Spray Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠄⠙⠲⢤⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠦⠤⡄⠀⠀⠈⠑⠦⠀⠀⠀⢀⡴⠶⠶⠶⠶⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⠄⠀⠀⣈⡉⠒⠲⠤⠒⠤⢸⡟⠻⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠤⠶⠖⠂⠀⠀⢀⡀⠀⠈⢉⣉⡥⠴⠀⢸⣧⣴⠇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠉⠀⡀⠀⠉⠀⠀⠀⠀⠸⠷⠶⣶⠶⣶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋⠀⠠⠒⠋⠁⠀⠀⠀⠀⠀⠀⣶⠶⠶⠻⠷⠻⠷⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠤⠤⠶⠤⠤⠤⠤⢤⣤⣤⣤⣄⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠚⠛⠋⠉⠛⠛⠒⠒⠒⠒⠒⠤⠭⠭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣤⣤⡟
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠶⠶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠠⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠈⠻⣿⠿⢿⣦⠰⡾⠇⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣼⡿⢘⣟⠟⠛⠿⠟⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠿⠿⠿⢿⣿⣦⡀⠀⠀⠀⢿⣏⠉⠀⠈⠛⠃⠶⣶⣦⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠙⢿⣦⡀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣷⣶⣶⣶⣾⣿⣶⣶⡟⢿⣦⡀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⢿⣦⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣤⣶⣶⣶⣶⣶⣤⣄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠛⠉⠀⠀⠀⠀⠀⠉⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡶⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡳⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣯⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠞⠒⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣿⣿⣛⣿⣭⣭⣭⣽⡟⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⡀⠤⠔⠒⠒⠒⠒⢢⣼⣿⣿⣿⣿⣿
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿

⢠⣴⡀⣖⣦⣦⡀⠄⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀
⢹⣿⠀⠈⠐⣹⣖⡖⠛⣲⡆⣰⣷⣀⣰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⡆⠀⣿⠀⢀⣄⠀⠀⠈
⢸⣯⠀⠠⠀⠁⠉⢀⡶⣿⣷⡈⣿⣝⢿⣿⡀⠀⢀⣤⣾⢻⣿⡿⢻⣦⡀⠀⠤⣿⡿⢊⣿⢃⣾⡍⠉⠁⠀⠀⠈
⢸⣿⠀⠒⠀⠀⡒⢸⣷⡛⣿⣿⡿⢿⣶⠛⢷⣠⣾⡿⠿⣿⣿⣿⡿⢿⣿⣞⣶⠟⢷⣿⣿⣿⣿⢺⣿⠀⠀⡀⣀
⢸⠉⠀⠀⠀⠀⠀⠈⢿⣷⣿⣛⡳⣿⣱⣿⡶⢿⣃⣀⣾⣿⣿⣿⣷⣤⣀⣿⢳⣿⣎⣿⢟⣻⣷⣿⡏⠀⠒⣓⡂
⠈⠃⠀⠀⠀⠀⠀⢤⣀⡻⣿⣷⣮⣳⣮⣍⣙⣸⣿⣿⣿⣟⡙⣩⣽⣿⢿⣟⣈⣭⡵⣞⣵⣿⣿⣯⣽⠤⠀⠉⠁
⢤⠶⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡼⢿⣷⣬⣙⣿⠸⠿⠟⣥⡝⠿⠟⢸⣯⣩⣿⡿⢷⣤⣾⣿⠿⢫⡄⠀⠞⢡
⠘⠓⠀⠀⠀⢠⣄⠀⠀⠾⣿⣿⣷⣧⣤⣌⣍⣻⣿⣿⣿⣄⣋⢃⣼⢿⣿⣿⣯⣭⣴⣾⣾⣿⣾⠇⠐⠀⠀⠠⣖
⢀⡆⠀⠀⠀⠞⠁⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣧⡻⣜⠿⡾⠿⢟⡼⣳⣾⣿⣿⣾⣿⣿⢧⠀⠀⠀⠀⠀⢄⠨
⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⣿⣿⣿⣿⣿⣿⣽⠿⣿⣷⣤⣴⣯⡿⣯⣿⣾⣿⣿⣿⣿⠀⠀⢀⠀⠀⠀⢘⢠
⢰⡖⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠯⢿⣷⣾⣿⣿⣿⣿⣿⣾⠿⠀⠈⠀⠀⠀⠄⠐⢎
⣾⣿⡀⡂⢟⡀⠀⠀⠀⠘⠘⠋⠈⠋⠙⠈⠻⣿⡟⢻⡽⢿⣿⣿⣇⣿⣿⣯⡿⠛⠛⡏⠃⠀⠉⠈⠙⠃⠈⢉⣥
⣾⣿⠁⠀⠠⠄⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠂⡟⠛⠟⠻⣿⢷⡾⠿⠿⠟⠉⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⡀⢘⢻
⢹⣿⠛⠚⠓⠒⠋⣠⡆⠀⡀⠀⠀⠀⠀⠀⢀⡧⠄⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠠⠶⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠈⠀⠀⠀⠀⠀⢀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⠈⣉⣉⡹⣏⣀⡂⣀⠀⠀⠀⠀⠀⡀⡠⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠈⠉⠙⡛⠋⠛⣩⣏⢀⡀⠤⠒⠈⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⢀⡀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢊⠴⣾⠉⠉⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠐⡁⢛⣿⠈⠑⠂⣁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠰⠀⠀⠰⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢿⡛⠧⣀⠀⠀⠨⡶⢉⣼⢷⡦⣀⠀⠈⠐⠠⢀⡀⠀⠀⡀⠀⠐⠀⠨⠀⠀⠂
⠀⠀⠀⠀⡀⠐⣂⠀⠓⠈⠃⠀⠐⠂⠀⠈⠋⠈⠿⠿⠿⠚⣙⡿⢹⢶⡄⠠⠄⡈⠑⠢⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠂⠀⠠⠀⢀
⠀⠀⢠⠎⢰⡾⠋⠀⠀⠀⠀⠘⠲⠄⣤⣤⣤⣀⣤⣤⠶⠾⢋⢻⠘⡀⠍⣻⡆⠈⢣⡀⠀⠀⠁⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠆⠀⠈⠳⣔⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠌⡌⠆⠆⠂⣸⡼⠃⠀⣼⡁⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⠀⠂⠀⢀
⠀⠀⠐⣤⡄⠀⠀⠈⠉⠓⠒⠶⠤⣤⠄⠀⠀⢀⠀⣠⣀⡴⢼⠴⠚⠋⠁⠀⣀⡴⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⢀⠀⠐⠀⠀⡀
⠀⠀⠀⠩⠘⠻⢴⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢠⢠⣠⣤⠶⠟⡉⡔⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠓⠰⠲⠖⠲⠺⢂⠋⡙⢩⠋⡝⢡⠁⠌⠜⡰⠡⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠰⢡⠂⡐⢡⠂⠌⡐⢁⢣⠘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢡⠊⠐⠁⢂⠃⠜⠃
⠀⠀⠀⠀⢠⠀⠀⢠⡤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⢁⡈⠠⢀⢂⠀⠘⠁
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⠀⠉⠙⠛⠉⠀⠐⠒⠒⠀⠀⠀⠉⠉⠉⢙⡀⢠⢂⠈⠸⠃
⠀⠀⠀⠀⠀⡇⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠠⢂⢎⠼⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠅⢨⡗⡡⢁⢎⠸⠃
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠰⠎⡠⢣⠂⢼⠁
⠀⠀⠀⠀⠀⢸⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠰⠎⡠⢡⢂⢺
⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠰⠊⠔⡡⡡⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠰⠂⢴⠕⡡⠟
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⢼⡣⢊⠜⡠⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣂⣿⠔⢁⠜⡰⡇
⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⠀⠀⠀⠀⠉⠁⠐⠛⠓⠒⠒⠀⢛⠟⡡⢃⠔⠡⢞⣼⡇
⠀⠀⠀⠀⠀⢀⡇⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠔⣡⢆⠡⡣⣫⣦⣀⡀
⠀⠀⠀⠀⠀⡌⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡡⠚⡡⢋⢌⡔⣵⣟⣷⠤⠤⠤
⠀⠀⠀⠀⠀⢣⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠔⡡⢔⡡⢊⢎⣾⠏⢹⣇⣀⠀⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠔⢉⣔⣵⠟⠃⢸⡟⠂⠐⡂⠐⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣀⠈⠁⠀⠀⠀⠀⠀⠀⠒⠄⠆⠃⢈⣀⡰⠖⠁⠄⠀⠀⠤⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣓⢓⣦⣤⣦⠴⢤⠅⢃⠓⣛⣉⣉⣉⣉⣉⣁⣈⣉⣀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⣀⣀⣀⣀⣀⣀⣀⣀⠐⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠛⠿⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣀⡤⠖⠋⢁⣀⠤⠖⣒⣋⡥⠤⠒⠀⠀⠀⠀⢀⣸⡀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠯⠷⠖⠚⢻⣀⡤⠖⠋⢁⠤⠒⠉⠑⠒⢉⣉⡥⠤⠤⠤⠤⠤⠄⠀⠀⠀⠤⠇⠀⠰⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢄⢠⣿⣋⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣤⠉⢹⠏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⢳⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣠⣴⠟⠀⠀⠀⢻⣄⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠛⠉⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠘⠷⣤⣀⣀⣀⣀⣀⣀⣀⣤⠴⠚⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⡀⠀⠉⠉⠉⢉⣉⣡⡤⠴⠚⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⡏⠋⠉⠉⠉⠉⠁⠀⡤⣄⣀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⣸⠁⠀⠉⡗⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡇⠂⠀⠀⠀⠀⣰⠃⠀⢀⠞⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⣰⠀⠀⠀⢰⠏⠀⠀⠛⢋⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⠀⢠⡯⠴⣶⠀⢠⠎⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⠀⠈⠀⢠⡇⡰⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢡⢸⡄⠀⠀⠀⣸⡸⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⢸⡇⠀⠀⠀⣿⠃⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠸⡈⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⠷⠦⠤⠶⠶⠶⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⢤⣖⣆⣄⡀⠈⠀⠈⠐⡒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠂⠀⠀⠀⢀⠈⢰⡑⠁⢀⠠⠈⣽⡄⠀⣼⢛⠓⣗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢾⡉⠉⢳⣌⢂⠐⠠⢐⣀⣺⠇⠀⣺⣦⣶⠷⠠⠌⣛⣓⣀⣜⠄⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠘⠚⠱⢀⣴⠇⠈⠀⠀⢮⡍⢛⡷⠀⠠⢀⣈⣁⣉⡐⠠⡉⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠰⣀⡀⣥⣮⡴⠛⠥⡊⠠⣀⣀⠩⠔⣎⠇⣠⠮⠍⣥⠑⣿⢛⣦⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣈⣥⡵⣺⠖⣀⠢⠁⠈⣉⠉⠉⢃⠜⠁⡐⠄⡛⠒⣋⢴⠇⠀⠁⢀⡀
⠀⠀⠀⠀⠀⢀⢀⠤⠒⢚⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠘⡠⠁⠄⠠⢀⣛⠒⢈⢾⠟⠋⠛⠗⠈⢻⣈⣁⠤⠀⢀⠄⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠉⠄⠀⠀⢫⣯⡄⠀⠀⡂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⢮⢅⠛⣶⠶⡞⠛⣻⣷⢼⣇⠘⡗⠚⠁⡐⠑⡈⠀⠀⡀⠄⠄⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⣣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⠄⠀⠀⢀⠀⠹⣝⢦⢹⣦⣤⠄⠟⠲⠻⢶⢤⣤⠥⣾⠵⡂⠆⣾⡁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣠⡿⠖⠒⣟⠽⠛⢀⠜⠒⢦⡀⠀⠀⠀⠀⠈⢡⣎⠁⢀⣼⠆⠀⣿⡮⡣⢁⠴⣀⢰⣦⠀⠈⡓⢇⠰⠥⠂⠀⠀⢈⠁⠷⠥⠤⠂⠄
⠀⠀⠀⠀⣆⣀⣉⣉⣀⣀⠤⠔⠊⢁⠂⠀⠀⡉⢠⠀⠀⠀⠀⠀⠉⡛⠋⢃⠀⢀⠏⠣⡊⢓⠉⠟⠛⢁⠀⢀⣎⠢⠀⠐⠉⠀⠀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡴⢡⡞⣃⠀⠀⠀⠀⠀⣧⡀⣀⡠⡪⠂⠀⣈⢀⣬⢽⡉⠀⢒⠛⠯⣑⡦⣄⡀⠀⠠⠀⠈
⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠚⢁⡤⠎⠇⠩⠀⠀⠀⠀⠀⠈⠁⡛⠉⠀⠩⠰⣇⠈⢀⢸⡧⢀⣴⡆⠐⠘⠙⢹
⠀⠀⠈⢦⣄⣀⠀⠀⠀⠀⣀⣀⡤⠔⠊⠁⠀⠀⠠⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠞⠁⡤⡻⣌⣒⡩⠄⢤⢀⠈⠁
⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣐⠀⣐⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⢀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢘⢄⠀⠀⠀⠀⠀⠀⢀⠀⡠⠴⣋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⣀⣀⣀⣉⡩⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠉⠉⠉⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣤⢶⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⣿⣿⣿⣿⣿⢸⣸⣿⣿⣷⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣥⣿⣿⣿⣿⣿⣿⣹⣼⣿⣷⣿⡿⣣⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⣿⣮⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡻⠟⠋⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠽⠒⠂⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢰⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣮⣻⣿⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⡿⢋⡰⡇
⠀⠀⠀⠀⠀⠀⠈⢰⣎⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠿⠛⢋⣡⡴⣻⡆⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣭⣽⣟⣛⡻⠻⠯⠭⠍⠭⠭⠭⣭⣥⣴⣶⡶⠿⠛⠉⠚⢹⢫
⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⢟⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠘
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⣿⣿⣿⣿⣿⣿⡷⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⣿⣿⣿⣿⣿⣿⣿⣿⣟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⣿⣿⣿⣿⣿⣿⣿⣯⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿⣿⣿⣿⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠛⠿⡿⠿⠛⠻⣴⠆⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣷⠶⡷⠲⢦⠆⠛⠟⡄⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡶⠾⠢⠒⠂⠐⠀⠀⠀⠀⠀⢀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⢻⣽⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⣯⣛⠿⣿⣿⣿⣽⣿⣶⣤⣤⣤⣤⣼⠿⠛⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠹⢳⣶⠤⣤⠴⠶⠿⠷⣖⡂⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉

⠀⢀⣤⣤⣄⣀⠀⠀⢀⣀⣀⣀⠀⣠⣤⣤⣄⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⡛⠛⢶⣄
⠀⠈⢿⣦⠉⠙⠻⣶⡟⠉⠉⠙⢿⠟⡅⠈⣿⡾⢛⠉⢀⣴⠟⠀⣤⣀⠀⠀⢰⣶⣤⣄⠀⠸⣿⣷⣶⣄⠀⠀⣿⡾⠀⠀⠀⣿⡀
⠀⠀⣾⡟⠋⠂⠀⢹⣇⠀⠀⠀⠀⠀⢃⣀⣿⠁⠀⠸⣿⣅⠀⠀⢻⣿⣿⣶⣄⠙⣻⣿⣿⠀⣷⣿⣿⠏⠀⢼⠉⠈⠀⠀⠀⠀⢹
⠀⠀⠈⠻⣶⡦⠀⠀⣹⣷⣄⠀⠀⠀⣨⣾⡃⠀⢠⣶⠾⠛⠀⠀⢠⣶⣿⣿⡋⣾⣿⠿⡇⠘⠿⠟⠉⠀⠀⠈⠷⣞⢓⢖⡛⣶⠋
⠀⠀⠀⠀⠹⠷⠾⢿⣉⡯⠛⢷⣤⡾⠋⠉⢻⠿⠶⠿⠁⠀⠀⠀⠙⠻⠋⠀⠈⠉⢁⡀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠻⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣼⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠘⢻⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⡟⠷⣴⠟⢛⣿⣿⠋⣉⠍⠀⠀⠀⢀⣤⣄⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢶⣾⣿⠀⠀⠁⠀⠀⠐⠋⠀⣿⠁⠀⠀⠀⠀⠀⠘⢿⣿⣦
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠅⠘⣧⡀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⢰⣟⡵⣦⢸⠙⢷⡄⠀⠀⠀⠀⠙⢷⣦⣀⣰⣟⣻⣷⣄⣀⣴⠟⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⢀⣼⢇⣍⠁⠀⠀⣸⡇⠀⠀⢠⣾⣶⠛⠉⠛⠛⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⢸⡇⠈⡛⠀⡀⠀⢹⡇⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠛⣿⢻⡛⠛⣿⠟⠁⠀⠀⢀⣴⣾⠟⢻⡿⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠸⣦⠁⢸⡟⠀⠀⠀⢰⡿⠉⠁⠀⢸⡇⠀⣿⠈⣿⡆⠀⠀⠀⠀⠀⣀⣸⣿⣿⣦⣶⣶⡦⠀⠀⠀⢀⣼⣿⣿⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢿⣅⣿⠁⠀⠀⠀⢿⣇⠀⢀⣴⣿⣷⣤⡙⠀⣿⡇⠀⠀⠀⠰⣿⣿⣿⣏⠉⣿⣿⠟⡁⠀⠀⠀⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⠀⢸⢿⣶⣟⡁⢸⡇⠉⢻⣷⡿⠁⠀⠀⠀⠀⠇⠉⣿⣿⣿⣿⣿⣧⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠈⠙⠻⠿⠿⠿⠟⠋⠃⠀⠀⠀⠀⠀⠀⠰⠿⠟⠁⠈⠏⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡖⠋⠉⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⡀⠀⠀⠀⠀⠀⠀⠠⠟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⣤⠀⠀⣀⡤⠖⢋⡡⢤⣒⡬⠥⠖⠒⠀⠀⠀⠀⠠⢦⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠉⣥⠼⠒⠉⠀⠐⠈⠀⠈⠀⠐⠒⠚⠛⠋⠉⠁⠀⠀⠀⠉⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣄⠛⢲⠒⠒⠒⠒⠓⠚⠛⠛⢻⠟⠒⠒⠲⠄⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣴⠃⠀⠀⢲⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣏⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⣏⡀⠀⠀⠀⠀⠀⠀⣀⡼⢋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠉⠉⠛⠛⢛⣋⣉⡥⠖⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⡏⡏⠉⠉⠉⠁⢰⠤⣄⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⡇⠀⠀⠀⢀⠇⠀⢀⠏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⢀⡎⠀⢰⠧⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⡇⠀⢀⣞⡤⡄⠀⡔⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⣇⠀⠈⠁⢰⢃⠎⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢀⢸⠀⠀⠀⣾⠏⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⢸⠀⠀⠀⠋⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠾⣼⣄⣀⣀⣤⡤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠒⠂⠀⢀⡼⠤⠀⠀⠀⢀⡨⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣠⠴⠟⠒⠒⡂⠋⠈⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣿⣀⣴⠞⠋⠀⠀⠀⠀⠀⠑⠒⠃⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⣠⣄⣰⠏⠀⠀⠀⠀⠘⣧⣰⠻⡶⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⡤⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⢀⣴⣿⠋⠳⡀⠀⠀⠀⢠⡞⢉⣯⠻⡄⠀⠀⣠⠶⠦⠞⠃⠸⣟⠎⠳⢄⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠙⣦⣤⡴⠋⢰⡿⣌⣢⠈⠳⢮⣁⣀⣀⠀⠀⠀⠹⡗⢀⣀⣵⢶⣷⡾⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⣠⠴⣿⠁⠀⠀⠀⠘⢧⣀⡈⠑⠀⠀⠀⠀⠈⠑⠄⠀⠀⠉⠉⠁⠀⠸⣅⠙⠲⠦⢤⡴⢺⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⢀⣿⡄⠀⠀⠀⠀⣤⣻⡍⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣹⠇⢸⡄⠀⠀⡞⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⢸⣿⣇⠀⠀⠀⠀⠸⣟⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⢠⠘⣆⣿⣇⠀⡼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰⠀⢹⠀⠈⠙⠢⠀⣤⠀⠀⠀⢀⣿⣷⠀⠀⠀⣿⡆⢸⣿⣿⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⢄⠀⠀⣾⡇⠸⡀⠀⠀⠀⢠⣿⡆⠀⠀⢸⣿⣿⡂⠀⢠⣿⡇⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⣿⣦⣿⣷⠀⠙⢂⠀⠀⢸⣿⣇⠀⠀⣼⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡷⢦⡴⢄⣀⣿⣿⣿⣀⣀⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣾⣧⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⡀⠘⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠁⠀⠀⠀⠀⢠⠀⠀⠘⠦⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠊⠸⠈⠃⠀⠀⠀⠀⢀⠰⠀⠀⠀⠠⡀⠀⠀⠠⠀⠀⡀⠀⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠐⠒⠀⠀⠀⠀⠀⠔⡀⠀⡠⠀⠁⠀⠀⠲⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠐⡀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠠⠈⠀⠈⠢⠔⠊⠁⠀⠀⡀⠄⠂⠀⠀⢎⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠄⠀⠀⠀⢀⠇⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠤⠀⠀⣀⡠⠄⠀⠈⠁⠀⠀⠀⠀⠀⠠⠀⠉⠐⠢⢀⠀⢧
