Language Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠋⠉⠉⠀⠀⠀⠀⠉⠉⠛⠛⠶⣤⣀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠛⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣰⣿⣄⣀⣀
⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠈⠉⠉⠛⠷⢦⣄⡀
⠀⠀⠀⠀⢀⡾⠁⠀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠾⠛⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢀⣿⣥⡶⠛⠉⠀⠀⠈⠉⠉⣿⠛⠉⠉⠉⠉⠀⠘⠛⢛⡟⠛⠛⠛⢛⣿⠛⠛⠂⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣄⡀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠛⢶⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⠶⠶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⣿⡇⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣾⡧⠀⠀⠀⢸⣿⣿⣉⣉⣉⣀⣉⡉⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⣾⡎⢿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠿⠿⠿⠿⣿⣿⡿⢃⠀⠀⠀⢸⣿⣿⣿⡙⣿⣿⣿⢃⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠘⠛⠛⠘⣿⣿⣿⡇⠀⠀⠀⡟⣴⣿⣿⣿⣷⣶⣶⣶⡜⠃⠀⠀⢸⣿⣿⣿⣷⡙⣿⢃⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣰⣿⣿⣿⣷⠹⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡿⢋⣤⡛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⣀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣴⣿⣿⣿⣤⣾⣿⣿⣿⣷⣬⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣦⣝⣛⣡⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⢀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡄⢠⡍⡄⠀⠀⠄⠀⢀⠀⠀⠀⠀⡀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡄⢀⠀⠀⠀⠣⡔⠈⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠈⠉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠐⣤⣁⣀⠔⠄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣄⡄⠀⣠⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⠸⠙⠞⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠟⠙⠻⠸⡊⠠⡀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠜⣔⠀⡀⠀⣰⣥⣠⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠈⠀⠂⠀⠀⠀⠡⠐⠒⢫⡆⠀⠀⠀⠀⠀⠀⠀⠉⠱⠶⠚⢉⡉⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠠⠀⢴⠘⢇⠀⠀⠀⠤⠀⠴⠾⠷⠄⠀⠠⠾⠶⠱⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠠⢀⠣⣀⠘⠀⠠⠀⠄⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⢉⣠⡆⠀⠀⠀⠀⠀⠁⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣃⣈⣀⣀⣀⣓⣤⡀⠇⠀⠀⠀⠀⠀⠈⠡⠤⠤⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠄⠀⠀⠃⠠⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠆⠀⠤⠸⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢃⠀⠤⠀⠀⠀⠘⠀⣀⠐⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠊⡀⠠⡀⠀⡠⢘⠇⠀⠄⠂⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣢⡀⠈⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⢠⠀⠀⠶⠥⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⣀⠁⠂⠍⢈⡉⡐⢄⠀⠁
⠀⠀⠀⠀⠀⢀⠐⢎⣪⢗⡋⢉⡱⠂⢄⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠉⠉⠈⠉⠈⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣌⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢣⣭⡸⣿⣶⣿⣣⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣭⣭⣿⢟⣫⣽⣶⣤⣄⡀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⢳⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣩⣿⣽⣿⣿⣿⠀⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⡹⣿⣿⡿⣿⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⡇⣿⢸⣿⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⣿⣾⣿⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠼⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣰⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣦
⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠠⣶⠶⣶⠟⠃⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⢠⣤⣤⣿⣤⣤⡀⣿⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⣀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⠟⣿⠙⠿⠀⣿⠀⠀⢸⣯⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⡆⠀⠀⠙⠋⠀⣿⠀⠀⠐⣿⣤⣤⣼⣯⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠛⠛⠛⠛⠛⣿⡄
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠿⠀⠀⠈⠛⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠙⢿⣦⣤⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢹⣿⡆⠀⠀⠀⠀⣠⡴⠒⠲⣶⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢠⣿⢃⣠⣾⠟⠉⠉⠉⠉⠉⠉⠉⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢻⣿⡀⠀⠀⠀⠻⠋⠀⠀⣸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠰⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⢿⣷⠀⠀⠀⣠⡶⠚⠉⢹⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠘⣿⣧⡀⢸⣿⡇⠀⢀⣼⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠀⠙⠛⠛⠉⠈⠛⠛⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠠⣤⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠄





⠀⡉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣐⡦⠾⠖⣷⠈⠉⠉⠃⡾⠻⡿⣻⠿⠡⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣵⣤⣀⡉⠛⠶⢿⠸⠠⡀⢠⣿⢽⠯⣒⠕⠋⣠⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢛⡉⠉⠡⠤⠤⠴⠈⠀⠰⠄⠭⠈⠤⠤⠬⠁⠐⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠡⠐⠂⣀⢊⣈⣀⠀⠀⠄⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣨⡤⣥⡄⠀⠀⡀⣁⠙⠀⠀⠈⢡⣀⠐⢖⣢⣶⣷⣾⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋⠁⡠⣒⣥⣶⡛⡇⠚⡁⠘⠸⡏⠑⢤⡈⠑⠟⠫⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡭⠶⠿⠿⠿⢶⣶⣢⣕⠿⠍⠿⠳⡿⠶⠇⢦⡥⠤⠧⠷⠷⠿⠛⠂⠁
⠀⠀⠀⠀⠀⠀⢶⠋⡔⠁⠀⢀⡤⣠⣀⠀⠈⠙⢝⢽⢦⠀⣄⠀⠰⠎⢀⡀
⠀⠀⠀⠀⠀⣼⠏⠘⢀⡼⢟⡩⠚⠁⠙⢷⣦⣇⠈⠆⢲⡇⠀⠈⡩⢂⠤⠊
⠀⠀⠀⠀⢠⣿⠠⠧⠔⠚⠁⠀⠀⠀⠀⠀⠉⠛⢤⢼⣀⡿⠀⠘⢢⠌⣒⠄
⠀⠀⠀⠀⣸⠃⠀⣆⠀⠠⣀⡀⠀⠀⢀⡀⣀⠀⢸⡄⠀⢱⡄⠖⠓⠊⠁
⠀⠀⠀⠀⠙⢔⠰⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡇⠔⡜⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠄⡀⠠⠔⠁⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⢲⢄⣀⣀⣀⡴⡿⢦⢄⡀
⠀⠀⠀⠀⠀⢀⠔⠡⠤⠀⢸⣄⣀⠑⠁⣰⡏⠐⠁⡈⠲⢄
⠀⠀⠀⠀⡰⠁⡀⢁⠀⠐⢽⡝⠂⠀⠐⢚⡯⢐⠠⡂⠄⠀⠣⡀
⠀⠀⠀⠐⠀⠄⡠⢀⠀⡀⣸⡇⠀⠀⠀⢰⡪⢐⠨⢂⠅⡂⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠐⠈⠈⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣸⡆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢠⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠉⠉⠉⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣆⣨⣿⣿⣿⣿⣿⣿⠟⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣠⣾⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠋⠛⠛⠿⠿⠿⣿⣿⣿⠿⠿⠿⠛⠛⠉⣀⣤⣶⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⣶⣾⣉⣉⣉⠉⣉⣁⣀⣉⣉⠉⢉⣉⣉⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⡿⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠟⠋⢁⣠⣶⣿⣶⣄⡈⠙⠻⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡄



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣄⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⢤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠧⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣀⣀⣠⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡆⠐⠀⠘⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⢿⡟⠡⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⣴⣂⠀⠀⠀⠀⢿⣿⣇⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⣄⢸⡇⢀⡀⠀⠀⠁⢀⢽⠇⠀⠀⠀⢘⣿⣦⣄⠀⢀⣠⣀⠀⠀⠥⠀⣴⣿⣿⠟⠁⣀⠀⠀⣿⣿⡇⠀⠀⢠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡇⠀⣀⣴⠀⠀⢠⣀⡆⠀⠀⠀⢰⣮⣟⡯⡱⠟⠉⣰⡀⠀⠀⢰⣿⣿⠃⢠⣶⠤⠀⠀⢿⣿⡇⠀⠀⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡷⣿⡇⣿⡿⣿⠀⠀⢸⣿⡇⠀⠀⠀⣼⡇⠨⣿⢠⠀⢐⣿⣿⡇⠀⠈⠻⢿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⠻⢿⡇⠉⠀⣿⠀⠀⢸⣿⡇⠀⠀⠀⡝⠋⠐⢿⠈⡀⢐⣿⣿⡇⠀⠀⠀⠀⢤⣾⣷⣤⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⣿⣸⣷⣾⡇⣾⣿⣿⠀⠀⠀⠉⡀⠀⠀⠀⣿⣷⠀⠀⠀⡂⠠⣿⣿⡇⠀⠸⢦⡄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠈⠠⠃⣿⢙⠻⢿⡇⠁⠀⣿⠀⣠⠰⠃⠑⠀⠀⠒⣿⣿⠀⠀⠀⣤⡾⣿⣿⡇⠀⠀⠀⡀⠔⠚⡃⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣷⣾⡇⣶⣿⣿⣾⣿⡷⠉⠀⠀⠀⠈⣿⣿⠀⠀⠊⠀⠀⠙⡿⡇⠀⠀⠀⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⠙⢻⣋⠉⠉⢤⣿⣿⡟⠀⣄⡀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢹⣿⣿⠀⢸⣿⣿⠀⠘⠻⡇⠠⣦⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡅⠿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣦⡤⠈⠀⢸⣿⣿⠀⢸⣿⡇⠀⠰⣿⣶⠖⠀⠀⠀⠈⠁⠀⠀⠀⢰⠈⣿⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⡇⠀⠀⠀⠙⠿⠀⠈⠙⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⣤⡞
⠀⢠⣄⠀⠀⠀⠀⢀⣀⢰⣿⡇⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⡇⠀⢴⣾⡿⠛⢹⣿⡇⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⣿⡇⠄⠂
⠀⢸⣿⡇⠀⠉⠁⣀⣤⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠐⠋⠁
⠀⢸⣿⡇⠀⣴⣿⠟⠋⠉⠉⠓
⠀⢸⣿⡇⠀⠈
⠀⠈⠻⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠒⠒⠛⠋⠉⠉⠉⠉⠙⠓⠒⠲⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⣀⣀⣤⠤⠴⠶⠒⠒⠒⠲⠶⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⣠⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠈⠉⠓⠶⣤⡀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⢀⡼⠁⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢳⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠻⣄
⠀⠀⢀⡾⠁⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢳⡀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠹⣆
⠀⠀⡾⠁⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⢷⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠹⡆
⠀⣸⠁⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠈⣧⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢹⡀
⢠⡏⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣀⣀⣀⢀⠀⣀⠘⣧⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⣧
⢸⠁⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠻⣆⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢹
⡿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠸
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣷⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣯⣤⣤⣤⣭⣷⡶⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣥⣤⣤⣤⣤⣬⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢰
⢸⡀⠀⠀⠀⣧⠀⠀⠀⣀⡤⢶⡶⠶⠶⠶⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⢶⡦⣤⡀⠀⠀⢠⡇⠀⠀⠀⣸
⠈⣇⠀⠀⠀⠸⡄⢠⠞⠁⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠄⠀⠉⢷⡀⡼⠀⠀⠀⢀⡏
⠀⢹⡄⠀⠀⠀⢳⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢻⠃⠀⠀⠀⣼⠁
⠀⠀⢳⡀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⣰⠃
⠀⠀⠀⢳⡀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣰⠃
⠀⠀⠀⠀⠻⣄⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⢀⡼⠃
⠀⠀⠀⠀⠀⠙⢾⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠂⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠓⠒⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⢀⡶⠚⠋⠉⠳⡄⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⢺⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠴⠶⠚⢷⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠸⡄⠀⠀⠀⣀⣠⢴⣫⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠴⣶⣚⣛⣓⣒⣚⣯⠽⠞⠋
⣶⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠠⠘⠀⠀⠀⠀⠀⠈⠋⠂⠈⠘⠀⠋⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⡀⠤⢠⠄⠤⡀⢀⢰⠀⡀⢀⣀⠀⠤⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⡆⠀⡀⢀⣠⡀⢀⣤⡄⢁⠀⡀⢀⣀⠀⣀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣖⣿⢸⠀⣯⢼⡈⡏⢸⣀⡇⢼⢺⠷⣥⣼⠸⣇⡏⠀⡈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠊⠂⡠⠇⠰⠀⠀⠀⠈⠡⠈⠹⠠⠉⠦⠐⠎⠀⠙⠗⠃⠀⠀⠀⠀⠀⠀⠒⠋⠀⠀⠁⠈⠁⠈⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠐⠆⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠦⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠛⠉⠁⠈⠀⠀⠁⠀⠀⠴⠀⠀⠔⠲⠀⠀⠀⠀⡳⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠄⠀⢄⠀⠀⣀⠀⢀⡀⠀⠀⠀⢀⠀⡀⣀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠂⠀⠀⠀⢠⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠊⠈⠀⠈⠀⠀⠀⠀⠀⠁⠸⡜⢣⠍⠀⣤⢤⡄⠀⠀⠀⠴⡠⠄⠄⠴⠀⠀⠀⠀⠾⠈⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠂⠰⠀⢲⢶⠄⠒⢀⠦⠰⠀⠀⠀⠀⠛⠤⡅⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⢀⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣤⣴⣤⣤⣶⣦⢠⡄⠀⠀⠘⠂⠠⠤⠀⠀⠰⢤⡀⠀⠀⠀⡔⠈⢈⠉⠉⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠢⠂⠀⠘⠀⠀⠀⠀⠀⠘⠛⠈⠈⠀⠀⠀⠙⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⠿⠃⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⣷⣶⣶⣶⣶⣶⣿⣿⡿⠿⠶⠶⠖⠀⠀⢻⣿⣿⣿⣿⣇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣾⣿⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠿⠿⣿⡿⠿⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣃⣤⣿⣧⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣴⣿⠉⠉⠁⠉⣿⡏⠁⠉⠉⢹⣷

⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡿⠿⠿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⡿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣻⣟⣛⣻⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠸⡆⣸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠐⣱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠠⠄⠀⠀⠀⠁⠀⣾⣿⣿⠟⠋⠀⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠰⠒⠊⠀⠀⠙⠛⠛⠀⠀⠀⠀⠚⠛
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠺⠃
⣶⣿⡆
⣿⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⢀
⣯⡤⠴⠄⠀⠀⠀⠀⠀⠀⠀⢰⡆⢠⣤⣤⣴⠏⠀⠀⣶⣶⣤⣤
⠀⠀⢤⣤⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠘⠛⠋⠛⠛⠇
⠀⠀⠘⣧⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⣤
⠄⠄⠈⠻⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⠀⠀⠲⠚
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⡉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁⠀⠂⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠾⠿⠿⠟⠉⠩⠏⠁⠀⠀⠐⡟⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡶⢶⣶⣶⠶⢲⣶⢢⡖⠒⠒⠒⠒⠒⢶⠶⡶⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣥⣿⣤⣴⣦⣼⣄⢠⡷⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠈⠛⠛⠿⠿⢿⣿⣿⣿⢿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⢀⣠⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⠟⠛⠉⠉⠉⠉⡥⠖⠒⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠄⠠⣀⣀⣀⣀⡀⠀⢀⡤⠖⢒⠉⠀⠀⠀⠀⠀⣼⣷⡤
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠀⡏⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠑⠒⠒⠢⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠢⠤⠄⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠀⠄⠤⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢠⣾⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠿⠛⠻⠿⣿⣿⣿⣆
⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⢸⣿⣿⡀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠸⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠹⣿⣿⣿⣷⣦⣤⣤⣀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⢀⣤⣤⣤⣴⣾⣿⣿⠏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣫⣴⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⡿⠿⠿⠿⠿⢿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⠤⠤⠤⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⣀⣠⠖⠒⠲⠤⢤⣀⠀⠀⠈⠉⠀⢒⠆⡠⠂⠀⠀⠀⠂⠐⠂⠈⠉
⠀⠀⠀⢀⡴⠛⠛⠛⢳⠖⠒⣦⡄⠈⣧⠀⠀⠀⢀⣊⠎
⠀⠀⠀⢸⡇⠀⡠⠤⠜⠀⣸⠉⠈⢢⡘⡆⠀⢠⡞⠁
⠀⠀⣴⠋⠀⠸⣁⡴⠶⠶⠃⠀⠀⢀⣧⠃
⠀⠀⣯⠀⠀⢀⣼⡇⢀⣬⡃⠀⢨⣼⡻⣤⡀
⠀⠀⠘⠦⡀⠘⣆⡀⠀⠀⠀⠀⠀⢀⣇⣼⠇
⠀⠀⠀⠀⢸⠃⣸⣷⠀⠀⠠⣤⠆⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠈⠙⢻⣿⠀⠀⠀⠀⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠌
⠀⠀⠀⠀⠀⢠⡂⡀⠉⠐⠢⣀⣑⣄⣀⠀⠀⠀⠀⠀⡀⠘⠁⠘⡄
⠀⠀⢀⡠⠖⠁⠀⠀⠐⠴⠋⠈⢇⠀⠀⣃⠀⠀⠀⢀⠀⠀⠀⠀⢱
⠀⠀⠘⠁⠈⠁⠐⠠⡀⠀⠀⠀⠀⠀⡀⠈⠀⡀⠀⢨⠀⠀⠀⠀⠂
⠀⠀⢰⡀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠈⡆⠀⢀⡀⠂
⠀⠀⠀⠧⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠃⠀⠊⠀⠀⠀⠁
⠀⠀⠀⠸⡘⠄⠀⠀⠀⠀⠀⢠⣄⢀⡞⠉⢂⠀⠀⠠⠁
⠀⠀⠀⠀⠱⡈⠢⡀⠀⠀⠀⠨⡇⠀⠈⠆⢈⢦⠄
⠀⠀⠀⠀⠀⠐⡀⠈⠢⡀⠀⠀⠁⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠑⠄⢀⣀⠠⠐⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀

⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠄⠀⠐⠒⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠶⠶⠶⠞⠻⠛⠛⠛⡏
⠀⠀⠀⠀⠀⠈⠓⠲⣤⣀⠀⠀⠀⠀⠀⠀⠘⠲⣶⣿⠿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠙⠲⠦⣄⡀⠀⠀⠘⣿⠄⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⣶⣶⣒⣺⣿⣯⣽⠇
⠀⠀⠀⠀⠀⠈⠁⠢⠑⠢⢄⡀⠀⠀⠉⠙⠲⠤⣿⣿⣶⣶⣖⣺⣭⡭⠷⠶⠚⠛⠋⠉⠉⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⣑⠢⢄⡀⠀⠀⠀⠈⣿⡯⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⡴⣖⣿⠂
⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠈⠑⠢⣌⠑⠢⢀⠀⣿⣄⠀⢀⣀⣀⣤⣴⣶⣺⣿⣽⠷⠾⠛⠛⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠉⠒⢤⠈⣙⣻⣭⡽⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠐⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⢠⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⠀⢸⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢸⣷⣆⣀⣠⣤⡤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠗⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⣄⠀⠀⠀⣤⣀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⣶⠀⠘⣿⠀⠀⠀⠈⠻⣦⣠⣾⠏⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢀⣤⡿⠿⣷⣄⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⠃⢿⡆⠀⠀⠀⠀⣽⠀⠀⣿⠀⠰⠾⠟⠉⠀⠀⠈⠙⠻⠶⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣾⣯⣀⣨⣿⡄⠀⠀⠀⢻⡇⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣸⡿⠉⠉⠉⠹⣷⠀⠀⠀⠀⠀⠀⣿⠀⠘⠛⠛⠛⠛⠛⠛⠛⠻⣶⣤⡀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡧⠀⣀⣴⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⢠⣼⡟⠀⠀⢻⣿⣿⣿⣿⣿⡇⣀⠘⣿⣿⣿⣿⣿⣿⣃⣀⢸⣿⣿⣿⣿⣿⡇⣀⠘⣿⣿⣿⣿⣿⣿⢁⡀⣿⣿
⣴⠛⡛⣶⠋⠙⠙⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠛⠛⠛⠛⠛⠙⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠋⠛⣿⣿⣿
⢉⣿⠆⢰⠀⠀⠀⠀⣶⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠘⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠒⣠⠀⠀⠀⢀⣶⣿⡏⠀⠀⠀⠀⣿⣿⣿
⠠⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣿⡗⢀⣼⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿⣿
⡄⠀⠀⢀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡏⠁⠀⠀⠀⠀⠀⣿⣿⣿
⢿⣿⣿⢨⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⣿⣿⣿
⣼⣿⣿⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⣀⣤⣬⡉⠛⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠟⢿⡿⠃⠀⠀⠀⠀⠀⠹⠟⠻⣿⣿⣿⠟⠛⠻⠋⠀⠈⠁⠹⣷⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠆⠀⠀⠹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⠢⢭⣙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣈⠒⠦⣌⡉⠒⠦⣍⡛⠿⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡐⠤⢍⣒⠦⢌⣙⠲⠤⣀⡀⠀⠈⠙⠻⣦⣄⠠⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠒⢬⣝⡲⠬⣉⡒⠤⢌⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⣙⣳⣦⣍⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣛⢻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⠉⠉⠉⠉⠛⠷⢶⣤⣄⣀⠀⠀⢀⣀⣼⠇⢀⣀⣠⣤⣶⠶⠞⠛⠉⠉⠉⠉⠉⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⢶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡟⢧⡛⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠾⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠸⠀⠈⢷⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⠀⠀⢀⣀⡈⣻⣿⣤⣴⣶⣶⣶⠶⠶⠶⠾⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣀⣠⣴⠶⠞⠛⠋⢹⣿⣴⠾⠟⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⠶⠶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣆
⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⣛⣒⣒⣒⣒⣒⡒⠲⠶⠶⠤⠤⠤⠭⠭⠭⠭⠉⠉⠉⠉⢉⣿⡉⠉
⢠⡿⠁⠀⠀⠤⣄⡀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣂⣀⣀⡀⠀⠀⠀⠀⠈⠭⠭⠭⣭⣭⣭⣭⣅⣒⠒⠀⠀⠀⠀⣾⠀⠉⠓⠦⣄⡀
⣿⠁⠀⠀⠀⠀⠀⠈⠉⠂⠀⠘⣷⡀⠀⠀⠀⠐⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⢭⣍⣉⣉⣉⣉⣒⡒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠙⠳⣦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣐⡒⠲⠶⠶⠶⠶⣤⣤⣤⣤⣄⣀⣀⣀⣒⡒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢷⣀⣀⣤⣤⡴⠶⣶⠟⠛
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠷⠶⠶⠶⠾⠿⠿⠿⠿⣿⣿⣽⣿⣭⣭⣤⣀⣀⣀⣀⣀⣀⣐⣒⣀⣀⣀⣙⣏⠁⠀⠀⢰⠇
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠠⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠐⠂⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡴⠶⠶⠶⠚⣛⣋⣉⣥⣤⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣌⣛⣬⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡴⠶⠶⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠉⠢⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⢀⡀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣀⡤⠒⠢⡀⠀⢇⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣀⠜⠀⠀⠀⠠⠽⡀⠘⡄⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢸⡁⠀⠒⠀⡄⠐⠀⡇⠀⡇⠀⠀⠀⠀⡎⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠘⣄⠀⢶⡲⠀⡸⠀⣠⠃⠀⠀⠀⢸⠥⠒⠉⠈⠳⣄⠀⠀⠀⠀⢀⠔⠉⠓⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢆⣀⣀⡼⠑⠒⠒⣏⣀⣄⡧⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠈⠉⠓⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠫⡀⠀⠀⠀⢈⡵⠒⠤⡀⠀⠀⠀⠀⢠⢦⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⢦⠀⠀⠀⡌⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⡠⢪⢗⢝⣷⠀⠀⠀⠀⠀⢣⣀⠞⠓⠤⡎
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢀⣴⠁⠀⠀⢹⠟⠀⠀⢸⠀⠀⢀⠏⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⠜⠁⠀⠉⢲⡎⠁⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢸⠈⠢⢄⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠚⢻⡏⠘⠀⠀⠀⢀⣤⠀⢀⣤⡀⠀⠀⠀⢀⡠⢒⡩⢿⣧⠀⠀⠀⢀⣀⣟⣿⣶⣤⡀⠀⠰⡯⠏⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡤⢺⡹⠃⠀⠀⠃⠙⠱⡉⠛⠛⠖⠛⠁⠻⢿⣿⠋⠯⣽⣿⠿⠿⠟⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⢄
⢀⢀⡀⠀⢀⡤⣀⡤⢚⠅⢘⡻⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢫⢳⡴⡯⠃⠀⠃⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⢰⣦⠀⠻⣟
⠀⠉⠻⢶⡾⣿⡿⠋⠁⡠⡱⠁⠀⠀⠀⠀⠀⠀⠈⠦⠘⠀⠀⠀⠀⠀⠀⠀⠻⣷⠁⠀⠀⠀⠀⢼⣿⠄⠛⡀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠹
⣦⣤⢖⠥⠚⠻⢶⣤⣰⣽⠇⢀⡠⢄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠹⣿⡀⠀⠀⠀⠀⡹⣿⣿⣦⣻⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⡼⢭⣽⡶⠰⠦⡾⡴⠶⠶⠿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠶⢰⡄⠀⠀⠀⠀⢛⠀⣶⡆⠀⠀⠀⠀⠀⣠⢄⣄⣄⠀⠀⠀⠠⠶⠯⠤
⣤⠞⠿⡷⣀⡜⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢪
⠟⠀⠀⠘⢿⡜⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠐⠦⣤⢿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡶⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠈⢡⡇⠀⠀⠀⠀⠸⠄⠛⣿⣦⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠤⠀⠀⠀⠀⠀⠀⠐⠂⠀⢻⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠋⠀⠀⠈⠉⠀⠀⠀⠉⢿⣌⣻⡷⣦⣀⠀⣀⡤⣊⠕⠉⡠⡟⢹⠆⠀⠀⠀⠀⠀⠀⠠⠆⠉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠳⢶⡴⠖⠓⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⢴⣾⡹⢛⣥⣋⠀⠀⣔⡜⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢟⠁⠊⠀⠈⠙⠿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠆⠀⠀⠀⢤⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢤⣆⣶⣦⣠⣀⣀⣀⣜⡍⠙⣿⣦⣄⠀⠀⠀⠀⠀⠔⡆⢀⠄
⠀⠀⠀⠐⠋⣿⠉⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢿⣏⠉⢉⣿⡟⠉⣩⠉⠉⠉⠉⠀⠀⠀⠀⠐⠂⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡤⢶⣿⠋⠘⢙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣷⠏⠟⢹⠟⠈⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⣀
⣦⠀⠀⠀⠀⠀⢀⡠⢶⠋⠘⡿⠁⠀⠀⠘⠇⠀⠀⠀⠀⠐⡽⢧⡴⠻⠀⠀⠀⠹⠏⠀⠈⠀⠀⠀⠀⠀⠀⠋⣴⣆⡤⠀⠀⠀⠀⠀⠀⢻⡟
⢨⠛⠷⢦⣤⢖⡯⠂⠁⡠⣱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠋⠒⠐⠀⠀⠀⠀⢀⡀⢻
⣆⣠⢔⡀⠘⠿⣦⣀⣠⣷⠃⠀⠀⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡤⢻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠞
⢻⣷⡁⠀⠀⠀⠀⣹⠿⢧⣄⠀⠀⢘⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⠀⣤⠀⠀⠀⠀⠀⢀⠰⠆⠉⡎⠁⠀⠀⠀⠀⠴⣀⠀⠀⠀⠀⢀⡠
⡞⣿⣷⡶⢶⣶⣶⣿⡶⠴⠿⠿⢶⡿⣉⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠳⠓⠹⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠀⠀⠠⠴⠯⠤
⠀⠀⣿⢷⣶⣞⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⣀⢀⣀⣤⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠘⠿⣟⡘⠈⠁⠀⠀⠀⠀⣶⠄⠀⠀⠀⠈⠈⣿⡾⣟⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡽⣻⣿⠎⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀
⣤⠄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠙⢿⡝⠻⢦⣄⡀⠀⠀⢀⡤⣚⠝⢈⢿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠉⢛⡶⣞⡽⠋⠁⠀⢦⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⢀⣠⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣿⣶⣞⡑⠊⠙⠻⣖⣤⣿⡋⠀⠀⢀⠀⢈⣙⡀⠀⡀
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡀⠀⠙⠒⠂⠀⠢⢄⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈⢰⢦⣄⢀⠀⡀⠀⠃⠙⠉⠙⠐⠢⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⣄⠂⠀⠀⠀⠀⠀⠀⡇⢤⣀⣀⡈⠈⠠⠀⠄⠀⠸⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠾⣆⠘⣿⠅⠀⠀⢀⠀⠀⠂⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣀⠀⠀⠀⠀⠄
⠀⠀⠀⠜⣰⠂⢐⡥⠄⡀⣬⠀⢠⠂⡄⡀⢠⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⠁⠀⠀⡌
⠀⠀⠀⠀⢪⣇⡀⠉⠀⠀⠀⠈⠀⠞⠋⠛⠻⠂⠐⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠑⠦⠌⠃⠤⢀⠀⠀⠀⠀⠄⠈⢀⠀⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠐⠁⠢⠴⠀⣸⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢠⢀⡄⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⣂⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠘⠟⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡄⠱⡈⠓⠀⠀⢰⣀
⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠤⢀⠙⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⠿⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣝⢉⣭⣥⣤⣶⣶⣿⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠉⠙⠃⠙⠿⣿⣿⡿⠿⠟⠻⣿⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢸⡏⢠⣶⡿⠷⣶⣄⠀⠀⣤⣶⣟⠛⠦⡀⢹⡿⣀⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⢯⠈⠛⠻⢿⠀⠀⠛⠟⠓⡶⠄⠀⣸⡿⢿⣿⣦⣩⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⢺⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⢦⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠙⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠛⠿⢿⡿⠋⠉⡁⠀⠈⠂⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠓⠒⠛⠀⠀⠀⠀⠀⡼⠁⠀⠠⢤⡤⠾⠋⢀⡀⢳⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠒⠲⠦⠤⠤⢤⣀⣀⢶⣶⣧⣀⣴⣃⣀⡀⠀⠀⠁⠷⠄⠙⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠈⠉⠑⢲⣶⣿⣿⣵⣾⢿⣿⣿⣿⣿⣯⡴⠊⠉⢒⡄⠀⠀⠀⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⡀⠀⠀⠀⢸⣿⢿⣏⣿⣿⠿⣿⣿⡿⠋⠈⠳⡄⠈⣽⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⡟⢿⣷⣦⣤⣤⣀⣤⣿⣿⣿⣿⠷⠘⠹⡀⠀⠀⠀⠹⡄⢛⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠐⡄⠃⠈⠉⠙⠛⠛⠉⠉⠉⠁⠛⠚⡿⠀⠀⠙⠦⢀⡀⠀⠙⢾⠄⠀⠀⠀⠀⡼
⠀⠀⢀⣀⣠⠤⠒⠋⠁⢿⡀⠁⠀⠐⠠⣤⡀⠀⠀⠀⠀⠀⡀⠼⣷⢆⠀⠀⠀⠀⠀⠉⠀⠈⢧⠀⠀⠀⢀⣇
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢈⣦⢤⣴⣾⣇⠀⠀⠹⣻⡆⠀⠀⠀⠀⠀⠀⣠⣚⣦⠶⠒⠉⠈⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⣼⣿⣿⣿⣿⠟⣆⢀⡀⠙⣿⣧⠀⠀⠀⠀⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠃⠙⣿⣿⡏⠀⠈⠉⠀⠀⠘⢿⣷⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠛⠉⠉⠉⠉⠉⠉⠙⠛⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⢸⣿⣧⣿⣿⣷⣿⣷⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠈⠇⠀⠸⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣼⣿⡆⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡆⠀⢰⠀⠀⠙⠁⠙⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠇⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⠀⠀⢀⣠⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⠀⠀⠘⢿⣿⣦⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⡇⠀⠀⢸⣿⣿⣿⣿⣄⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⣿⡇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⠃⠀⠀⠈⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⢻⣿⣄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⡀⠀⠀⠀⠉⠛⠛⠛⠉⠀⣀⣤⣾⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣶⣦⣤⣴⣶⣶⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⡠⠂⢀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠄⠀⠄⠁⠀⠀⠀⠠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⣽⣿⢿⣿⡟⣿⣿⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠨⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣧⣿⣿⡧⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⣿⣿⣿⣿⣮⡻⣿⣾⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡟
⠀⠀⠈⡀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠐⠀⠀⢄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠁⡀⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠢⡀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠀⠀⠀⠤⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠋⠉⠉⢯⠉⠉⠉⠙⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⢷⡄⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠆⠀⠀⠳⡄⠀⢻⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠁⠀⠹⡆⠀⢻⠀⠈⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠇⠀⣼⠀⢀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⣴⠃⠀⣼⠃⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⡞⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⣤⣀⣀⣀⣀⣤⠤⠖⠛⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡀⠀⣰⠃⠘⣧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡉⠛⠃⠀⠀⠘⠛⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⣤⡀⠸⡷⠶⠶⠶⠶⠶⠾⠶⢶⣶⠶⠀⠘⡇⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢿⣆⠀⠀⠘⣷⠲⠦⢾⠉⠓⢦⣄⣿⠀⠀⢸⡇⠀⠀⢸⣀⡴⠞⠉⢿⠴⠶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣳⡄⠀⠀⠀⠀⠈⠉⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⢀⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⣀⠀⠀⠀⠒⠦⣄⡀⠀⢸⡇⠀⢀⣠⡴⠖⠀⠀⠀⣀⡽⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠙⠳⢼⣧⠶⠋⠁⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢛⡛⠛⠛⠛⢻⡟⠛⡟⠛⠛⠛⢛⣛⠛⢦⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠓⠆⠀⢸⡇⠀⣟⠀⠰⠞⠋⠁⠀⠸⠶⢲⡟⢀⡀⢸⡟⠛⠛⠓⠲⠦⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠁⠀⠙⠛⠛⠛⠛⠛⠛⢛⡿⠋⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠒⠒⠒⠒⣒⠒⢲⡆⠀⡶⠒⢒⡒⠒⠒⠒⢿⠁⠀⠀⢸⠇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠠⠖⠛⠁⠀⢸⡇⠀⡇⠀⠈⠙⠳⠦⠀⣸⠀⠀⠀⢼⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠑⠀⠀⠒⠂⠂⠀⠂⠀⠒⠂⠐⠀⠒⠀⠀⠐⠀⠀⠒⠀⠂⠐⠒⠂⠐⠒⠂⠂⠒⠐⠂⠐⠚⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠠⠠⠠⠄⡤⠀⣀⠠⠀⢄⠀⠠⡀⠠⠀⠄⠀⠄⠀⢄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣤⣤⣼⣷⣤⣤⣤⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⡄⠀⣾⠇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⣻⣾⡋⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣴⠾⠋⠉⠻⠷⡦⠀⠀⠀⣠⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⢿⣇⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠉⠉⣿⢀⣴⡟⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠁⠀⠀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣾⠁⢻⡆⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣾⠧⠤⠤⣿⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣼⠇⠀⠀⠀⠈⢿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷⣄⠀⣿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠒⠒⠦⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡄⠀⠀⠩⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣲⠟⢻⡆⠀⡴⠶⠶⡷⠶⠶⠖⠋⠁⠸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠋⠀⠀⠻⠀⡇⠀⠀⡇⢀⣀⣰⣄⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢀⣤⠖⠛⠛⠛⡟⠛⠛⠁⠀⠸⣄⣰⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠛⠙⠒⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠫⣄⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡞⡨⡽⡸⢹⠸⡅⡎⡬⡇⡏⡮⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠁⠀⠉⠁⠁⠉⠁⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⡿⠋⢄⡈⠑⠤⣀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⡫⢈⠔⠁⠀⠀⠀⠀⠉⠒⠄⡀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡫⠊⠋⠀⠤⡁⠀⢰⠀⠀⠀⠀⡀⠀⠀⢈⠑⠢⣄⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⣠⡶⠶⣬⡑⠢⢄⠀⠠⠀⠀⠀⠄⠀⠈⡀⠀⠁⠀⢄⡀⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠋⠀⡒⣾⠛⠀⠀⠀⠍⠓⢦⣉⠢⢄⠐⠂⠀⠀⠊⠀⢀⠔⠠⠀⠈⠒⠤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⣠⡾⠋⠀⠀⠈⠑⢢⠀⠐⠤⠤⠌⠛⠦⣍⠀⢀⡐⠠⡴⡡⠊⠤⠀⠀⠐⠀⠈⣁⠀⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠋⠉⢀⣾⠏⠀⠀⠀⠰⣶⣤⠀⠀⠀⠠⠈⠁⠀⠀⠈⠑⠦⣉⠒⢄⡀⠀⠀⠀⠁⠠⢀⠞⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⣪⠆⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⠀⠀⠀⠈⠓⠦⡈⠑⠤⡀⢂⠔⢁⣼⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⠞⠁⣰⠟⡅⡆⠀⠠⠂⢀⡆⠐⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢈⣴⡿⣫⢞⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⣠⠾⣱⠐⠿⢀⠸⣄⠒⠺⣵⣮⠀⠾⠂⡨⠀⢄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣴⡿⢋⢜⣵⣿⠿⣷⡿⠂
⠀⠀⠀⠀⠀⢀⠞⣹⠞⠁⣠⠞⠁⠀⠛⠃⢠⣦⠠⡌⠁⡠⢄⢀⠅⣠⠘⡀⡠⠶⢆⣀⡘⠀⢠⠀⣀⢀⣤⠞⢁⣴⡯⢋⢔⣵⣿⣯⣜⣷⣾⠁
⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠈⠻⢦⣀⠠⢶⡂⠷⠖⢘⡏⢁⡤⠠⢤⢠⡶⡎⠀⠀⠘⠁⠈⠁⢻⢔⣵⠟⢁⡴⡯⢋⢔⣵⣿⡻⢝⢷⣽⡿⣻⡀
⠀⠀⠀⠀⠀⠈⣿⢨⡙⠳⣦⣄⠢⢝⠳⣤⡀⠘⠉⢻⡁⢞⣻⠁⠄⢿⡉⣳⠈⠑⠠⡄⢌⡴⣵⠟⢁⣴⠏⢀⢔⣵⣿⠻⢷⣌⣳⣿⠏⢈⣼⣿⠆
⠀⠀⠀⠀⠀⠀⣿⡀⢌⡑⢤⡙⠻⣦⣍⠪⢕⠒⠘⠟⠙⣼⡧⢂⠄⡼⢄⠢⠁⠐⠈⠀⣡⠞⢁⣴⢟⡵⠀⣰⣿⠿⣗⢍⣢⣽⣿⢇⢔⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣶⣌⡓⢌⡒⢄⡙⠳⣦⣉⠲⢄⡐⠋⡣⠉⠋⠃⢌⡗⠀⠀⢠⡞⠁⣰⣿⡿⠋⣠⣾⣿⣭⣿⢾⣿⣿⠟⠕⣱⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣌⡓⢬⡂⢌⠉⠙⠲⣍⡢⠀⠺⢦⣀⠀⠀⠀⡴⠋⣠⡾⠿⠉⣰⡾⠟⡧⣊⢝⣷⣿⡟⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣥⣬⣙⠻⣿⣿⣿⣿⣦⠈⠃⠉⣀⠈⠒⠝⣢⣌⠓⢍⡳⠆⠈⣠⡾⣫⠄⣠⣾⣟⡒⢕⡪⣠⣾⡻⢋⢤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⢸⣏⣉⠻⢶⣽⣢⣭⣻⢿⣦⣄⡈⡑⠢⣐⠬⡙⠻⣦⣌⣠⡾⣫⠞⣡⣾⠟⢓⠬⡻⣵⣾⢿⢟⢔⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠸⣏⠠⣉⠢⢌⡙⠿⣯⣻⢮⣝⠻⣿⣗⠤⡉⠢⢕⠀⡽⢋⠜⣡⣾⡟⢁⠪⣑⢉⣾⠟⠕⡱⣫⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣍⠢⢍⡢⢌⠙⠳⣮⣟⡒⠿⢿⣮⣕⠢⠀⡷⣡⣾⠟⡁⢌⡒⠙⣢⣿⠋⠀⣪⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣍⠒⢭⡒⢄⡁⠉⠻⢷⣮⢝⠿⣶⣾⠟⢛⣶⣮⣕⣭⣾⡿⠋⢜⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⡈⠑⠉⢀⠀⠑⠪⢿⣿⣦⢤⡲⣿⣿⣿⣿⣿⠏⡰⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⢉⠒⢤⡒⢬⡙⠿⣯⣄⣠⣿⠟⡡⣪⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⡢⣌⡑⠪⣕⠄⣽⠿⠃⢔⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣭⡓⠤⡁⡇⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡥⠴⠤⢤⣠⣀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⢉⡼⠟⠉⠀⠀⠀⣀⠀⠀⠈⠉⠲⢗⡀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠋⠄⠀⠐⠊⠉⠀⠀⠀⠈⠂⢄⠀⠈⠹⣆⠀⢀⠀⠀⠀⠀⢀⡀⠀⠤⠠⠄
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠒⠂⠀⢴⡞⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠞⠒⠀⠉⠁⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠖⠀⠀⢸⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⣀⡟⠀⠀⡇⠀⠀⠀⠀⠀⢂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⢧⠀⠀⠇⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⠀⠀⢸⡆⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠌⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠤⠄⠀⠀⠤⠄⠀⠀⠄⠠⣾⣦⡀⠈⠀⠀⠁⠀⠀⠀⠀⠀⢀⠄⠂⠀⣠⠞⢠⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠛⢦⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡼⠁⢀⠀⠑⠄⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠁⠀⠀⣈⣽⠶⠲⡠⠀⠴⡴⠶⠓⠤⣀⠀⠀⠈⠀⠀⠂⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠈⠀⠀⠰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠖⠒⠲⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⢄⡀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡇⣞⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⣠⡤⠶⠛⠁⠀⠀⠀⠈⢇⠈⢧⡙⠆⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡁⠀⣗⡒⠒⠒⠂⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠈⠉⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⡀
⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠦⣤⣀⣀⣤⠤⠦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡏⠉⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠈⠉⢹⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣀⣀⣤⣤⣀⢸⣇⣀⣤⣤⣀⣀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠛⠉⠁⠀⠀⠉⠛⠛⠉⠀⠀⠈⠉⠛⠛⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸
⠀⠀⠀⠐⠋⠁⠀⠀⠐⠒⠂⠤⠄⠀⠨⠥⠄⠀⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢧⠀⠀⠀⢸
⠀⠀⠀⠐⠂⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⢆⠆⠠⡁⠀⠀⠀⠀⠀⣀⣀⢠⠤⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠔⠚⠁⢀⠉⠁⠁⠀⠀⡀⠈⠀⠀⠢⡀⠀⠀⠀⠀⠂⠒⠘⠑⡄⠂⠀⠀⢸
