Vulcan Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣐⢣⢀⠐⣤⠛⢢⠈⠂⣜⡆⢮⡀⡟⠁⠀⡰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠠⠋⠀⠈⠃⠀⠣⠀⠀⠋⠐⠀⠂⠃⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣷⡄⠀⠀⠀⣠⠊⢀⠀⠐⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣾⣿⠇⠀⠈⠁⠀⠀⠀⣡⣴⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⣰⣿⣿⡏⠀⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⣿⣿⣯⣤⣤⣄⢰⣤⣤⣸⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣿⡆⠀⠀⠙⠿⣿⣿⣿⡿⢃⢹⣿⣿⣽⡿⣾⠟⠋⠹⢓⣤⠀⠀⠀⢀⣿⡏⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⢹⣷⠀⠀⠀⣠⡌⠙⠛⠇⣿⣼⣿⣿⣿⠟⠁⠀⠀⣠⣻⣿⡃⠀⠀⣼⡿⠀⡄⠀⠘⣧⠀⠀⠀⠀⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⣿⣆⢀⣴⣿⣧⠀⠀⠘⣿⣿⣿⣏⡵⡓⠤⠴⣾⢿⣿⣿⣷⠀⢸⡿⠁⢰⣿⠀⠀⢿⡆⠀⠀⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⣦⢸⣿⣾⣿⠋⣿⡄⠀⢀⣌⡫⢒⡵⠉⠉⠀⣰⠏⠈⠻⣿⣿⡄⠘⠁⠀⣾⣿⠀⠀⢸⣷⢀⣴⣿⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⢸⣿⠀⣿⣿⡏⠀⣿⣧⠀⢸⠃⠀⠸⠀⠀⠀⠀⠘⠀⢰⡄⠹⣿⣿⣶⣄⢺⣿⣿⠀⠀⠸⣿⣿⣿⠏⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⣾⣿⠀⢿⣿⠀⠀⣿⣿⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠈⠛⠿⣿⣧⣝⡛⣤⡀⠀⠿⠋⢀⣼⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⠇⢠⡿⣿⡇⢸⣿⠀⠀⢹⢇⡎⠀⠈⢴⡀⠀⠀⠀⡄⠀⠀⢿⣿⣿⣦⠀⠀⠀⠻⣿⣿⡟⠁⠀⠀⣠⣾⡟⠀⠀⢸⣇
⠀⠀⠀⠀⠀⣾⣿⡿⠁⢸⡇⣸⠃⣿⡇⠈⡟⠀⠀⠈⣼⠀⠀⠀⢹⠀⠀⠠⠊⠀⢖⠀⠘⢿⣿⣿⣧⠀⡀⠈⡹⢿⠀⢀⣠⣾⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⢸⣿⡟⠀⠀⢸⣇⡿⠀⣿⡇⠀⡇⠀⠀⢰⡏⠀⠀⠀⣼⠁⠀⠀⢀⡴⠿⢿⣂⣀⣠⣤⡠⠊⢠⠞⢀⡄⢀⣈⣉⠙⣿⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠁⠀⠀⢸⣿⡇⠀⣿⣧⠀⠁⠀⠀⠻⡇⠀⠀⠀⣿⠐⠦⢴⣾⣶⣶⣯⣯⣭⡏⠉⠀⣴⠋⢀⣾⣶⣞⣿⡿⢠⣿⣿⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣵⣿⣧⣼⡿⠶⠞⢉⠻⣿⣿⣿⣿⣿⡇⠀⠞⢁⡴⣻⣿⣿⠟⠉⣰⣿⣿⣿⠀⠀⠀⢸⣿⣿⡀
⠀⠀⣠⣄⣸⣿⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⢹⣿⣟⡿⢀⣠⣴⣿⣆⠘⠻⠿⣿⣿⡧⢤⠴⠷⢞⣛⠋⠁⢠⣾⣿⣿⠏⣿⠀⠀⠀⢸⣿⣿⡇
⠀⠀⣿⠈⠙⠿⠀⠀⠀⢸⣿⡇⠀⢸⣿⡄⠀⠀⠀⠀⢈⣿⣿⠁⢸⣿⣿⣿⣿⣶⡧⠀⠹⣿⣷⡘⣿⣿⣿⣿⠀⠀⢿⣿⣿⣿⠀⢿⡄⠀⠀⣾⣿⣿⣇
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠈⣿⡇⠀⠈⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⠀⣰⣶⣿⢣⣴⣿⣿⣿⡄⠀⢸⣿⣿⣿⠀⢸⡇⠀⠀⣿⣿⡿⡇
⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠹⡇⠀⠀⢿⣇⠀⠀⠀⠀⠘⣿⣿⡇⠈⠛⣱⣿⣿⡟⠀⣰⣿⣿⢃⣾⣿⣿⣿⣿⣿⣄⠘⢿⣿⠇⠀⠸⡇⠀⠀⣿⡿⠀⡇
⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⣰⣿⣿⡟⠀⣴⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⡇⠀⢰⣿⠁⠀⡇
⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⢀⣿⣯⠀⣰⣿⣿⡿⠁⣼⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠁
⠀⠀⠀⠀⠈⠻⣷⣄⣰⣷⠀⠀⠀⠀⠀⢻⡟⠀⠀⠰⠿⠋⠁⣰⣿⣿⡿⠁⣼⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⡆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠗⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠠⢤⣾⣛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡐⠧⠀⠀⠀⠀⠐⠀⠀⠀⡬⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠾⠃⢶⠇⠀⠀⠀⠀⠀⣀⠜⠛⠿⢛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢀⣀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠀⠀⠀⠀⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣬⣿⠟⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢨⡆⣾⠏⠀⠐⠺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣤⣀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠲⣶⠂⠀⠉⣥⡄⠀⢀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠷⠀⢳⡌⢳⣆⣞⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠈⢶⣈⣿⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⢛⠘⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⡀⢀⡿⣿⣼⣏⣴⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⢈⡻⠃⣤⣀⠀⠀⠀⣀⢀⣿⠋⢽⣿⡴⢯⣿⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⠀⣻⠀⠀⠉⠓⠀⠰⣿⢿⣧⣴⠟⠙⣦⢴⡝⢃⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣏⠈⡦⠉⢶⣄⡀⠀⠀⠀⣤⣾⡿⠁⠀⠀⠈⣦⣷⠎⢷⡀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⠃⠀⠀⠈⠀⠙⠀⢲⡿⠟⠁⠀⠀⠀⠀⠈⣿⠀⠸⢿⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠦⠀⠈⠁⠀⠀⠀⠀⠐⣿⢾⡅⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⣤⣄⢒⠟⣿⠿⡀⠀⠀⠀⠀⠠⡈⠲⣌⡀⠀⠀⢹⡄
⠀⠀⣅⣐⠄⠀⠀⠀⢀⠀⠀⠀⡲⠀⠐⢄⡀⠀⢀⠉⠀⠀⣤⠁⠐⣿⣿⣹⡆⠀⠂⠀⠀⠈⠑⢈⣳⣄⠀⠀⢹⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠉⠂⠠⠀⠀⢀⠀⠀⠀⠹⠂⣀⣈⠻⣤⣉⣻⠀⠀⠋⠀⠀⣿⣏⣿⣅⠀⠀⠀⠀⠀⠀⠈⠙⠻⣧⡀⠈⠆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠐
⠀⠀⠄⠄⡠⠀⠀⡀⠀⠀⠀⠀⣧⠀⠀⠙⠻⠷⣿⣿⡇⠀⠰⡇⠀⣿⣤⣿⣿⣿⣦⡀⠀⠀⠀⠀⠄⠀⠈⠛⠂⠀⣀⡀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠿⣿⡇⠀⢰⡇⠀⣸⣿⣿⡽⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣼⣿⣦⡄
⠀⠰⠀⡀⢂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢻⣷⠀⢸⡇⠀⢹⣷⣾⡁⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⣿⡄
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠘⠿⣎⣳⠀⠀⠀⠀⠀⠹⣇⠀⠀⢠⡄⠀⠀⠀⠀⢈⣿⠉⡀
⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠀⢸⣇⣀⠀⡀⠈⠹⠷⣤⡀⠀⠀⣤⡟⠆⢠⡌⠻⣯⡁⠚⠚⠛⢁⡴⢀⠀⠀⠀⠀⠂
⠀⠃⠀⠄⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⣿⠀⠈⠉⠉⢰⠇⠀⢀⠐⠈⢹⡆⠀⠀⠁⠀⢉⡤⢀⣈⠉⣿⣿⣻⣿⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣴⡾⢻⡃⠀⠀⠀⠈⣰⢏⠀⢰⠄⠀⠀⠙⠗⣾⠿⠔⣿⣿⣻⣾⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠄⠀⠀⣰⣿⠏⠀⠉⣵⠼⠀⠀⠀⠠⣼⠃⠀⠀⣹⠇⠀⠀⠀⠐⠂⠰⢇⡿⢿⣿⣽⣿⣿⠂
⠀⠀⠐⠢⠤⠤⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣲⣬⡿⠀⠀⠁⠀⣠⡤⠟⢾⣱⣿⠀⠀⢀⣿⡆⠀⠀⠀⡷⠀⠀⠈⠉⣽⣿⣿⡿⣟⡀⠀⠀⠐⠂
⠀⠀⠈⠑⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⡇⠀⢠⣠⣴⣿⣷⡾⣊⣽⡿⢠⡀⣾⢿⠀⠀⠓⠀⠀⠀⠀⠃⢐⣾⣁⢿⣤⣿⡗⠐⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠄⠁⠀⠀⠀⠀⢘⣻⣧⣸⡴⣻⡿⢫⠙⠇⣉⣽⢿⣬⣿⠏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣾⣽⣿⣿⡂⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢘⣻⡿⣿⢞⣿⠟⣋⠀⠲⢙⣿⠆⣿⠇⠐⡿⠀⢀⣶⠙⢣⠀⠘⠀⢀⣠⣽⣏⡿⣿⣿⠁⠀⠀⢀⠄⠠⢀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⢸⣿⡽⣿⣟⠠⠄⡍⣠⠟⣾⢋⣾⣿⠰⠾⠇⠀⢸⡧⠀⠀⠙⠆⢠⣟⡿⠿⢿⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣽⠋⣬⢰⡖⠾⠗⠻⠿⠿⣏⠉⠀⠀⠀⠀⠀⢧⠀⠀⠈⣣⡜⢿⣽⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣛⠀⠈⠹⣿⢀⠚⢸⡿⠶⣄⡀⠀⠀⠹⠆⠀⠀⠀⠀⢠⣯⢡⣤⣠⡟⢹⣮⣭⡿⡿⣿⡛⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣷⠘⠐⣿⡌⢤⣾⡗⣶⣾⣤⡿⢶⣤⠀⣀⠀⣰⣶⣿⣿⣿⣧⡙⣇⣼⡯⣬⡗⢹⡾⣿⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⢾⣿⣟⠛⢯⣽⣿⢟⣬⣿⣦⢹⣿⣿⡇⠉⣹⣿⣿⠛⣛⢳⡌⢿⡟⢼⢷⣿⢱⡿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡀⡀⢾⡿⠿⠟⠛⠛⠉⠉⠙⢛⠱⢟⡷⣿⣿⣿⡟⣃⣀⣴⢇⣛⣿⣿⣿⣻⣿⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠁⠀⠀⠘⠛⠙⠃⠛⠃⠀⠀⠀⠈⠀⠈⠀⠈⠁⠈⠀⠻⠛⠛⠙⠛⠛⠋⠉⠛⠁⠉⠛⠛⠛⠛⠉⠙⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⠃⠐⡇⠀⡇⠀⢸⠆⠀⠀⢰⡟⠙⠃⠀⣰⢣⡀⠀⡿⢧⣸⠂⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠻⠶⠋⠀⠸⠗⠶⠂⠈⠓⠚⠃⠐⠋⠉⠓⠀⠛⠀⠙⠂⠀⠒⠀⠀⠀⠀⠀⠘⣦
⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶
⣿⣿⣿⡛⠏⠛⢿⠋⢉⡛⠛⣛⡻⠋⣛⡿⣻⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⣶⣶⠶⢶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣾⣃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡖⠛⠗⠙⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣠⣤⡄⠹⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠑⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡽⢎⠻⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⣀⠈⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠘⣓⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡄⠠⠉⢈⠀⠄⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⡐⢶⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⠀⡆⠀⠤⠷⢄⠐⠇⠀⠈⠁⢂⡀⠀⣱⠀⢫⡽⣴⡟⠙⠇⢻⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡆⠁⠀⠆⠘⢫⠀⢚⣄⣠⣶⣾⣿⡦⠋⠀⠀⢰⣴⠁⠀⠀⠙⠿⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠈⠀⣀⢰⢴⡆⠐⢠⠉⠉⠉⠉⢻⠃⠀⠀⢀⡞⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡀⠘⠃⠀⢀⠴⢁⡜⠀⠀⠀⠀⢸⡀⠀⠀⢀⡝⠁⠀⠀⠀⢠⣤⣿⣦⣄⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠘⡸⠀⠀⠀⠀⢘⠱⠦⠶⢻⣧⣀⠀⠀⣀⣼⣿⣿⡿⠃⠀⢈⡻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠂⠇⠀⠀⠀⠀⠈⠢⢠⠀⠓⠀⠈⠛⠛⢿⣿⣿⡿⠁⠀⠀⠋⠁⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠚⣁⣀⣀⡀⠀⢀⡘⠀⡀⠄⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⡉⠒⢀⠀⠉⠩⣁⠉⡉⠁⠀⠄⠑⣌⠚⠤⠀⠀⠾⣿⣿⡇⠀⣠⠀⢀⣶⣿⠏
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆⠈⣡⣀⠈⡳⡄⠀⠉⢷⣄⡈⠀⠀⠀⠈⠓⢢⣞⠛⢻⣿⣿⠀⠀⠁⣠⡾⠋⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⣿⣿⣷⠸⡆⠱⠀⢹⠓⠊⠀⠀⠀⠈⡙⣭⡁⢲⣿⣿⠀⠀⠀⢹⡇
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⣿⣿⠀⣿⠰⠀⣹⣶⡴⠚⢁⣠⣤⣤⣬⣿⣿⣿⣿⡇⢠⣄⣼⣧
⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⣠⣿⣿⠀⠃⠀⠀⢹⣿⠖⠉⠀⠀⠀⠀⠀⢻⡿⠟⠛⠃⠈⠙⣏⡁⠐⠒⠒⠀⠀⠤⠄⠀⠀⣀⣀
⠀⠸⠿⣿⡂⠊⠀⠀⠐⢛⠀⢸⠀⠀⣿⣿⣿⠀⢀⠀⠀⣿⣿⠀⠀⠒⠄⠀⠬⣦⣔⣿⡄⠀⠀⠀⠀⠘⣿⣿⣿⣷⣾⠾⠿⠷⠞⠛⠛⠋
⠀⣀⠀⢺⣿⣿⣿⣶⡗⢿⡑⣿⠀⠀⢻⣿⣿⡄⢸⠀⢰⣿⣿⣧⠀⠀⠀⠻⣄⢌⣿⣿⠷⠀⠀⠀⠀⠀⢹⣿⣿⡏⢀⠀⢀
⠀⢸⣏⣿⢿⣿⣻⣿⡿⣮⠝⢠⠆⣀⠸⣿⣿⠃⡎⠀⠘⣿⣿⣿⣷⣄⠀⠀⢘⣿⣿⣏⠀⠈⠙⢟⠀⢹⣿⣿⣿⣷⡄⠉⠑⠄⡀
⠀⢀⣿⣿⢻⣿⣿⠎⠋⠀⡀⠐⠒⣻⠶⣿⠟⠂⡇⢸⡄⢿⡾⣿⣿⣆⠀⠀⠀⠙⣿⣿⡆⠀⠀⠂⠀⠨⣿⣿⣿⣿⡗⢶⣤⠀⡀⠁⠠⠀⠀⠀⠠⣄
⠀⢠⡛⢧⣤⡿⠃⠀⠀⠈⠘⠀⠐⠿⠘⠃⢀⠠⠀⠠⣇⠸⣿⣜⢿⣿⣧⠀⠀⠀⠸⣿⣿⡄⠠⠀⠀⠀⣿⣿⣿⣿⣯⣿⣤⣙⣧⣤⠀⠀⠀⣀⠀⠎
⠀⢘⣿⣷⠿⠁⠀⠀⠀⠀⡀⠤⣤⣶⣴⣄⠀⠀⠈⠐⡘⣧⠹⣿⣆⠉⠻⢿⡶⠀⠀⢸⣿⣧⣀⠀⠀⠀⢸⣿⣿⣿⣏⣽⣭⣍⡛⠛⠧⡀⠘⠣⠓⢤
⠀⢨⣿⣿⢷⡆⠀⠂⠠⢴⠁⠀⠀⠙⠛⣻⣷⣶⣶⣿⣿⣿⣧⣽⣿⣷⣄⠀⠀⢀⣀⢸⣿⣿⣿⣷⡄⠀⢨⣿⣿⣿⣿⣯⣀⡨⢝⣷⣶⡃⡀⠀⠛
⠀⢀⡷⣿⢿⡧⠈⠐⡇⢸⣷⡀⠀⠀⡘⢿⣿⣮⣻⣿⣿⣿⡿⡏⡻⢿⣄⢀⠀⢀⢻⣿⣿⣿⣿⣿⣧⠁⠀⢿⣿⣿⣿⣿⣿⣿⣧⣝⣿⣋⡀⠨⣶⣄
⠀⣬⡶⠻⠿⡷⠀⢀⠀⡀⠙⣿⡄⠀⠈⠀⠻⢿⣿⣿⣿⣿⣿⠿⠉⠀⠀⠉⠛⠛⢟⠻⣿⣿⣿⣿⣿⣶⡄⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿
⠀⠀⠀⠠⠤⠐⡆⢠⢀⣘⣫⡽⠃⣰⣿⡤⠤⣸⢿⣿⣟⠉⠁⠀⠉⠀⠀⠀⠀⠀⠘⠀⠸⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⢾⣿⣿
⠀⠀⠶⠀⠀⠀⣤⣘⠾⠋⠉⠁⠈⠁⠀⠀⠈⠀⠀⠉⠻⢿⣖⡠⠀⠀⠈⠉⠀⠀⠀⠰⡀⢾⡛⠿⠿⠿⣻⠽⠛⠈⠿⠿⠿⠟⠛⠛⢻⣿⣿⣷⣿⣿
⠀⠀⠀⠀⢀⡎⣀⠉⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡉⢻⣶⣄⡒⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠛⠛⠛⠿
⠀⠀⠀⠀⠘⠁⠻⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠷⣬⠉⠉⠙⠋⠛⠛⡛⠛⣛⡛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠂⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⡀⢼⠟⠶⠻⠋⠁⠀⠉⠓⠪⢝⣳⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⢍⣡⣶⡆⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⡷⠆⠁⡀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢠⣀⣴⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠐⠀⠀⠀⣨⡿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⣀⣠⡞⢣⣴⠆⠀⠀⠀⠀⠀⣀⡴⠟⢶⠞⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⢀⣀⣠⣴⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⢠⡄⠚⠉⠀⠀⠀⢾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⣀⣄⣻⣿⠟⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⣴⠖⠀⠠⢶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣏⡀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⢤⣤⠄⠈⠓⣋⡀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢦⡙⢧⡀⡼⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣥⠘⣿⡟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠄⠀⠀⠀⠀⠀⠀⠀⣿⢷⠹⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⢠⡿⢈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡍⠻⣦⡄⠀⠀⠀⠀⠀⠀⠘⣷⡿⢿⣿⢉⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡅⠰⡆⠈⠛⠶⣄⠀⢠⣷⣼⡟⢀⣼⠿⣉⢘⠻⡧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢄⠰⢷⡀⠀⠀⠀⠀⠈⠋⣽⣿⠟⠁⠀⠻⣍⢧⡰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠃⢸⠁⠈⠙⠶⠁⣤⠀⣀⣿⣿⠏⠀⠀⠀⠀⠻⣿⡅⠘⣧⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠁⠀⢧⠀⠀⠀⠀⠀⢀⣤⣸⠏⠀⠀⠀⠀⠀⠀⠀⠺⡅⠀⠉⢿⣄⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠁⠀⠀⠀⠀⠀⠀⠁⣈⣯⣽⣦⠀⠠⠀⠀⠀⠀⢠⣀⠻⠀⠀⠀⢻⡄
⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠠⠄⠐⠶⢂⠌⢀⣿⣉⣆⠀⠀⠀⠀⠈⠢⣍⠳⣤⠀⠀⠀⢷⡀⢀
⠀⠀⠒⢬⠒⠀⠀⠀⠀⠀⠀⠀⢸⣑⠀⠙⢦⡄⠠⣤⣄⠀⠠⡆⠀⠈⣿⣿⣼⡇⠀⠀⠀⠀⠀⠈⠑⣾⣳⣄⠀⠘⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠐⠀⠀⠰⠄⠀⠀⢠⠉⠠⢶⣤⣙⢶⣤⣭⠀⠀⢃⠀⠀⣽⠷⣿⣦⣄⠀⠀⠀⠀⠄⠀⠀⠉⠻⣦⡀⠘⠀⠠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠃⠐⠴⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠙⠻⣿⣿⠀⠀⢸⠀⠀⢿⣾⣿⣿⠿⣷⣄⠀⠀⠀⠀⠂⠀⠀⠈⠁⠀⣠⡤⣨⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢻⣿⡆⠀⢸⠀⠀⣾⡧⢿⠎⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣶⡄
⠀⠘⠄⠠⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣼⡀⠀⢹⣿⢿⡄⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣟⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⣾⠇⠀⠘⠻⢦⣵⠀⠀⠀⠀⠀⠈⢳⡀⠀⠠⣦⣀⣀⠀⠀⢀⣸⡏⢠⠂
⢃⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⣿⣦⣄⠀⡄⠀⠙⠛⢶⣄⠀⠀⠴⣏⠓⢀⡦⠘⢷⣤⠉⠙⢋⣉⣤⠾⣀⠀⠀⠀⠈⠁
⠀⠁⠀⠂⠀⢀⠄⠀⠀⠀⠂⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⡸⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢀⣤⢀⣠⡄⢸⣿⣏⡿⣧⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣶⠟⢻⠁⠀⠀⠀⢌⣰⢯⡀⢸⡆⠀⠀⠈⠛⢾⠿⠖⢻⣿⣏⣿⣯⣿⡿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠠⠀⠀⠀⣶⣿⠁⠀⡉⣶⠞⠀⢀⠀⠠⣼⠃⠀⠀⣸⡇⠀⠀⠀⠀⠒⠀⠟⣼⢟⣿⣯⣿⣿⡟
⠀⠀⠐⠂⠤⠤⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⢐⣶⣼⡟⠀⠈⠁⠀⣤⠶⠟⢮⣱⣿⠀⠀⢀⣹⡇⠀⠀⠀⠰⠄⠀⠀⠁⢩⣿⣿⣿⢿⣏⠀⠀⠀⠐⠁⠀⠀⠀⠄
⠀⠀⠈⠑⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⣠⣠⣶⣿⣷⡾⣊⣽⣿⢠⡀⣾⢿⠀⠀⠻⠀⠀⠀⠄⠘⠀⣲⣏⡸⣯⣼⣿⠇⠀⠠⠂⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠉⠀⠄⠀⠀⠀⢛⣻⣆⣩⠜⣳⡟⣯⠙⠅⡉⣽⢿⣬⣿⠏⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣷⣯⣽⣻⣷⠂⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠐⠄⠀⠀⠀⢛⣿⡿⣧⢾⣿⠟⡋⠀⠦⢹⣿⠆⣿⡏⠐⡾⠀⠀⣴⠘⠳⡄⠀⠂⠀⣀⣈⣿⣹⢿⣿⡏⠀⠀⠀⠀⠄⠡⢐
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢿⣷⡏⢡⠈⡁⣠⠷⣾⢃⣼⣿⠠⠴⠇⠀⢸⣿⠀⠀⠘⠆⠀⣼⢿⢿⠿⣿⣾⡿⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡾⢉⡁⢰⠦⠶⠿⠾⠿⠾⣛⠙⠀⠀⠀⠀⠀⢻⡀⠀⠀⠓⣀⢿⣿⣷⣾⣿⣿⡇
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠇⠀⠙⢿⣇⠀⠿⣼⣷⣤⡀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣼⡱⣀⣀⣾⠛⣦⣝⣹⣷⣿⣿⠇⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⡄⠓⢸⣧⠙⣤⣾⠦⣤⣭⣡⣶⣤⣄⠀⠀⠀⢀⣤⣾⣿⣷⣏⡙⢧⢀⣿⣉⣹⠞⣦⣿⡆⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠁⠠⣶⣿⡶⢾⡟⣻⣿⠿⣙⣷⣬⢻⣷⣿⡆⠛⢻⣿⣿⡿⠿⢳⣌⢿⣿⢎⣿⣴⡿⣴⡇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣿⣿⣿⡶⠿⠿⠛⠚⠻⡿⢫⡾⣯⣿⣷⣶⡾⠿⠙⢀⣟⠱⢿⣶⣿⡶⣿⣿⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠿⠾⠾⠰⠿⠃⠀⠀⠀⠢⠀⠒⠀⠐⠂⠹⠉⣿⠿⠿⠿⠾⠷⠿⠷⠿⠿⠹⠿⠿⠿⠷⠟⠻⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣴⠀⢠⡆⢠⡄⠀⣴⠀⠀⠀⢀⡴⠶⠆⠀⢀⣶⠀⠀⢰⣦⡀⡆⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠏⠀⠸⣧⣰⡇⠀⢽⣤⣤⡀⠘⢧⣤⡆⠀⡼⠚⢷⠀⠸⡄⠹⡧⠀⢠⡄⠀⠀⠀⠀⠈⢷⡀
