Insolent Emojis & Symbols

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣶⣶⣶⣶⣶⣶⣶⣶⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠉⠉⠉⠉⠉⠉⠉⠉⢳⡀
⣿⣿⣿⣿⣿⣿⣿⣩⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⡝⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⣆
⣿⣿⣿⣿⣿⣿⣿⢻⣿⣸⣿⣼⣿⣞⣼⣿⣼⣿⣔⣿⢇⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡀⠨⠂⠈⠁⠀⠭⠀⠘⠇⠀⠇⠀⡿
⣿⣿⣿⣿⣿⣿⣿⣷⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡅⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡎
⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⣭⣭⣭⡝⣿⢟⣫⣛⢿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⠒⠒⠒⡆⠀⡤⠶⢤⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⣮⣭⣭⣾⣿⢸⣿⣿⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠒⠒⠚⠁⢸⡇⠀⢸⡀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣬⣭⣭⣭⣭⣭⣭⣬⣭⣭⣬⣥⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠚⠓⠒⠚⠓⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣇⣲⣯⣭⣭⣭⣭⣩⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣦⠂⣠⣤⣄⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⠛⠉⠉⠙⠛⠿⣶⣧⣤⠀⠀⢀⠤⠐⢂⣀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠉⠻⣷⡔⣠⣴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⡀⠀⠀⠀⠀⠉⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡿⢇⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠓⢶⡾⢿⣿⣿⣦⠀⠀⢀⠀⠀⡼⠿⣡⡴⠋⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠛⠿⣧⠀⠈⠉⡀⠀⡋⠉⠀⠀⠀⠀⠀⢀⣠⣴⠻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⢀⢰⡵⠊⠀⠀⠀⠀⣀⣤⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⠀⠀⠉⠀⠀⠀⠄⠒⠻⠉⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢂⠀⠂⠀⠀⠐⠀⣀⡴⡋⠠⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠂⠋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠈⠓⠂⠀⡀⠀⠂⠤⠄⢀⣀⣀⣀⣀⣀⣠⣤⣦⣷⡆⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠭⠛⠛⠛⠯⠭⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣟⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠐⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠸⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡜⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣉⣴⣿⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⣥⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣾⠿⠿⢇⣀⣈⣛⣛⣋⣁⡇⠀⠰⠶⠶⢶⠿⠿⠷⣀⣀⣙⣛⣛⣃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⢻⣿⣿⣿⣯⡅⠀⠀⠀⠀⣿⠀⠀⠀⣿⡟⣿⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⢰⠀⠀⠀⠀⡆⠀⠀⠀⠀⣿⠀⠀⠀⣿⣷⡆⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⡿⡇⠀⠀⠀⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣀⠀⢶⣶⠆⢀⣴⣾⣿⣶⡄⠀⣶⣶⣶⣶⣶⣶⣶⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣤⣿⣿⣧⡀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢼⣿⡿⠿⠁⠀⠀⠘⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠖⠉⠁⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⠟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠆⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠴⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⠀⠀⠀⠀⠀⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣄⣀⣀⣀⡀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⡿⠿⠿⠿⠇⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠀⠀⠀⢿⣿⠇⢸⣿⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⣠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⢀⣤⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⣿⠃⣴⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿
⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠇⠘⠿⠿⠟
⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡖⠒⠀⠐⠒⠒⠒⣲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⡏⣯⠙⢦⠀⡇⠀⠀⠀⠀⠀⠸⣅⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠇⠇⣀⡼⠀⣇⣀⣠⣶⣄⣀⡴⠾⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠉⠁⠀⠀⠉⣿⠭⠽⢯⡥⣦⠀⠀⠈⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡆⠀⢠⣤⠀⡇⠀⠀⠀⠀⡿⢰⡄⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣋⣀⣠⢏⡟⠀⠃⠀⠀⠀⠀⡇⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⡇⠈⣥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠷⠏⠳⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣧⠀⠙⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡤⢤⡼⣧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠇⢸⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠉⠉⠙⠉⢋⠙⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠛⠉⠋⠙⠙⠋⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⢠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢆⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⢸⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⣿⣿⣿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⣾⠿⠿⠿⣤⣤⣬⣭⣭⣭⣥⣸⡇⠀⠸⠛⠛⠻⡿⠛⠛⠗⣤⣤⣬⣭⣭⣭⣥⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣹⣟⢻⣿⣿⣿⣿⣭⡅⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡟⣿⣿⣿⣿⣿⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⢰⠄⠀⠀⠀⢰⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡷⣶⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⢸⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠙⠋⠘⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠛⠃⠙⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠏⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣀⣴⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⢀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠟⠁⠀⢀⣴⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⠿⠋⢀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⠋
⢀⣀⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣟⢿⣿⠋⠁
⢹⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⠙
⠸⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⣏⢻⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀
⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣷⡀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠈⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⡀
⠀⠀⢻⣿⣿⣇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢩⣿⣿⠋⠉⠉⠉⠉⠉⠙⣿⣿⡉⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡟⠉⠉⠉⠉⠉⠉⢹⣿⣯⠁⣼⣿⣿⣿⣿⠏
⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣿⣿⣿⣿⡏
⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿
⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠁
⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁

⠀⠀⠀⡀⢀⣀⣄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠘⠒⠐⠂⠀⠀⠐
⠀⠀⣿⣷⣾⣿⠉⠁⠟⠺⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠐⣿⣯⣾⣿⣧⠀⠀⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡄⠀⣤⠀⠀⠀⠀⠀⠁⠀⠀⢠
⠀⠀⣻⣿⣿⣿⣿⠆⢄⡹⢿⣿⣿⣦⣀⠛⠛⠛⠉⠉⠉⠛⠛⠚⠷⣶⠮⠀⠀⠀⠀⠀⠋⠀⠁⠐
⢠⣾⣿⣿⣿⣿⣩⣤⣮⣿⣦⡭⠿⢿⡛⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⢻⠆⠀⠀⠀⠀⡆⠰⠀⠀⠀⠀⠀⠀⣀⣴⠖⢤⡀
⠀⠉⣿⣦⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⡀⠠⣴⠾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⡍⠓⠀⢀⣿
⠀⠀⠙⠛⠛⠛⠛⠉⠉⢉⣉⠉⣄⠀⠀⠀⠐⣇⣀⠊⢉⣾⡇⠀⠈⠉⠁⠀⢀⣠⠀⠁⠀⠀⠀⠀⢀⢀⡍⠓⢀⣀⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄⠹⠀⠀⠀⢀⠈⠻⣷⣼⣿⡇⠀⠔⠀⡁⠀⠘⣿⡆⠀⠀⠀⣀⣤⣿⠞⠻⠞⢻⣿⡛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⢶⣾⡏⢀⢀⡁⡰⡀⠀⠀⠀⠸⠇⠀⠰⠾⣻⠟⢉⣡⠀⢴⣾⣿⡿
⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⢤⠤⠶⠒⠻⠟⠋⠁⠈⠻⣷⣾⣿⣿⣿⣅⠤⠀⢀⠲⠄⠠⠄⠐⢁⣀⣐⡝⠶⣺⣿⡿⠁
⠀⠀⠀⠀⣿⣿⣿⠈⠡⠅⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⡁⠁⢿⣿⣿⣿⣤⣁⠚⠅⠒⣠⡶⣢⣤⣄⡂⠛⣷⣿⡿⠁
⠀⠀⠀⠀⠀⣿⣿⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢾⣿⣿⣿⣿⡿⠛⢲⣾⣯⢩⢹⣶⣿⣿⣷⣽⣿⠃
⠀⠀⠀⠀⠀⣿⣿⢸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⢺⣿⢿⣟⣋⠀⠀⠀⠉⡿⢿⣿⢟⣧⣼⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠘⠈⠁⠀⠹⣿⣿⡀⢴⣿⣷⣶⣶⣿⣿⣟⣿⡟
⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠁⠼⢧⡤⠈⠉⠋⠥⣤⠌⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⠛⣆⣠⡶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⠻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠛⢿⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢰⣇⠀⠀⠀⠀⣇⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠙⡖⠚⠳⣄⠀⢠⠏⠸⣆⠀⠀⣸⢉⢧⡀⠀⠀⣸⠘⢦⣀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⢢⠀⠀⠀⠀⢹⠀⠀⢈⠛⣋⠤⠔⠚⠛⠛⠛⠟⠉⠙⠲⠚⠓⠲⢷⡈⠙⠛⡏
⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠹⡆⠀⠀⠀⢸⠀⠀⠸⠊⠀⠀⠐⠗⠀⠀⠀⠰⠀⠀⠘⠃⠀⠀⠀⠁⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⠒⠛⢦⣄⣠⠟⠀⠀⢀⢦⡀⠀⠀⠀⠀⠀⡠⠂⡀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢳⠀⠀⠀⠀⠀⠈⡑⠈⠐⠀⠀⠐⠊⠀⠀⠈⠀⠀⠀⠀⠤⢞⠅⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⣶⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠼⠗⠋⠀⠀⠀⠀⠀⠀⣀⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠦⣤⣄⣀⣀⡀⢀⣀⣀⣀⣀⣤⡤⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠉⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠱⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣦⣄⡀⢴⠀⢀⡤⠠⠴⠿⠾⣿⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣶⣶⡶⣶⣶⡷⠶⣤⣿⣿⠿⣿⣿⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢿⠿⠷⠶⠿⡟⠁⠀⣿⣿⣷⣶⣿⣿⡿⣻⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠽⣿⠍⣅⠀⠀⢨⣬⣉⣛⣋⣉⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣾⡿⡿⢟⣻⣗⣚⣋⣋⣉⣉⣉⣛⣫⠩⣭⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠌⠀⠈⠈⠉⠋⠉⣿⡖⠀⠀⠈⠛⠿⣿⢛⣼⡿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠙⣷⣶⡞
⠀⠀⠀⠀⠀⠀⠀⣠⠞⣱⡇⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣿⣿⠶⢤⡀
⠀⠀⠀⠀⠀⣠⡟⢻⣿⣿⣷⣌⡻⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣋⣵⣿⣿⣿⣴⣿⣿⣦⡀
⠀⠀⠀⣠⣾⣫⣀⣿⣿⣿⣿⣿⣿⣿⣶⣭⠃⠀⠀⢀⡄⠀⠀⠀⣈⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⢠⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢼⡿⣿⣿⣿⣿⣿⣿⡝⡒⠿⠿⠟⣾⣿⣿⣿⣿⠿⠟⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠛⠛⠻⢿⢿⡟⢻⣋⣈⢿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⢸⡟⢻⣄⠀⠀⠀⠀⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢳⣄⠀⠀⢸⡇⠀⠙⢦⣀⠀⠀⠀⣸⠏⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠷⣤⣸⡇⠀⠀⠀⠙⠳⣤⣰⡏⠀⠀⠙⣦⠀⠀⠀⣠⡶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠙⠗⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠳⣤⡾⠋⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣧⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⣠⣤⣀⡀⠀⠀⠉⠙⠛⠷⠶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠹⠷⣭⣝⣳⣦⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⣴⣶⣶⣤⣄⣾⠏⠀⠀⢀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⢿⡟⠛⠻⠿
⠀⠀⠀⢿⣄⠀⠉⠛⠿⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣷⣄⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⣤⡶⠆⠀⠀⠀⠀⠙⢿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢰⣟⠁⢠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⣠⣾⠟⠀⠀⠀⠉⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠈⢳⡄⠀⠀⠀⠀⢰⣦⣄⡀⣿
⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠀⠀⢸⡇⠀⠀⠀⠀⣸⡏⠉⠛⠿
⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⣀⡾⠃⠀⠀⠀⠀⢸⠃
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⣻⡛⠛⠉⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠉⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣴⣶⣄⠀⠀⣾
⢻⣧⣄⣀⣀⣀⣀⣤⣴⣾⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣸⡏⠀⠙⢷⣶⡿
⠀⠙⠛⠻⠛⠛⠛⠉⠁⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣄⣠⡟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⢻⡿⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣼⡇⠀⠀⠉⠛⠳⢶⣤⣀⡀⢀⣴⠟⠁⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣶⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣟⢿⣯⣻⣷⡹⣷⡄
⠀⠀⢀⣀⣠⣿⠿⢿⣿⣿⣿⣟⢿⣯⡻⣷⣽⣷⣿⣧⣿⡇
⢀⣴⣿⣿⣿⣿⣿⣿⣷⣶⣯⣭⡛⠿⡿⠛⠋⠉⠛⠻⢿⡁
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣿⣿⣷⣄⢀⣴⣤⣿⡝⠩⢗
⣿⣿⣿⣿⣿⣿⣗⣶⣿⣿⣟⣿⣿⣿⠋⠘⠃⠤⠀⠐⢦⢸
⣻⣿⣿⣿⣟⣷⣿⣫⠚⢲⡝⣿⣿⠁⠀⠀⠀⣧⠀⠀⠈⢷⡳⡀
⣿⣿⣿⣿⣿⣿⣷⣻⠀⠘⣧⡘⠛⠁⠀⠀⠀⠙⠳⠦⠤⠾⠇⠙⠶⠤⡀
⢻⣿⣿⣿⣿⣿⣿⣿⣦⡸⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠓⢦⡀⠀⠈⠣⡀⠀⣀⣀
⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡀⠀⠀⠀⠀⠀⠀⣯⢰⡶⢦⣀⠈⣢⡴⡚⣯⡍⢭⣭⣭⡲⡀
⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠤⢚⣹⠃⢰⠞⢻⣟⠵⣏⡀⣀⠇⠀⠙⠻⠿⡝⣆
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⣿⡇⠀⠀⡄⢀⣴⡿⠏⠀⣸⡇⣟⠔⢀⠀⠉⢁⣀⣀⣤⣴⣶⣶⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠙⢛⡡⠴⢦⡶⢻⡃⢻⣇⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠑⢤⡈⠀⢻⡏⠀⠀⠁⠈⢻⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢠⡹⣿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣾⣿⣿⣿⣿⣿⣿⠳⣞⣿⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⣠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⣟⡛⢿⡷⢶⣤⣭⣃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⠙⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡎⡿⣞⠳⠶⠾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⢳⣴⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢹⣽⠿⠛⠒⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⣿⣈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡷⠋⠑⢒⣠⣤⣧⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣶⡶⠿⣞⣡⣚⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⣙⣛⡥⠞⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⣛⢻⠩⣿⠩⣽⢛⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣠⣤⣦⣤⠤⣤⣴⣤⣄⣀
⣿⣿⣿⣿⣯⡲⢛⡬⠼⠚⠛⠚⠧⢼⡏⢴⣾⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣿⣿⡟⢫⣿⠟⠁⠀⣀⣀⣴⣦⡀⠀⠈⠙⢿⡟⢻⣿⣿⠀⢠⣾⣿⣿⣿⣿⣿⣿⣛⣟⡟⡻⢿⣿⣿⣿⣿⣷⡄
⣿⡟⢭⡾⠁⠀⠀⠘⢿⣿⣿⣿⣿⠇⠀⠀⠈⢻⡭⢽⣿⠰⣿⣿⣿⣿⣿⡟⡿⠋⣠⣤⣄⠁⢿⣻⣿⣿⣿⣿⣿
⣿⢭⢼⠃⠀⠐⠰⠶⠿⠿⢿⡿⠿⠖⠀⠀⠀⠈⣧⡧⣿⠀⠈⠻⣿⠿⣿⢸⠀⠀⠾⠿⠟⠀⠀⠃⣿⠿⣿⠟⠁
⣿⢙⢽⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡿⡂⣿⠀⠀⠀⠀⠀⣿⡸⡀⠲⢄⣀⣠⠖⢀⢆⣿
⣿⡗⡹⣇⠀⠙⠿⠦⠄⣀⡀⠀⡤⠶⠞⠁⠀⢰⢏⢺⣿⠀⠀⠀⠀⠀⣿⣷⣝⡦⠉⠉⠉⢴⣪⣾⣿
⣿⣿⡖⡝⣦⡀⠀⢠⢞⣿⢶⢞⣽⠆⠀⠀⣰⢫⢓⣾⣿⠀⠀⠀⠀⠀⣿⣿⣿⣷⣷⣶⣷⣾⣿⣿⣿
⣿⣿⣿⣮⡨⣻⣦⠄⡉⠁⠀⠉⢁⠀⣴⣞⢥⣵⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⡓⢕⡇⡙⣉⢴⢱⠨⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠉⠀⠀⠈⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠙⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⡃⢀⠀⠀⣸⠄⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⡛⠿⣦⣴⣿⡟⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣳⣦⡀⠙⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⣻⣿⣿⣿⠟⠋⠀⠀⠀⠐⠒⠲⣾⡿⢧⣍⣀⠉⠛⠿⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⠦⢝⡳⣌⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣠⣿⣿⣿⣿⣶⣦⣙⣦⡀⠀⠀⠀⠀⠀⣿⣿⡿⣶⣍⢢⣄⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠀⠀⠀⠀⢀⣿⣿⡃⠪⣟⣿⣮⡳⣄⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣷⣬⣷⡜⠻⣾⡳⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣮⣻⣿⣿⣶⣙⣿⣿⣿⣿
⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⠟⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⡛⠛⠛⠛⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣴⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠏
⣠⡿⠋⠀⢀⣀⣀⣀⣀⠀⠒⠒⠚⠿⠿⠿⢿⣿⡦⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⢀⣁⠈⠀⠈⣿⢃⣿⣿⣿⣿⠏
⢿⡿⠾⠿⠿⠛⠛⠛⠛⠿⠿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠀⢀⣾⣧⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣟⣫⣴⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⣾⡟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣰⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣿⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣤⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠁⠈⠙⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣅⣺⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠶⢦⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠢⣆⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠨⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣾⣮⣷⡜⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣌⣲⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡻⢿⣂⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣕⡢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⡵⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠙⣉⡙⣏⢁⣿⠉⣹⡝⢉⣿⠟⣭⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⠀⢱⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠁⠀⠀⠀⠀⠴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠃⠀⠀⢀⠀⠀⠀⠀⣠⣼⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⣤⣄⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢽⢢⠄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⣾⣷⢆⠆
⠀⠀⠀⠀⠀⠸⡀⡶⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠘⡿⢤⡀⠀⠀⠀⠀⣽⣿⣀⡴
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⡴⠒⢢⣤⡀⠀⠀⣰⠓⠀⠿⣹⣿⡆⣸⣇⢀⣏⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠉⡁⠀⢸⡿⠁⠀⠀⠋⡀⠀⠀⠈⢿⣿⢿⣿⡿⠏⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠸⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠐⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠉⠐⠢⡀⠀⠀⠀⡐⠀⠣⠀⠠⡀⣀⠀⠀⠀⢚⠒⠂⣘⡈⠉⠛⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⢀⡠⢂⡊⠦⡼⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⡅⡀⢠⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠉⢀⠀⠀⡀⠀⠀⢀⠄⠂⠙⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⢀⣀⠀⣳⠘⠀⠀⠑⠠⢀⣀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡡⣤⠜⢹⠋⢿⠱⣆⣡⡀⠀⠀⢠⣄⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠳⠏⠂⠒⠊⠀⠘⣏⡘⠀⣀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠤⢂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠘⣆⠈⠀⠀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⡠⠐⠀⠉⠈⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢀⡔⡄⣤⡘⣧⠀⠀⠀⠀⠈⢀⠔⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡇⢰⡐⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⠉⠭⠤⠤⠀⠱⠼⠆⠠⠀⠀⡤⠁⠀⠀⠀⠀⢣⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⢀⣀⣀⣀⣀⣀⣀⠀⡠⠊⠀⠀⠀⠀⡠⠀⠈⠄⠰⠀⠀⠀⠀⠁⠈⠉⠀⠈⢹⠀⠀⠀⡘⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢄⡀⠃⠰⠀⠀⠀⠀⢀⠴⠞⠅⠀⠀⠀⠀⡴⠀⠀⠀⢹⡀⡃⠀⠀⠀⠐⡂⠀⠀⠐⠺⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠁⢀⡠⠀⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⣀⠊⠘⡀⠀⢀⣀⠇⠐⡀⢀⠀⢀⡭⠉⠉⠉⠉⠀⠀⢰⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠥⠼⠤⠬⠆⡀⠀⠐⠀⠀⠒⠂⠀⠤⠎⠀⣄⡊