⠀⠀⠀⠈⠀⠈⠤⠀⠀⡀⠀⠀⠀⠈⠉⠉⠉⠁⠀⢈⡀⢠⢰⠀⠤⠀⠐⠒⠒⠒⠒⢣⡅⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠰⡀⠈⠁⠀⠀⠀⠀⠀⠛⠂⠰⠦⠠⠀⠀⠰⡀⠈⣀⠀⠀⠈⠄⠀⠂⠸⠀⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠃⠀⠤⢀⠀⠀⠀⢀⡀⣀⣀⠀⡄⠈⠄⣀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠔⠄⠀⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠘⢣⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠒⠒⠒⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡀⡰⠉⠉⢢⣠⢤⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⣄⠀⠀⠀⠀⠙⢦⡎⠁⠈⡇⠀⠀⢸⠀⠀⢸⢀⣀⡀⠀⠀⠈⠳⡀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⠋⠈⠳⣄⠀⠀⠀⠀⠙⢦⡀⡇⠀⠀⢸⠀⠀⢸⠋⠀⠹⡄⠀⠀⠀⠙⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⠏⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⢧⡀⠀⢸⠀⠀⢸⠀⠀⢨⡅⠀⠀⠀⠀⠸⡀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡘⠀⠀⠀⠀⠀⣀⣈⠳⣄⠀⠀⠀⠀⠙⢦⣸⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⢃⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡎⠀⠈⢣⠈⡷⣄⠀⠀⠀⠀⠙⢦⡀⠈⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢱⠀⠀⠈⢦⡇⠈⠳⣄⠀⠀⠀⠀⠙⢦⡀⠀⢘⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢣⠀⠀⠈⠃⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⢦⡰⡇⠀⠀⠀⠀⠀⡌⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⡆⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢰⠁⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⢦⡀⣠⠃⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⣄⠀⠀⠀⠀⠙⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠤⠤⠒⠉⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠒⠂⠉⠁⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⠤⠤⠤⠤⠤⠔⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠀⠠⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶⠦⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠐⠒⠦⠤⢍⣁⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠬⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠋⠉⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠲⠤⢤⣤⣠⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀
⠀⠀⠀⠀⢀⡀⠨⠶⠒⠶⠶⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠉⠀⠀⠠⠤⢀
⠀⠀⠀⠀⠀⢻⠓⠒⠤⢤⣀⡀⠀⠀⠈⠉⠙⠓⠲⠶⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⠀⠈⠉⠻⣿⣆⣠⣤⣶⡾⠿⠿⠓⠒⠊⠉⠁⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠻⠶⢤⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⣤⣤⣀⣀⡀⠹⡆⣀⣤⣴⢾⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢦⣤⣤⣤⠴⠶⠒⠚⠋⠉⠉⠀⠀⠉⠉⠛⠛⠛⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⢶⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⣮⣄⣀
⠀⢠⣶⣖⣋⣭⣉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠐⠒⠊⢉⣁⣠⣬⡵⣿⡞⠉
⢀⣿⠏⠀⠉⠉⠛⠻⠶⣤⣍⣛⡳⢦⣤⣀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠖⠒⠚⣉⣉⣡⣤⠤⠴⠖⠒⠋⠉⠉⠉⠁⠀⠀⣾⠁
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣶⣤⣀⡜⢁⣆⣠⣤⣤⣶⠶⠾⠛⢛⣋⣉⣭⣥⣶⡶⠶⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣯⣥⣶⣶⠾⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣐⣈⣧⣤
⠀⠀⠙⠛⠿⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⢷⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡾⠶⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⢶⣤⣀⣀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠶⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣮⣤⡶⠶⠟⠛⠋⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠀⠄⠀⠀⠀⠠⠀⠄⠄⠀⠀⠀⠀⠀⠐⠠⠂⠀⠆⠄⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⣐⣒⡒⣨⠽⠿⠯⠍⡩⣭⣭⡥⣴⣒⣒⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⢫⣵⣶⡛⡉⡝⠘⡇⢠⠠⡢⠂⢀⢸⡇⡔⢝⢋⡴⢛⡥⢚⡺⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⣖⣤⣄⡀⠉⠙⠻⠶⣭⡇⢸⠀⡵⡡⢸⢸⣞⡹⢟⡥⢚⡡⠞⠉⢀⣴⣶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣼⣿⣿⣿⡶⠭⠵⠶⠄⠀⠀⠁⢸⠀⠎⡠⢸⠈⠉⠚⠑⠚⢉⠀⠀⠘⢛⣋⣉⣉⣉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠤⠤⠤⠄⠖⠂⠒⠒⠚⠂⠉⠙⠋⢀⠌⡔⡁⠉⠉⡉⠉⢉⢉⠉⡩⢉⢉⠄⡀⡀⠈⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡠⣠⠆⠄⠔⡠⢀⢊⠜⠔⡠⡡⡣⢪⢊⠌⠔⠕⠑⠐⠁⢋⣊⣐⣁⣁⣊⣨⣬⠤⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠀⡡⡡⠀⠏⠉⠉⠈⣤⡤⢄⡠⢖⡶⡶⣶⣶⡆⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣿⣛⢏⡽⠛⠁⠀⣠⢔⢊⣴⣷⡀⡄⢔⠔⠀⡆⢻⡶⣄⡀⠙⠲⢭⡓⠯⣻⠯⢫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡛⠁⠀⣠⠔⢋⣴⡾⠿⣩⢋⡇⡃⣣⠪⠀⢇⢸⡕⡠⡹⠦⣄⠀⠉⠓⡫⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⢲⣈⣭⣭⣭⣭⣯⣏⣶⣖⠦⠤⣉⠢⢝⣒⠛⠺⠿⢼⣼⣥⣾⡇⣆⣘⣁⣁⣸⠘⠮⠾⣴⣿⡿⠅⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣫⠟⠉⠈⠀⠀⠁⠀⠈⠀⠉⠛⢷⡾⢝⠲⣄⠀⠉⠁⢘⠃⠀⢀⠤⠊⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠎⣽⠃⡴⠁⠀⣀⣀⠠⡢⢎⣴⡥⡄⠀⠀⢀⠙⢦⠙⡾⢳⡀⠰⢅⣀⡀⠛⠒⠠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⢰⠃⢀⢼⡮⠕⣩⠖⠏⠀⠜⡻⣦⣧⣑⠅⠀⢣⠘⢶⣇⠀⠀⠀⢈⡩⠃⣀⡥⠊
⠀⠀⠀⠀⠀⠀⠀⡟⡇⠀⣌⠐⣚⡭⠖⠛⠀⠀⠀⠀⠀⠀⠉⠿⣯⣓⠀⢨⡄⠘⣿⡄⠀⠴⣉⡀⠺⠥⣀
⠀⠀⠀⠀⠀⠀⢰⢣⠗⠚⡋⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⢚⠓⠦⡇⣇⠀⠀⣀⠜⢀⣠⠜
⠀⠀⠀⠀⠀⢀⢼⠏⠀⢠⣦⠁⠀⢠⡀⢀⡀⠀⠀⠀⢀⡀⠀⢀⠀⠀⣺⢤⠀⠀⠹⣿⡰⠚⠒⠋⠉
⠀⠀⠀⠀⠀⠘⠻⡄⢄⢸⣿⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣺⣽⠀⣤⢁⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣨⣿⡇⠀⠀⠀⠀⠀⠰⠒⠂⠀⠀⠀⠀⠀⠀⣺⣿⣀⡡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⡜⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠔⠁⠀⢡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠳⡦⣀⠀⠀⠀⠀⣀⣴⣛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⡀⢀⠀⣏⠈⡝⡹⣩⠋⢩⢰⡥⢒⡡⠎⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠅⢊⠡⢒⣿⣷⠤⠤⠄⣁⠤⢺⢖⡡⠔⣉⠤⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⡴⠁⠀⡈⠔⣊⠕⣊⠡⣷⡉⠓⠒⠒⠒⠊⢹⡭⠖⣉⠤⢒⡡⠀⡀⠈⢣
⠀⠀⠀⠀⠀⡜⠀⢀⡩⠐⡩⠔⢊⠤⠒⣿⠠⠀⠀⠀⠀⠀⢸⠔⣊⠥⢒⡡⠔⣊⠤⠀⠀⢧
⠀⠀⠀⠀⠈⠀⠀⠉⠐⠉⠀⠊⠱⠊⠕⠻⠆⠀⠀⠀⠀⠀⠰⢊⡤⠒⣉⠴⢊⡡⢐⡭⣂⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠁⠀⠁⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠉⢁⣰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠛⡙⢿⣷⣄⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣷⣦⠾⠿⠿⠿⠿⣿⣿⣿⣿⣿⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡤⠀⠀⠀⠀⣠⣤⠀⠀⠈⠀⠀⠀⠠⠂⢠⣀⣸⣶⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡶⢚⣱⣿⠄⠀⠀⠀⡀⠀⠀⠀⣶⣺⠟⠻⠿⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡔⢡⣴⣿⠋⢀⠀⠄⠀⣀⣤⣴⣶⣷⣿⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢉⣾⡿⢻⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢶⠍⣡⣿⣿⠶⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠆⢰⣄⡠⣿⡿⣿⣶⣺⣿⣿⣯⣻⣿⡿⢷⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⢃⣰⣿⣷⠿⣿⡗⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣀⣀⣀⣀⣀⣀⣀⣈⣿⣋⣛⣐⣻⣳⣺⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣛⣛⣹⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠠⠤⠄⠶⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠋⣁⣀⠀⠀⠀⠀⠀⠈⠐⠬⡱⣤⡄
⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠒⠒⠶⣄⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣊⡄⠰⠉⠀⠈⠲⠒⠉⢢⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠛⣆⠀⠀⠀⠀⠀⠀⠉⠟⠒⡃⢐⡋⠃⠀⢶⡄⠀⠑⠤⠀⠀⠀⠐⡖⠳
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢀⢄⡴⡿⠁⠙⠐⠓⡆⠁⠀⠀⡇⠀⠀⠀⢠⢼⠄
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢦⠀⠀⠀⠀⣸⠞⠀⠧⣀⠠⠀⠀⡎⠀⠀⠀⢱⠀⠀⠀⡸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡏⠰⡄⠀⠯⠁⠀⠈⠀⠀⣀⠀⣎⠀⢉⡉⢣
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣓⣶⠆⠀⠹⣄⠈⠓⠦⢤⣤⣴⣶⠋⠀⠀⣀⠼⢤⡤⠞
⠀⠀⠙⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⠈⢳⣄⡀⠀⠙⠚⣁⡤⢴⡛⠁
⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⣠⠐⠊⢡⠋⠛⠙⠛⠉⠁⠀⠀⣈⣹⡶⡄
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⢛⡿⠉⠀⠠⡄⠒⠉⠉⣆⣰⠿⠋⠉⠓⢢⠀⠀⠀⣜⠟⢄⠈⠓⠤⣀
⠀⠀⠀⠀⠈⠙⠶⢤⣤⣤⡴⠞⠁⠙⠒⠶⠚⠉⠀⠀⠀⠀⠘⣄⠀⣆⡼⢃⣉⣭⢒⠦⠇⠀⠠⢼⣸⣀⢤⡷⣄⠀⠀⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡼⠁⠀⠀⠈⠑⠀⠀⢀⠔⠺⣍⢛⡩⠒⠁⠁⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⠁⠀⠀⠀⠀⠀⠀⢠⠜⠂⠀⠉⠁⠀⣀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣸⣦⣀⠀⠀⠀⠀⠀⠓⣤⠦⡀⠀⠉⡧⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⡁⠀⠈⠉⠉⡉⠉⠉⠉⠁⠀⠈⠓⠲⠶⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣤⡀⠀⡼⠻⡀⠀⢀⣀⡤⠤⣖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠒⢿⠛⠁⠀⠙⠚⠉⠀⠙⣄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢶⣒⣻⣿⡍⢉⣦⠀⠀⠀⠀⠀⠀⠀⠈⢿⣉⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠦⣀⣀⣈⣦⢽⡪⠽⠋⠀⠀⠀⠀⠀⠀⠀⣠⠋⢤⣄⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠬⠍⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠉⠉⠉⠓⠂⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⣁⣒⣒⣒⣩⠜
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣈⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣇⠀⣤⣠⣿⣿⣆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣇⠛⠛⠛⠛⠛⠘⠛⠛⠃⠀⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣤⣬⣷⣦⣀⢸⣿⣿⢸⣿⣿⢀⣤⣾⣧⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⡿⢿⣿⣿⣿⢸⣿⣿⡿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣮⣝⠻⠸⣛⣭⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣤⣤⣤⢠⣄⢠⣤⣤⡤⢤⡙⣿⣿⣿⣿⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠮⠽⠿⢸⣿⠸⠿⠥⠶⠿⠧⠭⠉⠀⢸⣿⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠟⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⢟⣫⣽⣿⢸⣿⢸⣿⣭⣛⢿⡇⠀⠀⠀⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠈⠁⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣷⣤⡀
⠀⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄
⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡍⠉⢨⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣷⣦⣄
⠀⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⡄
⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣶⣶⣶⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠻⢿⣷⣶⣤⣀⠀⢸⣿⡇⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⡇⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠸⠿⠿⠇⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠙⠿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢀⣀⣤⣶⣿⡿⠟⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⣋⣭⡤⠴⠶⠦⠤⣬⣍⡛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⠛⠛⠛⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⣴⠟⣡⠶⠛⠉⠀⠀⣠⠶⣄⠀⠀⠈⠙⠳⣤⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⢠⠤⣄⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⢠⡞⣁⠺⣧⡀⠀⣀⡞⠛⡇⠀⣿⠶⣦⠀⠀⠀⠈⠻⣦⠹⣆⠀⠀⠀⠀⠀⣰⣿⣿⣄⠀⠀⠀⡔⠁⠇⠀⢸⠔⢤⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⢠⡟⣰⠏⠛⢦⣙⣿⢹⡇⠀⡇⠀⣿⠀⢸⠀⠀⠀⠀⠀⠘⣧⠹⣆⠀⠀⠀⣰⡿⠁⠈⠻⣷⡞⠉⡇⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⣾⢠⡏⠀⠀⠀⠙⣿⢸⡇⠀⡇⠀⣿⠀⢸⠀⣠⡴⢤⡀⠀⠘⡇⢹⡀⠀⢠⣿⠃⠀⠀⠀⠀⢱⠀⡇⠀⠀⠀⢸⠀⢸⠀⢀⠤⢤⡀⠀⠀⢻⣷
⠀⠀⡇⢸⠃⠀⠀⠀⠀⣿⢸⣇⣠⣧⣤⣿⡀⢺⣴⠏⠀⣼⠁⠀⠀⣿⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⢸⠀⡇⣀⡤⠀⢼⡀⢰⣠⠊⠀⡸⠁⠀⠀⢸⣿⡀
⠀⠀⡇⢸⡄⠀⠀⠀⠀⣿⠘⠋⠁⠀⠀⠀⢀⠈⠁⠀⣸⠃⠀⠀⠀⣿⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⢸⠐⠉⠀⠀⠀⠀⠀⠈⠁⠀⢠⠃⠀⠀⠀⢸⣿⠁
⠀⠀⢿⠘⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠋⠀⠀⣰⣯⡀⠀⠀⢠⡇⣸⠁⠀⠘⣿⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡔⠉⠀⠀⢠⣇⠀⠀⠀⠀⣼⡿
⠀⠀⠘⣧⠹⣆⠀⠀⠀⢹⡄⠀⠀⠀⣾⠀⠀⠀⢠⠿⢮⡙⢦⣤⡞⣰⠏⠀⠀⠀⢹⣷⡀⠀⠀⠀⠘⡀⠀⠀⠀⡘⠀⠀⠀⢀⠞⠻⣷⣤⡀⣰⣿⠃
⠀⠀⠀⠘⢧⡙⢧⡀⠀⠈⢷⣀⠀⠀⠃⠀⢀⣴⠏⠀⠀⢙⣷⠈⣰⠏⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠣⡀⠀⠀⠁⠀⠀⡠⠊⠀⠀⠀⢙⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣦⡙⠶⣄⣀⠉⠛⠛⠚⠛⠋⠁⢀⣠⡴⢛⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠈⠁⠒⠒⠒⠉⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣍⣛⠓⠲⠶⠶⠒⢛⣋⣥⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣤⣤⣤⣤⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠠⣶⠛⠷⠶⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⢸⡇
⠀⠀⠀⠀⠀⣀⣴⡯⠂⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠛⠒⠒⠛⠛⠛⠛⠛⠛⠲⠶⠶⠶⠶⠶⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⡄⣸⠃
⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣆⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⣼⡴⠋
⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡤⡄⠀⠀⢀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠋⠀⠀⠀⠀⠀⡜⣾⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠰⠀⡇⠀⠈⠀⢸⠀⡰⠁⢀⢄⣈⠆⠀⢀⠞⠀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⣷⠏
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⠀⡄⠈⠀⠰⡀⠱⡀⠀⠀⠀⡌⢰⠋⡆⢣⠀⠀⠖⠒⢤⡠⢄⠀⠀⠀⠀⠀⣿⠀⠈⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠇⢀⠀⠀⠙⢦⡈⢦⠀⢠⠀⠈⣒⠃⠘⡀⠀⢣⡀⠈⠁⡸⠀⠀⠀⠀⠀⠅⠀⢰⡏
⠀⠀⠀⠀⠀⢻⡆⣄⠀⠀⠀⠀⠀⠀⢡⠀⠢⠤⠎⠀⢸⠀⠀⠀⠀⡡⠈⡇⡸⠀⡜⠉⡆⠀⢇⠀⠀⠑⢤⠜⠀⠀⠀⠀⠀⠀⣠⣲⠏
⠀⠀⠀⠀⠀⠀⠹⣮⡦⡀⠀⠀⠀⠀⠈⢆⡀⠀⠀⡠⠃⢸⠉⠒⠔⠁⣰⢁⡃⢠⠃⠀⢱⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣅⣄⡀⠀⠀⠀⠉⠁⠈⠀⠀⠈⠳⠤⠴⠚⠁⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⠉⠛⠛⠲⠶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⢀⡀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠉⠈⢹⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠿⠿⢶⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠋⠉⠀⠈⠙⠓⠛⢿⣦⡀⡘⣧
⠀⠀⠀⠀⠀⠀⠀⢾⡁⠀⠠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣦⣄⠀⠀⣴⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣠⠟
⠀⠀⠀⠀⠀⠀⢠⡾⢁⣀⡀⠻⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡠⠿⣇
⠀⠀⠀⠀⠀⠀⠙⠛⠋⠉⠙⠛⠲⠾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣶⡄
⠀⠀⢀⣼⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦
⠀⢀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡀
⢀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷
⢸⣿⣿⡿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠿⠟⠋⠀⢀⣀⣀⣠⣤⣤⣄⣀⣀⡀
⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣦⡀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣄⣀⣾⣿⣿⠏⠀⠀⠀⢀⣴⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣦⡀
⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡟⠁⠀⠀⠀⢠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄
⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⢿⣿⣿⣷⡦⠀⢰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣆
⠀⠈⢿⣿⣿⣷⡀⠀⠀⠀⠘⢿⡿⠿⠛⠉⠀⠀⠈⠙⠻⠃⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣏⣸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃⢀⣴⣿⣿⣷⣶⣶⣶⣶⣾⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢠⣶⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⠿⠿⠿⣿⣿⠿⠿⠇⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠉⠉⠉⠉⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣄⠀⠀⢻⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠻⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢈⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠑⠀⠁⠈⠒⠀⠘⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠋⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠸⠄⠀⢨⠀⢡⠄
⠀⠀⠄⠁⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⡀⢰⢂⡅⠒⠑⠂⠈⢺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀
⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⡦⠏⠀⢀⣀⠀⠄⠰⣇⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠁⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠛⢇⣀⠀⠉⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⢙⣖⣲⡂⡠⠨⠀⠀⠀⠀⠀⠁⠉⡇⠀⠀⠀⠊⠀⠀⠉⢶⡤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡀⠐⠠⠀⠀⠐⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠄
⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠉⠈⠀⠙⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⡄⠈⠙⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢠⣄⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡄⠀⠐⠂⠀⠀⠀⠰⣶⡤⠀⠀⠀⠀⠀⠑⠀⠠⠀⠛⠂⡀⠀⠀⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠄⠀⠀⢰⠀⠀⠀⠀⠀⠁⠀⣤⣀⣤⠀⠀⠀⡀
⠀⠀⠀⠠⠚⠆⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠉⠲⠤
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⡀⢀⡀
⠀⠀⠰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⢄⠆⠂⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢷⠦⣀⠀⡠⢚⠋⠀⠀⠀⠈⠀⠀⠀⠀⠤⠤
⠀⢀⡀⠂⠀⠀⠀⠈⡔⠂⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣷⡞⠙⢦⣎⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠐⠆
⠀⠘⠱⣀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⠀⠀⠀⠀⠀⠉⠉⢻⣹⠋⠉⠉⠀⠀⠀⡄⠚⠀⠀⠀⠄⠀⠤
⠀⠀⠠⢁⠀⠀⠛⡆⠁⠀⠀⠤⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⢰⡇⠠⠀⠀⠀⢀⠀⠀⠀⠁⠀⢀
⠀⠀⢀⡸⠃⠀⠠⠀⠀⠈⠀⠂⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠀⠀⠀⠀⢈⠆⠀⢛⠀⠀⠀⡁⠀⠀⠠⠾⡄⡀⠀⢄⣶⣀⢴
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠘⠅⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⢠⣤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⢸⡇⠀⣸⡿
⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠐⠟⠁⣀⣄
⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠿⠶⠶⠶⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣿⣷⣶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⢿⣇⠀⠀⣠⡀⠀⠀⣠⣴⠾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠈⠻⣶⣴⠿⢁⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⡄⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣠⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣀⣴⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠛⠛⠉⠉⠉⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣾⣿⡿⠟⠉⠉⠙⠛⠻⠿⠿⠷⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣠⣤⣤⣤⣤⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⠉⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⣴⣄⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣷⣤⣀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣷⣶⣶⣶⣤⣶⣶⣶⣾⣿⡿⠿⠛⠉⠛⠿⢿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⢁⠀⣤⠄⠀⡄⡄⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⣠⠬⡅⣶⡆⢰⡄⠀⣸⣇⡿⡇⣿⡆⣀⣏⠧⡄⣄⠷⣷⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⢘⠛⢻⢔⢲⡂⣾⣷⢠⡅⣍⣡⣝⢃⡓⢇⢻⡄⠃⡍⡋⡋⠂⢣⣤⠄⠀⠀⡄⠀⠀⠀⠀⠀⢆⣖⡃
⠀⠀⠀⠀⡞⠓⠒⣦⣤⡖⢁⡌⠙⠀⡆⡇⣾⠾⠂⡁⣁⣀⠀⠂⣪⣇⣀⡄⠽⠏⠀⢻⣥⠁⠀⢀⡀⠐⣈⠙
⠀⠀⠀⠀⢙⣛⣇⠛⠛⣧⡘⠬⠤⢴⠀⠟⢥⠚⣄⡀⣯⡭⢉⣓⡬⠝⠾⠇⣘⣀⠀⢰⣐⣲⣞⠘⠑⠀⠙⡆
⠀⠀⠀⠀⠈⢳⣾⡤⠛⣇⡇⠽⠿⣖⢂⣃⣨⠦⣋⢱⣖⢠⡆⣧⡆⣩⣄⢠⢸⢿⠀⠓⠣⢝⢁⠜⢠⣼⡄⠁
⠀⠀⠀⠀⠀⠀⢸⣷⣂⠈⣼⢷⡐⢓⣤⣭⡝⢘⣩⣊⣉⢺⢹⢋⢃⢣⠞⡼⠲⢀⣸⣘⡰⠀⠐⢰⣮⣿⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⣛⣿⣟⡃⣛⣛⣇⡙⡃⢹⡏⠞⢾⢻⣙⢠⣾⣾⢄⣠⠮⢭⠳⡼⣂⢎⠰⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠭⠍⠑⠤⠆⢸⠇⢶⣰⠢⡟⢦⠮⠤⠮⢻⠇⢢⡇⡝⣥⣆
⠀⠀⠀⠀⠀⠀⠠⠴⣶⣞⣆⠀⠀⠀⠀⠀⠀⠧⢴⣦⠼⢸⣯⠠⣍⡇⢻⣃⣹⡙⠉⠈⠁⠛⠉⠬
⠀⠀⠀⠀⠀⠀⠀⢴⡲⡆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡘⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡝⡀
⠀⠀⠀⠀⠀⠀⠀⠲⠟⠈⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠿⣿⠀⣿⡿⣿⠹⢙⣸⡇⢸⡍⣽⣭⢹⢘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡤⡄⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⢿⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣿⢾⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣴⣿⢝⢿⣦⠀⠀⠀⠀⠀⠀⠈⢿⣿⠏⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⢸⢸⣿⡇⡟⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣹⡇⠀⠀⠀⣿⣿⣿⣿⣿⠇⢻⣿⣦⣌⡿⠀⠀⠀⢠⠿⡄⠀⣿⣉⡷⢠⡞⠉⠳⠀⠀⢸⡟⢸⣿⢼⢸⣿⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠟⢿⡿⠛⠀⠀⠀⠀⠸⣿⣿⠟⠋⠈⠉⠉⠉⠁⠀⠀⠀⠠⡟⠛⢻⠀⣿⣨⡽⠈⢧⣤⡶⠀⠀⢼⡇⠸⠽⠾⠈⠿⠁⠷⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⠶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⡉⠁⠀⡁⠀⠤⠀⠀⢰⠀⠀⠀⠀⠀⠈⠟⣂
⠀⠀⠀⠀⠀⠀⠀⣿⢑⡙⣿⣿⠛⣼⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠈⣤⣌⠂⢐⡤⠂⠁⠀⠀⠀⠀⠀⢠⣌⢛⣁
⠀⠀⠀⠀⠀⠀⠀⠿⣿⠿⠿⠿⢿⣿⠇⠀⠀⠀⠀⠀⠈⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣼⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠶⢤
⠀⠀⠀⠀⠀⠀⠠⣶⡾⠗⠶⢲⠛⡿⠟⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣯⡉⡹⣿⠀⠀⠀⢀⣼⣿⢿⠟⠿⣿⣿⣤⡀⠀⠀⠀⠀⢠⣁⣘⣻⣛⣃⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡉⣫⢭⢩⣉⠅⠀⠀⠀⠀⠰⣋⣉⣉⣉⣹⡿⠛⠛⠛⠁⠀⠀⠀⠈⢹⣷⣶⣶⣶⣿⣿⣿⣷⠆⠀⠀⠀⢿⠏⢤⣖⡶⠌⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡉⠉⢳⡄⠀⠀⠀⠀⠀⠀⣠⣶⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⢨⣿⣧⣼⡿⡇⠀⠀⠀⠀⠀⠀⠀⢠⠎⡀⢀⠀⣈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⠃⠞⢣⡇⠀⠀⠀⠀⠀⣿⡿⣻⣶⡾⢿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠷⢫⡍⢢⠀⠀⠀⠀⠀⠀⢸⡐⠛⠈⠚⠄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠛⠻⣦⠀⠀⠀⠀⠿⢷⣦⣽⣥⡼⠿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠘⠤⠭⠾⣄⠀⠀⠀⠀⠀⠀⠙⣤⣄⡤⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠂⠐⠒⠒⠂⠤⠤⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢔⣾⠯⠷⠦⢰⠋⣤⣔⣒⣀⠀⠀⡀⠄⢀⣀⣀⣀⣀⣠⣤⣤⣔⣲⣬⣷⡿⠟⠃
⠀⠀⠀⠀⠀⢸⣼⠂⠀⠀⠀⠀⠀⠀⠤⠄⣉⡉⠉⠙⠓⠛⢛⡞⠛⢙⠩⠩⡫⡀⠀⠐⡨⡇
⠀⠀⠀⠀⠀⠀⡾⠶⢦⢤⣤⣄⣀⣀⣀⠉⠉⠉⠉⠀⠀⠀⢸⡈⢈⠀⣨⣪⣠⡦⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠁⠉⠉⠙⠙⠒⠒⠛⠲⠲⠶⣿⣿⣿⣛⣛⣛⣭⣉⣓⣦⡀
⠀⠀⠀⠀⠀⠀⠴⠁⢑⣦⣁⣀⣀⣤⣤⣤⣤⣶⡶⡶⠲⡻⡻⣻⠞⠝⠝⡝⠝⠝⢝⢍⠻⣧
⠀⠀⠀⠀⠀⠀⠈⢔⠄⢫⠙⡯⡉⠁⠑⠑⠁⢔⠕⢔⢌⢔⠩⠪⠊⠂⠪⠂⠢⠢⡢⡀⣩⣽⡇⢀⡀
⠀⠀⠀⠀⢀⠠⠔⠚⢮⡄⠀⡏⡨⣠⣢⣢⣢⣠⣠⣡⣵⣰⣵⣵⣵⣵⣵⣴⢷⣶⣾⣾⣯⣿⣴⡿⠃
⠀⠀⣠⣞⣑⣧⣤⣤⢀⡙⠛⡻⠿⠿⠟⠛⠛⢙⣛⠹⠭⣍⡉⠈⠉⠅⠀⠉⠠⠀⢂⣴⡾⠛⢻
⠀⢰⣿⠉⠁⢨⢉⡉⠉⠒⠒⠓⠾⠭⠶⣶⣒⣒⡠⠤⠤⢀⡀⠀⠀⠀⠀⠀⣠⣴⠻⠏⠂⡄⣬⣶⣄⡀
⠀⠘⢿⣄⣀⡳⣾⠀⠠⠤⠀⣈⠀⠰⠂⡀⠀⠀⠉⠉⠛⠒⠒⠿⠽⣦⣶⢿⡏⠀⡁⣀⣶⣾⡛⠁⠀⠀⢉⣴
⠀⠀⠀⠀⢉⣩⣿⣿⠷⠶⢶⣦⣤⣀⣉⢀⡒⠀⠉⠁⠀⠀⠀⠀⠀⡇⠁⠨⠁⣤⣷⠟⢉⡀⢉⣠⣴⠞⡿⠁
⠀⠀⠀⢠⣻⠛⠛⠛⠯⣔⡂⠠⢀⡈⠉⠛⠛⠛⠲⠲⠶⣤⣄⣀⣀⣷⣠⣶⠟⠛⢂⣠⣶⢾⡿⠫⠁⠁⣧
⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠌⡛⠓⠦⢅⣒⠠⠀⡀⠀⠀⠀⠈⠉⠉⠛⢋⡹⣖⡿⠟⢙⠍⡠⠁⠐⢐⣠⣿⠦⠍
⠀⠀⠀⢀⠻⣦⡀⠀⠀⠄⢀⠀⠑⠢⢀⠈⠉⠓⠤⢭⣐⡀⢐⣤⣴⢾⠻⠫⡁⠎⠀⠀⣂⣱⣴⣾⠟⠉⠁⠀⠭⠁
⠀⠀⠀⠀⠀⠐⠙⠶⣦⣀⠈⠀⠒⢤⣀⠀⠁⠀⠀⠀⠀⡝⠉⠩⡢⡁⡁⠀⢌⣂⣶⣻⡽⠚⠓⠂⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠒⠶⢢⣄⡉⠙⠀⠀⠀⠀⡟⡑⠐⠔⣔⣴⣾⠿⠟⠉⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠪⢤⣒⠤⢹⣤⣾⣾⠿⠋⠉⠁⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠁⢀⣀⠀⠈

