Embarassed Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠒⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠠⠀⣦⡀⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠄⠈⠀⠀⢹⠀⣡⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠐⠀⡒⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢢⡀⠀⠠⢡⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂⠀⢃⠀⠀⠀⠀⠛⠖⠒⠂⢆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠂⠀⠀⢀⠠⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠀⠀⠀⠉⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠰⡿⠷⠞⠃⠀⠀⡀⠀⠀⠈⠛⠛⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠀⣠⡀⠀⠀⠀⢸⠆⠀⠀⠀⣤⣤⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡉⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣦⣀⠀⠙⠛⠋⠀⠀⠀⢀⣴⠟⠛⠛⠛⠛⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⢿⠿⠿⠿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠉⠛⠓⠲⠶⠖⠒⠛⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡦⠔⠒⢈⡭⠯⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠋⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠠⠴⠊⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⢀⡤⠤⢤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠁⠈⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⢞⠀⠀⠀⢠⠃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠾⣄⣸⡄⢀⠔⠈⠉⢱⠂⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣽⠇⢧⣀⢠⣤⠞⠳⠒⠊⠈⠉⠉⠑⠒⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠚⠓⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠉⢺⢼⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⠉⠁⠉⠁⠐⠒⠒⠀⠂⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢹⡀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣇⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⢠⢄⠀⠔⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⡘⠀⠀⠠⠁
⠀⠀⠀⠀⢣⠀⠁⢇⢸⠀⡔⡠⢝⡢⢤⠤⣀⣸⠀⣉⠃⠀⡐
⠀⠀⠀⢏⠉⠳⢦⠙⠚⠜⣀⠀⠀⢉⡏⠀⠀⠉⣄⢈⡦⣤⣼⡦⠤⣀⣀⡠⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠉⠉⠉⠉⠑⠂⢈⠃⠀⠀⣬⢷⡂⢀⠊⠁⠑⠤⠉⡅⢀⠓⠊⠱⠘⠀⠀⡰⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡈⣢⡤⠾⡀⠀⠉⠉⠈⠉⠈⠉⠉⢉⠁⠀⠀⠀⡀⠤⠔⣁⡬⠔⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢩⡶⠬⣁⣐⣂⠀⠤⠤⠤⠀⠐⣣⠀⣀⠏⡠⢄⡰⠦⠤⠔⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⡀⠾⠀⣭⠸⠟⢩⡉⠿⢋⣭⠏⠸⠤⠔⠀⠀⠉⠲⠤⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⢈⡁⢴⠄⣉⣰⣦⣈⣁⣴⠆⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠉⢑⠶⣦⠿⠥⣞⡾⣗⢒⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠂⠀⠒⠚⠐⠃⠈⡆⡘⠓⠓⢒⡡⠜
⠀⠀⢀⣀⣀⣠⠠⠔⠒⠁⠀⠀⠀⠀⠀⠀⠠⠟⠁⠀⠀⠶⠄⢲⣄⠀⡀
⠺⠭⠤⠀⠀⠯⠭⠑⠦⠤⠤⠤⠤⠤⣀⣀⡀⢀⣀⠠⠴⠎⠩⠭⠁⣀⡀⠀⠬⠂

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

⠀⠀⠀⠀⠀⠀⢰⡷⢤⡀⠀⢸⡶⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠓⠢⠵⠄⠀⠈⠙⠒⠦⢄⣀⣠⠤⠤⠴⠖⠒⠒⠚⠛⠛⠛⢲⠀⢀⡀
⠀⠀⠀⢀⣀⡀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡭⠥⢽⠄
⠀⠀⠀⠈⢯⡉⠉⠓⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠙⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠘⠓⠲⠤⠤⣤⠴⢶⣶⠒⠋⠉⠑⠻⠶⠖⠒⠒⠉⠉⣼⢁⡄⠀⠀⠀⣀⣴⠥⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⡬⠷⢤⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠈⠋⠙⠦⠴⠚⢁⡠⠤⣄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⠏⠀⠀⠀⠙⣆⠀⢀⣈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠚⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣾⠤⠤⠤⢤⡴⢺⠾⣍⡉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠀⠀⠀⠀⠉⡏⠀⠀⠿⠗⢺⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠑⠦⠄⠤⠖⠙⣦⣄⠠⠴⠃⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢠⠒⠒⠠⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⡀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠉⠈⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢰⣤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡡⡞⠁⠀⠀⠀⠀⠀⢸⠖⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠤⠤⠴⠒⠋⠉⣰⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢸⢸⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠙⡻⠛⠂⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⣽⠟⢒⣷⣤⣀⣀⡀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠉⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⢸⠀⠀⠀⠀⠀⢹⠧⢄⠀⠀⠀⠀⠀⠀⠀⠸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠀⠑⡄⠀⠀⠀⡜⠀⠰⡅⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡠⠄⢀⡃⠀⠀⢠⠁⠀⠳⠄⠀⠀⠀⠀⠀⠰⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠀⠈⠉⠉⠉⠀⠙⠀⠚⣄⠠⠀⠔⠀⢠⠿⠇⠄⠁⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢠⠆⠀⠀⠀⠀⠀⠁⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⠀⠉⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠌⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢴⡉⠐⠀⠀⠀⠀⠀⠀⢀⠗⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢰⡁⣈⠢⣈⢀⠀⠀⠀⠀⠀⢠⠋⠀⠑⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸
⠀⠀⠀⠀⡠⠒⠂⠓⠪⠁⣩⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡔⢰⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠘⠀⠀⠀⡿
⠀⠀⠀⠀⠑⠂⠈⠳⠐⠁⠀⠀⣀⡠⡎⠀⠀⢀⣀⠤⠤⠒⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡆⣁⡽⣊⠵⠈⠉⠀⠀⢀⡀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠰⢀⠴⣿⠊⡏⠀⢠⠤⠔⠒⠈⠁⠀⡆⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⡤⡀⠀⠀⠀⠀⠀⣺⠤⡪⡀⢀⠗⠄⠀⠀⠀⠀⠀⠀⢠⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠃⢠⡀⠀⠀⢀⡐⢫⠐⠈⠈⠸⠗⠦⠤⣀⣀⠉⠛⠋⠉⠀⠀⠀⣢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⡟⠐⠂⠉⠁⠀⠐⢚⣾⢋⠋⡏⠀⠀⠀⠀⠈⡇⠃⠀⠈⠉⠉⣧⠀⢁⠔⠉⢀⠠⡄⠀⢰⠒⠠⢀⡀⡠⠀⠈⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠱⡄⠀⠀⠀⠀⢀⢞⠇⢬⣬⣁⠀⡠⠤⣀⠶⠢⡀⠀⠀⢀⠔⠁⠁⠒⠶⠖⠉⠠⢀⡖⠃⠀⠀⠂⠈⠦⠄⠐⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠹⡀⠀⠀⠀⠞⠸⡤⠊⠀⠈⠁⠣⠤⢰⠀⠀⠘⠊⢉⠁⠀⠀⠀⠀⠀⢼⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠑⡀⠀⡠⠂⠁⠣⡀⠀⠀⠀⠀⠀⡆⠀⢧⠈⢇⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠒⠚⠃
⠀⠀⠀⠀⠈⠊⠴⠀⠀⣒⣯⣅⣒⣀⣀⣀⣈⠉⠀⠁⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠆⣀⣤⣶⡶⠿⢿⣿⣿⣿⣿⣷⣶⣦⢤⣀
⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣾⣿⣿⣵⣶⣶⣠⣤⣔⢛⠛⠛⣛⣻⣿⣾⣝⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣀⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⠿⠿⠟⢿⠻⣿⣿⣵⣶⡶⠆⠀⠈⠭⣿⣿⣶⣕⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠉⠉⣹⣟⣋⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣷⡶⠄⠘⠻⢿⣿⣷⣙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠋⠉⣉⣙⣁⣀⣈⡋⠉⡿⢿⣿⣿⣿⣿⣿⣿⣶⣤⣐⠢⣽⣿⣿⣮⣳⡀
⠀⠀⠀⢀⣤⣤⣤⣤⣶⡶⠶⠀⠀⣰⡏⠀⡴⠞⠛⠛⠛⠉⠉⠉⠋⠙⡷⠷⢾⡁⠈⢿⡿⠿⣿⣿⣿⣦⣄⠦⣝⣿⣿⣿⣽⡄
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⡜⣹⠁⢀⣤⣤⠴⠖⠷⠒⢲⡟⠻⣶⡿⠤⠞⢀⣤⣼⠇⠀⠰⣿⣿⣿⣿⠂⣈⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⡏⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣾⡟⠻⣦⠀⠈⠉⠁⠀⢠⣿⣿⡟⠙⠻⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢡⣶⠶⠛⠉⠁⢨⡿⢷⡈⠻⣿⡿⣦⣈⡀⠀⠀⠀⢠⣿⣿⡿⠉⠀⡀⠢⣼⣿⣿⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣸⣿⡄⢤⡀⣶⡄⠀⠀⠓⠀⠈⠁⠈⠋⠉⠀⠀⠀⠸⣿⣷⡄⠀⠀⠻⣷⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⢴⣿⣿⠟⠈⠛⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⡃⣈⣢⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠰⣏⠀⢢⣀⣼⣿⣿⡯⠬⠭⣄⡀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⣠⣾⠃⢰⣿⣿⣿⢟⣛⣙⠻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢙⣿⣿⡿⠶⠶⠶⢦⣭⣦⡄⣿⠃⠀⠀⠀⢠⣴⠾⠛⠋⠁⠀⠙⢋⣀⣻⣿⣽⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢋⡿⠶⠶⠶⠦⠄⠈⠻⠁⣀⡄⠀⠀⣠⠞⠁⠀⠀⠀⢀⣤⡴⢾⣟⠿⠟⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠐⠶⠚⠋⠀⠀⠀⠁⠀⠀⠀⠀⣠⣾⠟⠷⠤⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠷⠿⠾⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣤⠶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠛⢿⣆⠀⠀⠀⢠⠞⠁⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢻⠀⠀⢠⠃⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⢀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⣶⣸⠀⢸⠀⠀⠀⠀⠀⢿⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠛⠏⠀⠸⡄⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠤⣄⣀⣠⠞⠚⠁⠂⠙⠤⣀⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡠⡠⠀⣨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡴⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⢤⣤⠤⠴⢲⠄
⠀⠀⣿⣠⣿⠀⠘⢻⡖⠓⣦⠖⢦⠀⠈⢻⠀⠀⠀⡴⢦⣤⣤⣤⢤⣤⣤⣠⢤⡀⠠⠶⡄⢀⠤⢤⡶⣠⠤⢤⡆⢀⣀⠀⠀⣀⣹⡆
⠀⠀⣿⠀⠛⢰⠄⢸⡄⠀⣿⠀⠈⣧⠀⢸⠋⠙⣦⡴⢒⡇⢸⠁⠀⠀⢸⠁⠀⠀⡤⠒⡟⣹⡒⠲⡇⣽⠒⢺⣧⣯⣬⣽⡾⠁⠘⡇
⠀⠉⠉⠉⠉⠉⠀⠚⠓⠐⠛⠂⠐⠛⠒⢸⠦⠖⠋⠳⠾⠷⠾⠶⠀⠀⠾⠶⠀⠀⠳⠶⠷⠛⠉⠉⠁⠋⠉⠉⠀⠳⠤⠖⠳⠤⠶⠧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡗⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⢻⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠞⠛⠛⠁⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠖⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠀⠀⠀⣿
⠀⠀⠀⣀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠈⠿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⣴⣾⣿
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠸⣿⣿⣿⠟⣿
⠀⠀⠀⠀⣾⡷⠀⠀⢀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⢻⡿⠁⠀⣿
⠀⠀⠀⠀⠙⠃⠀⠀⠺⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣆⠀⠀⣹⣿⠟⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣾⣿⡇⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠷⠶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⢼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣀⣼⣿⣿⣧⠀⠀⠀⠀⠀⣾⣷⠀⠀⠈⡿⠛⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿
⣿⣿⠟⠙⣿⣦⠀⠀⠀⠀⠻⡇⠀⠀⠘⠃⠀⠀⠙⢻⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣦⡀⠀⣿
⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠸⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠇⠀⠀⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠙⢻⣦⣄⠀⠀⠀⠀⠻⢿⣷⡀⠀⢀⣠⡄⠈⠉⠉⢻⣿⣿⣿⠆⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠃⠀⢀⣀⣈⠉⠑⠶⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⠀⠀⠀⡙⠋⠁⠀⠀⢻⣿⣿⠿⠂⠰⣿⣿⣿⣿⢿⣿⣷⣄⣀⠀⠀⣠⡄⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⣿⣿⡇⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢹⣿⣿⡇⠀⠘⠃⠀⠀⣰⡾⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⠊⠉⠉⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢸⣿⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢿⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠈⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⢸⡿⠋⠀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠆⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠻⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠶⠾⢿⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠶⠀⠀⠀⠀⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡖⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣀⡀⠀⠀⢀⣠⣿⣷⣶⣶⣶⣿⣶⣶⣤⣀⣀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠦⣉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠀⠤⠤⠠⠄⠤⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠉⠒⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡝⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠋⠉⠀⢠⠊⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⡰⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠢⡔⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⠀⠀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⢦⡦⠤⣤⣤⣜⣀⣀⡤⠴⢴⣲⣒⣒⣒⣶⣲⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⠭⢁⣐⣒⡒⠒⠒⠒⠒⠷⠤⠤⠤⠤⠶⠺⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠦⠤⠤⢀⣁⣀⣉⠉⠉⢻⣍⣉⣉⣉⣉⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⠒⠒⠀⠤⠀⠀⠠⠤⠤⠤⠤⡏⠙⠧⠄⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠬⢉⣉⡐⠒⠒⠒⠂⠀⠀⠀⠐⡇⠀⢷⣒⣒⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠈⢹⢹⠉⠉⠉⠉⠉⠉⡝⠀⠀⢡⠜⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢸⢸⡀⠀⠀⠀⠀⠰⡇⠀⢀⠐⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⢀⡀⠈⡆⡇⠀⠀⠀⠀⠀⠈⠙⠉⠈⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠮⠌⠁⠒⠄⠘⣙⣏⠙⠋⠁⠀⢀⣀⡤⠖⠋⠁⣹⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡴⣩⠚⣉⣡⠴⠒⠻⣕⡖⠤⣤⢀⣀⣹⠛⠛⠉⠉⠁⠀⣐⣀⣤⡴⣶⡛⠒⠒⢤⡑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠓⠺⠧⠼⠦⠤⠄⠒⠒⠒⠚⠷⠾⢼⡯⠭⠯⠭⠿⠭⠤⠤⠤⣬⣀⣏⣐⣀⣀⣱⢼
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠓⠒⠒⠤⠤⠤⠤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣶⣾⣿⣿⣿⣿⣿⣦⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠛⠋⡉⠻⠿⠿⠂⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠁⠀⢀⣾⣿⡄⠀⠀⢰⣿⣷⣀⣀⣀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⢠⣾⣿⣿⠟⠀⠀⠀⠈⠻⢿⣿⠿⠟⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡆⠈⠉⠠⣿⡆⠀⠀⠀⣿⠆⠀⠀⠀⠀⣿⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣇⡟⡇⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⢷⠀⠀⢀⠀⠘⠷⠶⠂⠀⢀⣠⣿⢦⠀⣟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡄⠚⢹⣗⠲⠶⠖⠚⣋⣥⣾⠏⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢿⡛⠛⠿⠛⠛⢹⡟⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠛⠒⠒⠛⠉⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢤⣀⣀⣀⣀⡠⠔⢹⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⣿⠀⠀⠉⠉⠀⠀⠀⢸⣏⡭⠉⠙⠓⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠐⠾⣤⡀⠀⠀⣀⣴⣾⡭⠴⠶⠒⠚⠋⠁⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠙⠛⠒⠒⠶⠶⠿⠷⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠠⠀⠀⣒⡒⠲⠀⠀⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠲⣄⠀⣀⣀⣀⣑⡀⠀⠀⠀⠠⢡⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⢑⠠⠀⠈⣿⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⡄⠀⠈⠓⢤⠀⠉⠃⠱⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠼⠃⠀⠀⠀⠀⠀⠰⢤⠇⠀⠃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠂⢰⠄⠀⠀⠀⡄⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠸⠢⣀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⣇⠀⠀⠔⠁⠀⠀⠀⢀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡴⠀⠀⡀⣸⠃⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠴⠊⠁⠀⠀⢈⡴⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠞⠋⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠞⠉⠀⠀⠀⠀⠀⠀⣠⠾⡁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⠤⠀⡀⠠⠤⠞⠉⠀⠀⢀⠀⠀⠀⠀⣠⡞⠁⠀⠈⠑⠒⠤⣀⡀⠐⠀⡟
⠀⠀⠀⢀⡔⠋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡁⠀⠀⠀⠀⠀⠀⠀⢸⠋⠛⠻⡓⠀⠤⣀
⠀⠀⠀⠀⠣⠤⠞⠉⡼⠂⣀⡀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠙⠢⣄⠀⠀⠀⠀⠀⡟⠀⠀⢀⠘⢦⣀⡼
⠀⠀⠀⠀⠀⠀⣠⠞⠁⣰⠿⣀⢶⠂⡣⢾⠁⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⡼⠀⠀⣤⠈⢳⡄⠱⣄
⠀⠀⠀⠀⠀⡰⠃⢠⠞⡍⢀⠼⡆⣍⠁⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢿⠃⢠⡄⠸⣧⠈⢧⠉⠛
⠀⠀⠀⠀⠀⠓⠴⠋⢰⠀⡘⠀⠆⢸⢧⡀⡇⠀⠀⠀⠀⠀⠀⣠⠔⠊⡡⡾⠀⣀⢇⢀⡏⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠇⠀⡇⠘⡆⠉⠁⠀⠀⢀⡠⠖⣋⠠⢔⠁⠀⠻⠶⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠀⠀⠙⢬⣁⠀⣠⡴⠞⡩⠔⠊⠙⣄⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢶⣁⡀⠀⠀⠀⠘⢢⡀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⢤⡀⠉⠢⣙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠴⠚⠁⠀⠀⠈⠑⠮⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣀⣤⣶⣒⣒⣲⣶⣶⡶⠶⠒⣛⡩⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠖⠚⠛⠛⠛⠛⠓⠲⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣴⠏⠀⠀⢀⣠⠶⠛⠛⣛⣛⣛⣲⣦⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣚⣛⣛⡛⠛⠶⣤⡀⠀⠀⠹⣦
⠀⠀⢀⡾⠁⠀⠀⣠⣿⠵⠖⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠳⠶⣿⣦⠀⠀⠈⢷⡀
⠀⢀⡾⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢷⡀
⠀⣼⠃⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠘⣧
⢠⡏⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⢹
⣸⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈
⣿⠀⠀⠀⢀⡏⠀⠀⠀⠀⣴⣿⣿⣿⣦⡀⠀⠀⠀⢿⠀⠀⠀⠀⣾⠀⠀⠀⠀⣠⣾⣿⣿⣷⡄⠀⠀⠀⢹⡄
⣿⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇
⢹⡀⠀⠀⠀⢿⡀⠀⠀⠀⠈⠛⠿⠛⠋⠀⠀⠀⢠⡟⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⡾⠀⠀⠀⢀
⠘⡇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸
⠀⢻⡄⠀⠀⠀⠀⠈⠓⠦⣤⣄⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⣀⣤⡴⠞⠋⠀⠀⠀⠀⢠⡟
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠦⠤⣤⣤⣤⣤⡤⠴⠶⠚⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⠴⠞⠛⠓⠲⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁⠀⠀⣀⣤⠴⠓⠲⠶⢦⣄⣀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠚⠉⠁⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠠⠴⣿⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⣿⣿⡟⠷⣦⣀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢠⣄⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡢⠖⠀⠀⠠⣀⣤⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠛⠶⣦⣄⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣶⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⠋⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⢀⡴⠶⠖⠲⢶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠶⠗⠶⣤⡀
⢠⡞⠁⠀⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠖⠖⠋⠀⠀⠀⠙⠲⠶⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠶⠀⠈⢻
⣾⠁⠀⠀⢠⣟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⣿⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⠟⠀⠀⠀⣰
⢻⣦⣀⣀⣀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠈⠉⠉⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣀⡀⠀⠀⠀⠀⢀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠳⢦⡀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⢻⡀⠀⠞⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⡟⠛⠛⠛⠛⢻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⠶⠶⠛⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⣀⠀⢀⡀⠀⢀⡀⠀⠀⣀⡀⠀⠀⣀⡀⠀⢀⣀⠀⣀⣀⣀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣇⠀⢾⣏⣉⡛⠂⢸⣇⣀⣸⡇⠀⢰⡿⣿⡀⠀⣿⣿⠀⣾⣿⠀⣿⣏⣉⣉⠀⣿⠉⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣴⣿⡄⣠⡉⠙⣻⡆⢸⡏⠉⢹⡇⢠⣿⣧⢾⣧⠀⣿⢸⣶⡏⣿⠀⣿⡏⠉⠉⠀⣿⡀⢀⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠟⢿⣯⢸⣿⣿⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⣀⣾⣿⣿⡿⠟⠉⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣀⣠⣿⣿⣿⣿⣷⣄⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠑⠊⠉⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⣀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠠⠿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⡐⠁⠀⠀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠈⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿⣿⣿⣿⣿⣟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣿⡟⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣦⣤⣴⣿⠟⠋⠙⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣥⡀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣟⠀⢀⣾⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣧⡘⠉
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠛⠛⠛⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣻⣯⠤⠴⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢯⣍⡀
⠀⠀⠀⠀⠀⢀⡴⠋⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣦⡀
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⢷⡀
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢷⡀
⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣧
⠀⠀⢿⠀⠀⢀⠀⠀⠀⠀⣴⠏⠀⠀⣤⣄⣀⡀⠉⣷⣄⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿
⠀⢀⣸⡆⣀⣼⡄⠀⢀⡾⢷⣠⠴⠋⠉⠈⠉⠛⠻⠇⠈⢷⡀⠀⢀⡞⢹⡄⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠋⠉⢹⠃⠘⣧⠀⣼⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⡾⠁⠀⠿⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿
⢿⣄⣀⣸⡆⠀⠘⢧⣿⠀⠀⢀⣤⣶⠶⠄⠀⠀⠀⠀⠀⠀⠘⠋⠀⠐⠛⠿⣶⣄⠹⡆⠀⢤⣀⠀⠀⠀⠀⢸⠷⠷⢶
⠀⠙⠛⠛⣇⠀⠀⠀⠁⠀⠀⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠉⠉⠉⠉⠳⣄⠀⠀⣿⠀⠀⢀
⠀⠀⠀⠀⣻⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⢀⣤⣶⣶⣶⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⢷⣸⣿⡶⠾⠛
⠀⠀⠀⡼⠙⣧⠀⠀⠀⠀⠀⢠⡾⢻⣿⣿⡏⠉⢳⡄⠀⢀⣾⣿⣿⣿⢻⣀⣸⣿⡄⠀⣠⣶⠶⠆⠀⢈⡿
⠀⠀⣸⠁⠀⠘⣧⠀⠀⠀⠀⣾⠀⠘⢿⣿⣾⣶⣿⣿⠀⢸⡟⠿⣿⣿⣿⣿⠟⢁⡇⠀⠁⢸⡆⠀⢠⣾⡁
⠀⠀⡿⠀⣸⠀⠘⣧⡀⠀⠀⢹⡄⠀⠀⠙⠛⠛⢋⡟⠀⠈⣧⡀⠀⠈⠀⠀⠀⣼⠇⠀⠀⢸⠇⣠⠟⠀⢻⡀
⠀⠀⣇⡀⣻⠀⠀⠈⠳⣄⠀⠀⠙⢶⣤⣤⣤⠶⠋⠀⠀⠀⠈⠻⢦⣤⣤⡴⠞⠁⠀⠀⠀⣼⡴⢋⣀⣀⣨⡇
⠀⠀⠘⡏⠻⣇⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⣼⠉⢩⡏
⠀⠀⠀⠱⡄⠹⣆⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣤⠆⠀⠀⠀⠀⠀⢹⣷⣤⣤⠶⠾⠛⠉⠀⠀⣰⠇⣠⠟
⠀⠀⠀⠀⠙⠳⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠉⠛⠶⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡶⠟⠋⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢤⣀⡀⠀⠉⠉⠙⣻⣿⠛⠛⠛⢿⣏⣉⠉⠉⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠉⠙⠛⠛⠛⠛⢻⡇⠀⠀⠀⣾⠀⠉⠉⠛⠛⠉⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣇⠀⠀⠀⠀⢀⣴⡟⠁

⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⡤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣀⣠⣤⢀⣀⣤⠤⣄⣀⣠⣤⣶⣦⣼⣿⣿⣿⣤⣀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡼⡶⣿⣿⣿⠋⠁⠀⠀⠀⠁⠀⣼⣿⣿⣿⡟⠁⠀⠀⠈⠳⣄⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢻⣿⣿⣶⠆⡄⡐⢀⠄⠀⢠⠀⠈⡌⠉⠙⣼⠀⠀⠀⠀⠀⠀⠘⣆⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠸⠿⢻⠃⠰⠀⠇⡾⠀⠀⠘⣇⠀⢰⢠⠀⠸⣧⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⡨⠇⢳⠀⠀⠀⢛⣅⣘⡮⢦⠀⡇⠱⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠠⠋⡼⣿⢾⢆⠀⠰⠛⡿⠹⣿⡤⠀⡷⢤⡟⢦⢀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢿⡄⠸⠴⡇⠁⣬⠀⠀⠂⠀⠨⠄⠈⠹⣿⠉⢸⠇⠈⣧⡆⢀⣀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⣸⠳⣾⡤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⣽⠔⠋⠀⠀⣿⡇⢸⡿⢹
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⠸⣿⠀⢰⣏⢤⣄⡀⠀⠀⠀⠀⢀⣤⣚⠻⣿⠀⠀⠀⠼⢋⡧⠞⠁⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠓⢿⠓⢈⣧⣑⣴⣛⣛⢿⣦⡀⢨⣿⣿⣷⡄⠀⠀⠀⠀⠈⠁⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠣⢀⣀⣀⡀⣰⡻⢥⠊⠙⢑⡐⣿⣿⣷⣿⣿⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠢⢄⡀⠀⠀⠀⣀⠴⠋⠉⠀⠀⡛⠋⠠⠔⢲⡄⣴⠭⠤⠤⠉⠛⢿⣛⣽⠾⠛⠉⠉⠉⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡰⢷⣿⣅⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⠉⢹⣦⡀⠀⠀⣨⣷⣄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⣄⣀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⠀⠀⣄⢲⢲⡄⠀⠀⢀⡀⠀⠀⢴⣛⣿⣱⠀⠀⢸⡆⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠹⣆⠀⠈⠉⠉⠀⠀⠀⠸⠃⠀⠀⠀⠉⠉⠀⠀⠀⣸⠇⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠈⠑⠚⠁⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠶⠦⠤⠶⠶⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣇⠀⠀⣤⠶⣶⣶⡶⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣯⣿⣿⣤⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠑⠦⠽⠯⠴⠞⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠻⢶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⡃⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⢀⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠒⠰⢶⢰⣦⠀⡀⢀⣀⢀⡀⠀⠀⣠⡀⡇⠀⠀⠚⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠁⠈⠉⠈⣻⠀⠀⠈⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣤⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢻⣿⡿⠋⠘⠋⠁⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⢿⣿⠏⠀⠛⠁⠐⠒⠲⠶⠄⠀⠀⠀⠀⠀⠴⠖⠒⠒⠋⠀⠉⠛⢿⠏⢉⠙⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⣇⠈⠏⠀⠀⠀⠀⠀⢰⣶⣦⠀⠀⠀⠀⠀⠀⠠⣶⣶⡂⠀⠀⠀⠀⠘⢀⢼⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⠈⣡⣤⠀⠀⠀⠀⡀⡈⠀⠀⠀⠀⢀⡽⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣶⣦⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⣀⡴⠀⠀⠀⠁⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠒⠒⠛⣿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠺⣿⣿⣿⣿⣿⣿⣿⣿⠟⢸⡇⠀⠀⠀⡿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⢹⠟⡿⢿⣿⣟⣁⣀⣨⡇⠀⠀⠀⣯⣀⡀⣠⣝⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢒⣶⣶⣇⣼⠁⠀⠀⡸⠁⠀⠙⠮⠷⠶⠯⠞⠁⠈⢳⠀⠀⠀⣇⢸⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠢⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⠔⠚⣿⣿⣿⣿⣿⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠓⠒⠦⠤⠶⠒⠒⠒⠒⠒⠲⠶⠦⠤⣄⣀⡤⣼⣿⣿⣿⣿⣿⣈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⡿⠛⠋⠉⠉⠙⠓⠒⠒⠒⠛⠉⠉⠉⠉⠙⠛⠓⠒⠶⠶⠶⣿⣿⣿⣿⣿⡟⠒⠒⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡅⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉⠉⠓⠒⠶⠶⠖⠒⠶⠶⠤⠤⠤⢤⣤⣤⣤⣄⣀⣀⣠⡤⠤⠶⠶⠬⠭⠭⠭⠤⠤⠤⠤⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠶⠦⠤⠶⠶⠶⠶⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠛⠛⠻⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⡿⠿⠿⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⡀⠨⠭⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⡉⡉⠉⠁⠩⠅⣀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡏⠀⢨⠀⠔⠈⠀⡀⠚⠛⠁⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠢⠀⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠊⠀⣠⠨⠐⠀⢁⣶⣤⠀⠀⣤⣶⡈⠀⠀⠀⣄⠀⠑⠂⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠴⠦⡍⠽⠀⠀⠯⢩⠴⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢾⣷⡄⢠⣿⡷⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢁⠝⠀⠀⠫⡈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠠⣀⠀⠀⣀⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠄



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣦⠴⠚⠛⠉⠉⠉⠙⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠳⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⠒⣦⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠙⠒⠒⠖⠚⠉⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⠤⠤⠶⠚⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠒⠦⣴⠚⠳⠜⠉⢳⣠⠊⠙⡶⠋⠙⡦⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠃⠈⠀⢀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⣁⠀⠈⠁⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠞⠉⠀⢀⣰⠋⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⠁⠀⠀⠰⠁⠘⠃⠀⠀⢠⠖⠢⠀⠀⠀⠀⠀⠀⠐⢯⠽⠳⣆⠀⠀⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡴⠋⠁⢀⡔⠚⠂⠀⠀⣴⠋⠙⠂⠀⠀⠀⡰⠀⠐⠲⢤⡀⠀⠀⠸⠓⢦⠀⠀⠀⣾⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡷⠀⡔⢹⣳⠀⠀⢠⠊⠻⠂⠀⠀⢀⣤⠞⣇⠀⠀⡀⣴⣷⢦⡀⠀⠀⠈⢷⡀⠀⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣿⠁⠘⠀⠀⠀⢀⠴⠲⠄⠀⠀⠀⠀⢀⡟⠀⠙⠶⢾⡇⠈⠁⠀⣽⠶⣄⠀⠸⠏⠳⡄⢀⣾⣤⡀
⠀⠀⠀⠀⠘⢿⣶⣤⣀⢸⣰⡇⠀⠀⠀⢀⢴⡷⡄⠀⠀⠀⠀⣰⠖⠋⠀⠀⠀⠀⠈⠷⡤⣶⠀⠀⡀⠈⢷⡀⠀⠀⣿⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⡿⠧⠀⠀⠀⠁⠈⡛⠁⢀⡠⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢺⡇⠀⠈⠋⠹⡄⠉⠈⢳⣴⣇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⣴⠒⠦⣄⣠⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠶⣶⠀⠿⠳⣆⠘⡇⠈⡇⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢠⠖⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢿⡀⣠⡄⠘⡄⠃⣸⣷⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠉⠹⣧⣀⡄⠈⢻⡿⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⣸⣷⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⢸⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⣹
⠀⠀⠀⠀⠀⠀⣼⠁⢹⣾⡗⠘⣇⠀⠀⠀⢰⣤⣶⣶⣴⣶⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣷⡄⠀⠀⠀⣸⡟⠙⢦⠉⢷⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠸⣏⣠⠀⢿⡀⠀⠀⠘⠛⠛⠛⠛⠛⠋⠀⠀⠸⠧⠤⠀⠀⠀⠀⠀⠀⠻⠿⠛⠛⠛⠋⠀⠀⢀⣿⢃⣄⣸⠀⣸⠇
⠀⠀⠀⠀⠀⠀⡸⠧⠀⢸⠃⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠤⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠸⣾⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠑⡆⢀⡘⠷⣃⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠛⡧⢀⣽
⠀⠀⠀⠀⠀⠀⠀⠛⠛⣧⡴⠛⢦⡼⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢀⣠⠞⠃⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⢻⡤⢶⣠⠷⠴⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣲⣢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡩⢟⡖⠶⠖⣶⠭⡒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣲⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠦⡿⢂⠇⠀⠀⠀⠀⠀⠀⠰⢡⣪⠞⠓⠒⠒⠒⠒⠛⢮⣪⢆⠀⠀⠀⠀⠀⠀⠀⠇⠺⡧⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⣇⣾⣏⠠⠤⠄⠀⠀⠤⠤⢌⣿⣎⡆⠀⠀⠀⠀⠀⠀⠈⠶⠤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡏⠸⠇⠐⠒⠀⠀⠀⠐⠒⠂⠿⠈⣧⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡆⠀⠀⠀⠀⠀⢸⠉⠲⡀⠀⠀⠀⠀⡄⠀⠀⠀⡰⠊⣹⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠸⡀⠀⠀⠀⠀⢸⠆⠀⠳⡀⠰⣖⠒⢒⡶⠀⡰⠃⠀⡻⠀⠀⠀⠀⠀⡸⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠉⠑⠒⠂⠤⠤⠦⣀⣀⣙⣦⣌⣉⡉⣤⣞⣁⣀⡠⠇⠤⠄⠒⠒⠉⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣀⡠⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠓⠈⠉⠉⠁⠓⠒⠋⢄⠀⠀⠀⠈⠉⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢘⠀⡆⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢉⣁⡉⠙⠀⠒⠀⢻⠀⠀⠀⠀⢸⣉⡃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠤⢬⠀⢸⠤⠤⠤⠤⣼⠀⠀⠀⠀⢸⢸⠂⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠯⠿⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿⢽⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠇⢸⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⢸⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⢲⠃⠀⢸⠒⠒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠺⣁⠀⠈⠳⡖⢎⡀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠚⠃⠉⠉⠒⠒⠚
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠒⠒⠒⠒⠲⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡠⢦⣠⠤⡀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠤⠤⣀⣠⢷⡀⠹⡄⠱⡄⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠃⠀⠀⠀⣹⠀⠱⡀⠱⡀⢹⢡⢾⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢻⡎⠀⠀⠀⢸⠈⢣⠀⢣⠀⢇⠈⣧⠘⣤⠗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢺⡇⠀⠀⠀⣈⣆⠀⢧⠘⠂⠀⠀⠛⠀⡿⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡇⠀⠐⠢⢄⣸⠀⠘⠀⠀⠀⠀⠀⢠⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣵⢢⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡀⠉⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠑⣄⠀⠀⠀⠱⣄⠀⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡏⠱⣄⠈⠩⠭⠵⡌⠁⠀⠀⠀⠘⢦⣀
⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⢧⠀⠀⠀⠀⠀⠈⢯⠉⠒⠲⠤⢄⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢣⠀⠣⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠚⠛⠋⠉⠙⠛⠓⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠦⣄⡀⠀⠀⠀⠀⠀⠈⠓⢬⡲⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⠖⠒⠒⢤⡀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⡶⠚⠛⠉⠙⢦⡀⢢⡀⠀⠀⠀⠀⠙⢮⡳⣄⠀⠀⠀⠀⠘⣷
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢳⡀⠀⠘⣇⠀⠀⢀⣴⠟⠁⠀⢀⡀⠀⠀⠀⠉⠳⢌⡒⣤⢬⣱⣤⡀⠙⢮⠂⠀⠀⠀⢰⣿⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀
⢤⣤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣽⡄⢠⡾⠁⠀⢠⠞⠛⠛⠃⠀⠀⠀⠀⠀⠉⠁⠘⠛⠶⡝⢦⡀⠳⡀⠀⠀⢸⡷⣤⠞⠉⠁⠀⠈⠉⠙⠒⠲⠶⢚⣿
⠀⠉⠓⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⢹⣀⣹⣿⡁⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣴⠦⣄⠀⠀⠻⣄⠀⠀⢠⣿⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣤⣤⣴⡿⠋⠉⠈⣧⠀⠀⠀⢰⠋⣿⣏⣉⣦⡀⠀⢀⣾⣿⣷⡶⢻⠀⠀⠀⢹⣆⣴⠟⠉⠉⠳⣦⣀⡀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢈⣧⡀⠀⠸⣄⠈⠙⠛⢋⡇⠀⠘⡄⠉⠉⢀⡾⠀⠀⢠⡾⡏⠁⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⢦⡀⠈⠓⠲⠖⠋⠀⠀⠀⠉⠓⠛⠉⠀⣠⡶⢻⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣧⠀⠉⠓⠦⣤⣤⣤⠀⠀⠀⣷⠤⠶⠖⠋⠁⡠⢺⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠸⣧⡀⠀⠀⠀⠀⠈⠳⠶⠚⠁⠀⠀⢀⡤⢊⢕⡟⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠙⣍⠲⢤⣀⡀⠀⠀⣀⣀⡤⠴⢚⣥⠞⢳⡞⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡈⠳⣦⣬⣍⣉⣉⡭⠤⠖⠋⠁⢟⡣⠊⣠⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠙⣄⠈⠳⣔⣂⠄⣒⡆⠠⠤⠖⢋⡤⡞⠁⡧⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢣⣀⡼⣳⢦⣀⣱⣋⣠⠖⡊⣩⠝⠁⢳⡈⠉⠁⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀⢀⡴⠃⠀⠀⠉⠣⣤⣀⡴⠟⠀⠀⠀⠙⠲⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠠⢴⡶⢶⣿⠿⠿⣿⣿⣍⣁⣰⣤⣤⣤⣤⣶⣿⣿⡗⠒⠀⠉⠀⠉
⠀⠀⠀⠀⣠⡾⠿⠛⠿⠛⠋⠉⠀⢀⣐⣶⣶⣶⣴⠧⢶⢾⣯⡿⠿⢻⣛⣿⣿⡷⢾⣿⣿⠗⠂⢀⠂
⠀⠀⠰⠞⠁⠀⠀⣀⡀⠰⠞⡁⢁⣉⠉⣻⡟⣡⣤⡴⠂⠈⠛⢳⣶⣦⣤⣴⣈⣤⣶⡖⡀⢀⠴⣶⠿⡖⣂
⠀⠀⠀⠀⠀⠀⠈⠉⣴⣶⡾⣿⣇⣠⣿⣿⣇⣉⣤⣤⣂⣠⣽⣧⣬⣿⣾⣿⣿⣿⣟⠢⣀⣀⢀
⠀⠀⠀⠀⠀⢀⣀⣐⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⢿⠿⠛⠛⠹⢯⢹⢏⣹⣴⣤⡴⠆⡀
⠀⠀⠀⠀⠀⠈⢹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢛⣱⣿⣷⣄⠸⡷⣶⣄⠀⠂⠚⠿⠛⠛⠛⠇⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠻⢿⣿⣿⣿⡿⠿⠟⠋⠀⢠⣿⣷⣿⣿⣿⣿⣦⢹⣿⣷⡄⡠⡀
⠀⠀⠀⠀⠀⣀⠀⠈⠁⢠⢿⣿⣶⣾⣿⠾⠛⢠⣆⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣿⣿⡟⠃⠀⠀⢸⣿⣿⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣹⣷⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⠀⠀⠀⣿⡄⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠠⣤⠦⣤⠄⠀⠀⠉⠀⢀⠀⣶⣿⣿⣿⣿⣿⣿⣏⣻⣿⣿⣷⣦⣄⡀
⠀⠀⠀⣾⣷⡆⣀⠠⠕⠊⠀⣿⠀⠀⠀⠺⠇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣠⡤
⠀⠀⠰⠀⠈⠤⣤⣤⣠⠀⠐⠿⠓⠋⣴⣶⣦⣀⣹⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⠂⠁⠐⠀⠀⠄⠀⣀
⠀⠀⣶⣤⣸⣿⣿⢻⣟⣁⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⡶⠚⠒⠂⠀⠀⠀⠉
⠀⠀⢈⣉⣭⣯⣭⡍⠀⠀⠀⠀⠀⠀⣌⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢤⠼⠟⠒
⠀⠀⢀⣩⣭⣿⣿⣿⣷⣶⠦⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠈⠷⠀⠀⠀⣀⣀⡀
⠀⠀⢙⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⣤⡄⠀⠀⠼⠿⠧⠿⠓
⠀⠈⠂⠻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⢀⡾⣾⣿⣿⣿⣿⣿⣷⣤⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣹⣿⣿⣿⣿⡇⠀⣤⣄⣀
⠀⠀⢿⡿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡆⠈⣷⣦⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢿⡟⠉
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠹⠇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⠀⢀⣀
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⢀⣼⣤⣤⣄⡀⢠⡴⠂
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⡿⠿⢿⠟⠿⠶⠶⠦
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢿⣿⣿⣿⣿⣿⣿⢷⣤⣶⣤⣴⣶⣄⡀
⠀⠀⠙⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣾⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⡉⢛⡳⣶⠦
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠿⠿⠿⣿⡿⠅
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣶⣾⣶⣏⣤⣷⡿⠻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⢸⣿⣿⣿⡿⢿⣿⣿⣿⡇⠠⣾⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡿⠿⡿⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠄⣩⣽⠿⠿⠍⠍⣉⠁⠀⠀⠀⠘⠛⠉⠀⠀⠈⠉⠉⠿⠇⠀⢹⣿⣿⣻⣿⣿⣿⣿⣿⣷⣟⣿⣿⡛⣿⣟⣻⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠻⠟

⠀⠀⠀⣰⠞⢩⣿⡛⢿⡷⣦
⠀⠀⢠⣏⣰⣿⣿⣿⣶⣿⡟⣷⣄
⠀⠀⢀⣟⣹⣯⡼⢻⣿⢻⣧⣿⢹
⠀⠀⢸⡟⢿⣟⣿⢿⣿⣾⣿⣷⡟⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠈⢷⡈⠻⢭⣽⣿⡿⠿⠿⠃⠀⠀⠀⣀⡴⠞⠋⠁⢀⣀⣤⣤⠀⣀⠀⣉⡙⠳⣤
⠀⠀⠀⠀⠉⠛⠛⠹⣇⠀⠀⠀⠀⠀⣠⠞⠉⠀⣠⣶⡿⠿⠛⠉⠉⢠⢚⣼⡿⡑⠄⣤⠘⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⡾⠁⣀⣰⣿⠟⡁⢅⠂⡥⠌⣠⣴⠿⠋⢂⢃⣼⣿⣦⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠠⢤⣾⢟⡠⡔⢁⣐⣬⣶⡿⠛⢁⠌⠒⣡⣾⡟⠀⡙⣿⣤⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣱⡿⠃⣪⣴⡾⢟⡛⢉⠑⠐⠎⣣⣬⡾⠋⠀⠻⣶⣔⠈⢿⣦⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠸⠟⣠⡿⢋⡁⢒⣠⣤⣶⠶⠶⠟⠋⠁⠀⠀⠀⠀⠈⢻⣮⠘⢿⣆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣌⢇⣾⢟⡔⣢⡷⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⢲⡀⢀⡀⠀⢻⣯⠸⣿⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇⣼⡏⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⢻⡇⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣹⣟⣼⠋⠀⣄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⠀⠘⣿⣿⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⣿⠁⠀⠀⢿⡀⠀⣿⠀⠀⠀⣠⠄⠀⠀⠘⣇⠀⢠⡟⠀⠀⠀⣿⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣿⠀⠀⠀⠈⠛⠛⠁⠀⠠⣾⣁⠀⠀⠀⠀⡈⠛⠋⠀⠀⠀⠀⢿⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠀⠀⠀⣴⠟⠃⠀⠀⠈⠁⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⣾⣿⣭⣙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⡇⠀⠀⢠⡾⠛⠙⠓⣦⠀⠀⠀⡗⠀⠀⠀⢠⣿⠐⡩⡻⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣿⡻⢷⠶⣄⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠁⠈⠛⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⠛⠛⠛⡛⣛⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⣍⠙⠛⠁⠀⠈⠋⢋⣽⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠦⣤⡀⠀⠀⠀⡾⠀⠊⠀⠙⠷⣦⣀⣀⣠⣾⠛⠁⠈⣠⠈⣷⠀⠀⠀⣠⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣬⠿⠛⠀⠀⢹⡆⠀⢀⡇⢨⣄⠀⠀⠀⣿⠉⠉⠈⣿⠀⠀⠀⣠⡀⢹⡄⠀⢰⡃⠀⠀⠛⠻⣯⣀
⠀⠀⠀⠀⢠⡾⠋⢀⣠⣤⢶⡞⢛⢿⡄⢸⡇⢸⡏⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢿⡇⢸⡇⢠⣿⡛⢲⡶⢦⣄⡀⠙⢶⡀
⠀⠀⠀⠀⠻⠴⠞⠋⠁⠀⠀⢿⡀⠊⣿⡄⡇⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⡇⢸⣇⣾⠣⠀⡾⠀⠀⠈⠙⠛⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠜⣿⡇⢸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢺⡇⢸⣿⠫⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠃⠘⣷⣸⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣾⣧⣾⢃⠇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⠀⠽⠛⠿⠄⠀⠐⢿⠀⠀⠀⡿⠀⠀⠂⠿⠛⠋⠐⠀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠉⠉⠉⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣀⣀⣀⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠲⢅⠀⠙⠢⡀⠀⠀⢀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⣄⠈⠑⢦⡀⠈⠲⣄⢁⡤⢤⠀⠀⠻⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⢸⡄⠈⠓⢄⡀⠙⠂⠀⠈⢿⠀⡏⠀⢠⡶⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡟⠦⣄⠀⠉⠀⠈⠀⠀⠀⠀⡇⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡼⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⢹⣦⡈⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡇⠙⠻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠡⣤⣤⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡄⠈⢳⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠳⣄⠀⠀⠀⠀⠀⠀⢿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⢀⡠⠔⢒⣉⣀⢀⣀⣀⢀⣀⠤⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄⠀⠀⠀⠀⢘⠷⡄⠀⠈⠉⠛⠛⡿⣿⣿⣿⣶⡄
⠀⠀⠉⢢⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⠤⠚⠁⠀⢹⠀⠀⠀⡰⠋⠀⠈⢻⣿⣿⡇
⠀⠀⢀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠞⠊⠁⠀⠀⠀⠀⠈⡇⢀⠎⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠙⣄⢣⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⠀⠀⠀⢀⣀⡀⠀⠘⠇
⠀⠀⠀⠀⠈⢆⡇⠀⠀⢀⠴⠒⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⣶⠿⠛⠓⠒
⠀⠀⠀⠀⠀⠘⡇⡠⠞⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⢆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⣷⣶⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⠀⠿⠿⠿⠿⠿⠈⠿⠿⠿⠿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣶⣤⠄⠀⠀⠀⠙⠉⠉⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⣉⣙⠻⢿⠀⠀⠀⠐⠒⣶⠖⢄⠀⠀⠆⢀⠤⣶⡖⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠠⠄⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⠀⠀⠀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⣀⣀⣀⣀⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⠶⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠿⠦⣀⠀⠀⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠉⠀⠀⠈⠋⠛⠓⠂⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠐⢄⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠂⠀⠀⢠⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢀⣠⠤⠤⠤⠀⣀⠀⠀⠀⠀⠀⠐⠀⠈⠧⠤⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠉⠑⠦⠤⣧⠀⠰⡤⠄⠀⠘⠖⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢠⠷⠤⢄⡀⠀⢀⠐⠳⠦⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⢀⠘⠓⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠰⠁⠀⠀⠀⠈⡉⠠⠀⣀⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠃⢸⢳⢸
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠰⡄⠄⢐⠁⢹⠀⢨⠁⢸⠛⣦⡀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠤⢋⠎
⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠘⣘⡖⣿⠂⢾⠒⢻⠖⣟⡇⠀⡇⠀⠀⠀⠀⠀⢤⡤⠖⠁
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠯⠍⠉⠀⢈⡑⠚⠟⠛⠁⠈⠀⠀⠀⠀⠀⠀⢸⡀
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠣⣤⢀⠀⣠⠳⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⢆
⢰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⡄⠀⠘⡄⠹⡈⠀⠀⠀⠄⠀⠈⠓⠦⠤⠤⠤⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠑⠒⢤⡀
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡁⠀⠀⠀⠀⠐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠃⠰⠀⠀⠀⠀⢀⠹⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡄
⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠂⠀⠀⠣⠀⡠⠂⠀⠀⢹⡀⠀⠀⠀⠠⠚⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⢳⡀⠀⠀⠀⠀⢀⠠⠀⠀⢀⣠⠺⡗⢢⡴⠀⠀⠨⢄⡀⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⠀⠀⠑⣄⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠉⠒⠖⠺⢏⣁⣀⣠⠴⢿⠁⠀⠇⠘⠢⠤⠄⠀⠀⠀⠂⠀⠀⠘⠀⠂⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠈⠉⠲⢄⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠗⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠄⠤⠀⠀⠀⠀⢤⡞⠉⠉⠉⠁⠒⠤⣀⠀⠀⢀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠼⢀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣠⠧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⡠⠤⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠙⠛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣷⡄
⠀⠀⠀⠀⠀⠀⣠⠴⠛⣩⠽⠖⠒⠚⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⢠⡞⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠻⣆
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠹⣆
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡄
⠀⠀⣇⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⣀⡀⠈⠙⣦⡀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣧
⠀⣀⣻⡀⣠⡇⠀⠀⡼⢇⣠⠴⠋⠉⠉⠛⠶⠇⠙⢦⠀⠀⣸⠻⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⠉⠉⣿⠀⢻⡀⣸⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠋⠀⠹⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⢠⢀⡟
⢿⣄⣀⣿⠀⠀⠳⣿⠀⠀⣠⣶⠶⠶⠀⠀⠀⠀⠀⠀⠙⠁⠀⠙⠛⢷⣄⠹⣄⣀⣤⣀⠀⠀⠀⣸⠛⠓⢶
⠀⠈⠉⢹⡄⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠹⠃⠈⠁⠀⠈⢳⡄⢀⡿⠀⣀⣼
⠀⠀⠀⣨⣧⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⠶⣄⠀⠀⠀⣴⣾⣿⣿⠛⠳⣆⠀⠀⠀⣀⡀⠀⠹⣾⠟⠛⠋⠁
⠀⠀⣰⠃⠸⣆⠀⠀⠀⠀⣼⠁⣿⣿⣻⣄⣼⣧⠀⢸⢿⣿⣿⣯⣶⣾⠿⡆⠠⠚⣏⠉⠀⣰⡟
⠀⢀⡇⠀⡆⠹⣆⠀⠀⠀⣿⠀⠈⠻⢿⣿⡿⣿⠀⢸⡄⠙⠛⠛⠛⠁⢰⠇⠀⠀⣿⠀⣰⠟⢦
⠀⢸⡇⠀⡇⠀⠘⢧⡀⠀⠘⢧⣀⣀⣀⣠⡼⠃⠀⠀⠻⣤⣀⣀⣀⣴⠟⠀⠀⠀⣏⡼⠃⠀⠘⡇
⠀⠀⢻⠚⣷⠀⠀⠀⠙⢦⣀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⡾⠋⢀⡟⢛⡟⠁
⠀⠀⠈⢦⠘⣧⠀⠀⠀⠀⠉⠛⠶⢤⣤⣤⡄⠀⠀⠀⠀⠀⣶⣤⣤⡤⠶⠛⠉⠀⢀⡾⢀⡞⠁
⠀⠀⠀⠈⠓⠾⢧⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣤⣴⡾⠟⠁⠀⠀⠀⠀⠀⢀⣾⡗⠋
⠀⠀⠀⠀⠀⠀⢸⡙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠳⣄⡈⠙⠓⠶⢤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⠶⠞⠛⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠲⢤⣤⣀⣀⣠⣽⡟⠉⠉⠉⡿⠦⣤⣤⣤⣤⣴⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠈⠉⠀⠘⡇⠀⠀⢸⡇⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⣤⠞⠁⠀⠀⠸⢧⣤⣀⣀⣤⠟⠁
⣿⣿⠶⢿⠿⠷⣿⡷⣿⡿⢿⣿⣶⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠂⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠚⠉⠁⠒⠄⢃⣠⡤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⣡⠐⠀⢀⠀⠀⣐⠳⡌⢸⣿⣶⠄⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠂⠀⠈⢀⡴⠛⠋⢉⣉⡀⠛⢤⣆⡤⠀⡀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠊⠀⠀⠀⠀⠄⠀⣠⣾⣿⣿⣿⣶⣤⣀⡈⠳⠌⠂⢄⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠊⠂⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠁⡨⣷⣄⠲⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠲⠀⠠⣠⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢀⠹⣆⠁⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠠⠃⠀⣾⠃⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⡆⡄⢩⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⣾⢀⡻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⣿⣿⣿⣿⣿⠇⡰⣧⣰⢰⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣿⠇⠠⠁⢿⣿⣿⣿⣿⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⢠⡟⣋⢡⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢄⠀⣍⠀⠃⡀⠐⣿⣿⢸⣿⣿⣿⠃⠀⢿⣿⣿⣿⣿⣿⠸⣇⠿⠘⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠋⢸⣿⣧⠀⠈⢻⠈⡿⣛⣿⡇⠀⠘⣿⣿⣿⣿⣿⡆⢿⣿⣤⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠀⡆⠈⣿⣿⣷⣰⣾⡄⠈⠛⠛⠁⠀⠀⣼⣿⣿⣿⣿⠃⣴⣮⣿⣿⡿⠫⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢔⠀⣿⣏⣻⣿⣿⣷⣄⣀⣠⣤⣶⣿⣿⣿⣿⣿⠇⠸⣷⣻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢻⡏⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡄⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⢸⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣾⣾⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠳⡀⠀⠀⠙⣿⣿⣿⡿⠟⡋⠐⣯⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⢀⡴⢒⣶⠶⣶⢦⡀
⠀⠀⠀⠀⠀⣾⣠⣿⣿⣿⣼⣿⢳⣦
⠀⠀⠀⠀⠀⣿⣹⣯⣞⣿⡟⣿⣸⢻⠆
⠀⠀⠀⠀⠀⣏⢻⣯⣟⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠙⢦⣝⣳⣾⠟⠛⠛⠁⠀⠀⢀⣤⠴⠖⠛⠛⠛⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⢀⡴⠋⢀⣠⣴⠶⠿⠶⢶⣤⡀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⣴⠟⠉⢀⣀⣤⣄⣀⠈⠻⣦⡈⠑⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣀⡀⣀⡴⠃⠀⠈⠁⣠⡾⠛⠉⠉⠉⠉⠻⣶⡘⠃⢀⣤⣤⣀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠈⡉⠉⠀⠀⠀⠀⣴⠟⠀⠀⣠⣶⣶⣶⣤⡈⠛⠀⠈⠉⠈⠙⣷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠸⢶⣤⣤⣤⡾⠋⠀⠀⣰⠿⢦⣍⠳⡌⠻⠄⠶⠶⠶⣄⠀⢹⡇⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⣀⠈⠉⠀⠀⠀⢀⣼⠋⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠘⣧⠈⢿⡀⣧⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠃⢨⡟⠶⠶⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠀⢸⣆⡈⠳⠦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠁⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⠀⠀⠘⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡼⠛⣿⠀⠀⠈⣇⠀⣸⠁⠀⢀⡴⠀⠀⠀⠸⡀⢀⡾⠀⠀⠀⣿⠛⢷⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⣤⣿⠀⠀⠀⠈⠉⢁⣀⠀⠛⠶⠀⠀⠀⣀⡉⠉⠀⠀⠀⠀⣿⣤⡼⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢿⡀⠀⠀⠀⣰⠋⠁⠀⢀⣀⣀⡀⠀⠈⠙⢷⠀⠀⠀⢀⣿⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡄⠀⠀⠘⠀⠀⠰⠏⠁⠀⠙⠦⠀⠀⠘⠀⠀⢀⡾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⣶⡶⢶⣶⡖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢶⠾⠏⠉⠉⠉⠽⠒⡶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⢀⡞⠁⢀⠹⣧⡀⠀⠀⠀⣠⡾⠁⠆⠙⣧⠀⠀⠀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣽⠷⠂⠀⠙⣆⠀⢸⡇⣠⡥⠠⠌⠛⠛⠾⠛⠋⠐⡄⢠⡁⢹⠀⠀⣾⠁⠀⠐⠿⣯⡀
⠀⠀⠀⠀⢠⠞⠉⣀⣤⠶⣾⠛⢻⣆⢸⡇⣿⠁⢣⡄⠀⠀⠀⠀⠀⠀⢈⢸⡇⢸⡆⣰⡟⠛⡶⠶⣤⣀⠙⢦⡀
⠀⠀⠀⠀⠘⠒⠋⠁⠀⠀⠘⣆⠀⢻⣼⡇⣿⡄⡆⡁⠀⠀⠀⠀⠀⠀⠈⢸⡇⢸⣷⡟⠀⣸⠃⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢿⡇⢿⡇⢁⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⡐⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⠌⢿⢼⡇⢈⠀⣀⠀⣂⡒⠂⠀⢂⣸⣷⡟⠐⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⠶⠶⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣾⠻⣆⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣙⠛⠿⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⡼⠀⠀⠀⡄⠀⠀⢠⠀⠀⠀⡇⠀⢰⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡼⠁⠀⠀⣸⠁⠀⠀⣾⠀⠀⠀⡇⠀⢸⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡼⠀⠀⠀⠀⡼⠁⠀⠀⢠⡇⠀⠀⢠⡇⠀⠀⢰⠇⠀⢸⠙⡆⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⢀⣼⡇⠀⠀⠀⣼⠃⠀⠀⢀⡞⠀⠀⠀⡼⠀⠀⠀⣼⠀⠀⡼⠀⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢸⡇⢷⣠⡴⠋⢸⡇⠀⣠⠞⣿⠀⠀⣠⡞⠀⠀⠀⣼⠁⠀⠀⣰⠛⠶⠞⠁⠀⢸⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢀⣀⣀⡀⠈⣇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠛⠋⠁⠳⣤⠴⠚⠹⣄⣠⠴⠃⠀⠀⠀⡀⠀⠸⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢸⡁⢨⡑⠦⠼⠀⠀⠀⠐⠳⠶⣤⣤⣤⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⡤⠴⠾⠁⠀⠀⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠶⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣄⠀⠀⠀⣀⣐⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠚⠀⠀⠀⠀⣀⣀⠀⠈⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⢠⠀⠀⠀⠈⠳⢦⡞⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢷⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣴⣏⠀⢀⣼⠀⠀⢠⡇⠀⢘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠸⠻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠛⠛⠉⠛⠛⠻⡧⠶⠋⣯⠉⠛⢲⡦⣄⠀⠀⢀⣀⣀⣀⣀⣤⡤⠶⡟⠙⠒⣆⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⠷⠟⢦⡈⠓⢦⣀⡤⠞⠁⢙⣦⣤⠞⠁⠀⣰⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠳⠴⠟⠉⠳⣤⠖⠋⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡋⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡍⠉⢹⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠾⠃

⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢻⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿
⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣾⣿⣷⣴⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠄
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣤⣄
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⡿⠛⠋⣩⣭⡽⢿⣿⣯⣉⣡⣬⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⢀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣭⣿⠿⠿⣿⡟⢋⣀⣤⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢟⣽⣿⣿⣿⠿⣣⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣟⠻⢄
⠀⠀⠀⠀⠀⠀⠀⣹⡿⣃⣼⠿⠟⣫⣿⣶⣿⠟⢿⣿⣿⣿⣿⡿⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠻⣿⣷⣝⡿⢿⣦⠀⠀⠀⠀⠀⠘
⠀⠀⠸⣦⠀⠀⣶⣿⣿⣿⣿⣿⣿⡿⠉⢿⠃⠀⣿⣿⣿⣿⣿⣧⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠹⣿⣷⡄⠿⣷⡀⠀⠀⠀⠘
⠀⠀⠀⠈⠻⣾⣿⣿⣿⡿⠛⣻⡟⠁⠀⠀⠀⠀⢻⣿⣿⣿⡏⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣿⢦⣈⣷⡀
⠀⠀⠀⠀⠀⠴⢿⣿⣿⣿⡖⠌⠃⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠀⠈⠙⢿⣿⣿⡟⠛⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠘⠉⠙⢿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄⡀⢀⡀⠀⠀⠀⠙⡄⠀⢸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠈⠻⢷⣾⣿⣿⣀⣾⣄⣴⣾⣷⠿⠋⠀⠀⢀⠀⢀⣤⣴⣶⣿⣶⣿⠿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⠿⣿⣶⣶⣦⣤⣀⡀⢀⣀⣨⡉⢉⣙⣛⡉⢉⣩⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⡿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢨⡏⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠲⣄⠀⣄⢹⡀⣘⣁⣉⢉⣋⣉⡀⠀⣀⣀⢀⣼⣿⣿⣿⣿⣿⡟⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠻⣿⣿⣿⡿⢩⣿⣿⣷⠶⠛⢿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⢺⠋⠀⠀⠀⠀⠀⠸⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠚⠀⠀⠀⠀⠀⠀⠀⠘⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣤⣤⣴⣶⣶⣦⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣆⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠁⢀⡀⠄⠀⠈⠉⠙⠛⠛⠛⠿⠹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⠐⠁⢀⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠃⠀⠐⠁⠀⠄⠀⠀⣠⠎⠀⣀⣀⣀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⡿⠁⠀⠀⢀⠄⠀⠀⠔⠈⡹⠿⠿⠛⠋⠉⠁⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡜⠀⠀⠀⢀⠂⠀⡀⠁⢀⠔⠁⠀⡀⠀⠀⠀⠸⣿⡛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⠀⠀⠀⠀⠀⠀⠘⠀⢀⠊⠀⠀⠀⠈⠉⠀⠀⠀⡿⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠠⠀⠀⠀⠀⠀⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢀⡴⣇⠀⠀⠀⠀⠀⠀⣀⠜⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠉⠁⠁⠀⠀⠈⠁⠀⠀⢸⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠣⢀⣀⢀⡀⠀⠀⠀⢀⣀⡠⠴⠊⠁⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠈⠈⡏⠀⠀⠐⠉⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠲⠤⢤⣀⠃⠐⢀⠃⠈⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⢰⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⣼⠀⠘⠄⠀⠀⡀⠄⠒⠚⠉⠉⠉⠁⠀⠈⠉⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠡⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠲⠶⠶⠶⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⡇⠀⠀⢀⡴⠚⠛⠓⢦⣄⢀⣠⡤⠖⠚⠛⠛⠛⠶⣄⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣠⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠓⠀⣴⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠙⠳⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⢀⣴⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇⠀⠀⣠⠶⠒⢶⠖⢶⠖⠶⢶⠖⢚⣛⣫⣭⣤⣤⣤⣤⣤⣤⠴⠶⠛⠁⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣀⠻⢤⣤⣼⣤⣼⠧⠶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⢀⣠⣴⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣋⣉⣉⣉⣉⣩⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⣠⣴⡾⣿⣿⡿⠿⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⢀⣼⣿⣵⣺⣽⢶⣟⣛⣛⣛⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⣾⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣏⣇⡤⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⢣⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⣿⡿⠿⠿⠟⠋⢻⣿⣿⣿⣿⠀⠀⠀⠀⣿⢋⡴⠋
⠀⠀⠀⠀⠈⠻⢾⣿⡀⠀⠙⠁⠀⠀⠀⢰⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠛⠁⠀⠤⠴⠂
⠀⠀⠀⢀⣀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢀⢼⣤⠀⠀⠀⢠⣴⣧⠄⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠦
⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⢿⠿⠃⠀⠀⠐⠾⠯⠗⠀⢸⣿⡇⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣣⣄⠀⠀⣸⣿⣿⠀⠐⠚⠛⠀⠰⡤⠀⠒⠖⠂⠀⢸⣿⡇⠀⠀⠀⣼⢶⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢊⠝⠇⠀⠀⣿⣿⣿⣧⠀⠀⢠⡀⠀⠀⢠⠀⠀⠀⢀⣾⣿⡇⠀⠀⠰⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⡀⠙⠒⠒⠁⠀⣀⣤⣾⣿⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣉⣉⣛⣛⣛⣛⣛⣛⣓⣚⣛⣛⣛⣛⣋⣉⡩⠭⠽⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣠⠤⠬⠿⢦⡀⠉⠉⡩⠭⢍⡀⠀⠀⣴⣛⣛⠒⠲⢄⡈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⡔⢋⡠⠖⢊⣉⠭⢷⡀⣞⠰⣿⡆⢹⣀⠼⠓⠲⠤⡉⠲⣄⠙⣄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠋⠀⠊⣠⠞⠉⣀⠤⠦⢬⠟⢦⣙⡡⠞⠳⠒⠛⠲⣄⠈⠳⠈⠁⠈⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢠⠾⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⢧⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠓⠒⠒⠒⠚⢳⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢦⠤⠽⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠿⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠢⣀⡛⣄⠀⣀⠸⡓⠀⠀⡠⠔⠒⠃⠀⠀⠀⠀⢠⡤⠒⠋⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠾⠱⣿⡄⠀⠧⢤⣈⠀⠀⠀⠀⠐⠺⣜⣠⠤⠖⠛⠒⠛⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠶⣖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⣀⣀⡤⠔⠊⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣒⠿⠛⠉⣀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠈⠠⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⠈⡙⢲⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⡙⠁⠘⡆⠈⢶⢤⣀⠀⠰⡶⢄⠀⡆⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢦⠀⢸⡀⠀⠇⠀⠘⣆⠈⠉⠐⠛⠈⢺⠥⠤⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡜⠀⠀⠃⠀⠈⠀⠀⠘⠆⠀⠀⠀⣰⡃⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡧⠤⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢾⣿⡀⢿⠿⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⣖⡀⠛⢁⡀⠳⡄⠰⠒⣻⠁⠀⠀⠙⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣇⣠⣅⠀⢠⠇⠀⠀⠀⠀⠈⠒⡛⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠦⣀⠀⠀⠈⢑⣿⠉⠬⠤⠞⠢⠄⠀⠀⠀⠀⠘⢅⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⣿⣿⣿⠟⠋⠀⣀⣴⣿⣤⡀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠉⠻⠟⠛⠋⠉⠠⢶⣿⣿⡛⠛⣻⣦⠀⠀⣼⣿⡟⢿⣯⠻⡆⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣶⣈⣉⣤⡿⠋⠀⠀⠙⣧⣜⣀⣉⣿⠆⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀⠀⢀⣠⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⢿⣿⣿⣿⣿⣤
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠈⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠈⠉⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⣣⠀⢿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠛⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣁⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣿⣿⠏⠙⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⠿⠟⠋⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠛⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠶⢶⣶⣤⡀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⣰⠏⠀⠀⠀⠀⠀⢀⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⠈⢷⡀
⢠⡟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⣧
⣼⠃⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⣻⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣠⠶⠟⣻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣳⣶⣶⣶⣶⣶⣶⣶⣖⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢰⠓⢤⡀⠀⢺⡒⠤⣄⡀
⠀⠀⠀⠀⠀⠘⣇⠀⠉⠓⠦⠷⠄⠀⠉⠙⠒⠤⢄⣀⡤⠤⠤⠶⠖⠒⠚⠛⠛⠛⠓⣶⠀⣀⡀
⠀⠀⢀⣀⡀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡭⢭⣽
⠀⠀⠈⢯⡉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡤⠤⢶⡖⠚⠂⠉⠑⠧⠀⠀⠀⠀⠀⠀⡞⢠⡄⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡬⠭⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠳⢤⠴⠋⣁⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠋⠀⠀⠀⠹⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣮⣤⡤⠤⢤⡤⢿⢾⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢧⠀⠀⠀⠈⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣤⣤⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣠⠞⠁⠀⠀⠀⠀⠀⠋⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠤⠀⢤⠄⠠⠀⠚⠉⠘⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⣀⡀⢀⣀⣀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⡸⠁⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠐⡟⠳⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢉⣽⠷⢶⣧⣀⣀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⠶⣿⠾⣿⣷⠿⠿⠿⠏

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠋⠉⠉⠀⠉⠀⠉⠛⠋⠉⠉⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿
⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿
⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡄⠀⢤⣀⣄⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡗⢦⣼⠏⢁⣤⣤⣭⣿⣿⡄⣤⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡀⠀⠀⠀⠀⠰⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⡍⣬⠻⡿⠁⠀⠀⠿⡟⣿⣷⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⡀⠠⠂⢄⡀⠀⠛⢦⠀⠀⠀⠀⠀⠀⠀⠈⠋⢆⠙⢷⠖⠆⠄⠀⢉⡼⣿⡟⢀⣀⣀⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢩⣈⠀⠙⣦⡀⠀⠀⠀⢁⣤⣿⡻⠷⠦⣤⣀⠤⠀⠀⢈⢣⡈⢦⣄⠀⠈⠈⠚⠉⠷⢧⠀⢹⡛⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠲⣦⣀⣨⣉⣉⠙⠶⠶⠶⠿⢾⡟⠎⠁⢸⣧⠱⣌⠁⠀⠀⠀⠀⠀⠀⣰⡏⢻⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣭⣶⣶⡭⣥⣤⣴⠶⠀⠚⠷⢭⣙⣦⡙⠆⠹⡆⣿⡶⠦⠐⣚⣽⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⣄⢰⣴⣧⣬⣁⣠⠖⣊⠛⢻⣿⣿⣦⣶⣿⣿⣶⠟⠋⠹⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣬⣿⢟⣿⣿⢻⣾⠿⣷⠾⣿⣝⡛⠶⡟⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡟⣺⡏⢸⣿⡟⢷⠘⣿⡲⠤⣵⣮⠃⠙⣾⣰⠏⠀⠀⠀⡀⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣧⣈⣣⡈⠷⣑⡶⠾⠿⠞⠋⢫⣿⡀⠀⢨⡃⡘⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⢀⣝⡳⣭⣬⣙⠛⠷⠖⠻⠿⢿⣧⣄⠀⠉⢳⣶⣿⡇⡄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⡤⠄⠀⠀⠠⣾⣿⣿⣆⣭⣽⠶⠞⣻⣿⣶⡀⠈⠉⣕⠀⠀⠈⣼⣷⡷
⠀⠀⠀⠀⠀⠀⠺⢻⣯⣽⣆⠀⠀⠀⡀⠀⠀⣀⣤⣾⣿⣿⣿⣶⣶⠶⢿⣿⣻⣿⢿⣿⡀⠈⢷⡐⣄⠹⣯⣧
⠀⠀⠀⠀⣰⠀⣰⡿⠛⢂⠹⡄⠀⠀⢱⡀⠀⣿⣛⣿⣿⡟⢿⡿⠿⣄⠀⢉⠛⠙⢻⣿⢇⠈⡤⢅⠙⣷⠈⠃⠀⠀⡀
⠀⠀⠀⠀⠁⢸⡿⠁⠀⠀⢠⠇⠀⠀⠈⠿⣄⣠⠾⠟⠉⠀⢸⠇⢂⠙⣧⠀⠱⡄⠀⠙⠸⠆⢹⡌⡅⢹⠀⠈⢠⣉⡁
⠀⠀⠀⠀⠀⡈⣧⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠏⢰⣆⠄⠸⡄⠀⣿⠀⡈⢰⡇⠸⠃⠀⠰⢿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⢳⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣩⣾⠟⠀⣠⢿⣿⢃⡾⠇⠀⠀⣰⣧⣿⢷⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠴⡿⠛⠉⠀⢐⣮⣽⣿⣾⠟⠁⠀⠀⣠⡟⢻⡟⣦⢾⡁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⢶⡀⠁⠀⠀⠐⢲⣾⣷⣿⡿⢉⠴⣚⡩⢉⣠⣤⡤⠖⠊⣠⠞⠛⠁⠘⣇⠘⣦⠽⠂⠠⠍⠁⢀⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⣀⣉⣤⡍⣰⣥⢊⣵⣿⡿⠛⠁⣤⣠⠞⣵⠎⠀⠀⠀⠙⣷⡄⡀⠀⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⡟⠀⣿⣥⠞⣽⠋⠀⠶⣴⣅⣿⠸⢇⡄⠀⠀⠀⠀⠈⠓⠈⠓⠒⠞⠂⠑⠊⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⢸⣯⠀⠀⠀⢸⣿⣷⠱⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⠀⠀⠀⢰⡿⠷⢦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢸⣆⡀⢘⠀⢠
⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢆⠀⠽⣯⣙⣆⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣦⣴⠞⢿⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠿⠃⠈⠀⠀⠈⠉⠀⠀⠀⠉⠉⠉⠛⠒⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⡆⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣰⣧⠀⣼⠇⠀⠀⠀⢸⡄⠀⠀⢸⠀⠀⠀⢸⡇⠀⢰⡆⣰⣿⡀⢀⣼⠀⠀⢀⡀⠀⣀⣠⡀
⠀⣿⡀⠀⠀⠀⣴⠋⠉⢷⠀⠀⠀⢸⠉⠉⠁⠀⠙⠁⢸⡀⠀⠀⣾⣃⠀⢀⡼⣇⠀⠀⣾⢧⣠⠟⠈⠁⠀⠙⠋⢹⠀⠀⢸⡃⢰⠋⡀⢹⠄
⠀⠘⢷⣀⠀⠀⢧⠀⢑⡀⠳⡀⠀⠸⠀⠀⠀⣶⡄⠀⠀⠻⣦⠜⠁⠙⠳⠋⠀⠙⠒⠚⠁⠀⠀⠀⠀⢠⡄⠀⠀⠸⣄⠀⡸⠓⠋⠰⢆⡼⢰
⠀⠀⠀⠙⢻⡆⠈⠳⢤⢥⡄⢿⠀⡆⠀⠀⠀⠘⢿⣦⣄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣤⣴⡿⠁⠀⠀⠀⠙⠛⠁⣠⠞⠛⠛⣵⠋
⠀⠀⠀⠀⠘⣧⣀⣠⣤⠀⠹⣎⠛⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⡞⣡⡶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣤⣀⠀⠀⠀⠀⠀⠛⠠⣤⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠉⠀⠉⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠊⠉⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠓⠠⠶⠦⠴⠶⠶⠒⠛⠋⠁⠀⠀⠀⠀⣠⡀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠚⠀⠀⠲⠂⣀⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⢠⣀⠀⠀⢀⠰⠆⠉⠁⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⠎⠀⠚⠲⠀⠀⢀⠶⠄⠈⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡀⠀⢘⡓⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠛⠀⠀⠀⢻⡁⠀⣸⣄⣀⣀⣀⣀⣠⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠶⠄⠀⠀⠠⠴⠆⠉⠙⠢⠐⠋⠀⠀⠀⠀⠀⠑⠚⠁⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠉⠀⠐⠒⠒⣠⣤⣤⠴⠶⠶⠶⠶⠤⢴⣶⣖⠈⠉⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣿⠂⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⡿⠛⠋⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠙⠓⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣈⣀⣀⣀⠀⠀⣿⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣽⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⠩⠿⣁⣩⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣎⠒⠶⢤⠘⣐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⡨⢠⠈⢀⡰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⠁⠀⠈⠑⠢⢤⣠⡏⠀⠀⠁⠃⠀⠨⠓⢄
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⢴⡾⡋⠢⠤⠄⠀⠀⢀⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⢀⣠⠖⠋⠁⣀⠐⢱⠀⠀⠴⠆⠑⠂⢄⡀⠀⠈⡌⠀⠀⠀⠀⠀⠀⠜⢀⡀⠀⠑⡀
⠀⠈⢻⣄⠀⠘⠄⠸⠃⠈⠢⡀⡀⢀⠤⡪⠀⣈⡴⢻⠁⠀⠀⠀⠀⠀⠘⠀⠀⠑⠀⠀⠈⠠⡀
⠀⠀⢸⠈⠓⢤⡀⠀⡀⠈⢆⣀⠍⠓⣨⠴⠚⠓⠂⠣⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢄⡠⠣⡀
⠀⠀⢸⠀⠀⠀⠙⠢⡉⠀⢀⣠⠖⠋⠁⠀⢀⠴⠋⠀⠀⠉⠓⠂⢀⡴⡀⠀⠀⠀⠀⠀⠀⢑⡢⡈⡆
⠀⠀⢸⠀⠀⠀⠀⠀⢹⠚⠉⡇⠀⠀⢺⠚⠁⠀⠀⠀⢀⣠⠔⠚⢁⣀⡜⠀⠀⠀⠀⠀⠀⠐⠠⠬⡀
⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⠀⢖⠈⣿⠀⠉⠉⣏⢸⡇⠀⠀⠀⠀⠀⠈⠒⠤⠥⠃
⠀⠀⢸⡀⠀⠀⠀⠀⢸⠀⠀⣧⠀⠀⡜⢆⠀⠀⠈⡄⣿⠀⠀⠀⡿⠈⡇⠀⠀⠀⢠⣶⣻⣛⡏⣻⢙⣓⡄
⠀⠀⠈⠳⡄⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇⠈⢆⠀⠀⠸⡇⠀⠀⠀⡇⠀⣷⠀⠀⠀⠈⣿⣟⣏⣹⣉⣯⣿
⠀⠀⠀⠀⠙⢦⠀⠀⢸⠤⠒⠋⠉⢹⠀⠀⠈⢧⠀⠀⢣⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⡀⠈⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠳⣄⢸⠖⠒⠋⠉⠁⠀⠀⠀⠈⢧⣀⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠈⠺⠇⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡿⠟⠛⠛⠛⠋⠉⠉⠉⠛⠛⠛⠻⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷
⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠿⢶⣄⡀⠀⠀⠀⠀⣠⠾⠉⠁⠈⠉⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠙⣷⡀⠀⠀⠀⣿⠀⠀⠀⢀⣤⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⡄
⠀⠀⣀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⣿⡇⠀⠀⠀⠻⣦⡀⠀⠈⢛⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠛⢿⣿⣧⡀
⢀⣾⡿⠛⠛⠛⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⢀⣀⣼⣏⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣦⡄⠀⢿⣿⡇
⣾⣿⠁⢠⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣷
⣿⣿⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏
⠸⣿⣦⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣾⣿⣿⠟⠁
⠀⠙⠿⣿⣶⣶⡾⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏⠁
⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣤⣤⣠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣷⣦⡀
⠀⠀⠀⠀⣿⣿⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⣼⣿⠋⠻⣷⡄
⠀⠀⠀⠀⢻⣿⡀⠈⠙⠻⠿⣷⣶⣦⣤⣄⣄⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢀⣠⣤⣾⠿⢿⣿⡇⠀⢻⣷
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠋⠉⠀⠀⢸⣿⡇⠀⠀⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿
⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⢸
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸
⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸
⠀⠀⠀⠀⢸⣿⡇⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼
⠀⠀⠀⠀⢸⣿⡇⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⢸⣿⣷⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣶⣶⣶⣿⣿⣿⠿⠋⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢸⣿⠃⠀⢠⣿
⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣾⡋⠛⣦⣠⣤⣶⣶⣿⠇⠀⠀⣿⡏
⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣷⣤⡼⠛⠉⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⣿⣿⠈⢻⣷⡄⠀⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⡿⠿⠿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠻⣿⣆⠀⠀⠙⠿⣿⣦⣄⡀⠀⠀⠀⣠⣿⣧⣿⣿⣦⣤⣤⣤⣶⣶⡿⠛⠙⣿⣿⣇⠀⠀⠀⢀⣀⣠⡶⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠈⠿⣷⣄⡀⠀⠈⠙⠻⠿⣷⣶⣾⡿⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣠⡿⠛⠻⠿⠾⠿⠟⢛⡿⠀⢠⡄⠀⠀⠀⣼⣿⡁
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠉⠻⣿⣶⣤⡀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⢀⣤⣄⣀⣠⣤⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⣿⠀⢠⣾⠁⠀⡐⠀⢿⡟⣿⡄
⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣶⣦⣤⡄⠀⠀⠀⠀⠀⠀⠉⠛⢿⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⡟⠀⢸⡇⠀⠘⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡇⠀⠀⠀⠀⠀⢠⣤⡀⠀⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣷⣤⣼⣿⣦⣀⣴⡿⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⢀⡀⠀⠀⠙⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠻⢷⠀⠘⣿⣶⣄⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣀⣾⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠈⠿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢻⣿⡋⠁⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⣽⣿⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣷⣄⠀⠸⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡿⠟⠀⣠⣼⣿⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⢀⣤⣴⣾⣿⠿⠿⠿⠛⠛⠛⠛⠃⠀⠈⠉⠙⠻⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣏⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣶⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⣀⣀⣤⣤⣴⣶⣶⣿⡿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣾⣆⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣰⣷⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠾⠇⠙⠃⠙⠉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠋⠘⠋⠸⠷⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⠀⠈⠳⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠞⠁⠀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠅⠀⢀⣀⣀⠈⠀⢀⣴⠶⠞⠳⠶⣦⡀⠀⠁⣀⣀⡀⠀⠨⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠉⠀⣴⠏⠀⠀⠀⠀⠀⠀⠹⣦⠀⠉⠻⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠆⢸⡇⠀⣀⣀⡀⢀⣀⣀⠀⢸⡇⠰⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⢻⡾⠋⠉⢻⡟⠉⠙⢷⡟⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣴⠟⠁⠀⢀⣼⣧⡀⠀⠈⠻⣦⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡾⠁⠀⠀⣠⡞⠁⠈⢳⣄⠀⠀⠈⢷⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠼⠋⠀⠀⢀⡾⠋⠀⠀⠀⠀⠙⢷⡀⠀⠀⠙⠧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣼⣧⣤⣤⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣤⣤⣤⣼⣧⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⣀⣠⣴⣶⡿⠛⠛⢿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⣸⡿⠟⠛⣿⣿⣿⡟⠁⠀⠈⠉⠉⠉⠙⢿⡟⠻⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⣴⠟⠉⠀⣠⣾⣿⣿⣿⠿⠶⠦⣴⣦⣸⣿⣾⣦⣽⣷⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⣼⠃⠀⣠⣾⣿⣿⣿⡿⠿⣿⣶⣤⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠃⠀⣰⠃⠀⢠⣿⣿⡿⠋⠁⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡾⢠⠇⠀⠰⠇⠀⢀⣿⠟⠋⠀⣰⣾⣿⣿⣿⣿⣿⣯⣙⡛⢿⣿⣿⣿⠻⣿⣿⣈⣛⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⠀⠀⠀⠀⠈⠁⠀⠀⣴⣿⣿⢿⡿⣻⠿⠛⠉⠉⠃⠀⠀⠈⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢰⠀⠀⠀⠀⠀⣸⣿⡿⢃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⡄⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢸⠀⠀⠀⠀⠀⢸⠇⣾⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⠟⠁⣠⠄⠙⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠼⠀⠀⠀⠀⠀⢻⣤⣿⠀⢸⣿⠀⠀⢀⣴⣾⡿⠿⠷⠦⠀⠀⠀⠹⣿⡿⠀⣸⣿⣤⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣶⣶⣶⣶⠒⠀⠀⠀⠈⠀⢰⣿⠟⢻⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⢹⠇⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠈⠛⠛⠛⠿⠿⠏⠉⠙⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣰⡇⠈⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠉⣽⣿⡆
⠀⠀⠀⠀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠠⠖⠻⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴⠾⠿⣿⡇
⠀⠀⠀⢰⣿⡏⠙⣦⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⣀⣀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⢸⣷
⠀⠀⢠⣿⣿⣷⡀⠘⢷⣄⠀⠀⢀⡟⠀⢿⣿⣧⣤⣶⣿⡟⠁⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿
⠀⢀⣾⣿⣿⣿⣿⣄⠀⠙⢷⣄⣸⠃⠀⠈⢿⣿⣿⣏⣉⣀⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠹⠆
⠀⣼⣿⣿⣿⣿⣿⣿⣷⣄⠀⠉⠻⠆⠀⠀⠘⣿⣿⡋⠉⠉⠉⠈⠉⠉⠛⢧⠀⠀⠀⠀⠀⠀⣦⡀⣀⣴⡟⠁
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠘⣿⣿⠶⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠛⠁
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠙⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣄⡀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠋⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿
⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣸⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣻⣷⡤⢷⡄⢠⣰⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⠋⠁⠀⢱⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣏⠠⠀⠀⢸⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡘⠀⠀⠀⠀⠀⠁⠀⠎⣰⡿⢏⠀⠀⠀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⡶⠛⣦⠀⠀⠀⢼⣰⡏⠀⠘⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠠⠛⠀⠀⠀⢀⣿⡿⡀⠀⡄⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣴⣾⣴⣶⣿⣯⣉⣵⣿⣋⡀⢰⣿⣋⣷⡆⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣷⣿⣿⣿⣿⣟⡛⣿⣿⠟⣿⣿⣠⣿⡿⠛⢸⣿⣶⣶⡉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⣿⣿⡿⠋⠉⣉⣉⠿⣿⣏⠛⠉⠃⢼⣿⡿⣿⣿⣄⣤⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⢻⣿⢿⣿⣿⠟⢋⣉⣁⠀⠻⣿⣷⣶⣿⣁⣐⣻⣿⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⣾⣿⣾⣿⣧⣾⠟⣿⣿⣿⣷⣆⠈⠉⠉⠛⠋⢙⣿⡟⢹⣿⠁⠀⢀⣤⣾⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⢿⣿⣷⣦⣤⢚⠛⢉⣁⣤⣾⣿⣿⣿⣿⣿⠏⠀⡄⠃⢰⠇⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⢠⣛⣿⡟⢘⣿⣿⣿⣿⣿⣿⣿⣷⢬⣿⣿⣿⣷⡤⠉⠙⢻⣽⣿⣿⠛⠛⠁⣀⣰⣶⣤⠂⠀⠀⠀⠀⢀⣠⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢋⣾⣿⡿⠋⣼⣿⣿⣿⣿⣿⣿⢿⣇⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣭⣤⣤⣶⣾⣿⣿⣟⣀⣀⣠⣤⣴⠾⣻⣿⡟⠋
⠀⠀⠀⠀⠀⠀⣤⣶⣿⣷⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢦⡀⠀⠐⠠⢦⣴⣿⣿⡿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠆⢀⠈⡉
⠀⠀⠀⠀⠀⠀⣰⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣹⣿⣿⣿⣮⣻⣶⡵⣄⡀⢠⣿⣿⡿⠫⣿⣿⣿⣷⢦⢤⡈⠉⠑⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠘⢻⣿⠀⢠⣿⣿⣿⣿⣿⡿⠋⠁⠋⠉⠀⣽⣿⣿⣿⣿⣿⣿⣞⣻⣿⣟⠛⠓⣲⠟⠉⠉⠃⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣰⣦⣾⡏⣀⣼⣿⣿⣿⣻⣿⣁⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⢿⣾⣟⠂
⠀⠀⠀⠀⠙⣿⣿⣷⣿⣿⡿⠋⠈⠩⢿⡭⠳⢶⣶⣿⠿⠛⢿⣿⣿⣾⣿⡟⢤⡍⠈⠏⠀⠉⠻⢿⣦⣤⣀⣀⣀⣀⣀⣀⣠⣶⣦
⠀⠀⠀⠀⠀⠈⠙⠛⠿⠋⠀⠀⠀⠀⠀⠀⢀⣶⣾⡿⣦⣤⣾⡿⠛⠟⢿⣿⣼⣧⡀⠀⠀⠀⠀⠀⠈⠙⠲⢯⣽⣿⣿⣿⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡷⠛⠛⣿⡛⠛⠿⢤⣿⠿⣿⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠞⠀⠀⠐⠀⣼⣿⡀⢰⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⣸⢻⠀⠸⡆⠸⡇
⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠸⡇⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿
⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠞⠛⠛⠛⠛⠳⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠁⣸⠖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢸⠀⡟⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢸⢸⠀⡇⠀⢀⡟⢳⠶⠶⠶⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣿⠘⠀⠀⠇⠀⢸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⠀⠀⠀⠀⠀⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠲⢄⠀⠀⠀⠀⠀⣰⢿⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣸⠃⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⣤⠞⠋⢀⡼⠃⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣀⣤⣾⣧⣤⣤⡶⢞⣋⣁⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠇⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⠀⠀⣀⣤⡴⠶⠞⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠳⣦⣤⣤⣤⣤⠶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠛⠛⠛⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠾⠟⣛⠥⠚⢄⠀⠀⡜⠁⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⣀⠈⢆⢰⠁⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⢻⣧⢸⢸⠀⠀⠀⠀⠰⢿⣄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠾⠟⢸⠘⡆⠀⠀⠀⠠⣾⠟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⠤⢤⣄⣀⠀⠀⢀⡎⠀⢳⡀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠖⠋⠀⠀⠀⠙⠦⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⡀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠉⠉⠉⠉⠉⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣀⣀⣀⣀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣭⡤⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠑⠂⠀⢀⢠⠀⠁⠀⠂⠰
⠀⠀⠀⠀⠂⠀⠁⢀⠄⠂
⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠛⠓⢂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⡠⡤⢀⣀⠄⠁⠀⠀⠀⠀⠓⠓⢦⣤⣄⣀⡀⠀⠀⠀⢈⠂⢣⠈
⠀⠀⠀⠀⠀⠐⠀⠄⠤⢀⡀⠠⢸⣁⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠟⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⡛⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⢋⣡⣴⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣤⣉⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡆⢻⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠃⣾⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⡿⠟⠋⠁⠀⠀⢻⣿⡿⠛⣉⣤⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⣌⡛⠻⣿⣿⠃⠀⠀⠉⠛⠿⣿⣿⣷⣦⣄⡀
⠀⠀⢀⣠⣴⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣤⣀
⣴⣾⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿
⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠋⣉⣉⣉⣉⠛⠋⣉⣉⣉⣉⠛⢿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢁⣴⣿⣿⣿⣿⣿⣿⣦⠙⣿⣿⣿⣿⣆⠙⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⢹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⣿⣇⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⣉⣉⣉⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢉⣉⣉⣉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⡿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢡⠀⡁⠀⠀⠀⠃⡄⠀⠜⠇⠀⠊⠀⠀⠉⢄⠀⠀⠀⢘⠀⢈⠀⠈⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⡁⠀⠃⠀⠀⠀⠀⠈⠎⠀⡇⠀⡀⠀⠀⠀⠘⠀⠃⠐⢂⠀⠀⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣴⣤⣷⣶⣶⣤⣤⣤⣤⣤⣧⣤⣤⣶⣶⣦⣤⣤⣤⣤⣤⣦⣴⣶⣶⣤⣼⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⡾⠿⢿⣿⣿⢿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠫⣝⣿⡭⠶⣶⡿⠟⠻⠿⠿⢶⣭⣍⠛⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢟⣡⣔⡲⠛⠉⠓⠂⠀⠀⠐⠦⡀⠀⠶⢶⣬⣉⠁⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣴⣏⣈⣥⣦⣤⣤⡤⣴⡶⣮⣿⣛⢶⣦⡀⣤⡀⠈⠁⠠⡄⡙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⣷⣾⠿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡶⢀⠀⠈⢳⡜⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠞⢹⣿⡅⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠦⠈⠛⠛⠛⣿⣿⣿⣄⠀⣿⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣷⠹⠀⢼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢰⣿⣿⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠸⣿⣿⣷⠀⣎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⣾⣿⡟⠠⠚⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⡄⠀⣧⢹⣿⣿⢻⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⠃⠀⠒⡉⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠁⠀⣼⣷⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠻⡿⠀⠀⠁⡀⣀⠐⣀⣽⠀⠀⠀⢰⡋⡤⠐⠠⠐⢄⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⡐⠀⠀⠉⠐⠺⠿⠟⠋⠁⠀⠀⠀⠀⠳⢶⣿⡆⣀⣠⡄⠀⠀⠀⢻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠰⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡍⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠞⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢴⠟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢞⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⣝⠳⠖⣿⠇⠀⠀⠀⠀⢀⢺⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⣷⣄⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣰⣭⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠐⢀⡙⢷⣤⣀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢋⠀⠀⠀⠀⠈⢻⣄⠈⠙⠳⢶⣶⣶⣾⢿⡿⠋⡐⣹⣿⣯⣻⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⣆⡀⠀⠀⠀⠀⠀⠙⠳⠄⠀⠀⠀⠀⠀⠉⠀⠀⡿⢁⣠⣿⣟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡿⠿⠀⠀⠉⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⠛⠛⡿⢛⠁⢀⣿⡿⠹⡈⢿⣿⢻⠻⢿⣿
⣴⠾⠛⠉⠉⠉⠉⠙⠛⠛⠻⠶⠆⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⡶⠚⠛⠉⠀⠀⠀⠼⠋⠀⢃⣌⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⣭⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠕⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠋⠉⢀⣬⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠤⠄⠀⠀⠀⠀⠉⠉⠀⠀⢀⣠⠴⠚⠋⠁⢀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⠚⠉⠀⠀⠀⣠⣶⣿⣿⣿
⣦⣴⣦⣦⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣆⡠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠥⠟⢶⣉⣓⠻⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠭⣑⠢⣄⠀⠔⠊⠀⠐⠊⢭⡈⠳⡄⠈⠳⣄
⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡑⣄⠀⠀⠀⠀⠀⠈⠑⢄⠀⠳⡈⢆
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢆⠀⠀⠀⠀⠀⠀⠀⠱⡄⡃⢸
⢰⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣄⠀⠀⠀⠀⠀⠀⠀⣠⢃⠏
⢘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⠳⣄⠀⠀⠐⠒⠋⡜⢣⠋
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢤⣈⡳⡄⣆⡆⣼⡵⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢉⠟⠁
⠈⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⠒
⠀⢣⠀⠀⠀⠀⡓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⡆⠀⠀⢸⣇⠀⠈⠉⠒⠒⠶⠶⠶⠖⠒⠚⠓⣤⣀⣀⠳⡀⠀⠀⡄⠀⠀⠀⠀⠀⡸
⠀⠀⠹⡀⢠⢸⣽⠖⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⣹⣿⡉⢱⡇⠀⠀⡀⠀⢠⠇
⠀⠀⠀⠱⡈⡆⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠁⣾⠁⢀⡼⠁⢠⠎
⠀⠀⠀⠀⠙⣧⢹⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡞⡜⣠⣾⠁⡴⠃
⠀⠀⠀⠀⠀⠘⢆⢫⠙⠲⠤⣄⣀⣀⣈⣁⣀⣀⡠⠤⠴⠒⠋⡿⣱⠕⢱⡧⠊
⠀⠀⠀⠀⠀⠀⠈⢦⢣⠀⠀⠀⠀⠀⢀⡝⠈⠂⣠⠠⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣰⣡⠈⠉⠉⠀⢤⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡹⠀⠀⠀⠀⠈⢆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⣀⡀⠀⠀⠶⠉⢻⣿⣷⣾⣿⣿⣿⣿⣿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠛⠛⠛⠛⣽⣿⢿⠝⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⣿⠇⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⡿⠛⠉⠛⠛⠓⢓⣿⣿⣿⣿
⠀⠞⠀⠀⠀⠁⠀⠀⠉⠉⠉⠉⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⠫⡤⣤⡿⠛⠁⠀⠀⢀⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣀⣀⣀⡀⠀⠀⠠⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁⠉⠀⠁⠈⠉⢻⣯⣅⣀⡡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠉⠁⠐
⠀⠀⠚⠀⢀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⡄
⠀⠀⠀⢀⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠿⠀⠀⠀⠰⠦
⠀⣀⣲⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠄⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣶⣤⣤⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⡀⠀⠈⠉⠐⣺⣿⣿⡟⠉⠙⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡬⠛⠉⣁⠠⠃
⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡶⢶⠤⠄⣤⣦⣴⢶⡀⢠⣤⣤⣄
⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣭⣟⣻⣶⡷⣾⣿⣹⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠿⠛⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⢀⡾⠛⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣸⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠓⠶⣼⢷⡀⠀⠀⠀⠀⠀⢠⠷⠤⣄⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢰⠇⠀⠙⠦⣄⡀⠀⣰⠋⠀⠀⢨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡟⠀⠀⠀⠀⠀⠉⣹⠃⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡾⠀⠀⠀⣀⣤⠤⣶⠃⠀⠀⠀⣼⡽⠟⠉⠛⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣼⠃⢀⡴⡟⠁⠀⣰⠃⠀⠀⠀⣰⠏⠀⠀⢀⡴⠞⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣰⣣⠶⠋⢸⡇⠀⣰⠇⠀⠀⠀⢠⠏⠀⢀⡴⠋⠀⠀⠀⠀⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⠟⠁⠀⠀⢸⡇⢠⡏⠀⠀⠀⢀⡟⢀⡴⠋⠀⠀⠀⠀⠀⣴⠃⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠠⠟⠀⠀⠀⠀⣸⣇⡞⠀⠀⠀⢀⣾⡴⠋⠀⠀⠀⠀⢀⣠⠞⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣿⠃⠀⠀⠀⡼⠉⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⣿⠀⠀⠀⠼⠁⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠶⠋⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣈⣉⠶⡄⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠥⠤⢑⡄⠀⢀⡐⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢉⡴⠓⣈⡉⢁⡒⠁⠀⠠⠠⠑⣹⣇⡠⠒⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡔⣫⠖⠉⠀⣀⡀⠀⣀⠀⠀⠀⠀⡨⣿⣧⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠌⣼⠁⠀⠀⠉⠈⠁⠈⠁⠀⠀⠀⣞⢀⡿⡧⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡈⢸⠃⠀⠀⠖⠲⠂⠀⠐⠒⠖⠀⠀⠈⠉⠀⡇⠃⢋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⢸⠈⡄⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠠⢀⡇⠸⠀⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠸⠀⠇⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⠋⠉⠱⡀⠀⠄⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠤⠤⣤⡔⠈⣠⠶⠁⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣴⠂⠀⠈⣇⠀⠉⠐⠀⠋⠁⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⣠⠏⠓⢦⣠⣀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠧⢠⠞⠁⢀⠀⠂⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡠⠆⠁⣠⡔⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣃⠈⠀⡠⠚⠄⠈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⢘⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠁⠀⢀⡌⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠛⣅⡠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠠⠃⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠰⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠱⡀⣰⡁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠑⣄⠈⠋⠁⠈⠋⠁⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠶⠉⠉
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠿⡄⠀⠀⠀⠀⢀⠤⢄⡀⠤⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠒⠒⠐⠢⡀⠀⡔⠁⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠟⠈⠑⠖⠉⠃⠀⠀⠀⠀⠈⠉⠳⡄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠈⣈⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠉⢀⣀⣀⡤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⢒⢒⡟⠉⠉⠩⣩⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠠⠀⣀⠠⠖⣒⠈⠃⠉⠉⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⠃⠀⠀⠉⠀⠀⠀⠀⣠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠒⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⢠⠁⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠀⠐⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠤⠂⠁⠀⢀⠼⠀⠑⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⡠⠂⠉⠀⠀⠀⠠⠠⣄⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⢀⠴⠋⠀⠠⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⢀⠠⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠨⠲⡄⡀
⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⠘⠀⠀⠨⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠉⠊⢀
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣀⢀⠀⠠⠀⢀⢀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠠⠀⠂⠀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⣀⠀⠀⠲⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣄⡀⠀⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⡄⠁⠀⠂⠀⠀⠀⠁⠀⠃⢀⡰⠍⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢈⢳⡄⠀⠀⡄⡁⠈⢡⠀⠀⠀⠀⠀⣄⠀⠀⠀⠘⠉⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠉⠀⠈⠉⠀⠀⠰⠀⠀⣷⠍⡐⡄⠂⠀⠉⠁⠛⠸⣦⠀⠃⠀⡘⠃⢀⡔⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠳⣄⠄⣰⠈⠀⢙⡓⠀⠓⠐⠛⠀⠁⣙⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠱⠀⠀⠀⠘⠄
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⣵⠙⢤⠀⢀⡆⠀⠀⢠⢀⠀⠀⢠⢆⡀⠑⣶⠂⣠⠂⠊⠰⡀⢰⡡⠘⠀⠀⢰⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠘⢁⠾⠳⡿⠘⠍⢸⡆⠠⣂⣍⠀⠐⣴⣃⢈⡵⣀⣿⢸⡅⠀⠘⠃⠉⠀⠀⠀⠀⠤⠜⢁⣠⣴
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣉⣠⣿⠁⣰⢐⢀⡴⢷⣮⢡⣤⣆⣩⡿⠀⣩⠃⠺⠂⠄⠙⠁⠀⠀⠀⠂⠀⠀⠀⠈⢀⠟⠂
⠀⠀⠀⠀⠀⠀⠄⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣋⡿⣿⡛⣹⢿⣶⢶⣿⢾⢟⠻⢟⡇⣀⣻⡀⠸⠧⠠⠄⠀⠀⠐⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⢀⡠⠈⠁⠀⠉⠀⢿⡇⣿⢳⣥⡠⠌⣡⣍⣴⠟⢳⡜⣷⠃⣘⡀⡠⠀⠀⠈⠀⠁
⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠈⣿⣧⠞⣴⢷⠚⠑⢸⡟⠂⢿⠓⣿⡀⠛⠁
⠀⠀⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⡾⠁⠀⠀⠀⢀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠏⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠒⠂⠀⠀⠀⠀⠀⠀⢘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⢠⣄⠴⠒⠉⠉⠉⠁⠀⠐⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣴⠞⣻⠏
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢁⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠷⠀⠓⠴⠃⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⠔⠋⢁⠤⡄⢠⠈⠒⢄⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣠⠏⠀⠀⡎⠀⡇⠈⢆⣀⣀⡑⣄⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠋⣄⠀⠀⠓⠢⡱⣄⠀⠁⠀⡹⠈⡆⠀⠀⠀⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸⠀⠈⠓⠤⠤⠚⠁⠀⠉⠒⠊⠁⠀⠸⡀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠈⢇⠀⠸⡐⠻⠿⠟⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⡇⠀⠀⣀⠀⣀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠎⠑⡼⠀⢀⡇⠸⣿⢀⠠⠀⠀⠀⠸⣿⠀⠀⠀⢀⠇⠀⡸⣡⠄⠀⢣⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠀⠀⡜⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡠⠴⣊⡠⠔⠁⠻⠂⡠⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠠⡀⠀⠀⠀⡏⢿⡧⠀⠀⡀⠀⠀⠀⠀⠠⣿⠏⠀⠀⠀⢀⠏⠙⠉⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⠜⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⢠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣥⠀⠀⠀⠀⢸⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣀⡙⠤⠤⠤⠤⢛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠸⣅⣀⣠⣄⠀⡨⠁⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠉⣀⡀⠉⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢀⡀⠀⠀⠀⠰⠆⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠷⣶⣄⠀⠀⠀⠀⠀⠀⣾⡏⠀⣀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⠻⣷⡀⠀⠀⠀⣴⡟⣀⠾⠁⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠶⠆⣼⣿⣤⣄⣤⣿⡇⠛⠀⠀⠀⣠⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢹⡏⠉⠉⠙⣿⣦⣴⣿⣿⣏⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣀⣠⣾⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⠿⣿⣝⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠈⠻⢿⣤⠻⣯⡙⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⠁⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠙⣿⣆⠙⣦⡙⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠷⣶⣤⡈⠀⠀⠀⠈⢿⣷⡘⢿⣼⢿⣿⣿⣿⣿⣿⣿⡇
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⡈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣀⡀⠀⠙⣿⡆⠀⠀⠀⠀⠿⣧⡼⢻⣼⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣁⡄⢸⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢼⣿⣏⣠⠀⢸⡧⠀⠀⠀⠀⠀⠻⣧⡼⢻⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡟⠿⣧⡀⠀⠈⠙⠋⢁⣼⠋⠀⢰⠆⠀⠀⠀⠀⣿⡄⠀⠈⠻⠟⠃⢀⣾⡇⠀⠀⠀⠀⠀⠀⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡇⠀⠙⠿⣦⣤⣤⣴⠿⠉⠀⠀⠸⢧⠀⠀⠀⠀⠈⠿⣧⣤⣤⣤⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⣿⣿⣿⣿⣿⣿⣿
⣸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠟⠛⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠛⠛⠿⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⠶⠿⠛⠛⠁⠀⠀⠀⢿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⡏⢙⡛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⠿⢯⠛⣦⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣤⣤⡤⠚⠻⡿⠛⢠⡜⠀⢸⣤⠘⠛⣥⠀⠀⠀⠀⠙⠛⠛⠓⠢⣤⣤⢀⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣶⣿⠟⠛⠛⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣴⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣄⠘⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⣦⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣏⠀⠃⠀⠞⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠃⠀⡄⠀⠀⠹⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⢀⡆⠀⠀⠀⢠⡄⠚⠀⠀⠀⠀⠀⠀⢠⡇⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡠⠏⠀⠀⠀⠀⠀⠀⢀⠈⢀⠤⠀⠠⠤⠀⠀⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠀⠀⠀⠀⠀⠀⠈⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⣧⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠿⠷⠶⠶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢉⠉⠉⠉⢙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⢿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⢸⠃⠀⠀⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⢠⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠘⠁⠀⠀⢠⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⢀⡜⠃⠀⠀⠀⢸⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡄⠀⠀⠀⠀⠤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⢀⡁⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠑⢦⣄⠀⢸⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠈⠉⢶⣿⡁⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠇⣀⠀⠀⠈⣿⡇⠀⠀⠛⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⡔⠂⠀⢻⣧⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⢅⠉⢤⡤⠙⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣤⣤⣤⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠀⠈⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⣷⠀⡸⡄⠀⠀⠀⣄⠆⠀⠀⠀⢄⠀⠀⢘⡂⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡃⠀⠱⡄⠀⢠⠟⠀⠐⠢⠁⢈⠀⣀⠊⠑⠂⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⠀⠘⠀⠰⠀⡗⠱⣤⣽⠆⡱⠁⠀⠀⠀⠈⠃⢺⠁⠘⣀⡴⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡀⡀⠀⠀⠀⠀⢤⣀⡙⠮⣋⡐⣇⠀⠀⠀⠀⡰⣡⠧⣖⣪⣥⣌⣏⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠍⠀⢡⡁⠔⠀⡅⠀⢀⡈⠻⡐⡂⠹⣎⢷⣤⠴⢚⡽⠟⠋⠉⠀⠀⠀⠀⠃⠆⡀⠀⡆
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠐⠀⠀⠉⠙⠳⣷⣘⣦⡟⢸⣷⢴⡋⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⡆⠁
⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⢀⡀⢐⢉⠄⠀⠈⠀⡈⣱⢀⣘⣽⣘⡤⣰⠀⠄⢀⠀⠀⠀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⠘⢻⠸⠈⠀⠀⠀⠉⢿⣣⣬⡾⣋⡜⣰⡇⡐⠉⠀⠀⠀⠈⠄⢃⠀⠰
⠀⠀⠐⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⡆⠃⠀⠀⣿⡿⠷⢚⡩⢞⣽⡇⠁⠆⢀⠀⠘⠈⡇⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠆⠀⠀⠀⠇⠀⠠⠈⠀⣸⠏⢒⣋⣥⣶⠛⣽⠁⠘⠀⣨⠀⠘⠀⢱
⠀⠀⠀⢀⡀⣶⣆⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠂⣸⡿⣿⣪⡽⢋⣡⡤⣿⠀⠀⠀⠀⠀⠀⠘⠘
⠀⠀⢀⣿⣷⣿⡿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⣯⡾⢣⣯⢾⡟⢁⣼⢷⣤⠶⠤⠀⠀⠀⠀⠀⠀⠀⢀
⣤⣤⣤⣿⣿⣿⣿⣾⣿⣶⣶⣶⣶⣶⣦⣤⣤⣠⣿⣻⣿⣶⣾⣯⣾⣯⣥⣾⣿⣯⣾⣿⣿⣓⠀⣀⣀⣀⣀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠉⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⠄
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠆⠀⠀⠀⠀⠠⠐⠋⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣀⠀⠀⢀⡄⡀⠀⠠⠀⢠
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡤⠚⠁⠚⠀⠀⠀⠈⠁⠀⠀⣬⠃
⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠈⠃⠀⠗⠁⠀⠋⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⡀⠀⠛⠢⠄⠀⠀⠀⠀⠴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣁⣡⠄⡀⠈⠢⠄⡀⠀⠀⠀⣀⠐⠁⠀⠀⠀⠒⠂⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠉⠁⠠⡄⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠖⠢⠤⠠⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⡐⠀⠀⠐⠀⠀⠀⠈⠀⠐⢂⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠤⠄⠒⠂⠈⠉⠀⠀⠈⠉⠉⠉⠉⠉⠑⠒⠀⠀⠄⠠⠀⠐⠂⠀⠀⠉⠉⠁⠀⠈⠉⠉⠀⠀⠀⠉⠒⠒⠒⠢⡀
