Pitcher plant Emojis & Symbols

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠧⢽⡀⣹⣿⣽⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣧⣤⣬⣇⡀⠼⡄⢻⡭⠛⡞⢾⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡚⢁⡀⠸⠔⠊⡉⢠⡀⠑⣞⢉⣳⣾⣾⣁⠰⡇⢸⣯⣿⢋⣜⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣁⡀⣴⣲⣦⠁⡀⠀⣨⣍⣩⠉⣹⣯⣻⣷⣧⣸⣏⣸⡏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣴⣿⣿⣿⣏⢛⣦⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣴⣿⢟⡙⠏⠉⠹⣷⡄⠀⠀⠀⠐⡀⣶⡄⡀⠀⠀⠀⢀⢀⣤
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⣜⢛⣻⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⣿⣵⣰⣆⠂⠾⣿⡄⠀⠀⠀⠐⠿⣿⣿⣿⣿⣷⣾⠿⠁
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢻⣿⣽⢿⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⡷⠚⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⡿⠋
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢯⠙⠋⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡃⠀⢀⣿⠃⠀⠀⠀⣶⣶⣿⣿⣿⣷⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠀⢀⠈⢿⣆⠀⠀⡤⠶⠖⣻⡇⠀⠀⢸⣿⡇⠀⣿⡿⠀⠀⠀⠀⣿⣻⢿⣿⡭⢽⣿⠁
⠀⠀⠀⣸⠀⠀⠀⢀⡀⠀⡀⠀⠀⠈⠙⠛⠶⣦⣷⡀⠙⣆⠀⠀⠀⢠⠏⠀⠀⠀⢸⣿⣇⢰⣿⠇⠀⠀⠀⠀⠸⢭⣽⣿⣿⣿⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⡟⠀⠀⠀⠀⠀⢶⡜⣿⣿⣿⣿
⠀⠀⠀⡇⠢⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎⢿⠁⠀⠀⠀⠀⠀⣸⠆⣿⣿⣿⡇
⠀⠀⢸⡇⠀⠀⠈⠙⢷⣶⣤⡤⠤⠄⠀⠀⠀⠘⢻⣿⢿⣿⡤⢰⡀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿
⠀⠀⢸⡇⠀⠀⠀⡀⣼⠉⣿⣧⣀⠀⣀⠀⠀⠀⠀⠻⡄⢻⣿⡀⢳⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⠇
⠀⠀⢸⡇⠀⡴⠃⣱⣿⣷⣿⣿⣿⣯⡿⡝⡷⠦⣄⠀⠹⡌⢻⡆⠀⠣⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⠀⣿⣿⡿
⠀⠀⠀⡇⢸⠅⢲⣿⣿⣿⡿⠛⠛⠃⠿⣿⣿⣴⠈⣷⠀⠱⡀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⡚⢸⣿⣿⠁⠀⠀⣶⣿⣿⣆
⠀⠀⠀⢳⣸⣦⠘⠿⣿⣿⣄⠀⠀⠀⠀⠀⠘⠛⠀⣿⠀⠀⢣⠸⣷⠀⢠⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⢠⣧⣿⣿⠇⠀⠀⠸⣿⠉⢻⡃
⠀⠀⠀⠸⡏⠻⣦⣄⡼⢻⠿⠂⠀⣀⠀⠠⣀⣤⣾⣿⡆⠀⠈⠄⢿⣧⣿⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢀⣾⣿⣿⠏⠀⠀⠀⣰⣿⠀⠘⠃
⠀⠀⠀⢠⣻⠀⠀⠉⠛⠛⠴⠴⣼⣿⣿⡄⢹⡟⠋⣿⡇⠀⠀⠱⠸⣿⣿⡀⠀⠀⠀⠀⠈⠀⠈⢹⣯⣽⣿⣿⠏⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠈⡟⣧⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠈⡇⢰⣿⠇⠀⠀⠈⡅⣿⣏⣇⠀⠀⠀⠀⠀⠀⠠⠸⣿⣿⣿⠏⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⢹⠸⡆⠀⠀⠀⠀⠠⣿⣿⠀⠀⠀⣇⣾⡿⠀⠀⠀⠀⢷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣌⣿⣧⠀⠀⠀⠀⣶⢿⠀⠀⡠⡤
⠀⠀⠀⠀⠈⢇⢳⠀⠀⠀⠀⠨⣿⣿⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣷⡄⠀⠀⣿⢸⡇⢸
⠀⠀⠀⠀⠀⠘⡿⡆⠀⠀⠀⠀⢹⡿⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠀⠀⠀⢠⣿⡟⠀⠈⢿⣿⣷⠀⢸⣿⢸⡇⡄
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⡄⠈⡇⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠸⣿⣿⠇⣸⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢄⣨⡇⢀⠇⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⣿⢁⣿⣿⠃⣠⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣥⣼⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠀⠀⠿⠁⠀⠀⠀⠀⣀⣿⣿⣿⡟⣁⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠉⠉

⠀⠀⠀⢠⡴⢆⣠⠖⣤⢒⠀⣤⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⢚⠻⠷⢦⣄
⠀⠀⠀⠀⠙⠚⢽⢿⡗⡾⡴⢼⡇⠠⣤⣼⡿⠟⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢀⡄⠰⠋⢁⠀⠀⠔⠁⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠋⢰⣦⣌⠲⣭⡷⡏⠀⠾⣆⠀⠀⣰⠋⠀⠀⠀⠉⠉⠊⠀⠄⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣕⡿⣷⢰⠮⠶⢧⣤⣷⡋⢀⠀⡏⠐⠀⠀⠄⠀⠀⡀⠁⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠊⠀⢠⡄⢈⣫⣌⡛⢸⡇⠀⠀⠀⠀⠀⠌⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⣄⣯⠙⣄⠃⠀⣀⢀⠀⡘⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠈⢧⠀⢸⠈⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠀⠤⠄⡀⠀⠀⠀⠀⠈⣷⡆⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢰⡦⢄⣀⠀⠀⢀⣤⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠟⠚⠿⣟⠶⣀⡀⠀⠁⠠⠀⠀⠀⠘⡷⠠⢠⠇⠀⠀⠀⠀⠀⠀⢀⡠⠴⣞⡥⠖⢛⡤⠴⠶⠎⢿⣅⡀⢿⣼⣯⡤⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠐⣯⡀⠀⢀⡀⠀⠙⢦⣉⡓⢤⣶⢄⡀⠀⠀⢹⡇⡘⠀⠀⠀⠀⠀⠀⢀⠎⣴⢟⡭⠞⣃⣀⣠⠔⠊⠁⠈⠙⢿⡞⡏⢃⠀⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠟⠋⢈⠷⠀⠀⠉⠛⠶⣍⢻⣿⡳⢄⢠⡇⡅⠀⠀⠀⠀⠀⠀⡸⢄⣵⡻⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡛⢗⡀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢯⣛⢿⣇⠀⠀⠀⠀⣠⠞⢀⠎⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣼⢓⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡿⠄⠀⠀⡰⠃⢁⢢⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⡥⠄⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠘⢠⡀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠐⠼⡛⢛⠛⠛⣟⣷⣆⠢⢤⣀⣀⣀⠀⠘⣿⠀⠀⠠⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠻⣦⠞⠁⢀⡇
⠀⠀⠀⠀⠀⣴⡮⠟⢒⣈⣀⡀⠥⠦⠖⠊⠉⠁⠈⠻⣦⡈⠻⠛⡿⠢⣿⣠⡰⢃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⠜
⠀⠀⠀⠀⢰⡏⡘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠐⢿⡏⠀⣨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣯⠈⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠘⡇⢀⠏
⠀⠀⠀⠀⢻⡇⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢻⢸
⠀⠀⠀⠀⠀⢿⡀⣾⡟⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⢠⢸⡀
⠀⠀⠀⠀⠀⠸⡇⣿⣷⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⣷⠀⠀⠀⢀⣠⠴⣂⠀⠴⣤⡤⠤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⢰⠇⣿⣿⠐⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠂⢻⠀⢀⠔⢁⠔⡩⢭⣉⠀⢀⠙⢮⡐⠌⠑⢯⡛⠶⣄
⠀⠀⠀⠀⢰⣏⣘⣟⣇⣀⠸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣾⠀⢎⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠙⢆⡈⠳⡄
⠀⠀⠀⠀⠀⠈⢿⣿⠀⣉⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠀⠑⢄⠘⣆
⠀⠀⠀⠀⠀⢰⢸⣿⣴⣷⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈⠁⠘⡄
⠀⠀⠀⠀⠀⠈⠺⠿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⣰⣤⣠⣴⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠉⠉⠙⠻⠿⣭⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠁⠀⠀⠀⢠⡈⢳⣟⣿⡴⣶⠶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⣾⠀⠹⣿⢫⠀⣼⠏⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢳⣾⠃⠀⢰⡻⢇⢰⣇⡼⠻⣇⣠⡶
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣀⣀⣠⠇⠀⢤⣎⠻⠄⠻⢹⣇⠀⠹⣏⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⢻⡗⠀⠉⡙⢹⠃⠀⠀⠸⢉⣷⠀⣹⣷⠟⠁⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⣬⣯⡅⠈⣑⣿⡗⠀⠀⠀⠀⠘⢉⣯⠁⣴⣿⣟⠁
⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠁⠀⣹⣧⠀⠘⢿⡃⠀⠀⠀⠀⠀⠈⠁⠿⣀⡾⠻⣿⠿⠗
⠀⠀⠀⠀⠀⠀⠸⣇⢸⠀⠴⠟⠛⢿⡀⠘⠻⣏⠀⠀⠀⠀⠀⠀⠀⢰⠿⣤⣴⡿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⣠⣮⣶⠁⣸⣿⡤⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡳⢶⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⠘⠋⠀⣽⣟⠉⠿⠤⠊⡀⠀⠀⠀⠀⢀⣼⣿⡷⣉⠻⣦⣻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠘⠉⢿⣣⣄⠛⣾⣤⣤⣤⣼⣶⠷⢹⣷⣿⡞⣧⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠃⠀⠀⠀⠀⠀⢾⣿⣿⣿⠷⢶⣷⣼⣦⣿⡶⢶⣿⣻⡿⠏⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⠄⠐⠆⢸⡇⠘⠃⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⠀⠀⠀⠀⠀⠈⣿⣿⣧⡀⠀⠀⣼⠇⠀⠀⠀⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⣿⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⣿⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⢸⣿⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⢈⣿⣀⠀⠀⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⢠⣨⣿⠿⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⣾⣿⣿⣾⣰⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡿⢀⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣴⣿⣿⣿⠛⢿⡟⠉⢻⣿⣽⣋⡟⠓⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣼⠁⣼⣿⣶⣿⣿⣿⠋⠀⠉⠁⢀⣾⠑⠎⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠿⡿⢿⣿⡿⢥⠀⠀⠀⣠⠞⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⠀⠙⠣⠜⣁⠖⠋⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢀⡠⣀⠠⠤⠴⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⢠⣖⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢀⡠⢶⠆⠀⠀⠀⠉⠳⣶⠂⠀⣠⣠⡾⡏⠠⢞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣯⠌⢀⣢⡄⠀⠀⡿⠀⡀⠚⠉⣀⠠⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣼⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠁⠀⠀⣸⣧⡀⡀⠂⣠⣶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣨⣿⣿⣿⣷⡞⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⣞⣿⠿⣷⣏⢩⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠘⢁⠇⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣿⡿⡿⠟⢏⠋⢰⠟⡞⡊⢹⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢻⡀⡧⠈⣶⠋⣠⢪⡜⢹⢀⡿⢣⡇⠀⠀⠀⠀⠀⣀⣴⠞⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⢀⠀⠀⠀⠀⢿⡌⢳⡷⠀⢄⠴⣡⢤⣡⣾⣿⣿⡟⠁⠀⠀⠀⣰⣾⣭⣷⣶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠀⠘⠂⠀⢀⡀⠀⠙⢦⣾⡿⠟⣿⠿⢿⣿⣟⡛⣿⠀⠀⠀⠀⠀⠙⠛⣿⠛⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢴⠇⠀⠀⠀⠀⠘⠳⢄⠀⠸⠹⣧⣀⡗⠑⢦⡏⡌⣷⣍⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢾⣀⠀⢾⠀⢠⠀⡂⠀⢠⠀⢾⡗⠀⣦⢻⣇⣹⣮⡶⠷⠞⠛⠉⠁⢀⣠⣄⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠻⣥⠐⠊⠹⣀⠀⣧⢸⣰⡇⠀⢀⡴⠟⠁⠀⢸⠀⣿⣿⣄⣙⣦⣎⣤⣶⣏⢯⢉⣿⣯⣶⣟⡯⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠋⠛⣅⣈⡰⠉⣳⢦⠈⢣⠄⣀⠿⠀⢰⡏⠀⠀⠀⠀⢸⡄⠸⡟⠹⣿⣿⣿⣿⣻⡮⢾⡀⠸⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡆⠀⡈⡖⠀⡀⡀⠁⠀⣠⣿⡦⠀⠀⠀⠀⠸⡇⠀⢛⠀⣿⣿⠟⠓⠚⠓⡀⠈⠀⣟⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡓⠊⢤⡠⠉⣖⠃⣼⠀⢐⣼⢻⣷⣦⡄⣤⣄⣀⣿⠀⢸⡇⢿⣏⠾⣰⣧⣠⣍⡤⢾⣛⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠶⠍⠀⡇⠀⣼⠘⣟⠻⣟⢿⣟⡧⣈⠛⢧⠈⣷⢻⢹⣷⣽⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⣻⣿⣆⢫⠀⢹⡷⢘⡟⣙⡖⣿⣿⣿⡇⣿⠸⠀⠉⠏⠹⣿⣿⢿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⠄⣏⣻⣾⣿⣾⣟⣿⣿⣿⡇⠊⠈⠀⠀⠀⠀⢹⣼⡎⣿⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⠀⠈⣿⣧⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣶⣾⣿⣿⣿⣿⠿⠁⠸⡇⠀⠀⠀⢠⢸⠀⠀⣿⣹⣿⣿⣿
⠀⠀⠀⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣏⣿⠿⣾⢩⣿⣿⠃⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠐⣥⠿⢿⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⣿⣻⣯⣬⣉⣿⢻⡿⣿⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠒⠺⠿⢿⣿⣿⠂
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣿⠿⡇⣼⡇⢹⣿⡇⣿⠀⠀⠀⠀⠈⣷⣀⣀⣀⣀⣀⡐⠣⠀⢘⣿⣿⡇
⠀⠀⠀⢀⡿⢛⣿⣾⡉⠉⠣⠀⠀⠀⠀⠀⠀⣿⡇⣿⣧⣟⠸⡟⣧⣿⡀⠀⠀⠀⢠⣼⣿⣿⡿⠿⠿⢿⡆⠀⣼⣿⠛
⠀⠀⠀⡞⡱⢿⠃⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⢹⡶⠃⢿⣿⡇⠀⠀⠀⢸⢻⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠒⠒⠛⠃⠈⠀⣄⠀⠈⠩⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⡜⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣎⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣰⡾⠃⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢸⡿⢷⡟⢿⣧
⠀⠀⠀⠀⠀⣤⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣘⠷⠀⠁⠚⢻⡆
⠀⠀⠀⠀⢰⣿⣶⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡟⠀⣾⠗⡌⢹⡄
⠀⠀⠀⠀⣾⠋⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⠀⠀⠀⡇⠀⠈⠀⠀⠀⢠
⠀⠀⠀⢰⡇⢸⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢢⣶⣅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣸⠀⠀⢀⡿⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢸⡇⣾⣿⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢼⠀⠺⡇⠁⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠿⠀⠀⢀⠁⠀⢠⠄⠀⠀⠀⢠
⠀⠀⠀⠸⣿⣿⣿⢹⣦⡄⢀⣠⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⡛⠃⠀⠀⠀⠀⠀⠀⠀⡾⢇⣠⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡏⢉⠉⠙⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣄⠘⠃⠀⠀⠀⠏⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠟⠁⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⢠⡶⠀⠀⠀⠄⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠈⠉⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⣠⠓⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⢀⣿⠄⠀⠀⠀⠶⠀⢀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⣸⡄⠀⠀⠀⠐⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⣿⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠈⠀⣆⠀⠀⠈⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⣸⡇⠀⠀⢄⠀⣀⡀⠐⠀⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡏⠀⠀⣀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠃⠀⢀⡉⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡄⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⢠⡇⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⣿⣷⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠐⠀⠀⠀⠀⣶⠇⠀⠄⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⣿⠀⠀⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⡘⠃⠰⠆⠀⠀⠀⠐⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠐⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣀⠀⠀⠠⣷⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣷⡄⢀⡀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠁⠀⠰⣾⡏⠀⠀⠀⠁⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⠁⢸⣷⠀⢀⣤⣴⣿⣿⣿⡟⠀⠠⠤⠀⢘⣿⡏⠁⡠⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠋⣤⠀⡄⠀⠈⣿⠿⠏⢹⡀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣤⣠⣴⣿⣿⣿⣿⣿⣿⡿⠋⢡⣤⡟⠀⠁⠀⢾⣿⠆⢰⡆⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⢘⣉⠶⠆⠀⣸⣿⠶⠄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⠏⢠⣦⢀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⢣⣴⣿⡛⠣⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠑⣾⣿⡏⣿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠲⣿⣿⣿⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢧⣀⣽⡇⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣶⠉⢹⣿⡏⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠹⠇⣿⣿⡇⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⣡⠰⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡾⣿⣶⣿⣿⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣤⣤⠀⠀⠠⠤⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢹⣿⣿⠉⢀
⠀⠀⠀⠀⠀⡀⠀⢀⣰⠾⠟⠿⠿⣶⣄⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢹⣼⣿⣿⣦⡜
⠀⠀⡀⠐⠂⢀⡴⠋⠁⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢸⣿⣿⣧⡙⠁
⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣿⡿⠃
⠰⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠹⣷⡄⠀⠀⠀⢀⣿⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠙⣿⣆⠀⠀⣾⡿⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠈⢿⣦⣼⠟⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⣿⠃⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠛⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠘⠉⣹⡟⠻⣿⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⡴⠾⠋⠀⠀⠀⠀⠸⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⣻⡆⠀⢠⣄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠻⡇⢀⡟⠙⠃⠀⠀⠰⡏⠀⣹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣄⡀⠀⢀⣀⣀⣿⣿⣤⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⣹⢿⣿⣿⠿⠟⠿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠙⠛⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣯⣈⠛⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣤⡿⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣿⣸⡁⠀⠀⠀⠀⠹⢦⠀⣠⣇