⠀⠀⠀⠀⣀⣀⠀⣀⣤⣄⠀⠀⠀⠀⣀⣀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⣿⣿⡳⣿⣼⣛⡇⠀⠀⠀⣿⣿⡳⢾⡧⠍⣿⠀⠀⠀⠀⠀⠟⠋⢙⠀⠀⠀⠀⠀⠀⠐⠛⠚⣿⡼⠇⠀⠀⠀⠀⠚⠓⢻⣧⣿⣤⡀
⠀⠀⠀⠀⣯⣾⡇⠙⠛⠛⠁⠀⠀⠀⣯⣾⡇⠈⠙⠋⣉⠀⠀⠀⢀⣠⡤⢈⢁⣤⣄⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠸⢯⡽⠃
⠀⠀⠀⠀⠿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠿⠉⠁⠰⠟⠉⠁⠀⠀⠀⠸⠿⠷⠼⠘⠾⠞⠀⠀⠀⠀⠰⠶⠶⠆⠀⠀⠀⠀⠀⠀⠶⠶⠶

⠀⠀⠀⠀⣠⣤⣤⣤⡤⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣿⣿⣾⠛⠀⠀⠀⠀⠀⠀⢸⣿⣺⡟⢀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠤⠠⠀⠀⠀⠀⢰⣿⡽⡟⣴⢺⢶⡄⠀⠀⠀⠀⠠⢶⣷⠆
⠀⠀⠀⠀⠻⠛⠻⠀⠌⠄⠀⠀⠀⠀⠈⠛⠛⠃⠀⠂⠀⠀⠀⠀⠘⠿⠾⠇⠠⠄⠀⠀⠀⠀⢨⣷⢴⡿⠹⠿⠞⠁⠀⠀⠀⠀⠀⠏⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣶⡶⣶⣶⡆⠀⠀⠀⠀⢠⡖⣦⣰⠶⣆
⠀⠀⠀⠀⣼⡿⢾⢿⡧⣿⡇⠀⠀⠀⠀⣦⣞⠀⣰⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣿⣋⡚⢻⡇⠀⠀⠀⢀⡾⣿⢛⣘⣻⠳⡄
⠀⠀⠀⠀⠹⣿⣻⣽⣟⣿⠇⠀⠀⠀⠈⠁⠈⢡⡿⢿⡆⠀⠀⠀⠀⠐⠃⠀⠘⠛⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠛⠛⢸⣿⠛⠚⠁
⠀⠀⠀⠀⠐⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⢀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣶⡀⠀⠀⠀⠀⢀⣴⣿⠷⣦⠒⠀⠀⠀⠀⠀⣠⠶⣶⢶⣄
⠀⠀⠀⠀⢠⣤⠾⠿⢆⣥⠀⠀⠀⠀⢀⢸⢷⡿⡷⣷⢀⠀⠀⠀⢸⣿⠹⠿⠾⢹⣿⠀⠀⠀⠸⣿⣯⣿⠟⠂⢀⠀⠀⠀⠀⣿⡭⠾⠬⣿⢲⡄
⠀⠀⠀⠀⣼⣿⡆⠀⣴⣿⡆⠀⠀⠀⠀⠘⠯⣯⠯⠋⠀⠀⠀⠀⢸⣿⠂⠀⠀⢺⣿⠀⠀⠀⠀⠈⠛⠃⠠⠡⠀⠀⠀⠀⠀⠛⠳⣶⣞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⣿⢿⠿⠿⡿⠟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣛⣟⣿⣿⣿⠟⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⠛⠛⠛⠛⠁⠈⠉⠙⣿⣿⣧
⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏⠙⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣷⣤⣤⣴⢀⠀⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢿⠁⠹⣿⠀⢈⣉⡀⠀⠉⢈⠹⣿⣿⢟⡅⣿⡟⢸⡆⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠘⠀⠀⠹⠇⢸⣿⣿⠀⠀⣿⡆⢻⠇⣾⣁⡘⢁⣿⡇⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆⠀⠀⣿⣦⣤⣿⣿⣿⡀⠀⣿⣿⣤⣠⣽⣿⣿⣿⣿⣿⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣶⣶⣮⣙⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⠉⠟⠉⡉⠹⣿⣿⣷⡄⢉⣉⣉⣀⢩⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⡆⠸⣿⣿⡇⢸⣿⣿⣿⡄⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢿⣿⣿⡄⢻⣿⠁⡈⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣌⠋⣸⣷⣌⣛⣁⣼⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⠄