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠸⡆
⠀⠀⠀⠀⠀⣎⠠⠂⠁⠀⡀⠀⠀⠀⠀⠈⠁⠀⠡⢄⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⣤⣄⣠⣬⣭⣷
⠀⠀⠀⠀⠀⡏⠡⠀⣀⣀⣀⣤⣄⣀⠀⠀⣠⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣇⠐⠋⠉⠉⠉⠀⠀⠀⣀⣀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⡿⠿⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡟⠛⠳⢦⣠⣄⡀⠴⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⠀⠀⠀⠀⣤⣾⣦⡐⠛⠉⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢷⣦⡀⠀⠈⠁⠀⣀⣤⣾⣷⣦⣤⣄⣀⣀⣤⣦⣴⣶⣾⣷⣷⣶⣶⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠻⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢿⠀⠄⠺⠦⠴⠶⠴⠶⣤⣤⡤⣤⣤⣤⣤⣠⣤⣄⣀⣀⣀⣈⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⠛⢻⠖⠊⠈⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣬⣭⣭⢽⠶⠶⠶⠦⢤⠄⢀⣀⠀⠀⠀⠈⠉⠛⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡤⣿⠶⠶⣦⣄⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠠⠅⠀⠀⠀⠀⠀⠀⠉⠙⢩⣙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢴⣶⡒⠶⣿⡙⠻⣿⣍⠉⠻⣿⡙⢿⡖⢶⡶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠺⢿⣭⣉⡛⠷⢮⣝⡲⢬⣙⠳⢤⡉⠳⢤⡈⠳⣄⠹⡌⢧⢹⢸⢇⡟⣹⠖⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⢯⣛⡿⠷⠶⢤⠤⠭⡙⠳⠦⣬⣙⠳⢮⣝⡲⠬⡓⠤⣙⠢⠈⠃⠈⠀⠈⠈⠈⠐⠡⢞⡡⠞⣋⣷⢤⡀
⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠈⢲⡀⠀⣰⠃⠀⣨⠉⠓⣤⣌⣙⣒⠦⢍⡓⠤⢁⠀⠁⠀⠀⢠⠞⠉⠉⠉⠳⡄⠀⠐⠈⠥⠴⠶⢛⣳⣄
⠀⠀⢀⣴⣫⠔⠢⣅⣠⡞⠉⠳⢄⡀⠉⠀⠠⠉⠁⢀⣼⠷⢻⣿⣷⢶⡌⠉⠂⠀⢀⡴⢲⡇⠀⠀⠀⠀⠀⢸⣀⣠⠞⠛⣯⡩⠭⠭⠭⣷⣄
⠀⢠⣾⣿⣶⣾⣿⣿⣿⣷⣄⠰⠞⣉⡦⠴⠞⣳⠖⠋⠁⠀⠘⣿⡏⠀⠈⠓⣲⡯⣍⡀⣨⢷⠴⠚⠓⢦⣤⣾⣾⡃⠀⣼⡿⠻⢏⣹⢷⠚⠙⢦
⢠⣿⣿⣿⣿⣿⣇⣤⡤⣉⡛⠳⢿⡀⣀⣴⣞⠁⢀⠀⠀⠀⣩⣿⡿⠀⢀⡟⠙⠲⡀⠉⠳⢤⣴⡶⠾⡿⠟⡻⠉⠀⠀⣘⣷⠀⠀⢙⣷⣯⣴⣾⣇
⣼⣿⣿⣿⣿⣷⠟⣽⣻⣯⣿⣷⣶⡉⠛⢿⣿⣿⣶⡄⠀⣴⣿⠓⠀⣠⣿⣤⣤⣠⠞⢀⣀⢤⣬⣻⣮⣄⠈⠀⠀⢀⣠⣿⣿⣤⣾⣿⡿⣿⣟⣿⣿
⣿⣿⡟⣷⡿⣿⢻⡼⣧⣷⣿⣶⣿⣿⣶⣄⣌⡙⠻⢿⣼⣧⣤⣤⣾⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠶⢬⣿⣿⢿⡟⣿⣾⣿⣿⣻⣯⣿⣿
⣿⣿⢿⣿⣷⣞⣚⣿⣿⣭⣿⣿⣿⢻⣿⣶⡬⠝⠳⣤⣈⣙⠻⢿⣿⣿⣿⣟⡷⠾⠿⠛⢛⣋⣭⣭⣷⣶⣾⣷⣿⣿⠿⣾⣝⣗⣯⣷⣿⣿⢾⣿⣿
⢹⣿⣾⣿⣷⣮⣿⣧⣿⣭⣻⣏⡽⣻⣿⣿⣿⣷⣄⡙⠻⣿⡷⢯⣤⡿⠉⣿⣀⡀⠝⠻⠿⠷⠶⠯⢿⣧⢾⣛⣿⣶⡿⣿⣿⣿⣷⣿⣿⣹⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡟⣿⣆⣤⣍⣳⣯⡳⣄⣈⠙⠛⠻⠶⣬⣵⣾⣿⠿⠷⠿⠯⠿⢶⣿⣟⣿⣞⢻⡿⣽⣿⣿⣿⠃
⠀⠀⠹⣿⡽⢿⣿⣿⣾⣿⣿⣿⣿⣿⣯⣗⣺⠭⣿⡷⣷⣿⣾⣳⣾⣿⣦⣽⡷⠀⠀⠀⠈⠹⣿⣿⣿⣯⠭⠽⢿⣟⣒⣶⣭⣝⣛⣿⣿⣿⡿⠁
⠀⠀⠀⠈⠻⣷⣾⣷⣺⣭⣿⣿⣿⣿⣭⢟⡻⢿⡲⢭⣓⢪⣽⣿⡿⠟⠋⠁⠀⠀⠀⡀⠀⠀⢸⣿⣛⣛⣯⣭⡿⢾⣛⣛⣿⣭⣭⣭⡽⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣞⣛⡻⣍⣯⣭⣝⣿⣮⡓⣭⣷⡾⠟⠋⠁⠀⠀⠤⠀⠀⠀⡀⠀⠀⢀⣼⡿⣟⣻⠽⢗⣛⣭⠭⠴⣶⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣿⣽⣹⣯⣷⡿⣿⣭⠁⠄⠀⠀⠀⠒⡤⠀⠁⠀⠴⠀⣠⡶⣛⠽⣛⣭⠶⣚⡭⢵⣲⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣯⣶⣶⣖⢐⣠⣶⠖⠔⠐⠀⢢⣾⣷⣻⣵⣟⣯⣶⠿⠗⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⢧⣀⡿⠀⠀⠀⠀⢀⣤⠶⢶⣤⡀⠀⠀⣸⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢉⣩⣭⣉⠛⢷⣄⠀⠀⠀⠈⢻⡇⠀⠀⠀⣴⢏⣴⠶⣶⡌⣷⠀⠀⠛⠁⢀⣤⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉⠉⠛⢶⣌⢷⡄⠀⠀⠀⠀⠀⠀⣸⠃⡾⠁⠀⠀⠀⣽⠀⠀⠀⢠⠿⠟⠛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠙⠷⠹⡆⠀⠀⠀⠀⠀⡟⢸⡇⠀⠀⠀⢀⡟⠀⠀⢀⡟⠀⢀⣴⠏⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡾⠁⠀⠀⣼⣇⡴⠛⠁⠀⢀⣾⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⣇⠀⠀⠀⢠⡾⠁⠀⠀⠀⠻⠟⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⡀⠀⠀⢿⠀⠀⠀⣿⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢞⣛⣫⣭⣭⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣸⡇⠀⠀⢸⡆⢠⡟⠀⠀⣤⣤⠀⠀⠀⠀⠀⢀⡴⢋⣴⠞⠋⠉⠀⠀⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣭⣿⣦⡄⠀⠀⠀⠀⠀⠙⠿⠗⠀⠀⠀⢷⣸⡇⠀⠀⣇⣼⠀⠀⠀⠀⣰⠏⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠈⢿⡇⠀⠀⣿⠃⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠈⠳⣿⣆⠈⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣋⣩⣥⡴⠶⠀⠀⠀⠉⠙⠛⠶⣄⡀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢉⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣀⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣴⠾⠛⠉⣡⣤⣤⢤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣴⠞⠋⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠷⣦⣤⣀⣀⣤⣤⠶⠾⠛⠉⠀⠀⠀⢰⡏⠛⢛⣃⣈⣙⣆⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⣠⣶⣋⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉⠙⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠿⠛⡿⢿⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠲⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣧⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⣠⣷⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣦⣆⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⢶⣿⣷⣠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣟⠛⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣓⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠉⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠒⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢐⡪⡆⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠠⢪⣵⠗⡄⠀⠀⠀⠀⢀⣿⣿⡄⠀⠀⠈⠿⡿⠀⠀⠀⣸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣻⣿⣿⡇⠀⠀⠀⠀⠀⠘⢴⠴⡺⠁⠀⠀⠀⠀⣸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣷⢿⣿⣷⠀⠀⠀⠀⠀⡰⡈⡎⢌⢎⠁⠀⠀⠀⣬⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢻⣟⣿⣿⠀⠀⠀⠀⢐⢘⢰⢩⢣⢂⣣⡄⠀⠀⠹⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⣟⣿⣿⣿⠀⠀⠀⠀⢘⠂⠡⢮⢎⠒⢘⡀⠀⠀⠀⠀⠉⠁⠀⠀⣴⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⣿⡿⠏⠀⠀⠀⠀⠀⠨⢈⡔⡆⡌⠹⠂⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⠀⠀⠹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣆⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠈⠛⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢀⣾⣿⣿⣿⡆⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠏⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⢹⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⡽⣤⡇⠀⠀⠀⠉⠋⠁⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⣳⣶⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣷⠶⠞⣣⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⢀⣽⡛⠞⢉⣧⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⡄⠀⠀⠀⣻⣿⣾⣯⣋⡆⠀⠀⠀⠀⢸⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢿⣿⣿⠀⠀⠀⣼⢿⣿⡇⠀⠀⢰⣿⣽⠿⣮⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⢿⣝⣶⢏⣿⠀⠀⠀⠀⠀⠀⠙⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠉⠙⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠠⠤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⢿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠀⢤⣄⡈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠛⢿⢧⠖⠢⣤⠿⠾⣷⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠹⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣴⣶⣶⡞⣠⣾⣅⣀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣿⣿⣿⣿⣓⣻⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡿⣿⣿⠡⣽⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠀⠨⠀⠍⠀⣽⠉⡇⠈⡉⠨⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣂⡇⣄⠙⠀⠀⠀⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⡖⣋⢹⠛⢀⣥⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣭⣿⡛⠛⠛⠻⠟⢃⣻⠞⠛⠻⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣉⣉⣙⣷⢶⣾⣒⣛⠿⠛⠶⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣁⡀⠈⠙⣒⣦⣤⡶⠒⠒⠲⠶⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣝⡛⠛⡟⠛⠃⢐⣒⡂⢀⡠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠉⠳⣄⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⡇⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡷⢤⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⡠⠚⠛⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠉⠉⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠘⢿⡿⠃⠀⠀⢀⣀⣼⣄⣀⣀⣧⣀⣀⣸⣇⣀⣀⣼⣀⡀⠀⠀⠀⠈⠻⡿⠃⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⢸⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⡇⠀⠐⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠁⠀⠈⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡇⠁⠉⠉⠁⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣅⣀⣀⣀⣀⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡧⠤⠤⠤⠤⠤⢴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠒⠒⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⡿⠿⠿⢿⡿⠿⠿⢿⠿⠟⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⣤⣶⣄⠀⠀⠈⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⣠⠞⠀⠀⣤⣷⣄⠀⢸⡇⠀⠀⠈⠉⠉⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠗⠒⠚⠁⠀⠀⠀⠿⣿⠟⠀⠘⢧⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠠⠄⠀⠀⠀⢤⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⣠⠾⠛⠃⠄⣀
⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡴⠟⠉⠀⢀⣾⣿⣿⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣿⣴⣷⣧⢀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠰⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠃
⣿⣿⣿⣿⣿⣿⣷⣿⣠⡀⣦⣷⢠⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠰⠶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣼⣧⣧⣼⣰⣇⢠⡀⣿⡆⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡆⣠⢸⣷⣿⣦
⣿⣿⣿⣿⢿⢿⡿⠿⠛⠛⠟⠿⠛⠋⠛⠉⠉⠋⠉⠉⠋⠈⠉⠛⠁⠀⠋⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡀⣸⣧⡆⢠⢀⣿⣾⣷⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⣾⡧⢀⡀⣸⡇⣰⣠⣧⣴⣰⣿⣰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣾⣧⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣽⣗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣸⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠻⣿⣿⢿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠈⠁⠈⠀⠁⠈⠈⠉⠈⠉⠙⠛⠋⠉⠉⠛⠻⠟⠿⢻⡿⣿⠿
⣿⡿⣿⣿⣿⣿⣿⡇⠀⠀⢠
⣿⠃⢹⣿⣿⣿⣿⣿⡃⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣸⡇
⡟⠀⢐⣻⣿⣿⣿⣿⣷⣼⣿⣧⡀⢀⣦⠀⠀⠀⢰⣿⠄⠀⠀⠀⠀⠀⢸⣿⣯
⠉⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⡅⣸⣿⡄⠀⢰⣾⣿⡇⠀⢰⡆⠀⠀⣿⣿⣿⡁⠀⠀⣾⠄⠀⠀⠀⢰⡧
⠂⠀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢺⣿⣿⣶⣤⣾⣿⠀⣘⣿⣿⣿⣶⠆⢸⣿⡇⠀⠀⠀⣿⣟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⣘⣿⣿⣿⡟⢻⣿⣿⣦⣽⣿⣿⣿⣿⡥⣿⡿⠿⠀⠀⣽⣿⣿⠇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⣶⣤⣿⣿⣿⣷⡤⠀⠀⢰⡇
⠀⠀⠀⣤⣽⣿⣿⣿⡧⠀⠀⢽⣿⣿⣿⡿⢻⣿⣿⠦⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠠⢽⣿⣿⣿⣿⣿⣧⠀⢀⣿⣿⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠤⠼⣿⡟⠓⠀⠀⠙⢿⣿⣿⡃⠀⣿⡟⠃⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠙⢿⣿⣿⣿⣿⣷⡖⣸⣿⣿⡅⠀⢀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠒⠚⣉⠁⠀⠀⢤⣿⣿⡟⠀⠀⠹⡏⠁⠀⠀⣤⣿⣿⡿⠁⢿⣿⣿⣏⠁⠀⠀⢤⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢁⣾⣇⠀⣸⣿⣟
⠀⠀⠀⠀⠠⠴⠒⠀⠀⠀⠀⠐⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠃⠀⠘⣿⣟⠀⠀⠀⠀⠐⢺⠻⣿⣿⠁⢻⣿⣿⡟⢻⣿⣽⣿⣧⣠⣿⣿⣿⣃⣾⠀⠀⣴
⠀⠀⠀⠀⠤⠦⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⢒⣿⠁⠀⢈⣿⣿⠀⠈⣿⠙⣿⣿⣿⠇⢸⣿⣿⣿⡆⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠉⣉⠅⠀⠀⠀⠙⠀⠀⠀⠋⠀⣿⣿⡿⠀⠈⣿⣿⣿⣷⣾⣿⣿

⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⡿⣿⣿⣷⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣠⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⡀
⣼⠟⢤⠀⠀⠀⠀⠀⣀⣀⣠⣴⣽⣿⣿⣿⠟⠻⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡟⢠⣯⡄⠀⠀⠘⠛⠻⠿⠿⣧⣀⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⢿⣄
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠿⣿⣿⣟⡉⢛⣉⡏⡀⣀⣸⣿⣿⡶⠿⠒
⠀⣾⡻⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣻⣿⣄⡄⠀⠀⠀⠀⠀⢸⡏⣿⣿⠟⢿⣿⠿⢿⣿⣿⣿⡀⢀⣠⣶⡀⣠⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣀⣄
⡟⠀⠙⠃⠀⠀⠀⠀⠀⢸⡿⢯⣿⣤⣀⣻⣆⣠⠿⣿⠟⣿⣯⣨⣿⣹⡟⣹⣿⣿⢿⣋⡀⣄⣽⢳⣻⡿⠁⠀⠀⠀⠀⣼⣇
⠅⠀⢰⡀⠀⠀⠀⠀⠀⠈⢄⣈⣭⣿⣿⣿⣣⡿⠷⣯⣟⣯⣅⠉⣻⣿⣾⣿⡿⠿⣫⡻⣿⠿⢻⣿⣼⣧⠀⠀⠀⠀⠀⠈⢿⣧⡀
⡟⠀⢺⡇⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⡼⡟⢶⣅⣄⣀⣿⣾⣿⣷⣿⣿⣿⣍⣴⣶⡿⠏⣥⣤⢿⣷⠖⢯⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠁⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢗⡋⢱⣐⣯⣶⣽⣿⠿⠿⣻⣿⣿⣭⣔⡻⠿⢿⣾⣷⣚⣺⣛⣸⣗⡋⣻⡖⠀⠀⠀⠀⠀⠀⠘⢿
⠀⣿⣿⡇⢠⡄⠀⠀⠀⠾⣷⡿⠷⠛⢫⣽⠋⠡⡶⡖⠀⠛⠿⣿⣿⣿⣥⣤⡀⠈⠉⢻⣿⣿⣿⣿⣵⣾⣷⠄⠀⠀⠀⠀⠀⠀⠈
⢸⣿⡿⣰⣿⡇⠀⠀⠀⠈⠛⠿⣿⠿⣿⣯⣤⣤⡍⢵⣶⣿⣿⣿⣿⣿⣿⡿⠟⠙⣛⠋⠛⠋⢻⣾⡿⠟⠛
⣿⣿⣧⣿⣿⠃⠀⠀⠀⠀⢀⠀⠈⠙⢿⣿⣿⣻⣶⣿⣿⣿⣿⡿⠋⠁⠀⣠⣤⣤⣤⡼⣿⣿⡫⢋⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⢿⠇⡀⠀⢸⣿⢿⣽⣿⣿⣿⣿⢿⣿⣶⡶⣾⠟⡓⣿⣫⣿⣿⠋⢀⣈⠛⠃⠀⠀⠀⠀⠀⠀⢰⣾⡇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣀⡀⠉⠀⠈⣿⠐⠙⣿⣥⣿⣇⣦⢻⠻⠐⡿⣾⣿⣧⣼⢿⣿⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⠀⢿⣰⣿⣿⣿⣿⣿⣷⣼⠀⣰⣿⣿⣿⣿⠀⢘⣿⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣴⣦⡉⠙⣿⣿⣿⢿⣿⣿⠀⣿⣿⡿⠛⠃⠀⢘⣿⠙⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡏⢀⣶⣦⠀⠀⠀⠀⠈⠀⠀⠻⠟⠁⠀⣿⣿⣿⠀⢻⣿⣶⣿⠏⠀⠀⡀⠀⢸⣿⠀⢀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝
⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣿⣿⣿⠀⢸⡟⠟⠡⣿⣿⡟⠀⠀⠘⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠛⠿⠿⠀⠘⠇⠀⠀⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⣦⣄⢨⣦⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⣲⠀⠀⠀⠀⠀⠀⢀⣀⣐⣒⣦⠀⠀⠀⠀⠀⠀⠀⡆⠤⡤⢻⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢻
⠀⠀⠀⢿⣤⠿⡇⠀⠀⠀⠀⠀⠀⠀⢩⣤⡤⠜⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⡞⠀⠀⠀⠀⠀⠀⠀⣿⣉⣁⣜⠀⠀⠀⠀⠀⠀⠀⠻⠦⠤⠟
⠀⠀⠀⢈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⢇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⠀⠀⠨⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀
⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠐⠀⠒⣦⡀⠀⠀⠀⢰⣦⣄⣀⣀⣀⣀⣠⣾⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⣼⠋⠙⠂⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⣄⡀⠀⠝⠀⢡⠀⠀⠀⠳⠒⠾⠟⠛⠲⠲⠋⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄
⠀⡟⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠒⠠⠬⠭⢤⠴⠖⠛⡆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃⠀⠀⠀⠐⠀⢀
⢀⣇⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠻⠟⠿⠛⠃⠀⢘⠓⠒⠶⠶⠶⠖⠒⠒⡃
⠈⡏⠉⠛⠉⠛⠋⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠄⣀⣀⣀⠠⠤⢴⡇⠀⠀⠁⠀⠀⣀⣀⠀⠀⠺
⠀⢻⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣦⣤⣤⣄⣤⣤⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⠀⠀⠙⡄⠀⣾⡇⠀⠀⠀⠀⡂⠀⠈⠆
⠀⠘⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠄⠀⣿⡇⠀⠀⠀⠀⠁⠐⠂⠀⠀⣰