⠀⠀⠀⠀⢹⣿⡇⠀⠠⠀⠀⣶⡆⠘⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣦
⠀⠀⠀⣴⢸⣿⣿⡆⠀⠀⠀⠀⢀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⢻⣿⡀⢀⠀⣀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶
⠀⠀⠀⠸⣀⣸⡿⣿⣿⡅⠈⣇⣠⡀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠠⠸⣿⠛⠛⠷⣶⣿⣶⡏⠙⠃⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣀
⠀⠀⠀⠀⠋⠀⠀⠀⠘⢉⣿⡍⠂⣄⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠰⠟⠛⠛⣿⣿⠿⣯⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣶⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠘⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⢰⣶⣶⠏⣿⠛⠳⣾
⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢉⡄⠀⠈⣿⡄⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠁⡄⣠⡀⠋⠀⠀⣿
⡋⠙⣿⠀⠀⣤⣤⠀⠋⠀⠀⠀⢹⣇⠈⢿⣤⠀⣿⡇⠀⠀⡀⠀⡀⠀⠀⣠⡄⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⢀⠀⠀⠋⠻⠶⢆⣰⡏
⣧⠀⢹⣷⣆⠈⢹⡄⢀⠀⠀⠀⠀⢻⠀⣾⣧⠘⣿⣿⠀⠀⠇⠈⠀⠀⢠⡿⠁⠀⢀⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢠⠌⠀⠀⠀⢰⣤⣾⡟
⠀⣷⣿⣿⣿⠀⠀⢿⣿⢧⠀⠀⠀⠀⠀⡹⣿⡍⣾⣿⣇⠀⠀⠀⠀⠀⢸⡇⠀⢀⣾⣿⣿⠃⠀⠀⠀⣰⡗⠀⠀⠀⠀⡀⠃⠀⢀⣀⣴⣿⣿⠏
⠀⣿⣿⣿⣿⠀⢀⠀⠠⣾⣧⣦⡀⠀⠀⠰⣾⠁⠙⣿⣿⡀⠀⠀⣿⡀⢸⣿⣃⣿⣿⣿⠇⠀⠀⠀⣸⣯⠽⠀⠀⠀⣤⢁⣠⣶⣿⣿⡿⠛⠁
⠀⠘⣿⣿⣿⣷⠊⣴⣷⣬⡙⠻⣿⣄⡀⠀⠈⢢⣤⣸⣿⣧⠀⠀⠻⣷⣿⣯⣽⣿⣿⠃⠀⠀⠀⣠⣿⠇⢀⣴⣶⣾⣿⣿⡿⠟⠋⠁
⠀⢠⣿⠟⠛⠘⠛⠋⠀⠉⠙⢢⡀⠉⠻⣷⣤⡀⢻⣇⠸⣿⡄⠀⡄⢹⣿⣿⣿⡿⠁⠀⠀⠀⣰⡿⢃⣰⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴
⠀⠘⠋⠀⠆⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⢿⣮⣻⡄⢸⣿⠀⣿⠈⣿⣿⡿⠀⠀⠀⠀⢰⡟⢱⣾⣿⠟⠁⠀⠀⠀⠀⣠⣤⣤⣤⣤⠔⠒⠚⠛⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡴⠘⣿⣧⡀⠀⠀⠙⢿⣷⠈⢹⣇⣿⠀⠋⣿⠃⠀⠀⠀⢀⣯⣴⣿⠟⠁⠀⣀⡶⠋⢁⡰⠟⠛⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠈⠘⠿⣿⣉⢻⣦⡀⠀⠈⢻⣧⠸⣿⣿⠀⢸⡏⠀⠀⠀⣠⣿⡿⠋⣠⣤⣴⣾⣿⠿⠚⠋⣀⠀⠰⠁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠈⠛⢿⣷⣤⣅⣤⣶⣄⣀⣀⣹⣿⣿⣷⣾⣿⡆⢻⣧⠹⣇⢠⣾⠇⠀⣠⣾⠿⠋⣠⣾⣧⡼⠉⠁⣠⣤⣠⡴⠋⣀⣀⣤⣴⠆⣀⢣⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⣿⣤⣾⣟⣛⣷⣚⣻⣭⣭⣶⣶⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⡏⠻⠟⢉⡿⠟⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠊⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠉⢙⠛⠋⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣼⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠙⣷⣦⡉⢳⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠰⢀⣿⣇⠀⠙⡙⠉⣩⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠁⠄⠀⣤⠀⠉⣿⣿⣿⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⢃⢀⡸⠗⢆⡂⣰⣿⣷⡀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⠿⠿⠟⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠛⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣤⠣⣄⠃⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣦⡄⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣿⡀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠹⣿⣷⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠿⣿⣿⣿⣿⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠇⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢀⣼⠋⠉⠉⣿⣿
⣀⣴⠒⠀⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢⡀⣼⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣿⣿⠀⠀⢀⢹⣿
⠀⠿⡀⠀⠙⠳⣤⢄⡀⠀⠀⠈⡇⠀⠀⠁⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠻⣿⡄⢠⣸⢻⣿
⠀⠀⠇⠀⠠⠀⠀⢸⣇⠀⠀⢠⡃⠀⠀⠀⠈⠙⣿⣿⣶⡀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⢸⣿⡌⠿⣾⡇
⠀⠀⢸⡦⠀⠴⣶⣟⣿⡂⠀⢸⠃⠀⠀⠀⠐⠀⠙⣿⣿⣿⣦⠀⠀⠀⢀⡞⢈⠁⠀⠀⠀⢰⣾⣿⣿⣿⣿⠃
⠀⠀⢸⠀⠀⠀⠘⠿⣿⣿⣿⣾⡄⠀⠀⠀⠀⠆⠀⢻⣿⣿⣿⡄⠀⠢⣾⠃⣸⡇⠀⠀⣠⣼⣿⣿⣿⣿⡏
⠀⠀⠘⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⡀⠀⢠⠀⠀⢡⣿⣿⣿⣿⣧⠠⠂⢻⣤⣾⡁⢀⣼⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⢢⡄⣾⣿⣿⣿⣿⣿⠀⠀⠀⣸⡇⣼⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⣿⣿⣷⣦⣽⣿⣿⣿⣿⣿⡏⠀⠀⢰⡿⢸⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⣿⣿⡗⠈⠙⣿⣿⣿⣿⡇⢰⣦⡿⠁⠸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⠀⣾⡿⠀⠀⢠⣿⣿⡿⠋⠀⠀⣠⣿⣶⣤⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢘⣿⣿⣶⡿⠁⠔⢀⣿⣿⠟⠁⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢨⣿⣿⣿⢁⡄⢀⣾⣿⣯⣤⠖⠂⣾⣿⣿⣿⣿⣿⡿⠙⢿⣿⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢯⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⣟⣿⠉⣿⣧⣿⡟⠉⠀⠀⣀⣾⣿⣿⣿⣿⣿⠏⠁⠀⠈⣻⣿⡅⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣦⡀⠀⠘⡿⣿⠾⡏⣿⠏⠀⢁⣤⣴⣿⣿⣿⣻⣿⢿⣿⣤⣦⣷⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢷⣤⡘⢻⣿⣷⣸⡏⣠⣾⠿⠛⠋⠙⠋⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⢹⣿⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡯⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⢻⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠛⠛⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⠔⠊⠉⠉⠉⠛⢦⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⠓⢦⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠀⠀⢻⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡴⠶⠶⠶⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡄⠀⣸⠀⠀⠀⢠⠏⠀⠀⠀⠀⣰⠃
⠀⠀⡤⠒⠉⠩⠩⠭⠭⠭⠤⠤⠔⠒⠒⠒⠂⠄⠈⠻⣆⠀⠀⠀⠀⠀⠀⢷⠈⣷⢠⠏⠀⠀⣰⠏⠀⠀⠀⠀⣰⠇
⢀⠞⠁⠀⠠⠄⠒⠒⠒⠒⠒⠒⠒⠉⢉⣉⣉⣀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠘⣇⠸⡏⠀⢀⣴⠋⠀⠀⠀⢀⡼⠃
⢧⡀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠒⢚⣋⣭⡤⠶⠶⠶⢤⣤⣀⠙⢦⡀⠀⠀⢀⣿⡄⢻⣠⡞⠁⠀⠀⠀⢠⡞⠁
⠀⠙⠲⢦⣤⣄⣀⣠⣤⡴⠶⠛⠋⠁⠀⠀⣀⣀⣤⠤⠤⠤⠀⠀⠱⡄⣠⠏⠀⣷⠈⣏⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣻⠽⠓⠀⠉⠀⠀⠀⠀⠀⢾⠋⠀⢀⣼⠀⣿⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣫⠖⠉⠀⠀⠀⠀⠀⠀⠀⣀⢈⠁⢸⡇⢠⡞⢸⠀⣿⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⠚⠑⠥⢊⣄⡤⠀⠀⠀⠀⡰⢅⠀⢀⣾⢁⡿⠀⡞⠀⡟⢀⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⣰⢏⡾⠁⠀⣀⢶⠟⠉⠀⠀⠀⣠⠞⠄⠀⣰⠟⡏⣸⠃⡼⠁⣸⠃⢸⡇⠀⠀⠀⠀⠀⣠⡴⢋⡵⠁⢠⡟
⠀⠀⠀⠀⠀⠀⢰⡏⣼⠁⠀⠀⠀⠀⠀⠀⣀⡴⠞⠑⢀⣤⣾⠇⢸⠃⣿⡞⠁⣰⠇⠀⠀⠻⢦⣤⡤⠶⠛⢁⡴⠋⠀⢠⠟
⠀⠀⠀⠀⠀⠀⢸⡀⠹⢦⣄⣀⣤⡤⠶⠋⠱⠀⣀⣴⡟⡙⢸⠀⣟⡴⠋⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⠀⠀⠂⣀⣠⠴⢛⠁⣸⠋⠃⡼⢰⠏⠀⣠⣟⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⢀⢿⣤⠸⢸⠀⡟⠀⠀⡇⢸⡥⠚⠁⠈⠓⠶⠶⠶⠶⠛⠋⠁⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠇⠀⠸⡏⠂⠈⠀⡇⠀⠀⣿⢸⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣧⠀⠀⠀⡇⠀⠀⣿⠌⡇⠈⠑⢦⣄⡀⠀⠀⠀⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢸⠀⠀⠀⢿⠀⠀⠀⣷⢠⠀⢸⠁⢸⡀⠀⠀⠈⠉⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢠⢾⢀⠀⡆⢸⠀⠀⠀⢹⡈⡇⠘⡆⠂⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠡⠄⠇⡘⠈⠀⠁⢸⡆⢀⠀⠘⡇⠁⠀⢱⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢧⠇⠀⠀⠀⢸⡇⢸⠀⠀⣿⠀⠀⠈⡆⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢦⡀⠏⠀⡿⠀⠈⠀⠀⢻⠀⠀⠀⢸⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⠙⠦⣤⡇⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⣄⠀⠀⠙⢷⣤⡀⠀⢸⠀⠀⠀⠀⢿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⣳⣶⡖⠀⠁⠉⠀⣾⠀⠀⠀⠀⢸⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡴⠃⠘⠻⠆⢠⡤⢠⠏⠇⠀⠠⢀⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⣈⣡⢋⣈⣠⣤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠠⣴⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⢰⠀⡠⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⢣⠈⣆⠼⡤⣇⣰⠁⣠⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⡤⠫⠁⠃⠐⠈⠀⠐⠫⠟⣡⢞⣁⡠
⠀⠀⠀⠀⠀⠀⡀⢄⠀⡀⠀⠀⠀⠀⠀⣈⠮⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⡧⠔⠀⢀⣄⡔⠀⠂
⠀⠀⠀⠀⠑⣦⣼⣾⣿⠟⡋⢀⠀⠀⠀⠘⢓⢀⢀⣀⣴⣜⣿⣿⣶⣀⠄⠀⡛⠃⠀⠀⠀⢻⡇
⠀⠀⠈⠛⣶⣿⣿⡟⢀⡎⠀⠄⠂⠀⠀⠀⠀⠢⢤⣽⣿⠿⢿⠻⢿⡿⢦⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠉⠳⣿⣿⡷⠀⣾⣇⠀⠀⠀⠀⠀⠀⣀⠰⣍⢻⡈⡇⢸⡂⢸⠀⡼⢡⠶⡠
⠀⠀⠈⠙⣿⣿⣏⠈⡿⠏⠀⠀⠀⡄⠀⠀⠨⣷⣼⠶⠿⠛⣿⡟⠛⠛⠓⠿⠾⠵⠃⡀
⠀⠀⠀⢉⣿⣿⣷⠀⡈⢂⠀⢀⣤⢧⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠺⢿⣞⢄⡀
⠀⠀⠀⠀⠀⠈⠿⠧⠈⣿⡿⠛⠉⠹⣷⠀⠀⠀⠀⠀⠀⢠⡛⠀⠀⠀⠀⡐⠿⣤⡄⢻⣷⠪⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠁⠀⢼⣷⠀⣿⡮⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠁⠀⠀⠀⢠⡇⠀⠀⢼⡷⢀⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡞⠀⠀⠀⢀⠗⠛⠣⠔⣻⡅⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠃⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⡵⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠑⠂⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣤⣴⣶⣤⣤⡀⠀⢿⡄⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠉⠻⣿⣿⣿⠿⠙⠈⠁⠈⣿⣆⠀⠐⢿⡶⠇
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠻⣿⡄⣼⣿⢿⠏
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⠀⠀⢹⣧⢠⣿⠿⠚⠂⠀⢀⣤⣴⣶⣶⣿⣷⣶⣲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠈⣿⣴⣌⡿⣣⣤⣶⣿⣿⣿⠿⠋⠍⣹⠟⠁⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⢻⣿⣼⣾⣯⣾⣏⠿⠋⠁⠀⢀⠞⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢠⣴⠖⠂⠀⠀⢉⡀⠱⡀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠘⢿⣦⣀⣤⣾⣿⣇⠀⠱⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢰⢿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⣼⣿⣿⣿⡿⠉⣿⡀⠀⢧⠀⠘⣿⣿⣿⡇⠀⠀⠀⠟⣾⣿⣿⡽⢃⣄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⢻⣿⣿⣇⠀⠀⣿⡇⣠⣴⡀⠀⢻⣿⣿⡇⠀⠀⢠⡞⣿⣸⣟⣤⠀⠈⢁⡠⠒⣨⠴⠒⠈
⠀⠀⠀⠘⣿⣿⡇⠀⠀⢸⣷⣅⣸⠉⢆⠀⢿⣿⡇⢀⠀⢨⣽⡇⢻⣏⠛⠀⠶⣿⣶⠞⠃⠀⠀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⣿⠉⠁⠀⠈⡆⢸⣿⣷⢸⡇⣼⣿⣡⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⢹⣇⠀⠀⠀⢷⠀⢿⣿⣿⢧⣿⡛⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣴⣾⣿⣷⠀⠀⠀⣿⡀⠀⠀⢸⡀⢸⣿⣿⣸⡟⣧⣟⠀⢠⣾⡏⡀⣠⠂⠀⠀⠀⠀⠀⠈⣿⣿⣿⠿⠁
⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⢻⡇⠀⠀⠸⡇⢸⣿⣿⡿⢡⣿⠏⣰⣿⣿⣿⠿⠃⠀⠀⢠⣶⣶⣶⣄⣘⣛⣠⣀
⠀⠀⠀⣹⣿⣿⡻⣿⣿⡄⢀⣿⡇⠀⠀⠀⡇⣸⣿⡟⣠⡿⠁⣼⣿⣿⡿⠋⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣻⣿⣿⣧⡚⢿⣿⣾⣿⡇⠀⠀⢸⡇⣿⢹⣱⡿⠁⣼⣿⣿⠿⣁⠀⠀⠀⠀⠀⠈⠙⣷⢻⣿⣿⣿⣿⣿⡟
⠀⠀⠀⣿⣿⣿⣿⣷⡈⠻⣿⣿⣷⣤⣴⠟⢠⡿⣿⡿⣠⣾⣿⣿⣯⢟⡥⠀⠀⠀⠀⠀⣠⡞⠁⣠⣿⣿⠉⠉⠁
⠀⠀⠀⢿⣿⣿⣿⣿⣷⡀⠱⠀⠉⠉⠀⠀⣸⣧⣿⣴⣿⣿⣿⣿⠷⠷⠒⠀⠀⠀⢀⠄⢉⡴⠞⠛⠿⣿⣷⡄
⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⡰⡄⠀⠀⠀⣸⡟⡾⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⣸⣄⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡇⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⢠⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠋⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⡸⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣀⣈⣿⠀⠀⠀⠀⣿⡟⠀⠀⢠⠃⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠁⠀⠀⠀⢸⣿⡇⠀⢀⠎⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠆⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠄⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⣾⣿⣯⣀⣴⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⣸⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢒⣢⣤⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⠎⠝⠀⠈⠳⡤⠦⠒⠊⠀⠈⢛⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡶⣷⣼⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⣎⢨⠦
⠀⠀⠀⠀⠀⠀⣠⣼⣷⡈⢻⣵⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢁⡘⢯⣀⡵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢨⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠟⢀⡈⣱⠎⢳⠞⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡔⢧⢽⡵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡵⠃⠸⠃⠄⣬⡇⢡⡇
⠀⠀⠀⠀⠀⠀⢠⡎⡰⠋⠀⠛⢞⠂⠀⠀⠀⠀⠀⣴⠆⠀⢀⢮⣏⠘⠀⠀⡐⠓⢯⣷⣮⠟⣿
⠀⠀⠀⠀⠀⢠⠓⣞⠁⠀⠀⠀⠈⢆⠂⠀⠀⠀⠀⣿⡇⠀⠘⡽⣤⣦⣄⡺⣳⣰⣾⠿⣻⣴⣿
⠀⠀⠀⠀⠀⡎⢹⠃⠈⡀⠀⠀⠒⢸⡦⠀⠀⠀⠀⣿⠅⠀⠀⣿⢾⣿⣯⣽⠽⠿⠞⡛⣉⣿⣿
⠀⠀⠀⠀⠀⡏⢻⠄⢀⠁⡀⠀⠠⠀⣷⠂⠀⠀⠀⣿⠀⠀⠀⣿⣾⣿⣧⢿⡂⣶⣶⣴⠍⣹⣿
⠀⠀⠀⠀⠀⢹⠘⡆⠀⠐⠁⠀⠀⢀⡏⠀⠀⠀⠀⣿⠀⠀⠀⢿⣼⣿⢿⣇⡇⢨⡿⠇⢂⣹⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡓⢢⡤⣲⣶⢾⣿⠀⠀⠀⠀⣿⡀⠀⠀⢸⣜⣿⣿⡼⠁⣿⡇⡖⣾⣽⠁
⠀⠀⠀⠀⠀⠸⡟⣿⣿⣯⣿⢉⣿⣻⡏⠀⠀⠀⢐⣿⠙⠲⢤⣀⡟⣷⢸⡇⣾⣿⠇⢰⣿⡟
⠀⠀⠀⠀⠀⠀⣿⢿⣿⡽⣿⣸⣿⢹⡇⠀⠀⠀⢸⣿⠀⠀⠀⠈⣻⡬⢘⡇⠙⡿⠘⣿⢿⠇
⠀⠀⠀⠀⠀⠀⢻⣙⣿⡇⣿⢉⠋⣼⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣳⠸⡇⣆⠃⠀⢧⣿⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣟⣇⣿⣿⠀⠄⡟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠃⠀⡄⣿⠰⠇⢀⡟⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⠀⢀⠇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠃⠈⠁⡟⠀⡶⢠⠁⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣽⣯⣿⢿⠸⠸⠇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠂⠀⡆⠀⠀⢡⠀⡀⠀⠀⣠⢠⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡁⡟⣿⠀⢰⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⠀⢣⠀⠇⢾⣸⠃⠀⠀⢻⣼⣇⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡇⢷⣾⠀⢘⡄⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡇⡊⢼⢠⡜⣇⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠘⡇⠀⠀⠸⠀⡟⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣤⡅⣸⠰⠄⣛⠀⠀⠀⠀⠀⣿⢳⠀⠀⠀⠀⠀⢇⠀⠐⠀⠀⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡄⢧⢛⡤⠀⣽⠀⠀⠀⣠⣼⣿⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⡇⠀⠀⠀⠀⢀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣆⡟⡁⠀⢸⣶⢾⡿⣿⡼⢣⠃⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄⠘⠛⣛⣋⠩⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠢⡤⠈⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡀⢀⣀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⣷⣄⠀⠀⣠⠏⠠⠀⢀⣤⣶⠶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡀⠸⣷⣼⠀⣿⣄⡧⢺⠀⣰
⠀⠀⠀⠀⠀⢀⣠⣤⣀⠉⠀⠀⢻⡆⠀⡐⣵⠟⠁⠀⠜⠃⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣝⣿⣿⢀⢹⡟⠁⢻⣷⡿
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢷⣄⣼⡇⢰⣸⡿⠠⠦⠀⢀⣴⢾⣿⠟⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⣿⣿⣾⣷⣶⣼⠏⠁
⠀⠀⠀⠀⠀⣀⡤⠶⠶⡤⢤⣉⡙⢿⡀⣿⠅⠀⢀⡰⠛⠁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⠿⢿⣿⣿⡏⠉⢰⣦
⠀⠀⠀⠀⠃⢀⡀⠀⠀⠈⠉⠉⠙⠉⠻⣿⠀⣠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⢇⡀⠀⢀⢂⢿⣧⣔⡠⡛
⠀⠀⠀⠀⠈⠃⣀⣠⠴⠲⡴⠟⠒⠾⠛⠙⢶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣟⣻⣽⣿⡏⢻⣿⣿⠿⠇
⠀⠀⠀⠀⠀⡠⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢀⣀⣤⣤⣠⠾⠟⠂⠀⠴⠛⠛⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣇⣬⡟⡀⢻⣿
⠀⠀⡤⠛⠋⠒⠐⠛⠛⠿⢉⣁⡀⣀⣤⡄⠀⢹⡀⢺⠟⠋⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⠋⠉⠛⣿⡆
⠀⡄⠀⠀⠀⡀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⢠⣌⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⣾⣿⡷⠐⢄⡀⠀⠀⢲⣿
⠀⠀⠀⠸⠞⠁⢀⣀⣠⣖⡶⠞⡷⠿⠟⠛⠿⢷⣶⣧⣿⡿⠃⠀⠀⢩⡇⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀
⠀⠀⠀⢀⣠⣴⡿⣻⠍⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⡿⠙⠁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣭⣭⣍⡉⠃
⠀⠀⣰⣟⡽⠷⠛⠁⠀⠀⠀⢀⡀⠈⠃⣄⠀⠀⠀⣧⠀⠀⠀⣈⣦⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠛⠉⠁⠈⠉⢙⣿⡟
⠀⠰⡏⠉⠀⠐⢀⠀⠀⠀⢀⣾⣵⠶⣦⣴⣆⡄⠦⣿⠀⠀⢠⡇⢈⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⡇
⠀⠀⠓⠀⠀⣾⠄⠀⢀⡔⠉⠩⠄⠈⠁⠈⢻⣿⡆⢨⡤⢖⢻⡇⠐⢷⠖⠒⠒⠀⠀⠲⠤⠄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠃⣅
⠀⠀⠀⠀⠀⢸⣧⠘⠉⢀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣘⠀⠀⠘⠷⠠⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢸⠄⢰⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠃⠀⢿
⠀⠀⠀⠀⠀⣼⡿⢀⡴⠚⠀⠀⠀⠀⠀⠀⢀⡀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⣿⡆⠀⠨⡀
⠀⠀⠀⠀⠐⠃⠀⡟⠁⠀⠀⣠⡄⡶⠖⠛⠉⠁⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠰⠃⠀⠀⣇⠀⠀⠀⠀⠀⠀⠘⣿⡇⢿⣿⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⢊⠠⠋⠀⠀⠀⠀⠀⠀⣿⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠠⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⡹⠈⠀⠀⠀⠀⠀⠀⠀⠀⣙⠢⡝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⣿⣿⠘⣿⡿⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢈⣉⣀⣰⣖⠒⠀⠀⠀⠀⠀⠀⣿⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⠀⣽⠀⠀⠀⠙⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡿⣿⣿⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣻⣀⠄⠢⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⣿⣿⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢻⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣷⣿⡿⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⣿⣿⠇⣸⣟⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⣿⡿⢰⣿⡏⠁⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣦⣴⣿⣵⣿⡿⠋⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣼⣄⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣯⣴⣦⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠛⠿⠟⠿⠿⠿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠤⠀⡀⢂⣤⠶⠚⠛⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⣀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⠂⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢛⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⢦⣀⡀⠀⠀⡀⠀⢀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢂⡀⠀⢀⡄⠉⠉⠢⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠌⠑⠛⠛⢙⠛⠛⣯⢨⡝
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣹⣶⡾⠷⠶⠶⢦⡝⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢰⠄⠔⠀⠠⡆⠀⣿⠀⡇
⠀⠀⠀⠀⣀⠀⣁⡠⠶⠚⠉⠀⠀⠀⠀⠀⠀⠙⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⣇⠀⡀⠐⠇⠀⣿⠀⡇
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⢹⠀⢇⠀⡅⠀⠂⢰⣧⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣠⢆⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⡀⢈⠀⠇⠀⡃⢸⠟⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⣦⡛⣚⣻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⠀⠀⠤⠁⢸⠐⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⣤⣴⣦⡶⠿⠋⢻⣹⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠐⠇⢸⢇⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⢁⠉⠘⢃⡆⠉⡴⣩⢟⡀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡂⠀⠀⠀⠰⡆⢸⡄⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢸⡀⠘⢃⠀⢣⠌⣛⡀⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡄⠀⠀⠀⠠⡇⠰⠄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠆⠙⠀⢁⡌⠹⣟⠉⡜⢠⣥⠈⠉⠁⠘⣀⠀⠉⠀⠀⠉⠋⠉⠉⢀⡀⠀⠀⠚⠁⡄⣈⣧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠂⠐⠀⡀⠀⠛⠀⠇⢼⡇⠈⢀⡈⠄⢳⣤⣾⠻⢿⡁⢤⡤⢤⡈⢉⡀⠀⠘⠁⢻⡇⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠈⠁⠀⣘⠋⣧⡴⢹⡛⣤⣵⠈⠃⣼⣿⡄⢺⠍⠀⠀⢀⡉⠛⠁⠀⠀⠈⠄⣽⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠠⣼⡏⣼⣿⠿⠁⠄⠺⡿⠙⡿⠟⣖⣡⡶⡿⢹⡎⠀⠀⠀⠀⠠⠶⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠐⢁⡼⡗⢸⡿⣦⠰⣾⢺⡇⠀⠀⢈⠉⠀⠰⢺⢎⡇⠀⠀⡀⠀⢀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠈⣠⠏⢸⣷⣩⡒⣏⣸⡇⠀⠀⠀⢀⠀⢀⣿⢰⡇⠀⠀⠀⠀⠼⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⡀⠲⢌⠣⢼⡯⣿⣬⡇⠀⠀⠀⠀⠰⢦⣿⡄⠃⠀⠀⠀⠀⠂⣸⠀⠀⠀⠀⢀⠀⠶⣤⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⣦⣤⣿⠈⠁⠀⠀⠀⠸⢘⠧⠌⣷⢼⣷⣿⢴⡇⠀⠀⠀⠀⠆⣸⣟⣧⠁⠀⠀⠀⠀⡥⡉⠀⠀⠀⠰⠖⠀⣨⣡⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣻⣿⡇⠀⠀⠀⠀⣠⠈⠄⠀⣧⢘⣿⡗⢻⡇⠀⠀⠀⠀⠀⡏⣿⠳⠀⠀⠀⠐⠂⠀⠇⠀⠀⣿⠀⠀⠂⠀⢀⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠌⠋⢻⣟⠃⠀⠀⠀⠀⠀⠀⡓⠀⣿⠘⣼⡟⣿⡇⠀⠀⠀⠈⠀⢄⣿⣡⠀⠀⠀⠀⠀⢸⠀⢰⢃⠿⣶⣶⣴⣶⠛⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢾⣭⢸⠀⠀⠀⠀⠀⠀⢡⠀⡟⢹⣿⢫⠟⣿⠀⠀⠀⠀⠀⣾⡟⣿⠀⠀⠀⠀⠀⠆⠀⡟⢸⠄⠀⠀⠀⢈⡅⢸⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⢤⢜⡇⢀⠀⠀⠀⠀⠀⣲⠀⣧⢸⣟⣇⣸⡿⠀⠀⠀⠀⠀⢿⡁⣹⠀⠀⠀⠠⢀⡦⠙⠀⠸⠀⠀⠀⠀⠤⠁⠘
⠀⠀⠀⠀⠀⠙⠣⡄⡀⠀⢸⡈⠢⣾⡇⠀⡀⠀⠀⠀⠀⠬⠀⣿⢸⠇⣿⢼⡇⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⣼⠃⡆⠀⠀⡀⠀⠀⣀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢈⣩⡄⠁⠀⢾⡀⡃⠀⠀⠀⠀⠀⠀⣿⡄⣤⣷⣾⠁⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⠀⠀⣼⠹⠀⠀⠀⠀⠀⠁⠀⠌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠸⠧⣿⠀⠀⠀⠀⠀⠀⡞⢱⡇⢻⣟⠀⡇⠀⠀⠀⠀⢸⢠⡅⠀⠀⠰⢠⡝⠁⠀⠀⢰⡀⣴⠆⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠱⠌⠀⢲⡿⡇⠀⠀⠀⠀⠀⣿⣻⣯⠸⣿⣯⡇⠀⠀⠀⠀⢺⡌⠁⠀⠀⠀⣾⠁⠀⣀⠀⠈⡠⠂⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠲⠸⠀⠀⠀⠀⠀⠋⠙⣾⠁⣾⡳⣧⠀⠀⠀⠘⠁⡄⠀⠀⠀⣲⡧⠔⢁⡢⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⣃⠀⠀⢃⡖⠀⠀⠀⠀⢀⠀⢻⡄⢠⣯⣇⠀⠀⠀⠀⢀⠁⠀⡠⠞⠉⡲⠀⠈⠀⠀⢸⠀⠀⢀⠆
⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⣀⠄⠚⠿⡙⠷⣄⠀⠸⣇⠀⠀⠀⠘⠀⠘⣷⠈⢠⠁⠀⠀⠀⢰⣙⡀⠂⠀⣠⡜⠀⠀⠀⠀⠀⢸⠀⠀⣈⠀⣀
⠀⠀⠲⠶⠛⠖⠒⠒⠈⠁⠀⠀⠀⠀⠑⡀⠀⠱⣄⢻⡄⠀⠀⠀⣠⠰⡿⢁⠐⣦⠀⠀⠀⠀⠏⠀⢀⣶⡿⠀⠀⠀⢀⣰⣾⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠶⠒⠻⠄⠀⠘⣎⢷⡀⠀⠀⠹⣾⠀⡟⡀⢿⡀⠀⢀⡕⠀⠖⢀⣼⠃⠀⠀⢐⡴⠋⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠘⡄⠀⠘⠌⢷⡀⠀⠘⣿⠀⢸⢣⣼⡇⠀⢸⢃⡿⠀⣨⡏⠀⠀⣠⠊⠀⣠⡖⢀⣶⣤⣤⣤⣤⣤⠶⠾⠋
⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡘⣄⠛⣄⠀⢻⠀⠀⠞⣹⡅⠀⣿⡜⠅⢀⠞⠀⢠⡴⠁⢀⡞⢩⣄⣄⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣷⣈⠳⡀⠃⡀⢰⣿⠃⣸⡯⠁⣀⡟⠀⡶⠋⠀⠀⣬⡿⠊⠉⠀⠀⠀⠀⠉⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢻⡷⣿⢶⣝⣾⣧⣴⣿⡿⣴⠿⢿⣿⡇⠀⠰⠿⠛⠣⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠐⠒⠤⣄⡀⠀⢀⣾⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⣿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠘⠶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⠙⠻⢷⣶⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠉⠉⠓⠒⠠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡿⠁⠹⣿⣿⣄⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣤⡤⠛⢤⣿⣦⣶⣿⣿⣿⣿⣿⡇⡇⡟⣤⢶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⡾⠗⢠⣹⣿⣿⣿⣿⣿⣿⣽⢷⣏⡇⡇⡇⠃⡾⡩⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢺⣧⠤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠹⠇⠿⣷⣷⣧⡿⣵⡆⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢿⡥⠁⢾⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠉⠛⢿⣿⣇⣀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣾⣷⠏⠊⠉⡻⣿⣿⣿⣿⣧⠤⠀⠀⠀⠀⠀⠀⠀⠸⠆⠐⡒⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⣾⣤⣤⡊⣿⠿⣿⣿⣿⣾⡀⢀⣀⡀⠀⣠⡀⢩⣑⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠉⠙⠻⢿⣿⣤⣟⠑⠋⠘⠯⠿⢿⠿⣿⡛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡆⠀⠀⠀⠀⠀⠙⠿⡷⠤⣾⣤⣴⣮⣶⣿⡧⠸⣿⣿⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⡀⠀⠀⠀⠀⠀⢈⠀⢀⣠⣤⣤⣾⣧⡀⣀⠀⢠⣤⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠹⣧⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣏⠁⠉⠀⢸⣯⢩⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡃⠀⠀⠀⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡞⣿⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⠍⠀⠀⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⢶⡀⠸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⢸⡇⠐⢸⣿⣿⠀⠀⢀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⣿⠀⠀⢿⡏⠀⢠⣼⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠐⠀⣿⡆⠀⢸⡇⢸⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢷⣄⣀⣿⡇⠀⣾⣧⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⡿⠟⣸⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣾⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⣴⣻⠙⡻⢝⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⣻⣿⣿⣦⠀⠀⠀⠀⢖⣶⣆⠀⠀⠀⠀⢀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⣡⡌⣿⣶⡀⠁⢿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣻⣿⣿⣿⣧⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣾⡿⠋
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠈⣄⠀⠿⣿⣿⣶⣯⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠘⣿⣆⠀⠀⠀⠀⠀⠀⢀⡩⠀⠙⠻⢿⠁⣐⣻⣿⠀⠀⠀⢰⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠠⠶⠚⣿⠇⠀⠀⣾⣿⠀⢙⣿⡏⠀⠀⠀⠈⣟⣟⣯⣭⣭⣭⠇
⠀⠀⠀⠀⠀⡌⠀⠀⠀⢀⠀⢀⠀⠀⠀⠉⠛⠒⢤⣶⡀⠘⢆⠀⠀⠀⠰⠏⠀⠀⠀⢻⣿⣧⣾⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠈⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⠁⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠀⠀⠀⠀⠀⠀⣟⢉⣿⣿⡿
⠀⠀⠀⠀⢸⠀⠀⠀⠈⠙⢶⣶⣤⠀⠀⠀⠀⠀⠀⠹⣿⢿⣿⡄⠸⡄⠀⠀⠀⠀⠀⠀⢸⣿⠈⠆⠀⠀⠀⠀⠀⡿⢿⡿⣿⠇
⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⡀⠸⣿⡄⡀⠀⡀⠀⠀⠀⠘⡇⠙⣿⠀⠹⡄⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣷⡟
⠀⠀⠀⠀⠸⠀⠀⠞⢁⣴⣿⣿⣿⡇⡀⡀⠀⡹⠒⢤⠀⠀⠄⠸⣇⠀⠑⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡆⢸⠀⣻⣿⣿⣿⠏⠁⠁⠈⠛⠷⢶⡄⢹⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⢰⣿⣿⣿
⠀⠀⠀⠀⠀⢠⢨⡄⠘⠻⣿⣿⡄⠀⠀⠀⠀⠀⠈⠀⣸⡀⠀⠀⠀⢿⡄⢀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⣀⣰⣿⡟⠀⠀⠀⢸⡇⠀⣧
⠀⠀⠀⠀⠀⠸⡀⠙⠲⣤⣏⡟⠉⠀⢠⣄⠀⢠⣴⣿⣿⡇⠀⠀⠀⢸⣿⣾⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⣰⣿⣿⡟⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠉⠁⠓⢳⣿⣿⠗⠀⣟⠁⣸⡇⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⢹⢀⣿⡇⠀⠀⠀⢃⣸⣷⣱⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘⢸⡇⠀⠀⢸⣸⡿⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡀⠀⠀⢰⡇⣧⠀⠀⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⡇⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⡾⠃⠰⣾⣿⡆⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢹⠃⠀⣼⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣟⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⢻⣿⣦⠀⢸⣇⡿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢰⠀⢸⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⣿⡏⠀⣿⣹⠃⢈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣾⠀⡬⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣿⣧⣾⡿⠃⣠⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠛⠛⠓⠀⠊⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠐⠒⠃⣀⣤⣤⣤⣤⣴⡶⢾⣿⣿⡶⠿⠛⠓⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⡿⠟⠋⣉⡤⠟⠛⠩⠛⣀⣴⣾⣿⡿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⣠⣤⣴⡾⠿⣋⡵⠾⠛⠋⠉⠁⠠⠴⠚⠉⣠⣤⣴⣶⣿⣿⣻⡿⠋⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠁⠀⠀⠀⠀⢀⣤⣶⣿⠿⠋⠁⠄⠂⠁⠀⠀⠀⠀⠀⠀⡄⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⠀⣠⣾⢟⡉⠛⠋⠀⠀⠀⠀⠈⠁⠀⣀⣤⣶⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⢀⣴⠾⠋⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⣴⣛⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⡶⠛⠛⢻⣿⣄⠀⣠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠃⠀⠀⢀⠲⣾⣿⢇⣀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⣠⣴⣿⣿⣴⣿⣿⡿⠿⠛⢛⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡀⠠⠦⠛⢁⡟⢁⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠿⠿⣿⣥⡀⠸⣷⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣠⣤⣤⣶⣿⣿⠿⣿⣿⣧
⠀⠀⢠⡄⠀⠀⠀⠋⣀⣤⣤⣶⣿⡿⠿⣿⣍⡉⠓⠶⣤⠛⢻⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⢃⣼⠇⠛⠋⠀⠀⠀⠀⠀⠀⢿⣽⣄
⠀⠀⢸⡇⠀⠀⣾⣿⣿⣿⡿⠟⠉⠀⠚⠛⠛⢻⣷⣶⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣨⣿⠈⠀⠀⠀⢀⣤⣤⣴⣶⣿⡿⠋
⠀⠀⢸⡇⠀⠀⠀⠉⠉⠁⠀⠘⢃⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣿⡿⠃⠀⠀⠀⠉⠉⠉⣿⣏⣉⣀
⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⣶⠟⣿⣾⣶⢂⣤⣤⠀⠰⣿⣿⣿⣿⡿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠶⣶⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠁⠐⠂⢶⣿⣿⡿⠉
⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠠⡄⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣿⡿⢀⠀⠀⣴⣦⣶⣿⣶⣿⡏
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠈⠙⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⠛⣧⣟⠀⣾⠟⠁⠀⠀⠀⠀⠀⠠⠉⠉⣹⠄
⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠖⠚⠛⠻⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⡙⢿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠺⡿⠛⠁
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⠟⠁⠀⠀⠀⠀⢀⣀⡐⣠⣼⡇
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⢀⠀⠠⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠠⣀⠀⢀⣠⣬⣽⡿⠿⠿⠃
⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣤⣤⠁⠀⠁⢀⣿⠿⡝⢿⣧⡽⠛⠉⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠇
⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⣰⡿⠟⠛⣿⣟⢃⣤⣶⣦⣬⣀⠀⠀⢩⡿⠁⠀⠀⠀⠀⠀⢄⡀⠠⣶⣿⣿⣿⣿⡿
⠀⢼⣿⠀⠀⠀⠀⠀⠀⢠⣿⢇⣶⣿⡿⠋⣾⣿⣿⣿⣿⣿⣿⣦⣼⠇⠀⠀⠀⣿⠀⢀⣺⣿⣿⣿⣿⣿⣿⣿⠇
⠀⣸⣿⠀⠀⠀⠀⠀⠀⣾⡿⢾⣿⠙⢠⣜⣿⣿⢿⠏⠘⢫⡀⠈⠻⣷⣴⠃⠈⢻⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢿⣿⠀⠀⠀⠀⠀⣸⡟⣠⣴⠏⢠⣿⣿⣿⡧⠀⠀⠀⠘⠁⠀⡐⢾⣿⠀⢸⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⡇
⢀⣿⡿⠀⠀⠀⠀⢠⣿⠀⣿⡗⢠⣾⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⢁⣿⣿⠄⢸⣯⣄⠀⢰⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠘⣿⡇⠀⠀⠀⠀⣾⢃⣀⡿⢀⣼⣿⠇⣸⠃⠀⠀⣀⠀⠀⢘⢀⣾⣿⡏⢠⣿⠅⣏⠀⠸⣏⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣿⡇⠀⠀⠀⢰⠃⢸⡿⠀⣴⣿⣯⠀⠉⠀⠀⠀⠁⠀⠀⣌⣸⣿⣛⠁⣸⠇⣰⣿⡇⡶⠉⢣⣿⣿⣿⠃
⢸⣿⡇⠀⠀⢀⣿⠀⡾⢀⣼⣿⣯⠁⠀⠀⠀⢘⣧⠀⣠⡦⣸⣿⣿⡏⢠⣿⠀⣾⡄⢀⣥⢀⣿⣿⣿⡟
⢸⣿⡇⠀⠀⣸⣿⣧⠀⠀⣽⣿⠉⠀⣤⣀⠀⠻⠋⠠⣾⣥⣿⣿⣿⠀⣼⡇⠀⣠⣶⠿⣿⣼⣿⣿⣿⠁
⣾⣿⡇⠀⠀⣿⣹⡇⢠⣼⣿⡷⠗⠘⠋⠉⣠⣄⠀⠛⠈⣿⣿⣿⡅⢰⡿⢃⣾⣿⠋⠘⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⡇⠀⢰⡟⣽⠃⣾⣿⡟⠀⣀⠀⠀⠀⠈⠉⣰⣷⣸⣿⣾⠉⢀⣾⣷⠀⠛⡛⢠⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⣿⣿⡇⠀⣼⣟⡇⢀⣿⣿⣧⠸⣏⠀⠘⠃⠀⠀⠼⢹⣿⣿⠟⠀⣼⣿⠋⠀⠛⣠⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸
⣿⣿⡇⢠⡿⠿⠀⣿⣿⡟⠀⠘⠁⢰⠀⠀⠀⠐⣷⣿⣿⠛⠃⢸⡿⢁⣴⠀⣸⣟⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣾
⣿⣿⡇⢸⡗⡟⢀⣿⣿⣿⠇⠀⠸⠟⠀⣠⠀⠐⣼⣿⣷⠆⣰⣿⠁⠘⠃⠘⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⣿
⣿⣿⡇⢸⣿⡇⣸⣿⣿⠀⠀⠀⠀⠁⠀⠁⣴⣶⣿⣟⠉⢰⡿⠃⠾⠁⠀⢰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⣴⡾⠆
⢸⣿⣧⣿⢻⢃⣹⣿⡏⠀⣼⠓⠀⠀⠀⢠⣿⣿⣿⠏⢠⣿⠁⢀⡀⢠⣀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠀⠆⠄⠘⠋⠁
⠘⣿⣿⣿⢴⡅⢹⣿⠃⠀⠁⠀⠀⠀⠀⣼⣿⡿⠁⣰⣿⡿⠂⠿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⡴⠛⣁
⠀⢻⣿⣿⣿⣇⣿⣿⠐⠂⠀⢂⡤⣀⣼⣿⠟⣠⡾⠛⠀⣠⣤⣾⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⣀⠴⠛⠁
⠀⠀⠻⣿⣿⣿⣿⣿⠶⠲⣾⢟⣼⣿⠟⣡⣞⣻⣄⣀⣴⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣤⠾⠂⠁
