀⢈⡁⣀⣠⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⡈
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈

⠀⡀⠀⡠⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⢀
⠀⢀⠁⠈⠉⠑⠀⠀⠀⠀⠀⣦⠂⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠠⠄⡈⡀⠤⠀⠀⠀⠀⠂⠂
⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠽⠚⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠉⠑⠂⠀⠀⠀⠠⠀⠀⢀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⣀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠈⠀⠀⠄⠠⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠠⠉⠉⠁⠀⠀⠀⠐⡄⢠
⠀⠀⠀⠀⠀⠡⠀⠀⠀⠢⡈⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠠⠈⠃
⣤⣤⢦⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣷⣤⣤⣤⣤⣤⣤⣶⣤⣤⣶

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⢀⣿⠀⠀⠀⠀⠀⠀⡠⠔⠒⢆⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠕⠋⠉⠝⢆⠀⠀⠀⠀⠀⡼⢋⠻⡄⠀⣽⠀⠀⠀⠀⢠⣯⣄⠀⠀⠀⣭⠆
⠀⠀⠀⢀⡴⠟⢦⣀⠀⠀⠀⠀⠘⢿⡀⠒⢒⡮⠗⠀⠀⠀⢸⠁⢻⠃⣷⠀⣿⠀⠀⠀⠀⠸⠿⣿⣿⣿⡿⠛
⠀⠀⠀⠙⠦⠤⠦⠋⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠀⠀⢸⡒⠚⠒⣿⣠⣿⣄⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢉⠉⠉⠛
⠀⢠⠖⡩⠍⡑⢢⠤⠤⠄⠀⠀⠀⠀⢀⡤⠔⡢⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣖⣒⣺⢷⣶⣶
⠀⢸⠐⠀⠁⠀⢸⠀⢸⠀⠀⠀⢠⠊⠡⢢⡟⠟⠈⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣤⠠⠄⠄⡏
⠀⢸⠀⠀⠀⠀⢈⠀⢸⠀⠀⠀⢸⡀⠢⡥⢐⣨⠖⠁⠀⠀⠀⠀⣀⣠⣧⡤⠤⣄⠀⠀⠀⠀⢸⣿⠀⠀⢡⡇
⠀⠈⠲⠤⠤⠔⠋⠉⠉⠀⠀⠀⠀⠘⠓⠛⠃⠀⠀⠀⠀⠀⠀⣿⢄⣈⣩⣥⡴⠟⠀⠀⠀⠀⠘⠿⠤⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⡶⣏⢿⣿⣷⡀⠀⠀⠀⠀⠀⢀⡠⠔⠓⠦⣄⠀⠀⠀⠀⠀⢀⡠⣴⡶⠀⠀⠀⠀⠀⠀⠀⠰⡛⢻⠂
⠀⠀⣸⡜⠂⠀⠹⣧⠀⠀⠀⢠⡒⠁⠀⠀⠀⠀⡈⢽⡂⠀⠀⢸⢳⠟⠛⠉⢉⣷⡀⠀⠀⠀⣀⠔⠋⠙⠢⡀
⠀⠀⠓⡤⡀⠀⠀⠣⡀⠀⠀⢸⡀⠀⠂⠠⠀⠈⣠⡾⠁⠀⣸⣍⡚⢀⣠⢞⡿⠛⠁⠀⠀⢸⠁⠀⠀⠀⠀⢱
⠀⠀⢀⡇⠈⡄⣀⣠⠗⠀⠀⠈⠙⠷⣤⣤⣤⠟⠁⠀⠀⠀⠓⠿⣿⡯⠟⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠊⠉⠁
⠀⠀⠀⠀⡤⠤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⢷⣴⡟⠅⠀⠀⠀⢾⠙⡦⠤⠤⢤⡤⣤⠚⡆⠀⠀⣴⣊⡉⠉⢉⣲⡄⠀⠀⠀⠀⡼⠉⠻⠯⠿⡆
⠀⠀⠀⣠⠞⠀⢆⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠘⡶⠃⠀⠀⢸⣌⡉⠉⣉⢼⠁⠀⠀⠀⠀⣧⠀⠀⠀⠀⣴
⠀⠀⠀⣏⠉⠀⢸⡆⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠚⠁⠀⠀⠀⠀⢧⣍⣉⣠⠎⠀⠀⠀⠀⠀⡯⢀⣀⣀⡀⢽
⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠒⠀⠉

⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣝⣄⡠⣤⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⢸⣽⠀⠀⠀⠀⠀⣠⠔⠊⠉⠈⠳⢄
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⢰⡉⠀⠀⠓⠵⣄⠀⠀⠀⠀⢀⡞⡁⢈⢳⠀⢸⢸⠀⠀⠀⠀⠀⣿⣶⣄⣀⣀⣀⣴
⠀⠀⣠⠞⠛⠳⣄⠀⠀⠀⠀⠀⠘⠺⣄⠛⢁⡵⠚⠁⠀⠀⠀⢸⠀⡑⠀⢸⡄⢸⡎⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⠟⠉
⠀⠀⠙⠶⠤⠖⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⡛⠒⠒⢺⣥⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠉⡙⠛⠛⠛
⣠⠖⢒⡒⠒⢤⣀⡠⡄⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠶⢦⠤⢤
⡧⠀⠀⠈⠁⢸⠀⠀⡇⠀⠀⠀⢀⠔⢊⣥⣢⢟⠎⢱⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⡉⢡⠈⡅⡏
⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢟⠠⡀⠘⢉⠄⢂⡽⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⢀⡇
⢧⡀⠀⠀⢀⡴⠤⠔⠃⠀⠀⠀⠸⢦⣄⣯⡴⠚⠁⠀⠀⠀⠀⠀⣤⠖⠪⠿⠿⠀⢙⡇⠀⠀⠀⠀⠘⡇⠀⠂⠈⢸⠇
⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠵⢤⠶⠖⠛⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠰⡻⣎⢻⠿⢿⣦⠀⠀⠀⠀⠀⠀⡠⠔⠚⠁⠑⠲⣀⡀⠀⠀⠀⠀⣠⢴⣺⠽⣃⣀⠀⠀⠀⠀⠀⠀⠀⢫⣉⡼
⢀⣵⡈⠀⠀⠸⣿⠄⠀⠀⠀⡶⢉⠀⠀⠀⠀⠀⡀⠔⢿⠀⠀⠀⢸⠙⡟⠋⠉⠀⣨⡷⡀⠀⠀⠀⢀⠔⠋⠉⠉⠢⡀
⠈⠲⡄⠀⠀⠀⠹⡄⠀⠀⠀⣯⡀⠈⠐⠠⠄⠈⢀⣴⠟⠀⠀⣸⣍⡙⠁⣠⡴⣻⠿⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢱
⠀⢸⠀⠙⡀⠀⣀⡼⠀⠀⠀⠉⠻⢶⣤⢠⡠⣲⠞⠁⠀⠀⠀⠓⢿⣟⣟⡵⠚⠁⠀⠀⠀⠀⠀⠰⡅⠀⠀⠀⠀⠀⢠
⠀⠈⠛⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠋
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠐⣇⣀⣼⡏⠀⠀⠀⠀⡴⢢⣀⡀⠀⠀⠀⠀⢀⢤⠀⠀⠀⢀⡤⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⢈⡟⢻⣧⡶⣦⡀
⠀⠀⠀⣹⠛⡇⠁⠀⠀⠀⠀⢳⣴⠏⠉⠉⠉⠋⠉⣧⢸⠁⠀⠀⢻⠒⠶⠶⠶⠶⣾⠃⠀⠀⠀⠀⡸⠀⠀⠀⠉⠉⢇
⠀⢀⡎⠁⠀⢱⡄⠀⠀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⡸⠋⠀⠀⠀⠈⡗⠤⠤⠤⠚⡏⠀⠀⠀⠀⠀⡗⠉⠉⠉⠉⠉⣿
⠀⢸⣇⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⠞⠁⠀⠀⠀⠀⠀⢇⠂⠀⠀⠀⠐⣺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⠀⠀⠀⢠⠤⠖⠒⠛⠉⠉⠉⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣿⡏⠉⠉⠀⠀⠀⣤⡀⠀⣿⣿⠀⠀⢀⣤⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⢸⣇⠀⣿⣇⠀⣿⣿⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⠀⣿⣿⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣶⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣫⣵⣾⣿⣿⣦⣶⣿⠟⠉⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠶⠶⠒⠚⠋⠁⠀⠹⣜⣻⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣤⠀⣄⡄⡄⢶⢠⣄⢰⠆⣄⡴⢄⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠃⠉⠃⠃⠘⠘⠙⠘⠀⠋⠓⠋⠋⠙


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶
⠀⠀⠀⠀⠀⢸⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡇⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻
⠀⠀⠀⠀⠀⠿⣿⠿⠂⠀⠀⠀⠀⠀⣤⣶⠾⣿⣻⠉⠁⠀⠙⠿⠀⠀⠀⠀⠀⣶⣶⣶⣾⣹⢇⣤⣤⡟⠉⠄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣹⢿⡇⠀⢠⠴⠃
⠀⠀⠀⠀⠀⣛⠯⣛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⡼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣿⣷⣿⣶⠀⠀⠀⠀⠀⣤⣤⣤⣤⣿⣶⣾⣿⣿⣿⣿⣯
⠀⠀⠀⠀⢀⣉⠿⣉⡄⠀⠀⠀⠀⠀⠀⠀⢸⢃⡸⢧⠀⠀⠀⠀⠀⠀⠀⢀⠀⢉⣻⠋⣉⡉⠉⣹⡏⣉⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢨⠉⠒⠏⠅⠀⠀⠀⠀⠀⠀⠀⡟⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠁⠀⠈⠀⢸⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠠⣛⢄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⡶⢔⢾⣤⡿⢰⢴⡆⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣺⡂⠀⠀⠀⣠⣀⣀⣀⣤⣤⣤⣤⣤⣤⣦⣴⣶⡇
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠃⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣿⣿⢺⠂⠀⠀⠀⠛⢚⣛⠚⠻⣿⠿⣾⣾⣾⣞⣾⣾⡇
⠀⠀⢀⠀⡀⣻⣿⣿⢀⡀⣀⠀⠀⠀⠀⠀⠈⠢⢠⡐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣒⣿⣰⣷⣿⣿⢽⠄⠀⠀⠀⢠⣯⣤⣽⣿⣷⣧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠘⠑⢋⣹⣿⣿⡙⠋⠛⠀⠀⠀⠀⠀⠀⢰⣾⣷⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡅⠀⠀⠀⠰⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢻⠿⠿⢿⢻⠻⢿⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣻⣿⣟⣛⣛⣿⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠊⠁
⠀⠀⠀⠀⣀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣶⣀⣀⡀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⣿⣏⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡃⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⡃
⠀⠀⢸⣿⣋⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣄⣄⣠⣤⣤⣿⣧⠀⠀⠀⠀⠀⠀⠉⣈⠙⣿⡥⠔⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁
⠀⠀⢸⣿⢫⡍⣭⠉⠉⣍⣤⢀⣤⢠⣤⡄⠘⠟⠘⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⢆⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⣻⡆
⠀⠀⢸⣿⣀⣁⣀⣀⣠⣏⣫⡬⠯⠬⠥⠧⠤⠤⠤⠤⢼⣿⠀⠀⠀⠀⠀⣀⠐⢸⠇⣣⢺⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣘⡙⡇⢠⡄⠸
⠀⠀⢸⣿⢬⣤⣤⣤⣤⣤⣶⣶⣶⠶⢶⣶⣶⣲⣶⣶⠛⣿⠀⠀⠀⠀⠀⠺⠇⠀⢀⠭⠻⠧⠒⠋⠀⠀⠀⠀⠀⠀⠠⢺⠋⠁⠔⢿
⠀⠀⠸⣿⢸⣋⣉⣉⣉⣁⣨⣭⣿⡄⢈⣭⡭⠭⢹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣷⠀⠀⠀⠀⠠⠂⢀⣾⠀⠀⢀⠘⠁
⠀⠀⠀⣿⣼⣶⣤⣤⣤⣤⣴⣶⣦⣴⢱⠠⡔⢶⣾⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠉⠀⠠⠀⠀⣤⡴⠿⠃⠀⠀⠼⠲
⠀⠀⠀⣿⣿⡿⠿⠻⠛⠿⠿⠿⠿⠿⠀⠀⢰⠈⠻⣿⣧⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠸
⠀⠀⠀⡿⠿⠇⠀⠀⠀⠀⠠⠞⢣⡐⣠⣦⡜⢁⣤⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡤⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⡇⢸⣀⣀⣘⣋⠁⠀⠀⠀⠳⢉⡾⠞⠉⢅⣽⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡋⠉⢁⠈⠁
⠀⠀⠀⣿⣈⣿⣹⣯⣉⣱⡀⠀⠀⠀⠸⡥⠘⣇⠀⢿⣈⣼⠇⠀⠀⠀⠀⠀⠀⢰⡆⠤⠬⠭⢱⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⣿⣿⣿⡏⠀⠀⠘⢦⡀⠀⢀⡙⠀⠙⣦⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⠨⡎⠁⠀⠀⠀⠀⠀⠀⠁⢴
⠀⠀⠀⠀⠿⣿⣿⠇⠀⠀⠀⠀⠑⠂⢸⠁⠀⠀⠿⡀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⡿⣀⠀⢤⡄⠀⠀⠀⠀⠀⠀⢧⠈⠀⡠⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⠀⠀⠀⠠⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡀⠇⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⣠⠘⣁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡹⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⢣
⠀⠀⠀⠀⠀⢰⣿⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⡀⢡⠀⠀⠀⠀⠀⠀⠀⡈⠀⢨⠏⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠐⠋⠀⠘⠃
⠀⠀⠀⠀⢀⣷⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣈⡡⠤⠄⠀⠀⠀⠀⠀⠀⢰⣼⣷⣤⣼⠂
⠀⠀⠀⠀⠸⣿⣿⡏⠻⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢰⣖⠓⠾⣿⢿⡽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢀⡿⠟⣽⠖⢪⡿⣿⡂⠀⠀⠀⠀⠘⠀⠀⠈⠁⠀⠀⡉⠘⠃
⠀⠀⠀⠀⢠⠃⢸⢁⠀⠀⣄⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠂⠀⡈⠁⠀⠀⣸⠃⠀⠀⣰⡟⠁⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠐⠏⠀⠈⠋⠀⠀⠿⠆⠀⠘⠁⠀⠀⠀⠀⠀⠘⠒⠀⠉⠀⠀⠀⢏⡀⠀⣰⣿⠁⢠⠓⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠳⠄

⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⢀⣴⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣴⡿⠿⢿⣆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣦⡀
⠀⠀⢠⣾⣿⣿⣿⡯⠉⢣⡀⠀⠀⢀⣾⣿⡟⠀⢿⣿⣷⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄
⠀⢰⣿⠟⣋⡍⣴⣾⣆⣸⣷⡀⢠⣿⣿⣿⡆⠀⢸⣿⣿⣿⡄⢀⣾⣿⣿⣿⡛⡻⣿⣿⣿⡄⢀⣾⣿⣿⣿⣿⣿⡟⠋⢸⣆
⠀⣿⣿⠼⣿⡳⣸⣿⣿⣿⣿⡇⣿⣿⣿⣿⣯⣒⣽⣿⣿⣿⣷⣾⣿⡿⠿⠏⠀⠁⠿⠿⣿⣿⢸⣿⣿⣿⡿⡟⡏⠃⠀⢸⣿
⠀⢿⣁⣸⣯⣽⣧⣿⣿⣙⣿⠇⢻⣿⣿⣿⣲⣭⣖⣿⣿⣿⠏⠹⣏⣀⣤⣶⣶⣶⣦⣀⣘⡟⠸⣿⣉⣀⣀⣀⣀⣀⣀⣸⡟
⠀⠀⠙⠿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠
⢰⣦⡤⢶⡞⠋⠁⢀⣀⣀⣠⠴⠆⠀⣿⣿⣿⣤⣄⣤⣠⣄⣤⣠⣼⣿⣿⣿⠀⣿⡗⠒⠒⣒⣒⡆⠀⣼
⠈⠻⠀⡼⢷⠀⢀⠸⠯⢱⢷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇⣿⣿⣿⣿⣿⡆⣿
⠀⠀⢠⠃⠸⡌⣿⡆⠀⡼⠈⡆⣻⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡇⣿⣿⣿⣿⣿⡇⢺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣭⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠿⡿⠿⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠾⢿⠷⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⢿⠛⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣀⣠⡄⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡾⡽⡆
⠀⠀⠀⠀⠀⠀⣤⣮⣿⣿⣿⣿⣼⣇⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢠⢃⠘⢣
⠀⠀⠀⠀⠀⠀⢿⡿⡿⠿⠿⠿⠾⠿⣿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣼⢋⠀⣹⡀⠀⣠⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢸⣟⣻⣾⣽⣧⣴⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣿⣛⣷⣯⣟⣤⣶⠏⠑
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠄⠘⡧⠐⠠⠄⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣣⣴⣾⠿⢛⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢣⣾⣿⡟⣡⣾⣿⡿⠟⢛⣛⣓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⡿⣴⣿⡟⣡⣾⣿⣿⠿⠿⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠷⠿⠿⣿⣶⣦⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⢟⢻⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣏⡽⣿⣷⣶⠾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣒⣒⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢿⠸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣟⣯⡗⢾⠀⠹⣿⣽⣿⣿⢛⣿⣛⡟⣯⢞⡷
⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⣛⣿⣷⠿⠛⠉⠀⢸⡆⠀⠈⠻⠿⠟⡟⠿⢿⠟⠋⣸⠇
⠀⠀⢀⣤⣶⣿⣿⣿⡿⠟⠻⡆⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠅⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡆⠀⠀⠀⣸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣷⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢿⠀⠀⢀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠦⣄⠀⠀⠀⠀⠀⠀⣰⠛⠛⠛⠛⠛⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⢳⡄⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⣀⣀⣀⣀⣀⣀⣠⣧⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⡀⠀⠀⠀⠀⣰⠋⠉⠙⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢰⠇⠀⠀⠀⣿⠀⠀⣿⣀⣀⣀⣀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡏⠀⠀⣾⠀⠀⠀⠀⣿⠀⠀⣿⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣴⠋⢳⡄⠀⠀⠀⣧⠀⠀⢿⠀⠀⠀⠀⣿⠀⠀⣿⣤⣤⣤⣤⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢀⡾⠁⠀⠀⠹⣆⠀⠀⢻⡀⠀⠸⡆⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣾⠁⠀⠀⠀⠀⢹⡆⠀⠈⢳⡄⠀⠹⣆⠀⠀⣿⠀⠀⣿⠶⠶⠄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢻⣄⠀⠀⠀⢀⣼⠃⠀⠀⠀⣿⠀⠀⠈⠳⣤⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠉⠓⠲⠒⠋⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠒⠒⠒⠒⠒⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠉⠉⠉⠉⠉⠙⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡀⠀⠀⠀⢀⣠⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⢹⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢟⣭⣶⣶⣮⣝⢿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⡄
⠀⠀⠀⠀⢰⡏⠀⠀⣿⣿⣿⣿⣿⡉⠉⠀⠀⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣷
⠀⠀⣆⢀⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡇⢀⡆
⠀⠀⣿⣾⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠋⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣾⡇
⠀⠀⢹⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠛⠛⠒⠒⠒⠚⠛⠛⠻⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢀⠎⠁⠒⢀⣨⠳⡀⠀⠀⠀⠀⣀⠾⠧⡚⠀⠀⠀⠀⢰⠁⠀⠀⢸⠋⡍⡆⠀⠀⠀⡣⠟⣤⢴⡖⠒⡄
⠀⢸⣀⣠⣧⣊⠋⠁⡇⠀⠀⠀⠰⣇⡀⠀⣡⡄⠀⠀⠀⢸⠀⠀⠀⢸⣦⡡⠃⠀⠀⠈⢣⠚⠩⢄⣦⣾⠇
⠀⠈⢦⡋⠏⠀⢀⡰⠁⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣶⣶⡿⠟⠁
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠟⠉⠉⠻⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠠⢤⣶⡡⠤⣬⣷⡀⠀⠀⠀⢰⡲⠒⢢⣀
⠀⠀⠀⢸⠘⠃⡇⠀⠀⠀⠀⠀⠀⢘⠙⠿⡇⠀⠀⠀⠀⠀⢨⠛⡄⠀⣿⣿⡇⠀⠀⠀⣦⣤⣶⣎⣸⣶
⠀⠀⠀⠘⣤⣤⠃⠀⠀⠀⠀⠀⠰⣀⠁⠈⣀⠆⠀⠀⠀⠀⠀⠀⠑⢄⣿⠟⠀⠀⠀⠀⠻⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠰⠆⣁⠀⠀⠀⠀⠀⠀⢰⠒⠒⡂⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⢸⣁⡤⣤⣤⣤⣤⡀
⠀⢰⣩⣵⣶⣾⣿⣿⡆⠀⠀⠀⠀⢸⣶⣤⡇⠀⠀⠀⠀⢰⣈⣉⡀⢀⣉⣑⡆⠀⠀⠀⠉⡇⣿⣿⣿⣿⠁
⠀⢸⣿⣿⣿⣿⠿⠿⡇⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⠈⠍⠉⠉⠉⠁⠀⠀⠀⠀⠀⡇⠻⠿⠿⡿
⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⢿⠞⠃⠙⠀⠀⠀⠀⠀⠀⠀⡔⢦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⣟⠀⠈⢇
⠀⠠⣤⣤⡤⢤⣤⣤⠄⠀⠀⠀⣀⣰⢓⢢⢠⠔⠀⠀⠀⠀⠀⢨⠟⠛⡅⠀⠀⠀⠀⠀⠉⠙⠒⠼⣆⣀
⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠹⠤⠤⠥⠎⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⣍⠁
⠀⠀⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠦⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡩⠷⠒⠒⢼⣏⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠀⠀⠀⠀⠀⠀⠱⡜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠘⡄
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢸⠳⣄⠀⠀⡎⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢧
⠀⠀⠀⠀⠈⠓⠬⣓⠢⢄⡈⢣⡈⠓⢶⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀
⠀⠀⠀⢀⡤⠤⠤⠀⣉⣦⠈⠉⠁⠀⠀⠙⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠌⡃
⠀⠀⠀⠀⢀⣉⣉⠉⠠⠦⠀⠀⠀⠀⠀⠀⠘⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⡟⠛⠛⠶⡒⢤⡀
⠀⠀⠀⠀⠙⢒⣒⣐⡒⠂⡄⡴⠒⠒⠒⠒⠒⠛⠒⣲⣦⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⣧⣠⣤⣴⠡⠾⠇
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢰⠋⠉⠛⠛⠛⠛⠛⠉⠉⣩⣿⡇⣯⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠷⣿⡟⢏⠉⣲⠛⠦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠉⠀⠈⣿⠁⠀⡰⠋⠲⢄⡀⠀⠀⠈⣧⠤⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠞⠀⠀⠀⠀⢘⣦⠞⠉⠀⠀⠀⠀⠀⠀⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⡜⡇⠀⢀⡤⠤⠄⠀⠤⠤⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣠⡄⢰⡇⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⡖⠒⠤⠤⠄⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣠⠎⠀⢠⠘⠇⡇⠀⢻⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡴⠁⠀⠀⢸⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣸⠁⠀⠀⠀⢸⠀⡄⡄⠀⢸⠀⠀⠀⠀⠀⠀⡸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⠀⡇⡇⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡆⠀⠀⠀⠀⡌⠀⢰⢁⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡁⠀⠀⠀⢠⠃⠀⢸⢸⠀⠀⢣⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢇⣀⣀⡠⠋⠀⠀⢸⠘⡀⠀⠀⠙⠒⠒⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⢫⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢇⣀⠀⠀⠀⠀⠀⠀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠀⠛⠋⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣭⣭⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠲⢰⡆⠖⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⢸⡇⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⣴⢸⡇⣦⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⢸⡇⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣍⢸⡇⣩⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣿⢸⡇⣙⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣴⣿⣿⢸⡇⣿⣿⣦⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣿⣿⣿⢸⡇⠿⢛⣩⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⢸⡇⠺⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣋⣥⣶⣿⢸⣇⣿⣶⣬⣙⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠉⠉⠉⠁
⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⣠⡀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣿⠀⢀⣄⠀⠀⠀⠀⠀⢀⣄
⠤⠴⠚⠁⠙⠲⠦⠤⠶⠚⠁⠙⠶⠿⠤⠶⠛⠉⠙⠶⠤⠤⠶⠋⠉⠛⠶⠤⠿⠶⠋⠈⠓⠶⠤⠴⠖⠋⠈⠓⠦

⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢶⢰⣒⡆⡆⣶⠀⢰⠀⢰⢲⣶⣰⢠⡖⣒⠀⢰⣲⠆⡆⡴⢒⡂
⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣯⣼⣭⣧⣧⣽⣶⣼⣶⣼⣼⣧⣿⣬⣷⣿⣤⣼⣽⣧⣧⣽⣾⣧⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣉⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣿⡤⠤⢤⣶⣶⣶⣶⡶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃⡼⠓⣛⣿⣯⡙⡍⣉⠉⡍⢍⠙⣓⠺⣝⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⣸⠿⢭⣤⣈⣏⣀⣉⣔⣲⣌⣭⡾⠤⢗⣫⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣰⣯⣍⣙⡒⠒⠲⠶⠶⠶⠶⠖⠒⠒⣋⣉⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⠏⣇⠙⢮⡉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⡟⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞⠀⢻⠀⠀⠉⠲⣄⣀⠀⠀⠀⠀⠀⢀⣠⠼⠗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢾⡁⠀⠘⡇⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠺⠿⡶⠚⠁
