Yuletide Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠐⣷⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠡⢠
⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠑⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢈⠃⠀⠀⠉⠉⠀⠀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠃⠀⠀⠀⠔⠈⠀⠀⠀⠀⠨⠤⠀⠀⠀⠀⡀⠀⠮⠌⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠛⠛⠛
⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢤⠈⣌⡁⣧⡀⣇⣀⠱⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠄⢀⠀⠀⠄⠀⢀⢠⡀⠠⡄⠀⠀⡄⠀⠀⡠⠀⠀⠀⣰⣿⠀⠀⠫⣶⠙
⠀⠀⠔⡃⠀⠀⠀⠀⠀⢀⠀⠐⠤⠀⠇⠳⠘⠒⠄⠂⠐⠅⠀⠳⠀⠇⠘⠐⠃⠐⠐⠒⠀⠀⠊⠟⠫⠀⠀⠀⣿
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⣀⠀⢠⠀⠠⠄⠐⠂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠿⠇⠀⠀⠀⠀⠈⣠⡄⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⡆⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠂⠀⠀⠀⠀⠐
⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠐⠀⠀⠂⠀⠁⠀⠀⠀⠈⠈⠄⠀⠘⠀⠀⠀⠈⠀⠀⠀⠁⠀⠠⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣭⣯⣽⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⢏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠶⣽⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣠⣶⣌⠻⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⢿⣿⠿⣿⣿⣶⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡴⠋⢁⣶⠿⣿⣿⣮⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣯⡁⢻⣤⣿⣿⣦⠞⢛⠲⣻⣾⡇⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿⣿⣾⣾⠛⡋⣖⠚⠳⠶⣉⣉⡠⠜⣻⠄⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡠⠞⠋⠀⣾⣿⡟⠳⠶⢞⣛⠛⢟⣾⣧⣴⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠆⠚⡛⠋⠈⣦⢠⣴⣾⣿⣿⣿⡷⠒⢾⡿⣿⣼⣿⣛⠛⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⡞⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣗⡐⠛⠳⠤⢥⣛⠻⠛⣻⣍⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠫⢟⣿⠟⠋⣱⡿⠿⣶⠟⠛⠻⣷⡶⢆⣠⣛⣁⡀⢠⣶⢦⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⠧⠀⠘⣆⠉⢀⣤⣴⠿⠓⠾⣿⣝⠓⠒⠊⢿⣥⡀⠙⠛⠿⠿⠥⠿⠄⣀⣠⡄
⠀⠀⠀⠀⡆⣶⣦⣠⠀⠻⢦⣦⣷⡦⢰⣼⡶⡟⠁⣠⣶⣻⣦⠀⠙⣿⣶⠀⠀⡨⠻⣿⣷⣦⣴⣤⡟⠚⠉⠛
⠀⠀⠀⠸⣷⣿⢫⠏⠀⠀⠀⠀⠻⣷⣿⠋⠁⢐⣿⣯⡿⠺⣿⡇⠀⠈⢿⣷⣧⣑⡰⠃⠈⠻⢯⣉⠀⠀⠀⠀⣠⡠⣶
⠀⠀⠀⢠⠏⢱⣏⡄⣾⣿⣷⣤⠾⠛⢁⣄⢿⣿⣿⠛⠛⠚⣿⣿⣇⣠⡀⢉⡛⣿⢽⡛⠲⣦⣤⣨⣕⠲⠶⣿⣿⣷⣿⠂
⠀⠀⠀⢨⠀⡾⠘⣷⡹⣿⣿⡇⣦⣤⠿⠿⣯⣿⣿⣿⣿⡾⠿⠷⠿⣿⡇⣿⣿⡷⢀⡵⠈⠁⠀⠉⣿⣟⣿⣿⠛⢀⣯⣭⠁
⠀⠀⠀⢸⡆⢳⡴⢻⠇⠉⣉⣁⣼⢯⣶⣿⠿⠿⣿⣿⠏⠀⣀⣀⡀⠿⠱⠿⣿⣧⣾⣴⡿⠿⣦⠀⣇⣼⣠⡟⡠⡸⠁
⠀⢀⡴⣾⣧⠀⢻⣿⣶⣿⡟⠉⢹⣿⣿⡧⠀⠀⢿⢻⣠⡞⠋⠉⡿⠓⠒⠲⣾⣿⣿⡇⠀⠀⡿⣠⠗⣛⣭⣾⡴⠧⢄
⠀⠀⠘⠻⠛⢢⣼⢹⠿⠋⢀⠀⣺⣿⣿⣌⡀⡀⣸⡧⣾⣇⠠⠀⣧⢀⠀⠸⢾⣿⡙⢃⠈⢁⠛⣿⣯⣵⠟⠓⢻⣶⢦⠄⠂
⠀⠀⠀⠀⡴⣞⠻⡟⠷⢿⣖⣭⠋⢛⣏⠁⠁⠒⠛⠁⣸⢳⣥⣤⡾⢏⢠⡤⢾⣿⣿⣦⣭⣁⠔⣉⠹⠆⣸⠥⠈⠁
⠀⠀⠀⢸⣃⢉⣄⣸⣦⣀⡈⣭⡚⠛⠿⣦⣤⣀⣤⣴⣿⣿⣿⣿⣄⣴⠉⣦⣴⣿⣿⠿⠟⠛⠈⠀⠀⢀⣷⣄⠀⢁
⠀⠀⠀⣼⢵⠋⣼⣿⣿⠟⠁⣯⡝⣷⠀⠀⠀⠉⠙⠛⠛⠛⢋⡙⡉⠉⠀⠉⠁⠀⠀⠀⠀⠀⣀⣀⠀⣸⠿⠋⣀⠜
⠀⠀⠐⣧⠿⡄⢈⡎⡉⠀⠀⠻⠼⣿⣼⢳⡆⢰⠠⡄⡄⢸⠈⠁⡧⡄⡤⢠⠴⣂⡇⣾⣿⣷⠿⠘⢠⠏⠉⡏
⠀⠀⢰⡏⠀⣿⠾⠀⠉⠓⠲⠤⠤⣄⣀⡈⠁⠘⠀⠓⠇⠘⠀⠀⠓⠛⠓⠈⠁⠉⠁⡉⣀⣀⠤⠒⠙⣆⡜⠁
⠀⠀⠀⢻⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⠤⠢⠤⠤⠤⠤⠤⠤⠄⠒⠚⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⠢⠀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠄⠀⠈⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣠⣥⠖⠖⠶⠿⠝⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣉⠀⣀⣠⣤⣤⣶⣤⢤⣶⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⢤⠀⠈⠉⠉⠠⣤⣉⠋⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⡀⠀⠀⠄⠤⠤⠺⣿⡉⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣶⣿⠿⢿⣨⣿⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣬⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠈⠛⠲⣴⣾⠿⠷⢬⣤⡖⠶⣤⣶⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠖⠶⣤⢴⠶⠶⠦⠄⠀⡄⠠⣰⣤⡤⠀⠀⠽⠻⠧⢼⡷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣧⡿⠟⠂⠒⠒⣾⠒⠒⠒⠬⠛⣿⡷⠦⡄⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢀⠅⠃⠀⠀⠉⠀⠀⢀⠀⢀⢀⠀⠀⠀⠀⡀⠀⠀⠉⠲⣤⣤⣤⠢⢀
⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⢀⢀⡠⠀⡄⠀⣂⣀⣾⠤⠥⠆⠒⠛⠛⠓⠂⠀⡀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠉⢉⡚⠻⢛⡛⠒⠉⠋⠙⠿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠦⣤⡄⠲⣶⣿⠗⠖⠂
⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⣀⡁⣀⣀⣀⣀⢀⣀⡀⣀⠀⠀⠈⠁⢀⣀⣀⠠
⠀⠀⠀⣠⣌⣀⣀⣀⣀⣀⣥⡄⢀⠠⠰⠤⣮⣍⣈⡿⢈⠿⠓⠋⠉⠉⠉⢁⣀⣤⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠈⠉⠉⠉⠉⠱⠦⠬⠉⠁⠀⣀⠈⠀⠀⠀⢀⠀⠀⠀⢀⣀⣄⣠⡀⡍⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⡤⣤⡤⠄⠀⠀⠀⠀⠀⠀⠤⠷⠶⣾⣭⣽⣿⣿⣣⠟⠈⠉⠉⠉⠀⠙⠉⠉⠙⠻⢴⡶⠶⠛⠒
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠂⠀⢀⣠⣀⣀⣀⡀⡀⠀⠄⠀⠀⠀⠀⠀⣠⣷⣿⣆⢄⣀⣀⣀⠴⠾⣴⣶⠾⠷⢆⣠⣤⣄⡀⠀⠠
⠠⠤⠖⠊⠋⠀⠈⠀⠛⠓⠛⠓⠶⣶⣾⢗⠶⠋⠀⠈⠉⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠙⢳⢶⣆⣀
⠀⡀⠐⠠⠀⠀⠀⠀⢀⠀⢀⣠⡌⠉⠉⢀⣀⣀⡤⣦⣀⢠⣄⡀⣀⣀⡠⠤⣤⡄⣀⣠⢤⣤⣤⡀⣀⣀⡀⣾⡍
⣤⣴⣾⣤⣭⣭⣥⣴⣶⣭⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣭⣬⣭⣤⣭⣭⣭⣭⣭⣭⣥⣤⣬⣭⣥⣬⣭⣭⣽⣥
⣿⣿⣁⣉⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠
⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⣴⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠥
⠀⠀⠀⣀⣭⣖⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣭⣿⣛⣷⠆⠀⠀⠀⠀⠀⠀⠨⠓⣀
⠀⠀⢰⣻⠽⠖⠒⠒⠲⢤⣀⣀⡀⠀⠀⠀⢾⡭⠤⠴⢒⣒⣦⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠤⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧
⠀⠀⠀⠀⢀⣀⣤⣶⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⢀⣼⣿⣿⣏⣁⣀⠀⠀⠀⠀⠀⠀⠀⠐⣊⣁⡒⢤⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠠⠒
⠀⠤⠚⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠘⠉⠉⠐⠢⠤⠈⠀⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠔⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⣀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠟⡄⢦⣘⠁⠀⣀⡞⡸⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⣴⣯⣾⠁⡹⡗⠂⠥⠰⢿⣽⣋⣰⡦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⣾⣿⣿⡍⢉⠴⠒⠦⠅⠐⢦⣀⠈⠿⠦⠙⢶⣖⢶⡈⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢉⣙⠫⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠛⣹⡟⠿⠚⢃⢾⠁
⠀⠀⠀⠐⣶⣀⡄⣀⢸⣫⣽⡄⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠟⠋⠘⣷⣄
⠀⠀⠀⠐⠾⢟⡰⢧⡼⠋⡿⢶⡀⠀⠀⠀⢰⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡣⡀
⠀⠀⠀⠀⠙⢲⡷⣸⣴⡇⢛⠀⠀⠀⠀⠀⢸⠈⡇⢸⢀⠠⠠⢤⠠⢤⢀⠀⠀⠀⠀⠀⠀⠀⡉⢸⡟⢦
⠀⠀⠀⠸⡀⠀⠿⠏⠀⢠⠏⠀⠀⠀⠀⠀⠘⠀⠇⠈⠂⠁⠀⠈⠀⠈⠈⠉⡆⠀⠀⠀⠀⠀⠘⡆⠛⠳⢄
⠀⠀⠀⠀⢉⡼⢂⠃⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠡⠵⡾⠌⠛⠢
⠀⠀⠀⠢⣤⡐⡾⢖⢂⡆⠀⠀⢀⠂⢀⢧⠂⠃⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢠⠡⡞⣁
⠀⠀⠀⠀⠀⠑⣦⣤⠸⡄⠀⠀⢸⠀⠀⠸⠔⡄⠛⡆⣆⠘⡠⢣⡀⡶⡄⡆⣄⣄⠘⡠⠀⣰⠃⢀⣠⣶⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠳⣼⡇⣈⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⢼⠟⡟⢾⠧⣄
⠀⠀⠀⠀⠀⠀⠈⢪⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣾⣢⡞⢳⠆⣵⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⣠⣴⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠘⣟⣯⡇⠉⠐⠉⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢡⡄⣶⣴⣏⣤⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣢⣍⣁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡈⠳⠽⠷⣄⠲⣶⡀⠉⠳⠄⢐⠲⠤⠖⣁⣘⣻⣿⡿⠋⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠲⢏⣩⣝⣷⡆⢒⠠⢼⣎⢀⡿⣹⠟⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⢿⡎⡼⠉⠀⢀⡍⠳⠘⣴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠉⠁⠀⠀⠐⠁

⠀⠀⠀⠰⠁⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡀⠂⠀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠛⣚⠚⡄⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠖⠀⠁⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠄⠂⣄⡐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠄⡀⠘⠀⠀⠀⠀⢠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠫⠄⠀⠀⠀⡀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡀⠀⠀⠀⠀⠢⣤⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠐⡄⠀⠀⠍⠇⠀⠀⠀⠘⠀⠀⠋
⠀⠀⠁⠀⠀⠀⠀⠐⠠⠀⢀⡠⠰⠠⣠⠀⠈⠀⠀⠈⠁⣀⢂⠀⠀⠀⡀⡀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣵⠛⣶⠐⣄⠀⠠⠄⠀⡜⠁⠀⠙⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⢛⢉⢬⠀⠀⠀⠀⠀⠱⠄⢀⠔⠀⠀⠀⠀⠀⠁⠀⠀⠈⠆⡀⣨⠡⡀⠞⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠠⢱⡟⢳⡡⠀⣤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣠⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠎⢿⠋⠎⡀⠀⠀⡀⠀⡀
⠀⠀⠐⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⢰⠂⠀⠙⠀⠀⠁⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⢆⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠈⢢⠤⣖⠅⠀⠀⠘⠀⠈⠈⠣⢆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢁⢞⠖⡃⡊⠀⠀⣀⡰⣄⡀⠀⠀⠉⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠗⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⠀⢀⠟⠉⠀⢠⡄

