Vintage Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠈⠋⠀⠒⠉⠉⠉⠘⠛⠛⠿⠿⠿⠿⠿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡀
⠀⠀⠀⠀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡖⠉⠉⣿⣿⣿⣶⣖
⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣿⣿⣿⣧⠀⢀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡀
⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⣻⣿⣿⣿⡄⠀⠀⣿⣿⠃⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠛⠿⠇⠀⢀⢀⣁⣄⣿⣿⡟⠀⠀⠀⢀⡀⠞⠻⣿⣿⣿⠏⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⠺⠋⣺⣿⣿⠉⠁⠀⠀⢀⣤⡻⡿⠶⠌⠉⠉⠿⠿⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⣻⣿⡄⠀⠀⠀⢈⣙⠛⠳⠀⠀⣀⠀⠀⢀⢙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢴⣿⣷⠀⠀⠀⠈⠛⠠⠶⠀⠀⠉⢳⠀⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⣦⣄⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣻⣿⣧⠀⠀⠀⠀⠀⢀⠀⡀⠀⠈⠉⢘⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀⠀⠀⠀⠀⠀⢈⣿⠆⠀⠀⠘⠿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢿⣿⡇⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣩⣿⣿⡀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣿⣿⣇⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠿⣿⣿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⠄⣽⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢛⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⢯⡘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢷⡟⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢟⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣈⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⢻⣿⣿⣿⣻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⣼⡿⠋⠀⣾⣿⠿⠎⣹⣿⣿⡇⠘⡿⣿⣟⠈⠻⡿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢠⣾⠿⠁⠀⣴⣿⠃⠀⢀⣿⣿⡏⠉⢹⣿⣿⣿⠀⠀⢻⣿⣿⡁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⣶⣿⠟⠀⢀⣼⣿⡏⠀⠀⣾⣿⣿⠁⠀⢠⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠂⢾⣿⡟⠀⠠⣾⣿⡿⠀⠀⢸⣿⣿⡟⠀⠀⢸⣿⣿⣿⡄⠀⠈⣿⣿⣷⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⢠⣿⣿⠁⠀⢲⠋⣻⠗⠀⠀⣼⣿⣿⡇⠀⠀⢸⣿⣿⣿⡇⠀⠀⢻⣿⣿⡆⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡗⠀⣿⣿⡇⠀⢠⣿⣤⣿⠀⠀⠀⣿⣿⣿⠄⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿⣾⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢠⣿⣿⠇⠀⣸⣿⣾⣏⠀⠀⢈⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠨⣯⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢈⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢐⣿⣿⠄⢸⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⢼⣿⣿⣿⡗⠀⠀⢰⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠘⣿⣿⠀⠀⢿⣿⣿⣇⠀⠀⢘⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⣿⣿⡗⠀⣿⣿⡄⠀⢸⣿⣿⣧⠀⠀⠀⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿⣿⡇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⣾⣿⣷⠦⠨⣿⣿⣿⡀⠀⠀⢸⣿⣿⡆⠀⠀⠸⣿⣿⣿⡇⠀⢀⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡶⢿⣿⣿⣿⠀⠀⠈⣿⣿⣿⠀⠀⠐⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⠚⣿⣿⣿⣧⢀⣀⣾⣿⣿⣦⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡠⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⣃⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢦⡈⡄⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠐⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹⣿⡟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠸⣿⡇⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣿⣥⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠄⠀⠀⡄⢀⡄⣠⣠⣤⣄⣀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡄⠀⠀⣿⣿⣧⣽⣿⣿⣿⡿⣆⠈⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠈⠻⠿⠿⠿⠿⠏⠀⠈⠀⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠀⠀⠀⠙⢻⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⡄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡖⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠀⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣤⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⢛⡶⠀⠘⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣅⠀⣠⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢠⠛⠀⠀⠈⠉⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠲⠶⠖⠦⠟⢧⣉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠰⠀⠈⠀⠀⠀⠀⠀⠐⡄⠃⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⢠⠴⠒⠈⠉
⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣶⣎⠁⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠲⠶⣤⠊⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠀⠀⠀⡂⠀⠃⠀⠀⠀⡀⠀⠀⠀⢠⠅
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤⣿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠄⣘⡿⠟⢠⣶⣄⣲⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠢⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡝⢿⠁⢀⣀⣀⡀⠀⠀⢀⣀⠀⠀⠰⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡏⢠⣿⣿⣿⡿⠀⠀⠈⣾⠆⠀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⣼⣿⠋⠀⠀⠀⠀⠛⢀⠎⠀⠀⡠⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣷⠷⣦⢀⠘⠁⠀⠀⠰⠀⠀⠑⠊⢀⡴⠒⠒⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠙⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣛⡛⠆⠀⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢾⡄⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠋⠉⠉⠓⠀⠀⢰⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⢿⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣦⠀⠘⠛⣛⡿⠶⣄⠀⢀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠀⠀⠀⢠⣾⡿⠟⢋⣟⠉⠋⢾⡻⣗⡀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⣶⠤⠘⢋⣶⢞⣻⣿⣁⠀⠀⠀⠈⠓⠀⠀⠂⠠⡤⠒⠦⠜⠃
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠾⠳⠒⣯⣿⣿⠁⠀⠀⢀⣉⠀⠀⠠⠀⢀⡀
⠀⠀⠀⠀⠹⢿⡉⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⣯⠟⠃⣠⣶⠞⡉⠹⠀⠀⡄⠀⢸⡷⠖
⠀⠀⠀⠀⠀⠀⠉⠙⠓⣄⠀⠀⠀⢀⠀⠀⡐⠀⠀⣠⢿⠁⣄⡟⠉⢫⠿⣧⣀⠀⠀⠀⣀⢈⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⢀⠀⠞⠁⠀⠀⣀⣀⠁⢾⠀⠀⠀⠀⣿⣿⠆⠳⣦⣿⣦⠈⠁⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣸⡿⠇⡎⠀⠀⠀⠀⠈⠻⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤⣄⣠⣿⣾⢶⡀⢸⣇⠀⢰⠷⣶⣤⡆⠀⠀⣠⣄⡀
⠀⠀⠀⠀⢀⣤⣴⢶⢶⡄⠀⣀⣸⣻⣿⣇⠙⢿⣦⠻⡄⡞⢹⣆⠁⠀⢻⡿⠃⠀⣿⣉⠋⣥⣴⡿⣓⣬⠶⠦⠄
⠀⠀⠀⠘⠋⠁⠀⠁⠉⠴⠟⠛⠛⠛⢛⣯⡤⠄⠩⡄⠠⡇⠀⣿⡄⠀⣨⢤⣦⣤⣴⣤⣄⠀⠉⠙⠻⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠷⠂⠀⠀⠀⢰⠀⠈⣣⠀⣀⣀⣼⠾⠽⡟⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⠀⠂⡆⢠⠁⠈⠙⠻⠟⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⡶⠛⠻⢿⣷⣤⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠠⣠⣾⡟⠁⠀⠀⢘⣷⣿⢿⡄⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⡃⠀⢁⣬⣿⣿⠅⣸⣿⣄⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣿⠛⠒⢂⠘⣿⣿⠀⣿⣿⣿⣀⡀⢀⢠⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠁⣰⠀⠀⠈⠉⠨⡀⠀⠀⠐⣨⠃⢲⡙⠛⠫⣽⣆⡀⢨
⠀⠀⠐⠆⠈⢉⠛⣋⠲⠖⠛⢋⣽⠀⠀⠘⣧⡩⡉⢛⢻⣟⣲⠿⠃⠉⠀⢸⣷⡄⠀⢾⣿⣾⣿⣦⠤⠤⠤⠤⠤⠄
⠀⠀⠀⡇⠀⠄⠀⠱⠈⠀⡹⠎⠀⠀⠘⡟⢿⣥⣿⣿⣿⣿⡑⠂⠀⠀⠀⠜⠀⠹⠀⠈⠁⣿⣿⣟⣐⠤⠄⠀⠁
⠀⠀⢸⢥⠤⠴⠤⢤⣴⠎⠀⠀⠀⠀⣰⣳⣿⣷⣿⣿⠏⠀⠈⠀⠀⣤⡀⠀⠀⢺⡧⠀⠈⢿⣿⣿⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠟⠀⢉⡳⠋⠁⠀⠀⠀⢀⣠⣼⣧⠤⠀⠀⠀⠀⠀⢼⣿⣧⢠
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢤⣥⡸⣬⡄⣿⣀⠀⣀⣠⣶⣿⣿⣿⣭⡑⠂⠀⠀⠀⠀⢀⣿⣿⣷⡒⠂
⠀⠀⢰⣷⣶⣾⣿⣶⣶⣶⣶⣶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣤⣵⢶⣶⣾⣿⣿⣍⠛⠐⠤⣄⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⠷⠖⢿⠛⢿⣿⣿⣿⡟⣷⣤⡈⠇
⠀⠀⢘⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⣿⡿⢛⡿⠉⠃⠔⠋⠀⠀⠘⠀⢰⢸⣿⣿⣿⣿⣿⣵⡆
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡏⠁⣠⡴⠖⠫⠉⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⠀⣧⢻⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⡦⠙⣿⣿⣿⣿⣿⢿⣹⢿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣵⣶⣿⣿⡿⠋⠀⡘⠁⠀⢠⣿⠃⠐⠀⢀⠀⠀⠢⠤⠀⡀⠀⡀⠀⡀⠀⠀⠂⣀⡍⢸⡟⣿⣿⣿⣧
⠀⠀⢨⣥⠼⣿⣿⣿⣾⠛⣴⠂⠁⠀⠀⣹⡝⠀⠀⠀⠈⠀⠀⠀⠀⠉⠱⡶⠟⠁⠄⠀⣠⣾⣿⣄⣴⣧⣿⣿⣿⡗
⠀⠀⢸⣿⣿⡷⢶⡟⠛⢿⣏⣓⣦⡾⢭⣽⣗⣤⠀⠀⣀⢀⣀⠀⠀⡴⠟⠏⣅⠀⢂⣿⣿⢷⣿⠟⣿⠉⢻⣿⠑
⠀⠀⢸⡟⣿⠀⠀⠛⠄⠀⠻⣿⣿⣿⣿⣿⣾⣿⣭⢟⣺⣭⣥⣥⣭⣵⣶⣶⣶⣴⣿⡿⣟⣻⣿⣿⣿⣿⡿⣿⠀⠁
⠀⠀⢸⡷⠋⠀⠀⣠⣇⣠⣤⣽⣣⡶⠟⠛⠛⠛⠛⠛⠛⠛⠈⠉⠉⠿⠟⠻⠟⠛⠛⠛⠙⠛⠻⠿⠟⢿⡇⣿⣷⡆
⠀⠀⠸⠁⣀⣩⣾⣿⣟⣁⢀⡘⣧⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠇
⠀⠀⢰⣾⡿⠟⠛⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠓⢦⣀⣀⣠⠔⠋⠁⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠀⣠⠾⢉⢙⡙⢶⡄⠀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠐⠀⠨⢍⠲⠦⣄⡀⠀⠀⠀⠀⢹⡆⢿⣷⣄⣶⣴⣿⠇⢠⡿⠁⠀⠀⠀⣀⣠⠔⠊⠀⠄⠀⠈⠈⠑⡆
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠁⠈⠁⠀⠀⠐⠀⠉⠓⠤⢄⡀⠀⢹⣿⣩⡿⣿⣻⠻⣿⠛⠁⣀⡤⠖⠋⢁⠀⡄⠀⠐⠀⠂⠁⠈⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡐⠐⠀⢀⢀⡀⠀⠀⠀⠀⠉⢙⡚⣷⣿⣇⠍⠁⠈⣛⢐⣋⠁⠀⠀⠀⣁⡀⠁⡀⠠⠂⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢒⠊⠔⠐⠒⠒⠐⠂⢂⣠⢼⣿⣿⣿⡾⣞⣾⣧⢄⣀⡀⠁⠒⠒⠢⠤⠀⠁⣤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⢚⣿⠋⢀⣙⣿⣿⣿⣟⣿⣿⣗⡈⠹⣯⠓⠂⠤⠤⠤⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠁⣸⡟⣿⣿⡿⣿⣿⣽⣿⣿⡟⢷⡉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⢰⣻⠀⣿⣿⣷⣟⣛⡛⣹⣹⡇⢸⣱⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣻⠀⢻⣿⣟⣷⣿⣓⣿⣿⠇⠀⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠃⠀⠘⣿⣿⡷⣿⣷⣿⡟⠀⠀⠀⠙⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⠿⣯⣿⡿⠟⠀⠀⠀⠀⠀⠀⠙⣧

⠀⠀⠀⠀⠀⠠⠄⠤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢙⣿⣿⣷⣶⣤⣀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣤⣤⣈⣻⣿⣿⣿⣿⡿⢲⣤⣾⣿⣧⡻⣿⣿⣿⡿⠁⣿⡿⠿⣿⣶⣶⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣛⡻⢿⣿⣿⣿⠟⣉⣴⣿⣿⣿⣿⣿⢇⣟⣿⣿⣷⠀⣏⢸⣇⢸⠟⠋⠁⣹⣯⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⢿⣿⣿⠋⢡⣶⣿⣿⣿⣿⣿⣿⡟⣌⢿⡿⠉⠯⠀⠙⠷⠖⠋⠀⠀⡴⣮⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢐⠲⠿⢏⣽⣿⠏⢡⣼⣿⣿⣿⣿⣿⣿⣿⣏⠐⠸⣉⠁⠀⠀⠈⠢⣄⠀⠀⠀⢀⠀⠀⠈⠝⠁
⠀⠀⠀⠀⠀⣀⡤⢴⣶⣿⡿⢡⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠈⠀⠀⠀⠀⠀⢸⡀⠀⠀⠫⢀⡠⠊
⠀⠀⠀⠀⠀⠀⢈⣨⣽⡟⢀⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢇⠀⠈⠃⣀⡠⠤⣄⠙⠲⢤⡴⠋
⠀⠀⠀⠀⠐⠚⠩⣻⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⠃⠸⣳⢄⡀⠀⠒⠀⠈⠝⠋⠁
⠀⠀⠀⠀⠀⣀⣴⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⡀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠈⠄⣻⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠠⠻⢿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣠⣦⣴⠞⠉⠑
⠀⠀⠀⠀⢀⣴⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⢿⠀⣰⣟⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠈⣽⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⡽⠯⢿⣿⡻⠜⣅⣸⣽⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠜⠙⡄⣿⣿⣿⣿⣿⣿⣿⣿⣯⠕⠳⠀⠻⠞⢿⠮⠞⠉
⠀⠀⠀⠀⠀⠀⠼⣗⢸⣿⣿⣿⣿⣿⣿⡿⣷⢈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⣄⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣆
⠀⠀⠀⠀⠀⠧⠀⢸⡎⡿⣿⣿⣿⣿⣿⣿⣧⢢⡌⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠛⠛⠛⠻⠿⢾⣝⢮⣻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠈⣲⠃⢸⡘⣻⣿⣿⣿⣿⣿⣿⡟⡠⠄⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡙⠟⢿⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠽⠨⢿⣿⣿⣿⣿⣿⡾⣽⢃⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⡗⡏⡇⠀⠀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⣿⡞⢻⣿⣿⣿⣿⣿⣧⠧⡤⠀⣤⡴⠿⠿⠋⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿⣿⠀⠀⠀⠀⠀⣴⢻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣿⡏⢰⣜⢿⣿⣿⣿⣿⣿⣳⡅⠈⠛⠟⠉⠉⠿⣦⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠹⣿⣠⣤⣴⡶⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢣⠈⣿⢆⣙⠻⣿⣿⣿⣿⣿⣧⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠊⣸⡈⣟⡧⠉⠿⣿⣿⣿⣿⣿⣥⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠈⠀⢀⣠⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠄⣤⣠⣀⣉⡙⠻⣿⠿⠿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣷⡾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠳⠾⠶⠿⠾⠿⠷⠞⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⡠⣶⣶⣿⣍⠉⠐⠒⠂⠤⣄
⠀⠀⠀⠀⠀⡀⠀⠈⠙⢿⣿⣿⣶⣦⣐⢀⡤⣠⠀⡄⠀⠀⠀⠄
⠀⠀⠀⠀⡒⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠒⠒⠒⠒⠆⠀⡀
⠀⠀⠀⣔⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠈⠩⠐⣂⡲⠆
⠀⠀⠀⢪⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣠⣤⢾⣿⡿⠃
⠀⠀⠀⠸⡀⠀⠀⠀⠐⠐⠒⠂⠀⢽⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣿⣿⠟
⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⣄⠂⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠈⢦⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣼⣻⡅⡮⣀⢘⢏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠹⣿⣛⡾⣿⡽⡒⡋⠀⡃⠀⠀⠀⠀⠀⡠⠞⠉⠈
⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⣀⣀⣀⣠⣄⣀⠉⠓⢾⢿⣿⣜⠧⢶⡠⠓⢠⠟⠀⠀⠄⣀⣀⣠⠄⠀⠂
⠀⠀⠀⠀⠀⠉⠉⠉⠻⠿⢿⣿⣿⣿⣿⠿⠛⠒⣾⣿⣿⣿⣿⣛⣁⣤⠟
⠀⠀⣰⡇⠀⠁⠂⢀⣤⣴⣾⣿⣿⡟⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⢸⣆
⠀⢰⣿⣃⣠⣴⣾⣿⣿⣿⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⢸⣿⣿⣿⣿⡿⠟⠋⢁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠱⡀⠈⠙⠲⠀⠀⠀⢀⣀⡀⠤⠤⠀⠚
