Hello kitty heart Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠲⢤⣀
⠀⠀⢀⣠⠤⠤⣰⠁⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠙⢦⡠⠖⠛⠛⠛⠻⢤⡀⠀⠀⠀⠈⢳⡀
⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣇⣀⣀⣀⣀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠉⠀⠀⠀⠈⠓⢆
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠈⠑⠢⠤⠤⠤⠤⡤⡏⠀⣠⠤⣀⠀⠀⢀⣘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⡴⠛⠒⠶⣤⣤⣼⣧⡞⠁⣀⣈⣶⡛⠉⠉⣯⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡁⠀⢧⡏⠀⠉⣯⡉⠉⡆⠈⠑⢦⣀⣀⣤⠤⠖⠋⠁⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋
⠀⠀⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠓⠦⢾⠓⠃⣰⠃⠀⠀⢫⡽⠀⠀⠀⠀⠀⠀⠀⢸⣭⠏⠙⠓⠒⠒⠚⠛⠉⠁
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⣇⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠤⠤⣷⠄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠒⣿⣻⠋⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⢠⢽⡆⠀⠀⠙⠁⠀⠀⢠⠶⡄⠀⠀⠘⠃⠀⠀⢸⡿⠓⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⣠⠿⢾⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡼⠛⠒⠄⠀⠀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠙⢓⡶⠶⠶⣤⣤⠤⡶⢒⡾⠋⠀⢀⣠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣴⠋⠉⡿⠛⢳⡾⠶⢾⡉⠁
⠀⠀⠀⠀⢀⣠⣤⡾⠁⠀⠘⢦⣸⠁⠀⣸⡇⠀⠀⢻
⠀⠀⠀⠀⢾⡀⠀⢷⡀⠀⠀⠀⠈⠉⠉⠉⡇⠀⣀⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⢀⣠⣤⣴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⣄⣀⢀⣀⣀⣤⣤⣴⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠉⠙⠛⠋⠉⠉⢠⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣶⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠻⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠟⠋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣷⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠛⠋⠉⣿⡉⠁⠀⠀⠐⣿⡇⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠈⣿⠋⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠛⠛⠻⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠈⣙⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⠛⠙⢷⡄⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⠋⢁⣼⣿⣿⣿⣧⣀⠀⠀⠀⣀⣾⣿⣿⣿⣧⠀⠀⠻⣆⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠉⠻⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣤⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠋⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠶⠶⠟⠋⠻⠶⠶⠶⠶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⠻⣦⡄⠀⠀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣀⠈⢻⣶⠛⠋⠉⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠛⠛⠻⢶⣤⣤⣤⣶⠶⣾⡇⠀⢰⡟⢉⣿⠞⠛⠻⣶⣤⡶⠾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢿⡇⠀⠈⠛⢾⡇⠀⠀⠀⢨⣿⣤⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠶⠟⢷⣄⣀⣤⡾⣧⡼⠇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣦⣀⣀⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠶⠾⣿⡿⠿⠿⠷⠆
⠀⠀⠀⠀⣤⣶⣾⣿⣿⠛⠂⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⣠⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣽⣿⣤⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⢠⣟⠛⢻⡆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⣽⡿⠙⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠛⢻⣿⣦⣦⣤⣤⣤⣄⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⣠⡶⠟⠛⠻⢶⣼⣿⡿⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⢠⣶⠟⣿⡿⠉⠀⠀⠀⢹⣿⠛⠛⠻⢶⣤⡀⢀⣤⡶⠟⠛⣿⡁⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣠⣶⡾⠃⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠈⠛⠿⣦⣀⣀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣛⣛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣿⣦⣀⡀
⠀⠀⠀⠀⣠⡾⠟⠋⠉⠉⠉⠙⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠉⠉⠉⠛⠻⣦⣄
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⢿⣿⣦⣤⣦⠀⠀⠀⠀⢠⣶⠛⣷⡘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣴⠟⢷⣄⠀⠀⠀⠀⢠⣦⣤⣴⡿⠁
⠀⠀⠀⠀⠉⠉⣿⣿⣤⣴⣶⡆⣸⣿⡻⣛⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣙⠿⣻⣿⠀⣶⣦⣤⣼⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⠿⢿⠿⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣄⡀⣀⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠟⠋⠁



⣀⣀⣀⣀⣀⣀⡀
⡙⢿⣉⣉⣉⣿⣉⣀
⣿⣾⣿⣿⣿⣿⣿⣿⣶⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀⠀⠀⠀⡰⠁⢀⡈⢶⠚⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠙⠒⠋⠉⡇⠀⠫⡜⠉⠉⢳⠒⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠑⠤⣤⠾⠝⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢻⠃⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠈⢉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢯⣀⠀⠀⠁⠀⠀⠀⢏⡱⠀⠀⠀⠉⠀⠀⣀⡝⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣍⡳⠴⠖⠒⠦⢤⡀⠀⠀⢀⡠⠴⠖⠲⠤⣎⣉⡓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢯⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⡷⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣈⣙⣢⣤⣀⡀⠀⠀⢀⣀⣤⣖⣋⣁⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠲⢤⡀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠛⠉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠿⣶⣦⣤⣠⣾⠟⠉⠙⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠈⢹⡿⠁⣀⡀⠀⠀⠙⣿⡿⠿⠿⠿⢶⣶⣾⠿⠛⠉⠁⠀⠈⢻⣧
⠀⠀⠀⠀⠀⢐⣿⣧⣤⣤⣤⣴⡶⠿⠿⣿⡟⢻⡆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⣾⠟⠉⠀⢉⣿⡏⠀⠀⠀⠘⣿⡿⠃⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠘⣿⡄⠀⠀⣿⣻⣧⡀⠀⠀⣰⡿⠷⢶⣶⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠀⠀⠀⠀⠀⢹⣷⣄⠀⠉⠛⠋⣻⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⣿⡏⠻⢷⣶⣶⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠛⠛⠛⢻⣿⠛⠷⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠶⢶⣿⣿⠶⠶⣦⠄
⠀⠀⠀⢀⣸⣿⣄⡀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠛⠛⠛⢿⣯⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠⣾⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⡿⠶⡶
⠀⠀⠀⠀⢀⣨⣿⣷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣾⡏
⠀⠀⠀⠺⠟⠋⠉⠙⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣴⠾⠛⠻⢶⣿⣛⣋⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⠋⣉⣩⣭⣭⣙⣿⠃⠀⠀⠀⠀⠹⠟⠉⠉⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⣼⠟⠉⠉⠉⣹⣿⠷⠶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠘⣿⡀⠀⠀⣰⣿⠃⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⡿⠁⠀⠀⠹⣷⣄⣰⣿⠃⠀⠀⠀⢈⣿⠀⠀⠀⢀⣠⣴⢿⣿
⠀⠀⠀⠀⠀⠀⠀⣴⣾⠟⠛⠛⣿⡇⠀⠀⠀⠀⠈⠛⠿⣷⣤⣄⣤⣴⣿⣇⣀⣤⣶⠿⠋⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠰⣿⡁⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠙⣿⡏⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣤⣼⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣦⡀⠀⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⠷⣶⣤⣤⣤⣶⣴⣾⠋⠀⠀⠈⢻⡟⠋⠉⠀⠈⢿⡄
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡏⠀⠀⣾⣿⡿⠛⠷⣦⣴⠶⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣹⣿⠀⠀⠀⣸⣦⡀⠀⢹⡆⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠳⢶⣾⠛⠛⠁⣠⣿⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠞⠋⢻⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣷⠤⡶⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⣿⠛⠃⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⢀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⣶⠂⠀⠀⠈⠉⠀⠀⠀⠀⣴⣶⣶⠀⠀⠀⠀⠉⠁⠀⠀⠀⢈⣿⢋⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣶⣖⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠈⠛⠷⢶⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⠟⠫⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⠛⠛⠿⣶⣶⡾⠛⠉⠁⠈⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠛⠻⢿⣿⣶⣦⣤⣶⣶⣾⣿⡿⠿⢿⣿⡿⠁⠀⠀⣀⡀⠈⣿⣧⣀⡀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⣼⡟⠀⠀⢠⣿⠟⣿⠟⠉⠉⠙⠻⣷⡾⠿⠿⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃⠀⠀⠈⢿⣶⡏⠀⠀⠀⠀⠀⠸⣧⣄⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣀⠀⠀⣈⣿⣄⠀⠀⠀⢀⣼⣉⣿⡇⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠙⠿⠿⠾⡟⠛⠛⠋⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣤⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⠉⠁⢹⣿⡇
⠀⠀⠀⠀⠀⣠⣶⣶⡿⢿⣿⠿⠷⠀⠀⠀⠀⢠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⣤⣾⣿⣷⣶⣦⠄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⢀⡴⠶⣦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠘⢿⣦⣤⣤⠀⠀⢀⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣄⠀⠀⠀⠀⢀⣴⣿⠟⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠿⣧⣤⣾⣿⠟⠉⠉⠉⠉⠻⣦⡀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠈⢙⣿⣤⣴⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠋⠀⠀⠀⢀⣿⡿⠟⠛⠻⣦⡀⠀⠀⠈⠻⣦⣴⠟⠁⠀⠀⠀⠀⢀⣾⠟⠉⠉⠙⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢈⡧⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣀⣠⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠿⣷⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠟⠋⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⡀⠀⠀⣠⡾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⣠⡿⠋⠀⠉⠻⣷⡾⠋⠀⠈⠻⣷⣄⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠿⢿⣦⡀⣀⣴⣾⡿⠟⠛⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣼⣿⠋⠀⠀⠀⠀⠙⣿⣿⠋⠁⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠛⠛⠛⠻⠿⣿⣿⣶⣿⣿⣿⣿⠿⠿⠿⠿⢿⣿⠃⠀⠀⢀⣤⣶⣦⣼⣿⣧⣀⣀⠀⠀⠀⢀⣀⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⢠⣿⠋⠉⣽⡿⠛⠛⠛⠿⣿⣾⣿⠟⠋⠉⠉⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢿⣦⣸⣿⠁⠀⠀⠀⠀⢸⣿⡟⢿⣦⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⢉⣻⣿⣆⠀⠀⠀⠀⣼⣿⠀⠀⣿⡇⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠛⠿⣿⣶⣶⣿⣿⡷⣶⠾⠟⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣄⠀⠀⣀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣨⣿⣿⣿⣿⣿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠿⠟⠛⠛⠛⠉⠉⠛⠛⣿⣿⣿⣷⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢻⣿⡿⠟⠛⠛⠉⠉⠉⠛⠛⠛⠻⠿⢷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠛⠉⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣟⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⢻⣿⣿⠉⠉⠉⠉⢿⣿⡇⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣀⣀
⠀⠀⠀⢠⣶⣶⠿⢿⣿⣿⡿⠛⠀⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠹⢿⡿⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⢀⣾⡿⠀⣴⠶⠒⠶⡄⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠟⠀⠘⣧⣀⣀⣠⠟⠀⠀⠙⠻⢷⣦⣤⣄⣀⣀⣠⣤⣴⣾⠿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠾⠟⠛⠉⢻⣿⣿⣦⣄⣀⣀⣀⣀⣠⣤⣶⡿⠟⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⠉⠉⠉⠀⣠⣾⡿⠋⠀⠈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⡿⠿⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠛⠿⢿⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠛⠁⢈⣿⠟⠁⠀⠀⠀⠀⠹⣿⣶⣿⣶⣶⣤⣄⣀⣠⣤⣶⣶⣶⣶⣶⣾⡏⠀⠀⠀⠀⠀⠉⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢠⣿⣿⠛⢿⣿⠛⣿⡿⢿⣿⠿⠿⠿⠿⠟⢸⡿⠿⣿⣿⣧⡀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣤⣤⣴⣿⣿⡇⢀⡈⠿⠐⠃⠘⣃⡁⢀⣶⡄⣾⡇⢸⣇⠰⣾⣿⡿⠿⠶⠶⠾⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠘⣿⣇⣼⣿⣤⣸⣧⣬⣴⣿⣾⣿⣷⣬⣥⣤⣭⣤⣾⠟⠀⠀⠀⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⢿⣿⣿⣿⣿⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠉⠉⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⣀⣀⣀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣀⡀⠀⢀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⢶⣤⡀⢀⣀⣤⣴⡶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣤⣀⡀⠀⠀⠀⢀⣀⣀⣼⡟⠁⠀⠀⠈⢿⡿⠛⠉⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠉⠉⠛⠻⠿⠿⠛⠛⠋⢹⡟⠀⠀⢀⡴⢦⣤⡷⢤⣄⣀⣀⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢧⡾⠁⠀⠀⠈⣿⡁⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣀⣤⡴⢷⡀⠀⠀⣠⣟⣹⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣏⠉⠁⠀⣠⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠛⠁⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣇⣀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⣿⠉⠁⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠙⣻⡿⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣦⣤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣉⣹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠈⠉⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⢉⣙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡾⠟⠉⢰⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣶⣶⡶⠶⠶⣿⣿⣿⡿⣿⠋⢁⣠⠴⠚⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣦⠘⠷⣤⣤⣾⣧⡀⠀⠀⠀⣠⣾⠃⠀⣿⠁⠀⠀⠀⠀⠀⠉⠀⠸⣿⣉⣀⣀⣀⣀⣻⣧⣀⡀⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣾⡏⠀⠈⠻⣿⠿⠛⠋⠁⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⣿⠁⠀⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣷⣤⣤⣴⠏⣀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⢀⣾⠃⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣭⣉⠀⢸⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⣶⠶⠛⢳⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟⢷⣄⠛⠉⠀⠀⠀⣾⣷⣶⡄⠀⠀⠀⠀⠀⢸⡆⠀⢸⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⠙⠷⣦⣤⣤⣀⠘⠟⠋⠀⠀⠰⣿⣶⣶⠈⣷⠀⠸⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⢻⣄⠀⠀⠀⠀⠈⠙⠷⣤⣄⣀⣀⡀⠛⠛⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⡿⠶⣦⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣿⣶⣄⣀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠚⠛⠁

⢠⣿⠿⠿⠿⠶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠾⠿⠿⢿⣧
⣾⡇⠀⠀⠀⠀⠀⠀⠉⣙⣻⣷⣦⣤⣤⣄⣀⡀⠀⠀⠀⢀⣠⣤⣤⣄⣤⣤⣤⠶⠛⠻⢯⡅⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⢀⣤⣶⠿⠛⠉⠉⠉⠉⠉⠉⠛⠻⢷⣦⣴⠟⠁⠀⣠⡾⠛⠉⠛⢶⣤⡀⠀⢻⣦⣄⠀⠀⠀⠀⣿
⣿⡇⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢎⣿⠀⠀⠀⢀⣰⠷⠃⠀⣸⠏⠻⣷⣄⠀⠀⣿
⢸⣧⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⢈⣻⠗⠚⠿⣥⣤⣤⠾⠋⠀⠀⠈⢻⣦⢠⣿
⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠉⢻⣶⡄⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣼⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢹⣇⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠘⣿⡀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠋⠀⠀⠀⡴⠛⠉⠛⢲⡄⠀⠀⠘⠻⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠹⣷⡀⣠⠴⠶⠶⠛⠛⠀⠀⠀⠀⠀⠀⠀⠳⢤⣤⡤⠶⠃⠀⠀⠀⠀⠀⠀⠘⠛⠳⠶⠶⢤⡀⣰⡿⠁