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣤⣄⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠛⠿⣿⣦⣄⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣷⣶⣤⡀⠈⠙⣿⣧⡀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⣀⣀⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀⠈⢻⣿⡄⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠋⠉⠙⠛⠻⠿⠿⠛⠉⠉⠉⠛⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⢿⣿⡀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠸⣿⡇⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⢿⠇⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠛⠁⠀⢀⣀⣤⣤⣴⣶⣶⣿⡿⠿⠿⠟⠿⠻⠿⠿⣿⣶⣤⡀⠀⢀⣼⣿
⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣧⣤⣶⣾⣿⡿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢰⣿⣿⣛⠛⠉⠉⠉⠙⠛⠻⠿⢿⣷⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣾⡿⠛⠁⠀⢸⣿⡆
⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⣶⡆⠀⢰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⢰⣿⡇⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⣼⣿⠁⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣷⣶⣶⣶⣶⣶⣶⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⢀⣼⣿⠃⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣿⠿⠛⠁⠀⣠⣾⡿⠁⠀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣠⣴⣿⠿⠋⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⠿⠟⠋⠁⢀⣠⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⠿⠟⠛⠉




⣷⣶⣶⣶⣶⣾⢶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⡿⣶⠶⡶⠶⠶⣶⡾⠷⠶⢷⠶⢶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⢀⠔⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠤⢸⠔⢱⠄⠀⣀⠀⡾⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⡖⢺⠀⢸⠽⢠⡇⡜⢘⢹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠈⣧⠘⣦⠞⠓⠉⠓⠙⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠻⠦⠀⠀⣤⠠⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢂⡄⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶
⠀⠀⠀⠀⠰⣦⡸⣼⠃⡄⣤⣠⠄⡄⢠⡄⢠⢦⡀⠸⡼⣠⡴⠀⠀⠀⢘⡀⠿⣿⡇⠀⠂⠀⠀⠀⡀⠀⠀⠀⠰⣶⡗⠀⠁
⠀⠀⠀⠀⢤⡶⣝⠁⡄⣇⣿⣿⡀⡇⢸⡇⣾⠀⡇⣄⢑⡵⣦⠄⠀⠀⠉⠐⠃⠈⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠻⠀⠸
⠀⠀⠀⠰⠮⡿⡄⢸⠃⣿⣿⣿⠁⣿⢸⡇⢿⠀⡇⢛⠀⣼⡯⠶⠀⠀⠀⠀⠀⠀⠄⠐⠈⢠⠀⡁⠀⠀⠀⠈
⠀⠀⠀⠰⠛⠥⠖⠉⠂⠙⠟⠻⠶⠻⠊⠷⠘⠶⠃⠊⠑⠶⠝⠳⠀⠀⠄⠀⢄⠀⠠⠀⠀⠀⠔⠁⠀⡀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⢹⡇⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠
⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢉⠉⠀⢀⠀⠀⠀⠀⢀⣸⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠐⠀⠙⠈⠀⠀⠀⣆⢰⠀⣶⠀⢠⠀⠀⠀⠀⣼⢀⢰⡏⡇⠘⢠⡏⢸⠒⡀⠠⠄⠀⠄⠀⠀⠀⠀⠀⢸⡧⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠈⠓⣃⣀⠛⠀⠈⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⠛⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡞⣻⣗⢳⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠑⢽⡿⠋⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢠⡜⣇⣠⡴⠀⠀⠀⠀⣤⠀⢀⣤⠐⢠⡄⠂⣠⠂⣤⣶⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⢀⣌⠻⣬⣵⠶⠀⠀⠀⠀⣿⠠⠀⣿⠀⢸⣷⠀⣯⠀⡿⣿⢶⣰⡿⠀⠀⠀⠀⠀⠀⠀⢠⡄⠙⡞⣦⢀⡟⢼⡇⣀⢀
⠀⠀⠀⢀⠻⣆⣀⣴⡶⠀⠀⠀⠀⠿⠀⠂⠛⠂⠈⠿⠀⠈⠈⠁⠛⠀⠙⠃⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠁⠀⠁⠉⣿⠰
⠀⠀⠀⠹⣷⡈⣉⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⣥⡿
⠀⠀⠀⢴⣤⣉⣩⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣀
⠀⠀⠀⠀⠀⢰⠇⡆⡴⣤⢤⢰⢠⡀⣦⡄⢀⠤⠆⢻⡁⢙⣦⠀⠀⠆⠀⠤⠀⠠⠠⠀⠀⠀⠀⠜⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠘⣇⠙⠒⢲⠄⣣⡾⣩⣖⣦⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⡄⠹⡄⠈⣧⠘⠆⠙⠋⠀⠉⠀⢙⠲⠏⣼⠛⣶⢀⡀⠀⠀⠀⠀⣠⡀
⢀⡤⠄⠀⢤⠖⠳⡄⢰⡋⣹⠄⠑⠀⠀⠀⠁⠀⠀⠒⠛⣏⠙⡇⠘⠋⠛⠡⣖⣾⠇⣠⣀⠀⠀⠀⠀⠀⠀⢀
⠸⠦⠴⢦⡈⢧⡠⠟⠀⠉⠀⠀⠀⠠⠀⠀⠀⠄⠀⠠⡤⠤⢆⠀⠀⠀⡀⠀⠀⠉⠸⠋⣼⢣⡤⣄⠀⠀⠈⠈⠅
⠀⠠⠤⠞⠁⠘⢂⠔⢡⠦⡄⠀⠀⠀⠀⠀⠀⠀⢰⡀⢀⢤⢬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠟⠰⣗⠄⢀
⠀⠀⠀⠀⠀⠀⠁⠐⠉⡟⣡⠀⡀⠀⠀⠀⠀⢈⠀⢥⢫⠘⡤⠃⠀⡁⢠⢰⠰⢰⠀⡇⡆⠂⡀⡆⠛⢻⢸⡻⣷
⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⢣⡜⠱⡒⢀⡀⠀⠀⠨⠈⠀⠶⠒⢲⡀⠈⠁⠀⠁⠀⠀⠀⢰⡆⠸⣂⣆⠀⠀⠉⠁
⠀⠀⠀⠚⢿⣹⠏⠀⠀⠀⠀⠀⠑⠣⣘⡌⡠⠀⠀⠊⡀⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⢨⠉⢌⠡⠉⡙⠉
⠀⠀⠀⠀⠘⢣⡴⢦⠀⠀⠁⠀⠢⠤⣀⠘⢱⣒⡕⢠⣅⢰⠋⠉⢳⣠⠃⡄⠠⡀⡔⠆⠅⠇⠂⠃⡆⢰⠢
⠀⠀⠀⠀⠀⠀⢠⡶⣽⡆⠀⠀⠀⡀⠀⠉⠀⠈⠐⢁⡜⠓⣇⣴⢾⡼⠖⢑⠐⢎⢀⢀⠆⠢⠣⠁⠘⠁⠉
⠀⠀⠀⠀⠀⠀⠈⠙⢋⣴⢾⡁⠀⠀⠒⠀⠰⠀⠂⠁⠁⠚⢀⣀⣤⠇⠀⠠⣀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠾⣣⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢩⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠺⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⢂⠄⠀⠀⠉⠀⠦⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢅⠀⠀⠐⠀⠀⠀⠄⠀⠀⠐⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠁⠀⢀⡠⠤⠴⠴⢄⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠁⠀⠀⠀⠀⠚⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡠⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢠⣀⣀⡄⠀⠀⠤⠈⠀⠈⠀⠀⠀⠀⠀⢇⠸⣒⡲⠀⡼⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡀⠑⠚⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠄⡄⠞⠀⠁
⠀⠀⠀⠀⣠⡶⢁⠉⠸⢛⡕⣦⢄⠀⠀⠀⠀⠀⠀⠀⢠⠠⠄⠀⠂⠠⠄⡀
⠀⠀⢀⣾⢸⡇⣼⠁⡏⢸⠀⣿⢸⡏⢆⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⢀⠑⢤
⠀⠀⠀⠉⠈⠁⠛⠀⠃⠸⠆⠿⠘⠇⠋⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠚⠐⠂⠀⠄