⠀⠀⠀⠀⠀⠀⡄⠀⢀⣤⣤⣀⠀⢠⣞⣉⣙⠺⡀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⣀⠀⢙⡶⣫⠶⠦⣌⣳⢚⣹⢹⡟⣿⣷⠦⣤⠼⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠂⠀⠀⣯⣍⡭⠞⡥⠖⠶⢤⡼⢋⡡⢫⣷⣿⣅⡠⢬⡛⣩⠴⢭⣳⡀⠀⠀⠀⠀⣀⡼⢣⡄⢀
⠀⠀⠀⠀⠙⢦⡤⠞⣱⠚⣹⡏⣰⢋⡶⢡⠋⡏⣽⡖⠲⠭⠗⣲⡶⠏⢝⢟⢛⡛⣫⠝⡴⡢⠢⠀⠑⠂
⠀⠀⠀⠀⠀⢀⣝⡻⢥⣞⡴⢧⣹⣼⣿⣜⣦⡳⢝⠻⣏⠙⣻⡥⣀⠀⠀⠐⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠐⠀⠐⠂⠁⠀⠒⠆⠉⠙⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⢈⢳⠀⠀⠀⠀⠀⠐⠀⣼⠁⠁⣿⠎⡠⡄⢀⠤⡄⢠⢤⡄⣠⢁⡄⣤⠀⡤⢠⠄⠐⡄⠀⠀⠉⠠⡀⢀⠐⠄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠻⠀⠘⠇⠸⠏⠀⠏⠘⠇⠋⠸⠑⠇⠚⠀⠇⠘⣣⡞⠀⠀⢁⠀⠀⠜⠁⠿⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠀⢀⡞⣸⠁⠀⠈⠁⠂⠀⠀⢰⢀⣰
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠸⠀⢀⠀⠌⠃⠁⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠙⠀⠀⣱⢰⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠂⠀⠰⢀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠂⠀⠀⠘⠃⠸⠉⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠄⢂⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⡰⠒⠀⠀⢠⠀⠀⠀⠀⠘⠀⠂
⠀⠀⠀⠀⠀⠈⠸⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠀⠀⠈⠢⣀⣀⡠⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⠃⠀⠀⠀⠀⠀⠸⠀⢠⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⢔⢄⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠘⠄⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⢀⣠⣤⡖⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠦⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠏⠿⠩⠑⠋⠁
⠰⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡠⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠀⠁⠀⡀
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠒⠠⠀⡆⠢⠐⣄⠲⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠑⠠⠀⠀⡀⡄⠀⠀⠀⡀⠓⡄⠀⠀⠀⠀⠁⠀⠁⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁
⠀⠀⠀⠀⠀⠁⠃⠀⡔⠁⠁⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡧⠁⠀⠀⠀⠀⠀⠀⠀⠂⡄
⠀⠀⡀⣄⣀⠈⠀⢰⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⠀⢠⠒⡄⠀⠀⠀⠂⠱⠘⠘⠐
⠀⠈⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢅⡀⢈⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠚⠀⢸⢠⠁⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠀⠀⠀⠀⠈⢠⠃⠀⠀⠀⠀⠀⠈⠀⠀⠐⡄
⠀⠀⠔⠑⡒⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠀⠀⠀⠀⢀⠆⢎⣀⣀⣀⣀⠀⠀⡀⠀⠀⠀⣧
⠀⠉⠒⠂⠀⠀⠑⠀⠐⠐⠒⠒⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠼⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣤⣄⣀⣴⣿⣿⣿⣿⣷⣦⣄⣀⣴⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠘⠓⠒⠖⠶⠠⠔⠋⢉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡙⢠⡀⠀⠀⠉⠃⠀⠀⠘⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣷⣆⠉⠰⢶⠖⣀⠀⠟⠀⠀⠹⣧⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠋⢀⡠⢤⡍⠛⣿⣶⣄⣤⣤⣀⣴⣆⠀⠀⢱⠊⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⢉⣠⡤⠶⠛⠁⠀⡌⠁⠈⠟⢿⣿⢻⣿⣿⣷⡟⠀⠀⠀⢶⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣶⣿⣿⠿⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠟⠛⠛⠁⠀⠀⠀⠈⢧⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⢫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣤⣶⣶⣤⣄⠀⢀⠀⣀⣀⠈⢆⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠈⠷⠀⠀⠀⠀⣠⣴⠖⣺⣶⣄⠈⢻⣧⠜⣟⣾⣿⣿⣿⠀⢻⣿⣿⡿⢷⣾⣆⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢣⠀⣠⠦⡌⠁⠰⠟⢸⣿⣶⡀⠙⣷⠸⡜⢿⣿⣿⠀⢸⣷⡞⣿⣾⣿⣻⡷⡆⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢀⢲⡟⣦⡄⠙⡖⢢⠙⣾⣿⣷⠀⠸⣁⣧⣨⣿⣿⠀⢾⣿⣷⣽⡷⣻⡿⢀⡏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠈⢿⣿⢇⠀⣋⡤⣰⣿⣿⣿⣧⠀⠻⣇⠸⣸⣿⠀⢸⣿⣿⣿⣷⣏⣤⣤⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⡽⣷⠀⢸⣿⡿⣋⢿⣿⣿⣇⠈⢷⣄⢧⡟⢠⣻⣿⣿⣿⣶⡿⠿⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⠜⣿⡉⠀⠉⠉⠀⠹⢮⡹⣾⣿⣆⠸⣇⣾⠁⣼⣿⠋⠛⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠰⠈⢳⣀⠀⠀⠴⣷⣎⢿⣿⡟⣿⡆⢸⡏⢰⣿⡧⠀⠘⠂⠙⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢡⠀⠀⢀⣠⠞⠛⠉⠀⠹⣿⣾⣿⡶⣇⣻⠥⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⢀⠟⢀⠀⢠⣾⣿⠳⣄⠀⠃⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠐⠁⣠⡾⢿⣿⣇⠙⣦⣠⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⣰⠟⠀⢸⣿⣿⠀⠈⠻⢿⣎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⣰⡏⠀⠀⢸⡟⣿⡀⠀⠀⠀⠹⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⢀⡿⠀⠀⠀⢸⡇⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⢷⠄⣼⠇⠀⠀⠀⠘⠁⡄⣿⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⡯⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⢸⣿⡀⠀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⡠⠈⣿⣷⡄⠀⢳⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠤⣤⣀⠀⢠⡇⠀⢸⡇⠀⣿⡏⣧⠀⠀⠙⠦⣤⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⠃⠀⢸⡇⠀⢻⣷⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⣾⣿⡀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⣤⢶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⡄⣿⡏⣻⣿⢶⣾⣿⢟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⢀⣿⠏⣼⢿⡷⠀⢠⣤⣴⡖⢲⣶⣤⣀⣘⣧⡾⢀⣿⣷⡹⢿⠏⡻⣿⣨⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣼⣿⠐⣙⣿⠃⠀⣀⣿⣇⢇⠘⢦⣿⠉⠏⡼⠀⡚⠸⣿⠐⣿⣧⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⢿⡀⠀⠹⣦⠻⡂⠀⢸⣿⠂⣼⠇⡘⢁⣶⠙⠧⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠿⠽⣦⣁⡔⠊⢹⣷⡞⠘⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⢧⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠟⣿⡟⠛⠁⠿⠇⠠⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⡻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣋⣵⣾⣿⣿⣷⣮⣙⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢟⣩⣶⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣶⣍⡻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⢛⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣷⣮⡛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣭⣻⣿⣿⣿⣿⣿⢸⡟⣭⣭⡝⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣃⣛⣛⣛⣛⣛⣛⣛⢸⡇⣿⣿⡇⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣭⣭⣭⣭⣭⣭⣭⣭⢸⡇⣿⡿⠷⠭⠭⠭⡍⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⢸⡇⣿⡿⠿⠿⠿⢛⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⣝⣛⣛⣛⣛⡛⢸⣇⣛⣛⣃⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣾⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⣠⣴⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿
⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠸⣯⠉⠉⠛⠛⠃⠶⢼⡿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠁⢠⡟⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⡿⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣰⡄
⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣤⣄⣰⣧⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣺⣿⣿⡀⠀⠀⠀⠈⠙⢻⠟⠉
⠀⠈⡆⠀⠒⠠⢤⡤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠀⠀⠉⣹⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠀⠀⠠⡦
⠀⠀⢻⠀⠀⠒⠤⠤⣤⣄⣀⣀⣠⣤⣤⠤⠤⠖⠀⠁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠛⠛⠿⡿⣿⡇
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢠⠀⠀⡇
⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠈⠀⠀⡆
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢼⡏⠀⠀⢀⣴⣶⣶⣶⠞⠃⠀⠠⢂⡄⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠙⠶⠦⣤⣤⣀⣀⣀⣀⣀⣀⣬⠤⠶⠋⠀⠀⢀⣬⠟⣛⠉⠀⠈⢀⠔⢀⠞⠀⠠⠀⠀⠀⠀⠀⠘⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣴⡶⢞⡭⠖⠉⠀⠀⣀⠔⠁⡠⠋⠀⢀⠃⠀⡄⠀⢠⠀⢃⠸⣿⣤⣤⣤
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣌⣛⣻⣿⣿⣿⣿⣿⣋⡴⣫⠔⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⠎⠀⢠⠀⠀⠘⠀⠈⡄⣿⣿⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣁⠀⢠⡞⠁⠀⠀⠀⠀⢀⢠⡏⠀⠀⡟⠀⠀⠀⠀⣠⣿⣿⣿⣿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⣠⣏⣾⣠⣤⣾⣧⣀⣠⣴⣾⣿⣯⣍⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣚⣋⡉⠉⠉⠉⣉⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣯⡟⢻⣶⠏⠀⠻⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠁⠀⣀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⠖⢻⣧⠘⢦⠀⠀⢀⣴⣿⠙⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⣿⠐⢧⡙⣦⣤⣤⡴⠟⣹⠏⠀⣷⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢿⠀⣿⠀⠀⣧⣘⣿⠤⣠⡾⠋⢠⡴⠿⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⣼⠧⠾⠦⡄⠈⠉⢻⡷⠯⠶⠾⠟⠀⠀⢀⣤⠄⠀⠀⢹⡇
⠀⠀⠀⠀⢀⡾⠋⠀⢠⡞⠁⠀⠐⢚⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠘⣧
⠀⠀⠀⢠⡟⠀⢠⣴⠋⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣿⡤⢶⡟⠁⠀⠀⣠⣴⢶⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⢀⡿⠁⠀⢸⣷⣀⣴⣿⣿⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠆
⠀⠘⡇⠀⠀⠸⣿⡿⢹⠏⢹⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⢣⡀⠀⠀⣀⠔⠁⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠇⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠸⣿⠀⢿⣄⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡟⠛⠂⣠⣤⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡧⠾⠗⠚⠉⠁⠀⠈⣿⡄⠀⠀⠀⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠀⠀⠀⠀⠀⢀⣤⣴⣿⡇⣀⣤⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⣀⣴⠟⠉⠉⠛⠿⠿⠿⠟⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⣴⣿⠟⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠉⣿⠀⠀⢠⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⠀⠀⣼⠃⠹⣇⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⣿⣿⠀⢹⡆⢰⡏⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⠀⢸⣇⡿⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇⠸⣿⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣾⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⣿⣿⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣯⠞⣹⣿⡇⠀⠀⠀⠀⠀⠀⡴⠚⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢰⡏⣰⣿⣿⣷⣶⣶⣶⣶⣶⡞⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠇⠀⠀⠀⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⡿⠛⢛⣿⣆⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣤⡶⠟⢻⣿⣿⣿⣟⣛⣛⠛⠛⣿⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠇⠀⢺⣿⣿⡀⠀⠀⠀⠀⠀⡾⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣉⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⠶⣶⣾⣿⣿⡷⣄⠀⠀⠀⠀⢳⣶⣶⣶⣤⣤⣤⣤⣤⣉⣉⣉⣿⣧⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣟⣮⠳⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⣿⡇⢹⡟⣿⢹⣿⣿⠸⣧⢹⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣼⣿⡇⢸⡇⣿⡘⣿⣿⡇⣿⣾⣿⠀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇⢸⣧⢸⡇⠿⢹⡇⣿⢻⣿⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⡿⢿⣿
⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠇⢸⡿⢸⣿⢰⠺⡇⣿⢸⡏⠀⢹⣿⣿⡟⠋⠉⠙⠛⠛⠛⠛⠿⣿⣿⣆⠾⠛⠁⠀⠈⠙
⠻⠿⠿⠿⠿⠿⠿⠛⠻⠿⢿⡟⣠⣼⠇⣸⣿⣾⡇⣇⣻⡘⣧⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣾⣥⣿⣾⡿⠿⠿⢿⣿⣿⣷⣿⡧
⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⣠⡾⢋⣽⢿⣿⡯⣉⠉⠉⠉⠉⠙⠛⠿⣿⣮⡓⢦⡀
⠀⢰⡏⣰⢟⠌⢰⡟⠻⢦⡀⠀⠉⠀⠒⠀⠤⢀⡙⢿⣄⢹⡄
⠀⢹⣷⣿⣊⣴⠟⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⢘⣿⣾⡃
⣴⠟⢻⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⠚⠛⠋⢩⡿⢹⡆
⣿⡀⢸⣇⠀⠀⠀⠀⡀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⣸⠇⢰⡇
⠈⢛⣾⣿⣆⠀⠀⠉⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠶⠋
⢠⡟⠱⣈⠻⢷⣄⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⣠⣴⣯⡀
⠀⢻⣶⣤⡭⣦⣍⡛⢶⣦⢤⣤⣤⣤⢶⣶⠋⠁⢈⣿⠙⢦⣄
⢰⡏⣤⣈⣷⠀⢹⢺⢘⣧⡀⠈⣻⣶⡿⠳⣤⣤⢾⣧⠀⠀⠙⣧⡀
⠈⠷⣬⣭⣥⣴⠏⣪⣾⠏⠛⠟⠉⠉⣇⠀⠀⠀⠀⠹⣆⠀⣴⠟⢧⣀
⠀⠀⠀⠉⣻⣿⣟⢉⡿⠀⠀⠀⠀⡤⠟⠀⠀⠀⠀⠀⠹⣿⠃⠀⠀⠙⢷⣄⡀
⠀⠀⠀⣰⣿⠃⠉⣻⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⣧⣄⠀⠀⠀⠙⢿⣦⡀
⠀⠀⣴⡿⠁⠀⢠⡿⣮⣠⣀⣀⣀⠚⣹⣆⣀⣀⣦⠤⠞⢉⣿⣟⠷⣄⠀⠀⠀⢹⡇
⠀⢸⣿⠃⠀⣠⣿⣧⣄⣀⣉⣉⣉⣉⣉⣉⣩⣤⣤⠶⠞⠋⠁⠙⢷⣽⡗⠀⠀⣿⡇
⠀⣿⠃⢀⣼⠏⠀⠀⠈⠉⠉⠉⠉⠉⠉⢉⣉⣀⣀⣀⣀⣤⣾⠀⠀⢻⣇⠀⢠⣿
⢸⡏⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⠻⣟⠙⡇⢻⡇⠀⠀⢿⣷⣾⣿⡀
⢸⡇⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⡄⢻⠀⢡⢸⡇⠀⠀⢸⣿⠀⠙⣿⡀
⠀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢇⠘⡄⢸⣾⠃⠀⠀⣸⣿⡶⣾⣏⡻⣦⣠⣴⠶⢦⣤⣄⣀⡀
⠀⠀⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣼⣆⣷⡾⠃⠀⠀⣠⢿⠛⣷⡄⠈⠙⣿⣿⣿⢿⣦⣑⠒⣩⡿
⠀⠀⢠⣿⣻⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣀⣤⣞⠟⠁⣰⠟⠀⠀⣸⡏⢿⡟⠻⣿⣝⣿⣅
⠀⠀⠘⠛⠷⢿⣽⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣒⣛⣻⣥⡵⠖⠋⠁⠀⠀⠀⣿⠙⠈⣿⠢⢈⣷⣥⣽⠷
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠙⠛⠛⠛⠛⠻⣿⠿⠛⠛⠛⠻⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠹⡔⠂⣿⣄⠢⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⢇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⠶⠯⠉⠛⠶⡷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⡿⢸⣤⡦⡀⠀⡀⠀⢀⠠⠁⡾⣢⠄⠀⡀⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢸⠀⠀⠀⠀⣸⠃⠸⠀⡇⢭⠻⠛⠣⢾⠁⠠⡇⠃⠠⡼⠇⢾⢤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠷⢶⣤⣼⠀⣀⡤⠶⠿⠶⣄⠀⠀⠀⢀⡀⡀⣀⣀⣀⢀⠀⣄⣀⢀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠈⣿⣿⢋⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠰⣿⠿⠴⠦⠤⠤⠴⢿⣿⠾⠶⠶⠖⠒⢶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⢰⣷⣠⣤⡴⣶⠛⠉⠉⠀⢀⡀⠀⠀⣀⣠⡤⠦⠀⠀⣿⣿⣿⣿⣇⣤⣄⣤⣄⣠⣄⣠⣄⣠⣤⣸⣿⣿⣿⣿⠀⢸⣿⣿⠓⠛⠛⠚⠛⠛⢳⠀⠀⢨⡇
⠈⢻⣯⠀⢰⡏⣇⠀⠀⠀⠸⣿⡚⠛⣽⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⢰⣶⣾⣿⣿⣿⣿⣶⡄⢸⣟
⠀⠀⠈⠀⣼⠁⢻⠀⣠⣴⠄⠙⠃⢠⡏⢷⠀⣀⣄⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣟⢸⣿⣿⣿⣿⣿⣿⣿⡇⠰⡗
⠀⠀⠀⢀⡇⠀⠈⡇⠁⣿⣦⠀⠀⡸⠐⠸⡆⢹⣷⡀⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣟⢸⣿⣿⣿⣿⣿⣿⣿⡇⣭⣇
⣶⣶⣶⣿⣷⣶⣶⣿⣶⣿⣿⣶⣾⣷⣶⣶⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⡏⣿⠉⣿⡍⣿⢹⢹⡏⣭⢻⠋⢿⠉⣿⣿⢩⡝⡏⣽⡋⣍⡏⣭⢹⢹⡏⡏⣭⢻⢩⣽⡏⣽⠩⣝⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⡀⡇⣃⢻⡇⣿⢸⢸⡇⡅⡿⢘⢸⠀⣿⣿⢨⢸⡇⣾⡻⡌⡇⠿⢸⠸⠇⡇⣄⢿⠸⢿⡇⣾⠳⠜⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⣶⠄⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣧⠀⠀⠀⢸⣿⣿⣿⠟⠛⠻⣿⣿⣿⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢸⣿⣿⣧⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⣠⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⠀⠤⠶⠿⠿⠿⠿⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⣤⣤⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢖⣫⣭⣝⡶⣄⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢚⣽⣾⣿⣿⣿⣿⣿⣷⣭⡓⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢠⣤⣤⣤⣤⣤⠀⠄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⢻⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠹⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣲⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠘⠟⠓⠒⠚⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡌⠉⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣤⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢷
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣫⢿⣇⣸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢠⠀⠸⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⠂⢢⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡄⢰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢨⠀⠸⡀⣀⣠⣤⣤⠀⠀⢾⣿⣿⣿⣿⢸⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣀⡀⠀⠀⠀⢸⠀⠀⢹⣿⣿⣿⣿⠀⠀⠈⠻⠿⠟⠃⣼⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⢸⠀⠀⢸⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⡄⠿⠿⠿⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⣤⣤⠀⢾⣿⣿⡆⠀⠀⢠⣤⡄⣾⣿⣿⡇
⠀⠀⠀⠀⡀⠙⣻⡁⠀⠀⠀⠀⠀⠘⠛⣃⡀⠀⠀⠀⣀⣛⣛⡀⠺⢛⠛⠃⠀⠀⣘⣛⣃⠛⠛⠛⠃⠀⢰⣿⡦
⠀⠀⣟⠀⠙⠒⣿⣿⠀⠀⢀⡏⢀⣠⡴⣿⣷⠀⠀⢰⣿⣿⣿⡟⠓⠛⠁⠀⠀⠀⣿⣿⣟⠻⠿⠂⠀⠀⠀⠉⣷⣦
⠀⢹⣿⣿⣿⡏⣽⡿⠀⠀⣿⣿⣿⣿⡏⣿⡿⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢀⣠⣾⣿⣿⠀⠤⡀
⠀⢨⠉⠉⠉⠀⣿⡇⠀⠀⢬⠉⡍⠁⠀⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⣿⡟⠛⠋⠀⠀⡇
⠀⠀⠁⠁⠀⠀⣿⡇⠀⠀⠈⠉⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢿⡗⡀⠀⠈⢲⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣹⣿⠷⠞⠛⣦⡀⠀⠀⠀⢀⣤⣿⢟⣠⡀
⠀⠀⠐⢦⡀⢀⣤⣄⡀⠀⠀⠀⠈⠛⠛⣻⠛⠛⠃⠀⢀⣄⡀⠀⠀⠀⠀⢰⣿⡟⠛⠛⣿⣤⡈⣿⠀⠀⠀⣾⣿⠉⠉⢁⡼
⠀⠀⠀⠈⢳⣞⣿⣋⣻⠄⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⣹⣿⡃⠀⠀⠀⠀⠘⣿⡇⠀⠀⢿⣿⠇⠀⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠻⠒⠛⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠻⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⣿⣟⣿⠀⠀⠀⠀⣸⡟⠀⢀⠀⠀⠀⡀⠀⠀⠀⢀⣿⠃⠀⠙⠃
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣶⣶⣶⠄⠀⠀⣿⣿⡏⠁⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠉⠋⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠛⠙⠋⠀⠀⠀⣠⣶⣄⣠⣴⣄⠀⠀⠀⠀⠀⣴⣦⣀⣴⣤⡀
⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣼⣯⣿⣽⣷⣿⣭⣷⡄⠀⠀⣠⣽⣯⣹⣿⣯⣻⡄
⠀⠀⠀⠀⢀⣼⠛⠃⠀⠀⠀⠀⠀⠀⢀⣬⣛⠃⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⠙⢛⠛⠉⠂⠀⢰⣿⣿⣿⡟⠻⡛⠋⠑
⠀⠀⠀⠀⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣭⠿⢦⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⣶⣶⣶⡆⠀⠸⢿⣿⣿⣧⣶⣶⣶⣦
⠀⠀⠀⠀⠻⢿⠿⣿⡄⠀⠀⠀⠀⠀⠻⠿⠶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠉⠉⠉⠁⠀⠀⢸⣿⣿⡏⠉⠉⠉⠉
⠀⠀⠀⣿⣿⣷⠀⠸⡷⠀⠀⠀⠀⠀⣿⣿⣷⠸⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡴⠦⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠙⠁⠀⠀⠈⠉⠁⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡞⠀⠀⠠⢦⠀⠀⠀⠀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣩⡀⠀⣎⣠⠶⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠷⠁⠀⠈⠁⢠⠀⠒⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⣀⠀⠀⠀⠀⠘⠷⠶⠋⠀⠀⠀⠀⠀⢀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣀⣀⣠⣤⠈⠆⠈⣹⣦⣤⣄⣀⡴⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠂⢈⣓⠁⢀⣿⣿⣿⡫⠯⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣿⣿⣿⡿⡹⡐⠓⢤⣀⡤⠔⢲⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⢏⢞⣠⠽⣄⠀⠙⡄⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢪⡽⠛⠛⡻⣱⡫⢿⣄⠀⠀⠉⢋⡅⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡎⠀⢀⢞⠎⠈⡇⠀⠉⢻⣦⠀⡾
⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⢀⣠⠴⠂⠀⠀⠀⠳⣄⡱⠊⡷⠁⠀⢠⠇⠀⣾⣿⡿⢾⠃
⠀⠀⠀⠀⠀⣰⠋⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⢸⢄⣵⠋⠀⣰⠃⢠⣴⣿⠟
⠀⠀⠀⠀⢰⡇⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⠋⠀⠀⡔⠁⢀⣨⣿⡟
⠀⠀⠀⠀⢸⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢀⣎⠞⠀⠀⢀⣾⣶⣤⣼⣿⣿⡇
⠀⠀⠀⠀⠸⡄⠀⡟⠀⠀⠀⠀⣠⠤⠄⣴⡿⠁⠀⠀⠠⠿⠛⢿⠛⢻⡏⠉⢻⡀
⠀⠀⠀⠀⠀⠑⠀⠘⠀⠀⢀⠊⢄⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠘⡀⢸⡏⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⡀⠀⠀⢀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠃⢳⠀⣿
⠀⠀⠀⠀⠀⠠⠔⢊⡡⠀⡨⠂⢀⢄⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠇⡏
⠀⠀⠀⠀⠀⠀⠩⠴⠚⣁⠤⣢⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⠀⠀⢶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⢸⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡀⠉⠉⠉⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⣿⣿⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⣠⣤⡈⢳⡀⢸⡿⠛⠛⠁⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠛⠓⠲⠶⠶⠶⠶⠶⠖⠒⠒
⠀⠀⠀⠀⢰⣹⣿⡟⠈⡇⢸⠇⢀⣀⣀⡀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⢤⣴⣮⣥⡠⣄
⠀⠀⠀⠀⠘⣽⣿⡷⢀⡇⢸⡾⣿⣿⣿⣂⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡹⣧⡀⠀⠀⠀⠀⠀⠀⢰⠀⢀⡆
⠀⠀⠀⠀⠀⠈⠉⣠⠞⠀⢸⣇⣿⣞⣿⣿⣿⣿⠀⠀⠀⢀⡆⠀⠀⠀⠀⣿⣿⣿⣿⣷⡹⣷⡀⠀⠀⠀⠀⠀⠘⡇⢸⠁
⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⢸⣿⢹⣿⣿⣿⣿⡏⠀⠀⠀⣸⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡹⣷⠀⠀⠀⠀⠀⠀⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠁⠀⠀⢀⡏⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡹⣇⠀⠀⠀⠀⠀⣏⣿
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡹⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠈⢉⣉⣛⣻⣿⣻⣇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⢟⡇⠐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠻⣿⣿⠸⠃⠀⠀⠇⠀⠐⠃
⠀⠀⠀⣀⣀⣤⣤⣤⣴⣄⣀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣠⣿⣿⣧⡀⠀⠀⠀⠀⠀⠨⠿⠈⠀⠀⠀⠈⠂
⠀⠀⠻⠿⠿⢻⣿⡿⠿⠿⠟⠇⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣶⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠆⠀⠀⢀⣀⣿⠀⠀⠀⠀⠛⠛⢿⠿⠿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠇
⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠄⠈⠀⠀⠀⠀⠀⠀⠈⣧⣤⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣤⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠙⢛⣛⣻⣿⣿⣦⣄⣀⣀⣀⣀⣄⣀⣴⣟⠛⠛⠀⠀⠀⠀⢐⣺⣿⣿⣿⣓⠀⢹⡄⠀⢻⣀⡀⠀⠀⢿⣇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⣿⣿⡿⠟⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢐⣚⣿⣿⣿⣒⢦⣤⡿⢄⠀⠈⠀⠀⠀⠈⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠿⠷⠒⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⣠⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⡄
⠀⠀⠀⠀⠀⢈⡷⠢⣔⠺⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡼⠀⢹⠇⠀⠀⣣
⠀⠀⠀⢀⡔⢡⡬⢋⣟⡏⠉⠀⠀⠀⠀⡤⠤⠖⡶⡏⣿⢻⢹⠒⠒⠚⣏⠀⢱⣂⣸⡀⠀⢸⠋⠑⠢⡀
⠀⠀⠈⠁⠀⢸⠀⠀⠓⣿⠀⠀⠀⠀⠀⣸⠀⠀⡇⣇⢹⢸⢸⠀⢀⣀⢸⡀⠀⠀⣠⠜⠋⠛⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠣⡀⠀⠀⠀⡏⠀⠀⠇⠸⠘⠘⠸⣤⢋⠞⢠⠇⢠⠞⠁⠀⠀⠀⣠⡀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⡀⠀⠀⠳⡀⠀⢸⠀⠀⠀⠀⢠⠏⠀⡾⠀⣞⠀⠀⠀⣀⡜⠀⠿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠘⠛⠻⠀⠀⠀⢀⠟⠋⠉⠙⠀⠸⣄⠀⠀⠛⠇⢴⡦⡆⠀⢀⠇
⠀⠀⠀⠀⠀⠘⠢⠤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⡗⠀⡀⠤⠄
⠀⠀⠀⠀⠀⢰⠁⠈⢉⣛⠳⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣓⣧⡀
⠀⠀⠀⠀⡄⠺⡏⡩⠭⠭⢭⣓⠲⢬⣆⠀⠀⠀⠀⡔⢋⣩⠭⡽⡪⠓⢄⠀⠀⢀⢜⣛⣫⠉⣟⡢
⠀⠀⠀⠀⡏⠈⠉⠀⠀⠀⠀⠀⠉⠑⢲⡟⠀⠀⢸⠁⣼⣠⠾⣎⠀⡠⠎⠀⠀⢸⢸⡅⠀⠀⢳⠇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⠏⠀⠛⢁⡴⡡⠊⠀⠀⠀⠀⢸⠘⡇⠀⠀⢸
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠏⠀⠀⡴⠯⠊⠀⠀⠀⠀⠀⠀⡸⠀⢇⣀⣀⣸
⠀⠀⠀⠀⠈⠒⠦⠤⣄⣀⡤⠤⠴⠒⠋⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢦
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⠀⠀⠀⠀⡇⢰⠆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠱⠗⡠⠊
⠀⠀⠀⠀⠀⠀⣰⠳⢦⣨⠇⠀⠀⠀⢳⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢀⡏⠀⠀⢧⣀⠀⠀⠀⠘⡆⠀⠀⣇⢀⡞⠉⠉⠙⠓⠒⠶⢦⣀⡴⠊⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠉⢦⠀⠀⢹⡀⠀⢹⡈⢳⠀⣠⢤⣤⣄⣀⡸⠊⠀⣀⡴⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⠀⡀⠈⡆⠀⣀⠷⠚⠺⣇⡏⢠⡟⠾⣄⡴⠋⢀⣴⡚⢧⡀
⠀⠀⠀⠀⠀⢸⡀⠀⡎⠀⠀⠈⢂⠹⡋⠁⢀⠴⠒⠋⠀⠸⣧⡾⢋⣀⣜⡧⠤⠽⠔⠛⠲⡄
⠀⠀⠀⠀⠀⠀⢧⠀⢣⠀⠀⠀⠈⡄⢳⠀⡎⡤⠀⠦⠤⣀⠈⢱⠉⠀⢀⣀⣀⣀⣀⣀⠀⢻⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⡆⠈⢆⠀⠀⠀⡀⠀⢿⢰⠁⠀⠀⠀⢀⠀⢸⠀⡎⠁⠀⠀⠀⠀⠈⡆⢸⠋⠁⠀⠈⠙⠲⣄
⠀⠀⠀⠀⢀⡔⠒⠻⡀⠈⠳⠤⠴⠃⠀⠈⠇⠀⠀⠀⠀⢸⠀⡽⠀⢰⠀⠀⠀⠀⠀⠀⢣⠈⡆⠀⢀⣄⠀⢠⠘⡄
⠀⠀⠀⠀⠀⢳⡄⠸⠥⣀⣀⣀⣀⣀⣀⠀⠈⣆⠀⠀⠀⡇⢰⢦⡆⢸⠀⠀⠀⠀⠀⠀⠘⡄⣧⠜⡅⠈⡇⢸⠀⢹⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠠⠤⢀⣀⣀⣀⣉⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⠔⠁⠀⡆⠀⢧
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⢢⠀⠀⣇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⡟⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠳⡤⠇
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⢠⣶⡀⣀⣠⣤⡶⠞⠛⠃⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⢰⣿⣿⣿⣿⡇⣀⠀⢀⡀⢀⡀⠀⡀⠀⣀⠰⣿⣿⣿⣿⡆⠈⣿⣿⡷⠶⠶⠶⠶⠶⠶⣦⠀⠀⣘⡇
⠘⣿⣟⠉⠉⡞⣧⠆⠀⠀⠀⣿⣤⠶⢾⣿⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣿⡇⣿⣿⢸⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⣿⣿⡇⣠⣤⣴⣶⣶⣶⣿⣄⠀⢼⡷
⠀⠈⠛⠀⢰⠇⢻⡀⠀⢀⡀⠙⢷⠀⣸⢻⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣧⣿⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⣹⣏
⠀⠀⠀⠀⡾⠠⠈⣇⠚⢻⣧⠀⠀⠀⡏⡄⢇⠰⢿⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡀⢺⡗
⣀⣀⣀⣶⣇⣠⣄⣹⣀⣿⣿⣇⣠⣼⣁⣅⣸⣀⣿⣿⣸⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣅⣹⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡿⢿⡟⢿⡿⣿⢿⠟⠿⣿⠿⣿⢿⣿⣿⠟⢿⡿⢿⡿⡻⣿⠟⢿⡿⡿⣿⠟⢿⡿⠻⣿⢟⡿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡙⢸⢣⢸⣿⢸⡇⣿⢸⠘⢃⡿⣀⢿⢸⣿⣿⠸⢣⡇⣻⣧⡳⡇⢸⢸⡇⡇⣿⠸⢣⡇⣷⣿⢘⣷⡙⢾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣧⣼⣰⣄⣿⣸⣧⣋⣼⣸⣎⣇⣶⣸⣈⣻⣿⣸⣜⣇⣛⣯⣣⣿⣘⣼⣧⣣⣿⣸⣜⣧⣋⣿⣘⣧⣙⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠉⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣤⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⡿⢁⣿⣿⣿⡇⢸⣿⣿⣧⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⠃⣼⣿⣿⣿⡇⢸⣿⣿⣿⣇⠸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠘⠛⠋⠀⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠂⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⢱⠀⠀⠀⢀⠀⠀⠀⠀⠀⢄⠀⠙⣉⠑⣦⡀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠑⢦⡀⠰⡄⠈⠷⠄⣤⡴⢧⠙⠚
⠀⠀⠀⠀⠀⠀⠀⡆⠱⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⡈⠀⠀⠀⠀⠀⠘⣧⡀⣟⠀⢟⢡⠀⢀⡀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⠑⠀⠐⠁⠀⠀⠐⠄⢀⣐⢀⡶⠃⠀⠀⠐⠙⠉⠀⠈⠉⠉⠁⠈⠀⠀⠀⠈⠀⠀⠑⠀⢀
⠀⠀⠀⠀⠂⠥⠀⠀⠀⠐⠀⠀⠁⠀⠁⠒⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠐⡀⠉⡄⠀⠀⠀⡀⢙
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢁⢀⣷⠀⡀⠀⡤⠞⠛⠛⠓⡦⣠⡞⠉⢂⡀⠂⠤⠌⡀⠔⠀⠀⡀
⠀⠀⠀⠄⠐⠒⠢⣀⠀⠲⠚⣧⠀⠀⠀⠀⠀⣾⢃⣾⠃⡜⡴⣫⡾⠀⡾⢱⡿⢰⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢄⠑⠀⠀⠀⠐⠈⠓⢉⠳⣊⢈⣉⠙⠄⢪⠗⣸⠁⠻⠴⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠈⠈⢠
⠀⠀⢦⡀⠈⠀⠀⠀⠂⠀⠠⠴⠖⡲⠁⠀⠄⢠⣟⣴⠯⣞⠾⢴⣇⡾⢢⢟⡶⠀⡦⢄⡀⠀⠀⢁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢚⠖⠁⠐⠫⡊⣦⣡⠏⢀⡤⠤⠮⠥⣭⣁⡈⠋⢁⡴⢋⡽⠀⠀⡈⠁⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⣸⣏⣤⠀⣡⢤⡤⢠⣄⣤⡄⣬⢭⡟⣩⢤⠀⠠⠀⠉⠀⠄⠀⠀⠀⠀⠁⣈⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢱⠃⠾⠵⣟⡱⠿⢫⣏⠼⠇⣾⢀⣣⠈⠀⡀⠃⠀⠀⠀⠁⠀⠀⢀⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠰⠂⠀⠄⠁⠀⠀⠙⠈⠁⠀⠀⠡⣄⢀⠀⠀⠀⠀⠀⢀⠪⡄
⠀⠀⠀⠀⢰⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠀⢀⡄⠀⠀⣀⣀⡀⠀⠀⠀⠠⠀⠀⠀⡀⠸⡀⠀⣄⠀⠀⢠⠀⠐⠂⠈⠢⠁
⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠋⠉⠠⠶⠛⠀⠀⡀⠉⠉⠉⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠡⠚⣳⠤⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠌⠁⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⢤⣤⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⣑⡏⠉⠔⠚⠢⠠⠄⢀⠀⠀⠀⠈⠠⠤⠭⠻⢷⣥⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⢀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠘⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠈⠟⠀⠀⠀⠀⢠⣖⠁⠈⠢⡀⠀⠀⢀⠣⠄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠈⠓⢄
⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠸⠀⠀⠀⠀⠀⠠⡛⠀⠀⢰⠃⠀⠀⠈⢄⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠒⠒⠁⠀⠀⠀⠀⠀⠑⠂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠒⠀⠀⠈⠢⡀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⢀⠔⠉
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⡠⠃
⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠈⠢⡀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⡀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠈⠑⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢠⡄⠀⡴⠄⠀⣴⡄⢠⡄⠀⢠⠐⣶⠆⣠⡀⠀⣤⢀⡴⠀⠀⠀⠀⠀⠀⡠⠟⠚⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⣧⡀⠀⠸⡇⢸⠹⡀⢸⠀⣿⠀⣟⢳⠀⡏⢸⠁⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⢰⣒⣒⡆⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⠀⡇⠀⢰⡇⡇⢸⠀⢣⣸⠀⣿⠀⣿⠈⢇⡇⢸⠐⢲⠀⠀⠀⠀⠀⣷⡒⠒⣒⡇
