Excited face Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⢤⣤⣀⡀⠀⠀⠀⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⣴⣾⡿⠛⠷⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶⣶⣤⣴⣶⣶⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠶⠶⠶⠒⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⠶⠶⢶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⡠⢀⡤⢒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠉⠛⠸⡆
⠀⠀⠀⠀⠀⠀⡜⠁⣠⡾⢞⣡⠖⢋⣠⠴⠒⠋⢉⣀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠛⠛⢿⡀⠀⢠⡇
⠀⠀⠀⠀⢀⠞⠀⠀⠩⠖⣋⠤⠞⠋⣀⡠⠄⠚⠉⠀⣀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠙⣆⠉
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠈⣐⣒⣊⣉⣁⣀⣠⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⢀⣿⠀⠀⡠⢂⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⠿⠿⠖⠀⠀⣿
⠀⠀⢸⡇⠀⠠⠔⠁⢀⡎⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⣀⡀
⠀⠀⠘⣿⠀⠀⢠⡿⢶⠄⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⠁⢹
⠀⠀⠀⢹⣇⠀⠈⣰⠟⡄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⢀⡿
⠀⠀⠀⠀⢻⣆⠐⢡⠞⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠁
⠀⠀⠀⠀⢀⠝⠓⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠽⠛⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢸⡀⠈⢹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠹⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣄⣀⢀⣀⣀⣀⣤⠶⢻⡟⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣶⠖⠀⠀⠀⠀⠀⠈⠀⠀⠙⢿⣍⠁⠀⠀⢀⣾⠃⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⢴⡿⠋⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠤⠤⠤⠤⠶⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠻⠿⣿⣿⣿⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣋⠉⡻⠛⠉⠀⠀⠀⠀⠀⠠⠔⠒⠛⠓⠒⠄⠀⠀⠀⠀⠀⠀⠠⠖⠒⠛⠓⠒⠄⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⣿⡷⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡙⠦⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣤⣤⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠶⣶⣶⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣶⣮⣭⣭⣭⣭⣭⣭⣤⣴⣶⣿⣿⠏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡋⠉⠉⠉⠉⠉⣻⡿⠋⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⠋⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣠⣤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠛⠋⠉⠉⠉⠉⠉⠙⠛⠛⠿⠿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠮⢍⡛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠑⠢⢈⠑⠊⠻⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠄⠀⢀⠞⠀⠀⣄⠻⣄
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡉⠛⠒⠒⠀⠀⠀⠀⣀⠀⠠⠄⠀⠀⢠⡾⠋⣠⣴⠖⠒⠀⠴⠃⠀⠀⠀⠈⢦⢹⣆
⠀⠀⠀⠀⣰⢯⣤⡶⢦⣤⣤⣄⣀⠀⠑⠢⢤⣉⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⡄
⠀⠀⠀⠘⠁⢰⡇⠀⠀⠉⠙⠛⠿⣿⣶⣦⣄⡈⠉⠓⠒⠤⠤⠐⢀⣀⣤⠴⠛⢁⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⣿⣧
⠀⠀⠀⠀⠀⢸⣇⡞⡠⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣶⣶⣦⣤⣤⣤⡤⠤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠸⡀⢹⣿
⠀⠀⠀⠀⠀⢸⣿⠰⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢧⢸⣿
⠀⠀⠀⠀⠀⢸⣿⡇⡜⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣄⠀⠀⠀⠀⠀⠀⠁⠀⢸⣸⣿
⠀⠀⠀⠀⠀⠸⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⡇⢸⣿⡏
⠀⠀⠀⠀⠀⠀⢿⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⡜⠀⠀⠀⠀⠀⠀⠀⡴⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⢸⠀⣴⣌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⣀⠀⠃⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⢰⡏⠙⣿⡄⠀⠀⠀⠀⠀⠿⣿⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⣀⠆⠀⠀⠀⠀⠀⣿⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠘⢿⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⢸⣿⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠞⠙⠻⣶⠲⠤⣤⣤⣀⣤⣤⣤⣤⣤⣶⠶⠿⡆⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠈⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠈⡙⠲⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠉⠛⠶⠆⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⠶⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢴⣶⣶⣶⣶⣶⣦⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦⠀⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠿⠟⠁⠀⢰⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⡐⡰⡀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⡠⡀⡤⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⣀⣀⣀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⡤⠤⠀⣀⣀⣀⣠⡀⠠⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄⠀⠀⠠⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠚⠛⠛⠛⠛⠓⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢠⠴⠒⠚⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⠲⠶⢤⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣴⣿⣿⣷⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠙⢿⣿⠟⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣿⣿⡟⠒⠲⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠶⣾⣿⡗⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣤⣤⣀⣀⣉⣉⣉⣉⣉⣉⣡⣤⣤⣶⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⢤⣤⣤⣤⣤⡤⠤⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⠛⠛⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⠟⣻⠏⠀⠀⠀⢀⣀⣀⠀⠈⢻⡋⣛⡶⠖⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⡼⠁⣰⠏⠀⢀⣴⠟⠉⠉⠉⢳⡄⠰⠛⣁⡤⠄⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣼⠁⢠⡏⠀⢠⡟⠀⠐⠒⠦⣄⠀⡇⣠⠞⣁⣠⣤⣀⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠃⠀⡾⠀⢠⡟⠛⠛⠛⠛⠒⠚⠳⠿⠟⠋⠁⠈⢧⠈⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠃⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡏⢠⡟⠀⠠⠤⠄⠀⠀⠀⠀⠀⠖⠦⠀⠀⠀⠀⣿⠈⠳⣄⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⡀⢀⠞⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⠀⠙⣦⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣼⠁⢠⠃⠀⢠⠏⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢸⡇⠈⢳⡀⠘⡇⢹⡆
⠀⠀⠀⠀⠀⠀⣼⠃⣰⠃⠀⢠⡏⠀⠀⠀⠀⠈⠿⠿⠁⠀⠀⠀⢿⡿⠃⠀⠀⠀⠘⣇⠀⢈⣇⣀⡟⠀⢻⡄
⠀⠀⠀⠀⢀⡾⠃⣼⠃⣠⡴⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠛⠉⠉⠙⣆⠀⢷⡀
⠀⠀⠀⢠⡟⠀⢰⠇⢸⢣⠤⣄⣷⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⣏⠙⠃⣿⠀⠘⣧
⠀⠀⠀⡾⠀⠀⣿⠀⢸⠀⢠⠏⣿⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠆⠀⠀⠀⢀⡴⠂⠀⢸⣿⠃⣿⠀⣰⠏⠀⠀⢻⡄
⠀⠀⢰⡇⠀⠀⡏⠀⠈⢧⡈⠀⢿⠀⠀⠈⠑⠲⣤⣤⣀⣀⣀⣀⣠⣤⠖⠋⠀⠀⠀⣼⡛⣀⣠⡾⠋⠀⠀⠀⢸⡇
⠀⠀⠘⡇⠀⠀⣧⠀⠀⠈⠛⠦⢼⡆⠀⠀⠀⠀⢻⣷⣶⣦⣤⣾⣿⠋⠀⠀⠀⠀⣰⠏⠉⠉⠁⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⢿⡀⠀⠘⢦⡀⠀⠀⠀⡀⠹⣆⡀⠀⠀⠀⠹⢤⣁⣤⠞⠁⠀⠀⠀⠀⣴⠏⢠⠶⠒⢶⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⢷⡀⠀⠀⠙⠶⣄⠀⢸⡄⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠸⣦⠐⠋⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠛⢶⣄⣀⠀⠀⣠⡾⣀⠀⠀⠀⠉⣳⠦⣤⣀⣀⣀⡤⠖⠋⠻⣄⡀⠀⢀⣈⡿⠶⢤⣤⡴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⠙⠓⠒⠚⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠔⠂⠐⠒⠉⠉⠐⠊⠁⠐⠤⠒⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⡀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀
⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⡠⠒⠒⠁⠀⠈⠁⠀⠀⠀⠀⠀⠑⠢⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⢀⠸⡀⠀⢠⠀⠤⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠤⢄⠀⠀⠀⠀⢠⡁
⠀⢠⠁⠀⡑⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⢔⠊⠀⢠
⠀⠄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡄
⡎⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘
⢣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣀⣠⠄⠀⠀⠀⠀⠀⢡⢀
⠘⠀⠀⠀⠀⠀⠀⠐⠤⠤⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⠀⠀⢀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠀⣀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⣀⠀⠤⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠿⠟⠛⠛⠉⠉⠙⠛⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡟⠉⣁⣠⡤⠤⡖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣻⣟⡥⠖⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⣯⣾⠛⣡⣶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⢠⡟⣡⢞⣭⠶⠚⠋⠉⣉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠶⢦⣌⢻⣆
⠀⠀⠀⠀⠀⢠⣿⠏⣠⠀⠈⣟⡵⠋⠥⠖⣚⡻⠭⠿⣛⣑⣀⡈⠉⠓⠦⠀⠀⠀⠀⠀⠤⠴⠶⠚⠛⠛⠛⠋⠉⠁⠂⠀⠙⠿⣿⣷⣄⣀⡀
⠀⠀⠀⠀⢠⣿⡏⣴⠃⢀⠀⠹⣄⣴⣶⡯⠭⠿⣿⣀⠀⢀⠈⠉⠙⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠰⠶⠶⣶⣿⡋⠉⠉⠁
⠀⠀⠀⠀⣾⣿⣰⠃⠠⠁⡄⠀⠈⠉⠋⠙⠛⠶⣤⡙⢷⣄⠙⠒⠶⠤⠤⠬⠉⠒⠲⠦⠴⠖⠂⠈⠉⠁⠀⠀⠀⣀⣀⣤⣤⣀⡉⣿⡆
⠀⠀⠀⠀⣿⣇⡏⢠⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠙⢷⣦⣤⣤⣤⣄⣀⣀⣠⣄⣠⠄⠀⠀⠀⣀⣤⡶⠛⠉⠁⠀⢄⠀⣇⢹⡆
⠀⠀⠀⠀⣿⣿⡇⣼⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣍⣁⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⠿⠛⠁⠀⠀⠀⠀⠀⠈⣧⢸⣾⡇
⠀⠀⠀⠀⢿⣿⡇⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⠇
⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⢀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠀⠀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⢰⠏⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣛⢶⡀
⠀⠀⠀⢿⡀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣦⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⣸⡇
⠀⠀⠀⠘⢿⣄⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠀⠀⠀⠀⠀⠀⣾⡏⠀⣰⡿
⠀⠀⠀⠀⠈⠻⣷⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣼⠟
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⣤⠶⠟⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠈⠓⢶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠿⠆⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⣍⠙⠛⠛⠛⠒⠒⠚⠟⢛⣻⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⡀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⣦⣤⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣤⣤⣴⡶⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣀⣀⣀⡤⠤⢤⣀⣀⡀⠀⣀⣤⣄⣀⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⢀⣀⣀⣀⣀⣀⣀⣉⢻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠾⠯⢍⣉⣉⠓⠒⠦⠤⣀⠉⠉⠛⠻⢿⣿⣿⡆⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢄⡀⠉⠒⠦⣀⠀⠈⠙⠿⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⡿⡗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣈⠙⠢⣄⠀⠈⠙⠢⣄⠀⠘⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢸⠁⠹⣄⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢝⡲⢤⡀⠀⠑⢦⡸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⡜⠀⠀⠈⢳⣄⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣤⣽⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⢀⡼⠁⠀⠀⡰⠃⠈⠻⢶⣤⣀⡀⠀⠉⠑⠲⠤⢄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣅⣠⢧⠹⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠀⠀⡠⠊⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠙⠻⠭⣖⣒⠤⠤⠤⠤⠤⠭⠭⠝⢛⣋⣩⣤⣒⣚⣋⣉⠤⢎⡀⠈⠻⠟⠛
⣿⡿⠋⢀⣠⠔⠋⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢌⡉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠲⠤⠬⠭⢤⠤⡤⠚⠁
⣿⣿⣿⡿⢲⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢸⣤⣀⣀⣀⣤⣾
⣿⣿⡿⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠤⢤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢿⣿⣿⣿⣿⣿
⣿⣿⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢨⣿⣿⣿⣿⣿
⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢾⡁⠀⠀⠀⠀⠀⠀⣠⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀⠀⠤⠤⠤⠤⠔⠀⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠄⠐⠂⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠶⠒⠦⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣐⣺⠄⠉⠁