⠀⠀⡀⠄⢠⠀⠐⠂⠐⠃⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠴⠦⠄⠀⠄⢀⠀⠀⠀⠀⣤⠚⠉⠁⠀⠀⠀⠄⠐
⠀⢮⠀⠠⠁⠀⠀⠄⠂⠀⣼⠂⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠀⠉⠉⠁⠀⠀⠀⠈⠉⠈⠙⠉⠉⠀⠀⠀⠁⠈⠁⠀⠀⠈⠀⠐⠂
⢀⠁⠀⢀⠀⠀⠀⠀⠀⡴⠹⠖⠒⡖⡶⠒⠀⠰⠶⠀⠀⠐⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠀⠠⠄⠠⠤⠤⠄
⠈⠸⠀⠀⠑⠂⠐⠢⠐⢁⠁⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠰⢉⠠⠀⡤⢀⡀⠒⠁⠀⠀⠀⡇⠀⠀⠀⠰⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠠⠀⡃⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢈⠶⠤⠎⠀⢉⠴⠀⠴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠸⠤⠤⠤⠆⢧⣄⠤⠄⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠛⠛⠛⠛⠛⠛⢧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣶⣿⠗
⠀⠀⠀⠀⠀⠀⣾⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⠟⠁
⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣾⣿⠿⠿⠿⠿⠷
⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠉⠀⠒⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠛⠉⠉⣿⡇
⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⡇
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⣿⡀⠀⠀⠀⢀⠐⠀⠀⠰⠆⢀⣤⣾⠟⠀⣀⣀⣀⣀⣀⠀⣴⣿⠀⠀⠀⠀⢀⡀⠾⠷⠾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠙⣿⡀⠀⣴⠁⠀⠀⠀⠀⣀⠈⠉⢿⠆⠀⠉⠉⠛⠛⠛⠓⠛⠛⠀⠀⠀⠀⠀⠀⠰⠖⢀⣾⣿⠀⠀⠀⠀⠀⠀⢰⣦⠀⣼⢿⣦⣀⣀
⠀⠙⢿⣦⡀⠀⠀⠀⠀⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠂⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠸⣿⠿⣦⡀
⠀⠀⠀⢻⣧⡀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡆⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢰⣿⢹⣿⡇⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠛⠷⠶⠶⢶⣦⡈⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠇⠀⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⣀⡀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⢻⡄⠈⠷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠷⠛⢿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠠⣶⣶⠿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⣶⡶⠤⠀⣀⡀⣀⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠙⠦⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⡇⠀⠀⠹⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⡏⠉⢻⡇⠀⠀⠀⠻⢷⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠀⠀⠘⢧⡌⠡⠄⠀⠀⠄⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠉⠀⠠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠃⠀⢰⡆⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠘⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⣄⣠⡄⠀⠃⠀⠀⠀⠀⠀⠙⢿⣦⣀⡀⠀⣀⣠⣾⠛⠀⠀⠀⠀⢀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣶⡏⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠙⠻⢿⣶⣤⣶⣶⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡏⠉⠁⠰⠶⠶⣶⣦⣤⣶⠿⢿⣆⠀⢶⣾⠿⠁
⠀⠀⠀⠀⠻⣷⠶⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠀⢸⡇
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣾⠃
⠀⠀⠀⣠⣿⠀⠀⠀⠀⡀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⢸⡇
⠀⠀⠀⣿⡇⠀⢹⣷⡆⠻⠶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢿⣿⣤⣈⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠙⠿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡿⠃⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣄⠀⠒⠚⠛⠛⠃⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⠀⠀⠐⠒⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣥⣿⠿⠿⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣭⣿⣄⡀⠀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠐⠶⣾⡟⠁
⣤⣄⣀⣤⣤⣤⣤⣤⣤⣀⣭⣿⣿⣅⣽⣿⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣯⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠁⠀⢙⠉⠉⠉⠁⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣅⣠⣀⣠⣄⣠⣠⣤⣄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠿⠿⢿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣾⠟⣡⣾⣿⣷⠸⠄⢾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣩⣍⡉⠛⠈⠻⣿⣿⣿⣷⣄⢸⡏⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣸⠟⠉⠀⠀⠰⣶⡙⣿⣿⣿⣿⡜⣿⡌⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠿⠟⢛⣡⡀⠀⠀⢀⣴⣷⣝⣿⡌⣿⣿⠿⠟⢈⣃⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣴⣶⣾⣿⡆⠹⣷⡀⣰⣿⣿⡟⣛⣭⣵⣶⣶⣾⠿⠛⢻⡿⢿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣰⣿⣿⡿⠋⠀⠈⠙⢷⠸⣿⡟⣰⡿⠻⠿⠛⠛⠙⣏⢿⣷⣅⣶⣦⣍⡙⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⣛⠻⠏⠀⠀⠀⠀⠀⠀⠀⠙⢰⣿⣱⡄⠀⠀⠀⠸⠞⣏⢿⣿⣿⣿⣿⣿⡷⠆⠉⣿⡇
⠀⠀⠀⠀⢀⣴⡿⠋⠰⠺⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣽⣷⡽⣆⠀⢀⣀⣤⡼⠆⠁⢠⣍⡉⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⣴⡇⠀⠀⠈⠻⢄⠀⠀⠀⠀⠀⠀⠀⠁⠽⠛⣿⣿⣬⣯⣭⣥⣀⣀⣀⣀⢸⣿⣿⠀⠀⢠⣴⣿⡇
⠀⠀⠀⠀⢹⡇⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⢛⣉⣁⡙⠄⣤⣙⣛⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⢸⣿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⢀⣠⣴⣿⣿⠟⣩⣿⣿⣿⣿⣿⣦⡈⠛⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠛⣉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⣿⡀⠀⠑⢲⣦⣄⡀⠀⠀⠀⠀⢠⣿⠋⣠⣶⣿⣿⣿⠀⢘⡻⠿⣿⣿⣿⡿⠃⠀⠀⠀⢀⠰⣿⡟
⠀⠀⠀⠀⠀⠙⢿⣆⠀⠉⠉⠙⠿⣷⣤⣀⣰⡿⢃⣼⣿⣿⣿⣿⣿⡇⢺⣿⣷⣶⡍⠉⠀⠀⠀⠀⡀⠢⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⢀⣤⣌⡙⠛⠛⢁⣾⣿⣿⣿⣿⣿⣿⡇⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⢭⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣴⣶⣶⣶⣦⣬⣉⣉⣉⣉⡉⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠻⣿⣿⣿⣿⣿⣿⠟⣉⣴⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣌⠻⢿⠿⢋⣤⣾⣿⠏⠀⣼⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣶⣿⣿⣿⡇⠀⢰⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢀⡠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠉⡀⠊⣠⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⣠⡗⠀⢀⠆⠀⠀⠀⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢐⣤⠀⢠⣆⣀⣬⡤⢐⣊⠭⣶⣚⠋⠀⠀⠀⠀⠀⠴⣾⡟⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣈⠛⠖⠄⣼⠟⠁⠛⠋⠉⠉⠉⠉⠛⠙⠃⢀⣀⣀⣄⣀⡌⣆⠀⠀⠀⠠⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⡰⠊⢐⠁⠀⠀⡠⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⢹⡷⠤⠤⠼⠀⠠⠄⠀⣈⡁⠢⡀
⠀⠀⠀⠀⠀⠀⡻⠂⠠⠞⠛⠒⣶⣇⣀⣠⣤⣶⣶⡦⠀⠀⠀⠀⠰⣿⣬⢕⣂⠀⠀⠀⠀⠀⠈⠑⢌
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀⠀⢁⣨⣿⣿⣄⣼⠟⠀⠀⠀⠀⠀⠀⠨⣷⣤⡉⠓⠀⠀⠠⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣤⡭⠤⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠁⠀⠉⠳⠑⠂
⠀⠀⠀⡄⠀⠘⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡀
⠀⠀⠀⢛⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⢀⣰⣴⣶⡈⠀⠀⠀⠀⣠⠃⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠙⠿⣷⢤⣤⣤⣤⣤⡖⠁⠠⠐⢡⡾⣿⣿⣷⠀⠀⠀⠠⠃⠘⠃⢦⡄⠀⠰⡄⠀⠀⡘⠀⠈
⠀⠀⠀⠀⠀⠀⠰⠄⠀⢀⡀⠈⡇⠀⠀⠀⡋⠀⠘⣿⣿⡇⠀⠀⢀⠊⠀⠀⠈⠳⠀⠀⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢰⣷⠀⢰⠲⠇⢠⠂⣼⢿⠇⠀⢠⠏⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠃⡆⠾⠤⠌⠀⢸⠟⠀⢀⡏⠀⠀⡀⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠦⣴⡶⠀⠀⢠⡂⡀⠁⢀⠀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠐⢋⡴⠯⡇⠈⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠶⠶⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢧⠀⢀⣆⠀⠀⢀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⣀⠉⠁⠈⠓⠒⠚⢧⣀⣀⠴⡇⠀⡴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠑⢲⠖⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠙⠦⠔⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣤⣉⠀⠀⠀⠀⠸⣄⠀⠀⣸⠁⠀⠀⡋⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠙⠲⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⢸⠀⣀⡀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡠⠉⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣏⡙⠲⢬⣙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡈⠉⢙⡗⠲⠤⣀⡀⠀⠀⠀⠀⣀⣠⢤⣶⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⣤⡀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⡏⠉⠛⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠒⠶⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⠀⠀⠀⠀⢀⣀⣨⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡴⠒⠒⠲⠶⠶⠶⠒⠲⢦⣀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣄⠀⣶⣶⣶⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠟⠛⠛⠛⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣚⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠤⠐⠲⠂⠀⠠⡘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⢡⠙⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣤⢠⠀⢑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢸⡟⠙⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠄⠀⠀⠙⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠺⣄⠺⠀⠀⠀⠀⠿⠇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠤⠠⠤⠄⢤⣤⣤⣀⣐⣐⠂⠂⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⢀⣁⣀⣀⣤⣤⣤⣴⣶⣶⣶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⣼⡿⠿⠟⠛⠛⠛⠉⠩⠉⠈⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠒⠒⠒⠶⠤⠤⠤⠤⢄⣀⣀⡀⢀⣀⣠⠤⠴⠒⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠑⠐⠛⠒⠐⠒⠉⠐⠚⠀⠀⠗⠒⠒⠒⠒⠚⠀⠀⠀⠛⠚⠒⠀⠒⠒⠒⠒⠀⠋⠐⠃⠆⠐⠐⠒⠁⠈⠀⠐⠒⠖⠐⠒⠐⠒⠀⠛⠒⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠴⠶⠒⠒⠒⠒⠲⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠒⡄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡤⠊⡴⠁⠀⠈⢳⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡞⢁⣀⣇⠀⠀⠀⠀⢩⠑⢄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣼⠀⢇⠀⠟⢄⠀⠀⠠⣀⠄⠀⠠⡧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡇⠀⠀⠀⠀⠈⠓⢤⡔⣄⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣀⣴⠀⢷⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠂⠀⠀⠀⠀⠀⠀⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢼⡆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡶⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⢄⢀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⣰⣅⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⢀⠉⠗⣄⡠⠁⡵⣀⣉⠒⠢⢤⠀⣀⠀⣀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠢⠜⠀⣀⣈⡋⠁⠚⠀⢩⠉⠘⠂⠄⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠐⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⠀⠀⢀⠀⠀⡄⠰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠺⠒⠛⢻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠁⠀⢸⠀⠀⠩⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠍⠂⠰⠛⠋⠛⠛⠷⠮⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠂⠉⠉⠉⠉⢻⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⢀⡀⠀⠀⢿⡄⠁⠁⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠰⠤⢐⡤⠖⠋⠀⠀⠀⠀⠁⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠋⣉⠟⠃⠈⠶⠴⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⡿⡀⢧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢠⠟⠷⠙⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠛⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⢀⠇⠀⣠⢞⣓⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⢸⢀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠴⠃
⠀⠀⠀⠀⠸⡘⠤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣤⢀⣤⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠙⢤⡚⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣰⣁⣀⣀⣀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣱⣿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⡾⢻⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠶⡙⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠸⣁⣉⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠛⠯⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠓⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣀⣤⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠿⣿⣦⣤⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠉⡀⡈⣉⣁⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⡘⢓⠀⣀⠀⠀⠀⡼⡷⣿⣿⣿⢣⣟⣈⠀⠀⠀⠀⠂⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣘⡻⠋⢀⠁⢹⠏⠀⠀⠀⠀⠀⠁⠁⠀⠈⠉⠀⠀⠀⠠⡀⡈⢠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⢤⠿⠉⣀⡀⠶⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡈⡎⣾⡉⠛⠷⠀⢀⣠⢴⣗⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⢴⣄⣷⠈⡖⠀⣠⣀⡀⠄⠀⠈⠹⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠄⠰⢿⠿⠇
⠀⠀⠀⠀⢀⠐⢻⡐⣷⣞⣼⡃⢹⣤⣅⢛⣫⡤⣷⡐⠟⠂⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣯⠱⢷⡌⠀⣀⠀⣧⠀⣤
⠀⠀⠀⠀⢯⣇⣵⣜⠢⣹⠿⠛⢻⣧⣀⣾⠿⣇⡿⠓⢠⡄⠰⠅⡅⠁⡤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⣀⡆⠉⣬⡁⠁
⠀⠀⠀⠀⢠⡄⡤⢸⢷⣶⣶⡶⢿⠟⣟⠵⡆⢈⣷⡀⢌⠀⠈⠦⢈⡀⣰⡆⠀⠀⠀⠀⠠⠀⠀⠘⠆⠀⠀⡏⠐⠒⠈⠸⠂
⠀⠀⠀⠀⠀⠀⠐⠞⠳⣍⠀⠰⡎⠀⣀⣀⡴⠋⠀⠁⠎⠀⠀⠘⠛⠃⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠘⠾⠇⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠉⠉⠙⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⡠⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣮⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣦⣿⣾⣿⣙⣿⣿⣝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⢿⣶⣶⠶⢤⣤⣤⡤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣝⣿⣿⣿⣿⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠚⠁⠀⠀⠀⣿⣿⡇⠀⠙⠉⠓⢠⡉⠉⢴⣾⣯⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣷⠀⠀⠀⠀⠀⠁⡀⠀⠀⠈⠉⣡⡖⢶⡘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⣀⣀⠠⠤⣤⣦⡀⠀⡸⠻⢿⣶⣤⣤⡀⠀⠀⠁⠀⠀⠀⠘⣿⡆⠘⢇⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢀⡠⠤⠀⠈⠟⠛⠛⠛⢛⠛⢻⣷⠀⠀⠉⠀⠀⠀⢉⢩⡍⡇⠀⠀⠀⠀⠀⠀⠸⠍⢃⠀⣼⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⢀⡔⠋⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⢌⣿⡀⣠⣤⠴⠖⠀⠁⠸⠁⠁⠀⠀⠀⠀⠀⢠⡀⢀⣤⠾⠋⠁⠈⢹⠋
⠀⠀⠀⡆⢀⣠⣾⠉⣻⣿⣿⠀⡰⡦⠄⠀⠀⣜⣩⣂⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠉⠀⠀⠀⢀⡠⠞⢷⣄
⠀⠀⠀⠀⢩⣿⣿⣾⣿⣿⣿⣿⣿⡟⢆⠤⢰⠉⢹⣿⠟⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⣠⣯⢟⣳⣄
⠀⠀⠀⠀⠸⣼⣿⣿⣿⣿⣿⣿⢟⠌⠢⠂⡜⠀⢸⡏⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠔⠊⠁⠀⢀⣴⣟⡱⠏⠀⢹⣷⣄
⠀⠀⠔⠉⠉⠙⠋⠹⠟⣿⣿⣿⡇⠀⠀⡰⠁⠀⠘⠷⢤⣶⣿⠁⠀⠀⠀⠀⣀⣴⣾⠧⠚⠁⠀⠀⠀⣠⣶⣿⠟⠉⠀⣀⣐⣿⣿⣿⣷⣦⣄
⠀⠀⢳⠖⠒⠚⠛⠉⠀⠀⢹⠏⢸⡗⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢶⣾⢟⡉⠁⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⢀⣤⢞⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣄⡠⠤⠀⠐⠆⠕⢪⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣼⢫⡀⠀⢀⣤⡶⣿⠟⠋⠀⣠⣾⡿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⠤⢤⣯⡀⣀⣀⣤⡤⣴⠤⡄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠀⣈⣥⣶⣫⡵⠛⠉⢀⣴⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠃⣀⠀⢤⡞⠋⠉⠉⠁⠘⠄⢀⣠⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠻⣟⣩⣿⠟⠋⠁⢀⣠⡖⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠣⡀⠋⠉⣷⢦⣤⣤⠀⢀⡍⠗⠀⠀⠀⠀⠀⢀⡼⠛⠻⢿⣧⠀⠙⠏⠁⠀⢀⠀⢠⣼⣿⣾⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⡀⠀⠁⠸⢹⣷⣠⡞⣶⡄⠀⠀⢀⣆⠀⠈⠁⠠⠀⠄⠀⠀⠠⣀⠳⣄⡨⠴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠈⣃⠀⢀⡼⣿⠋⢠⣿⣿⣀⠀⣮⠈⠀⠠⡤⡄⠀⠀⠀⠀⠀⠀⠑⠈⣶⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡴⢿⠀⣉⠜⠁⣠⣿⣿⣿⣿⡤⠘⠦⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⡠⣪⣾⠾⠋⠁⢠⣶⣿⣿⣿⣿⠟⠓⣄⠀⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⣼⠱⣋⣀⡤⢒⣵⣿⣿⣿⣿⣿⡇⢠⢫⡿⠜⠰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⣿⣠⣿⣿⣾⣿⣿⣿⣿⣿⣇⣿⣇⠀⣤⠠⠐⠀⡁⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣷⢘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣾⣷⡀⠡⠄⠊⡊⠀⣹⣿⣷⣶⡄⠀⠀⢀⠎⠀⣼⣿⣿⣧⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡻⣿⣦⣄⣀⠸⣿⣾⣽⣿⣿⣿⣦⡴⠃⢀⣼⣿⣿⣿⣿⣗⡀⠙⢿⡟⠛⠉⡿⠿⠟⠁
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠉⠉⠉⠙⠁⠁⠈⠙⠛⠿⠚⠙⠛⠛⠻⠿⠿⠋⠀⡰⠟⠻⠿⠿⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣾⢁⣤⣶⣿⣯⣶⠟⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⣐⣚⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣓⣀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣞⠛⢿⣿⣷⣾⣿⠛⠉⠉⠳⠙⣿⢿⡿⣿⠋⠞⠉⠉⠛⣿⣷⣾⣿⡿⠛⣱⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠉⠛⠛⢿⣿⠛⠛⠀⠀⠀⠀⠀⠈⠘⠃⠀⠀⠀⠀⠀⠀⠛⠛⣿⡿⠛⠛⠉⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣶⣾⣿⣀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⣿⣷⣶⣶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⣿⣿⣷⠟⠀⠀⠀⠀⠀⢄⠀⠀⡠⠀⠀⠀⠀⠀⠲⣄⠈⢿⣿⣿⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⣿⣿⣧⡀⢿⣿⣶⣦⣄⠀⠀⠈⢦⣦⣴⡴⠁⠀⠀⣠⣴⣶⣿⡷⢀⣼⣿⣟⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣭⠀⠐⣿⣿⠁⢘⣳⣄⢀⡀⢸⡇⢀⡀⣠⣞⡇⠈⣿⣿⠂⠠⣭⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⣿⡿⡋⠀⠀⠀⠙⠻⢷⣤⣥⡾⠟⠛⠟⠻⠛⠻⢷⣤⣤⡾⠟⠋⠀⠀⠀⠙⢿⣿⣿⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣵⠇⠀⠀⠀⠀⣀⡀⠙⢷⣿⣷⣦⣴⣾⣿⡾⠋⢀⣀⠀⠀⠀⠀⠸⣦⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⣞⠏⡹⠳⣄⠉⠙⠿⠿⠋⠉⣠⠞⣏⠹⣳⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⢸⣿⣿⣿⣿⣿⣧⡅⢤⠘⢷⣇⢠⡍⣙⢲⢶⡶⡞⣋⢩⡀⣹⡾⠃⣤⢠⣼⣿⣿⣿⣿⣿⡏⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⠏⣿⣿⣿⣿⡇⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡇⠀⠀⠸⣿⣿⣿⣿⠹⣿⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠰⣿⡟⣿⣿⢰⡆⠀⢸⡇⠹⣿⠿⡿⡿⢿⢿⠿⣿⠏⢸⡇⠀⢰⡆⣿⣿⢻⣿⠇⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢻⣿⣿⡇⡄⠈⠷⣿⡴⠦⠞⠓⠚⠳⠴⢦⣿⠾⠃⢀⢸⣿⣿⡟⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿⣿⣷⣷⣄⠀⢀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⣠⣾⣿⣿⣿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢿⣏⢻⣷⣿⣀⣆⣿⡄⢀⣿⣸⢠⣴⣾⡟⣹⡿⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢿⣿⣿⣿⣷⣼⣿⣿⣿⡿⠋⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⣿⣿⣿⣿⠟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⠉

