Safe Emojis & Symbols

⠀⠀⠀⠀⢀⣀⠠⠤⣄⠒⠀⠀⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⢀
⠀⠀⠀⠀⠈⢉⠄⣡⣀⣀⠂⠀⠀⠀⠤⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠⣀⣀⢀⠀⠀⢀⣀⣠⠤⠼⡛⡇
⠀⠀⠀⠀⠀⠀⠀⠆⡤⠀⠤⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠩⣁⢀⣒⡒⣞⢋⠉⡽⠃⠀⠑⡘⠁⣧
⠀⠀⠀⠀⠀⢰⠆⡇⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢒⣂⠠⠁⢠⡀⢹⣀⠈⢀⢀⠀⠈⡰⠃⢸
⠀⠀⠀⠀⢘⠸⠀⡇⠀⠀⠀⢀⡴⠋⠙⠛⠷⣄⠀⠀⠀⠀⠀⠀⠂⠆⠈⠀⢸⠊⡀⡠⠐⢀⠔⠁⠄⠸⡆
⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⣸⢰⣯⠝⢣⢀⢹⡆⠀⠀⠀⠀⢸⣙⢛⢰⡄⢸⠔⠀⠈⠐⠋⠄⠂⠜⢄⡇
⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠸⣇⡩⠭⠀⠠⣿⠉⠀⠀⠀⠀⠀⠉⡛⣿⠃⢸⠁⢀⠀⠀⠀⠀⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠝⡚⡮⠔⠀⠀⠀⠀⠀⠀⠀⠛⠛⢠⢸⠀⠘⠀⠀⠀⡀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⢄⠀⠀⠀⠀⠀⡀⢰⠂⠀⠀⠀⠀⠀⠀⠄⠑⢹⠀⠀⠀⠀⠀⠀⠁⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣦⣱⣀⡀⣖⣡⣴⢟⠡⠀⠀⠀⣐⠚⡗⠀⠀⢐⠄⠂⠀⠀⠀⠄⠀⠂⢰⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠰⣿⣭⣻⠹⣿⡍⠀⠀⠀⠀⠀⠀⢀⡫⠀⠁⡥⠀⠀⢈⠀⠀⡀⣔⡔⠕⡇
⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠻⣿⢿⡾⠗⠃⠀⠀⣠⠢⠁⣤⣸⣔⢀⠀⡧⢀⡀⣑⢄⠀⠈⠐⢈⣼⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠤⢀⠀⠘⠘⣿⠀⠀⢠⢞⠁⠀⠀⣤⣍⡚⡧⠃⠗⠅⠂⠂⠂⠀⠄⠠⠈⣸⣗⠂⠀⠀⠒⠂
⠀⠀⠀⠀⠀⢥⣣⠄⠀⠀⠀⠀⠀⠉⠧⡽⠧⠴⢓⠁⡀⠀⡀⠀⣯⣻⠇⠀⡃⡀⢀⠔⠂⠀⠁⠈⣠⡿⡛⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠠⠀⣀⡀⢨⣴⠖⡽⡺⠹⠻⠲⠾⢇⢈⠵⣠⠏⠀⠉⡁⠀⢀⣠⡞⠋⠁⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠛⠏⠥⣠⣄⠀⡠⠈⠐⢉⠵⣫⢘⠑⢁⢀⣠⡶⡋⠉⠉⣁⠛⡣⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠛⠺⠥⢼⣯⣸⣬⣶⣿⣝⣛⣤⠅⣭⠭⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠋

⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠚⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠙⠛⠓⢦⡀
⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⣰⣞⣁⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸
⡇⠀⠀⢠⡖⠒⠒⠒⠒⠒⠚⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠀⠈⣿⠀⠀⢸
⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸
⣧⠀⡴⢺⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⣿⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸
⣿⠀⢇⣸⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠚⠓⠲⣤⡀⠀⢸⡇
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡠⠶⠒⠶⣄⠈⠹⡄⢸⡇
⢿⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⡎⡴⠋⠉⠙⣎⢳⠀⣷⢸⡇⠀⢀
⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⣇⢧⣀⠀⣀⡟⣸⠁⡟⢸⡇⠀⢸
⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠳⠭⡭⠥⠞⣡⡾⠁⢸⠀⠀⢸
⢸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠶⠖⠛⠁⠀⠀⢸⠀⠀⢸
⢸⡇⢀⣤⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠸⡇⢸⡇⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⡇⠘⡇⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⣿⠀⠓⢻⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⢸⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⢸
⠀⢸⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠘⠧⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠶⠛

⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦
⠀⠀⠀⠀⠀⢀⣾⣿⠻⢿⣷⣶⣤⣤⣤⣴⣶⣿⡿⠟⠻⢿⣿⣶⣦⣤⣤⣤⣶⣾⡿⠟⢿⣷⡀
⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠈⠉⠉⠛⠋⠉⠉⠀⣤⣤⣤⣤⠀⠉⠉⠙⠛⠉⠉⠁⠀⠀⠈⣿⣷⡀
⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣿⣿⣿⣿⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⣠⣤⠄⢠⣤⡀⢠⣤⡤⢠⣤⡤⣤⣤⡤⣤⡀⣤⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⠀⣛⣷⡆⣾⣿⣧⢸⡷⠖⢸⡗⡂⠀⣿⠀⠈⣿⠃⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠉⠉⠀⠉⠀⠉⠈⠁⠀⠈⠉⠉⠀⠉⠀⠀⠉⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⢸⣯⡅⣿⣾⣽⡇⢿⣍⠉⣿⠁⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠘⠃⠀⠛⠙⠈⠓⠛⠋⠀⠛⠀⢀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣤⣤⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⠁

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

⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⣄
⠀⠀⠀⠀⠀⠀⡇⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡄⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⡼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⡧⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢳⠇⠀⠀⠀⠀⢀⡾⣫⠷⠶⠾⣝⢷⡀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⣼⢹⡇⡎⠉⢱⢸⡏⣧⠀⠀⠸⣿⡏⣿⡿⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠸⣞⢷⣕⣒⣪⡼⣳⠇⠀⠀⠀⠈⡇⡏⠁⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠙⠳⠿⠭⠿⠞⠋⠀⠀⠀⠀⠀⠧⠇⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⡗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⢸⢸
⠀⠀⠀⠀⠀⠀⣇⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞⣸
⠀⠀⠀⠀⠀⠀⠈⢹⠉⠉⠉⢉⣹⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣏⠉⠉⠉⠉⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠴⠃

⢰⠒⢖⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⢤⣀⡀
⢸⠀⠀⠒⠉⢑⡲⠤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠈⠉⢒⣢⣤⣀⡀
⢸⠀⠀⠀⣠⢞⢍⡟⠂⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⣉⢽
⢸⡇⠀⠀⢀⠄⠁⠀⠀⠀⡇⠀⡤⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣿⠀⠀⡄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⣨⠿⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠆⠀⡇⠀⡧⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠀⠀⠷⣤⣤⠄⡇
⠀⡇⠀⠀⠀⠀⠀⣾⣡⠀⡇⠀⡇⠀⠀⢠⡏⠀⢀⡞⢀⣷⠊⠁⠈⣽⢲⡀⠀⠻⡄⠀⠀⢠⣿⢠⣤⡇
⠀⡇⠀⠀⠀⠀⠀⠊⢸⠀⡟⠛⣧⠀⠀⢸⡁⠀⠘⣏⠷⣧⡀⠀⠀⢽⡽⠃⠀⠀⡇⠀⠀⢸⡟⠀⢸⡇
⠀⠃⠀⠀⠀⠀⠀⠠⢸⠀⡇⠀⣿⠀⠀⠀⢳⡀⠀⠀⠀⠀⠉⠻⠿⠏⠀⠀⠀⡾⠁⠀⠀⠀⡇⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢸⠀⣧⣄⣾⡄⠀⠀⠀⠙⠢⠄⣀⢀⡀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠙⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⢸⠃⠀⠊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠁⠀⢸
⠀⠐⣤⣀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠒⢺⡀
⠀⠀⠈⠛⠿⠶⣤⣀⠼⠀⡀⢀⢘⠇⠀⠀⠀⠓⠒⠒⠂⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣓⣖⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇⣙⠮⠤⡤⠤⠤⠤⠀⠀⠒⠒⠒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠛⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⢭⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⢰⣮⠀⠀⠲⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠁⠀⠐⠉⣠⣾⣯⣼⡤⠀⠀⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⡙⠓⢴⡄⢀⡴⠞⠁⠁⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⠶⠶⠚⠛⠛⠻⠿⠶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠶⢤⣀
⢠⣤⣶⣚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠋⠉⠙⡆
⢸⡀⠉⠙⠻⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣧
⠸⡇⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠛⠉⠁⠀⠀⢀⣀⣠⣾⠷⠿⠛⠛⡇⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠛⠋⠉⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⢿⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠄⣀⠤⠴⠖⠚⠋⠉⠁⠀⠀⣀⣠⠤⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⣿
⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠶⠦⣄⠀⣿⡇⠀⣿
⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣡⠴⠲⠶⣾⣆⣿⡇⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⡅⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⣸⢣⡏⠀⢀⣌⣿⣿⡇⠀⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠇⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠅⣿⣸⣷⠀⠸⡿⢸⣿⡇⠀⣿
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣅⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣧⡐⠘⢷⣝⡳⢦⣶⠄⢻⡇⠀⣿
⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⡀⠉⠙⢛⡡⠀⣾⡇⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⠿⠋⠀⠀⣿⠀⠀⢻
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠀⡶⠀⠀⠀⠀⠀⢀⣠⡴⠿⠁⠀⢸
⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢠⠆⠀⠀⠀⠀⠀⠀⠀⣀⣔⣾⣉⣁⣶⠿⠞⢋⣀⣠⣴⠶⠛⠉⠀⠀⣀⡀⠤⡿
⠹⣮⡛⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠋⣉⣩⣤⣶⠿⠟⠋⣉⣤⣴⣶⣶⠿⠛⠉⣀⣴⠃
⠀⠙⠿⣦⣉⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢧⡸⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠛⠋⣉⣤⣶⣾⣿⣿⡿⠿⠿⠷⢷⠿⠛⠉
⠀⠀⠀⠈⠙⠛⠋⠉⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡀⠀⢀⣀⣤⠶⠟⠋⠉⢀⣠⣴⣾⣿⣿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣤⡂⠀⠀⠀⠀⢸⡇⠀⠀⠸⠧⠞⠋⠁⠀⣀⣤⣴⣾⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⢦⣀⠀⠀⢸⠇⠀⠀⠀⠀⣀⣤⡶⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣉⠓⠦⢈⠀⢀⣠⠴⠚⢉⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⣶⣬⣋⣡⣴⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢀⡠⣴⡖⢲⣦⢄⡀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣰⠻⣷⣘⡧⢼⣃⣾⠿⣆⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⡿⠷⢾⠁⠀⠀⠈⡷⠾⢿⡀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠿⢻⣄⠀⢀⣠⡛⠻⣿⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠘⢾⡟⣰⡏⢹⣆⢻⡷⠃⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠈⠙⠓⠚⠋⠁⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣴⠟⠉⠉⠉⠉⠉⠉⠉⠉⠻⣦⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⢸⣿⣿⣿⢶⣴⣤⣶⣠⡲⣽⣿⣿⡇⠀⣼⣿⡯⢶⣤⣤⣴⣴⡶⣽⣿⣧⠀⢸⣿⣿⣯⢖⣤⣤⣦⣤⡶⣟⣿⣿⡇
⠀⢸⣿⣿⣧⣿⠋⣤⣌⢹⡇⣽⣿⣿⡇⢸⣿⣿⡇⣹⡏⣠⣄⢹⡟⢸⣿⣿⡇⢸⣿⣿⡯⢸⡏⣡⣄⠙⣿⢸⣿⣿⡇
⠀⢸⣿⣿⣟⣿⣦⣉⣁⣼⡇⣿⣿⣿⡇⢸⣿⣿⡏⢽⣧⣈⣁⣼⡗⢸⣿⣿⡇⢸⣿⣿⣏⢸⣧⣈⣉⣴⣿⢸⣿⣿⡇
⠀⢸⣿⣿⢽⣮⢭⣩⣍⡭⣿⠿⣿⣿⡇⠈⢿⣿⡿⣮⢍⣭⣭⡭⣽⢿⣿⡿⠁⢸⣿⣿⡿⣿⢭⣩⣭⡩⣵⠧⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠘⠋⠛⠛⠛⠛⠛⠛⠉⠋⠛⠛⠛⠁⠀⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠛⠋⠉⠁
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⡶⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⡀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢀⡤⠶⠷⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢴⡶⢤⣤⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠸⣇⣀⣀⣀⣼⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣸⣇⡀⠀⠙⢷⣄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠈⠉⡿⠉⠀⠀⠀⠀⠀⢠⡟⠀⢀⡴⠛⠉⠉⠙⢷⣄⠀⠹⣆⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣧⣤⣾⠁⠀⠀⠀⠀⠀⢿⣤⣤⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣄⠀⠀⠀⠀⢀⡿⠀⢀⡿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢀⣤⠶⠿⢦⣄⠀⠀⠀⠀⠀⢻⣄⠀⠙⠳⢶⡶⠾⠋⠀⢀⡾⠁⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⢸⣅⣀⣀⣀⣽⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣸⣇⣀⣤⠶⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠈⠉⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠙⠳⠶⢶⡶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⣶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⣴⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡞⠛⠁⠒⠲⠤⠤⣀⣀⣀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢺⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⡟⡇⡄⢐⠂⠆⠠⠀⡀⡀⠉⠉⠓⠒⠲⢇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⡇⠈⠰⠀⠀⠀⠀⠀⠀⡈⠈⠁⠂⠰⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠂⢈⠀⠀⠀⠀⡰⠋⠈⠳⡀⠀⢘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠂⠰⠀⠀⠀⢠⠃⡔⡹⠉⢳⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡆⢘⠀⠀⠀⢸⠀⡃⣧⠋⡾⢆⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠂⠠⠀⠀⠀⢻⠀⡅⣯⠀⡇⡸⠭⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠡⢘⠀⠀⠀⢸⠀⡇⣏⠒⢻⠁⠒⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠠⠀⠀⠀⠈⣇⠈⠚⢒⡏⠀⠅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡉⢐⠀⠀⠀⠀⠈⠢⠤⠊⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡁⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣇⡴⠋⡣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠂⠐⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⡤⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠄⢈⠀⠀⠀⠀⢰⠁⠀⠀⠀⢸⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠰⠀⠀⠀⠀⠘⢆⣀⣀⣀⡞⢩⡟⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡂⢈⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣼⠍⠒⠦⠬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⢸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡇⡇⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠄⢈⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⢉⢉⡇⡇⠄⠠⠀⠀⠀⢀⠀⠄⠂⣈⣤⢖⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⣇⡇⢀⠘⠄⠂⢈⣠⠤⠒⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣥⡠⠔⠊⠉