⠀⠀⠁⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡄⠀⠀⠀⠉⠁⠉⠀⠒⠀⠠⠂⠉⠁⠁⠁⠩⠀⠀⠠⢤⡄⢀⠀⠀⠀⠀⠀⣤⡀⣠⠄⠀⠀⠀⠀⣀⠀⠃
⠀⠀⡆⠀⢸⡇⠠⡤⠀⣄⢀⡖⠀⠀⢾⡀⠈⠁⠀⣰⠀⢀⡄⢠⠀⢀⠀⢠⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠃⢤⡀⠀⠛⠋⣥⢄⡀⢀⠀⠀⠘⠃⠀⠃
⠀⠀⠀⠀⢻⣧⠀⠀⠀⠁⣾⠁⠀⣤⡜⠀⢠⡀⠀⠟⢧⣾⣁⣿⠀⣿⡀⢸⣇⣼⣇⠀⣾⠀⠀⠀⠀⠈⣷⠀⠀⠸⣧⠀⠉⠺⠷⠄⠀⠁⠀⠀⢸⠇⠀⡆
⠀⠀⡆⠀⢻⣿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢹⣧⡄⢰⡆⢸⡟⢩⡟⢉⣷⣆⣼⠛⢻⣶⢈⣷⠀⣤⠀⠀⠉⠀⠀⠀⣿⡄⠀⢰⡆⢠⡾⠀⠀⣀⣤⠀⠀⠃
⠀⠀⠀⠀⠘⣿⣧⠀⠀⣾⣿⠀⢰⣆⠀⣀⠈⡏⠹⣿⠷⣸⠧⠈⣹⣏⢹⡋⢰⣶⠃⠘⠛⡻⢾⠏⠀⠀⠀⠀⠀⠀⡍⡇⠀⠄⠀⣼⠇⠀⠀⠉⠇⠛⠀⡆
⠀⠀⠀⣷⡀⢹⡏⠰⠀⠙⣿⠀⠈⢿⣷⣿⠷⢴⣤⣽⣀⣠⣤⠾⢛⠿⡷⢿⠛⢿⣷⣆⢰⣇⣈⣴⢃⣴⠀⠀⠀⢰⡇⠁⠀⠀⡘⠟⠀⠀⠀⢀⠶⠶⠆⡁
⠀⠀⠀⢸⣧⠀⢣⡀⠱⡆⠈⡀⠀⠀⠫⡉⠀⢠⣯⠙⣿⠋⠁⠰⢏⣤⡟⢈⠇⠀⠸⡿⠈⠛⢹⠟⣹⠇⠀⠀⠀⣼⠁⠀⠀⡰⡙⠀⠀⢠⡄⠈⢘⡇⠀⡅
⠀⠀⠁⠀⢿⣧⠀⠓⣀⣧⠀⢧⡀⠀⠀⠈⠢⣬⣿⣤⣿⡆⠤⠀⠈⠛⠐⠋⠤⠆⢀⢷⢾⣴⣧⡜⠃⠀⠀⠀⢀⠏⠐⢀⡾⡣⠁⠀⣠⡟⠀⠀⠈⠁⡀⡄
⠀⠀⠀⠀⠀⢋⠳⡄⠈⢻⡄⠈⣷⠀⠀⠃⢠⡼⠉⠟⣿⠟⠀⢰⣦⠰⠄⢠⣶⠀⣠⣶⡄⠻⠋⠀⠉⠙⠃⢀⡎⢀⡱⠋⠄⠁⢀⡴⠏⠀⢠⣄⣻⡏⠀⠁
⠀⠀⠂⠀⠀⠀⠑⠈⠲⣄⠑⡀⠈⢿⣦⡔⠞⠃⠀⣼⠃⣴⠈⠘⠛⠀⠠⠈⠋⢸⣿⡄⣀⡀⢾⠀⠀⢀⣰⣟⣴⠟⢁⢂⣤⠖⡱⠀⠀⠀⠈⠈⠹⠃⠄
⠀⠀⡦⣘⠀⠠⡀⠈⠒⢄⡉⠻⣄⣀⣹⣿⣦⣱⡾⣅⡶⢉⣴⠀⠀⢀⠀⠀⣦⡄⠉⣧⣍⢻⣼⣄⣴⣾⣿⣿⣿⠞⠛⣉⡤⠊⡀⠀⠀⠛⠁⠀⡀
⠀⠀⠀⠙⠁⠰⠶⠄⠀⠀⠁⠛⠿⠿⣿⢟⣵⡿⣷⢩⣷⢟⡴⡀⢀⠘⡏⠀⣏⢿⡄⢿⡟⠻⠿⣿⣍⣹⣿⠿⠷⠿⠛⡍⣻⡏⠁⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠂⠀⠀⡀⠀⠀⣠⠄⢀⠀⠀⠀⣼⣿⢿⡇⠉⠀⣿⣾⡁⡆⠘⢠⣧⡈⠉⠋⣁⣼⡇⠀⠐⡿⣽⢿⣿⣷⡀⠀⠠⣴⠠⡿⠀⠀⠰⣄⠀⠀⠀⢻⡄⠂
⠀⠀⠀⠀⣾⠀⠀⢠⡟⠀⠹⠀⢀⣼⣿⣿⠉⠁⠀⠊⣱⣋⡼⢀⣿⡌⠋⢈⡍⠀⢻⡁⠀⠀⣴⠄⣈⣀⢻⣿⣧⠀⠀⣤⢴⡷⣶⠀⠀⢹⣇⠀⠀⠘⣇
⠀⠀⠀⣸⠇⠀⢰⣿⠃⠀⡶⢀⣾⣿⠛⠋⣰⡿⢧⡄⠙⠋⠒⠜⣽⠇⢠⣿⡟⠁⠰⢇⡀⠀⡁⢀⣩⣹⣧⠉⠻⡄⠀⠘⠈⠃⠉⠀⠀⠀⢿⡆⠀⠀⣿⡆
⠀⠀⢀⣿⠀⠀⣿⣿⠀⠀⠁⣼⣿⠏⠀⣸⡿⠳⡄⠡⡆⠀⢴⣶⠏⣿⡼⣿⡿⡦⣀⠠⡷⠀⡇⠈⠙⢻⡹⢷⡀⠻⣄⠀⠀⠀⠙⢦⡀⠀⠈⣿⡀⠀⠛⡇
⠀⠀⣿⣇⣾⡃⠀⡏⠀⠀⠀⣿⠟⠀⣨⣿⢻⡴⢆⠀⠀⠀⢮⣿⢌⣿⣿⠸⡷⢶⣿⠂⠀⠘⢿⠀⠀⣼⡃⠿⣿⣤⠙⡄⠀⠀⠀⠈⢿⣄⠀⠀⠇⠀⠀⣿
⠀⢸⡏⢻⣿⡇⠀⠁⠀⠀⢀⡷⠀⣰⣿⢋⡙⢧⡀⠑⠀⢀⣸⣷⠐⣿⣇⣾⡖⢲⡟⠀⠀⠀⣸⠀⠀⠟⣷⠀⡿⢿⣆⠹⣄⠀⠀⠀⠀⢹⣄⠀⠘⡆⠀⣿
⠀⢸⡇⠘⠙⢇⡀⠀⠀⠀⡼⠃⢠⣿⣉⠛⠁⢸⣷⣄⡀⠈⠀⢻⣶⡿⣿⣿⣁⣷⠀⣠⣶⠾⢇⠀⠀⠘⠛⣴⡇⠘⣿⣧⡹⡆⠀⠀⠀⠃⢻⡀⠀⣇⢀⡧
⠀⠈⣧⠀⣄⠘⣷⣆⠀⣸⠇⢠⣿⣋⡜⠀⠀⡌⠁⠉⠛⠶⠀⠸⠿⣷⣿⣿⣿⡿⠀⠀⢠⠀⣸⠂⠀⠀⠀⢨⡷⠄⠈⣿⣧⡘⣄⠀⠀⠀⢸⡇⠀⠋⣸⡇
⠀⠀⠉⠆⠈⢦⣸⣿⣀⠟⢀⣾⡇⣹⣙⠀⠀⠘⣄⣀⣈⠀⠀⠀⢀⣉⣼⣍⠛⠀⠀⠀⢀⣾⠃⠀⣴⡦⠀⠀⠀⠲⣿⣰⠿⣷⡙⣆⠀⠀⢸⢇⣼⢣⡟⠁
⠀⠀⡀⠀⠠⣀⣻⣿⠋⢠⣼⡟⡟⠻⠉⠀⠀⠀⠈⣿⡟⣿⠿⣿⡏⠁⣴⣇⢸⣤⣶⡾⣿⠋⠠⠀⠀⠀⠀⠀⠀⠀⠈⡿⠶⠿⣷⡘⢦⣴⣿⣾⢇⡾⠁⠁
⠀⠀⡄⢀⠀⠈⠻⡿⣿⡻⣏⠰⣿⡀⠀⠀⠀⣰⠏⣨⣷⣤⡀⢙⣿⣀⠙⣈⣨⡿⠇⣠⣿⣿⣶⣅⣄⠀⠀⠀⣀⣀⣄⠁⢼⠘⢻⠇⡌⢿⣿⣹⡿⠁
⠀⠀⣁⠼⠥⠄⠀⢹⡿⠃⠀⠈⡿⠀⠀⢠⡾⣣⣼⡏⢛⣼⣿⣿⡋⠉⠉⠉⠉⣹⣿⣿⣿⣷⣟⢻⣿⡕⣤⠀⠼⠠⡤⠀⠀⠸⠉⡜⣿⡿⣿⠟⠶⡆
⠀⠀⣀⣀⣤⠀⠀⢸⡿⠀⠀⠀⠀⠀⢰⠋⣰⣿⣟⡉⢤⡿⣿⢻⡿⠷⠒⠲⣿⣿⠉⠃⠈⠈⠛⢠⣿⣿⣦⠁⠀⠀⣀⡆⣀⠀⢸⠇⠹⣷⠂⠀⠦⣴⠄
⠀⠀⠏⠉⠁⠀⠀⠀⣇⠀⢸⠀⠀⠀⣿⠀⢈⡁⡯⠙⠘⢁⡄⠘⣷⣤⠀⠠⣾⠏⠀⢸⣀⡀⠀⠀⠛⢻⡇⠀⠀⠀⠧⣤⡾⠂⢠⡷⣼⠟⠀⢠⣤⣤
⠀⠀⣤⠖⠛⣃⡀⠀⢹⡆⠀⡇⠀⠀⠘⢦⣄⣙⣧⠀⠀⢹⡇⠀⠿⣯⣀⢠⣿⠀⠀⠘⡿⠀⠀⠀⢀⣀⣠⡶⠀⠀⠀⠀⠀⠀⣼⢱⡏⠀⠀⠀⢒⣇
⠀⠈⡃⠀⠚⠋⢀⠀⠈⣷⠀⢣⠀⢠⡆⠈⢇⠉⠙⠳⡀⠈⠡⠀⠀⢿⡍⢀⣿⠀⠀⠀⢀⠀⠀⣠⠞⢋⣴⠂⠀⠀⢰⡄⠀⠀⣿⣿⠀⠀⠀⡘⠋⠉⠉⠁
⠀⠀⠁⣠⠂⠀⢾⠀⠀⢹⡆⠸⠆⠀⠃⠀⠈⢟⠛⠻⠧⡀⠀⠀⢀⡈⠹⣏⠈⣦⠀⠀⢨⡇⢰⡶⠟⠋⠁⠀⠀⣠⣄⠁⠀⢸⢁⡟⠀⠀⢸⣇⠠⡞
⠀⠀⠀⠙⠀⢀⣴⡆⠀⠘⣇⠀⣆⠀⢠⡿⠀⠈⢳⣤⣤⣿⡄⠀⣿⡇⠀⠈⠻⣌⣑⡀⠈⢁⣮⡴⠞⣇⣀⣤⠿⠛⠁⢰⠀⡄⢸⡃⠀⠀⣠⠀⠀⠁
⠀⠀⢷⣄⢰⡟⣿⠀⠀⠀⠙⡀⠈⠀⣿⣧⣤⠀⠀⢣⡈⠉⠻⣰⡟⠿⢿⣧⠀⠈⠉⠒⠓⣿⡥⠲⠞⠋⠉⣡⡶⠀⠀⠋⢀⠃⠸⠁⠀⣾⠋⠀⣰⡆⠀⠂
⠀⠀⣾⣿⠀⡇⠻⣷⠀⢷⡄⢣⠀⣠⣿⠙⢿⣄⠐⣦⡈⠻⣗⣿⣁⣶⣸⡏⣀⠀⢀⣄⠀⡽⢷⠁⢀⣷⣰⠿⢇⣠⣿⡦⠋⠀⡸⣀⡼⠋⢛⡛⢻⡇⢰⡧
⠀⠘⡏⢻⡿⠃⢠⡿⠀⣼⡿⢾⠀⣙⠻⢛⡀⣙⠿⠋⠻⠞⢿⡀⠙⢿⣿⣁⣿⡷⠀⢻⡏⢠⣌⣼⣿⠿⠟⢀⢈⣻⣿⣄⡀⣠⡀⣈⣷⣦⠈⠁⢿⣤⡿⠇
⠀⠀⣤⡀⠀⠀⢺⣧⡌⣉⣀⠈⢁⡁⢰⣶⢾⢿⡆⠀⠀⠀⣾⠃⡴⠊⠈⠋⣿⡁⣠⡟⢻⣦⣻⡟⠁⠀⠀⣼⣿⡏⣼⢹⡿⣦⣄⣈⣻⣷⣶⡄⠀⢀
⠀⠀⠛⣁⠀⢀⣼⣁⣾⣧⠛⣼⣱⢿⡀⣿⠏⠃⣑⠀⠀⠘⣷⠘⢷⣄⠀⠀⠈⠛⠋⠁⠀⣿⢿⡇⠀⠀⠀⢻⣿⠚⢛⣦⣸⠧⠈⢹⡟⠋⣸⡏⠀⠛⢠⡄
⠀⠀⠀⢉⡴⠋⠉⣡⣿⡃⠀⡿⣯⣿⣌⠻⣄⠀⣿⠀⠀⠀⠙⢿⡍⣻⠀⠀⠀⠀⠀⣠⡞⣁⠜⠁⠀⠀⠀⠸⣿⡴⠏⢸⣿⣇⢷⡄⢡⣷⡟
⠀⠀⠃⠀⠀⠀⠀⠀⠘⣧⠀⢸⡿⣿⣿⡆⠙⣦⡿⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⣾⢏⣸⣇⠀⠀⠀⠀⠀⠀⢹⡟⠁⢼⡿⣁⢸⡇⣼⠏⠀⠀⢀⣠⠆⡀
⠀⠀⣶⣦⣀⠀⠀⠀⠀⠘⣇⣸⡆⠆⠻⣿⠉⠸⣷⠀⠀⠀⣠⣤⣟⢡⠀⠀⠀⠀⠈⠢⣿⣽⣷⢦⡀⠀⠀⠀⠈⠙⠀⠂⠀⣝⣸⠟⠁⠀⠀⠀⠉⠉⣤⡄
⠀⠀⡏⠹⠃⠀⠀⠀⠀⠀⠙⢿⡄⠄⠀⠛⠃⠀⠚⠁⠀⠸⢟⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠂⠂⠀⠀⠀⠀⠀⠂⠐⠀⠛⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠈⠉⠛⠿⢿⣿⣿⣶⣦⣤⣀
⣾⣿⣿⣿⠿⠟⠛⡉⢡⣤⡄⢶⠾⠛⠃⢀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠰⠆⣀⠀⠀⠉⠛⠻⠿⣿⣿⣷⣶⣤⣀⡀
⠋⠉⠁⡆⢰⠆⠻⠓⠀⣁⣄⢰⡠⠶⠃⠙⢁⣀⡸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⢰⣒⣤⣈⠁⠸⠿⠀⣖⣄⡀⠀⠉⠙⠻⠿⣿⣿⣿
⡄⠀⢀⠄⣤⡴⠶⠛⠛⠋⢁⣠⣤⣦⠴⠿⠻⠏⠁⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠉⠉⠛⠳⢦⣄⡄⢁⡘⠷⠶⡤⠀⠀⠀⣸⣿
⣷⠀⡈⠉⣉⣤⣤⡶⠆⠑⠊⡉⣀⠉⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⣀⠀⠀⢰⣇⡄⢈⡁⠳⠰⢶⠤⣀⠀⠀⢰⣿⣿
⣿⣟⠇⠋⠛⠋⠀⠀⠸⠗⠋⠑⢉⣀⢠⣴⡄⠀⠀⢀⣸⣿⣷⠀⠀⠀⠀⠀⠀⣰⣿⣿⡃⠀⠀⠀⠘⠛⠧⣶⣤⣀⣉⠉⠃⠀⠀⠀⠀⠉⠀⢠⣿⣿⠇
⣿⣿⡀⠀⠀⠀⢠⣤⢠⣤⠶⠿⣟⣋⣈⣥⣴⣶⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠉⠛⠛⢠⣶⡄⠀⠀⠀⠀⣾⣿⡏
⢿⣿⣧⠀⠀⠀⠈⣋⣡⣤⣶⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠛⠻⠿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠁⠀⠀⠀⣼⣿⡿
⠘⣿⣿⣶⣶⣾⣿⣿⣿⠿⠟⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣷⣶⣤⣀⡀⢰⣿⣿⠁
⠀⢻⡿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⠷⠖⠶⠶⠷⠖⠶⡶⠀⠀⠀⠀⣿⣿⠀⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡆
⠀⣼⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠐⠠⠀⠴⠀⠦⠀⠤⠼⠏⠛⠋⠀⣿⣿⡇
⣼⣿⡟⠈⠙⠻⢿⣶⣤⣀⠀⠀⣿⣿⡇⠀⠀⠀⠰⠠⠶⠖⠶⠶⠶⠶⣦⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠐⠖⠒⠒⠶⠖⠀⠄⠖⠂⠆⠸⠀⠀⠲⣿⣿⡇
⣿⡟⠀⠄⢁⠒⠀⣈⠙⠻⢿⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠛⠒⠖⠒⠒⠒⠐⠓⠀⠀⠀⢀⣿⣿⡇
⣿⣀⠀⠠⠄⡈⠢⢤⡉⠲⠀⠈⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠀⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠉⠛⠿⣶⣤⣈⠐⠢⣄⠀⠠⢄⡛⠻⠿⠟⢻⣿⣿⠟⠛⠛⠛⣛⣿⣿⣿⣿⡿⠿⣿⣿⠛⠀⠀⠀⢰⣿⡟⠙⠛⠻⠿⠿⣿⣿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠙⠻⢿⣶⣤⣈⠃⠰⠍⠓⠦⣴⣿⣿⣯⣴⣶⣿⠿⠟⠛⠉⢉⠀⠤⠄⠸⣿⡄⠀⠀⠀⣼⣿⠇⠠⠤⢌⢀⡒⠒⠂⠬⢭⣉⠛⠛⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⡀⣰⣿⣿⣿⡏⢉⣀⠤⠐⠀⡂⠡⠀⠐⢂⣈⠄⢿⣷⠀⠀⢀⣿⣿⠈⣁⠒⠂⠀⠤⠌⢁⡐⠒⠐⠤⠤⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠟⣿⣷⠠⠀⢐⣈⠁⠤⠒⣀⡈⠭⠄⠀⠘⣿⡇⠀⢸⣿⡗⠒⠠⠨⢈⣁⡒⠂⠀⠤⠌⣀⣀⠒⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠈⠭⠄⠖⡀⠡⠄⠐⠂⠀⠀⢀⣹⣿⡀⣿⣿⠁⠀⣀⠂⠲⠀⠠⣀⣁⣐⠒⠢⠄⠤⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⠿⠟⠛⠱⣿⣿⣿⣶⣶⣯⣤⣍⣐⣒⠂⠨⢌⣁⠀⠀⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⠐⠂⠈⠉⠉⠁⠐⠒⠒⠤⠤⢤⣀⣀⣀⡀
⠀⠀⠀⢀⡤⠴⣒⣀⡉⠁⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠓⠒⠲⠦⠤⣤⣀⡀
⠀⠀⠀⡘⠀⠀⠀⠀⠈⠉⠑⠒⠒⠦⠤⢤⣀⣐⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⢙⡏
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠦⠤⣤⣀⣀⣀⣀⠤⠴⠚⠋⠉⠀⠀⠀⠀⠸⡄
⠀⠀⠀⣄⣭⠷⠶⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⡀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⢉⣒⡢⠤⡄
⠀⠀⠀⢸⡏⠙⠲⠤⣄⣀⠀⠀⠉⠉⠓⠲⠶⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠉⠙⣷⣦⣤⠶⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⢹
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠉⠙⠒⠦⣤⣀⡀⠀⠀⣀⣀⣉⣭⠽⠷⠶⠶⢤⣼⣤⡶⠟⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎
⢀⣴⣶⣏⡙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⢒⣒⣫⣭⣽⣿⠟
⣾⠇⠀⠉⠙⠛⠷⢮⣿⣿⣶⣤⣀⡀⠀⠀⡾⠐⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠶⠒⣚⣋⣩⡥⠤⠶⠖⠛⠛⠉⠉⠁⠀⠀⡏
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣶⣥⣧⡴⠶⢒⣛⣛⣭⣭⣴⠶⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠘⠿⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠛
⠀⠀⠀⠀⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣀⣀⣧⠀⠀⣀⣀⣠⣤⣴⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠾⠛⠛⠉⠉