⠀⢀⡇⠀⠀⠀⠀⠘⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠖⠀⠐⠒⠛⠃⠀⠀⠀⣸⠈⠓⠆⠀⠀⠀⠀⢻⡇⠀⠀⡀⠀⠀⠀⠀⢀⣄⣿
⢠⣾⠉⠉⠘⠁⠉⠀⣹⠀⠀⠐⠓⠀⠠⠦⠤⠄⠀⠛⠀⠀⠀⠈⢀⠀⠀⠀⠀⢀⣡⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠁⠀⠀⠉⠉⠉⠉⢠
⠀⠻⢟⣳⣒⡲⠆⠞⠃⠀⠀⠀⠠⢄⣀⣀⣀⣠⠤⠒⠀⠀⠀⠀⢀⣈⣙⣿⣿⠿⠃⠀⠀⠙⠶⣤⣀⣀⣀⣀⡤⠴⠃⠀⠀⢆⠘⢦⣀⢀⠼⠀⡨






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣋⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠟⠛⠛⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⡟⢿⣿⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⣠⡅⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠛⣿⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠋⠐⢉⠉⠉⠀⠀⠈⠻⣿⠉⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⠀⠀⠀⠀⠦⠀⠀⠀⠀⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡇⠹⣄⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⡹⢻⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠉⠀⠀⠈⠓⣤⣀⣀⠀⠀⣀⣀⣴⡚⠁⠸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⢀⣠⠴⠒⠉⠁⣸⣭⠽⠛⠉⠁⠀⠈⢦⠀⠰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣾⡉⠀⣀⣀⡤⢯⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡱⣴⣏⢷⡫⠵⠚⠛⠉⢁⣀⣀⣸⣆⣀⣠⠤⠶⠒⠚⠹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⡽⣭⡽⣀⣴⣶⣒⣿⡉⠉⡽⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⣿⡿⠧⠋⠉⠁⢀⡿⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡟⠉⠉⠙⠓⣲⠶⠋⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣳⡤⠴⠒⠋⠁⠀⠀⠀⠀⠹⣖⠒⠒⠒⢒⣖⡒⠒⢖⣒⠖⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠳⣄⠀⠀⠀⠀⠈⣣⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠘⠴⠃⠀⠀⣀⣄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⢶⣄⣠⣴⣿⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⡔⢦⠀⠀⠀⠳⠜⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⣇⣸⠟⠛⠒⠲⠤⢭⣍⣙⡛⠲⣄⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⡙⠉⠀⠀⣏⣳⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣟⠀⠀⠀⠀⠀⣀⡴⠯⢤⡯⠼⠦⢤⠤⠬⢤⠴⠿⠦⠿
⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠴⠖⠋⠁⡇⠀⢸⠃⠀⠀⢸⡆⠀⢸
⠀⠀⠀⠀⠀⠐⠁⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣀⣾⠀⠀⠀⠸⡇⠀⢸
⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⡴⠒⢶⣋⣉⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⠁⠀⡇⢀⣀⣀⣀⡏⠉⢹
⠀⠀⠀⠀⣨⠤⢤⡖⠒⣲⣵⠒⠙⣒⠋⢰⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢀⣀⣠⠤⠴⢺⡟⠉⠉⢀⣀⣀⡤⠼⠃
⠀⠀⠀⠀⢱⠦⠘⠭⠜⠩⡭⠟⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠈⠓⠚⠉⠉⠁
⠀⠀⠀⠀⠈⠒⠒⠉⠓⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢷⡄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⡄⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢳⠂⠀⠀⠀⠀⠀⠀⠡⡀⢲⡀⣰⢀⡅⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣙⠢⣑⣘⣜⡐⠉⠴⠂
⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢒⣽⢿⢿⠒⣒⠒⠂
⠀⠀⠀⠀⠀⠢⠆⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⡰⢃⠂⡇⠑⡌⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣠⡄⠀⠀⣏⠰⢿⣇⠀⠀⠠⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠂⠉⢓⠋⠁⠐⠀⠀⠀⣿⣿⠏⠀⣀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠨⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠂⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⣀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠠⠀⠉⠢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠄⠀⠀⠀⠛⠀⠀⠀⢀⡍⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⢸⡀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠐⠄⠀⡀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⣰⠀⠀⢸⡇⠀⠀⠈⢸
⠀⠀⠀⠀⢶⣦⡂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢧⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢪⠉⠀⠄⢸⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠇⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⣀⣭⡆⠀⡄⠂⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠰⠂⠈⣄⡀⠀⢸⡇⠀⠀⢀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣶⠆⠀⠀⠀⠠⡝⢻⡇⠀⠀⠆⠣⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣁⠀⢸⡇⠀⢀⠄⢠⣤
⠀⠀⠠⠀⠐⠠⠀⢀⡀⠀⠀⠀⠘⠀⠰⡿⠂⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠀⠀⣆⡄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠹⣄⣿⡇⢀⡎⠀⠀⠙
⠀⠀⠀⠂⠀⠀⠁⢸⠃⠀⠒⡠⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠻⣿⣇⡞
⠀⠀⠀⠀⠀⠀⠐⠚⠐⠂⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠂⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡆⠀⠀⢸⣿⢸⣿⠀⠀⠀⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣴⡆⣿⣿⢸⣿⣿⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣄⣴⣿⣿⣸⣿⣿⡷⠶⠾⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢿⣏⣽⣿⣯⠿⠟⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣰⣿⣼⡟⣿⣩⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠿⠿⣺⣧⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣹⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣷⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠰⣷⡖⣻⣿⡇⣿⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡇⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢿⡇⣿⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⡖⣶⣼⡏⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠹⣿⣶⣯⣿⣙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⢸⣏⣻⣿⣟⣶⣦⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡶⠛⠸⣿⣿⢹⣿⣿⡷⠶⢶⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⣿⠻⠇⣿⣿⢸⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠸⠃⠀⠀⢸⣿⢸⣿⠀⠀⠈⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡳⡄⠀⠀⠀⣴⢟⡭⠳⣄⠀⢀⡞⡆⠀⠀⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⣄⠙⢇⢡⠀⠀⣸⢧⠏⠀⠀⢹⠀⢸⣠⠃⠀⠀⡼⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠉⠘⠂⠀⠙⠀⠀⣿⠺⠀⠀⠀⢸⠀⢸⠃⠀⠀⠀⠀⠀⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⣳⠀⠀⣰⢞⣛⢶⣦⡀⠀⠀⢻⠀⠀⠀⠀⣼⠀⠈⢀⣔⡌⣶⠀⣠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣛⡓⢦⡀⠈⠃⠀⣿⢸⠉⠃⠙⢷⡀⠀⢸⡇⠀⠀⢠⡇⠀⡔⠛⢁⡴⠃⠀⢀⣤⣶⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠙⢦⠀⠀⠸⣆⠀⠀⠀⠈⢷⠀⠀⢧⠀⠀⡾⠀⢸⢁⠔⠋⠀⠀⢠⠞⢉⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢮⣣⡀⠀⠹⣆⠀⠀⠀⠘⣇⠀⠸⡆⣼⠁⠀⡿⠃⣠⣄⠀⢠⠿⠚⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠝⠓⠂⠀⠈⠓⠀⠀⠘⢦⡀⠀⠀⢹⡄⠀⢻⠇⠀⠈⢁⡾⡱⠎⡇⠀⠀⣠⠴⣒⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣚⣲⠶⣤⣄⡀⠀⠈⠻⣄⠀⠈⣇⠀⠈⡀⠀⠀⣾⡿⠁⢀⡇⢀⡞⠑⠉⠉⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣘⠋⠀⠀⠀⠀⠉⠛⠷⣤⡀⠈⢷⡄⢻⠀⢸⢿⡄⢰⡇⠀⢀⡼⢁⠎⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠦⢤⣄⣀⡀⠀⠈⠻⣦⡀⠙⢾⡇⠀⣼⠀⣾⠀⢀⡾⢁⡞⣀⡴⠚⠁⢀⣤⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⢈⡉⠙⠒⠦⣌⢳⡀⠀⠁⠀⠉⠀⡏⣠⠏⢀⣾⠞⠉⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠥⠖⠚⠉⢉⣀⣀⡀⠈⠙⢷⡐⠝⡆⠀⢀⡿⠃⠀⠚⢁⡤⠞⠋⠉⠑⠯⡍⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠟⠛⠉⠙⠶⣄⠁⠀⠳⠀⠘⠃⠀⣠⠞⠉⠀⠀⠀⠀⠀⠴⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠾⢷⡄⠀⠀⠀⢀⡴⠿⠖⠒⠒⠒⠶⠶⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣶⢀⡁⠀⠀⣨⡄⠀⣼⣿⢷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣦⣿⠟⠉⣹⢻⠟⢰⠿⡯⠊⠀⠀⢀⣠⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠈⠛⠋⢠⣷⠃⠀⠀⠀⢀⠴⠛⠛⠝⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠸⠋⡀⠀⢠⡶⢋⡀⠀⠀⢸⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢠⣼⣧⡀⣾⠁⠀⢀⣀⠜⢻⡟⠀⠝⠓⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⠀⠉⠉⢉⡳⠦⢶⣿⡀⠄⣈⠓⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢺⡋⠈⠐⠂⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⢦⡄⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣟⠁⠀⠀⢧⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣸⠁⠈⠀⠀⠈⠿⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⢐⠃⠀⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⠠⠀⠐⠈⠠⡄⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⣿⠀⠀⠀⠀⠐⢘⠉⠢⢄⢳⠀⠀⢀⢀⣀⠀⢠⠄⠨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⠁⢀⡇⠀⠀⠀⢀⡀⠀⠒⠀⠢⠘⢀⣠⣼⡏⠀⠁⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡋⠈⣿⡇⠀⠀⠀⠈⠀⠁⠀⢸⣶⣯⡍⢿⢺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠀⠀⣾⠇⠀⠀⠀⣀⠀⠀⠀⢸⣿⡞⢁⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⠀⣸⡿⠀⠀⠀⠀⠉⠀⠀⠀⢸⣿⣧⢨⡄⠀⣀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡯⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⡛⣁⣿⣾⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠯⡍⣛⣿⠟⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣟⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⢿⣟⣿⣿⣗⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡴⣦⡄
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⣿⣶⣶⣶⡄⠘⠛⠀⣠⣶⣶⣿⣷⣾⣷⢦⣶⠆⠈⠙⠛⠷⠶⠄
⠀⠀⠀⠀⢸⡿⠊⠉⠉⠙⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢰⣦⠾⠿⠛⠃⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⣶⡿⠀⠀⠀⢀⣾⠛⠋⠻⣿⣿⣤⣄⣀⣀⡰⠖⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢀⣤⣶⣤⣴⣿⠋⠀⠀⠀⠈⢻⣿⣿⣿⣿⡇⠀⠀⠘⢿⣿⣿⣿⠀⠀⠈⠻⣿⣶⣷⡆
⠀⠀⠀⠀⣻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⠏⠀⠀⠀⠀⠀⠻⡿⠁⠀⠀⠀⠀⠀⠙⢿⣄⡀⠀⣠⣤
⠀⠀⠀⠀⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿
⠀⠀⠀⠀⠸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⢀⡀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠀⠀⠲⣦⠀⠀⠀⠀⠸⠆⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠘⠛⠛⠀⠂⠙⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠀⠾⠐⠀⠆⠀⠀⠀⡠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠿⠶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡑⢄⣀⣀⡀⠈⠉⠙⠛⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢈⢀⣀⣈⠉⠑⠒⠦⢤⣀⠀⠈⠙⠛⠶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠖⠛⠛⠓⢊⠔⠉⠀⠀⠀⠢⢄⡀⠀⠉⠑⠢⢄⡀⠘⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣟⠋⠀⠀⠈⠑⠒⠴⡃⡀⠀⠀⠀⠀⢀⠀⢀⠉⠡⢄⠀⠀⠀⠀⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠑⠒⠤⢀⠀⠀⠈⠑⠲⠤⣀⣀⡁⠐⠁⢀⣀⠀⡀⠦⢄⡀⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢉⣁⠀⠀⠀⠈⠑⡀⣿
⠀⠀⠀⣠⣤⡶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢄⣀⠀⠀⠈⠑⠤⠐⠃⠀⡇⣿
⠀⠀⢸⡏⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠝⣦⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⢸⡟⡉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⣰⠁⣿
⠀⠀⢸⡇⢸⠑⠢⠄⡈⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠊⠁⡸⠀⣿
⠀⠀⢸⡇⢠⣍⠀⠀⠀⠁⠂⠄⣀⠉⠒⠤⣄⡤⢔⠂⢀⡄⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⣿⠀⢰⡟
⠀⠀⠈⣧⠐⢤⡙⠢⢄⡈⠐⠠⠀⠉⠒⠤⣀⠃⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⠶⠟⠀⠈⠛⠛⠁
⠀⠀⠀⢻⣤⣾⠙⠳⢦⣌⡑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣈⠉⠒⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡇⠀⡄⠀⠀⠀⠀⢀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣇⣤⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡿

⠀⠀⡀
⠀⡀⡇⠀⠀⡀⣼⡀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐
⢰⡇⣿⣿⠀⣧⣿⡇⢰⢾⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠛⠿⣿⣀⣿⣿⡇⣼⣾⡆⣷⢀⡇⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢿⡧⠇⢄⡀
⠀⠀⠀⠀⠓⠉⠛⠓⠿⠿⢯⣿⣤⣧⣾⣠⡆⠀⠀⠀⠀⠀⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⠇⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠇⠀⠀⢳⠀⢸⠄⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠑⠄⠀⡬⡆⢀⠂⢠⠈
⠀⠀⠆⠠⠒⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⠀⠀⠒⠣⣀⡤⡀⣸⣭⠆⠠
⠐⡆⢰⣛⣒⡀⣤⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⠌⢨⠅⣀⠹⣷⣥⣕⠂⢓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢿⡿⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢻⠳⢤⣖⠉⠀⠠⡇⠀⠉⣲⠤⠚⡟⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⢸⠀⠀⠈⢱⠀⠈⡇⠀⡏⠀⠀⠀⡇⠀⣤⣀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⣿⣿⣿⣦⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⣇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠢⣄⡀⢸⠀⢀⡇⠀⡇⠀⣀⠼⠿⠧⡈⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠔⠊⠀⠀⠀⠀⠀⢉⠿⠀⣨⣏⠰⣷⣉⠀⠀⠀⠀⠀⠑⠠⢄⠁⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢁⠀⠀⠀⠀⢀⡠⠔⠊⠀⣠⠔⠻⠟⠢⣄⠈⠑⠢⣄⡀⠀⠀⠀⠀⡨⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⠤⠒⠁⢀⡠⠖⠉⠀⣀⢴⣦⣀⠀⠉⠲⢤⣀⠉⠲⠤⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠄⠊⠁⢀⡠⠔⠈⠀⠐⡇⠈⠙⠢⢄⡀⠈⠑⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⡉⠀⠀⢠⠂⠁⠀⠀⠀⠀⣴⣷⡀⠀⠀⠀⠉⠲⡄⠀⠀⢙⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢼⠀⠀⠀⡠⠔⠊⠉⠉⠙⠢⢄⡀⠀⠀⡧⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⢀⠠⠐⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠈⠻⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡿⣛⣿⠏⠀⠀⠂⠀⠀⠈⠉⢸⣿⠁
⠀⠀⠀⠀⠀⠀⢧⠉⠈⠀⠀⠐⠇⠀⠀⣀⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠜⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⠀⢀⣀⡴⠖⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠘⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢰⣿⣿⣿⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣶⣶⣤⡂
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⢸⣿⣿⣿⣿⡯⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠈⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⣿⣿⣿⠃⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣾⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⢰⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠋
⠀⣾⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⠟⣿⠻⡿⠿⠿⠟⠁
⢠⣿⣿⣿⣿⠃⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏
⣼⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣷⣦⡀⠘⠛⠛⠻⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠂⠀⠌⠙⠻⠿⠿⠿⠆⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⡤⣠⣄⡠⣄⣄⣠⠀⠀⠀⠙⠒⠶⠶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠙⠙⠐⠛⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⢿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠛⠾⣭⣟⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣭⡛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟⠳⢦⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠙⠳⢮⣙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⠀⣠⣾⠟⡋⠹⠙⠛⠻⠷⣦⣄⡀⠀⠀⠀⠀⠀⠙⠳⣬⡻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢿⡛⠙⠳⣤⣀⠀⠀⠀⣿⣥⠂⠀⠀⠀⠀⠀⠀⠈⠙⡻⣷⣦⡀⠀⠀⠀⠀⠈⢻⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠉⠲⢄⡀⠉⠳⢦⣀⠈⠛⢷⣦⣄⡀⠀⠀⢀⠐⣨⠷⢚⣿⣿⠀⠀⠀⠀⠀⢸⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠉⠛⢦⣄⡈⠙⠻⣶⣴⣠⣟⣥⣶⠟⠋⠉⠛⢦⣄⠀⠀⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠈⠙⠶⣄⡀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠛⠶⣼⡇⣿⡇
⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⣀⣤⠶⠛⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⠛⡇⣿⡇
⠀⠀⠀⢸⡿⣥⡀⠀⠀⠀⠀⣴⡟⠙⢀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠈⠛⢦⣄⡀⠀⣠⡴⠛⠉⠀⠀⢀⡿⢹⡇
⠀⠀⠀⢸⡇⠈⠙⢦⣄⠀⠀⢿⡻⣷⢄⡉⠰⣀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⣨⣿⡟⠁⠀⠀⢀⣠⠶⠛⠁⢸⡇
⠀⠀⠀⢸⣧⠀⠀⠀⠉⠻⢦⣀⠙⢶⣕⣭⡲⣄⠘⢠⠀⠀⣀⣤⣶⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢻⠁⢸⠇⣀⣤⠾⠋⠁⠀⠀⣀⣼⡇
⠀⠀⠀⢈⣿⣦⣀⠀⠀⠀⠀⠉⠳⣦⡈⠛⢿⣮⣷⣤⣚⡯⣷⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠀⠀⢸⣠⡼⠟⠋⠀⠀⠀⣠⣴⡿⠛⠉
⠀⠀⠀⠀⣿⠀⠉⠳⢦⣀⠀⠀⠀⠀⠙⠳⣤⡙⠻⠿⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⢀⣠⡴⠛⠁⠀⠀⢀⣠⣶⠿⠋⠁
⠀⠀⠀⠀⣿⣆⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⢀⣤⠾⠋⠁⠀⠀⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠈⠳⣦⡀⢀⣠⠶⠋⠁⠀⠀⣠⡴⠞⠋⠀⠀⠀⢀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⡀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠈⢻⠋⠁⠀⠀⣠⡴⠛⠉⠀⠀⠀⣀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠈⠛⢦⣄⠀⢸⠀⣠⡴⠛⠁⠀⠀⠀⣀⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠈⠛⢿⠟⠁⠀⠀⠀⣀⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⠀⠀⢸⠀⠀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣼⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠰⡆⠀⠀⠀⢀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢏⠀⡗⠀⠀⡇⠀⢸⠂⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠳⡈⢢⠀⡇⡴⠋⡔⠋⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⠉⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⡔⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⣀⣀⣀⣀⣀⣿⣿⣟⡍⣭⢹⣭⡍⣭⣩⣩⡝⣿⣿⣀⣀⣀⣀⣀⡤⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⣀⣀⣀⣀⣿⣿⣯⣥⣭⣬⣭⣥⣭⣭⣬⡅⣿⣿⣀⣀⣀⣀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠉⠉⠉⠉⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠉⠉⠉⠉⠉⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠢⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⢸⠀⡇⡇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠔⢁⠜⠀⡇⠳⣄⠣⡀⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣇⠀⡧⠀⠀⡇⠀⢸⠄⢹⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠁⠀⠀⠀⠰⠇⠀⠀⠀⠈⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣬⠇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢠⡄⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐
⠀⠀⣻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠙⠁⠀⠀⠀⠀⠋⠠⡥⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⡤⠄
⠀⠀⠻⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⠀⠀⠀⢀⣀⣀⠀⡀⠓⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⣀⠀⠈⠰⠦
⠀⠀⠰⡀⠀⢠⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠨⣿⣿⣿⡿⠿⠟⠻⢿⣿⣦⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠃
⣦⠀⠴⠄⠀⠈⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⠀⢀⣄⣀⠄⠀⠙⠛⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⣿⠙⠒⣦⠀⡘⠀⠀⣬⠀⠈⠆⠀⠀⠠⠀⠀⠀⢀⡀⠘⠀⠀⠀⠀⠉⠁⠀⠀⠀⠜⠢⠀⠰⠀⠀⠀⠐⠀⢠⠀⠀⠐⠀⠀⠤⢀⠆⠠
⢻⢀⠀⠈⠀⠃⠀⠀⣽⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣷⣺⣶⠃⠀⠀⠀⠀⠀⢀⠄⠀⠀⠄⠀⠇⠀⠀⠀⠄⠀⢀⠀⠀⠰⠤⠇⡉
⠀⢯⣠⡀⠈⠁⠀⠀⣽⠀⠈⣠⡀⠀⠀⠀⠀⠀⠀⠘⣿⡽⢀⡠⡶⢀⠀⠊⠍⠀⠀⠀⣔⠀⢠⣰⣿⡀⠀⠀⠀⠠⣄⠠⠇⣀⣄⠘⠃
⠃⠉⢀⠙⠄⠰⡆⠀⢺⣁⣼⠀⠀⠀⠀⠀⠀⠀⠠⠄⠙⣇⠬⠴⠃⠸⢂⠀⠀⢀⡞⢡⠘⠚⠀⣏⡣⠀⠀⠀⠁⠀⠈⠶⣌⠉⠀⠀⢴
⠀⣰⠏⢳⡀⠀⠆⠀⣽⠛⣿⣧⠀⡀⠀⠀⠀⠀⠀⡘⢀⠃⠒⠚⢃⣸⠏⣰⣤⣻⠂⣻⠀⢦⢠⣿⠃⠀⣄⠀⠀⠀⠀⠀⠈⠚⢁⠊
⠀⠁⠀⠀⣷⢀⣶⠄⣿⠀⡟⢹⣏⠁⠀⠀⢀⠄⣾⠇⣼⠆⠀⠀⢎⢈⠓⠈⠠⣨⡁⠚⣥⠜⠮⠝⠂⠀⠈⢢⠄⡀⠀⣀⠀⣀⠈⠐⠆
⡴⡄⠀⠓⠮⠋⠦⠀⣼⠰⠁⠙⣿⠳⡄⢀⠌⢠⡟⠀⠀⠓⢦⡇⡠⠈⡀⠀⠀⠀⠁⣞⣾⣷⣄⣠⠀⠐⢀⡰⢠⢴⠐⠣⠠⠘
⠀⢸⣦⣆⣠⡠⡾⠿⣿⠦⠃⢨⣟⠀⢶⠀⢰⡾⡶⠄⡄⠤⣭⢜⣓⢺⡏⠉⢀⡠⠀⠘⣺⣏⡀⡦⠀⢀⣉⡀⣼⡜⡙⠃⠀⠁⠀⠶⡀
⠆⠈⣥⣄⠐⢛⣿⣀⣿⡀⠐⠢⡌⠀⡈⠰⢦⣞⠁⢠⠓⠂⠀⠀⠀⢾⣁⣠⡜⠱⢾⡠⠝⠛⠛⣰⠆⡙⠛⠁⠛⠁⠒⣤⡳⢄⣤⣤⠙
⠻⢎⣿⣿⣡⣬⣭⡙⣿⢛⠀⠀⠸⠀⣷⣶⣾⣿⣷⣷⣶⠌⢀⠀⢴⡟⡱⠾⢆⣟⣻⣀⣗⠘⠋⠝⢨⡅⠀⠀⢀⣄⣀⠬⠁⣹⠘⠓⠶
⡖⠺⣟⡽⠿⢋⣙⣫⠁⡀⠀⢤⡄⢨⣯⣾⠽⠿⢿⡿⠀⠒⠤⠀⠘⣝⠿⢛⠈⢻⠿⢿⠏⠉⢰⠓⠃⢩⠑⢂⣼⠾⢻⣤⣺⣭⠶⣄
⠉⠴⣿⣂⢶⡄⢀⣤⢞⣿⣿⠶⣲⢹⣏⡈⢥⣣⠛⣟⣀⣤⡗⢠⣦⣯⡟⣁⢀⠘⠸⡏⠃⢀⠘⡁⠻⢻⠄⠛⢣⢛⣜⣷⢷⢿⡙⠿

⠀⠀⠀⠀⠡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠔⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀
⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣿⣷⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⣩⡄⠀⣥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⢳⠀⠀⠠⠀⡀⠀⠀⣤⣀⠈⢸⠘⢻⠛⠟⠛
⠀⠠⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠐⠀⠀⠱⡄⢸⣿⡿⠟⢻⡆⠖
⢲⣾⣻⠯⢀⣀⡀⢀⣄⠀⠀⢀⠄⠂⠀⢠⠀⠀⠀⠐⡄⢠⡈⠑⠄⣀⠀⢹⢨⣤⣤⡈⢻⠐⣿⣷⣦
⠘⡇⢨⠀⠫⠀⠀⢹⣿⠀⠀⣠⠀⠀⠴⠋⠀⣾⠀⠀⡅⠀⠁⠀⠀⠻⣤⡀⢢⠀⠙⠑⢾⣠⠐⠈⠁
⠠⠶⠾⠶⠴⣤⡄⠉⡉⠀⠘⣿⠗⠀⠈⠀⣤⣤⣤⣦⡥⣤⣤⣤⣥⣴⣴⣶⣶⣷⣶⣶⣼⡷⠶⠛⢻⣯
⢰⣦⢞⣀⣠⡀⡀⣀⣀⣒⣸⣿⣧⣤⣤⡤⠬⣿⠭⢭⠽⠟⢻⣿⣿⣿⣻⣿⣿⣿⡟⡙⣹⡄⢠⡀⣾⣿
⣀⡶⢚⣻⢲⣿⠌⣱⣿⠟⣿⡿⠿⠿⠃⠀⠀⠠⠴⠫⣷⣖⠶⢿⢿⣿⣽⠟⠉⠈⢰⡟⣿⡥⣿⣄⣿⣿⠄
⠡⠺⠯⠖⠁⠀⠀⠿⠷⠏⠁⠀⠐⠂⢚⡁⢀⣉⡉⠉⡉⠙⠛⠿⠓⡬⣁⠉⠃⠀⣷⡀⣿⣗⠥⣸⣿⣿⣦
⠀⠀⢐⣒⣀⠀⠀⠀⠀⠀⠒⠀⠠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠸⠦⠉⠀⠀⢠⣿⠀⣼⠁⠀⣿⣿⣿⣧
⢴⣾⠏⣽⡿⢿⡶⣴⣏⣿⠦⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢟⡃⢹⠅⠈⣻⡿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⠿⠿⠛⠻⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠋⠁⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⢸⡇
⠀⢀⣀⣀⣀⡀⠀⢀⣠⣶⣿⡿⠟⠋⠁⠀⠀⠀⠀⢠⣿⠋⠉⠙⣷⠀⠀⠀⢸⣿
⣼⣿⣿⡿⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣿⠇⠀⠀⢸⣿
⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⣼⠟⠀⠀⠀⢸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠻⢶⣤⣸⣿
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⡶⠟⠋⠉⢀⣀⡀⠀⠈⠙⠻⢶⣄
⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⣤⣴⠶⠟⠛⠿⢾⣿⣥⣀⠀⠀⠀⠈⠁⠀⣀⣠⣤⣶⣾⣿⣀⡀
⣿⣿⣿⡇⢀⣠⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣶⣤⣶⠾⠟⠛⠉⠁⠀⠀⠈⠉⣛⣷⣦⣀⡀
⣿⣿⣿⡗⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠟⠋⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠛⠋⠉⠈⠉⠛⠿⢶⣤⣄⡀
⣿⣿⣿⣿⡿⠟⠻⠷⣶⣤⣤⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣤⣄⣀
⣿⣿⣿⣅⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣦⣄⣀
⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀⠀⠀⣀⣤⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⠛⢻
⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠟⠛⠉⠀⠀⠀⠀⢸
⠀⢸⣿⣿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢸⣿⣿⡇⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣀⣀⣤⣴⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠛⠉⠁⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⣤⣴⠾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⠶⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣲⡦⠀⠀⠀⠀⠀⢀⣀⣠⠈⣐⠖⠀⠀⠀⠀⠀⠀⠀⣀⣔⣾⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡿⠃⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣋⣿⣥⡿⠯⠀⠐⣀⣰⣦⣤⣈⣿⡍⠈⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⢠⣼⣿⣶⣶⣶⡆⠀⠀⠾⠿⠿⠋⠋⣰⣤⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⣀⡄⣠⣀⡉⠛⠋⡟⠉⠉⠀⠀⣺⣿⣿⡻⣟⣷⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡤⣮⣿⣿⣿⣿⣾⣿⣷⡀⠀⠀⠀⠀⣲⣿⣿⢿⣷⣿⣿⣿⣿⡁
⠀⠀⠀⠐⣰⣿⣛⣷⣽⣿⡆⠀⠃⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣷⣿⣿⠷⠀⠀⠀⠀⠀⢀⣾⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⠂⠀⠀⣩⣿⣯⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⢀⢿⣿⣶⣿⣷⣿⣿⣶⡄⠀⠀⠰⢹⣻⣿⡏⢈⣿⣯⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣿⣿⡿⠷⣟⡽⠛⣯⣿⡏⠋⠀⠀⢸⡣⢿⣿⣿⣿⠿⣿⣻⣿⣷⡆
⠀⠀⠀⡾⣿⣯⣝⣏⣯⣿⡿⠀⡇⠀⠀⠉⠛⣿⡆⡆⢀⣽⣷⣾⠿⠟⠉⠀⠀⠀⠀⠀⠀⢀⡹⣀⠀⢸⠀⠀⠿⣿⡇⠀⠀⠀⢸⠈⠀⠙⢹⡇⠀⠈⠉⠋⠉
⠀⠀⠀⠙⢿⠟⢸⡁⠀⠉⡿⢿⣿⣶⠀⠀⠀⠉⠁⡇⢸⠙⣿⡇⠀⠀⠀⣀⣀⡀⠀⠀⠀⠻⢹⣿⡇⢸⠀⠀⠀⠀⠀⠀⣀⣤⣼⣄⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡅⢩⣉⣿⠀⠀⠀⠀⠀⡛⠈⠓⠛⠓⢠⣶⣿⣿⣿⣿⣷⣶⡀⠛⠹⠿⠦⢾⠤⠤⠤⠤⠤⠠⠨⠹⣿⣿⡇⠀⢨
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠓⠛⠿⠷⣾⣿⣿⢷⣶⢿⣤⣀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⡤⣿⣶⣿⣷⣿⣦⠀⢰⡴⣿⣿⣀⣀⣸⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣀⢀⡀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡿⣃⣶⣶⣽⣲⣶⣿⣯⡿⣟⡡⠊⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡟⠁⡰⠄⣿⣿⣿⣿⣿⣦⣽⣿⣿⣍⣉⣿⣾⡯⣥⣴⣶⡶⣶⣶⣶⣿⣿⣷⣾⣶⣄
⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠳⣀⠉⢻⣩⠽⣉⡿⠬⠏⣀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣷
⠀⠀⠀⠀⣿⣟⣿⣿⣿⠟⠛⠛⠛⢛⣛⣩⣭⣵⣾⣿⣿⣿⣿⣿⣿⣷⣷⣶⣿⣶⣒⣻⣥⣾⣿⣿⣿⣿⣿⣷⣿⣿⢥⣟⣛⣻⣿⢿⣏⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠸⠯⠿⠿⠿⠤⢤⣴⣾⣿⣿⣿⣿⡿⠿⣿⣿⣿⡟⣻⢿⣛⣿⣹⡽⣯⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡟⣿⣶⣦⣌⠉⠙⢻⡇⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡛⢚⣿⣿⣿⣿⣷⣿⣯⣿⣿⢿⣿⣿⣿⣿⣟⣿⡄⢹⣿⣿⣿⣟⣿⣿⣿⢿⡆⠀⠀⠁⠙⠛⠛⠛⠛⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⡆⢹⣿⣿⣿⠷⠮⣿⣽⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠿⢿⣿⣯⣽⣿⣿⡽⠟⠋⠈⠓⠛⠛⠛⠛⠛⠒⠒⠒⠚⠛⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣯⣍⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠸⢄⢢⡅⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠙⠿⣿⣇⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⢞⢚⠿⡻⢆⠂⡀⠀⠀⠀⠀⠀⢀⠀⢀⢶⣀⠀⠐⠠⡀⠀⠀⠀⠀⠘⠂⠘⠀⢈⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⠏⢘⢣⣧⡡⢾⡈⠓⠌⠤⡚⠀⠀⠎⡊⣀⣀⢤⡈⡉⡁⠠⡀⠁⠂⣄⠀⠰⢤⡀⠀⢺⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⣟⠽⣀⠿⡗⡇⠀⡰⢆⠀⠀⡠⠚⠉⠀⠀⠀⢱⡛⠋⠂⠌⡐⠄⣼⠀⠀⠿⠀⠀⣩⡄⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⢸⠅⠀⠪⠿⢦⡀⠉⠉⠢⢉⠁⠀⠀⠀⠀⠀⠀⡗⠂⠄⡀⠀⠀⠐⠀⠯⡧⠙⠉⠀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢦⠀⣀⠀⠔⠊⠀⠀⠀⠀⠈⠐⠠⢀⠀⡆⠀⠀⠀⣀⠄⠀⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⢀⡀⢙⠁⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠈⠁⢀⠀⠙⠀⢰⡞⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠘⠃⡠⠈⠀⢘⠃⠸⡗⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠰⠂⠀⡌⠀⢿⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠀⠀⠆⠀⠀⠀⠸⣆
⠀⠀⠐⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠂⠀⢀⡄⠀⢺⠀⠀⠒⠉
⠀⠀⠀⣉⣻⣿⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣦⡤⣴⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⢠⠗⠀⠘⠀⠀⠀⠀⡆⠀⠀⠐
⠀⠀⢀⡀⢛⢽⡆⠈⣸⡆⠀⢰⡦⣠⠊⠀⠀⠀⠀⠀⢀⠀⠀⠈⠄⢉⣱⡵⡆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠦⠀⠖⠀⢘⠂
⠀⢀⡈⠱⠸⢟⡿⣏⢺⡔⢀⣏⡼⢋⠈⣰⣴⡖⠘⠂⠈⠷⠈⢀⢤⢄⢼⡿⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠂
⠀⢀⡹⣦⣀⣿⣿⣃⠰⢶⣼⣾⡟⢃⣴⠟⠡⣐⠖⠀⠆⢈⠴⠄⡀⠋⣻⠹⠿⠁⠡⠀⠈⠁⠀⠀⠀⠀⠀⠣⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠴⠋⠈⢪⢿⣿⣿⣿⣶⣿⣿⣿⡾⣯⣶⣔⢾⣄⠠⣀⡷⠀⠰⠈⠉⠀⢹⠰⠆⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣀⣹⡉⠉⠉⠉⣯⣴⣾⣷⡄
⠀⠀⠠⢡⠋⠛⣻⣷⣷⡻⠛⠻⡗⡞⡿⣯⣟⠃⣀⡨⣮⢊⠐⠷⢀⣧⣞⣤⣤⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⡴⠀⠀⠀⢻⠏⠈⠺⠿⠷⠳⠘⣰⠌⠀⠀⠈⠀⣻⡌⠃⠀⠀⢡⣤⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⠇⠀⠀⠀⠀⣽⣿⣿⡿
⠀⠀⠃⠀⠘⡄⠋⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠃⡠⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢶⡶⠶⣶⣶⣦⡤⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠿⠿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⡠⠄⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⡏⠀⠀⣅⣠⠀⠀⢹⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠤⠄⠒⠒⠒⠛⠛⠓⠒⠒⠤⠤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠠⠤⠴⠶⠶⠶⠶⠶⠤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣄⡀
⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄
⠀⠀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠀⠀⠀⠈⠛⠷⢶⣶⠶⠾⠋⠁⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆
⢀⣾⡿⠃⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⣠⣴⡾⠟⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⣠⣤⠀⠀⠀⠀⠀⠘⢿⣷
⣼⣿⠃⠀⠀⠀⣠⣶⠿⠋⢁⣤⣶⠿⠋⠁⠀⢀⣠⣶⠿⠛⠁⠀⢀⣠⣴⡿⠛⠁⠀⢀⣤⣶⠟⠋⠁⠀⢀⠀⠀⠀⠀⠘⣿
⣿⣿⠀⠀⠀⠀⠉⣀⣴⡾⠟⠉⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⣤⣶⠟⠋⠁⠀⣠⣴⡾⠛⠉⠀⠀⣀⣤⡾⠟⠁⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠰⠾⠛⠉⠀⠀⣀⣴⡾⠛⠉⠀⠀⣀⣴⡾⠟⠉⠀⢀⣠⣴⠿⠛⠁⠀⠀⣠⣴⡾⠛⠉⠀⠀⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠁⠀⢀⣠⣴⠿⠛⠁⠀⣀⣤⡾⠟⠋⠀⠀⢀⣠⣶⠿⠋⠁⠀⢀⣠⣴⠿⠋⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠠⣾⠟⠋⠀⠀⣀⣤⡾⠟⠋⠀⠀⣠⣴⡿⠛⠉⠀⠀⣀⣴⡾⠟⠉⠀⠀⣀⣤⣾⠟⠋⠀⢀⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠉⠀⢀⣤⣶⠟⠋⠁⠀⢀⣠⣴⠿⠛⠁⠀⢀⣠⣴⡾⠛⠉⠀⣀⣴⡾⠟⠁⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣶⠟⠋⠁⠀⣀⣴⡾⠟⠉⠀⠀⣀⣤⣶⠟⠋⠁⠀⢀⣠⣶⠿⠋⠁⢀⣠⣴⠿⠋⠁⢀⣠⡄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠁⠀⠀⣀⣴⡾⠟⠉⠀⠀⣀⣤⡾⠟⠋⠀⣀⣤⡾⠟⠋⠁⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣾⠟⠋⠀⠀⢀⣠⣶⠿⠋⠁⠀⢀⣠⣴⡿⠛⠁⠀⣠⣴⡿⠛⠉⠀⣀⣴⡾⠟⠋⠀⣀⣴⡆⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⢀⣤⣶⠟⠋⠁⢀⣠⣶⠿⠋⠁⢀⣠⣶⠿⠛⠁⢀⣠⣴⠿⠛⠁⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣴⡿⠛⠉⠀⠀⣠⣴⡾⠛⠉⠀⣀⣤⡾⠟⠉⠀⣀⣤⣶⠟⠋⠁⣀⣤⣶⠟⠋⠁⣀⣤⣶⠇⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠁⠀⢀⣠⣴⠿⠛⠁⢀⣠⣴⠾⠛⠉⢀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠟⠉⠀⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣤⡾⠟⠋⠁⢀⣤⣾⠟⠋⠁⢀⣠⣶⠿⠛⠁⢀⣠⣶⠿⠛⠁⢀⣠⣶⠿⠛⠁⢀⣠⣶⠿⠃⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠉⠀⣀⣴⡾⠟⠋⠀⣤⣴⣾⠿⠋⠀⣠⣤⣾⠟⠋⠀⣠⣴⣾⠿⠋⢀⣠⣴⣾⠿⠋⠀⣠⣤⠀⠀⠀⠠⣿
⢹⣿⠀⠀⠀⢀⣶⠿⠋⠁⢀⣤⣰⠘⠉⠉⢀⣀⡀⠿⠟⠉⢀⣠⣶⡿⠟⠉⢀⣤⣶⡿⠟⠋⢀⣠⣴⡿⠟⠁⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⠀⣀⣤⣶⠟⠛⠉⢀⣀⡿⠛⠛⠁⠀⣀⠾⠟⠋⠁⠀⣠⡾⠟⠋⠉⣀⣴⣾⠿⠛⠁⣀⣤⡶⠀⠀⠀⢀⣿
⢸⣿⠀⠀⠀⢠⡾⠟⠋⠀⣠⣴⡾⠟⠋⠀⣠⣴⡿⠟⠋⠀⣠⣴⡿⠟⠋⠁⣠⣴⡾⠟⠋⠁⣠⣴⡾⠟⠋⠁⠀⠀⠀⠈⣿
⢸⣿⠀⠀⠀⠀⢀⣠⣶⠿⠛⠁⢀⣠⣴⠿⠛⠁⢀⣠⣴⠿⠛⠁⢀⣠⣶⠿⠛⠁⢀⣠⣴⠿⠛⠁⢀⣠⣶⠇⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠰⠟⠋⠀⣀⣤⣾⠟⠋⠀⣀⣤⣶⠟⠋⠁⢀⣤⣶⠟⠋⠀⢀⣤⣾⠟⠋⠀⣀⣴⡾⠟⠋⢀⡀⠀⠀⠀⠀⣿
⢸⣿⠀⠀⠀⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⠀⣠⣴⡾⠛⠉⢀⣠⣴⡾⠛⠁⢀⣤⣶⠟⠃⠀⠀⠀⠀⣿
⢸⣿⡄⠀⠀⠘⠋⠁⢀⣠⣶⠿⠋⠁⢀⣠⣶⠿⠋⠁⢀⣠⣶⠿⠛⠁⣀⣤⣶⠟⠋⠁⣀⣴⡾⠛⠉⣀⣤⡆⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⣤⡾⠟⠋⠀⣀⣤⡾⠟⠋⠀⣀⣤⡾⠟⠋⠁⣀⣴⡾⠟⠉⢀⣠⣶⠿⠛⠁⣀⣴⡾⠟⠉⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠁⠀⣠⣴⠿⠛⠁⠀⣠⣴⡿⠛⠉⢀⣠⣶⠿⠛⠁⣀⣴⡾⠟⠋⢀⣠⣶⠿⠛⠁⣀⣴⡾⠃⠀⠀⠀⢸⣿
⠸⣿⣇⠀⠀⠀⠐⠿⠋⠁⢀⣤⣶⠟⠋⠁⣀⣤⡾⠟⠋⠀⣠⣴⡾⠛⠉⣀⣤⣶⠟⠋⢀⣠⣴⠿⠛⠁⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⡄⠀⠀⠀⠀⠀⠾⠛⠉⢀⣠⣴⡾⠛⠉⢀⣤⣶⠟⠋⢁⣠⣴⡾⠛⠉⣀⣤⡶⠟⠋⢁⣤⠖⠃⠀⠀⠀⠀⢠⣿⣿
⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠐⠿⠋⠁⣀⣴⡾⠟⠋⢀⣠⣶⠿⠋⠁⣠⣴⡾⠛⠉⠀⠤⠞⠋⠁⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠈⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠛⠉⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠈⠻⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣷⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣾⣿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠴⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⣁⣀⡤⢤⢴⠒⠒⠲⣄⢸⡆
⠀⠀⠀⠀⠀⡴⠊⢁⡠⢴⢺⠉⠁⠀⠀⢼⢸⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⣸⠀⣰⠋⠀⢸⢸⠀⠀⠀⠀⢸⢸⠀⠀⠀⣸⣸⡇
⠀⠀⠀⠀⡏⠀⡏⠀⠀⢸⢸⠀⠀⢀⣀⣼⠼⠖⠛⠛⠛⢮⡑⠤⡀
⠀⠀⠀⠀⡇⠀⡇⢀⣀⣼⣼⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⡆⠈⠓⢤⡀
⠀⠀⠀⠀⡇⢰⣏⠉⠀⠈⢿⡿⡲⣄⠀⠀⠀⠀⠀⣀⡤⢖⣚⡥⠴⠒⠋⠙⠳⢤⡀
⠀⠀⠀⠀⡇⢸⠀⠑⠦⡀⠀⠉⠪⢿⡦⠤⠒⣊⠭⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⡇⠸⣄⠀⠀⠈⠳⢄⡀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⡇⠀⡇⠑⢦⡀⠀⠀⠙⠺⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡭⠷⠒⠋⠉⠉⢲⡀
