Platoon Emojis & Symbols

⠀⠀⠀⠀⠀⠀⣾⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡆
⠀⠀⠀⠀⠀⠸⣿⣿⡇⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣻⣛⡏⠀⠀⠀⠀⠀⠀⠀⢠⠶⠛⠉⠉⠉⠙⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⢈⣟⡛⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣤⣤⣤⣰⣧⡀⠀⠀⠀⡀⠀⠀⢠⣶⠆⣤⣤⣤⡀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡇⣿⣟⣾⡇⠀⠀⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠆⠀⢠⣟⡻⣻⡇
⠀⠀⠀⠀⠀⠀⣿⡫⣾⣽⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣥⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⣿⣿⣿⣿⣿⡿⢾⣿⣿⣿⣟⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣾⣿⣿⣵⣾⢀⣿⣽⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡆⣿⣾⣟⣿⣿⣷⡇⣸⣿⡷⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣷⣿⣿⣿⣿⣯⣏⣷⡿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⣿⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣳⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣯⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡟⠁⠀⠀⠈⠻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡿⠿⠻⢿⣿⡿⠿⠿⢿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣄⠀⢹⡇⠀⣠⡀⠀⢻⠇⠀⠀⢸⣿⠀⠀⣤⠈⠉⡏⠉⢠⡀⣘⣿⣿⣿⣿⣿⣿⡿⣿
⣿⣿⣹⣿⣿⣿⣿⣿⣿⣄⠀⠙⢿⣿⠇⠀⠉⠀⠀⣾⠀⠀⠀⠘⣿⠀⠀⣿⣷⣶⡇⠀⠈⠉⠉⢸⣿⣴⣦⣿⣟⡓⣿
⣿⣿⣽⣿⣿⣿⣿⣿⡿⠿⢷⣄⠀⠙⠀⠀⣶⣶⣾⡿⠀⠰⠀⠀⣿⠀⠀⣿⠉⠉⡇⠀⢰⣶⣾⣿⣿⣿⣿⣿⣽⣾⣟
⣻⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠋⠀⣰⡆⠀⢸⣿⣿⡇⠀⢀⡀⠀⢹⡄⠀⠋⠀⠀⠃⠀⠈⠋⢹⣿⢯⣿⣟⣟⠿⠿⣿
⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣷⣶⣿⣿⣿⣷⣶⣾⣷⣶⣾⣿⣶⣦⣤⣽⠓⠒⠂⠲⣶⣿⣿⣿⣿⣿⣿⠿⠷
⣹⡇⠀⢀⠀⠈⢿⠀⠀⣿⣿⣿⡇⠀⠀⢹⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢡⡌⠁⠀⠀⠀⠀⢻⣿⣿⠀⠀⢻⠀⠀⣷
⢁⡅⠀⠘⠃⠀⠸⠀⠀⣿⣿⣿⠁⠀⠀⢸⣿⣿⠀⠀⣿⣿⡟⠉⠉⠉⠙⢶⣦⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⣿
⣿⠃⠀⢠⣤⣤⣿⠀⠀⣿⣿⣿⠀⢀⡀⠈⣯⣿⠀⠀⣿⡿⠁⠀⠀⠀⠀⠈⣿⢀⡀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣿
⣷⡄⠀⢸⣿⣿⣿⠀⠀⠿⢿⡏⠀⠀⠀⠀⢻⣿⠀⠀⣿⠃⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⣨⠀⠀⡀⠀⠀⣿
⢻⣧⣠⣼⣿⣿⣿⣄⣀⣀⣸⣇⣠⣼⣇⣀⣸⣿⣄⣠⠏⠀⠀⠀⠀⠀⣰⣿⣛⣿⣿⣄⣀⣀⣠⣾⣧⣄⣠⣷⣀⣠⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⢸⣇⢂⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠣⡀⠀⢀⠀⠀⠀⢀⠔⠩⠿⣶⡠⡄⠈⢦⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⡀⠀⠀⠀⠀⠙⡼⠛⢻⡄⠴⠉⡠⠀⠀⠉⠗⠃⢀⣬⡑⣄⣤⣄⣠⣤⡀⠈⠂⣀⡑⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣶⢟⢁⣀⠀⠀⠀⠀⣠⣀⣜⢔⡆⠈⠀⠀⠀⢠⠀⢢⣿⠏⠱⠟⠺⡽⠿⠋⠁⠀⠋⠀⠉⠈⢋⠋⡾⠛
⠀⠀⢀⣤⠄⠀⡆⣠⡠⠠⠀⢸⣿⠃⠀⠀⠂⢠⡁⠸⢹⣶⠀⠀⠉⠁⠀⡼⠔⠁⢀⠁⢀⡀⠉⠉⠈⠂⠀⠀⠀⠀⠀⠀⠘⠔⠁
⠀⠀⠈⠀⠀⠀⣰⡟⠁⠎⢠⡿⠀⠀⠀⠀⢀⠸⠇⠴⠛⣅⣢⡄⠀⠀⠀⠀⠖⡠⡔⠂⡜⠄⢀⡘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⣠⣀⠀⣰⠋⠐⠀⢀⣿⠇⠀⠀⠀⠤⠘⠒⠶⢶⡏⠙⠺⠃⠀⠀⠠⣿⣶⣦⣄⣈⠉⠀⠒⡇⠀⢁⣁⠈⠀⠀⠀⢥⠀⢠⣚⡟
⠀⠀⠻⢫⣾⣛⣆⡴⠾⠛⣿⠀⠀⠀⠠⠀⣤⣤⣤⣼⠗⠀⠀⡼⠿⣶⣾⣿⣟⣿⣿⣿⣻⢗⣦⡇⣞⣸⣟⡎⢳⡴⣅⡀⠀⣷⡎
⠀⠀⠀⠈⢹⣿⣿⣿⣷⣶⣾⠀⠀⠀⠀⠀⠸⣿⣿⡏⠐⠒⠀⠀⢰⡿⠿⣿⣿⣿⣟⣿⣿⣿⣿⡏⣷⣤⣿⣶⣟⣫⣿⣿⣂⣿⡇
⠀⠀⠀⠀⣼⣿⡿⠉⠻⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠉⠀⠰⣯⡟⣩⣿⡏⠉⣼⡿⠛⠉⠉⢉⣯⣿⣿⡿⠃⢿⣟⡇
⠀⠀⠀⣠⣿⣿⠇⠀⠀⢹⣷⣿⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⡀⣼⣿⢃⠁⠀⠐⣿⠿⠿⡿⠀⢠⣙⡟⠁
⠀⠀⢠⣿⣿⠏⠀⠀⠀⣿⣟⣿⠀⠀⠀⢴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⣏⡀⠀⢀⠀⠛⠀⠛⠧⡐⣿⡿⠁
⠀⠀⣾⣿⡏⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠻⠿⣷⠲⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠓⠂⠈⠁
⠀⢼⣿⣿⠇⢀⡀⠀⢰⣿⣿⣧⣄
⠀⠘⠛⠗⠚⠀⠀⠀⠈⠉⠉⠁⠀⠤⠀⠀⠀⠀⠀⣀⡤⠒⠆⠀⡌⠀⠂

⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⡀
⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣤⡄⣠⣦⣤⠀⠀⠀⠀⠀⠐⠾⠋⠀⠻⣷⣄⢰⣞⣿⣄
⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣾⣿⣶⣶⣿⣿⣟⣋⣿⣿⣶⣀⠀⢀⣀⣠⣤⠤⢤⡀⠉⢠⡿⢯⣿⣿⣏⠀⠀⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⠿⠞⡿⠀⠀⠈⢢⣾⣷⢦⡿⠛⠛⠁⠠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⡰⣾⣿⡟⠂⠀⠀⠀⠀⢶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠠⠞⢳⡏⠸⢓⠂⢤⡂⠀⠀⣼⣿⡟
⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⣷⣦⣴⢂⣿⠃⢈⠀⡀⠘⡟⢆⡼⠋⠘⠃
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠓⠛⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣰⣿⣿⣿⣿⠀⢿⣶⣿⣎⠁⠀⠀⣠⡴⠿⠿⠭⠉
⣏⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⢿⣿⡖⠙⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠙⠙⠛⣯⢰⠋⢿⡟⠛⠀⠀⠀⣁⣴⣶⢴⠒⠚
⣿⣏⣯⠀⠀⠶⣶⣶⣶⣦⣀⠀⠀⢰⡃⠀⠀⠀⢸⡆⠈⠛⠃⢀⣿⣿⣧⣘⣿⣿⣿⣿⣿⣿⣿⣤⣠⣤⡔⠊⠑⢈⠀⢀⠁⠀⣈⠂⠰⠛⠛⠉⠉⣀⣤
⣿⣿⣁⠀⠀⠀⢠⣾⡿⢿⣿⡭⢶⣾⣿⠋⠀⠀⠈⣿⣶⣄⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⢿⢿⠧⣤⡏⢈⣠⣬⣿⣾⣿⣇⠀⠀⠀⠀⠈⣿⣿
⣿⣿⡏⠀⠀⠀⠄⠈⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠸⣿⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠃⠀⠀⠀⣿⠀⢄⡤⢿⣿⢿⣿⡿⢁⠀⠀⠀⠐
⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⡀⠀⠺⡁⠀⠁⣴⣿⣏⠀⠏⢱⣤⣶⣦⡀
⣿⣿⣿⣿⣄⡀⠀⠀⠀⠠⠄⠀⣀⡀⠀⠀⠀⢀⣀⣄⡀⣸⣿⣯⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣷⣤⣀⣤⣀⣼⣿⣿⠀⠀⠾⠻⠿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣷⡶⠀⠀⠀⠀⠀⠈⢿⡶⠋⠁⠉⡉⢛⣽⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⠿⣱⣯⢿⣯⣽⢍⣵⢶⠿⡝⣁⠛⢤⣀
⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣧⣤⠀⠀⢿⣄⡷⠊⣠⣿⣿⣿⣿⣿⣿⣍⣡⣶⣶⣶⣤⣄⠀⠹⣇⣾⣷⣾⢁⣴⠀⢀⡄⠉⠁⢠⣨⣶⣀
⣿⣿⣿⣿⣿⣿⣶⠿⠿⠿⢿⡿⢿⠆⠀⣿⣧⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠉⠛⣿⣶⣸⣿⣯⣼⣿⣏⢣⣾⣧⣴⡶⠿⢿⣿⣿⣷⡀⠀⠀⣠
⣿⣿⣏⣿⣼⣿⠁⣠⣤⣤⣮⣵⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡉⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⠿⢧⡀⠀⠀⠀⠛⢷⣢⣾⣿
⣿⣶⣿⣿⣯⠃⠈⠉⠛⠋⠈⠻⠟⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠖⠙⠿⠛⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣉⠱⠆⠐⠈⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠑⠶⠀⠤⠰⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡤⠉⡤⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⣀⣀⢀⡹⢷⣦⡈⠃⢸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣆⣒⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠉⠀⢸⣿⣿⣶⣿⣿⣿
⣿⣿⣿⢿⣿⣿⣿⣿⣯⣤⣦⣼⣿⣿⣿⣭⡙⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠚⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣽⠰⣎⣿⣿⣿⣿⣿⣿
⣉⣀⣴⣾⣿⢿⣿⡙⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣤⣄⣩⠿⠟⠋⣩⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣬⡙⠛⠿⢿⣿⣿⣿⣿⣿⣾⣿⣐⠏⣹⣿⠳⠛⢻⣿
⣿⣿⣿⣯⣤⣿⡿⢃⣹⣿⢟⣻⣿⣧⡟⢠⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡏⠙⠛⠙⢻⠟⠉⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇⡿⢸⡟⢞⣦⣴⣧⣉
⡿⠿⢿⣿⣿⣷⣴⣿⣿⣵⠿⢿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠗⠷⣿⠷⣾⣿⣷⣾⣻
⣶⣶⣤⣼⣯⣭⣭⣥⣽⣯⣀⣀⣶⣠⣄⣀⣹⣿⣿⣯⡿⢿⣿⠿⠛⠿⣿⡇⠀⠀⠀⠁⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⠋⣼⣿⣿⣧⣼⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣴⣼⣿⠿⠛⠉⠹⠀⠀⠁⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⠏⠈⠿⣿⣿⣄⣘⣟⣀⣿⡿⠋⠉⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣥⡉⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠰⣿⣿⣿⣿⡏⣿⣿⢃⣀⠀⠰⣿
⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣻⣿⣿⣿⣿⠇⠀⣿
⣿⣿⣿⣿⣟⣻⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠈⡙⠃⠀⠀⠀⠀⠈⠙⠛⠋⢿⣿⠤⣾⣤⣴⣿⡘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣶⣾⡿⠃⠀⠀⠀⠀⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⡧⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⡀⠀⠠⣔⢐⡷⠤⠦⣤⡋⣼⣿⣿⣿⣿⠟⢻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⣴⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠁⢰⢇⣾⣶⣿⣿⣷⠤⢀⣿⣿⣻⣿⣿⣿⣿⣶⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣿⣿⣦⣴⣶⣤⣿⣿⣿⠿⠶⠾⠏⢻⣉⡁⢀⣠⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡷⢯⠶⡔⢠⠀⡄⠀⣤⡈⢻⠙⡗⢛⡿⢿⡿⣿⣿⣿⣿⠉⣤⠀⣥⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣷⣾⣶⣧⣾⣤⣿⣤⣿⣧⣼⣦⣷⣬⣧⠼⠰⠿⣿⣿⣿⣷⠿⢤⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⠄⠈⡇⠀⣿⣿⡏⠀⠈⣦⡄⠀⣤⡾⠉⠁⠉⠸⠏⠀⠀⠀⠀⢸⠀⠀⠡⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣤⣼⡇⠀⣿⣿⠁⢀⠀⢸⡇⠀⡿⠁⠀⠀⠀⢠⣿⡀⠀⠀⠀⠘⠀⢀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣸⣿⣿⡇⠀⠙⡋⠀⣠⣄⠈⡃⠀⣀⠀⠀⠀⢀⣿⣿⣷⣀⣀⣀⣰⠀⢸⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠿⠟⠻⠻⢿⡿⠛⠛⠿⠟⠻⢿⣿⡟⠲⠖⠻⠛⠛⢿⡿⠛⠿⢿⠛⠛⠛⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⣄⠀⡄⢸⡄⠱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⡄⣿⠄⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢣
⡀⠀⢾⣤⡇⠀⠃⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⡀⠀⠦⡀⠀⣷⢸⠀⠐⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡶⠀⢈⣢⡄⢹
⣚⡃⠘⢸⠃⠀⠓⢀⣉⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠷⠀⠐⣚⡁⠛⢸⠀⠘⢃⢀⡉⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠁⠨⠓⣠⠈
⠰⢿⠀⢺⣧⡒⠂⠩⠠⢟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠀⠰⣿⠀⢻⣄⠒⠀⠅⠸⢋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠉⠀⠀⠬⣿
⡈⠻⠄⠀⠙⠛⢳⣶⣶⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣡⣴⣄⠉⠳⠀⠈⠙⠛⣶⣶⣶⣟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣛⣷⠤⣈⠃
⣿⣾⠀⠀⠀⠀⠈⣿⣿⣿⣟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⡧⡞⠀⠀⠀⠀⠈⣿⣿⣿⣚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡟⠺
⠃⣾⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⠁⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⢀⡟
⣶⡧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣟⣆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣏
⣿⢳⡄⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠈⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⢣⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠷⠈⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⠹⡆
⣿⡈⡇⠀⣠⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⢨⡇⠀⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⣤⣀⣶⣾⣯⣿⣿⣿⣿⣿⣿⡆⣇
⣿⣧⣷⠀⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣧⠘⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⠿⠛⠛⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⠀⠂⠒⠲⠦⡴⠄⠠⡦⢤⠠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠠⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣶⣓⡟⠓⡀⣠⢀⠐⢤⡀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠂⠀⠤⠄⠵⠣⠄⣤⡅⢃⠂⡄⢢⠁⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠛⢴⡜⣀⣀⣤⢜⢛⡀⠐⠢⢀⡲⠓⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣘⡹⢿⣷⣦⣤⣀⠉⠼⠰⣠⢉⠳⢠⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠗⠲⢬⣙⣻⣿⣶⣶⣭⡅⣸⣂⡀⣤⣆⠀⠀⠀⠀⠰⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠙⠓⠮⣉⣒⣶⣬⣹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠂⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⢟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⣀⡀⢀⠀⠘⣉⡍⡛⠛⠲⣦⢀⠀⠀⠀⠀⠀⠀⣠⣼⣃⢠⡾⠉⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠉⡁⠀⠘⠋⠉⠀⣠⠘⠓⣠⣻⣷⡦⣤⢂⢀⣴⠟⣛⡻⢿⡾⣼⢃⣠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠣⠀⠀⠄⢀⠀⠀⠀⣰⣳⣷⣾⣿⠇⠀⣾⢯⣰⣿⣓⣫⣼⡿⢁⠞⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⠬⠥⠤⣥⣦⢶⠁⢁⢀⢰⣟⣿⡿⣿⠿⠆⠼⢯⠿⣿⣿⣿⠿⠿⠏⢾⣷⡚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢺⡿⠿⠿⠿⠾⠒⣾⣷⠒⠶⠶⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠠⠀⣈⠉⠃
⠀⠈⠡⠁⠁⢀⡄⠈⠛⣯⠂⣠⣴⣶⣿⣭⣤⣤⣴⣤⣶⣤⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤
⠀⢺⣿⡖⡞⢻⡿⡧⠀⣀⠹⠁⠉⠻⠿⡿⠻⠛⠛⠛⠿⠛⢿⣿⢻⣿⣿⣿⢿⣿⣿⡿⡛⢛⠿⣽⡟⣿⣿⠾⢿⠿
⠀⠀⠈⠑⢹⣟⡀⠀⠸⣨⠳⠤⠀⠿⠋⡈⢐⢴⡿⢴⠀⠓⢠⠫⣻⣇⣌⡝⢸⣧⣼⠟⠃⡻⢾⣿⠅⡷⢿⡉⣟⢠
⠀⢀⣀⢰⣿⣿⣿⣶⡉⠹⡇⠀⡀⢀⢀⣣⢯⡘⢟⡽⠊⠐⢿⠅⣿⣸⣛⡧⣜⡟⠳⡄⣜⣿⣾⡿⡷⣈⡟⠘⡁⣗⡇
⠀⡨⢠⣷⣿⣿⡿⠿⣿⣷⢤⠀⠀⣠⡉⢹⣯⣷⠛⠀⠀⢻⡖⢉⣿⣺⣯⢉⣻⣇⡃⠅⢾⣧⣿⡃⠀⣿⣬⠔⠋⠉⠁
⠀⣠⣾⣿⣿⣿⣿⡆⣿⣿⣾⡀⣦⣽⣿⡦⢍⣻⣷⡒⠒⠿⢿⠁⠘⠡⠉⠊⢉⣤⣴⣶⢸⣷⣿⠉⣷⣿⣉⣶
⠀⠿⠏⣿⣿⡿⣿⣯⣿⢿⣿⡷⠂⠀⠁⠀⠀⡞⣾⣷⣶⣚⢳⣷⡀⠄⠀⠂⠶⣶⣚⣁⣀⣧⣽⣏⢹⣻⣻⣿⣥
⠀⠀⡀⢸⡿⠁⢻⣿⠋⠸⠃⠀⠀⠀⠀⠀⠀⣶⡯⠁⠈⠙⠿⣷⡀⠀⠀⠒⠘⢛⡛⠛⠋⢙⠉⠉⠋⠃⢿⠋⡏
⠀⠈⠛⠸⠿⠚⠾⠇⠈⠀⠀⠘⠀⠠⠐⠶⠦⠸⠇⠐⠂⠀⠀⠈⠷⠃⠶⠶⠶⠊⠀⠉⠐⠄⠀⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⡀⠀⠀⠀⢀⢌⠀⠀⠀⠀⠠⢁⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡶⣒⣦⣶⡖⠀⠀⠀⠀⠀⠀⠐⠢⣶⣦⣖⡖⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣯⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⣯⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡣⠷⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠿⢜⠷⣄
⠀⠀⠀⢀⣠⡤⣤⣴⣟⣿⣱⣯⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣽⣅⣯⣻⣷⣶⠛⢲⣦⡀
⠀⠀⠀⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⡏⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣟⡇
⠀⠀⠀⠳⠞⠻⠟⠋⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉⠁⠉⠉⠉⠉