⠀⠲⠆⠠⠗⠒⠚⠺⠛⣇⡀⠀⠀⠀⠀⡴⢊⠀⠤⠄⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠜⠋⡉⣠⠀⢠⠠⠈⠄⠀⠀⠀⠀⠢⠀⠀⢀⡀⠠⠐
⠐⠏⠀⠀⠒⠀⠒⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂⠀⢀⠂⠈⡇⠀⠀⠀⠀⢸⠁⠀⠄⠠⠀⠈⠀⠀⡄⠀⠀⠀⡇
⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⡨⠀⡄⠀⠀⠀⠀⠀⠠⠐⠱⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢇⠙⠁⠀⠀⠀⢣⠀⠀⠀⠈⠀⠘⠁⠀⠰
⠀⠡⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡈⢄⠁⢳⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡤⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡈⠀⡄
⠀⠀⠈⠒⠢⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠠⠃⠀⠀⢀⠀⠈⠶⠌⠚
⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡦⡢⡀⠀⠀⠀⡠⠒⠉⣉⣁⠉⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⡤⠂⠁⠀⠈⠁⠀⠂⢄
⢆⡀⠢⠼⠶⠗⠶⠶⠂⢸⡘⠘⠀⠀⠈⠆⠈⠁⠉⠐⠁⠓⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠐⠀⠀⢿⣆⠀⠀⢀⠐⠁⠀⠀⠀⠐⠈⠆⢄⠀⠀⢵
⢨⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠃⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠀⣀⠀⠀⠀⢻⡞⡄⠀⠘⠀⡈⠄⠀⠀⠀⠀⠀⠀⢃⢀⠂
⠀⠀⠀⠁⠀⠈⠀⠀⢘⠈⠀⠈⠀⠀⠐⠄⠀⠀⠁⠀⢠⡀⢸⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠐⡌⡈⡇⡇⠀⠀⠀⢸⠀⠀⢀⠀⠈⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠸⣤⡀⡀⠀⠀⠀⠀⢰⢇⠃⠃⠁⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⢀⡼
⣀⣀⣀⣀⣀⣀⣀⣘⣶⣆⣛⣁⣀⣀⣀⣀⣈⣒⣤⣄⣒⣁⣀⣀⣀⣀⣀⣀⣀⣀⣉⣈⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣈⣂⣤⣀⣠⣒⣉⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠛⠋⠭⠭⣉⡉⠛⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠈⢙⡿⠋⠉⠀⠀⠀⠀⠉⠑⠢⢄⡀⠈⠑⠒⢤⡀⠈⠙⠻⢍⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠⣤⡤⠖⢉⢿⡒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠈⠑⠦⣀⠀⠉⠲⢴⣦⣄⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣼⣿⣿⠏⠀⠀⢸⠈⢳⡀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠈⠑⠢⢄⡀⠉⠻⣿⡄⠀⠀⠸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢰⣿⣿⠃⠀⠀⠀⡌⠀⠀⢙⣦⣄⠀⠀⠀⠉⠒⠢⢄⣀⠀⠀⠀⠀⠀⠉⠒⠢⠤⠄⣈⣁⣲⡾⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⠃⠀⠀⢀⠞⠀⠀⢠⠎⠀⠑⢭⡒⠤⢄⣀⠀⠀⠀⠉⠓⠒⠂⠤⠤⠤⣤⡤⠶⠚⢉⣀⠴⢳⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⡼⠃⠀⢀⠔⠁⠀⣀⠔⠁⠀⠀⠀⠀⠈⠑⠒⠤⣉⣉⠐⠒⠒⠒⠒⠒⠈⠉⢉⣉⡩⠽⠯⠥⢶⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠰⠭⣤⣔⣊⣁⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠐⢉⡀⠀⣈⠃⠉⠉⠀⠐⠂⠀⠉⠉⠓⠠⠤⠀⢠⣤⣤⡞⠁⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⠀⠀⠀⠀⠀⠀⡨⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠘⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⣼⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⣿⣿
⡟⠛⠋⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣄⠀⠀⠈⠛⠻⠿
⣷⣦⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⢄⣀⣀⠄⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢀⣠⣤⣤⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠁⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⡄⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠠⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⢸⠛⣛⠳⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢭⣉⡇⠀⠀⢠⢾⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠚⠈⢿⡿⠃⠉⠀⠀⣸⠫⠉⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠛⡇⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⡆⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⢿⡇⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠙⠳⠴⠚⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⡀⠀⠀⠀⢀⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠉⠉⠉⠉⠉⠉⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠁⠀⠹⣆⠙⠢⡀⠀⠀⠀⠀⠀⣰⠋⠀⣰⠃⠀⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠸⡆⠀⠉⢢⡀⠀⣠⠞⠁⠀⣰⠃⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠙⣶⡃⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢠⡞⡏⢣⡀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠋⠀⡇⠈⠳⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠒⠒⠒⠒⠲⠖⠗⠶⠒⠒⠒⠒⠒⠒⠒⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠾⠿⠿⠛⠛⠻⠿⠿⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠻⣦⠀⠀⠀⠀⠈⠙⢿⣦
⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠿⠷⢦⣄⠀⠀⠀⢀⣤⣶⡿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠉⠳⡄⣰⡿⠋⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣠⡤⣾⡟⠁⠀⠙⠶⣤⣀⣀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣀⡆⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠀⠀⠀⠀⢶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢀⡆⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠙⠻⢿⣿⣮⡙⠛⠿⣿⣿⣿⡛⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠈⠉⠛⠀⠀⠉⠛⠻⠎⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣄⡀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⢠⠾⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠂⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠀⠀⠐⠞⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠈⠙⠶⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣀⣒⠒⣒⣀⣡⣿⠇⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣋⠉⠛⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠤⢀⣀⡀⠤⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠴⠶⠶⠦⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠒⠲⠦⠤⢽⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠓⣒⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠋⠉⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⠀⠀⠳⡄⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⢀⡴⠃
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡋
⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠉⠙⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠆⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⣆⠀⠀⢀⣴⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠹⣄⣠⣾⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣵⣶⣎⢱⡀⢻
⠀⠀⠀⠀⠀⡴⠋⣀⣀⣈⠙⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢟⡼⠁⠸⡆
⠀⠀⠀⠀⢸⠁⡜⠁⠀⠈⣷⡀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡯⠞⠁⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⠉⠉⠀⠀⠀⢳
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠛⠉⠉⠉⠉⣉⣩⡿⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠉⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⡉⠁⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠋⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣀⣀
⠀⠀⠀⠀⠀⡐⠈⠀⠀⠀⠀⠉⠳⡀⠀⠀⢀⠆⠁⠀⠀⠀⠈⠑⢆⠀⠀⠀⠔⠁⠀⠀⠀⠀⠙⢦⠀⠀⠀⠠⠊⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢸⠀⢐⣤⠀⠀⣴⡂⠀⢱⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠘⡆⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⠀⠀⠀⡸⠀⠀⡆⠀⠀⠀⠀⠀⣠⠀⢠⡇⠀⢇⠀⢀⡠⠤⠤⢄⡀⠀⡟⠀⠘⡀⠀⠀⡀⣀⣀⡀⠀⢸⠁
⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠀⢀⠴⠁⠀⠀⠘⢦⡈⠑⠒⠚⠁⡠⠎⠀⠀⠈⠢⣀⠀⠀⠀⠀⣡⠜⠁⠀⠀⠑⢄⠀⠀⠉⠉⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠊⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉
⠀⠀⠀⠀⠀⠀⡠⠤⠴⠤⢤⣀⠀⠀⠀⠀⠀⢀⠠⠤⠦⠤⣄⡀⠀⠀⠀⠀⢀⠠⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠈⢢⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⡔⠁⠀⠀⠀⠀⠀⠙⢆⠀⠀⢠⠊⢀⡀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⡇⠈⠉⠁⠀⠀⠉⠀⠀⡇⠀⢸⠀⠘⠓⠀⠀⠘⠛⠀⢸⠀⠘⠀⠀⠶⠀⠀⠐⠺⠁⢸⡆⠀⡆⠀⠀⠘⠀⠀⠋⠂⠀⣷
⠀⠀⠀⠀⢇⠀⠀⠠⠲⠄⠀⠀⢠⠃⠀⠸⡀⠐⠢⣀⣤⠤⠆⠀⡼⠀⠘⡆⠀⢦⣠⢆⡴⡄⠀⣸⠁⠀⢣⠀⠀⠀⠀⠀⣠⠀⢀⡏
⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⣀⠴⠃⠀⠀⠀⠙⠦⣀⠀⠀⢀⡠⠞⠀⠀⠀⠈⠢⣀⠁⠈⢀⣠⠔⠁⠀⠀⠀⠑⢄⡀⠁⠉⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⢤⡀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⡠⠊⠀⠀⠀⠀⠀⠈⢦⠀⠀⢀⠊⢀⣀⠀⢀⡀⠈⢳⡀⠀⠀⡐⠁⣀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠺⠁⠀⢹⠀⠀⠃⠀⠀⠂⠀⠀⠂⠀⠈⡇⠀⡆⠀⠘⠟⠀⠈⠿⠁⠀⣷⠀⢰⠁⠈⢿⠇⠀⢺⡟⠀⢸⡀
⠀⠀⠀⠀⠸⡀⠀⢠⡤⢴⠀⠀⠀⡼⠀⠀⢇⠀⢀⣀⣀⣠⡅⠀⢠⠇⠀⢣⠀⠀⠀⠀⠀⣀⠀⢀⡏⠀⠈⡄⠀⣆⡀⢠⠀⣐⠆⣸
⠀⠀⠀⠀⠀⠑⢤⣀⠉⠁⢀⡠⠞⠁⠀⠀⠈⠢⣄⡈⠉⠀⣀⠴⠋⠀⠀⠀⠳⢄⣀⠈⠉⣁⡠⠊⠀⠀⠀⠘⠦⣈⡣⠬⢖⣩⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣄⡀
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠉⢦⠀⠀⠀⡠⠊⠀⠀⠀⠀⠈⠳⡄⠀⠀⢠⠊⠁⠀⠀⠀⠀⠙⢆⠀⠀⠀⡔⠁⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⡈⠀⠠⠀⠀⠀⠄⠀⠀⡇⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠃⠐⣤⠂⠀⢠⣄⠀⠘⡆⠀⠘⠀⡔⣦⠀⠀⡤⠄⠀⢳
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⣀⠀⠀⡏⠀⠰⡄⠀⠀⠀⠒⠄⠀⠀⢸⠁⠀⡄⠀⢀⠀⠀⠀⠀⠀⢠⠇⠀⢰⠀⠉⠁⣀⣀⣀⡀⠀⡼
⠀⠀⠀⠀⠈⠢⡐⠋⠁⠉⠉⣠⠞⠀⠀⠀⠑⢄⠀⠐⠂⠀⢀⡴⠃⠀⠀⠘⢦⡀⠈⠒⠊⢉⡡⠎⠀⠀⠀⠳⣄⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠟⠛⠛⠛⠉⠉⠛⠛⠛⠻⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⢀⡠⠄⠤⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢀⠞⠁⢀⣀⡀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢠⠊⠁⢀⣀⣀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢀⠃⢠⣾⣿⠏⠉⠁⠀⠀⠀⢡⠀⠀⠀⢀⠃⢀⣾⣿⠏⠉⠙⢦⠀⠀⢡⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣧⣀⣀⡷⠀⠀⢨⠀⠀⠀⢨⠀⢸⣿⣿⣧⣀⢀⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢃⠘⢿⣿⣿⣿⡿⠃⠀⢀⠎⠀⠀⠀⠀⢆⠈⠻⣿⣿⣿⡿⠋⠀⢀⠎⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡈⠉⠁⠀⣀⠔⠃⠀⠀⠀⠀⠀⠀⠑⢄⡀⠈⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠙⡆⠀⠀⠀⣿⠁
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠤⠀⠀⠸⡄⠀⢀⠇⠀⠀⢠⣿
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣤⣤⣤⣤⠴⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣶⣤⣤⣤⣤⣤⣤⣶⡶⠾⠟⠋⠉

⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⢰⠋⢉⡁⠀⣴⠀⠀⠠⢀⠐⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⠛⠘⠋⠀⠀⠀⠀⠀⢁⠀⢀⠀⠺⣆⠀⠠⡖⠻⠈⠉⠖⠳⠆⠀⠀⠂⠀⢠⠠⢤⣀⡄⠀⠀⠀⠀⠀⠈⡚
⠀⠀⠀⠀⠀⠊⠀⠀⠀⣠⡄⠀⠀⣀⠀⢠⠥⠆⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠠⡀⠀⠄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠛⠂⠀⠀⣨⠄
⠀⠀⠀⠮⠁⠐⠀⢢⠀⢐⠀⠀⠀⠀⡴⡀⠂⠐⠀⡀⠀⠀⠀⠀⡐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠉
⠀⠀⠀⠠⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⢧⠀⠈⠀⠀⢁⡀⠀⠀⠀⠠⠀⠀⠈⠂⠀⠀⠀⠹⠀⠂⠀⠂⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⠛⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠃⢀⢀⣠⠄⠐⣂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠖⠀⠀⠀⡄⠀⠀⠀⣠⡴⠶⢖⢀⠀⠀⠀⠀⠒⠀⠉⠁⠀⡀⠀⠀⠀⠀⢤⠆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⢤⠀⠀⠀⠀⠁⠀⠀⢰⠏⠀⠀⠈⡖⠆⠀⠀⡄⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⣀⡄⠀⠀⠀⠾⢦⠀⡀⢀⣆⠴⠀⠈⠀⠀⠄⠄⠀⠊⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠁⠀⠀⠈⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⢒⡀
⠀⠀⡦⠀⠀⠀⠀⣤⠀⠠⠊⠀⠀⠀⠘⠄⠀⠘⠂⠆⠀⠂⠫⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠁
⠀⠀⠅⠀⢀⠀⠀⡉⠀⠐⠀⠀⡠⠤⠆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠈⠀⠀⠀⠶⠒⠂
⠀⠀⡀⢀⠔⠒⣐⠒⣜⠀⠀⠠⣄⣀⠀⡤⢤⠑⠀⠀⠀⠀⠂⠈⡆⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠠⠇⠑⠀⠁⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠠⢀⠀⠀⠀⠘⠀⠀⠐⠂⠀⠁⣀
⠀⠀⠀⠁⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⡶⠶⠿⠿⠟⠛⠛⠿⠿⠷⠶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠋⠉⠉⠙⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠛⠛⠛⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠸⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣤⣤⣀⠀⠙⢷⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠁⠀⠀⠀⠈⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠰⣶⣦⣄⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⢹⣧