⠀⠀⠀⠀⠀⣠⠶⠶⠶⠄
⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠦⣈⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⢀⣤⣾⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡉
⠀⣤⠙⣿⣿⣆⠐⣶⡄⡒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢀⢴⣓⠀⡀
⠀⣿⣄⠉⢻⣿⣭⣿⣯⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢺⣯⣬⣶⣡⣾⣿⢣⠈⠄
⠀⠻⣬⡆⠀⠙⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⣿⣿⣿⣿⣿⣿⠏⣧⣄⢀⠀⢀⢀⢄
⠀⠀⡝⢿⣦⡀⠈⠻⣿⡟⣿⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⢩⣥⢀⣤⣬⣃⣾⣿⣿⣿⣿⣶⡆
⠀⢠⡧⠈⢻⣷⡄⠀⠘⢧⣿⠻⡇⠀⠀⠀⠠⢀⣀⣀⠀⣠⠏⠀⠀⠀⣀⠴⢷⣿⣼⣿⡯⠀⠀⣀⣬⡟⢉⣿⣿⣿⣿⣿⣿⡡⠂
⠀⢸⣿⣮⠄⠈⢡⣄⠀⠀⢿⡄⢧⣤⣤⣤⣷⣶⣦⣽⣽⣿⣤⣶⣤⣶⣤⣴⣿⣿⣭⣽⣿⣤⣨⣿⣿⣧⣿⣿⣿⣑⣘⣋⠁
⠀⠈⢰⢸⠀⠀⢰⠿⣿⢇⢨⡞⠈⠛⠛⠛⠛⠛⠛⠛⢋⠛⠉⠛⠛⠛⢛⣛⠛⡛⡛⢿⣛⠛⠛⡿⡶⠓⠀⠀⠀⠀⠀⠉⠉⠈⢹⠿⠂
⠀⠐⣠⡀⠀⠀⠀⠀⣟⣀⣩⠁⠀⠀⠀⠀⠈⠉⠉⢉⣉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡏⠉⡉⠉⠈⡹⣽⣟⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⣷⠀⠀⠀⢀⣿⠃⢹⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⣿⠦⠀⠀⢸⢿⣆⣈⣠⣤⣿⣶⣀⣤⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣾⡇⠆⠀⡟⣸⢯⢉⠁⢩⣿⣏⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣷⢸⡀⢸⣇⣛⠸⣾⣾⠚⣿⡏⠀⠐⣖⣠⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉
⠀⠀⣸⣿⣷⣾⣿⣿⣶⣧⣄⡈⢹⣿⡇⠉⠉⠏⠉⠉⠉⠙⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⢃⡀⠀⠀⠈⣿⡿⠿⠿⠟⠋⠙⠃
⠀⠐⠿⠿⠿⠟⠉⠉⠫⣿⣭⣽⣼⣙⠃⠁⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠒⠶⣶⣶⡦⠋⠉⠉⠋⠉⠈⠉
⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠈⠁⠀⠃⠀⠈⠁⠈⠀⠀⠀⠀⠁⠁⠑⠈⠁⠁⠃⠉⠁⠘
⠀⠀⡀⠀⢀⠀⠠⠀⡄⡀⢀⠠⠀⡀⢀⠀⢄⢀⠀⠠⠀⢀⠠⠀⠀⢀⠀⠄⠀⠀⢀⠀⣀⠀⠀⡠⠂⡄⡀⠠⢀⠀⡀⠀⠀⠄⠐
⠀⠀⠁⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠙⠋⠙⠛⠻⠟⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣟⡙⠛⢷⣶⡛⠻⣿⣿⣿⣷⣄⣴⣶⣦⣤⣴⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣾⣶⣿⣿⣿⣿⣿⠄⢹⣿⣿⣿⠉⠁⠈
⣿⣿⡿⠉⠹⣿⠟⠋⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡛⣿⣿⣿⣷⣽⣿⡏⠀⠀⠀⠀⠀⠀⠐
⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⡀
⣿⡧⠀⠀⠀⠀⠀⠐⠀⢀⣀⡀⢉⠿⣷⣤⣄⣼⣿⠌⢀⣶⠾⣟⣃⣀⣀⣀⡀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦⡶
⣿⣷⠄⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣮⣿⣽⢿⣧⣶⣿⡿⠛⠙⣿⣏⣙⣿⣥⣶⣤⠈⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⠛⠉
⣿⣿⡇⠀⠀⠀⠀⢰⠋⢤⣼⣿⣿⣿⣿⣿⣿⣷⣽⠧⡉⠳⣲⣴⣿⣿⣿⣿⣷⡞⢱⣿⣤⡀⠀⠀⣨⣿⣿⣿⣿⣿⡇
⣿⣟⠀⠀⠀⠀⠀⢁⠴⢶⣿⣿⣿⣿⣿⣿⣿⣿⣟⠢⡙⢄⣻⣿⣿⣿⣿⣿⣿⣟⡿⣿⣌⣱⣆⠀⠉⢻⣿⣿⣿⣿⠃
⣿⣿⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⣳⣟⣽⣿⡏⠂⢀⣽⣿⣿⣿⣽⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⢀
⣿⠋⠀⠀⠀⠀⠊⡩⢤⣿⣿⣿⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⢋⣿⣿⣿⡯⢹⣿⣿⡿⣾⣹⣿⣷⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣤⡼⠁
⣿⡆⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⡻⣟⣿⣿⣿⣿⣟⢷⢿⣯⣿⣿⠻⣿⠿⣺⣾⠿⢾⣿⣙⠀⠸⣿⠟⠛⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆
⣿⣇⠀⠀⠀⠀⠘⡡⠾⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣎⣧⣉⢻⣿⣾⣥⣾⣿⣿⣿⣿⠺⣿⡀⠐⠓⠀⠀⠀⣿⣿⣿⠿⠛⠋⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣤⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢿⣧⢹⡏⣿⣿⣿⣿⣿⡞⣿⣧⡈⢹⣿⣿⠓⠌⠟⠋⠁⠉⣦⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧
⣼⣿⣿⡀⠀⡄⣸⡿⠟⢻⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⡝⡿⣿⠀⣿⣿⣇⠸⣿⡿⢇⣄⠀⠀⠂⠈⣻⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣿⣿⣿⣿⣿
⣿⣿⡿⠁⠀⠁⠁⢀⣤⣾⣿⣿⣿⣿⣿⢿⣿⢹⣿⠟⣿⢀⠛⠀⢻⣿⣿⣷⣶⣶⣿⣏⣀⣤⡤⣾⣏⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣄⠀⠀⠀⢠⠟⠛⢙⣻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣂⣴⣶⢤⢼⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠁⠀⠀⣤⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⢟⣁⡤⣧⡴⠁⢀⣴⣿⣿⣿⣿⣿⣿⣧⣹⣾⣿⠿⣿⣿⣷⣾⢼⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⠛⠋⠉⣄⣤⣫⠏⠸⣿⣿⣿⣿⣯⣿⣿⣿⣿⣷⡘⣿⡋⢁⣿⣿⣿⣿⣿⣷⣄⣠⣤⣦⣄⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣿⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠤⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⣀⣀⠑⡀⠀⠀⠀⠀⠀⠠⠀⡠⠔⠉
⠀⠀⠀⠀⠁⡀⣀⡴⠉⠘⢶⣤⡀⠈⠀⠀⠀⠀⠘⣛⣛⠃⠠⢄⠛⣉⡽⠁⠐⠀⠁⠀⠠⠄⠀⠌⡡⠈
⠀⠀⠀⢀⠋⠛⣭⡀⠠⣤⠄⢻⠏⠀⠀⠀⡠⠀⠀⠀⠀⢀⠀⠀⠴⠀⠀⠀⠀⣀⠀⢀⡠⠐⠠⠊⠀⣀
⠀⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠣⠀⠠⣤⠄⣖⠒⣖⠒⣶⠒⡆⠀⠐⠂⠈⠀⠘⡀⠄⠊
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⠀⠹⣷⣿⠛⣜⠽⣫⣿⠃⠀⠀⠒⣀⡀
⠀⠀⠀⠀⡀⢆⡙⠰⠴⣺⣷⣇⠀⠀⠀⠀⠀⠀⠢⡀⠈⠚⠾⠩⠞⠛⠁⢀⠄⠐⠦
⠀⠀⠀⠐⠉⠈⠂⠀⠀⠀⠈⠈⠁⠢⡀⠀⠀⠀⠀⣤⡕⠦⣤⣀⢀⢴⠒⣧⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠊⠀⢸⠀⠁⠒⠠⡄⠀⠈⠁⠁⠌⠉⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠈⡆⠒⠲⣞⣙⡦⠖⠚⠀⠘⣏⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⢃⠀⠀⡟⠉⣇⠀⠀⠐⠠⠁⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠘⠀⠈⠀⠐⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡳⣾⡎⣿⡿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⣻⣿⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣘⣛⡻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣻⠿⠿⠿⣛⣿⣿⢸⣿⡿⣫⡶⢞⣛⡷⢾⣝⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡟⣼⢏⣾⣿⣿⣿⣷⡹⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⡇⣿⢸⣿⣿⣿⣿⣿⢇⣿⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⣵⣶⣍⣿⣿⣿⣿⣿⢸⣿⡝⣷⣝⠿⠿⢿⣫⡾⣣⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣛⣛⡻⣿⣿⣿⣿⡟⣬⣭⣭⣭⣭⡝⣿⣛⣭⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣟⢿⣿⠇⣿⢱⢖⣒⢶⣶⡶⠶⠶⣿⣶⣶⣶⣶⣶⡝⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣿⣿⢸⣮⣭⣾⣫⢞⣭⣟⢮⣻⣿⣿⢸⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣧⢿⢻⣿⣿⣻⢧⣿⣿⢸⣿⡇⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⢸⣿⣿⣿⣮⣓⣲⣞⣫⣾⣿⣿⣸⣿⡇⣿⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠛⠛⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⠠⠤⠄⣀⡠⠀⢀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠁⠀⠀⠀⠀⠀⠉⠀⠂⠀⠈⠀⠀⠀⠈⠈⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠁⠁⢈⢀⠀⣁⡉⠈⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⢠⠄⠈⠀⠁⠀⠉⠁⠀⠀⠊⠁⠂⠐⠋⠂⠀⠐⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⠀⠈⡑⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠄⠚⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢯⡀⡀⠈⡁⠀⡹⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⣿⠷⢺⠀⢻⠟⠀⢠⣧⣿⡞⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⡀⣸⢹⡆⡟⢀⣾⡀⠀⣇⡾⣿⠑⡂⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠞⠙⡜⣧⡇⣼⣿⡇⣸⡾⠁⣹⠲⡱⢡⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⢹⣷⣿⣿⢃⡿⡡⣀⣀⠀⡛⢮⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧⠿⠛⠁⣼⡁⠚⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃


⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀⣀⣀
⠀⠀⠀⠠⣶⣧⣗⣼⣖⡖⠈⠷⡝⣿⣶⣄⡀
⠀⠀⡰⠅⢶⠰⠟⠛⡶⠙⣷⢱⣇⣈⣇⡼⣿⠻⡿⣶⣦⣄
⠀⠀⠁⢀⠈⣿⣾⠿⣆⠖⠹⢸⣭⡙⠁⠏⡻⢀⠇⣿⢸⣿⣿⠲⣄
⠀⠀⠐⠒⠵⠃⣉⣴⡟⡠⣴⡿⠀⠀⠀⠀⡇⣿⠸⣳⣫⢿⡿⢶⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠐⠿⠛⠈⠀⠈⣿⠧⠵⣖⡒⠿⢻⠠⣿⣿⣾⣁⣴⡾⣯⢳⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠒⠂⢢⣳⣼⡷⣷⢸⡹⣿⡋⢠⣶⢬⡻⣿⣟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⣛⣻⣠⠿⣕⠉⠛⢤⣽⣾⣿⣮⣦⢞⣽⢽⠛⢖⣿⢧⡀
⠀⠀⠀⠄⠠⠔⠀⠐⠂⠤⠀⠟⠬⣍⠃⢠⣾⣿⣦⣘⣻⣿⣛⣾⣟⠾⢡⣾⠛⣒⣿⡶⣷
⠀⠀⠐⡂⣡⡤⢰⢀⠀⠀⠀⠀⠀⠪⠀⠸⡿⠯⢝⠺⢍⡛⢝⡯⡙⣷⣼⡣⡞⢋⣭⣵⣿⣌⣦
⠀⠀⠁⡁⠟⠱⠉⠉⠀⠀⠠⢀⡀⠀⠀⠀⡇⠀⠀⠙⠂⠉⠢⣷⣈⠋⠳⡷⣷⣽⢁⠴⢬⣆⣧
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⢧⠹⣦⡟⢧⡙⡏⠠⡞⠉⠉⠑⠒⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠙⠙⢿⡃⣄⢐⠀⠀⠀⠀⠀⠘⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⣿⠘⡏⢆⡀⢠⢄⣸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡟⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⢿⣿⣿⣿⣿⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠈⠙⠉⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⡟
⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇⠁⠽⡿⣶⠀⢰⡾⡿⠍⠁⡟⠃⠀⠀⠀⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠰⣄⣀⠀⠀⠀⠨⢳⡀⠀⠀⠉⠀⠈⠁⠀⠀⣠⠫⠀⠀⠀⢀⣀⡴⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠘⡀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠒⠦⠖⠂⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⡘⢀
⠀⠀⠀⠦⡀⠀⠀⠈⠢⠈⠁⠀⠀⠀⢡⡀⠀⠀⢄⠈⢙⣓⣛⠁⢀⠄⠀⠀⣠⠃⠀⠀⠀⠉⠠⠊⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠈⠐⠢⣶⣤⣍⡀⠀⠀⠈⠙⠢⣄⠀⠁⠀⣉⡁⠀⠁⢀⡤⠚⠉⠀⠀⠀⣈⣥⣴⡦⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⣀⠀⠀⠀⠙⢿⡶⢄⡀⣀⠴⣾⠟⠁⠀⠀⢀⣠⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣄⠠⠙⠂⢉⠴⠚⠁⢀⣤⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡶⠿⠛⠉⠀⠀⠀⠀⠀⠤⠔⢊⡵⠟⠙⠻⠷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠶⠒⢀⣴⣯⣤⣄⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⠤⠐⠀⠈⠉⠉⠉⠀⠀⠀⠔⠅⠀⠀⠈⠉⠉⠉⠀⠐⠠⢤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⣀⡤⠔⠈⠀⠀⠀⠈⠐⠤⣄⣀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠀⡠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⡀⠀⠀⡠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠟⠛⠛⠛⠛⠛⠻⠶⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⣠⡞⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡄⢀⣠⡄⢀⣠⠄⣠⠄⢀⡤⠀⣠⣄⣠⣶⢠⡿
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠰⣯⡴⢻⣀⣠⠖⣏⣠⠖⢿⡤⠞⠉⠓⠛⠙⠛⠋⠛⠛⠙⠛⠋⠙⠋⠉⠉⠁⣿⣸⠇
⠀⠈⢷⡀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⢈⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⣤⡀⢹⡏
⠀⠀⠈⢻⡄⢀⡴⠒⠺⠋⠀⠀⠀⠀⢠⡾⠋⠉⠉⠉⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠈⠃⠘⣇⣀⣤⢤⣄
⠀⠀⠀⠀⠙⣿⠀⠀⠷⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡶⠶⠋
⠀⠀⠀⠀⠀⠘⢶⣶⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣀⣀⣀⣤⣶⡃⠀⠀⠀⣶⣋⣱⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠈⠷⠈⢻⡄⠀⠀⠀⠉⠁⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⣸⡇⠀⠀⠀⣶⡶⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⢀⣠⡼⠷⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠿⠓⠦⠤⢄⣀⣀⣀⠀⠀⢀⣀⣀⡤⠴⣾⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⣿⣅⠀⠀⠀⣀⣤⡾⠋⠉⠙⢿⣍⡈⠁⠀⣠⠾⠿⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠈⠙⠶⠛⠉⣿⡀⠀⠀⠀⢀⡿⠛⠒⠋⠁⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣄⣴⠋⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢀⣴⠾⠛⣿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠙⣦⠀⠀⠀⠀⠀⠀⢻⡏⠻⢷⣄⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣠⡿⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠘⣷⡀⠀⠀⠀⠀⢸⣧⠀⠀⠙⣷⡀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⣸⡇⠀⢠⣿⠁⠀⠀⢰⡏⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣧⠀⠀⢿
⠀⠀⠀⠀⠀⢀⣿⣇⣀⣸⡇⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡆⠀⠀⠀⣿⠄⠀⢿⡇
⠀⠀⠀⢠⡾⠋⠉⠉⠙⠻⣦⡀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠸⣧⢠⣶⠛⠛⠲⠖⠿⣧
⠀⠀⠀⣿⣴⡟⢠⣶⢠⡶⣾⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⣿⡈⠛⠛⣿⠃⣤⢤⣌
⠀⠀⠀⠀⠘⠿⠛⠻⠿⠁⠀⠀⠀⠀⠻⣶⠶⣤⣤⣄⣀⣀⠀⠀⠙⣧⣀⣼⡁⠀⠀⠀⣀⣀⣤⣤⡾⠁⠀⠸⣷⡾⢿⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣸⡟⠛⠛⠛⠟⠋⠻⠋⠙⠿⠛⢿⡏⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣭⣤⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣽⡇
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⡍⠀⢸⣿⠋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣸⡇⠁⢀⣾⣿⣄⠿⣿⣏⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢻⣽⣟⣴⣿⣿⣿⣿⣶⣻⣯⣟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⣸⣿⡿⠇⠸⠿⠿⣿⣯⣯⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣾⣽⡇⠐⣦⣀⣁⣴⠊⠈⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣾⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣇⣀⣈⣻⣿⣁⣀⣸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣦⠀⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣠⣬⣥⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⣹⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⣟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣟⣛⡛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⢻⣿⣿⠿⠿⠿⠿⠀⠰⢤⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠟⠒⠀⠀⠠⣤⣄⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣶⣶⣶⣶⣦⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣥⣤⣤⣤⣤⣤⣤⣤⣙⣿⣿⣷⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠁⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠈⠙⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⢀⣤⣾⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣷⣦⣄⠀⠈⠛⢛⣋⣠⣤⣴⣶⣾⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⢠⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣭⣤⣶⣶⣿⡿⠿⠟⠛⠋⢉⣠⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⣰⣿⣿⠋⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⠿⠿⠛⠋⣉⣁⣤⡄⢠⣾⡿⣿⣮⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠘⢋⣉⣡⣤⣶⣶⣿⡿⠿⠟⠛⢉⣉⠀⢤⣴⣶⣶⣄⣿⡟⠋⠁⢸⣿⡀⠈⠁⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣈⣤⣴⣶⣾⣿⠿⠿⠛⠛⠉⢁⣤⣤⣶⡆⣾⡟⠻⠷⠸⣿⣄⣸⣿⢸⣿⡿⠿⠀⣿⣧⢀⣶⣷⠀⢹⣿
⠀⢠⣴⣶⣿⡿⠿⠟⠛⠋⣉⣤⣤⡀⣶⣿⠿⣿⡆⣿⣏⣀⣄⠻⣿⣶⣶⣄⢿⣿⠛⠋⠀⣿⣧⣴⣶⡈⠻⠿⠟⢁⣀⣤⣤⣶⣶⣧
⠀⠀⢉⣩⣤⣤⣄⠸⣿⡰⣿⡋⠙⠛⢸⣿⣤⣾⡟⢸⣿⡛⠋⠀⣤⣄⣸⣿⠸⣿⠄⠀⠀⠘⣉⣉⣤⣴⣶⣾⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠈⣿⡏⠙⣿⡇⢿⣇⠙⠿⠿⢿⣦⣿⣏⠹⣷⣄⣿⣷⡾⠿⠙⠛⢛⣁⣠⣤⣴⣶⣿⡿⠿⠟⠛⠋⢉⣀⣤
⠀⠀⠀⢹⣷⠀⢹⣿⠸⣿⡴⣶⣤⣾⡿⠸⠿⠀⠙⣉⣈⣤⣤⣶⣾⣿⠿⠿⠛⠛⠉⠉⢠⣤⣶⠆⠀⢀⣿⣿⠇
⠀⠀⠀⠈⣿⣷⣾⠟⠀⠛⠃⢈⣉⣡⣤⣴⣶⣿⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⢀⣾⣿⡟
⠀⠀⠀⠀⢉⣁⣤⣤⣶⣶⣿⡿⠿⠛⠛⠉⢉⣤⣴⣦⣄⣀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⠟⠁⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠈⠿⠛⠋⠉⠁⠀⠰⣶⣷⣦⡀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣀⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⣤⣶⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠞⢢⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠉⠛⠀⠘⠉⠁⠀⠀⠀⠈⠉⠉⠛⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢀⠀⣀⠀⢀⡧⣀⠀⢀⡧⣀⠀⣰⢄⣀⣸⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⠈⡗⠚⠉⠉⠀⠈⠉⠉⠀⠈⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠪⣉⣷⡀⠀⡁⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⢦⡀⢀⠞⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢸⠉⠀⠀⠀⠑⠀⠚⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⡀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡼⢣⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣠⠎⠀⠀⣀⠭⠶⠤⢄⣀⡀⠀⠀⠀⠀⠀⡀⣠⣤⠒⢩⣄⣀⡗
⠀⠀⠀⠀⠀⠀⠘⢤⡤⠞⠙⢆⣴⡋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⢹⢾⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣹⠚⠋⢹⠆⠀⠀⠀⠀⠀⠀⠀⠰⡏⠙⠓⣇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡽⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢯⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⣤⣤⣤⣤⣤⣀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢹⠃⠀⠀⠀⠀⠀⠸⡏⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣤⠏⠀⠀⠀⠀⠀⠀⠀⠹⢤⡵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠶⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⡇⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣾⣿⣾⣼⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⣿⣿⣿⣽⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⢹⣾⣿⣿⣾⣿⣿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣿⣿⡿⣷⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣿⣿⣿⣿⣿⣾⣇⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣽⣟⣿⣽⣿⣿⣿⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⢀⡀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠀⣤⠾⠻⢶⣠⢟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡾⠋⠉⠳⠎⠁⠀⠀⢠⣯⠟⠶⣶⠀⠀⣠⡆
⠀⠀⠀⠀⢀⣤⡶⡖⠷⠚⠛⢀⡤⢾⠁⠀⢴⣦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣼⢔⣫⠾⠃
⠀⠀⠀⠀⢼⡏⡌⣶⡒⠚⠋⠉⠀⣸⠀⠀⠀⢨⡟⠋⠀⠀⣷⠿⣄⣀⠀⠀⠀⠰⠏⠲⣏
⠀⠀⠀⠀⠈⠻⠇⢹⠻⡄⠀⠀⢀⡟⠀⠀⠀⠘⢷⠄⡠⠚⠉⠠⡌⠛⠒⣆⣀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⣆⠀⣼⢃⠀⠀⠀⠀⢠⡞⠀⠟⠰⠇⠀⠀⠀⠘⠟⡇⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠈⢧⣿⣸⠀⠀⠀⠀⢸⢠⣄⣑⣠⣄⠀⠀⠠⣄⣰⠃⢀⣹⠆
⠀⠀⠀⠀⠀⡀⢀⡿⠀⠀⠀⢸⣯⡼⢷⡀⠀⠀⠘⢦⡀⠀⠈⠉⠀⠶⣤⡾⠃⣤⠈⢷⡀
⠀⠀⠀⠀⣼⡏⠘⣧⠀⠀⢀⣏⡇⢹⡈⠳⣄⣀⠀⣤⡍⠳⠀⡤⢶⡞⠉⠲⠛⡉⢣⡠⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⢸⣿⢷⡛⡿⣇⠀⣼⠷⠤⣄⠙⠲⠤⣭⣉⡁⢹⡤⠖⠋⠉⠉⠙⠓⠶⡌⢆⡙⢌⢾⣧⡀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣀⣤⣀⠙⢦⡀
⠀⠀⠀⠘⠟⠛⠛⠓⣿⣰⠓⠲⡄⠙⣦⠀⣀⣠⠞⠁⠀⠀⠀⣰⠏⣉⣉⣹⡆⢹⠈⠣⠈⠛⠋⢥⣄⠀⠀⠀⠀⢰⢋⡤⣠⡌⠋⢛⡛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣠⠶⠷⢄⡈⠳⣯⠀⠀⠀⠀⠀⣴⣫⠎⠁⢀⢸⠏⣿⠙⢦⡄⠀⣿⣿⣿⠿⣷⣶⡀⢸⣄⣒⣈⣀⠀⠈⣻⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣫⠞⠁⣠⣀⠀⠉⠳⠘⢷⣄⣀⣠⠞⣱⠃⠀⢈⣹⣆⣀⣧⢀⣾⠇⠀⣿⣿⡿⢿⣧⣾⣷⡀⢳⡒⠛⠉⠀⠙⣽⠋
⠀⠀⠀⠀⣿⡀⡶⠋⢀⡴⢚⡁⠉⠛⠶⣤⣄⣀⡭⠟⢁⡼⠁⠀⠀⠈⠁⠋⠃⢁⣾⠟⠀⠸⣿⣻⣤⣿⢿⣎⠈⢿⣆⠉⢓⣦⠴⠦⠼⢦
⠀⠀⠀⠀⣿⣷⣳⠞⣩⡴⠁⣁⣂⣀⣀⣀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠉⠉⣿⠻⣷⣿⣷⠶⠛⣷⠋⠀⣀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠻⠟⠱⠮⢋⡶⠛⠉⠉⠀⠉⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠸⠀⣇⠹⡿⢋⡴⠚⣉⡷⢾⣇⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣳⡶⢦⣤⣄⡀⣠⣴⣶⠶⠶⢶⣶⣶⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣦⠙⢁⣠⠞⠁⢀⣤⠟⠒⠀⣤⣤⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠉⠙⢋⣾⠃⠀⠀⠀⢹⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⣠⠾⠋⣠⠖⠀⠀⠀⣀⡴⣋⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⣀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣇⣠⠞⠁⠀⣴⠚⠋⠉⣻⠳⡌⠧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠂⠘⠀⠅⢠⡿⠡⠌⠡⠄⠀⢤⠄⢻⠆⣿⠀⠠⠦⠀⠃⠠⠤⠀⠀⠘⠃⠰⠆⠑⠎⠠⠙⣧⣠⣤⣤⣤⣤⣀⣀⣿⣤⣝⠒⣲⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠶⠶⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡭⠶⠤⣀⣀⡤⡴⢯⣕⣆⠀⠈⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠸⡀⣠⣖⣿⣿⣽⡗⠫⣭⣿⣿⣷⣦⡄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣴⢻⡿⠛⢍⣻⠺⢿⣿⣴⣿⢿⣦⣧⣿⢢⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⣇⠘⢧⡀⠀⠙⣿⣏⡼⣓⣼⣧⡿⢷⡯⠠⠃⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⠤⣍⣹⠟⠷⢿⡋⠩⠭⠭⠗⢮⡇⠐⠦⣤⣀⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠊⠛⠒⠒⠒⠋⠀⢀⡴⠶⡎⠱⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⣰⣫⣭⡳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠘⣦⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣿⡇⢀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡧⠀⠀⣤⣶⣞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠈⢧⠠⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⢀⣤⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠧⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠒⠚⠁⠀⠀⠉⠀⠀⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⡀⠀⠀⠀⠒⠒⠶⠶⢾⣿⡆⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠐⢦⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠙⢾⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃





⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡱⢒⠤⣿⠶⢄⣨⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣰⢁⣀⠀⠀⠠⠔⠀⢣⠸⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⡏⠀⠀⠀⡀⠴⠀⠀⢸⣄⡇⠈⢏⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢳⡀⠀⠀⠀⠀⠀⢀⡼⠽⠉⣲⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣉⣓⠚⠒⠢⡀⠀⢰⢁⡠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠛⠻⢽⠁⠀⠈⡧⠒⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠑⠒⠊⠁⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢳⠢⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⠫⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⢠⠊⠑⢄⠀⠀⠀⠀⠀⠀⣰⡅⠈⣆⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡪⣶⡇⠀⠀⠀⠀⠀⢿⣫⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠯⠭⠤⠤⠤⠤⠤⠤⠤⠜⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠄⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠠⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⠐⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠂⠀⠀⠠⢁⣀⣀⠀⠀⠀⢀⣀⣀⠘⡀⠀⡀⠀⠀⠀⠰⢂⣠⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡀⢀⠜⠁⠹⣦⡂⠀⠁⠀⠀⢀⠀⢚⡛⠛⠃⠠⠀⢛⣋⣉⠁⠀⠀⠁⠀⢠⠔⠡⡤⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⡀⠚⠞⣋⠀⢀⣀⠘⢿⠂⠀⠀⠀⢠⠀⠀⠉⠉⠀⠀⠈⡀⠁⠀⠀⠀⠀⠀⠀⢀⡠⢊⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠐⢤⠈⠉⠀⠈⠉⠀⡈⠀⠀⠀⠀⡆⠀⠀⠀⣀⡨⢬⡥⣤⡄⣀⡀⠀⠩⡀⠐⠁⠀⠄⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠘⣯⣤⣷⡤⠷⢤⠧⣤⡇⠀⠐⣀⡀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⣰⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⢎⢀⣐⢋⣼⡟⠁⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⡀⣀⠄⠉⠉⠼⣼⣠⠀⠀⠀⠀⠀⠀⠁⢄⠀⠙⠓⠘⠛⠉⠀⡀⠂⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠈⠢⠀⠀⠀⠀⠀⣤⡝⠦⣄⣀⣠⠐⠋⣢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠣⠈⠐⠂⢄⡠⠀⠈⠁⠁⠈⠁⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠐⠂⢰⡤⠤⢠⡆⠳⣄⣤⠴⠀⠰⡴⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡗⠶⡏⠀⠀⢀⠀⠊⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠄⢒⠀⠠⠃⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⢰⡀⠀⠀⠑⠁⠀⠀⠈⠈⡙⠀⠀⠀⠀⠀⡠⠘⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡟⠇⠂⠀⠀⠀⠛⠂⠂⠀⠀⠀⠚⠀⠀⣤⣧⠀⠀⠀⠀⢠⠇⡆⠀⣀⠴⠋⠅⢰⡟⠋⠙⣦
⠀⠀⠀⠀⠀⠰⠀⡔⠀⡄⠠⡄⠀⠈⢲⡏⠛⠙⢻⡹⡶⣗⠘⠋⠀⢹⠀⣰⠀⠓⠶⠶⠒⠖⠛⠁⠚⠃⠀⠀⠀⠀⠀⠀⠉⠓⠐
⠀⠀⠀⠀⠀⠀⠸⣷⡄⢸⠂⠓⡿⣾⢻⣿⠀⢰⣦⠁⠤⣭⣢⡄⠀⣼⣀⠀⠀⠀⠀⠁⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢐⣟⠀⣠⣀⣬⡄⠋⢀⡉⠐⠛⠛⠛⠃⠋⢀⠁⠀⢙⣿⡅⠀⠀⠁⠉⢩⠀⡀⡇⠀⠀⠀⠀⠀⣻⣏⡼⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠈⠁⢰⠢⢎⣀⣶⡄⠀⠀⠀⠀⢸⣇⠀⠸⠟⠀⠀⠀⡀⠀⠀⢀⡅⠃⠀⠀⠀⠀⠀⠿⠻⢴⡟⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠘⠢⠀⠀⠀⠀⢸⢈⠀⠀⠀⣤⠠⢤⡁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠈⠉⠀⣀⣈⢎⣿⣶⣤⣀⣂⡄⠲⠄⠀⠀⡄⠀⡧⠀⠀⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⢀⣀⢀⡀⣰⣤⡘⣧⣿⣿⣿⣿⡟⠃⡰⠶⠒⠚⣷⣖⠀⠐⢈⣁⠀⠐⠃⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠤⠔⠞⠋⠘⠇⠈⠁⠣⣼⣟⣻⣿⣿⣿⣿⣆⠀⠁⠀⠀⠀⡾⠃⠀⠀⠀⠉⠀⠀⡀⠢⢴
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠞⡻⢁⢀⣤⣄⣤⣘⢦⣀⢠⣤⣴⣿⣿⣹⠉⠈⡟⠿⠘⠀⠀⠀⠀⠀⢀⣺⠁⠀⠀⠀⢠⠌⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠠⠋⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣷⠀⠀⣶⣿⣳⣿⡆
⠀⠀⠀⠐⢤⠀⠀⢀⡞⣰⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⢸⡿⣿⣿⣧⣀
⠀⠀⠀⠀⠁⠀⢸⣿⣧⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣥⠆⠀⠀⠀⢻⡿⠅⠀⠈⠀⠈⠻⣏
⠀⠀⠀⠀⠀⠀⣰⡟⣽⡇⣸⣿⡇⣿⣿⣿⣿⢿⠿⠛⠋⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠋⠋⠀⠀⠀⠀⠘⢿⣿⡦⠤⢤⡄⣀⠐⠋⣀
⠀⠀⠀⠀⡄⠀⢿⢣⣿⡀⣿⣿⠃⣿⣿⣿⡇⠀⠀⠀⠠⣶⣿⣿⣿⡿⠿⠿⢿⡿⣿⠿⠶⠖⠤⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⣾⣿⠀⠀⢺⣿⢰⣿⣿⢘⣿⣿⣿⣿⠀⠀⠀⠲⠼⢉⠈⡯⠛⠛⠷⠶⣤⣄⡀⠀⠀⠀⠤⣀⡀
⠀⠀⠀⢼⣿⡇⠀⢀⠀⠸⣿⡿⣼⣿⣿⣿⣿⡆⠀⢀⠐⠀⢠⠐⠇⠰⣷⣷⣄⣘⠙⠛⣿⣷⣶⣦⣌⣉⡙⠒
⠀⠀⠀⠸⣿⢇⠀⢐⡤⢳⣿⣿⡿⣿⣿⣿⣿⣷⠀⣾⣿⠀⢸⡆⠀⠀⠸⠁⠈⠙⠻⢶⠀⠀⢀⣀⣉⠛⠻⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠐⢿⣿⢀⡿⢀⣼⣿⡟⢁⡟⠻⣿⣿⠟⢀⣿⣿⡇⢨⡁⠀⠐⠀⠀⠀⠀⠀⠸⠆⠀⠀⢿⠛⠻⢶⣦⣭⡙⣏⠙⠿⣿⣶⣦
⠀⠀⠀⠀⠉⢳⣼⠀⢉⣴⡟⢠⠚⠛⣫⣼⠋⠀⢼⣿⣿⣿⣬⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠃⢹⠀⠙⢰⣶⣤⣆⡉
⠀⠀⠀⠠⠞⠸⢃⣴⣾⣿⠡⠃⣠⣾⡿⠃⠀⢀⣿⣿⣿⡯⠡⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⡀⠀⠀⢸⠀⠀⢘⣿⠉⠙⠋
⠀⠀⠀⠀⠀⡟⠀⣶⣿⡅⣡⠾⢋⣼⠃⠀⠀⠀⣿⣿⣿⣿⡲⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣿⢠⣰
⠀⠀⠀⠀⠘⢀⣼⣿⡏⢀⡴⣾⣿⠃⠀⠀⠀⣀⣽⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣛⠘⣻⠆
⠀⠀⠀⠀⣶⣿⣿⡿⢶⣶⣶⣿⣿⣀⣤⣤⣤⣿⣿⣿⣿⠇⠀⠀⠸⡆⠀⠀⠀⠀⠀⠠⠂⡐⢪⣅⣉⠐⠐⠫⠍⠉⠉⠉⠙⠛⢿⣶⡀
⠀⠀⠀⠀⠙⠛⠉⠙⠛⠛⠚⠿⠿⠿⠿⠻⠿⠛⠛⠛⠿⠷⢤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠒⢌⡙⠻⢤⣘⣂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⣒⣭⣁⠀⠀⢀⣄⡀⠀⠀⠀⠈⠿⢷⣦⣌⡙⠳⠦⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠈⠉⠀⠘⠃⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⠄⠀⠀⠀⠀⠠⠀⠂⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠚⠛⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢠⡤⡄⠤⡠⠦⠔⠖⠂⠒⠒⠐⠒⠒⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠄⠎⠀⠀⠔⠒⠒⠄⠀⠀⠀⠀⠚⠉⠑⢸⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣅⠀⠀⠀⠀⠀⠀⠀⢄⢠⢴⡀⠠⠴⠄⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠈⠦⣀⠇⠀⢶⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⡗⠒⠀⡤⢀⡀⠤⠰⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⠈⠙⠒⠹⢄⢀⠝⠒⠉⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡠⠚⡗⠀⠀⠀⡜⠙⢆⠀⠀⢹⠑⢦⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⢀⡇⠀⠀⡜⠁⠀⠈⡆⠀⠸⡆⠀⣇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣉⣹⡆⢸⠃⠀⠀⠑⣄⠀⢰⠃⠀⠀⣧⠾⣭⣘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠈⡗⠒⡶⠤⠬⠷⠧⠤⡔⠒⡏⠀⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⠃⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡞⠀⠀⠀⠀⠀⠀⠈⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠤⠇⠀⠀⠀⠀⠀⠀⠀⠳⡼⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⢤⡀⠀⠀⠀⠀⠀⣀⣀⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢀⡤⣤⢤⡀⠘⣆⠀⠀⡠⠊⡁⠀⠀⠀⢀⡉⠳⣄⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠘⢆⠀⣰⠃⠀⡿⠀⣸⠁⢸⡁⠀⠀⠀⠀⠙⠳⡜⣆⠀⠀⢰⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⢰⢒⡒⠓⠒⠒⠚⣇⣀⠀⠉⠁⣀⡴⠁⢠⡇⢀⡞⠙⠳⢦⣀⣀⠀⠀⠙⣿⠄⠀⢸⠀⠀⠀⠀⠠⡏⢓⡦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⣈⣋⣀⣉⣉⣀⣀⣈⣉⣟⠉⠁⠀⠀⠘⣇⣸⠁⠶⠆⠀⠀⠹⠿⠓⣆⣻⠇⠀⢸⡄⠀⠀⠀⠀⠛⠁⠀⠀⣦⡀⢠⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣀⠀⠐⠂⠀⠀⠐⠂⠀⣀⡟⠀⠀⠀⠓⠦⠤⠤⠤⠤⢴⠛⠛⠁⣉⠛⠙⢻⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⣐⣂⠀⠀⣸⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠞⠙⢲⠆⢸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠸⣗⣦⣍⣉⣤⠞⠁⠃⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠓⠒⠛⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⡇⠀⠀⢸⢦⡀⠀⢰⡇⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢸⡤⠤⠤⠤⠤⠤⠤⠤⠤⣿⠀⠀⠀⣀⡤⠶⢺⠏⢀⣇⣀⣀⣸⠀⠹⡖⠶⠿⣇⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⢸⠀⠒⠒⠒⠒⠖⠒⠲⠀⣿⠀⠀⢰⠁⠀⠀⢹⣿⢾⠐⠀⠀⢸⣷⣿⡏⠀⠀⠈⢇⠀⠀⠀⡇⢰⣒⣒⣒⠒⠂⢨⠗
⠀⠀⠀⠀⠀⢸⣀⣉⣉⣉⣁⣀⣀⣀⣀⡿⠀⠀⡇⠀⠀⣄⣼⣽⢿⣄⠤⣖⣾⠏⢿⣧⢠⠄⠀⢸⡄⠀⠀⠧⠤⠤⠤⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠇⠀⠀⢸⡇⣿⠈⠁⢺⠭⣿⠂⠸⣿⣿⠀⠀⠀⡇⠀⠀⠀⢀⡞⢛⣛⣛⣛⣛⣛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⢹⣀⠀⠀⢸⠀⠀⠀⢸⣧⠇⠀⠀⣀⡿⢿⣄⠀⢿⣿⠀⠀⠀⣧⠀⠀⠀⠈⢧⣬⣭⣭⣭⣭⣭⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠋⠹⡇⠀⠈⣿⠂⠀⠸⣶⠲⠤⣼⠏⠀⢰⡞⢩⠀⡡⢻⡄⠈⢿⣀⣀⣠⣽⠀⢀⡖⠒⠒⠒⠒⠒⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⡾⡟⠛⠢⠿⠀⠀⠀⢸⠀⠐⠃⠀⢠⡟⠀⠀⠈⢠⠈⣧⠀⠀⠉⠁⠘⠋⠀⢸⣇⣉⣉⣉⣉⣉⣋⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⣾⢁⣀⠶⠘⠤⢒⡸⣇⠀⠀⠀⣉⠀⠀⠀⠈⠈⠈⠉⠀⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠿⣷⣬⣥⣭⣭⣭⣥⣶⣿⣷⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣌⠝⣿⢿⡿⢾⣿⣿⡛⠋⡉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠁⣾⣿⣿⠇⡂⣿⣘⣷⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠃⡀⠌⣩⡛⠰⠃⡛⠛⡏⠀⣶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠢⠉⠀⠁⠈⠀⠀⠁⠁⠈⠈⠨⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⢄⠀⡇⠈⠢⣀⠀⡼⠻⡄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠳⠀⠀⠈⠙⠃⠀⠈⢦⡴⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⢶⣲⣄⠀⠉⠉⢟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠒⠶⣾⠁⠀⣄⠈⠉⠁⠀⠀⠈⠳⢄⡀⠀⠀⠂⠀⠀⠀⢹⡉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠀⠀⠀⠈⠓⠦⠴⠆⠀⠀⠙⢏⡁⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢤⣀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠈⡆⠀⠀⢰⠶⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣈⣡⡴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢀⣀⠤⠖⠊⠉⠐⠂⠀⠀⠀⠀⢸⠀⠀⣠⣄⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠴⠶⠚⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⢰⠇⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡟⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢻⠦⢤⣀⠀⠀⠀⠀⣠⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢺⠀⠀⠈⠙⠓⠶⠞⠁⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⣠⡦⠈⠻⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⡇⠙⠁⠀⠀⡀⠀⠀⠈⣧⠀⠀⢠⡇⠀⠀⠀⠻⠽⠏⢴⣶⡇⢠⣦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠸⡀⠀⡼⠀⣀⢀⡠⣄⠀⠀⠀⠀⠀⠘⠷⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠃⠀⠃⠀⣇⠀⠀⢘⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠴⠄⡄⠀⠀⢀⣤⣤⣤⡀⠀⠉⠲⠤⠜⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠁⠀⢠⣿⣿⣿⣿⣿⡆⠀⣀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡆⠀⠀⠀⠘⣿⡉⠙⣿⣿⠃⠀⠉⠁⠛⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣾⣿⣿⣷⣾⣯⣷⣿⣿⣶⣤⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⢉⣿⣿⣿⣿⣿⣿⣿⠛⠿⣿⣿⣷⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠙⠣⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⡿⣿⣿⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⣽⣿⣿⡇⣿⣿⣿⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢀⣄⢀⣀⢀⣀⡆⢀⢀⣀⢀⣀⠀⣀⡆⣀⢀⣀⠀⣀⠀⣀⡰⢀⡄⢀⠀⡀⢀⠀⢠⡀⠀⡄⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠇⢠⣬⢥⣺⣽⠀⡟⡈⢹⠀⠯⣸⠀⣇⡿⣀⢇⣸⠀⣑⠸⣊⠵⢇⡸⢅⠇⠐⠮⠂⢁⡭⡇⡇⣗⡉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣰⣾⣷⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⠟⠙⠿⠁⠁⠈⠈⠁⠉⠙⢻⡿⣖⢤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠿⠿⣿⣿⡆⠻⠀⠀⠰⣶⢤⣤⡤⠶⠄⠁⣿⢸⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⣀⠀⢰⣿⣿⠃⠀⠀⡀⠀⠂⠀⠀⠀⠂⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠈⠁⠀⢠⠟⢻⡆⠀⠀⣡⠐⠀⠀⠀⠀⠀⠋⠁⣸⣿⣿⠿⠿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢹⠀⠀⠀⢀⠀⠈⢻⡄⠀⠉⠀⢎⡽⣭⣌⠀⢸⣦⡿⠛⠀⠀⠀⠈⠉⠻⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣇⡈⠸⠀⡆⢸⠁⠀⠀⠙⣶⣾⡓⠸⠀⣉⠻⠀⣾⠉⠀⠀⠀⢰⡆⢫⣀⠀⠀⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠃⢠⡾⠁⢼⡀⠀⠀⠀⠈⢿⣷⣦⣀⣾⣤⣼⠏⠀⠀⠀⠀⢸⡇⠀⠻⣦⡀⠀⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠟⠁⢀⣼⣿⡄⠀⠀⠀⠀⠙⠿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠙⠀⠀⠻⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⢀⣿⣿⣿⣿⣶⣦⡐⠲⢤⡶⠿⠿⠷⠀⠀⣀⣤⣴⣾⣿⣿⣿⢿⣶⡄⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⡟⠂⠀⢠⣾⢿⣿⣿⣿⣮⡉⠁⠀⠀⠀⣠⡀⠀⠀⠀⠈⠻⢛⣿⣿⣿⣿⠀⠙⢿⣆⠒⠀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠸⠇⠀⣿⣿⣿⠉⠛⠒⠀⠀⠚⣭⡛⠓⠂⠀⠐⠚⣻⣿⣿⣿⠇⠀⠀⠀⠻⣧⡀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⣴⠀⠀⢹⣿⣿⣷⠈⠂⠀⠀⠘⣿⠛⠗⠀⠀⠶⢶⣿⣿⣿⡟⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⣷⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢰⣯⠀⠀⢨⣿⣿⣿⣶⠆⠐⠲⠶⠾⡿⠦⠀⠀⣀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠛⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⢹⣇⣠⣼⣿⣿⣿⣿⣿⣷⣤⣶⣴⣾⣦⣤⣦⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⠆⡀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⢠⡀⠀⢽⣿⠋⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢹⡇⣇⠀⠹⣦⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢄⣙⠀⠘⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢸⡇⠸⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⠌⠣⠀⠈⢣⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠛⠿⠿⠿⠿⣿⣿⣦⠀⠘⣿⠀⠀⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣿⣿⣅⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠈⣻⣿⣧⠀⣿⡇⣤⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢀⡿⣏⣿⡦⠀⠀⢼⣾⣏⠉⠀⠀⠀⠀⠉⠁⠀⠈⠙⠻⠿⣿⣿⣷⣷⠀⠀⠀⢀⣽⣿⣿⣧⣿⠓⣏⠇⢀⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣾⣿⣿⣇⠀⠀⠉⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣄⣤⣴⣾⣿⣿⣿⣿⠏⠀⠙⣦⡈⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠘⠢⠀⠀⠙⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⡀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⡴⢂⣤⠘⣽⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣰⣇⣴⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠮⠶⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⡠⢦⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⢩⣷⠋⠀⠀⠉⠀⠀⢀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠀⢀⡀⢤⠰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣁⣀⣀⣀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣦
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⣈⣉⠉⠻⢧⡀⠀⠀⠀⠀⣿⣿⣿⡷⠒⠉⠉⠉⠉⠱⢶⣦⣀
⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡋⢾⣵⣿⣿⣽⣿⢟⣶⢦⣹⣆⣀⣀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣯⣍⣉⣀⣙⣛⣽⣾⣿⣋⡿⢾⣧⣾⠛⣿⣿⣶⣭⣉⠋⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡿⠿⠟⡉⠉⣷⣧⣤⡿⠈⠛⠿⣿⠉⠻⠳⠒⠛⢷⣦⣤⣤⡀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠈⠉⠉⠁⠀⡀⢁⣴⣿⣿⣿⠇⠀⠀⢠⠃⠘⠋⠉⠁⠀⠘⠉⢁⡀⣱⣄⣀⡸
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣤⠀⠀⠀⠀⠀⠁⣸⣿⣿⣿⡟⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⣿⡿⠋⠀⠀⠀⠀⠀⢩⣄⣀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡋⠉⠉⢁⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠹⠿⠋⠁
⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⣴⣻⣿⡿⣦⠀⠀⠀⠀⠀⠀⠀⢨⠉⣭⢉⣀⣤⢤⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣣⣿⣿⣯⣽⣼⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⢫⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⣿⣿⣿⣿⣽⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢣⡆⠄⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠿⠈⣿⠿⣯⣿⡿⢿⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⡇⢀⠸⣿⠌⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠷⠀⠁⠀⠀⣿⣳⣦⣼⠆⠀⠀⠀⠀⢸⡿⣿⣿⡇⣿⠁⠈⠀⠹⣷⣌⣙⣻⠟⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡧⠶⠶⠂⠀⠀⠀⡿⣿⢇⣿⡇⠀⠀⠀⠀⣿⠠⠀⠋⣴⡿⠀⠆⠐⠀⣸⡏⣿⠁
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣶⡶⣶⢶⣷⣶⣟⣿⡇⠀⠀⠀⢸⡏⡁⠈⢃⣿⡇⠐⠀⠀⠀⢼⣇⡟
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⣾⣿⣟⣻⣾⠃⠀⠀⠀⣾⢡⠁⠊⢸⡏⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⠀⠀⠀⣸⠏⡀⠐⠀⣿⠁⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⡟⣿⢟⣿⣿⣿⣿⣿⣇⠀⠀⢀⡿⢀⠤⠄⢰⡿⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠏⠋⠛⠙⠋⠉⢻⣏⣻⣵⠇⠀⠀⢸⣧⣀⣀⠀⣼⡇⠠⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⠈⠋⠉⡅⠀⠀⠀⠀⠀⡏⠉⠛⣿⠉⠠⠀⠐⡆⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⢸⢀⠞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⣻⠶⠶⠶⢦⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣿⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⡏⠺⣄⣤⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⢸⡇⠀⠀⠈⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢰⣾⡌⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣿⣧⢿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠛⠛⠻⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣠⣄⣼⣿⣿⣿⣷⣾⣽⣄⠀⣠⣾⢠⡾⠇⠀⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⠙⣳⡆⠀⡆⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠙⠛⢻⣿⠛⠛⠛⠛⢹⣿⣿⣿⣿⣿⣏⠈⠉⠀⠀⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣤⠀⣧⣤⣄⣀⣀⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣬⠸⣿⣽⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣶⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢭⣿⣿⣿⣟⣋⣭⣅⠶⠇⠀⣏⠈⢿⣿⣿⣿⢟⣉⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠉⠉⠁⠀⠈⠉⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠙⠻⢿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣷⣦⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⣤⢀⡾⠉⠉⢉⣉⡟⣿⣿⣿⣿⣿⣟⣙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠚⠓⢒⣛⠛⠷⠒⠒⠒⠒⠛⠛⠉⠉⣽⣿⣯⢹⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣙⣛⣁⣠⣶⣾⣿⣿⣶⣦⣤⣤⣭⣭⣥⣼⡇⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢛⢿⣿⣿⢟⡽⢶⣾⣝⢿⣿⣿⠉⠉⣿⣿⡇⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⣀⣼⣿⣏⣿⣦⣾⣿⣿⡏⣿⣿⣦⣶⣿⣿⣧⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣝⢿⣿⣿⢝⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⠿⠿⠿⠿⠿⠟⠛⣿⣀⣠⣼⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⢶⣶⣿⣿⣿⠟⢻⣿⢿⠿⠿⠿⠿⠟⠛⠉⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⣿⣿⣟⣿⡷⢶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠻⢼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠁⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣿⣏⣥⣤⣤⣤⣤⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣟⣡⣤⣤⣤⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⢆⠀⠀⠀⣸⠠⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⢒⡖⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣀⠀⢇⡿