⠀⠀⠀⠙⣷⣄⠀⠀⣀⣤⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣤⣄⡀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠈⠻⣦⡚⠉⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠉⠙⣢⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄⡀⠀⠀⣀⣴⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠙⣆⣠⡤⠶⠚⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠓⠶⢤⣀⠀⠀⣀⣀⣀⣼⠁⠀⠀⠠⠄⢀⣸⣅⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⣟⠀⠀⠀⠀⡠⠁⠀⠀⠈⢳⠚⠛⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣧⠀⠀⠀⠀⠸⠓⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠓⠤⢤⡖⠁⠐⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣤⡤⠾⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⣀⣀⣸⣧⣀⡀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠘⠋⢻⠏⠉⠉⠙
⠀⠀⠀⠈⠉⠀⠈⢷⡀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⢤⣿⠤⣤⡀
⠀⠀⠀⡶⢶⢠⠞⣛⢿⡁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠜⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣼⡷⢤⡀⠒
⠀⣼⣓⣷⣿⠞⢀⣭⠞⢻⠟⠋⠉⠉⠳⣤⠤⠤⣄⣀⠀⠀⠀⠀⣀⣠⣤⡞⠉⠀⠀⠉⢳⡿⠛⠛⠛⠿⢤⣘⠂⣀
⠀⣀⣤⠟⠁⠀⠈⠀⠶⡏⠀⠀⠀⢀⣀⠇⠀⠀⠀⠈⠓⢦⡴⠋⠁⠀⠀⠣⣀⡀⠀⠀⠀⡇⢀⡀⠀⠀⠀⠘⠷⢦⡄
⠀⣏⠴⠀⠀⠠⠄⢘⢶⡿⠦⠤⠔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠶⠚⢷⢈⣠⡄⢀⠀⠀⠘⢇⡿
⠀⠉⠛⠛⢦⣤⣤⣤⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣉⠂⠉⣀⡴⠋⠉
⠀⠀⠀⢰⠋⠉⠉⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠉⣿⠛⠛⠻⡆
⠀⠀⠀⠈⠓⠶⠶⢿⣦⣘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣠⣿⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠳⠶⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡴⠋⠉⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠛⢦⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠀⠀⠀⠀⠀⠉⠳⣤⣀⣠⠖⠋⠀⠀⠀⠀⠀⠀⣀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠠⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣀⡀⠀⠀⠀⠀⢀⠄⠀⢠⣾⠟⠉⠉⠻⣷⣤⣴⡶⠿⠛⠛⢿⣆⠀⠀⠔
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠉⠙⠻⢷⣶⡶⠿⠿⠿⢿⡿⠁⠀⠀⣀⣀⠘⣿⡁⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢼⣿⣿⡿⠛⠛⢿⣦⣶⠶⠶⣿⣇⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢺⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠈⣻⣿⠀⠀⠀⠀⣿⣧⡀⠀⠈⢻⡆⢠⠄⠒⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⢻⣦⣀⣀⣴⣿⣿⠇⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣧⡀⠀⣀⣴⣿⣷⠀⠀⠀⠈⡠⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⢻⡇⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣶⣶⣶⠌⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠟⢿⣿⠛⠓⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡆⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣄⣤⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠛⣿⠿⢶⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣿⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣟⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⢟⠛⠻⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠈⠛⠿⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⠿⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠋⠙⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠀⠈⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠖⠒⠶⠶⣤⣀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠠⠂⠘⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠘⠤⠀⣀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⡀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠛⠛⣷⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠦⠶⠛⢄⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠛⠋⠉⠘⢦⡀⠀⣔⠈⠁⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠉⢷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⣀⠀⠀⠛⠀⣀⣽⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⣄⣀⠉⠀⠀⠈⠉⢁⡴⠒⣎⠈⠉⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢓⡶⠒⠒⠒⠻⣄⠀⠘⡷⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡀⠀⠀⠀⠀⠀⠉⠉⢧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⡴⣌⣃⣤⡐⢂⣤⣛⣹⠞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣀⣻⣀⠈⠁⢀⡏⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⣠⣤⣤⣄⣀⠀⣠⣶⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣦⣀⣴⣿⠿
⠀⢰⣿⣿⠿⠛⠻⠿⣷⣿⠟⠉⠉⠉⠉⠻⣧⣴⡾⠿⠿⣿⣷⣦⣀⣀⠀⢀⣾⣿⠿⠿⠿⣿⣷⣠⣶⣿⣿⣿⣶⣄⡀⠀⣰⡿⠋⠉⠈⠉⠛⣿⠋⠁
⣤⣿⡟⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠉⢻⣿⡿⢿⣾⠟⠀⠀⠀⠀⠀⠙⠿⠋⣡⣾⣿⠿⠟⠛⠳⠿
⣿⣛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠈⠀⣀⡀⠀⠀⠀⠀⠀⢰⣿⠟⠁
⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠿⢿⣦⣴⠿⠿⢿⣷⡀⣀⡀⠀⠸⣿⠀⠰⠀⠂⠰⠀⠀⠖⠀⠠⠀⠒⠀⠀⠂
⠀⠈⢻⡷⠿⠿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠷⠾⠃⠀⠀⠀⠙⠁⠀⠀⠀⠸⠟⠛⠻⢷⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣧⣤⣀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣄⣀⣀⣼⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠉⠛⠋⠁⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⢶⣤⣤⣴⠶⠿⠿⠿⠛⠛⠛⠛⠛⣿⠟⠁⠀⠈⠙⢿⣿⣤⣤⡶⠾⠟⠛⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⡀⠀⢿⡏⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣴⠛⢛⣷⠾⠿⢶⣤⣀⣀⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠛⠛⠛⢻⡇⠀⠀⠀⠹⢶⡟⠁⠀⠀⠀⠙⣿⠉⠉⠀⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣤⣾⣧⠀⠀⠀⠀⢠⡿⢻⡆⠀⠀⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣶⡶⠛⠷⠟⠃⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣀⣀⣀⣴⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠸⣧
⠀⠀⠀⠀⢀⣀⣀⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣿⣤⣤⣤⣤⡀
⠀⠀⠀⠛⠛⠛⠉⢻⡏⠙⠛⠀⠀⠀⢠⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠉⠁⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢘⣷⣤⣤⡀⠀⠀⠈⠿⠿⠏⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⠶⠶⠶⣶⣶⣾⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠲⠛⠛⠻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣀⣹⡗⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠙⣿⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢠⡶⠟⠋⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠟⠛⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠹⣧
⠀⣴⠾⠿⠷⣦⡀⠀⣀⣤⡶⠟⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⡇⠀⠀⠀⢸⣿⠟⠋⠁⠀⠀⠀⢀⣀⣻⣦⣤⣤⣤⣤⣤⣶⠛⠉⠉⠉⠙⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⢷⣄⣀⣠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⣰⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠟⠉⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⡟⠀⠀⠀⠀⠈⢿⡄⠀⢀⡾⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠉⠉⣿⠁⠀⠀⠀⠀⠀⠀⠻⢶⣿⠁⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣬⣿⣦⣄⣀⣀⣀⣀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣤⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⢰⣦⣶⣰⣶⣴⣆⣶⣰⡆⠰⣶⢢⣶⣦⠀⠀⠀⣀⣀⣠⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠦⢤⣀
⢸⣏⡟⢙⣿⡟⠻⢻⡟⢉⠙⢿⡾⣷⡋⡠⠐⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⣄⠀⠀⣠⠖⠉⠀⣀⣀⠤⢀⣀⠀⠈⠑⢄⢀⡠⠴⠒⠋⠉⠉⠉⠙⠲⣄
⢸⣿⣻⣿⣯⣿⢀⣿⣿⣿⣧⣿⣧⣾⡟⢦⡀⠀⡠⠚⠉⠉⠉⠙⠒⠢⣄⠈⢣⡜⠁⡠⠖⠉⠀⠀⠀⠀⠀⠉⢢⢀⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠸⠟⢻⠸⠟⠿⡾⠏⠻⠋⠿⠟⠿⠟⠀⠀⠙⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣠⢊⣀⣀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣹
⠀⠀⠈⠢⠤⢺⠁⠀⠀⢀⠄⠀⡰⠊⠉⢹⠀⠘⡆⠀⣠⢤⡀⠀⠀⠀⠀⠀⢈⡕⠉⠀⠀⠱⡤⠔⠒⢆⠀⢰⠃⠀⢠⠤⢄⠀⠀⠀⠀⠀⠀⠸⡏⠉⠁
⠀⠀⠀⠀⠀⠘⢦⣀⡠⢾⠀⠀⣇⠀⠀⠁⠀⣸⠃⡰⠁⠀⠈⠑⠤⠒⠒⠂⡎⠀⠀⠀⡰⠒⠓⢦⣀⠼⢄⣸⡀⠀⠁⠀⠈⡆⠀⠀⣷⠦⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⡸⢦⣄⡤⠞⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣉⣹⠀⠀⠀⠀⣣⠤⠀⠈⣷⢄⣀⣀⣴⡁⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢣⠀⠀⠘⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⡤⠴⠅⠀⠀⣠⠇⠀⡠⠃⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⠤⠚⢇⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣄⡀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆⠀⠀⠀⠄⡾⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⢖⠀⠀⠉⠀⠀⠀⠰⣉⡵⠀⠀⠀⠈⠛⠃⠀⠀⠀⡲⠓⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢶⡶⠤⠤⡤⢤⠤⠴⢲⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠈⠁⣠⢖⢁⢀⠀⢀⢀⢀⣀⢀⠀⠀⣀⢀⡀⠀⡀⢀⠀⡀⢀⢀⢀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠀⠸⠿⠹⠘⠸⠻⠿⠛⠿⠏⠊⠻⠻⠓⠃⠟⠿⠼⠛⠿⠏⠬⠽⠟⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠁⠀⠈⠛⣿⣆⠀⠀⠀⠀⣀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠈⢻⣷⣴⡾⠿⠛⠋⠉⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⢀⣤⣤⣤⡀⠈⣿⣏⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣶⡶⠿⠿⠿⢿⣿⡇⠀⠀⠀⣿⠁⠀⣨⣿⠟⠛⠛⠛⢷⣦⣠⣤⣤⣴⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠻⣦⣴⡿⠁⠀⠀⠀⠀⠀⢹⣿⠉⠁⠀⠈⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⢸⣿⠷⣦⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⣶⡶⠿⠿⢿⣦⡀⠀⠀⢀⣠⣿⡃⢀⣼⠇⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⡋⠉⠛⠛⠉⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣠⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁⠀⢻⣿⡄
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣷⣤⣤⣤⣤⣤
⠀⠀⠀⢀⣀⣨⣿⣧⣀⣀⡀⠀⠀⠀⠀⣰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⡏⠉⠉⠉⠉⠉
⠀⠾⠟⠛⠛⠉⢻⣿⡉⠉⠁⠀⠀⠀⠀⢹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠈⣿⣧⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⣿⡿⠛⠛⠛⠓
⠀⠀⠀⠀⣴⠾⠟⠛⢿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣶⠾⠿⠷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠉⠙⠛⢿⣦⣩⣿⡿⠛⠻⠷⢶⣦⣄
⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⣼⡟⠉⠀⠀⠀⠀⠀⠹⣿⠿⠿⠿⠿⢶⣦⣄⡀⠀⠀⠀⢀⣠⣴⣶⠿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠙⣿⡏
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣶⠿⠋⠁⠀⠀⠀⢻⣦⣀⣀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣀⠀⠀⢀⣠⡾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢷⣤⣀⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠛⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠙⣿⡄
⠀⠀⠀⢀⣠⣴⡶⠿⠟⠛⠛⠛⠿⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠾⠿⠿⠿⢶⣶⣤⣀
⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠘⣿⣄⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⢹⣧⠀⠀⡔⠀⠀⠀⠀⡀⢠⠀⢀⠀⡄⢀⣾⠏⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣿⠁
⠀⠀⠈⠙⠛⠛⢿⡏⠀⠀⠀⢀⠀⠀⢰⡿⢡⣄⣹⡷⢀⣿⠀⢠⣅⠀⠀⢧⡐⠃⢸⠀⠨⡇⠀⢸⣿⠀⣾⡇⣠⡙⣿⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠘⠿⣶⣤⣶⢿⣧⡀⢈⣿⣌⡙⢋⣡⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣈⠛⢋⣰⣿⠀⢀⣿⣷⣦⣤⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠛⠻⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⠟⠛⠿⠿⠋⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠀⠰⠄⠀⠀⠀⠀⢀⣤⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⠀⠀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣤⡀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠶⣦⣄⣤⣤⣤⣤⣾⠋⠀⠀⠈⢿⡟⠉⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⡏⠀⢠⡟⣻⠿⠛⢶⣤⡴⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣹⣏⠀⠀⠀⣿⣦⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠷⢶⡾⠳⠛⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠞⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡦⠤⠤
⠀⠀⠀⠀⠀⠀⠐⠖⠛⢻⡟⠓⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⠀⠀⠀⣀⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡶⠀⠀⠈⠋⠀⠀⠀⠀⢰⠖⢲⠀⠀⠀⠀⠉⠁⠀⠀⠀⢨⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣴⠆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣹⡿⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢋⡽⠛⠛⠛⠛⢦⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠞⠛⠉⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠀⠀⠀⠀⠀⢸⡟⠛⢿⣿⡛⢻⣿⡿⠛⠋⣿⠀⠀⠀⠀⠀⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠘⢷⣤⡴⠶⠶⠛⠁⠀⠀⠈⢿⡿⠁⠀⠀⠀⠙⠷⠶⠶⢦⠶⠟⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡆⢰⠐⠰⠐⠔⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠸⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠠⡄⠀⠀⡀⡂⠀⡀⡀⣀⢠⠀⠀⡀⡀⠀⡀⠂⡀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠑⠀⠄⠇⠄⠢⠀⠇⠸⠸⠀⠆⠅⠆⠢⠀⠀⠑⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⢀⡀⢄⠂⠇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⠳⣤⠔⠚⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠉⠲⣤⡤⠤⠶⡇⠀⢰⠒⡦⠓⠒⢄⣀⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣈⣹⡀⠀⠀⢸⢦⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢲⡉⠊⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⡗⠂⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠉⢹⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢷⡂⠀⠀⠈⠛⠀⠀⠀⢠⠖⢢⠀⠀⠀⠈⠛⠁⠀⠀⠐⡿⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠷⣞⢀⣀⣀⣀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣀⣀⠀⣰⠞⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢛⡶⠋⠉⠉⠈⠉⠉⠓⢦⣀⣀⡤⠖⠋⠉⠉⠉⠉⠓⢦⠶⢬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠈⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠬⠽⠶⠦⣤⣀⣀⣀⣀⡤⠴⠶⠯⠥⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠻⣷⣄⣀⣤⣶⣾⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠛⠻⠿⣷⣦⣤⣤⣤⣤⣶⣶⣶⣾⠏⠀⠀⠀⠀⠈⢿⣿⠋⠁⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠁⠀⣽⡟⠀⠀⢀⣴⠿⣷⣼⣿⣶⣤⣀⣀⣀⣀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⣿⣴⡿⠉⠀⠀⠙⣿⡟⠛⠛⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣀⣨⣿⡇⠀⠀⠀⠀⣾⡿⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠿⣦⣤⣤⣾⢿⣶⠟⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⣀⠀⢀⣠⣾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠋⠁⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣷⣤⣤⣤⡄
⠀⠀⠀⠀⣀⣠⣤⣴⣿⣧⣤⡄⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠉⠉⣽⡏⠁⠀⠉
⠀⠀⠀⠀⠉⠉⠉⠈⢿⣇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⢠⣤⣿⣧⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣶⠶⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣶⠟⢿⣦⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣸⡿⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣾⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣧⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣾⣟⡁⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠰⠟⢋⣴⡿⠛⠉⠉⠉⠙⠻⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⠟⠋⠉⠉⠉⠛⠿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⠛⣿⡏⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠟⢻⣧⠀⠀⠀⢀⣀⢀⣠⣾⠇⠀⠀⠙⠻⣿⣦⣴⣿⠟⠋⠀⠀⠀⠹⣷⣀⠀⢀⣀⠀⠀⢀⣼⡟⠻⣿⣦⡀
⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠛⢷⣶⡾⠟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⢻⡿⠃⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠛⠿⠾⠿⠋⠀⠀⠈⢻⣿⣄
⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣤⣀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣤⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠙⣷⣄⢀⣠⣤⣶⠾⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⢀⠀⠀⠘⣿⡟⠋⠁⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠉⠉⠛⠿⣶⣤⣀⣠⣤⣤⣴⣶⣶⣿⠇⠀⠀⢠⡟⠋⠛⣷⣶⠿⠷⢶⣄⡀⠀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠁⠀⠀⢸⣿⠀⠀⠀⠸⣧⣀⣾⠋⠀⠀⠀⠀⠙⢿⡟⠛⠛⠛⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⢸⡷⢦⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣶⣶⣶⠾⢿⣦⡀⠀⠀⢀⣰⣿⠁⢨⡿⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⠉⠙⠛⠋⠁⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣠⣴⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠙⣿⡆
⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⣿⡿⠿⠿⠿⠶
⠀⣠⣤⣶⠶⢾⣿⡶⠶⠖⠀⠀⠀⠀⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⣤⡀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠀⠀⠀⠀⠀⠶⣾⣿⡷⣶⣦⡄
⠀⠀⠀⢀⣠⣴⠾⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠁⠀⠀⠈⣻⣷⣶⣶⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣴⡶⠟⠛⠻⠶⣦⣘⣿⣿⠿⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠾⠋⣹⡿⠋⠁⠀⠀⠀⠙⢿⣶⠶⠶⢶⣶⣤⣀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣾⠋⠀⠀⠀⠀⠀⠈⢿⡟⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠉⠁⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠉⠛⢷⣦⣤⣾⠟⠋⠁⠀⠀⢻⣦⣀⣀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⢀⣴⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄⣀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣟⣙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⣿⣇
⠀⠀⠀⣠⣴⠾⠟⠛⠛⠛⠛⠛⠿⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠛⠛⠛⠛⠿⠷⣶⣄
⠀⢠⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠈⠻⣶⣤⣤⣶⠀⠀⠀⠀⠀⠀⢀⣶⠟⠻⣶⡀⢿⡆⠀⠎⠀⠀⡄⢰⠀⡖⠀⢤⠐⠀⣿⠃⣰⡞⠛⢷⡄⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠈⣿⣄⡀⣀⣴⣆⠀⢸⣇⠺⢶⠿⢁⣾⠃⠈⠃⠀⠀⠙⠀⠀⠋⠀⠙⠀⠀⣿⡄⠻⢶⠾⢨⡿⠀⢠⣤⣀⠀⢀⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⠿⣦⣶⠿⣷⣶⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣶⣶⡿⣧⣤⡾⠟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⣠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀⠀⠀⢤⠀⠀⠀⠀⠀⣀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⠿⠛⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠻⣷⣄⠀⠀⠀⢀⣀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠈⢻⣷⣶⠿⠛⠋⠉⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⣶⣶⣤⣀⠀⠀⠀⢀⣀⣀⣀⣾⡟⠀⠀⣠⡶⠶⣦⣤⣿⣷⣤⣀⡀⢀⣀⣨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠉⠛⠿⣷⠿⠟⠛⠋⢹⣿⠇⠀⠀⣿⣄⣠⡿⠋⠀⠀⠈⠻⣿⡟⠛⠋⠛⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⣿⠿⣶⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⣿⣤⣀⣀⣤⣾⣿⣤⣾⠇⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣄⠈⠉⠀⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢿⣿⣿⣿⣿⣿⡷⠆
⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣷⣶⡶⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠛⠋⠉⠛⢿⣿⡄⠀⠀⠀⠀⠀⠛⠿⠋⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠐⠿⣿⣿⢿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣼⠟⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣼⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢙⣿⣿⣶⣷⣶⡶⢶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⣀⣴⡿⠛⠛⠛⠻⢷⣾⣿⡿⠿⡿⢿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⢿⣿⡿⠉⠀⠀⠀⠀⠹⣿⠿⠿⠻⠿⣷⣤⣀⠀⠀⣠⣴⣾⠿⠿⢿⣏⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⣿⠀⠀⠀⠀⣀⣠⣴⡿⠀⠀⠀⠀⠀⠈⠻⣷⡿⠟⠉⠀⠀⠀⠈⠻⣷⣶⣄⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣤⣤⣶⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⠿⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿⣿⣬⣭⣍⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣾⣿⣶⣴⣤⣀⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠋⠉⠀⠀⠀⠀⠈⠉⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠋⠉⠁⠀⠀⠀⠈⠉⠛⢿⣶⡄
⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠠⣿⣿⣄⡀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣠⣶⣦⣄⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣰⣿⠃
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠀⠀⢀⣀⠀⢰⣿⣧⣤⣿⠇⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣿⣤⣬⣿⡆⠀⣀⡀⠀⠀⢸⣿⡿⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣶⣿⣿⣆⣬⣿⣯⣍⣥⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣬⣉⣵⣿⣇⣠⣿⣿⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠻⠿⠿⠿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡿⠿⡿⠿⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣤⡀⠀⠀⠀⢀⣠⣴⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣷⣦⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⠿⠋⠉




⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠉⢿⣏⣉⣉⣉⣹⣍⣁⡀
⣿⣤⣿⣿⣿⣿⣿⣿⣿⣯⣤⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⣤⣤⣤⣤⣤⣶⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠿⣿⣦⡀⠀⢀⣠⣤⣶⣶⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⢿⣿⣶⣦⣄⡀⠀⠀⢀⣀⣀⣀⣀⣠⣼⡿⠃⠀⠀⠀⠈⠻⣿⣿⡿⠟⠋⠉⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⠿⠿⠿⠛⠛⣿⡿⠁⠀⠀⠀⣠⣤⣄⢹⣿⣄⡀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣾⡏⢩⣿⡿⠟⠛⠿⣷⣤⣶⣶⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠻⣷⣿⠏⠀⠀⠀⠀⠙⣿⡉⠀⠀⠈⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣀⣀⣤⣼⣿⡆⠀⠀⠀⠀⣰⣿⢿⣧⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠙⢿⣶⣤⣤⣶⣿⣿⣾⠟⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣄⠀⠀⠀⣀⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣶⡿⠟⠋⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡾⢿⣿⡿⠟⠻⠿⠗
⠀⠀⠀⠀⠀⡶⠾⠿⠿⣿⣿⠿⠿⠆⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⢶⣿⣿⣷⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⡿⠓⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢿⣷⡄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣏⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⣿⠋⠉⠛⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢛⣥⣿⠿⠛⠛⠛⠛⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠿⠟⠛⠛⠛⠿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠛⠿⣿⣿⣿⡛⠛⣻⣿⣿⠿⠛⠉⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠈⢿⣧⡀⠀⢀⣠⣤⣤⣤⣾⡿⠁⠀⠀⠀⠈⠙⣿⣿⣾⣿⠏⠁⠀⠀⠀⠀⠈⢿⣷⣄⣠⣤⣤⣀⣀⣀⣼⡿⠉⠙⢿⣿⣦⡀
⠀⠀⠀⠀⣰⣿⡿⠉⠀⠀⠀⠀⠙⠿⠿⠿⠛⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠙⠛⠛⠛⠋⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀
⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣤⣀⣴⣾⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠈⠳⣄⢀⣠⡤⠶⠶⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣀⠀⠀⢹⡋⠁⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠈⠙⠳⠶⠖⠛⠛⠛⣿⠀⠀⠀⠀⠀⡀⠋⠉⠉⠑⢦⡤⠴⠶⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠰⡇⠀⠀⠀⠀⢈⣦⡀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠖⢧⣀⠀⠀⢀⠜⢀⡀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣇⠀⠈⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠹⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣀⣠⣿⣤⣤⡀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠉⠉⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠉⠁⠀⠘⣧⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⠀⠀⢀⠤⢀⠀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠠⣴⣿⣤⣤⡀
⠀⠀⠀⣤⢤⡄⣰⠶⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠠⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣾⣧⣄⠠⠦⠀⡀
⠀⢠⠶⣼⣴⣧⠿⠘⣫⡼⢿⡿⠖⠛⠛⠲⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠋⠉⠉⠙⢦⣽⡿⠻⠷⠿⣷⣀⠐⠄
⠀⠙⢛⣻⠟⠁⠀⠚⠁⣄⡟⠀⠀⠀⠀⠀⢸⠉⠀⠉⠉⠓⢦⣄⣀⡴⠞⠋⠉⠹⡄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠈⠱⣦⣈⡁
⠀⡾⢋⡉⠀⠀⣀⠀⣤⣈⣧⡀⠀⣀⡴⠒⠋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠛⠳⣄⣀⣠⣾⡀⠚⢀⠀⠀⠀⠀⢠⣌⢻
⠀⠻⣬⣥⣄⠀⠉⠀⠘⣹⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣷⠰⢟⠃⠴⠀⢀⡤⠴⠛
⠀⠀⠀⠀⣈⣿⠷⢾⡟⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⢦⣤⣤⣴⣿⡁
⠀⠀⠀⠸⣏⠀⠀⠈⣧⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⢀⣿⠀⠀⢈⡿
⠀⠀⠀⠀⠉⠙⠛⠛⢻⡷⢬⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⣀⣾⡧⠴⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠙⠛⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡴⠋⠉⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⢷⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠒⠒⠒⠒⠛⠛⠛⠳⠒⠶⠶⠶⠶⠶⠶⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠲⢤⣀
⠀⠀⢀⣠⠤⠤⣰⠁⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠙⢦⡠⠖⠛⠛⠛⠻⢤⡀⠀⠀⠀⠈⢳⡀
⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣇⣀⣀⣀⣀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠉⠀⠀⠀⠈⠓⢆
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠈⠑⠢⠤⠤⠤⠤⡤⡏⠀⣠⠤⣀⠀⠀⢀⣘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⡴⠛⠒⠶⣤⣤⣼⣧⡞⠁⣀⣈⣶⡛⠉⠉⣯⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡁⠀⢧⡏⠀⠉⣯⡉⠉⡆⠈⠑⢦⣀⣀⣤⠤⠖⠋⠁⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋
⠀⠀⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠓⠦⢾⠓⠃⣰⠃⠀⠀⢫⡽⠀⠀⠀⠀⠀⠀⠀⢸⣭⠏⠙⠓⠒⠒⠚⠛⠉⠁
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⣇⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠤⠤⣷⠄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠒⣿⣻⠋⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⢠⢽⡆⠀⠀⠙⠁⠀⠀⢠⠶⡄⠀⠀⠘⠃⠀⠀⢸⡿⠓⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⣠⠿⢾⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡼⠛⠒⠄⠀⠀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠙⢓⡶⠶⠶⣤⣤⠤⡶⢒⡾⠋⠀⢀⣠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣴⠋⠉⡿⠛⢳⡾⠶⢾⡉⠁
⠀⠀⠀⠀⢀⣠⣤⡾⠁⠀⠘⢦⣸⠁⠀⣸⡇⠀⠀⢻
⠀⠀⠀⠀⢾⡀⠀⢷⡀⠀⠀⠀⠈⠉⠉⠉⡇⠀⣀⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠚⠋⠙⠓⠲⢤⡞⠉⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠁⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣀⣀⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣔⠲⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠶⠒⠛⢇⠀⠁⡤⠛⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠉⠊⠉⠐⠊⢵⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⣄⠀⠀⣀⠀⠈⠃⣀⣼⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠝⣦⣄⡀⠀⠀⣁⣠⠒⠹⣄⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡼⠉⠉⠀⠈⠢⠤⢼⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣒⡀⠤⣀⢠⠀⡠⠼⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⢻⡉⠓⠋⢹⠋⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠋⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣠⣤⣤⣤⣀⠀⣴⣾⣿⣷⣦⣄⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣠⣾⠿
⠀⣼⡿⠟⠋⠙⠛⢿⡟⠁⠀⠀⠈⠹⣶⠞⠛⠛⠿⣿⣦⣤⣄⣠⣿⠟⠛⠛⠻⣷⣴⣿⠿⣿⣿⣦⣄⣀⡿⠋⠀⠀⠀⠙⠟⠁
⣶⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠻⠃⠀⠀⠀⠀⠀⠛⢡⣾⡿⠛⠉⠉⠉⠃
⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣄⣠⣴⣶⣦⡀⠀⠀⠀⢸⡏⠀⡀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠠
⠉⠛⣿⣴⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠦⣴⠋⠁⠈⠙⠋⠁⠀⠙⢷⠶⢶⣦⡘⢧⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠈⢻⣇⣄⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⣀⣀⡀⠀⠀⣠⡶⠖⠲⢶⣄
⢠⡟⠉⠉⠙⠷⡾⠁⠀⠀⠀⠀⢿⡄
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠈⠻⣄⡀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠈⠻⣦⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠙⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠿⣿⣦⡀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠛⠛⠻⠿⣿⣷⣶⣤⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠛⠋⠉⠁⠉⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡿⠿⠿⠿⠛⠛⠛⢻⣿⡏⠀⠀⠀⠀⢀⣀⣀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⣼⡿⠛⣻⣿⣿⣿⣿⣷⣦⣄⣀⣤⣤⣤⣼⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢿⣧⣴⣿⠋⠀⠀⠀⠈⠻⣿⡿⠛⠛⠛⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⣉⣿⣿⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⠿⠿⠿⣿⣆⠀⠀⠀⢀⣼⣿⠋⣿⡇⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠟⠻⠟⠛⠁⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⢀⣀⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠿⠛⠉⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⣀⣀⣤⣤⣿⣿⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⣿⡿⣿⣿⣷⡦
⠀⠛⠛⠛⠛⠛⢻⣿⡟⠛⠛⠁⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣇⣀
⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⠛⠃⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⠛⠛⠻⠿⠂
⠀⠀⠀⠀⠉⠉⠀⠀⢻⣿⣆⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠉⠙⠿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠋⠀⠀⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣶⣦⣴⣶⣿⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⠿⢿⣶⣦⣿⡿⠋⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣤⣴⣶⣤⣤⣀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣶⣿⣿⡟⠋⠉⠉⠉⠛⢿⣧⠀⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠈⣿⡇⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⣰⣿⠇⠀⠀⠹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⣷⣄⡀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠈⢿⣿⡄⣠⣴⡿⠏⠀⠀⠀⠀⠹⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠉⠻⢿⣶⣤⡀⠀⠀⢹⣿⣦⣀⠀⠀⠀⢀⣸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢻⣿⡿⠿⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠉⠛⠿⣿⣶⣾⠟⠛⠿⣿⣿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣦⣤⣄⣀⣠⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣿⣿⣿⣥⣤⣤⣴⣶⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠈⢷⠞⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⠻⠿⣷⣶⣴⣿⠋⠁⠀⠈⠻⣷⣄⣀⣀⣀⡀⠀⠀⠀⣀⣤⣶⣶⡿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢹⣿⠃⣀⣀⠀⠀⠀⠙⣿⡿⠿⠿⠿⠿⣿⣿⠟⠋⠉⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣦⣤⣶⡿⠿⠿⣿⣏⠹⣷⠀⠀⠀⢹⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⢉⣿⠏⠀⠀⠀⠈⣿⣿⠟⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣇⠀⠀⢰⣿⢻⣧⡀⠀⠀⢀⣿⣿⣶⣦⣶⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣆⠀⠈⠛⠛⠛⢿⣶⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⢰⣿⠿⣷⣦⣄⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⣴⣶⣶⣶⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣀⣀⣀
⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠈⠉⠙⣿⡟⠉⠉⠛⠃
