Won Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⢧⣄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣠⣤⡀⠹⣧⡀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡰⣿⣿⠇⢠⣿⠀⠀⠀⠀⠈⣿⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣦⠉⣶⣿⠇⠀⠀⢰⣿⣿⢸⣸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣿⠇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⣿⣽⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢻⣿⠁⠀⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⡿⠀⠀⠀⠀⠀⢸⣿⡁⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⠃⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣀⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣤⣤⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⢠⡞⢁⣬⣻⣌⣭⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣭⣡⣞⣥⡈⢳⡄
⠀⠀⠀⠀⠀⢸⡀⠘⠮⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠻⠵⠃⢀⡇
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠖⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠢⣄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣛⣛⣛⣛⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⣶⣻⣿⣿⣿⣿⣿⣳⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⣽⠛⠛⠛⠛⠻⠿⠛⠛⠛⠛⢻⡵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡛⣚⡛⣛⣛⣛⣛⣛⣛⣛⣛⡛⣛⢚⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣛⣛⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣛⣟⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡆⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣚⡯⠵⠒⠚⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠯⢝⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠸⣅⡰⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⡿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⣫⠽⡇⠀⠀⠀⠀⠀⠀⠈⠀⡏⢸⡇⠀⠀⠀⠀⠀⠀⢸⠟⠒⢌⢦
⠀⠀⠀⠀⠀⠀⢰⢡⠎⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⡏⡇
⠀⠀⠀⠀⠀⠀⠘⡞⢦⡀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇⠀⠀⠀⠀⠀⢸⠇⠀⣀⡼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣽⣿⣋⣽⡄⠀⠀⠀⠀⠀⣏⣁⣀⣩⠇⠀⠀⠀⠀⡿⠭⠭⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢏⣿⡇⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⡟⡿⡆
⠀⠀⠀⠀⠀⠀⢠⢯⠎⡏⡇⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⡇⣷⡙⡄
⠀⠀⠀⠀⠀⠀⠣⠏⢸⣿⡅⠀⠀⠀⠀⠈⠛⠶⢤⣤⣤⣤⡤⠶⠚⠉⠀⠀⠀⠀⠀⢷⢹⠹⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠯⠽⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣙⠲⠤⣤⣀⣤⠤⠴⣺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠒⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⢡⠤⠓⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠥⠉⢀⠀⡀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠀⠚⠁⠉⠁⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⢰
⠆⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠈⣟⡄⠀⠀⠀⠀⠀⠀⢀⠤⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠙⣷⠀⠀⠀⠀⡄⠐⡧⠠⠄⠂⠑⠒⢠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢰⢇⠀⠀⠸⠉⠃⠀⠀⠀⠀⡀⠀⣿⠆⠒⠠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣤⣐⠎⠁
⠀⠀⠀⠀⡾⡄⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⡇⠘⡔⠁⠀⠉⠙⠫⣙⠉⠀⠉⠈⠉⠉⠁⠀⠀⢀
⠀⠀⠀⠀⠐⡖⠘⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⠀⠨⠀⠀⠀⠀⠀⠀⠑⢌⠢
⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠂⠅⣀⡀⠀⢄⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⠇
⣿⣿⡿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢤⣠⣤⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⠀⢯⠀⠀⠈⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡰⠛⠚⠓⠀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠰⠯⣀⣨⣧⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣴⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷⣦⠀⠋⠀⢀⡀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⣾⣯⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⢾⣿⣤⠞⣋⣉⡉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣫⠥⠶⢤⠙⣿⠀⠉⠙⠛⠛⠿⠿⠿⠿⠟⠛⠛⠉⠉⠈⠀⣿⠃⡞⠁⠀⠈⣧⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣰⠃⠀⡀⠘⠁⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣁⣠⠀⠀⣽⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⠀⠀⠙⠒⠊⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⠉⠀⠀⣰⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣧⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢀⣀⠤⠚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⣆⠀⠀⠂⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠗⣋⣩⡤⠶⠚⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡓⠦⢤⣤⠤⠶⠒⣛⠛⢷⣄⠀⠀⠀⠀⠀⠀⣠⡿⠀⠞⢯⠀⠀⠀⠀⠀⠹⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠶⠞⠛⣍⠁⣨⠏⠻⣆⠀⠀⢀⡾⠁⠙⠦⠤⠞⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⠀⠈⠉⠁⠀⠀⠘⡇⠀⣾⠁⠀⠀⠀⠀⠀⣠⠏⠙⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠙⠋⢹⠀⠀⠀⠀⢠⣾⢻⡇⢀⣸⡟⡷⣦⠀⠀⠈⠓⣦⢀⠤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢧⣄⠀⣀⣙⣄⠀⠀⢠⡏⢻⣆⣭⣩⣥⣾⠞⢹⡇⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡼⠉⠀⠀⣀⠀⢸⡿⢶⣬⣉⣉⣉⣤⡴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⠂⢸⡇⠀⠀⠉⠉⠉⠀⠀⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠳⣦⣄⣀⣀⣠⣤⠾⠋⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢸⡑⢤⣀⡉⠉⠉⢉⣀⡤⠚⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣌⡉⠉⠉⣉⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣤⣤⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢰⣦⣤⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠿⠟⠃⠀⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⠀⠙⠿⠧⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠻⣿⣿⣶⡶⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⠠⣶⣶⣿⡿⢣⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣿⣶⣴⣾⠍⠀⠀⠈⣷⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⡿⠀⠀⠈⢭⣶⣦⣾⣿⢃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣙⣹⣋⠉⠀⠀⠀⠀⠘⠳⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣤⠞⠁⠀⠀⠀⠀⢙⣛⣋⣣⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡛⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⣡⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣙⣩⣥⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣬⣥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣹⣡⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⢟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠿⣿⣿⠟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣬⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣛⡛⢏⣵⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣷⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣍⠛⣛⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣫⡄⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣬⡻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣬⣴⣿⡟⣀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⡘⣿⣷⣬⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢋⣼⣿⢀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⠀⠀⠀⠀⠀⢸⣿⣎⢟⣛⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠏⣾⣷⢀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⢰⣿⡎⠻⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⣿⡟⣾⣷⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣿⡜⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣥⣶⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣷⣶⣭⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣫⣴⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣬⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠁⠑⠂⠂⠁⠂⠄⠀⠀⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠱⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡄
⠀⠀⠀⠀⠀⠀⠀⠰⡋⠘⣧⠀⠀⠀⠤⣀⠀⡀⣠⠶⢤⠁⡐⠁⠀⠈⠑⠂⠤⠄⢀⣀⣀⠀⡀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣼⠠⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣧⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⡆⢀⠤⣀⠀⠀⠀⠀⠀⠤⠦⡄⠈⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠉⠀⠀⠠⠃⠐⠷⡿⠀⠀⠀⠀⢔⢲⣦⠀⠀⢨⠀⠀⠀⠀⡰⠀⠀⠀⢤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢬⢉⠉⢂⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠀⠚⠛⠏⠀⠀⠈⠤⠄⢤⠊⠀⠀⡌⠀⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠈⣾⠆⠀⠀⠄⡀⠀⠡⡀⠀⠀⠀⢟⠐⠒⢒⡛⠀⠀⠀⠀⠀⠐⠋⢀⠃⡠⠊⠀⢀⡃⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠉⢖⠢⡀⠀⠈⠐⠄⡨⢐⡠⢄⣀⡁⠀⠉⢀⣀⡀⠄⠐⢀⣁⣄⠕⠉⠀⢀⠔⠡⣦⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠒⠒⠒⠢⠈⠀⢄⠀⠀⠀⠀⠀⠈⠁⠘⠃⢀⡠⠛⠀⠉⠁⠀⠀⠀⠀⢀⠐⠁⠀⠔⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠀⠀⠀⠀⢆⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠠⠀⠀⡀⡀⠀⠀⠀⠠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⣀⣀⣀⠀⣀⣀⣀⡀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠇⠀⠀⠀⠸⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠪⡙⢖⠒⠒⡆⠀⠀⠀⢠⠖⠒⢲⢋⠕⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⣀⠠⠄⡇⠀⠀⠀⢸⡠⠄⣀⠁⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠉⠉⠑⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠒⠉⠉⠑⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⣤⣤⣤⠄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡾
⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣉⣉⣉⠙⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠃
⠀⠀⠀⣤⠤⢼⡏⠉⠉⠉⠉⠉⠉⠀⠙⢧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡴⠷⠶⠶⠶⠶⢶⡄
⠀⠀⠀⣇⠀⢸⡇⠀⠀⠀⣀⡤⠶⢦⣄⠈⠻⣇⠠⡆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠓⠂⠐⠒⠂⠀⢸⣇⣀⡀
⠀⠀⠀⢿⠀⢸⡇⠀⠀⠀⠉⠀⠀⠀⣹⠀⠀⢹⡄⠻⡄⠀⠀⠀⠀⠙⠛⠉⠛⠋⠀⠀⠀⠀⠀⠀⢠⡟⣀⣤⢤⣄⠀⠀⠀⢸⡇⢈⡇
⠀⠀⠀⠈⢧⣸⡇⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠹⣄⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠁⠀⣠⡾⠀⠀⠀⢸⡇⢸⠇
⠀⠀⠀⠀⠈⠻⡇⣆⠀⠀⣤⣶⣯⣤⣤⠀⠀⠀⢀⡟⠳⣄⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⠀⠀⠀⠙⢷⡀⠀⠀⢸⣧⠏
⠀⠀⠀⠀⠀⠀⠻⣜⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠈⠙⠳⣶⢤⣤⣤⣤⡤⣶⠞⠋⠉⢿⣿⡀⠀⠙⠓⠒⠋⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣍⣁⠀⠀⠀⢀⣠⡴⠋⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⡆⠀⠀⠈⠳⣝⠦⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠛⠋⢻⡅⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠛⡶⠶⠶⢶⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⣸⠃⠀⠀⠈⢷
⣠⣤⣄⠀⢠⣤⣤⣤⣶⣿⣶⣶⣶⣶⣶⣾⣷⣤⣤⣤⣴⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣦⣤⣴⣿⣶⣶⣶⣶⣾⣷⣤⣤⣤⣤⣤⣤⣤
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⠊⠙⡦⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⠤⣤⣤⣤⣤⣤⣤⣦⣶⣶⣴⠶⠤⠤⠤⠤⠤⢤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢰⠋⠙⣆
⠀⠀⠀⠘⢦⢶⣷⣾⣿⣿⡯⣏⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠓⠒⠒⠾⡶⡴⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣾⣿⣀⣠⣤⣤⣤⡶⣾⡿⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣷⣶⣶⣾⠿⠿⠿⠿⢿⣿⣿⣇⣀⠀⠀⠀⠀⠀⢰⣷⠃
⠀⠀⠀⠀⠀⡟⡿⠛⠛⠋⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⣼⢻
⠀⠀⠀⠀⠀⡇⣷⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣄⣤⣤⣤⣤⣤⣤⣿⠿⠿⠿⠿⠿⣿⣿⣀⣀⣀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣼⢸
⠀⠀⠀⠀⠀⣿⢹⣀⣤⣤⣶⣶⣿⠋⠉⠉⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠿⠿⢿⣿⡏⣿
⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⠀⠀⢀⣀⣀⣀⣼⣿⣿⣿⣿⡿⠿⠿⠧⠤⠤⠤⠤⣤⣤⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢈⡇⡏
⠀⠀⠀⠀⠀⢸⠘⣿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠲⠦⢤⣄⣀⢸⠃⡇
⠀⠀⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⡇
⠀⠀⠀⠀⠀⢸⡆⣷⠀⠀⠀⠀⠀⠀⠀⠰⢮⣿⡧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣼⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇
⠀⠀⠀⠀⠀⠘⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠇
⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⡤⠖⠚⠛⠲⢤⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢠⠋⠖⠲⠀⠀⠐⠂⠙⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⡏⢀⡄⠀⢀⣀⣤⣺⡄⢸⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⡀⠙⢯⡉⡱⠋⣡⢇⣼⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸
⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡶⠦⡭⡟⠟⠿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⡇⢀⣣⡄⠐⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠙⠲⠶⠶⠛⢉⡜⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⣀⣤⠴⡞⠁⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⣸⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⢠⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣠⠏⣠⡿⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⠀⠀⢸⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠒⠖⠒⠚⠛⢷⡟⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠸⣄⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇
⢀⠀⠀⠀⠀⢾⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣷⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⡿⠿⠿⠟⠛⠛⠛⠛⣛⣛⣛⣛⣛⣻⣶⣶⢶⣶⣦⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣠⠞⣉⡉⢷⡀⠀⢠⣾⣿⡿⠿⣿⣟⣛⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠭⠭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣻⡶⣤⠀⠀⢀⣏⣉⠙⠳⣄
⠀⢠⡞⢡⣾⡿⢷⣌⢷⣄⠸⣿⣿⣾⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⣀⣒⢚⣻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠁⣠⢿⡾⠛⢷⣄⢌⢳⡄
⢀⡏⠀⣿⡟⠀⠀⠈⠷⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡯⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣏⣉⣤⣴⣾⠧⣾⣷⠟⠀⠀⠀⢻⣖⢇⠹
⣼⠁⢸⣿⠁⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⢿⢿⡿⠷⠶⠶⠶⠶⠶⢶⡶⡶⠶⠶⠶⠶⠶⠶⠶⠶⢾⣛⠛⠿⠉⠁⣿⣇⠾⣿⠀⠀⠀⠀⠀⠈⣿⢿
⣿⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⠉⣿⣿⣿⣿⣿⣿⣍⣤⣤⣤⣄⡀⠀⠒⠂⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣠⡴⢦⡀⢀⣷⢶⣦⣿⠛⠛⠋⠀⠀⠀⠀⠀⠀⣿⡈
⢿⣄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⣾⡿⢿⡛⠛⠃⠀⠀⠀⣴⡿⣉⠂⠀⠑⡄⠀⠀⠀⣾⠛⠁⠚⠇⠰⣿⣹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢱⢀
⠸⣷⡸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣻⣿⡶⠂⠀⠀⠀⠀⢰⣿⡭⠀⠀⠀⠀⡁⠀⠀⢸⠁⠀⠀⠀⠁⢸⣗⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣎⣾
⠀⢹⣧⠹⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣾⣿⣽⡅⠀⠀⠀⠀⢸⣿⣉⡁⠀⠀⠀⠙⠀⢀⡟⠀⠀⠀⠀⢀⡿⢄⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⠃
⠀⠀⠻⣧⡙⣧⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣟⡃⠀⠀⠀⠀⢸⡯⣄⡀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⣸⢸⡗⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⣼⠃
⠀⠀⠀⠘⢷⣜⢷⡄⠀⠀⠀⠀⠀⠈⣿⣿⣿⡷⣿⣿⣶⣒⠀⠀⠀⠀⢸⣯⢀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠆⣾⣁⣻⣿⠀⠀⠀⠀⠀⠀⠀⡰⠟⣡⡟⠁
⠀⠀⠀⠀⠀⠻⣷⣝⣦⡀⠀⠀⠀⠀⢹⣿⣿⣿⡍⣿⣿⣿⠄⠀⠀⠀⢸⣿⠒⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⢠⡧⠈⣼⠇⠀⠀⠀⠀⠀⡠⢚⣡⠾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⡀⠀⠀⣿⣿⣿⣿⣸⣿⣿⣇⠀⠀⠀⠘⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣾⢉⣳⡟⠀⠀⠀⢀⡴⣫⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⢶⣄⡘⣿⣿⣿⣏⢿⣿⣶⡀⠀⠀⠀⣿⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢰⡇⣌⣿⢁⣠⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⡜⣿⣷⣤⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡯⣰⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣻⣿⣿⣿⣿⠸⣿⣷⠆⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⡾⢴⣿⡿⢟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⠙⣿⣿⣇⢹⣿⣗⠀⠀⠀⣿⠀⠀⠀⠀⡀⠀⠀⠀⡶⣴⣿⠏⠳⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣏⢿⣿⣧⠀⠀⢹⠀⠀⠀⠀⠇⠀⠀⣼⣵⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣻⣯⡀⠀⢸⡇⠀⠀⢰⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣻⣯⡀⠀⣿⠀⣤⣮⣬⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⣤⣶⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣋⡉⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣟⣛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣹⠃⣇⠸⣿⣯⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⢿⣿⠏⣿⡄⢿⣄⡹⠞⢿⠟⠋⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠅⣻⣿⣿⣿⣶⣶⣾⣶⣶⣷⣶⣶⣶⣾⡽⠟⠁⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⣤⣤⣤⣤⣤⣬⣭⣽⣯⣭⣥⣤⣬⣭⣭⣥⣤⣴⡶⣶⣴⣦⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡟⠄⠀⢣⣾⠛⠛⠛⠓⠛⠻⠛⠛⠒⠒⠏⠁⠉⡽⠆⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠁⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡀⢸⡟⢴⠶⡶⠶⠶⢶⣶⡶⢶⠶⡶⡶⣾⠿⠗⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣾⣿⡿⠿⠿⠾⠿⠷⠶⠷⠾⠷⠶⠾⠶⠶⠶⠾⠶⠷⠒⠒⠶⠷⠶⠿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣖⣀⢒⣲⠶⠶⠦⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⢴⡖⢲⣀⣐⣶⣿⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠀⠃⠀⠀⠰⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠

⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⣀⡤⠶⠖⠒⠒⠒⠒⠒⡒⠒⠒⠒⠶⠶⠯⢴⣤⣄⢀⣤⣶⣭⣖⣄
⠀⠀⠀⢠⠞⠉⠥⣭⡉⠻⣾⣧⣄⣛⣛⣛⠛⠓⠛⠻⠿⠿⠫⠿⠿⢿⣓⣒⡼⣷⠋⠀⣠⣤⠙⣿⡆
⠀⠀⠀⣾⠀⡇⠀⠈⢻⣇⣿⠀⣉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⣠⠀⣿⡷⠋⠁⠈⡇⠸⣷
⠀⠀⠀⣿⡄⠱⠀⠀⠀⠻⣿⠀⢀⠀⠀⠀⠀⠀⠒⠒⠒⠶⠀⠀⠀⠀⠩⠄⢰⡟⠀⠀⠀⠀⠃⠀⣿
⠀⠀⠀⠹⣿⣦⡙⠦⣀⡀⢻⠀⠀⠀⠀⣠⡂⠀⢠⠀⠀⠀⠀⠀⠀⠀⡊⠁⣸⠁⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠙⠿⣿⣷⣌⠙⢿⡇⠀⠀⠀⠈⠙⠓⠙⠀⠤⠄⠀⠀⠀⠀⠂⠀⣿⣤⡤⠔⢂⣠⣦⠞⠁
⠈⠁⠀⠀⠀⠀⠀⣯⣉⢀⣸⣿⡀⠁⠀⠀⠀⠀⢸⠀⠀⠆⠀⠀⠀⠨⠀⢠⡟⠀⢰⠟⠛⠉⠀⠀⠀⠀⠈
⠀⠉⢡⠄⠀⠀⠀⠻⠟⠛⠁⢹⣷⡀⠀⠀⢰⠒⠊⠀⠀⠓⠒⢶⠀⡈⢀⡾⣷⡶⠏⠀⠀⠀⠀⠀⠠⡌⠉
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠙⠛⠛⠛⠛⠛⠛⠋⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣤⣀⣀⣀⣤⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⠹⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠖⠺⠀⠀⠀⠈⠛⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠠⣮⣤⣙⣒⠒⠂⠀⠀⠒⠒⣒⣚⣉⣿⠆⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⡤⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⢀⡀⠀⠐⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⣤⣠⡴⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣁⣤⠤⠶⠒⠈⠀⢀⣤⡈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠿⢥⣀⡒⣲⣶⣾⡿⢟⣋⡉⢳⡜⣧
⠀⠀⠀⠀⣴⠾⡟⢻⣦⣤⣄⠀⠀⠀⠀⣽⠀⠀⠀⡎⠀⣩⣉⠁⠀⠀⡏⣤⢹⡞⢤⡟
⠀⠀⠀⣸⣟⠳⣜⡿⠤⠄⢹⡆⠀⠀⠀⢸⡀⢀⣀⢣⢸⠁⣬⢧⢠⠔⢣⣀⣸⠁⢨
⠀⠀⠀⢿⣀⣹⡟⠀⠀⢶⣟⠀⠀⠀⠀⠈⢷⣟⢉⢻⣷⠳⣤⠞⠂⠀⠼⠁⠀⠁⢘⡆
⠀⠀⠀⠈⠉⠙⠳⢶⡄⠀⠉⠛⢦⣄⠀⠀⠈⠻⢧⣃⠀⠀⠁⠀⡀⠀⠀⠀⡖⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠙⢧⣠⣄⡀⠀⠁⢱⡀⠀⠀⠉⠒⠒⠋⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⠀⠙⢻⡟⠦⣤⣾⡇⠀⠉⣻⣶⣤⣤⣎⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣀⡼⠁⠀⠈⣿⣧⡀⢀⣸⡿⠙⠻⣭⣀⣀⣹⣧⣤⡤⠴⣶⣾⢟⣯⣅⠀⠈⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣀⠀⠀⠀⠈⠿⣿⡛⠋⠀⠀⠀⠀⠉⠉⠁⠈⠻⢶⣴⣿⣥⡾⠿⢿⣄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠈⠙⠛⠶⠶⣶⣦⣴⣶⡶⠶⠾⢛⣿⡏⠀⠀⢀⡟⢿⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⣦⣴⡞⠛⣧⠀⠀⡜⢳⠘⣷⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠙⠒⠬⢻⣦⢼⡉⡷⢾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣷⢻⠉⠆⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢀⣀⣀⣤⣤⣴⡶⠶⣿⠇⠀⠀⠀⠀⠘⢄⣡⡴⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠶⠿⠛⠛⠉⠉⠀⢀⠀⠀⢺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⢠⠴⠟⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣾⣄⣀⣀⣤⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⣤⣄⣠⠤⢾⣿⡏⠉⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢸⡟⣷⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣿⠀⢸⡇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⡌⠀⠀⣿⣀⣼⡇⠀⢀⣧⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢩⡇⠀⠀⢸⣇⠀⣿⣤⣼⠷⢤⡀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡎⠻⣄⣀⡼⢻⡄⡇⠀⠈⠂⠴⣍⡓⠲⡄⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣿⠁⢡⠀⠿⠼⠇⢸⠀⠙⠓⠒⠢⢄⣈⣀⣴⠏⠈⣿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⢻⣄
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠈⠳⣦⣤⡴⠟⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⣤⣴⣶⣶⣶⣶⡶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⢰⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠷⠶⠿⣷
⠀⠀⣿⠁⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⡿⠛⠛⠿⣶⡀⠀⠀⠀⣿
⠀⠀⠻⠀⠀⠀⠀⠿⠋⠀⠀⠙⠿⠆⠀⠀⠀⠻⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠘⠟⠁⠀⠀⠀⠛⠓⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⠄
⠀⠀⢀⡤⠊⠉⠀⠀⠉⠑⠦⡀⠀⠀⠈⢿⣿⣿⣆⠀⢀⠂⠀⠀⠌⠀⠀⠀⢀⠔⣊⡩⠉⣍⣑⠢⡀
⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠈⢿⣿⣿⣆⠂⠀⢀⠌⠀⠀⠀⡰⣱⠟⠉⠀⣀⠀⠉⠺⣌⢆
⠀⢘⠀⡞⠀⠀⠀⠀⠀⠀⢸⡀⢡⠀⠀⠀⠀⢻⣿⣿⣆⢀⠂⠀⠀⠀⢀⠡⡯⠀⠠⣶⣿⣶⠆⠀⢽⠘
⠀⠸⡀⢻⡀⠀⠀⠀⠀⢀⡼⢀⠆⠀⠀⠀⠀⢀⣿⠿⢿⡅⠀⠀⠀⠀⠀⡆⢿⡀⠀⠞⠛⠓⠀⢀⡿⢸
⠀⠀⠑⢄⠙⠒⠆⠠⠖⠋⣠⠎⠀⠀⠀⠀⠀⣠⣜⠃⣛⣤⡀⠀⠀⠀⠀⠘⢌⠳⢦⣄⠀⣠⡴⠎⡡⠃
⠀⠀⠀⠸⣿⣶⣶⣶⣶⣿⣣⠀⠀⠀⠀⢀⢮⢏⠕⠀⠐⢝⢯⢦⠀⠀⠀⠀⠀⣹⣶⣤⣤⣤⣶⣯
⠀⠀⢠⣳⣿⣿⡏⠸⣿⣿⣧⠆⠀⠀⠀⢸⣻⢀⠀⠀⠀⢸⢸⣿⠀⠀⠀⠀⢀⣿⣿⣿⠛⣿⣿⣿⡆
⠀⠀⠊⠁⠘⡟⠀⠀⢻⠃⠈⠘⠀⠀⠀⠀⠻⡧⣕⡀⣐⢥⡻⠋⠀⠀⠀⠀⠚⠋⢹⠏⠀⢹⡏⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⢸⣿⡄⣸⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠃⠞⠁⠟⠹⡆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠀⠀⠂⠀⠀⠀⠀⠃⠉⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣶⢦⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠶⡡⢿⡞⡏⠀⠀⠀⠀⢰⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡂⢀⠟⡉⠀⠁⣘⢳⢿⡄⢸⠀⠀⢨⠻⠻⣤⣇⢀⣖⠦
⠀⠀⠀⠀⠀⠀⣴⣥⠄⡀⠀⣹⡗⣡⣾⣿⠃⣰⡇⡸⠀⣿⠿⣄⣠⣿⢣⡀⠈⠹⣾⠁⠀⠈
⠀⠀⠀⠀⠀⠠⠋⠉⠛⠳⢾⣯⣾⣻⢳⠃⢠⣿⠇⡇⠀⢿⣾⡘⡈⢻⣇⢻⣤⠬⢶⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⠉⠭⠙⠒⠋⠛⢻⣿⡄⠚⠛⠓⠓⠘⠻⡏⢁⡤⠔⠈⣷⣀⠄
⠀⠀⠀⠀⠀⠢⡀⠀⠀⡀⠘⠇⠀⠀⠀⠀⠀⠀⢸⢸⡇⠖⠂⠀⠀⠀⠀⣹⣀⠀⠀⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠈⠑⠤⠷⠞⢀⣀⠀⠀⠀⠀⠀⡄⢸⠃⠀⠀⠀⠀⢀⠌⠀⢠⠋⠱⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠈⠉⠀⠉⠉⢉⡉⠉⢩⡀⠴⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣾⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⠿⠿⢿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠸⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⢸⣿⣇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⢿⣿⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣷⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢫⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢩⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⠒⠛⠉⠉⠉⠉⠉⠙⠛⠒⡦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠃⠀⠀⠀⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⡀⠈⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⠁⠀⢠⣤⣿⣦⡄⠀⠈⢻⣿⣿⠟⠀⠀⣤⣼⣷⣤⠀⠀⠊⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠁⠀⢀⠀⠀⣽⠿⣿⠀⠀⠀⠘⠉⠉⠃⠀⠀⢨⡿⢿⡇⠀⢀⠀⠈⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠠⣤⣾⣧⣤⠀⠀⢀⠠⠀⠀⠉⠀⠀⠀⠐⠀⢄⠀⠀⠠⣤⣾⣷⡤⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⢸⠟⢿⡅⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⡅⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⣰⣤⠀⠀⣤⡶⣦⠀⣶⠀⢰⡆⢰⡄⣶⡶⢶⠆⣶⠀⢠⡶⢰⡶⢶⡆⢰⣆⠀⣠⡆⢰⣶⠶⡦⢰⣦⠀⣶⠰⢶⣶⠶
⠀⢠⣿⣽⣆⠰⣏⠀⠀⠀⣿⠷⢾⡇⢸⡇⣿⠿⠂⠀⠘⣧⣾⠃⢸⡿⠇⠀⢸⡿⣶⡿⡇⢸⡷⠂⠀⢸⡟⣷⣿⠀⢸⣟
⠀⠛⠁⠀⠛⠂⠙⠛⠛⠁⠛⠀⠘⠃⠘⠁⠛⠛⠛⠃⠀⠛⠃⠀⠘⠛⠛⠓⠘⠃⠛⠘⠃⠘⠛⠛⠓⠘⠃⠈⠛⠀⠘⠛
⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠘⣎⠀⠀⠀⢸⣶⣾⡃⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢸⣶⣶⡇⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⣖⠀⠐⠛⢿⡟⠛⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⠀⠀⠂⠀⠀⠐⠛⢿⡿⠛⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠈⠀⠀⣻⣶⣿⠀⠀⠀⢠⣀⣠⡄⠀⠀⢈⣷⣾⡇⠀⠈⠀⢀⢔⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠘⠛⣿⠛⠓⠀⢀⣼⣿⣿⣄⠀⠀⠛⢻⡿⠛⠂⠀⢐⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢐⡄⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠁⣀⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠦⣄⣀⣀⣀⣀⣤⡤⠼⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⠀⠀⠀⢀⠀⢐⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⣝⡀⠀⢀⠁⡊⠀⠀⠈⢻⢦⠀⠴⢂⡿⡄⠀⠀⠹⡦⡐⠐
⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠈⠓⢍⠓⢤⣁⠀⠀⠀⠀⡟⢀⠀⢀⠞⠀⡟⠀⠀⢠⠀⠹⡄
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⢧⠀⠈⠳⡀⠐⣦⡙⢮⠢⢸⠁⢠⠃⠀⡠⠋⠀⠈⡇⠀⢀
⠀⠀⠀⠀⠀⠀⢖⠈⣀⣀⡀⠀⠀⢸⡤⠀⠀⢧⢠⣇⠹⡌⢳⠘⣆⡿⢀⡞⠁⠀⠀⢀⠆⢀⠎⢣⠀⡀
⠀⠀⠀⠀⠀⠈⠀⣈⠉⢳⡎⢢⡀⢸⠀⠶⡀⠈⠻⠂⠀⢱⠀⠀⡿⢽⣼⠀⢠⠀⢀⡌⢀⠠⡀⡏⠐⠁
⠀⠀⠀⠀⠀⠀⠈⢹⠙⠸⡇⢀⠱⣾⡄⢁⠈⠢⡀⢠⡄⡞⠀⢀⠇⡄⠋⡠⢁⢣⠏⠀⢸⢣⢰⠇⢀⢦
⠀⠀⠀⠀⠀⢀⣆⢺⡊⠀⠹⣇⠰⠠⢄⠀⠂⠀⠰⠂⢠⠃⢀⡞⡐⢸⠐⠀⢠⠃⠀⡰⠃⣸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠘⡏⢦⣀⠀⠀⠈⠳⣝⢤⡈⠲⠀⠀⡄⢸⢠⠟⠐⠀⢾⠀⠂⡇⣠⠏⠀⣀⡎⠀⣠⢀⡸
⠀⠀⠀⠀⠀⠀⠱⣄⠈⠑⠢⢤⡀⠘⡄⠱⠀⠀⠀⠆⠈⠿⠀⠀⠀⠀⠀⡆⢹⠇⡄⢠⠎⢀⡰⣻
⠀⠀⠀⠀⠈⢻⠆⠈⠲⢄⠀⣄⡙⠶⣗⠀⠁⠀⠀⠘⢄⠈⠂⠰⠀⠀⠀⠐⡤⠊⢄⣮⠖⢋⡜⠁⣦⡀
⠀⠀⠀⠀⠀⠘⠒⠠⠱⠀⠱⡀⠂⠑⠄⠀⡄⠀⠀⠀⠀⠉⠂⣤⠀⠀⠀⠀⠄⠐⠎⠥⢈⠏⡀⠀⠁⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠢⡀⡇⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢐⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠙⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠈⠨⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠓⢤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠀⣀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠉
⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠂⢀⠁⠀⠀⡀⠁⠀⠔⠀⡀⠀⡀⠀⠀⢐⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠈⠐⠀⠀⠈⠈⠀⠀⠁⠀⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠘⠃⠀⠙⠋⠀⠘⠃⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢰⡖⠒⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠒⢲⡦⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⣀⠸⣧⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣾⠃⣀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠀⠹⣧⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣼⠏⠀⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣌⠟⣷⠀⠈⠻⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠟⠁⠀⣾⠻⣡⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⡧⣿⢠⡀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢀⡄⣿⢼⠟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣮⣌⣿⣦⣤⡀⠀⠈⠙⣿⣿⠋⠁⠀⢀⣤⣴⣿⣡⣵⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣾⣿⣬⢿⠘⢿⣦⠀⢠⣿⣿⡄⠀⣴⡿⠃⡿⣥⣿⣷⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣿⣿⣭⣽⣶⣿⣿⣿⣿⣶⣯⣬⣿⣿⡿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣖⣖⣺⣻⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠁⠀⠀⠛⠁⠉⠉⠛⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠈⠋⠋⠉⠉⠉⠉⠛⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢉⡴⣒⠿⢗⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠐⠒⠒⠒⠒⠒⠒⠒⢂⡟⡞⣀⠀⣠⠝⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⢤⡀⠀⠀⠒⠒⠒⠒⠒⠊⣧⢣⡈⠛⠁⣠⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠈⣳⣯⣿⣯⣷⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣰⣿⣿⠃⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋⠀⠈⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤
⠀⠀⠀⠀⠀⣰⣷⢸⡿⠋⠀⠀⠀⣠⣴⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⣦⣄⠀⠀⠀⠘⢿⡇⣾⣆
⠀⠀⠀⠀⠀⣿⡟⣠⣾⡗⠀⠀⢰⣿⡏⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠹⣿⡇⠀⠀⢺⣶⣄⢻⣿
⠀⠀⠀⠀⣾⡜⠳⢿⡋⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢰⣿⡇⠀⠀⠀⠙⠿⠞⢋⣾