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠿⠟⠛⠛⠻⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⣷⣶⣿⣿⣿⣿⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠈⢿⣿⣿⣿⣿⣿⠟⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠉⠙⠛⠉⠁⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣄⣀⣀⣀⣠⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠈⠉⠛⠻⠷⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⡶⠾⠟⠛⠉⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⣼⠟⠉⠉⠉⠛⠻⠶⠟⠋⠉⠉⠉⢻⣆⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⣀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣛⣿⣷⣶⣶⣶⣶⣶⣶⣿⣟⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠛⠛⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠐⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠙⢦⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⣄⠈⠙⠳⣤⡀
⠀⠀⠀⠀⢀⣴⠟⠁⢀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⢳⣄
⠀⠀⠀⣠⡟⠁⢀⡴⠋⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⢻⡑⣆⠀⠀⢻⡆
⠀⠀⣰⡟⠀⢠⡟⠁⣴⠃⠀⠀⠀⢀⠶⠶⠶⠶⠶⣶⣶⣶⠶⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠘⡇⠘⣧⠀⠀⢿⡀
⠀⢠⡿⠀⢀⡟⢀⡞⣿⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⣷⠀⠘⣇⠀⠸⡇
⠀⣼⡇⠀⢸⠀⡾⠀⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠋⠙⠃⠀⠀⠀⠀⠀⠀⠚⠛⠆⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⣿⠀⠀⢻⠀⠀⣷
⠀⣿⠁⠀⠀⢰⡇⠀⡇⠀⠀⢠⡇⠀⠀⠀⠀⢀⡤⠶⠒⠒⠶⣄⠀⠀⠀⢀⡤⠖⠒⠲⢦⡀⠀⠀⠀⠀⢻⠀⠀⠀⡧⠀⠀⢸⡆⠀⣿
⠀⣿⠀⠀⠀⢸⠀⠀⢿⠀⠀⣸⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠘⡆⠀⡴⠃⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡄⠀⢠⡇⠀⠀⢸⠁⠀⣿
⠀⣿⡂⠀⠀⠈⠀⠀⠘⡆⠀⡟⠀⠀⠀⠘⠀⠀⣠⣤⣤⠄⠀⠀⠃⠘⠁⢀⣴⣶⡶⢄⠀⠈⠀⠀⠀⠀⠘⡇⠀⣸⠁⠀⠀⢸⠀⠀⣿
⠀⢻⣇⠀⠀⠀⠀⠀⢀⣹⣄⡇⠀⠀⠀⠀⠀⢸⣿⠟⠳⣴⡆⠀⠀⠀⠀⢸⡿⠋⢳⣾⠀⠀⠀⠀⠀⠀⠀⣇⣠⣇⠀⠀⠀⠈⠀⠀⡿
⠀⠘⣿⡄⠀⠀⠀⢠⠏⠁⢸⡇⠀⠀⠀⢂⠀⠀⠻⠦⠴⠟⢀⣀⣀⠀⠄⠈⠛⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⢷⠀⠀⠀⠀⢠⡇
⠀⠀⠹⣿⡀⠀⠀⣿⠀⠉⢿⠁⠀⠀⠀⠀⠁⢄⣀⣀⣀⣴⡾⠁⠀⠉⠂⠐⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣷⠋⠓⡼⠀⠀⠀⠀⣼⠁
⠀⠀⠀⢹⣷⠀⠀⠸⣦⡀⢸⠀⠀⠀⠀⠀⠀⡔⡉⡁⠀⠸⡇⠀⠀⠀⠀⠀⠀⠔⠐⠐⠀⠀⠀⠀⠀⠀⠀⣿⢀⠔⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢻⣧⠀⠀⠈⠉⢹⡄⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣿⡄⠀⠀⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⠀⠀⠀⣿
⢀⣴⠂⠀⠀⣾⠁⠀⠀⢸⡇⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⡇⠀⠸⣇⠀⠀⢿⣆
⢸⣧⣀⣠⡼⠃⣠⠀⠀⣿⠃⠀⢸⡏⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⠀⢸⣧⠀⠀⠻⣷⣄⣈⠻⣷⣶⠞
⠘⢿⣿⣥⣴⣾⠏⠀⣼⠏⠀⢀⣞⡴⠁⠈⠳⣄⠀⠀⠀⠀⠀⣀⠳⠞⠁⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⢻⣧⡀⢿⣦⣀⠀⠈⠙⠛⢛⡟⠁
⢤⣤⣶⣾⣟⣁⣤⠞⢃⣀⣴⠟⠋⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠈⠓⠂⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠙⠻⠶⠽⠿⠿⠿⠶⠛⠉
⣀⣈⣁⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣻⣶⣶⣶⣶⣟⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠤⠒⠢⢄⠀⠀⠀⠀⠀⠀⡔⢢⠀⠀⠀⠀⠀⢀⠔⠂⠐⠒⢄⠀⠀⠀⢀⠔⠒⠐⠒⢄
⠀⠀⠀⠀⠀⠑⠓⠀⠐⠓⠁⠀⠀⠰⡂⠨⠄⠠⠁⢐⠆⠀⢀⠃⠖⠀⠀⠸⠊⡆⠀⠀⡎⠸⠀⠀⠘⠃⢃
⠀⠀⠀⠀⠀⢄⡀⠂⠂⣀⠄⠀⠀⠀⢈⠂⢀⠐⢰⠁⠀⠀⠀⢇⠀⠠⠤⠂⣠⠃⠀⠀⢣⡀⠠⠤⠄⢀⠆
⠀⠀⠀⠀⠀⠀⠈⠐⠊⠀⠀⠀⠀⠀⠘⠂⠁⠈⠐⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⠉⠒⠒⠊⠁
⠀⠀⠀⠀⠀⣠⠔⠒⠒⢤⡀⠀⠀⡠⠒⠒⠒⠒⠒⠒⢄⠀⢀⡖⠒⠒⠒⠒⠒⡄⠀⠀⢀⠤⠒⠒⠂⢄
⠀⠀⠀⠀⢰⠁⠗⠀⠐⠇⢱⠀⠀⡇⠘⠃⠀⠀⠘⠃⢸⠀⢸⡇⠟⠀⠀⠘⠃⡇⠀⢠⠁⠷⠀⠀⠐⠗⢱
⠀⠀⠀⠀⠸⡀⠀⠠⠐⠀⡸⠀⠀⡇⠀⠰⢤⠤⠂⠀⢸⠀⢸⡇⠑⢄⣀⠜⠁⡇⠀⠘⡄⠀⠦⠀⠔⠀⡸
⠀⠀⠀⠀⠀⠈⠒⠤⠔⠊⠀⠀⠀⠙⠤⠤⠤⠤⠤⠴⠋⠀⠈⠣⠤⠤⠤⠤⠴⠃⠀⠀⠈⠐⠠⠤⠐⠊
⠀⠀⠀⠀⠀⢀⠤⠄⠤⢄⠀⠀⠀⠀⢀⡠⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠠⡀
⠀⠀⠀⠀⢰⠡⠆⠀⠀⠶⢱⠀⠀⢠⠃⠲⠀⠀⠆⠑⡀⠀⡴⠒⣶⠒⠒⢲⡖⢲⠀⠀⠌⢰⡆⠀⣶⠈⠂
⠀⠀⠀⠀⢸⡀⢄⠀⢀⠄⢸⠀⠀⠸⡀⠀⣀⣀⠀⢀⠃⠀⠑⢄⠀⢀⣀⠀⢠⠞⠀⠀⡄⢀⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠑⠠⠬⠤⠔⠁⠀⠀⠀⠙⠢⠤⠠⠔⠁⠀⠀⠀⠀⠑⠤⠤⠞⠁⠀⠀⠀⠈⠢⠈⠈⠠⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⠤⣒⣒⠂⠀⠠⠖⠚⣉⣉⣉⠉⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⣡⠖⠉⢁⣀⡀⠠⠖⠋⣉⣉⣀⣉⣉⠙⠒⢤⡈⠑⢦⡰⡄
⠀⠀⠀⠀⠀⠀⡔⡽⢡⠞⢀⡴⠊⠁⣀⣤⠤⠖⠋⠉⠀⠀⠀⠉⠙⠲⣄⠉⢦⡀⠳⡜⣄
⠀⠀⠀⠀⠀⡜⡸⢡⢏⡴⠋⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠳⡄⠹⡜⣆
⠀⠀⠀⠈⣿⣶⣷⣟⣭⢶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠱⡄⢳⡘⡄
⠀⠀⠀⠀⡏⡏⣿⣿⡄⡾⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄⠀⠈⣧⡹⣄⢳⡹⡄
⠀⠀⠀⢸⡇⡇⡟⣿⣿⡇⠀⠾⠋⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠁⠀⢸⣟⠮⣀⣿⣷⣀⡀
⠀⠀⠀⢸⢣⡇⡇⣿⣿⡇⠀⠀⡴⡖⡛⢳⣆⠀⠘⡆⠀⠀⠞⡏⠈⢹⢳⠀⠀⠀⡿⣿⡄⡀⣏⠙⣏⠁
⠀⠀⠀⢸⢸⡇⡇⣿⡏⣧⠀⠀⠀⠉⠀⠈⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢯⠁⢇⢹⠀⢸⡄
⠀⠀⠀⢸⢸⡇⡇⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡀⠸⡈⣇⠀⣧
⠀⠀⠀⢸⢸⡇⡇⡿⣧⣏⠀⢀⡀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣿⠀⣇⠀⢧⠸⡄⢹⡀
⠀⠀⠀⣸⢸⡇⡇⡇⣿⣇⠳⣿⠙⣄⡀⠀⠀⠈⠉⠂⠉⠀⠀⠀⠀⣀⡞⢸⡷⢁⣿⣇⠸⡄⠈⢆⠹⡄⢷
⠀⠀⢀⡟⢸⢣⣇⡇⣿⡸⡄⢹⡄⢿⠹⡄⢀⡴⠒⠉⠉⠢⡀⠀⡾⢹⠀⣼⠀⡼⡸⠹⡄⢱⡀⠈⢧⠙⣌
⠀⠀⣼⠁⡼⣸⢸⠁⡏⢣⣧⠘⡇⢸⡄⢻⣎⣴⣭⣭⣭⣶⡹⡼⠁⣾⠀⡏⢰⣧⠃⠀⢳⠀⠳⡀⠀⠳⡌
⠀⣰⠃⣰⢣⡏⡟⠀⡇⢈⡿⡄⣧⠈⣇⠀⠻⣿⣿⣿⣿⣿⡿⠁⢀⡇⢰⠇⡼⡏⠀⠀⠈⢧⠀⠁
⡼⢁⡴⢁⡞⣸⠁⢰⠃⡸⢻⡹⡜⣆⢹⡀⠀⢹⡄⠀⢀⡼⠁⠀⣼⢁⡟⡼⣼⠁⠀⠀⠀⠈
⠰⠋⣠⠏⡴⠃⢀⠏⣰⠃⣸⣧⠀⠀⠀⠁⠀⠀⢻⡉⣹⠃⠀⠀⠁⠈⠀⢰⡇
⠀⠀⠀⠚⠁⣠⠋⡴⠁⡴⠃⣿⠀⠀⠀⠀⠀⠀⠘⣧⡏⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⡡⠊⣠⠞⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⢀⣠⣤⣤⣄
⠀⢠⣾⠛⣿⣿⡟⣻⣆⢀⣾⡛⣿⣿⡟⢻⣦⠀⣼⡟⢻⣿⡟⢻⣧⠀⣴⡟⢻⣿⣿⢛⣷⡀⣰⡟⢻⣿⣿⠛⣷⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⠆⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣻⡇
⠀⠈⠻⣿⣿⣿⣿⡿⠃⠀⠻⣿⣿⣿⣿⡿⠋⠀⠹⣿⣿⣿⣿⣿⠏⠀⠙⢿⣿⣿⣿⣿⠟⠀⠘⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⢶⢶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠈⠣⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡎⠞⠛⠛⠳⠆⠀⠀⠀⠘⠛⠋⠉⠙⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠀⠀⠸⡄⢀⠴⣶⡦⠀⠀⠀⠀⠀⠔⣶⡲⡄⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢣⠈⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣠⠔⠚⠁⠀⠀⠀⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⣦⡀⠀⠀⠀⠀⠀⠀⠘⢄⣀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⠀⠀⣶⠆⣰⡶⡏⠀⠀⠀⠀⠑⠀⣀⡀⢿⠆⠀⢠⣶⡌⢱⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⢸⠀⢰⠉⠀⡏⠁⣿⢦⠀⠀⣠⡎⢰⠹⡏⠀⠀⠀⡌⠉⡇⡸⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⢸⠀⢰⡇⠀⡏⢸⠟⢻⣏⠀⡎⠀⡇⠀⠀⠀⠇⢠⠇⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⢸⠀⢸⠀⠠⠁⢸⣴⣮⢹⠀⡇⠀⠇⠀⠘⠸⠀⣸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⡀⠀⠀⠀⡇⠀⠈⠀⠀⠀⢸⠃⢸⠀⡇⠃⠀⠀⠀⢠⠇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠘⣆⠀⢠⠇⠀⠀⠀⠀⠀⠞⠀⡿⠀⠘⠄⠀⠀⠀⠈⠀⠀⡇⠀⢀⠔⠋⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⠢⠋⠀⠀⠀⠀⠀⠀⠀⢰⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⢠⠏⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⡤⣔⡉⠉⢉⣒⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡜⡠⠔⠛⠈⣙⠶⠄⠐⠚⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣏⣀⠤⠒⠉⢀⣠⢒⠘⣿⡶⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠤⠤⣼⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠤⠐⢲⡉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⠠⠐⠈⢈⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡊⣤⠀⠀⢀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠀⣠⠤⣰⠂⢉⣉⣉⠁⢻⡃⣁⠊⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠆⠈⠁⡐⠋⠉⠩⠳⡀⡉⠈⠀⡧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢇⠙⣀⠷⣇⣇⡄⣄⣄⡷⢇⡜⢁⠋⠢⠈⢒⠢⣄⣀
⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠱⡉⠢⢠⡜⡷⡟⣤⢠⠋⡳⠁⠀⢰⠀⠸⠀⠘⠀⠑⠆⠤⡀
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠑⠲⢌⡖⢅⠕⣯⠴⠊⠀⠀⢀⠆⢠⠃⠀⠸⠀⠀⠀⠀⠀⠑⢄
⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⠎⢠⠃⠀⠀⠆⠀⠀⠀⡇⠀⠀⠀⠡⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⠁⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⢡
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠊⠀⠠⠃⢠⠀⠀⢀⠃⠰⠀⠀⠀⠀⠀⠀⡆
⢃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢀⠔⠁⠀⡰⠁⢀⠃⠀⠀⡌⠀⠇⠀⠀⢰⠀⠀⠀⢰
⠀⠢⡀⠀⠀⠀⡀⠴⠚⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⡠⠒⠁⠀⢀⠊⠀⠠⠃⠀⠀⡐⠀⡌⠀⠀⠀⠨
⣤⣤⣬⣥⣬⣥⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣿⣶⣶⣶⣶⣷⣶⣶⣷⣶⣶⣾⣶⣶⣷⣶⣶⣶⣷⣶⣶⣶⣶
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⢀⣾⠛⣿⣿⡟⢻⣦⠀⣴⡟⢻⣿⣿⠛⣷⡀⢰⡿⠛⣿⣿⠛⣿⡄⢀⣾⠛⣿⣿⡟⢻⣦⠀⣴⡟⢻⣿⣿⠛⣷⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠐⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣟⣿⡇
⠀⠈⢻⣿⣿⣿⣿⣿⠏⠀⠹⣿⣿⣿⣿⣷⠟⠁⠘⢿⣿⣿⣿⣿⡿⠃⠈⠻⣿⣿⣿⣿⣿⠏⠀⠹⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣆⠀⠀⠀⠀⣀
⠀⣴⣶⣶⣤⣶⣦⠸⣿⣿⣿⣿⣦⣀⣾⡿⣿⠂
⠀⢻⣿⣿⡿⠿⣿⣷⣻⣿⠛⣿⡿⣠⠿⡀⠀⢀⣀⣀⣠⣤⣤⣄⣀⣀⣀
⠀⣿⣿⣿⣷⣄⠈⢿⣿⡇⣠⣿⣵⣯⣴⣶⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣦⣄⡀
⠀⠈⢛⣛⣛⣿⣷⡜⣫⣴⣿⠿⠿⠻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣷⣦⡀
⣴⣿⣿⣿⠟⠋⠉⣠⣍⠱⣶⡿⣷⣶⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄
⢹⣿⣿⣿⣄⣤⣾⣿⣿⣇⠈⢿⣦⡛⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⣾⣿⣿⣿⣿⣿⡿⢹⣿⣿⣶⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆
⠈⠛⠋⣽⣿⠟⠁⣿⣿⣿⣿⣿⡿⠋⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⣼⣿⡏⠀⠀⠻⠿⠿⠿⠟⠁⢀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⡀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢿⣿⡇⠀⠀⠀⠠⠾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡆⠀⠀⠀⠀⠀⣼⣿
⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⡀⠀⠀⠀⠀⢀⣀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟
⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠦⣀⡀⣀⠀⠤⢶⠶⣖⣶⣶⣷⣶⡶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠖⣿⡿⣫⣿⠟⠋⣠⢐⣨⣵⣾⣿⣿⣿⣟⣭⣶⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⡾⢛⡟⢡⣾⣯⣾⡟⣡⡨⣶⣵⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡿⢿⣿⣿⣿⣿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣾⣿⠀⣾⢁⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⠟⢋⣴⣿⠿⣿⣿⣿⣿⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢘⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣉⣤⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠿⠻⠟⠻⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠀⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣄⣠⣤⣄⣀⣠⣄⡀⠀⠠⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡯⠀⠇⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⢠⣾⣿⠟⠋⠉⠉⠉⢉⣭⣍⡉⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠉⠉⠙⠻⣿⣿⡆⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣴⣾⣿⣎⡸⢏⠀⢀⣤⣴⣶⣷⣿⣿⣿⣿⣶⣦⡀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⠀⠙⣧⠀⣿⣿⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡾⣍⣿⡏⠁⠿⣉⣽⢻⡟⢩⣽⣭⡙⢿⣿⡿⠿⠃⠀⠠⣿⡟⣹⣿⡿⠛⣩⣭⡉⠻⣿⣿⣧⠀⠀⠀⢿⣷⠟⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣽⠿⡇⢸⠀⠉⠳⣾⠇⣼⣽⣯⣥⣤⣿⠆⠀⠀⠀⠀⣿⠘⠙⠿⢠⣜⣛⢻⠃⠀⣿⡿⠛⠃⣤⠀⡼⠁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢇⠀⠀⠈⠀⠀⠀⠙⠾⢷⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣉⣡⣤⠾⠏⠀⠀⠀⠘⡶⢈⡆⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠈⢲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣘⣇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⡾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡅⠘⠀⠀⠀⠀⠀⠀⠀⣠⣀⢀⡀⠀⠀⠀⢠⡀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠔⢸⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢻⡟⢿⢿⣦⣤⣼⠿⣿⣼⣿⣿⠀⠀⠀⠀⠀⡀⠃⡜⠀⣽⣛⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡿⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣹⡿⣿⡿⣿⣿⣟⢿⣏⠀⠀⠀⠀⠠⠎⠀⠀⡼⠀⣤⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⡆⠀⠀⠀⠀⢠⣾⣷⢞⣇⢼⣴⠆⣜⡷⢾⣦⡀⠀⠀⠀⠀⢿⡶⠁⣔⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⢷⡄⠀⠀⣰⣿⣾⢷⣾⣿⠷⠿⢾⣿⣿⡾⣿⣷⣆⠀⠀⢊⠼⣿⢘⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠸⢧⡿⣶⣾⣷⣾⣿⣿⣿⣷⣤⣶⣶⣾⣿⣿⣿⣧⣺⣷⡶⣾⣾⣟⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⠛⠿⣿⣿⣿⡟⠁⠹⡿⠿⢿⣿⣿⡿⠉⢻⡿⣿⣿⠿⣻⡏⠂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⢠⣽⣿⣿⣿⡄⠀⠈⠁⠉⠈⠁⠀⢀⣾⠁⠰⠂⣰⠋⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣾⣿⣿⣿⡏⠛⠦⣄⣤⣤⣤⡤⠖⠟⣁⠔⠀⠌⠁⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡁⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⡁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⣀⣀⣂⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣶⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡖⠲⣯⣍⣉⣩⣭⣩⣉⣨⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⢀⡤⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠦⣄⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣴⣶⣦⡀⢷⠀⠀⠀⠀⠀⡾⢀⣤⣶⣤⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⣿⣿⣿⣷⢸⠀⠀⠀⠀⠀⡇⣾⣿⣿⣿⣧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠘⢿⣿⡿⢫⠏⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⠇⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣨⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹⢬⣉⣁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣶⠶⠦⢤⡤⠤⢤⡤⠴⠶⡶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⣇⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣿⠛⠛⣿⠿⠿⢿⠿⠿⢿⡿⠿⠿⡟⠛⠛⣿⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠹⢤⣀⣿⣀⣀⣸⣀⣀⣸⣇⣀⣀⣇⣀⡴⠃⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⢒⡆⠀⢖⡶⠀⠀⠀⠀⣿⡂⠀⣿⡃⠀⠀⠀⢺⢓⣄⢀⠖⣽⠀⠀⠀⠀⣀⠒⢚⠀⢘⠂⢂⡀⠀⠀⢈⣥⣄⠀⣠⣬