⠀⠀⠀⠙⠿⣿⣿⣿⣷⣤⣴⣯⣵⣶⣿⣿⣿⣿⣶⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⣿⣿⣯⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⡏⠀⡁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣽⣿⣿⣿⣿⣿⡇⣶⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡸⣍⠀⠀⠘⣥⡜⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢷⣿⣿⣿⣿⣿⣷⡾⣿⣿⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⡁⠈⠐⢢⡄⣳⣾⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠈⣿⣿⣿⣿⣿⡿⠙⠂⠉⢀⡄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⢰⣏⢿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡉⠀⠂⠀⢀⣬⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢺⣿⢶⣿⣿⣿⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠛⠃⢸⣿⣼⣿⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿⠇⠀⠠⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⢨⣿⣻⡿⠿⠿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠳⣿⣿⣿⣷⡀⠀⡀⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡛⣡⣤⣤⣤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⡀⠀⠀⠘⣻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠁⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⢀⡤⠜⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⣴⣿⠇⡀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠏⢸⡇⠀⠀⠀⠀⠀⠀⢻⠃⠀⣄⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢹⡀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠈⡀⠀⠀⠀⠀⠀⠀⢹⣿⠛⢃⠀⠀⠀⢸⣿⡇⢀⡴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣄⠀⢃⠀⠀⠀⠀⠀⠀⠘⣿⡄⢸⠀⠀⠀⠸⣿⠃⣼⣷⣿⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣾⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣿⠀⣿⢹⣿⢻⠀⠀⢀⠂⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠹⣿⡿⢿⣿⡀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⣸⡟⠀⣿⣏⠉⡞⠀⢀⡎⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢻⣿⡄⠹⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣿⢋⠀⣡⣤⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠈⢿⣿⡀⢹⡆⠀⠀⠀⠀⠀⣇⠀⠀⠀⢰⣿⠀⠀⣿⠈⠀⢉⡋⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠘⣿⡇⠀⣴⠀⠀⠀⠀⠀⠛⠀⠀⠀⣼⣷⠀⠀⣿⡲⠀⣾⠃⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠂⠀⠀⣿⣿⣿⡇⠀⠀⠀⢹⣿⡄⠈⠀⠀⠀⠀⠀⡖⠀⠀⠀⣿⣿⣷⠇⢸⡇⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⢀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢿⣧⠀⢀⠀⠀⠀⠀⡇⠀⠀⣸⡇⣿⡟⠀⣾⡃⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠸⣿⠀⢈⠀⠀⠀⢰⡇⠀⢠⡿⢡⡿⠁⠀⣿⠁⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⢿⡇⠘⠃⠀⠀⠸⠁⠀⣼⠁⣾⠃⡰⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠸⣧⠀⠁⠀⠀⡆⠀⣸⠇⡸⠁⠂⠀⢰⡏⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢻⣿⣇⡀⠀⠀⠀⣿⠀⠀⠀⢀⠃⣰⠏⢠⠃⠈⠀⠀⡞⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⢻⣿⣧⠀⠀⠀⢸⡆⠀⠀⡜⢠⡟⢠⡎⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢻⡿⣆⠀⠀⢸⡇⠀⠀⢁⡿⢱⡿⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠈⢻⣿⡄⠀⢸⡇⠀⢇⣼⢧⡟⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠈⣿⣷⠀⢸⡇⢀⣼⢣⡞⠁⠀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄⢹⣿⡆⢸⠇⣼⣿⡾⠀⣠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⣷⣿⣷⣿⣼⣿⣿⣵⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡟⣿⣿⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣼⣧⣿⣿⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠙⠛⠛⠛⠻⠛⠛⠛⠛⢛⠛⣉⡉⠉⢭⣉⢨⡤⣠⡄⠀⣀⣠⠄⠀⣄⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠙⠛⠛⠃⠀⠚⠛⠛⠛⠛⠛⠛⠛⠙⠿⠛⠒⠹⠿⠻⠟⠙⠟⠛⠛⠛⠗⠉⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣯⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣟⠛⣷⠀⠀⠄⠁⠀⠈⠀⠀⡀⠂⠀⠀⠀⠀⠀⠔⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣼⣻⣿⡋⠄⠀⠀⠠⢀⠠⢀⠀⠄⡁⠂⠈⠀⠀⢀⠀⠐⠊⠁⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠂⠐⠈⣉⣉⠛⠉⢁⠀⠀⠨⠕⠢⠀⠤⠔⠠⠄⠤⠆⠉⠒⠁⠀⠀⠀⣰⢶⠀⠀⠐⠀⠈⠑⠂⠤⢀⡀⣤⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠻⢞⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢷⡻⡧⣬⣤⣧⠌⣁⠂⢄⠨⢘⣿⡔⠟⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⡋⣰⣤⣀⣶⣿⣿⠴⠀⠉⠀⠀⣏⣻⣧⣤⣙⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠃⠐⣰⢃⣯⣠⣼⠛⠁⠀⠀⠀⠀⠀⢰⡯⣷⡄⠍⡁⠸⠅
⠀⠀⠀⠀⣠⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣴⠛⠓⣘⠛⣿⣡⣬⡿⠋⠀⠀⠀⠀⠀⠀⣠⠀⠀⠛⡳⢷⡆⠆
⠀⠀⠀⠀⠘⡀⠀⠀⠐⣚⠃⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⠿⡡⢠⣨⡁⣾⣷⣶⠏⠀⠀⠀⠠⠄⡐⠲⡌⢿⣧⠀⠀⣯⡈⡇
⠀⠀⠀⠀⠀⠈⣦⠀⠥⠏⠀⢀⡪⠁⡔⠊⠛⣛⠛⣻⢻⡓⠄⢲⣓⣾⡿⠷⠋⠉⠁⠀⠀⢀⠐⠁⠀⠀⠠⠗⢈⣾⠀⠀⠶⠁⠁
⠀⠀⠀⠀⠀⠀⠈⢔⣻⡁⠀⠰⠁⣾⠂⠀⠀⠿⢷⢶⡾⣡⣼⣾⣏⠒⡀⠀⠀⠐⠀⠀⠀⠀⠈⠐⠀⢨⠔⠀⠺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠰⠀⣷⢀⡀⠀⠀⢀⠈⣿⣿⢟⠛⡉⣴⣧⣤⣀⡐⢀⠀⠀⠀⠀⠀⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠉⠁⠀⠀⣰⣿⠁⠴⠇⢙⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣰⡟⠀⡁⠀⣈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠖⠓⠛⠦⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⣿⠤⢚⠦⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠄⠐⢶⣖⣴⣌⣷⡀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣑⢂⠠⢈⣿⡀⠀⠀⠀⠀⠀⣀⣤⣾⣸⡽⣧⡀⠀⢨⡿⡿⢿⣿⡟⠀⠆⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡎⣴⢧⣺⢿⣦⣤⣤⡴⠿⠋⠁⣡⣿⣧⢸⡇⠀⠈⠀⠀⠀⢹⣿⠶⠀⢠⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠲⠶⣤⣀⠻⣿⣶⣌⡿⣶⣿⡿⣥⢀⣀⢰⣟⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⡈⣬⡟⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣦⣴⣴⣶⣧⣡⠀⣉⡛⢮⡝⠻⢿⣿⣿⣷⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⠀⢀⣠⠠⡴⣿⣤⡔⣷⣿⣥⣤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡼⣿⠟⠁⠀⠈⠙⠻⣿⣿⠆⠈⡓⢦⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⡟⢻⣿⣿⡾⣿⠻⣿⣤⠶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠻⡀⠀⠀⠀⠀⠀⠈⠛⢧⣲⣍⡂⢈⠑⠦⢤⣀⣀⣀⣀⣄⣠⣤⡴⢘⣛⣤⣤⣰⢶⣋⣿⠟⠙⠷⣦⣤⣴⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⢳⣄⡀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣧⣀⣤⣤⣄⣨⠿⣿⣡⣿⢔⡠⠿⣿⢟⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⢻⡶⢤⣀⡀⠀⠀⠀⠀⣀⠉⠛⠿⠿⣷⣿⣽⣶⣷⣾⡶⠀⠀⠈⠿⣟⡁⠀⠀⠀⠆⠀⠓⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⣾⣿⣿⣿⣿⣿⣷⣿⠿⠿⠿⢿⣶⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣍⣾⢽⣿⣿⣛⠿⡿⠿⢻⣯⣡⣾⣟⣼⣭⡶⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣿⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣹⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡹⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣩⣿⣿⡟⠛⣩⠉⣹⢫⡮⢹⣿⣿⣿⣿⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣿⣿⣿⣿⣷⡝⢎⣹⣶⣿⣿⣿⣿⣟⣻⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣮⣿⣿⣿⣟⣭⡉⣶⠏⠉⣻⡿⣿⣽⣿⡶⣚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⣿⣿⣿⣹⣫⣿⣸⡕⢎⣶⣿⣴⣿⣿⣿⣿⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡿⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣮⣿⣯⣾⢿⣿⣿⣿⣷⣤⣴⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣻⣿⣿⣿⣿⣿⣥⣷⣾⣿⣿⣿⣿⣿⣿⡟⠿⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡿⣿⣿⣿⣿⢹⣿⣿⣿⣏⣾⣿⣿⡿⠵⠁⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣴⣾⣗⣽⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠷⢿⢿⡿⠿⠛⠿⠿⠦⢿⢿⣿⡟⠭⠿⢿⠷⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢠⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⡴⠀⠀⠐⠖⠠⠀⠀⠀⡄⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⡤⠀⠀⢿⡇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢿⣿⠟⠀⠀⠀⠀⠀⠀⠄
⠤⠄⠀⠀⢠⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⡤⠤⠤⣰⠰⠂⠀⠀⠀⠀⠀⠀⠀⠠⣂⡆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠉⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠠⢘⠀⠀⢴⡆⠀⠀⠀⢠⡾⠃⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠐⠓⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⢠⠄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⣀⠠⢄
⡤⠄⠂⡀⠀⠈⠓⡀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠌⡑⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⢤⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠋⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠄⠁⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢏⠈⠐⠊⠁⢈⠂⠀⠀⠀⠈⠲⠐⠊⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⢀⡀⡀⠀⠀⠀⠅⡀⢀⠠⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢠⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠆⠐⠊⠀⢁⠐⠁⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠐⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠢⡀⠀⠀⢨⠀⠀⠀⠀⠠⠂⠰⠀⠀⡀⠀⠀⠨⠙⠀⠀⠘⠠⠰⠊⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠊⠁⠀⠀⡀⠀⠄⠀⠀⠰⡿
⠛⠳⠢⣄⠀⠀⢈⠆⠀⠀⡆⡀⠐⠈⠐⠈⠀⠀⢀⠃⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠁⡀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠈⠀⢙⡀⠀⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠐⡇⠀⠀⡮
⠀⠀⠀⣸⠁⠀⠁⠀⠀⠀⠀⠌⠀⠑⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠁⠆⠀⠧⡈
⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠠⠁⠀⠈⠁⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠰⠀⠀⠈
⠀⠀⠀⢀⠀⠀⡀⠀⠒⠠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⢿⣆⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠈⠀⢈⠀⠀⠀⠀⠐⠢⢀⠀⠘⢿⣷⣄⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠈⢻⣆⠀⠀⢀⠀⠀⠀⠦⠈⠁⠈⠁⢠⠀⢠⠚⠁⠀⠀⠁
⠀⠀⠘⠁⠀⠈⠦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠗⠀⠀⠈⠟⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢀⠀⢴
⠀⠀⢠⠀⠠⡄⠀⠠⠀⠀⠀⠀⠀⠈⠀⢀⡀⢿⣿⣦⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⢀⡀⠀⠄
⠀⢠⠚⠀⢀⡄⢀⡀⠐⠀⠀⠀⠀⠀⠀⣼⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⢀⠀⠄⠀⠀⠐
⠀⠈⠀⠀⠇⢤⣌⠁⠀⠈⠀⠀⠀⠴⢀⠏⠀⠀⠀⠀⠀⠀⠀⠂⢰⠀⠀⠀⠀⠀⢀⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡀⠈⠐⠡⠁
⠀⠀⠀⠀⢀⠀⠆⠀⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠴⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠐⠃⠀⠗⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠶

⠀⠀⠀⠀⢀⡀⠐⠀⠐⠀⣤⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣀⠀⠀⠘⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠛⣝⠷⢚⣳⡀
⠀⠀⠀⠖⠓⠀⠀⠀⠀⠀⠀⠃⠀⠀⠓⠡⢿⣧⡀⢀⠀⣰⠿⠿⠾⡟⠁⠀⠄⠀⠀⠀⠀⠋⠿⡿⠁
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢶⠲⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠙⠀⠀⠀⠀⢸⠍⢠⠀⠀⠄⠀⠀⠸⡟⠘⡛⣷⠖
⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⠈⠐⠀⠆⠁⠰⠀⠀⠈⠁⠘⢦⣄⣿⠀⠀⠀⠶⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⢠⠀⠀⠀⠀⠐⠿⠋
⠀⠀⡀⠀⠀⠀⠚⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⢀⢠⠆⠐⠀⠀⡀
⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣼⡛⡷⣦⣀⠴⠂
⠀⠈⢀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠀⠀⣿⣷⣷⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣼⣥⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣄⠀⠀⠀⡀⠀⡠⠇
⠀⠂⠀⠀⠈⠳⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠾⠷⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠐⠀⠀⠐
⠀⠀⠀⠀⡄⠁⠀⠀⠀⢀⠀⠀⠐⢀⣹⣿⣿⣏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡀⢀⠀⡆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢬⡙⡆⠀⠀⡉⠻⠏⠁⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⢀⠀⢲⢃⠘
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠁⠀⠀⠤⠀⠀⠀⠀⠉⣿⣿⣿⠿⢀⠈⠀⠈⠆⠀⠘⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⣤⠐⠀⠀⢠⡶⠂⠀⠀⠀⠀⠀⠀⢀⣐⠂⠀⠀⠀⠐⠀⢀⠁⣰⡽
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠⠔⢚⠟⠁⠀⠀⠀⢠⣀⣤⠂⠀⠀⠃⠀⠀⠀⠀⠆⠐⠂⠀⠈⠀⢠⠞⠃
⠀⠀⠀⠀⢀⡞⠀⠀⠀⣴⣋⠤⠚⠁⠀⠀⠀⠀⠀⠘⡿⠿⠄⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⡎⢀⣠⣴⡴⠿⠓⠒⠒⠲⢤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⢶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡶⠶⣤⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⣾⣿⣿⣿⣹⢿⣿⡆⠀⠀⠀⠀⢀⣿⣿⣆⠁⣀⢈⣷⡄⠀⠀⠐⢽⣷⣦⣴⣤⣴⣴⠃
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢿⣟⣿⣿⣿⣶⡋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡦⢿⡇⠀⠀⠀⠀⢹⣿⣿⠟⠉
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠘⠟⠓⠀⠁⠹⣷⡀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡃⠀⣸⡇⠀⠀⢰⣶⣿⣿⣿⣦⡀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢀⠙⢷⡀⠀⠚⠋⣽⠃⠀⢰⣿⡇⢸⣿⠁⠀⠀⠸⣿⣿⣿⣫⣿⠃
⠀⠀⢰⠁⠀⠀⠐⠀⠀⠀⠀⠀⠉⠙⢻⣧⡀⠳⠀⠀⠀⠁⠀⠀⠀⣿⣧⣾⡏⠀⠀⠀⢀⡱⢺⣿⣿⡏
⠀⠀⠸⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢹⡽⡿⠀⠀⠀⠀⠀⡇⣻⣿⣿⠁
⠀⠀⡇⠀⠀⠉⠳⣦⣤⡤⠤⠀⠀⠀⠚⣿⠿⣿⡄⣆⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢸⣷⣿⣿⡏
⠀⠀⡇⠀⢀⣀⢰⣄⣿⣧⣄⣠⣄⡀⠀⠈⢧⢹⣧⠘⢆⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠸⢻⣿⣿⠁
⠀⠀⠃⢠⡞⣠⣿⣿⣿⣿⢿⢭⣬⣛⠓⢄⠈⢆⢻⡄⠈⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⡆⣾⣿⠇⠀⢀⣠⣄⡀
⠀⠀⢸⣸⣅⢻⣿⣿⡋⠀⠀⠀⠙⠻⠇⢸⡆⠀⡄⢿⡀⢀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢠⣰⣿⡟⠀⠀⣾⡟⢿⠃
⠀⠀⠈⣟⠻⣤⣨⢿⠿⢂⢀⡀⠀⣀⣤⣾⡇⠀⠠⠸⣷⣾⠀⠀⠀⠀⠀⠘⣿⡀⢀⣾⣿⡿⠀⠀⠀⣼⠇⠘⠂
⠀⠀⠀⣽⡀⠀⠉⠛⠒⢶⣾⣿⠆⢹⡋⢹⡇⠀⠀⠂⣿⣿⡀⠀⠀⠀⠀⠀⠸⣷⣾⣿⡟⠀⠀⠀⣾⠏
⠀⠀⠀⢯⢳⠀⠀⠀⠀⠸⣿⡇⠀⢸⣇⣿⡇⠀⠀⢸⣸⣿⡇⠀⠀⠀⠀⠀⠀⡻⣿⡟⠀⠀⠀⢠⡇
⠀⠀⠀⠈⡌⡆⠀⠀⠀⢸⣿⡇⠀⢸⣿⡿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣶⣿⣷⣄⠀⠀⣇⣷⠀⠆⠁
⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⣿⠇⠀⣼⣿⠃⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⣰⣿⠃⠘⣿⣿⡀⢰⣿⣿⢰
⠀⠀⠀⠀⠀⠹⣇⠀⢀⡀⢸⠀⣸⣿⠏⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⢠⣿⠃⠀⠀⢹⣿⡇⢸⣿⡏⠚⢀
⠀⠀⠀⠀⠀⠀⢻⣷⣼⡇⣸⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⣾⠇⠀⠀⠀⢸⣿⣧⣿⠟⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠘⠛⠿⠇⠒⠁

⠉⠂⠀⠘⣇⢀⣌⢘⠠⠆⠀⠁⠸⠀⡀⠸⡄⠿⡾⠓⠀⢀⡴⡶⣶⡻⠃⠐⢿⠆⠀⠀⠀⠓⢀⠇⢀⠈⠉⢇⣤⠉⠙⢶⡞⠃⠠⠂⢨⡮⠨⡅⠀⠀⠉⠁⣠⣡⡀
⣌⠒⠂⠈⢡⠷⡄⢠⡄⠘⡁⠀⠄⢠⡀⠈⠀⠩⠐⠀⠁⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠊⠀⠀⠹⠤⢦⡼⠓⠄⢀⠢⠂⠛⠃⠀⠀⣠⠐⢾⠙⣏
⠤⣱⢄⡄⢄⠐⢱⣤⣁⣠⠢⠀⠐⠚⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠠⣀⡡⠀⠄⣴⠅⠠⣌⡰⠁
⠒⢯⡜⢔⠀⠁⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠆⢀⢘⠇⠀⠀⠂
⡤⢄⠙⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡄⡄⣠⠄⢠⢤⢀⡤⣄⢀⠤⡀⡠⢄⢀⠤⠀⢠⡀⢠⡀⡀⡄⣠⠄⣄⠀⣠⢠⠄⠀⠀⠀⠀⠀⠀⠈⠉⣀⠤⠔⢦⣀
⠧⠜⠀⠀⠤⢄⡀⠀⠀⠀⠀⠀⠀⠸⠱⠊⠄⠇⢧⡄⠸⠻⠸⢄⠼⠻⣤⠌⠜⠢⠰⣄⠦⠟⠳⠸⠑⠇⠇⢉⠗⠏⠖⠱⢨⡷⠀⠀⠀⠀⠀⠀⠀⠀⡿⢦⣶⢾⠏
⠀⠀⢐⡠⠒⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠠⢴⠏⠀⡀⠔⠟⠁⠀⣄⠀⡦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢖⡚⠺⡄⠘⠃⠁
⠀⠘⠛⠋⣥⠎⠀⠀⠀⢸⠇⠛⠛⣧⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢔⠿⢧⠄⠀⠐⠝⣁⡁⠷⠂⢆
⣠⠎⠀⠊⢀⠄⠠⠀⠙⠋⡠⠲⠒⠉⡦⡄⠈⣄⡠⠉⣀⠀⠀⢠⠖⡬⠀⠀⠀⠀⣇⡀⢤⠧⢀⣀⡄⠀⠰⢎⠙⠄⠜⢸⡶⠀⣄⡤⠂⡂⣯⡆⠀⠁⠴⡂⢿⠆⡕
⣷⣦⣤⣤⣷⣶⣾⣧⣤⣼⣧⣤⣴⣾⣶⣤⣤⣤⣷⣮⣤⣼⣦⣬⣿⣧⣤⣤⣤⣾⣷⣴⣦⣴⣿⣧⣤⣴⣷⣴⣧⣤⣴⣾⣧⣼⣧⣤⣴⣷⣿⣤⣤⣤⣬⣴⣶⣾⣦
⣿⣿⣇⣀⣈⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡴⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢾⠆
⠀⠀⠀⠀⠀⠀⣂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⣀⠀⠀⠙⠒⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⠙⠛⠳⢦⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣠⡾⠁⢀⡴⠞⠋⠉⠙⢻⡟⠺⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡤⢀⠀⠘⠀⠀⠀⢸⡿⠁⠀⠀⣼⠿⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⢿⣧⠤⠀⠁⠀⠀⢸⡇⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢡⣾⡇⠀⠀⠀⣿⣇⠀⢠⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⣾⡏⠀⠀⠀⢀⣿⣿⠷⠾⠏⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⡏⠀⠀⠀⢰⣿⠣⣈⠀⢤⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⢈⢷⣄⠀⠀⠀⠀⠀⠀⣠⣾⣿⠛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⣿⠇⠀⠀⠀⠀⢸⡏⢰⡷⣯⠘⠀⠀⠀⠀⠀⣀⢀⡀⣼⣤⣢⣤⣠⣦⡆⣴⣿⣦⣤⣴⣶⣶⣿⡿⣿⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣣⣶⠀⠀⠀⣠⣴⡾⣿⣁⠀⠀⠀⠀⠀⢠⣵⠟⠛⠛⠻⢿⡇⢩⣿⣿⣿⠛⠟⠋⠉⠉⠁⠄⠀⠈⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⠀⣰⣿⡼⠂⠁⠀⠀⠀⠀⢀⣴⠿⣿⠿⠟⠛⠛⠛⠛⠙⠛⠉⠀⠀⠀⠀⠀⠀⣀⢀⣀⣨⣶⣿⣧⣾⣷