⠀⣀⡀⣠⡤⠐⠒⠂⠀⠉⠉⠉⠉⠉⠁⠀⠒⠂⠀⠀⠀⠠⠤⠤⠀⠄⣀⣀
⠏⣅⠂⡘⠓⠐⠤⠄⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣤⡶⣶
⠁⡣⢊⠏⣉⣁⡈⠀⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢔⡒⠂⠄⠭⢭⡼⠖⡲⣻⠟⢙⢄⢌⠞⣼⡄
⠐⣽⢸⠰⠒⠒⠀⠀⠉⠀⠁⠐⠒⠒⠂⠀⠤⠤⠀⠄⢈⣑⠊⡛⡀⣷⢄⠪⠪⠠⡢⡠⠊⡔⠨⡇
⡆⡅⠸⠀⠀⠀⠀⠀⣠⠴⠶⠶⣦⡀⠀⠀⠀⠀⠀⠐⠢⢸⠀⠒⠅⢰⠕⠅⢔⠔⠕⢀⡞⠀⠁⣷
⠃⡅⠀⠀⠀⠀⠀⣰⢁⡴⠶⣄⠈⢻⣄⠀⠀⠀⠀⠀⡤⢵⣄⣄⠀⠸⢢⠪⠪⠢⣶⢏⡤⢊⡔⢹
⡇⠅⠀⡀⠀⠀⠀⢿⣿⣿⡷⢓⠀⣶⣿⠀⠀⠀⠀⠀⠱⢦⢶⣽⠇⢸⠕⠔⢔⠎⡀⠔⢁⠈⢪⣺
⠇⠀⠀⠇⠀⠀⠀⠈⣿⣯⣭⣁⣴⣽⡽⠀⠀⠀⠀⠀⠀⢸⣬⣿⢠⢸⡠⣠⠔⢀⠈⠂⠠⠑⢝⢼
⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⠛⠻⠓⠉⠀⠀⠀⠀⠀⠀⠀⢸⠉⢁⠜⣸⠀⠪⠢⠀⠀⠄⢀⠂⠢⢱
⠀⠀⠀⡆⠀⣧⣙⠢⡀⢀⣀⠀⡀⠄⢪⣯⠄⠀⠀⠀⢀⣸⠐⡠⡋⣴⠑⢐⠀⠠⡂⡠⡑⡑⣰⢻⠄
⠐⠀⠀⠃⠀⠠⣛⢷⣼⣦⣄⣯⣔⣶⡏⠥⠀⠀⠀⢀⡌⢹⢋⠀⡂⣫⠪⠀⢁⠄⠔⢄⢌⠊⢢⢿
⢸⠀⠀⡆⠀⠀⠈⠘⣿⣤⣻⠸⣿⣆⠀⠀⠀⠀⢀⠀⢐⢸⠕⠐⠅⠗⢀⢈⢢⣪⡢⣢⣳⣵⡽⣸
⢸⠀⠀⠁⠀⠀⠀⠀⠙⢶⢷⣾⠓⠃⠀⠀⢀⡔⠂⢠⣷⣽⣣⢀⠁⡧⣣⠠⠀⠔⠄⣤⣼⣿⣽⣿⣟⣷⣦⡀
⠈⠀⠀⠈⠓⠹⠤⣀⡀⠘⡘⣿⡀⠀⠠⡴⡋⡠⡨⢺⣬⣕⣚⡆⠂⡗⢝⢯⠫⠪⡙⡝⠉⠼⢡⣿⣟⣉⡙⠛⠛⠂
⠸⣼⡢⡀⠀⠀⠀⠀⠈⠈⠧⣽⢧⢠⣺⡮⢌⠊⢐⢁⠔⣏⣿⠇⠀⡣⡓⡁⢕⢍⠊⠠⡐⣐⣽⣿⠥⠈⠉⠁⠠⠀⠤
⠀⠀⠉⠑⠢⠤⣠⡀⡀⠀⠀⢀⣵⡶⣾⢧⠻⡳⠶⠥⣖⣇⠍⣴⣰⠝⢌⠪⡢⡀⡐⣅⡴⠟⡁⠤⡄⠠⠤⠦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠊⠾⠟⣛⢅⢔⠜⢕⠁⢁⠕⢻⡿⣪⠺⠪⡑⡕⢁⣤⣾⡛⠚⠀⢶⣶⡖⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠛⠴⠷⣾⣫⣲⣟⢸⣔⣼⣾⡿⠏⢉⣉⡉⠙⠃⡘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣘⣒⣒⣒⡒⠒⠒⠂⠊⠀⣀⣀⣀⣀⡁⠀⠀⠀⢀⠉⠈⠁⠈⣀⣀⣀⣀⣠⣤⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢻⠉⠩⡄⢸⣿⣿⣿⡿⠛⠛⠉⠉⠁⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡍⠹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠠⠁⢸⣿⣿⣿⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡀⢸⣿⣿⣿⣟⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⢸⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠈⠲⠂⡶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠃⢸⡿⢉⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠄⢀⡈⠛⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠆⠀⠈⠋⠁⡎⠀⡀⠀⠀⠀⣠⡀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡆⢠⠀⠀⠀⢠⠀⣧⠀⠀⢠⠛⠳⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⣤⣴⣶⣎⠀⢿⠀⠀⢸⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⢸⣿⢿⣿⣿⠀⣿⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠁⢸⣿⣿⣿⣿⠀⣿⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⢀⡀⠀⠠⣄⠄⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠄⢾⢹⠏⠉⠉⠀⣛⠠⠀⡠⠄⠀⠀⠀⣇⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡂⢺⢡⣄⠀⣤⠀⣿⠀⠀⢠⡀⠀⠀⠀⠘⢶⢓⣀⡠⠄⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⡇⢨⡅⠠⠀⠀⠀⠿⠀⣷⠀⠀⢸⠋⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠘⣗⢠⠀⠀⠀⢠⠀⡷⠀⠀⠀⣀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⡇⡄⣿⢈⠀⡀⠀⠈⠀⡇⠀⠀⠀⠉⠀⠂⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢻⡇⢱⠀⣢⠈⠉⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⣸⡇⣸⡀⢼⣦⣤⣤⡄⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣀⣈⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⡿⣸⡇⡏⠃⢸⣿⡁⠀⠉⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⠐⠒⠈⠉⠀⠀⠀⠀⠀⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⣙⣷⣼⡇⠀⡅⢸⣿⣷⣶⣦⢀⣤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣽⡇⠀⠰⠮⠍⠉⢩⣿⣈⢿⡀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣶⡿⠿⠿⠭⠭⠭⠀⢀⣀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣤⣶⣶⣶⡖⢀⣉⣭⣙⣿⣿⣯⣭⣯⣿⣏⣒⣚⣻⣿⣛⠈⠉⠋⠙⠁⣉⣙⠛⠛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠁⠈⠙⠛⠃⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡇⠀⠀⠀⢀⡔⠒⠒⠒⠒⠒⠒⠒⠲⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣸⡁⠀⠀⠀⢺⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡿⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣴⠃⣰⡆⣩⠍⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠸⠇⡆⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢰⣏⢸⣿⣸⣇⠀⠀⢨⣿⢸⠀⠀⠀⠀⠀⠀⠛⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢿⣄⠻⣷⣽⣓⣺⡿⢋⡾⠀⠀⠀⠀⣄⣀⣀⣐⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣭⣩⣥⠴⠋⠄⠀⠀⠀⠀⠈⠙⣿⣭⣥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠈⠁⢰⡊⠑⡄⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢇⣀⠼⡀⡄⣀⡜⣹⡞⡁⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢨⠶⣌⣿⢟⣴⠿⡅⠀⠀⣠⠄⣄⠀⠀⢸⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⢸⠃⠀⠀⠀⠀⢀⣀⣀⣯⡤⢀⡜⠁⠀⠘⣯⣤⣶⠶⠶⠟⡀⠀⢼⠅⢉⡑⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⢸⠀⠀⠀⠀⢰⣏⠀⣹⣴⣶⣎⣹⢦⣤⡞⢿⡟⠉⠈⠀⠀⡀⠀⠀⠀⠤⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠛⠙⠟⠛⢙⠃⢸⡟⠉⠲⣍⠳⢤⡀⠀⠛⢿⣿⡖⠚⢃⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣿⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠂⢻⡄⠀⠀⠘⠧⠴⠃⠀⠀⠀⢸⣿⡿⠛⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣆⡠⣿⠀⠀⠀⠸⣆⣀⣀⣀⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⠾⠟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⠶⠶⠶⣶⣶⣶⢶⣶⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠀⠀⠈⠉⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠼⠷⡄
⠀⠀⠀⠀⠀⢠⠖⠛⠓⠋⠉⠉⠉⠉⣉⢉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠛⠛⠻⠭⠭⠭⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⠼⠤⢤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠘⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⢸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣄⡀⠀⠀⠀⠀⠀⢸⠀⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡎⠀⠘⠀⢀⠟⢦⠀⠀⠀⠀⢸⠀⢸⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⡎⠉⠳⠀⢀⣈⡄⠀⠀⠀⢸⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠓⠤⠚⠀⢀⡼⠀⠀⠀⠀⢸⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠴⠚⠉⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⢸⠇
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠑⠒⠒⠒⠒⠀⠤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠹⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⡉⠉⠉⠒⠒⠒⠒⠒⠒⠋⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣉⡏⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⢀⣀⣀⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠈⠙⠻⠿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣿⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠖⠋⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠆⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⠿⠋⠀⠀⠎⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠌⢿⣿⣿⡿⡿⠋⠁⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠶⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⢿⣿⣿⣿⣿⣿⠀⢸⣿⠿⠿⠿⠋⠀⠟⠱⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⣿⣿⡿⠿⣿⡀⠀⠀⠀⠀⠀⢀⣈⡀⢐⢸⡇⠀⠀⠀⢀⣤⣤⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⣿⣷⡄⠀⡅⢰⣶⣶⣾⣿⣿⣿⡇⠀⢸⡇⢀⠄⣀⢸⢱⣿⠳⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠹⣧⠀⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇⢨⣾⣍⣽⠷⠟⢠⢾⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠁⢸⣿⡿⣿⣿⣿⣿⢣⡄⠈⡇⠀⠙⣿⠇⠀⠀⢁⣸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⡄⠀⠁⠘⠋⠀⠀⠙⢿⣿⡏⠀⠀⡇⠀⣸⣿⠀⠀⠀⠙⠸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢳⠀⡇⠀⠀⠀⠀⣀⣬⣿⡇⠀⠀⡇⠀⠏⠃⠀⠀⠀⠰⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⡇⣴⠂⠀⠀⠉⢿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⡇⠓⠀⠀⠀⠀⠀⣿⡇⠀⠀⡇⠀⠀⠀⠰⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⡇⠀⠀⠀⠀⠀⢀⣀⡵⠦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⣇⣀⣤⣴⠖⠋⠉⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⡇⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢰⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠘⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠛⠛⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠉⠻⣦⣴⡟⠉⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⣴⠿⠿⣦⣀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣀⣀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢠⠷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠘⡶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠘⠲⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠑⠦⢤⡤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⢤⡤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣀⡤⠶⡻⢶⢤⢄⣤⡖⢟⡳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠶⠷⣷⣮⣳⡗⡐⡱⠡⢪⣞⢕⢹⡳⠲⢖⢶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣍⣿⠛⠽⣻⣴⣿⢵⡫⠉⠉⠉⢳⣛⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣌⣿⡄⢸⣺⠀⠀⣷⣥⡤⠔⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣎⣙⠷⢴⡿⣏⣰⣿⣿⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠽⣿⣶⣶⢭⡶⢻⣋⣭⣤⡽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣧⣼⣥⣮⣾⢿⣟⣿⡍⠝⡶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠝⣿⠛⠁⠁⠃⠀⠘⠆⠱⣄⠀⠁⠊⡜⡹⢲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠔⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠊⡰⠉⡹⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⡠⠰⠊⠀⠀⠀⢀⣀⣀⣸⣿⠝⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⢊⠜⢑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⡆⡇⠃⠀⠀⠀⡰⡫⢋⢅⣵⢵⠕⡫⣻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠔⡡⢎⡝⣦
⠀⠀⠀⠀⠀⣠⠚⠁⠀⠣⠉⠀⠀⠀⠀⢸⠋⣈⡵⠝⢳⡧⡪⣮⣏⣕⢉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⡴⢋⠔⣡⢷⡀
⠀⠀⠀⢠⠞⠁⠆⠢⠘⠀⠀⠀⠀⠀⠀⢿⡼⣿⣇⠀⢸⡪⡮⡟⠀⠙⢷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣡⠞⣡⢊⠝⡄
⠀⠀⡴⠁⢠⠺⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣕⢟⢷⠿⡪⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⠞⡡⢋⠜⡼⡄
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣕⡩⡪⣞⣷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠊⡠⢂⠞⡰⣳
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣪⣾⣵⠟⡻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⢁⢞⣔⣽⡀
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡄⠀⠀⡼⣊⣼⡇⠈⢿⣮⡪⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⡡⡫⢪⢾⡅
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡱⡻⣦⡀⣿⣿⣿⠁⠀⣸⣯⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⡡⢊⠔⡵⡱⣻⠇
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣊⡿⢻⡿⣯⢞⢟⢟⣽⡻⣮⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⢊⠔⡡⢪⢎⡔⣩
⠸⡆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣏⢔⢵⠵⠦⠛⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡫⢖⡡⢊⠔⡡⢊⠴⡏
⠀⠱⡀⢠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠔⢋⠴⣋⠔⡡⢊⠔⣩⡾
⠀⠀⠱⡀⠰⢘⢰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⢀⠀⠀⠀⢀⠀⡀⢀⠀⡰⢃⠄⣁⡴⢩⠞⡡⢊⡴⣡⢞⣽⢭⠉⢩⠉⠉⣠⣤⡄
⠀⠀⠀⠘⢦⠈⠸⢈⢠⠠⢀⠀⡀⢀⢀⢀⠜⡰⢃⠜⡡⢊⠤⢊⠴⢊⠴⢁⠔⡠⢂⠞⡱⢡⢮⠎⡰⢁⠞⡰⢋⢜⡵⠟⠛⠒⠒⠒⠒⠒⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⢙⣦⣈⠘⠸⠸⠄⠃⠂⠂⠘⠐⠀⠃⠚⠐⠃⠐⠁⢘⣁⠚⠁⠊⠐⠡⢊⡔⡱⢃⠞⡰⢃⣎⣴⠷⠋⠉⠉⠉⠉⠉⠉⠩⠭⠭⠭⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠁⠉⠉⢲⠦⣤⣤⡄⠒⣶⣶⣲⣶⡶⠾⢿⣿⣭⣤⣬⠭⢭⣉⣁⣰⣋⣞⣰⣧⠾⠶⠿⠭⠤⠤⠤⠤⠤⠬⠥⠤⠭⠭⠭⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⣒⣀⣀⣀⡭⠭⠍⠉⠭⠭⠍⠉⠉⠀⠂⠘⠉⠩⠿⠿⠿⣭⣭⣭⣽⠉⠉⠩⠭⠭⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡿
⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⣼⠁
⠀⠹⣟⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⣠⡾⢻⠇
⠀⠀⢘⣦⡀⠉⠙⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠉⠁⠀⣀⡤⠞⠉⣰⣯⡦
⠀⠀⠘⢯⡙⠲⢤⣀⠀⠀⠈⠉⠉⠛⠒⠒⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣶⠶⠚⢛⡖⢀⡤⠞⠉⠀⢀⣀⡤⠴⣒⣋⣭⠀⣠⡾⢋⡼⠁
⠀⠀⠀⢶⡿⣦⡀⠉⠉⣛⣒⡒⠶⠦⠤⣤⣤⣀⡀⠉⠓⢦⡀⠀⢠⣶⡿⠟⢋⣵⣿⣿⣾⣿⠿⠟⣿⣼⡵⠋⠀⣠⠴⠚⠉⠀⠀⠀⣀⠴⠋⠀⠁⣠⣞
⠀⠀⠀⠀⠻⣤⡉⠓⠀⠛⠬⣅⣀⡀⠀⠀⠀⠈⠉⠓⢦⡀⠙⢦⡻⣭⡀⢀⡙⣏⠋⠉⠁⠀⠀⠀⣿⡟⠀⣠⠟⠁⠀⠀⠀⠀⢈⣉⡭⠞⢀⡤⠞⣫⠟
⠀⠀⠀⠀⠀⢰⣿⢦⣄⡀⠐⠮⠭⣉⣀⣀⡀⠀⠀⠀⠀⠉⠢⡀⠹⣄⠙⠳⣝⣪⣳⣴⡄⢰⠀⢸⣿⠀⠞⠁⠀⠀⢀⣀⣉⡭⠽⠂⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠦⣭⣁⣀⠀⠘⠓⠒⠒⠒⠲⠆⣠⡶⢦⣌⣀⠙⡆⠀⠈⠙⣿⣿⣿⣄⣇⣠⣿⣿⡄⠀⠤⠔⠋⠀⠀⠀⣀⣺⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣭⣤⣄⣀⣀⡀⠈⠛⠻⠿⠿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⠿⣧⡀⠀⢀⣠⣶⣾⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⢲⣶⣤⠶⠟⠛⠩⢛⣿⣿⢿⡟⢹⣧⠹⣆⠈⠻⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⣠⡶⢶⡶⣻⡿⠟⢡⠟⠀⠈⡟⢧⠙⢦⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠘⠛⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⠁⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⢿⠿⠏⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⠦⢤⡀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠳⣄⠀⢀⠜⠁⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⠤⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡱⡄⠀⠀⠄⠀⠀⢀⠀⢀⡜⠁⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣦⠤⠬⠷⠶⠥⣤⣿⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⣀⣀⣀⠀⠀⠈⠉⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠖⠋⠉⠀⠟⡏⠉⠙⠒⢤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠀⠱⡀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⣿⠉⣟⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣿⢠⡇⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢻⡷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠃⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣸⠇⣿⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⢻⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠋⠲⠤⣤⣤⣠⣤⡤⠴⠚⠧⠤⠤⠔⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣄⠀⣀⣀⣀⣀⡀⢀⣀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡏⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠙⢷⣤⣤⣖⠊⠁⠀⠀⠈⠛⣔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⠋⠀⠀⢄⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⡤⠒⠊⠉⠙⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠑⠄⠀⡀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠈⠢⣀⡠⠒⠁⠀⣡⣀⠙⢿⣿⣷⣄⠀⢀⣴⣦⡈⠳⣄⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⡠⠖⠁⠀⣣⣀⠙⢿⣿⣷⣌⠻⡿⠋⠀⠈⢳⠊
⠀⠀⠀⠀⠀⠀⠀⢆⠀⡠⠊⠀⠀⡠⠊⠀⠀⡧⢄⠙⢿⣿⣷⣄⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⡠⠊⠀⢀⡤⠋⠀⢀⠇⠀⠙⢿⣿⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⣀⡴⠋⠀⢀⠔⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠔⠉⠢⣀⢀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⣊⣉⣍⣍⣭⣭⣙⣒⡰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡛⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢨⣀⣸⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠽⠋⠋⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⡈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠒⠐⠀⡀⢀⠀⠀⠂⢀⠔⠂⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠈⠀⠀⠠⠁⠀⢹⣿⣟⣽⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠰⠁⠀⠀⠔⠁⠄⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢼⢃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠂⠀⠘⡀⠀⢰⣽⠘⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢐⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⢸⣿⢈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡷⠊
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡇⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣟⠃⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⡶⠶⠲⠲⠶⠖⠖⠖⢿⠄⠀⢿⡶⡲⠆⠆⠠⠄⠦⠄⠄⠦⠦⠤⠤⢤⡴⢿⠭⠀⡇⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⠈⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡀⣸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢁⢸⡋⠀⣗⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣆⠀⡇⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠄⠀⠠⢸⡵⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢸⠖⠀⡇⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠨⠈⡧⠀⡇⡔⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠟⠙⠛⠛⠃⠀⠀⠸⡧⠀⣿⢸⣿⡄⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢍⡀⠀⠀⡁⠠⣿⢸⣿⠅⣿⣿⣿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡯⠀⣿⣃⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠈⠉⠁⠀⠒⠂⠠⠤⠤⠄⣀⡀⣀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⣻⡦⢯⠿⠙⣀
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡁⠐⠒⠠⠆⠀⠤⠀⡤⡤⠄⠀⠄⠈⠅⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿
⣿⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠀⠀⠀⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏
⣿⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⣿⠀⠀⠀⢾⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠂⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠇⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣹⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢺⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⠀⠀⠀⣸⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡿⠛⠿⠿⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠠⠤⠤⠒⠶⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠤⠤⠤⠤⠤⠤⣄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠀⠀⠀⠀⠒⠒⠒⠛⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡇
⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⡔⠐⠒⠀⠒⠒⠒⠒⡄⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⡱⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠰⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠔⠒⠊⠉⠀⢀⣀⡨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠒⠒⠉⠁⠀⣀⣀⠤⠔⠒⠂⠉⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢀⣀⡠⠤⠒⠒⠊⠉⠀⠀⠀⠀⠀⢀⠤⠤⠒⠒⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡒⠉⢡⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠰⠠⠀⠆⠀⠀⠀⠰⠀⠀⠀⠀⠰⠰⠰⠰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠂⠀⠍⠀⠂⠀⠈⠀⡀⠀⠄⢤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠥⢈⡑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠿⠷⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠉⢠⢦⣀⠀⣀⠤⠔⠒⠉⠀⢀⣠⣴⡶⡆⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠙⠻⠶⣶⣤⣀⠠⠜⠋⠉⠀⣀⣤⠀⢰⠟⠛⠉⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡇⢀⣤⡴⠞⠛⠉⢩⠀⢸⡇⠀⠀⠀⠀⡇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠅⡂⠀⠠⢀⣾⠀⢸⣅⠀⢀⡐⠁⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢢⡇⠀⡤⠂⠸⠀⢼⡗⠀⠝⠔⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⢸⠌⠇⠀⠁⠀⠀⠀⣸⠁⠁⠎⠀⠸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠅⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠁⣧⠀⠎⠀⠀⠀⣾⡆⠀⠀⠀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⡇⢸⣧⡷⠤⠦⠠⠆⠀⠛⠂⠒⠂⢈⠥⢤⡄⢸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⡂⢘⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡄⢠⣤⣤⣤⣤⣄⡄⠀⡴⠴⠶⠶⠾⣐⡾⠇⢸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⢸⠁⣦⡀⠀⠀⢀⠀⣷⠀⠠⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⢈⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⢸⢸⡆⠀⠀⠆⢸⠀⣿⠀⠀⠀⡐⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠈⡂⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸⠇⠰⠃⠄⠘⠀⣿⣈⠀⢬⠂⡀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠨⡀⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⢸⣤⠈⠀⢄⠀⠀⠀⣿⠴⠺⠣⠈⠁⠀⠀⢼⡗⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⢸⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢾⣿⡲⢀⣠⣴⡇⠀⣿⠁⡐⠀⠂⡆⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠓⣂⡇⠀⣿⠀⠀⢀⡀⡇⠀⠀⣋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢰⢨⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⠼⠾⠦⣬⣀⠀⠀⡀⠀⠁⠀⠛⠳⠖⣀⠤⡇⠀⠀⢵⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠕⠄⠰⢸⠀⠀⡀⢸⠀⠀⠀⠀⡀⠤⡐⠪⢉⢁⣈⠀⠂⠀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠉⠁⠀⢀⢿
⠀⠀⠀⠀⠀⠀⡀⠁⠰⢸⠀⠀⠂⢸⣀⣠⠴⠊⠀⠁⠀⠀⠀⠀⠈⣁⢀⡀⠀⠒⠐⠀⠀⠈⠉⠉⠒⣢⣄⣀⠀⠀⠠⢸⣽⠀⢀⠐⠊⠉
⠀⠀⠀⠀⠀⢀⠈⡀⢸⣸⠀⢐⣩⡼⠿⠟⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠤⠄⠈⠉⠛⠲⢍⣸⣿⠠⣀⣀⠠⢤⣆
⠀⠀⠀⠀⠀⢈⣡⡤⠚⠛⠛⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠈⠄⢐⡀
⠀⠀⠀⠴⠚⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⢠⠤⠠⡄⡄⠰⡄⡖⡖⢲⡴⢹⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠯⠃⣼⡺⠸⡧⢹⡇⡀⠘⠇⡧⠈⢰⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⣒⣈⣇⣰⡰⠤⠧⠿⠦⠤⠷⠼⠾⠾⡦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠉⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠰⡖⠒⠒⠚⠉⠙⠉⠉⠙⠉⠉⣹⠀⢸⠀⢀⡐⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡏⠀⡆⢰⠁⣖⠨⠆⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣸⠀⢹⠀⠈⠁⠀⠈⠀⠀⠸⠀⠀⡇⢀⡇⠀⠑⠠⡉⠂⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡇⢸⡄⠀⠅⠀⠠⠀⠀⠴⠀⢰⠇⢸⠀⠀⢏⠢⢂⠆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠣⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠉⠀⠉⠉⠁⠉⠀⢹⠀⡆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢇⠀⣀⠀⣀⠀⠀⡇⠀⣸⠀⡇⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠃⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠰⠀⠰⠀⠀⠆⠀⣿⠀⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡇⠘⠒⠒⠒⠒⠒⠰⠦⠤⠏⢸⠀⠀⠀⠀⠀⠀⠘⠒⠢⠤⠀⡀
⠀⠀⠀⣠⠤⠤⠂⠀⠉⠀⣀⡤⠔⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⢀⠤⠖⠒⠂⠈⠁⠀⠀⢠⠉⠉⠀⠒⠒⠂
⠀⢀⡞⠁⣀⣀⣀⣀⡀⢠⡏⢉⣉⣉⡉⠁⠀⣀⡀⠒⠒⢲⠉⠔⠒⡂⢠⠎⠀⠀⠀⠀⣀⣀⡀⢠⠃
⠀⠘⠀⠂⠀⠰⠀⠀⠀⠀⠇⢸⡀⣀⣸⠀⢸⣀⣀⠀⡇⠨⠀⠤⠒⠁⢸⠀⠀⠀⠀⠀⠁⠀⠘⢸⠀⠀⣀⠀⠠
⠀⠀⡄⡤⠤⠤⠤⠤⠄⠀⠆⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠈⠉⠉⠉⠉⠉⢺⠀⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡠⢄⡪⠔
⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠤⣄⣀⣀⣖⣀⣐⣠⡤⠕⠊⠁
⠀⠀⠧⠬⠭⠭⠭⠭⠭⠭⠧⢸⠀⠀⠈⠀⠀⠈⠀⠀⠃⠀⠀⢀⡠⠔⠉⠉⠁

