Grilling Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠛⢛⣛⣛⣉⣉⡛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢿⣿⣿⣿⣶⡛⠀⠈⢳⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠤⣿⢻⣽⣿⣿⣷⣟⠀⠀⢸⣏⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⣠⠴⣲⡿⠋⠉⣠⠖⠋⠀⢀⣿⢻⣽⣿⣿⣿⣟⠀⠀⣸⣯⠟⠉⣩⠟⠉⠉⣹⠷⠦⣄⡀
⠀⠀⠀⠀⣾⣡⠞⠁⢀⡴⠛⠁⢀⣠⠞⢉⡇⠻⠿⠿⠿⠷⢟⣠⡾⠛⠁⣠⠞⠁⠀⣠⠞⠁⢀⡴⠋⣷
⠀⣀⣀⣀⣿⠓⠦⣶⣋⡀⣀⡴⠋⠀⣠⠾⠛⠛⠋⠉⠉⢉⡽⠋⠀⣠⠞⠁⠀⢀⡼⠃⢀⣴⣯⠴⠚⣿⣀⣀⣀⡀
⡞⢩⡤⠤⢤⣄⠀⠀⠈⠉⠙⠛⠒⠺⠷⠶⠤⠤⠤⠤⠴⠯⠤⠴⠾⠥⠶⠶⠖⠛⠛⠋⠉⠁⠀⠀⣠⣤⠤⠤⣬⠙⡆
⠹⣌⡛⣋⣩⣭⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣭⣉⡓⣋⡼⠃
⠀⠈⠉⠉⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⣉⣓⣶⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣚⣋⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢩⡏⠀⠀⠀⠉⠻⡯⠭⣿⠛⠁⠀⠀⠹⡏⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠀⠀⠀⠀⠀⠀⡇⠀⣏⠀⠀⠀⠀⠀⢷⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⡇⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⢸⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢷⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠸⡆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⡷⠦⢤⣀⡀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⢀⣀⣠⠴⠾⡆⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠶⢤⣄⣀⠉⠙⠒⠶⢤⣧⡀⣿⠴⠖⠛⠉⢁⣀⣤⠴⢷⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⠀⠀⠀⠉⠙⠓⠶⢤⣄⣀⡽⣿⡤⠴⠖⠋⠉⠁⠀⠀⠸⡆⠘⣇
⠀⠀⠀⠀⠀⢠⣾⡿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⢿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣄⣠⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⣨⣿⠇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⡼⠃
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣴⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠻⠿⣿⣿⣿⣯⠉⠽⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢊⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠍⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⡉⠉⢻⡏⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡁⠀⠀⠀⠀⡛⠻⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⢤⣤⣤⣤⣤⣤⣤⣴⡶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠛⢦⠤⠤⠤⢷⠖⠛⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⠿⣿⣿⣿⣿⣿⣿⣾⣷⣦⣽⣭⣹⣛⣛⠿⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⣾⣳⣦⠀⠀⢸⠄⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣦⢭⣭⣛⣛⠻⠿⡿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠿⣿⠿⠁⠀⠘⣗⠀⠀⢀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢟⣻⣿⣴⣿⣯⣭⣿⣿⣷⣾⣭⣭⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢠⠶⠾⣧⡀⠀⠀⠠⠀⠀⠀⠠⢿⠄⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⠋⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠸⣄⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠺⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠟⠻⠛⠿⣿⡿⠛⣏
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠤⠂⠀⠀⢸⠇⣤⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⣇⠀⠀⠀⠀⠀⠈⣎⠙⢛⠛⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠽⠷⣤⣄⣀⣀⣀⣨⡤⠾⠗⠻⡅
⠀⠀⠀⠀⢠⡿⢻⡇⠀⠀⠀⠀⢀⣠⡤⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣠⣤⣾⣄⠀⠀⠀⠸⡆⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⣿⣹⠁⠀⠀⠀⣴⣏⣷⣿⣿⣿⣿⣿⣾⣹⢦⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣿⣯⣿⣿⣧⠀⠀⢀⣷⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⢹⡏⠀⢤⠤⡼⣷⢹⠓⠣⣷⠛⠹⡩⠚⢝⣾⢷⠤⣤⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠚⡿⣿⠿⠃⠂⡀⠈⢽⣃⠔⣀⠈⠀⢳⡀
⠀⠀⠀⠀⠀⢸⡇⠀⢸⢰⡇⢿⢸⠀⡃⣿⠘⢣⡇⣿⢸⢺⢸⡇⡇⠀⠀⣿⠀⠀⠀⠀⠀⣠⡶⢿⣿⣤⠈⠔⠂⠐⡡⠈⠀⠀⡐⠂⠼⣇⠀⢁⠐⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠸⠄⣧⣿⢸⡀⢃⣿⢘⣣⣇⠋⣘⣸⣸⠇⠧⠀⠀⣿⠀⠀⠀⠀⠐⣏⠀⠄⠘⣷⡇⠐⠁⠀⠠⠈⣀⠀⡀⠂⠠⢯⡴⠂
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢯⡦⣶⣶⣶⣷⣾⣯⣵⢛⣸⠏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠶⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠛⠯⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⣤⣤⣄⣀⣀⣀⠀⠤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⢿⣿⡷⠀⠀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢋⡀⠤⠖⠋⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡄⠀⠀⠀⠀⣀⠔⠊⡡⠒⠉⠀⠀⠀⠀⠀⠀⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⢿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢀⠀⣠⣴⣾⣿⣿⠟⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣠⠖⠁⠈⠛⠿⠿⠓⠀⠀⠀⣰⡏⠐⠻⠿⠟⠋⠀⠑⢦⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⣾⡀⢠⣿⡇⠀⠀⣠⠀⠀⠀⠀⠀⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢹⠇⣼⣿⣇⠀⠰⣿⠀⢠⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⡄⢠⣷⡿⠀⠀⠀⣿⣿⣿⡄⠀⠃⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡄
⠀⠀⠀⠀⠀⢀⣴⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣿⣿⡇⢸⣆⢰⣿⣿⣿⣷⢠⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣦⣄
⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣼⡇⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦
⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠿⢿⣿⡟⢻⣿⠟⣛⡛⣿⢛⣛⠻⡟⣛⣛⣿⢛⡛⢿⠻⣿⡿⡿⠿⠿⣷⣶
⠀⠀⠀⠀⢤⣴⣶⣶⣶⣶⣶⣿⣿⠀⣛⠠⣿⠱⠆⢿⡇⣭⠡⣿⢨⣭⠘⡇⢨⣽⡇⢺⣿⢾⠐⣿⡇⡇⢘⣻⣿⣿⣶⣶⣶⣶⣶⣶⡤⠄
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣀⣛⣥⣧⣿⣿⣼⣷⣿⣷⣿⣶⣶⣾⣷⣶⣶⣿⣦⣵⣾⣦⣭⣼⣧⣘⣛⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠴⠿⠿⠿⠟⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠿⠿⠿⠦
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣤⡴⢖⣚⣛⣉⣉⣉⣙⡓⠶⢦⣄⡀
⠀⠀⠀⠀⠀⣠⣴⠟⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⣴⣟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡟
⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⠟⣽⠆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢋
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⣫⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⣠⡾⠁
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⣻⣼⠟⠁⢀⣴⣶⣶⣶⣦⣄⠀⢰⡏⢠⣾⡿⠁⣠⣴⣶⣿⣿⣿⣷⣦
⢀⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡿⣫⣾⠟⠁⠀⢀⣿⣿⡿⠛⠛⠿⣿⣷⡈⢠⡿⠋⣠⣾⣿⣿⣿⠟⠉⠀⠀⠉
⣼⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣵⡿⠋⠁⠀⠀⠀⠈⢿⣿⣧⣠⠀⠀⢹⣿⡇⠘⢁⣾⣿⣿⣿⠟⠁
⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣭⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣸⡟⠀⣠⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣫⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣀⣾⣿⣿⣿⠟⠁⣀⣤⡴⠶⣒⣀⣀
⢹⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢟⣯⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣠⣴⣾⣿⡿⠿⠋⣡⣴⣿⠟⠉⠐⠊⠉⠉⢙⣿
⠈⢿⣿⣿⡿⣟⣋⣭⡶⢟⣻⣵⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠩⢭⣤⣴⡾⠿⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠈⡿⠿⠿⣛⣯⡵⠞⠛⣉⣀⣠⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠉⠛⢋⡩⣴⣶⡿⠿⠛⠛⣩⣟⠛⢁⣤⠖⠀⣠⡞⠀⢠⡾⠠⠀⣶⠀⠈⢯⡉⠉⢿⡟⠿⠿⢷⣦⣤⡀
⠀⠀⠀⠀⣏⢼⡋⠀⠀⣠⡴⠛⠁⢀⡴⠟⠁⠀⣴⠏⠀⠀⣼⠇⠀⠀⣿⡅⠀⠈⢷⡀⠀⠙⣦⡀⠀⠈⠙⢯⡂
⠀⠀⠀⠀⠹⣾⣿⣴⣿⣁⡀⣠⡼⠋⠁⠀⢠⡾⠋⠀⠀⢰⣿⠀⠀⠀⢹⡇⠀⠀⠘⣷⡄⠀⠈⢻⣦⣠⣴⣿⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣶⣦⣤⣾⣧⣤⣤⣤⣾⣷⣶⣶⣶⣾⣿⣿⣿⣿⡿⠟⢛⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⣭⣽⣛⣛⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣾⡇⢸⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡾
⠀⠀⠀⠀⠀⠀⠹⣎⢿⣿⣿⡉⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠻⣿⣿⣦⣬⣭⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠈⠉⠉⢹⣿⡏⠉⠉⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣤⣶⣶⣶⣿⣿⣇⣿⣿⣸⣿⣿⣿⣷⣶⣶⣦⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠈⠻⠿⣿⣿⣛⣿⣿⢻⣛⣻⣛⠿⠍⠁⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡅

⠀⠀⠀⠀⠀⢀⡀⢠⠄⣠⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣼⣱⣿⣆⠫⣆⠀⠀⠀⣰⣿⣿⢦⡀⠀⠀⢀⡾⢡⡀⠀⠀⠀⣠⣾⠆⠀⠀⠀⣠⡾⡀
⠀⠀⠀⠀⢿⡿⣮⡈⡼⢢⢟⡆⠀⠰⣏⡻⣴⡾⡷⠀⠀⢼⠿⣆⣿⠀⠀⠘⢿⣏⣠⠇⠀⠀⢿⡱⢹⠆
⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠖⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⢀
⠀⠀⠀⠀⢀⡖⣩⣦⣭⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢷⢀⣹⣆⢾⡄
⠀⠀⠀⠀⠈⢰⣿⣿⣿⣗⠘⠀⠀⠀⠀⠀⢀⣠⣾⣟⣿⣦⣀⠀⠀⠀⠀⠀⠀⢠⣿⢈⡸⢿⢻⢸⣷⣠
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⣰⣿⡿⣿⡿⡿⣿⣿⣷⠀⠀⢀⣀⢀⣟⣻⣎⣻⣆⣾⣾⣹⣟⡁⢀⡀
⠀⠀⠀⠐⣒⠀⠀⠀⠀⠉⣙⣓⠀⠀⠀⣿⣯⡇⢺⡇⢸⢱⢸⣿⡇⠀⠀⠘⠻⡍⢿⢹⣿⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢻⣷⣧⣵⣯⣜⠎⡰⣿⠃⠀⠀⠀⠀⠙⢮⣣⣻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠤⠠⠩⠉⠉⡀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣏⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢈⠟⠉⠉⠉⠛⡃
⠀⠀⠀⠀⠀⢰⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⡜⠒⠒⠲⠶⠒⢳
⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠠⢤⣿⢾⣿⣦⠄⠀⠀⠀⠀⠀⠀⢠⠄⠐⠀⠠⡀⠀⠀⠀⠀⢰⡉⢹⡆⠄⠀⠀⠀⠄⠀⠸⣄
⠀⠀⠀⠀⠀⠈⡿⠿⢿⠁⠀⠀⠀⠀⠀⣠⠔⣡⣶⡶⢶⣦⡈⠢⡀⠀⠀⠀⠉⠉⠀⠀⢀⣀⠀⠀⠀⠀⠉
⠀⠀⠀⡆⠀⠰⢱⣿⡍⠀⠀⢰⠀⠀⢘⠡⣴⣿⣿⣤⣼⣿⣿⣲⠉⠂⠀⠀⠀⠀⠠⡀⢸⣭⢀⣤⡀
⠀⠀⠀⠣⢠⣃⡈⠀⣉⣻⡇⠜⠀⠀⠲⣄⠙⣿⡏⠏⠻⣻⡿⢚⡴⠂⠀⢀⡀⠀⣀⣙⢦⣥⢚⣋⠀⠀⣀
⠀⠀⠀⢰⣰⣮⡅⠄⢸⣅⣦⡦⠀⠀⠀⠈⠀⣈⠛⠛⠛⢋⡠⠊⠀⠀⠀⠈⠉⠉⠉⣩⠞⡟⢺⣍⠉⠉⠉
⠀⠀⠀⠈⠀⠒⠶⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠐⠟⠁⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠇⠀⠙⠃

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

⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢾⢉⣤⣴⣷⣶⣶⣶⣤⣀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣭⣽⣥⣄⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⢄⣉⣠⠆⠀⠀⠀⠀⠀⠀⠀⣾⣧⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⢸⣿⣿⣿⠿⠛⢋⡅⢠⡆⢸⣾⠃⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⢚⣋⣉⣀⣀⢶⣾⢿⠿⡷⢾⡿⢿⣿⣂⠀⠀⠀⠀⠀⠀⠀⢠⣤⡴⢖⡞⣻⠏⡹⠋⡽⢋⡿⢛⡶⢦⣤⡆
⠀⠀⠀⠀⠈⢿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠋⠹⣿⣒⣷⠾⠷⠾⠴⠾⢶⣟⣺⡏⠙⠃
⠀⠀⠀⠀⠀⠈⢷⣌⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠿⠿⠿⠿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠙⠛⢛⡛⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠸⡇⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣷⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⣸⡿⠉⠉⠉⠉⠉⠉⠉⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⡶⢲⢞⡿⢻⣿⡳⢲⠖⣾⣧⡀
⠀⠀⠀⠀⠀⣿⣛⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⠶⠧⢮⣴⣥⣿⡴⠧⠾⠖⢿⡁
⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⣟⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠛⠉⠀⠀⠀⠀⢀⣀⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢀⣀⣀⠤⠔⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡭⢝⣛⣳⣬⣽⣿⣿⣯⣴⣾⣿⣯⣽⣒⣲⡦⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣫⠽⣗⣾⡿⣿⣷⣾⡿⣿⣷⣺⡽⣿⣷⣺⡿⠽⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣶⣿⣭⣿⣿⣯⣭⣵⣿⣯⣭⣽⣶⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠟⠻⠿⣿⠿⠟⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⣤⣿⣤⣤⣤⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠈⠙⠓⠦⣤⣿⡤⠶⠛⠉⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇

⠀⠀⠀⠀⠀⠀⠠⠾⣿⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠉⢢⡀
⠀⠀⠀⠀⠀⣀⣠⣴⡿⢠⣿⠇⠀⠠⢶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠀⠀⢀⠇⠀⠠⢶⠒⠦⢄
⠀⠀⠀⣠⣿⣿⡏⠀⠐⢿⣯⣤⣤⣴⠿⢹⡿⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠋⠀⠀⠀⠰⠯⠤⠤⠐⠉⠀⠀⠘⠄⡀
⢀⣠⣾⣿⣦⣿⣶⣶⣶⣷⡟⣳⣾⣿⣶⣶⣰⣾⣮⠇⣶⣠⣠⡀⢀⣀⢀⠋⠀⠀⠠⠒⠂⠐⠐⠠⠄⠀⠀⠀⠐⠀⢀⠅⠀⣀⣀
⠀⠈⡟⠙⠻⣿⡿⠟⠛⠛⠃⠛⠻⠿⠿⠿⠿⣿⠏⠉⣿⠳⠴⠃⠀⠈⠎⠐⠒⠀⠀⠂⠂⠀⠂⠐⠀⠠⠀⠠⠤⡀⠈⠈⠞⠁⠤
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⣿⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣶⣤⡀⣿⠀⠀⠀⠀⠀⢐⠀⠀⡤⠆⠨⡁⠀⠀⠐⣂⡉⣭⠙⠃⠐⠠⡀⠅
⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⡿⢿⣿⣷⠏⠀⠀⠀⠀⠀⠐⢦⡈⠤⣄⠀⢅⠤⠀⠀⠀⠊⡀⢂⠀⠀⠆⢰
⠀⠀⠈⠙⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠌⠑⠪⠄⠀⣀⣁⣀⣀⣀⡁⠤⠥⠒⡝⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⢀⠄⠀⠀⠀⡀⠀⡠⠊
⠀⠀⠀⠀⠀⢠⣯⡽⠿⢿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡙⠲⢼⢨⣤⡤⢴⣶⡋
⠀⠀⠀⠀⠀⣾⠋⠛⠛⠾⣿⢿⠟⠛⠋⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠐⠀⠸⢸⠭⠅⠘⠀⠁⣆
⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣿⢸⡆⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⣼⠁⠀⠀⠀⢸⣿⢸⡇⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⢆
⠀⠀⠀⢰⡏⠀⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⡆
⠀⠄⠠⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁
⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣀⣀⣀⣀⣠⣤⣤⠴⠶⠶⠗⠛⠛⠛⠛⠛⠉⠛⠉⢙⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⡿⢛⡉⠩⢥⠀⠀⠉⠁⠐⠂⠀⠒⣂⠈⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⡠⠊⠄⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢃⣀⡀⠠⠀⠐⠀⠉⠁⢀⣀⠀⠤⠀⠀⠀⠀⠀⠉⠭⠁⠀⣴⣿⣿⣿⠀⠀⠀⠀⢀⠔⠀⢠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠂⠠⠤⠀⠀⠀⠀⠈⠉⠀⢀⡀⠠⠄⠀⠀⠂⠠⠤⠄⠀⠀⠐⢩⣭⣿⠀⠀⠀⡰⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡶⠀⣉⡁⠀⠀⠉⠉⠄⠀⠀⠀⢀⣐⠐⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡟⠀⠀⣀⠠⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢟⣋⢉⡀⠀⠀⠤⠄⠀⠀⠤⣀⠡⠤⠄⠀⠈⠉⠐⠒⠒⠀⠀⢻⣿⣿⡇⢀⠄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠉⠐⠂⣀⠀⠀⠉⠀⠀⠐⠀⠀⠀⣉⣀⣠⣔⣂⠤⠒⠤⢢⣰⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣍⡮⠉⣀⣀⠀⠀⠀⣐⣂⣐⣤⣄⠔⠲⢭⣶⣬⡳⠯⣔⠊⠀⣩⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣴⣤⣾⣷⣶⣿⣿⡿⠿⠛⠛⠛⠛⠲⢦⣼⡄⠀⢀⢀⣿⣤⣾⡿⢃⣿⣿⣿⣷⣖⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣶⢾⣿⣿⣿⣭⣿⢶⡶⣿⣿⡿⣽⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣟⢟⣿⣿⡿⣿⣷⣶⠶⡶⣖⣺⣺⣭⣿⣷⣾⣿⣟⢟⡿⣻⣟⣿⡷⣜⡫⠿⠛⠉⠑⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣷⣝⡻⣽⡶⢶⠾⣿⢿⡿⣿⢿⣿⣿⣿⢿⠿⢏⣛⠿⠍⠚⠊⠁⠀⠀⢀⡶⣶⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡇⠇⡏⣿⣮⡙⢾⡑⠮⢷⣙⠯⠏⠝⠉⠐⠃⠈⠀⠀⣀⣶⣶⠀⠘⠃⠀⠘⢷⡿⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣦⢇⡌⡏⡟⣾⡟⠉⣀⣄⣀⠀⣴⣻⡆⠀⠘⠃⠀⠘⠿⠏⢀⠀⠄⠄⢴⣖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠓⠧⣀⢿⡇⠀⣿⣼⣿⠀⠙⠛⠁⢀⣀⠠⠄⢲⣶⠁⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡽⠀⠀⠀⠀⠈⠹⡇⠀⣈⡙⠥⠐⠂⠀⠈⠀⠀⠀⠀⣀⣿⣆⡀⠀⠀⠀⠀⢻⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢀⣀⡤⠤⠖⠚⠋⠉⢹⣿⠐⠍⠀⢀⠠⢸⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⠀⠀⠀⠀⠀⢀⣀⡄⣷⠶⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢈⠐⢗⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣧⠴⠖⠒⠋⠉⠉⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣃⠀⠀⠀⠀⠀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠭⣒⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠒⠈⠁⠈⣿⡄⠀⠀⠀⠀⠀⠘⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡸⠀⠀⠀⠉⠒⠤⣁⠒⠄⣴⣠⡠⠄⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⣿⣿⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⢿⣿⣦⠀⢻⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠈⢻⣿⣇⠀⠹⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣿⣿⠀⢀⣾⡄⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⢹⣿⡆⠀⠈⣿⣿⠀⠀⠀⣾⡇⠀⣿⣿⠀⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠹⣿⣧⠈⢿⣿⣆⠀⠀⠀⠀⠀⠀⣿⡟⠀⠈⣿⡇⠀⠀⢹⣿⠀⠀⠘⣿⣧⡀⠹⣿⣧⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠘⣿⣇⠀⠙⣿⣦⠀⠀⠀⠀⢰⣿⠇⠀⢰⣿⠇⠀⠀⣾⡿⠀⠀⠀⠈⠻⣿⡄⠙⣿⣇⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⢸⣿⠀⠀⢹⣿⠀⠀⠀⢠⡿⠃⠀⠀⣾⡏⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⣹⡇⠀⢸⣿⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣸⡏⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣼⡟⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢠⣿⠁⠀⠼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⢠⡿⠁⠀⠼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⠶⠶⠟⠛⠛⠛⠉⠉⠉⣉⣉⣉⣁⣁⡠⣤⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⠙⠛⠛⠻⠷⢶⣦⣤⣀⡀
⠀⠀⠀⠀⢀⣠⣤⡶⠟⠛⠉⣉⣁⡤⣴⠶⢂⣶⣾⡿⠛⠻⣿⡿⠛⢉⣩⣿⠟⠁⣉⡼⠛⢿⡉⢉⣭⡯⠟⠉⢀⣿⡾⠷⠢⣤⣤⣌⡉⠛⠿⣦⣄⡀
⠀⠀⣠⡶⠟⢋⣡⠤⠐⠛⣡⣽⡯⠛⢁⣴⡿⠛⠻⣷⣄⢀⡼⣠⣴⠿⢋⣡⡴⠛⢿⣦⡀⢠⣿⠿⠋⠀⣠⡴⠛⠿⣧⡄⢀⡼⠛⠉⡹⢿⣶⡀⠉⠻⣦
⢠⣾⠋⢀⡴⠋⠁⣀⣤⣾⣽⣧⣴⡞⠉⠛⣷⠆⠀⢀⣤⣿⣿⣿⣿⠛⠛⢻⣷⠀⠀⣉⣼⣟⣡⣴⡚⠛⢻⣷⠄⠀⢈⡴⠋⢀⣤⣾⠟⠁⠀⢱⠀⠀⢸
⣾⠇⠀⠸⣄⣠⣾⠿⠋⣹⠁⠀⠹⡿⠂⠀⣁⣤⣾⣿⢿⠟⠁⠈⢻⠷⠀⠈⣁⣤⡾⠋⠝⠁⠉⢹⡿⠀⠀⣁⡤⠞⠉⢀⣴⡿⠛⠁⢀⣀⠴⠃⢀⣠⣾
⣿⣆⡀⠀⠙⠛⠧⣤⣴⠿⢄⣀⣀⠤⠞⢋⣤⡿⠛⠁⠈⣦⣀⣀⣀⠤⣴⣿⠟⠁⢀⣴⡷⣄⣀⣀⣤⡶⠟⠁⠀⢀⣴⣿⡟⠤⠖⠛⠉⣀⣤⣶⣿⣿⣿
⢻⣿⣿⣷⣦⣄⣀⡈⠈⠙⠛⠛⠣⠴⡾⠿⠃⣀⣀⣤⣾⣋⣁⣀⣤⣾⣏⣁⣠⣾⣟⣁⣀⡀⣴⠿⠿⠤⠶⠐⠚⠛⠋⣁⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⣀⣀⣀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠈⠉⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⠋⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡏⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢠⣿⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⣿⡄⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠛⢻⣿⠛⢻⡿⠻⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢀⣿⠁⣀⣀⣤⡤⠶⠛⠛⠉⠉⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠉⠙⠛⠶⣤⣄⣀⠀⠀⠸⣷⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣿⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠦⣿⡆⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠇⢀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⠿⢿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⢰⡿⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⣷
⠀⠀⠀⠀⠀⠀⣾⠇⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢻⣇
⠀⠀⠀⠀⠀⢰⡟⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠈⣿⡄
⠀⠀⠀⠀⠀⣾⠃⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠸⣧
⠀⠀⠀⠀⢰⡟⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻⣆
⠀⠀⠀⠀⣿⡃⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⣿⡀
⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⡿

⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣤⡸⠋⠿⢳⣾⣀⡄⠀⠀⠀⠀⠀⣀⠸⣿⣆⢲⡄⠀⠀⠀⠀⢄⠙⢦⡀⠀⠀⠀⢰⣿⣿⣿⡿⠃
⠀⠀⠀⠀⢠⡴⢼⣦⣤⣤⣤⣤⣿⣤⣤⠀⠀⠀⠀⢿⣦⣿⣿⣿⡿⠀⠀⠀⠀⠈⠳⠞⢇⠀⠀⢀⡼⠿⠿⠋
⠀⠀⠀⠉⠈⣧⢸⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁⢠⣆⣸⣿⣿⡟⢹⣧⣼⡀⠀⠀⠀⠀⠀⠀⢑⣴⠋
⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⣿⡟⠻⠀⠁⠀⠹⡿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣸⠁⠉⠉⠉⠁⢱⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⡢
⠀⠀⠀⠀⣠⣮⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⢀⣼⣄⠀⢀⣾⡀
⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣰⣿⠟⠋⠁⠈⠙⢻⡿⣄⠀⠀⠀⠀⢸⣿⣿⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡆⢰⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⡄⢴⠀⠀⢸⣿⣿⠀⣿⣿⣿
⠀⠀⢀⠞⣤⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠗⠸⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⠃⠚⠀⠀⢸⣿⣿⠀⣿⣿⣿
⠀⠀⠛⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡦⣄⣀⣀⣠⢼⡿⠋⠀⠀⠀⠀⠘⠛⠛⠀⠛⠛⠛
⠀⠀⠸⡏⣿⣿⡿⠋⠀⢂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠋
⠀⠀⠀⢻⣌⡟⠁⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⣀⣻⡶⢲⣶⣶⣶⣶⣶⣶⣶⣶⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣶⣶⣿⣿⡿⠏
⠀⠀⠀⠀⠀⢷⡸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣟⡏⠉⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣠⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠉⠁⠀⠸⡄⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠧⠤⠤⠤⠤⠤⠤⢷⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡷⠶⠤⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⣴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠅⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⡀⠞⠁⡄⠀⠀⣀⣤⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃⠀⠀⠈⠈⢹⠟⠉⠀⠉⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠠⠞⠶⣆⠀⠀⠁⢛⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠈⠁⠒⠒⠒⠂⢤⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣴⣏⣽⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣧⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠢⢄⡀⠀⠀⠄⠂⠐⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣸⡧⡀⠀⠠⠐⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡥⠂⠒⠣⠤⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡿⠀⠙⢄⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠘⠿⣤⢸⠀⠀⠀⠀⠀⠠⡀⠀⢸⠁⢸⠁⠀⠀⠀⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⠄⠀⠀⠀⠀⡇⢸⡆⡠⠊⠀⠀⠀⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⢠⡤⠤⢤⣤⢀⣀⢀⣀⢀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢚⡇⠁⠀⠀⡠⠊
⠀⠀⣠⠴⠚⡃⢀⡹⣿⣿⡿⣿⣷⠆⠶⠶⠶⠶⠾⠟⣃⡉⠁⠘⠳⢤⡀⠀⠀⠀⠀⠀⠀⣏⡯⡀⣀⠴⠋
⠀⠘⣷⣔⣿⣿⣿⡿⠿⣀⣀⣤⣴⣴⣶⠷⠀⠀⠾⣿⣿⣿⠿⠆⢀⡴⡇⠀⠀⠀⠀⠀⢸⡿⣄⣸⣿⡀
⠀⠀⠃⠈⠉⠙⠓⠲⠶⠾⠭⠭⠥⠤⣤⡤⠤⠤⠦⠶⣒⣒⣾⣭⣷⢆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡇
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⣼⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡇
⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣸⣿⣿⣿⢟⣼⡿⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠈⠲⣄⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢟⡿⠋⠸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠓⠯⢿⣿⣶⣶⣶⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠉⠉⣻⡏⠉⠀⢻⡆⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⠤⠿⠇⠀⢀⣀⣿⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢰⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⢿⣿⣷⣆⣀⣀⣀⣬⣿⣷⠀⠀⠀⠈⠙⣾⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠑⠸⢿⣿⣿⣿⣿⠿⠛⠁⠸⣇⠀⠀⠀⠀⣿⠹⠿⢿⣷⣶⣦⣤⣤⠇⠀⢇
⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⢹⠀⠀⠀⠹⡏⠉⠉⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠈⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣧⣴⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢦⡴⠚⠉⠉⢙⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣴⠋⠀⠀⠀⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡇⠀⠀⢀⠸⠅⠈⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣠⡎⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣴⣃⣀⣀⣤⠀⣀⠀⡀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣻⣯⣯⣿⡅⠈⢠⣀⢀⣌⡞⡿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⣿⣿⣿⣟⢛⣡⣻⢆⢗⡼⣫⣿⣿⣷⣍⡷⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣻⠿⠷⣿⣿⣟⣵⣿⣿⣿⡿⠽⢛⣯⣞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣗⠖⠀⠰⠖⠒⠒⠒⠀⠛⠙⣿⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣄⣀⠁⠀⠀⠀⢀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⢿⣿⣿⣷⡾⣾⣿⡿⣏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⣿⡟⠛⢁⣻⡿⠤⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠙⠧⣤⣀⡽⠃⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢠⢖⣿⡓⢤⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⠃⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢀⡤⠖⠋⠛⠿⠛⠙⠓⠤⣀⡀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣦⣽⣷⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⢀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣾⡿⣼⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣛⣛⣟⣹⣛⣟⢛⡿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⣔⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠿⣛⣫⣭⣭⣥⣭⣭⣭⣛⠻⢿⣿⣿⣷⣴⠞⠋⢺⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣫⣴⠿⠛⣩⡵⠛⠁⣠⠘⠛⠙⣻⣶⣍⠻⣿⣿⣷⣄⠀⠙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢋⣾⠟⣁⣴⠟⠉⣀⣴⠞⢡⢇⣴⠞⢋⢄⣽⢷⣌⢻⣿⣿⣦⡴⠟⠛⠃
⠀⠀⠀⠀⠀⢀⣿⣿⡿⢡⣿⣥⠞⠋⢀⣤⠾⠋⢀⣄⣼⠈⢀⡀⡞⠈⢁⣠⣿⣧⢻⣿⣿⡆
⠀⠀⠀⠀⠀⣸⣿⣿⢃⣿⠋⣱⢯⣙⡒⢧⡠⡶⠛⠁⡙⣷⣶⠞⣡⡴⠛⠁⣠⣿⡆⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⢸⣯⢤⣿⣓⠲⢭⣿⣷⡀⠴⠟⢡⣏⣴⠞⠉⢀⣴⠞⠋⢸⣿⢸⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⢸⣏⢈⡓⠦⢭⣽⠿⠋⠉⢛⠲⣾⣬⣄⣀⡚⠋⠀⣠⠾⢻⡿⢸⣿⣿⠇
⠀⠀⠀⠀⠀⢹⣿⣿⡌⣿⡋⢁⣠⡶⠊⢀⣠⡶⠋⣼⢇⡶⠉⠛⠿⢿⣿⡦⣠⣾⠇⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣷⡘⣷⡟⠁⣠⡴⠟⠁⡀⣸⣿⠀⣀⡴⠟⠁⣀⣼⠞⣽⡟⣼⣿⣿⠇
⠀⠀⠀⣤⢤⡾⠛⢿⣿⣷⣌⢿⣾⡉⢀⣴⠞⢱⣿⡏⠞⠋⢀⣤⠞⠋⣠⡾⢋⣼⣿⣿⠏
⠀⠀⠀⠹⣷⡷⡀⠈⢻⣿⣿⣷⣝⠻⣿⣤⣠⡶⠉⢀⣠⡾⣋⣡⣴⠾⣋⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠘⢿⣿⣄⣴⠟⠻⣿⣿⣿⣶⣭⣝⣛⣛⠛⣛⣛⣛⣭⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣶⢶⣿⠿⣿⠿⣿⢿⡿⣿⠿⣿⠿⣿⡶⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⠟⣛⡿⢋⣽⠏⣡⡟⠁⣼⠃⢰⡏⢰⡇⢸⡇⢸⡆⢹⡆⠘⣧⠈⢻⣄⠹⣯⡙⢿⣛⠻⣿⣷⣶⣄⡀
⠀⣀⡀⠀⠀⣾⣟⣡⠟⢁⡼⠋⣠⠟⢁⡼⠋⢀⡾⠁⢠⡟⠀⣼⠁⢸⡇⠈⣧⠀⢻⡄⠘⢷⡀⠙⢧⡈⠻⣄⠙⢧⡈⠻⣌⣻⣷⠀⠀⢀⡀
⣾⣿⣿⣆⠀⣻⢿⣷⣴⣏⣠⡞⠁⣠⠟⠁⢠⡾⠁⠀⡾⠁⢀⡿⠀⢸⡇⠀⢿⡀⠈⢷⠀⠈⢷⡄⠈⠻⣄⠈⢳⣀⣙⣦⣼⡿⡿⢀⣰⣿⣿
⢸⣿⣿⣽⣿⢿⣦⣌⡉⠛⠻⠿⢿⣿⣶⣶⣿⣤⣤⣼⣧⣤⣼⣇⣀⣸⣇⣀⣸⣧⣤⣼⣧⣤⣤⣿⣶⣶⣿⡿⠿⠟⠛⢉⣠⣴⡿⣿⣯⣿⣿
⠘⣿⣿⡧⢻⡀⢻⣿⣿⣿⣶⣶⣤⣤⣄⣀⣉⣉⣉⠉⠉⠙⠙⠛⠛⠛⠛⠛⠛⠋⠋⠉⠉⣉⣉⣉⣀⣠⣤⣤⣶⣶⣿⣿⣿⡟⢀⡟⢾⣿⣿
⠀⠙⣿⠛⢻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡟⠻⣿⠇
⠀⠀⠙⠳⠞⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⠳⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⡛⠿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⠿⢛⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⣶⣬⣉⠛⠻⢿⣿⡧⣿⣿⣿⣿⢸⣿⡿⠟⠛⣉⣤⣶⠟⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣍⠛⠻⠶⣾⣿⡧⣿⣿⣿⣿⢸⣿⣷⠶⠟⠛⣩⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⢠⡞⣿⣿⣷⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣾⣿⣯⢳⡄
⠀⠀⠀⠀⠀⣾⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡆⣷
⠀⠀⠀⠀⠀⡇⢸⣿⣿⣟⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⣿⣿⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣟⣻⣿⣿⡇⢸
⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢸⣿⣿⡇⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⣷⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣼
⠀⠀⠀⠀⠀⠹⡄⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⢠⠇
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠿⠿⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⢈⢉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣤⡀⠀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠉⢿⡇⠀⣿⡇⠀⣿⢸⣟⣸⠇⣴⡆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠸⡇⠀⢿⣇⣰⠏⣸⢻⡄⢰⡿⣧⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠋⣠⣬⣭⣥⣤⣭⡀⠃⠟⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⢿⣿⣿⣿⣿⣿⠋⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣾⣸⣇⠀⠀⠀⣹⢛⡟⢿⠁⠀⠀⣸⡇⣧⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣹⡏⣿⠆⠀⢠⠻⣿⡿⠟⢇⠀⠀⣿⢹⣿⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⠋⠀⢀⡯⠀⠙⠁⠐⣾⣷⢀⡙⠮⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢲⡇⢠⣄⣘⠀⠀⡁⠀⢰⣿⠁⣾⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣼⠇⢸⠀⢸⡇⢸⡇⠀⣿⠃⢰⡿⠁⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⣿⠄⣾⠀⣾⠃⣸⡇⢠⣿⠀⣼⡇⢀⡏⣹⣦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠃⠀⠃⠀⣿⢠⠛⠷⠃⠻⠞⠈⠧⠞⠘⢤⢏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣉⣵⠛⡆⣳⣆⡇⣰⣯⣭⣭⠭⠭⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠙⠛⠛⠉⠋⠹⣠⠇⢽⠹⠇⢹⣟⣛⣓⣒⣒⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠉⠛⢿⣷⡄
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠿⣿⣿⡀⢀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⢸⣿⣷
⠀⠀⠀⣰⣿⣿⣿⠟⠋⠁⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⢀⣾⣿⡿⠋⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⢸⣿⣿⠋⠙⣿⡇⠀⠀⠀⠀⢀⣿⣿⠁⣠⡶⠿⣷⣦
⠀⢿⣿⣿⣧⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⠿⠃⢀⡿⠃⠀⠀⠀⣠⣾⣿⠃⢸⣿⠁⠀⠀⢹
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣴⠊⢁⣠⣤⣶⣿⡿⠟⠁⢠⣿⡏⠀⠘⠿⠿
⠀⠀⢠⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⡟⢁⣴⠿⢛⣋⣭⣥⣤⣤⣾⡿⠟⢠⣶⣷⡆
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣾⢠⡿⠁⠐⢋⠁⣴⡶⠶⠦⣤⣤⣄⣤⠴⠋
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠁⠘⠇⢀⡀⢻⣷⣍⠃⢀
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣀⠀⠀⢀⣾⣷⠈⣿⣿⣧⡈⢷⣄
⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣠⣾⣿⠛⣠⣶⣿⣿⠟⠀⣿⣿⣿⡇⠸⣿
⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⢀⣤⢰⣿⣿⠇⣸⣿⣿⠟⣡⠀⣰⣿⣿⣿⡇⠀⠏
⢹⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⣴⣿⡇⣾⣿⠏⠀⣿⣿⣧⡾⠁⣼⣿⡿⠋⣸⠇
⠘⣿⣿⠿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠐⠚⠋⠿⠟⠀⠉⠀⠀⠀⠻⣿⣿⠁⢸⣿⠋⢀⣾⠏
⠀⠘⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠈⠛⠀⠈⠻⠂⠈⠁
⠀⠀⠘⢿⣷⣿⡿⠿⠟⠋⠉⣀⣠⣤⣴⣶⣶⣿⣿⣿⡿⣿⣿⠿⣿⡿⢿⣿⠿⣿⡿⢿⣿⠿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢻⣿⠉⣿⡇⢸⣿⠀⢸⡇⠀⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣻⡇⠈⣿⠉⢻⣿⠻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠉⣿⡇⢸⣿⠀⣿⡇⢸⣿⠀⢸⡇⠀⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⠀⢸⣿⠀⣿⡇⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣀⣿⡇⢸⣿⠀⣿⡇⢸⣿⠀⢸⡇⠀⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⠀⢸⣿⠀⣿⣇⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣾⣿⣤⣿⣇⣸⣿⠀⣸⡇⠀⣿⡀⢸⣿⠀⣿⡇⢸⣿⠀⣿⣇⣀⣿⣄⣼⣿⣶⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢟⣻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣬⣭⣭⣭⣭⣭⣥⣤⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠁⠀⢠⣤⣤⠀⠸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⠀⠀⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⢸⣿⣿⣀⣀⣀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣾⣿⠿⠿⠿⠿⠿⠛⠛⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠛⠻⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣤⣀⡶⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣮⣭⣵⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷

⣤⠖⠚⠛⠛⠉⠉⠙⠛⠓⠶⣤⡀⠀⠀⠀⠀⠀⣤⠖⠚⠛⠛⠉⠙⠻⣿⠷⠶⣤⡀⠀⠀⠀⠀⠀⢠⡴⣒⠛⣛⡍⢉⡽⢛⡛⠲⢦⣄
⢽⣒⣲⠤⣄⠀⠀⠀⢠⡄⢠⡄⠹⡆⠀⠀⠀⠀⣽⣶⣶⣦⣄⠀⠀⠀⣯⠀⠀⠀⠹⡆⠀⠀⠀⠀⢨⣗⣲⣶⣥⡚⣫⡴⣋⡴⢋⡴⠊
⠀⠉⠉⠙⢿⣷⣄⣀⣀⣉⣉⣠⣼⡇⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣟⣛⣛⣆⣀⣠⣴⡇⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣦⣚⣉⣶⣋⣐⣯
⠀⠀⠀⠀⠀⠙⠻⠿⣿⡿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⡿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⢀⣠⣤⣴⢶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢀⣤⡾⣿⣛⣷⣽⣼⣿⣽⣽⣏⡝⣷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⣿⣿⣿⠟⠁
⠀⣠⡟⢯⣱⠾⠋⠁⠀⠀⠀⠀⠈⠙⠿⣦⣯⡽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢐⡔⣧⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠁⢀⣾⡀
⠐⣿⣿⣾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣃⣎⣿⣿⣶⡷⠀⠀⠀⠀⠀⢸⣿⠟⢁⣼⣣⣻⣾⣷
⣰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠰⣽⣯⣯⣿⣯⣯⣭⣷⠀⠀⠀⠀⠀⠐⢧⣦⣭⣭⣿⣽⣯⣽
⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟
⣤⣠⣶⣶⣦⣤⣴⣤⣤⣤⣤⣤⣤⣴⡟⠛⠛⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠘⡇
⣀⣀⣀⣉⣁⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿
⡷⠶⠶⠬⠭⠭⣿⣀⣀⣀⣀⣀⣀⣀⣤⣄⣠⣤⣤⡤⢄⡀⠀⠀⠀⠀⠀⣏⢎⡦⠤⠤⠤⢿⠀⠀⠀⠀⠀⠀⠀⢸⡱⡱⠤⠤⠤⢼⡆
⣟⣛⣛⣛⣛⣩⡿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣻⣦⡀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⡔⣩⣿⣿⣿⡿⠋
⠀⢀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡾⣫⡾⣻⠆⠀⠀⢾⣽⣯⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠀⡠⣇⡀
⠀⠀⠙⣆⡈⠳⣄⠀⠀⠀⠀⠐⢯⡴⣫⠞⣩⠞⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠸⣿⡿⠋⢠⡼⣔⣷⣿⣴⡄
⠀⠀⠀⠈⠳⠶⣿⡄⠀⠀⠀⢀⠞⠛⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⢹⠉⠀⠈⣯⠀⠀⠀⢠⣤⣾⡄⢿⣤⣬⣭⣭⣭⣭⣭⣭⡄⣰⣧⣤
⢀⣀⣾⣿⣿⣿⢿⣷⣿⢻⣿⣿⢻⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢹⡄⠀⠀⠀⣿⣿⣿⠈⢿⣿⣿⣿⣿⣿⣿⠟⠀⣿⣿⣟
⠈⣿⣿⡿⠿⠾⢿⣿⠿⠾⢿⣿⠾⠿⠿⠿⢿⣿⣟⠀⠀⠀⠠⡫⢝⠤⠤⠤⠬⣇⠀⠀⠀⠻⣿⠟⢦⣄⢸⠏⠉⠉⢻⠁⣠⡾⣫⣽⠟
⠘⠛⠛⠁⢀⣤⡾⠛⠀⠀⠈⠛⣷⣄⡀⠀⠘⠛⠛⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠳⣆⣽⢻⡷⠶⡶⣾⣿⣹⣵⠞⠁
⠀⠀⢀⡴⣿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢟⡛⣻⣿⣿⣛⣿⠉
⣤⣤⣼⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣷⣾⣥⣤⣤⣤⣼⣦⣤⣤⣤⣤
⣿⣿⣄⣠⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⣆⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢠⣾⣿⠟⠀⠀⣰⡟⠀⠀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠈⣇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⢰⣿⠁⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢿⣿⡇⠀⠀⠹⣿⡄⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⢷⠀⠀⠀⠈⢳⠀⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⡇⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢛⣛⣿⣿⣯⣽⣧⣴⣯⣤⣿⣤⣼⣧⣼⣤⣼⣧⣤⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠉⠉⠉⠉⠁⠀⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⣀⣀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠘⣇⠀⠀⠀⠀⠀⠀⣸⠇⠹⣦
⠀⠀⠀⠀⠀⠀⣸⠋⢻⣄⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠹⣄⠀⠀⠀⠀⣰⠏⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠙⢷⣄⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠹⣆⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠳⣄⢸⠃⠀⠀⣴⡀⠀⠀⠀⢹⡾⠃⠀⠀⢠⣦⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⡇⠘⢦⣀⠀⠈⠀⣀⣠⠔⠃⠈⣆⠀⠀⠀⠀⠈⢷⡀
⣤⣄⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠙⠓⠛⠋⠉⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⣧
⠈⢿⡛⠶⢦⣄⡈⣇⠀⠀⠀⢀⡀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣴⠒⠒⡶⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢳⡀⠀⠈⠙⠻⠆⠀⠀⠀⡏⠳⢄⣸⠁⠀⠀⠀⠀⢰⠃⠀⡼⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠁⢀⡀⠀⠀⠀⢸⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⠲⣄⡀⣸⠀⡟⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡄⠀⠉⣙⣻⣧⣀⡀⠀⠀⠀⠰⠿⠞⣷⠀⠀⠀⠀⢰⠇
⠀⠀⢰⡇⠀⠀⠀⠸⠓⠦⣤⣸⠀⠀⠀⠀⣠⠖⠛⠶⢤⣄⣠⠟⠉⠙⠳⣄⠀⠀⠀⠀⡏⠀⠀⠀⢠⡏
⠀⠀⢸⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡜⠁⢀⣤⣄⠀⣠⠶⠖⠶⣄⠀⠈⢧⠀⠀⠀⡇⠀⠀⠰⢿⣤⠴⠶⠛
⠀⠀⠈⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰⠁⠀⡏⠠⠎⢻⠁⠀⡶⠀⠈⡇⠀⠘⡆⠀⣾⡁⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠸⡆⠀⠙⠦⠤⠾⣧⡀⠀⢀⣰⠇⠀⠀⣷⠀⠈⣩⠗⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠁⠢⣀⠀⠹⣄⠀⠀⠀⣀⡈⠉⠛⠉⣀⡄⠀⠀⣽⠖⠋⠁⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠙⠒⠾⠷⢦⡀⠈⠉⣟⠛⡟⠉⠀⠀⠀⡇⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠹⣄⠀⢸⡖⣇⠀⠀⠀⣼⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠲⠶⠶⠖⢻⡄⠈⣧⢹⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⠿⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠠⠤⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠛⠲⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣶⠓⠟⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⣄⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⢦⣤⣤⣤⣤⡤⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠶⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⡛⠛⠓⠒⣀⣈⣭⡷⠤⠤⠤⠶⠶⠤⠤⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⣉⡭⠶⠛⠩⡭⠀⠰⠖⠀⣤⠀⠛⢁⠀⠀⡀⠀⠀⢀⠈⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠚⣁⠀⠘⠀⢤⠄⠀⠒⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣞⠷⢽⠆⠀⠀⣀⣈⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢒⡀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⣁⣤⡶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠴⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠈⢧⠀⠀⠀⠀⠛⠛⠛⠊⣩⠇
⠀⠀⠀⠀⠀⢀⡴⢻⡏⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠤⠒⠂⠀⠈⠁⠀⠀⠈⠃⠀⠀⠀⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⣾⠄⣸⠀⢀⡴⡒⠂⠀⠀⣀⡤⠴⠖⠒⠛⠳⣉⣄⣀⡴⠶⠶⠶⠶⠖⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠹⣮⣿⠀⠸⣥⣢⠔⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢰⠋⡷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢠⠇⠀⣌⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣀⡠⠔⠃⠀⢦⣈⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢄⠀⢦⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⣆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢒⡒⣐⡐⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⣛⣛⣛⣛⡃⠀⢈⣛⡲⠶⠶⠶⠶⣒⣺⣏⠉⠉⠉⠉⠉⠉⠉⠴⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣒⣒⣒⣒⣂⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣹⣶⣴⣶⣶⣶⣦⣴⣦⣶⣶⣶⣦⣶⣶⣴⣶⣦⣶⣶⣶⣴⣤⣶⣶⣶⣦⣶⣶⣶⣴⣶⣯⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢀⡠⠒⢒⡒⠢⢄⠀⠀⠀⠀⢀⡤⢤⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡤⠤⣄⣀⡀⠈⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡎⠀⣼⣁⠈⣷⠀⢣⠀⠀⠀⠸⣦⠈⣇⠀⠀⠀⠀⠀⠀⣿⣩⣩⣉⣉⡉⠡⢒⣒⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢣⡀⠈⠓⠒⠁⠀⡇⠀⠀⠀⠀⠹⣧⡈⠢⡀⠀⠀⠀⠀⠑⠒⠒⠒⠉⠉⠉⠙⠚⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠰⣿⠀⠀⠉⠒⠦⠤⠴⠊⠀⢀⣀⡀⠀⠀⠈⠻⢄⠼⠀⠀⠀⠀⣴⠶⠶⠶⠶⠿⠝⣒⣲⡄⠀⢸⡧⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡎⠉⣿⠀⠀⠀⠀⠀⠀⡠⠚⣩⠥⣄⠙⣆⠀⠀⠀⠀⡜⠉⣧⠀⠀⣗⣒⣒⣒⣶⡶⠦⣔⣲⠇⠀⢸⡏⠙⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⠸⣀⣀⣸⠀⣿⠀⠀⠀⢀⡇⢸⡟⠀⠀⢀⣀⣀⠀⣀⣀⢀⣀⠀⠀⠀⢸⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠣⠤⣿⠀⠀⠀⠀⠀⠀⠳⡀⠉⠉⠁⢠⠇⠀⠀⢀⠞⢀⣼⠃⠀⠀⣟⣈⣉⣛⣛⣋⡑⣚⣛⡇⠀⢸⡧⠴⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⣿⠀⠀⠀⣀⠤⠤⠤⣌⠒⠒⠒⠋⠀⡴⠲⣌⠶⠟⠁⠀⠀⠀⠓⠒⠒⠒⠒⠋⠁⠒⠚⠃⠀⢸⡟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⣷⠊⢓⡄⠀⠙⡄⠀⠀⠀⢻⣆⠘⢦⡀⠀⠀⠀⠀⣴⣲⣶⣤⣖⣲⣒⣒⢤⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⡝⠶⠞⠃⠀⣠⠇⠀⠀⠀⠀⠙⢷⣄⡉⢳⠀⠀⠀⣷⣒⣒⣒⣲⠶⠖⣒⣲⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣦⣀⣀⣈⣑⣒⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣉⣑⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠬⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠯⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⠋⠰⠟⠂⠳⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠐⠀⠂⠀⠀⠀⠀⠈⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠘⡍⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⣄⡰⠋⠙⢲⠀⠀⠀⣶⢸⠉⠀⠳⡎⠙⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡼⠀⠀⠀⠀⠈⡏⢹⠋⢸⠘⡀⠀⠀⢻⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⠃⠀⠀⠀⠀⠀⡇⢸⠀⠀⡇⡇⠀⠀⠈⡇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⠇⠀⠀⠀⠀⠀⢀⠁⢸⠀⠀⢣⢳⠀⠀⠀⠸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡞⠀⠀⠀⠀⠀⠀⢸⠀⠾⠀⠀⢸⠸⡄⠀⠀⠀⢣⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡰⠁⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⡜⠃⠉⠉⢦⠀⠘⡄⠸⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠸⣄⠀⠀⠀⡸⠀⠀⢳⠀⢳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡼⢤⡎⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠙⠓⠒⠊⠁⠀⠀⢀⠗⠈⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣄⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢶⣏⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣎⠡⠞⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠄⠁⠀⠀⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠠⢣⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠛⠋⢧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢱⣶⣶⣆⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣼⣀⣀⣀⣸⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣿⣿⠏⠼⠿⠿⠿⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⣧⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠳⣿⣿⣿⣿⡇⡻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦
⠀⣤⠖⠒⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡞
⠘⢧⠾⠻⢦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢷
⠀⠀⠀⠀⠀⠹⣆⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢧
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠉⠓⠒⠒⠒⠋⠉⢳⡀
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠤⠀⢀⡴⠃⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⡷⣄⣀⣀⣠⣤⠶⠭⣀⡠⠒⠀⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢋⣉⢳⠀⠀⡸⠓⠒⣲⡞⢑⡒⠐⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡘⠀⣤⠀⠈⡇⠰⠄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠋⠈⠉⠀⠀⡇⠀⠀⠉⣇⡈⠀⢀⠞⠀⡀⠸⡇⠠
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠱⡀⠀⠀⣠⠗⡤⡀⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡏⠀⠀⠀⠀⠀⠘⢲⣴⣚⣉⣉⣉⠥⠔⠊⡇⢠⡇⠄
⠀⠀⠀⠄⠀⠀⠀⠻⡄⠐⢷⠬⢥⣉⣁⣀⣀⡀⡤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢸⡇⠀⠀⠀⢀⠄⠀⠸⣀⢈⠀⠀⢀⣀⠠⢾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⣄⣀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢹⠉⠉⢉⠀⠀⠐⠉⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡼⠳⢤⣀⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⡟⠉⠉⠉⢻⠈⠉⢻⠀⠀⢀⡤⠒⠒⠺⣅⠀⣀⣈⣉⠉⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠐⠀⠀⠸⠋⠀⢀⠈⠱⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⠀⠀⠀⠀⠀⢸⢀⠀⠘⢄⡀⠀⣀⣧⠀⠀⠀⠀⢠⠃⢠⠏⠀⠑⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠣⣀⠀⠀⠀⡠⠎⠈⠀⠀⠀⠈⣹⡁⠀⠑⠦⠤⠒⣁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠙⠀⠀⠀⠀⠀⢉⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣈⠁⠉⠉⢉⣀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢿⡿⠿⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⢠⠋⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡄⠀⠘⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠚⠁

⠋⠀⠬⠘⠀⠀⠀⠀⠀⠁⠀⠀⢰⠂⡀⠀⠀⠀⠊⠀⠀⠀⠉⠀⠀⠁⠀⠁⠈⠈⠀⢈⠉⠀⠀⢮⣭⡷⠂⠀⠀⠈
⠏⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠨⡀⠆⠀⠀⠀⠀⠤⢀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁
⠀⠀⣀⡔⢒⠙⠛⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣀⠂⠀⠀⠀⠀⠀⠀⠀⢲⠄⠀⠀⠀⣠⠂⡸⠂⠀⠰
⠁⠀⠈⢁⡉⠀⢬⠀⡀⠀⠀⠀⡐⠀⢀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠘⠁⠀⠀⠁⠀⠀⠸⠁⠐⠂⠀⠀⠀⠀⠐⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⣑⢶⣄⡀⠀⠀⠀⠀⠘⠁⠀⣀⡀⠀⠀⠀⠀⠘⡅
⠃⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠂⠀⠀⠀⠀⠀⢀⠠⠝⠛⠖⠀⣀⠀⠀⠃⢀⠀⢰⠒⢰⡄⠀⠲
⠁⠠⠂⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢀⠀⠄⢀⠀⠁⠀⠀⠀⠉⣈⠁⠠⡌⠁⠀⠀⠀⠂⠙⠀⠀⠁⠩⠤⠀⠀⠁
⠀⢀⡀⠀⠀⠀⡆⠀⠉⠠⠄⠈⠇⠀⣠⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠠⠓⠀⠀⠀⠀⠠⣤⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰
⠰⠈⠀⠀⠀⠀⣁⠀⠀⠀⠀⢱⠀⠀⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠋⠙⢠⠁⢐⣀⡄⠀⠀⡀
⠀⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⢠⠴⣶⣧⠀⠀⠨⠁⠀⠀⠀⠂⠀⠀⠑⠘⠾⡌⠁⠙⠀⠀⠀⠋⠠⡐⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠖⠐⠀⠀⠠⠀⠀⠈⠙⠉⠁⠀⠀⠀⢠⠄⠀⠐⠀⠤⠄⠲⠄⠀⠀⠀⠀⢠⠆⠀⠀⠀⠡
⠲⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⣛⡂⡀⠀⠀⠀⣀⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠈⡀⠀⠈
⠄⠀⠠⠤⠒⠂⠀⡀⠘⠠⠐⠂⡀⠈⠁⠀⠻⢡⡗⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡅⠀⠀⠈⠀⠀⠀⢃⠀⡅⠀⠀⠀⠀⠶
⣄⠀⠳⠀⠀⠀⠙⠉⠁⠀⠂⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⡁⠀⠀⠢⠀⠀⠀⠀⠖⠆⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⠳⠀⠀⠀⢠⠁⡸⡃⠀⠀⠀⠀⠀⢠⡄⠘⠁⠀⠀⠐⠀⠀⠀⢀⣀
⠷⠃⠀⠀⣀⣤⠊⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢲⡀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢲⡀⠀⠀⠀⠀⡠⡐⢰
⠀⠀⠀⠯⠽⠞⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠑⢀⠀⠀⢀⠀⠀⠀⠀⠆⠀⣀⠀⠀⠀⠀⠈⢄⠸⠀⠀⢱⠈⠁
⠀⣶⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠈⠩⠡⢤⠄⡀⠀⠀⠲⠊⠁⠀⠀⠀⠉⠁⠀⠀⠠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠁
⢠⠋⠠⠂⠀⠀⠸⠸⢰⠋⠁⠀⠄⠀⠀⠰⠂⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⢠⡄⡀⠀⡄
⡮⢄⠢⠀⣤⡴⢒⢉⣢⠖⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠎⠸⠇⠀⠈⠠⠀⡀⠐⠚⢀⣠⠀⠀⠀⠠⡌⠀⠀⠀⠁⠈⠐⠆
⠁⠁⠀⠀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠂⠛⠠⠁⠀⠈⢃⣘⠆⠀⠦⠄⠀⠀⠀⠁⠀⠀⠄
⢀⠀⠀⠀⠀⢰⠂⣢⡀⠀⠆⠀⠀⠀⠀⠀⡀⠀⡒⡒⡄⢂⠀⠀⠀⠀⠀⠀⠴⡿⠞⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⣀⠁⠀⢀⠀⠀⢈⢉⡅⠁⠀⠁⠀⡇⠀⠀⠟⢋⠄⢁⠪⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣀⣠⠐⠂⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠠⣠⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡀⠀⢀⡀⠀⠘⣧⡄⢠⢰⠒⠒⠋
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢣⠀⣧⠀⠈⠁⠀⠀⠸⡿⠜⠚⠓⢤⡄⠀⠀⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⡀⠘⢧⡀⠠⠀⠀⡇⠀⠀⠀⠀⠀⢀⡀⣸⡿⠆
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡠⠄⢀⣀⠈⠙⠓⠘⠂⠛⠐⠚⠛⠛⠋⠈⠁⣀⠠⣀⠀⠀⠀⠈⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⣸⠤⠀⡰⢧⠀⣏⡁⣰⠋⠳⣤⡏⠉⢣⢹⠀⢘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠓⠂⠰⠉⠹⠀⠇⠀⠹⠤⠴⠃⠳⠤⠊⠈⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠛⠛⠛⠒⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣉⣉⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⡆⣿⣹⡆⡾⢳⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠿⣈⡷⡏⣨⡗⢧⡼⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⣦⡀⠀⠀⠠⣤⡤⢠⡄⣤⢀⣤⢡⡤⠀⠀⠀⠦⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡆⠀⠀⣾⠀⣸⢸⣿⡿⣿⣟⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣯⣤⣤⣤⣬⣤⣭⣤⣤⣥⣥⣭⣥⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠉⠉⠛⢻⠛⠋⠉⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⣀⣀⣸⣆⣀⣀⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⢻⠟⠛⠛⠛⠛⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉

⣿⣿⣿⢿⣿⠿⢿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣮⣶⣦⣴⣧⣾⣯⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⣠⠔⠚⠉⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⣴⠈⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠇⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡼⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⣄⣀⢀⣀⣀⣀⣀⠤⡆⠀⠀⠀⢀⣀
⠀⠀⠀⡇⠄⠀⡏⠀⠀⠀⢀⡠⣐⣂⣬⣧⢼⡯⢍⠠⠴⡆⠒⠖⠒⠒⠲⠢⠤⠠⠌⠉⠀⣀⡠
⠀⠀⠀⠜⡄⠘⠀⠀⠀⣾⣡⣾⣿⣳⡟⣷⣽⣛⣸⢃⣤⠤⢀⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣾
⠀⠀⠀⠀⠃⡇⠀⠀⠀⣈⡿⢺⠿⣿⣿⣿⣿⠋⠗⡉⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⡟⠁
⠀⠀⠀⠀⢂⣿⠀⠀⠀⠣⡝⠲⢯⠆⡄⠉⠉⠉⠛⠛⠒⠒⠒⠒⠐⠒⠒⠒⠉⠉⠀⠉⠀⠀⢀⠃
⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠈⠳⠺⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠘⣇⠤⠤⠤⢖⠚⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠈⠓⠄⣀⠀⠀⣻⠿⠗⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠒⠉⠁⠀⢰⠶⠶⡂⣒⠊⠈⠙⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠁⠀⠀⠀⢸⠀⠀⠉⠁⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⣀⣀⣨⣀⣀⣠⠀⡀⠀⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠘⠈⠉⠉⠁⠈⠁⠀⠀⠀⡄⠁⠀⠀⠀⠲⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠃⠄⡀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠉⠉⠃⠲⡄⠀⠀⠘⢸⠉⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡐⠀⠀⠀⠀⠀⠀⠀⠀⣄⡡⢀⡀⡀⡀⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠷⠟⠀⠀⠀⠃⠘⡂⠄⠠⠀⣀⠚⡈⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠈⠐⠀⠐⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣸⢀⡇⠀⢀⠀⠘⠁⣳⣰⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠄⢀⠈⠁⠀⠁⠀⢎⠠⠔⠃⠑⢌⣁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠠⢁⠖⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡰⠃⡤⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡼⢡⠞⡤⢊⡴⣉⠖⡢⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⢮⡀⣇⢼⠤⢋⠜⢡⢊⢔⡥⣢⢭⡢⡄⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⠻⣿⣿⣿⡫⠻⣟⣟⣿⡿⡾⢾⣷⣦⣔⡃⠀⣠⣦⣢⢀⠌⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⣀⣤⣾⡿⣻⢫⠂⢌⢯⣿⣿⠑⡁⢞⣿⣿⡥⡁⣡⣿⣿⣿⣿⡿⡻⣶⣿⣵⣫⣪⣶⡏⡂⠀⠀⠀⠀⠀⠀⢀⣤⢞⠤
⠀⠀⣀⡔⣺⢋⣠⡀⠀⠉⠑⠪⠦⠲⡻⢿⠙⢈⢜⣿⣿⣽⢉⠱⢻⣿⢿⣷⣜⡜⡹⣿⣿⡯⣹⣻⣦⣤⡤⣄⣤⣤⡤⣴⢻⡷⢓⡀
⣴⠞⢅⠐⠧⣩⠌⢀⠀⠆⠰⣤⡁⡀⡁⡀⠎⢎⢌⢾⣿⣧⢆⢄⠑⣿⣿⡿⡈⢬⢹⣿⣿⢟⠿⠿⣿⡿⠴⢿⣾⢿⢷⢽⢩⣿⠳⠔⠺
⠉⠲⡃⠙⣶⠘⠇⡻⠀⠀⠱⠒⡀⠈⢲⣴⡟⠂⠀⠂⡛⠔⡥⢣⢎⣼⣿⡱⠵⡱⡸⡿⡯⠪⢣⡣⣋⣥⡮⠵⠒⠘⠒⠦⣽⣿⣟⡑⠂
⠰⠤⢽⣟⣏⡫⠘⠱⢖⡋⠀⠀⠼⡐⣿⣍⡉⠀⠀⡰⣝⡬⡺⣔⢵⢟⣃⡌⡬⡪⠨⠈⣎⣶⡵⣾⠾⠻⢦⠔⠀⠀⠀⠀⠈⠳⠦⣳⣔
⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⣂⣄⣨⢚⡥⢂⢙⠣⢑⠡⢰⢡⢊⡊⣗⣬⣮⣬⠧⠿⠒⠋⠉⠋⢉⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠛⣛⣩⠍⣉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣻⣞⢿⠣⡉