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠀⢀⡶⠿⠿⢿⣦⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢠⡆⠀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠰⡄⠀⢰⣿⡀⠀⠀⠀⠀⠀⠹⠆⠀⠙⠃
⠀⠀⠀⢠⣾⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠰⠇⠀⡆⠀⣤⠀⢰⡄⠸⠆⠈⠃⠀⠀⠀⣼⣿⣇⠀⢀⡖⣶⢦
⠀⠀⠀⢈⣠⡤⠄⠤⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⡤⢀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⠈⠯⠿⠟⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠈⢁⡀⠁⠀⠀⢹⣿⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢡⢻⣆⠀⠀⠀⠀⠀⠀⢀⣠⡀⠈⠀⠀⠈⠉⠉⠉⠀⠒⠂⠐⠂⠐⣠⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⠀⠀⠀⣤⠀⠀⠈⢸⠈⠓⠀⠀⣶⡀⠀⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠠⢄⠀⠀⢀⠂⠒⣂⣀⣐⠂⠀⡀⠀⠀⠀⠀⢰⠿⠿⢿⠀⠀⠁⣠⣴⣶⣶⣦⠀⠀⠐⠂⠀⠀⢤⠂⠀⠈⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠂⠁⠀⠀⠈⠀⠟⢻⣿⡿⠻⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣁⡀⣀⠙⢧⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠄⠂⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⠀⡀⠀⠀⣀⣈⣁⣀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⢐⠀⠀⠰⠆⡃⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠄⠂⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠼⣩⠧⠀⠀⠈⠉⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠀⠈⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⠀⠀⠀⣠⡄⠀⠀⣤⡀
⠀⠀⠀⠀⣀⣤⣾⣷⠀⠀⢀⣤⣤⠄⠀⠀⠀⢁⠀⡀⠆⣡⠶⠿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢴⠍⠀⠀⠀⠸⠀⠀⠀⠀⣈⡀⠀⠀⠉
⠀⠀⠀⠈⠉⠙⠛⢿⠀⠀⠘⠛⢃⣦⣾⠂⢀⡀⠀⠀⠀⠀⠖⠀⠂⠙⠀⠀⠀⠀⣀⠒⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠸⠿⠾⠿⠗
⠀⠀⢠⣶⣿⣿⣷⣄⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⣴⡆⠠
⠀⠀⠀⠹⣿⣿⣿⣿⣧⡀⢀⣨⣿⠀⠀⠀⣀⣀⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⢈⡁⣿
⠀⠀⠀⠀⠙⠻⠿⠿⠿⡿⠿⠛⠁⡠⠀⠀⠼⠧⠀⠀⠘⢿⣿⣿⣿⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣄⠀⠰⠒⠀⠀⠠⣶⠀⣶⠀⠀⠀⢸⡇⠂
⠀⠀⠀⠀⠀⠉⠀⠁⠀⠁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢠⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠃⠀⠀⠀⠀⠀⠀⠀⠚⣛⣖⣄⡚⣃⠘⠐⠷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠘⠓⠈⠯⡖⣹⠁⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠺⣿⣏⣽⠿⣼⠿⣾⠁⡀⢖⡆⠀⠀⠀⠀⠀⠀⠖⠊⠙⢿⣾⠇⡀
⠀⠀⠀⠀⠀⢀⣜⠀⠀⠀⠀⠉⠇⠀⠀⠀⠀⠀⠈⠐⠠⡀⠀⠀⠀⠀⠙⢺⠛⢇⠀⢶⡆⠀⢀⠏⠣⢚⡇⠀⠀⠀⠀⠀⠀⠁⢀⠈⠘⡁⠀⢈
⠀⠀⠀⠀⢀⡞⠍⡍⠓⠢⡄⠄⠀⠀⠀⠀⠀⠱⣆⠤⠶⠖⠀⠀⠀⠀⠀⠘⠀⢨⠉⢛⡗⠈⠉⠀⠀⠀⡄⠀⠀⠀⠦⠖⠂⠐⠒⠒⠒⠒⠒⠓
⠀⠀⠀⠀⡼⠁⢀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠰⣀⣋⡉⠒⢒⡰⠀⠀⠀⠀⢀⣤⡀⠀⠈⡇⠀⠀⠤⣌⠠⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢰⠠⠁⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣍⠓⠢⠶⣌⡑⣄⠀⠀⠀⠀⠙⠃⠀⠴⡇⢀⠀⠋⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠘⣄⠁⠀⠐⠀⠐⠄⠀⠀⠀⠀⡀⠫⢤⣄⠠⠂⠀⢒⠧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠑⠢⣔⣠⡡⠀⠀⠀⠀⠀⣤⣠⠀⣠⠈⠁⠒⢒⣒⡒⠁⠀⠀⠀⠀⠀⠀⢀⣀⠶⠊⢣⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣩⡁⠒⠆⠠⢀⣀⣀⠀⠉⠤⠀⠀⠀⠀⠀⠶⠊⠁⠀⠘⠋⠀⠀⠀⢉⣀⡀⣀⡀⣀⢀⣀⢀⡀⢀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠄⠀⠀⣤⡀⠈⠀⠆⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠋⠉⠉⠉⠙⠙⠛⠛⢹
⠀⠀⠀⠒⢶⠄⠀⠚⠀⠖⢁⡀⢰⣆⠀⠀⠀⠀⢰⢯⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣠⡀⠀⣠⡄⠀⢰⡇⠀⠘⠊⠀⠀⠀⠈⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡜⡉⠋⠉⣽⠉⠉⢹⡩⡉⠟⠁⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠣⠁⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⡀
⠀⠀⠉⣀⠂⡛⠋⠀⠀⠈⠋⢁⠐⢺⠁⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠀⠀⠈⠊⠢⡀⠀⠊⠀⠀⠈⠴
⠀⠀⢸⡀⠃⢀⡄⠀⠀⠀⡄⠡⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠈⠀⠄⠡⠓⠀⠀⠀⠎⠀⡀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡋⠀⠀⠀⠠⠎⡀⠀⠀⠀⠀⡀⠤⡀⠀⠀⠠⠦⠤⠤⠄⠤⠤⠤⠤⠤⢤⣄⡀
⠀⠀⢰⡠⠊⢀⢤⠀⠀⡀⡀⠈⠕⠁⠀⠀⠀⠀⠠⡄⢀⡄⠈⠲⠁⠀⠀⠣⣀⠙⠥⠂⠀⠒⢠⡈⠀⠈⠀⠀⠢⣤⣤⣤⡤⢤⠤⡤⡤⢤⣬⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠶⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⢻⣟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡦
⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠀⠀⠀⡠⢟⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡦⠀⠀⠀⢠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠒⢲⡴⡖⠂⠀⠀⠀⢠⡾⣿⣆⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⢞⡓⠀⠀⠀⢛⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠋⠁⠈⠀⠀⠀⠐⠋⣴⣿⣎⠓⠄⠀⠀⠀⣼⣿⣿⡄⠀⠀⠰⢊⡟⠆⠀⢀⣾⣿⣿⡄⠀⠀⠀⢠⣷⠀⠀⠀⠀⠴⠾⣿⣿⡿⠶⠂
⠀⠀⠀⠀⠀⠀⢠⡤⣤⠴⠀⠂⠀⢠⠞⢩⣿⡻⢦⡀⠀⠀⣸⣿⣿⣿⣷⡀⠀⠰⢋⡝⠦⠀⠀⣠⣿⣧⡀⠀⠀⠀⣾⢹⡆⠀⠀⢀⣠⣾⣿⣿⣷⡶⠄
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡴⢻⣿⠷⣄⠉⠀⢰⣿⣿⣿⣿⣿⣧⠀⠔⢁⡍⠶⠀⣼⢿⣿⡿⠿⠂⠀⣸⣟⣸⣿⠀⠀⠀⣀⣼⣿⣿⣦⣤⡤
⠀⠀⠀⢀⢀⡀⠀⡀⡀⠀⠀⠀⠠⠈⣠⠟⡿⢧⠈⠓⢀⣾⣾⣷⣿⣿⣿⣿⣇⠠⠞⡻⠄⠀⢀⣼⣿⣷⡀⠀⢀⣿⣻⢸⣿⣇⠐⠛⠛⢻⣿⣿⣷⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠊⠁⠀⡇⠀⠑⠀⠾⠿⠿⠿⢟⠿⠿⠿⠿⠀⠀⡇⠀⠀⣾⣿⣿⣿⡿⠄⣼⣿⣿⣼⣿⣿⡄⣀⣴⡿⢿⡿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠺⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠂⠐⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣀⢀⠀⠀⠒⠤⠀⠀⠀⠀⠁⣀⡀⡤⠐⢒⠂⠐⠛⠀⠁⠀⠀⠠⡀
⠀⠀⠀⠔⠰⠂⠰⠆⠀⢀⠠⠛⠀⢀⠀⠀⠢⠀⠀⠀⠀⠁⠈⠀⠀⠈⠍⠀⠈⠄⡀⠀⠀⠠⠨⠀⠀⠰⠀⠀⠠⠀⠀⠈⠃
⠀⠀⠀⠄⠠⠀⠀⡀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠔⠀⠀⠠⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠀⠀⠀⠐
⠀⠀⠂⡀⠀⠀⠈⢅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⣩⠄⠈⢀
⠀⠀⠀⠀⠀⠘⢒⠘⠃⠀⠀⠃⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡉⢀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⣁⡄
⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠔⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠋⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠥⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣨⣭⣶⡽⣢⡴⢂⣼⣤⣷⣾⡖⠴⣂⣤⡴⣾⣤⣄⣥⢆⢦⣴⣾⣽⣾⣽⠟⣡⣿⣿⣿⡿⡡⢾⣛⣿⣹⣆⡔⣡⣶⣶⣲⢾⣤⣤⢑⣴⡄⣠⡄
⠀⠀⠿⠫⡜⢫⡶⡽⠛⠛⠻⠫⠄⠺⠿⠛⠿⠿⢹⡟⢣⣿⠾⠟⠛⠛⠉⠁⠙⠉⠐⠪⠍⡿⠷⠿⡿⠉⡟⠋⠜⢿⠌⠲⠜⡞⢿⠏⠾⣿⠵⠟⠛⠉⠉
⠀⠘⠀⠄⠀⠀⠂⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⠀⠄⠁⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠑
⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⢸⠀⠀⠀⠀⢁⠀⠀⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠰⠄⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡠⠘⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢛⣷⡀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣠⣼⡿⠀⠀⠀⠀⡀⣌⠀⠖⠙⡚⢹⠆⠆⣂⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠀⠀⠀⠀⠀⠁⠙⠂⠁⠀⠙⠧⢤⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⡿⣿⣻⡀⠀⠀⠀⢸⢤⣄⣴⠖⠚⢲⣄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢸⣿⣟⣷⡂⠀⠀⠀⠀⠈⠉⠿⠏⠽⠈⠻⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⣿⢾⣏⣿⣛⠀⠀⠀⢸⣽⠙⠉⡿⢻⠛⠛⠋⠛⡁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⣿⡃⠀⠀⠘⣫⢼⣿⣿⣿⣿⣿⠈⠉⠁⠀⠀⠀⠾⠀⠀⢀⡇⠀⠆
⠀⠀⠀⠀⢰⣷⣷⣿⣿⣧⣿⣻⡄⠀⠠⠞⢻⣿⠏⠀⠻⣿⠀⢴⠀⠉⡂⠀⡀⣠⣤⡿⠀⠐
⠀⠀⠀⠀⣝⣷⣿⣿⠏⠿⣽⣯⣴⡀⠀⠀⢸⣿⢠⣴⢠⣿⡃⠸⠀⠂⠹⠐⠟⠹⠔⡀⠴
⠀⠀⠀⠀⢽⡿⠛⢿⡶⣸⢿⡟⣧⢀⡙⠃⠈⣉⣉⣹⣉⣛⢃⡀⠀⠀⠀⠀⠠⠘⡆⠏⠀⠚⡉⣠
⠀⠀⠀⠀⣆⡀⠀⡀⠀⠋⠀⢰⡿⢼⠇⢰⡎⢁⣿⣿⣧⣉⣉⣅⣠⠆⠄⢀⢀⠠⠃⢃⣶⠸⢇⣼⡀
⠀⠀⠀⠀⠉⠒⠲⠇⠀⠐⠒⠛⠓⠚⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠛⠒⠚⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢄⠀⢀⣀⡤⠖⡀⠒⠁
⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⣀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡡⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠀⠀⡀⠀⢄⠀⢄⠀⠀⠀⠀⠖⠀⠀⠀⠀⢀⠀⠀⢀⠓⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠐⠖⠂⠀⠑⠄⠁⠀⠁⡲⠁⠀⠘⠂⠀⡀⠐⠁⠀⠄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠂⢀⠀⠀⠀⠀⠀⣴⡃⠀⠀⠀⠀⡤⠉⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣤⣤⣴⠹⣇⠀⠀⠀⡴⠀⢴⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠞⠛⠛⠛⠛⠮⣇⠀⢰⣧⡐⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣀⠀⣈⢷⣶⠘⢿⠟⠁⠀⠀⠀⠀⠰⠀⠀⠁⠀⠸⠀⠀⠀⠙⠂⠄
⠀⢄⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠘⠁⠀⠀⠀⠰⣿⣾⡿⣛⣿⣷⡿⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⠄⠚⠀⠀⠄
⠀⠈⠳⣄⠀⠀⣰⠀⠀⠗⠀⠀⠀⠀⡤⡇⠀⢰⣧⣴⡞⢁⣾⡇⠀⠀⠀⠀⠀⣠⠀⠀⡀⡤⠜⠃⠀⠀⠀⠈⣀⠀⢀
⠀⠀⠀⠘⠙⢢⢙⣆⠀⠈⠀⠀⠀⠀⠀⠱⡀⠀⡙⠙⢿⣾⢿⡇⠀⠀⠀⢠⢀⣡⣀⣈⢁⡀⠦⠀⠀⣠⣴⠶⡥⠰⠋
⠠⠄⠀⢠⠘⢦⡾⠟⠋⠀⠘⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢸⡇⢀⣤⠾⢛⠉⠉⠉⢉⠁⣀⣴⣦⣾⡿⢧⠆⣰⠁
⠀⠀⠀⠀⠢⠙⠇⠐⠀⠀⠀⡄⠀⠀⠒⠀⢈⠀⣄⣰⣄⠀⠸⡿⠟⠦⣀⠀⠐⠟⠀⠘⠛⠛⠁⣿⣿⡗⠀⢀⡟⠤⠤
⠀⠀⠀⠠⠀⠓⠰⣼⡀⠀⠀⠄⠠⠀⣴⣷⠄⠀⠿⠿⠋⣷⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠂⠀⠘⠁
⠀⠀⠀⠐⠀⠦⡒⣿⣷⣄⠀⡀⣤⣾⠟⠁⠀⠀⢠⣀⣸⣮⠿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠈⠙⣟⡛⢶⡅⠙⠃⠠⠤⠂⠀⠈⠛⠛⢹⣦⣠⣷⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠲⣤⠀⠀⠀⠀⠀⢡⡟⠛⠁⠀⠀⠀⠿⠀⣀⡀⠀⠀⠱⢆⠀⠀⠀⠀⢀⣠⠔
⠀⠀⠀⠀⠀⠀⣬⠟⡁⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠢⡤⠀⠩⠅⠀⠀⠀⢀⠤⠐
⠀⣠⡄⠀⠀⣴⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⢠⡞⠁⠀⢀⡀⠀⠀⣀⠀⢰⡿⠋⠆⣵⠶⠖⠂⡰⠤⠀⠀⠀⢠⠞⠁
⠈⠀⠀⠀⣰⠋⠐⠀⠀⡀⠀⠈⠀⠁⠀⠀⠀⠀⠀⣽⠁⠀⢾⣶⣿⠿⣶⣿⣠⠋⠠⠚⢀⠥⠀⠐⠉⠀⠀⠀⢀⣡⣤⡀
⠀⠀⠀⢠⠁⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡇⢀⣠⠴⠿⢛⠀⠒⠻⣇⠐⠀⠀⠀⢀⣀⣩⣧⠀⡊
⠀⠀⠀⣸⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⢰⠾⠋⡑⢠⡀⣬⢑⣦⠀⠀⠙⢦⠀⠘⠀⠀⣠⠀⠈⠉
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠀⠘⢿⣾⠇⠻⠿⠏⠀⠀⠀⠀⡀⠈⠲⢿⣱⣄⡀
⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠛⠓⠲⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠀⠀⠀⢰⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣸⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⣇⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠙⠟⣿⣿⣿⣟⢀⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡬⣿⢿⡷⢶⣿⡿⢿⣼⢟⣻⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⠉⠾⣅⢹⣿⣷⣾⣿⣫⣧⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⢀⣤⠆⣸⡯⠶⠋⢿⢾⣿⣿⣷⣿⡿⣿⣿⢶⡿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⣿⣀⣿⠗⠚⡧⣸⣼⣿⣿⣿⡛⣿⠓⢈⠘⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣤⣶⣿⣿⠟⢐⣟⢻⠟⣹⣿⣿⣿⣿⣶⢶⣾⣤⣶⡧⣤⠄
⠀⠀⠀⠀⠀⠀⠼⢹⣿⣿⣿⣿⣿⠟⢁⡴⢢⣥⣴⡇⡋⣹⣿⣿⣿⣿⠿⠚⢭⠉⠉⠛
⠀⣴⣾⣿⣦⡀⡀⠀⠈⣿⣿⣿⣿⠀⢜⢻⢮⣌⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻⠛⠁
⠀⢿⣿⣿⣿⡟⢻⣄⣶⢾⣿⣿⣿⣗⣺⢭⣛⠹⠏⢟⣿⣿⣿⣿⡏⢰⠁⣸⠉⣠⣶⣶⣄⢀⣤⣄⣀
⠀⠀⠉⢻⣿⣿⣾⣿⡇⣾⣿⣿⣿⡟⠄⢸⣭⣷⣿⣿⣿⣿⣿⣿⣿⣿⣯⠟⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠘⣿⣿⣿⡟⠀⣿⣿⣿⣿⣧⡴⢻⢛⡿⢾⣿⣿⠀⣾⠀⢸⣿⣿⢟⡂⠛⠛⠛⠛⢉⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣷⣾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡕⠞⠁⡔⠛⠛⣻⣿⡇⠀⣹⣿⣿⣿⡏
⠀⠀⠀⠰⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⡟⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⡇⠀⢀⠐⠿⠞⢛⠉⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢹⣿⣿⠀⢸⣿⣿⣿⡿⠟⢿⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣦⣼⣿⣧⣴⢿⡆⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⠄⠀⣿⠋⣿⡇⣎⡽⡿⠛⣹⠋⠉⣿⡟⢻⠛⠿⢿⣿⣿⠉⠀⡜⠉⠈⠘⡆⠀⢸⣿⡟⣿⣧
⠀⠀⠀⠀⣿⢹⣿⠀⣴⣿⠀⣼⡟⠃⢀⠀⠀⢻⠠⠀⡧⠄⠈⣎⠀⠀⢻⠉⠀⠀⡇⠀⠀⠀⠃⠀⠸⣿⠇⣿⣿⠆
⠀⠀⠀⠀⠛⢻⣿⣦⠈⠿⠄⠈⠿⠄⠀⠀⠀⠺⠀⠀⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠶⠟⠠⣴⡿⠆

⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣶⣾⡿
⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⡟
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡷
⣀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿
⣿⣿⣦⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⣿⠁⣹⣿⣿⣿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⣿⣿⣿⣧⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣿⣟⣿⣿⣿⣿⣿⣷⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣻
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣻⣿⣅⣤⣿⡿⠋⠙⢿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⢠⡽⠋⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠉⠈⢷⣆⣠⢤⣴⡿⢤⠶⡠⠎⢹⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠏⠀⠀⠀⣠⠟⠁⠀⠹⣀⣿⡷⢦⣠⡤⠆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣷⣶⡶⠛⢿⣇⡴⠶⠶⠒⠚⢛⣁⠌⣱⣄⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⠟⠁⠀⠀⢿⡀⠀⠀⠀⢀⣉⣉⡀⠀⠈⡛⢿⡃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⢹⣿⣿⣿⣿
⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣰⣿⠟⠁⠀⠀⠀⠀⠈⡉⠻⣟⣛⣉⠭⠔⠃⠶⠾⡀⢀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣾⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣁⠀⢠⣀⣀⣀⣾⣾⣷⣶⣾⣿⣷⢶⠶⢓⣆⣠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠛⠛⠉⡍⠰⠿⢿⣿⣿⣿⣿⣇⠀⣀⣽⣥⡾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠶⣾⣒⡤⡴⠷⢼⣤⣿⣗⢸⡍⠉⠑⣛⣻⣯⣽⠇⢀⠉⠉⣕⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢳⣿⣾⡿⣽⡇⠀⠀⣾⣷⠀⠍⠚⠳⢤⠀⠈⢉⡏⢠⣦⣸⣾⣧⢯⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⡅⠈⠙⣿⠄⡄⣴⠀⠠⡅⣧⢘⣷⣾⣿⣿⣿⣿⣼⣿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠟
⣿⣻⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣽⣿⣿⣿⣿⢠⡄⠀⢷⣿⣷⡄⠀⢈⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠚
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⣿⣿⣭⣿⣿⣿⣿⣿⣿⡿⣷⡀⠈⠛⠿⢊⠛⠺⢿⣿⣿⡿⣿⣿⣿⢿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾
⣿⣿⣰⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠷⢶⣿⣟⣹⣿⣿⣽⣻⣿⣿⣷⣗⠻⠈⢹⢒⣼⣛⣿⣽⣿⠧⣶⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⡟⢿⣽⠇⠀⠀⠀⠠⠴⠞⠛⠟⠻⠛⠛⠛⠛⠻⠖⠛⠛⠛⠃⠙⠛⠛⠛⠘⠓⠚⠋⠉⠈⠛⠻⠛⠛⠻⠟⠙⠻⠛⠻⠿⠿⣿⣧⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣷⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠠⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠐⣻⣿⡷⠄
⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠺⡷⢾⢿⣻⡂⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠀⣻⣿⣿⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣤
⠀⠀⢰⣿⣿⣿⣷⡀⠀⠀⠰⣿⣿⣿⡿⢿⣿⣄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢘⣿⣿⣿⡗⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⠁
⠀⠀⠛⠛⢻⡟⠻⠇⠀⠀⠛⠛⠿⢻⠟⠙⠛⠛⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⠛⢿⠛⠓⠀⠀⠀⠐⠛⠛⠻⡿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠰⠀⠀⠀⠀⢠⣦⠀⢰⠋⠙⡇⠈⣇⣤⡄⡤⣤⠤⠄⢀⠄⢀⣤
⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠄⠠⠓⠀⠁⠀⠉⠑⡋⠁⢁⣀⠜⣗⠐⠀⠉
⠀⠠⠖⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⣴⣶⠰⡇⠀⠀⠘⡖⣶⠱⣸⢼⣡⠒⠖⣤⣲⠈⢳⡄⣵⣾⠄⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠲⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠀⠈⠈⠀⠀⠐⠑⠀⠀⡀⠉⠉⠉⢀⠀⠀⠉⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣒⣒⡒⠎⠀⢠⡁⡁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⢴⡄⢠⡦⠀⢀⠐⡀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⢡⠀⠘⠠⠀⠀⠀⠐⠄⠀⠊⠀⠀⠐⠴⡦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠡⠈⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠐
⠀⠀⢰⣴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣀⠤⠤⠤⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣥⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⢰⡆⢀⡠⢲⠀⠠⠁⠠⠀⠁
⠀⠀⠀⠀⠀⠠⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⠞⠱⡎⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠃⢡⡉⠀⠂⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣕⡽⡁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠁⠀⠁⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⢠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁
⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⢄⡀
⠀⠀⠀⠀⠀⠻⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢃⣀⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⣶⣰⣷
⠀⠀⠀⠀⠀⠀⠈⠢⠹⢦⡀⠀⠀⠀⠀⠀⠀⠰⣿⠋⠛⠉⢻⠆⠀⠀⢀⡀⠀⠀⠰⣟⠉⡉⢉⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣀⡀⣀⣀⣀⣤⣠⣀
⠀⠀⠀⠀⠀⠀⣤⠀⣨⡾⠟⢦⡀⠀⠀⠀⠀⠀⢰⣼⡇⣶⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⡟⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡿⠻⠿⣿⣿⣿⣿⣿⢿⣿⡻⣽⣿⣳
⠀⠀⠀⠀⠀⠰⣿⣿⣷⣅⣀⠈⠙⢦⡀⠀⠀⠀⠸⠋⠃⠛⠀⠀⠀⣠⠟⠛⢦⡄⠀⣀⣁⣥⣤⣦⠀⠀⠀⠀⠀⠀⣾⡁⢰⡆⠀⠈⠙⠙⢻⡟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢈⡙⠿⢸⡿⠀⠀⢈⣿⣦⡀⠐⠟⠾⠛⠛⠻⡧⠚⠁⢠⣆⠀⠈⠓⢿⠀⠀⠀⢰⠀⠀⠀⠀⠀⣻⢹⡇⠘⠃⠐⢛⡀⠀⢸⡃⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣃⠀⠀⠀⠀⢰⣿⠙⡏⢀⣀⣀⣀⣀⣀⣀⣶⣆⣸⣧⣠⣶⣦⣼⣶⠶⠶⠶⠆⠀⠀⠀⣰⠃⠈⢠⠀⣤⢸⣿⠀⠀⢸⡇⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣮⠁⠀⠀⠀⠸⠏⠀⠀⠈⠉⠉⠉⠀⠈⡿⠉⠉⠁⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⢰⡏⣄⣠⢸⡇⣿⢸⣿⠀⠀⢸⡇⠜⠉⠥
⠀⠀⠀⠀⠀⢀⡀⣿⣿⢀⠀⠀⠀⢲⠦⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡶⡆⠀⠋⠁⣿⣿⢸⡇⠟⠈⠁⠀⠀⢸⡇⢠⣀
⠀⠀⠀⠀⠀⢸⠶⠿⡿⢾⡏⠀⠀⣸⢹⡇⡀⠀⠀⠀⣠⠀⠀⠂⠀⢰⣿⢿⣷⡄⠀⣀⠀⣰⡆⠀⡟⠉⠁⢲⠆⡿⠋⠈⠄⠀⢺⡇⠀⠀⢸⡇⢘⣿
⠀⠀⠀⠀⠀⠈⠒⠌⡳⠦⠀⠀⠀⣾⣸⠃⠇⠀⠀⠀⣿⠇⠀⠀⠀⠸⣿⣼⣟⠁⠀⡏⠀⢿⡇⠀⣷⣶⢻⣿⠀⢀⠀⡶⡇⠀⢸⠟⠀⠀⢸⡇⠰⠿
⠀⠀⠀⠀⠀⠀⠈⠛⠤⠀⠀⠀⠀⡀⢁⠀⠃⠀⠀⠀⠿⠆⢀⡀⣠⡖⠉⣍⠈⠓⡤⣡⡀⠈⠁⠀⠼⢿⣦⢻⡌⣿⠀⡯⠟⠀⠻⣏⠁⠀⠈⣇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡂⠀⠀⠀⢹⡯⠀⢀⣴⠒⠚⠛⠛⣻⡿⠷⠶⠐⠛⠒⠒⠓⣘⡿⠁⠀⠀⠀⢸⡏⢻⡄⢋⠀⠀⠀⠀⠀⢻⡀⠀⠀⣿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢸⡟⠀⢸⣿⡆⠀⣀⡀⠀⠏⠀⢉⣽⠷⠷⣦⡀⠀⣇⠸⠥⠃⠀⢰⣧⣾⣇⣤⣿⠀⣄⠀⠐⠜⠿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠠⠚⠛⠓⠲⠂⠀⠘⠉⠀⢾⣿⡇⠀⠀⠀⠀⠃⠀⣭⢳⣿⣿⡌⣷⠀⣇⠀⠀⠀⠀⠸⣿⣿⡇⣿⣿⠀⠀⢠⣤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⣴⣿⣶⡷⠸⠿⠃⠀⠀⠀⠀⠱⣤⣿⡆⣭⣭⢸⡿⢶⣇⠀⡇⠀⠀⠀⠉⠙⠃⣿⣿⠀⠀⢸⣿⠄⠀⠀⠈⣯
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⣿⣟⣸⠇⠀⢀⡾⠀⠀⢰⠆⠀⣿⣶⡇⣿⣿⠠⣷⡾⢿⠚⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠛⢤⡠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣫⡇⠀⠀⠙⠉⠀⠀⠀⡈⠀⡇⠀⢸⣶⠀⣯⡻⠇⠉⠉⠁⢻⡕⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠐⠿⠀⠰⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⠀⠀⢀⢼⠟⢻⡇⠀⠘⠟⠀⢿⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡽⠧⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈




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

⠀⠀⠀⠀⢀⣀⣀⣴⣾⣿⣦⣤⣴⣾⣷⠦⢤⡴⢾⣿⠶⣤⣴⣾⠭⠽⠦⠴⠾⠿⠶⠤⠶⣿⣷⣦⣤⣀⣀⡀
⠀⠀⢠⡞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡶⣝⢦
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠉⡎⣿
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡿⠴⢃⣿
⠀⠀⠀⠈⠙⢲⠖⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠼⠗⢺⢹
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣤⣤⡀⢠⡤⠄⢀⡤⢤⠀⣴⠄⠀⣤⢀⡄⠀⡴⢦⡀⢠⠤⠄⣤⢤⡄⠀⠀⠀⢸⢸
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⣿⡼⠃⢸⣿⡀⠸⢦⡤⠀⠯⠅⠠⡟⠋⡷⠠⣟⣻⠇⢼⣓⡀⡷⢺⡇⠀⠀⠀⢸⠘
⠀⠀⠀⠀⢸⠃⠀⠀⡤⠤⠤⠤⡤⠤⠤⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⠀⠀⣾⠀⠀⠀⡧⠭⠭⢤⡧⠬⣙⣀⣇⣒⣒⣂⣇⡐⠒⠆⢀⡏⠩⠭⠍⣹⢉⣭⡛⡳⢒⣒⠒⡆⠀⢸⢸
⠀⠀⠀⠀⣿⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⢠⠃⠀⠀⢈⠇⠉⠉⠉⢹⠉⠉⠉⠑⡟⠒⠒⠒⡷⠤⠭⢴⠇⠀⢸⢸
⠀⠀⠀⠀⡇⠀⠀⢸⠗⠶⠦⣼⠤⢤⣀⣸⣀⣀⡀⣸⠀⠀⠀⠀⡌⠀⠀⠀⢸⠀⠀⠀⢰⠃⠀⠀⢸⠀⠀⣾⣿
⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠉⡏⠉⠉⠉⢹⠋⠙⠓⠒⡞⠒⠒⠒⡾⠤⠤⠤⡏⠀⠀⡏⡿
⠀⠀⠀⢸⡇⠀⠀⣯⣀⣀⣰⠁⠀⠀⢸⠀⠀⠀⢠⠃⠀⠀⠀⡸⠀⠀⠀⢀⠇⠀⠀⠀⡇⠀⠀⠀⡇⠀⢰⢿⡇
⠀⠀⠀⣼⠀⠀⢠⡇⠀⠀⡝⠉⠉⠉⡟⠓⠒⠒⠺⠒⠒⠤⢤⡧⠤⠤⠤⣼⣀⣀⣀⣠⡃⠀⠀⢀⡇⠀⣾⣼
⠀⠀⢠⡇⠀⠀⣸⠀⠀⢠⠇⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⡸⠉⠉⠉⢹⠁⠀⣇⡿
⠀⠀⡾⠀⠀⢠⡏⠉⠉⢹⠉⠉⠁⣺⠒⠒⠒⢲⠇⠤⠤⠤⡧⢀⣀⣀⣰⣁⣀⣀⢀⡇⠀⠀⠀⡎⠀⢸⣿⡇
⠀⢸⠃⠀⠀⣼⠀⠀⠀⡇⠀⠀⠀⡏⠀⠀⠀⣸⠀⠀⠀⢰⠁⠀⠀⠀⣸⠀⠈⠉⢹⠉⠉⠉⢺⠇⠀⣼⣿⠇
⠀⡟⠀⠀⢠⡏⠉⠉⢹⠓⠒⠒⢲⠓⠒⠶⢤⡧⠤⠤⢄⣼⣀⣀⣀⢀⡇⠀⠀⠀⡞⠀⠀⠀⣼⠀⠀⣿⣿
⢸⠃⠀⠀⡼⠀⠀⠀⡞⠀⠀⠀⡞⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢹⠉⠉⠉⢚⠗⠒⠒⢲⡇⠀⢰⣿⡇
⡿⠀⠀⠐⠧⠤⢤⣴⣇⣀⣀⣰⣃⣀⠀⢀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡼⠀⠀⠀⣸⠀⠀⠀⣸⠁⠀⣼⣿⠃
⠙⡖⠦⠤⣤⣀⣀⣀⡀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠶⠧⠤⠤⠤⠧⠤⠤⠖⢁⡴⠊⣽⡟
⠀⠛⠿⠿⢶⣶⣶⣬⣭⣭⣭⣭⣍⣉⣉⣉⣉⣛⣓⣒⣶⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠖⠋⠀⣰⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣦⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⢿⢦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⣸⠳⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⣀⣴⣄⣾⣶⡴⣿⣿⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⡀⢀⣸⣿⢹⣿⣿⣿⣾⡅⣿⣷⠿⢿⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠓⠖⠒⠒⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⡟⣿⣿⣯⣿⠻⣿⣿⣷⣶⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣛⣛⣷⣿⣿⣯⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢳⣿⣿⣿⡿⢷⣹⣿⣿⡙⢿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠥⠤⠾⣿⣿⣿⣟⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣼⣟⣿⣿⣿⣿⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣃⣀⣉⣉⣉⣤⣤⣭⣭⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣏⣩⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠷⠶⠶⠶⠖⠒⠒⠒⢒⡒⣒⣒⠒⠒⢻⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⢷⣄⣿⠛⢻⣿⣷⡿⡿⠿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⢉⣩⣯⣽⢿⠻⣿⣭⣭⣭⣭⣭⣥⣤⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠘⣿⣄⡿⠐⠋⠞
⠀⠀⠀⠀⠀⠀⠉⠉⣿⡶⠲⢶⡾⣡⣿⣦⠘⢿⣿⣶⠶⠶⠶⠦⠀⣾⡇⠀⢰⣿⡆⠀⣸⣿⡀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣩⣿⡽⣣⣿⣿⣷⡈⢿⣯⣭⣍⣉⣉⣛⣿⡇⠀⢸⣿⣯⠀⣿⣿⡇⢸⣿⡗⠀⠀⢸⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⢿⡟⣰⣿⡟⣉⠉⣷⡈⣿⣛⣲⣶⣒⣒⣿⡇⠀⢸⣿⣿⠀⣿⣿⡇⠘⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣴⣆⣰⣄⣿⣿⣾⣧⣿⡟⣿⣽⣷⣿⡇⣿⠤⠤⠤⠤⢤⣿⠇⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣧⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣸⣿⣿⣿⣯⠿⣿⣿⣿⣿⢹⣿⡇⣿⣿⡏⣿⠄⣿⢭⣭⠛⠋⢙⣿⠀⠀⢸⣿⣿⠀⣿⣿⡇⠀⣿⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⣯⣿⣿⣶⣾⣷⣿⣿⠧⣿⣾⣿⣇⣿⣿⢧⣿⠂⣿⣿⣿⣛⠛⢻⣿⠀⠀⠸⠿⠿⠀⠿⠿⠇⠀⠿⠟⠀⠀⢸⣇⠀⠀⠀⢸⡇
⠀⣉⣿⡿⠿⣬⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣷⣿⣒⡒⠶⠖⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣀⢀⣾⡟
⠐⠛⠻⠿⠿⠿⠿⣿⣿⣟⣛⡒⢒⡀⢠⣤⣴⣿⣿⣿⣶⣿⣿⣿⣿⣿⣾⣿⣿⣶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣾⣶⡶⠿⠶⠾
⣀⣀⣢⣤⡄⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠃⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡀⠐⠒⠂⠒⠈⠉⠉⠉⠁
⠀⠀⠑⢀⠶⠊⠀⠀⠀⠀⠀⣤⣀⡁⡀⢀⠀⢤⣍⣀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣀⣨⠽⠵⠀⠀⠀⠀⠀⠲⠂⠠⠄⠤⠄⠀⠄⣀⡀⠀⠤⠰⠂⠉⠉⠉⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠤⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠒⠒⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⣀⣀⣀⣀⣤⣄⣠⣤⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠿⠻⣿⠟⢻⠿⢿⠿⢻⢿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣶⣿⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡄⢀⣀⣴⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠻⠟⠙⠁⠀⠀⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⢘⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣋⣤⣷⣄⣈⣉⣃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠙⣤⣠⠌⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⠀⠔⠀⠐⠧
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠈⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣾⡿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⡛⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠴⢆⣾⡿⣫⡌⢻⣦⡀⠀⠀⠀⠀⣸⣶⡙⠻⡇⠀⠀⢀⣴⣦⠀⢀⠀⠀⢰⣿⡀⢀⡀⢀⣴⣿⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠍⠀⣮⡀⣥⠙⢿⣶⡆⠀⠀⢸⣿⡇⠀⠅⢨⠀⢸⣿⣿⠀⠘⠓⠀⣿⣿⡇⠈⠉⠈⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⡿⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠙⠃⠻⠁⠨⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠈⢩⡅⠀⠀⢀⣶⠀⡀⢠⣿⢺⣇
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢸⡜⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡃⠀⠀⠀⣿⣿⠀⠀⠀⢀⣾⡟⠀⠒⢸⣿⠀⠉⢸⣷⢸
⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢱⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢸⠀⠀⠀⢸⣿⣿⠀⠀⢠⠀⢿⣿⡇⠀⠰⣆⢹⣿⠀⠀⠀⠈⢻⣏⠀⠀⢸⣿⠀⠀⢸⣿⢸
⠀⠀⠀⠀⠁⠀⠀⠀⡆⠀⠸⣿⣿⡇⢸⠀⠀⠀⠀⢸⣿⡇⢘⠁⠈⠀⢸⣿⣿⠀⠀⠘⠓⢸⣿⡆⠀⠀⠀⠀⣿⠄⠀⠀⠀⢸⡏⠀⠀⢈⣿⠀⠀⡠⣿⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⡿⢿⡇⣸⠀⠀⠀⢀⣼⣿⡇⠈⠀⢘⠀⢸⣿⣿⡄⠀⠀⠀⠈⠏⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢠⡰⠆⣻⠀⡆⠁⢂⣸
⠀⠀⠀⠀⣄⠀⠀⠀⢀⡀⠐⠇⠀⠀⢹⠀⠀⠀⠀⠀⠉⠉⢰⠀⢠⠀⠉⠉⠁⠀⢀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠆⠂⢸⣿⢸⡇⠀⠹⠂⣿⡄⢈⠱⠂
⠀⢰⡆⡆⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⢈⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢸⠀⠀⢘⢁⠰⠿⣦⡄⠀⠉⠙⠃
⠀⠀⠁⠀⣠⠞⣻⣄⠅⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⣿⠀⢘⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⣿⣿⠀⠀⠐⠚⠀⠐⠂
⠀⠀⢤⣴⠗⣫⠭⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣄⠀⠀⠀⠀⣿⠀⠠⠀⠀⠀⠀⠀⣸⣿⣷⠀⠂⠀⠀⠉⠛⠊
⠀⠀⠀⠀⠉⠓⠯⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠒⠿⠀⢠⡄⠀⠐⠀⠉⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠃⠀⠓⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⡇⠀⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠧⠼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣀⠤⢤⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⡤⠚⠁⠀⠀⠘⠢⣄⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢉⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣠⡔⠋⠀⠀⢀⡴⣺⠿⢭⠿⢿⣦⡀⠀⠀⠙⣦⡈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⢹⡆⠀⠀⢠⣯⡟⠁⠀⢐⠀⠀⢹⣿⡀⠀⠠⡇⡿⠦⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⡇⠀⠀⠸⡿⡇⠉⠉⠙⠀⠉⣨⣿⠃⠀⢸⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣩⢼⠀⡇⠀⠀⠀⠙⠮⣒⡶⠶⠶⣚⡿⠋⠀⠀⢸⢸⣯⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣉⠤⠒⠉⠁⢸⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⢸⢸⠀⠈⠓⠦⣍⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣋⡥⠖⠋⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠀⠀⠀⣀⡀⠙⠳⢬⣙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⢺⣥⢴⡛⡇⠀⠀⣠⡶⠿⢷⣦⡀⢸⡄⣇⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⡿⣿⠀⣰⡿⠻⠛⢷⡄⠀⠈⢳⢦⣉⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣧⠀⠐⣿⠀⠲⠆⣿⡇⢸⡇⣿⠀⠀⢠⢞⡿⠋⢹⠉⠻⣿⣦⠀⠀⠀⡇⣿⠀⢿⡇⢙⠃⣻⡗⠀⢀⡏⡟⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡀⠀⠙⠧⣴⣶⠿⠁⠸⡇⢸⠀⠀⡟⣸⠁⠀⢸⠀⠀⠘⣿⣧⠀⢠⡇⡏⠀⠈⠛⠾⠟⠋⠀⠀⣸⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⡇⣿⠤⠤⢼⠤⠤⠄⢿⣿⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⠀⢠⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⡇⡗⠀⠀⢸⠀⠀⠀⢸⣿⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡄⠀⠀⠀⠀⠀⠀⠀⣧⢸⡄⢸⡇⡧⠤⠤⢼⠤⠤⠤⢼⣿⠀⡿⢸⠇⠀⠀⠀⠀⠀⠀⢰⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣴⠒⠒⡖⠛⠛⡆⣿⢸⡇⢸⡇⡇⠀⠀⠸⠀⠀⠀⢸⣿⠀⡇⢸⢰⠖⠒⠶⠒⠲⡆⣼⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣹⡀⠀⠁⠀⠀⡇⢿⢸⡇⢸⡇⡇⠀⠀⢀⡀⠀⠀⢸⣿⢰⡇⢸⢸⠀⠀⠸⠀⢰⣷⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣏⠛⠛⠛⠛⠛⠁⢸⢸⡇⢸⡇⡏⠉⠉⠉⠉⠉⠉⢹⣿⢸⠃⢸⠈⠙⠛⠛⠛⠋⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⡀⠀⠀⠀⠀⠀⢸⠀⡇⢸⡇⡇⠀⠀⠀⡀⠀⠀⢸⡏⢸⠀⢺⠀⠀⠀⠀⠀⢰⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⣯⣋⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣶⠞⠝⢶⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠅⠄⠀⠤⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠠⣏⢘⠃⢱⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠉⡈⠔⠃⠀⠀⠀⠙⢷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⢀⣿⣆⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣾⣏⣿⡀⠀⠉⢱⠀⣠⣤⣤⠠⣾⠿⢽⠿⣿⢿⣿⣷⠄⢠⣄⣀⠀⢰⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⢠⣿⣏⣿⣷⡀⠀⢸⠀⣿⣿⠉⠀⠀⢀⡔⢂⢐⡒⣄⠀⡇⢸⣿⡿⡄⢸⠀⣤⣤⠄⢠⣤⣤⠄⢸
⠀⠀⢰⡿⣟⣯⣿⣯⡁⠀⢸⠀⣿⠋⠰⠀⠀⣮⠛⠛⠛⠛⠋⠀⣧⢸⠟⠀⠇⢸⠀⣿⣃⠀⠘⣇⣋⡆⢸⠀⠀⢀⡆
⠀⠀⠀⣴⣿⣷⣿⢿⣷⡄⣸⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠂⢸⠀⡿⠀⠀⠠⡇⠀⠇⢸⠀⠀⣼⣿
⠀⢀⣾⣿⣿⣏⣿⣿⣿⡄⢿⠀⠋⣶⠐⣠⠖⣯⠀⠀⠀⠀⠀⢠⣿⡎⢠⠆⠀⢘⡀⠃⠀⠀⠀⠃⠀⠠⢼⠀⣸⣿⣿⣧
⣠⣿⣿⣿⣿⣧⣷⣿⣿⣿⣿⠀⠀⢼⠋⢠⠤⠯⠤⠤⠤⠤⠤⠴⣿⡿⠦⣄⣀⣸⣁⣉⣉⣙⣉⣿⣿⣿⣏⣰⣿⣿⣿⣿⣆
⠉⠉⠉⠉⠉⠱⢿⣿⣿⣿⣿⣶⣶⣸⣦⣭⣭⣭⣭⣭⣭⣭⣭⣭⡘⣧⣀⣻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠹⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡿⠿⠛⠉⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣠⡿⠁⠈⢷⡀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠁⠀⢀⡄⠱⡄⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⠟⠀⠀⢸⡇⠈⡇⠀⠀⡗⠀⠀⠀⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣛⠛⠉⠀⠀⠀⢸⡇⠀⣿⠀⢰⠇⡄⠠⡀⠦⡀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⠤⠀⢄⢀⠀⠀⠈⠏⠀⠀⠀⠸⡀⠓⠀⠧⠀⠇⠙⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡼⠁⠀⠀⣸⡇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⢸⠃⠄⠀⡀⠀⠉⠀⠀⢠⠀⠀⠈⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠓⠃⢸⡇⢶⠀⠇⡄⣨⠰⣇⣸⠂⣶⢰⡄⡇⣰⢀⡄⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡴⠙⠃⠾⠈⠸⠇⠙⠂⡹⠻⠎⠛⠘⠸⠇⠉⠉⠧⠀⡽
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣿⣦⡄⠀⠀⠀⢀⣴⣾⣿⣆⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣾⣄
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠙⢿⠇⢹⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠐⠿⠻⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠘⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠳⠾⠛⠃⠀⠀⠀⠀⠸⠿⠿⠿⠿⠀⠀⠀⠀⠐⠛⠻⠛⠓

⠀⠀⠀⠀⠀⢀⣶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⣰⣶⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⠀⣰⣦⠀⣀⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠀⠀⠀⢹⣿⣾⣿⡃
⠀⠀⠀⠀⠐⠛⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠨⠿⣿⡿⠇⠀⠀⠀⠀⠀⠙⢿⠄⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⠀⠀⠿⠿⣿⡿⠿⠆⠀⠀⠀⠀⠛⠻⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠆⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣠
⠀⠀⠀⣀⢀⣷⣀⣀⣰⣿⣿⣿⣿⡆⠀⠀⠀⣾⡏⠀⠈⣿⡆⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠛⠀⠀⠀⢠⣨⣿⣿⣦
⠀⠀⠨⠔⢁⡛⢿⠿⠿⠿⠿⢿⠟⠁⠀⠀⠀⠘⢿⣶⣾⠟⠁⠀⠀⠀⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠈⢿⠋⠛⠛⠁⠀⠀⠀⠀⠘⢿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⢀⠀⡀⣿⣆⡀⠀⣠⣾⣿⣿⣿⡿⠀⠀⠀⠘⠶⣿⡷⠖⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣠⣿⣧⠀⠀⠀⠀⠀⢀⣴⣿⣿⢻⡆
⠀⠸⣐⠂⠘⣿⠿⠿⠿⠿⠿⣿⠟⠁⡀⠀⠀⢀⣴⠿⢷⡄⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀⠀⠀⠀⣾⣿⣿⣿⡞⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠁⠉⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣁⠒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠠⢀⠁⠀⠀⠀⠈⢀⣀⣄⠰⡎⡄
⠀⠀⠀⠀⠀⠀⠀⢔⣤⡄⠀⠀⠀⠀⠀⠀⠀⢇⡨⠂⠀⣼⣠⣿⣸⠃⡇⢻⡁⠸⣆⣽⡖⡈⠒
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⢸⠏⢸⡇⠀⠙⣆⢻⡀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡤⠀⣀⡈⠁⠛⠒⠋⠈⠁⠀⠻⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣠⠂⠀⠀⢠⣟⣀⡾⢡⡾⠋⠀⢀⣶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠃⠘⠇⠀⢀⣾⠃⣶⣀⠀⠀⠀⠘⠀⠀⢀⡀⠀⠀⠀⠀⠀⠴⢃⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠋⠁⠀⠀⠒⠃⠀⠙⠁⠉⠁⠐⣀⠀⠀⠀⠲⣬⠵⠋⠀⠀⠀⠀⠀⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠲⠦⣤⣀⡀⠁⠀⠀⠀⢀⣀⠀⠀⠀⠛⢷⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡼⠚⡁⠀⠀⠀⠀⠸⡅⠀⢠⠄⠀⠀⠉⠙⠛⠻⠿⠖⠛⠃⠀⠈⠀⠀⠃⠀⠀⢨⣔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⡀⠀⠀⠀⢀⡀⠀⠄⠠⠀⠀⠀⠠⠀⠠⠠⠀⠀⣀⠀⠀⣶⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⢻⡇⢠⣄⠀⠈⣿⠀⠀⡄⠀⠀⠑⣤⠀⠂⢰⡆⠀⣽⡇⠀⣿⠀⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⢸⡇⢀⣿⠀⠀⣿⠀⢸⣄⠀⠀⢠⣟⠀⢠⣿⡀⠀⢿⡦⠀⣿⡁⠀⠁⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⢸⣿⡀⠀⢿⡦⠜⠛⠉⠀⠈⠁⣀⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢧⡂
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠀⠀⠀⠸⣧⠀⠉⠀⠀⠠⠤⠤⠤⣀⠀⠂⣀⣼⣥⣤⣄⣀⣀⠀⠀⣿⡇⢠⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⢰⡆⠀⠀⢠⣤⠀⠀⠀⠹⠋⠁⢹⠀⠀⠀⠀⢩⣙⣷⠿⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⢸⡇⠀⠀⠘⠉⠀⠐⠂⢀⣤⣄⢸⠀⠀⢦⡶⢶⡿⣦⠀⠶⢶⡄⢠⣤⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠠⢂⠂⠒⠀⠀⠀⢸⣇⣄⡀⢴⡾⠻⢸⡇⠻⣦⡀⢸⠀⠀⢸⠀⢸⠀⢸⢠⡖⢛⡇⠘⢷⣅⠀⣀⡀⠒⠒⠋
⠀⠀⠀⠈⠀⠀⠀⢀⡁⠀⣶⡞⠀⠀⡀⢸⡏⠈⣧⢸⠁⠀⢸⡇⢀⣀⣿⠸⢦⠞⠺⠂⠘⠃⠚⠊⠛⠛⠓⠸⠦⠿⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠄⠙⠇⠀⠘⢷⣤⡤⠀⢸⡇⠀⠿⠼⠇⠀⠘⠁⣈⣉⣠⡤⠤⠒⢶⡂⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡊⠹⢋⠀⠀⠀⠠⠠⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠁⢠⠄⠐⠀⠀⠙⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⣀⡀⢀⣀⡴⠴⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡄⢸⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⡀⠀⣠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⡿⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠋⠉⢷⣄⠀⠀⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡟⠁⠀⠀⠀⠙⢧⡀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡇⠀⠀⠀⠀⠀⠈⠗⠲⠶⠶⠦⢽⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⢰⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠔⡩⢾⣿⣦⣸⠀⠀⠀⠀⢸⣯⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠊⠀⠀⠈⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠷⠶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢣⠞⠁⠀⣀⣄⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢁⡴⠃⠀⠀⢸⡅⠀⢱⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⢀⣴⣟⣋⣠⡏⠀⠀⠀⠀⠘⢧⣀⠞⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠳⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⠊⠉⠉⢦⠀⠀⠀⠀⢸⠀⠀⢀⠤⣄⠀⠈⠉⠉⢉⡙⠛⠒⠒⠶⠤⢤⣤⣙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡟⠛⢻⠛⠻⡀⠀⠀⠀⡜⠀⢠⠃⠀⢸⠄⠀⠀⡰⠉⢹⡆⠀⠀⠀⢀⡾⠋⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠁⠀⢸⠀⠀⡇⠀⠀⠀⡇⠠⠞⠒⠄⠾⡄⠀⡤⢇⣀⣼⡇⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⡀⠀⠘⡇⠀⢸⠀⠀⡇⠀⠀⢸⠁⠈⠛⠓⠒⠒⠁⠀⠓⠂⠤⠤⠜⠀⣰⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠘⠷⠶⢦⡷⠤⠾⠤⣤⣇⣀⣀⣼⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠤⣤⣤⣤⠋⠉⠉⠛⠻⢷⣄
⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⣏⡀⠀⠀⠀⠫⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢺⣯⣀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⣶⣦⣾⣋⠁⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣠⣴⡾⠿⠟⠛⠛⠛⠛⠻⠿⠿⠷⠶⠶⠶⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⠶⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠐⠒⠂⠀⠰⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠦⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠇⠠⢸⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡗⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢸⠀⠠⠦⠴⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡇⡀⠠⠞⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⠀⠀⠀⢸⡇⠀⠀⣿⠡⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⣿⢏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⢀⣀⣸⣇⠀⠀⠹⠴

