Motivation Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠠⣀⠀⢀⡄⢠⡀⢠⣤⡀⠀⢠⡄⡤⣠⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⣿⡇⠀⣿⠀⠹⣷⡀⠀⠀⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠟⠁⠻⠇⠰⠟⠠⠀⠹⠿⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡄⢀⡄⠠⣄⢠⣤⠀⠀⠀⣤⣄⠀⢠⡄⣤⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⣿⠸⣿⠀⠀⠀⠈⢿⣆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠆⠙⠇⠰⠛⠈⠻⠄⠀⠀⠀⠈⠻⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠴⢰⣦⢰⢠⣶⠀⣴⡆⢴⡆⠰⠄⠀⣶⡆⢠⣄⠀⣰⡄⠰⣶⡀⠀⣤⠀⠦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠂⣿⡇⢸⡇⠆⠀⠀⣿⡇⢸⣿⠀⠸⣿⡀⢹⣷⠀⠻⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠘⠛⠀⠛⠃⠚⠃⠐⠃⠀⠛⠃⠘⠁⠀⠀⠛⠃⠘⠛⠀⠓⠀⠘⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣆⠀⣶⡆⠀⣴⡄⠀⣶⡆⡀⠀⢰⣶⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⣿⡇⠀⠸⣿⡀⣿⡿⣷⡀⢸⣿⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠛⠃⠀⠀⠛⠃⠛⠃⠙⠓⠘⠛⠐⠃
⠀⠀⠀⠀⠀⠀⠚⢸⣷⠘⢰⣿⠀⣾⡇⢸⡇⠘⠂⠀⣿⡇⠸⣦⠀⢼⣆⠘⣿⡄⠀⣶⣀⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⣿⡇⢸⡇⠃⡀⠀⣿⡇⢰⡿⠀⠘⣿⡄⢸⣿⠀⡈⠛⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠉⠁⠈⠁⠈⠁⠀⠉⠁⠈⠀⠀⠀⠈⠁⠈⠉⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠈⢃⣾⠀⢸⣦⢸⡇⠀⠀⠈⢿⣆⠀⠘⢸⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢀⡜⣿⠀⢸⡟⢸⣇⠀⠀⠀⠀⢻⣦⠀⢸⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠈⠁⠁⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⣀⡀⢀⠀⠀⠀⠀⢀⠀⠀⢀⣀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⣸⣿⡏⠀⣿⣿⣿⣿⣿⣇⠘⣿⣷⠀⠀⣸⣿⡏⠀⣻⣿⣿⣿⣿⣷⠀⢸⣿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⣿⣿⠃⠀⣿⣧⣤⣠⠆⠀⠀⢹⣿⡇⢰⣿⡿⠀⠀⣿⣿⣬⣤⠤⠀⠀⢸⣿⣤⣾⣿⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⡿⠀⠀⣿⡿⠿⠿⠀⢀⠀⠀⢿⣷⣾⣿⠃⠁⠀⢹⣿⠿⠿⠂⠀⡀⢸⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⡇⠀⠀⣿⣿⣶⣶⣶⣆⠁⠀⠸⣿⣿⡇⠀⠀⠀⢸⣿⣷⣶⣶⣶⠁⢸⣿⡟⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡆⠀⠀⠀⢀⠀⠰⡄⠀⠀⠀⣶⠀⠀⠀⠀⣠⡦⠀⣠⠒⠉⡂⠀⠀⠀⣶⢴⢠⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⠁⠀⠀⢠⠃⢀⠜⠀⠀⠀⡴⠇⠀⠀⠀⣰⠃⠀⡴⠁⠄⠊⠀⠀⢀⡜⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡎⠀⠀⠀⣏⠔⠁⠀⠀⢠⠎⠸⠀⠀⠀⡠⠇⠀⢸⠁⠀⠀⠀⠀⡠⠊⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠇⠀⣀⠔⠇⠀⠀⠀⠀⠁⠀⢀⠀⢀⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⠀⠀⠲⠀⠀⠂⠀⠀⠀⠒⡒⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⢀⠀⠀⠉⠀⠀⠡⠀⠀⠀⠤⠀⠁⠐⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠰⠆⠀⠀⢠⡄⠀⠀⠀⠀⣄⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⠀⣀⣀⢀⣀⣀⣀⣀⡀⠀⠀⢀⣀⡀⠀⢀⣀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠛⠁⠀⠸⢿⣿⠿⠀⢹⣿⡀⢠⣿⠇⠀⣿⠿⠿⠿⠂⠀⠀⢸⣿⠀⠀⢸⣿⠀⣿⡟⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣀⣀⣀⣄⠀⢸⣿⠀⠀⠀⣿⣇⣿⡟⠀⠀⣿⣷⣾⠁⠀⠀⠀⢸⣿⡄⠀⣼⣿⠀⣸⣷⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣿⣿⣿⠀⣤⣾⣿⣶⡖⠀⠸⣿⣿⠁⠀⠀⣿⣦⣤⣴⡀⠀⠀⠈⢿⣷⣴⣿⠇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠉⠉⠉⠉⠁⠀⠀⠉⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣷⣶⣄⠀⠀⢠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣿⣿⠀⠀⢸⡇⠉⠉⠀⠀⠀⠀⣠⣤⠀⠀⠠⡡⠚⠳⠀⠀⠀⣓⠀⠐⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠋⠀⠀⠸⡿⠿⠛⠀⠀⠀⢸⡏⢹⡄⠀⣦⠀⠀⠀⠀⠀⢠⣿⡄⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡀⠀⠀⢸⣧⣤⣤⡀⠀⠀⣿⣧⣼⡇⠀⢡⠐⡀⠀⠤⠀⢸⣿⠁⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠹⣿⣷⡆⠀⠈⠉⠉⠉⠁⠀⢸⣿⠁⠈⣿⠀⠀⠁⠈⠁⠀⠀⠸⠿⠀⠠⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⣾⣥⡤⠄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⡿⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠁⠈⢇⠀⠀⠀⠀⠀⠀⢰⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⢸⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⣸⣿⣽⣆⠀⠰⣆⠀⠀⠀⠀⣿⣶⣿⣿⣿⠏⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣄⢻⡆⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠃
⠀⠀⠀⠀⠀⢀⡿⢲⢦⡀
⠀⠀⠀⠀⠀⢸⡏⠳⣞⣻⡆⠀⢀⣀⡀⠀⠀⢰⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⢰⣷⠀⠀⠀⣠⣶⣶⣧
⠀⠀⠀⠀⠀⢸⡿⠀⡿⣭⣿⠀⢸⡏⣻⡆⠀⢸⣦⣤⠄⠀⠀⢠⡿⢿⠀⠀⠀⣸⡿⢿⣤⣿⢿⣇⠀⠀⢿⣅⡀
⠀⠀⠀⠀⠀⢸⡗⢾⣱⣾⠏⠀⣿⣿⡛⠁⠀⢸⣁⡀⠀⠀⠀⣼⣧⣼⡄⠀⠀⣿⡇⠈⠿⠃⠸⣿⡄⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠘⠛⠛⠉⠁⠀⠀⣿⡟⢿⣆⠀⠘⠻⠛⠛⠀⢀⣿⠋⠹⣇⠀⠘⠛⠀⠀⠀⠀⠀⠛⠃⠀⣤⣤⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉


⣶⣶⣶⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢘⣘⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⣠⣾⣿⣷⠠⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⠟⢳⣿⠗⠶⣾⠟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣼⡇⠐⢛⡟⠀⢀⠀⠀⢸⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣿⢫⡘⣿⡇⠀⠈⠇⠀⠸⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⣱⡆⢟⡇⢀⣤⠶⡿⠟⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢀⠙⢀⣾⣷⣿⣿⣶⠆⣿⣧⣿⣟⣧⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣧⣾⢲⣾⣷⣀⣹⣿⣏⣀⣉⣽⣿⣿⣿⣿⣯⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⡍⣽⣿⢿⣿⠟⣿⢋⣽⡏⣽⣟⢩⣗⣿⡋⣽⡇⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡜⣿⠟⠻⣷⢋⢾⣿⡇⣸⣿⠘⣿⣿⡇⣿⣷⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣤⣾⣧⣤⣾⣿⣿⣃⣘⣿⣄⣉⣥⣧⣉⣩⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠴⠶⠶⠿⠿⠿⠛⠻⠿⠴⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡿⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⢷⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠈⠛⠲⠦⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠴⠖⠛⠁⠀⠀⢸⠆
⠀⠀⠀⠀⡀⠀⠀⠷⠶⠦⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⢀⣠⡀⢠⣤⡄⠀⣠⣤⢠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠴⠶⠾⠀⠀⢀
⠀⠀⠀⠀⢿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⣿⣿⡇⠘⣯⣧⢀⣷⡇⢸⢿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡿
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡇⠀⠀⣟⣿⢻⣿⠇⠀⢻⣾⣾⣾⠁⢸⢿⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠠⣄⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⣿⣿⡟⠀⠀⠘⣷⣷⠇⠀⣸⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣠⠄
⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣷⡶⡆⠀⢻⣿⠁⠀⠀⠀⢻⡾⠀⠀⣼⣻⣲⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠃⠀⠀⠀⠀⠈⠃⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠲⣦⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣴⠖
⠀⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢼⠀⡛⡄⣄⢠⡀⣠⡀⠀⢀⠀⠀⠀⠀⣤⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉
⠀⠀⢠⣤⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⠁⠘⠾⠃⢻⢇⡀⢻⡅⢰⡷⠀⢿⣀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣤⡄
⠀⠀⠀⠀⠀⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣠⠎⠉
⠀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣥⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠄⣿⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠠⠚⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠠⢈⣇⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⣸⡁⠀⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠁⠀⢁⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡈⠀⠈⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡇⣸⣀⡀⠀⢀⠀⡄⢀⣀⡀⢀⠀⡄⠀⠀⣀⢀⡀⣀⠀⣠⠄⣀⣠⡔⣿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⢸⣆⠀⠀⣾⢀⡇⣾⢸⣿⣼⢀⡇⣠⠀⣿⢸⡇⡿⣾⢰⡇⣿⠀⡇⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⢸⡄⠀⠈⢩⡟⠉⠉⠁⠈⠉⠉⠁⠀⠈⠉⠉⠁⠈⠉⠉⠉⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⠀⠀⠀⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡆⢀⡤⣄⠀⠐⣶⠀⠀⣶⠀⣠⣤⣤⢠⡤⣤⢤⡄⢶⠂⠠⣤⠀⣤⠀⢴⡆⠀⢴⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⣇⠘⡆⠀⣿⡄⢰⣿⠀⣿⠀⠈⠀⠀⣿⠀⠀⢸⠀⠀⡇⠀⣿⠀⢸⢧⠀⣼⠀⡞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⣿⠈⠛⡷⠚⡇⢧⡾⢸⡀⣿⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⡇⠀⣿⠀⢸⠘⡆⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠀⣧⢸⡀⢠⠇⠀⡇⠘⠃⢸⡇⣿⠉⣀⠀⠀⣿⠀⠀⢸⠋⠉⣇⠀⢹⠀⢸⠀⢹⡇⠈⢧⠘⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠁⠀⠉⠉⠀⠈⠋⠀⠀⠀⠀⠘⠛⠉⠀⠀⠛⠀⠀⠘⠃⠀⠉⠀⠛⠂⠛⠀⠀⠁⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⢹⡏⣿⢺⡄⡔⡧⣰⠞⡆⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠟⠸⠷⠿⠥⠿⠀⠧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⣀⣀⠀⠀⣀⣀⠀⠀⢠⣤⣴⠦⣤⣀⠀⠠⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⢸⣿⠀⢹⡏⠀⣼⠋⠙⢷⠀⠀⢸⡇⠀⠀⠙⡇⠀⢸⡇⠀⣸⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⢹⡇⣼⠀⠀⣿⣆⠀⢸⡇⠀⢸⡇⠀⢀⣼⠇⠀⢸⣇⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢠⡟⠸⣇⣿⠀⢰⡇⠈⠛⠶⣷⠶⠾⣧⡴⣟⠁⠀⠀⢸⡟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⣿⣿⠀⠘⣇⠀⠀⢠⡇⠀⠘⡇⠀⢻⡄⠀⠀⢸⡇⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢿⡇⠀⠀⢿⡀⠀⣼⠃⠀⠀⡇⠀⠀⠻⣄⣀⢸⡇⠀⠀⠻⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠘⠃⠀⠀⠈⠙⠛⠁⠀⠀⢴⣷⠄⠀⠀⠈⠁⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢆⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⢸⡗⢢⡶⣶⢸⢗⡆⠀⢰⡆⣿⠀⢠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡞⠧⠟⠚⠸⠷⠃⠘⠧⠿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠉⠉⠁⠛⠁⠀⠘⠛⠛⠛⠛⠛⠓⠒⠒⠒⠂⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢠⣤⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠻⣧⠀⢸⡿⠀⠀⣠⣤⣀⠀⣿⡿⠀⠀⢸⣿⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠹⣷⣿⠃⢀⣾⠟⠙⣿⡆⣿⣿⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡏⠀⢸⡏⠀⠀⣿⡇⣿⣿⠀⠀⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⣇⠀⢸⡇⠀⠀⣿⠁⣿⡟⠀⠀⢸⡟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠙⠛⠀⠀⠻⠶⠾⠋⠀⠻⣿⣦⡴⠟⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣦⠀⠀⢠⣴⣦⠀⠀⣠⣄⠀⢠⣶⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣾⠁⠀⠀⠉⠀⠀⣿⠃⠸⡇⠀⣿⡟⣆⢸⣿⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⣿⡀⠀⠀⢀⡀⢠⣿⣤⣤⡇⠀⣿⡇⠹⣎⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣧⡀⢀⣿⡇⣼⡿⠀⠨⣿⠀⣿⡇⠀⢻⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠛⠛⠋⠀⠿⠃⠀⠀⠿⠀⠿⠇⠀⠈⠿⠃⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⡶⢦⣄⠀⠀⠀⣀⡀⠀⢠⣤⣤⡄⢠⣤⣤⣤⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠹⣆⢠⣾⠛⢻⡆⠀⢸⣧⠀⠀⠀⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⡇⠀⠀⣿⣾⠃⠀⢸⡇⠀⠐⡿⠀⠀⠀⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣠⠏⢻⡄⠀⣸⠃⠀⠈⣷⠀⠀⠀⢻⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠘⠛⠛⠁⠀⠈⠛⠚⠋⠀⠘⠛⠛⠟⠀⠀⠸⠇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉






⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢚⣀⣓⡛⣀⠀⠀⠀⠀⠀⠂⠈⠅⠛⠁⠀⠀⠀⠀⠀⡠⡀⢲⢣⠆⠀⠀⠀⠀⣖⡄⠂⣆⣓
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠾⠿⠄⠀⠀⠀⣻⣽⣟⢴⣿⣽⣿⡁⠀⠀⢲⢆⣶⢲⢶⣾⠟⠀⠀⡀⠉⢉⠉⡉⠁⡀
⠀⠀⠀⠀⢴⡶⢂⢰⠏⣦⣄⠀⠀⠀⠘⠉⠉⠉⠉⢉⠉⠉⠁⠀⠀⣼⣫⣽⣬⣼⡽⠷⠀⠀⠳⢩⠊⠈⠘⠚⠥⠂
⠀⠀⠀⠀⠈⠀⠀⠋⠒⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠂⢉⣰⢀⢐
⠀⠀⠀⠀⢀⡤⣌⣤⣴⣶⣶⡆⠀⠀⠀⡄⠀⣀⢀⣀⠀⣠⠀⠀⠀⠘⠷⢶⡄⢳⢷⡇⠀⠀⠀⠆⠂⠐⠈⡂⡄
⠀⠀⠀⠀⡸⠠⡯⠭⠁⢀⣒⡂⠀⠀⠀⠳⢂⣿⣾⡟⢺⠏⠀⠀⠀⠀⢨⡭⡄⠂⣄⡄⠀⠀⠀⢻⡿⣿⣛⢿⡟
⠀⠀⠀⠀⠀⠀⣇⡀⠃⠀⠈⠁⠀⠀⠀⠰⠟⠉⠀⠃⠉⠀⠀⠀⠀⠀⠸⠷⠃⠀⠛⠀⠀⠀⠀⠛⠿⠟⠛⠊⠛⠂
⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⡄⠀⠀⠀⢠⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⡀⡀⣀⣐⣐⣒⣀⣀⣀⠀⠀⣸⣿⣿⣛⣿⣞⣵⣇⠀⠀⣠⢏⣚⠘⠆⡿⣩⠅⠀⠀⠀⠀⠚⠃⠛⠃
⠀⠀⠀⠀⢻⢽⠿⠙⢯⢽⣋⣏⡏⠀⠀⠉⠉⠩⠉⠉⠉⠉⠉⠀⠈⢁⣐⣀⣉⣼⣿⣷⡟⠀⠀⢰⢶⢶⣶⣶⢲⣶
⠀⠀⠀⠀⠿⠾⠿⠷⠿⠾⠿⠾⠷⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⢠⣧⣇⣿⣽⣿⣷⠄⠀⠀⠸⠿⠿⠷⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠄⠀⠀⡠⠀⠠⡔⠀⠀⣾⠃⠁⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⢐⣷⣦⠀⠏⠀⠀⡧⠃⠀⠃
⠀⠀⠀⠀⠀⠞⠁⠀⠀⠒⣺⣿⡿⠀⠀⠀⠸⠁⠞⢀⡅
⠀⠀⠀⠀⡔⠀⠀⠀⣤⣾⠋⣡⣦⣤⡶⠴⠛⠀⣐⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡄
⠀⠀⠀⡀⠀⠀⠀⠼⠛⠻⣿⣿⣷⠶⠤⡤⡦⠀⡤⣀⠈⠂⢠⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠘⠇⠘⡠⠶
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠰⢡⣿⠻⠓⠀⠀⠀⠀⠀⣚⠗⣼⠆⠈⣠⣴⡶⠶⠶⠛⠛⢩⣍⣤⣴⡶⢸⡟⢻⡇
⠀⠀⠀⠀⠉⠲⡖⠚⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠈⣁⡄⢨⡆⢰⡶⠟⠛⣿⠀⣿⡇⣿⣄⡄⢸⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⢿⣋⣀⣀⣀⣀⡠⠔⠚⠃⠀⣿⣷⣸⡇⢸⣧⣤⠀⢿⣧⣿⠀⣿⣋⣁⢸⡏⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋⠉⠉⠉⠀⠀⠀⢀⡄⠀⣿⠸⣿⡇⢸⣧⣤⡄⠸⣿⣇⠀⢻⡟⢩⣌⣷⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠤⠤⠀⠐⠒⠂⠉⢽⡇⠀⢿⣀⡙⣣⣼⣏⢁⣶⢺⡟⠋⠀⢸⡇⢸⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⢉⣋⣿⠙⣿⣸⡇⢸⡿⠇⠀⢸⣇⣸⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⣀⣀⣀⣀⣀⣴⠶⣟⢸⣧⢈⣿⣿⡆⢻⣿⠁⠸⠟⠛⠀⣀⠩⠭⠔⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠛⢛⣋⣉⣥⠤⠶⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠒⠒⠒⠒⠒⠋⠉⠘⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠁⠈⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠋⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢻⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣤⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣄⣀⣀⣀⣁⣄⣀⣠⣀⣤⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠸⠿⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠔⠐⠄⡠⠤⠤⣠⠈⠀⠀⣾⠇⠈⢀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⣼⣆⠀⠀⡿⠀⠀⢰⡰⠁⢰
⠀⠀⠀⠀⠀⠀⢀⣤⠴⠛⠁⢀⣾⣿⣧⠀⠀⠀⠀⡿⠁⡀⠈⡀
⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠚⢋⣿⣿⡟⠀⣀⠀⠐⠁⡜⠃⣠
⠀⠀⠀⠀⠀⡈⠀⠀⠀⢀⣴⡿⠋⣁⡀⢀⣇⣠⡾⠃⠠⠜⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⢠⣟⣿⣦⣶⣿⠛⠻⠭⠭⠀⣀⠒⠛⠄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⢀⡟⢀
⠀⠀⠀⢠⡁⠀⠀⢰⠟⠉⠉⣿⣿⣿⣿⠿⢷⢶⡶⠒⣢⢤⣄⠈⢀⠰⣦⣤⣤⣤⣤⣤⣤⡤⠤⠄⣀⣀⡀⠀⠀⠀⠘⠃⠈⡴⠟⠃
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠏⣸⡟⣿⠆⠐⠀⠀⠊⠀⠠⠟⢷⢸⠄⠈⠀⣠⣤⣶⣶⡶⠾⠟⢛⣉⣁⣤⣤⡦⣾⡿⢿⣷
⠀⠀⠀⠀⠱⢤⣀⣀⣠⣄⠤⠖⠛⠉⠀⠤⠐⠀⠀⠀⠠⠐⡇⠴⠟⠀⠀⠚⠋⣉⣥⣤⢴⣶⠀⣼⡿⣿⡟⠉⠁⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠻⣋⣉⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣶⡆⢸⣿⢸⣿⠋⠉⠸⣿⠀⣿⡇⣿⣷⣶⡆⣿⣷⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⣾⣿⠄⠀⠀⠀⢀⣀⡤⠖⠟⠀⢸⣿⣷⢸⣿⢸⣿⣤⣄⠀⣿⣧⣿⠃⣿⡏⢉⡁⣿⡏⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣴⣿⠛⠛⠛⠛⠉⠁⠀⠀⠤⠀⢸⣿⢻⣿⣿⢸⣿⠉⠁⠀⢹⣿⡿⠀⣿⣿⠿⢏⡛⣣⣼⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠀⢸⣿⠈⣿⣿⢰⣿⣿⠿⡀⣘⣯⣥⡄⢰⣿⠀⢸⣇⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠲⠶⠖⠒⠒⠉⠉⠉⠱⣿⠀⠸⢿⣀⡈⢻⣤⣶⡀⢸⡿⢿⡏⠉⠀⢸⣿⠀⢸⣿⣿⣇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⠀⢰⡿⠛⠿⢻⡇⣿⡇⣼⡇⣿⣷⣷⠀⢸⣿⠀⣸⡟⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡄⢸⡇⢴⣶⣾⡇⢸⣿⣿⠀⣿⣇⣤⡄⠀⠿⣷⠟⠁⣛⣃⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡦⣤⣄⣀⣀⣈⣤⣾⠟⢿⡇⢸⣇⢀⣿⢻⡇⠘⣿⡿⠀⠛⠛⠋⠁⣀⠤⠶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡇⠘⠿⠿⠋⠈⣁⣠⣤⡴⠖⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣄⣀⣀⣀⣠⣤⠴⠞⢧⣴⣶⠶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⠠⠉⠉⠉⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣫⡍⠉⣷⠀⠀⣀⠀⠀⢀⠀⢀⠀⠀⠜⠁⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⡿⢠⡞⠉⣷⠀⣿⠖⣻⠇⠀⠀⠀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣠⡴⠋⠀⢻⣀⡴⠃⣼⠏⠀⣯⠀⠀⠀⠰⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣠⣶⣄⢴⡄⣴⡆⣴⣦⣰⡄⢠⣶⡄⣦⢰⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣹⡇⣿⣾⣷⣿⡇⣿⠾⢻⡇⢸⣿⡇⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠙⠷⠋⠟⠸⠛⠇⠿⠀⠸⠷⠾⠋⠗⠟⠸⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⢀⣀⡀⢀⣀⡀⠀⣀⣀⣀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⢸⣿⡇⣾⣿⠻⣿⣷⠸⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢸⣿⡟⢸⣿⡇⢸⣿⡇⠹⣿⣷⣤⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣏⢨⣿⡇⢸⣿⡆⣤⣤⠙⣿⣷⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠃⠸⣿⣷⢿⣿⡇⠻⣿⣶⣿⠟⠀⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠴⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⢸⡇⠀⣠⢤⡀⠀⠀⠀⠀⢰⡗⢀⣴⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢠⡾⢁⣾⠁⢸⡗⠀⠀⠀⢀⣾⠁⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠘⠋⠀⠘⠧⠰⠟⠀⠀⠀⠠⠾⠁⠀⠸⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⡶⠶⠶⢚⣛⣓⠒⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⢋⠥⠖⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉





⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⢧⡀
⠀⠀⣇⠱⢄
⠀⠀⠘⡄⠀⠑⢄
⠹⡑⠒⠛⠀⠀⠀⠑⠒⠤⠤⠀⠀⠀⣀⣀⡀⠀⠀⠀⠠⠀⢤⡀
⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣄⣀⣤⣴⢿
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣯⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢈⣀⣀⣀⣀⣀⣀⠠⠤⠤⠤⠤⠤⠤⠔⠒⠒⠒⠒⠒⠒⠒⠂⠤⠤⣄⡀
⠀⠀⠀⠀⠈⠓⠄⡀⠀⠀⠀⠀⢿⡇⠀⢀⡤⠍⠈⠉⣉⣀⠄⡀⢀⠀⠀⢀⡠⠀⣄⠤⠒⢲⠄⠀⠀⢸⡿⡄⠀⡸⣿⠀⢰⡖⠀⠀⢢⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠘⣧⠀⣸⡇⠀⠜⠀⢸⡇⠀⢹⣇⠀⠀⢸⠃⠀⣿⠀⣀⠁⠀⠀⠀⢸⣧⠘⢾⠁⣿⠀⣸⡇⠀⠐⠊⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡄⢿⣇⠀⡀⣀⢺⡷⠀⠠⣿⡆⠀⡟⠀⠺⣿⠋⠉⢀⡀⠀⠀⠈⣿⠘⠞⠀⣿⠀⢹⠏⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢿⡈⢃⡟⠀⣇⠀⠀⠘⣷⣸⠇⠀⠀⣿⣄⠀⠀⠼⠀⠀⠠⠿⠄⠀⠀⠛⠀⠚⠂⠄⣈⡆⣯⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠠⠁⣉⠀⠈⠀⠀⠀⠀⠙⠉⠃⢀⣀⣀⣀⣠⡤⠤⠴⠶⠖⠒⠒⠒⠋⠉⠉⠉⠓⠒⠢⢴⡇⠀⠀⠈⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣍⠉⠉⠉⢁⡀⠀⢀⠀⣠⠤⡄⠀⢸⠀⣀⣀⣤⡤⠤⠤⠤⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢂⡤⣤⠀⢰⣿⠏⢰⡆⢀⣼⡇⣠⡏⠀⠘⢦⣏⠀⢰⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡛⠉⣠⠾⣇⢀⣼⣡⠎⠀⢉⣸⣥⡤⠖⠋⠀⠀⠈⠙⠓⠒⠒⠒⠒⠛⠛⢂⡒⣒⠀⣀⡐⠓⠒⠲⠤⢄⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣾⣥⣤⠀⠉⠉⠁⠀⠈⢉⡤⠔⠚⠉⣽⠏⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣽⣿⠋⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⣶⣶⠀⠀⣰⣏⣀⣠⠴⣻⡅⣰⣶⣶⠀⢰⣶⣶⣦⣤⡐⢶⣶⣶⣶⣶⡎⠛⠛⠛⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣀⣿⣿⠀⣀⣉⡉⢀⣀⣸⣿⣧⠀⢀⣀⣨⣿⡇⠈⢹⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠛⣿⣿⠀⢛⣛⣃⡘⠛⠛⢻⣿⣆⠘⣿⣿⡉⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠿⠿⠀⠀⠻⠿⠀⠿⠿⠿⠇⠀⠀⠀⠻⠿⠄⠘⠿⠷⠀⠀⠸⠿⠇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣪⣿⣿⣿⣿⡤⢤⣤⢦⣤⢦⡤⠤⠤⠤⠤⠤⠤⠤⠤⠠⣠⠔⢂⠲⢤⠔⠲⣤⠤⠖⠒⢲⣦⠰⠦⠀⢼⣿⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠿⠟⠛⠋⠀⣿⠙⠀⠀⠊⠱⣄⠀⠀⠀⠀⠀⢀⡤⠞⢡⣆⠀⡷⠀⠀⠀⠁⠀⠀⠀⢰⢰⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡄⠀⠀⠀⣰⣤⣍⠓⢒⣒⠉⠁⡴⠲⣄⠹⣦⣥⣀⣤⣤⣤⣴⡾⠁⠘⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠀⣀⡿⠫⣤⡿⠃⡼⣿⠀⢸⣇⠀⠀⠀⠙⢿⣏⠉⠉⠉⠛⠀⢀⠆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢱⡙⠦⣤⡿⠃⣾⡇⣿⡄⢸⣿⡄⠀⠀⠀⠀⠙⡷⠆⠀⠀⢀⠎⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣬⠈⠳⢄⡀⢴⠋⠀⢹⡁⠀⠹⣷⡀⢈⡆⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠉⡄⠀⠀⠉⠑⠢⠤⣚⠃⠀⠀⠈⠉⣉⡀⠤⠐⠊⠁⠀⠀⢄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠿⠿⣿⡿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠛⠉⠉⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⣠⠴⠒⢦⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⡜
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢀⣾⠃⠀⠀⠉⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣴⡿⠃⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠿⣦⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠿⠃⠀⢀⣾⡟⣀⠔⠁⠀⣀⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡤⠊⣹⣿⠆⢀⣶⠀⠀⢠⣿⡟⠊⠀⠀⣠⣾⠟⢁⡴⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⢀⣾⠇⡠⢊⣽⣣⠊⠀⣴⣿⠏⢀⣿⡟⠀⢀⣾⡟⠀⠀⠀⣼⡿⠧⠒⠉⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠘⣿⣿⠀⣼⡏⡜⠁⣾⡷⠁⠀⣼⡿⠃⠀⣸⡿⠀⢀⢾⣿⠃⠀⠀⣰⣿⡇⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⠀⠀⣿⡏⠀⣿⡟⠀⢸⡿⠁⠀⣼⡿⠁⠀⡔⠙⠷⠔⠁⢸⣿⠀⢀⡔⠁⠙⠓⠒⠋⠀⠀⣠⣶⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⣀⡼⠋⠀⠀⠙⠀⠀⠀⠀⠀⠀⠻⣧⡠⠊⢀⡴⢛⠂⠀⠘⠿⠗⠋⠠⠚⠛⠛⠛⠛⠲⣴⣿⡃⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⡁⣀⣀⠀⢤⡤⢤⠄⣠⢀⡤⣄⠀⠀⠀⣰⠟⡰⠃⠀⠀⠀⠀⣀⣀⡀⢀⣴⠖⠂⡂⣰⣿⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠘⣧⠁⠀⡼⢀⡟⢲⣷⠟⠒⡋⠀⠀⣰⡿⢊⡴⠶⣦⡀⣠⡾⢋⠠⠂⣿⣧⠀⠘⢡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠒⠋⠀⠀⠃⠘⠀⠙⠘⠲⠚⢁⡠⢶⣿⡵⠁⡀⠀⣿⢳⣿⠁⠀⢀⠀⠈⢿⣷⠀⣼⡏⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡠⠚⣿⡆⠀⢀⠴⣶⣄⠉⠁⠀⢸⣿⠁⢎⣀⠔⠃⠘⢿⣄⡴⠃⢄⣠⣾⠟⠀⠻⣧⠴⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⠞⠁⣼⡿⢁⠞⠁⣰⣿⢃⣤⣤⣄⡀⠁⢀⡴⢋⠝⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢀⣤⢀⣴⡤⠚⠙⣷
⠀⠀⠀⠀⠀⠀⠀⣾⣯⠊⠀⣰⣿⡱⠃⠀⣼⡿⠁⣉⡤⡀⠀⠉⣹⡟⣉⣥⣄⠀⣠⡶⠚⡂⢀⣼⡟⠁⢀⣾⢃⣾⠋⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⣰⣿⡟⠁⢀⣾⡿⣡⡞⠁⠀⣼⠇⣰⡿⠊⠁⣠⢯⣾⢋⡠⠔⠁⣾⠏⠀⢀⣾⠃⣼⠇⠀⣠⣾⠟
⠀⠀⠀⠀⠀⢀⣿⠏⠀⢠⣿⡟⠀⠀⣾⡿⣱⡟⠀⠀⣼⠏⢠⣿⠷⣶⣉⠀⢸⣿⣀⣀⠔⢰⡏⠀⢀⢾⡏⢰⡏⠙⠛⠉
⠀⠀⠀⠀⠀⢸⡟⠀⠀⣾⡿⠁⠀⢸⣿⠃⢿⡅⢀⢼⡟⢀⣾⡇⠀⠈⢻⣧⣀⠈⠉⣀⡴⠈⠣⠴⠋⢈⡣⢼⡖⠒⠒⡆
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠃⠀⠀⠸⣿⡀⠈⠉⠁⠈⠉⠁⢸⡇⠀⠀⠀⠉⠛⠛⠛⠉⣀⣠⠴⠚⠉⠁⠀⠈⠳⠦⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠁⠀⠳⣄⣀⣀⣀⣀⣀⣀⣤⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉⠁

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣸⣿⡀⢠⣤⣿⣟⣿⠁⢠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠀⠀⢹⣿⣿⢿⣧⠈⣹⡿⠉⣿⠀⣾⠏⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣟⢻⣷⠸⠿⠈⠈⠛⠀⠉⠁⠈⠛⠸⠿⣶⢄⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣤⠈⠻⠿⠋⠀⠀⣤⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⠷⢄⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣿⠃⡾⠀⠀⡤⠤⣼⡿⢄⣀⣤⡤⠞⠋⣠⣤⠀⠀⠀⠀⠀⠦⠹⣯⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣴⡷⠀⢀⣼⡿⠁⢀⣤⡶⣦⠀⢰⣿⡧⠒⢲⡄⠀⢠⣶⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢉⣿⡦⠊⣹⣿⠁⢠⣾⠿⠚⢁⣴⣿⡟⣀⣠⣿⠧⠚⠹⣿⡀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⣠⠖⠁⣿⡇⠀⣿⣃⡴⢻⣏⣀⡴⢋⣿⢿⣎⣽⠟⠁⣠⠖⠒⣿⡷⠲⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣀⣠⠿⠁⠀⠉⠁⠀⠀⠉⠁⢀⣾⡏⠀⠀⠀⠀⠀⢿⣀⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣀⢀⣀⣀⠀⢀⣀⣀⠀⣀⣾⣋⣀⡀⠀⣀⣀⣀⣀⠀⠀⣀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⢿⠀⢻⣿⠀⢸⣿⡏⠀⣿⣿⠛⠛⠃⠀⣿⡟⠙⢻⣿⡀⣿⣿⠀⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣤⠀⠀⢸⣿⡇⣸⣿⠁⠀⣿⣿⠾⠃⠀⠀⣿⣷⣶⣾⠟⠀⢸⣿⣄⣸⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠋⠀⠀⠀⣿⣧⣿⡟⠀⠀⣿⣿⠀⣀⣠⠀⣿⡏⠉⣿⣆⠀⠀⠙⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣀⣴⠀⠀⢻⣿⣿⡇⠀⠀⠿⢿⡿⠿⠿⠀⢿⡇⠀⣿⣿⠀⠀⣿⡇⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠀⠀⠈⠁⠀⠀⣠⣶⣿⡿⢿⣿⣶⣄⠀⠀⠀⠈⠙⠃⠀⠻⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⣉⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠈⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢿⡄⢰⡆⠀⣶⣦⣄⡘⠛⠃⠀⠀⠀⠀⠀⢸⣿⡄⠀⣾⣿⠀⢰⠆⣸⣟⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣭⠽⠆⠁⣼⣿⠟⢻⣿⣧⠀⣰⣶⣶⣶⣷⠀⣿⣷⠀⣸⣿⣇⠈⠺⢯⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣼⣿⡏⠀⢸⣿⡿⢀⣿⣿⣀⣿⣿⡄⠘⢿⣿⡿⣿⣿⡄⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣤⣴⣿⡿⠁⣾⣿⠟⠛⠻⣿⣷⠀⠀⣤⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠀⣼⣿⡿⠀⠀⠀⣿⣿⣧⠀⠻⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⠃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣾⠀⠁⢰⣿⠀⠀⠀⠀⠀⠀⠄⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠉⠒⠀⢾⡟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠃⠀⢀⡠⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣿⡇⠀⢀⡤⠄⠀⠀⣾⠃⢰⡄⠀⡔⠁⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢿⡆⠁⣿⡇⠀⣿⠁⢸⡏⠀⣿⠀⣸⣷⠎⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢹⡇⠀⢻⡇⠰⣿⢀⢿⡇⡘⠻⠴⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣸⠃⠀⠘⠷⠁⠙⠁⠘⠷⠁⠀⠀⡌⣿⠂⠀⠀⡠⠀⠀⠀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢰⢀⡿⠀⢀⠌⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢣⡴⢋⠁⢀⠀⠀⣴⠆⠀⢠⡅⢀⣿⠀⠀⠠⠙⠚⠁⢀⢋⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠎⠀⠀⡇⣀⠀⢃⠀⢻⣆⡀⣿⠃⢸⣿⠀⢰⡟⢀⡾⢀⢢⡟⠁⠌⠀⡀
⠀⠀⠀⠀⠀⢠⣀⠰⣿⠎⢹⣇⢸⡟⠀⠨⠀⠀⢿⡆⢿⠀⣼⡏⠀⢸⡇⢸⣇⡌⣿⠃⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡰⣿⠘⢧⠠⠂⠀⠇⡸⠃⠈⠛⢹⣇⢀⡏⠓⠁⠻⠀⠻⣆⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡔⠵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⡠⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣴⣿⡀⢀⡖⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢀⣴⣾⡿⣽⣿⣿⣿⣿⣿⣿⡿⣷⡞⠫⡏⠉⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⣿⣇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⠥⠀⡀⠄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⢸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠳⠶⠚⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠀⠠⡿⠿⠿⠟⠛⠋⠉⠉⠀⠀⢀⣀⣤⣴⡖⡀⠀⠀⠀⡄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⣤⣤⣶⣶⣿⣿⣿⣿⠟⢩⣗⠠⠀⢰⣷⠈⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⢠⣀⣿⠀⠀⠋⢀⣤⠀⣀⣍⠴⣍⡛⢿⣿⣿⡿⡟⢿⠿⢿⣿⠛⠍⠀⣼⣿⡀⠒⣸⣿⠀⣤⠆⢀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⢸⣼⡿⠞⠀⢈⠗⠲⣂⠄⢈⡅⠉⢁⡀⠆⣉⡩⠛⠉⠁⢀⠐⠂⡄⠑⠐⠭⣢⡀⡀⢨⠑⠣⠛⠻⢇⠁⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠊⠀⠀⠠⠖⠧⢀⡄⠀⠀⡀⠀⠁⠘⠋⠈⠁⠀⡀⠄⠂⠙⡀⢰⣷⠀⠠⡀⠀⠀⠈⠻⠗⡤⠀⠀⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠃⠀⠀⢀⠂⠠⢸⣇⣾⠀⠈⠁⠀⠀⠀⢠⡆⠈⢀⠀⠂⢠⣇⣸⣿⡀⣀⣀⠄⢀⠅⠀⠁⢠⡀⠀⢀
⠀⠀⡆⠀⠐⠀⠀⠀⠤⣀⣀⠀⢀⣠⣤⡤⣬⣍⠀⠀⠴⠄⠀⣠⡄⣺⣇⢒⠶⠖⠠⡚⠛⠙⠉⠉⢀⠄⠊⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠁⠨⠑⠠⠄⠉⠀⢀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠐⠆⠨⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠂⠈⠘⢿⣿⣶⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡰⢠⢤⣄⡄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⣤⡄⢠⣤⣤⣤⣤⡄⠀⠀⡀⠔⢄⠀⠀⢠⣤⡆⣦⣦⣴⣴⠀⣤⣴⣤⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠛⣀⣃⣛⣃⣘⣘⣛⠛⠛⢃⡀⠊⢀⣀⣀⣈⣢⣜⡛⠓⣁⢛⣙⣙⣀⣉⣙⣙⣋⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⢸⡇⣿⡯⢹⢹⠀⢺⣿⣿⡿⡟⣻⣻⣿⣽⣿⣇⠀⣿⣸⣿⣿⣿⠸⢿⠿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⡇⣿⣇⢸⢸⠀⢸⣿⠟⣿⠻⠽⠿⡿⢛⢿⣿⠀⣿⣿⣿⣿⣿⠀⢸⠀⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⢹⢸⡇⡏⣿⢸⢸⠀⢸⣿⣧⣫⡆⣧⣦⣧⣷⣾⣿⠀⡿⣿⣿⣿⣿⠀⢸⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣙⣉⣉⣁⣉⣉⣋⣀⣈⣛⣋⣉⣉⣉⣉⣉⣁⣀⣀⣀⣃⣘⣉⣋⣙⣃⣘⣁⣛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡫⢿⣟⣻⡿⡙⣿⣙⣻⡫⢻⡏⢿⣿⣿⣹⡏⣛⣿⠹⣿⣿⡟⣫⡟⡹⣯⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢙⣙⡋⡙⡛⣛⡙⢛⣋⡙⣙⣋⢛⣙⢋⢉⠋⣙⣙⠛⡛⣉⣉⢙⢉⣉⣙⣋⡛⣙⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢋⠛⠁⠛⠙⠋⢁⠘⠛⠀⡛⠙⠛⢙⣻⠛⡁⢛⡋⠛⣃⠛⡙⢋⡘⠋⣛⡋⢛⣟⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡋⡗⠀⠀⠀⢩⡎⣷⠇⠀⡏⠉⢀⡿⠛⢃⡟⢸⡇⣿⢹⡏⣿⣼⡇⢸⡏⠛⣾⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣧⡀⠀⣀⣀⣇⣿⡆⢰⠛⠀⢸⣇⣤⢸⡟⣿⢁⡿⢻⡇⢿⡟⣷⣾⣹⣿⣿⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣩⠅⣈⠁⣭⣤⣥⣬⣤⣬⣥⣭⣬⣭⣥⣭⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⡆⢰⣶⡆⡆⣶⣦⡆⣦⣶⡆⠨⢉⢈⡍⠀⢰⡖⠂⣶⢶⡆⣶⢶⡆⣶⠀⢰⡆⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡿⣧⣸⣧⣯⡇⡟⡏⢹⠁⣿⡇⢀⢸⢸⡇⠀⢸⡟⠀⣿⣼⡇⣿⣼⡇⣿⠀⢸⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠉⠙⠋⢙⣿⠁⠁⠃⣜⣤⣝⣥⡌⠈⠈⢡⣠⡜⡛⣣⢙⡘⣣⡟⡙⢣⡟⡛⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⠘⣻⡞⡕⠐⠻⣦⠘⣿⣿⣳⠀⠚⢷⡄⢿⢸⡇⣿⢸⠀⣿⣿⡇⢸⣜⡛⣷⢘⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣿⣇⢷⣄⣀⡉⣰⣿⣷⡻⣦⣐⣊⣀⣿⡼⣧⣈⡈⣠⣿⣿⣇⢸⣿⡇⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠬⠈⠍⠭⠡⠈⠅⠬⠩⠍⠍⠩⠤⠭⠨⠄⠭⠩⠄⠭⠨⠩⠍⠥⠡⠉⠍⠥⠬⠌⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡂⠀⣆⣶⢲⣶⢰⠀⡖⣶⣶⣶⠂⠀⣶⣄⢰⠳⣷⣶⣶⡖⠐⣷⣶⣶⣶⣶⣶⢲⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠁⠀⠾⠸⠼⠿⠼⠀⠏⠿⠽⠿⠁⢀⡟⢸⠸⡿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡁⢀⣛⣁⣀⣛⣀⠈⠛⣀⡈⠛⡴⣫⡆⡗⢩⠀⣾⡟⢫⣽⠛⠴⠲⢶⡖⢶⠶⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣾⢿⣿⠉⠉⣿⠀⠀⣿⠃⣾⡇⢻⣿⣇⣈⢣⢻⣽⣯⣽⣮⣛⣿⣜⣒⣾⠸⢟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣻⣿⣧⣸⣿⣠⣠⣿⣀⢷⡀⢠⣽⣿⡟⡸⢸⣿⣽⣟⣟⣿⣀⣿⣻⣯⣼⣿⡇

⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⢀⡀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⢀⣈⣥⣬⣍⣤⠀⠀⠀⢀⣈⣀⡙⣉⣁⣀⡀⠀⠀⠘⠸⠘⠏⢾⣀⠀⠀⠀⣿⣆⣹⣭⡅
⠀⠀⠀⠀⠘⢛⠛⣛⠛⠛⠀⠀⠀⢠⣿⢿⡝⢿⠿⠿⠄⠀⠀⢹⠼⡏⠛⣿⣏⠀⠀⠀⡀⡤⠰⠄⡰⢀
⠀⠀⠀⠀⠟⠉⢶⣃⠸⢳⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠛⠚⠚⠛⠚⠋⠁⠀⠈⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⡄
⠀⠀⠀⠀⠀⣀⢄⣀⣠⣤⡀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⢾⣯⣉⠠⣤⣤⠀⠀⠈⡔⠍⠍⠧⠈
⠀⠀⠀⠀⡎⣸⣛⡛⠉⠭⡅⠀⠀⠘⠧⢸⣷⣿⢶⡟⠀⠀⠀⠀⣲⢛⠡⣁⣁⠀⠀⠀⢦⣶⣶⢶⣾⠆
⠀⠀⠀⠈⠀⢸⡀⠂⠀⠉⠁⠀⠀⠀⠰⠟⠃⠻⠘⠁⠀⠀⠀⠀⢿⠜⠀⠼⠃⠀⠀⠀⠿⠷⠿⠿⠻⠆
⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⡤⡖⣒⢀⠂⡀⠀⠀⠀⡀⣠⢄⣀
⠀⠀⠀⢀⣀⣀⣘⣒⣒⣀⣠⣄⠀⢀⣿⣿⣯⣿⣿⣷⣧⠀⢀⡴⠧⣋⡽⢸⣩⢍⠀⠀⠀⠉⠙⠘⠉
⠀⠀⠀⠈⡟⣟⡃⢻⣜⣧⢧⡇⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣥⢤⡭⣿⣿⠿⠀⠀⢺⣷⣿⣿⣟⣿⡃
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠼⠾⠾⣷⠿⠿⠓⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣼⡇⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢀⣿⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠃⣀⠀⠀⠀⠀⠰⠋⠁⠀⡀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⠀⣬⠀⣤⠀⣤⡄⢀⡀⠀⣤⢸⣿⠀⢀⣤⢀⣤⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣿⠀⣿⣼⢹⣷⢛⣗⢸⡿⢸⣿⠀⣿⠌⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⠀⢰⠿⠌⠻⠇⠸⠟⢸⡇⣸⠷⠻⣿⠀⣿⣤⠞⠓⠺⣿⠀⡼
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠘⠧⠂⠀⠀⠀⠀⠀⠀⢸⣧⠃⣠⠀⠻⠞⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣿⠀⡀⠀⡀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢋⡀⠀⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢀⡤⢼⡇⢠⡆⠀⣿⠀⣴⡄⣶⠒⢦⣰⣷⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⠁⣿⡇⢸⡇⢸⡇⢰⣿⠀⣿⠀⣾⡉⢹⠸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡴⠻⠧⢿⡇⢸⣧⠞⣿⠀⣻⠞⠈⠑⠁⠀⠉⠀⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⢿⣆⠇⠀⠀⠀⠀⠀⠀⠘⣿⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⡇⢁⡀⠀⣸⠃⠀⣰⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⠀⡇⣼⢸⡇⣿⢀⡴⠃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⣇⠋⠈⢳⡏⢿⡀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⠀⠀⠈⠃⠀⠀⣾⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⢀⡀⠀⣰⠂⡄⠙⢦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢿⢸⡇⠀⣿⢸⠃⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡆⠁⠀⠈⢀⡄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠀⣠⡆⡸⣇⡠⣆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⡟⠀⢻⣠⡇⠈⢠⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠧⠀⠀⠀⠉⢀⠄⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡟⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⡇⠀⠀⠀⠀⢠⡞⠐⣼⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢈⡗⠀⢸⠀⡇⣼⠀⠀⣿⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠸⡄⠻⠼⡄⠠⣿⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢻⡰⢹⠁⣷⡇⠀⣧⡀⠀⠀⡟⠰⡇⢹⣿⠀⠀⢺⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠁⠁⠁⠀⠁⠈⠀⠀⣁⣀⠁⠁⠉⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⣀⠴⣄⠀⠀⠀⡔⢲⠀⠀⠳⣄⡎⠀⠀⠀⢠⡖⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠁⣿⠀⣿⠀⠀⣼⡝⣲⠗⠒⠊⠉⠉⡇⠀⠀⣾⠉⠉⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⠀⡏⠀⡼⠀⠉⠁⠀⠀⠀⠀⢸⡇⠀⡜⠘⣧⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠈⣇⡄⢽⣷⠈⡇⠈⢧⠘⡖⠘⣾⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠘⠀⠋⠓⠈⠋⠀⠃⠀⠘⠁⠈⠈⠈⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢢⠀⠀⣠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢾⠓⠒⢦⡀⠀⠀⠀⠀⢸⠁⠀⡸⠀⡼⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣠⠢⡄⠀⠀⣸⠔⠊⠀⠀⡧⢔⡪⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢧⡜⢣⣀⠔⢹⠀⠀⠀⡜⢳⢾⠊⠁⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⡰⠃⠀⠀⠀⠀⠀⠘⣇⠀⡜⠀⠀⠀⠑⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣾⡇⢲⣠⠀⣶⡆⢰⢸⡇⠀⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠛⠃⠚⠙⠀⠛⠇⠙⠘⠃⠀⠘⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢠⡄⡄⣠⠆⠀⠀⢠⡀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⡞⠋⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⡇⣿⠀⠀⠀⣾⢀⠸⡶⠀⡏⠀⠒⠒⠒⠶⠤⡄⠉⠉⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡧⣿⣿⠄⠀⡠⡏⢸⠀⠹⡄⡇⠀⡠⡄⣆⣀⠀⣠⠄⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠉⠙⠂⠈⠓⠧⠘⠂⠒⠃⠓⠊⠒⠃⠛⠉⠒⠓⠒⠛⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠂⠔⠒⠠⠒⠀⠔⠒⠄⠒⠢⠐⠂⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⢸⢲⢰⠋⢰⠉⠸⡇⣇⠐⡆⡏⠀⣇⢸⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⣈⡓⠢⠤⠀⠀⢸⡸⢸⠀⢸⠀⠀⡇⣿⠀⡇⡇⠀⡟⣾⠀⠀⠐⣒⠚⠩⣤⣤⡤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠸⠵⠸⡍⠸⣍⠀⠇⠻⢶⠇⢏⠀⠇⠻⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠢⠤⠒⠤⠔⠢⠤⠂⠤⠔⠠⠄⠂⠤⠐⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢴⣲⠑⠀⠀⢰⢲⡀⣖⡦⡖⠂⡔⡆⣶⣴⣆⡖⠦⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⣠⡄⠀⠀⠙⠏⢠⣄⠸⡯⡇⣿⠀⡇⠀⡇⠁⡇⠐⡏⢧⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣏⠏⠳⠴⡺⠏⠀⣯⠃⡟⡆⢟⡀⡷⠆⢿⠀⡇⣈⡇⠀⠀⠢⠀⡀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⠁⢀⠀⡀⠀⢀⠀⣠⢠⢄⡀⠀⠠⢠⢰⡆⠀⠠⡦⠀⠀⠀⡂⠑⠉⠀⠑⠅⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠢⠄⠔⠢⠀⠀⡏⡇⠷⡠⢸⢡⠈⠒⠋⠀⠀⠒⠀⢸⡇⣤⠀⡇⡀⢀⠠⠐⠀⠪⢬⢕⠢⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠌⠄⠥⠅⠀⠀⠷⠇⠇⠻⠸⠎⠀⡟⢢⠶⢆⡔⣶⢸⠁⣏⠀⣇⠉⢻⠀⠀⠀⠈⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠅⣒⡪⠅⠀⠀⠒⠂⠘⠐⠄⠂⠀⠁⠈⠉⠁⠉⠈⠈⠉⠉⠁⠉⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠒⠲⢤⠀⠀⠀⠀⠀⢠⣷⢸⡇⠀⠀⠀⢰⡆⠀⣤⠀⡠⣄⢤⠌⡅⣠⡄⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡆⠸⡅⠀⢷⠀⠀⠀⠀⢸⡏⣼⠃⠀⠀⠀⢸⠀⠀⡇⡆⡇⠙⢸⠀⡇⡗⣻⢸⣇⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣇⢀⣸⠀⣸⣖⢃⣔⣻⢸⣁⣿⣠⡶⢇⣴⣻⠀⠀⣇⡇⡇⡀⢸⠀⡇⡧⢺⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣈⣀⣡⠴⠃⠈⠉⠀⠀⠀⠉⠈⠁⠈⠁⠀⠈⠀⢠⣥⣥⣭⣥⣬⣥⣁⣨⣥⣬⣤⣅⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⣠⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣷⡀⠹⠇⢰⠏⢠⡆⢹⡇⢸⡷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡿⣿⣿⣿⣿⣷⠀⢠⣿⠀⠸⠇⣸⣇⠘⢃⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⣿⠀⠀⣤⣤⣾⣿⢉⣹⣿⣀⣼⣿⡷⢶⡞⢻⡟⠉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⣀⣸⣿⡤⠴⡿⢻⣿⣿⠋⢹⡏⠀⠐⣶⣇⠘⠁⣾⡇⠈⣉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⡏⢉⠹⣿⠀⢿⡇⠘⠛⣻⠀⣿⡇⠀⠙⢻⣿⠀⢰⣿⣇⠈⢋⣹⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠚⣶⣿⡖⢚⡿⠛⢿⡟⣿⢻⣿⣿⡿⠋⠹⡟⠹⣿⡋⢿⡏⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⡿⠀⣼⠁⣀⢸⠇⣿⠈⣿⣿⠀⣾⣶⡇⠀⣿⡇⠸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠃⢰⡏⢠⡟⢸⠀⣿⠀⣿⡇⢠⣿⣿⠀⠀⢻⡇⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠇⢸⠇⣼⠀⣿⠀⣿⡇⢸⣿⡿⢀⡁⢸⡇⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢹⡆⠈⢀⣿⡀⠀⢀⡿⣇⠈⢁⡇⢸⣷⢈⣇⣸⣧⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣿⡿⠶⢶⣾⡿⠲⣾⠛⣿⠚⣟⠛⣿⠉⣉⣻⠉⣿⠉⣹⠉⣠⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣄⠘⣿⢰⣦⣿⠀⠿⠀⣿⠀⣿⠀⠿⣿⡀⠙⠀⣿⠀⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣁⠀⡇⢸⡏⢻⠀⣤⠀⣿⠀⣿⠀⣶⣿⡇⠀⢸⡇⠀⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣏⣸⣿⣆⣸⣌⣁⣾⣀⣿⣀⣿⣀⣿⣀⣉⣹⣷⣀⣾⣇⣀⣀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉






⡿⠿⡿⠿⡿⢟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣛⣻⣿⣛⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀
⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣤⣴⣶⣶⣿⣿⣷⣶⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