⠀⠀⠀⠀⢧⣤⠇⠀⠀⠈⠳⢄⡀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⢉⣀⣤⠴⣞⣿⠷⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⢀⡔⠋⢁⣠⡴⣖⣋⣭⡶⢿⡀⡏⣁⣠⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠙⠷⣄⡀⠀⡞⠀⣸⡥⠾⡇⣟⢉⣀⡤⢼⡇⡏⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠈⠙⠿⡇⠀⣯⠠⠴⡇⣿⠉⠀⠀⣘⣇⡧⠔⣚⣹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⡇⠀⡇⠀⢀⣇⡿⠔⢚⣉⠥⠖⠚⠉⠉⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢴⡇⠀⣟⣉⠥⠼⠒⠉⠁⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡤⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⣀⣀⣀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣀⢀⡀⡀⣀⣀⡀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠠⣤⣼⣢⠞⠀⠀⠘⢦⣸⠃⠀⠀⠀⠹⡄⠀⡄⠀⣠⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⣀⣻⡀⠈⢿⡀⠀⢀⣠⡖⢷⡄⢻⠞⠃⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢦⣀⠈⡏⠉⣩⠍⠻⣄⠀⣷⠞⠋⠳⢤⠀⢳⣈⡆⠀⠀⢸⣀⣠⠀⠀⠠⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⣷⠚⠁⠀⠀⠙⢾⡇⠀⡀⠀⣄⠀⠈⣿⣀⣤⠶⠛⠙⠀⠁⠀⠀⢀⡤⠚⠻⣝⢩⠻⡏⣏⡙⠟⢦⣀⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢼⠄⠀⠀⠀⠀⣿⡤⠞⠁⠀⠘⣇⣸⠋⠁⠖⡄⠀⡃⠀⠀⠀⡴⢋⣀⣀⠀⠈⢻⣴⣬⣤⣧⣷⣧⣽⣦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⣇⣤⣦⣶⣠⣏⣀⣀⣠⣶⠀⣸⠏⠀⡄⠡⣧⣤⢶⡤⣄⣼⡇⣯⢉⢹⡇⠀⠀⡇⣶⢶⠲⣆⠀⠀⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣬⣤⣤⣤⣤⣤⣥⣄⣉⣉⠉⠉⠛⠛⠛⠛⠛⢳⠲⠶⠷⠦⠼⣯⣍⣇⣹⡈⠀⡇⠀⠀⡇⠿⠾⠴⠟⠀⠀⣿⣉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠉⠛⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠙⢓⠲⠷⠦⣤⣤⣀⡀⠀⣻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡶⠶⠶⠶⠴⠦⠶⠤⠤⣤⣤⣄⣤⣀⣀⡀⠠⢀⠀⠀⠈⠉⠙⠻⢦⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠒⠶⠤⣤⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡤⠤⠤⠤⢤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣶⡿⢳⡄⠀⠀⠀⠀⠀⠀⠀⠳⣌⠠⠀⠀⠀⢀⠀⠈⠉⠉⠉⠙⠛⡓⠲⠦⠤⢤⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢀⡞⠳⣬⡛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢶⣶⣶⠓⠚⡛⠓⠓⠺⠶⡶⢴⣤⣄⡀⠀⢂⠌⡃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⡀⠃⠀⢸⡳⣄⡞⣉⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡌⠁⣀⡀⠁⡀⢀⠈⡀⣵⠟⢳⡄⢍⡓⠦⣄⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣞⣯⣹⢦⡀⠈⢇⣸⢉⡽⠋⠀⠀⢀⣀⣠⣤⣤⣤⡄⠀⠠⠀⠙⠲⢤⣄⠠⠀⣀⠀⣀⠙⢳⡟⠂⠠⡸⠁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⡈⢹⡃⡴⠛⣿⣾⡟⣭⡄⢀⡤⢾⡏⠸⣇⡼⠿⡿⠁⠀⠀⠁⠀⢰⡟⠻⡟⢦⣠⣄⢀⡀⣹⡏⣈⡴⢦⣴⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠉⠛⣟⠁⡴⢻⠍⣿⡿⣠⢾⡇⢸⣧⣞⡉⣩⠟⠁⠀⠀⠀⠀⠀⠈⡿⠛⠁⠈⣧⡶⣄⣤⣼⣏⣉⠷⣴⠛⠖⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡙⠲⢿⠞⠁⡼⠀⢸⣠⢿⣠⠿⢯⣤⡤⠛⠁⠀⠄⠀⠀⡀⠀⠀⢰⣇⣤⡀⠀⢸⣿⠉⡄⢼⡇⢉⣨⣏⢉⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠙⠲⢬⣇⠼⡁⠘⢿⡿⠿⠛⠛⠉⠁⠀⠀⢐⠀⠀⣻⡟⠛⢦⠀⣸⢿⣬⡿⠀⢸⣿⠤⠴⢾⡇⣤⠼⣯⢠⣀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢴⣤⣿⣾⠷⣦⣄⣻⣠⣤⢀⣄⡀⠂⠈⡆⠈⠳⢹⣷⣊⡼⠀⢻⠀⣿⠀⠀⢸⢿⠄⠰⢸⡧⠄⠄⡧⠤⠉⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠈⠹⡇⠀⠀⠉⠻⣏⣿⠟⠋⢠⠀⢀⡗⠀⢠⠏⠠⢻⠀⠀⢸⠀⢸⠀⠀⢸⢸⠖⠒⢺⡗⠶⠒⡷⠲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠈⠀⠀⠀⣹⡀⠀⠀⠀⣷⠀⠀⣴⣀⣿⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣄⣠⠶⠛⠉⠁⠀⡀⠀⣿⣷⢠⡿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠸⣇⠀⣿⣿⡾⢡⡏⡿⠈⠙⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠞⡄⠀⢀⠔⠀⠐⠂⠐
⡄⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⢁⣿⣿⠃⠀⠁⠈⢈⡀⢠⣤⣤⣤⣧⡔⠁⢸⠋⡴⠁⡀⢸⡇
⠂⠀⠀⠀⠀⠄⠀⠀⠈⣿⢾⣼⣻⡏⣠⣴⡆⣪⣶⣗⠉⣉⠛⣯⣝⠳⢄⣾⢉⣁⣠⢀⠏⠈⠐⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⣹⣿⣿⣿⣿⣿⡟⠟⠛⠛⠒⠛⠛⢯⡛⠓⣾⢿⣍⣀⣠⠟⠀⡀⠀⠀⠀⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⡟⠁⢸⡀⠀⠀⠀⠀⠀⠹⢾⣃⠘⠓⡾⠉⠳⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠋⠀⠁⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢈⡄⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣄⡀⠀⠀⠀⠀⡀⣼⣿⣿⣿⠀⠀⠀⢸⠁
⠛⠒⠀⠘⠀⠼⢹⣿⣿⣿⡇⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠄
⡀⠀⢠⠀⢀⡀⣾⣿⣿⣿⠀⡀⢠⠈⠀⠀⠀⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠙⣶⠞⢣⠸⡇⡸⠋⠉⠙⣨⣡⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣆⠀⠀⢠⠶⠆⡒⣂⠀⢀⠀⣔⢉⠰
⡷⠛⣿⡾⢻⣶⣧⡄⣠⠃⡟⢁⡶⠂⠀⢠⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢉⠚⠀⠌⠘⠛⠉⠀⠀⠀⠀⠈⠉⠀⠉⠀⡐⠋⠀⠀⠉⠘⠻⡟⠛⠂
⣁⣤⣯⣧⣾⣿⣾⡖⣹⣰⢴⠋⠀⠀⠀⠀⢙⣆⠙⣄⣀⣀⣀⣀⢀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣈⣡⣄⣈⢀⣀⣀⣄⣀⡀⢀⣄⣀⠀⣀
⠉⡁⣿⢹⠗⣿⣷⣧⣽⣿⡏⣉⡔⠁⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂⠀⠤⢭⣝⣯⡙⠛⣛⠛⠛⠛⠧⠷
⠀⢰⢻⢸⡄⣿⠂⣿⢟⣸⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠘⡇⠸⣿⠃⠈⣯⣌⠃⠖⡾
⠲⣶⣼⣆⢻⢠⣾⡟⣿⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠
⣼⣾⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡠⠆
⣽⠿⠿⢿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⠔⠁⠈
⠀⠀⠀⠘⣋⢩⣿⡛⣯⠈⠙⢢⣴⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⡂⢠⠮⠂⣴⠤⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⡀⠀⠐⢒⠉⢤⣼⣿⣾⣷⣽⣿⡯⠭⢭⠹⣿⣿⡂⢀⣴⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡜⡩⠋⠀⠀⠿⠤⠄⠠⠀⠘⠀⠀⠀⢸⣻⡅⢀⣀⠺⠄
⡄⠔⠀⠀⠀⡀⢭⣥⣥⣨⣿⣷⣤⡶⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣠⢆⣀⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢡
⣄⣸⢀⣶⣭⣀⡨⠙⣻⣿⣯⣿⣇⣠⣼⣶⣾⠋⢻⠿⠿⢟⣿⣻⣿⡿⢿⣿⣿⣿⠋⣩⡿⠉⠀⠀⡀⢀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠁
⣯⢿⣿⣿⣿⣿⣿⣿⣾⠿⠻⠛⠙⣿⣾⣿⡿⣿⣿⠤⣴⢿⣭⣭⣿⣿⣿⣿⣿⣿⠀⠟⢿⣧⣤⠛⠉⠙⠺⠇⠀⠀⢈⣤⣄⡀⠀⠀⠀⠀⠀⢀⡀
⣟⣿⣿⣿⢿⣿⠋⡭⣄⠀⣀⣀⣼⣿⣿⣍⣟⠉⣃⡾⢣⣴⣶⠶⣶⡿⣿⣿⡏⠉⠉⠀⠀⠻⡇⠀⠠⣤⣤⢆⠀⠀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠈⠲⣤⣄⣀⣀
⣿⣿⣿⣷⠿⠁⢀⣀⣙⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⣾⡿⠛⠛⠛⠻⣿⣿⣿⣠⣤⣤⣤⣀⣀⣼⣶⣾⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠙⠛⠿⠿
⣻⣿⣟⣱⣟⠻⣧⡄⠀⠀⢈⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡶⠒⠒⢛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⣠⣤⣤⣤⣤⣴⣶⣟⠋⠉⠀⢄⣤⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣶⣄⣀⣀⠀⢀⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⡿⣯⡶⠿⢾⣝⢿⠠⣼⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀
⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠰⣷⠀⣿⢾⣷⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢈⣧⡿⢦⣤⡼⣿⣾⠁⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠻⣿⠛⠉⣋⡼⣿⡽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡃⢶⠄⣄⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢯⣷⣶⣾⡿⠂⢄⠀⠀⠙⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠁⠀⠀⠁⠄⠀⠘⢦⠀⠀⠀⢸⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠘⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⠄⠐⢃⣀⠭⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⠔⠚⡈⠁⠀⠐⠢⣶⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⠀⠀⠀⠀⠄⠁⠘⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢛⣷⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⠀⠀⠀⠉⠉⠛⠛⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡉⠉⠉⠓⠒⠦⠤⣄⡀⠀⠈⠉⠛⠷⢦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠧⣄⠗⡢⠒⠒⠒⠠⢀⠀⠈⠉⠒⠢⢄⡀⠀⠈⢹⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠛⠛⠛⠥⢀⡀⣰⠀⠊⠀⠀⠀⠀⠀⠀⠉⠲⠦⠀⠀⠀⠈⠉⠒⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠞⠋⠙⠣⠤⢄⡀⠀⠀⠀⠈⠑⠼⣀⡀⠀⠀⠀⡔⢁⠄⠂⠈⠙⠂⢀⠀⠀⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⠀⡀⠀⠀⠉⠑⠒⠦⠧⣨⣤⣤⣤⠤⠤⠈⠉⠒⠤⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠠⢄⠀⠀⠀⡀⠀⠀⢸⡇
⠀⠀⠀⣠⣶⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⣀⠀⠀⠀⠑⠆⠈⠀⠀⡁⢸⠃
⠀⠀⠐⡏⣉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⢹⡄⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⣿⢉⠙⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣠⠁⣸
⠀⠀⠀⣿⠈⡍⠓⠢⢄⠉⠐⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠊⠁⣸⠀⣿
⠀⠀⠀⣿⠀⣍⠐⠀⠀⠈⠐⠠⢄⡈⠉⠒⠤⣄⣀⡤⣒⠀⢀⣀⠠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣷⢶⣟⠁⢀⣿
⠀⠀⠀⣿⠐⢬⡙⠲⢄⡀⠑⠢⢄⠈⠑⠢⢄⡀⠙⠀⢸⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⡴⠏⠀⠙⠿⠟⠃
⠀⠀⠀⢸⣇⢰⡟⠷⣦⣈⠑⠢⢄⡀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋⠉
⠀⠀⠀⠀⠉⠋⠁⠀⠀⠉⠛⠶⣤⣈⠙⠲⠤⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⢹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⢸⠀⠀⠀⣀⣠⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠈⣷⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⠶⡭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⣁⣤⡁⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⠁⠈⢿⡁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⡒⠀⠋⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠀⠀⠁⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡄⠀⡠⠎⠹⣷⣦⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠓⡒⣾⡅⠤⠄⠒⣨⢋⠳⢤⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢚⠋⠀⠀⠀⠀⠶⣀⡎⠸⠀⠀⠀⢀⡜⠃⠀⠀⠀⢷⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠞⠐⠒⠒⠒⠊⠉⢙⢁⡀⠆⠀⣠⠏⠸⠂⠀⠀⠀⢸⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠤⠚⠁⢠⠃⠀⠀⠀⢠⡟⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⢀⡌⠀⠀⠀⠀⣼⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⡖⠉⢛⣦⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠤⠁⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡿⢣⡤⠈⠀⠘⠀⠒⠲⠖⠒⠛⠛⠛⠛⠃⠀⠀⠀⠐⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⢸⠤⠴⢶⡉⠉⠙⠒⠶⢤⣄⣀⡀⠀⠀⠀⢀⠀⠀⢸⡄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠁⠿⠀⠀⢸⠆⠀⠀⠀⠀⢰⠆⠉⠉⠛⠛⠛⠁⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⠂⠚⠁⠖⢀⡀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⢸⡀⡠⠞⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢸⠙⠳⠶⠶⠶⠞⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠉⢷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⡠⠤⠀⠀⠀⠀⠀⣿
⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⢩⡅⠀⠀⠀⠀⠀⠀⣸
⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢸⡇⠀⠀⠀⠀⢀⡟⠀⠀⠒
⠈⠓⠀⢀⡀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⠀⡼⠁
⢀⠈⠀⠸⠈⢆⠀⢸⠃⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⠓⠚⢅⠸⠂⠀⣇⢀⠠⠐⠀⠂⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⡀⠀⠀⡇⠀⣀⠤⢤⡀⠀⠀⣠⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠾⠀⠀⠀⠀⢧
⠀⠀⠐⠉⠈⠉⠉⠰⠋⠀⠀⠀⣹⠪⡁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡀⠀⠀⠀⢸
⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡇⠀⠀⠀⢸⠆
⠀⠈⠀⠀⢃⠀⢀⠃⠀⡜⠃⠀⠀⠀⠀⠀⣠⡴⠛⠙⢳⠄⢠⠀⠀⠀⢠⠏⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠂⠈⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠒⣤⡄⠀⠉⠉⠉⠉⠈⠁⣼⠃⠀⢀⡰⠏⠀⠘⠀⠀⠀⡎⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣀⠀⠠⡎
⣀⠀⠀⢀⡀⠘⠀⠈⣂⠀⠀⠠⠀⠀⢰⠇⣠⠖⠋⠀⠠⠀⠛⠛⠓⣤⠁⠋⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠰⠁⠀⠈⠁⠀⠛⠊⠀⠀⠐⠃⠜⠁⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⣀⠠⡤⠴⠊⠁⠀⢀⣠⡤⠞⠁⠀⠀⠀⡇⠀⠀⡝
⠀⠀⢤⡠⠔⠂⢀⠔⠓⡀⠤⠀⠀⠁⠀⠀⢀⠤⠒⠛⠓⠈⠁⠀⡄⢀⠔⠋⠉⡢⢀⣠⠖⠈⣷⣉⣁⣀⣀⠠⣴⡶⣾⠁⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠇⠀⠀⡁⠀⡠⠖⠒⠲⠦⢠⡀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⠉⠀⢾⠁⢀⡴⠛⢦⣤⠀⠌⠉⠀⣰⠏⠀⠀⠈⣿⠀⠀⠀⠀⠀⢀⠴⢄
⠤⠤⢄⣀⠀⠀⡄⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠔⠁⠀⢱⠀⠀⡰⠋⠀⠀⢀⡾⠟⣀⠤⣄⠀⢰⡏⠀⢸
⠀⠀⠀⠉⠙⠂⠶⡀⠀⠀⠈⠳⠤⢄⡀⠤⠶⠚⠀⡔⠀⠀⠠⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠜⠀⢀⡴⠃⠀⢸⢆⡠⠦⣰⠞⡡⠊⠁⢀⣼⠃⠀⠀⢀⡞
⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣭⣴⣤⣤⣤⣭⣭⣭⣤⣮⣤⣴⣶⣿⣮⣥⣤⣴⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠤⠤⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⣇⢀⠈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⡉⠁⢀⡇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢹⠀⠀⠰⠀⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⠀⠀⠀⠀⠰⠀⡄⠀⠀⠸⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡄⠀⠐⡆⢸⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠀⡆⢠⠀⠀⠀⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⡇⠀⣰⠀⠀⠀⠃⡆⠀⠀⠀⠀⡇⢠⠀⠀⠀⡆⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⡃⠀⠀⠀⣇⠀⢸⠀⠀⢀⡀⣇⠀⠀⠀⢀⡇⢘⠀⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠛⠛⠋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣉⣈⣉⣈⣉⣉⣉⡉⠉⠉⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⡰⠁⣀⡠⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⠒⠂⠠⠤⠤⢄⡀⠱⡀
⠀⠀⠀⠀⠀⠀⣡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡇
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣀⢀⣀⢀⣀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣀⡀⢀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣘⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⣠⣠⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡁⣛⣀⣈⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣄⣀⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢸⡟⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⣶⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⣿⣿⡟⠛⠻⠿⣿⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣿⠃⢰⣿⡿⠀⠀⠀⠀⠀⠻⡘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⡇⠀⣿⣿⠁⠀⠀⠀⠀⠀⠈⡤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢸⣿⣅⡀⠀⠀⠀⠀⠀⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⢸⣿⣟⠀⠀⠀⠀⠈⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠠⡇⠀⢸⣿⣇⡀⠀⠀⠀⠀⣈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡇⠀⢸⣿⣿⣷⠯⠭⠀⠾⠛⣱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⢸⣿⣿⡆⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡇⠀⢸⣿⣿⡀⠀⠀⠀⠀⠐⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠁⠀⢸⣿⣿⠀⠀⠀⢀⣀⣀⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡆⠀⢸⣿⣧⠀⠀⠀⠈⣽⠋⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡦⠀⠸⣿⡏⠀⠀⠀⢠⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⠀⠀⣿⣿⡿⠒⠐⠋⣻⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠀⠀⣿⣿⣷⡀⠀⠤⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⣿⡆⠀⣿⣿⣿⡇⢤⣖⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⢻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⢠⣾⣿⠟⢿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣰⣿⢟⣼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢙⡞⢳⡀
⠀⠀⠀⠀⠀⣿⣿⣏⡠⠁⠈⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠻⢫⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠋⠀⠀⠹⣄
⠀⠀⠀⠀⠀⢿⣿⣿⡁⠀⢀⠄⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⢸⡀
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣴⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣛⣡⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠤⢊⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉


⠀⠀⠀⠀⠀⠀⢀⡜⠱⣤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠱⡠⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢾⠀⡰⠃⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡔⠁⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⣠⢃⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡠⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠭⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠉⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⢛⣛⣛⣛⣛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣥⠶⠛⠉⠉⠉⠉⠉⠛⠶⣮⣙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣹⣇⢀⠀⠀⠀⠘⢷⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⢋⡿⢿⣤⠀⠀⠀⠈⢻⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡶⢋⡴⢋⡴⠋⠙⢶⣄⠀⠀⠈⢷⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⣠⠞⢡⡼⢋⡴⢋⡴⠛⠀⠀⠀⠀⠙⢧⣀⠀⠸⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⣠⠞⣁⡶⢋⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⢿⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢹⡄⣠⠞⢁⡼⢋⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣬⠟⢁⡴⢋⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⡼⢋⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⣯⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⡾⢋⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⡾⢋⣴⠟⣡⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣻⣗⠲⠶⠶⢦⣤⣤⣤⣀⣀⣈⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⡾⢋⣴⠟⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⠞⠉⠉⠛⠳⠶⢤⣄⣀⡀⠈⠉⠉⠉⠛⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⡾⢋⣴⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⢦⣤⣀⡀⠀⠙⣦⡀⠀⠀⠀⠀⣠⡾⣫⡾⢋⣴⠟⣡⡾⠋
⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⣿⣦⡀⣠⡾⣫⡾⢋⣴⠟⣡⡾⠋
⠀⠀⢰⣿⣯⡉⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣯⡾⢋⣴⠟⣡⡾⠋
⠀⠀⠀⢿⡉⠙⢿⡀⣿⠀⠀⢀⣤⠶⢛⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡟⣡⡾⠋
⠀⠀⠀⠀⠻⢦⣤⠿⢧⣾⢠⡟⣿⣴⠿⢷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣷⣟⣁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠈⠉
⣴⠞⠓⠚⠛⠛⠛⠓⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⢶⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠙⠶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠟⠁
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠙⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⡤
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠙⢿⣧⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣠⣿⠏⠀⠀⠀⠀⣤⠛⠙⣿⣄⠀⠐⣄⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⣶⣤⡤⠀⡀⣠⣿⣿⣿⠋⠀⡀⠀⣠⣴⣞⠁⡀⠀⠀⠙⣳⣶⣿⣿⣿⠷⠦⠀⠸⠇⣷⣤⡄
⠀⠀⠀⠤⠈⣋⣾⣿⣯⣿⣿⣿⣿⡟⣿⣿⣶⠿⢛⡿⠟⠁⢤⣾⣿⣿⣿⣿⣯⠱⠶⢶⣄⠀⢀⠈⠁⠀⠄
⠀⠀⠀⠰⣾⣯⣿⣿⣾⣾⣿⣿⣿⣷⣿⣿⣿⣷⣾⣷⣾⣾⡆⠉⢻⣽⣿⣿⠿⠁⠀⢀⣨⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣟⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠋
⠀⠀⠀⠀⠀⠀⠻⠿⡿⣿⣿⣿⣿⣿⣏⠙⠻⣿⣿⣿⣿⡛⠀⠂⢀⣀⣠⣬⣯⣭⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣶⣶⡀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⣿⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠿⠿⠿⠟⠙⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣿⣿⣿⣯⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢸⣿⣿⣿⣿⣷⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⣿⣿⣿⣿⣿⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠿⠿⠿⠿⠿⠟⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠲⠄⡈⠓⢄⠀⠀⡀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⣇⠀⠈⠢⣠⠇⠀⠀⠟⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⠀⡤⣑⢦⣸⣷⣦⣀⣀⣦⠎⢻⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠚⠇⠘⠛⢬⣯⠈⢿⣿⠇⠘⢿⡟⠁⠀⢠⢀⣤⠌⠀⠀⠀⠀⠀⠀⢠⣀⠀⠴⠒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠘⠀⠀⣹⠀⠀⠘⠀⠀⢠⣾⠟⠃⠀⠀⠀⠀⠀⢀⡖⢸⢃⠀⠀⠴⠛
⠀⠀⠀⣀⣀⣀⢡⡀⠈⠀⠁⠀⠠⣧⣃⡟⠀⢀⣄⣠⡶⠏⠁⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⠏⠰⠀⠐⣯
⠀⠠⠟⡉⠉⠉⠉⠉⠓⠐⠢⠤⣤⣤⣼⠷⠶⣿⠷⠤⠤⡀⠀⠠⠴⠂⠀⠀⠀⠀⣻⣴⢣⡆⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠂⠄⢂⡤⢦⡴⠎⠁⠀⠀⠐⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢾⣷⣶⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⣀⠀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⢨⠋⠀⠀⠀⡀⠂⠀⠀⠀⠀⠄⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣴⠃⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠚⠀⠖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣴⠀⠀⠀⠃⠀⠀⠀⢀⠀⢢⠀⡀⠂⢀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠇⠀⠀⠀⠀⠀⠀⠀⠐⠩⡶⢿⣷⣱⣶⣴⠦⠀⠀⠚⠂⠀⠀⠀⠒⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⡄⠂⠀⠀⠀⠘⠁⠶⠺⢧⠩⡭⣝⠀⠠⠀⠐⠖⠶⠤⢄⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠃⠃⠀⠐⢤⠀⠀⢀⣒⠐⠂⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠉⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⠀⠀⠻⣟⠣⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⠄⠀⠀⠀⠈⠙⠲⢴⣶⣆⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⢁⠀⠀⠰⢂⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿⠃⠀⠈⠉⠉⠋⠚⠛⠲⠤⠬⠦⠤⠵⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠠⡇⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⡘⠃⠀⠀⠀⣰⡏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣁⣀⣀⣀⣀⣠⣆⣀⣀⣀⣞⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠀⠀⠀⣼⣷⣷⡄⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣦
⠀⠀⠀⠀⠀⢰⡀⠘⠛⠁⢀⡆⠀⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠷⠀⠞⠋⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⠻⠿⠟⠁⠀⠀⠀⠀⠀⠘⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠆⢠⣶⡀⠀⠀⠀⠀⠀⠀⢀⡾⣿⡄⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⣿⣷⣿⣷⠀⠀⠀⠀⠀⠰⣿⡿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⣠⠀⠀⠀⠀⠀⣾⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⡧⠀⠀⠀⠀⠀⠘⣿⣟⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠐⢛⠟⡛⠛⠛⠁⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠀⠀⠀⠀⠐⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⣠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠄⠀⠀⠀⠀⢤⣦⣤⣄⣠⠴⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠂⠀⠃⠀⠂⠀⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠈⠀⠀⠀⠀⠈⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣴⣀⡀⠀⠀⠀⠀⠀⠀⣴⣠⡀⡀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡗⠀⠀⠀⠀⠀⠉⠿⣿⣿⡗⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠘⢿⣿⠿⠀⠀⠀⠀⠀⠀⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣤⣤⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡾⠿⠻⠛⠻⠉⠉⠉⠁⠁⠉⠙⠋⠋⢻⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠐⠛⣻⣠⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠁⡿⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⢀⣀⢀⣀⣒⣶⣶⡦⢀⢀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣠⣴⣦⣦⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⡦⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⢈⣿⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣄⣀⠀⠐⠟⠻⠀⠀⠀⣼⣶⣦⡄⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⢿⢿⣿⣷⣾⢿⣿⣻⡻⠈⠁⠘⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣟⢿⣽⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣧⣸⠁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣳⣿⡡⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠏⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⢤⡄⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡘⠁⠀⢀⣠⣴⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢿⠷⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⡄⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠃⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣇⠻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣧⠈⢧⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠷⣾⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⡙⠂⠀⢀⣠⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⣾⣿⣿⣿⡗⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⢿⣿⣿⣿⠆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⡄⠀⠀⠀⠀⢀⣾⣷⡆⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠙⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣤⠀⠀⠀⢾⣿⢿⣄⠀⠙⠇⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠾⠟⢀⠀⠀⠈⠁⠈⢙⢄⠀⠀⠁⠀⠀⠄⠱⢤⡀⠀⣤⡀⠀⢸⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢡⣀⠀⠀⢸⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⣆⠈⢀⣾⣿⣿⠀⢸⠀⠀⢀⡟
⠀⣀⡀⣶⣾⣿⠷⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠘⡄⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣨⡿⢿⣿⣿⠃⢸⠀⠀⠀⠁
⠀⠀⠂⣿⣿⣿⡟⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⢱⡹⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⡇⠐⠈⠙⡏⠀⢸
⠀⠀⠀⠹⡟⠑⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠀⣷⠀⠰⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⢀⠀⠀⡇⠀⣀⠀⠀⠰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⢸⡆⠀⠀⣙⣷⠀⠀⠀⠀⠀⠰⠀⠀⢸⡇⢆⠀⡃⠀⢾⣀⠀⠀⠀⠀⢀⠾⠁⣹⢿⡆
⠀⢸⡄⠀⣇⣰⣿⡆⠀⠀⠈⢉⡛⠆⠀⠀⠀⠀⠀⣀⣀⣴⣧⣾⣿⠁⣷⠀⢀⠿⠀⣴⠀⢾⠀⠀⠀⠀⠀⠈⣧⠘⡆⢿⠀⢼⠈⠳⡄⠀⠀⠀⠀⠀⡉
⠀⠈⣧⣸⡿⣿⣿⣇⣠⣴⣤⡾⠃⠀⠀⠀⠐⠴⣾⣿⢿⠟⣿⡛⠉⣧⠸⣆⡡⠐⢠⡇⠀⠈⠀⢀⣶⡚⠀⠀⣿⠀⢹⣼⡀⣺⠐⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⢿⡇⢤⣿⡿⣿⣭⢿⡇⠀⠀⠀⡆⠀⠀⠈⢳⣄⠀⠘⡇⠀⠸⡆⣿⠅⠀⣼⠃⠀⢐⠀⠈⠀⠁⠀⠀⢹⡇⠈⣿⡇⣿⠀⡇⢀⠻⡆⠀⠀⠀⠀⠀⣰⠃
⠀⠀⢸⠀⣇⢸⣯⣤⣴⣾⣿⣧⣤⣤⢦⣇⢀⣠⠀⠘⣿⣦⡀⣿⠀⠀⢿⢸⡄⠀⡟⠀⠀⢸⠀⠀⢠⠆⠀⠐⣼⡷⣤⣼⣿⣿⠀⣿⡎⠀⢻⡀⠀⠀⣠⣾⠇
⠀⠀⣾⡄⣿⡾⢻⣿⡛⢿⣿⣿⣦⡁⣾⣿⣿⣿⣿⡀⢿⣿⣷⣿⠀⠀⠸⡿⣇⢸⡇⠀⠀⡼⠀⠀⡼⠀⣠⠀⢸⣷⡀⠈⣿⣿⠀⢻⠋⡄⠈⣧⡀⠀⢻⣿
⠀⠀⡟⡇⡏⡇⠈⡏⣇⠘⣍⠙⢿⣛⠹⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⠀⣷⣿⢸⠇⠀⢠⢑⠰⣾⣧⡾⠋⠀⣼⡏⢳⢰⣿⣿⢀⣸⡀⢣⠀⢻⠀⠀⣼⡏⠄
⠀⢰⡧⣇⣷⣿⣠⣷⢸⡆⠘⣧⠸⡇⠀⣿⠿⣿⣿⠇⢸⣿⡿⢿⡇⢰⠀⣿⣼⣾⠀⣀⣼⣸⣶⣿⣉⢷⡀⡼⢸⡇⢨⣿⠀⣿⢻⣿⣆⡸⡆⢸⡇⣷⣿⠀⠸
⠀⣸⠀⣯⡿⣿⡉⣿⡀⣷⣿⣿⣧⣿⣴⣿⣷⡏⣿⠀⠸⣯⡇⠘⡇⢸⠀⢸⣿⣿⣿⣿⡟⣿⣿⠛⣻⠛⢰⠃⠘⡇⢀⣿⡄⣿⢸⡇⡇⠙⣧⡈⣿⡇⣿⠀⠀⡇
⠀⣿⠀⢿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⢿⡀⠀⣿⡇⠀⡇⢸⠀⢸⣿⣿⣿⠈⣷⢸⡏⣰⠃⠀⡿⠀⢀⣧⠦⣿⣇⣿⣾⡇⡇⠀⢹⢻⣿⡗⡏⡁⠀⢹
⠀⣿⠀⢼⣧⣾⡟⢻⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⢸⡇⢰⡿⡇⠀⡇⢸⠂⢸⡿⣟⣷⣤⣇⣸⢧⡏⢀⢰⣇⡴⢛⡟⠀⣿⣿⣿⣿⣧⡇⠀⢸⣈⣿⣧⡏⣿⠀⢸⡀
⠀⣿⠀⣾⣿⠨⡇⢸⡿⣿⣿⣿⣿⣿⣿⣿⡇⢣⣸⡇⡼⣷⣷⠀⣧⣼⡇⢸⠆⣿⣿⣿⢻⣿⣾⣷⣾⣾⠟⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⠴⢻⡅⣿⣿⡇⢹⠀⠸⡇
⠀⣿⣴⣿⣿⠀⡇⢸⡇⢻⣿⣿⠟⢹⡿⣿⡷⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⢸⠀⡿⠹⣿⢸⣿⣿⣿⣸⡿⠶⡀⢸⠃⢀⠹⣿⣿⣿⡿⣷⡄⢸⡇⣿⣿⡇⢸⡇⠀⡇
⠀⢿⡟⢹⣿⣆⡇⢸⣷⣾⣿⣿⣧⣾⣧⣿⠇⢸⢻⣿⡏⡟⠀⡏⢿⢻⠈⣿⠀⡇⠀⣿⣼⣿⣿⣿⣿⡇⠀⠀⣿⠀⢸⢀⣿⣿⣿⡇⢹⡇⢸⡇⣿⣿⡇⠘⡇⠀⡇
⠀⣿⠃⠈⣿⣟⣷⣿⣿⠿⣿⠙⡟⢿⡏⣿⠀⢸⣾⣿⠀⡇⡆⣇⣿⢸⠀⣿⡆⡇⠀⣿⡿⣿⠏⡟⢸⣇⣀⣀⣿⠀⣼⡞⢿⣿⣯⡇⠀⠀⠈⡇⣿⣿⣧⠀⡇⠀⣿
⠀⣿⠀⢸⣿⡿⡏⠈⡿⣧⣸⠀⡇⢸⡇⣿⠀⢸⣿⣿⠀⢧⠃⢻⣿⠸⠀⣻⣿⡇⠀⢻⡇⡇⢸⣿⣿⣿⡉⢩⣿⠀⣿⠇⢸⣿⡏⠁⠀⠀⠀⡇⣿⣿⢹⠀⡇⠀⣿
⠀⢸⠀⢸⣿⡇⡇⣦⡇⠀⢹⠀⡇⠀⡇⣿⠀⢸⡟⣿⠀⢸⡀⢸⣿⠀⠀⣿⣿⡇⠀⢸⠁⡇⣼⠋⢸⠈⠛⢸⣿⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⡇⠛⣿⢸⠀⣷⠀⣿

⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⡀⣀⣠⣀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣟⣿⣷⣿⡇⠘⠓⠛⠛⠚⠐⢠⣺⣷⡄⡀⠀⠀⠀⠰⢆⣂⣂⠈⠀⢀⣀⣠⡶⠋⠀⠀⠠⣀⠀⠀⠀⠀⠳⢎⡸⠀⠀⠀⠀⠄
⠀⠀⠈⠛⠿⠟⢿⠋⠀⠀⠀⣠⠶⢄⠀⠀⠉⠉⠀⠀⠀⠲⢣⣞⠁⠈⢻⣛⣆⠛⠋⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠸⠟⠻⠇⠀⣠⡄
⠀⠀⠀⣠⣴⣤⣌⡀⠀⣀⡤⣷⣤⡾⠃⠀⡀⠀⢠⠀⠀⠀⢾⣿⡄⠀⠈⣿⡏⡇⣰⠆⣠⡦⠀⡴⡄⠀⠀⢠⡶⣦⢠⣶⣦⡸⣧⣸⣿⡀
⠀⠀⢰⠻⣿⣿⡿⠟⠋⠑⠀⡼⠁⡀⠐⡤⡾⣿⡯⠑⢏⠀⠘⢏⡁⣠⣤⣍⡠⠃⠋⠊⠁⠀⠐⣃⡀⢠⣦⠘⢷⡿⢻⣧⣴⣷⠘⠋⠹⣧
⠀⠀⡏⠀⠘⣯⣠⣄⣀⣀⣾⣇⠀⣧⠈⠳⢤⣮⣦⠴⠋⠀⠀⠀⠉⠙⠛⡉⣠⡄⢠⣠⣦⢠⣿⠛⣿⡌⣿⣤⣴⡆⠀⠻⢿⠿⠀⣀⡀⢀⡄
⠀⠀⢿⠀⢤⠁⠉⠻⢮⣿⣿⣿⡇⠈⠷⣶⣆⠀⠀⠀⢤⣰⠚⢣⣤⣄⣺⣧⣸⣷⢻⣿⣿⣎⣿⣦⣿⠇⠻⠿⠛⠋⠀⠀⢰⣾⣿⢿⡇⠈⠇
⠀⠀⠈⢷⣤⡀⠀⠀⠀⣈⡽⠟⠁⠀⠀⠉⠀⣀⣤⣴⡄⠈⣀⣿⡏⠙⠛⣿⡏⢿⡞⡿⠙⠟⠈⡉⠁⠀⠀⠀⣧⠀⠀⠀⣸⣷⣤⣄⣧⣀⡀
⠀⣤⣶⣾⣍⣛⣻⣿⡛⠉⢀⢤⡒⠄⣇⡀⠹⠛⣿⡏⢺⣿⣽⡜⢿⣶⣿⠻⠗⠈⠀⠀⠀⡀⠛⠁⠲⠄⠀⠤⠞⠀⠀⠘⣿⣷⣾⣿⣿⣿⠃
⠀⣿⣟⠛⠛⠛⠛⠛⢻⠃⡜⠀⠀⠀⠀⠂⢀⠀⢹⣿⡘⣿⣷⠶⠀⠉⢁⠀⠀⠀⢹⡀⠀⠸⡄⢠⠤⠒⠀⣀⣠⣤⣤⣀⠀⠙⢿⣿⣿⠃
⠀⣿⠿⡀⠀⠀⠀⢀⡏⠀⢷⠀⠀⠀⠀⢈⡆⠀⠀⠉⠀⣀⡀⠀⠀⢰⣬⣟⣲⠤⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣟⣿⣿⡟⠀⢠⡀⠙⠁
⠀⠻⣧⣇⣀⣀⣀⡼⠁⡀⠀⠑⠂⠔⠐⠀⢀⣠⠤⠒⠉⠙⣁⣀⣤⣼⣿⣿⡿⢿⣿⣶⡆⠀⡁⠀⠀⠈⠙⢿⣿⣿⣿⠀⠰⡏⠘⠳⠆⣀⡀
⠀⠀⠀⠉⠉⠉⠁⠀⢄⠀⠠⣀⡀⠰⠞⠛⢉⣀⣤⣶⣺⣿⣿⣿⣻⣿⣿⣾⣿⣿⡿⠃⠈⡥⠁⠀⠀⠀⢀⡀⠈⣵⡆⠀⣼⣿⣦⣦⠐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣆⡔⠀⠸⣃⣤⣶⣾⣿⣽⣿⠿⠿⠟⠛⠛⠛⠉⠹⣿⡿⠋⠀⠂⡠⠔⠉⡁⠀⢤⣿⣷⡄⢸⡆⠀⠻⠿⠛⠁⠐⣿⠂
⠀⠀⠀⠃⠁⡔⠀⠀⠉⠉⠀⠈⠛⣛⠋⠉⠉⠀⠀⠀⠀⠀⢈⣶⠃⠀⠀⠉⠈⠆⡆⠀⠀⣀⣀⢿⡿⠄⢉⣩⠥⠔⠧⢄⡀⠀⠀⠀⠬⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠂⣤⡞⢉⣻⣅⣈⠑⣤⠢⠐⠀⠻⢷⣾⠋⠀⠀⠑⠈⠢⣠⠖⣆⠀⡇⠈⡎⠁⣰⠋⣠⣶⣶⣶⣦⣙⡄⠀⠠⣤⣄
⠀⠀⠀⠀⡀⠰⡆⠀⣾⣤⣾⣿⣽⠶⠶⠟⠾⠷⣉⡴⠠⠐⠀⠀⠀⣤⣦⣤⣤⡘⣦⣈⠛⣷⣤⠇⢰⡇⣬⣿⣿⣿⣿⣿⣿⣇⠀⢨⣷⣾
⠀⠀⣠⣾⣷⡀⠉⠘⡿⢋⠉⠀⢻⡀⠀⠀⢤⣤⢸⣷⡀⠐⡅⢰⣟⣛⣿⠻⡟⣿⣎⠙⢻⣿⣿⣆⠈⣧⠸⣿⣿⣿⣿⣿⠏⢹⡀⠘⠿⠟⠂
⠀⠀⠛⣿⡟⠓⠀⢸⣿⡾⠆⢠⣾⣿⣷⣦⣄⣀⠞⠙⣿⡀⢇⢸⣿⣛⣿⣠⣇⣿⣿⠀⠘⣏⡕⡻⣷⣿⡦⠄⣉⣉⣉⡤⠒⠉⢿⡵⣦⡀
⠀⠀⠀⣿⡇⠀⢠⣿⠉⠓⢦⣾⣿⣿⢿⣿⠟⢅⠀⠀⢸⣧⢪⢸⣿⢙⠏⡿⡏⣿⣿⠀⠀⠙⣾⣱⣹⣿⢴⢒⣍⣩⣭⡍⡭⣽⠸⣿⣒⣳
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠘⢯⣛⣛⠛⣋⡠⠆⠀⠀⣴⣛⢦⡀⣿⡞⡿⣷⣗⢿⣿⡆⢰⠄⠈⠛⠿⣿⡸⣙⠞⠙⢿⠇⢉⡼⠀⣿⣿⣾⡆
⠀⠀⢀⣿⡇⣴⠋⢿⣳⡄⠀⠀⢨⠏⠛⠂⢲⠀⠀⢸⣿⡦⡌⢷⢹⣷⣤⢿⡼⠼⣿⣧⠘⡄⠀⠀⠀⢸⣦⣄⣉⣘⣛⣋⣥⣴⣾⣿⡿⠙
⠀⠀⢸⣿⡇⢹⡀⡏⢿⡷⠀⠀⣸⣀⠀⠀⡇⠀⠀⠈⠛⠓⠘⠉⣿⣿⡆⣾⡿⠾⣿⣿⡙⡄⠀⠀⠀⠈⣿⠿⠿⢿⣿⣿⡿⢟⣹⡞⢷⣰
⠀⠀⣾⣹⣷⠀⠉⢡⣴⣶⠆⠀⠀⢸⠁⠀⢉⠇⢀⣀⣾⣀⡀⠀⣾⡿⣻⢻⡟⢿⢻⣿⡧⡇⠀⠀⠀⢀⣿⣭⣭⣭⣧⢸⣿⣿⣶⡷⣶⣄
⠀⢰⣿⣹⣿⣧⡀⢀⡏⠀⣾⠆⠠⠚⠚⣧⠎⠀⠀⢹⢿⡏⠁⠀⢻⣷⠹⢿⡇⣆⢘⣿⠇⡸⠀⠀⢰⣯⣛⣿⣿⣿⣿⠸⣿⣿⡿⠿⠿⢿⡆
⠀⠈⠉⠉⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠁⠀⠀⠈⠛⠿⠿⠿⠿⠿⠋⠀⠁⠀⠀⠘⠋⠋⠛⠛⠛⠉⠀⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠒⠒⠒⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠈⠉⠙⠉⠙⠉⡛⠉⠁⡅⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⣠⣀⣀⠀⣀⣀⣇⣸⣤⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠮⢭⣭⣭⣭⣭⡭⠥⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠡⠤⠔⠒⠒⠒⠒⠒⠒⠠⠼⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠑⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠉⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠐⠫⢶⣢⣤⣤⣍⣉⣁⣀⣀⣀⣀⣀⣩⠭⠕⣂⡠⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⣈⡉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢻⣾⣶⠃⣼⣷⠚⣷⠖⢰⣿⠃⢸⣭⣟⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠀⢀⡀⠀⠀⠀⠀⠉⠉⠀⠛⠀⠘⠛⠂⠈⠀⠀⠀⢀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠘⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠤⢄⣉⣉⡐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠁⠀⠰⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠤⢀⣀⡀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠒⠶⠶⠶⠒⠒⠋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣑⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠉⠓⠊⠉⠛⠓⠒⠒⠒⣒⣶⣖⣒⣚⣯⠝⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡂⢤⣀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⣀⣤⠖⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢮⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣋⣥⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⠶⠶⠶⠶⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣹⢿⣶⣦⣌⣒⢾⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⠀⠈⠉⠁⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠑⠢⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠖⢛⡇
⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠉⠒⠒⠶⠶⠦⠖⠒⠒⠉⠉⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠐⠠⠤⠀⠀⠠⠤⠐⠒⠂⠒⠂⠤⠤⢼
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠑⠀⠀⢀⣀⡀⠠⠄⠒⠒⠒⠂⠀⠤⠤⢰