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⠖⣌⣻⣫⡕⣦⣄⡂⡀⠲⠖⠒⠖⠂⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠈⠻⠋⠀⠘⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⢡⡄⠀⠀⠀⠀⠀⠀⢐⢙⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⢃⣤⡞⠁⠀⠀⠀⠀⠀⠀⠘⣧⡛⣠⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⢀⠀⠀⡀⣀⡤⢾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠘⣿⠉⠉⠀⢠⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠄⠀⠀⠀⣿⠀⠀⠀⠨⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢻⡆⠀⠀⢠⣿⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⣼⡇⠀⠀⣾⣿⣇⠀⠀⠀⡀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⢸⣿⡇⠀⠀⣿⣿⣿⡀⠀⠀⠃⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⣀⣹⣷⠠⢶⣿⣿⣿⣧⠀⠀⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣷⣿⣾⣿⣇⡀⠀⠀⠀⢠⣦⠀⢤⣣⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⠀⠀⠀⠀⠀⢻⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⡿⣿⡇⠈⠂⣈⠁⠐⠸⡀⡼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢸⣷⡿⠀⠀⠀⠐⠂⠀⠀⢣⠀⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇⣼⣿⡇⠈⠁⠀⠘⠉⠀⠀⠀⡇⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⠀⠀⠀⠀⠰⠀⠀⠀⠀⢹⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠙⣿⢇⣠⠤⠤⠄⠀⠀⠀⠀⠀⠈⣧⠞⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⠛⠓⢦⡄⠀⣀⡤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠶⢤⣀⠀⣠⡶⠛⠛⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⠋⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢏⠙⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⡄⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⠋⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢻⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⡏⠀⠀⠘⡆⠀⠀⣀⡤⠤⠶⠦⠬⢥⡀⠀⠈⠃⠀⠀⠀⠀⠀⢸⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣍⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠞⠁⠀⣞⣉⡙⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢀⠀⣸⣇⣀⡄⠀⡸⠀⠀⠀⠀⠀⠀⠀⢰⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⢹⠀⢳⡈⢳⡜⠁⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⢇⠀⠁⠀⢹⠀⠀⢀⡀⠀⣰⢋⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣠⠜⠀⠀⢻⣍⡾⠁⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡠⠤⣄⣀⠤⠤⠤⠤⢤⡀⠙⢷⣌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠻⠴⠚⠁⢀⣠⠴⣖⣛⠉⠉⠁⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⠴⠖⠒⠋⠁⠀⠀⠀⠈⠉⠓⢲⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢲⡖⠒⠒⠒⠒⠒⠲⠶⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡮⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣻⣿⣿⡇⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣿⣿⣿⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣦⡈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡀⠹⣆⠀⠀⠀⠀⠀⠀⢀⣼⢃⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⡀⠹⡇⠀⠀⠀⠀⢀⣾⢁⣼⣿⣿⣿⣯⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⠄⠵⠾⣶⣤⣄⣸⢃⣾⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⡿⠟⠁⠀⣀⢀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣩⡄⠀⠤⢶⣃⠅⢚⠃⣸⣿⣿⣿⣿⣿⠿⡿⠟⠁
⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠴⣮⠥⢲⢖⣿⣿⣾⣶⣾⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣷⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⣠⣤⠵⠞⠹⣿⣿⡿⢻⣿⣿⣿⣿⡟⠻⣯⡀
⠀⠀⠘⢿⣷⣾⣿⣿⣙⣿⣿⡿⣿⣿⢷⣾⣧⣦⣴⣶⣴⣿⢉⡁⣈⣻⣿⠿⠏⠀⣀⠈⣙⢳⣦⡤
⠀⠀⠀⠈⠻⣿⣯⣿⣿⣟⣿⣿⡿⣿⡾⣻⣿⡿⢋⣿⡿⣿⠟⣫⡿⣛⣿⣾⣿⣷⣎⣙⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠙⠛⠿⢯⣿⣿⣿⣿⣾⣿⣥⣾⣿⣿⣤⣶⠞⢋⣾⡿⣫⣿⠟⣿⣿⢻⣿⣿⡿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⢸⣯⣍⣩⣤⣾⠿⠻⠾⠟⠁⠈⠛⠋⠘⠋⠻⣷⠈⣿⣧⠹⣿⡛⠿⠶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⢹⡯⣽⡏⠽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣀⣀⣤⣤⣤⣴⣶⣶⣤⣤⠀⣀⠀⠤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡖⢛⠛⣿⣾⣿⣿⣿⣿⣿⣯⣿⣻⠙⠏⣛⣂⡗⢦⣜⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⣭⣿⡟⣼⣿⠹⣿⣿⢍⡀⠀⠁⢀⣤⣷⣿⣿⣿⣿⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⡴⢻⣿⠛⣿⢿⢤⣽⣽⡢⢤⣴⣾⣿⣿⠿⢛⣉⡥⠒⠛⢻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠻⠀⠈⠛⠷⠮⠍⠉⠀⠉⠉⢉⣤⣶⠏⠁⠀⠄⣠⣬⡻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣤⢤⡀⠀⠀⠀⠰⣯⣏⠙⠒⠦⠘⠳⣿⣿⣿⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⡀⠀⠀⠈⣁⣀⠉⠀⠀⠀⠀⠰⣿⣥⣄⣰⣾⣾⣝⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣷⠀⠀⠸⠉⣭⣿⠂⠀⠀⠀⠘⣿⡟⣩⣬⣍⣿⣝⣦⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠟⠀⠀⠀⠌⠉⠥⠄⠀⠀⠀⠀⠈⠁⢿⡢⢰⢿⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣧⡀⢀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣸⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠉⠁⠀⠈⠀⡙⢁⡦⡀⠀⠀⠀⠀⠀⠀⠐⢫⠚⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠲⢶⣲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⢢⣬⣽⡧⠋⠰⠃⠀⠀⠀⠀⢠⠿⠶⠃⢠⠟
⠀⠀⠀⠀⠀⡠⠒⠛⡁⠀⠀⠀⠀⠙⢿⣀⣍⡺⢖⡊⠵⠀⠢⠄⠀⠀⠀⠀⠹⠀⢭⣥⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⢠⠋
⠀⠀⠀⠀⢸⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠉⠻⣽⡒⣍⡲⢤⣴⠲⢐⡦⢄⠀⠰⣀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⣸⢄
⠀⠀⠀⠀⠈⣇⠀⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠙⠻⣽⡷⣮⣝⣲⢤⣿⣆⡀⡈⠙⠓⠒⠒⠚⠁⠀⠀⠀⠀⠀⣠⠔⠁⢸
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⣭⣽⡒⠤⣀⣀⡀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣶⣭⣟⣲⢭⣢⣀⠔⣣⠀⢀⡠⠴⠒⠉⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⢘⢻⢦⣀⠓⠥⣾⡳⢌⣲⡌⡙⢦⣄⡒⠤⣄⠀⠀⠀⠀⠀⠉⠉⠛⢿⣟⠳⡼⡜⢅⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡗⢄
⠀⠀⠀⠀⠀⠀⠀⢏⠸⣉⠳⣤⣄⠈⠲⢭⣯⣏⠉⣽⣿⣽⢶⣝⢦⣀⠀⠀⠀⠀⠀⠀⢹⠀⠹⡄⠀⠙⢦⡀⠀⠀⠀⣴⣿⣿⣿⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⢡⠀⠦⠈⢗⢶⣤⣀⠈⠁⠐⠒⠺⠓⢤⣕⣦⣬⡆⠀⡴⠀⠤⠮⢀⢠⠇⠰⣶⣻⣿⣶⣤⣾⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⡆⠸⡀⠘⡗⠈⡏⠙⠳⡶⢦⣤⣀⣉⡀⠀⠀⠉⠉⠉⠒⢠⣄⠀⠉⠲⢄⠀⠙⠿⢿⣷⣿⣿⣿⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣧⡀⢠⡀⠉⠒⠄⠘⡌⠻⣯⢝⢍⡉⠙⠛⠛⠋⠉⠉⠙⠓⠀⠀⠀⠑⢄⠀⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠁⠀⠀⠀⠀⠀⠀⠀⠑⢭⣋⠭⢭⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢹⠤⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠲⠦⢤⣤⠿⠤⡤⢤⡀⣿⠔⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠙⠛⠁⠀⣸⠃⡿⠀⠀⠀⢿⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠠⣄⠀⠀⢀⣀⣤⠾⠃⢰⡇⠀⠀⠀⡟⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠈⢹⣿⣿⠏⠀⠀⠀⣼⠁⠀⠀⢸⠇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⡰⠞⠛⠻⣿⣄⣀⠀⢶⡿⠀⠀⠀⡿⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠀⠀⠀⠀⠈⠙⠻⣿⣿⣏⢀⣀⣀⣧⢀⠀⠀⣼⡏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣭⡿⠿⠦⠿⠿⠋⠉⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⢀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⣀⣀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣋⠉⠀⠀⠀⠀⠀⠀⢁⣀⣀⣈⣙⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⠅⠉⠡⣀⡠⠤⠀⠉⠭⢍⣉⠙⠿⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠂⠛⠋⠉⠉⢁⣀⣀⡀⢀⣤⣄⡄⢠⣤⢀⣤⣀⣀⣀⡀⠉⠉⠙⠛⠑⢲⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠺⣿⣙⠃⢻⡇⣿⡇⢨⣧⢸⡿⢶⣿⠘⢩⣿⠹⢻⡷⠀⠀⠀⠀⠀⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠀⠀⠀⠀⠀⠀⡿⠦⠀⠚⠓⠈⠛⠛⠋⠘⠛⠘⠛⠀⠺⠿⠀⠾⠅⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠙⠉⢧⡀⠀⣀⣀⣀⣀⣀⣀⠶⠶⠆⠀⢠⠴⠤⠀⡄⠰⠶⠶⣄⣀⣀⣀⣀⣀⠀⠀⣼⠋⠙⢧⣀
⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠘⣿⣿⡏⠉⠀⢸⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⠀⠉⢹⣿⣿⠇⠀⠀⠀⠈⢣
⠀⠀⠀⠀⡸⠃⠀⠀⡀⠀⣠⠤⠿⠿⠀⠀⣾⡇⠀⠀⠀⠀⠀⣾⠀⠀⢸⠇⠀⠀⠀⠀⢻⣿⠀⠀⠾⣿⣤⣤⠀⠀⠀⠀⠀⢧
⠀⠀⠀⡔⢰⡤⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠤⣦⢡⡀
⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠿⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠛⠛⠛⢧⣤⡀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢁⡀⢤⣤⡀⠀⠀⠉⠻⣿⠏⠁⠀⠀⠉⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠛⠁⠀⢻⠇⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠘⠉⠑⠿⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠉⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠴⠾⢿⠿⠆⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⣤⣤⣾⡟⠃
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⣷
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠖⢠⣦⣀⣿⠁
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠶⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⢰⣞⠉⠀⠀⡀⢙⣿⡇
⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⡆⠀⠀⠀⣏⣀⣀⣀⣀⡀⢀⣴⡄⠀⠀⠀⠀⢸⣿⠀⠸⣿⠿⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠙⣧⣀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠛⠛⠛⠁⠈⡻⠃⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⣀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣄⣀⠀⠀⠀⠀⠙⠤⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢻⣷⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢶⠶⠉⠋⠁⠘⣿⡆⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣴⡿⠁⠀⠈⠙⠛⠳⣶⣄⡀
⠀⠀⠀⠀⠀⢀⣠⡶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⠶⠶⣄⣀⣠⣠⣴⡶⠿⣷⣶⣦⣀⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠀⣰⠆⠘⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠛⢷⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠈⠛⠛⠿⣷⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⡄⠀⢀⣀⠀⠀⠀⠀⣠⣤⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⠛⠛⠓⠀⠠⣤⣤⣤⣤⣤⣴⣤⣼⠃⠀⢸⡿⠛⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠃⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⡿⠿⠿⠶⠶⠒⠀⠐⠒⠒⠲⠖⠐⠺⠿⢿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⣶⡦⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠛⠓⠂⠀⣀⣤⣤⣄⠀⠀⠒⠊⠙⠿⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⣠⣤⣀⣤⣤⣀⣀⣀⣀⣤⣶⣦⡀⠀⠀⠀⠐⠲⠶⠶⠾⠿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⠿⠟⢻⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣅⣀⣀⣀⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⣀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⣿⠛⣷⠀⣾⠛⣷⠀⣿⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡉⣷⠀⣿⠀⣿⠀⣿⢀⣿⠀⣿⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠉⠀⠉⠀⠈⠛⠁⠀⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣡⣭⣭⣬⣭⣯⣽⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⡆⠀⠃⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⣀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣢⣄⠀⠀⡠⠠⠿⢿⣷⣿⣽⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠷⠄⠐⢶⡙⢿⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⢿⡆⠀⠀⠀⠀⣴⣞⣸⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡌⡓⠀⢀⣠⡀⠊⡽⢷⣽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⣠⠇⠈⡿⠍⠃⠾⠾⠷⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠉⠀⣀⢸⠇⢀⠘⣦⣴⣶⣶⣿⣿⣷⣤⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠛⠉⢄⣜⣀⣙⣿⣿⣿⡿⢲⡞⢛⣿⣿⣶⣶⣦⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⡆⣀⣀⣤⣨⣼⣿⣽⣾⣿⣾⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠉⡟⠈⣧⠞⣿⣿⣿⣿⣿⣿⣿⣿⣶⡂⠀⠤⢤⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⣠⡴⠏⠸⠐⡟⠋⢁⣁⣾⣿⣿⣿⠟⠻⠷⠦⠶⣤⡉⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢟⡟⠓⠲⣤⣶⣿⣿⣿⣧⣽⣿⣿⣷⣶⡶⠶⠦⠤⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣤⠶⠞⣛⢳⣟⣿⠟⣯⠙⠛⠶⣿⣿⢿⣿⣷⣶⡤⠶⢴⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠙⢷⣎⠓⠢⠀⣿⣿⠿⠛⠁⠀⠀⠀⠀⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠙⣷⣁⠸⢻⣏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡦⠀⠈⣿⣰⣿⣽⡧⠀⢠⣄⢠⣀⠤⡐⣦⠱⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⣾⢁⠏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⢦⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣿⣿⣿⠿⠛⠋⠉⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠉⢻⣏⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣠⣴⠿⠛⠋⠉⠀⠀⠀⠀⠻⡄⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡀⠀⠀⠀⣸⣿⡟⠁⠀⢶⡀⠀⠀⠀⠀⠀⠀⣁⣴⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣧⣀⢠⡿⠁⢠⡄⠀⠀⠒⠋⠉⠉⠉⠉⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠋⠁⠠⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣠⠟⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣶⣦⣄⣿⣿⣿⣿⣿⣿⣶⣦⣶⠶⠶⠖⢶⠆⠁⠀⠀⠙⠛⠛⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠉⠀⣄⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠉⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠀⠀⢀⡿⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠙⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣤⣤⣭⣤⣤⣄⣤⡄⡀⠀⣀⣠⣤⣤⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠋⢉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠀⠀⠸⠃⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⢀⣀⣀⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⢀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠰⠛⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡿⠂⠀⢻⣿⠗⠚⣿⣿⣦⡀⠀⢻⣿⠟⠺⣿⣿⣿⠃⠀⠀⢻⣿⣿⡟⠸⣿⣿⡟⢐⣿⡿⠛⠀⠘⣿⣿⡿⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⠀⠀⣹⣿⣿⣿⣷⣼⣿⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣷⣿⣿⡀⠀⠀⠀⣿⣿⣧⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⣸⣿⠀⠀⣽⣿⠈⠻⣿⣿⣿⠀⠀⣿⣿⣿⠀⢀⣤⢀⣿⣿⡇⠀⣿⣿⡿⢿⣿⣿⣄⠀⠀⣿⣿⡏⠙⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠋⠀⠰⠿⠿⠆⠀⠈⠻⠿⠀⠰⠿⠿⠿⠶⠿⠏⠼⠿⠿⠷⠰⠿⠿⠿⠄⠙⠿⠿⠷⠴⠿⠿⠿⠶⠾⠿⠃





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣶⣤
⠀⠀⠀⠀⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣴⡄⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠋⣵⣶
⠀⠀⠀⣴⣿⣿⣶⠀⠀⠀⠀⣾⣦⡀⠀⠀⠀⢠⣶⣿⣧⠀⠀⠐⠶⢾⣿⣿⣷⢰⣿⣿⣯⣤⠀⠀⠀⠀⣴⣿⣿⡇⣼⣿⣿⢾⡄
⠀⠀⠀⠙⣿⣿⡋⠀⠀⠀⣴⣿⣿⡿⣿⠆⠄⣸⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⠉⣼⣿⣿⠛⠋⠀⠀⠀⠀⢿⣿⣿⠁⣽⣿⡟
⠀⠀⠀⢺⣿⢻⡇⠀⠀⠀⠈⢻⣿⣧⣤⡄⠀⠿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠘⣿⡿⣦⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⣀⣼⣿⡧⢸⡇⠀⠀⣿⢸⡇⠀⠀⠀⠀⢸⣿⣿⠀⣰⡟⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠉⠈⠁⠀⠀⠀⠀⠈⠁⠉⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⠀⠀⠀⢳⡀⠀⣀⠀⢀⡾⠀⠀⠀⠀⢿⣛⣿⡁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠰⣿⣿⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠹⣾⣭⣷⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⢰⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡯⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢠⣶⣞⣻⠃⠀⠀⠀⠀⠀⢀⣴⠿⠟⠿⣦⡄⠀⠀⠀⢀⣀⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠇⠀⠀⠀⢠⠎⣿⣿⠁
⠀⠀⠀⣾⣿⠿⠛⢷⣤⠀⠀⠀⠀⣸⡏⠀⠀⠀⠸⣇⠀⠀⣀⣿⣿⣿⣿⣿⣿⣷⣂⠀⠀⠀⠀⢸⣿⡯⠀⠀⠀⠐⠳⠆⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⠄⠀⠀⠀⣀⡀⠠⠂⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⢠⣿⡹⣷⠀⠀⢠⡀⠀⢀⣀
⠀⠀⠀⠻⠟⠀⠀⢶⠞⠉⠀⠀⠀⠘⢿⢇⠀⠀⠀⣰⣿⡿⣶⣟⣇⣀⣶⠀⠀⠀⠀⠀⣴⣦⠈⠛⠛⠋⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣾⣿⣾⣿⣿⣿⣿⣷⠀⠀⠰⣿⣿⣿⠇⠀⠀⠹⣧⣾⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⣾⣾⣷⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⠀⠀⠀⢠⠀⠘⣿⣿⡇
⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢤⠀⠀⠀⠀⠀⢸⡿⣿⠛⠀⠀⠀⠀⠀⠀⢸⣿⢹⡇
⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⣿⠃⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠁⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣀⢦⡰⠢⠞⠢⠚⠤⠚⡤⠢⣠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⠢⠎⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠋⠳⡴⢄⢀
⠀⠀⠀⠀⠀⣀⡰⢦⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠣⣀⡄
⠀⠀⠀⠀⡴⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡄
⠀⠀⠀⡸⠁⠀⠀⢀⢀⡠⠤⠀⠐⠂⠀⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠰⡀
⢀⠤⠠⣇⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣄⡠⠴⠒⠂⠙⠉⠁⠀⠀⠀⠀⠀⠉⠑⠲⣄⠀⢱⡤⠤⡄
⠈⠰⠲⡌⡞⠀⠀⠐⠦⣤⡤⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⡠⢤⠤⣤⠶⠂⠀⠩⢱⠎⣠⢤⠘
⠀⢸⢠⣼⠃⠀⠀⠐⢿⣞⠛⠒⢻⣷⡤⣀⣠⡎⠀⠀⠀⢊⠀⠀⠀⢀⣆⠤⢞⢋⡿⣷⠚⢻⡄⠀⠀⠀⠈⣞⡁⢸
⡀⡄⣿⡟⠀⠀⠀⠀⠀⠈⠓⠮⣶⣯⠥⠤⠟⡿⠂⠀⠀⠀⠀⠀⠀⠙⣏⠒⠮⢿⡟⠓⠊⠁⠀⠀⠀⠀⠀⢿⢹⢸
⡇⡇⢹⠁⠀⠀⠀⠀⠀⠀⠐⠚⠫⠤⠤⣄⡞⢀⡤⠖⠊⠉⠙⠒⠢⢄⠈⣆⡜⠓⠚⠓⠂⠀⠀⠀⠀⠀⠀⠘⡞
⢣⠈⠃⠀⠀⢳⢄⠀⠀⠀⠀⠀⠀⠀⡠⣿⡮⠕⠒⠉⠉⠉⠉⠉⠁⠒⠳⢽⢧⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢃⠜⢠
⠀⢹⠀⠀⠀⠈⡆⠁⣀⠀⠀⠀⡤⢪⠏⠙⢄⣀⣄⡀⠀⠀⠀⢀⣀⡀⡠⠎⠳⡝⠢⣀⠀⠀⢀⡴⡛⠀⠀⠀⠘⠖⠁
⠀⠀⢦⡀⠀⠀⠰⡄⠈⠲⡤⠊⢀⠎⠀⠀⠀⠀⢶⡁⠀⠀⠀⠸⡄⠈⠁⠀⠀⠹⡀⠈⠳⠖⠁⢀⠇⠀⠀⠀⢰⠇
⠀⠀⠈⢮⡀⠀⠀⠱⡀⠐⠀⢀⡟⠀⠀⠀⠀⠀⠀⠈⠢⠤⠒⠉⠀⠀⠀⠀⠀⠀⢡⠀⠄⠀⢀⡜⠀⠀⠠⢂⡟
⠀⠀⠀⠀⠙⢤⡈⡨⠷⡄⢂⡌⠆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣠⠾⠣⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⢣⠀⠔⢘⣼⢱⠀⠀⣠⡔⠊⢹⢈⠣⠜⠦⠁⠌⠤⢙⠀⢐⡤⢄⡀⢸⢚⣇⠁⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠢⠘⢸⢠⡏⠠⡘⢄⢸⠀⠀⠀⠀⠀⠀⠀⠸⡠⢋⡂⠀⠙⠪⢈⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠈⡀⠘⠃⢣⠀⡇⠀⡛⠀⠀⠤⠤⠤⠄⠀⣘⠁⠸⡛⠀⡌⠀⢸⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠃⠀⠀⠈⡆⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢳⠂⣰⠀⠀⠸⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠸⣼⡄⢦⠀⠀⠀⠀⠀⠀⠀⠰⣄⡎⢠⠃⠀⠀⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢹⡟⠈⢦⢓⡔⣣⢓⡜⢡⠁⢹⣰⠃⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠳⢊⣉⣀⣀⣀⢀⡈⠩⠦⠜⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣀⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣜⡥⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠭⠥⠤
⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⢀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⢠⠞⢁⣠⠾⠭⢄⣠⠤⠤⢄⣀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⢀⣠⠟⠊⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠀⢀⠇⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⣆⠀⠀⡸⠀⠀⠀⠀⣠⠔⠁
⠀⡼⠁⠈⠑⢦⡀⢠⠞⠁⢀⠀⠀⠀⠀⠀⠈⣶⠰⠧⢤⣀⣠⠞⠁
⢰⠁⠀⠈⣻⡶⠄⠀⠠⠾⣛⠀⠀⠀⠀⠀⠀⢸⣧⡀⡠⠊⠉⠓⢤⡀
⡎⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢫⠀⠀⠀⠀⠀⠉⢢⡀
⢀⠀⠀⠀⠀⢀⡐⠂⠁⠲⣄⠀⠀⠀⠀⠀⠀⢸⡧⣄⣇⠀⠀⠀⠀⠀⠀⠙⢄
⢘⣄⡤⠖⠛⠉⣠⠄⠀⠀⣸⠀⠀⠀⠀⠀⢠⣿⡤⣸⡉⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠁⠹⣄⣠⠔⢋⡴⠂⠒⠊⠁⠀⠀⠀⢀⡴⠋⢸⡃⠘⡳⡄⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠐⠋⢓⡴⣏⣀⣀⣀⣀⣀⣠⡤⠖⠋⠀⠀⢐⡂⠀⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢠⠃⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⠸⡀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⣀⣀⠤⠔⢒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠧⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣦⡀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠚⠛⠻⠙⡌⠉⠉⠙⡄⠀⣀⡠⣤⠶⡖⠒⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢤⣤⣄⣀⣀⣤⣄⠤⢞⠗⠋⠁⠀⠈⠀⠉⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠆⠤⠤⠤⠔⠶⠖⠒⠛⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢖⡽⢶⣄⠀⠀⣀⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡎⢀⡀⠝⣦⢊⠁⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡿⡇⠀⠀⢠⠃⡄⠀⠀⠀⠀⠀⠀⠠⠵⢄⣀