⣀⣀⣀⣄⣀⣀⣀⣶⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣷⣄⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⡿⠿⢿⣿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣼⣧⣤⣶⣠⣴⣶⣥⣾⣾⣱⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣤⢤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣼⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡤⠶⡯⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣀⡀⠚⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠉⠪⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢻⡝⢿⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠛⠋⠉⣦⡙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠴⠲⠀⠀⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⢀⡀⣶⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠚⠀⠈⠀⠐⣆⠉⠈⣧⠀⠀⠠⣀⠀⠀⠀⡀⠀⣿⣽⠂⣽⢿⣾⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠈⠅⠀⠉⢉⣱⠀⣉⡀⠀⠀⡀⣴⣄⢀⡟⠀⠸⠛⠛⡿⠀⠉⢈⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠘⡇⠀⠀⠃⠀⠰⠂⡀⠷⠶⣾⣿⣿⣿⠎⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⠃⠈⠙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⡤⠹⠚⠟⢀⢁⡆⠀⠀⠀⠀⡿⠀⠀⠀⠠⣇⡁⠀⠀⠀⠀⠀⢀⡀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠠⠆⠨⠀⠀⠠⡞⠀⡞⡁⠀⠀⠀⠀⣵⠀⠀⠀⢠⡌⠀⠀⠀⠀⠐⣶⣾⣿⣶⠄⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠈⣡⠃⠀⠀⠀⠀⣋⠛⠖⠚⢻⢷⣄⡀⠀⢠⣴⣿⣿⣿⠏⠀⠀⣈⡻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠒⠛⠀⠀⠀⠀⠀⠈⠂⢠⠀⠙⠀⠀⠙⠛⢻⣿⣿⣿⠏⠀⠀⠘⠃⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠚⣁⣀⣠⣀⠀⠀⢀⠑⢀⠠⡀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⣲⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠙⠒⠀⠀⠉⠉⣉⠉⠉⠙⠈⠀⠀⢣⡈⠺⢄⠀⠀⠴⣿⣿⣿⠀⢀⡄⠀⢀⣶⣿⠟
⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⢸⡄⠘⣁⡀⠈⠳⣄⠀⠈⢳⣦⡀⠐⠀⠀⠀⠙⠒⢤⡔⠻⢿⣿⣿⡇⠀⠈⠁⢰⡾⠟⠁
⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠘⣀⠀⠹⣿⣷⣷⠘⡆⠐⡄⠙⡟⠤⠀⠀⠀⠀⠐⠛⣟⡃⢤⣽⣿⡅⠀⠀⠀⣿
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠀⠀⢻⣿⣿⡄⢿⠀⡄⠀⣷⣄⣠⠴⠂⠀⣀⣁⣉⣷⣿⣿⣿⣧⠀⢀⢀⣿⡄
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡇⠀⠀⠈⣿⣿⡇⢸⠁⠀⠐⣿⣿⠥⠒⠚⠉⠉⠉⠋⢿⣿⣿⡿⠿⠤⠿⢿⠓⠣⠤⠤⠤⠀⠀⣀⣀
⠀⢸⣷⡷⡄⠠⠀⠀⠀⡠⠄⠀⣧⠀⠈⣿⣿⣿⡇⠀⠀⠀⢠⣿⡏⠀⠀⣀⠀⠀⠀⣄⠀⣿⡀⠄⠀⠀⠀⠀⢷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⡷⠆
⠀⠈⠉⣛⣶⣤⣤⣤⣠⢠⡶⢤⡿⠀⠀⣿⣿⣿⡇⢠⡆⠀⣸⣿⣷⠀⠀⠀⠁⣤⠉⡻⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠋⠉⠉⠁
⠀⢰⣤⣴⣿⣿⣿⣿⣿⣆⡻⣤⢳⡀⠀⢸⣿⣿⡇⢸⠃⠀⣿⣿⣿⣷⡀⠀⠀⠈⢳⣽⣿⡿⠋⠠⢤⣠⡄⠠⣤⣽⣿⣿⣇⠐⠦⢄⡂
⠀⠘⣷⣾⣿⣿⣯⣟⣿⡍⠛⠁⢚⡑⣶⣈⣿⣿⡁⢸⠀⡀⢸⣟⣿⣿⣟⠂⠀⠀⠐⢿⣿⣷⡄⠀⠀⢈⠂⠀⣹⣿⣿⣿⣿⣇⣀⡀⠁
⠀⠠⣿⡿⠏⣿⣿⠏⠀⠀⡠⢄⠉⢉⣽⢋⡿⠋⠁⠸⠀⢷⠈⣿⡞⣿⣿⣧⠀⠀⠀⠀⠿⣿⣷⡀⠀⠈⠀⠀⠸⣿⣿⣿⣿⣧⣝⠷⢇⡤⠀⠀⠁⠀⠀⠀⠐⢦
⠀⠐⣎⡱⢶⣾⠃⠀⠀⠀⠀⠈⠀⠈⠛⠈⠁⠠⠀⠂⡀⢻⣄⢻⣿⡜⠻⣿⣧⣄⠀⠀⠈⢻⣿⡇⠐⠂⠀⠀⡀⣿⣿⣿⣿⡷⣿⣶⣦⣿⣷⣦⠀⠀⣠⢆⠈⠑
⠀⠀⡯⣿⡟⠂⠀⠀⠀⠠⢀⠔⠒⢶⣿⣾⣦⡄⠀⠀⢈⣡⣹⣆⢻⣿⣦⡀⠙⠻⠟⠀⠀⠈⣿⣿⣦⣄⠀⠀⠀⢽⣿⣿⣿⣧⣾⡧⢶⣌⢉⡙⢢⠀⠈⠂⡁⠆
⠀⠠⡿⣿⣏⣗⠀⠈⠀⠖⣎⠀⠀⠀⠈⢉⣛⣿⢿⣿⣿⣿⣿⣿⣧⣽⣿⣿⠆⠀⠀⠤⣀⣰⣿⣿⣿⣿⣷⡀⠀⣼⣿⣿⣿⣿⣿⣄⣐⡊⢹⣶⣭⣀⠄⠊⠉⠂
⠀⢠⡿⢿⢿⣟⠄⠁⠺⠀⣿⣷⡀⠀⠀⡌⢿⣯⣶⣽⣿⣿⣿⣿⢿⣉⡳⠿⣄⣀⣀⢀⠙⣿⣿⣿⣿⣿⣿⡆⠀⠐⢿⣿⣿⣿⣿⣿⣿⣷⣮⣿⣏⡠⢀⠰⣶⡄
⠀⣤⡷⠊⠿⢫⠆⠀⢄⢀⠀⢻⣿⠄⠀⠀⠀⠹⢿⣿⣿⣾⣿⣿⡿⠿⠉⠀⠀⠈⠉⠛⢿⠛⢿⣿⣿⣿⣿⣿⢦⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⣧
⠀⠀⠀⠀⠤⠀⢶⠀⡄⢀⣛⣏⡽⠁⣰⣿⣤⣤⣌⣿⣿⣿⡋⠉⠀⠀⠉⠀⠀⠀⠀⠀⠘⠇⠐⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡷⣾⣿⣿
⠀⠀⠤⠀⠀⠀⢀⣄⣳⠿⠋⠙⠁⠈⠉⠉⠀⠈⠁⠀⠈⠛⢷⣧⡄⣀⠀⠀⠉⠁⠀⠀⠀⠠⡄⠬⣝⢿⡿⣿⣟⣛⣿⠛⠙⠿⢿⠿⠿⠟⠛⠛⣿⣿⣿⣾⣿⡿
⠀⠀⠀⠀⠀⡠⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠉⠛⢷⣤⡐⠤⣀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡿⠿⠿⠻⠿⡟
⠀⠀⠀⠀⠀⠇⠸⡇⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠛⠛⠛⠛⠛⠓⠛⠛⠿⠷⣦⣙⠛⠒⠒⠒⠲⠶⠶⠶⠖⠶⠶⠶⠶⠶⠶⠤⠶⠤⠤⠤⠤⠤⠤⠤⢤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠠⡤⠄⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡶⣤⣴⡶⠖⠀⠘⠶⢬⣛⡲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠴⠦⠶⠒⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠛⠿⣛⠒⢒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠺⠿⠛⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠴⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣼⡇⠘⠁⣀⡰⡢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠃⠘⠿⠏⠀⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⡄⠲⢜⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣏⠳⢤⡀⣤⣼⢧⣠⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣴⠀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠦⠚⢦⠀⠚⠋⠀⠀⠄⠀⠀⠀⠈⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣧⡀⠁⢀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠸⡐⠉⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠀⠀⠑⢄⣠⠊⠁⠀⠀⢰⡄⠀⠀⠁⠹⡀⠀⠀⠀⠀⢀⡞⠉⠛⣦⣤⣀⡀⠀⠈⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⢦⡀⠀⢠⡇⠀⠀⢀⡴⠋⢧⡀⡀⠀⠀⡜⢄⠀⢀⣻⠞⠁⠀⢀⡟⠀⠈⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠦⣄⣀⡤⠶⠋⠀⠀⠈⠻⡗⠒⠀⠀⠗⢅⡀⠀⠀⠀⠐⢸⡇⡄⢀⠀⣀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠁⠈⠁⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢙⠠⡀⠂⠀⠀⠈⠉⠒⠶⠂⠈⠉⢣⡴⠚⠁
⠀⠀⠀⡤⠒⠒⠀⠀⠀⠁⠠⠥⠴⠆⠶⠓⠲⠶⠦⠤⡀⠀⠀⠀⠀⠀⠀⢢⣗⣠⣀⣠⣄⠐⣠⣾⠝⣄⡾⠛⠤⠄
⠀⠀⠀⢿⠀⠀⠀⠐⠒⠖⠒⠒⠒⠒⠒⠒⠒⠂⠶⣶⡇⠀⠀⣀⣀⣰⠟⠋⠀⠍⠉⡈⢢⡼⠃⠀⠀⠉⠀⠀⠀⢠⠳⣄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠛⠉⠉⠁⠀⠌⢁⣀⠀⠠⠤⠤⢘⣾⠑⣆⠀⠀⢤⡀⠀⠀⡖⠀⠘⣆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠢⡠⠀⡀⢀⡀⡠⠚⢧⡀⠀⠀⠀⢠⡏⠀⠳⡼⠳⡅⠊⠁⠀⢠⠀⠀⠀⠸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠁⠀⠙⣆⠀⢀⣿⠀⢠⣄⣧⢀⠀⠀⡾⠀⠄⣠⠿⣌⠈⠀⠀⣆⡼
⠀⠀⡠⠚⠄⣀⠀⠀⠀⠀⠀⠀⢰⠇⢀⠴⢻⡏⠉⠙⠢⣄⣿⡄⠸⢿⣿⠄⡆⢸⠁⠀⠈⠁⠀⠈⠣⣠⠀⠙⠁
⠀⠀⡗⠀⠤⠈⠀⠀⠀⠀⠀⠀⡏⠀⠏⠀⢸⡧⠀⠠⣤⢺⠉⡿⡄⠈⠻⣧⠹⣟⠀⠀⠀⣀⣠⠒⠶⠓⠲⡀
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡸⠀⠀⣼⠁⠀⠀⠀⢸⠀⠃⠀⡀⠀⢹⣿⡇⠀⠀⠸⠈⠡⠿⣿⣀⠀⠘⠧⡀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠁⠀⣰⠇⠀⠀⡀⣤⢸⠀⢸⡀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢮⣯⢙⢻⣀⠀⠉⣆⢧
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡏⠀⠀⣄⣶⣾⢸⠀⠀⢷⠀⠀⢠⡿⡇⠀⠀⠀⠀⣀⠼⠛⠢⠤⡘⢆⠀⣽
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠆⠄⢠⡞⠀⠀⠀⡀⡿⢹⠾⡀⠀⠈⡒⠂⢺⠇⣷⠀⠀⢀⠞⠁⠀⠀⠀⠀⠘⣦⠀⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡀⢆⡞⠀⠀⠀⡠⠆⠘⢯⣼⣧⣄⣀⣰⠀⣿⡀⢸⡄⡰⣣⡆⠀⠀⠀⠀⠀⠀⡿⠂⣾⠀⣸
⠀⠀⠀⠹⠉⡉⠉⠒⠒⡠⠟⢀⣹⠇⠀⠐⠋⠁⢀⠀⢀⠉⠉⠉⠉⡿⠠⣿⠀⢠⠋⠈⠁⠀⠀⠹⣄⠀⢀⡰⠟⢂⡇⣰⠃
⠀⠀⠀⠀⠀⢁⡦⠴⠊⣀⣤⣮⣥⣷⣶⣷⡿⡿⠛⠳⠝⠶⠄⠀⢠⡇⠀⣹⣄⣤⣤⣶⣾⣇⣿⠀⠻⣟⠉⠀⠈⣾⢿⡁
⠀⠀⠀⠀⠀⠘⠓⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣧⣬⠿⠛⠛⠛⠛⠛⠛⠳⠶⠼⠷⠶⠾⠿⠶⠿⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠴⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣼⡇⠘⠁⣀⡰⡢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠃⠘⠿⠏⠀⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⡄⠲⢜⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣏⠳⢤⡀⣤⣼⢧⣠⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⣴⠀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠒⠦⠚⢦⠀⠚⠋⠀⠀⠄⠀⠀⠀⠈⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣧⡀⠁⢀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠸⡐⠉⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠀⠀⠑⢄⣠⠊⠁⠀⠀⢰⡄⠀⠀⠁⠹⡀⠀⠀⠀⠀⢀⡞⠉⠛⣦⣤⣀⡀⠀⠈⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⢦⡀⠀⢠⡇⠀⠀⢀⡴⠋⢧⡀⡀⠀⠀⡜⢄⠀⢀⣻⠞⠁⠀⢀⡟⠀⠈⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠦⣄⣀⡤⠶⠋⠀⠀⠈⠻⡗⠒⠀⠀⠗⢅⡀⠀⠀⠀⠐⢸⡇⡄⢀⠀⣀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠁⠈⠁⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢙⠠⡀⠂⠀⠀⠈⠉⠒⠶⠂⠈⠉⢣⡴⠚⠁
⠀⠀⠀⡤⠒⠒⠀⠀⠀⠁⠠⠥⠴⠆⠶⠓⠲⠶⠦⠤⡀⠀⠀⠀⠀⠀⠀⢢⣗⣠⣀⣠⣄⠐⣠⣾⠝⣄⡾⠛⠤⠄
⠀⠀⠀⢿⠀⠀⠀⠐⠒⠖⠒⠒⠒⠒⠒⠒⠒⠂⠶⣶⡇⠀⠀⣀⣀⣰⠟⠋⠀⠍⠉⡈⢢⡼⠃⠀⠀⠉⠀⠀⠀⢠⠳⣄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠛⠉⠉⠁⠀⠌⢁⣀⠀⠠⠤⠤⢘⣾⠑⣆⠀⠀⢤⡀⠀⠀⡖⠀⠘⣆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠢⡠⠀⡀⢀⡀⡠⠚⢧⡀⠀⠀⠀⢠⡏⠀⠳⡼⠳⡅⠊⠁⠀⢠⠀⠀⠀⠸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠁⠀⠙⣆⠀⢀⣿⠀⢠⣄⣧⢀⠀⠀⡾⠀⠄⣠⠿⣌⠈⠀⠀⣆⡼
⠀⠀⡠⠚⠄⣀⠀⠀⠀⠀⠀⠀⢰⠇⢀⠴⢻⡏⠉⠙⠢⣄⣿⡄⠸⢿⣿⠄⡆⢸⠁⠀⠈⠁⠀⠈⠣⣠⠀⠙⠁
⠀⠀⡗⠀⠤⠈⠀⠀⠀⠀⠀⠀⡏⠀⠏⠀⢸⡧⠀⠠⣤⢺⠉⡿⡄⠈⠻⣧⠹⣟⠀⠀⠀⣀⣠⠒⠶⠓⠲⡀
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡸⠀⠀⣼⠁⠀⠀⠀⢸⠀⠃⠀⡀⠀⢹⣿⡇⠀⠀⠸⠈⠡⠿⣿⣀⠀⠘⠧⡀⢀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠁⠀⣰⠇⠀⠀⡀⣤⢸⠀⢸⡀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢮⣯⢙⢻⣀⠀⠉⣆⢧
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡏⠀⠀⣄⣶⣾⢸⠀⠀⢷⠀⠀⢠⡿⡇⠀⠀⠀⠀⣀⠼⠛⠢⠤⡘⢆⠀⣽
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠆⠄⢠⡞⠀⠀⠀⡀⡿⢹⠾⡀⠀⠈⡒⠂⢺⠇⣷⠀⠀⢀⠞⠁⠀⠀⠀⠀⠘⣦⠀⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡀⢆⡞⠀⠀⠀⡠⠆⠘⢯⣼⣧⣄⣀⣰⠀⣿⡀⢸⡄⡰⣣⡆⠀⠀⠀⠀⠀⠀⡿⠂⣾⠀⣸
⠀⠀⠀⠹⠉⡉⠉⠒⠒⡠⠟⢀⣹⠇⠀⠐⠋⠁⢀⠀⢀⠉⠉⠉⠉⡿⠠⣿⠀⢠⠋⠈⠁⠀⠀⠹⣄⠀⢀⡰⠟⢂⡇⣰⠃
⠀⠀⠀⠀⠀⢁⡦⠴⠊⣀⣤⣮⣥⣷⣶⣷⡿⡿⠛⠳⠝⠶⠄⠀⢠⡇⠀⣹⣄⣤⣤⣶⣾⣇⣿⠀⠻⣟⠉⠀⠈⣾⢿⡁
⠀⠀⠀⠀⠀⠘⠓⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣧⣬⠿⠛⠛⠛⠛⠛⠛⠳⠶⠼⠷⠶⠾⠿⠶⠿⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⢄⠀⠀⠀⠀⠀⢀⠃⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠚⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇⠀⠀⠀⠀⠸⠁⠒⠀⠇⠀⢸
⠀⠀⠀⠀⠀⡏⠉⢻⡄⠐⢺⡅⠀⠀⠀⠀⡆⠀⠀⠀⡗⠒⠻
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢧⠀⠀⠀⠀⡇⡠⠤⡀⡇⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠀⠚⡆⠀⠀⢸⡆⠀⠀⢀⠁⠀⠀⢸⡷⢶⡤⡇
⠀⠀⠀⠀⠀⠃⠀⠀⢸⡄⠚⠽⠃⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⢀⡴⠟⢧⠀⠀⠀⠈⠤⠆⠠⠤⠀⠸⠧⣀⡀
⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠤⠄⠀⢤⡄
⠀⠀⠀⠀⠀⠀⢣⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣠⡤⠤⠀⡆⠀⠀⠀⣰⠃⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⢁⣤⡾⠁⠀⠀⠀⠘⠀⢀⡴⣇⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⠂⠉⠀⠀⢀⠴⢋⠟⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠁⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡠⠔⠊⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⣀⣀⠤⠾⠶⣄⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿

⠀⠀⠀⠀⣀⡀⢀⣀⣀⣀⣀⠀⠀⠀⠀⢀⡀⢀⡀⠀⢀⡀⠀⠀⠀⢀⣀⣀⣀⠀⣀⠀⡀⠀⠀⢀⣀⣀⣀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡦⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠀⠀⠈⠁⠀⢀⠁⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣤⣀⣤⡁⠀⠀⣠⣄⠀⠀⠀⠠⠰⠟⣷⢀⣀⡀⣀⣠⣤⣤⠴⣦⣄⣀⣀⠀⠀⠀⣀⠀⢀⡀⠀⡀⠀⣀⠀⢀⣀⣀⣀⠀⠀⣀⠀⠀⠀⣀⣀⣶
⠀⠀⠀⠀⣯⠀⠈⡇⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣠⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠉⠛⡇
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣄⡀⢀⣀⣀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡉⠉⠉⠀⠉⠁⣶
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⡀⢠
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣯⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠿⢷⡦⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⣿⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⡏⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠁⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢳⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠘⠿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠙⠻⠦⠄⠠⠤⠾⠦⠤⠤⠤⢤⣶⣶⣦⣼⣿⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣤⣤⣄⣀⣄⣀⣤⣠⣶⣦⣶⣶⣦⣥⣤⣶⣦⣤⡟
⠀⠀⠀⠀⠸⣇⠀⠴⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⢤⠀⠈⠁⠈⠉⠉⠉⠁⠀⣠⡄⠀⠀⠀⠀⢈⣉⣉⠉⢉⣉⠁⠀⠀⠀⠉⠉⠁⠉⠁⠉⠉
⠀⠀⠀⠀⠸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠻⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⢀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⣀⣀⣀⣀⣤⡄⠀⠀⠀⠀⢀⠀⠂⣠⠒⡙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⠋⠉⠙⠛⠛⢃⠀⠀⠀⠀⠀⢀⣰⠗⠊⠀⠀⠰⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠈⣦⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠂⠀⠀⠀⠀⠀⣀⠼⠗⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇
⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠿⡄
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠟⣤⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠂⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⢀⣀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠸⢤⠶⢦⠶⠾⠯⠋⠤⣤⣴⠦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣀⠀⠄⢀⠀⣀⡀⠄⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡸⠁


⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣄⣠⠶⠚⠳⢶⠶⠛⠛⠳⣦
⠀⠀⠀⠀⠀⢠⡶⠚⠚⠋⠀⠙⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠶⣄
⠀⢀⣠⠖⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀
⢰⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣄⡀
⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠘⢧⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⢀⣠⣤⣴⠏⠉⠛⠓⠒⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⡄⠀⠀⠀⠀⠀⠀⣀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣶⣾⣶⣶⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢻⡟⢻⡿⣿⡿⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠃⠀⢘⡇⠀⣇⢿⡇⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⣠⡾⠁⠀⢿⠈⢻⡀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⠀⣼⡟⠀⠀⠀⠈⠀⠀⢷⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠆⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⠀⢰⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠤⢴⣿⡇⠀⠀⠀⠀⣴⣇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠈⢻⡿⠷⠶⠶⠶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡤⠶⠾⠛⠛⠛⠶⠶⢤⣤⣄⣀⣀⡼⠛⠛⠛⠛⠛⠛⠋⠁
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡄⠀⠠⠤⠠⣤⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⡀⢀⠀⠀⣄⣀⣀⠀⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠆⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⡶⠴⣦⠀⠀⠠⠦⠀⠀⠀⠐⠘⠋⠇⠠⢤⠠⠤⠶⠶⠖⠷⠦⠤⠤⠀⠀⠢⠄⠀⠤⠀⢠⠀⠠⠀⠀⠤⢤⣤⣀⣀⠄⠀⠀⣠⣤⣾