⠀⢰⡶⠶⠶⠶⠶⢦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢸⠃⠀⠀⢰⠶⠶⠶⠶⠖⠒⠒⠒⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⠀⠀⠀⣿
⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡆
⠀⢸⡀⠀⠀⢸⠀⠀⠀⠀⣟⡽⠯⠭⠿⠿⠛⠳⠙⡇⠀⢸⣏⡭⠭⠭⠭⠽⠭⢍⣶⠀⠀⢸⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⢰⣧⡄⠀⠀⠀⠀⠀⢸⣽⠀⠀⢸⠇⠀⠀⢸⡇
⠀⢸⡅⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡏⡇⠀⠸⣿⡇⠀⠀⠀⠀⠀⢸⣼⠀⠀⢸⡀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⣻⠇⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⠇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡀⠀⠀⠀⡇⡧⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⣿⠂⠀⠀⠀⠀⠀⢸⣾⠀⠀⠘⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⢸⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣿⡆⠀⠐⣿⡁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⡇⢼⠀⠀⠀⠀⠀⠀⣿⡁⠀⢰⣿⡇⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⡇⠀⠀⢀⡇
⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⢷⢸⠀⠀⠀⠀⠀⠀⣿⠇⠀⠈⣧⠆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⡇⠀⠀⠘⡇
⠀⢸⡇⠀⠀⠘⡇⢀⢤⡀⢸⣜⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⣻⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠘⡇⠀⠀⠀⡇⢼⠀⠇⠸⠯⠭⠿⠿⠷⠿⠟⠿⠆⠀⠀⠫⠿⠶⠶⠾⠶⠿⠿⠿⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⡜⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠳⠤⠞⢀⣄⣀⢀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⢤⣄⣤⣄⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⢀⠁⡆⢸⢹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠋⠛⠉⢻⢸⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⡇⠈⠒⠁⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⣧
⠀⠈⡇⠀⠀⠸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠎⡄⠀⠀⡇⠀⠀⠀⡿
⠀⢸⡇⠀⠀⢐⡇⠀⠀⠀⠸⢾⠭⠭⠿⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⠭⠭⠭⠭⢶⠇⠀⠀⣗⠀⠀⠀⣟
⠀⢸⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣟
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢰⣖⣒⣒⣶⣶⣶⢶⣶⣶⣶⣾⣷⡾⣶⠶⠶⠶⠶⢶⣶⠀⠀⠀⡗⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⡇⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠘⡇⠀⠀⢀⡇⠀⠀⠀⠀⡿⣿⠾⣿⣿⣿⣿⡿⠿⠾⠶⠿⠷⢾⣓⠺⠶⣖⠚⠛⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⣠⣧⣤⣤⡤⣷⠶⠦⡶⢦⣴⠴⠦⣤⣶⠶⣦⣶⣶⣶⣶⡶⢶⣶⡦⠶⠶⢶⠿⢿⠿⠗⠒⠧⠀⢤⡤⠷

⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⢸⡇⠀⣶⣷⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣷⢰⡶⠀⠒⠀⢀⣶⡤⠤⠤⠀⣿
⠀⠀⠀⠀⢸⣇⣀⣛⣛⣁⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠛⣿⣿⣿⠯⠭⠉⣿
⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠛⠉⠛⠤⠤⠠⣿
⠀⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢰⡿⠛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⢸⡟⢉⣉⡉⣉⠉⡉⠉⠉⢉⣩⠿⣷
⠀⠀⠀⠀⢸⡇⢠⣿⡦⠀⠀⡁⡀⠀⡀⡀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠐⠒⠀⠒⠒⢂⡤⠖⣉⣤⣾⣿
⠀⠀⠀⠀⢸⡇⢀⣀⣀⠀⣀⣀⣀⡀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⡴⠚⣡⣴⣾⠛⠛⣿⣿
⠀⠀⠀⠀⢸⣇⠐⠀⠂⠐⠀⠀⠂⠂⠘⠟⠀⣿⠀⠀⠀⠀⠀⠀⢸⣧⣴⣋⣡⣶⣿⣿⣿⣿⣤⣤⣿⣿
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⣤⣴⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⠉⠀⠀⠈⠁⠀⠁⠀⠀⠐⡤⠀⢸⣿⣿⣫⡭⣿⣿⣷⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⣧⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⢸⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⠄⠹⣄⠀⢠⡄⠀⠀⠢⡖⠀⠀⢸⣿⣿⢸⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡞⠉⠉⠉⡟⠉⠉⠙⡆⠀⠀⡇⠀⢰⠖⠒⠒⢲⠒⠒⠒⢲⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡟⠉⠉⠉⡏⠉⠉⠉⡇⠀⠀⡇⠀⢸⠒⠒⠒⢺⠒⠒⠒⢺⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⣀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠛⢻⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⣧⣤⣤⣤⣧⣤⣤⣤⡇⠀⠀⡇⣶⣼⣀⣀⣀⣸⣀⣀⣀⣸⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠁⠀⠀⢸⠁⠀⠀⢹⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡗⠒⠒⠒⡗⠒⠒⠒⡇⠀⠀⡇⠀⢸⠦⠤⠤⢼⠤⠤⠤⢼⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠧⠤⠤⠤⠧⠤⠤⠤⠇⠀⠀⡇⠀⢸⣄⣀⣀⣸⣀⣀⣀⣸⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣽⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⡀⡠⠤⠤⡀⡄⠰⡄⠖⠖⢰⡤⢂⠗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⠽⢂⡝⡇⢸⡆⢱⡅⠀⠘⠇⠀⠈⢡⢂⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠦⠎⢹⣇⣴⣲⣌⣧⣿⣤⣤⠤⠼⠦⠦⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢠⠤⠤⠴⠒⠒⠶⠒⠒⠒⠒⠚⢻⠀⢰⠃⠀⢠⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⠘⠀⠀⠀⠀⠀⢰⠀⠀⣼⠀⢸⠀⡜⠡⢌⢂⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⡆⠈⡇⠀⠀⠄⠀⠒⠀⠀⢸⠀⠀⡟⠀⡎⠀⠣⢈⠒⢄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢧⠀⠀⡀⠀⣀⠀⠀⣈⠀⠀⡇⠀⡇⠀⠔⢄⠸⢠⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⡄⢸⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠇⢸⠀⠀⠈⢂⡐⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡄⠀⠄⠀⠤⠀⠀⠦⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠆⠀⠲⠀⠀⡇⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠁⠀⠙⠀⠀⠃⠀⠹⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠄⠀⣸⡄⠈⠉⢉⡉⠉⠉⠛⠛⠒⠋⢀⠀⠀⠀⠀⠀⠀⢀⠉⣉⡓⠒⠠⣄
⠀⠀⠀⡠⠒⠒⠈⠀⠀⠀⣀⡴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠉⠀⢀⠖⠚⠋⠁⠀⠀⠀⠀⠀⡔⠀⠀⠈⠉⠉⡇
⠀⢠⠎⠤⠤⠤⠤⠤⠤⠤⢵⠀⡀⠀⠀⡀⠉⢉⡉⣉⣉⡉⠙⠀⠔⠉⠁⢀⠮⠀⠀⠀⠀⢀⣤⡄⡀⠌
⠀⠘⡀⠂⠀⠰⠆⠀⠀⠀⠘⠀⢇⠀⠀⡅⠀⠠⠀⠀⣀⡇⠈⠀⠆⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠂⢀⡀⠀⠀⢤
⠀⠀⡇⢠⠀⠀⠀⠀⠀⠒⠲⡄⠀⣀⣀⡀⠠⠤⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠰⢸
⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⡇⡏⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⢀⠠⢀⡪⠜
⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⣀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠄⠀⠤⠤⠤⠄⣀⣘⣀⠬⠐⠊⠁
⠀⠀⢧⣈⣋⣉⣉⣉⣁⠀⢸⡇⡇⠀⠀⠉⠀⠀⠉⠀⠀⡇⠀⠀⠀⣀⠤⠚⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠦⠤⠤⠴⠶⠶⠤⠤⠤⠷⠰⠒⠉
⠀⡁⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠛⠛⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡆
⠀⠀⠀⠀⠀⠀⢠⡞⠛⠉⠉⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣷⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⣿⣿⣿⠀⠀⠀⠀⢀⡀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⣿⠀⠀⢐⣁⣸⠆⠀⡄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠀⢸⠄⢼⠤⠤⠭⠭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿⠀⠀⠀⠑⠚⠒⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⠀⠀⠀⠆⠰⠀⠆⠰⠀⠀⠢⠘⠄⠐⠐⠂⠀⠐⠆⠶⠐⠀⠆⠀⠰⠀⠶⠐⠔⠒⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣶⣶⣶⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢸⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿
⠀⠀⠀⢰⣶⣦⣬⣍⣉⣛⠛⠛⠃⣿⣿⡟⠛⠛⣛⣉⣉⣥⣤⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⠉⠙⠛⠶⠶⠦⣤⣤⣼⠀⠀⢸⣤⣤⡤⠶⠶⠞⠛⠋⠉⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣛⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣥⠴⠶⠶⠦⣬⡛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣷⣞⣉⣀⣀⣀⣀⣀⣀⣙⣶⣾⣿⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⢸⠀⡇⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣧⣼⣤⣤⣤⣤⣇⣠⣀⣧⣤⣤⣤⣼⣤⣧⣤⣤⣴⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠁⠀⢹⣻⣟⡏⠁⠈⠹⣏⣉⣉⡿⠁⠀⠙⣟⣟⣻⠉⠀⢹⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡟⢻⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠛⣿⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠸⡇⢸⡇⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢿⠀⣿⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠘⢦⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⠞⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⣀⣠⠤⠒⣦⣀⣀⣀⡤⠖⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⠈⠻⣯⠥⠀⠀⠀⢿⠉⣽⠉⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⢳⡀⠀⠀⢸⣠⠇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠈⢀⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢴⣶⡷⠞⠛⠛⠉⠉⠉⠉⢉⣩⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣛⣿⣯⡤⣤⡤⠤⢴⡶⠶⠛⠉⠁⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⣭⡽⠋⡽⠀⢀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡟⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡸⠓⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠰⠃⢀⡄⡦⡀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠰⡀⠀⠓⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⡄⡀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⡉⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠱⠼⠘⠁⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢊⡈⠁⢨⡉⠉⠉⢠⡈⠁⢁⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠈⡇⠀⣸⡀⠀⠀⣸⡀⠀⣸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠠⠏⠉⢻⠋⠉⠉⢻⠋⠉⢻⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢠⡆⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢀⡗⠒⢻⠗⠒⠒⢻⠗⠒⢻⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⢸⠀⠀⢸⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢘⡇⠦⢾⡆⠶⠤⢾⠦⠤⢾⡇⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠸⣇⣀⣸⣀⣀⣀⣸⣀⣀⣸⣷⣶⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡄⢠⡀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠵⠌⠱⠠⠘⠆⠼⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠠⠀⠀⠄⠀⠒⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡴⠒⠋⠙⠛⠒⠲⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣔⡴⠶⣶⠴⢦⣠⣴⣲⠶⠤⡤⣿⣦⡤⠤⠤⢤⣤⣤⣴⣧⣤⣤⡴⣆⣀⣀⢶⣶⡤⠤⢤⣤⣤⡤⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⣠⠞⠛⠁⠠⠾⠋⠉⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠉⠁⢠⡾⠋⠡⡄⠀⠉⠁⠐⠢⡀⡆⢸⡆
⠀⠀⠀⠀⠀⢸⠑⢊⠔⠋⠁⠀⣇⠀⣀⣧⠀⠀⠀⠠⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⠀⢸⡇⠀⠀⠙⢦⡀⣈⡦⠀⣧⡇⢸⡇
⠀⠀⠀⠀⠀⢸⣀⠏⠀⠀⠀⠀⢧⠤⠤⣤⠀⠀⠁⠀⠓⢄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢤⡇⠀⠀⠀⠀⠙⢦⡀⠀⡿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠠⠀⠀⠀⣿⠀⠀⠀⠑⢀⠈⠐⠉⠀⠀⠂⠂⠀⠀⠀⠀⡃⠀⠀⡼⠇⠀⠀⠀⠀⢀⢠⣙⠐⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠀⠑⠔⠉⠀⠀⠀⠀⠀⠐⣠⢦⠀⠀⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠠⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠰⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⢀⡀⡀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠀⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⠀⢫⡀⠀⢀⠀⠀⠅⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⡈⠃⠀⠀⠀⢰⠀⠈⠁⠂⠀⠤⠤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⡳⠠⠀⠀⠤⣤⠆⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢡⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⡆⡀⠈⠑⠪⡥⢤⡌⠀⠀⠃⢸⠃
⠀⠀⠀⠀⠀⢠⠋⢢⠀⠀⠀⣡⡠⠀⠀⢸⠀⠀⠐⠀⠀⠀⠉⡀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠁⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠼⡄
⠀⠀⠀⠀⠀⢸⡀⢀⠑⣄⠉⠀⢸⠀⠀⢸⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⢀⠀⠂⡘⡄⡀⠑⡇⠀⠀⡆⠀⡘⠀⠀⠀⠀⠂⠈⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠸⣅⣨⣤⣈⣷⠤⢬⣀⡀⣸⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠁⠁⠀⠀⡇⠀⠀⡗⠈⠀⠀⢀⠀⠁⡀⠀⢟⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠧⠤⠀⠷⠦⠠⠴⣁⣀⣐⣁⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠶⠾⠛⠋⠉⠉⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠛⠉⠉⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⣀⡀⢠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠛⠋⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⡀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠉⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸⡇⢠⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄⣤⢤⡀⣤⠤⢠⡤⣄⢠⡄⣤⡤⠀⢀⡤⡄⢠⣄⢀⡤⣤⢠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡄⣿⢺⡁⣿⣃⢸⣧⡿⢸⡇⢸⡇⠀⠸⣄⣄⡿⢿⡬⡟⢿⢸⣧⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣅⠀⠈⠁⠈⠙⠳⠾⠛⢓⡶⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣄⣀⣀⣀⣞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠖⠲⠦⠤⠶⢦⠀⠀⠀⠀⠀⢀⣠⠼⠿⠿⠿⠿⠿⡿⠿⢤⡀⠀⠀⠀⠀⠐⣏⠉⠓⠚⠉⠙⢲⠆⠀⠀⠀⢀⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⣳⣤⣦⣤⣴⠃⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⢈⣦⣤⣤⣴⣴⡏⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠻⠛⠿⠛⠦⣀⡴⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢀⠜⠛⠻⠛⠛⠛⠻⣄
⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡴⠒⠛⠓⠲⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⡴⠃⠀⠀⠀⢀⡤⠶⠄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡴⠦⠄⠀⠀⠀⠀⠹⣄
⠀⠀⢰⠇⠀⠀⠀⠀⠸⣄⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⣆⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠸⣄⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⣱⠀⠀⠀⠀⠀⣸
⠀⠀⠈⣇⠀⠀⠀⠀⠀⠙⠞⠁⠀⠀⠘⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠓⠞⠁⠀⠀⠀⠀⣰⠃
⠀⠀⠚⡼⢦⣄⡀⠀⡀⠀⠀⠀⠀⠀⣀⣨⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⠀⠀⠀⠀⠀⠀⠀⣀⡤⢚⢁⢀
⠀⠀⠀⠀⠁⣠⠟⢉⡽⢋⡽⢛⡽⢛⡥⠋⡠⠋⣹⠗⡶⢲⡶⠒⡶⠒⢲⠖⢲⠖⢲⠿⢲⠟⡵⠋⡼⢫⠟⠲⢒⠖⡲⠋⡽⠣⠛⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠀⠈⠀⠈⠀⠀⠁⠞⠴⠋⠀⠚⠀⠐⠃⠐⠋⠐⠋⠐⠃⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠃⠀⠒⠒⠒⠂⢬⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠔⣠⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⡠⠒⠒⠂⠉⠁⠀⠀⣠⣾⣿⡿⠋⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠐⠤⣀⣴⣁⣀⣀⣀⣀⠀⣠⣾⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠒⣩⣾⣿⡿⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠀⠀⠈⠺⣿⡿⠋⡠⠎
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡀⠀⠑⢄⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠊
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦⡀⠀⠑⢄⠀⠀⡠⠊
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠑⢎
⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠐⠋





⣏⠉⣹⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣟⣛⣿⣟⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠐⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⡿⠀⠀⠀⠘⠣⠈⠉⠳⠲⡦⠤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⣐⣎⣹⣍⣙⣴⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠙⠒⠒⠺⠶⠦⠤⠤⠤⠃⠄⠂
⠀⠀⢀⢠⠤⣄⡈⠉⠉⠉⠉⢀⠀⡀⢠⡀⠀⠀⠀⠀⢄⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⣀⠤⠀⠒⣒⠀⠂
⠀⠀⠈⠛⠷⢿⣌⣑⢀⣤⡒⣩⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠐⢠⠴⠐⠒⠖⠻⣯⣀⡤⠔⠊⠁
⠀⠀⠀⡀⠠⣄⣀⠀⣽⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢝⡳⠤⠶⢒⣉⣉⣀⣀⡀⠀⢀⡀
⠀⠀⠀⠙⠷⣶⣏⣑⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⠤⣠⣽⣦⡤⠤⠤⠤⠊⠀⠈
⠀⠀⠀⠀⠀⢀⢤⠛⡷⠲⢄⣠⠀⠀⢀⠀⠀⡀⠀⠀⡀⠀⡀⢀⠀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠀⠀⠁⠀⠀⠁⠀⠀⠀⢸⣷⣾⣅⠀⣀⣉⣙⣯⣁⣀⣀⣈⡆⠀⡀
⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⡿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣼⣿⠇