⠀⠀⠀⠀⢸⡚⡧⣰⡞⡳⠀⠀⠀⠈⣿⣇⡀⣿⡇⠀⠀⠀⢀⠼⣿⡰⣼⣿⠀⡄⠀⠀⣋⠹⢋⠀⠼⢈⣛⡇⠀⠀⢸⠉⢙⠀⡋⠙⡇
⠀⠀⠀⠀⢺⡟⣴⢪⣜⡷⠀⠀⢰⠉⣍⠈⠀⢈⡊⠁⠀⠀⠁⠀⠠⠶⣅⠀⢥⠁⠀⠀⠀⣠⣶⡶⣷⣶⡀⠀⠀⠀⠚⠉⢙⣙⠋⠉⠂
⠀⠀⠀⠀⠀⠙⠻⠟⠛⠁⠀⠀⠈⠢⠀⠉⠉⠁⠑⠀⠀⠀⠀⠀⠀⠙⠋⠈⠈⠀⠀⠀⠀⠛⠛⠈⠚⠛⠁⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⢘⣎⣵⡄⢠⢊⣽⠗⠀⠀⠐⣿⡄⢸⣿⠀⠀⠀⠀⠀⣷⡶⠀⢰⡶⡇⠀⠀⠀⣠⡴⠟⢸⠷⣄⡀⠀⠀⠀⢸⠓⡄⢰⠺⠃
⠀⠀⠀⠈⣋⠉⠁⠥⢝⡙⠀⠀⠀⢸⠿⡁⢼⡿⠀⠀⠀⠀⠀⢿⣿⡦⠸⣿⠇⠀⠀⠀⢷⣾⣗⡘⣿⣴⠇⠀⢀⠤⠼⣿⠧⢾⡿⠤⢤
⠀⠀⠀⠀⠹⣿⡶⠾⣿⠋⠀⠀⠀⠀⠣⣤⣤⣴⣿⠒⠀⠀⠰⠰⡒⡬⡔⣶⠑⠀⠀⠀⠀⢉⠬⠡⢌⠁⠀⠀⠈⠂⠑⠤⣀⣁⠤⠋⠘
⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠉⠿⠟⠁⠀⠀⠀⠀⠀⠉⠑⠋⠁⠀⠀⠀⠀⠀⠁⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠤⠄⠠⠤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⣀⡀⠀⠀⠀⠀⢠⣤⠀⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⢠⣰⢣⢱⡞⡌⡆⣄⠀⠀⠀⠙⢧⡄⠤⢽⡇⠀⠀⠀⢸⣅⡃⢸⣅⡇⠀⠀⠀⠀⢸⣽⠀⣿⡇⠀⠀⠀⠙⡿⣬⡄⢰⢥⣿⠃
⠀⠀⢰⠤⠼⠔⠻⡜⠂⠧⠤⠆⠀⠀⠠⠾⠇⠦⡾⠃⠀⠀⠀⠈⠿⡛⠈⠿⠃⠀⠀⠀⠀⢸⣿⣀⣻⡇⠀⠀⠀⠀⢢⡿⢡⣼⣼⠏
⠀⠀⠀⠙⠰⠆⡴⢧⠤⠆⠛⠁⠀⠀⠈⠶⠯⣯⡷⠀⠀⠀⠀⠈⢳⡬⢤⡶⠋⠀⠀⠀⠀⠀⣀⡀⣻⠦⠀⠀⠀⠀⠘⠳⣦⣧⠞⠂
⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣠⣤⡀⣠⣤⣄⠀⠀⠀⠀⡠⠀⠀⠠⡀⠀⠀⠀⠀⣠⡀⠀⣠⡀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣄⠀⠀⠀⢠⡤⡄⢠⠤⡄
⠀⠀⠀⠀⠈⡇⠈⡇⢉⠀⣋⠀⠀⠀⢨⣲⣦⣠⣶⣮⠀⠀⠀⢰⣏⡇⢸⡏⡇⠀⠀⠈⣇⣤⣷⢠⡧⣤⣿⠀⠀⠀⢸⣉⡇⢸⣉⡃
⠀⠀⠀⠀⠈⢓⣲⠋⣳⣒⠞⠀⠀⠰⠿⠌⢿⠙⠌⠿⠷⠀⠀⠘⣿⠣⠼⣧⠁⠀⠀⠀⠱⠬⠟⠤⡧⠩⠃⠀⠀⠀⠼⢍⡧⠼⣍⠧
⠀⠀⠀⠀⠈⢧⡄⣈⠪⠀⠀⠀⠀⠀⠀⠠⣴⣶⣶⠄⠀⠀⠀⠀⠢⢍⣤⠧⠤⠂⠀⠀⠀⠐⡦⣤⠤⠀⠀⠀⠀⠀⠠⠴⣦⢬⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⣿
⢿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⢰⣿
⠸⣿⣧⠀⠀⠀⠀⠀⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿
⠀⢻⣿⣆⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⣤⣀⣀⡀⢀⣀⣀⣤⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣤⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⢀⡠⠤⠤⣀⠀⠀⠀⢀⡤⠤⢤⡀⠀⠀⠀⣀⠤⠤⢄⡀⠀⠀⠀⣠⠤⠤⣄⠀⠀⠀⢀⡠⠤⠤⣀
⠀⠀⢀⠎⢠⠀⠠⠌⢣⠀⢰⠃⡄⠀⢠⠈⡆⠀⡸⠁⠄⠀⠤⠹⡀⠀⡞⢀⡀⢀⡀⢳⠀⢀⠏⢤⠀⢠⠈⢆
⠀⠀⠀⢆⠘⠂⠒⢀⡜⠀⠸⡄⠙⠉⠋⢠⠇⠀⠱⡀⠊⠉⠂⡰⠁⠀⢧⠀⠤⠤⠀⡼⠀⠈⣆⠐⠀⠒⢀⠎
⠀⠀⠀⠀⠑⠒⠒⠉⠀⠀⠀⠈⠓⠒⠚⠁⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠈⠑⠒⠒⠉
⠀⠀⠀⢀⠤⠒⠲⢄⠀⠀⠀⣠⠔⠒⠢⣄⠀⠀⠀⡠⠖⠒⠤⡀⠀⠀⢀⠤⠒⠲⢄⠀⠀⠀⣠⠔⠒⠢⣄
⠀⠀⠀⠃⠰⢀⠠⠄⢣⠀⢸⠁⠖⠀⠲⠈⡆⠀⡞⢠⠄⠀⢄⢸⠀⢀⠇⠐⡀⡐⠂⢳⠀⢰⠁⣀⠀⣀⠈⡆
⠀⠀⠀⢧⡀⠙⠁⣠⠎⠀⠘⢄⠘⠒⠁⡠⠃⠀⠱⡀⠉⠈⢀⡜⠀⠀⢧⡀⠳⠋⣀⠎⠀⠘⢄⠀⠶⠀⡠⠃
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⡤⠒⠛⠒⢄⠀⠀⢀⡴⠒⠙⠒⢄⠀⠀⢀⠔⠚⠓⠲⡄⠀⠀⢠⠖⠚⠓⠢⡀⠀⠀⡠⠒⠚⠒⢦⡀
⠀⠀⢸⠒⠄⠀⠠⠀⡇⠀⢜⠐⡁⠀⡒⠈⡆⠀⡏⢐⣂⣀⡂⢸⠀⠀⡇⠐⠀⠀⠂⢹⠀⢰⠁⠒⡠⡐⠀⡃
⠀⠀⠈⢦⡈⠉⣁⡴⠁⠀⠘⢄⡈⠉⣀⠜⠀⠀⠱⢌⡛⠛⣃⠞⠀⠀⠳⣄⠀⢀⣠⠎⠀⠀⠣⣘⠓⢃⡠⠃
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢀⠔⠋⠉⠙⢢⡀⠀⢀⠖⠉⠉⠑⢦⠀⠀⣠⠞⠉⠉⠑⢄⠀⠀⡠⠊⠉⠉⠳⡄⠀⠀⡴⠊⠉⠉⠲⡀
⠀⠀⢸⠀⠑⠀⡋⠀⡇⠀⢎⠀⠃⠀⡉⢈⡇⠀⡇⢀⡃⢘⡃⢸⠀⠀⡇⠈⡠⣄⠁⢸⠀⢸⡁⢉⠀⢈⠀⡇
⠀⠀⠈⠣⣌⣉⣡⠜⠁⠀⠈⠢⣀⣀⡠⠞⠀⠀⠘⠤⣁⣈⡠⠊⠀⠀⠑⢄⣉⣉⠴⠃⠀⠀⠳⢄⣉⣁⠴⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢀⠞⡁⠀⢈⠳⡀⠀⢠⠊⠁⠀⠉⢢⠀⠀⡰⢋⠁⠀⡙⢦⠀⠀⡔⠉⠀⠈⠑⡄⠀⢀⠞⠉⠀⠈⠳⡄
⠀⠀⢸⠀⣥⣖⣬⠀⡇⠀⢣⠈⠡⠤⡉⢀⠇⠀⣇⠈⠀⡄⠁⢸⠀⠐⡅⠎⠄⠬⠁⡸⠀⢸⡀⣊⣀⣑⠀⡇
⠀⠀⠀⠑⠤⠤⠤⠊⠀⠀⠈⠳⠤⠤⠴⠊⠀⠀⠈⠢⠤⠥⠔⠁⠀⠀⠙⠦⠤⠤⠖⠁⠀⠀⠑⠤⠤⠤⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠉⠀⠀⠀⢀⣀⡀⠉⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⢚⣓⡀⠀⠀⢨⣭⠉⠀⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⢸⣿⣷⠀⠀⢸⣿⡆⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⢀⣄⠀⠸⣿⡏⠀⠀⠸⣿⠇⠀⠀⣨⣽⡆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡅⠸⢰⣄⡀⠀⠀⠀⠀⠀⠁⣀⣤⣾⡟⠉⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠻⣿⡷⣶⢶⡶⢾⠟⢿⣽⡿⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠘⠷⣿⣴⣧⣼⣷⣾⠟⠀⠀⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⡀⠀⠀⠀⠉⠁⠀⠉⠀⢀⣤⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣶⣦⣤⣤⣴⣶⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠿⠿⠟⠛⢿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⠒⠒⠲⠖⢸⠀⢱⠉⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⡁⠀⠉⠁⠀⠀⠀⣀⣀⠀⠘⠡⢷⡀⠀⠀⠀⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⠀⠀⠁⠂⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⢀⠴⠦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠤⠶⠶⠀⠀⠀⠀⠀⠀⠀⠐⢊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢁⠜⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⣀⠀⣀⣀⡀⠤⠤⠂⠀⠀⠀⠀⠀⠀⠈⠋⢁⢀⣀⠠⠤⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠉⣀⠀⠤⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠱⡸⠀⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠸⠃⡸⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠁⠒⠇⠀⠀⡀⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣄⠀⠀⠘⡥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣤⣀⡀⠀⠀⠀⡀⠀⠐⠤⣙⠋⢄⢈⡽⢦⡺⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣉⣀⠤⠒⠉⢳⠠⢞⠀⠀⠀⠈⢹⠀⠁⠁⠠⠾⠁⠀⠀⠈⠦⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⠠⠇⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡅⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡦⢀⣀⣀⡀⣤⣶⠚⢛⣭⣹⣯⡅⣨⣠⣀⠀⠀⢀⠮⠈⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠿⢛⣛⣯⡁⣿⣿⡀⣿⣿⣿⣿⡇⢸⣿⣿⡄⠐⣹⡆⢢⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄⠻⠿⣟⡄⢻⣿⡀⢿⣿⣿⣿⣇⠘⢿⣿⣿⣤⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⠹⣿⣿⣿⣿⣿⡿⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠸⣿⣿⡟⠉⣲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡹⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠐⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡡⠝⣱⣄⠀⠀⠀⢀⠀⡀⡀⢀⡀⠀⢀⡤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠞⠀⠐⠛⠛⠛⠋⠗⠈⠀⠈⠁⠘⠉⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠙⠛⠓⠶⢦⣴⠦⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⠀⠀⣀⠀⠀⠀⣀⣀⡀⠈⠳⢆⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⢀⡴⠋⠁⣠⣶⣿⣿⣿⣿⣿⣦⡄⢀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣠⠏⠀⠀⣼⠿⠛⠉⠁⠀⠀⠀⠀⠁⠀⠉⠓⢦⡀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢠⡟⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠀⠀⢸⡇⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡞⠀⠀⠘⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡼⠁⠀⠀⠀⢻⡄⣸⠇⢀⣴⣶⣶⣶⣶⣶⣦⡄⠀⠀⠀⢀⣴⣿⠟⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣠⡞⠁⠀⠀⠀⠀⠘⣷⠋⠀⠋⠀⠀⠀⠉⠉⠉⠙⠁⠀⠀⠀⠈⢉⣠⣤⣄⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠾⠋⠀⠀⢀⡎⠀⠀⠀⢹⡆⠀⠀⢀⡶⠋⠉⣯⢦⠀⠀⠀⠀⢰⠀⠉⠀⢿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⠟⠀⠀⠀⠀⠈⣷⠀⠈⠉⠓⠰⠖⠚⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⣄⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡇⠀⠀⢀⣿⡋
⠀⠀⠀⠀⢸⡏⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⢸⡇⠈⠳⡄
⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⣶⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⠀⠀⢸⠁⠀⠀⡇
⠀⠀⠀⠀⠘⣷⡀⠀⠸⣧⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⣼⠿⠛⠉⠹⣿⠀⢀⡿⢲⡦⠘⢧⡀
⠀⠀⠀⠀⠀⠈⠻⣄⡀⠹⣆⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣶⡖⢄⠀⠀⠀⢸⠇⢠⠀⢾⣁⡀⠀⣀⣴⡏⠀⠈⣠⠞⣣⣄⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⣿⡀⠘⢿⣿⣿⣿⣿⣿⣿⣧⠘⡇⠀⠇⠀⠀⡇⠀⢸⣰⡿⡿⢿⣿⣿⡿⢃⣠⣞⠁⢰⠟⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡄⡇⠀⢸⡀⢸⠁⢀⡾⠋⢠⠇⠀⠈⢁⠔⠁⣸⠏⠠⠁⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣣⠀⠀⣃⠃⣠⠎⠀⣠⠏⠀⠀⢠⠎⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠏⡴⠁⠀⡴⢃⣤⡶⣠⠋⠀⣰⠃⠠
⣴⣞⣉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠜⡜⠀⠀⣾⠵⠋⢀⡜⠁⠀⡰⠁⡴⠁⠀⠀⠀⠀⠀⠀⣠⡴
⠉⠉⠁⠀⠀⠀⠀⠀⠐⠛⣻⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡸⠀⣠⠞⠁⢀⡴⠋⠀⢀⡞⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡰⠁⠐⠁⠀⣰⠋⠀⠀⠀⠘⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⠉⠻⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⢰⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠛⠻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣠⣿⡇⠀⠀⢸⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⣿
⠀⢸⡇⠀⢠⢶⡖⠦⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠙⠃⠀⢸⡇
⠀⣿⠁⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣷
⠀⣿⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠈⣿⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣤⣤⣤⣤⡤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣿⣿⣟⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠶⣶⣶⣶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠲⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠤⡶⠶⢶⠶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠒⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⢉⣡⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿
⠀⠀⠀⠀⠀⣴⣶⣋⣩⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⣿⡅
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣀⣀⣀⡇⢨⣀⣤⣴⣶⣿⣿⡿⠿⠂⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⠀⠀⠀⠈⠉⠉⠙⢛⣛⣛⣛⡛⠇⠸⠿⠟⣋⣩⠭⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠖⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠛⣶⡿⠀⠀⠀⠀⠀⠀⠘⠟⠁⢀⡤⠋⠀⠀⠙⠶⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠂⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢰⠋⢸⠇⠀⠀⠀⠀⠀⠀⠀⣮⣀⡀⢀⣀⣀⣀⣀⣤⣴⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠞⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⣾⢀⡟⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⡟⠘⡇⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡃⢸⡇⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⠁⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⠤⠶⠶⠶⠦⠤⠤⠤⢤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠀⢸⡇⠀⠀⠀⠀⠀⣸⣶⠋⣹⠖⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⡏⠀⢸⡇⠀⠀⢀⣰⠞⠁⣀⠼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠄⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⠇⠀⠘⡇⢀⡴⠋⠁⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⣰⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⣼⠋⠀⠀⠴⡿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⠖⠁⣠⡾⠟⣽⠟⠛⠿⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡘⠃⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣉⡙⠛⠛⠋⠉⠉⠉⠉⢉⣁⣀⣤⣤⠶⠞⠋⢠⡾⠃⠀⠀⠀⠀⠀⠉⠛⠿
⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⠛⠳⢦⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⠿⠛⠛⠋⠉⠉⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠴⠞⠋⠉⠀⠀⠀⠀⠀⠢⠈⠙⠳⣾⣤⣤⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠶⡙⠻⡶⢦⣄⠀⠀⠀⠀⠀⠀⠀⣿⠳⡀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢈⡏⠀⠀⠀⠀⠀⠰⢀⡟⠀⠙⣄⠀⢀⣴⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⡞⢸⡇⠀⠀⠈⢳⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⠃⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡤⠖⣋⣩⠤⠤⠄⠀⠐⢋⣁⣠⠤⠤⠤⠤⢤⣀⡀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⢋⡴⠚⠁⢀⣀⣤⣤⠀⠞⠉⢁⣀⣠⣤⣤⣤⣀⡀⠉⠓⠦⣄⠈⠑⣄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣱⠏⣠⠏⠀⣠⠞⠉⠀⢀⣀⣤⠤⠞⠋⠁⠀⠀⠀⠀⠈⠉⠛⢦⣄⠈⠳⡄⠈⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠏⣰⠃⣠⠞⠁⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠙⣆⠀⢳⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣡⣏⣼⡧⠚⣁⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠘⣆⠀⢷⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⡿⢿⢷⠞⠏⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣆⠈⢧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡇⣾⢸⣆⢡⡇⠀⠀⣠⣶⣶⣶⠶⣤⣄⠀⠀⠀⠀⠀⢀⣤⠶⠾⠿⠿⢿⣦⡀⠀⠘⣷⡜⢧⡈⢧⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⢸⣿⣾⡇⠀⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣏⠲⣅⡘⣷⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⣿⢸⢹⣿⡇⠀⠀⢀⢤⠶⡒⢲⣄⠀⠀⠸⡄⠀⠀⠀⡴⡟⠩⠉⣳⢦⠀⠀⠀⠀⣿⣿⡄⣄⢹⡟⠛⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⣿⢸⢸⣿⡇⠀⠀⠉⠘⠦⠁⠘⠉⠃⠀⠀⣷⠀⠀⠀⠀⠉⠀⠀⠁⠈⠀⠀⠀⢸⡇⡏⠧⢸⠀⣇⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⢸⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⠀⠘⡇⢻⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⢳⠈⣧⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⢸⢸⣿⣇⠀⢀⣀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣸⣿⠀⢷⠀⠈⣇⠸⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡟⣼⢸⣿⡌⢳⣾⡉⢷⠀⠀⠀⠀⠀⠘⠘⠂⠘⠉⠀⠀⠀⠀⠀⢀⡞⢩⣧⠞⢹⣿⡇⠘⣇⠀⠘⣆⠹⡄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣸⠃⣸⢠⡇⣿⢸⡇⢷⠀⢹⡇⠈⣿⢶⡄⠀⠀⣀⡤⠤⠦⢤⡀⠀⠀⢠⡾⣿⠀⣸⠏⠀⡞⣼⢻⡀⠹⡄⠀⠘⣆⠙⣆⠹⠆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⡿⢸⢇⡏⢸⡿⡜⡇⠀⣷⠀⣿⠈⢷⣀⠞⢁⡠⠤⠤⢄⠙⣆⢀⡿⠀⡿⠀⣿⠀⣸⢡⠇⠈⢧⠀⢹⡄⠀⠈⢦⡈⢧⡀