⠀⠀⣠⠤⡤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⢳⠀⠀⡎⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠐⡇⢾⠒⠒⠒⠬⣑⠤⡀⠀⠀⠀⠀⣰⣿⣿⡿⢻⡈⣇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣊⠭⠒⠲
⠀⠀⠘⢮⡳⣄⠀⠀⠀⠑⢮⡒⢄⠀⠀⠛⣋⣀⣠⣴⣇⠈⠲⠇⣸⠀⠀⠀⠀⠀⡠⠚⡡⠖⠉
⠀⠀⠀⠀⠘⢁⢀⡱⠄⡀⠀⠙⢶⠋⣢⠸⣿⣿⣿⡿⠿⡄⠀⠀⢿⠀⠀⠀⡠⠊⠠⠊⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠙⠤⡰⠊⣶⠒⠤⣙⣤⡀⠹⡤⠿⠧⠤⢽⡄⠀⢸⠀⠀⠘⠀⠄⣀⣀⡤⠤⣀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢄⠀⢁⣴⠴⠓⠉⠁⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣑⠠⡈⠑⢆⡲⢦⠤⣶⡂⡸⢁⡊⠉⡙⠁⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡾⢦⣼⣦⡀⠑⢜⠦⣘⣯⣤⣌⣩⠋⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣧⣻⣿⣦⣌⠳⣄⠑⢸⡤⡬⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣀⣀⣗⣄⠙⠢⣐⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⠔⢉⠧⡙⣂⡈⠂⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣰⢋⠔⢹⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠈⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⡶⣾⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠀⠀⠀⠒⠀⠀⢀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢹⣿⠀⠀⠐⠟⠀⠀⠙⡉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠙⠓⠀⠀⠀⠀⠈⣀⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣌⡁⣀⠀⠀⠀⠀⠙⠛⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡟⠿⣻⣿⡇⠀⠀⠠⠤⠤⣤⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣿⡿⢶⣿⣿⡷⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠤⠤⣶⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⡿⠁⡀⠀⠀⡀⠀⢘⠻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⡐⠂⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢀⠀⠀⠀⠈⠉⠀⠀⢱⣤⣤⣑⣤⣼⡆⠈⠁⠀⠒⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⣿⣀⡭⠖⠁⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⣦⡄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠛⠻⣿⡀⢰⡯⠀⠀⡀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣼⡇⡜⠁⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⡴⠿⠛⠛⠁⠀⡇⢰⠀⠀⠀⠀⠀⢹⣏⣟⣢⠊⣿⣇⠈⠁⠀⠀⣷⢠⢦⡀⠀⠀⠀⠀⠀⠀⢠⠇⢠⡿⢰⠇⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⢠⠞⠀⠐⠒⠀⠀⣰⠃⡾⠀⠀⠀⠀⠀⣼⣿⣿⣾⣿⣿⣿⡄⠀⠀⠀⢿⣸⣎⡟⢂⠀⠀⠀⠀⢀⠞⠀⣸⡇⡞⢀⣤⠴⣆⣤⣶⣿⣷
⠀⠀⠀⡠⠃⠀⠀⠀⠀⢀⡜⣿⠀⡇⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⡿⣾⠁⠀⠓⢄⡀⠀⢠⠃⠀⣿⡋⠹⠿⠿⠟⣋⣉⣭⣭⣭
⠀⠀⡠⠁⠀⠀⠀⠀⡴⠋⠀⢸⡄⡇⠀⠀⡞⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⡇⠀⠀⠀⡄⣟⣶⡞⡄⠀⣿⢃⣠⣴⣾⣿⣿⣿⣿⣿⣿
⢀⡼⠁⢠⣤⣄⣀⠞⠁⠀⠀⠀⢷⣧⠀⡼⠀⡀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣠⠟⢤⡀⠀⠀⢃⠙⣿⠇⡇⠀⢿⣏⣛⣿⣿⣿⣿⣿⣿⣿⡿
⡾⠀⠀⠒⠶⡖⢛⣂⠀⠀⠀⠀⠸⣿⡼⢁⡜⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⡿⠀⠀⠙⢦⡀⠀⠀⠳⠀⠀⠰⠈⢿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠑⢦⡀⠀⠀⠈⠉⠘⢦⡀⠀⠀⠀⡛⠁⡼⠀⣾⣿⡿⠿⠿⢻⣿⣿⣿⣿⢀⠀⣼⠁⠀⠀⠀⠀⠈⠳⢤⣀⢀⡄⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠑⢄⣀⠀⠀⠀⠀⠈⠓⠤⣜⠀⢠⠇⠸⠟⠁⢀⢛⢛⡛⡛⠛⠛⠻⢨⡀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠙⠻⣟⡒⠒⠂⠀⣀⠍⠑⠋⡀⠀⠀⠀⠀⣒⣂⡨⠁⠀⠀⠒⠢⡄⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠓⡤⢉⡀⠀⠀⢠⠿⢀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠃⠆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠓⠒⡽⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄
⣶⣶⡶⠶⠶⠶⣾⣷⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣷⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠂⠉⠉⠉⠉⠉⠉⠐⠒⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢨⠀⠸⡀⡉⠘⠈⠃⢰⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠄⢤⢸⠀⢠⡀⠄⠄⠠⠀⠀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠂⠚⠘⠈⠃⠘⠀⠂⠘⠀⡙⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠤⡀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⢀⠔⠒⠃⠀⢡⠐⡜⢦⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⡏⠀⢌⠀⡃⠈⠈⠀⠁⠀⠸⠀⠀⠀⠀⢀⠜⠁
⠀⢀⡎⠁⣀⠤⠒⠊⠉⠉⠑⠲⢜⣆⡀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠈⠀⢀⡠⠖⠁
⢰⠁⠀⢸⠁⢀⣀⣀⣀⡀⠀⠀⠾⠛⣏⠀⠀⠀⠀⠀⠀⠀⠈⢑⠆⠀⠀⢀⣀⣀⣀⣀⣀⠠⠤⠒⠊⠁
⠏⠀⠀⡜⠀⠈⠉⠀⠄⠁⠀⠀⠐⠀⠈⡆⠀⠀⠀⠀⠀⠀⠴⠡⠤⠔⠚⠉
⢣⡠⣤⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡯⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠖⠒
⠈⠂⣔⢇⠀⠀⠀⠀⢠⠞⣀⣤⡤⠤⠔⡲⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠄
⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢀⣀⣤⠚⣱⣺⡅
⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⢀⠜⠁⣀⡠⠤⠒⣈⠥⠐⠛⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢈⡱⢦⠀⣠⠟⠊⢉⡠⠔⠊⠉
⠀⠀⠀⠀⠀⢀⡴⠁⠙⠋⢿⠛⠻⡎⠁
⠀⠀⠀⠀⢠⠊⢀⡴⠀⠀⢸⡗⠉⢱
⠀⠀⠀⡰⢁⠔⠁⡇⠀⠀⢸⡗⠂⠀⢇
⠀⠀⡘⢠⠋⠀⠀⢰⠀⠀⢸⣿⠤⠀⢸
⠀⠀⢱⢸⡀⠀⠀⡼⠄⠀⠀⠁⢀⠀⣸
⠀⠀⠨⣉⠌⣱⠀⣯⣀⣀⣀⠠⠤⠊⡙
⠀⠀⠀⠙⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠆⠀⠀⠀⠀⢀⡀⢀⣠⠆⢀⣠⡴⠀⠀⠀⠀⠀⠀⠐⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢹⠁⠀⠀⠀⠂⢠⣿⣠⣿⣿⣾⣿⠏⣀⣤⣤⡤⠄⠀⠀⠀⠀⢳⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣄⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⠀⢀⡞⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠈⠙⢛⣽⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⢁⠀⠀⠙⣿⣿⠹⣿⣿⡋⠁⠀⠀⠈⢭⡙⢿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠀⠁⡀⢀⠀⠈⠻⠄⠈⠻⢧⠀⠀⢀⠠⢀⠙⠆⢿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⣿⠇⠀⢘⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡁⠈⢸⣟⠛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⠟⢡⣿⠀⠀⠊⠛⣧⠙⠶⣄⣠⡄⣀⣄⣠⠾⢁⡿⠉⠁⠀⠀⣿⡌⠻⣿⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠸⣿⣿⠀⠀⠀⠀⠈⠛⣻⣏⣉⣙⣃⣈⣹⣛⠛⠀⠀⠀⠀⢀⣿⣿⠃⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⣀⠀⠀⣀⢹⣿⡀⠀⣀⡤⣶⠶⢦⣭⠍⣿⣿⢭⡥⠶⣶⡶⣤⣀⠀⢸⣿⡏⣀⠀⠀⣀⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠀⠈⢻⣿⢧⠀⣿⣇⣸⡀⣸⡁⠀⢸⠁⠀⡇⠀⣹⣀⣾⣿⠀⠿⣿⡟⠁⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣀⠀⠀⢀⠈⠻⡄⠃⢻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡏⠘⣰⠟⠁⡀⠀⠀⢀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢰⡏⠀⢰⣷⡀⠈⢧⢻⣧⣽⣻⣿⣿⣿⣿⣋⠤⡟⣸⠟⠀⢰⣿⡆⠀⢹⡆⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣿⣄⣼⣷⣀⣼⣿⣧⡀⠀⠙⠢⢼⣇⣀⣿⣀⣈⣧⠤⠛⠁⡠⢀⣾⣿⣧⣀⣾⣧⣠⣿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⡿⠿⠛⣿⣿⣷⣄⠂⢑⠲⢤⣭⣭⣭⡥⠴⠒⡠⠊⣠⣿⣿⣿⠛⠿⢿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣴⠁⠉⠙⢿⣷⣦⣄⣀⣀⡀⣀⣀⣀⣤⣴⣿⠟⠋⠁⠈⣆⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣷⠀⠀⠀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣆⡀⣸⣿⠻⠿⢿⣿⣿⡿⠿⠟⣿⣇⠀⣀⣤⠞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢏⣩⣍⣽⣿⣤⣤⣤⡤⠤⢤⣤⣴⣿⡏⢩⣍⠹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⡶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶⢀⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⣤⣦⠈⣷⠀⠀⢀⣀⣀⣄⣐⣶⣖⢂⡀⠀⠀⠀⠀⢀⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⡟
⠀⠀⠀⠀⠀⣰⣿⠃⢶⣶⠀⠀⠙⠛⠀⢸⣇⠀⠿⠿⢿⣿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠉⣻⣿⣿⣧⣀⣤⣶⠀⠀⠀⢀⣼⣿⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠉⠙⢷⡈⠉⠀⢠⣴⣶⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠴⠾⠛⠿⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢿⣿⣧⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁⠉⠉⠀⠀⠀⠀⠀⠐⠛⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⣀⣀⣤⣴⣶
⠰⣶⣹⣧⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢰⣿⣿⡝⣿⣌⣁⡀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠘⠿⠟⢻⣏⢻⣯⣿⠆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⣀⣀⣽⣷⡿⢿⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠿⢿⣿⠿⠿⠟⠛⠉⠁
⠀⠀⠈⠉⠉⠉⠻⠊⠁⠀⠀⠀⢠⣿⣿⣿⡟⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠰⠶⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⣀⡀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠾⣭⣿⠟⠀⠀⠘⢧⣬⡿⠟⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠉⠉⠀⠀⠰⠀⠀⠉⠁⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣁⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⢀⡀⣠⠶⠤⣤⣀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⠀⠀⠈⠉⠻⢿⣿⡽⠋⠀⠀⢰⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠲⢤⣄⣀⣀⣀⣀⣠⡤⠖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣀⣀⣀⣸⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠯⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡤⠤⡒⠒⠒⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡡⠖⣩⠴⠞⠛⢻⣧⠀⠀⠀⢀⣶⣶⡄⠉⠓⢦⣀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⣣⠖⣭⣦⠀⢹⡀⠀⢀⣾⡏⠀⠀⠐⠋⡿⠀⢀⣴⣦⡀⠙⢷⡀
⠀⠀⠀⠠⡞⢁⠀⢀⡴⠊⠀⢠⡟⣿⡆⠈⣧⣴⣿⡟⠀⠀⠀⠀⠀⠇⠀⠸⣧⡼⠃⠀⢸⣿⡄
⠀⠄⡰⢙⣀⡬⢟⡵⣆⣠⣄⢸⡟⠹⠇⠀⢛⣻⡯⠄⠒⠂⠙⠛⠻⠿⣷⣶⣦⣄⠀⠀⢠⣯⡇⠇
⢠⠠⡇⠀⠀⠀⣮⣼⠟⣡⠏⠘⢃⣤⠶⣿⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⣾⣿⠇⠺⠆
⠀⢻⣧⠀⠀⠀⠈⣿⣶⠋⣠⢾⣟⠁⣸⠁⠀⠀⠠⡞⠙⠀⠀⠀⠀⢀⠀⠀⠘⣿⣿⣿⣿⠋⠠
⠀⠀⢻⣷⣄⠀⠀⠈⢛⣾⣿⠏⠈⠤⠇⠠⠀⠀⠰⠉⠀⠐⠒⢦⠀⠀⠃⢠⠀⢣⣿⣿⣃⢀⡤⡀
⠀⠠⠀⠻⣿⣿⣦⣤⣾⠟⠁⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⢤⡆⢸⣇⠀⠈⠀⠀⠘⣾⣿⣇⠸⣠⠆⣤
⠠⠀⠀⠀⠈⠙⠻⣿⣯⣷⣶⣶⡀⠀⢀⢞⣿⣿⣿⠿⣿⡀⠃⢸⣿⡀⠰⣤⣏⠀⣿⡿⠎⠛⠛⢿⣷⢸
⢡⠄⢠⠀⡏⠙⡆⠸⢇⣿⣿⡋⣳⠀⢈⠸⣿⣿⣿⣀⣸⡇⠀⢸⣟⣿⢦⡿⠙⣷⠈⠛⠤⢠⣾⣾⡟⣾
⠇⠀⠀⢀⠙⠒⠁⠀⠀⣿⣿⣿⡿⣌⠸⣆⢻⣿⣿⣿⣿⢿⠀⠈⠻⠿⠋⠀⢒⡟⠀⣠⣾⡿⣿⠿⠇
⠀⡄⠀⠀⠀⠀⠀⠀⡸⠾⠿⢛⣽⣿⣆⢺⣷⢿⣿⣿⡿⣫⣴⢶⣏⠀⢀⡆⢎⠀⣾⢯⣫⠔⠂⠀⠀⠀⡆
⠀⠰⣀⠀⢀⣀⠀⢀⡇⠀⠀⣼⡿⠻⠿⣧⠙⠻⠿⠷⠋⣿⡏⠀⠈⢳⠀⠀⢈⠇⠻⣮⢿⠀⠀⠀⠀⠊⠐
⠀⠀⠀⠉⡛⠻⠓⠀⢿⣷⣶⡄⠀⠉⠉⠉⠀⠀⠀⣴⣄⢙⣿⠀⠀⣼⡀⢠⠏⡀⣴⠟⡘⠀⠀⣤⣤⡤⠖
⠀⠀⠀⠀⠈⠓⡀⠀⠀⠙⢻⠃⠀⠀⢀⠀⢰⣶⣤⣼⣿⣿⡿⢀⣼⣟⡤⢘⣄⣿⣧⣤⣿⣦⣴⣧⣀⠈
⠀⠀⠀⡀⠀⠰⢿⡆⠀⠀⣼⣦⣿⣤⣼⣷⣾⣿⠟⠛⡏⢋⣁⣌⣉⣭⣀⣉⣩⣼⣿⠿⠿⠿⠿⣿⣿⣄
⠀⠀⠀⡨⠇⠀⠈⠀⠀⢸⣿⣿⣿⠛⢿⠀⠸⣿⠃⠀⢠⢤⣼⣿⣿⡿⠟⠛⠉⠁⠀⢦⣾⢷⡄⠈⠙⢟⢧
⠀⠀⠊⠀⠠⡀⠀⢀⠀⢸⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣯⣿⡿⣻⠛⣷⣤⢤⣰⠛⣶⢸⡟⢸⣇⣴⢶⡌⢾
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⢻⣿⣿⣿⣿⣦⣼⣶⣿⡿⠛⠁⠀⠙⣾⠉⣠⡾⠉⠙⠁⠀⠁⠺⢻⡟⠛⠃⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢫⡙⠛⠛⠿⣟⠟⡁⠰⣋⡷⡀⠀⠸⣟⣭⠶⢛⣏⠉⠛⠳⣄⠘⣧⣄⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⠱⡄⢠⣤⡌⢿⡍⠳⡍⠛⢁⣴⠞⠉⠀⠆⠀⠁⠀⠀⠀⣽⠇⢀⡴⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢇⠀⢳⠀⠀⠛⣠⣦⡾⡿⠁⠀⢉⠇⠀⠀⢀⣠⣴⣟⡥⠖⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⢰⡘⢶⣬⣴⣶⣾⣟⡿⠏⠀⠀⠀⠋⠐⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣍⡙⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⢿⣷⣦⣄⠀⣴⡞⣛⣻⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⣡⣶⣶⣤⡹⣿⡟⠛⠛⠉⠁⠖⠀⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢻⡘⠿⢾⣿⡻⠁⡿⠀⠈⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠟⣡⡾⢧⡳⢤⣉⣁⡤⠞⠃⠀⠀⢀⣠⣤⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⡋⢀⣼⠏⢸⡆⠙⠛⠓⢀⣀⣤⠴⣖⣿⣿⣿⣿⡿⠟⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⢂⣴⠟⠹⣤⣼⣃⣠⠴⠞⣋⣥⣶⣿⣿⣿⣿⠟⠉⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢷⣾⣶⡿⠿⠞⢛⣫⣭⣷⠶⢿⣿⡿⠟⢋⣵⡿⠟⠁⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠐⠀⠉⢿⣦⣤⣶⠟⠋⢠⠤⠖⠛⣉⣠⣴⣿⠿⠛⠁⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⣀⡤⠊⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠄⠀⠳⡀⠀⠀⠀⠀⠈⢄⡀⣀⣠⡤⠴⠒⠋⠁⠀⣀⣴⡿⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠴⠀⠉⠙⠛⠛⠓⠛⠛⡉⠁⠀⠀⢀⣠⣤⣶⣿⣿⢏⡴⢋⣭⡷⠚⣉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠐⠀⠀⠈⠋⠉⠛⠲⢤⣀⣤⣴⠶⠛⠛⠋⠉⠙⠛⠛⠿⠶⣿⣫⣴⡯⠭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡀⠀⠀⠀⠀⠆⠀⠀⠀⠃⠈⠙⠳⢤⣄⣤⡤⢤⣀⠀⠀⠀⠀⠀⠙⣷⡞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠄⠀⢘⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠳⡄⠀⠀⠈⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠉⣉⣽⣶⣶⣴⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣼⣆⠀⠓⠦⢤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣾⣥⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣭⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⣴⠒⡖⣦⡖⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠁⢹⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⣀⣤⣶⠾⠿⠿⠿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠛⠉⠉⠛⢿⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠙⣷⣄⣠⣤⣤⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠉⠁⠀⠀⠀⠀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⡞⢻⣦⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣆
⠀⠀⠀⠀⠀⣰⡟⣰⠟⠀⠀⢿⠀⠀⠀⠀⠀⢀⣤⣾⡇⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⡆⠈⣿
⠀⠀⠀⠀⢠⣿⢠⡟⠀⠀⠀⠈⢷⣤⣤⣤⡶⠛⠁⠸⣇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣠⠶⢻⡅⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡏⢸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⣦⣀⣀⣤⠞⠉⢷⡄⠀⠀⠀⠀⢀⡴⠻⣇⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠰⣶⢦⣤⡀⠀⠀⠉⠉⠀⠀⠀⠀⠙⠷⣦⣴⣾⡏⠀⠀⠻⣦⣤⣤⡶⠛⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⢀⣀⣠⠟⠋⠀⠀⣤⣄⡀⠀⢶⡄⠀⠀⠙⢦⣼⠏⠀⠀⠀⠀⠀⠀⠀⢠⠖⠛⣩⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾
⠀⣟⠛⠻⣄⠀⠀⠀⠸⢿⣿⣷⣬⣿⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠞⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠙⣿⠗⠛⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣤⣤⣰⣧
⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠋⠉⠀⠀⠀⣀⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠻⠶⠟⠋⠉⠀⣈⡉⠉
⠀⠐⣿⠀⠀⠀⠀⠀⠙⠓⠛⠃⠀⣠⡶⢛⣛⣿⡿⠿⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⡛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⣠
⣴⠾⢿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣬⣭⡤⠶⠞⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠾⠋
⠙⠿⠿⣏⠁⠀⠀⠀⠀⠀⢀⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⣹⣧⡀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣸⡇
⠀⣾⣟⣋⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⢸⣧
⠀⠈⢩⣿⣃⣀⡈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣸⣦⣤⡿⠃
⠀⠀⠀⠉⠉⢿⣧⣶⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢿⣤⡿⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⡶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠈⠻⣄⠀⠀⠀⠀⣠⡾⠛⠋⠉⠁⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠘⢷⣄⠀⣴⠏⣀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⡀⠀⠙⠛⠁⠀⣿⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⢷⡄⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⡤⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⣿⠀⠀⠀⢸⡟⠀⠀⠀⢀⡾⠁⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⣿⣀⡀⠀⣼⠃⠀⠀⢀⡾⠁⠀⠀⠀⠀⣀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠾⣯⢹⡟⠛⠀⠀⢀⣾⠿⠿⢿⡛⠛⠛⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⢀⣿⣹⡆⠀⠀⢀⣾⠃⠀⠀⠘⣷⡶⠛⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠛⢉⣿⠿⠶⠶⠟⠁⠀⠀⠀⠀⠸⣷⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣿⡀
⣄⣠⣀⣠⣤⣄⣀⣠⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣸⣿⣍⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣉⣋⣠⣤⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣀⣤⣤
⣿⣿⣿⠟⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠙⠲⠤⠤⣤⣤⣤⣤⣤⢶⡾⠿⠟⠿⢷⣶⣶⠿⠿⠻⠿⢷⣶⣤⣤⣤⣤⣤⠤⠤⠖⠋⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡿⢿⣦⡀⠀⠴⠛⠋⠁⡐⣉⠤⠤⠶⠖⣚⡻⠶⠿⠿⠶⢟⣓⠲⠶⠤⠤⡉⢂⠈⠙⠛⠦⠀⢀⣴⡿⢿⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡦⠀⠀⠉⠂⠀⢀⣀⣾⢫⡵⠖⠚⠛⠉⢉⣩⣿⣷⣾⣿⣍⡉⠉⠛⠓⠢⢮⡝⢷⢀⡀⠀⠐⠉⠀⠀⠴⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⡀⢠⡄⠀⠀⠉⠀⣘⣡⠖⠊⠀⣴⡿⠛⠉⠉⠈⠁⠉⠉⠛⢿⣆⠀⠈⠲⣌⡂⠀⠉⠀⠀⢠⡄⠀⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠖⣲⣶⠖⠀⣠⣦⡿⠁⠀⠠⢚⣩⣴⣾⣧⣾⡄⢰⣷⣼⣷⣦⣌⡓⠄⠀⠈⢿⣴⣄⠀⠲⣶⣖⠲⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠀⣠⡾⢟⣛⠁⠀⠀⠚⣉⣉⣭⠄⠀⣈⠁⠈⣁⠀⠠⣍⣉⣉⠓⠄⠀⠈⣻⡿⢷⣄⠀⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠃⢠⡾⠁⠀⢾⣿⡇⢠⣄⢸⡟⢋⡔⢠⡾⠋⠀⠀⠙⢿⡄⢠⡉⢻⡇⣠⡄⢸⣿⡷⠀⠈⢷⣄⠘⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡷⠚⢠⣿⠀⢰⡇⢀⣼⡇⠻⠿⠿⣇⠟⠀⠿⣾⣡⠀⠀⣌⣶⠿⠀⠻⣸⡿⠿⠟⢸⣧⡀⣸⡆⠀⣿⡄⠓⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠆⢀⣿⡟⢀⡿⠃⠾⣉⣽⡛⣾⠻⣦⡁⠀⠀⠟⠛⠀⠀⠛⠻⠀⠀⢈⣷⠟⣷⢛⣯⣉⠷⠘⢿⡀⢻⣿⡀⠰⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠏⠀⠀⣾⡟⠸⡇⠀⠘⢯⣀⠙⠉⠢⠴⢿⡦⠐⠍⠀⠀⠀⠀⠹⠂⣴⡿⠦⠔⠳⠋⣀⡽⠇⠀⢸⠇⢹⣷⠀⠀⠹⡻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠆⠈⣽⡿⠀⢿⣧⠀⠠⠤⢠⣄⡀⣀⡀⠙⢦⠦⡀⠀⠀⢀⠴⡶⠋⢀⣀⣀⣠⡄⠤⠄⠀⣼⡿⠀⢿⣧⠁⠸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⡆⠈⢠⡷⠀⠈⠙⣧⣀⠀⣼⠛⠛⠛⠲⢌⠒⠀⠀⠀⠀⠀⠀⠚⡡⠖⠛⠋⠛⣇⠀⣀⣼⠋⠁⠀⣾⡄⠁⢰⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡄⠾⣿⣶⡄⣄⠹⢿⣿⣿⣄⠰⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠆⣠⣿⣿⣿⠏⣠⢠⣶⣿⠃⢰⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡇⣠⠛⢹⠿⣿⣷⣮⣿⣿⡏⠳⣌⠓⠀⢀⣠⣄⠁⠀⣠⣄⡀⠀⠚⣡⠞⢹⣿⣿⣵⣾⣿⠿⡏⠙⣆⢸⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⢾⡏⠀⢃⠻⠻⣿⢿⣿⣦⡙⠳⣄⠀⢠⡌⠓⠚⢡⠄⠁⣠⠞⢋⣴⣿⣿⣿⠟⠗⡜⠀⢱⡶⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢢⡿⣧⣰⣙⠆⠻⣷⣭⡑⠀⠀⠀⠻⣄⣠⠟⠀⠀⠀⢈⣭⣾⠟⠐⣋⣆⣼⢾⡔⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠣⣵⣽⡿⣿⠲⣶⣤⣶⣿⣿⣦⣤⣶⠖⣿⣿⣯⣯⠜⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⡈⣦⠙⡟⠋⠉⠉⠉⢻⠏⣸⢁⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡀⣄⠀⡀⠀⠀⢀⠀⣠⢀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠷⠌⣙⡛⠉⠉⢛⣋⠡⠾⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⠀⠀⠈⠁⠈⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠦⠴⠶⠞⠛⠉

⠀⠀⢀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠄⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿
⠀⠀⠀⠀⣿⡷⢀⡀⢶⣩⣾⣦⠀⠀⠀⠀⠀⢠⣤⡄⠀⢠⣿⣿⡞⠁⠀⠀⠀⠀⠀⠍⡅⢨⡁⠺⣿⣿⣿⣆⡀
⠀⠀⠀⢠⣶⠞⠉⠀⠀⢹⣿⣿⡇⠀⠀⠀⢀⣼⣟⣡⣄⣼⣿⣿⣿⠀⠀⠀⢀⣰⣿⣷⣧⣿⣷⣶⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠸⣿⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⣿⡟⠉⠁⠀⠙⣻⣿⡿⠀⠀⣰⢻⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⠂⠀⠀⠀⠈⣿⡏⠉⠀⠀⠀⢸⡇⠀⠀⠀⢀⡿⠁⣿⠀⠀⠀⣻⡿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠉⠀⠀⠛⠋⠁⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣠⣤⡈⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠟⠠⠉⠀⠀⠀⠀⠀⣐⠀⠄
⠀⠀⠀⠀⠀⢙⣏⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢏⣹⠺⡿⠇
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠘⠁⣿⠋⠀⠀⠀⠀⠀⠀⢛⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⣿⣦⣀
⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡟⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠉⠁
⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠙⣿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠾⣿⡟
⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠃⠀⢁⡖⠀⠀⠀⣸⣿⣄
⠀⠀⠀⠀⢸⣿⡟⣳⣀⣿⣿⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷
⠀⠀⠀⠀⢸⣿⡏⠁⠉⢿⢛⡟⢹⡇⠀⠀⠀⠀⢰⢾⣿⠟⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢸⡿⣇⠀⠀⠈⠈⠁⠈⠁⠀⠀⠀⠀⠀⣻⣿⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⠀⣠⣤⣤⣾⡷
⠀⠀⠀⠀⣿⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠈⡿⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠿⡿⠃
⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⠶⠶⠶⢶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⢀⣤⣤
⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⢠⣶⡄⢹⡆⠀⢀⣀⣀⣤⣐⣶⣖⣢⡀⠀⠀⠀⠀⢰⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⢠⣿⡇⠘⣿⡿
⠀⠀⠀⠀⢀⣼⣟⠀⣾⣇⠀⠈⠉⠁⠀⣷⡀⠺⠛⠻⡿⠿⠿⣿⣿⣿⣿⠀⠀⠀⢈⣽⣿⣿⣄⣤⣴⡆⠀⠀⠀⣼⣿⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠁⠙⢧⡈⠀⠀⣶⣾⡆⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⠞⠋⠻⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⣿⡿⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⠀⠉⠀⠀⠀⠀⠀⠈⠉⠁⠈
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣀⣠⣤⣴⣶⣿⣆
⠘⣿⣽⣷⡶⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀
⢸⣿⣿⣎⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠈⠉⠉⠹⣧⣻⣷⡟⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠶⣶⠾⢿⣏⠷⠀⠀⠀⠀⢰⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠈⠉⠁⠀⠀⠀⠉⠉⠛⠻⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢠⡴⣶⣶⠀⠀⢠⡖⢲⣶⡆⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠈⠻⠿⠁⠀⠀⠄⠙⠛⠋⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠶⠞⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⢀⣠⣀⣀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡇⠀⠘⠓⢾⣷⣶⣿⠟⠀⠀⢰⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⠠⣄⣀⠀⠈⠉⠁⢀⣀⠄⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠛⠛⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣀⠀⠀⣸⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠙⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢀⣼⠛⣿⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢿⡀⠀⠀⠀⠀⣠⡾⠁⠀⣿⠀⠀⠀⢀⡾⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⢀⡾⠃⠀⠘⣷⡀⣀⣴⠞⠁⠀⠀⠀⣿⠀⣀⣴⠟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠙⢿⣄⠀⣠⠟⠁⠀⠀⠀⠸⠟⠋⠀⠀⠀⠀⠀⢠⣿⠟⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⢠⣾⡟⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⣈⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣿⣅
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠋⠀⠀⠘⢿⣆
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⡶⠾⠟⠛⠋⠉⠀⠀⠀⣀⣠⣤⡄⠈⢿⣧
⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣠⣶⣛⣩⣥⠾⠃⠀⠈⣿⡀
⠾⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠻⠛⠋⠁⠀⠀⠀⠀⠀⠹⣷⣀⣠⣤⣴⣤⡄
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢀⣴⠞⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⢼⠿⠛⠉⠉⣹⡿
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣻⡿⠋⠀⠀⠀⠀⢠⣤⣀⣀⣤⣴⡾⠛⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿
⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣧
⢠⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠀⢰⣿⡀
⣼⣧⣤⡶⢿⡇⠀⠀⠀⠀⠀⣾⠁⠀⠰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠁⠀⠀⠀⠀⠀⠙⣿⣤⡀
⠈⠉⠁⠀⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣯⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠙⠳⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⣠⣴⣷⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠉⠛⠳⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢿⣦⡾⠟⠁⠈⣿⡄⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣷⣦⣤⣀⣀⣀⠀⣀⣠⣾
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠸⣷⠀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠉⠛⠻⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣦⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠙⢷⣄⠀⠀⠀⠀⠀⣀⣤⣶⠾⠛⠋⠉⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⢷⣤⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⣿⠿⠟⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣤⣤⣤⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠉⠚⠂⠠⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠂⠀⠈⠀⠀⠳⡀⠀⠀⠀⠂⠘⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠥⠀⠐⠀⢀⣀⠓⠮⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠖⠃⣀⠀⠐⠂⠀⠅
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢀⡵⠖⠂⠀⠀⡙⢾⠀⠀⠀⠀⣠⡶⠂⠀⠒⠶⢅⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠂⠀⠀⠀⠈⠢⡈⢍⠁⠀⠐⢀⢄⡠⠂⠀⠀⠀⠀⢂⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠧⢀⠀⠀⠀⠀⠙⢇⠀⠀⠉⠀⠞⠁⠀⠀⠀⣀⠀⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢹⣷⣄⠀⠀⡀⠀⠀⢀⠀⡀⠀⢀⣴⣾⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠟⢰⡿⢧⣠⠿⢄⣼⣈⡹⢧⣤⠿⣿⠈⠏⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⠤⠤⠀⠞⠉⠁⠀⠀⠀⠀⠉⠙⠇⠠⠤⠤⠂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠠⣄⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⠀⠀⡠⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠄⢈⣦⣡⣠⠏⡀⠄⠀⠀⠀⠀⠀⣀⢻⣥⢀⣞⡠⠔⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢀⠈⢉⠋⠈⠀⠀⠀⠀⠠⣀⠀⠠⣉⣣⠈⢀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢌⡝⠁⠀⠀⢀⡀⠀⠀⠀⠀⣀⡒⠀⠉⠙⣏⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢦⡀⠰⠇⢀⣀⢀⣀⡀⠀⠿⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢟⠋⠙⠛⠋⠉⠙⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⣸⡆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠁⢸⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡴⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢱⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣟⠋⠉⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣡⣤⣾⠇⠀⠀⠀⠀⠀⠀⣠⣤⡴⣾⠟⠋⠉⠁⠀⣶⠟⠇⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢾⣆⣀⣿⣯⠀⢠⣴⣄⠀⠀⠀⠀⠀⠉⠁⠀⣆⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⢹⡟⠙⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⢷⡄⠡⣿⠟⣰⣦⠀⠀⠀⠀⠀⠠⠋⠀⣤⠘⠀⠐⠛⠀⠀⠀⠰⠏⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⡎⠉⠛⢄⠀⠀⠀⠀⣧⠀⣈⡋⠀⣀⡀⠀⠀⠀⠀⠀⢠⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠀⠈⠢⠀⠘⠀⠀⠀⠀⣧⠀⠙⢉⣀⠈⠁⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠰⣶⠀⠀⠀⠀⢛⡀⠀⠈⠉⠀⠀⢀⣀⣀⣠⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠁⣀⣠⠀⠀⠀⠀⠀⠘⠿⠒⣶⣿⡿⠿⠋⠉⠉⢉⣁⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⡜⢿⡟⢣⣀⠛⠒⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⣀⣴⣿⡿⠁⠈⠁⠀⠿⢀⣼⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠀⠀⠀⣀⣴⣾⣿⣿⣿⠁⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠋⠀⢀⣤⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⢠⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⡇⠀⢀⡼⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠠⠎⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠀⠀⢀⣿⣿⣿⡇⠠⡿⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣠⣾⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢀⣴⣶⡀⠀⢀⣿⣿⣿⠋⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣶⣿⣿⣿⣿⣴⣿⣿⣿⣿⡄⠀⠀⠀⠀⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⡿⠿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠛⣧⡄⠒⠛⠓⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⢥⠀⣠⣤⣄⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⣇⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣷⢤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⣻⣿⣏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⢨⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⡇⠘⣿⣿⡟⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⠇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⠘⣿⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠟⠛⠁⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⢀⣤⣴⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⢉⡑⠢⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣶⣄⠀⠀⡝⡍⡏⢆⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⣤⡖⠺⠧⠑⣄⠹⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠋⠛⢿⠷⣏⠱⢁⠀⠈⠆⠀⠀⠀⠀⠀⢀⣀⠀⠸⡃⢱⢄⡄⠮⠀⠈⠛⡇
⠀⠀⠀⠀⠀⣿⣿⠏⢥⡤⡘⣂⡀⢠⡏⢆⡀⠃⣀⡆⣀⣀⣀⠤⠤⠬⠥⠄⠀⢇⠀⣌⠄⢄⠂⢀⠎⠁
⠀⠀⠀⠀⠀⢍⠁⠿⡴⠒⠚⠁⣸⣸⡿⠆⢉⠁⠀⢃⠀⠀⠀⠀⠒⢒⠉⠀⠀⠈⠢⠜⠶⡮⠔⡏⠉⠉⠉⠒⢄
⠀⠀⠀⠀⠀⠈⣿⣷⣼⣶⠮⠥⢾⡿⠡⠀⡸⠀⠀⠸⡀⠀⠀⠀⡀⠉⠀⡠⢖⢄⣇⠀⠈⡧⠔⠁⠀⠀⠤⡀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡩⠝⠉⠙⠤⣴⣿⣷⠀⠀⠀⠀⠀⢀⠅⠀⠀⠚⢀⡴⠊⠔⠁⡸⠈⠉⣻⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸
⠀⠀⠀⠀⠀⠎⠀⠀⣀⠤⠴⡆⠀⢹⠀⢰⠤⠤⠐⠊⠀⠀⠀⠐⠁⠀⠀⠀⠀⢇⣀⣠⠃⢃⠀⠀⠀⡠⠒⠚⠂⡤⠃
⠀⠀⠀⠀⠸⠀⠀⠉⠀⠀⠀⢡⡠⠊⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⠀⠀⠷⠷⠧⠊
⠀⠀⠀⠀⠀⠑⠤⠄⠤⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠞⠛⠛⠛⠛⠛⠛⠓⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣆
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⡿
⢸⡇
⢸⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠸⣇⢸⠳⠾⠷⢤⡽⢦⣌⡷⣬⡷⢤⣸⠶⣄⣲⢦⡀⢠⣄⡀⠠⣄⡀⠀⢤⣀⠰⡄⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠛⠁⠙⠲⠏⠙⠲⠖⠈⢿⡃⠀⠀⠀⠀⠀⠀⣰⠇
⠀⢀⡏⠀⠀⠀⡾⠛⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⣤⡀⠀⠀⠻⣄⣀⣀⠀⠀⣰⠏
⣞⡛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠙⠆⠀⠀⣨⡄⠉⢧⡼⠃
⣸⠋⠛⠛⠀⢀⣤⣄⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣠⡾
⢻⡄⠀⠀⠀⠛⠶⠞⠃⠀⠀⢠⠟⢻⡷⠒⣶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠉⠉
⠈⢷⣄⠀⠀⣀⣠⡀⠀⠀⠀⡟⠀⠈⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠙⢷⣞⣋⡿⠃⠀⠀⠀⠻⠦⠤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢉⣛⣛⣛⣛⡛⠛⠉⠉⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⣴⠟⣡⣾⣋⡽⠋⠉⠉⠉⠛⠲⠶⣟⠛⠲⢦⡈⢻⣄
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠉⠙⠳⣦⣄⣀⠀⠀⠀⠀⠙⠶⢄⠷⠀⣿
⠀⠀⠀⠀⠀⠸⣧⣀⣀⣤⣤⠖⠛⠉⠀⠉⠛⠲⢦⣄⣀⡀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠓⠒⠒⠒⠒⠶⠶⠶⠒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢘⡿⠒⠲⠶⠶⠶⣶⠒⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡾⠁⠀⠀⠀⠀⠀⢸⡇⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠰⣿⣥⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣤⣽⡷
⣿⣿⡿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣤⣦⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⡴⢒⣛⢣⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⢰⡖⣦⣶⠒⣶⠂⡆⣴⡆⣶⣶⢰⡖
⠀⠀⣰⢛⠃⡫⣬⢼⢇⣘⣧⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠧⠸⣧⡿⣿⡅⣿⠀⡇⡿⣷⣿⣿⢸⡇
⠀⠀⣿⢛⡋⣿⠼⠯⠀⣷⣸
⠀⠀⠸⣞⠣⠨⠦⠺⠒⠂⠇⠀⢀⣤⣤⣄
⠀⠀⠀⠀⢠⣶⣿⣷⣦⡄⠀⠀⢸⠟⠛⠙⠃⠀⠀⣠⣶⣶⣶⡄⠀⠀⣠⣤⣤⡀⠀⠀⠀⠐⠛⠉⣠
⠀⠀⠀⠀⢹⣯⣀⣀⣹⡇⠀⠀⠸⠶⠄⠶⠆⠀⠀⠿⣿⣟⣿⣿⠀⢰⣏⣥⢤⠹⠀⠀⠀⠲⠶⣿⣿⡇
⠀⠀⠀⠀⢰⡀⠁⠈⢁⠁⠀⠀⠀⣀⣤⠀⡀⠀⠀⠈⠛⢣⣟⠉⠀⠀⠉⣡⣤⡀⠀⠀⠀⠀⠰⣦⣟⠁
⠀⠀⠀⠀⠈⣃⡀⠀⣏⣀⠀⣴⣾⣷⣶⣾⣿⣿⣤⣴⣆⠙⠉⢶⣤⡄⣠⡏⠋⠉⣷⣤⣴⣾⣦⣺⣿⣶⣤
⠀⠀⠀⢰⣿⣿⣳⣾⣽⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠀⣠⣼⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠿⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠿⠿⠿⠶⠿⠿⠿⢿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠲⡤⠖⠋⠉⠳⣄⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⣴⠀⠀⠀⡜⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⠁⠘⠦⠴⠊⢇⠀⣀⡄⠀⠀⢠⡇⠀⠀⡠⠲⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠊⣀⠀⡀⠰⣢⠀⠉⠀⢈⢶⠾⠁⠑⠒⠋⠀⠀⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠗⠀⠈⠛⢿⠄⠀⠀⠀⠀⠘⠁⢀⡄⢀⠀⠀⠀⢠⣇⣀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠘⠃⢰⣲⣾⠷⠄⠀⠀⠀⠻⠟⠛⠋⠀⠀⠀⠀⢀⣂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣫⡬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠢⣀⠜⠉⠁⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢆⠀⢸⠀⠀⡀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⣤⠜⠀⣬⣤⠤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠴⡻⠤⠞⠀⠘⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⠁⠀⠀⠀⠀⠘⢾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⢠⡿⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡛⢿⣄⠀⠀⠀⢸⡇⠀⠙⢧⡀⠀⠀⠀⠀⢀⣾⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠙⢷⣄⠀⢸⡇⠀⠀⠀⠙⢶⣄⡀⠀⣼⠃⠀⠘⢧⡀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠙⠳⣾⣧⠀⠀⠀⠀⠀⠈⠻⢾⡏⠀⠀⠀⠈⢳⣄⠀⢀⣴⠟⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡾⠁⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠻⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⣿⡛⠶⢦⣄⡀⠀⠀⠀⠀⠈⠉⢹⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠈⠛⠳⠶⣮⣿⡆⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣻
⠀⠀⠀⣰⡿⠿⠿⣶⣦⣾⠋⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢿⡉⠉⠉⠛
⠀⠀⠀⠸⣷⣄⠀⠀⠉⠛⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣉⣻⣦⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣤⣴⡶⠆⠀⠀⠀⠀⠈⠛⢯⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⣴⡿⠃⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⢀⣿⡇⠀⠈⠉⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻
⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⠛⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠟⠷⣦⣼
⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉
⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⠟⠀⠀⠀⠀⠀⠀⣿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠃⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠋⠉⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣴⣶⣄⡀⠀⠀⣿
⢻⣧⣀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣸⡟⠀⠙⠿⣦⣼⡏
⠀⠙⠿⠿⡿⠿⠿⠟⠛⠉⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⢠⡿⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠻⢷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣄⣄⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⢿⣯⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡟⠛⠷⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣤⣾⡇⠀⠀⠀⠈⠉⠛⠷⣶⣤⣀⣀⣤⡾⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠷⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠭⠭⠍⠉⡏⠉⢉⠭⢭⡉⢉⣉⣉⣉⣉⡉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠸⣄⡠⠃⠠⠤⠤⠤⠤⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠒⠒⠒⠓⠒⠲⡄⠀⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⠋⣠⠖⠒⠲⢤⡙⢦⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠃⡼⠁⠀⠀⠀⠀⢹⠀⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡄⢳⠀⠀⠀⠀⠀⣸⠀⡏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⠄⠑⠦⠤⠤⠞⢁⠜⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢦⠀⠀⠀⠀⢀⣀⣀⣰⣋⣉⣉⣉⠉⣹⣆⣒⣊⣁⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⡼⠀⠀⠀⠀⡏⡤⠤⡄⠀⣀⡤⢤⣀⠀⠀⠀⣀⠀⢹⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡇⠉⠉⠁⡞⡵⠚⠓⢎⢣⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠘⡄⢇⡀⠀⠸⣸⠀⠀⣿⠀⢸⡧⠤⠤⠤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣀⣀⣀⣠⡇⠀⠀⠀⠘⠲⠭⠥⠖⠁⠀⠀⠛⠀⢸⡇⠐⠒⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠲⠀⠑⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠤⠀⡄⠤⠀⠀⠠⠀⢠⠀⡄⢠⠀⠀⢄⣀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠁⠀⠀⠀⠀⠁⠁⠈⠁⠀⠈⠉⠘



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠒⠀⠀⣀⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⣠⡾⠀⠀⠀⠉⣛⠿⣆⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢸⠋⠉⠉⢢⡤⡞⠁⠀⠀⢳⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣧⠀⢠⣤⣸⠃⢹⣄⣛⣫⠞⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢰⠉⠑⠒⠒⠁⠈⠉⢹⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠐⠚⠋⠀⠀⠀⠀⠀⠀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠹⡀⠀⠀⠀⠀⢀⠔⠋⠙⠃⠀⠀⠀⢸⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⢥⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠶⣦⣹⠀⠀⠀⠀⠐⠴⠶⠒⢛⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⢷⢶⣶⣤⣤⣤⣴⣶⢻⢳⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣽⣿⡽⠼⠽⢏⠁⣩⣿⢿⢼⣼⣼⢿⠿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣭⣭⢽⣿⣽⡯⣿⠽⢽⢽⢼⣽⣿⢟⡛⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠟⠉⣯⣯⡭⠽⠐⢻⣻⣇⣀⣈⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣤⡶⡟⣿⣿⡷⣿⣿⣿⣿⣻⣿⣯⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡗⡷⣷⣧⣷⣿⣟⣟⣛⣺⣼⣾⣛⣁⣿⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢻⠞⠟⠳⠟⠟⠛⢻⣿⣿⣷⣿⣿⠿⠾⠾⠿⠿⠟⠛⠛⠛⠫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⢀⣀⠀⠀⠀⠀⠀⠀⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠸⠉⡅⠀⠈⠓⠀⠘⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⡀⠀⠀⠀⠐⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠱⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⢠⡗⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡟⠉⣧⠀⠀⠀⣠⠴⠆⢸⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡟⠙⢳⡎⢀⣤⠀⠁⠀⠘⠀⢸⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠁⡜⠁⠀⡀⠘⠛⠋⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⢿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⣿⣿⣯⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠟⣛⣉⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠸⠇⠘⣿⣿⣿⣿⣦⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡏⠀⠀⠛⠛⠁⠀⠀⢠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿⣿⣿⣿⣿⣷⢰⣦
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⣿⠋⠀⠀⠀⠀⠀⢶⣏⠀⠈⠡⡀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠿⣦⣶⣾⡿⠀⠀⠀⠙⢧⡀⠀⠈⠢⠀⠀⠀⠀⠀⠀⡠⠐⠛⠁⣼⣿⣿⣿⣿⣿⣿⡿⠓
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢻⣿⣧⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⣀⡀⠀⠀⠈⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠋⠀⠆⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⣿⡿⣯⣿⣿⣿⣿⠋⣻⡀