⠀⠀⠀⢰⡇⠀⠸⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠶⢤⡤⠤⠶⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⣿⣿⠦⠤⠀⢠⡄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠈
⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⢨
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠘
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣦⣀⣀⣀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢃⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠘⣄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠘⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠛⠳⠦⠀⠤⠾⠧⠤⠤⠤⠶⠶⠶⠤⣿⣷⣦⣄⣀⣀⣀⣤⣤⣀⣠⣀⣠⣤⣤⣤⣤⣀⣤⣀⣤⣤⣶⣦⣾⣶⣬⣤⣴⣶⣤⣾
⠀⠀⠀⠀⢸⠄⠐⠂⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⢀⡀⢀⣄⣀⣁⢈⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⢰
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣦⣤⣤⣤⣾⡆⠀⠀⠀⠀⠐⠈⢠⣊⠄⠙⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⣿⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠆⠛⠁⠀⠀⠀⠣
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⢶⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⡤⠀⠀⠀⠀⠀⣀⣸⠷⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢈⠙⣷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣆⣀⣤⣈⣀⡴⠾⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠁⠀⠈⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⣿⣶⣾⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣾⣿
⣿⣿⣿⡇⠀⠉⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣤⣤⣀⣠⡴⠶⠳⠶⣴⠾⠛⠛⠷⣦
⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⠟⠉⠉⠻⠋⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠷⠶⣦⡀
⠀⠀⠀⣠⠴⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⢠⡶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣦
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢶⣄
⠈⠙⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠙⠶⠶⠶⣦⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⣀⣤⣤⣴⠟⠉⠉⠛⠛⠻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣴⣶⣦⣤⣤⣤⣤⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠋⢸⡏⠀⣿⢻⣿⡁⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⣨⡇⠀⢸⡘⣿⡇⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢀⣴⠏⠀⠀⢸⡇⠈⢷⡀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡡⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠆⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠙⠿⣷⣤⣀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠶⠶⠿⣿⡇⠀⠀⠀⠀⣰⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣄⡀⠀⠀⠀⢈⣿⠛⠛⠛⠲⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡤⠾⠿⠟⠛⠛⠛⠶⠶⢦⣤⣤⣀⣀⣠⡾⠋⠛⠛⠛⠛⠛⠋⠁
⣶⣶⣶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠈⢠⠀⠀⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠶⠆⠀⠀⠈⠀⢀⠐⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⢈⡀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠰⠆⠉⠀⠀⢰⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣠⣤⡾⠋⢠⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠛⢣⠄⠁⠀⠀⠀⠑⠀⡀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣤⠏⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢿⠀⠆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠏⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠇
⠀⠀⠦⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⡀
⠧⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢿⣿⡇
⠐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠉⠁
⠀⠀⠀⠀⠀⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘⠟⠁⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠀⠀⠀⠛⠀⠀⠀⠒⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠹⣦⣀⠀⠐⣤⡄⠂⠀⣠⡌⠀⣀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠛⠋⠛⢉⡟⠻⣾⠛⠓⠀⠈⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⡌⠁⣿⠁⡄⠀⠀⠀⠈⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁⠀⠀⠀⠀⢻⣿⠀⠀⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⡀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠉⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⢀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣼⠀⠀⠀⠄⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣄⢤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⢀⡀⡰⢆⣀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠹⣿⣿⡁⠀⠛⠛⠲⣾⡇⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠀⢼⢁⣠⣄⣙⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢠⡞⣷⠀⠀⠈⢿⣿⣧⣉⢿⡿⡤⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢸⣬⢿⣿⠃⠀⠀⠸⣿⣯⢻⡾⢣⣤⣸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠛⠃⠠⠀⠀⠈⠙⣿⣾⡀⢾⡿⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠤⠀⠀⠀⠀⠀⠀⠀⠸⠃⠙⠻⣷⢽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢸⣿⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⣿⣿⣶⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣬⣤⣭⣥⣤⣤
⣿⣿⣿⡟⠋⠹⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⣧⡀⠄⢀⡄⡀⢠⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢰⠸⠃⠘⠄⠘⠈⠉⠆⠀⠄⠏⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣧⠀⠀⠀⡰⢂⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠀⣶⣾⣿⡿⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⢸⣿⣟⣀⠀⣀⣀⠠⣦⣿⣿⠟⠁⠀⠀⠀⠀⠀⣸⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⠘⢿⣿⣿⣿⠎⣿⣿⣿⣿⣿⠖⠰⢢⡀⠀⠀⢰⡿⠁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢸⣇⠀⠀⣠⡈⠙⢣⣿⣿⣿⣿⠛⠁⠀⣠⣿⣇⠀⢀⣾⠃⣄⠀⢻⡀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⢿⣄⣼⡿⣧⠀⠈⢿⣿⣳⡾⠅⠒⣹⠻⣿⣿⡀⢸⠃⢰⣿⠀⠸⣧⠀⢠⣾⢿
⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣿⢸⣿⣿⠀⣿⡀⢰⠗⠁⠏⠀⠀⠘⠇⢠⠘⣿⣷⣄⢀⣾⣿⠀⠀⣿⣶⣿⠏⠘⡇
⠀⠀⠀⠀⠀⢀⣿⣿⠁⢸⣿⠀⣿⡇⠀⣿⡇⠞⠀⠀⠀⠀⠀⠀⠀⢸⣷⡈⠛⠿⣷⣝⡛⣄⠀⠿⠛⢡⡆⠀⣿
⠀⠀⠀⠀⢠⣾⡿⢻⠀⡾⢿⡄⢿⠁⠀⡿⡸⠀⠀⡦⠀⠀⣠⠀⠀⢻⣿⣷⡄⠀⠈⣿⣿⠏⠀⢀⣴⣿⠁⠀⢸⡆
⠀⠀⠀⢠⣿⡟⠁⢸⢠⡇⢸⡇⢸⠀⠀⢰⠇⠀⠀⡧⠀⠐⠁⣸⣦⡈⠛⠛⠛⡠⢀⡔⠙⠀⠴⢿⣿⣿⠀⠀⢸⣧
⠀⠀⠀⢸⡿⠀⠀⢸⣿⠀⣾⡇⠘⠀⠀⢾⠀⠀⢀⡇⢤⢠⣬⣭⣼⣿⣿⠛⠋⣠⠋⣠⣥⣴⣶⢂⣿⣿⠀⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⣿⣧⠀⠀⠀⠀⣶⣦⣼⠳⠖⢚⠻⣿⣏⣿⣿⠀⠜⢁⣴⣿⣿⠛⢡⣾⣿⡇⠀⠀⢸⣿⡇
⠀⢠⡶⣼⡇⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⣿⣯⠟⣤⣾⣿⣆⣈⠛⣿⣿⠐⣖⣻⣿⠍⠀⣴⣿⣿⠋⣧⠀⠀⣼⣿⣧
⠀⠸⣇⠀⠉⠀⠀⢸⣿⠀⠘⣿⠀⠀⠀⠀⣾⣿⡄⣿⣿⣿⣿⣿⠀⢈⣿⡧⣛⣿⣿⡄⠀⢿⣿⣿⠀⣿⠀⠀⣿⣿⣿
⠀⠀⣿⡄⠀⠀⠀⠀⢿⠀⠀⣿⡆⠀⠀⠀⢻⣿⡇⠸⢋⣿⣿⠇⢠⣿⡿⢱⣿⣿⣿⣿⡄⠸⣿⡟⠀⢸⠀⠀⣿⠏⣿
⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⢀⣾⣿⠏⢠⣿⣿⢡⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠘⠀⢰⡟⠀⠁
⠀⠀⠀⠹⣷⡀⢀⡄⠀⠀⠀⠹⣿⠀⠀⢀⣼⠿⢀⣾⣿⡟⣠⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠈⠻⣾⣿⠀⠀⠀⠀⠛⠀⠀⠈⠀⠀⠘⠉⠉⠀⠛⠛⠃⠾⠿⠿⠿⠿⠿⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠀⠓⠀⠁⡀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⠀⠀⠀⠷⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⠁⠀⠰⣦⠀⠀⠰⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠔⠁⠀⢀⠘⠂⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⣠⣼⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⣤⣤⠴⠶⠾⠋⠀⠰⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄⠀⠀⠈⠉⠶⠀⠀⠀⠀⠀⠀⠁⠤⠀⢀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⣰⡞⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⠇⠰⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢈⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡇
⠀⠀⣠⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇
⣆⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣾⣿
⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⢠⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡟⠟
⠀⠀⠈⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠆⢈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣰⠁⠀⠈⢉⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠙⢷⣄⣀⣈⣶⣶⣁⡀⣠⣶⠀⠀⣦⠀⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠁⠉⠠⡞⢉⣻⠏⠙⠁⠀⠀⢀⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⣤⠁⣼⠁⢰⠀⠀⠀⠀⢀⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠈⢻⣿⠁⠀⠀⠀⠂⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡀⠀⠀⠀⠀⠈⠻⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠻⢿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠋⠀⣭⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡀⢶⣴⣶⡞⠀⢻⣦⣤⣤⣤⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡇⠀⠀⠀⠀⠈⣿⠃⠀⠀⢾⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢀⣀⡀⠀⠀⢻⣴⣾⡷⢶⣦⣰⡆⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣿⣭⣏⠀⠀⠈⢿⡿⣷⣤⡘⣿⠛⠂⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠸⣧⠟⢿⣏⠀⠀⠀⠸⣿⣿⣈⠿⠛⠾⣦⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠉⠛⠁⠰⠀⠀⠀⠘⢿⡿⣇⣸⣿⢟⣣⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠣⠄⠀⠀⠀⠀⠀⠀⠀⠸⠋⠈⠛⠿⡶⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣿⣯⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡄⢐⣿⣿⣦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣯⣬⣅⣈⣋⣉⣁⣀
⣿⣿⣿⣿⡿⠻⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢂⡰⠀⠀⠀⠀⢸⣇⠀⠀⣩⢞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠘⣾⡿⠀⠀⠙⠋
⠀⠈⠁⠂⠀⠀⠰⣼⣿⠀⠀⢴⣯⠇⠀⠀⠀⡀⠀⠀⠀⠘⠛⠀⠀⠀⠾⣷⠄⠀⠀⠀⠙⡇⠀⠀⢀⣿
⠀⡴⣶⡄⠀⠀⠀⠻⡇⠀⠀⠀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣾⡇⠀⠀⢊⡴⡃⠀⠀⠀⠀⠃⠀⠀⢠⣿
⠀⢞⣵⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⡠⠁⡀⠀⠀⠀⢸⣇⠀⠀⣩⢞⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣫⢾⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠨⠮⡈⠀⠀⢠⣸⣿⠀⠀⢴⣯⠇⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⣵⣻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⢿⡟⠀⠀⢀⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⡆
⠀⠈⠉⠀⠀⠀⠀⠆⠀⠀⠀⠈⣉⡀⠀⠀⢋⠿⡃⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡐⠁⠀⠀⠀⠀⢹⡇
⠀⢸⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⢿⡇⠀⠀⡡⢚⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠊⠩⢎⡁⠀⠀⢠⣸⣧
⠀⢸⣿⠀⠀⡐⢓⡜⠂⠀⠀⠀⢸⣷⠀⠀⠔⣫⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠃⠀⠀⠸⣼⣿⠀⠀⠺⠷⠃⠀⠀⠀⡄⠀⠀⠀⠈⣉⡀⠀⠀⢚⡿⠂⠀⠀⠀⠈⡇
⠀⠈⠋⠀⠀⠀⠴⣷⠄⠀⠀⠀⠹⡟⠀⠀⠀⣤⠀⠀⠀⢀⠆⠀⠀⠀⠀⣿⡇⠀⠀⣩⢞⡁
⢀⠀⣿⡇⠀⠀⢚⡵⠂⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⠬⡠⠄⠀⠀⠀⢸⣇⠀⠀⠴⣫⡄
⠀⡀⢿⡇⠀⠀⣩⢞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠑⠄⠀⠀⠸⣼⣿⠀⠀⠺⡷⠃⠀⠀⠀⡄
⠀⢠⣸⡇⠀⠀⢴⣯⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣠⣦⡀⠀⠀⠀⠻⡇⠀⠀⠀⣤⠀⠀⠀⠀⠆
⠀⠈⢿⡇⠀⠀⠀⣁⠀⠀⠀⠐⠂⠀⠀⠀⠀⣤⡄⠀⠀⡴⣫⡄⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⢌⠠⠄
⠀⠀⠈⠇⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⠀⠐⠀⢻⣇⠀⠀⢞⡵⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠐⠑⠄
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠔⢣⡜⠂⠀⠀⢀⢸⣿⠀⠀⣫⣞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠇⠀⠀⠈⣾⣿⠀⠀⠈⠋⠀⠀⠀⠀⠆⠀⠀⠀⠀⣤⡄⠀⠀⡵⣫⡅
⠀⠀⠄⠀⠀⠀⠈⣉⡀⠀⠀⠾⣿⠆⠀⠀⠀⠘⡇⠀⠀⠀⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⢻⡇⠀⠀⢞⡵⠆
⠀⠰⠂⠀⠀⠀⠀⣿⡇⠀⠀⢚⡵⡇⠀⠀⠀⠀⠁⠀⠀⢠⣿⠀⠀⠔⠓⡜⠂⠀⠀⠂⢸⣧⠀⠀⣫⣞⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠛⠉⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⢁⣀⣀⡀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠞⠂⠈⢁⡤⠀⠘⣿⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠚⣰⠀⢀⠀⢀⠠⡉⠐⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢓⡶⢽⡲⡶⠑⠘⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣴⣶⣶⣤⣤⡀⠀⠘⢦⡓⠀⠚⢁⠀⠀⠀⠀⠀⢀⡷⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠋⠉⠀⠀⠉⠙⠛⠻⣷⣦⡀⠙⢦⡁⠁⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⠴⠆⠀⠀⠤⣄⠀⠀⢿⣿⣿⢀⡤⠟⠲⢤⣤⠴⠚⠁⠀⠀⠀⠀⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢴⢀⠀⠀⣄⠈⠀⠀⠸⣿⣿⢯⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣓⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠾⣡⢤⠀⠁⠀⠀⠀⠀⠙⠙⣷⢍⣻⠅⡉⠉⠒⠲⠤⣄⣀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠘⠿⣖⣩⠿⠂⠀⠀⠀⣼⡛⠛⠁⠉⠓⠶⠔⠲⣞⠒⠶⢴⡇⠈⠉⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⢋⠲⣦⣤⡴⠒⠒⠻⢤⣉⡽⠃⠀⠀⠀⢠⠔⠚⣦⡀⠈⠉⠛⠒⠲⠦⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⡀⠙⠒⠀⠒⠈⣀⠄⠀⠪⠭⣓⢦⡄⠀⠉⠀⠈⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠧⣄⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠀⠉⠒⠢⠤⠤
