Flatus Emojis & Symbols

⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡿⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢠⣿⣿⣟⡄⠀⢠⡾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡤⠀⠈⣼⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢃⠀⠀⠁⣾⣿⣿⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣄⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⠁⠀⠀⠀⠒⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⢀⣀⣤⣤⣤⣤⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡄⠈⠁⠀⣨⣾⣏⣠⣴⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠐⢾⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠀⣘⠂
⠀⠀⠀⠀⠀⣿⣧⣿⣿⣿⣿⣿⣿⠋⠛⠉⣿⡿⠿⠿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠀⠉⠁⠀⠀⠀⠀⠀⠊⠀⠀⠠⠀⠀⡀⡀
⠀⠀⠀⢀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣠⣿⣿⣿⣾⣿⣿⣶⣶⣾⣾⣖⡀⠀⠀⠀⠀⠀⠀⠄⠀⢄⠇⠀⢢⡄⠂⠀⠁⡀⢀⠀⠀⠀⠊
⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠻⢿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢒⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢁⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠈⠉⠓⠦⣄⡴⠚⠉⠉⠉⠙⠲⣄⣠⠤⠒⠋⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠉⠓⠢⡄⢠⡴⠚⠉⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠶⠶⠶⠶⠧⠴⠧⠶⠶⠴⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠀⠐⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡠⠐⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠳⠴⠴⠶⠶⠴⠤⠴⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠞


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣎⠻⣿⣿⣿⡿⠟
⠀⠀⠀⠀⣼⣿⣿⡿⠛⠛⠿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣠⣤⣤⣀⣀⣀⣀⣤⡀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧
⠀⠀⣠⣾⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⢻⣿⣿⣿⡆
⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣉⣥⣤⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟
⠀⠈⢿⣿⣿⣦⣀⣀⠀⠀⠀⠀⢿⣿⣿⠿⠛⠉⠁⠀⠀⢸⣿⣿⣿⡟⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁
⠀⠀⠈⠻⢿⣿⣿⣿⡇⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠈⢿⣿⣿⣿⣷⡀⠀⠀⠀⢰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣀⣠⣴⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠈⢻⣿⣿⣿⣷⡄⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⠙⠳⢦⣀⡼⠋⠀⠀⠀⠀⠐⠠⡀⠙⢦⡀
⠀⠀⠀⠀⠀⣠⠴⠖⠲⠶⣤⣀⣴⠋⠀⠀⣀⠀⠀⠀⠀⠈⠟⠀⣀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢷
⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠰⠿⠿⠀⠀⠀⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⡼⠁⠄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢸⠇⠘⠀⠃⠀⠀⠀⢠⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⡼⠷⣦⡀
⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠀⢸⠀⢀⡀⠀⠀⠀⣠⢠⠀⠀⠀⠀⠀⣀⠀⠙⣆
⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⢸⢆⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⣹
⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠁⠃⠳⣀⠀⠀⢀⡠⠃⠀⠙⠒⠒⠋⠀⠀⠀⠀⠈⡗⠀⠀⠀⠀⣠⣟⡀
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⠙⢦⡀
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣾⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢨⠳⣄⡀⠀⣀⡠⠖⠒⠢⢤⣀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣧⡀⠀⠀⠀⠀⠄⠀⠀⠀⡇⠀⠜⠉⢉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠈⠛⢷⡦⠀⠀⠀⠀⢀⡀⠈⠢⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⢀⡞
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣄⣀⣀⣠⡴⠟
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠘⢷⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠹⢆⠀⠀⠀⠀⣀⡀⠔⠂⢀⡴⢻⣧⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⢤⣤⡴⠾⠋⠀⠀⠉⠛⠷⠶⠶⠟⠋
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡷⠒⣆⡶⢶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⣯⣿⣀⣾⣿⣿⣬⣿⣯⡟⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣾⢿⠛⠛⠋⠉⠉⠛⠛⢻⢿⣮⣿⣷⡿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⡇⢂⣀⣀⠀⠀⠀⢀⣀⣀⠀⣻⡿⢿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣁⣽⣽⡅⢸⣿⣿⣷⠀⢰⣿⣿⣿⡀⢹⣿⣹⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣇⠹⠿⠿⠃⠀⠈⠹⠿⠿⢁⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡿⡿⠿⣿⣿⣦⣤⡤⠶⠦⣤⣤⣾⣿⠿⠿⣿⣿⡿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⢷⡿⢷⡀⠈⢻⣿⣿⣿⠯⢿⣿⣿⣿⠃⠀⣰⠿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣵⣤⣿⣦⣠⣿⣿⣿⣷⣿⣿⣿⣷⣀⣾⣧⣔⡝⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠌⠉⠻⠻⠿⣿⣿⣿⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠘⠭⠚⠛⠛⠋⠜⠇⠛⠇⠛⠀⠛⠇⠞⠆⠳⠃⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⢻⠇⡸⢛⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⡸⠃⡴⠀⠀⢸⠀⣇⣼⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⣿⡏⢠⠀⠀⠀⡄⢸⠆⢠⡏⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠻⢰⠇⣤⠀⡄⠀⡄⢠⡄⢠⡈⣷⡌⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡱⠇⣶⣹⡆⠰⢆⢱⡀⢠⠈⣧⡟⣡⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇⢻⣼⡇⣾⣀⠿⠀⣶⠃⣤⠹⢶⡌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣌⣷⠿⣌⠿⣀⠿⣉⣷⢈⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣏⣷⣟⣴⣿⣶⡿⣷⣾⣗⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡏⠀⠀⣤⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⢠⣾⠏⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⢀⡀⢹⣿⣿⣿⣿⠀⠘⢱⣿⣿⣤⣴⣤⣬⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣦⣤⣿⣥⡿⣿⣿⣷⣾⣿⣿⣿⡇⢰⣤⣿⠟⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⠿⠛⠛⠋⠀⠉⠉⠈⢻⣿⣀⡈⣹⢛⡁⣿⣿⠋
⠀⢸⠀⠀⠀⠀⠐⠾⠿⠿⠿⠿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠿⣿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠤⠖⠲⠖⢾⡿⣿⣶⣿⣶⣶⢾⣷⣶⡶⢶⣤⣀⣤⡶⠀⣴⣦⣶⣶⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣄⣀⣠⣤⣭⣅⢀⣤⣵⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⣦⣤⣴⣿⣏⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢠⡀⠀⠀⣀⣠⣴⠦⢤⣿⣾⠿⠿⠟⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠁
⠀⢸⠋⠁⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⣠⣴⣶⣦⣤⣀⣀⣤⣤⣄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣠⣴⣦⣤⢠⣏⣿⣴⣦⣵⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣷⣶⣶⣶⡄
⠀⢸⣄⣀⣀⣀⣀⣴⣦⣤⣀⣿⣿⣿⣷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⢸⣿⣿⣟⣭⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠝⢋⣩⠿⣿⠋⠀⠀⠀⠈⢻⠛⢿⣿⣿⣿⠿⠿⠿⠙⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⠿⠟⣛⣤⣴⣿⣽⣷⣿⣾⣷⣴⣿⣶⣿⣦⣀⣤⠀⢀⣠⣤⣀⣠⡄
⠀⢸⣿⣯⣽⣍⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠉⢉⣩⡇⠀⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⠉⠉⠉⠹⢯⣉⣿⠿⠟⠛⢻⠛⠛⣉⣙⢻⣥⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣉⡉⣷⠾⡇
⠀⢸⠀⠠⠂⠠⢶⠆⠀⣀⣀⣀⣘⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣛⣻⣿⣿⡷⠶⢿⣋⠉⠃⠘⠀⠁
⠀⢸⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠓⠛⣫⣭⣭⠀⠒⠚⠛⣉⣤⠀⠀⠐⠆
⠀⢸⣿⡛⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠟⣿⠿⣭⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⡛⠛⠛⠛⠋⣀⡨⠤⠶⠒⠒⠒⠂⠉⢁⡄⠶⠈⢉⣀
⠀⢸⣛⣛⣛⠓⠀⠀⠀⠀⠀⠈⣩⡍⠙⠒⣒⡶⠏⠹⢿⣿⣟⣿⠃⢤⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠶⠶⠂⠉⢀⠀⠀⠴⠶⠏⣀
⠀⠘⠀⠀⠀⠀⢀⡀⠀⠀⣀⣤⣤⣄⠀⢈⣭⣤⣤⡖⠒⠛⣼⣯⠀⣤⣀⣀⣉⣉⣀⣉⣀⣈⣉⣉⣀⣀⣤⣄⣀⣀⣉⡉⠉⠉⠀⠐⠛⢥⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣴⣶⣿⣿⣿⣿⣿⡿⠟⢟⣫⣿⣿⣿⡏⣸⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⡤⠖⣋⣬⠾⠛⣟⣯⣷⡿⠿⠟⠛⠉⠈⣠⣼⣿⣿⠞⠁⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡯⢋⣠⢶⣫⠵⠖⡻⠻⢛⡩⠄⠒⠈⢀⣠⠤⠒⣿⡿⠛⠁⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢋⡶⢫⠵⣫⠵⠚⠋⣠⠀⠀⠀⣠⡴⠊⠁⢀⣴⣾⡃⠀⠀⠀⣠⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣯⠞⣡⠞⠁⠀⣠⡾⠃⠀⣴⠌⢁⣠⣶⣾⣿⣿⣿⣿⣶⡶⣞⢷⢩⣿⡿⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣿⣼⠟⢡⣾⡷⣈⣴⠾⠉⢀⣭⣾⣥⠶⠋⣻⢿⠿⡎⢹⡄⠁⠹⣹⠸⡄⣿⠟⠁⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⢸⣿⢟⣴⡿⠋⣠⡾⠋⢠⣾⠟⠛⣩⣴⠖⢛⠱⡀⠇⢻⡄⢱⠐⢀⣿⣶⡟⠁⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⣸⣿⣧⣿⡿⠞⠉⠀⢸⡿⠁⣰⣿⠄⢠⡞⠃⠐⡄⠀⠓⡁⣈⣤⣷⢾⠟⠉⣿⣯⡇⠀⠀⠀⠀⠀⠀⠀⣸⣄
⠀⠀⠀⠀⠀⢠⣿⣿⣾⣟⡵⣿⠊⢠⡾⢁⣴⣿⡟⠉⢄⠳⣄⠶⢈⣦⡴⠿⠣⢌⡰⠏⠀⢀⣛⣛⣃⡀⠀⠀⠀⠀⠀⠰⠃⠈⢳⡄
⠀⠀⠀⠀⢀⣿⢿⣿⣿⣾⡟⣁⣴⣿⢳⣿⠟⠣⠹⢶⣀⠀⢮⡷⠏⠣⣌⢃⡕⠈⠀⠀⣠⡠⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠓⢤
⠀⠀⠀⠐⠋⢁⣼⣿⣿⣿⡿⢹⣟⣿⡿⠉⢂⠀⠑⠤⡙⣷⠟⣡⣄⣱⠾⠋⠀⠄⣀⣸⡇⠀⠀⠀⠀⠀⠁⠄⠂⠀⢀⡠⠀⠀⠀⠀⣰⣿⣧⡀
⠀⠀⠀⠀⠀⡞⣹⣿⣿⢟⣴⣿⣿⠟⠹⣄⠈⠑⢄⠀⣸⣿⣶⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⠿⠀⢀⣠⣾⡁⠺⠄⠒⠀⠀⠈⠀⠀⣸⣿⣿⠟⣉⣲⡄
⠀⠀⠀⠀⢈⣴⣿⡿⢃⣿⣿⣿⠛⣆⠀⠙⢦⡀⠀⢹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠐⣤⣤⣄⣠⣤⣤⣤⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠉⠀⣀⣭⣞⣿⣧⡳⡈⢣⡀⠀⠙⣶⣿⠋⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣻⣿⣟⣿⣿⣿⡿⠋
⠀⠀⠀⢀⣴⣾⢟⣽⣿⣻⡿⢻⣷⡀⠀⠙⢢⣴⡟⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣾⣹⡏⣺⣿⠿⠋
⠀⠀⠐⠛⣿⣿⢯⣿⣿⢟⣿⣿⢟⣷⣌⡀⠘⢿⣧⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠙⠻⠋
⠀⠀⠀⠀⠈⢻⣿⣿⣷⣿⣿⣯⣿⡿⠋⠀⠐⠿⣿⣯⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠑⢾⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠐⢶⣶⡤⣤⡤⢤⣄⣠⣦⣤⣤⣤⣤⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡶⣢⢤⡶⢦⣤⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠿⢷⣽⣿⠿⣿⣯⡻⢿⣟⣷⣿⣶⣖⢤⠀⣀⡀⢀⡀⠀⣰⢞⣻⣟⠿⣻⣽⣛⣧⣾⣼⣧⣯⠾⣫⡽⣛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣿⣶⣽⣿⣾⣦⣯⣿⠿⣿⡯⠻⣿⣿⡼⣷⣿⣜⣿⣽⣾⣗⠿⢻⡿⣫⢿⣹⣿⠿⢾⣿⣿⡿⣯⣭⣽⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⣽⣿⠫⣿⣿⡟⠛⠛⠛⠻⢿⣿⣿⣷⣮⣿⣶⣿⣿⣿⣭⣽⣿⡷⠟⠛⠛⠛⢛⣿⣡⣏⡿⣸⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⣟⠿⣏⣳⣻⣦⡀⠀⠀⠀⠙⣶⣯⢹⡯⢿⡜⣿⢶⣿⣹⠏⠀⠀⠀⢀⣠⣿⣿⣿⣿⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣜⣛⣿⣿⣟⣛⣛⣻⣿⣟⣛⣻⣷⣟⣛⣼⣾⣷⣾⣿⣛⢻⣟⠿⣿⣿⣿⣽⡾⣿⣟⣷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⢿⣿⠛⣦⣭⢿⣿⠙⢿⣭⣿⠟⣽⠏⠀⠀⠹⣷⣼⣷⡾⣋⢿⣶⣶⢾⡸⣧⡇⣿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣟⣡⡿⣿⣽⡿⢻⣤⡿⠃⠀⠀⠀⠀⠙⠿⣿⣷⣼⣷⣼⣿⣷⣿⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣯⣛⡛⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢛⣛⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⣰⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⡿⠿⠿⠟⠿⠿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⠷⠄⠀⠀⠀⠀⠀⠀⠰⠾⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣩⣭⣥⣤⣭⣿⣷⣦⣄⠹⠆⠀⠀⠀⠀⠀⠀⠰⠋⣠⣴⣶⣿⣯⣤⣬⣭⣍⣛⢶⣄
⠀⠀⠀⠀⠀⢦⣤⣾⣏⡚⠿⠿⠿⣿⣿⡿⠿⢿⣿⡿⠂⠀⡄⠀⠀⠀⠀⢰⠀⠐⢿⣿⡿⠿⠿⣿⣿⠿⠿⠿⢓⣹⣷⣤⡴
⠀⠀⠀⠀⠈⠁⠀⠀⠉⠙⠛⠛⠛⠗⠐⠛⠛⠋⠉⠀⠀⢰⡇⠀⠀⠀⠀⢸⡆⠀⠀⠉⠙⠛⠛⠂⠺⠛⠛⠛⠋⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢿⡇⠀⠀⠀⠀⢸⡿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢠⠞⠁⠀⣿⠇⠀⠀⠀⠀⠸⣿⠀⠈⠳⡄⢠⣄⡀
⠀⠀⠀⠀⢹⣶⣤⣤⣤⣤⣤⡴⠶⠞⠛⠉⠀⠁⠀⠀⠀⢿⣀⠀⠀⠀⠀⣀⡿⠀⠀⠀⠈⠀⠉⠛⠳⠶⠦⣤⣤⣤⣤⣤⣶⡏
⠀⠀⠀⠀⠀⢻⣷⡈⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢂⣍⣛⣛⣛⣛⣩⡐⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢁⣾⡟
⠀⠀⠀⠀⠀⠀⠻⣿⡄⠙⢿⣿⣄⡀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⠟⠻⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⢀⣠⣿⡿⠋⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡰⢄⡀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⡠⠖⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠚⠛⠛⠒⠒⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠊⠉⠁⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣼⠁⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⡯⠄⣀⣄⠀⠀⠀⢨⡇⠀⠀⠀⣴⣾⣔⠯⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣟⡫⠬⠽⣿⠅⠀⠀⣸⠇⠀⠀⠀⢹⡛⠛⠋⠛⠻⢿⣿⠟⠀⠀⣠⣶⣶⣤⡇
⠀⠀⠀⠀⢀⣶⣿⡿⣷⣄⠀⠘⢻⠋⠀⠀⠀⠀⢲⡄⢀⣿⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⣼⡾⠀⠀⡞⠛⠻⣷⡾⠃