⠀⠀⠀⠀⡮⠉⠀⢀⠮⣢⡀⠀⠀⠀⠳⣤⠸⣤⡄⢧⡄⠼⠀⠿⠸⠀⠈⠟⠀⠿⠄⠻⠀⠸⠇⠘⢦⠼⠀⠀⠀⠀⢐⠁⠀⠀⠀⠈⠆
⠀⠀⠀⠠⠁⠀⠀⠈⠢⠨⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⡄⢀⣄⢀⣄⠀⡀⠀⢠⢠⣤⠀⢀⣤⢀⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⢸⣼⠇⢳⠀⡏⠀⡇⠀⡏⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠐⠄⣀⣀⠀⢀⣀⠴⠁⠀⠀⠀⠀⠀⠀⠘⡆⢸⠉⢈⣟⠀⠸⣀⠃⠀⡇⠀⡇⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⢸⣀⢰⡜⡄⠀⣿⠀⢀⣧⠀⣧⣀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢀⠆⡰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢬⣍⣷
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⢀⠈⡄⢠⠁⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣾⠰⠇⡌⠀⠀⠀⠀⠀⠏⢆⠋⠄⠉⡀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠇⢻⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠃⠁⠀⠐⢀⠔⠆⠀⠀⠀⠸⡈⣾⠸⡀⢠⠸⠀⠀⠀⠀⠀⢸⡧⠉⠉⢭⡇
⠀⠀⠀⠀⠀⠀⠀⡰⠃⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠶⢁⠌⠀⠀⠀⢀⠀⢡⠈⠀⠇⢸⢀⠀⠀⠀⠀⠀⡌⠉⠉⠈⠉⢁
⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠⠊⠀⠀⠀⠀⠢⡑⢌⡆⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⡴⠋⠀⠠⠀⢀⡠⠤⠄⠈⠳⡄⠀⠀⠀⠀⠀⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⡈⠁⠀⠀⢐⠇⠀⠀⠀⠀⠀⠘⠒⠀⠀⠒⠉
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠠⣾⠒⢚⡆⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠐⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣦⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡏⢼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣟⡏⠀⠹⣿⣿⡀⠀⠉⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣽⡟⣻⠿⢛⣧⠀⠀⠹⣿⣷⡀⠘⡻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⡟⡆⠀⠀⠹⣿⣿⣦⢬⣿⣿⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⠀⠹⣿⣿⣧⣹⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⣿⡿⡿⣿⣿⣿⣇⢣⠀⠀⠀⠹⣿⣿⣷⡄⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣻⣟⢸⣿⡽⣏⠸⡇⠀⠀⠠⠹⣿⣿⣿⣆⠟
⣶⣦⣄⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⠋⢡⠈⢻⡃⣻⣆⣁⠀⠀⠀⠡⠹⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣄⣀⣀⣀⠀⠀⢰⣸⡏⠀⠀⠁⢸⣅⢹⣿⣿⣧⠀⠀⠀⢡⠹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣴⣾⣷⡼⣿⣿⣿⡆⠀⠀⠀⢣⠹⣿⣿
⣿⡿⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠰⠆⠀⢤⠀⢀⡀⢀⣀⠀⠋⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢰⣿⣶⣿⣿⣌⠧⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠅⠄⢸⣿⠀⢀⣰⣾⣷⣤
⠀⠀⠀⠂⢀⣴⣶⣴⣿⣿⣿⣿⣧⣍⣄⣠⣤⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢠⠀⠈⠈⣿⣿⡟⠙⠶⠄⠀⢀⠀⠀⠁⠀⢸⣿⠀⠚⠻⣿⠟⠛
⠀⠀⠀⢠⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⢐⣿⡇⠀⢰⣿⣿⣿⠀⠀⠀⠀⣾⣷⣾⣦⡀⢸⣿⠀⠄⠀⣿⠀⢠
⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠃⠀⠀⠀⠇⠀⣿⣟⣿⣿⣿⣿⣿⡏⠀⢰⣿⠁⠀⣾⣿⣿⣿⣷⣤⠀⠘⢿⡿⠻⡟⠁⢸⡇⠀⠀⠀⣿⠀⢨
⠀⠀⠊⠀⠀⢻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠃⠀⠀⢀⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⠃⠀⠸⠿⠀⢠⡟⣿⣿⣿⢸⣿⡅⠀⠀⠀⠀⠀⠀⣸⣷⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠄⠀⣾⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢰⣦⠀⢸⣿⣿⣿⣿⣼⣿⠀⢀⡀⠀⠀⢀⣾⣿⣿⣿⣆⠀⠛
⠀⠀⠀⠀⠁⠀⠀⠙⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠁⠀⣿⠀⠀⢠⠀⢿⣿⣿⡿⠿⠏⠀⠀⠈⠁⠀⠘⠛⠛⠛⠛⠛⠋⠀⣿⣽⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠁
⠀⠀⠈⠀⠀⠆⡀⢀⣾⣶⣀⢰⣿⣿⣆⣀⠀⠀⠀⠀⣿⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⢐⠀⠀⠀⣿⣿⠀⠀⠂⠒⠀⠀⡀⢀
⠀⠀⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣧⣄⢠⠀⣿⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⢀⠏⢠⣾⣿⣿⢶⣄⡀⠀⠐⠄⠀⢹⡟⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣠⣼⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⠀⠀⣠⣶⣿⣿⣿⣧⠀⠀⢈⠀⢸⣿⣿⣿⣀⣹⣷⠀⢀⣠⡄⢸⡇⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠆⠀⣿⠀⣼⣿⣿⣿⣿⣿⣿⠇⠀⣿⡇⢸⣟⣿⣿⣿⣿⣿⠈⠻⣿⡇⢸⡇⠐⢀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢏⠁⠀⢠⣿⠀⠀⢉⠛⠿⣿⣿⠟⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣸⣧⠀⠌⠀⠈⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠠⣿⣾⢿⡿⣿⣿⣿⣿⣿⡿⠿⠟⠠⡎⠀⠀⢸⣿⠀⠀⠀⢈⠐⠀⠉⠀⠀⠀⠛⠋⠀⠉⠉⠉⠉⠉⠁⠀⠀⠄⠀⣿⣿⠀⠀⠀⠀⢿⣿⡿⣿⢿⣿⠀⠢⠆
⠀⠀⠀⠈⠁⠀⠄⠘⠻⢿⣿⡏⠀⠀⠀⠀⣀⣀⣀⣸⣇⣀⣀⣀⠀⠀⠀⠂⣠⢾⣿⣿⣿⣷⣦⠀⠀⣘⣿⠀⠀⠀⠀⠀⢹⡟⠀⠠⠀⠀⠸⠿⠿⢿⣿⠇
⠀⠀⢀⠀⣀⣀⠀⠀⠀⢸⣿⡇⠀⠈⠁⢠⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⡀⣼⣿⣟⣿⣾⣿⣿⣿⣇⠀⣿⣿⡇⠀⠈⠀⠀⢸⡇⠀⠈⠠⠀⠀⠀⢤⠀⡄
⠀⠀⠉⢀⣿⠿⣿⡖⠀⠘⠿⠗⠀⠀⠀⠈⠉⠈⠈⠉⠁⠉⠁⠁⢁⠘⡀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⣿⣿⡆⠀⣀⠀⢸⣇⠀⠀⠀⡀⢠⡹⣎⣇⣷⠀⡄
⠀⠀⡀⠀⠀⢾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⢁⠀⠁⠹⣿⣿⣿⣿⣿⣿⠟⠀⠹⠿⠿⠿⠃⠀⠀⢰⣿⣿⡆⠐⢃⡀⠈⢻⣾⣿⣿⣧⣷
⠀⠐⠀⢀⣤⣼⣿⡇⠀⠄⠀⠀⣀⣤⣶⣾⣿⣿⣿⣶⡟⠀⠀⠀⠂⠳⠀⠀⣀⠉⠛⠛⠉⠀⣤⣤⣤⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠙⢿⣿⣿⣿⣾⣧
⠀⠀⢰⣿⣿⣿⣿⡇⠀⠁⣠⣾⣿⣼⣟⣿⣿⣿⣿⣿⣧⡀⠀⠀⠘⠀⠀⠆⢠⣶⠶⣦⣄⠀⠙⠿⠏⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⢤⢶⣾⣿⣿⣿⣿⡏
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠘⢿⣿⣿⣾⣿⣿⣿⣿⣏⡿⠟⠙⠛⠛⠷⠶⠀⠤⠈⣿⣿⣿⣿⣧⠀⠀⣀⣤⣴⣶⣶⡄⢰⢿⣿⣿⣿⣿⣿⡆⣠⢶⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠈⠉⠉⠉⠉⠀⣄⠀⠀⠙⢿⣿⣿⠿⠿⠟⢻⡇⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⡿⠏⠀⠀⣿⣿⡿⠟⠋⠁⠸⠟⣿⣿⣿⣿⣿⠇⠰⠟⡿⡿⠋⠈⠁
⠀⢀⠉⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠁⠀⢀⡄⠀⠀⢀⣶⣶⠂⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⢀⠀⠉⠁⠄⠀⢁⡀⠀⠀⣉⠉⠁⠀⠠
⠀⠀⢀⣴⣶⣶⣿⣿⣾⣷⣶⣦⣄⠀⠠⠈⠂⠀⢰⣿⣿⡧⠉⠀⠀⠀⠀⠈⠉⠛⠋⠀⠀⣴⣿⠗⠀⣀⣰⣿⣿⡿⠀⠤⠋⠀⠤⠀⠈⠋⠀⠀⠀⢀⣠⣴⣷
⠀⠐⣿⣟⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⠛⠛⠁⠀⠀⢀⣼⣭⣭⣴⣶⣶⠆⠀⠈⠁⣶⢿⣻⣿⠏⠁⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄⠀⠀⢸⣿⣿⣦
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠶⠶⣶⣶⣦⣄⠀⠀⠀⠻⠿⠿⠟⠛⠛⠀⠀⡀⠘⢿⣿⠿⠋⠀⠈⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠘⠟⠿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠋⠀⠀⠀⠀⢀⣸⣿⠆⠀⠀⠔⢈⣤⣄⠀⢸⣯⣭⣽⣷⠀⠀⠀⠀⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⣠⡀⠀⠄⠀⣠⣶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⡄⠀⣀⣸⣿⣿⠀⢸⣿⣿⣿⣿⠀⢰⣿⣿⣿⠀⠀⠀⠘⠿⠿⠿⠟⠛⠛⠁⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠿⣧⣤⣤⣤⣤⣶⠶⠶⠶⠶⠟⠛⠛⠛⠛⠛⢻⣿⣿⠀⢸⣿⣿⣿⣿⠀⠘⣿⣿⣿⠀⠀⣀⣀⣀⣀⣀⣀⣤⣶⣶⣦⠀⠘⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠐⠔⠈⠛⠋⠀⠈⠛⠛⠛⠋⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⢠⣆⣀⣤⣤⠶⠚⠋⠀⡀⠀⠀⢀⣀⣤⠀⠀⣿⣿⣿⣷⣀⡀⣀⡀⣀⢀⣀⢀⣀⣿⣿⣿⣷⠀⣿⣿⠶⠶⠶⠶⠶⠶⡄⠀⢸
⠘⢿⡍⠀⡎⡇⠀⠀⠀⣿⡖⢛⣿⡁⠀⠀⠀⣿⣿⣿⣿⣿⣏⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⢠⣶⣶⣿⣿⣾⣧⡄⢸⡇
⠀⠀⠁⢰⠃⢹⠀⣴⡖⠈⠃⢸⠛⡇⢀⡤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣺⡂
⣀⡀⣀⣞⣀⡈⣇⢸⣿⡆⣀⣞⣀⢸⣀⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣸⣿⣿⣿⣿⣿⣿⣇⣺⣒
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⢿⣿⢿⡿⡿⡿⡿⣿⢿⡿⣿⣿⢿⢿⢿⡿⡿⡿⡿⡿⡿⣿⠿⣿⠿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⡉⡿⠄⣿⢸⡇⡇⡇⡣⡟⠎⡇⣿⣿⠘⣼⢨⣷⡙⡇⡇⡇⡇⣿⠛⡇⣿⡇⣽⣮⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣷⣿⣾⣷⣾⣷⣷⣷⣷⣷⣾⣿⣾⣾⣾⣷⣶⣷⣶⣷⣶⣷⣶⣾⣶⣷⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠉⠉⠉⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⡄⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⣀⡀⠄⠤⠤⡤⠄⠈⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⢠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⡠⠀⠀⡠⠔⠂⠉⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢠⡃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠄⠀⢀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⢀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠈⢄⠈⢆⠀⠀⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠠⠁⠼⠒⠠⠄⠀⠀⠀⠈⢂⠉⠐⠂⠀⠐⢂⠀⠀⠀⢀⠀⠀⠀⠀⠠⠂⡀⠄⠂⠁⠀⠀⠀⠀⠀⠲⠑⢌⠀⠀⠀⠘
⠀⠀⠀⠀⠸⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠠⢆⠀⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⢀⢠⣀⠀⠀⠀⠈⡴⠀⠀⡆⠀⠀⠀⢠⣤⡀⠀⠀⠀⣀⠀⠀⠀⣄⣤⠀⠀⠀⢰
⠀⠀⠀⢨⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠈⠒⠁⠀⠀⠈⠁⠀⠀⠀⠈⠈⠀⠀⠀⢸⠀⠸
⠀⠀⠀⠀⠈⠉⢀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠌⠈⠁⠀⠈⡘⡄⠆⠀⠀⠀⠀⠀⡆⠀⠀⠆⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⢧⢠⣀⠀⠀⠀⠀⠈⠒⠈⠀⠀⠀⠀⢀⠠⠊⢸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢤⠄⠀⢠⠤⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠁⠒⠂⠤⡤⠀⠀⠤⠄⠐⠂⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠀⠀⠀⠓⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢆⠀⡀⡠⠔⠀⠀⠀⠈⠒⠠⢀⡇⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠐⠨⠨⣠⠀⠰⠳⠿⠟⠁⠰⠴⠐⠴⠆⠀⠷⠀⠘⣆
⠀⠀⠀⠀⢰⠱⠀⠀⢠⠀⠤⢦⠀⠀⠀⠀⠀⡀⠀⠀⡏⠀⣚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⢄⠀⠀⢤⢤⣤⣽⠀⠀⠰⡟⡇
⠀⠀⠀⠠⠈⠈⠢⠀⠈⠘⡁⠀⠡⢀⠀⠀⣀⠀⢀⠜⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢯⠀⠀⠤⣀⠤⣤⡤⢤⡠⠆⢀⠾⠀⠀⢀⠖⠛⠳⢆
⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠈⠒⠤⠄⣈⡁⣍⠈⣀⠤⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣒⠀⠀⠀⡓⠢⠠⣀⣉⢨⡁⣁⡠⠄⡂⠀⠀⠀⢸⠀⠀⠀⠨
⠀⠀⠀⠅⠀⠀⠀⠇⠀⠀⠠⠀⠀⠤⠤⠤⠤⠤⠤⠀⠨⠄⠀⠀⠀⠀⠠⠍⠡⠄⠀⠀⠀⠀⠇⠀⠀⠤⠤⠤⠤⠤⠄⠀⠥⠀⠀⠀⠸⠀⠀⠀⠠