⠀⠘⠟⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠂⣬⡶⠿⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⡄⠀⠀⠀⠐⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢸⠛⢛⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢘⡰⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣈⢀⣀⣠⣷⢿⣰⣧⣠⣠⢪⡦
⠀⠀⠀⠀⠐⠖⠶⠶⠤⣤⣤⣌⣀⡉⠈⠉⠀⣿⣿⣿⣷⣿⠃
⠀⠀⡄⠀⠀⠀⠀⠀⢀⣀⣈⣉⣉⣭⣽⣿⣿⣿⣿⠟⠁⣿⡄⠐⠄⡀
⠀⠀⠀⠀⠀⠐⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⠁⢀⠀⠈⠻⠷⣤⡄⠕⣀⡀
⠀⣾⣟⠁⠀⠀⠀⠀⡠⠾⠙⠹⣋⡼⠞⣿⣿⣿⣁⡈⠙⠲⠄⠀⠀⠀⣾⡭⠿⠔⡖⠠⢀
⣼⡿⠛⠃⢠⣤⡀⠾⢦⣠⣾⠞⠋⠤⡼⠋⠀⠈⠉⣹⡆⠀⠀⠀⠀⣀⠀⠄⠄⣼⣿⡤⠟
⠿⠃⠀⠠⣄⡉⠻⠿⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠽⠄⠀⣨⠈⠁⠀⠀⠀⡀
⠀⠀⠀⠰⠋⢻⣷⣤⣔⣶⡶⣶⣶⢳⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠉⠐⠂⠛⠛⠂⢷⣿⢧⣝⠋⠚⠃⠀⢰⣟⣧⠞⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠌⠉⠑⠀⠀⠛⠛⠻⣷⣤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⣠⣴⡦⣦⡀
⠀⠀⠀⠀⢰⣿⡟⠠⠘⣁⢀
⠀⠀⠀⠐⠸⣿⠟⠀⡀⠉
⠀⠀⠹⠈⠘⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⡞⢈⡙⢷⡄⠀⠀⠀⠀⢿⠛⠿⣿⣿
⠀⠀⠀⠰⠄⠀⠀⠀⢀⠀⢘⡄⠀⠀⠀⠀⠀⠀⣼⡟⠉⠀⢁⡅⠀⠀⠀⠀⠈⠀⠀⠘⠫
⠀⠀⠀⠀⠈⢴⠀⢊⠗⢋⠿⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⡄⢀⠀⠀⠀⢠⢄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣄⢀⢠⣾⣥⠃⠀⠀⠀⠀⠘⠃⠀⠠⣦⠓⠀⠉
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣥⠐⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠰⠂⣁⣤⣤⣤⡁⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢩⠁⣸⠋⣿⡏⠀⠀⠀⠉⠁⠀⠀⠀⠄⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠙⡁⠠⠏⢰⡇⢠⣿⠃⠀⠀⠄⠀⡄⠀⡀⢦⡀⠀⠀⢖⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠙⣄⢀⣼⡇⣼⣇⠀⠀⠀⠀⡀⠀⠀⠀⢨⣷⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⢏⣿⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⠈⡆⠀⠀⢐⡆
⠀⠀⠀⠀⢤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⣽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡦⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣽⣿⣿⡟⠀⠀⢶⡆⠀⣴⠂⠀⢀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⡻⣿⠁⢀⠴⠉⠐⠂⠀⣶⠄⢈⣂⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠥⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠛⠂⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⡧⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⡴⠚⢉⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠶⠋⠘⠓⠛⢷⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣿⠠⣴⣶⠶⠒⢺⣿⣿⣿⣿⡀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢀⣽⣶⡸⠇⠀⠀⠀⠻⣿⣿⣿⠃⢼⣇⠀⠂⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⣛⣟⡸⣷⣾⣄⠀⣀⠀⠀⠀⠘⣧⠈⣥⣞⡈⣻⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠸⣿⣗⢿⣿⣿⣿⣿⣶⣶⣆⣰⡷⠀⢀⠋⢡⣭⠤⠤⠴
⠀⠀⠀⠀⠀⢀⣀⣤⣠⣀⡐⢾⣃⠀⢸⣿⣮⣍⠻⢿⡿⠿⠻⣽⣋⢁⣰⠞⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠉⢀⡤⢒⡨⠽⠆⢻⡧⠀⠐⠉⠻⠟⠾⣶⣶⠖⠈⢻⣿⣿⠀⠀⠀⠂⠈⠂
⠀⠀⠀⠀⠀⠀⠑⠚⠁⠀⢀⡄⠘⡁⢈⡏⠀⠀⠈⠻⠟⠋⠀⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡠⠂⠀⠐⢼⡇⣀⠀⢀⡀⠠⠠⣤⣴⣿⣟⢃
⠀⠀⠀⠀⠀⠀⢄⣋⡠⠛⠀⠀⠀⠀⠘⣯⠀⠀⠀⠙⢶⣦⡀⠉⠛⠿⠿⠵⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠁⢀⡑⢵⡉⠒⠲⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠈⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⠈⠀⠊⣈⠁⠀⠈⢀⠀⣈⠀⠀⠒⠋⠀⢀⣀⡀⠀⠀⣦⣶⣴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠔⠋⠤⢁⡈⠀⠊⠀⠘⠊⠀⢘⡧⢐⣂⠀⢤⢿⡿⠃⠀⠀⣟⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠤⠤⠀⠉⡻⠧⡁⠾⢒⣤⡩⠂⠈⢾⣿⣿⡈⣵⠀⠲⣆⠈⠉⠀⢀⠀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⢠⠄⠛⠀⠀⠀⠀⠀⠒⠆⡛⢳⡄⠀⠄⠈⡙⠷⡀⠀⠀⠈⢛⠈⠳⠠⠄⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣠⡄⠳⠖⠀⠀⠉⠀⢠⠘⡐⢉⡦⣄⡀⠀⣠⠜⠉⡀⠀⢠⡄⠀⠀⠀⡀⠀⠛⢩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⡼⡇⢸⡶⣵⣠⡄⠀⡏⠼⠠⢸⡿⠃⠙⠛⢷⠀⠀⠀⠀⠺⢏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠋⠈⠿⠿⢏⠀⠀⠀⠀⠀⡘⠁⡀⠤⠀⠛⠀⠀⢀⠑⢶⡀⠰⡄⠀⠀⠉
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠠⠴⠆⠀⠆⠀⠀⠀⠀⢀⡄⠀⠈⠠⢄⡀⢈⡐⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡞⣦⡀⠀⠀⠁⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⠛
⠀⠁
⠀⠄
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣥⠶⢏⡠⡴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣼⣷⣶⣶⣾⣷⣿⠀⠀⡄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣴⣿⠿⢿⣿⣿⣷⣤⣥⡀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣧⣘
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣌⣿⣀⡹⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⣯⣋⣽⣯⡁⣉⠛⣋⣻⣿⣿⠃
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣯⡔⢊⡽⠟⠁⠙⠫⢤⢋⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾⣵⠃⠀⡀⠀⣤⠄⢋⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣟⣿⡄⠛⠁⠀⠀⠀⢸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡏⢻⣯⣷⣄⠀⠐⠀⢀⣮⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠉⢮⣽⡏⠀⠀⠀⠀⠿⡃⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠏⠁⢀⠰⠷⠊⠀⠀⠀⠀⠐⣷⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⢘⠀⠦⠞⠀⠀⠀⣀⠀⠀⠘⣅⢀⡀⡅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠂⠀⠙⠢⡘⠳⠄⠀⢹⣇⠀⠀⢿⡠⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣦⡀⠐⣤⠀⠀⢻⡷⣞⢻⣏⣿⣿⣿
⠀⠀⠀⠀⠀⠂⠀⢀⠀⠘⣿⣿⣿⣿⣿⣿⣶⡞⣧⠀⣘⣅⣿⣿⢿⣿⡿⣿⣧⡀
⠀⠀⠀⠀⠰⠀⠀⡊⠀⢃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⢷⣾⣿⠷⢾⣷
⠀⠀⠀⠀⠀⠀⣀⣤⠾⢷⣟⣨⣻⣿⠟⣿⣯⣿⣿⣿⠟⠁⠀⠀⠀⠹⣟⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠛⢰⣿⣿⣿⣿⣿⣿⣿⡿⠉⠙⠻⣿⣶⣄⡀⣀⣀⣀⡬⣥⣤⠄
⠀⠀⠀⡄⠀⠀⣰⣿⣿⢧⣾⣿⣿⢿⢿⣧⡀⠀⣤⠈⠙⠻⣛⣉⣡⣠⣴⣲⢿⡆⠀⠐
⠀⠀⠀⠀⠀⣠⣽⣿⣿⣾⣿⣻⣿⡀⣾⣿⣿⣷⣬⣞⠃⠀⠈⠙⢤⣈⠍⠁⡼⡁⢀⠀⡄
⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠛⠓⠄⠤⠾
⠀⠀⢀⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⣤⣤⣀⣿⡤⣤
⠀⣼⠛⡻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣯⣴⣾⣿⣿⣿⣿⢿⣿⣿⣿⣷⡄⢀⠀⣿⡿⢛⣿⠀⠈⣤⠤⠤
⠀⣷⣾⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣼⡟⠸⣿⠀⢠⣤⣀⣤⣀
⠀⠉⣻⣿⡟⠉⣿⣦⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠀⣿⠀⠀⣿⣿⣿⣦⣀
⠀⢀⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡈⢻⠀⣿⠆⠀⣿⣿⣿⣿⡯
⠀⢹⣿⣿⡏⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣗⠃⠈⣄⢸⡄⠰⣿⣿⣿⣿⣷
⠀⠀⢹⣿⡿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⠀⣿⡈⠁⠀⣿⣿⣿⣿⣟
⠀⠐⠛⢿⡇⠘⣿⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢗⣠⡄⣾⣿⣿⢟⣣⡆
⠀⠀⣶⣿⡇⠈⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇⣸⣛⠁⠘⣿⣿⣤⣽⡅
⠀⠘⣿⢿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢀⣼⣿⣿⣿⠆
⠀⠀⠀⠀⠃⠀⣿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠘⠁⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣮⣻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠚⢀⣹⠈⣻⢿⡏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣝⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠛⠻⠿⠿⠿⠿⢿⣿⣿⣇⡼⠿⠿⠿⠿⠛⠻⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣒⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣻⡟⠊⠛⢻⡛⠙⠻⡛⠻⣟⠛⠿⣿⠛⠿⢷⣄
⠀⠀⠀⠀⠰⠿⠿⠿⣿⠿⠿⢿⣿⠿⠿⢿⣷⠾⡾⡷⠶⣾⣿⣷⡶⢶⣧⡴⢾⣦⣤⠼⣷⣤⣤⣬⣷⣄
⠀⠀⠀⠀⠀⠈⠢⠀⢀⠑⡄⢄⣈⣦⠀⠀⢳⡀⠀⡁⠀⡇⠌⡖⠀⠀⠌⠀⣠⢚⠵⣪⠕⠁⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢧⣄⠀⠅⠀⢁⠀⡇⠀⠀⠀⣴⢁⣼⠯⠒⠁⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠉⠙⠬⡀⢠⠀⣷⣰⣶⡿⠇⠙⠁⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⠛⠁⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⡀⣴⢧⢮⣧⠀⡐⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣘⣿⣶⣲⣽⣷⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢕⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠻⣮⣱⣿⠏⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣴⢶⣦⡄⠀⢠⡆⠈⠁⢬⣿⠏⠀⢠⡏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠀⠀⠀⠀⠙⢿⢿⣿⡋⠀⠙⠗⢤⢦⣄⡀⠀⣼⡁⠀⣠⠏⡁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⢀⣤⡀⠈⢏⣛⣹⡆⠉⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣐⠀⠀⠄⣀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠹⢾⣇⠀⠈⠛⠻⣽⣶⡦⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⢅⠀⠙⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣠⣿⠀⠀⠀⢢⣹⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠢⡀⠀⠀⠀⠀⢀⣟⣿⣿⡇⣿⣿⣧⢀⣒⣸⣿⡯⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠀⠀⠀⠀⠀⣻⢻⣿⣇⣽⣿⣿⣇⢉⣿⣿⡋⡄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠛⠧⠀⠀⣀⠀⠉⠀⣿⠁⠋⠏⣿⢹⣷⣿⣿⡇
⠀⠀⠀⠀⠀⣠⡀⠀⢠⣤⣤⣤⣤⡤⡀⢰⡷⡟⣾⣻⡄⠀⠀⢠⢿⣷⣦⣄⣈⠀⠀⠀⢸⡎⢻⣿⣿⣇
⢀⣤⢶⡀⣰⣿⠘⢆⢟⣷⣇⣞⣻⣄⡈⣨⣯⣹⣿⡟⠁⣤⣶⢟⡛⠉⡸⠃⢹⣿⣶⣶⣦⣤⣀⣿⣿⣿⠀⢠⠀⢀⡿
⠨⣿⣿⣿⠋⢋⢠⢞⣬⡿⣍⡽⢛⣯⠞⠣⠊⢹⣇⣠⣶⣿⣿⡌⠑⣤⠈⠀⠘⠋⢿⣿⣻⣿⣿⡿⠿⠛⠁⠸⠟⠛⠀⠀⢂
⠀⠉⢛⣏⣴⣦⢟⣼⣻⣝⣊⣼⣦⡥⣾⣽⣿⣿⡟⠻⣶⣭⣸⣇⠙⠋⠀⠀⢀⡀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠀⠀⣧⣽⡟⠳⢦⡾⣿⣧⣦⢛⣹⢷⣿⣽⣏⣼⣴⣧⣌⡛⣿⢹⡗⢠⠀⠀⠀⠀⠨⡇⠀⠀⡀⠀⢤⣤⣄⣂⣻⣄⣤⠄
⠀⠀⠙⠿⢛⣻⣖⠛⣪⣇⣹⣾⠣⣤⣿⢿⡿⠿⠯⢛⣿⠹⢡⡀⠿⠿⠀⠀⠀⡀⠀⠀⠀⠾⠀⠀⠈⠛⠥⠔⣫⣈⡅
⠀⠀⠀⠀⢠⣾⣿⠃⡿⢀⣾⣩⠒⣾⣾⢿⣿⡶⢒⡐⠋⢶⢻⡍⠁⡀⠀⢀⣀⢙⣶⡤⠦⣄⠀⣀⡒⠀⢀⡈⠹⢿⠿⡷
⠀⠤⠄⠀⠹⠿⠿⠿⠧⣨⢿⣛⣿⣧⣿⣮⡯⣑⡿⠿⠷⠞⠓⠀⠀⠀⠀⠛⠛⠛⣿⣦⡴⠿⠾⠿⠗⠒⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⢀⠀⠀⠀⢤⡠⠀⠀⠁⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣠⡀⠀⠀⣠⠄⠩⢁⡔⠐⠷⢷⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣾⢇⡶⠃⠀⣠⠀⠂⢙⣶⣀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠹⠟⡹⢈⠀⠾⣿⣆⠄⠉⢉⣹⣿⣷⡟⠂⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢼⡇⢀⠀⠀⢉⡇⢠⠉⠘⠋⠀⠾⠿⠛⠉⠠⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢈⡳⢌⢦⡄⣼⢟⣤⡴⢶⡞⣧⠀⠀⢀⡔⠉⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣼⠤⠈⢧⢱⠈⡇⡾⠰⣿⠃⠐⣾⠳⣮⢹⡖⠊⡁⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠲⡫⠞⠃⢀⢹⣷⡄⢅⠀⠀⠀⠳⢀⠀⡁⢼⢗⢒⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⡢⢴⠞⠫⠜⣏⠛⢊⠙⠿⠷⢶⢟⢼⠅⠀⡈⠉⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠇⠨⢁⠈⠀⢀⠟⠛⢦⡀⣀⣴⣤⣤⠄⠂⠀⠄⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡲⠠⢀⠐⠋⢑⡆⠙⠟⠈⠀⠀⠀⠀⠀⠀⠾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⡀⢤⡾⢾⣯⡷⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠔⣫⣵⣏⢱⡅⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⡋⢈⣿⡿⣫⡝⠀⠀⠀⠀⠀⠀⠀⠄⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⡷⠩⣋⣁⣤⠀⠀⠀⠀⠀⡀⠀⢠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢏⠉⢉⣳⣿⠛⣰⣇⣼⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⡤⠀⠀⠀⠂⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⢤⡀⡀⠀⠀⣀⡀
⣀⣾⡞⣧⣀⠀⠀⢀⠀⣠⢹⡆⠀⢸⡄⠀⠀⠀⣶⠹⡦⣤⢀⣠⣧⡄⠄⣾⣆⣀⣷⢸⡇⠀⡄⣿⠉⠙⣿⠄
⠙⣿⡧⠿⠟⠶⠿⣾⣷⣿⣿⠃⣴⣾⣇⠀⣠⣼⢿⣾⡿⡋⠅⠀⠀⠀⠀⠀⠀⠉⠛⢾⣿⣷⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠟⢿⣿⡿⣋⠔⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣍⢹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇⡇⠀⣀⣠⠴⠶⠤⠤⡀⠀⠀⠀⠀⠀⠀⣠⡤⠄⠀⢉⠛⠋⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡢⣄⢀⡀⠇⡼⠉⢀⠃⣃⡝⡲⡙⢦⠒⣀⣀⣼⠏⠀⡠⣤⣤⡄⠁⢄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣖⠍⡂⢶⣿⠀⠛⡰⣯⣷⡒⣦⢸⠿⠿⢿⡇⠀⢘⠉⣟⣷⡘⠀⢸⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠺⠤⠼⢿⣆⠐⢆⣸⣈⡴⠏⡈⠀⣀⣈⣧⡀⠘⢄⢘⡉⠌⢠⠈⠁⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠲⠤⠄⠐⠛⠛⠿⠿⠟⠛⠛⠢⠤⠤⡤⢴⠾⠾⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⡠⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⣀⠀⠀⠀⠀⡈⠀⠁⠀⠀⢀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⠁⠀⡠⠽⣶⡀⠀⠀⠐⡄⠀⠀⠀⠈⠀⠀⠈⠷⢄⣀⣀⠀⠀⣀⢀
⠀⠀⠀⠀⢀⣀⣼⡇⠀⢛⡆⠀⠀⠀⠀⢸⣇⣤⣔⣛⣓⣦⣄⡀⠀⠀⡈⠀⡸⣷⣆⠈⠉⠉⣉⢻⣆
⠀⠀⣠⣾⢿⣿⡟⣷⣴⣾⡇⠀⠀⣢⡶⣿⡿⠛⣻⣿⠿⢿⣿⣿⣦⣼⠀⢨⣿⣿⢸⠀⠀⢠⠁⢸⣿⡆
⠀⠀⠉⠩⠺⣿⣧⣿⣿⣿⣿⠀⣄⣿⣿⣿⣶⡿⠉⠀⠰⢖⡊⢻⣿⣿⣇⢸⣟⠃⣸⣤⣤⣼⣷⣾⣿⠃
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⡟⠀⠀⣦⣾⡫⢷⠈⣿⠿⣿⣿⣿⣦⡿⠟⣛⠉⢽⣿⡽
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣻⣿⡇⠀⠀⣿⣿⢟⡇⢰⡁⠀⠀⠀⠈⠙⠋⠉⠀⠀⠈⠛⠃
⠀⠀⠀⢀⠀⠻⢿⣿⣿⣿⠿⠿⠿⠿⢿⣿⡇⠀⠀⡽⣿⡿⠀⠘
⠀⠀⠀⠀⠁⠀⠀⠒⠋⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠧⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠌⠢⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣁⣭⣿⣿⣾⣿⣿⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣉⢻⣿⣿⣿⡿⠟⠉⣙⡛⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡆⠻⠞⢛⣋⢠⣴⣾⣿⣿⣿⣷⣜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢀⠛⢻⡿⠟⠋⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠑⠽⠿⠿⠖⠁⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠙⠲⠀⠀⠀⠀⠤⠖⠢⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⣤⣤⡲⠀⠀⡴⣠⣤⣀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠈⠉⠀⠀⠀⠀⠀⠈⠉⠛⠁⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣁⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣏⣹⣷⡆⠀⠀⠰⣞⠛⠛⢳⣦⠀⠀⠀⣷⣿⣿⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡄⠀⠀⠉⠛⠛⠋⠁⠀⠀⣰⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣟⠻⣿⣶⣬⣝⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠙⣿⣧⠀⠀⠀⠉⠉⠉⠙⠋⠻⣿⣿⣷⣞⠙⢿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣽⣿⣿⣏⣸⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡻⢿⣇⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣶⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠉⠺⢿⣿⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⣄⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣯⡟⣿⠭⠩⣷⡀⣴⡟⠅⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⢿⡿⣿⠵⠁⢳⠖⠑⣿⣀⣿⣧⠴⢥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣛⡵⠒⠋⣩⣥⣄⠀⢸⣠⠴⣿⢟⠍⠀⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠁⠀⠀⠀⡼⠀⣀⠿⠓⠉⠀⢰⡏⡎⠀⣠⣦⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣠⡤⠤⢀⣀⠤⠛⠈⠀⠀⠀⠀⠀⢸⣇⣇⠀⠙⠟⠀⣸