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠐⡇⡀⠀⠀⠀⠀⠀⢀⡀⡀⠀⢀⢀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣀⠤⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠑⠂⠰⠶⠆⠐⠂⠉⠉⠁⠐⠀⢀⡝⠀⢰⣯⣀⠀⠀⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⡀⠀⠀⠀⠀⢀⣀⡤⠖⠋⣀⡀⢨⣧⡉⠉⠉⠉⠉⢉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠓⠂⠚⠁⠉⢫⠉⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣍⣬⣯⣠⣍⣽⣍⣽⣯⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡠⠤⠠⠤⠤⣂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠶⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣠⡾⠿⠠⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⢄⠈⢀⠀⠂⠀⠀⠡⠄⠤⠌⢤⣈⣀⠄⡀⠀⠀⠀⢇⠀⣀⣰⣽⡿⠞⠡⠰⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⢀⡂⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣢⠽⢟⠋⠄⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡴⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⡜⠀⠀⠁⠒⣅⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠐⠄⠀⠄⡀⠀⠀⠀⠀⠔⠀⠌⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⣂⣀⡠⠤⠖⢒⡋⠉⠉⠈⠂⢄⠀⠀⣂⡠⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⢁⣀⣁⡀⠔⠁⠀⠀⠁⠀⠀⠀⠴⡈⢋⠢⣤⣺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠄⡄⠀⠠⠀⠁⠀⠀⠀⠀⠀⠆⣠⠑⢢⢁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠄⠃⡄⡈⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠤⠐⠒
⠀⠀⠀⠀⠠⠒⠠⣀⢁⡒⡐⠒⣂⡂⡄⣀⢢⠀⠀⠀⠀⡄⢁⣤⣷⠚⢠⢠⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠂⢾⠃⠋⣄⣶⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⢁⡼⣗⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠿⢼⠷⠋⢠⠁⡀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠇⠠⡂⢧⠰⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⣀⡃⣸⢀⠀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠠⠤⠠⠄⠰⠆⠴⠀⠀⠙⠋⠠⠇⡈⢼⠏⠁⠀⣠⡀⣠⠀⠀⡀⠀⠀⢂⡠⠀⠤⡈⠉⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠉⡱⠀⢡⣤⠀⠀⠶⠀⠁⢠⡅⠀⢄⣄⠁⠘⡋⠌⣛⣁⠀⠂⠁⠀⠁⠀⠀⢀⠀⠀⢀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠀⠀⠒⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠉⠁⠀⠉⠀⠒⠒⠒⠒⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠖⠒⠒⢲⣍⠤⠦⠤⠄⠀⠴⣆⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⡇⢐⣰⠀⢀⠕⠀⡀⠄⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡠⠊⡇
⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⣀⣈⣑⣐⣈⡡⠂⠁⢀⠤⠂⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡡⠋⢀⠎
⠀⠀⠀⠀⡏⡉⠭⠭⠭⠄⠀⠒⠒⠒⠒⡆⣀⠔⣊⠥⠤⠄⠀⠉⠁⠒⠢⡉⠑⠄⠀⠋⢀⡐⠒⢒⠞⠁⠀⡴⡇
⠀⠀⠀⠀⢧⢃⠀⠀⠀⣎⣑⠀⢀⡠⢐⡉⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣠⠊⠙⠁
⠀⠀⠀⠀⢸⠸⠤⠤⠤⣤⣴⠾⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠃⢀⠔⠁
⠀⠀⠀⠀⠘⣆⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⠤⠤⠤⠤⡀⣠⢎⠌⠀⡠⠋
⠀⠀⢀⡠⠒⠉⢸⠀⠀⠀⠈⠉⠩⠀⠈⠩⠭⠥⠄⠒⠒⠀⠀⠐⠒⠒⣒⣒⠒⠚⠁⡘⢀⠞
⢠⠞⠥⠤⠀⠀⠀⠛⠒⠒⠒⠒⠒⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡟⠦⠔⡱⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠊
⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠤⢤⠤⠞⠁
⠀⠀⠈⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⡇⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⡼
⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⠒⡖⣢⠨⣡⣶⡔⠦⠃⢀⠀⠂⠀⡤⠀⠒⠒⣂⣀⡐⠒⠒⠂
⠀⠀⠀⠀⣀⠤⣺⠁⢠⣤⣤⣤⣶⣶⣶⣾⠀⠀⣾⣦⣴⣿⣿⡧⠀⠨⢎⠌⠜⡼⠠⣖⡻⠷⣔⠃⡎⠀⠀⡀⢰⣷⠀⠁⢻⣿⡇⠀⠿⠿⠿⠿⠗⠐
⠀⠀⠀⠸⢐⠥⠚⡁⢸⣿⣿⣛⣟⣻⣿⣿⠀⠀⠙⢿⣿⣿⡟⠁⠀⣐⣠⣂⣊⣾⡀⠢⠄⠉⣹⡶⣄⠀⢀⠁⠸⠿⠀⢠⠘⠛⣁⣤⣤⣦⣤⣄⠀⠀⡀
⠀⠀⠀⡋⠤⢂⠥⡇⠈⠛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠄⠙⠉⠀⠀⣸⡿⠿⠿⠿⢿⣧⣀⠀⢀⡟⠃⠙⠆⠘⠀⢠⣶⣦⠸⠀⣼⣿⢟⣩⣍⡝⠻⣿⣄⠁
⠀⠀⠀⡇⠊⡡⠔⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⢀⣼⡟⠁⠀⢀⡀⠀⡉⢻⣧⠈⠁⠆⠀⠀⠀⠀⠈⠻⠋⠀⢸⣿⡇⣾⣿⣿⣿⣇⢹⣿
⠀⠀⠀⡇⣈⠔⣊⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠀⠹⣿⣇⠀⠀⢘⡁⠉⠁⣸⡟⠀⠀⡃⠀⠀⠀⠀⠀⠀⢠⠀⠘⣿⣧⡹⠿⣿⠿⣣⣿⡟⡀
⠀⠀⠀⣋⡠⢊⠅⠘⠼⡄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⠀⠀⠙⢿⣷⣤⣨⣁⣠⣴⠟⠁⠀⠀⠃⠀⠀⢠⠀⠀⠀⠈⠀⠀⠈⠻⢿⣷⣶⣿⠿⠋⠀⠂
⠀⠀⠀⠧⢈⣤⢊⣠⡀⠀⠀⠀⠀⠀⢀⢼⠀⠀⡒⠄⡁⠠⠐⠀⠀⠈⢻⡻⠻⠟⣿⠀⠀⠀⠀⡆⠀⠀⠀⢄⣀⣀⣀⣠⡄⣠⡠⠠⠤⠤⠤⠤⢠⠲
⠀⠀⠀⠀⡀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠐⠊⠀⠀⠈⠢⣿⠤⠚⠀⠀⠀⠘⠒⠀⠒⠞⢀⡴⣤⠀⠘⠀⠠⡶⣦⠈⠓⠓⠚⠓⠒⠒⠂⠋⠂⠈⠈⠁
⠀⠀⠀⠋⠀⠀⠀⣀⠄⣆⣂⣖⡠⡀⠀⠀⠀⡌⠁⡀⡀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠿⠂
⠀⠀⠀⠀⠀⠠⡎⢰⠊⠉⠙⢿⣿⣮⣢⠀⠈⠀⠁⠠⣿⡴⠂⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣥⣥⣤⣤⡀⢱⡀⠀⡊⠁⠰⠈⠀⣧⢀
⠀⠀⠀⠀⣐⢃⡡⠁⠀⠀⠀⠀⠹⣿⣿⣇⠀⠀⣠⣤⣿⢀⡀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠉⠉⣉⣉⡉⠉⠙⠛⢻⣿⡇⢸⡇⠀⠀⡀⠀⠀⣰⠏
⠀⠀⠀⠀⡉⢀⠃⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠀⢸⣿⣿⣿⠁⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠿⢷⣆⠀⢸⣿⡇⢸⡇⠀⢀⠀⢠⡴⠃
⠀⠀⠀⢠⡁⣈⠀⠀⠀⠀⠀⠀⠀⡄⣿⣿⣿⠀⠈⠛⠛⠁⠀⠀⢀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠀⠸⣿⡇⢸⡇⢀⠀⠀⣸⣿⠀⠻⠃
⠀⠀⠀⠀⡁⢌⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣆⠀⠀⢠⣄⣰⡆⢀⠀⣿⣿⠀⠀⠀⠀⠐⠘⢿⣷⣿⣽⣿⡛⠋⠀⠀⣿⡇⢨⡇⢸⠀⠀⣿⣿⡆
⠀⠀⠀⠀⢅⠈⣭⣉⠛⢶⣄⠠⠐⠂⢩⣙⢻⣿⡄⠀⠈⢿⡿⠃⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⡇⢠⡇⠸⠂⠀⣿⣼⠇
⠀⠀⠀⠘⠂⠁⠈⢿⣷⣄⠻⣧⠔⢠⣿⣿⡆⣿⣷⠀⠀⠀⠀⠀⠀⢄⠻⠿⠿⠿⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣾⠿⠿⠿⢃⣸⠇⠀⠀⠄⠀⡀
⠀⠀⠀⠀⢰⣾⣿⣆⠻⣿⡄⢻⣇⣿⣿⣿⡇⣿⣿⠀⠁⠀⠀⠀⠄⠂⠀⠀⠈⠉⠁⠁⠈⠻⢿⣿⣿⣷⣾⣟⣃⣉⣀⣀⣉⣑⣒⡄⠀⠀⠸⠀⠰⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡆⢹⣷⢸⣿⣿⡿⠏⣰⣿⡟⠀⠀⠠⠀⠀⢷⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠠⠀⠐⠀⠈
⠀⠀⠀⠀⠻⣿⣿⣿⡏⢨⣟⣿⣧⣤⣴⣾⣿⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⣲⠄⣀⡲⠄⢠⠰⠤⠀⣀⣀⡀⠀⠀⠀⢀⠠⠐⣒⣒⣒⡀⠄⠁
⠀⠀⠀⠀⠄⠈⠛⠛⠴⠛⠋⠀⢀⠉⠁⠀⠈⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠘⠛⠀⠀⢹⣿⣷⠀⠁⠀⠓⣿⣿⠿⣿⣿⠿⢷⣾⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⢠⠂⠚⠀⠠⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⡆⠀⠀⠀⣿⡇⠀⣿⡏⠀⣾⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠡⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⠀⢹⣷⠀⢸⣇⣼⠏
⠀⠀⠀⠀⠀⠀⠒⣴⣶⣶⢽⣿⢿⡍⠃⣉⣶⣶⣿⡙⠻⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠀⢀⣿⠀⣸⣿⠀⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣼⣿⣽⡧⠧⣿⣿⣿⣿⡷⠀⠀⢐⢂⠀⠀⠀⠀⠀⠉⠉⠈⣈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣴⣾⣧⣀⣿⣧⣸⣿⣷⣦⡀
⠀⠀⠀⠀⢀⠀⠀⠙⠛⠛⡉⢩⣉⣠⡂⠙⠻⠿⠛⠁⢠⠀⠈⠀⠀⠀⡦⠠⠤⠀⠤⠤⠤⠀⢀⠀⠀⠀⢸⣿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠓⠒⠐⠂⠂⠒⠒⠒⠓⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣤⠶⠒⣛⣛⣙⣛⡒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⡶⢋⡤⠒⠉⢁⣀⣀⣀⡉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⡼⣱⠋⣴⢋⢔⡤⠚⠉⣉⣀⣉⠉⠳⢦⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢰⣽⠁⠀⡼⢱⠃⣼⢡⢎⠞⠀⣴⠞⠉⠁⠩⢽⣦⠈⢧⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⡏⡏⠀⢰⡇⡟⢰⠇⡾⡜⠀⣸⠃⠀⠀⣶⣶⣬⣿⠂⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⡇⠀⢸⠀⡇⢸⠀⡇⡇⠀⢿⠀⠀⠀⠻⠿⠟⠋⣠⠏⠀⠀⢀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⠀⡇⠀⢠⠀⡇⣧⠀⢸⡆⣇⠸⡄⢷⠸⡀⠘⢧⡀⠀⠀⠀⠀⠈⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⡸
⠀⠀⡀⠀⠀⠘⡄⣇⠀⢸⠀⢇⢻⠀⠘⣧⢿⡄⢳⠈⢧⡱⣄⠀⠙⠶⣤⣀⣀⣀⡠⠴⠋⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⠄
⠀⠀⡇⠀⠀⠀⣇⢸⡀⠘⡇⠘⡜⣇⠀⠹⣞⣿⡀⠳⡄⠳⣌⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠜⠁⠀⠀⢀⡜⠀⣠⠃⠀⠀⢀⡞
⠀⠀⢹⠀⠀⠀⠸⡈⣇⠀⢹⡄⠹⡜⣆⠀⠹⣯⣷⡄⠈⠲⣄⡙⠲⠤⣄⣀⣀⣀⣀⡤⠴⠚⢉⡠⠖⠁⠀⠀⠀⡠⠊⠀⡴⠁⠀⠀⢀⠞
⠀⠀⠀⢇⠀⢀⠀⢳⠹⣆⠀⠳⡀⠹⣜⢧⡀⠈⠻⣿⣶⣄⠀⠉⠛⠲⠦⠤⠤⠤⠤⠤⠒⠚⠉⠀⠀⠀⢀⡠⠊⠁⣠⠊⠀⡠⠀⣠⠊
⠀⠀⣆⠘⡆⠈⢆⠀⢣⡹⣦⠀⠹⣄⠘⢦⡹⣦⡀⠈⠻⢿⣿⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⠤⠔⣊⡥⠖⠉⢀⡤⠊⠁⣠⠞⢁⠔⠁
⠀⠀⠘⣆⠘⡄⠈⢆⠀⠳⡜⢷⣄⠈⢳⣄⠙⢮⣙⢦⣄⠀⠈⠙⠛⠻⠶⠮⠭⠭⠥⠴⠒⠚⠉⢁⣠⠴⠚⣁⣀⠴⠊⣠⠔⠁⠀⢀⠄
⠀⠀⠀⠘⣆⠘⢆⠈⠣⡀⠘⢦⡻⣶⣀⠈⠳⣄⡈⠳⢬⣛⠶⣦⣤⣀⣀⡀⠀⠀⠀⠒⢒⣛⣉⣥⣴⡶⠟⠋⣁⠴⠊⠁⠀⣠⠖⠁
⠀⠀⠀⠀⠈⢦⠨⣣⡀⠙⢆⠀⠙⢮⡻⢷⣌⡚⢿⣶⣤⣈⠙⠒⠻⠿⢿⣿⠿⠿⠿⠟⠛⠛⢛⣉⡡⠤⠒⠋⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠱⣝⢿⢦⡀⠙⢤⡀⠙⠲⣝⡫⣶⣬⣛⠿⢿⣷⣶⣤⠄⠀⢀⠀⠀⠈⠉⠉⠉⢀⣀⣤⣴⠶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣵⣽⢦⡀⠙⠢⣄⡀⠙⠲⢭⣛⡻⢷⣶⣭⣭⣛⣛⣳⠶⠶⠶⠿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⣺⢶⢄⡙⠲⢤⣐⠪⢭⣓⣒⠭⠭⣅⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣈⠑⠦⢌⣙⠓⠶⢭⣽⣛⣒⡶⠶⠶⠶⠶⠶⠶⠖⣚⡛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠻⢿⣶⣤⣌⣉⠓⠲⠦⠤⣍⣉⣉⣉⣉⠉⠉⡉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢌⣉⡛⠻⠿⢿⣶⣶⣤⣤⣤⣤⣤⣀⣀⣠⣤⡤⢤⣖⣒⡋⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⣬⣍⣉⣉⣙⣛⠛⠛⢉⣉⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⢚⣉⣉⣩⣉⣹⣖⣒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣡⠶⠓⠉⠉⠀⠀⠀⠀⠈⠉⠙⠓⠶⣝⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣤⠋⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡼⠁⠀⠀⠀⡴⣫⣴⣦⡹⡉⠒⠦⢄⣀⠀⠀⠀⠀⠸⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣺⠁⠀⠀⢀⣞⡾⣿⣿⣻⣧⠃⠀⠀⠁⠘⠹⢳⡶⢤⣀⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣿⠀⠀⠀⡞⣾⡽⣟⡿⡝⡜⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠷⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠸⡄⠀⢸⠁⠈⠙⠻⠿⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡘⢦⡏⠀⠀⠀⠀⡼⢁⠎⠔⡄⡄⡄⡀⡀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡉⣦⠆⣢⢞⡵⡣⣪⢮⢊⠜⡔⡱⢁⠃⠄⡴⠠⢀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠴⢮⣕⣍⡊⢔⣕⠵⡣⢊⠜⡵⢣⢎⠜⡰⢁⢮⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡟⠙⠓⠺⢾⣵⣪⣖⡵⣫⢎⣔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡽⠀⣀⠴⢚⣡⠵⠋⠉⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣼⡧⠊⣁⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣱⢣⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣶⠖⠒⢒⢒⠒⢏⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠈⣷⠈⢣⠹⣯⢆⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡎⠠⠀⠐⠀⠚⣧⠡⢳⢸⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠩⠀⠆⠀⠉⠰⠋⠆⡀⢳⠦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⡱⠰⠀⠐⠁⠰⠂⡰⠀⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⢠⠠⠀⠀⠂⠀⡄⠀⡄⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡄⡠⠀⠰⠀⢀⠀⠀⠄⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⢀⠁⡁⠀⠆⡀⢈⠀⠀⢀⢁⠀⣄
⠀⠀⠀⠀⠀⠀⢠⠀⠃⠀⠘⠀⠁⠀⠀⡀⠈⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⣤
⠀⠀⢀⡀⢰⢧⢀⠐⠀⠀⠘⠈⠁⠀⠀⡁⠈⠁⣶⣄⡜⠀⠈⡀⠀⠀⡶⣿⣤
⠀⠀⠈⠻⣾⠶⢀⠔⠒⠤⠴⠘⡥⣤⠤⡔⠒⡐⢻⣧⡛⠦⠤⢠⠤⣀⣷⡯⠜
⣠⢔⡉⠙⢡⣴⡦⣤⡤⠶⠦⣂⡴⠦⢳⢇⢦⠥⢶⢶⡅⡀⣤⣀⣦⣀⣀⢉⣑⡎⣍⡒⠦⠠⠄⡠⣠⢔⡤⣢⢔⠥⣊⢔⡠⢒⠠⡢⢀⢄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀⠀⢰⣦⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⣼⣿⠉⡇⠀⢸⢿⡇⠀⣀⣤⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣿⣇⣼⡿⠃⣰⡇⠀⢸⡞⠛⠀⠀⣀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢀⣾⣿⡿⠋⠀⣰⡿⠁⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⣿⠃⣠⣿⡿⠋⠀⣠⣾⡿⠁⢠⣿⣿⣿⡘⢧⣀⣤⡶⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣱⣿⣏⣼⡿⠋⠀⣠⣾⣿⡿⢁⣴⣿⣿⣿⣿⣟⢮⡻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢱⣿⣿⣿⠟⠀⣠⣾⣿⣿⡟⣡⣾⣿⣿⡿⠿⢿⣿⣷⡝⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⡇⣿⣿⣿⠃⢠⣾⣿⣿⣿⡟⣴⣿⣿⣿⡟⠀⠀⠀⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⣿⣷⣿⣿⠇⣰⣿⣿⣿⣿⡟⣼⣿⣟⠛⠻⠇⠀⠀⠀⠀⢧⢹⡇
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⡘⣿⣿⣿⣿⢠⣿⣿⣿⣿⡟⣼⣿⣿⣿⣷⡄⠀⠀⠀⠀⢠⣸⡇⠁
⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣄⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⡀⠀⠀⣠⡀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣴⣾⢿⣷⣦⣀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⠇⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⣽⣶⣾⣿⠇⢠⣿⡇⣰⡆
⠀⠀⠀⣠⣾⣿⣿⣿⣤⣙⢿⣿⣿⣶⣄⡈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣆⣠⣶⣿⣿⣷⢻⣿⠀⠀⠀⠘⠿⢿⣿⣿⣿⣇⢸⣿⣿⣿⣷⣤⡀
⠀⠀⢰⣧⣬⣉⣉⣉⣉⣉⣛⡛⢿⣿⣿⣿⣶⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣼⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⢻⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠺⠿⠿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡻⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠉⠛⠛⠛⢋⣽⣿⣿⣿⠟⣫⢟⣿⣿⣿⣿⣿⣿⡿⠝⠛⠿⢿⣿⡿⠗⠀⠀⠀⣰⣦⠀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⢁⡾⢡⣿⣿⣿⢿⡟⠉⢀⣠⣤⠶⠆⠀⠀⠀⣀⣠⣴⣾⣿⠟⣰⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣼⣧⡘⣿⣿⣿⣜⠃⢰⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣛⣩⣤⡾⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠿⠿⣿⣿⣿⣶⡶⠭⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⡙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠉⠉⠉⠛⠻⠿⠿⢿⡿⠿⠿⠯⠾⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⠤⣉⡑⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠓⠒⠲⠤⢌⡉⠒⠢⢬⣉⡒⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢴⡯⣄⡀⠀⠀⠀⠀⢀⣨⡷⠤⠤⠤⣉⡉⠒⠤⢍⣙⠒⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁⠀⠀⠀⠈⠙⠺⠷⣦⣤⣠⣿⠋⠀⠀⠀⠀⠀⠈⢓⣤⣀⠀⠙⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠭⣖⣤⣄⣀⣴⣊⠥⢖⣫⠽⠲⡇⢸
⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣥⡔⠊⠁⠀⢀⣠⠇⢸
⠀⠀⠀⢰⠉⠓⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣇⡠⠔⠚⠉⠀⠀⢸
⠀⠀⠀⠈⡦⢄⡀⠀⠉⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡅⠀⠈⠑⠢⢄⡀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠈⠉⠒⠶⠚⠉⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄⣀⣸⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⡀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣠⡾⢛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⠋⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⢲⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⢹⣾⣿⣿⣷⡄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡾⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⣼⣇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣽⣿⣿⡀
⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⡀
⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣿⣿⣿⣧⣀⠉⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣦⣀⣠⡄⠀⠀⡀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢹⣇⣠⡄⠊⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⣉⣉⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠟⠿⣿⡿⠿⠿⠟⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠛⠋⢛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠄⠀⠀⠀⠐⢻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢻⡇⠀⠀⠀⡀⣀⣤⣴⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣥⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⡇⣸⣤⣤⣶⣾⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣶⣤⣴⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡞⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⢀
⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⣦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⡶⠀⠈⠄⣠⣤⣀⣀⠀⠀⠀⠀⠀⠀⣠⣾⣆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣯⡵⣤⣴⡾⡅
⠀⠀⢀⠀⠀⠀⠀⠠⠂⣰⠿⣿⣿⣿⣿⢷⣶⣤⣤⣀⡣⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢹⡟⠀⠀⢀⠀⠀⠀⠀⢼⣧⠀⠀⠋⠀⢉⠙⠀⡅
⠀⠀⢈⠀⠀⣼⠀⡆⣼⣇⢾⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⠀⡄⠻⡇⠀⠀⠀⠀⢰⠀⣀⣿⣿⣦⠀⠀⠀⡏⣠⠈⡇
⠀⠀⠀⢳⠠⢿⣆⣴⠿⠻⣿⣿⡿⠿⣿⣿⢟⠾⠛⣿⣿⣿⣿⡄⠀⠀⠘⠀⢀⡀⠀⠀⣦⠀⠀⠀⠀⡀⡇⢘⡇⠀⠀⠀⣠⣿⣦⣿⣿⡛⠿⣷⡆⠀⠈⠀⠀⡁
⠀⠀⢀⡸⣷⣸⣿⣧⢿⣾⣿⣿⣧⣤⣽⣿⡟⠿⢿⣿⣿⡿⢿⣷⡄⠀⢨⠀⡌⠀⠀⠀⣿⠀⢀⢄⠀⢻⠁⣯⡇⠀⠀⣲⣿⡟⣅⢱⣀⣠⡀⢈⠻⠀⠈⠿⠁⠑
