Humour Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⣿⣶⡀⠀⠀⣀⣠⡤⠤⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠞⠋⠉⠀⠀⠀⠀⡇
⠀⠀⣀⣤⣶⠶⠿⣿⣿⣿⣿⢯⣿⣯⣭⣭⣽⣿⠀⠀⠀⠀⠀⣾⣿⣿⣶⡀
⢠⡾⠋⠁⠀⠀⠀⠹⣿⡷⠚⠻⣿⣿⣋⣀⣉⢻⡄⢠⠖⠚⠉⣹⣿⣿⣿⣿⣶⣄⣀
⠸⣧⣀⣀⣀⣀⣀⡀⣿⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡞⢦⣀⣰⣿⣿⣷⣽⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠉⠉⠉⠉⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣶⣿⣷⣄⢹⣿⣿⣿⣿⣿⣻⣿⣿⣿⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠻⢿⣿⡿⠛⠁⠀⠀⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⣿⡟⠉⠉⣀⡀⠙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⢟⣡⣴⡶⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⣰⠟⠉⣀⣀⣻⣧⣴⣿⣿⣿⣿⡟⣿⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⠁
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠁⢙⡟⢿⣿⣿⣿⡿⢹⣿⡇⠀⠉⠛⠿⠟⢛⣿⡿⠟⠁⠻⣦⣀⣤⠖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠀⠉⠉⠉⠀⢸⣿⣷⣀⢀⣀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠋⠁⠀⠀⡇⢸⠀⠀⠀⠄⢀⡀⡀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡆⡃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠲⠂⠐⠦⠆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠿⡏⠁⠛⠃⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣉⠉⠉⠉⣿⣿⣧⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠠⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠰⠂⠰⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣦⡀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢾⣿⢋⣤⣾⣿⣷⣶⣾⣋⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢸⠀⠀⠀⠀⠀⠀⠀⡄⠀⢤⠀⠁⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠘⠁⢺⡏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠉⢻⣿⣿⣿⣿⣿⣿⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠸⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠉⠁⠀⠀⠆
⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠚⠒⠒⠒⠒⠒⠒⠒⠂⠃⠘⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡅⢈⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠐⠂⠀⠀⠀⢤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⡷
⠀⠀⠀⢰⣦⣤⣤⣤⣄⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣤⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⢠⡄⢀⡈⡷
⠀⠀⠀⢠⣤⣿⣿⣿⡋⢙⣿⣿⣿⠛⠿⣷⣶⣶⣶⣶⣦⣄⣤⣄⣄⣀⣀⣀⡀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢶⡄
⠀⠀⠀⠘⠋⠉⠛⠛⠿⠿⠿⠿⠿⢿⣷⣿⣿⣿⣧⣤⣿⣿⣿⣍⣉⣻⣿⣿⡿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠒⠀⠀⠀⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠛⠛⠛⠿⠿⣷⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢸⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡈⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣟⣟⠙⠋⣿⣿⠋⣹⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠿⡿⠿⠿⣿⠿⢿⡟⠿⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣴⣶⣴⣤⣿⣦⣾⣷⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠷⠶⣤⣤⣤⣄⣀⣀⣴⣟⢁⣀⣸⣷⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠉⠉⠻⠛⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⠀⠀⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠃⠁⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⠀⠀⠀⠈⠉⠁⠈⠆⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡤⢀⠀⠀⠀⠀⢠⠀⠀⠤⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢀⡀⠀⡅⠀⠀⠀⢱⠀⠁⠀⠀⠀⠀⠁⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⡀⠘⡂⠀⠹⡀⠘⠀⣖⡃⢤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠹⡀⠀⠠⠀⢄⠽⠇⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⠀⠀⠀⠀⡀⠀⠀⠱⠀⠀⠀⠆⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⠀⠐⡍⠀⠀⢠⠀⡗⠀⢤⡄⠀⣀⣀⡀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⢄⡀⢐⣒⡒⢚⠒⠒⠒⡂⠀⠀⠙⣷⠐⠀⠀⠀⠘⠀⣼⣿⣿⣷⠀⢿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠓⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠉⠉⠉⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠻⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣹⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⣠⣴⠞⠛⠉⠉⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣽⣿⣧⠀⠀⠀⠀⢸⡇⠀⣿⣤⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣨⣿⣿⠏⠀⠀⠀⠀⣸⡇⠀⠻⢤⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⣾⢿⣆⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣸⠟
⠀⠀⡠⠒⠂⢀⡾⠋⠀⠀⠈⢻⡄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣨⡿⠁⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⢤⡀
⠀⢸⡁⠀⢀⡾⠀⠀⠀⠀⠀⢀⠁⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠈⠛⠿⠶⣶⣦⣴⡶⠶⠿⠀⠀⠀⡄⠀⠀⠀⠀⢣⠀⠉⢦
⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠉⠛⠛⠶⠶⠶⠶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠊⠀⠀⠀⠀⠀⠈⠣⠤⠞
⠀⠀⠀⠀⠀⢀⠔⠈⡇⠀⠀⠱⣄⠀⣠⠃
⠀⠀⠀⠀⠀⠘⠤⠜⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠟⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠻⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⡀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⡴⠞⠛⠉⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠛⠄
⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡯
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠲⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠐⠈⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⣆⠀⠀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⢀⡾⠛⠛⠛⠛⠉⠉⠛⠛⠒⢂⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠋⠉⠛⠛⠉⠀⠀⣀⣤⣴⠶⣾⣿⡴⠟⠋⠉⠉⠉⠉⠉⠉⠉⢉⣉⣩⣽⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣤⡾⠋⠁⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣯⣉⣉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⣀⠀⠀⢀⣠⠟⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠐⠋⠉⠉⣩⡿⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⢠⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⡆⢀⣀⣤⣤⣶⡶⠾⠿⠛⢿⣿⠟⠛⠻⣿⡶⢶⣤⣀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⢸⠉⠙⠳⣦⣄⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⣸⣿⣤⣴⣾⣿⣿⡏⠉⠙⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⡓
⠀⠀⠀⢸⡀⠀⠀⣄⠙⢷⣄⠀⠀⠀⣸⠉⠀⠙⣶⣿⣿⣿⠁⠀⢿⣿⣿⡇⠀⢰⣶⣶⣦⡄⠀⠀⠀⠑⡖⠦⣤⣀⠀⢈⡞⢷⣄⠀⠀⠀⠀⠧⢤⠆
⠀⠀⠀⠘⣇⠀⠀⠀⠑⢤⠹⣷⡀⣰⠏⠀⣀⣼⠛⣿⣿⣿⠀⠀⠈⠻⡿⠁⠀⣸⣿⠿⠿⡇⠀⠀⠀⠀⠑⠤⢤⡿⠻⠟⠀⠀⠙⣦⡀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠹⣦⠀⠀⠠⣌⣳⠾⣿⣿⠚⢻⣿⡇⠀⠙⢿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣿⠀⠘⠋⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⣼⠏⠀⠀⡠⢿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⠛⠛⢿⣤⣿⣿⣶⣶⡖⠒⣶⠶⠶⠖⠒⠛⠋⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⡿⠀⠀⢀⣴⠟⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠈⠏⠙⢿⣿⡟⣧⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⢸⣧⣤⠶⠛⠁⠀⠀⠀⢲⣬⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢻⣧⡞⢶⡀⣀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠳⡄⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⢸⠿⠧⠶⠛⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠙⣆⠀⠀⠀⠳⣤⣍⡙⠢⣄⡀⠀⠀⠈⠙⠓⠦⠤⠴⠋⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠤⠔⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣷⡀⠀⠸⣯⢍⡉⠀⠀⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⢦⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠹⣄⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠉⠙⢄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠙⠒⠖⠠⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠳⣤⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⢦⠤⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠉⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠟⠛⠛⠛⠛⠛⠛⠉⣡⠏⠀⠀⠀⠀⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⣶⣟⡀⢀⡴⠋⠀⠀⢀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣉⣙⣋⣤⣤⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣠⣶⣿⣿⣿⣶⣦⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⠋⠀⢠⣠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠋⢻⠟⠳⠀⠀⠀⠀⢁⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠀⢀⡾⠀⠀⢀⠀⠀⠀⡎⠀⠀⣾⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢢⠴⠊⠀⠀⠀⢀⡟⠒⠊⠀⢰⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢄⠀⠘⠫⠉⣶⢆⠀⠐⡀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢏⠀⠀⠀⠀⠀⠀⢁⡆⠀⠸⣿⠋⣁⠀⡧⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⢿⡧⣄⠀⠀⠀⠀⣠⣀⠀⢴⣯⡞⠀⠙⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⢶⣾⣿⣿⣿⣿⣷⣦⠁⠀⠀⣻⣼⠁⠒⠒⢈⣡⠤⠤⠾⠟⠓⢦⣤⣀⡀⣰
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣭⣿⣿⣿⣿⣿⢟⣛⣽⡏⠀⠀⠺⢿⣿⣿⣿⣿⡟⠁⣴⣾⠇⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⢿⡿⢑⡯⠛⢛⣽⣿⣷⣤⣀⠀⠀⣝⣿⠿⠟⠃⠀⡌⠉⠙⠓⠒⠤⣀
⠀⣠⣾⣿⣿⣿⣿⡿⢟⣫⠟⣩⠞⠉⠀⠰⣿⣿⠟⣯⣾⣿⣿⡟⡿⠷⠄⣤⠤⠒⠀⠀⠁⠀⠀⠀⠀⠁⠢⢄
⣰⣽⣿⣿⣟⢁⣠⣴⠛⣡⠞⢁⡰⠂⠀⢸⣿⠟⠋⠀⠉⠸⠋⠐⠁⠀⢀⠀⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠑⢤⡀
⣯⣿⡿⡵⣿⡿⠋⣠⠞⠁⣤⠋⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠡⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⢀
⢹⣧⢤⣾⠏⠀⠞⡡⢚⣾⡗⠀⣠⡞⠁⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⡀⠀⠀⠀⠀⠀⠱⡀⠀⡀⠀⠀⠠⡡
⠀⢻⡿⠁⠀⣠⡾⣳⣾⡟⢀⡴⠋⠀⠀⠀⡐⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢄⠈⢦
⠀⣠⣠⢀⣴⣿⣽⣿⠏⡠⡟⠁⠀⠀⠀⠌⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠆⠀⣆⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠈⢦⠀⢣
⠀⠈⠃⠾⣿⣿⣿⣯⠞⡜⠀⠀⠀⢠⠆⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠳⢀
⠀⠀⠀⠀⠀⠉⠫⢌⡀⠁⠀⠀⢀⠎⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠣⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠂⠈⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣝⠀⠀⠀⢀⣸⠧⠴⠶⠦⠖⠒⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢲⡶⣶⣶⡾⣿⠏⠈⠈⠁⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠴⣖⠶⡀⢀⣀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠒⣤⣰⣿⣿⣿⣿⣿⣭⣿⣦⡀⣸⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠛⠛⠛⠻⢿⣿⣿⣥⣽⣅⠀⠈⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣻⡷⠂⠀⠸⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣶⡤⠀⡇

⠀⠀⠀⠀⢀⣴⣦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡤⠴⠶⠂
⠀⠀⠀⠸⣿⢿⣻⣯⡿⠀⠀⠀⠀⠀⠀⠀⢰⢣⣎⣴⠶⠶⢶⣦⡀
⠀⠀⠀⠀⠐⠿⣿⠟⠁⠀⠀⠀⠀⣤⡀⠀⢸⣾⡿⡡⢊⣉⠓⢌⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢻⡆⠀⣀⣀⣀⣤⣬⣿⣷⣦⣿⣿⣧⣑⠐⢊⢦⠑⣌⢷⣄
⠀⠀⠀⠀⣀⣤⡾⠿⠛⠋⠉⠉⠉⠉⡽⠋⠉⢉⣙⢿⠙⠛⠿⣷⣥⡈⠢⡙⢷⣄
⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠙⠏⣸⠃⠀⠀⠀⠙⢿⣷⠘⣜⣿
⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠶⣾⡁⠀⠀⠀⠀⠀⠈⢿⣧⢸⡿⣄⡀
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠸⣿⣮⣷⣀
⠀⠹⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⢿⡝⠂⠀⢀⡴⠿⣿⣿⡿⠇
⠀⠀⠈⠙⠻⠶⠶⣦⣤⣤⣤⣤⣴⡶⠶⠀⠀⠀⢀⣴⠋⠀⣧⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⡤⠶⠛⠁⠀⣠⣿⠀⠀⠘⢷⣶⠾⠋⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣧⣀⣠⣴⠾⢻⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡟⣠⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠿⣏⣁⣠⠟⢀⡄⠀⠀⢸⡟⠀⠀⠀⠺⠿⣿⣿⣿⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠁⠠⠴⠛⠀⠀⠀⣼⠇⠀⠀⠀⢀⣴⠟⠛⠛⠁⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣙⣒⣒⠶⠶⢦⡀⣠⠤⠤⠤⠤⠶⢶⣿⢀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠉⠉⠉⠉⠙⠛⠛⠓⠒⠒⠒⢺⡟⠋⠉
⠀⣠⠞⢲⡀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⡟⠀⢸⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣠⣧⢠⠏⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠧⠴⠛⠛⠶⢶⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠘⠻⠶⢤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠷⠶⢶⡶⠶⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣙⢲⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡂⠀⠦⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠐⠂⠀⠲⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡈⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡁⠀⢀⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠁⠉⠉⠚⠒⠒⠒⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣀
⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠠⠀⠁⠀⡈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠇
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠆⣅

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣽⣯⣿⣯⣿⣭⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠊⢀⢈⣉⢉⡀⡐⠀⠀⡀⠀⠀⠂⠀⠅⠀⢄⠀⠄⠀⠈⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⣤⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣾⣦⡎⣢⣦⣾⣷⣾⣷⣾⣷⣾⣷⣶⣦⣬⣄⠀⢀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⣿⣿⣿⣿⣿⣿⣧⣗⣃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⠭⠍⠀⣶⠹⣿⣿⣿⠟⢻⣿⣿⣿⣿⠟⣲⠶⠀⣤⣈⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣇⣩⣤⣴⣤⣰⠿⠿⠣⡀⠈⢛⢿⣿⣿⡆⣤⣧⣴⠘⣸⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡛⠙⠉⢉⠽⠂⡀⠀⠀⠀⠀⡐⠉⠉⠃⢀⠀⡙⠰⠿⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⡀⠠⣸⣆⠈⠒⠀⠀⠀⠁⣠⠴⡦⢄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠈⠀⢿⣶⠊⠲⢨⣷⢖⣖⣶⣶⠉⡧⠂⢫⠉⠁⠉⠀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⢠⠞⠁⠀⣠⣾⠁⡀⡀⢰⣉⣤⡇⠀⠘⣇⠀⠄⢰⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠉⠀⢈⣴⣿⣷⠤⠀⠠⢬⢶⣿⠹⡀⠀⠘⡄⠀⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⡁⠀⣠⠞⠘⣿⣿⣆⡂⠀⣚⣿⣿⣦⠳⡀⠀⠰⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠁⠀⢠⣾⠀⡀⢀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠓⣄⠄⠈⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠀⣾⣿⣧⡂⢑⠀⣿⣿⣿⣿⣿⣿⣿⢠⢠⣿⣦⣀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣦⡁⡺⣿⣿⣿⣿⣿⣟⣀⡾⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⠶⠶⠂⢀⡾⡿⣿⠋⣿⠟⢶⡀⠀⠀⠀⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡸⣇⢀⡜⠃⢷⣞⣧⠁⣿⣆⣄⣠⣼⠃⣠⣶⣦⡠⣢⣤⣄⣤⠄⡄
⠀⠀⠀⠀⠀⠀⠀⢐⣩⣉⠁⠋⠤⠀⠈⠹⠀⢶⠈⢿⣿⣿⣯⣸⣿⣿⣿⣷⡩⠡⠙⠉⠉⠀⠆⠁
⠀⠀⢀⣤⡞⠀⢀⣿⣿⣭⠁⠶⠀⢀⠀⠀⠀⠀⡆⠀⣿⣿⣏⣽⣿⣿⣿⣿⠃⠀⠙⠐
⠀⠠⡀⠀⠸⢶⣿⣟⣛⣻⣀⣠⡔⠎⠾⠚⡺⢳⡲⠏⢿⣿⠙⣿⣿⣿⣿⡟⠋⣀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠿⡰⠆⠉⠿⠃⠀⢀⣅⡅⡈⣉⡴⢂⠀⢨⡍⠐⣻⣿⣿⡟⠁⠘⢿⡇⠨⠀⡈⠁
⠀⠀⠀⠀⣀⠀⠀⠠⠿⠻⠇⡀⠈⠁⡴⠛⠋⡀⣤⠀⠀⠀⣰⣧⣾⣿⡀⠀⠀⠀⠁⢀⡀⢁⡀⣐⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⢃⠀⠀⠈⠉⣀⡀⠀⠙⢿⣿⣿⡿⠆⠀⠀⠀⠘⢷⠩⠙⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣾⠀⠀⠀⠀⠀⣤⡞⠙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠀⠀⠓⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠙⢉⡗⠀⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⢸⠁⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣤⠆⠓⠀⠀⠀⠀⢸⣿⠻⣿⣶⣄⠀⠀⢸⠀⠸⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⣿⠀⠈⠻⣿⣿⣶⣼⣤⣤⣴⣿⣿⣧⣄⣀⣠⣤⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠛⠛⠛⠛⠋⣽⣿⠃⠀⢠⡤⠤⠶⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣶⣶⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⡀⠀⠘⣧⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⣠⠞⠋⠁⠀⠀⠈⠉⠓⣶⣿⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠃⠀⠀⣿⡇⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⣼⠀⠈⣿⣟⠄⣠⣴⣶⣶⣦⣄⣼⣯⣼⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⡾⠟⠛⠛⠲⠦⣄⣿⡵⡛⢦⡀⠀⢀⡏⠀⠀⠘⣿⢲⡟⠁⣶⠚⠉⠉⠁⠀⣿⣿⠀⠀⢻⣤⣄
⠀⠀⢀⡀⢀⡄⠀⠀⠀⠀⠈⠙⠻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⡎⠈⠻⣄⢸⠇⠀⠀⠀⠹⣟⣧⡀⢻⣄⣤⡄⠀⠀⣿⣿⠶⠒⠛⣇
⠀⠀⠈⡯⠁⢙⠀⠀⠀⠀⠀⠀⢠⣿⣷⡅⣴⡾⠟⠻⢷⣆⠀⠀⠸⣧⠔⠀⠈⠿⠀⠀⣤⠀⠀⢹⣎⠛⠛⠛⠉⠀⠀⣼⠇⣸⣇⣀⣀⣿⠿⠷⠶⠶
⠀⠀⠈⠉⠑⠞⠀⠀⠀⠀⠀⢀⣼⣿⣿⢾⡏⠀⠀⠀⠀⠹⡆⠀⠀⣿⣸⣦⡀⠀⠀⠀⣿⣆⠀⠀⢻⣤⣄⣄⣤⣴⠿⢣⡾⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⣠⠇⠀⠀⣿⣯⣿⢿⣄⠀⢸⡇⢻⣦⣴⣾⡟⠛⠉⠉⠀⠀⢿⡀⠀⠀⠀⠀⠀⢠⢀⣀
⠀⠀⠀⠀⠀⠤⠴⠾⠿⠿⠿⠿⢿⣿⣿⣿⣯⠛⠶⠶⠞⠁⠀⠀⣰⣿⣿⣿⠀⠹⠿⠟⠀⠀⠉⠀⠀⠀⢀⣘⣦⡤⢤⡬⣷⣄⠀⠀⠀⠀⠎⡀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢈⠿⣿⣿⣇⡀⠀⠀⣀⣠⣾⣿⡛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠳⣄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⢹⡿⠻⠿⠿⠛⠋⠉⠙⢿⣆⠀⠀⠀⠀⣠⡴⠲⠒⠲⢤⣸⠄⠀⠘⢦⡈⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢠⡞⠁⠀⣀⣤⣄⡀⠘⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠆⠀⠀⠀⠀⠀⠀⢘⡇⢠⡏⠀⣠⠟⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠀⢈⡺⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠶⢦⣼⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⠂⠃⠀⠀⠀⡔⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⣁⠀⠀⠀⠀⠠⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠘⠳⠊⠀⠀⠀⠡⡀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠺⢃⡀⠀⠐⡄⠀⠀⠀⠈⠑⠒⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠤⠤⠃⡀⠀⠀⠀⡜⠀⠀⢨⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠨⠜⠀⠀⠀⠀⠀⠈⢆⣠⠞⠠⠤⠒⠀⢣⣀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⠀⠀⠐⢄⠀⠰⠀⠀⠀⠀⠑⠢⣄⠀⠑⢄⢀⣀⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠕⠀⠈⠀⠀⢑⠜⠻⣌⠉⠉⠉⠀⠀⠛⠆⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠒⠈⠒⡀⠀⠀⠀⡞⠒⠒⠒⠀⠤⠤⠄⣤⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠾⠄⠁
⣶⣶⡶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠶⠄⣀⠀⠀⠠⠤⠀⠀⠀⠀⠀⢀⠔⠉⠁⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠠⣄⠀⠀⠘⠢⣄⠀⠀⠀⣀⣤⣴⡏⠀⠀⠀⠀⠀⠘⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠖⠀⠀⠀⠀⠉⠲⢦⣀⠈⠳⣤⠞⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣁⠠⠚⠀⠶⣆⡠⣀⠀⠀⠳⣤⠀⠀⠀⠀⠀⠈⠙⢆⣀⢀⣠⠦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠙⢯⡓⠀⠀⠉⢷⠖⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠺⡏⠀⠀⠐⠉⠳⡄⠀⠀⠈⠁⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⠟⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣎⠻⢃⡴⠂⠀⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠂⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⡶⢠⣾⠀⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⡴⣿⡋⠀⠛⡀⠀⠈⠳⣶⣦⠀⠀⠀⠸⡇⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⡇⠀⠀⠀⢈⣉⡔⠂⠀⢀⡄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠱⠶⠶⠖⠊⠉⠉⠉⠙⠻⣧⡀⠘⣇⣀⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⢸⡧⠀⠘⣷⠛⠛⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠟⠋⠁⠀⠀⣿⡆⠀⠀⠀⠈⠳⡀
⠀⢠⡞⠉⠉⠉⠐⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣿⠂⠀⠀⠀⠀⠹⡆
⢠⡏⠁⠀⢰⡀⠔⢛⣤⣴⣶⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣠⣤⣤⣤⣴⠿⠀⣇⠀⠀⠐⠀⠸
⣬⠁⠀⠀⢸⡿⠀⣟⡁⣀⡄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠉⣉⠍⠀⠀⠀⠀⣰⠇⢸⠀⠀⠀⡇⠀⠻⡀
⣿⠀⠀⠀⢸⠇⠀⠈⠛⠋⡄⠠⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠙⣤⡶⠋⠀⠀⠀⣠⠞⠁⠀⢸⠀⠀⠀⠁⠀⠀⢷
⢸⡇⠀⢠⡎⠀⠀⠀⠀⠀⠃⠀⠀⣹⡗⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢰⠏⠀⠀⠀⠀⡞⠁⠀⠀⠀⢰⡇⠀⠀⢸⡆⠀⠸⡇
⠀⠹⢦⣼⠇⠀⠀⡀⠀⠀⡆⠀⣤⠋⠰⣤⠎⣤⣤⣤⣤⠞⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⢹⡀
⠀⠀⢀⡿⠀⠀⠀⠇⠈⠀⢶⡶⠈⠉⠶⠇⠘⠁⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⢀⣀⡄⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇
⣄⠚⠋⠀⠀⠀⠀⠀⢰⠀⢠⠀⠀⠀⠀⠻⢆⠀⠀⠀⠀⠀⠀⣠⠞⣿⣤⣴⠟⠻⣍⠁⠀⠀⠀⠸⣇⠀⠀⠀⠀⣇
⠀⠑⠆⠀⠀⠀⣧⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠱⢦⣤⣤⠶⠋⠁⣀⠁⡸⠧⠘⠁⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠰⡆⠀⡟⠀⢸⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢠
⠀⠀⠀⠀⠘⢦⡟⠒⣿⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⣷⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣙⠀⠀⠘⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣿⡀⣠⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⠑⠶⢶⣀⣀⡀⠀⢀⣀⣀⣽⡇⠀⠀⠀⠀⢠⡿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠂⠢⠤⠤⢤⣤⡤⠤⠉⠉⠻⣤⣀⣀⣴⡏⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣍⣉⣼⣇⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣹⡶⠴⠶⠶⠶⠶⡞⠁⠀⠀⢈⣀⠉⣉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣶⠀⠀⠀⡰⠮⡙⠚⠉⠀⢁⡀⠀⢠⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⣴⠁⠀⠘⡀⠀⠀⠈⠁⠿⣀⠐⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠠⠾⡃⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢸⠅⠀⠀⠀⢻⡄⠀⠀⠀⠀⠘⠧⠾⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠠⣿⡀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡎⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⡇⠀⠀⠀⠀⢿⡀⠀⠀⠀⣽⠂⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡧⠀⠀⠀⠀⠀⠻⣆⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣿⠀⠀⣿⠃⢤⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠉⠉⠛⠛⠁⢸⠀⠀⠀⠀⠀⠀⣀⣸⣇⠀⢸⡟⠰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠆⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⢸⠀⠙⠛⠛⠛⠋⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠚⠁⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⣿⣤⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠛⠛⠒⣂⣤⣤⣤⣀⣀⣒⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢰⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⣦⣤⣤⣤⣤⣤⣤⠴⠶⠶⠾⣍⠙⠛⠛⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣦⣤⣤⣤⣤⣤⣤⣾⣶⣶⣶⣶⣾⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⡟⣿⣿⡿⠟⢿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣈⣥⣄⣸⣧⣀⣼⣇⣤⣼⣇⣧⣰⣿⣉⣼⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠿⠿⠻⠿⢟⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⣠⡾⠟⣭⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣴⣖⣤⠖⠀⠀⠚⠉⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣯⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣾⡁⡀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣀⣀⣤⡼⠿⠛⠋⠉⠉⠀⠉⣧⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠛⠋⠙⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⢀⡀⠒⠀⠄⢀⠀⠀⠀⠀⠀⠀⠄⠒⠉⠀⢀⣀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⢀⣴⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠻⠄⠀⢻⣿⠯⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡙⣿⠇⠀⡞⠁⠈⠉⠛⠻⢦⡀⠀⠀⠀⠀⢀⠐⠋⠁⠀⢀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⠀⠀⠁⠀⣠⢀⣤⡄⢠⡀⢀⡀⠀⠀⣼⠀⠄⠰⡦⠶⠘⠇⠀⠀⠀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢈⡁⠈⠙⠡⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢴⠀⠀⠀⠀⠀⢀⡴⠃⠀⠁⠀⠀⠀⠀⠀⠻⠳⢄⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠋⠀⠀⠒⠲⢄⡀⣀⠐⠉⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢠⠀⠀⠀⠠⢄⡀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠀⠐⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠢⡀⠳⣄⠀⠀⠐⠤⣀⣈⣁⡀⠤⠒⠀⠀⠀⠀⣠⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠳⣄⠈⠳⣌⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠈⠻⣶⣾⣿⣷⣤⣀⣀⣀⣀⣀⣀⡤⠔⠊⠀⢀⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠃⠀⠀⢀⡈⠻⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⣠⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⣠⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⢀⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡟⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⢀⣾⣿⣿⠁⠀⠀⠂⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠇⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠀⢀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢸⣿⣇⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠛⠛⠛⢷⣄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡀
⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⠀⠀⣠⣤⡀⣰⣿⡟⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠞⠲⣄⠀⠀⠀⢴⠀⢀⣴⣿⣿⡿⠟⠋⠉⠉⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣶⣄⡀⠀⠻⠿⠃⣿⣿⠀⢠⠀⠐⢄⣀⡤⠏⠀⣿⣇⠀⠀⠋⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠐⠂⠈⣻⣿⣿⣿⣷⣶⣶⣾⣿⣿⣦⣤⣄⡀⢀⣴⡄⠀⠀⢹⣿⣦⣤⣴⣶⣶⣶⣶⣾⣿⣿⣏
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⠁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠸⠷⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⡻⢿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣶⣦⣽⣿⣿⣿⣋⣭⣭⣿⣿⣿⡏⢸⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣤⣶⣿⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⣬⣿⣿⣿⣿⣿⡿⢛⠛⠻⣿⡿⠀⠈⣿⣿⣿⠟⢛⣻⣿⣿⣿⣿⣿⣭⣤⣬⣽⣿⡿⠋⠀⠀⠀⠀⠀⢹⣧
⠀⣰⡟⠉⢳⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠃⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⡟
⠀⢿⣧⡀⠀⣾⣿⣿⣿⣿⣿⣟⡿⣿⠿⣟⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⡿⠿⢿⡿⣛⣿⣿⣿⣿⣿⣿⡇⠀⠀⠑⠲⠖⠋
⠀⠀⠛⠿⢿⢿⣿⣿⣿⡿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠸⣿⣿⣿⣿⣿⣿⣿⡷⠾⠿⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠉⠙⠛⠛⠛⠉⢩⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠈⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⣀⠉⢉⡛⠿⣿⣿⣿⣿⡏⠀⠀⢀⣀⣀⣀⠀⢻⡿⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣤⣤⣤⣴⣿⣿⡟⠉⠛⢿⣷⡈⠁⣠⡤⠿⣿⣿⣥⣴⣾⣿⣿⣿⣿⣿⣿⠁⠀⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠰⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢹⣷⠀⣿⡀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⢿⣿⣿⣿⠟⠁⠀⡅⠀⠀⠀⢸⡟⠀⠻⣿⣶⣾⣿⣿⡿⢿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣠⣀⠀⠀⠀⠀⠙⠦⠤⠶⠛⠀⠀⠀⠀⢹⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡖⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡿⠃⠀⠀⠀⠀⠡⣀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢿⢀⠖⡏⠉⢶⡖⠋⠉⠡⠄⠀⠀⠀⠤⠤⠀⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠈⢯⣠⠟⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣤⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣴⣿⣿⣏⠿⠁⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠤⣈⠂⢄⠀⢱⡀⠀⠀⠀⢠⠞⠉⠀⣹⣿⣿⣿⡞⡄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⣿⣿⠋⠅⠀⠀⠘⣶⣶⠄⠢⢁⢀⢴⢏⡀⣀⡀⠀⠀⠁⡈⢷⣘⡗⠀⢀⡴⠁⠀⠀⠀⠳⣿⣿⢟⣷⣧
⠀⠀⢠⠁⠀⠀⠀⠀⣼⣿⠏⠙⢀⣯⡴⣶⡶⠛⠃⠀⣀⣼⣅⣸⣧⣤⣤⣤⣀⣸⣧⣾⡋⢿⣿⡴⠋⠀⠀⠀⠀⢀⡼⠟⠁⢙⣿⣿
⠀⠀⠀⠀⠀⢀⣲⣿⠧⠀⢀⠤⠚⠛⣠⣖⣀⠀⣤⣄⣉⠻⣿⣿⣟⠓⠀⠚⠛⠟⢿⣛⣿⣿⣏⣾⠀⠀⠀⣀⠔⠊⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠉⠒⠻⠛⠁⣴⠎⢀⣠⣤⣾⢛⠟⠃⡙⠻⣿⠇⠸⣿⣿⠳⡂⠀⠀⠀⠀⠈⣙⣿⠻⣷⣤⠶⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠐⢿⡿⢃⣄⢀⡀⠃⠸⠅⠀⠀⢈⣿⣷⣍⣣⠀⡀⣤⣰⣾⣿⡧⡏⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠻⢿⡞⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣰⣯⣉⣤⣤⣤⣤⣶⣶⣶⣦⣤⣽⣶⡟⠉⢻⣇⣿⣿⣿⣿⣻⠟⠓⠂⠀⠄⠤⠶⠐⠊⠀⠀⠀⠀⠈⢿⡈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⢲⣾⣿⣿⣿⣿⠿⢇⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣸⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢣⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⡕⠀⡐⣾⡿⠟⠋⢟⢿⠃⠆⠀⠀⠀⠀⠉⠛⠒⠒⠋⠀⠀⠀⠀⠀⢺⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡄⠀⠈⠻⠿⠿⠿⠿⠛⡡⠊⠀⣠⣸⠏⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡇⠀⡁⠀⣠⠶⠶⠶⡖⠀⠉⢀⣄⣷⠟⡀⠀⠀⠀⡀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⡤⠒⠒⠂⠄⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣯⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢾⠿⢋⢾⣷⣶⣿⣷⣿⢿⠟⠀⠠⣄⣂⣁⣸⣷⣿⡿⠒⢠⠟⢿⣷⣮⣶⡖⢀
⠀⠀⠀⠀⠀⠀⠀⢃⠸⣬⣳⣅⣷⢀⣀⠀⣀⡀⢀⣤⣶⣠⣖⣮⡿⠹⠋⣛⣉⢵⠏⠘⣛⣭⣵⣿⡿⠯⠘⣿⣷⠀⠘⣷⠀⠙⠛⠁⠃⠈⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⣿⣧⣼⠛⠛⠉⠉⠙⠒⠭⠟⠛⠋⠉⣙⣽⣿⣿⠀⡠⠂⠀⠈⠛⠉⠉⠀⠀⠠⣿⣿⠀⠀⢻⠀⠀⠀⠀⢠⢰⡛⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⢁⣠⠴⠛⠉⠿⠝⢻⡄⢀⣷⣄⡀⠀⠀⠀⠀⡀⢠⣿⣿⣂⠀⠸⣷⡀⠀⢪⡇⠋⠐⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⢀⠄⠈⠀⠱⠙⣷⣛⣿⣿⣗⠝⠁⣴⠻⣅⠈⠋⠀⠀⡧⢷⠀⠰⠀⣖⠆⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠤⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠣⠹⣭⣿⣿⣿⠀⢸⡇⠀⠈⠉⠁⡑⠚⠁⢸⡆⣸⣼⠋⠀⢀⣠⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣾⣴⣿⣷⣿⠄⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⢢⣿⣿⣿⣿⡆⠸⡇⢀⣤⣤⣤⣭⣤⢴⣝⠃⠈⠉⠈⠀⢉⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⡻⣿⣏⠀⢷⠟⢉⣉⣉⣉⣁⡀⠹⠆⠀⠀⠀⠀⣾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣿⢠⡈⠀⠈⠛⠋⠉⠉⠀⢠⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⡀⠀⠀⣀⣀⣠⠃⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠠⡁⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣽⣷⣈⣆⣠⣔⣾⣐⣯⣺⣗⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⢻⡏
⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⣀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠀⣠⣤⡇
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣀⣼⣿⣿⣿⣿⡿⠉⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⣿⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣀⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠀⢀⣀⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣷⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣷⣾⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠘⣿⠇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠋⢻⣿⠋⠻⢿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠏⠀⣀⣀⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣟⣰⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠈⠻⢄⡀
⠀⠀⣄⣀⣠⢴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿
⠀⠈⠉⠉⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⡀⠀⠀⠀⠀⠀⢼⡟⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⡀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠐⣿⣤⠴⠊⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣧⠀⠀⠀⠀⣀⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣡⣶⣿⣿⣿⣇⠀⣴⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣋⣭⣽⣿⣿⣿⣿⣋⣤⣿⣿⣿⣧⡻⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠟⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⢤⣀⢀⣀⣴⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣬⣿⣿⣿⣿⣷⣌⣻⣿⣿⡏⠀⠀⠀⠀⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⠻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣍⣉⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⣿⣶⣤⡈⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠟⠋⠉⠉⠉⠉⠻⢶⣄⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠛⠉⠁⠀⠀⣠⡶⠶⢦⡀⠀⠀⠻⣦⠀⠀⢀⠏⠠⠀⠉⠓⠶⠤⣀⣠⠞⣉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⠘⣆⠀⠙⣆⠀⠀⠹⣇⠀⡞⠀⠀⠀⠀⠀⠀⢀⡼⡁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠀⢠⡶⠾⢶⣄⠀⠀⠀⠈⢧⠀⠘⣧⣤⠼⠄⠀⠀⢹⣾⠁⠛⠀⠀⠀⢀⡴⠋⠈⠃⠀⠀⠀⠀⠀⣀⣹⠶⠒⢦⡀
⠀⠀⠀⠀⠀⠀⣿⠀⢀⡟⠀⡆⠀⢻⡄⠀⠀⠀⠈⠳⠟⠋⣀⡀⠀⠀⠀⠀⢿⡀⠀⠀⠀⡠⠋⠀⠀⠀⠀⣀⡤⠴⠚⠋⣥⠀⠀⠁⠀⢣
⠀⠀⠀⠀⠀⠀⣇⠀⠈⡇⠀⣿⣀⣬⡅⠀⠀⠀⠀⠀⠠⢋⣤⣶⡦⠀⠀⠀⠈⠻⣄⣠⠎⢀⣀⡤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠡⠞⠋⢉⣁⣀⣀⠀⠀⠀⠀⠀⠺⠋⠁⠀⠀⠀⠀⠀⠀⠙⢷⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣳
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢋⣵⣶⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⡌⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⡜⢀⣿⡏⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⠂⣠⠏⠀⣸⣿⠃⠀⠀⠀⠀⠠⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣀⡀⠀⠀⠀⠒⠲⠤⠃⠀⣀⣾⡿⠁⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡆⠀⢰⠋⣀⣀⣀⣉⣀⣀⣀⣤⣴⣾⣿⡿⠁⠀⣴⡟⡸⠀⠀⢀⣤⡴⠶⠟⠻⠷⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠙⢦⡘⢦⣄⡉⠉⠉⠛⠿⠿⠿⠿⠛⠋⣀⣴⡿⠋⢠⠃⢀⡼⠋⠁⠀⠀⠀⣠⠄⠀⠀⠉⠛⠷⣶⣤⣴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠓⠪⢝⡛⠶⠶⣦⣤⣴⡶⠶⠟⠋⠁⠀⠀⡞⢠⡟⠁⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⡀⠀⠀⠀⠀⠀⠀⠀⡰⢳⡾⠁⠀⠀⢤⣄⣛⣁⣤⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣮⡑⠢⢄⣀⣀⡤⠚⠁⣼⠁⠀⠀⢀⣀⣈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⣤⣶⣬⡳⠀⠀⠀⠀⠀⠀⢺⣆⠸⠏⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⢀⡤⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⠀⠀⠀⠀⣼⠃
⠀⠀⠀⡞⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣠⣶⠏⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣍⠢⡀⠀⠀⠀⣼⠏
⠀⠀⠀⠹⣄⠀⠀⡇⢀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⢿⡇⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠘⢆⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠼⡇⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⠀⣠⠚⠉⠉⠓⠒⠦⣵⣒⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠈⡆⢇⠀⠾⠿⢿⣿⣿⣶⣌⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⢀⡼⠈⠑⠲⠦⣤⣀⠉⠛⢿⣷⡄⠈⢷⡄⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡀⠀⣉⣵⠚⠁⠀⠀⠀⠀⢤⡄⣠⠋⠀⣇⠀⠀⠀⠀⠉⠛⠶⣄⡈⠃⢀⣼⠇⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⠚⠁⠈⠳⢄⡀⠀⠀⠀⢀⡼⠁⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠈⠉⣛⡿⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⡤⠋⠀⠀⠀⠀⠀⠙⢦⣀⡀⣀⣠⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⣶⣦⡀⠀⢀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣴⣤⣴⠺⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣶⣼⣿⣿⣿⡿⠟⠛⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⣀⣙⣿⣿⣿⣿⡿⠋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣄⠘⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣧⣠⣶⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⣿⣿⣿⣿⡏⠀⢀⣀⣈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⣼⢻⡇⠙⠻⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⣿⣿⢿⣿⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣷⠀⠀⠀⠀⠈⠉⠉⢁⣠⣾⣿⡿⠃⠀⢨⠏⠈⢻⣿⣿⣿⣿⣿⠟⠋⠉⣁⣼⡟⠉⠛⠿⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⣀⣀⣠⣴⣶⣿⣿⡿⢏⣀⢠⣶⠇⠀⠀⠀⠙⣿⣿⣽⣏⣴⣿⣿⣿⣿⠁⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠰⠏⠹⣿⣧⣄⣤⠀⠀⠀⠙⠻⣿⣿⣿⣿⡿⣇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠋⠀⠀⠀⠀⠀⠒⠁⠀⠿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠙⣄⠀⣠⠄⠒⠂⠀⠀⠐⠒⠒⠈⠉⠀⠀⠀⢡⠀⢀⠖⢉⠤⠄⣉⡑⡄
⠀⠀⠀⠀⠀⢀⠤⠊⠀⢀⣤⠶⠒⠂⠘⡖⢂⠀⠀⠀⣀⣠⣀⠀⢀⡤⠶⠤⣄⠀⢸⠀⢸⠀⠇⠀⡠⠚⣉⢝⠠⢄
⠀⠀⡠⠔⠊⠁⠀⣀⠀⠸⠀⢀⣤⠤⠄⠘⡌⡄⠀⠚⠁⠀⠈⠁⠈⠀⣀⣀⡀⠀⢨⠔⠁⡸⠀⠀⢧⡞⣊⠥⠒⠉
⠀⢎⠀⠀⠀⣰⠟⠉⠁⠀⠀⠸⠁⠀⢀⡀⠘⣷⠀⢠⠶⠛⠓⠄⠀⠘⠉⠉⠉⠀⡖⠒⠊
⠀⠈⠢⡀⠀⠃⠀⣴⠞⠃⠀⠀⡀⣠⠎⢱⠀⢹⢃⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠈⡶⢄⠀⠁⠀⠀⣀⡭⠞⠁⠀⠸⠀⢸⠈⢆⠀⣠⢶⣒⠛⠿⠟⠻⠄⠸
⠀⠀⠀⢠⠀⠎⠑⠤⡀⠙⠤⣀⣀⣀⠔⠁⠀⡌⠀⠈⢦⡉⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠈⣼⡀⠀⠀⠈⠒⠤⠤⢀⣀⣀⡠⠔⠁⠀⠀⠀⠉⠢⣀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠈⠫⣁⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣈⠆
⠀⠀⠀⠀⢐⣉⡁⠬⠄⠒⠊
⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠐⠀⠐⠂⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠄⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠄⢉⠠
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢢⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⣀⡀⠀⠀⠀⠀⢰⠀⡀⡠⠀⠀⢀⠃⠀⢐⡀⠀⠀⢀
⠀⠀⠀⢂⠀⠑⡀⠀⣀⠀⡸⠛⠃⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⣠⡂⠀⢀⠎⢀⠆⠀⠐
⠀⠀⠀⠀⠐⠂⠓⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠐⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠒⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣯⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣯⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⡦⢄⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡀⠀⠀⠀⢀⠔⠈⠀⠠⠈⠀⢀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠸⠀⢆⠀⠄⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠻⠒⡽⠃⠈⠂
⠀⠀⠀⡠⠂⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠤⠤⠤⢄⣀⠈⠁⠀⠀⠀⠄⠦⠤⠤⠤⠀⠀⡀
⠀⠀⢀⡁⠀⠀⠀⠂⢩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠐⠠⡀⠀⠈⠑⠢⣄
⠀⠀⢼⠀⠀⠀⠂⠐⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⢦
⠀⠀⠘⡀⠀⠀⠀⠈⢼⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⡈⠍⠛⠀⠀⠈⠀⠁⠈⠉⡉⠉⢉⠉⠉⠉⠁⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠂
⠀⠀⠀⠳⠀⠀⠀⣀⠙⠀⠀⠀⠀⠀⠀⠄⠈⠢⠠⠤⠂⠡⠔⠠⠀⠠⠀⠀⢄⡀⠐⠢⠤⠀⡀⡀⢤⣶⠀⠀⠀⣀⠤⠄⠒⠈⢀
⠀⠀⠀⠀⠉⠂⠀⢀⠀⠀⠀⠀⠀⡃⠀⢀⢬⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⠀⠈⠁⠈
⠄⠀⠤⠁⠀⠐⠂⣁⣀⡀⠀⠀⠀⠁⠊⠁⠀⠉⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠠⠄⠈⠍⠉⢀⡀⢀⣐⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⡠⠘⠣⣀⢀⣈⡍⠁⠈⠀⠀⠀⠀⠀⢰⠂⠐⠒⠒⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⡔⠋⡀⢀⠀⡬⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⠔⠂⠈⠒⠑⠊⠀⠀⠣⠘⠒⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡨⠄
⠀⠀⢀⠂⠤⠀⣀⡀⠀⠘⠀⡠⠔⣊⠤⠊⠀⠀⠀⠀⡀⠀⡀⠀⠐⠉⠉⠉⢨⡂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠈⠑⠒⠦⠬⣉⣹⣁⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⠐⣐⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠀⠜⠀⠰⠤⢀⣉⠐⠒⠠⠤⠤⣸⡄⠊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠢⠤⠆⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠖⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠛⠲⡀⠈⡇
⠀⠀⢹⡆⠀⢠⡴⠟⠛⠛⠻⣶⣼⡟⠀⠀⢰⣆⠀⠀⠀⠀⢷⣾⠋⠉⠉⠉⠉⠙⠓⠒⠤⢤⣀
⠀⠀⠀⣧⢠⡟⠀⢰⡄⢀⣤⣀⠙⠀⠀⣠⠜⠛⣒⣂⠀⡀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⢹⡈⣀⡀⠸⠋⠚⠿⢿⡄⠀⠛⣃⡾⠟⠛⠛⠃⣥⡘⣟⡥⠖⠛⠛⠛⡇⠀⠀⠀⢀⣤⣀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⣷⠿⡏⣴⠿⠛⠛⠞⠁⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⢿⣇⣀⡀⠠⣴⡇⠀⠀⠀⠎⠉⠙⠳⣄⠀⠀⡎
⠀⠀⠀⠀⢸⣀⡚⠛⠂⠀⠀⠀⢀⠀⠁⡠⣄⠀⠀⠀⠀⠀⠀⠸⣯⠛⠿⠿⡏⡁⠀⠀⣾⣉⡀⠀⡀⠈⠧⢸⠁
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠠⣏⣯⣩⠴⠞⠀⠀⢈⡑⠀⠀⠀⣿⣧⠀⠞⠀⠀⠀⠀⣟⡛⠻⠿⠖⠀⠀⡎
⠀⠀⠀⠀⠀⠸⡆⠀⢠⠆⠀⠀⠀⢀⡘⢂⣤⡴⢖⣿⠁⠁⠀⢀⣿⠇⢠⢔⡄⠀⠀⡄⠈⠀⣼⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢧⠀⠻⣌⠻⣿⣟⣋⣉⣯⣥⠶⣿⠃⠀⠀⠀⣼⣯⠄⠀⠚⠓⠤⠬⠸⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠙⠦⡈⠻⠿⣛⠉⠥⠄⠁⠀⠀⠀⣰⣿⢃⣀⣀⣤⣈⣀⠀⠀⠀⢀⡆⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⣼⡿⠿⠟⠋⢁⣀⣈⠉⠻⣶⣄⠈⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⣠⡾⣧⠐⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⣠⠂⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠖⠚⠋⠀⠘⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣀⣤⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠄⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢰⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⡿⠟⣛⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢟⣛⣛⣛⠿⣿⣿⡟⣴⣿⣿⣿⣿⣿⣿⡇⢰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡿⠋⠁⣀⣉⢻⣇⠘⣿⣿⣿⣶⣤⣀⣀⡀⢀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠉⠀⠀⠙⢿⣷⢻⣧⣶⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⠿⠟⣻⣿⣿⣿⢣⣶⣬⣛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⡇⣿⣿⣵⣶⠾⠿⠿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣠⣿⣿⢿⣿⣿⢡⣿⣿⡿⢁⣀⣀⣀⣈⣿⣿⢻⡏⣀⣀⡈⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣸⣿⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡈⠛⠻⠿⣿⣿⠿⠟⠋⠀⢀⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢈⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠈⠻⣿⣿⣿⣿⠟⠋⠉⠉⠉⠙⠻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣤⣤⣾⣿⠏⠀⠀⠀⠀⠀⠙⣿⣿⡏⢀⣠⣤⣤⣤⣀⠀⠘⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣿⣿⣿⣿⣿⣿⣧⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⠿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠰⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣦
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⢏⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣵⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⣿⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠉⠉⠉⠻⣿⣿⣿⣿⣿⢀⣤⣶⣶⣶⣿⣿⣎⢿⣿⡿⠛⠋⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⣴⣶⣾⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣯⣴⣶⣶⣦⣄⣹⣿⣿⣿⡿⠛⠉⠙⠻⣿⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣄⣹⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣾⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣣⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣍⠛⠻⠿⠿⠿⠛⠋⠀⢰⣿⣿⣼⣿⣿⠟⠋⠁⠀⠀⠀⠙⠻⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣾⣿⡟⣿⣿⠃⢀⣀⣤⣀⡀⠀⠀⠀⢹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⣄⠀⠀⠀⣰⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣷⡀⠀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣾⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⠿⠟⠋⠀⠀⠀⠙⠻⠿⣿⣿⠿⠟⠉




⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠿⣿⡟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣶⡤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢯⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠛⠛⠛⠙⠓⠒⠒⢛⣏⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠦⣄⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠖⠚⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣀⡀⠚⠳⣄⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡍⠒⠊⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣁⣤⠆⠘⣿⢁⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠰⣎⣉⡒⠒⢋⡽
⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⢸⠋⣤⠇⠀⠁⢾⣅⣀⣠⠏⠀⠀⠀⠀⠀⠀⡾⠁⠈⠉⠉⠉
⡠⠶⢦⡤⠚⠁⠀⠀⠀⠀⠀⠉⠁⢈⠀⠀⡿⠀⠉⠀⠀⠀⠀⠀⣤⠖⠊⠁
⢷⣀⣼⣃⣀⡠⣤⢤⣤⣀⣀⣠⠐⣑⣠⠞⢧⣤⣄⣠⡴⠖⠒⠉⠁
⠀⠀⠀⠀⣜⣠⠃⡞⠀⠀⠀⡨⢋⡟⠁⢀⠞⢸
⠀⠀⠀⠀⠀⠉⠉⠀⠀⢀⡞⢀⠎⠀⡠⠋⣠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⡀⣀⣤⣶⠿⠛⠋⠉⠉⠉⠁⠉⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⣻⣿⣿⠿⣭⣟⣛⣿⣷⠀⠀⣤⣶⣤⣄⡀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⢀⣀⡤⠤⣤⣬⣿⣫⠵⠛⠉⣸⣿⣧⣀⣈⣉⣙⣿⣄⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣶⣶⣦
⠀⣾⠋⠀⠀⠀⠀⢸⣿⣧⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⢻⣷⣤⣀⣀⠀⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣀⡙⣿⣿⡟⠉⠉⠻⣿⣿⠀⢀⣀⣀⣀⣀
⠀⠀⠈⠙⠛⠛⠿⠿⢿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠟⠋⠉⠁⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣇⣠⣴⣶⣷⣶⣶⡿⠁⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠉⠉⠻⣿⣿⣿⣿⣿⣿⠃⠀⠘⣿⣿⣇⠈⠙⠛⠿⠿⠟⠛⣹⣿⣿⡿⣱⡾⠋⠁
⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠈⣛⣛⡛⠛⠁⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⣰⣿⡿⢋⣾⠟
⠀⠀⠀⠀⠀⠈⠳⠦⠤⠤⠴⠶⠾⠛⠋⠙⠻⣷⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⣴⣶⣿⣿⠏⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡾⠟⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠃⠀⠀⠈⠻⠶⠶⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡈⢣⢸⢇⠎⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⣿⣿⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⣿⣯⣽⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢧⣟⡿⡾⢾⠶⣟⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⡾⢽⣭⣟⣹⡏⢹⣅⡼⡆⠀⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣏⣹⠓⡿⢤⣯⡼⣯⡽⢧⡤⢷⢀⡴⢋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢧⠀⠀⣸⠦⣯⣹⣿⣾⡟⠚⣟⠶⢻⠒⠛⡏⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣄⣟⢺⠧⢼⡇⠙⣏⢛⣏⠓⣻⡓⢚⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢹⣍⣽⠓⢺⡇⠀⢹⣏⣿⣙⣹⣏⣉⣼⢰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢦⣻⡀⢼⠢⣿⣝⣽⠇⠀⠈⡯⢾⣍⣉⣿⣭⢼⣾⡏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠻⡇⣾⣉⡿⢤⡟⠀⠀⠀⡷⢺⡯⡭⢿⠦⠾⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡷⢭⣽⣾⠁⠀⠀⢰⣟⣹⡗⠶⣿⡓⣾⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡖⢿⡏⠀⠀⠀⢸⣧⣤⣏⣋⣹⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠈⢿⣹⠇⠀⠀⠀⢸⠶⠾⣧⣉⡽⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⢿⠀⠀⠀⠀⢸⣓⢚⣷⠏⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠸⡇⠀⠀⠀⠸⣯⡟⠁⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢻⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⢳⣄⠀⠀⠘⡇⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣍⣷⣤⣼⡧⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠙⠛⠻⠿⠿⠿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⠛⠻⠶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣶⣦⣤⣀⡀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠈⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠶⠾⣿⣿⣶⣦⣤⣤⣶⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠉⠙⡙⣝⠆⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠘⠀⣈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠐⣦⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⢠⡀⡀⢹⢿⣛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠻⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣤⣄⣀⠀⢀⣠⡀⠀⠀⠀⢀⣤⡳⢧⠀⢦⣄⠈⠋⢛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠟⠛⠋⠉⢿⢿⡻⣿⣿⣿⣿⣿⡀⣄⠄⠀⣰⣿⣿⣾⣶⣿⣿⣿⣿⣦⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢾⡿⠛⠁⢀⣠⣤⠾⠟⠒⠶⠷⣤⡝⣟⣿⣿⣧⣨⡣⡘⣿⣿⣧⣽⡟⠍⢈⠇⣨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠢⢶⡝⠠⠞⢋⣁⣤⣶⣶⣿⣿⣷⣶⣽⣾⣯⡑⠙⠳⢄⢜⣿⣿⣟⣥⣶⣶⣶⣤⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠶⣿⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⢤⡀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡀⠜⠀⢠⠿⠿⣿⣿⣿⠿⠛⠁⠐⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠛⢋⣙⣿⠁
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠿⠓⠆⠈⢳⠀⠀⠀⢱⠀⠀⠀⠀⣀⠤⠊⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣧⠂⠀⠀⠀⡀⠀⡇⠀⠀⢸⣀⠀⠀⠉⠀⠀⣠⠴⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⢬⡂⢙⣿⣷
⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⡇⠀⠃⠀⡠⠊⠁⠀⣀⠤⠐⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⢪⣔⢽⣿
⠀⠀⠀⠀⠀⠀⣀⣿⣷⠀⠀⠀⡇⣤⡷⠊⢀⣴⡆⠉⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣀⣀⣤⣤⣤⣀⠀⠀⣼⣷⣬⡻⣿⣿
⠀⠀⠀⠀⠀⣼⣅⡈⠙⠧⡀⠀⣿⡏⢀⡔⠁⢸⡇⡀⠈⠐⠀⠀⠀⠈⠘⠀⠀⠀⠀⡇⠀⠀⡇⠁⠀⠈⠙⠻⠿⠿⠻⠿⣿⣿⣿⣶⣿⣿⣿⣍⣻⡿
⠀⠀⠀⠀⣼⡿⢻⠛⠦⡀⠘⢆⢹⠁⣾⠀⢀⣼⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⣿⣿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⣯⠇⣿⠀⠀⠀⠠⣌⣻⣶⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⣠⠔⠲⢶⣤⣤⣾⣟⠀⡑⢻⣇⣻⣿⠏
⠀⠀⠀⠀⢻⣿⢻⡄⠀⠀⠀⠀⣹⡁⣲⣿⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⣤⣴⣶⣿⣼⣶⣿⣦⣝⣱⣾⣿⣷⣿⣤⣿⡿⠁
⠀⠀⠀⠀⠘⣿⣾⡇⠀⠀⠀⠀⡏⠛⢿⣿⣷⣆⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠙⣿⡏⠉⢉⣿⡿⣼⡏
⠀⠀⠀⠀⠀⢸⣍⣿⣆⠀⠀⠀⡇⠀⠸⡏⣿⣾⣧⠀⠳⡄⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠰⣤⣀⣀⣀⣀⣩⣿⣴⣿⡿⣳⡿
⠀⠀⠀⠀⠀⠈⣿⣏⢻⡀⠀⠀⠹⠀⠀⡇⢸⣿⡇⢀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⢿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠹⣯⣱⢭⠀⣾⠇⢀⡼⠃⠈⣿⣧⡈⠳⣄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠂⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⠛⠉⠀⠀⠀⠘⣷⡙⣷⡬⣷⠄⠀⠉⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠇⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣀⣀⣀⣀⣀⣤⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠂⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡴⠗⠀⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠀⠀⠀⠘⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠂⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠁
⠀⠀⠀⠐⠀⠀⠠⠆⠀⠆⠆⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠶⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⡦
⠀⠠⢤⠀⠀⠀⠀⠠⠊⠑⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠠⠊⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠈⢢⢀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠉⠑⠂⠤⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⢄⠉⠀⠂⠀⠂⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠈⠹⢅⠈⠁⠈⡁⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠤⠀⢢⡀