⠀⠀⠀⠰⣿⡇⣼⣿⠇⠀⠀⠀⠀⠻⣿⣄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢠⣿⠟⠀⠀⠀⠀⠘⣿⣦⢸⣿⠇
⠀⠀⠀⢠⡙⠇⠿⣋⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣴⠿⠃⠀⠀⠀⠀⠀⠀⣙⢿⠸⢟⡄
⠀⠀⠀⢸⣿⡄⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢠⣿⡇
⠀⠀⠀⠈⢿⣧⣿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣸⡿⠃
⠀⠀⠀⠀⣷⣦⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣨⣾
⠀⠀⠀⠀⠹⣿⣧⢿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡿⣰⣿⠏
⠀⠀⠀⠀⠀⢨⣝⡋⢱⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡎⢑⣫⡅
⠀⠀⠀⠀⠀⠀⠻⣿⣦⢻⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡟⣱⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠉⣛⣓⠉⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡗⠈⢚⣛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣝⠿⢸⣷⣤⡀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣠⣶⡇⡿⣫⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢭⣥⣤⣙⡻⣿⡀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⢀⣾⠿⣋⣤⣬⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⢠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡴⠶⣏
⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⣠⣄⢹⡄⠀⢠⠔⠀⠐⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣀⣤⡀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡏⢠⠏⠀⣏⣠⣗⠀⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢸⣇⣻⠆⠘⡆⠈⣇
⠀⠀⠀⠀⠀⠀⠘⡇⢸⡀⠀⠀⠁⣿⠀⢠⠄⢄⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢳⡀⢧⡀⠀⠀⢹⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠳⣄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡴⢃⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢧⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣠⠎⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡙⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠘⢿⡌⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢸⠻⣄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⢸⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢋⡾⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠹⣜⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠓⠦⣤⠀⣤⡴⠞⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⣈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡎⣝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠋⠀⢈⣁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢍⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠁⠤⠈⠩⠟⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠙⠂⠀⠈⠠⠀⠂⠂⠒⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⢤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⠴⠿⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣴⣷⣦⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⣿⠏⣻⣿⣯⣬⣿⣿⣛⢹⣿⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣉⣽⣿⡿⢟⣛⣛⣛⣛⣛⠻⢿⣿⣯⣉⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢛⣿⡿⢋⣵⣾⠿⠟⠛⠛⠛⠻⠿⣷⣮⡛⢿⣿⡛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⣿⢏⣴⡿⠋⠁⢀⣠⠴⠋⠉⢿⠀⠈⠛⣿⣦⠻⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣡⣿⠏⣾⡟⠁⠀⠀⣏⣀⠀⠀⠀⢸⠀⠀⠀⠈⢿⣧⢹⣿⣈⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣿⢸⣿⠁⠀⠀⠀⠀⠈⡇⠀⠀⢸⠀⠀⠀⠀⠸⣿⡎⣿⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣦⣽⣿⢸⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⢀⣿⡇⣿⣧⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠻⣿⡌⣿⣇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⣼⣿⢰⣿⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⣷⡜⢿⣦⡀⠀⠀⠀⣇⣀⣀⣸⠀⠀⢀⣼⡿⢣⣿⣿⣶⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⣿⣮⡻⢿⣶⣤⣀⣈⣉⣉⣁⣤⣶⡿⢛⣴⣿⣋⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⠻⢿⣶⣭⣛⣛⠿⠿⠿⣛⣫⣭⣾⡿⠛⣿⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⡟⣻⣿⡿⠿⢿⣿⡛⢻⣿⣿⣿⡿⠃⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠻⣿⠿⠛⣿⣶⣿⡛⠿⣿⠟⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠛⠋⠉⠹⣄⠀⠀⣰⠃⠀⠀⠀⠘⣧⠀⠀⢠⡞⠉⠙⠛⠒⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢠⠏⠀⠀⠀⠀⠀⠘⣆⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠙⢷⣄
⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠙⢷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡾⠋⠀⠀⠀⠙⣷⣄
⠀⠀⢠⡾⠋⠙⢷⣄⠀⠀⠀⠈⠻⣯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠟⠁⠀⠀⠀⣠⡾⠉⠙⢷⣄
⠀⠀⠈⢷⣄⠀⠀⠙⢷⣄⠀⠀⠀⠘⢷⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡾⠃⠀⠀⠀⣠⡾⠋⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠙⣧⡀⠀⠀⠙⢷⣄⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⣠⡾⠋⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠙⢷⣄⠀⠀⠈⢷⣄⣠⣤⡤⠶⠶⣤⣤⣄⣠⡾⠁⠀⠀⣠⡾⠋⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠙⢷⣤⡴⠟⠋⠁⠀⣀⣀⣀⣀⠀⠈⠉⠻⢦⣤⡾⠋⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⣠⡾⠋⠀⣠⡴⠞⠛⠉⠉⠉⠉⠛⠳⢦⣄⠀⠙⢷⣄⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣴⠏⠀⢠⡾⠋⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠙⢷⡄⠀⠹⣦⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢰⡟⠀⠀⠀⠀⠀⣰⠏⠹⣆⠀⠀⠀⠀⠀⢻⡄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⠀⢠⣴⡶⠶⠾⠋⠀⠀⠙⠷⠶⢶⣦⡄⠀⣿⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻⡄⠀⠀⠀⣸⠇⠀⠀⠀⠀⠸⣇⠀⠀⠀⢠⡟⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣷⡀⠈⢷⡄⠀⠀⣿⣠⡴⠟⠻⢦⣄⣿⠀⠀⢠⡾⠁⢀⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠘⢷⡀⠀⠙⢦⣄⠋⠁⠀⠀⠀⠀⠈⠙⣠⡴⠋⠀⢠⡾⠃⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣋⣀⠀⠀⣠⡿⣦⣀⠀⠉⠛⠶⠶⠶⠶⠶⠶⠛⠉⠀⣀⡴⢿⣄⠀⠀⣀⣙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⢻⡏⠀⠀⠙⣳⣶⣤⣄⣀⣀⣀⣀⣠⣤⣶⣞⠋⠀⠀⢹⡟⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣠⡾⠋⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠙⢷⣄⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠘⣇⠈⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⣸⠀⠀⠀⠀⠀⠀⠀⠰⠋⡇⣦⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠰⡆⣨⢛⢥⡶⠀⠀⠀⠀⠀⠀⢸⢸⣄⣧⣠⠀⠀⠀⠀⠀⢀⠀⣲⠛⡏⠀⠀⠀⠀⠀⠀⡾⡿⠜⡄
⠀⠀⠀⣷⣼⢀⡾⣤⠦⡼⠀⠀⠀⠀⢸⣾⠈⡟⠙⢻⠀⠀⠀⠀⠀⠘⢟⢫⣿⣀⡀⠀⠀⠀⢀⠠⢇⠠⣤⢷⡄
⠀⠀⠀⠘⠦⠾⠇⠘⠃⠁⠀⠀⠀⠀⠀⠁⠰⠛⠖⠁⠀⠀⠀⠀⠀⠀⠀⠱⠐⢯⠃⠀⠀⠀⠘⣷⠛⠘⢆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠜⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡀
⠀⠀⠀⠀⠘⡿⡏⡅⠀⠀⠀⠀⠀⢀⢀⠀⣿⢣⣀⡀⠀⠀⠀⠀⠀⠀⠎⣼⣿⢰⡀⠀⠀⠀⠀⠀⡄⣾⡼⣃⡋
⠀⠀⠀⠀⠸⣧⠛⣇⠀⠀⠀⠀⠀⠈⠉⢷⠾⡌⠛⢛⡁⠀⠀⠀⠀⢰⣜⣇⢹⡾⣧⡀⠀⠀⠀⠀⡁⢏⢁⢹⠠⠇
⠀⠀⠀⠸⣆⣿⢶⣼⠀⠀⠀⠀⠀⠀⣻⡛⠀⡿⣆⠛⠇⠀⠀⠀⠀⠸⣿⠹⢰⡀⠻⠀⠀⠀⠀⠠⣼⡾⢿⡘⢾
⠀⠀⠀⠀⠀⠰⠏⠋⠀⠀⠀⠀⠀⠀⠈⠃⠻⠅⠙⠋⠀⠀⠀⠀⠀⠀⠙⠘⠋⠃⠀⠀⠀⠀⠀⠀⠈⠃⡎⠻⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣀⣀⢠⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⢩⡝⣿⣿⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡉⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠛⠛⣿⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠁⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⣿⡘⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⡀⠀⢸⡇⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠳⢦⣿⡸⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣴⠾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡿⢧⢳⠘⣿⣿⣿⣿⣿⣿⡿⢿⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣅⡘⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢹⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣾⠿⠿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠘⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠻⣿⡄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⡀⣰⣿⡿⠃⠀⢠⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡼⣿⠁⠀⠀⠸⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠘⣿⣿⣷⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣦⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⢤⡀⠀⢀⣿⣷⡀⠀⠀⣸⣿⣿⡟⠀⢤⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⡇⠀⠀⠀⢱
⠀⠀⠀⢤⡀⠀⣴⡿⠀⡀⠙⣿⣿⣿⣿⡆⠀⠀⢸⣷⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⠁⠀⠈⣿⣷⡄⠀⠀⠙⠻⣿⣿⡿⠁⢸⣿⣿⣿⣿⣿⣷⣤⣀⣤⣿
⠀⠀⠀⠈⣿⣶⠟⠀⠀⣿⣦⣾⣿⣿⣿⠇⢀⣴⣿⠟⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣿⣿⣿⡀⠀⠀⠀⠀⠉⣰⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⣇⠀⢀⣿⣿⠀⡀⠀⠘⢿⣿⣿⣿⢟⣴⣿⣿⠋⠀⢠⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣀⠀⣿⣷⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟
⢿⣿⣿⣿⣿⣇⣀⣽⣷⡀⠈⠙⠿⢋⢸⣿⣿⣿⣷⣤⣾⡇⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⣷⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⣿⣌⣿⣿⣿⣿⣿⡿⢡⣾⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⢠⠀⠙⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⡟⠀⣿⣿⣷⣦⣴⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣷⣶⣿⣿⣿⣿⡟⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣘⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⡟⠛⢛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠟⣫⣵⣮⣭⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣥⣀⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⣤⣉⠲⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣯⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠩⠽⠟⣻⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠒⢠⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣾⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡀⠀⠀⠀⡀⠲⣽⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢛⣵⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣯⣤⡄⠀⢙⣛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠰⠟⠛⠋⢸⣿⣿⢟⣩⣿⣿⣿⣿⣿⣿⣿⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⣀⣴⣿⣿⣣⣿⣿⠿⢟⣼⣿⣿⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⣽⠋⢹⠟⣡⣴⣾⣿⡿⣋⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠽⢀⠉⠀⣰⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⠛⠛⠛⠉