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠱⠎⡁⠀⠀⠀⠀⠀⠠⣶⣿⣿⣿⣿⣷⣶⠂⠀⠀⠀⠀⠀⠄⠀⡈⠴⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠁⠙⠻⢦⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠰⠀⡀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠶⠀⠀⠰⠄⢀⠈⠀⠀⠂⠀⠈⢀⠀⠰⠀⠀⠰⠆⠀⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⢰⣷⡀⠀⠘⠀⠀⠀⠃⠀⠁⠘⠀⠀⢰⣷⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠈⠁⠄⠀⠀⢾⣿⡇⠀⠀⠐⠄⢀⢁⠀⠔⠀⠀⠀⣿⣿⡆⠀⠈⠀⠉⠀⠀⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠆⠄⠰⠐⠀⠆⠀⠆⠀⠀⠰⠰⠀⠀⠆⠂⠀⠆⠀⠂⠀⠀⠆⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⡀⠀⠀⡀⢄⠠⢤⠉⠆⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠎⢁⠴⢀⠄⣀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠼⢀⡵⢘⡆⣹⠀⡅⠐⠁⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠘⠀⠇⢰⡁⢞⠱⣅⠘⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠈⠀⠁⠀⠀⢀⠆⠀⠀⢀⠀⠀⡆⢴⣶⠀⡆⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠀⠀⢀⠁⠁⢀⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠈⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠋⠀⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⢄⠀⠉⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠰⠘⠀⠀⠀⠀⠘⠃⢰⣶⠀⠛⠀⠐⠀⠀⠈⠐⠠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠆⠄⠄⠂⠁



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

⠀⠀⠀⠀⠠⢄⡼⢄⡤⠀⢸⠀⢸⠀⢸⡏⠑⠈⡏⠀⠀⢹⠁⠀⢠⠎⢣⠀⣤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠵⣺⢱⠥⠀⢸⠀⢸⠀⢸⡇⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⠈⠀⠀⣀⠀⡜⡝⡄⢀⡀
⠀⠀⠀⢀⣀⠀⠈⠁⠀⠀⢸⠁⢸⠀⢸⡇⠀⠀⡇⠀⠀⢸⠀⠀⢸⠀⢠⠀⠀⠙⢅⢺⣿⠂⡜⠁⠀⣺⠶
⠀⠀⠀⠀⠀⢀⠤⢤⠀⠀⠚⠀⠘⠀⠘⠃⠒⠀⠓⠴⠀⠚⠢⠂⠈⠣⠂⢀⠀⠀⣤⠄⠂⠵⡅⠀⠰⣶⡆
⠀⠀⠀⢠⣧⠏⠀⢸⡇⢀⡠⣄⠀⢠⠖⠦⠀⢀⡤⣄⠀⠀⡴⢫⡇⠀⠀⣾⠀⠀⠀⠀⢀⡤⡄⢀⡄⠐⠓
⠀⠀⠀⢸⡟⠀⠀⢸⣃⡞⣠⠋⢀⡏⠀⠀⢠⡟⡰⠃⢠⣾⠁⣼⣠⢤⠀⣿⢀⣤⠀⣠⡟⡰⠃⢸⣧⠄⣀⣀
⠀⠀⠀⠸⠇⢀⢴⡟⢸⡏⠀⢀⢾⡇⠀⢀⣾⡏⠀⢠⢾⡏⠀⠿⠁⢸⠀⣿⠏⢨⡟⢹⠉⠀⢠⢾⡏
⠀⠀⠀⠀⡔⢁⠞⠀⠀⠳⠔⠋⠀⢧⣀⠞⠈⠳⠔⠁⢸⡇⠀⠀⠀⠸⡄⡿⡰⢻⠃⠈⠣⠔⠁⢸⠇
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠷⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡛⢝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠊⣏⣸⢀⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠉⣹⡍⢹⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠭⠤⠼⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡐⢀⡠⠾⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣴⣿⣷⣦⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢫⠟⠱⢽⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⠔⠀⠀⠀⠄⠂⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢟⠅⠉⠁⠀⠀⠀⠀⠀⠊⠁⡝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢟⠑⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢽⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢋⠅⠁⣂⣀⠀⠀⢀⢀⣠⣄⡀⠐⠀⠈⠀⠀⣨⣙⢝⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⠕⠑⠂⠀⠀⠀⠀⠀⣠⣶⡫⠟⢧⣄⠀⠀⠀⠀⠀⠀⠀⠐⢙⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠕⠋⠀⠀⠀⠀⠀⠰⣤⡾⠫⠊⠀⠀⠑⢝⢷⣄⠀⠀⠀⠒⠀⠀⠀⠙⢿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣷⠁⠀⠀⠈⡀⠀⠉⣩⡾⡡⠊⠀⠀⠀⠀⠀⠂⠑⢝⢷⣅⠀⠀⠀⠀⠀⠀⠀⠗⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠄⠀⠀⣴⣶⡤⠘⠛⢺⡧⠭⠥⠍⠁⠀⠀⠀⠀⢽⡿⠛⠁⠀⣢⣖⠀⠐⠂⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢹⣽⣟⠀⠀⢸⣟⠀⢠⡤⠤⠬⠥⣄⠀⣿⡇⠀⠀⢘⣿⣿⣃⠀⠁⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠆⠠⠴⢺⣿⡷⠀⠀⢹⡷⠀⢸⡅⢠⡄⡂⡇⠀⣾⡇⠀⠀⠰⣿⣿⠂⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⠂⣀⣸⣿⣟⠀⠀⢸⣗⠀⠘⡄⢸⡇⡇⡇⢀⣻⡇⠀⠀⢺⣿⣿⡂⠐⠆⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠄⠀⠠⣾⣿⣯⠀⢀⢸⡯⠀⠠⠀⠸⠇⡈⡧⠄⣽⡇⠀⠠⢴⣿⣿⡇⠀⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡂⠀⠀⠛⠛⠛⠂⠀⢸⡷⠀⢰⡇⠐⠐⠆⡇⠀⣿⡇⠀⠀⠘⠉⠛⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣁⡀⠈⡉⠉⣉⣁⣹⣹⡶⠈⠉⢉⠩⠿⢽⣿⣏⣏⣀⣁⣉⣀⣉⣉⣀⣁⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡛⠀⠀⠀⠀⠰⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⢤⣀⠀⠀⠀⢀⢂⣒⡀⢠⣤⣍⡀⠀⢠⣦⡀⠛⠈⠁⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠚⠁⣴⣿⠇⢀⣿⡆⠀⠀⠀⠂⠂⠀⠈⠙⠛⠃⠀⠀⠙⠟⠀⠀⠀⠀⣠⡀⠣
⠀⠀⠀⠀⠀⠀⣼⡿⠃⣠⣾⠟⣠⡶⣲⢆⣤⡖⡦⣠⡶⣶⢀⣶⢄⣴⣶⣠⣶⠀⣴⡿⣁⠀⣠⡴⡤⢀⡖⣀
⠀⠀⢠⡾⠃⠀⢿⣡⡾⠟⠉⢸⣿⣛⠵⣿⣏⡠⠾⣿⣋⠥⢾⡿⠫⣿⠟⣿⣯⢼⣿⣫⡾⢻⣿⣛⡵⠋⣽⠟⣰⠆
⠀⠀⠀⠙⠒⠊⠉⠁⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⢈⢀⠀⠁⠉⠀⠀⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢒⡘⣐⠂⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⠀⠐⠂⠀⠐⠀⠀⠄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠀⢀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⣀⠄⠀⢠⠀⠀⡠⠀⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠠⠀⠂⢀
⠀⠀⠀⠀⠀⠄⣈⣁⠠⠀⠐⠀⠫⠛⠀⠒⠂⠀⠀⠴⢀⠀⣁⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢘⡃
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠂⠠⠀⠀⢁⠀⠀⡾⠀⠀⠈⠀⠀⠈⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⢲⡄⠀⠀⢤⠄⠀⠐⠀⠁⠀⠀⠀⠄⠀⠂⠈⠀⠘⠀⠀⠁⠀⠀⠂⠀⢀⠀⠂
⠀⠀⠀⡀⠃⠀⠠⠀⠀⡀⠀⠠⠀⠀⠈⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠐
⠀⠀⢠⣻⠀⠀⠀⠀⠀⠁⠀⠑⡂⠀⠁⠀⠘⠀⠀⠀⡆⠀⠉⠀⠀⠀⠀⠀⡀⠢⡀⠴
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⡇⠀⠰⠀⠀⠀⠁⠵⠶⠐⢠⠘⠀⠀⠀⡀⢠⠀⠀⠐⠁⢀⡀
⠀⠀⠀⠀⠀⠁⠀⢃⠉⠁⠀⠈⠀⠌⢀⡀⠁⠀⠀⠐⠁⠀⠀⠀⠊⠀⠀⠰⢤⣀⠈⢁⡤⠞⠁
⠀⠀⠀⠀⠰⠆⠀⠂⠀⠐⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⢀⠄⠐⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠧⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠚⢃⠀⠈⠓⠀⠠⡤⠒⠔⠀⠛⠒⠀⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⣀⢀⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⢀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠒⠛⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠡⠀⠀⠐⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⢖⠒⠒⠂⢒⠐⢒⡶⠆⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠀⢀⠀⠀⠈⠉⠀⠒⠻⣑⠭⠀⢀⣀⠀⡀⠀⠉⡒⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢁⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⠂⠀⠀⣰⠆⠀⠀⠉⠲⣄⠀⠀⢸⡗⠁⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢀⠆⢀⠁⠀⣛⣛⣬⣬⣭⣭⣭⣥⣼⣒⡀⠤⢈⡃⠀⠜⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⠀⠀⢠⡴⢓⣨⣴⠾⠋⠉⠉⠀⠀⠀⠀⠀⠉⡉⠛⠛⠻⣶⣮⣔⡠⡀⠀⠀⠰⠀⠘⠉⠙⢦
⠀⠀⠀⠀⠀⠀⡠⡁⠀⠀⠀⣠⠞⢃⢜⣥⣾⡟⠉⠀⠀⠀⠀⠀⠀⠘⠛⠂⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣦⣕⠄⡀⢀⡀⠄
⠀⠀⠀⠀⢀⠌⠀⠉⠩⡍⠀⢀⢜⣵⣿⣿⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣔⢾⠟⠀⠱⢆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠘⡤⣢⣾⣿⣿⡿⣤⣾⡿⠃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠹⣿⣿⣿⣧⡹⠿⡀
⠀⠀⠀⡜⠀⠀⠀⠀⠀⡰⣱⣿⣿⣿⣿⣵⣿⡍⠀⠀⠀⠻⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢳⡀⠀⢿⣿⣿⣿⣷⡐⡀⢡
⠀⠀⢐⠀⠀⠀⠀⠀⠠⢱⣿⣿⣿⡿⣿⣿⣿⣷⣦⣶⡀⢀⣹⡆⠀⠀⢀⣤⠶⠚⠛⠉⢹⠿⠷⢶⣮⣇⡀⢸⣿⣿⣿⣿⣿⡘⠻
⠀⠀⡎⠀⠀⠠⡆⠀⢣⣿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣷⣿⣿⣧⣴⣾⡥⠄⠀⠀⠀⢠⣠⡤⠤⠈⢻⡟⢿⣿⣿⣿⣿⣿⣿⣧⢹
⠀⢠⠀⠀⠀⠉⠀⣤⢸⣿⣿⣿⣿⣷⣗⢹⣿⣿⣿⣿⣿⣿⡟⠛⢿⠃⢀⡤⢤⣷⡀⠙⠇⠀⠀⠀⢈⣉⡉⠻⠿⢿⣿⣿⣿⣿⡈
⠀⠸⠃⠀⣦⠀⢀⠇⣿⣿⣿⣿⣿⣿⣿⡿⣧⣿⣿⣿⣿⣿⣷⡀⠘⠛⠉⠀⠀⠙⣿⡄⠀⠀⡆⠀⠊⠐⠀⠄⠀⠈⠙⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡌⠁⣿⣿⣿⣿⠿⠿⠫⠟⠐⠈⢺⣿⣿⡏⠻⢿⣿⡇⠢⠄⣀⡴⣁⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠸⠀⡀⠀⠘⠁⢠⢻⣿⣿⣿⣢⡐⠀⠀⣠⣶⣿⣿⣿⠃⠀⠀⠈⠀⠀⡬⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⠀⠀⠀⠀⢘⣿⠃
⠀⠀⡆⢇⠀⠀⠀⠀⠸⣿⣿⡟⡼⠁⠀⠰⠿⡿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢤⣤⠖⠛⠛⠉⠙⠁⠀⠀⠀⠀⠀⣿⣰
⠀⠀⢐⠀⢷⡀⠀⢀⣠⣿⣿⠋⠄⠂⠀⠀⠀⣀⢸⣿⣿⣷⡄⠀⠀⢀⢰⣿⣿⣿⡋⠻⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠈⠆⠀⢙⣴⣿⣿⣿⠟⠀⠀⠀⠉⠀⠀⠀⡀⢻⣇⢍⠟⣷⠀⡈⡀⠐⠤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣴⣾⡿⢟⠁⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⣴⣿⡟⣡⠏⠐⠀⠳⠀⠀⠀⠀⢀⠀⢀⣋⣠⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣾⡟⠟⠀⡉⠀⠁⠀⠀⠀⠀⠀⠰⠀⣠⣾⣿⣿⣿⣧⣴⣿⠃⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⣦⠀⡆⠙⠀⠀⠀⠀⢠⡴⠂⠀⠀⠙⣿⣿⣽⣿⣿⣿⣶⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⢺⠀⠀⠀⠀⠀⠀⠀⠌⠐⠊⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣿⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠀⠒⠒⠒⠒⠒⠒⠂⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⢩⡄⠀⠀⠀⠐⠀⢀⡄⠀⠈⠀⠀⠀⠀⠀⡀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠁⠀⣀⠔⠁⠀⠀⠘⠀⠀⣠⠃⠀⠀⠀⠀⠀⡀⠀⡰⠀⠀⠀⠈⢀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⣉⠀⡄⠀⠀⠀⠀⠀⢠⣇⢀⠀⠀⠀⠀⠀⠀⠴⢡⠄⢰⡀⠰⠋⠁⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠂⠀⠀⠁⢸⡇⠈⠂⠀⣉⠖⠁⠀⠆⠀⠖⠀⠜⠁⢀⠴⠊⠀⠘⢡⠀⠀⠀⢀⠞⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠤⠄⠀⠀⠒⠉⠀⢇⠃⠀⡀⠈⡠⠀⠀⠀⠇⠀⠆⠀⠀⠀⠛⠀⠛⠀⠀⠃⢂⠀⠎⣁⠀⢤⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠄⠀⡀⠀⠀⠂⠀⠀⠸⢸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠚⠁⢀⠖⠀⠸⠈⡄⠀⠠⠎⣁⠀⠀⠀⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣀⠉⠀⠀⠂⠀⠀⡇⡄⠀⠀⠖⠀⢁⡀⠀⠀⠀⣀⠤⠤⠀⠀⡖⠀⠛⡄⢠⠔⠓⠧⠀⠀⣘⣄⣀⣀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⢀⠀⣇⡠⠤⠀⡴⠊⠀⠈⠁⠀⡇⠀⠀⠁⠀⡇⠀⠀⠋⠉⠀⠀⠀⠀⣁⠃⠀⠀⢸⡎⠉⠁⠀⠀⢣
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠀⠀⡠⠖⠉⢡⣼⢠⠏⠀⡀⠅⠀⠀⠰⠀⠁⠀⡇⠀⠀⠀⣀⠃⠀⠀⡀⡇⢀⡀⠄⢀⣿⠀⢸⠀⠁⠇⠀⠂⠀⠀⡀⠆
⠀⠀⠀⠀⠀⡆⢰⠊⠁⠀⠀⠀⠊⠀⠠⠂⠁⠘⡇⠀⡌⠀⠀⠀⠀⠐⠀⢠⠀⡇⠀⠀⠀⠘⠀⠀⠀⢁⡆⠈⠁⠐⠋⠸⠀⠐⠀⢠⠁⢸⢀⠆⠀⡇⢸
⠀⠀⠀⠀⢰⠀⠈⡄⠀⡆⠀⡀⠀⠈⠌⢦⠀⢳⡄⢰⠀⠀⠀⢀⠀⠀⡀⠐⠀⠃⠀⠸⠀⠀⠰⡄⠀⢸⡇⢠⠀⠀⠀⣦⠀⢠⠠⠞⠀⠈⠁⢀⡼⠁⠀⡄
⠀⠀⠀⠀⢸⠀⠀⠧⠀⢰⠀⠸⠀⠀⠘⡄⡠⠚⡇⠈⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢁⠀⢸⡇⢸⢀⠀⠀⣿⠀⠐⠀⠈⠀⠀⠀⠃⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠰⡀⠀⢇⠀⠆⠀⠀⡇⠐⠀⢡⡀⠈⠀⠀⢾⡀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠀⠸⠀⢸⠇⠘⠀⢀⠄⡃⠀⠀⢀⡀⠀⠖⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠸⠀⠀⠀⢃⠀⠈⣄⠀⠀⢀⢇⠀⠁⠀⢹⠆⠀⠀⠸⢧⠀⠀⢀⡚⠂⡈⠀⠀⠛⠀⠀⠀⠐⠂⠀⠂⠁⠀⡇⠀⠀⢸⠃⠰⠀⡆⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡆⠀⠀⠘⡀⠀⠉⠀⡠⢊⠀⠦⠀⠀⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⡄⠚⠂⠘⠀⠣⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢣⢀⡠⠊⠁⠈⠀⠀⠀⠀⠁⠀⠀⢀⣠⠄⠀⠀⠀⠐⠒⣀⠀⠃⣀⠀⡀⠀⢠⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⠁⠀⠀⠞⡄⠀⠐⠉⢀⡀⠀⠀⠁⠴⠂⠉⣤⠀⠀⠴⠀⠀⠉⠀⠀⢁⢴⣥⡰⠀⠀⠀⠀⠋⠲⠀⠈⠉⠁⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢮⠖⡀⡐⠠⢀⠀⢀⡶⠀⠠⢦⠄⠖⠒⡂⠀⠀⠀⠀⠀⠀⢆⠖⠉⠀⠀⠀⢀⡠⢂⠀⢀⠀⢀⠠⠈⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⢀⠼⡁⠀⠀⠀⠀⣀⠂⠐⠂⣀⠴⣁⡠⢀⠆⠀⠀⠀⠄⠀⠈⡀⢈⠀⣸⢆⠀⠀⠔⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠂⠀⠀⠉⠐⠀⠀⠁⠊⠀⢀⡠⠀⡸⠇⠨⣏⠉⡥⢖⡝⠱⣎⠀⠀⣀⠤⠀⠄⠠⠄⠀⠒⠚⠈⠂⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠰⠀⠀⠈⠛⠞⡋⠉⠀⠀⢠⠀⠈⠁⠀⠀⠀⠀⠀⣀⢀⡀⠄⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡰⠇⠇⢼⠀⠀⢠⠀⠀⠠⠀⠀⠂⠀⠈⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡧⣴⡠⠎⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡠⠜⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⠤⠤⠀⠀⠁⠀⠠⠤⠄⠒⠂⠉