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡏⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢼
⠀⠀⠀⠀⣷⡄⠐⠢⠤⠤⠤⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠉⠁⠀⠀⠀⢨
⠀⠀⠀⠀⢣⠑⠤⢀⣀⣀⣀⠀⣀⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠀⠀⣀⠤⠖⠂⠉⠉⢉⠏
⠀⠀⠀⠀⠘⢮⢢⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠐⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠒⠒⠂⠀⠒⠊⠁⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠈⠙⢮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀⡼⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠖⠒⠲⠞⠉⠉⠲⠒⠛⠒⠦⡤⠴⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⢀⡀⢀⡠⠶⢤⣤⠤⠔⠒⠦⠤⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢄
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⣄⣠⠜⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢀⡤⢿⠀⠀⢰⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⠶⢄⠀⠀⠀⠀⠀⣯⡀
⠀⢀⡏⠀⢀⡟⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⢶⠚⠁⠈⣆
⠀⣸⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢹⡀
⡴⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢳
⢇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠞
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢴⡃⠀⠀⠀⠀⠀⠀⠀⠀⣨⠓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⢤⣀⣀⣀⠲⠤⠤⠴⣃⣀⣀⡤⠤⠶⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠈⠻⣿⣿⡿⠃⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠲⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠹⣿⣿⣿⣿⠟⡿⠟⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠟⠟⠿⠇⠀⠁⠀⠀⠰⠈⠟⣹⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⢻⡟⠁⠀⠀⠈⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠿⠟⠿⣿⡿⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠟⠛⠛⠛⠛⠛⠛⠻⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⣀⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⣍⣉⣉⣉⣽⡟⠁⠀⠀⠀⠀⠀⠻⣯⣉⣉⣉⣩⣿⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⡶⠶⠿⠿⠶⢶⣦⣽⣷⡀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⢶⣦⣤⣤⣤⣤⣴⡶⠾⠟⠛⠉



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠓⠶⠒⠉
⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⣤⣴⠟⠁
⠘⣿⣿⣧⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣤⣶⣦⣄⣤⣢⣶⣾⣏⠙⣿⡁
⣾⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⢀⣴⡿⢛⡹⠾⠟⠻⣿⢸⣿⡿⣷⣎⣿⠀⠉⡁
⠛⠻⢿⣿⣿⣿⣿⣿⣤⣳⣴⡶⠒⠒⠒⢒⣶⣿⢋⣭⣟⠆⢀⢘⣿⡏⣴⠏⣠⡾⢟⣻⣲⣤⡜⣷⣿⡋⠉⢀⣸⠃
⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣉⡽⠟⣉⢉⣿⣿⣽⣿⠁⠀⢀⣣⡿⣽⢫⣿⢰⣿⢡⡾⢯⣽⣧⣹⣿⣿⣇⠀⠉
⣿⣿⠁⠀⣀⣽⣿⣿⣿⣿⣿⣶⣮⣉⠠⣿⣿⡿⣿⣦⣬⣷⣿⡾⢋⣾⠿⣶⢿⣄⠁⠀⢙⡛⠛⢹⣿⡏
⣿⡟⠀⣾⠹⡁⠀⠀⠉⠙⠻⢿⣿⣿⣦⢹⣿⣧⣿⣻⡟⠛⣋⣴⠟⣹⠀⢻⣾⢻⣦⠔⠉⣀⣴⡿⠛⠂
⠋⠀⠀⣿⢠⡇⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣵⠾⠻⣂⣿⢿⣿⣿⣦⠞⣿⢿⡃⠀⡁⠀⠀⢀⡀
⠀⠀⢠⣾⡿⠁⠀⠰⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⡿⠿⠿⠛⢟⣫⣴⣿⣿⣿⣷⡄⠃⣈⢳⡄⠀⠀⠀⠈⢁
⣶⢾⣿⡟⢁⠂⣀⡤⠙⢿⣯⣤⡼⠋⡉⠋⢉⣠⠶⠋⢹⡀⠀⣀⣼⡿⠛⣉⣩⣽⣟⡙⠻⣦⣹⠟⠀⠀⠀⠀⠀⠘⠂
⣡⡾⠋⣠⣾⡿⠋⣀⣀⣤⣬⣭⣶⣯⠴⠞⠛⢁⣤⣤⡀⠑⣼⣿⠏⣠⣾⠿⣿⣿⣿⣿⡄⣸⣿⣄⡀⠀⠀⠀⠀⠀⠠
⢋⣤⡾⢋⣿⣶⣿⠟⠋⢩⣴⠟⠋⠀⠀⠀⢀⣿⠓⠻⣷⢠⣿⣿⠠⣿⡧⢀⣽⣿⣿⡏⣿⡿⠋⣽⠛⠻⠛⠋⡡
⠿⣋⣴⣿⣿⠟⠁⠀⠀⣿⠃⢠⠞⡛⠳⣦⠘⣧⣀⢀⣿⠋⣿⣿⡀⢛⣿⣿⣿⣿⣿⣼⡟⠀⠀⠀⠆⠀⠀⣼⠁
⣾⢿⡿⠛⠁⣴⠏⠀⠀⢿⡀⢿⡈⣹⠀⣽⠇⠈⠉⣸⣿⡆⠘⢿⣷⣽⠷⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣴⡟
⣵⠟⢀⣴⡾⠃⠀⠀⠀⠈⢷⣌⠉⣁⣴⡟⠁⢀⣴⣿⣿⢧⣀⠀⠀⢙⣷⣦⣤⣉⠑⠻⣬⡛⢿⣿⣿⣿⣟⡳⣦
⣥⣾⣿⢟⣡⣤⣀⣀⣀⣀⣠⣽⣿⣿⡏⣠⣾⣿⣿⠟⣡⣼⣿⣷⣦⣀⣉⡛⠻⣿⣿⣶⣄⠙⠇⢺⣿⣿⠂⠑⡌⢃
⠿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⠋⣠⣾⣿⣿⣿⠟⠉⢻⠋⠛⣷⠀⠹⣿⣿⡇⠸⣸⡿⠃⣼⠀⠇⢸
⣶⣿⣿⣿⣿⢟⣭⣿⣿⣿⡿⢛⣴⢿⣿⠟⣡⣾⣿⡿⠟⠋⠀⠀⠀⢸⠀⠀⠉⠀⠀⣿⣿⡏⣼⣯⢤⡎⠥⠊⡠⠂
⣿⡿⢟⣿⣾⡿⠿⣿⡿⢋⣰⣿⣵⡿⢃⣾⠟⠁⣀⣀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣠⣾⡿⢛⡄⣈⣽⣿⣥⡐⠊
⠉⠀⠘⠉⠀⢀⡼⠋⢠⣾⣿⣿⣿⠃⣼⠃⣰⣿⣿⣟⣛⡲⣦⡀⠀⠀⠹⣷⣤⣿⡏⠀⠀⢸⡟⠉⠉⠉⢻⣦
⠀⠀⠀⠀⣠⠞⠀⣀⣾⣿⣿⣿⣿⠀⢿⢠⣿⠋⠉⠉⠉⣿⣿⣿⣦⡀⡀⠈⠛⣿⣿⣤⣀⠀⠳⠄⠀⣠⣾⡟⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿
⠀⠀⢀⣴⠋⢀⣼⠇⣿⣿⣿⣿⣿⣄⣊⢸⣿⣄⣀⡀⠐⣽⣿⣿⣿⣷⡜⢦⡄⠘⢿⣿⣿⣷⣶⣶⣾⣿⡿⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿
⣤⣾⡿⠁⢠⣿⡏⣿⡜⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⣿⣿⢸⣿⣦⠈⢻⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿
⣿⡟⠁⢠⣿⣿⢹⣿⣿⣍⠿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣸⣿⣿⣿⣿⣧⠀⠈⠛⠿⣿⣿⣿⣿⣿⣟⣛⣋⣉⣀⣀⣀⣠⣽⣿
⠟⠀⢠⣿⣿⠃⣼⣿⣿⠛⢣⡈⠛⠿⣿⣷⣮⣭⣛⣉⣉⣀⣀⣠⣴⣿⣿⡿⢳⣿⣿⠀⣠⡶⠿⠿⣿⣿⣍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿
⠀⢠⣿⣿⡏⢀⣿⣿⣿⠀⠀⠈⠳⢤⠀⡉⠛⠻⠿⠿⠿⠿⣿⣿⣽⡿⠋⣰⣿⣿⡏⡼⣯⣶⣶⣶⣄⠈⠻⣿⣦⡀
⢠⣿⣿⣿⠁⢸⣿⣿⣿⠄⠀⠀⠀⠀⠨⠭⠶⣞⣛⣛⣻⣿⣿⡿⣯⣷⠟⣿⠟⠋⠀⠹⠃⢸⣿⣿⣿⠆⠀⢹⣿⣷⡄
⡿⠃⣿⡟⣰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⠿⠋⣰⣿⣏⣾⠃⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⣸⣿⣿⣿⣄⡀⢀⣀⣠⣴⣾
⠀⢀⣿⡇⣿⣿⣿⣿⣿⠀⠀⠀⠘⢷⣾⣿⣿⠿⠟⢛⣥⣶⡆⢿⣿⣾⡇⠀⠀⠀⠀⠀⣠⣿⣿⣿⠁⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⣿⡇⢻⠋⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣇⠈⢿⣿⣿⣄⣀⣀⣤⣾⣿⣿⠿⠋⣼⠁⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣤⣿⣧⡞⠀⢿⣿⣿⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡀⣤⡙⠛⠿⣿⡿⠿⠟⣋⠁⠀⠀⣿⡆⠀⠻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣿⣿⣿⡃⠀⢸⣿⣿⡆⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⠓⠀⢤⣤⣴⠶⠛⠁⠀⠀⠀⢹⣿⣦⣀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⠀⠀⣿⣿⣷⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⣿⣷⣤⣀⣀⣀⣀⣤⣤⣤⣀⣀⠀⠙⢿⣿⣷⣦⣄⣉⠛⠿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⡀⠀⢹⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣦⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢻⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤
⣿⣿⣿⣿⣿⣇⣀⣘⣿⣿⣷⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⣆⣰⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⢿⡀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⠋⠉⠛⠲⢤⣀⣼⠋⠙⢦⡀⠀⠀⢷⠀⠀⠈⣧⠀⣠⣤⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⢹⡄⠀⠈⢷⡀⠀⠈⣧⠀⠀⠸⡏⠁⢀⣬⣧⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣸⠃⠀⠀⠀⠀⠀⠀⣀⣤⣷⠀⠀⠈⢧⠀⠀⠘⣇⠀⠀⢻⢠⡏⠀⣿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⣷⢠⡇⠀⠀⠀⠀⠀⠀⢠⡏⠀⠘⣧⠀⠀⠘⣇⠀⠀⢹⡄⠀⠸⣿⡇⠀⢹⡀⢸⡧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠘⣆⠀⠀⢹⡄⠀⠈⣧⠀⠀⣿⡇⠀⠸⣇⡟⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠸⣆⠀⠈⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⡷⠓⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⣷⠀⠀⠀⠀⠀⠉⠉⠛⠓⢿⡀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢹⠀⠀⠀⠀⠶⠖⣄⣀⠀⠈⣧⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⣀⠀⠀⠀⠀⠀⠉⠉⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣌⡇⠹⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠷⣄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡆⠀⠀⠈⠃⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠟⣦⠀⠀⠀⠐⠶⠶⠶⠿⠶⣄⠈⠳⠤⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣾⠀⠈⠳⣄⠀⠀⠀⠶⠤⠴⠶⠞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡙⠳⠦⣤⣀⣀
⠀⠀⢀⣀⡤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠶⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠈⠉⠙⠛⠲⢦⣄⡀
⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠈⠉⠉⠉⠐⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠠⣞⡙⡿⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡄⠘⣇⢱⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢪⣿⣇⠀⠀⠀⠀⠀⢀⠞⣳⠀⠀⣸⠙⣆⢸⠯⡁⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⢀⡞⢿⠀⠀⠀⢀⡠⠔⠁⢠⡟⣠⠶⠃⢠⣷⡞⢦⡁⠀⢰⣜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣾⡀⢸⠀⠈⣾⣡⠀⠀⠠⣴⡆⠀⠀⠀⣴⠖⠋⢰⢻⡄⢷⠀⡼⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠓⠬⠧⠀⠘⠊⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠚⠀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠆⠀⠀⠀⠀⠠⠀⠀⢀⡼⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠀⠀⠄⠀⠀⠐⠉⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠁⡀⠀⠀⠀⠀⠠⠃⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢦⣀⠈⠲⠄⣀⠜⠁⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⠉⠂⣄⠀⢱⠊⠁⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠙⠛⠊⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠚⢻⠒⢻⠒⠋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡾⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡇⠀⢸⠒⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠉⣳⣄⣼⢀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢾⠗⠈⡿⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠚⠉⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠬⠷⠘⠙⠚⠉⠛⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡎⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠈⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛
⠀⠀⠀⠀⠀⢰⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢀⣐⣂⣀⡀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⣇⣤⠀⠀⢀⡀⠀⢸⢇⢖⡴⠋⠀⠋⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⣀⣈⣸
⠀⠀⠀⠀⠀⠁⠀⠀⣠⠌⢯⡑⠒⠈⠋⠀⠀⢀⣠⠤⠤⠄⠀⠀⠀⣧⠀⠀⡾⠠⠢⠙⡄
⠀⠀⠀⠀⠀⠀⢰⢸⡇⠉⢉⣀⡀⠀⠀⠀⠞⢉⡠⠔⢺⡍⠃⠀⠀⠸⡆⢀⢧⠐⣲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢳⡅⢼⣻⡉⢭⡀⠀⠀⠀⠉⠻⣿⣿⠇⠆⠀⠀⠀⠙⠦⠞⠸⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠛⠛⢁⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡠⠀⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠠⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠙⠛⠃⠀⠀⠀⢀⡴⠁⠀⠀⠀⣠⠎⡇⠀⠐⠒⠒⠋⠉⠉⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠂⠑⣄⠀⠠⢄⣀⡠⠔⠉⠀⠀⠀⣠⠞⠡⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⠑⢦⣀⡀⠒⠂⠀⣀⠴⠋⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⡀
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡉⠲⠙⠀⠄⠠⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠐⢹⢦⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠢⠑⣄
⠀⠀⠀⠀⢠⠇⠀⡀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⡴⡡⠒⠀⠁⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⡜⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢩⡝⡂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⡗⠾⠶⠖⠋⠀⠀⠀⠀⠀⠀⠈⢆
⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡣⡀
⠀⡄⠀⠀⠀⠀⠀⠀⣀⣠⡭⠴⢂⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡽
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡈⡎
⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡈⢃⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠠⠀⡰⠁
⠀⠀⠑⢄⠀⠀⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢱⡀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢤⡀⠀⠀⠀⠒⠒⣒⡒⠒⠪⠍⠀⠀⠀⠀⢧⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠦⣸⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠀⠀⠀⢌⣂⢪⠇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡠⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠒⠒⠊⠉⠀⠀⠁⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⢂⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡞⡩⢙⡄⠀⠀⢀⣀⠤⠚⠡⠀⢀⡡⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠈⠉⠉⠒⠂⠤⠤⠄⢀⠳⢬⡞⠡⠔⠚⠉⠄⠀⢈⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠄⣀⠀⠂⠀⠀⠀⠀⠀⠀⡆⠀⢀⣀⠥⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⠒⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠛⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⣛⣺⣿⣶⠤⠿⣿⣿⣿⣇⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠞⢅⣉⣥⠐⠑⠀⠘⢱⡖⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⣻⣟⣿⡄⠀⠀⢀⠀⠇⢌⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢿⠟⣹⣿⠁⣰⡀⠈⡆⠀⢠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠃⣐⢁⡏⣰⡟⠑⠐⡇⠀⢨⠀⠂⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠒⠒⠀⢾⣿⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡂⠁⢸⣇⣿⣧⣀⢀⡟⡀⢸⠄⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⣾⣿⣿⣽⣆⣿⣿⣿⣿⣿⣧⢰⡷⢾⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⣠⡐⠏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢳⡟⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠄⠀⡀⠘⠛
⠀⠀⠀⠀⠀⠛⠁⠀⠀⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣷⣮⡴⠇⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⣶⠀⠀⠀⠀⠈⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣼⣿⣿⣿⣽⡷⣿⣿⣿⣯⣽⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠁⠀⠀⠀⠀⠀⠃⠀⠖
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡶⢿⣿⣿⣿⡗⠿⣿⣫⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣄⣀⣴⣦⣈⣷
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢰⣾⣿⣿⡿⠿⣫⠉
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡛⢉⡉⠉⢁⠁⢈⠛⠘⢻⣿⣇⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣋⣿⡄⠀⣿⡀⠻⣶⠃⠀⠀⠀⠀⡟⠿⣯⡄
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠛⠉⡀⢀⣜⣧⢫⣧⣶⣼⣷⡿⠃⠀⠀⠀⠀⠰
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⢠⠉⢲⣆⢿⣿⢻⡄⣽⢿⠋⠔⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠈⠿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠘⠳⣾⣾⠷⣦⣼⣿⣿⡶⣍⡯⡦⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠿⡹⣯⣽⣿⠇⠀⠀⠀⠀⠀⠀⠠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣀⣤⣄⠤⠀⡀⠀⠀⠀⠀⠀⠀⠈⡱⠃⠀⠀⠀⠀⠀⠀⠠⠁⣴⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⣾⣿⣿⣋⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢿⠟⠏⠀⠀⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠂⠀⣀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