Spamming Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠴⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠐⠀⠀⠁⠀⠀⠠⡄⠅⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠈⢀⣀⣀⣀⣆⠀⠀⠀⠀⠀⠕⠊⠀⠀⠀⡀⠁⠀⠀⢀
⠀⠀⠑⠀⠌⢁⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠂⠀⢠⢨⠦⠤⠦⠴⠛⢂⠥⠞⠛⣋⠁⠈⡀⠀⠂⠁⠀⠀⠀⠈⠀⠈⠀⠜⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠘⠀⠀⠀⡀⠀⠀⠘⠰⠉⠄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⣤⢹⠀⠀⠀⠀⢐⡤⠙⠛⠋⠂⠠⠐⠂⠀⠀⠂⠀⠠⠀⠀⢈⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢑⢈⠢⠄⠄⠐⠂⠀⠉⠀⡆⠂⠀⠀⠃⠠⠀⠀⠈⠀⢨⠁⠀⠡⠀⠀⠅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠂⠀⠈⠀⠀⠀⠀⡀⢀⠀⢠⠠⡀⠆⢡⠀⠀⠀⡄⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠒⠄⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠈⠀⠀⠀⠀⠀⠐⠘⠈⠉⠑⠉⠈⠐⠀⡀⠀⠇⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⢂⠠⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠐⡀⠂⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠂⠀⠠⣤⠠⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢤⣧⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠐⡀⠀⠞⠃⠀⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣐⣲⡠⠭⠤⠴⠟⠓⡓⣛⢉⢩⣍⠒⣀⠴⠋⡈⡄⠀⠀⠀⢀⣀⠠⣤⠔⠁⠉⠙⣄⠀⢠⣄⡁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡞⢯⢫⡛⡞⠹⠉⠍⠹⠌⠦⣕⣀⠀⢑⠔⢡⠼⠾⡥⠙⠙⠘⠚⠁⠤⢤⣭⣍⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⠀⠀⠀⠒⠒⠂⠁⠀⠀⠀⠈⠉⠒⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠤⣀⣀⣨⢶⣶⣦⡤⠄⣀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁⡀⠀⣤⡄⠴⠶⠶⠖⠒⠓⠛⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣾⣿⣿⠿⠛⠟⠛⠛⠉⠉⠉⠉⠙⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⣶⣿⣿⡿⠿⠿⠛⠛⠋⠉⣉⣀⣠⡤⠤⠄⠐⠒⠚⠉⠉⠙⠳⠦⠀⠈⢿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣾⣿⡿⠿⠿⠟⠛⠛⠉⠉⣁⣀⣤⡤⠤⠶⠒⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠘⣶⣿⡆
⠀⢀⣤⣶⣶⣿⣿⠿⠿⠿⠛⠛⠋⠉⣉⣀⣠⣤⠤⠶⠖⠚⠛⠉⠉⠁⠀⢀⡀⢀⣰⣄⠀⢠⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣿⣧
⣰⣿⣿⣿⠛⠁⢀⣤⠴⠶⠒⠚⠋⠁⠈⠁⠀⠀⠀⠀⢠⣀⡀⠠⣿⡖⠀⠙⠃⠀⠘⠛⠀⠀⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⢻⣼⡀
⣿⣿⣿⠃⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠧⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣸⣿⣅
⢹⣿⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣤⣄⡴⣶⣆⠀⠀⠀⠀⠶⠀⢻⣿⣿
⠘⢿⣿⡄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠰⠞⠛⣿⣆⠀⣿⡇⠉⠻⡄⠨⣿⡀⠀⠀⠀⠸⡆⠀⠿⣿⡆
⠀⢿⣿⣧⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣾⡛⠙⠀⠰⡆⠀⠹⣿⢠⣾⡋⢩⣿⠀⢻⣇⠀⠿⠇⠀⠛⠃⠀⠀⠀⠀⢃⠀⣥⣈⣁
⠀⢸⣿⣿⡄⠸⠄⠀⠀⠀⠀⢠⠄⡰⢶⣦⠀⠺⡏⠉⢻⣧⠀⠀⠀⠛⠛⢻⣷⠀⣶⣦⣾⠏⠀⠙⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠸⣿⣿
⠀⠀⣿⣿⡇⠀⡆⠀⠀⠀⠀⠈⣧⡄⠈⣿⠂⠻⣧⣀⣼⠟⠀⠀⠀⠘⠛⠋⠁⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿
⠀⠀⢹⣿⡟⠀⢰⠀⠀⠀⠀⠀⠻⠇⠀⠛⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⣿⣿
⠀⠀⠘⣿⣴⡇⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠶⠛⠁⠀⣾⣿⣿
⠀⠀⠀⢿⣿⣇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⢤⠤⠴⠖⠒⠘⠋⢉⡉⢀⣠⣄⡀⢲⡾⣦⣿⠿⠋
⠀⠀⠀⢀⣸⣿⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠰⠴⠒⠚⠛⠉⣉⣁⣀⣤⣤⣤⣴⣶⣶⢿⡿⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⢿⣿⣧⠀⠙⢦⣄⡀⠀⠀⣀⣀⣠⡄⠠⠶⠲⠒⠛⠋⢉⡁⢀⣤⣤⣤⣶⣶⣾⣏⣿⠷⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⢸⣿⣿⣧⣄⠀⠉⠉⠉⠉⣉⣁⣀⣤⣤⣦⠀⠚⣻⣿⠿⠧⠞⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠻⢿⣿⣟⣿⣿⣾⣿⡿⠿⠷⠛⠛⠛⠉⠉⠁
⣶⣶⣶⢶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕
⠀⠀⠀⠀⠀⠘⠁⠀⠀⡠⠀⠈⠁⠀⠀⠀⠀⢠⠂⠀⠈⠀⢀⠄⠀⠀⠂⠀⠀⠂⠈⠉⢀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⡇⠀⠀⢠⢄⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢀⠀⠀⠀⠓⠨⢜⡇⠀⠀⠠⠀⠂⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⡆
⠀⠀⠀⠁⢤⣀⠀⠀⠀⡇⠀⠀⠀⡀⢠⣴⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢤⠖⠲⠣
⠀⠀⠀⠀⠁⠀⠋⠀⠀⡇⠀⠀⠀⠋⠉⠈⠀⠀⠀⢳⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⢀⠀⠀⠀⣀⡔⠃⠀⢀⣠⠀⠀⠀⣀⣀⡈⠈⠀⢀⡠⡄⠀⣀⡄⢀⣀⡄⢀⣀⠈⠀⣀⠀⡀
⠀⠀⠀⠀⠉⠀⠀⠁⠀⠈⠁⠀⠁⠀⠀⠉⠀⠉⠀⠀⠉⠉⠊⠙⢀⠁⠈⠀⠉⡉⠛⠉⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠄⢀⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡄⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠺⠷⠶⠶⠶⠶⠶⠶⠶⠶⣶⢸⡇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⣿⢸⡇⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢸⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⣾⣶⣤⣶⣶⣤⣤⣀⠀⠀⠀⠉⠂⠀⠀⠁⠂⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡉⠙⠺⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢀⠄⢀⣠⣽⣿⣿⣿⡟⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠁⠀⠀⠛⢿⣿⣿⣿⡿⠋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣤⣿⣦⣿⣿⡏⠁⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠀⢰⡟⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣀⡈⢁⣀⠀⣀⠀⣀⣀⣽⣿⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠟⠋⠈⣭⠉⠉⠉⠉⡉⡩⠉⠉⣭⠉⠉⣭⠉⠉⠁⠀⠀⠀⠈⠀⣤⠐⠀⠀⣴⣶⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠀⠀⠐⢀⣿⠀⠀⠄⠀⠀⢀⠀⢈⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠘⠀⠊⠀⠛⣴⣶⣦⣤⣤⣤⣴⣤⣿⠀⠀⣿⠀⢀⣠⣾⣿⣿⣦⡀⣿⢠⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢸⣿⣿⣾⠟⠛⠛⠓⣿⠛⠛⠛⠋⠁⠀⢄⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠒⠐⠀⠒⢦⣀⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⢀⣿⣿⡏⠀⠀⠀⠀⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢉⣀⢀⡀⣼⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠈⠻⠛⠁⠐⠀⠀⠂⣿⠀⠀⠀⠂⠀⢐
⠀⠀⠀⠀⠀⠄⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠄⠀⠀⣿⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠉⠉⠉⠉⠉⠉⠁⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠻⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣤⣶⠿⠟⠿⢷⣦⣄⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣿⠋⠀⠀⠀⠀⣠⣾⢿⣧⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⣿⡇⠀⠀⢀⣴⣾⠟⠁⠀⣿⡇⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢿⣿⡇⠀⠀⢿⣇⢀⣴⡿⠋⠀⠀⠀⢀⣿⠇⠀⠀⣿⣿⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣄⢸⣿⡇⠀⠀⠈⢿⣿⣋⠀⠀⠀⢀⣠⣾⠏⠀⠀⠀⣿⣿⣀⣬⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⢿⣿⣿⣇⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠁⠀⠀⠀⢀⣿⣿⣿⠟⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠛⢿⣷⣦⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠⣶⣿⠟⠋⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣴⣿⡿⣿⣷⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠟⠁⠀⠀⠙⠿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣤⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣿⣷⣶⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣶⣿⡇⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⢶⣦⣤⣤⣤⣤⣤⣤⣶⠶⠾⠛⠋⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣠⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠶⢢⠵⠂⢀⡡⣀⢁⠀⢘⣁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠄⠠⠤⢀⡀⠐⠢⡤⢀
⠀⠀⠀⠀⠠⠎⠀⠂⠈⢩⠀⠀⠀⡄⢀⢤⠇⢀⡄⣤⠤⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⡀⢰⡤⠂⠀⢎⠙⡢⣄⣀⣀⣀⡈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠘⠀⠀⠀⠀⠀⠘⠁⠢
⠀⠀⠀⠀⠀⠀⢘⠁⠀⠂⢾⠀⠘⡈⣧⠔⣂⠀⠱⠁⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢀⣀⡀⠀⠤⠀⠓⠒⠒⠘⢄⣀⠇⠈⢀⠧⡑⠚⣀⠸⡀⠀⠲⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠇
⡔⠉⠀⠀⠀⠀⠀⠀⢀⠠⠊⡀⠉⠀⣀⠈⢆⠈⠀⠀⠙⠫⢁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣀⠴
⠣⡀⡀⡀⠀⠀⠤⠂⠁⠀⠀⠰⣠⠗⢻⣦⠀⠑⢄⠀⠀⠀⣌⣰⠁⠀⠀⢀⡀⠠⠀⠂⢠⡄⢸⡐⠠⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⢀⢀⣀⠤⢄⠀⠀⠀⠀⠀⢀⠀⣿⣧⠀⠀⢳⠞⠋⠀⠀⢀⠠⠖⠀⢀⡀⣀⠤⠤⠤⡄⢉⡙⠻⠶⡦⣌⣸⡀⡀⢀⠀⠌⠀⡀
⠀⠀⠀⠅⠠⢝⠖⢆⠇⠀⠀⠀⠀⠈⠉⠢⠄⠄⠀⠢⢀⢀⡴⡈⠀⡤⢒⡖⢈⡭⡍⢉⠙⠯⣒⠾⠤⣌⡓⠦⢄⣁⠒⢠⠄⠒⠂⢉⡏
⠀⠀⢠⠂⢐⢖⠋⠉⣧⢐⣯⣑⠂⠴⠀⠀⠀⣀⠔⠁⠀⠀⠀⠈⠢⠳⣼⠀⠀⠻⣧⡼⠭⢛⠛⢟⠊⢓⠩⠓⠦⢌⡀⡀⠀⠔⠚⠉⠁⠈⠁⠒⠤
⠀⠀⠈⠐⠁⠘⠀⠤⢌⠃⢤⠀⠁⠢⣡⠴⠮⠁⠀⠌⠀⡈⠈⠈⣀⠀⠉⢈⠃⢀⡜⠈⠐⠠⢉⠂⠬⣚⢄⡆⡅⠍⣂⡭⠆⠀⠀⠀⢀⠠⠔⣂⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠴⠀⠀⠈⠃⡖⣇⠀⠀⠀⢀⠀⠓⠁⠀⣁⠲⠄⠀⣨⠁⠀⠀⠈⠐⠨⠐⠀⠑⠺⠖⠋⠀⠀⢀⡠⠐⡈⠄⠂⠙⢻⠁
⠀⠀⠀⠀⠀⡠⠒⠂⠠⠀⡀⠀⢠⣊⢈⠋⠈⣣⣀⣴⡁⠀⡠⠒⠉⢀⠄⢠⠊⠉⠣⡀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⡤⠂⡡⠐⠁⠀⠀⢀⠁⠇
⠀⠀⠀⠀⠈⠀⠠⠀⡀⠀⠈⢐⠄⢃⠨⡔⠉⣠⠏⢃⠀⠉⢀⡠⠔⠁⠈⡎⠧⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⢸⣷⠇⠈⠀⠀⠀⠀⠀⠌⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠊⠀⠀⠠⡀⠑⢿⡘⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⣈⢡⠀⠀⠠⠄⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠉⠉⠾⠷⠤⠷⠴⠃⠀⠀⠊⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠿⠛⠛⠿⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⣀⣀⣀⣀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⣀⠸⠋⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⢷⣄
⢰⡟⠀⠀⠀⠀⠀⠀⢠⣾⠿⠻⠿⡿⠂⠀⠀⣿⡿⠿⠿⠿⣿⣆⠀⠀⠀⠀⣰⣿⣿⡆⠀⠀⠀⠀⣿⣿⣆⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠈⢻
⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣶⣤⣤⣄⠀⠀⠀⣿⣧⣀⣀⣠⣿⡟⠀⠀⠀⢰⣿⡏⢻⣿⡄⠀⠀⠀⣿⣿⣿⣆⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣷⠀⠀⣿⡟⠛⠛⠛⠋⠀⠀⠀⢠⣿⣿⣶⣖⣿⣷⡀⠀⠀⣿⡧⠹⣿⣿⡟⢹⣿⠀⠀⠀⠀⠀⠀⢠
⠸⣧⠀⠀⠀⠀⠀⠀⠾⣿⣶⣶⣶⠿⠏⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠉⠉⠉⣿⣧⠀⠀⣿⣷⠀⠻⡟⠀⢸⣿⠀⠀⠀⠀⠀⢀⣾
⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣠⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⣀⣤⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠉⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣶⣶⣶⠾⠛⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀
⠀⠀⠸⠿⠿⠁⠀⠿⠷⠀⠀⠸⠿⠿⠀⠀⠀⠿⠀⠀⠀⠹⠏

⠀⠀⠰⡶⡶⠀⠀⠀⠄⠀⠀⠀⠳⠶⠀⠀⠀⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢠⣤⡤⠀⠀⣤⣶⡄⠀⠀⣄⢦⠀⠀⢠⣶⡄⠀⢀⣠⣦⠄⠀⠀⠀⢠⡖⠛⠳⡄⢀⡴⠛⠳⣄
⠀⠀⠈⠉⠁⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠈⠛⠃⠀⠀⠁⠉⠀⠀⠀⠀⠈⠀⠀⢠⠏⢸⠁⠀⠀⢸
⠀⠀⠀⣠⡀⠀⠀⢀⢀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡄⠀⢀⡀⡀⡀⠀⠀⠀⠀⢀⡴⠋⠀⢸⡄⠀⠀⣸
⠀⠀⠘⠷⠗⠀⠀⠻⠿⠃⠀⠀⠘⠛⠀⠀⠺⠿⠀⠀⠐⠊⠉⠂⠀⠀⠀⠐⠛⠒⠒⠒⠀⠙⠶⠞⠁

⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣠⣤⡀
⣴⣶⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⢀⡀⠀⢀⡀⠀⠹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣷⣴⣿⡇⠀⢸⣷⣦⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣾⣿⡿⠋⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠛⠋⠁⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣠⣴⣿⣿⠿⠛⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⠀⣴⣾⣿⣿⣷⣤⣤⣶⣾⣿⣿⡿⠿⠛⠛⠿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⡇⠀⢀⣼⣿⡿⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣁⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣆⠀⠀⣴⣶⣶⣶⣶⣄⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⢠⣶⣦⡀⠀⢀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠛⠁⠀⠀⣿⣿⡛⠛⣻⣿⡇⠀⠀⣿⣿⠛⠛⣿⣿⡇⠀⢸⣿⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣆⠀⠀⣿⣿⣿⣿⣿⡿⠃⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⠙⠛⠋⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣽⣿⡿⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢹⣿⡇⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇

⠃⢠⡟⠀⠀⠀⠀⢰⠀⠸⡇⠀⢸⣿⣿⠿⣿⠟⠙⠀⠈⠃⠀⡐⠆⠀⠀⢰⡄⠀⠀⠀⠀⣠⠔⠀⢠⡀⢈⡀⠀⢈⡉⠀⠀⠁⠈⠠⡴⠀⠀⠀⠛
⠆⠘⠃⠠⠃⠂⠂⠀⠀⢀⣷⣦⣼⣿⡇⠀⠁⠀⠀⠀⠐⢦⣀⣉⣰⡆⠀⠘⠃⠀⡀⠒⠙⢁⣠⣤⣄⡀⢸⡗⠀⠈⠛⠀⠰⣤⡄⠀⠀⠀⢀⣠⠒
⠀⢠⡀⠀⣠⡄⠀⠀⠈⠀⣙⠋⠉⢍⠀⢀⢤⡀⠀⠛⠛⣤⠉⠉⠉⠁⠀⡆⠀⠀⠀⡴⠀⠉⢻⡋⠉⠉⠀⠈⠒⠤⣀⠀⠀⢀⣿⠗⠒⠦⡀⠁
⠉⢸⡇⠀⠈⣀⠀⢀⠄⠋⠁⠀⠀⠘⣦⠀⠉⠀⣀⣴⡾⠿⠇⠒⠘⠚⠛⢣⠀⠀⢰⠂⠀⢀⣿⡇⠀⣀⣀⡀⠀⠀⠀⣀⣄⣿⣿⡦⠄⠀⠀⠙
⠀⢸⣿⡿⠿⠛⠃⢄⡀⠀⢴⣆⡠⣴⣾⡷⠒⠛⢿⣀⣀⣀⠤⠀⣶⣶⡖⠈⠒⠦⠬⢄⣀⣼⠟⣁⠈⠻⣿⣿⣤⣴⣿⣿⡏⢙⣿⠁
⢀⣿⡁⠀⠀⡀⠀⠈⠻⢄⣼⣿⣿⠟⣋⠀⠀⢠⠄⠈⠀⠀⠂⠀⠈⡟⠀⠀⠷⠂⠀⢰⣿⡁⠀⠈⠙⠢⡄⠉⠀⣨⠛⠋⢠⣾⡏⠀⠰⣿⠆⠼⠶
⣸⣿⣷⠉⠈⠓⣦⠀⠀⠀⣙⣿⡥⠀⠈⢷⡄⠀⠀⠀⠚⠀⠀⣠⣸⡷⢠⡄⠀⠀⢠⣻⣿⣿⣿⣆⡀⠀⠀⠴⠎⠀⠀⠀⡀⠀⠈⢣⡀⠓⡀⢀⣀
⣿⡿⢿⣦⣄⠀⠈⠛⣢⣾⠟⠉⠂⠀⣀⣼⣿⡶⠂⠀⠀⢄⠀⣿⣿⡀⠀⢹⣄⠀⠀⢙⣿⡋⠉⠉⣤⣄⠀⠀⢀⣠⠀⠀⢱⣷⡄⠀⠳⣄⢻⣿⡏
⠃⠀⠈⠟⠉⢀⡤⡀⠉⠀⢀⡴⠟⠁⢙⡟⠉⠁⠀⠓⠄⠀⠈⢿⣿⣷⡀⠀⠙⣠⡴⠟⠉⢻⣆⠀⠘⢿⡀⠀⠈⠓⠃⠀⡈⠿⠗⠀⠐⢿⣿⣿⡇
⠶⣷⡆⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠰⢾⣤⢤⣀⡀⠀⠀⠀⠀⠾⠉⠿⠷⢶⡟⠉⠀⣠⠴⠟⠛⠧⡀⣰⣿⡄⠀⣀⡴⠋⠀⠀⠀⠱⡄⠈⢿⣿⡇
⠀⠛⠛⠷⠦⣠⠀⢀⣴⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣶⣮⣄⣄⠉⠀⠉⠉⠙⣦⠋⠁⠀⠀⠀⠀⠙⢿⣿⣿⣿⡏⠀⢀⣴⠆⠀⠀⠫⡀⠀⠀⠈
⢤⣄⣀⠀⠀⢁⡀⠀⠁⠀⠀⢀⣾⣿⣿⠶⠖⠂⠀⣸⡏⠁⣀⠀⠉⠁⠒⡀⠀⠿⢷⣔⠀⠀⠀⠔⠦⠀⠈⢿⣿⣇⠰⠟⠉⠀⠀⠀⡤⠑⠲⣿⣷
⠀⠀⠉⠀⢀⠞⠀⠀⠀⠘⣾⣿⣿⠥⠔⠂⠁⠀⠀⣿⣿⡄⣿⠁⠀⢠⡿⣁⣀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠚⢿⣿⣶⡦⠀⠀⠀⠀⢻⡄⠀⣻⣿
⣦⠀⠀⢠⠊⠀⠀⢰⣄⠀⢸⡟⠀⠀⠀⠀⠀⠀⣾⠛⢹⣿⣿⠀⠀⡞⠁⠉⠉⠙⠚⢂⡾⠃⠀⠠⢞⡖⠀⠀⢠⣍⢹⣤⣶⣦⠀⠀⠀⠿⠄⠘⠻
⣿⣷⣶⣶⣤⣀⠀⠀⠹⣷⣾⣿⣄⠠⣶⣿⣶⠀⠀⠀⣼⣿⣿⣷⣤⣷⣤⣤⣀⣀⣀⣾⣧⣄⡀⠀⠀⠀⠀⣰⣇⣸⡾⠟⢛⣻⠶⣄⠀⠀⠘⣦
⠛⣛⡿⠿⠿⠿⠷⢦⣠⡟⠹⠿⠿⢦⣝⣻⣯⣾⢛⣛⠃⠠⠤⠄⠀⢹⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣡⡤⠖⠉⠀⢀⣀⣷⡄⠀⠈⢷
⠀⠀⠀⠀⢀⣴⣶⣀⠀⠊⠛⠉⠀⠀⠙⣧⠀⢀⡀⠈⠉⠉⣀⣴⣶⣾⠁⠀⠀⠈⠉⠁⣠⣆⠩⠭⠭⠹⠭⠝⠻⣷⣦⠀⢠⡞⠉⠉⠉⠒⠦⠤⠈
⣦⣤⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⠀⢸⡃⠀⠀⢸⡟⠉⠉⠉⠑⠦⢤⠀⠀⠀⣿⣿⡇⠀⠀⢠⠤⠀⠀⢸⣿⣇⣿⡇⠀⣿⡀
⡀⠉⠁⢸⣿⣿⣿⣿⡀⠸⠿⠛⠟⠛⠛⣿⡀⣶⣇⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⢨⡟⠀⠀⣼⣿⣿⣿⡧⢀⣿⠀⠀⠀⣴⣿
⠛⠛⠶⠿⠛⠛⠻⢿⣧⣀⣀⣤⣤⣠⣤⠿⠿⠿⢷⣶⣴⡿⠗⢸⣿⡀⠀⣤⣶⣴⡴⠾⣿⣿⣿⠀⠀⢸⣇⠀⠀⣼⠿⠛⠿⣿⣵⣿⣆⠀⠀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣟⠀⠀⠉⠀⢀⣀⣀⠉⠉⠛⠓⠒⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠞⠛⠛⠉⠉