⠀⠀⠀⠀⠈⠛⠻⠿⠿⠟⠁⠀⠀⠀⢀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⠟⠈⣽⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠋⠐⢧⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⡳⠀⣀⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢳⡤⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠈⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠣⠙⠻⣤⡀⢀⣠⡄⠀⠀⠐⠛⠁⠀⠀⠀⠀⢠⡦⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠀⠙⠛⠧⠀⠀⠀⠀⣀⡴⠀⣠⡆⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡈⠀⠀⠀⢀⡀⠀⠀⠀⠘⠛⣤⣾⣿⠿⣦⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⢻⣿⡿⣶⢿⣷⣶⡶⣶⡾⡟⠃⢉⠁⠚⢹⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣝⣻⣦⡞⢋⣃⡀⠈⢁⡀⠛⣩⠗⢉⠠⠚⠃⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⣿⣿⡣⣬⢭⣴⠒⢊⡀⠀⢀⡰⣿⣐⠾⠁⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠸⣿⠁⠈⣎⡙⠆⠡⢦⡁⢾⡿⠊⡉⠀⣀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡧⢚⡀⢿⣿⣌⣿⣾⠁⠀⣜⠋⡀⠂⠤⢀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣤⠑⠢⢿⣿⣿⣇⣠⠖⠈⠉⠒⢀⠄⠈⠐⣻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣻⣿⣿⠞⠁⠘⣿⣿⣯⠷⣆⡀⠀⣀⠶⠇⠀⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⠀⣼⣿⡯⢿⣤⠄⢀⠘⣣⡀⠠⢈⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢯⣷⠈⣿⡗⠊⢹⠒⣤⢀⣬⠛⠔⢺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⡇⠈⠀⢙⡷⡁⠀⠛⠰⠟⠇⡁⠐⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠭⠄⠀⠀⠀⠿⠁⠀⠀⠀⠠⠘⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠄⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡛⠋⠉⠷⢤
⠀⠀⠀⠀⠀⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠘⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠗⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⣀⡀⠌⠉⣀⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠁
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⠤⠍⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠤⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢟⣹⠏⢳⠀⠀⠀⠀⠀⢸⡿⠙⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⠱⠀⠀⠀⠀⠀⠲⢾⠗⠈⠜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠄⠀⠀⠀⠀⠰⢬⡼⠻⣄⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⡡⣠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠂⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⡀⠀⢀⠀⠀⠀⠀⢀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡒⠂⠀⠈⠀⠀⠀⠀⢀⣀⠁⠀⠈
⠀⠀⠀⠐⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠞⠅⠀⠃⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡟⠋⠁⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⢻⡆⠀⠀⠀⠀⠀⢀⡴⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡗⠀⠈⠳⡄⠀⠀⠀⠀⠈⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣷⣔⣤⣼⠇⠀⠀⠀⠀⢀⣼⣷⣦⣄⠈⠣⠀⠀⠀⠀⠀⣴⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣷⣶⣿⠟⠉⠀⠀⠰⠶⢶⣿⣯⣙⡙⠛⢷⡄⠀⠀⠀⠀⠀⢿⣿⡭⣓⢮⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⠟⠋⠁⠀⠀⠀⠀⠀⣠⣟⣥⣿⣿⣿⣷⣦⣹⣦⡀⠀⠀⠀⢸⣿⡿⣾⣕⣯⣌⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣼⣿⣧⡙⢿⢿⣿⣦⣹⣿⣿
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣽⡻⣿⣷⣍⢻⣿⣿⣿
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿
⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⢒⣒⠛⠛⠛⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⣶⣿⡿⣿⣿⢿⣿⣿⣿⡟
⣴⣟⣁⣀⣤⣤⣄⣀⣀⣛⣛⠛⠛⠿⠿⠆⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⣠⠀⠀⣸⢏⣾⣿⣿⠟
⠙⠛⠋⠉⠉⠉⠉⠉⠙⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⢂⣴⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠋⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣼⠆⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠲⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⣁⣀⣉⡙⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠣⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣼⣾⣷⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣞⣯⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⣓⠄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠄⢀⠰⠂⠠⠄⠀⠀⡆⠀⡄⢠⠄⠀⠀⠀⡀⠄⠀⠀⠀⠂⠀⠂⠠⠀⠀⠀⠐⠀⠀⠤⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⢀⣀⣤⣶⣾⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠴⠾⠛⠛⠛⣿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⠿⠿⠛⠻⣧⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣸⣿⣿⣤⣴⣶⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢿⡋⠉⣿⣿⠛⠻⠿⢿⣿⣿⡟⣿⣿⠿⠿⣿⠛⠻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⣼⠃⠹⣇⠀⠀⠙⠦⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⠀⠈⠛⢷⣤⣀⡀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⡞⠛⠛⠋⠉⠉⠉⠉⣻⠛⠛⠳⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢹⣷⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⡏⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣀⣀⣀⣿⣀⣀⣠⣤⣤⣤⣤⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠷⠶⠶⠾⠟⠚⠛⠛⣿⠛⠛⠛⠛⠛⠛⠉⠉⠉⢹⡏⠉⠉⠁⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⡀⠀⢸⣿⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣀⣀⣀⣀⣀⣀⣀⣸⣧⣀⣀⣀⣀⡀⠀⠀⣿⠇
⠀⠀⠀⠀⢰⡅⠀⢸⣧⠀⠈⢻⣄⢀⣀⣤⣤⡶⠶⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠶⠦⣤⡀
⠀⠀⠀⠀⠘⣇⠀⠀⠻⢷⣤⡶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠙⢷⣄⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢸⠀⠀⠀⡿⠀⠀⠀⣼⢸⡟⢉⣹⢳⣤⡀
⣿⣿⠟⠉⠙⠛⠛⠛⠛⠻⣿⣿⣿⡿⢦⣤⣤⣮⣷⡀⢀⡴⠚⠓⠒⠲⢧⡀⠀⣠⣏⠘⠛⢯⢴⣾⣷⣹
⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠉⠁⠀⠀⠀⢀⣩⡻⣿⣠⠖⢒⣒⢦⡀⠹⡴⢹⡟⠛⠓⠛⠿⣷⠞⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⡀⠀⠀⠀⠀⣀⣤⡘⢷⣝⢿⣍⠛⢿⣿⠀⡇⠀⢹⣾⠃⠀⠀⠀⣴⡟⠀⣀⣀
⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣶⣾⣽⠁⠀⠙⠻⣿⣷⣮⣓⠈⢧⡼⠁⢀⡼⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠘⠓⠾⠿⣿⣿⠿⠿⠛⠛⠻⢿⣻⡄⠀⠀⠀⠈⠻⣝⠪⣝⣾⣶⣾⠟⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠛⠙⣿⠀⠀⠀⠀⠀⠀⣹⣿⣿⠟⠁⠀⠀⠀⠀⠀⢰⡿⠿⣿⣿⣿⣿
⣆⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢸⠃⠀⢀⣠⡖⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠈⠉⠙
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⢾⠀⠀⠈⢁⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣀
⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠈⠀⠀⡠⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣦⣀
⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣦
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠗⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁
⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⡏⢙⠲⠦⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣶⣾
⣿⣿⣿⣃⡏⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠒⠒⠶⢤⣴⣶⣶⣼⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⣶⣤⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠟⣡⣤⣶⣶⣿⣶⣶⠖⣿⣶⣶⣤⣀⠻⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣶⣿⣿⣿⣿⣿⣿⣦⠀⠀⣴⠾⣟⣿⣿⣿⣾⠸⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢀⣼⠙⣿⣿⣿⣿⣿⠿⢊⣀⡀⠹⢻⣿⡿⣿⣿⠁⢨⣴⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⢘⠻⠿⠿⠛⣡⢋⡀⡉⣄⠐⠢⠍⠿⠋⡄⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣷⢸⣷⢶⡶⠋⠁⢈⣀⣀⠈⠙⠳⠶⢞⣧⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣧⣿⣆⣠⣄⠈⢋⡉⠝⠁⠀⡠⢂⡼⠿⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢠⠀⠀⡈⣴⠎⣀⢈⡆⠐⣨⣴⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣾⣿⣿⣿⣿⣷⣿⣮⣶⠞⠉⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁⠑⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠁⠀⠀⠈⢻⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢷⣜
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⢀⣀⡀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠀⣀⠀⠘⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢄⡀⢷⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢬⡙⠧⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⢈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⡀⠸⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢀⠔⠁⠈⢆⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⡇⢀⠞⠈⠁⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢄⢄⠀⠀⣱⡄⠀⠈⣀⠤⣀⠀⠀⠀⢠⠃⠇⠀⢦⠸⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢎⢆⠞⡡⠃⠀⢰⡁⢐⡾⠤⠐⢶⠁⢸⣤⠉⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠒⠊⠁⠀⠀⢠⢿⠁⠀⢀⡴⠋⠀⣾⡷⣊⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣠⠞⠁⠀⠀⣀⡠⠄⢚⡶⠒⠊⠁⠀⠀⠀⠘⠹⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡄⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⡴⠋⣀⠤⠒⠋⠉⠀⢀⡴⠃⠠⠀⠀⠀⠀⠀⠀⠀⢰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⡇⠀⠀⠀⠀⡴⠁⠀⢀⠔⠁⠀⠰⡁⡀⠀⣀⡤⠖⠉⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⢣
⢀⣠⡄⣢⠄⠀⠀⠀⢠⠃⠀⠀⠀⠀⡇⠀⠀⢀⡞⠀⢀⠔⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢧⢀⡠⣒⡆
⠋⠈⠑⠪⢕⢕⢲⣠⠇⠀⠀⠀⠀⠀⡇⠀⠀⠎⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠳⢯⣏⠒⠀⠈⢇⣠⢿
⠀⠀⠀⠀⠀⠑⢵⣟⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠈⠛⠀⢀⠜⢁⡎
⠀⠀⠀⠀⠀⠀⠀⠿⣧⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⡀⠀⣠⠔⠁⢠⠂⠣⣀
⠀⠀⠀⠀⠀⠀⠀⢨⠬⢷⡄⠀⠰⠶⠓⠐⠐⠒⠒⠒⠒⠒⠒⠒⢂⠖⠀⠀⠀⢀⠤⡤⠀⢄⠀⠀⠀⠀⢀⡀⠥⡞⣳⠎⠈⠑⠦⠃⠰⡄⠀⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘⣷⣦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡠⠃⠀⠀⠀⠀⠣⢔⢻⠿⢸⠀⣀⣤⣯⠁⢀⡴⠁⠛⠁⠀⠀⠀⠀⠀⣷⢀⣴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢎⠜⠫⡀⡭⠐⠈⠉⠹⢔⡝⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠟⠋⠙⢧⠀⡠⣫⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢰⡆⠀⠀⣶⠆⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡵⠁⠀⠀⠀⠀⢪⡀⡀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⡡⠛⢷⡬⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⡰⢩⠋⠁⠒⠒⠂⢠⠤⠃⠀⠀⠀⠀⢎⣸⠐⢈⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡀⠀⢻⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠖⠳⠧⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢭⢸⠖⠛⠃⠀⠉⠓⣮⢦⣵⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣱⣄⠸⠀⠀⠀⠀⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠡⠃⠀⠀⠀⠀⠀⠀⠻⠞⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠾⣛⣛⣷⠶⠖⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⡈⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢈⣀⣀⡀⠀⠀⠀⠈⠷⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠁⠀⠀⡠⠤⣄⠀⠾⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⣠⠀⠀⠀⢀⠎⠀⠀⠀⠑⣄⠒⠺⣄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⢊⠏⠁⠀⠠⠊⠁⠀⠀⠀⠀⠑⠈⢢⠀⢈⣧⠀⠀⠀⢀⠜⢁⡠⡙⢆⡠⢔⠂⠉⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡠⠀⡇⠀⠀⣀⣀⡀⣠⠒⠐⠀⢣⢀⠊⠀⠀⡠⠃⠴⠁⠁⣸⡤⠒⠁⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢘⠁⠀⢃⢰⠋⠀⠀⠈⠉⠁⠀⠀⠃⢺⣆⠀⡔⢡⠊⠀⣢⠴⡇⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⢄⣈⠉⠑⢄⡄⠄⠀⠀⠠⡶⠄⣀⠠⠃⠀⠓⠊⡔⡁⠖⠊⠁⠀⠈⢳⣈⣁⠀⠀⢀⡄⠈⠄
⠀⠀⠀⠀⠀⣀⣀⡀⡇⠀⠠⠒⠲⠤⣭⣐⡚⠷⠀⠀⠉⡉⠀⠀⠀⠈⠐⢄⣠⠊⠁⠀⠀⠀⠀⠀⠀⢀⡴⠁⢠⠊⠁⢁⠀⠠
⠀⠀⠀⠀⠘⠁⠀⠐⠛⠒⠲⢤⣤⣀⣆⣀⣸⣑⡒⢤⣔⠀⠀⠀⠀⠀⠀⣸⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⢱⠔⠋⠀⠀⠈⡆⠀⠇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣀⠕⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⣀⣠⣴⡿⠿⢿⡃⠀⠀⠀⠀⠀⠀⡘⢈⡠⠒⠁⠈⢣⠜⠀⡰
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣹⠉⢆⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠛⠛⠛⠁⠀⠀⠀⠈⡆⠀⠀⠀⠀⢠⠃⠉⠀⠊⠉⢁⠆⢀⠜⠁
⠀⠀⠀⠀⠀⠈⡆⠠⡤⠭⡀⠉⢳⡳⡀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢁⣀⡀⠀⣀⡔⢁⣀⣀⡀⣤⠎⣠⠣⢄⡠⠔⢋⡔⠁
⠀⠀⠀⠀⠀⠀⢡⠀⢇⠀⠘⠤⠜⠣⡈⠢⡀⠀⠀⠀⠑⣄⣀⣤⠚⠁⠀⠉⠙⠛⠡⠭⢀⡠⠖⠁⡰⣁⠔⠉⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠈⠆⠈⡆⢰⠚⢉⢑⣛⠂⠈⠲⢆⣛⣉⡱⠚⢀⡔⠀⠀⠀⠀⣀⠝⠓⠲⡤⠴⠚⢏⢀⡠⠚⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠙⢄⠘⠂⠀⠀⡝⠒⠤⠤⠥⠤⠒⠁⠈⠉⠉⢏⠁⠱⡤⠦⠴⠁⠀⠀⠀⢣⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠅⢂⡀⠚⠢⠤⠤⣀⣀⣠⠒⠒⠢⠤⠖⠋⣄⠠⠋⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⡐⠁⠈⢢⣀⣀⡤⠒⠀⡖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠠⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⡰⠁⠀⠀⢠⡇⣇⢘⠒⢤⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢲⠃⠀⠀⠀⢸⡇⡏⠁⠀⡸⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⢸⢽⠁⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⢀⠤⡀⣸⣿⡿⠀⢠⠃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡤⠤⠔⠒⠊⢫⠀⢆⠀⠀⢀⠔⠁⠀⢳⡜⢿⠃⡠⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢣⣈⡦⠒⠁⠀⠀⠀⠈⠧⣏⠔⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠑⢄⠀⠀⠀⠀⠀⠐⠚⠉⠱⡀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣠⢔⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠻⢿⣟⣛⠿⢿⡿⣿⣭⣭⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢘⡏⠒⠀⠈⡉⣉⣉⠛⠺⠿⠿⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠄⠀⠠⠤⠤⠄⠒⠒⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠢⣄⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⡁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⡫⠷⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⣵⡄⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⣩⠅⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡟⠩⣟⢦⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡻⢟⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣼⠃⠀⠈⠳⢝⣦⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢛⣩⠖⢋⣴⡟⠃⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠿⠏⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⡏⠡⢠⠚⡍⠁⠀⣠⠒⠛⠻⠟⠉⠸⠛⠛⠻⣆⠀⠀⠀⢀⣴⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⡿⠁⠀⣸⡇⠀⠐⣟⢻⢄⡀⠀⠀⠋⠁⠀⠛⢾⡟⠠⢄⡀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⠁⠀⡞⢹⣯⠀⢰⡿⣥⡈⠁⣀⢄⡄⢚⣀⣤⠾⠀⢀⢎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⠀⣠⠟⠻⣿⡄⠘⣄⢈⡻⠧⠂⠈⢣⠊⠰⠁⢀⣠⣞⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢸⠿⡌⠁⠀⠀⢸⣧⠀⢸⡁⠀⢀⣀⣠⣏⠀⠁⠐⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠸⣀⡀⠀⠀⠀⢈⡇⠀⣸⡿⠛⠋⠉⣹⣿⣃⣇⣠⡿⠿⠷⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⣤⣴⣿⣧⡀⢀⣠⣾⣷⡞⠁⢀⣀⣤⣾⣿⣿⣿⣧⣀⣀⣤⣤⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠤⣤⣤⣤⣤⣤⣤⣤⠤⠤⢤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⢶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡛⠛⠛⢉⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⠉⠛⠛⢛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢤⣀⡞⠀⠀⠊⠁⠈⠘⠛⠲⠤⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠠⠤⠚⠛⠁⠀⠉⠂⠀⠀⢳⣀⡤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⢻⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡟⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⢸⡇⢀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢀⣤⠤⢤⣴⠖⣲⡦⣄⡀⠀⢠⣿⣿⣿⣯⡄⠀⣀⣠⢤⣖⠲⣶⡤⠤⣤⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡟⠀⠀⠀⠀⠀⣦⠠⡹⣄⣿⣿⠞⠁⢀⣹⣿⡿⢿⣿⣏⡀⠈⠳⣿⣿⣠⢏⠀⣴⠀⠀⠀⠀⠀⢻⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⡆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢰⠀⣷⡇
⠀⠀⠀⠀⠶⣤⡤⣤⣀⣸⣿⣼⠀⢤⡀⠀⠀⠀⠀⠀⢀⠴⢊⠁⠀⢈⣹⣿⣿⣿⣿⣿⣿⣏⡁⠀⠉⡙⠢⡀⠀⠀⠀⠀⠀⢀⡤⠀⣧⣿⡃⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠛⢷⣶⣼⣿⣿⠀⠀⠙⡦⠀⠀⢀⡖⢁⠖⠁⠀⠈⠉⠘⠿⢿⣿⣿⡿⠿⠁⠉⠁⠀⠈⠳⡈⢲⡀⠀⠀⢰⠎⠀⠀⣿⣿⣃⣉⣭⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⣏⢧⠀⠀⠀⠀⡠⠛⠀⠀⠠⠀⠀⠀⢂⣒⣠⣿⣿⣿⣿⡄⣐⡐⠀⠀⠂⠀⠀⠀⡛⢄⠀⠀⠀⠀⡼⣹⢿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠈⠊⠃⠀⢀⡞⠁⡜⠀⠀⠠⠔⠚⣉⣡⣶⣾⣿⡿⢿⣿⣷⣦⣌⣉⠓⠢⠄⠀⠀⢣⠈⢳⡀⠀⠘⠐⠁⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⡄⢠⠀⠀⠀⠀⢼⢁⠎⠀⣠⣾⣿⣿⣿⣿⡷⠶⠶⠶⠶⣿⣿⣿⣿⣿⣷⣄⠀⢱⡈⡧⠀⠀⠀⠀⡄⢀⡏⣿
⠀⠀⣀⣀⡤⠴⠖⠒⠛⣿⡄⢳⠀⠀⠀⠀⠀⢸⠀⠀⣼⣿⠿⢿⣧⢯⡤⠶⡦⢠⡄⢴⠶⢤⡽⣼⡿⠿⣿⣧⠀⠀⡇⠀⠀⠀⠀⠀⡞⢠⣿⠙⠒⠲⠦⢤⣄⣀
⠀⠀⠉⠙⠛⠻⠿⢿⣿⣿⣧⢀⠣⠀⠀⠀⠀⢾⡀⠀⣿⣷⠀⠘⡇⢀⣇⣀⣄⣠⣄⣠⣀⣨⡀⢸⡇⠀⣾⣿⠀⢀⣗⠀⠀⠀⠀⠘⡀⣼⣿⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⢆⠀⠀⠀⠀⢹⡇⠀⣿⣿⣇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⣿⠀⢸⡏⠀⠀⠀⠀⣰⣱⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣎⠆⠀⠀⠀⢸⡇⠀⣿⣿⠿⣾⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⢹⣷⡿⣿⣿⠀⢸⡇⠀⠀⠀⠰⣱⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣻⣿⣿⣆⠀⠀⠀⠈⣇⠀⢻⣿⡄⢽⣷⡀⣻⣿⣿⣿⣿⣿⣿⣟⢀⣾⡿⠀⣼⡟⠀⣸⠁⠀⠀⠀⣰⣿⣿⡿⢦⣀
⠀⠀⠀⠀⢀⣴⣺⣥⣶⣿⣿⣿⣿⣿⣆⠀⡄⠂⢹⡀⢸⣿⣄⢀⣿⣿⣿⠿⠿⢿⡿⠿⠿⢿⣿⣿⣇⣀⣿⡇⢀⡏⠐⢀⠀⣰⣿⣿⣿⣷⣦⣌⡓⢤⡀
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢻⣿⣿⣦⡸⡌⠈⣧⠘⣿⣷⣄⢸⣿⠁⠀⠀⢛⠀⠀⠀⠀⣿⣟⢀⣼⣿⠃⣼⠁⢡⢇⣼⣿⣿⡟⠉⠉⠉⠙⠛⠛⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣽⡄⢸⡀⢿⣿⣆⠀⣿⠀⠀⠀⠸⠀⠀⠀⢀⣿⡇⢠⣿⡟⢀⡏⢠⢯⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣨⣿⣿⣿⣿⡌⣧⡞⣿⣷⣀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡿⣀⣼⣿⢣⣾⢡⣿⣿⣿⣿⣅⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⣾⡿⠛⠉⠙⠻⣿⣼⣿⣹⣿⣿⣦⢿⣄⠀⠀⠀⠀⢀⡾⣧⣾⣿⣏⣿⣧⣿⠿⠋⠉⠛⠿⣷⣜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣷⣻⣿⡇⢸⣿⣿⣶⣶⣾⣿⡇⢸⣿⣟⣾⣿⣿⠁⠀⠀⠀⠀⠀⠈⠙⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢠⣿⣿⣿⣿⣤⣼⣙⠛⠛⠛⠋⣉⣧⣠⣿⣿⣿⣟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠋⠻⣿⣿⣿⣿⣭⣛⠛⠛⣋⣭⣿⣿⣿⣿⠟⢿⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠈⠉⠛⠛⠛⠻⣿⣿⠟⠛⠛⠛⠉⠁⠀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣤⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢢⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢹⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢀⣤⡤
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⢿⣿⣿⣿⠳⠟⠁
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠉
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠠⠶⠿⠿⠿⣿⣿⣶⣶⣶⣦⡄⠀⠀⢠⡿⠋⠉
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⢙⠛⠻⠟⠛⠁⠀⢀⡤
⠀⠀⠀⠀⠹⣿⡿⣋⠛⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣀⡤⢂⡭⠻⡿⢰⡖⠀⠀⠐⣾⡋⠉⠁
⠀⠀⠀⠀⠀⠘⠻⠉⢿⡄⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢷⠚⡉⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣁⠀⠀⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠙⣌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⢀⡀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠙⣻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠘⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠙⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡈⠙⠿⡏⠀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢸⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣿⣿⡆⢹⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⣾⣿⣿⡇⠈⣿⠃⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⠀⠀⠀⠰⣾⣿⠀⠀⠟⠀⠀⠀⠙⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