⠀⠀⠁⠐⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠒⠀⠀⠠⠤⠤⠤⠤⠀⠀⠐⠒⠂⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⢀⠔⠒⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⠀⠀⠀⠀⠀⡐⠁⢀⣀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢧⠀⢀⠀⠀⡰⠃⠀⠈⠱⠉⠁⠔⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⡀⢀⠤⢊⣑⣤⠄⠐⠉⢲⡶⣘⣄⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠟⠄⡰⠋⣁⣀⠀⠀⠀⠀⠀⢈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠰⡁⢠⡇⠁⠊⣐⡄⠑⠀⠀⣇⣤⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡱⢃⣧⠘⡄⠀⠂⠿⠃⠀⠂⠀⠘⢎⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠘⠁⠉⠀⠀⠀⠀⠀⠀⠠⠄⠠⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠈⢢⡀⢁⠀⠀⠀⠫⢤⡄⢀⡀⠠⠤⠆⠀⡞
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠩⢿⡀⠀⠀⠀⠀⢝⠢⠰⠀⠡⠀⢰⠁
⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠑⠢⢄⡀⠀⠀⠐⠂⠀⠁⠸
⠊⢀⡀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠙⠢⠤⠤⠤⡤⠐⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠘⣋⠘⢦⡀⠤⢵⣄⠀⠀⠀⠀⠀⠀⢳⣄⠀⠈⢃⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡰⢄⠈⢈⠗⠀⠀⠀⡱⢭⡁⠒⠒⠒⠒⣡⠞⢆⠀⠀⢳⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡂⠁⡄⠀⠒⠒⠁⠀⠀⠀⡔⠀⢘⡏⠉⠉⠉⠉⠀⠀⠘⡀⠀⢠⢱⠘⢦⣀
⠀⠀⠀⠀⠀⠀⢀⠖⠁⠄⢀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠃⢱⠤⠄⠰⠖⠒⠒⠒⡇⠀⢰⡌⡆⠀⠀⠁⠢⡐⠀⠀⠲⡀⠀⠲⡀⠀⢢
⠀⠀⠀⠀⠀⢠⠃⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠀⠀⠀⠀⠀⠀⠀⣇⣤⣼⣷⠁⠀⠀⠀⣀⠀⢀⡀⠀⠈⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⣼⢰⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⢹⡘⣄⠐⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠤⠤⠤⠖⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠐⠈⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠙⠮⣦⡈⠒⠤⠄⠀⠴⡾⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣼⣿⣶⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⠀⣿⣿⣇⣀⣀⣀⣀⣀⣀⠀⠀⠈⡏
⣿⣷⠶⠛⣿⡍⠀⠀⠀⢠⣄⣀⣤⣤⠶⠛⠃⠀⢼⣿⣿⣿⣿⣶⣦⣴⣶⣴⣶⢴⣦⣴⣶⣼⣿⣿⣿⣧⠀⣿⣿⡇⠉⣉⣁⣭⣍⣹⡆⠀⢰⣇
⠈⠻⠄⢠⡏⣿⠀⠀⠀⠘⢿⡉⢰⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⠸⡗
⠀⠀⠀⣼⠐⠸⡄⠶⣿⡁⠀⠀⣼⠁⣧⢠⣶⠂⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⠬⡏
⠀⣀⣤⣇⣐⣀⣧⣀⣿⣷⣀⣤⣇⣀⣸⣀⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣇⣂⣇⣂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣛⣛⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣸⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠉⠀⠀⠀⣀⣤⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⣟⡳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣯⣽⣶⣦⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣐⠾⣭⡋
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⢻⠋⠙⢷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⡟⠉⠉⠉⠉⠉⠙⢻⡟⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⡇⠀⠈⢷⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣸⠇⠀⠀⠈⣧⠀⠀⠀⠀⣇⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠰⣇⢴⣯⡻⣦⢠⣴⡿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠀⠀⠀⠘⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⣾⣿⢿⢿⣿⣝⢷⣅⠀⠀⠀⢉⣿⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣈⣇⣀⡀⣠⣤⣤⣤⣽⣀⡀⣤⣭⣤⣤⣤⢀⣸⣃⣀⣀⣀⣘⣛⣻⣞⣙⣿⣿⣓⣙⣳⣀⣀⣸⣟
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠛⣻⣻⣻⣟⣛⣛⣽⣿⣭⣽⣿⣜⢃⣿⣿⣭⣿⣿⣙⣟⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⣛⣛⣛⣛⣛⣛⣻⡛⠋
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣿⢻⡿⢿⡟⣿⢹⡟⢻⡿⣿⠿⣿⣽⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⡟⢸⡇⢸⡇⡙⣾⡇⣾⡇⣿⡆⣿⣼⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⣵⢹⣇⢸⣇⣇⣯⠀⣿⢧⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣜⣻⡿⠇⣿⣝⣧⣛⣽⡹⢿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⣿⣭⢬⡟⢧⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢿⣦⣤⣳⠏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠻⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠲⡌⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡈⠀⢸⣉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣀⡷⠒⠒⠛⢧⡀
⠀⠀⠀⠀⠀⠀⣟⠉⣉⣙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣁⣀⣀⡀⢀⣀⣸⡀
⠀⠀⠀⠀⠀⢀⡌⡉⢩⠈⠛⠦⣌⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠚⠲⢬⣙⠳⠆⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⣠⣄⣀⡊⠀⠃⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⡇⠹⡄⠀⠈⠑⠂⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠊⠉
⠀⠀⠙⢌⠻⣟⠓⠖⠚⠛⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠻⣿⣦⣄⣠⡤⠤⣄
⠀⠀⠀⠀⠑⠮⣉⣉⣁⣠⣀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢬⡒⠤⠤⠄⠀⠀⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠉⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠾⠒⠿⣂⡀⠸⠗⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣴⡿⠛⢻⠷⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣟⣉⣹⣤⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠉⠀⣞⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣄⣤⣙⠛⠛⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⢈⠶⠴⠾⠿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡤⢤⠾⠤⠤⠤⢤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⣉⣉⣀⣀⠀⢀⣈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⢀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⢙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠽⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣠⣤⠀⠘⡆⠀⠀⠀⠀⠀⠴⠖⣋⡥⠶⠤⠤⠇
⠀⠀⠀⠀⠀⠀⢠⠞⢳⠀⠙⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢯⣼⠓⠀⡇⠀⠀⠀⠀⠀⠴⠋⠁⢀⡴⢰⡆⢲⡀
⠀⠀⠀⠀⠀⠀⢈⡵⠯⠤⠤⡏⠉⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣀⣤⣙⠓⠤⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣠⣶⡶⢺
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠉⠛⠁⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⡄⢄⠈⠙⠲⢄⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣬⠭⠤⢋⡴⠋
⠀⠀⠀⠀⠀⢹⡉⠉⠀⠉⠉⠉⠉⠉⣹⠃⠀⠀⠀⠀⢠⡤⣤⣬⣁⠈⠁⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡤⠒⠒⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠲⠲⠚⠁⠀⠀⠀⠀⠀⠀⠱⣌⠻⣉⣓⣋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⠭⠤⠤⣄⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⡗⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠸⣅⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢧⣄⠜⠉⠉⠉⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⡛⢛⡛⠛⠛⣛⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡥⠤⠤⠤⠤⠄⠠⠤⢬⡇⠀⠀⠀⠀⠀⠀⠋⣉⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠋⢙⡥⠤⣈⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣭⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⢩⡭⣽⢯⣥⢤⡸⢏⠷⠀⠀⣀⢴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣟⣻⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣼⣷⣿⣶⣿⣾⣏⣓⣋⣿⣯⣿⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣨⠗⢻⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣹⡙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢙⣮⣕⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣶⣻⣷⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⠉⠉⠉⠉⠉⠉⢻⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⡒⣒⡿⢷⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠰⡧⠖⠋⠙⠲⢼⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠉⠉⠉⠉⠉⠉⠉⠙⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠸⡇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠟⠋⠉⠀⠘⣿⣿⠀⠀⠀⠀⠀⠴⣾⡷⠶⣿⡷⠶⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⡖⣿
⠀⠀⠀⠀⠀⣴⠿⠛⠋⠉⢰⢻⡀⠀⠀⠀⠀⠘⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⣿
⠀⠀⠀⠀⢸⡻⡄⠀⠀⢀⣿⢿⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⡄⠀⡇⣿⡇⣿
⠀⠀⠀⠀⠀⣇⣇⠀⠀⡼⢷⣒⠿⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣄⡀⢘⣿⣿⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⣻⡄⣾⣿⣶⡇⣿⡇⣿
⠀⠀⣤⣤⢶⣿⠛⠀⢰⣧⣊⣉⣲⣵⠀⠀⠀⠀⣸⡀⠀⠀⠀⢀⣤⣿⣄⣹⣿⣿⣃⣴⣷⣄⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⡇⣿⡇⣿
⠀⠀⢿⣿⣿⣿⠀⠀⠎⠀⠀⠀⠁⠈⠧⠀⠀⠀⢿⡇⠀⠀⠀⠀⠉⠀⠉⢼⣿⣿⠄⠉⠈⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠿⢛⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⠛⠛⠿⠿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡉⣶⣿⣛⣉⣉⣉⣉⣠⠤⠤⠤⠤⠤⠬⠭⢭⣉⣉⣙⣛⣳⣶⠌⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣝⠻⠿⢶⣿⣭⣽⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣭⣭⣽⣶⠿⢛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣷⣶⣦⣬⣭⣉⣉⣙⣛⣛⣛⣛⣛⣛⣛⣛⣉⣉⣭⣥⣶⣶⢿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⠳⣄⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣤⣀⣀⣀⣠⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⢀⡤⣴⢲⣤⣤⣴⣶⡶⠿⠿⠿⠛⠛⠛⠛⠿⣶⣤⠀⠀⠀⠀⢀⣀⣠⢴⣿⠟⠛⠳⠶⢤⣄
⠀⠀⣀⣻⣶⣀⣿⡛⠯⣄⣀⣀⣤⠤⠤⠖⢒⣒⣚⣩⣽⣿⣋⣻⣭⣭⣭⣶⡾⠿⠋⠀⠀⢠⣤⣤⣌⡷⠀⢀⡀⣀⣀⡀
⠀⠀⠧⠜⣠⡼⣿⡿⣷⣦⣧⣴⣶⣶⣿⣿⣿⣿⢻⣿⠿⠏⠉⠉⠉⠉⠁⠀⣀⣤⡴⠶⠶⠞⠛⠛⠋⢡⡞⠋⡽⣤⠄⣹
⠀⠀⠀⣾⠋⠀⠿⢿⣿⣿⣿⣿⡿⠿⠛⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣀⣀⣠⣤⠶⠶⢶⣝⠋⠓⠒⠒⠋