⠀⠀⠀⢢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣯⡤⠀⢀⡏⠉⢉⣻⣿⣶⣶⣤⣄⡀⠀⠊⠉⣿⡄⠫⠘⢨⠋⣷⣏⣧⡼⣴⣿⣟⡙⢻⣶⣾⣻⣆⣿⣾⣂⠠⠄⠀⠁
⠀⠀⠈⡈⣿⣻⣿⣿⣿⣿⣿⣿⣿⠿⢹⣿⣟⣻⣦⣶⠐⢲⣾⣿⣿⣿⣯⣿⣿⣷⣶⣶⣿⣺⣦⡶⣾⢻⣿⣿⡿⣇⣦⣼⣇⠀⢀⣹⡉⣿⣿⣿⡿⣿⠀⡀
⠀⠀⠀⠁⢿⣾⣿⣿⠛⣿⣿⣿⣿⣶⣿⣯⡽⣝⣿⣿⣄⡈⣿⣿⣿⣿⡿⢻⣿⣿⢿⣿⣿⢿⢿⡿⠿⢾⣿⣻⣿⣿⠿⣿⡿⣴⣾⡿⢃⡷⡯⣝⣯⣝⣻⣿
⠀⠀⠀⠒⣼⡿⣟⣼⣷⣿⣿⡥⠸⣏⠉⢙⣿⢾⣯⡇⢸⣈⡿⢿⣷⣶⡟⣾⣷⠷⣶⣿⣿⣾⣟⣿⣅⣸⣿⠟⠻⣾⠞⣣⣿⠟⣻⣿⠿⢷⣾⣭⣽⣾⣟⣿⠀⠈⠃
⢠⡦⠀⠀⠀⠿⣡⣾⡟⣛⣻⡵⢶⣿⣿⣿⠿⡷⡼⣻⣿⣱⣆⣲⢿⣿⣻⣿⠛⢹⣿⣿⣿⣿⣧⣼⣎⣿⣿⣽⣇⢷⡼⣯⣧⠂⠉⢘⣿⣏⣣⣿⣿⡿⢷⣟⠀⢷⠂
⠈⠡⣀⡆⠘⣰⡧⢿⠛⣡⡿⣏⢛⡟⣇⣁⡃⣠⠜⢡⡀⣽⣩⢟⣘⠻⣿⣏⢈⣈⣻⣿⣿⣿⣿⣯⣿⢿⡿⣿⣏⠻⣿⡛⣿⡄⠀⣀⣿⣿⣿⢛⡿⣧⣾⣿⠀⠨
⠀⠙⢻⡁⡼⠁⠀⣂⡼⣿⣤⢫⢿⣿⣈⡯⢷⣅⣮⢠⡿⠇⣹⣤⣾⢕⣿⣿⣳⣿⡟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣭⣹⣿⣿⣟⣻⣿⣿⣿⠻⣟⣹⡿⢿⣽⡿⡇⢸
⠀⠀⠀⢣⣠⡆⣼⢿⣷⣟⣿⠰⢚⡽⢿⡿⢦⡋⡵⠾⠂⢶⣧⢹⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣿⣼⣷⣿⡙⠿⣿⢷⣾⣷⣿⣿⢻⣧⣼⣻⣿⣫⣠⣿⡇⢠
⠀⠀⠀⠀⠧⢸⢧⣸⣿⣾⣧⣷⣤⣾⡏⣳⢼⣿⣷⣿⣿⣿⣿⣸⣿⠟⡻⠛⢿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⠒⠤⣤⡌⢶⣿⣿⣯⣻⣏⣈⣇⣹⣿⣿⣿⣿⣧⠘
⠀⠀⠀⠀⠘⣾⣾⢿⣿⣏⣉⣉⢻⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⡼⣀⣰⣿⡿⢻⣿⠿⣼⠁⠉⢊⡁⠀⠻⠿⠷⡶⣿⣿⡏⢹⣿⣧⣿⢟⡿⠏⡹⣷⣶⡇⠉⡹⠘
⠀⡞⣁⡉⡑⣾⣇⣘⣿⣿⣿⣿⣷⣽⠹⣿⣿⣿⣿⣿⣿⣿⡗⣣⡿⢻⣿⣷⣾⣿⣦⡼⣀⣸⠋⣁⣸⣠⣄⣿⠀⣨⣾⣿⣾⣿⣿⠿⣏⡁⠘⣤⣌⣷⡏⢿⡦⠖⠋
⠈⠋⠀⠈⠹⣿⣦⡉⣿⣿⣿⣿⣟⠿⣦⡼⣿⣿⣿⣿⣟⣿⣾⣿⣦⣿⣿⣟⣻⣿⣿⣿⣿⣿⢿⡟⡄⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⢍⣢⠈⠁⣿⢷⣷⣤
⠀⠀⠀⠀⠀⢸⣿⣹⣿⣿⣿⠋⠀⢠⣿⣿⢶⣽⣿⣿⣿⣿⣿⣿⢡⣟⣽⣿⣻⣏⣭⣿⡗⢻⣾⡦⢶⣿⡻⡏⢛⠋⣿⢙⠀⣿⡏⣲⣾⣿⣿⣿⢶⣟⢌⣿⠋
⠀⠀⠀⠀⠀⢸⣿⣻⡟⣿⢣⡄⣰⣿⣿⣿⣞⣿⣿⣿⣿⣯⣻⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣼⣿⣵⣾⣿⡅⢠⡊⠀⣿⣭⣿⣿⣽⡁⠩⠏⢙⠀⠀⢹⣿⣿⡗
⠀⠀⠀⠀⠀⠀⢻⡇⠹⣿⠸⡰⠟⠃⠟⠻⡟⢿⣿⡿⡿⠟⣿⡿⣿⠻⠿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡿⣷⣤⣶⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠻⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠉⠉⠀⠙⠁⠉⠉⠈⠉⠈⠉⠀⠀⠀⠈⠃⠋⠙⠙⠙⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢹⡯⠈⠀⣾⣧⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⠇⣨⡝⠂⣸⣿⡏⠀⠀⠀⢋⠁⢤⠤⠒⢂⣷
⠀⠀⠀⢀⡀⠀⠀⠐⠀⢠⠥⠄⢹⣿⣿⣿⣿⣿⡿⠀⠀⠒⣞⠀⠈⡀⢠⣿⣿⠁⠤⢤⣄⡐⢠⢯⢐⣦⣍⡿
⠀⠀⠘⡄⠀⠀⠀⢀⣂⣀⣭⣵⣀⣿⡿⠟⠋⠁⣀⣀⣀⣀⠈⠉⠒⠋⠚⠛⠁⠀⠀⠀⠀⠫⠉⠭⠜⠛⠉
⠀⠀⠀⠙⠛⡛⢛⡻⠛⠉⠀⣹⠛⠛⢋⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠀⠀⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⢠⠉⣐⢎⠐⠁⠀⠀⠀⠀⡇⢠⣿⣿⣷⣶⣶⣶⣶⣦⣤⣬⣤⣤⡀⠀⢿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⢸⢰
⠀⡼⠣⢍⡁⢱⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⡿⠟⠻⣧⣐⣲⣆⡼⣸
⡘⢠⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣨⠛⣙⣤⣶⣿⣿⣿⢟⡠⠋⠔⠁
⠇⢸⡀⠀⠀⠀⠀⢀⣀⡀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⢾⣟⠁⣿⣿⣿⣿⡟⠉
⠈⠀⢱⢤⡒⣈⣭⣤⠄⠀⢀⠗⠘⠛⢛⡿⠿⠿⢿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣶⣿⣿⣿⢤⣤
⠀⡔⠁⠀⠚⠛⠋⣁⡤⠖⠉⠛⢶⣤⣤⣀⣈⣉⠉⠙⢌⠻⣿⣿⣿⣥⠖⣺⣿⣿⣿⢿⣫⣷⣿⣿⠀⣀⡀
⠀⠈⠒⠒⠒⠛⠛⠳⠤⠄⠀⣶⠻⠿⣒⣚⠛⠛⠛⠓⣚⣓⣍⠻⣿⣿⣷⠞⠛⣫⣵⣿⣿⣿⣿⣿⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣐⡒⠫⠉⠩⠉⣙⣓⠒⠓⠾⠛⠿⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠒⠠⠬⠥⢉⣑⣒⠒⠂⠨⠥⠄⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⡠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⠠⢍⣉⠑⠒⠒⠤⠬⣭⣉⣉⣉⠀⢻⣿⣿⣿⠟⠁⣀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣈⣉⠛⠓⠶⠤⣤⣄⣀⣀⠀⠀⠉⠁⠀⢸⣿⡿⢃⡴⠊⠑⣢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⡂⠭⢉⡓⠒⠦⠤⢤⣉⣉⡙⠛⠻⠶⠶⠾⠋⠠⡏⠑⢢⡾⠋
⠀⠀⠀⠀⠀⠀⠀⣚⠁⠀⠀⠈⢉⠚⠂⠀⠨⠉⠐⠂⠀⠀⠉⣍⣩⢿⠒⠒⢂⡄⠈⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣀⣀⡉⠒⠒⠀⠀⠀⣠⢮⢟⣰⢃⢞⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣷⣶⣦⣤⣽⣷⣶⡖⣡⡾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣛⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠢⡀⠀⢠⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡑⣤⠃⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠂⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠈⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠑⠀⠀⢀⠞⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣢⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢌⢦⡟⡠⠂⠀⠀⠀⠀⠀⠀⣠⣄⠀⠘⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠰⢀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠑⡄⠀⠀⡔⢱⠀⠀⠀⠀⠀⠀⠈⢿⣧⣷⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⣄⡆⢀⠄⠀⠀⠀⢸⣷⡄⠀⢸⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡧⠂⢀⡀⠀⠀⠘⣿⣇⢀⠂⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢄⠀⠃⢀⠇⡇⠀⠀⠀⢹⣿⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢣⡈⣸⢀⠇⣾⣦⡀⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣽⣿⠈⠀⠹⣿⡇⠰⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⢻⣿⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⡀⠀⠐⡀⠀⠀⠀⠀⠈⠁⢰⠀⠀⡀⠀⠐⠼⠁⠀⠀⠀⠀⢀⠂
⠀⠈⠄⠀⠰⡄⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⢸⡄⡞⠀⠀⠀⠀⢆⢤⠀⠐⠀⠆⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⢀⣦⠘⢄⠒⠲⠆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇⡄⠀⠀⠀⠊⠁⠔⠂⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠐⠀⡀⠀⠀⠈
⠀⠀⠀⡀⠸⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⡀⠠⠀⠀⡀⠀⢀⠖⠀⠊⢀⠁
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠄⠀⠀⠀⠀⡦⠃⠀⢠⠀⡇⢀⠀⢰⡁⠀⢸⠀⡰⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠱⠠⠀⠀⠘⠁
⠀⠀⠤⠀⡤⠤⠒⡠⠔⠀⠀⠀⠀⠂⠀⡀⢀⡒⢦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠀⠤⠠⠠⣄
⠐⢀⣠⡀⠄⠀⠀⡀⠀⠀⣀⠐⠁⢀⠔⠁⠠⠉⠋⠁⠀⠀⠀⠀⠀⢀⣻⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠰⠲
⠀⠘⠛⠉⠀⠈⠀⠀⠉⠁⠙⠶⠛⠛⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠢⣄
⠀⠀⠀⠀⠀⠤⢠⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠈⠀⠀⠀⠁
⠀⡄⢠⠀⠀⠀⢐⡀⣠⠀⠀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡾⠋
⠀⠻⠶⠘⡆⠀⠀⡼⠘⠛⠾⠧⠢⠀⠠⠄⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠄⠀⠘⠂⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡛⠙⠻⣧⣄⣤⣀⠤⠂⠘⢇⠀⠀⠀⢿⣶⣦⠀⢀⣼⣾⡇⢁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠈⠁
⠀⠀⠀⠀⠀⣠⡌⡵⠇⠀⠘⣿⣿⣮⣿⡦⢻⣁⡘⢷⣦⣠⣤⠟⠃⣼⠟⠛⠛⠁⠘⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠖⠃
⠀⠀⠀⣰⣰⣿⠟⠃⠀⠀⠀⠉⠀⠈⡀⠀⠚⠋⠁⠈⢋⡫⡀⠀⠾⣝⣴⣾⠿⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠨⠉⠁⠀⠀⠀⣠⠀⠀⠀⠀⢠⣶⡄⠀⠀⣸⡁⢀⣷⣦⣶⣿⣿⣿⣿⣀⠀⢿⣷⣶⡦⠀⢰⣁⠆⠀⢀⠰⡲⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠄⢂⢀⡚⠮⢿⣦⣀⠀⠨⠉⠛⣷⣶⣿⠃⡄⢀⣽⣥⣿⣿⡿⣻⣷⣄⣼⣿⢀⣬⡻⠽⢯⠄⡀⣀⠉⠀⠤⣄⠀⠄⠀⡀
⠀⠀⠀⢀⡀⢠⠼⠮⠛⠀⢨⣧⢈⣠⢀⣀⡁⠨⢁⣍⢓⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣷⣿⠋⢀⣟⢁⣤⣙⣿⣿⣧⣖⣶⣫⢀⡐⢠
⠀⠀⠀⠀⣽⠀⡐⠒⠶⠀⠀⣿⡽⢿⣿⣿⡦⣠⣾⣀⣷⣿⡿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢿⡿⠻⣛⣻⣿⣿⣾⣷⣮⣠⢡⡏⠄⠁
⠀⠀⡄⠀⡉⠀⡇⠀⠀⠀⢀⣿⠀⢸⣿⠿⠇⢁⠛⠛⣿⣿⣧⣾⣿⣿⣿⣿⣿⣷⣽⣿⣿⡟⠀⣿⡏⠘⠙⣿⣿⣿⠛⣻⣉⠿⠦⡤⠄
⠀⢀⡇⢀⣡⢨⡇⠀⠀⠀⢸⣿⡇⣸⡏⣠⣶⣾⣤⣀⣽⣿⣿⣿⣿⡿⢿⣿⣿⣿⣻⢿⣿⣷⣶⣿⣇⢀⣴⣿⣿⣿⣾⣿⣷⣿⡟⠐⠀⣀
⠀⠈⡇⠠⡦⢸⣷⣴⣾⢳⣿⣿⣇⣸⣷⣿⣿⣧⡿⢯⣽⣿⣿⣿⣾⣽⡿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⣿⣿⣷⣾⡿⠿⠿⠋
⠀⢘⣇⣀⠀⢸⡇⠉⠁⠶⠖⢺⣇⢸⡷⣾⣛⣗⣬⣵⣚⣿⣿⣿⡿⢿⣿⡏⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⣿⣿⢋⣼⡥⠚⠲⠁
⠀⠺⣿⣿⢀⠘⡏⠀⠀⠀⢠⣻⣗⣸⣇⢉⣉⣿⠛⠉⠙⢛⣿⣿⣿⣛⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠄
⠐⢶⣝⡧⠊⠈⠍⠀⠀⠶⡈⢻⣿⣽⣇⠀⠀⣟⢀⣠⣤⠄⠩⠤⣽⣿⣿⣄⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⡘⢴⠞⠀⠀⠀⠀⢀⣘⣾⡿⠿⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⣉⣩⣽⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡄
⠀⠷⣿⠇⠀⠀⠀⠀⠀⠈⠹⠿⣶⠂⡐⠀⠀⠿⠃⠀⠀⠀⢀⣀⣶⠤⠭⢿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡟⠁⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠚⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠀⠀⠢⠲⠂⠀⠀⠀⠘⠋⠃⠀⠀⠀⠁⠤⠈⠉⠉⠈⠉⠹⠤⠀⠉⠙⠿⠦⠁⠀⠀⠀⠈⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢻⡆
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢻⣆⠻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢀⡿⠛⢿⣿⠟⠉⢻⡆
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⣍⢢⡀⢀⡤⢒⡦⣄
⠀⠀⠀⣠⢴⣶⢤⠸⡄⠸⠤⠴⠃⢸⠀⠻⣍⡼⣀⠤⣄
⠀⠀⢰⡃⠸⠬⠼⠀⡨⠗⠒⠋⠉⠙⠓⠢⢄⣸⠁⡴⢦⢳
⠀⠀⠀⠳⠤⡤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠘⠂⠋
⢀⡔⡺⢲⠀⢰⠁⠀⠀⠀⠠⡄⠀⠀⠈⠀⠀⠀⠈⢄⠔⢒⡲⡄
⠸⡀⠑⠋⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⡇⠟⣹
⠀⠉⠓⠊⠁⠘⡀⠀⢀⣀⡀⠀⠀⠀⢀⣀⡤⠞⢀⡞⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠉⠉⠉⠉⠉⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡶⠶⠶⠖⠒⠚⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⡠⡖⠉⣀⣉⣩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡘⣽⠀⠀⠀⠀⠀⠀⠀⠀⢷⠙⠒⠤⠄⠊⠁⠈⠖⠛⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡠⠊⠑⠒⠂⠄⣸⠀⠀⢀⣄⡄⠠⠆⠀⠦⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠈⢢⠤⠜⠹⠄⠀⠐⠀⠠⠀⠀⢀
⠀⠀⠀⢀⡴⠒⠻⡄⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠤⠚⠤⠆⠀⠠⠀⠀⠀⠂⠀⡀
⠀⠀⠀⢯⡠⠾⠴⢣⣧⡄⠀⣤⣄⠀⣠⠀⠰⠆⠀⠉⠘⡆⠀⠙⠲⠤⠤⣀⣀⣀⣠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠉
⠀⠀⠀⠀⠀⠀⠰⣇⠤⠤⠄⠐⡶⠒⠊⠉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠉⢳⠀⠀⢠⠖⢢⡤⠠⣡⡖⠲⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⡜⢲⡏⠉⡞⠙⢣⠚⢄⢤⣧⠗⡔⢢⠴⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⡍⠈⣻⠑⡖⠈⣠⣁⡴⠻⣄⡽⠋⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⡄⠀⠀⢀⡼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⡟⠋⠁⠀⠙⠚⣷⠀⠀⠀⠀⠙⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⢆⠀⠘⢆⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡼⠀⠀⠀⠙⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠐

⠙⢹⡏⠉⠏⠀⠀⠀⠀⠀⠙⠁⢰⣬⣿⣿⣿⣿⢿⣿⣻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⠃⠊⠉
⠀⣼⠁⠀⢀⡀⠀⠀⠀⠀⠀⢠⡼⠗⣹⣿⣿⣿⣿⣿⣻⣻⣧⣾⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣯⣹⣿⣏⠳⠰⠆
⠀⢹⡀⠀⠘⠙⢿⡶⠄⣴⢰⣯⣦⣿⣋⡼⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⠟⢶⣀⠁
⠀⠈⠁⢀⣶⣶⡏⢿⣻⢋⣉⣙⣿⣯⣿⣠⣾⣿⣿⣿⣷⣿⣇⣀⠙⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠂⣿⡍⡆
⠀⠀⠀⠙⢣⡙⠗⢽⡉⠇⠈⢻⠟⠛⣟⡿⡟⠛⠛⠛⠻⣿⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠑⠉⠈⠹⣀⠑
⢀⠀⣍⣽⡇⠀⠀⢠⣉⣓⠶⣶⡶⣟⡽⡀⢀⠀⠀⠀⣤⣧⣀⣉⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠦⣶⣾⡍⠤
⠀⠛⣿⣿⣿⣿⣧⡶⢦⠋⠹⣿⣿⣿⢀⣁⠀⠒⠂⠙⠛⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⢉⣸⡿⠉⠐⡀
⠀⠀⣇⢹⣷⣿⣿⣷⡆⢳⣿⣿⣷⣿⢟⡉⠤⣿⣖⣀⣀⣀⣬⣷⣿⣿⣿⢿⠚⣻⣿⣾⣿⣿⣯⣼⡷⡆⠀⠀⠁
⠀⡄⢽⢹⡇⢻⣿⣿⡑⠀⠑⡻⣿⣿⡟⢱⢴⢾⠿⠧⠀⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠷⠃⡅
⠀⠻⣿⢺⡧⣞⣿⣿⣅⣤⣶⣦⣿⣿⣻⣿⣻⣤⣴⣶⣶⣿⣿⣿⡿⢧⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⡀⡇
⠀⠀⢿⣿⣇⠈⠋⣿⡛⢋⣀⢀⡩⣍⠛⢟⣛⣿⣿⣥⣿⣿⣿⣋⣵⠾⠟⠛⡉⣛⣽⣿⣿⣿⣿⣽⠃
⠀⠀⠂⣼⣿⣿⢇⣾⣇⠀⠙⠻⠻⠛⠾⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠁⠲⠭⢍⠁⠌⠉⢉⣻⣭⣀⠄
⠀⠀⠀⠈⠋⠁⠀⠁⠈⠀⠄⠀⠀⠀⠀⠁⢠⣿⣿⣿⡿⠀⠀⢀⠀⠀⢀⡈⡉⠛⣏⣠⣄⣻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠙⠷⠧⢾⣯⣻⣿⡟⣷⢶⡄⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠊⠛⢋⣛⡯⣓⣹⣦⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠉⠛⠛⠻⣯⠁⠛⠃

⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢀⣞⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⣾⣾⢿⣉⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠂
⠀⠀⠀⠈⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣯⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣿⣟⡣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠓⠚⠛⠒⠊⠚⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡀
⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⢀⣠⣤⣶⣆⣂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣷⣦⡀⠀⠀⠀⡼⢻⣿⣿⣿⣿⣷⡄
⠀⠀⠈⠉⠙⠻⢿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⣿⡇⠀⠀⠀⣧⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⠠⠤⠤⠴⠟⠛⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠻⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢻⣦⣤⣀⠀⠀⠀⠀⢠⡴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠁⠀⠈⠳⣄⠀⠀⢻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⢹⡷⢦⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠉⠉⠉⢷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⣄⠀⠈⠙⠋⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠸⣧⣤⡟⠀⠀⠀⠀⢀⣸⣀⣀⣀⣸⣀⣀⣀⣸⣇⣀⣀⣸⣃⡀⠀⠀⠀⠀⠘⢷⣤⡿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⣻⠀⠀⠀⡇⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦⠀⠀⠀⠀⠀⡇⠀⠀⢼⡋⢙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠀⠀⠀⠓⠒⠒⠒⠚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠃⠀⠀⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⢀⡀⢀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣅⣨⡷⠒⠒⠒⠒⠒⠒⠒⠲⣿⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⡘⠃⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠒⠒⣿⣀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠒⠲⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠒⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣻⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⣀⣀⠀⠀⠀⠀⢿⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠸⠧⠤⢴⡏⢉⣷⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣦⣀⣀⣠⠞⠁⠀⠀⣾⠋⢹⡆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠀⠀⠁⠀⠀⠀⠀⠘⠛⠛⠁⠀⠈⠳⣄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠺⣇⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠢⠐⠀⠀⠀⠀⢀⡀⠐
⡀⠀⠀⠀⠀⣀⠀⠀⢀⠀⢨⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡤⣀⣀⣀⡀⠀⠀⢀⡐⠠⠒⠚⠉⠀⠀⠈⠉⠙⢃⠀⠿⠂⠠⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠴⠤⠤⠤⠆⠀⠈⠑⢢⣤⣠⡄⣀⡀⠀⠀⠀⠠⡀⢀⠘⣧⡀⠀⠘⠐⠉
⠀⠀⠀⠀⠤⡄⠀⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠁⠈⠛⠚⠋⠁⠀⠀⠀⠛⠉⠑
⣀⣀⠐⠃⠉⣀⣤⣠⣤⣰⣤⣴⠜⠂⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠂
⣄⠴⠿⠒⠛⠋⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢄⠤⠰⣄⠀⣤⡴⠤
⠀⠀⠀⠀⠀⣀⡄⣠⠀⠘⠃⢀⡀⠀⡀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠙⠠⠦⠄⠈⠉⠁
⠀⠀⠀⠘⢳⠀⣤⣀⠠⠄⠀⠀⠀⠀⠀⠀⠁⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠄
⠀⠀⠀⡿⠿⡶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⢠⡄
⠂⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⡠⠶⠶⣶⣦⡀⠀⢠⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠈⠁⣠⣴⠞⠋⡋⠀⠀⠀⠀⠷⣤⡀⠛⣇⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣠⠮
⠀⠉⠀⠹⣆⠀⠀⠙⠏⢠⡜⢀⡄⠀⠀⣵⡀⢙⣿⣆⠸⠋⠓⣤⢻⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠌⠚
⡄⢠⣤⠈⣿⣄⠐⣷⣦⣼⠀⣼⡇⠀⠀⠻⣖⠄⢚⣿⠡⡎⠇⢠⣾⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠀⠈⠁⠉⠁⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠄⠤⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⢠⠀⠋⢐⡀⠁⠀⠀⠀⠘⡄⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠅⠀⡀⠀⠙⣄⠲⡆⠀⠁⠀⠀⠀⠆⠀⠀⠇⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢀⠀⠀⠀⠁⠀⠀⠈⠳⢄⣀⣀⣀⠠⠊⠀⣀⠄⠸⡀⠀⢛
⠀⠀⠀⠀⠀⢔⠢⠀⢒⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠀⠀⢬⡉⠁⠀⠈⠉⠈⡄⠀⢧⠀⢸⡄
⠀⠀⠀⠀⢀⠈⠠⠐⠀⠀⠀⢠⣐⠀⠀⠒⠀⠀⠀⠀⠀⡀⠠⡈⠁⠑⠀⡀⠀⠸⢻⡀⠸⡄⠈⡇
⠀⠀⠀⠀⠀⠘⠀⠀⢡⡠⠂⠰⠇⠔⠁⠀⠤⠀⢀⠞⠀⠁⠀⠹⡟⠀⠀⠁⢠⡏⠀⢧⠀⢳⡀⢹⡀
⠀⠀⠀⠐⠀⠑⠀⠀⣄⡃⡆⢠⡤⠃⠴⠂⠀⠀⣊⠉⠀⠀⠀⠀⠑⠌⠄⠀⠊⠐⠀⠈⢂⠀⢷⡀⢇
⠀⠀⠀⠀⠐⠁⠀⢦⠈⠁⠛⠀⠀⠀⢠⠠⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠘⠄⠀⢀⠀⢠⠘⡤⢄⠳⣌⢆
⠀⠀⠀⠀⢀⠔⠀⠈⠥⠴⠤⠠⠤⠤⡄⠀⠢⡃⠀⠀⠈⠢⡀⡠⠀⠀⠀⠘⠉⠸⠀⠈⠉⢡⠀⠉⠀⠈
⠀⠀⠀⠀⠘⠀⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠁
⠀⠀⠀⣠⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠠⠤⠀⠀⠀⠒⢒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢱⣀⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣬⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