⠀⠀⠀⠀⠀⢠⠀⠀⣠⣶⣿⣿⣿⣿⣯⡻⢄⡿⢁⡸⠧⢴⠄⠀⠀⢠⣿⣿⣟⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⠶⠄⢀⡀
⠀⠀⠀⠀⠀⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡾⢯⡘⠃⠀⠀⠀⠀⠀⢨⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⢿⠿⠻⣿⣿⣿⡿⣟⣉⣁⡐⣶⡶⣶⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣭⣿⣿⡙⠛⠁⢠⣗⠫⠽⠖⠀⠀⠀⢺⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠙⢛⣿⣿⣿⣯⠿⣯⠳⡄
⠀⠀⠤⠀⠀⣾⣿⣿⣿⣿⡟⣋⣭⠏⣡⣶⣾⠄⣩⠜⠂⠀⠀⠀⠀⠘⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃⠀⠙⠛⠻⠷⠋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⡙⢻⣿⡆⠙⠉⢱⠎⣿⣼⡦⠄⠀⢀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣠⠦⣄⡀⠀⠀⢹⡆⠀⠀⠀⠄
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣤⡙⠋⢷⣿⡷⣷⣿⢈⣀⢀⣼⣿⠋⠀⠄⢀⣦⠔⠂⠀⠀⠀⢀⣼⠁⢀⣤⣤⡾⠛⠀⠀⠘⣀⠀⢸⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⣹⣿⣿⣯⣴⠿⠛⠉⠉⠙⣿⠳⠷⠀⣴⡟⠁⠀⠀⠀⠀⢠⣾⣿⣴⣿⣿⣍⠁⠀⠀⠀⠀⠘⢿⣼⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡟⢻⣧⣼⣶⡟⠁⠀⢀⡀⠀⠀⠈⠓⠓⢾⡟⠁⠀⠀⠀⠀⠀⢳⣿⣿⣿⡟⠠⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⡄
⠀⠀⠀⠀⢹⣿⣿⣿⣍⣽⣿⡿⣿⠫⠘⣷⠀⠀⢸⠇⠀⠀⠀⠀⠀⢈⡇⠰⣀⣠⡀⠀⠀⡈⢿⣿⣿⣇⠀⢠⡅⠀⠀⠀⠐⠀⠀⣿
⠀⠀⠀⠀⠈⣿⡟⣿⣿⢿⢏⡝⠀⣴⠾⡏⠁⠀⢻⠀⠀⠀⠀⠀⠀⠾⠁⠀⠈⠁⠈⠉⠀⠀⠙⣗⣿⣿⡄⠘⠷⢎⢀⡀⠀⣸⣶⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣴⣾⡏⣾⠁⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⣿⢷⣦⡹⢯⣿⣻⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣿⡇⣿⠀⠀⠧⠀⡆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠀⠀⢷⠻⠀⠀⠙⠛⠟⠛⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣯⡏⢰⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣶⣼⡧⡴⠞⢠⣼⣾⣷⣄⠀⠀⣤⣤⣴⣶⣦⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣹⣿⣧⡟⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠉⠉⠁⠀⡀⠚⡟⠁⠉⠛⡇⠈⠉⠉⢿⡥⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠴⠀⠹⣿⣷⣿⣿⣷⣿⣾⡙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⢠⡶⠿⠀⢠⣿⠶⢡⣄⠀⢰⣄⠀⢶⣾⠟⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣟⣻⡿⠆⠀⠀⠀⠀⠀⢀⠤⠤⣾⠇⠃⠀⠀⠀⣤⣷⡀⠀⢀⣠⡀⢸⣧⡈⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⠀⠀⢀⠟⠀⠀⠀⠀⢠⡿⠏⠀⠈⠀⠙⠁⠸⣇⠉⠉⠉⣹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⡻⠁⠢⠀⠀⠀⠀⠀⠀⠈⠂⠀⢠⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣯⣿⡤⣿⣦⣀⡀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⣞⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⡄⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⣤⣤⡤⠲⡎⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣗⣻⣣⡀⠣⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢠⡾⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣏⣌⢀⡛⡉⠓⠀⢀⡄⠀⠀⠀⢰⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡐⢦⠀⡄⠀⢀⣠⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡄⠠⠀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡓⠃⡠⠀⠀⠈⠑⢄⢈⡀⠀⠀⠀⢄
⠀⠀⢀⡀⠀⠀⠀⢠⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⢄
⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠢⠄⠀⠀⠀⠀⠲⡀⠀⠆⠀⠀⠀⠀⢁⡀⠀⠀⣀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠂⠀⠀⢂⣀⠀⠀⠀⠐⠀⡀⠠⠶⠶⠶⢦⠄⠂⠀⠀⠀⠀⠠⠀⠀⠸⣁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠈⠀⠀⠿⠿⠇⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠉
⠀⠀⣠⠂⠀⠀⡀⠠⠀⠀⠀⠀⢠⡀⠰⡀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠠⡦⠀⠀⠀⠀⠈⠀⠀⢰⢀⠀⠀⠀⠀⠀⠑⠀⠠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠉⠀⠀⠠
⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⢀⣄⠀⠀⠁⠠⡀⠉⢂⠘⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠠⡀⠄⠀⢀⠄⠀⠈⠁⠘⡆⠀⢶⣤⣄⠀⠻⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⢀⡀⠀⢇⣀⡸⠀⠄⠀⡇⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⡤⠦⡄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠟⠁⠀⠐⠘⡆
⠀⠀⠀⠀⢠⠀⠱⠴⠃⠀⠀⠀⠀⠀⠀⠀⠐⠷⣤⡀⠀⠘⣇⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠃⠀⠈⠒⠆⠔⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠄
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠹⠤⣠⠖⠀⠀⠀⠀⠀⠀⠶⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠂⠀⠀⠀⣀⡀⠀⠀⡠⠀⠀⠀⢀
⠀⠀⠀⠀⠰⠄⠀⠀⡀⠀⠀⠈⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⣠⠊⡠⠒⠂⣉⡁⠀⠀⠀⠉⣤
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠑⠊⠀⠀⠀⠈⠉⠙⠒⠲⠤⡄
⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠰⡶⠉⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⣠⣤⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡔⠁⠇⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣰⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣂⠀⠀⠃⠀⠀⠙⠛⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠨⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⢁⣄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠈⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣠⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠁⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⡼⠀⠀⠀⠈⠀⠐⠂⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠐⡸⡀⠀⠀⠈⠉⠀⠀⠀⠃⠧⣛⡫⡰⠀⣠⠁⡃⠸⠃⠀⠀⠀⠠⠜⢢⡑⠊⠐⡸⡀⠀⠀⠈⠉⠀⠀⠀⠃⢣⣛⡫⡸⠀⣠⠁⠇⠺⠃⠀⠀⠀⠠⠜⢢⡑
⠀⡁⡉⢶⠀⠀⠀⠀⠀⢀⣧⡠⠤⠞⠁⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇⠈⡉⡈⢶⠀⠀⠀⠀⠀⢀⣧⡤⠤⠞⠁⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⢨⣶⣎⠀⡯⠄⠀⠀⣠⠟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠜⡡⡠⡍⢦⠀⠀⣉⡏⢨⣶⣎⠀⠧⠄⠀⠀⣠⠟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠞⡡⡀⡍⢦⠀⠀⣉
⢤⣉⡡⢜⠑⠂⠀⠀⡏⠀⠀⡔⢉⡠⠤⣉⠆⠀⠀⠘⣌⢜⢋⠆⢃⠀⠀⠁⡱⢤⣉⡡⢜⠁⠂⠀⠀⡏⠀⠀⡔⢉⠠⠤⣉⡆⠀⠀⠘⡄⢜⢛⠆⣸⠀⠀⠁
⠃⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠊⠁⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠚⠁
⠀⠀⠀⢤⢀⠀⠄⠀⠀⠀⠀⣀⡤⠦⠦⣌⡠⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⡀⠀⠀⠀⢤⢀⠀⠄⠀⠀⠀⠀⣀⡤⠦⠦⣌⡠⢀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡄
⢂⠀⠀⢢⡏⢙⡦⠄⠀⠀⠠⢜⣶⣕⣢⠀⢉⣧⠀⠀⠀⢜⢀⣹⡆⠀⠀⠳⡌⢢⠀⠀⢢⡏⢹⡦⠄⠀⠀⠠⠜⣶⣗⣢⠀⢉⣧⠄⠀⠀⢜⢀⣙⡆⠀⠀⠳
⠤⠀⠀⠀⠹⠙⠈⠃⢀⠀⠀⠀⠊⠝⡦⣶⡿⠻⣦⠀⠀⠈⠑⠉⠀⠀⠀⠀⠘⠦⠂⠀⠀⠹⠙⠈⠃⠀⠀⠀⠀⠊⠕⡶⣶⡿⠻⣦⠀⠀⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠠⡀⢨⠀⠀⠀⠀⠀⠁⠁⠈⠀⢹⢆⠀⠀⠀⠀⡄⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢸⠀⠀⠀⠀⠀⠁⠀⠈⠀⢛⢆⠀⠀⠀⢀⡤⠂⢢
⢀⠂⡄⠀⠀⠀⠤⣱⠼⢤⡎⡠⠔⠁⢠⠠⡀⠀⠀⠑⠓⠀⢀⠎⠀⢀⡟⠀⠀⢀⠒⡄⠀⠀⠀⠤⣱⠼⢤⡎⡠⠔⠁⢀⠠⡀⠀⠀⠑⠓⠀⢀⠎⠀⢀⡟
⡎⡸⠀⠀⠁⠀⣰⢱⣶⡆⢸⡖⠊⠀⠀⢧⠰⡀⠀⠀⠀⠀⡌⠀⢠⠎⠀⠀⠀⡌⡼⠀⠀⠁⠀⣰⢱⣷⡆⢸⠖⠊⠀⠀⢧⠐⡀⠀⠀⠀⠀⡜⠀⢰⠎
⠷⠃⠀⠀⢀⡡⠘⡦⣄⣨⡶⢍⡁⠀⠀⠸⣄⡇⠀⢀⣀⣼⡀⣤⠂⠀⠀⠀⠀⠧⠃⠀⠀⢀⡡⠘⡦⣄⣠⡶⢍⡁⠀⠀⠸⣀⡇⠀⢀⣠⣼⡀⣠⠂
⠀⡀⠀⠈⠁⠀⠜⠀⠂⢃⠘⠀⠈⠂⠀⠀⠉⠀⠶⠡⢊⢊⠍⡅⠀⠀⠰⡀⡐⠀⣀⠀⠈⠁⠀⠞⠀⠆⢡⠘⠀⠈⠂⠀⠀⠉⠀⡶⢡⢊⢊⠍⠁⠀⠀⠰⡀
⡛⡥⠆⠀⠀⡠⢤⡀⠀⠘⠀⢀⣀⣀⡀⠀⠀⢀⠔⣡⢃⠃⡰⠀⠈⢱⢊⣅⠭⡛⡥⠆⠀⠀⠠⢤⡀⠀⠘⠀⢀⣀⣀⡀⠀⠀⢀⠔⠡⢋⠁⡰⠁⠈⢱⢊⢍
⢃⢙⠉⠀⠀⠢⠾⠁⠀⠀⡔⡽⣷⡖⠸⠀⢀⠇⠔⣡⡎⠐⠃⠀⠁⢹⡘⢝⡣⢃⢻⠉⠀⠀⠢⠶⠃⠀⠀⡔⡑⣷⡶⢱⠀⠀⠆⠔⣡⡎⠐⠃⠀⠁⢹⡘⢟
⠠⠎⠦⣀⠀⠀⠀⠀⠀⠀⡄⠉⠒⣪⠎⠀⠚⢠⠃⠈⠀⠀⠀⠀⠈⠀⠜⠲⡄⠠⠮⠦⣀⠀⠀⠀⠀⠀⠀⡄⠉⠒⣊⠎⠀⠚⢠⠂⠈⠀⠀⠀⠀⠈⠁⠜⠢
⠠⣂⡢⠙⡄⠂⠀⠀⠀⣴⠟⠊⠉⠁⠀⠀⠐⠃⠀⠀⣠⠄⠀⠤⡀⠀⠀⠤⡆⠠⣂⡢⠙⡄⠀⠀⠀⠀⣴⠟⠋⠋⠁⠀⠀⠐⠃⠀⠀⣠⠄⠀⠤⡀⠀⠀⠤
⠘⠿⠑⢠⢏⡁⠀⠀⡾⠁⠀⢀⠤⠄⠀⠤⡀⠀⠀⢠⠁⢪⣦⡃⡙⠄⠀⠖⢆⠘⠿⠗⢠⠍⣁⠀⠀⡾⠁⠀⢀⠤⠀⠀⠤⡀⠀⠀⢠⠁⢪⣦⠃⡹⠀⠀⠖
⡝⠒⢊⠱⠀⠀⠀⠀⠃⠀⠀⠣⠘⠈⠉⠓⠃⠀⠀⠀⠣⢑⣐⣡⠜⠀⠀⠀⠊⡙⠒⢂⠱⠀⠀⠀⠀⠃⠀⠀⠣⠚⠊⠉⠓⠁⠀⠀⠀⠣⢑⣐⣁⠞
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡀⠠
⡀⠀⠀⡙⡼⣴⣁⠀⠀⠀⠀⢲⡋⢍⡉⠓⠮⡔⠀⠀⠀⢠⠒⢢⡀⠀⠀⢆⠢⡀⠀⠀⡙⡼⣤⡁⠀⠀⠀⠀⢲⣋⣍⡉⠓⠪⡔⠀⠀⠀⢠⠒⢦⡀⠀⠀⣇
⠘⡀⠀⠘⢧⢴⠣⡅⠀⠀⠈⠐⡻⢗⢛⣄⣴⣿⡁⠀⠀⠱⢄⠾⠃⠀⠀⠈⢣⠘⡀⠀⠘⢧⢼⠣⡅⠀⠀⠈⠐⡻⢗⢛⣤⣴⣿⡁⠀⠀⠱⢴⠶⠃⠀⠀⠈
⠉⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠁⡏⠟⠫⠈⠿⡀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠁⠏⠟⠫⠈⡿⡀⠀⠀⠀⠀⢀⡀⡀
⠀⡀⠀⠀⠀⠀⡈⢆⢸⠀⡠⠀⢀⠄⠀⠀⠀⠀⠐⢕⣄⠀⠀⡰⠋⠀⣴⠀⠀⠀⣀⠀⠀⠀⠀⡈⢆⢘⠀⡀⠀⢀⠄⠀⠀⠀⠀⠘⢕⣄⠀⠀⡰⠋⠀⣴
⠰⢠⠃⠀⠄⠀⢉⢞⣍⡙⢣⡊⡥⠀⠸⡌⢂⠀⠀⠀⠀⠀⠰⠁⠀⣠⠃⠀⠀⠰⢠⠃⠠⠄⡀⢙⢞⣍⡑⢣⣂⡥⠀⠘⡍⢂⠀⠀⠀⠀⠀⢰⠁⢀⡰⠃
⡇⡎⠀⠀⠀⠄⢺⡘⠛⠃⣘⠧⠄⠀⠀⢸⠀⡆⠀⠀⠀⢠⠃⠀⡘⠁⠀⠀⠀⡇⡎⠀⠀⠀⠄⢺⡘⠛⠃⣨⠧⠄⠀⠀⢹⠀⡆⠀⠀⠀⢠⠃⠀⡸⠁
⠉⠀⠀⠠⠔⠊⣠⠋⡓⠞⢫⠑⠢⡄⠀⠈⠧⠃⣀⠔⡪⡻⠮⠟⠀⠀⢀⠀⢀⠉⠀⠀⠠⠔⠊⣠⠋⡓⡞⢣⠑⠢⡀⠀⠈⠶⠃⣀⠔⡺⡻⠮⠟⠀⠀⢀
⣤⠆⡀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢋⠌⡘⡔⢁⠅⠠⢄⡠⠆⠢⣤⠆⡀⠀⠀⠀⠁⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⢉⠌⡴⠕⢁⠆⠠⢄⡨⠆
⡆⢯⠥⠀⠀⡊⣹⠆⠀⠀⢀⢴⣖⢒⢢⠀⠀⡔⢁⠞⡰⢀⡎⠀⠄⢼⢰⣷⡌⡆⣯⠥⠀⠀⡉⣙⠆⠀⠀⢀⢴⣖⣒⢆⠀⠀⡔⢁⠎⡰⢀⡎⠀⠄⢼⢰⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⡲⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⢷⠀⣉⣉⣛⡛⠻⢿⡟⠉⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠛⠟⠛⢿⣿⣿⣿⣿⣻⣿⡿⢻⡿⠟⣀⣴⣾⣻⣶⣄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⣶⣿⣿⣿⡿⣿⣿⠟⠀⠀⠀⣸⡉⠙⣙⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠙⠛⠋⠉⡆⢸⣧⢸⣿⡇⢀⣿⡧⣀⣺⡟⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣄⠀⠀⢧⣿⣿⠳⢿⣿⢸⠏⠀⣿⡿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⣿⣿⣿⡄⠘⢹⡟⢷⣀⣠⣝⠻⣤⣄⡸⣧⣏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣷⣄⠈⣷⣔⣟⢿⣿⡷⠀⣟⡃⢿⢻⣷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⢿⣿⣿⡿⠿⣿⣿⠉⠋⣿⡇⠀⠁⠉⠘⡆⢽⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠰⣿⡿⢹⠇⠀⠸⣿⡅⠀⢻⡇⠀⠈⠀⠀⠹⣞⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢰⡇⠀⢺⠀⠀⠀⣿⣧⣠⣿⠇⠀⠀⠣⠀⠰⠜⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠁⠀⣸⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣤⣬⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⡿⠀⠀⣤⣿⣿⣿⠁⠀⠀⣀⠀⣅⢿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠠⣜⠂⠀⣇⠀⠀⣹⣿⣿⠇⠀⠀⢸⣿⣶⣿⣇⣨⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣿⣾⡂⠻⣶⣾⣿⣿⠃⠀⠀⠛⠛⣿⠛⣿⣇⣠⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⢠⣄⠏⠉⠀⠀⡀⠀⠀⠀⠁⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠎⠀⠀⠈⠇⣠⡀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠛⠀⠈⠀⠉⠀⠀⢀⡀⠀⠀⠀⠀⠠⣤⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡀⢹⡶⢀⡤⠀⠂⠀⠀⣰⣿⣿⡆⠀⠠⠤⣴⠟⢹⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠟⡀⢸⣟⣥⠀⠀⠀⠀⠈⣿⠿⣗⠀⠀⠼⢏⣰⣾⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣗⣉⢿⣟⣶⣶⣴⡆⢘⣿⣿⡷⣠⣴⣶⣯⠿⠿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣦⣿⠿⣿⣿⣷⣪⣽⣿⡻⣿⣯⠽⢿⣯⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣈⣿⣏⣳⣿⣿⣿⣿⣾⣵⣾⣿⣿⣴⣿⣯⣥⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠛⠛⠛⠛⠿⠋

⠀⠀⠀⡀⢠⠀⡀⠀⠀⠀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠁⠀⠀⠀⠀⠀⢂⠀⠨⣀⠀⠀⠀⠁⠂⠀⠀⠀⠈⠀⠁⠀⠀⠰⠊⠁⠉⢡⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢠⠀⠌⠀⠂
⠀⡁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠠⠤⠀⠂⠀⠀⠐⠈⠀⠀⠘⣀⠀⠀⠔⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠂⢀⡀⠁
⠀⠀⠘⠀⡄⠐⠄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠒⠀⠐⢄⠀⠂⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⡀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠄⠁⠈⠂⠀⡃
⠀⠀⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠈⠈⠀⠀⠉⠀⢀⢀⠤⠞⠀⠀⠀⠠⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⣀⣀⣤⣤⣴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠘⠀⠄⠀⠀⠀⣠⣾⠟⠛⠫⠛⡛⡛⢉⣙⡻⢯⡀⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⠐⠢⠀⠈⠀⢠⣿⠘⢸⡄⡄⣌⣳⣜⣈⣯⣩⣼⠿⡿⡏⠉⣿
⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠁⠒⠂⢀⠁⠀⠸⡿⠶⢾⠛⠻⡏⣭⣭⡩⡇⠒⢓⡐⢠⡀⢰⣽⠄
⠀⡠⠂⠀⠢⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠻⣌⠌⠂⢦⣀⣝⣉⣷⣹⣌⢉⣣⣠⣥⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠐⠀⠄⠀⠀⠀⠈⠙⠛⠛⠋⠉⠉⠙⠉⠋⠛⠛⠛⠉⠁⠀⠀⢀
⠀⠀⠁⠀⠁⢀⠀⠀⠀⠀⠀⢍⢈⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⠱⠃⠚⠁⠀⠀⠀⠄
⠀⡊⠀⠀⠌⠀⠀⠀⢀⠀⠂⢀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠊⠀⡀⠀⠀⠐⠠⠛⠁⢀⡀⡀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠂⠀⠀⠀⠀⠈⠀⠀⠐⠰⠖⢲⢸⡣⢈⠄⠂
⠀⠈⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠘⠁⠁
⠀⢀⠂⢐⠈⠀⢃⠂⠀⠐⢈⠄⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡠⠐⠁⠀⢀⠃⠀⠀⠀⠊⠐⠆⠀⠀⠀⠀⠀⠀⠄
⠀⠠⠢⠀⠀⠀⠀⢄⠀⣀⢂⠐⠀⡜⡠⠐⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⠀⠄⠐
⠀⠈⠀⠡⠠⠐⠀⠊⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢤⣦⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇⠀⢇⠀⢸⡇⢠⡏⠀⣠⠞⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⣄⠀⢸⣄⠼⡦⠜⠧⠾⢇⣴⡃⠀⡴⠋⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⣄⡸⢉⡁⠤⠀⠒⠀⠓⠂⠠⠏⣼⢏⣡⢴⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠤⠋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣖⢛⡋⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠢⣌⣢⣄⣶⣄⣄⠀⠀⠀⠀⠀⠐⢶⣁⠀⠀⠀⠀⠀⠀⢀⡄⣀⣀⠀⠀⠀⠀⠀⢸⣩⡛⠁⠀⠀⠰⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠉⣷⣴⣿⣿⣿⠛⢉⡄⢭⡀⠀⠀⠀⠀⠘⡍⢀⠀⠀⣠⡠⢯⣜⣳⣧⣿⣝⣆⠠⠀⠀⣩⠉⠁⠀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠉⠛⢲⣴⣿⣿⣟⠿⠀⢰⡏⢠⡌⢻⡄⠀⠀⠀⠀⠈⠳⡅⠈⣻⣿⣿⣿⣿⠿⠿⡿⣿⣷⣖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠦⣄⣴⣿⣿⣿⣟⠁⣠⣾⠟⠉⠀⢰⠀⠀⠀⠀⠀⠀⢀⡠⠾⣭⠋⢧⡀⠘⡇⠀⢸⠏⠀⣿⠛⣀⡄
⠀⠀⠀⠀⣀⣽⣿⣿⣿⡉⠀⣾⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠙⣆⠘⣇⢈⣇⣠⣧⣀⣿⣄⣴⡇⣰⠏⣡⠞⣠
⠀⠀⠀⠀⠈⢻⣿⣿⣯⡁⠀⢻⠟⠁⠀⠀⠀⢀⡀⠀⠀⠀⠠⣽⣶⣾⠷⠿⠛⠛⣿⣿⠋⠉⠉⠙⠛⠿⠾⢧⠴⠃⢀
⠀⠀⠀⠀⠁⠛⣿⣿⣿⡄⠀⡘⢠⠀⠀⠀⠀⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠠⣶⣟⣵⠗
⠀⠀⠀⠀⠈⠉⠛⠿⣿⣧⡀⠁⡴⢧⣤⣴⡟⠻⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⠆⠀⠀⠀⠀⠀⢀⠤⡀⠀⠈⣱⣾⡔⣡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠂⠭⠿⠟⠉⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣬⠄⠀⠀⠀⠀⠀⠀⠌⠙⠻⠿⠆⠀⣾⣿⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣀⠀⠰⣿⣯⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣿⣿⠀⢰⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⠀⠀⣰⢻⣄⡀⠀⣀⣿⠏⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣘⡃⠀⠀⠀⠀⢠⡟⠉⠈⠐⠢⠍⣿⠦⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠀⠀⠀⠀⣋⡃⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠃⠂⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣧⠀⠀⢸⠠⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠈⠓⠀⠀⡰⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠁⠀⠘⠊

⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠟⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⣴⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡆⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠘⡆⠸⡄⠸⡇⢰⠃⢠⠞⠁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢦⡤⠷⠄⢱⠊⣕⠙⢒⠿⣤⣼⠁⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠈⡷⣊⠐⠀⠁⠀⠀⠀⠀⠉⠑⠋⠶⣋⣥⠴⠒⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢄⠠⣄⢠⡄⠀⠀⠀⠀⠀⠀⢤⠇⠀⠁⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⢹⠶⠶⠒⠂⠀⣤⣴⡊⠈
⠀⠀⠀⠀⠈⠑⣦⣬⣿⣿⡿⠛⠉⢐⡀⠀⠀⠀⠈⢛⡂⠀⠀⢀⢜⡌⣿⣿⣿⣇⣀⠄⠀⢘⠛⠁⠀⠀⠀⠙⣿⠇
⠀⠀⠈⠒⠲⣶⣿⣿⣿⠏⢀⠆⠀⣈⠕⠀⠀⠀⠀⠈⠪⣀⣌⣿⣼⣿⢿⠿⡿⣿⣿⣤⡀⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠐⠦⣴⣿⣿⣿⡍⠀⣼⠂⠁⠀⢠⠀⠀⠀⠀⠀⢀⡄⠻⣝⠹⡆⠈⡆⠀⣿⠁⢸⠟⢠⡄
⠀⠀⠀⠤⣼⣿⣿⣿⠀⢴⣷⡏⠀⠀⠀⠀⠁⠀⠀⠒⢄⠙⣆⣸⣄⣷⣤⣷⣤⣿⣤⣿⣰⢋⡰⢂⡄
⠀⠀⠀⠠⣼⣿⣿⡷⠀⠀⠛⠁⠀⠀⠀⢸⠀⠀⠀⠈⠚⠻⠛⠋⠉⠁⢹⣿⠀⠀⠀⠉⠙⠛⠓⣋⡤⢄⡀
⠀⠀⠀⠀⠬⠽⣿⣿⡂⠠⡉⣢⠀⢀⣤⣟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠀⠀⠀⠀⠀⠀⠈⠛⠻⣯⡕⡤⢀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠤⢘⣾⣿⠟⠁⠈⢻⡷⠀⠀⠀⠀⠀⠀⠀⢠⡙⠁⠀⠀⠀⠀⢀⠿⢷⣤⡀⢛⣿⣎⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠁⠀⠀⣻⡆⠈⣽⣶⣂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⡔⠀⠀⠀⣿⣇⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⣞⣷⡤⠀⣠⣿⠉⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⢠⡅⠀⠀⠀⠀⢼⠁⠀⠀⠑⠐⠻⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⠀⠀⠸⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⡼⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢄⠀⠳⠄⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠈⠀⠰⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣷⣿⣿⣿⣼⣯⡟⣻⠶⣄⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣽⣶⣄⣀⣀⣠⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢾⣻⣿⣿⣿⣾⣾⡏⠁⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀⠀⠀⠀⠀⠀⢄⠠⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣀⣀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠸⣿⡿⣿⣿⣷⣦⣤⣴⣶⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠿⣇⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⡿⣿⣿⡾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠙⠿⣶⣼⣿⣧⡀⠈⠟⠿⣿⣿⣿⣿⠿⢃⡾⠚⠟⠉⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠷⠶⠤⠶⠏⠁⠀⠈⠀⠠⠀⠈⠀⠀⠀⠀⠉⠀⣀⣿⣿⣿⣿⢏⣴⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣠⠀⣤⣄⣀⣤⣤⣤⣤⣴⣾⣿⣿⣿⣏⢸⣯⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣾⣿⣿⣿⣿⣏⢸⣿⡿⣯⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣦⡙⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠰⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢠⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢨⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣺⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⣿⣿⡿⠯⠁⠀⠉⠻⣿⣿⣷⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⣿⣿⣿⣿⡧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣱⣿⣿⠏⠀⠀⠊⠀⠀⠀⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣘⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣼⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⡙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣵⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⣿⣿⣷⣦⣤⣴⣶⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣀⣠⣴⣾⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣠⣴⣾⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠛⢋⣉⣉⣤⣤⣤⣤⣉⣉⡙⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⣁⣤⣶⠿⠟⠛⠋⠉⠉⠉⠉⠙⠛⠻⠿⣶⣤⣈⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠉⠻⢷⣦⡈⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣴⡟⢁⣴⡿⠋⠀⣿⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠹⣧⠀⠀⠀⠀⠙⢿⣦⡈⢻⣧⡀
⠀⠀⠀⠀⢀⣾⠏⢠⣾⠏⠀⠀⢀⣿⣿⣷⠶⠓⣠⣤⣤⣄⠀⠀⠹⣦⣼⠋⠀⢀⣤⣤⣤⡀⠙⣿⡄⠹⣷⡀
⠀⠀⠀⠀⣾⡏⢠⣿⠃⠀⠀⢀⡿⣹⡏⠀⠀⠀⢻⣆⣀⣹⢷⣴⠾⠋⠙⠷⣦⡾⣃⣀⣼⠇⠀⠘⣿⡄⢹⣷
⠀⠀⠀⣸⡿⢀⣿⠇⠀⠀⠀⣼⢷⡟⠀⠀⠀⠀⠀⠉⠋⠙⣷⠀⢈⣀⠲⠂⠀⣾⠋⠉⠁⠀⠀⠀⠸⣿⡀⢿⣇
⠀⠀⠀⣿⡇⢸⡿⠀⠀⠀⢰⣏⣾⠃⠀⠀⠀⠀⠀⣀⣤⣤⡿⠀⣏⣍⣳⠀⠀⣿⣤⣤⡀⠀⠀⠀⠀⢿⡇⢸⣿
⠀⠀⢀⣿⠀⣿⡇⠀⠀⠀⣾⢛⡟⠀⠀⠀⠀⠀⣼⠏⠀⣱⡟⠻⢾⡍⣥⡶⠛⢷⣍⠈⢹⡆⠀⠀⠀⢸⣿⠀⣿⡀
⠀⠀⠈⣿⠀⣿⡇⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⣸⠇⢻⣄⠀⠀⠙⠛⠛⠁⠀⠀⠀⢸⡿⠀⣿
⠀⠀⠀⣿⡇⢸⣷⠀⠀⠀⢻⡌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣿
⠀⠀⠀⢹⣷⠈⣿⡆⠀⠀⢸⡟⢿⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣾⡏
⠀⠀⠀⠀⢿⣇⠘⣿⡄⠀⠀⣿⣼⡇⠀⠀⠀⠐⣶⣼⡷⢦⣤⣤⠶⠾⠛⠛⠛⢷⣴⡟⠀⠀⠀⢠⣿⠃⣸⡿
⠀⠀⠀⠀⠈⢿⣆⠘⢿⣆⠀⢸⡇⣿⠀⠀⢘⣻⡏⢀⠰⠖⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠃⠀⣰⡿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣧⡈⠻⣷⣌⠛⠛⠀⠀⠉⣹⣷⣬⣀⡐⠀⠀⣀⣀⣠⣴⠾⠋⠀⠀⣠⣾⠟⢁⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡈⠻⢷⣦⣀⠀⠀⠉⠰⠟⢹⠟⠛⠛⠉⠉⠁⠀⠀⣀⣴⡾⠟⢁⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⡉⠛⠿⣶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣴⣶⠿⠛⢉⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⣌⣉⣉⠛⠛⠛⠛⣉⣉⣡⣤⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠿⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢔⠀⡀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣶⣠⢉⣠⣴⣤⣔⣷⣦⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠙⣿⣿⠟⠉⠻⣿⣽⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠀⠀⠀⠀⠴⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣴⣾⣿⣆⣤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⡶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⠀⠀⠀⠠⠄⠈⢻⣿⣿⣿⣷⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠖⠀⠀⠀⠐⠀⢀⣀⡀⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⢂⠀⠂⠡⠀⣰⣿⣿⣿⣶⣌⢻⣿⣿
⠀⠀⠀⠀⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⣾⡿⠋⠋⠀⠀⠉⠉⠉⠻⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠂⠀⡾⣿⣧⠄⠂⠀⡀⠄⡉⠤⠀⠈⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠶⣾⣽⡶⣆⣴⠿⠠⠀⠐⠀⠀⠀⠀⠀⠂⡄⠀⠀⠙⠻⣿⣿
⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣆⣀⡒⣚⡭⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻
⠀⠉⠺⢷⣀⠐⠂⠦⡤⢀⢀⠀⠀⣀⠀⢀⣠⣤⣾⡾⠛⠛⢛⠿⠉⠙⠁
⠀⠀⠀⠀⠉⠙⠛⠚⠅⠦⠖⠖⠀⣠⠤⠀⡀⠊⠊⠀⡠⠖⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⣋⡴⢚⣭⠀⡠⠀⠠⠄⠀⠔⠁
⠀⠀⠀⠀⠀⠀⣀⣴⣞⣯⠟⠋⠉⢠⣎⣑⡸⠁⠀⡖⠊⣧
⡀⠀⠀⣠⣴⣾⠿⠛⠉⠀⠀⠀⢀⣾⣹⣿⠂⠀⠀⠙⣯⠜⣇
⣿⣶⣬⣍⠉⠀⠀⠀⠀⠀⠀⠀⢸⣷⡼⠃⠀⠀⠀⠀⠘⠛⠿⠿⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿

⠀⠀⠀⠀⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⢰⡶⠀⠀⠀⠀⠀⡠⢶⡆⠀⠀⠀⣠⡶⣆
⠀⠀⠀⠀⢿⣯⣷⣄⠀⠀⠀⢁⡈⠒⠀⠴⠄⠰⣇⡈⠀⠀⠘⢧⠀⠰⣶⣶⡜⠉⡼⠃⠀⠀⡰⢀⢀⡻⠆
⠀⠀⠀⠀⢸⡛⡿⣿⣆⠀⢀⠈⠓⠂⠀⠀⠀⠀⠛⠁⠀⠀⠀⠰⣿⣴⡿⡍⠃⣩⠞⠀⡀⡘⢓⣄⣏⢻⣦
⠀⠀⠀⠀⠈⠳⠅⢈⣻⠏⣻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⣌⣛⣷⣾⣿⡿⢀⣿
⠀⠀⠀⠀⠀⠀⢠⣮⣿⠿⠛⢿⣷⣶⣦⣀⡀⠀⠀⠀⠈⢹⠋⠉⠀⠀⡒⠀⠙⣦⣴⣿⣿⣿⠻⣮⠁⠋⠁
⠀⠀⠀⠀⠀⠀⠟⠛⠁⠐⠦⢄⣩⡿⣿⣼⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠜⢿⣦⣉⣉⣭⠆
⠀⠀⠀⠀⠀⢀⠀⠀⠦⡘⠀⠀⠀⠀⢰⡤⠤⡀⠀⠐⠒⠚⠒⠒⠀⠀⣠⢤⣤⠀⣉⡉⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠲⡶⠂⠀⢀⣴⣿⡇⠈⢿⣄⠚⠶⢄⠀⠀⠠⢀⡠⠶⠒⣸⡏⠀⣿⣿⡄⠀⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣼⣧⡀⠀⢿⣦⠁⠈⢃⢀⣶⣌⠀⢀⣼⡷⠀⠀⣿⢻⣿⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⡿⠿⠃⠀⢐⡦⠀⠀⠘⠙⡿⠀⠀⠀⢰⠆⠀⢘⡇⢻⣿⣇⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠿⣦⣄⠀⢌⢉⠀⠀⠀⢂⠀⠀⢻⣴⡣⠀⠈⠀⠀⣠⣜⠋⠀⠀⣽⡇⣾⣿⡏⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠙⠟⠃⠈⠸⣄⣀⠀⠊⠀⠀⡼⠇⠀⠀⠀⠀⠀⠐⠿⡀⠀⢀⣾⡥⠛⠉⠀⠀⠀⠙⠛⠂
⠀⠀⠀⠀⠀⢴⣷⣤⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⢠⣾⣦⣀⠀⠀⠈⠂⢀⣄⣀⠀⠀⠀⠀⠀⢀⣤⣶⡄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣆⠀⠀⣀⣤⣶⣾⣷⠀⠀⠀⠀⣈⣋⣙⠛⢲⣄⠀⢹⡞⣷⢿⣦⡀⢀⣴⣟⣾⣿⡆
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠋⠀⠘⠶⡌⣻⣿⣿⣿⣿⣿⣷⣿⡁
⠀⠀⠀⠀⠀⢹⣿⣿⣿⢿⡟⣿⣿⣿⣿⠉⠀⣀⠀⠀⠈⠂⠂⠀⠐⠀⠀⠀⠀⠸⣷⣽⣿⡃⠛⠘⣿⣷⣿⠅
⠀⠀⠀⠀⠀⠀⠙⠋⠁⠈⠀⠻⣿⡿⠋⠀⠀⡿⢴⣦⣀⠀⠀⠀⢀⣀⢠⣤⠀⠈⠙⠛⠛⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣄⠀⠀⢠⣄⠀⢸⡟⣻⣷⡶⣶⣬⣿⣿⡭⠃⠀⣾⣤⣄⠈⠃⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡸⣷⠛⠋⠉⠀⢰⣿⣿⣆⠀⠉⠙⠃⠹⠃⠹⠿⠿⠃⣠⣴⣿⢟⣻⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠠⠰⠋⠀⢀⣀⠀⠀⠀⠸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣸⡿⢛⣵⣸⡏⠀⢈⠀⠀⠐⠩⣿⡆
⠀⠀⠀⠀⠀⢀⣀⢶⣿⡽⠃⠀⠀⠠⣮⡙⣿⣿⣿⣶⠀⠀⠀⣴⣿⢛⡱⣾⢻⡯⡿⠁⠰⠁⠀⠀⠐⣀⣹⡇
⠀⠀⠀⠀⠀⠸⣿⠟⠋⠀⠀⠀⢀⣴⣿⣿⣿⣼⢿⣿⠀⠀⢸⡟⣡⣾⣷⡟⠿⠟⢃⡀⠀⠀⠀⠀⠈⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣶⣿⣿⡎⠝⢆⠈⠈⠁⠀⢀⠀⣉⣉⣁⠀⠐⠒⠹⣿⣅⡒⣦⢤⣄⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢲⣶⣿⣿⣿⣿⢿⣿⣿⣆⠀⢀⡤⢐⣲⣤⡞⠛⠛⠉⣠⣶⣿⡉⢻⠟⠟⣛⣢⣩⣵⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⣽⠿⠛⢠⣾⡿⠟⠁⠈⠁⠀⠀⠀⠳⣶⣶⣶⡶⠏⠙⠛⠛⠛⠙⠛⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⢰⠀⠰⢶⠄⠀⠀⠀⠀⡁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠆⠠⣾⠇⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠀⠀⠀⠆⣰⡿⠀⠀⠀⡀⠀⠀⠀⠀⠀⣟⠀⠀⠀⣴⡀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉⠁⢀⣴⣿⣷⣄⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠐⠲⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡹⠆⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣰⣏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⣸⣷⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾⡇⠀⠀⠠⣶⣧⠀⠸⡇⠀⠈⢱⡄⢨⠁⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⣼⣿⣦⣎⣿⣄⠀⠀⠀⣀⡚⠂⠀⠀⠀⠈⠁⠀⠁⢰⣿⠁⠀⠀⠀⢀⠀⣈⠀⠊⠉⣿⡷⠆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠐⢿⣿⡟⣿⣿⣯⣿⠀⠀⠈⢻⣅⠀⠀⠀⠁⠰⠦⣤⡿⢘⣿⡷⣄⠀⠀⢠⣓⣻⣟⣷⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠐⠿⡿⢿⣿⡿⣰⡿⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠇⠈⠀⠀⢿⠁⢈⣿⣿⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠠⢤⠃⠀⠀⠑⠀⠀⡀⡅⡟⠀⠀⢠⡇⠀⣺⡇⠘⠁⠀⣘⢁⡀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢶⡟⠀⠀⠀⢧⡄⠀⠀⢸⡇⢰⠁⢘⠇⠀⠀⠀⠈⣿⡄⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠈⠀⠔⡖⠀⠚⡁⠠⠀⢸⡇⣿⠦⣼⠆⢀⠀⠎⠀⣽⠇⠀⠀⣷⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡄⠠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣧⠀⠀⠘⠳⠿⠀⠻⠀⠀⠀⢀⡾⠁⠀⠀⠀⠋⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣦⣄⡀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇⠀⢸⠀⠀⠀⠀⢠⠏⠀⠀⡼⠁⠀⠀⠀⠀⢶⣿⠊⠆
⠀⠀⠀⠀⠀⢀⠀⢠⣌⣦⣼⣡⣄⣡⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⠀⠀⠀⠠⠍⠀⠀⢀⡟⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠞⠛⢛⠷⠂⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠁⠀⢠⠀⢸⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣄⠀⠻⡀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠁⢠⠟⠀⠀⠀⠀⠀⠘⠃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢹⣷⡄⠀⠀⡆⠀⠀⣿⠀⠀⠆⠀⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠏⣠⣠⡀⠀⣻⡅⠀⠀⠸⣴⡤⠁⠀⢠⡄⢀⠇⢀⡼⠁⠀⠀⠀⠀⠀⠀⠠⣆⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣴⢿⣝⣿⣾⣿⡇⠀⠀⢀⡍⠀⠀⠀⣸⡟⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠳⠀⢠⡞⢥⠈⣻⣿⢫⡛⠄⠈⠹⣿⣶⡤⢠⠀⠀⠀⣰⣿⣁⡼⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⣤⠈⠉⠀⣀⠀⢠⠤⣶⡄⠀⢈⣿⠻⡦⡹⢷⣶⣿⣿⡏⠀⣾⣿⣤⣶⣿⣍⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠰⡄⠀⠀⢈⡟⠻⠈⠳⡉⠀⢰⡏⣴⣆⣶⣿⣿⣅⣤⣼⣟⢳⣄⡙⢃⣠⣌⣛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠠⠿⢤⡔⠲⠤⠒⢲
⠀⠀⠀⢠⣶⣠⡄⢈⣥⣤⣤⡴⠛⢾⠛⣷⠀⠉⠉⡩⢤⣤⡀⠓⣫⡛⢟⣁⣙⠛⢿⣧⠀⠀⠀⠀⠀⢀⡀⠀⠰⡄⢀⣉⣻⠒⠚⠟⠀⠀⣀⠠⠂
⠀⠀⠀⠸⣷⠞⠾⠋⠀⠈⠹⣧⠀⠚⢣⢾⡿⡶⠾⣤⠼⡿⠙⠿⠿⣥⠿⠼⠿⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠒⠲⠀⠐⠀⠁⠀⠀⠂⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⣼⣷⠦⠀⠐⠃⡜⠓⠖⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠄⢨⡀⠟⠆⠈⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠠⡀⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢬⡅
⠀⠀⠀⠀⠀⠀⡃⠂⠁⠐⠟⠀⠀⠀⠐⢦⠀⠈⠂⠀⢀⠀⠀⠀⡄⠀⢄⡰⣄⠉⠲⡘⠀⣠⠁
⠀⠀⠀⠀⠀⠀⢓⣠⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣄⠀⢳⣄⠀⠰⡀⢀⢳⣮⢠⠈⠄⢀⠔
⠀⠀⠀⠀⠀⠈⠀⠀⠈⡀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠘⡁⣄⠩⠃⠀⡘⢶⢦⡽⠇⠀⠅⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠃⠀⠀⠀⠐⢦⡄⠀⠀⠀⠀⠈⠀⡍⠣⠀⠠⡀⠓⢂⣐⠒⣀⣠⠶⠂
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠂⠀⠄⠀⠀⠀⠱⠋⠤⠀⠀⠀⠢⡂⡂⠀⠰⢀⣈⠈⡉
⠀⠀⠀⠀⠀⠀⢁⠐⠠⠄⢀⠀⠀⠠⠄⠀⢀⠀⠀⠁⠄⠀⠀⠀⠈⢠⡁⠁⠉⠀⠉⠈⠋⠙⠓⠂
⠀⠀⠀⠀⠀⠠⠀⠁⡀⢀⠀⠀⠁⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢀⢀⠌⠀⠀⠀⡈⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⢠⠈⠁⠈⡄⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⢀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠃⠈
⠀⠀⠀⠀⣀⠁⠈⠀⠄⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀
⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⠀⠀⠀⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⢅⠁⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠡⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⡀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢱⡀⠀⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡋⣀⠀⠉⠛⢒⣲⣷⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠦⠤⠤⠉⣻⣅⢹⣿⣿⣿⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠢⡐⠂⠈⠻⠿⠟⣫⣽⣶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠂⣿⣿⣆⠀⠀⢀⡤⠀⡰⢊⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡅⠡⢷⣾⡟⠋⠀⠀⠳⠌⢹⣂⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠓⠒⠒⠚⠿⠇⢀⣴⣧⡀⠀⢹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⣤⣤⢤⣄⡀⠀⠀⠙⡿⠋⣀⣴⣾⣿⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⢉⣿⣿⡿⠁⣐⣶⠋⣴⣾⣿⣿⣿⣇⣀⣠⣶⠏⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡻⠖⠛⠁⠁⠘⠿⠿⣦⡙⢿⣿⣿⣿⣿⠿⠿⠋⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠀⢀⣀⠀⣀⣀⡀⢻⣿⣶⣝⡻⢿⣤⣴⢦⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣾⣿⣿⣿⣁⢿⣟⢻⣿⣿⣿⣿⣿⠃⠨⢯⣤⡅⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠋⠽⠛⠁⠈⠙⡿⢿⣿⣻⣽⣶⣿⡿⠁⡸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⣠⡼⢀⣘⣿⡿⠿⠿⣿⢧⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢻⡽⠿⠇⢸⣿⡋⠀⠀⠀⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⢀⣴⡾⠃⣼⣿⠃⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⠀⠙⡿⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⣺⡁⠀⠀⢠⠏⠀⠀⠀⣠⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⡞⠀⠀⠀⣼⣿⡿⠛⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⣠⣅⠀⠀⠀⠹⣿⣿⣿⣿⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣶⣤⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠛⠛⠛⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⠟⠿⠿⢿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⡿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣍⡻⣿⡿⢸⣿⣿⣿⣿⣿⣷⣦⣄⣀⡀⠀⠀⠈⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣭⣴⣿⡿⢛⣛⣿⠿⠿⣿⣿⣿⣿⣶⣄⣀⠀⠀⢨⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣶⣿⣿⡏⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⣫⣥⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⢸⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣷⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠠⠀⠈⢿⣿⣿⣷⠀⠀⠙⣿⣿⣿⣿⡙⣿⣿⣿⠿⠿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠛⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⣻⣿⣿⣦⣀⣠⣿⣿⣿⡿⢃⢿⣯⣴⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠈⠀⠸⣿⣿⣿⣿⣿⣿⣧⣬⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠈⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣯⠀⠀⢀⣿⣿⠟⠉⡉⠙⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠿⢿⣿⡀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣾⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠂⠀⣿⣿⣷⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡀⢀⣹⣿⣿⣷⣶⡀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣷⣤⣤⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣄⣀⣼⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⡀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⢦⣶⠛⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠘⠀⠀⠀⠙⠘⡀⠀⠀⠀⠁⠉⣦⠂⠆⣔⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠂⠂⠀⠀⠀⠀⢎⣱⣿⠷⣶⠉⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡟⢻⣄⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠄⠈⢋⣿⠢⠁⢄⠈⢸⣏⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢠⣧⠘⠺⠅⠀⠀⠔⡀⠀⠄⠀⠀⠀⠀⠀⠀⠘⡛⠃⠀⠐⠚⠻⠏⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠈⠁⠀⠀⠈⠃⠀⠀⠀⡟⠀⠀⠠⠀⡗