⡆
⡆⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠆
⠆⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠘⠉⢀⣀⣤⣀⣀⣠⣿⢿⣿⡶⠐⠁⠀⠀⣀
⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠉⠉⠉⠛⣿⣿⡽⠻⠿⢭⣷⡄⠈⠀⠀⠀⠀⠰⠟
⠆⠀⠀⠀⠀⠀⠀⠀⢀⣉⣀⣀⡄⠀⢀⣤⣄⣀⣤⠤⠴⣽⣿⣗⠒⠿⠿⠏⣀⡀
⠆⠀⠀⠠⣤⣶⣄⣤⣤⡾⣿⣿⣿⣴⣿⣿⣿⣿⣿⠀⠀⠶⠦⣬⣉⣗⣒⠀⠤⠽⣷⣶⣤⡄⠀⠀⠀⢀⡄
⠇⠀⠤⠿⠿⠿⠿⡿⠿⢿⠿⢿⠿⡿⢿⣿⣿⣿⣿⣦⣀⣀⣂⠄⠠⠌⠉⠙⠓⠲⣶⢬⡍⠁⠀⠀⢠⠞⠁
⠇⠀⠀⣙⡛⠓⢲⠓⠂⢐⣑⢹⠤⣷⡀⣭⣽⠉⠛⠉⡷⠿⢿⣭⣕⣦⣖⣲⣾⣬⣥⣈⣿⣶⣦⣀
⠗⠀⠀⠽⠿⠖⠀⢉⠀⠉⠉⠁⠶⠛⠍⠿⢧⣾⡼⣾⣷⠆⠀⠀⠉⠉⠀⠀⠀⠈⠉⡁⠉⠨⠁⠀⠐⡀
⠛⠀⠀⠀⢀⣀⣀⠀⠀⠠⠴⠶⠌⠉⠀⣠⡈⠛⠛⡟⢿⡦⠀⠀⠛⠒⠶⠆
⣻⣄⣠⡀⠸⠋⢉⠁⣀⠀⠀⠀⠀⣀⣼⣿⢹⠁⠀⠀⣼⣇⠀⠀⠀⠀⠈
⣛⠸⣿⣧⣶⣴⣬⠌⣩⣶⠀⢠⣯⠑⣿⣿⠉⠀⠀⠠⣹⠏⢠⡀⡄⡀
⣛⣀⡾⣿⣿⣿⣿⠼⣋⣵⠇⣼⢉⠈⠙⠅⠀⠀⠀⠴⠋⠀⠈⠓⠀⠉
⣿⡍⠉⠉⠭⠤⠒⠊⠉⠁⣴⠋⠘⠉⣴⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣴⡀⠀⠀⠀⠀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢤⠀⠀⠀⠀⡠⢧⡤⢄⡛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡋⠀⠀⢢⣀⠃⠐⠋⠁⡈⠁⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠚⠛⢰⡄⢸⠟⠈⠀⠒⢨⣿⠄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢴⠟⠁⠸⠀⠅⠀⠀⠃⢬⡇⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡶⠀⠀⢀⡄⠀⢠⣁⢿⠇⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⠚⣆⣤⠈⣹⣶⣷⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠁⣤⣿⣿⣀⣿⣷⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⢘⣿⣿⣿⣸⣿⣿⣿⡏⠉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣽⡟⣽⣿⣧⣿⣿⣿⣿⣻⣾⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠛⣴⣿⣿⣿⣿⣿⢾⢸⣿⣿⣄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⡄⢒⣾⣉⣻⣌⣿⣿⣾⣾⣿⣿⣿⣿⣦⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠛⣋⣭⡿⠛⠉⠀⠀⠙⢿⣿⣿⣿⣿⣿⡄⢈⣥⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣿⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣏⣉⣤⠓
⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠺⠿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡿⣶⣶⠒⠒⠒⠒⠒⠒⠲⠶⠿⣶⣾⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⡆⠀⠀⠀⠀⣴⣟⣿⣿⣿⣻⣿⣤⣀⣤⣶⣤⣄⢀⣾⡶⣷⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⣄⠀⠀⠀⠀⠴⠟⠛⠛⢛⠛⠛⠛⠟⠛⣛⠻⠛⠻⠟⠻⡛⠒⠿⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⡆⠀⠀⢸⡟⢿⠀⠀⠀⣾⠻⡷⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠸⠿⠤⠄⠀⠀⠼⠇⠀⠀⠀⠀⠿⠧⠷⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣛⠾⣷⣶⣶⠥⠀⠈⠉⠉⠉⢉⣠⠆⣀⡀
⠀⠀⠀⠀⣀⣀⣠⣤⡄⠉⡿⣽⡟⠁⠀⠀⠀⠀⠀⠀⠴⡤⠓⢿⣿⢦⡬⣴⣟⠀⠀⠀⢸⣿⣿⢫⣤⣉⣹⡅⡀⠀⠀⡠⠠⣦⡤⠰⣄⣀⣀⣀⡀⠀⢶⠆⠰⠶
⠀⠀⠀⢻⣿⣬⣽⣏⣴⣶⣶⣾⣷⣶⠀⣤⣤⠀⠐⠖⠲⢾⠗⠸⠟⣠⣴⣿⠟⠂⣲⠒⠀⠿⣤⠞⠁⠸⠠⠝⠷⣤⡀⠀⢘⠀⢀⡛⢻⣄⣙⣿⡄⠀⠀⠗⠰⠿
⠀⠀⠀⢸⣯⣠⣾⠃⠹⠁⣾⠟⢿⣿⠣⢴⣿⣦⠀⠀⠀⢾⣧⠀⣦⠩⠀⠹⣦⣾⣿⣷⡀⠰⠒⢤⡌⠈⢱⣶⡖⠻⠿⣿⣿⣖⣈⠉⡎⢉⣨⡏⡝⠀⠀⠀⠂⢠
⠀⠀⣠⠠⠿⢫⣿⣧⣄⡸⢅⣼⣿⣿⣀⡰⣤⣅⡀⢀⣴⠲⠟⠉⡁⠀⠚⠃⠀⣀⠈⣿⡅⠸⠃⠀⠀⢐⣯⣬⠀⠀⠞⠉⢉⠻⢏⣿⡗⠄⢉⠷⢤⣄⡒⠀⠤⠋
⠀⠀⠃⠀⣀⡘⠛⠻⣯⠁⣿⠉⠉⠉⢹⢷⡌⠛⠿⢾⣿⡟⠀⠀⠄⡇⠈⠛⠄⢿⠰⣿⣧⠀⠀⢣⠈⠀⢻⣿⢃⣄⣠⡾⠉⠑⠈⢹⡟⠛⠛⠂⠀⠉⠀⢶⡤⠂
⠀⠀⠀⣠⠟⠑⣆⠀⠻⣄⠛⠀⠀⢀⣿⠈⣷⣰⠂⣸⣿⠁⠀⠀⠀⠀⠘⡁⠴⠋⢻⡄⢽⠀⣿⠘⢄⠲⣭⢾⠿⢋⣍⠀⠀⠀⠀⢸⡇⠀⠀⢨⠂⠇⡠⠈⡁
⠀⠀⢠⠀⠀⠀⡿⠆⢸⡿⣷⠀⠀⠰⣿⡄⢹⡎⣳⡛⣿⠀⢰⣟⠀⡀⢹⡏⢨⣉⣾⡅⢸⠀⣿⡂⠈⢉⡏⠈⠀⠀⣸⠀⠀⠀⠀⢸⠁⠀⠀⢻⡀⠀⡀⠀⠃
⠀⠀⡀⣀⠀⠀⣷⣤⣴⣇⣨⣗⣦⣶⠿⡇⠸⣏⣹⡇⣼⠀⣼⠛⠏⠀⠀⡁⠀⠟⢾⡇⠈⠐⣟⠟⠀⠘⡀⠀⠀⠀⢻⠀⠀⠀⣠⡏⠀⠀⠀⣾⡇
⠀⠀⣿⣟⠀⢼⣿⠛⢛⡟⠛⠋⢟⣿⣷⡇⢀⣿⠉⣷⣿⡆⣿⡦⠤⢤⣼⡗⠰⣶⢸⣿⢠⠸⣿⡤⠀⣼⠁⠀⠀⢠⡞⠀⠀⢠⠟⠀⠀⢀⣀⣿⡿⠆⠀⠐⡗⢰
⠀⠠⠦⢸⡀⠸⠃⠀⢸⠇⠈⠀⢸⠁⠙⣷⣰⡿⠃⢸⣿⣀⣿⡇⠀⠐⣿⡿⠀⠹⣤⣯⠸⣶⢿⡆⠰⡇⠀⠀⢠⡟⠑⠀⣴⣯⣤⣶⡿⢿⠿⣿⡇⠀⠀⡛⣻
⠀⠀⠀⠘⣿⡂⠀⣀⣿⠀⠀⠀⠈⡇⠀⣹⣿⠇⣸⣻⡍⣿⠛⣷⠀⠀⡿⠃⠀⠀⠿⠿⠿⣿⠈⡇⢠⠁⠀⠀⣾⣷⣾⣧⡈⠋⠉⠀⠀⠀⠀⠙⣿⣆⣹⣿⣿⣦
⠀⠀⢰⡀⢈⠛⠛⠉⠉⣄⣤⣀⣀⣈⣵⣿⣛⠀⢻⣿⡃⣿⡄⣿⠀⠀⣶⠀⠀⢰⡁⠀⠀⢸⠀⣇⠹⡄⠀⢰⢉⣿⣹⠏⣼⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣙⠛⡿
⠀⠀⢸⡇⠀⣦⠀⠀⠀⠀⣿⠉⠀⠀⣹⣿⣿⡀⢸⡧⡇⢹⡆⣿⠂⠀⣹⡯⠀⠘⡇⠀⢦⣾⡁⣿⠀⠧⠀⢸⣘⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣿⣅⢀
⠀⠀⢸⣿⡰⣿⡟⠐⠒⣨⣿⠀⠀⣶⡟⣷⣞⣉⡉⢱⠇⠈⣇⡻⢀⣞⣩⢷⣢⡀⠇⠀⠀⣿⡇⣏⠹⠣⢤⣆⣜⠋⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣹⡇⡯
⠀⠀⠀⣿⣧⣿⢣⣶⡀⢸⣿⢀⢘⣿⣿⡇⢸⣿⣿⠀⣿⠀⡇⣥⣸⣿⣿⢈⣿⡇⢀⣿⠀⣿⡇⣽⣦⢠⡈⣿⣿⣶⠒⠋⠋⢀⢀⣀⣀⣤⡤⠞⠛⣛⣿⣿⣀⡧
⠀⠀⣤⣽⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣇⣸⣿⠏⠀⣸⣾⡇⢸⣿⣿⣿⢾⣻⠃⢀⣿⣄⣽⣇⣿⣿⣧⡀⢨⡿⠁⠀⢀⣀⠨⣤⣿⢿⣧⣀⡀⠀⠉⢀⣿⣿⠁
⠀⠀⣉⡴⣿⣷⣿⣿⢠⣿⣶⣿⢺⣿⣿⣿⣿⡿⣶⣾⣿⠿⠛⣮⡉⣿⣿⣮⡿⠷⣿⣯⣛⣽⣿⡿⠿⣿⣿⠊⢷⣤⣀⠀⠀⣤⣾⣿⠟⣩⣿⣟⠉⢰⣞⠉⠉
⠀⠀⠻⡶⠿⠛⠙⠻⢦⣿⠋⢻⣼⡿⠛⠉⠉⠳⠤⠾⠀⠀⠘⢛⣷⣿⡿⠿⣶⣤⣾⠟⠉⠀⠈⠳⠶⠛⠉⠉⠂⠸⡇⢲⣾⠋⠐⠛⠾⠛⠉⢷⣦⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠈⣽⠇⠀⠀⠀⠀⢠⣴⣶⡿⠻⣟⣃⣶⠶⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣿⡇⠀⠀⠀⠀⠀⠀⠘⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⢰⣿⣄⣀⣠⣴⣶⣿⠟⠛⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⡿⣿⣿⣿⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠘⠟⠛⠉⠀⠀⠉⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣦⣼⣷⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡶⠞⣁⠀⢻⣿⣄⣤⣤⣤⣤⣶⣶⣾⣿⣏⡉⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠺⠿⣿⣿⣸⣿⣿⡿⠟⠛⠛⠉⠉⠉⠉⠷⣼⣿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⡀⠀⣀⣀⠀⠀⠀⢀⠀⢀⣀⣀⣀⢀⣀⠀⠀⣀⣀⣀⣀

⣿⣿⣿⣿⠿⣿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢎⡿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠿⠺⠟⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡀⠚⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣼⠹⠿⣿⠟⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠠⢲⣄⣠⣞⢁⣡⡀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⣤⠸⣿⣭⢿⡟⣾⠘⣧⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢻⣷⢹⡟⢺⡶⠟⠼⠟⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣄⠂⠐⠀⣠⠀⠄⠀⣃⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣤⣀⣠⣄⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣛⣿⣿⣿⣿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⠻⣿⣷⣟⣽⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣟⣾⡇⣷⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣁⠀⠀⠀⠀⠰⠘⠃⠀⠀⠀⠀⠀⢰⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⠛⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣤⣤⣶⣾⣿⣾⣿⣿⣿⣧⣠⠀⣶⣾⣿⣶⣄⣄⡴⢀⣶⣿⡟⢿⠟⡿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⡽⠇⠁
⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠙⣿⡿⣋⣺⣿⣿⣿⣿⣿⣿⣟⣿⣷⣴⠿⠿⠿⠟⣯⣿⣧⣿⣯⣭⣤⣤⣾⣧⣬⣿⣬⣽⣿⣿⣿⣿⣏⣯⣩⢸⣦⣤⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣸⣿⣧⢀⣤⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠗
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠛⠙⠉⠛⠛⠛⠛⠋⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣠⠊⠒⢤⣔⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⡤⠔⠊⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠯⣻⣶⣤⡈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⣠⣴⢾⡫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠫⢟⡷⣦⣄⡉⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⣁⣤⡶⣟⠯⠺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣦⣍⠓⠯⣻⢷⣤⣈⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⢉⣠⣴⢿⡫⠗⢋⣤⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣬⣙⠺⢽⡻⢶⣤⡉⠓⢦⣄⡀⠀⠀⠀⠀⢀⡤⠖⠋⣁⣴⡾⣻⡽⢚⣩⣴⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣍⡓⠯⣟⡷⣦⣄⠙⠳⢤⣴⠛⣁⣤⡶⣟⡯⠗⣋⣥⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣬⣙⠳⢽⣻⢶⣤⣈⠙⠻⢷⣛⣡⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠞⠛⣿⣶⣬⡙⠪⣝⡻⣦⣄⡈⠙⠢⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⠋⠁⠉⢲⣶⣿⡫⠟⣋⣿⣷⣦⣍⡓⠯⣟⣷⣦⠋⠉⠉⢓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣊⣥⢶⣫⠷⣀⣠⣴⣿⣡⣴⠾⠛⠉⠀⠈⠙⠻⢷⣤⣹⣷⣤⣀⡰⢯⣓⢦⣍⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣩⡴⠾⠿⣖⣯⣶⣿⠿⣟⣯⡶⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠳⣮⣿⡿⢿⣷⣯⣗⡾⠿⠶⣬⣑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣺⣥⣸⣏⢁⣠⣴⣾⣿⠟⣋⡥⣖⣽⣿⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢹⣿⣕⡦⣍⠛⢿⣿⣷⣤⣀⢉⣿⣠⣽⡂
⢀⡠⢒⣒⡒⠐⢺⢡⠃⠙⣿⣿⠿⠛⠁⢈⣿⣿⣿⣿⣿⡿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠿⣿⣿⣿⣿⣿⣍⠀⠙⠻⢿⣿⡟⠁⢣⢸⠒⠐⣒⣒⠢⡀
⠺⠟⠋⠉⠙⢧⣸⣾⣇⣀⠘⣿⣧⣴⣾⣿⣿⣿⠿⠛⢁⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠙⠻⢿⣿⣿⣿⣶⣤⣿⡟⢀⣀⣿⣼⣠⠟⠉⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠙⠻⢿⡿⣿⣿⡿⣿⡿⠛⡉⢀⣤⠾⠿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠿⢦⣄⠈⡙⠻⣿⡿⣿⣿⡿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠨⠏⠁⠀⠀⢀⣴⣿⠁⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢹⣷⣇⠀⠀⠀⠉⠷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⣴⡄⠀⠀⢠⣿⠁⠺⣧⣄⡀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣀⣤⡿⠂⢻⣧⠀⠀⠀⣴⣄⢦
⠀⠀⠀⠀⠀⠀⠀⡔⣱⣿⣿⣿⡄⢠⠟⠛⠷⣦⣤⣥⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣥⣤⡶⠟⠛⢧⠀⣼⣿⣿⣷⡑⡄
⠀⠀⠀⠀⠀⠀⡜⡼⣽⣿⣿⣿⣿⡆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⡜⡄
⠀⠀⠀⠀⠀⢰⢸⢳⣿⣽⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣽⣿⢻⢰
⠀⠀⠀⠀⠀⠈⣿⢸⣿⣿⣿⣿⣿⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⣿⣿⣿⣿⢸⡟
⠀⠀⠀⠀⠀⢰⢿⢸⣿⣿⣿⣿⣿⣧⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢧⣿⣿⣿⣿⣿⣿⢸⣻
⠀⠀⠀⠀⠀⠈⡾⣼⣿⣿⣿⣿⣿⣿⣞⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣾⣿⣿⣿⣿⣿⣿⡼⡌
⠀⠀⠀⠀⠀⠀⢱⣻⣿⣿⣿⡿⣿⣿⣿⣎⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⣾⣿⣿⡿⣿⣿⣿⣿⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⡿⣟⠵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢝⡿⣿⣿⣿⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠘⡹⣿⣿⡻⠝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠽⣻⣿⡿⡝