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠱⡐⠒⠢⠤⠤⠤⠤⠶⠒⠋⠁⠀⠀⢀⣠⡤⠶⠖⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠺⢧⣄⡀⠀⠀⠈⠑⠦⢤⣀⡀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠈⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣠⣤⡀⠀⢤⣤⣤⠄⠠⣤⣤⣄⠀⢀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠘⣿⣛⣷⠀⢸⣯⣽⣧⠀⢸⡷⠆⠀⢸⣿⣶⡟⠀⢿⣭⡉⠀⣿⣃⣀⣿⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡶⢿⣦⠀⠀⣼⣿⣧⠀⠀⣿⠉⣿⡀⢸⣷⣶⠿⠁⠸⠷⠶⠇⠸⠇⠸⠷⠀⢶⣼⠟⢀⣿⠙⢻⡇⢰⡟⠁⢹⡇⢀⣿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣻⡇⠀⠿⠉⠙⠃⠀⠉⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡁⠀⠙⠁⠈⠻⠶⠟⠁⣸⡟⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⣀⣠⠤⠴⠖⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠶⠦⢤⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⡀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣿⣿⡟⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⢀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠁⠀⠀⠑⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠊⠀⠀⠈⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⢘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠈⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠫⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠝⠁⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠑⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠨⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠈⠢⠄⠈⢠⠀⠀⡠⠗⠁⠀⠀⠀⢐⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠕⠀⠠⠪⠂⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣀⡀⣀⡠⠬⠭⣥⡄⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠠⠼⠓⡉⢠⡆⠀⠀⠀⠙⣴⠟⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⣀⣄⣦⠸⡧⠈⣻⠂⠀⠀⢀⣇⢐⠆⠈⠀⢠⣴⠘⢧⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠾⠟⠁⢀⡠⡄⠀⣿⣝⢿⠀⠛⠁⣠⡶⠶⠶⠟⠋⠀⣀⢰⡸⡇⠹⡄⠘⠁⠀⣠⣴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠠⣿⡘⣗⢷⠀⠋⢉⡤⠐⣫⡾⠋⠀⠀⠀⠀⠀⠐⣇⡈⠳⠛⠀⢀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠸⠃⠉⢀⣠⠔⣉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠠⢄⠉⠢⣑⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⢴⣾⠟⢋⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡙⠳⣿⡷⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⣈⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⣀⠔⠂⠀⠀⠒⠠⣀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠘⠀⣠⣤⣤⠀⠀⣤⣤⣄⠀⠁⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠉⠉⣡⣌⠉⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡀⢢⠀⠀⠀⠛⠛⠀⠀⠀⡔⢀⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⡌⠀⠈⡆⢅⢀⡇⢀⡀⢸⡀⡨⢰⠁⠀⢡⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠰⠀⠀⡠⢈⡑⠂⠤⢀⣀⠤⠘⢊⡁⢄⠀⢀⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢀⠜⠚⢅⡀⠄⠛⢒⠤⠌⣁⠛⠠⢀⡨⠓⠫⡀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠘⡄⠀⠀⡔⠂⠉⠀⠀⠀⠀⠉⠐⢢⠀⠀⢠⠃⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠑⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠊⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢟⣥⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣓⣧⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠀⣠⣤⠀⠸⣿⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠭⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡏⠋⠀⠓⠤⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⣈⡶⠁⠀⠀⠀⠀⠀⠀⡠⠊⢁⡫⢹⣿⣧⣢⠤⠀⠀⠀⠀⣸⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠓⠀⡇⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⣴⠂⢾⣿⣯⠁⠈⠀⣀⡾⢁⠏⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠠⢀⡀⠀⠋⠀⠀⠀⠀⠀⠀⣻⠀⠉⢻⠿⣷⡶⣞⡋⠀⢀⠀⠁⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⣴⡶⢿⠿⣿⣿⣶⡀⢀⣴⠃⢀⡀⠈⠀⠈⠟⠹⡇⠰⠃⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠻⠿⢿⠿⠿⠿⠛⣿⡋⠁⠀⠈⣹⠀⠀⠀⠀⠀⠛⠒⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠻⠃⢐⡆⠄⠀⢴⡀⠂⠈⣏⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⣨⣄⢀⡀⠤⠀⠀⠀⠀⠀⠀⢸⡾⣇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⠀⠠⣤⠔⠀⠀⢸⡝⢩⣟⠂⠁⠀⠀⠀⠀⠀⠀⠈⢷⢙⡀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡌⠀⢶⠂⠒⠚⢀⣿⣿⣎⣻⡇⠀⠀⣇⠀⠀⠀⠀⠀⢈⣟⡇⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⣀⣀⣀⣴⠋⣿⣿⣿⣽⢿⣤⣀⠀⠉⠉⠁⠐⠒⠚⠫⠴⠶⠶⢦⣄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⡻⠿⠟⠋⠁⣠⣿⢿⢻⣿⡸⣿⣿⡿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣇⠹⣯⠁⣀⠠⢥⡟⡟⠈⢸⡟⠻⢿⣿⡧⢸⢛⢻⣷⣦⣤⣀⠘⠚⠶⢆⣀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢶⡟⠹⡇⢻⡄⠹⡆⢠⡴⠋⡜⠁⠀⢸⢿⠀⠀⠛⠿⣾⣿⣿⡞⢻⢿⠋⠙⠓⠲⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠃⠆⢸⠇⠀⣧⠈⢷⠀⠙⠀⠀⠈⢠⠆⢠⠇⢸⠀⠀⠀⠀⠀⠙⢿⣗⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸⡄⠀⠘⠀⠈⠀⠀⢀⣠⠶⠁⢠⠎⠀⠸⠀⠂⠀⠀⠀⠀⠀⠙⣿⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⣧⠀⠠⣄⡀⠀⠺⠛⢁⠠⠊⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠉⠀⠀⠋⠁⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⢠⣤⣀⠀⢠⣤⣤⣠⣤⣤⡄⠀⠀⢀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡤⣤⣦⣴⡆⠀⢠⣶⣶⣶⣦⡶⣶⣦⣤⣤⣤⣤⣤⣴⢶⣴⣤⡤⠀⢀⡀
⠀⠀⠀⢠⣶⣤⣍⠀⣿⣿⡿⢛⣵⡜⠿⠿⠟⠋⠀⢀⣠⣌⠐⠠⢠⣿⣿⣿⣿⣯⣿⣬⣿⣛⣿⡀⠀⠈⠻⠿⣛⣽⣿⣿⣿⠻⣿⣯⣿⣟⢿⣷⡽⠟⢀⣿⣿⣧
⠀⠀⠠⣿⣿⡿⢁⣾⣿⣿⣾⣿⣿⣿⡀⠠⠀⡄⠀⠸⢿⣿⡠⠀⣿⠟⣩⣶⡘⢿⣿⣿⣿⣿⣾⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⡁⠐⢿⣿⣿⡛
⠀⠀⢠⠹⠏⠀⣿⣿⡿⣻⣭⣽⠻⠟⠀⠘⠿⢀⣶⣿⣦⣶⠙⠀⠀⣾⣿⣿⣷⣶⣿⣿⣿⢛⡛⠟⣁⠰⠐⠀⠻⣿⣿⣿⣏⣠⣿⣿⣾⣿⣿⣿⡷⠀⠘⣿⣿⡗
⠀⠀⢸⡇⠀⠘⣿⣟⣄⢿⣿⣿⣷⡄⠀⠀⣤⣾⣿⣿⣿⣷⡟⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⡆⢲⣷⡀⠈⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠠⠘⢿⡇
⠀⠀⠘⣷⡄⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⣿⢻⣿⣿⣿⣿⡿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣎⣾⣿⣿⣟⡿⡎⠟⠃⠰⣷⡜⢿⣿⣿⣿⣏⣥⣮⣝⠻⠀⠀⣴⣮⠃
⠀⠀⠀⣿⣧⠀⠘⢛⣿⣿⣿⣿⣿⠂⠀⢘⣿⣿⣿⣿⣿⣿⡇⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢩⣻⣿⣿⣿⣿⣷⣷⣿⠛⠁⠀⣾⣿⡿⢂
⠀⠀⠸⢿⡇⠐⠀⣿⣿⣷⣿⣇⠀⠀⡆⠈⣿⣿⣿⣿⣿⣍⠀⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠀⡿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣿⣿⣡⠏
⠀⠀⢰⣶⡀⠀⠀⢻⣿⢿⣿⡿⠀⠀⠁⠀⣿⣿⣿⣿⣽⡟⠀⡆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠳⢿⣿⣿⣿⣿⣿⡷⠀⠀⠙⣿⣿⣿⠇
⠀⠀⢸⣿⣿⡄⠀⠐⣿⣿⣿⠇⠀⡀⠀⢈⣿⣿⣿⣽⣿⡧⠀⡇⠘⣿⣿⣿⡟⣭⣼⣭⢹⢛⣛⡿⢿⣻⣿⠁⠘⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠀⠀⠀⠘⣿⣿⡆
⠀⠀⣾⣿⣿⡇⠀⢀⠺⠿⢏⠀⢸⡇⠀⠈⣿⣿⣿⣿⣿⠃⢸⠀⠲⣿⣿⣿⣿⣿⣿⣿⡆⣄⠈⠁⠉⠱⢿⠀⠴⡄⢀⣼⢠⣦⣈⢹⣿⣿⣿⢀⣠⠀⠄⠸⣿⣦
⠀⠀⢹⣿⢋⣼⡇⠛⣴⢯⡿⢃⣼⣿⣄⠀⡙⢿⣯⣿⣏⠀⠹⠀⢀⢿⣿⣿⣿⣿⠿⢋⠀⠀⠀⠉⠀⣷⣶⣾⣆⢡⣿⣿⣾⣿⣼⣷⣽⣿⡟⢨⠁⠆⣈⣳⣝⣯
⠀⠀⠈⠉⠛⠛⠓⠺⠛⠳⠘⠟⠛⠛⠿⠄⠙⠒⡈⢉⣓⣀⠠⡐⠃⠄⠙⣻⠿⡿⢡⠅⠸⠐⢰⡄⣷⠌⠝⠿⠿⠧⠿⠛⠛⠛⠫⠿⠿⠫⠴⠺⠤⠼⠿⠛⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣐⣶⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠑⠶⢄⡀
⠀⠀⠀⠀⠀⢀⣬⠉⠀⠛⠛⠉⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣁⣲⣶
⠀⠀⠀⠀⠀⢉⣤⣤⣴⣵⣤⣴⣶⣶⢾⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⡭⢾⡟⠋⠉
⠀⠀⠀⠀⠀⠸⣇⠆⢹⡄⠀⠙⠉⠋⡀⠇⠀⠀⠀⠀⠀⠀⢀⡄⣬⠀⠀⠀⠀⠘⠁⡴
⠀⠀⠀⠀⠀⠀⠉⠒⢶⣧⣀⣀⢀⣽⣽⠆⠀⠀⠀⠀⠀⠀⣸⡧⢧⢠⠖⡆⢶⣼⢰⡗⡶⣰⡯⠓⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣟⢟⡿⣶⣿⡆⠀⠀⠀⠀⠀⠀⣿⢀⠃⠺⠼⠇⢀⠟⠘⠛⠁⠛⠀⠀⠘⠃
⠀⠀⠀⠀⠀⣶⠃⠀⠀⠸⣿⣿⣷⣿⡿⠁⡀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢾⡁⢠⡿⠀⢠⠤⠀⡤⢤
⠀⠀⢂⣀⣾⣿⡀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣷⣽⢿⡇⣿⡴⢸⡧⠋
⠀⠀⣿⣿⣿⣿⡇⠐⠠⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⣤⠀⠀⠀⠀⠀⠀⢎⡴⠏⠉⠈⠀⠀⠀⣿
⠀⠀⢻⣿⣿⣿⣷⠀⠀⠐⠻⠻⣿⣿⣿⣿⡏⠀⠐⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠊⠁⠠⠞⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠄⠤⠀⠀⠤⠀⢖⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠐⠁⣤⠶⠦⠀⠀⠀⠀⠀⣤⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⢀⠂⢹⡠⠀⠀⠂⠐⠄⠀⠀⠀⠠⠄⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⡀⢰⡇⠙⠁⠀⢠⣾⣿⡆⠀⢠⢠⣴⣶⣼⠀⠀⠀⠀⢠⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠀⡄⠀⠀⠀⠀⠉⠁⠀⠀⡈⢳⠉⠉⡁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⡼⠁⠄⠀⠀⠀⠀⢠⡀⠀⡉⠉⠀⡰⠀⠀⠀⠀⢠⣬⠍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠀⠘⠀⠀⠈⠢⡀⠀⠀⠀⠀⠙⠁⢀⠜⠀⠀⠀⠀⠀⢩⠖⣒⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠸⡄⠀⢠⠀⠀⠀⠻⠐⡤⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠈⠯⣥⣴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡣⠖⠀⠲⡀⢁⠀⢀⣑⡒⣲⣡⣄⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⠁⠀⠀⠀⢱⠀⠞⠁⠀⠉⠁⠀⡏⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠀⡈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠃⠀⠀⠀⠀⠸⠀⠀⢁⠀⠀⠀⠀⠇⠀⠀⠀⠉⢢⠄⣤⠤⠀⠀⣔⡡⠿⠠⠧⠼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢆⠀⠀⡀⠀⠀⠀⠀⠠⠂⣰⠃⠰⣀⡙⢤⣡⠷⢛⣡⡌⡁⢘⠛⠳⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠁⠈⠀⠳⢄⠀⠐⢁⣴⠇⠀⠀⣢⢴⡶⠋⢔⠉⡍⣿⣿⠊⢗⢄⠑⠻⣷⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢃⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢀⡩⠖⠉⠀⠀⣀⣐⠖⣿⢡⠜⣡⠎⠀⣿⡝⠀⠈⠳⣄⠢⢨⠇⢂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡀⠀⢀⠔⠀⢀⠤⢇⠀⣀⡠⠔⠋⠁⠀⠀⠀⡠⠊⠀⠀⢸⣷⡷⠚⠥⠆⠀⣿⠇⡇⠀⠀⠾⠹⠂
⠀⠀⠀⠀⠀⠀⣀⣸⠀⠀⠀⠐⡄⠀⡠⠊⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠠⣇⣇⠀⣴⣿⠆⠆⠀⡇⣼⣿⠗⠀⠀⣶⡴⡆
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⢈⠞⠑⢦⡀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠈⣈⣿⠀⠉⠉⠄⢀⠄⠀⠈⠉⠀⠀⠀⡟⡄⠃
⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⢀⠔⠁⠀⠀⠀⠙⠲⠤⡤⠴⠚⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣆⠀⠀⠀⠈⣀⠄⠀⠀⠀⢀⡜⠻⣜⡀
⠀⠀⠀⠀⠀⠀⠘⠿⠆⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡟⡏⠑⣠⣀⣀⠀⠀⣀⡀⠰⣁⠀⠀⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠈⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⡴⣿⠋⣼⠆⠙⠛⠻⢤⣤⡼⠁⠀⠹⠓⣀⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠣⡌⢂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⣴⣶⠀⠀⠀⠙⠉⠀⣠⣤⠀⠈⢦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⡀⠃⠈⠄⠀⠀⠀⠀⠀⠀⠔⢠⣀⠈⠈⠀⢼⣶⠀⠀⠀⠙⠛⠀⢀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣡⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣁⣀⣀⣀⣈⡀⠀⠀⠀⠀⠜⠀⠹⠛⠀⠀⠀⠈⠀⠀⢤⣶⠆⠀⠀⠈⠛⠃⠃

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⣀⣼⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⢠⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⡟⠀⠐⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢿⣧⠀⣸⣿⣿⡄⠄⠀⠀⣠⣿⠃⠀⢀⣀⣠⣴⣦⠀⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⣇⠁⣿⡿⣿⡭⢀⣠⡿⣁⡠⢲⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣻⣷⣾⣿⣿⡟⠀⠀⠈⣿⣿⡏⠙⠻⣷⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣾⣷⣶⣤⣄⣈⢿⣷⣿⣧⣿⡯⣤⣀⡀⠀⣿⣿⡇⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣠⣄⣤⣧⣀⣠⣄⣤⣤⣤⣤⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡉⣙⣿⣿⣿⣿⣿⣿⣭⣭⣽⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠙⡻⢫⣴⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣯⣻⣿⣿⣿⠟⠿⡿⣿⠿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣦⣤⣴⣇⠸⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⡿⣿⣿⣿⣿⣶⣶⣮⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠿⢻⣿⣿⣿⣿⣿⢏⣿⣿⣟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⠙⠟⠺⠾⠿⠇⠈⠉⣟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣍⣸⣿⣿⣿⣿⣿⡟⡃⠒⠛⠁⢶⠿⢿⣿⣯⣿⣿⣿⣿⣿⣿⣶⣶⡶⠶⠿⢲⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⠛⠛⠉⠀⣟⠀⣿⡄⠠⠠⠾⠽⠛⠛⠏⣿⡏⠩⡏⠁⠘⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠋
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣼⠀⢀⠀⠀⠀⠀⢠⡆⣿⣇⠀⠁⠀⢸⠀⠀⠀⣮⣀⠀⠀⣇⠀⠀⡀⠰
⠀⠀⠲⠦⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠇⠀⠙⠀⠙⠀⠀⠀⠀⠘⠁⠈⠁⠀⠈

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢠⣤⣄⠀⠀⢀⢀⡀⢀⡤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠄
⠀⠀⠀⢰⠟⢋⣩⠀⠀⡇⠀⡱⢮⠀⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢋⣡
⠀⠀⠀⢰⡾⠟⢛⠀⠀⠀⢀⠀⠀⠡⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢰⡾⠟⢋
⠀⠀⠀⢨⣴⡾⢿⠀⠀⠀⠀⠂⢠⡀⠀⠀⠀⠢⠀⠀⠀⠌⠀⠀⠀⢠⠁⠈⠀⠀⠀⣚⣿⠀⠀⢸⣴⡾⠟
⠀⠀⠀⢘⣡⣴⠿⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠊⠉⠱⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣒⣾⠀⠀⢘⣡⣴⠿
⠀⠀⠀⠸⣋⣥⣾⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠤⣾⠀⠀⠸⢋⣥⣶
⠀⠀⠀⠸⠛⣉⣴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢰⠁⠀⠀⠀⠀⠠⠭⣽⠀⠀⠸⠟⣉⣴
⠀⠀⠀⠁⠈⠉⠀⠁⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⣀⣀⣀⣀⣐⣂⣩⠭⠀⠀⠀⠀⠀⠉⠭⠀⠀⠁⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⢀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⢰⡾⠀⠀⠀⠀⠀⢷⠀⠄⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⢠⠀⣦⠀⠀⠀⠀⠀⠀⠀⣾⡘⠀⠀⠀⠀⠀⢢⡞⠀⠀⠀⠀⠀⢀⠔⢋⣉⡙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠆⠀⠀⠈⢼⣷⠄⠀⠀⠀⠄⣿⠇⠁⠀⠀⠀⠀⣠⠀⣾⣉⡷⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢶⣄⠀⣤⠎⠡⠀⠀⠀⠀⠶⠁⠘⠦⡈⡉⢡⠴⠁⠨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠛⠿⠿⣿⣿⣿⣿⠿⠿⠛⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⢈⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣦⣴⣶⣿⣿⣿⣿⣶⣦⣤⣶⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠋⠙⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⠋⠙⠻⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⠀⢠⣿⣿⣿⣧⣴⣦⣼⣿⣿⣿⡀⠀⠀⠀⠀⣠⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣯⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣿⣿⣿⣿⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣶⣤⡀⢀⣴⣾⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟⠋⠛⢿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠂⠒⠶⠶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⡿⢹⣿⣿⣦⢀⣀⣠⣤⣶⣦⣤⣶⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⢸⣿⣿⢽⡿⢀⣈⡻⣿⣿⣿⣿⢿⡛⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠿⠀⢸⣿⣧⣀⣙⣻⣛⢳⡎⠙⣿⡏⠀⢠⣴⣾⣿⣿⣿⣿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠧⠀⣿⠇⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠋⠙⣴⣿⠃⠒⠋⠨⣽⣻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡿⠋⠠⠀⢲⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠘⠛⠀⡠⠔⠁⣴⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⡿⠀⠀⠀⡠⠊⠁⠀⠀⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⢀⠔⠁⠀⠀⣀⣠⣶⣿⣿⣿⣿⣷⡄⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠊⠀⢀⣴⣿⣿⢿⣿⣿⣿⣿⡜⠿⣿⣦⡀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⣴⠃⢀⣴⠿⠋⠡⣾⠘⣧⣿⣿⣿⣿⡄⣈⠙⢳⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⠉⢿⡿⠁⠀⠖⣴⣿⣦⣝⣿⢻⣿⣏⣿⡏⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⣸⣷⣄⣠⣾⣿⣿⣿⣿⣿⣷⣽⣿⣿⠀⣿⣿⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⠀⢀⣧⣼⣿⠀⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⡆⣼⠀⠀⠀⣰⠞⣿⣿⠛⠟⠃⠸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⠀⢀⠀⠀⠸⠋⡿⠀⢀⣷⢶⣶⠄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣤⢸⡇⠀⠀⠀⣠⣶⣿⠇⢾⡀⣔⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣧⠀⠀⠘⣿⠟⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠿⠿⠿⠿⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⢀⣀⢤⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⢴⠇⠀⢠⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠒⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢸⠀⠀⠀⠒⠐⠀⠀⠄⠄⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠤⠀⡸⠀⠀⠀⠀⠀⠀⢀⣄⢠⠆⠀⠀⠀⠀⠀⢘⠀⠀⠀⡚⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⡄⠀⡇⠀⠀⠀⠀⢀⠔⠙⠁⠞⠰⠧⣦⠀⠀⠀⡨⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⡠⠋⠀⡀⠀⠀⠀⠀⢂⡄⠀⢠⣧⣄⣀⢀⡇⠂⠀⠐⠀⢠⠀⡀
