Assert Emojis & Symbols

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠛⠛⢶⣄⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣞⣋⣀⣀⣀⣀⣀⣈⣛⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡀⢀⣀⡀⢀⡀⢀⣀⡀⣀⣀⠀⡀⠀⡀⣀⣀⡀⣀⢀⡀⣀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⢈⣓⣦⠀⡏⢀⣾⣷⠀⣿⠀⣿⣂⠀⡿⡾⡇⣿⣂⠀⡟⢾⡇⢸⡅⠀⠀⠀⠀⣿
⠀⠀⣀⣼⡇⠀⠀⠀⠀⠉⠁⠀⠁⠈⠀⠀⠁⠀⠀⠈⠉⠀⠁⠀⠁⠈⠉⠁⠁⠈⠀⠈⠀⠀⠀⠀⠀⣿⣄⡀
⣴⣞⠉⢸⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠈⢙⣶
⣿⠉⠛⠾⣧⣀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⣠⡿⠞⠋⢹
⣿⠀⠀⠀⠀⠉⠛⠶⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣿⡷⠞⠋⠁⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⡶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⢸
⣿⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣸
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⣀⣤⣶⣶⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣤⠈⠁⣠⣾⣿⣿⣿⡏⠉⠉⠙⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⣰⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⢿⣿⣿⣿⣿⡇⠀⠀⢈⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡟⠀⠀⠀⢰⣿⠿⠋⠙⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⣇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡉⠻⣦⣾⠋⢹⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣾⡟⢿⣷⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠟⠛⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢻⡟⣻⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠘⢸⠓⡆⢸⠁⣔⡂⠀⣰⢀⡄⣠⢠⠔⡦⢰⠀⡤⢰⠂⡖⠀⠀⠀⢀⣀⣀⣠⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈⠁⠉⠈⠈⠁⠀⠉⠈⠉⠁⠈⠈⠁⠸⠽⠃⠀⠈⠀⠀⠀⢴⠋⠀⠈⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⠞⢁⡤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠐⠣⠊⠉⠉⠛⠛⠉
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⢿⣿⣿⣿⡿⣦⡀
⠙⠛⠛⣿⣶⣶⣶⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡧⠿⢿⣧⣽⣿⣿⠀⠀⠀⠈⡆⠳⢄⣀⡀
⠀⠀⠀⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣀⣀⠻⣿⣿⣁⣀⣀⣀⣀⣨⣿⠏⠀⠀⠀⣚⣁⡀⢀⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣷⣮⣝⣛⣻⣿⣛⣛⣭⣀⠀⠀⠀⠀⠀⠀⠈⠆⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡄⠙⠿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣿⣿⣿⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣟⣋⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢀⣴⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣿⣿⣿⣿⣿⡟⠀⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⡿⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣿⣿⣿⣿⣋⠉⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⡿⠻⡇⣼⣿⣿⣿⠋⠀⠀⠸⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⢀⣤⣼⣿⣿⠟⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠻⣿⣿⣆⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⢀⣾⣿⡿⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⠀⣹⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⡁⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠟⠁⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⠀⠀⠙⠿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⠶⠛⢷⡉⣵⠟⠳⢦⣄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣠⣶⣶⣶⣶⣶⣶⣶⣦⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⣿⣿⣿⠟⢿⣿⣿⣿⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢸⣿⣿⣿⣶⣾⣿⣿⣿⠀⠺
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣌⡛⠛⠛⠛⠛⠛⠛⣋⣴⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⢻⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣴⠶⣦⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣾⠁⠀⠀⠀⣿⣀⣿⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠻⣯⡿⠋⠉⠉⠉⢹⡇⠀⢸⡇⠀⠀⣠⡴⢾⣉⡷⢦⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⠀⢸⡏⠀⠈⠛⠁⠀⢹⡇⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢷⡈⠛⠛⠛⠛⠛⠛⠛⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠟⢷⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠷⢦⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣀⣸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⡉⣩⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠁⠈⠻⠋⠀⢙⡇⠀⢸⡇⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣀⣀⣀⣀⣀⣨⡇⠀⢸⡇⠠⣤⠶⠞⠋⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⢸⡇⠀⣤⣤⣤⡴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⠿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠈⢻⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠋⠀⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣴⣾⣿⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣦⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠋⢀⡀⠀⠀⠀⢀⣿⠿⣿⡿⢿⣇⡀⠀⠀⠀⣀⠈⠛⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⣴⠾⠋⠙⢷⡄⣴⠟⠉⠛⢶⣤⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡈⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣿⣿⣿⣿⣀⣸⣿⣿⣿⣿⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢈⠃⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣙⣉⣉⣉⣉⣉⣉⣉⣙⣉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡟⠉⠉⠉⠉⢹⡟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠤⠤⠤⢤⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡰⠋⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣸⠠⡇⠀⣀⡀⠀⢠⡄⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠈⠻⢤⣷⡀⠛⢁⡀⣈⠃⢀⣴⣃
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠴⢦⠤⣄⣉⡓⠾⠷⠵⠖⠫⣄⠈⠳⣄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⢳⠀⢠⡞⠙⠚⠃⢀⡼
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡇⠈⠓⠤⠖⠚⡏
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡏⠀⢀⣀⠀⠀⣇
⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠀⠀⠀⠀⢸⠁⢀⡏⠈⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⠀⠀⢧⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢹⠀⠀⢸⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣠⠴⠋⠀⠀⠸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⠀⠀⣀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢀⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⠀⠾⠟⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠈⠻⣿⣿⣿⣿⡿⠃⢸⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠰⣶⡶⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣾⡟⠃⠀⠀⠉⠉⠉⠀⣴⡌⠻⢿⣿⡿⠿⠿⣿⣿⣿⣿⣿⠀⢴⣦⡀⠀⠀⢀⣤⣄⠘⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣤⣿⣷⣄⠀⠀⠈⠛⠿⠟⠀⠀⠉⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⢰⣶⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣠⣤⣴⣶⡿⠿⠛⠋⠉⢁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣶⡶⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠻⠟⠛⠉⢁⣀⣠⣤⣶⣾⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⡿⠟⠉⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠘⠿⠛⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠀⣠⣴⡄⠀⠀⠀⣿⡇
⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⣠⣴⡿⠟⠉⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⣠⣶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣧⣤⣀⣀⡀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⢹⣿⠉⠛⠛⢿⣿⠿⣶⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⠛⢛⣿⡿⠛⢻⡿
⠀⠀⠀⢸⣿⠀⠀⠀⣼⡿⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⣴⣿⠏⠀⢠⣿⠃
⠀⠀⢠⣿⠇⠀⠀⣰⣿⠁⡀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢀⣾⡟⠁⠀⠀⣾⡟
⠀⠀⣾⡟⠀⠀⢰⣿⠃⢠⣿⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣸⣿⠀⣴⡀⠀⢹⣿⡀
⠀⣸⣿⠁⠀⢠⣿⠏⠀⠈⣿⡇⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⢠⣿⠇⠀⢿⣧⠀⠀⢿⣧
⢠⣿⠃⠀⢀⣾⡟⠀⠀⠀⢹⣷⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣾⡟⠀⠀⠈⢿⣧⠀⠈⣿⣆
⣿⡏⠀⠀⣼⡟⠀⠀⠀⠀⠘⣿⡆⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢰⣿⠁⠀⠀⠀⠈⢿⣇⠀⠸⣿
⠙⠿⣶⣾⡿⠁⠀⠀⠀⠀⠀⢻⣷⣤⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣿⠏⠀⠀⠀⠀⠀⠘⢿⣦⣴⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢀⣠⣤⣤⣤⣾⣿⠀⠶⠀⠀⠀⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠛⠿⠿⢋⢺⣥⣿⣿⣿⣿⣿⠂⣛⠃⠀⢀⠹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢀⣸⣧⡼⢷⡄⠈⠙⠿⠀⠉⠀⣴⠋⠀⠀⠉⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠰⠖⠛⣋⣩⣤⠶⠾⠃⠀⠀⠀⠀⠀⠀⣿⣠⠶⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⣁⣴⠖⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠸⣧⡴⣿⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⣀⣀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⢹⡟⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⢉⣽⠋⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⡿⢀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢠⡟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢀⡾⠁⢹⡄⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣾⠁⢷⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣼⠃⠀⠘⣧⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣼⠃⢰⡇⠀⠈⢷⡀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠃⠀⠀⠀⠻⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠻⠶⠟⠀⠀⠀⠘⠷⠾⠃

⠀⠀⣀⣤⣀⠀⢀⣠⣄⡀
⠀⢸⣿⣛⣿⡇⣿⣿⣿⣻⠀⠀⢀⢴⣇⣤⣤⣤⣠⣤⠀⢺⢯⣄⣤⣶⣤⣤⣤⡄⢠⣤⣤⣤⠒⣿⢡⣤⣤⣤⣤⣠⣤⣤
⠀⠈⠛⠿⠛⠁⠙⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⣠⣤⡈
⠀⠀⢠⣦⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⣴⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃
⠀⠀⣨⣯⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣠⣾⣷⡄⠁⠀⠀⠀⠀⢰⣷⠄⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⢾⡷⠀⠀⠀⣿⠆
⠀⠀⠙⣿⣗⣛⠀⠀⠀⠀⠀⠀⠀⡴⠋⣿⣿⠙⣆⠀⠀⠀⢸⣿⣷⣶⡀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⣾⡷⣄⡤⠄⠀⠀⣿⡆
⠀⠀⠀⢹⡏⣿⠀⣀⣸⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣾⣿⡏⠉⠁⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⢿⣷⡀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⣼⠇⣽⣼⣿⣿⣿⡇⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠸⣿
⠀⠀⠀⣾⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠠⠀⠤⠄⠠⠠⠀⠄⠀⠀⠄⠠⠀⠄⠄⠀⠀⠀⠀⠤⠀⠠⠠⠀⠀⠀⠀⠄⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡀⣠⡌⠂⠀⠀⠀⠰⣶⡄⠀⠀⠀⠀⣇⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢣⢁⡠⢣⡴⡐⢂⢷⣶⣦⡎⡄
⠀⠀⠚⢛⣶⣷⡇⠀⠰⢴⢖⣿⣷⠀⠀⠀⠀⠀⠉⢻⣿⡿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣯⣿⣼⣴⣿⣿⣿⣿⣧⣿⡇
⠀⠀⠀⠨⣿⣿⣧⡄⠀⣬⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠉⠃⠀⣀⡤⢀⠀⠀⠀⠀⠈⣿⡿⣿⢿⣿⣿⣿⢻⣿⣟⣿⠁
⠀⠀⠀⢀⣿⢩⡟⠁⠀⠻⢤⡿⠁⠀⠀⠀⠀⠀⠀⣸⠏⢹⡇⠀⢀⣤⣍⣿⣿⡇⠐⠀⠀⠀⢻⡇⠉⠀⣿⡏⠛⠘⣿⠃⠉
⠀⠀⠠⠿⠁⠀⠀⠀⠀⠠⠿⠁⠀⠀⠀⠀⠀⠀⠾⠋⠀⠘⠷⠰⠟⠿⠿⠟⠋⠷⠤⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⣸⣿⠀⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣜⣿⣡⡷⠄⠀⠀⠀⠀⠀⠀⣠⣤⠉⠉⠉⢡⣦⡀⠀⠀⠀⠀⢀⣿⣧
⠀⠀⢠⣿⣿⠛⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣴⣾⣧⡀⣀⠀⣴⣿⣤⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠸⣿⣧⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇⠀⠀⠀⠀⠀⠀⠘⣾⣿⡏⠛⠉⢸⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿
⠀⠀⢠⡿⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⣷⠀⠀⠀⠠⠠⠄⠀⣿⣿⡇⠀⠀⠈⣿⣿⡏⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠘⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠉⠉⠈⠉⠁⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠈⢻⡆⠀⠀⠀⠀⠀⣀⣤⡄⢀⡀⠀⠀⠀⣠⡾⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⣀⣸⠇⣶⢲⡶⠛⠛⠉⢹⡅⢙⣁⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠉⠉⠁⣤⠛⢾⡟⠶⠶⣤⣼⡅⢴⡄⠀⢠⣤⠙⠷⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣀⣤⡾⠷⠟⢻⡆⠀⠀⠉⠁⠀⠀⢠⡟⠀⢀⣠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠐⠛⠋⣉⣤⣴⠶⠚⠛⠁⠀⠀⠀⠀⠀⠀⢸⣧⡶⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⢁⣤⡾⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⢯⣤⢾⡟⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢙⡿⠻⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⢉⣽⠟⢩⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣼⠃⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣸⠏⢸⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⠁⣦⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣰⡟⠀⠀⣿⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⣸⠇⠀⠹⣇⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡟⠀⠀⠀⢹⣆⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡿⠀⠀⠀⢹⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉




⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣝⢿⣿⢮⠻⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢮⡑⠌⠳⡀⠙⡄⢻⣿⠸⠟⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠯⣍⡙⠻⢮⡂⠀⠀⠀⠀⠀⢻⠀⠀⢿⡟⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣟⣛⡛⠛⠒⠢⠄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⢫⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⣫⣭⣭⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢣⠞⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣛⡫⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡄⣤⢄⠀⠀⠀⠀⠁⣠⡾⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⣀⣤⠔⠂⠀⠀⠀⠀⠀⢸⡏⡟⠻⠜⠧⠘⠋⠀⠀⠀⠀⠐⣉⣴⠿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣾⠟⣉⠀⠀⠀⠀⠀⠀⡄⢠⣶⣶⡇⢸⡇⠀⠀⠀⠀⠀⠠⠔⠋⠉⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡾⠋⢀⠀⠀⠀⠀⠀⠑⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣪⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⡴⢣⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣛⣛⣯⣭
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣣⡧⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠐⠢⠤⣤⣤⣬⣽⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣼⣷⠀⠀⣧⠀⠀⠀⠀⠀⠨⡳⣦⣬⣙⣲⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇⣿⣿⣴⡄⣿⣧⠘⣄⠈⢦⡐⢌⡳⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣼⣦⡳⣿⣷⣝⣾⣿⣿⣿⣿⣿⣿⣿⣿

⢿⣿⣷⣦⣀⠉⠪⡳⣤⡈⠳⣕⢄⠙⢿⣧⡀⠳⡱⡱⠱⢱⡱⢻⡀⠀⠇⡇⠘⡇⠀⢸⠀⠸⠈⢀⡟⢌⣎⠎⢎⢎⠞⢀⣼⡽⡿⡫⣪⠞⢁⣴⢞⠕⠋⣀⣴⣾⡿
⣄⡀⠉⠙⠿⣗⣤⡀⠙⠝⢦⡈⠢⡑⢀⠙⣟⣄⠑⠄⠀⠁⠳⠈⢇⠀⢠⢻⠀⡇⠀⠘⠀⡆⠀⣸⠁⠜⠈⠀⠠⠊⡠⣻⠫⠊⠈⠜⢁⡴⠫⠃⢁⣤⣺⠿⠋⣁⣴
⢈⠙⠳⢤⣀⠈⠙⠪⣖⠀⠀⠙⠲⣄⠀⠀⠈⠻⡢⡈⠀⠀⠀⠁⠘⡀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠃⠘⠀⠀⠀⢁⢔⠝⠁⠀⠀⣠⠴⠋⠀⠠⣲⠕⠋⠁⣀⣬⣾⡿
⢄⡀⠀⠀⠈⠛⠦⣄⡀⠉⠂⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠐⠈⠀⡠⠔⠪⠛⠋⠁⢀
⣦⣌⣉⠂⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣈⣥
⠀⠀⠉⠙⠓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠚⠉⣁⣠
⠷⠶⢶⣤⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⣚⣋⡭⠥