⠀⠀⣠⣤⣴⣿⣿⠶⠀⠀⠀⠀⠀⠹⣿⠟⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠀⠀⠀⢠⣤⣼⣿⣁
⠀⠀⠈⠁⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⣀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠛⠛
⠀⠀⠀⠀⣀⣤⣾⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣾⣿⣇
⠀⠀⠀⠘⠛⠉⠀⠀⠙⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣠⣴⣾⠟⠉⠈⠙⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠿⠛⠛⠿⣿⣟⣫⣭⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠏⢉⣉⣭⣭⣍⣹⣿⠁⠀⠀⠀⠀⠘⠿⠋⠉⠉⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⢰⣿⠛⠉⠉⠙⣻⣿⣶⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⣿⡇⠀⠀⠀⣰⣿⠃⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠸⣿⣄⠀⣰⣿⠃⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⣀⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡾⢿⣿⡏⠀⠀⠀⠀⠈⠻⣷⣿⣇⠀⠀⠀⣀⣼⡟⠀⠀⣀⣴⡾⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠟⠛⠿⣶⣿⠛⠉⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣀⡀⠀⢹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠁

⢰⣦⣶⣠⣶⣶⣆⣴⣠⡆⠰⣶⠖⣴⣶⡀⠀⠀⠀⣀⣀⣠⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠲⠦⢤⣀
⢸⣏⡿⠙⣾⡗⠻⢛⡟⠋⠛⠻⣧⣿⡟⢀⠤⠊⠉⠁⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⣠⠞⠉⠀⣀⣀⣀⣀⡀⠀⠈⠑⢆⠀⣠⠤⠒⠚⠉⠉⠉⠉⠒⢦⡀
⢸⣿⢻⣾⣿⣿⠆⣾⣿⣿⣷⣼⣿⣼⣿⠳⣄⠀⢀⠔⠂⠉⠉⠙⠒⠲⢤⡀⠙⣆⠞⠁⡠⠖⠉⠀⠀⠀⠀⠀⠙⢦⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠸⠿⢻⠹⠿⡿⣾⠟⠹⠇⢿⡿⢿⡿⠃⠀⠈⢦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣠⠊⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⠢⣀⢴⡇⠀⠀⠀⡀⠀⢀⡔⠉⠙⡆⠈⣧⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢙⡕⠋⠀⠉⢢⣀⠤⠶⣄⠀⢠⠇⠀⢀⡠⣀⠀⠀⠀⠀⠀⠀⠈⡗⠒⠚
⠀⠀⠀⠀⠀⠈⢇⡀⣀⣼⠃⠀⢸⠀⠀⠈⠁⢀⡟⢀⡎⠀⠈⠑⠢⡤⠄⠒⢒⠎⠀⠀⠀⣣⠴⠧⣄⢀⣸⣀⢸⡀⠀⠈⠀⠈⡇⠀⠀⢶⢤⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⡆⠀⢈⣦⣀⣀⣠⢾⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠒⢺⠁⠀⠀⠈⣇⣀⠀⠙⣧⣀⠀⠀⣠⠇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⢩⠁⠀⠈⢆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢦⣀⣀⣴⠃⠀⠀⢀⡞⠈⢉⠏⠉⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣀⡴⡏⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣸⣴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣏⠂⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⡉⣸⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⣏⠀⠀⠙⠛⠀⠀⠀⡠⠒⡄⠀⠀⠀⠸⣿⠇⠀⠀⠀⢤⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⠳⣄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣄⣀⣀⡀⠀⠀⣀⣀⣀⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢬⡉⠙⠯⠍⠉⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡴⠋⣦⣶⢶⠠⢴⣶⣴⣶⣴⡴⡤⢶⣶⣦⠄⣶⣴⣆⣶⣦⣦⠂⢶⣴⣶⣴⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣄⠀⢀⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠉⠉⠙⠻⠷⠿⠛⠛⠛⢹⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣤⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⡿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢾⣿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣧⣤⡄⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⢀⣸⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⡀⠀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⣽⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠾⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣶⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠉⠛⠛⠻⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠁⠈⠙⣻⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣴⣶⡾⠟⠛⣷⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⢡⣿⣿⣿⣿⣏⠉⠉⠉⠉⠉⣹⣿⣿⣿⣿⣄⠀⠈⢿⣄⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⢷⣄⣠⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡆⠀⣀⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⣀⣿⣄⠀⠀⠀⠀⠀⠀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠛⠛⠋⠉⠙⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⡀⠀⣠⣤⡶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⡀⠀⢀⣀⣀⣤⣤⣤⣴⣿⠟⠉⠉⠙⢻⣿⠋⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠈⠙⠛⠻⠟⠛⠛⠛⠉⠉⢹⡿⠃⠀⣠⣶⣶⡤⠿⠷⢶⣤⣤⣤⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢻⣧⡾⠀⠀⠀⠀⠘⣯⡀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠀⢉⣿⣄⠀⠀⠀⣠⣏⣿⠆⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠙⠿⠷⢿⡛⠛⠋⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⡴⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠾⠛⣿⣟⠛⠉⠉
⠀⠀⠀⠀⢀⣠⣤⣴⣿⣦⣤⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⣀⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠈⠉⠉⠀⠹⣇⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⢀⣤⠤⣄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢹⣿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠹⣧⣴⡆⠀⣀⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢶⣤⡀⠀⠀⣠⣶⡿⠋⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠛⠷⣾⣿⣏⣁⡀⠀⠈⠛⣦⡀⢀⣴⠟⠉⠀⠀⠀⢀⣬⠿⢶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢀⣼⡟⠋⠉⠉⠻⡆⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⢰⣿⠁⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⢀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢷⣶⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠿⠿⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⢀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⢀⣾⠟⠙⢷⣤⣴⠟⠉⠻⣧⡀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠴⠶⠋⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⠛⠶⠦⠴⠖⠋⠁

⠀⣠⣤⣄⠀⣠⠶⠒⠒⢦⡀
⢸⠃⠀⠈⠙⠋⠀⠀⠀⢈⡇
⢸⡄⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠙⢦⡀⠀⠀⠀⣠⠞
⠀⠀⠀⠙⢦⣀⢰⠃
⠀⠀⠀⠀⠀⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⢿⣶⣄⡀⠀⠀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠻⠿⢿⣶⣦⣄⣀⣀⣀⣀⣤⣤⣤⣤⣾⠟⠁⠀⠀⠈⠻⣷⣶⡿⠿⠛⠋⠉⠙⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠛⠛⠛⢻⣿⠏⠀⠀⢀⣠⣤⣄⣹⣿⣀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣾⡏⣩⣿⠟⠛⠛⢿⣶⣴⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠘⢻⣿⠇⠀⠀⠀⠀⣻⣧⡀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⡶⠿⢿⣦⠀⠀⠀⣠⣿⢛⣿⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡟⠛⠛⠋⠀⢀⣾⡟
⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣤⣤⣶⡿⢻⣿⡀
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠈⣿⣇
⠀⢀⣀⣀⣠⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣶⣶⣶⣦
⠀⠛⠋⠙⠙⣿⣟⠉⠉⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⣸⣿⣤⣤⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠰⢶⣿⣷⣦⣤⣄
⠀⠀⠀⠚⠛⠛⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣁⣨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣹⣿⣾⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣧⣄⡀
⠀⠀⠀⠀⢠⣾⠟⠋⠉⠙⠿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋⠀⠈⠙⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣍⡛⢿⣷⡿⠟⠛⠿⣦⣤⣤⣠⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠉⠙⠻⠿⠋⠀⠀⠀⠀⠈⣿⣟⣛⣛⣛⣛⠛⠛⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⠛⠋⠙⠛⢿⣆⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠘⣿⣆⠀⠀⠀⠈⣿⠄⠘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠘⣿⣆⠀⢀⣼⡿⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠛⢿⣦⣄⠀⠀⠈⢿⣆⠀⠀⠀⠀⢹⣿⣶⡿⠏⠀⠀⠀⠀⠈⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠈⠛⠿⣶⣤⣾⠟⠿⣶⣶⠾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠈⠉⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⢀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣶⣶⣿⣷⣦⣶⣶⣦⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣤⣄⡀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠉⠙⣷⣤⡶⠟⠛⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠉⠛⠿⠿⠛⠛⠛⣻⡏⠀⠀⣠⣤⣌⣿⣄⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣯⣼⠋⠉⠉⢻⡿⠛⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣤⣼⣿⡀⠀⠀⣰⡿⣦⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠷⣾⡛⠛⠃⢀⣴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠟⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣧⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢾⡷⠶⠆⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠈⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⡀⠀⠀⠸⠿⠃⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠚⣿⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣯⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⣯⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣏⡉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⡾⠋⠁⠀⠀⠉⢻⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣴⣶⡟⠉⠀⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⡇⠀⠀⠀⠀⠀⣸⡏⠉⠛⢿⣿⣉⣹⣿⠟⠋⠁⢹⣇⠀⠀⠀⠀⠀⢨⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠈⠻⢶⣶⠾⠷⠾⠋⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠛⠷⠾⠻⠶⠶⠟⠁⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢼⠀⢠⠀⡄⠢⢠⠐⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠘⠐⠚⠃⡇⠀⢸⠂⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡆⢸⢀⠐⢰⠆⡆⢹⠀⡆⠆⠂⢀⠐⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁⠈⠈⠁⠈⠁⠁⠈⠈⠁⠁⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢀⢸⡀⢦⠁⡃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⡀⠀⠀⢀⣤⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⡾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⠶⣦⡀⣀⣤⣴⠶⠶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣤⣄⡀⢀⣀⣀⣤⣤⣾⠟⠀⠀⠀⠈⢿⡟⠉⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠉⠙⠛⠛⠋⠉⠉⢸⡏⠀⠀⢠⠖⢲⡾⠷⠶⣤⣤⡤⠶⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠘⢳⡏⠀⠀⠀⢸⣧⡀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠶⠞⠳⣄⣀⣀⣼⣧⠟⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄⡀⣀⣠⠞⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠘⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠾⠛⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣴⣿⣦⡤⠀⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢠⣤⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⣠⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠛⠛⠷⢶⣄⡀⠀⠀⠀⠀⠀⠀⠛⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠶⢶⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣾⠿⠛⠁⣀⡴⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⡘⢷⣀⠀⣀⣤⡀⠀⠀⠀⠀⣸⡟⠙⣿⠟⠉⠉⠉⠙⠛⠟⠉⣿⡤⠖⠋⠁⠀⠹⣷⠀⠀⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⣀⣉⣿⠛⠙⠷⣦⣤⣴⠾⠋⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣞⠒⠛⠚⠛⣿⡟⠓⢸⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣽⣿⣄⠀⣀⣿⠃⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣿⠇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣈⣛⠛⠋⠡⣶⣦⣦⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣤⡤⠾⣇⠀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠙⡿⢶⣄⠻⠟⠉⠀⠀⣴⣦⣤⡀⠀⠀⠀⠀⠀⢿⠀⠀⣼⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣷⠀⠙⢷⣤⣀⣀⠀⠹⡿⠟⠁⠀⢰⣶⣤⣄⢸⡇⠀⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⣆⠀⠀⠈⠉⠉⠻⢦⣄⡀⠀⠀⠀⠿⠟⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡙⢧⡀⠀⠀⠀⠀⠀⠉⠉⠛⠛⣿⡶⣤⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣿⣶⣄⡀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠙⠻⣶⣤⣾⠟⠋⠉⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⣶⣶⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⠏⠀⠀⣀⣀⣈⣿⣇⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠈⠉⠛⠋⠉⠉⠀⠀⠀⢸⣿⠀⠀⣼⡟⢉⣿⠟⠛⠻⢷⣶⣶⠟⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠹⣧⣼⣏⠀⠀⠀⠀⣿⡟⢳⣆⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⢸⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⣠⣤⣿⣿⣄⣀⣀⣴⣿⣁⣠⡿⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠛⠛⢻⣿⡉⠉⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣻⣶⣾⣿⣿⢷
⠀⠀⠀⠀⠀⠀⣾⣅⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠟⠛⠛⠉⠉⠉⠉⢻⣿⣿⡷⠶⠶⠶
⠀⠀⠀⠀⠀⠀⣿⣹⡿⠋⠉⠁⠀⠀⠀⠉⠉⠉⠛⠻⣿⣿⡿⠿⠿⢿⣿⡟⠁⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⣀⣤⣴⣿⣿⣷⠆⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⢸⣿⠃⠀⠀⠈⢿⣇⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠰⣿⣿⡷⠶⢶⡤
⠀⠀⠀⠉⠁⠀⢹⣿⣧⠀⠀⠀⠀⠀⢿⡿⠃⠀⠀⢀⣾⠏⣠⠴⠶⢤⠈⢿⣆⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠻⣤⣠⠼⠁⠀⠉⠻⠶⣦⣤⣤⣤⣤⡶⠾⣿⣿⠿⣶⣤⡀
⠀⠀⠀⠀⠘⠋⠉⠀⠹⣿⣿⣶⣦⣤⣤⣶⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⣠⣾⠟⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⣿⡿⠋⠉⠉⠙⣷⣤⣤⣤⣄⡀⠀⢀⣀⣤⣤⣤⣤⣾⠋⠁⠀⠉⠻⣿⡁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣰⡏⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⠀⢀⣰⣿⡏⠙⣿⢹⠟⣛⠟⠛⡛⢋⡉⢸⠛⣻⣿⣷⣄⣀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠙⠛⠛⠻⣿⣁⣶⣈⢠⣀⣒⣶⣼⣿⣘⣃⣘⣃⣹⡿⠉⠉⠉⠉⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠻⠿⠿⠟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣶⣶⣶⣤⣬⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⣀⣠⣾⠟⠋⠉⠁⠀⠈⠉⠉⠛⠿⣷⣄⡀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉

⢠⣾⠿⠿⠿⢷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⠿⠿⠿⢿⣦
⣸⡿⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣷⣦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⠶⠿⢿⣏⠉⠁⠀⠀⠀⠀⠀⠸⣿
⣿⡇⠀⠀⠀⠀⠀⢀⣤⣶⡾⠿⠛⠛⠛⠛⠛⠛⠿⢷⣶⣤⣀⢀⣾⠟⠉⠉⢉⣿⠿⠿⠿⣤⣀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⣠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠁⠀⢠⢶⡟⠀⠀⠀⠀⠈⣿⣱⠀⠀⢸⡿⣷⣦⡀⠀⠀⠀⣿
⢻⣧⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠚⢷⣄⣀⣀⣤⡾⠋⠀⠀⣠⡿⠁⠀⠙⣿⣆⠀⢠⣿
⠘⣿⣤⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣤⠿⠋⠉⠙⠛⠷⠟⠛⠋⠀⠀⠀⠀⠈⢿⣧⣼⡟
⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠁
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣴⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠃⠀⠀⢀⣤⠶⠖⠶⣤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠹⣿⡀⠀⠀⣀⣀⣠⣤⣬⡿⠛⠋⠁⠀⠀⠀⠀⢸⣅⠀⠀⠀⣀⡿⠀⠀⠀⠀⠈⠙⠛⠻⣭⣤⣤⣀⣀⡀⠀⠀⣼⡿⠁
⠀⠀⠀⠹⣿⡔⠛⠉⠉⠉⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠉⠉⠙⠓⣼⡿⠁