⠀⠀⠀⠀⠀⢠⣤⠤⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⡀⠀⡀⠈⡑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠈⠁
⠀⠀⡠⠄⠀⠉⠉⠀⠀⠀⠀⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡑⣠⣤⣤
⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⠄⠀⠀⠀⠀⠤⠄⠀⡈⠻⣿⠟⢡⡙⠻⠛⠀⠀⠐⠒
⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠁⠈⠁⠀⠀⠀⠀⠀⠈⡄⠀⠈⠀⠀⠂⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠀⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⢠⣄
⠀⠀⠀⢤⣤⣤⣀⡀⠀⠀⠀⣀⣠⣤⠶⡦⠀⠀⠀⠀⠀⣠⠴⠞⠋⢠⠜⠓⠦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠐⡀⠈⠛⠛⠛⠿⡛⠻⠁⠀⢀⠇⠀⠀⠀⠀⠸⡀⠐⠲⠂⠀⡀⠒⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠘⣠⡀
⠀⠀⠀⠀⠉⠂⠀⠐⠊⠀⠈⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠊⠁⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡰⢀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⡀⠄⢰⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈
⠀⠀⠀⠀⡰⠯⠤⠤⠘⠀⠔⠒⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠇⠄⠄⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷
⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠌⠉⢉⡁⠀⠒⢆⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠉⠸⠀⠃⠈⡃⢀⣠⠈⠁⢊⠲⢄
⠀⠀⠀⠀⠀⡐⡑⠁⢢⠤⠀⠀⠀⠀⠀⠀⠈⠣⠤⣭⠔⠶⢦⠀⠀⠀⠀⠠⠄⢀⠀⢀⠀⡀
⠀⠀⠀⠀⠐⠉⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡐⢎⡀⠴⠅⠈⢀⠀⠀⠚⠀⠇⢲⠐⠂⢠⡀
⠀⠀⠀⠀⡦⠋⢉⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⣀⡀⠀⠉⣿⠴⡌⣓⠠⠀⠉⠉⠁⠀⠀⠄⠀⡀⠐⠌⢆
⠀⠀⠀⠀⠡⣛⡘⠀⠂⢀⣀⣤⡴⢠⠤⠄⠛⠻⣛⠛⣻⠂⠸⣶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡠⣬⠓⠂
⠀⠀⠀⠀⠀⠀⡇⠠⠞⢛⡋⠿⢄⢺⠀⠘⡇⢯⣤⣴⡾⠀⢸⡇⠂⠀⢈⣀⠀⡠⠀⠀⠀⠀⡀⠀⠹⡖⠺⢤
⠀⠀⠀⠀⠀⠀⢰⡀⠐⠧⣤⡄⠼⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⢱⣚⠋⣻⡍⠉⡄⠀⠠⣦⣀⠀⠐⢆⣏⡐⠘
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠰⡆⠐⢠⡸⣆⡀⢈⠃⠸⠀⢘⢂⡈⢛⡻⢦⡀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⣄⢠⣶⠎⠐⠳⠦⣶⡋⠀⠀⠜⣧⣿⠇⠀⠉⠁⠀⠀⠀⠘⠈⢖⣄⢌⠷⠉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢜⠎⢟⠀⠀⠀⠀⠉⠁⠀⢀⠄⡾⡿⣤⠀⠂⣀⡀⠀⠀⠂⡀⠀⠉⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⠦⠐⠄⢀⣀⠠⠄⡂⠀⠰⣿⠁⠀⠀⠈⠳⢦⠶⠒⠙⢲⣦⢀⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠒⠒⠒⠋⠁⠀⢸⡏⠂⠁⠀⠀⠈⠉⠐⠠⡀⢲⣿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⣀⣀⣶⣶⠔⠁⢂⠀⠀⠀⣀⣐⡠⢄⣀⢴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠸⡀⠀⠀⠈⠁⠉⠁⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢶⣾⣶⣤⠤⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠋⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢀⣀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣰⠟⠉⠀⠀⠘⠳⡄⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢠⠖⣶⣄⡔⣶⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠳⠀⠀⢸⣄⣛⡿⢇⣉⡽⠀⠀⣀⣸⣧⡶⠶⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⢄⠀⠀⠀⠈⠁⠀⠀⠤⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⡽⠓⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢰⡇⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇⠀⠀⠀⠀⠈⠙⠶⣤⣀⣀⠀⠀⢀⣀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⡄⠀⠀⠀⡸⠀⠀⠀⠉⠉⢻⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷⣄⡀⠀⠙⠶⠤⠞⠁⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣼⡇⠀⠙⠓⠾⠷⢾⣥⣤⠶⠶⠟⠋⠁⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣡⠶⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡙⢦⡀
⠀⠀⢫⣑⠦⠤⠖⠒⠋⣠⠞⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢦⡙⢦⣀
⡸⠭⠭⠭⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢦⡉⢳⣤⢤⡔⣨⠇⡆
⠉⣖⣂⣉⠤⠔⠊⢳⠐⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⠝⣀⠀⠀⠀⠐⢛⣒
⠀⠈⠁⠀⠀⠀⠀⠈⠣⣬⠟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠣⠤⠔⠉⠑⢦⣈⠛⠻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢦⠀⠀⠀⠀⠈⠙⣿⢳⠶⠶⠶⠶⢶⡶⣶⠶⠟⠋⠁⠀⠀⣀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢷⢤⣤⣤⣤⣤⣽⠸⡇⠀⠀⠀⠀⡇⢸⠀⡀⠀⣀⣠⠞⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⠀⠀⠈⠁⢳⡄⠀⠀⡾⠁⠉⠉⠉⠉⠉⠉⢀⣠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣶⣶⣾⣿⣷⡴⠛⠉⠁⠀⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⠻⠿⣿⡀⣴⣶⣤⣤⣀⣠⣔⣃⡀
⠀⠀⣠⠖⠒⢂⣾⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠉⣿⣯⣤⣾⣿⣿⣾⣿⣷⣿⡿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⢧⡀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣶⡞⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠦⢿⣿⣿⠟⠉⠉⠻⣿⣿⣿⣶⣶⣿⣿⣤⣬⣿⡤⠖⠒⢲
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠿⣤⣾⣿⣶⠏⢻⣿⡿⢿⣿⣿⣿⣿⣿⣇⡤⠖⠋
⠀⠀⠐⢸⣿⣿⣿⣿⣿⣿⣿⡿⠁⠙⠛⣫⣿⣤⣵⡀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣈⣻⣟⠋⠀⠀⢻⣧⡀⠀⢀⣴⡿⣻⠏
⠀⠀⠀⠈⣿⣿⣿⣿⡿⢟⠏⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⣄⠤⠤⡀⠈⠉⠉⢉⣉⣀⣼⠇⠀⠀⠀⢻⣿⣿⣿⠏⠀⠻⠤⠤
⠀⠀⠀⠀⢘⠛⠿⢋⡀⠘⣶⣉⣉⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠳⢤⣿⡉⠑⣬⠁⣾⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣶⣿⣿⣷⣶⣤⣤⣄⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢀⠤⣌⠉⠀⢹⣿⣿⣿⣿⣿⣿⣿⠏⠀⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠿⠿⠿⠿⠟⠋⠁⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣩⣽⣿⣯⣶⣾⣿⣟⣍⣻⣿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⡤⠤⣄⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠻⠿⠋⣿⣿⠟⠛⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢀⣠⠴⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣼⣿⣷⣿⣿⣦⠟⠁
⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠋⠁⠀⣾⣿⣶⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⠛⠁⠙⠻⠛⠁
⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⣻⣄⣴⠉⢸⣿⣿⣿⣿⣿⣿⣿⠛⠛⣦⠀⣤⣄⣀⣀⣤⣤⣶⣿⡇
⠀⣴⠞⠉⠉⢻⣛⣿⣿⣿⣿⣿⣯⣩⣿⣶⣽⣿⣿⠿⣿⣁⣤⠾⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣠⣤⣤⣤⣤⣄⣀
⠘⣧⣤⡤⠖⠒⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⣠⠤⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⡿⢿⡏⣠⣴⣾⣿⣷⠈⠛⠛⣻⡟⠁⠉⠉⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣞⣥⠖⢿⠿⣿⠿⠉⣿⣷⣶⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠔⠋⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡿⠗⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⣀⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⡗⠀⢸⠀⠀⠀⠀⣴⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣆⣸⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠟⢋⣼⡿⠟⣽⡟⣿⠟⣽⠏⠙⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⠟⠁⠀⠋⠉⠀⠰⠋⠀⠉⠀⠁⠀⠀⠀⢻⣿⣿⣿⡆⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⡟⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠂⠀⢿⣿⣿⣿⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣛⡀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⢀⣼⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠟⠛⠁⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⠘⠙⣿⡇⣠⣴⣾⣿⠀⠀⠀⠐⠀⠀⠀⠀⢰⣶⣶⣶⣿⡿⠁⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⠀⠐⠃⠉⣿⣿⣿⣿⡦⣄⡀⠀⠀⢀⣠⠴⢺⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠉⣷⣏⣉⡀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠢⢀⠤⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠴⡚⠁⠀⠀⢀⢀⠀⠀⠀⠣⠄⠤⣴⠂⠀⠀⠀⠀⣠⠀⠀⠀⡰⠻⡀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠈⠳⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⡰⠛⣄⢀⠜⢀⠀⠉⢉⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⣸⠤⠤⢄⠀⡔⠁⠀⠀⠀⠀⠤⠀⠀⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢀⣀⡔⠉⠙⣲⠁⠀⠀⠀⣿⠂⠓⢤⠀⠀⠀⣀⠤⠒⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠉⠲⠒⠁⠑⠤⠤⠚⠈⠢⠤⠚⠢⣀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⡠⠚⠁⠈⠢⡀⢀⡠⠊⠁⠈⢦⠀⠀⣀⣀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠀⡉⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⡀⠀⠑⠊⠀⠀⠉⠒⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠘⠋⠀⠀⡀⠀⠀⠀⠀⠷⠀⠀⠈⠁⠀⠀⢠⡀⠀⠘⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⢀⡄⠀⠀⠀⢀⡄⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠰⠆⠀⠀⠀⠠⠀⠀⠠⡄⠀⠀⠀⣀⠀⠁⠀⠀⠦⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣀⠠⠇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣀⣀⣀⣀⣀⣀⣀⣀⠛⠀⣀
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠐⠊


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

⣿⡿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣿⣼⣷⣿⣶⣿⣿⣿⣾⠷⣿⣷⣾⣷⣿⣾⣿⣿⣿⣾⣷⣿⣷⣿⣾⣷⣾⣿⣼⣿⣼⣿⣶⣿⣷⣿⣾⣿⣷⣾⠷⢿⣼⣿⣶⣿⣾⣷⣷⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣀⣸⣿⣿⣿⣿⣿⣿⣿⣖⣊⡭⠍⠉⠉⠉⠉⠩⠭⠭⣽⣾⣿⣿⣿⣿⣿⣿⣛⣒⣒⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⢸⡇⡟⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⢿⢻⢸⡇⣿
⣿⣿⣼⢱⢷⡇⡿⢸⣻⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡇⣿⠘⣿⢸⣾⡜⣇⣿
⣿⡿⣿⣸⣼⢸⢇⣯⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⣿⢸⡇⠋⡿⢸⠉⡏⡏⢹⡏⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡘⣇⢻⡎⡇⡇⣿⢿
⣿⣿⡇⣇⣇⡟⡼⡾⣱⣿⣿⣿⣿⣿⣿⣿⡿⣿⠻⡇⢸⠁⡿⠘⡇⠀⠇⢸⠀⡇⡇⢸⠇⠃⢸⠘⡏⣿⣿⣿⣿⣿⣿⣿⣿⣗⡹⡜⡇⣇⣷⢻⢸
⣿⣿⣻⢹⡾⣸⣽⣵⣿⣿⣿⣿⣿⣿⣿⣿⠇⢹⠀⡇⠸⠀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠸⠀⠀⢸⠀⡇⣿⢹⣿⣿⣿⣿⣿⣿⣿⣧⣷⣻⢹⢹⢸⣾
⣿⣿⡟⣿⢣⣿⣿⣿⣿⣿⡏⣿⣿⣿⢿⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⢻⠈⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⡞⣿
⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣿⢻⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⣾⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⢻⣿⣿⣿⣿⣿⣿⣻⣾⣿⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣿⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣸
⣿⣿⣿⢹⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿
⣿⣿⣿⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿
⣿⣿⣿⣾⣸⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠚⠐⠃⠚⠀⠛⠚⠂⠃⠒⠐⠃⠓⠚⠚⠂⠃⠚⠚⠃⠚⠘⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣼⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣶⣦⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⣛⣯⣭⣭⣭⣭⣝⣛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠋⠹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠏⠁⣠⣙⢿⣿⣿⣿⡟⣉⣀⠉⠻⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠃⠀⠀⠀⠀⢿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠀⠰⢿⡿⢸⣿⣿⣿⡔⢽⡿⠂⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⣿⣶⣤⣴⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⠉⠻⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⢀⣠⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠘⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠏⠀⠀⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠛⠉⠉⠉⠉⠉⠙⠒⠲⠤⣀
⠀⠀⠀⠀⠀⢀⣠⠄⠀⠉⠀⠀⠀⠀⠀⠀⠈⠓⢦⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄
⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢹⡾⠀⠀⠀⠀⡀⢤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⣾⣶⡄⠀⠀⢰⡇⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣷⡄⠀⠀⠀⣀⠀⣧
⠀⠀⠈⠀⠀⣠⣤⣤⡀⠀⠙⣿⣿⠗⠋⠉⠻⢁⠀⠀⣾⡁⠀⠀⠀⠹⠍⠀⠀⠀⠉⢿⡏⠀⢀⣴⣿⣿⣻⡟
⠀⠀⠀⢄⣸⡿⣛⠿⢿⡆⠀⠀⢧⡀⠀⠀⠀⠈⣷⠠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠁⠀⣼⣁⠈⢹⡋
⠀⠀⠀⠈⢻⣿⠁⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠏⠉⠁⠉⡇
⠀⠀⠀⠀⠈⠉⠁⠀⢰⠁⢄⠀⠉⢱⠶⣤⡀⠀⠀⠈⣿⣇⠀⠀⠀⣴⠚⠛⠛⠿⣠⣤⣄⠀⣠⣀⠀⣸⠃
⠀⠀⠀⠀⠀⣄⠀⠀⣼⣤⣴⡿⠟⣃⣀⣈⠙⣆⠀⠀⠸⣿⡄⠀⠀⠁⢸⠀⠂⠀⠀⡀⠉⠉⢉⣽⡞⠁
⠀⠀⠀⠀⠀⠘⢦⡀⢸⠋⢡⡶⠋⠁⠀⣈⡉⠚⢧⠀⠈⡿⠹⡄⠀⠀⠀⠳⣀⠀⠀⠀⢀⣴⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢮⡃⠈⠀⣠⠖⣋⣩⣤⢤⠎⢀⣼⠃⠀⠹⣄⠀⠀⠀⢈⣛⢛⣛⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣈⣡⠞⠋⠉⠀⠀⣠⡾⠁⠀⠀⠀⠈⠢⣀⠀⠀⠈⠉⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠑⠲⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⣴⠃⢀⡴⢖⣶
⠀⠀⠀⠀⠀⠀⣿⡀⢸⡅⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⠀⠀⠀⠀⢀⣠⣬⣿⣦⣝⣶⣦⣴⣶⣾⣿⣿⣶⣾⣿⣿⣿⣯⣤⣤⣤⣀⣀
⢀⠀⢀⡴⢋⣴⣶⣿⣿⣿⠿⠛⠉⠁⢠⡿⠋⠀⠀⠀⠈⠻⣧⠀⠈⠉⠉⠛⠛⠷⢶⣦⣤⣀⡀
⠙⠶⠋⣰⡿⣻⣿⣿⠟⠁⠀⠀⠀⠀⣿⡇⣾⡆⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄⡀
⣀⣀⣴⠟⣱⢿⣿⠋⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⣠⠼⠁⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣶⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣤⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⢰⣿⠏⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠰⣶⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠃
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠉⠉⠉⢻⡿⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠻⣿⣶⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⠤⠶⠾⠛⠋⠻⣦⣀⠀⠀⠀⣠⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠉⠉⠉⠛⢿⣦⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⡄⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⢈⣿⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠉⠻⣷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡶⠾⠿⠛⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⡠⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⡐⡱⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⢴⣦⣈⠏⡴⣤⣤⣤⣤⣤⡶⠶⠞⠛⠉⠁
⠀⠀⠀⢰⣿⠁⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠹⢿⣦⡀⠀⠀⠀⣠⣿⠋⠊
⠀⠀⠀⣸⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠿⣷⣶⣿⡿⣷⣷⣤⡄
⠀⠀⠀⣿⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣟⣾⣷⣯⣭⡶
⠀⠀⢠⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢠⢊⠟⠿⣾⠏⠉
⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠁
⠀⠀⢸⣿⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⠿
⠀⣀⣨⡍⠉⠉⣿⠋⠉⠉⠉⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠁
⠀⠻⣿⣯⣤⣤⡿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⢉⣡⣤⣤⡤⠤⠤⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣧⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢰⡏⠁⠀⡄⠀⢀⣀⡄⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣤⣀⠀⠀⢰⠀⠈⣻⡆
⣀⣀⣀⣈⣻⣶⣶⣷⣚⣋⣉⣷⣶⣛⣀⣀⣀⣀⣀⣀⣀⣀⣘⣷⣶⣏⣙⣛⣷⣾⣶⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣉⡖⠢⠄⡀⢠⣦⠄⠀⣤⣤⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⠀⠀⠀⢀⠀⣠⣴⡄
⠀⠀⠀⠻⠃⠀⢀⣶⣵⡦⣀⣈⣛⣋⡤⠔⠋⠙⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠗⠢⢿⠘⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⡟⢱⠈⠁⠄⢶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡝⢄⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣸⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡷⢀⣶⠻⠧⣌⣇⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡇⠘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠁⠀⠀⠀⠈⠓⢮⣅
⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⡾⠿⠿⢿⣶⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠋⡉⡁⠀⠀⢈⢉⠙⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠿⢤⣀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⡆⠀⠾⠆⠰⠷⠀⠠⠄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢎⡁⠀⠀⠀⢠⡾⠟⢆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠘⢿⣶⡀⠀⠀⡁⠀⠈⢣⠀⠀⠀⠉⠀⠀⠀⣸⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣈⡁⠀⠉⠉⢢⡀⠈⡄⠀⠀⠀⠀⠀⢐⢻⡾⣷⡄⠀⠀⠀⡀⠀⠀⠀⣼⣟⡞⠇
⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠠⡄⠀⢀⡘⢣⣼⣧⣶⣶⣿⣶⣷⠀⠀⠀⠀⠀⠸⣸⡇⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣸⣇⡄
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⢻⠄⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠻⣼⡟⠀⠒⠀⠀⠀⠀⠀⠀⢻⣧⡟⠁⠐⣶⡄⠀⠀⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣹⠇⢾⣆⡛⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠤⡀⠀⠀⠀⠀⠀⠀⢀⠠⠋⠀⠀⠀⢀⣤⡤⢘⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠹⣿⢡⡟⢀⠘⢿⣿⣶⣬⣭⣛⣉⣋⣭⡍⠠⣶⣶⣦⠀⠀⠀⠀⠈⣶⣤⣤⣠⣶⣶⣇⣀⣀⣀⣀⣀⣀⣈⣤⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡟⠿⠇⠀⠀⠀⠙⠻⡿⣿⣿⣿⣿⣿⠇⠀⣈⣛⣯⣵⣶⢶⡶⠾⠿⠛⠛⢛⣻⠛⠛⠉⠉⣉⣉⡉⠉⠉⢉⡩⠝⠋⢉⣭⡉
⠀⠀⠀⠘⠿⣿⠧⢀⠀⠀⣀⡜⠀⠀⠀⠀⠀⠀⠀⣐⣀⣠⣵⣿⣡⡴⠞⢻⡏⠉⠁⠀⣴⣿⣷⠀⠀⠀⠀⠘⠟⠁⠀⠀⠘⠿⣿⠟⡤⠒⠉⢾⡆⠀⣿⣿⠃
⠀⠀⠀⠀⢀⣤⣦⣤⡝⠲⠿⣿⣿⣂⠘⠀⠉⢉⣉⣉⣥⣿⡿⠛⠁⠀⠀⣼⠀⠀⠀⠀⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣿⣾⣷⠀⠀⠁⠀⠈
⠀⠀⠀⢠⣾⣿⣷⡿⠀⠀⠀⠀⢈⣽⡋⠙⠛⠋⢉⣽⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠄⠔⠁⢸⡍⠿⠋
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢿⡿⠃⠀⣠⡾⠋⣾⣷⠀⠀⠀⠀⣠⣤⢿⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⡄⠀⣈⣇⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠀⠀⠙⠛⠀⠀⠀⢸⣿⡿⠛⡏⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠁⢺⡿⠛⠙⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢐⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⢀⢀⣀⡂⢀⡀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠐⣒⣲⠶⢖⣒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⡭⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣀⣀⡾⠓
⢬⠬⢭⠃⠠⠡⠡⡍⠉⡅⠀⠀⠀⠀⠀⢠⣈⣯⣟⣫⣭⣁⠄⠀⠀⠀⠀⠀⠀⠀⢰⡷⢀⣀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠁⠈⠁⠋
⠈⠁⠉⣤⣶⣶⣄⠉⠉⠁⠀⠀⠀⠀⠀⠈⣹⣏⢀⡀⣹⣏⠁⠀⠀⠀⠀⠀⠀⠀⠈⢡⠿⠿⠦⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠶⡆⡶⠲⢶
⢠⣶⣆⠙⠋⠛⢁⣴⣶⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⢰⣴⣶⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣔⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠀⠦⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠀⠒⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣤
⠀⠘⡏⢻⢹⠋⡟⢻⠁⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⡿⠟⣿⠀⠀⠀⠀⠀⠀⠀⡤⠁⠘⢷⢀⣵⠄⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⠳⡄⠀⡀
⠀⢀⣇⣸⣸⣀⣇⣸⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⣬⣧⠀⣩⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⣄⣨⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠇
⠀⠀⠉⢩⠉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠚⠃⢀⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⣁⣀⣀⣈⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⣤⡤⠶⣄⣤⠶⣤⡤⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠙⠳⠶⠛⠻⠶⠛⠁⠀⠀⠀⠀⢸⠉⠡⠾⣷⡀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⢊⢉⠈⣉⣘⠲⣄⠀⠀⠀⠀⢀⡄⢈⠇⠄⢀⢺⣗⢦⣀⡀
⠀⠀⡒⣶⡆⡆⣶⣶⠀⠀⠀⠀⠀⠀⠳⣄⡀⢈⣙⡛⢁⣀⠀⠀⠀⠀⠀⠀⠰⠋⠐⢮⣿⣥⣷⡷⠂⠈⣧⠀⠀⠀⠀⠄⠁⠀⠀⢈⠀⠉⠙⢯⣽
⠀⡇⣤⣤⢥⡄⣤⢠⡀⠀⠀⠀⠀⠀⠘⠉⠙⠈⠈⠁⠈⢀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠤⠈⠛⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠚⠿⠘⠎⠿⠂