⠀⠀⠀⠙⢾⣿⡔⠋⢉⣿⠀⠀⠈⠑⠀⠀⠀⠀⠘⠿⢨⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⠷⠦⠠⠻⡄
⠀⠀⠀⠀⠈⡿⠤⠀⠀⡿⠀⠀⠠⢤⣀⡀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⣀⣤⣤⣴⠞⠉⠀⠀⠸⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⣢⣤⣤⠝⠙⠳⣄⠀⠀⠀⢻⣷⠀⠀⠀⠺⢡⠀⡀⠈⠒⠛⠻⠤⠤⠵⡆⣠⠟
⠀⠀⠀⠀⠀⢺⡄⠀⠰⣿⡾⠟⣋⡥⠶⠶⠤⢌⣣⡀⠀⠸⣿⣇⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢀⣴⠿⠃
⠀⠀⠀⠀⠀⠀⠙⢦⡀⣏⠀⠞⠁⠀⢀⣠⠤⠀⠘⣳⠀⢀⡟⠘⣦⠀⠀⠀⠈⠢⣄⣀⣀⣀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢠⠞⣩⣤⡶⠂⠊⠁⢠⡾⠁⠀⠈⢧⡀⠀⠀⠀⠤⣭⣭⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⠜⠋⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⠾⠿⠷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⢀⣠⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢟⣿⣿⣿⣿⣭⣟⣛⡿⢿⡇⠀⠀⣠⣴⣤⣄⣀⠀⠀⠀⠀⠀⢀⣠⡼⠋
⠀⠀⠀⠀⣀⣀⣀⣀⣈⣿⡿⣫⡶⠟⠁⣸⣿⣧⡈⠉⠉⠛⢻⣿⡀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣷⣶⣶⣶⣄
⠀⣠⡾⠋⠁⠀⠀⠈⢹⣿⣞⣁⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢿⣇⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠙⢿⣿⣿⣿⠿⠿⠿⣿⣿⣿⠁
⠀⠈⠻⠿⣷⣶⣦⣤⣽⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣧⣤⣤⣤⣘⣿⣧⣠⣴⠶⠶⠒⠒⠶⡄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠻⣿⣿⣿⠁⠀⠀⠠⣄⣀⠀⢹⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⠏⠀⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣤⠾⠛⠉⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⢿⣿⣿⡀⠈⠉⠛⠛⠛⠛⠋⢩⣿⣿⣿⢟⣴⡿⠋
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⣛⠛⠁⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⣠⣿⣿⠟⢁⣾⠟
⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣤⣤⣤⡶⠾⠟⠛⠛⠻⣷⡄⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⣤⣤⣴⣾⣿⡿⠃⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠙⢷⣤⣤⠴⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣆⠀⠀⠐⠄⠀⠀⠀⠀⣠⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠠⣄⣀⣀⣤⣿⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠀⠀⠀⠈⠛⠛⠛⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠦⢄⡀⠀⠀⠀⣠⢊⡈⡑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠰⠦⣀⠈⠐⡄⠀⠘⠦⡅⠰⣔⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⡈⠉⠀⠀⡀⠠⠐⠈⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣄⠀⠇⠀⠀⠄⢠⡄⠀⠘⠀⠀⠀⠙⢢⡀⠂⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⠔⢉⠀⠀⠀⠀⢀⠄⠊⣀⢀⡀⠄⠀⠀⠂⡀⡀⠄⠤⡑⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠴⠷⠤⠤⢄⡤⢠⠀⠊⠀⠀⠀⠤⠤⢒⡉⠀⠂⠀⠀⠀⠀⠐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⢐⣃⠀⠀⡥⡵⠀⣀⠀⠀⠀⠀⢀⣩⠀⠔⢊⠀⠀⠀⢠⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠂⠉⠉⡄⠸⠤⠤⡄⠈⠑⠒⠒⠋⠁⠣⣭⡤⠔⣀⠔⢊⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠀⢠⡀⠀⢠⠀⠀⠂⠠⡭⠉⠈⠀⠞⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣦⣬⣹⣯⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⣾⣽⠿⠿⠟⠋⠀⢈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠐⣿⠅⠀⠀⢸⠀⠐⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡄⠀⠀⠈⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⣾⡇⠀⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡈⢹⡇⠀⠀⣸⠀⣸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⣁⣼⣇⢀⡤⠿⣶⡇⠀⣤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠄⠀⠠⡆⣈⡹⠉⠉⣉⣹⣿⡟⠂⠀⣄⠀⠀⡀⠸⠟⣻⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⢠⡀⠀⠃⠀⠀⠀⢛⣛⣳⣀⣤⣿⡿⠟⠿⠷⠾⠿⠀⢿⣥⡤⠶⠖⠋⠈⠁⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣠⢀⡀⢀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢒⡬⠤⠐⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠘⢄⡀⠀⠀⠀⠀⠀⠀⣠⠂⢠⠬⠭⠥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠉⠢⡀⠀⠀⠀⠈⠐⠊⠭⢒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⢠⠄⡶⠓⠊⠀⠀⠀⠀⣀⠤⠀⢖⣊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣐⠑⠁⠀⠀⠀⠈⠀⠡⣒⢤⠀⠀⠀⠈⠢⢘⡥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢉⠱⣈⠆⠀⠀⠀⠀⠀⢀⢀⡦⠃⠀⠀⠀⣀⠤⣘⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠣⠯⠀⠀⠀⠀⠀⠃⠏
⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⣤⣤⡀⢀⡤⠤⠤⠤⠤⡄⠀⠀⢀⡤⠤⡴⠦⠤⠤⠤⡤⠤⠤⠤⢤⣠⢤⡤⢤⠤⢤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⣀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠳⢤⡔⠋⣀⠔⠁⡤⡄⠀⠀⡇⢀⢔⠁⡠⢉⢍⡠⣪⠆⠘⡡⡢⢉⢸⠃
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠘⠬⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⡀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠀⠀⠀⡇⢐⠅⡨⢒⠕⢅⡞⣱⣟⡝⣾⠔⣡⣎⡳
⠀⠀⠀⠀⠀⠈⢻⡻⡄⠀⠀⠀⠀⠀⠫⠂⠀⣦⠶⠃⠁⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡲⠷⠦⣈⢔⠕⣡⣠⠆⢿⡉⣀⠼⢖⢕⠕⡇
⠀⠀⠀⠀⠀⠀⢀⠗⡇⠀⠀⢀⡀⠀⠀⠀⠀⠑⠤⢤⡴⠞⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠻⠇⣀⣠⡧⡢⢊⠝⡺⢊⠔⡡⢀⢔⢡⠗⢁⡇
⠀⠀⠀⠀⠀⣠⣮⠜⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢤⣄⠀⠀⢠⢄⠀⠀⠀⡸⢊⢈⠔⡡⢊⡰⣡⢊⢔⠕⣱⣡⣾⠁
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠌⠊⠈⠈⢱⠀⠀⠁⠀⠀⠀⡇⠔⡡⢊⢰⡙⠝⠉⡇⡡⠪⣪⢜⢆⣳
⠀⠀⠀⠀⠘⢷⡄⠀⢀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣢⠀⠀⢀⡠⡰⠀⠀⠀⢤⢶⢲⢷⣾⣴⡡⡠⠙⠒⡲⢋⢔⣟⢕⡑⡰⣹
⠀⠀⠀⠀⢠⠎⠀⠀⠻⣍⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠐⣉⠔⠁⠀⠀⢸⡦⠃⠁⠀⠀⠀⢹⡔⡡⠊⡠⣳⠛⡹⣏⡱⢈⢼
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠻⢇⣀⠐⠒⣊⢽⠖⡡⠪⣪⣪⡾⢊⢄⢔⡕⡏
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⡴⡖⡶⢦⡀⠀⢾⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡹⢝⠍⠐⢡⡊⢎⠝⡠⣫⣖⣻⡼⠿⠗⣧⣤⡶⠶⠶⠶⠶⠄⣀⣀⠐
⠀⠀⠀⠀⢸⠋⠀⠀⠀⢰⢏⢜⡞⢋⣉⣱⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⡗⡧⡯⠪⣪⣪⣎⠮⠴⠯⣐⡛⠻⠿⢿⡷⠖⣲⣖⣛⡛⠓⠋
⠤⠐⢒⣒⠸⠦⠤⠶⠶⠟⣛⣛⠋⠹⠿⠽⢒⣒⠶⢿⣶⡿⠶⠖⣛⢒⠒⠶⢶⡶⣶⡶⠾⣷⢿⣿⣿⣽⣿⣟⠛⢩⣯⣬⣷⠶⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢛⣿⣯⣥⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡴⠊⠙⠋⠉⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⣀⣀⣀⣀⡤⣶⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠠⠐⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠁⠉⠂⢹⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⢉⠳⡀⡇⠀⠀⠀⠀⠀⠤⠤⠭⣕⣆⠀⠀⠀⠀⢀⣴⣂⣀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢣⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠉⠀⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡴⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠴⡇⠀⠀⠀⠀⠀⠀⢴⣖⠒⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⡀⢹⡄⠀⠀⠀⠀⠀⠸⡿⠿⣷⣶⠶⠤⠤⠭⡿⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣋⠀⢀⣱⣄⠀⠀⠀⠀⠀⠹⣄⠈⢻⡀⠀⢀⡼⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠢⣄⠀⠀⠀⠀⠈⠙⠒⠓⠒⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⣀⣀⣠⡤⠴⠚⠁

⠀⠀⡀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⢀⢀⠀⢀⢀⡤⠆⠀⠐⠒⠚⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣄⡅⠰⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠀⠀⠀⢱⠶⠒⠲⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⢀⣀⡴⠊⠉⠀⠀⠀⢠⠆⠀⣰⠃⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⢠⠃⣀⣀⡇⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣀⠀⠀⠘⠐⣆⠀⠀⠀⠘⢡⠋⠉⠉⠳⡄⣀⠀⠀⠀⢘⣇⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠈⢦⡀⠀⠀⢸⡄⠀⠀⣴⣏⠉⠀⠀⣠⣾⡽⠋⠁⠀⠉⡆⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣑⣄⣱⠀⠀⠀⠙⠦⠤⣺⠏⠑⠒⠊⠉⠀⣄⣀⣀⠤⠚⠁⠀⠀⠀⠉⠐⢒⣶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⣉⣉⡉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢰⣋⠉⣠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠛⠚⠋⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣤⠼⠓⠙⠒⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠤⠴⠚⠉⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⣸⠃⠀⠀⠀⠀⠘⢧⡀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢰⠚⠉⠉⠤⠛⠚⠙⠧⡄⠀⠀⠀⡸⢉⠂⠈⢲⡌⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈⠇⠀⠀⠠⠀⠤⢤⡶⠃⢠⠒⠧⣍⡛⣄⠀⠘⠁⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⣯⠜⠉⠀⠉⠉⠀⠀⢸⠀⢦⣴⣿⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡹⡁⠀⠀⠀⠀⣀⡇⢸⣀⣀⣤⡀⠀⠀⢀⣀⠴⡠⣈⠀⡽⢏⠒⠄⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⡄⠀⣴⠛⢻⠀⡞⠁⠀⣶⠟⠒⣋⠭⠒⠉⠀⠈⡏⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢜⡦⠃⢠⡏⢠⡃⠀⠀⣟⡏⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢉⣶⢰⠃⡜⠉⠑⠒⠿⠀⠀⠀⠀⠀⠀⢠⠞⡇⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⢠⡿⠻⡜⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠇⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢿⠝⢦⡀⡸⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠁⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠔⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠒⠒⢻⠉⣩⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⡏⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠰⢾⠶⠇⢀⣇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⢻⡟⢶⠶⣲⡾⣿⣽⠿⠫⢿⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣷⣴⡶⠞⠋⠀⠈⠛⠷⠶⣴⣧⠾⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡿⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠲⠀⠀⠈⠀⠀⠀⠀⠀⠳⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠄⢀⣤⠀⠀⠀⠀⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠣⣈⡿⠏⠀⠀⢀⠔⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⢀⡤⢶⠁⠀⠈⡍⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠊⠀⠔⠊⠀⠠⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⡉⠡⠀⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣇⢀⣤⣤⣛⠁⠢⠡⠄⠠⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⠅⠀⣽⣿⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⢀⣀⣀⡀⠀⠤⠐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠐⢐⠒⠒⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡴⡇⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⢁⣘⣿⡷⠴⠇⠀⠀⠀⣠⠀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣛⣛⣹⣋⣎⣛⣟⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣿⣭⣭⣽⣿⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⡏⠀⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠂⠸⣿⣶⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣈⣆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣄⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⡀⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣽⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡇⠈⠉⠙⠛⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡠⠐⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣧⣤⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⠊⠱⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠴⠠⠒⠒⠉⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⣡⠀⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠒⠋⠀⠀⠀⢙⡶⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢠⠏⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠁⢠⡀⠀⠀⠀⠀⠈⠀⠈⠑⣄⢀⠀⠀⠀⠀⢇⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠀⢣⣄⠀⠀⠀⣄⠀⠀⠬⣿⠁⠀⠀⢀⠔⠀⠋⠀⠀⠈⡷⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣦⣘⡆⠀⠀⠈⠒⠒⣒⠏⠀⠀⠀⠀⠀⣄⣀⢠⡴⠚⠁⠀⠀⠀⠈⠁⢙⡳⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⡤⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠰⣏⡁⣠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠋⠘⠉⢀⡠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡤⠶⠛⠈⠓⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠚⠉⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⢀⡰⠃⠀⠀⠀⠀⣈⠳⣄⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡂⠀⠀⠐⠛⠋⣉⠗⣤⠀⠀⠀⣘⢉⠇⠈⠳⡄⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡶⠃⢠⠞⠓⠬⡙⣦⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡠⠜⠁⠀⠀⠀⠀⠀⠸⠂⢰⡴⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⢠⣇⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠉⢹⠃⡜⠀⠀⣴⡜⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⣵⠂⠀⡞⢀⣇⠀⠀⢏⡏⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⢀⡵⢆⠃⣸⠀⠈⠐⠚⠁⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⣄⣾⠛⠾⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠹⠘⠓⠄⣠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢀⣀⣀⠠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠂⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠹⡇⢸⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⡜⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠈⢛⣟⣋⣀⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⢹⠿⠽⢟⣭⠿⠿⣾⣿⠗⠻⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣿⡶⠾⠛⠁⠀⠀⠈⠙⠛⠶⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠠⠀⠀⠠⠀⠀⠀⠄⠄⠄⠄⠀⠀⠀⠀⠀⠆⠆⠀⠄⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⢢⣀⣀⣀⠤⠤⠤⠤⠤⠤⠤⠒⠒⠈⠉⠀⠀⠸⠀⠀⠠⠊⢉⣉⣉⡑⠢⡀
⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⢀⣤⠤⠤⠀⢣⡤⡇⠀⠀⠀⠀⣀⡀⠀⠀⣀⣤⣤⣀⠀⠀⡆⠀⡇⢰⠁⠀⣠⠜⢁⠇⣀⡄
⠀⠀⠀⡠⠔⠊⠁⠀⠀⠠⠏⠀⢀⣀⣤⠀⢣⢰⠀⠀⠔⠋⠉⠉⠂⠘⠁⠀⠀⠈⠁⢀⣁⠔⠁⡌⠀⠰⣁⡾⠕⣉⠤⠚
⠀⡰⠉⠀⠀⢀⡴⠚⠓⠀⠀⠀⡟⠁⠀⠀⠀⢣⡆⠀⢀⣤⠶⢦⡀⠀⠠⠞⠛⠓⠂⢸⠤⠤⠊⠀⠀⠀⠈⠈⠉
⠀⠘⢄⠀⠀⠼⠁⢀⣤⠤⠄⠀⠀⡀⢠⠞⡆⠈⡾⡄⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸
⠀⠀⠀⠑⢄⡀⠀⠞⠁⠀⠀⠀⣲⡴⠋⠀⡀⠀⡇⠰⡀⠀⢀⣀⣤⣭⠤⠤⠤⡄⠀⡇
⠀⠀⠀⠀⠇⢨⠦⡀⠀⠠⡖⠋⠁⠀⠀⡠⠃⠀⡇⠀⠱⡀⠾⠒⠊⠉⠉⠉⠉⠉⡰
⠀⠀⠀⠀⡄⠇⠀⠈⠑⢄⡈⠑⠒⠒⠊⠀⠀⡠⠁⠀⠀⠘⠦⡀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠱⡗⠤⣀⠀⠀⠈⠉⠑⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠐⠁
⠀⠀⠀⠀⠀⠈⠒⠤⣈⡑⠒⠄
⠀⠀⠀⠀⠀⡀⠤⠤⠤⠬⢳⠞
⠀⠀⠀⠀⠀⠧⠔⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠛⠉⠉⠉⠉⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠂⠀⠀⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⣤⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⢀⡞⠁⠀⠂⠁⠀⣱⠏⡴⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢸⠁⠎⠀⠀⠀⠊⡏⣸⠃⠀⢀⠤⠂⠈⠁⠀⠀⣀⣈⣐⣠⡀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⢻
⠀⠀⠀⢸⡌⠀⠀⠀⠀⠀⠻⡇⡠⠂⠁⠀⠀⢀⡠⠖⠋⠉⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄
⠀⠀⠀⠀⢻⡀⠀⠀⠀⡆⠠⡏⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⣀⣀⣠⣤⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠂⠀⣇⠀⡠⠐⠂⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠒⠀⠀⢸⡇⢸⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠀⢹⡎⠀⠀⠀⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡾
⢀⣀⠀⠀⠀⢸⠀⠀⠀⠁⠀⢸⢷⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⠀⢸⡽⠁
⡞⠻⣄⣠⡴⠃⠀⢀⠈⠀⠀⡞⠀⡷⠋⠙⠛⠂⠀⠀⠀⠀⣀⣠⠆⠀⠀⠀⠠⣄⠀⠀⠈⠉⠀⠀⠘⡇
⢳⡀⠀⠀⠤⠀⠂⠀⠀⢀⡞⠁⠸⡇⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠀⠀⠀⠀⣿
⠀⠙⠦⢤⣤⣤⣤⠤⠖⠋⠀⠀⠀⠙⠲⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣠⣤⣴⣾⡆⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢿⡿⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠇⠖⠒⠋⠀⢀⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠴⠦⢄⣀⣀⣀⣀⣤⡤⠤⠖⠛⢻⠉⢹⡗⠒⠒⠒⠒⠊⢉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⢀⠀⢠⠁⠀⠀⠠⠤⠤⠒⠀⠀⠈⣧⣼⠦⠶⠒⠒⠒⠢⢄⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⢦⠏⢀⡴⣄⣀⣄⠀⠠⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠈⠉⠋⠀⠉⠴⡇⠈⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣉⣙⣲⣌⣿⣁⣀⣤⠤⠤⠴⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠉⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣈⣹⣏⣁⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠚⠒⢺⡗⠓⠚⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣸⣇⣀⣰⣇⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠂⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⠀⠈⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠚⠀⠀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠑⠘⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⢺⡆⠀⠀⠀⢀⠀⠰⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢒⡗⠀⠀⣀⡄
⠀⠀⠀⠘⢆⠀⠀⠹⠗⠀⠀⠹⠀⠀⠀⠀⠥⣀⠀⢀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⡀⠀⣠⠆⠀⠀⠀⠈⠋⠀⠀⠺⠏⠀⠀⡴⠃
⠀⠀⠀⠀⠈⣃⠀⠄⢠⡈⠒⠀⠡⣀⠠⢄⣀⣀⣥⣤⣤⣤⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀⡠⠄⣠⠌⠀⠐⢁⡄⠠⠀⣘⠁
⠀⠀⠀⠀⠀⣿⠓⠶⢤⣬⣠⣤⣦⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣴⣤⣄⣥⡤⠶⠚⣿
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣰⣆⡀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠙⠛⠶⣄⠀⠈⠿⠿⠁⠀⣠⠶⠛⠉⠉⠉⠉⠉⠛⠲⢤⣀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⣀⣀⣠⣤⠾⠻⠋⠀⠀⠙⠟⠷⣤⣄⣀⣀⣀⣠⡤⠶⠋⠁⠀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⣀⣀⣀⣀⣀⣤⡴⠶⠖⠒⠛⠛⠒⠲⠶⢦⣤⣀⣀⣀⣀⣀⣀⣀⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠋⠉⠢⡀⠀⠀⠀⠀⢀⡴⠚⠉⠓⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⡌⠀⡀⠀⠀⠹⡄⠀⠀⣠⠋⠀⠀⠀⠀⢸⠀⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⡇⠀⠀⠀⢀⠀⢳⠀⢰⠃⠀⡀⠀⠂⠀⡸⠀⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱⠀⠀⠐⠀⠀⢸⡄⡇⠀⠂⠀⠀⠀⢠⠃⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⠀⠀⣿⠀⠀⢀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡤⡌⢣⠀⠀⠀⠀⡇⠀⠀⠀⠂⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⢦⠀⠀⢸⢁⠇⠈⢢⠉⠀⢸⡇⠐⠈⠀⢀⡞⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠃⠓⠒⠛⠘⠛⠉⠉⠑⠒⠚⣄⠀⠀⢀⠌⠉⠙⠋⠉⠉⠳⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⢀⡴⠉⠀⠀⢀⣀⠠⠀⠀⠀⡀⠀⡠⠄⠒⠢⢄⠀⠈⢦⡴⠃⠀⠀⠀⠐⡀⠀⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠈⠙⡄
⠀⠀⠀⡾⠀⢠⠦⡤⠤⡤⠵⡀⠀⠀⠁⠞⠒⠲⠰⠒⠘⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠁⠀⠀⠀⠀⠀⠓⠈⠒⠲
⠀⠀⠀⣧⠀⢇⣀⣐⣚⣁⣀⡇⠀⠀⠀⢦⠐⠒⠒⠒⢒⠇⠀⠀⠀⡀⠀⠂⢀⡎⠀⠂⠀⠀⣀⣀⣸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠤
⠀⠀⠀⢧⡀⠈⠢⣤⣠⡤⠊⠀⠀⠀⠀⠈⠒⠤⠤⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⢹⡀⠀⢀⡠⠄⠀⠀⠀⠀⠀⣀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠉⠁⢀⡞⠉⠙⠓⠲⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⢀⡤⠛⠋⠉⢀⣀⣤⣀⣄⠼⠉⠛⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⣀⣀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⡤
⠀⠀⠀⡼⠀⢀⡔⠋⠉⠳⠤⢤⡄⢀⠀⠀⡤⡶⢋⠀⠀⡏⢹⠀⡆⣷⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢠⠇⠀⡼⠀⠀⠀⠀⠀⠈⠳⠋⠋⠛⢡⠃⡎⣼⠀⡇⢸⠀⡇⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠋⠁
⠀⢀⡞⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇⣿⡀⡇⢸⣠⢧⡎⢣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⡤⢤⠀⠀⠀⠀⢀⡖
⣞⠉⠄⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⡼⠷⡄
⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠎⠀⣠⣇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠞⢫⠟⠛⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡴⠃⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⡏⠀⢀⡿⠒⢲⠀⠀⠀⠀⢀⡟⠀⠀⠀⠠⠎⠙⠓⣲⠖⠲⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠾⠀⠐⠋⢀⠴⠚⣓⣦⣀⣴⣯⢷⣦⠀⠀⠀⠀⠐⠚⠁⣀⡴⠋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠛⠒⠒⠚⠒⠒⠒⠒⠂⠙⣋⣿⣠⣷⠖⠛⠂⠀⠀⣀⣤⣤⣭⡥⠴⠋⠁
⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⢻⣀⡼⠻⠄⠀⢀⡞⠁
⠀⠀⠀⢀⡞⠀⠀⣤⣄⣴⣶⠄⠀⠈⢷⣄⠀⢠⡀⠀⠈⠉⠛⠀⠀⠀⠀⣼
⠀⠀⠀⣼⠀⠀⠀⣸⣿⣿⣷⡄⠀⠀⠀⠻⣧⠈⢻⡄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⡇⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⢻⠀⣸⠇⠀⢤⣄⣀⣤⣤⡟
⠀⠀⠀⣧⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣾⠀⠋⠀⠀⠀⠠⣤⣀⣤⣿
⠀⠀⠀⠹⣷⣞⣋⣀⡀⠈⢻⡀⠀⠀⢀⣼⠏⠀⠀⠀⠑⣦⡀⠠⢤⠤⣾
⠀⠀⠀⠀⠀⠉⠉⠉⠉⣧⣸⠇⠀⠀⠈⠀⠀⣀⣠⢤⡞⠁⠉⠛⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠖⠒⡟⢿⡁⠰⣄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡷⠶⠚⠋

⣿⣷⠿⠷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⠋⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣷
⠀⠀⠀⣠⠴⠋⠀⠀⠀⢀⣀⣤⠴⠖⠒⠛⠋⠉⠉⠉⢙⠓⠲⢤⣹⡀
⠀⢀⣠⠟⠀⠀⠀⣠⠖⠋⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠹⡇
⣠⣟⠁⠀⠀⢠⡞⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢉⡟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡤⠜⠂⠀⡇
⠘⣧⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠹⣆⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠆⠀⠀⠀⠀⠀⠙⣦
⠀⠀⢹⡄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸
⠀⠀⠀⠻⣄⠀⠈⣻⠄⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⡄⠀⠀⡾
⠀⠀⠀⢰⠏⠉⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⡇
⠀⠀⠀⠘⠷⡤⣦⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠋⠙⣋⡡⠟⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠧⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠲⣤⣤⡤⠴⠶⠶⠚⠛⠉⠉⠈⢹⣆⠀⠀⡏⠉⠉⠉⠉⠉⠐⠒⠉⠁⢀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠞⠋⠉⠉⠉⠉⠉⠉⠁⠛⠶⣤⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⡷⠤⣤⡞⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣿⣁⣀⡀⡴⠃⠱⠖⢊⡦⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⡀⠀⠀⢀⣉⣙⠆⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠒⠒⠤⣩⠇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⣶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠛⠛⡟⠛⠛⡿⠛⢻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢼⡇⠀⠀⣿⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡇⠀⠀⣿⣿⣿⣷⣤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⡿⠭⠥⣤⢛⡛⠒⠂⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⣿⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⣿⢸⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣦⣀⠀⠙⠿⣿⣿⡇⠀⠀⣿⢸⣿⠀⠀⠀⠀⣠⣴⣿⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣎⠻⣷⣄⠀⠈⠙⠃⠀⠀⣿⣸⣿⣠⣤⣶⣿⣿⠟⢋⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣝⣿⣦⣄⣠⣤⣄⣀⠿⠿⠿⣿⡿⠛⠁⠴⣋⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠻⠿⠿⠿⠿⠿⠿⣒⣂⣭⡤⣠⣦⡺⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠀⡀⠀⣿⡿⣿⣾⣿⢻⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣷⣤⣴⣿⠂⣿⡀⠙⠛⠃⠀⡙⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣤⣼⣿⠇⠀⣿⣧⠀⡚⠛⠛⣡⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⠁⠀⢠⣿⣿⣿⣿⡆⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢸⢸⣿⣿⣿⣧⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣠⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣲⠭⢉⣹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠋

⣿⡯⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⢽⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⠽⠋⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⢝⢦⡀⠀⢀⡤⡺⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⣽⣿⡵⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⢔⡫⠚⠛⢮⡢⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣠⣾⠗⠉⠀⠀⠀⠀⠙⢮⣢⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣯⣄⡉⠁⠀⠀⠀⠀⠀⠀⠙⠪⡲⣄⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠨⠉⠙⠓⠶⢤⣄⣀⠀⠀⠀⠈⠪⡳⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⣈⠪⡳⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⣇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜⡿⡟⢙⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣤⣶⣵⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠚
⠀⠀⠀⠀⠀⠀⢰⡿⣻⣯⣵⣿⣿⡏⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣠⣃⣤
⠀⠀⠀⠀⠀⠀⠸⣿⠻⣿⡿⣿⣿⢃⣾⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠏⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡾⠿⠿⠿⠊⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠤⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠴⠃⠈⠓⢢⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢁⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡞⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡟⡗⠚⡇⠀⠀⠀⠀⣠⡶⠶⠂⠐⣧⢒⡶⠃
⠀⠀⠀⠀⠀⠀⠀⢄⣏⠸⣇⣇⣀⣙⣲⡄⢸⣾⢟⣶⣄⣀⠚⢉⣀⣀⣀⣀⣀
⠀⣁⡀⠤⠒⠚⠃⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠀⠸⠖⠠⢀
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠖⠒⢲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠖⠋⠉⠙⠢⣼
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⡠⠒⠈⠁⠀⠀⠈⠙⠲⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡤⠤⢷⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⡄⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣴⣾⣿⣿⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣋⠉⠈⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⡀⠀⠀⣀⣀⣀⠤⠖⠚⡅⠠⡤⠤⠤⠄⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢈⠉⠉⠉⠀⠀⠀⠀⠀⢹⣤⡧⠤⠤⠤⠤⣈⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠲⣞⢀⡴⣄⣀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⡈⣉⠀⢀⢲⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠤⠵⠶⠗⠒⠒⠒⠛⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠠⣤⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⠶⠶⣿⠶⠶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⣀⣿⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠁⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠉⠉⠉⠉⠙⠻⣦⣄⢀⡼⠋⠀⠀⠀⠀⠀⠙⠒⠢⣄⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⣠⡴⠋⠉⠀⠀⠀⠈⠙⠳⣦⣾⠁⠀⠀⠀⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠐⠛⠋⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⣠⡟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⢠⡿⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠛⢷⣄
⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡇⡆⠸⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⣧
⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⣰⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹
⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢣⠹⣄⠀⠀⠀⠀⠀⠀⢀⠆⠀⠓⢤⣤⣤⣤⠞⠃⠀⠀⠀⠀⠠⠴⣤⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⢀⣹⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠐⠻⠷⣤⣀
⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⢰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⣿⠃⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠑⢦⠄⠤⢤⡴⠚⠁⠀⠀⠉⠛⠒⠶⠖⠋
⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⢰⡆⠀⠰⠃⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰
⠀⠀⠈⠻⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⢀⣾
⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠈⠀⣠⡾⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣴⣶⡶⠟⠋
⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢕⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠶⠿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠳⣄⡀⠀⠀⠀⠠⠤⠴⠚⠁⢀⣠⣾⠋⠙⣿⣦⣄⠀⠀⠀⠀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⡤⢤⣤⣤⣶⡾⠟⠋⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠐⠂⠀⠀⠀⠀⠂⠂⠀⠃⠂⠁⠀⠀⠐⠀⠐⠀⠀⠀⠀⠈⠀⠀⠘⠀⠂⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣻⣄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠝⢽⠭⡢⣱⠑⠀⢤⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠒⠒⠒⢄⠴⠊⠁⢱⠊⠀⢈⣳⣄⠔⢭⠎⣰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⠽⢷⡾⠦⢤⣠⠎⠀⠀⠸⢋⣉⢳⢄⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠉⠉⣡⠙⠶⠊⠀⢠⣶⠆⠀⣤⣴⠿⠋⠉⢉⠐⠢⡄⠀⠀⠀⠰⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⡀⠀⠀⠀⣀⠔⠁⠀⠀⢀⣔⡧⢁⣠⠴⠿⣀⠀⠀⢀⠆⠀⢀⠇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠈⠉⠉⠉⠀⠀⢀⣠⠶⣉⢎⡔⠁⠀⠀⠀⠈⠳⡄⠘⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢩⠼⠊⢁⡎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢺⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⢤⣈⡬⠄⠉⠁⢇⠉⠀⠀⠀⠀⢀⣀⡠⠔⠊⠁⠀⠸⣄⣂⠀⠀⠀⠀⠀⢸⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡜⠁⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠿⠆⠀⠀⠀⠀⣇⢀⠎
⡠⢮⠿⣳⢊⡱⠀⠀⣃⣤⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⠄⠐⠒⠚⠀⠀⠀⣸⠁⠈⡻
⢩⠁⠀⢠⢃⡠⠐⠊⠁⠀⠈⢉⣗⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠒⡶⠀⠒⠈⠁⡃⠀⢀⢃⠀⢰⡁
⢈⠃⠀⡜⠇⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠉⠙⠓⠶⠖⠠⠤⣄⣿⠀⠀⠀⡘⠀⠀⠀⠘⠅⠲⠀⢰⣺⠁⡜⠁
⠸⠀⡠⠤⣞⣀⠔⠊⠉⠚⣄⠀⠀⠉⢐⣒⣶⡖⠛⠛⠛⠛⠛⢋⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⣉⣀⣤⠇
⠈⠊⠁⠀⠉⠀⠀⠀⢠⢤⣌⡩⠍⠬⠷⠌⠙⠓⣶⠠⠤⣄⣀⠀⠀⠀⠨⠀⠀⣀⡠⠤⠐⠊⢯⣸⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣡⠤⠦⠴⠖⠒⠂⠈⠀⠀⠀⠀⠀⢉⠃⠒⠖⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⣉⠉⠉⠛⠛⠛⠛⠛⠛⢋⣽⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠂⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢸⠇⠀⠀⠀⠀⢀⣤⠀⠀⠘⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠛⠋⠛⣯
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣶⠾⠋
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠰⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠲⠤⣀⡀⠀⢀⣀⣠⠤⠶⣟⠁
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⢠⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣧
⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡿⠀⠀⠀⠀⡠⠀⠀⢄⣀⡤⠖⠒⠀⠀⠀⠀⠠⡀
⣼⠁⠀⠀⠀⠀⠰⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡼⠁⠀⠀⡐⠁⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡞⠁⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⢿⠠⠀⠀⠀⠀⠁⠈⠀⠉⠈⠉⠀⠒⠂⠤⣀⡀⠀⠀⡈⣰⠏⠀⠀⠀⠀⢀⣈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢇⡀
⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⡟⠁⠀⣀⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠂⠁⠀⠀⠈
⠀⠘⣧⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⠀⠀⠀⠀⠀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⡠⠂
⠀⠀⠸⣆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠉⠉⠉⠉⠀⠀⠀⠈⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠙⣆⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠋⠙⢦⣀⠀⠀⠀⢀⠠⠔⠋
⠀⠀⠀⠀⠘⣦⠀⠀⢱⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠈⢧⡀⠀⢃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠈⣳⡄⠘⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣤⡴⠥⣌⡀
⠀⣠⣤⠴⠖⠛⠋⠋⠁⠀⠙⠀⠀⠉⠛⠛⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡀⠙⠉⠁⠀⠀⠀⠀⠈⠳
⠾⣥⣄⣀⣀⣀⣀⣀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣋⠤⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠉⠙⠛⠛⠒⠒⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠐⠛⠒⠒⠉⠈⠓⠢⠤⠒⠊⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣠⡾⠛⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⡾⠋⠀⠀⢀⣀⣀⣀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣴⡟⠁⠀⣤⣾⣿⣿⣿⣿⡷⠀⠀⠉⠛⠳⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⠏⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⡛⠿⢶⡿⠛⠛⠛⠛⠻⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⢷⣀⠀⢀⣀⣄⠀⠀⢀⣠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠙⠛⠛⠋⠁⢀⣴⠟⠁⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠈⠋⢠⡏⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠻⣦⣀⠀⢀⣀⡀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢀⣠⣤⣤⣾⠁
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣴⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⢛⣉⠀⠈⠹⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣷⣄⠀⠀⠀⠀⠀⠀⣴⡟⠉⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘⣯⡉⠙⠛⠛⠷⠶⠶⠿⠿⠟⠛⠛⣉⣽⣿⠈⠁⠀⠀⢀⡀⠀⢸⣧⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢿⠙⠳⠶⣤⣤⣤⣤⣤⣴⣶⣶⠟⠋⠉⣿⠀⠀⠀⢀⣿⡇⠐⠛⠁⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣰⠏⠀⢠⡟⠁⠀⠀⣸⡟⠀⠀⠀⣾⠿⣷⣄⣀⣠⣴⠟⠃
⠀⠁⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⢻⡄⠀⠀⢰⡏⠀⠀⠈⠀⣀⣤⡾⠟⠀⠀⢀⣾⠟⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠙⠶⢦⣼⣷⡶⠶⠾⠛⠋⠉⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣄⣀⡀⠀⠀⠀⣀⣀⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⣤⡴⢶⣶⡄⠀⠀⡀⠀⠀⠠⣰⣆⠄⠀⠀⢀⠀⠀⢠⣶⡶⢶⣤⣄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠧⢲⣿⣏⠁⣆⡜⣩⡓⣤⡲⠯⢙⣆⣿⣿⣰⡃⠽⢖⡤⠚⣍⢣⣰⠈⣹⣿⡆⡽
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣄⢁⡽⣃⠈⠙⠛⠷⠿⣙⡟⣉⠛⠙⠪⠛⣉⢻⣋⠿⠾⠟⠋⠀⣙⢯⣈⣠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠘⣇⠙⣛⣡⠂⠉⠉⠒⢬⠫⡳⠜⣀⣶⣾⣀⠣⠞⢝⡥⠒⠉⠉⠐⣌⣛⠋⣸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠸⣷⣿⣧⠀⠰⠿⠀⠀⡹⠞⠅⠘⠉⠉⠁⠩⠳⢏⠀⠀⠿⠆⠀⣼⣿⣾⠇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⣿⣌⠻⣷⣄⣀⡠⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⢄⣀⣠⣾⠟⣱⣿⡟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠄⠩⢧⠙⢏⣁⡴⠟⠋⠉⢳⡀⡀⣀⢀⡞⠉⠘⠻⢦⣉⡹⠋⡼⠝⠤⡀⠀⢱
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡎⠠⣤⠶⢧⠀⢻⡒⠂⣢⠴⠒⠚⠒⠒⠓⠒⠮⣔⠒⣒⠟⠀⣼⠴⣤⠄⠱⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡄⢻⡀⢮⣧⠀⢹⡏⣁⠀⠀⡀⠈⠁⢀⠀⠀⣌⢹⡏⠀⣼⡵⢀⡟⢀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣌⡢⣑⣦⠟⡆⠀⢷⠁⣸⠀⡁⠀⠀⠈⠀⣇⠈⡞⠀⢰⠻⣴⢋⢔⣡⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠁⣔⣿⣄⡌⢰⣷⣶⣿⣿⣿⣿⣶⣿⡆⢣⣰⣿⣢⠈⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠉⠤⢶⣿⠁⢹⡿⠁⣸⣿⡿⠿⠿⠿⠿⢿⣿⣗⠈⢿⡏⠈⣿⡶⠠⠉⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢾⠃⣼⠙⢿⣀⠀⠀⠀⠀⣀⡿⠋⣧⠘⡷⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠊⢉⠀⡈⡿⠽⠒⠒⠮⣿⢁⠀⡉⠑⢁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡄⠀⢯⠀⡀⠠⡢⠘⠋⢔⠄⣀⡀⡽⠀⢰⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠁⠒⠈⢀⣀⣀⡀⠁⠐⠈⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢔⠶⣤⣴⣾⠟⠛⠛⠻⣷⣦⣤⠶⡢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠀⠁⠀⠀⠈⠐⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠈⢳⣤⣤⣤⣤⣤⣄⣴⠾⠿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠙⠳⣄⠀⠀⠀⠙⢿⣄⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠖⠂⠀⣠⣴⠶⠿⠿⠿⠿⠿⠿⢿⣷⣦⣄⣀⣼⠟⠋⠉⠀⠀⠈⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣧⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠀⠀⠀⠀⠉⢻⡟⢁⣤⠶⣦⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⣾⣡⣦⠘⡧⠀⠀⢀⡾⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⡼⠿⣿⣇⠀⠀⣸⣵⣶⡀⣸⠃⣀⣀⣀⣀⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣠⡶⠶⠶⠶⠶⣤⣿⠿⠛⠳⠶⠤⠀⠉⠙⠻⢏⣰⠿⠋⠉⠉⠀⠈⠉⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⢀⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢋⡽⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣄⣤⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⡶⠋⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠒⠶⠦⠤⠤⠤⣤⣤⡿⢱⣶⣶⡄⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⣆⠀⠀⠀⢀⣀⡤⠶⣻⡽⣿⣿⡇⣴⡶⠶⠿⠛⠋⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡬⠿⣷⣖⣋⣩⡴⠖⣋⣡⢟⡶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠒⠒⠒⠚⠛⠋⠉⠉⢿⣆⠀⣀⡬⠟⠛⠛⠟⣻⠟⢁⣞⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠿⣧⣤⣀⣀⢀⣼⣁⣀⡼⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠙⡻⠋⠛⠉⢹⠇⠀⠀⠘⢿⢿⣷⣶⣤⣤⣤⣶⠟⠋⠉⠛⠿⢿⣷⣶⣦⣤⣤⣤⣤⣶⡾⠟⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣾⠀⠀⠀⠀⠈⢣⡈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⣎⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⣡⠃⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠀⠐⣜⡀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⢀⣣⠂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠰⢄⠘⠂⠐⢀⣈⡀⠀⠀⠀⠀⢀⡁⠀⠂⠐⠂⡀⠆⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠦⠤⠴⠒⠉⠉⠉⠉⠉⠒⠚⠉⠉⠉⠉⠙⠒⠦⠤⠴⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣤⠒⠊⠉⠑⠲⢤⠀⢀⡤⠒⠋⠉⠑⠲⣄⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⠓⠦⠤⠤⠤⠚⠁⠈⠓⠤⠤⠤⠴⠊⠁⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢤⣄⣠⠤⠤⠖⠒⠒⠲⠤⢤⣄⣠⠤⠔⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣲⣯⠤⠶⠒⣒⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠊⠉⠀⠀⠘⠉⠉⠉⣉⣩⡭⠽⠶⠦