⠀⠀⠀⠀⠙⢿⣦⠀⣀⣤⡶⠟⠋⢁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠉⠛⠷⣶⣄⡀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣍⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢈⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⡀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⣴⣾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢦⣤⡴⠋⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⣀⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣤⣤⣠⡴⠾⠷⠶⠶⠶⠶⢺⡿⠛⠉⠙⠳⣿⣄⣤⡴⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣀⣀⢹⡏⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢷⣄⣀⣀⣀⣤⣤⣼⠃⠀⠀⢾⣉⡽⠛⠛⠻⣦⡤⠶⠶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣤⠄⠀⠀⠀⠈⠉⠉⠁⠀⠀⠘⣧⣀⢀⣈⣹⡇⠀⠀⠀⢸⣧⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠳⢦⣤⡴⠿⠼⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣠⡴⠻⣆
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿
⠀⠀⠠⠴⠶⠾⣷⠶⠦⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠚⠛⣿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠘⢿⡿⠁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠸⣿⣧⣤⡤⣤⣤⣤⣿⣄⣀⡀
⠀⠀⠀⠀⠒⠛⠙⢿⡁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣩⡿⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠈⢻⡇⠈⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣼⠿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠚⠉⠀⠀⠈⠙⣳⣦⣤⣤⣤⣤⣄⣄⣀⣀⣀⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠈⠙⢷⡄
⢀⣤⠶⢦⣄⠀⢀⣠⡤⠖⠛⢹⡏⠀⠀⠀⠀⠀⠀⣀⣀⡀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢼⡁⠀⠀⢹⣟⣋⣁⣀⣀⣀⣀⣷⣦⣤⠤⠤⣴⠞⠉⠉⠙⢻⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠘⠷⣤⡴⠟⠉⠉⠉⠉⠉⠉⠁⢀⡾⠁⠀⠀⢿⡀⠀⠀⣰⠟⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣾⠁⠀⠀⠀⠈⢳⣄⡼⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⣤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣬⣷⣤⣤⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⡇⠀⢀⣀⣤⡴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⢲⡄⠀⠀⠀⠀⠀⣤⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢸⡟⠛⠛⠛⠋⢰⡏⢰⠇⣰⢂⠀⣠⡄
⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⢹⠀⠀⠀⢰⠇⡼⢣⡾⢱⡆⢀⣿⢀⡿⢸⢳⡟⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠈⠀⠸⡷⠊⠀⢸⡇⣼⢁⡟⠘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠀⠿⠀⠈⣷⣀⡀⠀⠀⠀⠁⠘⠇⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⠀⠠⠟⣸⡤⠖⠚⠛⠛⠛⠶⢼⢃⡶⢰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣯⠁⡀⡆⢸⡆⢰⡆⢀⡿⠃⠛⠛⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⣴⠃⢸⡆⣿⠸⣇⠇⣿⠀⣾⠁⣾⠃⣼⢠⡆⢀⣤⢀⣀
⠀⠀⠀⠀⠀⠀⢠⡆⠁⠲⣄⠀⠘⣇⣿⠀⣼⠀⢿⠘⣇⠰⠏⢸⡇⣾⡃⣼⠃⣼⠋⣹⢻⡏⢰⡇
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⢷⡄⡏⢿⡄⠙⠂⠘⠇⠉⠀⠀⠀⠀⠈⠁⠙⠺⠃⣼⠃⣼⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⠇⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠓
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⢿⣦⡀⠀⠀⠉⠒⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⠀⠈⠉⠻⣶⣄⠀⠀⠀⠀⠁⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢸⣦⣶⣤⠞⠉⠙⠃⠀⠀⠐⠀⠺⢿⣷⣄⠤⠒⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⣿⢻⣷⣤⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢐⣦⣄⠀⠀⠀⠀⠀⠀⢶⣶⣤⣿⣮⣿⣿⣿⣦⡀⠀⠀⡆⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠀⠀⠀⢀⡀⣀⠀⢉⠤⠈⠁⠈⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠘⠀⡀⠀⠀⠄⠨⣿⣨⠶⠆⠂⠀⠀⠀⠀⢸⣿⣦⣅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠘⢿⠀⡄⠀⢀⡄⠁⠰⣞⣏⡻⠀⠀⠀⡾⠻⠝⢿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠘⣷⠀⣤⠁⣀⠀⠀⢺⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠐⠆⠀⠀⠰⠏⠀⠀⠄⣸⣆⠀⠘⠏⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠰⣿⣏⠈⠙⠲⣦⢄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠷⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⢀⣀⡀⡿
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⢠⣤⡴⠂⠀⠀⣿⠁⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠃⠀⠋⠀⠀⠰⡾⠏
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⡗⢚⠉⢋⡡⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠰⠞⠂
⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⣀⣠⣴⣶⣾⣿⣶⣶⡶⠄⠀⠀⠀⣠⠀⠀⠀⡠⠉
⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢱⣤⣀⣀⣀⡀
⠀⠀⠀⠀⢠⠊⠉⠉⠑⢾⡏⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠤⣤⣾⠋⠀⠀⠀⢙⠆
⠀⠀⠀⠀⠀⠑⢦⡀⣀⣼⣿⣖⢶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣱⡦⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠫⠞⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀
⠀⣠⠶⠛⠛⠋⠙⠛⠳⢦⣄⣠⠞⠋⠉⠀⠀⠉⠉⠛⢦⣄⢀⡴⠟⠉⠉⠀⠀⠉⠙⠳⣦⣀⣤⠶⠞⠛⠛⠶⢦⣄⡀
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⢤⡀⠀⣀⡀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠘⢧⣤⣤⠀⠀⠀⠀⠀⠀⣀⡀⠹⣆⠀⡰⠒⠲⢤⣤⠤⡴⠁⡤⣼⣏⣁⣹⡀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣾
⠀⠀⠀⢻⣄⣀⣤⠀⠀⡞⣉⣹⠆⢹⠀⡇⠀⠀⠀⠀⠀⠣⣀⣻⡀⠀⣹⡄⢹⢰⡏⢰⠟⠳⡆⠀⢀⡀⠀⣸⡏⠉
⠀⠀⠀⠀⠉⠉⠹⢦⣤⣿⣬⣩⡴⠏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣍⠴⡇⠘⢧⣈⠛⣡⣧⣀⣼⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀⡀⠉⣿⠏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡤⢷⡀⠀⠛⠀⠀⡠⠀⠀⠀⢰⠆⠀⣀⣼⣀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣍⠳⣄⣀⣀⡀⠀⠁⠀⠀⠀⡠⠊⠁⠀⢈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⠙⢧⣼⠉⠀⡟⢲⠒⠲⢖⡺⣁⣀⣤⣴⠟⠁
⠀⠀⠀⠀⢀⠠⠤⢤⡎⠀⢐⠀⠀⠀⠀⠈⠻⣶⣾⠁⠸⣄⡰⠊⠑⣄⣽⠟⠁⠀⠀⠀⠀⠀⠔⠢⢄⣀⡀⠤⢄
⠀⠀⠀⠀⠐⠤⡀⣀⡿⠦⠧⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠦⣄⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠣⡀⢀⣿⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⠁

⠀⠀⠀⠀⠀⣠⠀⠀⣠⠀⠀⠀⠀⣤⢶⡄⣠⠶⡄⠀⢀⠀⠀⠀⠀⢰⠀⠀⣤⠀⠀⢠⡄⠀⡄
⠀⠀⠀⠀⢀⣿⣀⣀⣿⣀⡞⣻⢾⡇⣸⠇⣏⢠⡏⡾⣏⣷⠆⠀⠠⣾⣤⣞⠁⢘⠑⢿⠓⢺⡟⠛⣰⠂⢰⠆
⠀⠀⠀⠀⠈⡟⠀⠀⡏⠘⠿⣥⡴⠿⢯⡤⠿⠯⣴⠷⡴⠏⠀⠀⠀⠿⠀⠘⠇⠿⠀⠻⠦⠞⠧⠶⠻⣶⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⢰
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢨
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠁⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠶⣤⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⠻⣾⡟⠁⠀⠀⢠⠦⣄⠀⠸⡇⢀⣠⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠃⠈⢧⠀⢿⣁⣀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⠀⠀⠀⠉⢉⣤⠀⠀⠀⡽⠗⠚⠛⠛⠓⠶⣤⡀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠚⠋⠙⠦⣤⡀⠡⢈⣐⠄⠀⠀⠀⠈⣷⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢦⡄⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢹⡀⠀⠀⡶⠋⠀⠀⠀⠀⠀⠀⢸
⠀⡇⣀⣀⡀⠀⠀⠀⡔⠁⠀⠶⠿⣿⠀⠀⠀⠱⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠇⠀⣰⠇⠀⣀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠈⠢⡀⠔⠁⠀⠀⠀⣠⡿⣷⡀⠀⠀⠙⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠙⣳⡛⠻⠒⠂⠀⠀⠀⠀⠀⣀⣨
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡞⠉⠉⠉⠃⠉⢹⡀⠀⠀⠯⠽⠀⣀⡀⣀⠀⣻⣟⡀⠀⢀⣴⠟⠓⠶⠀⠈⠆⠀⠀⠠⠋⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⠔⢒⣳⣄⣀⠀⠀⠀⢀⡿⣉⠢⠔⣀⣬⠶⠖⢶⡟⠉⠉⠙⢾⡛⠻⢦⠀⠀⠀⠀⠘⠖⠉⠁⠀⠀⢸
⠀⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⡸⡴⠋⠀⠀⠉⠉⠙⠋⠉⠀⠈⠳⠞⠉⠀⠀⠀⠈⠙⠛⣆⣀⡼⢣⡘⢄⠀⠘⠀⣀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⡎⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢳⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⣠⡴⠶⠇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⡆⠶⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⣸⠏⠀⠀⡇⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⢻⡄⠀⠀⠐⠤⠝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣸⣋⢉⣱⠄⡀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠘⣷⡀⠀⠀⢀⣸⣄⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣤⣾⣡⠞⠁⠀⠀⠙⡆⠅⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠉⠱⢀⣙⠳⣤⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠏⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣇⠂⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠹⡶⣄⣠⡴⡾⠟⠉⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣀⡴⡫⠁⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠁⠀⠀⠀⠈⡁⠀⠀⠀⠀⠈⠒⢛⠳⢦⡥⣤⣄⣠⠾⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠑⠖⠚⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⡔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⡇⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠧⠠⠀⠼⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⢤⠀⠤⠠⠀⠄⡤⠶⢤⠰⠠⠤⠤⣤⠄⠀⠀⠠⠤⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣦⣄⣀⠀⢀⣤⣄⠀⠀⠉⠛⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠋⠉⠀⢠⣿⣿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡀⠀⠀⠀⣩⣿⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠴⠾⠟⠋⠀⠀⠀⠘⣧⠀⠀⠀⢨⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠘⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢻⣿⣿⣿⣿⣸⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⠏⠹⡇⢸⣿⣿⣿⣿⣿⡇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⡉⢄⠀⢷⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠻⣿
⠀⣶⠀⠀⠀⠀⠀⠻⣼⢦⠘⡇⠀⠒⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣠⣿⣿⣿⣿⣿⣿⣿⡁⠁⠀⠀⢸⣿⣇⠈⠉⢛⣫⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿
⠀⠘⣧⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⠀⠘⣿⣿⣆⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠹⣷⣦⣀⠀⠀⠀⠀⠈⢻⣿⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟⠋⠉⠀⠉⢻⣿⣿⣿⣿⡆⠀⢿⣿⣿⡆⠀⠈⢿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿
⠀⠀⠀⠹⣿⣿⠁⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠱⢦⣀⣠⣤⠀⠀⠈⠛⠿⣿⣿⣇⡜⡘⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿
⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⣄⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣄⠀⠀⠩⠿⢿⣶⣶⡇⠀⠀⠀⠙⢿⡼⣷⢿⣿⣿⡷⠀⠀⠘⣿⣿⣿⣿⡿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣧⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠈⣿⣹⣤⣿⠿⡥⠃⠀⠀⠸⣿⣿⣿⡇⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣤⣤⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣈⠻⣷⣦⣤⣴⣿⣿⠟⠁⡆⠀⠀⠀⠀⣰⣿⣿⣿⡇⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⠀⠀⠄⠀⠀⠀⠀⢸⣿⣿⣿⣧⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣼⣷⣼⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⡇⠀⢰⣾⡟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣼⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⣦⡀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⡴⠛⠿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣦⠀⠀⠀⠀⠀⠲⣌⠻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⢻⣿⣿
⠀⢸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠹⢶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣵⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢿⣻⣿⣿⣿⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋⣀⡼⣿⣹⣿⣿⡆⠀⠈⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣧⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡝⣯⡇⠈⢻⣿⣿⣿⣿⣿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣼⣷⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣱⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣤⣄⡀⠀⠀⠀⠀⢀⣀⣰⠟⠉⠉⠙⣷⣤⡶⠟⠛⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠈⠉⠛⠿⠿⠛⠛⠛⣻⡏⠀⠀⣠⣤⣌⣿⣄⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣯⣼⠋⠉⠉⢻⡿⠛⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣤⣤⣼⣿⡀⠀⠀⣰⡿⣦⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠷⣾⡛⠛⠃⢀⣴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠟⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣧⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢾⡷⠶⠄⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⡀⠀⠀⠸⠿⠃⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠚⣿⠟⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣯⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣯⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⣏⡉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⡾⠋⠁⠀⠀⠉⢻⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣴⣶⡟⠉⠀⠀⠀⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢿⡇⠀⠀⠀⠀⠀⣸⡏⠉⠛⢿⣿⣉⣹⣿⠟⠋⠁⢹⣇⠀⠀⠀⠀⠀⢨⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠈⠻⢶⣴⠾⠷⠾⠋⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠛⠷⠾⠻⠶⠶⠟⠁⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢼⠀⢠⠀⡄⠢⢠⠐⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠘⠐⠚⠁⡇⠀⢰⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⡆⢸⢀⠐⢰⠆⡆⢸⠐⡆⠆⠂⢀⠐⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠁⠀⠁⠁⠈⠈⠁⠈⠁⠁⠈⠈⠁⠁⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⢀⢸⡀⢦⠁⡃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⡀⠀⠀⢀⣤⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⡾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠙⢻⣦⠀⠀⠀⣀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠹⣷⡾⠟⠋⠉⠁⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠙⠛⠻⢶⣦⣄⠀⣀⣀⣠⣤⣤⣾⡏⠀⠀⣠⠶⠲⢦⣤⣿⣧⣤⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠈⠙⠟⠛⠉⠉⠉⠉⣿⠀⠀⠀⢿⣄⣰⡟⠁⠀⠀⠈⠻⣿⠛⠛⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠉⣿⡀⠀⠀⠀⠀⠀⣿⠶⣄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠶⠾⠿⢷⣄⡀⢀⣠⣾⣏⣀⡿⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣯⡀⠉⠉⠀⢀⣾⠋
⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣶⠾⠛⢿⣧
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣷⣶⣦⣤⣤
⠀⣀⣤⣤⣼⣿⡶⠶⠄⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠉⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠈⠻⠛⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⡀⠀⠀⠀⠀⠈⢿⡿⠃⠀⠀⠀⠀⠀⠶⣾⣿⢶⣦⣤
⠀⠀⠀⢠⣴⠾⢿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣽⣷⣾⣷⠶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⣶⠟⠛⠙⠛⢷⣮⣿⡿⠛⠷⠶⣦⣄
⠀⠀⠀⠀⢠⡶⠛⠁⣼⠏⠀⠀⠀⠀⠀⢻⡟⠛⠛⠛⠷⣦⣄⠀⠀⣀⣤⡶⠿⠛⢿⡇⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣠⣤⡾⠃⠀⠀⠀⠀⠀⠙⢷⡾⠛⠁⠀⠀⠀⠘⠷⢦⣤⡀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢷⣤⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠾⠛⣿⡄
⠀⠀⠀⣀⣤⣶⠶⠿⠷⠶⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠾⠿⢶⣦⣤⣀
⠀⢠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠘⠿⣦⣤⣴⡆⠀⠀⠀⠀⠀⣴⠟⠛⢶⡈⣿⠀⠰⠁⠀⡀⢰⠀⡆⠀⡄⠂⣼⡇⣰⠞⠻⣦⡀⠀⠀⠀⠀⠀⣤⣤⣤⡼⠏
⠀⠀⠀⠀⠀⠻⣧⣀⣠⣶⡄⠀⣿⡘⠷⠟⣠⡿⠀⠉⠁⠀⠈⠀⠀⠁⠀⠁⠀⢹⣇⠙⠶⠟⣼⠇⠀⣶⣄⣀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠛⠶⠛⠻⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢻⣶⡿⠛⠶⠾⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀⠘⠀⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡠⡜⢄⠉⣇⣀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⡠⠚⠢⢠⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⠳⢘⡌⠢⡨⠊⠀⠀⠰⠝⠊⢀⠀⠀⠀⠊⠈⢃⡀⠤⠆⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⡏⡭⡢⡀⠙⠢⠅⢀⡴⢊⠁⠀⠰⠇⠀⠈⠀⠀⠀⠀⣠⡄⠀⠀⢀⠀⠀⠀⠰⡹⠁⠹⠂
⠀⠀⠀⡖⠢⡀⠘⢄⡚⡀⠀⢀⠔⠉⠀⠣⠐⠀⠀⠀⠀⠀⠐⠂⠀⠄⠻⢀⡠⠐⠁⠀⠀⠀⠛⠃⢀
⠀⠀⠀⠘⢄⠑⠄⡄⠀⠀⠘⠀⠀⠀⠀⠀⣠⡾⢑⢄⠀⠈⠐⠁⠀⠐⠂⠁⠀⠎⠐⠀⠀⠀⠀⠀⠃⠁
⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⠀⠀⠀⣰⢯⣲⠃⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠞⢛⣼⠃⡐⠀⢀⠀⡰⡡⢀⠀⠀⠀⠀⠛⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠂⠀⠀⠀⠀⠠⡒⠁⠀⠀⠀⠉⢤⡾⠋⢉⠔⠀⠀⠈⢄⣔⢲⠇⠀⠂⠂⠢⡀⢀⠾⠆⠀⠞⢂⠄
⠀⠀⠀⠀⡀⢄⡀⠀⠈⠈⠁⠀⠀⢀⠀⠀⠀⠁⠀⢀⢤⡀⠀⠉⢻⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠁⠀⠀⢀
⠀⠀⡄⠊⡤⠊⠑⡀⠀⠀⠀⠀⠒⠀⠆⠀⠀⠀⠰⡟⠋⡰⠀⠀⠈⢇⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠈⢎⠀⠀⠅⠅⠀⠀⠀⠀⠀⠆⠐⠄⠀⠀⠀⣵⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡁
⠀⠀⠀⣊⡇⠀⠀⠀⠀⢀⠊⠀⠉⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠊⠉⠉⠀⠀⢠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⠀⠀⣈⠀⠀⠀⠀⠂⠀⠀⠠⠒⢀⠀⠀⠒⠀⠀⠠⡀⠀⠀⠀⡞⠂⠂
⠀⠀⠀⠀⠀⢶⣀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠐⠀⠀⠐⢄⣀⣶⣠⡔⠁⠀⠀⠀⠁⠀⠘
⠀⠀⠀⠀⠄⠄⠌⠍⡒⠀⠀⠀⠀⢀⠴⣗⢄⠀⠀⠀⠀⠀⠀⠀⠌⠍⢿⡇⠀⡃⠀⠀⠀⣀⣀⡀
⠀⠀⢠⢆⠀⢊⠈⠌⠀⠀⠀⠀⣰⡁⠀⠈⡪⠇⠀⠀⠄⠀⠀⠀⡂⠀⢺⣇⣀⡁⠀⠸⠛⠃⠉⢁
⠠⠀⠀⠀⠁⠆⠆⠀⠀⠀⠀⠀⠀⠀⢠⢊⠀⠀⠀⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢆⠄⠀⠘⠃⠀⠀⠂
⠀⠀⠄⠆⠶⠲⠀⠀⠀⢀⠀⠀⠀⠀⠈⠈⠀⠀⠘⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠤⠴⠶⠦⠤⠤⣀⡀
⠀⠀⠀⠀⠀⢀⡄⢀⡤⠖⠋⠉⠀⢀⡠⠴⠒⠚⠛⠛⠛⠒⠀⠉⠛⢦⣄
⠀⠀⣀⢶⡒⠈⡷⠉⠀⢀⠀⠀⠈⠁⠀⠀⣀⣠⠤⠴⠒⠒⠀⢀⣀⣠⠽⢚⡥⣰⠃
⠀⠀⠉⢢⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⡤⠶⠊⠉⠠⠒⣩⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣤⣤⡤⢤⣤⣀
⠀⠀⣰⠃⠄⠀⠀⠀⠀⠐⠂⠤⠤⠤⠤⠤⠤⠐⠒⠉⠀⠀⠀⠀⣠⡴⠞⠁⡌⣇⠀⠀⠀⠀⢀⣀⡠⠔⣉⠤⠒⠉⠀⠀⠀⠀⠀⠀⢉⣲⣄⣀⡴⠋⡙⠛⢶⡄
⠀⢰⠃⡜⠀⢠⠂⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣤⣤⠤⠶⠚⠋⠁⠀⠀⠀⢱⣹⠀⠀⣠⠞⠩⠎⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⢿⣀⡠⠀⠹⡄
⠀⡟⢠⡆⠀⡇⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣆⢷⡠⠔⠀⡇
⠀⡇⢸⡇⠀⢀⠄⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠆⠀⡇⢰⢻⠸⡀⠀⠀⠀⠀⠀⢈⣉⣉⠉⠑⠒⠤⣄⡀⠀⠀⠀⠀⣠⠟⡾⠇⠀⣸⠃
⠀⣧⠀⠇⣠⠋⣠⣤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠸⢀⠓⢦⣄⣦⠤⣄⣀⠀⠈⠙⠲⢤⡀⠀⠉⠓⠶⠶⠾⣫⠀⡷⠴⠚⠁
⠀⢻⡀⠀⡏⣾⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⠀⠀⣸⠁⠘⣇⣸⠰⠀⠀⣤⠀⠀⠉⠙⢦⡀⠀⢀⠙⣆⠐⠀⠀⠀⠊⠁⡔⡇
⠀⠘⣷⠀⠀⠙⢮⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡼⣰⠃⠀⠀⠈⢿⠀⡄⠀⠈⠀⠀⠀⠀⠀⠃⢸⡍⢱⡈⠀⠀⠀⢀⠄⡼⢡⡇
⠀⠀⠸⣧⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⢞⡵⠃⠀⠀⠀⠀⢸⡘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⠼⠁⠀⠀⠂⣡⠞⢁⣾⠁
⠀⠀⠀⠈⠳⣄⡙⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⢉⡴⠋⠀⠀⠀⠀⠀⠀⠀⢷⣄⣀⡤⠎⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠒⠉⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠈⠙⠓⠺⠷⠦⢄⣀⡀⠀⠀⠀⠀⢀⣀⣀⡤⠖⠋⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠂⠤⠤⠖⣋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡄⠈⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠙⠒⠤⣄⡀⠀⠀⣀⣠⣷⣤⠤⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠓⠤⠍⠓⢤⡀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣦⣀⣀⠀⠀⠀⠀⠀⢀⡨⢆⣀⢎⡀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠰⡄⠀⠀⠀⠀⢡⠈⠧⠀⠀⠀⠀⠹⣿⣿⣿⡿⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠀⠀⠀⠀⠀⡆⠣⢄⡀⠀⠀⠀⠀⠙⠛⠁⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠛⠃⠀⠀⠀⠀⠀⠀⢡⠀⠀⠉⠙⠒⠶⡶⠜⣦⠤⠴⠚⠉⢈⠀⠀⠀⠀⠀⠀⠸⣄⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠳⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡞⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⠲⡄⠀⠀⢀⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢠⠇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠦⠤⠿⠗⢒⠋⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢑⣠⠀⡀⠀⣀⣠⣼⢰⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⠞⠉⢂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠾⢦⢰⣴⡉⡿⠾⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠃⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠤⣤⣄⣀⣤⡤⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣄⡤⡀⠀⣏⠋⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢮⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠜⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠈⠰⠩⠽

⢀⣤
⢸⢛⠀⠀⠀⠲⣀⠀⢀⠀⠀⠀⠀⠀⣐⠀⠀⢰⠆⠤⢦⣤⣤⣄⣀⣀
⢸⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⢀⣀⡆⠸⠃⠀⢀⠙⠚⡻⢯⡙⠲⠨⣧⠈⣠⡂⠀⠀⠀⠀⠀⢠⡆
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⣿⡇⠸⠇⠀⠀⡤⢤⠁⢸⡟⠙⠙⠿⣷⠊⠁⠀⠀⠀⠀⠀⣼⡁⢀⡇
⢸⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠀⠁⡹⡇⢰⠀⠀⢀⠀⠀⠀⢿⢿⠀⠒⢰⡘⢷⠀⠀⠀⠀⣶⣷⡿⠀⣩⡇
⢸⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⣠⣤⣿⡇⢸⡇⠀⠷⠄⠀⠀⣾⣿⠷⡄⠠⣉⣻⣆⣤⡰⣆⣿⣿⠁⠀⣿⣇
⢸⣿⣦⣄⠀⠀⠀⠀⢨⡄⠀⠀⠈⠙⢛⣾⡇⢸⡁⠀⢐⣶⡄⠀⢹⣿⠠⠆⠀⠉⢹⣿⣿⠷⣿⡿⡟⡇⠀⢼⣤
⢸⣿⡿⣿⠨⠀⠄⠀⣼⢇⠀⠀⠀⠀⢠⢿⡧⢸⠆⢠⠤⠹⠇⠀⣾⣿⠃⣤⡀⠀⢸⣯⣟⡿⣛⠀⠷⠁⣴⣿⣿
⢸⣿⠀⢈⡁⠀⠀⢠⣃⡘⡀⡀⠀⠀⠀⠈⣇⢸⣷⣟⣾⣶⢀⠀⢿⣿⣶⣿⣿⣿⣿⣏⣿⡇⠉⠀⢐⣻⣿⣽⣿
⢸⣿⣴⣬⣿⡀⠀⠾⠟⠽⢷⡇⠀⠀⠀⠀⣿⢸⡿⣿⣿⡁⠀⣲⢾⣯⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⡽⣿⣿⣿⣿
⢸⣿⣿⣿⡃⢀⣦⣀⣀⣀⣼⡇⡄⣤⣴⡴⡿⢸⡇⠐⣇⣟⠀⠉⣿⡏⠙⠛⠋⠛⢹⡟⠙⣸⠟⠛⠿⡿⣿⣿⣿
⢸⣿⡿⣛⣿⣿⣇⠀⠄⠀⢻⡇⠰⣮⠔⣯⡇⢸⡇⢀⣿⣿⠦⠀⢿⠓⠂⠀⠀⠀⢸⡆⠀⢸⠄⠀⠀⠀⠼⢿⣿
⢸⣿⣿⣥⣾⣿⠛⠀⠀⠈⢻⣿⠷⣴⣾⣿⡿⢸⣿⣗⣯⣉⣁⡤⠯⠥⠤⠤⠤⠄⠘⠗⠒⠒⠒⠛⠓⠀⠈⠉⠁
⠘⠿⠿⠿⠿⠛⠂⠒⠒⠒⠛⠛⠋⠉⠉⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣳⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣄⣀⣤⣶⠟⠃⠀⢿⣿⣿⣿⢻⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠈⣿⣿⣿⣇⣷⢹⣿⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⢹⣿⣦⢠⣤⢀⡀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⢠⠦⣠⣶⣦⣤⣤⣤⣵⣦⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣏⣾⣿⣿⣶⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⢠⣀⣴⣿⣿⣿⡿⢉⠀⠀⠀⠰⣦⣤⣄⣀⠀⠀⠉⢩⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣧⣶⣴⣸⣿⣿⣿⣿⣶⣶⣿⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠛⠇⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠
⡄⠀⠀⠀⠀⠀⠀⠀⠉⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣼⣿⣿⣿⣿⣿⣿⡇⣹⡁⢀⣴⡀
⣿⣶⣄⣠⣾⣿⣿⣿⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠛⠉⠿⣿⣏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣯⣷⣿⣷⣿⣿⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⣄⣼⡿⣷⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⡏⢟⡇⠻⢿⠿⣇⠊⢳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠋⡿⡿⠿⢶⢷⡌⢀⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠶⠿⡏⠂⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠙⣻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⡿⠂⠁⠀⠀⢠⡀⣀⢂⠀⠀⠀⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣬⡙⢋⠇
⠀⠀⠀⠀⠀⣠⣶⡿⠿⢫⠙⠿⠶⠒⢂⠐⠛⣄⠻⠿⠻⠁⠀⠀⠠⣆⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⢦⢢⣦⣼⡇
⠀⠀⠀⢀⣾⣷⣮⡀⣠⣀⡄⣀⠀⣰⣺⣠⣀⢿⠋⠀⠀⠡⠀⠀⠀⠈⠝⠀⠀⣿⡛⠶⣤⠀⠀⠀⣼⣿⣶⣿⣿⡿
⠀⠀⢀⣾⣿⣿⡟⠁⠈⣛⣾⡿⠒⠺⣿⣿⣉⠀⠀⠀⢀⡴⡆⠇⢠⠀⠀⠀⠀⠈⠛⠂⠈⠁⠀⢸⡿⢿⠿⢿⡿⠁
⠀⠀⢸⣿⣿⣟⠂⠀⣼⢿⡿⠁⠀⢠⣿⣿⡇⠀⠀⢀⠊⢠⠇⠀⠠⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⢹
⠀⠀⢸⣾⣿⡏⠀⠀⢻⣸⣄⣼⠯⠛⢱⣶⣬⠀⠀⠈⠀⠁⠀⠀⠄⠈⠒⠦⠄⠀⡀⠀⠀⠀⠀⠀⠒⠃⡠⠂
⠀⠀⠘⣿⡜⡿⠁⠀⠸⣷⡜⣿⡷⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠁
⠀⠀⠀⢿⡷⣘⣴⣦⢀⠀⣤⣿⠃⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠘⠬⢀⡀⣀⢀⡀⠤⠄⠐⠊
⠀⠀⠀⠘⣿⡹⣿⣷⢀⢸⣿⣇⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⢀⠈⢀⠘⣿⣷⡍⢻⣶⣿⡉⠑⠚⠁⠀⠀⢀⣠⠤⠖⠊⠁
⠀⠀⠈⣾⣶⣿⡇⣾⣿⣽⣿⣿⡿⠇
⠀⠀⠀⠙⠝⠟⠿⣿⣦⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⣶⣆⣤⠶⠂⠀⠐⠂⡒⠀⢀⣠⠤⠆⠤⠤
⠀⠀⠀⠀⢸⣿⣿⣿⣦⣤⣪⡷⢾⢿⣥⣿⣦⠹⣿⣦⢀⠀⠃⠀⠰⢿⡀⡀⠄⠀⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⣤⣤⣤⣀⣠
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣻⠿⣻⣧⣶⣿⣿⣿⣿⣷⠒⠀⠌⡉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠉⢹⣋⣙
⠀⠀⠀⠀⢸⣿⣿⣟⣣⡿⢿⣿⡏⠉⠐⠘⡟⠛⠟⠁⠉⠻⠷⡶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣿⣿⣻⢟⡿⣶⡆⣀⣉⡉⠉⠋⠠⠀⠀⠊⠒⡖⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⣿⣿⣿⣯⣼⣦⣽⣿⣿⣢⠌⠀⢀⣄⠀⠀⢘⡁⢀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⢸⣿⣿⠀⢝⣯⣶⣬⣿⡿⣿⣶⣶⣿⣿⣿⣷⣌⣟⠻⢿⡀⠀⢠⣄⣄⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡀⢀⡅⠀⠻⠤⠨
⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣿⣄⡸⠿⣿⠟⢻⡌⠛⠁⡈⢘⢿⠏⣄⠀⠈⠉⠩⠭⠉⠁⠀⠀⠀⠀⠀⠀⢀⡷⣆⢈⡉
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣻⣟⣽⣷⡤⠀⣎⣁⡀⣀⣄⠀⠀⠈⠊⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣛⡘⡇
⠀⠀⠀⠀⢸⣿⣿⡛⣿⣿⣿⣿⣿⣼⠉⡡⣇⡤⢁⠀⠛⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⢰⣠⣿⣿⣿⡆⢘⠿⠟⠾⢿⣧⣶⣶⣰⣖
⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⠟⢷⠓⡶⠏⠀⠃⢠⠄⠀⠀⠂⢀⣶⣦⣽⣷⣶⣶⣤⣤⣽⣿⣉⣀⠀⠁⣿⣀⠀⢴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠻⣿⡿⠿⠽⢶⣄⠈⠤⠀⣀⣶⣤⣯⣿⣦⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣦⣬⣿⣾⣼⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣠⣿⣿⣴⡖⠀⣠⠁⢀⣠⡉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡛⣿⣿⡛⠛⠛⢻⡟⠛⢛⣛⣛
⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣯⣄⣷⣷⣶⣿⣿⣿⣿⣿⣶⣾⣿⡿⠿⡿⢿⣟⣿⠟⠛⠻⣿⢿⣽⡿⠛⠛⠛⠛⠛⠛⢹⡟⢉⣉⣉⢉
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣉⣉⣙⣛⡛⣋⣀⣀⣀⣉⡉⢁⣀⢀⡇⠀⠉⠁⠀⠀⠀⣾⠠⣿⢣⡀⠀⠀⠐⠂⠀⢸⡧⠽⠛⠋
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡯⠿⠋⠉⣻⡟⢻⢿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⣠⡶⣦⣤⣤⣼⣧⣤⣤⣤⣤
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⣡⣴⡶⠟⠋⣉⣵⡦⠻⠿⠿⠛⠛⠉⠙⠻⢻⣿⣿⡿⣟⡛⢻⠛⣻⣿⣾⡛⣟⠛⠛⣿⣿⣿⣷⣿⣿⣭⣿
⢀⣀⣀⣀⣸⣿⣿⠿⢿⣿⣯⣥⣤⣴⣾⣿⣿⣿⣿⣯⣆⣀⣀⣠⣤⣴⣿⣿⣋⡙⠫⣝⣳⣿⡻⢿⠿⠁⢧⠈⠹⠿⠿⠏⠛⠛⡻⠿⠟
⠀⠉⠻⢿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠏⠒⠉⠓⠀⠁⠉⠉⠫⠖⡿⠿⠾⠟⠋⠈⠀⠐⠋⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠈⠻⢿⣿⠋⠉⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡛⠘⢏⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠡⠀⠈⠙⠢⢤⣀⠰⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢱⣄⠀⠀⠀⠙⣷⣾⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⣠⡆⢀⠤⠀⣐⣈⡀⠐⢿⡇⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠰⠖⠉⠒⠓⠳⢰⠋⠉⣩⡶⠛⠁⠙⢦⠀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠃⠀⠀⠀⠀⢨⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠞⠃⠀⣼⠀⠀⠀⠀⠀⡸⠃⢸⡥⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠞⢃⣀⣀⣤⠶⠛⠁⠀⣠⣵⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣴⠖⠲⠤⠔⠒⠒⠒⠐⠂⣀⡤⠖⠛⠉⠉⠁⠀⢀⣀⣠⣤⣿⣷⣟⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠉⠢⣄⡀⠀⠀⠀⣠⣴⠟⠁⠀⠀⢀⣀⣤⣴⣾⡿⢟⡭⠏⠹⡋⣋⢱⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠻⠟⠗⠖⠻⠟⠻⠋⠉⠁⢀⠈⢰⣞⣴⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡦⣤⣄⣠⡄⠀⣀⠀⣀⢀⣀⣀⣁⣘⢠⣂⣠⣤⣼⣾⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢣⢹⡄⢹⡟⠈⢿⠛⣿⠛⢻⡟⢻⠟⢻⡟⢻⡏⢩⣇⣸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡎⣧⠀⣿⠀⢸⡀⢿⠀⣸⡁⢸⠃⢼⠀⢺⠇⢺⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⣸⡆⢿⠂⠸⡇⢸⢢⣼⡇⢸⠂⣿⡀⣿⢓⣿⣿⣿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣿⣇⢺⣧⠀⣷⢾⡧⣿⣧⣿⢲⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡸⣿⣿⡧⣿⡎⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⣿⣿⣧⣿⣿⣷⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣶⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣾⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢟⡛⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠿⠟⠻⠛⠛⠙⠉⠉⠉⠁



⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⠶⠶⢶⣶⡶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⣰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣄⢀⣠⣤⣶⣶⣴⣏⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡏⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⣇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⠈⢿⣧⣀⣠⣿⣿⣿⠋⠉⢉⣿⣿⣿⣿⣿⠀⢀⣀⣀⡀⠻⠚⠛⠛⠓⠦⣄
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠻⢿⣿⣿⣿⣿⣷⣶⣿⡿⢟⣽⣿⣿⠀⠙⠛⠛⢡⡶⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣶⢀⠀⠈⠙⠙⠛⠛⠙⠉⣰⣿⣿⣿⠇⠀⣀⡀⠀⠸⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣈⣉⡙⠛⠛⠛⢿⡇⢠⣶⣿⣶⣄⣀⣤⡤⣼⢿⣿⣿⣿⣄⠀⠀⠀⢀⣠⣴⣾⣿⣿⡿⠃⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠈⠉⠛⠶⠖⠚⢻⣿⣿⣿⣿⣿⣥⣀⣀⣸⣇⠙⢿⣿⣿⣾⣼⣾⣿⣿⣿⡿⠟⠻⡄⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠉⣻⣿⠿⠿⠛⠋⠁⠀⠀⠀⣹⣄⣀⣀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⣿⣿⣿⣟⠀⠀⠀⣿⠿⠀⠀⠀⠀⠀⠀⠀⢈⡷⠼⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢸⡄⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠠⠤⢾⡦⠀⠀⢤⣮⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠐⢺⡟⡽⠃⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⢀⣼⣧⠄⠀⠞⠛⠉⠀⠀⢠⠀⡄⠀⠀⠈⠋⠀⠀⠀⣻⠟⠓⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢀⣸⢷⣆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡽⠛⠦⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠓⢶⣤⣤⣤⣤⣤⣤⣤⠤⢶⡾⠋⠀⠀⣀⣠⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠶⢿⣴⣫⣤⡖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⠇⠀⠈⡇⠀⠈⠹⡆
⠀⠀⠀⠀⠀⠀⢠⡞⠉⢹⣿⣿⣿⣿⣿⣿⣦⣤⣾⣇⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠿⠿⣿⣿⣿⡿⣿⡿⠿⠤⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡆⠀⢀⣀⠀⠀⣴⡆⠀⣠⡄⠀⢀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡗⢰⡿⣿⡇⢸⣿⡇⠀⣿⡇⠀⢸⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣷⡿⠁⢸⣿⡇⠀⣿⡇⠀⢸⣇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢸⣿⢻⣯⣤⣶⢸⣿⣷⣶⢿⣿⣿⠏⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠈⠛⠈⠙⠋⠁⠀⠛⠛⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⢀⣦⠀⠀⣤⡄⠀⣰⣶⣆⠀⣴⣶⡀⢶⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⠉⠿⠻⣿⣠⣄⢻⣿⢰⣿⣿⡇⢸⣿⣿⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⡀⣿⣿⣿⣼⣿⢸⣿⡿⢀⢸⣿⡃⢀⣠⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡇⢿⣿⣿⣿⡟⠈⢿⣿⡿⠉⠻⣿⡿⠟⠉⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣋⠀⣬⠉⢈⡉⠁⠀⠀⣤⣄⠀⣠⣶⣶⡄⣠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⢠⣿⣿⡇⠀⣸⣿⣿⡄⢹⣿⢸⡇⠈⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⢸⣿⣾⠃⠀⣿⡿⢿⡇⢸⣿⣾⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡎⢿⣧⣤⡀⣿⠃⢸⣷⢸⣿⢻⣷⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠿⠇⠈⠙⠋⠘⠟⠀⠈⠋⠘⠛⠀⠛⠁⠸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⣷⣦⡀⠀⠀⣀⣠⣤⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⣶⣦⣤⣀⠀⠀⠀⢀⣀⣀⣀⣀⣼⡟⠁⠀⠀⠀⠙⣿⣶⡿⠟⠋⠉⠁⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠉⠛⠿⠿⠿⠛⠛⠛⠛⣿⡟⠀⠀⠀⣠⣤⣤⣘⣿⣄⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢰⣟⢁⣿⠟⠛⠛⠻⣷⣶⣾⠿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠛⣿⡏⠀⠀⠀⠀⢸⣿⣀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⡶⢿⣷⡀⠀⠀⢀⣼⠟⣻⡆⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⠛⠛⠋⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣴⡾⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⡷⠶⠶⠶⠄
⠀⠀⠀⠀⣤⣴⠶⢾⣿⡷⠶⠦⠀⠀⠀⠀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠰⣾⣿⣷⣤⣤
⠀⠀⠀⠀⠀⠀⢠⣤⣼⣿⣿⠟⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡟⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢿⣦⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢿⣿⣧⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣾⣿⣁⡀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠰⠟⢉⣴⡿⠋⠉⠁⠀⠉⠙⢿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠟⠋⠁⠀⠈⠉⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⡿⠶⠿⢷⣿⣿⣿⠿⠟⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⣀