⣉⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠈⠁⠀⠀⢀⣀
⠓⠶⠶⠭⠥⠄⠂⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠉⠁
⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤
⠶⠶⢶⣒⡒⠬⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⣀
⠉⠉⠀⠀⠀⢀⡀⠀⣀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉
⣶⣖⣾⣿⡯⠷⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶
⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢄⡀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠋⠁⠀⠀⢀⡤⠖⠋⠀⡀⠄⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣦⡀⠀⠀⠢⢄⠀⠀⠂⢄⡀⢀
⢁⣠⣴⡚⠉⢀⣠⢖⠏⠂⠀⣠⠖⠋⠀⠀⢀⣴⠕⠀⠀⠀⠀⡄⢠⠁⠀⢸⠀⠀⠀⠀⡃⠀⠈⡄⢁⠀⠀⠀⠀⠨⣦⡀⠀⠈⠻⣶⣄⠀⠀⠹⡲⣄⡀⠈⢓⠯⣷
⡿⠟⢉⣠⣶⡯⠛⢁⢠⣪⠞⢁⠔⡡⠀⣰⡻⠫⡢⠂⠀⠀⡼⢀⡏⠀⠐⣼⡄⡇⠀⢠⣇⠂⠀⢹⡀⢣⠀⠀⠀⢔⠝⣽⣕⢄⠈⢮⡫⣷⣕⡄⡈⠛⢽⣶⣤⡉⠻
⣤⣾⡿⠟⠉⣠⢔⡵⠛⢁⡴⡻⠊⣠⣾⡟⡕⡵⡱⠀⠀⡼⡱⣼⠁⡀⡆⡇⣿⡇⠀⢸⢸⢰⢀⠈⣧⢎⢧⣆⢆⠈⢮⢪⢻⣷⣷⣄⠙⢮⡻⣟⢮⡢⣄⠉⠻⢿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢰⣮⡥⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢖⡎⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⣳⡚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡮⠭⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠭⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣭⣭⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣀⠩⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠔⣛⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠬⠽⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣎⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠭⣟⡉

⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣭⣿⣷⣵⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣆⠀⠀⠶⢿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠉⠁⠀⠀⠀⠀⠈⠙⠃⠀⠀⠈⠉⠙⠻⠿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠇
⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⢠⣤⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢛⣿⣿⣿⣿⣶⣶⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⣀⣰⣦⣄⡀⠀⠀⠀⣠⣴⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⢿⣿⣿⠿⠛⠀⠀⠘⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⣼⣿⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⢠⣿⡿⠿⠛⠛⠿⢃⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠈⣿⠿⠟⢋⣉⣤⣤⣶⣾⣿⣷⡀⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⣿⣿⣿⣿⠇⠹⣿⣿⣿⣿⣿⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣈⣉⣙⢋⣠⣄⡙⠛⠻⠿⠟⠀⠹⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣦⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣧⠀⣉⣉⣉⣙⣛⠁⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⠛⠛⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡛⣛⣛⡛⢛⣻⣿⣿⠿⠿⢿⡿⠿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣟⡻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⢿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣦⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣸⡿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⡸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠗⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⡀⢀⣠⣴⣶⣿⣿⣿⡿⣿⣷⣶⣤⣙⠿⢋⣴⣶⡝⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢋⣴⣿⠿⣋⣭⣶⣶⡆⣶⣶⣦⣍⡛⢿⣷⣌⠻⣫⣾⣿⡟⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡙⢿⣷⡙⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣼⣿⣿⣿⣿⣿⣿⡏⣿⣿⡿⣫⣾⣿⣿⡎⢿⣷⡸⣿⣷⡀⠀⠀⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⡇⠿⣫⣾⣿⣿⣿⣿⣿⠸⣿⡇⣿⣿⣿⣷⣶⣶⣿⣿⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢨⣭⣽⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣯⣭⠀⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⡟⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣌⠻⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⢛⣴⣿⠟⣴⣿⣿⣿⣿⣿⣿⠿⣫⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣭⣛⣛⣃⣛⣛⣫⣥⣶⣿⠟⣣⣾⣿⡿⠿⠿⣛⣫⣵⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠛⠉⠀⠘⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣀⠀⢀⣾⠟⠻⣷⡀⠀⣀⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⠋⠀⠀⠙⣿⡿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⣼⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣧⠀⠀⢀⣴⡦
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡿⠀⠀⠛⠋
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣈⡁⠀⢈⣉⠀⠀⣉⡀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⢀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠛⠃⠀⠘⠛⠀⠀⠛⠃⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣶⣤⣤⣶⣾⣿⣁⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⠿⠟⠛⠻⣿⣄⣉⣉⣠⣿⠟⠛⠻⠿⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢰⣿⣅⡀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠀⠀⠈⣿⡆
⠀⠀⠀⢀⣾⡟⠛⠛⣿⣆⠀⣿⡇⠀⠀⠀⣴⡿⠛⠛⢿⣦⠀⠀⠀⠀⠀⠀⣰⡿⠋⠙⢻⣿⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⡀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠇
⠀⠀⣀⣤⣿⣷⣶⣶⣿⣧⣤⣶⠿⠿⣶⣤⣽⣿⣶⣶⣿⣿⣤⣶⠿⠿⣷⣤⣼⣿⣶⣶⣿⣿⣤⣴⡿⠿⣷⣄
⣴⡿⠛⠉⠁⠀⠀⠀⠀⠉⣿⡇⠀⠀⢸⣿⠁⠀⠀⠀⠀⠈⣿⡏⠀⠀⠘⣿⡍⠀⠀⠀⠀⠈⣿⡏⠀⠀⠈⣿⡆
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣄⣠⣾⣟⠀⠀⠀⠀⠀⠀⣻⣷⣄⣠⣼⣿⠀⠀⠀⠀⠀⠀⢹⣷⣄⣀⣴⣿⡁
⣿⡇⠀⠀⠀⠀⠀⣠⣶⠿⠛⠋⠙⠋⠙⠛⠿⣶⣤⣠⣶⠿⠟⠋⠙⠋⠉⠛⠿⣷⣤⣠⣶⡿⠟⠋⠙⠛⠉⠛⠿⣷⣦
⠿⠃⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣶⣶⠶⠶⠶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⣤⡄⣀⣀⠀⣤⠀⣠⡄⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠻⠆⣯⡅⢿⡼⠃⣿⠸⢧⡄⡾⠿⠼⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⣀⢀⣠⡀⣀⣀⢀⡀⣀⢀⣠⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠇⡗⢯⠸⣔⡶⢸⣏⠸⠟⠿⠸⣄⠄⢹⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⡤⡤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣰⠒⣆⢸⢰⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠹⠤⢸⠯⠭⠭⠭⠭⢿⠭⡿⢿⢸⠉⠉⢹⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁⠀⠀⠸⣶⡦⣏⣳⠤⠬⠥⠯⠼⠤⣎⣳⡺⠁⠀⠈⠓⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⢬⢧⡤⣤⣤⣤⣤⡤⡬⣄⠀⠀⠀⠀⣀⣤⣬⣀⠀⠀⢀⣠⣼⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⠀⠀⢀⢞⠕⠋⠉⠲⣕⢖⡵⠊⠉⠓⣝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢀⡤⣄⠀⠭⣒⢤⠀⢸⢸⠀⠀⢸⢼⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⣎⠀⣼⠁⠙⡎⣇⢧⢸⢸⠀⠀⠘⣾⢶⣦⠴⠭⠭⠭⠭⠷⢤⣶⢮⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈⢦⡈⠳⣄⠤⣄⠀⢸⢸⠀⠀⠀⡇⠀⣀⣤⣀⠀⠶⠲⠶⠶⠶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠙⠦⣄⡠⠞⠀⢸⢸⠀⠀⠀⡇⢸⣏⡀⣹⡇⠀⠤⠿⠤⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⢸⠙⢦⡤⡇⠀⠉⠛⠉⠀⠀⠩⠥⠍⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠀⠀⠀⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠀⠀⠀⠀⠄⡌⡸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠀⢀⣤⣤⡄⠀⢀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠃⠀⡼⠛⠿⠓⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⡶⢾⡀⠀⢰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠠⠀⠀⢸⠇⠘⡇⠀⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⢀⠄⠀⠎⠀⠀⠃⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢠⠤⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠸⠇⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠒⠊⠉⠁⠀⣠⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢐⡂⠤⠐⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⠛⠁⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣋⠔⠀⣠⠔⠊⠁⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣠⣶⠗⠋⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡿⢛⡡⢔⡠⢔⠮⠐⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣟⣉⣔⡩⠦⠥⠬⠤⠤⠥⠤⠤⠤⠤⠤⠤⠀⠀⢡⣀⣀


⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣽⣯⣿⢷⡄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⢀⣀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣿⣿⡏⢘⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣿⣿⣧⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⡿⢿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣾⣿⣿⣿⡆⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠾⡿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠙⠿⡿⠟⠁⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠒⡗⠂⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣼⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠀⠀⠰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣉⣀⣈⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣿⣿⣿⣿⣦⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⣿⣿⣿⣿⣿⣿⣿⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⡿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠹⣿⡿⢀⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣻⣿⣿⣿⣿⠟⠋⠉⠉⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣤⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⣿⡀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠈⠻⠶⣦⡶⠾⠋⠀⠀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⣠⡶⠟⠛⢿⣶⠶⠶⣾⠟⠛⠶⣤⡀⠀⠀⠀⠈⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠻⣦⣾⠃⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣠⡾⠛⠋⠛⠻⣦⡀⠀⠀⠀⢠⡿⠀⢀⣾⠀⠀⠀⠙⠁⠀⠀⢸⡆⠀⠸⣧⠀⠀⠀⠀⣠⡾⠛⠋⠛⠻⣦⡀
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢸⡇⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠈⠻⠶⣶⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢶⡶⠾⠋
⠀⠀⣠⡶⠟⠛⢿⡶⠶⠶⣾⠟⠛⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⢿⣶⠶⠶⣾⠟⠛⠷⣦⡀
⠀⣼⠏⠀⠀⠀⠀⢻⣦⣾⠋⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⣦⡀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⢻⣆⣼⠋⠀⠀⠀⠈⢿⡄
⢸⡏⠀⢀⣶⠀⠀⠀⠹⠃⠀⠀⢰⣆⠀⠈⣷⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢰⡏⠀⢀⣶⠀⠀⠀⠹⠃⠀⠀⢰⣆⠀⠈⣷
⣾⠃⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠃⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻
⠿⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠸⠇⠀⠀⠈⠻⣦⣀⣀⣀⣠⡾⠋⠀⠀⠀⠿⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣌⣉⣉⣉⣉⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠈⠻⣏⠉⢉⣿⠋⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠹⣧⡾⠁⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣼⠇⠀⠀⠀⠘⠁⠀⠀⠀⢿⡄⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣆⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡀⠀⠀⠘⣿⡟⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⣿⡇⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⠀⠈⠻⠃⠀⠀⠈⠀⠀⠀⠻⠋⠀⠀⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣷⡄⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀⠀⠀⣴⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⡟⠉⣠⣤⣴⣶⣶⣿⣿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⡄⠀⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣤⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢠⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣛⣉⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣉⣛⣛⣋⣙⣉⣙⣙⣉⣉⣛⣙⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠬⠀⠀⠀⣀⡀
⠀⠀⢠⣄⣾⣦⡀⣠⠤⡹⠻⣆
⠀⢀⣠⣼⣿⢿⠄⠂⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠖⠓⠘⠛⣿⢰⡄⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢈⣧⣿⣞⣋⠉⠈⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠐⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠾⢻⡙⣿⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⣿⡃⠉⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸
⠁⠀⠸⢳⣿⢳⡦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠚⢙⢹⣧⡄⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠒⢰⣾⣖
⠀⣀⣴⣶⣿⠻⠇⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⠯⠅⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢈⣽⡟⠙⠢
⠛⠁⢘⠹⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠰⢸⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠘⠀⠉⠨⣿⣿⣤⡀
⠀⠀⡝⢂⣿⠄⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⢴⣾⣟⠇⠀⠙⠆⠀⠀⡠⠖⡀⣠⣤⠀⠀⠀⠀⢀⣤⢤⣾⡶⠀⠈
⠀⠀⡄⠘⣿⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⢹⡏⠃⠀⠀⠀⠀⠀⠀⣀⡀⣹⡿⠲⢄⡀⠀⠁⠀⢘⣽⡿⠖⡤
⠀⠰⠈⢤⣿⠃⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡇⠃⠀⠀⠀⠀⠀⠘⠉⠉⢿⣿⣤⡀⠀⠀⠀⠀⢀⢸⡇⠂
⠀⠐⠀⢸⣿⡂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⡇⠤⠀⠀⠀⠀⠀⠀⡠⠀⢺⡏⠀⠈⠀⠀⠀⠀⡀⢼⡇
⠀⠁⠈⣺⢿⠉⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣸⡇⢈⡀⠀⠀⠀⠀⠀⠀⠀⢾⡟⡉⠂⠀⠀⠀⠀⠁⣸⡇⠰
⢸⣳⣄⢼⣾⡧⣀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⣭⣧⡀⠂⠀⠀⠀⠀⠀⠀⠀⢊⡇⠀⠀⠀⠀⠀⠠⠁⣼⡇⢀⡄
⣻⣀⣈⣻⣻⣃⣉⣹⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣏⣈⣿⣗⣉⣵⣀⣀⣀⣀⣀⣀⣀⣠⣇⣀⣀⣀⣀⣀⣸⣉⣻⣟⣉⣇
⣿⣿⠋⠛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣷⠀⠀⣀⠹⢦⣄⣀⣤⠞⢁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡄⠀⠒⠒⠒⠒⠒⠒⠒⠀⣿⠀⣾⠋⠁⠀⠈⠉⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠷⠟⠁⠀⣛⣛⣛⣛⣛⣛⣛⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⣤⡼⠃⠀⠶⠶⠶⠶⠶⠶⠦⠀⣿⠀⠻⣦⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠶⠶⠶⠶⠶⠶⠖⠀⣿⠀⠀⠻⡶⠶⠶⠶⠶⢶⡞
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡶⢶⡄⠀⣉⣉⣉⣉⣉⡉⠉⠀⣿⠀⠀⠀⡇⠀⢰⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⠾⠃⠀⣭⣭⣭⣭⣍⠁⠀⠀⣿⠀⠀⢰⡇⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠘⣧⣀⣸⣇⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣛⣟⢻⣛⣿⣷⣿⣻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⡿⣻⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⢡⣟⢻⢿⠟⡇⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣸⣷⣾⣾⣼⣧⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡟⡿⡟⣿⢛⢟⡿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣷⣵⣿⣾⣿⣧⣧⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣀⠀⠀⠀⢀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⣡⣤⠶⠿⠿⠿⣿⣶⣄⣀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠟⠋⠉⠉
⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠠⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢲⡶⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠎⠀⠘⠉⠀⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠃⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡇
⠀⠀⠀⠈⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣷
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣿⣿⣇
⠀⠀⠀⠀⢀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡇⣿⣿⣿⡀
⠀⠀⠀⠀⠘⡀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣽⣷⠾⠿⠛⠛⠁
