Caucasian Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠒⠚⠛⠓⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠖⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⣀⡤⠔⠒⠒⠂⠤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣀⡀⣾⠀⠀⠀⢰⡋⠙⠒⠒⠲⡄⠀⢀⠤⠤⠤⣤⣀⢸⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⣡⠄⠉⠁⠀⠀⠀⠀⠉⢉⣉⡉⠉⠀⠀⠈⠒⠦⠤⠤⠤⢻⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇⠀⠀⢀⠀⢠⣾⣷⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠤⠒⢳⣄⠑⠂⠀⠀⠀⠀⠀⢶⣶⡀⠠⠂⠀⠀⢸⠀⢦⡉⣁⡀⠀⣾⠀⢀⡞
⠀⠀⠀⠀⢰⠃⠀⠀⢸⣠⣭⡉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠐⠛⠛⠀⡗⠒⠋
⠀⠀⠀⠀⣸⠀⠀⠀⠀⠻⢍⣀⣻⠀⠀⠀⠀⠀⠀⠀⠠⡤⠶⢦⣄⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠚⠁⠀⠀⠀⠀⠸⢿
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡀
⠀⠀⠀⠀⠈⠢⣄⣀⣀⠀⠀⠀⠀⠀⠀⠈⡗⠒⠤⠤⠤⣀⣀⣀⣀⡠⠤⠖⠋⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣀⣀⣀⣀⡠⠴⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⠶⠶⠭⣭⣭⣀⡑⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠿⠟⠒⠒⣀⣠⠴⢶⡒⠒⠈⠙⠊⢿⣧⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣛⣉⣁⣤⣤⣴⡶⠚⢋⣩⣶⣾⠗⠀⠀⣀⣈⣷⢸⣿⣸⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⠿⠿⣛⠋⣁⣴⣿⣿⠋⣀⠀⣠⣾⣿⣿⣿⣿⣿⡏⣰⡯⣉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠶⠶⠞⠋⠵⠚⣁⡿⢟⣡⣾⣿⣿⡿⠿⠟⠃⠈⠙⢯⣾⣿⣇⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣤⣤⠴⠢⠴⣚⣩⣴⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣝⢧⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣋⡭⠴⢾⡟⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⣇⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣁⣴⣾⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣋⣠⣴⣀⣤⡶⢶⣶⣤⣤⣤⣤⠀⠀⠀⣠⣴⣶⣶⠾⠶⣤⣼⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⡏⠈⠀⠀⠴⡻⠿⠉⢿⡿⠿⣿⣿⠫⠥⠤⠄⠀⠀⠈⣿⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡧⣿⠇⣇⠀⠀⠀⠀⠉⠀⠀⡿⠀⠀⠸⡿⠀⠀⠂⠀⠀⠀⢰⣿⡠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣥⢹⠀⠸⢄⣀⣀⣀⣀⣠⠜⠁⠀⠀⠀⡷⢄⣀⣀⣀⣀⣠⠞⣿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢺⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⢰⡏⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠠⠴⠊⠀⠐⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⢿⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⡀⠀⠀⠈⢀⣀⣀⣀⡈⠁⠀⠀⣰⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠱⡀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⣉⠉⠙⣿⠿⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠛⠁⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⠿⠿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠠⠀⠀⠀⡤⣦⣤⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠤⠒⠀⡠⣶⠖⠐⠉⠻⠷⣄⠀⠀⠈⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡐⠀⠁⠀⠀⢔⠤⠤⠘⠳⠀⣠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡼⠣⠞⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠇⠀⠀⠀⠀⠀⣀⣧⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢹⣷⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠠⠀⠀⠀⠀⡀⣿⡟⡇⠀⡰⢡⠏⠀⠀⠀⢀⡄⠀⠀⡀⣻⠾⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢰⠃⡄⠀⠀⣀⣤⣿⣿⢳⠇⡼⣡⠏⠀⢨⠁⠿⠀⠀⠀⠈⠛⠁⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠄⢸⠀⠃⠀⠀⣿⣿⣿⣿⠈⢼⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡈⠀⠸⠀⠀⠀⠀⠀⠸⢿⡇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠁⣆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠸⣿⢇
⠀⠀⠀⠀⠀⠀⠁⠀⢀⠆⣸⡟⠀⠀⢸⠀⠀⠀⠀⠀⡰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡎
⠀⠀⠀⠀⠀⡀⠀⠀⡌⢠⣿⠇⠀⠀⠘⠀⠀⠠⠀⢠⠁⡐⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡉
⠀⠀⠀⠀⠀⠀⢠⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢙⠀⠡⠀⠀⠀⠀⠀⠀⠀⠐⣠⣶⠔⡏⡐
⠀⠀⠀⠀⠸⢀⠂⡴⣹⣿⠅⠀⡀⢰⠀⠀⠀⠀⠀⣇⠀⠀⢆⠀⠀⠄⠠⠀⠀⠀⠀⠀⠙⣱⠞⡴
⠀⠀⠀⠀⣰⢢⠞⣰⢿⣟⠀⢀⠃⠈⡄⠀⠀⠀⠀⠘⣆⠀⠀⢆⠀⠆⠀⠀⡄⠀⠁⠀⠀⠀⠀⠁
⠄⣀⡠⠚⠃⢀⡠⠟⠚⠁⠀⡜⠀⠀⡇⠀⠘⠀⠀⠀⡆⠀⡄⠈⢻⠀⠀⠚⠁⠀⠀⠠⠂⣰⠃⠀⠀⠀⠀⠀⢀⣼⡶⠂
⠀⠁⠀⠊⡰⠁⠀⠀⠀⠀⣰⡡⠀⠀⠀⠀⢸⠀⢠⠀⢸⢠⡁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⠀⠀⠀⠀⢀⣾⠟⢣
⠀⠀⠀⡰⠀⠀⠀⠀⢀⡴⡫⠀⠀⠀⡄⠀⡞⠀⣟⠀⠘⢸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢨⣯⠀⠈
⠀⠀⢠⠃⠀⢀⠤⠒⠉⡰⠁⠀⠀⢠⠀⠰⠁⢰⡙⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⡌⠀⠀⠀⠀⠀⠐⠁⠀⠀⡆⡆⠠⢁⠃⡾⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠄⡌⣰⠃⠀⠀⢸⡾
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠎⣘⢔⠏⡐⠁⢄⡾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢹⠏⣴⢋⠎⠜⢀⠎⠘⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠕⣋⡼⢃⡞⠁⡰⠋⠀⡴⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠃⠁⠀⠉⡴⢋⡤⢊⣀⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠱⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣵⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⢠⣴⢶⣦⣀⡀⠀⣠⣤⣤⣄⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⣸⠁⠡⢴⣶⡶⣹⠁⢸⣟⣿⣶⢢⡁⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠹⠀⠀⠀⠉⠋⣼⠀⢸⢙⠹⠉⠁⠀⠄⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡆⠀⠀⠀⠀⢠⣨⠀⠨⡉⠀⠀⠀⠀⠁⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡧⠀⠀⠀⠀⠐⠡⠆⠴⠤⠄⠰⠞⢠⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠰⠀⠀⠴⣖⠰⡗⢬⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣰⡄⠀⠈⠡⠉⣼⠃⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣦⣀⡤⠿⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣿⣿⣧⡀⠀⠙⠿⣷⠞⠧⠀⠀⣿⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⠻⣏⠉⢉⡟⢿⡉⠉⣿⠻⣏⠉⢹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢸⣿⣿⣿⣧⣤⣼⣧⣿⣤⣬⣷⣼⣥⣤⣽⣶⣿⣿⡇⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣾⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣇⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⢻⣿⣿⣶⣄⠀⠀⠀⠀⣸⣛⢛⣓⣀⠀⠀⠀⠀⣴⣶⠶⣆⠀⠀⠀⢀⣶⣾⣶⣦⠀⠀⠀⠀⠠⢒⣆
⠀⠀⠀⠀⠀⢻⠿⠿⠿⣿⠀⠀⠀⠛⠋⠀⣀⠈⠛⠃⠀⠀⠀⡟⠀⠀⢻⠀⠀⠀⣾⡧⠀⠠⢿⡇⠀⠀⠀⡖⠋⠋⢲
⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄⠀⠀⠀⠀⢀⣠⣄⣀⣦⡄⠀⠀⠹⣷⡄⠀⣾⠃⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠐⠄⠀⠀⠀⠄⠀⠀⠂⠀⠀⢀⠀⠀⠈⠗⠻⡛⡗⠃⠀⠀⠀⣀⠁⠀⢁
⠀⠀⠀⠀⠀⡀⠀⡀⠁⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⣷⠹⠄⠀⠀⠀⠈⠂⠔⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠁⠈⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣶⣶⣶⡄⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⡿⡿⠟⢿⠀⠀⠀⠀⣾⠋⠈⢿⠀⠀⠀⠀⣾⠛⠛⢻⠀⠀⠀⠀⠸⡇⠀⠉⡓⠀⠀⠀⠀⡿⠛⠋⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣄⠀⠀⠀⠰⣤⣤⡼⠁⠀⠀⠀⠀⠁⡀⢠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⢿⡟⠀⠀⠀⢀⠉⠉⠁⡀⠀⠀⠀⠀⠴⢠⣀⠦⠀⠀⠀⠀⣄⠈⠀⠡⠆
⠀⠀⠀⠀⢠⣾⣿⣷⣷⣿⣷⡄⠀⠈⠀⠂⠀⠂⠈⠃⠀⠀⡙⠄⠀⠼⡃⠀⠀⠀⠀⠳⠀⠠⠇
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠠⠶⠀⠀⠰⠆⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⡄

⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⡟⠛⠛⢿⠀⠀⠀⢠⣿⡿⠋⣿⡆⠀⠀⠀⢐⣋⣉⣙⣓⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⣴⣿⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠌⠉⠉⠡⠀⠀⠀⢸⣇⠁⠀⢠⡇⠀⠀⠀⠘⠁⠀⠀⠋⠀⠀⠀⠐⡟⢛⠛⣿⠀⠀⠀⢸⣿⠃⠐⢉⣧
⠀⠀⠀⠀⠀⠀⢀⠂⠐⡀⠀⠀⠀⢸⣿⠇⠀⢿⡇⠀⠀⠀⠀⠴⢀⠴⠄⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠘⢿⡦⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠁⠐⠃⠁⠀⠀⠀⠀⠋⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢐⣤⡐⠀⠐⢁⣴⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⡮⢵⣿⣿⡇⠀⠀⠀⠘⡿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠛⠛⠃⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣋⣭⣭⣻⣷⣌⠻⣿⣟⠇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠔⣋⣉⣉⠙⢿⣿⣿⡆⠹⠹⡄⢰⢸⡔⡌⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡟⢁⣼⣿⣿⢿⣿⣆⢻⣿⡇⣶⠀⣧⣮⣾⢃⡇⣸⣿⣹⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣶⣿⡿⠿⠭⡷⢌⢻⣞⡇⢁⣿⣼⣿⠟⣡⠞⣴⡋⠁⢈⡻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣭⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣮⡳⣤⣀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠯⣭⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡦⣍⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣠⣤⣄⣄⡀⠀⠀⠀⢀⣠⣤⣴⣶⡄⡀⠘⣏⢿⣦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠁⢉⣉⣙⢻⡄⠀⠀⠈⠉⣉⣭⣀⠀⠀⠀⠻⣧⠡⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣄⢸⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢸⣾⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡗⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠀⠀⠔⠊⠀⠀⠀⠀⠊⠀⠲⢄⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢄⣀⣀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠀⠀⠈⢿⠿⠿⠿⠿⢿⠏⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠌⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠉⠛⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣦⣀⣠⣴⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⢻⠏⠀⠘⣿⠋⠉⠉⠉⠉⠙⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠑⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣤⡀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣿⣿⣿⣿⣿⠛⠉⠁⠈⠉⠉⠉⠛⢿⣦⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡾⠅⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⡀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣯⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠃⢻⣿⡵⠖⠲⢶⣤⡄⠀⠀⢠⣤⣶⠞⢻⣿⢹⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⠀⣾⣿⠇⣴⣶⣿⣿⡧⠀⠀⠼⢿⣿⣟⠿⢿⣿⣟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⡇⢻⣿⡆⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢿⣿⣧⠀⠀⠀⠈⠓⠐⠚⠁⠀⠀⠀⣿⣧⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠿⣿⢸⣿⣿⡆⠀⠀⣤⣶⣶⣶⡶⠂⠀⢸⣿⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⢹⡀⣿⣿⣿⡀⠀⠀⠉⠛⠉⠁⠀⣠⣿⣿⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⢸⡇⣿⣿⣿⣿⠷⢄⣀⠀⣀⣤⣾⣿⣿⢿⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⡇⢹⣿⣿⣿⠀⠀⠈⠉⢁⢿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢹⣿⣿⠀⠀⡇⢸⣿⣿⡿⠀⠀⠀⠀⠀⠈⣿⣿⣿⠸⣿⣿⣿⡘⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢸⣿⣿⠀⠀⡇⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⣿⣿⣿⡇⣿⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⢸⣿⣿⠀⠀⡇⢸⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⢱⣿⠀⣿⣿⣿⣧⢸⣿⡸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣸⢹⣿⠀⠀⡇⢸⣿⣿⣿⠀⠀⠀⠀⢠⣿⡿⢸⣿⠀⣿⡇⢻⣿⡘⣿⡇⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣿⠇⠀⣿⢸⣿⠀⢠⠃⢸⣿⣿⣿⠀⠀⠀⠀⠘⣿⠃⣿⣿⠀⣿⣧⢸⣿⡇⣿⣿⠸⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣼⡿⠀⢀⡏⣸⣿⠀⣸⠀⣸⣿⣿⣿⠀⠀⠀⠀⣸⡿⢠⣿⠇⠀⣿⣿⠸⣿⣧⢸⣿⡇⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⢱⣿⠃⠀⢸⠇⣿⡏⠀⡏⠀⣿⣿⣿⣿⠀⠀⠀⣠⣿⠇⢸⣿⡄⠀⠘⣿⠀⣿⣿⠀⣿⣷⠸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢣⣿⡏⠀⠀⣾⢸⣿⠁⢸⠃⢰⣿⣿⣿⡿⠀⠀⢀⣿⣿⠀⣿⣿⡇⠀⠀⣿⡇⢹⣿⡇⢹⣿⡆⢿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏⣼⣿⠁⠀⢰⡇⣾⠇⢀⡟⠀⣾⣿⣿⣿⣧⡀⢀⣼⣿⡇⠀⣿⣿⢳⠀⠀⢿⡇⢸⣿⣷⠈⣿⣇⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⢱⣿⡇⠀⠀⣼⢱⡿⠀⢸⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣸⡄⠀⢸⣿⠈⣿⣿⡀⢿⣿⡜⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡏⣾⣿⠃⠀⢀⡿⣼⠇⠀⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⡇⠀⠈⣿⡄⣿⣿⡇⢸⣿⡗⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⠁⣿⡿⠀⠀⢸⡇⣿⠀⠀⡇⠀⢸⣿⣿⣿⢿⣿⣿⣿⣿⡇⠀⢸⣿⣿⠏⣷⠀⠀⣿⡇⢻⣿⣷⠘⣿⣇⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⡿⢸⣿⡇⠀⠀⢸⢸⡟⠀⢰⡇⠀⢸⣿⣿⡏⢹⠿⠋⣼⣿⡇⠀⢸⣿⣿⣷⣻⠀⠀⢿⣇⢸⣿⣿⠀⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⠀⢘⢸⡇⠀⠸⠁⠀⢸⣿⣿⠇⢸⣿⣦⡙⣿⡇⠀⠈⣿⣿⣿⣿⠀⠀⢸⣿⢸⣿⡿⠀⢿⣿⢸⣿⡟⣿⡇
⣀⣀⣀⣀⣀⣸⣿⣇⣼⣿⣧⣤⣀⣼⣾⣇⣀⣠⣄⣀⣼⣿⣿⣧⣼⣍⣻⣷⣿⣧⣀⣀⣿⣿⣿⣿⣄⣀⣸⣿⣼⣿⣧⣀⣬⣿⣧⣿⣧⣿⣿⣀⣀⣀⣀
⣿⣻⣛⣛⣻⣫⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣿⣯⣯⣿⣿⣭⣿