⠀⠀⠀⠀⠀⠀⠐⠁⠀⣀⠔⠫⡄⡇⡔⠉⠀⢀⣀⠤⢒⣊⣭⢴⣞⣿⢿⣮⣢⠀⢀⣴⣇
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠀⠀⠇⡇⣇⣐⣩⣵⣶⡿⢿⣛⡯⢿⡿⣥⣤⣭⠿⠿⠯⠥⠬⠷⢤⣤⣤
⠀⠀⠀⠀⠀⣈⣭⣿⣿⣿⡄⠀⣀⣧⢼⣟⣿⣽⡶⠋⠛⣝⡇⢸⡇⢿⣇⡠⠀⠀⠀⠀⠀⠀⠸⢧⣠
⠀⠀⠀⠀⠈⠭⢭⣿⣿⣿⣷⢹⣷⣿⣿⠁⠈⣿⣇⣤⡄⢸⢸⣆⣹⡉⡉⠉⣹⠉⠉⠉⠉⡉⢉⣉⣉⣉⡇
⠀⠀⠀⠀⠀⠰⠶⠾⣿⣿⣿⣿⣿⣭⣧⠘⠏⣹⡄⠹⠀⢸⣼⠿⠟⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠦⢴⣿⣷⣀⣠⣿⣧⣀⣘⣿⣇⣿⢻⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⣿⣿⣯⣾⠧⢿⣿⣿⣟⣷⣽⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠦⠤⣍⣉⡛⠛⠻⠿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠢⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⣲⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢟⠿⠟⠋⢡⣴⣞⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠠⡑⡄⢈⠿⠋⠙⣷⣄⠀⠀⠀⠀⠀⠌⠀⠀⠖⠀⠐⢛⣻⢯⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠈⠀⠀⠀⢀⣾⠿⠆⠀⠀⠀⠀⣀⡀⠀⠀⠐⣤⣀⢈⠉⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠀⠀⠀⠸⢫⠉⠀⠀⠀⠛⠋⠐⡄⢸⣿⣿⣿⣿⡏⡇
⠀⠀⠀⠀⠀⢘⠁⠨⡽⡆⠀⣠⣤⣄⣸⣿⣿⡟⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣹⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣸⡋⠑⠚⢿⣿⣿⡆⠀⠀⠀⢈⣀⠔⢀⡀⠀⠀⠀⠀⢠⡝⠛⠉⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⠀⠘⠀⠠⣯⡇⠀⣠⣿⣿⡿⠀⠀⠀⠀⠀⠈⠠⢤⠄⠀⠀⠀⠀⣼⠇⠀⠀⣿⣿⣿⠿⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢩⣄⠋⠄⣸⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣠⣿⡏⠀⠀⠀⣿⠿⠋⣰⡿⠿⠦
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠐⠿⡿⢱⣦⣀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⠾⠿⢿⠁⠀⠀⠘⠃⠀⣰⡿⠁
⠀⠀⠀⠀⣀⣤⠀⡎⠏⠀⠔⠺⠛⠁⢸⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠑⢤⠀⠀⠀⣰⡿⠁
⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⡸⣼⠆⠀⣰⡿⠁
⠀⠀⠀⣯⣿⣇⠀⠀⠀⠂⠠⠀⡠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢰⣷⣷⠀⣰⡿⠁
⠀⠀⠀⠉⢻⣿⠀⠀⠀⠀⢀⣾⣣⣿⠷⣿⢯⠯⢭⣵⣿⢿⣫⣿⣿⣿⠟⠋⠀⣼⣿⣿⣷⡿⠁
⠀⠀⠀⠀⠈⠉⠀⠀⠀⣠⣾⢝⣾⡟⣸⣇⠃⠠⠿⠟⠖⠋⢱⣟⠟⠁⢀⠀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢆⠑⠀⢀⣶⠛⣯⣿⣿⠃⢹⡽⡣⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣸⢻⣻⡿
⠀⠀⠀⠀⠀⠢⡀⠸⣏⣾⣿⣇⢟⣴⡷⠉⠈⠁⠀⠀⠀⠀⡄⢀⠀⡐⠀⠀⡏⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢦⠈⠉⠿⢿⢶⠫⠿⠁⠀⠀⠀⠤⠊⡀⠂⠈⡆⠀⠀⠀⠛⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠉⢳⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢤⣤⡤⠤⠤⠤⠤⢤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⠀⠒⢒⡒⠒⠛⡏⠛⢓⣒⡒⠒⠒⠲⠾⢶⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡰⠋⠀⠈⢳⡀⡇⠔⠁⠀⢹⡄⠈⠑⠄⢺⠀⠀⠀⠀⠀⠀⢳⣄⣠⣼⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⠀⠀⢸⠄⠀⠀⢰⣾⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⠀⠀⠀⢠⠚⢻⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⡇⡇⡀⠀⠀⠸⣆⣀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢸⢤⠤⣿⠀⠀⠀⢸⠁⢸⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⡇⠤⣾⠟⠋⠑⠒⠀⠤⠔⢚⣽⣷⠶⢖⠒⠚⠛⠛⣟⣴⡀⣤⣮⣀⡼⠥⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣆⣖⡒⠒⠊⠉⡇⡇⢰⡇⠀⠀⠀⠀⠀⠀⢠⡿⠛⠒⠂⠠⠤⠤⠤⢭⠅⢠⠞⡩⠒⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠂⠭⠽⡏⢉⡇⡏⡩⠉⣉⡛⣛⠒⢲⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡗⢠⠞⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⡇⡇⢰⠀⢲⠒⠀⡇⠈⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⡇⠀⠀⡖⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⡇⡇⠌⡀⠚⠀⠀⡇⠠⢸⡇⠀⣀⣀⡀⠀⠀⠀⠀⢸⡿⠀⢇⠀⠀⢷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣇⠀⡇⢉⢐⠒⠠⠤⠄⠀⣀⣸⡇⣎⠀⠀⠁⠀⠟⡄⠀⢸⢻⡣⡀⢁⡺⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣆⠀⠠⢈⣃⣀⠡⠥⡄⠈⠁⠖⠀⠀⠄⣀⠁⠓⠶⢭⣀⡺⣄⠃⠀⢸⢀⣑⠬⠕⠒⠉⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⠀⠁⢹⣯⣤⠛⢄⡤⢦⢌⠁⠐⠆⠠⠀⠀⡀⠐⠖⠠⠍⢁⣉⣉⣙⣛⣛⣛⣛⣛⣋⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢸⣄⠿⢄⣯⠦⣄⡜⡇⣠⠞⢉⠳⡎⢉⣙⡲⣦⠀⣸⠉⡋⢫⠉⠹⡉⠉⠍⠃⢱⡹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠤⢹⣖⠛⠀⣸⢺⣨⣿⡒⡇⠀⠀⡇⣿⠀⣿⠀⠈⠄⠅⠀⠀⠀⠀⠐⡐⣝⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠣⢾⡄⠁⢀⠂⠀⠀⠀⣿⠔⡯⠀⠀⠑⡊⠄⢤⠀⠀⠀⠘⠼⢺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠉⠑⠒⠤⢥⣇⠤⣟⣀⢀⠀⠰⡕⡈⣧⡣⠀⠀⢪⢳⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⣝⣓⠶⠷⠶⠾⠾⠶⠶⠒⣋⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠁
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⢬⡆⠀⠀⢀⡾⠉⠱⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⢶⢴⠂
⠀⠀⠀⠘⠲⠶⢶⠂⠒⢒⠖⠲⣺⣗⡆⠀⢀⡇⠀⠸⡇⠀⠀⢰⡄⠀⣶⠉⢳⠀⢸⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠘⠛⠋⠹⡏⠀⠀⠀⠛⡿⠒⣾⠣⠀⠃⠀⠘⢦⡼⣠⡤⠼⢴
⠀⠀⠀⠀⠀⠀⢸⣐⠂⣀⣘⣳⡄⠀⠀⠘⡏⠠⣀⠀⠀⣇⠀⣿⠀⠀⠀⢀⡀⠈⠀⠀⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢸⡿⠗⠒⡓⢒⣶⣒⠒⠐⡗⣐⣒⣐⣒⣷⣂⣿⣤⣀⣀⣀⣲⣶⡄⢀⡈⠶⣴⠃
⠀⠀⠀⠀⠀⠀⠸⣴⣆⣤⣰⣻⣶⣶⡙⣾⣿⣑⣶⣬⣳⢳⢯⣶⣶⣝⣿⡍⡛⢻⠃⠀⢻⣏⠉⣿⠒⠒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⡘⠿⠿⣳⠿⢷⡻⠿⢟⡼⠛⢿⡻⢟⡹⠋⠉⣏⣛⡷⠼⢿⡉⣳⠈⢿⣻⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠉⠈⠀⠈⠉⠉⠉⠁⠀⠉⠉⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⢠⠌⠉⠉⠛⠓⠒⠢⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣿⣄⢠⣄⡀⢚⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣉⣽⣿⡹⣄⣿⣿⢈⣿⣇⣢⠤⢀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠇⣐⠉⠙⢿⣶⣌⣻⣮⡿⣧⣿⣏⢸⣿⣯⣀⣀⣘⣿⣷⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⣿⣶⡦⢸⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⡉⠛⠣⠑⡆⣘⣻⣿⣿⣿⣴⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⠿⢿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠐⡧⣿⣿⣿⣿⣿⣿⡿⣽⣿⣛⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⢀⡄⠄⠀⢀⡀⡀⣠⡄⠀⣤⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⣖⠂⠁⣿⡭⠙⢻⣿⣛⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠤
⠀⠀⠀⠀⣠⣀⣀⣸⡏⣾⢖⣾⣗⣭⣼⡿⣿⣹⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣉⣤⣄⣻⣀⣀⣀⣀⣀⣟⣈⣀⣛⣋⣹⣋⣛⣛⣻⣋⣽⣿⡟⣿⠇
⠀⠀⠀⢠⡙⣿⣿⣿⡇⣧⡾⣟⣝⣼⣿⣷⣿⣿⣿⣿⣿⣿⡏⣿⣽⣯⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣶⡿⠟⠛⠛⡛⢛⣛⣉⣉⣩⣏⣭⣭⣯⣿⣿⣟⣉⣠
⠀⠀⠀⠀⠙⣿⣿⣿⣯⣻⣵⣽⣼⣼⣿⣿⣿⣿⣿⣿⣿⣆⣾⣿⣯⣿⣿⣿⣿⣿⣿⣾⢿⣿⣟⣟⣟⣿⣿⣯⠿⠿⠷⠾⠿⠿⠛⠛⠛⣛⣛⠉⠩⠭⠉⠀⣒⡀
⠀⠀⠀⠐⠛⠛⠛⠛⠋⠉⡙⢽⣏⣿⣿⣿⣿⡿⠿⢾⢿⡷⠿⠟⠿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠀⠀⢀⣀⣀⣠⡤⠤⠤⠖⠒⢒⣉⣉⣩⠤⠴⠖⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠤⠴⠶⠖⠒⠚⠛⠉⠉⠠⠤⠤⠤⠤⠖⠶⠺⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠈⠑⢄⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣠⠔⠒⠄⠀⠀⠀⠀⠋⠁⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠒⠲⣤⡀⠀⠐⠄⠀⠀⠐⠀⠀⠀⠀⠀⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⣴⣶⠶⠴⢾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⣶⡌⣏⠉⠉⣽⡿⠃⠠⠤⠤⠖⠠⠤⢂⣀⣠⠀⠀⠁⠑⢀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠼⠿⢯⡇⠸⢦⣾⣿⠃⣰⡆⠀⣠⡤⠶⣤⣤⣅⣀⡈⠉⠉⠢⠬⠄⠤⠈⢦⣀
⠀⠀⠀⠀⠀⢠⢎⢖⡀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠄⠸⠇⠈⠉⢹⣦⣦⣴⣤⣄⡀⠀⠉
⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠸⣀⣀⣀⣠⣀⣀⢀⡇⠀⢀⠀⠀⢸⣿⢾⠀⡏⠀⠉⠉⠉⠙⠓⢂⡀
⠀⠀⠀⠀⠀⢹⣿⣧⣀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡷⠤⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⢻⣧⡄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣿⣭⣭⣤⣤⣤⣶⣶⠂
⠀⠀⠀⠀⠀⢀⡼⠟⠛⠛⠛⠛⡿⢿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⡾⣟⡓⠒⠒⠒⠒⢒⠷⣚⣟⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠛⠁
⠀⠀⠀⢨⣷⣟⣁⢀⣀⣀⣀⣘⣶⣋⣿⣶⣶⠛⠻⣿⣿⣿⣿⠟⠿⠟⠁⠈
⠀⠀⠀⣿⡏⡇⣿⣿⢱⡿⢹⡏⣿⢻⣿⣿⣿⣶⣿⠏⠉⠉⠁
⠀⠀⠀⠙⢿⣇⣿⡇⣾⣇⣿⣷⡿⠾⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⢷⡶⢤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⠁⠀⠀⠈⠀⠉⠒⠌⠀⠀⠉⡓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠘⠀⠀⠀⠀⠀⢀⣠⣶⠿⢛⡷⢾⣆⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠃⠀⠀⠀⢀⣀⠤⠒⠻⠁⠈⠀⠀⠀⠜⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢀⣠⣴⠦⠘⠫⠁⠀⠀⡖⠀⠀⠀⠀⠀⣀⣤⡶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢄⣿⠀⠀⠀⣴⣾⣿⡿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⣀⡀⢀⣀⣤⣤⣤⣴⣾⡎⠙⢳⣤⣾⣿⣟⡃⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣉⠈⣀⣶⣿⣿⣿⡇⠀⢸⣿⣟⣩⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠒⠟⠺⡇⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠈⠘⢿⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡇⠀⠀⠀⠁⠀⠓⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣾⠀⣀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⣠⡿⠉⠁⠻⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣤⣄⣦⣌⡗⠤⡀⠈⠄⠀⠀⠸⠀⣼⠏⠀⠀⠀⠀⠈⢅⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠽⣿⠁⠹⣶⣿⠷⢾⠞⠀⠀⣿⣿⡔⠊⠈⡁⠐⠐⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠈⠀⠀⠀⠀⠹⠜⡶⣶⡿⠉⢹⣶⣞⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠃⢠⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⡀⠀⠰⠀⠀⠀⠀⠀⠀⢰⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠙⢦⡀⠆⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⡇⠀⠀⠀⠀⣦⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢹⡀⠀⠀⢸⣧⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠳⠦⠴⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⣤⢀⡖⢰⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⣸⠀⠘⣄⣻⣾⣿⠟⠛⠉⢵⢼⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣿⠟⠛⠉⠁⠀⠀⠀⡀⠀⢸⣿⢡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠉⠀⠀⠀⠤⠖⡾⢿⣷⢸⡿⠀⠘⣻⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⠤⢤⡒⢶⠂⠰⡟⠉⢉⣀⣠⣀⡄⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢘⣇⠀⠠⠄⠀⠐⠀⠔⢁⠻⡿⠿⠷⢖⣚⣛⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣤⢤⣤⣴⣶⣶⣦⣼⡟⣷⣿⣦⣯⣽⣿⣿⣿⣿⣿⣿⣿⣴⣶⣹⣦⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣗⣯⣽⣯⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁⠀⢀⠈⢀⠈⠀⠀⡀⠁⡀⠀⠀⢐⠈⢐⠀⠀⠀⡀⠁⡀⠀⠈⢐⠈⢀⠀⠀⠁⡀⠁⡀⠀⠨⢀⠈⢀⠀⣀⣁⣠⣤⣦⣤⣬⣀⣨⣀⣥⣀⠁⡀⠁⠈⠀⠨⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢈
⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠄⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠄⠠⠀⠠⠀⠠⠀⠀⠀⠄⠀⠄⠀⢀⢠⣼⣿⠿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣯⢻⣆⠀⢐
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⢀⣾⡿⣣⣾⣿⣿⠟⠋⠉⠉⠙⠛⠛⠛⢿⣿⣝⢿⣰⣿⠀⠐
⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡠⠤⡀⠀⣼⢻⣳⡿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢾⣿⣿⡇⠠
⠄⠀⢐⠠⢐⠠⠀⠄⠀⠄⠀⠀⠀⢐⠠⢐⠀⠀⠀⠂⠄⠀⠀⠠⢐⣀⣐⠀⠀⠸⣄⣀⡸⢀⣿⣼⡿⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⠃⠠⠀⠠
⠀⠀⠐⠠⠐⠀⠀⠀⠀⠀⣀⡴⠶⠒⠛⠛⠛⠒⠂⠂⠄⡠⠖⠉⠉⠉⠉⠉⠒⢄⡂⠁⠀⠰⣿⣿⢧⣿⣿⠠⠦⣄⡀⠀⠀⣠⠴⠛⠃⠀⠀⢿⣿⢿⣿
⠀⠀⠠⠀⠀⠐⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢨⣿⣿⣼⣿⡷⠖⣾⡟⠣⠀⠈⠑⠺⣿⠟⠗⠀⠈⡅⠉⡟
⠀⠀⠈⠀⠈⠐⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡟⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣷⠀⠐⠀⠐
⠀⠀⠀⢀⡠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣵⣿⡆⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢿⣿⣆
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⣾⣷⣿⣿⣿⠨
⢱⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠻⢿⢿⠟⠁⠀⠀⣠⣾⣿⣿⣿⣇⠛⠈⠀⠈
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⢿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣠⢾⣿⣿⣿⣿⣿⣿⣦⢈
⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⢰⣝⣯⣻⣿⠟⠻⡎⢳⠒⠒⠋⠀⢸⣿⣯⢿⣿⣹⣿⣿⣷⣨⣀
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠁⠀⠀⠀⠀⠉⠉⠛⠻⠟⠃⠀⡴⠁⠀⠁⠀⠀⠀⠀⠙⠻⣧⣿⣷⣿⡿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠒⠒⠒⠒⠒⠒⠈⠁⠁⠀⠘⠟⠀⠀⠀⠀⠀⠀⣈⣿⢹⣿⣧⠟⣿⣷
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⡀⢀⣀⠀⠤⠤⠔⠒⠒⠈⡅⠀⠀⠀⠀⠀⠀⣠⡴⢁⡼⢿⡏⠀⠛⣿
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢀⣀⣀⠄⠴⠈⠀⡄⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⠀⢾⣿
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⣀⠠⠠⣶⣷⠀⠀⠀⠘⠄⢔⡈⠒⠀⠀⠀⠀⠀⢀⣴⣿⠋
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠉⠉⠀⠀⠀⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷
⠀⠀⢀⠈⢙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⠉⣿⣿⠛⠢⣀⠀⠀⠀⠀⡄⢀⠞⣿⣿⠋⠙⣿
⠄⠀⢐⠠⢀⠠⠀⠍⡉⠍⡃⠘⠊⢐⠠⢘⠢⣄⡀⠀⠀⠀⠀⢀⣠⠠⢀⠤⠤⠤⡤⠶⣷⠀⠀⢸⠀⠀⠀⠀⠀⢳⣿⣿⣷⣶⣿⣷⣄⢣⣸⢠⣷⣾⣿⣿⣷⣾⣿
⠀⠀⠐⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠉⠛⠛⠛⠙⠉⠐⠀⠐⠀⠀⠀⠀⠀⠘⡀⠀⢸⠀⠀⠀⠀⠀⠈⡋⠉⣿⣿⠋⠉⣿⣦⣿⠉⣿⣿⠋⠙⣿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠚⠛⠛⠋⠙⠛⠛⠶⠒⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠈⠱⡄⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠒⠉⠀⠀⠀⠀⡴⠋⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡠⠃⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢀⡤⠔⠒⠒⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⢤⡀⠘⣧
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠹⡀⠘⣧
⠀⠀⣰⠃⣠⠖⠉⠉⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢰⠁⠀⠸⣆
⠀⢠⡏⢰⠁⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢄⠤⠖⠁⠀⠀⠀⢹⡀
⠀⣼⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⢠⡇⠀⠈⢦⡀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠃⠀⠀⠀⠉⠒⠂⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡞
⠘⣧⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠀⠀⢳⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⡇⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠈⠳⣤⣀⡀⠀⠀⣀⣈⣳⣤⡤⠤⠴⠶⠚⠛⠛⠛⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠛⠓⠒⠒⠶⠦⠶⢮⣤⣀⣠⣴⡋⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠉⠛⠻⠿⠭⢤⣤⣤⣤⡤⠤⠤⠤⠤⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠤⢤⣤⣤⡤⣤⣤⣤⣤⢤⠭⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠦⠿⠶⠤⣤⣤⣄⣀⣤⣤⠤⠞⠛⠒⠒⠛⠉⠁

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⣿⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⣠⢶⡿⠇⣠⡔⠶⣖⠀⠀⠀⠀⠀⢀⣤⣶⡶⠆⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠈⣁⠉⠚⢠⣦⠈⠀⠉⠀⣶⣿⣿⡻⠆⠀⠙⠃⠀⣶⠟⠋⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⢠⠈⠉⠠⡿⠀⠀⠀⠠⠀⠉⠉⠁⠰⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠈⠄⠀⠈⡧⡀⠀⠀⢀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠈⠃⠰⠃⠈⡀⢀⣬⠀⠀⠠⡄⠀⠁⠀⠰⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡟⠁⢁⣤⣾⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣴⣾⣿⢟⣿⣿⣿⣿⣿⡟⠂⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠛⠛⠁⠚⠟⠋⠁⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢠⢠⡄⠤⡄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠗⠈⠛⠘⠇⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⢨⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠐⢠⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠠⣤⠉⠙⠋⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣦⣤⣤⣼⡇⠀⢠⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⠀⠀⢁⣼⣿⡶⢾⣯⠟⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠞⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠦⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⠴⠦⠶⠶⠾⠿⠓⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⢿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠸⣣⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠋⠀⢁⡄⠀⣷⠀⠀⠀⢠⡤⠘⠉⣀⣭⢴⠀⠀⠀⢰⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⢷⣄⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⠄⠀⠀⢺⡇⠀⣿⠀⠀⠀⢀⠀⢠⣾⣿⣦⡇⠀⠀⠀⢸⡟⠀⠀⠀⣠⣄⠈⠛⠛⠃⣤⣶⣤⣤⣄⠘⡃⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠠⣄⠸⠣⠀⣿⣶⣄⠀⠀⠀⠀⠸⠋⠁⠀⠀⠀⠀⢸⠃⠀⠀⠀⠛⠉⠁⠀⢀⠀⠀⠛⠛⠉⠻⡧⡇⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡄⠀⢠⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠁
⠀⢸⣿⠀⠀⠀⠀⣤⠀⠀⠸⠧⠀⠰⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⣬⣍⣄⣀⣉⣧⡀⠀⠀⢸⣠⣤⣀⣤⣀⣶⣦⣼⣿⣿⣿⣿⣦⡀⠙⢷⣆⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠟⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠏⠀⠠⠐⢂⡀⢀⣀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠲⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡆⠀⠀⠀⠈⠀⠈⡀⢠⣈⠻⣶⣤⣤⣤⣤⣦⣴⣤⣴⣾⠏⠁⠀⠈⠃⠀⠀⣿⠄
⠀⢸⣿⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠀⢰⣷⠀⠁⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠇
⠀⠘⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣿⣽⣿⣿⣿⣿⣿⣿⣿⣟⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢻⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣤⣤⣤⣠⣤⣤⣤⡄⠀⠀⢀⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣼⣇⣼⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⣴⣧⣄⣿⣿⣿⣿⣿⣿⣿⡷⢀
⠀⠀⠀⢰⣜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣠
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⢀⠆
⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢀⢰⢳⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢂
⠀⠀⠀⠀⠀⡸⣿⡿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠬⠐⠄
⠀⠀⢀⡴⠊⠁⣀⡀⠦⠡⢧⣶⠄⠈⢷⡀⠀⠀⠀⠀⣤⡄⠐⠒⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣻⡷⢦⡄
⠀⠀⠈⠀⠺⠟⠛⠋⠁⠂⠹⠟⣇⣀⣨⣷⣀⣠⣤⣖⠿⢟⣄⣀⣀⣀⠀⠀⠀⠀⣐⣀⠀⠀⣄⣤⣤⣄⣀⡀⣀⣠⣿⣥⣤⠐⠀⠒
⠀⠀⠀⠀⠀⠀⠲⠶⠴⣶⣤⣄⣀⣀⠀⠛⣯⠉⠉⠉⠓⠦⠀⠀⣉⠀⠀⠀⣀⣨⠿⠇⠀⠀⠀⠀⠀⡀⡤⠤⠋⠉⠀⠒⠉⠁⠀⢀⡀⡐⠃
⠀⠀⠀⠀⠀⠀⣐⡀⠐⠞⢣⡉⠻⢦⠉⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣔⣊⣉⡉⣁⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢿⣿⡶⠓⠀⠙⠲⣤⢌⣉⣒⠲⣶⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⢿⣿⣿⣍⣩⡤⣤⣽⠛⢛⣓⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠹⡇⣰⣷⣤⠄⢉⠬⠟⣛⡛⠹⠟⢦⡀⠀⠀⠀⢀⡴⡏⠀⠀⢠⡄⠀⠉⠁⠀⠈⠀⠀⠀⠀⢀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠥⠀⠠⠄⠖⠀⠀⠚⠀⠀⠙⠄⠀⣴⣉⠁⠀⠀⠀⠀⠙⢴⡀⠈⠠⠀⣏⡤⢤⠠⠟⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣋⣽⣿⣧⠴⣶⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣰⣿⣆⣀⣠⣤⣁⣤⣠⣶⣄⣀⣄⣉⣈⠁⢀⣀⠐⠂⠈⠛⣦⣄
⠀⠀⠀⠀⣰⣏⡴⡛⠛⠉⠀⠈⠪⣻⠁⠙⡭⡹⢹⣹⢹⡙⡏⣉⢋⢋⠟⡯⢴⣦⣀⡉⡤⠈⢣⡀
⠀⠀⠀⠀⡟⠁⠀⠀⠄⢀⠂⠄⡐⠈⢶⣄⠌⣱⡾⣏⣧⣕⣼⣏⣡⣪⢊⡴⠓⠉⡹⠶⣗⣄⠀⢱⡀
⠀⠀⠀⠀⢷⡀⡀⠀⠀⠠⠬⣒⣬⣑⣦⣜⡷⠿⠷⢿⣿⣿⣿⣿⣥⣿⣥⢤⣴⣮⣴⣵⣮⠿⣌⣋⠡
⠀⠀⠀⠀⠀⠻⣤⠒⣒⠺⠷⢶⣬⣽⣻⣿⡇⠀⠀⢰⡟⢿⣿⣶⣦⣤⣄⣀⡤⠍⠉⢉⡬⠟⢻⣥⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠒⢬⣥⠶⢒⣛⣻⣿⠀⠀⠀⢸⠀⢸⣧⣙⠡⡉⠙⡻⠽⢍⠛⡻⠶⠄⣼⡛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢾⣿⠀⠀⠀⠀⡆⢻⣿⣿⡀⠀⠀⠈⠂⡀⣑⣌⣢⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢠⡇⣼⠻⢽⣧⠴⠖⠚⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠀⠠⣶⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⣠⠂⣤⡀⡻⠗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣰⣵⣿⣾⣿⣿⣷⣮⣬⣢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣀⡀⣤⣀⣼⠟⠋⠁⠸⠸⡟⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠇⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢈⣗⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣠⡏⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⣀⣻⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠶⠶⠶⠶⠶⠶⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠙⢦⠀⠈⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⡊⠉⢓⡦⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠒⠋⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⡥⠶⠶⠤⣄⠀⠉⠛⠉⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⢡⠊⠁⠀⠀⠀⠀⢈⡇⠀⠀⢀⡔⠋⠁⠀⠀⠈⠉⠉⢳⠀⠀⠀⢀⡖⠉⠀⠀⠈⠉⠳⣄⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠸⣄⣀⣠⣤⠤⠖⠋⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠴⠶⠒⠛⠉⠁⠀⠀⠀⠀⠀⠢⣄⣀⣀⣠⠴⠚⠁⠀⠀⠀⣀⠙⣦
⠀⠀⠀⠀⢰⠇⠀⠀⢀⣠⠖⠚⠋⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠹⡜⡆
⠀⠀⠀⠀⡿⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠘⠦⠤⠇⢿
⠀⠀⠀⠀⣧⠀⠀⠀⠳⢄⠀⠀⢀⣀⡤⠖⠛⣉⣀⡤⠴⢶⡖⠒⢲⠒⢲⠒⡖⠚⡟⢛⠖⠒⢲⡶⠦⣤⣀⠀⠸⡁⠈⠙⢦⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⣴⡋⠉⠑⢤⡀⠀⠹⡄⢸⡆⢸⠀⡇⡼⢠⠎⢀⡴⢋⡴⠊⣠⠜⠛⢦⣌⠓⠒⠋⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⢧⡀⢸⠉⠙⡆⠀⣠⠛⠥⢤⣈⣉⠒⠤⣌⡓⢤⡘⢆⠇⠘⣀⣠⣣⠋⣰⡫⢞⡡⢖⣉⡤⠖⠋⢁⣈⣷⡀⠀⢠⠒⣹⣴⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⠁⢸⡃⠀⠒⠒⠒⠬⢭⣷⣦⣭⣝⣿⠶⠛⠋⠉⠉⠉⠛⠛⠶⣟⣋⣭⣤⣶⡾⠭⠭⠥⠐⣷⠀⠧⢴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⢯⡉⢉⣉⣭⣭⠭⠭⢽⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠵⠒⠒⠒⠒⢒⣒⣰⣏⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠽⣶⣤⣤⣒⣚⣉⣡⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠶⠶⠶⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠉⠉⠣⡈⠉⠀⠀⠉⠉⠀⠀⠁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢐⠀⠀⠀⠀⠨⢐⡢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠠⣚⠄⠀⠒⠒⠈⠉⠉⠙⠉⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡀⠠⠤⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠈⠉⠀⠈⠇⢸⠀⢨⠀⠀⠀⠀⠀⠀⠀⠉⠱⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡁⠀⣀⣀⣀⡀⢀⠝⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠈⡇⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠙⠆⠀⠸⠀⡀⠀⠀⠀⠀⢸⠀⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠼⠤⠄⠠⡒⠓⠒⠒⠒⠒⠒⠒⠲⠾⠖⠶⠶⠶⠶⠤⣀⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⡠⠰⡟⣍⠪⡠⡀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠈⠌⠂⢀⡀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⠀⠐⠂⠀⡄
⠀⠀⠀⢠⠀⠀⠰⢸⠀⢡⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠢⠤⠦⠞⠀⠀⠀⡄⠀⠀⠀⠘⠀⣤⠀⠀⠀⠀⠀⠀⠀⠇⠀⠤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠆⠀⠀⠀⠀⠀⠀⡧⠼⠤⠤⠤⠄⠀⠤⠤⠤⠄⠀⠀⠀⠒⢤⠀⣀
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⠰⠀⠀⠀⠀⠐⠠⠆⠀⠀⠀⠀⠀⠀⣰⣰⡀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠼
⠀⠀⠀⠘⡄⠀⠀⠀⡠⠁⠂⠀⠀⢀⢀⡂⠀⠀⠀⠀⢀⠔⢈⠤⠐⠖⠂⢌⠑⢄⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣆
⠀⠀⢀⠈⠈⠐⠰⢚⠒⠃⠀⠖⠋⠅⠀⠤⣑⠢⠀⢰⠃⣔⠉⠂⠀⠁⡈⠀⣱⡄⢲⠓⠒⠲⠖⡀⠀⠀⠀⢤⠒⠒⢒⠶⠂⢀⡐⠲⠚
⠀⢀⠂⡐⢀⠀⠀⠀⠀⠀⠎⡘⠃⢤⣄⡀⠉⡃⢰⢸⠀⠁⠈⣁⣳⢻⣶⠡⠄⢧⠈⠀⠐⢁⠈⠀⣀⡀⠈⡄⢃⠰⢡⠊⢁⣀⠈⠢⠈⡄
⡠⠃⠀⠀⠂⠀⠀⠀⢀⠸⠀⡍⠀⠻⠷⢟⠠⡆⡘⠈⡀⢋⡠⠀⠈⠉⠡⡐⢠⠆⡔⠀⠀⠠⡀⠈⠳⠇⢀⠇⡘⠀⠠⡀⠘⠿⠃⡠⢀⠃
⠑⠔⠀⡀⠀⡐⠀⠄⢸⠀⠡⡐⠰⢤⣸⠬⢊⠔⠁⠀⠘⢄⠑⠴⠅⢄⠀⠕⢈⠔⠀⠀⠈⠢⡁⠂⠀⠐⡁⠔⠀⠈⠢⡉⠒⠐⠊⡠⠃
⠀⠀⠀⠀⡠⠄⠈⠉⠀⠀⠀⠈⠑⠂⠐⠒⠁⠀⠀⠀⠀⠀⠉⠒⠐⠒⠐⠊⠁⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⠉⠉
⣀⣰⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣝⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣴⣿⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣶⣄
⠀⠀⠀⠛⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡟
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⢀⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⢠⡀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣧⣤⣄⡀
⠀⣸⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⡿⠁⠀⠀⠁⣰⠀⠀⢀⣤⡤⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢿⣇
⠀⣿⠁⢻⣿⣿⣿⣿⠀⠀⠀⠀⣰⡟⠁⠀⠀⡀⠐⠻⣇⣰⠟⠀⠀⠀⠀⣿⡟⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢰⡿⠉⣿⣿⣿⡏⠈⣿
⠀⢿⡄⠈⠟⣛⣿⣿⡆⠀⠀⠀⠘⣷⠀⠀⠁⠙⢰⡖⠻⠁⠀⠀⠀⠀⠀⣿⡁⢹⣿⣿⣧⣀⠀⠀⠀⠀⠀⢸⣷⠀⣹⣿⡟⢃⡀⣿⠇
⠀⠈⣿⠀⢼⣿⣿⣿⡧⠀⠀⠀⠀⠈⢧⣀⠀⠀⠸⡇⢠⠀⠀⠀⠀⠀⠀⢹⣷⢰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠹⣯⠀⠋⠉⠠⡐⡟
⠀⠀⣿⡆⠈⣽⣿⣿⣿⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣁⡀⠁⠀⠀⠁
⠀⠀⠈⠋⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠨⣀⡀⠘⠒⢠⠀⢢
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⠀⠂⠉
⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠃
⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢰⡟⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⢀⡾⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⣿⠃⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠰⣿⠃⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⢀⣴⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⣴⠃⠀⠀⣿⣻⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣠⣤⣤⣴⣦⣶⣶⡞⠀⢢⡿⢠⠝⠒⣿⠛⠋⠉⠉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣾⣿⣿⣏⡿⠝⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⡅⢸⣧⣤⠐⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣙⣩⣭⠻⠿⣿⠿⢿⣀⡀⠀⣀⣠⣤⣴⣶⣶⣄⣶⡾⣷⣜⠛⠛⣁⣬⡦⠄
⠀⠀⠀⠀⢴⠶⡚⠛⠛⡍⠉⠑⢸⣿⡿⡟⡀⢘⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⢿⣿⠁
⠀⠀⠀⠀⠰⠀⠹⠀⠀⠇⠀⠀⢸⢿⣗⣁⣥⣔⣻⣤⣼⢭⣿⢟⣿⣍⡛⣿⠵⠶⢿⢹⣿⣿⣿⣿⣿⣿⢾⣿⡇
⠀⠀⠀⠠⠤⣴⠤⣶⡶⠶⡶⡶⠾⣿⣿⣟⣿⣿⣿⣟⣻⣿⡗⣿⣿⣿⣿⣧⣦⣧⣸⡟⢿⠋⠿⣿⣿⣿⣆⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠉⠹⠃⠤⠈⠹⠻⠿⠤⠭⠤⠤⠶⠿⠟⠿⠿⠷⠸⠦⠬⠩⠭⠭⠟⠙⠻⠟⠃

⣿⣿⠛⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣎⣛⡶⡎⣤⡐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣭⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀⠀⠀⢄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣭⣿⣿⣻⣿⣿⣿⣿⣿⣿⡿⣿⣷⣦⣝⡻⣧⣟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⢿⣿⣿⣿⡟⠁⠀⠀⣻⣿⣿⣿⣿⣾⣻⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⠁⠓⠑⣿⣿⣿⣿⣿⣿⣿⣿⣾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⢘⣶⣤⣦⣠⠏⠀⠀⢀⡾⠿⢿⣿⣿⣝⣻⢋⣿⣼
⠀⠀⠀⠀⠀⠀⣴⡺⢄⣀⣤⣤⣤⣤⣼⣿⣷⡈⠀⠠⠄⣄⠸⠀⠀⠀⠀⠈⠙⠛⠛⠋⠊
⠀⠀⠀⠀⠀⢀⣡⣶⣿⣿⣿⡟⠛⢹⣿⣿⠿⢿⡿⠛⠞⢻⠿⠟⠳
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⡆⠀⢨⠀⠀⠀⠘⠇⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⣾⣄⡀⠀⣾⠟⠶⠾⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠋⠙⠟⠀⠀⢸⠇⣶⣶⣶⣤⣤⣶⣶⣶⣶⣦⣤⣤⣘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢢⠄⡼⢢⣝⠻⣿⠛⠛⢿⠋⠙⢾⣿⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⣸⣆⢸⠏⠉⢀⢻⡆⣀⡾⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠹⡗⠀⠸⣿⠀⠀⠀⢘⣥⢈⠿⠛⠋⠙⡏⠡⠢⠂⠀⢀⠰⡎⠭
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡅⠀⠀⣀⢹⠀⠈⣿⡀⠤⠰⠋⣿⢨⡗⠀⠀⣠⡿⠁⠀⠀⠀⠘⠓⣟⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠈⠒⣪⠀⡥⣾⠣⢠⣿⠂⠀⠀⠀⣾⣮⣄⠀⠀⢸⣧⡀⠀⣄⣶⡼⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣆⣊⠩⣇⠏⣷⣸⡋⠀⢀⡦⣼⣿⡏⠙⠀⠀⡞⢀⡟⢻⣿⠯⡐
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠺⣻⣖⠟⣕⣿⣿⣟⠴⢋⢛⣿⣜⠀⠃⠀⣸⣧⣶⣪⢿⡫⣤⣴⠴⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⣌⢻⣦⣿⣷⣿⣻⡖⣻⡿⣾⢿⡆⣦⠴⣾⣽⣿⣿⣽⣷⡻⠋⠊⠂⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⡧⣿⣿⣿⣿⣿⣿⣽⢶⣯⣷⢶⠎⡦⢛⣿⣿⣿⡟⣿⡃⣁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠀⠙⠿⠿⠿⠿⠻⠿⠹⠿⠟⠀⠼⠿⢿⡏⠨⠟⠿⠘⠋

⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣴⣾⣿⣷⣤⣴⣿⠿⢿⣷⣶⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠛⠁⠈⠛⠉⠀⠉⢉⣁⡀⠈⠉⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣷⣄⠀⠀⢈⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⣄⡀⢀⡀⠀⠀⠈⠙⢿⡿⠈⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣸⠛⠛⢦⠞⠉⠙⠛⠋⠀⠀⠀⠈⠙⠻⣷⣤⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠐⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣴⣿⡿⣻⣿⣿⣿⣿⣿⣿⣓⠁
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠾⠛⠉⠀⣼⢹⢏⣾⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠦⠐⠋⠀⠐⠀⣀⣀⣤⠏⠼⢸⣿⣿⣿⣿⡏⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠛⠉⠀⠀⢀⣠⣤⣶⠿⣯⣥⣄⣴⣼⠘⣿⣿⣿⣝⡻⢶⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⣀⣠⠴⠖⢛⣉⣥⣶⣾⣿⡿⣿⣷⢹⣿⣧⡘⢿⣿⣿⣿⣿⣿⣿⣧⣤⡄