⠀⠀⠀⠀⠀⣠⣾⡿⠟⢻⣇⠀⠀⠀⠀⡀⠀⢀⣰⡿⠀⠀⠈⠙⠿⣿⣦⣴⣿⠟⠋⠀⠀⠀⠘⣿⣄⠀⠀⢀⠀⠀⠀⢀⣾⠟⢿⣷⣆
⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠙⢷⣶⣶⠿⠛⠿⠟⠋⠀⠀⠀⠀⠀⠀⠘⢿⡿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠿⠷⠾⠟⠁⠀⠀⠘⢿⣷⡄
⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣄⡀⠀⠀⢀⣠⣴⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣶⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢛⡻⢿⣿⠇⠀⠀⠀⠘⣿⣿⠿⠿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⠀⢿⣷⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠛⠀⠀⣄⣠⡄⠀⠈⠻⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⣉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠉⠉⠐⠢⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⡠⠊⠁


⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⠶⠶⠶⠶⢶⣶⠶⠶⣶⠶⠶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣤⣀⠀⠀⠀⠀⣀⣀⣠⡾⠋⠀⠉⢻⣦⣶⠾⠛⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠉⠙⠻⠿⠛⠛⠛⢻⡿⠁⠀⢀⣤⣄⣹⣧⣀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣤⡟⠉⠉⠙⢿⡟⠛⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣤⣿⣇⠀⠀⢀⣾⢷⡄⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠙⠷⢶⣟⠛⠛⠁⢀⣾⡃
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣶⠾⠛⠹⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣤⣤⣤⡀
⠀⠀⠀⢠⣤⠶⢾⣷⠶⠄⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⠀⠀⠀⠉⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣷⣀⡀⠀⠀⠸⠿⠇⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠈⠿⠏⠀⠀⠀⠀⢻⣿⠷⠶⠄
⠀⠀⠀⠀⠀⠐⠛⠻⣯⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣾⣃
⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣥⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⡉⠙⠻⠆
⠀⠀⠀⠀⠀⠀⠚⢉⣾⠛⠉⠁⠈⠉⠻⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⠟⠉⠀⠀⠉⠻⣧
⠀⠀⠀⠀⠀⣠⣶⣿⡇⠀⠀⠀⠀⠀⢀⣿⠙⠻⢿⣿⣛⢛⣿⡿⠟⠋⠉⣿⠀⠀⠀⠀⠀⠀⣸⣷⣦⡀
⠀⠀⠀⣠⣾⠟⠁⠈⠿⣦⣤⡶⠶⠶⠟⠁⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠙⠷⠶⠶⢶⣤⡶⠟⠀⠙⢿⣦
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⢀⣿⠇⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⢸⣿⠀⠀⠀⠀⢂⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⡴⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⢸⣿⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⡄⠁⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⣼⡏
⠀⠈⣿⡆⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠚⠲⣄⠓⡤⢲⠐⠀⡈⠀⠀⠀⢨⠀⡆⠀⠉⠱⡌⠙⡐⠱⠀⢠⣿⠃
⠀⠀⠘⣿⣄⠀⠀⢀⠀⠀⠃⠘⠉⠙⢶⠐⠂⡈⢆⠀⡘⢀⠈⠔⠂⠀⠀⢸⠀⡇⢠⠘⢠⠳⡀⠃⡀⢠⣾⠏
⠀⠀⠀⠘⢿⣦⡐⠞⠀⠀⠘⠦⠤⠔⠃⠲⠚⠁⠈⠖⠁⠈⠒⠚⠁⠀⠀⠀⠓⠁⠀⠑⠋⠀⠑⠚⣹⣿⠋
⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⣀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠉⢉⠱⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣆⢴⣏⣀⣀⣠⣨⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡾⢿⠻⠿⠃⠁⠝⠉⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠛⠻⠟⠁⠀⠀⠀⠀⠑⢐⠀⠀⠀⢀⡈⠳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣆⠦⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡗⣤⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⢿⣿⡁⠬⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⠲⣶⡀⠀⠀⠀⠀⠀⠀⢺⣤⠋⠀⠀⢀⢻⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣦⡝⠁⠀⠀⠀⠸⠘⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡇⠀⠀⠀⠀⠀⠢⢹⡄⠀⠀⢀⡤⠃⠀⠀⠀⠀⠀⢀⣾⡄⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠁⠀⠀⠀⠀⠀⠀⠻⢷⡀⠀⠉⠀⠀⢀⠀⠀⠀⠀⣠⠟⠛⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠖⠛⣳⠀⡤⠀⠀⠀⡀⡤⠔⣻⠟⠰⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⣤⠶⠉⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢄⣀⠀⠀⠀⠀⢀⣀⣤⣾⠃⠀⠠⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠛⠛⠛⠋⢻⣿⣏⠀⠀⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣾⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠁⡄⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⠿⠉⠛⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠏⠁⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⣼⣧⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⡀⢸⣿⠘⣇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢻⣿⡀⣿⠀⠀⠀⠀⠀⠀⢸⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢩⠂⠀⠀⠀⠀⠘⢻⡇⣿⠀⠀⠀⠀⠀⠀⢸⠃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⡇⠿⠀⠀⠀⠀⠃⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⢻⠿⡿⠟⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠃⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⢀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠋⠀⠀⡀⠈⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⠂⠀⣤⡀⠈⠁⠀⢀⣀⠀⡗⢄
⠀⠀⠀⠀⠀⢠⠎⠀⠑⠄⣀⣀⣀⣀⡀⠀⠄⣀⡀⣉⠰⣡⢀⠦⡀
⠀⠀⠀⠀⠰⠁⢄⠐⠃⠀⠀⠀⠒⠀⠀⠠⠤⠀⠀⠀⢀⠤⢊⣀⠈⢆
⠀⠀⠀⠀⡆⠀⢀⠉⠒⠀⠤⠀⠀⠀⠀⠤⠄⠀⠒⠈⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠈⠁⠀⠀⠋⠀⠀⠀⠐⠒⠀⠀⠘⠁⠀⠉⢁⡀⡴⠁
⠀⠀⠀⠀⠀⠸⠈⠉⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁⠀⠉⠁⠉⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠩⠶⠦⠀⠀⠀⠀⠤⠶⠭⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠢⠴⠃⠀⠀⣀⢄⢶⠋⢉⠔⠖⢄⣀⡀⠤⠐⠈⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡤⠠⠤⠄⠀⢀⠔⢠⢂⡆⢀⠊⠘⠀⠱⠱⡤⠄⠒⠁⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⠧⠀⠀⠀⠤⡜⠀⠂⢻⠃⣬⡀⠿⠆⠀⠀⡇⠠⠤⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⢄⢰⠀⠸⠈⣻⠏⢀⠀⡜⡰⠁⠒⠠⢄⣀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠈⢀⡠⠘⢢⡀⠀⠄⠰⡀⠈⣎⠔⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠒⠁⠀⠀⠀⠈⠑⠚⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠙⠷⠞⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠾⠁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢦⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢤⡴⠏⠻⣦⡀⠀⠀⠀⠀⣰⡟⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢰⣿⠶⠶⢶⣤⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠋⠛⠳⢦⣄⣀⣀⣼⣷⣤⣤⡟⠁⠀⠀⠀⠘⣷⡾⠛⠋⠉⠉⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⣿⠀⠀⢀⣴⠞⣦⣼⣦⣤⡀⠀⢀⣀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣼⠋⠀⠀⠈⢻⡟⠛⠋⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⡴⢿⡀⠀⠀⢀⡼⢻⡆⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣟⠙⠋⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡴⠞⠋⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⡤⠾⣿⠶⠆⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠘⠉⢹⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠉⠁⠀⠀⢿⣆⣀⠀⠀⠀⠻⡿⠃⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠻⡿⠇⠀⠀⠀⠀⠶⣾⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡏⠀⠀⠉
⠀⠀⠀⠀⣠⠶⢦⣀⡴⢶⣬⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠻⢶⣄
⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⢸⡇⠈⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠁
⠀⠀⢀⡶⠋⠀⡶⠉⢳⠀⠘⢶⡀⠀⠀⠀⠀⢉⣩⣿⠟⠛⠛⠛⠒⠛⢿⡟⢛⣿⣋⠉⠉⢙⣷⣄
⠀⠀⠸⣧⣀⡀⠙⠒⠋⠀⢀⣸⠇⠀⣀⣤⠞⢫⡿⣹⣿⣷⡄⠀⣀⠀⣸⣷⢿⣿⣿⠃⠀⣿⣿⡿⣷⡀
⠀⠀⠀⠈⠉⢻⣄⣀⣠⠟⠉⠻⣾⠋⠉⠁⢀⡿⠀⠘⠻⠛⠀⣾⣿⣿⣿⠙⣧⠉⠁⣀⡀⠈⠉⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢸⠃⠀⠀⢸⣿⣿⣦⠀⠀⠀⠈⠛⣹⠇⠀⠘⢷⣾⣿⣿⡆⢀⣴⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⣸⡟⠿⠋⢀⣤⣄⠀⣰⠟⠀⠀⠀⠀⠙⢿⣋⠀⠈⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠙⣧⠀⠀⠻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⣀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⠉⠛⠒⠶⠶⠶⠤⢴⡶⠒⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⠶⠶⠶⠶⠶⠶⠟⠛⠛⠛⠛⠋⠁

⣟⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⠟⠁⠀⠀⣠⣿⢃⣾⠃⠀⣾⠇⠀⣠⡿⣧⣼⡿⠛⠁⠀⠀⠀⠀⠀⠀⢠⡾⠃⢀⡜⠁⠀⠀⢠⣿⣧⣃⣤⢾⡟⠀⢀⡆
⠛⠁⠀⢀⡴⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⣠⡾⠛⠻⠟⠁⠀⠀⢿⣤⡾⠋⠀⠀⢻⣀⣀⣴⠖⠀⠀⠀⠀⢀⡿⢁⡴⠋⠀⠀⢀⡴⣿⣏⡟⠉⠀⣾⠁⢠⠚⠷⣴
⣀⣀⡴⠞⠀⠀⠀⠀⠀⠀⠸⣿⡀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣾⡷⠋⠀⠀⢀⡴⠋⠀⠻⠛⠀⠀⠀⠿⠖⠁⠀⠀⠸
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄
⠀⠀⠀⢠⡾⠛⣦⠀⢀⣠⣾⠆⠀⠀⠀⠀⠀⣴⡶⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⢀⣿⣿⣰⣿⡶⠿⣿⡏⠀⢀⡄⠀⢠⣾⠋⢠⣾⠇⠀⠀⠀⠀⠀⠀⣠⡟⠁⢀⡼⠁⢀⠄⠀⠀⠀⢴⡆⠀⠀⢀⡀⠀⢀⣴⢿⡆⢠⡀⠀⠀⠀⣠⣿⣿⠁
⠀⣴⣿⣿⢋⣿⠋⠀⣸⣿⠀⢀⣾⡇⢠⣿⣃⣴⠿⠃⠀⠀⠀⠀⠀⢀⣾⡿⠁⣰⠟⠀⣠⢷⣠⡤⠊⣠⠏⠀⠀⠀⣼⠃⣠⣞⡡⠋⠀⢸⠃⠀⠀⣰⣿⡿⠁
⡾⠋⢿⣇⡼⠃⠀⠀⣿⡇⢠⣾⠉⠓⢾⡿⠋⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣡⡞⠁⢠⢾⣏⡠⠀⠀⢠⠏⠀⠀⣠⠎⠻⣴⣿⠋⠀⠀⣠⡟⠀⠀⢠⣿⡿⠁
⠁⠀⠀⠉⠀⠀⠀⠀⠻⢷⠿⠁⠀⠀⠸⣇⣀⣀⣤⡴⠂⠀⠀⢀⣿⣿⠟⠃⢀⡴⠋⠈⠉⠀⠀⠀⣏⠀⢀⡼⠁⠀⠀⢾⣿⣀⣠⣾⠏⠀⠀⢠⣿⡿⠁⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢸⣿⣟⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠿⠶⠋⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⣼⡿⠁⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⣠⠟⠀⠀⡠⣄⠀⠀⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀
⡆⠀⢸⡆⠀⠀⠀⠀⠀⢀⣾⣿⢃⡾⠋⠀⣠⣶⡇⠏⣠⣾⠏⠀⣴⠂⢠⣾⣿⣿⠋⠀⠀⡀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⢠⣿⢻⡆⠀⣠⣿⠃⢀⡀⠀⣠⡿⠋
⠀⠀⣸⠇⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⢀⣴⣿⣏⡟⠋⣹⡟⠀⡴⣿⣤⣿⡿⠋⠁⠀⣠⠎⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣼⡿⣿⣧⣾⣿⠃⠀⣸⡇⢰⣿⣡⣾
⠀⣴⠏⠀⠀⠀⠀⠶⣾⣿⡏⠀⢀⣴⠏⠸⠿⠟⠀⠀⢿⣧⠞⠁⠀⠹⣿⣄⣀⣤⡾⠃⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠛⢹⡏⠀⣰⡟⠙⣿⠟⠋⠁
⡾⠋⠀⠀⠀⠀⠀⠀⣿⣿⣥⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⣠⣿⠛⢿⠟⠀⠀⣿⡇⣴⡿⠁⠀⢿⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠙⠻⠛
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⠀⣠⡖⢶⡄⠀⢀⣴⡷⠀⠀⠀⠀⠀⣠⣾⠟⣶⡀⢠⡏⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⢠⣿⡇⣼⣧⣶⢿⣿⠃⠀⣀⠀⠀⣴⡟⠁⣴⡿⠀⢸⡇⠀⠀⣠⣾⠟⠀⠀⠀⣠⠖⠀⣸⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀
⠁⣶⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⣰⣿⡿⢻⣿⠉⠀⣾⡏⠀⢠⣿⠀⢰⣿⣠⡾⠛⠀⠀⠀⠳⠶⠿⠋⠁⠀⠀⢠⣾⠏⢀⡼⠁⣠⠈⠀⡠⢀⡟⠁⠀⢀⡖⠀⣰⢏
⢠⠇⠀⢠⣿⡿⠃⠀⠀⠀⢠⣾⠟⣿⢃⣾⠃⠀⢰⣿⠃⢠⡿⠉⠳⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⣴⠋⢀⣴⠋⡙⠋⢀⡞⠀⠀⣠⢾⣇⣼⡿⠋
⠏⠀⢀⣾⣿⠃⠀⠀⢀⣴⡟⠁⠀⠉⠛⠁⠀⠀⠈⢿⣴⠟⠁⠀⠀⠹⣇⣀⣠⣴⠖⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠁⣴⠋⠘⠛⠀⠀⡼⠀⢀⡴⠁⠀⢸⣿⠀⣀
⠀⠀⣸⣿⠃⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢁⣠⠟⠁⠀⠀⠀⠀⠀⠿⡶⠋⠀⠀⠀⠈⠛⠛⠋
⠀⠀⣿⡏⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠁
⠀⠀⠻⢧⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⣠⣾⠟⠁⢠⣤⣄⠀⠀⣠⣶⠄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢷⠀⠀⠀⢀⡀
⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣠⣾⡟⠁⠀⢠⣿⣧⣿⠶⢻⣿⠃⠀⢠⡄⢀⣔⢋⣼⠃⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⣠⠏⠀⠀⢠⡏⡃⠀⠀⣠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⡠⠐⢄⠀⠀⠸⣧
⢀⣶⠟⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠊⠀⠀⠀⠆⠀⠀⣿
⣾⠃⠀⠀⡀⢀⠈⠻⢷⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⢶⠶⠶⠿⢤⣤⣁⠀⠀⠀⠀⢠⠀⠀⣿
⣿⠀⠀⢰⠀⠀⠑⢄⠈⠻⣦⡀⣀⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠘⠀⠀⣿
⣿⡀⠀⢸⠀⠀⠀⠀⠑⢀⣸⠟⠋⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⢰⡟
⢸⡇⠀⠈⡀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠻⠿⠦⠀⠀⠘⣦⣾⠃
⠈⣿⡀⠀⢡⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⠀⡀⠀⡀⠀⠀⠈⢿⡀
⠀⠸⣧⠀⢀⡾⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔⠁⠀⠀⠀⣠⣖⡊⠙⠓⢴⠁⠀⠀⠀⠘⣷⡀
⠀⠀⢻⡆⢸⠁⠀⢠⡿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣠⡆⠀⢇⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢿⣾⠀⠀⠈⠀⠀⢀⣱⢄⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⢿⣿⠿⠃⢀⡎⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠈⣿⠀⠀⠀⠐⠢⡞⣿⣿⣄⣠⡄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠠⠴⠊⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⡇⢿⣿⣿⡿⠃⢀⠇⠀⠀⠀⣀⡠⣤⣄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠘⢦⣈⣁⣀⡠⠊⠀⠀⣀⠾⣷⣶⣿⠏⠀⠀⢠⠀⠀⠉⠑⠢⡀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⣧⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠘⢤⡀⢀⡴⠃⠈⠉⠉⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠠⠂⠀⠀⣀⡤⠞⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣤⣄⣀⣁⡀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠶⠚⠋⠙⡄⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡏⡙⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠸⣧⣀⡀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡏⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⠟⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣇⣴⠟⠁⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠋⠁⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠛⠒⠒⣾⣤⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢰⣃⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⡀⠀⠀⣰⢋⠁⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣠⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⣿⣦⣤⣅⣀⣠⣤⣇⠀⠀⡄⠀⠠⠀⢀⣇⣀⣀⣤⣴⠾⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣿⣶⣷⣶⣿⣾⣟⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠓⠦⠤⠖⠋⠙⢦⣀⠆⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣄⠈⠀⠀⠀⠀⠀⠀⠀⢀⡔⠩⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠙⠲⣦⡀⠀⠀⠀⣰⠏⠀⠀⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⣠⠞⢳⡝⠲⣤⣼⠃⠀⠀⠀⠠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠚⠒⠂⠘⠁⠀⠀⠙⠛⠛⢻⡀⠀⠀⠀⠀⠈⠛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⢠⡀⠀⢀⡴⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡤⠤⠤⣤⣤⣼⠳⣖⠋⠀⠀⠀⢠⠞⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⠀⠀⠈⢹⡟⠛⠳⣤⣀⠀⠘⣷⠶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⠛⣏⠁⠰⣞⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠃⠀⠀⠀⢰⣏⠀⠀⠀⠀⣀⠀⣾⣄⠀⠈⠙⠚⠦
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣤⡞⢻⣄⠀⡼⢻⣉⠛⠀⠸⣄⣠⡄
⠀⠀⠀⠀⣀⠀⣰⠏⠉⣿⢣⠾⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣰⠀⡜⣼⠛⠡⠚⣻⡗⣾⡓⠀⠀⠑
⠀⠀⠀⢀⠋⠉⠁⠀⢠⣿⡆⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⣼⡉⠀⠀⠀⡙⠀⠈⠙⢆⠀⠤⣄⣀
⠀⠀⣀⣈⣣⡀⠀⠀⣺⣿⣥⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠝⠀⠀⠙⣇⣀⢄⠁⠀⠀⠀⠀⢻⡤⠞⠁
⢀⡴⠋⠀⠉⠻⢦⡾⢻⣯⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠒⢲⡶⠚⠑⡟
⠈⠻⣄⠀⠀⢠⡟⢁⡾⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡤⠀⠀⠀⠐
⠀⣠⠟⠀⣴⢋⡴⠛⡁⢺⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡑⢾⣷⠂⢈⡁⠐
⠰⡏⠀⠀⣻⣸⡇⠈⠀⢈⡇⠀⣠⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⡀⣀⡹⡄⠈⠂⠈⠃
⠀⠙⢦⠄⠹⠏⠀⣀⣀⠀⣇⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⠼⠀⠉⠻⣏⠛⢻⣄⡀
⠀⠀⠀⢀⡠⠖⠚⠁⠀⣬⠁⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⣀⠇⠀⠀⠛⠛⢧
⠀⠀⠐⢋⠀⠀⠀⠀⢀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⠀⠀⠀⢀⣻⠆⠀⠀⢀⠠⠬⣄
⠀⣟⠉⠁⠐⣀⠀⢀⣼⠀⠀⣠⣎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡁⠀⠀⠈⠉⠛⡞⢻⣶⡶⠋⠀⠀⣠⡿
⢀⡞⠁⠀⠀⠈⣳⣮⡟⠀⠘⡇⠈⠀⢀⡴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄⠀⠀⠀⣸⠋⠈⢷⣄⠀⠘⣯
⠈⠳⠳⣄⡀⠛⠙⠏⠀⠀⣠⠇⠀⢀⣜⣳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⠜⡌⠀⠀⠀⠛⢦⠄⠚⠃
⠀⠀⠐⠈⣷⡐⠠⣦⣤⣞⢙⡆⠀⠀⠋⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢦⣤⠴⢦⣄⣼⠃
⠀⠀⠀⠀⠀⣀⡀⠀⡼⠀⠈⠑⠀⠀⠚⠉⢻⣄⠀⢀⣀⡀
⠀⠶⡤⠀⢀⡟⠈⠉⠁⠀⠀⠀⢀⡴⠒⠢⠼⠋⢱⠎⠀⠘⢧