⢄⣀⠄⢀⡀⠠⡀⠁⠀⠀⢀⣀⣤⣀⣀⡀⠉⠉⠁⠀⣀⡀⠀⠀⠀⠐⠄⠀⡤⣄⠀⠘⢠⠀⠌⠀⠁⠀⠀⠀⠁⠀⠹⠂⠀⢀
⠀⠀⠀⠚⠂⠤⠈⡄⠀⢰⠋⠀⡚⠻⡿⠞⡧⠀⠀⡐⠉⠉⠑⡄⠀⠀⠀⢸⠁⠀⢱⠀⠀⠁⠀⠐⢀⠀⠋⣀⣀⣀⣠⠁⠀⠘⢣
⠀⠀⠀⠁⠀⡀⠀⠁⠀⡀⠀⠀⢀⡀⠀⠀⠂⠀⠀⠃⠀⣀⠀⠇⠀⠀⠀⠘⠀⡀⠘⠀⠀⡁⠀⢾⠉⣁⠉⠗⠀⠀⠘⠄⢀⡀⠸
⠀⠀⠀⠀⣀⣀⡀⠀⠀⡃⢀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⢀⠈⠉⠉⠉⠉
⠀⢀⠢⠀⠉⠉⠁⠀⣀⣑⡜⠄⠀⠀⠩⠴⠈⠡⠀⠀⠐⠒⠓⠀⢀⡀⠀⠂⠀⠀⠀⠂⣤⣤⠀⠀⡦⢀⡴⢤⣀⡠⢀
⠀⠘⡂⢀⠀⠀⠀⠀⠈⠂⠛⡆⠀⠀⢀⡐⠀⠂⠀⠀⠀⠀⠀⠠⠁⠁⠚⠂⠀⠀⠀⣸⠀⠂⢸⡀⠀⠀⠀⡜⠉⠓⢺⡀
⠀⡀⠀⠂⠀⠀⠀⠘⡈⣉⢉⠃⠀⠀⠈⢧⠀⠀⠰⠀⠀⠀⠀⠃⢀⣄⠀⠇⠀⠀⠀⢿⠀⠀⢠⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠂⠀⠀⠀⠀⠀⡄⠀⣀⠀⠀⠀⠀⢈⠆⠀⢾⠀⠀⠀⠀⢠⠂⠀⠂
⣄⣀⣤⣄⣤⠠⠤⠈⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠑⠂⠈⢠⠄⠀⢄⠈⠀⠀⠈⠁⡀⣀⠀⠀⠐⠂⠀⢀⡀
⠀⠇⠀⣥⢄⢠⢰⠅⠀⠀⠀⠀⠑⠀⠀⠄⡄⠀⠀⠐⠁⠈⢹⠀⠀⠀⠀⡛⠐⠂⢀⠆⡠⡠⠄⠀⠐⠁⠀⢀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠐⠀⠈⠄⠀⡀⠀⠀⢁⠗⡀⠀⠈⢀⣠⡀⠘
⠀⠀⠀⢀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠤⣄⡀⠀⠂⠀⠀⢀⠀⠄⡀⠀⠀⠈⠀⣠⠀⠠⠄⠁⠀⠀⠀⢀⠄⣀⠀⠈⠀⡀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⢀⡀⠀⠀⣀⠀⢀
⠉⠈⣒⠀⠀⠀⢄⠙⠄⢀⡀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢚⡁⠀⠀⠀⢠⣟⠁⢹⣆⠀⠀⠀⠀⡁⠀⠜⢢⠈⠀⠀⡿
⢄⠀⠇⠀⠀⠀⢻⠀⡀⠀⠣⠀⠀⠀⢸⠀⢀⠀⡄⠀⠀⠀⠸⠋⠁⠉⡛⠀⠀⠀⢸⠟⠤⠸⠻⠀⠀⠀⠀⡆⠀⡀⢸⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢆⣀⣾⠁⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠰⡐⠀⢰⡑⠀⠀⠀⢘⠆⠒⠀⠐⠀⠀⠀⠀⠀⠈⠁
⠀⠐⠀⠀⠀⣀⠀⠉⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠂⠀⣠⢀⣘⠀⠀⠀⠌⠀⠀⡀⠀⠘⠒⠒⠀⢀⣤⡤⠔
⠆⠀⠀⠀⠀⠀⠈⠂⠀⢀⣀⠎⠑⠃⠶⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⡀⡏⠁⠈⡇⢰⠂⠀⢐⣊⠈⢶⢂⠂⠒⣶⡩⠂⠦⠤
⠀⠀⠀⢸⡀⠀⠀⣞⠀⠀⠙⠆⠀⡀⠀⠂⠀⠀⠀⠘⠄⣀⢀⠇⠀⠀⠀⠀⢃⢀⢀⠃⠈⠀⠀⣼⠿⠄⠞⡆⠀⠀⠙⢿⠀⠀⢀
⠀⠀⠀⢀⠱⠈⣴⣏⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⡆⡄⠀⢰⠇⠀⠀⠀⠀⠁
⠀⠀⡀⠈⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠁⠀⡤⠀⠠⠀⠀⠠⢄⠀⠀⢤⡀⠉⢁⠀⡘⠁⠀⠀⠀⠉
⠀⠰⠈⠠⠠⠵⢦⡁⠄⠀⠁⠆⠀⠀⢠⠀⠀⠘⢷⣆⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠩⡆⠀⠀⠀⠀⠾⠀⠀⢀
⠀⠀⢈⠜⠀⠀⠀⠱⡀⣤⠰⠂⠀⠀⠀⣀⠀⡀⢀⠇⠀⠀⠀⢉⠀⠂⠀⡇⠀⠀⠈⠱⠀⠀⠀⠃⠀⠀⠀⡎⠀⠈⢘⠚⠀⠀⠐
⠀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⠀⠀⠀⡀⠀⢈⠀⠆⠀⠀⠐⡄⠃⣤

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠈⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⣀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣶⣦⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢿⣿⢻⣷⣦⡀⠀⠀⠀⠀⠀⠀⢻⣿⢋⡀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠿⣟⣿⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣶⣶⡶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⢉⡩⠐⡀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⡀⠬⣉⠃⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢁⠔⠋⠀⠀⠀⠀⠀⠠⣀⣤⠾⠦⣤⡤⠀⠀⠀⠀⠀⠈⠲⣄⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄





⣿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣻⣿⣿⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣿⡄⠄⠀⠀⠀⠀⠀⢀⡼⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠑⢸⣿⣿⣄⢦⠀⠀⠀⡠⠋⡄⣻⣿⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡶⢭⣸⣿⣦⣄⣀⠀⠀⠀⠀⡴⢫⠖⡂⠸⣿⣿⣿⣿⣷⣮⠭⠀⠊⠐⣱⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠬⠷⠛⢉⣜⣧⡝⡻⢦⣀⢀⡠⠃⠃⠀⠄⠘⢿⣿⣿⣼⣿⠁⠀⠀⠀⣴⡛⠋⢛⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣠⠞⣉⣩⣍⢻⣦⣿⡟⠀⢠⡄⠀⠀⠀⠀⣿⣿⣿⣿⣧⡞⠋⠛⠁⠁⠘⠉⣿⣿⣿⣿⣵⣳⡀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠒⢲⡃⠨⢻⣷⠟⣻⠁⠀⠠⠿⣦⣀⠤⢤⣤⣿⣷⣿⣿⣿⣆⣀⠁⠐⠊⠄⡛⢿⣿⣿⣿⣿⣷⣿⣿⡄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠀⢐⠝⠉⣰⡏⣠⡄⢀⡀⠀⠨⢑⠤⡀⠉⠉⠙⢹⣿⣿⣷⣤⡄⠀⠀⠀⠀⠉⠒⠻⣿⣿⣿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠐⠁⠀⠀⠁⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⣺⣿⣿⡿⠿⣿⡆⠀⠀⠀⠀⠀⣐⣿⣷⣾⣿⣯⠅
⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⢀⣠⠄⡀⠀⠀⣀⠀⠰⠿⠛⡛⠓⠸⠯⠿⠽⠓⠒⠖⠻⡍⠛⠉⠉⠁⠐⠚⠒⠱⠶⠦⠆⠲⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⢀⡉⢀⢀⡀⠀⠁⠀⠉⠁⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠲⠂


⣿⣾⡾⠿⠾⠾⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⣾
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠀⠛⠶⣦⣀⢀⣀⡴⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⣸⣴⣿⣿⣿⣿⣿⣷⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⣿⣿⢻⣻⠁⠻⣿⢻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⢿⣿⣾⠿⠀⠸⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣶⣿⣿⣿⡄⠉⢁⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⢿⣿⣦⣀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⢸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⣿⣿⡯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣾⣿⠧


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⡀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⣧⣾⣷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡠
⠛⠻⠿⠿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⣹⣧⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠞⠉⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣀
⡄⠀⠠⡤⠀⢉⠁⣀⠀⠙⢧⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢨⡷⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⢁⡆⠀⠀⠀⠀⠀⠀⢠⣷⢠⣦
⣷⡀⠀⢁⡤⠀⣀⣿⡀⠀⠀⠀⣸⣧⠀⣰⡀⠀⣀⣠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠾⠱⠦⠀⠀⠀⠁⠀⠀⠀⢠⡜⠃⠀⠀⠁⠘⢦⠀⢀⠀⠀⢸⣿⣿⣿
⣿⣇⠀⠸⠁⣰⣿⣿⣖⠀⠀⠀⠿⠿⢦⣿⡅⠠⣿⡍⠀⣞⠋⠘⠃⠀⢠⡀⠀⠸⠃⠤⠀⠀⠀⠀⢀⠀⢡⣀⠀⡴⠣⢀⡖⠀⠦⠐⠀⣄⢀⠀⠚⠀⠀⠀⠀⠿⠟
⠉⠀⡀⠀⠀⣿⣿⠇⠀⠀⠈⠁⠀⠀⠸⠿⠛⠀⠈⠁⠈⠉⠀⡆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠒⠛⠠⠆⢨⣤⣵⣄⣠⣄⣆⠰⠇⢱⣿⠈⢠⠂⣴⠀⢀⠀⣤
⢀⡼⠀⡀⢸⣿⣿⡡⠀⠀⠀⠀⠀⠢⣄⢠⣶⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠛⠁⠀⢰⣿⣢⣿⠀⠻⡆
⡜⠁⢠⣿⠀⢀⡀⠀⠀⠛⠷⠖⠆⠤⠀⠈⠛⠓⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠠⠄⠀⠙⠛⣿⣿⡧⠈⠡
⠀⠀⣿⠭⣅⠀⠹⠅⠀⢀⠀⠀⠀⣄⡀⠀⠀⠀⠀⢄⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⣤⣀⡁⠲⠤⠀⠀⠀⢤⡄⠀⠀⠀⢒⡀
⠀⢠⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣣⡀⢀⠀⠀⢤⣀⡀⠈⣿⣄⣀⡀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠀⠉⢭⣦⣀⣁⣴⣶⣶⢾⡇⠀⠀⠤⢀⣄
⠀⠬⠭⠍⠉⠀⠀⠀⣀⣤⣄⣀⣠⣤⣾⣷⠶⠴⠖⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠲⠿⢳⣤⣄⣀⠉⠀⠦
⠀⠀⡀⠀⠀⠀⠁⢀⣀⣤⡾⣟⠛⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠶
⢀⣀⣤⣤⣤⣶⠿⠿⠟⠟⠛
⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⣉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠛⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣷⣄⢳⡀⠆⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣿⣧⢹⠟⢿⠀⢀⠴⣌⠉⠉⢉⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠙⣿⡻⣿⣾⠳⠀⠀⢿⣻⠏⠀⠐⣛⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠃⠀⠀⢀⣀⡤⠶⠟⠓⠉⠉⠁⠚⢋⣉⠛⠀⠀⠀⠀⠁⠐⠊⠀⣨⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠈⠁⠒⠠⠦⣈⣴⣆⠀⠀⠈⢃⣶⢤⡀⠀⠀⢀⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠄⠊⠁⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠈⡢⠿⠀⠀⠀⡰⠽⠋⠸⠀⠀⠟⠛⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣮⠀⠀⡤⢸⡗⠀⢀⠀⠠⠗⠀⣶⢠⣀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢠⠀⠉⠀⠀⢷⠈⡁⠀⣼⡁⠠⣌⠀⢁
⠀⠀⠀⠀⠀⡆⠀⢂⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⡀⢱⡛⡀⠸⠀⢼⠿⠄⠈⠁⠛⠺⠶⡦⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠰⡀⠀⠂⠀⠀⠀⠀⢀⠊⠀⠀⠠⠀⠀⠀⠹⡄⣇⠁⢀⡆⠈⢀⠀⠀⠀⠒⠄⠠⣠⡄⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢧⢲⡀⣆⢳⠹⡆⠘⢷⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣄⠈⢢⠘⡼⣌⢳⠙⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⠁⠀⠘⠜⠀⠀⣐⣀⣠⣄⠀⡀⠀⢀⣴⣶⠶⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⢿⠔⠒⡛⡃⠀⠨⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠤⠤⣀⡀⠀⠈⣷⣿⠿⠟⠋⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠤⠤⣀⡀⠀⠀⠀⠉⠉⠛⢿⠶⡴⠶⡆⠀⣶⠂⢀⡟⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⣀⡀⠘⣆⠀⠀⢀⣇⠻⣷⠞⠀⠀⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⢛⠓⠺⠛⠉⠙⡌⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⢻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠀⠀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⣾⣦⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⠁⠀⠹⣿⣿⣶⣄⠀⠀⠀⠀⢿⣿⣿⠟⠁⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⡿⠃⠀⠀⠀⠁⠹⣿⣿⣦⣀⠀⠀⠈⠛⠁⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⠀⠀⠀⠀⠀⠉⢿⣿⣷⡀⠀⠀⠀⠀⠀⣘⣿⣿⣁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠀⠀⠀⠀⠀⠈⠿⢿⣿⣦⣄⠀⠀⢨⣿⣿⣿⣯⣈⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠧⠀⣤⣽⣿⣿⣥⣼⣿⣿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿
⣠⣤⣤⣆⣿⣷⣾⣾⣶⣿⣿⣷⣷⣷⣦⣶⣶⣾⣷⣶⣦⣤⣤⣶⣤⣤⣦⣤⣤⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢒⣲⣖⣒⣲⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠇⠀⡀⢹⠀⠀⠀⠀⡼⢷⣶⣧⣾⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣈⣾⣯⠻⠀⠠⣽⣸⠼⣿⣿⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⢻⢿⣿⣿⣿⢇⡄⣼⣿⣧⣿⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣤⠀⣀⣰⣤⣿⣿⣿⣿⣿⣿⣦⣼⣿⠛⣿⣿⡷⠧⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣶⣿⣷⣿⣧⣽⣿⣿⣿⣍⣿⣿⣿⣽⣿⣅⣿⡏⠀⠀⠀⢀⠀⠀⢅⣠
⠀⠀⠀⠀⠀⠀⡄⡂⠀⠀⣴⠯⠻⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣦⣴⣤⡜⣤⣿⠟
⠀⠀⠀⠀⠀⠀⢠⣿⠶⢏⣁⡄⢻⣾⣿⣿⣿⣿⣿⣿⣯⠛⢿⣿⡿⣿⣟⢻⣏⣼⣎⣿⣿⣿⣤⠾⠀⣀⠀⢀
⠀⠀⠀⠚⣄⢂⢿⣿⣮⣉⡛⣿⣿⣿⣿⣶⣿⣿⣿⣝⢻⡗⠸⣏⢳⣸⠋⠙⢛⢿⣿⡿⡛⣹⡋⣀⢀⠡
⠀⠀⠀⠺⣽⣧⣸⣿⣿⣽⣯⣽⡿⢿⣿⣿⣿⣿⣿⣮⣿⣏⣠⢋⣼⣇⠀⠖⣹⣯⣎⡠⣞⣙⣓⠍⣍⠁
⠀⠀⠀⠠⠍⠙⠿⢿⣿⣿⣿⣿⡾⠯⠶⣿⢿⣿⢿⣿⣿⣯⡟⠁⢿⡿⠳⢤⠁⢸⣣⡞⠛⠈⠛⠛⠂⡤
⠀⠀⠀⠀⠀⠈⠀⢦⢼⣎⣙⣳⢬⣷⣺⣶⡘⠙⣦⡽⣿⣿⠀⠀⠀⣇⠀⢸⠷⣾⣹⠛⣇⠀⣜⡀⠀⠀⠀⠀⠀⠀⡠⣤⣴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣎⡛⠙⢹⣛⣿⣿⣟⠻⣿⣿⠀⠀⠀⢿⡰⠋⠀⢸⣏⠠⠀⠀⠘⢹⠛⡁⠀⠀⠀⣼⢿⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠈⣿⣦⡤⠞⠛⠳⣿⡛⢿⣿⡆⠀⠀⣼⡁⠄⠀⢸⢧⠀⠖⠙⣳⡯⢾⣇⣻⠶⢛⣿⣏⣉⣮⣻⢧⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⣧⡼⠀⠈⠳⡄⣿⡇⠀⠀⣿⠁⠄⠀⡞⢸⣧⣴⣿⣿⡟⠻⣯⣉⣴⣿⣿⡟⠛⠚⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠿⣦⠉⢔⠿⣿⣿⣂⣼⣿⡀⣹⣿⣷⣾⣿⠵⠀⢉⡁⣠⣾⣿⠿⣷⡦⠶⣤⣶⡛⠑⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢻⣖⢨⣿⣿⠴⣿⣿⣟⣽⢋⣘⣿⡒⢀⣶⠋⢠⣂⣀⣴⢻⠞⣪⣽⡇⠉⠊⠕⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣇⣼⣿⣧⣀⡽⣿⣿⣆⣿⣿⡿⠟⠋⢙⣯⣵⠏⣁⠠⡶⠛⠉⠚⢯⢉⠉⠃⠪⢯⠐⠆⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣛⠻⠟⠛⠿⣿⣀⠁⣦⣹⣿⣿⣿⣿⣷⣴⠂⣲⣿⣷⣞⣡⣾⣡⣴⣾⢦⣾⣬⠭⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠊⠁⣀⡀⠸⢢⡌⢿⣏⣹⠿⣻⣿⣿⣿⣅⣤⣹⣿⣿⣿⣿⣿⣿⣿⣾⣷⡿⣋⣨⠀⣀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠁⠀⠀⠘⡩⠄⣿⡇⠀⢸⣿⠎⠛⣯⣠⣿⣻⡿⠿⠻⠟⠻⠟⠻⠻⠛⠿⡿⠿⢶⣍⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠤⣶⠁⠂⠸⣧⠀⠺⣿⡄⣴⣿⣿⠟⠋⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠱⡑⠤⠽⡬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⣿⣿⣿⡟⠀⠀⡠⢶⠚⠛⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⡀⠰⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣿⣿⣿⣁⣠⣀⡤⠼⣀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⣴⣿⠤⠿⠿⠷⠶⡿⠿⣿⣿⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠿⠾⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣄⣀

⠀⠀⠀⣀⡠⠄⡀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠤⢍⣩⠩⣽⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣣⡄⠀⠀⠀⠀⣾⣷⣾⣂⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠨⠿⠞⠛⣟⣻⣿⣿⣧⡤⣀⡀⠀⠀⠒⣻⣿⡿⠅⠀⠀⣰⣾⣿⣿⣿⣿⣶⣦⠀⠀⠼⣿⣿⣟⣻⡿⣿⣿⡉⠃
⠀⠀⠀⢀⣀⣤⣴⣿⣿⣟⢿⣿⣿⣟⡿⢷⣤⣀⠈⠇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⢸⡏⠁⠉
⠀⠀⠐⠋⠉⠉⡬⣭⣿⠿⣟⡯⢿⣿⣷⣾⠛⠁⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠘⢃⣀⡀⣐⣦⠤
⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⢠⡄⢀⣀⠀⠉⠛⠉⠉⢹⠏⠉⠙⠋⠁⠀⠀⠀⠀⢠⣿⣿⣿⣧⣿⣯⣵⡆⡻⢶⡀
⠀⠰⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⣠⣤⣶⣿⢐⣬⠶⢉⡇⠀⠀⠀⠀⠀⠸⠇⠀⠀⣀⣴⣶⣦⡄⠀⠀⠋⠛⠛⢿⡻⡿⠿⠿⠃
⠀⠘⡿⣽⣿⣿⡿⠄⠀⠀⠀⢠⣿⣾⣿⣿⣏⣿⣿⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣜⢻⣿⣿⡟⠛⡃⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠘⠉⣉⣉⠁⠀⠀⠀⠀⠾⠘⠋⠿⢿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢯⣈⣻⣿⣿⣭⣤⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠠⣽⠦⠀⠀⠀⣠⣤⣤⣤⣤⣠⣀⡀⠈⠉⠱⡿⠉⠀⠑⠀⠠⣤⣶⣦⣤
⠀⣠⣴⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣏⣻⣇⠤⠄⠀⠀⠀⠀⠀⣾⡿⢿⣿⣿⣿⣷⣦⡄
⠀⣿⣯⠽⠿⠿⠾⣿⣿⣿⣿⡟⠿⣿⣿⣿⣀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣄⣀⠀⣴⢶⡾⣿⣧⣾⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠈⠀⠀⠀⠀⠿⠿⠯⠟⣿⠧⢴⣿⣿⣿⣿⣦⠀⠀⠉⠈⠉⠻⣻⣧⡿⠋⠉⠈⠉⠉⠁⠀⠀⠀⠈⠁⠈⢻⡿⢩⠋⠻⠟⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁

⠀⣰⠲⠋⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠲⣄⣀⣤⣤⣤⣤⣿⣿⣷⠁⠀⠀⢀⡀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣷⣀⠁⠀⠀⠀⠀⠀⠁⢀⠀⢠⡄⠀⠰⡆⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⣾⣿⠀⠀⠀⣾⠏⠘⢿⠃⢹⣿⣿⣶⡀⠀⢠⣿
⠛⠀⢀⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣻⣿⢿⠿⠻⣿⣿⢿⡟⠀⠀⠀⠉⠙⠛⠉⠀⠀⠉⠙⠉⠁⠀⠘⠿
⠠⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠻⠂⠁⠀⠀⠉⠉
⠄⠀⢀⡄⠀⠀⣼⡿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣄
⠼⣿⣿⡿⠂⠀⠉⡀⠈⠉⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣷⠉⠀⢰⢶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠸⣿⣿⡧
⠀⠻⠟⠀⠀⠀⢶⣦⠀⠀⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⠣⣄⡠⠄⠀⠀⠀⠀⠀⡄⠀⠀⠉⠛⠛⠋⠁⠀⠀⡐⠀⠀⠀⠀⠙⠉⠁
⣀⣀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠐⠲⠢⠠⠤⠠⠠⠠⠰⠁⢀⣄⠀⣀⣀⣀⣀
⣿⣿⡿⠀⠀⠀⠀⠀⠠⢶⡶⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣤⠄⠀⠀⠀⠀⠀⢰⡿⠋⠀⠻⣿⣿⣿
⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⡀⠀⠀⠀⠀⠀⠀⢠⣤⡄⢠⡟⠀⠀⠀⠀⠀⠉⠉
⣤⣤⣤⣤⣿⣿⣷⠁⠀⠀⢠⣀⢀⣠⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣰⠒⠋⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠲⣄
⣿⣿⣿⣿⣷⣾⣷⠀⠀⠀⣾⠏⠘⢿⠃⢹⣿⣿⣶⡀⠀⢠⣿⣿⣷⣀⠁⠀⠀⠀⠀⠀⠁⢀⠀⢠⡄⠀⠰⡆⠀⠀⠀⠀⠀⣠⣿
⢿⡟⠻⣿⣿⢻⡟⠀⠀⠀⠉⠙⠛⠉⠀⠀⠉⠉⠉⠁⠀⠘⠿⠿⠛⠈⢀⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣻
⠂⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢰⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣄⠀⣠⠄⠀⣠⡄⠀⠀⣼⡿⣿⣧
⠀⠀⠀⠲⣿⠉⠀⠰⢶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠸⣿⣿⡧⠟⠼⣿⣿⡿⠂⠀⠀⡀⠈⠁⢀⡀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠉⠛⠛⠋⠁⠀⠀⡘⠀⠀⠀⠀⠙⠉⠁⠀⠀⠹⠻⠀⠀⠀⢶⣦⠀⠀⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⠣⣄⡠
⠀⠀⠀⠀⠈⠐⠐⠲⠢⠢⠤⠤⠠⠠⠔⠁⢠⣄⠀⣄⣀⣀⣀⣀⣀⣀⣀⡄⠀⠀⠀⠁⠀⠀⠀⠈⠛
⠀⢀⣀⡀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⣰⡿⠋⠀⠻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠠⢶⡶⠆⠂
⠰⠿⠛⠛⠀⠀⠀⠀⠀⠁⠀⢠⣤⡄⢠⡟⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⣀⠀⠀⡀⡀

⠑⠦⣴⡞⣳⠄
⠀⠀⢿⠽⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣤⣀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⡀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⡿⠿⢿⣿⣿⣦⣤⣾⣿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣀⣈⠙⠛⠻⢿⣿⣿⣿⢶⣶⣶⡲⣤⡢⣤⣀⠀⠈⠻⣿⢹⣿⡟⠻⢌⣛⣻⣿
⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣭⣗⣄⡀⠀⠻⠿⣿⣕⡮⠿⣿⣿⣿⣿⣿⣷⣦⣴⠿⣾⡟⠋⠀⠀⠛⣿⡇
⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣳⣦⣤⣀⠙⠿⣷⣗⡻⡛⠻⢿⣿⣏⠁⠀⠈⠉⠓⠦⡴⠞⠉
⠀⠀⠀⠀⠀⠈⠙⠷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣷⣝⣷⣦⣄⣹⣿⣿⣷⣤⣉⠙⠻⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⣾⡿⠿⠭⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣾⣿⢿⣿⣻⣿⣿⠟⠁⠀⠈⠙⠻⢦⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣄⡀⠀⠀⠀⢀⣤⢖⣫⣙⠛⢿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣶⣿⣷⣾⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡉⠛⠷⣦⣤⡤⠤⠄
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣾⣵⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⠴⠂⠀⠀⠀⠀⠉⠋⠁⡄⠀⡀⠀⠀⠀⠀⢠⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⣧⠞⠛⢫⠄⠀⣄⠀⡠⠀⠓⠀⠙⠖⠦⠀⡀⠀⡷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢨⠀⠀⢀⠀⠀⠹⣧⣣⡀⠀⠀⠠⡄⢀⠀⠻⡦⢀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠇⢀⣨⣻⡀⠘⡞⢠⡀⣰⠋⠉⠉⠀⠙⠶⡿⠋⠀⡰⠋⠀⣀⠀⠀⢸⡇⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠺⡄⣠⠀⠀⠀⣑⣄⠀⠳⣄⡀⠈⢻⡇⠀⠀⠠⠀⠀⠀⡇⣠⠞⢠⠆⠀⣹⣀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⢴⠶⢼⡟⠀⠀⠀⠀⠈⠑⣤⣙⣷⣄⢸⠀⠠⣄⠀⢷⡀⣠⣿⡧⠶⠚⠋⡙⠉⠀⠈⠓⠀⠀⠈⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠱⣄⠀⢀⡄⠀⠛⠁⠀⠀⡉⢻⣇⢠⠁⠀⢨⣿⠟⡁⠀⠀⠘⢦⡷⢚⡀⠀⢰⡇⠔⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⣈⣳⣼⣟⢠⣄⡀⣹⠙⠁⢀⠹⣷⡀⣰⣿⢁⠠⠇⠒⠀⢀⡜⠀⢈⡇⠔⠊⠀⢤⡌⣒⣀
⠀⠀⠀⠀⠀⠀⠴⠀⠀⠐⠋⠀⠠⠾⠛⠦⣄⡈⢏⠉⠀⠘⠢⣹⣷⣿⠋⠀⠀⣿⣀⣤⠾⠖⠚⠉⠛⠒⢰⣄⠀⢘⠋⠁
⠀⠀⠀⠀⠀⠚⠲⠛⠀⣰⠄⠀⠀⠘⠆⣀⠀⠭⠿⠷⣦⡀⠀⠘⣿⣿⠀⣠⡴⠟⠡⠤⢀⠠⠋⠀⡄⢀⠜⠁⢲⠌⢀⡀
⠀⠀⠀⠀⠀⠀⡘⠣⡐⠚⠣⡀⠀⠀⠐⠃⢀⠀⠀⢠⡬⠿⣦⡀⣿⣿⡾⠋⠀⠀⡤⠀⠈⠃⠀⣀⣿⣫⡤⠶⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣩⡷⠶⢬⣶⣤⣤⣤⣬⣧⣤⣄⣀⡀⠹⣿⣿⣿⢁⣤⣴⡾⠿⠿⠿⣛⠋⠉⠉⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠊⠀⠀⢀⠴⣎⣉⣠⠖⠛⠛⢿⣿⣿⣿⠟⠉⠉⠉⠻⡶⠞⠋⠑⢄⣀⠀⠀⢡⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡦⢠⡀⠀⠒⡿⠁⠀⠀⢱⠁⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠘⡄⠀⣤⠀⠹⣟⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡞⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣷⣀⠈⠀⠀⠚⠁⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠂⠀⠉⠀⠀⡀⡰⠋⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠈⠣⡴⠂⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠹⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣧⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⢀⠔⢀⡠⢤⠖⡔⢢⡄⡀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣀⢠⡖⢢⠶⡤⢄⡀⠢⡀
⠀⠀⠀⡀⢁⣴⠫⡜⠣⢛⠌⡡⢎⡰⣉⠦⡁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠴⣉⢆⡱⢌⠠⣛⠜⢣⠝⣦⡈⢀
⠀⠀⠀⠀⣜⠶⣙⢎⡱⣉⠶⡑⢪⡑⣥⠓⣜⢢⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡔⣣⠚⣌⢪⡵⢊⠶⣉⢎⡱⣉⠶⣣
⠀⠀⠀⠀⣏⠒⣥⢢⡜⢢⢛⡬⡣⢜⠴⣉⠎⡱⢧⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡼⢎⠳⣙⠞⡣⢜⠤⡛⡔⢣⡔⣬⠲⣹
⠀⠀⠐⠒⢧⢙⠤⣑⢎⡱⢉⠶⡑⢎⡑⢩⠺⡔⢣⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⡜⢢⠖⡭⢊⡱⢊⠶⡡⢎⡱⣊⠤⡃⡼⢐⠂
⠀⠀⠀⠐⡀⢊⠶⣍⢊⡕⢫⠛⢤⢣⢜⡢⣛⠼⡱⢞⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠀⡳⢌⠣⣛⢄⡣⡜⣤⠛⡝⢪⡕⣉⠶⡑⠀⠂
⠀⠀⠀⠀⠈⠢⡘⠤⡓⢜⡱⢍⠶⣑⢎⡱⢭⠒⡕⠪⠀⠞⠀⠀⢀⣠⡶⠛⠁⠀⠓⠀⠕⢪⠖⡩⢎⡱⣊⠶⡹⢎⡡⢚⡤⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠈⡚⠱⣮⠒⣥⢣⡜⣣⡛⡤⠃⣰⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⢆⠘⢤⢛⣔⢣⡜⣬⠒⣵⠎⠓⠡⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠌⠉⢈⣁⡡⠐⠊⢰⠀⠀⢿⡀⠀⠀⢀⣠⡶⠶⡇⠑⠂⢌⣈⡁⠉⠡⠅⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠈⠻⠶⠾⠛⠁⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢤⠤⠤⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠄⠀⠀⠀⠈⠠⡀⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢷⣄⠜⠁⠀⠀⠀⠀⠀⠀⠈⠣⣠⡾⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡺⣦⡹⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢏⣴⢗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈




⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⡿⢿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢻⣄⠀⣤⡄⠀⣤⡄⢠⣼⠏⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣿⣧⣠⣿⣷⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣉⠉⢻⣿⣿⣿⢋⡉⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⢀⡀⣀⡀⢀⣀⢀⣼⡀⢸⣿⣿⣿⣿⣧⣴⣿⡿⠿⣿⣿⣥⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠸⠇⠀⠁⠀⠈⠁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠂⠀⠀⠀⠶⠷⠂⠀⠁⠀⠉⠉⠃⠀⠀⠛⠛⠛⠛⠣⠠⠄⠜⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣦⣴⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢻⣿⠁⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠁⠀⠀⠉⠐⠂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⠐⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⡤
⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢐⣿⣾⣷
⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠣⣽⣿⣿⣷⠀⠀⠀⠀⠰⣄⣀
⠀⠀⠀⠠⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⡇⡀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠗⣿⣿⡛⠓⠀⠀⠀⠀⠀⠉⢻
⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣿⣿⠛⠂⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠘⣃⠀⠀⠀⠀⠀⠀⠐⠀⠰⢿⣿⣿⣿⣿⣼⣂⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣠⣏⣿⠌⠙⠃⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢀⣷⡁⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠻⣿⣿⣿⡿⠶⢀⠀⠀⠈⠋⠙⢿⣿⣦⣶⣦⣄⣾⣿⠏⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠸⠟⢡⣌⠹⠙⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡤⠄⣤
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠈⠩⠭⠟⠩⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢦⣾⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢰⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡏⠁⠀⠀⠀⠀⠀⠀⠰⠞⠉⠀⠀⢠⠀⠃⡀⢮⢃⡕
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⡀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡅⢠⣤⣿⣿⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⡆⣰⡷⠉⣿⡟⡆
⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢩⣿⣾⣿⣿⡏⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣷⢸⠃⣀⡿⣿⡇
⠀⠀⢰⣼⣿⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡟⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣼⣿⡆⠀⢹⣿⣿⡇⠄
⠀⢠⣿⣿⣿⣷⣾⣇⣠⠄⠀⠠⠀⡀⠀⠀⢀⣠⠀⠀⢸⣿⡿⣿⡭⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢠⣏⡿⢃⣶⢈⣿⡿⡇
⠀⠈⠿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣤⡤⠷⠛⠋⠁⠀⠀⠹⢻⣟⠻⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠃⢠⡿⣻⣇⠀⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠛⠘⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡃⣼⣾⡷⢰⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⡏⠁⠄⠀⠀⠰⣴⣤⡆⠀⠀⠀⠀⠀⠀⣀⠆⢰⡄⣧⠀⣿⣽⣿⣿⠁⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⡀⢠⣶⣯⣿⡿⠀⠂⠀⠀⠀⠀⣽⠇⡈⣴⠏⢰⣿⣿⡟⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠁⠀⠀⢠⠵⡋⣗⠀⢀⠦⠊⣠⠟⠐⣹⠊⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠂⠀⢀⠀⠀⠁⠻⠴⠛⠒⢀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠀⣆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢖⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⢉⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠿⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣿⣿⣟⣦⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣆⡀⠀⢰⣷
⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⣾⣿⣿⣟⠉⠻⢿⣗⣦⡀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⠀⣤⣾⣿⣿⣿⣿⣿⣿⣦⣾⣿⣼⡇⢀
⠀⠀⠀⠀⣧⣿⣧⣦⣾⣿⣻⣛⣽⡿⢷⣄⣀⣰⣼⣿⣷⣾⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠾⣿⣭⣉⡙⠛⠿⠟⠙⢺⣿⣿⣷⣿
⠀⠀⠠⣤⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠈⠉⠛⠛⠿⣿⣽⣿⣿⡷⠦⠀⠀⠀⢀⣚⣏⣽⣿⠿⠿⠿⠿⠷⠶⠤⠤⣤⡤⢤⠴⠶⠿⠿⠿⠿⠿⠿⢦



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣄⣀⣴⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢀⣠⡀⢀⣴⣷⣦⣤⣤⣴⣿⣵⣾⣿⣿⣿⣦⣀⡀⣀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣄
⠀⠐⠛⢿⣿⣿⠉⠙⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠉⠋⣹⣿⣿⣿⣷⠴⠀⠀⠀⢀⣾⣷⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⣷⣄
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⡠⠔⣁⣻⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠏⠱⣴⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣛⣋⣳⣄⠀⠀⠀⠔⠋⠹⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠹⡶⡀⠀⠀⠀⠀⠀⠘⠁⠀⠂⠀⠀⡠
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠶⠀⠠⠞⠃⠀⠰⠧⠄⠀⠀⠀⠀⠀⢀⠀⢀⣀⣴⣂⢀⣤⣄⣀⣀⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⢀⡉⠀⣤⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠣⣀
⠀⠀⠀⠀⠀⠀⠃⠀⣈⡏⠀⠀⣒⡀⠀⠀⠀⠀⠀⠀⠒⢂⣀⠀⠈⢷⡐⠀⠀⠈⢢
⠀⠀⠀⠀⡎⠀⠀⢀⣿⠀⠀⠾⣿⠿⠀⠀⠀⠀⠀⢁⡴⢿⣾⠗⢀⢸⣿⠆⢀⡄⠈⠣
⠀⠀⠀⢰⢇⡄⠀⠘⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⢿⡇⠘⠂⠀⠀⠀⠄
⠀⠀⠀⣾⣋⠀⠀⠀⣿⠀⠒⠁⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠁⡁⣻⣿⡀⡀⠀⠀⡈⢚⠂
⠀⠀⠀⣿⣿⣿⣤⣤⢸⣾⠁⣦⠀⠀⢰⣿⣿⣿⣿⣿⠆⠀⠀⢰⣴⢸⣯⣿⡏⣿⣧⣼⡏⣾⠟
⠀⠀⠀⢿⣟⣏⣾⡾⢸⡜⠦⠹⡗⠀⠈⠙⠿⡿⠟⠋⠀⠀⢠⠟⠠⢾⠟⢿⡇⣻⣿⠄⡜⡇
⠀⠀⠀⠀⠘⣿⣿⡇⡼⢣⡀⠀⢰⣄⠀⡀⣰⣤⣀⣀⠀⣀⢟⡄⠠⠢⠐⣾⣿⢹⣿⣖⡽
⠀⠀⠀⠀⠀⠈⠿⣿⠛⠋⠀⠰⠌⢻⣿⡟⠉⠙⠛⢙⣿⡕⠉⠀⠀⢀⠀⠐⢽⡎⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠘⣷⣄⣀⣠⡿⠏⠀⠀⠀⠀⢄⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣆⢀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢸⠀⢈⣦⣼⡿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠙⠋⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠈⠈⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠹⢶⣄⣷⣠⣀⣤⣴⠄⠘⠃⠀⠀⠀⠀⠉⠀⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⠀⠀⠀⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔

⠀⠀⠀⠀⣤⠔⢠⠀⢀⡄⣀
⠀⠀⠀⣦⠅⠀⡀⡖⣞⡘⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⣊⠠⢆⠐⢶⠹⣠⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠄
⠀⠀⠘⢄⡢⣦⢠⡾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢆⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠓
⠀⠀⠀⠀⠑⣤⣤⣻⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠉⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠠⡄⠀⢀⣤⢄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣄⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣌⠘⣈⠳
⠀⠀⢲⡉⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠸⠦⠶⢢⡝⠘⢐⡀⠀⠀⠀⠀⠀⡄⡀
⠀⣅⡙⠡⡿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠦⢢⣱⡁⣢⣌⣛⢎⡀⠀⠀⠀⠀⢼⡏⢦
⠀⠼⢛⡏⣓⣍⠀⠀⠀⠀⠀⠀⢠⢾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢅⠨⠃⣓⠍⡸⡄⡀⠀⠀⠀⠀⣿⡇⡿⠆
⠀⢃⣻⡇⡇⢸⠀⠀⠀⠀⠀⠀⢘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡕⠀⠐⠂⠈⠀⠀⠀⠀⢈⡿⠏⣽⣱
⠀⠈⢾⡇⣟⠛⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢌⢾⠣⣉⢵
⠀⠁⣸⡶⡇⠚⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⢞⠺⠡⠘⢏
⠀⠀⠙⡆⠉⠀⠀⠀⠀⠀⠀⣽⣯⣿⣷⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢲
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣾⣟⣷⡯⢿⣷⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣚⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⡀⢼⠀⠀⠀⠀⠀⠀⠀⢐⣀⠀⠀⠀⠀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠛⣁⣁⠀⠀⠀⡤⠰⠐⡀⠄⠀⠀⠀⠀⠈⠋⠈⠀⠀⠀⠉⠁⠄⠀⠀⠀⠀⠀⠀⠻⠓⠀⠀⠀⢤⡴
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⠀⢀⠘⠊⠀⢠⠗⠂⠀⠀⠀⠀⢄⠀⠁⠀⡄⠈⠀⣀⠆⠀⠀⢰⡧⠀⠠⡄⠀⠁⢀⠐⠀⢐
⠀⠀⠀⠀⢠⠄⠀⢈⠖⠤⡤⢤⣮⠤⠈⠉⠀⠀⢀⡀⠀⠀⠉⠃⠠⢤⢥⢤⡔⠋⠀⠀⠀⡊⠁⠀⠀⠙⠀⠀⠉⠀⠴⡊⠌
⠀⠀⠀⠀⠀⣦⡅⠀⠀⠐⠀⠀⠀⠀⠢⠀⠀⠀⠘⣃⠀⠀⠀⡐⠀⠀⠁⠀⠀⡀⠀⠀⠀⠛⠃⠀⠀⠴⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠠⠄⠀⠀⡍⠀⠀⠀⠀⠀⢁⠀⠀⠀⠉⣠⠀⢀⡃⠀⠀⠀⠀⠀⠱⠀⠀⠀⠶⠀⠀⣴⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠰⠀⠠⠄⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠈⠺⢺⠁⠀⠀⠀⠀⡄⠀⠇⠀⠀⠰⢆⢤⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠂⡓⠠⠀⠀⢀⢀⡀⢀⣀⠜⠀⠀⠀⠀⠀⠈⠂⢸⠀⠀⡀⣇⡀⠂⠀⠀⠈⠼⠆⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠄⠀⠧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠧⠇⠀⠀⠀⠀⠀⠀⠑⠲⠂⠀⠀⠶⠖⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠛⠑⠂⠀⠀⠙⠋⡀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡐⢠⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠉⠀⠀⠀⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠳⡐⠂⠀⠂⠀⡠⠂⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠁⠄⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⠀⢠⢀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢘⠂⠀⠁⢰⡈⠀⠈⢥
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣴⣿⡄⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣸⣿⣿⢃⠀⠀⠀⠀⠑⡀⠀⠀⠀⠸⡇⠀⣉⠓⠦⠄⠀⠀⠀⠚⠁⡁
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠿⠟⠙⠁⠀⠀⠀⠀⣴⡀⠀⠀⢘⡅⢀⠉⠓⠂⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⢀⣆⠈⠀⠀⡰⠀⠐⠀⠘⠇⠀⠀⠀⠀⢣⣦⠀⠀⠀⠀⡀⠀⠀⣨⠄⡀⠀⠈⣞⠺⠙⠓⠀⠀⠀⠀⠀⢐⠲⠇
⠀⠀⠀⠀⠀⢸⣿⡐⢢⣠⠠⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣀⡐⠀⠲⠠⠁⢀⠄⠠⠀⠀⠀⠀⠛⢥⠀⡄⠀⢠⠀⡨⠃
⠀⠀⠀⠀⠀⠈⠻⢿⣷⣶⣦⣬⣄⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⠀⠀⠀⢶⠶⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⣀⠀⣿⡿⣲⠀⠀⢀⣀⠀⠀⠀⢀⢴⣥⣃⠁⢀⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣹⣿⡿⠶⡞⢻⣟⠩⢰⣿⠸⢿⣠⣷⡂⣼⢿⣯⣄⣀⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣍⣅⡄⠔⡶⠙⢄⡘⠋⣞⠺⣿⣿⡉⢻⠚⡏⠰⢾⣿⡩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠙⠆⠁⠀⠺⠀⡄⢠⢸⣿⣿⢦⡏⡁⢠⣯⠛⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⢌⡒⠾⠘⣷⣿⡟⢾⠳⡇⢸⢋⡐⢛⠛⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠛⣫⣀⣷⣘⠙⣷⠀⠀⢸⢣⡞⣉⡥⡶⢲⡶⣶⣿⣌⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠁⠀⠀⠀⠛⠿⣿⣏⣹⡆⠀⢈⣾⣧⣿⢶⣴⣯⣶⠷⠎⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡉⠑⠀⠀⠀⠙⢻⣿⢿⡇⠀⠀⣾⣿⣿⡿⢿⣿⣿⡗⠾⢿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⠦⠀⠀⠀⠋⠁⠉⠋⠴⣶⡿⠿⠿⣿⣆⡙⠹⠶⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢓⡀⠀⠀⠀⠀⠀⣈⡋⠀⠀⠀⠛⠋⣹⣷⣿⣿⣟⣶⣟⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠢⣄⣀⠀⠀⠀⠉⠉⠓⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣟⠻⣶⣤⣤⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢷⣾⣿⣿⣿⣿⣾⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣱⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢠⠎⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣸⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠿⣿⢟⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠘⠁⢈⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⣸⣿⡏⣿⣿⡿⢿⣿⣿⣭⣿⣿⢿⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠹⠛⠻⢻⣥⡾⣿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡺⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⣿⠑⠰⢭⣽⣷⣴⣶⣤⣤⣄⢀⣀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣮⠳⠀⠚⠋⠉⠉⠉⠉⠉⠁⠀⡿⠿⣿⠂
⠀⠹⠟⠦⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟⣩⠀⣿⣿⡄⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠉⠑⠀⢤⣄⣀⠀⠀⠀⣠⣄⠀⠀⢸⠛⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠿⠿⠂⠀⠈⠑⠯⠄⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⣀⡭⠵⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣤⣶⣶⡤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣼⣿⡿⣿⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠾⠃⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠤⢖⡂⠠⣿⣯⡿⠐⢯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣟⡳⣿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣤⣤⣄⣀⠐⠀⠤⠤⣀⣠⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣦⣤⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢛⠖⠯⡿⠶⣤⡙⣦⠀⠀⠀⠀⣰⡿⢫⣉⡉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠉⡉⠚⠛⡒⠶⣝⣾⠂⠀⠀⢸⡿⣉⡟⡾⣽⡃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣼⣾⠀⠁⠀⠀⠁⠀⠩⣻⠀⠀⠀⣿⡇⡾⣽⣟⡟⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⡠⠖⠒⠒⠛⣄⡀⠐⠃⣀⡴⠋⠁⠀⠀⠀⢸⡱⣌⣛⣚⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠓⠊⠀⠀⣠⣴⠁⠀⠙⢿⣻⡅⠀⠀⠀⠀⠀⠀⠀⣿⣋⣿⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⡞⠙⢄⠀⠀⢠⠾⣧⠀⠀⠀⠀⠀⢀⣀⣟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠞⠀⠀⠀⠙⠒⢧⡀⠈⠓⠒⠒⠒⠚⠁⠈⠁⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⣇⠀⠀⠀⠀⠀⠀⠀⡹⠢⢤⣀⡤⠴⠒⢒⡶⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣀⠀⠈⠑⠦⢤⣀⣀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠁⢠⠆⠀⠀⠀⠀⠀⠁⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠙⠒⣧⣀⣀⡟⠀⠀⢧⠀⠘⣎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣻⡛⠒⠢⣤⣠⡇⠀⠈⠉⡷⢲⡚⠛⠉⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡯⣿⡿⠢⣴⠇⠙⠀⠀⠀⡼⠁⠀⠳⣄⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⡹⠁⠀⠈⠑⢦⣠⠴⠚⠀⠀⠀⠀⠈⠳⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠞⠉⢳⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⢙⣶⣿⠶⠖⡻⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠻⣍⣀⣀⢴⡯⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠉⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⠘⠓⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣿⣾⣿⡿⣿⣦⠀⣠⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⣛⣿⣿⣟⣿⣶⣿⡟⣯⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣟⣻⣿⣿⣽⣿⣿⣿⣧⣴⡟⠈⠛⢛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠿⠍⡻⢿⣿⣿⡿⣿⣿⡆⣵⢿⣿⣏⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣟⣩⣾⣿⣿⣿⣿⣿⣿⣧⣜⣟⠿⣿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣥⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣿⣿⣿⡿⢻⣟⣛⣻⣟⢻⣿⢿⣿⣾⣿⣿⣽⣿⣯⣅⣤⢌⣋⣠⣄
⠀⠀⠀⠀⠀⠀⢠⣤⣖⣛⣉⣭⣽⠟⢻⡽⠿⣿⣿⣷⣿⣿⣯⣿⣿⣿⣿⣯⣿⣿⣝⡒⠛⠛⠁
⠀⠀⠀⠀⠀⠠⠚⠿⠿⠿⢲⣾⡭⢩⣿⣷⣿⣿⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣉⢙⡁⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣶⢿⣾⡿⠿⠿⠷⣿⣙⣻⣿⣿⡏⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢶⡾⠿⠛⣫⣭⣴⠞⣿⣿⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠒⠓⠚⠛⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣻⣿⣿⣿⣷⣶⣬⣑⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⠉⠋⠛⢻⣿⢿⡉⠛⠙⠻⠿⠿⠟⠂⠛⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠟⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣚⠛⠛⠻⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠉⠄⠉⡍⠙⠛⠛⠉⠉⠋⠉⠁