⠀⠀⠀⠀⠀⠀⣼⠁⢰⠇⡾⢸⠃⢸⡇⢳⣻⠀⣿⠀⢸⡄⠈⢯⣴⣿⣶⣶⣾⣿⣷⣜⡿⠁⢰⡇⢀⡇⢀⣏⡞⠀⠀⠘⣆⠀⠹⡄⠀⠀⠳⠄⠙
⠀⠀⠀⠀⠀⣼⠃⢠⠏⣸⠃⡟⠀⢸⠀⢸⡟⣇⢸⡆⠘⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⠀⣸⠇⡼⣹⠁⠀⠀⠀⠹⡄⠀⠉
⠀⠀⠀⢀⡾⠁⣠⠏⣰⠏⣸⠃⠀⡾⠀⡼⢻⡜⣆⢻⡀⢻⡄⠀⠈⢷⡈⠉⠉⢀⡿⠁⠀⢠⡏⢠⡏⣸⢣⡏⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣠⠞⠁⣴⠋⣰⠃⠀⣸⠃⣰⠃⢠⣷⠈⠈⠃⠈⠳⠀⠀⠘⣷⠒⠒⣿⠁⠀⠀⠞⠀⠘⠀⠁⡟
⠀⠀⠀⠀⠀⠀⠚⢁⡼⠁⠀⣴⠃⣰⠃⢠⡟⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠞⢁⡼⠁⣰⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠋⣠⠞⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁

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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣿⣿⠟⠛⠉⠀⠀⠀⣀⠀⠀⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⣙⣛⣻⣷⡄⠀⢀⣀⣴⡟⢻⡇⢀⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠷⠟⠛⢛⣻⣿⠿⣷⣿⠛⣿⡄⢸⣯⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣾⠟⠉⣤⣶⣿⣿⡄⠻⠇⠈⠉⣠⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⠻⣦⠄⢿⣏⠀⠻⠃⠀⠀⠀⢸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠸⣷⣾⣿⡄⠀⠀⠀⠀⣠⡟⠛⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⢀⣈⣿⣷⣄⣀⣀⣸⠏⠀⢀⣴⠦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠛⠋⠛⣿⠟⠛⠛⣟⣠⣶⠟⠁⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⢰⣿⣀⠀⢸⣿⠿⠃⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢻⠿⠿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡇⠀⠀⢸⣷⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⢀⣼⡟⠀⠀⠀⢸⣿⠀⠀⢀⣶⡆⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢠⣾⣿⠁⠀⠀⣠⠿⣿⡀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⡆⠀⠰⠇⠀⠸⣧⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⢀⣿⣧⣾⣿⠀⠀⠀⠀⠀⣠⣾⣿⠆⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⡿⠂⠀⢠⣿⣿⡟⠀⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⣼⡟⠀⠀⠀⠸⠃⠀⢹⣿⡿⠛⠋⠁⠀⢠⣾⡿⠋⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠃⠸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⣠⣾⣿⠋⠀⠀⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⡄⢁⣽⡏⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⡀⠉⢀⣾⡟⠁⠀⠀⠀⠀⠀⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⡀⠀⠀⠀⠀⢀⣾⢻⣦⣀⣠⣄⡀⠀⠀⢀⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⢸⣿⣿⣿⡏⠈⠉⠁⠀⣠⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⢤⣤⣀⡀⣿⡇⢹⣷⣤⣤⣼⣿⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣷⣜⠻⢿⠀⣿⡟⠀⠿⠛⢉⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣸⣿⣷⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⣿⡇
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣀⣩⣿⣿⣷⠄⠉⣩⣿⣯⣿⣿⣷⡀⠀⠀⢿⡇
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⡀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⢸⣿⠿⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠆
⠀⠀⠀⢀⡄⠘⠉⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠠⡟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⠋⡄⠈⠁⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠈⣿⠟⠁⠀⠀⠀⠀⣻⣿⠟⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⠀⢼⣿⣿⣿⣿⣿⣿⡿⠋⢀⣀
⠀⠀⠀⠀⠀⢰⡦⠀⠀⢀⣴⣦⠶⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠁⢀⠰⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠀⣒⡚⠙⠁⠀⢀⠡⣦⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⢠⢄⡴⠀⠀⣀⣸⡐⠀⠀⠀⣸⠀⢏⠀⠉⠀⠀⠀⠑⢉⠂⠀⢄
⠀⠀⠀⠀⣀⣤⠠⢀⠀⡀⠙⠉⠠⠀⠀⠀⠀⣜⡈⠀⢂⠂⠈⠙⡏⠀⠀⠀⠀⡙⢀⠀⢀⠀⠀⠀⢀⡄⠀⠀⠹⣴⡈⠃⠀⠀⠀⠀⠀⠈⡁⢐⣀⡀
⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠉⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠌⠀⠀⠀⠀⠐⠋⠀⢀⠀⠨⣠⠄⢀⠌⠀⠀⠙
⠀⠀⠀⠀⢐⢀⠀⡐⠃⢀⢔⣂⡴⠇⠀⠀⠀⠀⠀⠀⣀⣙⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠖⠀⠲⠤⡄⢠⠄⠀⠀⠀⠀⠠⠀⠀⡀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⢀⠈⠁⠄⠒⠘⢺⡇⠀⠀⠀⢆⠀⠖⠀⠃⠒⠂⣦⠀⠀⠀⠀⢀⠄⠀⠠⠀⠄⠒⢀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠐⠢
⠀⠀⠀⠀⠀⢸⠧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡈⠀⠐⠀⠀⠀⠀⣏⠀⠀⠀⠠⠚⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠁⠀⠀⠡
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⣌⡠⠀⢀⠁⠀⠀⠀⠀⠈⠀⠀⠠⡠⠀⠀⡈⠀⠀⠀⠀⠀⢠⠀⣀⠀⠌⠁⣰⣁⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⢀⣀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⡀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢⠀⠀⠀⠀⠠⢐⣆⡀⠀⢀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠐⠠⠀⠤⣀⠘⠀⠀⠀⠀⡄⠀⠐⠒⠐⠂⠐⢈⠀⠀⠀⠀⣶⠛⡒⠑⢻⠻⣤⣤⡖
⠀⠀⠀⠀⠀⢀⠻⠁⢐⠀⠀⠈⠀⠘⠐⠀⠀⡴⠆⠐⣂⠀⢐⠀⠈⡇⠀⠀⠀⠀⠻⠀⠤⠀⠀⠀⠀⡎⠂⠀⠀⠀⠈⠀⠄⠀⠀⠀⢀⣩⠁
⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⢀⡀⠀⠨⠀⠀⠀⠁⠆⠀⠀⠄⠀⠀⠀⠗⠀⠀⠀⠀⠐⡀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠐⠂⠀⢀⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠉⠉⡀⠔⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠢⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⡀⠀⠀⠀⠠⠂⠀⠀⡀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣀⠄
⠀⠀⠀⠀⠀⢀⣀⡴⠶⠂⡴⠴⢠⢀⠀⠀⠀⠀⠀⣴⣐⠶⢤⣤⡤⡄⡀⠀⠀⠀⢶⠀⢀⠀⠀⢁⠀⣄⡳⠀⠀⠀⢰⠘⠡⠜⠸⢄⣠⣄⡨
⠀⠀⠀⠀⠀⠈⠛⠦⠀⠀⢂⠀⠀⡄⢤⡃⠀⠀⠰⠍⠰⠀⠀⠂⠀⠐⠃⠀⠀⠀⢠⠀⢘⠀⠐⠃⠀⠸⢓⠀⠀⠀⠀⡀⠀⠂⠀⠈⠀⢹⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢢⣤⣤⠄⠀⠈⠀⠀⠀⠀⠀⠡⠀⠠⡒⠂⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠴⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠊⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⢨⡆⢀⠀⠀⠀⢀⠊⠈⠁⠀⡈⠉⡓⠄⠀⠀⠀⠀⢀⡔⠲⠄⣠⠀⢰⠀⠀⠀⡍⠀⠝⠛⠂⠰⢘⢲⡙
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠁⠀⢱⣼⠃⠀⠀⠘⠠⠀⠀⠀⠀⠀⢑⢬⠀⠀⠀⢠⠾⠃⠀⠈⠀⠀⢉⠄⠀⢀⠀⣊⠀⠀⠀⠀⠀⠘⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣏⠄⠀⠀⢏⠁⠀⢡⠀⠃⠀⢀⡄⠀⠀⠀⠘⡁⠀⢈⡀⡖⠀⡀⠀⠀⠀⠣⡌⠀⢀⠀⡀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⠂⠂⠀⢀⠌⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠔⠀⠀⠀⠀⠀⠂⢄⡀⢉⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁

⠀⢀⡀⠄⣀⠀⠀⠀⣀⠠⢀⡀⠀⠀⢀⡀⠄⣀⠀⠀⠀⣀⠠⢀⡀⠀⠀⢀⡀⠄⣀⠀⠀⠀⣀⠠⢀⡀
⢀⠋⠄⠀⠌⠃⠀⡘⠡⠀⠠⠙⡀⢀⠋⠄⠀⠌⢃⠀⡘⠡⠀⠠⠙⡀⢀⠋⠄⠀⠌⢃⠀⠘⠡⠀⠠⠙⡀
⠀⢦⡘⠛⣠⠆⠀⠰⣄⠒⢂⡴⠀⠀⢦⡐⠒⣠⠆⠀⠰⣄⠒⢂⡴⠀⠀⢦⡐⠒⣠⠆⠀⠰⣄⠻⢃⡴
⠀⠀⠀⡁⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⢈
⠀⣾⢿⣿⢿⡆⠀⢰⡿⣿⡿⣷⠀⠀⣾⢿⣿⢿⡆⠀⢰⡿⣿⡿⣷⠀⠀⣾⢿⣿⢿⡆⠀⢰⣿⣿⣿⣷
⠈⣿⣟⣙⣿⡏⠀⢹⣿⣻⣻⣿⠁⠈⣿⣟⣛⣿⡏⠀⢹⣿⣛⣻⣿⠁⠈⣿⣟⣛⣿⡏⠀⢹⣿⣛⣻⣿⠁
⠀⠈⠉⠋⠁⠀⠀⠀⠉⠙⠉⠁⠀⠀⠈⠉⠋⠉⠀⠀⠀⠉⠙⠉⠁⠀⠀⠈⠉⠋⠉⠀⠀⠀⠈⠙⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⣼⢠⡟⣽⣿⣡⣾⣵⣶⣾⣶⣶⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⡻⢿⣿⣾⠳⡇⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣫⠵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠋⣩⣽⣶⣶⣶⣾⡿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣮⣍⡳⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣭⡓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣌⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⡶⠟⠛⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠉⠀⣀⣀⡀⠙⢿⡂⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣶⣤⣄⠀⠀⠀⠀⠀⢸⣷⢸⡿⣮⡛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣩⢿⡿⠿⣷⣄⢹⡄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠸⣿⡏⢿⣆⡙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠸⠧⠂⠟⠙⡿⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄⡀⠀⠀⠙⠀⠀⠀⠀⠀⣿⢿⡌⠻⣟⡂
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠫⣴⣏⠄⢸⠿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⠻⣄⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢠⡟⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣌⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣙⣦
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢧⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡻⣍⣠⣤⣄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢉⣠⠄⠘⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠿⠷⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⢸⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣼⣽⠟⠙⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣸⠇⢉⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⣰⢿⡇⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢶⢉⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⡯⣷⡀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣄⣀⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣿⡇⡏⠛⠦⠤⠤⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢹⣷⣟⡒⠒⠶⢶⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⢿⣿⣿⣷⣶⡿⠏⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⡀⠀⠀⠀⠀⠉⠉⣉⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⣤⡴⠀⠀⠀⠀⠙⠀⠟⠀⠀⠀⠁⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⣁⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛

⣶⣶⡶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀
⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣾⣿⣿⣿⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣫⣵⣶⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣻⢦⣄⠀⠀⣠⠊
⠀⠀⠈⠱⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢓⣽⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣗⢻⣷⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠒⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⣿⠇⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣼⣿⣿⣿⡿⠉⢁⣀⡚⢥⣀⣀⣠⠶⠶⢶⣤⡀⠀⣿⣿⣹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢱⠀⠀⠀⢼⣿⣿⣿⣿⠃⠀⠛⠉⠛⡸⠃⢻⠁⢴⡟⢶⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣀⡀⢊⣾⣿⣿⢹⡟⠓⠦⠤⠤⠚⠁⠀⠘⢆⡀⠀⠀⣀⡴⠁⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡨⠂⢿⣿⣿⢰⡇⠀⠀⠀⠀⠀⠂⠘⠀⠀⠈⠉⠉⠁⠀⠀⣿⣿⣿⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠀⠁⣹⣿⣧⣾⢀⡀⠀⠀⠀⣴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠈⠁⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠼⠀⠋⣿⣿⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⠀⠀⢰⠀⠀⢸⣿⣿⣿⡟⠀⠀⢰⡀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⢿⣇⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣮⣿⡜⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⣼⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⣨⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣫⡖⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⣱⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣯⣻⣿⣿⣷⠃⣿⠇⠀⠀⠀⢀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⢣⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣿⡿⣿⡿⣿⣿⢟⣾⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣾⡿⣿⣿⠷⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡶⠿⠿⠿⠿⠿⠿⠿⠷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠚⠻⢶⣦⡀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠉⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠾⠿⢶⣦⣄⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠙⠳⣄⠀⣠⣾⠟⠉⠀⠀⠀⠈⠙⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡆⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⢀⣠⣾⡿⠁⠈⠳⣄⡀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠛⠋⣿⡟⠀⠀⠀⡀⠈⠙⠓⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣤⣤⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠐⣻⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣦⣤⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢤⡤⠀⠄⠤⠤⠤⠤⠄⠤⠄⠠⠄⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⠉⣿⣿⡝⠻⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠈⢿⣿⡆⢘⢏⣠⣤⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⡦⠜⠛⠁⠀⠈⠉⠉⠉⠙⠛⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣬⣽⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⢦⡀⠀⠀⠀⠴⠖⠒⢄⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡖⠦⠈⠀⠀⠀⢐⣶⣦⠀⠀⢠⠟⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡄⠀⠀⠀⠀⠀⠉⠈⠀⠀⢸⣬⣷⣭⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠐⠛⠂⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠤⠤⣾⡇⠀⢸⡿⠛⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⣡⣠⣿⣟⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠂⠀⠀⢿⡀⠈⣿⣿⡿⣿⡇⠀⠀⠀⠀⢠⣤⣾⣏⣉⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠠⠀⠈⠱⠀⣿⠀⠀⣸⡇⠀⠀⣠⠔⠾⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡐⠀⠀⠀⠀⠀⠀⢆⢠⠻⣦⡴⠟⣀⡄⠀⢹⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢳⣀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⡅⠀⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠓⢄⠀⠀⠀⠀⢹⠟⠉⢏⠀⠀⠀⠀⠀⣠⣦⣾⣿⣿⣿⣿⣿⣴⠏⠀⠀⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢸⠃⠀⠀⠀⡄⠀⠀⠈⢢⠀⠂⠈⠉⠉⢧⢈⣿⠃⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡏⠀⠀⠀⣰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡾⠁⠀⠀⠀⢀⡤⠶⠚⠛⠛⠳⢦⡀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢀⡞⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠏
⠀⠀⠀⠀⠀⠀⢠⡟⡆⢀⠞⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠎⠀⢱⡀⠀⠀⠀⠀⠱⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢠⡏⡴⠋
⠀⠀⠀⠀⠀⠀⡼⠀⢹⠏⠀⠀⠀⠀⠀⡟⢧⠀⠀⣠⠏⠀⠀⠀⢣⠀⠀⠀⠀⠀⠱⣀⠆⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠃⡰⠃⠀⠀⠀⠀⠀⢀⡇⠈⢆⣰⠃⣀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠋⠁⠈⡇⠀⣿⠓⠶⠤⠤⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢁⠀⢻⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀

⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣶⣾⣿⣶⣶⣄⡀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⠉⠀⡀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡈⠀⠊⠀⠀⠀⠐⠀⠀⠨⠀⠀⠀⠄⢼⣯⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠿⠿⣿⡿⠿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠄⠀⠀⢀⠀⠀⠠⠙⠿⠟⠿⠻⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡀⠙⣇⠀⢹⡄⡐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠋
⠀⠀⠰⠶⠀⣠⠀⠀⠀⠀⠀⠀⠀⠂⣠⣏⣨⢀⡼⢀⣠⡇⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠐⠉
⠀⠀⠀⠀⢀⣤⣤⣀⣤⡏⠀⠀⢀⣴⡄⠀⠘⠃⠀⣀⠀⠘⠁⠀⠀⠀⠀⠘⠻⠏⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⡁⠀⠀⢼⣿⠀⠀⠀⠀⠘⠛⠀⠀⢀⣤⢄⣠⣀⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠸⣿⣿⣿⣿⣿⠃⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⠠⣤⣾⣷⣿⣿⣿⣷⣶⣤⠀⠀⠀⢻⡿⢿⡿⡶⠀⠀⠀⠀⢀⠤⠔⢮⠒⠄⠀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⣀⣿⣟⣿⣿⣿⣿⣿⣿⣿⣟⣧⣄⠀⠈⠃⠘⠀⢧⠀⠀⠀⠀⢻⠀⠀⠀⠀⠸⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⢀⣠⣄⠀⠀⠉⠁⠉⠉⠈⠉⠉⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⢤⣀⡀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠜⠁⠁⠐⠒⢀⡀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⢀⠠⠀⠄⢈⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠐⡀⢀⠀⠀⠀⠀⠀⢠⠆⠀⠀⣤⡀⠈⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠛⠛⠛
⠀⠀⠀⠀⠘⠀⠃⠀⠐⠀⠀⠀⢠⠛⠃⠀⣴⣿⣿⣿⣶⡀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣙⠚⠀⠸⣿⣿⣿⣿⣿⡇⠀⠘⠄⠀⠀⠀⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠊⠉⠀⠀⠀⠙⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣍⠩⣭⣽⡏⢐⠷⠌⠃⠙⣿⣿⣿⣿⠿⠁⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣫⣽⢆⣀⠉⢡⡔⣶⡀⣽⣿⠘⣷⣄⠻⠟⠁⢀⡀⠀⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⣿⣿⣿⣿⣿⣿⣿⡗⣿⡾⢛⣩⡴⢈⣷⠘⣧⣶⣿⠀⣿⡉⠛⢶⠌⠙⠻⣦⡀⠿⠀⠶⣶⣾⣶⣿⣿⣷⣶⣦⣄⡀⢰
⣿⣿⣿⣿⣿⣿⣿⠀⠛⢿⣿⡿⢰⣦⡉⢻⣿⣿⡏⠃⢈⡀⠀⠈⣷⠄⠀⣿⣷⢰⣦⠲⠦⠤⠉⣩⣻⡿⠿⠿⠿⡇⢸
⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠘⣧⠀⠛⠿⠶⢿⣿⣇⠀⡘⠃⠀⣰⡟⠀⠀⣸⠟⢀⡈⠁⠀⠀⠀⣈⣻⡇⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠⣽⣧⣀⠀⣴⡄⣶⡅⠀⢴⣤⡼⠋⠀⢀⣠⣷⣤⣿⣿⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⢾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣬⡉⢛⠿⣷⣼⡁⢸⣶⣿⡿⠟⣀⠀⢀⣼⣿⣿⣿⣿⡿⠷⠦⠀⠀⠀⠙⣷⡇⣰⠂⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠿⢃⣿⡇⠈⠁⢺⣿⣿⣟⣷⠆⢹⣦⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⡿⢁⠟⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣾⣿⡇⠀⠀⠀⣿⣿⡿⠁⠀⠀⢻⣿⣿⣿⣿⢷⣦⡀⠀⠀⠀⠀⠀⢰⡞⡿⠀⠀⠀⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⡇⠀⠀⠀⣿⣛⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⠙⠋⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⡿⠀⠀⠀⢀⡛⣿⣧⠀⠀⠀⠀⣿⣿⡿⢿⣷⠄⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠃⣴⡆⣀⢀⣿⠀⢸⣿⠀⠀⢠⣾⡿⣿⡇⠀⢹⣷⠄⠀⢀⣾⡿⠁⠀⠀⢸⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣿⡆⢿⣥⡇⣸⢃⣴⢸⣿⡄⠀⢺⡿⢠⣽⣿⡄⠼⠃⠀⣰⣿⣿⡅⠀⠀⠀⠈⡇⠀⠀⣿
⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⣿⣟⢸⣿⠐⣳⣾⣿⣿⣿⣧⡀⣆⡠⣾⡏⠀⠀⢀⣴⣏⣈⣿⣿⣇⠀⠀⠀⠀⢳⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠤⢶⣟⠛⢻⣦⣷⣾⣯⣿⡿⠋⣻⣿⣿⣟⠛⠷⠆⣠⠶⠋⠁⠈⠙⠛⠿⠿⣤⣀⡀⠀⠐⡇⠀⣿
⠀⠀⠀⠀⢀⣴⣦⣤⣤⡈⠙⠓⢬⣉⣽⡿⡋⢀⣴⣿⣿⣿⢋⣄⣴⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣷⣜⠻
⠀⠀⠀⠀⠀⠘⠛⠭⠛⠿⣷⣷⣶⡿⠫⠊⣠⠿⣦⣄⡙⠿⣷⣭⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠛⠛⠀⠀⠾⠿⢧⣌⠙⠻⢷⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠹⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠁⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣭⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⣱⣶⡌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⡘⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢀⣾⣿⣿⣆⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠘⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢿⣦⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⡄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢀⣿⠀⠀⢿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡟⠀⠀⠠⣤⡄⢿
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⡀⠀⣸
⠀⠀⠀⠀⠀⣾⡟⠀⣴⠶⣦⡄⠈⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿
⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣧⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠉⠉⢹⡟⠙⢻⣷⡀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢻⡆⠀⠘⣷⠀⠀⢿⣧⠀⠀⠀⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠹⠇⠀⠈⠁⠀⢸⣿⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠿⢿⣶⣤⣤⣄⣀⣀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠈⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣷⡀⠀⠀⠀⠀⢻⣿⠀⢀⣠⣴⡿⠟⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⢿⣧⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⣿⣦⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠹⢷⡄⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⡆
⢀⣴⣶⣶⣶⣶⣶⣶⣶⠿⠿⠋⠀⠀⠀⠀⠀⣠⣾⠟⠋⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣀⠀⣀⣠⣾⠋
⠀⠙⣿⡿⠿⠿⠿⠿⣿⣿⣆⠀⠀⠀⠀⢻⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⡟⠉
⠀⠀⠹⣿⣦⣠⣤⣶⠿⠛⢿⣧⠀⠀⠀⣸⣿⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠻⢿⣿⠿⠟⠃⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⣏⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣿⠇⣿⣿⣤⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⢙⣿⡆⢸⣿⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠚⢿⡏⠙⣿⣶⣿⣷⣾⣷⣿⣿⠀⢘⣿⣷⣿⣶⣾⣷⣾⠟⢉⣾⠗⢲⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⣽⡷⣠⣿⣦⡾⠁⠀⠀⠀⠀⢹⣿⠀⢸⣿⠀⠀⠀⠀⠀⠹⣦⣼⣷⣤⣾⡿⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣀⣀⣨⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⣾⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣀⣀⣀⣀⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⣿⣯⣉⣉⣉⡉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤
⣿⣿⣿⣿⣿⠿⠿⢿⣿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⢀⠴⠒⠚⢉⡵⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠉⡩⠛⠉⠲⢄⠀⠀⠀⠘⢧
⠀⠀⠀⣼⠏⠀⠀⠀⢀⡜⠁⠀⠀⡔⠁⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠎⠀⠀⠀⠀⠀⢳⡀⠀⠀⠈⣧
⠀⠀⣸⠏⠀⠀⠀⠀⡞⠀⠀⠀⣸⠁⠀⠀⠀⠀⢠⠞⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡎⠀⠀⠀⠀⠀⢠⠊⣷⠀⠀⠀⠸⡆
⠀⢠⡿⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⡿⠀⠀⠀⠀⢿
⠀⣼⡇⠀⠀⠀⠀⠀⢣⡀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠑⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠸⡄⠀⠀⠀⠀⠀⣱⠃⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠙⢄⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠘⠦⣀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠘
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠤⠿⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠓⠒⠋⠉
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠸⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣇⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⡇⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⡜⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⢀⡜⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠉⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠊⠁⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠈⠉⠉⠒⠒⠒⠒⠉⠉⠁⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠖⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠄⠀⠀⡁⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣄⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⢟⣿⣿⡟⢻⣿⣿⣿⣿⣷⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⠹⢿⡿⠁⢸⣿⣿⡿⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡜⡇⠀⣿⣿⣿⠁⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⡀⠂
⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣾⣷⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢡⠴⣿⣿⣿⣿⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣯⣉⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠁
⡀⡀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡿⠋⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠋⠁⠉⠉⠛⠀⠀⠀⠀⡰⠛⠉⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢏⠻⡆⠀⠀⢀⣾⣿⣝⠂⠀⠀⠀⠀⠑⠺⣟⢷⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣯⣴⣛⣰⡏⠀⠀⠈⠁⠉⠁⠀⠀⠀⠠⠄⠀⢠⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⣼⣧⠀⠀⠘⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠈⠁⠁⠁⠀⠀⠀⠀⠀⣀⣀⣠⡿⠁⠻⢿⣿⣿⣿⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣤⣄⠀⠀⢠⣿
⠀⠀⢀⠂⠀⠀⠀⢠⠀⠚⠛⣹⣿⣷⣬⣶⣭⣽⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣉⣉⢻⣿⠀⠀⣸⣿⠷⠒⠒⠒⠂⢠⢀⠀⡀
⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡋⢉⣿⣧⣀⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⢀⣿⠠⠄⠈⠀⡤⠀⠈⡦⠀⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣤⣀⠀⠀⠀⠀⢻⡏⠀⠀⠈⣿⠆⢀⠾⠃⠀⢀⠀⠠⣀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⡈⠁⠈⡷⠂⢀⠀⠘⣿⡀⠀⠀⣿⠀⠈⡀⠀⠀⠀⠘⠀⠈⠃⠈⡖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠛⣷⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠃⠀⠀⡁⠀⠈⡧⢤⡉⠛⠿⠟⠋⢾⣀⠎⠀⠀⠀⠀⠀⠀⠀⢈⡴⠟⠂⢀
⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⠀⠀⠀⣼⠛⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠂⠐⠈
⡄⠀⡀⠁⣠⡾⠛⠁⠀⠀⠀⠀⠘⣿⣈⣻⣿⣿⣿⣿⣿⣿⣦⣠⡀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⡝⠁⠀⠀⠀⠀⠀⢠⠴⢿⣷⡄⠀⠀⠈⠀⠄⠀⠀⠀⡄⠠
⡀⠠⠄⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡯⠻⣿⡿⠛⢹⡿⠶⠶⢦⣀⡀⠀⢠⡞⠀⠀⠀⠁⠀⠀⠀⠀⢀⣴⠁⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠐⡀
⠁⣤⠾⠛⠁⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣀⡟⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢁
⢻⡏⠀⠀⠀⣠⡶⠛⠉⠉⠉⠙⠛⠳⢦⣀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠹⣇
⠈⢿⡄⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⡼⣆⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⣸⡏
⠀⠀⠻⣄⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡸⠁⠘⢆⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⢷⡀⠀⢠⣿⡇
⠀⠀⠀⠘⢧⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢠⠃⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⢦⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⢳⡀⡼⠈⣿
⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣰⠃⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢷⡄⠀⠀⣸⠃⣇⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠸⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢻⡆⢰⠏⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠋⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⣠⣴⣿⡇⠀⢿⠏⠀⠒⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠉⠁⠀⠀⠀⣿⠇⠀⡸⠀⠀⠀⠀⣧
⠀⠀⠀⡄⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡇⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⡀⡀⢀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣷⣆
⠀⢀⣤⣤⣤⣀⣠⣄⡀⢼⣿⣿⣿⣿⣿⣦⡀⢠⣤⣶⣿⣧⠄
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⡷⢳⣿⠟⠹⠛
⠀⢘⣿⣿⣿⡿⠛⠻⣿⣿⣞⣿⡟⠀⣿⣿⢃⡾⠛⠆⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀⡀
⠀⢾⣿⣿⣿⣿⣦⡀⠈⢿⣿⣿⠀⣰⣿⣧⣾⣥⣶⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠈⠻⠿⠿⠿⠿⣿⣦⡘⠟⣣⣼⣿⣿⣿⡿⢿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣶⣤⡀
⢀⣤⣶⣿⣿⣿⠷⠿⠛⠃⠛⠟⣋⣭⣤⣤⣤⣀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⠃⠀⠀⣠⣾⣿⣆⠙⢿⣭⡛⢿⣿⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⢈⣿⣿⣿⣿⣦⣴⣿⣿⡟⣿⣿⡆⠀⢻⣿⣷⡙⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀
⣾⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣶⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠘⠻⠿⠛⣿⣿⣿⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧
⠀⠀⠀⣼⣿⣿⠏⠀⠀⣿⣿⣿⣿⣿⣿⡗⠀⠀⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧
⠀⠀⢰⣿⣿⡏⠀⠀⠀⠈⠛⠉⠙⠛⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢸⣿⣿⡆⠀⠀⠀⠀⢠⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠉⠙⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⠛⠁

⣠⣴⡿⠉
⣿⣿⠁⢠⡀
⣿⣿⣿⣿⣷⣶⣶⡶⠆
⣿⣿⣿⣿⣿⣷⣶⣦⣤⣶⣤⣄⣀⣄⣀⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢀⣤⡄⣤⠀⠠⠄⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⣥⠀⠤⠄⣈⣀⣀⣠⣌⡀⢄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣤⡄⢤⣼⡿⡟⠉⣽⣿⡛⠻⣮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣇⢷⣾⣏⣀⣛⣀⣸⣯⡙⣦⣿⣧⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⣀⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣰⣿⣿⣿⣿⣷⣈⣿⡟⢀⠀⣀⠀⣀⢀⣀⣀⣀⣀⢀⣿⢛⣿⣦⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⡿⣟⣇⣼⣿⣁⣤⣤⣤⣀⣀⢿⣍⢛⣹⠏⣼⣟⠡⣉⣤⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡧⢿⣿⣾⣿⣿⣿⣽⣿⣿⣿⠻⣷⣆⣬⣹⣿⣾⣿⣿⣿⣼⣟⢉⣾⡟⠁⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⢞⣭⣿⢿⣿⣿⣿⣿⣿⣿⡃⣸⣿⣾⣿⣿⡿⠋⠋⡉⠘⠛⢿⣍⢀⣿⢻⣆⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣾⣿⢴⣿⣿⣿⣿⣿⣿⡟⢹⣿⣿⢹⣿⠛⠰⠋⠀⠀⠂⢀⠀⠻⣿⣭⣫⣽⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣾⣿⣿⣟⣿⢟⢻⡥⣋⣿⣏⣿⠋⣿⣧⡉⠀⠀⠀⠈⠀⠁⠈⠉⣿⣤⠒
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⣻⣿⣿⣿⣿⡏⠘⠈⠻⡿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣛⠻⠶⣦⣤⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⠣⠀⠐⠀⠀⢰⡘⠻⠦⣤⠤⠔⠛⠋⠙⠛⢦⣀⠀⢙⠛⢦⣴⠳⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡟⣿⣿⡙⠘⠀⠀⠀⠀⢺⡇⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⣤⡿⠅⠀⢘⠷⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⡿⣹⠧⠝⢻⣷⣧⠀⠀⠀⢂⣸⣿⠐⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣇⣀⢠⣿⡅⢈⠳
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⠚⣮⣿⢻⣎⢂⣴⡿⠻⡧⠴⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⣿⠀⣄⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣞⣵⣿⢯⣷⣕⣧⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⣿⢟⣿⣿⣿⡏⣁⡀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣿⣽⣾⣟⢻⣯⣡⡍⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣛⣿⣧⡨⠁⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣏⠼⠀⠘⠂⠀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣾⡻⣷⣄⠘⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡞⢿⣶⡆⢠⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣷⣼⣿⣾⣹⣷⣬⡇⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣿⣄⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣷⣙⡎⠹⣿⣶⣶⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⢿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣷⡰⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣹⣿⣼⣿⣿⣿⣿⣷⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣍⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢀⣃

⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⢈⣢⡀⠀⠀⠈⠐⠒⠊⢁⡠⠤⠴⠶⡇⠀⠀⠀⠘⠻⠖⠁⠀⠀⠀⠈⠛⠉⠀⠀⠀⡇⠀⠀⠀⠉⠁
⠒⠒⠒⠢⣀⡇⠀⠀⠀⢸⣿⠛⣻⠀⠀⠀⢰⡿⢿⣿⠑⠢⠤⣀⡴⠊⠁⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⣀⣀⠀⣀⣀⣸⡀
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠘⢦⣼⠟⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⢘⡿⠛⠉⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠈
⠒⠒⠒⠲⠮⢧⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⣢⣀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣤⡤
⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠈⠦⡀⠀⠀⠀⣠⠖⠀⠀⢰⠃⠀⠀⠀⣰⣿⣿⣷⠀⠀⠀⢠⣿⣿⣿⡍⡞⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⡀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠈⠉⢉⣉⠤⠤⠤⢄⣸⠀⠀⠀⠀⠹⣿⣀⡿⠀⠀⠀⠘⣇⣨⣿⠃⠂⠀⠀⠠⣿⠿⢿⠀⠀⠀⠰⢛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠔⠊⠁⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠹⠶⠊⠀⠀⠀⠀⡞
⡄⠀⠀⣾⣿⢿⣿⣿⡇⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢄⡀⠀⠀⠀⢇⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠘⢧⣤⣿⡿⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣶⣷⣦⠳⡀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠙⠆⠀⠀⠈⣦⣀⣀⣀⣘⣤⣀⣀⣀⣼⠀⠀⠀⠻⣿⣧⣤
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⢃⣶⣷⣆⠀⠀⠈⢯⣉⡿⢀⣈⠦⠤⠴⠠⠤⠤⣀⡀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠈⠉⠲⢄⢸⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⡞⠘⢿⣅⡽⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠠⡀
⢄⣀⣀⠤⠞⠁⠀⠀⠀⢀⠞⠐⠂⠤⣉⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠈⠑⢄⢀⠏⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⠦⣀
⢤⣤⣤⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣿⡿⢿⡇⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠲⢶⠶⠀⠀⠀⢸⣀⠀⠈⠑⢲
⠚⠁⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠙⠷⠞⠁⠀⠀⠀⢧⣨⣿⠃⠀⠀⠀⢸⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⡇⠀⠀⠈
⠀⠀⡅⠀⠀⠀⠛⠒⠀⠀⠀⠀⠐⠚⠃⠀⠀⢸⠀⢀⡠⠤⠒⠒⠒⠒⠒⠢⢄⡀⠉⠀⠀⠀⠀⠀⡎⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⢀⠞⢛⣚⠁
⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⣼⣅⣀⣀⡀⠀⠀⠀⠀⣠⠴⠒⠉⠉⠀⠀⠈⠉⠓⠢
⠙⢦⣸⢆⠀⠀⠀⠢⣀⠀⠀⢀⠴⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣠⠞⠀⠀⠀⠀⠀⠉⠒⢶⡊⠁
⢀⡜⠁⠈⠳⢄⠀⠀⠀⠈⠉⠀⠀⠀⢀⢴⠃⠀⢠⣶⣦⣶⣦⠀⠀⢠⣶⣦⣴⣦⡀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⡞⠀⠀⠀⠀⣀⡉⠒⠂⠤⠤⢠⣶⣮⡁⢸⠀⠀⠘⢿⣿⣩⡿⠀⠀⠈⢿⣉⣿⡿⠁⠀⠀⡇⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠈⢧⣤⣄⠀⠀⠀⢀⣾⣿⣷
⠀⠀⠀⠀⣾⣻⢿⣆⠀⠀⠀⣾⠟⢿⣷⠸⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣷⣦⡀⠀⠀⠸⣏⣹⡇⠀⠀⠀⠘⡟⢻⡆⠀⠀⠘⢏⣈⣿
⠀⠀⠀⠀⠹⣧⣤⠏⠀⠀⠀⠘⠶⠾⠋⠀⢣⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡼⣿⣉⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠷⠚⠀⠀⠀⠀⠀⠉
⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠑⠲⠶⠖⠊⠀⠀⠀⣀⠞⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣀⣀⡀⠤⠤⠤⣴⡆
⠵⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⡔⠉⠑⠤⢀⡀⠀⠀⠀⢀⣀⠤⠚⠉⢢⠀⠀⠀⠀⣀⠠⠤⠴⠶⠤⠤⢄⡰⠁⠀⠛⢷⣶⣶⣿⡿⣋⠤
⠀⠙⢆⡀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⣶⣄⠉⠉⠉⢀⣤⣀⠀⠀⠀⢃⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠙⠢⢄⣀⠀⠀⠀⠀⠀⢀⡠⠴⠛⡇⠀⠀⠀⢿⣏⣹⠀⠀⠀⠈⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⠞
⠀⠀⠀⢀⡄⠀⠀⣩⠟⠉⠉⠉⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢀⣠⣄⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⡝
⠀⠀⠒⠛⠃⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠢⡀⠀⠀⠀⢀⠄⠀⠀⠀⡇⠀⠀⠀⢾⡿⠿⡇⠀⠀⠀⡟⠛⣿⠀⠀⠀⠃⠀⠀⠠⣄⣀

⠀⠀⠀⣀⣀⡀⢀⣀⣀⣀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠠⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠸⣿⣿⣶⣭⡻⣿⢿⠿⢿⣿⡋⡀⠀⠰⣇⡀⠀⣿⢀⠀⣹⡟⠏⠁⣀⣴⡤⠲⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⢶⣾⠿⠟⢿⣶⢦⡹⢷⠀⠁⠖⠼⣉⣴⣶⣋⣯⣀⣿⣶⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⠋⠀⠰⢶⣾⣿⣿⣶⣬⣤⣄⣠⣽⣯⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁⠀⢀⣀⡓⢲⣿⣿⣿⣯⣹⣿⣿⣿⣻⣿⣿⣿⣿⡏⢿⣿⡟⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⣠⣿⣿⣿⣿⡏⠿⠿⢿⡿⣿⡿⢿⡿⠿⢻⣧⢸⣷⠸⣿⠁⣿⢏⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢻⣿⣟⠟⠀⠀⣠⣿⡄⠿⠂⠛⠘⠏⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣈⡀⢀⣰⣿⣿⣧⠀⢀⣀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣇⠛⠀⠈⠍⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⢛⠋⠉⠀⠀⠈⠀⠈⠁⠀⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠬⠛⠿⢿⣿⡄
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠹⠋⠓⠚⠻⢷⣤⣀⠀⠀⣀⣠⣴⡶⢶⠤⢤⡀⠐⠲⣅⠀⠀⠀⢠⣤⣤⣤⣤
⠀⠈⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⣰⣿⡇⠀⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⣿⣇⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡻⣶⣤⡤⠀⣠⣄⣀⣀⣀⣀⢀⣠⣴⣶⣾⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣿⣍⡉⢋⣴⣿⣿⣿⣿⡿⣯⣿⣭⣿⣿⣿⣮⠿⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡟⣰⢿⣿⣷⣿⢯⣭⣭⠝⠟⢿⣿⠟⡿⢿⣿⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠙⡞⠻⣟⣿⠠⡉⠛⣿⣄⠈⢹⡶⠃⠘⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣀⠀⠈⡙⠈⣃⣛⣁⠟⠀⡀⠃⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠄⠀⠉⠉⠋⠀⢠⠄⠀⣶⡆⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣧⡄⢀⣀⣀⣀⠈⢠⣴⡿⠁⠀⠀⣄⣀⣀⡤⠤⣄⢶⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠈⢛⣿⠁⢀⣰⠟⠀⠀⠀⠀⠈⠛⠿⣦⢸⡿⣙⡾⠶⠀⠁⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢘⡇⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉⠀⠘⠂

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣅⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢀⠀⣄⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠠⡾⠶⣶⡷⢬⣻⡿⢛⡂⠀⠀⠸⣿⣿⣿⣿⣿⣿⣧⡴
⠀⠀⠀⠀⠀⠤⣤⣈⠉⣽⣿⣣⣾⣿⠿⢿⡇⠀⠀⠁⠞⣿⣿⣿⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠉⠁⠉⠙⠛⠛⠛⣷⣾⣿⣿⣤⡀⠀⠸⡏⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠙⠷⠼⣿⣿⣾⢿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢀⣶⣻⣿⣿⣿⡿⠿⣿⡟⢧⣦⣤⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦⣀
⡀⢀⣠⣼⡿⢟⣿⠿⡛⢻⣿⣒⣒⠓⠂⠸⣿⣆⠀⠁⠨⣭⣹⣿⣿⣧⣤⣼⣿⣿⣿⣿⠿⣧⡀⠀⠀⠀⢀⡀
⣾⣏⣨⣠⣾⣿⣿⣿⡷⠴⠶⢤⣬⣅⢀⠀⠉⣿⣿⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⣿⡿⢄⣖⡀⠾⣿⡟⣂⡀