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢻⡟⠳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠁⠀⠀⣼⠀⣬⠱⣌⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⡆⠘⠆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡇⠀⠘⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠁⠀⠀⠸⣿⠀⠀⠀⢀⡄⠊⠀⠀⢸⡀⢻⣀⡎⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠰⠈⠀⠀⠀⢻⣄⠻⣄⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⢯⣹⡀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣷⡄⠀⠀⠀⠢⡀⠀⠀⠀⠹⣦⠉⠛⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⢀⣠⡤⠞⠋⣁⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢼⠿⣦⣀⠀⠀⠀⢀⣀⣠⣤⠞⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⣿⢋⣥⣴⠾⢛⣩⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣸⣶⠴⠞⠛⠙⠛⠻⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣽⠟⣩⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠛⠉
⢠⡶⠶⢦⡀⠀⠀⢀⡄⣾⡁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡟⠋⣡⣴⠞
⢿⣠⡴⠞⢧⡀⣶⢭⣀⠹⣇⠸⣧⣶⢀⠀⠀⠀⢀⣠⠤⠖⣏⢉⡝⠉⢸⡏⠉⢢⣄
⠀⢉⡵⠖⡛⢻⣿⣦⡽⠀⢙⣧⡈⠻⠿⠀⠀⠈⠀⠀⣠⠶⣼⣯⣽⣿⣿⣥⣤⠾⠿⠿⢳⡀
⠀⠈⠷⣦⠶⠞⠛⠲⠶⠚⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⣬⣄⢀⣀⣀⣀⣀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⡆⠀⢹⣟⠉⠉⠉⠙⣯⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠞⢻⡷⠤⠴⣶⢿⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⢷⣶⣶⣿⠀⠋⠉⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⠇⠀⠀⠀⣾⠀⡄⠀⠀⠀⣧⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡞⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸⠃⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢿⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢰⣏⣠⣤⣤⣤⡀⣿⠀⣿⠤⠤⡤⠤⠼⣇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣾⠛⢷⣶⣿⠀⠀⣿⠀⣿⠀⠀⣀⡴⠞⢿⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⠇⠀⠀⠪⢝⠻⣦⣿⠀⢻⣠⠞⠋⠀⠀⠈⣇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡟⠀⠀⠀⠀⠀⠁⠂⣿⠀⢸⠁⠈⠀⠀⠀⠀⢸⡆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣷⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠶⠶⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡤⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣤⣭⣿⡄
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣾⠃⣀⠀⢀⡂⢐⣄⣀⣀⣤⡀⠀⠀⠀⣾⡇
⠀⠀⢀⡀⠠⠄⠤⠄⠀⠀⠀⠄⠒⣾⣯⣿⣯⣯⣽⣿⣿⣿⣷⣶⣶⣶⣿⣿⣤⣤⣴⣤⣶⣤⣴⡬⠿⠿⠿⠿⠿⠷⠶⠾⠯⠟
⠀⠀⡃⢀⣀⣀⣠⣤⣶⣶⣷⡿⠟⠛⠛⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⡍⠀⠈⠉⠉
⠀⠀⢻⣿⣿⣯⢿⡿⠟⠋⠁⠀⠀⠀⣸⣿⣿⠁⠀⠉⠀⠀⢹⣿⣿⣿⡄
⠀⠀⠐⠭⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣼⡁⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⢄
⠀⠀⡴⠚⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⣇⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⢄
⠀⠀⠀⠉⠉⠉⠉⠁⠈⠑⠲⠤⠤⢤⡄⠀⣀⣠⠤⠤⠤⠤⠤⠤⡤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⢢⡀
⠰⠶⢶⣚⡉⠙⠒⠲⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠉⠉⠛⠒⠲⠤⠤⣈⣉⠛⠒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⣀⣀⠤⠤⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⢬⣍⣑⡒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⢀⣀⣠⡤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠢⠭⢍⣒⣢⡤⠤⠒⠒⠲⠤⣤⡖⣋⡡⠤⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠴⠚⢳⣤⣰⠉⣷⣀⣤⣟⣒⡲⠦⠤⠤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠴⠖⠒⠒⠋⠉⠉⣀⣀⡤⠴⠒⠚⠉⠁⠀⢸⣀⣇⠀⠀⠀⠀⠉⠉⠙⠒⠒⠶⠤⠬⣭⣉⣓⡒⠒⠶⠤⠤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠣⠤⠤⣀⣀⣀⣀⡠⠤⠖⠚⠉⠉⠀⠀⢀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠙⠳⠦⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠒⠤⠤⠤⠛
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⡤⠤⠶⠮⠿⡶⢤⣄⡀
⠀⠀⠀⣾⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣀⣤⠤⢒⠚⡉⠅⢆⢰⠀⠀⠀⠀⠀⠈⠢⡙⢍⠳⣤
⠀⠀⠀⢻⠸⡆⢀⠖⠉⠑⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣀⡤⠔⢚⡉⡅⠰⠁⢸⠀⡃⠀⠘⠈⡀⠀⠀⠀⠀⠀⠀⠐⡄⢣⠈⢳⡄
⠀⠀⠀⢸⡀⢷⡎⠀⠀⠀⠈⡇⠀⠀⢀⣀⣤⡤⠴⠖⠛⡤⠖⠉⠁⠐⠉⠀⠀⠀⡄⠀⠸⠀⡇⠀⠀⠁⠡⣀⠤⠐⠂⠀⠀⠈⠉⠈⠁⠂⢷⣄⡀
⠀⠀⠀⢸⡇⢸⣅⣠⣤⠴⠶⠛⠛⠋⠉⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⠠⠄⢁⠀⠀⠨⠀⠀⠀⣀⠀⠠⠀⠀⠀⠠⡄⠀⠀⢀⣿
⠀⠀⠀⢸⡇⣾⣉⣀⣀⣀⣠⣤⣤⡤⠤⠤⠤⠶⢶⡶⢾⠀⠀⠀⠀⠠⠤⠄⠂⠁⠀⠀⠀⠀⠀⠂⠐⠂⠁⠀⠀⡁⠀⠀⠀⠀⡀⠜⠀⢀⠔⢁⡼
⠀⠀⠀⣸⢰⠏⡏⠉⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣸⠁⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠈⠀⠀⣦⣁⡴⠛
⠀⠀⠀⡏⡿⠀⠹⣄⠀⢀⡼⠁⠀⠀⠀⠀⠀⢠⣏⣰⠃⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢒⣉⣩⠇⣀⣀⣀⣤⡏⠉⠉⣷⠛⠉⠉⠁
⠀⠀⢸⡿⠃⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⢝⣻⠿⠖⠒⠒⠒⠒⣲⢋⡶⣟⡛⠉⠉⠉⠉⠀⠀⢧⣀⣠⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠎⠀⠀⢀⣀⣀⠤⠴⠷⢋⡭⢧⡙⠦⡀⠀⠀⠀⠀⠀⠈⠁⠙⢦⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠾⢒⣛⣚⡭⠭⠥⠔⠒⠒⠋⠉⠁⠀⠀⠈⠳⡌⢦⠀⠀⠀⠀⠀⠀⠀⠀⢣⣘⣆⣀⡤⣺⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⣧⠤⠤⣴⣒⣲⡾⠿⠏⠛⠛⠋⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣠⠀⢸⠀⢠⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⡀⢸⡇⠀⡦⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⡄⠸⠆⠘⠀⠸⠀⠾⠁⣴⠃⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣦⠈⠧⠘⠃⠘⠇⠰⠇⢠⠞⠀⣠⠟
⠀⠀⠀⠀⠳⢦⣄⠈⠳⠄⢁⣠⠤⠶⠶⠶⠦⣄⡈⠁⠴⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣄⠀⠹⠆⠈⠀⢀⣀⣀⣀⣀⡀⠀⠉⠠⠞⠁⣀
⠀⠀⠀⠀⠀⣄⡈⠛⠀⣴⠋⣡⡴⠖⠓⠀⠀⠀⠙⢦⡀⠘⠋⣁⣤⠶⠂⠀⠀⠀⠀⠀⠀⢠⣀⠙⠃⠀⣠⡶⠛⣉⣭⣽⣿⣿⣷⣄⠀⠀⠛⢁⣠⡴⠖
⠀⠀⣀⣀⣀⣀⡉⠁⣼⠁⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣉⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⠁⠀⢰⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⢉⣁⡀
⠀⠀⠀⠀⣉⣉⡁⠀⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣉⣉⣀⣀⡀⠀⠀⠀⠀⢉⣉⣉⠀⠀⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢉⣉⣁⣀⡀
⠀⠀⢀⣀⠉⠉⠁⠀⣷⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣸⠃⠀⠉⠉⣉⣉⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⢻⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡿⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠈⠉⠙⠳⠦⣤⣼⣧⡴⠖⠛⠉⠉⠉⠙⠳⠦⣴⣯⣤⡴⠞⠋⠉⠉⠀⠀⠀⠀⠉⠉⠓⠦⢤⣠⡈⠛⠋⠁⠀⠀⠀⠀⠈⠉⠛⢁⣤⡤⠴⠚⠉⠉
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠉⠙⠶⠤⣤⡤⠶⠛⠉⠉⠉⠙⠳⠦⣤⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⠤⣤⣤⠤⠒⠉⠉⠉⠉⠒⠦⢤⣤⠤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠶⠛⠉⠀⠈⠙⠶⠤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⠤⠖⠋⠁⠀⠉⠲⠤⠤⠖⠉








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

⠀⢀⣀⣀⡀
⠀⡞⠤⠤⠬⠭⠭⠭⠷⠶⠶⠶⡖⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠱⣤⣤⠶⠦
⠀⡧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠁⠀⠀⢰⡏⠉⠩⠭⣭⣉⡉⠉⠉⢹⠛⢢⡀⠘⢇⣀⡀
⠀⢱⠉⠉⢰⠤⠉⠉⠉⠁⠠⠰⣦⣶⣶⣾⡇⠀⠠⠤⠤⠔⠋⠀⠀⢸⠀⠀⡇⠀⠀⠙⠻⠤⣄⣀
⠀⠈⠒⠀⠋⠀⠀⠀⠀⠀⠀⠀⢣⠉⠉⢈⡇⠠⠤⠤⢤⣤⡰⣶⣶⣼⠀⢠⡇⢀⣀⣀⡀⠀⠀⠰⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠸⣇⣀⣀⣀⣐⣒⣳⣿⣿⣻⣶⠟⠁⠣⠼⣂⡙⡦⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣈⣵⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣠⠊⠁⠀⠀⠈⢻⡏⠉⠱⡄⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣸⠃⠀⠀⣇⣤⣶⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠵⣦⣤⣈⣁⣀⣠⣴⠟⠉⠉⠉⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠠⠍⠘⡄⡀⡍⠀⠀⠠⠍⡄⡅⠀⢼⠀⠀⠐⠌⢠⠀⠀⠅⠀⠎⠠⠍⡐⡅⠀⠀⢹⣿⣷⣻⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠠⠤⠤⠤⠬⠤⠤⠥⠤⠤⠬⠤⠤⠤⠤⠤⠬⠤⠤⠤⠥⠤⠤⠥⠤⠤⠬⠄⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠊⠀⠀⣀⣤⣴⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠀⠀⣀⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠁⠀⠀⠙⣦⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁
⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡌⠈⠀⠀⠀⢀⣀⣤⣾⠛⠉
⠀⠀⠀⠀⢾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣨⣷⡄⢠⣤⣶⠟⠋⣡⡇
⠀⠀⠀⣀⣼⣧⠀⠀⠀⠀⠀⠀⣀⠤⣂⣥⣴⠾⠟⠋⠀⠿⠋⠉⡟⠛⠛⠋
⣠⣴⠟⠋⠀⠈⢷⡤⠤⠄⠂⣩⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⡶⠷⠶⠶⠾⢶
⠹⣄⠀⣀⣠⣴⠞⢷⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠘⠿⠛⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⡤⠤⠤⠤⠤⠼⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⢨⠀⠀⠀⢀⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⡄⠀⠀⢸⡇⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣼⡇⠀⠀⢸⣇⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⠋⡇⠀⠀⢸⠻⣆⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣸⠇⢨⠇⠀⠀⢸⡇⢹⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢰⡏⠀⢸⠀⠀⠀⢸⡇⠀⢻⡄⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⡟⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⢷⡀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢠⡟⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠈⢷⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⢠⡿⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⠈⢷⡀⠀⢀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⢙⡷⠟⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡆⠀⠀⠀⠀⠈⠳⣞⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡾⠁⠀⠀⠀⠀⠀⠀⠈⢻⠛⠛⡟⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢳⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⡶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⡇⢠⠹⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇⢸⡆⠱⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢠⡇⠀⠀⠀⠀⡠⠀⢳⠈⣧⢀⠻⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠉⠀⠀⠀⠈⣿⠀⠀⠐⢣⠀⠀⠀⢣⡘⣟⠀⣠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⡻⡄⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠳⡌⠓⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⡶⠋⣁⣤⠶⢛⣡⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠿⢷⣄⣀⣀⣀⣀⣤⡴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣵⢶⣟⡩⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠼⠷⠚⠋⠉⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁
⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⣠⡆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⢛⣩⠄
⠀⠀⠀⢿⣠⠶⢷⡀⣶⣅⠘⣧⠸⣇⢆⠀⠀⠀⢀⠠⠴⠞⢫⡛⠉⡶⠙⣥⡀
⠀⠀⠀⠀⣭⠶⠋⢹⣷⡬⠇⣈⠧⠙⠻⠃⠀⠀⠀⢠⠶⣾⡯⠿⠿⠧⠶⠿⠟⠻⡀
⠀⠀⠀⠀⠘⠳⠞⠋⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣆⠻⠷⣤⣤⣤⡴⣶⠶⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠃⢀⣿⣦⣀⣀⣸⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠂⠀⣯⣭⣽⣧⠻⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡐⠀⠀⠀⠀⠉⠀⠀⡆⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠃⠀⠀⢸⠀⣧⠀⠀⢸⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡞⠀⠀⠀⢸⠀⣿⠀⠀⠀⣇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⠁⠀⠀⠀⢸⠀⣿⠀⠀⣀⣸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⡿⢯⣽⣏⠉⢹⠀⣿⠈⠉⣩⠼⣧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⡸⠀⠀⠋⢛⠧⣼⠀⣿⡴⠟⠁⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢸⠃⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⢣⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡟⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠘⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⠁⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⢣⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⠀⠈⣄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡷⠀⠀⠀⡄⠀⠀⠀⠀⠀⢴⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣏⡀⢶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⢀⡿⠀⢠⡞⢁⡴⠆
⠀⠀⠀⠀⠀⢀⠴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠘⣇⠘⠀⡸⠃⡰⢋⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠈⢠⠀⠀⠂
⠀⠀⠀⠀⠀⢡⡘⢗⠤⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⡙⡄⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⢀⠀⡴⢰⠂⠀⢀⠀⠀⠀⠶⢶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠑⠈⠀⠀⠀⠀⠀⠀⠤⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢄⢠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠘⠧⠠⢤⣀⣤⡀⣀⡔⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⡿⠇⠀⠀⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠤⠊⠰⠐⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠙⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠇⠈⠉⠑⠀⠀⠀⠀⠀⡄⠀⠈⠁⠀⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡁⠙⠃⠀⠁⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠢⠀⠤⢀⠀⠄⠛⠁
⠀⠀⠀⠀⠀⠀⢀⡞⣠⣄⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠸⠻⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠦⠤⡄⣴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⣄⣹⣶⡀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁⠀⠂⠀⢠⠄⠀⠈⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠙⠢⠝⢀⣀⠀⠐⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠠⡿⠶⠶⠶⠀⠀⠀⣿⣴⢷⣤⠐⠀⠛⠙⠀⠀⠀⠀⠠⡶⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠤⠀⡄⠸⠛⠃⠘⠀⠘⠙⠣⡁
⠀⠀⢰⡶⠐⡒⠈⠁⠐⡂⠀⠿⠈⠀⠀⠀⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠑⠄⣡⡙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠶⠁⠀⠈⠂⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠈⠉⢉⣠⠖⠀⠴⠦⠐⠀⠒⠻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢀⠀⣠⡄
⠀⠀⠈⠁⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠁⠀⠈⠉⠂⠈⠤⠉⠁
⠀⠀⠀⢀⣀⡀⢆⡠⠄⠆⠐⠒⠢⡀⠤⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡠⠚⠃⠀⣤⣀⠀⠀⠀⢀⡀⠀⠊⡀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠇⠀⠀⠀⠙⠋⠀⠀⠀⠈⠁⠀⢀⠇⠀⠀⠀⠀⠶⠀⠀⠀⢀⡀⠀⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠜⠀⠆
⠀⠀⠀⠀⠀⢦⣀⣀⡀⠀⢄⠠⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⢀⠀⠀⠤⠄⠀⠀⠀⠀⠤⠤⠀⢀⣀⣀⣀⣄⣒⣢⣀⣀⣀⣀⣀⣀⣀
⠐⠂⠂⠀⠁⠀⠀⠀⣀⠀⠀⢀⣀⣀⣀⠤⠤⠒⢋⣻⠿⠧⣄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠂⠀⠀⠐⠒⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠠⠔⠿⠓⠉⠁⠈⠀⠀⠀⣠⠢⠂⣠⠔⠒⠮⠥⠦⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢘⣈⣤⠤⠤⠶⠖⠋⠀⠨⠁⠀⢰⡐⠀⠢⡌⠀⠀⠈⠫⡫⡐⢄
⠀⠀⠀⠀⠀⠈⠕⠛⠉⢑⠒⠦⣤⡀⠀⠀⠀⢀⠀⠀⠸⠁⠀⠀⠎⠀⠀⠀⠀⠈⠌⠢⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠙⢦⡀⠀⠈⠂⠀⠀⠀⠀⣀⡘⠌⣄⡀⠀⠀⠈⡄⠈⢢⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠓⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⢁⠀⡀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈⠀⣰⠾⢷⣶⡄⠀⠤⢦⠷⢤⣶⣴⡮⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠉⠙⠒⠛⠲⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠑⠒⠤⢀⠠⠀⣩⠉⠓⢒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣨⢱⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠑⣴⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⢀⠄⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠈⠁⠐⣻⡿⠻⠾⣾⡟⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠚⠋⣀⠀⠠⠼⠇⢻⠷⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠉⠀⠉⠀⠀⠀⠀⠛⠉⠠⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐


⠀⣆⣂⡀⣄⠀⡀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑
⣿⣿⣿⣿⣿⣶⣷⣿⣶⣷⣴⣿⣷⣶⣎⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠂⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣤⣤⡀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⣴⣶⣶⣶⣶⣾⣉⣀⣽⣿⣵⣴⣦⡀⠀⢈⠿⣏⠀⠀⠀⠀⠀⡘⠉⢛⣛⣯⣿⣶⣶⣶⣶⣤⣤⣤⣼⣽⣿⣿
⠀⢸⣿⣿⣿⣛⣿⣶⣶⣶⣿⣿⣿⣷⣾⣶⣶⣶⣾⡿⠃⠀⠰⣄⣁⡀⣀⣼⣿⣿⣗⣒⣒⣒⣲⣷⣲⣶⣖⣿⣿⣿
⠀⠀⣿⡿⠿⣿⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠺⠟⠹⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠸⣿⠁⠀⣿⡇⠀⠀⠀⠀⢰⣧⠄⠒⠲⡗⠰⣦⣤⣤⡖⢤⡤⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣸⣿⠋⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠈⣿⣛⣛⣿⣟⣺⣿⣶⣾⡒⣈⠤⢀⠀⠀⠀⠀⠈⠛⠻⠷⠻⣿⣿⣿⡄⠀⠀⡴⠋⠹⣇⣿
⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠈⠉⠁⢘⣧⠿⡿⠶⠶⠎⠁⠀⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠣⣀⣰⠿⣻⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡖⢺⡏⣿⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⠀⠀⡠⠋⡫⢸⣿⣿⣦
⠀⠀⠀⢠⣴⣯⣺⣿⡛⣻⣿⣊⣷⣤⠀⠀⠀⠘⣇⢸⣷⠋⢾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠚⠻⠛⠛⠛⠋⠀⠀⡀⢳⠗⢺⡟⣛⡟⣧
⠀⠀⠀⢾⣶⢶⡏⢉⣥⣍⠉⣳⢶⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢰⡷⢾⡶⢾⣷⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⢉⡙⠛⠁⠀⠀⠀⠀⣀⣀⣀⣤⣤⡀⠀⡁⢿⣿⣾⣿⠇⠀⠲⠶⠶⢶⠶⠀⠀⠀⠀⠸⠷⠼⠧⠽⢿⣿⣿⣿⠃
⠀⠀⠀⢀⣀⣀⣀⣄⣸⣷⣤⣤⣤⣤⣤⣼⣧⣴⣶⣶⣿⣷⡀⠀⠁⠉⠉⠀⠀⠀⠈⢻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠣⠗⢸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⠉⠙⠛⠛⠛⠋⣿⣿⣿⢿⣿⡿⠛⠛⠻⠿⣿⣿⣿⠀⠀⢘⡟⣿⣿⡈⡏⠙⠛⡂⠀⠀⠀⠘⠛⠛⠁
⠀⠀⠀⠀⠀⢸⣳⣏⣯⣵⣺⠀⠀⠀⠀⠀⣿⣿⡟⠚⠛⣿⣄⠀⠀⠀⠀⠈⠉⠀⠐⠋⠀⢹⣟⡃⠙⠛⠚⠓⠶⠶⠶⠖⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠘⠧⠤⠜⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⠟⠂⠀⠀⠀⠀⠀⠀⠀⠚⠦⣾⠿⠧⠤⠤⠤⠤⠤⠤⠤⠤⠭⢭⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣻⣆⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⠿⣷⣤⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢟⣤⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣫⣶⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢛⣥⣾⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣩⣴⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢋⣤⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣩⣴⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣿⡿⢋⣴⣾⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⢧⠹⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⣫⣴⣿⣷⣞⣿⣿⣿⠿⡛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⠹⣥⣾⣿⣿⣿⣿⣿⡷⣟⡿⢷⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⣋⣴⣷⣌⣻⢿⣿⣿⡟⣡⡴⠋⠦⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢟⣡⣾⣿⣿⣿⣿⣿⣾⡿⢛⣿⠟⣫⣴⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⣫⣴⣿⣿⣿⣿⣿⣿⠟⠉⠑⣝⢓⠼⣟⡔⢻⡛⠿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢟⣡⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠻⠋⠻⣿⡄⠀⠙⢦⡀
⠀⠀⠀⠀⣠⣴⣿⠿⣫⣶⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡟⣷⣶⣦⠀⠙⢦⡀
⠀⠀⣤⣾⡿⢟⣵⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣸⣿⡏⢹⣿⣎⢧⠀⠀⠙⢦⣀
⠀⠈⣿⣫⣴⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⡗⡇⢠⡟⢿⡎⢧⠀⠀⠀⠙⢷⣄
⠀⠀⠉⠻⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⡏⣽⡇⣀⡇⠈⢿⡜⣇⠀⠀⠀⠀⢹⣯⡻⣄
⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣿⠃⣿⡇⣦⡇⠀⠈⢿⡘⣆⠀⠀⠀⠀⠙⠿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⡏⠀⣿⡇⠋⡇⠀⠀⠈⢷⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⠁⠀⠙⠓⠃⠃⠀⠀⠀⠈⠗⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡇⠘⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠰⠄
⠀⠀⠀⠀⡂⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡇⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠
⠀⠀⠀⠀⡇⠀⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡃⠈⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⡷⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡇⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠢⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠄⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⡀⠀⠀⢀⣀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡶⠀⠀⠀⠛⠛⠀⠀⠀⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠒⠀⠤⠂⠚⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⣤⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠰⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣁⠠⢀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠁⠐⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢀⠀⠀⠉⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠂⠀⠀⠉⠉⠁⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢀⠘⢠⡾⠋⠀⠀⠹⠋⠈⣷⣤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠢⡁⠀⠀⠀⠰⡆⠘⢹⠋⣀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⡌⠀⠀⠀⠀⠀⠐⠀⢀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢰⠛⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⠀⢀⣀⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⣆⠀⠀⠈⣀⣾⠏⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠣⠀⠀⡴⠂
⠀⠀⠀⠀⠀⢸⢂⣠⣾⣁⠙⠂⠴⠶⠿⢉⣦⣶⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣇⡀
⠀⠀⠀⠀⠀⠈⠋⠉⠛⡻⠟⠓⠶⣶⠛⠛⠁⠙⠃⢢⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡞⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⡟⠀⠀⠀⡄⠀⢈⡶
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠋⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⠉⠀⠀⠄⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣠⢈⠁⠀⠀⠀⠠⠀⠤⢲⡆⠸⢁⠄⠠⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⢠⠁⠌⠀⠀⣄⡀⢀⢤⡼⢉⡁⠀⠀⠄⠦⠀⠀⡇⠀⠀⠀⠀⠀⢄⡢⠀⢀⣀⡀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀
⠀⠀⠀⠀⠀⠋⠘⠀⠀⢰⢈⣅⣼⡟⠁⠀⠀⠀⠀⠈⢘⡄⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠟⢀⠈⠀⠁⠀⠀⠀⠉⠀⠀⡌⠙⠛⠻⢛⠏⠁⠉⠋⠁
⠀⠀⠠⠘⠁⠌⢰⠀⠀⠀⢾⠞⠋⠁⢀⠀⠀⠀⠀⠀⠈⠄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠂⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠦⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢈⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠲⡃⠀⠀⠀⠀⠀⠐⠂⠀⠀⣠⠎⠐⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⣼⡆⠉⠀⢗⡀⠀⠀⢀⠤⠀⠒⠒⠞
⠀⠀⠀⠀⠀⠢⣀⠀⠈⠐⢤⡬⣤⠀⡠⠀⠔⠈⠁⢀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⢠⣠⣆⠈⣠⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠈⠃⠀⣀⢄⣀⡀⠀⠀⠀⣰⣧⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡼⠀⠁⠛⠁⠀⠉⠁⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡤⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⢀⡀⠄⠀⠀⠀⠁⠀⠀⠐⢦⡤⠾⠋⠀⠘⣁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠂⠀⠀⣼⠫⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⣠⠌⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠶⠤⠀⢄⣴⠋⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠈⢡⡄⠀⠀⠀⠀⠛⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠀⠄⣼⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠿⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠐⠀⠈⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠲⣌⠁⡰⠛⠋⠉⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣦⣀⠀⠈⠀⠀⣼⠀⢄⡀⠀⠀⣀⠀⠐⢦⣶⡾⠋⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣾⣧⡧⢾⣦⣼⣶⠿⠷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡆⢾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⠀⠀⠀⠀⢺⣿⠓⢶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠂⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⣿⠏⠁⣤⠆⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣛⣿⡿⠀⠀⣾⠠⣤⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣤⣀⠀⠒⠀⠀⠔⠀⠀⠀⠳⠒⠶⠟⠷⢄⡀⣹⠀⠉⠀⠰⠄⠀⢀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⣽⣽⣿⣷⣿⣤⣽⣤⣴⣀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠀⠇⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠁⠙⠛⠖⠺⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠒⡀⠐⠒⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣌⡙⠛⠛⠛⠛⠛⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠉⠉⠉⠉⠁
⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣶⣶⣦⣾⣦⣾⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣧⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣼⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢻⡀⠀⠸⣇⠀⠀⣼⠇⠀⢠⡏⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠈⣷⠀⠘⣇⠀⠀⣿⠀⠀⣿⠀⠀⣸⠃⢀⡾⠁⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠘⣧⠀⠻⠀⠀⠛⠀⠀⠋⠀⠀⠟⠀⣼⠁⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠦⠤⣤⣀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⡀⡄⡆⢰⢠⢀⠀⠉⠙⠶⣄⠀⠀⠀⠀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠰⢦⣄⡀⠀⠀⠀⠉⠳⠀⠀⣠⠞⠁⢀⣦⠈⠇⠃⠃⠁⠈⠘⠘⠸⠑⣴⡄⠈⠳⣄⠀⠀⠞⠉⠀⠀⠀⣀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⠀⠀⢠⡞⠁⢀⡤⠋⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠉⢤⡀⠈⢳⡄⠀⠀⣤⠶⠛⠋⠁
⠀⠀⠀⠈⠉⠛⠛⠲⠶⢤⣤⡀⠀⢠⠏⠀⡈⠛⠀⠀⠀⣠⡢⠓⢉⣁⣈⡉⠚⢔⣄⠀⠀⠀⠛⢁⠀⢹⡄⠀⢀⣤⡤⠶⠖⠛⠛⠉⠁
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡟⠀⠠⠭⠁⠀⠀⢬⠊⣠⠞⠋⠉⠉⠙⠳⣌⠑⡥⠀⠀⠈⠭⠄⠀⢻⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠀⢸⠃⠠⠭⠍⠀⠀⠸⠇⢰⡏⠀⠀⠀⠀⠀⠀⢹⠆⠸⠆⠀⠀⠩⠭⠄⠘⡆⠀⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⠀⢸⡄⠐⣒⣂⠀⠀⢰⡆⠘⣇⠀⠀⠀⠀⠀⠀⣸⠇⢰⡆⠀⠀⣐⣒⠂⢠⡇⠀⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⣧⠀⠐⣒⡀⠀⠀⢚⢄⡙⢦⣄⣀⣀⣠⡶⠋⣠⡓⠀⠀⢀⣒⠂⠀⣼⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⢀⣀⣤⣤⠶⠶⠚⠛⠁⠀⠘⣆⠀⢑⣤⠀⠀⠀⠙⠕⡤⣈⡁⢉⣁⢤⠪⠋⠀⠀⠀⣤⡂⠀⣰⠃⠀⠈⠛⠓⠶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠀⠀⠈⢧⡀⠈⠓⣄⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⣠⠚⠁⢀⡼⠃⠀⠀⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠰⠞⠋⠁⠀⠀⠀⣀⡴⠀⠀⠙⢦⡀⠈⠟⢀⡆⡄⡄⡀⢀⢠⢠⢰⡠⠻⠁⢀⡴⠋⠀⠀⢦⣀⠀⠀⠀⠈⠙⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠁⠃⠇⠸⠘⠈⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠉⠛⠒⠶⠤⠤⠶⠒⠛⠉⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⢠⡟⠀⣴⠀⠀⣤⠀⠀⣤⠀⠀⣦⠀⢻⡄⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⢀⡿⠀⢠⡏⠀⠀⣿⠀⠀⣿⠀⠀⢹⡄⠀⢿⡀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⠁⠀⢰⡟⠀⠀⢻⡆⠀⠘⣧⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡟⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢻⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⣤⣴⣶⣿⣶⣦⣀⣀⣀⣀⣠⣤⣤⣤⠤⠤⠤⠶⠖⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠒⠐⠐⠒⠛⠛⠛⠻⢿⣿⡟⠋⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢰⡦⡀⠀⢰⠉⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣗⣿⣛⣛⣻⣿⣏⠛⠇⠋⠀⠆⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠃⣰⠁⠈⢿⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠐⡿⠛⠀⠀⣸⣿⣿⣿⣿⣿⢿⣿⢹⣿⢹⡿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣼⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣸⣿⣼⣿⣼⣷⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢿⡿⠿⠿⠛⠛⠿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣩⣿⣍⠉⠈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀
⠀⠀⣠⣼⣿⣯⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣛⠿⢛⣛⢛⣛⡋⣛⣛⠛⠛⣛⠛⢓⣒⣒⣒⣀⡀
⣾⣤⣬⣤⣷⣤⣤⣤⣾⣿⣿⣿⣷⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣾⣷⣾⣿⣾⣷⣦⣶⣷⣖⣿⣷⣦⣭⣄⢀⣀⣀⡀⠀⠀⢀⣀⡀
⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⡿⡿⢍⣉⣛⣿⣬⣴⣿⣿⣷⣤⣤⣭⣉⣉⣉⢉⣉⣻⡟⣿⣿⣿⣿⣯⠛⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⠉⠉⣹⠟⣿⣽⣿⣿⣿⠟⡛⠻⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⣿⡗⣲⣿⣿⣿⣿⣿⡇⠽⣯⠷⣶⣶⣾⣿⣿⣿⣿⣿⡞⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣶⣒⣋⣉⣩⣥⡤⠶⠶⢯⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠛⣛⣫⣩⣅⣠⣤⣤⡽⣿⣿⣿⣷⠛⠾⠿⠿⢿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⣠⣭⠭⠽⠭⠻⠿⣿⣿⣭⣽⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣾⣿⣾⡟⠉⠉⠹⠁⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻
⢀⣀⣈⣉⣤⡤⠤⠖⠞⠻⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠛⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧
⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠋⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠋⠀⢰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⠀⢸⠀⢀⡀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⡇⠀⡀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠸⡆⠸⠄⠼⠀⡾⠁⣤⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣧⠀⠇⠠⠇⢰⠇⢠⡄⠀⣰⠆
⠀⠀⠀⠀⢦⣄⠀⠳⡄⠙⢀⣠⡤⠤⠤⣤⣀⠘⠁⡴⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡀⠘⢦⠈⠁⠀⠀⣀⣀⡀⠀⠀⠋⢠⠞⠁⡀
⠀⠀⠀⠀⢀⣈⠙⠂⢠⡞⢋⣤⠶⠖⠀⠀⠉⠳⣄⠀⠞⢋⣠⡤⠆⠀⠀⠀⠀⠀⠀⣀⡙⠓⠀⢀⣴⠞⣋⣉⣽⣿⣷⣦⡀⠀⠰⠋⣁⣤⠴
⠀⠀⣀⣀⣀⣉⠁⢰⠏⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢋⣁⡀⠀⠀⠀⠀⢀⣀⣀⣈⡙⠀⢀⡾⢡⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠙⣉⣀
⠀⠀⠀⠀⣈⣉⠀⢸⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢉⣉⣁⣀⡀⠀⠀⠀⠀⣉⣉⡁⠀⢸⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢈⣉⣉⣀⣀
⠀⠀⣀⣀⣀⠀⠀⢸⡆⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⡟⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠘⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⠇⠀⠀⠉⣉⣉⡁
⠀⠀⠀⠀⠉⠛⠲⠶⠿⠶⠛⠉⠁⠀⠀⠉⠛⠲⠾⠷⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠠⠴⠖⠋⠁
⠀⠀⠀⠀⢤⣤⣀⠀⠀⠀⠀⣀⣤⠤⣤⣄⡀⠀⠀⠀⢀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠉⠛⠓⠚⠋⠁⠀⠀⠀⠈⠙⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠈⠙⠛⠓⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠲⠦⣄⡀⢀⣠⠴⠖⠶⣤⣀⢀⣀⡴⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣀⡀⣀⣀⠀⠢⢀⣀⠀⢀⣠⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⡀⡰⠢⣄⡀
⠀⠀⠀⠀⠀⢠⠛⠺⠷⣮⣧⡀
⠀⠀⠀⠀⠀⣯⣀⡀⠀⠀⠉⠛⠻⢶⣦⠤⣀
⠀⠀⠀⠀⢠⠃⠀⠉⠓⣆⢀⠀⠀⠀⠈⠙⠒⠭⣕⣆
⠀⠀⠀⠀⠘⠦⣄⡀⢠⠏⠉⠛⠳⢆⣀⠀⠀⠀⠀⠉⠙⠲⠦⣴⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣂⠤⣀⠀⠀⠀⠉⠒⢠⢀⡤⢀⡀⠀⠀⠉⠒⠮⣕⣢⠤⣀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠭⡒⢦⠀⠀⠻⣄⣼⡳⠺⢵⣆⠀⠀⠀⠂⡈⡝⡶⠽⢶⣥⠮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠓⡤⣄⣀⠈⠉⢓⠋⢸⠉⠙⣬⣠⠧⢇⡇⡇⡆⠈⠉⢻⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢰⠃⠀⠀⠉⢙⠞⢤⡀⠉⡢⠕⠈⢦⣙⠓⣧⣧⣷⠀⡼⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⠀⠀⢀⣤⣯⡤⠶⡇⠀⡇⠀⠀⠀⠀⠉⠀⠈⢛⡍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⣷⣄⣀⠀⠀⠀⠀⣿⢠⠋⢰⡾⣳⠀⠀⣠⢖⣙⣃⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣦⠴⠏⡯⠗⠀⠉⠁⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠳⡄⠀⠀⠀⠀⢸⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡾⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢠⡇⠀⣔⡻⣦⠀⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢭⣝⣳⢤⣈⣛⡁⠀⣸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠶⠧⠞

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⡉⠈⠀⠐⢣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠒⢄⠀⠀⠈⢑⠆⡠⠄⠀⢜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⡀⠘⠄⠈⢐⢀⠀⠡⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⣀⡀⠀⠣⡀⠳⠎⠁⣈⠆⢀⡷⠁⠀⠀⠀⠀⠀⠀⠤⣐⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡈⠧⠤⠚⠀⠘⣓⢈⠑⡆⠀⠀⠉⠈⠐⠂⠒⠒⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⢠⠀⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⡖⠀⠉⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⢢⡄⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢄⠈⡌⠀⠁⠢⠜⡄⠐
⠀⠀⠀⠀⠀⠀⢀⠠⠆⠀⠄⣈⠢⡁⠺⡄⠀⠀⠀⢁⢧⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⠄⠐⠂⢈⢠⣈⠆⠈⠐⠂⠉⠁⡀⠀⠀⡞⡩⠃
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠝⠁⠀⠀⠀⠀⠀⢀⠃⠮⠄⠁⢅⢄⠒⡄
⠀⠀⣾⠉⠑⣄⠃⠈⠐⠄⣴⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⠣⠕⠂
⠀⠀⠈⠛⢶⣿⡣⡀⠠⠑⠨⢃⣀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠙⠣⣒⠒⠤⠴⢏⡰⠳⠤⣦⣄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀⠙⢄⠀⣿⡥⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⠼⠀⠁⠒⠊
⠀⠀⢀⣀⡀⣀⠀⣀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠒⠢⢄
⠀⠀⠀⠀⢀⠔⠒⠒⠒⠓⠀⠀⠀⠀⠀⠱⠤⢄⡀
⠀⢀⡤⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢸⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄
⠀⠀⠈⠉⠉⠉⠀⠑⠒⠠⠤⢤⣀⠤⠤⠐⠒⠒⠠⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⣀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠊⠉⠳⡀
⠈⠉⠑⠒⠢⠤⢍⣉⡒⠒⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⢀⣀⡠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⢍⣑⣒⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠁⣀⣀⠤⠴⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠭⠵⢒⡴⠒⢒⡒⠢⢴⡪⠥⠒⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠴⠒⢒⣈⠭⠽⠒⡇⡸⠖⠚⠯⠭⠥⣒⣒⠲⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠶⣒⣂⠉⠉⠉⠀⣀⡠⠤⠐⠒⠉⠁⢀⣠⠤⠴⠓⠓⠲⠤⣄⣀⡀⠀⠀⠈⠉⠉⠑⠒⠂⠤⠭⠍⣉⣐⡲
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⣀⣀⣤⠤⠭⠽⢦⢤⣀⡀
⠀⠀⢰⢻⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣀⡠⠤⠒⡊⡉⠐⡀⠂⠀⠀⠀⠀⠀⠉⢝⠢⡄
⠀⠀⢸⡌⡇⡜⠁⠈⢣⠀⠀⠀⠀⠀⠀⣀⣸⢁⡠⠴⢒⠉⠀⠂⠈⠀⡆⡇⠀⠀⠠⠀⠀⠀⠀⠀⠰⠄⠦⠘⡆
⠀⠀⠈⡇⢻⠀⢀⣀⣠⠧⠴⠖⠒⠋⠉⠁⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⡀⠃⠁⠀⠀⠀⠀⠀⡀⠀⠉⠳⡄
⠀⠀⠀⡇⣞⣉⣉⣀⣀⣀⣤⡤⠤⠤⠤⡤⢾⠀⠀⠀⠀⠠⠀⠂⠈⠀⠀⠀⠄⠀⠂⠀⠀⠁⠀⠀⠀⡀⠂⢀⠔⢨⠇
⠀⠀⢰⢳⠻⡉⠀⠀⢀⠇⠀⠀⠀⠀⢰⠃⡼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢁⣀⣀⡀⣀⣲⣥⠔⠋
⠀⠀⣾⡞⠀⠑⢤⠠⠊⠀⠀⠀⠀⠀⠾⠼⠁⠈⠦⣤⣀⣀⣀⣀⣀⣀⢔⣭⣴⣾⠤⠤⠖⢺⠀⠀⣯⠁
⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡩⠊⠉⠀⠀⢀⣠⡫⢞⣽⠢⡀⠀⠀⠀⠀⠓⠚⠣⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⣐⣓⣒⠢⠭⠍⠒⠒⠒⠉⠀⠑⢌⢢⡀⠀⠀⠀⠀⠀⠙⡌⣆⣀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣷⣴⡶⠶⠶⠞⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⢻⣿⣿⣷⣶⣶⣶⣤⣤⣄⣀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠹⣿⣿⣿⣿⠻⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠛⢻⡿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡿⠿⢻⣿⡿⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠹⣿⣿⠏⠀⢻⡇⠀⠀⠈⣿⢹⣿⣿⣿⠟⢿⣟⠁⠈⠀⠀⠀⠀⠀⠀⠀⢙⡿⠻⣿⣿⣿⡇⢹⡏⠀⠀⠀⣾⠃⠙⣿⣿⡟⠁
⠀⠀⠀⠀⠘⣧⠎⠀⠈⢧⠀⠀⠀⣿⢸⣿⣿⣷⢀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣀⣸⣿⣿⡇⢸⡇⠀⠀⢰⠇⠀⠀⣩⡟
⠀⠀⠀⠀⠀⠘⢧⠀⠀⠈⣆⠀⠀⣿⢸⣿⣿⡿⠋⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⢿⣿⣿⡇⢸⡇⠀⢠⡏⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠘⣷⣾⣿⢸⣿⣿⣿⣤⣠⣧⠄⠀⠀⠀⠀⠀⠀⠀⠐⢸⣄⣤⣼⣿⣿⣧⢸⣿⣷⡟⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⢿⣿⣸⣿⣿⣿⣿⣿⣷⡀⡀⣠⡀⡄⣀⢠⡀⣀⣸⣿⣿⣿⣿⣿⣿⣸⣿⢏⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣤⣊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠻⣉⡄⠀⠀⠀⠀⠀⢰⡾⡄⠀⠀⣀⡤⢶⡾⠂⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⡿⣧⡀⠂⠀⣾⢹⡏⠀⠀⡌⢀⣴⠟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜⢇⣰⣾⣿⣇⣓⢀⣪⣰⡿⢋⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣈⠺⠉⣟⣿⡶⣋⣾⢙⡫⢔⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢏⡿⣽⡄⡸⣆⣹⣯⣤⢵⣤⣼⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠶⠶⣲⡟⢛⡛⢛⡅⠘⣋⠉⠛⢉⡋⡉⢀⣀⢀⡀⠐⢶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⠛⡆⠙⣷⠲⣡⣳⡎⡀⣿⣀⡀⣏⠀⡉⢼⡿⢾⡇⢰⠞⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⡧⣤⠬⣁⣉⠁⠈⡁⢉⠉⢁⠈⠉⠀⠈⠁⢈⠁⢸⠄⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠐⠖⢄⠀⠀⠀⣀⣠⣤⢤⣄⡤⣤⣤⡀⠀⠀⠀⡴⠲⡆⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠀⢀⡘⠶⣾⣿⣿⣿⠧⠉⠃⠉⠈⠿⣨⣿⣷⠜⢃⡄⢀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡜⣿⣷⣤⣉⠛⢤⣀⠀⠀⠀⣀⡤⠟⣉⣤⣾⣿⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣙⣇⠹⣿⣿⣿⣿⣶⣬⣿⠶⠾⣥⣶⣬⣾⣿⣿⡏⣸⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⢿⡀⣎⣿⠿⠝⢈⣤⡶⠶⢤⠀⠉⠻⢻⣿⣿⠁⡿⣿⣷⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣭⠎⠀⣧⣸⠃⡀⠀⣾⡃⢀⢺⠂⢹⣦⠐⢀⠉⣿⣠⠀⠙⣏⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡄⠀⣸⡇⠘⠀⢀⣯⣄⢸⡼⢀⠾⣩⣧⠀⢀⢚⣇⠀⢰⣧⡿⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡿⢻⡥⢞⣿⠄⡇⠀⢰⣿⣯⡥⠀⢰⠃⠀⡧⠀⢸⠘⣿⡳⢬⣿⠇⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢋⣴⣿⣿⡧⠡⢠⣿⣏⡤⠀⠰⠇⣠⠴⠂⠀⠈⣠⣿⣼⣤⣙⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠒⣫⣶⣿⣿⢿⡿⣷⡆⠘⢷⣬⠀⣠⣾⣼⣿⣳⡤⠀⢠⣾⣿⣿⣿⣿⣷⣌⠳⠊⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⣀⠭⠭⣭⣭⡤⣤⡭⠽⠮⣀⡹⡛⠿⠻⢛⠛⢩⣐⡼⠟⠫⣭⡭⣭⡭⠉⠨⡄⠀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⢷⡿⣾⣧⡀⠀⠹⠹⣶⣿⣿⣿⢿⠏⡟⠀⢀⣼⣿⣾⣳⠗⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⣿⣯⣽⡒⢷⢹⣿⣿⣿⠉⣼⣷⣊⣼⣿⡍⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢾⣭⣻⣇⢻⣿⡟⣸⣿⣿⡹⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠿⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡀⠀⣀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⡿⠛⢲⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠉⠙⠁⠈⠉⠀⠀⠟⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡼⠀⣦⣿⣿⣿⣿⣿⣦⣤⣜⣃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⡴⣦⡀⠀⡀⠀⣀⣼⡿⠁⠀⢀⣠⡾⢫⢿⢿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⣋⣾⣿⣿⣋⣠⠶⠋⣿⣧⣴⣾⠿⠋⢀⡀⣴⠁⣸⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠁⣀⡈⠩⢉⡛⢏⣩⣤⣤⣼⣿⣿⠿⣫⣵⢀⣀⣽⣷⠿⠚⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠚⣻⣿⣿⣿⣷⣤⣹⣿⣯⣳⡾⣾⣿⣿⣿⡿⠛⠋⢀⣀⣤⣤⣠⣤⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⣠⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣻⣶⣶⣶⣤⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⢿⣿⣿⣯⣍⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⢁⡀⣾⣴⣷⣼⣿⣿⣿⣋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⢙⣉⣄⣿⣾⣿⣾⣿⣿⣿⣿⡿⠿⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠻⠟⠛⠿⠟⠛⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⡀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠛⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠰⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠆⠀⢀⣿⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠛⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠛⠁⠀⡀
⠀⠀⠀⠀⠀⢸⣷⡀⠀⠘⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⢀⣾⡇
⠀⠀⠀⠀⡀⠈⠻⠇⠀⣠⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠸⠟⠁
⠀⠀⠀⠀⢳⣤⡀⠀⠘⠛⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠓⠀⢀⣤⡞
⠀⠀⠀⠀⠘⢿⣿⡄⠀⣠⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣄⠀⣠⣿⡿⠃
⠀⠀⠀⠀⣄⡀⠈⠀⡆⠿⠋⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⠙⠟⢠⠀⠁⢀⣠
⠀⠀⠀⠀⠘⣿⣿⣦⡘⡄⢰⣿⡇⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⣿⡆⢠⢃⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠛⠛⠃⠱⡜⠋⢠⣿⣷⢀⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⡀⣾⣿⡄⠙⢣⠎⠘⠛⠛⠁
⠀⠀⠀⠀⠀⠙⢷⣶⣶⣤⣘⢦⡘⣿⠏⢸⣿⣇⢸⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡇⣸⣿⡇⠹⣿⢃⡴⣃⣤⣶⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁⠙⢦⣀⠘⣿⡿⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⢿⣿⠃⣀⠴⠋⠀⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠶⣶⣶⣶⣶⣶⣮⠑⠺⢤⣀⣙⠻⠁⠀⠀⠀⠀⠀⠀⠈⠟⣋⣀⡤⠗⠊⣵⣶⣶⣶⣶⣶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⢛⣁⣤⣴⣶⢶⡉⠉⠉⠉⠛⣷⣾⠛⠉⠉⠉⢉⡶⣶⣦⣤⣌⡛⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠿⢿⣿⣿⡿⠟⠀⠀⠀⠀⠘⠃⠘⠃⠀⠀⠀⠀⠻⢿⣿⣿⡿⠿⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟
⠀⠀⠈⠛⠻⠿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⣾⣿⢿⡆⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠉⠀⠀⠀⣿⣇⣸⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢶⣶⣶⣶⡴⠞⠃⡌⠀⠀⠀⠀⣀⡀⠀⠀⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠚⠋⠁⠐⣻⣿⣄⠀⠡⠠⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⡄⠉⠙
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠈⠀⢀⣀⠀⠶⢦⣤⣄⣀⡀⢀⠔
⢠⣤⣤⣶⠶⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠇⠀⠀⠀⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣢⠞⣩⣷⣿⡗⠒⠒⠒⠒⠒⣲⢶⣏⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢃⡴⠞⠁⠐⠒⠳⢤⣠⠔⠋⠀⣀⡽⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠚⠉⠀⠀⠀⠀⠀⠀⠀⢳⣤⠴⠋⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠉⠠⣀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠤⣀⠀⠀⠐⠦⡀⠈⠁⡀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢒⣂⠤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄⠋⠉⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠀⠘⠀⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⣠⡴⠚⠉⠉⠀⠈⡁⢀⣀⠂⠠⠖⢀⠀⠀⠀⠀⠀⢤⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢺⠁⠴⠋⠀⢠⠂⠀⠀⠉⠉⠑⠀⠀⠀⠱⠀⠀⠀⢀⡈⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠴⠃⠀⠀⡆⠠⠐⠒⠀⠀⠀⠈⠉⠑⠄⠀⠀⣤⠣⠀⠀⡴⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡾⢠⠀⢰⣇⡀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠜⠉⡘⠘⠿⢠⠀⠒⠀⠘⠃⠤⠄⠴⠀⠀⠀⡐⢀⡄⠴⣼⠀⡠⠤⢄⢀⡔⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡘⠁⠀⠀⠀⠀⠂⠀⠀⡤⡀⠀⠀⠀⠠⠞⠟⠉⠀⠀⢀⠋⠀⣠⠋⠀⢀⣗⣈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠈⠁⣨⠀⢠⠇⢀⡔⠋⠀⢀⠗⠂⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⠀⠀⢈⠐⠀⠀⡀⠀⠀⠀⣠⢊⠁⠉⠉⢡⠀⠀⠀⠈⠀⠴⠚⠡⡤⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠜⠉⠉⠉⠙⢷⠀⢿⡀⠁⡒⣉⣋⡁⣸⠓⠂⠀⠙⠀⠀⠀⠀⡠⠀⣤⡂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠒⢴⠒⠊⠉
⠀⠀⠀⠀⡀⠄⢈⠀⠀⠀⠀⠾⠰⡀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⢤⠀⣤⠃
⠀⠀⡠⠊⠀⠐⠠⠀⠀⡀⡠⠂⠰⡇⠀⠀⠀⠀⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠌⠀⠀⠀⠀⠀⠘⢠⠞⠁⠀⠀⠈⢢⡀⢰⠈⢀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠊
⡎⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠈⡱⠇⠀⡠⠦⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⣀
⠳⠤⣄⣀⡠⠔⠥⡎⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠤⠄⡀⠀⠀⠀⡜⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠉⠐⠒⠶⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⠄⠒⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⠠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⣁⣔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⢀⡤⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⣀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡄⠀⠀⠀⠀⠀⡔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⣀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠄⠀⠀⠀⠠⠄⠦⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⠒⠒⠀⢉⢍⡵⠋⠉⠉⠉⠑⠰⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⡀⡄⠄⠐⠀⠀⢸⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⡔⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠄⢂⠩⢇⠀⠁⠀⠀⢀⠠⠁⢁⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠁⠀⠀⠀⡀⡁⠀⢀⠰⠐⠐⡀⡄⢆⢡⠸⡈⢆⠡⡰⡀⠣⣮⠧⡁⠠⢃⠂⠀⣤⠜⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠐⡛⣀⣀⣜⣔⡀⠠⠰⠀⢣⢡⣆⢂⢂⢢⠣⡡⡑⡘⢌⠢⡑⢌⢪⡣⣍⠪⣒⣌⡛⣾⡥⠶⠶⠟⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠉⠀⠈⠙⠛⠛⠛⠉⣿⣏⠙⠳⠷⠿⠷⠾⠶⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠊⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢻⣾⣷⡿⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⡾⢿⣿⡗⢔⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣸⠁⠂⢸⡇⠘⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢠⠇⠀⢀⢸⡇⠀⢁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡎⠀⠀⢸⢸⡇⠀⠈⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡼⠀⠀⠀⢠⢸⣧⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣸⠃⠀⠀⠀⢸⢸⣿⠀⠀⢲⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⣤⡞⠀⠀⠀⠀⢸⢸⡯⡄⠀⢈⡄⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢋⡟⠃⠀⠀⠀⠀⣸⣸⣿⡇⠀⠀⠑⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢀⣤⣧⣼⣿⡇⠀⠀⠀⢣⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠃⠀⠀⠀⣠⠞⠁⢸⢻⣿⣷⡄⠀⠀⠈⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡞⠀⠀⣠⡞⠁⠀⠀⠈⠚⣿⢼⠻⣄⠀⠀⠅⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣸⠁⢀⡾⠋⠀⠀⠀⠀⠀⠀⢹⣿⠀⠙⣥⡄⢲⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣘⢀⣇⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠈⢻⣼⡀⡇
⠀⠀⠀⠀⠀⠀⠀⢐⣭⣽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠀⠀⠙⡟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⡇⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠈⠉⢨⣿⣇⠭⢂⡀⠀⢁⠐⠈⣁⠈⠀⡀⠈⠁⠹⠷⠈⢌⠰⠒⡠⣯⣷⡀⡀⠂⡀⠄⠀⢀⠂⢐⠀⠀⠀⠄⢀⣀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠐⠀⠐⠈⠀⠂⠁⠀⠟⠋⠅⠂⠑⠊⠁⠈⠁⠁⠐⠈⠐⠈⠑⠈⠐⠁⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠠
⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠁⠀⠀⠠⠀⢀⡀⠁⠀⠀⡄⠀⠔⠁
⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⢀⠀⠀⠠⠤⠠⠆⠉⠀⡀
⠀⠀⠀⠀⠀⠐⠁⠠⠄⢀⠄⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡀
⠀⠀⠈⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣠⠄⠀⠀⠀⠐⠀⠄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠐⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⠇⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃⠀⠀⠃⡀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠰⡄⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡠
⠀⠠⠄⠀⠂⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢖⡉⠀⠀⠀⠀⠀⣀⠉⠤⠔⠉
⠠⠄⠀⠀⠀⠀⢰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠂⠐⡀⠂⠀⠀⠀⠀⠀⢄⣀⣈⣉⡀⠥⠤⢦⡠⠖⠋
⠀⠀⠀⠀⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⢀⠃⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠁⠒⠂
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠠⠀⠜⠁
⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠉⠦⢠⠊⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⡀⡠⠒⠀⠀⠀⡀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠣⢀⡠⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠐⠤⠀⢀⡀

