Faucet Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⢀⣤⣤⣄⡀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⣛⣛⣛⣛⣛⣋⡉⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⢉⣿⣛⣿⡉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣟⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⣄⠈⢦⠉⠉⠉⠀⠀⠉⢳⡖⢾⡟⠛⠛⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠘⣷⠘⡆⠀⠀⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣽⠀⡇⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣠⣴⡟⡰⠁⠀⠀⠀⠀⢀⣼⣧⣼⣥⣤⣤⣤⣤⣤⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⣿⣙⠛⠓⠶⠶⠶⠶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣌⠷⣤⣤⣤⣤⠿⣩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣬⣙⡒⠒⠒⢒⣋⣡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡍⠉⠉⠙⠉⢉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢸⡇⠀⡄⠀⠀⣿⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸⡇⠀⡇⠀⠀⢸⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡇⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠀⠇⢸⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢸⡇⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡇⢸⢸⠀⠀⢸⡇⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⢸⡇⣼⠈⠀⠀⢸⡇⠘⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠃⢸⡇⠁⠀⣰⠀⢸⡇⠀⣈⡁
⠀⠄⠀⠀⢀⣀⡀⠀⠀⣠⠞⠛⠆⠸⡇⠀⠀⠋⠀⠘⣇⠀⠛⠛⣀⠀⠀⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠃⢀⡴⠀⠀⢀⡀⢤⡀⠚⡀⠚⠲⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠓⡀⠀⠇⠀⠀⠈⠈⠣⠄⠀⠀⢈⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠷⠴⠾⢧⣀⣀⡾⠀⠸⣦⣤⠾⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠉⠉⠓⠲⠦⢤⣤⣤⣀⣰⡋⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⣤⠤⠴⠒⠋⠉⠁⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡀⠀⠀⢀⣀⣤⠴⠖⠚⠋⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⢀⡷⠃⠈⠉⠙⠛⠲⠦⢤⣀⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡶⢶⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠷⠾⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠋⠉⠉⠉⠉⠉⠉⠻⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣩⣹⣉⣋⡏⢹⠛⡟⢻⠛⡟⢻⠛⡏⢻⢙⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠋⠉⠉⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⡴⠞⠛⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⢠⡿⠁⠀⠀⠀⠘⣷⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⣼⠃⠀⠀⠀⠀⢰⠏⠀⢈⡿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡖⠲⠶⠤⢤⣤⣀⣀
⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢦⣄
⣿⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⢸⡇⠀⠀⠀⠀⠸⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠈⢿⡀⠀⠀⠀⠀⢻⡤⠖⠛⠛⠛⠒⠶⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⢷⣄⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠲⠶⠦⠴⠶⠖⠒⠛⠋⠉⠉⠀⠀⠀⠉⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⠲⠤⣤⣄⣀⣀⣀⣤⡤⠶⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣤⣤⣤⣤⡴⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣤⡀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣉⣉⣹⣇⠀⣹⣋⣉⣉⣉⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣿⢙⡇⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠳⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⢤⡀⠀⠀⠀⠀⢸⡧⠶⠶⠦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡟⠛⠛⠛⢻⠏⠀⠀⠀⠀⠈⢻⡛⠛⠛⠛⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⡇⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⣤⠟⠒⠒⠒⠛⢧⣀⠀⠀⢀⣠⠞⠓⠒⢲⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⢀⣇⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⢸⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣹⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠒⠒⠒⢰⠋⠉⢱⠄⠠⠤⠄⠀⠀⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣠⣀⣀⣤⡦⠼⠰⢽⣻⣶⣄⣀⣤⠀⠘⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠉⠉⠑⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷⣼⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢸⠌⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠚⢦⡶⠾⠶⣾⠷⢶⣄⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠾⠟⠛⡋⠉⠙⠛⣻⠁⠀⠀⠀⠀⣀⡎⠄⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣀⢜⠌⠀⣀⠀⠀⠀⠤⠈⠽⢧⣀⣀⣀⣾⠈⣷⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⢀⠔⠂⠉⠀⠀⠀⠀⠐⠒⠉⠁⠀⠊⠁⠀⠀⠀⠀⠀⠈⠐⠉⠉⠘⢹⣆⣿⣾⠀⠆⣠⣶⡴⡆⠀⠈⢿
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣈⠏⠃⠠⡄⠀⢠⣾⣿⣿⣤⠀⡀⣀⠀⠀⠀⠀⠀⢸
⠈⠀⠀⠀⠀⠀⠀⢀⣀⡯⣤⠦⣕⡀⠐⠀⠡⠉⠄⠋⢀⢂⣡⠷⠤⣼⣸⣿⣿⢿⢀⠡⠀⠀⠀⠀⠀⠀⡿
⠀⢀⠀⠀⠀⢂⢲⠋⠁⠀⠀⠀⠈⠙⢦⣄⣤⢀⣠⣀⡦⠋⠀⠀⠀⠈⠳⣿⣏⡏⠔⠒⢈⣁⠀⠀⠠⠀⡇
⠀⢸⠀⠀⠈⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢟⡠⡥⡀⠤⠶⠂⢀⣀⣼⠃
⢰⢸⠀⠀⠋⢙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢿⠻⠛⡇⢐
⡎⡚⠄⠀⢉⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠁⠨
⠣⣗⣀⢀⣠⣯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐

⠀⠀⡠⢣⡀
⠀⣐⠁⠈⠈⢆
⠰⠗⠀⢠⠤⣻⢇
⠐⠂⠀⣼⣶⣾⡻
⠈⠶⣠⣄⠍⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠰⣙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢹⠞⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠛⠻⠻⣧⡀⠀⢀⣤⣤⣄⣀⣠⣤⣄⣀⣀⣤⡾⠋⠘⠇⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣛⣿⡅⠉⠙⠛⠛⢸⣿⣿⠿⣤⣤⣀⠀⠀⢀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⣶⡶⠛⠉⠉⠉⠉⣷⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⠿⣶⣤⣷⣄⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣿⠀⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣶⣶⣾⣶⣶⣶⣶⣶⣿⣷⡟⠛⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⠯⠭⠽⠿⠟⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣭⣵⠀⠀⠀⠀⠀⠀⠀⠠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠶⣶⣿⠺⢿⣿⣒⠀⠀⠀⠀⠀⠀⣲⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠉⠛⠻⠿⢶⣶⣤⣤⣤⣴⡶⠿⠟⠛⠉⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣧⡿⠀⣀⣀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⣾⠿⠏⠀⠉⠉⣉⢉⣉⣉⡉⠉⠉⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠚⠉⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⡆⡀⠀⠐⣒⣲⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠛⠛⠛⠻⣛⡛⠿⣉⠉⠉⠁⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢿⡇
⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀⢸⡇
⠀⠀⠀⠀⠸⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡶⠟⠛⠿⣷⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣸⡇
⠀⠀⠀⠀⠀⣿⣁⠀⠀⠀⠀⠀⠀⠀⡄⣀⣠⣿⠋⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣰⠛⢿⣯⣀⣀⠀⠀⠀⡆⢐⣂⣛⣛⣿⡄⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠹⣦⣤⡀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⢉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣶⣤⣤⣤⣤⣶⠾⠟⠋
⠀⠀⠀⠀⠈⠙⠻⠷⠶⣦⣤⣤⣤⣤⣤⣾⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢻⠀⡄⢧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠘⣇⢳⣀⡁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⡇⠀⢻⡄⠉⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣷⣴⠘⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡏⠂⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣧⣦⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⠀⢀⡴⠖⠶⣄⠀⢀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⡶⠦⣤⡀⠀⠀⠀⠀⣼⠁⠀⠉⠉⠉⢻⠁⠀⠀⢸⡏⠉⠉⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⣦⠀⠀⠀⠘⢦⠤⠴⠶⠒⢻⠀⠀⠀⢸⠓⠲⠶⠦⢤⡼⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣀⣼⣤⣤⣤⣼⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⣧⠀⠀⠀⠀⠀⢠⡷⠶⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠻⢤⣤⣤⡼⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠴⠞⠁



⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠿⢷⣶⣤⣤⣤⣶⠿⠋⠁⠀⠀⠈⠛⢿⣶⣤⣤⣴⣶⡾⠿⠛⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⣀⣀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠾⠿⠟⠛⠛⠉⠉⠉⠉⠛⣷⣦⣤⣤⣤⣤⣴⡿⠛⠉⠉⠉⠉⠛⠛⠻⠿⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠉⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣤⣤⣤⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡿⠛⠉⠉⠉⠉⠉⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡛⠛⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠿⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡿⠻⠿⠿⠿⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⣤⣤⣤⣤⣶⣶⣶⣶⣶⡶⠶⠾⠿⢿⣿⣦⣄⣀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢶⣶⣦⣤⣀⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣶⣤⣀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠿⢷⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣸⣿⣴⡶⠿⠿⠿⠿⢶⣶⣤⣤⣄⣀⣀⣀⣤⣤⣤⣶⣶⠾⠿⠿⠟⠛⠛⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⣤⣤⣤⣴⣶⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣴⡶⣦⣼⣷⣷⣦⣤⣴⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣤⣬⣭⣍⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠉⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⠀⠀⠀⠘⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⣂⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣀⣤⣤⣤⠀⠀⡤⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠛⠛⠂⠉⠉⠀⢿⣯⣽⣿⣿
⠀⠀⠀⠀⢀⡴⠋⣠⣴⣿⣿⣿⣿⣿⣷⣦⣷⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣧⣿⣿⣿⣿⣿
⠀⠀⠀⢠⠏⢀⣼⣿⠟⠉⠉⠉⠉⠛⠛⠿⠿⠿⠿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠻⠿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⡞⠀⣾⣿⠃
⠀⠀⠀⠇⠸⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠛⠒⠶⣤⠴⣾⡝⡟⠀⠉⠉⠉⠉⢙⠟⢿⠶⢤⠴⠖⠚⠉⠉⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⢻⠀⠀⠀⠀⠀⢸⢠⡆⠀⠆⠀⠀⠀⠀⠀⠀⠀⢰⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣇⡀⠠⠠⠤⠔⣒⣭⠽⠶⠺⡏⠸⡆⠀⠀⠀⠀⡇⢈⡗⠲⠾⣭⣑⡒⠤⠄⠀⠀⣨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠹⣄⣧⣀⣀⣀⣸⢀⡞⠀⠀⠀⠀⠈⠉⠙⠓⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣏⣉⣉⣉⣉⣟⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣉⣼⣁⣀⣀⣀⣀⣙⣄⠙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢹⡟⠀⠀⠀⠀⠀⠀⣿⠁⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣼⣧⣤⣤⣤⣤⣤⣤⣿⣄⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⣠⡟⢉⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠒⠛⣉⣉⣉⠩⠍⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⡇⢸⠀⠀⡀⡀⠀⢰⢸
⠀⠀⠀⠀⠀⠀⣠⠖⠋⣁⡤⠖⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⡇⡇⡇⢸⢸
⠀⠀⠀⠀⣠⠞⢁⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡇⢸⢸⢸⡇⡇⡇⢸⢸
⠀⠀⠀⣰⠃⣰⠋⠀⠀⠀⠀⡠⠂⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⢛⣛⣿⣿⣿⡇⢸⢸⢸⠃⡇⡟⢸⣼
⠀⠀⢠⡏⢰⠇⠀⠀⠀⠀⣸⠀⢀⡴⠚⠉⠉⠉⠉⠙⠛⠳⠶⠦⣤⣄⣀⣀⣀⣀⠀⠠⢄⣒⣉⣭⡴⠖⠛⠉⠁⠀⠈⠙⣇⠸⢸⣻⢰⣷⡇⣿⣿
⠀⠀⣼⠁⡟⠀⠀⠀⠀⠀⡜⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣈⣛⣛⣛⣛⣛⡏
⠀⣠⡟⢸⠇⠀⠀⠀⠀⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⢸⡏⠅⣘⣒⠒⠂⢀⣀⣚⣓⣚⣻
⠸⣅⠀⢒⣖⣠⡬⡭⠭⠍⣶⣶⣿
⠀⠈⠉⠓⠒⠲⠶⠶⠖⠛⠛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡼
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠛⠒⠦⢤⣄⣀⣀⡀⣰⠃⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣤⠤⠖⠛⠉⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢰⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⢀⣀⣤⠴⠖⠚⠉⠉⠉⢻⡀⠀⠀⠀⠀⠀⢀⡟⠉⠉⠉⠙⠓⠲⠤⣤⣀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢶⠓⡦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠶⠶⠶⠶⠶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⢾⢿⠥⡤⡤⣤⣤⣤⣤⣤⣽⡶⡦⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠚⠒⠶⠶⠶⠶⠶⠷⠾⠶⠶⠷⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣠⡶⠒⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣰⠏⠀⠀⠀⠈⢳⣤⣀⣀⣀⣀⣀⣀⣤⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⣿⠀⠀⠀⠀⢠⡏⠀⢠⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠛⠒⠲⠶⠤⢤⣄⣀⡀
⢸⡇⠀⠀⠀⠀⣼⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠸⡇⠀⠀⠀⠀⢿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⣿⠀⠀⠀⠀⢸⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⢹⡄⠀⠀⠀⠀⢧⣀⣀⣷⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢻⣄⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠶⠤⣤⣄⣀⣀⣀⣠⣤⠤⠶⠖⠚⠛⠛⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠓⠦⣤⣀⣀⣀⣀⣀⣀⣤⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣤⣤⣤⣤⡴⠶⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠞⠛⢻⣿⠿⣿⠛⠛⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠾⢿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣰⣟⠉⢻⡙⠻⠿⠟⠛⠙⠻⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠟⠋⠉⠙⢷⡄⢳⡀⠀⠀⠀⠀⠀⠈⢿⡄⢻⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠘⣷⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡿⢸⠇⠀⠀⠀⠀⠀⠀⣾⠃⣼⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢲⣿⣾⣡⣟⣀⣀⣀⣀⢀⣠⡾⠿⠾⠷⠶⠶⠶⠶⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣿⠀⠀⠀⠀⢸⣿⣬⠉⠙⠉⠉⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⢦⣙⠷⠶⠶⠶⠾⣫⠼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣌⣙⣛⣛⣛⣋⣡⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⡏⠉⠉⠉⠉⠉⡉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⡇⠀⣤⠀⠀⠈⣿⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣿⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⠀⠀⠀⣿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⣟⠀⢰⡆⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡿⠀⢸⠀⣿⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⢀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢸⡇⢀⣸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢸⡇⢸⢸⡇⠀⠀⣿⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢸⡇⣸⠀⠀⠀⠀⣿⠀⠉⢀⡤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣀⠀⠁⢸⡇⠉⠀⢸⡇⠀⣿⠀⢀⣤⣀
⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⣴⠛⠛⠇⠸⠇⠀⠀⠘⠁⠀⣻⠄⠘⠋⢉⣤⡀⠀⠚⠉⠉⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠁⢠⡴⠄⠀⢀⣀⠠⣄⠘⢃⠀⠛⠦⠀⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠓⠀⠀⠟⠀⠀⠈⠁⠹⠦⠀⠀⠀⢉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⠾⣧⡀⠀⣴⠆⠀⣷⣀⣴⠿⣦⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠉

⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣥⣦⣧⣤⣦⣤⣷⣤⣷⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⠤⠤⢤⠤⠤⠤⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⢠⣾⡆⠀⠀⠀⠀⠀⠀⠉⠉⣽⣧⠉⠁⠀⠀⠀⠠⠊⣶⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠆
⠀⠀⢠⣄⣠⣿⣇⣀⣤⠀⠀⠀⠀⣶⣾⣿⣿⣷⡆⠀⠀⠀⠀⠀⡤⣿⣷⣦⣀⣠⠀⠀⢀⣄⣶⣾⣿⣶⣄⣄
⠀⠀⢸⡿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠛⢻⣿⣿⡟⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠸⠿⣿⣿⣿⣿⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠦⠤⠤⠦⠤⠀⠀⠀⠀⠠⢠⣤⠤⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⡠⠄
⠀⠀⠈⠉⠙⢿⠏⠉⠉⠀⠀⠀⠀⠀⢠⢲⠒⡄⠀⠀⠀⠀⠀⠀⣤⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎
⠀⠀⣤⣀⣰⣾⣦⣀⣠⠀⠀⠀⠀⣀⠀⣮⣷⠀⡀⠀⠀⠀⠀⠀⣼⣿⣧⣼⣦⠀⠀⠀⠀⠀⠠⣾⣿⣷⣄⣀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⡟⠁⠀⠀⠀⠀⠀⣿⣿⣿⠟⠻⢿⣿⡧
⠀⠀⠈⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢽⡍⠁⠀⠀⠀⠀⠀⠀⢀⣄⣽⡇⣠⡀⠀⠀⠀⢰⣾⢸⣿⣿⣇⣿⡆⠀⠀⠀⠀⠀⣿⣿⡇
⠀⢰⣤⣀⣶⣶⣀⣤⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠁⣿⣿⠁⠉⠀⠀⠀⠀⠀⠀⢈⣯⠁
⠀⢸⠿⠚⣿⣿⠳⠿⡇⠀⠀⠀⠛⢻⣿⠿⠿⣿⡟⠛⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⣿⠶

⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⢠⣏⡁⣸⢀⣙⣦⠀⠀⠀⢰⣾⣅⣸⣇⣽⣷⠀⠀⠀⠀⡼⠉⢨⣇⠈⢳⡄⠀⠀⠀⣰⢯⣄⣀⣠⡿⡄
⠀⠀⢸⡀⢩⠿⣏⠁⡽⠀⠀⠀⢸⣧⡴⢿⡿⢤⣼⠃⠀⠀⠀⣧⡤⠿⠿⢦⣼⠇⠀⠀⠀⢷⡀⠹⣿⠋⢠⡇
⠀⠀⠀⠛⠯⣤⡼⠟⠁⠀⠀⠀⠀⠙⠦⣼⡧⠞⠋⠀⠀⠀⠀⠘⠳⢤⡤⠴⠋⠀⠀⠀⠀⠈⠳⠤⠿⠴⠋

⠀⠀⠿⠛⠸⠛⢾⠀⠪⠁⢯⠖⠳⠘⠶⢸⠐⠍⠸⠑⡽⠺⠟⠀⠿⠘⠶⠨⠇⠘⠆⠾⠼⡀⠆⠹⠀⠸⠮⠁⢯⠊⠳⠘⠆

⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣈⣯⣀⣀⠀⠀⠀⠀⢸⣿⣀⣼⣆⠀⠀⠀⠀⠀⢉⣻⣋⠁⠀⠀⠀⠀⠈⠉⢻⡏⠉⠀⠀⠀⠀⠈⠉⢻⡟⠉⠁
⠀⠀⢀⣿⡛⠟⠛⠛⠀⠀⠀⠀⢸⣿⠋⠁⠈⠀⠀⠀⠀⣶⣶⣿⣷⣶⡆⠀⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⢠⣼⣯⣄
⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠙⠉⠋⠀⠀⠀⠀⠀⠘⠛⠛⠛

⠀⠀⠀⠀⠀⣀⡤⠤⠄⠀⠀⠀⠀⠀⢀⣠⣠⡄⠀⠀⠀⠤⠤⣄⡤⠤⠄⠀⠀⠀⠀⢀⣄⣠⡀⠀⠀⠀⠀⠀⠀⠐⣶⡖
⠀⠀⠀⣀⣀⣿⣀⣀⠀⠀⠀⠀⣀⢀⣿⣅⣀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⢀⣤⣸⣇⣤⡀⠀⠀⠀⢠⣤⣤⣿⡇
⠀⠀⠘⠿⠿⣿⠿⠿⠃⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⠿⡿⠿⠿⡿⠇⠀⠀⠀⠛⠿⠛⠻⠿⠃⠀⠀⠀⠀⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⢀⠀⣶⡆⠀⠀⠀⣤⣦⣤⣤⣦⡄⠀⠀⠀⠀⣿⣇⣴⡿⠀⠀⠀⠀⠀⢸⣿⡇⣀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣶⠿⠿⠿⠛⠀⠀⠀⠀⠉⠉⣿⡏⠉⠁⠀⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿
⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃

⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⣼⠻⣧⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⡀⠀⠀⠀⠀⢺⣥⣽⠀⠀⠀⠀⠀⠀⠑⣿⣷⠋⠀⠀⠀⠀⠘⣿⣟⣀⣤⣄
⠀⠀⠀⠀⣱⣿⣎⠀⠀⠀⠀⠀⢿⣿⠃⠉⠉⠁⠀⠀⠀⣤⣦⣿⣧⣦⡄⠀⠀⠀⠀⡴⣿⡟⢄⠀⠀⠀⠀⢠⣿⣧⠈⠛⠃
⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⢼⣿⡆⠀⠀⠀⠀⠀⢀⣴⣶⡦⡄⠀⠀⠀⠀⠀⠀⣀⡠⠒⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠃
⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⢸⣶⡀⣴⡟⠀⠀⠀⠀⣤⣠⣽⣤⣠⡄⠀⠀⠰⣿⣿⠁⢈⣿⣿⠀⠀⠀⠀⢈⠀⡱
⠀⠀⠀⠀⢺⣿⠇⠀⠀⠀⠀⠀⢦⠙⠓⠛⢴⠆⠀⠀⠀⠋⠙⠛⠛⠙⠃⠀⠀⠀⠈⠀⠉⠁⠀⠁⠀⠀⠀⠀⠀⢿⣥⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⣄⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣆⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⡄⠀⠀⠀⠄⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣀⡦
⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠋⣉⡤⠞⠋
⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⢢⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠇⠀⠀⠀⠀⠀⠀⠐⢳⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢇⡀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢠⠀⠸⠀⠀⠀⠀⠀⠀⠃⡸⠁⢀⠶⠞⠁⠀⠀⠀⠀⠀⣀⡤⠔⢂⣀⣀⡄⢀⣀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠠⣈⠦⠤⡀⠀⠀⠀⡷⠈⢳⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣠⢧⠇⠀⠀⠀⣀⡤⠖⠋⣁⠔⠁⢀⡰
⠀⠀⠀⠈⠙⡆⠀⠀⠀⠙⢆⠈⢣⡀⠙⠊⠉⠒⠃⠀⠚⠓⠢⣀⡇⠀⠀⠀⢨⠤⠔⠒⢁⡈⠁⠈⠓⠒⠒⠋⠁⠀⢢⠎
⡖⢄⠀⠀⠈⠀⠀⠀⣀⢀⠼⢀⠆⡉⠉⠓⢤⠴⠂⢉⠒⠒⠐⢀⡇⠀⠀⠀⢐⠔⠀⠠⡄⠀⠀⠀⠀⠀⠈⠛⠉⠧⣜⠓⠦⣀⡠⠤⣄⣀⣀⣤⠤⠤⠀⠀⣀⣀⣀
⠈⠢⣉⠐⠒⠂⠀⠒⠃⠀⠀⡠⣘⣦⠤⠤⢤⣀⠢⠠⠁⠸⣇⣾⢧⣄⠀⢀⡌⢀⣴⠀⣧⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢀⠠⠒⢈⣉⣉⣀⣀⠔⠓⠶⠖⠐⠤
⠀⠀⠀⢹⡄⠲⠴⠖⠋⠉⠉⠉⠁⠀⠀⠐⠢⢌⡙⠓⠒⠚⠁⠀⠀⠀⢑⠺⠗⠋⣸⠄⠘⢏⠒⠒⠒⠚⠩⡀⠀⠈⠛⠒⠚⠡⠔⣋⠿⠛⠋⠉
⠀⠀⠀⠀⠑⠶⢤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠤⠄⠶⢿⣇⣀⣠⠞⢡⠀⢀⠀⠁⠚⠧⠒⠋⣉⠁⠐⠉⠉⡀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠰⢤⣶⡄⠴⠊⠀⠀⠀⠀⠀⠈⠉⡀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡠⠀⢀⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⡀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠰⠀⠀⠘⠦⢄⣈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⣀⡀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠈⠀⠈⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠉⠉⠈⠀⠀⠀⠂⠁⠀⠀⠐⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡦⢠⣤⣔⣒⣲⣦⣴⣯⣭⣤⣤⣤⣤⡟⠙⣧⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢩⢤⢷⠶⢿⣿⣿⣯⡿⠟⠛⠙⣉⡉⠩⠭⠭⠭⠇⣸⠧⠤⣬⣅⣀⠀⢀⣁⡛⣷⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠐⠸⢴⣊⢙⣩⢗⣛⠛⣋⣩⠤⠒⠚⣉⡩⠤⠄⠔⢒⣒⡂⣺⡁⠀⠈⠒⠒⠤⠭⣙⡒⠤⣝⠩⢝⡻⣭⣛⠍⠒⢀
⠀⠀⠀⠀⣠⠞⠃⠐⡐⡀⠀⢀⡊⠋⣀⡴⠞⠉⠁⢒⡈⠉⣉⣉⣉⣁⠃⢉⠉⠉⠑⠒⠢⢤⣄⠈⠁⠂⠄⠀⠈⠑⠉⠑⠢⣀
⠀⣰⢃⡞⠁⠀⠄⠀⠀⢀⡀⠀⢀⡾⠉⠀⡠⠀⠀⣴⣶⣾⠿⡿⠕⠁⠀⠀⠽⣿⣷⣶⡦⢄⡀⠓⢤⣠⣀⠀⠀⠀⠀⠐⡆⠈⠂⠀⠀⠀⠠⣄
⠰⠃⣸⠐⠀⠘⠀⡀⠀⡏⠈⠋⠿⠀⡂⢸⠀⠀⣶⢸⠇⠁⠀⠀⠀⣀⣀⣴⣮⣮⣿⠈⠂⠀⠀⠀⠀⣈⣿⣷⣤⠀⠀⠀⠈⢣⠀⠀⠀⠀⠛⡌⢦
⠘⠀⢀⢰⢀⢰⢡⢡⠀⣇⢀⣀⡀⠀⢇⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⡠⠆⠀⠈⠀⠀⢨⣽⢿⣛⠄⠀⠀⢀⠈⡷⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠈⢎⢧⢻⢧⠱⡈⢮⡻⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠈⠉⠁⠀⣀⣀⡤⠤⠶⠀⢀⡾⠉⠀⣠⣾⠏⠀⠀⢀⡄⠀⠐⢸⢄⣷
⠀⠀⠀⠀⠈⢦⠳⣙⣴⣌⠂⣙⠪⣝⡷⣦⣀⠀⠀⠐⠢⠄⠀⡀⠀⢈⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠴⠋⢀⣤⠾⠋⣀⠴⢋⠤⠋⠀⠀⢀⠎⣼⠃
⠀⠀⠀⠀⠣⣀⠑⢌⠳⣗⠑⠢⣔⠠⣙⠻⢭⣟⡶⢤⣤⡀⠀⠀⠀⠀⠬⡭⠭⠭⠄⠀⣀⣀⣤⠤⠶⠚⣋⣉⠤⢖⣫⠵⢚⣡⠔⢠⠄⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠑⠂⠈⠙⠲⢭⣒⡬⢽⡓⠲⠬⣭⣽⣛⠓⠚⠛⠛⣛⣛⣩⠭⠥⠖⣒⣛⡩⠥⠖⣊⡡⠄⠚⠁⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⠀⠈⠻⢡⡀⠀⡄⠀⠀⠒⠒⢈⠉⠩⠥⠴⡒⠊⠉⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠈⠑⠈⠑⠊⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⠀⠰⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡈⣠⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢾⡠⠄⠊⣡⠞⠁⠀⠈⠀⠀⠀⠀⠀⢧⡄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣔⠀⠏⠀⢈⡶⠁⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⢳
⠛⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣿⠁⠀⡸⠂⡼⣁⡀⠀⠀⠀⣴⠂⠀⠆⠀⢄⣠⡄⠸
⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠉⠁⠀⠀⢡⣴⣿⣿⠗⢀⠙⡄⠊⠉⠀⠀⠀⠀⠀⠀⠁
⣷⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡂⠀⢫⣿⣷⠛⣣⡛⠈⠀⠀⠀⠀⠀⢀⣀⣆⠀⠈
⢼⠗⢄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡾⠟⠀⠀⠀⠈⠁⠀⣿⣿⣶⢾⢽⣷⡇⠀⠀⠀⣀⣠⡟⠉⡈⠀⢠
⣾⣷⣾⣷⣤⣤⣄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠃⠀⡰⠂⢠⠀⠀⠀⢠⣿⡿⢡⡟⣼⣽⡇⠀⠀⠀⠀⠹⣧⢽⢣
⣿⣿⣿⣋⣿⡿⠿⠿⣧⣾⣿⣿⡗⠠⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⢹⣿⣿⣆⣽⡿⣿⡁⠀⠀⠀⠀⠀⠘⠶⢧⡀⡀
⣿⡿⠛⠉⠀⠀⢲⣶⣿⣿⣿⣿⡇⣀⡨⣿⢿⣶⣬⣯⢶⣾⡿⠋⢋⣀⣀⣀⣾⢟⣟⣮⡖⠀⠀⠀⠈⢿⣿⣿⣿⣧⡈⠷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣾⣿⣇⣀⡉⠻⠿⢿⡦⠰⢯⣉⡿⠿⡿⢋⣾⣿⣿⡇⠀⠀⠀⠀⠈⢻⣏⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣿⣿⣿⣿⡿⣿⣇⡈⠁⠀⠤⣀⠒⠿⣞⠛⣾⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣧⣿⣿⣏⡳⢶⣶⠮⣯⣷⡴⠶⠏⣲⣿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠯⣭⣀⠈⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠶⠿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⢞⣛⣡⣿⡳⠻⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣷⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢿⣿⣿⠿⠛⠫⠭⢭⣿⣿⣟⠚⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⠟⠒⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⡳⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠻⠿⡿⣿

⠀⠀⠀⠀⠀⠴⢶⣶⠶⠄⠀⠀⠀⠀⠀⠀⠰⠶⠶⡶⠶⠦⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠓⠓⠓
⠀⠀⢀⡄⠀⢀⣼⣿⣄⢀⠀⣄⠀⠀⢀⡀⡀⢀⣼⣧⣄⠀⢀⡄⠀⠀⢠⡀⡀⢀⣽⣏⡀⠀⢀⡀⠀⠀⣠⣀⣀⣰⣾⣷⣄⣀⣠⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⡟⠛⠛⠿⠿⠟⠛⠛⡿⠀⠀⠸⠟⠛⠻⠿⠿⠟⠛⠻⡇⠀⠀⢸⠟⠛⠻⠿⠿⠟⠛⠻⠇⠀⠀⠻⠛⠛⠛⠿⠟⠛⠛⠻⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣖⣒⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣬⣯⣄⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠻⠿⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⢀⠁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⡀⢀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠐⠺⡀⠀⠀⢐⠅⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢀⠀⠀⠐⢂⠀⠀⠀⢀⠀⠀⡠⠇⠁⠤⠤⠔⠀⠀⠀⠀⠄⠀⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠣⠈⠂⡀⡄⠁⠀⠀⠀⢀⠐⠂⠀⠺⡍⡀⠀⡄⠀⠐⢢⠐⠁⢀⠢⠂⠀⠀⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⡔⠀⠀⠀⠀⢱⢐⣄⠀⠴⠻⠽⠀⠀⢀⠑⡌⢄⠆⠠⠃⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⡀⠀⡀⠀⠀⢠⣇⠄⠀⢀⠀⠀⠈⡀⠀⠀⣈⠀⡱⠀⠀⠠⣁⠃⠜⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡂⢄⡀⢈⠐⠈⣔⠄⠄⣠⠟⠊⠰⠤⠄⡤⠏⣂⡀⠀⣀⣤⣊⣀⣀⡨⢐⡏⢐⡂⠀⠀⠊⡠⡈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡳⠖⠲⠄⠀⠊⣉⠈⠑⠂⡧⠀⠑⠒⠂⢈⠉⠉⠠⠁⠡⡠⠀⠘⣏⡏⣸⣿⣏⡀⢁⠄⠄
⠀⠀⠀⠰⠆⠀⠀⠀⠀⠉⢑⢤⡀⢀⣜⠀⡝⢯⡉⠁⠀⠤⠜⠚⠛⠤⠴⠈⠷⠊⢱⠀⠁⠉⠄⡙⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠰⠀⠮⠃⠐⠀⠁⠀⠀⠑⠩⠁⢠⣶⡛⢀⠀⣉⣠⠆⠉⢘⡰⠒⠈⣁⠈⠁⠄⠀⠂⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠂⠐⠉⡀⠀⠀⡀⠀⠀⡃⠀⠀⠁⠈⠉⡀⠠⠿⠿⠲⠛⠀⢐⠉⠈⠀⠀⠁⠀⠀⡁⠀⢀
⠀⠀⠀⢀⢀⠁⢻⠍⢀⣀⢀⡠⣎⠀⠀⠀⠀⡘⢠⣥⠼⢂⡄⠠⠐⠁⠑⠀⠀⠢
⠀⠀⠀⠐⠘⡛⢏⠀⠀⠉⠁⢐⠀⠀⠀⠐⡫⠔⠉⠀⠀⠀⠀⠀⠀⠈⠠⠁⠥
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠉⠀⠒⠈⠀⠀⠀⠂⢀
⠀⠀⢂⠀⠀⠀⣀⢜⡍⡀⠀⠀⡀⠀⠀⢑⠁⠀⠀⠄
⠀⠀⠀⠀⣀⡖⠁⡈⢀⠀⠀⠀⢀⢀⠒⠁⡁⢠⠂
⠀⠀⠂⠀⢉⠃⠀⠀⠂⠀⠀⠀⠈⠀⠀⠡⡒⠁⠁

⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠾⠛⢒⠒⣒⣋⣉⣤⡴⠶⠛⠓⣒⣒⣚⣉⣡⡴⠶⠚⠛⣒⣒⡒⢛⣉⣤⡴⠶⠚⠛⣓⣒⣚⣋⣡
⠀⠀⠀⠚⢛⣉⡭⠭⠭⣍⣠⠤⠶⠛⣋⡩⠭⠭⢭⣀⠤⠶⠞⢋⣩⠥⠬⠭⣭⣠⠤⠶⠞⢋⣩⠥⠤⠬⣄
⠀⠀⠀⣩⣉⣀⠛⠛⠶⢦⣬⣉⣩⣉⣉⡙⠛⠲⠦⣤⣍⣩⣭⣍⣈⠙⠛⠶⢦⣬⣉⣩⣭⣍⣉⠙⠓⠶⠦
⠀⠀⠀⠶⢤⣉⣛⣓⡲⢖⣒⡒⠲⠤⣍⣙⣛⣒⠶⣒⣒⠒⠶⢤⣉⣛⣓⣲⢶⣒⣒⠒⠦⢤⣉⣛⣓⡶⣖
⠀⠀⠀⠳⠶⠤⠭⠤⢤⣬⣉⠙⠛⠶⠦⠤⠶⠦⣤⣄⡉⠙⠳⠶⠤⠭⠶⠦⣤⣌⣉⠙⠓⠶⠶⠶⠶⢤⣤
⠀⠀⠀⢚⣉⣭⣝⠒⠦⢬⡭⠭⠛⣋⣭⣭⡟⠲⠤⣬⡭⠭⢚⣉⣭⣭⡓⠲⠤⠬⠭⠝⣋⣩⣭⠝⠒⠦⠤
⠀⠀⠀⢉⣁⣤⡴⠶⠒⣒⣒⣚⡋⣉⣠⣴⠶⠚⢛⣒⣒⣛⣋⣉⣠⣴⠶⠚⢋⣒⣚⣛⣉⣀⣤⡶⠖⠛⣒
⠀⠀⠀⠭⠁⠲⠶⠖⣛⣩⡤⠶⠬⠍⠒⠶⠖⢚⣩⡥⠶⠖⠉⢑⠲⠶⠖⣛⣩⡥⠶⠖⠉⢐⠒⠒⢚⣋⣥
⠀⠀⠀⠛⣉⣉⣭⠭⢭⣤⡤⠶⠟⠋⣉⣭⠭⠭⣡⡤⠶⠞⠛⣉⣩⡭⠭⠭⣥⡤⠶⠞⠋⣉⣩⡭⠭⢥⡤
⠀⠀⠀⢋⣩⣐⣒⠒⠒⠦⣤⣐⣚⣋⣁⣒⡒⠒⠶⢤⣍⣚⣋⣩⣐⣒⠛⠒⠶⣤⣐⣚⣋⣩⣐⣚⠛⠒⠦
⠀⠀⠀⠶⢤⣌⣉⠛⠓⠶⠶⠒⠶⢤⣄⣉⡙⠛⠲⠶⢒⠒⠶⢤⣄⣉⠛⠓⠶⠶⣒⠲⠶⢤⣌⣉⣛⠓⠒
⠀⠀⠀⠛⠲⠬⠭⠭⠴⣬⣍⣉⠛⠲⠦⠭⠭⠴⠦⣭⣍⣉⠛⠲⠤⠭⠭⠴⠮⣭⣍⣉⠓⠲⠦⠭⠥⠶⢭
⠀⠀⠀⢛⣛⣓⣛⠲⠶⣤⣄⡉⢙⣛⣋⣉⡛⠲⠦⣤⣈⡉⢛⣛⣋⣙⡛⠲⠦⣤⣌⠉⣙⣉⣉⣛⠓⠶⣤
⠀⠀⠀⣉⣤⡴⠶⠛⠓⣒⡒⠚⣉⣡⣤⠶⠚⠛⣒⣒⠒⢚⣉⣡⡴⠶⠚⠙⣒⣒⣒⣋⣡⣤⠶⠖⠛⢓⣒
⠀⠀⠀⢍⡠⠤⠶⠚⣋⣩⠭⠭⢭⣀⠤⠶⠞⢛⣉⡭⠭⠭⢍⣡⠤⠶⠞⢋⣩⠥⠤⠭⣄⣠⠴⠶⠛⣉⡭
⠀⠀⠀⢛⣈⣉⣉⣉⣡⣤⠶⠛⠛⣀⣉⣉⣉⣩⣤⠶⠞⠛⢉⣈⣭⣝⣉⣩⣤⠶⠞⠛⣁⣈⣭⡭⣍⣥⠴
⠀⠀⠀⣉⣩⠤⣖⡒⠒⠶⣖⣚⣛⣉⣥⢴⣒⠒⠶⢶⣒⣛⣋⣡⠴⣖⣒⠒⠶⢖⣒⣛⣋⣡⠴⣒⣒⠒⠦
⠀⠀⠀⠤⠤⣤⣍⡉⠛⠳⠦⠬⠭⠤⣤⣍⣉⠙⠓⠶⠦⠬⠷⠤⣤⣌⡉⠛⠳⠶⠬⠭⠷⠦⣤⣌⣉⠛⠓
⠀⠀⠀⠓⠶⢤⣭⡭⠽⠚⠉⠉⠛⠲⠤⣬⣭⠭⠛⠉⠉⠉⠓⠶⢤⣭⣭⠽⠓⠋⠉⠙⠓⠲⠤⣬⠭⠝⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣪⣭⣔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⠻⢿⣷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⠀⠀⢀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠖⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡔⢽⣿⣷⣦⡀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⢠⠀⠀⠈⠁⠀⠱⣄⠀⠙⢻⠇⠴⠀⠀⡐
⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⠄⠀⠀⣀⣂⣀⡨⠤⠤⠤⠤⠭⠄⡇⠢⡇⠠⠤⠖⣶⣀⣀⡀⠀⠀⠦⠤⢤⡠⠄⢀⢀
⢀⠤⠒⢈⠔⠂⠁⠀⢀⠤⣴⠖⠉⣉⣤⠴⠒⠊⡩⠑⠂⣀⠜⠁⠔⢋⠘⠒⠒⠶⢤⣄⠭⠙⢦⣄⡀⡀⠀⠈⠀⠢⣌⠁⠒⠠⢀
⠀⠀⢠⠏⠀⠀⠀⠀⠕⢸⡇⠀⠀⣻⡀⠀⠀⠀⠢⢀⡀⠀⠀⠀⠲⠦⡝⢖⠂⣈⠆⢹⠻⡄⠀⠉⠿⡦⠀⠀⠀⢀⠐⡱⡄⠀⠀⠉⠢
⠀⠀⠀⠳⡅⡀⠀⠀⠁⠚⢳⡄⢀⠈⠙⠲⠤⣀⣀⢠⣉⣙⣓⡀⣤⣤⣦⣀⣀⡠⠰⣋⠴⠃⢀⡠⠃⡝⠀⠀⠀⡐⢅⡼⠃
⠀⠀⢀⠀⠈⠙⠒⠈⠀⣄⢀⠉⠓⠕⠐⠆⠀⢀⡬⣭⣉⣉⣛⢛⢙⡙⠛⠛⠍⢉⢡⠄⠔⡊⡣⠔⠊⢀⠔⣐⡤⠔⠋
⠀⠀⠀⠑⠢⠄⣀⡀⠀⠀⠉⠈⠒⠒⠦⠦⠤⢄⣉⣉⢉⣚⣻⣻⣟⣟⠛⢒⡚⢋⣉⣍⣅⠤⠤⠼⠒⠊⠈⢁⡀⠤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠀⠀⠀⠀⠠⠤⠄⠀⠉⠉⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠐⠒⣀⣁⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠚⠆⠄⠀⣀⠐⠂⢄⠠⠄⠀⠀⠀⠀⠌⠂⠀⠀⣢⣠⡤⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣴⣯⣶⡄⣷⣸⢿⣄⣤⣴⣷⠆⠊⣛⣍⠱⣿⣶⡘⡒⢢⢤⣄⡆⡤⠀⡀⢁⢺⣁⣴⣹⡇
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣊⣶⣷⣻⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣶⣿⣽⣿⣿⣿⣿⣯⣦⣤
⠀⠀⠀⠉⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠙⠋⠉⠁⠁
⠀⠀⠀⠀⠐⠀⠘⢁⣠⣴⣖⣀⣠⠀⣦⠀⠀⢠⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣧⣦⠈⣳⣴⡆⣯⣦⣶⢸⣯⡀⣤⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠿⣿⣟⣛⣙⣿⣿⣿⠀⢀⣰⣦⣆⢀⣾⢿⡇⢰⢼⡄⠀⠀⠸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠿⠉⢉⡛⠻⠆⠀⠀⠏⣿⣿⣿⣿⣷⡠⠼⠇⠀⠀⣆⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⠀⠀⢳⡶⢖⠪⡿⠿⠶⠶⣿⣁⣈⣛⣻⣦⣤⠤⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡜⠿⢷⠦⠴⣠⣴⣦⣶⣾⣿⣿⣦⠀⠈⠉⠉⠙⠛⠓⠒⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠄⠹⣿⣿⣿⣿⣿⠿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠥⢄⡉⠂⡇⣑⣒⣆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⠃⠳⠆⡿⠛⠈⠁

⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⢀⣤⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣧⣄
⠀⠈⠙⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⡏⠁
⠀⠀⠀⣟⢿⡇⠀⠀⠀⣠⣶⣶⠶⠤⠀⠀⠀⠀⠀⢀⡄⠀⣀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠰⣿⣿⣿⡃⠀⠀⠿⠛⢿⡃⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣈⣧⡀⠀⢸⡻⣧⣸⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠈⢹⣿⣿⣷⡀⠀⠀⠀⠘⣇⠀⣀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠏⠳⡄⣸⣽⣿⣿⢻⡄⠀⠀⢀⠀⣴⣿⣟⡃
⠀⠀⠈⣿⣿⣿⣷⡄⠀⠀⠀⢻⡇⠈⢱⣄⠀⠀⠀⢠⣿⣿⠃⠀⠀⣹⣿⠋⠉⠀⢸⡇⠀⣴⠋⠀⠘⣿⣿⣿
⠀⠀⠀⡷⣿⣿⣿⣿⣄⠀⠀⢰⣿⣶⣾⢿⣤⡀⠀⢸⣿⣿⠀⠀⣰⣿⢋⣣⣦⡀⢸⣧⠞⠁⢀⣠⣾⣟⣻⣿⣷
⠀⠀⠀⣿⣾⣿⣿⡿⡿⠲⡄⠸⣿⡯⣿⠞⠋⠻⢦⣼⣿⡏⠀⣼⣿⠃⠀⠸⣿⣿⣮⠏⠀⠴⠿⠋⢹⣿⣟⠿⠋
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠂⠽⠶⣶⣦⡭⠉⢉⣉⠛⠉⣀⣈⣛⣈⣉⣠⣤⣾⣷⣽⣦⣬⣭⣉⣀⣠⣶⠶⣲⣾⣻⡓⠿⢿⣴⣶⣤⣤
⣭⣛⣧⣤⣤⣤⣤⠈⣉⡴⠛⠉⠀⢀⣒⣉⣹⣟⠉⠉⣽⣿⣿⢿⣿⢿⡷⠯⠛⠓⠈⠈⠛⣙⣣⡔⢶⠏⠹
⠮⣭⣭⣭⠍⣀⣠⣬⣽⣿⠽⢭⣭⢩⠁⠛⣻⣟⠉⠙⠉⠉⠁⠉⠃⡄⠀⣠⣁⣰⣆⣤⡤⠴⣶⣧⣉⣐⣀
⣉⣉⣀⣿⣿⣥⡴⠟⠿⠿⠶⠾⢿⠏⠀⠀⠙⢻⣷⣦⡀⠀⠀⠀⣀⣀⠀⠥⠌⣁⣙⡻⠶⢖⣛⣻⢏⡉⢴
⢙⣿⣯⠁⠀⠀⠀⠙⠛⠒⠶⠦⠤⠀⠐⠒⠞⠛⠉⠉⠉⠀⠐⣛⠻⢿⣤⣤⣶⠟⢃⣀⣤⠶⢞⣫⣤⢈⡵
⠀⠻⣷⣄⠀⠀⠰⣶⣶⣾⡿⠿⠛⣛⣃⣀⣤⣤⣤⣤⣤⡤⢤⣭⡄⢈⣻⣛⠛⣛⠋⢭⡴⠞⠋⠉⠤⠷⠞
⢄⣄⢀⣙⣷⡦⠀⠀⣤⣶⣛⠛⠛⠛⠛⢿⣿⣿⣿⠷⢖⣒⣺⣭⠴⠿⠶⠛⠛⣻⣥⣴⣿⣶⡶⠒⠚⠻⠿
⣤⣬⣷⡦⠉⣤⡶⠶⠦⢬⠭⢭⠭⣽⣛⣛⣛⡉⠁⣀⠭⠷⠶⠶⠶⠶⠚⠻⣿⣿⣿⣿⣿⣿⣷⣶⡶⠚⢲
⠀⠈⠉⢁⣤⣬⣍⣀⣂⣘⠧⠤⢄⣀⣒⣀⣩⣴⣿⣁⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⣉⡁⠀⠘
⠀⠀⠰⣿⣿⣿⡉⠉⠉⠉⠉⢙⣿⣿⣿⣿⣿⣯⣭⣭⡿⣿⣇⠀⠀⠀⢀⠈⠉⠛⠁⠀⠑⠀⠸⠁⠀⠘
⠶⠦⠄⠈⠉⠛⢛⣿⣶⠶⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⣠⣴⣿⣿⣏⣉⣛⣛⣻⣿⣿⣿⣷⣿⣿⣷⣶⣾⣷
⠒⠒⠒⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣁
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣤⣤⡴⠶⠟⠛⠉⠉⠉⠉⠉⠩⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢦⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡉
⠀⠀⠀⠀⣾⣿⣿⣿⣷⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠶⠶⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⠸
⠀⠀⠀⠀⠀⠉⠻⣿⣿⡃⠀⠀⠀⠈⠉⠈⠀⠒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⣻⣿⣯⣤
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠀⠀⠀⠒⠒⠒⠛⠛⣛⣿⣿⣷
⣶⣤⣀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠚⠿⠷⠶⠶⠃⠀⢀⣠⡤⣤⣴⣿⠟⠋⠈⠁
⠉⠛⠛⠻⠶⢦⣤⣤⣤⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣼⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠶⠶⠶⣶⣶⣶⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠶⢶⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⣿⠿⠟⠛⠁⠀⠀⠀⣀⣴⣿⣿⣿⡃
⣀⣀⣀⣀⣀⣀⣀⣤⣶⠶⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⡀
⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠉⠁⡀⠈⠑⠢⢀⠀⠈⠣⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠠⠊⠉⠉⠙⣦⠀⠀⠑⠢⢄⣀⣠⠾⠃⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⡄⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢳⣀⠈⠛⠋⢁⣠⣤⣄⡀⠀⠀⠀⢀⣠⣤⣄⡑⠦⠤⠤⠤⠀⠀⠠⢤⣀⠀⠀⠀⠀⢀⣠⡤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠙⠓⠶⠞⠛⠉⠤⣌⡙⠓⠒⠂⣠⡴⠶⠦⠍⠃⠐⠶⠶⠶⠶⣦⣄⣈⡛⠶⠶⠞⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠙⠛⠛⠛⠋⠙⠛⠲⢬⣉⠓⠐⠋⠁⠀⠴⠶⠦⢤⣌⡙⠳⢦⣤⣤⣤⠴⠖⢋⣁⣀⣉⡙⠓⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠠⣈⠙⠒⣛⣛⣋⣉⣉⡓⠂⢄⣈⠁⣀⣤⡶⠞⢛⣉⣂⣀⡈⠙⠶⢤⡄⠀⠠⠶⢚⣋⣉⠉⠉⠙⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠉⠙⠓⠦⣌⡉⠉⠁⠄⠊⠉⠀⠀⠈⠙⠳⢦⣤⣤⣤⡴⠟⠋⠉⠉⠛⠲⢤⣄⣀








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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠐⠒⠒⠂⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠦⢤⣀⡀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢧⡀
⠀⠀⠉⠉⠉⠉⠉⠛⠒⠲⢤⣉⠙⠲⣄⡈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠂⠀⠀⢄⡁⠢⣀⠀⠀⢇⢀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠙⢦⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⢪⢂⠈⠳⡀⠘⡈⡄⠀⢧
⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠒⠲⢤⣌⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠇⢆⠀⠹⡄⡇⣷⠀⠸
⠀⠀⠀⠠⠁⠂⠈⠉⠉⠉⠓⠢⢄⡀⠈⠙⢦⡀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠆⡀⠀⠀⠀⠀⢸⢸⠀⠀⣷⡇⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠙⢦⡀⠘⣎⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⢀⡇⣸⠀⠀⣿⠃⣹⡇
⠀⠀⠀⢀⣠⣤⣤⡀⡀⠀⠀⠀⡀⠀⠀⠹⣄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡜⢠⠇⠀⢀⡟⢠⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢳⡈⢦⡀⠀⠙⡄⠀⠀⠈⢧⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢠⠎⠀⠀⡜⠀⡎⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠑⢆⠀⢹⡀⡀⠀⠀⠙⣦⡀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⢀⣀⡤⠚⠁⡴⠋⠀⢀⠞⠀⡜⢠⡇
⠀⠀⠀⠀⠄⡀⠀⠀⠀⠙⠒⠋⠀⠈⣇⠐⢄⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⢀⡴⠊⠀⢀⡴⠋⢀⡜⠁⡾⠀⠀⣸
⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠸⡄⠀⠑⠤⣀⡀⠀⠉⠓⠶⢤⣤⣀⣀⣤⡤⠴⠚⠉⠀⢀⡴⠋⠀⣠⠎⠀⡼⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠙⢦⡀⠀⠀⢉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⣠⠞⠁⢠⠞⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣄⣀⠀⠉⠙⠚⠉⠀⠀⠀⠀⠀⢀⣀⣠⠴⠖⠋⠁⠀⠀⢠⡎⠀⣠⠴⠋⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠶⠶⠶⠒⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⢤⠴⠖⠶⢿⡗⢲⣶⠒⠶⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣳⡇⠀⣁⢈⣁⣠⠤⠤⠨⡿⠃⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⣿⣿⡟⣛⢁⣾⣿⣿⣿⢸⣾⣿⡏⠁⠀⠀⠀⣶⣤⠀⡀⠹⣖⢤⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⢿⣿⣿⣿⢿⣧⣭⣻⡟⣷⣿⣿⡃⠀⠀⠀⠀⣼⣹⠀⣰⠠⢼⠄⢔⣌⠁
⠀⠀⠀⠀⠀⢀⣴⡾⠿⡟⠁⡉⠺⣿⣿⠶⠭⠿⢟⣡⣿⣿⣿⣁⡀⠀⠀⠀⣿⡍⢘⠇⠀⢨⢠⣾⡏⢢
⠀⠀⠀⠀⣰⡟⢻⡁⠠⡇⠀⠍⠁⠀⠀⠈⠙⠟⠉⣫⣿⣿⡿⡙⠁⠈⠻⣄⠏⣟⡈⣁⣉⡃⠀⠀⣰⢿⣷⡀⡀
⠀⢀⠀⣾⣿⠃⠀⠁⠀⣈⣁⡠⠄⠀⠀⠀⠀⠀⠐⢿⣿⣟⣱⠟⣦⣀⣈⢟⢤⣾⣿⣿⣿⣿⣿⠛⠲⣄⠰⣷⣅
⠀⠉⣿⣿⡆⠁⣠⡶⠟⠋⢩⣈⣉⣙⣩⣲⢄⡀⠀⠐⢯⣿⣿⣉⣉⡽⣻⣿⣿⣿⣝⣿⣿⢿⣿⣿⣷⣝⣿⣏⣿⡄
⠀⠀⣿⡋⢇⡾⠁⠀⢀⣿⣿⡟⣻⣿⣿⣿⣿⣯⣖⢍⠲⠿⣾⡿⢿⣯⣿⣿⣿⠛⠿⠿⢿⣷⣟⣿⣿⣿⣯⣿⣿⣿
⠀⢸⣿⡇⣿⠀⠀⢠⣿⡿⣿⡿⠟⠿⣾⢿⣿⣿⣿⣮⠶⠖⣪⣶⣿⣷⡙⢿⣦⡤⢴⡆⠀⠈⡛⠻⣿⣿⣿⣹⣿⣿⡇
⠘⢨⡟⠇⣺⠠⣰⣾⣿⣿⣿⣿⣷⣄⠈⠹⣿⣏⢹⣿⡘⣼⣿⣿⣿⣿⡇⡀⠻⢷⠿⣄⢀⠀⣈⣗⠀⠪⢽⢹⡿⢿⣇
⠀⢸⣿⡀⣿⠀⣰⣿⣟⣛⡘⠻⣽⣿⡿⠿⢸⣿⣿⣇⣽⣿⣿⣿⣿⣻⣿⡏⣀⣀⣉⣉⣉⣬⣋⠉⢁⡍⣩⢸⣷⣶
⠀⠆⣿⣇⣷⡘⣿⣿⣿⠯⣿⣷⣴⣤⣽⣿⣿⣿⠝⣿⢿⣿⣿⡜⢻⣿⣿⣝⡛⠛⠙⠛⠋⠁⠀⠀⠸⠀⡏⣿⣿⡿⡄
⠀⠀⠘⣿⣿⣷⡜⢿⣏⠀⠀⠀⠉⠉⡁⠀⣻⡧⣸⡟⠀⠨⣿⡟⠁⠉⠛⠛⠛⠃⠀⠀⠀⠀⡄⠀⠁⣀⣼⣿⣿⠃
⠀⠀⠀⠘⢿⣿⠯⢧⣤⣀⡀⠀⡀⠀⠀⢚⣹⠗⢿⡇⠀⠀⠹⣿⣤⠇⡀⢀⢀⣀⢀⣄⣄⣤⣄⣶⡾⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠉⠻⣶⣌⣙⠛⠙⠛⠛⠛⠒⠲⢲⠈⡻⠷⠶⢶⢐⡾⠶⠷⠿⠿⠿⠛⢛⣋⣋⣵⣶⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣶⣶⣶⣶⣷⠷⠶⠾⠷⠿⠿⠿⠿⠿⠺⢿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣃⣀⣀⣰⣀⣀⣀⣛⣀⣀⣈⣁⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠣⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⢠⡀⠀⠀⠀⣀⣄⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠠⡄⠀⠀⠀⢺⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠰⣷⠀⠀⠀⣆⣀⠀⠀⠀⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠸⢠
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠉⠉⠀⠀⠀⡬⠅⠀⠀⠀⣾⠆⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⣠⣸⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠆⠀⠀⠀⠰⡇⠀⠀⠀⠘⠀⠀⠀⠀⢸⡄⢿
⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡏⢣⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣽
⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢓⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠁⠠⡇⠀⠀⠀⡇⠀⠀⠀⠀⣄⡄⠀⠀⢰⠁⠛⢀
⠀⠀⠀⠀⠘⠀⠀⠀⠈⣄⠀⠀⠀⠀⣀⣠⠀⠀⠀⣀⡇⠀⠀⢀⣠⠀⠀⠀⣀⣄⠇⠀⠀⠀⢿⠀⠀⠀⠀⠉⠁⠀⠀⢸⣀⠴⠃
⠀⠀⠀⠁⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠚⠋⠉⠉⠛⠳⢶⠶⠛⠛⠛⠓⠒⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠶⠒⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⣤⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡁
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣤⠤⠲⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⢠⡞⢻⠇⠀⠀⠈⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣷⡇⢀⠀⠀⣴⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⢻⠓⠛⠋
⣿⠀⣸⠀⠀⠀⠀⠀⢨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠹⠃⠘⢀⣾⠁⠀⠀⠙⠶⠶⠶⣄⣀⣀⡴⠋⠀⠀⠀⠀⢠⣤⣄
⠙⠶⠋⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢀⣤⠤⠶⠦⠶⠞⠋⢹⡄⠀⡀⠀⣼⠃⠀⠀⢀⠀⠀⠀⠀⠈⠉⢁⡀⠀⠀⠀⠀⠀⢸⡄⢸
⠀⠀⠀⠀⠀⠀⡴⣶⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⢰⠁⢰⠇⠀⠀⠀⠸⡿⢦⡀⠀⠀⠀⠈⡟⠳⣦⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡟⠀⠀⠀⠀⠀⣇⢀⡵⠀⠀⠀⠀⠻⣄⣸⠇
⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠇⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣴⣧⣤⣼⣿⠂
⣿⣿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠔⠃⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠘⢤⠌⠄⠐⠂⠀⡈⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢃⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⢈⠀⠀⠀⠀⠀⠁⡌⠀⠀⠀⠀⠀⣴⠃⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠠⡄⠀⢀⠀⠑⢤⠀⠘⠀⡀⠘⠀⠀⠀⠀⢸⢤⣴⢲⡶⢾⢿⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠦⣀⠠⢶⠀⠀⠈⠑⠐⢌⠀⠀⠀⡁⠐⠛⠠⠁⠌⠴⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⣂⣐⣂⡨⣤⡤⠀⢤⠽⣳⠾⢛⡀⢂⣄⠠⠤⠤⠆⠼⢤
⠀⢀⣀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⡉⠂⠀⠃⡀⠉⡉⠉⠈⠁⢀⣠⣀⠴⠀⠂⣀⠰⠛⣟⣶⡓⠓⠀⠀⠀⠀⠁⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠉⠉
⠀⠊⠁⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠍⠀⠐⠀⠄⠈⠉⠉⠍⠋⠁⠀⠄⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠩⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠂⠈⠁⠀⠉⠁⣀⡀⠀⡁⠀⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠤⠴⠀⠀⡀⣀⠹⠿⠄⠈⢀⣠⠀⣤⠤⠁⡙⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⢆⣀⡀⠀⢀⣤⠖⠀⠙⢚⣙⠀⠀⠄⠀⠠⣤⣀⡁⠀⣀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣠⢤⣾⠁⠀⠁⠯⡿⠀⠀⠀⠀⠀⠲⢂⣾⣶⣤⠄⢠⣽⡀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠀⠀⠀⢐⠘⠃⣹⢷⡄⠀⠀⠀⣻⣧⠀⡀⠀⠂⠉⠱⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡀⡸⠮⠲⡇⠒⠼⣷⣿⠏⠀⠉⣰⣦⣺⢿⣯⢏⠛⠰⠆⠤⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⡀
⠀⠀⠠⠀⠀⠀⠀⡙⠈⢡⠄⠈⠈⡒⠄⢈⣻⠇⠀⠀⠊⠉⢼⣾⠟⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠆⡈⢹⢿⢿⡗
⠀⠀⠀⣀⠀⠚⠁⠙⠚⠣⠇⠀⡀⠷⠌⠋⢉⣇⠀⠀⠀⠀⡼⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⣤⠄⣀⡀⡀⠠
⠀⢀⣍⣻⣿⡏⢀⣓⣶⢷⣶⡏⣯⣤⣤⣶⠂⣤⢶⣾⣷⣾⣶⣀⣔⡀⠒⠀⣀⣀⣀⡀⠀⣀⣀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠈⠉⠁⠈
⠀⠸⡷⢶⠄⠻⠻⣿⣿⠏⡟⠿⠐⢚⡷⠶⣿⣟⠙⠛⠀⠛⠻⣟⣩⣐⠂⠀⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⣀
⠀⠀⠤⠤⠤⠥⠤⠭⠝⠛⠛⣁⣘⣥⠖⣖⣵⠀⠁⠀⠁⠀⢀⡘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠤⠤⠦⡄⢀⠀⠀⠀⠊⠁⠀⠀⠛⠠⠉⠁⠂⠀⠀⠀⠈⠃
⠀⠀⣀⡠⠎⠁⠀⢉⣤⡍⠀⠤⠢⣀⣀
⠀⢈⣍⠁⠚⠀⠀⠘⠓⢢⣤⡴⢠⣬⡍⢉⡂⠀⠄⠀⠀⠀⠈⠉⠁⢀⠄
⠀⠈⠁⠰⠄⠀⠢⠀⠈⠁⠀⠂⠀⠒⠘⠉⠀⠉⠁⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢷⡀⠠⠃⠸⡀⠘⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢿⣄⡀⠀⣷⡖⢸⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢰⣶⠀⠀⠀⣀⣰⡇⠀⠈⠻⣷⣤⣽⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠘⣿⠇⢀⣠⣿⢻⠁⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣴⣿⣿⡿⠃⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣸⣿⣿⡟⠀⠀⣰⣿⣿⣿⣿⣶⣿⣯⣿⡿⠛⠉⠀⢀⣤⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢨⣿⣿⣿⠃⠀⣰⣿⣿⠉⣿⡿⠋⠉⠉⠉⠀⠀⠀⢀⡞⢩⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣯⣿⡛⠛⣹⣿⣿⣧⠹⣇⠀⠀⠀⢿⣧⡶⣿⠏⡀⠀⢀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣃⣄⣰⠏⢋⡙⠛⠳⢀⣿⣿⠿⣿⡇⣿⠀⠀⠀⢾⣏⡔⠁⠈⠁⠈⣠⡷⠀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣿⣟⠋⢹⡿⣿⣦⣄⠘⢳⣾⣿⣿⣿⡿⣷⠉⠀⠀⢐⣿⠉⠀⡀⠀⠤⠾⣏⣀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⢿⠟⠁⠀⠀⠀⠈⠛⢻⣇⣼⡿⣿⣿⠟⡕⠁⠀⠀⠀⣠⣿⣷⣿⣿⡟⣿⣧⣄⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣧⣤⣤⡶⠉⠀⠀⣼⣿⣿⣷⠟⢁⠞⠀⠀⠀⠀⣠⣿⣿⠿⠋⠁⠀⠙⠿⠟⢹⣄⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣯⣉⠉⣉⢵⣦⠀⠀⣸⣟⡻⠛⢀⣴⠀⣠⡆⠀⣠⣴⣿⣤⣿⣦⡀⠀⠀⠀⠀⠀⠀⢻⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⣿⣟⠁⠀⠉⠀⣰⡿⢿⣤⣶⠛⢡⣼⠏⢰⢾⣿⣏⢈⡼⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⡇⠸⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⡄⣠⣾⢏⣤⠑⠘⠻⢿⣋⠛⠳⣶⠺⣇⣩⣯⠠⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⣸⣿⣿⠿⢋⣴⡇⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⢀⠠⠠⠄⠀⠀⢐⠂⠉⠁⢀⠀⠈⠀⠖⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⣿⠟⠁⣤⢻⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⠟⠃⠀⠈⠉⠀⠉⠛⠛⠃⣀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⣼⣧⣼⢁⡀⠤⠐⠓⠶⡶⢤⣀⣠⡆⠀⠀⡇⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⠏⠀⠀⠀⠀⢀⡄⠀⠀⠙⣿⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⢀⠀⠴⠦⠀⠀⠈⣿⣿⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠⡿⠀⠀⠀⠹⡌⢿⡆⠀⠀⠀⠁⠀⠀⣀⡄⠀⠀⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠆⠀⠀⠀⠀⠀⠠⠀⢀⢸⡇⠀⠀⠀⠀⠀⢸⡇⣤⡶⠀⢄⣰⡀⢳⡄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⠆⠀⠄⢀⢐⠁⠀⠺⡀⠈⣿⡁⠀⠀⠀⠀⠰⠄⢷⡜⠛⣄⠈⠈⠃⠘⠃⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠈⢠⣤⣄⡤⠂⢩⡧⠀⠀⠀⠀⠀⠰⡀⣿⡿⣏⣹⡄⢶⣧⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⣈⢿⣣⠞⠋⠀⠀⠀⠀⠀⠀⠀⠻⡘⣷⣿⡋⠃⢠⡼⡏⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⢿⠄⡹⣄⣀⡀⣀⠀⠀⠀⠀⠀⠀⢡⠈⢿⣿⡇⣸⣡⣌⠟⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠋⠳⣦⣾⣧⡽⢝⣦⡀⠀⠀⠀⠀⠲⡈⠻⣾⣧⣿⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠈⠻⣵⡶⢀⣥⣝⢀⠀⠀⠀⠀⠙⣦⡈⠙⠁⠀⠀⢀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢄⠀⠉⠶⣽⣯⣉⣟⣷⠀⠀⠀⠀⠈⠛⠀⠤⠤⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁

⠀⠀⠀⠀⠀⣠⣴⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠂⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⡖⣆
⠀⠀⠀⠀⣸⠏⠀⠀⠀⠹⡆⠀⠀⠀⣦⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⢸⠁⢹
⠀⠀⠀⠀⠸⣇⠀⠀⠀⣸⠇⠀⠀⠀⣿⣇⠀⠀⢀⣴⣿⣿⠀⠀⠀⠐⠀⠀⠀⣃⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⠀⢸⡀⣸
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢠⡀⠀⠀⠀⠈⠉⠛⠻⠿⠄⣀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡿⣏
⠀⠀⠀⠀⠀⢸⡿⠛⢿⡏⠀⠀⠀⠀⠀⠙⣷⣶⣿⣿⡟⠀⠀⠀⠀⠈⢿⣷⣶⣦⣴⣤⣦⣀⠀⠀⢀⠃⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢹⡇⠀⠀⠀⠀⠀⠈⢻⣿⣿⣏⠀⠀⣙⣿⡷⠁⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⡉⠛⠛⢻⣿⠁⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⣶⣷⣿⣶⠀⠀⠀⠀⠀⣀⣇⣿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⢰⣿⠋⢻⣷
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⣿⡄⢰⣿⠀⠀⠀⠀⠘⣿⠀⢸⡿
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣦⣾⡇
⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣿⠋⠹⣿⡆⠀⠀⠀⢀⣿⠟⢿⣧
⠀⠀⠀⠀⠀⢸⣧⣀⣼⡇⠀⠀⠀⠀⠀⠀⣼⣷⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡏⠀⠀⢹⣷⠀⠀⠀⣼⡟⠀⠈⣿⡄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣿⠃⠀⠀⠸⣿⡄⠀⢀⣿⠃⠀⠀⢹⣧
⠀⠀⠀⢠⣾⣽⣿⣿⣿⣾⣗⠀⠀⠀⣾⣿⣿⣿⠿⠿⠿⣿⣷⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣄⠀⠀⣰⣿⡇⠀⢸⣿⡀⠀⠀⣸⣿
⠀⠀⠀⢸⣿⡟⠋⠉⠉⠹⣿⡇⠀⠀⣿⣿⡟⠁⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣿⡾⠟⠻⢷⣿⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⣿⠃⠀⠘⣿⣿⡶⣾⣿⡿
⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⢻⠇⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⣿⡁⠀⠀⢈⣿⠀⠀⠀⠀⠀⣿⣯⠀⠀⣽⣿⠀⠀⠀⣿⣏⠀⢈⣿⡇
⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣴⡿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃⠀⠀⠀⠘⠿⣿⣿⠟

⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣥⣿⣶⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣐⣬⣲⣾⣿⡿⣶⢶⠷⠖⠒⠒⠒⢺⢶⣶⣶⣶⣶⣶⣶⣶⠶⠛⢉⣙⣋⣈⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠙⠛⢿⡿⡟⢀⡀⠀⠀⠀⢸⣮⣿⣿⣿⣿⣿⡿⣿⡴⠛⠉⠉⠉⠉⠙⠢⡄⡝⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⡁⠀⠀⢶⢾⣿⣿⣿⡿⢿⣿⠋⡔⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠎⠁⠀⠀⢀⣬⠀⣿⣿⣿⣿⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠦⣄⠘⢿⣴⢿⣿⡿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠛⢅⠀⢙⣺⠢⠈⣻⣿⣽⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣀⠀⠀⠑⢿⣿⣀⢄⠈⠳⣿⣿⣷⣥⡀⠀⠀⠀⠀⠀⠀⠞⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠁⠀⠀⠀⠈⣿⣿⣧⡀⣹⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⡸⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠭⠭⠭⠠⠤⢄⢤⠤⠄⡤⠀⢽⢿⡿⠿⢯⣭⠭⢭⠍⢭⠹⠭⠩⠶⠄⣧⣷⣥
⠀⠀⠀⠀⢀⣤⡖⢉⡀⠠⢶⣀⡠⡀⠀⡠⢈⣠⠴⠋⡄⠀⣸⠾⡉⢢⠀⠈⢷⡄⢡⠀⡄⠀⢀⡔⠶⠀⠈⠙⠳⢄
⠀⠀⣠⡶⠋⠯⠙⠯⢹⣾⠋⠙⠡⠃⡐⢻⡟⠁⠀⠀⠄⣼⢻⠀⡈⠀⢳⠦⠀⢻⡘⣿⡇⠀⠈⠻⣶⡶⣾⣷⣤⣤⣁⠄
⠀⡴⠻⠇⠀⠀⠀⡼⢿⠿⠇⠀⠀⡼⢟⡟⠀⠀⠠⡞⣿⠷⢾⠚⠁⠀⠀⡄⠀⠘⡇⣿⣿⠀⠀⠀⢙⣿⣿⡿⢿⣟⠛⠿⣦
⠰⢁⡀⠠⢲⠄⠀⠀⠀⠀⠀⠀⣼⣶⡟⣀⣀⣀⠀⡸⠉⠆⢸⣀⠀⠀⠀⡇⠀⠀⡇⢿⣿⠀⠀⠀⢘⣿⣟⢠⣽⡿⠷⣦⡘
⡆⠈⣹⣾⡯⠄⡰⠀⠀⠀⠀⢰⣿⣿⠃⣽⠃⢈⣷⠃⠀⠀⣸⣿⠀⠀⢠⠁⠀⢰⡅⢸⣿⡂⠀⠀⢸⣟⢧⣽⢫⣾⣿⣌⣳
⠇⠀⣤⣿⣧⠀⠀⠀⠀⠀⠀⣸⣿⡟⢠⡿⠀⠉⡾⠀⠀⡤⢽⣿⠀⢀⡎⠀⢀⡿⠁⢸⣿⡇⠀⠃⣾⣿⣣⣿⠞⢿⣿⣿⣷
⢸⡄⠋⠙⠋⠀⠀⠀⠀⠀⠀⣿⣯⡇⠚⠃⠘⣹⣦⠴⠞⠀⠸⠛⢢⡎⠈⢐⡿⠁⠀⣸⣿⣷⣾⣶⣿⣯⣷⡻⣷⣿⣤⣿⡟
⠀⠹⡀⣰⢀⠀⠀⠀⠘⠀⢀⣿⣿⠁⡤⠞⢁⡬⢙⡓⠃⠀⢸⣤⢂⠀⣰⠟⢁⣠⣌⣽⣿⣿⣿⣿⣿⣿⣿⣯⣠⣌⠉⢛⡵
⠀⠀⠹⣧⡤⢀⠢⣀⣠⣤⣾⣻⠏⠀⢡⠄⠀⡷⠦⢜⣠⣴⣻⣧⡽⠚⣁⣤⣶⡿⠿⣿⣟⡋⣽⡟⢿⣵⡿⢟⠛⠁⢀⠝
⠀⠀⠀⠈⠻⢿⣿⡏⢠⣞⣋⣥⣴⠞⢀⣀⣤⣴⠶⣿⣿⠾⢛⣩⣴⣾⣯⣿⣿⣿⣿⣿⣿⠿⢿⣿⣾⣿⣛⣩⡤⠐⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⡿⠿⢿⡿⠿⠿⠿⡿⣿⠟⠻⣍⡉⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢿⣿⣟⣛⣛⡷⢿⣷⣶⣶⣶⣶⡲⡶⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠟⠛⠛⠛⢻⠿⣿⣿⣿⠏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢨⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠇⠀⠀⠀⢰⠐⣦⣿⣿⣧⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠶⠶⢶⣶⣾⣿⣿⣿⣿⣧⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣤⣀⠂⠸⣿⣿⣿⣛⡟⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⣡⣤⠀⠻⣿⣿⡿⠁⢠⢿⣿⣿⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠟⠉⠉⠱⠀⠱⠞⠀⠆⢹⡟⠐⠆⠹⠼⢫⣿⣆⠙⠻⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢿⣿⠿⠿⢯⡯⠉⠇

⠀⠀⣀⣠⣤⣤⡄⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀
⠀⠀⠉⠉⠈⢁⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⠀⠀⠉⠀⠀⠀⢀⣠⣔⠒⠾⠿⠳⠲⠦⢠⡤⡤⠤⠤
⢀⡄⠒⣀⣀⡉⠉⠁⠁⠀⠈⠶⠶⣿⣿⣿⣿⣿⣿⡶⠶⣂⣠⣄⡄⠀⣑⣏⣀⠀⠐⠲⠿⢿⣿⣿⣿⣿⠗
⢶⣤⣤⣼⣯⣿⣦⣄⣀⣄⡀⠀⠀⣀⠠⠼⠡⠤⣤⣔⣛⡙⠩⠥⣾⣿⣿⡿⠿⢷⡿⠶⣯⡥⠥
⢹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣠⣺⡿⠁⢀⣴⣠⡀⣀⡈⠉⠉⠓⠶⢦⣄⡀
⢛⣽⣿⣿⡟⢫⡽⠛⠋⠠⠺⣟⣡⣾⣾⣿⣿⣿⣿⣿⣶⣄⡀⢀⣀⣀⣀⣀⣀⣤⠀⠀⠀⢀
⣶⣤⣤⠔⠑⢤⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠏⣛⡻⠟⠛⠛⠛⢁⣠⣴⣿⣷⣤⣄
⠙⠋⣀⠄⠴⠀⠉⠓⠶⢤⡄⠘⠯⣿⡏⠉⠉⠀⠉⠙⢓⣀⣀⣀⠬⠇⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷
⣠⢈⣥⣠⣴⣤⣶⣶⣀⠀⠈⠑⠢⠤⣄⣀⢄⡀⠐⠛⠿⣿⣷⣆⣉⣉⡐⠉⠉⠛⠻⠿⠿⠿⠿⠟⠛⠁
⣿⣾⣿⣿⣯⣛⣙⣿⣿⢀⣷⡴⣦⣄⣀⠉⠛⠲⠄⣀⠉⠀⠛⠻⠿⠷⠤⠄⠀⠐⠒⠀⠀⠀⠀⢀
⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣮⠿⣿⣿⣶⣦⣤⣌⠙⡳⠶⡤⣤⣤⣶⣦⣤⣤⣄⠀⢠⠴⠒⠉
⠘⣯⣿⡿⠿⣿⣿⣿⣿⣿⣿⣭⣭⣤⣛⠟⢩⣤⣶⣦⣈⣟⡿⠿⠶⢤⠀⠈⠉⠉⠀⠀⠐
⢿⣮⡿⠿⣷⣮⠻⢿⡟⣿⣿⣿⠻⣿⣿⣷⠘⣿⣿⣿⣿⠿⢿⣿⣿⡶⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⣀⣠
⢲⣦⣿⣶⣿⣿⣷⣶⣶⣦⡙⠉⠛⠲⢭⣤⡀⣈⠁⠉⠀⠀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣛⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠬⢭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⠳⠄⠀⠀⣤⣤⣾⣯⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⡖⠀⢰⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⢀⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡾⠁⠀⠋⢠⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠸⣇⠀⠶⠀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⢿⡆⠀⠀⠘⠈⠃⠀⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣯⠉⠁⠀⠀⠀⣰⠃⠀⡆⠀⢀⣴⠀⠀⠀⠀⠀⢉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣌⣋⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⢀⣀⣠⣤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡀⠀⠉⣉⣉⠐⠒⠓⠒⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⢸⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣀⣈⣻⠛⠃⠀⢰⡇⠠⡄⢠⣆⠀⠀⠀⠀⠼⢫⡿⠀⢸⡇⢠⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠏⠉⠙⢢⣤⠤⡄⠹⡄⢀⡄⠉⡇⠀⣠⠶⢶⡄⠀⠀⢸⣷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⠴⠶⣄⠀⠀⠐⠃⣠⠇⠸⠁⣼⢁⡾⠀⢈⣸⠇⠀⠀⢸⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠺⣧⡶⠂⠘⣷⠀⠀⠐⠋⠀⠀⠀⣉⡴⠃⠰⠟⠁⠀⣰⠆⢸⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⣸⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⠖⠛⠁⠀⢸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠆⠀⠀⠀⠀⣠⡾⠏⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⣸⡆⢸⣿⡛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⠛⠳⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣦⣤⣴⠏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸⣿⠁⠀⠘⡄
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠤⠦⠞
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠈⠀⠒⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠉⠙⠲⠶⢤⣤⣍⣉⣙⣛⣛⣀⣀⣬⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣭⣭⣭⣥⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡆
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣤⣤⣤⢸⣿⣿⡇⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣿⣿⣿⢠⣤⣤
⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿
⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿
⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿
⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿
⢿⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⢏⠇⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠡⠄⠐⠒⣒⣒⣒⣒⣒⡒⠂⠀⠤⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣅⢀⢚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠠⢂⠭⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡆
⠀⠀⠀⠀⠀⢀⢁⠂⠀⠏⠭⠀⡀⠀⠀⠀⠠⠤⠀⠠⠤⠀⠀⠐⠀⡂⠨⠍⠇⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇
⠀⠀⠀⠀⠀⠌⠈⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁
⠀⠀⠀⠀⢀⢀⠁⠀⠉⠀⠂⠀⠂⠉⠡⠤⠀⠀⠀⠀⠀⠠⠤⠉⠁⠂⠐⠀⠈⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠄⢠
⠀⠀⠀⠀⡄⠁⠀⡁⠔⢢⠀⠀⠠⠒⠀⠀⢀⠤⠀⠀⠀⡀⠀⠀⠀⡠⠄⠀⠀⠦⡁⠀⠀⠀⠀⡈⠈⠘
⠀⠀⠀⢀⠘⠀⢠⠁⠀⠀⢄⠔⠁⠀⢃⡠⠀⠀⢃⢀⠎⠈⠄⠀⠔⠁⠘⣀⠎⠀⢡⠀⠀⠀⠀⠯⠀⣚
⠀⠀⠀⠘⢜⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠠⠀⠩⠀⠐⠒⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⠒⠀⠀⠁⠄⢠⠀⠀⠰⠠⢠⠇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡋⠉⠹
⠀⠀⠀⠀⠀⢒⠀⠀⠤⠄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⢐⡀⠜⠊⡴⠃
⠀⠀⠀⠀⠀⠠⠀⠈⠁⠒⠀⠀⠀⠀⠠⠤⠤⠀⠭⠥⠤⠤⠄⠀⠀⠀⠀⠒⠀⠉⠀⢸⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠑⠂⠄⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠁

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲
⡰⠒⠢⡄⠀⠀⣀⠀⠀⠀⠀⠸⢱⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠒⡄⠧⠴
⠑⠤⠠⠎⠀⡘⠂⠃⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣩⠆⠰⠅⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢎⣠⠇
⠀⠀⠀⢠⡏⠉⠑⢦⡤⡀⠀⠀⠀⠂⠣⡀⠀⠀⠘⢍⢧⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⢀⠴⠤⢔⡺⠛⢉⡭⠝⡩⠿⠽⢦⠀⡠⢤
⠀⢀⠀⠀⠙⠛⠛⠲⣄⠙⣄⣀⣀⡼⠆⣈⡉⡒⠤⠒⠃⠉⠁⠀⠈⠉⠉⠢⠴⠊⠑⢒⣁⡴⣤⠤⡝⣯⡛⠒⢒⣉⣰⡴⠋⠀⠀⡿⢀⡾⠔⠊⠉⠁⠀⠈⠉
⠀⠈⠀⠀⠀⢀⡴⠒⠈⢳⣉⠲⢷⡖⠲⣆⣏⣁⣠⠦⣀⣀⡤⠀⠀⠒⠲⠴⠂⠀⠀⠀⠠⠀⠀⡜⢵⠋⠙⠒⠋⠀⡜⠀⠀⠀⣠⠇⢸⠁
⢀⣀⠀⢠⠒⠉⡠⢄⠤⢌⡳⣧⡀⠉⠓⠏⠙⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠢⠤⠦⠶⠤⠀⠀⣸⠁⠀⢸
⠣⠔⠁⠈⠁⠋⠀⠠⠀⠀⠳⢌⣻⣦⡀⠀⠀⠀⠀⠙⠓⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠦⣶⡦⣄⠀⠀⠀⠀⠙⠦⣤⡤⠤⣄⣀⣄⣠⠖⠶⠃⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⠀⠁⢿⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠈⠓⠶⠛⠛⠋⢩⠉⠛⠉⠅⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⠶⠤⣤⠴⠋⠳⠦⠴⠒⠐⠦⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠘⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⠀⠀⠁⠒⠂⠤⠤⢤⡀⠀⠀⠀⠀⠀⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠂⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠂⠤⠤⠤⠤⠤⠤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡴⠶⣷⡤⢤⣼⣆⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⣲⡸⠏⠀⢠⣸⣧⡄⠒⢻⣷⡆⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠛⢋⣵⣾⣷⡐⣧⠀⠈⠶⢿⣷⣚⣩⠟⠂⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢛⣿⣿⣿⣿⡇⠀⠐⠿⣿⣿⣾⢄⠀⠀⣾⣿⣉⡽⢧⣤⣶⣤⠀⠉⠓⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠸⣿⣿⣿⣿⣾⣇⠀⠘⢻⣿⣿⣮⢆⠀⠈⣿⠟⠿⣷⣿⣭⣷⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⢿⣿⣿⣿⣿⣆⠀⠘⢿⣿⣿⣦⠃⠀⠹⠀⠀⠀⠀⠀⠉⠛⠿⣟⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄⠀⠘⢿⣿⣿⣷⡰⣸⡀⠀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡄⠀⠀⠻⣿⣿⣷⡽⣷⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⡄⠀⠀⣻⣿⣿⣿⡒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣅⠀⠈⢱⣿⣿⣷⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⣆⡀⠀⣩⡿⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⣚⣭⠾⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⢠⡀

⠀⠀⠀⠀⠁⠀⠁⠳⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢡⠃⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⡠⢔⣠⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⢀⣀⣀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢰⣀⣴⠛⣹⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣂⣛⣃⣸⡆⢾⣧⠽⠏⠑⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠈⠙⠛⠉⠳⡀⠀⠀⠀⠀⠀⠀⢀⣶⠄⠀⠀⠀⠀⣼⣹⡿⠀⠠⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⣓
⠀⠀⠀⠀⠀⠙⣦⣴⠢⠤⠔⠛⢻⣿⠀⠀⠀⠀⢐⣿⣿⠍⠀⠠⡇⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⣴⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠔⢦
⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⣲⠀⠀⠒⠲⢦⣤⣭⠅⠀⠀⣧⠤⠀⠀⠀⠀⠀⠀⢸⠃⠀⡿⠀⢠⡇⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡄⠁⠀⡗⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠈⠓⠀⠙⠀⠀⠃⣠⠞⠀⠀⠀⠀⠀⠐⠂⠀⢠⠚⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⡰⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢒⡂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠃⠚⠦⠤⠧⠤⠭⠷⠒⠛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠡⡓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠣⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊
⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⢳⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠠
⠀⠀⠀⠀⠀⠠⠽⠇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢢
⠀⠀⠀⠀⡼⠆⠀⠾⠀⢀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢐⠀⠀⡇
⠀⠀⠀⠠⡇⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⢸⡇⠀⠀⣆⠀⠀⠀⠀⠀⠠⠀⢈⡤⠄⠁
⠀⠀⢀⣶⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠐⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢨⠇⠀⠀⠀⠀⠀⠖⢼⡇⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀
⢠⢀⣾⠀⠀⠀⠀⠀⠀⠘⠤⡟⠀⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡄⢈⡇⠀⠀⠀⠀⠀⠀⠀⢵⠹⡇⠀⠀⠀⠀⠨⠀⠇⠀⠀⠀⠀⠀⠀⢀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠂⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⠀⡄
⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⠀⠘⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢻⣿⣿⡟⣿⡟⣿⣿⣿⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⣿⡇⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⣿⡇⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠾⠿⠿⠷⠿⠷⠿⠿⠿⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⣴⣶⣶⣶⣶⡆⣿⣿⢰⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣶⣶⣶⣶⣶⣶⢸⣿⡇⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣿⣿⣿⣿
⠈⠉⠉⠉⠉⠁⣿⡿⠈⠉⠉⠉⠉⠉⠉⠉⠈⠛⠛⠛⠛⠛⠛⠛⠋⠈⠉⠉⠉⠉⠉⠉⠉⠸⣿⡇⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠯⡀⠀⠀⠀⠴⠚⠛⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣟⣯⣥⣤
⠀⠀⠀⢀⡞⠳⣤⢦⡀⠸⣶⡋⠀⠀⣠⠞⠁⢹⣆⠀⠀⠀⠀⢀⣦⡀⣀⠀⠀⠙⠵⣿⣿⣿⢟⣾⣷⣄
⠀⢀⡴⠋⠀⢢⡈⠻⡀⠀⠀⢈⣦⡞⠁⠀⢀⡏⠸⡆⠀⢀⣴⣿⣯⣻⣯⡀⠀⠀⠀⠘⢟⣵⣿⣿⢧⣿⡄
⠀⠘⢧⡀⠀⠀⣹⠖⠃⢀⡴⣫⠎⠉⢢⣠⠞⠀⢀⡇⠀⠘⢿⣿⣿⡷⠉⠁⠀⢀⡴⠋⠀⠈⢿⣫⣿⣿⠇
⠀⠀⠀⠙⠒⢾⢳⢀⡴⣫⠞⠁⠀⠀⠈⠛⠢⡀⡈⠀⠀⠀⠀⠙⠉⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠉⢻⡟
⠀⠀⠀⠀⢀⡞⣯⡻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⢀⣼⣦⡛⠁
⠀⠀⠀⠀⠈⢧⣛⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠕⠁
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠠⢤⠍⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠙⠒⠿⠛⠋⠛⠋⠉⠋⠛⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣠⣶⣤⠶⣀⣀⣄⣀⣀⣤⡄⠀⠀⠐⠂⠐⠒⠄⠨⣯⣠⣤⣤⢤⣤
⠀⢀⡀⢤⣤⣤⡀⢀⣀⠀⠀⡀⠀⣀⣠⣤⣂⣤⡤⠀⠤⠴⠖⠒⠒⠀⢀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠁⠈⠉⠉⠉⠙⠛⠛⠛⠿⠿⢄⣀⣀⣀⣀⣀
⣭⣥⣤⣤⣄⣐⣒⣠⣤⣶⣶⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠓⠒⠈⠭⠍⠉⠉
⣛⣛⣛⣻⡿⠟⠛⠛⣿⣿⣿⣿⣿⡿⠒⣶⣶⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢐⣢⣤⣤⣤⣤⣤⣤⣀⣀⣀
⡉⠉⢸⣿⣿⣿⣯⣭⣭⣭⣭⣭⡟⠋⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⣉⣩⣭⣉⣉⣉⣉⣉⣩⣿⣿⣿⣶⠶⠶⠶⠶⣶⣖⣒⣐⣒⣒⣒⣒⣺⣿⣿⣿⣿⡿⠿⠿⠿⠿
⣿⣿⣿⣿⠿⠟⠓⠶⠶⠶⠶⠶⠶⣦⣤⣀⣀⡐⠒⠒⠒⢚⠛⠛⠛⠋⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠿⢛⣛⣛⣉⣉⣉⣉⣛⣛⣛⣻⣿⣶⣶⣤⣤⣐⡒⠒⠒⠒
⠋⠉⠙⠒⠢⢤⣤⣀⣀⣀⣠⣤⣤⣤⣄⠀⠀⠀⣀⣀⣠⣤⣶⠶⣛⣋⣉⣉⣩⣥⣄⠀⣤⣤⣶⣶⣶⣶⣶⣾⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣻⣿⣛
⣒⣀⣀⣀⣀⣉⡉⠉⠉⠉⠉⠙⣛⣛⣿⣿⣿⣿⠿⠿⠿⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿
⣿⣿⣿⠿⠿⠿⠷⠒⠂⠀⠉⠈⠉⢉⣉⣛⣻⣷⣶⣶⣶⠶⠶⠶⠶⠒⠛⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡥⠴⠶⠀⠀⢀⡀
⠉⠉⠀⠐⠓⠒⠒⠒⠒⠶⠶⠲⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣈⣉⣛⣀⣀⣛⣻⣿⡿⠛⠛⠛⠛⠋⠉⠉
⠛⣻⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠻⠿⠿⢿⣟⡓⠒⠀⠒⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠀⠤⠤⠀⣀⣀⣀⣀⣀⣀⣀⣈⠛⢛⣛⡓⣒⣒⣛⣿⣷⣶
⠿⢿⣿⣷⣶⣿⣿⣿⣿⣿⣟⣿⣿⣿⣭⣤⣤⣤⣤⣤⣶⣶⣶⠿⠿⠿⣿⣿⣿⣿⣿⣉⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⣉⣉⣉⣤⣤⠠⣤⣤⣤⣤⣬⣭
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠉⠉⠙⠋⣉⣭⣅⣀⣀⣈⣉⡉⠙⠛⢛⣂⣀⣀⣉⣉⣉⣙⣛⣻⣿⠷⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⠿⠿⠿⠛⠛⠛⠋⢉⣉⣁⣠
⣿⣿⡟⠛⠛⠟⠛⠿⠿⢿⣿⣿⣯⣭⣍⣉⣉⣉⣉⠉⠉⠉⠁⠀⠀⢀⣀⣉⣉⡉⠭⠭⠥⠾⠟⠛⠛⠋⠉⣉⣉⣉⣉⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠙⠛⠛⠛⠋⠉⠉⠁⠀⠈⣉⣉⣩⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟
⣉⣀⣀⣀⣀⣤⣀⣀⣀⣠⣤⣤⣶⣶⣿⣿⠿⠿⠟⠛⠛⠛⢛⣛⣛⣿⣿⣯⣭⣭⣭⣭⡽⠿⠿⠿⠿⠟⠛⠛⠛⠉⠛⠛⠛⠛⠋⠀⠉⠉⠉⠉⠁
⣶⣶⣶⣶⣦⣶⣦⣤⣤⣶⣶⣤⣤⣤⣴⣶⣾⣿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣾⣤⣄⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⣀⣀⣀⣀⠀⣀⣠⣀⠀⠀⣀⡀⣀⣀⣀⣀⣀
⣭⣭⣭⣭⣤⠤⠤⣤⣤⣶⠶⠶⠶⠖⠒⣛⣻⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠏⠉⠉⠙⠛⠚⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠀⠀⠀⠚⠉⠩⠉⣁⣒⣀⣀
⣿⣿⣟⣉⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠉⠉⠉⠁⢀⣐⣒⣒⣒⣚⣛⣓⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠠⠴⣶⣶⣷⣿⣻⣛⣿⣟⣿⣛⣟⣿⣉⣉⣛⣛⣛⣻⣿⢿⣿⣿⣟⣟⣛⣛⣛
⡶⢶⣾⣾⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠶⠷⠾⠿⠿⠿⠟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣄⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⠠⠤⣶⣶⣶⠶⠖⠒⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠈⣹⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢉⣉⣙⣛⣛⣛⠛⠛⠻⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣶⣾⣿⣷⣶⣶⣶⠶
⠀⠀⠀⢀⣀⠀⠀⠤⠤⠤⠤⠴⣾⣿⡿⠛⠛⠛⠻⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀
⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠒⠚⠛⠓⠒⠚⠓⠚⠛⠿⠭⠭⠿⢿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣺⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣤⣤⣤⣤⠤⢤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣻⣿⣿⣿⣿⣿⠿⣾⣿⣿⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡄
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢻
⠀⠀⣸⠙⢦⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢸⡇⢰
⠀⠀⣿⠀⠀⠉⠓⠶⠖⠋⠁⠀⠀⠀⠀⣸⡇⠸⠸⠁⠀⠀⠀⡠⠠⠔⠐⠤⠐⢄
⠀⠀⢹⡀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣰⠟⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠱
⡀⣄⠈⠳⣤⣀⠀⠀⣸⠇⠀⠘⢶⡛⠁⠀⠀⠀⠀⠀⠀⠈⢒⠀⠀⠄⢂⠀⠀⠀⠐⢀⠄
⠳⡈⠑⠀⠈⠉⠉⠉⢹⠀⠀⠀⠀⠙⠲⢤⣀⣀⠀⠀⠀⠀⠀⠙⠀⠘⠆⠉⡠⡀⠤⠂⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠒⠒⠛⠓⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠲⠶⠶⠶⡆⢠⠶⡶⠶⠒⠒⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣠⠀⠀⢊⠄⠠⡀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢟⠋⠀⠀⠀⡇⠆⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢤⣤⠤⠤⠖⠚⠉⢷⡄⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡌⠈⢆⠀⠀⠀⠰⠦⠔⠂⠀⠀⠀⠀⠀⠀⢸⡗⠢⠤⠤⠒⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠤⠦⣌⡻⣦⡀⠀⠀⠀⠈⠳⠶⠤⣤⠄⠀⠀⠀⠀⠈⢤⠠⠊⠀⠈⢐⠆⠠⠀⣀⣀⣀⣀⣠⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣤⠐⠁⠀⣀⠔⠁⣠⣞⣁⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠬⠓⠓⠤⣀⣀⣀⣀⣠⢤⡀⠀⠀⠐⠳⠤⡀⠈⠁⡩⠀⠀⣰⠏⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠙⠒⠉⠉⠙⠢⠤⠴⠒⠛⠧⢄⣀⣀⣀⣀⡀