⠀⠀⠀⠀⠀⠀⢶⡀⢐⠖⢀⡔⠀⢢⡀⠐⡖⠀⠐⡆⠀⠀⢀⡔⠀⢰⠀⣠⠂⠒⣄⠀⢲⠂⠀⠲⠀⢰⠆⠀⠀⢰⡆⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢳⡎⠀⢸⠀⠀⢈⡇⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⣹⠀⢸⠀⠀⠘⠀⢸⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠈⠃⠤⠚⠁⠀⠓⠤⠔⠁⠀⠀⠈⠳⠤⠀⠀⠙⠤⠔⠋⠀⠘⠦⠤⠂⠀⠸⠂⠀⠎⠘⠇⠀⠚
⠀⠀⠀⠀⠀⠀⢤⡀⠀⣤⠀⠀⢠⡀⠀⢠⠄⢠⠄⠀⡄⠀⠄⠀⠀⠀⣠⠀⠀⠀⢠⡄⢠⡄⠠⠄⢠⠀⠠⠀⢤⠀⣤⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠈⢧⠀⢸⠀⢠⠀⢧⠀⢸⡖⢆⠀⠀⡇⠶⠀⠀⠀⠠⠈⣇⠀⠀⠀⢧⡈⢧⡈⠀⢸⠀⠙⠲⢦⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠓⠈⠁⠚⠀⠒⠀⠘⠃⠘⠃⠈⠓⠀⠃⠀⠊⠀⠀⠃⠀⠘⠂⠀⠀⠘⠁⠘⠁⠀⠘⠀⠛⠀⠈⠀⠛⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠀⠀⣿⣿⣿⣿⣿⡇⠀⢐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⣀⠀⣀⠀⢀⡀⠀⠀⡀⠀⠀⠀⣀⠀⣀⠀⢀⡀⢀⡀⠀⣀⠀⠀⢀⠀⢀⡀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⢦⡐⠁⢸⠃⠀⠘⡇⠀⡇⠀⠀⡇⠀⠀⢸⠃⠀⠈⠀⡟⠀⠀⢻⠀⢸⠀⠀⢸⠀⢸⠄⠀⠀⠀⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢆⠀⡰⠃⠀⢧⣀⡠⠃⠀⠀⠘⢆⡀⠠⠀⠳⡀⢀⠞⠀⠸⣄⣀⠜⠀⢸⡅⢀⡴⢀⣇⠀⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢸⡇⠀⢀⢻⡄⠀⠈⡇⣀⠋⠁⠀⡏⢀⠙⠀⠀⠀⢹⠀⠛⠈⡇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡌⢸⡇⠀⠔⠀⢷⡀⠀⡇⠙⢦⠀⠀⡏⠙⢀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠁⠈⠁⠀⠈⠁⠈⠁⠀⠈⠁⠀⠉⠀⠉⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠠⣤⠀⠀⠀⠀⣤⠀⠀⠀⠠⣤⠄⠠⣤⡀⠀⣤⠄⠠⢤⡀⠀⢤⡄⠀⠤⣄⠀⢤⣄⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⠀⡘⠹⣇⠀⠀⠀⣿⠀⠀⣸⠇⠀⣿⠀⠀⣸⡇⠀⢸⡇⣠⠄⠙⠀⢸⠙⢷⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⣿⠀⠀⠰⠧⠤⢿⡄⠀⠀⣿⠉⠉⠁⠀⠀⣿⠉⠉⠁⠀⠀⢸⡇⠙⠄⢠⠀⢸⠀⠀⠻⣆⡃
⠀⠀⠀⠀⠀⠀⠠⠿⠄⠀⠠⠿⠄⠠⠧⠀⠀⠠⠿⠄⠠⠿⠤⠀⠀⠀⠠⠿⠤⠀⠀⠀⠀⠼⠧⠀⠤⠿⠀⠼⠦⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠛⠻⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣤⠶⠶⠶⢤⣤⡀⠀⠀⠀⠀⠈⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢀⣾⠏⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠻⣷⡀
⢐⣾⣦⣧⡸⡄⢷⡀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠀⣠⡿⠁⠀⢀⡞⠁⠀⣀⣤⡶⣤⣤⡀⠀⠀⠈⢿⡀⠀⠀⠀⠐⡄
⠉⢉⡵⠟⣻⡿⣾⣷⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣰⡟⠀⠀⣠⡟⢀⣴⠞⠉⠀⠀⠀⠈⠻⣦⠀⠀⠈⣧⠀⠀⠀⠀⣧
⠀⠀⠀⠈⠡⠞⠋⠉⠉⠓⠶⣤⣀⠀⠈⠳⠄⢠⡟⠀⠀⣰⡟⢠⡾⠁⠀⢀⣠⣄⡀⠀⠀⢹⡆⠀⠀⣿⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣈⡀⠀⢰⡟⢀⡿⠁⠀⡴⠋⠁⠈⢿⡆⠀⠘⣇⠀⠀⢹⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⢠⡌⠙⠳⠦⣤⣈⠁⠀⠀⠁⠀⠀⠀⢸⡇⠀⢀⣿⠀⠀⢸⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠒⠃⠀⠀⠀⣿⠀⠀⢸⡇⢰⡏⠙⣲⣾⣦⡚⡆⠀⢸⡇⠀⢸⡟⠀⠀⢹⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠁⣼⠁⠀⢀⠀⠰⣴⠇⠀⣾⠇⠀⢸⡇⠀⠀⣿⠃⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢿⠀⠀⣿⠀⠀⠀⠀⢰⡟⠀⢀⣿⠁⠀⢸⡿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡀⠀⢻⡀⠀⠀⢠⡿⠁⠀⣼⠇⠀⢀⣿⠃⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿⠀⠘⡇⠀⠈⠿⢶⡶⠟⠁⠀⣴⠏⠀⠀⣼⠏⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡄⠀⠑⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣼⠏⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢸⣇⠀⠀⠀⠲⠶⠶⠶⠟⠋⠀⠀⢀⣾⠏⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣰⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠷⣤⣤⣀⣀⣤⣤⠶⠛⠁⠀⠀⠀⣠⣾⡁⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⢿⣷⡄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⣷⣦⣄⣀⣀⣀⣀⣤⣤⣶⠿⢿⠉⣇⠀⠈⢻⣿⡄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣰⠏⡰⠋⠉⠉⠉⠉⠉⠀⢷⠀⢸⡇⣿⠀⠀⠀⠹⡽⣆⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢀⣼⠋⠜⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⣷⢸⡇⠀⠀⠀⠈⠘⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢿⡌⣧⠀⠀⠀⠀⠀⠘⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣭⣯⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣍⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠙⢻⣦⡙⣦⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣿⠀⠀⣿⡇⢸⠀⢀⣾⡝⠈⠀⠀⠀⠀⠀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣿⡆⢰⣿⠁⠛⠀⣼⡿⢁⣤⠀⠀⠀⠀⠘⣵⢶⠈⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⣧⡿⠁⣴⣦⢀⣿⠃⢀⣉⠀⣠⢶⠄⣦⠻⣿⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⢻⣆⠀⣿⡏⢳⣰⣿⠃⢸⡿⢀⣿⡏⢠⡿⠋⢰⣿⢠⡏⣿⠃⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣳⠉⠃⢻⣿⢸⢿⣇⢀⣾⡇⡼⣿⢃⣾⣇⣰⠏⣿⡞⠀⣿⢀⣼⢃⠀⢋⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡵⠂⣀⣈⠻⠟⠘⢿⡾⠙⠟⠀⠻⠟⠈⠙⠁⠀⠉⠀⠀⠈⠉⠀⠈⠀⢉⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣴⡿⢋⣿⠃⠀⣀⡀⠀⣾⠃⠀⠀⠀⡛⠀⣠⡶⡦⢸⡇⢰⣏⡇⢀⣾⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣡⣾⡟⠀⣾⡟⠁⢰⣿⢠⣷⠀⣼⡇⢰⣿⡴⠃⢸⡇⠈⣻⠷⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠷⠃⣿⡇⣸⣿⠀⣠⢻⣿⠇⣿⢠⣿⣇⣼⣿⣀⣴⢿⣇⣴⠏⠀⢿⣄⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠂⢘⣛⠉⠿⠶⠁⠀⠉⠀⠙⠋⠈⠋⠀⠙⠛⠁⠘⠟⠁⠀⠀⠈⡟⠩⣬⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢃⡞⢰⣿⠛⠃⣦⠀⢠⣦⠀⣴⡿⠗⢠⡶⠦⢠⡞⣷⢠⠶⢷⡄⡠⣤⣿⠐⣦⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠘⣿⡄⢠⣿⣠⢿⣿⢰⣿⠃⠀⣿⡇⠀⣿⡶⠃⣿⣦⠖⣰⡗⢸⡿⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡎⠻⠋⠸⣿⡼⣿⡄⢠⣿⣇⡼⢿⣄⡴⢿⣧⣴⠿⠟⣿⡇⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣼⡗⣠⠀⠀⠉⠀⢘⡳⣋⡄⠉⠀⠤⡭⠀⠀⠉⠁⠀⢦⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠛⣛⡵⠋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⣿⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⣿⠀⣿⡇⣿⣿⣿⣿⣿⢠⣶⡆⢀⣀⣴⣾⣿⣿⣯⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢸⣿⣾⣿⣶⣿⣶⣿⡇⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣼⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⢸⣿⡇⠻⠿⢿⣿⣿⣿⣿⣿⣿⣩⣩⣭⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠋⠉⠉⠉⠋⠀⠀⠀⠀⠀⢀⣍⣉⣉⣉⣉⣉⣉⣉⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⣉⣉⣉⣉⣉⣉⣉⣙⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡀⣰⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣛⣙⣛⣿⣿⣿⣿⣿⣿⣷⣶⣦⢸⣿⡇⣿⣿⣿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡟⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿⢸⣿⠿⣿⠿⣿⡿⣿⡇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⣻⣿⣿⡿⠟⠉⠁⠸⠿⠃⣿⣿⣿⣿⣿⢸⣿⠀⣿⠀⣿⡇⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⣿⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⣤⣤⣴⡶⠀⠀⠀⠀⣠⣴⣶⣶⣤⣴⣶⣦⢄⣄⣀⣧⣀⣀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⢠⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠈⢿⠿⠇⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⡟⢁⡀⠀⢠⣿⣿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣏⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⣰⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣤⣽⣿⣿⣿⣿⣧⣤⡄⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣾⠃⠀⢠⣿⣿⣿⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣸⣿⣿⠿⢿⣿⣿⣿⡁⠀⣠⣴⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢧⣼⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⠀⣰⣿⣿⣿⣿⡷⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠀⠀⢹⣿⣿⣷⣀⠈⠻⠋
⠀⠀⠀⠀⠀⠀⠀⣸⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⠁⣰⣿⣿⣿⣿⣿⡇⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣦⣼⣿⣿⣿⣛⠋
⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠿⠇⠀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣯⡍⠠⠿⡟⠛⠻⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠺⢶⡄⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⠛⠛⠋⠈⠛⠿⢿⣿⣿⡿⢿⠋⠀⠀⠀⠉⠀⢸⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣾⣿⣿⣿⣿⣧⡈⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠉⣿⣧⣼⣦⣄⠀⠠⠀⠠⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⠏⣸⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⡃⣼⣿⡇⠈⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢀⠀⡀⠀⠂⢻⣿⣿⣿⣿⣿⣴⣦⣾⣿⣿⡇⠀⠙⠋⠙⠿⣿⡿⠀⣿⣿⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⡟⣡⣴⣖⣿⢰⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣀⣠⣾⣿⣿⡿⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣦⣻⣿⣿⣻⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢫⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠒⣶⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⡟⠉⠋⣏⣽⡷⣾⣿⠟⠀⠂⠰⣤⡀⢀⣿⣿⣿⣶⣷⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⣽⣯⡿⠻⣟⡀⠀⣿⣿⣿⣿⣿⠀⠀⢀⣲⡿⢁⣾⣿⡿⣿⢹⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠸⣧⡀⠉⢨⣿⠛⠁⢰⣶⡟⠋⠀⠘⣿⣿⡇⢸⣿⡀⢻⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⠀⠈⣤⣰⣾⣿⣄⠀⠀⠀⣿⣿⣥⣾⣿⡆⠈⠉⢙⠟⠁⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢻⡀⠀⢠⣿⣿⣿⣿⣿⣿⣦⠖⠁⠀⢀⣎⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣿⣿⣦⣠⣿⣾⡿⠟⠋⠉⣤⡶⢠⣶⣿⣿⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠈⠧⣤⣿⣿⡽⢤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⠀⠀⠀⠀⠀⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣹⣿⣿⣿⡟⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡯⠙⣿⣋⣭⡄⠀⠀⠐⢿⣿⠻⠇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⢠⣿⣿⣤⣤⡀⠀⠀⢀⣻⣴⡶⢻⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣤⣶⣿⣽⣿⠓⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣟⣻⣿⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡄⠘⣯⡿⣿⢿⣿⠟⢺⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡀⠽⢀⠋⢸⡿⠘⣿⣿⡿⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⠇⠁⠀⠀⠀⡘⣣⣴⣿⣧⡴⠂⣠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣆⣣⡀⠀⢀⢠⠉⣾⣿⡏⠀⣾⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠊⠛⠉⠀⠀⠹⣷⣷⢿⡿⠁⢀⣠⣾⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠈⠙⠀⠡⡀⠙⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠳⢿⣾⠿⠋