⠀⠀⢀⣀⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣴⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠤⠤
⠐⡶⢻⡟⣿⠻⡟⢲⠂⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢱⡀⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠻⡃
⠀⠒⠶⠆⠰⠶⠲⠶⠀⠀⠀⠀⠀⠀⠀⠊⠻⠓⠾⠗⠃⡼⠆⠀⠀⠀⠀⠀⠙⢷⠀⢀⢀⣀⠀⡀⡀⣉⡁⠀⠀⠀⠀⠀⠀⠈⠀⣼⠷⡄⡈
⠀⠷⠾⠷⠿⠶⠷⠾⠆⠀⠀⠀⠀⠀⠀⡄⣤⣤⠀⡄⡄⣤⣄⠀⠀⠀⠀⠀⠀⣰⣷⡈⠈⠉⠀⣉⣁⢉⡁⠀⠀⠀⠀⠀⠀⠐⠲⠆⠲⠲⠶
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣭⣥⣤⣥⣥⣭⣥⣤⣤⣤⣤⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⡱⢀⠆⠀⢀⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢎⠰⡇⠀⠀⢆⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠁⠈⢱⠀⢀⠕⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡴⠁⡎⠀⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⠉⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠔⠁⠀⠀⢈⠐⡔⠁⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠐⠈⠀⢀⡉⠑⠢⣀
⠀⠀⠀⠀⠘⠀⠐⢃⠐⢈⠰⠀⠁⠀⡁⠀⠀⠀⠙⡄⠀⠀⠀⢀⡎⠀⠀⠠⠈⢍⠟⠙⢯⠉⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⣀⡌⠳⣄
⠀⠀⠀⠀⠠⠀⠀⠟⢸⠏⠠⠀⠿⠻⠊⠀⠀⠀⠀⢹⠀⠀⠀⠸⠀⠀⠀⡷⢶⣾⢢⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣀⣀⣸⣀⣐⡀⠈⢢
⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⡇⠠⠀⠀⠈⡀⠀⠁⠀⠈⡇⠀⠀⡇⠀⠀⠀⠑⠖⠻⣄⡩⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠙⠅⠜⠋⠈⠁⠀⠀⢧
⠀⠀⠀⠘⠀⠀⠀⠀⠀⣸⠁⠀⠴⠁⠈⠀⠀⠀⢂⠂⢁⠀⢀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⠀⠀⠀⠀⢻⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⠤⠤⠄⠀⠘⡧⡀
⠀⠀⢠⢀⠀⠀⠠⢄⠀⡏⠠⡀⠄⠀⠀⠀⠀⢀⠠⠂⢸⠀⢸⡇⠀⠛⠉⠉⠉⠙⠒⠒⠀⢠⡀⡆⠀⠀⠣⠸⣇⠀⠀⠀⡇⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣇⢃
⠀⠀⠐⣈⠂⢨⠀⠀⠈⡇⠀⠀⠐⠀⠀⠂⠢⡀⠄⠀⡌⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠇⢀⠀⠀⠀⣿⠀⠀⢄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠆
⠀⠀⠀⠙⡄⠀⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠄⢀⣸⠃⠀⢸⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣄⠀⠀⠀⠙⢦⡀⠀⠀⠔⠛⣦⣄⣀⠀⠀⢀⡠⠞⠁⠀⠀⢸⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⢀⣠⡏⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠠⡉⠢⣀⡐⡠⠉⢛⡒⢒⠛⠁⠀⠩⢫⠋⠙⡔⠤⡐⠄⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⡉⠡⡅⡞⠆⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⣀⣠⠞⠋⠀⠀⠀⡴
⠀⠙⠢⢀⡐⡁⠀⠀⠦⢀⡈⠑⠒⠁⠃⠀⠀⠈⢁⠷⠈⢄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠉⠚⠓⠀⠰⠀⠀⠀⠀⠀⡖⢈⣋⡋⠉⠉⠣⡳⡀⠤⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠒⠚⠀⠀⠀⠀⠑⠂⠀⠉⠀⠀⠀⠈⠓⠶⢤⣄⣀⣀⣤⣤⡶⠚⠉⠳⡒⠖⠁⠀⠀⠀⠀⢠⠁⠸⠀⠀⠀⠀⠀⢁⠀⢀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠇⢸⠀⣀⣿⢸⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠲⠄⠤⠭⠽⠂⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⡉⢉⡤⠖⠠⠶⠋⠁⠀⠉⠀⢱⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡀⠀⣰⣿⣿⡿⠁⠙⣿⣧
⠀⠀⢀⣠⣴⣶⣶⣦⣄⠀⠀⣿⣿⣟⣿⣷⡀⣿⣿⣿⣇⠀⠀⢹⣿⣾⣿⣿
⠀⠀⣾⡟⠀⠙⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿
⠀⠀⣿⣷⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠫⡍⠙⢿⣿⣿⣿⣯⣿⣿⣿⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢈⣿⣶⣿⣿⣿⣿⠋⠰⣉⠉⠛⡟⡇⠀⠉⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄⣠⣾⣿⣿⡿⠛⣿⣦
⠀⢠⣿⡟⢻⢿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⢺⣿⣿⣿⣿⡿⣿⣷⣿⣿⣿⡟⠀⠀⢸⣿⠆
⠀⠀⢻⣿⡄⠀⠈⠛⢿⡄⠀⠀⠈⠂⠄⠀⠀⠁⠀⢀⠄⢨⠙⢿⣿⣿⡿⠿⠻⠿⣿⣿⣧⣤⣤⣿⡟
⠀⠀⠀⠉⠻⡀⠀⠀⠀⠡⠀⠀⠀⡀⠀⠀⠀⠠⠈⠀⠀⠄⡀⠠⣏⡏⠀⠀⠀⠀⢹⣿⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠤⠁⠀⠐⣊⠀⠁⠀⢸⣿⡿⠋⠉⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠤⠄⢆⠁⠁⠄⠘⠀⠘⠀⡀⠄⠋⠰⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠈⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣤⣀⣤⣄⣄⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣄⣷⣾⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣥⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣤⣧⣿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⣟⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⣭⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣶⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣤⣄⡀⠀⠀⠀⢻⣿⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣽⣿⣿⡿⠁⢀⣠⣴⣶⡶⠶⠤⢄⣀⡤⠖⠋⠁⠀⠀⠀⠈⠻⣷⣄⠀⠀⢿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣿⣿⣿⠁⣴⣿⣿⠏⠁⠀⢀⣀⣀⠀⠀⠀⣀⡤⠴⠶⣤⡀⠀⢹⣿⡇⠀⠘⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⠇⢸⣿⡟⠁⢀⣴⠟⠛⠉⠉⠻⣦⡾⠋⠀⠀⠀⠀⠹⣆⠀⠉⠁⠀⠀⢻⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⡟⠀⠀⠁⠀⢰⡟⠁⠀⢠⣶⡄⠀⢸⣇⠀⠀⢾⣷⠀⠀⣿⡆⠀⠀⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⢿⡇⠀⠀⠀⠉⠀⠀⢀⣿⣄⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⣠⣴⠟⠁⠙⣿⣶⣤⣴⡾⠋⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⡿⣿⠀⠀⠀⠈⢿⣿⣿⣒⠒⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠛⠀⣿⠀⠀⠀⠀⠘⣿⡤⣌⠁⠀⠀⠀⠀⠀⠬⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⣿⠀⠀⠀⠀⠀⣿⡇⢸⣧⠀⠀⢀⠀⠂⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣽⣧⡀⠀⢀⣰⣿⣧⣼⡿⠀⠀⠂⠀⠀⠀⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠉⡁⢙⠻⡿⣿⠛⠁⠀⠂⠀⠀⠀⠀⠀⠀⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣽⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠂⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢘⣿⣿⣿⣿⣆⠀⠀⡀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠄⠂⠀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⠻⠿⢯⣀⠀⠀⢀⣀⠀⢀⣱⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣍⡀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠲⡀⢹⡎⠀⢀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠓⠻⠿⠶⢶⣦⣴⣦⣶⣶⣾⡿⠿⠟⠛⠉



⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⠆⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣪⠊⢹⡧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠟⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣹⡀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⡝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠦⢄⠀⠀⠀⠀⠀⣿⠇⢀⠠⠄⠀⠀⠒⠒⠒⠒⠀⠤⢅⡂⠁⠀⠀⠀⠀⢀⢧⡎
⠀⠀⢀⠂⠄⣀⠀⠀⢀⣀⣀⡤⣤⡶⠞⠋⠁⡘⡂⢣⠀⠀⠀⣾⡽⡰⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⢰⣯⠛
⠀⠀⠀⠀⠘⠩⡙⠛⠈⠉⠉⠈⠀⠀⠀⢀⠀⠈⡢⠈⡄⠀⣸⡽⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢕⡕⡄⠈⢻⣦
⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠖⠀⡙⣩⣅⣀⣨⡀⡇⠀⠹⣧⡇⠂⣠⣴⣦⡀⠀⠀⠀⢀⣤⣤⣤⡀⠀⢂⢕⠁⡱⠀⠀⡗⡆
⠀⠀⠀⠀⡅⡨⠀⠀⠀⠀⣀⣠⣴⣟⠅⠀⠈⠉⢩⡅⢣⠀⠀⢹⠀⠈⣁⣄⡈⠛⠀⠀⠐⠋⣁⠀⠉⠿⠀⢐⠩⠀⡇⠀⣰⣧⠃
⠀⢀⢤⣴⠀⡪⠀⠀⠀⠠⣍⣀⣉⣡⡆⢰⣹⠼⠿⢅⡈⣆⠀⢸⠀⠼⡛⠂⢬⡄⡆⠀⣴⠿⢉⡉⢢⠀⠀⠀⠊⠀⣤⡾⠛⠁
⠀⣿⡏⠙⠨⡂⠀⠀⠀⠀⠑⠤⠴⠟⠇⠀⢳⡁⠀⠉⠀⠘⡄⡆⠂⠀⠀⠉⠀⢠⠁⠈⠁⠤⠤⠌⠲⠀⠀⠀⠀⠀⡇
⠀⢻⣷⢠⠸⡂⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⢙⡄⠀⠀⠀⡿⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠅⠀⡇
⠀⣸⡏⠸⠈⣂⠀⠀⠀⠀⠀⢠⠀⠀⣈⣽⣷⣛⡈⢢⡺⡣⣏⠈⣕⠒⠄⠸⣄⣀⣤⠴⠀⢀⣀⡀⠀⠀⠀⠀⣀⠀⡇
⣰⡿⠀⠀⢇⢑⠀⠀⠀⠀⠀⠇⢠⠊⢀⣀⣀⣈⡙⣷⠁⢰⡇⠀⣿⠦⡀⠀⠈⠚⠁⠀⠀⠀⣠⠆⠀⠀⠀⡅⡂⢠⠁
⣿⡁⠀⠀⠈⢦⠣⠀⠀⠀⠀⠀⣧⡾⠋⠀⠐⠒⠈⡉⢀⡟⡂⠀⠹⣧⡈⠁⠒⠒⠒⠒⠒⣩⠎⠀⠀⠀⠐⠑⢀⠎
⢹⣧⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⡌⠀⡘⢻⠄⠀⠀⠙⠿⣦⣤⣀⣀⠤⠖⠁⠀⠀⠀⠐⡐⡠⠊⡔⡠
⠀⣿⡇⠀⠀⠀⠀⠈⠳⢄⠠⢄⡀⠀⠀⠀⠀⠀⠅⡰⠁⠈⢆⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠰⢂⠜⠁⠊⠈
⠀⣿⢣⣶⠀⠘⠻⢷⡄⠀⠉⠒⠬⣄⡒⠢⠄⢊⡴⠁⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⣩⠔⠁
⠘⠛⠚⠋⠀⠀⠀⠈⠿⣦⣤⣴⠾⠋⠁⠉⠉⠁⠢⠔⠀⠀⠀⠀⠈⠓⢌⡒⠤⠂⠒⣙⡨⠔⠉
⠀⠀⡀⠀⡄⢄⠀⡠⢀⠀⡀⠀⠀⡀⢀⠀⡀⡀⢀⠄⡀⠀⠄⠠⡀⢀⠄⡨⡩⢉⠍⡠⠠⢀⠔⠠⡂⢀⠔⡠⡠⠂⢄⢄⢄⠄⡠⢄⢀
⠀⠀⠁⠊⠂⠑⠩⢒⠅⢊⢔⡡⡪⢀⠅⡪⡨⢐⠁⡨⢊⠔⡡⢊⠔⡡⢊⢀⠔⡡⢊⢀⢔⠕⡡⢊⠔⡡⡤⢊⢔⠔⠕⠕⠡⠺⠓⠑⠉

⠀⠀⡀⢀⡠⠤⠤⠄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⡤
⠀⠀⢠⠋⠀⠐⠃⢠⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣁⣬⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢠⡤⠂⠐⠒⠒⣤⡄⠀⠐⠂⠠⢢
⠀⠁⡞⠀⠀⠀⣤⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⠂
⠀⠐⠆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⣿⠿⠛⠛⠉⠉⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢀⣴⠒⣲⡀⣠⣄⣤⡤⢤⡤⣀⡀
⠀⠰⡀⠀⠀⡠⣆⣽⣿⣦⠀⠀⠘⠀⠀⠀⠀⠀⢸⡇⠂⠀⠀⠀⠀⠀⡀⠀⠀⣸⡟⠀⠀⠀⠀⢀⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⢧⠀⠀⣽⣿⣿⣿⡯⠃⢀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢀⣴⣾⣿⣿⠿⠂⠀⠈⠀⠀⠀⠀⠀⠀⠂⢁⠚⠃⠐⠀⠈⠋⠉⠄⠀⠓
⠀⠀⠈⠳⢄⡈⠉⠁⠈⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣤⣀⡀
⠀⢶⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⢒⠀⠒⠀⠄⣀⣀⡤⠀⠀⠀⠀⠀⠀⠠⣮⣽⢻⠿⣾⣽⣿⣦⡀
⠐⠖⠉⠉⠈⠉⠉⠉⠉⠉⠉⠁⠀⢀⠀⠀⠀⠀⠀⢛⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⠀⠠⣿⣶⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⢀⡤⢶⠖⡖⢲⠒⡖⢲⢰⣶⡄⠐⠀⠀⠀⠀⠀⢠⣀⣈⠉⠉⠉⠉⠙⠛⠛⠋
⠀⠀⠀⠶⠷⡷⣷⣷⣦⣧⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⠿⠷⠶⠶⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠻⠿⢻⣿⣿⣿⡿⠯⠛⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⣤⣤⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉

⠀⠀⠀⠀⡠⠒⠊⠉⠑⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡐⠀⠀⠐⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠉⠀⠢⠀⠀⠉⢢
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠈⣋⡟⠛⠉⠀⠀⠈⠈⠉⠃⠢⡀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠨⠁⠀⠀⣤⣤⡄⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠷⠒⠚⠈⠋⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣄⣶⣰⣤⣦⣖⣤⠀⠸
⠀⠀⠀⣷⣚⣋⣛⣻⣒⣖⣲⣤⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠃⡄
⠀⠀⠀⠘⣿⣛⣛⣟⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠛⠋⠉⠉⠉⠀⠰⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢒⠒⠒⡂⠠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠔⢠⠄⠦⠘⠷⠀⠀⠀⠓⠉⣒⣄
⠀⠀⠀⠀⠀⠀⡠⠨⢄⠀⠀⠉⠀⠀⠀⠐⠀⠒⢣⡀⠠⠶⢠⣑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡰⠵⠄⠀⠹⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢫⣅⠉⠹⡀⠀⠠⢀⡔⠉⠀⠂⡤⠐⡄⠂⢐⡠
⠀⠀⠀⠀⢀⢁⡌⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⣥⠄⠁⣘⡀⠀⠦⢀⣈⠁⠈⢁⠸⠠⠁⠀⠠⡀
⠀⠀⠀⠀⢸⢋⡀⡜⠁⠀⠀⠀⠈⠂⠀⠀⠀⢠⣤⣤⣤⣤⣀⠀⢻⠚⢗⡤⠐⠊⠀⠀⠀⠀⠀⠀⠐⠠⡤⠀⠑⣌⠄
⠀⠀⠀⠀⠀⠪⣥⠃⠈⢀⣠⣤⣴⢞⢠⠖⠒⣀⠉⣋⠭⠭⢽⠉⠘⡧⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠢⠌⠁
⠀⠀⠀⠀⠀⠀⠸⡄⠴⠫⣋⠭⠽⢤⡙⡄⠀⠻⠘⣤⣔⣒⡿⠀⠘⡇⠈⠀⣀⣀⡠⢂⠄⠀⠀⠀⠀⣄⠀⠈⢯⠋⠳⡆
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠘⠥⣄⣒⠾⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢿⣚⠛⣛⡿⠛⠡⠄⣀⠰⢦⣀⡀⠐⢇⢾⣑⢂⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢤⣶⡄⢰⣼⢇⠤⠄⣹⠂⡇⠀⣰⢄⣘⣛⡻⢶⣄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀⢀⣄⡜⠁⠶⣦⣤⡼⠋⠀⠀⡘⣇⣾⡗⠀⠉⠉⠁⠀⠀⠀⡄⠩⣦⠄⠈⢢⠉⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠱⡌⢿⠠⠀⠀⠀⠐⠛⠀⠀⠀⢄⣼⣿⣀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠚⢤⠢⡀⠀⠀⠀⠀⡠⠊⠈⢁⣷⠘⠀⠁⠈⠻⣤⣀⣴⠌⠳⢤⡀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠁⠀⠩⣉⣡⡤⠔⠁⠀⢠⡟⠀⠔⠀⠒⠀⠼⠧⢀⡀⠀⣌⡏⣠⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⢀⣀⢀⢀⠜⡃⠀⢂⠠⠄⢀⡀⠀⠀⠈⣂⣸⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠴⠶⠾⠿⠛⠁⠀⢱⠀⠀⠀⠲⢶⠲⠭⠒⠜⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡀⠀⣀⠀⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠷⠖⠒⠁