⠀⠀⠀⠙⠷⢤⣤⣤⣍⠉⠉⠉⠀⠀⢀⣠⣤⡴⠦⣤⣄⣀⣶⣖⣚⣛⣷⣶⣫⡿⠸⢯⡍⡁⢀⡀⠀⢸⣇⣀
⠀⠀⢠⠤⠖⠒⡦⠤⡝⠷⣿⣟⣿⡴⠛⣥⠶⠶⢦⣤⣄⠉⠉⠉⠉⢠⡽⠿⣧⡀⠀⠀⠻⣿⣯⣥⣤⣀⣬⡍⢻⣆
⠀⠀⠈⠓⠒⠶⠤⠼⠗⠀⠀⠈⢠⣤⣤⣝⠛⠻⣷⣀⣈⣛⣛⣶⠄⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣾⣿⣿⣿⣿⠿⢷⣀⠀⠀⠀⠀⠀⣠⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠃⠛⠟⠙⠋⢱⣿⣿⡽⣷⠀⠀⠀⢰⠃⢠⠏⠀⡔⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠄⢀⠄⠀⣀⡈⢿⣷⣿⣀⠀⠀⡎⢀⠃⠀⡘⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠚⠛⠶⠀⠐⠛⠉⠁⢸⡿⢩⠁⡇⠀⠁⢈⡀⠴⣱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡺⠀⠇⠀⠘⠃⠀⠀⠀⣀⡼⠃⠀⢀⣿⠥⠃⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣴⠆⠐⠶⠀⣘⠷⠆⢀⡏⠀⠀⠀⢸⠛⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠁⠒⠒⠢⡥⠀⠀⢀⠎⣀⡀⠀⠀⢈⠏⣽⣀⡴⠏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⣄⣀⣀⣤⠔⢹⣏⠉⠉⠓⢦⠎⡼⣡⠏⠀⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠒⢾⣿⣧⡀⢿⡗⣾⢻⣀⡆⠀⡜⣸⠋⠁⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠠⣀⢻⣏⣿⣼⣷⠏⣸⠿⢦⡔⢰⣃⣀⣀⣠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⠀⠀⠈⢸⣿⣾⣿⢟⢤⡿⠀⠰⢡⠏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡀⠀⠀⠀⠘⣷⣿⣿⣴⠟⠁⢠⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡻⣿⣧⡀⠀⠀⠀⠘⢿⡋⠀⡠⢤⢆⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⣤⡀⠀⢀⡎⠉⠋⠀⠈⠛⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠶⣼⣷⣆⠀⠀⠀⠀⣾⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠉⠉⡓⣶⠶⠞⠁⢀⠞⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣸⣶⠇⠀⠀⠀⠋⠀⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠊⠉⠉⠻⣄⠀⠀⠀⠀⠀⠀⣠⠆⠈⠉⠉⣧
⠀⠀⠀⠀⠀⢠⠖⠒⢦⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣠⣿⣻⠇⠂⠀⠀⠀⠀⠇⠀⠀⠀⢀⠀⣱
⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠘⣿⣿⣿⣿⣿⣿⡏⠹⣿⡟⠀⠀⠀⠉⠉⠉⢙⣿⠸⠀⠀⠀⠀⠀⠘⣁⠀⢀⠜⠶⠃
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣹⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠀⠀⠀⣀⠤⠂⠈⠉⠉⣠
⠀⠀⠀⠀⠀⠈⢦⣞⡀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢿⠀⡆⠀⡄⠀⠀⢸⢸⡇⠀⠀⠀⠈⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠸⡆⢧⠀⠐⣄⠀⠘⣎⢿⡦⣤⣤⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣶⡾⠟⢿⠿⠋⠁⠀⠀⠙⠦⠳⠦⠈⠦⠀⠈⠶⠝⠒⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣲⠌⢵⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣶⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⢶⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠳⢄⢀⣈⡉⠀⠀⠁⠀⢀⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣼⠀⠀⠀⠀⠈⠉⠀⠙⠒⠉⠒⢤⠇⠀⠀⠀⠙⠖⠲⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡶⠧⠄⠀⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠠⠽⠤⡄
⠀⠀⠀⠀⠀⠀⠘⢶⠤⠤⠄⠀⠀⠀⠀⠀⠧⣄⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡰⠈⠉⠑⢆⠀⠀⠀⠘⡄⠀⢰⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠣⣈⢀⡰⠈⢂⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠉⠹⡄⠀⠣⡀⠀⣇⡀⢸⠘⠋⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠘⢄⠀⠈⢦⡀⠸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠑⣄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠈⠓⠦⢤⣀⣀⣀⡠⠤⠞⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀⣀⣀⣀⡠⠴⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣐⢐⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠆⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡠⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⡶⢾⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠯⣀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡜⠁⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⡿⣾
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠀⣰⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣙⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠠⠛⠀⢘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠠⠖⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡀⣏⠸⢧
⠀⠀⠀⠀⠀⠀⢰⠛⢄⠀⠀⠘⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠤⠎⠄⠤⠶⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣨⣯⡀⠸⡀
⠀⠀⠀⠀⠀⠀⣨⠃⠀⠛⢀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣁⡀⠀⠀⢀⡀⠀⠀⢀⡈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠊⠀⡇⠈⢲⠇
⠀⠀⠀⠀⡐⠀⠉⠈⠁⠀⠈⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣉⣉⣉⣉⣉⣉⣉⣉⣁⣄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠘⠃⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡶⣶⡿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠒⠒⠒⡤⠤⠤⠐⠆⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⢥⡠⢬⢹
⠀⠀⠀⠀⠀⠀⠀⣾⠀⢺⠃⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠈⣷⣀⣶⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡥⡋⠟⠢⡄⡄
⠀⠀⠀⠀⠀⠀⢰⡇⢈⢽⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠤⢿⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠈⢳⠁
⠀⠀⠀⠀⠀⠀⣸⠇⠀⢸⡀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢧⣀⡸⣷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⢈⡶⠝⢂⠈⠈
⠀⠀⠀⠀⠀⢀⡏⠀⠨⢺⠇⡀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡬⠒⢧⣸⡆⠀⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⡿⠈⠀⠀⠀⠡⣧⠆
⠀⠀⠀⠀⠀⣸⣇⣁⣀⣸⣀⣀⣱⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠹⠷⡄⠀⢨⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣤⠀⠀⣤⠀⠀⢠⠄⠁
⠀⠀⠀⠀⠀⢲⠂⠒⡖⠀⢲⠆⠐⡶⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⡿⠤⠤⣿⠤⠀⢿⠤⠄⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡔⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠑⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣙⣃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢂⡀⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢲⣶⣶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠆⣞⣻⣻⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⢬⡄⠚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠑⠲⠀⠈⡄
⠀⠀⠀⠀⠀⠀⡘⢀⣭⢿⣧⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⣀⣀⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠀⢀⠘⢻
⠀⠀⠀⠀⠀⢀⠁⠸⣄⣸⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢌⡀⢀⠱⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⡠⠊⠉⠢⣀⡀⡀
⠀⠀⠀⠀⠀⠆⠀⡦⠞⢻⠓⢴⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡄⠐⢉⡁⠢⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠄⠠⠤⠤⠄⠠⠤⡔
⠀⠀⠀⠠⠔⠀⠐⠀⠀⠚⠀⠀⠃⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠈⠁⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠘⡛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠈⠈⠀⠀⠀⠐⠀⠂⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠛⣿⣿⡀
⠀⢠⣾⣿⣿⡟⢻⠷⠆⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠈⠛⣿⣷⡄
⠀⠈⠻⣿⣿⡇⠨⠀⠠⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⠇⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣇
⠀⠀⠀⠀⠀⠀⣰⣦⣥⠀⠀⠀⠀⠀⠀⣰⣿⢿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢠⡿⡗⠰⣿⡁⠀⠀⠀⢀⡴⠚⠓⢦⡀⠀⠀⠀⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⣿⡏⣿⡇⠀⠀⠀⠀⠸⠰⠧⣤⡇⠕⠀⠀⠀⢸⡀⠀⠀⢀⡇⠀⠀⠀⣿⡟⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⢿⣿⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠋⠀⠀⠀⢠⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⢿⠇⠸⡿
⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⢀⣴⣿⡟⢻⠶⠄⠀⠀⠀⠀⠀⠀⣴⣶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⡟⣿⣿⣦⡀
⠀⠀⠀⢿⣿⣿⣷⠬⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠛⠃⠀⣿⡿⠀⠀⠀⠀⠐⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⠀⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣘⠻⠗⠀⠀⠀⠀⠀⠀⢨⣷⡆⠀⠀⠀⠀⢈⡃⠀⠀⠀⠀⠀⣬⣵⡆
⠀⠀⠀⠀⠀⠀⠀⣱⣿⣿⡆⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣧⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢹⣝
⠀⠀⠀⣀⠀⠀⢰⣿⠀⢸⣿⠀⠀⠀⠀⠀⣼⢿⣿⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣿⡇⠈⣿⡀⢀⣀
⠀⠀⠀⠛⢧⠄⠘⠛⠀⠘⠛⠂⠀⠴⠄⠐⠟⠘⠟⠀⠀⠀⠀⠸⣿⣿⡿⠀⠸⠿⠁⠀⠿⠃⠸⢿⡿⠦

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⢠⣤⣀⡤⢤⠀⠀⠀⠀⠀⠙⢧⣦⡌⠁⠀⠀⠀⠀⠀⠈⠉⢳⣽⠁⠀⠀⠀⠀⠭⠈⣉⠁
⠀⠀⠀⢀⣹⠹⡧⠟⠀⠀⠀⠀⠀⠂⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠠⠇⣿⡇⠀⠀⠀⠀⠀⠈⠻


⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⢄⡀⠀⠈⢀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠁⠀⠀⠀⠀⢀⣀⣀⠇⢸⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠀⠀⠀⠸⢿⣿⠿⠀⠀⠀⠀⠀⢞⢿⣿⠗⣸⡀⠀⠀⠀⠐⠂⠈⠠⡄⠒


⠀⠀⠀⠀⣴⣶⡶⡄⠀⠀⠀⠀⠀⢠⣐⣒⣀⠀⠀⠀⠀⠀⢀⣤⣦⣤⣄⠀⠀⠀⠀⠀⠀⣴⡖⡄
⠀⠀⠀⠈⣐⣀⣐⡈⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣤⢸⣶⣾⠀⠀⠀⠀⠀⠀⡗⠠⡜
⠀⠀⠀⠀⠛⠿⠟⠃⠀⠀⠀⠀⠀⠘⠛⠻⠛⠀⠀⠀⠀⠀⠸⠿⠾⠾⠷⠀⠀⠀⠀⠀⠈⠙⠚⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠹⠆⠀⠀⠀⠀⠀⢀⠀⢼⣷⡀
⠀⠀⠀⢀⣠⣄⣀⣀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣃⡇⠀⠀⠀⠀⠠⣾⡿⠶⢿⡦
⠀⠀⠀⠈⠉⠉⠋⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