⢾⣿⣿⣿⣿⡆⠀⣿⠀⡆⠰⡟⢿⡶⠀⢠⠀⢸⡇⠀⠉⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠛⠛⠓⠀⣤⠀⠠
⣿⡿⠿⠛⠛⠃⠀⡟⠀⡇⠀⠈⠁⠀⠀⣿⠀⢸⣧⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠
⣟⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⠀⠀⠀⠀⡿⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠡⡤⠀⠀⠐⠀⠀⠀⠀⣿⠀⢸
⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈
⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⠃⠀⠀⠸⡦⠀⢀⠀⠀⠀⠀⣿
⠀⠀⡇⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠖⠀⠀⠸⣿⠀⠀⠀⠾⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠡⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⢀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⡀
⠀⢿⣄⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣭⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⢿⡏⣠⣴⣞⢦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⣹⣇⣿⣿⣿⣎⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠉⠁⠈⠛⠛⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠒⢀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⠄⠀⠀⠈⠁⠐⠢⢄⠀⠐⠀⠐⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠚⠃⢱⡄⠀⠀⠀⠀⢀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠄⠀⠀⠀⠀⠀⡀⠀⢁⠀⠀⠀⡀⠇⠀⢀⡀⠀⠀⠀⠀⣨⠆⠀⠀⠀⠀⠳⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢠⡿⠀⠁⢀⠉⢀⣀⠴⠞⠛⢢⡈⠓⠈⡄⢱⡎⠀⢀⡀⠀⠀⠀⢦⡈⢻⡄⠀⠀⠀⢀⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠘⠃⡌⠀⣡⠞⠋⠀⠀⠀⠀⠁⠀⠋⠀⠁⠃⠈⠁⠊⠉⠳⠄⠀⠈⢱⡈⢷⣄⡀⠀⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠒⣀⡁⠰⢎⡌⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⠛⠦⠯⡝⣦⡀⣠⠞⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⠔⠈⠀⠀⠐⡉⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⣠⣤⣶⢿⡏⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⣷⣯⠄⠀⢐⡉⢤⠄⠒⡛⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠂⢀⠀⠀⠠⠀⠈⠻⣆⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡋⠠⡔⠁⠀⠀⠀⠈⡀⠀⠘⠋⠗⣉⣉⡉⢉⡳⡄⠀⢠⡾⠛⠉⣀⣈⣙⠒⠀⢠⠀⠀⠄⡀⠀⠀⠀⠀⠀⢹⠀⠉⠑⠂
⠀⠀⠀⠀⠀⡠⠄⠚⣿⡇⠀⠀⠀⠀⣀⠀⣴⠇⠀⠀⠐⠪⠿⠿⠇⠐⠀⠀⠀⠸⠇⠐⠹⠧⠿⠉⠛⠀⠀⠀⢀⠀⠀⠀⠀⢀⠈⣼⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣎⠦⠀⠀⠫⣄⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⠓⠀⠛⠀⠀⠀⠘⠀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠄⢣⡀⠈⣷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠔⠀⢈⠀⡠⠃⢀⣼⠃⠀⠠⠀⠀⠸
⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⢸⣐⣴⠇⠀⢻⢾⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⢁⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢰⢃⠎⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠆⠀⠀⢠⡔⠶⠛⠻⠋⠚⠻⠋⢀⠇⠀⠀⠀⠀⢀⡤⠀⠈⠁⢤⡔⠁⠠⠀⠀⠀⠀⠀⠃⠀⠐⠀⠐⠸⡦⢤⣜⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠲⣶⣤⣴⠾⠳⢄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⡄⠐⠒⠿⠋⠙⡆⠀⠈⠀⠀⢸⢠⡖⠺⢋⣉⣉⠝⢳⠀⠀⠀⠀⠀⠀⢸⡆⠀⠤⠤⢤⠤⠀⢈⡙⢦⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⢠⠲⡃⠀⠀⠀⣆⡄⠀⠈⠛⠀⠁⡄⠀⠈⠀⠁⠀⠸⡀⠀⠀⠀⠄⠀⠘⠤⡀⠀⠈⠉⠉⠉⠀⠀⢠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠘⢆⠀⢀⡣⠀⠀⠀⠈⢥⡀⠀⠀⡀⢠⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢀⡀⠀⢲⡄⠀⠙⢠⣴⡥⠈⠄⠀⠙⠀⠀⠐⢃⠜⠀⢀⡴⠾⠛⠁⠀⠀⠀⡀⠄⣐⠭⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡛⣀⠈⠱⠀⠁⢱⡀⠀⠀⠈⠁⠀⢸⡧⡾⠂⠸⣀⣤⣿⠶⠊⠁⠀⠀⠀⠀⠁⠘⠀⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁⠀⠀⢸⡟⠁⠀⠀⣀⠄⢠⠎⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠚⣗⠣⠠⠤⠴⠊⠠⠄⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠴⠀⢀⡴⣍⣩⣿⣾⠿⠋⠉⠒⠀⠤⢀⡁⠀⠀⠀⠀⠐⠤⠤⠉⢀⡀⠨⠅⠀⠀⠀⠈⠳⣤⣴⡾⣷⠀⠀⢀⡼
⠀⠀⠀⠀⠀⢰⣶⠁⠠⠤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⣤⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢷⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉⢁⣨⣅⡈⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠻⢿⡿⠟⠃⠀⣺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣦⣼⣧⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⣽⣯⣍⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⢿⣿⣿⣿⣿⡿⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁

⠀⢽⣶⣶⣦⣬⡍⠉⣉⠙⠋⠃⣀⣈⣃⣈⢉⣠⣤⣴⣿⣬⣷⣤⣭⣍⣁⣀⣰⢶⡶⣶⣾⣻⡓⠾⢿⣴⣶⣦⣄
⣙⣛⣂⣀⣠⣤⣤⡀⠉⣡⡶⠛⠉⠀⠀⠨⠉⢹⣟⠛⠋⣻⣽⣿⡿⢿⡿⣿⠿⡭⠿⠗⠁⠉⠛⢛⣃⡠⠶⡿⠉
⠭⣭⣽⣯⡭⠉⠀⣀⣛⣿⣿⣷⣍⣉⠩⠉⠿⢷⣿⠓⠒⠓⠚⠋⠈⠙⢁⡀⠀⣤⢀⣠⡀⣀⣠⣬⣤⣏⡐⠡⢄
⠒⠋⢁⣶⣿⣯⡭⣽⠿⣿⡧⣤⣾⣿⠟⠁⠀⠛⢿⣧⣄⡀⠀⠀⠀⠀⠈⠀⠘⣉⡙⠉⠛⢿⣦⣲⠾⢿⣟⠛⢩
⠻⣿⣿⠋⠉⠉⠉⠀⠲⠦⢤⣤⣀⣀⠀⠠⢤⡶⠾⠟⠛⠛⠂⠀⠶⠿⣿⣿⣀⣠⣤⡿⠏⠁⣀⣤⡽⠟⣀⡙⢒
⠈⢿⣿⡀⠀⠀⢀⣤⣤⣤⣶⣶⣶⠶⠶⠆⠀⠀⠀⣀⣀⣀⡀⠀⢐⣃⠀⠹⡿⠿⢭⠶⠞⢛⣫⣥⠶⠛⢋⣰⣭
⣀⠀⠉⠻⣦⣄⠀⠈⠉⣉⣭⠵⠶⠶⠾⣿⣿⣿⣿⣿⣿⣉⣉⠭⢽⣛⣤⣟⣹⢶⡶⠷⢅⣘⣵⣤⣤⣤⡬⢽⣶
⠀⠉⢵⣒⠛⢋⣀⣀⣘⣛⡛⠶⠶⠶⠤⠤⠾⠿⠿⠟⠉⠴⢖⣋⣛⣉⣀⣠⣥⣤⣶⣛⣻⣿⣿⣿⣏⣠⣤⣬⣭
⠛⠛⠿⠟⠀⠘⠿⠥⢀⡉⣯⠉⡉⠉⠛⣓⠺⠷⢂⣤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠿⠿⠋⠀⠔
⠀⠀⠀⣠⣿⣿⡿⠶⠶⠶⠮⠭⢭⣵⣶⣶⣶⣾⡿⠿⠷⢶⣶⣤⠀⠀⠀⠀⠀⠤⣤⣤⡄⠀⢄⠀⠺⡆⠀⢠⡄
⡀⠀⠈⠻⠿⣿⣷⣤⣀⣀⣀⣤⣾⣿⡿⠿⠿⠿⠗⠒⠚⠛⠛⣳⣦⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣈⣁⣀⣀⣠
⣛⣛⣂⣀⣀⣀⣤⠽⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⠷⡶⠶⣶⠤⢾⠿⠿⠿⠿⠿⠿⠿⢿⣿⠿
⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠠⣾⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⠿⢿⣿⣿⣷⡶⠶⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⢀⣀⣀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⣾⣿⡟⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠳⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡬
⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⡀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠒⠒⠂⠀⠀⠒⠒⠈⠉⠛⣛⣭⣿⣶
⣶⣤⣀⠀⠀⠀⠀⠀⠀⠈⢉⣽⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠟⠃⠀⢀⣤⡤⣤⣴⣿⡟⠋⠁⠁
⠙⠛⠛⠿⠶⢦⣤⣤⣀⣤⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣴⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠰⣿⣏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠷⠶⣶⣦⣤⣤⣤⣤⣤⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⢤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣷⠶⠆⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠾⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣆
⠛⠳⠶⠶⠶⠶⠶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠋⠉⠉⠙⠓⠶⠖⠛⠋⠛⠓⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠒⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⢶⣄
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⣠⣤⡄⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡏⡟⠋⠉⠛⢛⣦⣤⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡤⠞
⣼⠁⣸⠀⠀⠀⠀⠙⢶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢿⡇⠰⠀⣰⠏⠀⠸⣄⣀⣀⡀⠀⠀⠀⣸⠃
⢷⣀⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⡼⢿⡀⠈⠀⠀⣰⠏⠀⠀⠀⠀⠉⠉⠙⠲⠖⠛⠁⠀⠀⠀⠀⢸⡟⢳
⠀⠉⠀⠀⠀⠀⣠⡄⠈⠳⠦⣤⡴⠞⠋⠉⠉⠉⠁⠀⠸⡇⢀⠆⢠⡏⠀⠀⠀⢶⣤⡀⠀⠀⠀⠀⣶⣦⣀⠀⠀⠀⠈⢷⣼
⠀⠀⠀⠀⠀⢸⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⡾⠀⠀⠀⠀⢸⡏⠹⡄⠀⠀⠀⢸⡄⠉⡇
⠀⠀⠀⠀⠀⠸⢦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢰⡇⠀⠀⠀⠀⠘⠷⠞⠁⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⣩⣇⣀⣸⣷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣇⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠉⠛⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠛⠉⠙⠛⠳⣤⡀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⣀⣀⣀⣀⣀⡀⢀⣼⠋⠀⠀⠀⠀⢀⣤⣄⡈⠻⣆⠀⠀⢸⡇⠀⠀⣿
⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣧⠀⠉⠉⠉⠉⠉⠀⢸⠇⠀⠀⣀⡀⢀⡾⢡⡟⠀⠀⢻⡀⠀⢸⡇⠀⠀⣿
⠀⠀⣾⠉⠉⠉⠙⢿⣍⡉⠉⠉⠈⠉⠉⠉⢹⡆⠉⠉⠉⠉⠉⠁⢸⡀⠀⠺⣏⠳⡾⣡⡟⠀⠀⠀⣸⠃⠀⢸⡇⠀⠀⣿
⠀⠀⡏⠀⠀⠀⠀⢰⡏⠛⠶⠶⣤⣤⣤⣴⠞⠁⠚⠛⠛⠛⠛⠂⠘⣧⡀⠀⠈⠷⣴⠏⠀⠀⠀⣠⡟⠀⠀⢸⡇⠀⠀⣿
⠀⠀⣿⠀⠀⠀⣞⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀⠀⠀⣀⣠⡾⠋⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠸⣧⠀⠀⠙⢻⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢘⣷⣄⡀⢸⡇⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⣠⡾⣯⡙⢿⣟⠛⢷⣄⣀⣀⣀⣠⡾⣯⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠲⢦⣄⠀⠀⠀⠀⢸⡇⠀⠀⣿
⢰⡏⠀⠈⢷⡀⠹⣆⠀⠈⠉⠉⠉⠹⣆⠈⢷⡀⢻⡄⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⣠⣀⠉⢷⡄⠀⠀⢸⡇⠀⠀⣿
⣿⠀⠀⠀⠘⣇⠀⢹⣆⠀⠀⠀⠀⠀⣿⠀⠘⣧⠈⣿⠙⠛⠛⠁⢸⠇⠀⠀⠀⠀⠀⣼⢋⡿⠁⠀⢿⠀⠀⢸⡇⠀⠀⣿
⣿⠀⠀⠀⣰⡇⠀⠀⠙⠶⣤⣤⣤⡴⠟⠀⠀⠙⢶⣿⠛⠛⠛⠂⢺⡀⠀⢴⡟⢷⣼⢣⡟⠁⠀⠀⢸⠇⠀⢸⡇⠀⠀⣿
⣷⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠄⠘⣧⠀⠀⠙⢦⣡⡟⠀⠀⠀⢠⡿⠀⠀⢸⡇⠀⠀⣿
⣿⠀⠀⢸⡇⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⣄⠀⠀⣿⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠁⠀⠀⣀⣴⠟⠀⠀⠀⢸⡇⠀⠀⣿
⣿⠀⠀⢸⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠚⠛⠉⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⣿⠀⠀⢸⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⣿⠀⠀⢸⡇⠀⠀⠀⠛⠶⠶⠶⠶⠶⠶⠶⠋⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⣿
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⣿⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠿
⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠋⠉⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠋⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢦⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢰⣏⣁⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⢴⡶⠶⢶⡶⠚⠁
⠀⠀⠀⠀⠀⠀⣠⡭⣿⠿⢿⢸⡇⠀⠀⠀⢉⡀⠀⢀⡁⠀⣼
⠀⠀⠀⠀⠀⢰⠏⠀⢻⡈⠹⡆⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠳⣤⢿⣿⡀⠀⠀⠀⢀⠀⠚⠀⢀⡟
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣿⠀⢻⣿⣄⡀⠀⠈⠉⢀⣤⠞
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡿⣰⠻⣇⠉⠙⠛⠒⢺⡟⢷⡀
⠀⠀⠀⠀⠀⠀⣿⣀⣤⠼⢿⡇⠀⠙⢷⣄⣀⣀⣼⠃⠀⢿⠶⢤⣤⣀
⠀⠀⠀⣠⡴⠞⠋⠁⠀⠠⣾⢧⡀⠀⠀⣈⡳⣤⣏⣀⣀⣸⡇⠀⠀⠉⠙⠲⣄
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠙⢷⡙⢶⡏⠉⠁⠀⡄⠁⠈⠙⠃⠀⠀⠀⠀⠀⠘⣧
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣙⣦⣤⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢀⡤⢤⣤⣤⣤⣀⣀⡀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⣥⣶⣶⡚⢧⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣴⣯⠃⠀⠀⠀⠀⢀⡼⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢩⠭⢽⡿⣦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⠟⠁⠀⠀⠀⠀⣰⠏
⡏⠀⠀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⠐⠚⢻⡇⠈⠛⠾⠇⠀⠀⠀⠀⠀⡀⢀⣀⡼⡳⠋⠀⠀⠀⠀⢀⡾⠁
⣷⠀⠀⠀⠀⠀⠀⠸⡇⠸⡇⠀⠀⠀⣿⠞⠁⠀⠀⡇⠀⠀⠀⠀⠀⢀⡟⢿⡉⠻⣄⠀⣿⠳⣄⣴⠋
⣿⠀⠀⠀⠀⠀⠀⣀⣧⣴⡃⠀⠀⢠⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⡻⣄⠻⣦⡈⠳⣼⣧⠘⡇
⣿⠀⠀⠀⠀⠰⠞⠉⠀⠹⣿⣦⡾⠋⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣴⣯⠳⣜⢷⡌⠙⠀⠈⠉⠀⣿
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⢞⠟⠉⠷⣌⠃⠀⠀⠀⠀⢀⣴⠋
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠃⢀⡼⡷⠃⠀⠀⠀⠈⢛⣶⢤⣤⠶⢋⡿
⠀⠘⠷⣤⣀⡀⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠛⠲⠶⠶⢤⣤⣤⠞⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⠶⠋
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠶⣿⠆⣿⣿⣿⣿⣿⣿⣿⣿⠲⣿⠶⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢺⣗⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢶⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⣿⣿⣿⣿⣷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣿⣽⣿⣿⣿⣯⣦⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠿⡿⠿⠿⠿⠿⢿⠿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣿⣿⣿⣿⣿⣿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡷⠆⣿⣿⣿⣿⣿⣿⠰⢾⣿⣿⠆
⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⣿⣿⡿⠿⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⠗⠂⠈⠛⠛⠛⠛⠁⠐⠺⡿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡎⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢄⣤⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⣀⣀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠠⠤⠤⠤⠀⠀⣀⠀⠈⠀⠋⠉⠉⠙⠛⠋⠛⠂⠙⠒⠒⠲⠶⠶⢶⣮⣉⣉⠉⠉
⣥⣤⣤⣄⣷⣤⣤⣶⣿⣿⣿⡿⠿⣛⣙⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠋⠍⠉
⠭⠭⣽⣯⣭⡉⠿⠿⠿⡿⠯⠽⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⢀⡀⠀⣀⣀⣸⣶⣷⣶⣶⣶⣶⣶
⣶⣶⣾⣿⠿⢭⣭⣭⣭⣥⣀⣀⠀⠤⠤⠤⠤⠶⠶⠿⠛⠛⠓⠒⠚⠛⠛⠛⢽⠽⠿⠿⠷⠖⠲⠶⠶⣾⣿⣏⣉⡉⠡⠤⠤
⠛⠉⠓⠂⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣖⣒⣉⣭⣤⣄⣠⣤⣴⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⡀⠀⠉⠭⣽⣿⣿⣟⣛⣛⣉⣉⣀⣀⣁⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⢛⡛⠿⠻
⠛⠛⠩⠭⠭⠥⣤⣤⡄⠀⠶⠶⠾⠿⠿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⣼⣤⠿⠭⠵⠶⠶
⢖⣶⣶⣿⣿⣿⡶⣶⠶⠶⠶⠒⠲⠶⠶⠶⢶⡖⢂⡀⠐⠂⠀⠀⣀⣀⣠⣤⡄⠁⠀⢀⣀⣉⣉⣉⣉⣙⡛⠓⠒⣒⣲⣶⣶
⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⠶⠶⠶⢾⣿⣿⣛⡛⠿⠿⣿⣿⣭⣍⣀⣀⠀⣀⣀⣀⣀⣈⣉⣉⣩⣤⣴⣶⣲⣶⣶⡶⠶
⣤⣤⡤⠤⠤⣴⣦⣤⣤⣤⠴⠶⠒⠒⠶⠶⠤⠀⠀⠐⠒⠒⠲⠶⠖⢺⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣶⣶
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⢿⣷⠶⠶⠶⠶⠾⠛⠟⠛⠾⠿⠗⠒⠂⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠥⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠿⢟⣛⣛⣛⣛⣛⣿⣿⣿⡿⠯⠭⠽⠟⠛⠛⠛⠛⠉⠉⣉⣛⣉⣉⣀⣈⣉⣉⣀
⠿⠿⠿⠿⠿⠿⠾⠿⠿⠶⠾⠛⠛⠛⠋⠉⠉⢉⢁⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉
⣒⣒⣂⣠⡤⣄⣠⣤⡤⠤⠤⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⠲⠒⠒⠒⠒⠂⠛⠋⠀⠒⠀⠈⠙⠛⠓⠂
⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀
⣀⣀⣀⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣭⣭⣽⣿⣿⣭⣥⣬
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀⠀⠈⠉⢉⣩⣭⣭⠭⠭⠴⠶⠚⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠉⠛⠛⢛⣛⣛⣛⣛⡛⠻⠿⠷⢶⣶⣶⡶⠶⠶⠶⠶⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠛⠛
⣀⣀⣀⣲⣀⣈⣭⣭⣭⣿⣿⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣯⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣵⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⠅⠀⠀⠀⠀⢀⡀⠂⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠆⠸⣿⡆⠀⠈⠀⠀⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠱⡘⢄⠀⢻⣿⡀⠆⠀⠀⠉⢀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⢎⢂⠀⢻⣧⠀⠀⠀⠀⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡉⠉⠐⠢⢕⡀⠱⣕⢄⠻⣦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⠶⣤⣌⡐⠌⠓⠤⡙⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠈⠙⠓⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡈⡀⡌⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⠁⣥⠁⠀⠀⠀⢀⣀⣀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢷⢠⠇⠀⣠⣴⢿⣿⣿⣿⣿⣿⢧⣠⣤⡀⠀⠀⠀⠀⠀⢴⡿⢿⣶⣾⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⢸⢠⣾⣿⣿⣿⣿⣿⣻⣷⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠀⢸⣿⣿⣷⣿⣿⣿⣿⡿⠟⢋⣉⠀⠀⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠯⣙⣿⠿⢟⡋⠵⠒⠉⠁⠀⠀⠀⠀⠈⠳⣎⠉⠉⠛⠛⠋⠉⠉⠀⢀⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⢷⡐⢄⠀⠀⠀⠀⢀⣮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡢⠀⠘⣸⡸⡀⠀⠀⠀⠀⢀⡠⣈⠔⠀⠀⡼⠀⢰⠀⠀⠀⢣⠈⢷⡈⠳⠄⢀⣀⡾⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠰⠁⠀⢂⠁⠳⣍⣢⣤⢔⡢⠕⠊⠀⠀⠀⡰⠁⠀⢨⠀⠀⠀⠈⡆⠘⣧⠀⠀⠛⠉⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⡀⠄⠈⢄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⢸⣴⠀⠀⠀⠀⠰⡀⠸⡆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠓⠋⠉⠉⠉⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⡄⡁⠀⠀⠀⠀⠀⠡⠀⠹⡀⠀⢀⣪⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠡⣀⠠⡈⠳⣄⠀⠀⣀⠠⠔⠊⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠁⠀⠱⣶⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡒⢄⡉⠐⠂⠐⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣤⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⣀⠀⠀⠈⠒⢬⡓⠤⣈⠁⠒⠒⠒⠁⢸⠀⠀⢈⠉⠐⠒⠂⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⣀⡈⠑⠢⣑⠢⣄⠀⠀⠀⠣⠔⠉⠑⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣺⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠑⠀⠙⠢⢍⡒⠄⠀⠀⠀⠘⠀⠀⢀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢏⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⡄⢀⣀⣸⠘⠜⡆⠀⣀⠤⢤⡀⠀⠀⢀⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠛⠛⠉⠉⠀⢱⠃⢸⠁⣂⣆⣧⠤⠴⠗⡒⣮⢣⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣐⣢⣄⠀⠀⠀⠀⠑⠢⣤⣍⠑⡄⠹⡖⢚⠉⠉⠁⡄⡆⣆⣽⣿⡟⢸⢀⢹
⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⢠⠏⠀⣸⣧⡽⣀⣤⠤⠒⠚⡉⠉⢹⢣⡠⡖⠿⠿⢟⣴⡶⢷⢩⡷⢲⣿⡖⢸⠀⣺⡄
⠀⠀⠀⠀⠀⠘⢦⢦⣘⣦⠤⠤⣀⠤⠜⠒⡟⠀⠉⠙⢶⣨⠷⣗⡓⠓⠦⣌⡬⠵⡇⠀⠨⢹⣯⢸⢸⣾⠁⠘⠿⠷⣶⣧⣾⢻⣷
⢀⣤⡀⠀⠀⠀⠈⣾⢉⠬⣝⣦⠼⣦⢀⢸⣍⣖⡀⠀⠀⠀⡀⢀⠉⠲⢄⠀⠑⢤⡇⠀⠀⣸⢧⡎⣾⠃⢾⣯⣷⣒⡒⢤⢀⣾⣏
⢣⡤⠙⢆⡀⠀⠀⠘⢸⡒⡾⢸⢀⢽⠶⡞⢚⣛⣻⠀⠀⢐⠯⡒⢽⣦⣄⠑⣄⢦⡱⡀⡐⢺⣇⠻⡿⠀⣀⡉⢙⣛⢛⢋⣼⡏⠘⢆
⠀⠙⣿⡂⠙⠢⣄⡁⠀⠈⠀⠸⠀⢿⠀⡞⠰⠶⡏⢳⠤⣄⡉⣺⡿⢋⣈⢧⡈⣎⣿⣧⢘⣺⣷⡂⢱⠤⢤⠞⠋⠉⠉⣽⡿⡙⣄⢜⣆
⠀⠀⠈⠙⢾⣔⣤⡇⠀⠀⠀⡆⠨⢽⡀⠙⡖⣖⣁⡆⠀⠧⡟⣛⡽⠋⣽⡿⣿⢸⣿⠟⣐⣺⣯⡊⣇⣨⡴⢚⠂⠀⢬⣷⣽⣄⢸⡄⣺⡄
⠀⠀⠀⠀⠀⢈⡿⠁⠀⠀⠀⢡⢐⢽⣿⣉⣛⠻⣶⠁⠨⢱⣟⢽⡇⢾⣿⣿⡾⠋⣹⣶⣾⠿⣿⡿⠀⣿⣽⠧⠴⣶⣾⠽⠗⡞⣹⠀⢚⡅
⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠘⡄⠙⢻⡿⡿⣷⡇⠀⠥⣾⣷⢷⡷⡀⣈⣛⣁⣴⣿⣇⡽⠶⢛⣋⣽⠿⡶⢊⠟⡔⢡⢊⠎⡔⣹⠨⢽
⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠙⣇⣿⠀⠨⢽⣯⢮⣿⣌⣠⡿⢿⣿⡿⢯⠗⣲⢋⡝⣡⢃⠜⡠⢫⠎⡰⣣⢃⠞⡐⢹⢒⣽
⠀⡇⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠈⠳⣀⡭⠿⢷⣿⡯⡽⣲⢋⢝⡝⣡⢊⡔⣡⢊⡔⡡⢋⠔⡱⢁⠞⡴⣡⢋⠜⡰⣫⢐⣺⢄⣀⠀⠀⠀⣀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠂⣿⠲⢫⢫⢊⢞⡔⡡⣫⢊⡜⣡⢋⠔⣡⢊⠜⡡⢊⠞⡰⢁⠞⡰⢡⢎⠜⡔⡿⠀⣾⣒⣒⡀⠭⢅⣐⡒⠦⠤
⠀⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡇⡉⢽⠔⡵⡡⢣⢊⠔⡕⣡⠊⡔⡡⢋⡔⡡⢋⠔⡡⢃⠞⡠⢃⠜⡰⢣⢊⠜⡐⡯⢕⣿⣂⠀⠬⠅⢒⣒⣀⠉⠉⠥
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣒⠍⡏⡔⡱⣡⢪⢊⡔⢡⢊⠜⣡⢊⡔⢡⢊⠜⡠⠃⡴⣡⢃⠜⡰⢡⣊⣾⣿⠇⣽⣗⠒⠬⠭⠤⠀⣐⣒⠒⠪⠍
⠀⠀⣀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢒⣨⢋⠜⡔⡱⣡⠊⡰⣡⠋⡔⣡⠋⡴⣡⢊⡜⢁⠎⣴⣡⣮⠞⣗⣯⢗⣛⣛⣛⣂⣘⣂⠈⠩⠭⠤⠄⠐
⠀⢠⠉⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇⠕⣺⢃⢎⠞⡴⢁⢞⡔⢡⢎⡔⢁⠎⡴⢡⣊⡴⣗⣞⣭⡜⣚⣛⣛⣁⣈⡉⠭⠤⠤⠶⠶⠶⠒⠀⠁
⠀⠘⠢⢌⣓⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠥⢺⣠⢋⠎⡰⢣⠎⡴⣡⢊⣰⢧⠞⣟⣯⠷⠟⠋⠉⠂⠀⠀⠐⣒⣐⡒⠒⠚⠛⠓
⠀⠀⠀⠀⠈⠉⠒⠤⣀⠀⠀⠀⠀⠀⢀⠀⣇⡨⠝⡱⢃⠞⣰⡧⢾⢲⣻⣽⠼⡾⠏⠩⠭⠭⠥⠬⠭⠭⠭⠭⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠈⠀⡇⢒⡷⢶⣻⣭⢷⣶⣟⣛⣂⣐⣀⣶⣦⡀⠠⠍⠭⠭⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⠯⣷⣟⣛⣒⣒⡋⠭⠭⠭⠭⠭⠤⠶⡖⠒⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣴⠛⠻⣎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣧⠻⣦⡼⢯⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⢛⣛⣛⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣛⣛⡛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣾⠋⠉⠉⠙⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠏⠉⠉⠙⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢀⡶⠟⠻⣦⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣿⠀⣿⣰⡟⠉⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠸⣇⡀⢀⣽⠃⠘⢿⣤⣤⣤⣤⣤⣤⣤⣤⡼⠃⠀⠀⣿⠀⣿⠏⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠋⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⣠⡶⢶⣄⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡿⠃⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢸⡇⠀⠀⣻⠆⠰⣿⣉⣉⣉⣉⣉⣉⣉⣉⣹⠆⢠⡿⠁⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠛⠶⠾⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⣰⡟⠀⣠⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⣰⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢠⡟⠋⠙⢷⡀⢀⣴⠶⠶⠶⠶⠶⠶⠶⣾⠏⢀⣼⠏⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠘⣧⣄⣠⡾⠁⠈⠻⠶⠶⠶⠶⠶⠶⠶⣿⣴⠟⠁⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠹⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠻⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠟⠛⠛⠛⠷⠶⠑⠊⠉⠙⠙⠋⠉⠉⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⣤⣤⡀⢀⣀
⢠⣤⣤⣀⣀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣿⣿⣿⣭⠤⠤⠤⠀⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⡀⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠈⠉⠉⠉⠉⠀⠀⠀⠉⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉
⣿⣿⣿⣿⣟⣻⣳⣶⣀⣐⣶⣖⣶⣶⣦⣤⣄⣤⣤⣤⣶⣤⠀⠀⠀⣤⣤⣶⣶⣶⣶⣤⣶⣶⣦⣤⣤⣤⣿⣿⣿⣤⣼⣿⣿⣿⣯
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠐⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠿⠿⣿⢿⣿⣿⣿⣿⠿⠿⡿⠿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠺⢿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠿⠟⠿⠛⠛⠛⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠐⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⠀⠀⠐⣚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠯⢥⣀⣍⠹⠋⣛⡟⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠐⢚⠿⠻⠷⠶⢷⣶⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣿⣶⣤⣝⣲⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣭⣙⣛⡿⢿⣟⣛⣋⢻⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣷⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠰⠲⠶⠖⠶⠶⣶⣤⣴⣶⣬⣍⠉⢿⣿⣿⣿⣿⠟⡛⠋⠉⡛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣛⣋⣼⣟⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠠⡔⣩⣤⣀⣿⣷⣬⣭⣤⣦⣤⣀⣉⣿⣿⣿⣿⣷⠦⠌⠙⢿⣿⣿⣧⣤⣬⣛⠿⢿⣾⣿⣯⣤⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⠁⠀⣉⣁⣙⠛⠿⠿⠿⠛⠓⠒⠂⠀⠐⠛⠛⠻⠿⠿⠿⠿⢿⣓⣶⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⢗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣄⣤⡀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⣴⡖⢂⣤⣄⡀⠀⣤⣌⠀⠀⠀⢀⣀⣀⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠤⣴⠆⠒⣤⣤⣨⡑⠄⠠⢆⣴⣿⣴⣿⣿⣿⣿⣼⣷⣲⠈⠁⡁⣺⣾⣿⣿⠛⠛⣋⣭⣭⣽⣻⠖⠛⣩
⠀⠀⠀⠀⠀⠀⠈⠛⠷⠾⠽⠿⠬⠭⠉⠭⣿⣿⣷⣯⣿⣿⣽⣿⣿⣿⢼⣿⣸⣿⣖⣩⡵⠾⠛⠛⠿⠟⠿⠛⠉⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣿⡩⠽⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣹⡆⠀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠤⠶⠶⢦⡄⠄⣠⣄⣤⣤⣀⢀⣀⣀⣤⣴⣷⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⣿⡿⠛⠋⠉⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠤⠄⠤⠤⠤⠤⠤⠀⠀⠀⠀⠬⣿⣿⣿⣥⣤⣤⣤⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⢀⣁
⠀⠀⠀⠀⠀⠁⠶⠷⠤⠴⠶⠤⠭⢯⠶⠶⠶⠶⠶⠶⠶⢶⡶⣶⣿⣿⣿⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣷⣿⠿⠷⠶⠖⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠐⠒⠒⠒⠒⠲⠿⠛⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⣿⡿⠷⠶⠶⠖⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣈⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⡆⠀⠀⢀⠄⠄⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡛⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠈⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣣⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢄⡀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡇⠀⠀⠘⠄⣀⠀⠀⠀⠀⠀⣀⠤⠃⠀⠑⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠠⠄⠄⠀⠠⠀⠠⠀⠄⠀⡄⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⢒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡿⠻⣿⣷⣦⠆⠀⠀⠀⠀⠀⠐⠀⢠
⠈⠂⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡁⠀⠀⡇⠀⢹⣿⣿⠀⢀⠄⠆⠀⡤⠌⢄⠀⡀⠀⠀⠀⠀⡠⣤⣤⣶⣀⠄
⠀⠘⡄⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⡆⠀⠀⡇⠀⢸⣿⣿⢠⠃⠘⠀⠀⠀⠀⠂⠁⠁⠀⠀⠀⠐⣼⠋⠀⠀⠹⣯
⠀⠀⠘⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⣷⣶⣶⣶⣶⣶⣶⣶⣶⡇⠀⠀⠀⠀⢸⣿⣿⠸⡀⠈⠒⠐⠃⠀⡌⠀⠀⠀⠀⠀⠷⡻⣆⣀⢀⣴⢏
⠀⠀⠀⠱⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⡇⠀⢠⠀⠀⠸⣿⣿⡆⢳⠠⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⡭⠟⠑⠁
⠀⠀⠀⠀⠡⠀⠀⣍⠐⠤⣀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢸⠀⠀⠀⣿⣿⡇⠀⠂⣤⠀⠃⠀⠀⠀⠀⠀⠀⡜⣰⡟⡰⠁
⠀⠀⠀⠀⠀⢃⡘⠀⠁⠂⠤⢈⡁⠒⠠⣀⠈⢃⠀⢠⠃⠀⠀⠀⠀⠘⠀⠀⠀⣿⣿⡇⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠌⣼⡟⡐⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⢸⣿⠈⠀⠀⠀⠀⢀⠌⣼⠟⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⢰⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⣿⣧⢸⢸⡏⠀⠀⠀⠀⢀⠎⣼⠏⡔
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠚⠀⠀⠀⠀⠀⠃⠀⠀⠀⠚⠛⠛⠒⠚⠃⠒⠀⠀⠀⠒⠚⠛⠘⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣄⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⢦⡀
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣿⠉⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⡇⣀⣠⡴⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⡇
⠀⠀⠀⠀⠀⠘⡇⠈⠙⠳⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣬⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢔⠒⡒⢐⢂⡂⡀⣒⠄⠀⠀⠀⠠⠤⠀⡖⠲⠂⠠⢄⠀⠀⠀⠀⠀⡠⢤
⠀⠀⠀⠀⠀⠜⠘⠄⠀⠀⠀⠀⠀⠀⠐⠀⢀⠧⠠⠆⠀⠘⠀⠀⠀⢀⢌⡴⢣⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠈⠉⠁⠂
⠀⠀⢀⣀⠈⠶⠰⠆⢀⣀⠀⠀⠀⠀⢠⠤⠎⠀⠀⠑⡄⠄⠀⠀⠀⠁⠁⣑⡿⢋⠜⢄⠀⢀⡀⠀⠀⠀⢀⠄⠷⠂⠶⠲⣒⠒⠃
⠀⠀⡇⠀⠋⠁⠁⠉⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⡗⠁⠀⠀⠉⢹⢸⠀⠀⡇⠨⠀⠀⠀⠀⠀⡏⠈⡇
⠀⠀⠇⠠⠆⠀⠀⠀⢧⠀⠇⠀⠀⠀⠈⠈⠢⠀⠀⠤⠃⠁⠀⠀⠀⠀⠀⠇⠧⠀⠀⠀⠀⠼⠸⠀⠀⠃⠰⠀⠀⠀⠀⠀⠧⠀⠃

⠀⠀⢀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣉⣀⣀⣈⡁⠰⠀⠀⠀⠀⠐⠠⡄⢀⠄⠐⠀⠀⠀⠀⠐⠀⠀⠠⣂⣄⡤⡢⠀⠀⠐
⠀⠀⠑⠀⠰⣖⢒⡶⠀⠐⠁⠀⠀⠀⠀⠀⡼⢽⠩⠷⠀⠀⠀⠀⠀⠀⠀⡤⠿⠾⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⣟
⠀⠀⡤⡀⢠⣚⣘⣢⠀⡠⠄⠀⠀⠀⠀⣀⢱⠽⠬⡗⢀⡀⠀⠀⠀⠀⠀⡨⠀⠀⠿⢧⡹⡄⠀⠀⠀⠀⠀⢠⠓⠒⠒⠓⢄⣀⣀
⠀⠀⡇⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠋⠁⠀⠉⠁⠆⠀⠀⠀⠀⠀⡅⠀⠀⢀⠔⠓⠁⠀⠀⠀⢰⠁⢙⠒⠐⢄⠀⠈⠁
⠀⠀⠧⠖⠢⠄⠀⠤⠒⠧⠄⠀⠀⠀⠸⠤⠂⠀⠀⠐⠦⠅⠀⠀⠀⠀⠀⠞⠒⠒⠳⠀⠀⠀⠀⠀⠀⠸⠀⠸⠤⠐⠁⠁⠒⠢⠔