⣿⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔
⠀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠃⠀⠁⠀⠁⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢠⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⠾⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠲⠄⠘⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠁⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤
⣿⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⣠⠄⠀⠀⠁
⠀⠀⠀⡀⣀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠍⣶⠐⠆⠀⠀⠘⡄
⡔⠈⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡈⠁⠀⠀⠀⠀⡅
⠑⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⢿⠆⠂⠀⢴⠀⠃
⠀⠀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⢈⠣⠀⢀⣀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠈⠤⡀⠀⠀⠀⡌⠀⠄
⠀⠀⠀⠄⠀⡀⠘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠰⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⡄⠀⠀⠀⠢⠤⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡀⠀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠑⠒⠒⠀⠋⠭⠀⡁⢀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣔⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣦⣤⣀⣀⣰⣤⣤⠀⠀⠀⢀⣤⣶⣿⠃
⠀⠀⠀⠀⣀⠀⠀⣀⡀⢀⣀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟
⠀⠀⠀⠀⣿⠀⠀⢸⡇⢸⡷⠦⢸⡷⠶⠀⠀⠀⠀⠀⠀⢀⡀⢾⣿⣿⡟⣧⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠛⠛⠀⠛⢃⠘⠃⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣭⣤⣿⣷⣛⣠⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠖⠖⠶⠻⠸⠱⠲⠵⠲⠴⠹⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁
⠀⠀⠀⢀⡤⠄⢠⡄⠠⡤⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣠⣶⣷⡦
⠀⠀⠀⠈⠓⠀⠋⠙⠀⠃⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠰⣿⣿⠏
⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⣯⠀⢴⣻⢰⣒⡆⡖⡆⠁⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⢀⡀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣰⣿⣿⣿⣿⣿⣿⣿⡄⣠⣄⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠇⠏⠖⠺⠱⠻⠆⠨⠖⠧⠇⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢉⣉⣉⣢⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠢⡄⠀⠀⡇⢰⠉⠀⢠⠃⡴⡵⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠘⡄⠀⠇⠈⠀⠀⢣⣸⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠉⠀⠀⣀⣤⠤⠤⠀⠱⣀⠆⢀⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢄⡀⠀⠀⣀⠔⠈⠁⠀⠀⣀⣀⡀⠰⠃⠀⢀⣀⣀⠀⢇⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⠲⢅⡸⡄⠀⠀⢠⠞⠉⠀⠀⠀⠀⢰⠋⠉⠁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⡌⠀⠀⣠⠶⠂⠀⠀⠑⢌⢢⠀⠉⠀⣠⡶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡜⠀⠀⢸⡇⠀⢠⡶⠋⠀⠀⠑⢵⢄⠀⠋⠀⠀⠀⠀⢒⣡⠶⣚⠳⠆⠰
⠀⠀⠀⢠⠊⠀⠀⣀⡈⠁⠀⠸⠁⠀⠀⡞⢄⠈⢻⠳⣄⠀⠀⢠⢞⠭⠒⠉⠀⠀⠀⡇⠀⠀⠀⠀⢀⠔⢖⠀⠀⠐⠢⣄
⠀⢀⠔⠁⠀⢀⡞⠁⠀⣀⡄⠀⠀⢰⣰⠃⠈⢇⠀⢃⠈⠓⢄⠘⠁⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡰⠁⢣⠈⢆⠀⠀⠀⠈⠆
⠀⢈⠀⠀⠀⠸⡇⠀⣸⠋⠀⠀⠐⣵⠋⠀⠀⡘⠀⠘⡀⠀⠀⠉⠐⠂⠤⠤⠤⠚⠁⠀⠀⢀⡴⢣⠀⠀⠡⡈⢣⠀⠀⠀⡙
⠀⠀⠉⠒⠤⣀⡀⠀⠙⠀⠀⢤⡚⠁⠀⣀⠔⠁⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠐⡄⢣⡀⠀⠑⡄⠱⣄⡰⠁
⠀⠀⠀⠀⠀⠀⡇⢙⠒⠢⠤⠄⣀⣉⣁⣀⠠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠘⣦⡱⠤⠤⠬⠶⠊
⠀⠀⠀⠀⠀⠀⠡⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢁⠠⢤⡀⠀⠀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠿⣔⠒⠒⠒⠂⠐⠢⡀⠀⠀⠀⠀⠀⣠⠒⢁⡠⣒⣵⢞⡳⣊⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠥⡻⠃⠀⠀⣀⠔⠊⠉⢆⢷⠹⠯⡢⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⣁⠔⠊⠀⢀⡤⡊⠀⠀⠀⡠⠔⠓⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⢀⠠⠒⠁⠳⣘⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡠⠔⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣀⣀⣀⣠⣤⣶⣿⡿⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⠿⠛⠋⠀⢨⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢹⣿⣿⡿⣿⣿⣛⣻⣇⣿⣿⣿⣶⣦⣤⣤⣤⣤⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⣀⠀⠠⢤⠄⡼⠀⠀⠀⠉⠛⠛⠛⠛⢛⣿⣿⣿
⠀⠀⠀⠈⢿⣷⣾⣿⣿⣿⣿⢸⣿⣷⣬⣿⣿⣛⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⡀⠈⠁⠀⠀⠀⠀⡇⠀⠉⠓⠀⢠⣤⡄⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠈⢿⣿⡟⣉⣉⣻⡼⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠀⣠⠶⠶⠄⣇⠀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠿⢿⡿⠟⠀⢿⣷⣬⣭⣽⡿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢛⣂⣁⣀⣀⣰⣿⡄⠘⠷⠶⠂⣠⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣽⣯⣯⣿⣿⣿⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣸⡇⠀⠐⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢳⠀⠀⠀⢠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⣿⣏⠠⣀⡘⠂⠀⣾⡿⠻⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢯⣿⠇
⠀⠀⠀⢹⣿⡇⠀⢡⢢⡀⠙⣿⠃⣸⠁⠀⠀⠀⠀⠀⠀⠀⣴⠆⠘⠢⠀⡀⢀⢀⡆⢀⡀⣷⣄⡀⠀⠀⠀⠀⠸⣍⠿⠙⠀⠀⠀⢀⠀⢐
⠀⣾⣿⣿⠿⢟⠒⠺⢿⣷⡀⡙⠺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠣⣿⣏⣈⠃⠙⠀⣠⠶⠃⠀⠀⣴⠝⠘⠳⣶⡄⠀⠉⠃⠀⠰⠆
⠀⢹⣿⣿⡀⠔⠒⠖⠲⣍⠉⠉⢉⠀⣄⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡇⠀⠀⠀⠀⠄⠀⠀⢀⣉⠇⠀⠠⠠⡄⠀⡀⠠⠄⣀
⠀⠀⠙⣿⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⢻⣧⣤⣀⣀⣀⡊⠐⠃⠀⣝⡠⣤⠀⠀⢠⠉⠉⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢈⣇⡠⠤⢤
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⢿⡟⠉⠉⠒⠚⢽⡢⡀⠸⣧⣿⡁⠈⡆⠠⠀⢀⠂⠀⠴⢤⡁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠘⠿⠛⠛⠉
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠤⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠉⠜⡎⢢⢉⣧⡸⠃⡔⠛⠛⠛⢗⢤⡀⢣⠀⠀⠀⡀⠀⠙⠒⠶⣶⣦⣤⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠤⠤⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⣳⡄⠛⠂⠀⠈⠀⠀⠀⠀⠀⠑⡱⡘⣶⠀⢻⠇⠀⢀⣔⡀⠀⠀⠀⣹
⠀⠀⠀⠀⠐⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠴⠖⠒⠁⠀⠀⠀⠀⠀⠀⠀⣧⡤⠄⠚⠃⡂⢸⡿⠉⠉⠓⠒⠋
⠀⠀⠀⠀⠠⠄⢀⡈⢳⠦⠤⠄⠒⣒⡐⡞⣁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠤⠴⠒⢻⡇⢸⡤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣀⠤⠄
⠀⠀⠀⠀⠚⣃⡐⠀⠙⠒⠢⣜⠛⢻⡗⣦⣌⠙⠶⠤⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠠⠲⡆⠄⠈⡉⠙⠒⠓⠉⠀⠂
⠀⠀⠀⠀⠸⠏⠉⠉⠀⠀⠀⠀⣽⡆⠀⠀⠀⢩⡃⠂⠀⢨⡛⢚⠃⠊⣵⣀⣀⡈⠶⠤⣀⣀⣀⠀⣀⣀⡠⠤⠒⠉⠛⠻⠛⢉⣭⣵⠶⠟⠡⠅⠂⠪⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⣀⡤⠤⠤⠿⠀⠈⢀⢌⠀⠉⠉⣓⢀⣀⣈⠉⠉⣉⡠⢶⣦⣤⣶⡶⠂⠀⠸⠇⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠋⣠⡤⠥⠂⠒⠭⢁⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠘⠲⠀⠒⠒⠚⠀⠗⠒⠂⠓⠒⠂⠀⠶⠒⠃⠃⠐⠒⠲⠀⠒⠚⠰⠀⠒⠒⠒⠒⠚⠀⠒⠃⠘⠐⠐⠒⠒⠒
⣶⣶⡶⢶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⡆⢰⡿⣿⠇⢀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠉⣿⡀⠀⣿⡜⣿⣿⣿⣋⣴⡿⠃⢀⣅⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣿⣤⣤⡘⣿⣿⡆⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠉⣀⣾⠇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⢡⣾⠿⠁⢀⣀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠿⠿⠻⠟⠛⠛⠛⠋⠉⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣶⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠴⠀⡏⠀⣠⣭⡀⠙⠋⠀⠀⠀⠉⠉⠛⣋⣤⡀⠀⠀⣿⣿⠋⠀⠀⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠰⠽⢧⣠⢧⠀⠀⠀⠀⠀⠀⢤⠄⠒⠆⠀⠀⠉⠉⠀⠀⢸⡿⠃⠀⠀⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠀⠊⡕⠀⠑⢤⣀⠀⠀⠀⠈⢆⡰⠀⠀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⠒⠒⠊⠉⠉⠒⠶⡆⠀⢀⡠⠟⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣠⠔⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⢤⣨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⡟⠉⠉⠉⠉⠀⠀⠀⠈⢧⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣀⠠⠀⡀⠀⠈⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⢀⡀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⡋⠀⠀⠀⠀⠇⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠘⡀⠀⠘⠀⠘⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢁⠀⠡⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⢀⠂⡐⠁⡠⠁⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠣⠀⠀⢡
⠀⠀⠀⢀⠂⡐⢀⠊⡐⠀⠔⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠡⡀⠡⠐⠄⠡⠀⠄
⠀⠀⠀⠀⠌⠠⢀⠌⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠐⢄⠐⡌⢂⠡⡈
⠀⠀⠀⠈⠰⢁⠀⠐⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠢⡈⠄⢡⠐
⠀⠀⠀⠃⠃⣆⡈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡈⣀⠄
⠀⠀⠀⠐⠂⠠⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠄⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈
⠀⠀⠀⢸⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢰⠀⠇
⠀⠀⠀⢸⠀⡄⠘⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⡀
⠀⠀⠀⢈⠀⠁⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢰⠀⡇
⠀⠀⠀⠸⠀⢠⠀⠀⠀⣷⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢾⠃⠀⠀⠈⠀⡇
⠀⠀⠀⣰⣤⠼⢒⠛⠊⠁⠀⠀⢀⡀⢀⣀⠀⢀⠀⣀⢀⠀⡀⢀⣠⣤⣤⣄⣶⣶⣶⣄⣤⣤⣤⣀⢀⠀⢀⢀⡀⣀⠀⢀⠀⢀⡀⠀⠀⠀⠁⠛⠒⠶⣤⣥
⠀⠀⠀⠀⠀⠀⠀⠒⠀⢠⠀⠀⠀⠓⠒⠒⠒⣆⢠⠒⠒⠒⠒⠙⢫⢡⣼⠋⠀⠀⠀⠙⢷⡍⡩⠋⠘⠒⠓⠒⢀⢨⠒⠒⠒⠒⠂⠀⠀⣠⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢷⠀⡐⠀⠉⠉⠉⠉⠉⠉⢇⢈⡾⠉⠁⠀⠀⠀⠈⠉⢹⣇⢰⠁⠉⠉⠉⠉⠉⠁⢁⠀⣸⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠓⠒⠀⠀⠀⢢⠀⠀⢸⠋⠛⠀⠒⠒⠒⠒⠒⠂⠘⠉⡏⠀⠀⣰⠀⠀⠀⠐⠘⠓⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣠⡞⠚⠓⠒⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠴⠒⠚⠒⢒⣄⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠉⠀⠀⠀⠀⠀⠰⣶⣿⣧⠀⡅⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠠⣀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⣀⠀⠀⠀⢤⣤⣖⣒⡛⠛⠂⠀⢘⣿⡿⢠⠁⠀⠀⡀⢰⣇⠀⡴⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠜⠁⠤⠭⣿⣠⣀⣐⣿⣏⠤⠌⡙⣷⣤⣤⣾⣿⠃⡇⠀⠀⢰⡇⢸⣿⡎⣿⣀⠀⠀⠀⠤⣤⣼⣆
⠀⠀⠀⠀⠀⠈⡎⠀⣠⡵⠋⢀⣼⡿⣿⣇⣀⣤⠼⡿⣿⣿⣿⠦⠀⢇⠀⠀⣿⡇⣼⣿⣷⠟⠁⠀⠀⠀⠀⠀⠙⠻⠀⢀⣤⠆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠋⠶⠋⠀⠀⠀⠾⢿⣅⠉⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⢸⠀⠀⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡄⢻⠀⠀⠀⢀⣼⠃
⠀⠀⠀⡎⢠⣾⡄⠆⠒⢿⣶⣴⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⣸⠀⣠⡿⠀⣹⣿⣿⣿⡓⠒⠳⢶⣤⡐⢿⡝⢁⣼⠇⠀⢀⣩⠇
⠀⠀⠀⣧⣀⣞⠀⢀⠀⠘⠀⠈⢻⣿⣷⣄⡀⠰⢶⣾⣿⣿⣿⡟⣠⠃⣰⣿⡥⠀⠀⠉⡻⢥⣤⣤⡾⢻⣿⣿⣿⠁⢸⣇⠒⠛⢻⠃
⠀⠀⠀⠘⣿⢏⠀⠉⠀⠉⠀⠒⠚⠿⢿⣿⡟⠆⠀⢹⣿⣿⡟⡔⠁⢰⣿⣿⢡⣶⠀⠐⠿⠦⠤⠴⢣⡞⠘⢻⡏⠀⢸⠯⠉⠉⠇
⠀⠀⠀⠀⠀⢋⣴⣴⣷⣶⣶⣶⣶⣿⢿⣿⠁⡄⠀⢸⣿⠟⡼⠀⠀⢸⣿⣿⣿⣿⣆⡀⠀⠀⠀⣀⣾⣇⠀⠀⡇⠀⢸⠀⠀⢺
⠀⠀⠀⠀⠀⡌⢿⣿⣿⣿⣿⠿⠛⣡⣾⠁⣸⠃⣰⡿⣫⡾⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣻⣟⠃⢀⠃⠀⢸⣄⣴⠏
⠀⠀⠀⠀⠀⣷⣤⣍⣉⣠⠤⠔⣊⣿⡟⣰⢏⡼⢫⡾⠋⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⢟⣵⡾⠿⠿⢿⣤⣿⣤⠀⢸⠏⠁
⠀⠀⠀⠀⠀⢠⡤⠴⠒⠶⠜⠛⠛⠋⠈⠁⠘⣰⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⣥⣄⣀⡒⢿⣽⠿⡟⠃
⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⣠⣤⣿⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡟⠓⠦⠄⠉⣉⡩⠴⠃
⠀⠀⠀⠀⢀⣬⣶⣿⣿⣷⣮⣙⠿⠟⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡏⣀⣤⣤⣄⠈⡗
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠃⠀⠊





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠆⠐⠂⠀⠀⠀⠐⠐⠀⠀⠐⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠘⠀⠀⠀⠚⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠄
⠀⢀⠀⠄⠀⠀⢀⡌⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠸⣇⠁⠀⠀⠀⠢
⢐⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢄⣀⡀⠀⠀⠀⡄
⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠛⠏⠀⠀⢘⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡂⡠⠐⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⢐⢀
⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠐
⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠉⠁⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡑⠢⡄⠀⠀⢤⠤⠲⠁⠀⡀
⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⢀⠀⣀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁

⣿⣟⣻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡇⣇⠀⢠⠀⢀⡀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢀⡱⠞⠚⠉⠀⠃⠀⠀⠀⠀⠈⠈⠉⠞⣠⢦⡄⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣦⠊⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⣇⠰⣌⣢⠔⢈⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⠀⠀⠀⠂⡤⠤⢤⣀⠀⠀⠀⠀⢠⣒⣒⠢⠀⠀⠀⠀⠈⢢⡈⢇⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡲⣄⠀⠀⠰⠁⠀⠀⠀⠖⠋⢉⡑⢢⠁⠀⠀⠀⢀⣀⠀⠘⡄⠀⠀⠀⠀⠀⢳⡀⢀⡀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⡃⠀⡟⠀⠀⠀⡀⠀⠰⣤⡇⠀⡇⠀⢠⡀⢾⡶⠀⠀⡜⠀⠀⠀⠀⠀⠀⣧⠜⠀⢀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⣈⢹⠁⠀⠀⢀⠳⣀⠀⠈⢀⠜⠀⠀⠀⠑⠤⣀⡠⠔⢁⠔⠀⠀⠀⠀⠘⡏⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠸⡇⠀⠀⢠⠑⠈⠉⠉⠁⠀⠰⢀⠄⠀⠀⠀⠒⢊⡡⠀⠀⠀⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢦⡈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢊⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠃⠀⠀⠉⠲⣄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣈⠥⠒⠁⠀⠀⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⣰⠃⠈⠉⠁⠒⠂⠀⠤⠶⠒⠒⠚⢯⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠁




⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠖⠂⢸⠙⠉⠋⡇⠐⠲⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠺⠉⣘⡀⠖⠒⠚⠈⠈⠁⠒⠓⠦⣰⡡⠉⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡛⢄⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣎⠊⢑⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢎⠢⡸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡡⠚⠄
⠀⠀⠀⠀⠀⠀⢀⢆⠀⡝⠀⠀⠀⠐⣉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡠⢺⡄
⠀⠀⠀⠀⠀⠀⡌⠑⠽⠀⠀⠀⠀⡈⠀⠩⡢⠀⠠⡂⣭⠄⢐⣒⣂⠤⢀⠀⠀⠀⠀⢳⠁⢐
⠀⠀⠀⠀⠀⠀⠯⣢⠧⢒⠂⠂⠒⠕⢀⠀⢈⡀⢁⠎⣔⠌⠂⠀⠀⠪⡢⡑⡀⣀⣀⣀⡔⡨
⠀⠀⠀⠀⠀⢰⠄⣈⠀⠪⣢⣀⢀⡔⡡⡂⠀⡪⠺⢔⡒⠢⢀⠀⠤⠠⢀⡰⠉⠀⠀⠈⢻⠀⡆
⠀⠀⠀⠀⠀⢈⠑⠚⡤⢒⠩⡄⣸⠀⠈⠀⠀⠐⠉⠂⠓⡕⡈⠆⠣⠀⠀⠀⠀⠀⠂⠀⢰⡫⠇
⠀⠀⠀⠀⠀⠀⢇⡒⢷⠖⠁⠀⠈⡄⠘⡀⠀⠀⠊⠑⠢⢦⢡⢧⠤⠓⠈⠉⢢⠛⠀⠀⠉⠲
⠀⠀⠀⠀⠀⠀⡸⠄⠘⠂⠤⠄⠤⠴⠘⠒⠀⠒⠓⠀⠀⠠⠄⠀⣀⣀⣀⣀⡬⠀⠀⢄⡀⠖
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⢣⣠⠒⠉⠀⡀⠀⠀⠁⠂⢄⡀⠀⠀⠀⠀⢀⠠⠄⠒⠂⠀⠤⢀⣀⠠⠄⡀⢸
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠠⠄⠀⠀⠈⠑⠒⠒⠈⠡⣀⠄⠀⠀⠤⠤⢀⠀⠀⠀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠉⠀⡀⠀⠀⠁⠊⠀⢄⣀⠀⠀⠀⠠⣀⡄⠀⠀⠀⠉⠁⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⡀⡈⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠠⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠂⠠⠤⠥⠤⠤⠀⠒⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⢀⣀⣠⣈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣟⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⣿⣜⣛⣓⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣻⣟⣿⣛⣿⡀
⠀⠀⠀⠀⠀⠀⢀⠔⢿⣷⣦⣤⣽⣿⣿⣿⣟⣉⣤⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢾⣿⣿⣿⣿⣿⣿⣷⠠⠤⣀
⠀⠀⠀⠀⠀⢰⣧⡀⠀⣬⣉⣛⡛⠛⠛⠛⠛⠛⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠁⠀⣸⣿⣿⣿⣿⣿⣿⣏⠐⠀⠁⢡
⠀⠀⠀⠀⠀⠀⠙⢿⣶⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣀⣤⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣅⡠⣴⣿⡿⣿⣿⣿⣿⣿⣿⡖⠀⠀⣌⣆
⠀⠀⠀⠀⡾⢷⣄⣀⢿⣦⣤⣤⣄⣀⣀⣀⣀⣀⣼⠛⠛⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣤⣤⣵⣿
⠀⠀⠀⠀⠇⠀⠉⠁⡎⠉⠛⠛⠻⠿⠿⠿⠟⠛⣻⣦⣀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠸⡶⠶⠿⠿⣷⣶⣤⣤⣤⣤⣤⣤⣼⠀⠉⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⡟⡋⠉⠠⠁⠂⠐⠈⠟⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⣤⣶⣦⣄⣠⣼⣏⠻⣶⣤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⣌⣱⡖⠓⠀⠃⠒⣶⢈⣩⣽⠟⠋
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠋⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠉⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⡏⠀⣸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡁⠀⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠛⠋⠙⠻⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⡿⠇⠙⠿⠿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠄⠀⠀⠀⠍⠁⠠⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠘⠂⠀⠀⠀⠀⠘⠀⠐
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠤⠀⠒⠒⠒⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠛⠛⠛⠛⠀⡇⠀⠀⠀⠀⠈⠛⠿⣷⣤⣀⡀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠛⠁⠀⣀⣤⣤⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣠⣬⣉⠉⠙⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⣇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠏⠁⠀⠀⠈⢻⣧⢹⡀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠙⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠯⠄⠂⠒⠂⠒⠤⡿⢸⡇⠀⠀⠀⢸⠥⠐⠐⠂⠒⠂⠼⠂⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⡀⠀⡦⣀⠀⠀⠈⠛⢢⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠰⡈⠑⠤⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠑⡄⠀⠀⠀⠉⠉⠁⠂⢄⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⢘⠋⠉⠁⠐⠢⢈⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⢸⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠀⠀⠐⠚⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⠂⠀⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢓⠲⠲⠆⠀⠀⠀⠉⢀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⡀⠀⠀⠀⠎⠁⠑⢄⠡⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠬⢒⠏⠁⠀⠒⠀⠀⠀⠀⠂⠡⡀⢨⡄⠀⠀⠠⢊⡉⡑⡄⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠆⠈⠀⡀⠄⠤⣴⠀⠀⡕⣐⠁⠀⠔⡐⠃⠀⡴⠚
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠈⠐⠤⣀⠈⠀⠀⠀⣀⠀⠀⡄⠘⣦⢊⠔⡀⠔⠈⠢⡤⠄⠀⠀⠀⠁
⠀⠀⠀⠀⢀⡀⡈⠀⡠⢬⠅⣂⠘⠶⠀⠀⠀⠁⠀⠀⠠⡀⡠⠋⠀⠀⠀⠀⠀⢈⠒⠠⠒⠙⠀⢀
⠀⠀⠀⠈⠀⠀⠛⠒⠂⢤⠀⠄⠨⠁⠒⣤⡃⠀⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⠀⡏⠖⠃⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡤⡎⠀⠀⠀⠀⠐⣿⣷⣤⣤⣴⠾⠛⠛⠣⡀⠀⠀⠀⢀⠡⠔⠁⠤⢘⠊⢠⠃
⠀⠀⠀⠀⢰⠀⣀⡐⠓⢬⣆⠀⠀⠀⠀⠙⠏⠁⠀⠐⠀⠀⠀⢀⠀⠀⠀⢀⠌⡠⡀⢀⡠⢂⠔⠁
⠀⠀⠀⠀⠈⡄⢡⠀⠢⠠⠣⠑⢄⠀⠀⠀⣄⣀⡤⠊⠉⠁⠚⠫⠭⠈⠥⠋⡐⡡⠊⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠈⢆⡎⢡⡉⠅⢀⡑⠪⠥⠴⢊⠤⢀⢀⣀⢠⠘⠁⢒⠒⠚⢃⡠⠊
⠀⠀⠀⠀⠀⠀⠁⠄⢘⠀⡀⢀⣄⡈⠁⠀⠀⡁⣀⡀⣀⢥⠀⠫⠙⠁⠀⠀⢀⡂⠀⠀⡠⠢⡄⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡚⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠇⡀⠀⠰⠁⠀⢸⣷⡋⠦⢀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢲⠃⠀⠀⢰⣿⠉⠀⡈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⣸⡿⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⢀⡀⠤⠒⠅⠀⠀⢀⠌⠱⡘⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠉⠀⠀⠀⠀⠡⣈⡤⠐⠁⠀⠀⠫⣞⠔⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠈⠉⢱⡀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣤⣤⣤⣴⠦⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢱⠨⠟⠓⠛⠛⠿⡿⢿⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠈⠀⠀⠄⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣽⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢹⣿⡟⠉⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠰⠶⠿⠿⠗⠘⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣈⡁⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⣿⠙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢠⣾⠟⠿⠄⠀⠀⠀⠀⠀⠟⢣⣾⡿⢛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣿⣿⣶⣤⣄⠀⠀⠰⠶⣶⣾⢁⣠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡟⠘⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣶⠛⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣮⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⣀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⣀⠴⡆⠀⠀⣀⣀⠤⠖⠋⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠓⠲⠦⠤⠤⠤⠼⠧⠞⠛⠉⠁⠀⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣖⣒⣲⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢠⡴⠒⢦⡀⠀⠀⢀⡴⠒⢦⡀⠀⢸⠿⠶⠀⠀⠚⠉⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⡤⠴⠖⠒⠦⢄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠸⡗⠒⠶⠤⠤⠶⠒⢚⠇⠀⢰⠇⣰⡯⠒⠋⠉⠉⠉⠉⠓⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠹⣄⠀⠀⠀⠀⢀⠞⠀⢀⣾⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠈⠓⠲⠖⠚⠁⠀⢀⠞⠀⠙⢦⣀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠉⢧⠀⠀⠀⠀⣀⡀⠀⠀⢸⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⡀⠀⠀⢌⠀⡇⠀⠀⢸⢰⠇⠀⠀⠀⡐⠐⡄⠀⠀⡜⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠘⡇⠀⠀⠘⠀⢇⠀⠀⢸⢸⡀⠀⠀⠐⠇⡰⠁⠀⠰⢡⠃⠀⠀⢀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢧⠀⢀⡠⠶⠚⠓⠉⠉⠉⠉⠉⠙⠛⠚⠷⢤⣤⣁⡏⠀⠀⢠⠋⣠⠎⠀⠀⠀⣀⠤⢤⠱⡄
⠀⠀⠀⠀⠀⢀⠴⣄⡀⠀⠀⠀⡆⠲⣄⠀⠀⣈⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣴⡡⠞⠁⠀⠀⡠⠊⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠸⡄⠀⠙⢦⡀⠀⠘⠢⣌⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⣄⡀⠀⠀⠈⠛⢦⣀⡤⠊⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠱⣄⠀⡠⠋⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠀⠀⠀⠀⢀⠤⠒⠒⠲⢤⡀⠉⠀⠀⠀⠀⠀⠻⣄⠔⠉⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠤⠤⣄⠀⡔⠁⢀⣠⡀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠘⣦⠠⠒⢉⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣤⡸⡆⡇⠀⢾⣿⠗⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠀⠀⠀⠀⢀⡠⠤⠠⢄
⠀⠀⠀⣀⣀⠀⠀⠀⠓⠤⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⠀⠉⣠⠇⠻⣄⡀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣲⣶⢦⣒⣉⣀⣀⣀⠤⠊
⡆⢠⠊⠀⠈⠑⢦⣀⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣈⡉⣉⡁⠀⠀⠠⣍⣓⣒⣒⡭⠄⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⡇
⠁⠈⠑⠲⠤⠤⠤⠤⣬⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠴⡶⢲⡋⡽⠁⠀⠀⠀⡗⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⡤⣤⢤⠤⣤⢤⡤⢤⠤⡴⢲⡶⡖⠒⡾⣏⠉⡏⢹⡀⡼⠀⢳⢀⠇⠀⠻⠁⠀⠀⠀⢰⠟⠓⠒⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⢷⠀⠀⠀⠀⠘⠇⠘⣦⠇⠀⣧⡸⠀⢇⡸⠀⢹⡼⠀⠸⣼⠀⠀⢳⠃⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⣸⢄⡀
⠀⠀⠀⠀⠀⠀⢮⡀⡠⠚⣆⠀⠀⠀⠀⠀⠀⠙⠀⠀⠘⠃⠀⠘⠃⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠲⢭⣓⠒⠒⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⠀⠈⠙⠲⣄⣀⣀⣀⠝
⠀⠀⠀⠀⡠⠤⠔⠒⠊⢁⡤⠞⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠣⣌⡉⠓⢦⡀
⠀⠀⢠⠀⠀⠀⢀⡤⠊⠁⠀⢀⠤⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠁⠀⠀⠉⠓⠒⠁
⠀⠘⠌⠳⢤⠤⠋⠀⠀⠀⠀⢧⣀⠴⠃⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢬⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠊⠉⢳⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⡀⠀⠀⠙⣌⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⡜⠁⠀⡠⠊⣰⠀⠈⠙⠟⣶⠶⠤⠤⠤⣤⡤⢤⡤⢶⠚⠉⠀⢲⡀⢱⡀⠀⠀⠈⠳⣄⠀⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣠⠞⠀⠀⠠⣇⡠⠃⠀⠀⡸⢀⡇⠀⠀⣶⠀⡞⠀⠀⣇⢸⡄⠀⠀⠈⠳⠴⠁⠀⠀⠀⠀⠈⠙⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⡰⠁⢸⡇⠀⠀⠙⠖⠁⠀⠀⢸⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠞⡠⠃
⣿⣿⡟⠻⠟⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⠬⠶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢀⡀⢀⡀⠀⢀⣿⠁
⠀⠠⠖⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠸⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢳⠀⢀⣠⡀⠀⠠⡀⠢⡀
⠀⠁⠀⠀⠸⡉⢢⡀⢀⠴⠲⡆⠀⠀⠀⠀⠀⠀⣠⡴⠶⠿⠆⠀⠠⠶⠿⣥⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡇⢸⣴⠋⣠⠃⠀⠀⢸⡄⠹
⠀⠀⠺⣉⠳⣝⡦⠱⣼⢠⠎⠀⠀⠀⠀⠀⠀⡾⢡⡖⠀⠀⠀⠀⠀⠀⠀⣤⡙⢷⡀⠀⠀⠀⢸⣡⠤⡝⠳⣄⠀⠸⠵⠒⡦⠀⠀⡇
⠀⠀⢀⠨⠑⡎⢀⡞⠁⠈⡆⠀⠀⠀⠀⠀⢸⡇⢠⠻⣄⠀⠀⠀⠀⠀⠀⣠⣟⠈⢷⠀⠀⠀⠀⠀⠀⡆⠀⠈⠇⠀⠶⣉⠁⠀⢠⠇
⠀⠀⠙⠒⢲⠆⠈⠀⠀⠰⣇⠀⠀⠀⠀⠀⣼⠀⢸⡇⠈⠓⠶⠤⠤⠴⠞⠁⢸⠀⢸⣇⠀⠀⠀⠀⢀⡽⢀⣄⣤⠞⠒⠾⠃⠀⠈
⠀⠀⠀⠐⠧⠊⠙⠒⠚⠳⡈⠳⣄⠀⠀⢰⡋⠁⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⣸⠄⠘⠉⣹⠀⠀⣠⠎⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠣⡀⠈⢻⠁⠀⣿⣿⣷⣶⣤⣤⣤⣶⣾⣿⠀⠈⣿⠉⢀⡞⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠢⣼⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⡴⠋⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⠀⠀⠈⣿⠿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠘⢧⡀⠀⠉⢉⡽⠋⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣄⠀⠙⠒⠚⠋⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢤⣀⠈⠙⠶⢤⡤⠶⠋⠀⠀⣀⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠙⠓⠲⠶⠶⠶⠶⠚⠋⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡠⠂⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢁⠞⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡖⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠑⡄⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡇⢸⡄⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣰⠋⣻⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠳⣄⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠴⠾⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢦⡀⠀⢈⡟⣷⠶⢤⣤⣤⣤⣤⣤⣤⡶⠶⢾⠻⡁⢀⡴⠋⠁⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠙⠲⠞⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠋⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣠⡴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣤⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⢈⠠⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⠀⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠉⡆⠐⣂⠀⠀⢠⠄⠀⠙⠺⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢐⣆⡀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣷⡆⠀⠀⠀⠀⠀⠀⠘⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⢾⣿⣿⡄⢸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠀⠀⠀⠀⠀⡔⢀⣿⣶⣌⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣼⣿⡜⠟⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠠⣤⣤⠀⠉⠁⠀⠢⠀⠉⢀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣾⣿⡇⣰⢿⣯⡋⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡺⡄⠀⠀⠀⠁⣾⣿⣿⣿⡔⣄⢻⣿⣦⡴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⡿⢰⣿⣿⠛⣿⡍⠍⠙⠂⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠰⢃⣜⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢻⣿⣷⣿⢸⠋⡉⡙⠡⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⢾⣿⣿⣿⣿⠿⡗⣻⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣼⣏⣿⡬⢰⣧⠚⢀⣀⢟⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⣀⣩⣿⣿⡿⣿⡇⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣦⣹⣿⣷⡄⢻⣽⣤⣹⡒⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠐⠀⠀⠉⠉⠁⠀⠠⠤⣾⣿⣿⣿⣿⣿⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠻⠭⠁⠙⠘⣿⣎⠻⠙⢿⡷⠄⠀⣸⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⣿⣿⣹⣧⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣼⣿⣦⠀⠀⠀⠀⠀⠀⠰⣶⣦⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠉⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠷⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠁⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠈⠀⠈⠀⠀⠈

⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⢻⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⡀⠀⠀⠀⠀⢠⠂⠀⠒⡄⠀⢸⡏
⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⢤⣴⢇⣶⡶⢶⣶⣾⡓⠤⣀⣾⣶⣾⡇
⠀⠀⢸⣿⣿⢀⡀⣹⡟⠁⠈⢹⣿⠋⠉⢙⣿⡿⠁⠀⡜⢿⡇⠀⠀⣼⡏⠀⠀⠙⣿⣷⣦⣿⣿⣿⣿⣷
⠀⠀⣸⣿⣉⠠⠤⠄⠣⣄⣠⠜⡔⠒⠒⠒⣌⣿⣶⣖⣷⣊⠅⠤⠀⣬⣵⣤⣤⢾⣩⠑⠈⠉⢩⣹⣿⣣
⠀⠀⡷⠠⡀⡇⠀⠊⡭⠭⠬⠭⡑⠆⠀⠁⠖⠒⠒⠒⠀⡌⡅⠀⡎⡠⠄⠀⠀⠠⡈⡆⠀⡌⠡⠀⠀⡏
⠀⠀⢇⠀⡇⡧⣤⠀⡄⠀⠀⠀⡅⡄⠀⢃⢃⠀⠀⠀⠀⠃⣇⢀⣇⢃⠀⠀⠀⢠⢣⠃⠀⢡⠰⠀⠀⡇
⠀⠀⠀⠉⠁⠉⠉⠉⠁⠐⠒⠒⠒⠓⠒⠚⠚⠒⠒⠂⠚⠘⠉⠀⠉⠉⠁⠀⠂⠚⠚⠒⠒⠚⠃⠃⠉