⠀⠀⢀⠀⠠⠄⠀⣀⣀⠀⠀⠀⢰⠀⢃⠀⠀⠀⡇⡰⠊⠉⠀⠈⠛⢤⢨⠃⡄⠘⠦⣅⡀⢦⡇
⠀⠀⠁⠀⠀⠜⠉⠀⠀⠙⡄⠀⢸⠀⢸⠀⠀⢐⠱⡁⠀⠀⠀⠀⠀⠀⡿⠄⢇⠲⠀⢨⢁⡇⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⢉
⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠸⠀⢸⠀⠀⠀⠻⠁⣖⣒⡒⣒⣢⠌⠋⠀⠀⠣⡑⣇⠞⠀⠀⠁⠁⠀⠀⠀⠀⢠⡃⠀⡄
⠀⠠⠀⣦⣼⡀⠀⠀⠀⢀⠆⠀⠀⡆⢸⠀⠀⠀⠀⠑⢮⠁⠀⠀⡠⠊⠀⠀⠀⣠⢉⢌⡢⢢⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⢀⠀⡁⢄⣀⡤⡎⠀⠀⠀⠿⣿⠤⡀⠀⢀⢀⡄⠁⠀⠈⠢⠄⡀⠀⢞⣁⣽⠓⢶⠋⠀⠈⠉⠛⠒⠤⡴⢉⡁⠈
⠀⠀⠀⠀⠀⠀⣼⠀⢀⠇⢀⣀⠀⢷⠠⣏⢾⣶⣁⠀⠀⠚⠀⠄⠀⠀⠀⢢⡴⠁⢘⠄⠛⠉⠓⠢⢄⡀⠀⠀⣆⢁⣀⠈⠁
⠒⠀⠀⢀⢓⣋⠠⠬⠥⠤⠤⠈⠕⠣⠀⠙⠳⡌⠛⡓⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡰⡮⠃⠸⡂
⠉⠉⠠⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠎⡀⡉⢷⠀⠀⠀⠀⠀⢠⡗⣤⢠⡜⠀⠀⠁⠀⠆⠀⠀⠀⢰⠀⠀⠀⠐⢹⡶
⠤⠤⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣸⠒⠒⠒⢺⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢄⠀⠀⢠⠊
⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢅⡀⠠⠤⢸⡀⠀⠈⠀⠀⡸⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠴⠢⠄⢀⡠⠃⠠
⠀⢀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢈⠀⠀⠀⡸⠀⠀⠁⠁⠄⢾⠀⠀⠐⠀⠀⠉⠒⠦⢀⡀⠀⠀⠀⠘⠁
⡀⠈⢦⠙⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠉⠀⠀⡀⠀⠀⠹⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⡁⠆
⠀⠀⠈⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠂⠀⠀⢀⠀⠀⠐⠂⠀⠀⡠⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠌⠀⠀⢀⠄⠀⠀⣿⠀⠀⠀⡦
⠰⠀⠀⠀⠈⠁⠱⠲⣀⠉⠉⢉⡼⢹⠀⠀⠀⠀⣨⠀⠀⠀⡻⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⡁⢀⡀⣘⠁⢀⣀⠤⢼⠀⠀⠀⡇⠇⠀⠀⢸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣈⡶⠉⢁⠸⠀⢹⡍⡀⠁⠀⡆⠀⠀⠀⠇⢸⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠒⠒⠚⠉⠙⢉⠁⡴⠊⠄⠸⠀⠆⠊⠀⠈⡂⠀⠃⠀⠀⢸⠂⠈⡇⠀⠀⢃⠀⠀⠀⠀⠀⠁⠀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⠀⠀⣀⡠⠞⣠⠊⠉⠉⠁⢻⠀⢀⡼⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠘⠛⠛⠛⠋⠀⠘⠧⠤⠤⠦⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡈⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢐⠂⢈⡠⣌⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢅⠘⠿⠿⣦⣕⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⣬⡐⠤⣀⠈⠻⢶⢿⡖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢸⣿⣄⠀⠁⠐⠠⠠⠔⠹⣟⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠺⣿⡄⠀⠀⠀⠀⠀⠀⠀⠑⠲⣽⢷⣦⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⢀⡀⠤⢐⣢⣍⣤⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣄⣀⣠⡤⠵⠚⠛⠛⠠⣭⣤⣶⣾⣿⣿⠿⠛⠋⠉⠉⠋⠙⠖⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠀⠀⢀⣠⣉⣠⣤⣼⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢦⣤⡾⠶⠶⠶⠿⠛⠛⠛⢫⡀⠀⣺⣿⣿⠆⠀⠀⣀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡼⠉⠁⠀⠀⣼⠏⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⣸⣿⣠⠊⠀⠀⠀⣠⣴⡿⢳⡄⣼⡿⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣼⢁⠀⠀⢀⣤⡤⢼⣿⠀⣴⣿⢁⣿⡏⠀⣿⣿⠟⢉⣿⠀⢰⣿⠟⣠⠞⠀⡿⠷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠞⠋⢻⣇⢠⣿⡏⠀⣿⣿⠀⣸⠿⢻⣿⠁⢀⣿⡿⣦⣼⡏⢀⣿⣿⠋⠁⣠⠞⠀⢰⣿⠇⡠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣠⡿⠁⢸⣿⡇⢠⣿⡏⡠⠁⠀⣾⣿⣠⠚⣿⣇⡼⠋⠀⠈⠿⣿⠤⠞⠁⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠞⣸⠃⠀⠸⣿⠷⠋⠿⠋⠀⠀⠀⠙⠋⠁⠀⠉⠁⢀⣀⠀⠀⠀⢀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠉⠁⠀⣴⣿⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⣸⣿⣯⠔⠁⠀⠀⣴⣾⠷⣆⠀⢠⣿⡖⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⣀⠴⣿⣿⠅⣴⡦⠀⣸⣿⡇⠀⣿⠀⣼⣿⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡅⢸⣿⡟⢸⣿⠁⣠⣿⣟⢷⣠⡧⠈⣿⡟⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⣿⡇⣼⣿⠇⣾⣿⠜⠁⠻⢿⡶⠟⠀⢸⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣼⡿⠃⠻⠟⠀⠀⠁⠀⠀⠀⠀⠀⠐⠊⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⠿⢿⣟⡙⠿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣻⣿⣿⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣃⡄⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢀⣀
⠀⠀⠀⠀⠀⠀⠨⢤⡿⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡠⠂⣠⡋⠁⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⡤⡔⠁⠀⠀⠀⠀⠊⡔⠁⠀⢄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠈⠉⠁⠀⠈⡇⠀⢀⣴⡮⠀⠀⠀⢠⢧⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠞⠑⠒⠋⠃⠇⠀⢀⡴⠀⠀⠑⢼⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠂⢸⠀⠊⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⡠⠐⠚⠊⠀⠈⠀⠠⠔⠚⠛⠲⣤⡇⠀⠀⠀⠹⡀⡀
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠙⣄⠀⡠⠂⠀⠀⢠⠘⡄⠀⠀⢹⡇⠀⠀⠀⠀⠱⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠀⡇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⠀⢰⠇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠠⡀
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⢸⠀⢸⡀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢻⠓⠒⠒⠒⠲⠁⠀⠀⠀⠀⢁⢸⠀⠀⠱⡜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢓⣒⠒⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⢀⠣⣀⣸⣏⣉⣉⣲⣤⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡀⢠⡟⠉⠉⠉⠉⠈⠉⠢⣄⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠧⠥⢀⣀⣀⡀⠀⠀⠀⡌⠙⠦⣄⡀⠀⢹⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢿⣿⡿⡟⠋⠉⠉⠉⠙⠢⣼⠀⠀⠀⠀⢸⠀⢂
⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡍⠀⠀⠀⠀⢸⣤⠊⡀
⠀⠀⠀⠀⠀⠀⠀⡷⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢸⠉⠻⣮⡑⠤⡀
⠀⠀⠀⠀⠀⠀⣾⢧⡤⠯⣦⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⢼⠀⠀⢀⡇⢀⣀⠀⠀⢻⡀⠀⠀⠀⠀⢀⣮⣤⣤⣾⠿⢷⠆
⠀⠀⠀⠀⠀⠀⠹⢒⠲⠤⣌⣉⣙⡆⠀⠀⠀⠀⢠⡊⠁⠀⠀⠀⠀⠈⠙⠉⠉⠀⠀⠀⠉⠑⢢⡈⠲⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⠤⠤⣤⠤⠤⢤⡤⠤⠤⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠛⠛⠻⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⣀⣤⣴⡿⠉⠉⢿⣦⣤⣀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣿⣇⢸⡇⣸⣿⣿⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⣄⢸⣿⣿⣿⣿⣿⣿⡇⣠⣿⡆⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡀⠀⠀⠀⢻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣦⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⢿⣿⣿⣷⣤⣀⠙⠻⢿⣿⣿⡿⠟⠋⣀⣤⣾⣿⣿⡿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣄⠀⠀⣠⣴⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣾⣟⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⠟⠻⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⣿⡿⠟⠋⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣤⣦⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢀⠁⣸⣿⣿⠇⠀⢀⣶⣿⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠨⣿⠿⠃⢀⣰⣿⡻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⣀⣀⠀⠐⠄⣠⣶⡟⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣣⠾⢿⣯⣎⢰⡏⠉⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠛⣄⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢴⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⡀⠀⠀⠀⠀⠀⢀⡄⠚⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠉⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⡀⣠⣟⡿⠶⠶⣤⣀⣀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠐⢦⡤⠀⠰⡿⣿⣿⠆⠀⠀⠀⠀⠀⣾⣴⣿⣿⣿⣓⣀⣨⣍⣛⢿⣗⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⣦⡀⠸⢿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠘⢿⡋⠙⣿⣿⣆⢻⣿⣿⣶⡀⠀⠀⠀⠀⠀⢰⠀⢨⣿⢿⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢛⣿⣿⣿⣿⣿⡻⡄⠀⠀⠀⠀⠈⠀⠘⠻⣪⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢾⡇⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣸⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⢠⣜⣻⣿⣿⠀⠀⠀⠀⠉⠐⣥⣀⡀⠆⠀⣹⣯⠙⣿⡿⢋⣿⣟⣽⣧⣿⡿⠀⠀⠀⠀⣃⠀⢠⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢸⣿⣷⣆⠘⠀⠀⠀⠀⠀⠠⠜⠁⠁⠀⠀⠘⢿⣦⠈⢿⢸⠌⣿⣼⢃⣿⠀⠀⠀⠀⠀⣷⡀⢸⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⢀⠒⣀⠀⠀⠀⠀⠀⠘⣳⠀⠀⠀⡼⢿⣿⣾⠟⠀⠀⠀⠀⠀⣿⡇⠈⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⢻⣿⣿⠀⠀⠀⠀⢺⣗⠚⢧⣄⠀⢀⣠⣼⡉⠀⠘⣿⣧⣞⢋⠁⠀⠀⠀⠀⠀⠀⣿⠃⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢀⣶⣿⡟⢻⠀⠀⠀⠀⢘⡻⣿⠿⠟⠀⢾⣟⠟⠃⠀⢸⣿⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⡧⠀⢠⣿⣿⡟⢹
⠀⠀⠀⠀⠀⠀⠀⢈⠈⠁⠀⢻⣿⣿⣿⡇⠀⠀⠀⢀⡛⠁⠀⠀⣾⡿⠗⠀⠀⢀⣼⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠉⢹⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⡿⠁⠀⠀⠀⠈⢧⣶⠀⣤⣭⢴⣦⣶⣿⠿⢿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠘⠂⠒⠲⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⢰⣿⣿⣷⣤⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠼⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠁⠀⠸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣤⣀⡀⢀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣦⡀⠀⠀⠀⠀⣀⣤⣴⣆⠈⠀⠠⣤⣶⣤⣤⡿⠟⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣜⢿⣿⣼⡇⢠⣴⣶⣿⣿⣿⣿⣿⣷⣮⣤⣈⣹⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣾⢟⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠉⠀⠙⢿⠗⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠹⢟⣵⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠛⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠘⣿⡧⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⡾⠋⣴⣶⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢴⣶⡘⢷⣄⠀⠀⠀⠀⢀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠰⡽⠇⣼⡟⢠⣿⡇⠀⢠⣶⠄⣤⠀⣤⡄⢤⣤⠀⠀⣾⣧⠸⡦⡈⢿⡷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠾⠿⠀⠀⢸⡇⢰⣟⠀⢽⡇⠀⣷⠀⠀⢹⡿⠆⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⣡⣾⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠿⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠼⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⡀⠠⠴⠂⠀⠸⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⢄⣀⡀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠐⡞⠀⠀⠀⠀⠀⠀⣀⠔⠁⡀⢀⠘⠸⣧⠄⠀⠀⠀⠀⡠⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⡟⠀⠘⣿⣇⣀⣀⣠⠞⠁⠈⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠒⣂⣠⣤⣤⣤⣀⠀⠀⢀⡇⠀⠀⠟⠙⢻⡩⢶⠀⠠⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣴⣿⣿⣿⣮⣽⣿⣿⣦⡴⠋⠀⠐⣤⠀⠀⠈⠀⠸⡄⠁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⣿⡀⠉⠉⠉⠉⠉⠀⣿⡇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢺⡟⢠⣄⡀⠀⢀⣀⠀⢻⡧⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠚⠋⠀⠈⠛⠂⢸⠃⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⢀⡀⠀⠀⠸⠊⢒⠘⠐⠀⠀⠀⠀⠀⠀⠀⠃⡼⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠤⠉⠤⠔⢀⣼⣧⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢱⢦⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⠀⠀⠀⣰⠊⣿⣿⣾⡸⣧⣀⠀⠙⠒⠐⠀⠀⠀⠿⢋⡤⠤⣤⣀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡻⠿⠿⠟⠁⢠⣿⣿⢿⣷⢻⣿⣻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣷⠘⣯⠐⣐⠀⢠⡟⡟⠈⢸⡟⠿⣿⡿⡀⠽⠛⢷⣦⣤⣀⠀⠒⠲⢤⣀⣠⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣾⠋⣿⠸⡆⠘⣆⢠⡴⠏⠐⠁⠀⡸⠃⠀⠈⠻⢧⣶⣺⠆⠙⠧⡟⠉⠓⠲⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⡇⠀⢸⠀⢳⡀⠈⠀⠀⠀⢠⠄⢀⠁⠘⠀⠀⠀⠀⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡧⠀⠈⠃⠀⠀⠀⢀⣠⠖⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⢻⡄⠀⠤⣀⠀⠒⠋⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠀⠉⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⢂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠡⠀⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢶⣄⡀
⠀⠀⠀⠀⠠⠐⠚⠋⠓⠂⠀⠀⠀⠀⠀⠀⠀⡀⠂⢉⠀⠀⠀⡲⡀⠀⠀⠀⠀⠀⠠⠆⢀⡦⣤⣴⣴⢊⣶⣤⡾⡀⠀⠀⠀⠀⠀⡠⢦⡴⠤⠄⠀⠛⢿⣦⡀
⠀⠀⠀⣰⣦⣦⣀⣤⣠⣄⣄⣄⡀⠀⠀⠀⠀⠁⠖⠚⠉⠓⠛⠁⠈⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⣸⠉⡜⠂⠠⠤⠀⠡⡞⣙⠇
⠀⠀⠀⣡⡉⠈⠉⠉⠈⠉⠈⠀⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⠀⠀⠀⢹⡕⠀⠀⠀⠀⠀⠀⠘⢫
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠒⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⣀
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⡄⣈⡆⠀⠀⠀⠀⠀⠀⠀⠉⣉⡁⠘⢉⢣⡀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠠⢾⡿⣷⡄
⠀⠀⠀⡆⡟⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⣌⠗⠋⠈⠛⠋⠛⠏⣹⠀⠀⠀⠀⠀⠀⣾⣷⠶⠛⠷⠗⠚⠷⣇⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⢐⠏⡁⢿⡇