⠀⠀⡀⠀⠀⡀⠀⠀⢀⠀⢀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣿⣿⡿⠿⠿⠛⠛⠛⠛⠻⠿⠿⣿⣿⢷⣶
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣶⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣾
⠀⠀⠀⠀⠀⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣉⣉⣉⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣡⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠛⠛⠛⠛⠃⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠇⠒⠒⠒⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⠒⠺
⠀⠀⠀⠀⢂⣀⣀⣁⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣈⣀⣀⡐
⠀⠀⠀⠀⠀⠹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠏
⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠘⣂⢀⢀⡂⠀⠀⠐⣀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⢀⡀⡂⠀⠀⠀⣂⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⠲⠶⢤⣿⣿⣿⢠⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⡄⢸⣿⣿⣤⠤⠤⢤⣸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡀⢸⣿⣿⣿⢸⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⡇⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡀⢸⣿⣿⣿⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⣿⣿⢿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⠿⠶⠿⠾⠃⠀⠀⠀⠀⠀⠸⠷⠿⠶⠇⢸⣿⣿⠿⠶⠿⠶⢿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⣀⣀⣀⣿⣿⣿⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢸⣿⣿⢀⣀⣀⣀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⢸⠀⠁⢸⠀⠀⠀⡤⠠⠀⠀⡇⠉⠀⡇⢸⣿⣿⣿⠀⠉⢸⣾⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠁⢸⣿⣿⣿⢸⠀⠁⢸⠀⢸⠀⣤⠀⡇⠀⡇⠈⠀⡇⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣼⣤⣦⣼⡄⢸⠀⠤⠀⠆⢠⣧⣴⣤⡇⢸⣿⣿⣾⣤⣦⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡿⠿⠿⡀⠀⠀⠀⡿⠿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠿⡁⠀⠀⠀⡸⠿⠿⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠋⢹⡄⠀⣀⡤⠖⠋⠉⣉⠛⠷⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⣀⣀⠀⢷⣉⠀⣀⠴⠉⠊⠍⠀⠁⠀⠀⢀⣠⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣰⠞⠉⠀⢘⣧⠤⠾⠋⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠇⠀⢹⡠⠴⠚⠉⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⣸⠤⠴⠒⠋⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⣉⡤⠒⠺⢉⡍⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⣶⣏⣁⣀⡀⠼⠂⠡⠐⠊⠁⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠉⣟⠙⠛⠓⠶⢤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠲⢿⣄⣀⠀⠀⣼⠀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⣤⣰⠉⠲⡆⣀⡀⠆⡤⠀⠀⢂⠄⡀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡀⠀⠀⡄⠀⡀⢸⡤⠣⢽⠄⣶⡿⣠⣖⢋⣬⣩⡨⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⣟⡛⠛⠛⠛⠛⠛⠻⠤⠒⠚⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠤⢴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠉⠻⢻⡟⠉⠉⠉⢹⠟⠉⠁⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠶⠶⢦⣤⣤⣤⣤⣤⣤⡴⠶⠶⠶⠚⠛⠋⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⢀⡀⣿⣿⢈⣀⣀⠀⠀⠀⠀⢀⣀⡀⢸⣿⣀⣀⠈⠋⠀⠀⢠⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠁⠤⠤⠀⠈⠛⠶⠄⠘⠛⠓
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⠀⠀⣿⣿⢈⠈⢀⠀⠀⠀⠀⠀⠁⠀⢸⣿⢀⠈⠀⠀⠀⠀⠘⠳⠂⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡠⡀⣿⣿⣠⣀⣀⠀⠉⠁⠁⣀⣀⡀⢸⣿⡇⠀⠀⢰⡞⠳⠆⠀⠀⠀⠀⢿⣌⡉
⠀⠀⠀⢀⡤⡀⠀⣷⣧⣤⠅⢠⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⢈⡙⢻⡆⠀⠀⠀⠠⣦⣈⡿
⠀⠀⠀⠈⣙⡦⠀⣸⣿⠯⡜⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣿⣯⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠭⠭⠿⠿⠿⠿⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢻⠗⠛⠛⠛⠛⠒⠒⠒⠒⠒⣒⣶⣶⠶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠤⠄⠒⠒⠈⠉⠉⣀⠀⢸⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢠⠒⠒⠂⠉⠉⠉⢸⠀⢸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠘⠒⠒⠒⠒⠒⠒⠉⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢠⠖⠒⡆⠠⠖⠒⢲⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠈⠀⠀⢸⠀⢸⡇⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣇⣸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡛⢻⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⢸⠀⠀⡇⠀⠀⠀⢸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣇⠈⠒⠒⠃⠠⢄⣀⣸⠀⢸⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠉⠁⠒⠒⠤⠄⣀⣀⠀⢸⣇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⣄⣠⣤⣶⣶⣿⢿⣷⡄⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣛⣻⣿⣿⣶⣶⣾⣿⣍⠉⠁⠀⣴⣿⠵⠞⠛⠉⠁⢀⣠⣬⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⡉⠉⠉⣴⣿⡛⠋⣹⣦⣤⣾⡿⢿⣿⣶⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣄⠘⢿⣿⠿⠿⠛⠋⠁⣰⠏⠀⠀⠀⣠⡿⠋⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣬⣿⣦⣀⡀⠀⠰⠏⠀⣀⣠⣾⣯⣴⠶⠛⢛⣛⣩⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠿⣯⣭⣭⣉⣉⣙⣛⡛⠛⣛⣭⣿⣿⠟⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠁⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠋⠉⠀⠘⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⣠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣀⣬⣷⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⢀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠛⠋⠙⣧⠀⠀⠙⠛⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁⢻⡄⠀⠀⢸⣆⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠿⠟⢿⡛⠉⠙⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⣷⠀⢀⣴⡟⠁
⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣤⡀⠀⣷⣀⣀⣠⣼⣷⠿⠋
⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⣿⠟⠛⠛⣿⡀
⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠹⡇
⠈⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣶⣶⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⡾⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠒⠻⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠐⠚⢻⡿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠚⢿⠟⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠐⢺⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠂⠐⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⣰⡄⠘⠃⠀⠀⠀⠀⠀⠀⠄⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠤⢾⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠠⠴⢾⣷⠶⠤
⠀⠀⠀⠀⠀⠁⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠰⡦⠈⢻⡿⢁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡴⠊⠉⠉⠙⠓⠒⠂⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⢤⡀
⡇⠀⠀⠀⠤⢄⠄⠒⠄⡠⠄⡄⠤⢄⡠⠤⢤⠀⠀⢄⠀⠀⠤⡀⣀⣀⠀⢀⡀⡀⢀⠀⠀⢱
⡇⠀⠀⡇⠀⢏⠀⡁⠆⡁⠸⡅⠀⠀⡭⠀⢇⠁⠀⠊⠀⠀⠰⣁⢐⠂⢄⢈⣼⡀⠆⡁⠀⢸
⡇⠀⠀⠈⠐⠊⠑⠋⠊⠙⠒⠂⠒⠈⠑⠒⠂⠀⠊⠀⠀⠈⠐⠉⠀⠈⠉⠉⠈⠁⠉⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⣶⠀⠉⠟⠃⡆⠀⠀⠀⠄⡄⠀⠠⡄⢠⡀⡀⠀⡄⠀⡀⠀⢄⢀⡀⢀⠀⠀⠀⠀⢸
⡇⠀⠀⣧⠀⠀⡀⠀⡇⠀⠀⠀⠁⠁⠀⠈⠁⠀⠁⠈⠀⠀⠈⠓⠀⠃⠘⠃⠉⠃⠀⠀⠀⠸⣄⡀
⡇⠀⠀⠋⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡄⢀⠒⠀⠠⠀⢄⠄⢄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄⡀⠀⠉⠒⠤⣄⡀
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⣅⣈⠁⣀⡁⠅⢸⠸⠈⣄⠑⠁⡔⣔⣢⣒⡄⠀⠈⠚⠧⣨⠒⠂⠀⠀⠀⠈⠑
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠓⠀⠐⠓⠀⠚⠀⠂⠀⠈⠀⠉⠉⠁⠉⠁⡀⠀⠀⠀⠀⠉⠚⠌⠘⠀⠀⢀
⠀⠓⠦⠤⠤⠤⠤⠤⠤⠤⠤⡄⠠⠤⢤⠤⠤⠤⠤⢠⠀⡤⠤⡤⠀⠤⠤⠤⠤⠄⠠⠀⠊⢐⡐⠠⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⠈⠉⠙⠃⠢⡘⡐⠐⢂⡄⠀⠀⠔⠢⡀⠀⠀⠀⠀⠀⠀⠓⠐⡟⢠⠄⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢤⣀⡀⠈⠉⠁⠀⠀⡄⠘⢀⠴⢈⡵⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣄⡀⠈⠉⠀⢄⣛⠀⡀⢠⢀⠂⠠⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⡀⠀⠙⠒⠂⢀⠁⠤⠈⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⠈⠉⠀⠀⢠⠎

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣾⣷⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣼⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣷⠤⠀⠀⠹⠉⠀⠀⠀⠀⢠⣿⣧⡀
⠀⠐⠒⠺⢿⣿⣿⣿⣿⡷⠶⠆⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠰⠶⠾⣿⣿⣿⣿⡶⠶⠆
⠀⠀⠀⠀⠀⠹⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠻⠋⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣀⠀⠀⠀⠀⠀⠀⠈⢿⡏
⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣇⠀⢘⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⣠⠀⠀⢠⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢈⠿⠿⠛
⠀⠀⠛⣿⠟⠀⠀⠹⠃⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠈⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣼⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠙⣿⠛⠁⠀⠀⠀⠐⢦⣠⣿⣧⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⡉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠈⢿⡏⠉⠓⠀⠀⣰⣿⣶⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⠻⠓⠠⢴⣿⣯⠄⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⠈⠛⠁⠀⠀⠀⢀⣀⣼⣧⣀⡉⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠂⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⢡⣤⣾⣧⣤⠄⠀⢤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⠦
⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⢹⡿⣿⡇⠀⠀⠀⠀⠀⠉⠉⣿⣿⠏⠉
⠀⠐⠒⠒⠿⠿⠿⣿⣿⠿⠿⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣤⠀⠀⠀⠀⠀⠚⠻⠿⣿⣿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠰⠶⢾⣿⣿⡶⠶⠆
⠀⠀⠀⠀⠀⢰⠀⠀⠤⣤⣿⡤⠤⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⠀⠀⠀⠀⠀⠐⠶⢾⣿⣿⠷⠆⠀⠀⠀⠀⠀⠀⠲⠾⢿⣿⡿⠿⠒⠂⠀⢸⡇
⠀⠀⠀⠤⠤⣾⡦⠤⠄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡄⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣿⣿⣦⡤⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠠⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠏⠀⣀⣤⣾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠶⠶⣿⣿⡷⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⢀⣀⣀⠀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣯⣾⣿⣿⣆⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡟⠉⠀⢀⡀⠀⡀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⡇⢸⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠻⠇⢸⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡇⢠⣤⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⣿⡇⢸⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠙⠃⠘⠟⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⢒⡶⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠄⠀⣠⠔⢋⣴⣶⣤⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡤⠊⢁⠴⠋⠀⠙⠻⣿⣦⣄⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⡈⠳⢄⡀
⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⢾⣷⡄⣰⣆⢰⣄⣷⢸⡾⣁⠀⠈⠛⢿⣷⣄⡙⠦⣄
⠀⡴⠋⠀⠀⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⡴⠋⠀⠘⠯⠷⠏⠙⠚⢈⢋⡈⠠⠤⠐⣒⣂⣀⣉⣭⠿⠆⠈⠓⢦⡀
⠀⠉⠓⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣴⠯⠤⠤⢀⣒⣀⣨⡭⠭⠴⠖⠒⠚⠋⣉⣉⣉⣭⣤⠤⡴⠖⠒⢲⡋⠉⠁
⠀⠀⠀⠀⡇⠈⠉⠓⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠰⠿⠒⢒⣋⣉⣩⣤⠤⠤⠶⠒⠒⠛⠋⠉⠉⣏⢠⣄⣤⣈⡇⢧⣷⠻⢼⠃
⠀⠀⠀⠀⡇⣂⠔⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⣾⣥⠤⠖⠒⢞⣻⢩⣁⣀⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⠈⢹⠀⢸⢸⢀⢸
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⣅⣽⠛⣯⡼⠘⡶⠏⢈⢹⠋⠀⢀⣴⢿⡿⣿⣷⣄⠀⠀⢸⠀⠀⢸⡆⢸⢸⢸⢸
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇⠀⡇⠀⢸⢸⠀⢰⠏⠀⠀⡆⠈⢻⣿⡆⠀⢸⠀⠀⠘⡇⢸⢸⢸⣸
⠀⠀⠀⢸⡇⢀⣶⣷⡆⠀⠀⣀⣀⡀⠀⠀⡇⠀⢸⢸⠀⠀⡇⠀⡇⡀⢸⢸⠀⡾⠀⠀⠀⡇⠀⠀⣿⣷⠀⢸⠀⠀⠀⡇⢸⢸⢸⣿
⠀⠀⠀⢸⡇⢸⣿⠀⡇⠀⣰⣿⠟⢷⠀⠀⡇⠀⢸⢸⠀⠀⡇⠀⡇⡇⢸⢸⠀⡇⠀⠀⠀⡇⠀⠀⢻⣿⠀⢸⠀⡀⠀⡇⢸⢸⢸⣿
⠀⠀⠀⢸⡇⢸⡏⠀⡇⠀⣿⡟⠀⢸⠀⠀⡇⠀⢸⢸⢰⠀⡇⠀⡇⡇⢸⢸⠀⡇⠀⠀⠀⡇⠀⠀⢸⣿⠀⢸⡄⡇⠀⡇⢸⢸⢸⢿
⠀⠀⠀⢸⠀⠸⢧⣀⡇⠀⣿⡇⠀⢸⠀⠀⡇⠀⢸⠈⠈⢰⡇⠀⡇⠁⠸⢸⠀⡇⠀⠀⠀⡇⠀⠀⢸⣿⠀⠈⡇⡇⠀⡇⢸⢸⢸⢸
⣠⡤⠴⢾⠀⠀⠄⠈⠁⠀⠿⣇⡀⢸⠀⠀⡇⠀⢸⠀⠀⢸⡇⠀⡇⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠀⢸⣿⠀⠀⡇⡇⡀⡇⢸⡞⠘⢸
⡏⠉⠓⠾⣄⡊⠈⠂⠀⠀⠀⠀⠉⠛⠀⠀⡇⠀⢸⠀⠀⢸⡇⠀⡇⠀⠐⢸⠀⡇⠀⠀⠀⡇⠀⠀⢸⣿⠀⢀⡇⡇⠅⡷⢾⡁⠀⢸⣗⠦⣄
⠳⣄⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢸⡇⠀⡇⠀⢸⢸⠀⣇⣀⣠⠤⠷⠶⠒⡚⢟⡫⣹⣗⣷⣞⡿⢺⣓⣒⣫⠟⠉⠢⢝⠲⢤⡀
⠀⠀⠙⠦⣄⠀⠀⠀⠀⠉⠓⠲⢤⣀⠀⠀⡇⠀⣸⣀⡀⣸⣧⢴⣇⣀⣸⡼⡟⢍⡣⡀⠀⠀⠐⠨⠄⠀⠈⠪⣃⣍⣁⣀⣀⣤⡤⠤⠴⠶⠒⠒⠛⡏⢹
⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠈⠙⠲⢷⣞⠳⢿⠿⠗⠋⠈⠓⠛⠛⠋⢁⣀⣈⣪⣲⡤⠤⠴⠶⠒⠒⠛⠉⡏⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⠀⡌⠙⠲⢭⡦⡤⠶⠒⠚⠋⠉⠉⠉⠀⠀⠁⢠⠀⠀⠀⣀⣀⣀⣤⠤⣶⠶⣒⠚⠋⠉⠉⠉⠁⠀⠀⠀⠈⠊⢝⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⢡⠀⠃⢸⡇⣇⣀⣀⣠⠤⠤⠴⠶⠒⠒⠛⠻⡛⢍⠉⠁⠀⠀⠀⠀⠀⠉⠒⢁⣒⣤⣀⣠⠤⠤⠤⠴⠶⠒⠚⢻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠙⠯⣍⠢⢄⠀⠀⠀⠀⢀⣀⣀⣀⣠⣵⣵⢦⠤⠖⠒⠒⠚⠋⠉⠉⠉⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀⠀⠀⠙⡖⢻⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⣀⣀⣀⡤⠴⠼⠶⠖⠒⠒⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⣇⣘⣀⣀⣠⣤⠤⠤⠤⠖⠒⠚⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉⠀⠀⢀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠉⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠞⠛⠉⠁⠀⠀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⣀⣠⣼⣇
⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠁⠀⠀⣀⣠⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⣀⣠⣤⠶⠚⠋⠉⠀⠀⢻⡄
⠀⠀⠀⣴⠟⠋⠉⠀⠀⢀⣀⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⢀⣀⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⢻⣄⣤⡴⠖⠛⠉⠁⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠈⣯⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠸⣧⡤⠶⠚⠋⠉⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡴⠚⢛⣷⠞⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣿⠀⠀⠀⡟⠀⠀⣾⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⣿⠀⠀⠀⢳⣄⠀⠹⣆⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⣀⣿⠀⠀⠀⠀⠉⠛⠛⠋⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠋



⠀⡁⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣫⣴⠾⠛⠛⠻⣿⣿⣤⠤⠴⠶⠤⢃⠈⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠟⠁⠀⠀⠀⠀⠈⣏⢹⠦⣀⢀⠀⠠⠀⢙⣢⠀⠰⠊⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠋⠀⠀⠀⠀⠀⠀⠠⢈⣁⡠⢤⠳⠒⠃⠉⠻⣿⠠⠰⠰⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠤⠈⠁⠀⠀⠀⠀⣀⠁⠀⢐⣀⣠⣱⡶⢻⠛⢹⠸⠃⠀⠀⠀⠀⢀⠴⣿⠀⠀⢀⠴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⢀⠀⠄⢐⣢⣄⠀⢱⣶⢿⡛⠋⠉⣈⠡⠀⠀⠘⠀⠀⠀⣤⠀⠀⠈⢦⣻⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⢀⠔⢈⣥⣶⣿⣿⡿⡿⠐⠸⣏⠐⠁⠄⢈⣴⡶⠈⠀⠀⠀⠀⠀⠂⠑⠄⠀⠀⢹⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⠀⠀⠀⠎⢠⣿⣿⡿⣯⣿⣧⣿⠠⠀⡏⢀⠀⣰⣾⡿⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠸⢸⣿⣿⣟⡿⠋⢿⣷⠶⠚⡇⠀⠀⠟⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠸⠀⣿⡿⡿⠇⠀⠈⢿⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⣇⣟⡠⠒⠦⢼⠆⠰⠃⠀⠀⣟⣃⡒⠤⢀⠀⠀⠀⠉⡅⠀⠀⠀⡀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⣿⣿⡜⢱⡁⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠑⠃⠀⠀⠓⠀⠢⢄⠘⠆⢸⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⣿⢻⡆⠮⡁⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⢸⠀⠀⠀⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⢻⣨⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢈⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡁⢸⡇⡀⡀⠀⠀⠠⠀⠀⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣴⣿⣿⣟⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⢸⣿⢁⡇⠀⠀⣰⡆⠀⢺⠉⠢⡄⠠⠐⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠟⠏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⡀⠀⠀⢸⠈⣿⣿⣿⡶⠀⡿⡇⠀⢸⡄⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣶⣤⣀⠸⠅⣿⣻⣿⣇⡆⡇⣿⠀⢸⣷⣀⣠⣴⣶⣿⣿⣯⡛⠲⢭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠛⠿⢿⣦⣿⣿⣿⣩⣛⣧⣽⣤⣾⣿⣿⢿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠦⢶⠝⠻⠿⣿⣿⣿⣿⣿⣿⡿⢿⡻⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⢁⠀⠀⠀⠉⠋⠛⠙⠢⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣴⣶⣶⣶⡶⠶⠶⠶⠿⠶⠾⠿⢿⠿⠿⠿⠿⠿⠟⠿⠿⠻⠛⠛⠛⠻⢿⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⡀⠀⠀⣀⣀⡀⠀⢀⣀⡀⠀⠀⠀⠘⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠻⡇⠀⢸⣿⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣯⢿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣧⠀⠠⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣆⢺⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢹⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣻⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣛⢽⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡹⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⠀⠀⢻⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠤⢢⡄⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⢀⠀⣿⣸⣧⠀⣀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠐⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠑⢿⣆⡬⢤⣤⣥⣭⠂⢀⠀⢰⠀⠀⢸⡇⠀⠀⠰⠂⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣽⠀⢰⣿⣠⡸⣷⣀⣀⣀⣩⣠⣾⠀⠀⢸⣗⣒⣒⣒⣒⡷⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢻⠰⠞⠛⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣻⠀⠓⠤⠛⢠⣦⣶⣶⣶⡶⣶⣶⠀⠀⢶⣶⠶⠶⠶⠶⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢨⠀⠀⢸⡗⠀⠀⠀⠀⡃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠂⠄⠀⠀⠸⣿⠀⠀⠀⢀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⡇⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣵⡂⣀⠀⠀⠀⣿⠀⠀⠀⢀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡄⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠘⣿⡇⠀⠀⠀⠀⣿⣄⣀⣀⣉⣀⣸⠀⠀⢸⣷⣀⡀⠀⠀⣤⡇⠀⠀⠀⡄⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡃⠤⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠀⠀⠈⠉⠈⠉⠉⠉⠉⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⡀⢿⣇⣀⢀⠀⡀⡀⢀⠀⠀⡀⣀⣀⡀⢀⠀⠀⡀⣀⣀⢀⠀⠀⠴⠖⢠⡇⠀⠀⣀⣘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣭⡭⠱⠛⠛⠛⠛⠛⠻⠛⠿⠛⠿⠿⠽⠧⠽⠬⡧⢤⠶⠶⠶⠾⠶⠶⠶⠿⠾⠷⠶⠤⠭⠬⠴⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠉⡍⠉⠁⠛⢛⠙⢋⠜⢋⠗⠠⠖⠠⠖⡰⠆⡰⠆⠶⠲⠆⠴⢢⠦⢠⠄⡤⢀⢄⠀⢠⠤⠴⢠⠄⠶⠰⠖⠰⠂⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡔⣠⢂⠐⡀⠐⡡⢊⠥⢊⠠⢊⠄⢊⠤⢊⠔⡢⠀⡠⢚⠔⡡⠒⡡⢊⠔⡨⠊⢔⠤⠂⠀⡡⢊⠔⡡⠂⠀⡠⣚
⠀⠀⠀⠀⠀⡀⠤⠔⠊⠁⠁⠈⠉⠉⠙⠃⠚⠒⠛⠒⠓⠶⠃⠶⠃⠶⠅⠾⠤⡾⠶⠷⠞⡴⠿⠶⠷⠶⠶⠳⠗⠶⠒⠷⠲⠧⠾⠖⠺⠚⠛⠙⠢⣄
⠀⠀⠀⠀⠛⠩⠂⠦⢤⠄⣤⣠⡄⣠⣄⣠⣀⣀⣀⣀⣀⡀⢀⡀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡄
⠀⠀⠀⠀⢸⠑⠡⠒⠤⠢⠔⡡⢒⠤⢂⠤⠊⠄⠈⠠⢊⠡⢋⠝⣉⠝⡩⢋⠉⠉⠝⡉⢝⠉⠋⠝⡋⠋⠙⠋⠜⠁⠟⡡⢋⠉⡩⢉⠍⡩⢋⠍⡩⠉⢋⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠉⠀⠁⠐⠁⠊⠀⠀⠔⠁⠈⠀⠊⠐⠁⠐⠀⠂⠐⠀⠊⠁⠂⠠⠊⠐⠀⠂⠀⠀⠈⠀⠁⠈⠀⠁⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⣀⣤⠀⢀⠔⠚⢩⠂⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠈⠁⠀⡁⠁⠀⠀⠀⢩⠕⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣁⣀⡏⣠⣡⣴⡖⣓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠘⡄⢀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⣤⣬⣿⣛⣛⣉⣭⠽⠿⣕⠤⣀⠀⠀⠀⠀⠀⡠⠰⡄⠀⠀⠀⠰⠁⠀⠀⢀⠛⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠛⠀⠒⠐⠒⠖⢺⠒⠳⣟⠷⠀⠅⣈⠀⠀⠀⠀⡎⠀⠀⢡⠀⠀⠀⠆⢀⠀⠀⠌⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⢐⡇⠘⣀⡀⠀⠈⠈⢣⢱⡀⠀⠱⠀⠀⢰⠀⠠⠀⢸⡀⠀⢀⡀⠐⡂⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡔⢉⣀⡀⠀⠀⠈⠳⡄⠐⠈⢣⡵⠀⠀⠑⢄⠘⡄⢔⠀⠀⡇⠀⠸⠀⠅⠒⠠⠀⠥⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢪⡀⡃⢸⠉⠑⣀⣁⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⢐⠀⠀⢰⠀⡠⠀⠭⠰⠆⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠘⠢⡈⠁⠘⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠱⣯⣄⠢⠀⢸⠰⢁⡒⡨⠋⡒⠂⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡤⡄⠀⠀⡜⠒⢄⠙⡆⠀⠀⢀⡡⢂⠔⠠⠂⣀⠄⢹⣎⣀⣶⠥⠂⣔⡁⠤⠖⠂⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠘⢄⠘⠒⠀⠀⠀⠊⠀⡘⠀⠀⡠⠒⠡⢊⢔⠞⡡⠖⠉⠀⢀⠤⠒⠒⠒⢤⡎⠀⠀⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡂⠀⠀⠀⠈⠑⠒⡄⡐⠒⠀⠂⠁⠀⠀⣀⠄⡂⢑⠔⠋⠀⠀⠀⠀⠘⢔⠢⣀⡠⠾⠁⠂⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠡⠂⢀⠀⠀⠀⠉⠁⠀⠀⠀⡀⠔⡈⠔⣈⠔⠁⠀⠀⠀⠀⢀⠀⠑⠀⣽⠫⠭⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⣰⡥⠞⠚⠛⠛⠉⠁⠀⠋⠓⠒⠂⠈⠁⠀⠀⠀⠀⢠⠒⠀⣀⠴⣙⠲⠈⠁⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⣠⠺⢅⡒⠠⠈⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⢀⡜⠭⣑⠂⠈⠁⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠈⢀⣤⠪⠌⠁⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡀⠀⠀⠀⠀⢀⡀⡀⠐⠁⠔⡡⠞⠁⠔⡀⠀⠠⠔⠺⠉⠂⠁⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⠄⣀⡀⠀⠨⠊⡠⠊⣠⠊⡀⠊⢡⠊⠐⡡⠀⢀⣀⠤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠉⠐⠒⠠⠾⣡⣊⣀⠔⡡⠊⢔⠐⠊⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠈⠉⠐⠂⠤⠄⣀⡀⠀⠀⠀⠈⠈⠁⠐⠒⠢⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡁⠄⠂⢀⡀⠀⠉⠑⠒⠠⠄⢄⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢁⠠⠐⠀⣀⠤⠒⢉⡠⠔⢀⡠⠄⢀⠉⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠤⠂⢁⠄⠂⣁⠄⢊⡡⠐⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠁⠐⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⢀⡠⠀⢂⡀⠄⠒⡀⠄⢒⡀⠤⠐⣈⠠⠐⣀⠡⠐⢈⠡⠐⢂⠡⠀⢂⠨⠐⠂⠡⠄⠂⠀⠀⠂⠁⠐⠀⠡⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡐⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠂⠠⣿⣿⡇⡖⠒⠒⠀⠀⠀⠀⠀⠀⠐⠒⠒⡂⢸⣿⣿⠐⠒⠒⣶⣿⣿⠆⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⠀⠂⠰⣿⣿⡇⡇⠐⠐⠀⠀⠀⠀⠀⠀⠐⠐⠀⡇⢸⣿⣿⠐⠐⠀⣿⣿⠁⠀⠀⠀⠀⣀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣗⣀⣆⣸⣿⣿⣇⡇⠤⢀⡄⠀⠀⠀⠀⠀⡠⠠⡄⣇⢸⣿⣿⣄⢠⢄⠀⠀⠀⠀⠀⢠⣿⠛⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢠⣤⣤⣤⠀⠀⠙⠿⢶⡤⠀⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠀⠁⢠⣿⣿⡇⡆⠒⠐⠀⠀⠀⡀⠀⠀⠐⠐⠂⡆⢸⣿⣿⠐⠐⠂⠀⠀⠀⠀⠀⠰⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠘⣿⣿⡇⡇⠀⠐⠀⢀⣀⣀⣀⠀⠐⠐⠀⡆⢸⣿⣿⠰⠀⠀⠀⠀⢀⣤⣀⠀⠀⠉⠀⠀⠀⢀⣴⠾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣗⡀⢆⣰⣿⣿⣧⣃⣤⣀⡄⠀⠀⠄⠀⠂⣄⣠⣄⡃⢸⣿⣿⡄⠀⠀⠀⢰⣿⠉⠛⠃⠀⠁⠀⠀⠀⠈⢿⣦⣄⡀
⠀⠀⠀⣀⡀⠀⠨⣭⣽⣤⣠⠆⠀⣀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⠀⠀⠙⠻⢷⣦⠀⠀⠀⠀⠀⢠⣄⠀⢹⣿
⠀⠀⠺⣭⣉⠀⠀⢻⣿⣿⢏⢂⠣⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣤⣼⠟⠀⠀⠀⠀⠀⠀⠙⠿⠛⠁
⠀⠀⠰⢤⠿⠀⢀⣾⣿⣿⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣤⣤⣤⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠠⠤⠄⢀⠀⣀⡀⠀⠀⠀⠀⠔⠒⠔⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠒⠖⠶⠶⠤⠤⢤⣀⣉⣀⣁⠀⠐⠒⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣠⠆⠠⠀⣂⠀⠀⠀⢀⣈⡉⠉⠙⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢻⠀⠀⠀⠀⠸⣿⠈⠉⠀⢻⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢸⠀⠠⠂⠀⠀⣿⠀⠀⠀⢸⠀⠀⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠀⠀⢸⡇⠀⠔⠀⠀⣿⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣄⢄⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡼⢻⢸⡇⠀⠀⠀⡄⣿⠀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠷⢮⠞⠿⢳⣶⣦⠇⣿⣤⢄⣀⣨⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⡇⠁⠁⣄⣀⡀⠀⠀⠉⠈⠉⠛⠚⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢸⡇⠀⠀⡏⢈⣐⡉⠓⠶⠤⢤⣤⣀⡄⠀⠀⡇⢠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢰⡂⢸⡇⠄⠉⠡⠒⠂⢤⠈⠃⠀⠀⠃⠘⠀⢤
⠠⣄⣀⠀⠀⠀⠄⠀⢸⠀⣹⡇⠀⢸⣇⠸⠷⢦⣤⣀⣀⠀⢸⠀⠀⠀⠠⠀⠸⠀⠹⠂
⠀⠀⠈⠙⠓⠢⢤⣄⣸⠀⣿⡇⠀⠈⠉⠛⠻⠶⢶⣤⣭⣉⣛⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠑⠲⡢⠤⢄⣀⠀⠀⠀⠀⠈⠉⠙⠃⠀⣸⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠕⠃⠒⠦⠤⣄⣀⣘⣻⢀⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠂⠜⠖⡲⢄⣀⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠫⠲⣶⢤⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡠⣄⠀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠀⠀⢠⡀⠸⠀⠀⠀⠀⢀⣸⠀⣀⠔⠋⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⢇⠦⠀⠀⠀⠳⡀⡇⠀⠀⢀⡄⠈⠛⠭⣀⢄⠀⠀⠐⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⢆⣰⠎⠍⠀⠙⣿⣟⣁⠈⠀⠀⠠⠓⡼⠦⠑⠎⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢻⣄⠉⢀⣤⣤⡶⣶⡁⠈⠙⠪⢲⡀⡀⡐⣰⣇⣠⣿⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⢷⣤⠃⣤⣄⠎⠀⠫⣄⣠⡀⠀⠑⣷⡼⠟⣫⡴⠞⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠢⢌⡓⠿⠵⣦⣤⣄⣀⠤⠼⠧⠤⠚⠛⠋⣁⣠⠖⠋⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠐⠒⠚⠋⠉⠁⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠂⠀⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠒⠦⠤⠤⠤⠴⠖⠋⠉⠉⠁⢠⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠃⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⢀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⠤⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⠡⣄⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠒⠒⠀⠀⠐⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠖⠶⠲⠖⠲⠖⠶⠲⠖⠶⠒⠶⠲⠖⠶⠲⠖⠲⠖⠶⠲⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠥⠤⠤⠬⠤⠥⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠲⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⣢⢴⠀⣦⡆⠐⢰⠀⡦⠐⢐⢀⡄⢴⠰⣴⠀⠂⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣧⣀⣀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡗⡀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢤⡀⠀⠀⠀⡀⠀⠀⠠⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠻⣾⣄⠀⣧⠀⢂⡀⠀⠀⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⢸⣿⣷⣽⣶⣿⣾⡀⠰⣄⡀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣀⡴⣶⣾⣿⣿⡿⣿⠛⠙⠻⣶⣾⣿⣶⣶⣶⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠉⣾⢏⡟⣸⠑⠃⠉⣀⠀⠀⢆⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⣿⡾⠱⠇⢀⡀⠀⠀⠀⢀⣬⣄⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⢀⣀⢤⡤⠤⢠⠄⠤⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢰⣷⣤⣤⠖⠛⢋⠉⠉⠉⠉⠉
⠀⠀⠀⢀⣴⣦⣶⣿⣾⣷⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠓⠈⠀⠀⢠⡍⠉⠀⠀⠀⣼⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡟⠁⠀⠀⢀⣤⣿⣷⣤⣴⣤⣼⠿⠛⠛⠛⠛⠛⢅
⠀⠀⠀⣿⡿⠋⠟⠿⠟⢻⡇⣇⠀⢸⡶⠖⠂⠀⠀⠀⣀⣤⣾⡟⠀⠀⠀⠀⣾⡿⢿⡏⡟⠋⠁⢂⢺⡆⠀⠀⠀⠀⡨⡇
⠀⠀⠀⡀⢁⠀⡀⣀⢰⢸⣇⢸⠀⠈⣧⣶⣤⣄⣐⣶⣿⣿⠏⠀⠀⠀⠀⠀⢿⣧⣼⡇⢳⣦⣀⣼⣾⣶⣶⣿⣿⣿⣿⡇
⠀⠀⢸⣧⣸⣼⣧⣿⣾⣿⣿⠸⡆⠀⢸⣿⣭⣷⣶⣶⣿⡟⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣾⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⣇⠀⠀⣿⣿⠿⠿⠿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡄⠀⢸⡇⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠠⣤⣄⢸⣇⢸⡇⠹⣿⡇⠀⠀⠀⠀⠀⠤
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢧⡀⠀⠿⠀⠀⢸⣯⣄⠀⠀⠀⠀⠀⠀⠻⣿⣾⣿⣼⣧⣾⡿⠇⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⠀⠀⠀⠀⠈⢸⣿⣿⣶⣤⣀⠀⠀⠀⠈⢹⡿⢻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠘⠈⠛⠿⠿⠿⠿⠛⠛⠛⠛⠋⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣶⣤⣄⣀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣴⠤⠶⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣊⠥⠂⠁⠈⠑⠦⣅⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⡨⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢌⡂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢂⠡⠒⠁⢰⠀⣦⠀⠠⣄⠀⠰⣄⠀⠂⢲⠀⠆⠈⠓⠤⡑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡊⠔⠊⠁⠀⠀⠀⢸⠈⡳⢀⠂⠸⡄⢀⠈⠳⡄⣸⠓⣄⠀⠀⠀⠀⠁⠂⣅⠢⢄
⠀⠀⠀⠀⠀⠀⢐⠉⠰⠶⠶⠶⢶⣶⣶⠶⡶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⢀⠒⠒⢈⣢
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣛⣳⣋⣉⣻⣍⣀⣁⣂⣈⣙⣒⣀⡈⢰⣀⡀⣈⣉⣐⣊⣡⣤⣤⠤⢴⣶⣀⣉⣈⣛⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢀⠉⠛⠃⠀⠘⢻⣯⠉⠻⠛⠀⢀⣀⣄⢀⡀⠘⢻⠛⠋⠻⠊⠀⠀⠻⠛⠉⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡇⠀⠤⠄⡼⢷⡘⠆⠀⡼⡭⢳⣷⠙⠿⠀⠘⢀⠰⠊⠀⠤⠤⠀⠓⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠈⠀⠀⠀⢨⠀⠉⡇⠉⠅⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠂⠈⡇⠀⢸⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⠤⠤⠝⠐⠀⠈⠀⢐⣀⣀⠃⣀⡘⡇⠀⠀⠀⠀⠤⠤⠤⠕⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⢧⠀⠀⠀⡀⢤⣤⢤⡀⢹⠂⠈⠌⠀⠀⡇⡀⠤⣀⢤⡀⠀⠀⠀⢄⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣒⣒⣒⣒⣒⣓⣒⣒⣒⣒⣒⣒⣚⣚⣒⣒⣓⣒⣒⣛⣒⣒⣒⣒⣚⣒⣒⣒⣒⣒⣒⣚⣲⣀
⠀⠀⠀⠀⠀⢠⠚⢒⡒⠒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⣒⠒⢒⣒⣒⣚⣢
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⡜⠃⣤⡄⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠂⠀⡀⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠀⣆⠐⠀⠀⠀⠀⠀⢀⣠⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠛⢠⡔⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠙⡿⠋⠀⠀⠀⠀⠛⠛⠿⣿⡿⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⡐⠈⠐⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⠈⠉⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⠟⠁⠀⠀⢀⣼⣿⣄⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀
⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⣄⠒⢿⣿⣿⣿⠟⠂⠀⠙⠁⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠴⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠷⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠁⣰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⢍⠁⠀⠈⠉⠀⠁⠀⠀⠘⡤⠀⢸⣿⣿⣫⡭⣿⣿⣿⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⢷⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⣿⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⢄⠹⣄⠀⢠⡄⠀⠀⠣⠖⠀⠀⢸⣿⣿⢹⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠠⠐⡢⠒⠠⠀⠀⠀⡀⣀⢀⠀⠀⡀⠀⠀⠀⢀⣀⢀⠀⠈⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠰⠄⠵⠁⠔⠀⠀⠀⠉⠊⠘⠀⠁⠁⠀⠀⠉⠋⠉⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠰⠀⠆⠆⠠⠰⠀⠂⠆⠰⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠵⠞⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⡷⣿⡷⣿⣿⣿⣹⡭⣯⣿⣻⣽⣿⣿⣋⢋⣉⣿⣿⣿⡇⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠉⠉⠉⣿⣿⣿⣿⣿⡿⣿⣿⢿⢿⡿⣿⣿⢿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣷⣿⣿⣾⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢛⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⣩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⣍⠂⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⠬⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠆⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠘⣽⠛⠛⠛⢯⠁⠈⡽⠛⠛⠚⣮⠃⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⡀⠀⡇⠀⠀⠀⣿⢀⣀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠠⠝⠒⠒⠒⠊⠄⠠⠑⠒⠲⠒⠣⠯⠬⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠐⣭⠉⠉⠉⢭⠃⠘⡭⠩⠈⠍⣍⠂⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⣿⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠰⣝⣀⣀⣀⣚⡄⢠⣓⣀⣀⣀⣫⠄⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠒⠢⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⢾⡫⠭⣉⣉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠙⠢⣀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⢰⠉⠉⠙⢦⠈⢗⢦⠈⡷⠤⠬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠸⣄⡤⣖⣉⡇⣌⠈⠣⣘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠹⡎⠁⠀⠣⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡅⠀⡠⠤⢬⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣾⠠⡴⠓⡹⢾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⣷⠶⠚⠉⠚⠮⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⢉⡠⠔⠊⠉⠓⠦⣈⡙⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢉⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠖⠡⠀⠀⠠⠄⠀⠀⠀⠀⠤⠠⠄⠠⠀⠄⠂⠝⠖⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢤⠤⠤⠤⣤⢤⡤⠤⠤⣤⠤⠤⠤⠤⢤⡤⠤⠤⢤⠤⡤⠤⠤⢤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠒⠲⡇⠀⣷⠒⠒⠎⡤⠤⠤⠤⣄⣷⠒⠒⡞⠀⢳⠒⠒⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣿⠀⠀⠁⠄⠀⠀⠀⢸⣿⠀⠀⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣿⠀⠀⡄⠋⠉⠉⠉⠉⣿⠀⠀⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣿⠀⠀⡇⣠⠒⠒⠢⡄⣿⠀⠀⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣿⠀⠀⠃⡇⠀⠀⠀⢸⣿⠀⠀⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⣿⠀⠀⠀⡇⠀⠀⠀⢸⣿⠀⠀⡇⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⣤⡤⡇⢀⡿⣤⡤⣤⡇⠀⠀⠀⢸⡿⣤⣤⣧⠀⣼⣤⣤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠒⠒⠒⠛⠘⠒⠒⠒⠚⠧⠤⠤⠤⠼⠓⠒⠒⠚⠒⠓⠒⠒⠚⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠤⠤⠴⠒⠒⠒⠒⠒⠲⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢩⠤⠤⠴⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠉⠛⠒⠒⢢⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⡆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣄⠧⠤⠤⠤⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠚⠋⠉⠉⠉⢹⠉⡇
⠀⠀⠀⠀⡏⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹
⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡸⡇
⠀⠀⠀⢀⡿⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⢠⠇⣧
⠀⠀⠀⢸⡄⡇⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⢿⣿⣀⣀⣀⣤⠤⠴⠒⠚⠉⣁⡤⠤⠚⠁⠀⢸
⠀⠀⠀⢸⡇⠙⢤⣀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⡇⠀⠀⣿⣇⣀⡤⠤⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⡇⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⣷⣤⣤⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⠏
⠀⠀⠀⠀⢻⣻⡟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠖⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠙⠳⠿⠷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠰⠒⠒⠒⠊⠉⠉⠉