⣉⣙⡟⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠓⡆⠀⠀⠁⠀⣀⠤⠚⣁⡤⣶⡯⠒⠉⣀⢤⣶⣿⢰⣿⡿⠛⠉⠉
⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⢄⣀⠔⢊⡠⢔⠏⢵⢊⣡⠴⠚⠉⠀⡘⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠁⠀⡈⠠⠒⠋⢈⣹⠀⠀⠇⠘⢢⠄⠀⢾⣴⣿⡿⠏⠀⠀⠂
⠀⠀⢀⣀⣄⣀⡀⣠⣄⠀⠁⠜⠀⠀⠀⠀⠁⠀⠀⡞⢅⠈⡀⠈⠀⢰⠁⠀⠀⠈⠻⣿⠇⠆
⣴⣾⣿⣿⣿⣿⣿⣿⣯⡈⢳⠂⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠃⠐⡀⢀⠀⠀⠀⢿⠆⠀⠀⠓⠀⠀⣀⣀⣀
⣿⠟⠛⠉⠉⠛⠿⣿⣿⣿⡆⢹⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠚⡔⢲⡀⠀⠀⠀⢀⠤⠤⠞⠁⢀⣀
⠉⠓⢤⣤⣄⠀⠀⠸⠿⠟⢿⠐⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠤⠖⠊⠁⠀⠈⠲⠤⣀⡇⠀⣀⣴⣾⣿⣿
⠀⠀⠈⠻⣏⠀⠀⠀⠀⠀⠘⢤⢱⠀⠀⠀⠀⠀⠀⠀⡶⠋⠀⠀⠀⠀⣴⣾⣷⣦⣀⣀⠈⣺⣿⣿⡿⣿⣿
⠀⠀⠀⠀⢈⠵⠀⠀⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⢸⣿⣿⡿⠋⠁⠰⠀⠀⠀⠀⠀⠀⠉
⠤⢀⡔⠃⠓⡄⠀⠀⠀⠀⢀⣴⣶⣬⠀⠀⠀⡤⢠⠖⠛⠒⢀⣠⣤⡘⣿⡟⠀⠀⣄⡤⡀
⠀⠀⠌⠢⡈⠃⢠⡄⠠⠄⠛⠉⠻⣿⣇⠀⢼⠁⠀⠀⠀⠀⠈⠹⣿⣧⣿⡇⠀⠲⡋⠈⡓⠀⠀⠒⠀⠙⠂
⣾⡇⠀⠀⠉⠁⢠⣶⣀⣀⣤⣴⣶⣿⡿⠞⠢⠀⢀⠀⠀⠀⠀⠀⠈⠛⠛⢧⠀⠀⠇⡌⠀⠈⠀⠐⡦⠤⠞
⠁⠀⠀⠀⠰⢶⣿⣿⣷⣼⣿⠋⡙⠁⠀⠀⠀⠀⡀⠀⠀⢀⢀⠀⠀⠀⠀⠀⣇⡸⢰⠦⠤⣄⡀⢈⡀
⠀⠀⠀⢀⠎⢀⣿⡿⢿⣿⡟⠀⡇⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⣘⠀⠀⠀⢹⡁
⣉⠦⠒⠛⠋⠉⠉⠙⠛⠛⠒⠤⣵⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠒⢼⡠⢿⣿⣦⡀⠀⢉
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⢀⣴⡴⠖⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⠀⠁⠸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⣤⠀⠀⠀⢸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⣿⡍⠉⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣠⣬⡀⡢⠀⠀⠀⠀⠻⠀⠀⠀⠰⢾⡟⢚⣿⡗⠒⠂
⠴⠖⠂⠓⠒⠢⢄⠀⠐⠂⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⡿⢋⠙⠦⣤⠦⢀⡠⠤⠤⠔⠡⢼⣟⣛⣿⡇
⢀⡉⠉⢑⡆⠀⠀⣆⡀⠔⠈⢰⣶⣤⣿⣿⣿⣿⣿⠟⠐⠀⣠⠊⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⣏⣿⡇
⠤⠄⢐⠋⣂⣐⠈⢩⡄⠒⠁⣸⠟⠋⠉⠀⠀⠉⠉⠢⢄⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠶⣿⡇
⠀⠀⠈⠉⠀⠠⠀⠀⢣⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⡄⡀⠊⠀⠆⠈⠠⠀⠀⢸⡟⠛⣿⡇⠄
⠀⠀⠀⢰⣦⠀⢀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡁⠘⠄⠂⠀⠀⠀⠀⠀⢸⣯⣰⣿⡇⢀
⠀⠀⠀⠈⢉⠔⡡⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢿⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣙⣿⣷⡀
⡀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠂⠀⠀⠀⠀⠐⡇⠁⠂⠀⠀⠠⠴⠀⣼⡟⢐⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠈⢦⠀⣧⠀⠀⣾⠈⣇⡞⢠⡇⢀⣀⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⢦⡀⠀⢀⠀⢄⡹⡟⢧⣀⠈⢷⣿⠀⢹⡇⠀⣿⠁⢸⡇⠸⢋⡟⡴⢃⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡙⣦⡈⠳⣄⠙⠿⠀⠹⠆⣌⠻⠀⡜⠃⢀⠀⠀⠙⠃⡀⢸⡟⠁⠞⣰⠟⣫⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠑⢦⡈⠓⢦⣄⡈⣛⠆⠨⡀⠀⠢⣆⠀⠸⠀⠀⡷⢀⠘⠞⢀⠀⢀⡇⠈⠀⡠⠀⠁⠀⣴⣯⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣦⣴⣶⠛⠛⠈⢳⡄⠙⠀⢠⡀⠀⠐⠆⠀⠀⠘⠃⠀⠘⠀⠀⣠⠂⠈⠁⡠⠂⠈⡿⠋⣁⠤⠂
⠀⠀⠀⠀⠢⢤⣀⠀⠀⠑⠲⢿⣧⠀⣍⠁⠰⣄⠀⠀⠀⢦⠀⠉⠀⠀⠄⠀⠀⡀⠀⠃⠀⡄⠀⡀⠀⡄⠀⠀⠀⠠⠶⠿⠓⢚⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⠬⠓⠶⣤⣈⠃⠀⠈⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⡄⠀⢠⣇⡴⠀⠀⠀⠀⠀⠀⠭⣭⣤⠄
⠀⠀⠀⠀⠀⠀⠓⠒⠲⠶⠤⡄⠀⣀⠀⠀⠀⠘⠆⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⢿⠻⡄⠘⡿⠁⠀⠀⠀⠀⠀⠲⣾⡍
⠀⠀⠀⠀⠀⣀⣶⣶⣆⣀⠀⠤⠤⠼⠷⠄⠐⠒⠀⠀⠀⠁⠀⠐⠀⠀⠀⠀⢀⠀⢰⡀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠭⢤⣤⡤⠀⠀⠀⠀⠐⠃⠀⡀⠀⠀⠀⠀⠀⠀⠘⣿⡙⠛⠀⠀⠀⠀⢀⠾⠃⠀⠀⠈⣛⠆⣿⡇
⠀⠀⠀⠀⠀⠒⠒⠚⠋⠀⠀⠊⣁⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠀⠀⠀⢀⡙⡆⠀⢀⣤⠃⣿
⠀⠀⠀⠀⠀⠀⠉⣙⣿⠿⢖⣀⣤⣶⠄⢨⠟⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⢍⢁⡇⠀⠸⣥⢇⣟⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠊⣁⠀⠈⠉⠁⢠⡄⠀⢁⠀⠠⠆⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠠⣄⣀⡤⠀⠈⠉⠀⠠⠒⠋⠉⠉⠉⠙⠻⡟⣹⣿
⠀⠀⠀⠀⠈⠉⠉⣡⣴⡿⠀⠐⠙⠁⠐⠋⠀⣀⠀⠴⠀⡀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠉
⠀⠀⠀⠀⠀⠴⠛⢉⣾⣡⣶⠂⢀⣄⢀⣄⠀⣁⠀⡆⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣠⡴⠟
⠀⠀⠀⠀⠀⠀⢠⡿⠛⠁⣿⡴⠛⠟⠉⠈⠛⠁⠀⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠐⠈⠙⢓⣦⡀⢨⣭⣍⣹⠟⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⣇⣠⡀⣀⣰⣤⣼⠋⠛⠛⠛⠃⠻⣦⣤⣄⣤⣤⡀⠀⠀⣄⣨⣿⠋⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠷⠿⠟⠉⠉
⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣴
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠀⠄⠠⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣀⡠⠄⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⡼⠒⠒⠂⠀⠁⠉⠀⣀⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢰⢃⠀⢀⡤⠒⠲⠄⠚⠁⠉⠉⠀⠥⡱⠂⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⡇⠀⣆⠈⢀⡤⠤⠄⠀⠞⠉⠉⠀⠀⠆⠀⣀⣀⡀⠉⠐⠂⠀⠠⠄⢀⠀⠀⡇
⠀⠀⠀⠀⠀⣇⢊⠤⠆⠈⠀⠀⠀⣀⣀⡠⢴⠀⢾⠀⠈⠉⠉⠙⠆⠠⠤⣄⠀⠀⢀⠀⠀⣇
⠀⠀⠀⠀⠀⣷⡆⠀⠈⢆⠘⡉⠉⠉⠁⢀⠌⢀⢻⠀⠈⠉⠓⠀⠀⣀⣀⠈⠑⢀⠎⢀⠎⡡⠬⣁⠢⡀
⠀⠀⠀⠀⠀⢸⢌⠢⡀⠀⠢⢈⠑⠂⠒⠁⢠⠂⢸⠀⢀⡀⠠⠀⠀⠀⠈⠓⢀⡚⠂⣁⢼⠁⡠⠄⠓⡄
⠀⠀⠀⢀⡄⠠⢧⠑⠌⢢⠀⠀⠑⠂⠀⠒⠁⠀⠘⡐⠛⠋⠻⠷⡦⡄⠀⡠⠃⠈⠉⢀⠇⠀⢖⠋⠛⠂⡂
⠀⠀⠀⠈⠙⠢⡒⣪⡙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⣁⠔⠁⠀⠀⣠⠋⡠⠐⠉
⠀⠲⡂⠈⠀⠀⠌⠀⠉⠂⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠐⠈⠀⣀⠠⠀⠂⠁⠩⡀⠀⠈⠀⡒⠆
⠀⠀⠈⠂⠄⠪⣀⠀⠀⠀⠀⠀⠀⠁⠁⠐⠂⠀⠀⠀⠀⠀⠒⠀⠉⠁⠀⠀⠀⠀⠀⢀⡰⣀⠄⠊
⠀⠀⠀⠀⠀⠈⠙⠳⠖⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠶⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠂⠀⠀⠀⠀⠀⠒⠂⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⠴⠶⠒⠚⠛⠓⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣀⣀⣤⠴⠖⠛⠁⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠋⠉⠛⠛⠲⢤⣀⣴⠋⠁⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠖⠚⠋⠉⠉⠀⠀⢀⣠⠴⠖⢒⣻⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⣀⣤⡤⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠴⠖⠛⠉⠁
⠤⠖⠒⠚⠛⠋⠛⠓⠲⠤⣤⠞⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢊⡉⠉⠉⠁⠀⠉⠉⣹⡆
⠀⢀⣀⣤⣤⠤⠶⠚⠛⠉⠉⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢷⠿⢿⠛⠛⢛⣛⣛⣛⣛⣓⣦
⠋⠉⠁⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠚⠛⠉⢉⣉⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢶⠃⠈⠙⣇⢻⡉⠉⠉⢻⡍⢻⡀
⢀⣀⣤⠤⠶⠒⠛⠉⠁⢀⣀⣤⠴⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢺⣆⡀⠀⠸⣆⠹⢶⡆⠘⣇⢸⡇
⠉⠁⠀⣀⣠⡤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣸⠋⠀⣹⠀⠀⠘⠶⣌⣷⣤⢿⣀⡇
⠶⠚⠋⠉⠀⠀⠀⠀⢀⣀⣠⡤⠤⢶⡏⣿⠽⢧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣾⠃⢠⡟⠁⠀⠀⠀⠀⠀⠸⡇⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠐⢯⣤⡴⠖⣛⠉⡷⠘⢠⡀⣾⠳⡶⠦⣄⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⡟⠀⠀⠀⣀⣤⠶⠋⠀⣿⡄⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠗⠊⢸⣤⣷⠴⢺⡧⠿⣄⣿⢠⣯⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⠀⠠⠶⠛⠉⠀⠀⠀⠀⣿⣳⡀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡾⢋⣀⣽⠀⠘⣧⠤⣌⠙⢿⣍⠳⣄⠀⠀⠀⠀⠻⣤⡾⢻⣶⣶⣶⣶⣶⣦⣤⣤⣤⣟⠉⠙⢦⡀⠁⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣴⠋⣇⡇⠀⠀⠘⢦⣙⣻⣦⣀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠈⠢⡀⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣠⡟⣿⠷⠟⠃⠀⢀⣠⠀⠈⠈⣧⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢠⣴⡄⠀⠀⠀⣿⠀⠀⠀⠀⠐⢾⣾⠇
⠀⠀⠀⢀⣠⠶⠶⢤⣀⠀⠻⣿⠀⢠⠴⠚⠉⠀⣀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢄⠀⢸⡏⣇⠀⢰⡇⣿⠀⠀⠀⠀⠀⢸
⠀⣠⡴⠋⠁⠀⠀⡀⢈⠀⢀⣧⣤⣴⠶⠶⠛⠛⠛⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⢸⡇⢿⠀⢸⠀⣿⠀⠀⠀⠀⠀⡏
⠚⠃⢀⠔⢦⢀⡔⠉⠉⢀⠎⠁⠀⠀⢠⡤⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢸⡇⠘⠀⢸⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣰⠃⠀⡼⠁⠀⡟⠀⠀⣰⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠈⡇⠀⠀⢸⡇⠀⠀⠀⢸
⢀⡀⠛⠞⠉⠀⠀⢀⠎⢠⡟⢀⡾⠁⠀⠀⡇⠀⢠⡏⠀⣿⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣼⡇⠀⢿⣶⣶⣾⡇⠀⠀⠀⣾
⢃⢨⡶⠀⠚⠃⢀⡌⠀⠘⠁⣾⠁⠀⠀⠀⡇⠀⠚⠁⠀⣿⠀⠀⠉⠓⢦⣀⠀⠀⠀⠀⠀⡿⠿⠿⢿⡇⠀⢸⡟⠛⠛⣧⠀⠀⠀⡇
⠀⠀⠠⠖⢆⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⣇⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠀⠀⠀⣠⢾⡿⠀⠀⢸⡇⣤⠄⢷⡀⠿⣽⣧⠀⢰⡇
⠀⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠘⢧⠀⠀⣠⣾⡿⢧⠀⠀⠀⠀⠀⠀⢰⡏⠋⢁⣀⣤⡜⠁⠀⠀⠘⠛⢳⣮⣹⡇⠈⠷⠶⣶⣶⣶⣶
⠸⠃⠘⣴⠶⡤⢸⡇⠀⠀⠀⣿⠀⠴⠢⡀⠀⠈⢳⣾⡿⠋⠀⠘⣇⠀⣂⠀⣀⡴⠊⠙⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⠛⠶⣄⠀⢀⣀
⠀⠀⠀⠁⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠘⠋⠃⠀⠹⣄⢀⠀⢠⡟⣁⢈⣴⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⢧⠻⠿⢿
⠛⠛⠛⠛⠛⠃⣀⠿⠀⠀⠀⢻⠰⠶⠦⠤⣤⣤⣄⠀⡼⠛⣱⠛⣡⡴⠊⠁⠈⠉⠳⡴⠒⠛⠲⠦⣄⣀⣀⣴⠞⠋⠉⠉⠉⠛⠳⣄⠀⠈⠳⠤⠤
⣤⣤⣤⣄⠀⢾⣿⣦⣤⡴⠶⡏⠀⣤⣤⣤⡀⠀⠀⡀⠉⢉⣵⠟⠁⣀⣠⠤⠤⠤⣤⣀⡀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠲⠷⠶⠖⠒⠊⠁
⠉⠉⠉⢯⣤⣤⣤⣁⣈⣀⣀⣀⣀⣈⣉⣩⣤⣴⢶⣶⣦⡞⣡⡤⢤⡉⠙⠓⠛⠓⠀⠀⠉⠓⢶⣾⠛⠋⠉⠛⠛⠳⠦⣤⡀⠘⠉⣉⣛⣧⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⠶⣄⠀⠀⣧
⠀⠀⢸⡏⠀⠀⠀⣠⣴⣶⣶⣶⣄⣀⣾⠟⠁⠀⢀⠀⠀⠀⠱⠀⢹⢀⡤⠶⠶⠶⠶⠦⠤⢤⣄⣀
⠀⠀⠀⢷⠀⢠⡟⠋⠀⠀⠀⠈⠙⢿⡏⠀⠀⠀⣼⣧⣀⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⢸⡆⠟⠀⠀⢸⣦⢠⣷⣶⡄⠁⢠⣠⠞⣩⣤⣶⣦⡄⠆⠀⣿⣷⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⢧⣠⣦⡠⠈⣁⣭⣭⣛⡇⠀⠈⠩⣼⣛⣉⡀⠀⠁⠗⠦⢻⣵⠊⠉⠁⠀⢉⡇⠀⠀⠀⠀⡴⢶⣤⡀⠀⠀⠈⢱
⠀⠀⠀⠀⢸⡟⠻⡡⡿⠛⠉⠉⠙⠁⠀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⣿⣤⣤⣄⣙⣿⡇⠀⠀⠀⣜⡀⠀⠈⠙⢦⠀⠀⡎
⠀⠀⠀⠀⠀⢧⣠⠉⠉⠁⠀⠀⠀⢀⠀⠀⡀⠲⣄⠀⠀⠀⠀⠀⠀⠀⢿⡎⠙⠛⠛⠯⡄⠀⠀⣾⣭⣄⣀⠠⡀⠀⠳⢸⠁
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢯⣵⠍⡔⠚⠋⠀⠀⢀⡉⠀⠀⠀⢸⣿⡇⠀⠐⠀⠀⠀⠀⢿⣉⠛⠻⠟⠃⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢷⠀⠀⣠⠆⠀⠀⠀⠀⣀⠛⣀⣤⡴⢞⣿⠃⠀⠀⠀⣸⡿⠅⣠⢢⠄⠀⠀⢀⠀⠁⢠⡇⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠘⡇⠀⢷⡘⠳⣿⣟⣛⣛⣙⣉⣥⣶⣿⠇⠀⠀⠀⢠⣿⡧⠀⠈⠚⠷⠤⣤⠸⡆⠀⠈⠁⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⢦⡈⠻⣿⡿⠟⣋⡁⠠⠋⠀⠀⠀⢀⣾⡷⠁⠀⣀⣀⡚⠀⠀⠀⠁⠀⢰⡆⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠁⠀⠀⠀⢥⡤⠄⠀⠀⠀⠀⢠⣾⢿⣷⠿⠛⠉⠉⠙⠛⢶⣦⡀⠀⠻⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠉⡈⠀⠀⠐⠛⠛⠓⠀⠀⠙⠟⠆⡀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⣀⣀⣴⠾⠃⠸⣆⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣒⣂⠀⠤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⠤⠀⠀⠤⡀⠀⠀⢀⣀⡀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣤⠤⠴⠖⠒⠛⢛⡷⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⢠⠒⠢⠠⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⡋⠓⢸⡓⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠑⠰⠀⠳⠤⠪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⡇⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⢠⠶⢀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⢸⡇⡸⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠠⠄⠈⠈⠀⠈⢀⢠⡜⡵⢄⠀⠀⢠⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⢣⠃⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⢠⠔⠲⠶⢴⣆⠈⠀⡼⢰⡄⢼⠀⠀⠈⡄⠠⡇⠀⠉⠐⠀⠀⠀⠀⠀⠀⠠⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⢀⡏⠀⠀⠀⠀⠈⡗⠉⡇⠀⠀⡄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠖⠒⠀⠠⡰⠒⠒⠒⠒⠻⢄⡀⠀⠀⠀⠀⠀⢠⠇
⠀⣸⠀⠀⠀⠀⠀⢠⠃⠀⢧⡸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠂⡾⠀⢧⡀⠊⠙⠢⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣲⡄⠀⠀⠀⡞
⢀⠇⠀⠀⠀⠀⠀⡞⠀⠀⠀⡏⠀⡁⡀⢀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⠁
⢸⠀⠀⠀⠀⠀⢠⠇⠀⠀⢼⠀⠌⢁⠼⣀⠀⢀⠰⠀⠀⠐⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡏
⡟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠓⠛⠁⠀⠀⢦⣨⠖⠛⠶⠶⠶⠶⢾⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁
⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿
⠳⢄⡀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡇
⠀⠀⠈⠉⠓⠚⠉⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠲⠤⠤⠼⣇⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣀⣀⣠⡤⠤⠖⠚