⣿⣿⠻⠿⢻⠿⡿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠃⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⢀⡔⣛⣳⣦⠀⠀⠀⠀⠀⠀⠀⠨⢏⡀⡄⠀⠀⠀⠀⢠⡞⣭⣭⢳
⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⣾⢰⣿⠿⡾⠀⠀⠀⠀⠀⠀⠀⡀⠀⣷⡿⡄⠀⠀⠀⣼⡸⢿⣟⡾
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢠⡯⠷⠚⠋⠁⠀⠀⠀⠀⠀⠀⢸⣧⠀⢁⠀⢄⠀⠀⣠⠷⠛⠋⠉
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠛⠦⣄⡁⠐⢂⣺⠟
⠀⠀⠀⢠⣿⣿⣿⣿⠉⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠂⠀⠠
⠀⠀⠀⢿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣀⠀⠐
⠀⠀⠀⠘⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡁⠈⢻
⠀⠀⠀⠀⠘⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘
⠀⠀⠀⠀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡟⢀⠂
⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠀⠌
⠀⠀⣤⣿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⢿⢀
⠀⠀⠙⠻⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⣍⣓⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠉⠙⠫⢉⠉⠁⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠩⡀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠉⠒⢄⡀
⠀⠀⠀⢀⠄⣀⠁⢠⡠⠦⠄⣀⣀⠀⡀⠀⠀⠀⠀⠀⠠⢊⠰⠄⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠙⠂
⠀⠀⢀⡀⢣⡀⠳⡀⣡⠄⠛⣀⠰⡜⠂⣕⠀⠀⠀⡔⢁⠎⠀⢠⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢠
⢀⢀⠎⠋⡀⠱⠀⠙⡀⠰⠂⠉⣄⠘⢎⠈⡷⢡⠎⢀⠀⠉⠁⠈⠁⠐⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠀⢈⠉⠁
⡈⡌⠰⠀⢁⡤⠚⠀⠀⡄⠘⢅⠠⠖⠈⣁⠀⢾⡁⠈⢁⠀⠀⠀⠠⠄⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠠⠄⠀⠀⠀⡨⠁
⠇⣳⡄⠚⣀⠰⠄⠉⣄⠰⠆⠉⣤⠔⢊⣠⢖⣁⣀⡀⣩⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠙⠂⣠⠶
⠇⣄⠰⠆⠉⣄⠘⢆⠀⣦⠘⢃⡀⢶⠈⢡⡀⢖⣧⠔⠁⢀⣈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⠁⠀⠛⠉⡛⡅
⠠⢹⡄⠘⢃⡈⠆⠈⢃⠀⠖⠈⣠⠀⠓⡀⠰⢟⢱⠂⠁⠹⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠱⠀⠀⢈⢀⠇
⠀⠡⠠⠆⠉⣤⠐⢋⠀⠤⠀⠃⣀⠴⠊⢁⡀⠸⠋⢆⠀⡇⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⢠⢠⠏⡏⠀⠀⠀⠀⠀⣀
⠀⠀⠁⠚⢃⡀⠶⠊⢡⡀⠶⠈⢡⡤⠚⣁⡠⢼⠀⠈⢄⣅⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⠀⡇⠀⠀⢀⠀⠉⠀⠀⠁
⠀⠀⠀⠐⠌⠣⡀⠓⠀⠠⠄⠛⣀⠴⠆⠉⣄⡼⠀⠀⠈⡟⠑⠢⠄⡀⠀⠀⠀⠀⠑⠀⠊⠀⠀⠀⠀⣀⠠⠔⠛⡈⠀⠀⠰⡀⠀⠸⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠠⣐⠡⣄⡰⡂⢉⣤⢘⠵⢯⣀⢀⣠⡜⠀⠀⢀⠤⠄⠛⠖⢖⠒⠂⠀⠒⠒⠲⠚⠓⠀⠄⠀⠀⠸⡦⣀⢀⣨⡦⠀⠠⡀⢀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠩⠭⠀⠐⠀⠁⠀⠰⢊⠀⠀⠀⠁⠀⡇⠀⠀⠠⠁⡀⠀⠀⠄⠐⠁⠀⠀⠀⠀⢄⠀⣈⠖⠀⠁⠀⠀⡠⠀⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⡉⠋⠉⠀⠀⠃⠀⠀⠀⠐⠄⠤⢈⠊⠀⠀⠀⠀⠀⠀⠀⠉⠤⠂⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⣠⠔⠋⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠘⠂⠠⠄⢀⣀⠀⠀⢀⣀⠀⠠⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢒⠀⠤⠀⠀⢀⣀⣀⣀⣀⠀⠀⠤⠄⡐⠢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠐⠒⡒⠒⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢃⣀⡠⠄⡏⠢⢀⢀⠌⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⣠⠿⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⣀⡴⠃⠀⠙⢦⡀⠀⣀⡼⠃
⣶⣶⣤⣶⣦⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣤⣶⣿⣷⣴⣶⣶⣶⣦⣾⣿⣶⣤⣶⣶⣶⣶⣶⣶⣶⣦⣶⣴⣶⣶⣤⣶⣶
⣿⣿⣇⣀⣁⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠯⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⣀⠀⠉⢳⠢⠄⠄⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠚⠋⠙⢷⣌⣀⣀⣀⣤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡒⡪⢊⠍⢠⠋⠀⠀⠈⠛⠲⣒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡖⢳⡀⠘⢄⠀⠀⠀⠀⠀⡀⠑⢍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⢄⡋⠒⠄⣉⠉⠉⠁⠐⠢⡄⠈⠆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠙⠒⠤⢄⡉⢴⢤⠔⠒⠓⠒⡼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡄⠀⠀⠘⢣⣼⠞⠴⠐⢺⣝⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠁⣠⠞⠀⠀⡰⠡⠊⡌⠀⠀⡻⡏⠉⢹⡤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣙⠏⠁⠀⠀⠀⡰⠱⠁⢰⠀⡠⠊⠀⠈⠁⠊⠀⠀⠀⠉⠉⠉⠙⠛⠒⠈⠋⠓⠲⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠰⠁⡇⢀⠆⠀⠣⡀⠀⠀⠀⠀⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡇⠀⠐⣎⠀⠀⠀⠈⠂⣄⠤⠊⣁⠔⠊⠀⠀⢀⡠⠔⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠃⠀⡘⠀⠑⢢⣤⠔⢉⡠⢓⣍⠀⠀⣀⠴⠊⣁⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢸⠀⠀⣃⡠⠒⣁⠤⠛⠓⣶⠋⣨⠖⢉⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⢠⢾⣶⠾⣁⠔⠊⠁⠀⢀⡮⢓⣽⠡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢀⡰⠟⡫⠕⠈⠀⠀⠀⡠⠒⣁⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⡡⠔⠉⠀⠀⠀⡠⠒⣩⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⣁⠔⠉⠀⠀⢀⡠⠐⡩⠔⠉
⠀⠀⠀⠀⠀⢠⠒⠐⠂⣁⠔⠊⠀⠀⢀⡠⠒⢁⠔⠊
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠨⣁⣈⡡⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡈⢹⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⣤⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠒⠂⠀⢠⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠰⠀⡀⡀⠀⢆⡀⠉⠐⠈⡄⠓⠂⠄⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠔⠀⠀⢧⠀⠊⠀⠀⠤⠀⠀⠠⠠⡀⠀⢘⡄⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠀⢰⠀⠀⠀⠀⡀⠀⠀⠀⢁⠗⠐⠃⠀⠙⠦⡤⢤⡈⠀⠀⠀⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠉⠀⢎⠀⠀⠈⠃⠑⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢫⠉⠢⠤⣄⡠⣄⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈⠀⠀⠀⠀⠠⢀⡀⠌⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⠢⠈⡀⢈⡕⡆⠏⢑⣃⡄⠀⠀⠀⡀⣀⡇⠘⠊⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠟⠂⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠈⡇⠀⠀⠀⠁⢘⢠⠇⠳⠙⠊⠉⢙⣒⣲⠁⠧⠀⡀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠈⠂⡐⠀⠀⠀⠀⠀⠤⢄⠀⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠲⣀⢀⠤⡇⠉⠀⠁⠀⠀⠀⠉⢇⠀⣹
⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠓⠃⠀⠈⠨⠀⠀⢠⣀⢌⡠⠃⠀⠂⠠⡀
⠀⠀⠀⠀⠀⠈⠂⠐⠆⣨⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠀⠊⠁⠀⡀⠀⠀⠀⠀⠀⠠⢀⠀⠁⠀⠀⠀⠀⠀⠂⠆⠀⠀⢀⡀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠄⠠⠀⠄⠀⠉⡑⠀⠀⠀⠀⠠⠀⠀⠠⠀⠝⠐⡃⡀⠀⠁⠀⢢⠈
⠀⠀⠀⠀⠀⠀⠤⠚⠀⠁⢄⠀⢀⡁⠠⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⠀⠉⠉⠶⠚⢛⠄⡔⠠⡆⠀⠀⠀⠀⠀⠉⠁⠀⠈⡦⢀
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠸⠀⠀⠀⡈⠀⠀⠀⣤⡰⠄⠎⠉⠉⠇⠀⠈⠉⠙⠐⠘⡔⠉⠀⢧⡀⠰⠄⡖⠦⠤⠄⠀⠠⠦⠈⠔⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⢠⡀⢀⠀⠀⠀⠈⠙⠆⠶⡑⡄⠴⠒⠂⠀⠀⠀⠀⠀⠀⠈⠂⠑⠚⠤⢰⠇⠄⠐⠣⠄⡀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⡀⢀⠄⠠⠙⠁⠘⣀⣤⠀⠄⠦⠐⠁⠀⠈⠢⣀⠰⡄⣄⣀⣄⡞⣄⢠⣰⣂⣂⠔⠉⠋⠈⠀⠠⢌⠑⠒⠒⠘⢆
⠀⠀⠀⠀⠀⠈⠀⢂⡀⢀⠤⠀⢄⡨⠅⠀⠀⠀⠀⠀⠀⠀⡄⣁⣀⡀⠀⢁⡊⢀⡅⠬⠂⣉⠅⢠⣠⢄⡠⡠⢀⠘⡢⠀⠐⠀⢀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠐⠓⠊⠒⡈⢒⠆⠠⡀⠎⠈⠁⠋⣠⠒⠩⡀⣄⣠⠀⢄⢠⡄⠛⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠁⠀⠁⠀⠀⠀⠢⠈⠀⡘⠀⠑⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠘⠂⠇⠐⠐⠲⠦⠋⢂⠧⠘⠛⠃⠿⠤⢴⣠⡄⣰⢶⡏⠦⠸⡔⢴⡶⡰⡧⠠⠼⠶⡼⠆⢲⠶⠶⢤⠤⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠐⠒⠊⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠶⢠⠶⡶⢦⡄⢠⠆⣴⣦⢰⡴⣄⢠⢤⠄⢠⣶⣤⠦⣤⠄⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉⠉⠁⠈⠈⠉⠉⠉⠉⠁⠈⠈⠁⠈⠉⠉⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠁⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠠⡔⠀⠀⠀⣀⠀⠀⠀⠀⣠⣄⣀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⠊⠀⠈⠁⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸
⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠁⠒⠒⠂⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠈⠀⠀⠚⠛⠀⠀⠈
⠀⠀⠀⡠⠊⠀⠀⠀⠑⢄⢀⠠⠀⠒⠒⠒⠒⠀⠄⣀⠊⠀⠀⠀⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢐⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⡈⠙⢦⡀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠒⠠⠤⠀⠄⠀⠄⠒⠀⠈
⠀⠀⠀⠀⠈⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⢀⡠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠁⠈⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠠⠐⠈⠰⠶⠀⠐⠶⠀⠁⠀⠄⢀⣀⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⡁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⡁⠀⠀⢃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣶⣾⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠑⣀
⠀⠀⠀⠀⠀⠈⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠊⠁⠁⠁⢈⠒⠠⠄⠂⠁⠚⠀⢄⠀⠀⣠⣾⣿⠿⠛⠛⠛⠛⠛⠛⠻⠿⣿⣦⡀⠀⠀⠀⠀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⢀⣀⡄⠂⠀⠀⠀⠀⠀⠀⠘⠡⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠈⠻⢿⡄⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠙⠉⠉⠉⠀⠀⠈⠂⠀⠀⠀⠀⠴⢀⠀⠀⠀⢀⠐⠒⠠⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠱⣄⡀⠀⠀⠀⠀⠀⠀⡄⠔⠉⠀⡀⢀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣠⠂⠠⠁⠀⠀⠀⢀⡀⠠⠤⠀⠀⠀⠤⠤⠀⡀⠀⠀⠀⠡
⠀⠀⠀⠠⠤⠊⠁⠉⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠤⠱⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⡶⠂⠁⠀⠀⠀⠶⠆⠀⠶⠆⠀⠀⠀⠈⠀⢦⡴⠀⠀⠀⠨
⠀⡀⠂⠀⠀⠢⠀⠂⠀⢀⡀⠰⠀⠀⠀⠀⠀⠀⣶⡀⡄⠀⡀⠀⢠⠀⢠⠀⢠⠀⠲⠆⠀⡆⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⠀⠀⠀⣠
⠘⠋⠈⠋⠉⠉⠉⠉⠀⠉⠉⢹⠛⠛⠛⠛⠛⠛⡏⠁⠉⠀⠉⠀⠙⠀⠸⠁⠘⠁⠈⠃⠈⠃⠀⡍⠑⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠋⠉⠙⠉⠙
⠘⠃⠀⠃⠀⠃⠀⠛⠀⠘⠀⠸⠓⠚⠓⠒⠓⠓⡏⠛⡗⠀⠛⠀⠘⠀⢘⠀⠘⠃⠘⠃⠀⠃⠘⠤⠀⠛⠈⠳⢠⣀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠛⠀⠘⠀⠘⠀⠘
⠒⠂⠀⠂⠀⠒⠀⠒⠀⠒⠒⢺⠀⢰⠚⠒⡆⠐⡗⠂⠓⠒⡖⠀⠚⠀⢘⠂⠘⠂⠐⠃⠀⡇⠀⠓⠒⠀⠀⠶⠀⠀⠈⠙⠀⠀⠋⠁⠀⠑⠖⠀⠒⠀⠒⠀⠘⠀⠐
⠠⠆⠀⠆⠀⠆⠀⠶⠀⠰⠀⠰⠀⠰⠄⠠⠆⠀⡆⠀⠦⠀⠶⠦⢴⠀⢸⠀⠰⠄⠠⠆⠀⡆⠀⠆⠀⠼⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⠶⠀⠰⠀⠰⠀⠰
⠤⠤⠤⡤⠄⠤⠄⠤⠠⢤⠠⢤⠤⢤⠤⠤⡄⠤⡅⠤⡤⠠⠤⠤⢼⠤⢤⠤⢠⠄⠤⡤⠤⡥⠤⡤⠤⠤⠠⢬⢼⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡤⠤⢤⠤⢤⠤⠤
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠂⠀⠂⠀⠀⠀⠐⠀⠘⠀⠀⠀⠀⠂⠀⠂⠀⠂⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢀⡤⠓⢄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢈⠟⠛⠟⣦⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⡠⡠⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡘⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣄⠀⢀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠙⠒⠶⠤⣄⣤⣤⣶⣶⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣭⣭⣛⡻⠟⠛⠋⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⡿⠋⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠨⣽⣛⣲⠦⣤⣄⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡒⠤⢭⣙⣛⠻⠟⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠾⢿⣿⣛⡶⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣍⣀⠐⠤⢌⣙⡛⠿⠿⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠾⢭⡽⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣍⣒⣒⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣳⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⠿⣷⣀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣅⣬⣿⣿⣧⡄⠀⡀⡀⠒⢻⢀⠀⠉⠙⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣾⣿⠿⢩⠡⡇⠈⠁⣧⠀⠁⠀⠀⠛⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣟⣩⣼⡂⠠⠀⠀⢨⡾⣿⡿⣘⠃⢤⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⣏⠿⢙⣿⡰⠂⠀⠬⠓⠂⠁⠉⢀⣠⣽⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣟⣥⡴⠴⠂⡏⢃⠀⣰⠀⠀⠀⠁⣠⣬⢿⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣷⣯⠀⠈⢻⡁⠀⠀⣨⣵⣿⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣶⣿⣿⣿⣷⣶⠛⠉⠀⣹⣿⣿⣝⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣴⣿⠟⡾⣯⢡⡿⠟⠟⢿⣿⢿⢯⢙⠁⠀⠰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣒⣲⠿⠏⠙⠻⠇⣀⠈⣷⡄⠑⠉⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣩⣴⣂⠀⠀⠀⠀⠀⠁⠙⠋⠀⠀⠀⠀⠀⢦⠁⠀⠀⢀⣈⢉⠁
⠀⠀⠀⠀⠀⢈⣽⣿⠇⠀⠀⠀⣈⡓⠦⠈⠂
⠀⠀⠀⠀⠀⢸⡟⠀⠀⣀⣄⣀⡀⢁⢀⠎⠁⠢⡄⠠⡄⠀⠀⠘⡀⢠⢸⡀
⠀⠀⠀⠀⠀⡼⠋⠉⠋⠉⠉⠉⠙⣷⢷⣤⠄⢠⢹⣆⠀⢀⠀⠀⣧⣤⣴⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣦⢸⢨⡀⢸⠆⣰⡿⠛⠁
⠀⠀⠀⠀⠀⠺⢃⠀⠀⠀⠄⠀⠁⠀⠀⠙⠛⠿⠋⠘⠳⡌⣾⠛⣤⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⢻⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡽⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⠄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣁⣰⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⠀⡆⠀⢀⠀⢴⠀⠀⠀⠈⠛⢿⡦⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢧⠤⣾⡤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⢊⣿⣦⣀⠀⣠⣤⣴⠏⠁⠠⣟⣿⣿⠟⣦⠤⢤⣀⡠⠚⠛⢩⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢐⣩⣿⣯⣭⠿⠗⢃⣀⣠⣤⣄⠈⠻⠿⠅⠀⣠⣿⣿⣟⡢⠀⠻⡿⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠛⠋⠉⠀⠀⠐⠋⠉⢠⣾⣿⠧⠀⠀⠀⠘⠻⠿⣏⣻⡯⠀⢀⠀⠀⣾⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢶⠄⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⠜⠂⠁⠀⠘⠿⣿⣿⣟⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣆⣀⣠⣠⣠⣀⣠⣸⠀⠀⢀⣠⣤⣄⣀⣠⣤⣀⣀⣀⣀⣀⢀⣰⣷⣀⠀⠀⡀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠉⠉⠉⠉⠉⠹⠯⠭⠭⠽⣛⣛⣿⠿⣟⡛⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠻⠯⠭⠭⢝⣓⡚⠻⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒⠒⠪⠭⠭⣛⡳⠦⢄⡀⠀⠀⢈⣉⡉⠉⢉⣍⣑⣒⡂⠤⠄⣀⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠉⠹⠗⠶⠎⠉⠒⠀⠀⠀⠈⠉⠉⠉⠛⠛⠭⠍⣉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣄⠀⠠⠔⠒
⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠙⠶⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⡴⠤⣀⣰⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡤⠿⠷⡖⠛⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠚⠀⡀⠀⢸⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠁⠀⠊
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣨
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡀⠠⢂⣴⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⣀⠄⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠀⣸⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣡⡀⠀⠀⠀⠀⡄⠛⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣁⠉⠉⠋⠐⠒⠒⠴⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⠉⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠢⣀⠀⠀⠀⠀⠀⠉⠲⢌⡈⠸⣶⡀⠀⠄⠠⠞⢧⣀⣠
⠀⠀⠀⠀⣀⡼⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⣀⠀⠀⠀⠈⡙⠿⣧⡀⠀⠀⠀⠀⠈⣿⠆
⠀⠀⣤⡾⠋⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠤⣀⡏⠱⢦⡀⠀⠀⢸⣿
⣴⠿⡃⠀⠀⠀⠀⠘⣷⣴⣖⣼⣛⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⠤⢤⠤⠶⠛⠳⠶⢿⣷⣤⣾⣏⠆
⡏⠈⠀⠀⠀⠀⠀⠀⠘⢦⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢉⠟⢿⣿⣿
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠹⣿
⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠹⣄⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⢳⣄⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡘⠀⠀⢀⣄⠀⣠⡴
⣖⠀⠀⠀⠀⠀⠀⣹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠈⡄⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣼⡿⠛⠁
⣿⣆⢆⠀⠀⢠⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢤⣤⣧⡀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⢀⣀⡤⠼⠒⠛⠉⠉⠉
⠙⣿⣦⣄⠀⣼⠏⠀⠀⠀⢀⡀⠀⠀⠑⢀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠊⢿⣶⣄⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠈⠉⠁⠀⠀⠀⣀⣠⣤⣴⣶⣷⣶⣤⣄
⠀⠘⣿⣿⣿⡿⠦⢶⠏⠩⢩⣷⣈⠤⠒⠈⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣤⠀⠀⠀⠀⠀⠀⣀⣤⣠⣤⣴⣤⣶⣶⣾⣿⣿⠿⠿⠟⠛⠛⠛⠿⣿⣿
⠀⠀⢸⣿⠏⠙⠒⢶⣤⣤⣶⣿⡷⠤⣔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠿⠶⣾⠿⠿⠛⣿⠛⠛⠛⡟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣾⡟⠀⣀⣤⡾⠟⠛⡏⠀⠀⠀⠀⠹⢶⠤⠀⠠⠤⠄⣀⢀⣀⣠⡴⠞⠋⠁⠀⠀⢰⠇⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⣼⣿⣷⣾⠿⠋⠀⠀⡼⠁⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⣤⠞⠉⠉⡀⠀⠐⠀⠀⠀⠀⠈⡆⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄
⣾⣿⡿⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠈⢇
⡿⠋⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⠀⢀⠔⠁⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⢦
⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠈⢦⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠚⠁⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠇

⠀⠀⠀⠀⠀⡀⠀⠀⢀⠤⣀
⠀⠀⠀⠀⠀⢡⠄⠀⢨⠄⠈⢀⡄⡀
⠀⠠⢔⠀⠀⠈⠁⠀⠨⣄⣠⠞⡅⠈⠀⡀
⠀⠈⠑⠀⠤⢔⣮⠽⠀⠈⠁⢸⡇⠀⢀⣽⠛⣶⡀⣤⣤⣤⣀⣀⣠⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠐⠒⠂⠀⠈⢠⡀⠀⢾⠅⠀⠞⠁⢰⣿⡇⠈⣿⡿⠓⢤⡥⡴⢶⠦⠤⠀⠉⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⢀⠌⣿⠇⠀⠹⠀⠀⠟⠁⠈⠀⠀⠀⠀⡰⠃⠀⠑⡄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠁⠀⠀⠀⠀⣄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠸⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠸⢦⣄⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢴⣝⠀⠀⠜⡗⠀⠀⠀⠀⠀⠀⠀⠸⡏⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣸⠇⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠙⣿⣦⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡹⣿⣷⣄⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠙⡏⠘⠄⠙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠁⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠐⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢦⡤⠋⠠⡤⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣑⡾⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠉⠉⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡀⠀⠙⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⢱⠀⠹⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠄⢠⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⡈⢻⣦
⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣆⢿⡄
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠐⠉⠀⠈⠉⠉⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣸⣷
⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣭⣓⣶⣤⣀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢿⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⢿⣧⡈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⡿⢻⡆
⠀⠀⠀⠀⠀⠙⢿⣦⣧⡐⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣌⠀⠀⠀⠀⠀⣾⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⣰⡿⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣽⣿⣿⣿⣿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣨⣵⣿⣿⣿⣿⣿⢟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠛⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⢿⠿⠛⢁⣴⣾⣿⠛⠋⠹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⡄⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠑⠀⠐⡜⠝⢸⠇⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⢳⡀⠀⠀⠀⠑⠀⠀⢀⠠⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢳⡀⠀⠀⠀⠀⠀⠀⠱⡔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣄⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠠⢂⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠜⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋⣷⡖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠓⠚⠉⠁⠀⠀⠀⡄⠂⣈⠽⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣟⣿⣿⣿⣿⣻⣿⣿⣿⠋⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡉⡀⠠⠀⠤⣶⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⢿⣿⣿⣿⣿⡿⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⢠⠉⠾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣨⢉⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⣤⠈⢒
⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⢯⠄⢀⠀⠀⠀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠼⠟⠄⠒⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⠹⠄⠀⠀⠀⢀⣤⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⠋⠀⠐⡢
⠙⠙⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⣾⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣾⣿⣿⣿⣿⣿⡿⠻⠛⠛⠃⠀⠀⠐⠁⠀⠀⠀⠀⠘⠉⠠⡾⠊⠀⢀⣴⠡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⠟⢠⡄⠀⣰⣶⣆⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀⢀⡀⠀⠀⠀⠀⠀⠴⠃⠈⠉⠀⠈⢁⢩⠛⢟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠁⢩⠿⣿⣿⣿⢿⣷⠀⠈⠝⠲⠆⣠⠀⠣⢄⣥⣤⣾⣴⣿⣟⡷⣮⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠃⠀⠈⢻⣿⡾⣿⣔⣂⠀⠰⠂⣀⣘⣧⣬⣽⣶⣿⣿⣿⣿⣿⣿⣋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⡀⠀⠀⠀⢀⣹⣷⣽⣿⣿⣷⣿⠿⠟⠛⠛⢻⢿⠋⣿⣿⣻⡿⣿⡛⣷⣿
⠀⠀⠀⠀⠀⢐⣀⠀⠀⠀⠀⠀⠀⢖⡞⣨⣤⣶⠶⠿⣛⣿⣿⢿⡛⠛⠉⠁⠨⢧⠀⠚⠺⠤⣧⠰⣿⣞⣽⣄⣽⣯⣿
⠀⠀⠀⠀⣀⡻⢿⣤⣤⠃⣀⣀⣀⡮⢸⣙⣯⠿⠿⠛⠋⠁⠀⠈⠃⠀⠀⠄⠀⠀⠀⠀⡀⠤⢤⣄⠙⣭⢿⣿⣿⠏⢁
⠀⠀⠀⠀⠉⠁⠂⣀⡠⠔⢛⡠⠁⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠐⠚⠂⠤⣬⡯⠟⣻⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⣶⣠⡆⡆⢀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠰⢶⣶⡆⣲⣀⣿⣿⣿⠿⠿⠳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣥⡀⡓⢻⣭⠀⠘⣷⣿⣿⠓⠹⡿⠿⠋⠀⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⣻⣿⣷⣏⣿⣿⣏⠹⣉⡝⣿⣶⠿⠷⢶⡞⠉⠻⠿⣮⣶⡀⢍⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⡿⣿⣿⣿⣶⣚⢻⣿⣧⣦⣴⣿⠇⢀⡀⠀⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠋⠀⢏⣉⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⣀⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣾⣷⣾⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢻⣿⣿⣍⠻⠻⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣟⢉⣨⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣺⢦⣶⣿⣧⣼⣿⣍⡃⣹⣦⣉⠙⣽⣿⣿⣿⣳⣿⡟⡋⠡⠌⠉⠈⠿⠙⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⡗⠺⡿⣿⣿⡟⣿⣿⣿⣿⣿⢟⣽⣿⣿⣧⣶⡞⠄⢀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠊⠋⢀⣹⣮⣿⣿⡇⣿⣿⢿⡿⢟⡠⠀⣤⣸⣷⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢾⣿⣿⡿⢷⡇⢻⣿⣿⣿⣿⣟⠒⢻⣿⠛⠛⢿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣦⣿⣶⣾⣐⢲⣟⡏⠃⠈⣿⣘⣍⠹⢛⣻⡟⠒⠈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡾⢺⣯⡏⢛⣿⠛⢻⡶⠀⣿⣽⠿⡵⣏⣿⣿⣿⡧⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡚⠉⠀⠈⠁⠀⣿⡇⠀⠀⠈⠫⠥⡍⠈⠛⣿⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠁⠀⠀⠀⠀⢀⣟⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣚⣛⣛⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢻⡿⣻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠐⡿⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢶⠶⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣦⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⢡⣤⠁⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣶⠂⠀⠀⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⣸⡿⠤⠤⠼⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣆⠀⢠⣿⣿⣿⣿⣦⠀⠀⠀⣴⣾⣿⡄⠀⠃⠀⣼⣿⣶⡄⠀⠀⣠⣶⣶⣿⣶⣤⣤⣤⣾⣿⣶⣶⣄
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠀⠀⠠⠿⠿⠿⠿⠦⠤⠴⠿⠿⠿⠧⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠙⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣧⠈⠉⠁⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⠶⠲⠾⢿⡇
⠀⠀⠀⠀⠀⠈⠻⢿⣿⡀⠶⠀⣼⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢤⡔⡒⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠴⠒⢆⣸⡇
⠀⠀⠀⠀⠀⠀⢀⣀⣉⠛⠓⠚⢉⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠙⠛⠋⢀⣀⣀⡀⠀⠀⠀⠀⢀⣿⠿⠶⠶⠾⢿⣷⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣆⠀⢀⣿⣿⣿⣇⠀⠀⠀⣾⣿⣿⣿⣦⡀⠀⣀⣾⣿⣿⣿⣇⠀⢠⡄⣿⣿⣇⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠘⠀⠛⠛⠛⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢐⣭⣶⣶⣶⣮⡑⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠉⠁⢸⣿⠄⠀⠀⠀⠀⠀⢠⣿⣿⡟⠛⠛⢿⣿⣿⣿⡅⠀⠀⠀⠀⢀⣼⣿⣼⠿⠇⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⡀⢈⣿⠄⠀⠀⠀⠀⠀⠸⣿⣿⡁⠀⠀⠀⢨⣿⣿⡇⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠿⣷⣮⣠⣡⣾⣿⠆⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⢻⣿⣷⠀⠛⢠⣿⣿⡇
⠀⠀⠀⠀⢀⣠⣤⣵⠛⠍⣻⣋⠟⣾⣤⣤⡀⠀⠀⠀⣠⣤⣽⣟⠀⠀⠈⣿⣯⣤⣀⠀⠀⠀⢠⢿⠿⠟⠀⠀⠘⠿⠿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠉⠀⣰⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣆⢀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠈⣆⠀⠀⠀⠀⢀⣴⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠙⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠂⠀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⠄⠈⠃⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⡈⠟⠀⠸⡄⠠⣄⠀⣀⠀⢀⢀⣀⢴⡆⠿⠇⢻⣀⠀⢙⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⡤⣤⣠⡌⠀⠁⠀⢃⠀⡇⠐⡇⢸⡓⠀⢀⠀⠈⠒⣾⣿⣿⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣫⡴⠛⠀⠀⠀⠀⢸⠀⣧⠀⢄⠘⣅⠀⠀⠀⠀⠀⠀⠙⠛⣁⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⢎⣛⡿⠟⠀⠀⠀⠀⠀⢀⡼⡇⠈⡐⡟⢀⡏⢣⠄⠐⠀⠀⠀⠀⠀⠹⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⢻⡄⠻⡀⠇⠇⢠⠃⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⢡⣼⣷⣼⠀⠐⠃⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠘⢋⡙⠋⠀⠀⡜⢻⣿⣄⠹⣿⠛⠓⠶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠔⣿⡇⢠⣾⣿⠙⡆⠀⠀⠀⠀⠀⠐⣧⡘⠿⠋⣰⠟⠀⢠⡄⠀⠀⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠁⢀⠀⠘⢷⣌⣙⣫⡼⢷⠄⠀⠀⠀⠀⠐⢯⣙⣛⣋⣅⣠⣶⣣⣄⣄⣀⣀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢈⡱⢦⣤⣤⣭⡤⠖⠛⠀⠀⠀⠀⠀⣀⠈⠀⠈⠻⣿⣯⡭⠍⢁⠩⣥⠤⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠠⠶⠿⠿⠟⣩⣯⣿⡟⠁⠀⠀⠈⠻⣟⠚⡻⠋⠀⠀⠀⠐⢈⠀⢶⡚⠀⠀⠛⠀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⠂⠀⡈⣀⢨⣿⠀⠁⠈⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⢀⣷⠉⣴⡶⠢⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠔⠚⣙⣡⠔⢉⣤⡄⠀⠀⢀⡀⢠⠴⠿⠿⠶⠄⢤⣴⣤⡾⢿⣿⡿⢋⣩⡟⠻⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠶⠞⠉⠉⠾⣿⣿⣿⢻⣿⣏⡁⠀⠀⠀⠀⠀⠀⣠⣾⣏⣿⣿⣩⡽⠿⠁⠀⠀⣘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⠀⠁⠀⠀⢀⡼⠋⢹⣿⢿⣹⠙⣷⣦⣤⣦⣴⣾⣿⣼⡿⠿⠉⠇⠀⠠⣜⠲⠆⠁⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⡄⠀⠀⠊⠀⠲⢯⡄⠈⠉⠛⠿⠟⠾⡿⠿⠋⡟⠉⠰⠂⠐⣴⡿⡟⠺⠁⠀⠀⢠⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣤⣀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠂⠀⠀⠀⠁⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠄⠙⠉⠀⣥
⠀⠀⠀⠀⠀⠀⠀⡿⢧⣋⣈⠉⠱⠦⣀⡀⠀⡄⠀⠀⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⡜⣀⢀⠀⠉⠉⠓⠂⠉⠽⢶⣤⣤⣄⠀⢀⡀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠹⠇
⠀⠀⠀⠀⠀⢠⣼⠛⠉⣀⣤⣤⠒⠉⠉⠀⠐⠛⠿⣿⣄⣸⡧⣾⣵⣤⢈⣠⣤⣄⢀⣰⣄⣀⣀⡀⣀⡁⡠⡆⢀⣀⣄⣲⣿⣏⡀⢨
⠀⠀⠀⠀⠀⢻⣅⠀⣠⣄⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⣿⣿⣿⣿⡟⠚⠻⠛⠛⠉⠉⠉⠙⠋⠁⠀⠀⠀⠈⢏⡁⠈⠙⠛⢶⣥
⠀⠀⠀⠀⠀⠘⣦⡖⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠐⠻⡆⡛⢦⣶⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣾⡃⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣿⡟⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣒⠀⣿⣶⣭⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠴⠀⠀⠀⠀⠀⠰⡀⢀⡄⠀⣷⣿⣤⠞⠁⣰⠂⠀⠀⣰⠀⠀⠲⠀⢤⡀⢀⣹⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠄⠀⠀⠘⣦⠀⠀⣀⠀⡇⢀⣼⣿⢿⣧⣾⠁⡀⣠⡜⠁⠀⠀⣸⡿⠾⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢸⣦⣴⡿⠶⠟⠛⠉⠀⠀⠉⠙⠛⠛⠛⠛⠛⠂⠈⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠏⡛⠟⠉⡟⠛⣟⠛⣛⡿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣠⠖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣷⠀⡑⠀⠙⢴⡕⠁⢀⠶⢁⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⢌⢀⠠⠜⢺⣤⣄⣄⡗⠟⠁⡠⠄⢈⠤⣲⣀⡀
⠀⠀⠀⠀⠀⢦⡀⡀⠀⢠⠀⠶⡀⠀⠛⠀⠤⢤⡱⣿⣮⢳⣇⡞⠀⠠⣨⠒⡋⠉⠠⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠐⠱⣿⡄⡀⠂⠀⢸⡦⣠⡀⠀⠀⠀⢽⣿⣿⡯⠀⣠⡞⠛⠍⡄⠁⠀⠀⠀⣾⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣓⣳⡄⠠⠥⣽⣶⡀⠀⣸⣹⣿⣧⡽⢯⠗⡀⠌⣁⢀⣠⠶⠊⣀⡠⢞⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⢀⣻⣦⡱⢿⣝⣷⣤⣈⣹⣿⠏⠀⠈⣶⠿⠶⠄⣠⣖⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣤⠶⠛⠿⢿⣿⣿⢿⣿⣽⣿⣤⣶⣶⡟⢀⣴⣾⡅⠈⠉⡉
⠀⠀⠀⠀⠀⠀⠈⠹⠟⠊⠉⠈⡉⠰⠲⣶⠶⠿⣧⠻⣿⣿⣻⣿⣿⣿⠛⣽⡿⠷⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⡶⠖⠾⠷⣾⣿⡏⣿⣀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣲⠶⠶⠥⠼⠾⣝⣂⡀⢿⣷⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠳⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⢤⣤⠠⢤⠼⠷⠦⠐⠀⠀⠈⠀⠉⢙⠩⠍⠙⠙⠛⠛⠛⠓⠲⠲⠦⢀⡀⠠⠌
⠀⠂⠡⡄⡐⠀⠀⠘⠃⠀⠀⠠⢁⠀⣀⣀⢀⠦⠤⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢧⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⣴⡿⠖⠃⠀⠀⠀⣤⡄
⠀⣠⣴⡾⠳⢂⡘⢻⡛⠁⣠⣆⣘⡳⣶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⣴⣿⣿⠟⡄⠀⠀⠀⠻⠇⠀⠀⠀⢀⣀
⠀⠀⠀⠶⠛⡻⠛⠛⠛⠀⠀⠀⠈⠃⠊⠁⠀⠀⠀⠀⣠⡄⠰⣏⠼⢛⣱⣿⣿⢿⡟⢀⡖⠀⠀⠀⠀⠀⠀⠠⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢟⡟⠁⡀⠀⠈⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣞⠀⣸⡿⠞⢀⠀⠀⢠⣤⠂
⠀⠀⠀⠀⠀⠀⣰⡟⠟⣴⣿⡷⠷⡄⠀⠀⠀⠙⢇⠀⠸⠅⠀⣸⣆⣴⣤
⠀⣠⠄⢴⠇⠞⣛⣵⣿⣿⣿⠇⢠⠀⠀⠀⢠⠀⡀⠶⢤⣰⣄⡿⠟⡏⣀⠀⠀⠀⠀⠀⢀⡠⠢
⠀⣀⠀⡆⠀⢠⢿⠋⠀⠐⠁⠀⣼⣇⢰⣦⣿⣦⣿⣿⣤⠿⠿⠷⣾⡿⠋⠀⠀⠀⠀⢠⠃⠀⠀⢁
⢀⣿⣴⣷⡄⢀⣄⣼⣧⣦⣾⣷⣿⡿⠻⠿⠟⠋⠉⠀⠀⠴⠟⠛⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢆
⠈⠉⠉⠙⠓⠪⢥⣀⡀⡐⢨⡉⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠃⠀⠀⠀⠀⠀⠈⢢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠺⠷⠤⠿⢷⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⠁⠀⠀⠀⠀⠀⢂⠤⠊⠀⠀⠀⠀⠀⠀⠈⠑⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣷⣦⣀⣀⣤⠤⠤⠤⠤⠤⠤⠤⡾⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢠⠔⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡏⠀⢀⣄⢤⡀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠈⠉⠀⣼⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⣿⠹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠋⢻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⡴⢚⣛⣛⣦⡀⢠⠇⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣆⠀⠀⠀⠀⢠⡀⠀⠀⠀⢀⣠⠤⠶⠤⣀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠹⢶⣶⣎⣉⣤⣤⣶⠖⠊⢛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠦⡀⠀⠀⠀⠈⠉⠛⠛⠛⠉⢀⣠⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⣦⣀⣠⣄⣀⣤⣴⣿⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠿⡄⢀⠀⠀⠀⠀⠀⣴⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣾⣿⠟⣿⣇⠀⠈⠲⠈⠁⠀⠴⣾⣿⢻⣿⣦⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠋⣿⣿⢸⣿⣿⣆⠀⢀⠀⠀⠀⠀⠀⠘⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠈⠃⠀⠀⢺⣷⠀⡏⢹⣿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣴⣿⠟⠇⠀⠀⠀⠀⠁⠀⠀⢾⢟⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠹⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡄⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠚⠏⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣯⣿⣿⣿⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠣⢿⣿⣿⣿⠋⠛⠿⣿⣿⣿⣿⣷⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣟⠀⠀⠀⢿⣇⡄⠀⠀⠀⠈⠉⠉⠛⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣽⣲⣶⣦⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣵⣿⣿⣟⣽⣯⣿⣄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠻⡿⣿⣆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠻⣿⣿⣿⣷⣽⣯⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠈⠻⣿⣿⣻⣯⣟⣿⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦⣄⣀⠀⠀⣿⣿⣿⣷⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣶⣿⣿⣿⣿⣎⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣯⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢽⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣷⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣃⣴⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣾⣿⣫⣿⣿⣟⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣟⠉⠉⠉⠛⢻⣿⣿⣿⣿⣿⣿⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡻⣿⣀