⠀⠀⢰⣦⣂⢀⣀⣀⡠⠘⣀⠀⠀⢰⡀⠀⢠⣤⣴⣶⣿⣿⣷⣦⡀⠀⠠⠀⠀⢀⡠⣶⣟⡷⢲⡦⣀
⠀⠀⠘⠏⡠⢨⡡⠰⠴⠞⣋⡄⠀⠀⢧⠀⠐⢍⠻⢿⣿⣿⣿⣷⠇⠀⡤⣢⠄⠀⢀⣏⡟⣵⢏⡾⣻
⠀⠀⠀⠈⠒⠪⠦⠮⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠁⠲⡿⠟⠋⠁⠀⠀⠐⠃⠀⠀⠈⠙⠿⢿⣶⠿⠋
⠀⠀⢀⠀⠄⠀⣀⣀⡀⠀⢈⠀⢀⣀⠀⠀⢀⣤⣀⣀⡀⠀⣤⣄⡀⠀⠀⠠⠀⠀⠀⠀⠄⠀⢀⣀
⠀⠀⠀⠑⠂⠘⠿⢿⠯⠀⣨⣃⠋⠁⠀⠐⠀⠉⠛⠃⣁⣈⡀⠀⠀⠀⠀⡇⢇⡳⠾⠀⡜⠀⠹⠮⠇
⠀⠀⠀⢀⡄⣠⢀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠃⠛⠛⠚⠛⠒⠋⠉⠀⠀⠀⠈⠲⠤⠛⠋⠀⢀
⠀⠀⢴⣫⠎⠴⣫⣬⠇⢠⠀⠀⢴⣩⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠆⠊⠉⠁⠁⡸
⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠯⠂⠀⡠⠐⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⣄⠀⠀⠀⣭⠄
⠀⠀⠀⢀⠔⡺⢔⢢⣤⣤⣀⡀⠀⠀⠀⣠⣤⣶⣾⣁⣠⠄⠒⠉⠀⠀⠀⠀⠀⠱⠴⠫⠛⣓⠔⠚⠁
⠀⠀⠈⢿⡾⣴⣵⣋⡴⠋⣩⠼⠂⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠔
⠀⠀⢀⠀⠈⠙⠛⠝⣋⠿⠁⠀⠀⠀⠀⠘⠻⣟⠛⠛⠛⠛⠛⠉⠀⠠⠤⠶⠂⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠃⠔⠀⣀⠀⠀⠀⠀⢀⣶⣿⣻⣧⠀⠀⢨⠋⠀⠐⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣰⣿⣻⠦⡄⠀⣿⣿⡿⠛⠁⠀⢀⠞⠰⠶⢷⣞⢛⠀⠀⢠⠀⠀⠁⠀⠀⢻⡇⠀⡄⠀⢸
⠀⠀⢀⣜⡿⣭⣻⣷⠁⠀⠈⠁⠀⠀⢀⣀⠎⢀⣀⣄⠸⣿⠟⡄⠀⠀⠀⠀⢰⠰⠀⠈⠀⠀⢣⠀⢠
⠀⠀⠈⠙⠷⣍⠝⠁⠓⢦⠀⠀⠀⠀⠸⣿⠟⠓⠒⠒⠒⠒⠚⠓⠀⠀⠀⠀⢘⡇⠀⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠃⠈⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⢈⢀⡃⠀⠀⠀⠀⠸⢤⡀⠀⠐⡄⢿⠂⠀⠀⠞⠭⠇⠀⠀⢐⢠
⠀⠀⠀⠀⠀⠐⠞⣮⢫⢧⡀⠀⠀⠀⢂⠀⣿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠢⢀⡀⠀⠠⣄⠠⠤
⠀⠀⠀⠀⠀⠀⢠⡹⣇⣩⣆⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⡄⠠⣤⠀⠀⠀⠁⠲⠦⠤⠶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠶⠇⢠⠀⠀⡀⠀⠀⠀⠀⠉⠰⠬⠀⠀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠸⠒⠒⠚⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠀⠀⠀⠀⠆⠐⢢⡀⠀⡀⠈⢲⠑⢲⣀⡐⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠄⡀⣠⠔⠀⠀⢠⠞⠁⠀⠀⠀⣀⠔⠚⠓⢲⣷⡀⡉⠫⢿⣤⡧⠉⣹
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⠀⢠⠟⠘⠁⠀⠀⠆⠀⠀⣰⣦⣆⣿⣿⣿⣠⣴⣿⣤⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠴⣴⣻⠖⠁⠀⠀⢟⣃⠈⠀⠀⠀⠀⢶⣶⣿⠿⠛⠋⠉⠙⠛⠛⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠠⢀⣈⣥⡴⠶⠚⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠚⠻⣦⣀⡀⠀⠀⢀⣄⣹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢃⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣵⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠋⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⢿⠿⢿⣶⣶⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⡇⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢤⣬⣭⠤⠶⢴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⠀⠀⠀⢠⣤⣀⣤⡄⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⣄⣀⣉⣠⣄⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⡇⠀⠀⠀⠀⣈⣉⣩⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠟⠀⠀⠀⠀⠈⠉⣉⡀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠘⠉⠉⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡀⠀⠀⠀⠀⠀⠒⠛⠉⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⣼⠃⠀⠀⠀⠀⣘⣋⣉⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠃⠀⠀⠀⢀⣀⣩⣭⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⡏⠀⠀⠀⠀⢈⣉⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡟⠀⠀⠀⠀⠀⢬⠍⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⣠⠏⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⠁⠰⠋⠀⠀⠀⠛⠓⠚⢒⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⣀⣘⣛⣛⣓⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠤⣤⣉⡉⣩⣭⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠶⢦⣤⣤⣠⣤⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⢀⣀⣀⠙⠛⠶⠦⣴⣾⡿⠋
⠀⠀⠀⠀⣀⣠⣶⠿⠋⢀⡀⠴⢤⣀⣉⣉⣙⣛⣿⣿⠟⠁
⣀⣤⣾⡿⠟⠋⠁⠠⣤⣀⠙⢻⣶⣤⣽⣿⠿⠛⠋⠁
⠛⠻⠿⣶⣶⣴⣿⣿⣿⣿⡿⠿⠛⠛⠉
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⣘⡻⢝⠻⡙⣌⠦⠰⠒⠋⠋⡇⠉⠉⢉⣀⣂⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠒⠦⢈⠓⢓⡙⠁⢠⣇⠀⠀⠀⠁⠀⠀⡟⢟⢉⡇⠀⠀⢐⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠼⠷⠭⢍⣢⣄⣀⠴⠋⣢⠔⠃⢹⡄⠀⢀⣄⠀⠀⠙⠒⠛⠁⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠠⣞⣭⣍⣑⡶⠋⢀⣤⣶⠛⠁⠀⠀⢰⡇⢠⠞⠉⠳⣄⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡯⠖⠒⣲⠏⠀⠀⠐⠋⠛⢷⡀⠀⢀⡾⡿⡧⡀⠀⠀⠈⣇⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠉⠁⢠⠟⠒⠒⠶⣤⣄⡀⠀⠿⡤⠋⠈⠀⠁⢳⣠⠖⠀⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠙⣏⠢⡄⢻⠀⠀⣀⡤⢺⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡧⢴⠶⣾⠲⠒⠢⢜⡄⠘⣼⣠⠚⠁⠀⢸⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠱⡀⠀⠀⢡⢀⡟⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⠦⣧⣄⡀⣼⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⣾⣿⣦⣑⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⢻⢿⢿⡺⣷⣤⣤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠟⡇⠃⠈⢫⡳⡑⠢⡉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇⠘⠀⣀⣳⡈⠢⣈⠂⠌⣢⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡜⠀⠀⡸⠀⠈⠑⠤⠤⠌⠭⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢻⠜⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⣏⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣷⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⣷⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠟⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣤⣤⣤⣤⣌⣁⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠦⠤⢤⣤⣭⣭⣭⣭⣍⣉⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⢃⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⣠⣙⠳⢦⣤⣀⣀⣀⣀⣀⣀⣤⠶⢛⣡⡘⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⠏⠉⠛⠶⢦⣭⣭⣉⣩⣭⣥⠶⠞⠋⠉⢷⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⡏⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠈⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠇⣸⠁⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢹⠀⢧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢋⡴⠶⣌⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⣌⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⠷⠴⠏⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣤⠀⠀⠀⠀⠀⢻⡄⠀⠰⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠙⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⢛⣛⣯⣭⣭⣿⣛⡛⠿⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣫⣥⣤⣭⢨⣭⣭⣭⣭⣭⣭⡅⣭⣤⣬⣝⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣑⣛⣛⣛⣛⣛⢘⣛⣛⣛⣛⣛⣛⡃⣛⣛⣛⣛⣛⣊⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢣⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡜⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣱⣶⣶⣶⣶⣶⣶⣶⣶⢰⣶⣶⣶⣶⣶⣶⡆⣶⣶⣶⣶⣶⣶⣶⣶⣎⢿⣇
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⢣⣭⣭⣭⣭⣭⣭⣭⣭⣭⢨⣭⣭⣭⣭⣭⣭⡅⣭⣭⣭⣭⣭⣭⣭⣭⣭⡜⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠸⠿⠿⠿⠿⠿⠿⠇⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣿⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣿⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⢰⣶⣶⣶⣶⣶⣶⡆⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣿⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡜⣛⣛⣛⣛⣛⣛⣛⣛⣛⢘⣛⣛⣛⣛⣛⣛⡃⣛⣛⣛⣛⣛⣛⣛⣛⣛⢣⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡹⠿⠿⠿⠿⠿⠿⠿⠿⠸⠿⠿⠿⠿⠿⠿⠇⠿⠿⠿⠿⠿⠿⠿⠿⢏⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡜⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⢣⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⡩⣭⣭⣭⣭⣭⢨⣭⣭⣭⣭⣭⣭⡅⣭⣭⣭⣭⣭⢍⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣝⡛⠛⣛⢘⣛⣛⣛⣛⣛⣛⡃⣛⠛⢛⣫⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣶⣬⣭⣿⣛⣛⣻⣭⣥⣶⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣾⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠹⣿⣦⡀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡄⠀⢻⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠋⠀⢀⣼⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⢴⣶⣶⣶⣶⣶⠶⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠁⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠉⠙⠛⠛⠻⠿⠛⠛⠛⠉⠀⢺⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⣿⠏⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⣿⣟⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⠛⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠘⠛⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⠿⠟⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠱⡄⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⢻⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⡇⠀⡜
⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⣷⠈⢯⠘⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠹⣄⠙⢶⣴⣿⣷⣄⡀⠀⠲⣄⠤⢤⡀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠑⢬⡙⠻⢭⣕⢲⡈⠑⠶⣦⣤⣀⠀⠀⠈⠁⠒⠖⠠⠤⠤⠀⠀⠀⣼⠀⠠⣤⢤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣬⣽⣒⡛⠳⠦⣭⣒⢦⣰⠟⣻⣅⠒⠂⠀⡀⠀⠀⠀⠀⠀⢀⠡⣦⡀⠀⠀⢁⡃⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢛⣛⡛⠿⠷⣿⣷⣿⣞⢁⣖⠀⠙⠿⣷⣷⡾⢿⣿⣶⣿⣿⣿⣿⠘⣿⣶⣠⣄⡴⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢩⣭⣿⣿⣷⣾⣯⣿⣿⣇⠜⠢⣀⠈⠁⠛⠚⠛⢛⣿⣿⣿⠯⢸⣿⣿⣿⣿⣶⣦⣄⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡠⠖⢫⣄⣦⣬⣭⣭⢭⣦⢤⣬⣽⣶⣷⣶⣶⣴⡦⠶⣾⣿⢟⣀⣠⡾⠿⡋⢸⡏⠀⠉⢈⣥⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡾⡡⣪⠞⢋⣵⡶⠟⣛⣯⣽⣿⡿⠿⠛⠟⣛⣫⣭⠿⠟⠁⣩⠟⣫⣤⡴⠟⠶⣏⣀⣠⣎⠱⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠋⣥⣴⠟⠉⢀⡜⣿⠉⣴⣦⡶⣛⣿⠟⣻⡿⠊⠀⢀⡜⠁⡠⠞⠁⣀⣦⠴⢿⣿⣅⣿⠂⢀⡴⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⠕⣺⠅⠀⠀⢀⠌⠀⣿⠀⢇⢄⠟⣉⡴⠋⠁⠀⠀⢀⠟⢀⡞⢀⡴⠋⠀⣠⠖⠛⠛⢛⢿⣶⡎⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣨⠃⠘⠁⠀⠀⠀⠋⠀⠀⢧⠀⣼⡋⠀⠀⡇⠀⠀⠀⠀⠋⠀⠈⠀⠊⠀⠠⠊⣠⠴⠒⠚⢛⢷⡂⠁⢠⣤⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠁⠀⠀⠀⣿⡰⠟⠳⢦⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠴⠛⠉⠛⠛⣇⣤⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠘⣿⠀⠀⠀⢹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠇⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣑⠀⠀⠘⢦⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣆⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣻⡿⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⣾⠟⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣾⣿⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄
⠀⠀⠀⠀⠀⠀⢀⡀⣿⠓⣿⢧⣴⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⢰⡄⠀⠀⠀⠀⢀⠈⠲⣄⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⢀⣀⣈⣛⣁⣀⣀⣈⣉⣟⣃⣀⡀⠀⠀⠀⠀⣷⡈⣿⣿⣾⣿⠀⠀⠀⠀⠀⠱⣤⣾⡀⠀⠀⠀⢈⣿⣿⡿⠋
⠀⠀⠀⠐⠒⢿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⠒⠀⢠⠀⢻⣿⣿⣿⢻⡟⢠⡄⠀⠀⠀⠀⠀⠀⠙⢦⣀⡴⠋
⠀⠀⠀⠀⠀⠘⢷⣹⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢺⣷⣿⡿⠙⠇⠈⢿⣿⡇⠀⠀⠀⠀⠀⠀⢠⡴⠛⢢⡄
⠀⠀⠀⠀⠀⠀⠀⣹⠻⠿⠿⠿⠟⢿⠁⠀⠀⠀⠀⠈⠻⠁⠉⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⣠⣴⠋⠁⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⡞⠒⠒⠒⠒⠒⠒⠒⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠁⠀⠀⢀⠀⠀⠀⢀⡀⠀⠈⠋⠃
⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣠⡴⣾⣛⣛⣶⢦⣄⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⢀⣼⣧
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡾⡿⠋⠁⠀⠀⠈⠙⢿⢷⡀⠀⠀⠀⠀⣾⣿⣿⡆⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢰⠀⣾⢿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣷⠐⡆⠀⠀⢻⣿⣿⡇⢸⣿⣿⡇
⠀⠀⢀⡴⢣⣴⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠸⠀⢿⣼⡀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠐⠇⠀⠀⣼⣿⣿⡃⢸⣿⣿⡇
⠀⠀⠚⣴⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⣤⡀⠀⠀⢀⡤⣮⡿⠁⠀⠀⠀⠀⠻⠿⠿⠂⠸⠿⠿⠃
⠀⠀⠀⣿⢿⣿⣿⡿⠋⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠿⢿⡿⠿⠞⠋
⠀⠀⠀⢻⡌⢿⡟⠁⠀⠠⡘⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠐⠄
⠀⠀⠀⠀⢻⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣤⣤⣤⣤⣴⣶⣿⣿⠟
⠀⠀⠀⠐⠒⢻⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠈⢷⣜⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡧
⠀⠀⠀⠀⠀⠀⠀⢹⠛⠿⠿⠿⠟⢻⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡾⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡼⠒⠒⠒⠒⠒⠒⠒⠺⡄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢠⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢔⣚⣳⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠂⠀⠀⠀⠉⢿⡿⠉⠁⠀⠀⠀⠀⣀⡤⠔⣊⣵⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣴⡞⠁⠀⠀⠀⠀⠀⠀⠔⠞⠟⡳⣄⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁
⠀⠀⠀⠀⠀⠸⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠕⡦⣀⠀⠀⠀⠀⠹⠠⡀
⠀⠀⠀⠀⠀⠀⠹⣎⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⢒⢕⡢⡀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡓⣠⠙⢦⣀⢄⠀⠀⠀⠀⠀⠀⠀⠑⢬⡲⣄⢵⣮⣪⣷⢦⣐⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠪⡳⢦⠉⠳⢤⡀⡀⠀⠀⠂⣀⣄⣠⡨⣿⣿⡿⣻⣵⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠻⠆⠙⢦⣦⣴⣴⣮⣿⣿⣿⣿⣷⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣌⣽⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉


⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠁⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⣉⣉⣉⣉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣴⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣝⡛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⣻⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠

⢫⣵⢤⣤⣤⣤⣤⠀⣠⣬⡙⠿⢀⠙⠿⣷⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠀⠀⣀⠐⠟⠁⠀⢠⠊⠁⠀⠀⠡⣀⣀⣸⡿⠀⢠⣤⠈⠉⠀⠀⠀⠈⠨⢀⢁
⠀⡝⢟⣿⢵⣿⣿⣿⣿⣷⡶⠀⠀⢰⣄⠘⠃⠍⠑⡀⠈⠇⠀⠀⠀⠀⢤⠀⠁⢀⠇⠀⡩⠤⠄⠄⠀⢀⠠⠀⠸⢦⠀⠈⣉⠉⠀⢠⣶⠿
⢦⣀⣀⠈⠙⠇⢻⣷⠈⣿⣿⡦⠀⠀⢑⡙⠄⠸⡀⠀⠀⠀⠀⢀⠀⠀⡈⠃⢠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⡐⢒⠉⠀⣀
⠻⠿⠟⠁⠀⣀⠀⠈⠀⠙⢋⠀⠶⠀⠈⢴⣤⠄⣀⣈⠀⠐⣤⡀⠀⡼⢇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢠⠘⢇⣴⡀
⢀⡆⠰⡄⠺⠿⢣⠀⠀⠀⠘⠁⠀⠀⢴⡄⠙⣀⣴⠋⠀⠀⠹⣤⣶⣤⣮⠤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠑⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣡⠆⠚⠛⣦⢾⣿⣿⡛⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⢯⠿⣦⣀⡠⠤⠄⠀⠀⠀⠀⠀⠀⣠⣾
⣶⣲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠞⠫⠰⠀⠚⠋⠈⣿⡿⠁⠲⠷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣽⡿⢷⣿⢋⣠⡤⡠⠀⢤⡶⠄⣀⣴⣿⢋
⡧⣷⣞⡷⠀⠀⠀⠀⠀⠠⠀⣸⣿⡆⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠂⠀⠁⠀⠀⡀⡈⠙⠋⠁⠈⠛⠀⠀⠘⠉⠀⠈⠉⠉⠀⢸
⠛⠚⠋⢀⣀⣠⣤⠶⠆⠁⠀⢈⢉⢁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣶⣾⡇⠀⣤⡀
⠚⠛⠛⠛⠉⠁⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠑⠂⠀⠀⠀⠈⠉⠀⢠⣽⠗
⠀⠰⠂⣀⡴⠟⠉⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠰⠃⠀⠀⠀⠉⠢⠀⠀⠀⠈⠁
⠀⠒⠛⠉⢀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠈⢻⠓⢄⡛⠁⠐⠒
⠰⠵⠀⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⢀⡐⣷⠄⠒⠲⠈⡆⠀⠉⢀
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣦⠉⠀⠀⠀⠀⠀⠀⠔⠂⢀⣈⣿⣿⣤⣿⣿⢷⣟⢷⢛⠁⠀⢠⠃⡀
⠀⠀⠒⠰⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠚⠀⡀⠀⣄⠀⣠⣽⣿⢋⣽⡿⢾⠏⠋⠁⠘⣠⢀⠠⠂⠀⠀⠁⠀⠰⠆⠐
⠀⠂⠀⠀⠘⠒⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⠀⠀⠀⡈⠆⠙⠻⠟⡘⠋⠁⠀⠀⠐⢾⣶⡷
⠒⠀⠀⠀⠀⠀⠰⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠠⠖⠂⠀⠀⠷⠀⡰⣧⠀⠀⢀⡀⠀⠀⠉⠀⠀⠠
⠀⠀⣉⣀⡀⢀⣤⡄⠈⠁⠀⠀⠀⠲⠶⠤⠰⢂⠀⠀⠠⡤⠀⠀⠠⣤⠆⠀⠀⠀⣤⣀⠀⠀⠀⠐⢶⠶⢤⣇⣤⣔⣨⠟⠀⢆⡀⢧
⣿⣿⢿⣿⣷⣿⣽⣶⡖⠀⠧⡀⠇⢀⡈⠑⠚⠁⠀⠚⠀⠀⠐⠈⠢⡀⠀⠀⠀⠢⠛⠛⠀⡯⢸⡆⢀⠔⠚⣿⣿⣿⣁⣤⣄⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⢀⡠
⠚⠙⠽⣟⢿⣷⡌⣿⣿⡗⠀⠀⠀⠈⠘⠐⣶⣄⣠⠞⠀⠀⡄⠈⢢⢑⡄⠀⢠⡄⠐⠂⠀⢤⡍⠐⠛⠓⠈⣿⣟⠏⢿⡌⠁⠐⡦⠀⠀⠀⠐⠦⢤⣤⡴⠶⠚⣁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⡀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⣀⡤⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠠⠖⠁⠀⠀⠈⠀⡀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⡀⠈⠀⠀⠀⠑⠦⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⡥⠤⠠⠤⢠⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⠠⠤⠠⠤⡅
⠀⡰⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠰⠃⠠⣀⡠⠐⠀⠀⠂⠀⠀⠀⠀⠒⠀⠐⠀⣀⡀⠀⠳⠀⠀⣠⠄
⠀⢀⠀⠀⢀⠀⠀⠀⠀⢰⠂⠁⠀⠠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⠀⠀⠀⢲⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠩⡉⠀⠀⠀⠀⠀⠈⢀⡀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⣀⠈⠀⠀⠀⠀⠈⠈⡩⠁
⠀⠀⠀⠈⠀⠀⠠⠀⠄⠀⠀⠈⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⠀⠀⠀⠄⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠢⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⣀⡠⠒
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉⠉⣀⠢⡀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡠⢂⡈⠉⠉⠀⠀⢀⡀
⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣀⠀⠐⠂⠳⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡴⠓⠒⠀⢀⡀⠀⠀⠀⠀⣠
⠰⠂⠈⠋⠀⠀⠀⠈⠉⠉⠀⠀⢀⡉⠆⠛⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠃⠎⢁⠀⠀⠈⠉⠉⠀⠀⠀⠈⠋⠀⠲
⠀⢀⠀⠀⠀⠈⠣⠐⠀⠀⠀⢀⠁⠀⠤⠊⠁⡈⠒⠀⠤⠤⠤⠔⠐⠊⡄⠈⠢⠄⠀⠁⠀⠀⠀⠐⠠⠋⠀⠀⠀⢀
⠓⠉⠀⠀⡀⠔⠃⠀⠀⠀⠤⢀⣀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠀⠀⠀⠀⠀⠓⠄⡀⠀⠈⠑
⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⡀⠀⢀⠸⠀⠀⠀⠀⠀⣄⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢒⡄⠀⠀⠀⠀⠸⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠛⣆⡀⣤⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣍⡫⣅⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⣾⣷⠷⠚⠢⢤⣤⣼⠢⣀
⠀⠀⠀⠀⠀⠀⠠⠀⠈⢳⠺⠿⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⢷⣜⣿⣷⣤⣄⣀⣼⠃⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣤⣰⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢽⢛⢿⣿⣿⣿⣧⡀⣠⡖⠛⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⣽⡿⣿⣿⡦⣀⡉⠻⣿⠉⠑⠂⡀⠰⠃⠀⠑⠷⣭⣕⣛⣿⣿⡿⠃⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠈⢳⡙⢿⣷⣦⣿⣿⡟⠀⠀⠠⡿⠄⡀⠠⠤⣦⣉⢝⢺⣿⠿⡾⠶⣶⢃⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⡠⠿⡻⠻⣿⠟⣿⣧⣠⣾⠏⠀⠀⠀⢩⡉⢻⡪⡳⣽⣿⣶⣷⡂⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡝⣼⠃⠀⠀⠲⢌⠛⠿⣙⣿⣯⣀⢀⣀⣀⣸⣿⣾⣷⣿⡏⠛⠻⣿⢩⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⣼⡀⢀⣀⣠⣬⣛⠦⣽⢻⣿⣻⣿⠿⠛⠛⠺⣿⣏⠈⣧⣀⠀⣸⣤⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣿⡗⠢⣄⣉⢙⠉⡩⢟⣾⣧⡀⠀⠀⠀⢀⣴⣿⣿⣤⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣽⣀⠀⠙⡿⠓⢿⣾⠎⠙⢿⡭⣴⣿⣿⠃⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⠀⠈⠻⣆⠀⠀⠈⢺⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀⣰⣿⣶⣦⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣮⣽⣿⣿⣿⣾⣿⣸⣥⣿⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠆⠙⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⢀⡠⠚⣩⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢋⣡⡴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠓⢀⢐⡉⠚
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠵⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣴⠁⣀⠔⠈
⠀⠀⠀⠀⠀⠀⢠⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠠⠴⠞⠋⠀⠀⠀⠀⠀⠀⢀⣒⣼⢦⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠳⠇⠿⠇⠟⠁⠀⠀⠀⠀⠀⠀⢠⡠⠀⣐⡤⠤⡂⠀⠀⠀⠀⠀⢀⠀⣀⡀⡀⣀
⠀⠀⠘⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢌⣐⠀⠀⠀⠁⣈⠀⠀⠀⠀⠰⠈⠀⠁⠀⠁⠀⠁⠐⠦⢀
⠀⠀⠀⠀⠢⡀⠀⠀⠀⠱⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠉⠚⣻⠧⡀⠀⠀⣈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⢿⠋⠀⠀⠀⢣⡌⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣯⠏⠁⡀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠁⠨⣯⣆⠹⣆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⡆⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠹⣧⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠙⢿⣾⣆⢄⢀⠀⠠⠖⠂⠈⠀⠀⠀⠀⠰⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⡠⠈⠛⠣⣂⣀
⠀⠀⠀⠀⠀⠄⠐⠁⠁⠐⠆⡀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⢿⣷⡀⠀⠀⠀⠀⢠⠀⠠⠀⠄⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠤⣀⠀⠈⠉⡁⡀⣀⣹⠀⠀⠀⠰⠇⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠩⠈⡅⠱⠀⠀⠀⠀⠀⠀⠰⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠦⠀⠀⠀⠠⠠⠧⠚
⠀⠀⠀⠀⠀⣰⠤⠀⠀⠀⠀⠀⠀⢰⣰⡶⠄⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢰⠋⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡈⠒⠄⢶⣦
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢄⣀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡁⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠬⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠛⠒⠾⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣄⣠⡦⢀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠰⡼⠋⠙⠹⠿⣴⠀⠀⠀⠀⠀⢠⡈⣿⣧⣽⡄⠀⠀⠀⠰⣄⢱⡄⠀⠀⠀⣾⣿⣿⡿⠋
⠀⠀⠀⠤⣿⠹⣿⣿⣿⣿⣿⣿⣿⡧⠤⠀⠀⢸⣷⣿⣿⣿⠇⡀⠀⠀⠀⠈⠉⠳⣄⢀⡴⠋⠉⠉
⠀⠀⠀⠀⠘⢧⣻⣿⣿⣿⣿⣿⡿⠁⠀⠀⣿⣶⣿⠛⡇⠹⣷⣿⠀⠀⠀⠀⠀⢀⡬⠛⣄
⠀⠀⠀⠀⠀⠀⡟⠛⠿⠿⠛⢏⠀⠀⠀⠀⠘⠏⠈⠀⠀⠀⠘⠃⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⠀⠀⠈⠳⡆
⠀⠀⠀⠀⣠⡋⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⠀⠀⠈⠀⠀⠀⣰⡀⠀⢀⣆⠀⠀⠈
⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣴⡿⠛⠉⠉⠛⢿⣦⡀⠀⠀⠀⢸⣿⣷⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⠄⣾⡟⠀⠀⠀⠀⠀⠀⣿⣧⢠⡄⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⡠⢳⣶⣿⣿⣿⡿⠃⠀⠀⠀⠘⠂⢻⣧⠀⠀⠀⠀⠀⠀⣿⡟⠘⠃⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⢱⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⣀⣴⣾⠟⠁⠀⠀⠀⠈⠛⠛⠀⠙⠛⠁
⠀⠀⢸⡜⣿⣿⠋⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠄⠄⠄
⠀⠀⠀⢳⡜⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣶⣀⣈⣀⣀⣤⣶⣶⠆
⠀⠀⠀⠂⢻⡉⣿⣿⣿⣿⣿⣿⣿⡟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠈⠳⣽⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣀⣰⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⠆
⠀⠀⠀⠀⠀⠀⡞⠙⠛⠛⠋⢣⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⣸⢀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⣤⡇⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣄⠒⣶⢤⠤⡴⢶⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⠞⠫⠚⢡⠮⢂⠕⣡⢄⡡⢊⠔⠩⢫⡞⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡵⠏⠀⡀⢐⠍⡠⢀⡠⢀⠔⡠⢀⠄⡀⠐⠀⠐⢀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣺⠅⡢⢑⠬⢂⠔⡨⢒⠥⡪⡔⠝⡪⠕⠊⠄⢐⠅⡨⠚⡠⠂⠀⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠕⠠⠀⢡⡠⠀⡁⠢⣁⠼⠋⠔⡠⠐⡡⠊⣰⠀⢂⠔⢁⠔⡡⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠴⢊⢴⠎⢊⠔⡣⢲⡁⠡⠄⠀⢊⠔⠞⠡⠊⡡⠔⡡⢊⠄⢊⠔⠂⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢕⡡⢊⠔⠠⡊⡜⠆⢐⠠⢊⠄⡢⡄⢴⡈⠔⡡⢐⠡⢊⡠⠒⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡁⢈⠁⡠⠊⡔⠱⡑⣤⢈⠔⣡⠔⡹⢆⠑⢎⡐⠁⠔⡁⠒⠁⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡖⠡⠊⢔⡡⣊⡠⢁⢚⡡⢊⠔⠈⢔⡢⢻⡀⢩⢔⡈⡂⠀⠀⠀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⣊⠐⠁⡨⠈⠁⢞⡡⢂⠁⡊⠔⠙⢊⡶⣡⠊⠑⡩⠀⠀⠀⠀⠄⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣎⢂⠐⡡⢘⠈⡷⠁⢊⠀⠡⢀⠍⢰⡉⠀⡠⠠⠂⠔⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣦⡩⣪⠃⢞⠥⡊⢐⠅⣊⢄⠆⠀⠸⠅⠀⠁⡀⠂⣀⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠉⠙⠿⣤⡀⠹⣊⡐⠁⠊⠁⠁⠁⠀⠀⠀⣀⠠⠀⠀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⡉⠲⠷⠿⢦⣴⡾⢛⠀⢈⠻⠛⠋⡀⠛⠁⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣐⣂⣭⠟⣁⣤⣤⢶⡲⠟⠿⣧⣒⢚⠓⣶⢾⠜⠓⢚⣱⣟⡖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡗⣼⣏⡀⠠⠤⠈⢽⣌⠋⠉⠒⠶⠤⠉⠛⠋⢡⠄⠒⣠⣿⢫⡿⣽⣧⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣣⡿⢯⣞⢟⣦⣁⣒⢲⣛⣛⣺⣷⣦⣀⣀⣀⣀⣀⣴⣿⣟⡟⣿⢳⣛⣽⠿⢿⢿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣸⣻⠿⢞⡭⣘⣶⠋⠿⠿⢻⣿⠏⣛⣩⣿⣛⢟⣯⡽⢿⠿⠟⠘⣃⣋⡩⢥⡲⠍⠩⠍⠱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⡠⠈⠉⠓⠚⠛⠋⠽⡭⠦⠤⠤⢼⠭⡮⢭⠉⠟⠟⠟⠘⠲⠂⠈⠁⡄⣞⣛⣋⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠂⠐⠀⠀⡔⠰⢁⠌⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠁⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡀⠀⠆⡀⠠⠀⠄⠐⠰⢀⠃⠄⠐⠠⢀⠁⠂⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⡦⣀⡁⠁⠰⣱⠃⠞⠾⠲⠷⢷⣆⣤⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡟⠁⠀⠉⠐⠗⠀⠲⠒⢢⣄⢸⣼⣧⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⣠⣦⡄⣄⣀⣀⣼⣽⣄⣨⣅⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠂⣼⢋⠉⠀⠉⣖⠽⠬⡌⠁⢭⡤⠭⡘⣟⡙⠫⢨⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢰⡟⢀⠖⠉⢀⠠⠐⠠⠚⠀⠀⠀⠀⡇⢧⢘⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣀⠟⡁⢫⠐⠠⠢⢄⠠⠐⠀⠀⠀⠀⠀⢀⠘⣇⢀⣴⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡵⢄⡉⠀⠡⠒⠂⠀⠠⠤⠤⠤⠀⠐⠒⠻⢄⡟⠋⣙⣿⡟⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢦⢤⡜⢰⠁⠀⠀⠉⣴⣤⣤⠀⠀⢀⣀⣀⣀⣀⣠⢤⡏⣹⣨⣿⠆⠛⠛⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡔⣥⣶⢀⣇⢤⣔⠾⠝⠃⠉⠀⠀⠠⠍⠩⠭⠀⠀⠀⠀⣣⢸⡷⣟⣉⣁⠉⠁⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⡣⣾⣧⣬⣬⣭⣥⠖⣆⣒⣒⣄⣤⣥⡬⠭⢭⣭⣭⣁⡀⣽⢐⠞⠁⠠⠛⢆⠐⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣷⣝⣻⣟⢵⡦⠦⠄⢀⣀⣀⣁⠀⠰⠛⠳⠗⠒⠲⠶⠒⣯⣼⢀⣴⡆⣿⠸⠆⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠛⠛⠛⠛⢻⠀⢠⡛⠏⡶⠓⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠳⠞⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣤⣤⠤⠤⠶⠖⠶⠒⠒⠋⠉⣙⠉⠉⠉⠀⠁⠈⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣯⠔⠂⠈⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⣿⣿⡆⠀⠀⠀⠀⡠⠁⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⡀⠀⠀⠀⢀⡀⠀⠀⠁⢛⣻⡇⠀⠀⢀⠜⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠺⣿⠇⠀⢀⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠀⠀⠐⠒⠀⠀⢹⣿⣿⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⣄⠠⢄⠄⡐⠀⠒⡰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠆⠅⠀⢀⣤⣄⣠⣤⣤⣴⣶⣄⠉⠲⠻⠂⠙⢲⡁⢀⣴⡿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⣿⠟⠉⠉⠁⠀⠉⠉⢾⣆⣀⡀⣬⣷⣾⢯⣜⣛⣻⣭⣿⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣟⢿⣿⣿⣿⣿⣤⣤⣀⣄⡤⠤⣴⣛⣻⣿⣿⢷⢾⣿⣿⣿⣝⢿⡻⠿⡛⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⣷⣭⡻⣟⡭⣭⢽⣷⣾⣿⣿⣿⣿⣿⣮⡻⠷⠗⠛⠽⠉⠊⠁⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⡋⠻⠿⣦⣝⢾⡳⠝⢎⡻⠕⠋⠉⠑⠀⠉⠀⢀⣰⣦⠀⠺⠀⠀⢷⡿⠁⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⢀⢩⠻⣿⡇⠈⡀⣠⡀⢤⣼⡆⠀⠛⠀⠈⠻⠋⢀⠠⠄⢰⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠀⠑⠠⣸⡇⠐⢷⣿⠃⠈⠉⠀⠠⠄⠒⠘⣿⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⢣⣤⠀⠐⠈⠁⠀⠀⢀⡀⣀⠤⢿⣧⠀⠀⡀⠀⠘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠘⣃⣀⠠⠤⠤⠒⠉⠁⠀⠀⠀⠈⣿⠀⠀⠀⠂⠄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⣀⡠⠤⠒⠒⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠘⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡮⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠁⠀⠀⠀⠂⠀⠐⠀⢠⣠⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝

⠀⠀⠀⠀⠀⠛⠛⠓⠶⣦⡄⢀⡤⠀⠀⠀⠀⠀⢠⣂⣴⣶⣶⣲⣀⢠⡆⠀⠀⠀⠀⠀⢀⣀⣤⣔⣆⣄⠀⣠⡀
⠀⠀⢀⠀⠀⠀⣀⣀⠰⠄⡀⢸⡇⠀⠀⠀⠀⣶⣿⣿⢿⣛⡽⠏⠁⠻⠀⠀⠀⠀⣠⣠⠀⢠⣤⠤⠒⣚⠄⢵⠇
⠀⠀⠈⠀⠀⢀⣌⠉⠐⠛⠁⠀⠙⠀⠀⠀⠂⠈⠉⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡁⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣄⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠈⠐⠂⠈⠻⠀⣠⠀⠀⠀⠀⠀⠀⠔⠎⣿⣿⣿⣿⡿⠷⣺⡇⠀⠀⠀⠀⠀⢀⠈⠙⣿⣿⣿⡆⣀⡄
⠀⠀⠀⠀⠀⠐⠓⠉⣠⠈⣿⠀⠀⠀⠀⠀⠘⠒⠈⠀⢽⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠤⠌⠀⠀⢾⣿⣿⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠇
⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡄
⠀⠀⠀⠀⢾⡛⡟⣿⢹⠀⢀⡄⠀⠀⠀⠀⠀⣠⡞⢩⣍⢹⢇⣠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠠⣄⡀
⠀⠀⠀⢳⠸⡧⣿⣸⢸⠆⢽⡇⠀⠀⠀⠀⠀⣽⣧⣼⣼⣾⣼⠉⠳⠀⠀⠀⠀⠀⠀⠈⠃⠰⠻⠉⡿⣛⠓⢲⡿
⠀⠀⠉⠚⠪⣿⡃⠚⠻⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠻⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣶⣶⣦⣤⣄⠀⣠⠄⠀⠀⠀⠀⠀⢀⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣶⣴⣶⡄⢀⡀
⠀⠀⠀⢀⠨⢿⣿⡟⣛⠉⠉⠿⠀⠀⠀⠀⠀⠀⠘⢀⠀⠐⠈⠑⠀⣤⡀⠀⠀⠀⠀⡴⠿⠈⠏⣯⣿⣿⠷⢿⡗
⠀⠀⠢⠀⠀⣬⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠠⣀⡟⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡑⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠈⠀⠀⣠⡧⢱⠍⠲⣾⠄⠀⠀⠀⠀⠀⠀⠀⡭⢽⣿⣿⠗⠲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⠙⠻⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⣤⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠕⡹⡣⡄
⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠘⠛⠻⠛⠛⢛⠛⡛⠛⠛⠛⠛⠀⠀⠀⢀⡀⠀⡀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢄⠀⢰⡕⡡⡪⢊⠔⠁
⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠘⢿⠿⠿⠿⠶⠶⠤⠶⠾⠿⠿⠁⠀⠀⠸⣦⣄⣀⠀⠀⠀⠀⠀⠀⠁⢐⣦⠀⠀⠀⠀⠀⠀⠐⢣⣴⠛⠮⠞⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⡿⠿⠶⠦⠴⠶⠾⠿⠿⠿⡟⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠓⠖⠛⠛⠛⠁⠀⠀⠀⠀⠀⢀⣴⠏⠉⢶⣄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠛⠞⠓⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⠉⡋⡟⠂⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⢘⠀⠀⠂⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣆⠀⠀⠀⢠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠲⣶⠖⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⢿⠿⡇⠀⠀⢠⠪⢅⣛⣂⠀⠀⠈⠉⢲⠤⢄⡀⠀⠀⠀⠈⢿⣿⣿⣦⠀⢠⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿⣻⣯⣿⡭⠷⠿⣶⣿⣿⣿⡧⠀⠀⠈⠛⠶⠦⣤⣭⣭⣭⣽⣥⣤⠴⠟⠀⠀⠀⠀⠈⠻⠿⠛⣷⠛⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⡿⠷⠶⠖⠒⠚⠓⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⢀⣀⣀⢀⢀⢀⣀⣀⡀⠀⠀⠀⣼⡟⠛⠛⠛⠛⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⠀⢀⣼⣄
⠀⠀⠈⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⢰⣿⣷⣤⣤⣤⣤⣶⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣿⣿⣿⡿⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣾⡿⠃⠀⠀⠀⠀⠸⠋⠉⠙⢿⠋⠉⠙⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿⣉⣿⣿⡏⢠⣄⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡠⣶⣼⣿⠿⠋⠀⠀⠀⠀⠀⠀⢸⣶⣴⣾⣿⣷⣶⣶⡇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣿⠆⣾⣿⣧⣺⢹⣣⣿⡇⠀⠀⠀⠀⣴⣾⣷⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣷⣤⣤⡀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢸⣻⠛⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠿⠿⡟⠉
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠹⣿⣿⣴⣦⣙⡛⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣧⠁⠀⠀⢘⡄
⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⡎⠻⢻⣿⠆⠀⠀⠀⠀⠀⡜⠌⠉⠉⠉⠁⢱
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⠝⠃⠀⠀⠀⣀⣼⣴⡄⠀⠀⢀⡀⠀⢃
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠉⠁⠀⠀⠀⠀⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⡄⠠⠠⢤⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⣄
⠀⠐⢀⠀⠀⠀⠀⢴⣿⠉⡉⠛⢻⣖⠀⠀⠍⠁⠒⢄⠉⢣⠀⠀⠈⠓⢤⣄⡀⠠⠂⠀⠀⠀⠀⡄⠀⠀⠀⠠⠂⢀⡁⠤⡤⠬⣀⠈⣆⠀⢰⡀⠱⡀⠀⠈⢀
⠀⠀⠀⠀⠁⠁⠀⠈⠛⠿⢴⠢⢭⣀⠀⠀⠀⠀⠀⢺⣣⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⣤⠐⠒⠈⠀⠀⠔⢀⡤⢤⠼⠦⠌⠉⢙⣎⠱⢈⠀⣾⠷⣀⠈⢆⠀⠤
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⡂⠄⠀⢀⠽⠋⠀⣀⠀⠀⠀⣀⣽⡆⠀⠀⠀⠀⠀⠀⠀⠘⢀⢌⡵⠋⣠⠀⠀⣠⠟⢀⠔⠃⠀⢹⡄⠈⠀⠀⠠
⠀⠠⠞⠀⠀⠀⣤⡖⣻⢭⡑⠩⡉⢁⠀⠀⠀⠒⠒⠒⠒⠊⠁⢀⣄⣭⣭⣭⡸⠀⠀⠹⠀⡤⠀⠰⡇⢲⠃⠁⠉⣀⠤⠔⢁⠤⠂⠀⠀⠀⠈⢷⡀⠀⠈⡇⣃
⠀⠀⠀⢀⣤⠔⠂⠀⠙⠀⢀⡤⠁⠈⠀⠀⠠⠶⣤⣀⠀⠀⢠⡾⣿⣿⣿⣻⠃⠀⠂⠒⠛⠂⠀⠀⢟⣬⠿⢫⣭⡴⠒⠁⠀⠐⠒⠛⠂⠒⠀⠀⠙⠷⠾⠵⠃
⠀⠀⠴⠁⣠⠦⠒⠚⣀⣌⡥⠀⠚⠀⠀⠀⠀⠀⠈⢻⣷⡈⠒⠛⠅⠀⠊⠁⠀⠤⠀⡀⠀⢀⠀⠀⠈⠀⠋⠁⠀⠀⢀⠀⠤⠄⠐⠂⠠⠀⠀⠀⢀⣀
⠀⣂⣤⣿⠷⠚⠻⠛⠋⣩⣔⣤⣄⣴⣄⠀⢦⠄⠀⠀⢻⣿⡄⠀⠀⢀⠠⣤⠠⣔⠛⠻⢷⣌⠀⠀⠀⠀⢰⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢄⠑⡀⡀
⠀⠛⠛⠁⠀⠀⠀⣤⣶⣿⣛⣿⣿⣷⠟⠀⢸⠀⠀⠀⠀⠻⢿⡄⠀⠄⠀⢰⡇⠀⡀⠀⡀⣆⠁⠀⢤⠀⢸⠀⠀⠄⠂⠀⠀⠀⢀⡀⠀⠈⠠⠀⠁⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠓⠀⠀⠀⠀⡇⠀⠀⠀⢠⣘⣷⠀⠀⠀⠘⠷⣤⣤⣤⣴⡿⠁⠀⡚⠀⠈⠀⠀⠠⣴⠗⠈⠉⠈⠀⠒⢤⡀
⠀⠀⠀⠘⠦⠀⠀⠙⠈⠈⠁⣀⡀⠀⠀⠀⠀⠘⣆⠤⠀⠀⢿⢽⡇⠀⢠⡀⠀⢸⠉⠁⠁⠁⣀⠆⠁⠀⠀⢀⠀⠀⠹⣄⣀⠀⠀⠀⢠⠞⠠
⠀⠀⠀⡀⣀⣀⣀⣀⣀⡀⠤⠄⡀⠀⠀⠀⠀⠀⠈⠒⠠⠀⠀⠉⠀⠀⠀⠀⠙⠚⠲⠆⠓⠀⠀⠀⠈⠘⠁⠀⠂⠀⠁⠀⠙⠻⠟⠿⠕⠂
⠀⢀⣙⣉⡑⠂⠑⠁⢉⣀⠀⠐⠮⡳⡄⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠈⣋⣰⠀⠀⠀⠀⢀⠀⠀⠀⠒⠰⡤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠈⢿⠿⣁⠀⠁⠢⠤⢼⣆⠰⠠⠐⠀⠀⠀⠀⠀⠋⠀⠀⣀⣀⡤⠴⠛⡫⠁⠀⢀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢙⠂⢀⠀⣴⣾⣿⣿⠟⠛⠻⠿⠶⢿⣟⡛⠁
⠀⠀⠀⠑⠈⠑⠂⠭⣀⡈⠙⣷⣆⠀⠀⢠⣀⣤⡤⠶⠶⠿⠏⡁⠀⠔⠋⠀⡀⠁⠈⠈⠂⠀⠀⠀⠀⠀⢀⠠⠂⢀⡘⠀⠀⠉⠛⠿⠷⣦⣄⣀⠀⢀⣏⣿⡆
⠀⠀⠀⠀⠀⠈⠓⠒⠠⠭⠽⠟⠋⠀⠀⠀⠉⠀⠐⠒⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠷⠒⠂⠀⠶⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠂⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠳⠁⠀⠀⢀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⠁⠀⠀⠀⢤⡶⠏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣼⣤⡀⠀⠀⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠂⠠⠤⠭⠀⡄⠀⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⣠⢞⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡇⢰⡀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠠⣄⣀⠀⠀⠂⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡷⣀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⢀⠘⢑⣂⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⡸⠁⠈⠣⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠆⡇⠀⠀⠀⠀⠐⠄⠀⢰⠀⡇⠀⢀⠀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣀⡯⡰⠁⠀⠀⡠⠊
⠀⠀⠀⣀⣤⠤⢶⣶⣶⣖⣾⣓⣂⡀⣛⡂⣲⣶⠀⠦⠤⢄⡀⠀⠀⠀⠀⠀⠀⢰⠥⠟⠀⢀⡠⠊
⠀⢰⡏⢡⣶⣶⣶⣯⠛⠋⠛⣁⣈⣭⡍⠉⣩⣿⣶⣦⣤⠀⢀⣦⠀⠀⠀⠀⠀⢸⡆⠐⣶⡉
⠀⠘⠉⠛⠻⠯⢭⣥⣸⣛⣛⣛⣉⣉⣀⣀⣈⣍⢭⢭⣄⣲⣭⣰⠀⠀⠀⠀⠀⠈⠘⠛⠛⣇
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢧⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⡟
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣹⣿⣿⣿⢯⣿⠇⠀⠀⠀⠀⠀⠀⡀⠀⢰⡇
⠀⠀⠀⠀⠑⢤⣤⣀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠟⠁⠏⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠺⠿⣿⣷⣶⣿⣿⣿⣿⣿⠿⠊⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⡏⠀⠀⠸⣆⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣖⣚⠋⠀⠂⢤⢿⡄⠀⠀⠀⣄⣀⡄⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠛⢝⢻⣿⣿⣦⣶⣶⣶⠟⢷⡀⠀⠀⠀⢿⢷⣶⣤⣀⡀⠀⠀⢀⠀⢿
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠉⠙⠛⡟⠛⠉⠁⠀⠘⣧⠀⠀⠀⢸⠀⠉⠙⢿⠿⠿⠿⠏⠀⢸
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⡄⠀⠀⠘⡆⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⢠⣾⡆⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢠⣿⠏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢠⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣠⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠛⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢀⣤
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⣴⣿⣿⣿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⡄⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⢸⣿⣿⣦⡀⠀⢿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⡀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⢠⡶⠶⣦⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⡶⠶⣦
⠀⠀⠀⢻⣄⣀⣼⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢻⣇⣀⣼
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠉⠉⠙⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡧

⠀⣀⠀⠀⠀⠈⠐⠂⠠⠄
⠀⠀⠉⠉⠒⠠⠀⠀⠀⠀⠀⢤⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣄
⠀⠀⠀⠀⠀⠀⠈⠂⢤⡴⠤⠴⠀⠀⠈⠈⠐⠒⠤⢄⣀⣀⣀⣠⡤⢤⠦⢖⠶⠒⣋⣋⣙⣭⢿⣿⣃⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠻⣍⠻⢍⠻⣽⠷⠿⠶⣫⣳⣅⢬⡽⡿⢿⣾⣿⣿⣿⣿⡏⠨⢐⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠙⠲⣾⠚⢟⠻⢿⣷⣛⣿⠿⣛⠢⢑⠩⣛⣸⣣⣥⠦⠵⠖⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠓⢦⡀⠀⠀⠙⠲⣭⣶⣌⠶⠬⠷⠖⠛⡋⠉⠩⢹⡅⠂⠄⡀⡈⠓⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⣀⠈⠳⢤⡀⡆⡟⠁⣠⠀⠀⠀⠀⠀⡀⠤⠀⠀⠈⢁⠀⢈⠑⠠⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠓⢦⡀⠙⠲⣿⠂⢀⠀⠀⠀⠤⠀⠀⣀⠀⠀⠀⠀⢑⣴⡄⠀⠀⢽⠀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠘⠀⠀⠀⠀⢙⠲⣄⣻⢀⡈⠀⠀⠀⡀⠀⠖⠀⠀⠖⠀⠀⠀⠈⠁⠀⠀⢸⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠤⠔⠈⠁⠐⠋⢀⡬⣹⠾⠇⠐⠀⢢⣁⠀⠄⠉⠀⠲⣀⠀⠀⠀⢀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠈⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢈⣙⠻⢦⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣶⣟⣉⣈⣭⣿⣿⣿⣿⣯⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⠟⠉⢭⣽⣿⣿⣿⣿⣿⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃⠀⠀⠐⣺⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⠁⠀⠀⢤⣼⣿⣿⣿⣿⡭⠪⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠀⠀⢒⠺⢿⣿⣿⡋⠐⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⡀⣸⣿⣿⠿⡉⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠈⣙⣿⡿⠓⠚⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢀⣀⣂⣈⡉⠉⠁⠀⠀⢴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣴⣿⡿⠇⣠⠄⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠉⢀⣤⡾⠃⠀⣠⣴⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⣀⣤⣶⡿⠟⣡⣶⣿⣿⠏⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁⢀⣾⡿⠛⢉⣐⣿⣿⣿⠿⣡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠚⠉⠀⠀⢀⣤⠞⠋⠀⣀⣼⣿⣿⣿⣣⠞⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠖⠒⠋⠉⠀⠀⠀⢀⣠⣤⡤⠈⠉⠀⠠⢀⣨⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⣤⣾⡿⠋⠉⣀⣠⣤⣶⣶⣾⣿⣿⣿⣟⢛⣋⣀⣤⣶⣾⡿⠟⠋⠁
⠀⠀⠀⠰⣿⣏⠠⠚⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠈⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠬
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⠀⠀⠈⠁⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠠⢄⠀⠀⠀⠂⠀⠐⠀⢈⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢵⣖⠒⠁⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠠⢶⢀⡀⡀⡀⠀⠐⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣄⣀⣀⠀⢀⣀⡤⠞⠁⠀⠀⠂⢤⣤⠀⠀⠀⣠⡤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣤⠄⠒⠈⠁⠀⠈⠉⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠘⠃⠑⠤⣀⠀⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠠⠄⠤⠔⠒⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠄⠀⢀⣾⠋⠁⠀⠀⠀⠈⠉⠢⣀⠀⠀⠁⠐⠠⢄
⠀⠀⠀⡀⠀⠀⢤⣀⣸⣿⠦⣄⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠑⡄
⠀⠀⢰⡁⢀⣤⣄⡉⠙⠛⠻⠶⠮⣽⣾⣿⣶⣶⣶⡿⠀⠀⠀⠀⣀⠠⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠈⣇⣾⠀⠀⠈⠁⠀⠒⠤⠀⠀⠈⠻⣿⠿⠯⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⢀⠔⢁⠄⠀⣠⣄⣴⠏⠃⠒⠀⠀⠀⠀⠘⠀⠀⠄
⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⠀⢲⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠐⠛⢠⠽⠋⠻⠶⠒⢤⣤⣤⣤⣤⣤⣶⣤⡆⠀⡀
⠀⠀⠀⠀⠹⣿⣿⣦⣀⠀⠀⠀⠀⢸⠇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠐⠈⠀⠀⠀⠀⣴⠖⠷⠿⠿⣑⠫⠋⢭⣿⣧⡎⢀⣀
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣽⣧⣥⡎⠀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁⠠⠐⠀⠈⠁⠀⠀⠈⡥⢿⡟⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠂⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢁⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⡤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠤⢛⡕⠁⠀⠀⠀⠀⠀⠀⠀⠁⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡂⢶⠚⠈⠀⢀⠎⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠎⠀⠃⠀⠀⠀⠀⠛⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡤⢌⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠀⠀⠀⣤⠙⡄
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠄⠀⣈⣀⠄⠀⠀⠉⠀⢹⡀⠄⠂
⠀⠀⠀⠘⣧⡀⠀⢠⣶⠀⣴⠛⠉⠀⡀⠀⠀⠀⣰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⠶⣖⠀⠐⢧⡀⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠉⠛⠶⣾⡇⣸⣗⠈⠤⠊⡰⢀⢼⠁⠁⠀⠀⠀⢀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣤⣘⣶⡤⠓⠷⢤⣝⣫⣥⠄⠐⠀⢀⠃
⠀⠀⠀⢦⠀⠀⠀⠀⠑⠛⠂⠀⠀⢰⡆⠚⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠡⠤⠤⠤⠤⠍⣉⠉⠉⠉⠛⠿⢿⣶⣛⠓⣀⣀⣴⣶⠏
⠀⠀⠀⠈⠀⠀⠖⠊⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠢⣤⣀⡄⠀⢻⣿⣿⣯⠾⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡐⠤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠜⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣬⣥⣤⣤⣄⣁⣀⣀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠒⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⢟⣡⠞⠁⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣿⣿⣿⣿⣿⢟⡵⠋⠀⠀⠀⢸⡃⠸⣄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣿⣿⡿⢛⡵⠋⠀⠀⠀⠀⠀⢀⡗⢠⠟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣫⣴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣼⠋⣿⠉⣿⠙⣇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣰⠇⠀⣿⠀⣿⠀⠹⡆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⡏⠀⠀⣿⠀⣿⠀⠀⢻⡄⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⡞⠀⠀⠀⣿⠀⣿⠀⠀⠀⢷⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡼⠁⠀⠀⠀⣿⠀⣿⠀⠀⠀⠈⣧⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⠃⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠘⣇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⢿⣀⣿⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁



⣀⣀⣀⣀⣀⡀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⠞⣇⠀⠀⠀⠀⡼⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠦⣄⠀⢰⠁⠀⠘⢦⡀⠀⡸⠃⠈⠳⣄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠳⡇⠀⠀⠀⠀⠙⡾⠁⠀⠀⠀⠘⣦⠀⣠⢻⠀⠀⠀⢀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠃⠀⢧⣀⡴⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣀⠀⠀⠀⠀⠤⠀⠀⠀⡠⢄⠀⠀⠀⠀⢀⡀⠀⠀⠛⠁⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢠⠉⠐⢄⣠⠂⠀⠈⠰⡔⠁⠀⠡⡄⠀⠈⠀⠠⠀⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠂⠀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠠⣀⠀⠀⣠⠖⢤⡀⠀⣠⠖⠢⣄⠀⢀⣠⣄⠀⠀⠀⣠⠃⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡼⠉⠳⠶⠁⠀⠀⠙⠋⠀⠀⠀⠈⠒⠋⠀⠈⢣⣀⠴⣏⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣁⣀⡠⠤⠤⠤⠒⠒⠒⠒⠒⠒⠒⠢⠤⠤⣄⣀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⣀⣄⢀⠀⠀⣀⡤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⡀⠀⣀
⠀⠀⣠⠎⠁⠀⠈⠗⠉⠀⠀⣀⠤⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠤⠤⠄⠀⣀⡀⠀⠉⠿⠉⠳⣄
⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠓⠛⠋⠉⡁⠤⠤⢤⡄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠘⢆
⡼⠅⡀⢀⡴⢦⠀⠐⠀⠀⠰⠆⠐⠐⠀⠀⠤⠄⠀⡀⠀⠀⠀⠈⠀⠀⠀⠒⠒⠂⠀⠀⠘⠃⠀⠀⠀⣦⣄⣀⣸
⠇⣀⡴⠏⠀⢸⣧⠀⠀⠴⠐⠒⠂⠀⠀⠀⡀⠈⠉⠁⠀⠀⠓⠒⠀⠁⠠⠤⠄⠐⠒⠠⠀⠀⠀⠀⠀⢻⠿⣤⡼
⠀⠉⠀⠀⠀⢸⡆⠑⢄⠀⠀⠁⠀⠐⠂⠊⠙⠃⠀⠀⣤⠀⠀⠀⠠⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠎⢸
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⠐⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠒⠒⠐⠒⠁⠀⠀⡀⠄⠊⠀⠀⡇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠦⠶⠤⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢰⠃⡼⢁⡉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡼⠁⠈⠀⠞⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣇⠲⠀⡈⡘⠆⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⡶⠀⢀⠀⠚⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⣼⢃⡴⠋⠙⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠶⠚⠁⠀⠀⠀⡜⠁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⠤⣤⡄⢠⡤⣤⡤⠛⠀⠀⠀⠀⠀⠀⠀⡸⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⠟⡆⢸⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⣼⠃⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡿⠀⡇⢸⢻⡄⢣⠀⠀⠀⠀⠀⠀⣸⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡸⠃⠘⡇⢸⠀⣇⠘⡆⠀⠀⠀⠀⠀⣧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⠃⠀⠰⠀⢸⠀⢻⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⡎⠀⠀⢸⠀⢸⠀⠘⡆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡜⠀⠀⠀⢸⠀⢸⠀⠀⢧⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣴⠁⠀⠀⠀⢸⠀⢸⠀⠀⢸⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣰⠃⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠃⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠸⡀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡞⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⠀⠀⢧⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧⣴⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠃⠉⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠴⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠔⡂⠄⠈⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡡⠀⠠⠊⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⡐⢁⠄⠀⠀⠐⠈⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⣮⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠿⣾⣶⠶⠶⣵⣾⣥⣴⣎⠀⠀⠀⠀⠀⠛⠂⡀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⢞⣩⠃⠂⠀⢈⠹⢁⡬⢬⢻⠋⣍⢙⣛⢶⡞⣀⠀⠀⡀⠀⠀⠀⢀⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⣀⡀⠀⠀⢰⣿⣾⠶⢟⣱⠀⠀⠀⢃⠀⠐⠀⠓⠘⢄⠖⠀⢘⠏⠫⣻⢋⠀⠀⠀⢠⣟⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠈⠉⠀⣠⣴⣿⣛⣬⣤⡟⠁⣸⣿⡡⠀⠇⠀⠀⠀⠀⢊⡀⢀⢀⡈⠂⠠⠀⠈⣲⣾⣿⣿⣏⠕⠀⠀⠀⠀⠀⠀⢀⠜
⠀⢸⡶⠁⣀⠾⡿⠿⣻⡭⠉⠙⠛⠒⠋⢈⠛⠀⠀⢰⣠⡠⡃⠀⢷⢱⡋⠀⠀⠀⠀⠀⠛⠄⣿⣿⣿⡄⠀⠀⠀⢀⡴⠊⠁
⢠⠋⠀⠈⡔⣩⡍⠑⡌⠃⠀⣨⠟⠀⠀⠄⠀⠀⠲⠈⢐⢃⠒⠀⠑⡚⠃⠄⠀⠀⠀⠂⠢⣴⣿⣿⠅⠀⠀⠀⠀⠸⡅
⡏⢳⡀⠀⢇⡛⠓⣠⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠬⠂⠀⢕⣐⠡⣠⢀⡴⠞⠛⠳⣾⣿⡿⣧⡀⠀⠀⠀⠀⠀⠙⡄
⠐⠀⠙⠀⠀⣉⣉⠥⠀⠀⠀⠀⠀⢀⠐⢁⠆⠀⠀⣠⠄⡞⣠⣜⣧⣠⣴⡏⠙⡀⠀⠀⢿⣿⡿⠌⠓⠄⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠑⢆⠀⠀⡄⢀⠀⠀⠀⠠⠘⠁⣴⡟⢄⠀⠄⢀⠆⢢⣾⣿⣿⣿⡟⠀⠀⠈⠐⠒⢺⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠉⠓⠦⠀⠀⣀⣤⣒⣀⢴⣿⣧⡀⠃⠀⠀⣰⢾⣿⣿⣿⠟⠀⠀⠀⠀⠀⢀⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⢿⣽⣿⣿⣿⣿⡽⣿⣾⣿⣿⡷⠀⠀⠀⠀⠄⠀⣸⠁⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠳⣽⠿⠿⠿⠿⢿⡿⢯⣁⠀⠀⠀⠐⠊⠀⠀⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠙⠲⣤⡀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡀⠀⠀⠀⠟⠦⣀⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣰⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠐⠄⠀⠀⠅⠀⠀⠉⠑⠀⠀⠀⠀⠀⢘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣽⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢳⣿⢿⣿⠯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⣿⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣄⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠉⠁⠀⠀⠀⠀⠀⣸⣷⣤⡀⠀⠀⠀⠀⠀⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⣀⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠛⠀⠀⠀⣀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠈⠻⣿⣿⡇⠀⠀⠀⣴⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠈⠙⢿⡄⠀⠀⠈⠛⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⠿⠟⠛⢛⣿⠛⠉⠉⣩⡿⠋⠉⢉⡽⠋⠉⢉⡿⠋⠛⢻⡟⠻⠿⢿⣶⣶⣤⣄⣀
⠀⣠⣶⡿⠟⠋⠉⣡⡾⠃⠀⢀⣴⠟⠁⠀⢀⣼⠋⠀⠀⣠⠟⠁⠀⣠⡟⠀⠀⣰⡟⠀⠀⢀⣾⠁⠀⠉⣽⠟⢿⣶⣄
⣼⣿⠋⠀⠀⣠⣾⠋⠀⢀⣴⡿⠉⠀⢀⣴⡟⠁⠀⣠⡾⠃⠀⢀⣶⠃⠀⢀⣴⡏⠀⠀⢀⣿⠋⠀⠀⢠⡟⠀⠀⠙⣿
⢿⣿⣄⣴⣾⠟⠁⢀⣴⣾⠏⠀⢀⣴⡿⠋⠀⢀⣴⠟⠁⠀⣴⡿⠃⠀⢠⣾⡏⠀⠀⢠⣾⡟⠀⠀⢠⣿⠃⠀⠀⣠⣿
⠈⣿⣿⣿⣷⣤⣴⣿⡟⠁⠀⣠⣿⠟⠁⢀⣴⣿⠋⠀⢠⣾⡿⠁⠀⣠⣿⠏⠀⠀⢀⣿⡿⠃⠀⢀⣿⣯⣤⣶⣿⣿⣿
⠰⣿⣏⠻⠿⣛⡛⠿⠿⢿⣿⣿⣷⣤⣴⣿⣿⣅⣀⣴⣿⣏⣀⣀⣴⣿⣯⣤⣤⣤⣾⣿⣷⠶⠿⠟⠛⠛⠉⣉⠀⣽⣿
⠀⢻⣿⡙⠒⢿⣿⣿⣷⣶⣦⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣥⣤⣤⣤⣶⣶⣿⡗⠀⠀⠀⣸⣿⠇
⠀⠀⢿⣿⡆⠘⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⢤⣴⣿⠏
⠀⠀⠀⢻⣿⣷⣶⣤⣬⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⢂⣶⡿⠃
⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠹⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣸⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⡛⠛⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣤⣼⣿⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠛⠻⠿⠿⢷⣾⣿⣿⠿⠟⠛⠛⠉⠉⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡿⣿⠃⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣹⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⠐⠒⠒⠒⠒⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣊⠀⠤⠒⡁⢀⢄⣔⣤⠶⣯⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣰⠞⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⣀⣀⣤⡾⠟⠛⠉⣀⣠⠼⢀⡗⠀⠀⢀⣤⣶⣿⠿⣶⣦⡈⢳⠀⠀⠀⠀⢀⡞⠁⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠯⠭⣭⣉⣉⣉⣩⣭⣭⠥⠖⠛⢉⣠⣴⣾⠿⠛⠉⠀⠀⢨⣿⠃⡸⠀⠀⠀⠀⡞⠀⣀⣠⣼⣇
⠀⠺⣿⠉⠁⠀⠀⠀⠀⠐⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⡴⣾⠟⠛⠉⠀⠀⠀⠀⢀⣠⣞⠅⠊⠀⠀⠀⠀⢸⢡⠊⠁⠀⠹⢿⡆
⠀⠀⠈⠳⡄⠀⠀⠀⠀⢀⠀⢄⠀⠑⢢⡀⠀⠀⠀⠀⠙⠓⢶⣤⣤⠒⠒⠒⠺⠿⢭⣤⡤⠤⠤⢄⡀⠀⢸⢸⠀⠀⠀⠀⠂⢿⡄
⠀⠀⠀⠀⠈⠢⡀⠀⠲⠊⠄⡸⣠⠀⠀⢱⠀⠀⠀⠀⠺⣧⡤⣿⣿⣶⣦⣤⣤⣀⣀⣠⣼⣦⡴⠞⠁⠀⠘⡎⢆⠀⢀⠀⡀⢘⣷
⠀⠀⠀⠀⠀⠀⠈⠒⢤⣒⡛⣉⣠⢜⣄⡜⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⠤⣤⡬⠤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠹⣌⠲⠤⠚⢀⣜⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣉⣁⣀⣀⠀⠀⠀⢀⡀⠀⠀⢀⠄⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠑⢭⣖⣺⡥⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠈⡆⠀⢠⠇⠀⠀⠀⠐⠲⡄⠀⠀⢇⠆⠀⠤⠤⠄⠈⠉⠲⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠁⠈⡵⠀⠀⢠⠃⠀⠠⡠⠀⠀⠀⠀⢃⠀⠀⠀⢀⠀⠈⠐⠄⢁⠠⡀⠑⡄
⠀⠀⠀⠀⠀⠐⠀⢀⠈⠀⠀⠘⠂⠀⠀⠀⠀⡠⡲⠁⠀⠀⢡⠁⠀⠀⠀⠀⠘⡄⠀⠈⢦⡑⢄⠀⠀⠀⣳⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠈⠢⠀⠁⠀⠀⠀⠀⠀⢀⠀⠠⠊⠀⠀⢀⠂⠂⠄⠀⠀⠀⠀⠀⢁⠀⠀⠀⠙⢄⠁⠀⠌⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⢨⠀⠀⠘⢄⠀⢀⠆⠀⠈⠀⠀⠀⠀⠀⠁⠄⠤⠄⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠠⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠠⠤⠄⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣤⣈⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠰⡟⠘⠃⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢎⢃⡀⠉
⠀⠀⠀⠀⠀⡠⠤⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠸⢄⣰⣦⡤⠂

⠀⠀⠀⠀⠐⢄⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠛⠾⠿⠇⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣆⣹⠄⠤⠀⠀⠀⠀⠀⣾⣟⣧⣬⣀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢉⢉⣽⣤⣴⣶⣾⣿⡿⡻⢿⣿⢿⣿⡿⣿⠟⠇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠤⣬⡽⣿⣿⡟⢩⡟⠚⠉⢸⡇⣘⣀⣦⠀⠀⠀⠀⠀⣀⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⢸⠀⢀⠁⠈⢘⣧⣬⣷⣼⡿⠿⠛⠋⠉⠘⡄⠀⠀⠀⠘⠿⡿⠁
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠮⣁⣿⡼⠷⠸⠷⠾⠛⠛⠉⠁⠀⡇⠀⠀⠀⠀⠀⢱
⠀⠀⠀⣶⡄⣧⠀⠀⠀⠀⠀⠀⠀⡾⠙⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⢈⣿⡈⣇⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢹⣷⣹⡆⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠹⣷⣿⡀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠘⣌⠳⡄⠀⠈⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡵⠶⠶⠶⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢿⡿⣿⠿⠷⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠿⠿⣿⣗⡂
⠀⠀⠀⠀⠀⣠⡶⣿⠛⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣄⣀
⠀⠀⠀⠀⠘⣿⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠦⣼⡏⢹⡿⠧⠾
⠀⠀⠀⠀⡀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠟⠉⠋⠀⠀⢀
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠒⠲⠶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⣀⡘⢻⣿⣿⣷⣦⢤⣠⣶⣟⣿⣿⣷⡶⢰⣶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣉⣛⣿⣿⣿⡿⢋⣥⣾⣿⣿⣿⢸⣿⣿⡇⢸⡱⣆⠹⠛⠋⣻⠿⠃
⠀⠀⠀⠀⠀⠀⣉⣛⣻⣿⡿⢉⣴⣿⣿⣿⣿⣿⠟⢼⠿⠉⠃⡈⠛⠚⠁⠀⠘⠧⢤⡄
⠀⠀⠀⠀⠀⢈⣉⣵⡿⠋⣴⣿⣿⣿⣿⣿⣿⣷⡄⠑⠆⠀⠀⠀⠰⠀⠀⠠⠀⢀⠔
⠀⠀⠀⠀⠈⢁⣺⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂⠀⡀⣄⠢⠄⣠⠖⠉
⠀⠀⠀⠀⠚⢙⡿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠷⠤⣀⠀⠂⠈⠛⠋⠁
⠀⠀⠀⠀⠴⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠁⠀⠁
⠀⠀⠀⢀⣴⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣀⣀⡠⠤⠄
⠀⠀⠀⠀⣰⡇⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠀⣴⣤⣶⣿⠟
⠀⠀⠀⠈⢨⡇⢸⣿⣿⣿⣿⣿⣿⣻⠯⢿⡯⢚⢤⣮⡿⠛⠋⠁
⠀⠀⠀⠀⠉⣵⢸⣿⣿⣿⣿⣿⣿⠆⢙⠈⠉⠙⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡄⠙⡌⣿⣿⣿⣿⣿⣿⠅⡀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⢿⣿⣻⣗⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠐⡖⠰⡘⣿⣿⣿⣿⣿⡟⡡⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠉⢯⡻⠿⣗⣄⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠹⣇⠛⡸⣿⣿⣿⣿⣷⡿⡁⠀⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⢡⠀⠘⣫⡇⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠹⠆⣿⢈⡻⣿⣿⣿⣿⣽⡂⠰⢯⠯⠥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡯⠀⢀⣀⣴⡏⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠿⡈⢷⣝⠿⣿⣿⣿⣿⣆⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡀⠀⢠⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠋⢌⣱⡮⠙⠿⢿⣿⣶⣷⣤⣀⠀⠀⠀⠀⠀⣀⠐⠀⢀⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠤⣤⣬⣙⢛⡛⣿⢿⣿⣿⣿⣿⡿⠿⠷⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⡐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⣤⡼⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠏⠀⠀⠀⠀⠉⠱⣆
⠀⠀⠀⠰⠾⢋⣤⠄⠀⠀⠀⠀⠀⠀⡀⣀⠒⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠈⠙⣤⠀⢹
⠀⠀⠰⠃⠈⠡⠁⠀⠀⡴⢂⡈⠀⠀⠀⠀⠈⠁⢢⣤⠀⡤⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⠀⠀⠚⠁⠀⠋
⠀⠀⢀⣀⡤⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠡⠂⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⡄⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠶⡀⢀⠀⠀⠀⢰⠄⣀⣤
⠀⠁⡈⠀⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⠀⢀⠀⠈⢰⠀⠀⠇⠈⡻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⢀⣉⡱⢤⠒⠶⠠⣄⣀⠀⣀⡠⠞⠉⠀⠀⠀⠱⠀⠀⢀⠂⠀⡆⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠜⡁⠓⠉⠩⣍⠅⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠃⠃⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⠠
⠀⠀⠈⠀⠁⠀⠀⠈⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠊⠁⠀⠀⡰⠋⠁⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣠⡀⠀⠀⠀⡀⠀⠄⠀⠠⢠⢠⣴⡒⠚⣉⣦⡀⠀⠀⢀⣼⣷⣶⡿⠷⠶⣄⡱
⠀⠀⠀⠀⣤⠁⣠⠶⠖⠀⠁⠀⠀⠀⠀⡀⠈⡀⠀⠀⢸⠈⠣⠤⠀⠀⠈⠙⠲⢤⡾⠅⠀⠈⠉⠉⠉⠙⠁
⠀⠀⠈⠀⠀⠀⠀⠐⠠⠀⠒⠀⠀⢀⡠⠜⠓⠉⠙⠐⡒⠒⠃⠄⠀⢀⡀⠀⡠⠊⠀⠀⠀⠀⠀⢀⡀⢄⡀
⠀⢠⠒⢄⠀⠀⣀⠄⡔⢸⠁⠀⡏⣰⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⢈⣩⣭⠯⠤⣀⠀⠀⣠⠔⢊⡤⠚⣡⠞⡁
⠀⠺⡀⢸⢒⠂⡌⢀⠁⠀⠀⠀⣅⢸⡀⠀⠀⠉⠀⠈⡄⣀⠀⠀⠊⠀⠁⠀⠀⡀⠋⠪⣯⣴⣯⠴⠚⠁⠠⠁
⠀⠀⢆⠷⠸⢦⠳⡄⢧⡀⠆⠀⠘⣂⢳⣀⣀⢀⣀⣠⣤⢖⡀⠁⣖⠀⠐⠀⠈⠀⠀⠀⠀⠑⠤⣀⢀⠔⠁
⠀⠀⠎⢀⠏⠁⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⡟⠉⠁⠀⠠⠉⠢⣽⡦⠤⠔⠠⠅⠀⠀⠀⠀⠒⠉⠁⠂
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠁⠴⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢶⣰⣿⣏⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⢒⣮⣭⣿⣶⣶⣮⣕⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠋⣁⣤⣤⣴⣾⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠏⠉⠀⠁⠀⠀⠙⠿⣿⣿⣿⠿⠙⠋⠗⠉⠲⢄⡀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⣁⣴⣿⡿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⡠⢤⡖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢈⣛⣿⡏⠀⠀⠀⠀⠀⠀⠁⠌⡙⠛⠛⠛⠛⠯⣁⣀⣤⣾⡿⠟⠉
⠀⢀⣠⣤⣶⣽⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⣟⠹⢍⡛⢯⡳⢤⡀⠀⠀⠀⠀⠀⠡⠀⢠⠀⠀⠀⢨⣿⣿⠋
⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠲⣀⠑⢬⡐⠄⡑⠢⣙⠲⣍⡳⢦⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡙⢿⡁
⢳⣝⢿⣿⣿⣿⣿⡏⠀⢠⠀⠒⠈⠁⢀⠀⠀⡁⠀⠑⢦⣈⠲⢌⡁⢄⡙⠦⡙⠢⣍⡳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣍⠢⣄
⠀⢻⣧⡙⢟⣋⠐⢄⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠓⢤⡉⠢⢌⡐⢌⠓⢤⡙⠲⣍⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠼⣮⣗⢄
⠀⠀⠻⣧⡁⠌⠳⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡘⠦⣌⠢⣌⠒⢤⡙⠢⣍⠳⣄⡀⠀⠀⠀⠐⠢⣻⡿⢇⠇
⠀⠀⠀⠈⠻⣶⠂⠉⠻⠷⣌⡙⢦⡀⠀⠨⠀⢄⠀⠀⡀⠀⠀⠀⠀⡀⠀⢙⠲⣌⡳⢦⣕⠢⣙⠲⣬⡓⢦⡙⠶⣤⡀⣀⠴⣫⣶⠏
⠀⠀⠀⠀⢰⡅⠰⣿⡁⠈⠻⢿⣦⡈⠳⢤⠀⠀⠀⠀⡀⠀⠀⡈⠀⠀⠀⢄⠁⢪⡙⢦⡙⠷⣬⡛⢮⣝⣷⣭⣶⡮⣛⣵⣾⡿⠃
⠀⠀⠀⠀⠀⠙⢶⣈⠻⣷⣄⠀⠌⠻⣾⣄⣑⢦⡈⠢⠀⠀⢤⡈⠳⡄⠑⢄⡀⠢⣀⢰⣍⡳⣮⡻⢷⣽⠿⣋⡵⢞⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡉⣉⣀⣀⣀⡉⠻⣦⡙⠲⣀⠁⠀⠀⠢⣀⢳⣯⡻⣦⣝⣷⣬⣻⡾⣛⣯⣶⣿⣵⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠙⠿⣗⠁⠈⠛⢶⣜⣷⣟⠻⣤⣾⣷⣌⣻⣿⡿⣟⣯⣷⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄⢶⣶⣶⣿⣿⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠈⠛⠟⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡔⠀⣼⣃⠀⢀⠈⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠛⢅⣺⣤⣴⡿⢿⣿⠭⠭⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠄⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⣄⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡏⠀⢠⣤⣤⣄⠀⠑⢆⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠋⠉⠙⠳⣤⡴⠄⣇⠀⠈⣿⣿⣿⣷⣆⠀⠑⠆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢰⡈⠛⢿⣷⠀⠀⠙⠙⠛⢿⣿⣷⣄⠀⠉⠠⣀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄⢘⣇⠸⠁⠀⠀⠀⠀⠀⠀⢻⣿⠿⣳⣤⡀⠈⠙⠾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡘⠎⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠈⠈⠻⣿⣷⣤⡀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠀⠀⣀⣀⠀⠀⠀⠀⢀⣾⣿⣷⣦⣄⡀⠉⠑⠻⣷⠀⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⣴⣾⣿⣿⣿⣶⠾⣿⣽⣿⣿⣿⣿⣟⣿⣿⣷⣤⣿⠂⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠛⠿⠿⢿⣿⣦⡀⠀⠀⠸⠙⣿⣿⣿⣾⣽⡿⠏⠀⣴⣿⣇⠀⠀⢀⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⠀⢀⡀⠉⠙⠷⣦⣴⣄⣫⣿⣿⣿⣿⠀⠀⠞⠁⠉⠉⠉⠀⠁⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢀⠉⠙⠛⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠒