Drumstick Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⡾⠿⠿⠟⠻⠿⠿⠿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡛⠿⢶⣶⣤⣤⣤⣀⣀⣀⣠⡤⠤⠒⠚⠉⢀⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣴⣮⣭⣉⡀⠀⣀⣀⣀⣀⣠⣤⣴⡾⢿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠋⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡼⠁⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⢹⣿⣿⣿⣿⡟⠀⢻⣿⣿⣿⢿⠁⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢺⢿⣿⡐⠀⠀⢿⣿⣿⡿⠀⠀⠈⣿⣿⣿⠋⠀⠀⠈⣿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠈⠐⣷⣄⣁⠂⠸⣿⣿⠁⠀⠀⠀⠸⣿⡿⠀⠀⠀⣠⡟⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢌⡛⢿⣯⣄⣻⣇⠀⠀⠀⠀⠀⢻⣇⣠⠴⠚⢁⡗⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠈⠻⢦⣈⡉⠙⠛⠛⠛⠛⠋⠉⠉⠁⢀⣠⠖⠋⠀⠀⢀⣄⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠒⠒⠒⠒⠚⠉⠉⠀⠀⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡦⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠹⠿⠿⠿⠿⠿⢯⣿⣶⣄⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢧⠄⠀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠉⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⢀⡠⠴⠚⣉⠤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⣶⣺⢳⡞⠉⠙⠿⠉⢹⡤⠔⠊⠁⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡏⠛⠋⠹⡆⠁⠀⠀⣇⠀⠀⠀⠀⣸⢁⡠⠖⠋⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢷⠀⠀⠀⢹⡀⠀⠀⣹⠞⠛⠷⣤⡿⠓⠒⠒⠒⠛⠉⠉⠉⠉⣁⣀⡤⠤⠴⠒⠦⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠤⠼⣧⠀⠘⣆⡤⣤⣸⣧⣴⡞⠁⠀⠀⠀⠘⣧⠤⠤⠤⠒⠒⠚⠉⠉⠉
⠀⠀⡠⠒⠒⢚⠋⠉⠉⠉⠀⠀⣀⣀⣀⣀⣸⣦⣚⣹⣿⣹⡀⣀⡤⠃⢀⣠⡀⠀⠀⣿⡀
⠀⠀⠑⠤⠴⠒⠒⠒⠒⠉⠉⠉⠁⣠⠴⠚⠁⠈⣉⠽⠿⠿⢯⣥⣤⡴⠋⠙⠃⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⣀⡤⢶⡉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡷⠚⠁
⠀⠀⠀⠀⠀⢀⡠⠔⠋⠥⢀⡠⠖⠋⠁⠀⠀⠙⢦⠀⠀⠀⢹⡆⠀⠀⠀⢠⡞
⠀⠀⣠⠴⠚⡁⢘⣫⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⡇⠀⠀⠀⢻⠁
⠀⠘⣅⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣇⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠁⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⣛⡛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⡵⠖⢳⡄⢸⣏⣙⣶⠄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢰⣏⣤⠶⠋⠀⠀⠈⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡾⠁
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠿⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⢀⠀⠀⢀⡾⠁
⠀⠀⡟⠉⡍⠛⠓⠒⠚⠛⠉⠁⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⣹⢦⠇⠀⠀⠀⣠⡴⠚⠛⠉⠛⠛⠳⠦⣤⣁⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢁⣀⣤⠞⠋
⠀⢰⡇⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠒⠛⠛⠋⠉⠁
⠀⠀⢳⣄⠀⢀⡾⠁
⠀⠀⠀⠉⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣠⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⡜⠁⢀⠠⠐⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠠⠤⠶⠃
⠀⠀⠀⢀⡜⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠘⢧⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⠚⠉⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡄⢀⣀⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠒⠓⠦⣄⣀⣀⣀⣁⣬⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⡤⠶⠒⠒⠚⠛⠛⠛⡒⠒⠲⠶⠤⢤⣄⣀⡀
⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⣠⣴⡾⠋⠁⠀⠀⣀⣤⡴⠂⠈⠉⠙⠒⠦⣤⣀
⠀⠀⢠⡏⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⢀⣴⣾⠟⠁⠀⠀⢀⣠⣴⠞⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⣾⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⣴⣿⠟⠁⠀⠀⣀⣴⡟⠀⠀⠈⠙⠦⣄⣀⣀⣀⣀⣀⣠⡴⠚⠉⠉⢳
⠀⠀⡇⠀⠀⠀⠠⠏⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⣴⡿⠋⠀⠀⠀⢀⡴⠋⠁⠀⠀⠉⠉⠀⠀⠀⠀⢀⡆⡟
⠀⠀⢿⠀⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⠠⠤⣤⣤⣤⣤⣀⠀⢸⡇⣇
⠀⠀⠈⢧⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠉⣀⡬⠷⠒⠒⠒⠒⠒⠶⣭⡛⠶⠟⣹
⠀⠀⠀⠈⢳⡘⢧⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⠴⠶⠖⠛⢋⣉⣡⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠃
⠀⠀⠀⠀⠀⠙⢦⣉⣛⣛⣛⣛⣛⣋⣉⣉⣉⣩⣤⠤⠴⠶⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣶⣶⡶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⠛⠛⠛⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⣠⡴⢋⠁⠀⠀⠀⠢⣠⡀⠀⡀⣌⠛⢷⣤⡀
⠀⠀⠀⠀⣴⠋⠀⢀⠀⠀⠀⠀⠀⠙⡟⠀⠋⠈⢿⠀⠻⣟⢦⡀
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢿⣆⣹⣆
⠀⠀⠀⠘⣿⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠀⠈⠀⠛⠈⠳⣄
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠛⠛⢦⡀
⠀⠀⠀⠀⠂⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣶⣽
⠀⠀⠀⠀⠀⠉⠚⠻⢶⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⢤⡀⠀⠘⢯⡉⠳⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢛⠒⠒⠟⠛⠋⠁⠀⠀⠀⠀⠀⠉⠓⠂⠼⠿⣶⣄⣀⠉⠙⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠑⠂⠄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⠀⠉⢳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⠀⠀⠀⢽⣦⣤⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠶⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠠⠤⣀⠛⠶⣴⣦⣀⡉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⡰⢋⡤⠬⠍⣙⠲⢌⠑⠄⠘⢆⢻⣹⡶⣄⣀⠙⢷⣄
⠀⠀⠀⠀⢀⡞⡡⠴⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⡗⢯⣻⣄⡄⠻⣦
⠀⠀⠀⢠⡟⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⣈⣧⣷⢀⡘⣷⡀
⠀⠀⠀⣾⠞⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣜⡇⢸⠘⡇⠈⣧
⠀⠀⢸⣇⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠆⡞⠀⡇⠀⠸⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠃⢸⠀⢻⡄
⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢰⠘⣧
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⢠⢻⡄
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡾⢸⣷
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠂⡆⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⢀⡴⠒⠦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠸⣳⣴⠋⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠇⠈⢿⡀⠀⠀⠀⠀⠁⡟⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠁⠃⡟⠓⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠁⠃⠿⣹⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢳⠀⠀⠀⠀⠀⠜⡾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐⣻⣦⣀⠀⠀⠀⠀⣺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠛⣯⡟⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠛⠛⠉⠉⠉⠉⠙⠛⠻⠿⣿⣷⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠭⣝⣛⠻⠷⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⡟⠁⠀⠀⢸
⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠀⠈⠉⠙⠛⠻⠷⣶⠚⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⢠⣿
⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣠⣤⣤⠶⠶⠶⠶⠶⠶⣶⣤⡀⠀⠀⠀⣿
⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢀⣠⣤⡶⠟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣴⣿
⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⢀⣴⡿⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⣠⣿⠋
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠋⠀⠀⣠⣾⠟⠁
⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠈⠛⢷⣤⣀⡀⠀⠀⠀⢤⣀⣀⣀⠀⠀⠀⢀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠲⠖⠛⠛⠛⠛⠛⠋⠉⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⣾⠋⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⣶⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠶⣄⡀
⠀⠹⣧⠀⠀⠉⠙⠛⠓⠒⠒⠲⡶⠶⠒⠛⠉⠉⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄
⠀⢠⡟⠀⠀⠀⣀⣀⣀⣀⡀⠀⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⢸⣧⣀⣠⡾⠋⠉⠉⠉⠉⠉⠛⠛⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠶⣾⣤⡤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠜⠀⡄⠀⢠⡶⠛⠳⣄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⡇⠀⢻⡔⡄⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⠀⠈⣧⠀⠀⠛⢿⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠤⢤⣤⣀⠀⠀⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠹⠀⢀⡀⡼⠀⠀⣰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠈⡑⠸⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⢄⠀⠀⣀⣴⠎⣀⣥⠤⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠃⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠉⠁⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⣠⠔⠋⠀⠀⠀⣀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢳⡀⠀⠒⠀⠀⢀⡀⠀⣠⡾⠋⣀⣴⠟⠁⠀⠀⠀⠀⠀⠈⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠘⣧⠀⠀⠀⠀⠃⠁⣴⠟⣠⣾⠟⠁⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠹⣆⠀⠀⠤⠀⣸⠏⣰⡟⠁⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡙⢦⣀⠀⠀⡏⢰⡿⠀⠀⠀⠁⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠑⠀⠃⠸⡇⠀⢀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠶⠶⠒⠚⠋⠀⠀⠀⠉⠓⠀⡀⠀⢻⡀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⢀⣀⣀
⠀⠀⠀⢠⣴⣾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⡄⠀⠀⢹⡄⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠈⠉⠛⠻⢶⣦⣄
⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⢀⠤⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠿⠋⠁⠀⠈⠉⠛⠲⠤⡀⠀⠀⣈⣿⠇
⠀⠀⠀⠀⠙⠻⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣶⣶⣶⣶⡶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⡻⠿⢿⣶⣶⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣴⣶⣾⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢶⣤⣍⣉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⢛⣋⣩⣥⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣷⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⡾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠒⠒⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⣾⡶
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⣠⣄
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⣰⣿⠏
⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠦⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⣀
⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⣰⡿⠃
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠶⠦⢤⣄⡀⠀⠀⠀⠼⠋
⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⡀⠀⠀⢀
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠛
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠄⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣠⣀⣴⢄⡟⠙⠇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢘⡇
⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡟⢶⡿⠈⠉⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⢧⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⢀⡾
⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠⠹⣄⣤⣄⡀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡴⠋
⣀⣤⣤⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠀⠀⠀⡀⠀⠱⢠⠇⠀⠀⣟⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⢀⣀⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣰⣧⡤⠴⠋⠀⠀⠀⠛⢁⡷⠞⣧⣠⡀⠀⠀⠀⣠⠔⠉
⠀⣤⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠈⢉⡷⠚⣧⠞⠁
⠀⠈⠉⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡞⠀⠀⠉
⠀⠀⠀⠀⠀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⡞
⠀⠀⠀⠐⢿⠟⠋⠀⠀⠀⠀⠀⢀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡾⠋⠙⠁⠀⠀⠀⠐⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⣏⡀⠐⠎⣠⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⠀⠀⠄⢴⣧⣤⣤⠄⠀⠈⢓⣒⣶⡖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⠀⠀⠀⠐⠀⠀⣵⣟⡈⠀⠀⠀⣀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣷⠄⠀⠀⠀⡘⠀⣰⣿⠟⠀⠀⠀⠀⡅⠀⢸⣶⣖⠂⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⣹⠁⠀⠙⢻⠗⠁⠀⠀⢰⡄⣤⣿⣿⠂⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⢹⡟⠒⠀⠀⠀⠐⠉⢹⢃⠀⠀⣸⣶⢲⣶⠶⠆⠀⠀⠀⣹⣿⣃⡀⠀⠀⠀⢰⣾⠃
⠀⠀⠀⠀⠀⢀⣬⠇⠀⠀⠀⠀⠀⢀⣼⠴⠒⠟⠿⢿⡞⠃⠀⠀⠀⠀⣤⢿⠉⠛⠒⣶⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠐⣾⡆⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠘⡦⡄⡀⠴⠂⠀⠈⢄⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠻⢧⠄⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢀⠀⠀⠀⠀⢸⣆⠀⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⣯⠈⢂⣦
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣠⣾⣿⢉⡤⡾⡿⠲⣄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠣⠏⢱⢳⣇⡴⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⡋⠀⠀⠀⠀⠾⠀⠁⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠛⠉⠁⠀⠙⠦⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⠝⠋⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⠵⠊⠁⠀⠀⠀⠀⠀⣠⠔⡫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡽⠚⠁⠀⠀⠀⠀⠀⢀⡤⣊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⢤⣤⢖⣩⢾⣥⣀⣀⣀⣀⣀⢀⡠⢖⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⢉⠔⠛⠓⣯⠔⠋⠀⠀⠀⠀⣀⣠⣠⠝⢋⠴⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠸⣄⠀⣀⡜⠀⠀⠀⠀⠀⣼⠁⠀⠈⡟⠁⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠘⠦⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢸⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⡇
⠀⠀⠀⠀⢸⡷⣌⠙⠲⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠖⠚⠉⢀⣠⠔⡇
⠀⠀⠀⠀⠀⡇⠈⢑⡦⢄⣀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⡠⠤⡖⠊⠉⠀⢠⡇
⠀⠀⠀⠀⠀⡇⠀⡆⢡⠀⠀⠉⠉⢉⠗⠒⠒⠒⠒⠒⠒⡚⢏⠉⠉⠉⠉⠀⠀⠂⢱⠀⠀⠀⡎⡇
⠀⠀⠀⠀⠀⡇⣸⠀⠘⡆⠀⠀⢀⠎⠀⡇⠀⠀⠀⠀⡰⠁⠘⡄⠀⠀⠀⠀⢸⠀⠀⢣⠀⠀⣇⡇
⠀⠀⠀⠀⠀⣧⠇⠀⠀⢣⠀⠀⡌⠀⠀⢃⠀⠀⠀⡰⠁⠀⠀⠱⡀⠀⠀⢀⠇⠀⠀⠈⡇⢰⢹⡇
⠀⠀⠀⠀⠀⢻⢦⡀⠀⠘⡄⡸⠀⠀⠀⢸⠀⠀⢰⠁⠀⠀⠀⠀⢳⠀⠀⡘⠀⠀⠀⠀⠸⣸⢸⠁
⠀⠀⠀⠀⠀⠘⢦⡙⠢⣄⣁⠁⠀⠀⠀⠀⠇⢠⠃⠀⠀⠀⠀⠀⠈⡆⢠⠃⠀⠀⠀⠀⣀⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣉⠑⠲⠤⢄⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⠤⠤⠒⣉⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣌⣉⣉⣉⣉⣉⣉⣉⣉⣥⠤⠴⠖⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢿⣿⢰⣿⣿⣦⣿⣿⢰⣿⣿⣾⣿⣿⢹⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣿⣿⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⣶⣶⡆
⠀⠀⠀⢳⣶⣿⣎⣁⡄⣤⣤⣥⣤⢿⣿⢸⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣸⣿⡟⠿⠿⢿⡿⣿⠿⠿⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡟⠻⠿⠿⢿⡿⡿⢿⣿⣿⡛⠿⠛⠋⠉⠉⠀⠐⣾⢾⠿⢶⠶
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠁⣰⣤⣿⠿⠿⣧⣴⣀⠀⡀⠀⠀⠀⡇⣸⡀⢸
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡃⢰⡶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠳⣶⠀⠀⠀⣿⣛⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⡿⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣴⡿⣷⠛⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡷⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡯⣿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠫⢀⣀⣁⢜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢜⡇⢀⡙⠆⢀⠀⠁
⠀⠀⠀⠀⣀⠴⢊⠁⠀⠀⠀⠀⠁⢸⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⡏⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠁⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⡄⠉⠛⠶⠶⠶⠶⠶⠾⠋⢁⡄⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠟
⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠐⣧⡈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠈⠛⢦⣄⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣌⠙⢦⣤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠓⢲⣶⠋⣠⣾⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠑⠦⣉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣡⠞⠁⠀⠀⠉⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠑⠦⣙⠲⣄⡀⠀⠀⠀⢰⠟⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡙⠓⣆⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠉⠛⠓⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠉⠀⣠⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⠙⠳⢦⣄⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⣤⠶⠟⢩⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡄⠀⢀⠌⠉⢟⠛⠒⠶⠶⠶⣶⠶⢶⡶⠶⠶⠶⠶⠶⢻⠟⠛⢍⠀⠀⢀⡎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢱⡴⠁⠀⠀⠈⢦⠀⠀⢀⡔⠁⠀⠀⠑⢄⠀⠀⠀⡰⠃⠀⠀⠈⠳⣄⡼⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠊⢣⠀⠀⠀⠀⠈⢧⡠⠋⠀⠀⠀⠀⠀⠀⠱⣄⡜⠁⠀⠀⠀⠀⠀⣸⢧⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⣀⠀⢧⠀⠀⠀⣠⠞⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞⢦⡀⠀⠀⠀⠀⣰⠃⠀⢙⡶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠙⠳⢦⣧⣴⠊⠀⠀⠀⠱⣄⠀⠀⠀⢀⡴⠁⠀⠀⠙⢆⠀⠀⣰⣁⣤⠶⠋⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠉⠙⠳⠶⢦⣤⣬⣦⣀⣠⣎⣀⣀⣀⣤⣤⡤⠷⠞⠛⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢤⣄⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠲⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉

⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⢀⣴⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁⠀⢀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣴⣾⠟⠋⠀⠀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠾⠿⠿⠛⠛⠛⠛⢛⣻⣿⡿⠛⠛⠿⢿⣶⣾⣿⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⢉⣁⣤⣤⣶⣶⠶⠾⠿⣿⣿⡿⠿⠿⠿⠿⣶⣾⣿⣿⣥⣤⣈⡉⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⢀⣴⠟⢉⣤⣶⠿⠛⠋⠉⠀⠀⠀⢀⣠⣾⠟⠋⠀⠀⢀⣴⡾⠟⠁⠀⠀⠀⠉⠙⠛⠿⣶⣤⡉⠻⣦⡀
⠀⠀⠀⠀⠀⢸⣯⣾⠟⠉⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣽⡇
⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢰⡾⠋⠁⠀⠀⠀⢴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⢸⣏⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⣽⡇
⠀⠀⠀⠀⠀⢸⡿⢷⣦⣈⠙⠛⠿⢶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⡶⠿⠛⠋⣁⣴⡾⢿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⠻⠷⣶⣤⣤⣄⣉⣉⣉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⣉⣉⣉⣠⣤⣤⣶⠾⠟⣿⠁⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⣠⣿⡄⠀⠀⠀⠉⠉⠙⣻⡟⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⢻⣟⠋⠉⠉⠀⠀⠀⢠⣿⣆⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣧⠀⠻⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⠟⠀⣼⡇
⠀⠀⠀⠀⠀⢸⡿⣷⣤⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣤⣾⢿⡇
⠀⠀⠀⠀⠀⠘⢿⣄⡙⠿⣶⣤⣀⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⢀⣀⣤⣶⠿⢋⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣈⡙⠛⠻⠷⢶⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡶⠾⠟⠛⢋⣁⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣶⣤⣤⣤⣄⣈⣉⣉⣉⣉⣉⣉⣁⣠⣤⣤⣤⣶⠶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠠⢈⠉⠉⢓⣶⣶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⢾⠷⠐⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡾⠁⠁⠀⠉⠢⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣈⡑⠒⠒⢺⠒⠒⠉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠉⠀⠀⠀⠀⢠⠀⣾⡹⠀⠤⠄⢻⢈⡆⠀⢸⣀⠠⠤⠧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⣿⣧⣤⣤⣾⣼⣧⠀⡚⡂⠀⠘⠋
⠀⠀⠀⠀⠀⠀⣤⢤⡮⢤⡤⣤⣤⣄⠀⠀⠀⠀⠀⣸⠀⠹⡁⠀⠀⠒⣿⣿⠟⠈⣯⡇
⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⡟⣸⡗⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⣾⣏⢙⡃⠀⠰⣿⣿⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⣄⣿⡇⠀⢘⠿⠫⠉⠀⠉⠹⢆⠀⠿⠿⢿⣿⣿⡟⣿⠅⠀⠙⡂
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⣿⣿⣧⡾⠉⠀⠀⠀⡀⠀⠀⠀⠈⢲⠀⠀⢉⢿⠀⢹⠀⠀⣴⣧
⠀⠀⠀⠀⠀⠀⢹⣟⠋⢺⠋⠋⠛⡿⠁⠀⠀⠐⢧⡀⠠⡂⠀⠀⠀⢃⠀⢀⣏⠀⣿⢆⡔⣾⠂⠱⡀
⠀⠀⠀⠀⣠⠴⢻⢻⢧⣸⠀⠀⠀⣇⠀⠀⠀⠐⠉⠀⠀⠃⠀⠀⠀⣸⠀⠀⠏⡸⢹⠞⢳⡟⠂⠒⠙⡄
⠀⢀⡤⠚⠣⣠⠎⢸⡸⣿⣄⠀⠰⢅⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢉⡦⠤⣶⣤⣸⣀⡼⣿⠁⠀⠀⠈⢄
⡴⠉⠀⠀⠘⠀⠉⠙⠀⠈⠾⢢⠃⠀⠙⠈⠂⢄⣀⣀⣀⣤⡄⠘⠋⠀⢱⠈⣹⠛⠼⠛⠒⠛⡆⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠒⠀⠤⠴⠒⠒⠋⢀⣀⣀⣠⡄⣾⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡍⠉⠋⠋⠉⠑⠤⢄⣀⠠⡄⠀⠰⠤⠤⢤⡤⠖⠒⠀⣀⠀⠀⠈⢻⣷⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠉⠀⢀⣤⣇⡀⠀⠀⠀⠀⠈⠰⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡴⠛⠁⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⣤⣀⣀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⣿⣄⠀⠀⠈⠸⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⠀⢹⠀⠙⠢⢄⠀⢂⡤⠆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣤⢒⡟⣉⣧⣄
⠀⠀⠀⠀⢀⡠⠚⠁⠃⠀⠀⠙⠁⠀⠀⠀⠉⠺⠇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⡨⢠⣃⣤⠟⠫⠄⡇⠝⠦⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠑⠳⢦⣀⣀⣀⣸⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣀⣀⡀⣤⠄⠠⠤⠤⠤⠤⣤⡄⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⡤⠤⠤⠤⠤⠤⠤⠤⠄⢀⣈⢙⡛⠉⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣦⣄⡈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⣤⣄⣀⣴⣾⡥⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡽⡩⠷⠢⣄⣽⠢⣄⣠⣤
⠀⠀⠀⠀⠀⢸⣿⣋⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠂⡠⠃⠀⠀⠀⠙⠶⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⡈⡨⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷
⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠄
⣼⣿⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⣴
⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣿
⡼⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣇
⢹⣿⡏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡏
⠈⣿⣷⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⢴⣻⣿⣆⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋
⠈⣿⣿⣿⣾⣹⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡟⠁
⠚⣿⢹⣿⣿⡟⣧⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠣⠻⡇
⠈⢻⡆⣿⣿⣼⡧⢿⡟⡦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣿⠚⠁
⠀⠀⡜⡘⣿⠿⣿⣮⡓⢇⡀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⡡⣸⣯⣏⣴
⠀⠀⠰⡰⡘⠞⣿⣿⣿⣦⣕⣤⣀⠀⠉⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣠⢴⣾⣿⡴⣪⣴⣿⢏⢨⠇
⠀⠀⠀⢱⣹⣆⡿⣿⣿⣿⣿⠣⣈⠓⠦⣄⡀⠀⠈⠰⣉⡲⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⢠⣼⣿⣿⠼⢛⣭⣾⣿⣿⣏⣞⣽
⠀⠀⠀⠀⠻⣾⣿⡛⣿⣿⣿⣷⣼⣷⠦⢼⣯⣍⡐⢾⣿⣷⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⣁⣿⣿⣇⠛⢿⣫⣵⣾⣿⣿⣿⡿⡫⣻⠟⠏
⠀⠀⠀⠀⢀⢇⣿⣿⣮⡻⢿⣿⡟⡏⣷⣶⣿⣿⣿⣿⠶⣿⣶⣬⣭⣥⣤⣤⣀⣀⣠⣤⣤⣭⣭⣶⣶⠾⢓⣿⣿⣿⣿⣿⡿⣫⠎⢠⣿⣻⠇
⠀⠀⠀⠀⠀⠺⠛⠛⢧⡏⠦⢹⠶⡇⠿⣹⣿⣿⣿⣟⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢨⣿⣿⣿⠟⣩⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠛⣷⣼⡻⢿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⡹⢟⣩⣴⡏⣿⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢷⠉⡄⠁⠂⠭⡛⣰⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⣛⣹⢂⣻⣿⢣⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡟⠐⠠⢀⣑⢛⡀⠀⠒⠒⠒⠒⠀⠀⠀⠒⠀⠀⡏⣿⣿⣟⣹⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣻⠗⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠐⠛⠋⠙⢦⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⡑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢰⣆⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣤⡀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⡿⠯⠽⠿⠛⠿⠿⠿⠿⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠋⠁⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣬⡙⠛⠿⣿⡷⣶⢶⣦⣤⣤⣤⣤⣤⣤⠴⠶⠚⠛⠉⠁⠀⣡⣾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣶⣄⣒⣘⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠿⣿⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡏⢹⠛⠀⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⢿⣿⣿⣿⣿⣿⣗⡋⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠘⣿⣿⣿⣿⣟⡺⠀⠀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⢻⡿⣿⡏⠀⠀⠄⠸⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣟⠃⠀⠀⠀⠸⣿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣹⡇⠊⠀⣿⣍⠐⠂⠠⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⠀⠀⠀⢀⣿⠀⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠀⠀⠀⠘⣿⣷⢶⣤⠄⠈⣿⣿⠁⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⣀⡴⠛⡅⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠀⠸⣤⣉⠛⠿⣾⣤⣽⣧⣄⣀⣀⣀⣀⣀⣀⣹⣧⡴⠶⠛⠉⢀⣴⠇⠀⠀⠀⠀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⠷⣧⣀⡑⠈⠉⠙⠩⠍⢉⡉⠉⠉⠉⠀⠀⠀⢀⣤⠶⠛⠁⠀⠀⠀⣠⢄⢳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠶⠤⠤⠶⠴⠶⠶⠖⠒⠋⠉⠀⠀⠀⠀⠀⣠⣾⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⠦⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠃⢹⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢧⠠⢀⣠⣤⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣄⣰⣷⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⢠⡏⢤⡟⠀⠀⠀⠀⠀⠀⢀⣴⠞⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⠟⠀⠀⠀⠀⠀⢀⡴⢋⢱⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⠏⠀⠀⠀⠀⣠⡶⢋⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⠃⠀⠀⠀⣠⡞⣩⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡟⣼⣇⣀⣀⣴⢞⣥⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠞⠛⠋⣉⣉⣉⣥⣤⣤⣤⣤⣤⣿⣾⣥⣤⡽⣿⡷⠛⠛⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⣤⡶⠚⠋⠉⠉⠀⠀⠀⠀⠀⢸⡋⣸⢧⣟⣁⡼⠋⠉⠛⣳⠶⣤⣈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠳⠶⣤⣤⣀⣀⣀⣀⣀⣈⣙⣁⣀⣉⣭⣠⣆⣼⣧⡿⠶⠋⠁⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣤⣄⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣴⣿⠞⣩⡽⢛⡿⠟⣶⠶⢶⡶⣦⠀⢠⡶⢶⡶⠖⣲⠟⢛⡿⠋⣹⢿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢟⣥⠞⣩⡴⢋⣠⠞⢁⡴⠋⢀⡽⠦⠾⣣⠟⢁⡼⠋⢠⡞⢁⡼⠃⢠⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢋⣠⠞⢋⣴⠟⢁⡴⠋⢀⣴⠋⠀⣠⡞⠁⣠⠟⠁⣰⠏⣠⡟⠁⣰⠏⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⡴⠋⣠⡴⠋⢀⣴⠟⠁⣠⡾⠋⢠⡾⠃⢀⡾⢁⣴⠋⢀⡼⠃⣰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣿⣠⠾⠋⢀⡴⠟⠁⣠⡾⠋⢀⣴⡋⢀⣴⠏⣠⠞⠁⢠⡞⢁⣼⡿⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠉⠛⠲⠶⣯⣄⣠⣾⡋⢀⣴⣿⠉⢻⡞⢁⣼⠋⣀⣴⣏⣤⡾⠷⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣉⠉⠉⠙⠛⠋⠀⠈⠛⠋⠉⠉⠉⢉⣁⣀⣠⣤⠴⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⡄⠀⣠⠀⠀⣀⠀⢀⡀⠀⢀⠀⠀⣀⠀⢀⣄⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣡⡾⠋⣠⠾⠋⣠⠞⢁⣴⠏⢀⡴⠋⢀⡾⠃⣠⠞⠁⣠⡞⠁⣠⠟⢁⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠚⠁⠀⠘⠁⠀⠚⠁⠀⠛⠁⠐⠋⠀⠘⠃⠀⠀⠋⠀⠈⠁⠀⠈⠁

⠀⠀⠀⠀⠀⣻⠜⣾⠹⣭⠄⢸⡇⡿⢛⣯⠀⡿⠃⠷⡾⢾⣥⠀⣏⡇⡽⡇⡿⠀⢿⠗⣯⡗⡿⠀⢸⢸⠟⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠸⣉⡉⠒⠦⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠤⢍⣑⡒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠭⠽⠖⠲⠭⢕⣒⠶⣒⠶⣒⠒⠒⠲⠤⠤⣀⣀⠀⠀⠀⢀⣀⡠⠤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⢀⣀⣀⡠⠤⠤⠤⠭⠷⠤⠼⠤⢤⣀⣀⣀⡠⠤⠛⢒⣉⣡⠤⠔⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⡠⠤⣤⠤⣒⣒⡭⠥⠖⠒⠚⠛⠥⣀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⣈⠉⢓⣶⠤⠤⣄⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⠤⠤⣴⣒⠋⣁⠴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⢳⢼⠧⠤⢄⣀⣀⣀⣀⣜⢳⣀⣀⣀⣀⣠⠤⠤⠷⡞⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠲⢄⡀⢸⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣠⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡈⢟⣹⠶⠤⠤⠤⣀⣀⣀⣸⣽⣀⣀⣀⠤⠤⠤⠶⣏⣹⢉⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⠤⠤⢄⣀⣀⣳⣾⣀⣀⡠⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⢀⠀⠄⠀⠀⢤
⠀⠀⠈⠀⠢⡀⠀⠀⠒⠀⠀⠩⢂⠐
⠀⠀⠀⠀⠀⠈⠙⠋⣩⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⠒⢲⣾⣿⣿⡟⢻⡆⠠⢤⣶⣶⣾⡏⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣌⠀⠀⢸⣿⣿⠛⡇⠘⢓⣢⠄⠠⠻⣿⡇⡆⣿⣿⠀⠀⠀⠀⢠⣶⣒⣚⠡⠤⢤⣼⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹⠀⠀⠈⣿⣿⠀⡿⢸⣾⠁⣾⢠⣿⣿⣇⣧⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⢀⣀⣸⡿⣧⠿⠼⢿⡇⣿⣠⣿⠿⢟⣚⣾⡿⢃⣀⣀⣀⣀⣀⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢸⡀⣀⣈⣃⣈⡑⠛⠒⠚⠓⠚⠉⠀⣿⣈⣯⡉⠀⠀⢸⠷⣷⡶⠶⠶⠶⢾⢿⣿⣾⡇
⠀⠀⠀⢸⣿⣿⡟⣿⢉⡏⠩⢭⣭⣿⡇⠀⠀⣠⠤⠒⠒⢚⡷⠞⠛⠛⠒⠲⢤⡀⣿⡇⠀⠀⣇⠀⣾⣿⣿⡇
⠀⠀⠀⢸⣿⣿⡇⣿⢸⣇⣸⠘⠻⣿⣇⣴⣯⣄⠮⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⢿⣻⣻⠟⠛⠋⢉⡧
⠀⠀⠀⢸⣿⣿⡇⢿⢸⣿⣿⣰⠀⠏⣿⣿⣿⣿⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡅⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣿⣿⣷⠀⢸⣿⣷⣼⡆⣸⣿⣿⣿⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⢀⠴⡷⣄
⠀⠀⠀⠈⣿⣿⣿⣀⣸⣿⣿⣿⣇⣿⣿⣿⣿⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⠀⢠⠏⠀⡇⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⣰⠯⢤⣀⣇⣠⠼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢸⣷⠃⠀⠀⠈⡇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⡀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡼⠁⠀⠀⠀⣴⢃⡄⠀⠀⠈⣆
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠤⣀⣀⡠⠤⠖⠋⠀⠀⠈⠁⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⣯⣭⣄⡀⠈⣉⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠋⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠉⠁⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣄⠀⠀⠀⠀⠀⠀⠾⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⠦⠀⠈⠁⠀⢀⢠⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤⣀⠰⠊⠀⡈⠀⠀⠀⣀⣴⠋⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠈⠀⢺⣿⠟⠃⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡠⠐⣠⣧⣤⡀⠀⠀⠈⠹⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠢⠄⢁⣈⣈⣓⡀⠀⡐⠀⠞⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⠉⠀⢸⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣻⣤⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢹⡄⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠢⢄⡀⠀⠀⠀⢀⠤⠀⠠⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢀⣿⡄⠀⠀⢀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠊⡤⠠⡴⣄⣀⠈⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢾⡏⠈⠳⡀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠄⡤⣤⣖⠊⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⡟⠀⠀⠀⠈⣤⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠂⡇⠀⠀⠀⠀⠀⠈⠒⠀⠀⢸⡆⢠⡇⢀⠀⡠⠂⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⣇⠈⣟⢠⠞⠁⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⣤⡄⠠⠤⠤⠤⢤⢤⠠⣤⠀⣀⡀⣀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⡷⠈⠀⠀⠀⣠⠞⠁
⠀⠀⢀⣠⠶⠚⠃⠈⠘⣿⣿⣿⣿⣿⣷⡦⢰⣶⣶⣶⣶⣶⠿⠿⠈⠉⠉⠘⠳⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣏⣸⣁⠀⢀⡴⠋⠁
⠀⠀⣿⣄⢴⣿⣿⣿⣿⣿⡮⠁⠀⢉⣀⣄⣤⣴⣦⠀⠀⢠⣾⣿⣿⣿⣷⡆⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⢸⠿⡇⠈⣿⣧
⠀⠀⢿⠉⠛⠻⠯⣭⣍⣁⣀⣻⣛⠛⠛⠛⠉⠁⠁⡀⢀⣀⣉⣉⣉⡭⣥⣐⣒⣭⢠⡇⠀⠀⠀⠀⠀⠀⠈⠁⠛⠻⠟⢻⡄
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⢉⣤⣶⣾⣿⣿⣿⡟⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⣱⣿⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣤⣤⣾⣿⣿⣿⡟⣵⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠳⣔⢤⣀⡀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⢟⡳⠋⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣛⣷⣶⣶⣤⣴⣾⣿⣿⣿⣿⣿⣟⣫⠕⠋⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠛⠛⠻⣿⠟⠛⠋⢻⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣶⣞⣉⠉⠁⠀⠐⠲⡾⣿⡀⠀⠀⠀⠀⣄⣀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⡛⠿⣿⢿⣿⣦⣤⣤⣤⣴⣾⠿⣧⠀⠀⠀⠀⠀⢸⣿⣧⣤⣄⣀⠀⠀⠀⠀⠀⢁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠙⠢⢿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢹⣇⠀⠀⠀⠀⠈⣏⠹⠿⠿⣿⣷⣶⣶⣶⣤⠾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣿⠀⠀⠀⠈⢿⠛⠙⠛⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⢸⡀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠉
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢀⣤⣤⠠⣄⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠂⣴⣿⣟⣿⣷⠈⠉⠀⠲⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄
⠀⠀⠀⠀⣸⣷⠂⠀⠀⠲⠿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠅
⠀⣀⠤⠚⢿⠾⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶
⣾⣁⣤⣤⡶⠶⢶⡿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠉⠳⠠⢤⣤⣬⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠺⠦⠂⠀⠂
⠀⠀⠀⠀⠀⠈⠙⠻⣷⡀⢸⡷⢦⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣦⡰⣬⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠿⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣀⣀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣦⡦⣀⣴⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⡎⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢂⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠁⠀⠠⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠈⢉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣳⣦⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⢡⣿⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡦⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡟⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⢤⣤⣶⡾⠇⠀⢿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣴⣯⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⡉⢩⣤⠴⠶⠦⢤⣤⣴⣦⡴⠶⠚⠿⢿⣦⡀⢀⣄⣴⣶⣶⣶⡾⠿⠿⢿⣯⡉⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠻⠿⣷⠀⠀⠀⠀⠀⠻⠋⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠙⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⢀⣴⠾⠿⠆⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⣰⣟⠋⠀⠀⠀⠀⣿⡇⠀⠀⠀⠰⠂⠀⠀⠀⠀⢸⡇⠀⣠⡄⠀⣤⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠈⠁⠀⣿⠀⠀⠀⠀⠀⠀⣿⣡⣴⢿⣧⢸⡆⢀⣴⠿⠷⢸⣷⣟⠋⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠀⣿⡠⠀⠀⠀⢀⠀⣿⣿⠃⢸⣿⢸⡇⣾⣋⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠃⠀⠘⠳⣧⣴⣾⠟⠃⢸⡟⠀⠸⣿⠸⡷⠘⢷⣶⠆⢸⣿⠙⢷⣄⡀⠙⠻⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠒⢶⣴⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⣀⣸⠁⢈⣀⣉⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡝⢁⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠁⠀⠀⠀⠀⢱⡞⠁⠂⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⡾⡯⠀⠻⠛⡳⠆⠉⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠤⠤⢼⡇⠀⠐⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⡴⣿⣅⣀⣀⣠⢤⡶⠞⣍⠉⠉⠛⠻⢲⡧⣭⠤⠀⣀⣐⣒⣒⣤⣤⡄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣸⠀⠀⢠⠀⠀⡇⠀⣿⠀⣀⠄⠀⠨⡇⠀⠀⢨⠀⠀⠀⠀⠉⢹⠇⢠⣼⣃⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄⠀⢸⠀⢠⡅⠀⢉⣀⣻⣀⣟⢸⠋⠉⠀⣸⠀⠀⠀⠀⡇⢸⡧⠤⠼⡧⠀⠀⢒⣛⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢸⠀⠀⢠⡆⠀⠙⠿⣷⠿⢿⣿⣂⣀⣀⣤⣤⣀⣠⣤⣼⠁⠀⠀⣇⠀⠀⠁⢘⡂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⣄⠀⢸⣇⡀⠀⠀⣇⢀⣀⣘⣀⣠⡇⠀⠀⠉⠉⠙⠁⠉⠀⠈⡬⠒⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡄⢀⡾⡿⡄⠀⠉⠙⠛⣸⢿⡍⠉⠁⠀⡨⠙⠓⣀⠀⠀⢠⠀⠆⠀⠀⡇⠀⠀⠀⢀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣼⣁⣧⣨⣆⣀⡀⠀⠑⠚⠷⣂⠀⠀⠀⠀⠀⠙⢿⣆⠰⡄⣇⠀⢀⣟⠀⠀⠀⢸⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠋⢉⣿⡷⠾⠟⠯⠭⣤⣀⠈⠁⠀⠀⠀⠀⠀⣠⠆⢻⣇⠢⣿⡀⢈⣉⣀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢰⡿⠉⠀⠀⠀⠀⠀⠀⠉⠙⢦⡑⣄⢀⡤⠀⠀⠀⠈⣿⡄⠬⠀⠈⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⠹⣯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⠎⠀⠀⠀⠀⠀⡿⠃⠀⠀⠴⡆⢀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠍⢐⠃⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢃⠀⣀⠴⢁⡟⠀⠂⣠⣘⣻⠈⠉⠁⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⡄⠉⠉⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⡀⠉⠀⠈⠃⢠⠚⠁⡹⣷⠤⠧⠂⢀⡀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡾⠏⠀⢠⠿⠛⠒⠶⠫⠾⠒⠀⠀⠉⠓⠤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃⢀⠴⠁⠀⠸⠆⠀⠀⢠⣦⠀⠀⠀⠀⠀⠈⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠱⢿⣶⠀⠀⠀⡀⡀⢀⣴⠟⠉⢀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⠉⠐⠈⠛⠊⠈⠀⠀⠀⠘⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡀




⣿⣟⣛⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⣤⡶⠾⠽⠟⠛⢟⠛⠛⠛⠛⠛⠻⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠳⢄⡀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣧⡀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠣⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⡿⣶⣬⣔⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⢟⡽⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠷⢬⣉⣛⠛⠻⠿⠽⢻⠽⠿⠛⣓⣛⣯⠿⡟⠉⠚⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠆⠀⠀⠉⠉⠉⠛⢛⠚⠛⠉⠉⠉⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⣃⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣟⡲⠬⣤⣀⣀⡀⣠⣅⠀⢀⣀⣀⡶⠶⢾⣭⠧⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠿⠿⢷⡾⠭⠭⠥⠔⠒⠊⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠲⠞⢒⣀⣈⢁⡠⣫⡑⠓⠐⠶⠦⠄⡀⠀⠀⢀⡀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣉⠤⠐⠊⠉⠁⠀⢀⡴⠚⠁⠀⠀⠀⠉⠀⢐⠢⠄⣉⠨⡔⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡗⠉⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⣀⣠⠤⠖⠒⠉⠁⠀⠀⠀⠑⢌⣶⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡼⠑⠠⣀⠀⢀⣤⣞⡥⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⡂
⠀⠀⠀⠀⠀⠀⠀⢸⣗⠢⣀⠀⠉⠛⠁⣽⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠠⠔⠊⠁⢀⠿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠑⠢⢍⡒⠠⠤⣭⡟⠀⠀⠀⠉⠉⠉⠉⠁⠸⣍⡇⣀⠠⠔⡊⠕⢣
⠀⠀⠀⠀⠀⠀⢀⠛⡇⠀⠀⠀⠀⠉⠒⣾⠤⠌⢉⡉⡀⢀⣀⠀⠈⠉⣹⠥⠐⠂⠉⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⢐⠀⡆
⠀⠀⠀⠀⠀⠀⠰⣌⠆⠀⠀⠀⠀⠀⡸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡆⠀⠀⠀⠀⡈⢠
⠀⠀⠀⠀⠀⠀⠀⣝⠀⠀⠀⠀⠀⠀⡇⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡇⠀⠀⠀⠀⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠹⢍⠂⠄⡀⠀⠀⢣⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡦⠃⠀⠀⢀⠄⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣁⠒⢤⣤⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣇⠠⠐⢊⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠂⠠⠤⠤⠀⠀⠀⠠⠄⠐⠒⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠤⣤⣤⣤⣤⣤⣶⠶⠚⠛⢛⢛⠛⡛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠩⠅⠂⢠⡀⠰⠀⡰⠄⠀⠀⠀⣨⠑⠀⢺⠨⠰⠃⡆⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⡋⠁⠀⠀⠂⠄⠂⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠹⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣞⡿⠟⠻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢝⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡘⡄⠀⠀⠀⢰⢱⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢎⡓⠂⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠃⠀⠀⠀⡇⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⡜⠀⠀⠀⠀⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠶⡌⠅⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠠⠃⡄⠀⠁⢠⣧⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠞⠄⠀⠀⠀⠰⠀⠀⠈⢊⣾⠟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠀⠀⠀⠀⠂⠀⠀⠂⠖⢠⣿⡿⠛⠻⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠠⠐⠉⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⡆⣿⡿⠋⠀⠀⠀⠸⣿⣀⠀⣤⠦⠴⠤⢀⡥⠖⣊⠁⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⠠⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠹⢿⣾⣡⣼⠿⢋⠥⠀⠈⠁⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠁⣽⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣤⣀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⣕⣲⣶⣶⣶⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢠⡟⠀⠀⣴⢺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣠⣿⣶⡯⠤⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⣼⠛⣛⠁⠀⠁⣁⠺⣿⡇
⠀⠀⠀⠀⠘⣧⠀⠃⠀⢺⣶⡆⣀⣽⡗
⠀⠀⠀⠀⠀⠈⢳⣶⡶⣄⢉⡀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣤⣭⣠⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣶⣿⣿⣶⡀⠀⠀⠀⢀⣾⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡿⠋⠉⠉⠃⠀⠀⢀⣼⡇⠀⠄⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⢠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⣄⠀⠀⠀⣀⣤⡀⠐⠚⢡⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣷⣦⣴⣾⣿⣶⣸⣿⣿⣿⣷⡶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠉⠉⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⣁⠀⣤⠄⠈⠉⠻⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡀⠀⠠⠀⠈⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⡀⠸⠃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴⣿⣿⡏⠋⠀⠐⠖⠂⣹⣿⣿⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡀⡀⠀⠀⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡔⠀⠀⠀⢀⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡤⠤⠸⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠙⠶⠚⠉⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢀⣾⣿⣿⣿⣷⣦⣀⣀⠀⣀⣼⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠘⠻⠿⠿⠿⡟⢿⢿⣬⣦⣯⣯⣧⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⢆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠑⠀⠂⠀⠀⠀⠀⠐⠒⠀⢂⣤⣶⣶⣶⣦⣀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⢿⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠁⠈⠉⠛⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠀⢶⠀⠰⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⣾⣿⡿⠦⠀⠀⣀⡀⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣧⠀⠀⠀⠁⢀⣼⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠈⠁⠀⠀⠀⠀⣀⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣆⠐⠀⢀⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠈⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢿⣋⣠⣿⣇⠄⠀⠹⠯⠉⠯⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡀⠀⠀⠀⠀⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣄⡀⠀⠉⠁⠀⢉⣷⣦⣤⡄
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣦⣤⣀⢀⣴⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⠛⣛⡛⢟⡟⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠉⠙⠛⠘⠯⠿⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⣀⣤⠤⠴⠶⠶⠶⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠶⠶⠶⠤⠤⣤⣀
⠀⠀⠠⣽⡉⠀⠀⣠⢀⣀⠤⢾⣥⡤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣬⡳⠤⣀⡀⣄⠀⠀⢉⣿
⠀⠀⠀⠀⠈⢉⡟⠁⠀⠀⠀⡌⠀⠀⠀⠸⠉⠙⠓⢦⡀⢠⡴⠚⠋⠉⡅⠀⠀⠀⢡⠀⠀⠀⠈⢻⡉⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠤⠄⠀⣀⡇⠀⠀⣾⣆⠃⠘⣸⡇⠀⠀⢰⣀⠀⠠⠬⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡸⡄⠀⡘⠛⠛⠒⢦⣤⣀⠈⠁⠒⣯⡟⠀⠀⢯⡵⠒⠉⠀⣀⣤⡴⠚⠛⠛⢂⠀⢠⢇
⠀⠀⠀⠀⠀⡇⡇⠐⣃⠐⠂⠠⢎⡀⠈⠙⢯⡀⢻⡁⠀⠀⢸⡇⢀⡽⠋⠁⢀⡡⠄⠐⠂⡸⠂⢸⢸
⠀⠀⠀⠀⠀⢹⠁⠀⠀⠑⠂⢴⣆⡈⠑⠢⡜⠀⢸⡃⠀⠀⢸⡇⠀⢢⠔⠊⢁⣰⡦⠐⠈⠀⠀⠈⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠀⣠⠞⠛⠉⠉⠙⠳⣤⡀⠉⠉⠁⠘⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡜⡆⠀⠀⠀⠀⢸⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠸⡇⠀⠀⠀⠀⠰⢣