⣿⣿⠿⠿⡿⠻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠞⢉⣉⠛⣦⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⣿⠛⢦⣄⠀⠀⠀⠀⠀⠀⣿⠀⠙⣿⠀⢸⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣿⠀⣿⠀⠀⠙⠳⣦⡀⣀⡴⠟⠙⠷⣤⣥⡴⠛⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⠀⣠⡶⠋⠙⠋⠻⢦⡀⠀⢸⡇⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⣤⠞⠉⠀⠀⠀⠀⠀⠀⠙⢷⣼⡇⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⣿⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⢳⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢠⡟⢀⡀⠻⣆⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⣠⡟⠀⢸⡇⠀⠹⣆⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣰⠏⠀⠀⢸⡇⠀⠀⠙⣧⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⣿⠀⠀⠀⣰⠏⠀⠀⠀⢀⡁⠀⠀⠀⠘⣧⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡧⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⠾⠃



⡟⠉⢻⡟⠿⢿⠿⠻⢿⠻⠟⠻⠿⠿⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⢿⣿⣿⠿⡿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣿⠃⠀⠀⠀⣤⣤⣶⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡿⠀⠀⠀⢀⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⡿⠋⠀⠀⠀⠀⠰⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣤⡿⠟⠿⢿⣿⣶⣶⣶⣶⣿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⣀⣵⡿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢀⣶⣿⡿⠟⠋⡉⠉⠉⠉⠙⢻⣿⣷⡄⠀⠀⣐⣼⡟
⠀⠀⠀⠀⣰⣿⡿⠋⢠⣾⣿⣿⣿⣶⣿⣿⡟⠘⣿⣿⡄⢀⣿⣿
⠀⠀⠀⢰⣿⡟⠀⣰⣿⣿⠏⠀⠀⢻⣿⣿⡇⠀⢸⣿⣇⢸⣿⣿
⠀⠀⠀⣿⣿⠃⢠⣿⣿⡏⠀⠀⠀⣼⣿⣿⠀⠀⣼⣿⡇⢸⣿⡿
⠀⠀⠀⣿⣿⡀⢸⣿⣿⡇⠀⠀⢀⣿⣿⣿⠀⢄⣼⡿⠁⠜⣿⡇
⠀⠀⠀⢻⣿⣧⠀⣿⣿⣷⣤⢤⣶⣿⣿⣿⣵⣾⠟⠁⢀⣼⡟
⠀⠀⠀⠈⢻⣿⣷⣌⡻⠿⠿⠟⠋⠉⠛⠛⣉⡠⢐⣴⡿⠋
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣶⣶⣶⣭⣯⣿⣶⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠙⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠐⠶⠷⣀⠀⠀⠈⠉⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣄⣠⣿⢿⣦⣀⡠⠀⠙⠃⠤⠤⢤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠶⠋⠉⠁⠀⠈⠉⠳⢾⣥⡔⠂⠀⠐⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣟⠶⣤⣄⣠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⠦⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡾⢣⡄⠀⠰⠶⠶⠤⠄⠀⢀⣴⣿⣤⣈⠷⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠸⣿⣄⠙⢦⠀⠠⠤⠤⠄⣠⣾⣿⣿⠟⠉⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠙⢷⣄⠑⡄⠀⠀⣰⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣀⣀⣀⡀⠉⢛⣿⣦⣾⣿⣟⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠻⣟⠛⠛⠛⢻⣿⣟⢛⣿⠟⣟⠛⠛⠛⢻⣿⣿⠛⣿⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠳⣤⣤⣤⣬⣽⡟⠁⠀⠘⠳⣤⣤⣀⣩⣭⠟⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣠⠋⠉⣿⡇⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠛⣦⡀
⠀⠀⠀⠀⢻⣄⣀⣿⠇⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣸⡄
⠀⠀⠀⠀⠀⠉⠉⣿⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⡀⠤⠴⠶⠦⠴⠶⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠲⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣷⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠟⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣶⠶⠶⠛⠛⠛⣿⣷⡀
⠀⠀⠀⠀⠀⢠⡜⢫⠀⠀⠀⣸⠉⠀⠀⠈⢻⠉⠀⣻⡇⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷
⠀⠀⠀⠀⠀⡎⠀⡸⠀⠀⢠⡏⠀⠀⠀⠀⢸⠀⢠⡟⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⡇⠀⢿⡀
⠀⠀⠀⠀⣶⠁⢰⡅⠀⠀⢸⣧⣴⠖⠀⠀⠈⠁⣸⠇⠀⠀⠐⣶⣤⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⢠⠁⢀⣘⠁⠀⠀⠘⠉⢿⡀⠀⠀⠀⣸⡇⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⢸⡇
⠀⣀⠐⣾⠶⠊⣩⣤⣦⠀⠀⠀⠈⢻⡄⠀⢰⣿⠃⠀⠀⣰⠇⠀⠀⠀⠀⠀⠉⣉⣨⣿⠂⠀⠘⣇
⠀⢳⡶⠃⢀⣴⠋⢁⣼⣇⠀⠀⠀⠀⣹⣆⣀⣿⣀⣠⣴⣷⣤⣤⣤⣤⣤⣜⠛⣉⠉⣿⡆⠀⠀⣿⡇⢐⢀⡀
⣠⠏⠀⣠⠞⠁⢠⠟⠉⢹⡦⠴⠶⠞⠉⠉⠉⠉⠉⠀⠀⠀⢀⣠⣤⠶⠟⠋⠉⠉⠉⢿⠇⠀⠀⢻⡏⣉⠉⠳⠆
⠻⣤⣴⡏⠀⣰⠃⠀⣴⠟⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⡟⣋⣉⣁⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡧⠈⠛⢺⣆
⠀⢸⡇⠀⣶⠋⢀⣴⠿⠶⠶⠶⠶⠶⠛⠛⠛⣿⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⢿⣧⠈⢶⣄⣼
⠀⠘⢷⣤⣽⠟⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠶⠉⠁
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠢⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣿⠃
⠀⠀⠀⠀⠻⠶⢶⣶⣶⣤⣤⣤⣤⣤⣄⣠⣠⣿⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⢶⣿⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⠤⢾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠸⡿⠖⠊⠙⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡞⠋⠀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⣀⠀⠀⠙⠛⢷⣦⣄⣀⡀
⠀⠀⠀⠀⣠⠴⠖⠋⠁⠀⣠⣤⠾⢿⣄⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣤⣝⠙⠶⢦⣤⣄⡀⠀⠈⠉⠙⢶⣶
⣀⣘⣿⣦⣤⣤⣤⣴⣛⣛⣉⣁⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣈⣉⣉⣙⣛⣛⣉⣀⣉
⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠇⠀⠈⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⠙⡳⢦⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⠅⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡠⠀⠀⠀⠀⠀⢀⣤⠀⣠⠎⣠⠞⣧⠀⡆⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⣠⢔⣡⡴⠋⣯⣶⠷⠋⠁⠀⠘⣧⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣤⡴⠿⠛⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⠘⣷⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡆⠀⠘⠛⠲⣤⣀⠀⡀⠀⢀⣠⠴⠋⠉⠀⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⢸⣷⣶⠀⢸⠁⠀⠈⢡⣶⣶⠀⠀⣿⢰⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠬⣉⠥⢀⡟⠀⠀⠀⠈⣛⣫⠄⠀⢿⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⠓⠶⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠘⠈⠉⠉⠩⠍⠉⠁⠈⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⠒⡶⠒⠒⠒⠻⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⢺⡏⠙⡇⠀⠑⠀⠀⠀⠀⡷⠈⣿⠲⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⠀⠀⠀⠀⢸⠀⠀⠹⡄⠀⠀⠀⠀⣰⠇⠀⢸⠀⠀⠀⢀⠈⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢢⠀⠀⠀⢸⠀⢀⣀⠘⢦⡀⢀⡼⠃⣀⡀⡾⠀⠀⠀⡎⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠃⠀⠀⠈⠛⠉⢻⣀⠀⢹⠋⠀⣠⡏⠉⠁⠀⠀⠐⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⢻⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢱⡖⠛⠛⠛⠛⢦⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠸⣏⡙⠲⠤⣄⣀⡄⠘⡇⠀⠀⠀⢀⡏⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡤⠶⠋⣁⡿
⠀⠀⠀⠀⠀⠀⠀⣸⠛⠳⠶⢤⣭⣷⠀⠈⠓⠲⠚⠉⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⠴⠶⠛⡏
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢀⣀⣸⣧⡤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠙⠛⠛⠶⢦⣤⣤⣶⠶⠶⢤⣤⣼⣧⣇⣤⣤⣴⡶⠛⢋⡉⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀
⠒⠒⠒⠒⠒⠒⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢲⣶⣟⠛⠋⠉⠉⠉⠙⠛⠿⣿⠆⠀⠀⠀⠀⢀⡼⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣟⠀⠀⠀⠀⠀⠀⢤⣓⢮⣷⣤⣤⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠳⠶⣤⣤⣒⣮⣳⣵⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠲⡄⠀⣀⣀⡀
⠀⠀⠀⠀⢀⡀⠻⣦⣀⣠⡾⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠹⠊⠀⠀⠈⡆⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠙⣧⠀⠉⠁⢠⡞⠁⠀⠀⠠⣄⡀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡜⠁⠈⠖⠊⠓⡆
⠀⠀⠀⠀⠀⣠⡟⠀⢀⠀⠸⣧⡀⠀⠘⢶⣌⠛⠶⠞⢡⡾⠃⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠀⢀⠔⠚⠀⠀⠀⠀⠴⢥⡀
⠀⠀⠀⠀⠘⠋⣴⠞⠉⠛⢦⡈⠋⠀⠀⢀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⢀⡴⣄⢸⠉⠉⠀⠀⠈⠢⢤⡄⣀⠀⠀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣡⡶⠲⢦⣌⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⢦⡇⠀⠀⠀⠀⠀⢸⠖⠙⢄⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⣠⠤⠤⠤⢤⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⢄⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⠎⠀⠀⣠⡀⠀⠈⢦⠀⠀⠀⠞⣛⣓⣒⠾⠽⢶⡀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⣤⠀⠀⠙⣄⠀⠀⠠⢟⣛⣓⡲⠯⠽⢦
⠀⠀⠀⡏⢀⡠⠞⠁⠙⠢⣄⡀⡇⠀⠀⣠⡇⢐⡄⠀⣔⠀⢳⡀⠀⠀⠀⠀⠀⢰⠃⣀⠤⠊⠀⠑⢤⣀⠘⡄⠀⢀⣼⠀⢢⠄⠠⡔⠈⣇
⠀⠀⠀⣧⢻⠀⠾⢁⠘⠇⢠⢷⡇⠀⠀⠺⡇⠈⠁⠂⠉⠀⡼⠎⠀⠀⠀⠀⠀⢸⡜⣇⠀⠏⡀⠹⠀⣼⣢⡇⠀⠘⢾⠀⠈⠠⠀⠁⢀⡧⠃
⠀⠀⠀⠹⡉⢣⣀⣔⢠⣠⠎⣩⠃⠀⠀⠀⠙⢄⡢⠀⣁⡴⠁⠀⠀⠀⠀⠀⠀⠀⢧⠙⢆⡀⠀⢀⡴⠋⡼⠀⠀⠀⠀⠳⣄⠐⠂⣠⠞
⠀⠀⠀⠀⠙⢦⡜⢋⠛⣦⠔⠁⠀⠀⠀⠀⠀⣴⠛⡍⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠟⣉⠻⡤⠊⠀⠀⠀⠀⠀⢀⣞⢩⡍⢷⡀
⠀⠀⠀⠀⠀⡾⠑⠊⠒⠹⡆⠀⠀⠀⠀⠀⣸⠁⠉⠀⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⠂⠉⠂⢱⡀⠀⠀⠀⠀⠀⡞⠀⠁⠈⠀⢳⡀
⠀⠀⠀⠀⡼⠀⠡⠀⠠⠁⠸⡄⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢂⠀⡐⠀⢡⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣇
⠀⠀⠀⢀⡇⠀⠀⠑⠁⠀⠀⣇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⢱⠀⠀⠒⠀⠐⠘⡆⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡾⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡟⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⡄⠀⠀⠀⠀⢸⢸⡆
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⡀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠃⠁⠀⠀⠀⠀⠈⠀⠃
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣠⢾⣃⠜⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠠⡖⣀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠚⠀⠀⠀⣤⠒⢋⠏⠀⠀⠀⠀⢠⠎⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠉⠀⣣⣀⣠⣄⡀⠀⠀⠀⠀⠀⢠⡮⡤⠀⠀⠀⠀⢀⡴⣁⠀⢏⠀⠀⢐⡥⠤⠶⣆⠀⢠⡇
⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠁⠀⠈⢱⠀⠳⡞⠁⠀⠀⣠⠛⠔⠚
⡎⠀⠒⠉⠈⠁⠘⢿⣿⣿⠿⠛⠛⣆⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠙⢢⠘⠯⣀⡤⠞⠉⠐⠒⠂⠀⠤⠤⠤⠤⠤⠤⠄⣀
⢃⠀⠀⠀⠀⡀⢀⡘⡉⢀⠀⡠⠺⡏⢈⠦⡀⠀⠀⠀⠀⠀⡶⢴⠒⠃⠀⠀⠠⣮⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠘⡄⠴⣯⡽⣿⡿⠿⠰⠊⢛⡀⠀⣸⡅⠀⠈⣦⠀⠀⠀⠀⢿⣂⣀⣀⣀⣸⡶⣀⢶⠀⠀⠀⠀⠀⢠⠏⠒⠤⠄⣀⣀⣴⠤⠖⠋
⠀⠈⢖⠋⡀⠀⠀⠀⠀⢗⠁⠘⡁⠈⢏⠈⠉⣽⣀⠀⠀⠀⠀⠀⠀⠀⠈⢏⣤⡾⢸⠀⠀⠀⠀⢀⡞
⠀⠀⠘⢆⠱⡀⠀⢱⠀⢸⣤⣤⠝⠉⠻⣤⡤⣡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠠⠠⢀⠞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠦⠓⠲⣾⣏⠉⠹⡆⠑⠒⣒⡾⡅⠁⢀⡴⠊⠉⠂⣄⠀⠀⠀⢠⢤⣀⣸⠀⠀⣠⠋⠀⠀⠀⠀⢸⠉⢻⠁⠉⠑⠒⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢋⠉⠁⠀⠀⠘⡄⠀⠙⣄⠰⢯⣦⠠⡀⠀⡞⢣⣨⠃⢀⠞⠁⠀⠀⠀⠀⠀⢸⠀⡎⡤⣤⡄⢰⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠸⡄⡤⠤⣒⡦⠎⠀⠀⠌⢧⠀⠀⣠⠜⠀⢧⠀⠱⡔⠉⡆⠀⠀⠀⠀⠀⢠⠒⠚⢄⠙⠻⠗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠈⠈⠓⠉⠀⠀⢀⠟⡤⠂⠟⣩⠁⠀⠀⠀⠀⠀⠈⡦⢷⣊⣀⠴⢾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠀⠀⠐⣼⠉⠲⣤⠀⠀⠀⠔⠤⣅⡲⠚⡥⡀⠈⢣⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢱⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⣎⠉⣤⣧⢸⠀⢻⠀⠈⢣⣴⣿⢄⡀⡸⠀⠑⠫⣄⠀⣾⡏⠀⢀⡹⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢶⣉⣉⠰⣗⣀⡁⠫⢼⠔⠚⠒⢺⠅⢹⠀⢘⣀⣴⣿⠧⢙⣪⣅⣲⠳⡀⠑⣌⢳⣤⡶⠬⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⡚⠁⠚⢍⠉⢀⡨⠒⠹⠷⡶⠾⠭⢑⣞⠀⠈⠉⠀⢾⠊⡵⠠⠤⣥⣎⣱⣰⡈⡿⠋⠉⠲⢎⡉⠖⠈⠑⠂⣴⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣈⡎⠓⢒⡀⠀⣠⠽⠒⢶⡀⢰⣀⡀⠾⠟⠉⠐⠢⡀⣀⣿⠓⠢⢼⠡⠄⣹⢄⣉⡳⣄⡾⠾⠂⠉⢒⣤⣐⡞⠤⢤⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢠⢼⡁⣈⡏⢥⡤⡸⢦⡀⡙⠿⢀⡀⢈⡩⠓⠒⠤⢄⣏⠀⢾⡥⠶⣊⠁⢰⡖⠂⠀⢣⠀⠑⠢⢤⠔⢁⣈⣙⠃⡴⠛⠒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣈⢩⣇⡈⠀⠀⢰⠒⠦⠤⡄⠉⠀⠀⠀⠈⡡⠊⠀⠀⠄⠀⠴⠚⠁⠀⠀⢶⡎⠁⡸⠏⠉⠉⣄⠀⠈⠢⢄⠀⢄⠀⠉⢀⣀⡾⠂⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡀⢀⣀⠄⢸⣀⡀⢀⡩⠽⠋⠉⢒⠶⠁⠒⠤⡴⠀⠀⠀⠱⠀⡀⠉⡒⠴⠊⠐⢤⡀⡴⠋⠳⠤⠤⢞⡃⠀⠱⡤⠊⠀⣀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢍⡁⣀⡄⠀⠀⢀⠈⠉⠒⠤⠔⠁⠀⠀⠀⠀⠀⠓⠒⠒⠒⠃⢣⠤⠚⠤⢤⠄⢒⠟⠀⠀⠀⡶⣀⠀⠈⠂⣄⠀⠠⣀⡀⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠑⠒⠞⢉⡞⠁⠀⠀⣀⠀⠀⡄⠀⠀⠀⡐⠀⠀⢰⣠⣂⠀⢀⠎⠀⠈⠙⠒⡆⠼⠀⠈⠶⠒⢋⡉⠀⠀⣈⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠊⠸⠄⣀⣰⠦⠤⡷⠮⢭⠉⠉⠀⣯⡍⠉⠀⢰⣆⣀⡤⡼⠂⠀⢧⡤⠤⠼⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠃⠈⠉⠉⠀⠁⠀⠓⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⢀⣿⣇⣤⣴⣶⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁⢀⣙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠛⠛⠉⠁⢀⣀⣤⣤⣶⣾⡿⠿⠟⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴⣴⣦⣴⣦⣴⣴⣶⣾⠿⠿⠛⠛⠉⠁⢀⣀⣤⣤⣶⣶⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠉⢹⣿⠟⠉⠁⢀⣀⣤⣤⣶⣾⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⠄⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⣶⣶⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⣴⣿⢿⣦⠉⠹⣿⡆⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡍⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠰⣿⡌⣿⣿⣩⡀⢻⣿⠈⣥⡄⠀⢻⣿⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣧⠀⣤⣴⣦⣄⢸⣿⣟⡋⠀⠀⠀⢿⣷⢸⣿⣟⣡⣌⢿⣿⡿⠃⠀⠈⠉⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠸⣿⡆⢿⣿⣽⣿⠈⣿⣿⠛⣀⢰⣷⣾⣿⠂⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⣀⡀⢻⣷⠘⣿⡿⣿⣦⠹⣿⠿⠟⠃⠉⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⠛⠛⠛⠃⠘⣿⣇⠙⠋⠈⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⡿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⣀⣠⣤⣴⣶⣿⠿⠿⠛⠋⠉⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣄⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣠⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣄⣈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠉⢁⣈⢩⣿⠉⢉⡉⠙⢿⣿⡏⠉⠉⢻⣿⣿⠉⠉⢻⣿⡏⠉⠉⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣀⠈⠙⢿⣿⠀⠘⠋⠀⣼⡿⠀⢰⡆⠈⣿⣿⠀⡀⠈⣿⠀⡀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠿⠀⢀⣿⠀⢰⣶⣾⣿⠃⠀⣀⣀⠀⠸⣿⠀⢸⡀⠁⢸⡇⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠷⠶⠶⠶⠿⠿⠶⠾⠿⠿⠿⠶⠾⠿⠿⠷⠶⠿⠶⠾⠷⠶⠿⠷⠶⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉









⡿⠿⠿⠿⠿⢻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣴⡾⠋⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠛⠿⢶⣤⣄⡀⠀⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠃⠀⠀⠀⠀⠈⠉⠛⠷⣶⣤⣈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢋⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⣼⠏⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⣼⡏⠀⠀⢰⣿⣭⡛⣱⡿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣦⣀
⠀⠀⠀⠀⢸⣿⣿⣿⠛⠛⠛⣿⡟⠀⠀⠀⢶⡶⠟⣿⠟⠛⣹⣿⣿⠏⣴⣶⢆⣤⡄⠀⠀⢰⡿⠛⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣷⠈⠙⠻⣶⣤⡿⠁⠀⠀⠀⠀⠀⠈⠉⠠⠾⢿⣿⣯⣾⢿⣿⣿⡟⠀⠀⢠⣿⣷⠟⠋⠁⢸⡇
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠡⠟⢽⡿⢀⣤⡾⠟⠉⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡀⢀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⢿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⠀⠉⠻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄⡀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⢸⣇⢀⣤⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀⣸⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠛⣛⣛⡻⠶⠿⢛⠛⠏⢩⡙⠻⣦⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠛⠋⣉⣩⣤⣽⣿⣿⣿⣷⣼⣿⣶⣌⣿⣷⣌⢻⠟⠛⣙⡛⠻⢶⣤⣄
⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⣛⣭⣵⡾⠿⢿⣛⣭⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣶⣶⢗⣠⣿
⠀⠀⠀⠀⠀⠀⠹⣷⣬⠛⢛⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⡻⣦⡀
⠀⠀⠀⠀⠀⢀⣴⠟⢡⣾⣿⣿⣿⣿⣿⡿⠿⠯⠙⢳⣶⣶⣿⣿⣿⣿⣋⣘⣿⣿⣿⣿⣿⡛⠛⠿⣿⣿⣿⣮⣛⠛⢷⣦⡀
⠀⠀⠀⠀⠀⣼⠇⣴⣿⣿⣿⣿⣿⣿⣍⣀⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⢸⡇
⠀⠀⠀⠀⢰⡟⠼⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⣠⠔⣠⣿⣿⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢠⣿⡃
⠀⠀⠀⠀⠸⣧⡀⣾⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⣿⣿⣿⣿⣿⡿⠿⠆⣸⡇
⠀⠀⠀⠀⠀⢿⡄⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⢿⣿⣿⣿⣿⣿⣿⠷⠦⠘⣿
⠀⠀⠀⠀⠀⠸⣧⠛⠁⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡆⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠙⢻⡇⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⠛⢿⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⠋⠉⠻⣿⠇⢿⡿⢿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⢿⣧⠀⠉⠃⠀⢹⡏⠰⠀⡜⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⡀⠀⠀⠹⠀⣈⣃⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠀⠀⣠⡔⠉⠀⠀⡃⣠⢎⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡱⡄⠀⠠⠀⠀⠉⢛⣶⣶⣤⣤⡤⠀⠀⠀⠠⣤⣤⣶⡿⢯⡄⠀⠀⠀⠣⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢹⠈⢻⠀⠀⠀⠈⠑⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⡸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣜⠤⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢂⣀⠀⠀⠀⠀⠀⠀⠀⢰⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⢣⠀⠀⠀⠀⠀⠀⢰⣦⣄⣀⣉⣠⣴⣿⡄⠀⠀⠀⠀⣠⠣⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⠔⠉⢣⠀⠀⠀⠀⠀⣾⣿⡿⢿⡿⠛⢿⣿⡇⠀⠀⢀⠔⠀⠉⠢⡙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡱⠃⠀⠀⠀⠁⢀⠀⠀⠀⡿⢹⣧⣀⣀⣠⣾⡆⠀⢀⠴⠁⠀⠀⠀⠀⠘⡜⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⠁⠀⠀⠀⠀⠀⠀⢙⠲⠤⠔⠈⠙⢛⣛⣛⠩⡰⠿⠉⠀⡄⠀⠀⠀⠀⠀⠘⡸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠠⡈⠑⠂⠤⠄⠊⢁⠃⠂⠀⠁⠀⡀⠀⠀⠀⠀⠁⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⢎⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠒⠀⠤⠀⠂⠅⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⣰⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡎⠓⢤⣀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠤⢶⢫⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠉⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠸⢷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⢸⠉⠢⠭⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⢀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⠶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠰⣿⣿⣿⣾⠿⠂⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⡄⠂⠀⠀⠀⠀⣠⣼⡆⠀⠀⠀⠀⠀⢀⣴⣿⠿⠝⠿⣳⣴⡆
⠀⠀⠀⠀⠀⢀⣤⣷⣾⣷⣦⣤⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡟⠁⠀⠀⣌⣀⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⡀⠉⠁⠀⠐⠿⠿⠟
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠿⣿⣿⣿⠟⣶⣧⡼⢿⣿⣿⡿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠻⣿⣿⡄⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣄⣛⣟⢁⣼⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣷⠉⠓⠀⠀⠀⠀⡄⠀⣿⣟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⠿⣿⣶⡆⠀⢠⣾⣷⣶⣿⠏
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠘⣿⣿⣿⣟⣿⡿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠻⡟⠉⠁
⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠂⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⣀⣤
⠀⠀⠀⣠⢴⣞⣉⣁⣀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⢻⣛⣛⣛⣛⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠛⣙⣿⣿⣿⠇
⠀⠀⠀⠱⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⢹⣿⡿⣿⣿⣟⢹⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⡀⠀⠀⠉⠉⠙⠋⢀⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⢼⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣈⡀⠀⢀⣴⣿⣴⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠝⠋⠛⢛⠟⠋⡇⣀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⡷⢸⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠀⠺⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⣿⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡵⠋⠁⠀⠀⠀⠀⠀⣼⠂⠀⠉⠉⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿


⠀⠀⠀⣤⣶⣾⣿⣿⣶⣶⣄⠀⠀⠀⠀⢀⣠⣤⣤⣦⣵⣤⣤⣀⠀⠀⠀⠀⠀⣴⠶⠶⠚⠛⠛⠛⠛⠷⠶⢦⡄
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢈⢛⣟⣛⣛⣛⣛⣛⢛⡁⠀⠀⠀⠀⠸⠶⢶⣶⣤⣤⣴⣤⡶⠶⠾
⠀⠀⠀⢸⣿⣿⢛⠛⣿⣿⠀⠀⠀⠀⢏⣻⠐⡇⢸⡇⢸⠀⣿⢰⣏⡇⠀⠀⠀⠠⣴⣄⣄⣀⣀⣀⣀⣀⣤⡄⠀⠀⠀⠠⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⠀⠀⢸⣿⠉⣿⠿⢙⣿⠀⠀⠀⠀⠀⢹⠈⡇⢸⡇⢸⠀⣿⢸⡇⠀⠀⠀⠀⠀⣀⡀⠈⠉⠉⠉⠨⠀⣀⠀⠀⠀⠀⠀⢹⣶⣾⢻⣶⣾⣻⣷⣾⡟⣷⣶⡏
⠀⠀⠀⢸⣿⣷⣾⣶⣿⣿⠀⠀⠀⠀⠀⢸⡀⣷⢸⡇⢸⠀⣿⢸⠇⠀⠀⠀⠀⠀⢉⡋⠛⠛⠓⠛⠛⠉⣉⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⢹⣿⣿⡇⣿⣿⡇
⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⢸⣇⣿⢸⡇⣸⠀⣿⣸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣶⣶⣾⣿⡏⠀⠀⠀⠀⠀⠶⠿⠿⠾⠿⠿⠾⠿⠿⠷⠿⠿⠶
⠀⠀⠸⠟⠿⠿⠿⠿⠿⠟⠿⠀⠀⠀⠀⠀⠺⠭⠽⡧⡿⠭⠽⠒⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⣴⠶⢍⡇⢠⡎⠉⠃⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡌⠠⡶⡤⣟⠄⠁⢠⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⢘⡄⠀⠀⠀⠀⠀⠀⠆⠀⠡⣀⣙⠁⠎⢈⣒⠮⠐⢐⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣠⠀⣼⠀⠀⠀⠀⠀⠀⠀⠰⡀⠤⠀⠂⠀⣁⣡⡤⠖⢛⣩⡥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠈⠈⡱⠂⠀⠀⠀⠀⠀⠰⠶⠒⠛⠉⠉⠰⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠒⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⢉⣡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠚⢉⣁⠤⠴⠒⠚⠉⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠒⠊⢉⣁⠤⠴⠖⠒⠉⠁⣀⣠⠤⠴⠒⠋⠁⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⡠⠤⠶⠒⠚⠉⠁⣀⡠⠤⠶⠒⠋⠉⢀⣀⡤⠤⠖⠚⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡤⠤⠖⠒⠊⠉⢀⣀⡤⠤⠖⠒⠉⠁⠀⣀⣠⠤⠴⠒⠉⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣀⠤⠴⠒⠒⠉⠁⠀⣀⡠⠤⠖⠒⠋⠉⠀⢀⣀⠤⡤⢖⠿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡤⠤⠖⠒⠊⠉⠀⢀⣀⣤⠴⠒⢊⣉⣥⠴⠊⠉⠀⣀⣀⠀⢹⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⡠⠤⠴⠒⠚⠉⠁⢀⣠⡄⠀⠀⠁⢀⣀⣤⡴⠺⣿⣹⣇⠀⢻⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⢀⣠⡤⠔⠚⠉⠁⠀⣤⣦⡞⠻⣏⠙⣿⣫⡹⠿⠟⠀⠀⢻⡆⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠁⠀⠀⣤⢰⣟⣁⢹⡄⢤⡄⠻⠆⠙⠉⠀⣀⣄⠶⠚⠉⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠉⠀⣀⣄⠰⣾⣃⠀⢹⣆⢻⣥⠆⠛⠋⠀⢀⡀⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⣿⣽⣅⢻⣯⡼⠛⠛⠀⠀⣀⣤⡴⠜⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠘⠷⠙⠃⠁⠀⣠⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠄⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣄⠴⠚⠈⠉⠀⡀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠉⠀⠰⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠁

⣿⣽⡅⠉⠀⣎⣿⢛⣤⡀⠘⢻⣶⣾⣷⢾⡇⠀⠙⠟⣩⢆⠀⠀⠀⠀⠀⠀⣾⣿⠂⠀⠈⠻⡍⠀⠠⢶⡶⠇⠀⠀⣾⣿⣿⣿⠟⠁⣼
⢿⠏⢀⣶⢦⠀⠑⠟⢙⣫⣴⠀⠀⢈⠁⠉⢀⣤⣶⡼⠋⣠⡖⠀⠀⠀⠀⠀⣿⡿⠒⠀⠀⠀⠀⠀⣖⣆⠙⠀⢀⣶⣿⣿⠿⠃⠀⣼⣿
⠀⠀⠻⠿⣣⣟⡦⠀⢀⡼⠋⠀⠀⠩⢀⡾⡝⣫⣿⣄⠀⢉⡬⣀⣄⠀⠀⠀⣿⠀⠀⠿⣠⢤⣀⡀⠉⠁⠀⠀⠈⠉⠉⠁⢰⣆⣾⣿⣿
⢿⡵⣠⡀⠉⠰⢂⡀⠀⠂⠘⠃⠀⠀⠀⠹⠿⢿⣿⣋⣀⠀⠛⡉⣶⢢⠀⣸⡿⠀⠀⡀⠁⠉⠼⠃⠀⢾⡰⣀⠀⠀⠀⠀⠀⠉⠉⠁⠁
⡉⠿⠛⢛⣠⡄⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠀⣠⣿⠃⢀⡴⣿⢠⡦⠀⠀⠉⠚⠵⠛⢒⣠⡄⢀⡀
⣿⣦⣈⠉⠿⡀⢸⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠈⠁⠟⠀⡰⣇⢂⡄⡀⠀⠁⠘⠁⢸⡇⣴⠆⢀⣀
⣫⣿⣿⣦⡀⠀⠉⠼⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠞⠑⡃⣴⠀⣀⡀⠀⠘⠃⢠⡘⣷
⢋⣭⣟⣿⢃⡶⠄⠀⠀⠃⠿⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠈⠀⠛⡄⣿⢤⡾⠀⣀⡀⠉⠙
⠈⠓⡋⣿⣿⠅⣀⣾⡀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢠⡙⣷⢡⣆⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣴⡆⠀⣄⠀⠀⠛⠁⣠⢿⡞⣴⢦
⣆⠀⢑⡋⠀⠘⣿⡿⣾⣧⣋⣠⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠀⠤⠚⣿⠀⢺⡇⢠⡆⠀⠀⣼⡏⢀⡥⢚⣿⣴⡖⠀⣀⠈⠉⠻⠏⣋
⢉⣠⠉⣴⠃⠀⠙⢞⣿⣿⡟⠉⣽⠳⣴⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⡀⠉⠘⠘⠃⠾⠁⠀⠀⣿⠘⠂⠗⣠⢖⡏⠀⠘⣯⢛⣶⣄⠀⠉
⢟⣽⢆⡉⢺⡟⣀⠀⠺⢟⣶⣤⡀⠘⠻⣿⣿⣿⣿⣿⣦⣶⣿⣿⣤⡀⠀⠀⢀⠀⠀⠀⢀⣼⡿⢄⡀⠈⠁⠨⢰⠆⢀⡈⠈⠙⠋⠰⠇
⠘⠋⢡⡶⠀⠹⢟⠁⡀⠘⣿⠿⢇⣴⠀⡀⠉⠻⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠵⠋⠾⢀⠀⠀⠙⠄⣸⢃⣤⠄
⠀⠀⢉⣄⢁⡀⠀⣻⡽⢄⠀⠘⠫⣿⣰⣿⣠⣦⠀⠀⠙⠏⠛⢩⣿⠿⣻⡿⠟⠁⠀⠀⠀⠀⠀⠀⠠⡎⠀⣴⠀⠀⠀⠛⠁⠠⣖⠄
⣁⡀⠈⠚⢋⣡⣾⣿⣿⡿⢇⣤⠀⠀⠁⠋⠼⠃⢀⢿⡱⣢⢀⠀⠁⣴⠿⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢫⡼⠋⢀⡀⠀⠐⢫⡴⢯
⢩⣷⣤⣶⣿⣿⣿⣿⣿⡷⠛⣿⢠⣿⣀⣤⠀⠀⠈⠚⠣⡿⠉⠀⠀⠀⢞⡦⢖⡀⠀⢷⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣠⢄⠀⠀⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣬⣭⣿⣽⣯⣤⣾⣴⣤⣤⣤⣥⣴⣦⣤⣤⣦⣿⣼⣤⣤⣤⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣾⣭⣿⣥⣤⣤
⣿⣿⣅⣭⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠐⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠀⠀⠀⢀⠆⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠁⠀⠀⠐⣨⠹⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡀⡔⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⢀⠨
⠀⠀⠂⠈⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⢀⡲⣒⣀⣌⡬⠥⠄⢂⡴⠘⠏⠋⠈⠀⠀⠄⠐⠂⠀⠈⠀⠃⠀⠑⠀⠂⠀⠁⠀⠀⠀⢀⠀⠠
⠀⠰⡀⠀⠀⠀⠀⠀⠀⠃⠔⠒⠈⠀⠠⠄⠀⠐⠀⠀⠀⠀⠉⠀⠀⠴⠒⠂⠈⠀⠀⢀⢸⠠⠀⠠⠀⠀⢀⠔⠦⠶⠂⠀⡀⠤⠀⠀⠀⠄⠈⠄⠀⠀⠔
⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠦⠰⢄⠈⣌⠢⠄⠀⠒⠂⢃⠀⠀⠀⠘⠀⠀⠀⠀⠐⠀⠑⠀⠘⡄⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠰⣁⠀⠂⠂⠁⠙⠈⠁⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠄⠄⠀⠀⠀⠀⠀⠄⢀⠀⠪⠂⠳⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⠈⠐⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠁⠠⠀⠀⠀⢀⠀⠀⢀⠀⠀⡄⠀⠂⠁⠀⡁⠈⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠂⠀⠁⠠⠄⠀⠀⠀⠀⠀⠀⡄⠂⡀⠐⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⠀⠀⠀⣰⡃⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠤⠤⣤⣲⣲⣶⣶⡾⠵⠀⠀⠀⡠⢐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⠀⢡⡀⠀⠀⠀⡀⢌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠴⠿⠖⠓⠒⣈⣉⠩⣡⢅⣂⣒⠴⠠⢬⠶⠙⠀⠐⡆⢀⡀⢀⣠⡄⢂⡈⢋⣁⡀⠊⠙⠢⣠⣾⣷⣠⢄⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢠⢘⡜⠎⠴⠉⠃⠒⣐⡀⠘⠂⠑⠪⣤⠀⠨⠊⠠⠟⢛⠧⠉⢀⠩⢉⣀⣹⣿⡻⠶⠆⠀⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠂⠀⠀⠀⠀⠐⠀⠤⠀⢀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⣀⣀⢄⡄⣤⣬⠿⠿⠶⠶⠆⠠⠄⠀⠀⠀⢀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠰⠶⠰⠴⠟⠓⠋⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠰⣌⣿⠀⣶⢸⡟⢠⣦⠘⣿⢀⡙⣁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡇⢸⡏⠲⠄⣿⠀⣶⣾⡇⢠⣤⠀⣿⢸⣿⣿⢸⡇⢸⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡙⢿⡇⢸⣿⣷⣿⣿⣷⣿⣿⣿⣾⣿⣾⣿⣾⣿⣿⣿⡇⢸⡿⢋⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣄⠸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠇⣨⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⡙⠿⣿⣿⣤⣤⣤⣤⣤⣤⣿⣿⠿⢋⣴⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣌⡛⢿⣿⣿⣿⣿⡿⢛⣡⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣠⣉⠻⠟⣫⣄⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⢋⣴⣿⣿⣿⣷⣾⣿⣿⣿⣦⣙⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⢰⣐⢠⣶⠐⢲⠒⠀⡝⣤⢻⢀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⣒⣶⣖⣒⣶⣖⣖⣲⣄⠀⠀⣤⣤⣤⣤⣤⣴⣖⣒⣒⣒⣒⣒⣒⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⠇⠀⠘⣯⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠈⢿⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠁⠀⠀⠀⠉⠀⢰⡇⠀⠀⠀⠀⠈⣇⠀⠀⠁⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠠⣶⢲⣦⠀⠀⢸⠇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⣿⠀⠀⠰⣷⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠛⠸⠛⠁⠀⣸⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣄⡇⠀⠀⠀⠀⠀⠀⠀⣏⣤⡀⠀⠀⠀⠀⢸⣧⣄⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⣧⡀⢀⠀⡀⢀⠀⣀⣿⠀⡇⠀⠀⠀⠀⡏⡔⢲⡽⡆⢀⠀⢀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⡼⠻⣧⣼⣤⣷⣬⣦⡽⠻⣤⠇⠀⠀⠀⠀⠳⣽⣯⣼⣧⣬⣦⣼⣦⡽⠃