⣀⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣇⢸⣦⡙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢋⣴⡆⣸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠾⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠟⠛⠛⠛⠻⠿⢿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⢐⣤⡶⠿⠛⠛⠛⠛⠛⠿⢶⣤⡂⢀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⡀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠐⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⠂⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⢈⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⡣⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠈⠛⠷⣶⣤⣤⣤⣤⣤⣶⠾⠛⠑⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⣴⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠙⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠈⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠙⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠈⠻⠃

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⢩⣍⠯⣆⠀⠀⠀⠀⠀⠸⢏⣭⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⠒⢦⡏⣿⣿⣿⣿⣿⣿⣦⠖⢲
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣸⣿⣷⢿⠀⠀⠀⠀⠀⢰⣿⣿⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⣿⣿⣿⠁⠀⢸
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⢻⣿⢸⠀⠀⠀⠀⠀⣸⣿⢣⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢧⡀⢸⣼⣿⣿⣿⣿⣿⠀⢠⠏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣭⣋⠆⠀⠀⠀⢀⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠲⢟⣿⣿⣿⣿⡧⠖⠁
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠟
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣶⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢇⣤⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⢠⠟⠙⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠧⡀⢠⡆⠀⠀⠀⠈⣧⠄⠐⠸⠀⡤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⠀⡰⠉⠷⠀⠐⠑⠀⠃⠀⣤⡠⠞⡷⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠉⠸⢤⡇⠀⠄⠀⠀⠀⠀⠀⠀⠉⠀⠐⡁⠘⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠈⠁⡐⠂⡰⠁⠀⠀⠀⠤⠂⡳⠀⠇⠆⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣦⡱⡄⠲⠇⠘⠀⠀⠀⠀⠀⠀⢀⠀⠞⠀⠔⢁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠌⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠊⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠈⡤⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢥⡀⠀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠮⠴⠖⠖⢤⣼⢶⡿⣷⣾⣷⠋⢻⣿⣷⡖⠦⢤⡄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣤⣤⣴⠞⠋⠀⣸⠁⢀⣿⢻⣆⠀⠙⢯⣟⠉⣷⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢹⣁⣀⣠⣶⡇⠀⠸⣿⠄⣻⡆⠀⠈⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠈⣿⠷⠏⠛⠉⠘⠶⠶⠞

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣠⡇⠀⢠⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠋⠀⢀⣾⠟⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⢀⣴⣿⡏⠀⠀⣠⣴⡏⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⢳⣿⢀⠠⠾⢻⣿⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⣿⡿⣱⣿⡿⠃⠀⠀⣼⣿⣼⠠⠄⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣽⣿⣿⣿⣿⣆⣠⣾⣿⢿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢟⣿⡿⣿⣿⣿⣿⣿⢣⣼⣿⡎⢷⣇⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⢏⣾⡟⣸⣿⣿⣿⣿⡇⡾⣿⣿⢻⡌⠻⣆⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⢠⠁⣿⠟⣿⡿⣰⣿⣿⣿⣿⣿⡘⣇⢸⣿⣿⣿⠀⠙⣷⣄⠠⡱⣄
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣿⣾⣿⡇⣿⢸⣿⣿⣿⡿⠧⠀⣼⡈⣿⣿⡇⢢⠘⣿⣧⠙⣿⣦
⠀⠀⠀⠀⠀⠀⢸⡀⣼⢿⣿⣿⠀⣿⣼⠳⠏⢠⣷⢀⣼⣿⠃⣸⢹⡇⣸⡄⠸⣿⡇⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠘⣧⢹⡜⣿⣿⡄⢹⣿⠀⠀⢸⣾⡈⠻⣿⢀⡟⣹⠇⣿⣥⠀⣿⣷⠀⢻⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠹⣿⣿⣿⣦⡿⣷⣄⡈⠻⠏⠀⡹⣾⣱⠏⣼⣿⣧⠀⣿⣿⠀⢸⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⣻⣯⣛⣿⣿⡝⢷⣂⠠⣼⣷⣿⢏⠼⠁⢸⡇⢰⣿⡟⠀⣸⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠀⠈⠋⣉⣿⣿⣿⣷⡙⠿⠿⠯⠁⠀⢀⠴⠋⣴⣿⠟⠁⣰⢃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⣿⣿⣿⣿⣷⣦⣤⣤⣤⣷⣼⣿⣿⠗⢀⡴⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣹⣿⣿⣿⣿⣿⣿⣿⠿⢛⣡⠴⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣧⣴⣿⡭⠤⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠿⠟⢻⠟⢿⠿⢻⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣤⣄⣀⡤⣶⠶⠶⠾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠶⠶⢶⡦⣄⣀⣤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢹⡛⠛⠻⠷⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠿⠛⠛⣻⠁⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⠖⠚⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⠒⠒⢦⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣏⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢈⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⣇⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⡏⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⢦⡀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣠⠎⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠉⠳⠤⣈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡠⠴⠋⠁⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡡⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠦⠤⠴⠆⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⣤⡤⠤⠾⠦⠀⠀⠸⠦⠤⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠒⠒⠛⠒⠒⠒⠒⠒⠒⠛⠒⠒⠚⠓⠒⠚⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠗

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⢋⡝⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⠿⠟⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢀⣼⣻⣂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⣽⣿⣻⣿⣻⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣠⣶⣾⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢸⣿⣿⣿⣿⣸⠇⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⡝⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⢙⡿⠿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣾⡇⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢻⠿⠛⠛⠛⠃⠈⠉




⣏⣈⣯⣭⣟⣯⣿⣹⣽⣿⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣦⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣾⣿⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣾⠟⠀⠀⠀⠀⠹⣷⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⢠⣿⣿⣿⣿⣧⡈⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣅⠐⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣠⣴⠾⠷⣦⣄⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠿⠛⣉⣀⡙⠻⢿⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠾⠛⠉⠀⠀⠀⠀⠉⠛⠷⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡈⣉⣤⣶⣿⣿⣿⣿⣷⣦⣌⡉⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿
⣿⣿⡿⠛⠉⠀⠄⢠⡶⠒⠀⠀⠀⣠⣴⡿⠿⠙⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠫⠉⠁⠁⠈⠁⠀⠀⠁⠀⢠⣾⣿
⣿⣿⠀⠀⠈⢡⠶⠁⠀⠀⢀⣤⠾⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢰⠋⠀⣿
⣿⣿⠀⣄⠖⠁⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⢀⡞⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⡆⠀⠀⣿
⣿⣿⠐⠁⠀⠀⠀⠀⡴⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠄⣳⣇⠀⠀⠀⠀⠀⠀⠀⠾⠉⢠⠞⠁⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠰⠀⠀⠴⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠟⠀⠈⠁⠀⠀⠈⢿⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠁⠀⠐⠂⠀⠀⠐⠀⠀⣿
⣿⣿⠀⠀⠀⢀⡀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠇⠀⣿
⣿⣿⠀⠀⢰⣾⡇⠀⠀⠀⠀⠘⢳⣄⡀⠀⠀⡠⢴⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠴⡶⡾⣆⠀⠀⣿
⣿⡟⠀⠀⣿⡏⣷⠀⠀⠀⠀⠀⠉⠈⢏⡽⠋⠑⠾⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠰⢷⣤⠀⠀⠀⣿
⣿⡟⠀⢀⣹⣷⢹⡄⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣿
⣿⣿⠈⠈⠋⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠠⢄⠀⠀⣀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⡀⠀⢠⠇⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠹⢿⡾⣇⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠉⠀⠀⠈⠁⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣄⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠘⠀⠈⢣⡀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⣰⠀⠙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡈⠀⠀⠀⠀⠀⣰⡤⠊⠀⠀⠀⠀⠀⢀⠀⢀⣴⣶⣦⡀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⢂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⣀⠾⠋⠀⠀⠀⠀⠀⠀⢀⠀⠠⠏⢀⣀⠈⢿⠀⠀⠀⣿
⣿⡇⠀⠀⠀⢰⠀⠀⠂⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠐⠄⡀⣁⣠⠎⠀⠀⠀⣿
⣿⡇⠀⠀⠀⡎⠀⠀⡚⠂⠀⠀⠈⢹⡆⡀⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠈⠉⢸⣧⠀⠀⠀⣿
⣿⣿⠀⣀⣼⡇⠀⢠⠃⠀⠀⠀⠀⠀⠩⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠠⠀⠀⣦⡇⣦⠀⠀⣿
⣿⡟⠛⠋⠁⠀⠀⠄⠀⠀⠀⣀⡠⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⣀⣀⡀⡘⣏⢧⠹⡇⣿⣦⠰⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⣫⣿⣿⡆⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢸⡟⣿⣏⢻⡘⣎⢧⢹⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠤⠤⠴⠛⡉⠙⠋⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⡞⢫⡟⣿⠂⢳⡘⣎⢧⢻⡷⣿⣿
⣿⡇⠀⠀⢤⡤⠀⠀⣠⣼⠖⠁⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⣴⡏⢷⡸⡆⡆⢳⡹⣎⢧⢻⡯⣿
⣿⡇⠀⠀⠈⠀⠀⠰⠿⣿⡔⠀⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠈⠀⢀⡎⠀⠀⠀⠀⠀⠘⡻⣇⠘⣧⢻⡜⡄⢷⢹⣾⡏⣷⣿
⣿⣇⠀⠀⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⢠⣶⡆⢷⢹⡄⢹⣆⢷⢱⣼⢸⣿⡇⣿⣿
⣿⣧⠀⠠⠦⠘⠁⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⡴⠄⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠾⠿⡇⠘⣇⢻⡀⢻⡜⡧⣿⢸⣿⡇⣿⣿
⣿⣿⠀⢠⣀⢀⡤⠀⠀⠠⣤⠀⣠⣴⣿⣿⣿⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡄⠸⡎⣧⢸⡇⡇⣿⢸⣿⡇⣿⣿
⣿⣿⠐⠛⠻⠟⠛⠛⠛⠿⠟⠻⠿⠿⠿⠿⢿⣤⣤⣀⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣹⠄⠿⠸⠸⠇⠧⢿⢸⣿⡇⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣆⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⡶⣦⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣯⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⡆⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣤⣿⣿⣤⣶⣶⣶⣯⣴⣷⣭⣿⡆⠀⠛⢲⡀⢛⣒⣶
⠀⠀⢀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⡏⢿⣿⣿⠀⢀⢀⡐⢾⢈⣦⣠⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⣰⣶⠟⣤⢿⡏⢈⡷
⠀⢀⣾⣿⣿⣧⣷⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠋⠀⢀⡌⣦⣠⢷⣏⣿⣯⣯⣧⡐⠀⠀⠉⢈⣍⣉⣩⣍⣭⣥⣤⡄⠀⣜⣿⣶⣼⠟⢹⣏⣸⡅
⠀⣼⣿⠏⠀⢿⣿⣇⠀⠀⠸⣿⣿⢿⢿⣿⣿⠃⠀⠀⢀⣜⣾⣷⣿⣿⣿⣛⣿⣿⣼⣿⣿⣷⣦⡰⣼⣿⣍⠓⢺⣟⡋⠉⠉⠀⠘⠿⠞⢿⣶⣿⠿⠛⠁
⠀⠻⢧⠀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⣁⣸⣿⣿⡟⠛⠈⠑⠀⠀⠀⠀⡀⠀⠘⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⡀⣴⢠⡀⠀⠀⠀⠀⠀⠀⠆⡐⠂⡆⠐⠀⠀⠀⠀⢰⠀⠀⠀⣿⣿⣿⣿⠇⢀⠀⠀⠀⠀⣠⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⣸⢼⣠⢧⣟⣳⢼⡰⣅⣷⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠉⠁⡄⣑⣘⣄⠀⣀⣙⣦⡴⣃⠀⠀⠀⢠⡀
⠀⠀⠀⢸⡆⠀⠀⣯⣻⣾⣾⣿⣿⣿⣾⣷⣿⣷⣿⡄⠀⠀⠀⠀⠀⢀⣼⣧⣶⡤⣄⠀⠀⠀⠀⢀⠙⠐⠀⠑⠿⠟⠀⢠⣞⣿⡻⣞⡽⠁⠀⢠⣤⣽⠆
⠀⣰⢠⡿⢁⡀⠀⠈⠛⠛⢉⠉⠉⠉⠙⠛⠛⠛⠋⠀⠀⠀⠀⢠⣼⣽⣿⣿⣿⣿⣿⣤⠀⠀⣀⡴⠛⢦⡞⠀⠀⠀⠀⠀⣹⣟⣿⣏⠀⠀⢰⣛⣦⣹⣦
⠀⢿⡈⢷⣼⣄⠀⠀⠀⣠⢿⣆⣤⠀⠀⠀⠀⣦⣷⡄⢣⣀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣞⠀⡄⡟⢷⡜⣮⣷⣴⠁⠀⠘⠛⠛⠛⠛⠛⠛⠛⠀⢈⣽⣿⠁
⠀⢸⡿⢸⣿⣿⠇⠀⣰⣯⢾⣿⣏⣦⠀⠀⠲⣬⡟⣷⣦⡙⣷⠀⠁⢸⣿⣿⣿⣿⡟⠃⠸⣧⣿⣿⣵⣿⣟⣿⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢰⣿⡟⠋
⠀⣾⣷⣾⣿⣿⠀⠀⣿⡇⠀⠋⢹⡿⠀⠈⣧⣿⣷⣿⣼⣷⣡⣧⠀⠻⣿⡟⢽⣿⡅⠀⠀⣽⣯⣿⣿⣿⣿⣻⡇⠀⠀⣀⣤⣴⣆⡀⠀⠀⠀⠉⢻⣿⣦
⠀⠈⠛⠛⠛⠋⠀⠀⠈⠛⠀⠐⠋⠁⠀⠀⠙⠛⠉⠋⠋⠙⠛⠋⠀⠀⠘⠀⠘⠋⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠃⠀⠾⠿⠿⠿⠿⠟⠛⠗⠀⠚⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢴⠆⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢾⣿⡗⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠐⣶⣶⠂⠈⠙⢦⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⣟⠁⠀⠀⠀⠐
⠀⠀⡜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⠀⠀⠀⠀⡞⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⢳⠀⠀⠀⢠⣬⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠠⣧
⠀⠘⡇⣇⠀⠺⡰⣶⣶⣶⠖⠁⠀⣸⢸⡃⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⠀⠀⠀⠸⣿⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠻⣌⠳⠆⠈⢉⣿⡉⠁⠰⠞⣡⠟⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢻⣧⠀⠈⠀⠀⠀⠀⠀⠃⢀⣼⠏
⠀⠀⠀⠉⠛⠀⠰⠾⠿⠷⠆⠀⠛⠉⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⣀⡤⠚⠀⠀⠀⠀⠀⠀⠹⢗⣤⣀⣀⠀⣀⣠⣴⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠃⠀⠀⣀⣯⣀⠀⠀⠸⡴⣄
⠀⠀⡊⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢨⢡⠁⠀⠀⠀⠛⠛⠛⠀⠀⠀⠸⠜⡄
⠀⠀⢲⠘⢄⠀⠄⣿⣿⣿⠀⠀⡠⠇⡖⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢹⡸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠅
⠀⠀⠈⠣⣌⡙⠀⠈⣿⠁⠀⢛⣡⠜⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠈⢳⣌⠻⠀⠀⠀⠀⠀⠀⠐⠟⣡⠎
⠀⠀⠀⠀⠁⠀⠰⠿⠿⠿⠆⠀⠈⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⣠⠾⠁⠀⠀⠀⠀⠀⠈⠁⠈⠀⠂⠂⠐⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠂⠐⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣀
⠀⠀⠀⠀⠀⠀⡀⢹⣿⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠘⣿⣿⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⠇⠀⡀
⠀⠀⠀⠀⣔⡏⠁⢀⣯⡀⠈⢩⣢⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣹⣅⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⡔⡠⠈⠀⢀⣿⡀⠀⠑⣐⡆
⠀⠀⢀⣎⡎⠀⠀⠈⠉⠀⠀⠀⢱⣱⡀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠨⠠⠁⠀⠀⠉⠉⠁⠀⠀⠨⠌⠅
⠀⠀⢘⡓⡇⠀⠀⠀⠀⠀⠀⠄⢸⢚⡃⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢭⠘⠀⠀⠀⠀⠀⠀⠀⠄⢘⢂⡍
⠀⠀⠐⢳⡘⣄⠀⠀⠀⠀⠀⣠⢣⣞⠂⠀⠀⠀⠀⠿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠘⣎⠱⣀⠀⠀⠀⠀⠀⡠⠏⡼⠁
⠀⠀⠀⠀⠻⢏⡗⠀⠀⠀⢾⡨⠍⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣶⣦⣴⣲⣿⠿⠉⠀⠀⠀⠀⠀⠀⠘⠳⣭⣘⡆⠀⢰⣃⡥⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠁⠈⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣤⣤⣦⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⢤⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⣤⣴⡄⣶⣶⠀⠀⠘⢷⣄⠀⠀⠀⢀⣤⠞⠋⠁⠀⣀⣀⠀⣀⡉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢀⣿⣿⣷⣿⣿⣦⣄⠀⠀⢻⣆⠀⣠⡟⠁⠀⠀⠀⢀⣿⣿⣴⣿⡇⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣴⣿⡿⠛⠉⠉⠙⢻⣿⣆⠀⠀⢻⣶⠏⠀⠀⠀⣠⣾⣿⠿⠿⠻⠿⣿⣦⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠸⣿⣿⣀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠈⣿⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠠⢿⣿⣤⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠈⠻⣿⣷⣤⡀⠀⠀⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣷⡄⠀⠀⠀⢸⡇⠀⢀⣄⣀⡀⠈⠙⠿⣿⣷⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢻⣿⣿⠟⠁⠀⠀⠀⠙⣿⣿⠀⠀⠀⣾⠇⠀⠈⣿⣿⠛⠃⠀⠀⠈⢻⣿⡆⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠘⣿⣿⣄⣀⣀⣠⣤⣾⡿⠋⠀⠀⢀⣿⠀⠀⠀⢿⣿⣦⣀⣀⣠⣤⣾⡿⠃⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠈⠻⢿⣿⡿⢿⣿⡏⠀⠀⠀⠀⣼⠻⣆⠀⠀⠈⠙⣿⣿⢿⣿⡟⠉⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣤⣾⠿⠇⠀⠀⠀⠘⢷⡄⠀⠀⠀⠸⠿⠇⠸⠿⠃⠀⠀⢀⣾⠃⠀⠙⢧⣀⠀⠀⠻⠟⠘⠿⠃⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾
⠀⣰⡿⠛⣿⡄⠀⠀⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠙⠷⣦⣤⣀⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⣰⠋⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⣶⡶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠙⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠈⠙⠛⠻⠿⢶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⣶⣶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⠟⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⣈⣿⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣄⠀⠀⣴⠟⠉⠉⠉⠓⢎⠉⠁⠉⢷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣾⣇⣀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢶⣶⣶⣶⠟⠋⠀⠀⣼⣇⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣶⡧
⠀⠀⠀⠈⠛⢶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠤⠴⠿⣶⡀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠤⢠⣭⣥⡄⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣈⠉⠙⢿⣆⠀⣿⡿⠁⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⡗⣩⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠟⠛⠉⠁⢀⣾⣧⡙⣯⣙⠛⠻⠿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠈⠻⣟⣛⡁⠀⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣏⡀⠀⠉⣿⣭⣤⣀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣯⣤⠄⠀⠈⢿⣦⣶⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠿⣽⣖⠄⠀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠟⠁⠀⠀⠈⠙⠶⣶⡄⠀⠀⠘⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣸⡯⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⣿⡗⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣾⣿⡇⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢰⣿⣿⡧⠀⠀⠀⠀⢿⡄
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡟⠀⣽⣿⣿⡧⠀⠀⠀⠀⠘⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠙⠿⣿⣏⡛⠙⠋⠋⠙⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⢴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⣉⣿⡿⠟
⠀⠀⠀⠈⠻⢿⣦⣀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⢂⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡐⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡶⠋⣡⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⠈⠛⢷⣄⠰⢤⡀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠋⡵⠟⢁⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠙⠻⢦⡙⢷⣤⡀⠀⢀⠈⠋⠀⠀⣀⣰⣖⣡⣾⡿⢫⠔⠋⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠈⠙⠳⢬⣉⠓⠀⣠⣿⣦⠀⠀⠀⠉⠋⠀⠀⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠤⣍⡐⠂⢀⣼⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢉⡓⠢⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣀⠲⠤⢠⣾⣿⣿⣿⠿⠛⠃⣿⣿⣿⣷⣄⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠬⢁⣴⣿⣿⡿⠛⣁⣴⣾⣶⣌⡉⠿⣿⣿⣦⡀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠠⠌⣡⣾⣿⠟⢉⣴⣾⠟⠋⠀⠈⠛⢿⣶⣌⠙⢿⣿⣆⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣀⣾⠟⢋⣤⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣌⠛⢷⡀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠰⢋⣥⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣙⠂⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠙⠻⣿⣇
⠀⠀⠀⠀⠀⠐⠐⠢⠞⠔⠿⠫⠖⠖⡷⠾⠾⠖⡷⠺⠶⠕⠷⠶⢚⠶⡷⠲⠞⡵⠳⢞⠖⠣⢺⠖⠷⢺⠖⠷⠢⠾⠟⠶⠲⠂⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢇
⠀⢠⣤⣤⣤⣤⣤⣼⣿⡇⢹⣤⣤⣤⣤⣤⣤⠀⠀⣠⣤⣤⣤⣤⣤⢿⡇⠀⢧⣤⣤⣤⣤⣤⡀⠀⢠⣄⣀⣀⣀⣀⡤⣿⣇⠘⢄⣀⣀⣀⣀⣤
⠀⠀⠈⠳⢶⣤⣄⣀⢻⠇⢀⣀⣤⣴⠾⠋⠀⠀⠀⠀⠙⢦⣄⣀⠀⠈⠃⠀⠀⢀⣀⡴⠞⠁⠀⠀⠀⠈⠳⢦⣤⣀⣤⣽⣿⣤⣄⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠙⡟⠛⣡⣦⡙⠛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠁⡀⡀⠉⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡯⢫⣿⣿⣯⠉⠛⠉
⠀⠀⠀⠀⠀⡸⢀⣼⣿⢿⣷⣄⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⣬⡷⣧⣄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⣼⣿⠿⣿⣆⠘⡀
⠀⠀⠀⠀⢠⣷⠟⠋⠀⠀⠈⠛⢶⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡴⠛⠁⠀⠀⠙⠳⣼⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠁⠀⠈⠙⢷⣥
⣀⣀⣀⣀⣚⣁⣒⣒⣒⣒⣀⣀⣀⣙⣂⣀⣀⣀⣀⣀⣀⣒⣋⣒⣒⣒⣒⣒⣒⣒⣚⣓⣀⣀⣀⣀⣀⣀⣀⣚⣓⣒⣒⣒⣒⣒⣒⣒⣛⣂⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⡀⢀⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠟⠋⠉⠉⠙⠻⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡟⠁⠀⠀⠀⠀⣀⡀⠈⢻⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠁⠀⣴⣶⣤⣾⣿⠟⠀⠈⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡆⠀⠈⠻⣿⠟⠁⠀⠀⣰⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⢿⣿⡿⠟⠻⢿⣿⡿⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣤⣉⣤⡞⢠⣤⣉⣤⣶⣶⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⣿⣿⡟⠀⠀⢻⣿⣿⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠏⠀⠀⠀⠀⠹⠿⠁

⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⣠⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⢀⣼⣇
⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣤⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣄
⠀⠀⢰⣷⣿⠇⠀⢻⣷⡆⠀⠀⢰⣷⣿⠿⣿⣿⣆⡄⠀⠀⠀⠀⢻⣿⣿⣿⣶⣇⠀⠀⠀⠀⢸⣿⣿⠻⣿⣶⠀⠀⠀⠀⡄⣿⣿⠏⢿⣿⡆
⠀⠀⠸⣿⠁⣰⣦⠈⣿⡇⠀⠀⢸⣿⠿⠀⠘⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠐⣶⡿⠁⡀⠛⣿⠀⠀⠀⠀⣿⡿⠁⠀⢸⢿⡿
⠀⠀⠀⠙⠦⡙⢛⠴⠋⠀⠀⠀⠀⠻⠄⠀⠀⠠⠟⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠁⠀⠀⠀⠀⠹⠄⢾⡿⠀⠋⠀⠀⠀⠀⠙⠧⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣠⣿⣷⠀⠀⠀⠀⠀⠀⣆⢸⣿⡄⡄⠀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⠀⢠⡄⣾⣿⢿⣿⣀⡄⠀⠀⠀⠀⣶⣻⣿⣷⢀⡄
⠀⠀⠀⢸⣿⣿⣿⣇⣷⠀⠀⠀⢈⣿⣿⠏⠻⣿⣿⣶⠀⠀⠀⣼⣿⣿⠋⢻⣿⣧⠀⠀⠀⣿⣿⣿⠟⠈⣿⣿⣿⠀⠀⢰⣦⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣷⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⡇⢀⣦⠀⣿⠇⠀⠀⠀⢻⣿⠁⠀⣼⢿⡟⠀⠀⠀⠸⣿⠉⢀⣆⠉⣿⠏⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠛⠻⠟⠛⠁⠀⠀⠀⠀⠈⠁⠘⠟⠁⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠛⠟⠀⠉⠀⠀⠀⠀⠈⠙⠚⠛⠛⠋

⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠀⠀⢀⣴⣆
⠀⠀⠀⢀⢀⣼⣿⣷⡄⠀⠀⠀⠀⡀⠈⣿⣧⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⠀⡀
⠀⠀⢀⣿⣿⣿⢿⣿⣿⡄⠀⠀⢀⣿⣄⣽⣿⡟⢀⡀⠀⠀⠀⡀⢻⣿⠛⢿⣿⣆⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⡀⣿⣿⣿⣾⣿
⠀⠀⠸⣿⡟⠉⠀⠹⣿⡇⠀⠀⢸⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⣿⣾⠋⢠⡀⢻⣿⠀⠀⠀⢠⣸⣿⡏⠿⣷⣾⡆⠀⠀⢸⣿⣿⣿⠛⢿⣿⣿
⠀⠀⠀⠻⣧⡀⠀⣰⠟⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⢿⣄⣿⣧⡼⠃⠀⠀⠀⠀⠻⡏⠁⠐⢻⡟⠀⠀⠀⠈⢿⡿⠷⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⣠⣾⣿⣷⣆⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣻⣿⣿⡿⠛⢿⣽⢿⣟⣿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣿⣁⣠⣽⣿⣿⣶⡶⣶⣿⣿⣯⣅⣀⢿⣿⣿⣿⣛⡛⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⢉⣹⣿⣿⣿⣿⠿⠾⠛⠛⠛⠛⠛⠻⠿⣯⣿⣿⣿⣯⡉⠉⣷⣿⡆
⠀⠀⠀⠀⣴⢿⣿⣿⣿⣿⣶⢿⣿⠞⢋⣡⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣌⡙⠻⣿⣻⣶⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠈⣿⣿⡍⢉⣽⢟⣵⠟⣡⣶⣟⣻⠿⠛⠋⠉⠉⢉⣉⣉⡙⠛⠿⣿⣻⣷⣄⠙⢮⣿⣯⡉⢉⣿⣿⠇
⠀⠀⣀⣤⣿⣿⣧⣾⣿⠟⢡⣾⣿⡿⠋⠀⠀⠀⠀⣀⣴⠟⠉⠉⢹⡆⠀⠀⠙⢿⣽⢷⣄⠻⣾⣿⣼⣿⣿⣤⣀
⠀⣾⢿⡿⠿⢛⣿⣿⠏⣰⣿⣽⠋⠀⠀⠀⢰⠶⠛⠋⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣿⣿⣦⠹⣿⣿⡟⠻⢿⣟⣷
⠀⠻⣯⣷⡄⣾⣿⡟⢠⣿⣿⠃⠀⠀⠀⠀⢿⣄⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⣿⣆⢹⣿⣿⢀⣾⣷⡟
⠀⣠⣿⢿⣿⣿⣿⠁⣿⣿⡏⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣿⣿⣿⣿⣿⣄
⣾⣿⡿⠋⢸⣿⣿⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠙⢿⡿⣷
⠹⣮⣝⣦⣸⣿⣿⠈⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⣿⣿⣇⣴⣫⣵⠏
⠀⠈⣿⣿⡿⣿⣿⡄⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⢀⣿⣿⣿⡿⣿⡁
⠀⣾⣿⡟⠁⢻⣟⣷⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣿⣿⠃⣼⣿⡿⠀⢻⣟⣷
⠀⠻⣿⣿⣿⣶⣿⣾⣧⠘⢿⣽⣦⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣴⢿⣾⠋⣴⣿⣿⣷⣾⣿⣿⠟
⠀⠀⠀⠉⣿⣿⡏⢻⣿⣷⣌⠻⣿⣿⣦⣄⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁⠀⣀⣴⣟⣽⠟⢁⡾⣻⡿⢹⡿⣿⠉
⠀⠀⠀⠠⣿⣿⣥⣤⣽⣿⣟⣷⣌⠻⢿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⡿⠟⣁⡴⣿⣿⣏⣤⣤⡿⣿⡇
⠀⠀⠀⠀⠙⠿⠿⠿⣿⣿⡛⢿⣿⣻⣦⣌⣉⠛⠻⠿⠿⠿⠿⠿⠿⠛⢋⣡⣴⣾⣿⡾⠛⣿⣽⡿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣼⣿⡿⢿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⢿⣿⣯⣤⣤⣿⣿⠃⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⡿⠾⠿⣿⣿⡄⣈⣭⣿⣿⡛⠛⢛⣿⣿⣯⣅⢀⣿⣿⠿⠿⠾⠾⠛⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠹⣯⣿⣿⡾⠿⣿⣿⣦⣟⣿⠟⢿⣿⣟⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠻⠿⣿⠻⣆⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢰⡟⢀⣀⣀⣠⣤⣤⣤⣦⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⣴⣤⣤⣤⣀⣀⣀⠀⢿
⠀⠀⠀⠘⠛⠛⠉⠉⠉⠀⠀⠀⠘⣷⡀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠉⠉⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣰⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⠀⠀⢠⠉⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠡⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⢸⠀⠀⢂⡀⠀⣀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠂⠐⠉⠱⠀⠀⠀⠀⠀⠃⠘⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠄⠀⠀⠀⠄⠐⣄⠖⠀⡀⠂⠠⠀⠀⠀⠀⠄⢄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡤⠀⡀⠀⠀⠀⠀⠹⢂⡞⢁⠠⠁⠀⠀⣃⠸⢠⠀⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⡱⡐⠀⠀⠀⠀⠈⠀⠄⠀⠀⠁⠀⠀⠀⠘⡘⢦⡶⢠
⠀⠀⠀⠀⠀⠀⡀⠀⣈⠹⠎⠃⠀⠀⠀⠀⠀⠐⠄⠀⠐⠆⠃⠀⠀⠀⠀⠄⢐⡁⠐
⠀⠀⠀⠀⠀⠀⠈⠂⢀⠐⠺⡀⡀⠀⠀⠀⠀⢀⡦⠤⢤⠆⡃⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢄⠐⢇⠀⠀⠀⠀⢘⡇⣰⣦⣁⡇⠀⠀⠀⢰⡂⠀⠐⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡨⡔⠘⡆⠀⠀⠀⠈⠉⢹⠈⠁⠈⠀⠀⠀⢠⡇⢠⠒⠃⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⡼⠖⠚⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠛⠙⠣⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⢤⡀⢰⣤⡔⡄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠄⢆⣄⠀⢇⣤⠀⠀⡄⠀⠉⠒⠈⠊⠓⠚⠉⠸⠀⡤⠠⣤⢼⢠⡄⣤⢀
⠀⠀⠀⠀⠀⢀⠀⠈⠃⠚⠋⠓⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠉⠓⠚⠉⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠒⠒⠀⠒⠒⠒⠂⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠈⠛⢿⣯⣭⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢛⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠈⢻⣿⣿⠀⢀⣾⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠙⣿⣠⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣿⢳⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠟⠛⠉⠁⠀⠀⢠⣾⣿⠀⠹⣿⣿⣿⣿⣷⣶⣤⣤⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⣦⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠉⠀⠀⠙⠷⣤⡀⠀⠀⠀⠙⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣴⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠈⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⠀⠀⠈⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⢀⠄⠀⡆⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠰⠳⣄⡎⠀⠀⠁⠠⠀⡜⢆⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠁⠀⠈⠆⠀⠀⢀⠀⠰⠁⢸⠈⠰⠃⢰⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡏⠲⢦⠀⠀⠀⠀⠀⠈⢄⠁⠀⢸⠀⠀⠀⡌⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⡘⠘⡐⢚⡇⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⡀⠀⠳⠉⠀⠌⡰⠀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢤⣄⡀⠀⠀⠀⠀⠂⠄⠐⠂⠀⠀⠁⠄⠂⠀⠀⠤⠊⠉⠈
⠀⠀⠀⠀⠀⣼⠹⣭⢍⣠⢴⠚⢠⣀⣤⡤⡖⠢⣤⢼⣐⠒⣦⠀⢲⢢⡀⠘⠛⣻⣻⡄⢠
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠻⠼⠛⠛⠊⠳⣾⡴⠃⠰⣌⣠⡼⠷⣤⣸⡖⠲⠮⠯⠟⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⢸⡟⠀⠀⣀⣤⣤⣶⣶⣦⣤⣤⡀⠀⢸⣿⠀⣠⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡙⠛⠀⣠⣴⣿⣿⣿⣿⣿⢉⣿⣿⣿⣿⣷⣦⡀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⢀⣾⣿⣿⣿⣿⠋⣡⣤⠀⣤⡉⠻⣿⣿⣿⣿⣆⠘⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇⢸⣿⣿⠰⣿⣿⣦⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣀⠛⠿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⣤⡉⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⣿⣿⡆⢸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣄⠘⢿⣿⢠⣿⡿⠃⣼⣿⣿⣿⣿⠇⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣤⣤⠀⣥⣴⣾⣿⣿⣿⡿⠋⣠⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠛⢿⣿⣿⣿⣿⣼⣿⣿⣿⣿⠿⠋⣠⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣌⣉⡉⠛⠋⠉⢉⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡶


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⢤⡶⠒⠒⠒⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠉⠉⠑⠖⠛⠛⠓⢲⠆
⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⢀⠶⢤⣄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣀⠐⠢⣤⠄
⠀⠀⠀⠀⠀⠀⠈⠓⢀⠴⠦⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢠⠂⠀⢀⣀⡀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠉⣆⠄⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠞⢆⠀⠀⠀⠙⠶⠶⠦⠤⣧⠀⣮⠤⠤⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠠⠄⠀⠘⠀⠈⠙⡻⠆⣀⣠⡤⠤⠤⡿⠀⡗⠒⠒⠒⠶⠤⣄⣀⠀⠀⠶⡄⠀⠀⠀⠀⠀⢤⡀⠀⠜⠀⠳⢤⣄
⠀⠀⠀⢀⡀⠈⠉⠂⠜⠀⠀⠀⠀⠁⢀⠤⢀⣮⠴⠋⠁⠀⠠⠀⠀⠑⠒⠋⠀⠀⠀⠀⠀⢀⣭⠷⠶⠀⠘⢂⡴⠃⠀⠀⠀⠈⠂⣀⠀⢀⠐⠉
⠀⠀⠀⠈⠀⠀⠠⠀⢦⠀⠀⠀⠰⠟⠁⣠⠋⠉⢀⠀⠚⢀⠠⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⡆⢀⣀⡐⡄⠀⠀⠀⠠⠶⠊⠁⠀⠙⠂
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠈⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⠣⠀⢧⠉⠀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠮⡀⠨⡏
⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠒⠻⡄⣀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠀⠃⠀⠀⠀⠀⠀⠀⡴⡄
⠀⠀⡡⠑⠔⠈⠇⠀⠀⠀⠐⠋⢠⡄⠳⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⡀⠀⠐⠁⠘⠲⢶
⠴⣮⣠⡄⠀⢈⣃⣄⠀⠀⠀⠀⢸⠁⠀⠁⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠧⣰⠀⠀⠀⠉⠰⠀⣀⠀⣘⠁
⠀⠀⠀⠘⣴⠁⠀⠀⠀⠀⢠⠀⡟⠀⠀⠈⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⠁⠀⢀⣀⣧⠀⠀⠀⢠⡴⠊⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠷⠐⠔⡆⠃⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠷⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠄⡄⠀⠀⢼⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡎⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠰⣾⠧⡀⠀⠀⠀⠀⠈⠐⠢⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡜⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⢢⡈⠦⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣠⠔⢃⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠉⠲⣌⡓⠤⢄⣀⠀⢀⣈⡀⠀⠀⢀⣀⡠⠄⠊⠉⣀⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠤⣄⣈⣉⣤⡈⠉⠉⢀⣀⣀⡤⠶⠊⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡈⠀⢀⡄⠉⠀⠈⠋⠉⠉⠁⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠄⣧⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⠆

⠀⣤⡶⠂⠰⣦⣦⠀⠀⠀⠀⠀⠀⢠⢞⣭⡕⡄⠀⠀⠀⠀⠀⠀⢸⠉⠭⠭⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠇⡰
⢶⣿⠁⠀⠀⠸⣷⠆⠀⠀⠀⠀⠀⠸⣜⡿⣫⠏⠀⠀⠀⠀⠀⠀⢸⢐⣒⣒⠂⠃⠀⠀⠀⠀⠀⠀⠒⢿⣿⣿⣿⠷⠒⠀⠀⠀⠀⠀⣈⠊⠄⠀⠀⢈
⠺⢿⣦⠀⢠⣾⠿⠂⠀⠀⠀⠀⠀⢰⣷⠩⣿⣇⠀⠀⠀⠀⠀⠀⢸⣐⣒⣂⣀⣀⠀⠀⠀⠀⠀⠀⢰⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠊⡄⠀⡎⠂
⠀⠙⠋⠓⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏


⠀⠘⣿⣧⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢘⣻⣏⡀⠀⠀⠀⠀⠀⠀⢻⣦⣤⣟⣧⣤⣿⠀⠀⠀⠀⠀⠤⡴⡇⢴⠈⣄⣀⠀⠀⠀⠀⠀⠸⣸⣿⣿⣿⣰⠃⠀⠀⠀⠀⣶⣶⢲⣯⣍⣷⣶⣶
⠀⢰⣿⠉⢹⣷⠀⠀⠀⠀⠀⠀⡽⠞⠛⠛⠛⠛⢿⡀⠀⠀⠀⠠⠴⠧⠧⠬⠤⠤⠧⠄⠀⠀⠀⠀⠀⠈⢻⣿⡋⠁⠀⠀⠀⠀⠀⠉⠙⠘⢷⣵⠟⠛⠋
⠀⠀⠛⠷⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠭⠷

⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣻⣦⣄
⣿⠀⢀⣤⣀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣿⡇⠀⠀⠀⠀⠀⠀⠑⠛⠊⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣶⣶⠀⠀⠀⠀⠀⠀⣯⢿⢾⡁⠈⣛⡄
⣟⠐⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⣶⡆⣿⡇⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⢶⣿⡶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠓⢦⣌⠽⣄⠞⠁
⠻⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⡒⠛⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣖⣒⣦⣄⣄
⡇⠠⠤⠀⠶⠶⠆⡇⠀⠀⠀⠀⣤⠚⣿⣛⢿⡌⡸⠆⠀⠀⠀⠀⠀⣴⡾⢛⢷⣦⠀⠀⠀⠀⠀⠀⠀⣴⣮⢩⣷⣦⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡏
⠳⠤⠤⠤⠤⠤⠤⠃⠀⠀⠀⠀⠙⣜⣙⡿⠞⠉⠀⠀⠀⠀⠀⠀⢠⣿⡷⠮⢙⣿⡇⠀⠀⠀⠀⠰⠿⠧⠼⠿⠆⠾⠷⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣶⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢀⠀⠈⢉⡽⢯⣍⣛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠟⠶⠶⠟⠉⠁⠉⢀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠉⠛⠛⠻⠿⠿⠿⠿⠿⠟⠛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠒⠶⠶⠶⣶⠶⠆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣦⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣍⢉⡿⠿⠟⠓⣦⡀⠀⠀⠈⠉⠉⢁⣀⠀⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣀⣉⣛⣛⣁⣀⣤⣴⣿⣷⠈⠉⠛⠛⠛⠛⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣉⣉⣉⣉⣉⣍⠉⣿⣿⠀⠒⠶⠶⠶⠾⠃⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⣉⣉⣉⣭⡄⠀⣿⣷⠐⠲⠶⠶⠶⠶⠟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠠⢤⣤⣬⣤⠄⠀⣿⡇⠐⠶⢦⣤⣴⣶⠶⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠠⠤⣤⣤⣤⠄⠀⣿⣇⠀⣤⣄⣀⣀⣠⣄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢤⣤⣤⣤⠤⠀⣿⣿⠀⠀⠈⠉⢉⣉⣁⣠⣿⡧⠶⠖⠂⠠⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠠⣤⣤⣤⣤⣦⣀⣿⡿⠶⠶⠾⣿⣉⡉⠁⠀⠂⣂⢤⣒⠠⠀⣤⠌⣙⠻⠶⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣠⡤⠤⠶⠲⣾⣋⡉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣥⣄⡀⠈⠙⠛⠋⠐⠠⢕⡀⣄⣭⣙⠛⠿⠷⣶⣤⣄
⠀⠀⠀⡼⢻⢛⣋⣯⡥⢦⣴⣲⣮⣽⣵⢿⣛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣤⣀⠀⠂⣀⡽⠶⠮⠾⢀⣀⣠⡤⠄⢩⡄
⠀⡄⠐⠤⡀⠀⠘⢬⡙⣿⣎⡧⢀⠀⠈⠀⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣻⣯⡤⠶⠖⢚⣋⣩⠭⠶⢒⣋⣹⡇
⠲⡇⠀⠀⠈⠒⠄⠀⠉⠉⠙⠲⢤⠟⡒⠲⠦⡄⠀⠠⠔⢒⣈⡉⠒⠄⠀⠀⣀⣀⣤⠴⠶⠚⠋⠉⠁⢸⠀⠀⠉⠩⠤⠒⠚⢉⣁⣀⣿⡇
⠀⡇⠙⠶⣄⠀⠀⠈⠳⢤⣀⠀⠀⢦⡙⢷⢆⣹⣇⡤⠚⠉⣀⣠⣤⡶⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡄⣀⣠⠤⠖⠚⠋⠉⢁⣸⣿⡇
⠀⣿⠲⣄⡈⠛⢦⣀⠀⠀⠈⠓⠦⣄⠉⠛⠛⣉⣤⣶⠾⠛⠛⠉⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣁⣠⣤⡴⠶⠞⠛⣋⣹⣿⠇
⠀⣿⠦⣀⠙⠦⣄⠉⠳⢦⡀⠀⠀⠀⠁⣶⣛⣭⠥⠒⢊⣉⣤⠤⠒⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢉⣀⣠⣤⠶⣞⠛⠋⣹⣿⣷⣤
⠀⠈⣄⡈⠳⢤⡈⠑⠤⠀⠉⠳⢄⡀⠀⣿⡴⠶⠚⣛⣩⡥⠶⠚⠋⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠭⢭⣒⣮⣭⣶⣶⣿⡿⠟⠋⠉
⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠉⢲⣿⡶⠾⠛⠉⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣿⣿⣿⡿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣤⣴⣶⠿⠛⠀⢹⣀⣠⣤⣶⣾⣿⡿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⡀⠀⠀⠀⣿⡖⠚⠩⠛⣋⡥⢀⣠⣤⣶⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠸⣿⣧⠶⢞⣫⣵⡾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣯⡴⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⣠⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠉⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠀⠀⢿⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠙⢦⣌⢷⣄⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠙⣆⠙⠷⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢛⡓⠦⡀⠈⢷⡀⠀⠀⠀⣿⠀⠀⠘⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⡄⠀⠻⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⣰⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡆⠀⣠⠟⠀⠀⠀⠹⣄⡀⢻⠀⠸⣇⠀⠀⠀⠀⠀⢰⡇⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⠟⢰⡏⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠘⢷⣄⠀⠀⠀⠀⠻⣄⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⡉⠀⠀⠻⢦⣄⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠉⠳⢦⣄⠀⠀⠈⠳⡄⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⠀⠈⠉⢻⠓⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠃⠀⠀⠀⠀⢠⣸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢠⡞⠀⠀⠀⢀⣤⠟⣿⠃⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠀⠀⢹⡆⢀⡤⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢁⡖⠉⠀⣀⡴⠞⠋⠀⣰⠏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠅⠀⠘⡇⢸⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠘⢧⣠⠞⠁⠀⠀⠀⠘⠁⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢰⠇⢸⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊⠁⠀⠀⠀⠀⠀⢠⢀⠞⠁⠀⣰⠋⣠⡾⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠃⢀⣠⠄⣀⣴⠆⢀⠀⠀⠀⠞⠀⠀⠀⠀⠟⠉⠁⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠰⣶⣤⣛⠀⠈⢁⣾⣿⣷⣿⣿⠃⠀⠀⣀⠀⠖⢀⣀⡀⠀⠀⠀⠀⠀⣰⣯⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣍⣻⣦⣄⡀⠀⣀⠐⣀⣈⠛⠻⣟⣻⣿⣿⡿⠟⣋⠀⠀⣠⣤⣤⣤⡶⠿⠛⠁⠀⠀⠀⣀⡼⠯⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠴⠶⠛⠻⢶⣤⣄⣈⣁⣰⣿⡿⣯⣥⣤⣀⡛⣿⣿⣶⣬⣍⠀⠻⠷⠶⢶⣆⣤⣴⣾⣿⣶⣶⣶⠤⠴⠶⠖⣶
⠀⠀⠀⠀⠀⠀⢸⣁⣀⣀⣤⡤⠤⠶⠶⠚⠋⠉⠉⠉⠉⠁⣀⡤⣿⣿⣿⣿⣿⣿⣿⡿⠿⢶⠲⠶⣿⣿⣭⢤⣭⣍⣛⡛⠛⠶⠦⢤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠋⢑⣻⡿⠟⠋⢁⣀⡈⠉⠉⠉⠋⠉⠛⠒⠾⠭⣅⣀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⣯⣉⣀⠤⠖⠚⠉⠉⠀⠠⠖⠛⢉⣀⡙⠚⠙⠛⠶⠶⢦⣄⣠⣤⡴⠶⠯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⠀⠀⣀⣠⣴⣶⠟⠻⠿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠘⢿⣿⠗⠀⣼⡟⢛⠁⣀⣤⣤⣤⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⠟⠛⣿⡶⠀⠀⢸⣿⠀⠀⢻⣧⣿⣤⡿⢿⣦⣉⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣿⠀⠀⠀⢸⣿⠀⠀⢸⡏⠀⠀⠀⣠⡿⠋⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⢸⣿⠀⠀⠸⣷⡄⠀⠀⠿⠁⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⣿⣀⣀⣀⣀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⣾⠛⠛⠛⠛⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠶⠶⠶⠶⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⠶⠶⠶⠶⠶⠾⠇⠀⠀⠀⢰⡟⠉⠉⠉⠉⣿⡀⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠁⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⢸⡇⠀⠀⠀⠀⣿⡆⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠿⠀⠀⠀⠀⣸⡏⠀⠀⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⡁⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠛⠛⢻⣦⠀⣻⡇⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⠀⢾⡇⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⢻⡇⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⠂⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠿⠷⠶⠶⠶⠾⠿⠶⠿⠷⠶⠶⠶⠾⠿⠶⠾⠿⠶⠶⠶⠶⠿⠷⠶⠿⠶⠶⠶⠶⠿⠷⠶⠆