⠰⢿⣿⣿⣿⣿⣿⣏⡉⠈⠑⠌⠉⠙⢿⣿⡷⠶⣄⣀⡀⠉⠉
⠀⠸⣿⣿⣿⡉⣹⢿⢿⣶⣶⣦⠀⠀⠀⠈⠙⠛⠀⠉⠛⠿⣶⣀⣀⡀
⣠⣴⣿⣿⣿⣿⣏⡀⠉⠙⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠦⠄⠀⢀
⡉⣹⣿⡿⣿⣿⠿⠿⣷⣦⡠⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⣄
⣿⣿⣿⡇⠁⠈⠷⢶⡄⠙⠷⣿⣿⣶⣾⣂⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟⣽⣷⣶⡀⡀⠀⠀⢀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣧⡉⣡⠀⠀⠀⢀⣤⣠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣶⣤⣼⡀⠀⠀⡆⢀⣀⣤⣬⣧⣭⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠙⠿⣿⣿⢿⣿⣟⠻⠏⣉⣉⠛⠂⢀⣠⣤⣤⣦⣀⡀⠀⠀⠀⢠⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡿⡿⣿⡿⣿⣿⡿⠟⠛⠋⠉⠉⠉⠙⠛⠛⠷⡆⠀⠀⠉⠉⠛⠉⣴⣷⣾⣛⠛⠿⠿⠿⠻⠿⠿⠿⢿⣿⠿⢳⡄⢸⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠉⠀⠀⠀⠈⠈⠛⠻⠿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠉⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⣾⣧⣿⣿⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠀⣾⣰⠀⠀⣾⠀⠀⠀⠈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⣿⣧⣿⣿⣿⣿⣿⣇⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡄⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣾⢰⣤⣷⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣧⣼⣇⣿⣾⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠴⠾⠿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⠻⠟⠉⠉⠛⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠛⣿⣿⣯⣽⣭⣽⣿⣿⣯⣷⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠿⠿⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣴⣤⣄⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿
⣿⣿⣿⣿⣿⣿⣿⡟⠿⢻⣿⠿⣿⠟⠉⠹⠛⠋
⠏⠈⠻⣿⠛⠿⠿⠇⠀⠸⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⣴⡆⠴⣮⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢢⣿⡁⠀⠀⠀⠀⢀⣶⣤⣀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡿⣿⣧⡀⢀⠉⠀⣙⡿⠿⣿⣾⣏⣶⣛⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠻⢶⣟⡛⢳⣴⣿⣦⢈⠈⣀⣿⡤⠀⠉⣉⠟⠛⢛⠿⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⠿⠛⠚⠿⠥⠼⠟⠹⠿⠿⠲⠿⠧⠖⠚⠀⠈⠀⠀⠈⠚⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣤⣴⠀⡖⣆⢰⡆⡄⢲⣆⣰⣴⣷⣖⢰⡆⢺⡶⠾⣿⠤⠾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠿⣿⣼⠇⠇⠘⣤⡾⢸⣿⣿⡉⢡⢸⡚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠱⣷⣿⣿⣶⣷⣸⣏⣿⢸⣿⣿⠁⡄⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠸⠗⢿⠿⠿⠿⠝⠩⢿⠹⠿⠿⠓⠟⠹⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠚⠀⠀⠀⠀⠀⠄⠀⠸⠇⠀⢰⠀⠀⣙⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣬⣴⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣤⠀⠀⠠⠶⠀⠐⠂⠀⠰⠆⠀⣰⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄⠀⠀⠀⢠⣾⣿⡄⠀⠀⣨⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣆⣀⣄⠀⠀⢣⣄⣀⣠⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣄⠀⠈⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣀⡀⠛⠿⢿⣿⣿⡿⢷⣦⣄⠙⢿⠟⠋⢁⣀⣠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣷⠦⠄⠀⠉⠃⠀⠉⢉⣤⣈⣿⣶⣿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⡦⠀⠀⢠⣿⣤⣦⣤⣿⣿⣿⣿⣦⢍⣛⠻⣿⡿⣃⠀⠀⢀⣤⠴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡿⣧⣤⠘⠊⠃⠸⠟⢏⣹⡶⣿⣬⡿⢾⣏⠿⠷⠘⠛⠂⠀⠼⡟⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⣿⣿⣷⣤⣶⣠⠀⡀⠐⠛⠄⠙⠋⢀⠠⠀⣶⣦⣤⣶⣾⣋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⣿⣿⣮⣹⡿⣿⣿⣦⢶⣤⢦⣤⢯⣥⣴⣾⣿⣟⣿⣿⣿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣿⣿⣿⣿⡟⣿⡥⠿⠞⠛⠚⣿⠾⠯⣽⡿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⣿⣿⣯⣙⣋⢀⠀⠀⠀⠀⢀⣉⡄⠀⠈⠙⠿⣹⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠿⡶⢯⣿⡶⢿⣶⣦⠀⠀⠀⠘⣷⠀⠀⠀⠀⠘⢧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡛⡿⢁⠘⠋⠙⢻⡏⢸⠛⢣⣴⣦⣼⡆⢀⡄⣠⡄⠈⢷⠿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡰⢃⣾⠛⡇⠈⠳⣾⠀⢦⡙⠘⠀⢸⡟⠡⣿⣿⠈⠁⣹⡁⠀⢸⡿⢿⡄⣄
⠀⠀⠀⠀⠀⠀⣠⠊⣴⣿⣿⢛⡇⠀⠀⠆⠒⣉⡁⢶⠀⠀⠸⣿⠟⠛⣵⡖⢹⠇⠀⢸⡟⢻⣿⣮⡳⣄
⠀⠀⠀⠀⢠⠞⣥⣾⣛⣿⣿⠟⡃⢠⣤⡠⠀⢙⣤⡾⣧⠀⠀⣹⠂⢀⢿⡗⣹⠀⠀⣸⢻⣿⣿⣿⣿⣬⣦⡀
⠀⣰⠶⢶⣯⣿⣿⣿⣿⣿⣿⣿⣛⡌⢻⡁⠀⠛⠛⠳⣌⠷⣤⣸⣇⡈⣿⣷⣤⡶⣀⣛⣿⣿⣿⣿⣿⣿⣿⣏⠦⣀⣀
⢿⣤⣘⡤⢯⣉⣉⡙⠛⠿⠿⢿⣿⣩⣦⠁⠀⢀⡁⠈⠁⠀⠈⣉⣯⣅⣜⣿⢋⣵⣉⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣿⣿
⠀⠀⠉⠀⠀⠀⠈⠉⠉⠓⠒⠖⠀⡝⢷⣿⢳⣯⣉⡋⠉⠉⠋⠉⢉⣉⣽⡟⣿⣼⢿⣯⡿⠿⠿⠟⠛⠋⠉⠁⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡛⢷⣾⣏⣽⣯⣽⣯⣹⣧⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠻⣿⣿⣿⣿⣯⣿⣿⣫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⣿⣿⣿⣿⣿⠡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠘⢿⣿⡟⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡸⠏⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⡱⢤⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⢻⡌⠉⠛⠲⠦⣤⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠒⠒⠒⠲⠶⠦⣤⣀⡀⠀⠀⠀⠀⠀⢀⡤⠶⠚⠋⠉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⡀
⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠉⠛⠶⣤⣀
⠀⠈⠛⠶⢤⣄⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠲⠶⠶⢤⣤⣀⣀
⠀⠀⣀⣤⠴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⢤⣄⡀
⠴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠖⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠲⠶⠦⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠳
⠀⠀⠀⠀⠀⣠⡤⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠦⣤⡀
⣤⣤⣤⣤⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣦⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠉⣿⣇⣀⡀
⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⡿⢩⣙⣛⠿⠷⣶⣦⣤⣀⣀
⠀⠀⠀⠀⢸⡿⠀⠀⠀⣿⠇⠀⠀⠉⠉⠛⠒⠶⠭⡝⣿⣇⣀⡀
⠀⠀⠀⠀⣿⠇⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠛⠛⠿⠷⣶⣦⣤⣀⣀
⠀⠀⠀⢸⣿⣀⡀⢀⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣿⣂
⠀⠀⠀⠈⠉⠛⠛⠿⠟⠈⠉⠙⠛⠿⠷⣶⣦⣤⣼⣿⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⣸⡟⢻⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣆⠀⠉⠉⠛⠻⠿⢶⣶⣤⣤⣿⣷⣾⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣿⡏⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⣿⡇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⣿⡇⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣿⡇⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠿⠿⣶⣿⣷⡶⠿⠟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⡀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠊⡠⠤⠤⠤⠤⠤⢄⣀⡎⣤⠔⠊⠡⠄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠘⢀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠄⠙⠃⠋⠀⠀⠒⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠠⢡⠰⡇
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⣡⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠦⠄⠠⠄⠊⣁⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠊⢀⠄⠂⠐⣠⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠔⠒⠒⠆⢰⠉⠀⠀⢂⠀⠀⠣⠀⠀⠈⠀⠀⠠⣔⢁⣀⠠⠍⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠊⠀⠀⠀⠰⠀⠀⠀⠀⠀⠆⠀⠀⢡⠀⠀⠀⡀⢐⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⣆⡠⡪⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⡀⣇⣀⡀⠤⠒⢃⡀⢔⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⠉⠉⠉⠉⠉⠉⠀⣁⠠⠤⠒⠊⢩⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠁⠀⠀⠀⠀⠀⠉⣉⣁⠀⢄⣴⠞⢓⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣈⡧⠤⠤⣧⠴⠚⠉⠀⠀⠈⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡸⠀⠀⠣⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⠙⢒⢂⡀⢀⣀⣀⣂⣠⣤⣀⢠⣀⡀⠀⠀⠐⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⢸⠤⠈⠉⠀⠉⢻⣦⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⢾⡄⠰⠏⠋⠙⢿⣿⣷⣸⣿⣥⣤⣬⣭⣉⣉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⡀⠀⠀⠘⠀⠀⠀⡰⠀⢻⠶⠄
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⢷⡼⠀⠀⠈⡀⠰⠞⠋⠈⠙⢢⣀⠜⠂
⠀⠀⠀⠀⠀⢠⡊⠀⠀⠀⠀⠀⠐⠁⠀⣇⠀⠠⡀⠀⢀⡴⠋
⠀⠀⠀⢀⡜⠉⡀⠀⠀⠀⡰⠊⠀⠁⠈⠉⠑⠒⠒⠚⠋
⠀⠀⠀⣾⢀⡦⢏⣤⠄⠠⠀⠀⠘
⠀⠀⠀⣿⣭⣿⡿⢛⠀⠀⠀⠀⢠
⠀⠀⠀⠹⠟⠉⣎⢀⡸
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠌⠘⠃⠉⣩⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡃⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠪⠅⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠑⣄⠀⠀⠀⠀⢸⠋⠁⠒⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠈⠇⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣠⠜⠃⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠛⠓⠒⠒⠦⠀⠀⠘⢦⡀⣇⠀⠀⠀⠀⠀⢀⣂⣀⣀⣀⠀⠀⠀⢀⣤⡀⠀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠀⠀⠀⠀⠀⠀⢤⣒⠄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡀⡀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⡠⠠⠤⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⢤⡤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣀⣠⠁⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢄⡀⠹⡈⠀⠈⠑⠣⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠉⠢⠱⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⡈⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢈⣁⣰⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠘⠤⠔⠀⠢⡀⢄⡘⢲⡄
⠀⣴⡋⠙⠲⡀⠘⠀⠉⠂⢄⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⢦⣱⠤⠃
⠀⠈⢷⠀⢀⣰⣧⡀⠀⠀⡦⠘⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠐⠝⢣⣕⣄⡰⠁⠀⠙⠂⢠⢤⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠂⢈⠑⠒⠀⠚⠇⠰⠛⠛⠀⢵⡆⢀⠂⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡀⠀⠈⢦⠀⠀⣿⣓⡰⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡑⣄⠃⠙⠤⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⢀⣤⣤⡤⠤⢀⣰⣤⡀⠠⠄⢀⣤⡤⠄⣀⣤⣄⡀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⢉⣁⣠⣤⣀⣉⠙⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡈⢻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠙⠻⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣷⡀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡄⢹⣿⠀⢼⣿⡟⠻⣿⣿⣿⡿⠃⣰⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⢀⣿⡆⠸⣿⣿⣄⠘⢿⡿⠁⣼⣿⣿⣿⠃⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠋⢀⣾⣿⣿⡄⠙⣿⣿⣷⡀⢀⣾⣿⣿⡿⠋⣠⣿⣿⡟⢁⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠙⣿⣿⣿⣦⠈⢿⣿⣿⣿⣿⣿⡟⢀⣼⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⡄⠹⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠈⠋⠉⠉⠉⠉⠀⣸⣿⠟⢉⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣤⣤⡄⠹⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠿⠿⠿⠀⠙⠿⠿⠟⠋⠀⠿⠟⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⠀⠀⠀⣄⣀⣠⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡾⠋⠀⠦⠴⢂⣀⣉⣭⣭⣴⣶⣿⣻⣷⣴⣶⣯⣿⠂⠉⢻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠿⠿⠿⠿⠃⠀⠛⠻⠛⠁
⠀⠀⠀⠀⣠⣤⣴⣶⡦⠦⣀⣒⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣾⣿⢹⡇
⠀⠀⠀⣰⣭⣭⣶⣶⣿⣿⣿⣿⣿⣿⠛⢿⡿⢿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡿⠾⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠝⠋⠉⠉
⠀⠀⠀⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⣿⣿⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠈⠛⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣤⣤⣾⣙⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⠙⠛⠋⠉⡉⠀⣀⡀⣠⣤⣤⣤⣶⣾⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣴⣶⣿⣶⣼⣿⢰⣿⢿⣿⠋⣿⡇⢸⣿⠇⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢰⣿⣇⣙⠋⣿⣷⣼⣿⢸⣿⠀⣿⡇⢸⣿⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⢈⣛⠛⣿⡇⣿⡏⣹⡿⣾⣯⣸⣿⠀⣼⡿⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣶⡿⠻⠿⠁⠿⠇⠙⣛⠛⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡅⢀⣤⠀⠴⠆⠸⠯⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠛⠛⠛⢛⣛⣙⠇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣠⢢⡞⣰⣿⣿⣿⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⣠⣶⣄⠀⠀⠀⠈⠀⠀⢠⣴⣾⣿⣿⣾⡇⣿⣿⣿⣿⡀⢈⣧
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⢄⣴⣺⣽⣿⣿⣿⣿⢹⣷⠻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣠⣴⣳⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣧⡙⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠁⠀⠀⠀⣀⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⡿⣶⣬⣍⠉
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠓⠒
⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⣿⠇⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣿⣿⣿⣦⣝⠻⠿⠿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣿⡀⢀⣠⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⡇⠻⣿⠿⠟⠻⠿⠛⠁⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸
⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢀⡏
⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⢠⣿⡿⠁⠉⠛⠋⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣠⣆⡀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠘⣿⣷⡙⣿⣄⠀⠀⠀⠀⠙⠏⠁⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣼⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣧⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠙⢿⡀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣤⣀⣀⣀⣈⣻⣧⣀⣀⣠⣿⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠠⠀⠰⠆⠰⠆⠰⠦⠀⠀⠈⠀⠀⠀⠀⠀⠈⡀⡀⠁⠈⠀⠀⠅⠀⠁⠀⠀⠰⠆⠀⠆⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠲⠶⠦⠤⠤⠤⢴⣶⡖⠀⢀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠶⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢳⠙⠲⣄⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⢀⠇⠀⢸⠀⠀⠀⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠈⣿⠉⠙⠒⠢⠦⣀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⡠⠤⠠⠤⡀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢠⠊⠀⠀⠀⠀⠈⢆⠀⠀⠸⠀⠐⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⡌⠀⠉⠉⠉⠐⠒⠂⠤⠄⣀⣀⡞⠀⠀⡜
⠀⠀⠀⠀⠀⣀⣀⣸⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⠀⠀⢰⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣷
⠀⠀⢀⡔⠉⠀⣿⣷⠉⠛⠿⠷⢶⣶⣶⣶⣶⣾⠟⠁⠀⠀⠀⠐⢄⠀⠀⠀⢀⡴⠁⠀⠀⠸⠀⢰⣀⠠⠤⠤⠤⠤⠤⠤⠔⠒⠒⠲⡒⠂⡟⠁
⠀⠀⢸⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠁⠀⠀⠀⠀⡟⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣰⠁
⠀⠀⠈⢦⣀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠙⠻⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⢀⡿⠓⠂⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠘⠷⠶⢤⣾⣧⣤⣤⣄⣸⣇⣀⠈⠉⠁⠒⠒⣶⠄⠤⠤⠤⠤⣤⠔⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠙⣿⠉⠛⠛⠛⠛⠿⠿⠶⠶⠄⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠾⠿⠶⠶⠶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣟⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢦⣼⣿⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⡗⡂⣸⣧⡴⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠿⢿⣿⣿⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣟⣛⣿⣿⡶⠿⠶⣭⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⢠⣶⠄⣿⣿⣟⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣽⣿⣿⣿⣿⣿⢻⣿⣿⠠⣶⡄⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣿⡀⠸⢿⣿⣿⢿⡟⣿⣿⣿⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠟⢾⡿⣿⣿⡿⠇⢀⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⣧⡀⠀⠈⠀⠘⣿⣿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⡛⣿⠃⠀⠁⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣀⠀⠀⢻⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⢰⡟⠀⠀⣀⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣼⣿⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⣿⣿⣿⣃⣼⣧⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣯⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠟⠹⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⡟⣾⠏⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠀⠀⢠⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣶⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠀⠸⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⣤⣤⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣭⣭⣭⣭⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡋⣉⣉⣉⣻⣿⣿⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⢶⣾⣿⣿⣿⣭⣍⣽⣿⣿⣿⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⠀⣠⣴⣶⣶⣦⣄⠀⣠⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡏⣸⣿⣿⣉⡍⣿⣿⣧⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣷⣿⣾⡇⢿⣿⣿⣯⣴⣿⣿⡿⢸⣷⣿⣾⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⣼⡿⣿⣿⣿⡿⠟⠈⠻⢿⣿⣿⡿⠟⠁⠻⢿⣿⣿⣿⢿⣧⠀⣠⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣻⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢙⣻⣟⡋⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⣿⣿⣿⣿⣿⣿⠻⣿⣿⣦⡀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⢠⣶⣿⡿⠋⣿⣿⣿⣿⣿⣿⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠻⠿⠋⠀⠀⣿⣿⣿⣿⣿⣿⠀⠈⠻⠿⠇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⠿⠋⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⡿⠋⣿⣿⣿⣿⣿⣿⠙⢿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⣿⠀⠀⠀⠙⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠙⠻⠋⠀⠀⠀⣿⣿⡏⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡠⡀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢠⡆⠀⠀⠊⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡸⠊⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⡀⠀⠀⠀⠀⠀⢀⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣰⠀⠀⠀⠀⣠⡊⠭⠊⠀⡆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡌⠀⡰⠉⢆⠈⢦⠊⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠪⡐⡡⠀⢆⠀⠀⠀⠈⠈⠀⠀⠈⠢⡀⠀⠀⠀⣀⢴⣉⣈⡈⠐⠂⢄⡀
⠀⠀⠀⠡⠱⢴⡖⣸⡑⠒⠀⠰⠄⠀⡀⢀⡰⠚⠉⠉⠁⠀⠀⢃⠀⠀⠈⠑⠠⣈⠢⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠁⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠑⢤⠃⠀⠀⠀⢄⠙⠄⠀⠀⠀⢀⣀⠠⠄⠂⠒⠒⠂⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢸⠐⢍⢧⡈⡢⠝⠀⠀⠀⠉⠉
⠦⡀⠀⠀⠀⢸⠁⢠⠀⠑⠤⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠨⣀⡄⠀⠹⡄
⠐⡌⠠⡀⠀⡜⠀⠘⠣⡀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢄⠀⠸⠤⠤⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠈⠢⣈⠲⠀⢀⠃⠀⠈⠑⠄⣀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢣⠠⡇⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⢑⠑⡼⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠠⠀⢀⢀⣀⣀⣀⠔⠠⡀⠀⠀⠄⡈⢳⡈⠢⡀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠀⠉
⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠢⣤⢀⠀⠀⢣⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠊⠀⠀⠀⠱⣀⠂