⠀⠀⠀⠀⡀⠀⠠⡢⠤⠤⠴⠀⠀⠀⠀⢎⣉⠉⠉⣉⡩⠀⠀⠀⠀⠀⣠⢤⡤⣐⣂⢤⢤⣄⠀⠀⠀⠀⠀⠀⡤⡤⡤⡄
⠀⠀⠀⠀⢫⣩⡝⠀⠀⠀⠀⠀⠀⠀⣀⠔⠲⡏⠉⡗⠒⡀⢀⠀⠀⠀⡀⢸⡱⡄⣠⢎⢿⣇⠀⠀⠀⠀⠀⠀⡇⠇⠇⡆
⠀⠈⡇⠴⢯⠿⡽⠦⠚⠁⠀⠀⠀⡇⠀⠀⠠⠑⠌⡇⠀⡅⠈⠀⠀⠀⠙⠚⢱⠀⠐⡋⠛⠋⠀⠀⠀⠀⠀⠀⢸⠖⢾⠁
⠀⠀⣇⣛⡧⣠⣼⣛⡀⠀⠀⠀⠀⠇⢀⠀⠐⠀⠀⡇⠀⡇⠠⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⡎⠩⠀⠀⠀⠀⠀⠀⡋⡂
⠀⠀⠁⠀⠁⠛⠈⠀⠈⠃⠀⠀⠀⠀⠀⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠔⡉⢤⠠⠌⡑⢄⠀⠀⠀⠀⡠⠔⡩⠄⢠⢌⠐⢄⠀⠀⠀⠀⠠⠒⡡⡄⠠⢍⠢⡀⠀⠀⠀⠀⢠⢒⠭⠀⠠⢅⠢⡀
⠀⠀⡌⢈⠁⠒⡐⠂⢋⠈⠆⠀⠀⠀⠇⡤⢉⢂⡘⣊⠤⠀⡄⠀⠀⠀⠃⠎⢠⢃⡘⡄⢱⠘⠀⠀⠀⢠⢁⠄⡉⢀⠈⢁⢤⢡
⠀⠀⢃⠰⡜⢁⢅⠙⡤⢁⠃⠀⠀⠀⡆⠁⡡⠄⢡⢌⠉⠀⠁⠀⠀⠀⡄⢋⡡⠤⠤⢌⠘⠰⠀⠀⠀⠘⡄⢂⠱⠈⠠⢁⠆⡘
⠀⠀⠀⠂⠄⣑⢒⣁⠄⠃⠀⠀⠀⠀⠈⠢⢍⡃⠈⡉⠔⠈⠀⠀⠀⠀⠈⠢⢉⡒⢒⡉⠔⠁⠀⠀⠀⠀⠈⠢⢑⣁⣐⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢿⣿⡗⠀⠀⠀⠀⢺⣿⡷⠀⠀⠀⢸⠉⠹⢧⣬⣤⡽⠉⢹⡆⠀⠀⠀⠀⠀⠀⣀⣄⣴⣾⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣷⣾⣦⣀⣀
⠀⠀⢀⣤⣶⣄⣴⠶⠶⢦⣤⢶⣤⡀⠀⠀⢸⠀⣶⣰⠂⣤⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡏⢀⣉⠉⣉⣉⡁⠈⡇⠀⠀⠀⠀⠀⢸⡇⢈⡉⠉⣉⣉⠁⢸⢇⡀
⠀⠀⣿⣿⣳⡟⠀⣾⣿⡆⠙⣿⢿⣷⠀⠀⢸⠀⣭⢉⢀⣤⠶⢦⡜⠃⠀⠀⠀⠀⠀⠀⡇⢈⣉⠀⡉⠉⠁⠀⡇⠀⠀⠀⠀⠀⢸⡇⢈⡁⠈⠉⠉⠁⢈⡾⠁
⠀⠀⠀⠀⢻⡄⢀⣬⣭⣀⠀⣿⠀⠀⠀⠀⢸⠀⣛⡋⢸⡁⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⡇⠈⡉⠀⡵⠚⠛⢦⡁⠀⠀⠀⠀⠀⢸⡇⠈⠃⠈⠉⠉⢡⣟⡀
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣾⣇⡀⠀⠀⠀⢸⠀⠝⠾⠀⠛⠶⠞⢧⡀⠀⠀⠀⠀⠀⠀⡇⠈⠋⠸⣇⠀⠀⢨⡇⠀⠀⠀⠀⠀⢸⡇⠘⠃⠈⠉⠁⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠙⠿⠂⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠎⠛⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠉⠓⠚⠋⠳⣦⡀⠀⠀⠀⠈⠳⠶⠶⠶⠶⠶⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⣠⣾⠟⢫⡭⣝⠛⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⣠⠙⢷⡀⠀⠀⠀⠀⠀⣀⣠⡶⠋⠙⢶⣄⣀
⠀⠀⠀⠈⣿⠋⢿⡏⢻⡝⠙⣿⠀⠀⠀⠀⢰⣿⠱⣶⣶⣶⣶⣶⡆⢻⣧⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣠⠋⠀⢸⡇⠀⠀⠀⠀⠀⡇⠨⠭⠭⠭⠉⣤⢽
⠀⠀⠀⠀⣿⠀⢾⡇⠘⣣⠶⠶⢤⡀⠀⠀⢸⣧⢰⣮⣍⠐⣩⣥⣶⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⢷⣜⣁⣶⣤⠟⠀⠀⠀⠀⠀⠀⡇⢐⣒⣒⣒⡂⣓⢸
⠀⠀⠀⢀⣿⣀⣿⣇⠀⡇⠀⠀⠈⡇⠀⠀⠈⢿⣮⡉⠉⠉⠉⠉⣡⣿⠃⠀⠀⠀⠀⠀⠀⣒⠋⢰⡎⠉⣿⠀⢦⡄⠀⠀⠀⠀⠀⡇⠠⠤⠄⢨⠴⠶⢾⡀
⠀⠀⠶⠿⠿⠿⠿⠿⠷⠙⠶⠶⠚⢷⣄⠀⠀⠀⠙⠻⠷⣶⠶⠿⠛⠻⣷⣄⠀⠀⠀⠀⢨⣿⠀⢸⡇⠀⣿⠀⠈⢿⡆⠀⠀⠀⠀⢧⣬⠉⠁⡏⠤⣷⠄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠰⠿⠶⠾⠷⠶⠿⠆⠀⠀⠁⠀⠀⠀⠀⠀⠙⠶⠶⠿⢤⣥⠴⠋

⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⢰⣶⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢟⣵⣮⡙⢿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠓⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠯⠭⣍⠢
⠀⠀⠀⢀⣠⣤⣤⣿⣤⣤⠀⠀⠀⠀⠀⠀⣴⡿⣋⣴⣿⣿⣿⣿⣶⣝⢿⣦⠀⠀⠀⠀⠀⡇⠠⣤⡼⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⠂⡰⠊⠙⡢
⠀⠀⢠⣾⣷⡄⣴⣿⡆⡴⠋⠙⢷⠀⠀⠀⠈⢰⣿⣿⣿⣟⡵⠒⠲⡝⠇⠁⠀⠀⠀⠀⠀⠘⢦⣈⣁⣠⣎⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠂⠀⢇⠲⠞⣡⠃
⠀⠀⠸⠿⠿⠇⠿⠿⠷⠹⢤⣠⢾⣄⠀⠀⠀⢸⣿⣿⣿⣜⢧⣀⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠘⠤⠤⠤⠤⠟⠉⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠉⠉⠉⠉⠀⠉⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⣠⣄⣤
⠀⠀⠀⢀⣴⠟⠛⠉⠙⠻⣧⣀⠀⠀⠀⠀⠀⣠⡞⠋⢁⣤⡀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⡞⠛⠀⠐⠒⠶⢸⠀⠀⠀⠀⠀⠀⢸⣟⠉⢻⡷⢠⣦⣾⣿⣶⣶
⠀⠀⠀⣽⠇⠀⠀⠀⢀⡴⠈⣿⡀⠀⠀⠀⢰⡟⠀⡴⣫⣴⣝⢷⡄⢹⡆⠀⠀⠀⠀⠀⣶⡇⠀⠀⠤⢀⠠⢸⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠋⢜⡛⠋⠀⢹⣿⣶
⠀⠀⠈⢿⡄⠀⠳⣴⠏⠀⠀⣾⠃⠀⠀⠀⠸⣇⠀⠰⡯⢿⢽⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⢠⣾⠟⠛⢮⠀⠀⠀⠀⠀⠀⢠⣶⣾⠇⡼⠋⠉⠳⡄⣾⣿⡇
⠀⠀⠀⠙⢷⣄⡀⢀⠀⣀⡼⠛⠀⠀⠀⠀⠀⠙⣦⡀⠛⠛⠛⢃⣴⣟⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⠃⠀⠀⢸⠂⠀⠀⠀⠀⠀⢾⣿⡁⠀⢧⣀⣀⣴⠃⠃
⠀⠀⠀⣰⣿⣽⠉⠛⠋⢩⣿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠹⣷⣄⠀⠀⠀⠀⣿⣛⣉⣈⣓⣦⡖⠻⣦⡀⠀⠀⠀⠀⠘⠿⢿⡿⠷⠈⠉⠘⢷⡄
⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣾⣷⣾⣶⡷⢾⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣾⡾⠿⠿⠿⠿⠾⣷⡶⠶⢶⡿⢷⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠂⠀⠀⠀⠀⠀⠀⢐⡄⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠴⠃⠀⠂⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⣓⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⢀⢴⠁⠀⠀⠀⢠
⠀⠴⠾⠄⠀⠀⠀⠀⠎⡤⠄⠀⣜⠀⠀⠀⠰⠄⠀⠀⠀⣌⢠⠤⠀⠠⡜⠀⠐⠚⠧⠠⡄⠀⠠⠒⢢⠄⠀⢠⡃⠀⠀⠀⠐⠆⡀⠀⠀⡂⡤⠄⠀⢤⠃
⣄⡠⣒⠴⠷⡄⠀⠀⢳⡏⠀⣧⠐⢦⣀⡤⡢⠼⢦⡀⠀⠀⢺⠃⢈⡆⠐⢤⣀⢔⡢⠾⢦⠃⠀⠐⣾⠁⢰⡄⠲⣄⡀⢖⠤⠷⣄⠀⠀⠐⡞⠁⣱⠀⠲
⠡⢍⠁⠀⣰⣡⡤⠰⠆⠀⢠⣈⠀⠀⠹⣉⠀⢀⣸⣥⠄⠰⠄⠀⣠⣁⠀⠈⢌⡉⠀⢀⣎⡤⠄⠶⠀⠀⣄⡁⠀⠈⢎⡁⠀⣀⣋⡤⠀⠦⠀⢀⣈
⠀⠁⡄⢀⣽⠀⢠⡀⢘⡅⠀⢀⡀⠀⠈⢁⡔⢀⣇⠀⢄⠀⢸⡀⠀⢀⠀⠀⠈⢡⠆⣨⡅⠀⣀⠀⣻⠀⠀⡀⠀⠀⠉⣠⠀⣸⡄⠠⡀⢀⣇⠀⠀⡀
⠀⠀⢛⠉⠉⢡⡀⣀⣋⣀⠀⣀⢯⠁⠀⠐⡏⠋⠉⣤⠀⣠⣃⡀⠀⡘⠏⠁⠀⠘⡏⠋⠁⣄⢀⣸⣁⡀⠀⡘⡍⠀⠀⢚⡹⠉⢡⡄⢀⣜⣀⠀⢀⠻⠉
⠀⡀⠲⠀⠀⠀⠐⠈⡱⠇⣀⡄⢙⡃⢀⡐⠢⠀⠀⠀⠂⢈⠹⠁⣠⠈⣛⠀⢀⠐⠂⠀⠀⠀⠂⢉⠾⢀⣠⠈⣛⠀⣀⠒⠀⠀⠀⠐⠊⡉⠟⢀⡀⢙⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣏⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡦⢤⣤⣀⡀⣀⡴⠛⠛⠋⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⡀⠀⢀⢀
⠀⠀⠀⠀⠀⣠⣾⣿⣈⣿⡷⣾⣿⣽⣿⣿⡧⠈⠀⠀⠀⠀⠀⠃⠀⠰⠶⠄⠠⠶⠖⠶⠺⠅⠶⢿⡿
⠀⠀⠀⠀⢰⣿⢿⣭⢋⡀⠐⠀⠧⢢⠙⣿⣿⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠚⢹
⠀⠀⢀⣠⣿⣿⠉⡗⢛⠄⠀⡀⠀⢈⡁⢸⣿⣶⣿⡇⠀⠀⠐⣾⠀⠐⣦⠛⠀⠀⠘⠃⠀⠀⠀⢀⣿
⠀⠠⣿⣿⣿⣟⣶⣧⡀⠄⠚⠡⢀⠀⣡⣾⣿⣿⣟⡇⠀⠀⠀⣿⠀⠀⣿⡇⢠⡀⠀⠀⣠⡄⠀⠸⢻
⠀⠀⡙⠛⢿⣿⢿⣧⡔⠤⣶⢆⣄⡧⠈⢿⣿⠹⠟⡇⠀⠀⡸⡇⢀⠀⣿⡇⠈⠉⠀⠀⠈⠁⠀⠀⠘
⠀⠀⠈⠀⠘⣿⣞⠙⣷⡿⠀⢀⠀⠁⠂⣸⣿⣆⣃⡇⠀⠀⠐⠷⡦⠶⠿⠷⠀⠀⠠⠦⠀⠀⠀⠈⠂
⠀⠀⠀⠀⡀⠹⣮⣗⣻⣧⣀⠐⢂⣇⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⣷⡀⠘⢿⣿⣿⣿⣿⣿⣯⣹⠏⢙⢛⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⢿⣟⠠⣀⠀⠉⠙⠛⠋⠉⢀⠀⢸⣋⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⣺⣿⠿⣿⡿⣦⢶⣶⡶⢾⣏⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢴
⠀⠀⠀⠀⢩⣥⣆⢿⣷⣴⣽⣟⢲⡄⢉⠀⢰⣾⣷⡆⠀⠀⡖⢂⣶⢰⠀⡆⡖⡶⣦⣖⠠⡆⠀⢀
⠀⠀⠀⠀⢰⣮⣉⣽⣿⣉⣹⣿⣈⣑⣾⠀⢈⣴⡛⡇⠀⠀⠳⡌⣽⢾⡠⡧⡧⢿⠏⣯⠸⠇⠀⠘⣧
⠀⠀⠀⠀⢸⡿⠿⣿⣷⠿⢻⣿⢿⠋⢠⣤⡀⣿⣿⡇⠀⠀⣀⣀⡀⣀⣀⢀⢀⣀⣀⡀⡀⢀⡀⢠⠹
⠀⠀⠀⠀⢸⡇⣿⢿⣿⣷⣿⣷⢾⡦⠀⣸⠳⢻⣏⡁⠀⠀⣟⣿⣿⡏⣿⢸⢸⣉⣈⡀⠉⠘⠃⠀⢰
⠀⠀⠀⠀⢸⣧⣽⣿⣿⣭⣻⠟⢁⣠⡞⠋⢰⡬⡿⠁⠀⠀⠟⠟⠛⠳⠟⠘⠚⠟⠛⠇⠛⠚⠂⢰⣿⡆
⠀⠀⠀⠀⢹⡟⡿⣿⣷⠟⠁⣴⢿⡿⢆⣀⣰⠟⠁⠀⠀⢠⢤⠄⣤⠤⡤⠠⣤⣤⣤⡤⠀⡤⠀⢸⠻⡇
⠀⠀⠀⠀⢱⢶⣿⣻⡟⠀⣼⠷⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠚⠘⠙⠘⠛⠊⠉⠉⠉⠁⠉⠉⠀⢸⣀⡇
⠀⠀⠀⠀⢸⣷⣶⡿⠇⢀⣅⣀⠁⠀⠀⠀⣲⠆⢐⢒⣖⡒⣲⠒⡒⢲⣶⣶⡖⢲⣂⠀⠀⠰⠆⠾⠟⠃
⠀⠀⠀⠀⢸⡟⣃⠀⠰⠾⠿⠋⣀⡀⠀⣀⣀⡈⡀⠀⠈⠁⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⡀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠸⠟⠻⠷⠖⠤⠶⠾⠿⠿⠦⠬⠿⠦⠥⠾⠶⠤⠬⠷⠿⠿⠷⠦⠼⠿⠿⠿⠿⠦⠼⠿⠿⠇

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

⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠀⠀⠘⠁⠀⠀⠀⠀⢠⠾⠃⠀⠀⠀⠀⠀⠀⠖⠓⠲⣜⢦⠀⠀⢀⡀⠀⠀⡠⠒⠈⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠈⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠘⢧⡀⠀⢹⠋⠀⢠⠀⡁⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠒⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⡂⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠄⠀⠂⠔⠀⠀⢠⠀⠀⠁⠊⠁⠀⠚⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢀⠁⡘⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⡀⠠⠤⠄⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢘⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠐⠀⠀⠉⠀⠈⠈⠂⠀⠀⠄⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢿⠃⠀⠰⠁⠀⠀⠀⠀⠀⡀⠐⠊⠀⠀⢀⣤⣆⠀⠀⠀⠀⠀⠀⠀⣀⠈⠑⠒⠋⠁
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠆⠀⠀⠀⠀⠀⠈⢂⠀⠓⠀⠀⢻⣿⠀⠀⠀⠀⠀⡤⠊⠠⠆⠀⠀⠅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢸⠀⠀⠀⠀⠈⠀⠀⠀⠣⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠓⠀⠀⠀⢸⠋⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠑⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠈⠒⠒⠁⠀⢀⡠⠤⠄⠰⢀⡤⣀⠀⠀⠀⠀⠀⠂⡼⠁⡀⠀⠀⠇⠈⡄⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⡀⠀⠈⠁⢀⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡟⠃⠀⠀⠀⠀⠐
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠀⠉⠳⠀⢠⠂⠀⣸⡟⠀⠀⣀⠀⠀⠀⡀⠀⠀⠁⠀⠠⡒
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢀⠄⠀⠇⠀⠀⡞⠀⠠⠏⠀⢀⠀⠈⠣⡀⠀⠀⠀⢀⠀⠰
⠀⠘⠲⠶⠖⠶⠀⠸⠉⠀⠀⠀⢠⠁⠀⠀⠀⠠⠴⠑⠋⠀⠀⢠⡇⠀⠀⢀⡀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠁⠀⠀⠠⠁⠀⠀⠀⠂⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠖⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠁⠀⢀
⠀⠀⡀⢀⠀⠀⣰⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣿⠆⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢠⣄⡀
⠀⠀⠀⡀⠀⠀⠀⠈⠹⠃⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⡄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠈
⠀⠀⠀⠁⠀⠈⠀⠠⠂⠀⠀⠀⠀⢀⣾⠏⠀⣀⣤⣉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⡐⠈⠑⠢⣅⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠙⠂⢤⡰⠃⠀⠀⠘⠀⠙⠳⠀⠀⠀⠀⠀⠀⢀⣾⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⢤⣄⣠⡤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⣧⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⢦⡀
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⣹⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⢀⣀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡶⠒⠓⠖⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠋
⣰⠋⣿⠀⠀⠀⠙⢶⡤⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢹⡇⡔⢀⡼⠁⠘⣆⣀⣀⡀⠀⠀⣸⠋⠉
⠹⣤⠏⠀⠀⠀⠀⢺⡀⠀⠀⠀⣀⣠⣄⣀⡤⠟⡇⠀⠀⠀⡾⠁⠀⠀⠀⠉⠉⠛⠒⠛⠁⠀⠀⠀⠀⡏⠳
⠀⠀⠀⠀⠀⣠⡶⠀⠙⠒⠒⠚⠉⠀⠀⠀⠀⠀⣿⢠⠃⣸⠃⠀⠀⢸⡶⣄⠀⠀⠀⢸⠷⢤⠀⠀⠀⠙⠞
⠀⠀⠀⠀⠀⣇⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⠀⠀⠸⣇⣸⠂⠀⠀⠘⢧⣤⠇
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠐⣼⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣻⣀⣸⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠚⠛⠛⣛⣛⡛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⣠⣾⠋⠉⠀⠈⠑⢄⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣆⣀⡤⠤⠤⠬⢦⣀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⠑⠒⣶⠒⢒⡾⠛⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠎⠁⠀⠀⠉⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡀⠀⠀⠀⠤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣄⠙⢦⣄⣀⣹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠘⡆⠈⠑⢦⣿⠛⠁⡽⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⣸⠁⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠹⣆⡾⠳⡏⠱⣴⠃⠀⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢃⣀⠀⠀⠀⠀⠀⢰⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⠀⠀⢸⡦⢤⣀⣹⡶⠤⣤⣀⡀⠀⠀⢸
⣿⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣀⠴⠚⠓⠦⢼⡉⣩⠟⠀⠀⠀⠉⠿⢶⠧⠾⣁⡀⠀⢸
⢹⡄⠀⠀⠀⣸⠇⠀⠀⠀⣀⣼⣟⣀⣀⣀⣠⣤⠖⠋⠁⠀⠀⠀⠬⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣼
⠈⣧⠀⠀⠀⣿⠀⠀⠐⢻⡿⠛⠋⠉⠉⠀⢰⡇⠀⢀⣀⣠⣤⣉⣶⠋⠀⠀⠀⠀⠀⠀⢠⡂⠤⣞⠁⠀⠀⢠⡏
⠀⠘⣆⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠘⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢘⠈⠉⢀⠊⠀⠀⢀⡞
⠀⠀⠹⣆⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣱⣤⣶⣿⣿⣿⣿⣿⣏⣀⣀⠀⠀⠀⠀⠀⢀⣵⠆⠐⠁⠀⠀⢀⡞⠁
⠀⠀⠀⠘⢷⡈⠳⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠟⠁⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