⠀⠀⠀⢠⣄⣴⣤⣄⣀⣀⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣴⣤⣤⢤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⡿⠋
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢠⣾⣷⡀⠀⠀⢀⣀⣿⣿⣿⣿⡄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⡇⢀⣠⣾⣿⣿⣿⡿⠉
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⠏⠿⠿⣿⢿⣿⣷⡀
⠀⠀⠀⣨⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠙⣿⣿⣿⡿⠟⠉⠉⠉⠙⠁⠀⠀⠀⠀⠀⠰⣾⣷
⠀⠀⠀⠙⣿⠛⠛⠟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡡⠒⠋⠉⠉⠙⠲⢍⠲⡄⠀⠀⠀⡇⠀⠒⠒⢒⡶⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⢊⣀⣀⣀⠔⣣⠦⡀⠀⠱⡜⣆⠀⠀⡇⠀⢀⡴⠻⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⡏⠰⠤⠥⠖⠊⠁⠀⠉⠲⠤⢹⠉⢳⠀⡇⠀⠈⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣸⠀⡟⠒⠒⠒⠒⠒⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢗⠢⠴⣚⡆⠀⠀⠀⠀⡰⠋⠉⠁⠀⡇⠀⠀⢠⢆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡤⢄⣀⣀⡠⢴⠊⠀⠀⠀⠀⠀⡇⠀⢠⠧⠬⣆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢇⠀⠀⠀⠀⣸⣤⣀⠀⠶⣖⠶⡇⠀⠋⠀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⠀⠑⣄⣠⠞⠁⡇⠀⠙⢦⡈⠓⠤⠤⠤⠤⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣇⡤⣞⣱⢤⣸⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠈⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠸⠁⠈⠇⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠄⠀⢠⠀⠠⠀⠄⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣀⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠦⠄⠆
⠀⠀⠀⠀⠀⣿⣿⣤⠘⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠸⢿⠾⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠠⠀⠀⣄⠀⢀⠀⠄⡀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠈⠀⠀⠀⠀⢂⠀⠀⠀⠀⠈⣀⣸⣅⡀⣡⠀⠛⠠⠤⠀⠀⢀⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⣴⢺⠀⠀⠀⠂⠀⢀⡀⠀⠘⢿⣿⡋⠀⠈⡀⠀⢄⠒⠖⡄⠀⠀⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡀⠀⠀⢸⡇⡀⠁⣳⠦⠀⡀⠀⢨⡾⠟⠀⠑⣾⣷⣿⣷⠄⠰⢿⣁⣴⣤
⠀⠀⠆⠘⡆⠀⠀⠠⣀⡇⠈⠙⠒⠚⠠⡈⠂⠁⠀⠀⠁⠀⠀⠀⠚⠂⡀⠈⢉⡿⣿⣶⣷⣿⣿⣏⡿⠈⠃
⠀⠀⡄⠀⣧⠠⠐⢻⣽⣷⡰⠶⠤⠤⠴⠶⠶⠠⠦⠤⣴⣾⡷⠷⣶⣶⡾⠿⢾⣷⣶⡾⠟⠛⢿⠊⠀⠀⡃
⠀⠀⠈⠀⡿⠋⠀⢒⣹⢿⣷⠎⠁⠀⠀⠀⠀⠀⣠⣾⡿⠛⣿⡖⠒⡶⠒⢶⡾⠿⡿⠶⠂⢀⣨⢷⣤⠎
⠀⠀⢀⣀⡿⣄⡀⢐⢺⣿⣿⡄⠀⠀⠀⠀⣴⣾⣿⣿⠇⠀⠈⢁⡀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠁⠛⠁
⢐⣤⣤⣿⣿⣟⣿⣷⣾⣿⣿⣥⣤⣤⣶⣿⣿⣿⣿⣿⠂⠀⠑⠄⡆⠄⢀⠄⠀⡠⢀⢂⠀⠓⠚⠐
⠀⢿⣿⣿⣿⣹⣿⣷⢺⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣀⡀⠦⠐⠑⠀⠓⡀⠛⠀⠛⠈⣷⡶⠃
⠀⢸⡟⢻⡿⣿⣿⢿⣎⣿⡟⠛⢻⡿⠋⠀⠀⠙⣿⣿⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠸⠇⠈⠇⠏⢚⠘⣇⣿⡇⠀⢸⠀⠀⠀⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⢀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣄⣀⢠⡟⠈⢧⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡈⠉⠀⠀⠈⠉⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣘⣏⠉⠉⢉⣉⣉⠉⡿⢯⡀⠘⡇⠀⠀⡄⠀⠀⣼⠀⣀⡴⣷⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠻⣍⠉⠁⠀⠉⠳⠇⠀⠀⡇⠀⠀⠿⠚⠁⠀⠉⠉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣟⠀⠀⠀⠠⣄⡀⠀⠀⡇⠀⠀⣀⡤⠄⠀⠀⢈⣳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠉⠳⢦⣧⠴⠚⠁⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢛⣛⠛⠛⠛⣿⠛⣿⠛⠛⢛⣛⠛⢳⠀⠀⡀⠀⠇⣀⣀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣈⣙⣃⣀⣿⠀⣿⣀⣘⣉⣀⣀⣘⣛⡿⠃⣤⠘⡏⠉⠉⠙⠓⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣠⠟⠀⠀⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⠴⠂⠀⣿⠀⣿⠀⠰⢦⣄⠀⢸⠀⠀⢀⣇⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠿⠶⠿⠶⠶⠶⠾⠶⠞⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠄⠐⠀⢂⠠⠆⠀⠠⠀⠐⠀⠂⠠⠀⠀⠂⠄⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠈⠉⠉⠀⠀⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠠⠲⠠⠒⡂⠀⠄⡀⠀⠀⠉⠢⣄⡄⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠠⠄⠐⠂⠠⠀⡀⠀⠀⠀⠑⠀⠀⡐⠋⠀⢀⣠⢤⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⣀⠀⠀⠂⠈⠴⣂⠥⣂⠍⡢⠄⠀⠀⠜⠀⠀⠘⠁⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡼⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠠⠁⠀⠂⠤⢐⡂⠄⡀⠀⠁⠀⠀⢀⠎⠀⠀⠈⠀⠈⠉⠈⠁⠠⠄⠀⠈⠁⡐⠂⠀⠀⠀⠀⡜⣡⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⡀⢀⣀⠀⠤⢈⡑⠂⠅⠂⠤⡑⠪⡑⠀⠀⢀⠊⠀⣀⠨⠀⠀⠀⠒⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡜⣰⢿⣿⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠄⢐⡂⠤⢁⠁⠂⠍⡒⠤⡀⠈⠑⠀⠀⠠⠁⠀⠠⠐⠒⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⢀⢞⣼⣯⣿⡏
⠀⠀⠀⠀⠠⢎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⡂⠀⠈⠀⠈⠂⠀⠀⡐⠁⠀⠀⢁⡀⠠⠤⠄⠉⠉⠁⠰⠆⢀⡂⠀⠀⠀⠀⢠⢮⢮⢯⣾⣿⢫⡇
⠀⠀⠀⠀⠠⣮⣌⣉⣉⠒⠢⠤⣀⠀⠤⠀⠀⠀⠈⠑⠄⠀⠀⠀⢀⠜⠀⢀⡄⠰⠀⠤⠤⠤⠤⠀⠀⠀⠉⠁⠒⠀⠀⠀⠀⡰⢣⢣⣣⣿⡿⣣⡻⠁
⠀⠀⠀⠀⢴⢥⡤⠤⢭⢭⣍⣒⡢⠝⡒⣤⡀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⡱⢳⣿⡿⣱⡗⠁
⠀⠀⠀⠀⢹⠥⡆⢀⡀⠀⠉⠁⠛⠿⠷⢿⣾⣷⣤⡀⢀⠀⡠⠗⠀⠀⣀⣀⣤⣤⡤⡤⡤⠤⣤⣤⣀⣀⠀⠀⠀⠀⠀⡰⡱⡽⣱⣿⡟⣴⠟
⠀⠀⠀⠀⠀⢱⢘⣕⠯⠕⡖⠚⠓⡒⠷⣶⣟⢛⣻⣿⡷⣴⣶⢶⡿⢟⣳⣾⣭⣭⣥⣤⡴⠍⡠⡔⠂⠌⣉⠓⠶⣬⡴⣼⡝⣵⢿⠏⣾⠿
⠀⠀⠀⠀⠀⢯⣙⣨⣭⢤⡴⣶⣶⣮⣷⣾⡿⣿⣛⣻⣛⣛⡛⡻⣿⣿⣟⢛⣿⡿⡭⣭⡴⣒⣠⠀⡉⠛⠓⢦⠴⣠⢜⣟⣾⣿⢋⣾⡟⠄
⠀⠀⠀⠀⠀⡎⠟⢹⣧⣼⣏⠁⣑⡁⠀⢉⠛⠒⠒⠲⠲⠭⢭⠭⠿⠿⠻⠿⢿⠟⣫⣭⣍⣑⣨⣍⣛⣲⣤⣄⣈⣦⣭⣿⣼⣡⣿⣓⣀⣅⣂⣬⣐⡠⢀⡀
⠀⠀⠀⠀⠀⣤⠱⠂⢻⡗⡇⢒⣚⡿⠯⠭⠽⠿⠿⠟⠛⠋⠉⠉⠙⠓⠒⢲⣖⠿⣿⣿⢟⡵⢿⣃⣠⣌⣁⠨⣍⢻⣭⣿⣷⣿⣿⣿⡽⣛⣯⣽⣷⣾⣿⠇
⠀⠀⠀⠀⠀⢿⠀⡟⠸⣏⢏⣥⡤⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⣸⣾⡟⣡⣟⠀⢽⡤⣤⡥⣼⣵⠤⣅⣬⣤⣭⣭⣤⣼⠿⠿⣿⢿⠿⠏
⠀⠀⠀⠀⠀⠹⡄⠋⠀⢺⣿⠻⠛⠿⠿⠶⡶⠾⢶⢴⣤⣀⣀⡤⢤⣒⡃⣽⡿⣣⠾⠷⡾⢀⠡⠂⢁⠩⠁⡀⠌⠀⣊⠍⠑⡀⠅⡀⠠⢁⠐
⠀⠀⠀⠀⠀⠀⣣⡀⠀⢀⠽⡬⢴⣖⣂⡤⠾⠾⠛⠻⠿⠋⠉⠉⠛⠛⡱⠶⠚⠉⠉⣼⢁⠀⠀⢄⠂⢉⡤⠶⠖⢃⡠⢤⣂⣤⣂⠜⠊
⠀⠀⠀⠀⠀⠀⠘⣯⢲⢪⢿⠉⠁⠤⠔⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⢹⣧⠆⠁⠐⢊⠰⣒⠧⠖⠋⢩⢁⡠⠍
⠀⠀⠀⠀⠀⠀⠀⠙⣭⢿⣽⠀⢒⣒⡊⠉⠀⠀⠀⠀⢀⣀⠀⠀⢀⢀⣀⡀⢒⠐⠉⠡⣟⠷⠀⢊⡉⠠⡐⣪⡤⠆⡅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣛⡼⠶⠶⣒⢒⢂⣛⣙⣉⠭⠬⠽⠫⠖⠔⠃⢘⠗⡫⢑⠩⠑⡂⠌⢀⠌⢁⠭⢀⠠⢁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣠⣤⠶⠚⠛⠉⠉⠉⠉⠉⠉⠛⠓⠶⢤⣄⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡾⢦⡀⠀⠀⠈⠻⣶⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣤⣀⠀⠀⠈⢳⡄⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣰⠏⠀⠀⠙⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠹⣆⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⡏⠀⠀⢀⡾⠉⠙⢦⡀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⢻⡄⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⠀⣾⠁⠀⠀⠀⡹⢦⡀⠀⠀⠈⠳⣄⡀⠀⢠⡀⠀⣠⣄⠀⠀⣠⠀⠀⠈⣷⠀⠀⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢠⡇⠀⠀⢰⡇⠀⠀⠀⢶⠷⣾⠻⣦⡀⠀⠀⠈⠻⣄⢸⠇⠀⣷⡼⠁⠀⢻⠀⠀⠀⢹⠀⠀⢸⡆⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠈⣧⠀⠀⠀⣾⠒⡗⠂⣏⢻⣦⡀⠀⠀⠈⠻⣄⣾⡁⣹⣾⠀⢘⠀⠀⠀⣾⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⡇⠀⠀⠀⠹⣆⠀⠀⠁⠀⠀⠀⠘⠦⢍⣻⣦⡀⠀⠀⠈⠻⣍⠁⠈⠁⠈⠀⠀⣰⠏⠀⠀⢸⠃⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢿⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠳⣄⠀⠀⣠⡾⠃⠀⠀⠀⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠈⠳⣾⠋⠀⠀⠀⠀⣸⠃⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠀⠀⠀⠀⢀⣤⣤⠼⠿⣦⡀⠀⠀⠈⠳⣄⠀⠀⣰⠏⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⣠⡴⠛⠁⢀⣀⣤⠶⠋⠁⠀⠀⠀⠈⠳⣦⠀⠀⠀⠈⢳⡼⠃⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠈⠻⣦⡀⠀⠘⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠈⠙⠛⠶⢤⣤⣀⣀⣀⣀⣀⣀⣤⡤⠶⠚⠋⠁⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣤⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠉⠉⠉⠉

⠀⠀⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠸⣭⣝⣻⣻⣶⣶⢶⣦⣤⣤⣤⣀⣀⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⡀
⠀⠀⠀⢠⠻⢾⣿⣯⡟⣿⣯⣶⣤⣼⣟⣙⠹⣿⣛⣻⣻⣿⡯⣹⣟⣁⣴⣀⣤⡄⠀⠀⠀⠀⠀⣠⣤⡶⣸⠞⠁
⠀⠀⠀⣿⣶⣿⣥⣼⣿⠻⠻⢿⢻⣿⣿⣿⣟⣿⣿⣿⣿⣿⣇⣿⣿⣻⣏⡏⣠⢻⣂⠀⠀⢀⣤⡟⣻⡧⡛⠄
⠀⠀⢸⣿⢽⡷⢿⠿⣿⣟⣿⣟⣮⣯⣭⣉⣿⠿⣶⣷⣷⣏⣿⠿⠿⡽⣾⡇⢨⣯⣿⣀⣤⠾⢿⣟⣯⠛⠁
⠀⠀⠘⣿⡿⡿⣿⣶⣤⣿⣿⣷⢿⣿⣿⣿⣯⡍⣷⣶⣯⣽⡟⣶⣶⢻⡿⣿⡯⣽⣯⣭⣿⣟⣟⣿⠁
⠀⠀⠀⢿⣿⠻⣞⣿⣿⠿⢾⣷⣿⣿⣾⣥⣇⣷⣟⣟⣻⣿⣷⡳⠟⣿⣿⣿⣿⣿⣾⣿⣳⠿⣿⣿⣆
⠀⠀⠀⠘⠻⣧⣽⣿⣱⢿⣿⣿⣷⣿⢿⣿⣯⣱⣿⡿⣿⠿⢯⣽⣶⢿⣟⣼⣟⣿⡿⢿⣿⣻⣸⣿⠏
⠀⠀⠀⠀⠀⠘⠷⠟⣶⣾⣧⣤⣭⣽⣟⣿⣋⣿⣿⣷⣾⣶⣷⢿⣺⣻⣷⣿⣾⣿⣿⣿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠛⠛⠛⠛⢿⣿⣿⣯⣏⣿⣹⢹⣽⣿⣟⡿⣿⣻⣿⣽⡽⣯⣽⡟
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣟⡇⠀⠀⠀⠀⠻⣿⠿⣧⣷⣽⣾⣧⡿⠶⠾⣿⠍⠉⠈⠙⠋⠻⡿⡄
⠀⠀⠀⠀⠀⠀⢶⣝⣾⣿⡇⠀⠀⠁⠀⠀⢀⠀⠈⢿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⡀
⠀⠀⠀⠀⠀⠀⠔⠛⠀⠀⠉⠲⡄⠀⠀⠀⠀⠷⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠁⣂⣠⡴⠞⠁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠌⠛⠓⠊⠒⠦⠆⢀⠀⠦⠥⠤⠄⠀⢸⠀⢀⣈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢧⡀⠄
⠀⠀⠀⠀⠦⡄⣤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠄⡀⠀⠀⠂⢨⠀⠀⢧⠐⡆⠀⣈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣁⠈
⠀⠀⠠⠀⠀⠀⠄⠀⠀⠉⠓⠒⠀⠀⣀⣠⠴⠒⠀⠐⠒⠆⠂⠆⠀⠠⠆⠀⡈⢇⠁⠀⠘⡀⠁⠀⠀⠀⠀⠀⠄⣑⢋⣉⠀⡭⣜⠃⣠
⠀⠀⠀⠀⠤⠀⠀⡶⠀⠀⠀⠀⠀⠉⡈⠀⠀⠀⡀⢤⢠⣤⣠⣀⢀⠀⠐⢦⡀⢀⡤⠄⠀⠙⠳⠀⠀⠄⠂⠒⠀⠀⠀⠀⡀⢩⠜⠂⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠖⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⢰⠀⢳⠈⠰⠄⠄⠈⠙⠙⠛⠁⠉⠉⠁⣶
⠀⠀⠀⠀⠀⠲⡀⠀⠀⠉⠀⠀⡈⠀⠀⠀⠀⠀⠛⠛⠓⠰⠴⠦⠀⠆⠀⠀⠀⢣⡀⡀⠸⠀⠀⠀⠠⠈⠤⠄⠄⡑⠂⠻⠀⢸
⠀⠀⠀⠀⠀⠀⠈⢠⠄⠀⠀⠀⠰⠄⠁⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢌⡁⠄⠉⠈⠁⠉⠁⠀⠂⠀⠀⠲⠒⠒⠐⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡀⣄⡈⢁⠉⠀⠀⠀⠀⠀⡈⠓⠘⠋⠛⠃⠈⣧⠀⣈⠉⢉⠁⠐⠒⢀⠀⡀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠺⠄⠁⠉⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠂⠀⢦⠸⠓⠘⣆⠀⠘⣥⡀⠀⠲⡀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠄⠀⠀⠀⠀⠹⣤⡀⠈⠆⠀⠈⠳⠄⡀⠈⠢⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⠊⠂⠒⠂⠀⠀⠀⠀⠄⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⢆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱

⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⢸⡆⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢀⠠⠊⢡
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠯⣕⣦⡀⢀⡄⣊⣀⣀⡀⡠⠂⠁⠀⢀⡾⠃
⠀⠀⢸⡷⠦⠤⢤⣀⠀⠀⠃⢀⣠⣤⣴⣒⡈⣿⠀⠀⠀⠀⠀⡩⠛⠿⠿⣟⡒⠤⣁⠀⠀⢀⠔⠉
⠀⠀⠺⠿⠿⠷⠶⠶⠿⠷⡾⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠓⠾⡦⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣬⡓⢤⡀⠀⠀⠀⠀⠀⢑⠥⠆
⠀⠀⠀⣶⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⣞⡂⠀⠙⠲⣌⡒⣤⡦⠐⣈⠤⠒⡀⠒⠂⠀⠠⠤
⠀⠀⢀⠱⡈⠑⢄⠀⠀⠙⠢⣄⠀⠀⠀⣴⣿⠽⣮⡄⠀⠀⠈⠙⠷⠛⡍⠀⣠⠞⠁⠀⠀⠀⠀⢀⡰⡯
⠀⠀⠙⢇⠈⠢⡀⠀⠤⠄⠀⢈⣻⣆⣼⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⡠⢊⠞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⠰⣾⣻⠿⠟⠛⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠦⢉⣁⠒⠂⠤⢎⠔⠁
⠀⠀⠀⠀⠀⠈⠳⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣫⣾⣟⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣾⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢰⣶⣶⣶⡶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿⡿⣫⣾⣿⣿⣿⢟⣵⣿⣿⣝⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿⣿⣮⡻⣿⢟⣵⣿⣿⣿⡿⣫⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣽⣿⣽⣿⣽⣽⣯⣯⣽⣽⣯⣿⣯⣾⣽⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣯⢿⢟⣼⣧⣝⠿⣫⢸⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⡿⣵⣜⢿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣾⣧⣭⣭⣽⣿⠇