⠀⣠⣞⣛⠛⠛⠛⠒⠶⢤⣄⣀⡀
⠀⠀⠉⠉⠙⠛⠲⠶⢤⣤⣀⡉⠙⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⣤⣌⡙⠓⠦⣤⡤⣤⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣦⠐⢨⣭⣥⡤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣬⣉⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠀⠘⠓⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠳⣄⠀⠀⠀⠀⠀⣠⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠳⣄⡀⢀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⡀⠀⠀⠀⢙⣦⣄⠈⠻⠏⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠠⠀⠀⢼⡃
⠀⠀⠀⠀⣀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠈⣦⡀⠙⠳⣄
⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠚⠛⠛⠉⠉⠉⠉⠉⣰⣏⣀⣀⡼⠁⠙⢦⡀⠈⠛
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠉⠛⠛⠓⠓⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠿⠿⠇⠰⠶⠶⠶⠶⠶⠿⠿⠇⠸⠏⠷⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠷⠶⠶⠆⠰⠶⠶⠶⠶⠶⠶⠶⠶⠰⠶⠶⠶⠶⠶⠾⢧⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⡀⣀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⠀⠀⢀⠀⡀⢀⡀⢀⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠂⣿⠘⡏⢽⣿⢸⣏⢙⡇⢹⠀⣿⢸⡟⣿⡛⣿⢸⡟⡏⢹⣿⢈⡇⢸⢹⡟⣿⢸⣿⣿⢻⡏⢸⡇⣹⡏⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠲⣧⣰⡇⢰⣿⢰⡇⢸⡇⣼⢀⣏⣸⠇⣿⠁⡟⢸⡇⣷⢸⣿⢸⡇⣿⢰⡇⣿⣼⣿⣿⣸⣷⣼⡿⣿⡇⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⠀⣿⢸⡏⣿⣿⢸⣧⣸⡇⣿⢸⠋⣿⢀⣿⣷⡇⢿⡀⡟⢸⣿⢠⡇⢿⢸⣷⣿⣿⣟⣿⢹⡗⣹⡇⣿⣇⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣿⣸⡷⣿⣿⣿⣾⡿⡇⣿⢸⠀⣿⠸⡏⢸⡇⣿⣤⡇⣿⣿⢸⡇⢿⣿⣞⠣⣿⣿⣿⢼⢇⣿⣧⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣿⣿⣧⣧⣿⣼⣤⣯⣤⣧⣬⣤⣿⣿⣧⣿⣿⣼⣷⣾⣿⣿⣶⣿⣾⣿⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠈⠉⠉⠁⠹⣧⣤⣽⣾⣏⠉⢹⣿⣥⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣿⣛⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣰⣾⣿⡿⢻⣷⣀⣺⡶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣿⣷⣿⣽⣿⣿⣽⣿⣴⣿⣿⣿⣿⣷⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄⣿⣷⣾⠷⠿⠋⠋⠉⠈⠁⠀⠘⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⢹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣇⣻⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣏⠻⣧⣿⣤⣶⣤⡤⢦⣾⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣿⡿⠿⡟⢫⡶⢿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⠻⠟⠋⢰⠛⠋⠙⢻⡏⠉⠉⠉⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣧⡀⠀⣾⣿⡀⠀⠀⣿⣿⡦⠀⣠⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢳⡈⠻⣦⣿⡿⠛⠛⠛⢛⣿⡿⠛⠁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⢋⣿⢾⡇⠀⠆⠙⣷⣤⣤⠞⠛⠀⠀⠀⣴⣿⣴⠷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⣆⠀⢠⡟⢳⠟⣇⡾⢁⡾⢿⠀⠀⠀⠉⠁⢠⡬⠀⠀⠀⢠⣿⣆⠉⠀⢸⣿⣿⣤⡀⠀⠀⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⠳⣄⣧⠸⠈⡟⣠⣾⠷⢾⡇⠀⣤⡄⠐⠂⣄⠈⠀⠀⣾⡏⢿⣀⠀⠀⠀⠹⠿⣿⣤⠞⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⠀⠙⢿⣶⣿⣿⣉⣽⡻⣿⢻⣰⣿⣿⣏⡀⠀⠀⠀⣼⣿⡿⠆⠙⢿⣷⡀⢠⡴⠋⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠳⣮⡽⠟⣫⣥⣾⠟⠓⠛⠟⠛⠛⠿⣶⣿⣝⣻⣿⠀⣀⣹⠟⠁⢀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⠀⠄⠴⠤⠈⢳⣾⡿⠏⠁⢀⣠⠤⠀⢤⣤⣄⣀⠉⠛⢿⣿⣿⠛⠁⠀⠀⠬⠁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⡁⠀⠀⠀⠀⣀⣰⣿⠏⢄⡴⠆⣙⡥⠶⢶⡶⠞⣻⡙⢷⣄⠀⢻⣿⣇⡶⢄⡈⠀⠒⢿⣄
⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠀⣴⣶⣿⡏⠀⠜⢡⡶⠋⠀⠀⢀⣰⠟⢙⣧⠀⠹⣆⢀⢻⣿⡟⠉⠀⣀⢀⣄⡿⢷⣄
⠀⠀⠀⢠⣾⣯⣭⣤⣦⣤⣤⣤⣤⣤⣤⣬⣿⣿⠀⢰⠀⣾⠁⠀⠀⡸⠟⠋⠀⠛⢻⡤⠀⣿⠐⠈⢿⡿⠀⠈⠋⠘⠻⠀⠂⠙⢷⣄⣠⣄
⠀⠀⠀⠀⠙⠉⠉⠉⢩⡏⣿⡍⢉⠉⣽⡍⢿⣿⠀⢸⣧⠙⣦⣄⡀⠁⠀⠀⠀⢀⣾⠁⠄⣿⡇⢀⣾⣿⠶⣶⣶⡶⣾⢶⣶⠶⠶⢿⣭⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣌⢷⣄⠀⡹⣷⠸⣿⡄⠈⢿⣆⠘⡿⠃⠀⢀⣶⣶⣾⣿⠛⢠⡿⢃⠀⣼⡿⢦⠙⣠⠀⢁⣼⡏⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣌⠀⣿⠀⡰⣿⣦⣿⣿⣆⠈⠻⢦⣳⡖⢶⡾⠇⠀⠁⣀⣴⠟⠁⢛⣼⡟⢷⣼⢧⣿⠐⢋⡿⠃
⠀⢀⠀⠀⠀⠀⠀⠀⠻⣤⡈⠉⣿⡴⠛⠋⠐⢿⠻⣿⣷⣄⡀⠉⠛⠳⠶⠶⠷⠟⠋⠁⠀⣠⣾⡟⠳⣤⣥⡞⠽⢛⡿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⣹⡶⠞⠁⠀⠀⠉⠀⢠⡙⠛⠻⢿⣿⣦⣤⣠⢀⡀⣀⣀⣤⣶⡿⠛⠉⡀⠀⠈⠙⢷⣟⡉
⠀⠈⠀⠀⠀⠀⠀⣴⣟⣁⣤⣤⣄⣀⣀⣀⣀⠀⠂⠀⠀⠀⠙⣿⣿⡿⠿⠿⢻⣿⣿⠆⠀⠀⠆⠀⠀⠂⠀⠀⣹⠟⠳⢦⣤⣄
⠀⢸⠀⠀⠀⠀⠀⠙⠛⠋⠉⠉⠉⠉⠛⠛⢿⣆⠀⠄⡀⠀⣼⣿⣿⣷⣀⣾⣛⣿⠀⠀⠀⠀⠀⣲⣤⣴⣶⣶⣷⠶⠶⠾⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡤⠄⢰⢼⣿⣉⣿⣿⣿⣿⣟⣹⡇⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⣾⡟⣸⣿⣿⣿⠻⣿⢧⣷⠀⠀⣴⡟⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢰⣿⠟⣻⢉⡿⢿⡆⠙⣦⣹⡀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡇⠀⢿⣜⣇⢸⣿⠀⠈⢻⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠿⠛⠻⠇⠀⠀⠈⠉
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠁⠀⠈⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡠⢦⠤⠶⠒⠐⠲⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣀⠩⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣤⡿⢀⣆
⠀⠀⠀⠀⠀⠀⢰⣿⡀⠸⠆⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠀⠀⠀⠉⠉⠉⠓⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣇⠃⢒⠃⢄⣠⠃⠈⠀⠀⠈⠁⢀⠀⠀⢐⠻⡳⡄
⠀⠀⠠⡄⠀⠀⠀⠉⠻⠿⣮⠼⠘⠁⠀⢀⡄⢀⠀⠀⠀⠀⠀⠈⡇⣃⢱⠀⠰⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡐⠊⠉⢨⢰⣍⣷⢄⢈⣇⠀⠳⡹⣿⣸⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⣸⣥⡴⠷⣶⣾⠿⠷⢧⣸⠟⢻⡿⠛⠿⠿⠒⠁⢀⣾⠐⠂⠀⠈⠘⡋⠀⢀⡿⠂
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠉⠉⠀⠀⠀⣀⣄⠀⠀⠙⠓⡗⠁⠀⠀⠀⣀⣴⣾⣯⡀⢀⣠⡴⢞⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠙⠻⠀⠀⣦⣼⠍⠀⠀⠀⢾⢿⣧⠘⠛⣻⡭⠒⠋⠝
⠀⠀⠀⠀⠀⠀⢀⠀⢐⠈⠙⠳⢶⣄⠀⠀⠀⠀⢉⣼⣄⢦⡀⠀⠀⠀⠀⠀⡀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⢠⡟⡈⠁⠀⠀⢀⡴⢚⢧⠀⠀⢖⡎⢙⣓⠂⠴⠂⠀⢠⡾⠁⠀⠁
⠀⠀⠀⠀⠀⣾⠧⠄⢀⣠⣴⡾⠁⠀⢺⠆⠀⢠⡟⢸⣿⠀⢻⠀⠀⠸⣷⣀⠁
⠀⠀⠀⠀⠀⢹⣧⠾⠝⠟⠉⠀⠀⣠⡟⠀⠀⢾⠃⢸⣿⠀⠘⡇⠀⠀⠌⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⢀⣀⡾⠛⠀⠀⢠⠟⠀⠸⡿⢄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣠⡤⠊⠀⠃⠀⠑⢬⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⡀⠀⠀⠀⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠇⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠰⣄⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡢⠀⢠⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠀⠠⠀⣦⣠⠔⠀⡀⠀⠘⠆⠉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡛⠀⠼⠀⠀⠤⠀⠸⠛⢀⡀⠀⠁⠀⠀⠔⣈⠁⠀⠀⠀⠠⢖⣀⣰⠜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠁⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⡀⠀⠉⠈⢱⣌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠒⠰⢿⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⣛⣓⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡄⣸⣷⢸⣇⠷
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⣬⠍⠀⠀⠈⠀⠀⠀⠆⠒⠋⠀⠂⠰⡤⠴⢚⡁⢰⠀⠩⢾⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠐⣏⠀⠀⠀⠓⣚⠁⠙⠂⢑⣂⠀⢁⠀⢤⠴⢖⣼⣌⡐⠆⢠⣿⡿⠿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠁⠠⣿⡈⢉⡳⢬⢟⣻⢿⡿⢿⡿⣶⢶⡿⢿⣿⣿⣶⠀⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣀⣨⣿⠖⠙⣿⣷⠟⠷⠾⣰⠄⠸⣿⣿⠛⠷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡧⠒⠒⣧⡇⢤⣌⣼⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⢳⢶⡦⠁⣀⡌⠙⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⡗⡎⢈⣷⠘⡄⣠⣿⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢢⣿⣿⢹⠀⣅⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠻⠃⠺⠂⠀⠀⠉⠉⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡀⠀⢀⠀⠀⠀⠀⢀⣀⣠⡀⠀⠀⠘⣠⣤⣀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠤⠤⠴⠤⠴⠾⠿⠧⠴⠌⠛⠓⠲⠷⠾⣿⣟⡡⠞⠒⢚⣾⣿⣿⣶⣆⣀⣛⣛⣺⣷⣶⣶⣶⣶⣤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢠⡶⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣄⡴⠋⠉⠉⠛⢶⣄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠉⠁⠀⣠⡟⠁⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡤⠖⠛⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢿⡄⣀⠔⠊⠁⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿
⠀⢀⣀⣀⣸⣿⣁⢀⣀⣀⣤⠴⠖⠛⠉⠁⣀⣤⠴⠖⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⣴⠏
⢰⠟⠁⠀⠈⠉⠉⠉⠉⠀⠀⢀⣠⡴⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⣿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣀
⢹⡆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠶⡟⠋⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠞⠛⠉⠙⠈⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⠂⡡
⠀⠀⠀⠉⠛⠛⠓⠚⠛⠛⠛⠉⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⠈⠉⠙⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⣿⠟⠁⣼⠿⠟⣫⡿⠿⠛⠁⠀⠰⡏⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠈⠛⠄⠀⠀⢹⣿⠇⠀⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⠀⢠⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠏⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣯⠈⢣⠀⠀⠀⠀⢀⣏⠈⠇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠼⠀⢀⣾⣿⣿⣿⣿⣿⠋⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⣤⣾⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠋⠉⠀⠀⠤⠭⠋⠀⠀⠀⢻⣇⠀⣀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡆⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢦⡈⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⢀⡼⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⢰⡟⠋⠉⠉⠀⢙⠁⠀⠀⢀⣴⠏⠀⢻⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠉⠻⣦⣀⠀⠀⠀⠀⠦⠄⠒⠀⠀⠈⠀⢀⣴⠟⠁⠀⠀⢀⡙⠓⠰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠁⠀⠀⠉⠒⢤⡙⠻⠶⣤⣤⣀⣀⣀⣀⣠⣴⠾⢿⠁⠀⠀⠀⠀⣸⠋⠁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢯⡈⢢⡤⠚⠃⠄⠀⠀⠈⠲⣄⣀⣈⣉⣉⣉⠟⠛⣤⠠⣄⠁⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⢹⠃⠀⠀⠀⠀⠈⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠠⠴⠦⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⠞⠚⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣠⡶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢦⠀⠀⠀⠐⣿⠙⠛⠿⠿⠷⣶⣶⡶⢿⡀⠀⢨⠛⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⢳⡄⠀⢠⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣠⡾⠃⠀⠈⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣄⢀⣤⠿⡶⠋⠀⠀⣠⠶⠷⠶⠶⠶⠶⠶⠟⠛⠉⠙⠛⣶⣖⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠘⢳⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⣴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣸⡇
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠊⢿⣦⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡄⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠈⠟⠉⠙⣦⡀⠀⣀⣴⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢳⣄⠀⠀⠀⢰⣾⣿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠇⠀⠀⠀⢀⠀⠈⠀⠀⠀⠹⣷⣄⣀⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡾⠃⠀⠀⢀⠔⠻⣦⠀⠀⠀⠀⠀⠈⣛⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠉⠓⢶⣎⠀⠀⠀⠙⢷⣤⣤⣆⡤⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠳⣄⠀⠀⠐⠉⠛⠈⠀⣠⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠙⡦⢀⠐⠚⠛⠀⠀⠀⣰⠇⢰⣿⡿⠀⣿⡆⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠵⠋⠀⠀⠀⠀⠀⠀⠀⢀⠋⢱⠆⠸⡟⢻⠁⢿⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠁⠀⠀⠀⢸⠀⠀⢃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡰⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠠⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣶⣄⣼⠀⠀⠀⠀⠀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡴⢴⠾⠿⣦⣿⣷⣤⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢦⣴⡠⠤⠒⠉⠁⠀⠀⠀⠹⡽⠃⠉⠉⠁⠉⠐⠠⣿⡎⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢠⡴⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⡀⠀⠀⢸⡀⠀⣧⠀⠀⠀⢀⡴⢦⡀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⡆⠀⠈⡇⠀⢸⠀⠀⠀⣾⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢹⡤⠤⡇⠀⠘⡧⠤⢲⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⢠⠖⠒⠒⢒⠚⠛⠉⠹⡄⠀⢣⠀⡇⠀⠀⡇⠀⡸⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠰⡏⠙⢦⣀⣀⡇⠀⠈⢇⢸⠀⠀⠃⢀⠃⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠉⠙⣄⠀⠱⡀⠿⡄⠀⠈⠈⠀⠀⠐⠀⠀⠀⣸⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠘⣆⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡴⠋⢉⡳⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⠋⢉⡞⠁⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠙⠋⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⢀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣠⠴⢻⠁⡤⠊⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠦⢤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡤⠖⠋⠀⠀⣠⠏⠀⠀⣠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠎⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⣄⡀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣠⢾⡁⢀⡞⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠀⠀⠀⠙⠲⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡴⠃⠀⢣⡞⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣄⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠴⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡟⡦⣀
⣙⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠶⠶⢶⣶⣶⣶⠒⠲⣖⢲⡶⠒⠒⠒⠒⠒⢠⠄⢠⠀⢠⡝⠓⠾⠍⠉⠑
⠈⢶⣶⣶⣶⣶⣶⣶⣶⣶⣧⣤⣤⣼⣿⣿⣿⣦⡤⠭⢼⠦⣤⣤⣤⠤⠤⣬⡥⠬⠥⢬⡥⠤⠤⢼
⠀⠘⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⢿⣿⣿⣧⠀⠘⠀⢩⣿⣿⣶⣶⣿⣷⡜⠛⠛⠋⠀⣠⡞⠛⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⠟⠛⠛⠿⣿⣿⣿⣿⡝⠟⠛⠢⠄⠀⠠⠒⠻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣧⠏⠀⢹⣿⡌⠄⠀⠀⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⣘⣁⣀⡠⠞⠿⣧⠠⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢡⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠃⠀⠀⠀⠀⠠⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡈⡄⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⢁⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⡿⠟⠛⠛⠿⠿⠛⠛⠻⢶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⢟⠅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢛⠋⠙⠃⠀⠀⠀⠀⢀⠀⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⡶⣋⣄⡀⠀⢀⣀⣠⣁⠀⠀⠀⠀⢈⠶⠀⠀⠤⡀⠀⠠⡀⠀⠀⠹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣿⠿⠘⢛⡿⠯⣤⠔⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⡀⠂⡀⠂⠉⠻⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣣⠁⢀⡔⠉⠀⢰⡏⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠑⢄⡮⡻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣏⡾⢣⠀⢸⡇⠀⢠⣘⣿⣄⡀⠰⣄⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠈⠈⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣺⣿⣿⣷⣤⣿⣶⣶⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣦⣤⣤⣤⣶⣤⣄⣀⣤⣤⡤⠀⠀⠀⠁⠀⠐⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⠿⠿⣿⣯⣭⣿⣿⣿⣿⠟⠛⠛⠛⠛⢻⢿⣿⣭⣭⣭⣿⠿⠿⠿⣷⣶⣶⢂⣠⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠷⣝⣛⣯⣬⡻⣿⣶⣶⣿⣿⣿⣿⣿⢀⠀⠀⢀⡀⠀⣼⣿⣿⣷⣶⣶⣿⡿⠿⡿⣫⣶⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠛⠀⠀⢸⣿⣿⣿⣿⣜⣿⡟⠛⠻⣿⢿⣿⡏⠑⠚⠁⠈⢹⢿⠋⣿⡿⠛⢛⣿⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠀⠀⠀⠀⠀⠈⡁⢈⠿⢻⡟⢿⣷⡄⠀⠹⢸⣿⡇⠀⠀⠀⠀⣼⠀⢰⡟⠀⣠⣿⣿⡿⢋⣴⣿⠋⠉⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣬⡻⢆⡙⢿⣆⠀⢸⣿⡧⠀⠀⠀⠀⣿⠀⠈⠀⣴⣿⠟⣡⡴⢛⠋⠙⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⡻⢿⣷⣤⡙⠢⢕⠄⠙⠇⣸⣿⡇⠀⠀⠀⠀⢸⡄⢀⡾⠋⡡⢚⡥⠤⠍⣤⣤⣤⣤⣤⣤⣤⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣈⠁⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⢧⠈⠀⠀⠀⠁⠀⢐⣉⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣯⣭⣟⣛⠛⠁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠐⠚⠛⣋⣩⣭⣽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⣸⣿⣿⣿⣷⡀⠔⠒⠒⢀⡠⠼⢆⡀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠐⠀⠀⠀⠀⠀⠰⣿⣿⠿⠿⢿⣿⣤⣤⣎⠉⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠊⠀⠀⠀⠈⠒⢶⣦⣤⣤⠌⠀⠀⠀⠀⠙⠿⠟⠿⠿⣶⣴⣿⠃⠚⠦⢤⣴⣄⣀⣀⣀⣰⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⠿⠿⢿⣶⣭⣭⣭⣽⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣽⣿⣾⣷⣾⣶⣿⠿⠿⠶⠶⠚⠛⠋⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢿⠛⡻⠻⣟⠟⣿⠿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⣶⢶⣶⠶⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣷⣾⣶⣷⣾⣿⣾⣶⣾⣷⣷⣾⣶⣶
⣿⣿⣶⣷⣷⣾⣶⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣬⠒⠂⡀⠁⠀⠀⠁⠠⡀⠀⠀⠄⠀⠀⢄⠀⠈⠲⡀⠀⢀⠀⠈⠀⠀⠈⢇⠀⢹⡏⢻⣮⠹⣿⣴⣧⡿⠏⢉⡏⠀⡱⢋⠡⠂⢴⢁⠼⠋⠟⠀⣡⠞⠃
⣈⡀⠀⠀⠈⠠⠀⠀⠐⠀⠐⣄⣀⣀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠶⢾⣎⡀⢣⡎⣼⠶⡏⠅⠀⢻⣆⡜⠀⡀⠁⠘⠀⠀⠀⠈⠀⢀⠠⠊⠀⠀⠀⠀⠀⢀⡴
⣍⡉⠓⠠⡀⠀⠀⠀⠄⠀⠀⣾⠛⠀⠈⠉⠉⢻⡶⢦⣤⡀⠀⣀⣸⡍⠂⠀⠛⠙⢠⣵⣁⡀⠁⠀⠠⢼⠻⠔⠿⠉⠛⠙⢳⠞⠋⠙⡟⠛⢳⣦⠀⠀⢀⠤⠚⠙⠋
⠾⠈⠀⠀⠈⠀⠄⠀⠀⠀⠀⣷⡚⠀⠑⣞⠉⠛⠁⠈⠀⠈⠘⠙⢹⡦⣤⡀⠀⠀⠈⠀⠽⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠛⠀⠀⢰⠇⠀⡁⢿⠠⠚⠁⠀⠀⢀⠨
⣍⡀⠀⠚⠒⠤⠄⠀⠀⠀⠀⣯⠁⡀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡥⣄⣀⣀⡀⠀⠀⠀⠀⢤⡼⠋⠀⠀⠀⠀⢀⠀⣰
⠉⠉⠓⠠⢀⠀⠀⠀⠀⠀⠀⠈⠻⡷⠂⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠐⢠⠈⠀⠒⣿⠉⠀⢹⠆⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠈⠉⠉
⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣤⠷⣤⣄⠀⠀⠀⠈⠀⠈⠃⠰⠀⠄⠀⡀⠀⠀⢢⡔⢉⠠⠚⠁⠀⠉⠀⠀⠀⠀⠀⠘⠿⠀⠀⣤⠖⢓⡡⠀⠀⠀⠀⠀⠀⠀⠐
⣠⣤⣤⣀⡀⠈⠀⣢⠶⠛⠷⠂⠸⡄⠻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠹⠗⠛⠓⣦⣠⣤⡀⠈
⣀⣀⣀⡈⠉⠀⢠⡿⢖⠀⠀⣀⣠⠄⠀⢧⡀⠀⠀⠀⢀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠁⠙⠀⢹⡶⣖
⣤⣈⡐⠀⣉⣡⣾⠃⠀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⡀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠠⠶
⢉⣁⡶⠻⠀⠀⠙⠄⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠐⠂⠐⠀⠀⠠⠜⠛
⠀⢺⡁⣀⠀⠀⢀⡀⠀⣀⠀⠀⢀⡀⠐⠒⠀⠀⠀⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠄⠀⢀⠀⠲⣶⣦
⠲⠒⠻⣭⡴⣥⡟⠂⠀⠻⠦⢠⡎⢁⡀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⡀⢰⠀⠀⠀⠀⢀⠀⠀⠤⢀⠈⠂⠁⠀⠀⠀⠀⠀⠈⠁⢐⡀⠀⡤⠀⠀⠀⣀
⠡⠀⠀⠀⠀⠀⠙⢧⣀⢀⠀⠀⠉⠉⠀⠀⠀⠀⠔⠀⠀⢀⠁⠈⠀⠀⠨⠀⠀⠀⠀⠈⠃⠀⠀⠀⠁⠂⠄⢄⠀⠀⠀⠀⠀⠀⠀⣠⣰⡇⠀⠠⠴⠿⠛⠅⢒⠒⠦
⣴⡶⠓⠀⠀⠀⠀⠀⠉⠛⠿⠚⢤⡀⠀⠀⠀⠀⠘⠤⡤⠶⠤⠂⠀⠀⠀⠀⠘⡄⠀⡀⠀⠀⠀⠠⠄⠀⠀⠠⠽⠀⠀⢀⠀⠀⠀⣸⡽⠀⠈⡇⠀⠠⢀⣀⠀⠉⠙
⡩⡁⠐⠀⠀⠀⠀⡠⠄⠈⠁⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠰⣀⣠⠆⠀⠀⠲⣤⠤⡀⠘⢆⠀⠀⠀⣠⡀⠀⢀⠀⠑⠒⠉⢙⣷⢛⠉⢀⣤⣾⠁⠀⢀⠀⠀⠁⠀⠈
⠔⠀⠀⢀⡠⠔⠁⠀⠀⢀⣴⡿⠚⠷⣴⣤⡴⠞⣆⣀⣀⣾⡍⢩⣁⡀⡀⢀⣿⡆⠡⠸⠾⣤⡴⣶⠋⠙⠖⠞⢧⣤⣤⡽⠟⠀⠉⠙⠛⠉⠀⠀⠀⠀⠈⠓⠄⠈⠂
⣞⡭⠊⠁⠀⠀⣠⠴⠛⡁⠊⠁⠀⠀⠀⠀⢀⠀⠈⠉⢁⡾⠀⡸⣏⣠⣤⣞⠙⢶⡆⠀⠀⠀⣴⠇⠀⠀⠀⠈⠂⠀⠉⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠈⠛
⠁⠀⠄⢀⡴⠋⣠⡔⠋⢀⠀⣀⠔⣠⠄⡠⠛⣠⠂⠀⡞⢀⣶⡠⣎⢿⣜⡼⣼⣇⢙⠧⢹⣛⠁⠀⠀⠀⠨⡂⠀⢴⡀⠀⠢⠀⠀⠠⠀⠀⠀⠄⠁⠀⠀⠁⠢⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⣠⣤⣠⣤⣤⣤⣤⣤⡀⠀⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⢴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⡞⠁⢸⠉⠀⣀⣀⣀⣹⡇⠀⣿⣛⣛⣿⣿⣭⣭⣭⣭⣿⣿⣿⡏⠙⠛⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣶⣾⣿⣿⣿⣶⣶⣾⣿⡿⠋⠉⠉⠉⠉⠈⠈⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠘⠻⣿⡿⠿⠿⠿⣿⣿⡛⠃
⠀⠀⠀⠀⠀⢀⡤⢞⠛⢻⣿⡄⢀⣤⠴⠒⠚⡏⠁
⠀⠀⠀⠀⣠⡟⢠⣿⣿⣿⠟⠁⢯⠹⡀⠀⢀⠃
⠀⠀⠀⢠⡿⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣾⠇⢸⣿⣿⡏
⠀⠀⣰⣿⠀⢸⣿⣿⣧
⠀⢰⠿⠿⠿⠿⠟⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡠⠓⠄⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣀⠄⠊⠀⢀⣞⠀⠀⠀⠀⠀⢈⠀⠈⠐⠂⠄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠉
⠀⠀⠀⠈⠁⠒⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡀⣀⣤⠊⡙⠁⠑⠠⡀⢀⡠⠁⠈⠐⠂⠠⡤⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠠⠄⠀⡀⠀⠀⠈⠑⡋⠉⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠀⠀⡁⡀⢀⣀⠄⠠⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⠀⠀⠚⡀⠀⠀⡔⠞⠁⣏⡄⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠒⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠱⣀⣀⡇⢨⡆⢻⢠⠆⠤⠊⠀⡌⠀⠀⠀⠀⠀⠀⢈⣄⠀⠀⠀⡀⢀⡀⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠄⢀⠐⠂⠠⢀⠀⠀⠀⡐⢉⠈⣼⠈⡇⢸⠀⣤⠰⠇⡐⠀⠢⡀⡀⠔⠂⠈⠀⠈⠢⠄⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⡀⠔⠊⠉⠂⠔⢀⣯⠀⢶⠀⠇⠀⠄⠻⠀⡎⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠈⠈⢄⠀⠡⠀⠀⠀⠀⠀⠀⠈⠐⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡨⠒⠢⣀⣀
⠀⠀⠀⣀⠠⠄⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⢡⣀⣀⣀⠁⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠰⢀⡀⠀⢀⠊⠀⢀⠔⠀⠀⠁⠢⢀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠈⠀⢀⠌⠀⠀⠀⠀⠀⣢⠀⠀⠀⠀⠀⡇⠀⠈⠑⠢⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠂⠀⡠⠐⢷⠠⠀⠀⡀⠔⠈⠀⣡⠠⠄⠀⠤⠇⡀⠀⠀⠀⠑⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠁⠀⠀⠀⠀⢀⠎⠀⠀⠀⠑⣔⠆⠀⠀⠀⠸⠀⣀⠀⠤⠤⠀⣠⠀⠀⠀⠀⠈⠐⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀
⠀⠀⡀⠠⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠐⠒⠶⠶⠤⠤⢤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠓⠶⠶⠶⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⡿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣹⣏⣉⡉⠀⢀⣀⣀⣀⣀⣤⣾⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣟⣉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⠾⠿⠿⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⢸⢀⣽⣯⠘⠉⠉⠉⠛⠻⠿⠷⢶⣶⣤⣤⣀⣀
⠀⠰⠶⠾⠿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣤⣾⣶⣿⣿⣾⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⣼⣿⣿⣆⠀⠀⠀⠉⠉⠉⠙⠓⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠂⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢼⣿⠿⠿⠷⠿⠿⢟⠿⣄⣀⣀⡀⠘⠲⠂⣶⠐⢸⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⡥⠤⣶⡿⠤⠤⠤⣄⡀⠀⢀⠤⠤⠤⠦⠬⢉⣑⣒⣤⣄⣟⠿⠿⡿⢿⡟⠋⠁⠀⠀⠀⠀⠀⠀⢠⣧⣴⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠎⡜⣠⣾⣿⣁⣀⠀⢀⢠⢻⡀⠘⡰⠐⠒⠦⢰⠤⠤⠤⡈⣁⣉⣋⠒⠳⢆⡙⣧⡀⠀⠀⠀⠀⠀⢠⠇⢨⠏⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⢎⣞⣼⣿⡎⠀⢠⣦⢙⣾⡏⣼⠀⣧⡆⠀⠀⢸⢘⡀⠀⠀⣃⢁⠀⠈⠇⢰⢢⡈⠻⣼⣄⠀⠀⠀⠀⡌⠀⢸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⢏⣾⢯⣿⠏⠀⠀⣸⣿⣾⣿⢹⠃⢠⣿⠀⠀⠀⢸⠎⡇⠀⠀⢹⢺⠀⠀⢸⠈⡂⢻⢀⡃⠈⠙⡒⠂⠴⠤⣤⡆⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠿⢿⠶⠾⠷⠾⠿⣿⣿⣽⣿⣇⠃⠀⡘⡏⠀⠀⠾⢺⡆⡇⢠⣲⠜⢘⡄⣀⢸⠂⢷⣼⠀⠑⢰⠀⢳⣀⣀⣀⣹⣿⣏⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡔⠁⠀⠀⠀⠀⠀⢀⣀⣉⣙⡋⠐⠿⣇⣑⣒⣒⣒⣚⣀⣙⣚⣛⣚⣀⣛⡛⡚⣀⠀⠀⠀⡀⠘⠃⣸⠉⡏⠀⠘⡏⠙⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⣾⢰⣶⡄⠀⠀⠀⣼⣿⣿⣿⢿⣿⠋⢩⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⠉⢉⢋⡉⣩⠋⠉⠀⠀⠀⢹⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⣿⣼⣿⣧⣤⣤⣤⣼⣿⣿⣿⡛⠁⠀⢰⠀⠀⠀⠀⠘⠛⠛⠃⠀⠛⠃⠀⠈⠛⠀⠀⠀⢀⠎⢸⡟⠁⠀⠀⠀⠀⠀⠀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠂⠓⣄⡀⠀⠀⠀⠂⠁⠄⠀⠀⠀⠀⢀⠀⠀⣠⠊⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣭⣉⣒⠓⡖⢦⣤⣤⣤⣦⣤⣶⣖⣋⡤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠿⠛⠛⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⣼⠏⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣦⣄⡀⠀⣼⡇⠀⣀⣰⣯⣤⣤⣴⡶⠾⠿⡾⠃⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⣀⣀⣼⣷⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠹⣿⣿⣦⣤⣤⣤⣾⣷⠶⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋

⠀⠀⠀⣠⣴⣞⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣲⣶⣶⣶⣶⣶⣶⣿⣿⣶⣦⠀⣀⡀
⠀⠀⠀⡏⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠂⠀⠈⠉⠉⣭⠞⠋
⠀⠀⠀⣷⡟⠛⠛⠛⠛⠛⠓⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠤⠴⠦⠤⠦⠤⠤⣇
⠀⠀⠀⠙⠷⢶⠶⠶⠶⠶⠶⢾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣤⣤⣤⣤⣤⣤⡀⠀⣴⣶⡶⣾⣷⡀⠀⠀⠈⢑⠖⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢸⠋⠉⠉⢳⡈⢹⣿⠀⢻⣿⣧⣾⣿⣧⠀⢠⣾⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣄⡀⠀⣸⢃⢼⡃⠀⠈⣿⣿⣿⣿⣿⣇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠿⠿⠷⠶⠾⢿⡳⣦⠸⣿⣿⣿⣿⣿⡞⡾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢿⣿⣿⣿⣿⣿⡼⡻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡈⣿⣿⣿⣿⣿⣷⣱⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢹⣿⣿⣿⣿⣿⣧⢣⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⢿⣿⣿⠙⣿⣿⣎⢎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠸⠿⠿⢿⣛⣛⣛⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢾⣿⡏⣤⠈⣿⣿⠿⠋⠀⠀⠀⠐⠂⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣗⣂⣽⣿⡶⠞⠋⠁⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣶⡆⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡄⠘⣿⣿⡀⡀⠤⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣶⣶⡶⠿⠟⠛⠙⣿⣷⠠⠙⠉⠁⠀⠀⠐⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⣀⣀⣀⣤⣤⣤⣤⣴⣆⠀⠆⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣶⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⢡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠈⠉⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡍⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠑⠠⠀⣀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠤⠚⠁⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠈⠡⠀⠀⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⢀⠀⠒⡉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡠⠔⠊⠐⠒⠀⠠⠃
⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠂⠊⠐⠀⠂

⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⢀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠟⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⣿⣿⢿⣿⠿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣶⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠻⠿⠯⣿⣿⡿⠿⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⣀⡀⠀⠀⠸⠀⠀⠀⠀⠹⣷⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠰⡆⠀⠀⠀⠀⠈⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⢴⣶⣿⣶⣶⠆⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢈⣻⣿⣿⣭⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⣀⣀⣀⡠⢄⣀⣀⣀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠈⢱⠊⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠒⠒⠂⠉
⠀⠀⠀⠀⠀⢤⠀⠠⡀⠀⠀⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⠤⠶⠿⠿⠶⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠳⣀⢽⣾⠭⠦⠭⠤⠤⠤⠤⠤⠤⠤⠤⠤⠉⠀⠁⠀⡀⠭⠤⠬⢉⠀⢉⠁⠉⣐⣒⠢⢄⡀
⠀⠀⠀⢀⡠⢔⣮⠛⢩⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠉⠒⢌⢢
⠀⠀⠀⠘⠂⠱⠧⠤⠃⠀⠀⠀⠉⠉⠁⠒⠒⠲⠤⠤⢄⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢣⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢸⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠸⡀⠀⠀⠀⢀⠇⠈⠒⠒⠒⠒⠒⠒⠂⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠈⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⢉⡠⠊⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢲⠆⠄⠄⠀⠠⠠⠀⢶⢲⠊⠁⠀⢀⣴⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⣦⢧⠀⣀⣀⠀⢀⣀⣨⠮⠆⣀⢴⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⢀⣠⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⢻⣿⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⡿⠟⣿⣿⣿⣦⡀
⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆
⠀⠀⠉⠙⠻⠿⡿⠿⠛⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⢹⣿⣿⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⢸⣿⡿⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⢸⣿⡇⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧

⠱⠄⠀⢁⣀⣸⡞⠿⢾⠧⠀⠀⠀⠀⠀⠀⠀⡀⢿⠃⠀⢀⠀⡀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠊⢢⣤⠀⠀⣠⠋
⠀⣴⣔⢄⠙⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠋⠀⠛⠁⠀⡀⠀⠀⠀⢠⠂⡄⠀⠀⠀⠟⠁⢀⠞⠁⠹⡆
⣝⠋⠙⠁⣠⡄⠐⠄⠀⠀⢀⡄⠶⣖⢶⡄⠀⠀⠀⣴⠀⠲⢠⠀⠀⠘⠁⠀⠀⠀⠀⠘⢠⢰⠀⠀⠀⠀⠀⡄⠀⠀⠊
⠙⠄⠀⢘⠉⠁⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⡴⠰⠤⠀⠀⠀⠀⣡⠤⢄⠀⠀⠀⠀⠀⠀⠁⠀⣀⡀
⠁⠀⠀⠀⠀⠀⠀⣀⢠⡄⠆⠀⠀⠀⠀⠘⠋⠁⡄⠀⠀⠀⠊⠀⠀⠘⡤⠦⠀⠀⠀⠀⠀⠐⠉⠁⠔⠀⠀⠀⣠⠔
⠁⠀⢀⠀⣀⠀⣖⠈⠀⠀⠀⡴⠢⡖⠂⢀⠀⠀⠈⢀⠆⢠⠀⠀⠀⢀⣤⠀⠄⢈⡦⠠⠄⡀⣄⠀⠀⠀⠀⠈⠀⠀⢀
⢠⠐⠺⠄⠀⠀⢎⣀⢤⣀⣀⡃⣀⢀⡀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠘⢁⠀⠊⠁⠀⠀⢀⡇⠈⡆⠀⠈⠂⡀
⣧⣄⠀⠀⠀⠀⠚⠀⢀⣀⡨⠄⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⠀⠀⢐⠃⠀⢀⡠
⣀⠀⠀⠀⠀⡴⠚⠃⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠉⠀⢲⠀⠂⠠⠶⠖
⠋⠡⠆⠀⠘⠀⡠⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠦⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢐⠆⠀⠤


⠛⠋⠉⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠛⠋⢉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀
⢾⡏⢉⣀⡤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠾⠟⣋⣥⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢶⣤⣤⣀
⣺⠟⣋⠥⠂⠁⢀⡤⠊⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⣌⠓⢤⣈⠛⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠹⣧⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠙⣷⡀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⢿⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡄⠀⠀⠀⢻⣆⠀⠀⣠⣴⣶⣦⣴⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠉⢻⣆⠀⠀⣀⣽⣦⣼⡟⠁⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⢀⣠⣴⠿⠻⣶⠾⠋⠁⠙⢿⣇⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣴⠾⠋⠁⠀⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⣶⠿⣿
⠀⠀⠀⢀⣠⣴⠾⠋⠁⠘⣷⣀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣿⠀⠹⣧
⠀⠀⠀⠻⣯⠀⠀⠀⠀⠀⣨⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣿⠀⠀⠹⣧
⠀⠀⠀⠀⠹⣧⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣿⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣤⣬⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠶⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⠶⠾⠿

⣿⣿⡿⠿⣿⡿⢿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣿⣿⣿⣿⣿⢟⣫⣷⣾⣿⣿⣿⣶⣮⣝⡻⣿⣿⣿⣿⣿⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠈⠉⠑⠢⣄
⠀⠀⣿⣿⣿⡟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⣿⣿⣿⠃⣿⢙⢩⠋⡍⡏⡙⢹⢩⢹⠉⢹⣿⢸⣿⣿⣿⠀⠀⣿⠐⣧⡖⣖⢶⢰⣶⣶⡞⡆⣾⡆⠈⡇
⠀⠀⣿⣿⣿⣇⢻⣾⣶⣷⣶⣷⣿⣾⣶⣾⣶⣶⠟⣼⣿⣿⣿⠀⠀⢻⡌⠁⠉⠉⠉⠈⠈⠁⠉⠁⠉⠉⣸⠃
⠀⠀⣿⣿⣿⣿⣷⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣾⣿⣿⣿⣿⠀⠀⠀⠙⠠⣀⡀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣾⣭⣭⣭⣷⣯⣟⣳⣝⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠚⠛⠓⠦⢝⣦
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⡿⣲⠶⠤⠤⣤⣤⡴⠒⢶⣶⣶⣶⣶⣦⣤⣶⣶⠦⣴⢦⠤⠦⠤⠤⠤⠤⠤⠤⠶⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣸⡿⢋⢐⡁⠀⠠⠒⠛⢻⣷⠶⢾⣿⣿⡿⠿⠿⠿⠿⠏⢼⠇⠮⠀⠄⠤⠤⠤⠤⠴⠶⠶⠤⠤⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠊⢅⣠⣶⡼⣧⣤⣤⣼⣵⣿⣿⣿⡟⠛⠛⠘⠓⠚⠃⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⠀⠀⠙⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠢⢈⡡⣶⣦⣼⡏⠙⣿⠃⠀⢙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⣠⣿⡟⠉⠉⠣⢄⣈⠳⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⠀⠒⣼⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢾⣿⡗⢂⢊⣑⠏⣹



⣄⣄⣀⣠⣄⣄⣠⣄⣤⣠⣄⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣄⣀⣄⣀⣀⣀⣤⣀⣀⣀⣀⣀⣠⣀
⣿⣏⠖⣟⣋⣛⣛⣛⣛⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠄⣀⣀⢀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡄⠰⠋⠟⠿⠷⠿⠟⠻⠖⠀⡤⠀⠀⠀⠀⠀⠐⠀⠀⠔⠀⠀⠀⠀⠀⡴
⠀⣠⠴⠶⡆⠀⠖⠈⠐⠲⢄⠰⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⠄⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⡄⢀⡀
⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠘⠚⠉⠀⠀⠐⠈⠿⠀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠰⠢⠖⠂⠖⠦⠀⠂⠭
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣓⣀⡈⠀⠀⠀⠀⠀⠀⠀⡀⠵⠓
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠁⠉⠀⠀⡀⠀⢀⠀⠀⠘⡀⠀⡀⢨⣄⡄⠀⠄
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠈⠀⣀⠁⠄⢠⠁⠀⠀⠁⠂⠁⣤⠀⠀⠂⠐
⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠘⣿⣶⣙⣷⣴⣖⠒⡀⠀⠀⠀⠘⡄⠄⠀⠀⠀⠀⠀⢀⣄⣠⣄⡀⠀⠀⠀⠀⠁⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠁
⠀⠀⠀⠒⠘⢋⠠⣀⢀⠙⠏⠁⠀⠀⠀⠐⠐⠀⠀⠀⡀⠸⣿⡿⠾⠳⠿⢹⡿⠒⠀⠀⠀⠀⣰⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⡁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣤⡏⠀⠀⠀⠀⠀⠀⠤⠄⢀⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⡆⢀⢀⠀⢀⡠⠂⠀⠈⠀⠀⠀⠈⠀⠈⣸⠣⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣁⡠⠀⠈⠀⠠⠀⠀⠀⠂⢀⣀⣤⡞⠁⠀⠀⠀⢠⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠛⠦⠠⣦⣤⣤⠟⠻⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠠⣄⡀⠀⠀⠀⢀⣀⡀⢀⠀⢀⠀⠒⣀⢄⣀⣠⣐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠹⡟⠧⠿⠶⠟
⠀⠀⠀⠉⠲⠶⠖⠀⠀⠉⠉⠉⠁⠀⠀⠋⠀⠋⠙⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠓⠋⠉⠀⠀⠉⠉⠋⠁⠈⠛⠓⠀⠰⢤⣀

⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣷⣼⣼⣷⣷⣼⣼⣭⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣶⣿⣿⣿⣶⣶⣶⣿⣷⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⣉⣛⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣷⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢋⣿⣿⣿⣿⣧⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣹⣿⣿⣿⣋⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠐⢿⠟⠁⢸⣿⣿⣿⣿⠿⠿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⣰⣿⣿⣿⡟⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⡏⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣿⣿⠛⠛⣿⣿⡅
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠠⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⣤⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠘⣆⠀⠀⠀⡇⠀⢠⡏⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠈⠳⡀⠈⢣⡀⢀⠘⡆⢰⠀⡇⡇⡼⣰⠂⡰⠃⠀⢀⡴⠋⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠈⠢⡀⠑⢄⠣⠘⠘⠀⠁⠡⠡⢁⠞⡠⢊⠔⠋⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠑⠢⣑⠈⡀⠀⠢⠠⠀⠐⠒⠀⠠⡋⠊⠐⢁⡤⠚⡁⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠉⠙⠒⠦⢍⡢⠁⠢⡔⢁⠔⠉⠉⠉⠑⠢⡁⠄⠂⠡⢒⡉⠤⠒⠊⠉⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠒⠒⠢⠬⠤⢄⣀⠘⢠⠁⠀⠀⠀⠀⠀⠀⠰⠀⠄⠈⠀⠠⠤⠖⢒⠿⠵⠒⠛⠚⠁⠀⠉⠉⠉⠉⠉⠉
⣠⡤⠤⠤⠤⠖⠛⠉⠉⠉⠓⠲⢄⡀⠀⠒⠒⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⣘⣀⣂⡄⡀⠀⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⢤⣀⣀⠤⠵⠒⠒⠒⠒⠒⠋⠁⠀⠀⠀⣠⠼⢁⣠⠚⠁⠀⠉⠉⢀⡠⠤⠄⠀⠒⠀⠃
⠀⠀⠀⠀⠐⠀⠈⠀⠲⠀⣀⢀⡀⡀⠢⡄⠀⠀⢌⣶⣔⠒⠒⠶⠦⠶⠶⢲⠖⠉⢉⠀⠒⠝⠁⠐⡶⡀⣶⡉⣉⢀⡀
⠀⠠⠤⠰⠶⠀⠆⠀⠠⠀⠒⠒⠂⠒⠒⠒⠒⠒⠒⠚⠛⠛⠃⠀⠀⢠⡴⠋⣄⢨⡀⡖⢤⠀⠀⠠⠖⠁⠜⡀⠈⠥⣬⡔⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢒⣀⣰⣫⡴⠚⠛⠛⠋⠉⠛⠶⠶⠤⠤⠴⠒⠶⠧⠤⠤⠶⠤⠵⢦⡀
⠀⠀⠀⠀⢀⣀⣀⣠⣄⣠⣤⣄⣀⡤⠤⠶⠒⠚⠩⠭⠭⣛⣉⢉⣰⠖⠒⠒⠒⠊⠉⠁⠀⠉⠉⠉⠉
⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠀⠀⠀⠀⠛⠛⠓⠒⠒⠒⠒⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⢀⣠⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣀⠀⢻⣦⡀
⠀⠀⠀⠀⠀⠀⢠⠄⣸⣿⢏⣾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣎⢿⣿⡀⢤
⠀⠀⠀⠀⠀⢰⣿⠀⠿⣃⣼⣶⣿⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣾⣷⣾⣄⠻⠇⢸⣷
⠀⠀⠀⢀⠀⣿⡟⠀⢀⣉⣉⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⣉⣁⠀⠘⣿⡇⢀
⠀⠀⠀⣿⠀⣿⢁⣴⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣄⢹⡇⢸⡇
⠀⠀⢸⣿⡇⠀⢈⣡⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣉⠀⠀⣿⣿
⠀⢠⠘⣿⡇⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⣿⡟⢠
⠀⢸⣆⠘⠁⠛⣡⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡙⠃⠙⢀⣾
⠀⢸⣿⡆⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⣾⣿
⠀⡄⠻⣷⢸⡿⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⣿⢰⡿⠃⡄
⠀⢻⣦⡈⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠈⣠⣾⠃
⠀⠈⢿⣷⡄⢸⡿⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠹⣿⠀⣼⣿⠏
⠀⠀⣄⠙⠻⠘⠁⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠙⠸⠛⢁⡄
⠀⠀⠈⢿⣶⣄⠀⢿⣿⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢸⣿⠇⢀⣴⣾⠏
⠀⠀⠀⠀⠙⠿⣷⡌⠏⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠈⠏⣴⡿⠟⠁
⠀⠀⠀⠀⠘⢶⣤⣤⣀⠀⢻⣷⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⢰⣿⠃⢀⣠⣤⣴⠞
⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣶⠝⠀⠻⣿⣆⠙⣶⣄⡀⠀⠀⠀⠀⣀⣴⡾⢁⣾⡿⠃⠘⠵⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣶⣦⣤⣬⣛⠆⠙⠻⣿⣦⠀⢠⣾⡿⠛⠁⢚⣫⣤⣤⣶⣦⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣋⣁⣤⣤⣶⣶⠾⠀⠸⣶⣶⣦⣤⣄⣉⡙⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣯⣽⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣤⣶⣾⣿⣶⣦⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣰⣿⠿⠋⠉⠀⠀⠈⠙⠻⣿⣦⣠⡄⠀⠀⠀⠀⢀⣠⣠⣤⣤⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⣽⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠀⢠⣴⢞⣭⣷⣶⣶⣮⣝⡳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⣏⣉⡉⠻⣶⢠⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡄⢠⡿⣵⣿⣿⡿⠋⠙⢿⣿⣿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣟⠉⢹⠶⢻⡟⠉⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠁⢼⣧⣿⣿⠋⠀⠀⠀⠀⠙⣿⣿⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣄⣸⠛⠋⣠⡾⠃⢀⣼⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡄⠀⢘⣏⢿⣿⣷⣄⠀⠀⣠⣾⣿⡟⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⡟⣩⡁⣀⡀⠁⠈⠻⣿⣦⣤⣄⣠⣤⣴⣾⡿⠋⠀⠁⠀⠀⢛⣮⡻⣿⣿⣷⣾⣿⡿⣟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣘⠋⠉⠛⢻⡟⠛⠉⣙⠃⠀⢀⣰⣆⡀⠀⠈⣙⠳⢯⢭⣭⠷⡞⣋⣠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⠛⠉⠁⠀⠀⠉⠙⠻⣿⣿⣇⡀⠀⠀⠁⠀⢀⣽⣷⣿⠿⠿⠿⠿⣿⣾⣯⡀⠀⠀⠀⣴⡿⠟⠋⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠀⠀⣠⣤⣤⣤⣤⡀⠀⠈⢿⣿⣿⡀⠀⣶⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣶⣸⣿⠀⣰⣾⣶⡄⢸⣿⡆
⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⣰⣿⣿⠿⢿⣿⣷⡀⠀⠈⣿⣿⡏⠀⣸⣿⠟⢷⣦⣤⣶⠿⠿⣶⣤⣴⡾⠻⣿⣇⢸⣿⠀⠙⠿⠿⠁⢸⣿⠅
⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠘⢿⣿⣧⣀⣰⣿⣿⠇⠀⠀⣿⣿⡿⣤⣿⣿⠀⠀⢸⣿⠁⠀⠀⠈⣿⡇⠀⠀⣿⣿⣤⠹⣿⣶⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠈⢿⣿⣿⣿⣿⠏⠀⠀⣰⣿⣿⠗⠀⢿⣿⡀⠀⠈⢿⣦⣀⣀⣴⡿⠃⠀⢀⣿⡿⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠀⠀⣸⣿⣷⡀⠀⠀⠙⢻⡟⠋⠀⠀⢀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣶⣶⣤⣴⣶⣾⣿⣿⠿⠃⠀⠀⠀⠉⠈⠻⣿⣶⣤⣀⣸⣇⣀⣤⣶⣿⠟⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠿⠟⢿⠟⠿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠿⢿⡿⠿⠛⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⢰⣾⡟⠓⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢸⣿⡇⠠⠤⠤⠤⠭⠭⠭⠉⠉⠭⠭⠭⠭⠭⠍⠉⠽⠿⠿⠿⠝⢻⣇⣀⣀⣤⣤⣤⣤⣤⣤⣤⠸⣶⢒⣛⣷⣦
⠀⠀⣴⣿⡶⣗⣒⣒⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠋⠙⣿⠀⠈⣿⠛⠛⠛⢛⠏⠂⠰⣿⠟⠋
⠀⠀⣿⡘⢿⢹⣄⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⢶⡿⠶⢶⣿⣿⣶⣶⣶⣶⣆⣇⡉⠳⠤⣄
⠀⠀⠻⣷⣤⡟⣿⡿⢿⣿⣟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⢡⣀⣀⣖⣒⣒⣺⣿⠉⣈⣿⣏⠉⠉⠉⠙⣻⠋⠁⠀⠀⣨⢳⡀
⠀⠀⠀⣿⢷⣿⡍⠉⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣾⣿⠿⠿⠿⢿⣾⡏⠋⢹⣿⣿⣴⣶⣶⡶⠏⠀⠀⠀⣼⣿⣷⡳⡄
⠀⠀⠀⢻⣧⣽⣷⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠈⠙⣉⣉⣀⡀⠀⣴⣶⡆⢸⣿⣿⣻⣿⣟⣆
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣿⣶⣦⣿⣿⣿⣧⢸⡿⠋⣷⣾⣿⣿⡿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠹⣷⡘⣿⣯⣾⡃⢠⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣬⡭⠥⠖⠋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠟⢸⣿⣿⣩⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣿⣿⣿⣿⡿⣫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⠭⠽⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⣹⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣦⣤⣤⣤⣄⣀⡀⠀⠈⠻⣷⣄⣴⡤
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠿⠿⠻⠶⠶⠶⢦⡤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢈⡟⣽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢞⠱⠀⠃⠛⠂⠃⠀⠃⢀⢳⣛⣋⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣠⠏⣰⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⣳⡠⠤⢄⣸⢀⣀⣀⣀⣀⣀⣀⠀⢀⡎⠀⡈⡉⢉⡉⡉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠳⠶⢷⣤⡄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣁⡀⢀⡔⡽⠁⠀⣼⣧⠃⠀⠀⠀⠀⠀⢈⡣⡸⠀⠀⠁⠉⠈⢁⣁⣀⣠⣤⡤⢤⢶⡖⠒⢻⡉⢹⡍⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠉⠉⠀⠉⠉⠉⡹⠈⠉⣉⣉⠉⠉⠉⡉⢴⣣⠴⠒⠛⠋⠉⠉⠉⠀⠀⠀⠀⠈⠛⠁⠀⠀⠹⣄⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⣤⣀⣀⢀⣀⣀⣀⢠⠃⠒⠘⢝⠨⠃⠈⣉⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢺⡯⡝⠶⠦⠼⢿⠓⠒⠒⠒⠒⠛⠋⢉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠙⠦⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠾⠧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣖⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠻⣿⣎⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠈⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠁⠀⠉⠛⢦⡀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠖⠛⠢⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣓⣛⣓⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⣠⡾⣿⠿⠟⠛⠛⠫⠿⣶⣤⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⢀⣼⡿⠋⠀⣠⣤⣶⣤⠀⣀⠈⠻⡷⣌⢳⡄⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⣾⡏⠀⢠⢺⠟⠋⠉⠁⠸⠤⠃⠀⠘⢿⣆⢻⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⢸⣿⡇⠀⢯⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡇⠀⠀⢀⡴⠋
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⡇⢀⡴⠋
⠀⠀⠀⠘⣏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⣿⡷⠋
⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢫⡾⠋
⠀⠀⠀⠀⠀⠙⢿⣻⠷⣦⣤⣤⣤⣤⠴⢖⣫⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠛⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡀⠀⠀⣀⣀⣀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠚⠀⠂⠀⠉⠱⠟⠃⠈⠉⠋⠀⠛⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣞⠉⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡔⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣋⡄⠀⠠⠆⠀⠀⠀⠀⠀⡂⣺⠦⠋⠄⠀⠄⠀⠉⢍⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⣀⣑⠂⠀⠀⠀⠀⠾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⠲⠀⠀⠀⠀⠀⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⠂⠐⠒⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠐⠀⠀⠙⠁⡀⡀⠄⣰⠦⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠤⠖⢇⣠⣀⡤⣧⠀⠀⠀⠀⠀⢠⣅⣀⡤⠐⠛⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠑⠊⡀⢿⠃⡀⠀⠀⠀⣿⠉⠑⠋⠉⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣀⣀⡀⠀⠘⠋⠙⠆⢉⠒⠛⠐⠁⢀⡄⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠴⡾⠹⠤⣀⣰⣠⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣀⠀⠀⠀⣠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠤⢤⡖⠒⠖⡇⠀⢀⠀⣿⠈⠑⠒⡋⠁⠈⠉⡦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠕⠀⡠⠒⠊⠉⠀⠔⠓⡴⣉⡊⠁⠃⠀⠒⠀⠀⠀⠐⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠰⠀⠀⠔⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠋⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠂⠒⠐⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠈⠒⠒⠁