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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡀⠀⢀⣀⣀⡀
⠀⠀⠀⣠⠤⠶⠶⠚⠉⠀⠀⠀⠀⠈⠛⠋⠉⠉⠉⠉⠛⢦⡀
⠀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄
⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟
⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⡴⠦⣤⣀⣀⣀⣀⣀⡼⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡓⣶⡄⠈⠉⠉⠁⠀⠀⢀⣰⣤⣤⣤⣀⣠⣄⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢛⣀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢀⣿⣿⣿⡿⠟⠛⠙⠻⣿⣿⣿⡿⠃⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⠀⢀⣀⠀⠀⠈⠉⠁⠀⣀⣀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣻⣿⣷⣦⡀⢀⣴⣿⣿⣿⡂⠀⢸⣧⠀⠀⡒⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡸⠿⡇⠀⠘⢠⢊⣝⡟⠁⠀⠛⠻⣭⡆⢹⠀⠈⡼⢿⠁⢤⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣁⠸⠲⠱⡀⠀⠈⠮⠾⠭⠄⠀⣀⠘⠦⡠⠄⠋⠀⠀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⡤⠟⠀⠉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⢄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠠⠤⠤⠤⠤⠔⠒⠚⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⣶⣒⣒⠒⠒⠒⠚⠛⠛⠉⠙⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠶⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠂⠐⠂⠐⠒⠚⠋⠉⠉⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⡤⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡏⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠐⠒⠒⠒⠒⠒⠲⡟⠙⠓⠒⠒⠂⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢰⣿⠀⠀⣿⠀⠀⣰⣶⠀⠀⢰⣦⠀⢀⣴⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⣿⠷⠾⣿⠀⢠⣿⢹⣧⠀⢸⡟⢷⡾⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠸⡿⠀⠀⣿⢠⡿⠛⠛⢻⣇⢸⡇⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣀⣁⣈⣁⣀⣀⣈⡁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⢿⣿⣥⣮⣿⣭⣽⣿⣭⣻⣭⣹⣟⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠛⠋⠀⡇⠀⠀⢀⣀⣀⠠⠤⠄⠒⢢⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⣤⢴⡆⡆⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠛⠘⠃⢁⣁⣠⠜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠲⠤⠭⠭⣭⣝⣒⣒⣒⠂⠤⠤⠤⢤⣄⣀⣀⣀⡇⠀⠀⠑⣒⣋⡭⠥⣔⣒⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠒⠒⠦⠬⠭⠭⠭⠖⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢶⡒⠒⠶⣶⡶⠞⠋⠉⠉⠉⠙⠓⠶⠦⣤⣤⠶⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⠀⠈⠷⠀⠀⠋⠀⠀⠰⡄⠀⠀⠀⠀⠀⠘⠋⠀⢀⡀⠀⠀⣿⣷⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣟⠳⢤⡈⠛⢦⣀⣴⢦⣤⡦⠶⣶⠿⣤⣄⡀⠀⠀⠠⠀⣠⠞⠀⢀⢠⡿⠉⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠉⢳⡦⠤⣤⣀⣿⠠⣞⠁⢤⡀⠸⣄⠀⢨⡟⠻⣟⠛⢾⡗⠶⢦⣾⠈⠁⠀⣠⡟⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⣴⠋⢤⠀⢀⡴⠋⠀⠀⠈⠉⢻⣤⡼⠷⢦⣍⣀⣈⣀⣼⣃⣀⢌⣉⣈⣉⣷⣀⣹⣟⣄⣠⣯⠀⠀⠄⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢈⣤⠀⠀⠀⠀⣴⠟⠙⢧⢀⡀⠀⠉⠉⠉⠉⡿⠙⠻⠿⠿⢯⡍⠉⠉⠉⠀⠀⠻⡄⠀⠀⠀⠀⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣀⡴⠋⠀⠀⠀⠀⠚⠁⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠘⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠻⣆⠀⠀⠄⣤⠀⠀⡀⠀⣷⠀⠀⠣⠂⣤⠀⠀⠀⠀⠠⠀⠈⢧⠀⠀⣄⠀⠀⠀⠀⡆⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢤⡄⡇⠀⣈⣤⡤⣼⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⠞⣧⠀⣀⣤⠴⠦⣄⣿⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠻⠟⠋⠉⠀⠀⠈⠛⠛⠛⠛⠓⠒⠒⠖⠖⠒⠋⠀⠀⠈⠉⠉⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠀⠀⠰⡂⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⢿⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⣼⢃⠀⠘⠉⠓⠶⡄⠀⠀⠀⢸⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⣟⠂⠀⠀⠀⠁⠀⢹⡄⠀⠀⠀⠻⣟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡇⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠙⠲⢦⡈⠂⠀⣠⡟⠀⠀⠀⠀⠀⠈⠳⢮⣟⡲⠦⢤⣤⣤⣤⣀⣀⣀⣠⣤⠴⠖⣛⣭⠞⠋⠀⣠⢶⠛⢻⠉⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠙⠒⠚⠋⠀⣠⡴⠞⢦⡀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠤⠤⠤⠤⠶⠶⠚⠋⠉⠀⠀⠀⠀⣏⠈⠆⡀⠀⣈⣥⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⡈⢿⠛⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣤⣷⣋⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⡳⣤⣙⠀⠁⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣍⣛⣻⡶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⣠⣤⣤⣤⣀⣸⣷⣦⣴⣾⣿⣦
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠘⠿⠿⢟⠋⡙⢛⡉⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠸⠙⠁⠼⠇⢸⣿⡿⠛⣿⣿⣿⠛⠉
⠀⠀⠀⢱⠄⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠈⠇⡀⠀⢀⣀⣤⣼⣿⣍⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⣤⡄
⠀⠀⠀⠀⠀⣰⠟⠛⠉⠉⠁⣨⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⠟⠻⠿⢿⣿
⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⢹
⠀⠐⣠⡗⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣿⣿⣿⠀⠂⣹⡇⠀⠒⣀⠀⠄⠸
⠈⣼⠋⠀⢀⣴⡀⠀⠀⢼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣤⣹⡏⠛⢛⠇⢠⣿⡇⠀⠁⠿⠋⠀⡄
⠰⠛⠄⣠⠏⢹⡷⠀⠀⢺⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠉⠉⠛⠿⠛⣷⡐⠃⠀⠀⠉⠁⠀⠀⠀⠀⢰⠁
⡆⠀⠀⠁⢀⢼⣇⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢠⡟⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜
⠧⠠⠐⠊⠀⣸⠏⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⣿⠃⠀⣸⣿⠟⠀⢀⣀⣠⣀⠔⠁⢂⡀⠃
⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⣼⠗⠀⠀⠀⢠⡿⠀⣴⣿⠁⡘⠂⠚
⠀⠀⠀⢀⣾⠁⡶⢠⣄⣀⣀⡠⣤⢶⢾⡆⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢀⡼⠁⠐⣿⡷⢠⣿
⠀⠀⠀⢸⡏⢰⡇⣾⢹⡇⢻⡇⠹⡎⣇⢻⡀⠀⠀⠀⢀⣿⠁⠀⠀⢀⣾⠁⠀⣠⣤⡄⠠⢿⡀
⠀⠀⠀⢸⡄⣾⠀⣿⢸⣿⠘⣷⠀⣿⢹⢸⡇⠀⠀⠀⣸⡇⠀⠀⠀⢸⣿⠀⠀⣀⣀⡀⣷⢸⣷
⠀⠀⠀⢸⡇⣿⠀⣿⠈⣿⡄⢻⡀⣿⢸⢸⡇⠀⠀⠀⣿⠁⠀⠀⠀⢼⣿⠀⠈⢉⡉⣱⠋⢸⣿⡄
⠀⠀⠀⢸⡇⣿⠀⣿⡆⢻⡇⢸⡇⣿⠸⢸⡁⠀⠀⢸⡏⠀⠀⠀⠀⠈⣿⡀⠀⢸⡇⢿⠀⣸⣿⣿⡄
⠀⠀⠀⢸⠁⣿⠀⢸⡇⠘⡇⢸⡇⢸⡄⢞⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⢸⡆⢿⣿⣿⠇
⠀⠀⠀⠧⢠⣿⡠⠾⡁⢰⡇⢸⡇⢸⡇⢸⠅⠀⢸⡏⠀⠀⠀⠀⠀⢘⣿⠀⠀⢸⣿⠊⣧⠈⢿⡏
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠸⣿⣇⣀⣼⡿⢧⡿⠄⠈
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠸⣅⣀⡀⠀⠀⠀⠀⠀⢹⣿⣷⡖⠒⠒⠠⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⡄⠀⠀⠆⠀⠀⠀⠸⠳⡀⠉⠒⠢⠤⠄⣀⣈⣙⡃⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠆⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢀⣀⡹⢆⣀
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⡸⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠶⣌⣠⠐⠢
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠐⠂⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠲⣎⣸⡒⢠
⠀⠀⠀⠀⠀⠀⠀⠸⢀⣶⣶⣴⡆⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠡⠘⠬⠋⠤⣀⠀⡄
⠀⠀⠀⠀⠀⠀⢠⣤⡿⢻⣿⣿⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⢰⠈⠦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢲⠚⣛⠠
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠋⠉⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡏⠛⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⣴⣀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⡄
⠀⢀⠀⢀⣉⠄⡠⠊⠀⠀⠀⠈⠀⢉⡝⠶⢄⡀⠀⠀⡜⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠐⠸⣍⡵⢃⠀⠀⠀⠀⠐⠠⠁⠀⠀⡀⠈⠐⠶⠁⠀⠀⠀⠀⠀⠇⠀⠀⢀⣀⡤⠐⠸⠉⡁⠀⠢
⠀⠀⠐⠀⠤⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⡀⠀⠣⠀⠀⠀⠀⠀⢛⠉⠱⠒⠒⠂⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⢉⣉⣉⣹⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣉⣉⣉⢹⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡟⠛⠻⠿⠿⠾⠿⠿⠷⠿⠿⠿⠾⠿⠿⠛⠛⣿⣾⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠰⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠐⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢠⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣿⣾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⢟⣭⣶⣶⣾⣭⡻⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⢱⡿⣉⣿⣾⣿⠿⠛⠘⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣇⠻⠟⠛⢉⣁⣤⣴⣾⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢀⣀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⡄⢶⣿⣿⣻⣿⣠⡿⣣⡿
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡿⣶⣽⣛⣛⣛⣭⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣻⣿⠉
⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣿⢻⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣼⣇⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣁⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠛⠁⠽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⢛⣒⣒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣥⣤⣴⣶⣾⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⢋⣤⣿⡟⢉⠀⠀⠙⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⣿⡿⠛⡻⠟⠉⠀⠀⠁⣹⣯⣿⡿⠿⢦⣤⠀⣠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡶⠾⠷⠈⠀⠀⠀⠀⠠⢞⡋⠉⠁⠀⠀⠀⠀⠛⢛⣋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣴⡶⠚⢻⣆⠈⠀⠀⠀⢀⡼⢤⣒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⠿⣿⡶⠖⠶⠖⠒⠒⢿⣦⠀⢀⣾⡿⠛⠬⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⣟⣻⢻⣿⣿⣿⡷⠀⠀⠀⢀⠀⠈⣿⣷⣼⣇⠀⣤⠀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣿⢿⣷⠛⢷⡄⠻⣧⠘⠹⣷⠋⣿⡆⣽⡟⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣽⣿⣿⣿⣿⡟⣿⡀⢸⡇⠀⣿⠀⠀⣿⠀⣿⠃⣽⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⣿⠀⠀⣿⠀⣿⠂⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇⠀⣿⠀⢀⡏⠀⣿⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢻⣿⢸⡇⠀⣿⠀⢸⡇⢰⣿⢸⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢈⣿⠀⣿⡀⢸⡇⢸⣿⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⢷⣼⣿⣿⣿⣿⣿⣯⣿⡍⣿⠀⢸⡇⢸⣧⢸⣿⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⢸⡇⢀⣿⢸⣯⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠔⢫⡿⣻⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢿⡆⠈⡇⠈⣿⢸⡗⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠊⣴⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡇⠀⠁⠀⠟⢸⡧⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣻⣿⣿⣿⣿⣿⣿⣛⣻⡿⠿⠸⠃⢠⡄⣀⣙⣨⡥⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣽⣿⣿⣿⡟⣿⣿⣿⣿⠿⠀⠀⠀⢰⣧⠈⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣟⣿⣿⣿⣿⣿⣆⡀⠀⠀⣸⣿⣤⣼⡷⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣤⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠿⠿⣿⣦⠀⢀⠴⠛⠉⠉⠉⠁⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡏⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣀⣀⠀⠀⠀⠹⡤⠴⠒⠋⠱⣄
⠀⠀⠀⠀⠀⠀⢸⣧⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⡿⠟⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⠟⠃⠈⠛⠿⣿⣿⣿⣷⣶⣤⡷⣄⠀⠀⠀⠀⠙⠲⣤
⠀⠀⠀⠀⠀⡤⠖⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠛⠛⠉⠁⠀⢠⣿⣿⡿⠋⢀⣾⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣿⠗⢤⡉⠓⠒⠒⠒⠋
⠀⠀⠀⠀⡞⠀⠀⣿⣿⣯⠉⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠠⠿⠛⠁⠀⠀⠸⠛⠋⠉⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠹⡄
⠀⠀⠀⢸⡇⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⢈⣩⣭⣿⣿⣷⣤⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⣶⣿⣿⣭⣍⣉⠀⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⡇
⠀⠀⠀⠈⣇⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠉⠻⠾⠏⠀⠈⠃⠀⠀⠀⣸⠃⠀⠀⠀⠘⠉⠀⠙⠾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⢠⠇
⠀⠀⠀⠀⠈⠢⢤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠦⠤⣤⣤⣤⣤⣤⣤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⢀⠈⠐⠆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠀⠀⠉⠈⢀⠀⠈⢀⠀⠀⠂⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠀⠀⠀⠀⠀⢀⠔⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠉⠁⠀⠠⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠒⠠⠄⠚⠠⢀⠀⡇⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⢡⠈⠁⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣂⣀⣀⣀⣀⣤⣤⡤⠄⠻⡃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠣⢄⡀⠀⠐⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠚⠉⠀⠀⠚⠹⠄⠀⠀⠰⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠚⠋⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠶⠀⠀⠀⠈⠂⠤⠤⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⢀⠀⠀⣉⠒⠆⠀⡶⢤⠠⡀⢀⠐⢄⠀⡩⠙⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠒⠳⠰⠀⠀⣀⢀⠀⠀⠐⠌⠀⠀⠤⠏⠀⠀⠀⠀⠔⢅⠐⠂⠄⠀⣀⠀⠒⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠁⠀⠀⠀⠠⠄⠀⠤⠤⠑⠀⠀⠈⠀⢀⠢⡀⠀⠀⠀⣀⢩⡂⠀⠀⠀⢭⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠒⠒⠂⠀⠀⠀⠀⠀⠀⠘⠡⣄⠀⠀⠈⠁⠙⡀⠀⣞⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢄⢀⣠⠀⠀⠱⢄⡀⠀⣀⠀⠄⠚⠀⠀⠀⠀⢈⡐⠶⠂⠠⠤⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⣫⠷⣄⠀⠓⠠⠄⡀⠀⠀⡀⠈⠆⠐⠁⠀⠐⠀⠴⠋⢳⠁⠒⠩⠛⠻⠿⣟⣯⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠷⢉⡤⠊⠁⠀⡆⠉⠉⣶⣄⢔⢯⢻⡇⠀⠀⠀⠀⠀⠀⢨⣁⢀⠀⠀⠀⠀⡔⠒⣲⡿⣯⢿⣆
⠀⠀⠀⠀⠀⢀⡠⣴⡒⣋⢡⣛⡉⠀⠀⡠⠄⢀⣠⣾⢟⣵⠏⠁⢸⡇⠀⠀⠀⠀⢀⢀⣨⣭⣼⢶⣦⣤⣤⣌⣦⣿⣋⢹⡎⢿⡟⣷⠄
⠀⠀⠀⠀⠀⠈⠂⠉⠉⢺⣿⣿⣷⡶⠐⠂⢶⡵⢟⠵⢋⡮⠀⠀⣽⠇⠀⠀⠀⠀⢨⡅⣾⣾⣿⣤⣤⣤⡬⣄⣥⣤⠤⢿⣧⡀⢩
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠄⠈⡉⠅⠀⠀⠀⠀⡇⠀⡈⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠓⠂⠂⠀⠈⢻⣿⡿⠷⢶⠇⠀⠀⠀⠐⢹
⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⡀⠰⠔⠀⠀⠀⠀⠀⡋⠀⣁⡴⠋⠀⠀⠀⠀⠀⠀⢀⡀⡃⢀⠀⠠⠐⠀⠍⠠⠸⣉⠀⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠤⢁⣈⠐⠀⠀⠀⠀⣃⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠇⢀⠑⠐⠀⠀⠁⠀⠀⠀⠀⠐⡞⢰⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⠊⠀⠂⠀⠀⠊⠀⠀⠓⠐⠐⠂⣸⡠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⠀⠒⠂⠈⠀⠤⠀⠐⠁⠂⠀⠀⠀⢀⠔⠂⠐⠂⠀⡠⠀⠂⠐⠀⠀⠡⠀⠐⠑⡀
⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⠀⠀⠀⣄⡀⠀⠀⠀⡌⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠺⢁⣂⢼⠀⠀⠀⢙⢈⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠰⠀⠀⠀⢰
⠀⠀⠘⠢⢀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⣀⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣄⣤⡼⡀
⠀⠀⠀⠀⠘⠛⣲⠀⠀⢸⠀⠀⠀⢰⠤⠜⠹⠀⠀⠀⣔⡖⠀⠀⠇⠀⠀⠀⠆⠀⠀⠘⠀⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢰⠀⢀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡤⠀⠀⡦⠀⣠⡆⡀⢀⣾⡦⠀⠀⣠⠄
⠀⠀⠀⠐⠤⡀⢐⠴⠞⠀⠤⠂⠒⠎⠀⢰⣴⠒⠴⠂⠀⠄⢔⣖⢽⣗⠲⠖⢢⢚⠛⠓⠛⠻⠛⠓⢷⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠛⠉⠛⠋⠙⠋⠉⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⣿⣯⣭⣭⣭⣭⠙⣿⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⠉⠉⠠⠾⠛⠀⠀⠀⠀⠀⠀⠛⠧⠀⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣶⣤⣤⣤⣀⣐⣒⣒⣒⣒⣀⣀⣤⣤⣤⣶⣾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠽⠟⣒⣲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣙⣛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠾⠿⠛⢛⣉⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⢻⣿⠛⠻⢷⣶⠶⠶⢶⣾⠟⠛⣿⡟⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡏⠀⠀⣿⡇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⠇⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⠀⠀⢸⣿⠀⠀⣼⡇⠀⠀⣿⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣿⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⡄⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣧⣤⣼⣿⣀⣀⣿⣧⣤⣼⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠲⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⢳⡀
⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣧
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠰⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡌⠀⠀⣀⡀⡀⠀⠀⠀⠈⡇
⠀⠀⠈⠙⠒⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⡇⢀⡜⠀⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⡸⠓⠲⢤⣤⣴⠆⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢸⠃⡼⠀⠀⡇⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⣠⣌⡉⠓⠚⠹⣅⡀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠸⠺⠶⣇⠀⠀⢹⡀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣄
⠀⠀⠀⠀⡌⠀⢀⡤⠖⠋⠀⠀⠈⠳⣄⢀⡷⣌⠉⠙⠒⠀⠀⠄⠀⣀⣀⠀⠀⠀⠈⠳⡄⠈⢧⡀⢰⣄⠀⠀⠀⠣⠤⠄⠉
⠀⠀⠀⠰⠀⠈⢃⠀⠀⠀⠀⠀⠀⢀⣨⡏⠀⠈⠳⣄⠀⠀⠀⠀⠀⡿⠂⠀⠀⠀⠀⠀⣿⠀⠀⠉⠛⠈⠳⢄⡀⠀⢀⡠⠔
⠀⠀⠀⡆⠀⠀⠀⠳⣤⣀⠀⠀⠀⠈⢹⡆⠀⠀⣾⣾⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠉⠓⠒⠋⠉⣇⠀⠀⡠⠊⠀⠀⠀⠀⡄⢀⣀⣀⣀⣠⠞⠁
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠀⠀⠀⠀⠀⠀⠉⠠⠀⢄⠈⡁
⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣀⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣤⣄⣀⣐⣀⣀⣤⡤⠤⠶⠒⠋⠉⠀⢳⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣀⣀⡀⣀⣀⣀⣀⣀⣀⡠⠤⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⣈⣒⡫⠭⠛⠳⠶⢖⣒⣚⣿⣋⣈⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⢀⣠⣴⣾⣿⣿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⡿⠿⠶⣤⣀
⠀⠐⠚⠉⠉⠉⣿⠛⠿⠿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⣤⡄⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⡿⠿⠛⠋⠉⢹⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⣿⡿⢿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣾⣿⣿⣿⣿⣿⣟⠋⠁⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘⠧⠴⠿⠋⠀⠈⠉⠙⢻⡏⢿⡋⠉⠹⣇⣀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⢻⡏⠉⠉⠀⠀⠀⠀⢸⡏⠉⠉⠙⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡍⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣥⣤⣤⠿⠤⠴⠶⣶⠶⠶⠾⠷⠶⠒⠒⠒⣶⠟⠛⠛⠛⠋⠉⠉⣹⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢰⣿⠹⣦⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣀⣀⣀⣀⣀⣀⣿⣄⣀⣀⣀⣀⠀⢰⡿
⠀⠀⠀⠀⣆⠀⠘⣿⠀⠈⢷⣄⣀⣤⠴⠶⠞⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠲⠦⠤
⠀⠀⠀⠀⢻⡄⠀⠙⠳⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠙⠶⢤⣤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠆⠉⠉⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠄⠘⠋⠉⠑⠻⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⣳⣤⣄⠀⢀⣴⣎⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢻⡿⠇⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄⠀⠀⠀⢠⠁⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠈⠑⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠆⢤⣤⣤⠀⠶⠆⢸⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡇⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⡀⠀⢠⣥⠀⠀⠒⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠎⠁⠀⠀⠈⠑⢂⡀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣿⠟⢣⡀⠂⢀⣤⠰⠂⠠⣀⠈⢢⡀⠀⠀⠀⢀⠘⣹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢪⡑⠦⡀⠀⠀⠉⠑⢦⡄⠀⠀⢠⠌⠀⢠⣿⠀⠀⢹⢠⡉⠀⠀⠀⠀⠈⠳⣀⠻⡀⢠⠶⠉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡰⠁⠀⠀⠈⠱⢮⡑⢂⠀⠀⠀⠈⠑⢦⡁⠀⢀⣾⣿⠀⠀⠘⣾⡁⠀⠀⠀⠀⠀⣠⣷⠀⢃⣤⠷⠊⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⡄⠒⠢⡀⠀⠀⠈⠓⢬⡑⢠⡀⠀⠀⠘⠳⣬⡿⣿⡇⣠⠈⠀⠀⠀⠀⠀⢀⣄⠸⠄⠀⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠰⢡⡟⢀⡔⢀⠔⡀⠈⡆⢀⡀⠀⠀⠈⠑⢮⡔⢠⠀⠀⠀⠙⠷⣦⡈⠁⠀⠀⠀⣀⡘⠓⠒⠒⠈⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⡌⢠⠋⢠⠂⠀⠑⠂⠂⢠⡑⠀⠀⠁⠢⡀⠀⠀⠈⠑⢪⡕⣠⠀⠀⠀⠉⠓⢢⣄⢸⡟⠘⢷⡀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⢀⠘⡰⠃⡰⠂⡄⢻⠀⠴⠊⡑⠁⢴⠉⠒⠠⠂⠀⢀⣄⠀⠀⠈⠑⢮⡑⠂⠀⠀⠀⠈⠙⢦⡀⠀⠙⣦⡀⢀⠆
⠀⢀⠂⠉⢫⡴⠁⠀⠓⠒⠀⠈⠀⡀⠀⢀⢤⡄⢉⡄⠀⠀⡰⠂⣈⠳⠆⣄⠀⠀⠈⠑⢮⡕⠢⡀⠀⠘⠈⠑⢠⣈⠻⠎
⠀⢼⡀⠀⡜⠓⢤⡀⠀⠀⠀⠒⢰⡏⠀⠂⡄⠈⣁⡀⠀⡰⠁⣰⠆⠀⠀⠀⠁⠀⠀⠀⠀⠈⠑⢮⡑⠢⡀⠀⠀⠈⠑⠂⡀
⠧⡀⠉⠁⠀⠀⠀⠈⠐⢦⣄⠀⠀⠈⠒⢤⢀⡿⠁⢸⠆⢀⠲⠉⠀⠐⠀⠀⢀⠄⠂⠀⠰⠀⠀⠀⠈⠑⢮⡑⠢⠄⠀⠀⠀⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄⠀⠀⠀⠘⢤⡄⠐⠠⠃⡐⠁⠂⣄⠾⠁⡰⠈⢱⠀⣤⠂⠐⢄⠀⠀⠈⠑⢦⡉⠲⣄⠀⠀⠉⠁⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠘⠣⡀⠈⠒⠀⡘⠁⠞⠁⠀⠈⠈⠁⡤⢀⡀⢉⡶⢠⠀⠀⠈⠑⢦⡉⠲⣄⡀⠀⠀⠙⢢⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠙⠲⣤⡀⠀⠈⠑⠒⠀⡁⠘⠄⠐⢂⠀⢀⠜⠀⠎⢀⡛⢠⡄⠈⠒⡄⠀⠀⠈⠑⢦⡉⠂⢄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⡀⠀⠀⠑⠂⠀⠒⠁⢠⠂⠠⠁⢀⠎⠀⠣⡈⠑⢺⠂⠉⠒⢀⠀⠀⠈⠑⡌⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠀⠀⠐⢦⠔⠀⡀⠀⡀⠒⢤⠀⡐⠀⡄⠁⠄⢸⠀⠀⢠⠞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠗⢤⡀⠀⠀⠀⠐⢅⠀⠑⠂⣀⠐⢠⠌⠀⠘⢠⠄⠀⡰⢃⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠑⠦⣀⠀⠀⠉⠒⠰⠏⠀⠃⠄⠂⡠⠁⠀⠞⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠈⠒⠀⠐⠊⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠢⣄⡀⠀⠀⠀⣴⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠉⠲⣤⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠬⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠤⠤⠤⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣂⣐⠒⠒⣢⢱⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡂⠁⠀⡠⠀⠀⡠⠂⠀⠀⠐⢄⠀⠠⢀⠀⠀⢂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢛⠁⠒⠂⠀⠀⠀⠤⠤⠬⠤⠤⠤⠤⠤⠤⠥⠤⠤⠀⠀⠐⠒⠒⠈⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠠⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠄⡀⠀⠀⣀⡀⠀⠀⠐⠀⠐⠂⠀⠀⣀⣀⠀⠀⢀⠤⢤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡀⠂⢤⠀⠀⠀⡇⠀⠈⠀⠀⠇⠀⢸⠀⠀⠆⠀⠀⠀⠀⠃⠀⠀⠀⢠⠀⡤⠐⢂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⠈⠲⠀⡀⠀⡇⠀⠀⠀⠀⠄⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠖⠁⡘⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡑⢤⡀⡇⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠁⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇⠀⠀⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡇⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⡇⠀⡁⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡘⠤⠇⠀⠀⡇⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠠⠄⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⣁⠒⠤⠀⠁⠒⠁⠀⠀⠄⠸⠀⠀⠈⠒⠁⡀⠤⠒⣈⠤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢈⡑⠂⠀⠠⠌⠉⠉⠁⠈⠉⠉⠥⠄⠀⠐⢈⠡⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠖⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠰⠒⠙⠁
⠀⠀⠀⠀⠀⠀⡠⣴⠦⠤⣀⠀⠀⠀⠀⠀⠀⢠⢤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣄
⠀⠀⠀⠀⠀⢸⡞⠀⠀⡄⠀⢹⠋⠉⠉⠉⠂⡜⠀⠀⠀⠀⠸⠙⠉⠉⠑⡈⠉⠉⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠈⢇⠀⡰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠈⠠⠀⠀⠙⢦⡀⠀⠐⠇⠀⠀⠸⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠠⠀⢰⠀⠀⠇
⠀⠀⠀⠀⠀⢘⠛⠉⠙⡆⠀⠐⠀⠀⠠⠀⠒⠀⠀⠀⡀⠀⢰⠀⠀⠸⠀⢀⠀⢸⠀⠀⢐
⠀⠀⠀⠀⠀⠘⣍⠀⠀⠃⠀⠨⠀⠀⠀⠀⣆⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠋⠁⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠂⠃⠀⠁⠀⠀⠂⠃⠄⠀⠀⠀⠀⠂⠐⠈⠐⠂⠐⢅⡀⢠⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠂⠙⠀⠀⠀⠀⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⢸⣿⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣷⣦⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠈⠉⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⣀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣦⣀
⠀⠀⢀⣠⣾⠿⠋⠀⢸⣿⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣿⡇⠀⠙⠿⣷⣤⡀
⠀⠀⢸⣿⣷⣄⠀⠀⢸⣿⠀⢸⣿⠀⠀⢀⣀⡀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⣀⡀⠀⢀⡀⠀⠀⣿⡇⠀⣿⡇⠀⠀⣠⣾⣿⡇
⠀⠀⢸⣿⠉⠻⣿⣦⣸⣿⠀⢸⣿⠀⢰⣿⡟⠿⡷⠀⣿⡟⠛⣿⣧⢰⣿⠟⢻⣿⡆⣾⣿⣷⣴⣿⣿⡇⠀⣿⡇⠀⣿⣇⣴⣾⠟⠉⣿⡇
⠀⠀⢸⣿⠀⠀⠈⠛⢿⣿⣄⢸⣿⠀⢈⡛⠿⣷⣦⠀⣿⡿⠿⠟⠃⢸⣿⠿⠿⣿⡇⣿⣿⠙⠿⠃⣿⡇⠀⣿⡇⣀⣿⡿⠛⠁⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠘⠿⠿⠿⠋⠀⠿⠇⠀⠀⠀⠸⠿⠀⠀⠿⠇⠻⠗⠀⠀⠀⠻⠇⠀⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⢀⣠⣶⡿⠿⠛⠛⠿⢿⣶⣄⡀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣄⡀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠐⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠂⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⡄⠀⠀⠀⠀⠸⢲⠂⠀⠀⠀⠔⠇⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⣉⠁
⠀⠀⠀⠀⠀⠀⠀⣹⠛⠉⢀⢴⠟⢂⡀⠀⠀⠀⠀⢠⡤⠹⠀⠀⠀⠀⠀⠀⠉⠀⠠⡀⠀⠀⠀⠀⠉⠉⠉⢁⠆
⠀⠀⠀⠀⠀⠀⠼⠇⠖⢙⡁⠀⣻⠁⠈⣄⠀⠀⠀⢾⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⠄⣀⣀⡠⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠘⡉⠘⣀⣘⡀⠈⡵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢡⡠⢷⡖⢋⣈⣓⡖⠜⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠜⠀⠀⠀⠠⢄⡤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠐⠚⠳⡀⠘⣅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⢀⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠃⠂⠘⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠄⠰⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⠘⠀⡀⠱⡇⢑⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⡄⡠⠤⠔⡦⠠⠤⣤⢐⠣⣐⣞⢲⠀⢹⢀⣬⣟⣿⠦⠐⣅⠀⢔⢢⣐⣢⢄⣒⡩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠘⠺⢖⠓⠂⡢⠔⡧⡤⢔⡛⢼⠀⠘⠋⢹⠄⡂⠉⢀⣠⠂⠈⢢⡏⠣⢤⠐⠀⠐⠠⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠒⠂⠁⠈⠷⠒⠀⠰⣀⠠⠟⠀⠒⠄⣀⡾⠒⠤⠊⠍⠁⠈⠻⢄⠂⠀⠀⢐⣄⡀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠠⢎⠀⠸⠄⠀⠠⠤⠀⢠⠀⠂⠀⢀⡀⠘⠓⠂⠀⠀⡂⠀⠂⠀⣐⠆⠀⢐⠄⠀⠀⠔⠁⠬⠍⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠐⠂⠀⠈⠉⠀⠀⣤⠀⠀⣰⠀⠀⠀⠀⠀⠉⠀⠀⢌⡓⡠⠄⠀⠀⡰⣄⠀⠑⠀⠐⠀⢀⠄⠊⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠺⡓⠒⢈⠀⠒⠂⠀⠦⢄⣀⠘⠁⠀⠀⠀⠤⠤⠤⠆⣄⡠⢌⣀⡀⢠⠄⠀⠈⣅⠀⠀⢄⠀⠁⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠒⠠⠔⢀⠖⠀⠀⡀⠀⢀⠀⠀⢀⠀⠀⠐⣠⠀⠀⡜⠀⠘⠂⢤⠴⠀⠳⠤⢒⡉⠀⢈⡿⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠊⠉⠀⠈⠒⠀⠒⠉⠉⠉⠈⠋⠉⠀⠐⠀⠀⠘⠂⠐⠀⠀⠐⠑⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣠⡾⠁⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠓⠶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡏⠀⢹⠀⠀⠀⠀⣸⠃⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣄⠀⠀⠀⡾⠀⠀⠈⢧⡀⠀⠀⣿⡤⠞⠁⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⡗⠤⠾⠃⠀⠀⠀⠀⢻⡄⠀⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⣰⠟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⢷⡄⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢾⡋⠁⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠈⣷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠆⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡞⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠠⠟⠒⠛⠉⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠳⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣤⣤⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡄⢸⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣠⢃⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⣴⡿⠛⢯⡉⠻⠏⢸⡇⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢰⠃⠀⢡⡇⠀⣿⣻⣦⡤⢷⣤⣾⣟⣷⠀⠈⡧⠀⠀⢸⡇⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⡟⠀⠀⠈⠓⠦⢤⡥⠟⠀⠀⠈⠷⠬⣅⣤⠞⠃⠀⠀⢸⡇⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠇⠀⠀⠀⠀⢀⡤⠶⠲⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻⠶⠾⢷⠀⠀⠀⠀⢀⡏⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣷⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠚⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣶⠷⠖⢲⣢⡀⠀⢰⠁⠀⠀⠀⠈⡂⠀⠀⠀⠠⡶⠤⠀⠀⠁⠀⠀⠀⠀⢀⡔⠁⠈⠢⠀⠀⠀⠀⡷⢺⣄⠀⠚⢺⡆
⠀⠀⠀⠀⢈⣻⠕⡶⡒⡦⡽⣓⠀⢸⠀⠀⠀⢀⡔⣓⡢⡀⠀⢀⣠⣤⣤⣄⢤⣤⡀⠀⠀⠘⢆⣂⣴⣶⠷⡀⠀⠀⡿⣀⣤⣤⣄⣸⣇
⠀⠀⠀⠀⢈⣭⠘⡙⣉⠀⢥⣭⠀⢸⢀⣴⣿⣏⢃⡥⢠⡇⠀⠈⠻⠉⡒⢈⣠⡠⢃⠀⠀⠀⠀⢹⠷⠦⠟⠁⠀⠀⡿⢈⣿⣿⣟⣻⣭⡇
⠀⠀⠀⠀⠈⠻⢗⡤⣤⣶⠗⠁⠀⢸⠀⠀⠀⠈⠛⡟⠋⠀⠀⡀⠘⣛⣃⠉⡥⠐⠊⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣷⣈⠉⠉⢉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣈⣀⣁⣈⡀⠀⠀⠀⣀⣈⣉⣉⣀⣁⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡀⠈
⠀⠀⠀⠀⠀⢠⠔⣂⢄⠀⠀⠀⠀⠀⡏⢉⣀⣀⠀⣡⡀⠀⠀⠶⠒⣚⡒⠒⠒⠒⠲⠀⠀⠐⠷⠆⠉⠛⠒⡆⠀⠀⠀⠀⣾⠒⠙⡆
⠀⠀⠀⠀⠀⠘⠀⡀⡏⢀⢀⣀⠀⠀⡗⣾⡷⢺⣟⣌⠼⠀⠀⠀⠐⠛⢃⣭⡙⠁⠀⠀⠀⡀⣰⡞⢋⠛⣿⠃⠀⠀⠀⠀⣫⡦⣶⣅
⠀⠀⠀⠀⢀⠀⠾⢿⢛⣷⣛⠛⠇⠀⡟⢿⠿⠞⣋⡃⠀⠀⠀⠀⢯⣸⣟⣽⣭⡿⠀⠀⠀⡇⠚⠷⠤⡶⡣⡀⠀⠀⠀⡮⡽⢖⡠⡇⠇
⠀⠀⠀⠀⠥⠤⠨⠿⠟⠁⠬⠷⠄⠀⠇⠀⠀⠘⠧⠇⠀⠀⠀⠤⠯⠠⠯⠽⠯⠽⠤⠀⠀⠣⠠⠀⠸⠼⠏⠚⠀⠀⠀⠢⠤⠨⠱⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⡀⠀⢀⡀⣀⣀⣀⠀⣀⣄⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢾⡏⢸⡷⠶⠀⡀⢸⡇⣿⡷⠶⢸⣏⠈⣉⠈⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠛⠘⠛⠛⠃⠙⠛⠁⠙⠛⠛⠀⠙⠛⠉⠀⠘⠁