⠀⠀⠀⠀⠀⣀⡀⠀⠉⠀⢸⣷⣊⣭⣥⣴⣴⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢀⢤⣾⣿⠋⠁⠀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⣿⢹⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢰⣿⣿⣿⣿⠀⢠⣄⢈⣠⠼⠟⠿⣿⣿⠿⠛⠛⠉⠉⡟⠀⣀⣶⣽⡞⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⡟⠾⠋⠉⠁⠀⠀⠀⢰⣿⣷⣿⣭⣒⣀⣤⡎⠉⠉⠿⣿⣷⣿⡿⣿⠙⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⡿⠃
⢸⣿⠁⠈⠀⠀⠀⠀⠀⠀⢀⣀⣚⣿⣿⣟⣛⣋⣴⣿⣅⡀⣁⡀⣿⣿⣾⡇⣿⣷⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⡇
⢸⣵⣿⣶⣤⣾⣿⣦⣼⣷⣿⣿⣿⡿⣿⣽⡿⣿⣿⢻⡝⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⢻⡛
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠙⠛⠛⠛⠛⠛⠿⠿⠻⠿⠤⠿⠿⠿⠯⠭⠿⠿⠿⠭⠭⠭⠭⠭⠤⠤⠤⠤⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠄⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⢀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣆⠀⣠⡄⠀⢀⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢀⣾⣿⣶⣿⣿⣿⣿⣿⣆⣴⣿⣆
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⠿⠿⢟⣫⣶⣿⣿⣿⢭⣽⣛⡝⠂⠩⣭⣛⡛⠻⠿⠿⠿⢟⣛⣫⣵⣿⠟⣿⡿⢟⡪⣔⣮⠳⢍⡙⠛⢫⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣑⣃⣀⣠⠄⣐⣬⣽⢟⣋⡭⠷⠦⢶⣢⣤⡉⢔⣲⣭⣿⣿⣷⣮⣭⣿⣿⣿⣯⣙⣶⠾⠫⢾⣿⣿⣭⣖⣿⡶⠿⢛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠙⠋⣉⣉⣭⣭⣅⣛⠻⢭⣛⣫⡴⢮⣽⠷⠾⠿⠿⠶⡾⢿⣿⠿⣭⣷⣶⣮⠭⠽⠿⠛⠛⣒⣙⣫⡵⠾⠛⣉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣼⠻⠷⠶⢖⡫⠽⣛⣴⣬⠭⣿⣛⠤⣶⠾⢝⣟⣭⣝⣿⣿⣾⣷⣽⣍⣋⣙⣀⣈⣊⣛⠽⣵⣒⠴⢞⣋⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣩⣭⣭⣭⡭⢽⣓⠶⢿⣻⣶⣬⡵⢞⣛⣯⡥⢶⣿⣥⠯⠽⣟⣿⣶⠭⢭⣭⡉⣉⣩⣭⠭⢴⣞⡯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢭⣿⣶⣢⢌⣉⣑⡒⠒⠂⢘⣫⢭⠒⣉⣽⣖⣛⣩⣭⣽⣭⣭⣽⣛⠤⠮⣌⡕⠀⠚⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣭⣛⡻⠿⣙⣻⣤⣤⠈⠶⣯⣝⡻⣛⠭⣑⣻⣧⣶⣾⣿⡿⣛⡿⢾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠴⢿⣤⣞⠿⠯⠉⣭⣯⣿⡶⠾⠧⣬⣽⣿⢥⣼⣟⡻⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠟⠛⢭⡴⣬⠴⠾⠿⠟⠟⠋⠛⠈⠁




⣶⣶⣶⣶⣶⡾⢶⣶⣶⡾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣾⢿⡾⠶⡶⠶⢾⣶⡾⠾⠾⠶⠶⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⠛⠳⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣤⣄⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠫⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⢁⢀⡀
⣴⣶⣿⣿⣿⣷⣶⣄⢠⣶⣶⣿⣿⣿⣿⣶⡄⢠⣶⣾⣿⣿⣿⣷⣶⡄⣾⣿⣿⣿⣿⡆⠀⣠⠀⢸⣿⠁
⣿⠛⢻⠛⢻⠛⢻⡇⢰⣿⠛⣿⠛⣿⠛⢻⡆⢸⡟⠛⣿⠛⣿⠛⣿⡆⢸⡟⠛⠛⣿⣶⣶⣿⣶⣾⣿⣷⡆
⣿⣤⣾⣤⣾⣦⣾⡇⠸⣿⣤⣿⣤⣿⣤⣾⡇⢸⣷⣤⣿⣤⣿⣤⣿⠇⢸⣷⣤⣴⣿⢿⣿⣿⣿⣿⣿⣿⡿
⢛⣿⣿⠛⢻⣿⡟⠛⠚⢻⣿⡟⠛⢻⣿⡟⠛⠛⢻⣿⡟⠛⣿⣿⡟⠓⠛⣿⣿⣿⠿⣿⣿⠛⠛⠛⣿⣿⠛
⠀⠉⠁⠀⠈⠉⠁⠀⠀⠀⠉⠁⡀⠈⢉⠀⡀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠈⠉⠁⠀⠈⠉⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⠄⠀⠀⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣶⣾⣿⡁⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣶⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢿⣿⣿⣿⣿⣿⣧⡥⢠⣆⣴⡆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣱⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠙⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠦
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀
⠀⠀⠀⠀⠀⣤⢾⡀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⢸⡇⠸⣤⠀⠀⠀⣠⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⡼⠀⠀⠃⠀⢀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡘⢱⠻⠿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣧⢀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠄⠀⢉⠓⣿⣿⣿⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠁⠀⠘⡇
⠀⠀⠀⠀⠘⠆⠀⢀⡏⠀⠶⠀⢸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⡁
⠀⠀⠀⠀⠀⠄⠀⢸⣿⣆⢰⡀⢿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣜⣧
⠀⠀⠀⠀⠀⠆⠀⢈⣿⡿⠿⠁⠀⠀⠓⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠎⡉⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠉⠒⠠⣍⡒⠤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣠⣤⣄⡀⠀⠈⠑⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⠀⣤⢄⣀⡜⠁⢀⡴⠋⠁⠀⠈⠙⢦⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢥⡇⠀⠀⠀⠀⡔⠉⢆⠀⠀⠀⠀⠀⡠⡫⠊⠁⠀⢀⡠⢘⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⣠⠔⣉⠉⠭⣉⣿⡲⡀⠀⠀⠹⠛⡟⠀⠀⠀⠀⠀⣿⠁⠀⢀⠔⠁⢀⠎⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⡾⠁⠀⡌
⠀⠀⠀⠀⢰⠁⡜⠁⠀⠀⠈⡇⢸⠄⠀⡠⠃⡰⠁⠀⠀⠀⠀⠀⣾⠀⢠⠋⠀⠀⡌⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠋⠀⠀⡰
⠀⠀⠀⠀⣾⠟⢆⠀⠀⠀⡠⠃⡜⢆⠉⣀⠔⠁⠀⠀⠀⠀⠀⠸⡆⢠⠃⠀⠀⡸⠀⠀⠀⠀⠀⡠⢄⡀⠀⠀⠀⠀⠀⢠⢻
⠀⠀⠀⠀⠑⠵⢤⣉⣉⣉⣤⠮⠒⡌⢷⠻⠼⢖⢄⠀⠀⠀⣠⡳⢁⠏⠀⠀⢀⣇⡀⠀⠀⠀⡰⠁⠰⠁⠀⠀⠀⠀⢀⠇⠈⡄
⠀⠀⠀⠀⠀⢀⠊⢣⠰⡊⣴⣺⢷⢘⡌⢆⠀⠀⠑⠵⣖⠮⠛⠁⡘⠀⠀⣠⠋⠑⡦⠕⡒⡴⠁⣰⠁⠀⠀⠀⠀⢠⠋⠀⢠⠁
⠀⠀⠀⠀⠀⢸⠀⡇⠱⡑⣌⠥⠒⢉⣠⡼⣆⠀⠀⠀⠀⠀⠀⠀⢧⠀⢠⠁⠀⠀⠉⠒⠼⠁⢠⠏⣒⠤⣀⠀⡰⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠘⡄⠸⡄⢱⣠⣴⠾⠛⠉⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢰⠁⢠⠋⠒⠢⠟⠒⠿⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠘⣶⠛⢹⠻⡀⠀⠀⠷⣄⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⢠⠃⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠱⡀⠀⠸⡈⠢⡀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢃⠀⠘⣦⠜⠓⢒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠜⢆⠀⠈⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡡⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠁⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢀⣀⣼⣷⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⣴⠿⠿⣶⡶⢿⠿⠛⠉⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⢴⡺⢿⣷⠖⠶⡶⠒
⠀⠀⠀⠀⡸⠀⠀⣀⣀⣘⣶⣤⡀⠐⢚⠁⢘⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣰⡯⢶⣿⡅⠈⡟⠉⡠⠋⠀⢸⡴⠺⡷⠀⣼⣅⣀⣀⣠⣤⣼⣄⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⢀⣦
⠀⠀⠀⠛⠤⠤⠄⠇⠀⢘⠉⡐⠈⠁⢪⡅⠀⠀⠸⢭⣉⣿⠛⢻⠯⠭⠙⠀⠀⠀⠀⢀⣠⣮⣿⣶⣷⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠠⠴⠚⠻⢣⡀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡖⢲⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⡈⢷⣶⠒⠿⢿⠀⠀⠘⠀⠀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠠⡆⠠⠀⠐⡀⠾⠞⠛⡅⠀⠀⠸⡶⢶⡾⢸⡴⠀⠀⣠⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣀⣀⣉⣆⣄
⠀⠀⠀⠀⠀⠀⠀⢁⠈⣦⠀⠑⣦⡇⠀⢻⠀⣧⣀⠋⢹⠇⢸⡽⠍⡿⠟⢐⠀⠀⠀⠰⣷⣶⣿⣿⠓⢻⢿⣿⠿⠟
⠀⠀⠀⠀⠀⢀⡀⠸⡎⢙⣷⣴⣎⣧⡆⣸⣆⣿⡟⠀⡼⢀⣸⡇⢀⣿⣣⠀⠀⠀⠀⠀⠀⠀⠬⢀⠤⠲⡆
⠀⠀⠀⠴⠋⣉⣽⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣴⣿⣿⣿⣿⣿⣯⣿⣿⣶⣤⡀⠀⠀⠈⢷⣤⠀⠀⠀⣰⡀⣀
⠀⠀⠀⠀⠘⠿⡿⣷⣿⡟⡛⣿⠯⠿⣋⣀⣀⣸⠋⣙⡟⡙⠛⢋⣛⡛⠹⠻⠿⠖⠀⠀⠀⢶⡿⢿⣷⣾⣿⣿⣽⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠋⠛⠂⠈⠛⠋⠉⠉⠉⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠓⠚⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠘⠀⢿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣆⣤⣤⠐⣲⡿⠁⠁⠀⠀⠀⠀⠀⠈⠻⣷⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠠⢠⣀⣀⠀⢸⣷⣮⣿⡯⣹⠋⠀⢀⢠⣤⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣾⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠢⡤⡘⣿⣿⣷⣌⢻⣿⣿⢽⢿⣤⣌⠁⣈⡿⠆⠀⠀⠀⢨⠁⠀⠀⢠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠫⣫⡿⣿⣿⣷⣿⣿⡅⠁⠉⠉⠉⠉⠉⠉⠁⠀⠈⢸⠳⢴⢃⡘⠈⠻⠋⠑⠉⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣻⣿⣿⠟⠉⠓⠒⣲⠖⠒⠒⠒⠒⠲⡖⠒⠚⠢⡀⠙⠷⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡁⠀⢀⡴⠞⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⢄⣀⢝⢶⣫⣤⣾⣿⣶⣤⣤⢀⠀⢀⠀⢀⠙⢦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⣿⣿⡇⠄⣩⣵⣶⢻⣻⣷⣾⢣⣿⣿⣿⣷⣷⣷⣷⣾⡏⣽⣿⣶⡆⣴⠀⡦⢤⡄⢠⡶⠀⠀⠀⡀⠀⠘⢿
⣿⣿⣿⣿⣿⣿⣿⣛⠿⠟⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠾⠁⠺⠦⠻⣅⠠⠔⠈⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⡧⠤⠽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⠀⠀⠀⡞⣻⣏⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⢠⠀⠀⠰⣴⣶⡴⠀⠀⢰⣶⡦⠀⠀⠀⠀⠀⠐⢛⡴
⠀⠀⠀⠀⠉⠙⣿⠂⠉⠀⠀⢀⣾⣶⣿⣤⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⠻⠟⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⢧⠘⡀
⠀⠀⠀⠀⠀⠀⡜⢰⠀⠀⠀⠀⣿⣿⣿⢽⠄⠀⠀⠀⡤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠁⠂⠀⠀⠀⠀⠀⠀⢿⠈⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠩⠛⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⡄⠀⣲⣬⠃⠀⠠⢠⣰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣤⣴⡇⠀⢀⢸⠀⡄⢀⡀⠀⠀⠄⠀⠀⠀⠉⠕⠃⠀⠁⠀⠀⠀⠀⠈⠁⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠏⠉⠁⠀⠀⠈⠀⣘⠿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠲⠤⠀⠀⢀⣠⡀⠀⠀⡄⠠⣤⠆⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠃⠀⠀⠀⠈⠃⠲⠆⠀⠘⠛⠁⠀⢄⣷⠠⠁⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠠⣤⣄
⠀⠀⠀⠀⠀⠘⢠⡧⠖⠀⠀⣾⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣷⠃⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⣦⠠
⠀⠀⠀⠀⠀⠀⠘⠶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠈⠁⠉⠛⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⣓⠀⠈⠁⠀⠀⠀⢄⠀⠀⣀⠀⡀⠀⢀⣀⡀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢉⡸⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠙⠊⠁⠀⠈⠀⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⡉⠁⠀⠀⣠⠀⢴⡇⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢷⠷⠦⠤⢿⡀⠸⠇⠘⢿⡏⣿⠒⢠⣴⣿⣷⣦⣀⣀⣲⣦⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⢠⠱⣤⣠⢤⣤⢤⣠⢠⡄⢹⣿⠚⣿⠩⠉⠉⣿⣏⣿⣹⣿⣹⣩⣿⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠘⢠⣿⣿⣿⣶⣿⡿⡿⠿⠿⠿⠀⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣀
⠀⠀⠀⠀⠀⣰⢦⣔⣲⣶⣶⡿⣽⣿⣿⣧⣶⣖⣶⡞⣾⣿⣿⣧⣄⣒⣲⣶⣿⣿⠿⠿⣻⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⢸⡟⣀⣈⣛⣛⣛⣐⢉⠉⠉⢻⡿⠏⠉⠓⠋⠀⠉⠁⠀⠛⠋⠋⠁⠀⠀⠀⠁
⠀⠀⣐⣶⡟⣠⢿⣸⡿⢿⠿⠿⢰⣷⡰⠿⣓⣦⣤⣄⣤⡀⠢⣀⡤⠀⣠⡮⠗
⢀⣬⣽⣯⣤⣤⣤⡤⢶⠒⠂⠀⠂⠀⣀⡠⠤⢒⣀⡤⠔⢒⣩⠤⠖⠋⠁
⠈⠉⠉⢛⣛⠩⠝⠛⣉⣡⠤⠖⢊⣩⡤⠖⠚⣉⣤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⡢⠙⠛⡫⠹⠀⠚⠒⠒⠒⠦⠤⠤⡯⠄⣄⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠶⠯⠤⢴⢾⠤⠾⠤⠴⢤⠶⣖⠄⠚⠉⠳⢦⡐⠲⡞⢲⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠰⠛⠛⠒⠒⣶⠮⠀⠉⠀⠈⠀⠀⠀⠀⡀⢀⣈⣡⣈⣧⣀⠀⣤⣼⣷⠶⡷⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣨⠖⠋⠁⠀⠀⠀⠈⠉⠅⠀⠢⣴⣚⣲⣿⠀⠃⠁⠁⠀⢻⢹⠏⢿⡿⡀⠉⢹⡇⢰⡇⢨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⡿⠀⠔⠓⠓⠈⠈⠐⠀⢘⣸⠀⠀⣼⢃⣸⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠤⢄⠀⠐⠒⠒⠒⠓⢋⠃⠒⠀⠀⠀⠀⠀⠀⠀⢁⠀⡼⠸⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⣤⣼⠇⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⣈⣄⣩⣟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⡤⠒⢄⠢⡀⠁⠀⠀⠀⠀⠀⢀⣀⠀⠸⠀⠀⠀⠀⠀⠀⣀⢤⣾⣿⣿⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠷⠟⠀⠈⠀⠀⠀⠀⠀⢸⣠⣨⡆⠀⠠⡀⢀⣀⣵⣾⣵⣫⠿⠋⠉
⠀⠀⠀⠀⠀⠀⢀⠄⠭⡁⠀⣒⣀⣐⠀⠀⠀⠤⠍⠛⠂⢀⣠⣶⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠉⣶⣦⠔⣏⣛⣯⣿⣧⡀⡀⠄⣃⢀⣴⡯⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢈⡹⠖⠋⠉⠉⠉⠒⠒⠓⠘⠿⠿⣟⣭⡛⠁
⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣀⣀⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢴⡻⣏⢹⡿⣿⢿⠉⢉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣶⢲⣿⣿⣦⣄⠙⠋⢀⣼⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣾⣿⣿⣿⠛⠂⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⢿⣿⣿⣿⡿⣿⣳⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠙⣿⣿⣦⣤⣴⡀⣩⣽⡇⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁⠡⣾⣻⣿⣿⣿⣿⣿⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⢨⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢀⣘⡀⠴⠟⠁⠑⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡈⣸⣿⡶⢶⣿⣷⠠⣄⡀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠐⠃⠐⠂⠀⠀⠀⠀⠀⠀⠁⢻⣿⣿⣶⣶⣿⣿⣿⣿⢷⣄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⢻⠛⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡛⢻⡟⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣀⡈⡀⠀⠀⠀⠀⣀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣾⣿⣟⣉⠉⠀⠉⢙⡖⠁⠀⢩⣿⣾⣿⡄⠢⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠰⣾⣦⣀⠀⠀⠀⣀⠛⠛⠛⠿⠻⠷⠦⠀⠀⠼⠦⠿⠿⠿⢿⣿⣆⢦⡀⠀⠀⠀⢘⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⠀⢠⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢿⡟⠉⠉⠉⠉⢹⡏⠉⠙⠛⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⣴⣿⡏⠁⠀⠀⣿⠀⠀⠀⢀⣈⠛⠶⠶⠞⠃⢀⣽⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠈⠳⣦⣤⡾⠋⠀⢠⡾⠋⠉⠻⣦⠀⠀⠀⢾⡁⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢶⡄⠀⠀⠀⠀⠀⣀⣀⣸⣇⣀⣀⣀⣿⡀⠀⠀⠈⠳⢦⣴⠾⠋⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣀⣠⣿⣶⣶⠿⢻⡟⠛⢻⠋⠉⢻⠋⠉⠛⡟⠛⠻⡷⠶⢦⣤⣀⡀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠉⢷⡀⠘⠆⠈⠃⠀⠈⠀⠀⠘⠀⠀⠘⠁⠀⠸⠃⠀⡼⠋⢉⣿⠷⣮⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⠄⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠋⡁⢀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣴⣏⣀⠀⣴⠀⠀⣿⡇⠀⠀⠀⠘⣿⠀⠀⢄⢀⣰⣤⣤⣿⡿⠟⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⠁⠀⠀⠀⡄⣿⡿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⣧⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⡟⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⢀⢀⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⡜⡘⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⣸⠀⠇⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢠⠇⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠉⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣤⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⡠⡔⣒⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡄
⠀⠀⠀⠀⢀⡀⠤⢠⠀⣀⠀⠀⠀⠀⠰⠒⣟⣏⢭⠅⠃⠀⠀⣀⣀⣤⡤⠚⡁⠀⠊⠁⢲⠶⢤⠤⣄⡀⠀⠀⢍⠀⠈⠁
⠀⠀⠀⢸⢼⢶⢺⠻⠉⠉⣀⠀⠀⣀⡄⠀⠀⠀⠀⠀⢠⡖⠋⠹⣉⡀⠀⠀⠁⢀⢀⣀⣉⠤⠖⠁⣢⣿⠂
⠀⠀⠀⠘⠚⠉⠈⠀⠀⠀⠘⢻⡿⢛⠁⠀⠀⠀⠀⠀⠈⠻⠶⣤⣄⣈⣉⣉⣉⣉⣡⣤⣤⣤⡶⠾⠋⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠉⠉⡤⠑⠀⠀⢀⡀⡔⢢⠀⠻⠇⠀⠀⠀⠀⠀⠠⠀⠀⡤⡤⣦⢤⢤⡀⠺⠿⣀⡖⡗⢟⢻⢹
⠀⠀⢀⡠⡤⠴⠀⠀⠀⡈⠀⠀⠀⠁⠀⠀⠀⠀⠛⠀⠈⠐⠃⣠⠀⣇⢀⢹⣸⡼⠋⠀⠀⠏⠳⠶⠼⠼⠼⠄
⠀⠀⠀⠀⠀⠀⣀⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠼⠀⠠⡻⠀⠈⠉⠉⠀⠀⠀⢐⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠄⠖⠈⠀⠀⠀⢀⢀⢀⠀⠀⣈⠋⠀⠀⠀⠸⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠈⠑⢦⣀⣷⣦⡀⠀⠀⢠⣶⣶⣄⣴⠀⢰⣷⠿⣿⡟⠃⣿⣯⣩⢹⣿⣹⠗⠀⠀⠀⠈⡗⠒⠂⢰⠂⠐
⠀⠀⠀⠠⢤⣀⣀⠀⠀⠀⠉⠃⠀⠀⣿⣁⣿⡏⢿⣧⣼⠏⠀⣿⡇⠸⣯⣤⣤⠸⡿⠙⠷⠀⠀⠠⠀⢹⠋⠓⣾
⠀⠀⠀⠀⠀⠈⠃⠀⠀⡤⠶⣄⠀⠀⠘⠛⢉⣤⣄⢩⡤⢤⡀⣴⢶⡀⢀⣴⢶⢠⡶⠖⡂⠈⠀⠀⠀⠾⠲⠶⠻⣤⠐⠐⠂
⠀⠀⠰⠖⠶⢶⡀⠀⠀⢧⣀⠞⠀⠀⠀⠀⠘⠳⢦⣼⣷⠾⠛⣿⠾⣧⢼⣇⣠⠜⣟⢋⣁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠻⠷⠖⠀⠀⠀⠀⠀⢈⠠⠂⠈⠛⠛⠁⠛⠀⠀⠉⠀⠀⠀⠉⠁⠀⠛⠉⠉⠀⠀⠀⠀⠉⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢀⡰⠀⣔⢈⠀⢀⡀
⠀⠀⠀⠀⠉⠰⠶⠀⠀⠀⠀⢀⠖⠉⢠⠄⡀⠀⢁⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⢦⣀⡀⠉⠀⠄⠈⢁⣸⠔⠋⠀⠀⠇⠀⠀⠄⠀⠈⡷⠀⠀⢀⠀⠄
⠀⠀⠀⠃⣄⡀⢀⡀⠤⢾⡁⠀⠀⠀⠉⠋⠍⠀⠭⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠁⠀⠀⣌⠀⠄⠀⢀⡀⠀⢠⠐⢀
⠀⠀⠀⣴⠇⠀⠈⠇⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠋⠀⠐⢦⣤⠄⠒⠾⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢅⠀⠐⠀⠀⠀⡀⠀⠔⢀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠢⠀⠀⠀⠀⠀⢀⠘⢀⠀⠀⠀⠐⠁⠠⡀⠀⠄⠐⠠⢄⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠄⠀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠑⣀⠌⠀⠀⠀⠈⢀⣤⣴⣶⣶⣿⡿⣷⣶⡶⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠴⠶⠿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷⡿⠷⠿⠿⠿⠷⠤⠄⠀⠀⠈⠉⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⡀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠀⠀⠀⠀⠈⠀⠉⠀⠀⡐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⢀⠀⠀⠀⢀⣀⠀⢀⠀⠀⠀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⡏⠙⣷⢸⣷⠀⡖⣿⠁⢠⣿⡇⠀⡦⠉⣿⠁⣾⡇⠀⣸⣼⣿⠉⢷⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣷⠀⣿⠈⣿⣴⠇⣿⠀⢸⣿⣿⢸⠇⠀⣿⠀⣿⡇⢀⡇⣿⣿⣠⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣿⢸⣿⢀⡟⠀⢻⣿⠀⣿⠖⢸⡏⢻⣿⠀⠀⣿⠀⢻⣧⡾⠁⢿⣿⠻⡄⢸⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠋⠉⠀⠀⠈⠉⠀⠙⠃⠈⠁⠈⠁⠀⠀⠉⠀⠈⠉⠁⠀⠈⠁⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣀⣀⣀⣀⠀⠔⢢⠀⠀⣀⠔⠂⠀⡃⠀⠠⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠈⠑⠑⠀⠁⠊⠁⠀⠊⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⣄⣀⣀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡠⠒⠓⠓⠒⠉⠀⠀⠁⠀⠀⠘⠣⡄⣼⠒⠒⠖⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡾⠄⠀⠠⠒⠀⠀⢤⣤⠄⠀⠀⠀⣠⣤⠀⠀⠀⣀⣀⣀⠀⢱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠀⠀⠀⠀⠀⣀⣀⣀⣄⠀⢀⠠⣢⣱⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠸⢵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋⠁⠀⠀⠘⠳⣄⣀⣤⣾⠟⢛⡟⠉⠳⣷⣿⣾⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣶⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣆⠈⢢⣤⣴⣾⠋⠹⣧⣠⣿⣷⣄⢴⣾⣿⣿⣿⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠺⠛⠛⠓⠚⠛⠻⠿⠓⠚⠛⠉⠉⣽⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⣞⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡔⠛⣦⡀⠀⠀⠀⣤⡀⠀⠀⢀⡀⠸⡂⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡒⣈⣥⣾⣿⣿⡟⠋⠉⢻⣿⠀⢠⣖⡉⣷⣄⣴⠿⣿⡄⢣⡀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠚⣛⣃⢹⣿⢿⣿⣿⣧⠀⠀⢸⣿⠇⢸⠄⠀⢼⣿⣾⣤⣾⣇⣸⠀⠀⢺⣯⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣷⣿⢸⣿⢸⣿⠿⠃⡤⣠⠖⠛⠓⠚⣠⠞⠉⠉⠙⠉⠉⠉⠀⠀⣴⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⡶⠒⠂⠀⠙⠀⠀⠉⠀⣿⣿⠈⠉⠁⢸⣿⢺⠅⠐⣶⣶⠓⢲⡖⣢⢱⡏⠀⠑⠖⠀⠱⠈⠉⠉⣹⣿⡿⢋⣿⣿⣿⡿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣀⣀⠘⠛⢘⣋⣙⣹⠉⠉⠭⣭⣭⣼⣷⣷⣿⣶⣷⣷⣶⣾⣴⣿⣿⡷⣾⠘⣿⣿⠇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡝⣤⡐⢤⣿⡿⢋⣴⣍⢢⢈⢻⣿⣿⣿⣿⡯⠭⠄⣿⣿⡿⣿⣿⣿⣦⣷⣿⣿⣦⠄
⠀⠀⠀⠘⣧⢠⠤⡤⢤⠄⠤⠠⠰⣿⣿⣿⣟⢸⠛⢿⠠⣻⡇⢾⠛⢿⡄⣂⣿⣿⣿⣿⡟⡇⠲⢲⡿⡛⣳⣿⣿⣿⣿⣿⣿⣿⢋⡳⣄
⠀⠀⠀⠀⠋⣽⠉⡽⡟⢻⡟⠋⣝⣿⣿⣿⣿⢠⠀⢬⢁⣭⡅⢈⠁⢬⡅⣤⢤⣤⣤⢤⢸⣤⣤⣼⣼⣷⣯⣽⡭⣿⣿⣻⣿⣿⣿⣿⣿⣷⣶⡢
⠀⠀⠠⣤⣟⣿⣞⣿⣷⢾⣷⣴⣏⣿⡿⢿⣿⡘⣾⡟⣸⣿⣧⢻⣶⣿⢁⣿⣿⣿⣿⠊⣯⢻⣿⣿⣿⣳⣽⢵⢓⡆⣹⢿⣿⣿⣿⣿⣿⡍⢌⢳⣄
⠀⠀⢸⣿⣷⣶⣾⣿⣽⣿⣟⣷⣭⣭⠭⣟⣛⣛⣂⣚⠿⠿⠿⠦⢍⣥⣾⣿⣿⣛⣿⣢⢏⣾⣿⣿⣿⣿⣿⣸⢸⡇⣿⣿⣿⣿⣿⣿⣯⠙⣦⢣⡹⣧⡀
⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣶⣿⣭⣭⣭⣟⣛⣒⣲⠶⠦⠭⢭⣥⣍⣙⣛⣛⣛⠒⠿⠷⠭⠾⠿⢿⣿⣿⣿⣿⣽⣷⣼⣷⣹⣼⡾⢆
⠀⠀⢸⣷⣾⣟⣻⣭⣽⡿⠷⣿⣿⣻⣿⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣼⣭⣭⣉⣝⣛⣛⡒⠶⠶⠶⠦⣤⣬⣭⣭⣉⣉⣙⣛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⢀⣄⠀⠀⠀⠰⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠻⠑⠀⠀⠈⠉⠀⠀⠀⠀⠀⡴⠎⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠶⠾⠛⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⣀⠀⠈⠙⠒⠒⠀⠤⠄⠀⢀
⠀⠀⠀⠀⢀⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⣿⣧⣀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠟⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠤⢴⡟
⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠴⠒⣛⣋⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢮⣯⣛⠛⣿⡶⠶⠶⢶⡶⠶⠶⠶⠶⢶⢶⠶⠒⢒⠛⢛⠛⣋⣉⣭⡀⠨⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⣦⣿⣦⣤⣤⣤⣾⡆⢂⡶⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⠀⢸⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⢷⢦⣄⡄⡤⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢿⣦⣴⣠⡤⠼⠳⡏⠀⠀⠀⣰⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠐⡄⢠⠀⢸⣿⠋⠁⠀⠀⠀⢹⡀⠀⢠⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⢄⠱⣞⠀⣸⡗⠀⠀⠀⠀⠀⠀⢣⠀⣸⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠈⣎⢶⡟⢶⣿⠁⠀⠀⠀⠀⡀⠀⠈⡄⡿⢰⠃⡼⠀⠀⣠⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢿⣼⢷⣠⣿⣦⠀⠀⠀⢀⠁⠀⠀⠠⡇⡎⣠⡗⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣾⠋⣿⣿⡏⣿⠀⣀⠀⡈⠀⠀⠀⠀⣷⠏⣼⠀⡾⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢰⡇⣿⡇⣽⠂⡇⢀⠆⠀⠀⠁⠀⣿⢠⠏⣸⣃⣠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣷⣿⡇⣿⣴⠀⢸⠀⡆⠀⢴⢰⣿⣯⣴⢟⣉⠥⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣆⢸⡀⣇⢰⣼⣾⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⣀⣀⣤⣄⡀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠤⣤⢤⣤⣤⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀
⠀⠀⠀⠐⠒⠾⢿⡁⠀⠀⢀⡿⠟⠁⠉⠀⠀⠀⠀⠀⠈⠐⢶⣁⠀⠀⠀⠀⢀⠽⠟⠀⠀⠀⠀⠀⠀⠀⠹⠶⣧⡀⠀⠀⢈⡹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠚⠁⠈⠁






⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠁
⠀⠀⠀⠀⠀⠰⣦⠀⠉⠙⠷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⣁⠀⠀⠀⣠⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠱⠂⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢲⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠰⠀⠇⠀⠤⠄⠀⠀⠀⠀⠀⠤⠄⠇⠰⠇⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠆⠤⠀⣼⠁⠀⠀⢹⣄⠀⠤⠶⠄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠉⠉⠁⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠀⡠⢴⣶⢶⣤⡀⢠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠴⠀⠘⡄⣠⠟⠋⠛⢆⡁⡜⠀⠰⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡄⣠⠀⠀⠀⠘⠈⠂⠀⢀⡀⡤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠈⠀⠀⠙⡟⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠐⠒⠂⠲⠀⣶⡇⠀⠀⠀⠀⠀⠀⣷⡆⠰⠂⠂⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣾⣿⡷⠒⠒⠒⠒⠒⠲⣿⣿⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⠿⠁⡇⠀⠀⠀⠀⠀⠀⡏⠹⢯⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠲⡞⢣⣤⣧⠤⠤⠤⠤⠤⢤⡧⣤⢛⡶⠒⠒⠒⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠾⠉⡈⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⡘⡹⢦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⢞⡥⠊⠀⠇⢠⣇⣀⣀⣀⣀⣀⣀⣇⠀⢇⠈⠢⣙⢦⡀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠴⠋⢀⠔⠉⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠀⠈⠑⢄⡈⠓⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⠔⠉⣠⠜⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠑⢤⡀⠓⢌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠔⠁⠀⢸⠀⠀⠀⠀⠀⠀⢈⡯⠉⠉⣩⡉⠉⠨⣏⠀⠀⠀⠀⠀⠀⢨⠀⠀⠑⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⡎⠀⠀⠀⣿⡇⠀⠀⠈⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠛⠛⠛⠛⢻⠛⠛⠛⠛⠛⠛⡇⠀⠀⠀⣿⡇⠀⠀⠀⡟⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠲⡞⠀⠤⠄⠀⣀⠀⡇⠀⠀⠀⣿⡇⠀⠀⠀⡇⢀⣀⠀⠤⠄⠐⠶⠂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠤⠤⠤⠬⡇⢹⡶⠶⢾⣿⡗⡶⠶⢺⠃⡻⠤⠤⠤⠤⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢁⠀⣿⣿⣿⡿⢿⣿⣿⣿⠀⠃⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣰⣄⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣴⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠉⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⠉⠉⠉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠅⠀⠀⠀⠀⠀⠄⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠄⠤⠤

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡿⢟⡿⠛⡇⡀⢰⣤⡀⠆⣴⡷⣾⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠉⠀⠀⠈⠑⠀⠀⡀⠀⠉⠃⠸⠀⠀⠉⡰⡄⠸⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⡧⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠐⠀⠀⠀⠠⡄⠀⠀⢻⡄⠀⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢸⣿⣧⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢻⡄⢃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⠀⢈⠈⠉⡿⠎⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠈⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⡇⣸⡀⣀
⠀⠀⠀⠀⠀⠀⣤⢠⠐⡏⠀⡁⠀⣷⠀⠀⠀⠀⠀⢠⡊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠀⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⢧⡟
⠀⠀⠀⠀⠀⠀⢹⣟⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⢆⡀⠀⠀⠀⠀⠠⢠⠀⢸⣧⣄
⠀⠀⠀⠀⠀⣴⣿⠏⠃⡀⠈⡃⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠰⠆⠀⠀⣀⣴⣷⣶⣿⡖⠒⠒⠀⣤⠀⡇⠀⢘⣿⣿⡀
⠀⠀⠀⠀⠀⣿⢿⠃⢄⠁⠀⠃⠀⠀⠀⠀⢠⣶⡤⠤⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣾⣟⢿⢯⣟⠛⠃⠀⠈⠀⡠⠁⠀⢸⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣶⠀⢘⠀⠀⠀⠀⠀⠀⢀⣾⠐⠋⣀⡴⢾⣿⣯⣙⢧⣸⣼⣷⣿⣿⣿⠕⢛⡩⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⡄⠘⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡄⢀⠀⠀⠀⠀⢀⣴⣿⣷⣶⣛⣋⣭⣼⣿⣿⣿⣿⣿⣿⡿⠟⠛⢲⣾⣿⣿⣿⣿⣾⠖⠁⠀⠀⠀⠀⠀⠇⢷⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⢠⠀⠀⠀⢠⣿⣿⣿⣿⠏⠚⠉⠉⠑⠒⢩⡾⣫⣤⣤⣤⣀⣼⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣷⠈⠁⠀⠀⠘⠻⢿⣿⣿⣿⣶⡀⠈⣀⣾⣿⣿⣿⣿⣿⠁⠙⠸⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠉⠉⠙⠁⠀⠀⠀⠀⢹⡔⠀⠀⠀⠀⠀⣌⠀⡂⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠤⣄⠀⠂⠀⠀⠀⠀⠀⠘⠀⠀⢁⡀⠀⠀⢠⠘⡇⠀⠀⠀⢀⡠⠎⠀⠉⠀⠀⠀⠀⠅⣠⣸⡿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⢈⡓⢀⡀⠑⠄⠐⡄⣠⠀⠀⠀⠀⠀⠠⡆⠀⣇⠀⠐⣴⣿⣇⠀⠀⠀⠀⠀⠀⠱⠞⢉⡿⠋⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡅⠀⣾⠆⣀⠠⢡⣷⣆⠀⠀⠀⠀⠀⠇⠋⢻⣦⡀⠹⣿⣿⣧⣀⣀⠀⠀⠀⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡞⢶⣂⣘⣧⣼⣿⣷⣶⢿⣿⡦⣄⠆⠀⠈⣿⠀⠀⠈⡊⣹⠿⣿⡻⠒⠒⣢⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢉⣽⣟⣿⣴⡿⢿⣧⠆⠉⣙⣿⣄⡀⠘⣁⡴⠈⠀⣰⡏⠠⣿⣥⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⡰⠈⠁⣷⣶⣿⣿⣿⣞⣧⡄⣠⣶⣿⣧⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠻⠿⠿⠟⠛⠛⠓⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠶⠾⠿⠿⠛⢻⠶⠶⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⢿⠛⢇⠀⠀⣀⣀⣀⡌⠀⡀⠰⣉⠘⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢏⠐⠀⣤⣸⣶⣿⣿⣿⣿⣷⣾⣿⣿⣍⠁⢀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⢈⡱⣶⣽⣿⡿⠿⠛⣿⣿⣿⣿⣶⣶⣾⡇⣾⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡀⠈⢣⣭⣿⣿⡇⠀⠀⣿⣿⣿⣉⡟⠛⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠉⠉⠀⠳⠟⢸⡇⠀⣰⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣶⣦⣄⡀⠀⠀⠀⠀⠀⠒⣨⠻⢧⢤⣽⣿⣷⡹⠎⠙⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢤⠀⠀⣠⣶⣿⡛⣿⣙⢋⣭⣻⡛⣦⠀⣤⠆⢰⠞⢡⣿⣿⣿⣿⣿⣿⣿⡵⠂⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣾⡿⢿⣯⣿⣿⣿⣷⣿⡿⠿⠟⠉⣀⠀⠀⢤⠀⣄⣴⡿⢻⣛⣋⣩⣍⣿⣿⡷⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡼⠋⠡⢼⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠓⠂⠒⢀⠙⠀⠀⢀⣤⣿⣷⠿⢟⣿⣯⣁⡙⠟⠻⢿⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠲⢿⣿⣿⣿⢻⣿⣻⠆⠀⠀⠀⠀⠉⠀⠀⢀⠾⠏⣿⣿⣿⣿⣿⣿⣿⣷⣦⡠⠐⠊⢹⣿⣷⣿⣈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣭⣿⡗⠈⡿⠻⠄⠀⠀⠀⠀⢀⣶⣦⡈⠒⠈⣩⣿⣿⣿⡏⣿⣿⣿⡏⠀⠈⠁⣼⣿⣯⣭⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⠀⠀⠀⠀⠀⣄⣤⣶⣿⣿⣿⣧⡀⠉⠀⠀⠀⣿⠁⣽⠏⠀⠂⠀⣠⣾⣿⣟⡛⠋⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠛⢫⣿⣿⣿⣿⠿⣿⡿⠖⢠⣤⡄⢨⡔⠋⢀⣤⣶⣾⣿⣿⣿⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠄⣹⠙⢿⣿⡏⠉⠁⠀⣼⡿⠀⠀⠀⠀⠀⠙⠿⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠸⠇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣷⣦⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢼⡷⠀⠀⠀⠀⠠⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⣾⣧⠀⠀⠀⠀⡄⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠐⠀⠀⠀⣾⢻⡇⣀⠀⢀⡺⠀⠀⠀⠀⠈⠙⠻⢿⠻⠯⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⢀⡀⠠⣄⢀⣿⣿⡷⠁⣀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡀⠀⢀⣿⠀⣾⣇⣾⣏⢸⣿⣿⡇⣼⣿⠇⠀⠀⠀⠀⠀⠀⣤⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⢨⣿⣧⢸⡿⢻⠉⣸⣿⡿⢲⣿⣿⣤⠌⠀⠀⠀⠀⠀⠀⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⣿⣧⡸⣷⣿⣿⢷⣿⣿⣿⡷⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣤⣼⣿⡽⠋⠙⢻⢿⡏⠈⣿⡏⠳⣼⡻⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠚⠿⠛⢛⣼⠿⢿⡏⢹⡏⠀⠙⡿⠛⢻⣿⠻⣿⣿⡿⣿⠋⢠⣞⠁⠀⢠⠁⢸⡇⠀⡿⡇⠀⠘⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠜⢫⠁⠀⢨⠗⣾⠀⠀⠐⠓⡀⠀⠻⡄⠘⣿⡀⠁⠀⠻⠋⠀⠀⡾⢿⠛⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠙⢃⠀⠀⠀⢠⡜⠢⠁⠀⡆⠀⠼⠉⠀⢰⠇⣤⠋⠹⠀⠀⡄⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠘⡄⠠⢃⡈⠑⠆⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠁⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠹⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣇⣦⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣼⣿⣿⡷⣄⡘⣦⠻⠿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢉⢻⢿⣿⣿⣿⣡⡱⣾⡝⢦⡈⠛⠛⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠈⠫⠳⣄⠻⢿⣿⣝⣾⣿⣷⠙⠲⣴⣷⣦⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣍⣛⠛⠛⠉⣠⣴⣾⣰⣤⡴⠶⣤⣀⠀⠉⠛⠿⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣿⣿⣿⣿⡿⡋⣠⣦⡈⡛⠿⣷⣾⣴⣴⣴⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⣀⣼⣷⠶⣄⡀