⠀⠀⠀⢀⡴⢳⠷⣄⠀⢀⡴⢿⠷⣄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⢶⡿⣆⠀⠀⣠⠾⡞⢦⡀
⠀⢀⡴⠋⠀⠘⠀⠈⣷⡞⠁⠘⠃⠈⢳⣄⠀⠙⢦⣄⣀⣀⣀⡤⠞⠁⣠⠞⠁⠘⠃⠈⢳⣾⠁⠀⠃⠀⠙⢦⡀
⠀⠈⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡎⠀⠀⠀⠀⡠⡶⠀⠀⠞⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠙⠢⡀⢀⡚⠰⠀⠀⠀⠳⡀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⣙⣈⣅⠀⠦⣀⣀⣀⣈⣱⢈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⣶⣞⣻⢿⣻⢿⣿⢿⣯⡿⣯⡷⣦⢸⣯⣽⢿⣿⢿⣽⢿⣻⣿⣒⣶⡶⢤⣤⣄⣀
⠀⠀⠀⠀⢀⠔⣺⣿⡟⣽⡿⣱⣿⢣⣿⢇⣿⡟⣾⣿⣸⣿⢸⣿⡇⣿⡞⣿⣿⢸⣿⠸⣿⡏⣿⡇⣿⣷⢹⣿⡹⣾⡟⢿⡐⢄
⠀⠀⠀⠀⠸⣄⠙⠻⢼⣿⣱⣿⢇⣿⡟⣼⣿⢱⣿⡇⣿⣿⣸⣿⢧⣿⣧⣿⣿⢸⣿⡇⣿⣷⢻⣿⡸⣿⣇⢿⣷⣹⠿⠞⢁⡰
⠀⠀⣀⡠⠤⠌⣿⣶⣤⣤⣉⣉⡙⠛⠓⠛⠓⠿⠿⠼⠿⠧⠿⠿⠼⠿⠯⠿⠿⠼⠿⠷⠿⠿⠚⠛⠓⠛⣉⣉⣡⣤⣴⣾⡯⠤⢄⣀
⢀⣾⣿⠇⠉⢁⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⠸⣿⣷
⣾⣿⣛⠩⠭⠀⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠭⠍⣛⣿
⠉⠉⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠉⡟⠛⠛⠛⠿⠿⠿⠿⠿⣷⠙⠋⢸⠟⠿⠋⠋⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⡏⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠹⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢡⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣈⠅
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣴⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡠⠊
⠓⠤⢀⣀⣈⠁⠒⠒⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠂⠤⠬⠅⣐⠂⠀⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⠨⠀⠀⠀⠀⠀⠀⡰⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠁⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣀⡤⠴⠖⠚⠋⠉⠉⠉⠉⠉⠉⣩⠝⣴⣓⣒⠦⢤⣀⠀⠀⠀⠙⠁
⠀⠀⠘⠛⠒⠀⠀⠀⢀⡔⢋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⠊⠀⠀⠈⠉⠲⢬⡑⠄⠀⠀⠒⠚⠃
⠀⢠⣤⣄⣀⡀⠀⢀⡎⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠊⠀⠀⠀⠀⠀⠀⢀⡷⢸⠀⠀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⣁⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠤⢖⣚⡩⠔⢻
⠀⠛⠛⠒⠒⠀⠀⠈⠳⣄⣀⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠁⣠⣤⠚⠀⠀⠐⠒⠚⠛
⠀⠀⠀⣀⣀⣀⠀⠀⠀⡇⢏⠉⠑⠒⠒⠀⢠⠤⠤⠄⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⢠⣿⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠛⠉⠀⠀⠀⠀⡟⡈⡄⠀⠀⠀⠀⡎⢠⡄⢃⠀⠀⠀⠀⡸⠀⡄⢡⠀⠀⢸⣿⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠲⠋⠀⠀⡇⢱⠸⠀⠀⠀⢸⠁⡏⢣⠸⡀⠀⠀⢀⠇⡸⠹⠈⠄⠀⢸⢻⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢇⠀⠀⡇⡸⠀⠘⡄⣇⠀⠀⡜⢰⠁⠀⢣⢸⡀⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡸⡀⢸⢡⠇⠀⠀⢣⢸⡀⢠⠁⠇⠀⠀⠘⡄⢧⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⢇⢧⡎⡌⠀⠀⠀⠸⡄⣇⠎⡸⠀⠀⠀⠀⢱⠘⢀⡽⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡧⣍⡒⠤⠼⣌⣰⣃⣀⣀⡀⠀⣳⠈⣠⣃⣀⣀⣀⡤⠴⠓⣉⡤⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣈⣁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠂⢀⣉⡥⠖⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣷⣶⣶⣶⣶⣦⣤⣶⣶⣶⣶⣶⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠚⠉⠉⠉⠉⠉⠓⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀
⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⡔⠉⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀
⢀⡤⠒⠉⠉⢢⠀⠀⠀⠀⢧⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⢏⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠉⠉⢲⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢲⡀
⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠚⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣱⠄⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⠉⠁⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠸⡒⠤⠄⡀⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠔⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠒⠲⠦⠶⠶⠒⠒⠒⡋⠉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡜⠀⠀⠀⠀⠀⢠⠁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠊⠀⢸⠀⠀⠀⢀⣀⠴⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠪⠭⢄⡤⠄⠀⢀⠤⣬⠖⠊⠉⣉⠄⠀⢀⠤⠤⠑⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠥⠤⠔⠊⠁⠀⠋⠉⣠⢋⣁⡠⠖⠁

⢿⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿
⠀⠀⠈⠙⠛⠿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠟⠋⢁⣀⣤⣶⣾⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⡿⠿⠛⠉⣁⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⣀⣠⣴⣶⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣷⣶⣤⣀⡀
⣾⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢉⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣋⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢋⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢋⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⡴⠋⣡⣴⣿⣅⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠛⢋⣭⣭⣤⣄⡴⠛⢁⣤⠞⠉⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⣻⡶⠛⠁⣠⣾⣉⡀
⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠀⠀⠀⠀⣴⠋⠀⠀⠀⠘⣷⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⠞⠁⣀⣴⠟⠁⠀⠉⠉⠛⠶⢦⣄⡀
⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠙⣷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣿
⣿⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⣿
⣿⣆⠈⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⠛⠁⠀⠀⢀⣴⣿
⣿⠈⠳⣄⠀⠈⠉⠛⠳⠶⠶⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡴⠶⠞⠛⠋⠉⠀⠀⠀⠀⢀⣠⠴⠋⢸⣿
⢸⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠉⠀⠀⠀⡾⣿
⢸⠀⠀⠀⢀⡏⣇⠉⠛⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠚⠋⠍⢦⠀⠀⠀⠀⠀⠀⢠⡇⡿
⢸⠀⠀⠀⣼⠀⢸⡀⠀⠀⠀⠀⠀⠉⠉⠉⣹⢻⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⢯⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣆⠀⠀⠀⠀⠀⢸⢰⡇
⢸⡄⠀⢰⠇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣰⠃⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⣆⠀⠀⠀⠀⡟⢸⡇
⢸⡇⠀⡟⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢹⡄⠀⠀⠀⡇⢸⡇
⠸⡇⢸⠃⠀⠀⠀⠈⣇⠀⠀⠀⠀⣰⠃⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢳⠀⠀⢸⡇⢸⠇
⠀⣇⡏⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠏⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⠁⣿
⠀⣿⣦⠀⠀⠀⠀⠀⠘⡇⠀⢀⡞⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⠀⣿
⠀⢻⡌⠳⢤⡀⠀⠀⠀⢹⠀⡼⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡿⢀⡿
⠀⠈⢳⣄⠀⠙⠲⣄⡀⠘⢀⠇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⣻⠇
⠀⠀⠀⠈⠳⢤⣀⠀⠙⠓⠾⣤⣀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⡏⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠈⠙⠓⠶⠦⣤⣀⣀⣈⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣼⣁⣤⡤⠴⠶⠚⠉⠁⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠒⠒⠓⠒⠒⠒⠚⠛⠛⠛⠉⠉⠁⠀⢀⣀⣠⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠲⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠖⠛⠛⠋⠉⠁

⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣶⣾⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠿⣿⣿⣿⡿⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⠴⢿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡦⡷⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⡷⠀⠀⠀⣿⡂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⡷⢶⣶⣶⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠛⠛⠿⣯⠀⡇⠀⠀⠀⣿⣿⣿⠿⠟⠛⠛⠿⣧⠀⡇⠀⠀⠀⣽⠋⠁
⠀⢠⣶⣿⣁⣀⣀⣿⣿⣿⣷⣾⣿⣶⣶⣿⣤⣷⣶⣶⣴⣿⣿⣿⣷⣾⣿⣶⣶⣷⣤⡷⠀⠀⢠⠿
⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⠿⠟⠛⠉⠈⠁⠉⠉⠛⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠺
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⢟⠝⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠈⠪⣻⣿⣿⣿⠿⠿⠿⠿⢇⡀⢸
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⠃⠀⠀⠀⠀⢀⣴⣿⣶⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣶⣶⣶⣾⣧⡼
⠀⣿⣿⢿⠛⠛⠛⠛⢻⣿⠃⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰⣿⡉⠉⢹⡟⠉⠉⣿
⠠⣿⣿⣿⣷⣷⣿⣷⣾⡇⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠰⣿⡇⠀⢾⡷⠀⡰⡿⣆
⠀⠙⠿⣿⣿⣿⣿⡿⢻⣿⡄⠀⠀⠀⠀⠀⠘⠠⣟⣿⠠⠇⠀⠀⠀⠀⠀⢠⣿⡧⡀⣸⣧⠞⠁⠿⠈⠳
⠀⠀⠀⠀⠀⠈⠉⡄⠀⢻⣶⡀⠀⠀⠀⠀⠀⠐⠿⠿⠀⠀⠀⠀⠀⠀⢀⣷⡟⢈⣮⠛⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⣞⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣵⠟⣴⠟⢉⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠀⠈⠻⣷⣢⠄⣀⠀⠀⠀⠀⢀⢠⣰⣽⠟⠁⠚⠋⠀⣰⣦⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⠾⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣤⣤⣤⣤⣄⠀⠀⣀⡤⠖⠋⣡⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡛⠿⠝⠹⡏⠀⠈⠉⠀⣿⠒⠉⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⢿⠈⠀⠀⢧⠀⠀⠀⢧⣀⣀⡀⠀⣿⠤⢚⣁⣀⣀⣀⡠⠤⠤⠤⠤⠤⠤⠴⠊⠙⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⡆⠀⠀⠸⡆⠀⣠⠞⠉⠉⠙⢷⡏⠉⠁⠀⢀⣀⣀⡠⠤⠤⠔⠒⠚⠉⠉⠉⠉
⠀⠀⢀⣀⣀⣀⠤⠤⠤⠴⠒⠒⠒⠊⠉⠹⡄⣀⢻⣞⣿⠖⠓⠛⡇⠀⠀⠀⠀⠘⣗⠉⠉⠉⠁
⠀⠀⢎⠀⢀⣠⣀⣀⠤⠤⠤⠤⠔⠒⣒⠶⠛⠷⠼⣿⣯⡦⠖⠋⢁⡴⢾⡆⠀⠀⣿
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⣀⡤⠒⠉⠈⠉⠉⠛⠛⠋⠀⠀⢠⣄⡴⠋
⠀⠀⠀⠀⠀⠀⢀⡠⠔⡊⠁⢀⡠⠖⠋⠈⠳⣄⠀⠀⠀⢲⡄⠀⠀⠀⠀⡼⠋
⠀⠀⠀⣠⢴⠊⠡⢔⡨⠔⠊⠁⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⡇⠀⠀⠠⢾⠁
⠀⢰⡋⠀⣠⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣷⠀⠀⠀⢸⡀
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠋⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣧⣺⣯⣛⣿⣋⣋⣿⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⢉⡉⠛⠛⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⠉⠀⢠⣦⠀⠀⠀⢸⡇⠀⠀⢀⡀⠉⠻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⢠⣦⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠃⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢠⣄⠘⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⢃⡀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠘⣿⣿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢸⣿⣿⣿⡟⢸⡇⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣏⠀⠀⢻⣿⣿⣿⡇⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⡇⢸⡇⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⢸⣿⣿⣿⣧⢸⡇⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡷⠀⠀⣼⣿⣿⣿⡇⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⡟⠁⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣷⠀⢠⣿⣿⣿⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠸⡗⢠⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⠈⠋⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⣴⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡙⢿⣿⣿⣦⣀⠀⠘⠟⠀⠀⠀⢸⡇⠀⠀⠈⠃⣀⣴⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢿⣿⣿⣿⣶⣤⣤⣀⣀⣈⣁⣤⣤⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢬⣙⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠾⠟⠛⠻⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠉⠉⠉⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣆⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣴⠋⣠⡼
⠀⢀⣤⣶⡶⠾⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠈⠉⠛⠻⣦⣄⣀⣀⣀⣠⣤⣴⡾⢋⡾⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⢀⡾⠀⠟⢡⡞⠉
⠀⠻⣿⣤⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡇⠀⠀⣀⣤⣾⠿⠛⠋⠉⠉⠀⠀⠀⠀⠰⠟⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣡⣶⢻⡇⣀⠀⢸⡇
⠀⠀⠀⠈⠉⢻⣷⣶⠶⠀⣀⣀⣠⡄⠀⠀⢻⣄⠀⢀⣿⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢻⢗⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⣳⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠋⣉⣁⡤⠀⠀⠀⣹⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠵⢂⠀⠀⠀⠀⠈⠋⠀⠀⠚⠋⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣋⣩⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣉⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⣀⣠⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠩⣽⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⠽⢖⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⣿⣧⣼⣷⠀⣤⡀⢀⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠉⡿⠋⠶⠂⢶⣠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣇⣸⡏⢀⣶⠀⣿⠀⣿⠀⢰⡄⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠠⢤⣤⣤⡶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣾⡏⢸⣿⠀⣿⡀⢸⡇⠀⢸
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣀⣿⡇⢸⡇⠀⠈
⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⢢⣶⡤⢀⠀⠀⣼⡿⠁⠈⠙⠛⠿⢾⣷⡀⣰
⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⢟⣵⣿⢏⣼⡟⠁⠀⠀⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠶⣫⣴⣿⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣶⣦⣤⣽⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⣿⡟⠉⠛⠿⠷⣶⣶⣶⣶⣶⣦⣬⣝⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡎⠉⢉⣩⣿⣿⢿⣿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠙⢿⣤⣀⠀⠀⠀⠀⠀⢀⣴⣾⣯⡴⠞⠛⠛⠶⢽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠿⣧⠀⣾⠷⠾⠿⠿⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠁⠌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠁⠀⠀⢠⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣨⡀⠀⠀⠀⠀⠀⠀⡉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠑⠘⣻⠃⠀⣀⠀⢤⣠⣚⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⣒⣒⣚⣙⣟⠍⠓⡒⠒⠒⢶⡦⠤⣤⣤⣤⣠⣤⠖⠚⠍⠀⠀⠀⠀⠀⢘⢁⡴⣈⠭⠬⠽⠿⠭⠒
⠀⠀⠀⠀⠀⠀⡠⣴⡪⢷⢾⡛⠉⠋⢉⣉⠍⠁⠀⠀⠀⠀⠀⡀⠀⠛⠠⠶⠟⠛⠑⠂⠀⠀⠀⠀⠀⣀⣠⢀⣰⡬⠞⠁
⠀⠀⠀⠀⢄⣠⡾⠋⠀⠀⠀⡀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠠⢞⣁⡴⠋⠁
⠀⠀⠀⣠⡟⠉⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⢀⡌⢠⣠⡟⣠⠋
⠀⠀⡆⣯⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡰⢪⡿⡴⠃
⠀⢸⣨⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠘⢁⠀⠀⠄⠀⢬⣾⡵⠁
⢠⢿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⣠⣽⢷⠃
⠘⣾⣷⡔⠄⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠂⠋⠀⠀⠦⠀⣼⡿⣣⠃
⢀⡏⣾⠁⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⢀⣀⣠⣴⣴⣫⠟⠁
⠘⣇⠻⠧⠶⡚⢧⠤⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢟⡽⠟⠉
⠀⠀⠛⠛⣿⣿⡌⣷⡦⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⢾⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠄⡴⡀⠀⠀⢀⢀⡔⡀