⠀⠀⠀⠀⣒⣒⣲⣒⣒⣒⣒⣒⡄⢀⠤⡠⠤⠤⠤⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠲⢒⠒⠀⠀⠀⠀⢀⣀⣔⣢⣀⣀⣀
⠀⠀⠀⠀⢡⣋⣉⣉⣉⣉⣉⣦⡁⢸⠊⠁⠂⣀⣀⠂⠑⡆⠀⢄⡀⢀⠀⠀⠀⢀⣠⠀⠀⢸⠸⠾⠌⢀⡀⠀⠀⠀⠸⡕⢾⣻⣏⠢⢸
⠀⠀⠀⠀⢸⠩⠛⠍⣩⣩⣉⢹⡀⢸⠀⢴⣾⢿⣛⣥⡀⡇⠀⢁⠄⢄⠀⠤⠤⠤⠈⠀⠀⢸⢩⢭⡿⣟⣫⠄⠀⠀⠀⡀⠢⠉⠫⢁⡍
⠀⠀⠀⠀⢨⠐⠒⠂⣈⣉⣉⢸⡃⠘⠈⠋⠉⠙⠻⣿⡗⠂⠀⠘⠌⠔⠀⠉⢡⠤⡄⠀⠀⢸⢸⣖⣿⢿⠋⠀⠀⠀⠀⢁⠐⠉⠊⢁
⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠀⠀⠀⠈⠉⠈⠀⠀⠈⠉⠈⠁⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⡀⠀⠀⣀⢀⣀⠀⢀⠀⢠⣶⡶⡄⠀⢀⢴⣶⡄⠀⢀⡴⠼⠟⡛⣧⡦⡀⠀⠀⡴⢑⡶⣴⡂⠒⣶⡀⢠⠀⠼⢄⠔⢢
⠀⠀⠀⠀⠂⡤⡉⣭⠍⠉⠠⠐⡇⠀⢿⠀⡈⣤⠁⠀⡸⠀⠀⢲⠁⣼⢀⠮⠻⠗⡔⠀⠘⢫⣣⡏⢩⡶⡴⡎⠀⠸⣀⠉⠩⠉⠩⢈⡍⠉
⠀⠀⠀⠀⠀⠤⠤⣠⣴⣾⣭⠕⡁⢸⠰⠌⠋⠀⡿⠣⠔⠃⠀⠜⢴⡯⠓⠫⠀⢀⡣⠀⠀⢸⣿⡠⠿⠄⡛⠄⠀⠀⠀⢄⣼⣶⣌⡁⣀
⠀⠀⠀⠀⠈⠉⠁⠛⣽⣰⠟⠉⠀⠸⡀⡀⠔⠈⠢⢄⢀⠄⠀⠈⠢⢺⣧⣤⡿⠕⠁⠀⠠⢿⠟⠀⠀⠈⢣⠖⠀⠀⠀⠘⢵⣲⠇⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣤⣤⣄⢠⣤⣤⣄⠀⢠⣤⡄⠀⣤⣄⠀⠀⣤⡄⢰⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⢷⣤⣅⢸⣧⣴⡿⢀⣿⠙⣷⠀⣿⢿⡄⣼⢿⡇⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠰⢶⣼⠟⢸⡇⠀⠀⣼⠟⠛⢻⡧⣿⠈⢿⠏⢸⡇⢰⡆⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣀⣴⣶⣤⡀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⣾⣿⣿⣿⣿⣿⣶⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣤⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⢿⣿⣿⣿⣿⣿⣿⠃⣼⢣⣶⡹⣆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⡿⢁⡾⣳⣿⣿⣿⡝⣦
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⡾⣵⣿⣏⠀⣿⣿⣜⣧⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣟⣼⣿⣿⣿⠀⣿⣿⣿⣎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣾⣿⣿⣿⣿⣴⣿⣿⣿⣿⣯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⣾⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣧⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠤⠤⣴⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠚⡓⠲⠚⠁⠠⣤⣴⣶⣿⡷⣸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⡴⠾⢿⡿⠟⣡⠔⣂⡀⠉⠉⠰⠶⠿⠟⣋⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢠⢏⡔⠂⠀⠀⣼⠇⣾⣿⣿⣶⣶⣶⠄⡀⢤⣭⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠁⠀⠀⢿⢸⡄⣿⣿⣇⢻⢸⣿⣿⡿⠿⢿⣿⡀⠙⠲⠾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠰⠖⢦⣀⠘⠇⠙⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⢤⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⢀⣼⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢹⠀⠀⣿⣄⣹⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⣧⠀⢮⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣀⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⢙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠙⣿⣿⣶⡗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣏⣿⡇⣠⣿⡀⠀⢀⣤⡤⢤⣤⣀⣀⠀⠀⠀⠀⢠⡶⢋⣡⣶⠿⠷⠖⠀⢻⣯⣽⣳⣼⣿
⠀⠀⠀⠀⠀⠀⠀⣉⣿⠠⣿⣿⣧⠀⠋⠀⠠⢤⣤⣬⣉⠛⢶⡀⠛⣾⠀⠋⠙⠻⠶⠀⠀⠀⠈⣿⣬⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡿⢻⣷⠙⠿⠛⠀⠀⠸⠿⢛⣿⡧⠌⠁⠀⠁⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠸⡇⠘⢯⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣶⣥
⠀⠀⠀⠀⠀⠀⠀⣽⣦⣼⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣀⣀⢀⡶⠇⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⣾⡿⢹⡿⣠⣽⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⢸⣿⣏⣿⠁
⠀⠀⠀⠀⠀⠀⠹⣷⣜⣧⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠾⣿⡷⠶⠀⠀⠀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⡽⣦⠀⠀⠀⠀⠀⠀⠉⠁⠲⠶⠿⠛⠁⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⡌⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⢀⣀⣴⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉⠁⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠿⣫⣷⣶⣶⣯⣽⣛⣛⣟⣛⡋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⢨⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⢸⣿⣿⣿⣷⣶⣶⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣾⣿⣴⡎⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢟⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣱⣿⣿⣿⡇⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣧⣿⣿⣿⡿⣱⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠝⠛⠛⠛⠽⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀



⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣻⣿⣟⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠠⠀⠒⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⡎⡁⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠘⠁⠂⠂⠉⠂⢀⣰⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠬⢤⢤⣤⠤⠴⢶⡖⠂⠫⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠈⠰⠤⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠈⠉⠀⠀⠀⠉⠸⠂⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠀⠖⠀⠀⠀⠈⠄⠀⢠⢀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡈⠃⠀⠀⠀⠈⠐⠒⢂⠀⠀⣄⢀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⢀⠠⠌⠘⠀⠘⠟⢄⠢⠄⠀⠰⢂⠔⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠨⠁⢁⡁⠀⠠⠄⣆⠀⠀⠁⠂⠀⠈⠓⢀⠀⠀⢀⠊⠢⡀⠁⠐⠂⠦⠀⠉⠀⠒⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠉⠈⠀⢀⡀⠠⠐⠚⠢⠀⢠⠄⠒⣰⠀⢀⠀⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⠈⠁⠀⢒⠀⠀⠁⠒⠠⢀⡀⠀⠀⢂⡐⠃⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡇⠂⢠⡊⠀⠀⠉⠂⠀⠠⡂⠈⠀⠀⠀⠀⢀⡴⡂⠉⠀⠠⠭⠄⠄⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⢊⡵⠋⢙⠠⠤⣀⡁⠐⣠⠤⣤⠀⠠⠄⠀⠀⠀⢠⠌⠀⠘⢆⠀⠀⠀⠀⢉⡆⠣⢁
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⣫⡥⠞⠁⠀⠀⠘⠀⢀⣼⡿⣬⠏⠂⣿⠀⠀⠀⠀⠀⠀⢈⣶⣤⣀⠀⠀⠈⠡⠀⣼⠉⢧⢀⢦
⠀⠀⠀⠀⠀⢐⠘⠻⠶⣶⣽⣶⡄⣀⣐⣈⣠⣲⡿⡫⢊⠎⠀⢠⡷⠀⠀⠀⡀⠀⠆⣸⣽⣿⠑⠋⠓⠒⠒⠚⠉⠂⣼⡆⠜⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⠿⠿⠁⠀⠈⠛⢹⠪⢃⠎⠀⣠⡿⠁⠀⠀⠀⠀⠈⢏⠙⠛⠛⠛⢿⣿⣯⣿⣹⠏⠉⠙⠋⢁⢇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢈⢌⠁⠀⠀⠀⢸⠈⠘⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠿⠋⠠⠍⠀⠤⠀⠤⠀⢸
⠀⠀⠀⠀⠀⠘⠂⠤⢀⡐⠐⠌⠀⡀⠀⠀⠀⢸⢀⡰⠋⠀⠠⠀⠀⠀⠀⠀⠀⠄⠆⠈⠢⢈⠀⠀⠀⠀⠀⠁⠀⢠⠰⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⠀⠠⠼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠠⠄⠀⠀⡀⠀⠰⠈⠀⠇⠂⢸⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡡⠚⠉⠉⠉⠉⠒⢤⡀⢰⡕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠁⠆⠀⢀⠖⠚⠓⠢⢵⠀⠐⠫⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠸⡀⢠⠎⠐⠃⠀⢉⠀⡇⠠⡄⢒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⠀⡟⢳⠇⣤⠘⠀⠀⠀⠀⡇⢈⢡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠦⠄⠈⣆⡼⢲⢄⣰⢣⣠⠒⢲⡦⡃⡏⢿⢸⢾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⣤⣧⠙⢘⠜⡻⢹⠉⢯⠙⢦⠈⢘⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠦⠀⠸⠀⡧⠊⠀⢸⢦⣘⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠢⠤⢾⠁⠀⠀⡜⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠇⠀⠀⠀⣇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠫⣑⡒⠃⠀⠀⠀⢀⣼⠑⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣒⠯⠤⢭⡭⠭⣗⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⡞⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣇⠀⠀⡇⠀⠀⠀⡇