⠠⢁⠜⡿⡕⠌⣟⣵⡚⣷⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⠥⡾⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠁⠰⣯⡓
⠀⠘⠦⠅⠱⡀⠈⠀⠞⠓⠀⠄⠀⠀⠀⠀⠀⠖⠚⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀⠄⠀⠀⠀⠀⠀⣡⡄⠀⠀⠀⣸⠀⠐⠶⠋⠠
⠀⠀⠀⠀⠀⠈⠓⠤⣀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⠁⣤⡀⠀⠀⠙
⠀⠀⢀⠤⠄⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⢈⡈⠀⣉⡈⡀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⡂⠀⣰⠂⡄⠀⠀⠀⠀⠀⠀⠸⠔⡲⡛⠉⣻⠑⠜⠀⠀⠀⠀⠀⠀⠀⠘⡂⠀⠀⣀⠀⠀⠀⠀⠀⠴⢊⠃
⠀⠀⠀⠀⠀⠀⠰⠘⠁⠁⠀⢈⣬⠤⢤⡀⠀⠀⠀⠀⢀⠎⠀⣔⣺⠆⠀⠀⠀⠀⠀⠀⠀⢠⠚⠓⢄⡈⠘⠁⢀⡀⠀⠀⢀⠀⣻⡅
⠀⠀⠀⠀⠀⢰⠉⣉⢽⡷⠚⠻⡇⢕⠶⠁⠀⠀⠀⠀⢸⠀⠠⢬⣽⠆⠀⠀⠀⠀⠀⠀⠀⢈⣉⡭⢶⠿⣤⣀⣈⠁⠂⠀⠸⠄⠈
⠀⠀⠀⠀⠀⠘⠊⣡⠋⠞⢰⣤⡑⠂⠡⠀⠀⠀⠀⠀⠾⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠂⣀⣁⠄⢠⠏⣩⠉⡆
⠀⠀⠀⠀⠀⠀⢰⠁⠀⢼⣭⣽⡙⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⡀⢸⣽⡹⣯⠈⠂⠊⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢤⣸⣹⡽⠀⢠⠀⢆⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠿⣄⠀⠀⡱⣯⣎⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠠
⣑⠄⠀⠀⠀⠀⢸⠁⠀⠀⠸⠙⠀⠀⣉⠀⠞⠀⠀⠀⠀⠀⣈⣉⠛⠀⠀⠀⠀⠀⠈⠑⠤⣀⠁⠀⢀⠆⢀⠂⠀⠀⠀⠀⢀⠠⠀⠉⣤
⠕⠠⡀⠀⠀⠀⠘⠓⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠰⠁⠰⠽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠄⠠⠎⠀⠀⠀⠀⠀⣂⠄⣶⡞⠉
⠆⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡜⡙⠑⣰
⣫⡷⠀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⣤⡰⠁⠀⠀⠀⠀⠤⢄⠀⠀⠀⠠⠀⠶⠒⡏⠄⠀⠀⠀⠀⠀⠀⠀⡤⠚⠀⠀⠀⠀⡇⠀⠛
⠉⢂⠀⠊⠁⠀⠀⢀⡀⠀⠀⠀⠠⠀⢀⡿⠉⣫⣤⠄⣀⠀⠲⢀⠀⠀⠀⢀⠁⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠸⡄
⠂⠁⠀⠀⠀⠀⢠⡤⠖⠔⠀⠀⢨⢀⠋⣿⡐⠘⣧⠿⢱⡧⠀⠌⠀⠀⠀⣬⠀⠙⠋⠀⠀⠀⠀⢀⡀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⢹⣤
⠀⠀⠀⠀⠀⠀⠍⢠⣦⠀⠀⠀⠀⠘⠤⠇⠱⣀⠀⠀⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡤⠈⠠
⠀⠀⠀⠀⠀⠀⡄⠐⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⢀⠠⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⢠⢀⣦⢻⡕⢻⠆⠀⠘⣦⠀⢀⠈⠁⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢂⣤⣠⣤⡆⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣷⡎⠈⢋⠓⢋⠀⢀⡴⠋⠁⠀⠀⠰⣾⡁⢸
⣻⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣨⠞⠍⣀⠸⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠆⠀⠙⡶⢷⣒⠒⠊⠁⠀⠀⠀⠀⠠⡐⢸⠀⠺
⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢮⣝⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠘⠤⣈⠢⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠎⠁⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡼⠀⠐⢦⡬⠇⢀⠈⡀⠀⠀⠀⠀⠀⠃⣤⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠐⠤⢠⣄⠼⠀⠀⠀⠀⠀⠐⠳⠀⠄⠈⠀⠀⠤⠀⠁⠀⠀⢀⠡⣴⡞⠙⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢀⠀⢈⡈
⠐⠤⠀⣰⠏⠑⣂⣄⠀⠀⠠⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠔⢱⠀⣼⣛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠟⢉⠹⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⢉⣀⣬⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣭⣵⣶⠿⠛⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⣁⣠⣤⣶⣿⣿⡿⠟⢛⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣛⣫⣭⣶⠾⠟⠛⠉⠁⠀⢀⣤⣤⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⡉⣀⣴⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⠀⠀⣿⣿⣿⡏⣶⠿⠋⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⢿⠿⢿⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⡄⣀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣷⢹⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣼⠀⠈⠉⠉⠀⠉⠉⠀⠉⠉⠀⠉⠉⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠇⣿⣷⣶⣾⣿⣶⣶⣿⣶⣶⣿⣷⣶⣿⣿⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⡶⠾⠛⠁⠈⠙⢛⣛⣛⣻⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠥⠶⢶⣦⣀⣴⠖⠻⠶⠛⠉⢉⣽⡿⠁⣀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠈⠁⠀⠀⠀⠀⠀⣰⣿⣿⡾⠟⠛⠋⠉⠉⠙⠛⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⡀⠀⣀⣀⣤⣴⣿⣿⠿⠶⠶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠿⠿⣿⡭⠾⠛⠙⠿⣷⣤⣀⠀⠀⠀⠈⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄⡀⠀⠀⠀⠀⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⣤⣄⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢻⣿⡀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⣷⣦⣄⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⠀⣿⣇⠙⢿⣷⡄⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⢸⣿⡇⣿⣿⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢸⣿⡇⢿⣿⠀⠠⠿⠋⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣘⣿⡇⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⢻⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣿⠀⠀⣿⣿⠈⠻⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⢸⣿⡇⠀⢸⣿⡆⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠀⣿⣷⠀⠀⢿⣷⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⣸⣿⡆⠀⣸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⠛⠋⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄
⠀⠀⠀⠀⠀⠘⣿⣧⣄⣀⣈⣻⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠉⠛⠛⠻⠿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⠶⠾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⣀⡴⠚⠁⣀⡀⠙⠲⣄⡉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⣠⠴⠛⢁⣠⣶⣿⣿⣿⣶⣄⠀⠙⠲⣄⡀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠴⠋⢁⣠⠶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠙⠲⣄⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡴⢋⣠⠖⠋⢁⣠⠖⠋⠀⣶⣦⠀⣴⡀⣴⡄⡆⣦⡴⠀⠈⠙⢦⣀⠀⠙⠲⣄⡹⣆
⠀⠀⠀⠀⢀⡾⠗⠋⢀⣠⠖⠋⠀⠀⠀⠀⢯⡽⠰⠛⠧⠿⠹⠇⠏⠳⠀⠀⠀⠀⠈⠓⢦⣀⠀⠙⠿⣆
⠀⠀⢠⠚⠉⢀⣤⡖⣿⠒⠒⠒⡖⢲⡖⠒⠒⣶⠲⡖⠛⠛⠛⢻⠋⢿⠛⠛⢛⡟⢻⠉⠉⢉⡷⣤⡀⠈⠙⡆
⠀⠀⠀⢳⡞⠉⠀⡇⢸⣀⣀⣀⣷⠈⣇⣀⣀⣿⠀⣧⣀⣀⣀⣼⠀⣾⣤⣤⣼⠀⣾⠤⠤⢼⠁⣿⠉⠳⢼⠁
⠀⠀⠀⢸⠀⠀⠀⣧⢸⡇⠀⠀⣿⠀⣇⠀⠀⢹⠀⣿⠀⠀⠀⣾⠀⡟⠀⠀⣾⠀⡏⠀⠀⣾⠀⡇⠀⠀⢸⡆
⠀⠀⠀⢼⠀⠀⠀⢹⠀⡇⠀⠀⢸⠀⣿⠀⠀⢸⠀⣿⣶⣶⣶⣿⠀⡇⠀⠀⡇⢠⡇⠀⠀⡇⢸⠇⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⢸⡄⢿⠀⠀⢸⡄⢸⠀⠀⢸⠀⢹⣿⣿⣿⡏⠀⡇⠀⢠⡇⢸⠃⠀⢀⡇⣸⠀⠀⠀⢸⠇
⠀⠀⠀⠸⡇⠀⠀⠈⡇⢸⡀⠀⠘⡇⢸⡄⠀⢸⡄⢸⣿⣿⣿⡇⢀⡇⠀⢸⠃⢸⠀⠀⢸⠇⣿⠀⠀⠀⣼
⠀⠀⠀⠀⢻⡀⠀⠀⣷⠘⡇⠀⠀⡇⢸⡇⠀⢸⡇⢸⣿⣿⣿⡇⢸⡇⠀⢸⠀⣾⠀⠀⢸⠀⡇⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢷⣀⡤⢼⠀⣿⠭⠭⣷⢘⡯⠭⠽⣇⣸⣯⣿⣿⣇⣸⣿⣿⣿⢀⣿⣿⣿⣿⢀⣧⣄⢠⠏
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠑⠋⠀⠀⠈⠉⠁⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⣸⠋
⠀⠀⠀⠀⡖⠛⠛⠙⠋⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⢹⡀
⠀⠀⠀⣴⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡇
⠀⠀⠀⢹⣀⣀⣀⣀⣀⣠⣤⣤⣤⠤⢤⣤⣤⠤⠤⠤⠤⠶⠶⠶⠶⠶⣶⣶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠼⠇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢛⣿⡛⠛⠛⠛⠛⠛⢳⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣵⡶⠒⢶⣶⣶⣶⣶⣶⡝⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⢀⡇⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣋⣴⡿⢸⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⣠⠿⠿⠿⠿⠿⠿⠿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢟⣩⣾⣿⢟⣀⣀⣀⣀⣀⣀⣀⡀⣿⣿
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣸⠥⣤⠤⣤⠤⠤⠤⠤⠞⠀⠀⢸⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⢇⣛⢛⣛⠛⣛⣛⣛⣛⣣⣿⣿
⠀⠀⠀⠀⢀⡞⠀⠀⠀⣠⠗⠾⣥⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣱⣿⣿⣿⠟⣨⣑⡛⢡⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣴⣿⣷⣄⠀⠰⠚⠁⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠀⠙⠿⢿⣯⣥⣾⣿⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠻⣿⣿⣷⣄⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⠘⠿⡿⢛⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⣛⡛⠃⠀⠘⣛⣛⣛⠛⠀⠘⢛⣛⣛⡛⠀⠀⠛⣛⣛⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣭⣥⡄⠀⢤⣭⣭⣭⡤⠀⠠⣬⣭⣭⣧⠀⠀⢀⣽⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠛⠉⠁⠀⣀⠀⢀⣀⣤⣤⣶⡾⣤⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠚⠉⠉⡀⡀⣀⣀⣤⢶⡼⣿⡿⢷⢻⡹⣞⣧⣿⡟⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠚⠋⠉⠀⣰⣶⣷⢋⣿⣝⡟⢷⢿⣽⡰⣇⢹⣹⣼⣾⢋⡝⠙⠉⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⢠⢟⠛⣵⣍⢯⢱⣜⣤⢬⡟⠛⠵⠋⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⣀⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⡐⡀⠀⠀⠀⠀⠀⢎⣆⣒⡽⠼⢞⢟⠃⠁⠈⠁⠀⢀⠠⢠⠂⡰⡡⡡⡓⡀⠀⠈⠀⠀⠀⠀⠘⣇
⣠⠚⠉⡸⡠⢂⢌⢔⠀⠀⠀⠀⠀⡔⣐⣧⡀⠀⠀⠀⠀⠈⠋⠙⠈⠀⠀⡀⠀⠀⠀⠀⠀⠐⠡⠁⡔⡔⠕⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠃⢀⠜⠔⡡⡣⡣⢂⠄⠀⠀⠀⠘⠀⠈⠈⠑⠀⠀⠀⠀⠀⡀⢠⢀⠆⠜⡔⡡⠠⠠⠄⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠈⢊⢜⢔⠔⢵⡅⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡜⡠⢡⢃⢎⠜⡰⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢞⡀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠪⠪⠊⠃⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠎⡔⡠⠁⠆⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠠⠘⠐⠀⠀⠀⠰⠂⠀⣠⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠀⠀⠣⠀⠒⡀⠀⠀⠠⠃⠘⠀⠀⠀⠀⠀⠀⠀⡄⡄⡶⠄⠠⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠲⠀⠀⢹
⠀⠀⠐⠀⠀⠀⠀⠀⠆⠀⠡⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠁⠀⠀⠂⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠄⠁⠀⢀⠄⠁⠠⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠆⠀⢃⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠁⠀⠐⡁⠀⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⡀⣨⠅⠀⠀⠀⠀⠀⣀⠀⠄⠈⠁
⠀⠀⠀⠀⠀⢀⡐⢶⢤⢤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠆⡘⢌⢶⠱⠑⠈⠊⠠⠀⠐
⠀⠀⠀⠀⠀⠀⢺⢃⡦⢣⢊⠌⡐⣤⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⠋⠄⠜⡜⠌⠘⠈⠈
⠀⠀⠀⠀⠀⠀⠀⣾⢡⢃⢎⠌⡔⡡⠈⠢⡀⠀⠀⠀⢲⠏⢼⠃⡌⠈
⠀⠀⠀⠀⠀⠀⠀⠰⢂⢎⢂⠌⠜⡐⡑⡠⡀⠀⠀⢀⠈⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢡⢂⠆⡌⡜⣐⡰⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠁⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⡀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠔⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢂⣀⣰⣦⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠁⠀⠙⠉⠁⠀⠀⠀⠀⠀⠈⠣⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠁⠀⠀⠀⠠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠸⣝⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣶⠀⠀⢤⠀⠀⠈⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠄⠣⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣤⠧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣀⣠⠄
⠀⠀⠀⠀⠰⡄⠀⣀⣀⡀⠤⠀⠀⠈⠷⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣠⠄⠀⣜⠀⠈⠉⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠤⣤⡤⠞⡸⢻⡇⠀⠀⢸⠦⢱⡖⣚⣯⡇⠀⢰⠆⠀⢀⣀⣤⡶⠒⠲⠶⠋
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⢀⣤⡆⠀⠉⠚⠳⠠⡤⠛⠂⠀⠀⠀⡈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠶⠶⠞⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡤⠒⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⣿⠏⠀⠀⢱⠀⠀⠀⠀⠀⠑⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⣿⣿⠀⠀⠉⠹⠀⠀⠀⠹⡆⠀⠐⠀⠀⠀⠸⡆⠓⢖⣠⣄⣤⠌⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠳⣀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣛⣿⠀⠀⣤⠀⣈⢄⢀⠀⠀⣃⣀⣅⣀⣠⣾⠛⠀⠈⠼⣿⣇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢻⡏⠀⢀⣀⡠⠼⠻⠖⠲⠋⠉⠀⠀⠀⠄⠈⠉⠺⣠⡌⢿⣿⡇⣅⣈⣁⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡿⠭⢤⠜⠉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠻⣆⠛⡀⠆⠀⣀⣈
⠀⠀⠀⠀⠀⠀⠀⢨⡟⠉⠙⡻⠃⠀⠀⣠⣶⡿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠹⣷⣽⡶⢁⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⡸⠋⠉⠉⠁⢀⠀⢸⠇⣸⠇⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢹⣿⣿⢆⣂⠀⢅
⠀⠀⠀⠀⠀⠀⠠⠿⠄⣀⣀⡠⠴⡀⠘⣧⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠋⣻⣦⠐⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠁⠀⠀⣿⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡿⠋⣄
⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠈⠀⠐⠀⢠⡄⢸⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡇⠈⢀⣼⠿⠿⢶⣤⣀⡀
⠀⠀⠀⠀⠘⠠⠤⠅⡀⠀⠀⠀⠀⠀⠰⡙⠶⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣰⣅⣰⢟⡇⢤⣍⡀⣿⣿⠋⠠
⠀⠀⠀⠀⠘⠀⠬⢽⠐⠠⠤⠀⠀⡀⠀⢷⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡠⣾⣿⣿⣿⣷⣮⣗⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠡⠈⠉⠉⠁⢶⣤⣼⣿⣷⣄⠀⡀⠀⠀⠀⠀⡀⢺⣿⠟⣀⣾⠟⠻⢿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠧⢀⠀⠀⠀⠀⠘⠁⣰⡿⠵⢆⣀⠀⢈⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠠⣿⣿⡡⣘⣛⡎⣞⣿⠟⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣮⣭⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠙⠻⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢴⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠊⡀⠀⢤⡤⠤⠤⣤⡤⠤⢄⠆⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⣀⢀⣸⣀⣀⡀⣸⣀⡀⣸⡄⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠛⠛⢻⠛⠛⠛⢻⠛⠛⢻⠁⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠄⠀⢸⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠟⠛⢿⠟⠛⠛⢿⠟⠛⢻⠂⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⢸⠃⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡶⠶⢾⡶⠶⠶⢾⡶⠶⢾⠁⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡀⠀⣸⡀⠀⠀⣸⡀⠀⣸⢡⢶⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣹⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⠋⠋⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢰⠶⡆⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⣠⠏⠀⠹⡄⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣠⠴⠞⠋⢀⡴⠋⠀⠀⠀⠙⢦⡀⠙⠳⠦⣠⡀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢠⣴⣦⣄⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⣿⡅⢀⣿⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠈⠛⠛⠁⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⣟⣹⣯⣹⡾⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⡇⠀⣟⢻⣟⢣⡿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⡇⠀⣿⠋⠉⣿⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⣿⠀⠀⣿⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⡶⡿⢏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣶⠋⣙⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣋⡙⢲⡀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⡇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠆⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠻⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡼⠃⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣼


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠛⠿⠿⠿⠻⡿⢫⡝⠿⢿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣛⣛⣻⣻⣿⣿⣛⣛⣛⣛⣛⣻⣿