⠀⠀⠀⣀⠀⠐⠈⠀⠀⢀⣔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢄⡂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⡀
⠀⠀⢸⣿⠀⠀⠀⠀⣤⡼⠟⠒⠒⠀⠰⠂⠀⠀⠀⠀⢰⡀⠚⠀⠀⠀⠀⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⡇
⠀⠀⢸⣿⡄⠀⠀⢤⡏⠁⡀⠀⠀⠀⠀⠘⠗⠁⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⠀⠀⡆⠀⠀⢂⣀⠃⢉⡽⡀
⠀⠀⢸⣿⡇⠀⠀⠈⠛⠚⣁⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣆⠀⠀⣤⣦⣞⡜⠁
⠀⠀⢸⣿⣷⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠿⠟⠻⢿⡿⠇⠀⠀⣽⠝⣸⣃⠄⠀⡀
⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⠄⠒⠢⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠁⢬⠀⣴⠞⠶⣿⡿⣿⣶⠋⠀⣿⣿⠆⣸⠃
⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⠀⠀⢰⡟⡰⠁⣄⡠⠄⠀⠀⠀⠀⠀⠀⢁⡂⠀⠈⠙⠻⠆⠠⣈⠋⣈⢫⣤⣤⣿⣿⣿⣇⣆
⠀⠀⠈⢿⣿⣿⣿⣄⣆⠀⢀⣤⣶⠀⠀⣿⣷⡇⠀⡀⣰⡷⠀⠀⠀⠀⣤⣦⣄⢀⠀⢀⡄⠀⠀⠀⢻⡿⠿⠿⣿⡿⠿⠿⢛⡠⠝⠁
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣧⣡⠍⠀⢸⣿⡇⣠⡘⢛⣻⣿⡀⡀⡄⠍⠁⠉⠻⠀⣣⡜⡂⠀⠀⠀⣾⡇⠀⠠⣶⡄⠒⠒⠁⠐⠀⣀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣛⢙⣇⣼⢻⡟⣀⣸⣿⠛⠉⠀⠀⠙⣯⠀⠀⠀⡀⠸⢿⣿⣷⣶⠄⡀⣿⣧⣾⣦⡛⠂⠀⠀⢠⡶⠋⢁
⠀⠀⠀⠀⠀⠉⣶⣿⣿⡏⠋⠻⢿⣷⣾⠃⠀⠈⣷⠀⠀⠀⠀⢰⣧⣤⣀⣤⡇⠀⣤⣷⠚⡛⠊⢳⣿⡿⠟⢻⣿⡆⠀⠀⡎⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢀⢸⣿⣿⠈⣷⣤⠏⠀⠀⠀⠀⠈⢻⠿⢫⠥⠁⠀⠛⠉⠀⢛⣛⣤⣶⣷⠆⠀⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠈⠘⠛⢿⡈⠿⠿⠦⣀⣤⣤⣤⡤⠜⠀⠀⠀⠀⠤⠴⠶⠿⠿⠛⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣤⣄⠙⣿⣿⣿⡟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⣿⣿⣿⢨⢽⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣿⡇⣿⣿⣗⢻⣾⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣿⣿⣌⢿⣿⣿⣿⣿⣭⡿⣿⣿⣿⣿⢿⣛⣉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⢝⣿⣿⣿⣷⡙⢿⣿⣿⣿⣿⢻⡏⣿⣿⣶⣿⣿⢿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣿⣿⣿⡿⢿⣮⣿⡿⣿⣿⡌⣿⣿⣿⣿⣿⣿⢚⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⣿⡟⣽⣿⣧⣿⣿⣿⣿⣿⣿⣿⣛⠟⣼⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⢷⣝⠻⣿⣿⣿⣿⣿⣿⣽⣷⣝⢻⣿⡻⠝⣏⣾⣻⣿⣿⣯⡿⣫⣉⣛⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣝⡻⢮⣟⣿⣿⣿⣿⢿⠿⢻⠿⢿⣿⣷⠘⢻⣿⣿⠿⢋⣡⣾⣿⣟⣽⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣯⣛⣛⣛⣛⣛⣿⣛⡦⢻⡟⣠⣼⣷⣿⣧⣿⣿⣿⢟⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠝⢁⣟⡙⢁⣬⣭⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⣶⣆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡄⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⢹⣦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠸⣤⣭⣦⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠋⠉⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⣾⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣡⣄⣤⣤⣤⡀⠉⠁⠀⢳⠀⠀⢀⣶⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡧⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠏⠁⠈⠛⢻⣂⡀⠀⠲⣷⠀⢼⠁⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠀⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡆⠀⠀⢀⣀⡈⠻⣗⠀⠀⠜⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⢶⠶⠞⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⣀⢀⡴⠞⠋⠀⠀⠀⠀⠀⠀⠿⣿⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠟⠉⠀⠑⣄⠀⢀⢀⣠⣠⣄⢤⡈⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠺⡟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⣽⣿⠟⢻⠉⠛⠯⠟⠳⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⣴⡾⠤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⢀⠀⠀⣤⣤⣤⢀⣠⣒⣢⣼⠶⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⢳⡀⠀⠀⣴⣆⠀⠀⠀⠀⠀⢠⡴⠊⠻⠋⠛⢰⠋⠘⢀⠁⠈⠉⠉⠉⠛⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠐⠀⠀⠀⠀⣠⠆⠀⠐⢲⡟⣧⠀⠀⠈⢶⣌⡀⡀⣀⡤⠏⠁⠀⠀⠀⣀⠘⠀⠀⡈⠀⠀⢀⣠⣰⣾⡿⠋
⠀⠀⠀⠄⠀⠀⣤⣰⣶⣶⣶⣶⣿⠙⠛⢿⣷⡶⠿⠦⠞⢇⣀⡀⠴⠿⠻⠋⠀⠀⠀⠀⠉⠙⠙⠛⠀⠀⣀⣄⣠⣤⣼⣷⣷⣾⡿⠾⠿⠿⠿⠟⠉
⠀⠀⠀⣀⣿⣶⠿⠛⣫⣾⣿⣿⣀⡀⠀⠀⣿⠶⠦⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⠆⠴⠶⣦⣌⡁⠀⠀⠙⠉⠹⡏
⠀⠀⠸⣾⣿⣁⠀⢠⣿⠟⡉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠯⠽⢜⣦⣅⣀⠁⢈⣷⣄⠀⠀⠁
⠀⠀⠀⠙⠉⠛⠛⠛⠁⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣶⣿⣿⣿⣿⣤⣔⣛⣙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣐⡆⠀⠀⠈⠙⠇⠀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠀⠽⠿⠿⠿⠿⠛⠁
⠀⢀⠀⣠⣤⠀⠀⢀⠀⡄⠈⠈⠁⠀⠀⠀⢀⡄⠀⠀⢠⣔⠻⣿
⠀⠀⠁⠋⠉⠁⠉⠛⢋⠀⠀⠀⠀⠠⠀⠉⠘⠁⠀⡹⠊⣿⡒⠻⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠄⠀⣠⠼⢃⡀⠀⢲⣤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠒⠚⠋⠀⠀⠁⠀⢭⣿⣴⡆⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣄⣦⡖⣷⡀⣠⠇⢀
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠆⣀⣤⢴⣦⠾⢷⠷⠾⢿⢛⡗⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⣀⠀⠘⣺⠯⣿⣽⠟⣈⡿⠿⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠐⠋⠟⠢⠜⠆⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠛⠛⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣾⣷⣄⣽⣤⠤⢀
⠀⠀⠀⠀⠀⢠⠂⠐⢢⡸⣿⣿⣿⣷⡿⢿⣿⣷⡄⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣷⠮⠿⠿⠿⠋⢿⣷⣤⣄⠑
⠀⠀⠀⠀⠀⠈⠙⠢⡀⢹⠈⠛⠛⠋⢸⠘⣿⣿⡇⠀⠏⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠉⠉⢡⡿⢿⠁⠀⠀⡘⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠈⠚⡀⠀⠀⠀⠸⣦⢻⡏⢀⠞⠀⢀⡔⣃⣀⡀⡄⠀⠀⠀⠀⠀⠀⣀⢠⣀⣼⣶⠿⠏⠐⢻⡀⣰⣾⣅⡀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢀⣁⣤⣉⡄⠙⠆⠀⠀⡇⣿⣄⡤⢁⠤⠞⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡄⠀⢤⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄⢄⢀⣏⣿⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠠⢿⠉⢡⠀⠈⢿⣿⣿⠿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢷⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢾⡳⠾⣿⠃⠀⠳⠆⢼⣿⣇⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⣀⠄⠀⠀⣏⠙⠻⠏⠉⠉⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢀⣀⣹⡇⠀⣀⣀⠏⠀⠀⡆⠀⠀⠛⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃⠈⠛⠋⠀⠀⠈⠁⢠⣀⣠⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣀⣤⣤⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠗⢿⣿⣿⠂⢼⣭⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠁⣤⣼⠾⠷⠤⢠⡿⣃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢋⠳⣴⡏⢛⠁⣰⣶⣽⠗⢰⣶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠸⠋⢉⡓⢰⣇⣸⡏⢉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠠⢀⣈⡀⠀⣱⣼⣀⠀⠉⠡⠾⠨⠥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡇⠀⣾⡇⢸⠙⣿⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣀⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⠀⣿⡇⢸⢀⡏⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠛⠛⠁⠀⠀⠀⢶⣦⡄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⢻⡇⠸⢸⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢇⣀⣀⠀⠀⠀⠈⠓⠺⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀⣠⡀⣀⠀⣀⡀⣠⢀⠀⡀⢀⣀⢠⠠⣀⢀⠀⣀⠀⢄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠁⠈⠀⠀⠁⠈⠈⠀⠀⠀⠁⠀⠈⠁⠁⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡆⢀⠀⡀⠀⡄⠘⣆⣘⣄⣸⣤⣧⣴⣥⣎⠎⡠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⢰⠸⡀⢠⠀⣇⢈⣆⣡⣤⣧⡼⠶⠷⣚⠛⠻⠉⡉⢩⠄⠀⠀⠀⠈⢝⢿⣴⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢱⠀⢇⣀⣱⣬⣦⡴⠶⢷⡛⠛⠋⠉⠉⠑⡀⢀⠦⢠⠄⠀⠀⠶⠀⠁⡄⠀⠀⠀⠀⣘⠃⠁⢹⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⣼⠞⠛⠩⠉⣅⠀⠀⣈⣙⡤⣄⡀⢀⢀⠁⠀⠁⠀⠀⠄⠀⠙⠀⢠⠄⠀⠤⠀⠀⢠⡀⠀⠠⠆⠀⣷⣒⠂⠈⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢄⣹⡞⡑⠙⠆⠀⠀⠀⠀⠀⠐⠢⣌⠁⠮⠋⡣⠑⠀⠘⡃⠀⠀⠁⠀⠀⠀⠀⠋⢀⡀⠀⣀⡀⠀⠖⠐⣼⡓⠤⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠤⡤⣿⠀⠸⠃⠛⠀⠀⠀⠀⠀⢸⣀⣼⣧⠶⠠⠃⡜⠀⢀⠀⠠⠠⠀⠀⠀⠀⠖⠀⠆⠀⠲⠈⠠⣐⣥⢾⡓⠬⠁⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠐⠒⣿⡀⡠⡄⠉⢀⡄⠶⠀⠀⠀⠓⠀⠉⠤⠍⠙⠁⠤⠈⢐⡀⠫⠄⣀⣀⣀⣤⡤⣶⠶⡖⢛⠻⡙⠄⠣⠉⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⡨⢷⣜⠀⢃⠀⠀⠶⡀⠀⠀⠘⠀⣀⣒⣠⣤⣤⡴⠶⡖⠺⡟⢻⠉⠋⠃⠘⠀⠉⠀⠁⠈⣠⣴⡶⠶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢊⡝⢻⠶⡶⡷⢶⠶⡞⢻⠛⢏⠹⡉⢣⠸⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⠀⠃⠘⠀⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢶⡶⠾⠋⠀⠀⠀⠀⠀⠂⠀⠂⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀⣤⡀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⢀⣐⠂⠐⠂⢺⡌⠀⠈⢧⡀⣶⣶⡉⡞⠀⢀⡦⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠛⣷⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠘⠁⢠⣮⣷⣷⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⣿⠾⢷⣦⣤⣀⡀⠀⠀⠐⠖⠂⢀⡄⠐⣮⣽⣿⣿⢿⡉⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡘⠂⠞⠛⠾⠛⠀⠀⣀⣀⡀⠀⠉⠋⠛⢾⣭⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠈⠀⢠⣤⠈⢷⡕⣦⡀⠀⢀⠀⣠⠚⣿⠃⣶⣄⠀⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆⠘⠷⠄⠈⣆⣴⡎⠈⢼⠇⢀⣿⣿⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡿⠟⠛⠁⠀⠹⣄⡤⠈⠟⠠⡐⡼⠀⢸⡏⣿⡷⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠆⠳⣆⡀⠰⠀⢀⡟⠁⠀⠀⠈⠻⡀⠀⣾⡿⠟⠁⠀⠰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣀⠀⠀⠁⠀⣀⡀⠀⠀⠸⣷⣄⡀⠈⠀⣤⣄⡀⠀⠀⣀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣧⣤⣶⣿⣿⠃⠀⠀⠀⠒⠢⠉⠇⠀⢯⠻⣿⣦⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⢿⣿⣿⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡏⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠟⠋⠀⠘⣽⣷⣄⣀⢀⣤⢴⠆⢀⠉⠉⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠶⠛⠀⣼⣿⡄⠛⠻⠹⠟⢿⡿⠇⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⢀⣄⠀⠀⢻⣿⣿⣄⠀⠀⠀⢀⣀⡸⢟⣽⣽⠁⠈⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠿⠉⠀⠀⣸⣮⣿⣿⣷⠀⢠⡿⢫⣼⣿⣿⡇⠀⠁⠀⠠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣤⣄⣤⣾⣿⠻⡍⠙⠋⠀⢈⣙⣉⠉⠀⢳⣿⣢⣤⣄⣀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠿⠿⣿⣿⣿⣧⠀⣴⣬⠶⠞⠉⠉⡴⠾⠏⣟⣛⣭⣮⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠘⠋⠁⠀⠀⠀⠀⠈⠛⠛⠋⠀⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣶⣷⠈⠋⣉⠈⠛⠿⠿⣿⣿⣿⡿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⡜⠀⠟⢀⣉⡛⣧⣶⢶⡶⠇⣨⣽⣦⣈⣄⣹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠆⣀⣄⣤⣴⣿⣿⡿⣿⡿⠁⠀⢀⡎⠙⣻⣾⣿⠛⣿⠉⠝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠾⠿⣿⣟⠈⠉⠀⡆⣽⣆⣿⡇⢸⠿⣤⣿⡍⢿⠀⠻⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠘⣿⣿⣿⡆⠀⢾⣿⡏⠙⢷⣌⠀⢹⣿⡇⢸⠀⠀⣈⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⣿⣿⣿⣀⠘⣇⣿⢾⣷⡌⢻⠟⣿⣷⡌⡀⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠉⣿⣿⡿⠻⣿⡏⠋⣿⠀⠈⠀⠸⡬⡃⢇⣄⣿⡘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢐⡿⢋⡇⠀⢸⣧⠀⣿⠀⠀⡆⠀⠹⣅⠘⢿⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣿⣿⡿⠀⠘⠁⠈⡇⠀⢸⣿⣾⡿⠀⠀⠈⠀⢉⣙⣏⠛⢿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠿⣉⠀⠄⠀⠀⢰⠃⠀⣸⣿⣿⠃⠀⢀⠀⡔⣿⣿⣿⣤⢲⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣯⠐⡆⣬⡀⢸⡀⢀⣽⣿⠏⠀⣀⣿⣷⣿⠄⠏⣉⡹⣄⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⡷⠛⠛⠈⠛⠿⠿⠇⠀⠀⠈⠼⠉⢿⣶⣾⣯⣽⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⢄⠃⠀⠀⠀⠄⠀⠉⠇⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⢂⠀⡀⠀⠀⠀⠂⠈⠀⠉⠀⢀⡀⠀⠀⠀⠠⣄⢡⡨⢹⡛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠋⣄⢀⣧⠀⢇⡴⠀⠀⠀⠠⣿⣿⡄⠐⢒⡾⣋⡟⢡⣗⢻⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣷⡈⠺⣯⡄⢸⣷⣆⣀⣀⠀⣾⣿⡇⣀⣨⡵⣿⣿⡷⢷⣦⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢠⣭⡟⠹⡿⣿⣧⣹⡿⣿⣿⣖⣻⣿⢾⣯⣽⣿⣳⣿⣿⣿⣭⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣼⣅⣽⡖⢿⣦⣨⣿⣱⣎⣻⣿⣾⣵⣿⣿⣶⣿⣯⣽⣶⣿⣈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠀⠀⠀⠀⠀⢠⡄⢤⣤⢤⣤⡄⣤⣤⢠⣄⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣇⣾⣿⣿⣿⣇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢸⣿⠀⠀⠀⠀⠀⢠⣤⣤⣤⣬⠀⠀⠤
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣽⣿⣿⣿⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠄⣿⡇⢸⣿⠀⠀⠀⠀⠀⢸⡏⠉⠉⣿⠀⠀⠀⠀⠀⡄⠀⠠
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣗⣿⡿⠀⠀⠀⠀⠀⠀⢰⡆⠈⣿⠀⢻⠇⠘⡟⠀⠀⠀⠀⠀⢸⣿⣶⣶⣿⣠⣾⣿⣷⣄⣿⣿⡧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠉⠀⠈⠀⠀⠉⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠉⠉⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡡
⠀⠀⠀⠀⠀⠀⢠⡿⣟⠽⠋⢹⣿⣿⣿⠟⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣶⡂
⠀⠀⠀⠀⠀⢀⡿⢊⠁⠀⢀⣿⣿⣿⠃⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⢀⡤⠒⠒⣀⠒⠻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠐⠻⠃
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣿⣿⡿⠀⠀⠀⠀⠀⠀⣼⣴⣿⣿⣿⣿⣷⣼⡆⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣥⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠃⠀⠀⠀⠀⠀⢰⡿⣛⠿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁⠀⠁
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠀⠀⠸⣇⡁⢰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⡷
⠀⠀⠀⠀⠻⠥⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠈⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣤⢠⣀⡄⣀⣠⢀⡀⡄⡀⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠰⠇⠎⠼⠰⠡⠆⠮⠠⠠⠠⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣴⣶⣿⣶⣦
⠀⠀⠀⣰⣋⡘⣘⣁⢋⡋⣘⢁⢀⢀⠈⠀⠐⠁⠀⠀⢀⡔⠄⠠⠁⠀⠀⠀⠉⠀⠠⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⢰⠆⡴⠰⠤⠆⡶⠴⠠⠀⠄⠤⠠⣀⠁⠀⠀⠀⠈⠁⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡉⠻⣿⣟⢷⣶⣶⣤⣤⡀⠀⠀⣀⠄⠀⠀⠀⠀⠀⢀⣷⡄⣧⠤⢴⠴⣾⡐⣓⣤⡄
⠀⠀⠀⠀⢠⣤⣼⣿⡿⣿⣿⠿⠉⣀⠛⣿⠛⣮⣿⣩⠄⠛⢿⣇⢸⡅⢀⣠⣴⣦⣨⣦⣼⣷⡄⠷⢤⣄⡁⠈⠁⠓⠘⠋⠛⠻⠖⠦⢄⡀
⠀⠀⠀⠀⢰⣿⣿⡦⣴⣉⢛⣳⣿⣴⣶⣿⡿⠻⢿⣷⠤⡀⢸⣇⠀⠙⠈⢻⣧⣿⣿⣿⣿⣿⣿⠄⠉⠁⠉⠂⢀⣀⣤⣄⣀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⣿⠩⣿⣿⣶⣯⣬⣿⣿⢿⣛⣋⢀⣠⣄⠉⢸⣿⡼⢟⣛⣄⠀⠀⣿⣻⣿⡤⠿⣅⠉⠘⠂⠀⠀⠀⢻⣷⣶⣧⣿⢀⢤⡴⣿⣶⣦
⠀⠀⠀⣾⠑⣶⣿⡿⠿⡟⡯⠛⣻⣿⡿⠋⡌⠛⠋⠀⢸⣿⢾⠿⣛⣻⠛⠙⡿⠟⠂⠀⣀⡀⠀⠀⠀⠀⢳⡀⠀⠿⢛⡛⠇⣾⠰⣿⡿⢻⣿⣷
⠀⠀⠀⣿⣾⣿⣿⠇⢀⣯⣤⠆⣿⡁⠇⢠⣟⣄⡄⢠⣋⣼⡟⣾⡏⠀⠀⢀⣷⡀⠀⠸⣿⡇⢀⠀⠀⠀⠈⣇⠀⠶⠟⢻⣧⣽⡿⣿⣿⡏⢿⣿
⠀⠀⠀⣿⣿⣿⡿⠀⣹⣿⠋⠀⠉⠏⡀⠀⠈⠻⣧⣾⠟⢻⣦⣙⢛⣀⣤⣾⣿⣿⣿⡇⠈⣧⡘⢠⠀⠙⠀⠋⠀⢀⠀⠀⢻⡟⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⡇⣴⣿⡿⢀⣶⡄⣾⠃⢀⢐⡇⢀⡄⠀⢺⠋⡇⢸⣯⡛⢹⣿⡏⠛⢇⠀⢹⣇⠸⡆⠀⠀⠀⠀⢉⠀⠁⢸⣿⣠⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣷⣿⣿⡇⡞⠉⢳⣷⣦⢿⣿⡷⠸⠇⢡⣼⡄⠀⠘⡇⠄⠄⢿⣿⠀⠘⣆⣸⣯⢀⣿⡔⠀⢘⣶⣤⡔⢶⣿⣿⠙⠈⣿⣿⣿⣧
⠀⠀⠀⣽⣿⣿⡝⢿⣟⢷⡃⠀⠠⣿⡟⠠⠛⢻⣷⡆⢾⡅⡀⠀⠀⣿⡀⣠⣾⡿⣾⣿⡛⠻⣅⢸⣿⠁⣶⣿⡟⢹⡗⡈⢿⣿⣇⣿⣿⣻⣿⣿
⠀⠀⠀⣿⣿⣿⣯⣼⠁⢾⣧⣶⠀⠀⢹⣷⣤⡿⣟⣿⣿⡇⡆⠐⠀⢿⣤⣿⣿⡷⠎⠋⠁⡀⢙⠛⢛⢲⣶⡄⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣦⣾⣿⣻⣃⣴⣾⣿⡏⠹⣾⡍⠛⢿⣷⣴⡇⣿⣿⣿⣿⣿⣷⣧⢾⣷⡨⠐⣿⣿⠙⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣟⢿⣿⣿⣿⣿⣿⡿⠉⢻⣿⣤⣤⣿⠛⠀⠉⠙⠛⠿⠟⠛⠉⠉⠉⠉⠉⠀⡷⠛⠚⠛⢻⣳⠻⠿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣟⣏⣀⣤⣾⣿⡿⠛⠉⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⣄⣷⣞⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⠀⠀⠀⠀⠀⠰⣿⣿⣀⠀⠀⠀⢀⣀⣶⣿⣿⣿⣿⣯⣉⣉⣻⣿⣿⣿⣿⣿⣿⣯⣝⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠄⢠⠀⠀⠀⠠⣿⣿⠇⠀⠀⠰⣞⣵⡞⢉⣿⣿⣿⣿⡉⠉⠻⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⢀⣘⣿⣿⠇⠀⠀⢰⣽⣿⣵⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣶⣿⣶⠿⡿⠿⡿⠇⠀⠀⠈⠉⠉⣉⠉⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⡀⠀⠀⠀⢀⠀⠀⠀⠄⠀⡀⠄

⠲⡎⠱⢠⡔⢤⡔
⣧⠍⠈⠀⠔⢲⣆⡀
⠛⠳⢄⡂⣰⠟
⠀⠀⠞⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⣀⢀⣀⣤⠀⡀⠀⢀⡀⠀⣀⢀⣄⣄⢀⡀⠀⢀⢀⣄⣠⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡿⣄⣿⢸⣧⠤⢸⡇⠀⢸⡇⠀⣿⢸⡧⠄⢸⡿⣆⣸⠀⣠⡟⠁⢠⡛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠃⠈⠟⠘⠇⠀⠘⠷⠦⠈⠳⠾⠃⠸⠷⠦⠸⠃⠈⠛⠐⠿⠶⠀⠟⠁⠙⠇