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⠭⠝⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣞⣀⣀⠀⠀⢀⣀⣀⣀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣞⣉⣡⡤⠤⠤⠭⢽⡯⠤⠤⠤⠤⢵⠄
⠀⠀⠀⢀⡤⠒⠒⠢⣄⡇⠀⠀⠐⠓⡒⠒⠠⠤⠄⠒⠊⠙⠒⠒⠲⣖⡭⠖⠲⣄
⠀⠀⢀⡏⠰⠞⠓⠆⠈⡁⠀⠙⣆⢙⠦⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠀⠠⠾⣁
⠀⠀⠘⢄⠠⠤⠒⠈⠀⠈⠲⣄⠈⢿⡀⢰⠁⢠⠏⠉⡽⠉⠉⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠉⠳⠤⠮⠤⠴⠛⠇⠀⠀⠀⢀⣀⡤⠖⠋⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⡀⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠇⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡤⡾⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠲⢍⡑⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠳⠴⠋⠽⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠦⡀⣱⠚⠀⠰⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠄⠀⠀⠀⠀⡄⠉⠀⠀⡿⠧⠤⢄⣀⣀⣀⣀⣀⠀⠤⠇⠰⠋⣿⢄⠀⠀⡜⢰⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⡿⠂⠀⢸⠲⢄⣀⣿⠒⠒⠒⠒⠒⠂⠀⠀⠈⢉⠁⢄⡴⡇⠀⢑⡄⢙⠎⢆⠀⠀⠠⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⣳⠀⠀⠈⡇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠀⠀⡇⠀⢸⠀⡎⠀⠎⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⣿⣟⠁⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠪⡀⠀⡇⠀⡼⠀⡇⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡇⢻⣤⢄⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣴⠇⠀⠇⠈⠁⠀⠀⠀⠀⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡨⠝⠧⠼⠍⠉⡗⡶⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠂⠀⠘⡄⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⢣⠃⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⡇⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠎⡞⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠀⠶⠶⢾⠾⡃⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠺⡄⠀⠀⠀⠀⠀⠑⢢
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⡇⠀⠀⣮⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢠⣰⠇⠀⠀⠀⠀⠀⣴⢶⡢
⠀⠀⡖⠋⠉⠉⢹⣟⡟⢛⣛⡿⠿⢿⣿⣛⣛⣫⣋⠉⠉⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠛⠿⠶⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠉⠉⠓⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠆⠀⠀⠀⠀⠀⠀⠤⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣁⣣⡤⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⢾⣯⡽⠿⡒⢛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠙⠒⠛⠁
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡜⢿⣿⣿⣿⠟⠀⠀⠀⡁⢀⢈⠀⡀⡁
⠀⠀⠀⠀⠀⣾⣿⡿⠛⠛⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠉⠀⠀⠀⠀⠀⠂⠤⠠⠀⠀⠂
⠀⠀⠀⠀⢸⣿⣿⡁⠀⠀⢀⣴⣤⣤⣤⣤⣤⣄⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢐⠀⠀⠂
⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠟⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠈⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠅⡄
⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⡆⢸⣿⣿⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂
⠀⠀⠀⢿⣿⣧⡀⠀⠀⠀⠀⠰⣿⣿⡿⠟⠛⠉⠁⢸⣿⣿⡟⣿⣿⣿⣧⡀⠀⠀⠀⠀⢀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⡅⠀⢨⠈⡁⡅
⠀⠀⠀⠈⠻⢿⣿⣿⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣾⣿⣿⡇⠈⢿⣿⣿⣷⡀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣤⣴⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠈⢻⣿⣿⣿⡄⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠈⡁⡁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣀⢈⠀⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠠⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢘⠀⠂⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠨⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠭⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠉⠻⠟⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡴⠊⠀⠉⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠓⢀⡀⠀⠀⠀⠀⢀⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠀⡀⠀⠀⠀⠀⠀⡀⠈⠂⠀⠀⠀⠄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡾⠁⠀⠀⠀⠐⠚⠹⠆⠀⠀⠀⣠⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠙⢃⠀⠀⠀⠀⣶⣄⠒⠂⢠⡀⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢀⠟⡀⠀⠀⠀⠹⡙⢿⡆⠒⠚⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡉⠀⠘⢤⡀⠀⠀⠑⠰⠧⠄⠂⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠀⠀⠠⡁⠀⠀⠀⠽⠶⠦⢤⣤⣤⣤⡐⠋
⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠦⠴⠤⠤⠤⠤⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢸⠁
⠀⠀⠀⢀⠸⣄⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⡠⠊⠀⢀⠗⠐⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠰⠁⣀⡔⠁⠀⠀⠀⠃⠀⠀⠀⠀⢀⣤⠄⠀⣀⠀⠀⠀⢠
⡌⠀⠀⠈⠀⠀⠀⠙⠛⠋⠁⠀⠀⠜⠁⠉⠻⣿⣿⠆⠀⠀⠈
⠓⠀⠀⠒⠒⠂⠀⠀⡞⠉⠉⠁⠀⠈⠁⠀⠚⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⠒⠶⠒⠀⠘⠲⠒⠒⠒⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠘⠀⠀⠀⠀⠀⡆⠘⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠋⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣸⡆⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠑⢄⡀⠀⠀⠀⠀⢠⣌⠙⢩⣶⣶⠆⠀⢴⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⠎⠙⢂⠀⠀⠀⠀⠉⠛⠛⠿⠻⢤⣤⣌⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⣀⡚
⣿⣿⣿⣿⠿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣠⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠐⠒⠁⠂⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠈⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⠀⠀⠀⠀⢈⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⢀⡠⠰⢀⠤⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠄⠀⣠⠂⠁⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢌⠀⢠⠃⢜⡂⠀⠀⠠⠀⣠⠖⠊⠀⠂⠀⡀⠈⠠⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠢⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠉⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⢀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐
⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠣⢿⣿⢲
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡎⠌⢡⠀⢒⠲⠟⣫⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠃⠑⠂⠸⠇⠅⣤⣹⣯⡿⢀⣀⡠⣄⣤⣀⡀⠀⠀⢠⡾⣿⡾⢻⣦⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⣯⣍⣒⣤⡤⠀⠛⣿⣽⢛⠛⠀⠀⠀⠙⠻⢿⣷⣠⠾⠟⠛⠙⠯⢄⠙⡿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⠠⡇⠀⢧⣈⡩⠋⠀⢀⠞⠁⠻⢬⠔⠀⢀⣘⢳⡄⠀⡼⣵⡘⠀⡇⢀⡔⠄⠘⡿⣻⢛⣿⣇
⠀⠀⠀⠀⠀⠀⢧⣀⣱⣀⠀⠀⠀⠀⣠⠃⢸⣒⠢⡈⠀⠀⠻⡀⠀⠁⣿⠁⡈⠱⠈⠀⠘⠓⠁⠀⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⣦⡀⢠⠃⢠⠟⢻⣷⣌⣢⣰⣄⠀⠀⣤⡿⠘⣷⣒⣒⣒⣢⠗⠒⠂⠈⢹⠋⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⢸⠀⠘⠤⠤⠔⠋⠉⠀⠀⠀⠀⡬⢷⠀⢸⠿⠯⡽⠃⠀⠀⠀⢀⣾⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠎⢆⠀⠀⠀⠀⠀⠀⠀⠀⣇⣈⡠⢾⣆⠀⠑⠋⠀⠀⠀⠀⣠⢻⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠣⡀⠀⠀⠀⢀⣠⠞⠉⠁⠀⠰⠋⠑⢦⣀⠀⠀⠀⣰⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢶⣴⠊⢉⡴⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠈⣇⠀⠀⡴⠓⢦⣀
⠀⠀⠀⠀⣀⣀⣀⣤⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠁⠀⠀⠈⠙⠲⢤⣀⡀
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⣀⣠⣄⡀⠈⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠈⢧⠀⠀⠄⠒⠀⠀⠀⠀⠀⠀⠼⣿⡿⠿⠃⠀⢸⠃⢀⣠⣄⡀⠑⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠈⢳⡀⠀⣴⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⠿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣷⣦⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⠀⢸⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⢠⣾⣿⡿⠿⠿⠷⠘⡀⠀⢸⣿⣦⣀⠀⠀⠀⠀⣀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⢸⠏⠁⠀⠀⠀⠀⠀⡇⠀⠈⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⢀⠜⢹⡀⠀⠘⠿⠿⠿⠟⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠋⠁⠀⠀⠳⡀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠀⠘⠳⣤
⠀⣠⢤⠴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⠀⠀⠀⠀⠀⠘⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢤⢤⡀
⠀⣧⠀⠀⠘⢦⠴⠞⢛⠋⠙⠛⠻⠶⠶⠗⡀⠀⠀⠠⠀⠀⠀⡐⠷⠶⠾⠛⠛⠉⢛⠛⠶⢤⠞⠀⠀⢠⡇
⠀⠈⣗⠀⠩⠀⠀⠁⠀⠀⢄⣀⡀⡀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⡀⣀⣀⠄⠀⠀⠁⠀⠨⠁⢰⡏
⠀⠀⢻⠀⠁⠀⣀⣠⣭⠞⠋⠉⠉⠛⠶⣄⣀⠀⢀⡀⣀⠀⢀⣀⡴⠞⠋⠉⠉⠛⢮⣥⣀⡀⠀⠁⢸⠇
⠀⠀⠘⣧⢈⠀⠀⠙⠳⢦⣤⣀⣀⣤⡴⠛⠙⠁⠀⠀⠀⠀⠘⠙⠳⣤⣄⣀⣠⣤⠶⠛⠁⠀⢈⢠⡟
⠀⠀⠀⠘⣧⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠈⠳⣄⡠⠀⠀⠀⠀⠀⠀⠀⠠⢀⡴⠚⠋⠛⠲⣄⠢⠀⠀⠀⠀⠀⠀⠀⠠⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣶⣒⣀⣀⣒⣭⠾⠋⠀⠀⠀⠀⠀⠈⠻⢮⣔⣂⣂⣐⣲⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠳⡐⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣉⠛⠲⠦⣌⣢⡀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠉⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠾⠳⢦⣀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢚⣋⣉⣀⠀⠀⠀⠀⠀⢶⡬⣷⣄⠀⠀⠀⠀⠹⡄⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⡡⠖⠉⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠙⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠒⠒⠒⠶⠤⠈⠳⢄⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠋⡔⠒⢄⠈⢇⡀⠀⠀⠀⣿⠗⢃⠤⠒⠂⠈⠉⠁⠀⠀⠀⠀⢀⡽⠒⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠣⡀⠈⠃⠀⠉⠙⠛⠉⠁⠀⠉⠙⠒⠒⠤⠀⠀⠀⠀⠀⠴⣥⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⢤⠀⠀⠊⠁⢀⠔
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⢀⡄⡘⠀⠀⠀⠀⡎⢃⣰⣁⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⣠⠎⢻⢡⠀⠀⠀⠀⢈⠎⡠⢤⡧⢀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⢀⡖⠚⠛⠓⠲⢖⠒⠒⠒⢦⣀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠲⢞⠁⠀⢸⠉⠁⠀⠀⠈⠉⠐⠧⠒⡗⠠⡎
⣀⣐⣇⠱⣀⡠⠐⢈⣉⡇⠀⠀⠀⠀⢠⠙⠢⢄⡀⠀⠀⠘⠂⠀⠀⠀⠀⣀⡀⠈⠓⠹⠦⠀⢹⡀⢈⡉⠀⠀⠀⠀⠀⠑⡤⣚⣥⠵⢥
⠑⣤⡬⠀⢀⡐⠊⠥⡄⠙⣄⣀⡠⢤⠇⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠒⠉⠁⠈⢳⠶⠤⠤⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠣⣽⣡⣐⣚⣢
⠈⠃⠄⠊⠁⠈⠉⠉⠁⠀⠈⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠔⠣⡑⣤⠚⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣢⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⡤⠠⣠⣀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣼⠀⠀⠀⠀⠲⠓⠒⠳⠶⠆⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡹⢮⡧⡽⠒⠤⢄⣀⠄⠀⢸⡆⠀⠀⢀⣀⠴⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⠭⢄⣐⣒⡒⠒⠀⢸⠄⠀⠀⢏⣉⠉⠉⠀⠀⠀⠈⢀⠔⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢶⣦⣿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡶⠚⢋⣿⣿⠶⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢰⡋⠀⠰⣿⣷⣄⡀⠀⠀⠐⠀⠀⠀⠀⢠⠁⢠⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠀⠀⠀⠈⣉⣛⣿⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢀⡰⠋⠉⠁⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⢻⣦⠀⠀⠁