⠀⠀⠀⠀⢲⣿⣿⣜⠿⣧⡥⠧⠄⠴⠤⢆⣀⡀⣀⠴⢛⡩⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠞⠸⠱⠁⠎⠨⠾⠈⠕⠁⠾⠸
⠀⠀⠀⠀⠀⠐⠛⣻⣿⣶⣶⣾⣿⠿⠷⠶⠾⠛⠓⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⢀⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠿⠛⣛⣛⣛⣛⠛⠿⢿⣿⣿⣿⣷⣦⣀⣠⡾⠯⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⣩⣴⡶⠿⢛⣿⠟⠛⠛⣻⡿⢶⣬⣙⠻⣿⣿⣿⣯⡀⠀⠙⢽⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢋⣴⡿⠋⢁⣤⡾⠋⠁⣠⡴⢣⠀⢀⣤⡿⠛⢷⣬⠻⣿⣿⣿⣆⠀⢈⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⣡⡿⢋⣠⡶⠛⠁⢀⣴⠞⠋⢀⡏⡶⠟⠁⣰⢰⠾⠛⣷⡌⢿⣿⣿⣷⠛⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⣰⣿⡵⣛⣉⠀⢤⡾⠋⠁⣠⣴⢸⣇⠀⡤⣠⠇⠀⣀⣴⠟⢿⡌⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⢡⣿⠋⢠⡿⠬⣍⣓⣶⡰⠾⠋⠀⣨⢻⡷⠾⢋⣴⠾⠋⠀⣠⣼⣿⠸⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⡄⣾⣉⣓⠲⢬⣿⣷⠀⠴⠟⠉⣾⣠⡾⠋⠁⣀⣴⠟⠋⢀⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⠈⣓⠦⢬⣍⣿⠿⠋⠙⠛⠶⣾⣧⣅⣀⡰⠾⠋⠀⣠⡴⠟⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡘⣿⡟⠉⢀⣠⡶⠂⠀⣀⣴⠟⢡⡟⣠⡭⠛⠿⢿⣿⣷⣮⢀⣰⣿⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠹⣷⡶⠟⠁⢀⣴⠾⠋⠀⢠⣿⡇⠁⢀⣤⠾⠂⠈⣹⡵⢟⣿⢃⣾⣿⣿⠇
⠀⠀⠀⠀⣀⢀⣴⢿⣿⣿⣧⡙⣷⣤⡾⠋⠁⣀⣴⢢⣿⣿⢠⡾⠛⠁⣀⣴⠟⠋⣠⡿⢃⣾⣿⣿⡟
⠀⠀⠀⢺⣮⡻⡁⠀⠻⣿⣿⣿⣌⠻⣷⣤⡞⠋⠀⣘⠿⠋⠈⢀⣤⠾⠋⢁⣤⡾⢛⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠙⣿⣟⣄⠀⠘⣿⣿⣿⣿⣦⣝⠻⠷⣿⣯⣤⣤⣾⣿⣥⣶⠾⢛⣩⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣷⡟⠋⠉⠻⣿⣿⣿⣿⣷⣶⣤⣭⣭⣭⣭⣤⣶⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⣴⣴⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣴⣤⣤⣄⣀
⠀⠀⢷⠐⠸⡆⠀⠀⠀⠀⢀⢄⣦⣷⠿⠟⠛⠉⠉⠁⠀⢠⣶⣶⣶⣶⣶⡀⠀⠈⠉⠉⠛⠻⠿⣾⣖⡄
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠠⣱⠟⠉⠀⠀⠀⠀⠀⢀⣤⣄⠈⠓⠲⠯⠷⠚⠁⣀⣤⣄⠀⠀⠀⠀⠀⠙⢿⡕
⠀⠀⢸⠀⢰⡇⠀⠀⠀⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⣓⣲⡤⠶⣖⣋⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢣
⠀⠀⢸⠀⢸⡇⠀⠀⠀⡼⠷⠀⠀⠀⠀⠀⠀⠀⠲⣶⠞⠛⠉⠀⠀⠀⠀⠈⠉⠛⠻⠷⠂⠀⠀⠀⠀⠠⠿⠶
⠀⠀⢸⠀⢸⡇⠇⠀⠀⣸⣄⡀⠀⠀⢠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡄⠀⠀⢀⣴⡇
⠀⠀⢸⠀⠈⡇⠀⠀⠀⡇⣿⣿⣶⣤⣄⣈⠉⡿⠛⠛⠛⠿⠿⠷⠶⠿⠿⠿⠛⠛⠛⠿⢉⣀⣠⣴⣾⣿⣿⡅
⠀⠀⢸⠀⠀⡇⠀⠀⠀⢁⣿⣿⣿⣿⣿⣿⣿⣟⠷⣶⣶⣶⣶⣦⣤⣴⣶⣶⣶⣶⠾⣿⣿⣿⣿⣿⣿⣿⣇⡇
⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠘⡈⠻⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢂⣿⣿⣿⣿⣿⣿⠟⢡⠁
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠈⠢⡈⠙⠻⢿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⡿⠟⠋⣁⠔⠁
⠀⠀⠸⠄⠐⠧⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⣹⠏⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠹⣇⣠⠤⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠒⠒⠒⠚⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⢿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣫⢟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡫⢋⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡟⡟⠝⡟⢹⣇⠙⢿⣯⠻⣿⣿⣿⣿⣾⣿⣿⡱⢋⢞⡵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡫⢡⠀⠈⠀⠈⠁⠀⢻⣦⡅⢻⣷⣿⣿⣿⣿⣿⣿⣿⣵⣫⢊⠞⡡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡏⠃⠁⠀⠀⠀⠀⠀⠀⢀⡀⣿⣧⢠⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣏⢜⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠇⡇⠙⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡞⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠳⡤⠀⠀⠀⠀⠀⡄⣤⡯⣰⣏⢹⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠰⣇⣴⣿⡴⢴⣿⣾⣿⣿⣿⣿⣿⢺⣿⡿⠿⠿⠿⠿⠛⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢿⣿⣼⠁⠈⣿⣻⠧⣿⣿⡿⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠞⣸⡟⣧⠀⠙⠿⢷⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠟⢸⣝⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣞⠀⠀⠀⠀⠰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢿⠄⠀⠀⠀⣸⣿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢺⠀⠀⠀⢠⢿⡿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⠃⣄⠐⡱⡷⢻⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠃⢊⢈⢄⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠸⠀⠰⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⠀⠀⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢸⠀⡄⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢰⠇⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⡼⠀⣧⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠀⠀⠀⠁⢸⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠐⣇⢻⡆
⠀⣀⣀⣠⣴⣾⠻⠋⠀⠀⠀⠀⣀⣀⡀⠀⠰⠊⡋⠘⣷
⠘⢯⣭⣅⣈⢀⢀⣤⣤⣴⠿⠷⠻⠿⠿⢿⣶⣦⣤⣴⣿⠆

⣤⣶⠦⢤⣤⣤⣀⣠⡶⠾⠿⠛⠿⠷⢶⣦⣄⡀
⢿⡟⣦⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀
⢸⡇⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆
⠀⢷⠀⡈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠈⣿⠃⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⢠⡏⠀⠀⢶⣦⣬⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⣸⠁⠀⣤⢸⣿⣿⣿⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠷⣶⣤⣀
⠀⣿⠀⠀⠹⣦⡙⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡜⠋⠀⠀⠀⠈⠙⢷⡄
⠀⣿⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠦⠙⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⡶⠶⢶⣤⣄
⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠿⣦⡀
⠀⢸⡇⣄⣀⣤⢰⡰⡖⠈⠃⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⢸⡇⣿⣿⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣧⠟⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⢸⣇⢀⣠⣤⣠⣴⣴⣴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⢀⣹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶
⠘⣿⠘⣇⢿⢿⡻⠃⠉⠀⣷⣄⡀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣠⡿⠛⠉⠉⠁⠀⣸⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣭⣭⡟
⠀⠹⣧⡘⢎⠳⠁⠀⠀⠀⠹⣎⠛⠶⢤⣴⠷⣤⣈⠉⠉⠉⣹⣷⠶⠶⠶⠖⠛⠁⣀⡼⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁
⠀⠀⠈⢿⣎⠃⠀⠀⠀⠀⠀⠙⠷⣦⣄⣽⣄⠀⠉⠛⠛⠛⠋⠹⣷⠶⠶⠶⠶⠛⠉⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠈⠙⠳⠶⠤⠤⠴⢾⡟⠶⠦⠴⠤⠶⠖⠋⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⣄⠀⠙⠻⢶⣦⣤⣤⣤⡤⠾⠷⣤⣤⣤⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣄⡙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⣤⣴⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣦⣠⣤⣤⣄⣠⠾⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⣠⡿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⢀⣴⣿⣿⠟⢡⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣿⣿⣟⢶⣶⣿⣿⡿⢁⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⣿⣿⡿⠿⠿⠿⠟⠁⣿⣏
⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣛⣛⣭⣭⠅⢀⣠⣀⣠⣤⣌⣍⣟⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣟⣿⣶⠿⠟⠛⠋⠉⠁⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠠⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠻⠛⠓⠚⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠊⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣶⣾⣶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣦⣤⣤⣴⣶⣶⣴⣶⣿⡷⠾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣷⣶⣿⣷⣶⣦⣿⣿⣤⣤⣤⣴⣿⡷⣤⣾⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⣷⣿⣿⣿⣿⣿⣿⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣰⣿⡏⠀⠻⣿⠏⣩⠝⠉⢹⣿⠉⠙⢻⣿⣙⣿⣿⣿⣿⣿⣬⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠃⢹⣿⡇⠀⠘⣿⠇⠁⠀⠀⢸⣿⠀⠀⠽⢿⣩⡿⠟⢹⣿⣿⣯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠸⣿⡇⠀⠈⣿⠀⠀⠀⠀⢸⣿⠀⠀⢼⣿⣤⠴⠚⢹⣿⣿⢋⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠄⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⢸⡿⠀⠀⣸⡟⠀⠶⡶⢼⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡤⠀⣿⡇⠀⠀⣿⠀⠀⠀⠀⢸⣇⠀⠀⢹⣿⠒⢶⡦⢸⣿⡿⣦⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣧⠀⣿⡀⠀⠀⠿⡇⠀⠀⠀⢹⡿⠀⠀⢸⣿⢰⠞⠀⣼⣿⡇⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢉⣿⠀⠀⢰⡇⠀⠀⠀⢸⡇⠀⠀⢸⡟⢀⣿⣴⣿⣿⣤⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠸⣿⡄⠀⢸⣧⠀⠀⠀⣾⡏⠀⢀⣿⣿⣋⣿⣿⣿⣿⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢹⣷⠀⢹⣷⠀⠀⠀⣿⡇⣤⣸⣿⠏⣛⣩⣿⡟⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠸⣿⡄⢨⣿⡅⠀⠀⣿⣿⡂⣾⣿⣰⣿⣿⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣹⣷⣤⣽⣧⣤⣤⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠻⢶⣿⡏⠹⢿⣿⣿⣿⣿⣏⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠤⠀⠉⡿⠟⢿⣿⣿⣟⡵⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⠀⠀⠀⠉⠀⣛⣛⣿⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠳⠛⠶⠶⠶⠦⠾⣿⣿⣿⣿⣿⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡤⠰⠿⣿⣿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣿⣃⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⠟⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⡿⢂⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠺⣟⣿⣏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠻⣶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣧⣶⣿⣿⡿⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠙⠯⣹⣿⣿⣶⡁⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢲⣤⣿⣯⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠈⠋⠐⠲⠛⠋⠹⠟⣻⣧⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠶⢦⣤⣬⡥⠾⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣛⣉⡉⣉⣉⣉⣹⣉⣯⣿⣿⣿⣟⣛⡖⠶⢦⣦⣤⣀⡀
⠀⠀⠘⠦⢴⣯⣭⣔⠀⢠⣤⣶⠋⢹⣿⣿⣿⠿⠿⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠒⠠⠹⣿⣿⣶⣦⡄⠀⢄⠀⢳⣍⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣻⣷⣶⠤⣤⣍⣉⣛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠤⠄⣒⣀⣩⣽⠿⢿⣿⣟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⣿⣿⠷⠒⠲⠼⢋⡹⣛⣒⠶⢿⣶⣤⠤⢤⣀⣐⣶⣤⣬⠶⠶⠖⢒⠛⠯⠽⠳⠖⠚⠋⠙⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣟⣿⡛⢛⣀⣀⣀⣀⣤⣤⠤⠶⠶⣚⣛⣫⠭⠽⠟⠻⠿⢬⣭⣵⣦⣤⣀⣀⡀⠀⠀⠀⠀⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡛⠳⢛⡛⠉⠙⠒⠒⠒⠀⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠷⠶⣚⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⡟⣿⣿⣶⣶⣤⣉⣙⢲⡦⠤⠀⣀⣀⣀⡀⠀⢀⡀⠀⠀⠀⣀⣀⣠⠤⠤⠶⣖⣋⣩⣴⣾⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⢯⣙⡛⢿⣿⣿⣿⣷⣶⡶⢶⣦⣤⣥⣤⣬⣯⣬⣥⣤⣤⡶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡜⣇⣿⣿⠀⠐⠉⠙⠒⢿⡿⣯⣽⣶⡀⠛⠿⠿⢿⢿⣿⠿⠿⠿⢟⡛⢛⣋⣿⣿⡧⢴⠚⠉⡇⣄⣿⣿⢈⡇
⠀⠀⠀⠀⠀⠀⢀⣥⣿⢸⣿⠐⠀⠀⠀⠀⢸⡏⡇⠀⠀⠈⠉⠉⠈⠙⣿⣿⠀⣿⠉⠉⠉⠉⠀⢸⡷⡟⢸⠀⠀⡇⡿⢸⣿⡌⢧
⠀⠀⠀⠀⠀⠀⣰⡏⣿⠘⢹⣠⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣿⢈⠀⠀⠀⠀⢸⣿⣧⢸⠀⠀⡇⢿⡞⣾⡇⣾⡆
⠀⠀⠀⠀⠀⠀⠈⠀⢸⡆⡈⡧⠀⠀⠀⠀⢸⣯⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢈⠀⠀⠀⠀⢘⣿⣿⢸⠀⠀⡇⢸⠁⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣧⠀⠀⠀⢀⠇⢀⣧⠀⠀⠀⠀⠀⠀⠀⡟⢻⡆⣿⢸⠀⠀⠀⠀⢸⠃⢻⡄⠈⠀⡇⢸⣿⣿⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⢈⣀⡈⠘⡇⠀⠀⠀⠀⠀⡜⠁⡀⢷⣻⢸⠀⠀⠀⠀⢌⢀⣈⣿⠀⠀⡆⣼⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣽⡆⠀⠀⣼⠏⠛⢻⡇⠀⠀⠀⠀⠀⣧⣴⣯⣾⣿⢘⠀⠀⠀⢠⣾⠋⡟⢿⡄⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⣿⠦⣴⣏⠀⠀⠀⢳⡀⠀⠀⠀⣰⠏⢠⠘⢹⢿⡜⠀⠀⠀⣼⠃⠀⠇⣘⢷⡖⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⢧⣿⣿⣿⣭⣝⡒⠤⣜⣧⠀⠀⣴⠇⠀⠸⠀⢸⠈⣿⣄⣀⣠⡯⠴⢒⣉⣥⣶⣿⣿⡿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣭⣿⣿⣿⣟⡟⢿⣶⣼⣿⣽⣯⣾⣘⣛⣛⣉⣉⣉⢿⡤⣾⣶⣿⣿⣿⡽⠟⢛⣻⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠧⢨⡉⣹⣿⡿⣿⣿⣿⣤⣬⣀⣀⣄⣠⣤⡬⢿⡿⣛⠋⢉⠤⠂⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠶⠦⠤⠳⠤⠤⠤⠐⠂⠘⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣖⠛⠛⠛⠛⢛⣛⣛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡴⣿⣭⣭⣭⣭⣽⠗⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⣾⣛⣛⣛⣛⣻⡷⠀⠀⠈⣷⠒⠚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠚⢛⣿⠟⠋⠹⣇⣼⠿⠿⠿⢿⣿⣦⠀⠀⠀⡿⢶⡶⠶⠶⢶⡶⠶⢶⣶⣒⠛⠛⠚⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⣠⠶⢛⣽⠟⠁⠀⣠⠾⠋⠀⠀⣠⣼⡇⣩⡯⠿⠭⠽⠿⣥⠀⢀⣼⡷⠋⠀⢀⡴⠋⠀⠀⣠⠞⠉⠙⣶⣄
⠀⠀⠀⠀⠰⣯⡶⠋⠀⢀⡴⠟⠁⠀⢀⡴⠞⠁⣸⡃⠛⢛⣛⣛⣛⣛⣫⣴⡿⠋⠀⣠⡴⠋⠀⠀⣠⠞⠁⠀⣠⠞⠁⣸⡇
⠀⣀⣀⣠⣤⣿⠛⠶⣶⣏⣀⠀⣠⠶⠋⠀⢀⡴⠟⠉⠉⠉⠉⠉⠉⣩⠞⠁⠀⣠⠾⠋⠀⠀⢀⡾⠋⠀⣠⣾⣥⠶⠛⣿⣤⣤⣀⣀⡀
⣾⠉⣥⣤⣤⣤⣄⠀⠀⠈⠉⠙⠛⠓⠶⠾⠿⠤⣤⣤⣤⣤⣤⣤⣾⣥⣤⣤⣾⣥⡤⠴⠶⠶⠿⠚⠛⠋⠉⠁⠀⠀⢠⣤⣤⣤⣤⣌⢹⡆
⢻⡀⠻⠶⠞⠛⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠛⠛⠲⠶⠋⣸⠇
⠀⠙⠳⠶⠚⠋⠉⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠛⠳⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣉⣹⡷⠶⠤⣤⣤⣀⣀⣀⣀⣠⣤⣤⡤⠶⣾⣛⣉⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡿⠀⠀⠀⠀⠈⠛⣷⠶⠶⡾⠋⠀⠀⠀⠀⢿⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠸⣇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⢻⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⠗⠲⢤⣄⣀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⢀⣀⣤⠴⠾⣇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡿⢦⣤⣀⡀⠉⠙⠛⠶⢦⣤⣿⡀⠀⣧⣤⠶⠖⠛⠉⠁⣀⣠⣤⢿⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⠇⠀⠀⠉⠙⠛⠲⠦⣤⣀⡀⠉⢙⣿⣇⣀⣠⣤⠶⠚⠛⠉⠁⠀⠘⣇⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠟⠁⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⣾⣧⣄
⠀⠀⠀⠀⠀⠀⣼⣷⠏⠉⢳⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠹⣟⡷⣤⣞⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣞⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⡼
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣴⡶⢶⣄⣴⡾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⢿⣿⣁⣤⡽⠿⠴⠾⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⢤⡀
⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⢷
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢳⡀
⠀⠀⠀⢀⣸⡇⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⢀⡠⠞⠉⢸⠃⠀⠀⠀⢸⣿⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠺⠶⣤⣤⣸⡆⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⣴⠋⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠘⠛⠛⠛⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠈⠙⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢦⣄⣠⣾⣯⣵⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠦⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⢶⣤⣤⣤⣴⡶⢿⡆⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⣍⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣎⣿⡛⠛⠃⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠘⢦⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣠⣾⠟⠈⠛⠃⠀⠀⠀⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣌⣳⣄⡀⠀⠀⠈⢛⣀⣀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣄⣀⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠙⠛⠛⠻⠶⠶⠿⣷⠶⠶⠶⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣙⣻⣶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠈⣿⠀⠀⠀⠀⠀⠀⠤⠶⠶⠤⣤⣀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⢿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⢀⣤⠶⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡰⠃⣇⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⡼⠁⠀⢹⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⡤⠀⠀⠀⣸⠁⠀⠀⠀⢧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⠇⠀⠀⢰⠃⠀⠀⠀⠀⣸⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠃⠀⠀⣏⠀⠀⠀⡇
⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢰⠃⠀⠀⠀⠈⠁⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢸⡇⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢸⠁⠀⠀⠀⠀⠈⠓⠦⣤⣀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠳⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣀⣃⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣈⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣏⠉⠓⠤⣀
⠀⠀⠀⠈⠙⠶⣤⣀⠙⠢⢄⡀
⣀⠤⢄⣀⡀⠀⠀⠉⠓⠦⣄⡉⠒⠤⡀
⠳⢤⣄⣀⣈⠉⠑⠒⠢⠤⢄⣉⣳⣦⣈⠑⠢⣄⣠⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣈⡭⣽⣻⣷⡶⠦⠤⣤⣈⣉⡙⠛⠲⠤⠭⣻⣟⡛⠛⠛⠛⠿⠿⠿⣿⣶⣶⣤⣄⡀
⠀⡴⣫⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠲⠦⠤⣍⣉⣒⠒⠤⠤⣀⣀⠀⠉⠙⠳⢯⡳⡄
⢸⠣⣣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⣝⠳⢶⡲⠭⠭⣒⣒⡴⠷⣿⣀⣀⡀
⠘⢦⡈⢹⡿⠶⢦⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣲⣬⣓⠶⣾⣿⠉⣩⠟⠛⠦⠬
⠀⢸⣿⣿⣧⣤⣄⣀⡈⠉⠉⠉⣿⠛⠛⠓⠒⠒⠒⠒⠒⠛⠛⢻⡏⠉⠉⢁⣀⣈⣽⣮⣝⡻⢇
⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠫⡒⠢⡀
⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡇⠀⠈⠉⠁
⠀⣸⡇⣿⣷⣭⣙⡛⠿⠿⢿⣿⣿⣗⣻⣿⣿⣿⣿⣿⣿⣿⣓⡿⣿⣾⠿⠿⠟⣛⣩⢵⣿⣿⣧
⢰⢻⣧⣿⣿⣿⣿⣿⣿⣗⣶⣶⣿⣥⣭⣭⣭⣭⣭⣭⣭⣭⡥⡏⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡏⢷
⠸⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⡿⠃⡾
⠀⠹⣤⣾⠟⠿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⢿⣿⣛⣽⣿⡾⠟⢿⣤⠞⠁
⠀⠀⠀⠙⠲⢤⣀⡈⠉⠛⠛⠻⣿⠷⠿⢾⣿⣾⣿⣾⡿⠿⠷⢿⡿⠛⠛⠉⠉⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠶⠿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡽⠷⠶⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠛⠛⠻⠿⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠁⠀⠀⠠⡀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢡⠄⠀⠀⠀⠀⠀⠈⠢⣵⣆⠀⠀⣄⠳⢦⡈⠙⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠹⣟⠃⠘⠋⠀⠈⣯⠃⠈⢻⣿⠿⣦⡀
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⠆⠀⠀⢿⣧⠈⢳⣄
⠀⠀⠀⠀⠀⣿⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⡿⠄⣶⡙⢧⡀
⠀⠀⠀⠀⠀⢸⣿⢀⠀⡄⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡆⠀⠸⠀⠀⠁⠘⠈⠛⣦⣀
⠀⠀⠀⠀⠀⠈⢿⣼⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠈⠀⠀⠀⠀⠀⠘⠆⠀⠀⠸⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠈⢿⣆⡆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⡈⠑⠚⠻⣧⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠘⠰⠀⠀⠀⢰⣟⣷⣾⠂
⠀⠀⠀⠀⠀⠀⠉⠐⠓⠺⢿⣷⣷⣣⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⡀⠀⠐⠀⠘⣿⡙⠉⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢾⣾⣤⣴⣤⣄⣀⣀⣤⣤⣤⡶⠶⠛⠛⠉⠉⠁⠀⠈⠉⠳⣤⣀⠀⠈⢿⣦⡀⡀⠉⠙⠻⠶⣦⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⢍⡉⠉⠉⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠉⠛⠻⠿⣶⣤⣀⣀⠀⠀⠉⣀⣀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠈⠑⠢⠤⡀⠀⠈⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⡀⠀⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀⠀⠀⠀⠀⠀⡀⢿⣄⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠀⠀⠀⠈⠙⠻⣿⣛⢉⡀

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣠⣤⠶⣒⣛⣋⣉⣉⣉⣛⠲⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⡶⠟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠀⣾⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃
⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⠟⣼⠇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⢁
⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⣫⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⢀⣴⡏⠀⠀⠀⣀⣀⡀
⠀⣰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⡾⣫⣾⠟⠁⣠⣶⣿⣿⣿⣷⣦⡀⢸⠃⣰⣿⠏⢀⣴⣾⣿⣿⣿⠿⢷
⢠⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢟⣽⡾⠛⠁⠀⠀⣿⣿⡟⠉⠉⠻⣿⣷⡀⢰⠟⢁⣼⣿⣿⣿⠟⠉
⣾⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣷⠟⠋⠀⠀⠀⠀⠀⠹⢿⣷⣴⠀⠀⢸⣿⠇⠈⣰⣿⣿⣿⡿⠃
⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣵⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⢀⣼⣿⣿⣿⠏⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣵⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⣁⣴⣿⣿⣿⠟⢁⣠⣴⠾⢛⣩⠥⢤⣤
⠸⣿⣿⣿⣿⣿⣿⠿⣛⣽⣿⢟⣫⣵⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡴⠶⠿⠿⠟⠛⣉⣤⣶⡿⠟⠁⠀⠀⠀⠀⢀⣽
⠀⠹⣿⣿⣿⣿⡶⢟⣻⣽⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠁
⠀⠀⢯⣭⣭⠶⢞⣋⣭⣤⣶⣶⣶⣿⡿⠿⢿⡟⠛⢻⡛⠛⢻⡛⠛⢻⠻⠿⢿⠷⢶⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⡴⢺⡿⠛⠋⠁⣀⡴⠚⠂⣠⡾⠋⠀⣰⠏⠀⢀⣿⠁⠀⢹⡇⠀⠹⣦⠀⠙⢧⡉⠉⠛⠻⣷⣆⡀
⠀⠀⠀⠀⢧⣹⣄⣠⡴⠛⠁⠀⣠⡾⠋⠀⢀⣼⠏⠀⠀⣸⡏⠀⠀⢸⣇⠀⠀⠹⣧⠀⠀⠻⣦⡀⠀⢀⣽⠂
⠀⠀⠀⠀⢈⣿⣿⣿⣿⣶⣶⣿⣭⣤⣤⣤⣿⣁⣀⣀⢠⣿⠁⠀⢀⣸⣿⣀⣀⣀⣹⣧⣤⣤⣾⣿⣾⢿⡏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣭⡄⢸⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣯⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⠁
⠀⠀⠀⠀⠀⠈⢿⠻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠈⢷⡙⣿⣿⣦⣙⣛⡛⢋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠉⠉⠙⠛⠛⣿⡟⠛⠛⠉⠉⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣥⣤⣶⣶⣶⣶⣶⣸⣿⣗⣿⣷⣶⣶⣶⣶⣤⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠉⠿⢿⣿⣟⣻⣿⣛⣟⣻⣟⠻⠍⠉⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⣄⣐⣲⣶⣶⣶⣶⣶⣶⣖⣂⣀⣤⣄⠀⣀⡀
⠀⠀⢀⠔⢀⠀⡔⠢⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣢⣬⣷⣾⣿⠿⠿⠿⠛⠛⢛⣛⣛⣛⣛⡛⠛⠛⠻⠿⠿⢿⣿⣾⣯⣴⣢⢄⡀
⠀⠀⠈⡗⠜⠀⢻⠐⠀⠀⠀⠀⠀⡠⢒⣵⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⣿⠿⠿⣧⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣮⣕⢄
⠀⠀⠀⡇⠀⠀⣼⢀⠀⠀⠀⠀⡜⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠈⠙⠻⠿⠬⠿⠛⠋⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠉⠻⣧⢱
⠀⠀⠀⡇⠀⠀⣿⢸⠀⠀⠀⢠⠇⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠦⣤⣀⣤⡴⠶⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢆
⠀⠀⠀⡇⠀⠀⣿⢸⠀⠀⠀⡇⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡶⠶⠛⠋⠉⠈⠉⠛⠳⠶⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⢘
⠀⠀⠀⣇⠀⠀⣿⢸⠀⠀⠀⠸⡉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠋⢉⡏
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⢰⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡆⠀⠀⠀⢀⣴⣿⡄
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⢸⢸⣿⣿⣷⣦⣤⣀⡀⠉⢹⣯⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢵⡏⠉⠀⣀⣠⣴⣾⣿⣿⡇⡇
⠀⠀⠀⣿⠀⠀⣿⢸⠀⠀⠀⠘⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣦⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⡇
⠀⠀⠀⣿⢀⠀⣿⢸⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⣟⣗⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⠱⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡜
⠀⠀⠀⣿⢸⠀⣿⢸⠀⠀⠀⠀⠀⠑⢄⡀⠙⠻⢿⣿⣿⣿⣿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⡿⠟⠋⠁⡠⠊
⠀⠀⠀⣿⠸⠀⣿⢘⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠈⠉⠛⣻⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⠛⠉⠁⣀⡠⠔⠋
⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢭⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡭⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠁⠀⠀⠀⣀⣤⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠻⢦⡀⢀⣀⠀⣠⣾⣿⡿⠋⠀⠀⣠⣴⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢿⣿⡀⠀⠀⣨⡿⠟⠛⠻⣿⣿⠟⢀⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁⠀⣰⡿⠿⢛⣿⠉⠀⠀⡀⠀⠈⠻⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⣷⡄⠀⢿⣟⠛⠋⠁⠀⠀⢸⡇⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠈⢿⣆⠀⠙⠻⠶⣦⣤⣤⣟⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⢷⣦⡀⠀⠀⠻⣆⠀⠀⢀⣾⠟⠙⢿⡛⠃⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠃⠀⠀⠀⠀⠹⣷⡄⠀⠀⢹⣷⡾⠛⠁⠀⠀⠈⢻⡄⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣦⡀⠀⠀⠀⠘⣿⣤⣴⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣾⣿⣆⡀⣀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠟⣹⣿⣿⣿⡿⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⠿⢻⣧⣀⣾⣿⣿⡿⠋⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⠋⠁⠀⢀⣽⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢦⣤⣤⣿⡏
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⣠⣿⣿⣿⠿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⣀⣤⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠈⠻⣷⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠘⣿⣧
⣴⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠘⢿⣦⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠙⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢳⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⣰⠃⢻⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⡆⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⠃⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣰⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡴⠃⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠈⠲⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⢢⡐⣅⡦⡵⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠲⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⢿⣤⠤⠤⣤⠤⠤⠤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣤⣖⡺⠭⠷⠶⠟⠉⠻⣏⠉⠉⠉⠉⠍⠛⢛⠚⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠛⠿⢦⣄⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠈⠙⠲⢄⣀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠄⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣦⣬⣑⠲⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⢀⣠⣴⣾⠟⣻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢯⣾⠛⠿⢯⣿⣒⣲⠦⠤⠤⢤⣤⠤⠤⠤⠤⣖⣒⣾⡭⢷⡏⣿⡴⢛⣥⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢘⡏⠙⠳⠶⣤⣌⣉⣉⣉⠙⢻⢸⡛⠋⣉⣉⣉⣤⠧⠿⠛⢻⡃⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠉⠩⠭⢹⢻⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢞⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠖⣦⠀⢐⢂⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢠⣽⣠⠴⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠶⣏⡛⠲⠦⣭⣔⣀⡀⠀⠀⣠⣄⠀⠀⠀⠀⣀⣐⣲⠭⠷⢿⣀⡷⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠥⣭⣩⣿⣿⣇⣸⢟⣛⣋⣉⣉⣤⠤⠴⠒⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⣤⡤⢄⣒⣾⠟⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣐⣽⣿⣟⣽⠫⠏⠁⢀⠠⠤⠤⢄⣠⣒⣀⣺⣭⣭⣭⣵⣶⣖⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡻⣋⣉⡀⠀⠘⠓⠋⣙⣻⠽⠟⠋⠉⠀⠀⠀⠀⣀⠀⢉⣙⣏⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡎⢸⠛⠓⠲⠾⠶⠧⠤⠤⠤⠤⠤⠴⠶⠒⠚⠆⡏⠉⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⢸⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣧⣤⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⢹⡟⠛⠋⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⣿⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢿⣿⣼⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⣿⣿⡿⣋⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣽⣿⠿⢾⣶⢤⠀⣀⠀⠀⠀⢀⡀⡤⠴⠿⣟⡇⡿⠑⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣇⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⢠⡇⣀⡬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⣶⣭⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⣤⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣤⣀⣀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⡿⠿⠿⠿⠟⠻⠿⠿⠿⠿⢿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣮⣙⠛⠿⣿⣷⣶⢶⣶⣤⣤⣤⣤⣤⠤⠶⠖⠛⠉⠁⠀⢐⣴⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣬⣶⣏⡍⡉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⢿⣿⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡋⡿⠉⠸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⢻⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⠁⠀⠚⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⠀⠸⣿⣿⣿⣿⣗⡖⠀⠀⠀⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣿⢿⣿⠇⠀⠂⠀⢸⣿⣿⣿⣿⠇⠠⠀⠀⢻⣿⣿⣿⠿⠁⠀⠀⠀⢸⡿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠋⠰⣿⣝⠁⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⣠⡇⠀⠙⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠹⢿⣿⣶⠦⣀⡘⣿⡿⠀⠀⠠⠄⠀⠀⠘⣿⡏⠀⠀⢀⣠⠾⢋⡀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠸⣤⣌⠛⠿⣷⣧⣿⣧⣤⣀⣀⣀⣀⣀⣠⣽⣧⠶⠞⠋⠁⣀⡾⠁⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠿⢦⣝⢈⠉⠍⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⡴⠟⠋⠀⠀⠀⠀⡀⢊⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠲⠶⠲⠶⠶⠶⠶⠖⠒⠚⠉⠁⠀⠀⠀⠀⢀⣴⣾⣧⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡷⠦⠄⠀⢀⣀⣀⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠌⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠳⠄⢀⣤⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠺⣿⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡋⠉⠲⡀
⠀⠀⠀⢀⣠⣤⠾⠟⣸⡿⠀⠀⢒⡿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁⠀⡠⠃⠀⠐⡋⠒⢢
⠀⠀⣠⡾⠿⡏⣀⣈⣛⣶⡶⣶⡛⠙⢟⣹⢶⡄⠀⠀⠀⠀⠀⠀⡊⠈⠀⠀⢀⠀⢓⠒⠂⠁⠀⠀⠀⠓⠄
⠀⢸⡿⣿⣿⣿⣿⣿⣿⣃⡿⣿⣿⣿⣽⣿⡿⢽⣧⡒⡆⠀⠰⡸⢀⡀⠀⢁⠀⠀⡀⡈⠀⠀⠀⠁⠀⠰⠠⣠⠒
⠀⠘⡇⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⡇⠈⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠈
⠀⠀⡇⠀⣀⣤⣴⣶⣾⣿⣿⣿⣟⣶⣶⣤⣀⢰⡇⠀⠀⠀⠀⠀⠀⢀⡠⠀⠒⠀⠀⠭⡁⢒⡒⠶⠠⢀⠀⡇
⠀⠀⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⢿⣿⣾⠃⠀⠀⠀⠀⠰⣄⠠⠤⠀⢈⠤⠀⠀⠡⢂⠐⡀⠀⠄⣱
⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⠐⡉⠒⠥⠀⣀⣁⣀⣀⣈⠠⠥⠒⡞⠁
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣀⠀⠀⠀⠀⠀⠠⢀⡠⠊
⠀⠀⠀⠀⢀⣿⣾⣛⣻⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠙⢚⢰⣶⠒⠾⢯
⠀⠀⠀⠀⣸⠃⠀⠉⢹⡟⣿⠉⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠘⠒⠉⠀⠀⠹⡀
⠀⠀⠀⠀⡟⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀
⠀⠀⠀⣸⠃⠀⠀⠀⢸⡇⠛⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠋⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡏⠻⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠀⠀⠀⠼⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⢀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⣀⣴⠟⠉⠀⣤⡀⢀⡀⠀⣀⡼⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠶⠛⠛⠉⠉⠉⠉⠉⠉⠛⢿⡀⠰⣤⣷⠾⠛⠉⠙⢶⣭⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⢁⣀⡀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠈⢷⣾⠟⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠠⠤⠤⠬⠭⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠰⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣿⠁
⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⠇
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣶⠏
⠀⠀⠀⠀⠀⠀⢿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣦⡟
⠀⠀⠀⠀⠀⠀⢸⣇⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣤⠟
⠀⠀⠀⠀⠀⠀⠀⢿⣎⠅⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠦⠒⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡓⠂⣄⢀⢀⣠⠀⣀⠀⡔⡂⣤⣸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣮⣍⡀⠩⠰⢣⣄⣰⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡲⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠿⠟⠿⢷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠚⣦⠽⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠲⣼⡏⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⣼⣤⣶⡾⠆
⠀⠀⠀⠀⠀⠀⠀⢾⡳⢿⣧⡘⢿⡘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠘⢿⣿⠟⠁
⠀⠀⠀⠀⠀⢠⡦⢼⣿⣌⠻⣷⡘⣷⡸⡄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⢠⣼⣇
⠀⠀⠀⠀⠀⠈⢿⣦⡘⢿⣷⣜⢿⣮⣷⡗⠄⠀⠀⠀⠀⠀⠁⠀⢂⠀⣨⣤⢀⢿⢶⣯⣿
⠀⠀⠀⠀⠀⣶⠶⢿⣿⣦⣙⢿⡷⠿⣄⡁⠰⣀⠀⢐⠀⠀⣅⡠⢼⣟⣩⠟⣿⣾⣾⣼⡿
⠀⠀⠀⠀⠀⠙⢷⣦⣈⠙⠻⣷⡿⢦⣈⣣⣄⣀⡀⢼⡶⠖⣾⣴⠞⣩⢿⣾⣥⣋⣟⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⠿⣿⣶⣤⣝⣦⣘⣧⣄⣉⡉⠽⣶⡶⠚⢻⣿⣇⡴⢉⠟⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣭⣉⣛⡻⠿⣿⣤⣉⣉⠩⠽⣷⣖⢉⣨⠟⢿⣿⣾⣵⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣯⣿⣙⣟⣙⡿⣽⠋⣹⢧⣶⣡⡞⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⣼⣭⣿⣤⡾⢛⣯⣴⣏⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣶⣷⢾⣟⣻⣧⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⠛⠛⠻⠺⣟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢀⣀⣀⠘⢯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠭⠤⠄⠀⠿⠄⢈⡗⠀⠠⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⠭⠤⠤⠤⠅⠽⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣀⠤⠶⠒⠚⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣀
⠀⠀⠀⢻⣒⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⡟
⠀⠰⣏⣻⣿⠉⠀⠀⠀⢉⣉⡉⠛⠒⠒⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⢒⣒⠛⠉⠩⣖⣺⢭⡏⢹⡆
⠀⠀⢻⡟⢻⠳⠦⢤⣀⣸⣉⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣉⣇⣀⣀⡤⠾⢿⢻⣿
⠀⠀⢸⣷⣿⠀⠀⠀⠀⠈⡇⡟⠑⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⡬⣷⢶⡧⢤⣤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠒⠋⣧⡏⠀⠀⠀⠀⣿⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⡇⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⣾⡇
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢠⡇⡇⡇
⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇⣿⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⣧⣷⡇
⠀⠀⢸⢿⢿⡀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣨⢿⠟⡇
⠀⠀⠘⢺⡆⠉⠒⠦⢤⣀⡃⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⣧⠴⠒⠉⠀⣼⠋
⠀⠀⠀⠀⠙⠦⣄⡀⠀⢸⡛⢻⠛⠓⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⣤⣇⣿⣤⡤⠤⠤⠤⠶⠶⠖⠒⠒⠛⠉⠉⢹⢉⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠻⠯⠤⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣉⡼⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠬⠗⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠉⠓⠛⠉⠉⠉⠉⠉⠉⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣷⠤⢬⣯⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠐⠝⠂⠀⠈⠛⠈⠁⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠄
⠀⠀⠀⠀⠀⠀⢬⡙⠪⣽⣀⣍⡙⠿⢶⣦⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣰⡾⠗⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠴⢫⡁⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠒⠀⠀⠉⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⠦⠀⣀⡀⢀⠀⣀⣀⡠⠤⢴⣚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣈⠤⠈⠄⠄⠀⠀⡀⠈⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠐⠂⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢦⡀⠀⢰⡴⠀⠀⠤⠄⠀⠠⠗⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠈⠁⣿⠷⠤⣤⣤⣀⣀⣀⣀⡠⠐⢹⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡹⠁⠀⠀⣸⣽⠀⢀⣴⣾⣧⡶⣄⠀⠀⠀⠀⢟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⢀⡏⡇⢰⣏⣿⣡⢿⣷⡜⡇⠀⠀⠀⠸⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡼⠀⠀⢀⣸⢹⢀⢸⣧⣷⠟⢺⣻⣥⠇⡀⠀⠀⠈⣧⠧⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡗⠮⡥⠄⡏⣬⡈⠒⢷⣛⢛⣯⣭⠵⠦⢄⠉⣰⣾⣿⢟⣥⡦⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⡞⠠⠀⠈⢸⢡⡇⠘⠦⡀⠉⣣⣤⠈⠓⠶⠈⢙⣟⣺⣏⡟⣿⣄⡇⣿
⠀⠀⠀⠀⠀⠀⠀⡸⢸⠃⠀⠈⢫⡏⡼⠑⠦⡀⠈⠑⡄⠠⠁⠐⠀⠀⠸⡇⢿⠸⡋⠟⣿⢱⡟
⠀⠀⠀⠀⠀⠀⠀⠃⠋⠀⠀⠀⢸⢱⠓⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣧⣉⣈⡵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢴⢦⡴⠛⠻⠟⠛⡆⢀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠛⠛⠙⣇⠉⠁⠀⡇⠀⠀⠀⠀⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠈⡇⠀⠀⠸⡄⠀⠀⡹⠞⠳⢦⣼⣧⣤⣤⣤⣶⣶⣶⣶⣶⠿⠿⠟⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣼⡇⠀⢹⡠⢤⣤⡧⠴⡋⠀⠀⠀⠀⢹⡟⠛⠛⠉⠉⠉
⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣿⠿⠿⠿⠛⠛⢻⣾⣁⣿⣏⣆⣠⠔⠃⣠⣤⡄⠀⠈⣇
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠋⠁⠛⠓⠶⠶⠞⠁⠈⢀⡀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠛⠉⠱⣄⠀⠀⠀⢰⡄⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣇⠀⠀⠠⣾⠁
⠀⠀⠀⣴⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠙⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠶⠦⠤⠄⣀⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⡑⠦⣄
⠀⠀⠀⣠⠎⠉⠀⠀⠰⠀⠀⠀⢀⡴⢋⠁⠀⠁⠉⠙⠲⢄⡀
⠀⠀⢰⠃⠀⠀⢀⠄⠀⢮⠀⠀⢸⡇⠢⠤⡀⠀⠀⠀⣀⠔⠉⠢⡀
⠀⠀⢹⠀⠀⠀⠘⠲⡀⠀⠀⠀⠀⢷⡀⠀⡇⠀⠰⣏⠁⠀⢀⡴⠞⠂
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠙⠓⠳⠀⠀⠀⠉⠀⢸⠁
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⣀⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠋⠈⠉⡖⠀⠀⢻⡄
⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⡴⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢦⣀⠀⠀⢰⢤
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠘⠦⢴⡄⣱⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢠⣍⣛⣲⣎⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢈⡇⠀⠉⠉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠞⠁⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠶⠤⣄⡀⠀⠀⠀⠀⠀⡸⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠈⠔⠕⡹⡳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠑⢃⢦⣌⣓⠦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢌⢷⣌⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⣟⠏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⢧⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠳⡄⠀⠀⡝⠲⣄⡀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢉⡦⠀⠈⠉⠁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠈⢦⠒⢲⢄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠀⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⢠⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⠀⠀⠀⠀⢸⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⣼⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⢾⣷⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⢻⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⢘⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⢠⠶⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⠀⠀⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠋⠁⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠃⠶⠤⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠧⠄⠀⠐⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡟⠕⢁⡴⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⡄⢠⣴⠲⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠺⠵⠷⠤⠤⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠧⠼⠦⠗⠓⠁

⠀⠀⠀⠀⡤⠤⣄⣀
⠀⠀⠀⠀⠓⠢⠤⣌⣉⠓⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠦⠬⣉⣓⠲⠤⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠛⠉⠑⠒⠮⢭⣛⡾⠛⢯⡉⠉⠉⠓⠒⠲⠤⢤⣀⠀⣀⣠⠤⠴⠒⠊⣱
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⣀⣀⣠⠤⠤⠴⠖⠒⠒⠒⠛⠒⠚⠳⠦⠤⣤⣤⠤⠴⠒⣋⣉⡥⣤⠖⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⢰⡇⣠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⢔⣒⣪⠭⠴⠒⠒⠉⠉⠓⠦⣄⢸⡆
⠀⠀⠀⠀⠀⠀⢠⡟⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢻⡇
⠀⠀⠀⠀⠀⠀⢨⡗⢄⡈⠙⢒⣦⠤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⣴⡒⠋⢁⡠⢺⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠑⢻⣬⢧⣀⣀⣀⠈⠉⠉⠉⡽⢯⠉⠉⠉⠁⢀⣀⣀⡼⣥⡟⠊⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠈⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠁⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠧⡀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⢀⢼⡇
⠀⠀⠀⠀⠀⠀⠈⢧⡈⠓⠢⣼⢼⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡧⣧⠴⠚⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣽⣴⡋⠉⠉⠑⠒⠒⠒⠒⡟⢻⠒⠒⠒⠒⠚⠉⠉⢉⣦⡯⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠲⠶⠿⠿⠶⠖⠒⠒⠒⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⡀⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡀⠀⠀⢸⠀⠀⠀⠀⠇⢸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⡄⠈⠀⡠⢤⣀⠀⠈⠆⢄⠀⢸⢀⠀⢳⢀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⡴⣤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢁⣴⡿⠊⠀⠹⢗⢄⡀⠸⠀⢸⢸⠀⠘⠸⠀⠀⠀⠀⠀⢀⡞⠀⠀⠜⠋⣋⠳⢴⡀
⠀⠀⠀⠀⠀⠀⢀⠇⠣⠀⠀⣰⠞⠁⣠⠞⣛⡓⠠⢕⣝⢦⡀⠀⠘⠀⠀⡇⠀⠀⠀⠀⠀⠘⡃⠀⠀⠀⠀⠁⠀⣸⡇
⠀⠀⠀⠀⠀⢠⢸⠠⠀⡠⠚⠃⢄⡠⣄⣐⡦⢬⣛⢂⣷⠑⢝⢦⡀⠀⠀⡇⡆⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠈⠀⢀⣈⠋⣀⠠⠮⠷⢌⠙⠏⠓⣨⣬⠗⢤⢁⠕⢿⣦⡀⠀⠁⢂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠚⠉
⠀⠀⠀⠀⠀⠀⢰⡟⠀⡏⠠⢬⣛⠶⠀⡗⢨⡟⠛⣡⡀⠴⠽⠦⡎⠙⠻⣦⡀⠘
⠀⠀⠀⠀⠀⠀⠸⡷⣄⢣⠘⠶⢬⣛⢦⡜⣇⡨⡟⠡⣬⣉⠉⠓⠞⣦⠀⢈⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⢌⠒⢠⣍⠙⠂⣼⣠⢃⡒⠦⣄⡙⠛⣰⣸⠕⠣⣀⢏⡇⠀⠀⠀⠀⠀⠀⢀⠄⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⢆⡨⠛⡿⢿⠃⢦⣍⡛⠂⣠⣠⠄⠁⢀⡴⡣⠋⠀⠀⠀⠀⠀⠀⢀⡞⠂⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣕⢦⣀⢰⣘⠷⠮⠍⣸⡿⠉⠐⡤⡫⠊⠀⠀⠀⠀⠀⠀⠀⢀⠞⠢⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡺⣶⣾⠋⠀⠑⠯⡢⣙⠷⢦⡬⡋⠀⣠⣮⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⡄⢀⣾⠃
⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⣼⡿⠀⠀⠀⠀⠈⠪⢗⢆⣀⡨⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠣⠀⣰⡟⠁
⠀⠀⠀⠀⢀⣴⠋⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠻⡂⣠⡾⠋
⠀⠀⠀⣰⠟⢁⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂⠉⠻⣀⣴⠞⠋
⠀⠀⠸⣁⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠤⠶⠞⠋⠁
⠀⠀⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠
⠀⠀⠈⠳⣤⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡴⠏
⠀⠀⠀⠀⠀⠳⣌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡴⠋
⠀⠀⠀⠀⠀⠀⠘⢦⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⢦⠀⠀⠀⠀⢀⠎⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠑⢄⠀⡔⠁⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡨⠋⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⣠⠞⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡼⠻⣄⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣴⠟⠀⠀⠈⢷⡄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⡞⠁⠀⠀⠀⠀⠀⠙⢦⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⢢
⠀⠀⠀⠀⠀⠀⢀⡔⠁⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠑⢄
⠀⠀⠀⠀⠀⡠⠊⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠢⡀
⠀⠀⠀⢀⠜⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⢆
⠀⠀⢰⣇⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣀⣷
⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡛⣙⣻⣉⡟⣛⣛⣙⡿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠶⠶⠶⠒⠶⢪⠗⡶⢶⡶⣶⣦⡤⣤⣤⣄⣀⡀⡀
⡔⠒⠂⡤⢄⡀⣀⠀⠀⠀⠀⣀⣠⢴⣶⣾⣿⣿⣿⣿⣿⣾⣶⣶⣥⡴⠤⠶⠷⠚⠒⠃⠘⠛⠻⠿⠷⠴⢴⣄⣈⣛⠋⡟⢲⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⡀⠤⡤
⠙⠲⠿⠷⣶⣶⣉⡀⣸⣶⡿⠅⠀⠸⢿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠚⠸⣿⣿⣿⣿⣶⡄⠀⢳⣀⣀⣷⣤⡴
⠀⠀⠀⠀⠀⠀⢈⣟⣿⣹⣿⣶⡶⢦⣤⣤⣉⣉⡋⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠔⢒⣊⣉⣡⣽⡶⠿⣿⣿⣟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⣸⣿⣟⣛⣛⣟⣛⣶⠶⠶⠭⠭⢛⣛⠲⠶⠦⣬⣔⣒⣀⣤⣤⠀⠀⠤⢶⣐⣂⣠⣭⣤⠶⠦⠶⠚⣛⣛⣩⠭⠴⠶⠖⠛⠛⢹⣻
⠀⠀⠀⠀⠀⠰⣿⡹⣟⢽⡿⣿⣯⡭⠉⠀⠀⠀⠈⠀⠉⢉⣐⣒⣩⣭⣭⠴⠶⠖⢾⣿⣿⣿⣏⡓⠲⠖⣀⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣷⡝⣧⣖⣶⣄⣴⣶⡶⠶⠶⢒⣛⣛⣉⣉⠭⠤⠶⠒⠒⠋⠉⠉⠀⠀⠉⠉⠙⠛⠻⠶⣶⣶⣾⣶⣤⣤⣤⡀⡀⢀⣞⢿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⡈⠉⠠⠬⣉⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⡉⠛⢛⣡⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣿⢿⠏⢻⣿⣿⣷⣶⣦⣄⣈⣑⣰⡖⠤⠤⠄⣀⣀⣀⣀⠀⠀⢠⡀⠀⠀⢀⣀⣀⣀⣤⠤⠴⠖⡓⣞⣉⣥⣴⣶⣿⣿⣿⣿⣽⢛⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣇⢸⣤⡉⡛⠿⢿⣿⣿⣿⣿⡷⣶⠶⢤⣤⣤⣤⣁⣉⣁⣱⣯⣉⣉⣠⣤⣤⡤⢶⡶⣾⣿⣿⣿⣿⣿⣿⢿⣟⣿⣟⣧⣼⢨⡇
⠀⠀⠀⠀⠀⢀⣿⠀⣏⣿⠀⠉⡟⠓⠶⠤⣽⣿⣿⡛⠺⢅⢈⢸⢷⣿⣻⣏⣿⣿⣿⣿⣿⣿⣿⣾⣿⠾⠿⢿⡿⣟⣟⣁⡭⠾⢻⠩⢁⢸⣿⣿⠘⡇
⠀⠀⠀⠀⠀⠈⠸⡇⡿⢺⡆⠀⡇⠀⠀⠀⠀⣿⢸⡉⠛⠛⠒⠆⠤⠤⠠⠬⢏⣿⣿⠤⣤⢤⠶⠶⠦⠖⠖⠛⢹⡏⣿⠀⡆⠀⣨⠐⢸⠈⣿⣟⠐⣧
⠀⠀⠀⠀⠀⢣⠾⡇⣿⣿⡇⠀⡇⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣿⠀⡗⣼⠀⠀⢀⠀⠀⠀⢸⡇⣿⠀⡄⠀⡏⠀⢸⠀⣿⢻⠠⠸⡄