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣹⣿⣿⠋⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠟⠏⠀⠀⠀⠀⠀⣹⣿⣿⠋⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠟⠏
⠉⠀⠁⠀⠀⠀⠀⠀⠴⣾⣿⣯⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠴⣾⣿⣯⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⣀⣴⣄⡀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⣀⣴⣄⡀
⠀⠀⢠⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠙⣿⠿⠁⠀⠀⠀⠀⢠⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠙⡿⠿⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⣶⣦⠄⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⣶⣦⠄⠀⠀⠀⠀⠀⢠⣶⡀
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⢘⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⣤⣶⣆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠘⠋⠙⠀⠀⠀⠀⠀⣤⣴⣆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠘⠋
⢀⡀⠀⠀⠀⠿⠿⠏⠀⠀⠀⠀⠀⢹⣿⣿⡖⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠿⠿⠏⠀⠀⠀⠀⠀⢹⣿⣿⠖⠀⠀⠀⠀⠀⠀⣠
⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⢘⣿⣿⡶⠀⠀⠀⠀⣸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⢸⣿⣿⡶
⠀⠁⠀⠀⠀⠀⠀⠀⢐⣿⣶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋
⠀⠀⢀⣄⠀⠀⠀⠀⠛⠻⠟⠛⠀⠀⠀⢀⣀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠛⠻⠟⠛⠁⠀⠀⢀⣀⣶⡄
⠀⠲⣾⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢶⣾⣇⡀⠀⠀⠀⠲⣾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢶⣾⣧⡀
⠀⠀⠿⠋⠙⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠻⡏⠁⠀⠀⠀⠀⠿⠋⠙⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠛⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡏⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡏⠀⠀⠀⠀⢀⣤
⣀⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⢤⣴⣄⡀⠀⠀⠀⢀⣀⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⢤⣴⣄⡄
⣹⣿⣿⠋⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠟⠏⠀⠀⠀⠀⠀⣹⣿⣿⠋⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠿⠏
⠉⠀⠁⠀⠀⠀⠀⠀⠰⣾⣿⣯⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠴⣾⣿⣯⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣹⣿⣿⠇⠀⠀⠀⠀⣀⣴⣄⡀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⣀⣴⣄⡀
⠀⠀⢠⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠙⣿⠿⠁⠀⠀⠀⠀⢠⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠙⣿⠿⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⣶⣦⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⣶⣦⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⢀
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⢘⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⣤⣶⣆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠘⠋⠙⠀⠀⠀⠀⠀⣤⣶⣆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠘⠋
⢀⡀⠀⠀⠀⠿⠿⠋⠁⠀⠀⠀⠀⢹⣿⣿⡖⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠿⠿⠏⠁⠀⠀⠀⠀⢹⣿⣿⡖⠀⠀⠀⠀⠀⠀⣤
⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⢸⣿⣿⡶⠀⠀⠀⠀⣹⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⢸⣿⣿⡶
⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⣶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋
⠀⠀⢀⣄⠀⠀⠀⠀⠛⠻⠟⠛⠁⠀⠀⢀⣀⣶⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠛⠻⠟⠛⠁⠀⠀⢀⣀⣶⡄
⠀⠲⣾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢶⣾⣧⡀⠀⠀⠀⠲⣾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢶⣾⣧⡀
⠀⠀⠿⠋⠉⠀⠀⢀⣠⣶⣀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠛⠻⡏⠁⠀⠀⠀⠀⠿⠋⠙⠀⠀⢀⣠⣶⣀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠛⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡏⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡏⠀⠀⠀⠀⢀⣤
⣀⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡏⠀⠀⠀⠀⢤⣴⣤⡄⠀⠀⠀⢀⣄⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⢤⣴⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠞⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠦⠈⠡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣂⣄⡯⠉⠀⠀⠀⠰⠙⡄⠀⠀⠖⠙⢴⠉⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠂⠠⣉⡭⠋⠀⠀⢀⡀⠀⠈⠁⠀⠙⠀⠠⡄⠐⠂⠀⣐⠀⢠⡄⠀⠀⠐⢖⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⡠⠬⣳⡀⠲⠐⢊⠰⣄⠀⠰⡆⠺⢀⠀⠈⠢⠀⡠⢼⢥⠉⠀⠀⠀⢀⣈⡠⡎⢶⠛⠁⠀⠀⡀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢠⡥⠮⣒⠁⠀⠰⠕⢵⠋⠁⠀⢸⡠⠚⠑⢦⡇⠀⠀⠀⠈⠁⠀⠈⠺⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⣒⡒⢤⡀⠀⠈
⠀⠀⠀⠁⠀⠀⡞⡵⠒⠢⡙⣄⣀⠈⡉⣉⠉⠉⠈⠀⠀⠀⠁⠀⠀⡈⠉⣀⠉⢠⠠⢠⠀⠅⢁⣀⣠⠷⠃⠀⠀⠈⢆⢳⠀⠀⠉
⠀⠀⠄⠀⠀⠘⣸⠁⠀⠀⠓⠛⠛⠓⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⢒⠒⠒⠐⠂⠒⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⢸⢸⠁⠀⠒
⠀⠀⠄⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⡆⢉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠡⢊⠔⡡⢐⠄⡡⣊⢸⠀⠀⠀⠀⠀⠀⠀⡌⢸⠀⠀⠄
⠀⠀⠀⠀⠀⠘⢸⡀⠀⠀⠀⠀⠀⠃⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⢊⠔⡡⢊⠔⡁⠂⠀⠀⠀⠀⠀⠀⡜⢠⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠡⢇⠀⠀⠀⠀⠀⠸⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢚⠔⢈⠔⡡⢊⠔⠃⠀⠀⠀⠀⠀⠜⡠⠏
⠀⠀⠀⠀⠀⠀⠐⢟⣆⠀⠀⠀⠀⠀⠆⠀⡪⠀⠢⡀⠀⠀⠀⠀⠀⠀⠐⡡⢒⡡⢊⠔⡡⠆⠀⠀⠀⠀⠠⢂⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢎⡷⡀⠀⠀⠀⠈⢆⠐⠄⠀⠐⠀⠀⠀⠀⠀⠀⣪⠔⡡⢊⠄⡊⡐⠀⠀⠀⢀⠔⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡢⣄⠀⠀⠈⢦⠀⠠⠀⠀⢀⠀⠀⠀⠠⢔⡡⢂⠔⢪⠊⠀⢀⡠⢐⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢵⣢⢤⣀⡑⣄⠀⠂⠀⠀⠀⠀⢐⣡⡖⣻⣭⡥⣴⣶⠷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠷⠦⣄⠀⢀⠨⢛⡡⠚⠛⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠑⢐⡺⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⡠⠖⣝⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠸⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠢⠒⠔⡈⣹⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⢀⠕⡊⠢⢛⢛⣞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠺⠻⠯⠤⠞⠁⠭⠤⠿⠿⢿⠿⠿⠿⢿⠿⠧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡁⡟⠛⠛⠒⠒⠒⠒⠒⠒⠚⠚⠚⠚⠓⢺⠌⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣼⣶⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⣀⠸⡀⠀⠈⠀⠀⠀⠐⡛⠛⠛⠓⢒⠂⠖⠖⡶⢋⡞⣸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⠙⠉⠛⠐⠛⠓⠓⠓⠓⠓⠟⠺⠞⠗⠻⠛⠟⠩⠑⠨⠃⠩⠂⠝⠊⠐⠚⠊⠓⠉⠊⠂⠒⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⠖⠋⠙⠲⢤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⡀
⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⡴⠒⠲⠦⠴⠋⠁⠈⠙⠦⠴⠖⠒⢦⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣀⣠⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⣄⣀⠀⠀⠀⠀⢹⡀
⠀⠀⢀⡤⠞⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠳⢤⡀
⠀⢰⠋⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠙⡆
⠀⢸⡀⠀⠀⠀⢰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⠀⠀⠀⢀⡇
⠀⢸⠇⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠸⡇
⢠⠏⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠚⠀⠀⠘⠒⠒⠒⣦⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠹
⡏⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠘⣆⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⣰
⠀⢸⡆⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⠴⠋⠉⠳⢤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⢰⡇
⠀⢸⠁⠀⠀⠀⠸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠈⡇
⠀⠸⣄⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⣠⠇
⠀⠀⠈⠳⢤⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠛⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠉⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠖⠦⣄⠀⠀⣠⠴⠲⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠉⠉⡿⠉⠉⠳⢤⣀⣀⡴⠞⠉⢻⡉⠉⠉⠁⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢰⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⣠⣞⣁⣠⣤⠤⠶⠞⠋⠙⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⢰⠟⠛⠒⠶⠶⠤⣤⣄⣘⣧
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦
⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠿⢿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀
⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣷⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣾⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠁
⣿⣿⡿⠻⠟⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣵⣷⣮⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣤
⠀⠀⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡏⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠓
⠀⠀⠀⠀⠀⢹⢿⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢑⣶⣿⣶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣤⣔⠁
⠀⠀⠀⠉⠉⠉⠉⢹⣿⣿⣿⡋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⡀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠿⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡟⠀⠑⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣷⣄⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀
⠀⠀⣀⣠⣿⣆⡀⠀⠀⠀⠀⢀⡰⠣⡠⠀⠀⠀⠀⣀⠀⠂⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⣀⣀
⠀⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⠀⢠⠂⠣⠀⠀⠀⠀⠈⣃⠤⣄⠁⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢈⡟⢯
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠂⢘⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⡄⣀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠇⠀⠛⢀⡉⠀⠀⠀⠂⠀⢤⠄⠈⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠈⢳⡄⣚⠃
⠀⠀⠀⠈⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠐⠂⠉⠃
⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠂⠺⠠⠀⠃⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⢠⢠⡾⢤⠤⠀⠀⠠⣶⣿⣶⣀⣠⠔⠁⠀⠀⠄⠈⠃⡔
⠀⠀⣠⡀⠡⠂⣴⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⣙⠶⣻⡀⠀⠀⠀⠟⠙⠳⠀⠀⠀⠀⠀⠀⡤⠐⠨⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⠀⠂⠀⠀⠀⠁⠀⡀⣀⠘⡀⠀⠀⠀⠀⠀⢀⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣷⣤⡄
⠀⠐⣾⣿⡆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠁⠀⠀⠀⠉⣹⠿⣶⡀⠀⠀⠀⠰⠶⠴⠂⠀⠀⠀⠀⠀⢙⣿⣿⣇
⠀⠀⠋⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉
⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⣶⣼⠧⠖⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠙⠳⠄⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⢀⡀⠁⠠⠀⠀⠀⡀⡀⠸⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠤⠀⠀⠀⠠⡖⠂⠡⡶
⠀⠠⠂⠄⠀⠤⠄⠀⠀⠀⠀⠈⢓⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠇⠊⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣾⣿⠃⠀⠀⠀⣠⣤⣤⣤⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⣤⣤⣤⣄⠀⠀⠀⠈⣿⣷⣄⠀⡀
⠀⠀⠀⠀⠀⠀⣰⣿⡄⣿⡿⠃⠀⠀⠀⣼⣿⠟⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠻⣿⣷⠀⠀⠀⠘⢿⣿⢠⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⠃⣵⣶⡖⠀⠀⠀⣿⣿⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⣿⣿⠂⠀⠀⢰⣶⣬⠘⣿⣿
⠀⠀⠀⠀⠀⣦⠹⡟⣼⣿⠟⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣰⣿⡟⠀⠀⠀⠀⠻⣿⣧⢹⡟⣴
⠀⠀⠀⠀⢸⣿⣇⢀⣭⣶⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⣶⣍⡁⢸⣿⡇
⠀⠀⠀⠀⠸⣿⡇⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⢸⣿⡇
⠀⠀⠀⠀⣶⡝⠇⠿⣋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠿⠸⢋⣶
⠀⠀⠀⠀⣿⣿⡆⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢠⣿⣿
⠀⠀⠀⠀⠹⣿⣧⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⣿⠏
⠀⠀⠀⠀⢰⣮⡙⠘⢡⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡌⠃⢛⣥⡆
⠀⠀⠀⠀⠀⢿⣿⣆⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠈⠿⣿⡌⣿⢃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⣿⢣⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠰⣦⣍⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⣩⣵⡆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣌⢿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡿⣡⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠧⠙⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠛⠵⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣤⡙⢿⣿⢠⣤⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⢀⣠⡄⣾⣿⢟⣠⣶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠶⠍⠈⢿⣿⣷⡄⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⢀⣴⣿⡿⠃⠉⢵⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣷⣶⣭⣛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢛⣩⣴⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣀⠀⠀⠀⠀⠀⢀⡊⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠅⠀⢀⠀⠄⠀⠏⠁⠀⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡾⠈⠆⠀⠀⢀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠺⠀⠰⠉⣇⠀⠀⠀⡰⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠸⠀⠀⠘⠀⠀⢤⢀⠞⠙⠢⡀⠀⠘⠀⠀⡀⠀⡇⠀⡛⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢇⠀⠀⡄⠀⠀⠀⢁⢀⠃⠀⠠⠀⠀⠆⠀⠀⠀⠀⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⣄⠀⡆⠀⡇⠀⠀⠘⡀⠀⢱⡘⠋⠁⣠⠎⠀⢠⠃⠀⠀⡄⠀⠀⠆⠐⡄⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⣄⠀⢀⠁⠀⠀⠀⠁⡀⠘⠷⢂⡾⠉⡀⠠⠃⠀⠀⠀⢃⠀⡇⡠⡄⢸⠃⢀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠃⢷⡀⡌⠀⠀⠀⠀⠀⢰⠀⢈⣈⠁⠀⠑⠃⠀⠀⠀⠀⠘⡆⢳⣌⣠⣄⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢⣶⣄⣠⡏⠙⠀⠀⠀⠀⠀⠀⠈⡀⠀⠐⠂⠀⢠⠀⠀⠀⠀⠀⠀⢰⠀⢛⠿⠁⢋
⠀⠀⠀⠀⠀⠀⠀⠀⢔⠁⠀⣤⠐⠛⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠙⢸⠀⠀⠀⠀⠀⠀⠘⠀⢨⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢄⠒⠒⣛⠠⡄⠀⠀⠀⠀⠀⠀⠀⡤⠤⡤⢤⡴⠘⡀⠀⠀⠀⠀⠀⡈⠁⠀⠈⠳⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⠀⣁⡇⠀⠀⠀⠀⠀⠀⠀⣗⢨⢐⡅⠹⢠⡇⠀⠀⠀⠀⢀⣅⠀⠀⢀⡀⠰⡆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡌⡪⢠⢀⢘⣆⠀⠀⠀⠀⠀⢠⢼⣴⣼⢻⡵⢞⢣⠀⠀⠀⠀⠸⢲⠄⡒⠀⣢⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡩⢂⠔⡯⡀⣽⠀⠀⠀⠀⠀⠘⠀⠐⢸⠈⠀⠀⠀⡄⠀⠀⠀⢠⣞⠨⢔⠙⡨⢊⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⢴⣅⢿⠬⠖⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⡇⠀⠀⠀⢸⠓⠚⠺⢣⠌⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⠘⠀⡇⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡆⠀⠌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠀⡀⠀⠀⢀⠀⠀⠒⠦⣤⠰⣒⡧⣴⠚⣦⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄⣀⠁⢀⡇⣀⡄⢀⡀⣀⠀⠀⠛⠲⠆⠚⠊⢛⠶⠶⢋⣁⡒⢀⣀⡄⣤⣤⣀⡇⣠⠀⣁⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢴⡕⣋⡷⢤⣌⣻⡇⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣩⠾⠡⠷⣍⣡⡿⠍⠸
⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⣁⣀⠀⣉⠀⠈⠀⠁⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣄⣀⣀⡀⣀⣀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠬⠤⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠤⠥⠤⠽⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⢄⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡊⠸⢿⡧⠏⣰⠈⢍⣇⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣸⠛⣷⣹⣦⣿⣶⢘⢟⡳⠴⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡲⡗⠈⠀⠛⠛⠛⠛⠈⠉⢻⡿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢤⡀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠈⠓⠐⠂⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣷⠀⠓⠤⢀⠀⢀⣀⠀⠈⠀⣂⡄⣀⣀
⠀⠀⠀⠀⠀⢀⠀⣤⡆⣲⣬⣤⣙⢼⣿⡆⠀⠀⣠⡯⡯⢀⡀⠀⢸⣭⣠⢿⠽⣁⣄⠠⢄
⠀⠀⠀⢀⣤⣥⠺⢻⡍⢭⢻⣭⡿⢢⢸⣦⡴⠚⠹⡏⢭⠘⠙⣄⣸⣾⡗⢘⢈⠿⣿⢠⡼⣥⢨⣶⡀
⠀⠀⠀⠘⠝⣏⣦⠈⢹⡏⡿⠥⠀⣁⣿⢿⡀⠀⢀⡟⢛⠀⠀⠈⣲⣹⠠⣾⣶⡛⠳⢋⢃⠂⠾⠫⣧
⠀⠀⠰⡷⠄⣈⠭⡸⣟⢩⠻⣿⢣⠝⠾⣺⡃⠀⠸⢯⢰⠄⠀⠸⣧⠼⠀⠪⡿⢇⢤⢸⣶⡆⣷⣻⣷
⠀⠀⠠⡷⣾⢃⣗⣼⣻⢿⢹⣍⡿⢭⠱⢒⠁⠀⣤⣦⣴⣦⠀⡡⣫⣻⢿⢿⣷⢠⢥⡎⠏⣵⣸⣿⡛⡄
⠀⠀⠠⣶⡉⡝⠵⣟⢶⣧⠷⣷⣿⢿⣆⣘⣃⢰⣿⣿⣷⣌⡠⡗⣵⣻⣼⣿⣿⢘⣉⣏⣈⠿⠾⡿⣉⠄
⠀⠀⠀⢽⣀⡕⣠⣜⠉⣿⢘⣮⢵⣆⣽⣩⣬⢶⡷⡘⣷⣼⣷⣳⢙⣰⣖⡶⢷⣹⡿⣭⢹⣟⣓⣆⣻
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠒⠻⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠺⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠐⠚⢻⡿⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣰⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠚⢿⠟⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠐⢺⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠂⠐⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⣰⡄⠘⠃⠀⠀⠀⠀⠀⠀⠄⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠤⢾⣿⣿⡶⠄⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠀⠠⠴⢾⣷⠶⠤
⠀⠀⠀⠀⠀⠁⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠰⡦⠈⢻⡿⢁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠇


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

⠀⠀⠀⠀⢀⣤⠶⠒⠛⠛⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⢈⣿⣇⢀⢀⣰⢿⣿⣛⣿⣷⠂⠀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣴⣫⡀⢻⣾⡜⠙⣸⣿⣿⣟⠿⣴⣗⠛⠂⠭⠭⠭⠻⢿⣟⠛⢛⣛⣛⣛⢛⡿⠯⠭⣽⣿⣿
⠀⠀⠀⠈⠙⢿⣮⠿⠒⠾⠁⠀⠀⠈⣆⣈⣿⡌⡿⠋⠁⠀⠀⠀⠀⠀⠐⠒⠒⣀⡲⣶⣶⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠂⠀⠀⠀⠛⢿⡿⢷⢸⣥⠰⣆⡀⠀⠀⠀⠀⢀⡀⣿⡅⣿⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⡀⠀⠀⠀⣠⡴⠋⠸⡇⣷⣀⣺⠁⠀⣠⣤⣄⢸⡇⠟⠃⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠉⠲⡴⠾⣧⡀⠀⠀⣇⣿⠻⢤⣉⠀⠿⠿⠟⢸⡇⢸⡦⠟⣇⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣰⢯⠈⠳⢤⡴⠁⢀⠈⢻⠀⠀⣿⣻⣄⠀⢨⡗⣠⠀⠀⣜⢿⡅⠀⣠⣏⣸⣿⡿⣧
⠀⠀⠀⠀⠀⠐⣇⠈⠳⢤⠶⠳⣶⠁⢠⢿⡆⠀⢹⠀⠀⠁⠀⠛⠉⠉⠉⠉⠛⠀⠈⢉⡵⠛⠋⡦⣬⣷⡄
⠀⠀⠀⠀⠀⢀⣼⠗⠖⠋⠀⠀⠙⣖⣡⠞⠧⣄⣸⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⡴⠒⠚⠳⣌⣧⣼⣟⣿⡀
⠀⠀⠀⠀⣰⠟⠁⠀⢀⡀⠀⠀⠀⠘⣿⠶⣤⣾⣿⠒⢒⠚⠃⠈⠉⠁⣂⠖⠛⢃⠀⠀⠀⠸⠾⢵⣾⡟⠃
⠀⠀⠀⣼⠁⢠⣶⡿⣿⣷⣤⠀⠀⠀⠘⣦⠀⠉⣿⢰⡈⠲⠶⣷⣦⣤⣼⣄⣀⠀⢀⣀⣀⣀⣠⣾⡀⣙⣦⡀
⠀⠀⢠⡇⠀⣈⡻⢿⣿⣮⣍⠁⠀⠀⠀⣽⠀⠀⡿⣼⠇⢠⡄⠀⠉⠉⠛⠋⣽⠋⠉⠁⠈⠉⠉⠙⠻⡿⣿⣷⡄
⠀⢠⡾⠁⠀⠹⢿⣾⣷⣿⡟⠀⠀⠀⠀⡿⠀⢀⡇⣿⠀⢸⡇⢠⣶⣽⣷⢼⣿⡠⠀⠀⠀⠀⠀⠀⠀⢹⠘⣿⣷
⠀⠈⠛⢦⣀⠀⠀⠛⠀⠀⠀⠀⠀⢀⣀⣿⠀⢸⢣⣟⠀⣾⢧⠄⠉⠉⠁⠀⣟⣇⢀⠀⠀⠀⠀⠀⢀⡼⢠⡿⠃
⠀⠀⠀⠀⠙⠳⢦⣤⣀⣠⣤⡴⠖⠋⠉⠁⠀⣿⣬⣭⡥⠤⠤⠤⠤⠤⠤⢤⣭⣭⣇⠀⣤⡤⠤⠴⠚⢶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣀⣴⣧⣤⡀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠰⣿⣯⣿⣽⣏⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣶⣯⣿⣽⡿⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠈⠉⠛⠁⠀⠀⠀⣠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣤⣤⣤⣤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⣾⣷⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣼⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣷⠤⠀⠀⠹⠉⠀⠀⠀⠀⢠⣿⣧⡀
⠀⠐⠒⠺⢿⣿⣿⣿⣿⡷⠶⠆⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠰⠶⠾⣿⣿⣿⣿⡶⠶⠆
⠀⠀⠀⠀⠀⠹⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠻⠋⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣀⠀⠀⠀⠀⠀⠀⠈⢿⡏
⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣇⠀⢘⣶⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⣠⠀⠀⢠⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢈⠿⠿⠛
⠀⠀⠛⣿⠟⠀⠀⠹⠃⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠈⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣼⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠙⣿⠛⠁⠀⠀⠀⠐⢦⣠⣿⣧⣴⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⡉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠈⢿⡏⠉⠓⠀⠀⣰⣿⣶⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⠻⠓⠠⢴⣿⣯⠄⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⠈⠛⠁⠀⠀⠀⢀⣀⣼⣧⣀⡉⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠂⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⢡⣤⣾⣧⣤⠄⠀⢤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⠦
⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⢹⡿⣿⡇⠀⠀⠀⠀⠀⠉⠉⣿⣿⠏⠉
⠀⠐⠒⠒⠿⠿⠿⣿⣿⠿⠿⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣤⠀⠀⠀⠀⠀⠚⠻⠿⣿⣿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠰⠶⢾⣿⣿⡶⠶⠆
⠀⠀⠀⠀⠀⢰⠀⠀⠤⣤⣿⡤⠤⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⠀⠀⠀⠀⠀⠐⠶⢾⣿⣿⠷⠆⠀⠀⠀⠀⠀⠀⠲⠾⢿⣿⡿⠿⠒⠂⠀⢸⡇
⠀⠀⠀⠤⠤⣾⡦⠤⠄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡄⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣿⣿⣦⡤⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠠⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠏⠀⣀⣤⣾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠶⠶⣿⣿⡷⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢂⢠⡀⠀⠀⠀⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⡀⠀⠀⠈⡠⠁⢴⢤⣀⡈⠻⣅⣀⣄⣠⣠⠵⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠱⢄⠀⢀⠁⠀⠀⠀⠛⣡⣿⣿⣿⣿⣿⣿⣦⣠⣦⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⢀⠀⠀⡠⣦⠝⠉⠉⠁⠀⠙⢿⣿⣧⣰⣠⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠃⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣹⡗⠀⠉⠛
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦⣤⣶⣾⣛⡁⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠘⠃⢀⠀⠀⠂
⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠋⠉⠙⠛⠛⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣦⣠⡄⠀⠠⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⣠⡼⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠇⠄⢠⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⣿⡿⢿⣿⣿⣧⡀
⠀⠀⢠⢞⡽⢦⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢀⠈⡄⠀⠀⠀⠀⠀⠈⠵⢛⡄⠀⠈⠀⠀⠀⠙⢿⣿⡇
⠀⠀⠀⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⢺⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠤⠤⠚⠀⠀⠀⠀⠀⠐⢤⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⡀
⠀⠀⠀⠀⢀⣠⠔⠊⡀⠀⢰⠋⢳⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⣀⣀⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠⠀⠀⠀⠀⠀⠀⢻⠙⠯⠳
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⡧⢴⠆⡀⠐⠒⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⠄⠠⠆⣁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢀⠀⠀⠀⠀⠀⠀⣉⠠⠀⠃⢠⠉⣸⡆
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⡠⠤⠴⡖⠁⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⠉⠉⣻⠀⠀⠀⠀⠀⠀⠐⠃⠤⠤⠤⠋
⠀⠀⠀⠀⠀⠀⠘⠓⡎⠉⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡜⠉⠉⠒⠒⠤⣄⣱
⠀⠀⢀⡤⠤⠦⠤⠴⠤⠤⠤⠤⣤⣤⣤⣤⣵⣤⣤⣤⡿⠀⠀⠀⠀⠀⠀⣤⢼⠷⠖⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⣀⣸⠁⠀⠀⠀⠀⠀⠀⢘⠃⠀⢀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⣴⠆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣶⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠛⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠼⠆


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣆⠀⠀⡞⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⡜⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡬⠭⠭⠭⠭⠭⠭⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⣦⠀⠀⡶⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠟⠀⠀⠳⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⣡⣞⡶⠒⠂⠐⠒⢆⣳⡍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠰⣧⠎⢉⡤⠶⠒⠒⠦⣄⡉⠳⣿⢌⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⡷⠂⣰⠋⠀⠀⠀⠀⠀⠀⠙⣆⠰⣾⠇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣷⡶⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢶⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣮⣶⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣮⡷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡨⣥⣆⠘⢦⣀⠀⠀⠀⠀⣀⡼⠁⣴⣭⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⢭⣷⡄⣈⡙⠓⠚⢋⣡⢰⣷⡍⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣐⠛⠛⠀⠀⠛⠓⣂⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⠁

⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢘⡿⡏⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡇
⡀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠂⣀⣤⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⠀⠀⠀⣀
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠼⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠟
⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⣠⣦⣀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀⠀⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠈⡿⠻⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⢲⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⢺⣿⡂
⡀⠀⠀⢒⠷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⣠⡀⠀⠀⢙⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⣠⡀⠀⠀⠠⣾⡤⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠰⠿⡟⠁⠀⠀⢀⣠⡀⠀⠀⢠⣷⡄⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠸⠿⡏
⠙⠃⠀⠀⠀⠉⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠈⠉⠁⠀⠀⠀⡄
⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡿⣏⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢺⠿⡍⠀⠀⣄
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠇
⡀⠀⠠⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠄⠀⠀⣀⣤⡀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡦⠄⠀⠀⣀
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠻⠁⠀⠀⠀⠀⠀⠀⠀⠐⠿⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠟
⠀⠀⠀⢰⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⡶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣤⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠹⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠽⠛⠧⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠛⠃⠀⠀⢲⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢺⣿⡍
⡄⠀⠀⠐⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢐⠿⡏
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⡄
⣰⡀⠀⠀⠠⣾⡶⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠰⠛⡟⠁⠀⠀⢀⣠⡀⠀⠀⢠⣿⡖⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠼⠻⡏
⠙⠃⠀⠀⠀⠁⠉⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠈⠁⠁⠀⠀⢀⣄
⣀⣤⣀⣀⣀⣀⣀⣀⣀⣘⣿⣏⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣦⣀⣀⣀⣀⣀⣀⣀⣹⣿⣅⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⢷⣾⣻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣟⣷⡾⠷⢶⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣿⠀⠘⠻⠛⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠟⠃⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣽⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣯⡶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣯⠉⠉⢻⡀⠀⠀⢀⡟⠉⠉⣽⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢷⠀⠀⡾⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠇⠘⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠈⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