⠀⠀⠀⡀
⣰⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣢⡀
⠎⢳⣿⢲⡆⣠⠎⠀⠀⠀⠀⠀⠀⠙⠋
⠠⠈⢀⡁⢠⣿⡄⢀⡤
⠠⠄⠈⣀⠉⠉⣘⣡⠄

⡆⠠⡄⠀⠀⠴⣯⡅
⠀⠠⡄⠀⠑⢆⢀⡋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢲⠀⣄⠀⣴⣠⡿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⢅⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⣤⣄⣹⡰⢿⡉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⢁⡄⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠙⠃⠸⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀⣠⣜⠓⠂⠀⠀⠀⠀⠐⠶⢤⣴⢧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠿⢹⡈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰⠗
⠀⠀⠀⠀⠀⠀⠀⠠⢄⣷⣶⠾⣶⣶⠜⠀⠀⠀⠀⠀⠀⠠⣄⠈⢛⠀⢸⠃⢰⡆⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠛⠋⠁⠈⣦⡄
⠐⠾⠇⠀⠀⠀⠀⠀⠙⣹⣧⣤⣿⡛⠳⠆⠀⠀⠀⠀⣤⠀⣘⡶⠾⠛⠛⣟⠿⠃⢴⡎⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠘⠉⡿⠟⢋⠀⠀⠀⠀⠠⣄⣩⠾⠇⠘⢗⠀⠀⠉⠀⠀⠰⠞⠛⣶⡞⠉⠀⣀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠂⠀⠈⠀⠀⣀⡀⠀⢨⡟⠘⠳⡄⠈⢀⠀⡿⠁⢀⠴⠃⠶⠟⢷⠖⠛⠁
⠀⠀⢀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⢙⣋⠀⢈⡁⠈⠳⠈⠀⠰⠂⢀⣄⠰⠶⠿⢶⡗⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⣨⣄⣠⣃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡋⠀⡈⣁⢀⡀⠀⠷⠀⡀⠀⠀⡀⠀⠠⣴⡇⠄⠀⠀⠀⠀⠀⠀⠀⣀⢰⡭⢱⡄⠀⠀⠀⠈⠷⠘
⢻⣌⠁⠈⠀⠿⠄⡶⢀⢀⡤⠚⠁⠀⠀⠀⢠⣤⠄⢿⡴⠟⠉⠀⠀⣠⠀⠀⢁⠐⠶⡉⠳⣶⡾⠷⠄⣠⠄⠀⠀⠀⠀⠀⠈⠳⠴⠛
⡀⠙⠀⢀⡔⢀⠘⣧⠈⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣀⣰⡆⠀⠉⠀⣤⠀⣧⠄⠀⣦⡸⠷⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠲⠀⠈⠀⠀⠀⠀⢤⠿⡞⠁⠀⠀⠀⠀⠀⠀⠚⠋⠀⣘⡿⢅⣠⣜⣇⠀⢀⣼⣀⠴⢾⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠛
⢀⠤⠀⠀⠀⠉⠛⠰⠖⢀⡷⠄⠀⢤⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠿⠉⢻⠛⡛⠙⠇⠀⠀⠻⠖⠀⠀⠀⠀⣿⢀⠤⢤⡄⢻⡁⠀⠀⠀⠀⠀⠘⢋⡰⠖
⠋⠠⠄⠸⡏⠐⣤⠔⠶⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠉⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⠁⠀⠀⠷⠀⠀⠐⢷⣆⡀⠀⠀⠀⠈⡷⢠
⡄⠦⠀⣤⠤⠦⡁⢠⠤⡀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠃⢀⠁⠀⠀⠀⠀⠀⣠⠀⠠⢈⣡⡞⠃⠀⠈⠁
⠱⠶⢆⡻⠀⠄⠉⠲⠀⠈⠣⡤⠀⠀⢠⣄⠀⣱⣄⡄⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠁⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠳
⠇⠀⠘⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣁⢈⡁⠶⠈⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⡀⠀⠈⠴
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠓⠻⡧⠌⣠⠀⠴⠒⢴⣧⣀⠀⠀⠠⠼⠃⠀⠀⠀⠀⠘⣭⠀⠀⠴⠀⠀⠀⠰⠆⠀⠀⠈⠀⠄⡴⠀⠀⠀⠋⠓
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣶⡀⠰⠂⡰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡵⣄⠀⠀⠀⠀⠀⠀⠀⠀⣦⠔⠁⡴⢤⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠉⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠿⢧⠄⠐⡦⠀⣀⠀⢠⠿⠾
⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠠⡬⠁⠐⠈⠁⢡⡄⠀⠀⠀⠀⠀⠀⡀⠀⢠
⣰⢦⡗⢀⣸⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣧⠸
⠉⠉⠁⠁⠩⠶⠒⢄⣰⠃⠀⠀⠀⠀⠀⣎⣷⠀⠀⠀⠀⠐⢢⡄⠀⣀⢨⡄⠈⠀⠉⠀⠙⣿⠶⣆⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠁⠘⠆
⠀⠀⢌⠁⠀⠀⠈⠙⣃⣢⠀⠀⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⠋⢠⡀⠁⠀⠐⡂⠀⠀⠁⠀⠉⣱⣖⠁⠀⠀⠀⠀⠀⠀⣀⡱⠀⠀⠈⠉⣻⡉⢁⡄
⠀⠀⠀⢀⠈⠋⠀⠀⡡⠙⡶⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠛⢀⠀⠀⣠⠁⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡯⣤
⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣦⣤⣶⣤⣴⣶⣶⣶⣶⣾⣶⣤⣶⣶⣶⣶⣶⣶⣶⣴⣾⣷⣶⣷⣶⣿⣶⣶⣶⣶⣶⣦⣤⣴⣶⣶⣶⣦⣶⣿⣦⣶
⣿⣿⣿⠻⠍⠛⠟⠉⠙⠛⢻⡟⠏⣹⡿⢻⠝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠋⠀⠀⠀⠀⣤⣖⣲⣴⣲⣰⣒⣦⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠶⠊⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⡟⡏⠙⢹⣩⢹⣿⢹⡇⠀⠀⠀⠀⠂⣾⣿⣿⡆⠀⠀⠀⠀⠈⢧⡚⠟⠂⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠠
⠀⠀⠀⢀⣎⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⡇⠻⠟⠸⠿⠘⠿⢸⡇⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⢷⣄⠀⠀⠀⠀⠀⠀⠃⠀⢨⡇⠀⢀
⠀⠀⠀⠸⠏⠻⠇⠿⠸⠇⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠀⠀⠀⠨⠇



⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⡄
⠀⠀⠀⠀⢀⣸⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠉⢸⣿⣿⡎⢒⠂⠀⠀⠀⠀⠀⠂⠘⣿⣿⣿⠂⠐⠀⠀⠀⠀⢠⣾⢻⢯⣽⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠳⡀
⠀⠀⠀⠀⢰⡾⠿⠿⡁⠂⠃⠀⠀⠀⠀⠀⠀⠄⠻⠿⠟⠀⠀⠀⠀⠀⠀⠈⢿⣯⣾⣹⣾⠷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⣷⢳
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉



⠀⠀⠀⢀⡴⠚⠉⢉⡓⢤⡀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠛⠢⣄⠀⠀⠀⠀⠀⣠⢖⠛⠉⠛⠢⣄⠀⠀⠀⠀⠀⠀⣆⠶⢆⡤⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⡀
⠀⠀⠀⡏⣀⣄⠀⠿⠟⠀⢳⠀⠀⠀⠀⣸⢁⣾⣿⣯⣠⠄⠘⡆⠀⠀⠀⢸⠁⠸⡅⢄⢠⠴⠚⡆⠀⠀⠀⠀⠀⠈⣓⡺⣅⠀⠀⠀⠀⠀⠀⠀⢿⠟⠛⢻⡿
⠀⠀⠈⡇⠻⠟⢀⣤⣶⡄⢸⠀⠀⠀⠀⢻⠀⣻⠿⣿⣿⡄⢀⡇⠀⠀⠀⢹⠀⠀⢸⠴⠋⠀⠀⡇⠀⠀⠀⠀⠘⣆⠣⠇⠘⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡾⠁
⠀⠀⠀⠙⢦⣀⠈⠻⣛⡵⠃⠀⠀⠀⠀⠈⠣⣄⠀⠟⢉⣠⠞⠀⠀⠀⠀⠈⠳⣄⡘⠦⢀⣠⠞⠁⠀⠀⠀⠀⠀⡼⠍⠙⣆⠀⠀⠀⠀⠀⠀⠈⢘⣻⣯⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⡤⠠⡄
⠀⢄⣀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠄⠛⢀⡿⢷⡄⠀⠀⢸⣿⡏⠁⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⣿⣧⣄⠀⢸⡇⠀⠙⣦⣰⡏⠀⣿⣶⠏⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⢿⡄⠈⠻⣾⠇⠀⠀⠘⠛⠀⠀⣿⠁⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⣀⡀⠀⠀⠀⠀⣰⣎⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠋⠀⠀⠀⠀⡄⠛⣏⠉⠉⠿⡿⠃⠀⠀⠀⣀⣀⣀⠾⢷⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠈⣿
⠠⣶⡄⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⢹⠀⠀⢸⡇⠀⠀⢠⡞⠀⠙⠛⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⢸⡆
⠶⠈⠙⠶⣶⣏⣄⣀⠀⠀⠀⠀⠀⢀⠘⣿⠑⠶⠾⣧⠀⠘⡇⠀⠀⠘⡇⠀⠀⠀⠀⠘⣧⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣧⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠻⡀⠀⠀⠀⠀⠀⢱⣤⣤⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠹⣦⣄⡀⠀⠀⠀⠀⣀⣰⠾⠻⠶⣦⣠⣧⠀⠀⠀⠀⠀⠈⠁⠉⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠈⠉⠛⠻⠿⠟⠛⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣜⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣾⠁
⣴⠛⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡦⠀⠀⠀⠀⠀⢀⡀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣾⠏
⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣿⠛⠿⠛⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⢀⠀⠀⢸⡿⣦⣤⠎⠁⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠀⠀⣀⡀⠀⠀⢀⡀⠀⠀⢠⣾⡛⢶⡟⠛⠢⣾⣇⠠⠋⠀⠀⠀⢀⣠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿
⠀⠀⠀⢼⡋⣧⠀⠀⠀⠀⠀⠀⡰⠶⠒⠒⠶⢯⡀⠘⣠⡴⠟⣷⡄⠀⡾⠻⣆⠀⠟⠈⠻⣦⡄⠀⣀⠘⠉⠀⠀⢀⣴⠞⠉⠉⠉⠳⣦⣄⣀⣀⣀⣤⣴⠞⠁⠀⣿
⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⢸⣷⢤⣀⠀⠀⠀⠀⠘⣿⠁⠀⠈⣿⣶⠀⠀⠸⣿⠆⠠⠀⢽⠃⠀⠀⣠⣤⡤⠴⢾⣅⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⠞⠁⠀⢀⣤⣶⣾⣷⠶⠈⠉⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⣀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠀⠂⢠⠈⠉⠙⠛⠻⣦⣄⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡹⠛⢹⣧⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠙⠛⢀⣠⡏⠀⢀⠀⠀⠀⠀⠙⣦⣀⣀⣀⣀⡀⢀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠈⠁⠀⠀⢉⠹⣷⡀⠀⠀⢰⡞⢿⠿⢿⣤⣠⣤⡶⠟⠻⣧⡀⠈⠱⡄⠀⠀⠀⣟⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣗⢰⣾⠃⠀⠹⡇⠀⠀⢸⠃⢀⣴⠟⠁⢈⠉⠀⣠⡾⠛⣧⠀⠀⠐⠀⠀⠀⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡟⠀⠄⠠⣷⠀⠀⢀⣀⠾⣿⠀⠀⠀⠀⣼⠋⠀⠀⠙⢧⡀⠀⠀⠀⠀⠸⣷⡜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣤⣿⠀⠀⠸⠋⠀⢻⣄⠀⠀⠀⠋⠀⠀⠀⠀⠀⠙⠷⣤⣤⡀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⣽⠀⢀⡆⠀⢀⡞⠛⢦⣀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠀⠀⣿⠄⢸⡇⠀⡿⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠸⣶⡼⠃⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⣠⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⡄⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠇⠀⠀⠀⣀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠠⡄⠀⠀⠸⢷⣄⠀⠀⠀⠀⠀⢀⣠⣄⣠⣤⣤⣄⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⡐⠻⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢠⡾⠛⠛⠉⠉⠉⠉⠙⠛⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣀⣀⣤⡶⠛⠋⠙⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⠋⢀⣀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠙⠛⣳⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠉⠉⠀⠀⠀⠀⣀⣀⣀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡾⠛⠰⠶⢦⡀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⠀⠀⠀⠀⠀⣀⣤⣤⣤⣼⣿⣷⣄⣀⣀⠀⢀⣤⠶⠛⠛⠻⢷⣄⠀⠀⠀⢹⡆⠀⢀⣴⠟⠛⠓⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⠶⢶⣤⡾⠋⠁⠀⠀⠈⢙⠿⠛⠛⠛⠻⣿⠃⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢿⣶⣋⣁⣀⠤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠒⢲⣾⠏⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢸⡿⠺⠶⣤⡀⠀⠀⠀⠛⠀⠀⠀⠻⢿⣭⣄⣀⣤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⣏⣠⣶⠶⠛⠳⣦⣤⣀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠿⠟⠉⠀⠀⠀⠀⠀⠈⠉⠻⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠙⠻⠶⣶⣶⡶⠶⠿⠛⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢀⡤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⡰⠀⠙⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠁⠀⠉⠀⡄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠣⣄⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣄⠀⠀⠀⠻⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠔⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣶⣶⠾⠿⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡉⠀⠀⠀⠀⢀⣀⢂⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⢀⣔⣱⢿⣷⣮⣥⡀⠀⠀⢸⣷⣄⣶⣿⡞⡇⠀⠀⠀⠀⠋⠑⢠⡿⣻⠅⠀⠀⠀⠀⠀⢠⣿⣷⣤⣾⣿⣷⡀⠀⠀⠀⠀⠣
⠀⠀⠀⠸⠋⢁⣽⣿⣭⣄⣹⠆⠀⢸⣿⣿⣧⣻⣿⣇⠀⠀⠀⠀⡀⠀⢈⣆⡨⣰⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣶⣏⠆
⠀⠀⠀⣴⡖⢿⣿⣿⣯⡋⢿⡇⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠢⡿⣺⠇⡂⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿
⠀⠀⠀⠟⣠⡾⡿⣿⠇⢉⡷⡗⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⡀⠀⠀⠀⣿⣿⠇⠈⠉⠀⠀⠀⠸⣿⣿⣿⣿⢿⣿⣿⣿⠀⠀⠀⠠⣿
⠀⠀⠀⣇⣴⠟⢹⣿⠷⢎⣁⠀⠀⠀⠸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢌⢺⠗⠂⠄⠉⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⡏⠀⠀⠀⠈⢣⡄
⠀⠀⠀⠟⠀⣒⣿⢟⡂⠀⠉⠀⠀⠀⠀⠹⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⠋⠄⠈⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢼⡇
⠀⠀⠀⢀⠜⠋⣿⣗⠉⠿⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡟⠁⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⢠⢿⣿⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠃⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣂⣤⣄⣾⡿⠛⠃⠀⠀⠀⠀⢸⣿⢿⣷⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣣⣿⣈⣿⣿⠇⠀⠀⠰⠄⠀⠀⢸⣿⣿⣿⠀⠀⠀⢀⣾⣿⣷⡄⠀⠀⠀⠀⠀⢿⠎
⠀⠂⠀⠈⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣫⣿⣯⡀⠀⠠⠐⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢠⡘⡿⠂⢀⡤
⠀⠀⠀⢀⡰⣿⣿⣧⣽⠖⠀⠄⠀⠀⠀⢿⡟⠁⣞⠣⡄⡄⠦⠌⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠸⣿⣿⡿⠁⠀⠀⠀⢦⣟⣇⣠⣾⠃
⠀⡂⠀⠀⠀⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠙⠘⢻⣤⠙⠈⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠁⠈⠻⠉⠁
⠀⡀
⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⠖⠻⣗⣶⣾⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢿⠁⠰⣆⡀⠈⡉⠀⢹⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠉⠉⠀⠀⢳⡬⠛⠉⠠⠘⣧⣧⡀⠀⠀⠀⠀⣠⡄⠀⠀⢠⢦⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠻⡖⢤⡀⢀⣄⡈⠀⢀⠉⣏⠁⠀⠀⠰⣒⡋⡷⠖⠓⠚⣇⣩⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡛⣅⠀⣸⠎⠀⣸⡤⠚⠛⣂⣰⣧⡀⣤⡀⣴⠉⠉⠁⠀⠀⠀⠈⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠟⢦⣤⢀⠄⠛⠀⢀⣴⣧⠎⠀⠀⡱⠙⣯⠀⢀⡤⠜⡇⠀⡼⡂⠀⢬⠶⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠟⠣⢵⡖⠚⠋⠉⠛⠀⠀⠚⠊⡿⠁⣠⣄⡈⠙⠁⠀⠓⠚⠃⢸⡟⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠸⣅⠌⣱⡼⠀⡰⣩⠵⠆⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡤⢼⣢⠉⡼⡳⠦⣄⠀⢠⡄⠀⠀⠀⠘⠷⣍⣄⠄⢙⠁⣠⣰⠻⢥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠸⡏⠀⢊⡁⠙⠀⣀⠀⡑⣷⠎⠧⣀⡀⠀⠀⣴⡫⡸⡋⠉⠉⠉⠸⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠋⠁⣀⡜⢹⠀⢰⠙⢦⡄⠈⡺⣯⠉⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡏⠀⠀⠉⠑⠋⠀⠀⠉⠁⡄⠠⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢈⡽⠀⠾⣁⢒⠄⠀⡔⢒⡦⠀⠀⣤⠿⡲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠘⣤⢠⡈⠚⠀⠀⠷⠁⠀⠃⠀⣿⠳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠚⣦⣄⠀⣊⣵⢶⡶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢣⡇⠉⠉⠉⡝⡤⠽⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⢀⠀⠠⠀⠂⡄⢀⠀⢀⠔⠒⠒⠤⢀⠀⠀⠀⠀⠀⢄⡀⠀⠀⢀⣠⠠⣄⠀⠀⠀⠀⠀⣠⠀⡀⠀⠀⠀⠀⠒⠂⡀
⠀⠈⠀⠀⠀⠈⠤⠄⠘⠂⠄⠤⣀⠀⠀⠈⡀⣀⣀⠀⡸⠀⠀⣌⠀⠀⠀⢃⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⢀⠇⠱⠄
⠠⠀⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠦⠠⠄⡃⠀⠀⠀⣀⠀⠀⢠⠂⠀⠀⠀⠐⢶
⢀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠁⠄⠠⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠤⢀⠀⠀⠈⠁⠉⠁⠂⠀⠐⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠈⠀⢈⠄⡀⡐⠀⡀⠀⢠⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠨⠠⠀⠀⠀⠄⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⢀⠀⡀⠁⠀⠀⠈⠢⡀⠈⠓⢦⠀⠀⠀⡀
⠀⠁⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠀⠀⠀⠀⠠⠀⢀⡴⠀⠇⠀⠰⠀⠁⠀⠀⠀⠂⠈
⠈⠤⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⡀⠤⠐⠁⢠⠞⠀⠀⠈⢀⡗⠀⠀⠀⠀⠀⢀⢠
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣀⠀⠀⠒⠈⠉⡁⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⣠⣴⣶⠶⡶⣶⠶⠶⣾⣿⡟⠛⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⡀⡀⠁⠠⠀⠀⣾⢹⢡⠀⠈⠰⠎⠦⢰⡶⠎⣻⣮⣼⠟⢿⡆
⠀⠀⢠⠀⠀⠀⠀⡀⠀⠘⣀⠸⠀⠠⠀⢸⣿⣀⣼⠦⣶⠗⢾⡒⢺⠛⣻⡏⠍⡈⠆⣼⡇⠀⠀⠈⠈⠀⠂⠐⠀⠀⠄
⠀⡀⠤⢄⠀⠀⠀⠀⠀⠄⢀⡀⠐⠂⠀⠈⢯⠀⡹⠀⡌⠘⠿⢿⡜⣇⠰⠶⠀⡓⢀⣿⡏⠀⠀⠀⠀⠀⠠⠀⠄⠀⡀
⢐⠀⠀⠀⢱⠀⢀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠓⠧⠶⠿⠾⠷⠾⠿⠿⠶⠶⠿⠿⠟⠋⠀⡀⠀⢢⠀⠀⠐
⠈⠢⠄⠤⠃⠀⠀⠀⡀⠁⡐⠒⡄⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⡐⣰⠈⠾⠄
⠀⠀⡄⢠⠊⠉⠁⠢⠀⠤⠈⠂⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢄⠀⠀⠀⠄⢠⠂
⠈⠒⠁⢂⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⠁⠑⠀⠀⠀⠘⠠⠜⠀⠀⡠⣤⣠⡔⢶⢰⡖⠢
⠀⠂⡀⠀⠑⠒⠂⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠃⠝⠛⠀⠁
⠀⠠⠠⠔⢀⠀⠀⠆⡡⡀⡂⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢠⠢
⢄⠁⠊⠁⠈⠀⠈⠀⠀⠈⡢⠅⠀⢀⡤⠒⠉⢑⡄⠀⠀⡀⠒⢠⠀⠀⠀⠀⠀⠈⢠⠁⠀⠀⢈⠀⠀⠀⠀⠄⠀⠀⡀
⢂⡀⣀⠀⠀⠀⠀⠀⢀⠈⠐⠀⡰⢃⠤⠒⠂⠈⡀⠠⠂⠀⣀⠜⠀⠀⠈⠀⠀⠀⠀⠂⢠⠀⠔⠀⠀⠀⠈⠀⠄⠀⠂
⠀⠂⠂⡁⡉⠂⠊⠀⠁⠀⠁⠀⠈⠁⠀⠀⠀⠈⠒⠒⠂⠈⠀⠀⠀⠀⠁⠂⠀⠀⠀⠐⠀⠐⠀⠀⠀⠂
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠉⢉⠛⠒⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠖⠤⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠠⠖⠟⠋⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⢦⡀⠀⣬⣠⣶⣾⣿⣷⣶⣄⣀⠇⢀⡀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⣦⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⠿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣽⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣿⣿⣿⣿⣿⡿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