⠀⠀⠀⢿⠂⠀⠀⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⢸⡿⠟⠓⠚⠛⠋⠁⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⠥⠵⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣘⢤⣬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠘⠛⠁⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠁⢂⣠⡔⠀⠀⠻⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⡿⢷⡤⢀⣀⡴⠂⠀⠉⠓⠠⢄⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⢵⣶⣿⠥⠤⠀⠀⢄⠀⠀⠈⠉⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣽⠉⠀⠀⠺⠂⠆⡄⠈⠀⢁⠀⠀⠀⠐⣀⢀⠐⠁⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢉⠐⠀⠀⠀⠀⢤⠀⠁⠀⠀⠀⠈⠢⠄⠀⠀⠙⢄⠀⠀⣴⢳⣀
⠀⠀⠀⠀⠀⢠⠖⡙⠀⠤⢀⠀⠐⡳⠀⢧⠀⠀⠀⠀⢀⡀⠈⠢⡀⠀⠉⠲⡤⡄⢩⣆
⠀⢀⣀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠉⠛⢦⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⢃⠘⠄⢻⡆
⠀⠈⡿⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠖⣮⠐⣤⠼⡧
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠚⢻⡟⠁⠀⠁⡼⠁
⠀⢸⠀⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡄⠀⠀⠀⢠⠟⠀⡀⡠⠂
⠀⣾⠀⡥⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⢄⡀⠀⢰⠃⣠⠖⡹⣧
⢀⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠱⠤⠁⠀⡀⢘⠃⢹⣆⡀
⢸⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠄⠁⠀⠀⢛⡀⠘⠳⡈⠁⠀⠈⡁⠈⠉⠙⢦
⣼⡇⠰⠀⡆⣠⠤⠤⠤⠤⠄⠀⠀⡀⠤⠴⠖⢩⠀⠀⣀⣀⠴⠛⢧⣀⡰⠚⠀⠉⠉⠀⠀⠀⠀⠀⠳⣦⡀
⠀⢅⣀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠉⠁⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡀
⠀⠀⠸⠋⢲⡷⢿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⢟⡰⢰⣻⣿⣿⣿⣿⣿⠏⣴⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠹⣟⣿⣧⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⡟⢲⢠⣤⣄⡀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣏⠸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣶⣶⣯⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⡏⠛⠛⠁⠛⠛⠛⠛⠛⠛⠛⠛⢳⡆⠛⠛⠛⠛⠋⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⢸⣿⡇⢀⡾⢧⣄⠀⠀⠀⠀⠀⠀⠀⠈⠃⡆⠀⠀⠀⠀⠀⣀⣀⣀⣿⡏⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢸⣿⡇⠸⠧⠒⠛⠓⠒⣶⣶⣶⣶⣶⣶⣄⣻⣲⠖⠿⢯⣿⣟⣿⣛⣿⠀⠀⠙⢿⣿⣿⣷⣀⡴⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⢩⢿⡇⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⡿⠆
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠉⢀⣠⠖
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢑⠂⠉⠉⠉⠹⣃⣹⣿⣿⣿⣿⣿⣿⣿⣏⣿⠀⠀⠀⠀⠈⣿⠿⠛⠉⣱⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⠿⠟⠛⣉⡠⢿⣶⣶⣠⣤⠚⠁⡄⢀⣼⣿⠿⠏
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⣿⣇⡠⡄⠒⠈⠋⠘⠄⠒⠈⠀⠍⢀⠰⡟⠉⠉⠁⡌⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣶⣞⣿⠀⠀⠀⠀⠀⢐⢸⢇⡐⠈⠆⠂⠈⢀⡠⠀⢀⣀⣤⣾⡷⢇⣒⣂⠠⠚
⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⡠⠽⠟⠛⠟⠀⠀⠀⠀⠀⠀⢂⣁⣌⣹⡟⣓⣦⡀⠀⢀⠠⠄⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠁⠈⠉⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣤⣄⣈⡉⠃⠀⠀⠀⠀⠐⠂⠀⠀⠈⠉⢀⣀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣷⣶⣾⣿⣿⣿⡿⠿⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠟⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢸⣿⣷⣄⣀⣀⣠⣾⠟⢻⣷⣄⣀⣀⣠⣿⣿⡇⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣷⣌⠙⠛⠻⣿⣿⣿⣾⣷⣿⣿⣿⠟⠛⠋⣡⣾⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣟⠛⠛⠓⠀⠀⣿⢻⡟⣿⣿⢻⡟⣿⠀⠀⠚⠛⠛⣻⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⠷⣦⣀⠀⠀⠀⠈⠁⠉⠉⠈⠁⠀⠀⠀⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⣉⠳⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⣀⡄⢰⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⡆⠀⣿⣷⠀⢰⣶⡀⣶⡆⢠⣶⣾⣦⢸⣿⠛⠃⢸⣟⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣼⡇⣸⣟⣿⡄⢸⡿⣷⣿⡇⣿⡏⣶⣿⢸⣿⠿⠇⢸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠁⠿⠋⠻⠷⠸⠇⠘⠿⠃⠙⠿⠿⠿⠸⠿⠿⠆⠸⠟⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠷⢆⣠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠐⠈⠉⠀⠀⠀⢀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⣷⠆⠤⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⢀⢊⢠
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠈⠦
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠂⠀⠀⠐⠂⠡⠤⠀⠣⠐⠀⠀⠀⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠕⢾⣿⣿⡿⣫⡿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢰⣤⣄⣤⣴⣾⣿⣿⣿⣿⢿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡍⠙⠛⠛⠋⣈⠉⠉⠉⠉⣹⢿⣿⡿⠃⠀⢻⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠉⠁⢰⠶⠶⣦⢶⣮⣿⣯⡟⠈⢻⣿⣦⣄⣈⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠀⠀⠟⣿⣿⣿⣿⡇⠀⠀⣿⡿⢟⣉⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠸⡟⠛⠋⠍⠛⠲⠆⡤⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠶⠏⠀⠀⠀⢋⡀⣠⠀⠀⠐⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣀⡀⠄⠀⠀⠀⠀⠉⢀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢁⠀⡶⠀⣶⣶⠀⢺⠃⢰⡆⠀⣾⠀⢰⣴⡆⢰⡇⠀⣶⡆⠈⠀⢾⡀
⠀⠀⠀⠀⢀⣤⣤⣤⣄⠈⠉⢠⣭⣤⡅⠉⠀⣬⣤⣤⣍⠀⠉⣥⣬⣤⣥⡈⠉⣤⣥⣥⣤⡀⢩⣤⣤⣤⣀
⠀⠀⠀⠀⠀⣿⡿⠿⣿⡇⠀⢸⣿⣿⡇⠀⠀⣿⡟⠛⢿⣇⠀⢸⣿⠉⢻⣿⠀⣿⣿⢉⡉⠁⢸⣿⠋⠙⣿⡆
⠀⠀⠀⠀⠀⣿⣇⣀⣿⡇⠀⣸⣿⢹⣷⠀⠀⣿⣇⣀⣾⡟⠀⢸⣿⣿⣿⣿⡀⣿⣿⠿⠏⠀⢸⣿⣶⣾⡿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⣿⣏⣸⣿⡀⠀⣿⡿⢿⣿⠀⠀⢸⣿⢀⣸⣿⠇⣿⣿⣤⣤⡄⢸⣿⠀⢻⣷⡀
⠀⠀⠀⠀⠀⣿⡏⠉⢻⣿⢠⣿⡿⠿⣿⡇⠀⣿⡇⠈⣿⣧⠀⠼⠿⠿⠟⢋⣀⣽⣿⣿⣭⣥⣈⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣿⣇⣠⣾⡏⢸⣿⠁⠀⠿⠗⠀⠛⠁⡀⡈⢠⡴⠶⠶⣶⡄⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡿⠟⠀⠉⠉⣀⣀⣐⣚⣛⣛⣚⣙⣓⣘⣃⣤⠴⠛⠁⢻⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠈⠉⠀⣠⢶⠟⢿⣿⠉⣩⡍⠉⠉⣉⣉⠉⣴⠞⣷⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⡞⠁⠸⢦⡾⠟⢠⣿⣶⣆⣾⢁⣿⣸⡏⣸⠟⢠⣿⣄⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⣀⣤⣶⡄⣸⡇⣿⢹⣯⡼⢃⣿⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣷⣿⠿⠛⣿⠃⣿⠘⠛⠀⣀⣀⣬⣥⣤⣤⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢛⠋⠁⠀⣰⠃⠈⠥⠖⠚⢫⣭⣀⣀⣀⣼⡟⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢰⣿⢀⣠⠞⠁⠀⠀⠀⠛⣛⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠹⠟⠋⠁⠀⣀⡤⠶⣚⣫⡭⠆⠀⠠⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀⠐⡇
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣉⣚⡻⠖⠀⣠⣶⠚⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⢻⣿⣿⣿⣿⣿⣿⡇⢀⣄⢡⠇
⠀⠀⠀⠀⠀⢰⡟⠀⠘⠉⢩⡟⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⡟⠛⠓⠀⣿⡧
⠀⠀⠀⠀⠀⠘⠿⣶⠶⠞⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⣿⣿⣿⣿⡯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⡧⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠐⣆⢲⠂⠀⠈⠫⣭⣙⣛⣛⣛⣛⠛⠛⠛⠛⠛⠻⠿⠿⠛⠟⣡⣿⣿⣿⠟⢻⠂
⠀⠀⠀⠀⠀⠀⣤⣨⢤⣁⡬⣄⣤⠀⠀⢀⣤⣯⣾⣤⣤⣟⣙⣉⣉⣉⣻⡗⠃⠶⢿⠿⠛⠋⠓⠒⠸⡅
⠀⠀⠀⠀⠀⠀⠿⢹⡿⠋⣿⠟⠼⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡾⠀⢸⢾⣰⣿⣸⣸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠀⡇⠀⢨⠶⡤⡬⢶⠦⣴
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⣿⣿⣿⢰⠇⠀⠈⠂⠙⠃⠘⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣟⣻⡿⠡⢓⣴⠊⠉⢛⣿⣁⠸⠃⠒⠒⠖⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⣤⣭⡉⠉⠻⠖⠀⠀⣿⠁⠉⠀⢶⣦⣐⣀⠀⣀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢶⣤⣄⣀⣐⣿⡇⠀⠀⠰⠿⠿⠿⢟⣘⣃⣠⣤⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠁⠄⠁⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣿⣷⡆⠀⠀⢀⢀⠁⠁⡶⠳⠢⠶⡆⣠⠤⢶⠴⠤⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡟⠘⡇⠀⠀⠃⣚⢃⠀⡁⣴⣟⣈⣹⡇⢦⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⣰⢤⣾⣋⡴⢤⡗⣸⡁⣩⡟⠉⣻⡇⢸⣆⢠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡶⢿⠒⢤⡙⣷⣿⠿⠁⠉⠀⢻⣿⣿⠁⠀⠹⣿⣷⢸⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⢟⢸⠉⠢⢬⣿⠉⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⣸⡗⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⡚⠘⠆⣽⣿⣿⡇⠀⢺⡆⠀⠀⠰⠀⠀⠀⢠⣯⣵⠇
⠀⠀⠀⠀⠀⠀⠠⠒⢻⣷⡟⢦⠄⡾⠙⠛⡿⠶⣞⠛⠛⠟⢛⣷⣶⣶⣿⡇⠘
⠀⠀⠀⠀⠀⠀⢠⢳⠚⢿⠀⠈⠀⢀⡴⢆⠟⠤⠂⠠⠌⠡⠽⣿⢿⣿⠙⢌⡒
⠀⠀⠀⠀⠀⠀⢘⣋⠅⠈⠂⠀⠀⢸⡏⠓⠲⠁⢠⠀⠀⠀⡆⠰⠖⢿⣓⠄⠙⢺⡄⣤
⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⠀⠀⠀⢰⠇⠀⠀⠄⢸⡄⠀⢠⣿⡙⠀⠈⣿⠀⠀⣸⢻⠟⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⢀⣤⣬⠂⠀⠀⢸⣿⠿⢠⣿⣿⡁⠀⠀⢻⣧⣴⣿⢻⢺⠁
⠀⠀⠀⠀⠀⠀⡛⢀⡴⢣⣶⣿⣿⣿⠀⠄⠀⢸⡟⠀⠀⠹⣿⣇⠀⠀⠀⢻⣿⣯⡰⠟
⠀⠀⠀⠀⠀⠀⠙⢾⣵⠟⢻⡟⠁⢿⠀⠀⠀⢾⡇⠀⠀⠀⣸⣹⡆⠁⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣸⠀⠀⢸⠓⠶⣶⢿⣿⠀⠀⡼⡿⢛⢻⡟⠛⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠸⡏⣱⣿⣾⡏⠰⣶⣗⣡⠊⢠⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣇⠙⢹⣾⣿⣶⣧⠭⢥⣠⣾⣞⡛⠲⣆
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣴⣶⣿⣿⣿⡿⣤⠯⠞⣿⣿⣿⠀⠀⠈⠈⢿⣿⣿⣷⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠶⣯⣿⣿⣿⣿⣿⣿⡿⠀⠁⠀⠀⠸⣿⣿⣦⣀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠂
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣦⣤⣼⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣤⣾⣿⣿⠿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⡀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠉⠙⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠛⠛⠟⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡙⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠤⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠚⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠁⠀⠀⣈⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⢿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⡤⣦⡀
⠀⠀⠛⠁⠀⠈⢷⡀⠀⢠⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠈⢷⣶⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⢀⣿⣷⣦⣄⡀
⠀⠀⢀⣴⣧⡀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⣴⣿⣿⣿⣷⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠙⣦⣀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣂⠒⠁
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠘⠿⠇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣩⣶⣿⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠉⠀⠰⠾⠿⣿⡿⠉⡁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢹⣿⡀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣿⠃
⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⣒⣲⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠒⣒⣲⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣶⣒⡋⠉⠛⠿⣯⣟⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⢾⣵⣒⠋⠉⠛⠻⢿⣛⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠉⠙⠷⣦⡰⣌⢻⣷⡜⠞⡆⠀⠀⠀⠀⠀⢠⠏⠈⠛⠛⢿⣶⣤⡰⣄⠹⣷⡘⠺⡇
⠀⠀⠀⠀⠀⠀⠀⢞⣤⡄⠀⠀⢰⣟⣀⣙⣿⣧⢻⣿⠀⡗⠀⠀⠀⠀⠀⢘⣀⡀⠀⠀⢼⣟⣠⣙⣿⣧⢹⣷⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠛⠀⠀⠀⢀⣿⠟⣛⣻⣿⣾⣿⢠⡇⠀⠀⠀⠀⢀⠌⠙⠀⠀⠀⢸⣧⠶⣛⣻⣿⣼⣿⢀⡏
⠀⠀⠀⠀⠀⠠⢇⠀⠀⠀⠀⠀⠀⠁⠊⣵⠏⣿⣿⣿⡼⠀⠀⠀⠀⠀⣆⠀⠀⣀⣤⡐⣿⠁⠘⣡⠟⣿⣿⣿⣼⠁
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣘⢿⡿⠁⠀⠀⠀⠀⠀⢸⣭⣟⣻⣿⣷⣽⣷⡆⠸⣆⣙⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⣀⡠⠎⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⢻⢹⣿⡿⢟⡿⠁⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⣿⣤⣤⣤⣤⢤⡶⠟⠁⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣭⣁⣀⡴⠶⢾⣿⣿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣴⣫⣍⣀⣠⠶⠶⣿⣿⡿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠛⠛⠲⢶⣤⡀⢈⠻⣷⣌⠳⣳⡄⠀⠀⠀⠀⠀⣹⠛⠛⠓⠶⣦⣄⠀⡙⢿⣦⡙⢟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⣿⣿⣮⣷⡜⣿⣧⠀⣇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣸⣿⣷⣽⣦⢻⣿⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠉⠀⠀⠈⣿⣿⡿⣿⣿⣸⣿⠀⡇⠀⠀⠀⠀⠘⢸⡏⠀⠀⠀⢻⣿⣿⢿⣿⣇⣿⡇⢸
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠐⠛⠡⢞⣾⣿⣿⣿⣸⠃⠀⠀⠀⡠⠁⠀⠀⠀⠀⢀⡿⠋⠶⣱⣿⣿⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⣦⣀⣀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠀⠀⠀⡠⠀⠈⠙⢿⠛⠁⠀⠀⠀⠀⠀⠀⠋⠈⠉⠀⠀⠀⢀⠀⠀⠉⠻⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠔⢶⠚⠉⠁⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠑⠠⠴⠲⡖⠋⠁⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⡿⠿⠿⠟⠛⠛⠛⠿⠿⢿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠋⠉⢀⣀⣤⣤⣴⣶⣶⣶⣤⣤⡄⠀⠈⠙⠻⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣄⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⢀⣀⣄⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⢹⣟⣷⣠⣤⣤⣤⣀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⢟⣫⠽⠚⠉⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣛⣛⡿⠿⠿⢟⣛⣻⣭⡷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠹⣿⣝⣻⣿⣿⣿⣿⣛⣽⡿⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡈⠉⠛⣻⠃⣿⠛⠛⣋⣀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠞⠢⣀⣡⠟⠎⢿⠿⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⢠⠆⠤⢽⡯⣤⣴⢆⡔⢹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⡿⠨⣪⡀⠶⠛⠛⠿⢋⡾⡰⠁⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⢀⣠⣾⣿⣿⠃⣇⠀⠙⢿⣦⣤⣶⣶⣿⠟⠁⢠⣿⢻⣿⣿⣿⣿⢿⣶⡒⢢⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤⣶⡾⢫⣿⣿⣿⣿⡄⣿⠄⠀⢀⣽⣿⣿⣋⠁⠀⣠⣿⡿⣼⣿⣿⣿⣿⣥⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣿⣿⣿⣿⣇⣿⡈⠶⣿⣿⣿⣿⣿⠷⡼⢡⣿⢷⣿⣿⣿⣿⠟⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣿⣿⣿⣿⣼⣇⠀⠹⣟⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣆⣴⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⣤⠶⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⣠⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⢀⣴⣿⣿⣿⣿⠟⠀⠀⠀⡐⠄⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢰⠈⢂⠀⠀⣠⣾⡟⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⣾⣧⠈⠻⣿⣿⠟⠀⠀⠀⠀⠀⢰⡃⠀⠀⢣⣼⣿⡟⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⢀⣴⣿⣿⣿⣿⣧⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣇⢀⣴⣿⣿⢏⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠗⠄⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠟⠁⠈⠑⢄⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⠠⣤⣀⠀⠀⠀⣾⣿⣿⡿⣇⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣜⢿⠟⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠉⠛⢳⠀⠀⠀⠈⠳⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⢸⣿⣀⣨⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⣥⣿⣷⣦⣀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉⠙⠃⠈⠻⠿⠿⠿⠷⣦⣄⡀⠀⠈⠙⠒⠲⠤⠼⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣠⠞⠋⠉⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠉⠻⣦⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡾⠁⠀⠀⢀⣼⡇⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣷⣶⣦⣀⠀⠀⠀⢿⣦⡀⠀⠀⠈⣷⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣯⡀⣀⣴⣿⠟⠀⡤⢴⣾⣿⣿⣿⡿⡿⢿⢻⡿⣿⣿⣿⣷⣶⢦⡄⠙⠿⣶⣤⣴⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠋⠀⠀⠀⡇⢸⠈⡇⢿⢸⠀⡇⢸⠘⡇⡇⢸⠀⡇⢻⢸⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠃⠘⠀⠇⠠⠸⠀⠇⢸⠀⠇⡇⠸⠀⠇⠐⠘⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠛⠛⠓⢦⣙⣿⣿⣿⣿⣋⡴⠚⠛⠛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⡿⢿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢩⣝⣦⡀⠀⠀⠀⠀⢀⣿⣷⣄⣀⣀⣀⡴⢋⣼⣿⠁⠈⣿⣧⡙⢦⣀⣀⣀⣠⣾⣿⡀⠀⠀⠀⠀⢀⣴⣫⡍⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣬⣭⣴⣾⣿⣿⡇⠀⠀⢸⣿⣿⣷⣦⣭⣥⣾⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣿⣿⣿⣧⡀⠀⠀⠘⢿⣿⣿⣿⢿⣿⣿⣿⣿⣤⣴⣦⣤⣿⣿⣿⣿⡿⢿⣿⣿⣿⠃⠀⠀⢀⣼⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠉⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠉⠉⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠸⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠇⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣶⣄⡈⠉⠀⠈⠉⠁⠈⠉⠁⠀⠉⢁⣠⣶⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⣦⣀⠾⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢿⣿⣿⣿⣿⣿⣶⣍⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣿⣿⣷⣬⣙⠿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣵⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠖⠀⡀⠐⢂⡀⠉⠏⠉⠃⠀⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠒⠄⠀⠀⠀⡠⠄⠁⠘⠁⢀⠚⠡⠂⠊⠀⠀⢠⠘⠀⢡⠀⠀⠀⠙⠲⢄
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡌⠁⠁⠀⠀⠠⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠀⢲⣄
⠀⠀⣴⢷⣦⣤⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⢀⠌⠀⢣
⠀⠀⠃⢤⡌⠙⠻⢶⣼⣄⠀⠀⠀⠀⠀⠀⣠⠄⠃⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠘⡆
⠀⠀⠀⠀⡀⠉⠂⠄⣈⠛⠷⢶⣆⠀⠀⢻⡇⠄⠀⠀⢰⡀⠘⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⡰⠀⠀⠀⠀⠈⠉⠣⠄⡉⢧⠀⢸⣷⡈⠆⠀⠀⢃⠀⠘⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⡀⠀⠀⢹
⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⢇⢸⡞⡖⢸⡀⠀⠀⣇⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡠⠂⠉⠉⠙⠂⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡈⣇⢹⣼⠀⠄⠀⠸⢤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠱⡅⠨⢇⠀⠘⣜⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠶⣤⣸⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⢨⠠⠤⠉⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⡎⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⢀⠀⠀⠀⠀⢨⠀⢀⠆⠀⠸⢡⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠳⠀⠀⠀⢘⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⠖⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⢀⡠⠴⢒⣲⠶⠒⠀⠈⠁⠀⠀⠀⣀⠴⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⢀⡠⠖⠁⠒⠊⠁⠀⠀⠀⠀⠀⠀⠔⠈⠁⠀⡸⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠀⠒⠒⠒⠀⠀⠒⠊
⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣀⠔⡊⠁⠀⠀⠐⠒⠒⠂⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⡏
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢄⡜
⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡉⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⣀⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠀⢰⣿⡿⠿⠀⢀⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣾⣄⠀⣿⡇⠶⣿⡇⢸⣿⣛⣛⠀⣼⣿⣦⣿⠇⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⣆⠙⠻⠿⠛⠁⠈⠉⠉⠛⠀⠛⠃⠿⡇⠀⣿⡏⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⡀⠹⣷⠄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⠿⠋⣠⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠁⢀⣴⡦
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡈⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣾⣏⡀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣿⡟⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⠃⠀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⡀⠀⠘⢷⣿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡆⠀⠀⣰⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⠈⣿⣿⠁⠀⠀⠈⣿⣿⠁⠀⠀⠈⣿⣿⠁⣠⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⢿⣿⡗⣸⣿⣦⣤⣤⣼⠟⠻⣧⣤⣠⣴⣿⣇⢸⣿⡿⠿⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⡿⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⣿⣿⣿⣿⣿⣿⣩⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⢽⣉⣉⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠆⠻⣿⣾⣶⣶⣷⣿⠟⠰⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣶⣿⣿⠿⠋⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠙⠿⣿⣿⣶⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣠⣾⣷⡆⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⢰⣾⣷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⢿⣧⣿⠃⢀⣿⡟⣿⣧⢸⣿⣟⣻⡀⠘⣿⣼⡿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⢸⣿⢻⣿⠋⠀⣿⣯⣭⡅⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠐⢄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⣨⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⣯⣤⡀⠀⠀⠀⠀⢻⣓⡀⠀⡠⣤⣍⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣆
⠀⠀⠀⡟⠤⢤⡄⠀⠀⠀⣸⣿⠋⢠⡖⠂⢹⡆⠘⣿⣿⣿⣿⣿⠟⠀⠀⠈⣿⡆
⠀⠀⡼⠀⠈⠁⠀⠀⠀⠀⣿⣷⣄⣸⡆⠀⡼⠁⠀⣿⣿⣿⣿⢿⣦⡀⠀⠀⢻⣿
⠀⠸⢅⣀⡲⠀⠀⠀⠀⢠⣿⣿⣿⠋⢀⠞⠁⠀⢰⣿⣿⠟⠀⠀⠙⣿⣦⡀⠸⣿⣇
⠀⠀⢰⣿⣿⣶⡄⠀⠀⣼⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠈⠻⣿⣦⣿⣿⡄
⠀⠀⠀⠁⢹⣿⣿⡆⣰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡧
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣿⣦⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠻⣦⣀⡶⠒⠶⣄
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣟⣉⣀⣀⣀⣀⡤⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢻⡇⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⢿⣆⠀⠀⠀⠻⢦⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⠢⡀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⢀⢠⡄⠀⠈⠂⠰⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢡⣤⠃⠠⠉⡇⠸⢱⠀⡎⣠⠀⡮⠈⡐⢉⠂⡔⠐⠆⠁⠰⠋⡃⢠⠒⠂⢀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠤⠀⠁⠂⠸⠤⠁⠐⠘⠒⠉⠈⠂⠊⠀⠐⠊⠡⠤⠄⠐⠀⠸⠴⠇⠼⣥⠃⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⣠⣾⠾⠟⠿⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣧⣤⣀⣾
⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠉⣰⣿⡿⠛⠛⠉⠉⠛⠛⠻⣷⣦⣄⠀⠀⠀⠀⠀⠀⡆⢸⠇⡿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣸⢹⡿⠷⠦⠄⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⡇⠈⠀⠘⢹⣿⣿⡟
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠙⠞⡇⠀⢰⣿⣶⣤⣀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⢿⣿⣶⣶⣴⣿⠏
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠙⠀⠘⣿⣿⣿⠿⣿⣦⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠛⠛⠁
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠐⡁⠀⠹⣿⣷⠀⠀⠙⠻⣦⣀⣼⡿⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⡄⠀⠹⣿⣦⣀⠀⣸⣯⣿⣿⠁⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠈⢢⣀⢈⣿⣿⣿⣿⡿⠋⠙⢷⡀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠀⠹⣿⣄⠀⠀⠀⠀⣱⣤⣾⡿⣿⣍⠀⠀⢀⡈⢻⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⠶⠾⢿⡶⠶⠿⠿⠛⠛⠳⣤⣈⣿⣷⣀⢋⣾⡾⢟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⣀⣀⣈⣿⣿⣿⣿⣿⡿⣷⡿⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠟⠻⣿⡿⠻⢷⡹⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠑⢦⢀⢂⢀⢀⢀⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠈⠃⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠀⠊⠀⠘⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢀⣾⣿⣷⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⡠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠀⢀⡾⣟⣽⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡼⠡⠔⢁⠎⡀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣴⢶⣾⡟⠋
⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠠⠜⠀⠀⠜⠊⠰⠡⠠⠡⠃⠎⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠋⠄

⡾⡟⢹⣾⣿⢿⡿⠿⠿⠿⠛⠛⠃⠉⠉⠉⢠⣾⣽⣿⣿⡟⣡⣴⣾⣿⣿⢪⣴⣮⣿⡎⢪⣾⣿⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⢿⡟⢉⣿⣿⣿⣿⣿⣋⣷⣿⣿⣧⡿⡎⣿⡿⠿⣿⠿⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⡶⢿⠟⠋⠁⠸⣴⣿⡿⠛⠹⠛⠁⠛⠛⢻⣿⣿⣿⣠⠸⣿⣶⣦⣤⣄⠉⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂⡀⢀⣴⣿⠛⠁⢰⡇⠀⠀⠀⣿⣮⣙⢛⠟⣿⣤⣶⣯⠉⢻⣿⣷⣆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠛⣷⠀⢻⡄⠀⠀⢸⣿⠋⣠⠊⠀⢉⠀⠀⠀⢀⠈⠘⢉⣈⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⣤⣿⣷⡰⣿⡀⠒⢦⡎⣠⣿⠇⢾⠃⠐⢴⣿⣷⡄⠀⣸⣿⣷⣎⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣛⠾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟⠛⠟⠿⣿⣧⣸⣷⢿⣿⠇⢸⡀⠀⢘⣿⣿⡇⣾⡿⣿⢩⣍⡉⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿
⣽⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⣶⣦⣌⠙⠷⢠⡟⠀⠺⣿⠀⣸⠋⢛⠁⡈⠃⢸⣀⣈⡻⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠤⣤⡀⠙⢿⣷⣶⡿⣿⣿⣿⣿⣷⣄⡌⢠⣵⠈⣤⣁⣠⡾⠃⣿⠐⣿⣿⣿⡉⠓⠢⢼⣿⣿⣿⣿⣷⣿⣿
⣽⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣡⣄⡀⠈⢿⣿⡹⣿⣿⡿⢉⣿⣄⠈⣿⢃⡈⠛⠋⠀⣾⠁⠀⡽⠈⠙⢿⣷⠆⠜⢿⣿⣿⣿⣿⣿⣿
⣿⢿⣟⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠈⢿⣿⣦⣴⣶⣿⣿⣿⠛⣥⡜⣿⠆⠀⠀⠉⠀⠀⠃⣰⣦⠀⠙⠧⠀⢸⣝⣿⣿⣿⣿⣿
⣿⣻⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠙⢧⣠⣽⢻⡿⢛⠻⣿⣿⣿⠛⠋⠀⠀⠀⠀⡀⠀⠀⠈⣃⠈⠁⠀⠀⣰⣺⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣯⣟⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡄⠀⠙⠳⣾⣧⣸⣧⡘⣿⣿⡆⠀⠐⠀⠀⣰⠇⠀⠀⠀⠉⢻⡇⢀⢺⣿⢟⣿⣿⣿⣟⣿⣿
⣽⣟⣿⣿⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣶⣦⣀⠈⠛⢻⣷⣸⣟⢿⡄⣴⠀⢠⡟⠀⠀⠀⠀⠀⠈⢷⢘⢿⡯⠃⢸⣿⣿⣿⣷⣿
⣿⣿⣿⡿⣿⣟⣧⠀⠀⠀⠀⠀⠀⠐⠀⣤⣀⠀⠈⠈⠀⠀⠉⠐⢀⣿⡟⠉⣿⣿⡟⠀⣿⡇⠀⠀⠀⠀⢀⠀⠈⠀⠉⢀⣾⣷⣿⣿⣿⣽⣿
⣿⣿⣿⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⡈⠁⠲⣬⣀⣀⣀⣀⣾⡇⢀⣼⠟⠙⠁⠀⢻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⢻⣿⣿⣿⣿
⣏⣿⣿⣽⣿⣷⣾⡆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠉⢿⡟⣰⡿⠁⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡼⠃⠘⣿⣿⣿⣿
⢷⣿⡿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣦⣄⢀⡈⢳⣿⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⣤⣾⣿⣿⣿
⣿⣽⣻⣿⣏⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢿⣿⣿⣿⣿⣿⣦⡀⠀⣠⠆⠀⠁⠙⢐⡄⠀⡀⠒⠒⠲⠶⠶⠚⡩⠛⢻⣿⣿⣿
⢽⣿⣿⣷⣿⣿⣿⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠙⢻⣿⣿⠿⣆⡀⠀⠀⠘⣷⣜⠁⠀⠻⠷⣮⣤⠀⢰⣿⠟⣿⣿⣻
⡿⣾⣸⡷⣼⣍⣳⡳⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡀⢧⡈⠙⢄⣀⠀⠙⢻⡟⠛⡟⢷⣹⣿⣷⣿⠇⠀⣿⣿⣿
⣿⡿⣿⣿⣿⡿⣟⣿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠈⠀⠘⢢⠹⣦⡄⠈⠁⠀⠀⠀⠀⠀⠀⠙⠻⡏⢡⢄⠀⠹⣿⣿
⣻⣛⣿⣿⠛⢂⣙⣻⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣀⡉⠛⠲⠖⠒⠊⠀⠀⠀⣠⣶⠿⢶⢶⣶⣶⣿⣻
⣿⠷⢲⡟⢙⣛⣿⣭⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡷⠲⠺⠷⠤⠴⠶⠛⠛⠁⠀⠘⠿⠀⣷⢸⣿
⣷⡾⣽⣿⡼⣿⣿⣯⠛⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠘⠓⠀⠀⠈⣿
⢿⣶⡿⣿⣶⣲⣴⢟⣿⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⣿⣼⣛⡷⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠉⠁⠀⢘⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠆⠀⠀⠀⠀⠀⠘⣦⠀⣦⠄⠀⣻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠎⠀⠀⠀⠀⠀⠀⠖⠁⠀⠈⠀⠀⠘⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢠⡏⠀⠀⠀⠘⢦⡀⠀⠘⣗⡲⠄⠀⠀⠀⠰⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢹⡿⡍⡄⠀⠀⠀⠈⢳⣄⠀⢆⣀⣀⣤⣴⠼⠚⠁⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⠊⠉⢁⣹⣷⣶⣶⣖⢢⣤⣿⣚⠲⣏⡟⣉⡠⠤⠒⢉⡴⠐⡩⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢠⢫⡦⣀⣤⣩⣍⡁⢍⣽⡂⠅⠈⢃⠀⠴⠋⠀⠀⠀⠘⠁⡠⠔⠋⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠶⠸⣷⣍⠀⠀⠀⢆⠀⠀⠀⠀⠀⡼⢁⣤⣖⣢⣤⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⡀⠀⢒⡒⠀⠨⣗⠵⢻⠳⣄⠀⠈⠑⢤⡀⠀⡇⡇⠁⠈⠡⡀⠀⡠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢊⠸⠁⠈⢧⡀⠀⠀⠈⠑⠦⣉⣀⡀⠀⠀⠁⠘⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠑⠀⠀⠀⠀⠀⠈⠹⣤⡁⠀⠀⠙⢦⡀⠀⠀⠀⣾⠘⢝⠂⠀⢠⠞⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠐⠋⠀⢀⣴⡥⢜⡝⠱⣀⠀⠀⠀⠙⠷⣤⣀⡟⠀⠀⠁⠀⢸⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⣶⣶⣾⣻⢯⣠⢾⠀⡴⡩⢋⠓⠶⣄⠀⠈⣿⣇⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠉⣙⣹⡷⢡⡿⢠⢁⠎⡸⡅⢡⢂⠔⡈⡻⣦⠀⠉⠙⠳⠶⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⠌⠀⠌⡼⡕⡴⠫⢠⢁⠊⠌⠔⣱⣡⢋⢌⠜⡔⠘⣳⣤⠴⠒⠀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⣰⢫⢌⢂⢂⢎⢂⠄⢜⠌⡔⡀⡠⢁⢂⢎⠌⡴⢡⢃⢂⠎⠜⠄⠜⡴⣹⣧⡀⠼⠒⣧
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢀⠏⢀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⢳⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣾⣾⣿⣿⣿⣿⣷⡀⢠⣿⣿⣿⣿⣿⣷⣧⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣿⣿⣿⣿⣿⣿⡿⣥⡜⣿⣿⣿⣿⣿⣿⣿⣼⡇
⠀⠀⠀⠀⢠⠤⠤⣄⡀⠀⠀⠻⣼⡿⢿⠿⠿⠋⣸⣿⣿⡌⠛⠿⠿⠿⢟⣽⠟⠀⠀⣠⠴⠒⠒⢦
⠀⠀⢠⡖⠛⠀⠀⠀⠙⢦⣀⢰⣿⣿⣷⡆⠀⠀⠻⠟⠿⠃⠀⠀⣴⣾⣿⣿⣀⣤⠞⠁⠀⠀⠐⠻⡄