⣶⣶⣶⣶⣶⡾⣷⣾⣿⡿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣶⠿⡶⠿⠿⠿⠷⣾⡶⢾⠶⣾⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠛⣋⣉⣡⣦⣴⠍⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡇⠈⠣⡈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇⠸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣯⣯⣽⣿⣿⣾⣷⠄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⣰⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣷⡌⠳⣄
⠀⠀⠀⠀⢀⣼⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣦⡙⠦⡀
⠀⠀⠀⣠⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢦⡀
⠀⠀⣼⡯⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠻⣆⠀⠀⣀⣀⠄
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟⠛⠋⠉⠉⠀⠀⠀⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⡄⣻⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣿⠂⠀⣀⣠⣾⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣶⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⢹⣿⣿⣿⣿⣿⣧⠀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⡿⠿⠿⠿⠿⠷⠤⠤
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⣾⠁
⠀⠀⠀⠈⠉⠛⠛⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⠃
⠀⠀⠀⠀⣀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠿⠿⠿⠆⠀⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠙⠛⠛⠶⠶⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠷⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣥⣀⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣀⣘⣿⣁⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣾⡟⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⡛⣷⠶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠹⣇⠿⢤⣸⠓⠒⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⣤⡀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⣿⣿⡧⠀⠀⠈⢳⣴⡤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⠲⠷⠶⢤⣤⡔⣂⣀⣀⡐⠲⣶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⢸⠀⠀⢰⣿⠀⠯⢭⣭⡭⠭⢽⡿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠿⢿⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡟⡇⠀⢸⣿⠀⠘⡈⠳⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⡇⠀⢸⡯⡇⠀⢸⣿⠀⠀⣏⡙⠛
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⢤⣇⣶⣾⠀⠀⠀⠀⢸⠀⠀⠘⠓⠀⠀⢸⣿⠇⠀⢸⣻⠂⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣴⡏⣇⣾⢻⣿⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠸⣼⣟⡁⣿⡞⠁⢰⠀⠀⠀⠀⠘⠒⠒⠂⠠⠤⠤⠤⠤⠤⠀⣀⡀⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣾⡛⣷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠉⠉⠛⠉⣿⢿⣍⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠳⣧⠀⠀⠀⠀⠧⠤⠟⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⡞⠀⠀⣷⣄⡀⠀⠀⠀⠀⠀⠀⢳⡘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢮⣀⣽⢷⢂⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠉⠛⣲⣄⣀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣹⡞⠲⢽⣦⣞⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣾⠈⣽⠀⢈⠑⣄⠀⢀⠠⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠅⠲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠒⠸⠏⢁⡠⢎⣀⡀⠀⠤⠤⢄⠀⠀⠀⠀⢤⣠⠄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣴⡉⠟⠢⣀⡋⠀⠀⠀⠀⠄⠀⠀⠀⠂⠠⠤⠀⠄⠁⠘⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⣀⠀⠀⣀⡀⣀⢀⣀⣤⢠⣤⡆⠸⠋⠑⣏⢉⡧⠀⠀⣉⣀⣀⣀⡀⠀⠀⠀⠉⠀⠀⠐⠒⠒⠂⠴⣧⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠃⠋⡹⢠⡿⢿⡐⠉⠉⠈⠁⠀⣿⠀⣻⠀⠀⠀⠉⠈⠁⢀⠜⠁⠀⠀⠀⠀⠁⠂⠄⢀⠀⠀⠀⠀⠀⠀⠀⠙⠚⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡈⢴⣆⣴⢛⣁⣸⡦⡴⠇⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄⠀⡤⠖⢡⠖⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⣡⣿⢇⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠯⠄⠀⡀⠀⠶⠀⠀⠀⠀⠀⠀⠀⡃⠀⢠⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠉⠙⢳⣷⣶⡱⣤⡔⣄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠐⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⠛⠃⠀⠀⠀⠀⠽⣤⡼⠀⠀⢠⠟⢻⠊⠛⢫⡷⢶⡰⠦⢆⡺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠐⠛⠒⡶⣵⠇⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠚⠀⠀⠸⣍⣩⠃⠀⠀⢰⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢄⡀⠀⠀⠀⠐⠀⠀⠀⣀⠀⠤⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠗⣲⣤⡔⣠⢤⡄⣤⣤⠀⣤⣤⣂⡤⢔⡲⠴⢮⣴⠿⡥⢋⠝⠓⠠⠤⠄⠉⠉⣉⡉⣉⠤⠐⠒⠖⠀⠈⠁⠢⡀
⠀⠀⠀⠀⠀⠀⢠⡴⠃⢴⡾⣏⢈⣽⣶⡆⠀⠀⠀⣜⠋⢳⠀⠀⠀⠀⠀⠀⠷⢦⡞⠁⠀⠀⣶⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⢢⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣏⣹⣼⠤⣄⣀⣀⡈⠛⠉⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠡⡀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠀⠙⠀⢸⡏⠹⡖⠋⠛⠔⠒⠮⣱⣶⣆⠰⢴⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠐⡀⠀⠀⠈⡄
⠀⠀⠀⠀⢀⡀⠔⣁⡴⠆⠀⠀⠀⣀⢀⠘⣳⠞⡃⠀⣀⠀⢰⠆⣿⠀⣯⠀⡀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢠⠀⠀⠀⡇
⠀⠀⠀⠈⠉⠉⠡⠶⠑⢒⣀⠤⠪⠗⠹⠔⢫⠜⠓⠊⠀⠲⠁⠈⠉⠁⠙⠒⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡆⠀⠡⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒⠂⠁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⠄⠀⠀⢀⠀⠠⠤⠐⠊⠁