⠀⠀⠀⠰⣦⣴⡆⠀⠰⡀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣼⡟⠀⠐⠁
⠀⠀⠀⠀⠘⠛⠀⠀⠀⢰⠀⠀⠀⠀⣘⣷⠞⠳⠶⠶⠖⠒⠀⠀⢤⠴⠶⠾⠾⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠓⠛⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⣿⣷⣀
⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⡿⢿⣷⣶⣴⣤⣶⣶⣶⣤⣴⣴⣶⣶⣶⣴⣶⣶⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡄
⠀⠀⠀⠀⠀⡆⠀⣧⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣴⠛⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡇⢀
⠀⠀⠀⠀⠀⠙⣦⠘⣷⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⣷⠀⠢⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡾⠀⣸
⠀⠀⠀⠀⠀⠀⢸⠀⣼⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠙⣾⣇⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇⣰⠃
⠀⠀⠀⠀⠀⠀⠋⢰⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⣏⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢧⡀⠙⣦⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⣦⡢⠔⣣⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠳⠄⢰⡄
⠀⠀⠀⠀⠀⠙⢶⡄⢹⡆⢳⡄⠀⠀⠀⠀⠀⠀⠀⢼⠉⠉⢻⡻⣼⡧⣂⡽⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢀⠀⣟
⠀⠀⠀⠀⠀⠀⡼⢱⠞⡵⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠀⠃⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⠃⠈
⠀⠀⠀⠀⠀⠀⠉⠈⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄


⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠘⢦⡐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡀
⠀⠀⠀⠀⠀⢠⡀⢻⡆⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡅⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⠇
⠀⠀⠀⠀⠀⠀⢹⡄⡿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⡜⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠸⠃⢃⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡀⠀⠀⣼⠀⠀⠀⣀⡠⠒⠉⠀⠉⢳⡀⡀⢀⡀⠤⢄
⠀⠀⣀⠠⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠠⠠⠄⠀⠀⠀⠉⠀⠀⢀⡨⠀⢠⠊⠀⡡⠤⠤⡀⠀⠈⠁⠈⡇⠀⠀⠀⠗⠢⡀
⠀⠉⠉⠁⠀⠀⠂⠀⠌⡑⠂⠤⢀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠪⣀⠠⠔⠀⠀⢸⠀⠀⠓⠀⠀⠹⠀⠀⠀⠴⠃⢀⣀⠀⠀⠀⠚⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⢀⠈⠒⠠⠄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣣⣀⠠⠄⠐⠁⠀⠀⠀⢀⠀⠈⡸⠀⠀⠐⠀⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⣐⠀⠀⠄⢀⡀⠉⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠈⠉⠉⠀⣠⣤⡤⠶⢧⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠒⠒⠤⠤⢌⣁⡀⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣒⡒⠒⠬⣔⠄⡀⠀⠀⢠⠀⠀⠀⠈⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠢⡀⠀⠀⠀⠀⠈⠉⠉⠁⠒⠒⠒⡶⠶⠦⠤⣄⡀⢀⣸⡄⠀⠈⡆⠀⠉⠉⠁⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠐⢤⡀⢀⠤⠶⠦⣄⡀⠀⠀⢱⠄⠀⠖⠀⡹⠁⠈⠳⠤⠖⠁⠠⣄⠀⠀⠈⡧⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠤⠉⠒⠀⠀⠒⢿⡀⠀⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣆⠀⢀⡴⠄⠊⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠴⠦⠀⠤⠞⠉⠉







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

⠀⠀⠀⠀⠀⢀⣴⠟⠙⢶⣦⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄⠄⠀⠠
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢻⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⢸⡇⢠⣰⣤⠇⠀⢿⠀⢤⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄
⠀⠀⠀⢐⢨⡇⠀⠀⠀⠀⠀⢰⠐⠚⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⠤⣀⠀⠀⠀⡘⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⢠⠃⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⢠⠀⠀⠀⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢢⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢄⠀⠀⡠⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠊⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠈⢄⠀⠀⡀⢀⠀⠀⢀⣀⡀⢀⡠⠆⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠤⠂⠀⠀⠠⣤⠄⣀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⡔⠁⡀⠀⠀⠀⢀⡂⣰⡇⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣦⢄⣀⠀⠓⡁⠓⠉⠋⠀⠀⠀⠀⠀⣤⢤⣄⣀⣀⠠⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠟⡞⠁⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢚⠘⠃⠙⠒⠀⠀⠀⠂⡖⡒⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⢢⡄⠚⠀⠦⢀⣀⠀⡐⣆⠰⠃⠳⠄⠄⡀
⠀⠀⠀⠀⡀⠀⠂⠈⠀⠀⠀⠉⠀⢀⠰⠒⠚⠃⠁⠈⠂⠀⠤⠤⠄⠋⢢⡀
⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⣀⣀⡀⢀⣀⣀⢀⡀⢀⣀⣀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⡶⣦⠖⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⢿⡛⢯⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡟⢿⡉⢧⡈⣧⠈⣧⠹⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⣴⣾⡿⢿⠹⡆⢻⡄⢳⡈⣧⢘⣆⣼⣶⠿⠛⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⡏⢳⡘⣇⠹⣆⣻⣤⣷⡾⠛⠋⠉⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡌⢷⣼⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⢀⣄⣠⡀
⠀⠀⠀⠀⠀⠘⢯⣻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠘⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⣇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡛⢦⡈⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠙⢦⣙⣷⠿⣿⣧⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡿⠋⠀⠘⣿⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠸⣿⡆⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⠀⠙⠂⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⠇⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠈⣿⣇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣿⢷⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢸⡷⠆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣀⠀⠀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣼⣿⣿⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⡇⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⣿⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣿⡟⠀⠀⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣿⡟⢀⠀⠀⣿⣿⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠉⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣸⣿⣤⣴⡶⠾⠷⢶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢻⣿⠫⠄⠐⠋⠀⠓⠀⠿⢉⡻⣷⡄⠀⠟⠀⠀⠀⢀⣾⡿⠁⠀⠾⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠁⣌⢻⣆⠀⠀⠀⢠⣿⡿⡿⠿⡿⠷⣦⣀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⡄⣀⣴⣿⠋⠀⠁⠀⠁⠒⢊⡻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣄⣀⣀⣀⣀⣥⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣶⣦⣤⣴⣶⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢩⣳⣀⣀⣤⣀⣀⣞⡍⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⣤⣧⢀⠴⡏⣻⠛⣟⢹⠦⡀⣾⣤⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠐⣷⠆⢋⣹⡋⢸⠟⣶⠻⡇⢙⣏⡙⠰⣾⠂⣸⡀
⠀⠀⠀⣷⣤⣤⣤⣤⡶⢾⣟⣽⣿⣿⣷⣾⣯⡁⣾⠓⠉⠚⣷⢈⣽⣷⣾⣿⣿⣯⣻⡷⢶⣤⣤⣤⣤⣾
⡤⠀⣤⣽⣿⣿⣟⣿⢟⣿⡿⠷⠿⠿⣽⣿⣿⣿⣿⣢⣀⣔⣿⣿⣿⣿⣯⠿⠿⠿⢿⣿⡻⣿⣻⣿⣿⣯⣤⠀⢤
⠣⡉⠁⠻⣿⣶⣶⣶⣿⣅⠀⠀⠀⠀⠀⠙⢯⣽⣿⣿⣿⣿⣿⣯⡽⠋⠀⠀⠀⠀⠀⣨⣿⣶⣶⣶⣿⠟⠈⢉⠜
⠀⢀⡥⢾⣿⣟⣻⠿⣿⣿⣿⣶⣦⣤⣴⣶⣏⣿⣷⣿⣿⣿⣾⣿⣹⣶⣦⣤⣴⣶⣾⣿⣿⠿⣟⣻⣿⡷⢤⡀
⠀⠘⠮⠃⠙⠢⣿⣿⣿⣿⣿⣿⢿⣿⣻⣿⣿⡿⠋⠁⠀⠈⠙⢿⣿⣿⣟⣿⡿⣿⣿⣿⣿⣿⣿⠔⠋⠘⠵⠃
⠀⠀⠀⠀⠰⡏⠿⠉⠓⠻⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠟⠚⠉⠿⢹⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣿⣸⢿⣸⣷⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⣾⡧⣝⣁⣈⣛⣩⣯⣁⠀⠀⣀⠤⠤⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠓⠀⣉⠛⠒⠒⠚⣚⠉⢹⡇⠸⣕⡛⠿⠿⠿⠿⣊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠸⡄⢸⠉⠳⠄⠀⠚⠉⠃⣸⢿⡄⠘⠯⣍⠙⣉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡠⠀⠀⣰⢻⣦⠀⢰⠋⣷⡀⠈⣷⠀⠀⢸⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠙⠀⠀⢧⠸⣽⠀⠘⣞⣿⠇⣰⡇⠀⠀⡼⣛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠁⡀⠀⠀⢭⠁⢀⣱⡄⣡⠀⣼⢃⣠⡼⢡⣗⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣴⠇⡄⢀⣼⣤⣀⣉⡤⣧⡀⢿⠋⢁⡴⢻⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣴⡿⣯⠀⠆⢀⠀⢻⣶⣶⣾⠇⢀⣿⡴⠋⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠻⣿⠳⠀⢼⠀⠀⠙⠲⢫⠀⠊⡿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡝⢧⡀⠈⠃⠀⠠⡀⠘⠁⣸⠇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⣵⡟⠁⢀⠇⠀⠠⠃⠀⠆⢻⡄⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢋⡿⠇⣀⠈⢀⠠⡄⠀⠀⣠⡟⠁⠀⠀⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⡇⠀⡏⠀⢸⠀⠁⠀⢈⢩⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠁⠀⠀⠀⣇⠀⡿⠀⢀⢀⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠛⣠⠆⣼⠀⣿⢰⡇⣠⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡍⠀⣄⣁⣀⣀⣠⡏⢸⡟⣿⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠋⠉⠉⠉⣷⢸⢀⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣛⢿⡒⠿⣴⣄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠥⠤⢭⣝⣡⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⠛⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠟⢋⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣾⣿⡿⠋⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣯⡾⣽⣯⣿⣽⣿⣶⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣇⢸⣿⠇⢻⣿⣿⣿⣿⣟⣧
⠀⣀⣤⣄⣀⡀⠀⠀⠀⣀⣀⣠⡿⣿⣾⣟⣙⣼⣽⣿⣿⣿⣿⣿⠇
⠘⣿⣿⣿⣿⣿⣿⣻⣿⣽⣿⣷⣿⣿⣷⣿⢿⣿⣿⣿⣿⣫⡾⠋
⠀⠈⠻⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣵⣟⣹⣿⣿⣿⡏
⠀⠀⠀⠹⣮⡿⣿⣿⣿⣿⣿⣿⣻⡷⠟⠋⠉⠉⠛⣿⣻⣿⣾⣿⠄
⠀⠀⠀⠀⠈⠛⠷⠿⠿⠷⠿⠛⠉⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣻⣿⣷⡿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⡟⢹⣿⡟⣿⣿⣿⣿⢿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⢀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣿⣟⣿⣿⣿⣷⣟⢿⣶⣻⣿⣿⣿⣽⣻⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⢿⣿⣿⣻⡾⣿⣿⣿⣿⣿⣿⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣷⣿⣼⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠇⢿⣿⣽⡿⣿⣿⣿⣿⣷⠟⠁⠀⠀⢀⣤⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⢾⣿⣀⣠⣭⣿⣿⣷⣯⣿⢿⣅⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠛⠛⠿⠿⠶⠾⠶⠞⠛⠛⠀⠀⠈⠛⠻⠿⠿⠿⠟⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠈⠙⠄⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠲⠦⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⣠⠴⢦⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣬⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠐⠒⠚⠋⠉⠉⠉⠉⠀⠀⠀⠀⠙⠳⢦⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠶⠶⠶⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠂⠰⠤⠤⣤⣤⣀⣀⣀⠀⠀⠀⢀⠤⠊⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢣⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠋⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠤⠤⠀⠹⣦⣀⠀⠀⠀⣀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠛⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⢛⣋⡉⠀⠛⢁⣀⠙⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⠟⣠⠄⠀⠀⠈⢽⡻⢿⣿
⠀⠀⠀⠀⠀⢠⠿⣇⠀⢶⣆⣀⣀⣰⡖⠀⣿⠷⡄⠀⠀⠀⠀⠀⡼⢯⠀⢸⣇⣀⣀⣐⣿⠀⣸⠿⡄
⠀⠀⠀⠀⠀⠸⡀⢼⠻⠝⠒⠂⠐⠒⠫⠟⡧⢀⠇⠀⠀⠀⠀⠘⣆⢰⠫⠝⠒⠂⠀⠒⠚⠽⢳⢀⠇
⠀⠀⠀⠀⠀⠀⠈⠻⡐⡒⠀⠀⠀⠀⢒⢢⠟⠁⠀⠀⠀⠀⠀⠀⣸⣿⣆⣉⠀⠀⠀⠀⢈⣁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠭⣄⣠⠭⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠫⠤⠤⠭⠓⠛⠋⠉

⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⢠⠎⡕⠢⠀⠈⠙⠳⣄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠁⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠀⡹
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠎⡁⢤⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢧⠐⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠇⠀⠀⠀⠀⢀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⠄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⡑⠢⢤⣄⣀⣀⡤⠔⠀⠀⠀⠀⠀⠀⠀⠁⡠⠀⠀⠀⢀⠀⠀⠙⡄⠊
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡼
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⠠⠔⠁⠈
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠉⠻⣿⠻⣿⠟⠛⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⣀⣄⡀⠀⠻⠿⠶⠠⢼⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠴⠶⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢴⠀⠀⠀⢀⡴⠋⢠⡄⠘⠋⠀⠀⠀⠀⠀⠓⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠆⠀⣾⣿⡇⠀⠀⢰⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣴⣤⣄⣀⣤⡄⠀⠀⠀⠀⢿⠟⠁⠀⠀⢼⣿⠟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧⣠⣿⣶⣄⠀⠻⠁⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠀⢀⡝⠛⠀⠈⠉⠙⢳⠀⠀⢀⣀⣰⠷
⠀⠀⠀⠀⢠⡄⠀⠀⣀⠈⠁⠀⢠⣿⡿⠶⠶⠶⢀⣠⡌⠀⠀⠀⠈⢀⡀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢸⣇⠀⠐⣿⣀⣀⡀⠀⣹⣷⣄⠀⠀⠀⣿⡀⢀⡰⠶⠀⠙⢻⣶⠀⢶⡆⣀⣀⠀⠀⠀⠀⠠⠴⠶⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⡷⠀⠀⠀⠈⠙⠁⠀⠀⠙⠛⠛⠻⠛⢿⣿⢷⣶⣾⣿⡿⠋⢀⣤⠄⠀⠄⠉⠀⣠⠀⠉⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⠟⠛⠛⠁⠘⠋⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⠖⠀⠀⠀⠐⢿⡄⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠀⢿⣷⡄⠀⠰⠊⠁⠀⠀⠈⠙⠿⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡷⠀⠠⣦⠀⠀⠀⠠⢤⣄⠙⠛⠷⣶⣶⣦⣀⣀⣀⣀⣀⣀⠤⣶⣶⠎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⠀⢰⡀⢸⣿⠀⠁⣤⡄⠙⠀⠀⠀⠀⠀⠀⠘⠿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠸⣷⡄⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠃⠀⠈⠻⣆⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣧⡀⣀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⣿⣇⠛⠁⠙⢳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠙⢿⣷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⣿
⠀⠀⠀⠀⢀⡀⠀⢠⡶⠀⠘⠛⢂⣀⡀⣾⡷⠀⣾⡏⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⢻⡇⠀⣿
⠀⠀⠀⠀⠸⠇⠀⠀⠙⠷⠶⠶⠟⢻⣿⣿⣇⠀⢸⣿⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⢠⠈⢳⡆⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣸⠏⠹⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⠛⠀⢸⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠜⠃⠀⠀⠉⢸⣿⡷⠈⢡⡄⠀⠀⠀⠀⢀⠛⠀⢠⡌⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⡏⠀⠀⣼⡇⠀⠀⠀⠀⡿⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠹⠀⠀⢻⡇⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠀⠀⠀⠘⢇⠀⠀⣤⠁⠀⠀⣰⡆⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠟⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢿⡆⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡾⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⢀⣀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⣀⣀⣀⣀⣀⣀⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⣁⣀⣀⣀⣀⣀⣼⡗
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⢿⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣦⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⢈⣤⠦⠀⡁⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠈⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠁⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠈⣠⠟⠛⠺⠖⠒⠻⡆⡄⠀⠀⠀⠀⠀⠀⠠⠁⣔⠮⠞⠲⣄⡠⠀⡆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⣟⠀⣄⠀⠀⣤⠀⣿⡃⠀⠀⠀⠀⠀⠀⢰⡞⠁⣤⡄⢠⣤⠙⣄⡇⠀⠀⠀⠀⠀⣀⣄⠀⠒⡀⠀⣀⠒⠂⡠⢦⠄
⠀⠀⠀⠀⠀⠀⢏⡈⠂⠀⠀⠀⠢⠊⣑⠀⠀⠀⠀⠀⠀⠀⢸⠊⠖⠊⠁⠑⠀⢰⠹⡇⠀⠀⠀⠀⠀⢆⠈⠅⠁⠀⠠⠀⠁⠞⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠴⡀⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⢐⠄⢘⣿⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣬⠱⢦⢥⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠺⡶⢤⠨⢶⡏⠘⡀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⣈⣀⠥⢞
⠀⠀⠀⠀⡀⠀⠠⠘⢄⠑⢂⠀⠈⡠⠤⠀⡀⠀⠀⠀⠀⠀⢠⠀⠼⠂⠀⠀⠌⠳⠷⠴⣀⠀⠀⠀⠀⠀⠜⠢⢀⡀⠀⠀⠀⠐⢆
⠀⠀⠀⠈⠰⠀⠀⠂⠰⠁⠪⠀⠇⠀⠄⠀⠄⠑⡄⠀⡠⠈⣀⠀⠈⠢⠀⡀⠄⠊⠀⠀⠀⢣⠀⠔⠀⠈⠁⠒⠤⣄⡑⣮⠀⠠⠊⠁⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡤⢂⡄⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠥⣈⣁⣠⠾⢦⣝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⢋⣉⠉⠀⠙⠒⢦⡀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⡀⠁⠀⣀⠀⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡗⠽⢟⡷⣾⢻⡛⠦⡦⡇⠀⠀⠀⠀⠀⠀⠠⡞⢧⣈⠩⠄⠠⣉⣁⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡈⠂⠀⠀⡄⠑⠰⠋⠀⠆⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⢨⠐⠀⢉⡠⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡁⠀⠀⠁⡀⠈⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠸⠂⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⢲⡈⠉⠚⠙⡹⠤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠙⢄⡀⠀⠀⣈⠑⠂⠐⠠⡀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠓⠶⠀⠀⠀⢄⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠐⠀⠀⠀⠀⠀⡄⠀⣰⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡆⠀⠀⠀⠀⢀⠀⠀⠹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠙⠟⠛⢿⠁
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠖⠋
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠉⠑⡦⠀⠀⠒⠊⠉⠹⣆
⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢠⠞⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⣰⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠄⠀⠀⢤⡴⠒⠉⠀⠀⠀⠀⠄
⣿⠀⠀⠀⠀⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠
⣧⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⢄⣠⠞⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⡄
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⠒⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⢹⡄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⢀⡀⠂
⠀⠀⢻⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠀⠁⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⢳⡄⠀⠀⠆⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠦⠀⣀⠀⠀⠐⠊
⠀⠀⠀⠀⠻⡄⠀⢠⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠙⣆⠀⢂⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠲
⠀⢀⣀⣠⡤⠴⠞⠃⠀⠆⠀⠸⢧⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠑⠦⠒⠉⠉⠉⠉⠲⡄
⢾⣉⣁⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⠲⠶⠶⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠒⠢⠔⠋⠑⠢⠤⠄⠒⠉⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠛⠓⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠴⠶⠒⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠙⢳⢏⣠⡄⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠞⣉⠁⠀⢀⣤⣉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠀⠀⠀⢀⡎⠈⡇⢠⡏⠀⠘⡆⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣠⠖⠒⠚⢿⣆⡇⢸⣦⣄⡠⠟⠋⠉⠁⠀⠈⠉⠂⠀⠀⠀⠐⠋⠉⠀⠀⠀⠈⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠉⠀⣇⠀⠀⠀⠀⠩⠤⢬⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠘⢦⣀⡀⠠⢤⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠒⠀⠲⠤⠤⠀⠄⠀⡀⣾⣀⡀⠀⠀⣰⠛⠓⠚⠛⠃⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⣉⣡⡽⢀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡚⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠤⠤⢶⡀⠈⠁⠈⠳⠶⠛⠦⣤⣀⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⡴⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣶⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⢦⢤⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠀⠀⠀⣀⡤⠟⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠶⠛⠁⠀⠈⠙⠲⠶⠒⠛⠉⠀⠀⠈⠓⠶⠤⠤⠶⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢂⣀⣤⡤⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠡⠞⠋⠉⠀⠴⠶⠿⠷⠶⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⡛⢭⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠸⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠒⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣸⣀⣒⠶⠒⠀⠀⠐⢰⣛⠋⠉⠉⠙⡸⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡇⠀⢀⡀⢨⡥⠍⠉⠓⠊⠉⠀⠀⠈⠉⠉⠙⠣⠒⠈⠁⡲
⠀⠀⠀⢰⠁⠀⠁⠉⠀⠀⢄⡀⠀⢳⡀⡟⠲⢦⠤⢤⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢄⣀⠇⠀⡜⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠙⠢⠴⠧⠔⠚⠉⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⣀⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢼⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠯⣉⠻⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠛⠦⠞⠩⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⡀⢙⡖⠃⠀⢄⠐⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢀⠀⠉⠉⠉⠿⠄⠀⢀⣀⠀⣀⣀⡀⠀⠀⠤⠀⠚⢻⣦⡀⠀⢀⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⠀⠀⠀⠦⢀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡄⠀⠈⢂⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⢶⣶⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢰⡀⠀⢠⠀⠀⡀⢸⡀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⢹⣀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠃⢀⣸⠀⢸⠁⠘⠁⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡆⢺⢘⡏⠈⢧⣄⣀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⠁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠈⠉⠁⠀⡾⡸⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⢇⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠀⠀⠀⣀⣂⣼⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢀⠀⣧⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠉⠙⢹⠋⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡸⡀
⠀⠀⠀⢀⣀⠤⢤⣾⣤⣄⣀⣀⣀⣀⣏⣀⣀⣬⣦⢤⡤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣧⠟⠀⠀⠀⠀⠀⠀⠾⠻⠟
⠀⠀⢰⣇⣀⣀⣀⣸⣦⣟⡦⠬⠟⠋⠛⠿⢶⣮⠥⠾⣄⣀⣠⡼⠃
⠀⠀⠀⠉⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠺⠂⠀⠀⠀⠀⠀⠀⠒⠓⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠀⠄⠠⠠⠀⠠⠐⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠂⠠⠀⠆⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣄⣼⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣿⢿⣿⣿⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⣠⣶⣿⣇⣀⣿⣷⠀⠋
⠀⠀⠀⠀⠀⠀⢠⡐⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠞⣽⣿⣿⣿⣿⣿⡟⠠⢤⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⣇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠞⢡⣼⣇⣄⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢐⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⠁⢩⣿⡆⢰⡖⠀⣀⣴⠟
⠀⠀⠀⠀⠈⢲⠃⠀⠐⡀⣴⣦⡿⠂⠀⠀⠀⠀⠀⠀⠉⠋⠸⢿⣿⣿⣿⣿⠷⠋⠠⠞⠉⣡⡶
⠀⠀⠀⠀⠰⣷⢀⢰⣴⣿⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣼⣷⣮⣤⣶⣾⠏
⠀⠀⠀⠀⠀⢻⣦⣷⣽⣿⣷⡍⠀⢠⡶⠞⠛⣳⠀⠀⠀⠾⠷⢶⣶⡌⢸⣿⣿⣿⠛⣋⣽⠃
⠀⠀⠀⠀⠀⠀⣹⣿⡹⣿⣿⣿⢀⣏⣤⠷⢾⠟⠀⠀⢀⢒⡷⠿⣄⣹⠀⢻⣿⣿⣿⣻⠇
⠀⠀⠀⠀⠀⠀⢻⣿⡟⠻⢿⡏⠈⠉⠓⣾⠾⠇⠂⠀⠀⡜⢺⡗⠊⠉⠂⠸⠿⢛⡛⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠊⢟⣦⡷⠀⠀⠀⠈⡠⠄⠀⠀⠀⢠⡈⠀⠀⠀⢀⢇⣴⡋⠥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠆⠀⡔⠊⠓⠿⠦⡴⠾⠶⠓⠦⠀⠰⠃⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣿⠈⠀⢦⣤⡤⠤⠤⠤⠤⠤⣤⡤⠀⠀⠀⣼⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢿⣿⣷⣶⣶⣶⣿⡟⠁⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠻⣄⡉⢋⣩⠟⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠢⢭⡭⠞⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠛⠉


⠀⡀⣀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣯⢹⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣀⠀⠀⠀⢀⡖⠈⠑⠒⠢⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⡔⢉⣀⠤⠤⠤⣄⠈⢣⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⢀⠀⠰⡛⠉⠣⣄⠀⠀⠈⡆⠀⡆⠀⢸⠀⠀⣀⡤⠶⠾⠶⣦⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⡇⠀⠀⡰⠋⢀⣀⣀⠉⠑⠢⡀
⠀⠰⢇⣉⠑⠪⣕⢦⡈⢣⠀⠀⢱⠀⠱⢄⡘⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠙⠢⠀⢰⠀⠀⢰⠃⢠⠋⠀⠀⠉⢓⠤⣸
⠀⠀⠀⠈⠉⠒⠤⢭⠿⠃⠀⠀⠀⠳⣄⠀⠉⡇⠀⠠⠶⠟⠻⢦⡀⠀⠀⢀⣤⣤⣤⡀⠀⠀⣎⡠⠔⠃⢠⠏⠀⠀⡴⠊⢁⡠⠼⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠀⣸⣀⣀⣀⠔⠃⠀⠀⠀⣦⠾⠕⠚⣉⣀⠤⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠐⠤⠤⠄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣠⠖⣒⣛⠓⠶⠶⠶⢦⡄⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠉⠁⠀⠀⠀⠉⠉⠉⠒⠻⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⢂⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷⠀⠀⠀⠀⠀⠠⠄⠀⠐⠲⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣏⣤⣤⣴⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠛⠛⣩⣼⡿⢋⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢿⣿⠛⡻⠏⠉⠀⠈⢀⣿⣿⡿⠿⠧⢤⠀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣗⣚⣿⣀⠀⠀⠀⠀⠐⣫⡉⠁⡀⠀⠀⠀⢙⠩⢍⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⠤⠤⠿⠟⠉⢿⡄⠀⠀⢀⣠⣞⢲⠽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⣿⣆⠀⡾⠟⠉⠂⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣭⣿⣿⣿⣏⣀⡀⢠⣤⠀⣼⣿⣻⣷⡀⢶⣅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣻⣿⣿⣿⣿⣽⣿⠀⢻⠀⢹⡇⠀⢻⡁⢸⡇⢿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⣿⣷⣿⡄⢸⡀⢸⡇⠀⣸⠁⣿⡅⣿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⢸⡇⠀⣿⠀⣿⡀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢿⡧⢸⡄⢸⡇⠀⣿⠀⣿⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⢸⡇⠸⣇⠀⣿⢀⣿⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⣦⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⣿⠀⣿⠀⣿⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⡗⠀⣿⠀⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠞⣹⢟⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⠀⢹⠀⣿⠰⡟⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢞⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⠀⠈⠀⠛⢸⡷⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⣭⣿⣛⡟⠚⠀⠴⢀⢭⣥⡶⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⣿⡂⢸⡗⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣽⣿⣿⣿⣿⣧⣤⣤⣤⣽⣷⡾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢤⣼⣦⣼⣤⣦⣤⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠼⡡⣩⡢⡺⡆⠀⠀⠀⠀⠀⠰⡿⢻⡎⠋⣼⠿⡷⠀⠀⠀⠀⠀⠐⠛⣛⠉⡛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⡻⢿⠀⠀⠀⠀⠀⠀⠀⣮⡵⣯⡷⣵⣆⠀⠀⠀⠀⠀⠐⢱⡿⠻⠸⢻⣷⠍⠀⠀⠀⠀⠀⢀⠻⡟⠈⠷⢟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡍⣴⣶⠀⠀⠀⠀⠀⠀⠀⡿⡷⣿⡿⠿⣯⠀⠀⠀⠀⠀⠀⡦⠁⠀⠀⠀⢩⡆⠀⠀⠀⠀⠀⢨⣽⣛⠓⣟⣯⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⠂⠒⡩⠀⠀⠀⠀⠀⠀⠀⠹⡿⠻⠿⢿⠇⠀⠀⠀⠀⠀⠀⠠⡌⠁⠋⢋⡴⠃⠀⠀⠀⠀⠀⠈⣿⡽⠶⢩⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠿⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⢀⢠⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣸⣃⡄⠀⠀⠀⠘⣶⣿⣿⡄⠀⡄⢠⠀⣰⣿⣷⡾⠃⠀⠀⠀⢾⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⣘⠳⠶⢋⢅
⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⠛⡿⣓⡀⠀⠀⠘⠛⠿⢿⠻⣿⡏⣿⡿⢻⠟⠛⠛⠁⠀⠀⠀⣏⣸⣿⣉⠂⠀⠀⠀⠀⠀⠀⠀⣷⠷⠀⠶⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠀⠙⢿⡇⠀⠀⠀⠀⠀⠈⣮⡋⠃⢉⣫⡿⠀⠀⠀⠀⠀⠀⢰⠛⣽⣯⡍⢳⠀⠀⠀⠀⠀⠀⠀⣟⠛⢲⠛⣻⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠦⠭⣤⡭⠶⠿⠀⠀⠀⠀⠀⠀⠘⣯⣥⣭⣽⠅⠀⠀⠀⠀⠀⠀⠀⢿⣙⢋⣻⠁⠀⠀⠀⠀⠀⠀⠀⢿⣛⠛⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⠫⠾⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣸⣦⣿⣦⣿⣁⠀⠀⠀⠀⠀⠀⣰⣰⣭⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⢘⣤⢧⣜⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣾⣦⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣻⣿⡟⠋⠀⠀⠀⠀⠀⠀⠾⢾⡇⣵⠾⠆⠀⠀⠀⠀⠀⠀⠀⣤⢺⣖⣤⡅⠀⠀⠀⠀⠀⠀⠀⣉⡉⠯⢏⣉
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠓⣩⡘⢃⠀⠀⠀⠀⠀⠀⠀⡿⠶⠇⠳⠺⣧⠀⠀⠀⠀⠀⠀⠀⣽⢰⢱⡮⠇⠀⠀⠀⠀⠀⠀⠀⣯⡿⠀⢿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠻⠋⠖⣆⠀⠀⠀⠀⠀⢰⣛⣛⢳⡛⣋⢹⡆⠀⠀⠀⠀⠀⠀⣾⡏⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⣌⠀⡀⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣨⠤⠅⣩⠟⠀⠀⠀⠀⠀⠈⠓⠦⠼⠤⠷⠚⠁⠀⠀⠀⠀⠀⠀⠻⢯⣉⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠖⠃



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠸⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠀⣀⡀⠀⠀⠀⠀⠑⠉⠁⡝⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠒⠲⣀⠀⣀⠀⠀⠉⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠂⡍⠀⡱⠒⠀⠀⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣾⣚⣛⡛⠓⠦⣄⠀⣠⠔⠋⠁⠀⢹⠀⢺⠀⢀⠖⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠈⠉⠒⣬⣿⣧⠀⠀⠀⢀⡿⠀⡤⠃⠈⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠒⠒⠒⠒⠒⠛⠛⠂⣀⠉⠉⢇⣀⣀⡠⠔⠛⠀⠸⠒⣋⣉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠐⢲⠤⠤⡤⠔⠒⠉⣀⠀⠀⠈⠑⠀⠀⠀⠀⠀⣤⣈⣉⠡⠔⣇⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠇⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⠋⠀⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢖⣂⣉⡦⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣓⠭⡎⠑⠢⠄⡈⠢⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠉⢡⡀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠸⡖⠒⠁⠀⠀⠀⠈⠒⢄⠀⠀⠠⣴⡒⣉⠀⠀⠀⠀⠀⠉⠐⠊⠁⢠⣤⣶⡏⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠢⡈⠀⠀⠀⠀⠀⣠⣳⠀⣰⠕⢱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣦⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⣂⠴⡖⠚⢂⠖⠙⠋⣠⠔⠉⠉⠉⠉⢉⠿⠧⠄⠀⠀⠀⢰⠠⡼⠔⣣⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣗⣈⠗⠀⠀⠀⢀⡏⠙⠒⢦⣀⣮⠗⠀⠀⠈⠉⠁⠑⠤⣭⣭⡇⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠃⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⠓⠒⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣣⢣⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⣀⠖⠊

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡀⠀⡐⠊⠀⠑⠠⡀⠀⠀⠀⠠⣄⣀⠀⠈⠀⠈⠂⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠒⢄
⠀⠀⠀⠀⠑⠀⢈⠀⠀⠈⢂⠀⢰⠀⠀⠀⠀⠀⠉⡠⠁⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠢⠀⠊⠀⠀⠀⠀⡄⠀⠀⠀⠒⠀⢀⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠐⠄⠊⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠂⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠒⠁⠀⢱⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⡀⠀⢠⡤⠀⠘⠒⠄⠀⠀⠀⠀⡀⠀⠀⡁⠀⠀⠂⠒⠢
⠀⠀⠀⠀⠉⠉⠨⠀⠀⠀⠀⠒⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠈⠡⡄⠀⠀⡀⠄⠀⡀⠀⠀⠀⠈⠢⢀⡀⠀⠀⠇⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⣀⠀⣠⠃⠀⠀⠀⠀⡈⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⢂⡈⡀⠉⢀⠀⠐⠁⠀⠀⠀⠀⠀⠠⠁⠐⠀⡀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠐⣯⠀⠀⠀⣀⣠⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢰⡏⠁⠙⣧⠀⢻⠀⠀⠀⠉⠁⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢘⣇⠀⠀⠘⣧⣸⠇⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⣴⠟⠁⠀⠀⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣭⠰⠞⠛⠷⠞⠛⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣸⡏⠐⠞⠻⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠉⠛⠉⠀⢠⣤⠄⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠤⢄⡀⢸⠵⠲
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡤⠄⠼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣣⠀⠀⡜⠉⠈⢦
⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⡠⠒⡞⠁⠀⠚⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀⠷⡀⠑⣴⠋⠙⠣⡴⣠⣆⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣜⣄⠘⣏⠉⠉⡏⠥⡄⣀⣀⣀⠤⠴⠊⠁⠀⠀⠔⠁⠀⠉⢢⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣾⡄⠈⠒⠰⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣓⣢⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⢳⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣼⡢⢄⠠⡂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⡎⠢⡀⢳⡝⡳⠂⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡮⣹⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⡶⢦⡇⠀⣸⠀⢀⠤⢦⣤⣤⡴⠶⠞⠛⠳⣋⣽⢆⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⠒⠢⣤⠭⠎⣀⠀⢸⠁⠀⠀⠀⠀⠀⢐⠉⠇⣤⣇⣉⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠊⠈⠀⢣⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⢮⡉⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣬⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡤⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡇⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡔⠒⡶⢼⠒⠙⣰⣖⡖⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠠⡈⢶⡊⠑⠂⠋⢹⠤⠤⡘⠐⠯⢝⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠪⠽⢽⣹⡞⠀⠀⠰⢲⠋⡇⠀⠉⠊⠕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠕⠒⠒⠒⠚⠪⠽⠬⠃⠀⠀⠀⠁⢎⡑⠤⠤⠀⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣁⡠⠤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠝⠂⠁

⠀⠀⠀⠀⠀⠀⠀⢾⣛⣛⠛⠛⠒⠶⠶⠶⠶⢶⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣙⡆⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠸⢿⣿⣭⡽
⠀⠀⠀⢀⣄⠀⠀⠀⠀⣏⠉⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⢿⡾⠀⢀⣀⣀⣈⣹⠆⠀⠀⠰⢶⠛⠁⠀⠀⠀⠀⢿⣉⠉⠙⠛⠛⣧⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⢿⣍⣉⣉⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡞⠻⠿⠀⠀⠘⠷⠤⣤⣤⣤⣤⡀⠻⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⠀⢠⡞⠋⠀⠀⠀⠀⠀⠈⠷⣤⣤⣀⡀⠀⠀⠀⠀⠀⣠⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡿⠆⠀⠀⠀⠸⣧⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠙⢦⣤⡤⠖⠛⠳⣄⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢉⡇
⠀⠀⠀⠀⠀⢀⡶⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣤⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠙⠛⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣩⡿
⠀⠀⠀⠀⠀⠀⢀⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠁
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠈⢷⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠇⠀⠀⠀⠀⠘⠷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠿⠾⠃
⠀⠀⠀⠀⠀⣸⡧⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⢠⡞⠛⠁⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⢰⠏⠀⠀⠀⠀⢀⣠⣤⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡷⠶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡄
⠀⠀⡜⠀⠀⠀⠀⣴⠋⠉⠈⠉⠙⠛⠶⠤⠤⠤⣤⠤⠤⠴⠶⠶⠚⠛⠉⠉⠀⠀⠀⠀⠙⣦⣤⣄⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⡃⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠈⡇
⠀⢠⡧⠀⠀⢠⡾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠛⠒⠒⠒⠚⠛⠲⠶⢤⣄⠀⣧⣀
⢠⠋⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡉⢳⡀
⣸⠀⠀⠀⢠⡟⠋⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⡇
⢹⡀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇
⠀⢙⣦⣤⠶⠾⢿⣦⡤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁
⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣰⡟⠶⠶⣤⣤⣤⣤⣤⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠙⠛⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣴⠾⠻⢦⣤⣤⣤⣤⠾⠋⠛⠷⠶⠶⠞⠋
⣿⣿⣿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