⣏⣈⣯⣍⣟⣯⣿⣽⣽⣽⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⡄⠠⢆⡲⠆⠀⠀⠀⠀⢈⣰⣿⣣⡐⠀⠀⠈⡁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠒⠒⠀⣴⣴⠀⠀⠀⢐⡂
⠀⠉⠀⠈⡥⢤⡤⠠⣄⠀⠀⠀⠢⣿⠷⠃⠀⠀⠀⠀⠀⠀⠀⠄⡀⠼⠫⠀⠂⡆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠰⠓⣹⣧⢤⣮⡀⠀⠀⠀⢸⠀⠀⢪⡐⠀⠀⠀⠀⢠⡵⠋⠀⡀⠀⠀⢴⣀⠀⠀⠀⠀⠑⡀⠀⠋⠀⠀⠄
⠀⠀⢆⠀⠄⡿⠛⠂⠐⣶⡄⠀⠀⢈⠀⠀⠠⠀⠀⠀⠀⠀⠈⠲⣀⠀⠁⠀⠀⡘⠈⠀⠀⠀⠀⠀⡄⢠⠀⠄
⠀⠀⢰⠿⡴⢧⣤⣤⠤⣼⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢘⣗⣰⡀⠄⠄
⠀⠀⠰⠷⣟⠋⠉⠉⠉⠉⡿⠀⠀⢘⠟⢁⡀⠠⢠⣤⣤
⠀⠀⠀⠣⢻⣶⣻⠒⡶⢦⡄⠀⠀⠀⠀⢸⣿⣿⣶⣧⣸⣿⣧⡀⣀⡀⠀⠰⠊⠀⠀⠀⠞⣯⡭⢼⠃⠀⠀⠀⠈⠲
⠀⠀⠀⠀⢽⣿⣿⣻⢿⣶⠀⠀⠀⣠⣼⣿⣿⡿⣿⣿⣷⣿⣯⣿⣯⣧⡀⠀⡄⠀⢀⡔⣠⢰⢫⣙⢀⢡⡀⠀⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢧⣾⣿⣹⣿⣿⣿⣿⣷⣾⣿⠀⠀⢨⠖⢸⡟⢫⣹⣿⣐⢀⢻⡇⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠐⣻⡴⣾⣿⣿⠿⣿⣿⣷⣿⣿⣿⣟⠃⠀⢰⠀⠀⡇⠈⢻⢿⡿⡉⢸⠃⠀⠸
⠀⢀⠀⠲⠐⠒⢒⡛⠒⢒⠂⠀⠀⠙⠁⣛⢹⠟⣞⣹⣻⣿⢏⠹⠿⠃⠀⠀⠀⠀⠀⣧⠁⢿⣨⣻⠤⢸⠀⠀⡀
⠀⠲⢆⡠⡆⢰⠀⢆⠂⣁⠄⡿⠀⠀⠀⠁⠁⠀⣨⣿⣿⡀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣿⢀⠱⠉⣀⠀⣸⠀⠀⠃
⠀⠀⠈⠁⠈⠑⡎⡗⠈⠉⠀⠀⠀⢦⠀⠀⠀⣾⣯⣿⣟⢻⠀⠠⢂⣠⠗⠀⠀⠀⠀⢸⠀⢴⣺⣿⠇⡏⠀⢀
⠀⠀⠀⠀⠀⠀⡧⠇⠀⠀⠀⠤⠐⢠⠴⠀⠀⣿⣿⣿⣿⣼⠀⠀⠈⠀⠀⠀⢀⡀⠀⢸⣴⣾⣿⣿⣦⣏⠠⣬⡀
⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣼⠂⠀⠀⢀⠀⠀⠀⠚⠊⠉⠀⠀⣉⠅⠈⠉⠙⠟
⠀⠀⠀⢀⠈⠀⠀⡤⠁⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠂⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠁⠛⠀⠈⠀⠐⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣶⢀⣤⣤⣠⢤⣰⢠⡄⣶⣄⠀⣠⡤⠘⢿⠉⠹⣦⠀⠀⠠⠤⠤⠤⠄⠤⢄⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠄⠀⠀⠁⠀⠉⠈⠈⠈⠈⢈⣉⠉⢯⠘⠧⠴⢦⠈⢧⣼⢟⡰⢤⡀⠀⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢲⡋⠈⢧⠈⢧⠀⠶⠋⠈⠛⠁⠸⢯⣿⢃⣶⢦⡄⠀⠈⠀⠀⠀⠂⠀⣀
⠀⢀⣀⠀⠀⢀⠤⣄⠈⣡⠔⢳⡀⢳⠀⠈⠁⠀⠀⠀⠀⣤⠤⡶⠒⡆⠡⡾⠳⠾⢃⣼⣲⡄⠀⠀⠀⠈⠈⢀⢀
⢰⡋⢁⣀⠈⢧⠀⢘⡇⠙⠶⠫⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠛⠀⠀⡀⠀⠘⠳⠿⣡⡟⢳⡆⠀⠀⠀⠀⠀⠃⠴⡄
⠈⠉⠉⢉⡷⠈⣿⠋⢀⠔⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠙⠋⠉⠹⠄⠀⠀⠐⠀⠀⠀⠀⠈⡀⠟⢱⡿⢽⡆⣀⡀⠀⠈
⠀⠀⠛⠉⠀⠀⠈⠔⢁⠼⣉⡇⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡴⠻⡛⣦⠀⣠⠄⠄⢐⡀⢀⠀⣀⡀⡀⠙⢂⢤⣻⡌⣡⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⡔⠁⡄⠀⠀⠀⠀⠈⢐⠀⢣⠹⠀⠓⢃⠀⠣⠀⠇⠳⠈⠸⠀⠇⠅⠸⠰⠸⠀⠍⠇⠻⣝⠛
⠀⠀⠀⠀⣀⣀⣤⡀⠤⠤⠤⠕⠎⡀⢏⠃⣀⡀⠀⠀⠒⠄⠀⠛⠉⠙⡇⠀⠀⠀⡀⠀⠀⠀⠀⢀⣿⢐⣯⢸⣆⠀⡀
⠀⠀⠠⠘⠛⣯⡿⠁⠀⠀⠀⡀⠀⠈⠉⢄⡳⢀⠆⢀⡀⢁⡠⠒⠒⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⡅⡆⢱⠱⠘
⠀⠀⠀⠀⠀⠛⣴⠾⢦⠀⠀⠀⠁⠢⠤⢄⡀⠁⢞⣲⠇⣠⣄⠰⡏⠉⠙⣆⡝⢡⡀⠤⡀⡆⠆⠧⠕⠑⠊⢸⠂⡞⠆
⠀⠀⠀⠀⠀⠀⠀⣰⠦⣹⡄⠀⠀⠀⡀⠀⠉⠁⠀⠈⠘⢁⡜⠓⢇⣠⣾⡥⠷⠊⣣⠓⣇⢀⢀⢐⠰⠸⠨⠀⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢋⣤⣾⡃⠈⠁⠂⠂⠀⠼⠐⠃⠃⠃⠚⠈⣘⣀⡼⠀⠀⢇⣀⠐⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡼⢇⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⣘⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠢⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢸⡏⠁⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢠⣾⣿⡿⢻⣧⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇
⠀⠀⠀⠀⢰⣿⣿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⡇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠻⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣴⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⣆⠀⢹⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣧⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⡿⠀⠘⡝⢿⡇⠀⠸⣧⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣻⠟⢋⠙⠻⣿⡿⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠁⠀⠀⣄⣼⠇⠀⠀⣿⣂⣴⠟⠁
⠀⠀⠀⠀⠀⡟⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠻⣦⣀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⢀⠀⠀⠀⠀⣿⣶⣦⣤⡀⢀⣀
⠀⠀⠀⢀⣤⣶⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣠⡼⠃⠀⠀⢰⣿⣿⣿⣿⡿⠋⠙⢷⡀
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣛⣿⠋⠉⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⢠⡿
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⣉⣉⣙⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠉⠉⢳⡴⠛⢷⡄
⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⣿⠃⠲⢤⡾⠁⢀⡞⠙⣷
⠸⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡋⠀⠀⠀⢻⣆⠀⠀⠻⠶⢿⣀⣰⠏
⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠙⠛⠿⣶⣶⣶⣾⣿⣿⣿⠟⠉⠀⠈⠻⣦⡀⢀⡾⠋⠀⠀⣀⣄⣤⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⣆⠀⣰⠿⠟⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠘⢷⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠋⠀⠀⠀⢀⣼⠋
⠀⠀⠀⣀⣀⣀⣀⣠⣤⠟⠁⠀⠀⢀⡴⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⣤⣤⣤⠞⠁
⠀⠀⣾⠉⠉⠉⠉⠉⠀⠀⠀⢀⡴⠋⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠉
⠀⠀⠙⢷⣤⣤⣤⣤⣄⠀⠀⠸⣇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠘⢷⣤⡴⠟⠻⣆⣀⣠⡿⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠀⣿⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡿⣦⣄⣀⣄⣿⠀⣿⣄⣀⣠⣠⡿⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⡵⠸⠦⠜⠉⠉⠉⢻⡀⣿⠉⠉⠉⠹⠴⠇⢼⠟⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⢾⡇⣿⠒⠒⠒⠒⠒⠒⠚⠒⠒⠒⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⢀⡀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣰⡿⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣾⠃⠀⣼⡟⠀⢀⣾⠟⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡿⠀⢠⣿⠀⠀⣾⠇⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⠀⢰⣿⠀⠀⢰⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⢸⣇⠀⠸⣿⠀⠈⣿⡄⠀⠀⠉⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠈⣿⡀⠀⢿⣆⠀⠘⣷⣄⠀⠀⠘⢻⣷
⠀⠀⠀⠀⠀⠀⠸⣷⡀⠈⢻⣦⡀⠈⠋⠀⠀⠰⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠙⠃⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠶⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⠀⠀⡀⠀⣀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠞⠁⠜⠁⠀⠌⠀⠰⠁⠀⠸⠀⠇⠈⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡽⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠤⠤⠀⢀⣹⡴⠟⣫⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣴⡿⠋⣠⠤
⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⣆⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢁⣴⠿⢛⣵⠂
⠀⠀⠈⠉⠉⠉⠉⠉⠛⢻⣿⣾⣶⡭⠄⡀⢀⠀⡀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠛⢋⡡⠖⣉⣤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠘⠋⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⣠⡿⠛⠁⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠉⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠈⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠿⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⢻⡿⢷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⣿⠃⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣼⡏⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢰⣿⠁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢀⣤⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣾⡇⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⡿⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣼⡟⠀⢻⣧⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⣿⠃⣸⡿
⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⣰⡿⠀⠀⠀⢻⣇⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⢰⣿⠁⣼⡏⢀⣿⠃
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⢠⣿⠁⠀⠀⢀⠀⣻⣆⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⣾⠇⢰⣿⠀⣼⡏
⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠸⣷⡀⠀⠀⣸⡿⠀⣾⠇⢰⣿⠁
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⣿⣷⣾⣿⣤⣾⠇
⠀⠀⠀⢰⡿⠁⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⣿⡇⠀⣿⣧⣤⣉⣿⠟
⠀⠀⢠⣿⠃⠀⠀⠀⠀⣾⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡄⠀⠀⣼⡟⠀⠀⣿⡏⣹⡿⠃
⠀⠀⣾⡏⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⣸⡟⠀⠀⠀⣿⣿⡟⠁
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠀⠀⠙⠋


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

⠀⠀⡴⢶⠦⠄⠤⢀⠤⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠠⠄⠤⠄⠤⠶⠶⡄
⠀⠀⢲⠴⢄⢰⣶⣄⡿⣶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡾⢇⣴⣞⢀⠶⢴
⠀⠀⢠⣇⢩⣍⢴⣁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⣀⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⣱⢈⡼⢁⣧
⠀⠀⠠⡄⠂⠸⠎⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⣶⢶⠆⢠⢷⡀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠾⡁⠀⡦
⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⢠⠶⠛⠣⠀⠀⠀⠘⠛⠁⠀⢈⠥⣌⠀⠀⠀⠋⠉⠀⠀⠀⠀⢀⠠⠀⠐⠊⠉⠁⠀⠀⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⢘⠤⠏⠀⢀⠀⠀⠀⠀⠀⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠂⡘⠂⠀⠈⠷⡀⣀⡠⡤⠾⠋⠽⢀⡚⠙⢖⡀⠀⣤⠤⠀⠀⠀⡴⠃⠀⠀⠀⠐⠰⠋⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⠀⢳⡛⡀⠀⠈⣦⣀⡗⣾⢀⢠⣼⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠚⢁⠛⣷⠛⠶⣤⠀⢤⡆⠀⠹⣯⠞⠉⠘⠉⠿⢏⠁⠀⠀⠀⠀⠉⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡖⢁⠁⢀⠈⢁⣈⣀⣋⣥⡴⠨⣅⣨⣯⣷⣹⣿⠀⠄⢴⡤⣀⣀⠠⠀⡁⢀⠐⢀⣰⠂
⠀⠀⠀⠀⠀⠀⢠⠜⠁⠀⠉⠁⢠⢟⣯⠉⡟⠣⠀⠀⠈⠉⢹⠛⠿⡥⠀⠀⠀⠀⠸⠀⠀⠀⠈⠂⣉⡈⠙⣦
⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠠⠞⣫⣾⠀⠃⠀⢡⡀⠀⠀⠈⠀⡇⠈⠳⢦⣀⠀⠀⠀⠀⠀⡀⠔⠋⠑⠀⢘⡂
⠀⠀⠀⠀⠀⠀⠘⢧⣀⡄⠀⡒⠩⢾⡇⡄⠘⠆⠀⠈⠢⣄⣀⠀⡀⠷⠀⣤⠋⢻⠁⠀⣀⠈⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡟⡵⢼⡇⢡⠀⠀⡆⠀⠀⢸⡝⡾⢷⠂⢤⣅⣴⡆⠀⠈⠉⠙⠃⠂⡤⡀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣈⡹⡃⠀⠀⠀⢠⣶⠶⠛⢃⢧⣼⡇⠪⠀⠉⠃⠀⠀⠀⢀⣀⡤⠴⠿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠵⢠⠡⢿⡀⢀⣬⠁⠈⠁⠀⢸⡄⠙⢁⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⢠⡷⠀⠀⠘⠳⠤
⠀⠀⠀⠀⠀⠠⢧⡘⠃⠀⠀⠋⠀⠲⠞⡁⠀⠘⠇⠀⠀⠀⢸⠀⠀⣄⠡⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⢀⠀⣹⠁⣠⠏⠠⠀⠀⢸⠃⡀⠐⠒⢦⠀⠀⣇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⢀⡲⠀⠀⠀⠀⠀⠀⠐⢡⡔⢳⡶⠀⠀⣀⢸⠆⠀⠀⠀⠀⡇⠀⠙⢣⢄⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠆⠋⠀⠀⠀⠀⠀⠀⠀⠠⣜⡷⣄⢠⠀⠀⠀⢲⠀⢀⡠⠔⠊⠀⠀⠀⠀⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡍⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣉⡆⠋⠛⠀⢀⣠⠏⠀⢼⠀⠀⠀⠀⠀⠀⣀⡸⠅
⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠈⢥⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⢸⠿⠿⠒⠚⠋⠁⠀⠀⠀⠑⣤⣠⣤⢤⠴⠎⢫⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠘⠰⢦⣆⡀⢠⡅⠀⠀⣠⣤⢤⢾⠆⠠⣄⡀⠀⠀⣠⡴⠛⣩⡇⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⠆⠀⣀⣨⣿⢟⡀⠀⠀⠓⠀⡆⢐⡄⠲⠼⠀⠀⣘⠁⠀⠠⠋⠀⠀⠚⠃⠀⠆⠸
⠀⠀⠀⠀⠀⠀⠀⠘⢶⠀⣾⠀⢾⡟⠀⠀⠈⢿⡄⢤⡄⡀⠰⠟⠗⠀⣀⡴⠶⠇⠀⠀⠀⠀⣤⣄⠀⢘⠃⡠⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⢳⠏⢠⡈⠙⠁⢀⠠⠋⠀⡈⠁⠀⢹⣄⣐⠁⠈⢁⠀⠀⠙⠦⠀⠠⠋⣉⠀⠈⠛⠀⠀⠈⠋
⠀⠠⠀⠂⠀⠀⠀⠀⠀⢘⡄⠈⠁⠀⠐⠿⠀⢰⠞⠉⠀⠀⢸⠗⠿⠀⠀⠉⢳⡀⠀⠲⠆⠀⠈⠁⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠒⢀⡈⠓⢄⡀⠀⠀⠈⢷⡀⠀⢀⠀⣀⡠⠏⠀⠀⠀⠀⠈⠴⠂⠀⠀⠀⠀⠙⠒⡦⢤⡴⠋⠉⠁⠉⠫⠀⠀⠀⠀⣀⠐⠛⣄
⠀⠀⢠⡥⠀⡼⢹⠦⠀⠀⠀⠁⠉⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠹⡆⠰⣥
⠀⣰⢠⡗⣻⠲⣌⣦⡘⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠖⠈⠁⠉⠐⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡞⣠⣎⣴⠾⡓⣧
⠀⣠⣍⡌⣁⡜⣉⢥⣻⢍⡁⢀⠀⠀⠀⠀⢻⡍⣍⣉⡀⠀⠀⠀⡋⠠⠀⠀⠀⣈⣉⠍⣽⠃⠀⠀⠀⣀⡁⣈⢙⡣⢍⡙⠄⡩⣈⡇
⠀⣾⠈⠛⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠢⣤⠀⢀⡄⠺⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠋

⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣀⣤⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⠉⠛⠷⣤⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣴⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⢰⣿⣇⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⢀⣿⣿⣿⡄⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⣀⣁⣀⣀⣀⠀⠸⣿⣿⣿⣿⣿⣿⡇⢠⣦⣦⣤⣤⣤⣤⣤⣦⣦⣴⣦⣤⣤⣄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⢠⣿⣿⣿⣿⣿⣇⠀⢹⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣿⣿⣿
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⢠⣾⣿⠛⢻⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣾⣿⣿⣿⡆⠘⣿⣿⡟⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡀⠹⡿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡶⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⠛⠉⣀⣴⣷⣤⡈⠙⠛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣧⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⡔⠒⣒⣒⣒⣒⠒⡄⠀⠀⠀⠀⠀⠸⠟⢛⢛⡛⡛⡻⠇⠀⠀⠀⠀⠀⢰⢒⣒⡒⣒⢒⡒⢢
⠀⠀⠀⡇⠀⣿⠊⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⢰⢸⢸⡇⡇⡇⠀⠀⠀⠀⠀⠀⢸⢈⣇⡿⣸⢻⣄⢸
⠀⠀⠀⠑⠂⠉⠒⠊⠁⠐⠃⠀⠀⠀⠀⠀⠀⡸⠸⠨⠥⠇⢁⠄⠀⠀⠀⠀⠀⠘⠒⠀⠒⠂⠀⠒⠊



⠀⠀⠀⢀⠀⢀⠀⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⡙⢠⢖⠦⠉⠤⠀⠀⠀⠀⠀⠀⡎⢠⣶⡶⢰⣖⠆⢱⠀⠀⠀⠀⠀⢸⡟⠛⢻⣟⢻⠻⣿
⠀⠀⠀⠀⠒⡘⠮⠖⢠⡁⠀⠀⠀⠀⠀⠀⢇⠰⠿⠟⠸⠯⠆⡸⠀⠀⠀⠀⠀⢸⣷⣤⣾⣧⣿⣥⣿
⠀⠀⠀⠀⠀⠐⠇⠘⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡠⠤⠤⠤⠄⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⡾⠋⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠠⢱⣤⣭⣥⡁⠄
⠀⠀⠀⡇⢹⢧⡏⣾⢹⡆⡇⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⡿⠀⠀⠀⠀⠀⠀⠀⢀⠡⣟⣛⣛⣛⣧
⠀⠀⠀⠣⠬⠤⠥⠬⠭⠤⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠹⠿⠿⠿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