⣶⣿⣾⣷⣶⣿⣾⣿⣶⣿⣶⣶⣿⣶⣿⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣿⣷⣶⣿⣿⣿⣷⣿⣶⣿⣶⣷⣷⣿⣿⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣻⡟⣿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠤⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡔⢋⠭⡙⢦⠀⢀⡠⢤⡀⠀⠀⠸⣈⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⡇⠀⠖⢦⠀⢣
⠀⠀⠀⠀⠀⠈⠀⠘⢄⡁⠜⠀⠸⡀⠰⠃⠀⠀⠀⢀⡉⠑⠓⠀⠉⠉⠁⠒⢠⣤⠒⠋⠉⠉⠉⠉⠛⠲⣄⡀⠈⠒⠒⠁⡠⠃⠀⠀⠀⡀⢀⡀
⠀⠀⣠⠤⢤⡀⠐⠤⠄⣀⡀⢠⠄⠘⠁⠀⠒⠀⢀⠋⢀⡀⡀⠀⠀⠀⠀⠀⡞⠀⢱⠀⠀⠀⠀⢀⠤⣄⠈⢫⠁⠀⠙⠓⠤⢀⡀⡠⠈⢀⣠⠇
⠀⠐⣇⠐⠖⠃⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⡏⠠⠈⡂⠀⠀⠀⠀⢧⣀⣼⠁⠀⠀⠀⡃⠠⡀⠃⢸⢆⠀⠀⢀⠀⠀⠈⢋⠉⠁⠀⠰⣁⡄⢹
⠀⠀⠈⠑⠒⠂⠉⠀⠥⡱⠀⠀⠀⠀⡜⢠⠀⡸⠀⠓⠬⣤⠚⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠦⡤⠖⠋⠈⠢⢐⣂⡕⠀⠀⠈⡁⠒⠤⢤⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⣄⠀⠀⠀⠘⠦⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣇⣀
⠀⠀⠀⠀⠀⠀⠉⢻⣿⣏⠈⠀⠀⠀⠀⠈⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠞⠋⠄⠀⢰
⠀⠀⠀⠀⢀⡧⠀⠺⠀⠈⠃⠀⠀⠀⠀⣠⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠁⠀⠀⠀⠀⢀⣀⣀⣤⠀⠀⠀⠀⢨⣅⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣌⠻⣶⠀⠀⠀⠀⠐⠆⠀⠀⢿⣿⣻⠃⠀⠀⢀⠀⠀⠀⠀⢠⡄⢀⣧⣀⠤⠀⠀⠀⠈⠀⡩⠤⠀⠀⢀⣴⣾⣿⡆⣄⠀⠀⠀⠊⢺⡹
⠀⠀⠀⠀⠙⡿⡖⠀⠀⢀⣠⣾⣷⠄⡀⠀⠉⠁⠀⠀⠀⠛⢤⠀⠀⠶⣔⡃⠞⡃⣡⡴⠤⠄⠀⠀⠒⠁⠀⠀⠀⡾⠛⣧⡜⠛⣾⡇⠀⠀⠙⠀⠁
⠀⠀⠀⠀⡀⠿⠀⠀⣰⣃⢤⣄⣤⣤⡟⡄⠀⠀⠀⠀⠀⣰⡀⠀⠀⢠⣈⡿⢹⠟⢴⣖⢣⡀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣟⣳⣾⣻⠃⠀⠀⢀⣶⣄
⠀⠀⠀⠐⠤⠄⠀⠀⢿⣭⣧⣟⣳⣎⣹⡇⠀⠀⠀⠀⠐⣯⢕⠀⠀⠀⢠⡵⢟⢹⠦⢻⡀⠀⠀⠀⢠⣾⣤⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣠⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣏⣹⡽⠟⠁⠀⠀⠀⠀⣠⠗⠋⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠖⢶⡿⡿⣓⣤⠀⠀⠀⠀⠀⠀⠀⢠⣮⣭⣭⢭⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⠐⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠛⠿⠶⠧⠵⠗⣣⠀⠀⠀⠀⠀⠀⠸⣿⣻⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠠⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠝⣒⢒⢒⣒⣪⡽⠀⠀⠀⠀⠀⠀⠀⠑⢿⢭⡦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠃⢊⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣁⣁⣡⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⠠⠤⢀
⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠠⡀⠀⠀⠀⡠⠐⠒⠠⡀⠀⠀⠀⢀⠖⢈⣉⠢⡀⠀⡔⢁⣠⡌⣢⢠⠂⣀⠴⢶⡀⢣
⠀⠀⠀⠀⠀⣠⢋⡔⠋⡉⣣⡾⡄⢀⡞⢀⣞⠙⣧⣾⣄⡀⣠⠃⢰⣏⣹⣦⠧⣸⠀⣿⣠⡽⠾⡇⠰⡧⢤⡤⠛⢾⡄
⠀⠀⠀⠘⠀⡇⢸⡁⢹⠋⠀⠀⠀⠨⠀⢸⡁⢉⠉⠀⠀⠀⢀⠀⢿⣀⠦⠁⠀⢸⠀⣧⣌⠀⠀⢣⠈⠧⣚⠀⢀⢸⣿⣧⣄
⠀⠀⠀⠰⠀⠱⣤⠉⡆⠀⠀⠀⠀⠀⠣⣀⣛⡇⠀⠀⠀⠀⠀⠣⣤⡿⠁⠀⠀⠀⠳⡾⠃⠀⠀⠀⠻⠿⠃⠀⠀⠈⣿⣽⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⠀⣀⣀⡀⠀⣀⣀⠀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣹⡇
⠀⠀⠀⠀⡁⠀⠀⠀⣁⢍⠂⠈⡠⠈⠃⡔⡪⠵⠀⡉⡋⠋⢘⠈⢆⡎⠀⠀⡁⠶⣸⠀⢠⠚⠁⠡⡢⢱⠀⠀⢀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⡇⠀⠀⠀⣇⡴⠀⡂⣝⣛⢠⠸⡄⣀⠀⡧⣛⠀⠐⢱⡈⢱⡇⡆⠇⣒⢝⠄⢨⣟⠀⠰⡆⢇⠀⠀⠄⠀⣿⣇⣿⡇
⠀⠀⠀⠀⡇⡄⠀⠀⠂⠒⠁⠑⠒⠚⠀⠑⠂⠘⠁⠃⠄⠚⠐⠆⠑⠒⠳⠃⠒⠐⠊⠀⠂⠘⠃⠓⠉⠪⠂⡠⣀⠀⣿⣷⣿⡇
⠀⠀⠀⠀⡧⠤⠭⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠈⢈⣀⣀⣿⣿⣯⡏
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣷
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠔⣁⠔⠠⡪⢀⠔⠀⠠⠑⢆⠀⠀⠏⢩⢅⠄⡠⡠⠀⣤⠤⠄⣩⢍⡇⠀⠀⠀⠀⠀⠀⣿⣟⣷⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⢜⠔⡡⣪⡴⠵⢤⠪⡢⢂⡽⡆⠀⠆⢁⠄⡨⡪⠐⢑⢕⠠⠞⡠⣨⡇⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢃⠁⣠⣪⠾⠀⠀⠀⠋⡸⢋⢔⡇⠀⡆⠑⣀⡺⡏⠉⢉⡉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣽⡿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣰⡡⡾⢑⢥⡇⠀⡇⠙⡡⠚⡋⠡⡠⡠⢀⢉⠒⢄⠀⠀⠀⠀⠀⠀⡏⣽⣻⡶⡍
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⢤⢒⠸⠛⢁⢠⡠⢋⠝⠀⠀⠧⠷⠶⠧⠤⣪⡊⢔⠕⠁⠠⡤⠱⡀⠀⠀⠀⢀⡷⢿⣾⣟⡇
⠀⠀⠀⢠⠀⠀⠀⠀⠀⡔⡫⡰⢟⢇⣾⠾⠑⠚⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣅⡢⡪⠁⡀⡇⠀⠀⠀⢸⡴⣸⣷⣦⡇
⠀⠀⠀⢸⠀⠀⠀⠀⢈⠴⡪⡢⠆⠿⠀⠠⠤⠤⠄⢤⡀⠀⢀⡠⢖⠉⠢⢄⣀⣀⠠⡦⢊⠴⢋⢠⠁⠀⠀⠀⣸⣇⠶⡟⢿⠇
⠀⠀⠀⡈⠀⠀⠀⠀⣙⠨⠶⡊⢐⢔⠁⠢⠂⡰⢂⢰⡇⠀⠘⢔⡥⠂⣠⠂⡠⡠⢊⢔⢕⠔⢷⠆⠀⠀⠀⢠⣿⠿⢿⣿⣷⡆
⠀⠀⠀⠃⣊⠄⠀⠀⠉⠀⠒⠒⠒⠣⠤⠬⠬⠤⠵⢼⠇⠀⠀⠀⠑⠪⢄⣂⣈⡔⣕⣑⡱⠖⠁⠀⠀⠀⠀⠺⠻⠾⠺⠿⠿⠁
⠀⠀⠘⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡟⡟⡟⣟⡷⣻
⠀⠀⠀⠀⠀⠒⢨⡭⣭⢉⣉⣛⡓⠒⠒⠒⠠⠤⠤⠤⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡼⠀⣇⣇⡇⡗⡍
⠀⠀⠀⠀⠀⠀⠘⠲⡦⠤⠤⡤⠭⣭⣉⣉⣉⣉⡐⠒⠒⠒⠀⠐⠀⠈⠩⠭⠭⠭⠭⠭⠥⠄⠀⣉⣁⣉⣀⡸⢹⢸⢱⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⢛⣒⡒⠲⠶⠤⠤⠤⢭⣭⣭⣉⡉⣉⣉⣉⣙⣓⣒⣒⠒⠒⠐⠒⠠⠀⠠⠤⠤⠤⠊⢸⢸⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠤⠭⠭⣉⣑⣒⣒⣒⠀⠠⠠⠠⠤⠄⠀⠀⢡⣭⣩⣉⣉⣉⣉⣀⣒⣒⡒⠒⠒⠁⣼⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠓⠚⠚⠒⠂⠒⠒⠶⠮⠭⠿⠯⠭⢭⣭⣭⣭⣽⡿
⠀⠀⠀⢀⠀⠀⡀⢀⠄⡀⠀⡀⢀⠄⣀⢀⠀⢀⠄⡀⠄⡀⠀⠀⢀⠀⠀⢀⠀⢀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠊⠄⠀⠀⠔⡡⠀⡡⢊⠔⠡⠊⠠⢊⠄⠨⠐⡡⢂⠄⠠⢂⠁⢊⠔⠀⢈⠔⡡⠀⡡⢊⠄⡠⢀⠁⣈⢔⠄⣈⠔⡩⢂⢔⠀⡊⠐⢁⠠⢂⠔⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠿⡷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣷⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⢏⡟⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⠸⡗⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡙⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣴⣧⠗⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⡁⣔
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⠺⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡤⣀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢺⡿⢿⡿⠞⠿⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱⢝⠫⠖⣀⢚⣛⡀
⠀⠀⠀⠀⠀⠀⡈⠦⠤⠌⣀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⡇⣉⣤⣾⣶⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⡓⣻⡇⢌⠉⠻⣭⡄
⠀⠀⠀⠀⠀⣀⠛⠲⠀⠤⠁⠐⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣷⡟⣿⣟⡻⣷⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⢉⣵⡖⢹⣄⠻⠆⠖
⠀⠀⠀⠀⢠⠈⠁⠈⠘⠃⠁⠈⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣽⣿⣷⣿⣿⣿⣿⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣋⣋⢉⣁⣠⠠⠀⢡⠚⠋
⠀⠀⠀⢀⡀⠙⠓⠶⠒⡄⠙⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⣻⣿⣿⣿⣿⣿⡿⣆⢀⡀⠀⠀⠀⠀⠀⠀⠈⠋⢶⢁⣉⠁⠐⠒⠳⠿⠓⢚⡁
⠀⠀⠀⢨⠇⠠⠄⠰⠶⠰⠆⠀⠀⠞⠀⡀⠀⠀⠀⠀⠀⠀⢤⣤⡤⣭⣿⠋⠵⣿⢩⣯⢹⣟⡟⢋⢠⠀⠀⠀⠀⠀⠀⣫⠖⣩⢿⣷⠞⣽⣄⠑⠎⠞⣁⢀⣀⡀
⠀⠀⠠⠀⠀⢠⢀⠀⠀⠀⠀⠐⠴⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢛⣷⣿⣶⣿⣸⣿⣶⣿⣻⡿⠉⠀⠀⠀⠀⠀⠰⠭⢙⠃⣘⡋⣀⡅⠉⢴⠒⣈⡄⠉
⠀⠀⢤⡀⠀⠀⠈⠁⠒⠀⠐⡀⠀⠄⠄⠂⡠⠀⠀⠀⠀⠘⢦⣢⡼⢟⣭⠟⢩⣿⡎⣿⣷⣿⣿⣤⣆⣤⡖⠀⠀⣈⣰⣠⡆⣀⡘⠉⠹⠗⠂⠀⠜⠣⣿
⠀⠰⢌⠀⠃⠒⠂⠰⠂⠀⠂⠉⠀⡀⠀⠀⠀⠓⠀⠀⠀⣀⣄⣈⣴⣿⣿⢿⣺⣿⣿⣉⣹⣿⠦⣟⡿⣍⠀⠀⠠⣌⣛⣻⡍⢩⣥⡀⣶⡇⢒⡶⣦⡀⣰⢶⣆⠠⠄
⠀⠀⠀⠀⠀⠉⠉⠐⠐⠒⠘⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣹⣿⣏⣤⣿⣿⠻⣿⣷⣲⣦⢼⡿⠒⠋⠀⠀⠀⠉⠋⠀⠃⠺⠞⠟⠃⠂⠀⠈⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠈⠙⠷⠜⠛⠛⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠋⣧⠀⢠⢷⢗⠂⠀⢠⠏⢹⠀⢰⠏⢱
⠀⠀⠀⠀⠀⠀⢄⡄⡀⠀⠀⢀⡏⢀⠇⠀⠀⠉⠈⠀⠀⡿⠀⡜⠀⡟⠀⡜⠀⢀⣄⡀⠃⠀⢀⢰⡰
⠀⠀⠀⠀⠀⠀⢪⠷⣣⢵⡄⢸⡧⢪⢺⡆⠀⣼⢉⡇⠀⡷⠊⠀⠀⡷⠊⠀⢀⡽⢄⣷⣀⡾⢜⠺⠽
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠉⠀⠀⣷⠇⠸⡇⣰⢿⡁⢀⡜⢻⡀⠀⡼⢷⡀⢀⡜⢧⣀⠞⠀⠉⠈⠁⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣀⠈⠀⣀⡉⢁⣀⣉⡁⢀⣀⠉⠉⢀⣀⠉⣉⡀⠀⠀⢀⣀⣀⠀⣀⣈
⠀⠀⠀⠀⠀⢸⡏⢢⠀⢸⡇⠈⠀⣼⠁⠀⠈⡏⠈⠁⢸⣿⠀⠀⣸⡇⠀⢹⠉⣧⠀⢸⡇⠈⠀⢹⡇⢱
⠀⠀⠀⠀⠀⢸⡇⠘⡆⢸⡇⠀⠀⣿⠀⠀⠀⣇⠀⠀⢸⡏⡇⢀⡟⡇⠀⢸⣤⡏⠀⢸⡇⠀⠀⢸⡇⣸
⠀⠀⠀⠀⠀⢸⡇⢰⠃⢸⡏⠀⠀⣿⠀⠀⠀⡏⠁⠀⢸⡇⢹⣼⠁⡇⠀⢸⠀⠘⡄⢸⡏⠀⠀⢸⡷⡅
⠀⠀⠀⠀⠀⠼⠧⠊⠀⠸⠧⠤⠀⠙⢤⠄⠠⠧⠤⠆⠸⠧⠀⠃⠠⠇⠀⠼⣤⠴⠁⠸⠧⠤⠀⠼⠇⠸⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⣀⣻⣿⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠿⣻⣿⣿⣟⣳⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⠀⢀⣽⣿⣿⣿⣯⣥⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣯⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⢀⣿⣯⣬⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡈⢡⣴⣾⣿⣿⣿⣿⠦⡙⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⣿⣿⣿⣿⣽⣿⡛⣿⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢋⣭⡉⠉⢼⣯⣽⣿⣿⣿⣴⣾⣿⣿⣯⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⢻⣿⣬⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢶⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣟⠛⢻⣽⣿⢻⣿⣿⣿⣿⣿⣯⣛⠻⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⠾⢟⣱⣤⣼⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣷⣦⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⢿⣦⣤⣿⡛⣿⣿⣿⣿⣿⣿⣿⠻⣿⣙⠿⢿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣼⣿⣿⣿⣿⡿⢻⣿⠛⠿⠯⠙⠆
⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⡿⠿⣿⣿⣹⣿⣿⣿⣿⣿⣿⠛⠏⠋⣻⣿⣿⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠉⠁⠀⢻⣿⣦⣼⣾⣿⣿⣏⣹⣿⣿⣿⣿⣁⡆⣀⣿⣿⣿⣍⠻⠿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠙⣿⡏⠋⠿⠇⠘⠿⠻⣿⡌⠉⠻⠏⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠁⠀⠀⠘⠐⠀⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⢀⣦⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⠀⠀⠀⢀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⡄⢀⠀⣶⡧⠀⠀⢀⣿⣿⠟⢳⣄
⠀⠀⠀⠀⠀⠀⠐⡾⢁⠀⠀⠀⠀⠀⠀⢰⢋⠉⠀⠀⠀⢰⡀⡀⠀⠀⣤⠏⠓⡂⠀⠀⠀⠠⣗⢚⡄⢉⠃⡀⠐⢩⣿⣷⡖⠁⠛⢃
⠀⠀⠀⠀⠀⠀⢀⡷⢻⠀⠀⠀⠀⠀⣰⠿⠟⠀⠀⠀⠀⠤⠓⠃⡀⠀⢠⡄⠘⠟⢀⡄⠀⠀⣻⢼⡃⠀⠰⠀⡘⣿⡟⠉⠀⣀⡴⣿⡀⢀⡄
⠀⠀⠀⢀⢀⣰⣿⣃⣀⠁⢠⡄⠀⢀⣁⡄⠀⠀⠐⢶⠀⢼⠃⣿⠁⣶⠈⠁⢀⣀⠀⢳⡴⠞⠓⠐⠛⠦⣄⠀⠈⠛⣡⡶⠛⠁⠠⠸⠃⢀⡧⡀
⠀⠀⡰⠀⢼⣿⡿⢅⣠⡀⡨⢦⡀⠸⡿⠃⣀⡀⣿⡿⠀⣾⠶⣿⠀⠉⠠⣦⣾⡟⢲⣿⣤⣤⣤⡤⠤⠶⠞⣛⡀⢞⠙⡾⠀⣠⠶⠁⢀⣼⠄⢷⡀
⠀⠀⢧⠸⣿⣀⢶⡎⠀⢈⠃⠄⡇⡆⠈⠐⠿⠷⠈⠁⠸⠁⠀⠸⣧⠀⠠⠃⠘⠃⣾⣤⣤⣠⣤⣀⣤⢠⡶⣿⡧⠈⢢⡿⠋⢀⠄⠀⢀⣅⠀⢠⣅
⠀⠀⠘⡇⣾⣽⣔⢑⣄⣡⢈⣤⡠⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢻⣄⠀⠀⠀⠀⠹⣿⣼⡷⣦⡿⠴⠌⠿⣾⠃⠀⠀⠁⢴⠉⠀⠀⣸⣄⣀⣀⣸⣆
⠀⠀⠀⠈⠻⢿⣿⠿⠤⠀⠉⠞⠁⠀⠀⢀⣼⣤⣀⣀⡄⠀⠀⠀⢸⡟⢀⠀⠀⢀⠄⠙⠳⣶⠶⠶⠖⣲⡞⠋⠀⠀⠀⠀⠀⠀⣀⣼⣯⣥⣤⣤⣤⣽⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⠉⢡⡀⠀⠀⣼⠁⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠟⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠋⠙⠒⠀⢿⡀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸⢐⠃⠀⢀⠠⡀⠄⡀⡀⠀⢠⠀⠀⡄⡄⡀⢀⠀⡄⠠⢀⣠⣀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠰⠸⠁⠘⠀⠃⠘⢘⠁⠀⠠⠀⠀⠁⠀⠁⠈⠀⠅⠆⠘⠈⠀⠐⠸⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⢄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣟⣻⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠋⠧⠽⠈⢪⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢵⣿⠀⠀⣿⠀⢿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣌⣛⣀⣀⣛⣃⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⡏⠉⠉⠉⠉⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇⢰⣯⣿⣦⠀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢒⡭⠚⣿⡀⠘⠾⠽⠋⠀⣿⡓⢯⣒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡮⠕⠊⠁⠀⠀⣹⣧⣦⡀⢀⣴⣄⢿⡇⠀⠈⠑⠺⢵⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣪⡵⠚⠁⠀⢀⡀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣸⡇⠀⠀⢀⡀⠀⠈⠓⠮⣕⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢹⡅⠀⡴⢻⣿⣯⠳⡄⠀⡏⣿⣿⡇⢸⣿⣿⢹⡇⢠⠞⣹⣾⡝⢦⡀⠀⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡇⢸⢁⣀⠉⢁⡀⢸⠀⢧⡟⠉⢃⡘⠉⠛⣿⡇⡇⢀⡈⠉⣀⡀⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇⢸⣾⣿⣧⣿⣽⢸⠀⣼⠇⣴⠋⡏⢦⠀⣿⡇⡇⣿⣿⣴⣯⣷⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⣷⣿⣦⣾⣿⣼⠀⢿⠠⡇⢰⡇⢸⡆⣾⡇⡇⣷⣗⣽⣻⣆⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣺⡇⢸⣿⣿⡟⣿⣽⢻⢰⣿⡶⡇⢸⡇⢸⢷⣿⡧⡇⣯⣯⣿⢽⡏⡇⢰⣷⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇⠸⠽⠿⠥⠿⠯⠼⠀⡛⣶⡇⢸⡇⢸⡆⣻⡇⠧⠿⠿⠭⠿⠯⠇⠀⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡼⣤⣧⣤⡤⣤⣤⣤⣤⣤⣴⣷⣿⣿⣿⣿⣿⣷⣾⣧⣤⣤⣤⣤⣤⢤⣤⣴⡧⢧⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣀⣿⣏⠀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡈⢿⣿⣿⣶⣤⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣹⣷⣽⣿⣯⣴⢛⢀⡀⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣴⡾⠿⣿⣿⣗⠉⠉⠛⠛⠉⢩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣍⣯⠀⢀⢽⣿⡿⠦⠄⣁⣀⠀⠛⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⣤⡄⠽⢻⣿⡷⣾⠟⠑⢰⣼⣿⣛⣤⡶⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠓⠓⢓⡋⠰⢿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⣠⣄⠐⠏⣿⣿⡅⠀⠙⣿⣍⢁⣁⣈⣀⣤⢶⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠒⢿⡿⠿⢷⣶⣽⡳⢦⠾⠿⢏⣿⣿⣭⣻⠍⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢂⣀⣩⣿⠀⠀⢰⣟⣿⣿⣿⣿⣷⣿⡛⠙⠻⠏⠈⣤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠛⠿⠿⠲⣿⣭⣿⣿⣿⢿⣝⣿⣟⣶⡴⠾⣻⠿⢷⠶⡦
⠀⠀⠀⠀⠀⠀⠰⢽⣦⣶⣶⣾⣴⣶⡿⠟⢿⣿⢀⣟⣶⣿⣿⣿⠁⠀⠀⡸⣦⠴
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡤⡀⣉⡐⣤⣥⡼⣟⣯⣿⣏⣳⣮⣿⣷⣀⣈⣽⣧⣶⣯⣭⣀
⠀⠀⠀⠀⠙⠗⣶⣦⣄⣉⡉⢁⣭⣍⣛⣷⣿⣿⣿⣬⣋⠻⡟⣿⣯⣹⣿⡷⣶⣤⣬⣍⠙⠓
⠀⠀⠀⠀⠈⠀⠀⣈⠹⠙⢙⣿⢿⣿⠋⠉⠉⢿⣿⣷⣿⣟⣧⣸⣽⣿⠿⢿⣞⣂⣉⣉⡀
⠀⠀⠀⠀⢀⣠⣤⡤⠶⠚⠋⢀⣽⣿⠀⢠⢄⣹⣿⣿⣿⣿⣯⣀⣼⡁⣀⠀⣿⣿⠿⢿⣢⣦⣤⡀
⠀⠀⠀⠈⠉⠀⠀⠀⢀⣠⣶⣿⣍⣿⡿⣛⣿⠻⣿⡿⡏⣿⣏⠿⣼⡿⢶⠾⠻⡷⠄⡀⠈⠉⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠙⠋⠉⠀⣉⣽⠿⠛⠁⣰⠟⠈⣷⡜⢿⣆⠀⠛⢷⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⢹⡧⠀⠈⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠊⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⣈⣏⣹⣋⣏⣫⣙⣏⣙⣉⣋⣟⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣭⣿⣿⣯⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⡶⠊⠉⠉⠙⠻⣕⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⡰⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠈⠉⠉⠒⠦⣤⠋⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢀⣴⠠⠤⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠒⠒⠒⠒⠢⠤⣀⢀⠃⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠉⠀⠀⣀⡠⠤⠤⠤⠭⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣗⠀⠑⣄⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢠⠏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢻⠢⡀⠈⢆⠀⠀⠀⢀⡜⠀⠀⡠⢾⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡆⠑⡄⠀⢂⠀⠀⡼⠀⠀⡜⠁⡈⠀⢠⠁⠀⠀⠀⠀⠀⠀⣀⣀⠀⣸⡄⣃
⠀⠀⠀⠀⠀⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⠀⠘⡄⠘⡄⠈⡄⢠⠃⠀⡜⠀⡰⠁⠀⠃⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠲⢷⠦⣽⠀⢰⣼⠀⢸⠁⡰⠁⢀⠎⠀⠀⠀⡠⠊⠁⠀⣀⠤⠂⣺⠛⠂⡟⠒⠢⣌⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⡹⠆⠚⠁⠀⠛⠐⠦⣀⣠⡄⠀⡶⢄⡀⠈⢣⡈⢱⠀⢸⣴⣡⠴⠛⠻⠁⢀⠞⠁⢀⡴⠋⠁⢀⡴⠁⢀⠞⠀⠀⠀⠀⠙⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠟⠁⢀⣀⣀⠀⣀⠀⠀⢻⠁⠀⢳⠀⠉⢦⠀⢁⠀⡗⠁⠈⡵⢾⠀⡇⢠⠏⠀⡰⠋⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⢀⡠⠄⣒⠶⠟⠋⣀⣠⠔⠋⠁⠉⢀⡈⣙⣂⣄⣣⡀⠘⢦⡀⢸⠀⢰⢆⡀⠀⡎⠀⡘⢠⣷⠇⣀⠞⢀⡠⠖⠁⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢰⠃⠀⠛⠠⠄⠀⠀⣭⠀⠀⠀⣀⠽⠃⠀⣠⠤⠘⠻⠿⢦⣀⠙⢺⣤⠺⣮⣃⢠⡣⠞⠁⣠⣃⠀⠙⠺⢭⣤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⢸⠀⠀⠀⣠⠔⠂⠉⠉⣉⢭⡟⠁⢀⣴⣊⡀⢀⣀⢀⡀⠤⠽⣻⣋⡵⠒⢿⣈⣿⣤⡴⠋⠀⠀⢙⠶⠀⠀⠈⢛⡁⠒⠂⠤⢄⣀⠀⠀⠀⠀⡠⠃⠀⣸
⠀⠀⠀⢳⠀⠀⡇⠀⠀⡰⠊⣴⡊⠀⣀⣀⠀⠀⠁⠁⠈⠁⠀⢀⣠⣼⣉⣷⣄⡼⢥⣯⡁⠈⣉⣉⠉⠉⠳⡄⠀⠀⠘⣥⣀⡀⠀⠀⠀⠉⠢⡤⠚⠁⠀⡠⠃
⠀⠀⠀⡜⠀⠀⠈⡆⡜⠁⠀⢀⡠⠊⡡⠊⠿⣁⣀⣀⣠⠔⡾⠊⡱⠊⡹⠧⠼⢶⠾⠉⠢⡉⠁⠀⠈⠳⢄⠘⣖⠀⠀⠘⣆⠈⢑⡦⢄⠀⠀⢱⠀⣠⠞⠁
⠀⡠⠊⠀⠀⢀⡜⢠⠁⠀⢒⡏⢀⠔⠀⠀⠀⠀⠀⡜⠁⣾⠀⠀⣰⢰⠀⠀⠀⢀⠄⠸⣦⢈⣉⡈⠉⣉⣉⣋⠘⠧⢄⠀⣼⠗⠂⠀⠠⢷⠀⠘⠁
⠾⠒⠙⢆⡠⠋⠀⢳⠂⢀⡜⢡⣌⣀⣀⣀⡀⠀⢀⡄⠀⡇⢀⣠⠋⡆⠀⠀⠀⣎⠀⠀⢹⡆⠀⠈⠙⠃⠈⢣⡉⠻⠊⢳⡈⢢⡀⠀⠀⢸⠀⠸⣄
⠀⠀⠀⠈⠁⠀⠀⠺⠀⠀⠀⢼⡔⢛⠶⠖⠛⢫⡭⡇⠀⠘⠁⢀⠎⢇⠀⠀⠀⠘⢷⠀⡟⠇⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠁⠀⢣⠀⠀⢀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠠⠛⣶⣀⠷⣇⣀⣶⡔⠉⣆⡟⠀⣠⠴⣟⠂⢼⣄⣤⡀⠀⠀⢣⡃⠰⠀⠀⠀⠀⠀⠀⠀⠈⢻⠑⠢⣄⠀⠀⠀⠈⠢⡀⠀⠀⣈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠟⣡⠞⠛⠛⠒⠚⠓⠠⠽⠏⣉⣻⡧⢴⡖⠉⡍⢻⣍⣧⣵⢄⠀⠈⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠙⢷⡄⠀⠀⠈⠢⣀⡁
⠀⠀⠀⠀⠀⢀⣤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠿⣙⢻⣍⣧⠹⠆⠀⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡐⠋⠉⠁⠐⠒⠂⠀⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⣱⡾⠀⠀⠀⠱⠇⠀⠀⠀⠀⠀⠀⡠⢶⡤⢮⢀⣄⣼⣦⡀
⠀⠀⠀⡼⠛⠉⠉⠉⠉⠛⠒⠲⠦⣄⣀⡈⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣿⠶⣦⢀⣤⣠⣿⡹⠋⠙⠢⣄⡀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⡈⠙⠲⢄⡀⠀⠀⢱⣀⡀⢀⣠⠄⢶⣏⣩⠟⠛⠷⢀⣄⣠⣟⠩⠋⠀⠀⠀⠀⠀⠉⡳⢤
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⠢⣄⠈⠓⢤⣘⣿⡥⣼⠟⢳⢎⠀⡙⠓⢶⣤⣬⠋⠟⠉⠀⠀⠀⠀⠀⠀⠀⠐⠁⣸⠃
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠙⢦⡀⠘⠿⢣⣏⢳⣮⣠⡼⢟⡾⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⢀⠴⠶⠃⠸⡀⠀⠀⠀⠀⠙⢆⡄⠈⡭⠿⠐⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠄⣀⣸⠃⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⡼⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡷⠲⠖⠚⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡤⠤⠴⠒⠚⠉⠀⢀⣠⠴⠚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠤⠤⠔⠲⣎⡁⠀⢀⡰⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⣠⣤⡀⠀⢀⣀⣀⡀⠀⣠⣤⣄⠀⣠⣶⣶⣤⡀⠀⣀⣀⣀⠀⠀⣀⣀⠀⠀⢀⣤⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⠀⣿⠉⣿⡄⢸⡏⠹⣿⠀⢿⡉⢹⡇⠸⣧⠀⢸⣇⠸⣿⠛⢿⡆⢸⡟⠛⣿⡀⢸⡏⢿⡀⣿⠻⣧
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⠀⢹⠿⢿⠀⢻⡷⢾⡇⠀⣿⠷⢿⡇⠘⡿⠿⣿⠀⠀⣿⣶⣿⠀⢸⣿⢾⡇⠀⢹⣷⣾⡇⢸⡿⣿⠀⣿⣤⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣤⣼⡆⢸⣇⣸⠄⠀⠿⠶⠿⠀⠈⠷⣴⡇⠀⢸⣄⣀⣹⠀⢸⣤⣤⡇⠀⢷⣤⣼⠇⠸⣧⣼⠃⢸⣧⡿⠉⢿⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠛⠛⠛⠿⣷⣦⡀⠀⢠⣶⠿⠿⠿⠿⢷⣶⡄⠀⠀⣠⣶⡾⠿⠿⠿⢿⣦⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⢺⡇⠀⠀⠀⠀⠀⣿⠇⢠⣾⠟⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣤⣄⠀⠀⠀⠈⣿⣿⠁⠀⠀⠐⠾⣿⡿⠀⣿⠃⠀⠀⠀⠀⣀⣤⣴⡿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⣽⠂⠀⠀⠀⣿⣿⠀⠀⠀⢰⣾⣿⣤⣼⣿⠀⠀⠀⠀⢸⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠁⠀⠀⠀⣰⣿⣿⠀⠀⠀⠈⠉⠈⣽⡿⣿⠀⠀⠀⠀⠀⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢀⣴⡿⢹⣟⠀⠀⠀⠀⠀⠀⣿⠇⢻⣧⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⡿⠟⠋⠀⠘⠿⣿⣶⣶⣶⣶⣾⡿⠀⠀⠙⠛⠿⣶⣶⣦⣤⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣴⠶⠶⡄⢠⡶⢦⡀⢠⡶⠶⣦⡀⠀⣠⣤⣤⡄⠀⢠⡶⠶⢦⡀⢠⡶⠶⠛⢷⠀⢠⡶⠶⢶⡄⣰⠶⠟⣦⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣀⣀⡇⢸⣀⣈⡇⠸⣇⣀⣸⡇⢸⣏⣀⣀⣿⠀⣼⣀⢀⣈⡇⢸⣇⣀⣠⡾⠀⢸⣇⣀⣠⡇⢻⣤⣤⣼⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠁⠀⠉⠉⠉⠀⠈⠉⠉⠉⠉⠀⠈⠉⠋⠉⠀⠀⠙⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⣄⠀⣤⣤⣤⡄⠀⢠⣤⣤⡄⠀⣴⠶⠒⣆⠀⠀⡴⠶⢶⡄⠀⢠⡶⠶⢶⡀⠀⢠⡶⠶⢶⠀⢰⡟⠓⠻⡆⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣇⣸⡆⣿⣁⣀⣿⠀⣿⣀⣀⣿⠀⢿⣤⣤⡿⠀⠸⣧⣤⣴⡇⠀⢸⣧⣤⣾⠀⠀⢸⣇⣠⣼⠇⠘⣧⣤⣼⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⡇
⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⠉⠉⠉⠛⠛⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣦⡀
⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠙⣦⡀
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠠⠬⠷⣄
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠻⢷⣶⣶⣾⠂
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⠤⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⣼⡟
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⢀⣴⠶⠋⠀⠀⠀⠀⠲⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⣠⠞⠉⠙⢦⣠⠞⠉⠑⢦⡀⠀⣀⣀⣤⣽⣷
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠠⠀⠀⠀⠀⣠⠟⠁⠀⠀⡼⠁⢠⣶⣤⠈⠁⠀⣤⣶⡄⢳⡀⠘⠻⠿⢿⣥⡀
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢶⡇⠀⠀⣴⠏⠀⠀⠀⢸⠇⠀⠈⠛⠁⠀⠀⠀⠈⠋⠀⠀⢷⠀⠀⠀⠀⠙⢿⣦
⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠻⢦⣴⣿⣶⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠘⢧⣀⠀⠀⠀⣀⡴⠛⠉⠀⣾⣿⠿⠾⠟⠛⠁⠀⠀⠀⠈⢧⡀⠀⠀⠀⣠⢦⡀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠻⢿⣿⣿
⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⢀⣈⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠚⠁⠀⠙⠲⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡿⠃
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠟⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⢿⣿⣛⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠓⠞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⡀
⠀⣠⣤⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢫⡧⢤⠄⠀⠀⠀⠀⠀⠀⠀⣤⢬⣏⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⡆
⠀⠘⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢧⡀⠀⠀⠀⠀⠀⠀⠀⡨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⡿⠃
⠀⠀⠀⠈⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠙⡄⠀⠀⠀⠀⠀⡞⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⢠⣶⠿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⠶⣸⠉⠙⠒⠃⠀⠀⠀⠀⠀⠓⠊⠉⣇⡴⡆⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠙⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿
⠀⢠⣿⣋⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠋⠉
⠀⠈⠛⠛⠛⠛⠛⠛⠛⣿⣷⠆⢀⣠⣤⣤⣶⠶⠶⠶⣶⣤⣀⣀⣀⡀⠀⢀⡀⠀⢀⣤⣶⣶⣦⣄⠀⠀⢀⣀⣀⣠⣤⣦⣤⣶⣶⣶⣶⣶⣶⡶⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠿⠛⠋⠉⠛⠁⠀⠀⣀⣽⠉⠉⠉⠙⠿⣦⣿⡿⣿⠿⠃⠀⠀⠈⠻⣷⣶⡟⠋⠙⠛⠃⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠈⢹⡇⣿⠀⠀⠀⠀⠀⠀⠈⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠓⠒⠒⠚⠃⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠀⣤⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣽⠀⢿⡤⠠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⡼⠹⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢰⠁⠀⢰⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡰⠃⠀⠀⠀⣦⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣀⠃⣀⣀⡀⠀⠀⡆⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⠁⠀⠈⠉⠀⠀⠀⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡀⠁⠀⠒⠂⠀⠀⠋⠀⠀⢳⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣠⠁⢀⣀⣀⣀⡀⢀⣀⡀⠀⠀⣦⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣰⠁⠀⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⡄⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠸⠇⠀⠲⠖⠒⠐⠒⠒⠒⠲⠶⠶⠃⠀⠰⠆⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⣠⡷⠀⠤⠤⠴⠦⠤⠤⠦⠤⠤⠴⠆⠠⠶⠶⠾⣀⣸⠧⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣀⠠⢠⡄⢀⣀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡋⠉⠉⠉⠀⠀⠀⠈⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣉⣉⣉⣉⣀⣁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⠀⠀⣾⠁⠀⠀⠀⣨⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⡀⠀⢀⡀⢁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠤⠐⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⣠⠶⠋⠉⡄⠑⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⣠⡴⠋⠀⠀⠀⢀⡇⠀⠈⠛⢤⡀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠊⠉⠀⠀⠀⢀⡴⠞⠻⡄⠀⠀⠀⠙⢦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢀⡰⣿⠛⠋⠉⠉⢻⠶⢆⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⠀⠀⠀⢀⠏⠀⠉⠀⠀⠀⢰⡆⠀⠀⢳⠀⠀⠀⠈⠣⡄⠀⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⢠⠀⠀⠀⠟⠀⠀⠀⡇⠀⠀⡌⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⡠⠀⠀⠀⣴⠚⠻⢆⡰⠏⠀⠀⢸⠶⡀⠀⠀⠃⠀⠀⠇⠀⠀⢀⣴⣿⠀⡆⠀⠀⣀⡀⠙⢆⡀⠀⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠘⣦⣠⠴⡟⠀⠀⠀⠀⠛⠠⠖⠋⠁⠀⠀⠀⠈⠀⠹⣄⠀⢸⡆⣨⠀⢀⡴⠋⠀⠀⠀⠹⢀⠘⠋⣁⠀⠀⠙⢤⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⣤⣧⣇⡴⠋⠀⠀⠀⢀⡄⠀⠈⠁⠀⠉⠀⠀⠀⠀⠙⣤⡀⠀⠀⡰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⡆⠙⠋⠙⠑⡆⠀⠀⠈⠛⠀⠀⠀⠀⠙⠛⠛⠤⠦⠄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣿⠙⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⢀⣀⡀⢀⣀⣀⣀⠀⠀⢸⡏⠀⠀⠈⠉⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⡾⠉⠉⠉⠉⠉⠉⢹⠋⠉⠉⠉⠉⠉⠉⢹⡇⠀⢸⠃⠀⠀⠀⠀⡇⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢨⡇⠀⢸⡀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡄⠀⠀⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⠀⣠⡄⣤⠀⣿⠀⢀⠀⡀⠀⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⠷⠃⠀⣯⠐⢯⣤⠃⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠳⠆⠀⠀⠘⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣀⣀⣀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈
⠀⠀⠀⠐⣷⣄⣸⡇⠀⢠⣦⣀⡀⠀⠒⠒⠒⠒⠒⠂⠘⣿⠀⠐⠒⠒⠒⠒⠒⠛⠶⠖⠀⠀⣠⣄⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⠀⠤⠀⠉⠁⠉⠉⠋⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⠉⠉⠛⠋⠀⠀⠠⠶⠦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠒⠛⠉⢹⡆⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣤⣤⣤⡤⠀⣤⠉⠑⠒⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣤⡀⣴⠊⢳⠀⠀⠉⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢻⣤⢀⣈⣄⠀⣼⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠛⠋⢀⡴⠞⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⢰⡏⠀⢀⡀⠴⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠓⠄⠺⢆⡀⡨⠁⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠘⠛⡆⠀⠀⡶⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠘⠲⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣽⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣿⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣻⣿⣿⣟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣛⣿⣧⣾⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⣿⣿⣾⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⢿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣯⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⠿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣰⡇⣠⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣿⣿⣼⣿⢧⣿⣩⢿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣟⣿⣿⣿⣿⣿⣾⣧⣿⣿⡿⣁⣤⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢿⡿⢿⣿⣿⣿⣯⣴⣷⡿⣿⣾⣷⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣿⡿⣾⣿⣿⣿⣷⣟⣿⡿⠟⣿⣋⡭⠴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⠯⣵⣿⣿⣿⣯⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣝⣷⣽⠹⠃⣸⣟⣻⣿⣷⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠀⠁⠀⢀⣿⣟⣻⣯⣿⡷⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣍⣻⣿⣿⣟⡛⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣟⣿⣿⣿⣿⣭⣹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣦⣝⠿⣿⣿⡻⢷⡬⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣾⡻⣿⣿⣿⢿⣯⣿⣻⡶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⣤⣾⣿⣿⡙⣿⣿⣿⣿⢾⣧⣹⡟⠛
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡀⢀⣀⣤⠀⣀⣀⣀⢠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣧⣽⠙⠁
⠀⠀⠀⠀⠀⢻⣦⣼⡝⣾⢳⣞⣿⣇⣿⢹⣿⣏⡿⣽⣷⣿⣿⣯⣿⣿⣧⢹⣿⣿⣿⣬⡟⠿⠈⠁
⠀⠀⠀⠀⢤⣶⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣭⣷⣙⣿⣟⣿⣿⣿⡘⣯⢻⠻⠟⠛
⠀⠀⠀⠀⠀⠀⠰⠯⢾⣿⣿⣿⣿⡟⣽⡿⣟⣿⣿⣿⣇⣹⣷⣿⡞⠳⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠙⠿⠃⠛⠛⠋⠋⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⢀⣠⡀⠀⣠⣤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣿⣿⣿⣤⣶⣾⣽⣿⣶⣾⠅⣂⡠⠼⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠟⠉⢶⣿⣷⣿⣟⣿⡿⢿⣯⣿⣿⣥⡟⢺⡤⣗⠒⠒⠐⠂⠛⣤
⠀⠀⠀⠀⠀⠀⠀⢰⢃⣜⣢⣼⡿⣯⢿⣿⣏⣹⣇⣿⡿⣟⢯⠻⣿⣿⣤⣿⣷⣶⣀⣤⣞⢿⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠸⣯⣼⣿⣿⣿⣿⣿⡿⠿⠉⠙⣿⣿⣜⡎⢃⠣⢸⣻⣿⢿⣽⣿⣷⣼⠘⡏⣷⡄
⠀⠀⠀⠀⠀⣴⣦⣜⣿⣿⣿⣿⣿⢛⣂⣇⠀⢀⠼⠛⢿⣿⣿⠛⠒⠓⢿⣿⣿⣿⣿⣻⣿⣧⣇⣲⣿
⠀⠀⠀⠀⠀⡯⣷⢻⢷⣿⣿⣿⡟⡛⠉⢙⠖⠉⠀⢀⠔⠉⢳⣄⠀⠀⠀⣿⣿⠏⣸⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⡗⣿⢼⢊⣿⣅⣝⢳⠄⠀⣠⠃⠀⠀⠸⠀⠀⢸⡿⠀⠀⠀⢿⡿⣾⣏⣿⣿⣯⣿⣿⣿⣿⠷
⠀⠀⠀⠀⣰⠖⠛⢾⣸⣿⣿⣯⣨⣶⣾⣯⠀⠀⣠⡃⠀⠀⠘⣇⢣⡀⠀⠀⠛⠻⣽⣷⢻⣯⣅⡼⣌⠙⡄
⠀⠀⠀⠀⣽⢂⠀⠀⠈⢯⣿⣿⣿⣽⡟⠁⠀⢰⣯⣤⣤⠄⣭⣿⣯⡵⠀⠀⠀⢰⡟⠃⠁⢺⡤⢶⠃⠀⠨⡂
⠀⠀⠀⠀⠘⡄⢰⡀⡀⠸⣷⣿⣿⡿⢟⣧⡀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⣀⣤⣾⣟⠀⢀⣉⣷⣤⣁⠀⢀⠇
⠀⠀⠀⠀⣤⠛⠋⠉⠃⣢⣼⣿⣿⣏⡙⠲⢷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣽⣿⣿⢿⣿⣿⣽⡏⠃
⠀⠀⠀⠀⠘⢶⡂⢀⣀⠯⣨⣿⣹⣿⣿⣏⣺⣿⣻⣷⣦⣀⠀⢀⣤⣺⣿⣟⢙⣿⡿⣿⣿⡿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠈⠱⣿⣾⣿⣿⣷⣿⢿⣿⣽⣁⣽⣿⣿⣿⣭⣵⣶⣾⣿⣿⣾⣯⣿⣿⣛⠄⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡟⠋⠉⠙⢿⣽⣻⣿⣿⣿⣿⡟⣠⠿⣿⣾⣿⣿⣿⣿⣿⡆⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⠀⣄⣘⢼⣹⣍⣿⣿⣿⣿⣿⣿⡽⡋⣟⣺⣿⣾⣽⡗⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⢿⣿⣿⣿⣝⡻⢭⣩⠿⠬⠤⣥⡽⢻⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠉⠹⡿⠿⠛⠁