⠈⠹⣯⣽⡟⣿⣿⣿⣧⡀⢀⣼⡿⣿⡿⠀⠐⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣛⣛⣿⣟⣯⣸⣿⣷⣶⣦⠀⠤⣀
⠀⠀⢹⣿⠀⠀⠹⠛⠛⠏⠉⠉⠈⠁⠀⠀⢰⡟⣾⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣶⣿⣭⣽⣻⣿⠻⠷⠾⠇⠉
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠁⠀⢈⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣷⡝⣷⡍⠩⣜⣻⣿⠟⣀⡀⠤
⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⢸⣿⣿⣿⣿⣯⣶⣿⣿⣿⣯⣛⣻⣽⣶⣿⡀⡍⣤⠀⣽⣌⣛⣛⢛⣂⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠈⣿⣿⣏⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡻⣷⢠⣼⣶⣾⣿⣿⣿⣷⠿⣻⠋⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡅⠈⢹⣿⣿⣿⣿⣿⣿⣿⣽⣯⠙⣻⣿⣿⣿⡟⠛⠛⣇⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠚⠛⠉⠉⠉⠉⠉⠉⠉⠛⠓⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⢦⡴⠛⢦⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣀⣤⣄⣀⠀⢀⡤⣧⠀⠀⢻⡀⠀⢳⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣰⠃⠀⠀⠈⠑⢺⡀⠈⢳⡀⠀⢳⠀⠀⢷⠖⠉⣳⡄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢰⠇⠀⠀⠀⠀⢀⣤⣧⠀⠀⢧⠀⠈⢧⠀⠘⡆⡞⠹⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⣇⡟⠀⠀⠀⠀⠀⢸⠀⠘⣆⠀⠈⣇⠀⠸⡆⠀⢿⡇⠀⣇⢸⠗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢠⣽⠃⠀⠀⠀⠀⠀⠘⡆⠀⠘⣆⠀⢹⡀⠀⠃⠀⢸⡇⠀⢹⣯⡄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢹⡀⠀⠀⠀⠐⠒⠲⢿⡀⠀⠸⡄⠈⠁⠀⠀⠀⠀⠀⠀⣸⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⡇⠀⠀⠐⠒⢤⣀⣀⣧⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣷⢠⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⠈⠳⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⠈⠁⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠙⣄⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠈⣷⣄⠀⠀⠀⣄⣈⣳⣄⡀⢧⣀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⠀⢀⡇⠈⠳⡀⠀⠀⠤⠤⠤⠿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀
⠀⠀⠀⠀⢀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠦⣄⡀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠙⠒⠦⢤⣄⣀⡀
⣠⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⠹⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣣⠀⠀⠀⢸⣂⣀⣀
⠀⣀⠤⠤⠀⢶⡧⡈⣻⣵⠞⠙⠉⠀⠀⣀⣭⣵⣦⣀
⢸⡱⠉⠀⠀⢹⡿⡞⡏⠀⡀⣄⠀⢤⣀⡈⠉⠉⠉⠛⠷⣦
⠀⠙⣆⠀⢀⡞⣁⣻⡷⠶⠿⠿⠾⢦⣬⣭⣥⣤⣄⡀⠀⠈⢳⡀
⠀⠀⠈⢻⡟⢤⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⢦⠀⠀⡇
⠀⠀⠀⢼⡌⢾⡏⣠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⠀⡟
⠀⠀⠀⠘⢞⣻⢀⣀⣀⣀⠉⠃⠀⠀⠐⠚⠋⠉⠙⢺⣿⣧⡞⠁
⠀⠀⠀⠀⠘⢹⠈⠛⢍⡿⠁⠀⠀⠀⠀⢐⣟⢻⡶⠀⣿⡏⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⠀⠉⠀⠀⠀⠀⣴⠟⠋⠉⠉⠓⠦
⠀⠀⠀⠀⠐⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢆⠀⢠⡴⠒⠚⠃⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠲⣠⠉⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠠⠙⠛⢋⠀⠈⢢⠀⠐⠀⠀⠀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢀⣼⣟⣿⡇⢄⠀⠀⠀⠀⠠⡄⠘⠀⠀⡀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⣾⠸⠘⡿⣇⠀⠁⠒⠀⠒⠃⠉⠀⠀⠀⠃⠀⠀⠉⠈⠉⠉⠢⢤⣀⣀⠤⠒⠤⠤⠴⠶⠛⠁
⠀⠀⠀⠀⣿⡀⣀⠈⠚⢧⡀
⠀⠀⠀⠀⡈⢿⠂⠙⠳⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀
⠠⠐⠂⠁⡀⠈⣷⡄⠀⣲⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠌⠂⠂⠄
⢀⠀⠀⠂⠂⢹⠟⣡⡾⠯⣀⣀⡀⠀⠀⠄⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠄
⠨⠌⠀⠐⠀⠘⠦⡽⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠡⠁⠠⠀⠀⠈⠀⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠋⠉⠉⠉⠉⠉⠉⠉⠓⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢳⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠶⠶⠶⠾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⠛⠉⠉⠁⠀⠉⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⣫⠅⢀⣀⣠⣤⣤⠤⠤⠤⣤⣤⣤⣄⡀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⠋⢶⣿⠭⠷⠶⠒⠒⠒⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⡉⠙⠀⠈⠉⠐⠚⢛⣛⡛⠲⠶⠶⠶⠦⠤⠤⠤⠒⠃⠀⠀⠀⡀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⢀⣴⠏⢀⡾⢡⢃⢀⣠⢰⣦⣙⠳⣶⣭⣭⣛⠒⠲⠦⠤⠄⠀⠀⠀⠀⢀⡴⠁⣰⠀⡼⢻⣆
⠀⠀⠀⠀⠀⠀⣾⠇⣠⡟⡱⠃⡞⡼⡟⣾⠉⠻⠷⣦⣍⣙⠻⠿⣿⣒⣶⠤⠤⠤⠤⠒⠉⣀⠴⢁⡞⠁⠈⣿⣧⡀
⠀⠀⠀⠀⠀⢰⡏⣴⢏⡞⠁⡼⣼⣿⡽⠃⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣍⣉⣉⣛⣛⣋⣩⡴⠞⠁⠀⠀⠀⣿⠸⣷
⠀⠀⠀⠀⠀⣼⣿⠏⡞⠀⡼⣱⣿⣿⠁⠀⠀⠀⠀⠴⠚⠳⣦⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠞⠳⣄⠀⠀⣿⡆⣿⡇
⠀⠀⠀⠀⠀⣿⡟⢸⠁⣸⢡⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿⣇⢻⣿
⠀⠀⠀⠀⠀⣿⡇⣿⠀⡏⣾⣾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠹⣅⢻⠀⠁⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀⣿⢻⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⢸⣿⢹⣿⡄⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠁⠀⠀⢸⣿⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠋⢧⢻⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⣿⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⣿⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⣥⣾⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠦⣄⡀⠀⠀⠀⠀⠸⣇⣠⠇⠀⠀⠀⠀⢀⣤⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡇⠀⠀⠉⠓⠲⢤⣄⣓⣂⣀⣀⣠⣤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⢷⡀⠀⠀⠀⠀⠀⣿⠯⡈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠙⠷⣤⣄⣀⣼⠏⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⣿⠟⢿⣤⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣾⠏⠀⠀⠈⠻⣦⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⡏⠀⠀⠀⠀⠀⠀⠙⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿

⠀⠀⠀⠀⠀⠀⠀⡠⣤⢤⠤⣀⣀⠀⠀⠀⠀⠀⣠⡤⣤⢤⣀⡀⠀⠀⠀⠀⢀⢤⢤⡤⢄⣀⠀⠀⠀⠀⠀⡠⣤⢤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠁⠋⠘⠐⠃⢸⠀⠀⠀⠀⢸⠘⠃⠃⠘⠃⠁⠀⠀⠀⠀⡇⠛⠈⠁⠛⠀⡇⠀⠀⠀⢠⠑⠋⠈⠐⠃⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠂⢸⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⡇⠀⠀⠀⠀⡇⠠⠀⠀⠠⠀⠃⠀⠀⠀⢸⠀⠁⠀⠀⠁⢸
⠀⠀⠀⠀⠀⢰⣚⡀⢉⡀⢈⡁⢸⣲⠄⠀⣖⣺⠀⠃⠀⠋⠀⣗⡢⠀⠠⣒⡇⠐⠂⠀⠒⢸⣒⡆⠀⢰⣺⠀⢛⣀⣘⠃⣘⣲
⠀⠀⠀⠀⠀⠀⠀⢧⠈⠙⠉⢀⠏⠀⠀⠀⠀⠘⣄⠉⠒⠁⡰⠁⠀⠀⠀⠀⠱⡰⠗⠒⢁⡜⠀⠀⠀⠀⠀⢇⠙⠿⠟⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⠒⠈⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠁⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⡴⣶⢢⠒⡤⢤⠀⠀⠀⠀⢀⢔⡶⣶⢒⡤⢄⠀⠀⠀⠀⢀⢔⡖⣦⢢⡤⡄⠀⠀⠀⠀⡔⣶⢢⠒⡤⢄
⠀⠀⠀⠀⠀⠀⢠⠀⠃⠈⠈⠃⢸⠀⠀⠀⠀⢸⠈⡃⠀⢘⠁⢸⠀⠀⠀⠀⠘⠘⠁⠁⣘⠁⡀⠀⠀⠀⢰⠈⠃⠈⠘⠃⢸
⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⢀⡁⢸⠀⠀⠀⠀⠠⠀⣀⠀⣀⡁⠀⠀⠀⠀⠀⠀⠀⣒⠀⢀⠀⡏⠀⠀⠀⢸⠀⣂⠀⢀⡂⢸
⠀⠀⠀⠀⠀⠸⢩⡄⡤⢄⠠⡄⢸⠹⠂⠀⠪⢹⠈⣨⣤⣅⠁⡯⠽⠀⠀⠮⢽⠀⢄⠀⣨⠀⡏⠕⠀⠺⢹⡀⠉⣀⡈⠁⣬⠹
⠀⠀⠀⠀⠀⠀⠀⠣⡑⠲⠒⡡⠃⠀⠀⠀⠀⠀⢣⡙⠛⢋⡴⠁⠀⠀⠀⠀⠀⢦⡀⠉⢛⡜⠀⠀⠀⠀⠀⠣⡀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠂⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠢⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠰⠋⢉⠉⠓⢄⡠⠊⢁⡠⢀⠙⢆⠀⠀⠀⠀⠀⡠⠊⢁⠈⠑⢆⢀⠜⢉⠄⣈⠑⠆⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢃⠀⠐⢅⡠⢈⠁⢀⠥⢂⠡⢂⢸⠀⠀⠀⠀⠀⠁⠠⠕⡨⢀⠀⠛⢀⠅⣊⢔⠂⡐⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠈⢢⠀⠁⢐⡡⢊⠡⢊⠁⡊⢡⠂⠀⠀⠀⠀⠀⢇⠐⠡⡊⢔⡠⢊⠕⡪⠐⡡⢠⠃⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠁⠂⠕⡨⢀⠉⡰⠁⠀⠀⠀⠀⠀⠀⠈⢦⠈⠔⡡⣊⠔⡨⢐⠥⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡈⢈⡥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠊⠔⡡⡚⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠌⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⣀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡀⢂⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠀⡀⠆⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠂⠀⠒⠀⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠐⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠡⠁⢁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄⣀⣉⡐⠁⠊⠐⠁⠊⠐⠀⠃⠈⢀⣁⡠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⡀⠀⡠⠄⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣀⠠⠄⣀⠤⠐⢂⠠⠐⢀⠠⠐⡀⠄⢀⠠⠐⣀⠤⠂⣀⠄⢀⡠⠔⡂⠔⢂⠠⠐⡈⠔⣊⠄⠂⡩⠔⣁⠄⡊⠔⠨⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠉⠠⠒⠈⡠⠒⢈⠠⠐⣈⠄⠊⡡⠔⢊⠠⠐⣉⠄⢂⠡⠒⣩⠔⠊⡡⠐⠉⠄⠊⠀⠔⠉⠀⠊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠄⡄⠐⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣀⠉⠉⠀⠀⡀⡀⡀⠀⠩⢦⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠞⠑⠓⠢⣉⠀⠀⠀⠀⢱⡀⠈⣚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠉⠀⠂⠀⠀⠀⠔⠉⠓⠢⠔⠉⢂⢀⣠⠜⠀⠀⠤⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠒⠘⠂⣤⣤⠀⠀⠀⢸⣷⠠⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢈⡀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄⠀⠐⠂⠸⠀⠀⢀⡄⠈⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⡄⠀⠀⠃⠀⢀⡀⠀⠠⠸⢠⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣻⠀⠀⠀⡐⠩⠆⠬⠃⢀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠢⡀⠀⠀⢀⢄⡀⠀⠀⣦⡀⠀⠀⠶⣔⠁⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⣁⣙⠈⠐⢥⠚⠫⢑⡒⠃⠉⣦⣤⠎⠈⢀⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠥⠐⠈⠁⠰⠁⠀⠀⠈⡇⠉⠀⠸⠃⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠊⠀⠀⣊⠀⠀⠀⠀⢀⣠⣤⣶⣿⣷⣄⠀⠀⠀⠀⠐⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡤⣾⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡏⠀⠉⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⢿⣿⣿⣿⣿⠃⠀⠀⠀⠟⢻⡁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠝⢛⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⠼⠀⠰⠲⠒⠒⠄⠈⠐⠊⠐⠁⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠄⠀⠠⠤⠤⠀⢀⡀
⠀⠀⠀⠀⢀⠀⢀⡀⢄⠀⠀⢀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀⢀⠄⣢⠄⠀⠄⠀⡆
⠀⠀⠀⢆⡈⡀⠀⠙⣆⡱⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢼⡁⠀⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠒⠒⠒⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠴⠒⠀⠀⠀⠀⢧
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣿⣿⣻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⣿⠀⡠⠀⠀⠀⠠⣀⡹⣿⣿⣿⣿⡷⢤⣀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠁⢀⡠⠄⠀⠀⠐⠢⣄⣿
⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠉⠻⢿⣭⡁⠄⠀⠉⠉⠀⠀⠉⠉⠁⠉⢛⣩⠶⠚⠉⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⡏⠀⠠⠴⢶⡶⠦⠄⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢴⣶⠶⠄⠀⢸⠃
⠀⠀⠀⠀⢧⠀⠀⠀⢸⡗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡟⠀⠀⠀⣸
⠀⠀⠀⠀⠘⡄⠀⠀⠈⢇⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⡸⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⢯⡃⢀⡈⠉⠛⠳⠶⢶⣤⣤⣤⣤⣤⣤⣤⠶⠶⠛⠋⢁⣀⢔⡿⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠑⢦⡉⠀⠸⠄⠀⢀⡀⠀⢀⠀⠀⠀⣀⣀⢸⠄⠒⣩⡴⠋⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠓⠶⣤⣀⣈⡉⠉⢙⠉⠉⢉⣁⣠⡴⠶⠛⠁⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠷⠾⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠉⠉⠉⠉⠉⠛⠒⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⡠⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⡉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⡠⠚⢁⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡈⠓⢄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠠⢊⣠⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⢷⣤⡑⠄⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠐⠋⠁⢀⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⡀⠀⠙⠂⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣠⡼⠛⠋⠉⠙⠛⢶⡆⠀⠀⠀⠀⠀⠀⢰⡶⠛⠋⠉⠙⠛⢶⡆⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⠏⠀⠀⢀⣠⠄⠀⠀⠹⡳⡀⠀⠀⢀⢞⠏⠀⠀⣀⣀⠀⠀⠀⠹⣦⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠠⢿⠀⠀⠀⢾⢿⣤⡶⠀⠀⡃⠀⠀⠀⠀⢸⠀⠀⣾⢿⣤⣴⠀⠀⠀⣿⠀⠀⣧
⠀⠀⠀⠀⢀⠴⢼⡀⠀⠀⠘⣄⠀⠀⠈⠙⠋⠁⠀⡠⠁⠀⠀⠀⠀⠈⢄⠀⠈⠛⠛⠁⠀⠀⣠⠃⠀⠀⢸⠶⢄
⠀⠀⠀⠀⡆⢤⡈⡇⠀⠀⠀⠈⠳⠤⣀⣀⡀⠤⠂⠁⡠⠂⠀⠀⠠⣄⠨⣐⠤⣀⣀⣀⡤⠞⡁⠀⠀⠀⢸⣠⠌⡆
⠀⠀⠀⠀⢇⠰⠋⡇⠀⠀⠀⠀⠉⠓⠒⠂⠀⠀⠀⠸⡅⢠⠀⢀⡄⢨⠆⠈⠙⠲⠶⠶⠒⠋⠀⠀⠀⠀⡟⠹⢀⠇
⠀⠀⠀⠀⠈⠦⠤⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⠳⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⠁⠙⡇⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠳⠤⠴⠤⠴⠃⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⡄⠀⠀⠠⠤⠄⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⣶⣤⣀⣀⣀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