⠀⠈⠐⠐⡺⣁⣀⡀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⣨⠁⠀⠀⠀⠉⢱⠆
⠀⠀⠀⣼⠁⠀⠀⠉⠋⠰⢾⣅⠀⠀⠈⠑⢦⣠⣿⣀⣀⣀⣠⣤⣞⣀⢀⣠⡀⣦⠢
⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⣇⡞⠀⠀⠀⠀⠰⠿⣾⡋⠉⠉⠁⠀⠈⠙⡆
⠀⢀⠀⠀⠀⡟⠀⠀⠀⠛⠻⠤⣄⠀⠀⠈⠀⠀⠸⡇⠀⠀⠀⠀⢀⣼⠁
⠀⠈⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⢳⡀⢀⣴⠟⠉⠁
⠀⠘⠀⠀⠀⠀⠀⠀⠘⠦⣤⠖⠢⢬⡷⠛⠓⢦⡴⠋⠉⠛⠁
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⢄⣀⡀⠀⠀⠀⣀⣀⠤⠤⠤⢄⣀⠔⠋⠀⠈⠑⣤⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠉⠑⠚⠉⠁⠀⠀⠀⠀⣰⠋⠀⠀⠀⣀⠀⢱⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡎⢩⠟⠚⠓⢦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠙⡇⠀⠀⠀⠀⢹⠢⠤⠾⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠳⣄⠀⢀⡠⠛⠲⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠓⠶⠚⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⠤⠖⠉⢧
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢾⠖⠶⠤⠤⣀⠀⠀⠀⠀⢀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⢀⡠⠤⠄⠒⢾⠦⠤⢄⣀
⠀⠀⠀⠀⠐⠊⠉⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢄⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠈⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠙⡄⠀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⡼⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠛⣍⣀⡤⠖⠒⠉⠓⠒⠢⣄⡀⠀⠈⠁⠈⠉⠀⣀⡤⠒⠒⠉⠑⠒⠦⣄⣈⡝⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⣀⡤⠤⢞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⠦⠤⣀⡀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⢀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠈⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣀⡴⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣀⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠂⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡤⠔⠒⠲⢤⣀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠤⣄⡴⠋
⠀⠀⠛⠒⠖⠒⠆⠀⠀⠀⠀⠀⣀⣰⠓⣆⡰⠉⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠋⠀⠉⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣰⣅⠀⢀⣵⡀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢰⠖⠒⢦⡀⢀⣀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣀⡠⠞⠁⠈⠙⠉⠀⠙⠦⣀⣀⣀⣠⣴⣋⡠⠴⠞⢳⡀⠀⠀⠀⢠⠦⣀
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠟⠁⠀⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠈⠁⠀⠀⠈⠹⠀⠀⢀⣎⠀⠀⠷⢤⡀
⠀⠀⠀⢀⡞⠉⠁⡎⠉⡆⢀⡠⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡉⢰⠒⡆⠀⣹⠃
⠀⠀⠀⢸⡀⠀⢀⡌⠉⠀⢀⠇⠀⠀⠀⠀⠀⣠⠎⠙⠏⠀⠋⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣄⡌⠉⠁⢻⠁
⠀⠀⠀⠈⠉⠙⠉⠙⠒⠒⠚⠆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣴⣶⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣾⣷⣤⣤⣤
⣠⡄⠈⠃⠉⠘⠁⣦⡄⠀⠀⠀⠀⠀⠀⠀⡸⠏⠀⠻⠙⠉⢧⠀⠀⠀⠀⠀⠀⠀⠸⢉⣤⡀⢀⣤⡙⠃⠀⠀⠀⠀⠀⠀⢻⡿⠋⠁⠀⠀⠈⠻⢿⣤
⢸⣏⠁⢐⣀⡀⢈⣹⠁⠀⠀⠀⠀⠀⠀⠠⣯⣁⣤⣤⣤⣄⣽⡆⠀⠀⠀⠀⠀⠀⠇⣾⠟⠃⠈⠻⣿⠸⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣀⣀⠀⠀⢸⡟
⡀⠲⠂⠀⠉⠁⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⣿⣿⣾⠟⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⠛⠀⡠⠃⠀⠀⠀⠀⠀⠰⣿⣷⣀⠀⠈⠉⠀⣠⣾⠿
⢛⠀⣘⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣷⣶⣶⣿⠛⠃
⠘⣴⡇⢠⠄⡤⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠭⢭⢭⠽⡄⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⣶⢰⡆⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠠⠃
⠈⠛⠆⣸⡀⣇⠰⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⢉⣻⣛⣼⡉⠷⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⢿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⢿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⣆⢀⣶⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠈⠁⠀⠉⠙⢿⠟⠀⠀⠀⠀⠀⢀⣠⣄⠀⣀⡀
⠀⢀⠀⠐⠒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣠⣤⣤⣤⣀⣈⠀⠀⠀⠀⠀⢠⠋⣸⣾⠿⠿⠿⣷⣄
⠣⠁⠀⠀⠀⠀⠀⠡⠇⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢰⣿⢻⡟⠁⠠⠀⠀⠀⢻⡛⣷
⠠⠀⠀⢿⣿⠿⠀⢀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⢩⠉⠀⠀⠀⠀⠀⠀⠀⠹⣏⠳⢄⠀⠀⠀⡀⠈⢛⡯
⠀⠑⠤⠀⣀⠠⡔⠁⠀⠀⠀⠀⠀⠸⢿⣿⣷⣄⠀⠀⠀⣠⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠐⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡏⠉⠹⣷⣄⢿⣞
⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠈⣬⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⢀⣿⣜⡞
⢠⠂⠀⠀⠀⠀⠀⡓⡄⠀⠀⠀⠀⠀⠀⠀⢸⡿⢃⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢘⠀⡄⣤⢚⠎⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣤⣾⣿⣿⣿
⠈⠉⠺⠭⠶⠭⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠖⠲⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠓⠻⠆⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⣀⠀⠀⡀⠀⢠⣤
⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠳⠶⠛⠓⠙⠨⠄⠇⠀⠰⠿⠖⣇
⠀⠀⠯⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠄⢀
⠀⠀⠄⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢞⣵⣛⡮⠥
⠀⠀⠘⠎⣘⢓⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣻⣝⢮⠛⢁
⠀⠀⠀⠀⠀⠉⠉⠀⠁⢀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀⣾⡟⢻⣷⡀⠀⠀⠺⣿⣯⠟⠀⠀⠀⠀⠈⠊⠀⠀⠘⠺⠖⡶⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣴⠛⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⣿⠟⢻⣿⣿⣿⣏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⡟⢟⣿⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣪⡾⣣⠀⠀⠀⠀⠀⠈⠓⠺⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡀⠀⠀⢸⣥⡾⠃⠀⠐⠉⠁
⠀⠀⣤⡄⠀⠀⠃⠀⠀⡴⣫⠞⣡⢖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣀⣀
⠀⢀⣿⣿⠿⠀⠀⠀⠀⠀⠁⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⡿
⠀⠈⠉⠀⠀⠀⠀⠀⢀⣴⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠉⠛⠛⠁
⠀⠀⠀⢀⡄⢄⠀⠀⢸⣿⠻⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠁⠀⡐⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⣠⠺⠘⡾⠀⠀⠀⣿⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡿⠁⠀⠀⡀⠀⠉⢠⠖⣦
⠀⠀⠛⠗⠓⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣄⡀⠀⠋
⠀⠀⠀⠀⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⣶⡦
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠋⠀⠀⠂⢠⢀⠀⡤⠀⢠⡠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠄⢀⣠⠎⠉⢷⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡰⡞⠈⠀⠀⠀⢀⠈⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢧⢱⠀⠒⠂⡀⠈⠀⢀⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡀⠈⠓⠂⠀⠀⠀⡀⠀⠉⠀⠙⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡀⠐⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⣀⣹⠀⠀⠀⠀⠀⡀⠤⠀⡂⠱⠌⡰
⠀⠀⠀⠀⢀⠔⠊⠒⠒⠀⢄⡀⠀⠰⠈⠈⠉⠀⠀⠀⠁⠀⠈⠈⠀⠉⠁⠠⠀⠀⠀⠀⠘⠰⠈
⠀⠀⠀⠔⠃⠈⠀⠀⠀⢀⠀⠀⠁⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡄⣆
⠀⠀⡎⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠡⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡔⠈⠠⠀⠀⠀⠑⢢
⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠨⠀⢀⡴⠒⠒⠒⠲⢄⡀⢠⠀⢸⠀⠀⠈⠀⠀⠀⠂⠀⡇
⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢸⡧⠀⠀⣿⠆⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠘⡀⠀⣤⠀⠀⠀⠀⠹⠃⠀⡇⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠙⡒⠒⠠⠤⠤⠤⠴⢤⠂
⠀⠀⠀⢱⠀⠉⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠈⠲⠤⠤⠤⠴⠚⢡⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠤⠄⠒⠒⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠙⠛⠛⠁⠀⢀⠀⠀⠀⠀⠘⠑⠛⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⠙⠛
⠀⡀⣀⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣠⠄⢙⡇⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢂⠀⠀⠈⠁⠀⠀⠀⠀⢠⣀⠄⢙⡇⠀⠀⠀⠐⣶
⠀⠱⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡄⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠈⠓⠇⠀⠀⠀⠻⠲⡰⠦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠇⠀⠀⠀⠻⠲⡰⠶⠀⠀⠀⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠢⠤⠴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠦⠤⠴
⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠂⠈⠁⠀⢀⡀⠠⠀⠀⠀⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠂⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢤⠤⠤⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⡁⠀⠀⠀⠀⠀⠀⢐⠀⠀⡀⠀⢀⢤⠤⠤⣄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠋⠁⠀⡋⢠⠂⠂⠀⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠁⠀⡋⢠⠒⠂⠀⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡠⠃
⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢠⢚⠶⢀⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢰⠾⠶
⠀⠀⠀⠸⣦⣀⢀⣡⠟⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠠⠀⠐⠀⠈⠁⠀⢸⠁⠀⠀⠸⣦⣀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠠⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠘⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠀⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠉⠁⠀⠐⠀⠀⠸⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠘⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣤⢤⣼⠁
⠀⠀⠀⡀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠡⠆⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⣀
⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣡⠰⠇⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣡⡰⠆⠀⠀⠀⠀⠟
⠀⠀⠀⠑⠀⠀⠀⠠⣄⡄⠀⠀⠀⡯⣑⠀⣘⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⢀⣄⡄⠀⠀⠀⣯⣁⠀⣘⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠐⢦⣀⡀⠁⠀⠀⠀⠈⠈⠘⠁⠀⠀⠤⡄⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠁⠀⠀⠀⠈⠈⠊⠉⠀⠀⠤⡄⠀⠃⢀
⠶⠀⠄⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢀⠄⠀⠀⠃⡀⠠⠍⠉⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠠⠉⠉⠁
⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⠤⠆⠀⡴⠉⣉⡉⠃⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⠤⠇⠀⡴⠉⢉⡉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠄⠀⡅
⡆⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⢶⡽⠒⠀⠀⠀⠀⠘⠀⠤⠰⠊⠀⣠⣀⠀⠂⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⠶⠽⠖⠀⠀⠀⠀⠈⠀⠤⠰⠊⠀⣠⣀
⠃⠀⠀⢸⡀⠃⠀⠃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⠸⡄⠉⢰⠇⠀⠀⢸⡀⠃⠀⢃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠸⡔⠉
⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⢰⣿⡇⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⠀⣿⡇⠀⠀⠀⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢿⠟
⠀⠀⠀⠤⠄⠀⢀⣀⠀⢸⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⢰⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋⠀⠀⠀⢀⣴⣶⣄⠀⣀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⣄⠀⠀⠙⠛⠋⠁⣠⣴⣾⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣄⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣠⣤⣴⣦⣤⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⡿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣿⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣀⣀⣹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡟⠋⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠙⢻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠐
⠀⠀⠀⠀⢀⣀⣀⠌⠀⠀⣀⣀⡀⠉⠙⠛⠿⣿⣿⣿⣟⡁⢄⠀⣾⣷⠀⠀⠀⠀⠀⠀⣾⣷⠀⡠⢊⣻⣿⣿⣿⠿⠟⠋⠉⢀⣀⣀⠀⠀⠡⣀⣀⡀
⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠂⢸⣿⡷⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⢸⣿⡇⠔⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣴⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⣤⡆⠀⠰⣿⣿⣿⣿⣿⣶⣶⣶⣶⠂⠠⣤⣤⣄⣀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡷⣄⡀⣥⠄⢿⣃⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡈⢿⣿⣉⣁⣸⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⡋⠃⣥⡀⢈⣭⠀⠀⡀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣟⠳⣤⠜⠀⠸⣿⠀⠠⣤⣴⣿⣿⡟⠋⠁⠈⣨⣿⣿⡿⣾⣿⡆⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣵⠀⠀⠀⠀⢀⣿⠀⢠⠜⠋⠉⠻⡟⠀⣠⣴⣿⠛⣿⡟⠻⢿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣸⣟⡳⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣶⣟⠻⠟⠁⠀⠁⢀⡈⣏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠻⠇⠀⠀⣤⠀⠀⠀⢨⣿⠀⠀⣀⣀⣠⣤⣿⡟⠿⣏⡗⠀⣀⠀⠀⠛⢹⡀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣤⣤⡀⠀⠀⠀⢺⣿⠀⠈⣉⣽⣿⠋⠉⣡⣾⣿⣧⡅⠉⡀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠋⠛⠋⠂⠀⠀⣿⣿⠇⠈⣋⣘⡉⠠⠀⠙⣻⠉⠈⠀⠚⠅⢤⠇⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣥⡜⠂⠀⠀⠀⠀⠀⣿⠘
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣴⣾⣿⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣉⣹⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠉⠉⢩⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠘⠿⠿⢏⡙⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢻⣿⢄⠀⢠⣾⣿⣿⣿⠋⢁⣤⣤⣾⣿⣿⣿⣿⣷⣶⣔⠄⣤⣦⣶⣿⣮⡑⠦⣼⢇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣌⣿⡇⠃⠘⠛⠛⠛⠛⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣀⣿⣼
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠛⣿⣷⣄⣤⡄⢠⣶⣶⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣹
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡸⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣀⣵
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠊⠱⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⡿⠟⡋⠁⠀⢀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠶⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢀⠲⡄⠀⠀⡠⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠢⡜⢱⠀⠣⠤⠧⣤⠎⠙⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⣧⠈⢧⠀⠀⡴⠙⢄⣀⡎⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢫⢣⡀⠑⢌⡲⢤⣀⡼⠀⠀⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠳⣝⠢⢄⣉⠑⠚⢒⣚⡩⠥⢺⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣀⣠⣀⠀⠀⠀⠀⠉⠒⠂⠬⠝⠓⠚⠉⠉⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⢤⣀⠀⠀⠀⠀⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠼⠶⠠⠤⡀⠀⠀⠀⠑⠐⠷⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⡇⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠈⠳⠀⠀⣀⠄⠒⠒⠢⢄⠀⠀⠀⠀⠀⠀⡼⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⡠⢾⠀⡰⠃
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡷⡖⡏⠀⢸⠀⡧⣂⠢⣄
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⣇⠀⢸⠀⠳⠽⠃⠈⡆
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠉⠒⠚⠀⠀⠘⡆⠈⢧⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠵⢖⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠲⠤⠄⠀⠀⠀⢸⠤⢄⣀⣋⣻⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡦⠤⠤⠖⠚⡗⠒⢶⠒⠋⠀⠐⢲⡏⠣⣌⡉⠓⠚⠉⢉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢀⡸⠁⠀⢸⡀⠀⠀⢀⡜⠙⢦⣀⠈⠉⠉⣉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠙⠲⠖⠋⠀⠀⠀⠈⠉⠛⠉⠁