⣿⣷⣶⣦⣤⣀⣀⣉⣀⣀⣀⣈⣙⣷⣶⣦⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡿⢿⡾⣿
⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⠿⣿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠦⠠⢤⣴⣶⣶⣶⡦⠀⠀⠀⠀⠀⢠
⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⢀⣴⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⢳⣄⠀⠀⠀⠀⠀⢀⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣿⢿⠛⢹⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡓⡎⠀⠀⠀⡀⡆⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⠿⠒⠀⠀⠀⠀⠺⠿⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠄⠀⠀⠱⠀⠀⠀⠴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣆⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢻⣿⣿⣿⡄⠀⠀⠀⣸⣿⣿⣿⡟⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡼⣿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⢳⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⢻⣿⣿⣿⡄⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣟⣿⣿⣿⢃⣧⢸⣿⣿⡿⣹⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡌⠛⠉⢼⣿⠈⠉⠛⣱⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠟⠙⠻⣷⠀⠀⠸⣿⠀⠀⢠⡿⠟⠋⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡄⠀⠀⠀⢰⡗⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠒⠈⠉⠐⠒⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⢺⣦⠙⣴⡾⠀⠀⢠⣆⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠰⣶⡜⠟⣁⣀⠀⠀⠉⠁⠉⠀⣰⡄⠘⢏⣯⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⠶⣋⣋⠀⢀⠉⠉⠘⠛⠋⠀⠀⢀⡇⠀⠀⣟⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠈⠒⢦⡀⠀⠀⠀⣼⡥⠚⠉⠀⠀⠀⠀⠀⠀⣄⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⣀⢀⣙⣆⠀⢰⡟⠀⢰⣷⢀⣤⠀⠀⠀⠀⠻⠫⠷⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢫⡾⠀⠀⠀⣴⡿⠈⠹⣧⣿⠁⢶⣦⡟⠿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⣤⡀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⢻⣿⠀⠀⠉⠀⠀⢀⣀⠤⠤⠖⠀⠀⠀⣉⢽⣦⣄
⠀⠀⠀⠀⠘⠿⠿⣎⣙⠀⠀⠙⠇⠀⠀⣤⣤⡻⢇⣶⠆⠀⠀⢸⣿⠀⠀⢀⡶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠻⠇⠉⠁
⠀⠀⠀⣤⡀⠀⠚⠛⠁⣀⣠⣤⣬⣴⣤⣈⣛⡛⠈⠁⠀⠀⠀⢸⣿⡄⣰⠏⠀⠀⠀⣀⢼⡇⣀⠀⠀⠀⠀⣴⣠⣶⠇⠀⠀⣀⣤
⠲⣶⣤⠛⠃⢀⣠⣀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣄⡀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠻⠇⠟⠋⠀⠀⠀⠀⠙⠱⡶⠆⠀⠀⢛⠣⣶⣶
⠀⢀⣾⡟⠀⢸⠏⠻⠀⠀⠀⢦⣤⡀⢀⡠⠒⠚⠛⠛⠿⣿⣦⡀⣿⣿⠀⠀⠀⠀⣀⣴⣶⡶⠶⠶⣶⠒⠲⠒⠒⣒⠈⠀⠀⠘⢿
⠀⠀⠁⠀⠀⠈⠀⠀⠘⠻⣳⡄⠉⠉⣴⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡆⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣄⣻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠉⠑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠤⠤⠤⠤⠀⡀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠔⠙⣇⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠢⠀⠀⢀⠃⠀⠀⠈⠙⠓⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⢴⠚⠛⡏⠉⠉⠙⠒⠒⠦⢤⣀⠀⠀⠈⠑⠄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢴⡇⢈⣀⠀⢈⢦⣀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢿⡖⡢⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢾⡇⠃⡸⡿⠓⠢⢄⣀⡀⠀⠀⠀⠱⡀⡠⠃⢀⠤⠒⠈⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠻⣾⠗⣒⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠳⡀⠁⠀⠀⢀⠤⠹⠆⠀⠀⠈⠑⢦⠀⠙⠤⢄⣁⣀⡀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠚⠛⠛⠒⠚⠟⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢦⡀⠒⠴⠾⠇⠂⠀⠀⢀⣴⠋⠡⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠫⢁⠀⠀⠀⠀⠀⢀⠄⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣈⡑⡖⠢⠤⠤⠶⠞⠋⠀⠀⠀⢙⡦⡀⠀⠀⠀⣀⠀⠀⣀⣀⡀⠀⠉⠐⠂⠄⠊⠀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢀⡠⠤⠒⠋⠁⠀⠀⠑⠢⢤⡄⢀⣀⣀⠀⠈⠉⠒⠠⠤⢠⡤⢞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠘⠋⠉⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠳⣄⠤⢀⠑⠄⣀⡠⠄⠀⠀⣁⡠⡃⡧⠀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣳⡀⠀⠀⢹⡳⣖⠨⠥⠐⢖⡉⠈⠀⠉⠲⠖⠁⢈⢤⡁⢊⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠁⣡⠄⠀⡤⠓⠂⢀⠰⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⣨⠧⠬⢤⠤⠴⠦⢒⠒⠂⠀⠀⡠⠀⠙⡈⠀⢜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡒⠈⠙⠊⡉⠷⠚⠒⢊⠇⠀⢣⢢⣄⣀⠀⣀⣀⡠⠴⠒⠋⠁⠀⠀⠈⢆⠀⠀⠀⠈⠀⠒⠈⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⡥⠤⡖⠒⠈⠁⠀⣀⣤⡁⠈⠡⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠙⠒⠖⠒⢋⠁⢀⣡⠶⠚⠉⠉⠉⠉⠉⠙⠒⠒⠲⠦⡤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⡉⣡⣖⠐⠋⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⢀⠀⢷⢄⣀⡠⠎⠀⠀⠤⡔⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠀⠀⠑⢠⡀⠀⠀⠀⢀⣈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⢄⣀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⣀⠀⠀⠑⠢⡉⠁⠀⠀⠈⠲⣀⢀⠠⠄⠀⠀⣁⠠⠄⠒⠀⠁⠀⢀⣀⠬⠗
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠛⢠⠀⠄⠄⠒⠁⠀⠀⠀⠈⢦⠄⠂⠉⠁⣀⣀⡤⠤⠒⠚⠉⠁
⠻⣍⠒⠢⠤⠤⠤⠄⠀⠀⠀⠠⠤⠄⠐⠒⢒⣘⡦⠤⣤⠊⠀⠐⠂⢈⣁⣠⡤⠤⣔⣢⡥⠴⠒⠚⠉⠉
⠀⣌⠛⠒⠶⠶⠶⠶⠶⠒⠒⠒⣚⣋⠩⠭⠔⠒⠒⠀⣉⣉⣉⠭⠭⠷⠒⠚⠋⠉⠁
⠀⠙⠦⣍⡉⠐⠒⠈⢉⣉⣉⣉⣠⡤⠤⠴⠒⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣻⣿⣷⣖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣍⣚⣿⣯⣥⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣚⣿⣿⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⢿⣿⣿⡷⢞⣫
⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⡷⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠠⣄⣛⣿⣿⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠠⣙⣻⣿⣿⣟⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠢⡀⠀⠀⠀⠀⠀⡉⣽⣿⣿⣿⣿⣿⣏⣡⣤⠄
⠀⠀⠀⠀⠀⠀⠠⣭⣿⣿⣿⣿⣯⠔⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⣌⠛⢷⣿⣿⣿⣿⣿⣿⠟⢁⣀
⠀⠀⠀⠀⠀⢀⣄⣾⣿⣿⣿⣏⣥⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣞⡉⠻⣶⣿⣿⣿⣿⣿⣿⡿⠛⣩⡀
⠀⠀⠀⠀⠀⢠⣭⣿⣿⣿⣿⣷⣦⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠬⣿⣿⠾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣷⠶⠛⠓⠛⠒⠒⠐⠒⠒⠛⠉⠀⢀⣄⣰⣷⣿⣷⣞⣙⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⡥
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣯⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠈⠩⣽⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣽⣶⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠤⠴⢾⢿⣿⣿⣿⣿⣿⣶⣤⡤⠖⠀⠀⣆⠀⡄⠀⠀⠀⡄⢰⣇⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⣯⣍⡀⠀⠀⢰⡗⢠⡇⠀⡗⠀⣯⢸⣷⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣶⣤⣿⢀⣟⣾⣿⣭⣷⣿⣿⣤⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⣸⣏⣟⣿⣫⣋⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠅⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣼⣿⡿⢠⠀⠿⣿⣿⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠑⢀⣾⣿⡁⣽⣿⡇⠀⠠⠀⠈⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃⠀⢠⣿⣿⣿⣶⣿⣿⣧⠀⠀⠖⠀⠀⡀⠀⠀⠀⣤⣶⣯⠀⠀⢀⡀⣠⣴⡾⠋⠀⠆
⠀⠀⠀⠀⠀⠀⣀⣴⠀⡀⠀⠀⣠⡞⠀⠰⣦⣾⣿⣿⣿⠃⣴⣿⣿⡿⣿⣿⣽⣿⣿⣦⣀⠀⠀⠀⡀⢀⠀⢀⣼⣯⠏⠀⠀⠋⠈⢹⡾⠃⠀⠐
⠀⠀⠀⠀⢰⢿⣿⣇⠀⢀⣴⣿⣿⡟⠀⠀⠀⠙⣿⢻⣿⡖⠻⣻⡿⡃⠛⠋⡵⢟⠿⢿⣿⣿⣶⡀⠀⠀⣀⡈⠼⠟⢠⢤⡼⠁⢀⠘⠃⠀⣰⡄
⠀⠀⠀⠀⢙⢿⠟⠻⢠⣾⣻⢹⣋⡷⢸⢁⢸⡆⠘⣇⠹⠏⢤⡁⠠⣸⡄⢀⠀⠀⠀⠈⠁⠛⠹⣿⢶⣦⣌⢁⠋⣰⡎⣸⣯⡀⢸⣧⢰⣦⢼⣿⣀
⠀⠀⠀⠀⢀⣆⣶⣶⣿⣿⣶⣾⣷⣷⣿⣾⣼⣷⣶⣿⣧⣿⣿⣧⠀⣽⣿⢠⡆⠀⠀⠀⠀⠀⢀⣀⣴⣹⣇⣨⣤⣿⣇⣸⣿⣧⣶⣿⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣀⣁⡒⠺⠿⠿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡋⡿⣿⣿⣿⢿⣿⣯⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠆⠀⠀⠈⠙⢻⣿⣿⣾⣯⢻⢹⢸⣿⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣾⣿⣿⣷⣿⣿⣟⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠉⠉⢀⣄⠀⢀⠀⣷⣸⢫⠻⢿⣄⣾⣌⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣡⣿⢿⡿⢻⣟⣿⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⡄⠀⠀⠀⠀⠀⣰⡀⣸⣿⣦⣴⣇⣿⢾⣷⡦⠊⠉⢡⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣏⢹⢸⣷⠋⣿⣿⣿⣿⣯⡀⠀⣾⣆⠀⠀⢼⣯⠀⠀⠀⣤⣤⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣇⣤⣽⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣸⠉⢾⣬⣿⣿⣿⣭⣠⠌⣿⢗⡀⢈⣾⣿⣞⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣶⣦⣝⣿⣿⣿⣋⡥⢒⣿⡿⠃⡀⣹⣟⣯⡆⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⢃⣿⣿⣯⣶⣃⡄⠀⠀⠀⣾⣟⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣦⠀⠰⣶⣿⣿⣵⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠴⠲⣾⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⣷⣆⣹⣾⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢳⡀⠀⠀⠀⠀⢠⠞⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠉⠉⠉⠉⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣴⡇⠀⠀⠻⠇⠀⠀⠺⠟⠀⠠⣼⣦⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⢶⡿⠆⠀⠀⠀⠀⣠⡄⠀⠀⠀⠠⠶⢿⡶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⡿⠓⠃⠀⠀⠀⠀⢸⠁⠀⠀⠀⠐⠓⠚⢷⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⡀⠀⠀⡄⠀⣠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⡇⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⡇⠀⢺⠀⢸⡀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣁⣾⣱⣇⣼⣈⣧⣹⣀⣀⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠦⣤⣀⡀⠀⠀⣀⣀⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣈⠉⠉⠉⣿⣿⣿⠿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠒⠤⢄⣀⠀⠀⢰⡒⢲⣿⠉⠛⠟⠛⢃⢻⡇⠀⠀⠀⡠⢠⡀⠀⢀
⠐⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⣿⠛⠿⢤⡃⠉⢰⡦⡀⠄⠊⠀⡀⡀⡰⡡⡑⡡⡡⢁
⠀⠀⠈⠃⠉⠢⣀⠀⠐⢤⣆⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣀⠶⢉⠍⡸⠀⠀⠀⠀⠜⡐⠑⡔⢉⠜⠔⢠⢃
⠀⠀⠀⠲⢄⠀⠈⣈⠂⠈⠀⠉⠒⠤⣀⣸⣿⣿⣿⣿⣇⢨⣟⡳⠈⠀⠀⠀⢀⠎⠈⢈⢎⢀⢊⠎⡠⠁⠢
⠀⠀⠀⠰⢄⡈⠂⠁⠑⠀⠢⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣏⡇⠀⠀⠀⠀⠠⠂⠂⠀⣁⣀⡀⠔⡠⡑⡡
⠀⠀⠀⠀⠀⠈⠓⠆⢄⠐⠀⢄⢠⣠⣴⣾⣿⣿⠏⠙⢿⣿⣿⣷⡀⢀⣀⠤⠤⠤⠖⠋⠁⠀⠉⠒⠥⣴⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠝⠛⠛⠋⡉⢀⣼⣿⣿⠿⢿⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠉⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡉⠲⢤⣀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⣛⠶⢥⣄⣀⡀⠀⠀⠀⠠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠠⡈⠉⠠⣄⣀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠲⢄⠉⠉⢍⡁⠀⠘⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠈⠲⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⡠⠖⠋⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣀⣤⡞⠀⠀⠀⠀⠀⣀⣰⢦⣀⡈⢇⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣴⠋⢸⢿⠀⠀⢀⡞⠉⠉⠁⠀⠀⠀⠙⡮⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢳⠀⢸⢸⡜⢙⢾⡀⠀⠠⣴⠂⠀⢠⠦⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⢆⡀⡇⢷⡸⠈⡏⠓⠒⠲⠶⠖⠒⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢃⠇⠘⢿⣦⡇⠑⠢⠄⢀⡀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣯⠳⣅⠂⠄⢀⡀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠈⠛⠶⠦⢴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⣠⡤⢶⣾⡉⢷⣄⠀⠀⠀⠀⠀⣸⠟⢳⣶⡦⣤⡀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠋⠀⠀⠀⢻⣿⡄⠈⠛⠲⠶⠶⠚⠁⠀⠈⣿⣷⠀⠹⣆⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⢀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⣸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢣⡀⣙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⣰⡟⠻⡌⢳⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣫⣝⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⡰⠋⠀⠀⠈⠀⠙⣿⣿⣿⣿⣦⣄⠀⢀⣤⡾⠛⠕⠊⠁⠈⢀⣙⣷⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠹⣯⠻⢿⡿⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠙⢦⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠉⠙⠻⢶⣄⡀
⠀⠀⠀⠀⢀⠴⠋⠀⠀⣦⠀⣾⣆⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠁⠀⣀⣠⡴⠿⠞⢛⣋⣉⣉⣉⣭⣤⣀⣀⣀⣀⣈⠉⠉⠉⠙⠛⠛⠿⠷⢶⣤⣼⣿⣄⣾⡀⠀⠀⣿⡧⢰⡄⠀⠀⠀⣦⠀⣾⡀⠀⠀⠀⢀
⠀⠀⢀⠤⠖⢋⠭⠴⠒⠛⠋⠉⠉⣭⣴⠶⠿⠿⠭⠭⠉⡉⠉⠙⠛⠓⠶⠤⣄⣀⠀⠀⠀⠉⠉⠛⠛⠻⠶⢾⣿⣿⣿⣧⣀⣀⣰⣿⣴⣿⣧⡤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠐⠒⠦⢤⣄⣀⣀⡈⠉⠉⠉⠉⠉⠉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠶⣴⣃⣀⣴⣾⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⢡⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡿⠃⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠷⠾⣥⣤⠾⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣇⡀⠀⠀⠀⠀⠀⠀⢀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠀⠉⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣶⠟⣟⠉⢹⣿⣿⣿⡿⠛⢿⡟⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠟⢿⣿⠇⢠⣾⠿⠿⠋⠀⠀⠈⠀⠙⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠛⠉
⠀⠀⠀⠀⢀⣴⣿⣿⣿⡷⣶⣿⣤
⠀⠀⠀⢀⣿⣿⣿⣿⣯⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣠⠄
⠀⠀⠀⠈⣿⣛⠿⠛⠋⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠛⣷⡀
⠀⠀⠀⠀⠈⠉⠛⠷⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣭⣽⠿⣡⣾⣿⣿⠛⢷⡄⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢿⣿⣿⡦⠁⣾⣿⣿⣧⣼⣿⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⣛⣀⣀⣠⣽⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠸⣿⣇⣀⣀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠐
⠀⠀⠀⠀⠠⠀⠈⠁⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠤⠂⠀⠀⡀⠀⠀⠐⠤⠄⠀⠀⣦⡔⠀⠄⠀⠀⠄⠀⢤⡾
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠿⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠆⠀⠀⠐
⠀⠀⠀⠀⠈⠠⠠⠈⢠⠠⠘⠊⠀⠀⠀⠀⠀⠀⠀⢈⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠒⠂⠀⢀⠀⠀⠀⠀⠈⠧⠀⠀⠂⡐⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠁⠒⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠁⠉

⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠄⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠂⠄⠀⠄⠂
⠀⠀⢀⠰⣀⠌⠀⠀⠀⠀⣀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠈
⠀⠀⠀⠀⡁⠤⠀⣀⡀⠤⠀⠀⠀⠀⠀⠠⣀⠁⠐⠀⠀⠐⠂⠀⣀⠄⠀⠀⠀⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡄⠈⣛⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠩⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠬⠉⠧⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⢁⠀⣀⠀⠐⠁⠀⠀⠀⠀⠐⢄⠀⠂⠑⠀⡀⠀⠀⠀⠀⣇⡀⠀⣀⣀⣀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠣⠦⡉⠀⢉⢴⠜⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠈⡀⣀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠔⠀⠀⠁⠀⠁⡀⠢⡀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⡄⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠲⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠈⠀⠉⢀⠘⡟⢀⠈⠀⠊⠀⠀⠀⠀⠀⢀⠄⠉⣄⠉⢠⠀⠀⠀⠀⠀⢀⠀⢠⠀⠰⠆⠀⡄⠀⡀⠀⠀⠀⢷⠀⠁⣀⠈⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠟⠒⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠃⠀⠈⠀⠀⠀⠀⠀⣧⡀⢘⠀⣠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠠⠤⠄⠀⠀⠂⠀⠀⠀⠀⢈⠀⠀⠀⡈⠀⠠
⠀⠀⠀⠀⠀⢠⡀⠀⢠⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠺⠆⠀⠠⠟⠂
⠀⠀⠆⡀⢀⡀⠀⣛⡀⢀⡀⢀⠰⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢴⠀⠐⡄
⠀⠀⠀⠀⠀⠁⠂⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⡈⠀⠄⠀⠀⠀⠀⠀⠀⠑⠄⠈⠅⠀⠐⠀⠀⠀⠀⠀⠸⠤⣀⡘⣀⣤⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⢻⣦⣀⠀⠀⠀⠀⠀⢀⣴⣿⠅⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠐⠀⣀⣽⡿⢿⣿⣿⣿⣿⣿⣤⡀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠿⠛⠋⠉⠈⠁⠋⠈⠻⠛⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠀⠀⠀⠀⠀⣶⣶⣦⠀⠀⠀⣴⣿⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠊⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠹⣭⣤⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⠀⠈⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⣜⣂⣀⠠⠥⢀⣴⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠳⠤⣴⣾⣥⣴⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⢿⢿⣿⣽⣿⡿⢻⣿⣷⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡿⣿⣿⣽⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⢻⣭⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢈⡿⡿⠇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡎⢷⡆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⡀⠊
⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢰
⠀⣾⣿⣿⣧⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠛⠛⢳⠀⠀⠀⠘⢧⣰
⠀⣹⣿⣳⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠀⠄
⢰⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠀⠀⢀⣀⣠⣼⣷⣄⣀⡀⣀⣀⣴⣶⠶⠆
⠀⠉⠙⠛⠛⠛⠋⠻⠛⠛⠓⠂⠤⠤⠤⠤⠄⠐⠖⠚⠛⠋⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⢿⣿⣷⣤⣀⣤⣶⣾⣿⣿⣆⠉⢠⣖⣺⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⠒⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣷⡄⢠⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⣴⣿⡟⠀⠀⠀⢰⣿⠻⣿⣿⣿⣿⣿⣷⠀⠸⣡⣤⡈⢉⡉⢻⡿⣾⣿⣿⡟⡉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⠿⠿⠿⣦⠀⠈⠁⠀⠻⣿⣿⣿⣿⣿⡆⢠⣙⣛⣁⠻⡿⢚⡃⢿⡿⢹⣏⣁⣌⠁
⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣶⣋⣤⡀⢰⣶⣾⣷⣿⣿⡿⠓⠈⠀⠀⠉⠉⠙⠈⢿⣯⣽⣶⣶⣿⣿⣿⣷⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⢸⡿⠉⣀⣤⣽⣿⠙⠉⣡⣤⣀⡠⠟⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⠿⠈⠙⠻⢿⣧⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣟⡋⢹⢀⣻⣿⣾⡿⠁⠀⠀⠀⠀⠀⠒⠀⠀⠤⣀⣤⣿⣷⣶⣶⣶⣦⠈⠁⠀⢳⣾⣿⠋⠙⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢞⣛⣡⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⡿⢿⡿⠿⣿⣿⡟⠿⠀⠀⠀⣾⡿⠋⠄⠀⢠⠘⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣷⣿⠛⠁⠀⠘⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⡀⠸⣿⣧⣶⣶⣶⣦⣿⣷
⠀⠀⠀⠀⠀⠀⣿⡿⠟⢿⣿⣿⡿⠀⠀⠀⠀⣿⣿⠀⠀⠀⣠⣴⣦⣄⠀⠀⠰⠂⠈⠁⠈⢟⡟⠙⠻⠟⢻⣿⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠹⣿⠃⠀⣴⣶⣿⣿⡿⢇⡀⢸⡟⠋⠉⠙⢶⣦⣀⣤⣤⣤⣤⣸⡑⠂⠀⠲⢸⠋⠻⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠈⢀⣸⣿⣿⣿⡇⡀⠠⣍⣹⢀⡀⠀⣄⠸⣿⡟⠛⠿⣿⣿⣿⣷⣾⣽⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⠀⠀⠠⣶⣶⣤⡴⣦⣤⣴⣿⡇⠀⠰⢿⣿⣿⡿⠉⠉⢹⡀⠀⠀⠀⢠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠶⠶⠶⠾⠿⠿⠟⣽⣿⣷⣤⣀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣷⠒⠒⠉⠠⣤⠀⠉⠉⢛⣿⣧⣄⣠⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣧⠀⠀⢀⣿⡄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡏⠀⢋⣼⡉⠢⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠇⠀⠀⠀⠀⠀⣩⠿⠋⡀⣠⡿⠟⠁⠀⡀⠀⠀⠀⠹⣇⠀⠀⣀⣀⣠⣴⢾⡁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡿⠿⠛⡦⠀⠀⢂⣴⣫⣵⣶⣿⠁⠈⠳⢤⠀⠀⠸⣶⠀⠀⠀⠈⠳⣀⠙⠙⢿⠟⠀⠷⣄⡀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠊⠉⠉⠉⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⡿⠿⠋⠻⣦⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⣀⣀⣈⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒
⠀⠀⠀⢀⠀⠀⠀⢀⠀⣴⡤⠴⠯⣿⣦⣚⡿⣽⣶⠘⢷⡶⢶⣶⣄⣤⡄⠰⣽⣀
⠀⠀⣀⣈⣷⢤⣠⣿⡀⣦⣴⣧⣼⣻⣟⠿⣶⣧⣟⣺⣿⣅⡈⢛⣿⡿⡑⣲⡿⠿⣛⡄⠰⣶⠆
⠐⡰⣶⣿⣷⣈⣽⣿⣥⣿⣿⣟⣿⣿⡏⠀⣼⣿⣟⠛⣝⡛⠻⣿⣿⣷⠩⢿⠹⣷⡽⢿⣾⣿⣦⡖⣿⡁⠀⠠
⠀⣸⣷⣯⠟⢋⣿⣟⠋⣙⣾⣿⣄⡙⣣⣞⢀⣴⣴⢾⡿⠷⣽⣟⠋⣡⣄⣰⣫⡿⠻⢈⣳⣿⠿⣷⣶⢿⡇
⣏⠘⠿⣿⣴⣿⣊⣟⣾⣿⠟⠛⣿⣛⣶⣷⣄⣾⣴⠿⢳⣶⣿⣿⣾⣿⢧⣼⣿⣼⣶⣿⣿⣿⣾⣟⣾⡋⣦⣢⣠
⠘⠆⣀⣹⡄⠝⢰⡿⠋⡸⠀⠙⣷⣋⣼⣿⡿⠷⠛⢺⣿⣿⡿⣞⠟⢙⣁⢉⣬⣼⣻⡿⠈⠙⠈⠻⣿⣽⡋⠉⠁
⠀⠀⠀⠉⠙⢦⣾⠉⠁⢀⠀⣠⡟⠉⢹⣷⣲⣦⠖⢙⣛⡿⠣⣤⣆⢡⣩⡢⠿⠟⠿⠳⣧⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⢸⣏⠀⢀⡞⠋⢁⣠⠴⠞⠛⠋⠀⢀⣦⡅⣰⠬⠑⠁⠈⠛⠋⠃
⠀⠀⠀⠀⠀⢸⣷⡾⠛⠉⠉⠉⠙⠒⠲⠦⠤⠤⠤⠴⠊⠹⠓
⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠠⣿⣇
⣀⣀⣀⣀⣀⣹⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣙⣻⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⡿⠈⠻⣿⣿⣦⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡃⠀⠀⠈⠉⠛⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢀⣼⣿⣿⡿⣻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⣾⣿⣿⡏⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⢀⣿⣿⣿⡟⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠿⠿⠿⣿⠛⠛⠛⠛⠃⠠⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡽⣿⣿⣿⣿⡟⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⢹⣿⣿⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⣴⠾⣋⢿⠙⣤⠝⢍⡻⣶⣄⠀⠀⠀⠘⣿⣣⣿⣿⣿⣿⣏⢿⠏⠀⠀⠀⣠⣴⢟⡩⠛⣤⠋⡿⣉⡷⣦
⠀⠀⠀⠀⠀⣾⡽⢎⢑⣤⠛⣄⠷⡉⣣⡜⢫⡳⣄⠀⠀⠀⠻⠿⣿⣿⡿⠟⠁⠀⠀⣠⢞⡙⠣⣔⢉⠾⣀⠛⣦⡊⡱⢏⣷
⠀⠀⠀⠀⢸⣏⡱⡞⢣⡬⠻⣉⢶⠛⣤⠔⢎⣱⠜⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢣⣎⡱⠢⣤⠛⡶⣉⠝⢥⡜⢳⢎⣸⡇
⠀⠀⠀⠀⢸⡏⣣⡜⢣⡸⢖⢙⣤⠛⣄⠵⡊⢣⡜⢫⣷⡀⠀⠀⠀⠀⠀⠀⠀⣾⡙⢣⣜⢑⠾⣠⠛⣤⡋⡰⢇⡜⢣⣜⢹⡇
⠀⠀⠀⠀⠀⢿⣦⠜⢏⡱⡖⠻⣤⠷⣉⢦⠚⢧⡼⢎⣹⣧⠀⠀⠀⠀⠀⠀⣸⣏⡱⠦⡨⠓⣴⣉⠾⢥⡜⢣⢎⡹⠣⣤⣿
⠀⠀⠀⠀⠀⠘⢧⡵⠎⢡⡜⢫⡨⢶⠙⣤⠛⣅⡱⡎⢣⣿⠀⠀⠀⠀⠀⠀⣿⡜⢱⢎⣨⠛⣤⠋⡲⢅⡜⢣⡜⢑⢮⣼⠃
⠀⠀⠀⠀⠀⠀⠈⢳⣞⢣⡼⢎⣱⣔⠻⣀⠷⡉⣢⡞⢣⣿⠀⠀⠀⠀⠀⠀⣿⡜⠓⣴⣉⠾⣄⠟⣢⣎⡱⢧⡜⣳⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣵⡎⢡⡼⠫⣈⢶⠙⣤⢝⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⡻⣤⠛⡶⣁⠝⢣⡌⢱⣮⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣼⣮⣑⣤⣿⡴⠟⠁⢸⣇⠀⠀⠀⠀⢸⡇⠈⠻⢦⣿⣤⣎⣵⣧⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣤⣀⠈⢿⡄⠀⠀⢠⣿⠁⢀⣤⡶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣼⣷⣦⣴⣾⣧⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄⢀⣤⣤⡄⣀⣤⡄⢠⣤⡄⢀⣠⡀⢀⣀⣀⣀⣠⣤⡄⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡿⠿⢸⣿⣯⣭⡅⣿⣿⢿⣼⣿⡇⣿⣿⣷⢸⣿⡟⠻⣿⡟⠱⣿⣿⣿⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⢸⣿⣯⣴⡆⣿⣿⠘⣿⣿⡇⣿⣿⢻⣿⣿⡇⣸⣿⣧⡄⣥⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠷⠀⠘⢿⠿⠿⠃⠿⠛⠀⠹⠛⠃⠿⠿⠀⠻⠿⠃⠿⠿⠿⠇⢿⣿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⣰⣿⡿⠁⠀⠀⠀⠀⢀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⢠⣾⣷
⡀⣸⠆⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⠀⣠⣞⡿⠃⢰⣿⡇⠀⠀⣦⠀⠀⠀⢀⡁⠀⠀⢸⣿⠀⠀⢀⣠⡀⢀⡉
⣇⣿⠀⣰⠄⢀⢻⣧⠀⠠⠀⠐⠃⠀⣿⡿⣀⣴⣾⣏⠁⠀⠀⠹⠀⠀⠀⠀⢠⣷⠀⠞⠁⣠⠔⠛⠟⠛⠀⠡
⢮⣿⠐⢿⣐⣯⢀⠹⣷⡄⠀⠠⠀⠀⠘⠻⠿⠟⡿⠛⠛⡆⠀⡀⠀⢀⣴⠀⠀⠀⠀⠀⠈⢀⡆⠀⢠⡇⡀⣼⢰
⣿⣿⡀⣸⣷⣇⢺⠀⠈⢀⢰⣄⢄⠀⠀⠀⠀⢀⣀⠀⠀⢀⠀⠃⠀⢈⣀⣀⣢⡆⣾⣴⣶⣼⠧⣿⣾⣿⣧⣿⢸
⠉⠉⠉⠙⠻⢿⣾⣶⠿⠺⠲⠟⠛⠀⠘⠃⠀⠘⠙⠛⠛⠛⠿⠂⠀⠛⠿⠿⣿⣿⡿⠿⢿⣿⡄⠿⠿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠉⣀⣲⣶⠂⠀⠀⠀⠀⠀⣀⣀⠀⠂⠤⠀⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⢉⣁⣀⣴⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⢤⣤⣄⠀⠀⠀⠀⠉⠁⠀⠀⠀⣀⣀⠀⠉⠉⠉⠉⠉⠹⠿⠏⠉⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠔⠒⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢀⠀⠀⠀⢠⣄⡀⠙⣿⣦⣄⣤⣼⣿⣤⣀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠂
⠀⠀⠀⢀⣼⣶⣤⣽⣿⣿⣿⣿⠟⠛⠻⢿⣿⣿⡛⠉⠁⠀⠀⣤⣦⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒
⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣬⣿⣷⣤⣤⣤⣤⣬⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣴⠟⡋⣅⡠⡰⢤⡢⣄⢭⡙⠲⣤⡀
⠀⠀⠀⠀⣠⡟⢡⢮⢮⢞⢝⡾⣕⡽⡳⣵⠪⣳⢄⡙⢦⡀
⠀⠀⠀⢀⡿⠰⡣⣱⠯⣪⢟⢮⢞⢝⡾⣕⡽⡣⣱⢫⣂⠙⣦
⠀⠀⠀⢸⡇⢘⡾⡕⣵⡫⣳⢯⣪⢟⢮⣞⢝⡼⡵⣵⡫⣳⠌⢧⡀
⠀⠀⠀⠀⣷⠀⠎⠾⢝⠽⡳⣕⡫⣳⢯⣪⣞⢯⡞⣕⡽⡳⣵⡈⣧
⠀⠀⠀⠀⠘⣇⠁⢢⣂⢜⢮⣕⠝⡲⣵⡯⡣⢥⢪⣚⢟⠾⣕⠥⢸⡆
⠀⠀⠀⠀⠀⠘⢧⡑⠣⣳⢯⣪⢟⢮⣞⢝⡿⣱⡭⡫⣳⢮⣪⠏⢸⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠑⠫⣡⢯⣪⢞⢎⡞⣕⡽⡳⣁⠈⣠⡏⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣑⣉⠃⠛⠊⠚⠙⣊⣡⠾⠋⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠻⠷⠤⣤⣄⣀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠉⠙⠓⠦⣄⣹⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⡻⣿⣿⣿⣇⠀⠀⠀⠀⠀⠉⠾⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣮⣾⣯⣉⠛⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣠⣿⠀⢈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡶⠲⠄⣠⠞⢁⡼⠉⢈⡇⠀⠀⠙⢄⠀⠀⠀⠀⣠⡴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡫⢻⠅⢀⡼⡯⠒⠛⢁⡤⢾⠁⠀⠂⠸⡀⠑⣄⡤⣞⣡⢷⣄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢰⠒⠉⠊⠰⠃⠐⡩⠊⠀⠀⣀⡟⠀⣸⠃⠀⢐⠀⠁⠄⠈⠻⡑⠁⢠⣿⡀⠃⠉⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠀⢸⡂⠀⠀⠀⠀⠀⠀⢀⡴⡋⠋⣠⢜⠃⠀⠀⢘⠳⠄⠀⠀⠀⠈⠂⠼⠁⠁⠀⠀⠀⠀⢀⣾⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⢰⣿⡀⣽⡀⣿⡇⢸⡀⠀⠀⠀⠀⠁⠀⢀⠔⠁⢸⠀⠀⠀⠀⠢⠈⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢸⡀⢰⣿⡂⣸⣇
⠀⠀⠀⠀⠀⠀⠀⢰⣷⣄⣻⣿⣗⣿⢧⣿⣟⣾⡆⠀⠠⡇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⠀⠠⣿⣇⣽⡿⠷⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠚⠿⠿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣸⣿⣇⠀⣸⣿⣅⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠦⠿⠿⠦⠤⠤⠀⡀⠀⠀⠀⠀⢀⡆⢀⣾⡆⢀⣿⣿⡷⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡝⣲⠂⠒⠚⠛⠊⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⣚⠥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠏⡀⢀⣿⡇⠀⡠⠔⠒⡢⠎⡩⠔⠊⠉⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⢀⣻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⡑⠢⢬⡦⣘⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠀⢐⣿⡂⠀⠀⡅⠀⠀⠀⢠⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠂⠙⢮⡙⠒⢤⡀⠀⠤⢀⣰⣾⣿⡄⠀⠀⢾⣿⡥⠀⢠⣷⠀⠀⢀⣿⣿⣿⠧⠔⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⡽⠀⠀⠙⢆⠀⠀⠀⠈⠉⠉⠒⠞⠿⠿⠿⠆⠾⠿⠗⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠀⢀⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣦⠀⢀⠀⠀⢀⢤⣶⠤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⠟⡀⢐⡛⠀⠀⡘⠃⠈⣠⣴⣦⡦⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠷⡟⠀⣳⢠⡌⠳⠟⠘⠿⣦⢈⡹⣃⠀⣴⠘⠃⢻⣿⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⡇⠲⡏⠸⡟⠰⣿⣤⡄⢠⡶⣠⢠⣍⣬⣄⠻⠂⠀⠷⣠⠊⠀⠘⠃⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠈⠁⣠⡄⣦⠙⣤⣾⠁⠐⠛⠳⢇⡀⣽⠻⠧⢀⡶⢾⡤⠛⠻⣿⣤⢘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢯⣤⣯⡀⢠⣤⣿⠋⠙⢷⡆⣶⢸⢿⠇⣤⠷⠙⣷⢾⡄⣤⣤⡄⠋⢻⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠰⠬⣷⣟⡿⠿⣼⠇⠈⡀⠰⠀⢸⡄⠁⠀⣠⡏⣀⣀⠉⡹⢇⡏⠘⠋⠾⠛⠁
⠀⠀⠀⠀⠀⠀⡀⢀⢠⡤⠟⢁⣄⣀⠀⣄⢰⣶⠇⣨⠻⢾⣿⣧⡆⣾⣇⣠⡀⠻⣧⠹⠿⢎⣴⣯⢰⣧⠀⣴⣶⠆
⠀⠀⠀⠀⠀⢠⣿⣿⠇⠁⠸⠏⣛⠃⢶⣿⡧⠉⡘⢻⣶⣦⡙⣿⡜⢿⣿⡿⣀⠻⢿⣧⢷⣾⠉⠉⢀⡓⠀⠉⡉⢸⣦⠄⢠⣦⠄
⠀⠀⠀⠀⢀⡰⣿⡁⠀⢶⡆⠁⠙⠀⠸⠋⠀⢴⠠⣦⠙⢋⣠⡌⢿⣿⢃⣀⠙⠀⣤⣿⣿⠃⠿⠸⣿⡆⢀⡶⠻⠈⠁⢀⠈⢹⣶
⠀⠀⠀⠴⠾⠇⠀⠸⣷⠀⡷⣾⠆⢁⠀⠸⠃⠾⣧⢉⣦⠘⠋⣤⡘⣿⡆⢙⣧⡤⢠⣿⣷⣶⠦⣠⣿⣯⣼⡆⢸⣀⡴⠾⠿⣌⢉
⠀⠀⠀⠐⠿⠂⡒⢺⡿⠲⡇⢠⣄⢸⣆⠀⢹⡿⢻⠈⠉⣸⢠⣯⠀⢹⣿⠊⣠⣴⣿⠛⠛⣳⣤⡟⣠⡈⢋⣄⢸⠛⢛⢸⠇⠻⠏
⠀⠀⣠⣿⣦⠸⣇⡠⣤⢀⢹⡄⢀⡿⠿⡁⠀⣴⣿⣧⠞⣅⠙⠛⢠⣿⣿⣿⡿⠟⣉⠀⢄⢈⢸⣿⠿⠃⢈⣡⠟⢿⡟⠄⢀⣻⡏
⢤⣿⠆⠉⣭⡀⢛⣤⡄⢴⡌⢻⡞⠀⢼⠿⠀⠢⡅⢸⣿⣧⣴⡆⠈⢻⣿⣿⣶⡿⣫⡄⠸⡟⣹⣿⣾⠛⢿⣵⠀⣨⡧⠀⠘⠟⠃
⠀⣠⣄⣦⠬⠧⣤⡷⠞⠛⢻⡿⣿⣦⣰⣷⣴⠆⢹⠀⡀⢾⣝⣿⣷⣾⣿⣇⢠⣔⠉⠰⢿⣿⣿⠃⣴⡆⢨⣠⠎⠀⠀⠿⠆⢰⣧
⢰⡿⠉⠫⣤⣠⡛⢃⣜⠷⠙⠃⠙⠿⠛⠻⠿⣶⣼⣿⠃⠈⣹⡏⠈⢻⣿⣿⣴⣿⣶⣿⣿⣿⣷⣶⣿⣿⣋⠁⠀⠀⣆⡄⠂⢠⣤
⠀⠀⢴⣶⡄⢙⡃⢀⡋⠀⢠⣶⣦⣰⣦⣴⣶⣶⣿⣿⣷⡀⠙⢻⡄⣿⣿⣿⡿⠿⢿⣿⣍⣽⡇⠀⠀⠾⢫⡟⢟⡋⢡⣤⡐⢿⡿
⠀⠀⢠⣍⠀⢈⣿⣿⣈⣦⣈⣿⣷⣿⠛⠉⠹⣿⣿⠿⣿⣷⣄⢸⣿⣿⣿⡟⠀⠀⠀⠙⢻⣷⠲⣥⠀⡠⠀⢙⡘⠿⢰⣿⡇
⠀⠀⠿⣷⣶⡆⠘⠋⠸⠙⣿⢿⡆⠻⠃⠀⠈⠋⠉⠀⠈⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠘⣿⣿⡷⣶⡟⣷⠲⠛⠁⠀⣤⡌⠁
⠀⠀⠀⠀⠋⠴⣶⡄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⢿⠇⠈⢀⣐⣶⢰⣦
⠀⠀⠀⠀⠀⠀⠉⠀⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⡿⠃⠑⢆⣤⣀⠛⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠰⡞⠷⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠊⠙⠛⠛⢻⣿⣿⠿⠿⠿⠿⠿⠛⠛⠓⠲