⣿⣿⣿⣿⣿⣿⡿⠟⠃⠾⠟⣿⣻⣿⡿⠿⢿⣿⣿⣧⢹⠻⠷⣏⣛⣦⣍⠐⢰⣨⡯⣯⣽⣏⣿⣷⣿⠗⠸⣗⣞⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣿⣷⣾⣿⣷⣏⠻⢿⡿⠇⠘⠙⢃⠈⠻⠿⢞⡧⣾⣿⣶⡉⣿⣿⣿
⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡋⣯⣭⣿⣾⣶⣦⣿⠹⢳⣾⡸⢋⣄⠻⣷⢸⣷⣿⣶⡈⣿⡿
⠃⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣍⢻⢻⣿⣻⣿⣿⣿⣁⢘⢫⣝⣻⣍⣻⣦⠀⠀⣽⣏⣿⡌⢹
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣹⢿⣿⣟⣿⡿⣵⣿⣸⡇⣮⣿⣿⣿⣭⠘⢶⣌⡛⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⣿⣟⠿⣶⣤⣿⠹⣿⣿⣿⣿⣾⡟⢸⣷⣾⣟⢮⡛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⢟⣰⣷⣿⣿⢿⣷⣿⣽⣿⣿⣿⡖⣽⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠶⣶⠶⡶⠀⢀⣼⣿⡳⢻⣿⣿⣻⡟⢿⣿⣬⣽⡿⣿⢻⣿⣥⣿⣿⣷⣽⠛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢸⣿⡿⢰⣾⡿⣋⣻⡿⣷⣘⠻⢿⣿⡿⣼⣿⠛⢿⣿⣇⣴⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣧⠀⠀⠀⠸⠏⣴⢺⣿⣿⣿⣿⣿⣷⣼⣃⢘⡿⢧⣻⣿⠁⣿⣾⣏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⢿⣿⣿⢿⣿⡛⣿⢿⣧⣸⣷⣾⠀⡽⠒⢻⣿⣯⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡟⣿⣻⣏⠙⣨⢹⣧⣦⡈⢻⣿⣷⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣉⣣⣹⣷⠿⢾⡵⠂⣰⣿⡙⢿⣧⣾⣿⡛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢀⡀⢀⣼⣿⣿⡟⢫⣶⣾⡇⣼⢿⣿⡟⣦⠙⣿⣿⣿⣦⣉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣾⣿⡿⢿⣷⣿⡎⣿⠈⡆⣿⣿⢾⣅⡿⣿⢿⣿⣿⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣫⣴⢿⣽⣿⣧⡀⢾⠰⠿⢃⣽⡽⠇⣀⡰⣿⣾⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣊⡿⣿⣷⠷⣾⠿⡛⠋⢠⠠⠀⠸⠋⣥⣾⣏⢻⣿⣽⠏⠙⣛⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠁⠼⢛⣚⣉⠁⠀⠸⣀⣟⡀⠀⠀⠶⠆⠾⢨⣯⣏⢨⣭⣼⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣻⣋⠃⢠⣞⠿⢻⣩⣤⣴⣶⣯⠘⠇⣀⣹⡁⡆⣀⡘⣻⡿⠘⠿⣧⡶⢡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢈⠹⢷⠔⢻⠟⣾⢻⣽⣿⣾⣦⠟⣿⡆⠉⠸⣣⣼⣿⣦⣷⠆⠀⣾⡿⡇⢨⣦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠁⡜⢻⠟⢼⣟⣭⡙⣿⡾⣇⣾⣿⣷⡏⣿⣮⣛⢹⣷⡆⢞⢠⣷⣿⠇⢘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣦⣿⡆⠉⠩⣿⣿⣿⣯⡙⠣⢾⣿⡏⢇⣿⣿⠟⠀⠉⣾⠉⢸⣧⣿⣷⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡿⣿⠰⠆⠉⣋⣥⣾⡧⣔⢻⣿⡇⡀⢄⢰⣶⣾⣤⢰⣤⣼⣯⣿⡇⣿⣟⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣙⣠⣭⣤⡝⠶⢿⢿⡽⠛⣡⣦⣼⣿⣿⡟⣡⢸⣿⣿⣿⡧⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠉⣣⡀⠐⠨⠈⣵⡿⣿⠿⣛⣿⣿⣿⠙⡄⠻⣛⣥⡄⣾⡷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣭⣝⣷⡀⠚⢷⡟⠿⢗⣭⠹⠿⠟⡀⠃⡀⢿⣿⠃⣿⣱⣾
⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣯⣽⡲⣿⣿⣿⣿⠿⠿⡗⣈⠂⠛⠀⠉⠐⣒⡛⢃⠀⣼⣌⣿⣅⣿⣾⣿
⣞⣿⣿⡏⣿⢷⣄⡀⠀⠀⠀⢀⢠⣴⣯⣛⣛⣫⣭⡤⣄⠀⠈⡁⠤⠶⣶⣷⣬⣰⣿⣷⣽⣛⣿⣤⣿⣿⣾⣹⣿⣿⣿
⠛⢛⠛⠓⢙⣛⣹⣿⣶⣶⣾⡿⢿⣿⣿⣻⣿⠶⠛⢩⣔⣿⣷⣶⣶⢦⣙⣛⠿⣿⣿⣟⢰⣿⣧⠙⣿⣿⢿⠏⢻⣿⣿
⢠⢈⡆⣠⣻⢸⡿⣿⣿⣿⣿⣼⣿⠿⣛⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢳⣿⣿⣛⣿⣽⣿⡄⣷⣾⡘⢻⣿⣿⣿
⣄⣨⣀⣋⠉⣾⡾⣿⠿⠟⡁⢭⣶⣞⣿⢟⣾⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⡛⣿⢳⡾⣿⣿⣿
⣈⣉⢁⣿⠀⣼⡆⣴⡶⣴⣶⣿⣿⣿⣝⣽⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣆⣾⣦⣟⣿⣿
⣿⣒⣞⣿⣧⣿⣷⣜⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡆⣽⣿⣭⣭⣽⣭⣩⣭⣿⣹⣿⣷⠰⣎⣷⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣾⣿⣿⣷⣿⣾⣦⣾⣿⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡛⠉⠉⠙⢿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠈⠻⣿⣿⣿⠟⠉⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⢀⣤⡀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⠀⠀⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠐⣿⣿⣿⣦⣀⠀⠀⠀⣠⣶⣿⣿⣿⡇⠀⠀⠈⣿⡇⠀⠀⣠⠴⠢⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⡰⠛⣛⣿⣿⣿⡷⠀⠸⣿⣿⣿⣟⡋⠳⡄⠀⠠⠟⡳⣄⡠⠕⠀⠀⠀⠚
⠀⣀⣠⠴⠤⡀⠀⢀⠜⣛⠻⡀⠀⠀⡇⢾⢠⣴⣮⢏⠀⠀⠀⢫⠀⢷⣾⠇⠀⣿⠀⠀⠸⠏⣹⠊⠉⠯⣤⣴⠟
⢾⣁⠀⠀⠀⠉⣀⢽⠲⡋⡇⠀⠀⠀⠙⢬⣿⣯⣥⠎⠀⠀⠀⠘⢷⣄⡀⠤⠺⠃⠀⠀⢰⠚⠁
⠀⢧⣀⡀⠀⣹⠆⠈⠓⠶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⣤⣤⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠁⠐⠒⠂⠬⠿⠟⠢⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠈⠙⠿⣶⣤⡴⠂⠀⠙⠻⣿⣄⠀⠀⠀⢈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡈⣿⣿⣁⣀⣀⣀⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠿⡿⠿⠿⠿⠿⠟⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⡠⢴⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡖⢉⣁⣰⠶⠶⢞⣖⠀⢀⣴⣶⣦⠀⠏⡀⡈⡣⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⢱⠀⠀⠻⣜⡾⢐⠂⠈⠅⠈⣿⢄⠈⣟⣯⡸⢰⠤⠥⠥⢄⣀⠁⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠘⠤⠐⣜⠆⢀⣠⠴⠿⠥⡄⢠⢒⣢⣄⣼⣋⣴⡁⠀⢹⡥⠔⣚⢦⢤⡈⠳⡄
⠀⠀⠀⠀⢠⠒⠒⢦⠀⢀⠔⣉⡤⢄⣀⣀⣩⠞⢉⣄⠀⣀⡀⠀⠉⢇⣸⠁⠐⠊⠀⣐⠙⢦⣽⠀⢠⠔⠲⣄
⠀⠀⠀⠀⠻⠀⠔⢊⣀⡜⢀⠇⢀⠀⠋⠄⢣⡔⠉⠬⠙⠃⠭⠏⢹⣈⠿⡀⢾⡛⢢⠀⢀⡼⡯⠀⠀⠔⡄⠸⠂
⠀⠀⠀⠀⠀⡠⠊⠁⢀⢙⣜⡄⠑⠀⢀⡀⢇⣇⠀⠉⣀⣀⡈⠀⢸⡯⠂⠈⢆⣉⣥⡴⠛⠛⠓⠒⠦⣴⠄
⠀⠀⠀⠀⡼⠀⡞⣉⡛⠽⡘⠿⣦⣀⣉⣠⢴⠺⢄⣸⣍⣩⣽⣀⠾⠲⣄⣾⠋⢣⡼⡧⠖⢊⡹⣤⣄⠈⠳⣄
⠀⠀⠀⠈⢳⣸⠁⠰⠀⢀⣤⡄⣸⣟⣿⠋⠀⠀⠈⢏⠑⠀⢫⠇⠀⠁⠈⢻⣧⡤⢻⠁⢀⣁⡀⠐⡊⢱⡄⡏
⠀⠀⠀⠀⠀⠹⣷⣄⡸⣟⣭⡾⠚⡽⠁⢀⡴⡆⠀⠀⠉⠋⠁⠀⠀⡗⣄⠀⠙⣆⡸⢦⡯⣙⠛⡆⠀⣰⢻⠁
⠀⠀⠀⠀⠀⠀⢀⣴⡉⠻⡃⠀⣠⡇⠀⣎⠀⢹⠀⠀⠀⠀⠀⠀⢰⠃⠈⡇⠀⡾⠀⠀⠧⡤⠿⢥⡞⠉⠁
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠶⠬⢯⣤⣿⣀⣿⠷⠾⠤⠦⠤⠶⠦⠶⠼⠦⢶⣇⣰⢧⣤⡤⠤⠤⠦⣀⡇
⠀⠀⠀⠀⠀⡠⠛⠁⠀⠀⠰⣿⣿⣦⠤⣜⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠭⠥⣜⣿⠷⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠛⣋⣭⠉⢡⡦⣤⠈⢭⣭⡙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢟⠉⠀⢿⡄⢻⡶⠂⢸⡇⣤⠀⣸⡇⢀⣿⡛⠆⢙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠰⣏⣥⠀⣬⣿⢘⣛⣯⣬⣿⣯⣤⣿⣀⡾⣯⠉⣴⠟⣿⠌⠹⣷⣄
⠀⠀⠀⠀⠀⠀⣴⡟⣱⢶⣆⣙⢷⢞⣴⡾⠛⠉⠀⢀⠀⠀⠀⠉⠉⠛⢶⣮⡓⠾⠃⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⣼⡟⠀⠙⠻⣯⢙⣴⠟⠁⠀⢠⣤⠀⣾⢿⠀⠾⠇⠀⠀⠀⠈⠻⣦⣀⣀⣀⣠⣤⣿⣧⣴⣶⠆
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⡿⠃⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⢛⣛⣉⡉⣩⣥⣤⣄
⠀⢀⣀⣠⣿⣧⣤⣴⣶⠶⠿⠿⠟⠛⠛⠋⢋⣉⣉⣉⠀⣠⣶⣶⣄⠰⢶⣾⠿⠃⣿⡟⠛⠁⢸⣿⠉⣿⡇
⠀⠈⠉⣉⣭⣤⣄⠀⣴⣶⡶⠆⠀⠀⣿⡇⢸⣿⡋⣉⠀⣿⡇⠘⠛⠀⠈⣿⡇⠀⢿⣿⠿⠃⢸⣿⡀⣻⡇
⠀⠀⠀⢹⣿⢙⣿⠇⢻⣷⣤⡆⠀⠀⢹⣯⠈⣿⡟⠛⠀⢻⣧⠀⣶⡆⠀⣿⣇⠀⢸⣿⣶⣶⠀⠿⠿⠟⠃
⠀⠀⠀⢸⣿⠛⣿⡆⢸⣿⣀⣀⠰⣶⣸⣿⠀⣿⣷⠶⠆⠘⠻⠿⠛⢀⣀⣈⣉⣀⣤⣤⣤⣤⣶⠶⢶⣾⠟⠛⠃
⠀⠀⠀⠈⠿⠇⠻⠿⠀⠛⣛⣛⣁⣉⣩⣥⣤⣤⣶⡶⠶⠶⠟⠻⠛⠛⠛⠋⠉⢉⣿⠏⠀⠀⠀⠀⣸⡏
⠀⠀⢀⣶⡶⣶⣾⠿⠟⠛⠛⠛⢿⣍⠉⠉⠀⠀⢠⣤⠀⣦⣖⠀⠾⠂⠀⠀⣠⡾⢣⣤⣄⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⣠⡙⢷⣤⣀⠀⠈⠁⠈⠙⠋⠀⢀⣀⣴⡾⢋⡤⠻⣦⣙⡷⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⢀⣴⢯⣽⢆⣬⣝⠛⠷⠶⠶⠶⠶⠿⠛⢫⣥⡖⠻⣶⣃⡀⣉⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣅⡿⢁⣾⠷⠌⠀⣸⠇⣾⣛⠀⣿⠙⠂⠈⣷⡀⢘⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣭⣓⡐⣧⡟⠀⣿⣭⠀⠻⣶⣟⣀⣬⡷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠶⠶⠶⠾⠿⠛⠛⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠓⠒⡾⢤⡀⠀⠀⠀⣀⣀⡤⡄⠀⠃⣸⠃⠞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠉⢺⠀⠉⢁⠀⢀⡔⢺⡈⠁⢸⠘⠀⣿⠌⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⣆⠀⠀⣀⡠⢼⡴⠻⡙⠄⠙⣶⢾⠉⠹⣯⣬⣙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠉⠂⠈⠓⠾⡟⠓⠉⠇⠀⠀⢀⡔⠁⠈⢤⠬⢄⣀⡔⢳⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠋⠓⢶⠀⠠⠗⠒⠀⣺⣶⠶⠤⠧⠤⠤⠤⢤⣐⣚⠯⠧⣄⢙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⢰⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⣀⣀⡀⣤⡤⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠣⣀⡀⢉⡑⠃⠀⠀⠶⣶⠖⣺⢚⡯⡱⢰⣯⣼⠹⣻⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠤⣈⣉⡿⠯⠀⠀⠀⠿⠀⠙⠋⠉⠚⠉⠀⠀⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠉⠒⠀⠀⠒⠒⡆⠀⠀⠀⠒⠒⠂⠄⠀⠐⠒⠈⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃⠉⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡁⡘⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠂⠙⡭⡝⣾⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠋
⠀⠀⠀⠀⠀⢀⣀⣀⣬⠤⣀⠀⡀⠁⠈⠙⢤⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠄⣀
⠀⠀⠀⢰⠉⠉⣁⣀⠉⠉⠉⠂⠀⢸⣤⣀⣀⠬⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠸⠉⠉⠉⠉⡸⠛⢎⠉⠁
⠀⠀⠀⠘⡲⢞⠛⠀⠉⠋⠉⠈⠑⢬⣾⢨⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡙⠛⠛⠫⢭⣉⡩⢦⣄⣈⡢
⠀⢀⠤⠴⢡⢄⡰⡀⠀⠘⢆⠀⠐⠀⠟⢻⠀⢄⣀⠇⠀⠀⠀⠀⠀⢸⠀⠀⢀⡠⠔⢈⡧⠬⣁⡒⠒⠋⠀⠛⠚⠤⢀⡉⠑
⢏⣁⣴⣋⣁⣀⠿⠚⢄⢀⣀⠧⣀⡀⠀⠾⠗⠲⣤⣤⣤⣀⣈⣉⣉⣀⣤⣤⡤⠶⠖⠋⠁⠀⠀⠈⠉⠓⠊⠉⠉⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣗⡠⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠋⠈⠩⢛⡶⣻⣇⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣾⡷⠁⠀⣀⡸⢟⣯⡿⠏⣿⣼⡟⢦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣭⡀⠀⣀⣴⣷⣾⡿⠫⠀⣰⡿⢻⣷⣿⣧⣥⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢢⡼⢿⣿⣯⣿⡵⡟⢿⠏⣴⣶⣿⣷⡾⠏⠱⣿⣿⣭⣭⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡴⣟⣳⣟⣻⡏⢱⢼⣿⢳⡿⣷⣷⣿⣿⣿⣿⢃⣼⣤⣼⣿⣿⣿⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢴⡟⢾⡿⠃⣿⣙⠷⣼⢻⣿⣿⣱⣷⣾⣭⣚⣿⣿⣷⣿⣿⣦⡤⢸⡿⠿⢿⣿⠯⡀⢠⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠏⠙⢿⣤⣇⡄⠠⣶⣿⣭⣟⣛⣿⣏⣭⣴⣨⣝⣻⡿⢝⣿⣿⢁⣀⣽⣇⠀⣾⠇⠀⠀⠀⣧⡼⢿⣿⣿⣦
⠀⠀⠀⠀⢠⣾⣛⡀⠀⣠⠞⢹⣿⡇⣿⣿⡿⢷⣶⣭⣿⡿⡌⠉⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣶⣤⣣⣿⠿⣴⣤⣭⠃
⠀⠀⠀⢠⣤⣿⣿⣿⣯⡋⢀⣨⡟⢳⣹⣿⣿⢿⣿⣷⣿⣿⡇⠀⠛⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣇⠂⠈⠿⣦⡀
⠀⠿⠻⠟⠿⠶⠶⠿⠟⠛⠛⠿⠗⠛⠛⠛⠁⠘⠙⠉⠉⠙⠈⠛⠐⠋⠛⠉⠻⠷⠾⠟⠻⠗⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠿⠿⠿⣿⣶⣦
⣿⣿⠿⠿⡿⠻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠖⠒⠒⠒⠒⠶⠤⢤⣄⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢯⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠐⠺⠥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠡⠖⣀⡴⠛⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣠⠴⢏⣉⠓⠒⠢⠤⠄⣈⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡤⠖⠋⠀⠀⠀⠈⠳⡀⠀⠠⡀⠀⠈⠉⠉⠛⢒⡶⠂⠀⠀⠀⠸⠧⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢧⠀⠀⠀⠀⠀⢠⣀⠌⠀⠀⠈⠳⡀⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠹⡓⢄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⢰⣶⣼⠂⠀⢸⠃⢀⡤⠤⢤⣄⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠒⠮⡙⠒⠒⠂⠘⣄⠀⠀⠈⣡⠏⠀⠀⢉⣉⠉⠐⢦⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⠤⠤⠄⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⢀⡀⠻⠽⠆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀⠚⠉⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠠⢄⠀⠀⠀⠀⢀⣠⢤⡎⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⣀⣤⠤⠴⠚⠉⠀⡾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢾⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠤⢤⡤⣀⣤⣤⠤⠤⠤⡃⢸⠤⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⠀⣬⠀⠀⠀⠀⠈⠉⠋⡀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡗⠛⠂⠀⠀⠀⠀⠐⠋⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⣧⣄⣠⠼⣇⣀⣀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣯⠈⣾⡀⣰⠛⡏⢫⢹⡙⠓⣒⣒⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠋⠩⠭⠭⠭⠭⠷⠖⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⡤⠖⠚⠉⢩⣟⣥⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⣀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣀⡤⢶⡦⣄⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠇⠀⢠⠀⠀⢀⠀⠀⠀⠤⠤⠒⢀⡾⠋⠀⠀⠘⢦⠀⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠠⡀⢈⣆⣱⢄⣉⠙⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠈⡇⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣬⠽⠛⠁⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢸⡀⢠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⡄⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠈⡇⠘⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⠻⣤⡇⠀⠀⠈⣷⣤⠖⠁⠀⠈⠓⢶⣾⠅⠀⠀⠀⡿⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠈⢢⡀⠈⠀⠀⠀⠀⠉⠁⠀⡠⠤⢤⡀⠀⠀⠀⠀⠀⠀⣧⠞⢉⡉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢷⢰⡏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡼⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣝⡂⣠⠀⠀⠀⠀⣠⣴⣚⣋⣉⣉⣉⣉⣿⡄⠀⠀⠀⠘⢀⠇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⠀⠀⢰⣿⣿⠛⣛⣻⣿⣿⠿⣿⠃⠀⠀⢸⣇⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠘⣯⠭⠽⠿⠒⠒⠒⠒⠾⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⢤⣤⣤⣤⣤⡤⠴⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⡀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠃⠀⠀⠀⡀⠀⠊⠗⠀⠀⠀⣠⠧⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣰⡁
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣶⣾⣷⡀⠀⠀⠀⠀⠙⢰⠀⠀⠀⠀⠒⠀⠒⠄⠱⣄⠔⠂⠉⢀⡽⣀⠜
⠀⢠⠤⠤⠔⠒⡋⠱⣿⣿⣿⡿⠿⣇⣀⠀⠀⠀⠀⠀⠉⠐⠂⢄⡀⠀⠀⠢⡀⢸⣂⣀⡤⠚⠠⠤⠀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⠀⠀⠀⠀⠀⠀⣹⠋⠁⢀⠔⡟⠙⣤⡀⠀⠀⠀⠀⢠⢤⠤⠗⠀⠀⣠⣃⠀⠈⠉⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆
⠀⠀⢆⠰⢾⡗⣿⡿⠇⢶⠙⣃⠀⢘⡏⠀⠙⣄⠀⠀⠀⠸⣒⠀⠀⠀⣸⣦⡡⡆⠀⠀⠀⠀⢀⡟⠢⢄⣀⣀⣀⡤⠴⠚⠁
⠀⠀⠈⠲⠋⠀⠀⠀⠀⠸⡊⠈⠃⠀⠃⠀⠉⡧⡀⠀⠀⠀⠀⠉⠉⠉⢿⣤⠇⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢠⠐⡄⠀⢀⢀⡁⠀⡜⠉⠐⢤⡔⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⠚⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠓⠛⢹⣽⡇⠀⢀⣈⡁⠀⠿⡅⠀⢰⡂⠀⠀⠠⡀⠀⠀⠔⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⠀⠀⡀⠀⠀⠀⡂⠀⠀⠙⡄⠀⠰⠈⠀⠀⡄⠰⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⡰⠰⣣⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡄⠀⠯⠥⠀⠔⠂⠀⠀⢀⠀⢄⣠⠔⠀⠀⣁⠀⠀⡠⢬⠀⠀⠀⠀⠀⢠⢀⣁⡄⠈⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠈⠋⡈⠁⢇⡀⠀⠀⠀⢀⢀⢏⣓⠶⠊⠙⢖⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⢂⢘⠀⣆⠈⠣⣀⣽⡃⠀⢀⠃⢭⡗⠁⢨⣷⠀⠈⢫⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠇⢤⠴⠒⢺⠦⠄⠒⣖⣎⠽⠦⢽⡏⢾⠀⢸⢀⣴⣿⡻⠭⣒⢨⣄⠀⢊⠑⣌⣩⣔⣢⠭⠬⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡤⠋⠹⡓⠋⢉⡱⠖⢷⢤⠤⣊⡓⡜⠀⠘⠃⠨⡗⢲⠍⠉⣋⣜⢃⣂⠳⡾⠛⠢⡔⠒⠢⠒⠂⠤⡔⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡯⠆⢀⠊⠁⣈⠶⢴⡄⠐⣄⠀⢬⠟⠐⠢⢄⢀⣱⠗⠢⣴⠋⣉⠁⠉⡙⢄⡰⠆⠀⠀⠀⡀⠀⠈⠥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣇⡀⠈⢡⡄⠀⠀⢀⠀⠀⢀⠈⠉⠀⠀⠀⠘⠉⠐⠁⠀⢀⡁⠀⡀⠀⠈⡄⠁⠀⠠⠔⠉⣘⣓⡀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠅⠀⠤⠀⠀⠒⠂⠀⠈⠀⠀⠀⠀⡩⠂⠀⠀⠀⠀⠒⠉⠀⠀⠲⡦⢉⡀⠀⠀⢡⡄⠀⠐⢄⠀⢄⠀⠀⠀⡰⠃⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠠⠄⠀⡠⠄⠀⡘⠋⠉⠀⠔⠃⠀⠢⠎⠀⠀⠀⢃⠀⠈⢉⠒⠁⠀⠀⠀⠊⠘⠆⠀⠈⠁⠀⠁⠀⠀⠠⠄⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠤⣖⡀⠀⠀⠀⣈⠐⠒⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⡐⠒⠉⠓⡲⠊⢰⡀⠀⠀⡘⠢⡀⠀⣐⡦⠀⠀⢤⣄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠲⠅⣀⣀⢴⠀⠀⢐⠀⠀⡠⠥⠤⣠⠰⢧⢤⠤⠃⢀⠀⠀⠙⡘⠁⠀⠉⠉⢰⣄⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⢿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠐⡻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢠⠔⠋⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⠥⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠠⢧⣀⠀⠀⠈⢷⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠃⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠧⠤⢴⠄⠀⠈⠳⡄⠀⠀⢧⠀⠀⢀⣋⣀⣀⣤⣄⣼⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣧⢀⡤⣍⡦⡀⠀⠳⣞⠉⠀⠀⠀⠀⠀⡀⣹⢀⣀⣰⣃⣀⣀⣠⡤⠤⠶⠖⠚⠋⠉⠉⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⡟⡾⠀⠈⠳⣿⢆⠀⠈⠓⠒⠦⠤⠴⠚⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠴⠚⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡴⠙⠉⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠀⠤⠀⢀⣉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⣠⠜⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠉⠉⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⡴⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢌⡓⠠⠧⢤⣀⣀⣀⣀⣀⡀⠀⠊⢀⢀⣀⣀⣀⣸⣓⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢬⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⣼⡙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣇⣀⡉⠉⠐⠒⠠⠤⢤⣀⡀⠀⠀⡜⠉⢧⣿⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⣥⣄⣀⡀⠀⣀⡉⠉⠉⠉⠉⠓⠐⠒⠒⠘⠳⠶⠤⠤⢸⣇⣣⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣿⡿⢸⣿⠀⣿⡿⠙⠇⠀⢰⣶⠆⢰⣶⠠⣤⠀⣠⠄⣀⢀⡀⣀⠈⠉⠉⠉⠐⠒⠒⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⡏⣿⣦⢀⣿⡗⠟⠀⠀⢸⣿⠀⣾⣿⣰⠁⣼⣿⠀⠹⠺⢹⣿⠋⠇⣿⡿⠸⡆⢲⣶⢠⣄⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⠛⠃⠻⠟⠸⠿⠇⣼⠺⡟⣿⡿⢀⣿⡇⢈⡄⣿⡇⠀⠀⠀⢸⣿⠀⠀⣿⡇⡎⠀⣿⡿⢸⣿⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠁⠘⠓⠒⠁⠀⠿⠿⠀⢸⣿⣇⣴⢀⣿⡇⣾⡟⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⠉⠁⢛⢲⠒⠒⡶⢤⠤⠤⠤⠄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠈⡌⢧⡼⢁⡼⠃⠀⣀⣀⣀⣀⡍⢸⣍⣩⡭⠭⢶⡶⠒⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣛⡉⠉⠉⠁⠀⡼⠛⠃⠉⠉⠉⠁⡏⢰⣷⡌⢱⡄⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⣀⡀⠀⣀⠔⠁⠀⠀⠀⠀⠀⡼⢁⣏⠙⢇⡼⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡟⠉⠉⠁⠀⠀⠀⠀⠒⠛⠛⣛⢋⣉⣑⣂⣾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠧⠽⠿⠿⠷⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠓⠛⠛⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠍⡉⠉⠙⠛⠛⠛⠛⠛⠛⠓⠒⠲⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣶⡴⢶⡶⠶⢶⢤⣤⢤⣀⠀⠉⠳⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡟⠁⠁⠀⠀⠀⠀⠙⠀⢸⡇⠀⠀⠈⢇⠀⠹⣭⣉⣉⣉⠉⠉⠉⠙⢳⡄⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠇⠀⠀⠈⠀⠀⠀⢸⡇⠀⢸⣦⠘⣇⠀⢻⡤⠀⠤⠤⠤⠤⠤⠤⢷⡀⠸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡇⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠘⣧⠘⡆⠀⢳⣶⣴⣤⣤⣤⣤⣤⣤⣷⡀⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠛⠚⠛⠐⠓⠒⠒⠒⠒⣺⡇⠀⣿⠀⢸⡄⠹⡀⠈⢿⢿⠉⠛⠛⠿⢿⣿⣿⣧⠀⠸⣧⠀⠀⠀⢀⡶⠶⠖⠲⠶⢶⡶⠲⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡇⢰⡿⠙⠇⠖⠖⠖⢶⡶⠶⣾⡇⠀⣿⠀⠘⡧⠀⠁⠀⠘⡞⣇⣶⢾⠦⣾⣿⣿⣿⡄⠐⢿⡆⠀⣀⣸⢨⡭⠉⣉⣙⣺⡼⡄⣻
⠀⠀⠀⠀⠀⢀⣶⣋⠤⠼⡇⢸⡇⠀⠀⡠⠀⠀⠈⠇⠆⢿⠁⠀⣿⠀⢰⡇⣤⠘⠀⠀⢳⠙⣧⠘⢱⡜⣿⣿⣿⡇⠀⣼⣧⣴⣿⢽⢺⡿⠭⠿⠶⠒⠛⠛⣿
⠀⠀⠀⠠⣶⣸⣿⠀⣶⣶⡇⠰⡇⠀⠀⠃⠇⠀⠀⠀⠀⢸⠀⠀⣿⠀⣿⢠⡟⡄⢤⠀⢸⢇⣿⢤⣤⣉⣻⡅⣸⠛⢾⣿⣤⠟⠋⠘⢿⢯⡀⠀⣰⢿⣧⠀⣿
⠀⠀⠀⠀⣿⢾⣿⣠⣷⣿⡇⠀⣧⣤⣤⣄⣀⣀⣀⡀⣀⣿⠀⠀⡟⢰⠇⣼⠀⡇⢸⠀⠀⣿⡀⢉⣤⠿⠈⢻⣿⣿⡟⠃⢻⡄⠀⠀⢸⣏⠹⢦⣿⣿⣥⡴⠛
⠀⠀⠀⣀⣴⣻⣿⣵⡿⠋⠇⠀⣀⡀⢤⠄⠠⡤⡤⡄⣤⣽⠀⠀⣇⡸⢀⡏⠀⣧⠘⠀⠀⢸⠛⡋⠠⢠⣴⡿⣷⣿⡇⠀⣼⡇⠀⠀⠀⠈⢷⠀⠀⠀⠀⠈⡿
⠀⠀⣴⠋⢀⡀⠀⣹⠀⠀⠀⠀⣧⠸⢠⠀⠀⠀⠁⠁⠙⢸⠀⠀⣿⡇⢸⣡⡴⠟⠀⠀⠀⠘⡿⠷⠺⠿⠿⠶⠾⢿⡇⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⡇
⢀⣴⡇⠉⠁⠀⠀⢸⠀⢸⠀⠀⠘⠀⠈⠀⠀⠀⠀⢀⠀⢸⠀⠀⠋⠀⢉⣁⠄⢲⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⢠⣼⡇⠀⢺⡇⠀⠀⠀⠀⣸⠟⠃⠛⠻⢿⡇
⠈⢻⠁⠈⣴⣶⣆⢺⠀⢸⡆⠀⠀⠀⠀⢠⠖⠉⠁⢨⡤⢿⡀⠀⠀⠘⠉⠙⠓⠞⢰⠂⠀⠀⠧⠷⠾⠴⢦⠶⠿⣿⠇⠀⣻⡿⠀⠀⠀⠀⣿⢀⣠⣤⣤⣰⡇
⣤⣾⡀⣼⣿⣿⣿⣼⣦⣾⠷⢶⣶⣶⣶⣾⣔⣶⣄⣼⣳⣾⣷⣶⣶⣶⣶⣶⣶⣶⣾⠓⠒⠒⠒⠒⠒⠒⠶⠶⠶⠤⠤⠤⢼⡇⠀⠀⠀⠀⣟⠀⣻⠙⣯⢹⡇
⠻⠶⢿⣿⠰⢾⣿⠟⢻⣯⢼⢺⡏⣿⢶⢹⢻⡿⠿⠿⠟⢻⣿⠱⡎⣿⠃⡆⣹⣏⡿⠛⢻⣿⣿⣿⣿⣟⣿⣿⣿⣶⡾⠛⠛⠃⠀⠀⠀⠀⣿⠀⣿⠀⣿⠈⢷