⠀⠀⠀⠀⣀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠛⣡⠞⠁⠈⠢⡀
⠀⠀⠘⠤⡐⢂⣉⡵⠦⡀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⣀⣠⣴⣾⡿⠛⠉⡠⠞⠁⠀⠀⠀⣠⣾⣆
⠀⠀⠀⠀⠈⠢⢔⢛⣮⣍⠶⣄⠀⠀⣜⣀⣠⣤⣶⣾⣿⠿⠛⠋⠁⠀⠔⠉⠀⠀⠀⠀⣠⣾⡿⢋⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠑⢶⣊⣻⣭⠵⠴⡿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⢋⣴⠟⠁⠘⡀
⠀⠀⠀⠀⠀⠀⠄⠒⠒⠻⣦⣀⡛⣮⠡⠤⣐⢖⠐⡤⢄⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⣁⠴⠋⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⠟⡆⠘⢸⡟⣿⠃⡇⢸⠚⣀⣤⣴⣾⣿⠿⠛⣁⠤⠊⠁⠀⠀⠀⣠⡾⢋⡅
⠀⠀⠀⠀⢀⡇⠤⣐⣒⠆⠀⢰⣿⢀⣧⣥⣬⠷⢷⡛⣿⡿⠛⠋⣁⠤⠐⠈⠀⠀⠀⠀⢀⡠⠞⢁⣴⣿⠃
⠀⠐⠽⠋⠉⠹⡀⠀⠀⠀⠀⡼⠉⠀⢛⣉⣁⣲⣉⠽⠦⣚⣺⡉⠀⠀⠀⠀⠀⢀⡠⠒⠉⢀⣴⠟⠋⠎
⠀⠀⠀⠀⠀⠀⠈⠢⠄⠤⠚⠁⠀⠀⠀⠻⡉⠁⠈⠒⢍⡩⠷⣜⣓⣄⡠⠔⠊⠁⣀⣤⠾⠋⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣤⠴⠷⠮⡩⠕⠐⠛⢢⠶⠛⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣤⣶⠾⠒⠋⠉⠀⠀⢀⡀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡆⠀⠀⡐⠀⢀⠀⠀⠀⠐⠀⠀⢀⠂⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠋⠀⠀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⡠⠊
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠂⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢐⡗⣶⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡀⠠⠂
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠒⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠂⠀⠀⠀⠀⠀⠀⠄⠠⢀⡀⠀⠐⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠈⠁⠐⣎⣿⡖⡆
⠀⠀⠀⠀⠀⠈⢁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⢀⡂⡌⠀⢀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⡿⠙⠰⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡀⢀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡥⡇⠀⠀⢀⠠⠈⠀⠤⠄⠐⠈⠀⠀⠀⢀⠀⠀⠀⠲⠄⠸
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⠀⠀⠀⠀⠄⢀⠀⠀⠁⠀⠀⠚⠀⠀⢠⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠠⠀⠀⠤⡀⠀⠀⠁⠀⠠⠐⡇⠈⣂⠐⠠⠆⠒⠐⠀⠈⠁⠒⡀⠀⠠⢴⠀⢤⢀⣀⡅
⣠⣴⡖⢲⡄⠰⠤⠤⠀⠤⠀⠙⠀⠐⠀⠀⡀⠀⠄⠀⠆⢀⠀⢀⠇⡀⡈⠁⢰⠄⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠
⠈⠟⠗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠃⠀⠀⡀⠀⠤⢀⠀⡀⢠⠀⡀⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠐⠄⠠⠄⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠖⠀⠀⠀⠀⠀⣠⠊⠀⠂⠠⠂⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠠⠀⠀⠀⠈⠀⢀⠀⠀⠡⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠄
⠀⠀⠀⠀⠀⠴⠒⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠂⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡗⠀⠈⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢳⡧⣕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢇⠣⠉
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠂⠀⠀⠀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⣤⠶⣶⠒⢾⣒⢲⣶⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⢷⣿⣟⣹⠀⢸⣿⣿⡿⣟⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢢⡀⠀⠀⠉⠒⠺⠭⣿⠀⢸⠗⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠃⠀⠀⢻⣄⠀⠀⠀⠀⠀⡏⠀⢸⠀⠀⣀⣤⠞⠉⢳⡀
⠀⠀⠀⠀⠀⠀⢸⢁⣤⣤⣤⢤⣤⣤⡈⢳⠀⠀⠀⣸⠁⢀⡏⢰⠟⠁⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠯⢽⡏⢸⣛⣻⠥⠊⠀⠀⢰⠃⠀⡸⠀⠈⠣⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⢠⠏⠀⢰⠃⠀⠀⠀⢠⠟⣷⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢧⠀⠀⠀⠀⡼⠀⠀⡏⠀⠀⠀⣠⠋⢠⠇⣠⠤⠶⠋⠉⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠘⣆⠈⢧⠀⠀⢠⠇⠀⢸⠃⠀⠀⣰⠃⢠⠏⠘⣇⢶⣢⠤⣀⠈⢧
⠀⠀⠀⠀⠀⣠⡾⠁⠀⠙⢦⡜⣆⠀⠳⡄⢸⠆⠀⢸⠀⠀⢰⠃⢀⡏⠀⠀⠈⣱⠏⣸⣿⣗⡼⠁
⠀⠀⠀⠀⠐⣏⣀⠀⠀⣀⡠⠟⠘⣆⠀⠹⣾⠀⢀⣿⡀⠀⣿⠀⣼⠀⠀⣠⠖⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠉⢹⠍⢻⡀⠀⠀⠘⣆⠀⢻⡦⡽⣿⣇⢸⣯⡇⣿⡴⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠙⢶⣶⢦⣹⡂⡈⣿⢊⡙⢻⢸⡿⣣⢾⣇⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠿⠟⠉⣧⠂⢸⡷⠶⠛⠈⢷⣐⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⣱⣤⣄⡄⠀⠀⠉⠚⢦⣐⣀⠐⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠑⡀⠀⢀⣽⣿⣿⣿⣿⣾⣦⠀⠀⠈⢿⣟⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⣀⢭⣇⠸⢟⣿⠹⠤⠤⠄⣼⠅⠄⠠⠐⠒⠒⠂⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠥⣤⣴⣮⡆⠀⠀⠈⠑⠀⠐⣖⣻⣷⣶⣆⢀⡈⢱⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢀⠀⠀⣻⢸⡟⠀⠀⡀⠔⡈⠀⠈⠻⡄⣦⠶⡛⣃⡴⣮⣤⣄⠉⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡏⠲⣾⢰⠄⣐⡉⠙⠢⡄⠁⠐⢲⠁⠓⠒⠃⠤⡇⡬⠍⠠⠀⠂⠁
⠀⠀⠀⠀⠀⠀⣀⠀⢠⠄⢀⣈⠀⡇⠀⠀⠐⠐⠀⠖⢀⣠⣤⣤⣄⡀⡜⠀⠀⠀⢀
⠀⠀⠀⠀⠰⢉⠠⢴⡶⢶⡂⠨⠀⢣⠀⡄⠀⠀⠀⠘⠉⠰⡞⣿⣾⠁⣁⣠⠴⠶⠶⠦⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠘⣄⠇⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠃⠀⠘⡏⡟⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⢆⠘⠃⠀⠀⠀⡆⠀⠀⡇⡀⣈⠠⠄⠂⠚⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠞⣳⡶⣽⠄⡆⢠⠀⠀⠇⠀⡘⣀⠿⠽⠟⣶⡶⢶⣶⣶⠯⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣹⣔⢷⡀⢲⡀⢸⠌⡴⠁⠀⠠⢊⣵⠆⡐⡡⠍⠠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠀⠓⠾⠎⣠⠶⢚⡋⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⢀⣧⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠟⠛⠿⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⢀⢤⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠃⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠛⠀⠀⠘⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡼⣿⣿⣷⠀⠀⠀⠀⢻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢀⡼⡇⠀⠛⠁⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠯⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢠⣤⣶⠾⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠉⣁⣀⣤⡞⠓⢤⣀⠀⠀⢀⠴⠖⠘⣿
⠀⠀⠀⠀⢠⡀⡀⡠⠤⢈⡀⢉⠉⠻⠘⠿⠋⠁⠀⠀⠀⢹⣟⠷⠋⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⢶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠚⣁⣘⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⠀⠀⠀⠀⠀⣿⣹⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠠⠀⠉⠉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⣸⣟⠉⣤⠀⠀⣰⠊⡱⠆⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠁⠀⠀⠀⠐⢠⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣇⠀⢿⣀⠶⠙⠦⣽⠀⠀⠀⠘⢫⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣰⡿⣻⣿⠋⠐⠀⠀⠀⠀⠀⠀⣎⣻⢷⣾⣇⣠⣤⣿⠁⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⠀⠀⠀⣼⣿⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠺⠝⠛⠋⠉⠀⠀⠀⠳⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⢻⣿⢿⡟⠀⠀⠀⣸⣿⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⢠⠀⠀⠀⠠⣄⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⢨⣿⣿⡀⠀⠀⢀⣹⡟⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡄⠀⠀⠀⠀⠀⠀⠀⠰⡀⠈⠀⠀⢀⠀⠘⣇
⠀⠂⠀⠀⠀⠀⣾⣿⡟⢡⡀⣀⣿⣉⢰⣿⢤⠀⡤⠀⠀⢘⣃⠀⣰⠄⢐⢿⠙⣧⡤⡄⠀⢀⣆⠀⢃⠀⢱⡀⠀⢀⡸⢆⠀⢹⣧⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠠⣿⣿⠁⠀⣸⡿⠁⠀⣿⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⢳⠀⠀⠁⠀⠃⠀⢹⡆
⠀⠀⠀⠀⠀⢸⣿⡿⠂⢠⡿⠁⠀⢰⠏⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢀⣼⣿⠇⢀⣾⡟⡾⢠⡿⢠⠇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠈⣿⣿⠀⣼⡿⢹⠃⣿⡁⠊⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⢠⣿⣿⣶⣿⠃⡎⣰⡟⠳⡇⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⠀⠀⢸⣇⠀⠀⠀⢠⡀⠀⠤
⠀⠀⠀⠀⠀⣿⣿⣿⡿⣼⠃⣿⠀⠀⠇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠻⢛⣻⣛⠋⢸⡇⠀⠀⠀⣼⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⡇⠀⠀⢰⣦⣀⠀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢢⣾⡿⣿⡿⠀⣾⣁⠀⠀⢠⡟⠁⠀⢀⣮⣄⠀⠀⠀⠀⠀⠀⣼⡇⠉⡶⣦⣀⡀⠀⠀⣤⠹⡀⠁⠒⠀⣀⡄⢀⡠⣄⠀⠙
⠀⠀⠀⣼⠨⣿⣏⠻⣿⣿⣟⣿⣆⡀⠈⢗⣤⠴⣿⠁⢸⠗⢤⣄⣀⠀⢀⣿⣿⠀⢣⠈⣏⠻⣿⡂⢻⡄⢳⡀⠀⠀⠈⡿⠇⠀⠀⠀⠀⠀⢻
⠀⠀⠀⣼⣧⡙⣿⣷⣿⣿⣿⣿⡟⢡⣴⣿⡏⢰⡿⠀⢸⠆⠀⣿⠉⠙⠛⢻⣿⠀⢸⡄⢹⠀⢏⠹⣌⣷⠈⣧⠀⠀⠀⢐⠂
⠀⠀⠀⢿⣿⣷⣦⣉⣙⣛⣋⣥⢀⣿⡇⣿⠃⢸⣇⠀⢸⠀⠀⣾⠀⠀⠀⢹⢹⡇⠈⣇⠀⠀⠘⠆⠉⢻⣇⠸⡄⢀⠀⢈
⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⢻⡟⢸⣿⢰⣿⠀⢸⡟⠂⢸⠄⠀⢸⡀⠀⡀⢸⠀⣇⠀⢻⡄⢰⡀⠑⠀⢻⣿⠀⢳⠀⡆⠸⡆⠀⠇
⠀⠀⠀⣿⣿⣿⠿⢱⡟⠀⣿⠁⢸⡟⢸⡿⠀⣿⡃⠀⢀⡇⠀⢸⣇⠀⠀⠀⠀⣿⠄⢿⣿⠈⡇⠀⠀⠀⢿⡁⠈⠀⠙
⠀⠀⠸⣿⣿⣤⠀⠀⠀⢸⠃⢀⡞⠀⡾⠃⠀⣿⣶⠀⠸⠿⠀⠀⠉⠀⢸⠀⠠⣧⠄⠸⠅⠀⢃⠀⠀⣀⣸⣧⣀⣤⣄⣠⣤⣤⣠⣤⣶⣶⡶⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣶⣾⣴⣦⣤⣤⣀⣠⣾⣁⣀⣀⣦⣀⣀⣴⣶⣶⣆⣠⣴⣧⣤⣴⣶⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡉⠁
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠻⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⢿⣍⣹⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡿⠟⠛⠉⣉⡠⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠏⠛⠁⠀⣄⣠⢦⣿⣿⣿⣿⣿⣿⡿⠿⢛⡛⢿⣿⣿⣿⣿⠋⠄⠠⠀⠀⢽⣦
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠊⢴⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡆⡛⢿⢣⣻⣿⡏⠃⠀⠀⣤⠀⠁⢸⣿⡇
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠿⢿⣿⣷⡤⡺⣿⣿⣿⣿⠟⠋⠉⠈⠀⢀⠀⠀⠃⢧⠈⡆⢿⣿⣇⣄⠀⠃⢬⢀⣸⣾⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣥⣠⣼⣿⣿⣿⡿⠿⠛⠉⡏⢰⠻⣦⠀⠀⢸⡀⠀⠰⡘⣤⣇⢸⣾⣿⣮⣻⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣸⣿⣿⣏⣽⣿⡟⢻⠟⠋⣹⣀⢀⣀⣬⣧⣺⡟⢿⣴⣴⣾⣷⣤⣤⣵⣌⣛⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⣿⢻⣿⣿⣷⣾⣷⣿⣿⣿⣤⣿⣿⣿⠟⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢭⡏⠈⠢⠤⠌⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⢿⡿⠫⢉⡉⠉⠉⠛⣿⣿⢿⣿⣿⣿⣿⡏⣾⣛⡄⢉⣿⣿⣦⢹⣟⢇⣠⣿⣯⠂⠀⠀⠀⠀⠀⠄⢻⠿⣿⣿
⠀⠀⠀⠘⠟⣽⢛⠀⠉⠉⠀⠂⠀⠀⠀⠻⣷⡽⣿⣯⣿⣿⣿⣿⣇⡸⣿⣿⣿⣷⣬⡠⣉⡛⠿⢦⣥⣀⣀⠀⠀⠀⠘⠓⠈⠙
⠀⠀⠀⠀⣼⣛⣊⠈⠁⠀⠀⠀⠀⢠⡾⠟⣉⢹⣽⣿⡟⢯⠍⠛⠾⠦⠉⣽⠛⣿⣿⣿⣿⣿⡗⠲⣤⣍⣙⠛⠻⠿⠁⠀⠘⠀⣱⡄
⠀⠀⠀⠀⣿⣅⣈⣀⣤⣤⣤⢶⣦⠟⠁⠀⠁⣼⣿⣿⣿⢋⣶⣶⣶⣶⣶⣶⠐⣿⣿⣤⣿⣿⣿⣤⣀⣠⣍⣝⣿⣿⣶⣤⣶⠛⠓⠚
⠀⠀⠀⠀⣿⡆⠛⠮⠙⠷⠟⠋⠁⠀⠀⠈⣠⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⡅⢿⣿⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣉⣿⣿⠏
⠀⠀⠀⠀⠘⢧⣤⡀⣀⠀⠀⠀⠐⢀⣤⢾⣵⣿⣿⣿⢃⣷⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⡿⡃⠀⠁⣉⣾⣿⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠙⠳⠶⢆⣀⣂⣈⡿⢞⣿⣿⣿⣿⡏⡸⣻⣿⣻⣿⣿⣿⣿⣿⡆⣿⣿⣟⣫⣅⣤⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣶⣶⣿⡿⣾⣿⣿⡿⢃⣷⣿⣿⣿⡟⣿⣿⣿⣿⣧⡸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡷⣻⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣷⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢹⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠉⠉⠉⢧⡙⠉⠉⠉⠉⠈⢿

⣿⣿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠤⠄⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⢄⣀⡈⠢⡙⠦⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⠄⢙⠂⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡈⠀⠄⢈⠘⠐⠣⠰⡀⢠⡀⠀⠀⠀⡄⠀⠀⠠⢀⢀⠀⠀⠀⠀⡐⣆⢀⢢⣎⣾⣇
⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⠁⢰⣦⣤⣤⣉⢀⡁⢀⡁⣀⣀⡠⣀⣀⣀⣁⣈⠀⠉⢉⣈⣀⣉⠠⠃
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⣤⠠⢠⠠⠆⠖⠠⠂⡴⢀⢄⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢸⠉⠉⠘⠛⠷⢽⡍⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⠗⡌⢠⣠⡦⡴⣤⣠⣦⡤⢂⠡⢢⢐⠀⠀⠀⠀⠀⢇⠀⠀⢸⡗⠦⠠⡀⠀⠘⠐⢟⡇
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠬⢥⡿⠟⠀⠀⠀⠀⠸⣿⡷⠖⠁⣀⡐⣀⠀⠀⠀⠀⢸⡄⠀⢸⠀⠀⠀⠙⠖⠳⠞⣾⡟
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣰⡃⢽⡇⠀⠀⠀⠀⠀⠀⠻⣷⣆⠀⠀⠀⢰⡦⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⡼⠀⠀⠀⠀⠀⠀⠀⣰⣦⠥⢿⣇⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠈⠉⠑⠻⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢱⠀⠀⠀⠀⠀⠀⠀⠹⢶⣮⢻⣿⠀⠀⠠⣄⣀⡀⠀⠀⣹⣦⡍⡉⣐⡛⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢟⢔⢿⣤⠀⠀⠀⠀⠈⢢⣾⣿⠝⠻⠦⠀⠀⠀⠀⠀⡼⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠌⡾⡆⡿⠀⠀⠀⠀⠀⠀⠀⢺⠂⠁⠀⠀⠀⢀⡠⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣆⡀⠤⠔⠂⠉⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⣀⠀⠀⢀⣀⡌⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⢀⠆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠲⣤⡀⠆⢣⣿⠀⠀⠀⠀⠀⠀⣾⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⠞⠋⠁⠀⠀⠈⠀⠐⠨⣽⣼⣼⣿⠎⠰⠂⠀⢠⠀⢻⣿⣼⡅⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⣣⠞⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡀⠸⢠⢀⠘⣀⣿⡁⠟⡄⡘⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢀⠟⢸⡇⡿⠏⡷⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⢂⠀⠠⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣹⡀⡆⠙⡇⡇⢱⠂⠄