⠀⠀⠀⠀⠐⠲⡖⠂⣤⠀⢠⡆⣴⠒⠒⠀⠀⢦⠀⣶⡀⣰⢀⠔⠒⢦⠀⡖⠒⢦⠀⡆⠀⠀⡖⠒⢦
⠀⠀⠀⠀⠀⠀⡇⠀⣿⠚⢺⡇⣿⠚⠓⠀⠀⠸⣼⠁⢧⡇⢸⡀⠀⣸⠀⡗⠒⣏⠀⡇⠀⠀⡇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠈⠁⠉⠉⠉⠀⠀⠀⠉⠀⠈⠀⠀⠉⠉⠁⠀⠁⠀⠈⠀⠉⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠐⣶⠒⠲⣄⠀⢀⡔⠂⠒⢦⠀⢀⡴⠂⠒⢦⡀⠀⣶⠂⢀⠖⠂
⠀⠀⠀⠀⠀⠀⡸⠈⣧⠀⠀⠀⠀⠀⣿⠠⢴⣏⠀⣿⠀⠀⠀⢸⡇⣼⠃⠀⠀⠈⣷⠀⣿⣴⡁
⠀⠀⠀⠀⠀⣰⠉⠉⠙⣧⠀⠀⠀⠀⣿⠀⢀⣿⠀⠹⣄⠀⢀⡼⠃⠘⣇⠀⠀⣰⠏⠀⣿⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣀⢀⢀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⠀⣴⣿⣿⣦⢠⣀⣀⣀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠁⠀⠉⠉⠉⠉⠈⠉⠈⠉⠉⠉⠉⠀⠻⣿⡿⠏⠈⠉⠁⠁⠁⠉⠉⠉⠉⠉⠉⠉⠉⠈
⠀⠀⠀⠀⢀⣀⣀⢀⠀⡀⠀⣀⡀⠀⣀⡀⢀⣀⡀⠀⢀⡀⣀⠀⡀⣀⢀⡀⠀⣀⡀⠀⣠⢄⠀⡤⡄
⠀⠀⠀⠀⠈⣿⠉⢸⣤⣿⢸⡏⣿⠸⣯⡉⢸⣏⡁⠀⢸⡇⣿⣸⡇⣿⣸⡇⣼⠉⣷⠀⢿⡸⠘⢇⡿
⠀⠀⠀⠀⠀⣿⠀⢸⠁⣿⢸⣇⣿⢠⣌⣿⢸⣇⡀⠀⠈⣿⠙⣿⠀⣿⢸⡇⢻⣄⡿⠀⢧⣰⡏⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠂⠃⠋⠐⠙
⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⡀⠀⣶⠀⣶⣰⠀⢰⡀⠰⡄⠀⣶⠀⢰⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠘⠀⠀⠓⠀⠁⠀⠋⠀⠘
⠀⠀⠀⠀⢠⡴⠤⢄⠀⡤⠤⠆⠀⢠⣄⠀⢠⡤⠤⣄⠀⠀⢀⡴⠦⣄⠀⣤⠀⢠⡄⣤⠀⠀⣄⠀⣠
⠀⠀⠀⠀⢸⡗⢶⡋⢈⡗⠒⠂⢀⡯⢼⡆⢸⠀⠀⢸⠀⠀⢾⠀⠀⢸⠆⡏⢳⣸⡇⣿⠀⠀⠈⣶⠁
⠀⠀⠀⠀⠘⠁⠀⠙⠈⠓⠒⠒⠘⠁⠀⠙⠘⠓⠒⠋⠀⠀⠈⠓⠒⠋⠀⠃⠀⠙⠃⠛⠒⠂⠀⠋
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⢀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⢠⡎⠉⠙⣦⠀⡿⣄⢸⡇⢸⡏⡉⠀⠀⡇⠀⣹⠀⠀⠈⢧⠀⠀⡾⠁⠀⠘⠀⠀⡇⠀⡈
⠀⠀⠀⠀⠸⣇⠀⢀⡟⠀⡇⠘⢾⡇⢸⡇⠁⠀⠀⡏⠈⠀⠀⡀⠀⠘⣆⠀⢷⡀⠀⢲⠂⠀⡇⠀⢁⡀
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠁⠀⠈⠁⠈⠉⠉⠁⠈⠉⠀⠀⠀⠉⠀⠀⠉⠀⠀⠉⠀⠁⠀⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠛⣆⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢻⡀⢸⡇⢸⡴⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⡇⠸⡇⠘⣇⠀⣷⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢷⠀⣷⠀⢿⠀⢻⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡁⠀⠰⡏⠳⣸⡀⠘⠀⠈⠀⠈⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣄⠻⠃⠀⠀⢻⡀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠳⣌⢧⡀⠀⠀⠀⢳⡀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠈⢧⡙⢦⠀⠀⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⠷⠖⠉⠀⠙⢲⣀⣀⣀⣤⠤⠤⠿⣄
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⣼⠭⠤⠶⠒⠒⠛⣉⡉⡇
⠀⠀⠀⠀⠀⠀⠰⣇⣤⠞⠁⣠⠶⠀⠀⠀⠀⢀⡀⠀⡴⠳⢤⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠛⠃⢿
⠀⠀⠀⠀⠀⠀⢠⡞⢁⣴⠞⠁⣠⠞⠃⣠⢶⠏⢀⡞⠁⠀⠀⣸⢳⣄⡤⠶⠟⠶⠶⠤⠤⠤⣤⣄⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠈⡽⠋⢁⣴⠟⢁⡤⠚⢱⠇⣰⠋⠀⠀⠀⢰⢃⡞⠀⠀⠀⠀⠀⠰⠶⣤⣤⠤⠞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠳⠞⢿⣁⡴⢯⣀⡀⠈⠉⠁⠀⣀⡀⢀⡏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠓⠦⢤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣄⣛⠃⣼⣰⣃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠙⠲⠶⣤⣌⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠁⠈⠳⢤⣀⠀⠀⠀⠀⢀⡈⠙⠓⠶⣤⣄⣈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣀⡉⠛⠶⢦⣤⣈⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠴⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣄
⠀⠀⠀⠘⣧⠀⠀⠲⠀⠀⣠⣄⠀⠀⣧⡙⠣⠀⠀⠀⠀⠀⢰⡆⣰⢠⡄⣤⠀⣤⢀⡴⢦⠀⠀⢰⣦⣶⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣻⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣶⣦⢺⠀⠘⢿⢾⣧⠀⠀⠁⠀⢠⡀⡀⠀⠀⢸⡟⢿⢸⡀⣿⣀⣿⣈⣷⣾⠃⠀⠛⣿⠟⠃⢰⣿⢩⣿⡇⠘⠁⣿⢾⣿⠀⣿⣿⠉⣽⡏
⠀⠀⡀⠀⠛⠛⠁⠘⠛⠁⠷⠾⠏⠀⠀⠀⠀⠀⠤⣼⡄⠀⠈⠁⠈⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣑⣸⡇⣻⡇⣿⣘⣿⣀⣉⣹⣌⣿⡇
⠀⠀⣿⣀⠀⣿⠃⣾⣉⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢷⣥⠐⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢿⣿⠇⠀⠀⠙⠛⠋⠀⢸⣷⣾⡃⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣷⣦⡄⠈⡿⣦⣄⠀⠀⠀⠀⠰⠿⢧⠀⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡟⠉⠀⣰⣿⢟⣻⢻⡟⣟⢙⣿⠏⠙⣿⣆⠀⠈⠉⠛⢿⠾⠃⠐⠷⡆⠀⠀⢠⡄⠘⠛⢃⠀⢀⠀⢠⡄
⠀⢠⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣬⣽⣌⣡⣿⣌⣩⣄⣊⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠸⣇⡀⣅⢺⣀⢻⣁⢸⣅⢸⡀
⠀⠘⠂⠈⠉⠈⢘⡃⠀⠉⡛⠉⠁⠀⠀⠀⠻⣿⣬⣉⣉⣏⢸⡼⣏⢨⣠⣿⠏⢀⠀⠀⢀⡀⠀⠀⠀⠦⣆⠀⠀⢠⣉⡁⡀⢀⣉⢀⢠⣌⡉⣀
⠀⠀⢿⡀⣿⡷⢸⣧⢰⡟⠋⠀⡖⠀⠀⠁⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⠈⠀⠆⠈⢀⠀⠀⠀⠚⠋⠀⠀⠸⠉⡿⠶⠛⠿⠱⠘⠃⠃⠿⠁
⠀⠀⠙⠁⠛⠀⠛⠘⠀⠛⠛⠀⠓⠀⠀⡴⠀⠠⡦⠀⠈⠉⠉⠉⠁⠀⠀⠀⠟⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡄
⠀⠀⠀⢠⣤⡀⠀⢀⡀⠀⡶⢖⠒⠶⠚⠃⣄⢀⣀⠀⠀⣴⢤⠄⡀⢀⠀⣄⣤⢀⡘⡘⢣⠀⡄⠀⡀⠀⠀⠀⠀⠀⣀⠀⣀⡀⣠⡉⢁⣤⡄⢀⡀
⠀⠀⠀⠈⠉⠀⠀⠻⠟⠘⣄⡠⠃⣤⠀⠀⠧⠤⠼⠃⠀⣿⠂⠀⣇⣸⠀⣿⣿⢸⡿⡇⠈⣳⡇⠀⡀⡀⢀⢈⡀⠠⣱⠄⡟⠃⡿⡇⢻⣅⡀⢿⡀
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠈⠉⣉⣉⠉⠙⣇⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠁⠉⠀⠀⠀⠀⠀⠀⠛⠁⠛⠸⠎⠛⢀⣀⢀⡀⢀⠁⠀⠀⠉⢡⡄
⠀⠀⠘⣿⣶⡄⢸⣍⣵⠀⢸⣇⣸⡇⠀⠈⠁⣿⣦⣿⢠⡞⢲⡄⣶⣰⡆⡶⠖⠻⡿⠃⣿⠱⣶⢲⠴⣟⠂⠀⠀⠸⠃⢸⣇⢰⣰⢿⡾⠿⠸⢧⠄
⠀⠀⠀⠛⠋⠁⠀⠉⠁⠀⠈⠙⠋⠀⠀⠀⠀⣿⠉⢹⡿⣷⣾⠇⡟⢿⡇⣌⡷⠀⠧⠀⠷⠌⡿⢭⡰⠮⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣶⠀⠀⠐⠆⠀⠀⢠⡄⠀⠀⠀⠉⠀⠈⠁⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⠈⠋
⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣉⠃⠰⠤⠶⠀⢀⣴⣿⠀⣿⣿⣻⡟⢻⡿⠙⣿⡇⢸⣿⠿⣆
⠀⠀⢯⡀⣿⠿⢸⣿⠀⣿⠉⠁⣾⠁⠀⣿⣿⣿⡿⡟⢻⠟⣻⢿⣿⠶⢶⣶⡆⠀⢤⡄⠘⠳⠖⠀⣾⣿⣿⡀⠛⣿⡙⣧⠸⢻⡘⢛⡇⠙⢻⣸⣿⡆
⠀⠀⠉⠁⠛⠀⡘⠉⠃⠛⠛⠃⠙⠂⠀⢹⣿⣤⣷⣿⣘⣦⣽⣮⣴⣴⣾⣿⡇⠀⢤⡄⠸⠿⠒⠀⢿⣿⣿⣿⡿⢿⡿⠿⢹⡟⣟⢻⠛⡿⡟⠻⣿⡇
⠀⢰⠀⣶⣰⠖⣿⢺⣼⠆⡇⣖⣾⠀⠀⢸⣿⢻⠿⢿⣿⢿⢛⠋⡟⢭⡟⣿⡇⠀⠐⠀⢸⠛⢳⠀⠘⢿⣿⣼⣇⣼⣥⣾⡘⣣⣤⣾⣤⣧⣧⣼⠟
⠀⠄⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⡀⠀⠀⢸⣿⣿⣿⣿⠿⠿⠿⠿⠿⠾⠟⠛⠃⠀⠠⠄⠰⠲⠆⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣀
⠀⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠛
⠀⣿⣿⢻⣿⣿⢻⣿⣿⡟⡿⠿⣿⣷⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠄⣀⠀⢰⣴⢠⡄⣄⣠⠤⡄⣤⡄⠈⠁⠀⠀⣠⡤⢰⡆⢰⡆⣼⡧⠀⢠⡶⣶
⠀⣿⣿⣾⣷⣷⣾⣷⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠉⠀⠀⠘⠘⠈⠛⠋⠘⠛⠃⠉⠁⠀⣀⠀⠈⢿⣤⠘⣧⣼⠃⠸⣧⣴⠸⣿⣯⠄
⠀⣿⣋⣽⡏⢙⡇⠸⡟⢡⣼⡟⢻⣿⠀⢠⣀⣤⠀⠀⠀⢰⡆⣴⡆⣠⡆⠀⢠⡀⣶⢠⠖⠀⡟⠀⠀⠀⠉⠀⠀⣤⣀⣀⣀⢀⡤⢠⡄⢁⣤⣨⣥
⠀⠿⣯⣼⣥⣿⣅⣧⣿⣬⣭⣧⣽⡿⠀⠸⠿⠟⠀⠀⠀⠘⠉⠈⠁⠈⠀⠀⠘⠁⠋⠈⠛⠐⠛⠃⠀⣤⠀⠀⠀⠉⠛⠚⠙⠳⠿⠊⠷⠞⠿⠸⠓
⠀⠀⠀⠀⠉⠉⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢀⡤⠤⠤⣀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⣶⣶⣦⡀⠊
⠀⠀⠀⠀⠀⠀⣀⣀⢠⡏⠀⠀⠀⠈⡆⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣼⣿⣿⣿⣿⣿⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠼⢧⠀⠀⠀⣠⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠹⣿⣿⣿⣿⡿⠡⣄
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢳⣀⣸⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡠⠂⠘⣿⣿⡟⠀⠲⡌⢳⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⣶⡖⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢒⣒⠂⠀⠀⠀⠀⢷
⠀⠀⠀⢀⠖⠛⠓⢆⠀⢀⡤⠤⠤⡀⠀⣰⠋⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣤⡀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣴⣶⣦
⠀⠀⠀⢸⡀⠀⠀⣸⠀⡟⠀⠀⠀⢸⠀⣳⣀⠀⣠⣏⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠀⠀⣼⣿⣿⣿⡄⠀⠸⣿⣿⣿⠃
⠀⠀⡴⠋⠉⠛⠋⠉⣳⠿⢤⣀⡠⠾⢏⠀⠈⠉⠁⠀⢳⠀⠀⠀⠀⠀⠀⣠⣦⣭⣯⣵⣦⠀⡻⣿⣿⠿⡃⢰⣾⣶⣿⣶⣷⡄
⠀⠀⡇⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⣾⣿⣷⣶⣾⣿⣦⢻⣿⣿⣿⣿⣿
⠀⠀⠧⣤⣀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠼⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣀⣀⣀⣠⣤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣶⢶⣆⡀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣾⣛⠃⢈⡃⠛⡋⢻⠏⢱⣹⣇⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣠⣬⣿⠀⠿⠁⢸⠇⢈⡀⢛⣽⣿⣀⠁⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣫⡭⠀⣹⣦⣶⣷⣾⣷⣿⡿⣿⣿⣯⣝⡿⠃⠀⣿⠶⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡁⣿⡟⠁⠔⣛⠉⠉⠙⣿⠟⠁⢰⡾⠱⢣⡞⣵⠀⢰⡶⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣾⡷⠈⣴⣾⣿⡿⠀⠀⣠⣾⣇⠀⠹⣛⣤⠀⢸⣿⠀⢸⡟⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⡆⢻⡿⠋⣀⣴⣿⣿⢿⣿⣷⣾⣿⣧⣧⣼⣿⠀⢸⡿⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠀⣾⠟⠀⣴⣾⣿⣿⣿⣧⣤⣿⣿⣿⣿⣏⣹⣿⣿⣷⣤⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢽⣟⣻⠟⢻⣿⣿⢻⡿⣿⣿⠿⠹⢻⣿⡿⠋⣉⣥⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠗⠀⣼⠋⣯⠀⠿⢻⡇⣼⠀⠙⠀⡄⢸⠏⣼⣿⠀⡿⢀⢈⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢿⣀⣫⡖⡄⣸⣆⣘⡀⠰⣄⣱⣤⣼⣿⣿⣤⣤⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢀⣽⠿⠿⢿⣿⣿⡿⣵⠀⣿⣿⣿⣿⠛⢿⣿⣿⠟⠀⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠁⣠⣤⠀⠘⣿⣇⣃⣴⡿⢿⣿⣿⣶⣿⣿⣿⠃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣼⣛⣋⣠⣾⡿⣿⢿⣿⣿⣾⣿⣿⣿⣯⣿⣷⣶⣄⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⡟⠀⢸⠇⢫⠁⠹⠟⠿⠋⠙⣿⡿⠛⡻⣦⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢻⣿⣿⣿⡇⠀⢰⡿⢸⠀⠀⣾⠀⣿⠀⠛⠀⢨⣿⡿⢋⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠈⢿⣿⡟⣴⣄⡀⢃⣿⣇⣀⣻⣶⣿⣀⣴⣄⣘⣫⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠙⣼⠏⠙⠛⠛⠁⠙⠛⠋⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠟⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣠⣀⠀⣠⣄⡄⠀⠀⠀⠀⡄⣤⡀⢠⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣤⡄⣄⢠⣤
⠀⠀⠀⠀⠤⠤⠠⠤⡦⠄⠴⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⣧⡛⠘⠘⣽⣩⡇⠀⠀⠀⠀⠘⡿⠃⠀⠀⠀⠀⠈⠀⣉⡍⣉⢈⠈⠀⣀
⠀⠀⠀⠀⣿⠐⣿⡾⡆⡖⣿⠀⠀⠀⠀⠀⢋⣙⠙⠋⠛⠈⠋⠀⠀⠀⠀⠀⡙⡁⡙⣋⠙⠛⡁⠀⠀⢿⢿⢿⡇⡟⢿⠃⠀⠀⠀⢀⣯⣍⣍⣌⢀⣿⡟
⠀⠀⠀⠀⠛⠘⠛⠀⠛⠀⠛⠀⠀⠀⠀⠀⠸⠿⠟⠛⠚⠛⠃⠀⠀⠀⠀⠀⣉⢃⠓⣙⣈⢛⡁⠀⠀⠶⢸⡇⣿⢾⠐⠖⠀⠀⢀⡈⡅⣉⢩⣁⣠⠉
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠸⠛⠟⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠃⡉⡘⣙⢙
⠀⠀⠀⠀⠀⠉⣬⣩⣡⡉⠀⠀⠀⠀⣸⡇⠁⠻⠿⠿⠺⠇⠹⠿⠷⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠘⡘⢛⠛⣠⡀⠀⠀⣠⣌⢉⢊⣃⣃⡋⠉
⠀⠀⠀⠀⢠⣤⣦⣦⣴⣦⣤⠀⠀⠀⠿⠟⠸⠸⠀⠨⠷⠜⠇⠇⠇⠀⠀⣀⡀⠀⣀⡀⠀⠀⠀⣀⣹⢠⣋⣘⢃⣛⡇⡀⠀⠈⠋⠈⠘⠉⠁⠙
⠀⠀⠀⢀⣨⣉⠉⢥⣭⣈⣵⠖⠀⠀⠀⠀⠀⢀⣀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣄⣤⢠⠀⠀⠀⠀⠋⢘⣈⢙⣙⢘⡋⠃⠁
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⢩⡭⢩⡄⠀⠀⠀⠀⣶⣶⣶⣰⣶⠴⡆⠀⠀⣰⣮⣉⡈⣩⡌⠉⠀⠀⠰⣰⣦⢸⢰⣆⣾
⠀⠀⠀⠀⠀⢰⣰⣤⣤⣦⡤⠀⠀⠀⠀⠀⡀⣍⠨⣅⢩⢀⡀⠀⠀⠀⢀⣰⡄⢀⣠⣤⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠈⢠⢨⣮⠠⣬⣶⣀⡀
⠀⠀⠀⢀⡼⠦⢰⡖⣶⡆⢰⡄⢠⠀⠀⠀⢠⣭⣥⣬⣬⡤⠀⠀⠀⠀⠀⠋⠋⠙⠉⠁⠀⠀⣤⣤⡄⣤⢠⣤⣤⣤⢤⣦⡄⠀⠘⠈⠩⡰⠿⢿⡍⡐
⠀⠀⢰⣿⠀⣆⢰⢰⢰⢶⢰⠠⣌⠀⠀⠀⢈⣌⡈⣁⣬⡄⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⣭⠁⢀⣉⣈⣁⣱⣿⣭⢯⡏⠀⠀⠀⠀⠁⠃⠂⠃
⠀⠀⠈⠋⡶⢾⢰⢶⠶⡾⢰⢶⠀⠀⠀⠀⠈⠉⠉⠙⠉⠁⠀⠀⠀⠀⣠⡄⢩⣉⣉⣤⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡆⠀⠀⠀⠀⠀⠆⠠⣤⡤⡦⠄⢰⠄⠀⠀⡤⣤⣤⣤⣄⣤⡀⠀⠀⠀⠀⠀⡾⠛⣇
⠀⠀⠀⠀⠀⢠⣤⣤⣄⣄⡆⠀⠀⠀⠀⠀⠀⢀⣈⣛⣃⡀⠀⠀⠀⠀⠶⠲⠶⠀⠶⠆⠆⠀⠀⠀⠉⢩⣉⣉⣍⠉⠀⠀⠀⠰⠒⠆⠰⠶⠷⠶⠲
⠀⠀⠀⢰⢰⣠⣶⠀⡆⡆⡦⡤⡆⠀⠀⠀⠠⢨⡉⣩⢩⢠⡄⠀⠀⠀⢺⣸⠄⡇⣶⣼⡇⠀⠀⠀⠀⣨⣙⣋⣟⠀⠀⠀⠀⢰⠷⣶⢠⣾⣶⠂⣾
⠀⠀⠀⠀⠀⠀⢴⣤⣤⢴⡆⠀⠀⠀⠀⠀⠀⢡⣭⣬⣽⠌⠁⠀⠀⠀⠀⢰⣤⣀⣴⠄⠀⠀⠀⠀⠀⠙⢯⣹⠏⠀⠀⠀⠀⠘⠀⣙⣈⣉⣭⡌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⢹⠀⡇⢾⢸⡯⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢠⣿⢁⠞⠀⠀⠀⠘⠃⠃⠘⠸⣒⠀⢀⣾⣟⡽⠣⢧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⠸⣿⠃⢀⢠⣄⠢⣠⣴⠈⣇⡦⣩⡝⣿⣏⣵⢦⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡔⠻⣶⠁⢰⣟⣼⣏⢾⡜⣿⣿⡏⠼⣿⣹⣇⢀⢀⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠙⠃⣾⡟⠈⢡⡄⣠⢉⣀⢉⡀⡉⠀⠉⠀⢘⡛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠀⠀⠻⠇⠛⠀⢹⠁⣿⣸⢿⣷⣇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣀⢠⢠⠀⡀⢠⡙⠃⠈⠁⠈⠁⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢆⠻⣴⡏⠘⠈⠀⠘⢿⣧⢡⡴⢤⡿⣶⡄⣿⠃⡄⣶⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⡇⠀⢸⣷⡀⠈⡿⣿⡇⡸⢠⣿⣡⣿⣾⣾⡁⠈⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢻⣇⠀⡟⢙⠿⠖⠁⣬⢻⡄⠸⠿⠃⠘⢛⣹⣷⡋⢩⢆⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠢⠌⠁⠀⠀⠁⠀⠀⡻⣮⣤⣤⣤⣴⠾⠛⠁⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⢨⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⢹⡇⢸⡇⢺⡈⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢔⣴⠁⢀⣿⢻⢡⡆⠀⠘⠃⠈⠁⠀⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡎⢹⣇⣾⠏⠀⠀⠀⠀⣡⣴⢶⣿⠤⣾⠗⢛⣿⢅⠰⣶⣀⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠁⠀⠸⣷⣼⣷⠉⢰⣿⢀⣿⣇⣾⡟⠉⣠⢏⣾⡇⠙⢁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⣸⡿⠈⠻⢦⠃⠙⠋⠙⢟⡌⠛⠚⠁⢸⡿⢁⡴⠋⠞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠭⠴⠞⢋⠄⠀⡄⢠⢄⣀⡀⣀⢠⠀⠀⠀⠑⠈⠉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢺⣯⠇⣿⡾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢀⡖⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⣄⠀⢀⣀⡀⠒⢶⣿⠷⣶⢶⠆⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢰⣿⡏⠀⣿⣿⣴⣿⡯⠿⣢⣿⡟⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⣌⠛⠛⢋⣿⣿⠙⢿⡤⠞⠋⠻⣧⡴⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣿⡟⣀⠀⠀⠩⠀⠀⠀⠒⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣠⡾⠋⠜⠵⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣀⣀⣼⠀⠀⠀⠀⣠⠴⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠀⠀⠀⠀⡾⠁⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠚⠉⠉⠉⠉⠉⠉⠓⣆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠉⢹⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡏⠉⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⠀⢸⠉⢹⡇⢸⡇⠀⡇⠀⡇⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⠀⢸⠀⢸⡇⢸⡇⠀⡇⠀⡇⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣿⣀⣸⣀⣸⣀⣸⣇⣸⣇⣀⣇⣀⣇⣀⣿⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇

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

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠐⢲⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⠈⡌⢫⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣖⣒⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⢔⡳⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⣄⠀⢘⡙⠛⡛⡋⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⣀⢍⠉⣡⢠⠄⡄
⠀⠀⠀⠀⠀⡯⢸⠇⠣⠅⠹⠻⠃⢱⢘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠟⠡⡘⠄⠏⠀⡇⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣄⣤⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⡿⣿⢻⡿⣿⢻⡿⣿⣻⣿⢿⡟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣽⣽⣷⣿⣸⣿⣶⣾⣷⣿⣾⣷⣽⣾⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠻⠟⠛⠻⠿⠿⠟⠋⠈⠉⠉
⠀⠀⠀⠀⠀⣿⣿⣿⣥⣽⣿⡇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣯⣷
⠀⠀⠀⠀⠀⣀⠈⠀⡀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢿⣷⣿⣾⣭⠟
⠀⠀⠀⠀⠀⠎⠰⠴⠳⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢶⠖⠓⣦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⣀⣶⣙⠀⠀⠀⠀⠀⢸⡎⣿⡀⣼⣀⢺⢣⠹⡄⢹⢔⡦⣿⡀⣼⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡍⠿⠉⠀⠀⠀⠀⠀⠘⠁⠁⠉⠀⠉⠈⠈⠀⠁⠈⠀⠀⠁⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠒⢰⡄⣀⢠⡔⠐⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⠢⠻⠃⠸⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠛⠛⠛⠛⠓⠶⢦⣤⣀⣀⠀⠀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠚⠛⠶⣦⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⢤⠶⠛⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⢤⣤⡀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣿⠋⠉⠉⠈⠉⠉⠙⠛⠶⢦⣤⣄⣤⣤⣤⠾⠁
⠀⠀⠀⠀⠀⢠⡟⠀⢀⣶⠀⠀⠀⠀⠀⣶⣄⡀⠈⠳⣤⣿
⠀⠀⠀⠀⠀⣾⠁⠀⣾⣿⠀⠀⠀⠀⢀⡇⠈⠛⢦⣄⠈⠻⣆
⠀⠀⠀⠀⣸⠇⠀⣼⠃⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠛⣶⠋
⠀⠀⠀⠀⡟⠀⣸⠇⢰⡇⠀⢠⣦⠀⠘⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠛⠋⠀⢸⠇⠀⣸⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡿⢸⡇⠀⢹⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢸⡇⠀⣿⠀⠸⡇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⡾⠀⠀⠹⣦⣤⠇



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢈⡁⡐⢄
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠁⠘
⠠⠀⡶⠉⢫⠑⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡚⠛⠒⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⢼⣉⠀⠀⢖
⡄⢻⡤⠔⠚⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠴⠊
⠘⢦⣈⣓⣂⡤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⣜⣂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠠



⠀⠀⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠄
⠀⠀⢸⣄⣬⡀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠒⡂⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⡆
⠀⠀⠸⣿⣇⠇⠀⠀⠀⠀⠀⠀⢧⣀⣈⣉⡁⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠳⡅⠀⠜⢀
⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠘⠛⣿⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡶⠄



⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠃⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡍⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡘⢢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠻⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣠⢔⠂⢹
⠀⠀⠧⠤⠭⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣒⣒⣒⣲⡤⠇⠀⠀⠀⠀⠀⠀⠀⠀⣞⠡⠨⠤⢌⣳⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡀⠀⣴⠶⠷⢦⡀⢀⣤⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢸⣏⠀⠀⢸⡇⠘⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠻⢦⣴⠾⢁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⣀⣴⣄⠀⠀⣀⣠⡿⣷⣀⡀⠀⢀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣿⠀⠸⣯⠁⢹⡿⠀⢿⣍⠀⠈⣽⠟⠐⢿⡉⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⣿⠀⠀⠻⠛⠿⠃⠀⠸⣧⡶⢦⣿⠀⠀⠙⠟⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡆⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣿⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣷⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣯⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠙⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠋⠉⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣦⣄⠉⠻⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠋⢀⣤⣾⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠘⠟⠻⠛⠛⠛⠛⠻⠛⠻⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠛⠛⠛⠛⠛⠛⠛⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠛⢿⣿⣏⠉⠛⠿⠿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣤⣀⣀⣰⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠹⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠘⢿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⢀⣀⣛⣛⣛⣛⣀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣦⠈⢿⣿⣿⣿⣿⡟⢀⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⣄⡀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣧⠈⢻⣿⣿⠏⢠⣾⣿⣿⣿⣿⡿⠿⠋⢀⣴⣿⣿⠛⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠉⠙⣿⣿⣿⣷⡀⠻⠏⢠⣿⣿⣿⡟⠋⠁⠀⠀⠀⣸⡟⠛⠛⠀⠛⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⣷⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠉⢻⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣄⠀⢀⣬⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣤⣾⣿⣿⣿⡟⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⣼⣿⣿⣿⣦⣼⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⣉⣤⣶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⢋⣡⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠂

⢀⣤⡴⣦⠀⢀⣠⣶⣤⡀⠀⢀⣤⡄⠀⢠⣤⠀⠀⠀⠀⠀⣤⣤⠀⠀⣠⣴⣶⡄⣠⡄⠀⣤⡄⢠⡄⢠⣤⣤⡄⢠⣄⠀⢠⡄⢠⡄⢠⣤⡀⢀⣤⠀⣀⣤⣤
⣿⡏⢠⣤⡄⣾⡏⠀⠘⣿⠀⣾⡋⣿⡀⢸⣿⠀⠀⠀⠀⣸⡏⣻⡆⢸⡿⠁⠀⠀⣿⣷⣴⣾⡇⢸⡇⢸⣿⣤⡄⠈⣿⡀⣿⠃⢸⡇⢸⡟⢿⣤⣿⢰⣿⠁⣤
⠙⢷⣄⣿⡇⠙⢷⣤⣼⠟⢰⡟⠛⢻⣇⢸⣿⣤⡄⠀⢀⣿⠛⠛⣷⠈⢿⣦⣤⡀⣿⡇⠀⢹⡇⢸⡇⢸⣿⣤⣄⠀⠹⣿⡏⠀⢸⡇⢸⡇⠈⢻⣿⠈⠿⣦⣠
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⢀⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⢀⣿⡟⠀⡰⠁⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢿⡀⠀⣴⡿⠋⣉⠇
⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣸⣿⠤⠊⠀⣾⡿⠀⠀⢀⣤⣦⡄⠀⢠⣿⠀⠀⢠⠃⠈⢹⣿⠉⠉⠁
⠀⠀⠀⢠⠃⠀⠀⣤⡶⢄⠀⠀⠀⠀⢠⣿⠃⠠⠎⠉⠀⠀⢸⡿⠀⠀⠸⣿⠃⠀⠀⣾⣟⡠⠃⠀⡜⠹⣇⡴⠁⠀⠀⢸⡇⠀⢀
⠀⠀⠀⢈⠀⠀⠀⠈⠀⢈⠀⠀⢀⣴⣿⠃⠀⠀⣰⡾⢣⠀⢸⣿⠀⠀⠀⣿⡀⠀⡠⣿⡅⠀⢀⠜⠀⠀⠀⠀⠀⢀⣄⠀⠛⠒⠊
⠀⠀⠀⠘⡄⠀⠀⠀⢀⠜⢀⣄⠘⠟⠓⢤⡀⠀⣿⡧⠃⠀⠈⣿⠀⠀⢰⠀⠈⠉⠀⠘⠳⠒⠁⠀⠀⠀⠀⠀⢰⡿⠁⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠢⠤⠔⠊⠀⠀⢻⣧⠀⠀⡠⢿⡏⠙⠛⠊⠀⠀⠹⣆⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢠⠃⠀⢀⡮⠔⠁
⠀⠀⠀⠀⠀⣖⠲⠄⠀⠀⠀⠀⢿⡆⡎⠀⢸⡿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⠔⢲⣦⠀⠀⠀⠀⠀⢸⣿⢀⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠑⠤⠞⠁⠀⠀⠀⣠⡾⠒⠀⠀⠀⢀⡀⡜⠀⠀⣾⡏⠀⠀⠀⡀⠀⢸⣿⠎⠀⠀⠀⠀⢠⡔⣊⠐⡄
⠀⠀⠀⢀⠄⠒⣠⠄⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⣰⣶⠀⢸⣿⡄⠀⣼⡿⠀⠀⢠⣿⠋⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠳⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡴⢹⣿⠀⢸⠈⠀⠀⢿⡇⠀⣀⢸⣿⢀⡔⢹⣿⠀⠀⠀⠀⠐⠢⡑⢢
⠀⠀⠀⠀⠀⢠⢚⠃⠀⡠⠔⢤⡀⠀⠀⣴⠋⠉⠑⣆⠀⠀⠀⣀⠀⠙⠦⠃⠀⠀⠀⠈⠙⠈⠁⠀⠉⠉⠀⠀⣿⡄⡰⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠘⠊⠀⡜⠁⠀⠘⣷⠀⠀⠋⠀⠀⢀⠏⠀⣠⠞⠉⢱⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠘⢧⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣿⠀⢀⣀⠤⠒⠁⠀⢰⡿⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣀⠤⢼⠒⠒⠀⢙⣿⠋⠁⠀⢀⣴⠦⠀⣸⡇⠀⡰⠃⠀⠀⠀⠀⠀⣰⡷⠀⠀⣠⡾⠛⡆⠀⣴⡆⠀⢣⡇⠀⣰⡿⢋⡇
⠀⠀⠀⠀⠈⠀⠀⠀⡆⠀⠀⠸⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⠜⢁⠔⢹⣷⠀⠀⢠⣿⠃⠀⣰⣿⠥⠞⠁⢀⣿⡇⠀⡰⠉⢰⣿⠓⠉
⠀⠀⠀⠀⠀⣀⠀⠀⡇⠀⠀⠀⣿⡇⠀⠀⢻⣇⣀⡠⠜⣿⣦⠎⠀⣾⡏⠀⠀⡸⣿⠀⡴⢹⡇⠀⠀⢀⠎⠘⢧⠔⠁⠀⢸⡇
⠀⠀⠀⠀⢸⠋⠀⢀⠇⠀⠀⠀⠹⡷⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⣸⡟⠀⠀⢰⠁⠈⠉⠀⠈⢿⣀⡠⠊⠀⠀⠀⠀⠀⠀⠸⣇⠀⡠⠃
⠀⠀⠀⠀⠈⠢⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠂⠀⠀⠂⢀⣄⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⢃⣤⣦⢀⣶⢄⣤⡄⣇⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣈⣠⡟⠮⠞⣮⡼⢡⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠒⠂⢀⡀⠒⢀⢌⢰⠒⣆⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡞⠀⢸⠋⣽⢰⠸⢸⡞⠦⣇⡽⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⢤⣄⣀⠀⣙⡭⠤⠒⢛⢉⡇⢚⡉⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣀⡻⠤⣤⣤⠤⠵⠊⠀⢈⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇⣀⠤⢿⠀⢰⣀⡠⠀⢰⡿⠭⠽⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢅⣈⣀⡓⠶⠿⠄⢸⠻⠦⣄⠸⠧⠴⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢉⠛⢀⣀⠈⢀⡀⠄⣠⢤⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢨⡀⣼⠙⣷⡟⣿⢸⡗⢊⠈⣙⣲⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⠀⠸⠄⠉⠀⠿⠇⠙⠛⠁⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣴⣶⣦⠀⠀⠀⠀⠀⠀⣴⣿⣅⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⠉⠁⠀⢀⣠⣀⠀⠀⠉⢉⡹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢠⣤⠏⠉⠉⣷⣄⠾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠘⠛⣧⠀⢰⡟⠋⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⡆⢀⡀⢿⣦⠿⣶⢀⠀⠘⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⢼⡇⠁⠁⠀⠀⠀⠀⠈⢐⣀⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠶⠿⠿⠇⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠦⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⣄⣸⠂⡀⠶⢤⣄⠀⠀⠀⠀⠀⠀⠉⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⣥⣀⣉⡆⢸⠑⠲⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠃⠙⢶⡀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠄⠈⠓⣄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡸⠋⠙⠓⠊⢀⠆⠈⠳⡄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣴⡆⠀⣼⡆⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⠁⠀⠀⠀⠃⠀⠀⠀⠀⡿⣦⠒⠋⠀⡐⠀⠉⠙⢦⠀⢀
⠀⡄⠀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢣⣀⠀⠀⡸⢁⠃⠀⠀⢠⠂⠀⠲⣄
⠀⠧⠀⢀⣀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠒⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠘⠷⣄⠴⠋⢀⠆⠀⠈⡆⠀⠀⡈
⠀⠘⢆⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢼⣤⣤⠔⠁
⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⡖⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠃⠲⢄⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠤⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢤⠤⠤⠃
⠀⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡀⠀⠀⠀⠀⠀⠀⡇
⠱⡄⠀⠀⠀⠀⠀⡇
⠀⠙⢆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠢⣀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢌⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃
⠀⠀⠀⠀⠈⠁⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣧⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣷⣦⣄⠀⣿⣿⣿⣿⣿⠀⣀⣤⣜⡻⢟⣣⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⡜⣿⣿⣿⣿⡇⣶⡍⠛⢫⣶⢸⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠁⠿⣿⣿⠿⡃⣵⢞⣛⡳⣮⢘⠿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣔⡻⠇⣿⠸⣿⡟⣼⠸⠟⣢⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡆⣭⣛⠶⢟⣭⢰⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠿⣫⣶⣶⣶⣍⠻⠇⣫⣵⣶⣮⣝⠸⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⢫⣭⣤⡀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢹⣿⣇⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠘⣿⣿⣶⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠟⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠀⢤⠉⠐⡄⠀⠀⠀⠀⠨⡉⡉⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡆
⠀⠀⠤⠤⠀⢀⠐⠂⠀⠀⠀⠀⠀⠀⢈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠞⠉
⠀⠀⠢⠄⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⠚⠃⠐⢹
⠀⠀⠀⠀⢤⢄⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣄⡀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⡼⠀⢠⣾⠟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢆⡀⠶⠂⣠⡉⠁⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⠈⠛⠃⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠸⠇⠀⠀⠘⣱⣄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣅⠂⠀⠀⠀⣤⣶⣦⠀⠀⠀⠉⠁⠀⣤⣤⡐⠚⠉
⠀⠀⢀⡤⠀⠀⠀⢡⢼⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢰⡀⣷⣄⠀⠀⠀⠀⣘⣛⣛⣿⣿⣿⣿⣿⣦⣠⣾⣿⣿⣿⡇⠀⠀⠀⠀⣰⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⣐⣿⣿⣿⣧⠀⢀⣿⣿⣿⢿⣿⣿⠃⡜⠻⠁⠈⠢⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⠃
⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣆⣼⣿⠟⠁⣼⣿⡟⣠⠤⡇⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠻⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⠂
⠀⢸⣿⣿⣿⣿⣿⣿⢿⠿⠿⣿⠟⠁⠀⢠⣿⣿⠃⠀⢒⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⢠⣿
⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⡿⠠⢂⢴⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠛⠁⠀⣰⣟⣿⠆
⠀⠀⠹⠿⠿⠿⠿⠿⠟⠀⠀⠀⠄⢀⣀⣩⣙⠋⠉⣷⢚⢀⠀⠀⠛⠛⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⣰⣷⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⡴⢳⡸⣿⣿⡇
⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⡠⠄⠀⣊⠀⠀⠀⠈⠋⠀⠀⡀⠀⠀⠀⠁⣸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⡌⠑⡄⣷⣿⣿⡇
⠀⠀⠀⠀⢀⣻⡄⠠⠆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠄⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⢀⡼⢇⡀⣻⣿⣿⣿⡇
⠀⠀⠀⠈⠁⠀⠀⠀⢀⡖⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠞⠀⠂⠴⠀⠀⠈⠉⠉⠑⠚⡶⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⢀⡮⢤⡀⣱⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢲⡆⠉⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣤⣀⠀⠀⠀⣾⣿⣿⡏⠀⢀⡾⣿⣦⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠠⠦⠻⠆⢡⡀⠠⣆⠀⠀⢀⣄⡁⠀⠀⣠⣤⣦⠐⠁⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣿⣿⣿⠇⢀⡎⢸⣸⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡘⠀⣠⡄⢸⡇⠈⠁⠠⠀⠉⠁⠒⠊⠉⠀⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⡾⠠⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⣿⣿⣷⣮⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣃⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡞⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢘⡁⢠⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣶⣶⣶⣿⣿⣿⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣰⣬⣾⣗⠋⠟⡿⣿⣿⣿⡇⢾⣿⣻⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠰⣿⣿⣯⣭⠁⠤⣾⣿⣼⣿⣿⡇⣾⣿⣿⠀⢰⣼⣿⣿⣿⣿⣿⠋⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠁⠀⣽⣿⣿⣿⣿⣿⠃⡧⢛⣿⠀⢈⣽⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⡇⠛⠁⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠻⢿⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠛⠛⠛⠛⠀⠃⠒⠛⠀⠋⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠿⠿⠻⠛⠛⠓⠛⠻⠻⠛⠛⠲⠓⠚⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡟⢷⡶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡟⢿⡇⢹⡇⢸⠆⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⣿⣭⣭⣭⣭⣭⣭⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⠻⠃⠘⠁⠘⠁⢸⡏⠉⣉⠉⠉⠉⠉⠉⠉⢻⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⣷⠟⠛⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠉⠁⢀⡾⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⡇⠀⠀⠀⠀⠀⠀⠀⣠⣿⣥⣤⣤⣤⣤⣤⣤⣾⣦⣄⡀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢶⣄⠀⠙⢷⡄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠉⠻⠶⡾⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡶⠶⣦⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣤⣤⣤⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⢷⣄⠀⠙⠷⣾⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠃⠀⠀⠈⠛⠻⡿⠛⠛⠛⠛⠛⠛⢛⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿
⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⣠⡾⠁⢀⣀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⣄⣴⠿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⢠⣧⣀⣀⣀⣀⣀⣀⣉⣀⣸⡇⢠⡄⢀⡄⢰⣦⢸⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣛⣛⣛⣛⣛⣛⣿⡟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡇⢸⡇⢸⣇⢸⣷⣼⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⢷⣼⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠈⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡛⠁⠀⠀⠀⠀⠀⠛⣿⣟⠋
⠀⠀⠀⠀⠄⠁⠀⠀⠀⠐⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣷⡄⠀⠀⢀⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣦⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢣⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿⣿⣧
⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣭⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⠎⣀⣀⣀⠤⠤⠶⣚⡉⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⠛⠛⠉⠽⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠦⠤⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣴⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡂⠀⢹⡆⠰⡆⢀⡞⠑⣰⠆⠠⣴⠋⠅⠙⢦⡤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣿⢠⡇⢸⡇⠀⣿⠀⡄⡟⠀⠀⠀⣾⠃⠀⢶⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠟⠸⡇⠘⠧⠀⠘⠷⠀⠙⠀⢀⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⢰⡆⠙⠀⣴⠀⠀⠀⠀⠀⢀⡼⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡘⢷⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⠀⠈⢷⡐⣿⠀⢀⡀⠀⢸⣷⠀⠀⣼⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠐⠀⣷⢸⡇⣿⠀⣸⡇⠘⢸⡧⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠼⣸⠃⠹⡤⠙⠧⠀⢸⡇⠀⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡥⠀⠀⠀⠀⠀⠀⢸⠇⠀⣆⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⠀⠀⢀⡄⢀⡀⠸⣇⠀⠀⠀⠀⣄⢀⠀⢤⠀⡟⣰⠑⡆⢀⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠁⣾⢰⡇⠀⣿⠀⢹⠃⡟⣾⠀⢸⠀⣇⠻⠈⣷⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠁⠈⠈⠣⠀⣟⠀⠈⠀⡇⠈⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜




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

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠰⠩⠀⠠⠑⡄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣿⣾⣷⡆⠀⠀⠀⠀⠀⢠⣤⠶⡶⢶⡶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠐⠂⡒⠲
⠀⠀⠀⠀⡐⡄⣦⡄⠀⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⡇⠀⠀⠀⠀⠀⠈⢩⣿⠩⡯⣻⣎⠉⢱⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⠀⢸
⠀⠀⠀⠀⠡⠁⠈⠁⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⢀⡇⠀⠀⠀⠀⠀⠀⠿⠿⠷⠾⠿⠿⠖⠃⠀⠀⠀⠀⠀⠀⡸⠅⠤⠤⠼⣀⠌
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣠⣶⣾⣿⢶⣤⢀⠀⠀⠀⠀⠀⠀⢰⠶⡐⠂⠒⢂⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡍⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⢿⣿⣭⣿⣽⣿⠃⢱⠀⠀⠀⠀⠀⢸⠀⠈⠲⠞⠁⣠⣾⣎⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇⣀⣭⣯⣄
⠀⠀⠀⠀⠀⠿⠟⠫⣡⠒⠁⠀⠀⠀⠀⠀⠘⠏⠀⠠⢀⡸⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⡸⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠆⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃



⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⡄⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠠⠐⠒⣦⡀⠀⠀⠀⠀⠀⠀⡖⠂⠒⣀⠂⠒⢲
⠀⠀⠀⠀⣿⣿⣀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⣘⣟⣁⠀⢃⣙⣟⣁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠀⠀⣽⡇⠀⠀⠀⠀⠀⢀⡥⡄⠠⠿⠄⠀⢸
⠀⠀⠀⠀⣍⣉⣉⣉⣉⣉⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠇⠺⠿⠿⠿⠂⠀⠀⠀⠀⠀⠿⠷⠶⢤⡤⠤⢼⠅⠀⠀⠀⠀⠀⠸⣿⠟⡿⢿⠿⠟⠛



⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠰⡶⠀⠼⠧⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠿⠿⢿⠀⠀⠀⠀⠀⠀⠀⡛⢍⠉⠉⣉⠝⡇
⠀⠀⠀⠈⠉⠀⣈⣀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⣤⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⡘⡐⢛⡘⢰⠀⠀⠀⠀⠀⠀⠀⠃⡀⠑⠚⢡⢖⡲⡄
⠀⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⣭⣭⣼⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠀⠂⡛⠛⣰⠀⠀⠀⠀⠀⠀⠀⠛⠀⠂⠂⢣⣛⡓⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