⠀⠀⠈⠓⢤⣄⣀⣀⣀⠀⠈⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⣀⣤⣤⣤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣾⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠛⠒⠦⣭⣍⣉⣉⣉⣉⣩⣥⠴⠒⠛⠦⢤⣀
⠀⠀⠀⢀⡴⠒⠛⠛⠛⠉⠀⢀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⠀⠈⠙⠲⠖⠒⠶⢤⡀
⠀⠀⠀⠹⣤⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢆⠀⠀⠀⢀⣀⡽
⠀⠀⠀⠘⠧⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠴⠃







⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠲⠲⠔⠆⠢⠲⠐⠖
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣄⡀⠤⣀⣀⠀⠀⡠
⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⡿⠿⠿⠿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣽⠛⠂
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠛⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠘⡟⠋⠁⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣀⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣤⣧⣄⣖⣢⣤⣆⣄⡄
⠀⠀⠀⠀⠀⢿⣿⠿⠛⠻⠿⠛⠻⠟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠛⠃⠀⠀⠀⠀⢸⣿⡿⠟⠻⢿⣿⠾⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⡦

⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⣤⣷⣤⣭⣷⣶⢤⣬⡖
⠀⠀⠀⠀⠀⢿⡿⠋⠙⠛⠛⠛⠛⠿⠀⠀⠀⠀⢿⣿⣿⡿⠿⢿⣿⠟⠛⠃⠀⠀⠀⠀⢸⠿⠋⠁⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠛⠿⠿⠿⣿⣿⣿⣿⠿⠿⠿⠟⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣆⠀⠀⠀⠀⣠⣴⡿⣿⣦⣄⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⢼⣿⣿⣿⣷⣶⣶⣾⣿⡿⠁⠈⢿⣿⣷⣶⣶⣾⣿⣿⣿⡧⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⡅⢀⡄⢨⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⣿⣿⣦⡈⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠉⢁⣴⣿⣿⣅⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠿⠿⠛⠂⠀⠀⣿⡟⣿⠛⣿⣿⠻⣿⢻⣿⠀⠀⠐⠛⠿⠿⢿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄⡀⠀⠀⠀⠈⠀⠛⠀⠛⠛⠀⠛⠀⠁⠀⠀⠀⢀⣠⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡀⠀⠈⠙⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠁⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⠀⠀⢠⣤⡈⠙⠆⠀⠀⠀⠀⠀⠀⠀⢀⣰⠋⠁⣠⣤⣶⡆⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠙⣿⡇⠀⣸⣿⣧⠀⢸⣿⣧⠀⣿⡇⢀⣾⡿⢿⣷⠀⣿⡟⢉⡁⢸⣿⣀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⢰⣿⣻⣿⡄⢸⣿⢿⣷⣿⡇⢼⣿⠠⣶⣶⠀⣿⡿⠿⠇⢸⣿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠁⢾⡿⠛⣿⣧⠸⣿⠀⢻⡿⠇⠘⢿⣷⡿⣿⠀⣿⣷⣶⡆⢸⣿⠀⠻⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠬⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠁⠀⠀⠀⠙⡆
⠀⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠉⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠼⠦
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⣀⡠⣀⠀⣠⠞⠉
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⡀⠀⠀⠀⢀⡴⠛⠃⠀⡠⠚⠉⠀⠀⠀⠀⢰⡏
⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣿⣶⠖⠛⠉⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠘⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠈⠻⣦⡀⠀⠀⢀⣀⣀⠀⣠⡴⠶⢄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠼⠷⣄⠀⠀⠀⢀⣠⠜⠁
⠀⠀⠀⠀⠀⠈⣻⢶⣀⠉⢫⣿⠛⠉⠀⠀⠀⠈⠢⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠽⠶⠖⠛⠉
⠀⠀⠀⠀⠀⡔⠁⠀⠻⣿⣚⣧⡀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠱⣀⠀⠀⢩⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⢸⣿⠇⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠑⢦⣆⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣙⢿⡏⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⠏⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠟⠘⢠⣶⣜⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠛⢖⣥⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣩⡟⠁⠀⠀⠀⠙⢿⣿⣯⣻⣿⣿⣿⣷⡀⠀⢀⡠⢊⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠰⣷⣀⣄⠈⠛⢿⣿⣿⣿⣿⣿⣷⡉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣶⣾⣫⣻⣿⣿⣿⣿⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿⣿⣿⡘⢿⣿⣿⣿⣿⡿⢋⣵⡿⠋
⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣯⣟⠿⣿⣿⡿⢋⣴⡿⠋
⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣾⣯⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡏⠀⢀⣤⣤⣄⠀⢀⣤⣤⣤⠀⠀⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣏⢈⣿⣿⠀⠸⢿⣿⡿⠀⠘⣿⣿⡿⠁⢠⣿⣿⣉⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠚⠂⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⢿⣿⣿⠏⠀⠹⣿⣦⣴⣤⣴⣤⣼⠏⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣏⠀⢀⣀⡀⠉⠛⠛⠛⠉⠀⣀⡀⠀⣸⣿⣿⣹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠙⠛⠉⠀⣀⣤⣀⠀⠉⠛⠋⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡄⠀⢰⣶⣿⣿⣿⣿⣿⣶⡆⠀⢀⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⠂⠀⠘⠇⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣯⣨⣿⣿⣅⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡀⠀⠉⠀⠀⠀⠀⠀⣰⠊⠁⠈⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣾⣿⣿⣿⣿⡿⢫⣾⣿⣷⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡠⠤⣀⡼⠉⢑⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⣛⡻⠟⢱⣷⢹⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⢣⣀⡽⢸⡶⠊⠀⣀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣧⡻⠿⣸⡎⣩⣾⡿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿
⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⡀⠈⢳⠘⡇⠀⢀⣿⣿⣷⣶⠂⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣷⢹⡇⣿⣿⠇⠀⠀⠉⢩⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠃⠀⠀⣿⣿⣿⣿⠀⠘⣳⡇⢠⣿⣿⣿⣿⣿⠀⠈⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣴⣿⣿⡇⠀⠀⠀⢸⣿⡧⠃⣿⠏⠀⠀⠀⠀⢸⣿⣦⣾⣿⣿⣿⣿
⠀⠀⠀⢀⣤⠀⠀⢸⢿⣽⣿⡆⠀⣿⡇⢸⣿⣿⣿⣿⡟⠀⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢻⣿⣿⠀⠐⠀⠈⣿⡇⠀⣿⠀⠀⠀⠀⠀⣼⡟⢻⣿⣿⣿⣿⣿
⠀⠀⠀⣀⣥⠴⠆⠘⣮⡛⣿⣷⠀⣿⡇⢸⣿⣿⣿⡟⠐⠦⢤⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⢻⡟⢛⠋⠛⠓⠀⠻⡗⢸⣿⢧⠀⢀⡆⣾⣿⣿⣿⢿⣿⣿⣿⣿
⠀⢀⣀⣿⣆⠀⠀⠀⢸⡿⣿⣿⡄⣿⠇⣾⣿⣿⣿⠀⠀⠀⢀⣿⣅⡀⠀⠀⠀⣿⣿⣿⡿⠗⠀⢻⣿⣿⣛⠛⡛⠂⠘⡟⠘⣿⠂⠀⡰⣼⣿⣿⣿⣿⢸⣿⣿⣿⣿
⠒⠯⣀⠘⣿⣠⡤⠄⠘⡿⣎⣿⣇⢹⠀⣿⣿⣿⠃⠤⠤⣤⣻⡏⢀⡨⠗⠀⠀⣿⣿⣅⡺⢿⡆⠘⠛⣛⣿⡏⠙⠃⠀⣿⢸⡇⠁⠀⢁⣿⣛⡛⠛⠁⣿⠿⣋⣽⣿
⠀⠀⡎⣀⡽⠟⠂⠀⠀⠓⠛⠉⠁⠈⠀⠙⠛⠉⠀⠀⠀⠚⠿⣄⠸⡀⠀⠀⠀⣿⣿⣿⡿⣼⠟⣀⣼⣿⣿⣧⣤⣴⣶⣿⣾⣷⣤⣴⣿⣿⣿⣯⣄⡘⢿⣼⣿⣿⣿
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣶⡾⣡⣶⣶⣦⣍⡙⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣇⠀⠀⠀⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⣽⡿⣿⣿⢷⠘⠻⢿⣧⠻⠇⠾⣛⣿⢿⡋⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣀⡠⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠙⣟⠉⡟⠁⠀⠀⠀⢹⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⠀⠀⠀⢠⠏⠀⠹⣄⠀⠀⢀⡼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿⣎⠙⠂⠈⠁⠀⠀⠀⠈⠁⠉⠉⣼⣿⡇⠀⠀⠀⠀⢀⣤⣄⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢳⡀⠀⠀⢀⣿⣿⣄⢀⣀⣴⣾⣶⣷⣦⣄⣀⣼⣿⣿⡷⠁⠀⠀⢠⣿⡟⠀⠈⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢷⡀⠀⠀⣿⣿⣿⣿⡿⠛⠛⠉⠛⠛⣿⣿⣿⣿⣿⠀⠀⠀⢠⣿⠏⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⠶⢦⣬⣿⡀⠀⢸⣿⣿⣿⣿⣆⣵⣶⣧⣴⣿⣿⣿⣿⣿⡄⠀⣰⣿⠏⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠈⠉⢻⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⠋⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠓⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠇⠈⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠉⠉⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⢀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠒⠲⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡟⠁⠀⠀⠀⠳⠀⠀⡠⠒⠉⠀⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⡐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⡀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠒⣴⣾⣿⡷⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⣀⣀⠤⠒⠁
⠀⠀⠀⠀⢠⡈⠉⠁⠀⠠⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠈⣿⡠⢀⣾⣡⠆
⠀⠀⠀⠀⠈⢧⣀⠀⠀⣀⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠓⠛⠇⠚⣁⣴⠀⠀⣤
⠀⠀⠀⠀⠀⠀⣿⢁⠘⣿⡿⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⡼⠁⣸⠆
⠀⠀⠀⠀⠀⠺⠘⢿⡀⠋⢀⣼⣿⡀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠃⠺⣧⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠁⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⠀⠀⠀⠰⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣵⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣶⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡿⢛⣤⣍⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⣻⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⣼⣿⣿⣿⡄⡤⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠛⠁⣀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⣶⣿⣿⠀⠀⠀⠀⠀⢤⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠻⠟⢻⠄⠀⠀⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡸⠛⠚⠋⣡⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣄⡠⠀⣽⣄⣀⣾⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣁⠀⠀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢍⠉⢉⣹⠟⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣽⣿⣾⣶⡶⢮⣭⣛⠲⣌⡳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣭⣝⣛⡻⢿⣷⣯⡛⢦⡙⢎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠀⠉⠛⠿⣷⣮⣻⣿⣶⡝⡈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⢸⣦⡙⣿⣿⣷⢿⡜⣄⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡘⣿⣿⢸⣿⢸⣿⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⣿⡏⣾⣿⢸⣿⢃⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣼⣿⣯⣿⢏⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣷⡟⢡⠞⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢏⣾⣿⠇⣴⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡏⣿⣼⣿⠁⠐⣂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠸⣯⡙⣿⣿⣄⢠⣿⣟⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣜⠳⡙⠿⣦⣙⢿⡷⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠡⣿⣿⣦⣈⣾⡝⢆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣭⣾⡿⠃⠙⣫⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠻⠉⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣏⠤⣔⠴⣶⢴⣶⣜⠵⡶⠀⠠⢶⠴⡶⠴⡞⠀⢠⡧⢶⣦⡦⠾⡧⡮⡶⠦⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠒⠒⠒⠒⠒⠶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⣇⠀⢀⠀⢀⠔⣻⣶⣤⣴⣤⡀⠒⠲⠂⢀⠀⠀⢀⠀⠲⠒⠂⣠⣴⣤⣴⣾⠓⠄⠀⢀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡄⡎⡤⠈⢰⣿⣿⣿⣿⣿⣿⣷⣶⣶⡟⠀⠀⠘⣷⣶⣶⣿⣿⣿⣿⣿⣿⣷⠈⢠⡘⡄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡹⣅⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡠⢦⡀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⢀⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠌⠆⠐⣄⠙⢿⠿⢿⡿⢟⡟⢁⡞⣰⣷⠘⡄⠙⡛⠿⣿⠿⠿⠟⢀⠄⢀⠏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠁⠀⠀⢀⡠⠏⠀⠊⣴⣿⣿⣷⡌⠂⠈⠦⣀⠀⠀⠀⠁⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢤⣀⣤⡠⠀⠀⠀⠀⠀⠀⠀⠛⠿⡿⠟⠃⠀⠀⠀⡀⠀⠀⠤⣠⣄⣠⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⢉⣠⣴⣿⡄⠉⠀⠀⠀⠈⠙⠞⠉⠀⠀⠀⠈⠁⣾⣷⣄⡈⠉⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢿⣿⣿⡟⠁⢀⠀⢀⠀⣀⣀⣀⣄⡀⢀⠀⠀⠀⠻⣿⣿⣿⢫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡌⣿⠋⢹⠏⡟⠉⢻⠁⠈⡏⠀⢻⠋⢹⡏⢿⠉⢿⠇⡎⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⢸⡇⢾⡤⣷⡤⣼⣀⣠⣇⣀⣼⡤⣼⡧⣼⠂⣿⠀⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣄⠈⢧⠸⡆⣯⠀⣸⠁⠈⡏⠀⢹⠀⢸⣇⡾⣠⠏⢀⠄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⡦⠀⠈⠉⠳⠉⠢⡰⠳⡠⠊⠳⠋⠉⠀⠡⣚⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠙⠦⠤⠄⠢⠤⠤⠚⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠴⠤⠤⠤⠤⠤⠶⠚⠉

⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠠⠀⠠⠈⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⣁⣀⣀⣂⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠠⡤⠤⡤⠀⠀⠀⠀⢄⠀⢀⠄⠀⠀⠀⠠⡤⡤⡤⠀⠀⠀⡘⠾⠛⠛⢛⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⣤⡤
⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠈⠀⠀⠀⠈

⠀⣞⢩⣹⢏⣍⢹⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⠐
⠀⠈⠀⠀⠀⠄⠈⠀⠀⠀⠐⠒⠲⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠤⠴⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠛⠇
⠀⠀⢀⡀⠀⡀⠀⠀⠒⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠐⠐⠀⠀⠀⣿⠀⠉⠀⠈⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⠂⠀⠖⠀⠀⠀⠀⡘⠺⠘⢃
⠀⠀⠀⠲⠖⠀⠀⠀⠈⠀⠐⠀⠐⠁⠉⠀⠀⠀⠘⠃⠨⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠱⠀⠘⠂⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠘⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⡀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠈⢁⠀⠁⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠈⣀⣈⠄⠀⠀⠀⠙⢋⣈⢛⠁⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⣢⠀⠀⠀⣔⠒⠒⢒⠄⠀⠀⠄⡀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⢠⠀⠀⠀⢰⣶⣤⢤⣴⣶⠀⠀⠈⢁⠀⢀⠉⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠈⢁⡀⠁⠀⠀⠀⠀⠛⢀⠘⠁⠀⠀⠀⠉⠁⡀⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠉⠀⠀⠀⠀⢬⣀⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢠⡀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⣌⠀⠀⢩⡜⠁⠀⠀⠠⠀⠀⠀⣨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⡑⠀⠀⠀⠀⢉⡉⠛⢉⠉⠀⠀⠊⠢⠀⠠⠎
⠀⠀⢻⣋⠉⡭⠃⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⣬⣴⣶⣬⡤⠀⠀⠢⡲⠀⠸⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