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

⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠖⠋⠇⢠⡤⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡄⠀⠀⠀⠈⠣⣀⠀⢀⣚
⠀⣤⣤⣿⣤⡄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⢸
⠀⠀⡿⠿⣯⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣜⢿⢹⣆⠀⠀⠀⠀⠀⠀⠘⡿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣿⣧⠀⢀⣠⡄
⠀⠀⣸⣿⡇⠀⠀⠀⠀⠺⣻⣿⣿⣾⣿⠛⠢⠀⠀⠀⠀⢸⠏⠈⠊⢫⠀⠀⠀⠀⠀⠀⣠⣖⣦⡀⠀⠀⠀⠀⠀⠰⢎⡟⠢⠀⠈⢻⠏⠀⠈⣿⠁
⠀⠀⣿⣿⣧⠀⠀⠀⠀⣾⣿⣿⣿⣿⣮⡓⠦⠀⠀⠀⠀⠎⣰⠘⡇⢧⢱⠀⠀⠀⠀⣼⣿⣿⣯⣿⠄⠀⠀⠀⠀⠒⣫⣏⡒⠆⠀⠀⠀⠀⣠⢿⣆
⠀⢰⣿⣿⣿⡆⠀⣤⣤⣶⣿⣿⣿⣿⣮⡛⠷⠶⠀⠀⢸⡴⠙⠴⠉⠋⡙⢇⠀⠀⢀⣽⣷⣿⣿⣶⣤⣀⠀⠀⣘⣯⠴⡧⣭⣁⠀⠀⠠⣾⣿⣿⣿
⠀⣾⣿⣿⣿⣿⠀⠰⣤⢾⣻⣿⡿⣿⣿⣿⣅⠀⠄⢠⢋⡆⢠⢺⡀⣧⠹⣌⢂⢾⣿⣿⡿⣿⣿⣿⡷⠟⠀⠀⠠⢤⢚⡟⠲⠤⠄⠀⠀⣼⣿⣿⣿
⢠⣿⣿⣿⣿⣿⡆⠴⣒⣿⣿⣿⢿⣿⣿⣷⣭⣉⠀⣮⠞⠥⠻⣨⠓⠎⠑⠋⢻⣠⡴⣾⣿⣿⣿⣿⣷⡦⣀⠀⢶⠚⣉⣏⣟⢒⣖⣠⣾⣷⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⠛⠿⢿⣻⢿⣾⣿⣿⣿⣿⣯⣿⢋⣴⢸⢀⠀⢰⢰⡘⣆⣿⣿⣿⣿⣷⣿⢿⣷⣿⣿⣟⣷⣒⣉⠥⡧⢌⣋⣻⣿⣿⣿⣿⣿⣽
⣿⣿⣿⣿⣿⣿⣿⡗⣿⣯⣷⣿⣿⣿⣿⡟⣯⣕⣗⢻⡝⠛⣼⢄⡞⣬⠑⠋⣁⣭⣾⣿⣽⣿⣿⢿⣾⣯⣭⡏⢤⡴⠒⡗⠲⠤⢤⣿⣿⣿⣿⣿⣾
⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⡛⣿⣿⣿⣍⡻⣦⣭⣉⣛⡓⠀⣀⣸⣇⣀⢰⣿⣿⣻⣿⣯⣿⣿⣻⣿⣿⣾⣿⣻⡦⠀⠀⣇⣠⣴⣿⣿⣿⣯⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣻⣿⢿⣿⡟⢿⣳⣯⣭⣍⣡⠆⣿⡿⢿⣿⠁⠠⣄⢤⠀⠀⠀⣿⠀⢀⣉⣉⡉⠁⠀⢈⣿⣯⣿⠿⢷⣿⣿⣿⣿⣿⣿
⠛⠛⠛⣛⠛⠛⠛⠛⠒⠛⠉⠀⢸⣿⣇⣀⣀⣭⣭⢥⣤⠀⢹⣷⣾⡏⠀⣿⣿⣿⡆⠀⠀⣿⠀⣸⣿⣿⣿⣿⠀⠘⣿⣿⡿⠀⠀⠀⠉⠙⠛⣿⡟
⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⠀⠈⠉⠉⠁⠀⠻⠿⠿⠃⠀⠀⠀⠀⠻⠿⠿⠿⠏⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁

⠀⠀⣀
⠀⠀⠉⠁⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠃
⠀⠀⠀⠀⠀⠀⠈⠋⠀⠲⠀⢴⠆⠀⡄⠀⣴⠄⠠⠤⠀⠺⠇⠀⠉⠀⠀⠀⠙⠀⠤⠀⣀⡀⢀⡀⢠⣤⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⢤⣠⠗⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣮⣉⣩⣉⣭⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⣶⣤⣾⡿⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣄⢉⣁⡈⣉⣤⢒⡹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢛⡛⢋⣭⡥⠂⠀⠀⠀⠀⠀⠀⠀⠈⢉⡋⠳⠛⠚⣁⣠⣤⠂⠀⠀⠀⣠⠄⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠋⠁⠀⠀⠀⠀⠠⠀⠄⢀⣄
⠀⠀⠀⠀⠘⠋⠙⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢻⠶⠒⠒⣋⠩⠄⡀
⠀⠀⠌⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣤⣀⣈⣀⣠⠴⢟⡁
⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠄⠨⠄⠐⠈⠁
⠀⠘⠂⠀⠀⠀⠀⠀⠀⡹⠁⠀⠀⠀⠀⠀⠀⣀⠉⠀⠙⠁⡈⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠈⠙⠢⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⢷⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣐⣂⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⠿⠿⠿⠿⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣲⠀⠒⠒⠒⠒⠒⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣤⡶⠶⣦⣤⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠉⠀⠀⠄⡀⠈⠉⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠰⢞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⣠⡴⠶⣤⣀⡀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠈⠀⠀⠀⣠⠶⠛⠋⠁⠀⣀⠀⠉⠛⠳⡦⠀⡀⠀⢠⠀⠀⢠⣦⣀⡀
⠀⠀⠀⠀⢴⠖⠛⠉⢁⠀⠀⠉⠀⠈⡇⠀⠀⠀⢠⠊⠀⠁⠀⠀⡀⠀⡄⠀⠁⠀⠈⠀⠀⠘⠀⠈⠉⡟
⠀⠀⠀⠀⠸⡀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⢰⠀⠀⡆⠀⠀⠀⡄⠀⠐⠀⠂⠀⢰⠀⡇⠀⠇⠀⠰⠂⢠⠁⠀⠀⠀⡇
⠀⠀⠂⠀⠀⢇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠇⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⢸⣀⣀⣀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠒⠒⠒⠲⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣲⠶⠖⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣇⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡭⢫⠄
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢿⣷⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡓⠾⢿⡖
⠀⠀⠀⠀⠀⡒⠢⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣛⣞⠀⠀⠀⠀⠀⠀⠀⠀⣸⣮⠑⠦⣽⠅
⠀⠀⠀⠀⠲⠌⠁⠂⢁⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣷⣿⡿⢿⡥⠀⠀⠀⠀⠀⠀⠀⠐⠒⣻⠴⣄⢯⡥
⠀⠀⠀⢄⣁⠀⠁⢀⠉⠀⠀⠀⠀⠀⠀⠀⠠⢤⡼⣿⣾⣿⣿⣃⠄⠀⠀⠀⠀⠀⠀⣉⡭⢩⠤⢄⣈⣟⢉
⠀⠀⢰⡆⡀⢩⠁⡀⠀⠠⠀⠀⠀⠀⠀⢀⣀⣸⣷⠟⣻⢿⡟⣿⡷⠂⠀⠀⠀⠀⠠⠠⣧⣶⢴⡍⢬⡭⢐
⠀⠀⢄⠠⢠⡀⠀⠀⠀⠂⠈⠀⠀⠀⠀⣀⠙⢛⣿⣶⣾⣼⣷⣿⡿⠋⠀⠀⠀⣀⠮⢙⢁⣋⣬⠉⠔⣊⣅⢉
⠀⡘⠀⡀⠀⢀⠁⠀⠄⠀⠀⠐⡄⠀⠀⠘⠺⣫⣿⣥⣾⣧⠿⣿⣿⣿⡞⠁⠀⣸⠿⠷⢦⠀⣩⢁⣈⡉⢛⣀⢀
⠀⠈⠁⠀⠂⠀⠂⠐⠐⠀⠀⠀⠀⠀⠈⠙⠛⣻⣿⡭⣿⡻⣿⣭⡤⠿⠟⠂⠀⠉⠛⠑⠙⠿⠛⠀⠉⠁⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠑⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⠉⢩⣭⣭⡉⠈⠉⠙⠛⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢀⡀⠀⠀⠀⠄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠛⣛⣡⣼⡶⢶⣾⡶⠛⠓⠲⣦⣀⣀⠀⠀⠀⢙⣿⡿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⣶⣾⡿⠛⣿⡶⠶⣋⣷⡄⠀⢀⣤⣈⣿⣶⣤⣱⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣶⣤⣄⡀⠙⠋⢼⣿⠇⢀⣾⠋⠙⠏⠛⠻⠇⢠⣿⣯⣉⡁⢹⣿⣿⣁⣀⣀⣴⣾⣿⣿⣿⡶⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⣻⠁⠀⠸⣟⣀⣀⣀⣤⣦⣀⣈⡉⠉⠉⠁⢘⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠚⣿⠟⠛⣿⡿⣿⣿⣿⣿⣿⣿⡿⠃⣤⣄⣤⣼⣿⣛⣯⣁⣸⣏⢛⣱⣦⣤⣤⣿⡽⣿⣿⣿⣿⠏⠉⢉⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠀⢄⠙⠻⢿⣿⣿⣿⣇⠀⠶⠟⢁⣙⡿⢋⣻⣛⠻⠿⢿⣿⣷⣄⠼⢿⣤⣿⣿⣿⢇⣀⡀⠷⠶⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⣶⣦⡀⠀⠙⠿⠋⢀⠀⠈⠁⠀⠚⠻⣿⣿⠿⣿⣿⣿⣿⣽⠛⣿⡭⠟⠻⡟⠋⠉⣅⠁⣴⡆⢈⣿⣿⣿⡛⠉⢠⣄⣀⠈⢻⣿⣿⣿⣿⣯⡀⣀⣴⣄
⠀⠀⠸⠳⣾⡿⣦⣀⠀⢀⡌⠁⠀⠀⢠⣴⠿⠋⢀⣴⣿⣧⣸⠀⢻⣤⠉⡀⠀⠀⣿⡆⢀⠙⣶⠛⢳⣾⣿⣿⣿⣧⡀⠛⠛⠋⣠⣼⡿⠟⠿⡿⢿⣿⡉⢹⣅⣠⡀
⠀⠀⢠⣀⣹⣿⡾⠛⠋⠉⢷⣶⣶⣶⣿⠋⠀⣴⡿⠃⢸⣿⣏⠀⠚⠁⠘⣿⠀⢸⡿⠃⠈⠁⡀⢠⣌⣿⣿⣏⠉⠛⢻⣶⣶⡾⠛⠁⠀⢀⣠⣴⡾⢿⣭⡄⠉⣿⠇
⠀⠀⠀⠉⠉⠀⠓⢀⠀⡀⠀⠀⠀⠉⠛⢻⣾⠋⠀⢀⣸⣿⣯⠄⢠⣀⣤⠙⡀⠘⠻⣄⠀⢠⣟⣾⣿⣿⣿⣄⡰⣆⣸⣿⣏⢁⣤⣤⡾⠛⢹⣿⣧⣀⣉⣹⣶⠁
⠀⠀⠀⠀⠀⠀⢤⡾⢿⣷⠀⢤⡀⣈⡀⢀⡁⠀⣠⣿⣿⢿⣿⣦⠉⣟⠁⡀⢹⡔⠛⣿⡤⢀⣿⠻⣿⡿⠟⠛⠛⠋⠙⢉⣍⠉⢉⣻⠀⣦⡼⠀⢋⣿⣿⢿⣟⡆
⠀⠀⢀⡄⠀⠀⠀⠀⠘⢛⣿⣶⣷⣬⡿⢿⣿⣿⣿⣿⣿⡌⣿⣿⣷⣿⣼⠷⠾⡷⣂⡏⣰⡿⠛⢁⣤⣴⣦⣄⠀⣠⣤⣾⣿⣌⠹⡿⠀⣻⣿⣾⣿⣽⣿⣾⣿⣿⡄
⠀⠀⡄⠀⠀⠀⡀⠀⣾⠛⠁⠀⠀⡴⠶⠾⠟⢀⡸⢿⣿⡤⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⠁⠀⠦⣿⡿⢿⡍⢻⣿⡛⢷⣤⣤⣿⣇⠘⣿⣏⣻⡟⢿⣽⡃
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣴⣤⣤⣤⡔⠂⠀⠀⢰⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠰⠉⢁⣨⠄⠀⣿⣧⠂⡙⣿⣿⣿⣾⣏⢙⢿⣥⡿⠹⡇
⠀⠀⠀⠀⠂⠘⢦⡘⣿⣄⡀⠀⠀⠀⢴⣦⣼⠿⠷⢶⣶⣶⣴⣿⣿⣿⡗⠒⠈⢉⣭⣭⣛⣿⠟⡩⠛⠋⠀⢠⠞⢃⡤⠀⣿⣿⣾⣧⠸⣯⣼⣿⣍⠀⠀⠉⣷⡆
⠀⠀⣀⣀⣀⠔⠛⠇⢨⡿⠛⢿⣷⣶⣾⣿⠁⠀⠀⠀⠁⢀⣼⣿⣿⣿⣷⣠⡿⢿⣿⡾⠛⠟⠂⠀⠀⡀⠀⠀⢠⣼⣀⡼⠟⠻⣧⣈⠂⢉⡉⢿⡇⣴⡀⡀⣿⣇⣀
⠀⢠⣶⣿⣥⣤⣴⣶⣿⠃⠀⠀⠀⢁⣴⣿⡇⠀⠀⠀⠀⣿⡿⠛⢛⣻⣿⣿⣶⣿⢯⣶⣾⣷⣤⣜⢻⣟⣙⣿⠁⢸⣿⣶⣶⡾⠛⢛⣻⡿⠃⠸⠿⢿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣎⣿⣿⣿⡇⠀⠀⠀⢾⣿⠃⣿⣿⠟⣽⣿⠟⢁⣼⣿⣿⣿⣿⣟⣿⣧⡈⣿⡗⠿⠿⠻⠋⣠⣾⠟⣉⣤⣾⣷⣦⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧⣶⠶⣶⣾⣿⣀⣿⣿⠀⣿⣧⡀⢻⡿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣁⣰⣧⡼⠿⣿⣿⣯⣭⣅⣤⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⢻⣯⢻⣿⠀⣿⣿⣿⣿⣿⡷⢌⣻⣿⣿⣽⣿⣸⣿⣻⣿⣦⡁⠀⠛⢻⠿⣿⣿⣿⣿⣿⣿⣿⡛⡯⢹⣤⣿⣿⣿⣍⡙⣿⣌⡹⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡏⢰⣿⣿⣿⣿⣿⣧⠘⢻⣿⣿⣿⣿⠛⠻⢟⣫⢬⣷⣤⣤⣴⣄⡈⠉⠙⠿⠉⣷⣿⣿⡿⢋⣿⡇⢻⣿⣿⣷⣦⣉⣷⠃
⠀⠀⠀⠀⠀⣠⣴⣶⠛⠀⠨⢀⣿⣿⣿⣿⡟⠛⢿⣇⣻⣿⣿⣿⣿⣿⣶⣈⠁⠀⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣬⣽⣿⡖⠛⠁⠀⠘⠋⠹⣧⣶⡿⠃
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⣄⡈⢿⣿⣿⣿⣿⡷⠀⣿⡏⠉⢻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠟⠿⣿⣿⣿⣿⠿⣿⠟⣶⣶⠘⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣶⣾⣿⣿⣿⣿⣿⠶⠋⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⣠⡶⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣏⣽⣟⣻⣽⣿⣾⣿⣿⣿⡿⠗⠀⢁⣤⣿⡿⢿⣿⣶⡶
⠀⠀⠀⠀⠀⣼⠃⠰⠀⠀⠌⠉⢛⡿⠿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣽⣿⡿⣿⣿⣿⣿⠿⠖⠚⠙⠛⠁⠀⠀⠀⠙⢿⣶⡀
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⡟⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣿⡇
⠀⠀⠀⠀⠀⣶⣠⠄⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣼⠃⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⢹⣿⡿⠃
⠀⠀⠀⠀⠀⢻⣇⠀⠀⡀⠀⠀⠀⢢⠙⠄⠀⠀⠀⠀⣿⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⡃⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠛⢿⣤⣆⠀⠀⠀⠀⠤⠀⠀⠆⠀⠀⢿⡋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⣷⣶⣤⣴⣦⣄⢘⣿⣇⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⣀⣀⣄⣀⣄⣀⣀⣠⣀⣤⠜⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡛⠻⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡄⠀⢰⣿⡿⠿⣿⣿⣿⣿⣿⣿⠟⠋⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠉⠁⢀⠀⠸⢿⣿⡇⠀⠀⣯⣿⠆⡀⠈⠉⠉⠉⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⠀⠀⠀⠐⠛⢁⣀⣴⣾⡟⠀⠀⠀⠈⠛⠿⣦⣤⣤⣤⣤⣴⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣉⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣤⣼⣿⣿⣿⣿⡿⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠁⠀⢤⠤⡴⠲⣶⣦⡴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡀⢴⢾⡟⠙⣇⠀⡷⢶⣿⡆⠀⡀⠀⠀⠀⠀⢀⣄⣀⣤⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⡀⡙⠻⡎⠀⠘⠃⣈⠇⡙⢻⣿⣄⡀⠀⠀⠀⠀⢸⣿⣿⠟⠿⢿⣿⣿⣧⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠘⠀⠰⡞⢉⠁⠦⢀⠿⠁⢐⠇⢠⡉⣿⣿⣿⣗⣀⡀⠀⣘⠂⢠⠈⣀⣀⣰⡟⠉⠥⢀⠁
⠀⠀⠀⠀⠀⢸⡦⠀⢠⡀⠀⠀⠃⠀⢀⡗⠀⢀⣿⣈⣑⣆⢈⠄⠨⡂⢀⡘⠹⠋⠁⠈⠉⠉⠛⠛⠿⠆⠀⢴⡏⠠⠉⣷⣔⡀⠈⠁
⠀⠀⠀⠀⠀⠸⠙⣄⠀⠉⡂⠰⣅⠀⣼⢿⠟⠉⠀⣠⠴⠚⣿⠃⢀⠝⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡷⡦⠀⠈⣛⠀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠡⠘⢦⣀⣰⡄⠈⣸⢿⠀⠆⠠⠞⠁⢀⡞⠓⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢼⢿⣦⡀⢀⣀⡀⢀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⣥⠀⠉⣛⢻⣤⣿⣾⣶⣾⠧⣤⣤⣈⣀⡤⢶⢖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⢾⣿⣿⣿⣅
⠀⠀⠀⢀⡔⢀⣀⣀⣭⢷⡶⠷⣿⣿⣿⣿⣿⡷⠿⠯⣍⣠⠴⢱⣾⣥⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⡀⠐⠻⠿⣿⣷⣦
⠀⠀⠀⣢⠒⠉⠈⠀⣀⠀⡠⡴⠋⡽⢻⣿⢿⠦⣤⣀⣀⡨⡟⠉⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡷⠶⣠⠋⣿⠉⠗⣠⡄
⠀⠀⠼⠉⠒⠖⢨⠀⣷⣊⢀⣤⣶⠃⠸⣿⣏⠳⡤⣭⡄⠀⢙⠉⠀⠀⠀⠀⠀⠀⢹⠎⡟⢹⢀⡀⣀⢀⣀⣀⣀⠀⠀⠀⠀⢸⣿⣟⡙⠙⢇⡀⠁⢰⠂⠢⠄
⠀⠀⠀⠀⠀⢇⣀⠸⣯⠉⢉⡿⠹⡀⠀⢸⡍⠀⡇⠀⠑⢄⠠⠀⠀⠀⠀⠀⠀⠀⠚⠐⠃⠙⠛⠚⠋⠙⠉⢻⡗⠀⠀⠀⠀⠘⣿⡁⡅⠙⠀⠉⢀⠉⠈⠷⣆
⠀⠀⠀⢀⠀⠙⠁⠖⢈⢀⡟⠀⠀⣱⡄⠸⡇⢠⠀⠀⠀⠈⠓⠤⢀⡔⣩⢆⣏⠀⡀⢠⠀⣠⣾⡀⣀⠀⣀⠙⡁⠀⠀⠀⠀⠀⢛⠃⠀⣙⣀⢇⠈⠦⠘⠂
⠀⠀⠀⠈⣀⠀⢻⣖⢿⡼⣀⠀⣰⣏⣘⣦⣧⣉⠁⡉⠀⠀⠀⠀⠸⢇⡤⠾⠻⠾⠙⠯⠾⠗⠧⠟⠟⠾⠿⠾⠷⠀⠔⠀⠀⠀⢈⠁⠁⠀⠉⠂⠐⢲⡆
⠀⠀⠀⠀⠘⠁⡀⠁⠘⢧⠉⣹⣿⣧⡛⠛⠫⠹⠇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡰⠆⠑⠈⠉⠳⢿⠀⠈⠃⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠁⠈⣠⠔⠒⠋⠛⡛⡛⠿⠴⠰⣖⢛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠻⣠⢰⣉⣛⢳⠊⢳⣄⠈⢰⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠫⣶⣻⣀⡶⠲⣤⣤⠶⢶⠟⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣣⠗⢄⣀⡻⢄⡾⠻⡟⠶⣀⢸⡄⠀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣽⠐⠣⠌⣳⣶⢶⣾⠓⣼⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣲⣄⢨⠉⠳⠆⣜⠁⣉⡟⠁⢠⠖⠄⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠲⢦⣤⣄⣀⡉⡿⣡⣼⡿⠻⢤⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⢀⣀⣠⣤⡠⠀⠈⣿⣿⣧⡀⠐⠆⠒⠤⢩⡭⢠⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣾⣿⣾⠿⠛⠹⠋⠉⠛⠙⢿⣿⣿⣿⣿⣷⣿⣿⡿⣿⣿⢧⣤⡀⡉⢹⡏⠀⠀⡇⢠⡄⠀⢢⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⢻⠁⠀⠀⠀⠀⡌⢀⣉⣰⣿⠿⠿⡟⠉⢻⣿⣿⠟⠃⠈⡯⠓⠀⠾⠇⢬⣾⡁⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⢄⣀⣤⣶⣿⠀⠀⢻⣿⠀⠠⢤⠈⠘⢿⢟⠀⠀⢨⣧⣤⡄⠄⢠⣿⣿⠏⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠿⠷⠿⠿⠿⠟⠟⠓⠖⠈⠀⠙⠟⠛⢦⡕⠋⢦⣡⣤⣿⡿⠃⠀⢠⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢷⡴⠶⠀⠀⠀⠀⣠⡾⠻⠛⠉⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢧⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣁⣹⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠂⠀⠿⠆⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢈⣽⠛⢻⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠂⣴⡄⠀⢀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⣿⡏⠙⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⡍⢁⡀⠉⠁⠀⠀⣴⣶⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢻⣿⡟⠛⠚⠛⠛⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⡄⠀⢰⣼⣭⣦⣮⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠿⠈⠃⠄⡠⠤⢁⠀⡈⡁⠀⠀⠀⢀⣤⣀⠤⣾⣷⡦⠀⣆⢀⠀⠀⠀⠉⠃⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠀⣁⣦⡤⣀
⠂⠀⡴⢂⣇⢠⠀⠚⢁⡄⣀⣼⠄⠰⠆⡊⠉⠚⢁⡀⣴⠺⡿⠃⣻⠀⠀⣀⠖⢿⡆⠄⠈⠉⠁⠀⣀⠀⠠⢀⠀⡄⠑⡛⠛⠛⢍⣶⣻⡟⢦
⠠⠀⠀⠸⣯⣄⣉⢰⠄⠄⠘⠃⠞⣾⢰⠀⠋⠀⠀⠈⠁⠀⣿⠀⢠⠖⠂⣠⡄⠀⡄⠀⠔⠀⠆⠀⠉⠀⠁⠈⠀⠀⠀⠀⠠⣿⡋⣿⢻⡅⣤
⡃⠀⠠⠘⠀⠀⠉⠯⠀⠀⠘⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢃⣀⡉⠈⠻⠟
⡁⠘⠆⣠⣀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢿⠦⠀⠱
⣧⠂⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡄⠀⢀
⠏⠕⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢲⠾⣃⠂⠘
⠏⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠈⢶⠳
⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠨⢙⣀⡀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢰⠄⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠄⠀⣀⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⠡⢞⡂⠁⡉
⠀⠠⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢
⡄⠀⠀⢉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⡀
⣷⣶⠘⠻⠃⠘⠦⠸⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠛⠀⢠⠀⠐⡤⠤⣏⠙⣠
⡦⣿⣻⣷⣾⡗⠀⣩⠗⠀⡀⠀⠀⠀⠀⠀⠁⠀⢶⠣⣤⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⡀⠀⠀⠉⠉⠁⢷⡟⢦⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇
⠀⠀⠀⠀⢠⣄⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣭⠁
⠀⠀⠀⠀⠈⢁⡙⠛⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⡛
⠀⠀⣤⠐⠻⠿⣿⣷⡦⢤⣤⠀⠀⠀⠀⠀⠀⣀⠠⢤⣴⣶⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⢀⣁⣤⠄
⠀⢀⣼⣴⣶⡄⣀⠈⠻⣏⡀⠀⢀⡀⠂⠉⢁⣤⣖⣋⣩⠄⠀⣿⣿⡇⠸⠀⠀⠐⢤⣾⣟⠋⣽⠷⠉
⠀⠈⠉⣹⣵⣄⠄⣭⢀⡈⢉⣰⣾⣶⣶⣿⣿⣿⣿⣿⡟⠀⠀⣼⣯⡄⢰⡄⠓⠒⠛⣋⡉⣿⠨⠟⢲⣮⣿
⠀⠀⡽⢿⣿⣿⣿⣿⠁⢠⣆⣿⣿⣿⢸⣿⣿⡇⢶⣿⡇⠀⣤⢸⡿⠇⢸⠀⢸⠀⣾⣇⣸⣧⣤⣌⣺⣿⣿⣀
⠀⢶⣷⣾⣿⣿⣿⢋⣰⣿⣿⡿⠟⣻⣿⡿⣿⣿⣿⣿⣧⠀⣉⣼⣿⣷⡘⠶⠌⣴⣝⢿⣿⣟⣟⣿⣿⣿⣿⣿⡇
⠀⣭⣭⣭⣥⣿⣿⡿⠏⠉⠉⠈⠛⠛⠂⠒⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠃⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠀⠈⠻⣿⣿⣿⣿
⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠙⣿⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠛⠉⠙⢿⡏⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⣶⣿⣿
⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⠄⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼
⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿
⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⣿⠀⠘⠟⠛⠃⠀⠀⠀⠄⠀⠀⠀⠀⠛⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡯⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⠋⢏⠏⠻⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡇⣿⠀⢸⡇⠙⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣧⣿⣤⣬⣇⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣤⣄⠀⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣲⣯⡇⠸⣿⣿⠇⢨⣷⣢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣐⠮⠒⠉⠀⡈⣇⣄⠀⢀⣄⠈⡇⠈⠑⠪⢗⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣨⠗⠊⠀⣀⡀⠀⠀⡇⣿⣿⠀⢸⣿⠇⡇⠀⠀⢀⡀⠈⠓⢯⣐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣥⠀⡴⠻⣿⡏⠢⡀⡇⣿⣿⡀⣸⣿⡇⡇⣠⠊⣷⣯⠓⡄⠀⡏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⣇⣤⠀⣤⡀⡇⡇⡏⢉⣥⣍⠉⠁⡇⡇⣠⣄⢠⣄⠀⠀⡃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⣿⣿⣿⣻⡇⡇⣇⡇⡞⢸⠈⡆⢠⡇⡇⣿⣿⣟⣟⠇⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠀⣿⣿⣿⣿⡟⣇⣽⣇⡇⢸⠀⣇⣼⣇⡅⣿⣿⣿⣿⢲⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⠀⣿⡿⠛⣿⣇⠇⡏⡏⡇⢸⠀⡇⠉⡏⣇⣿⣟⠻⣿⡰⠈⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡧⣼⣤⣤⣤⣤⣤⣤⣤⣧⣷⣷⣾⣶⣷⣦⣧⣤⣤⣤⣤⣤⣤⣤⡤⢧⣄

⠀⠀⠀⡤⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡠⡄⠀⠀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⣡⢿⡈⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠃⠀⠀⠀⢩⠄⡠⠹⠄⠀⠀⠀⠀⠀⠨⠥⠀⠠⠬⠀⠀⠈⠁
⠀⠀⠀⠉⣺⠈⠒⠀⠀⠀⣠⣾⣿⡀⠀⠠⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⢸⠿⣿⣿⠿⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠁⠂⠁⠀⡤⠀⠀⠀⢠⠄⠀⣼⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⣲⣆
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⣀⣀⣀⠀⠀⣴⣶⠀⠀⣤⣤⣤⣤⣤⡄⢀⣿⣿⠀⠀⠀⠀⠸⠂⣠⣤⣤⣌⠁
⠀⠀⠀⣴⣿⣿⣧⠀⠀⠀⠈⠉⡧⠀⠀⠈⠀⠀⠀⠻⢿⣿⣿⣿⠀⣸⣿⣿⡇⠐⣞⣳⠂⠘⠋⣉⢉⠙⢧
⠀⠀⠈⣉⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠀⠀⠉⠉⠀⠀⠙⣀⠂
⠀⠀⠀⠈⠠⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⣠⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⢿⠿⠆⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠴⠿⢿⡇⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣶⠆⠀⠈⠛⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⠃
⠀⠀⠐⠿⠾⠿⠂⠈⠀⣀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠘
⠀⠀⠀⠸⢆⡀⠀⠀⠀⠉⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠸⣶⣀⢀⡀⠀⢠⡄⠀⠀⣠⣶⣿⣿⡆
⠀⠀⠀⢰⡌⠃⠀⠀⣠⠀⣭⠀⠀⢀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠀⠀⠘⢷⣤⣴⣿⣿⣿⠟
⠀⠀⠀⠘⠃⠀⠀⠀⠉⠀⠛⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⠃⠀⠐⠀⠋⠙⠉⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⠀⢀⣾⠏⠘⣷⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠈⢳⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣥⠀⠀⠀⡇⢀⢙⠀⠌⠀⡇⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⣿⡿⠃⠀⠀⠀⡇⠀⢸⡀⠀⢰⡇⠀⠀⠀⠀⠶⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠯⠍⠁⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⣾⢠⡆⠘⡄⠘⣄⢰⡠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⣀⠴⠒⠐⣆⢀⠤⠀⠀⠛⠀⠀⠀⠀⢻⣀⣉⡀⠗⠁⠓⠈⡡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠃⠀⠀⢀⣿⠇⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢀⡇⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡄⠈⢸⠀⠌⠀⡀⠀⠈⠁⠀⠀⣠⠀⠀⠈⠛⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠸⠺⠁⣿⠂⢤⠀⡧⣀⢈⡁⣾⠀⣿⠀⣶⢠⡄⣷⢠⡆⣀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⡴⠹⠇⣼⡐⠀⣯⠘⠇⡘⡿⢿⠜⠿⠸⠁⣧⠈⠋⢻⡄⠐⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⣷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣟⡁⠳⣌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣿⣾⠛⢦⡙⢷⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣾⣿⡟⢢⡈⢻⣌⠻⣕⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣱⢿⣿⡟⣧⡀⠹⣆⠙⢗⢬⡳⣝⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣫⠟⣡⡾⢣⣿⠟⣼⡏⠳⣄⡈⠻⣦⡑⢍⠪⢷⣄⡙⠳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣊⡵⣾⠟⣴⡿⢫⣾⠋⣧⡀⠙⢿⠦⡘⢯⡛⢷⣤⡉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣡⠞⣡⣾⢟⣵⡿⠃⣰⡇⠱⣄⠀⠑⢌⠳⢍⡲⣌⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢞⡽⣣⢞⡟⢁⣼⠋⡟⣦⣌⠻⣦⣈⠳⣄⠉⠺⢽⣦⣄⡉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢁⣴⡿⠟⡵⠋⣴⡿⢁⡶⣧⡈⠻⣷⣌⠻⣗⢦⣑⢦⡀⠀⠉⠙⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⣠⠞⣠⣾⠏⣠⡿⢰⡧⡙⢦⣈⠻⣿⣮⣓⣮⣙⠻⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⣠⣞⡥⣺⠟⣡⢞⡝⣱⢻⠇⠈⠪⣿⣷⢤⡙⠪⣝⠻⢷⣦⣌⣉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠋⣡⠞⣡⠞⡱⢋⠞⠁⢸⠀⠀⠀⠈⠛⢷⣮⡓⠲⠽⣶⣤⣉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣫⠵⠊⣡⢞⡴⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡈⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠋⠉⠀⣠⠞⡵⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⡵⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠻⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠛⠛⠓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⢨⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣯⡀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⣀⣤⠶⠛⠁⢹⡇
⠀⠀⠀⠀⠀⠀⣼⠿⠻⠛⠶⣤⣄⣀⡀⠀⠀⠀⠈⠙⠳⣶⣤⣤⣤⣤⠖⢿⡗⢶⣾⣿⣥⣤⣴⡤⠶⠖⠛⠉⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⢻⡇⠀⢈⣧⣤⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠟
⠀⠀⠀⠀⠀⠀⠘⠻⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣷⣤⣾⣋⣡⣿⣿⣿⠶⠦⠤⣤⣤⣤⣤⠤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⢉⣩⡽⠋⠀⣸⡿⣿⡿⢿⣇⠀⠈⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠾⠛⠁⠀⠀⣠⠏⠀⠀⠙⢦⡉⠳⣄⠀⠀⠈⠙⠳⠦⣤⣀⣀⣀⣀⣀⣀⣤⣤⡶
⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠶⠶⠶⠖⠒⠛⠋⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣆⠘⠻⣦⣄⠀⠀⠀⠔⠈⠉⠉⠉⢉⣩⡴⠋
⠀⠀⣀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠈⠛⠷⣦⣌⠀⠀⠀⠀⠙⠿⣥⣀⣀
⢠⡾⠿⠖⠒⢶⣦⡄⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠙⢳⡦⣤⣀⣀⣀⣀⣈⣩⠿
⠀⠀⠀⠀⣰⠟⠁⣀⣠⡤⠶⠒⠛⠋⢹⠏⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠈⠉⠉⠉⠉
⠀⠀⠀⢸⣃⡴⠛⠉⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡟⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⣶⣤⡀
⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⢀⣴⣞⣋⣁⣀⣀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣶⣄⣤⣤⠶⠒⠛⠉⠉⠙⢛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡻⠩⠉⠉⠉⠉⠛⢻⣷⣷⣤⣀⠀⠀⠀⠀⠀⠀⡿⠞⢻⣿⣁⣸⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣆⡀⠀⠀⠀⠀⠹⣧⣀⣼⠋⠙⠲⢦⣄⡀⢸⡇⠀⠀⠉⠉⠁⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠹⣷⣿⣄⣠⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⣤⣤⣄⣀⣀⣤⣤⣴⣿⠏⠁
⣶⣶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠁⢠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢪⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠰⠀⢀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⠀⠀⡀⠀⠠⠀⠀⠀⠈⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠀⢁⠀⠁⠉⠁⢀⠕⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠈⠂⠀⠀⠀⠀⠐⠁⠀⡀⠢⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠊⠀⠀⠀⠂⢀⡄⠐⠠⠃⠀⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢱⢃⢴⠤⠀⠀⠺⠕⢀⠀⠃⠀⠀⣀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢆⠰⠂⠀⠀⠀⠀⣣⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠴⠢⢀⡄⡠⠄⠀⢀⢀⠄⢀⠔⠙⠠⠤⠄⠁⠀⠀⠄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠔⠁⠄⠁⠂⡁⢐⠁⡀⠋⠀⠀⡰⠞⠣⠉⠀⠀⠀⠀⢈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡪⢠⠄⠠⡌⢠⠁⠁⠁⠉⠘⠄⠀⠐⠁⠀⠀⠀⠈⠀⠄⠀⠈⠆⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢈⠆⣉⠐⠀⠤⠀⠀⠀⢀⠄⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠒⠀⠠⠄⠒⠀⠤⡀⠀⢀⠀⢀⢖⡒⠀⢀⡒⠂⠀⠀⢀⠀⢀⡈⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣀⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠰⠔⠁⠀⠀⠀⠤⠀⠀⠨⠀⣀⠀⠠⠃⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣉⠐⠠⡀⠀⠀⠀⠀⠀⡀⠀⠠⡈⠁⠁⠐⠂⠃⠀⠀⡀⠀⠀⠀⠀⣅⡄⠀⠤⠘⠡⠀⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠜⠋⡠⠊⠣⡄⡁⡐⢨⡄⣀⠊⠀⠠⠄⠰⠀⠀⠀⠐⠀⠀⠀⢀⠂⢈⠀⠀⠠⢀⡁⠀⢆⠄⢈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡣⡄⠀⠀⠀⠁⠈⠉⠁⠘⡀⠀⠀⡐⠀⢠⠀⢀⠁⠠⠄⠈⠑⠀⠀⠀⢠⡀⠀⠀⠟⠐⠄⠀⠀⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠛⢃⠴⠲⠲⠛⠠⠂⠠⠰⠏⢀⠀⠁⠀⠀⠴⠂⠀⠑⠂⠀⠀⣄⠀⠀⠀⡀⠘⠐⠁⠀⠀⠀⡄⠔
⠀⠀⠀⠀⠀⠢⠁⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠤⠀⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠂⢀⡀⠁⡀⠀⠡⠐⠁⠄⠀⠂⠁⠀⠀⠀⠀⠁⠀⠀⠁⠠⠉⠢⠀⠀⠀⠬