⠀⠀⠀⠀⠀⠀⠈⠀⡀⣿⢸⢸⡀⠀⠀⢰⠀⢀⠀⠀⢠⠀⢳⠂⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠳⠙⣇⠘⡃⣷⠘⠖
⠀⠀⠀⠀⠀⠀⣃⠐⠳⡜⣰⢸⠀⡇⠀⢻⠀⢨⠄⣶⡞⡀⣿⠀⠠⠀⠀⠃⠁⠂⡀⠀⠀⠀⣰⢃⡗⢃⠸⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠉⠈⠀⠓⡿⠞⡔⣇⢻⢸⣧⠉⣆⣿⠰⣿⡛⠃⠾⠀⠂⠀⡀⢠⣼⣤⡼⠛⠁⠀⢳⠁⠀⠀⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠀⠇⠘⠀⡀⠃⠆⢬⠈⣸⠀⣿⢼⣇⣄⠼⠀⡀⡔⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠀⠁⠁

⠀⠀⠀⠀⣤⣤⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣿⣿⣧⣤⡀⣠⣤⣽⣿⣿⣤⣤⣤⣤⣤⣤⣤⣿⣿⣧⣤⡄⠀⠘⣿⠛⠛⠛⠛⣿⠀⠠⣤⣶⣶⡄
⠀⠀⠀⠀⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⠁⠈⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠁⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢻⣿⣿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢀⣿⡇⠀⠀⣿⣿⠀⠀⣠⣿⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢘⣟⣛⣛⣻⣛⣛⣟⣛⣛⣛⣛⣛⣛⣻⡃⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣟⣛⣟⠀⠀⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣟⣛⣛⡛
⠀⠀⠀⠀⢸⡿⣻⣻⢿⣻⡻⣿⣿⣿⣻⣻⢿⣟⡿⡟⠋⠋⣿⣟⣟⠿⣟⣻⣿⣿⡿⣿⣻⢿⣟⣻⣿⠙⠛⢿⣿⣯⣿⣿⣝⣿⣿⣿⡿⣿⣿⣯⡻⣿⣄
⠀⠀⠀⠀⠈⢹⣱⣿⣿⣳⣹⠈⠋⢾⣱⣹⣯⣶⣿⠁⠀⠀⠹⣟⣎⣿⣱⣺⠏⠛⠹⣷⣮⣿⣱⣻⠏⠀⠀⠀⠁⢻⣧⣼⡿⠙⠛⠛⠛⣿⣤⣾⠏⠛⠛
⠀⠀⠀⠀⠀⠐⠛⠓⠒⠛⠓⠒⠒⠒⠛⠓⠚⠛⠓⠒⠒⠒⠒⠚⠛⠒⠛⠓⠒⠒⠒⠛⠛⠒⠛⠓⠒⠒⠒⠒⠒⠒⠛⠛⠒⠒⠒⠒⠒⠚⠛⠓⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠠⠒⠶⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⠀⠀⠀⠀⠀⠀⢀⣵⡼⣶⣿⠀⠀⠀⣤⣤⣀⣀⣀⣄⣀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠒⠲⠶⣾⢟⢭⣴⣶⣿⣏⠁⠒⢶⡟⠀⡫⠅⠉⠛⡶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢉⠃⣾⣿⠟⠙⠓⠈⠐⢄⢻⡄⠋⠀⠀⠘⠞⣦⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠂⢀⣒⣤⣄⡌⢠⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⡷⡀⢰⠚⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⣉⡠⠤⠐⠛⡭⡙⢀⢎⠎⠀⣦⡀⠀⠀⠀⣼⡟⠛⢻⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢁⠴⣻⣿⠀⠀⠀⠀⢠⠁⣌⠃⠀⠀⠈⢿⣄⠀⠀⠙⠇⢀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠰⠁⡀⣿⡏⣀⠄⠀⠀⣼⡞⣤⡴⠈⡲⢟⣛⠿⠄⠐⢲⣾⢎
⠀⠀⠀⠀⠀⠀⠀⣿⢀⠇⠀⢸⣿⠏⢁⣀⢠⣤⡿⣿⠩⠔⠁⠀⠉⠉⠀⢢⣄⠀⢧⠸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⣼⣿⠟⠉⠀⠀⠹⢷⡝⣥⠀⠀⠀⢠⣦⠚⠀⠹⣄⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡀⢠⢊⠄⢀⠀⠀⡅⡟⠂⠀⠀⢹⠏⠀⠀⠀⢻⣷⠀⠀⠀⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⣿⣧⣤⠁⣠⠎⠀⠀⠀⠀⣤⣼⠿⣦⣀⡀⠀⠈⠓⠤⣀⠀⠀⠉⠻⢶⣖⡈⠒⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢝⠻⣧⣾⣧⣄⣀⣀⢀⣀⠿⢏⡀⠈⠓⢮⣳⠶⢤⣀⠀⠈⠁⠒⠢⢤⣈⠻⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡉⠀⠀⠀⢘⣶⣦⣴⡞⠁⠀⠀⢀⡞⠀⠀⠀⣽⢧⠀⠀⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠐⣹⢿⡇⣿⣷⡄⣀⣴⣿⣡⣤⣴⠤⠉⣸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠟⡘⠻⢿⣿⢀⠏⠈⠓⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠃⠇⠀⠀⠈⠁⠀⠀⢶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠟⢿⡟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣠⡾⢧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡿⢄⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠠⣷⣤⣤⣤⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠩⠁⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠐⣀⣡⢴⠶⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠒⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡿⣷⢸⠀⠀⠈⠹⠷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⢠⠆⢀⣀⣀⣴⢨⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠛⠛⠛⠛⠋⠉⠉⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠘⠿⠷⢖⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⣂⣈⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠀⠈⠉⠉⠉⠉⠁⠂⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣀⡀⠠⠤⢀⣀⣀⡀⠤⠄⢀⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠉⠙⢿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡟⣿⣿⣧⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣄⠙⢿⣿⣯⣝⣛⠛⠻⠿⠿⠟⠛⣛⣫⣽⣿⡿⠋⢠⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣉⠀⣿⣷⣄⠉⠛⠿⠿⣿⢿⣿⣿⡿⢿⠿⠻⠛⢉⣠⣶⣷⠀⣈⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣿⣿⣿⣿⣿⣸⠑⠒⠂⠤⠄⠄⠀⠤⠤⠐⠒⠋⣿⣿⣿⣿⣿⣿⢿⡋
⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⢷⠸⠘⡀⠈⠙⠻⢧⣖⣀⣀⣀⣀⣀⣀⣀⣀⣲⡬⠟⠋⠁⢀⠃⠆⡾⠉⠒⠢⢄⡀
⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⢸⠀⡆⠃⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⢠⠀⡇⠀⠀⠀⠀⠙⢢⡀
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⡄⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⢠⡇⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡔⠉⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠉⢢⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠉⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠠⢤⣤⣦⣔⣰⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣀⣀⠘⢷⣦⣤⣄⣠⣤⣅⡀⠀⣀⣀⣭⣵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠙⣿⣿⣿⣿⣿⣆⠈⠉⠻⠿⠿⣿⣿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡠⠀⢀⣠⣤⣤⣤⣤⢠⣀⡞⠁⢰⣿⠿⢿⡿⠛⣻⣷⡀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣟⢻⣿⣿⡿⠹⡇⠀⠞⢁⡴⠞⢻⣿⠿⡁⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠟⠻⠿⢻⣈⠋⠁⠀⠀⠀⠀⣴⣿⣿⣿⡾⠏⠀⠉⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⡏⠂⠀⠀⢠⠀⠀⠁⠁⠈⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⣢⠀⢡⠀⠀⠀⣾⠀⠀⠄⠀⠀⠀⠀⠀⠊⢡⠄
⠀⠀⠄⠀⢀⠀⡀⠀⠈⡐⢿⣿⣿⣦⠆⢀⠀⣿⣿⡄⢸⡇⣀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⢰⣦⣼⣧⣄⣠⣿⣿⣧⡛⠻⣿⣿⣿⣿⠿⣧⡿⠂⠁
⠀⠀⠀⠀⠀⠀⢈⣧⡀⠸⣿⣯⡟⣻⣿⣿⣶⣿⡟⠔⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⣯⣄⠀⠀⠃⣽⠞⢏⢀⠈⠍
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠠⠆⠚⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⢾⠀⠀⣷⣄⡘⣄⡀⠀⠳⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⣿⣿⠿⠿⠿⠿⠀⠺⠿⠿⠿⢿⣿⡆⠀⢀⣿⣿⡆⠀⠀⢸⣿⡿⠿⠿⠿⠇⠀⠿⠿⠿⠿⠿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⣧⠀⠀⢸⣿⡇
⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡟⢸⣿⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⢸⣿⡿⠿⠿⠿⠃⢰⣿⡇⠘⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⣼⣿⢠⡆⣿⣿⠀⢸⣿⡇
⠀⠀⠀⣶⣶⣶⣶⣿⡿⠀⢸⣿⠀⠀⠀⠀⠀⣿⡿⢸⡇⢸⣿⡄⢸⣿⣷⣶⣶⣶⡆⠀⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡛
⠀⠀⠀⠀⠀⢀⣤⣄⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⣤⠀⢰⣶⣾⣿⣿⣷⣤⣄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣧⣤⣀⣠⣄⠀⢀⣤⣀⣸⣷⠀⢀⣀⣼⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⣶⠶⠆⢲⣄⡶⢰⡶⢶⡆⣶⠀⠀⡶⠶⣦⢰⡶⢶⡄⢰⣶⠀⠶⣶⠖⢰⠀⣶⠶⣶⢰⣆⢰⡆
⠀⠀⠀⣿⣦⠄⠈⣿⡃⢸⣇⣸⡇⣿⠀⠀⡇⠀⣿⢸⣧⡼⠇⣼⢻⡄⠀⣿⠀⣼⠀⣿⠀⣿⢸⣿⣼⡇
⠀⠀⠀⣿⣀⡀⣸⠟⣧⢸⡏⠀⠀⣿⣀⡘⣧⣀⣿⢸⡏⢷⢀⡿⠺⣇⠀⣿⠀⣿⠀⣿⣀⣿⢸⡇⢿⡇
⠀⠀⠀⠈⠉⠁⠉⠀⠈⠈⠁⠀⠀⠉⠉⠀⠉⠉⠁⠈⠀⠈⠀⠁⠀⠁⠀⠁⠀⠈⠀⠉⠉⠁⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠂⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⢂⠉
⠀⠀⠀⠀⠀⠀⢰⡇⣰⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠓⠘⣋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣷⠆⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡼⠃⢸⡆⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⡿⣇⠉⢃⣨⠐⢪⣴⣶
⠀⠀⠐⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣠⣀⡠⣶⣤⡄⠀⢀⡙⠛⡟⠛⠁⢠⡆⠘⠃⠀⠀⠀⠀⠇⢀⣤⠀⠀⠄⢾⠷⣼⡶⠂⠨⡀⢘⠿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡀⠀⠀⠀⣼⣧⢛⠈⠑⢻⠁⡰⢒⠀⠡⢤⠁⠀⠠⠾⠇⠀⠀⠀⠀⠀⢠⡗⢸⠃⠀⠀⣀⣼⣶⠤⠠⠄⢳⣷⣮⣄
⠀⠀⠀⢀⠀⠀⠀⠀⠈⢐⣚⣶⠠⠄⠉⡁⠈⠀⠀⠻⡿⢂⡘⠋⠲⢾⡀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠙⠡⠂⠀⠀⠀⠘⠿⠀⠂⠀⠀⣾⢡⣤⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠿⢁⣨⣷⡀⢀⣀⣤⣆⠘⠀⡇⠀⠀⠐⠲⣸⠁⠀⠃⠀⠀⠀⡴⠲⡆⠀⡀⠀⠀⠀⠒⡖⠀⠠⣤⠼⠁⡼⠻⣼⣏⣩⣀
⠀⠀⢠⠀⠀⢀⠀⣠⠦⠀⣼⡇⠀⠀⠸⣿⣿⣯⡄⠀⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠘⣀⡇⣼⣵⣆⣶⣶⠹⣟⡦⠀⠃⠀⡾⠀⣤⣭⣽⣿⣯
⠂⠀⠀⡝⠻⣿⡀⠉⠀⣠⣶⠰⣶⡆⣰⣿⡿⠉⠆⠀⣰⠁⠘⢧⡀⠀⠀⠂⠀⠀⣷⠀⠐⣿⣧⣼⣿⣿⣿⣦⡀⠨⠯⠿⢀⡞⠁⣀⡉⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠚⠇⢠⣴⠟⠸⠃⠙⣷⣿⡿⠣⠀⢢⡀⣿⠀⠀⠀⢷⡀⢀⣀⠀⠀⠂⠀⠘⢉⣿⣿⣿⡟⣹⣿⣿⣦⠀⢀⡼⢶⡴⠻⠀⠈⢸⡅⠔
⠀⠶⢀⡄⠀⠀⢸⣿⠁⢠⠇⠀⠈⠱⠼⠃⠀⢂⢀⣤⣿⠀⠀⣤⣤⡅⢿⢧⣶⠀⠀⠃⣰⣿⣿⣿⣿⡏⠋⣿⣿⢻⣄⣾⣻⠟⠁⠐⠃⠀⣨⣴⡧
⠀⠁⣬⣴⠀⠀⠎⢩⡧⠰⢆⣀⠀⠶⣿⢿⣶⣾⣯⣟⡿⠁⠀⠉⢹⣧⣨⠈⣿⡇⠙⠚⢯⣟⠛⣿⠟⢷⣾⡇⠁⢈⡿⢻⢿⡳⣴⣾⣶⣿⣟⣋⣻
⠀⠀⠸⡱⢜⠛⡲⣿⣷⢀⣿⡯⠀⣸⡏⠀⠈⠙⡟⢻⡟⠀⠀⠀⠘⢋⣩⠼⠻⣿⡆⠀⡌⠉⠀⠁⠀⠸⠟⠀⠠⢋⣽⣟⣫⣿⣿⣿⣿⣿⣟⣹⣟
⠀⠀⠀⠀⠸⠀⠸⡿⢇⠘⢿⡇⢸⣿⠃⠀⠀⠀⢸⡀⡇⠀⠀⠀⢀⠟⢀⡶⠀⠉⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⡬⠛⠛⣻⠿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠑⣄
⠀⠀⠀⣠⠴⣶⣦⡂⠀⠀⣴⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⡀⠠⢀⠀⠀⣴⢶⡷⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣭⡽⠛⢿⠀⠈⠁⠀⠘⠀⠀⠀⠀⠁⠄⠀⠢⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⢠⣷⣆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢳⠀⠙⠀⢰⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⣆⠀⠀⢆⠀⠀⡀⣿⣦⡀⢠⡄⠀⡾
⠀⠀⠀⠀⠀⣮⠀⠿⠀⠀⢸⡇⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⢄⠀⠐⡄⠀⠀⣤⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠈⠂⠀⠓⠻⢿⡁⠘⠃⣘⠃
⠀⠀⠀⠀⠀⢸⡇⠃⠀⠀⠸⡗⠀⠀⣿⠀⠀⠀⠀⡄⠀⠀⠀⢄⠐⠀⢹⡀⠀⢿⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠂⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⡻⡆⠀⠀⠀⠀⠀⠀⣿⣄⣆⠀⠀⡂⠀⠀⠀⠀⠳⠄⠀⣷⠀⣼⣿⣠⡾⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠘⠛⢦⣄⣤⣤⡀⣀⣤⣻⠿⠋⠀⢀⠃⠀⠀⠀⠀⠀⠘⠎⠼⡇⢹⢹⡟⠁⠀⠀⣀⣴⣿⣿⣿⣿⠳⢶⣤⡀⠀⠀⠀⢀⣷⡾⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡉⠀⠀⠀⠘⠛⠀⠀⠀⢀⠀⠀⠒⠾⠟⠛⠛⠂⢀⣴⣿⣿⠏⢹⣿⠹⠞⠓⢂⣉⠛⠆⠀⠀⠀⠀⠀⢸⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢩⡏⠓⡀⠀⠀⠀⠀⡠⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣴⣿⠳⡿⢀⠈⢋⣄⣶⣿⣿⠿⠛⠀⠀⠀⠀⠐⠂⠘⠇
⠀⠀⠀⠻⢶⢶⣤⡀⠀⠀⠘⠠⣶⣿⣎⠀⠆⠀⠀⠀⠀⣉⠉⠙⠛⠋⠙⠁⠀⠀⠀⠶⢧⣼⢷⣾⣿⠭⠇⢙⠋⠁
⠀⠀⠀⠀⢠⠋⠉⠀⠀⠀⠀⠀⠉⢸⡇⠀⠀⠀⠀⠀⠛⠛⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠋⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⠃⠈⠋⠀⠀⠀⠀⠀⠀⠀⣤⡆⣴⢄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣤⣤⢿⣿⣯⠀⢸⡿⠋⠀⠀⠈⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠞⠋⠀⠀⠀⠀⠀⠀⢴⣦⣄⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⣀⠀⠸⣧⠀⡀
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⠁⠺⣤⡀⢀⠀⠀⠀⢀⣄⠀⠣⠀⢠⡿⠉⢹⣿⣶⡄
⠀⠈⠛⠚⣷⠈⠀⠀⠰⢦⣤⣀⣄⣀⣤⣤⡤⠶⠆⠋⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⠀⠀⠀⠀⠈⢛⡛⢿⣅⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡟⠀⣶
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⠀⠀⠀⠀⠀⠈⢠⡀⢠⡾⠃⣿⠿⠿⠗⠀⠤⠀⠀⢈⡁⣀⣤⡴⠂⠰⠀⢀⡤⢀⠀⠀⠀⣇⠀⣿⢀⡄⣶⠄
⠀⠀⠀⠀⣸⠁⠀⢀⠐⠀⢠⠂⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⡧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⡀⢻⣄⡿⠈⢣⡄
⠀⠀⠀⠀⠏⠀⠀⣂⠀⢶⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠐⠂⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⢀⣿⠘⠛⣁⠈⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⡧⣤⣤⣴⣷⣶⣶⣾⠗⠀⠀⢠⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠠⢦⣤⡀⠀⢄⣤⣤⡀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠺⡿⠿⠷⠜⠛⠛⠈⠉⠀⠉⠀⠀⡄⠘⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡶⣤⡤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠘⢿⡁⠈⠁⠞⠁⠀⢠⠀⢀⠀⠀⠀⠀⠀⠈⠉⠉⣇⠸⠃⢠⠆
⠀⠀⠀⠣⠄⠈⠐⢼⣿⠛⠋⢛⣿⣶⢶⣀⡠⣄⡀⠀⢀⠀⠘⠄⠠⠀⣿⣷⡀⠀⠀⠈⣧⠀⠀⣽⠀⠀⠈⢀⡞⠀⢰⠇⠀⠀⠀⠀⠀⡏⠀⠀⠘⢀⡀
⠀⠀⠀⠀⠀⠈⠉⠁⢻⠃⠀⡾⠛⠒⠓⠛⠑⠚⠃⠀⢺⡀⢸⢰⠇⠀⢻⣿⡇⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⡾⠁⢀⡀⢂⡄⠀⠀⠀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣠⡤⢤⣿⠀⠀⣗⡄⠀⠀⠀⠀⠀⠀⢀⣘⡁⠈⠈⠀⠀⠈⠛⠃⠀⠀⣼⣇⠀⠀⠀⣀⠀⢀⣱⣶⡍⠩⡧⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇
⠀⢰⡔⠶⠿⠿⠀⠘⢻⠀⠀⡏⠁⠀⠀⠀⣠⠴⠾⢻⣿⣷⣦⡀⠀⠀⠄⠀⠀⠀⠀⠈⠓⠈⠃⠛⠋⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠘⠃⠀⣿⠀⠀⠇⠀⠀⠀⢈⣀⣄⣀⠀⢹⡏⢹⣿⠳⡀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⢀⡀⢀⣀⣄⣠⡴⠀⠀⢀⣼⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠸⡀⠀⠀⠀⠉⠛⠻⠷⠦⠴⣮⣥⣦⣥⡄⠐⠀⠀⠀⠘⠋⠻⠇⠀⠰⠾⠟⠑⠛⠉⠉⠁⠀⠀⢰⣾⡟⠀⢠⠘⠀⣠
⠀⠀⢰⣦⡔⣆⠘⡐⠇⢠⡀⠀⡽⠇⠀⠀⠀⠤⠀⠀⠀⠐⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠋⠶⠀⡏
⠀⠀⠈⠉⠀⠈⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠘⠀⢀⣶⣶⡆⠀⠈⠛⠚⠿⠿⠋⠀⠀⠀⠀⠀⠲⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠤⠠⡄⠀⠀⠹⠿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⣠⣀⣤⣀⣤⣶
⠀⠀⠀⡤⠶⠄⢀⣤⠻⡧⢨⣤⣀⡴⢧⠀⠆⠀⢀⠀⠀⠻⠟⠀⠐⠃⠲⢴⣤⠀⠀⢀⣴⡾⠛⠃⠀⠀⠀⠀⡄⠀⠀⠀⣀⣀⣠⠀⠈⠁⠉⠉⠀⢀
⠀⠀⠀⠀⢸⣅⣤⠀⠃⠈⠀⠀⠀⠉⠘⠀⠀⠉⢭⡓⢄⡀⠀⠀⠀⠀⢀⡀⠈⠁⠀⢸⣿⠀⠀⠀⡀⠀⠀⠸⣷⢤⠄⠀⠉⠹⠉⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠘⣯⠙⠀⣠⠀⣀⣠⣶⣄⣠⣄⣀⣰⡿⣤⠂⡵⠀⠀⠀⠀⠙⠃⠠⠀⠀⢸⡏⠀⣤⡖⠙⠀⠀⣻⠀⠀⠀⠀⠀⠀⢆⣰⢴⡖⢠⡦
⠀⠀⠀⠀⠂⠛⠀⠋⠉⠐⠉⠉⠉⠀⠀⠉⠁⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣻⣇⢰⣿⠏⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣸⠿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠱⣄⡤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⢳⡆⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⡒⠲⢧⡄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡠⠄⡀⠄⢳⡀⠉⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡈⢄⡠⠇⠐⢸⡇⡇⠀⠀⠀⠈⠁⠐⠦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡳⣄⠀⠄⢐⣠⠟⣩⣵⠖⣀⣀⣀⠀⠀⠤⠄⢹⣷⠈⠉⠒⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢫⣾⣭⣭⣭⣷⢠⣿⢏⣈⣉⣉⣙⣛⡛⠳⠦⣾⣿⠀⠀⠀⣿⠈⠉⠂⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣥⣬⢿⢯⢭⣭⣅⣛⠟⣈⡏⠉⢻⣵⣦⣹⡟⣩⡰⣾⠀⠀⠀⣿⠀⠀⠀⠈⠉⣿⣩⣛⢲⣶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢯⢣⢯⡏⡞⡸⡰⠸⣹⢁⢉⣇⣀⣸⣤⣤⣭⣍⣉⣉⣡⠦⣤⢤⣿⣄⣀⣀⣀⣀⣸⡏⠉⠉⠘⠓⠻⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣏⣏⣎⣾⣸⣱⣧⣧⣧⡇⢟⣯⡻⠟⣵⣙⠟⡱⢣⡛⢟⠞⠻⢇⠿⣿⣷⣿⣾⣿⣻⡿⣷⣶⣶⣾⣶⣾⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠿⠿⠿⠶⠶⠾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⠁⠁




⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