⠀⠀⠀⠀⢸⡿⠀⡇⠏⢸⡇⠀⠇⠀⠀⠀⠀⣿⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠗⢹⠀⠀⠰⠀⠀⠀⣸⡇⣿⠀⢀⠀⡷⠀⢸⢸⢻⣿⢀⡷⡷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡗⠀⡄⠀⠀⠀⠀⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⢩⣧⣿⡄⢐⠀⡇⠀⢸⡈⠈⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡯⣿⠷⣿⠀⡅⠀⠀⠀⢸⠉⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⢿⡄⢳⢸⠀⠀⢀⠀⠀⠀⢸⠇⢸⡅⠀⠀⡇⠀⠸⣧⣶⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⡆⠇⠀⠀⠀⠃⠀⠘⠻⡄⠀⠀⠀⠀⠀⠀⠀⡾⠀⠘⣧⣸⢸⠀⠀⢸⠀⠀⠀⡏⠀⠘⣿⢸⠀⡇⠀⢸⣿⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢹⡇⠀⠀⠀⠀⣦⣤⣄⣠⠇⠀⠀⠀⠀⠀⠀⣸⠁⠐⣆⢹⡆⢸⠀⠀⠈⠀⠀⠀⣧⡴⢶⣼⡀⠀⡇⠀⢸⡿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣮⣿⠀⠀⠀⣶⠏⠀⠉⢸⡇⠀⠀⠀⠀⠀⠀⢸⡶⠾⣷⣾⣷⢸⠀⠀⠀⠀⠀⣼⡟⠀⠀⠄⣧⠀⡇⣀⣾⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣾⣼⣿⠦⣼⣟⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢠⡟⠁⠀⠘⠉⢿⣷⠀⠀⠀⠀⢠⡟⠁⠀⠀⢀⡹⣷⠂⢉⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⣸⣿⣾⣿⣌⣛⠲⠤⣄⣘⢷⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⣼⣁⡤⠶⠒⢋⣤⣼⣿⢿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣾⡛⣻⣿⣿⣿⣿⡟⠶⢤⣍⣙⣧⣒⢲⡿⠥⠤⠴⠤⠴⠶⠶⠶⠻⣞⢋⣹⣏⣤⣶⣶⣿⠟⣿⣿⠿⣾⣯⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣟⡟⠓⠾⢤⣸⣖⣳⣿⣿⣾⢛⠛⠓⢲⠲⠒⠚⠓⠒⠋⢻⣧⡿⣾⣿⣿⠿⠺⠋⠁⡠⠶⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢄⣬⣫⣛⠻⣿⣿⠷⠶⠶⢼⡤⠤⠦⠶⠶⠶⠚⢻⡟⣩⡅⠀⠀⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣛⣛⣲⣶⣶⣤⣦⣴⣶⣶⣦⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⣄⣀⣀⣀⣤⣤⠤⠤⠤⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⣀⡬⠂⣀⠤⠀⠒⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⣠⠔⢊⣡⠔⠋⠀⠀⠀⠀⠀⠀⠈⠑⠒⠢⠤⠤⠤⠤⠤⠤⠔⠒⠒⠒⠒⠀⠐⠒⠒⠢⠤⠤⠇⠀⠿⠀⠀⠀⠀⠀⠀⢀⡇
⡰⠋⠀⡔⠋⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣸⠁
⡇⠀⡈⡀⠐⠂⠀⠉⠉⠀⠀⠀⠀⠈⠉⠒⠢⠠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⢀⠁
⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠂⠤⠤⢀⣀⡀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢠⠃
⠘⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⢒⡇⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡎⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢻⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠒⣄⠳⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢆⠙⢄⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⢣⡀⠙⠢⠭⠷⠒⢦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣙⣢⣤⣤⣤⣴⣂⣁⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠞⠛⠻⠷⢶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⣴⠟⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⢰⣿⠀⠀⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠈⣿⡀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⠀⢀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⢸⡇⠀⠸⣧⣀⡀⠀⠀⢴⣄⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠸⣿⡀⠀⠈⣙⠋⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷
⠀⠀⠀⠀⠀⢻⣧⠀⠀⠉⠉⠁⠀⠀⠀⠲⣦⠄⠀⠀⠠⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢸⣷⣄⣀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠙⠛⠛⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠈⠛⠷⠂⠀⠀⠹⢷⣄⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠘⢷⣶⡶⠞⠀⣀⡀⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⡀⠀⠀⠀⠙⠛⢀⣀⠈⠀⠀⠀⢸⡷⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⡀⠀⠈⠁⠀⢰⣦⣄⠘⣧⠈⠳⡾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢶⣾⣇⠈⠛⠿⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣴⡶⠶⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⢀⣴⠾⠿⠖⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣿⠁⠀⠀⢀⣤⣤⣶⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠙⠀⠀⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣴⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠢⢤⡀
⠀⠀⠀⢻⡒⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⡟
⠀⠰⣏⣻⣿⠋⠀⠀⠈⢉⣉⡙⠓⠒⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⣒⠚⠋⠩⢕⣒⢭⡏⢹⡆
⠀⠀⣿⣿⣻⠲⠤⣤⣀⣸⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⡇⣀⣀⣤⠿⢿⢻⣿
⠀⠀⢸⣿⣾⠀⠀⠀⠀⠈⡏⡟⠓⠒⠒⠶⠶⠤⠤⠤⠤⠤⣤⣄⣬⣷⢶⣧⣠⣤⡤⠤⠤⠤⠤⠴⠶⠖⠒⠒⠛⡏⡏⠁⠀⠀⠀⣿⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⣼⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⣿⡏
⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢀⡇⡟⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⡇⡧⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⣧⣷⠇
⠀⠀⣸⢿⣿⡀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣨⢿⠟⡇
⠀⠀⠘⢾⡆⠙⠲⠤⣄⣀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣧⠤⠖⠋⠁⣸⠋
⠀⠀⠀⠀⠑⠦⣄⡀⠀⢸⡓⢻⠛⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⣤⣇⣿⣤⡤⠤⠤⠤⠤⠴⠶⠖⠒⠚⠋⠉⢹⢉⡇⠀⢀⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠻⠯⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣹⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠼⠽⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠒⠛⠓⠒⠒⠛⠛⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡿⣧⠀⠀⠀⠀⠀⠀⣿⢿⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⣀⣴⠟⢡⡏⠀⠀⠀⠀⠀⠀⢻⠈⠻⣦⣀⠀⢻⣦⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⡰⢻⣯⡴⠛⠋⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠙⠳⢦⣿⡿⣄⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⡘⢡⠟⠁⠀⠀⠀⣠⡾⠁⣠⠆⠀⠀⠀⠀⠲⣄⠘⢷⡄⠀⠀⠀⠉⢻⡌⢆⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣿⠁⠋⠀⠀⠀⠀⣴⣯⢴⡾⠁⠀⠀⠀⠀⠀⠀⠙⢷⣦⣻⡆⠀⠀⠀⠀⠃⠘⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⠀⠀⠀⠀⠀⠀⡿⢡⡟⠀⣰⡆⠀⠀⠀⠀⢱⣄⠈⢷⡙⢿⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⣇⡾⢻⣇⠀⠀⢀⠀⣾⠟⣷⣸⡇⠀⠀⠀⠀⠀⠀⠀⡏⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⣿⠃⠀⠻⣿⡀⢨⡿⠏⠀⢸⡿⠁⣠⠞⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡳⢄⠈⠀⠀⠀⠈⠁⠈⠀⠀⠀⠈⣡⠞⠡⠂⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠀⠀⠀⠀⠀⢀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢗⢦⡀⠀⠀⠀⠀⠀⠑⢄⠙⢦⡀⠀⠀⢀⠞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠢⣀⠀⠀⠀⠀⠀⠳⣄⠉⠢⣀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠳⣄⠀⠀⠀⠀⠈⠳⠀⢹⡇⠊⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠳⢄⠀⠀⢀⣰⣤⣼⣷⣬⡀⠈⠀⠀⠀⠀⠀⢀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⣶⣿⣿⣿⣿⣿⣿⣯⣳⡀⠀⠄⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣟⣷⣾⡇⣿⣿⣿⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣬⡼⣿⣿⡇⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣿⣿⣿⣷⣾⣷⣿⡿⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⠁⠈⠛⠿⠿⠿⠛⠉⠀⠈⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⠛⠛⠛⠛⠛⠳⠟⠛⠛⠛⠛⠛⠛⠛⠶⣤⡴⠟⠛⠛⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣷⣦⡀⢾⣿⣿⣿⣿⣿⣶⡄⠉⣠⣶⣾⣿⣿⣶⣄⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⢸⣿⣿⣀⣸⣿⡿⠇⠀⣿⣿⣇⣀⣿⣿⠏⢰⣿⣿⠏⠀⠙⣿⣿⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣼⠀⢸⣿⣿⠛⠻⣿⣷⣆⠀⣿⣿⡟⠛⢿⣿⣦⢸⣿⣿⣶⠶⣶⣿⣿⡇⠀⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠔⢹⡏⠀⣼⣿⣿⣤⣴⣿⣿⡏⣠⣿⣿⣧⣤⣾⣿⡿⠈⠻⣿⣿⣶⣿⣿⡿⢁⡀⢹⡦⡙⢦⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣷⣯⠔⠁⠀⢸⡇⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠀⢀⣤⠈⠉⠉⠙⢿⣿⠿⠁⣸⠇⠈⠢⣽⡾⠓⢦⡀
⠀⠀⠀⢀⣴⣿⣿⠟⠁⣸⠃⠀⠀⢸⡇⠀⣶⣶⣶⣶⣦⡀⢶⠾⢿⣶⡄⠰⣶⣶⣾⠀⣾⣿⡷⢶⣶⣶⠶⣶⡶⠀⢸⡇⠀⠘⣿⣿⣦⡀⠙⢦⡀
⠀⢀⣴⣿⣿⠟⠁⣠⠞⠁⠀⠀⠀⠈⣷⠀⣿⣿⡁⢸⣿⡇⣾⡿⢻⣿⣇⠀⣿⣿⠀⠀⢸⣿⠀⠀⠹⣿⣶⡿⠁⣰⠟⠀⠀⠀⠈⠻⣿⣿⣦⡀⠙⢦⡀
⣴⣿⣿⠟⠁⣠⠞⠁⠀⠀⠀⠀⠀⢠⡟⠀⣿⣿⡛⠿⠛⠀⠛⠿⠟⠛⠛⠚⠛⠟⠃⠀⠘⠿⠿⠀⠀⢹⣿⠃⢰⡏⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠙
⢻⣿⣁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠃⠀⣾⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠳⢶⡄⠐⠿⠟⠃⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⡄⢨⡝⠽⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠈⢣⡱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠸⣧⣵⡷⢶⣵⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉⣀⢢⡉⠱⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠂⠀⠀⠀⠹⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⠀⠉⢃⢣⠘⡙⠆⠘⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡁⠂⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢆⠀⡌⢧⠂⠸⡄⢀⢺⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⠂⡄⢀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢧⣸⠎⢃⢶⣛⡛⡉⣻⠗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢦⢀⡤⣤⣄⡴⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠋⠀⡙⢷⠇⣐⣒⡶⠣⠭⣐⡙⢃⣤⠄⡀⠀⠀⠀⠀⠀⣠⠘⠻⡏⢷⢴⢼⡄⡴⢉⡘⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢤⣘⡃⣘⣇⣀⣴⣟⠨⣝⣐⣖⠺⢿⡀⡌⠛⣻⠝⢿⡆⡘⠗⣞⢷⡈⢞⣂⡵⣇⡈⣹⠉⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⢠⣤⠟⢁⣆⣰⠊⢐⣾⣃⣭⣏⠡⣶⠺⠂⣩⠻⣧⠈⣙⣷⢊⠷⠎⠅⣏⡄⡐⠉⢿⠁⣌⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠟⠺⢧⡉⡽⠁⣉⣩⣅⠔⢳⣖⣩⡗⠒⣻⣿⡏⣤⣴⠋⣿⠞⢯⣥⣉⠉⠙⡿⠞⠛⠖⢣⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣏⢈⣡⠚⠚⠏⣹⠐⡦⠴⠦⣦⣤⡊⣩⡿⠒⣹⢛⡉⠡⢧⡜⣹⠛⡩⢈⠼⣆⣷⣟⡇⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣾⣿⢿⣠⢶⡅⢹⡄⢣⢾⣤⣉⡝⠙⢳⣏⡼⠯⢉⣢⣥⢝⢋⡞⠣⣎⡄⢩⢪⣉⣘⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡭⢶⣉⣷⣱⡙⡧⢥⣿⣿⣷⠶⣦⡶⠿⣏⣠⡏⢺⡃⣵⡀⠘⠿⣇⠜⣡⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⣿⣻⠟⣛⡛⢿⣿⣿⣿⣽⣯⣵⣚⣛⣱⣒⡟⣻⣧⣿⣶⣋⣷⣿⣐⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⣾⣏⣭⠤⣾⣫⠙⢋⠛⠿⠿⠿⠯⠹⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣾⣿⣷⣬⣡⠄⣠⣄⣄⣹⢤⣽⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣱⣯⣏⣉⣀⣩⣽⡟⢏⣉⡉⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣿⣡⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⣿⡿⠿⣿⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡽⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⠵⠋
⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠧⣀⣀⡈⠉⠉⠒⠒⠤⠤⣀⣀⠀⠀⣠⠖⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠢⠤⠤⣀⡩⠋⡡⠚⠡⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⡠⠊⠉⠁⠒⠒⠢⠤⢌⣉⣁⠒⠒⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠒⠤⠭⢍⣑⡒⠠⠤⣤⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣄⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⡠⠞⠁
⠀⠀⠀⠀⠀⡠⠚⠁⡠⠊
⠀⠀⠀⣠⠊⢀⡤⠊
⠀⠀⠀⠈⠒⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⣴⠋
⠰⣟⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢀⡾⠁
⠀⠈⠳⣄⡈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⠟⠁
⠀⠀⠀⠈⠛⢦⣄⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣴⠋
⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠉⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣳⣦⣙⠷⣄⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡾⣡⣞⣁⡀
⠀⠀⠀⢀⣀⣤⠤⠴⠒⣚⣛⣻⣯⣭⠭⠭⠿⠷⠿⠷⣿⠉⠙⠳⣶⠲⠶⠶⢶⡾⠛⢻⡿⠿⠭⢭⣭⣽⣛⣛⣒⠲⠤⢤⣀⣀
⣠⡶⠛⠉⣡⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⣷⠀⢰⠏⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠶⢦⡈⠙⠳⣦
⣿⣄⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⢻⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠿⠀⢀⣼
⣿⠈⠛⠶⢦⣭⣝⣛⣲⠶⠦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠶⢶⣚⣛⣭⣥⠶⠞⠋⢸
⣿⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠓⠲⠶⠿⠿⠿⠯⠭⢭⣭⣭⣭⣭⣭⣭⣭⣭⠭⠭⠿⠿⠿⠶⠶⠒⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢸
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿
⣿⠀⢿⡙⠛⠲⠶⢤⣤⣤⣀⣀⣀⣀⢠⡟⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢷⡀⢀⣀⣀⣀⣠⣤⣤⠴⠶⠚⠛⢉⡿⢸
⡟⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡟⠙⠛⠛⢿⡛⠛⠛⠒⠒⠒⠒⠒⠚⠛⠛⣿⠛⠛⠛⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸
⣷⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸
⣿⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸
⣿⠶⣄⣀⠀⠹⡄⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣸⠃⢀⣀⡴⢾
⣿⠀⠀⠉⠙⠛⢿⡶⠤⣤⣾⣅⣀⣀⣀⣀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣧⣤⣤⠴⣶⠟⠛⠉⠁⠀⢸
⣿⠀⠀⠀⠀⠀⠀⢷⣀⡾⠁⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⢻⡟⠛⢻⡟⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠹⣆⠀⣰⠏⠀⠀⠀⠀⠀⢸
⣿⡀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏⠀⠀⠀⠀⠀⠀⣸
⠈⠛⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠶⠶⠦⠤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠶⠶⠖⠚⠛⠋⠉⠁
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣶⡿⠿⠿⠛⢛⣛⣛⣉⣉⣭⡭⠍⠩⠭⠭⠭⠉⠉⠉⠉⠉⣉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠐
⠀⠀⢀⡀⣴⣾⠟⠛⣉⠤⠔⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⡙⠛⠿⢿⣿⣦⣄
⠀⠀⣾⣿⣟⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠌⡙⠿⣿⣦
⠀⠀⢻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⣿⣷⣧
⠀⢠⣾⣿⣿⡇⠘⢿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣟⣿⡟
⠀⠸⣿⣿⣷⣇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⢠⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣶⣶⢿⠃⢨⣿⣿⣿⣶⣄
⠀⠀⠙⠿⢿⠟⠘⠃⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢈⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠿⠏⠹⢿⣿⣟⣛⣛⣻⣯⣭⣿⣶⣿⡿⡇⢸⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⠿⠿⢿⣿⣛⣻⣯⣭⡿⠂⠙⠃⢺⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣰⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⢀⣾⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣸⣿⣿⡿⠟
⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠛⢿⣿⣿⠋⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⣾⣿⡉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠹⣿⣿⣿⠃
⠀⠀⠘⢻⣣⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠿⣿⣿⠆
⠀⠀⠀⠘⣷⡅⠸⣶⣽⣻⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⠀⣠⣾⡟⠛
⠀⠀⠀⠀⠘⠁⢰⣮⣝⡻⢿⣷⣯⣽⣿⣿⣿⣅⠀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣟⣯⣷⠀⣟⣽⠃
⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣾⣭⣿⣿⡿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣯⣽⣿⣾⣿⡿⢿⣻⣍⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⣿⣿⣿⣿⣿⣽⣿⣾⣿⠿⠟⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⢆⡸⠿⠿⠟⠛⠛⠋⠉⠁