⠀⠀⠀⠻⢷⣿⠟⠀⠘⢿⣶⣿⣷⢿⣶⣾⡿⠃⠀⠀⠀⠀⠿⣷⡾⢿⣷⣾⣿⠿⠁⠀⠀⠹⢿⣿⡿⠿⣿⣿⣿⡽⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⠀⢹⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⡟⠀⣸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠴⠶⠶⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣼⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⡏⠔⠉⣠⠖⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠉⠀⠀⣠⠴⠖⠛⠋⠉⠉⠛⠲⢤⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠸⢠⠟⢁⣴⣯⡥⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⢀⣤⠖⠻⣇⣀⣔⣡⡤⠤⠶⠶⠶⠶⢤⣄⡀⠀⠀⠀⠀⢹⠀⠀⠀⢠⡀⢸⡟⢧⡀
⠀⠀⣰⠟⠀⠀⣠⡞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄⠀⠀⢸⣇⠀⣦⠘⡿⠟⢳⣄⢳⡄
⠀⣸⠃⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⠞⢹⡼⠛⢷⠃⠀⠀⠹⡆⠻⡀
⢀⡏⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⢹⡀⣷
⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠶⢦⠄⠀⠀⠀⠀⠈⣇⢸⡆
⠸⡇⠀⠀⣿⠀⠀⠀⠀⣠⡶⠿⠿⢿⣦⣄⠀⠀⠀⠀⠀⠀⣰⡿⠋⣠⠖⠲⢦⡀⠀⠀⠀⠀⠿⢼⣇
⠀⣧⠀⠀⣹⠀⠀⠀⠀⢀⡴⠒⠙⠦⣌⠛⢷⣄⠀⠀⠀⣰⡟⢁⡴⠃⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⣿⢻⡄
⠀⣹⣤⡞⠁⠀⠀⠀⢠⡟⠀⠀⠀⣠⣬⠳⢄⠉⠃⠀⠀⠋⣰⢫⣾⣿⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⣿⢸⡇
⣾⠉⣇⠀⠀⠀⠀⠀⢨⡅⠀⠀⢰⣿⣿⣶⡄⢠⠀⠀⠀⠀⠻⡈⠷⣾⣿⣀⣠⠤⠟⠒⠋⠀⠀⠀⣿⠈
⢻⡀⢻⡀⠀⠀⠠⢤⠈⣧⣠⣄⣀⣙⣛⣋⣠⡏⠀⠀⠀⠀⠀⠉⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠛⠾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠛⠋⠉⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⣤⡤⠀⠀⠐⠶⠶⠾⢷⡄⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⢠⠋⠁⠀⠀⢀⣀⣀⣀⣀⢀⣀⣁⠀⠀⠀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠚⠒⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⢀⣤⠞⠋⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠷⠦⣤⣀⣀⣀⠀⠀⠀⢀⣀⣠⣤⠶⠚⢻⣆⠀⠀⠰⠞⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⢻⡉⠉⠉⠉⠛⠛⠉⠀⠀⠀⢸⡟⣇⠀⢀⣤⠾⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠙⠲⣤⣤⣻⣿⣿⡦⠀⢠⡞⠋⢻⣤⣤⣄⡴⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠉⠛⠛⠛⠛⠒⠋⠀⠀⠀⣤⣤⣤⣽⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠙⠓⠲⣶⣶⣶⣶⣶⣦⣤⣤⠥⠞⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠉⠉⠉⠉⣽⠛⠉⠋⠁⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣀⣀⣤⣿⠋⠀⠀⠀⠀⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠧⠤⢤⣄⣀⣰⡏⠀⠀⠹⡄⠀⠀⠀⢀⡼⠿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢈⡟⠀⠀⠀⢸⡷⠶⠶⠊⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡿⠲⡶⠶⠶⠾⠃⠀⠀⠀⠀⠷⠶⠶⣶⠚⠻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠉⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⣶⣶⣾⣿⣛⣷⣿⣿⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣵⣿⠿⠟⠛⠛⠉⠉⠉⠉⣉⠙⠛⠿⠻⠿⣿⣿⣷⣶⣆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣶⠿⠛⢉⠀⠀⠀⢀⠘⣧⡀⠀⠰⢦⣿⠠⠄⠀⣀⣴⠀⠀⠈⠉⠛⠿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⠋⠁⠀⢀⣤⡷⣆⠀⠀⠸⠛⠃⠀⠀⠚⠛⠆⠀⠀⠞⢛⠋⠀⢰⣤⡄⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁⠀⣳⣴⡆⠀⠀⠇⠀⠀⣀⣠⣤⣤⠶⠶⠶⠶⢶⣤⣤⣄⣀⡀⠀⠚⠻⠇⠁⠀⣄⣠⠈⠙⢿⣷⣄⡀
⠀⠀⠀⠀⠀⢀⣶⣾⠏⠁⠀⡀⠈⠉⡟⠋⠀⣠⣴⠾⠛⠉⠉⠑⠲⠶⠦⠤⠄⣀⣀⠀⠈⠉⠛⢶⣦⣄⠀⠀⠴⠿⠿⠄⠀⠀⠙⣿⣥⠆
⠀⠀⠀⠀⣼⣿⡿⠁⠀⣻⣾⣇⠀⠀⢀⣴⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠲⠶⠭⢿⣦⡀⠐⠀⠀⢰⣀⡤⠈⠻⣿⢃⡀
⠀⠀⠀⣼⡿⠋⠀⢀⠈⠉⠉⠀⣠⣶⠏⠀⠀⠉⠉⠐⠂⠀⠀⠤⠤⢀⣀⠀⠀⠀⠀⠀⠀⠈⡉⠐⠀⠀⠀⠀⠹⣇⠀⠉⢹⠟⠦⠀⠀⠹⣿⡅
⠀⢀⣼⡿⠉⠐⢶⠿⠦⠄⠀⣰⢟⣀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⢀⡀⠀⠀⠀⠈⠳⣄⠀⠀⢀⣰⣀⡀⠹⣶⣦
⠀⣼⠟⠃⠀⠀⠉⠙⠂⢠⡼⠃⠀⡦⠈⠉⠀⠒⠐⠀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢉⡿⠿⡄⠀⠸⠿⣀
⠀⢻⡶⠀⡀⢴⡀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠢⠤⠤⢄⣈⡁⠐⠘⣦⠈⠀⠀⣠⠀⠀⢻⣿
⠀⢠⠁⠀⠸⠿⡄⠀⠀⣿⡖⠲⣆⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠀⠤⠤⢄⣀⡀⠀⠀⠀⠠⠤⠄⣉⣁⢻⡆⠀⢻⣿⡗⠂⠈⣿⡇
⠀⣾⠀⠀⡀⠀⠀⠀⢀⣿⣀⡠⠿⠃⢨⡟⠉⠙⠗⠀⠀⠀⡖⢂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠀⠀⠀⠀⠉⣷⠀⠉⠀⠉⠀⠀⢹⣗
⢀⡛⠀⣤⣷⡶⠂⠀⢸⡏⠰⣧⠀⠀⣼⠃⠐⠶⠀⠀⠀⢘⠃⢰⡟⠀⠉⠏⣠⠖⠙⡷⢆⡦⢠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠐⢦⣾⣄⡀⢨⣿
⢸⡇⠀⠈⠏⠙⠂⠀⠉⠁⠀⠛⠗⠠⣿⣀⣀⣤⠰⣷⣀⣾⠇⣾⠃⠒⠾⢠⣉⠀⠀⠀⠀⠀⢬⠃⠈⠃⣻⠓⠒⢶⠀⣤⠤⣄⠀⠀⠀⠛⠩⠁⠀⢸⣿
⢾⡇⠀⠠⡀⢀⡀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠰⠿⢄⣠⣴⠘⣿⣤⣠⡖⠀⠀⢠⠀⠀⢠⡿⠠⢤⡄⢠⡛⠀⢸⣧⠀⠀⣀⣠⠀⠀⢸⣿
⣸⣿⠀⠴⣿⡿⡄⠀⢿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠼⢬⠀⠀⣼⡇⠀⢀⡀⣾⡇⠀⣾⡿⠀⠠⠿⡟⠢⠀⠺⠋
⢨⣽⡇⠀⠘⠀⠀⠀⠸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠑⠿⠧⠶⠛⠁⢀⠀⠀⠀⠀⢀⡂
⠈⠉⣿⠀⠀⢠⣴⡆⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⣜⠀⠀⣸
⠀⠀⠸⣇⠐⠛⢿⠋⠀⠀⠡⣤⠄⣀⣀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠐⠒⠠⠄⡤⡀⣀⡀⠀⠀⠀⠀⠨⣭⡉⠉⠒⠒⠂⣶⠃⠀⠀⠀⠛⠉⠁⣠⠛
⠀⠀⠀⣿⣆⠀⠀⢤⣠⡆⠀⠻⣦⡄⠤⢀⣉⠀⠐⠀⠀⠀⠄⠀⠉⠁⠒⠒⠳⠤⠤⢄⣉⡁⠒⠂⠀⠀⠀⠀⠀⢀⡾⠃⢀⣸⣶⡖⠀⠀⣰⣿⠃
⠀⠀⠀⢻⣿⣇⠀⠤⠶⡝⠋⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠐⢠⡶⠉⢀⠀⠀⠸⠏⠛⠀⣸⡟⠃
⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⢠⣤⣧⣀⠈⠻⢦⣄⡀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠄⢠⣀⠀⠀⠀⠀⠀⣀⣠⠾⠋⠀⠠⣌⢳⡖⠀⠀⣠⣞⠏⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠠⠟⠻⠀⠀⢠⡄⠈⠛⠷⢦⣤⣀⣉⣀⠀⠀⠀⠀⠀⣀⣈⣉⣴⠶⠟⠉⢁⣀⣶⠀⠀⠚⠉⠃⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⢶⣀⠀⠀⠘⣶⣾⡋⠀⠠⣄⣀⠉⠉⠉⠋⠛⠛⠋⠋⠉⠈⠁⠀⢀⡄⠀⠀⣽⢼⡏⠀⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢤⣀⠀⠀⠀⠀⠲⣿⣯⠀⢀⣹⣄⡃⠀⠀⣷⣾⣄⠀⠀⠸⠃⠂⠀⠀⠀⣀⣤⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀⡈⠁⠀⠀⠈⠹⠏⠀⠀⠈⠉⠋⠁⠀⠀⠀⢁⣀⣤⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠤⣤⣤⣤⡤⣤⡤⠴⠶⠶⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠤⠤⠴⠿⠿⠦⠤⠦⢤⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢛⣛⣛⣛⣋⣀⣈⣛⣛⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣷⠰⢿⣰⣟⠇⣶⡄⣷⣰⡇⣧⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⢸⣏⡩⡏⢍⡻⠉⣿⠙⠏⣧⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⢾⣿⣟⠈⠚⠉⠁⠘⠉⠑⠉⠉⠁⣿⣿⡷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⡁⠀⢸⣷⣵⣷⣿⣷⣿⣾⣶⣶⣏⠀⠀⣙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣷⣍⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣵⣾⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠙⠲⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠴⠋⠁⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⢄⣀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢙⡿⢿⣿⣿⣿⣿⠿⡋⠁⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⢟⣵⣿⣿⣷⣦⣴⣾⣿⣷⣝⢿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⢀⡴⠋⠀⠀⠉⠛⠋⠀⠀⠈⠢⡀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠀⠉⠁⠀⠉⠁⠈⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣀⣀⠀⣀⣀⠀⣀⡀⢀⣀⠀⢀⣀⠀⣀⣀⠀⣀⣀⢀⣀⡀⢀⣀⠀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣯⣤⣤⣽⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙
⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⢀⣀⠀⣀⠀⡀⢀⠐⡀
⣿⣿⣿⣿⣿⣿⣿⠟⢱⣯⡒⠯⢐⣹⠡⢻⠅⠏⢹⠈⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⡇⢙⣶⢸⠚⢱⠿⡄⡿⣾⠀⣿⣦⡀
⣿⣿⣿⣿⣿⣟⣁⠀⢘⣷⣶⣿⣿⣷⣿⣿⣿⣾⣾⠀⠀⣈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣜⡻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⣻
⣿⣿⣿⣿⣿⡇⠈⠓⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠋⠁⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⢈⡽⠿⣿⣿⣿⣿⠿⢏⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⣴⣤⡀⠀⢀⣠⣶⡙⢿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⢀⡴⠋⠀⠀⠈⠙⠋⠀⠀⠀⠑⢄⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⢿⣿
⣿⣿⣿⣿⣿⡇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠛⠵⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⠶⠶⢶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⣭⠴⢶⣒⣲⣶⣶⣐⣂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⡶⠿⠛⠛⠋⠉⣉⣉⣉⣙⣛⣛⣻⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣭⣶⠶⠾⠟⠛⠛⠛⠛⠋⠉⣉⡉⠙⠛⠛⠛⠿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⢏⠉⠀⢀⣠⣀⣀⣠⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣄⣀⣂⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣡⣤⣶⠾⠿⣿⣿⣿⣿⢿⣿⣽⠿⣷⣟⡝⣡⣋⣽⢿⣟⢿⣿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣏⠘⣛⣿⣶⣿⣿⡟⣇⣁⢹⣏⣷⡿⠛⣟⠛⢻⣷⣴⣶⣶⣿⣶⣥⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢃⢎⣸⣶⣿⣭⣶⣀⣙⣿⣥⣹⣾⣏⠹⣯⣵⢩⠙⢁⣦⣿⡿⢛⣋⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⢿⣿⠄⠀⠈⣉⣉⡉⠛⠛⣛⣻⣧⣬⣭⣽⣿⣿⡿⠛⠛⠟⡟⢻⡯⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠛⠛⢁⣠⣼⣿⠁⠿⢸⡿⣌⣿⠀⣸⡏⠙⣏⠀⢠⡖⢶⡄⠂⣰⢶⣦⠀⢸⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⣯⣤⣤⢸⡇⠀⣿⠀⣿⠇⠀⣿⠀⢸⠀⢸⣷⠀⣿⢸⣿⣶⠾⠿⠟⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⣶⠶⠿⢸⡇⠀⣿⠀⣿⠀⠀⣿⠀⢸⠀⢸⣿⠀⣿⠸⣿⠛⣷⡶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⢸⡇⠀⣿⠀⣿⠀⠀⡇⠀⢸⠀⢸⣿⠀⢹⠀⣿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡇⣺⡇⠀⣿⠀⣿⢠⡄⡇⠀⢸⠀⢸⣿⠀⢸⠀⣿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⡇⠀⣿⠀⣿⠀⠀⡇⠀⢸⠀⢸⣿⠀⢸⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⣿⡟⠀⡏⠀⣿⠀⠀⡇⠀⢸⠈⢸⣿⠀⣾⠄⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠢⠀⣿⡇⠀⡇⠀⣿⠀⠀⡇⠙⣿⠀⢸⣿⠀⣿⠀⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢰⡇⠀⣿⠠⢀⡇⠀⢿⠀⢘⣿⠀⣿⢰⡿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣄⡀⠈⠛⠋⠀⠀⢿⣀⣸⠃⠀⠸⣦⣼⠟⣀⡈⠋⠁⣸⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣏⣽⠲⢽⣷⡦⣤⣝⣒⣀⣉⠀⠀⣀⣀⣀⣀⣬⡭⠵⣖⣻⢵⢛⣿⠀⠀⠀⢀⣤⣤⡀
⠀⠄⠀⠀⠀⢰⣿⣿⢶⠾⣤⣙⢷⣄⡒⠈⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⢶⠶⠶⠚⠛⠉⣀⣬⣼⢏⣴⡾⢿⣾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠻⣷⣮⡠⢎⠽⠿⣯⡛⠻⠿⢷⣦⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⡶⠶⢟⣛⠉⢿⣿⣛⣛⡹⣙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠜⢿⡔⢡⢖⣽⣫⣻⡆⠓⢀⣠⣤⣀⠈⠉⠉⣩⣿⣷⣶⣶⡆⢾⣿⣿⡗⠀⣍⢛⠛⡛⠛⠋
⠀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣾⣮⣵⠟⠀⠀⢿⣿⣻⣿⠆⠈⠉⠻⢷⣤⣤⡾⠇⠀⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠛⠁⠈⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠿⠋⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠛⠁⠀⠀⠀⠀⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⡀⢀⣠⣶⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡄
⠀⠀⠀⠀⠀⢰⣿⣿⠀⣴⣿⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣽⣿⢿⡿⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠻⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣾⣥⣶⠎⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠿⡏⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣿⠂⣀⣼⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠟⠉⢡⠀⠀⠁⢀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⣻⠀⠀⡀⣘⣠⣤⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣰⣷⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡋⠁
⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀
⠀⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢻⣿⡟⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣧
⠀⠀⠀⣿⣿⡙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⡆⣾⣿⡇⣸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⢹⣿⡟
⠀⠀⠀⣿⣿⣡⡾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠉⠀⡁⣿⣿⣿⢻⡟⣼⣿⣿⣦⡀⠀⢀⣠⣾⣿⣿⢻⠋⡀⠸⣿⣿⡇
⠀⠀⠀⢻⣿⣿⣰⣯⣼⣿⣿⣷⡄⠀⠀⣠⣾⣿⠟⠋⢸⡀⣠⣿⣿⣿⣿⣾⡿⢻⢏⣻⣿⣷⣶⣿⣿⡟⠋⡞⢸⢀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠙⣿⣿⣏⣼⠇⣹⣿⣿⣦⣾⣿⣿⠋⠀⢀⣸⣴⣿⣿⠟⠙⢿⣿⣧⣿⡿⣿⠙⣿⣿⢻⠏⣠⠀⡇⠆⣸⣴⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⡾⢻⠇⢻⣿⣿⢛⡏⢀⡆⣸⣿⣿⠟⠁⠀⠀⠀⠻⣿⣿⣤⣿⢰⣿⣿⠀⡀⡇⣰⣷⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣾⠀⢸⣿⣿⠀⡇⣼⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣼⣿⡟⣀⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣸⣿⣿⢠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠁⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠂⠠⠀⣀⣀⡈⠀⠀⠀⠄⠀⢀⡠⢀⡀⠀⠀⠈⠀⣁⢈⠁⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠂⡜⠀⠀⠀⠀⠀⠀⠈⠀⠄⡔⠁⠀⠀⠈⡀⢰⠃⠀⠀⠀⠐⠀⠔⠁⠀⠀⠀⡤⠈
⠀⠀⠀⠀⠁⠀⠀⣠⡀⠀⠀⡇⠀⠀⠀⣧⢌⡢⠀⠀⢰⠀⠀⢠⠀⠀⠀⠰⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠛⠬⢭⡜⡇⠀⠀⠀⠆⠀⠇⠀⠀⡘⠀⠀⢸⡀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇
⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⣴⠁⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣠⣤⣤⢇
⠀⠀⠀⠀⠀⠛⠛⢢⠐⠀⠀⡇⠀⠀⠀⢰⡬⠼⠏⠇⠀⠀⠀⣔⡲⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⠀⠀⡜⠁⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⢠⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢴⠆
⠀⠀⠀⠀⢄⡀⠀⠠⢄⡴⠊⠁⠠⠠⢤⡼⠀⠀⠇⠤⠤⣤⠄⠀⠄⠠⠤⠞⡴⠀⣤⡤⠁⠠⢤⢾⠠⠤⣠⠄⠀⠠⠄⢤⠄
⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⢀⠈⠁⠈⠂⠀⠀⠀⠉⡉⠉⠁⠀⠀⠉⠉⢉⡋⠉⢉⡉⠀⢉⡉⠁⣈⠙⠫⠅⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⢀⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠄⠂⠀⡀⠠⠀⠉⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠖⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠙⣿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⢼⣿⠉⠛⠿⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⠁⠀⡰⠋⣳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣴⡴⠎⠁⢀⡴⠋⠙⢂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠉⠛⠳⣦⣄⠀⠀⠀⠻⠀⡼⠋⠀⣠⣞⣉⡤⠴⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⠀⠀⠀⠐⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠴⠒⠚⠁⠁⠀⠀⠘⠓⠢⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⣀⠤⠒⠂⠉⠉⠉⠁⠀⠉⠉⠐⠒⠤⣀⣠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⢀⡾⠁⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⢀⡟⢁⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⢠⣾⡴⠋⠀⠀⠀⠀⢠⣤⠶⠞⠋⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⠞⠁⠈⠉⠳⣄⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠈⠳⣦⡀⠙⢷⣄⠀⠀⠀⠀⢠⡾⠁⢀⣾⣡⣶⡆⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢹⣦⣄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣠⣤⣤⠙⢦⡀⠙⠃⠀⠀⠀⠀⢀⡴⠋⢼⣿⣿⣿⣶⡆⠀⠀⣹⠀⠀⠀⠀⠀⠀⠸⣧⠙⣧
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣴⡄⠙⢷⡀⠀⠀⠀⠀⠸⡇⠀⠈⠿⢿⡿⠟⠀⠀⢀⣯⢠⠴⠆⠀⠀⠀⠀⣿⠀⣿
⢠⡶⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠈⠿⣧⣭⠝⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⢤⣄⣀⣤⡴⠶⠚⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠇
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠐⠒⠶⠼⠷⠶⠴⠶⠤⠤⠤⣤⢤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁
⢻⣄⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠙⠷⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠉⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣀⠀⠀⠀⠀⠤⠶⠶⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢀⣤⠉⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⣀⣠⣤⡤⠶⠶⠒⠒⠒⠒⠒⠚⠃⠀⠀⠀⣠⠞⠁⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣇⠀⠀⠀⠀⠐⠛⠋⠙⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⢿⡏⠀⠙⢷⣄⠀⠀⠀⠀⣀⣠⣼⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⠷⠶⣤⣤⣀⣀⣀⡀⠀⣀⣀⣀⣀⣤⣤⣶⠞⠛⢉⡴⠃⣼⠃⠀⠀⢈⣻⡄⠀⠀⠛⠋⠁⢀⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠈⠻⢿⣏⣹⠛⠙⢛⡛⢻⡏⠀⠈⠢⣤⠊⠀⢠⡟⢀⣴⠟⠋⠙⣷⣤⣤⣤⣤⣤⣬⡉⠙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀⣘⣁⠀⠀⠀⠀⠀⠀⣸⣇⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡅⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠳⠞⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠉⣁⣼⠏⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠉⣙⠻⣷⣶⣶⣶⣶⠶⠶⠶⠶⢾⣿⣉⢉⣿⡶⠛⠛⠛⠋⠉⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠾⣛⠻⢿⣿⣿⡷⠀⠀⢀⣴⠾⠛⠛⠛⠉⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⢀⡄⠸⠁⠉⠛⠛⠲⣿⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣤⣀⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⠛⠛⠉⠙⢦⣄
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣾⣿⣿⣾⣿⣾⣶⣷⣿⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣽⣶⣶⣶⣶⣶⣤⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡼⠮⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠙⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠔⡖⠀⠀⠀⠀⠀⠧⠀⡴⡶⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡴⠊⠉⠈⢉⡶⠊⠀⢰⡧⠔⡞⠉⠙⡆⠀⠸⡄⢣⡀⢾⡙⣿⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠹⣠⠚⠉⠙⠁⠀⢀⡤⠯⢄⡀⠀⠸⡀⢈⣧⠀⠀⡿⠚⠉⠉⣆⠈⠉⠉⠁⢧⠀⠀⠈⡷
⠀⠀⠀⠀⠀⠈⢦⠀⣰⠁⠀⠈⠑⠢⣄⠀⠴⠁⠀⠀⠀⠈⠳⣄⣙⣧⠤⠴⠚⠡⠒⠉⠉⠙⠢⡀⠀⠀⣼⣀⣀⡘⠀⠀⠀⢰⡲⣄
⠀⠀⠀⠀⠀⢰⠃⢰⠃⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡶⣄⠀⠀⠀⠀⠀⠀⠀⠹⠒⠒⠣⠄⠀⢱⣠⠴⠶⠁⡟⠁
⠀⠀⠀⠀⠀⢈⡕⠛⠚⠋⠛⠓⠢⢤⠔⠊⠉⠉⠉⡣⠤⠖⠢⠤⣀⡼⠙⢎⢾⠉⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⡼⠁⠀⠀⠀⢹
⠀⠀⠀⢠⠖⠉⠀⣀⡤⠤⠀⠀⠀⠀⠀⣀⠀⢀⠴⠦⡀⠀⠀⠀⠈⢣⡤⠬⠷⠇⠀⠀⠀⠀⠀⡟⠒⠚⢹⠡⠚⠉⠉⠉⠓⢦⡿⠤⣄
⠀⠀⢰⠃⠀⣠⠎⠁⡠⠤⠖⢦⣀⠐⠋⠉⠱⢄⣀⣀⠇⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⠀⠀⠐⠋⠉⠙⢺⠀⢀⡤⠤⢄⣀⣀⣁⡤⠼
⠀⠀⠘⠶⠾⠓⠋⠉⠑⢄⢀⡤⠜⠓⠲⠒⠒⠒⠢⢤⣀⣀⣠⠴⠒⠉⠉⠀⠀⠀⠀⠈⠉⠒⠦⠤⣤⡤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⠉⠓⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⣀⡤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡞⠉⢳⡞⠁⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠚⠹⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⢸⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢹⡄⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⡇⠀⢸⠀⠀⡇⠀⠀⠠⠶⠖⠶⠄⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⡇⠀⢸⠀⠀⣇⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⣀⡀⠀⢀⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡠⠆⡆⠀⢸⣈⣉⣉⣉⣁⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⢉⡽⠉⠉⠉⠉⠉⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡿⢤⠀⡀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⣄⠀⠀⠈⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠈⠓⠦⠤⠤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄



⡏⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏

⠀⠀⠀⠀⠀⠀⠀⢸⠥⢸⠁⢙⠀⠃⢹⠁⡜⡄⢸⠭⡃⡇⠀⡯⠄⠀⠸⢄⠀⡏⠈⡇⡕⡜⠁⢱⢸⢔⠁⡯⠄
⣤⣤⣤⣤⣤⣤⣤⣴⣶⣼⣶⣮⣤⣤⣼⣼⣤⣼⣼⣶⣥⣷⣦⣷⣶⣤⣴⣾⣤⣧⣤⣧⣴⣼⣶⣮⣼⣬⣦⣷⣦⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣾⣬⣭⣭⣤⢤⣤⣘⣹⣷⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⡤⠖⣚⠋⢉⠡⠤⣒⠺⠛⡋⠉⠛⠉⠀⡠⠂⢸⠀⠱⣌⠂⢍⠛⠫⣄⡂⠄⣀⠀⠭⢉⡒⠦⠤⣤⡀
⠀⠀⠀⣼⣯⣩⠐⡀⠀⠂⢈⠠⠂⠉⠀⠔⠀⢀⠔⠁⠀⢘⠀⠀⠘⣆⠀⠈⠢⡀⠉⠒⠤⡉⠓⠢⠄⠉⢒⣼⣷⠄⡀
⠀⣴⣄⡹⠩⠙⠛⠻⠶⢾⣤⣤⣤⣈⣀⡀⣀⠊⠀⠀⠀⢸⠀⠀⠀⠈⢆⠀⠀⠘⣦⣀⣠⣬⣷⣶⠶⠿⢛⣛⠿⢀⣷
⠀⢹⡍⠉⠛⠲⠦⠤⣀⣀⡀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⢀⣀⣠⣬⡴⠶⠿⢛⣝⣷
⠀⠀⠙⢳⣶⣤⣤⣀⡀⠉⠉⠙⠓⠒⠶⠶⠦⠤⠤⣤⡤⠤⣤⡤⠤⠤⠤⠤⠠⠶⠖⠒⠛⠛⠉⠉⢁⣀⣤⣤⡾⠟⠟⠋
⠀⠀⠀⠈⣇⠀⠈⠉⠉⠛⠓⠲⢶⣾⣛⣻⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠶⠶⠶⠗⠛⠛⢻⡏⣿
⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠁⠀⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣿
⠀⠀⠀⠀⢹⠗⣤⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢤⣾⢻⡇
⠀⠀⠀⠀⠘⡇⢻⠀⢸⡋⢱⣤⠖⢢⠀⣤⠤⡀⠀⣀⣄⠀⠀⣠⣄⠀⢀⣤⢄⠀⣴⠒⣄⡞⢨⡇⢘⡇⣸⠃
⠀⠀⠀⠀⢠⡇⢸⡁⢸⡇⠀⣿⡀⠀⣼⡇⠀⢰⣼⠁⠀⣷⣸⠁⠀⢶⣿⠁⠈⣿⡇⠀⣿⡇⢸⡇⢸⡇⣿
⠀⠀⠀⠀⠀⣇⢸⡇⢸⡇⠀⣿⡇⠀⠿⡇⠀⢸⡇⠀⠀⣽⣿⠀⠀⢸⡟⠀⠀⣿⡇⠀⣿⠀⣸⠇⢸⠃⡏
⠀⠀⠀⠀⠀⢶⠸⡇⢸⡇⠀⣻⡇⠀⣶⡇⠀⢸⡏⠀⠀⢹⡇⠀⠀⣼⡇⠀⠀⣿⠁⠀⣿⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠌⠀⣷⠈⡇⠀⣿⡇⠀⣹⡇⠀⢸⣧⠀⠀⢸⡇⠀⠀⢿⡇⠀⠀⣿⠀⢰⡿⠀⢸⠀⣾⢸⠁
⠀⠀⠀⠀⠀⢸⡆⣿⠀⣿⠀⢻⡇⠀⢻⡇⠀⢸⣿⠀⠀⢼⣧⠀⠀⣸⡇⠀⢠⠟⠀⢈⡧⠀⣿⠀⡟⢸
⠀⠀⠀⠀⠀⠸⠇⢿⠀⣿⠀⢸⡇⠀⢸⣷⠀⠸⣿⠀⠀⢸⡿⠀⠀⣿⡇⠀⢸⣦⠀⢸⡇⠀⣿⠀⡇⡞
⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⠀⢻⡇⠀⢸⡿⠀⠀⣿⠀⠀⢻⡇⠀⠀⢹⡇⠀⢸⡟⠀⢼⡇⢀⡿⢰⡇⡇
⠀⠀⠀⠀⠀⠀⣀⢸⡇⢹⠀⢸⣷⠀⢈⡇⠀⢰⣿⠀⠀⢾⡇⠀⠀⣼⡄⠀⢸⡇⠀⣼⡃⢸⡇⢸⢡⡇
⠀⠀⠀⠀⠀⠀⢻⠸⡇⢸⡇⢸⣿⠀⠸⡇⠀⢠⣿⠀⠀⢼⡇⠀⢸⣿⡇⠀⢺⡇⠀⣿⠇⢸⡇⣸⢸⠁
⠀⠀⠀⠀⠀⠀⢸⡄⡇⢸⡇⠘⣿⠀⠰⣷⠀⠸⣿⠀⠀⢠⡇⠀⠸⣿⡆⠀⣿⡇⠀⣿⠀⢺⠃⡿⢸⠇
⠀⠀⠀⠀⠀⠀⢰⡆⣇⢸⡇⠀⣿⠀⢀⣿⠀⠀⣿⠀⠀⣽⣿⠀⢰⣿⡇⠀⣿⡇⠈⣿⠀⣿⢀⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠘⡇⣿⠀⡏⠈⣿⠀⠸⣿⠀⠀⣿⠀⠀⠀⣿⠀⢸⣿⠀⠀⣿⠀⢰⡿⠀⣿⢠⡇⣿
⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⣿⠀⣿⠆⢀⣿⠀⠀⣿⡆⠀⢾⡏⠀⠸⣿⠀⠀⣿⠀⢸⡇⠀⡟⢸⣇⡟
⠀⠀⠀⠀⠀⠀⠀⣇⢸⡅⣿⠀⣿⡇⠀⣿⠀⠀⣿⡇⠀⣼⣇⠀⢠⣿⠀⢰⣿⠀⣸⡇⢀⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⣿⡆⠀⣿⡀⠀⣿⡇⠀⣸⣿⠀⢸⣿⠀⢸⣏⠀⣿⡇⢸⡇⡙⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠈⡇⢸⠁⢻⡇⠀⣿⡄⠀⣿⡇⠀⢙⣿⠀⢸⣿⠀⢸⣿⠀⣿⠃⢸⠇⣇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠷⣸⡀⢸⡇⠀⣿⡇⠀⣿⠇⠀⢸⡇⠀⢸⡟⠀⢸⡇⠀⣿⠀⢸⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡁⠻⠧⠀⣽⡇⠀⣿⡆⠀⠻⡟⠀⢸⣷⠀⣼⡧⠤⠟⢂⣩⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⠀⣀⠀⠙⠁⠀⠛⠋⠀⠈⠉⠀⣀⠤⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣿⠀⠀⠀⡴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⢤⢤⣄⠀⠀⢸⡇⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⢀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠻⣄⣉⠀⣿⡉⣉⡷⠀⡰⠟⣦⠀⠀⢾⣆⠀⣴⡷⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⣄⣀⣹⠆⣿⠉⠉⠁⣰⠛⠒⠻⣇⠀⡆⠉⠿⠃⢻⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⡗⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠁⠀⠀⠀⠸⠃⠀⠀⢸⡇⠀⠸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠻⠦⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣿⠀⠀⠀⠀⣴⠊⠁⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠀⠳⣤⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣾⠁⠀⢀⡔⠒⠢⡀⠀⠀⢷⠀⠀⠀⠀⠀⣼⠃⠀⠀⡵⠀⠀⠀⠀⠈⢧⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠛⠀⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⠎⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠹⣆⠀⠀⠁⠀⠉⠳⠄⢠⠎⠀⠀⠀⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⣿⠀⠀⠀⠀⠀⠉⠳⠶⠶⠶⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠘⠳⢦⣀⣀⣀⡴⠞⠁⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠳⠴⠦⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⠤⣤⣤⣤⣤⠤⠾⠃⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣴⠖⠈⠉⠉⠉⠉⠙⠳⠲⣄⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠻⣦⣤⣀⣀⣀⣀⣀⣠⣤⠾⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠏




⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣤⣀⣀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⡀⠀⠀⠀⠀⢀⡉⢻⣿⣿⣿⣿⣷⡄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠋⠭⠁⣀⠀⠀⠂⠠⠄⡀⠈⠉⠛⠿⢿⣿⣷⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢀⠀⠁⠂⠄⡀⠀⠈⠛⠇⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⢻⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⡈⠑⠄⡀⠀⠈⠐⠄⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⠀⠳⡀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⢑⠢⢀⠀⠈⠐⢄⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠄⠀⠀⠘⢦⣀⠀⠀⠁⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠤⣤⣅⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⢀⠌⠀⠀⠀⠔⠀⠈⠓⢦⠄⢀⠀⠀⠉⠐⠂⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠔⣨⢀⠴⡈⢿⣿⣿⣿⣿
⣿⣿⣿⠟⠀⠀⠀⡠⠂⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⠐⠨⠄⢐⠂⠠⠄⠀⠀⠀⠩⠄⠘⠉⡉⣡⡄⠒⠋⢉⠀⠴⢁⠀⠙⠛⠛⠛
⣿⠟⠁⠀⡀⠔⠁⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⢀⠄⠁⠈⠁⠀⠀⠉⠁⠁⠉⠒⠀⠠⠀⠀⡠⢤⠀⠊⠀⠀⠀⠀⢀
⣿⣿⣿⠆⠰⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢦⣀⣀⣀⣠⣴⣿
⣿⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣿⣿⣿⣿⣿⣿
⣿⣿⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⣿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠄⠀⠀⠠⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⣠⣶⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠛⠛⢿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⡗⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⣿⣧⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⢀⣬⡍⢩⣥⡀⠀⠀⠀⣀⣬⡉⣩⣄⡀⠀⠀⠀⢸⣿⣿⣿⡿⠟⢶⡶⠻⢿⣿⣿⣿⠿⠳⣶⠖⠿⣿⣿⣿⣿
⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠘⠿⠿⠿⠿⠿⠿⠃⠀⠸⠿⠿⠿⠿⠿⠿⠀⠀⠀⢸⣿⣿⣧⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣿⣿⣿
⠀⠀⠀⢠⣶⣄⢀⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢸⣿⣿⣿⣿⠛⢛⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿
⠀⠀⠀⠀⢙⣿⣿⣏⠀⠀⠀⠀⠺⣷⣄⣴⡿⠋⠀⠀⠀⢸⣿⣿⣿⣿⠧⠀⠀⢼⣭⣿⣽⣿⣅⠙⠿⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠘⠿⠋⠙⠿⠃⠀⠀⠀⠀⠈⠿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⣠⣾⣦⣠⣿⣿⣿⣿⣿⣷⣄⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