⠀⠀⠀⠀⠀⢧⠀⠀⠱⣤⣤⣶⢶⣿⣿⣿⣽⠿⠿⠛⠛⠉⠉
⣀⣠⣴⣶⣤⣼⣟⣀⣴⣾⣷⣿⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣊⣤⣶⣾⣿⣿⣿⣿⣥⠈⠉⠒⠚⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⣫⣴⣿⣿⠿⣻⡿⠛⠋⠉⠉⠉⠛⠢⠀⠀⠀⠈⠐⣶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢥⣾⣿⡶⠛⠁⠞⠁⠄⣒⠒⠒⠂⣀⣀⢀⠀⠀⠀⠀⠀⠈⠃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⣾⣿⣟⠁⠀⠀⣠⡴⠟⣛⣻⣶⣜⢿⣿⢎⠳⡷⠀⠰⠶⣷⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡀⢸⣿⣿⠃⠀⣠⣿⠋⣰⠟⠉⠉⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠈⣿⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣇⠈⠐⣶⣆⣰⠃⢡⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢷⣿⣿⣿⠋⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠈⠻⠻⠷⣾⡷⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣷⣄⣷⣦⣾⣟⠁⠀⠀⠈⣀⣀⣀⣉⠙⠛⠃⠀⠀⠀⠴⣛⣉⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⣿⣿⡇⣙⢿⣷⣪⠂⠀⠀⠙⠻⣟⣿⠏⠳⠄⠀⠀⠀⡴⠻⡟⠉⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⡿⡟⠱⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣅⢻⣄⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⣿⣿⣷⣦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⡠⠞⠙⣿⣿⣿⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠶⠾⡦⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠩⠷⠶⠖⠀⡰⠁
⠀⠀⠀⠀⠀⠀⣸⣿⡄⢀⡴⠞⠀⢰⣯⣾⣿⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠘⣿⣏⡀⣨⣿⠁⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣻⣿⠃⢀⣨⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡿⢷⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⠟⡀⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠛⠋⡀⠀⠀⠁⠘⠢⠤⣀⠀⠈⠂⠀⠈⠻⠮⡳⣌⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠘⠻⠧⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠑⠀⠀⠈⠪⡓⡙⠛⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣾⣿⣿⣷⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⣶⠀⢰⣶⠀⠀⢀⣉⣉⣉⣉⣉⣉⣉⣉⡀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢈⣿⠀⠀⠘⠉⠋⠉⠉⠉⠉⠉⠙⠁⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠿⠀⠘⠿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⢶⠀⠠⡶⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡄⠛⠶⠶⠃⣠⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠹⣿⠀⣿⣿⣿⠏⢡⣤⣄⠹⣿⣿⣿⣿⣿⠏⣠⣤⣄⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⢀⣿⣿⠿⣿⣿⡀⢿⣿⡿⠀⣿⣿⠿⣿⣿⡀⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣤⣈⠉⢠⣴⣄⠉⣁⣤⣀⣤⣄⠉⣠⣴⣄⠉⣁⣤⣀⣤⣀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠻⣿⡿⠀⣿⣿⣿⣿⣿⡀⢻⣿⡿⢀⣿⣿⣿⣿⣿⡀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⢋⣁⣤⣤⣤⣄⡉⠛⠿⠟⢋⣁⣤⣤⣤⣄⡉⠻⠿⠟⢉⣡⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⣴⣿⣿⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧




⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⣀⣆⣈⣿⣿⣿⣿⣿⣖⡀⠀⠀⠀⠀⡠⠚⢹⣿⣿⣿⣿⣿⣥⣬⠤⡄⠀⠀⢀⣔⣀⣽⣍⡀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣯⡭⠍⠩⠈⠁
⠀⠀⢿⢿⣿⣿⣿⣿⡿⣿⠿⠿⠛⠀⠀⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⠷⠖⠲⠀⠀⠾⡿⣿⣿⣿⣿⣭⣁⠚⠁⠀⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠑⠊⠉⠉⠉⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋

⠀⠀⠀⠠⠀⢤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣿⠂
⠀⠀⠀⣀⣜⢂⣀⣏⣀⣹⡄⠀⠀⠀⠀⠒⢶⣶⣶⣶⣶⣶⣶⢰⣶⡒⠂⠀⠀⠀⠀⣀⣀⣾⣷⣤⡀⠈⣧⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣯⡝⢦⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠏⠀⠀⠒⠀⠾⠿⣿⣿⣿⣿⣿⣁⣠⣿⣷⣆⠀⠉⠉⢽⢿⡾⢿⣿⣿⣼⣁⣾⡷⢦
⠀⠀⠈⠛⠾⠋⠉⠉⠙⠷⠟⠁⠀⠀⠙⠼⠋⠉⠉⠉⠙⠧⠋⠈⠁⠀⠀⠀⠀⠀⠀⠿⠦⠟⠛⠉⠉⠉⠀⠧⠞⠀⠀⠀⠀⠁⠑⠚⠉⠉⠉⠉⠈⠓⠋

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣦⣤⣤⣤⣄⡀
⠀⠀⣐⣉⢹⣧⣿⠛⣷⠶⣤⣤⢀⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡂⠀⠠⠤⣤⣬⣭⣿⣿⣿⣿⠛⠛⣿⣷⠄
⠀⢰⣿⣿⣿⡹⣿⡛⢻⣦⣠⣷⠴⠛⢋⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⡅⢀⣀⢂⣰⣶⣿⣾⣿⣿⡟⠂⢀⣿⣄⡀⠐⠶⠾⠿⢿⣿⣿⣿⣶⣾⣿⡟
⠀⠈⠻⣉⡟⠃⠛⠛⠉⠁⠸⣉⡗⠀⠀⠠⠴⢾⣿⣟⣻⠛⠛⣟⣻⠃⠀⠀⠀⠀⢩⣿⣛⣿⠿⠿⠶⠟⢹⣛⡧⠀⠀⠀⠀⠀⠙⠿⡿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡀⢀⣄⡀⠀⠀⠀⡠⠀⢄
⠀⠀⣀⣀⣀⣠⡞⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡄⣶⠖⠀⠀⢠⣘⣿⣿⣿⣿⣿⣷⣿⣿⣿⠂⠀⣾⣶⣶⣶⣧⢀⣀⡀
⠐⣒⡿⢿⣿⣿⠀⠐⠆⠀⣿⣶⣶⠄⠀⠀⣀⣠⣴⣾⣿⣯⡤⣯⣤⠟⠁⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⢸⣉⣹⣄⣀
⠀⠀⠀⠀⠛⣹⣦⣄⣀⣴⣿⡿⠏⢀⡀⠬⢷⡾⣿⣿⣿⣿⣤⠟⠁⠀⠀⠒⠚⠛⠻⣿⣿⣿⡿⠿⣿⣿⣿⡿⠀⠀⣰⠿⣿⡿⢿⣿⣾⣿⣿⣿⢿⡿
⠀⠀⠀⠀⠀⠧⠊⠈⠉⠸⠤⠁⠀⠈⠀⠁⠐⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⠶⠛⠛⠋⠁⠀⠀⠘⠒⠉⠓⠋⠀⠀⠀⠀⠙⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣿⣵⣭⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣾⣿⣿⣽⣿⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣤⣴⣶⡟⠉⠹⣿⠀⢨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⢊⣉⣩⡍⠙⠛⠻⣆⠸⣿⣧⣼⣿⣗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢁⡒⠛⠛⠿⢿⣧⢠⣆⣿⡛⣯⡭⠏⠛⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡈⣽⣵⣁⡃⢀⣸⣿⣦⡿⣿⣿⣷⣶⣶⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣿⣿⣿⣿⢿⢺⡇⣿⣯⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⣸⣘⣿⣿⣿⣿⣿⣿⣿⣿⠻⠀⠉⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠈⢁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡟⡕⡻⢻⣿⣿⣿⣿⠙⢿⣍⠢⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣹⢏⠌⠀⠉⣟⢅⠙⣆⠻⣢⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡎⣀⡀⢀⣬⣺⣶⣌⣷⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⣶⣾⣿⣿⣿⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⡿⣿⢸⣿⡟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣷⣿⣽⡿⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠹⡇⣿⣿⢹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢿⣿⠀⢘⡇⠻⠋⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣾⡇⠄⢸⠇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⠇⠀⢸⡄⡄⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⢸⠀⠀⠈⡇⡇⠀⣿⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⣷⡇⠀⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠘⣿⠀⠘⣷⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢹⠀⠀⢸⡼⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣃⣀⣀⣀⣀⣀⣈⣀⣀⣀⣓⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣞⢑⠄⣀⠀⡎⢱⠀⣀⠠⡊⣳⠀⠀⠀⠀⣿⣿⣿⣿⣻⡯⣿⣿⣿⢹⡏⣿⣿⣿⢽⣟⣿⣿⣿
⠀⠀⠀⡤⠚⠓⠸⣄⠗⡺⢗⠺⣠⠇⠚⠓⢤⠀⠀⠀⣿⣿⣿⢿⣭⣬⣯⡿⣺⢇⡸⣗⢿⣼⣥⣯⡿⣿⣿
⠀⠀⠀⡇⡇⢰⡉⠉⢁⣳⣞⠈⣉⡉⣂⡸⢸⠀⠀⠀⣿⣿⣿⢿⣿⣯⣷⣷⡾⢜⡫⣷⠶⣾⡽⢿⡇⣿⣿
⠀⠀⠀⠈⡇⡸⡄⠀⢡⠀⢠⠊⢠⠔⢤⠈⢣⠀⠀⠀⣿⣿⣿⣷⣿⣟⢿⣿⣾⣿⡟⣵⡿⣫⡻⣷⣝⣿⣿
⠀⠀⠀⠀⠇⠇⡅⢠⢲⠀⢎⠀⢈⠓⢢⠀⢀⠇⠀⠀⣿⣿⣿⣿⣿⣼⣿⣿⣟⣿⣿⣿⣿⣯⣟⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠈⠦⣈⠳⠊⣠⠞⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢷⣯⣷⢿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠄⠀⡔⠂⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠸⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⣁⠂⠱⠄⠀⢵⠉⣡⡰⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⣤⣤⠀⠠
⠀⠀⠀⠀⠀⠙⠛⠢⠀⢀⠠⠀⠸⠌⠿⢁⠀⠀⡔⠐⠀⠑⠠⠀⠉⠐⠀⠀⢀⠀⢀⠀⢆⣂⠀⢠⣤⣤⡀⠈⢡⠈
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢸⣾⡷⡄⠈⠁⠀⠀⡀⠀⠀⢀⢀⠀⠀⠀⠘⠀⠉⠉⠁⠀⠈⢿⣄⡀⢘⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠃⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣏⠁⢺⠄
⠀⠀⠀⠀⠀⡑⠈⢆⠰⡇⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠤⠀⠀⠀⠈⢀⣉⣽⣧⣄
⠀⠀⠠⠐⠀⡈⠁⠀⠀⠙⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠄⠀⠀⣒⣦⡀⡤⡀⢂⠀⠙⠒⠂⠼⣿⣿⢿⡟⠂⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠓⠁⠀⠀⠙⠎⠀⠠⠄⠀⠀⠀⠀⣬⣤⡇⠀⠀⠀⠀⠁⠀⠈⠀⠀⡀⠀⠀⠀⠀⠆⠆⠀⣿⣿⠛⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⠂⠀⢠⠀⠀⣠⣿⡇
⠀⠀⠀⠐⠐⠄⠀⠠⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠀⠀⠀⠐⣿⣿⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡌⠀⣄⣠⣤⠐
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇⢠⡟⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⠟⠛⠀⢸⣏⡽⢞⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠉⠛⠀⢨⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⣤⣀⠀⣀⣀⣀⣄⡀⣀⣀⣀⢠⡀⣀⢀⡀⡀⡄⡀⣀⣀⣀⡀⢀⣀⣀⡄⠀⣀⣠⡄⠀⣀⣀⣠⢄⣠⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⢰⡦⠀⠀⣤⣴⣶⣦⠀⢠⣦⡀⠀⣴⣶⣦⡄⠀⣦⣾⣿⡄⣴⣶⣦⣄⠀⣴⣶⣄⠀⣴⡄⣴⡆⢠⣦⢠⣦⣦⣤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣯⣭⡅⠀⣾⣿⣇⠀⣿⣿⣿⣿⠀⣿⣭⣭⣿⣿⣏⣹⡿⣿⣿⣬⣉⠀⣿⣿⣿⡇⢸⣿⢸⣿⣉⣿⡇
⠀⠀⠀⠀⠀⢸⣧⣀⡀⣿⣏⣉⣁⢀⣿⣾⣿⠀⣿⣿⣼⡿⠀⣿⣉⣉⣿⣽⡟⢻⣷⣹⣿⣙⣿⠀⣿⣿⣿⡇⢸⣿⢸⣿⠛⠛
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠙⠛⠛⠋⠘⠋⠀⠛⠃⠛⠛⠛⠁⠀⠛⠛⠛⠃⠛⠃⠘⠛⠋⠛⠛⠋⠀⠛⠃⠛⠃⠘⠛⠈⠛







⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢋⠙⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠰⣿⣿⠆⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣾⠿⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠿⣷⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⡇⠀⢿⡶⠶⣶⠶⠶⠶⠶⠶⠶⣶⠶⢶⡿⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⡉⢹⡇⠀⢸⡇⠀⣿⠀⠀⢘⡃⠀⠀⣿⠀⢸⡇⠀⢸⡏⢉⡉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠃⢸⡇⠀⢸⡇⠀⣿⠀⢰⠋⠙⡇⠀⣿⠀⢸⡇⠀⢸⡇⢈⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠃⢸⡇⠀⢸⡇⠀⣿⠀⢸⠀⠀⡇⠀⣿⠀⢸⡇⠀⢸⡇⠘⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠀⠀⠀⢀⣠⡤⠖⠚⠛⠉⠉⠙⠛⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⣀⣠⡤⠤⡤⠤⣤⣀⡀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⡼⠁⠀⢀⡴⠛⠉⠀⠀⠀⡇⠀⠀⠈⠙⠳⣄⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⢠⠟⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⣧
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡞⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠋⠀⠀⠀⠹⡆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣤⠧⣴⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣅⢀⡽⠒⠒⠒⠒⠒⠒⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠸⡇⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⣠⠤⠶⠶⡶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⢳⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣰
⢠⠏⠁⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠋
⡏⠀⠀⡄⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠙⠲⢤⣀⣀⡀⠀⣀⣀⣠⠴⠚⠁⠀⣠⠞⠁
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠛⠦⣄⡀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣀⡤⠞⠁
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠉⠛⠒⠶⠶⠶⠶⠖⠚⠋⠁⠀⠀⠀⢠
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡴⠃⠀⠀⣠⣄⠀⣏⠀⠀⢹⠀⣠⣄⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⣰⠃⠈⠙⠃⠀⠀⠘⠋⠁⠘⣆⠀⠀⠘⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⣀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⣤⠖⠋⠀⠀⠀⢿⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⣶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⣶⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠘⣆⠀⠀⠘⣦⣠⠶⡇⠀⠀⣸⠶⣄⣴⠃⠀⠀⣰⠃⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠘⢦⠀⠀⠈⠀⠀⣧⣤⣤⣼⠀⠀⠁⠀⠀⣰⠃⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⠻⡄⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⢠
⠀⠙⠲⣤⣤⣤⣧⣄⣀⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣀⣄⣼⣤⣤⣤⠖⠋
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣄⡈⠛⠛⠁⢿⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⣽⣿⠿⠉⠛⢃⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣠⣾⣿⣿⣿⡶⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⡆⠀⠻⢿⣿⣿⢻⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣴⣧⣴⡿⠛⠁⠀⠀⣼⣿⠁⠀⣼⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⣄⠀⠀⠀⠀⠉⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣦⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠈⢛⣿⣿⣿⠩⣿⣿⡟⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠔⡻⢿⡿⢿⣿⣿⣿⣿⣷⠀⣧⠀⢸⣿⣿⣿⡇⣿⣿⡇⠀⢠⡇⢰⣿⣿⣿⣿⣿⠿⣿⠿⡓⠼⠿⠿⠛⠁
⠀⠀⠀⠀⣘⢧⢶⡀⠀⠀⠀⢶⣾⣿⣿⣍⠺⣿⣛⣿⡿⣻⣿⣶⣿⠀⢸⣿⣿⣿⣷⣿⣿⣷⠀⢸⣷⣾⣿⡻⣿⣝⣿⡷⢊⣽⣿⣿⣶⠆⠀⠀⢀⡶⡼⣃
⠀⠀⠀⠀⠹⣗⣪⢷⡄⠀⠀⠀⠈⢉⣥⣶⣿⣷⢌⣽⡒⢻⡼⢫⣟⠀⣸⣿⣿⣿⣿⣿⣿⣿⡄⢘⣯⠻⣽⠓⣺⣍⢼⣿⣷⣦⣍⠉⠀⠀⠀⢠⡾⣑⣲⠏
⠀⠀⠀⠀⠀⠘⢷⣉⠟⣦⡀⠀⠀⠀⠈⠉⠉⣴⣿⡿⢣⣿⡿⢹⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⠹⣿⣧⠻⣿⣷⡌⠉⠉⠀⠀⠀⢀⣴⠻⣉⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠛⣯⣜⡹⢷⣤⣀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠿⠋⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠈⠻⠇⠙⠛⠃⠀⠀⠀⠀⠀⣀⣤⡾⢏⣣⣽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣧⣎⡏⡛⢳⠶⠀⢀⣀⣀⡀⠀⣀⣀⣀⠀⣀⠀⣀⡀⢀⡀⣀⣀⣀⡀⣀⣀⣀⡀⠀⠀⡶⡞⢛⢫⣱⣼⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠳⠎⠀⢸⣿⣽⡿⢰⡟⠉⢻⡆⢿⣶⡿⣧⣿⠁⣿⡯⡭⠀⣿⣯⣽⡇⠀⠀⠻⠞⠋⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠀⠀⠈⠻⠶⠿⠃⠘⠿⠁⠻⠿⠀⠿⠶⠶⠆⠿⠇⢻⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠲⠦⠤⠤⠤⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⠂
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠡⠄⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠶⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠐⠙⠀⠁⠒⠈⠉⢮⡂⣂⣰
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⡌⢀⣾⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠰⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣒⣶⣶⣶⣒⣖⣲⣷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠋⠋⢉⡉⢉⣹⣧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠛⠛⠛⠛⠻⠟⠛⠻⡷⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣤⣦⣤⣄⠀⠀⠀⠀⠀⠀⠘⠛⠒⢲⣶⣾⣿⣷⣿⡿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠙⡇⢹⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⣿⢸⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣽⣿⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢿⠘⢹
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠉⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠀⡟⡆
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⡇⡅⢡
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⢀⣆⣀⣀⣀⢠⡀⣂⡀⠀⣄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡇⠫⠄⠀⠀⡄⠀⣤⠄⢞⠘⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣴⣿⡟⣿⣿⣿⣿⣿⣷⣄⣀⣀⡠⡼⠀⠀⠀⠀⠠⠃⠈⣯⡆⠈⡄⠡
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣻⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠘⠆⠱⡀
⠀⠀⠀⠀⠀⠀⢸⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡤⠠⣄⠤⢤⣤⣠⣤⣤⣤⣴⡆⣴⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⠛⠛⠛⠛⠓⠛⠟⢻⣿⣿⣿⡄⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠁
⢀⣀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⢀⡀
⠀⠈⠙⠲⢤⣀⠀⠀⠀⠠⣄⡉⠲⢄⠙⠢⣄
⢄⡀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠉⠲⢤⡉⠂⠈⠑⢄
⠤⣄⣁⢦⣤⣀⠀⠀⠈⠙⠳⠦⣄⠀⠈⠓⠄⢀⡀
⠀⠀⠀⠉⠓⠛⠙⠒⠦⢄⡀⢀⣈⠙⠻⠿⣶⣤⣉⠲⢄⡀⢀⡀
⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⢄⣀⡈⠉⠒⠢⢄⡉⠛⠷⢦⣈⠐⠉⠢⢄
⠤⠬⢍⣉⡙⠛⠓⠶⠦⠤⣤⣀⡉⠉⠒⠢⠤⠀⠀⠐⢤⣌⡑⠢⠑⠢⢁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣀
⠉⠉⠉⠉⠉⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⣒
⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠂⠐⠒⠒⠒⠩⠭⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⠀⠄
⠤⠤⠤⠶⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⢊⣉⡉⠤⠀⠈⠉⠀⢀⠤⠔⠂
⠔⠒⣒⣀⡀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⠽⠟⠒⠉
⠉⠉⠁⠀⠀⠀⠀⢀⡴⠖⠛⠉⠉⠁
⠀⣀⡤⠴⠒⣊⡩⠥⠒⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡠⠔⡀
⠀⣤⡶⠞⠋⠁⠀⠀⠀⠀⠀⣀⡤⠒⠉⢀⡠⠖⣩⠔⠋
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⠐⠊⠁
⠀⠀⢀⡠
⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠇⡟⢳⢼⠋⢸⡏⠁⡟⢻⠀⢸⠂⢸⡇⣿⣸⢸⡋⠱⣏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣙⡆⡟⠋⢸⡉⢸⡏⠁⣇⣸⠀⢸⡀⢸⡇⡏⣿⢸⡋⢠⣝⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠈⠉⠈⠉⠁⠉⠉⠀⠈⠉⠉⠁⠁⠈⠈⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⣀⣠⣤⣤⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣉⣉⠉⠉⠉⢁⣴⠾⠋⣉⣤⣤⣤⣤⣤⣉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⡀⠀⣰⡟⢁⣴⠿⠋⠉⠀⣿⠀⠉⠙⠻⣦⡈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⠉⠁⣸⡟⢠⡿⠃⠀⠀⠀⠀⣿⠀⢀⣴⠄⠈⢿⡄⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⣿⠁⣾⠃⠀⠀⠀⠀⣠⣿⣴⠟⠁⠀⠀⠘⣿⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⠀⣿⠀⣿⠀⠀⠀⠀⠀⢿⣤⡿⠛⠛⠛⠛⠛⣿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢿⡆⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⣿⠉⠉⢹⡏⠉⠉⠉⠉⠁⠀⠈⢿⣄⠹⣧⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⢠⡾⠃
⠀⠀⠀⠀⠀⠀⢰⣿⠀⣶⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢀⡀⠀⠈⠻⣦⣌⠙⠿⢶⣤⣤⣤⡶⠾⠋⣁⣴⠟⠁⣶⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠀⠀⣠⡾⠟⠁⢀⣀⣀⡈⠙⠻⢶⣦⣤⣤⣤⣴⡶⠟⠋⠁⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠀⣼⠟⠀⣴⣤⣿⡏⢹⣿⣤⣦⠀⠀⢀⠀⢀⡄⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⢸⡟⠀⣼⣏⡈⠛⠃⠘⠛⢁⣹⣧⠀⢻⡇⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⢸⡇⠀⣤⡿⠟⠃⠀⠀⠘⠻⢿⣤⠀⢸⡇⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠈⣿⡀⠘⢷⣶⣿⡇⢸⣿⣶⡾⠃⢀⣿⠁⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠀⠈⢿⣄⡈⠀⠻⠿⠿⠟⠀⢁⣠⡿⠃⠀⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠀⠀⠀⠙⠻⠶⣦⣤⣤⣴⠶⠟⠉⠀⠀⠀⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣄⠉⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠉⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⣀⣀⣀⣤⣴⣶⣶⣿⡿⠋⢻⡇⠀⢀⣤
⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣿⡇⠀⢾⣿
⠉⠛⠿⠿⣿⣿⣿⣿⣿⣷⣽⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢸⣟⣀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠻⠿⠿⢿⣿⣏⡁⠀⠈⢻⣷⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣾⣃⣔⣀⣼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⣰⣆⠀⠀⣀⣠⣆⠀⠀⠀⠀⠀⠀⠙⣤⣈⣀⡴⠃⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣤⣧⣤⣠⡴⠃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠐⡦
⠀⠀⠀⠀⣹⣿⠀⠐⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣾
⠀⠀⠀⠀⢹⣿⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢿⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⢹⣧⠺⠛⠒⣾⣿⣿⡷
⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢼⣿⠀⡟⢸⡇⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⡆⠁
⠀⠀⠀⠀⠾⠿⠀⠀⠀⠼⠇⠀⠀⠀⠀⠀⠀⠀⠸⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠙⠠⠃⠘⠇⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣤⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠐⠟
⠀⠀⠀⠀⢼⣿⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠟
⠀⠀⠀⠀⠘⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⡿⢿⡿⠃⠀⠀⠀⠀⠀⠈⣿⡏⠀⣿⢻⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠘⢻⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⢸⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⡟⠀⠀⠀⠀⠀⠀⢰⣿⣧⢸⣿⣹⡇⣸⠻⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠹⠃⠿⠁⠉⠀⠀⠀⠀⠀⠀⠸⠿⠟⠸⠿⠾⠇⠻⠛⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣬⠀⣀⠉⠀⣘⢁⣤⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⡄⠐⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣁⣰⠆
⠀⠀⠀⠞⢹⣿⣶⣿⣧⣸⣿⣯⡄⠀⠀⠀⠀⠀⠈⠿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢾⣻⣿⡿⡴⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣷⡟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣏⢻⣿⣅⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠰⡆⠦⠰⠆⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⢠⡞⠁⣸⠏⣹⡿⠋⠉⠙⣆⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⣿
⠀⠀⠀⠋⠀⢾⠻⠟⠉⠧⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⢼⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠠⠄⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⡌⠙⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣶⡀⠀⢰⣶⠶⠖⠂⠀⠀⠀⠀⠈⢳⣶⡞⠁⠀⠀⠀⠀⠀⠠⠄⠠⣿⣿⣿⡇⠀⠀⠙
⠀⠀⠀⠀⠀⢻⣶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⣶⣄⡀⠹⣿⣿⠓⠀⢀⣾⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣿⡇⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⠀⠀⠀⠀⠀⢺⣿⡟⠿⠶⡻⠟⠈⠾⢻⣿⣧⠁
⠀⠀⠀⠀⠀⢸⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡗⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢀⡿⠺⣇⠀⠀⠀⠀⠀⠘⠛⠛⣿⡿⡇⢹⢀⣿⠛⠛⠛
⠀⠀⠀⠀⠀⣸⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⣿⡇⠀⠰⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣼⣷⡄⠀⠀⠀⠀⠀⠀⣿⡇⠃⠘⢻⡿⠄
⠀⠀⠀⠀⠀⠁⠀⠈
⣿⢿⠿⠿⡿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣛⣿⣟⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠈⠒⠄⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠈⠂⢄⠀⠀⠈⠠⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠈⠂⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠑⠄⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠶⠒⠒⠈⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⡀⠀⠠⠤⠀⠒⠂⠈⠉⠉⠉⠀⠀⠀⠀⠀⡀⠤⠐⠊⡡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠔⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⡠⠖⠉⠀⡠⠂⠀⠀⢀⠞⠁⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢀⡤⠚⠁⠀⢀⠔⠋⠀⠀⢀⠔⠁⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣿⣷⣶⣾⣾⡿⣾⣶⣶⣾⣿⣾⣾⣶⣶⣶⣶⣿⣶⣾⣷⣶⣾⣶⣶⣿⣶⣷⣾⣷⣿⣶⣶⣶⣾⣿⣶⣾⣷⣶⣶⣶⣾⣷⡷⣷⡾⢾⣾⢷⣶⣷⠾⠶⡿⠶⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⣦⣤⣤⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢀⣀⢸⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠿⠿⢿⣿⡿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣠⣤
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣷⠀⠘⢿⣿⣿⣿⣿⠟⠀⢰⣷⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠈⠉⠉⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦


⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉⠉⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣤⣄⣀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣀⣸⣿⣇⣀⣸⣿⣀⣀⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⣇⣀⣸⣿⣀⣀⣿⣿⣀⣸⣿⡇
⠀⣿⣿⠿⢿⣿⡿⠿⣿⣿⡿⠿⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⡿⠿⣿⣿⡿⠿⣿⣿⠿⢿⣿⡇
⠀⣿⣿⣀⣸⣿⣇⣀⣼⣿⣄⣀⣿⣿⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⢸⣿⣇⣀⣼⣿⣄⣀⣿⣿⣀⣸⣿⡇
⠀⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠟⠀⢸⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⣿⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⡇⠀⢠⣤⣤⣤⣤⣤⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿
⣙⣉⣋⣋⣙⣁⣀⣸⣿⣿⣿⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣸⣿⣿⣿⣀⣀⣙⣉⣋⣉⣉
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⠿⠛⠛⠛⡛⢛⠟⠿⠿⢿⣿⣷⣦⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣽⠿⢛⣻⣯⣷⠾⠗⢛⡛⡛⠛⠛⠛⠛⠛⠷⠶⣦⣌⠉⠛⠷⣧⣗⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣫⣴⠞⢛⣉⣥⡶⠖⠛⠛⠉⠋⠀⠀⠀⣛⠒⠒⠶⢤⣌⡙⠻⢦⣤⣉⠻⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢆⣽⡿⢋⣴⠟⣋⣤⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠑⠲⠿⢶⣦⡈⠙⠷⣀⡙⠿⢇⣄
⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠟⣁⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠈⠻⣦⡈⢻⣧⡀
⠀⠀⠀⠀⣀⣾⠟⣰⡟⢁⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠈⢣⣄⠈⢿⣆
⠀⠀⢀⣾⡿⠋⣼⠏⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠙⢦⠀⢹⣦
⠀⠀⣼⣿⢃⣾⢋⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠘⣷⡀⢿⣆
⠀⠐⣿⠇⣸⡇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠘⣧⠈⣻⡆
⠀⣴⡟⢸⡟⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⡄⢿⡆⠈
⢰⣿⡇⣼⢃⡇⢠⣤⣤⢠⣤⣤⢀⣤⣤⣤⣤⡀⠐⠀⣀⣤⣤⣀⠀⢀⣀⣀⣀⠀⠀⣀⡀⢠⣤⡄⣤⣤⣤⣤⣬⡀⣤⠤⡄⢸⣇⠀⣶⠀⢦
⣶⠟⠀⣿⣸⠁⢸⣿⣿⢸⣿⣿⢸⣿⣿⠛⣿⣿⠀⠘⣿⡟⣿⣿⡇⢸⣿⣿⠿⠂⣤⣿⣧⢸⣿⡇⠛⢿⣿⣿⠛⠃⢸⣯⡅⠀⣿⠀⣿⠀⢺
⣿⡆⠸⣯⢹⠀⢸⣿⣿⢸⣿⣿⠀⣿⣿⣶⣿⡟⠀⣿⣿⡇⣛⣛⡃⢸⣿⣿⣤⠀⣿⣿⣿⣿⣿⣇⠀⢰⣾⣿⠀⠀⢸⣿⠀⠀⣿⠀⣿⡀⢸
⢻⠟⠀⣿⢸⠀⢸⣿⣿⢸⣿⣿⡄⣿⠿⡯⣿⢿⡆⣿⣿⡟⣿⣿⡇⢸⣿⣿⠉⠀⣿⣿⡿⣿⣿⡇⠀⢸⣿⣏⠀⠀⢸⣿⠀⠀⣿⠀⣿⠁⣸
⠈⢶⡆⣿⡘⡇⠘⢿⣿⣾⣿⡿⠁⣿⣷⡄⣿⣿⡀⢿⣿⣧⣿⣿⡇⢸⣿⣿⣶⡆⣿⣿⠇⢿⣿⡇⠀⢰⣿⣿⠀⠀⢰⣶⡆⠀⣿⢀⡿⠀⣿
⠀⠀⣄⢸⣧⢻⡀⠀⠉⠉⠉⠀⠀⠉⠈⠁⠉⠁⠀⠀⠉⠉⠁⠉⠀⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠁⠀⠈⠉⠉⠀⠀⠀⠉⠁⢸⡏⣸⠃⢰⣿
⠀⠀⠈⡄⢿⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡏⠀⣾⡏
⠀⠀⠀⣭⡈⣷⡌⠳⠈⠁⠠⠄⡥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⠁⣼⡿
⠀⠀⠀⠈⣷⡈⢻⡄⠀⠀⠘⠀⠀⠀⠐⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⡾⠃⣰⠿⠁
⠀⠀⠀⠀⠹⣷⣄⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣰⠟⠁⣼⠃
⠀⠀⠀⠀⠀⠙⣻⣧⡘⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⠊⣠⡾⠉⣰⣾
⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⡙⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⠀⠉⠚⢉⡤⠾⠋⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⠘⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⣀⣤⠾⠋⢁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣬⣙⠛⠿⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⠶⠞⠛⢋⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣷⣶⣤⣬⣍⣉⣉⣉⣉⣋⣉⣉⣥⣤⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠒⠛⠢⣄
⠀⠀⠀⠀⢀⣠⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⢀⡞⠉⢀⢰⣶⣠⡀⠀⠀⡰⢅⠀⠀⢀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣼⠀⠀⠀⠀⠀⢙⣿⣀⣤⣶⣀⠲⡠⠁⣠⣢⠤⠀⠀⠀⠀⠘⠳⢦⣄
⠀⠀⠘⣄⠀⠀⠐⠋⠋⠉⠉⣙⣻⣿⡷⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⠶⣄
⠀⠀⢀⡞⠱⠦⠤⠒⠊⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠝⠳⡄
⠀⡰⡯⠐⠲⣆⠀⠀⠀⠾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢰⡄⠀⢀⡃⠒⠀⠀⠀⣈⣴⡶⠶⠶⠾⢤⡄⣠⡤⣴⢂⠠⠲⠆⢀⠀⠀⠀⠀⠀⠀⠃⠠⡌⢳⡶⠾⠟⠒⠦⠴⢿⡓⣦⣄⣀
⠀⠙⣺⠛⠳⠶⢒⣋⣉⣉⣀⣀⢀⣤⣴⠿⣿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠘⣷⣆⢸⠀⠀⠀⠬⢷⠁⠍⠏⠉⠓⠲⠦⢤⣤⣤
⠀⢀⣇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⡔⠀⠘⣉⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣋⢸⠀⠀⠀⢀⣸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠻⣧⣸⣠⣄⣀⢀⣐⠀⡉⠠⣩⣴⡼⠟⢫⣤⡙⢿⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡧
⠀⠀⠀⠈⢹⡋⠛⠛⠛⠛⠒⠛⠛⠛⢩⠀⠀⣼⡯⠔⠉⠉⠓⠆⣤⣶⣤⣀⡀⠀⠄⢄⡀⢀⠀⠀⠀⡞⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢷⣦⣀⡀⡀⠀⠀⣄⣀⣷⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠰⠀⠀⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠁⠈⠙⠿⢼⣐⡜⣁⢀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣠⣀⠀⠀⢠⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠿⢧⣤⣥⣶⠾⠶⠛⠛⢶⣙⡀⡏⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⠀⠀⠀⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣄⣀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣻⣿⡟⡇⡀⡄⠀⠂⠀⠂⠀⡐⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣱⣹⡡⡐⣐⡀⡀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣵⣿⣿⣿⣟⡙⠐⢡⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠮⢼⣔⣐⣃⣩⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢫⠜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⡠⣴⣒⢖⡈⡀⠀⠄⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⢬⡭⠃⡉⢶⡇⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢟⡊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿
⣶⣾⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⡀⠀⠀⢸⣧⠀⠀⠀⣸⡏⠀⠀⢠⢾⠟⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣠⠴⠒⢳⣷⢤⣴⣼⠿⡄⠀⢰⢻⠃⠀⡴⢣⠏⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠷⠚⠁⠈⠉⠛⢿⣽⠿⠟⣟⢻⡸⢠⠞⢠⠃⠀⣠⣴⡿⠋
⠀⣠⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⣠⣴⣴⣀⠀⠘⣿⣀⣀⣀⣸⡷⠁⢠⣣⠔⢋⠝⠉
⠀⠘⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠘⢿⣟⢿⠀⠖⡟⠛⠹⠻⢿⡄⠀⠉⢀⠔⠁
⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣿⣿⡿⠁⠀⠀⠀⠀⣀⣾⣿⢮⣀⣀⣇⡀⢠⣄⣼⠇⠀⠀⠈⣉⣉⣩⠭⠽⠿⠛⠛⠋⠉⠉
⠀⣠⡴⠾⠛⠫⠭⠭⠭⣹⣿⠁⣿⡦⠀⠠⠤⣿⣿⣿⣟⣛⠈⠉⠉⠙⠻⢻⣏⠀⠀⠈⠉⠛⠲⣤⣄⣀⡀
⠀⠀⠀⠀⢀⠀⠐⡶⣶⣾⣿⣦⠊⠁⠈⢁⠞⢻⡿⣿⣿⠿⣧⣤⣠⣴⣄⣤⠟⠀⢀⡈⠻⢍⠉⠉⠉⠉⠉⠉
⠀⢲⣒⣋⣭⣽⣼⣴⣵⣷⣿⣿⡰⣄⠀⠀⠀⠀⠀⠹⣯⢀⡀⠉⠁⠉⠛⢣⠀⣦⠀⢿⠲⢄⡙⢦⡀
⠀⠙⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠷⣭⣓⣲⣤⠤⠤⣤⣈⡾⢭⣤⡶⣶⡣⢾⡀⡇⠳⣌⢧⠀⠉⠲⢌⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡬⠟⠉⠉⠉⣹⡿⠛⠛⠓⡞⡿⠁⠘⣧⡇⠀⠘⢿⣧⠀⠀⠀⠈⠚⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠀⢀⡾⠋⠀⠀⠀⢰⡿⠁⠀⠀⣿⠇⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⢧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠢⡁⠀⠀⠀⠀⣀⠀⠀⠀⠈⡉⠰⠄⢀⡰⡈⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⢦⢫⠻
⢀⡀⠄⠂⠀⠀⠀⠀⣀⠐⡄⠀⡅⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⢚⣛⡧⢄⠐⢀⠀⢀⠀⠙
⠈⢆⠀⠀⠀⠀⠀⠀⢘⡂⠱⡀⠘⠦⢤⣀⣀⡤⠤⠀⠀⠚⠀⠀⠀⠀⠀⠴⠂⠒⠤⠓⠊
⠀⠘⠀⠀⠀⠀⠀⡀⠤⢈⣂⣱⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⡀⢘⠧⡦⠚⠁⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠂⠀⠀⠀⠣⡡⣀⣠⣄⠀⣀⡤⢠⠎⠀⠀⠀⠀⠀⢀⡌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡦⢃⣀⠉⡩⠙⢧⠀⠀⠀⠀⠀⢠⠮⠀⠈⠠⠃
⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠃⡀⠀⢠⠀⠀⠀⠠⠄⠐⠀⠠
⠀⠀⠘⣼⡤⠤⠄⣀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡤⠤⣠⠁⠀⠀⠀⠄
⠀⠀⠀⠈⢷⠀⠀⠀⠈⠊⠑⠲⢤⠀⠀⠈⠢⡀⠀⠀⠘⣣⠀⣀⡤⠤⢄⠐⡤⠆⢀⡡
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⣃⠢⢤⢤⡜⠂⠀⠀⠁⠘⡟⢶⠓⠐⠊
⠀⠀⠀⠀⠀⠀⠓⠊⠁⠀⠀⠀⠀⠈⠁⢢⠁⡤⠀⠀⠀⠀⠀⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣈⠀⠀⠀⠐⠊⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢤⣄⠀⣀⣀⣀⣀⢈⣉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣦⠀⠀⣀⣀⡠⠘⠂⠀⠀⣴⣷⣿⣀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁⡀⢀⣸⡏⠀⠀⠀⣠⡖⠺⣿⣿⣿⣿⣿⢙⣿⣧⠀⠀⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡛⣣⣬⣿⡎⠿⢿⡆⠀⡴⠞⢻⡦⡞⣿⣿⣿⣅⣠⣿⣿⢤⢀⣾⠇⠀⠀⡴⠚⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢇⣾⣿⣿⣯⣭⢠⣼⣷⣤⣇⣿⣿⣿⣶⣸⣿⠛⠉⢉⣿⣿⡲⠾⢟⣃⣄⠀⠁⢀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣦⣿⣿⣿⣿⣿⣿⡿⣿⣏⠿⢿⣿⣿⣿⣿⠄⠀⣸⣿⣿⡃⢠⣾⣿⣿⢧⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡿⣷⣿⠋⡿⣡⣿⣾⣿⣿⣾⣿⣿⣈⣿⠛⠉⠛⡟⢿⡿⣿⣿⣧⣿⣿⣿⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢻⣿⡞⠒⡿⢛⣿⣿⣿⡿⠻⣿⢿⠇⣶⠀⣼⣿⡷⢾⣧⢿⣿⣧⠋⠙⡇⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣾⠈⡄⢸⡇⢿⡏⡆⢸⠛⠛⠻⣧⣀⢈⣿⣾⣿⠠⠿⢿⣧⠀⣿⣿⡘⠛⣾⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣟⣠⣼⡄⢿⣿⣷⢿⡇⠀⡄⠀⢻⣿⣿⣧⣿⡟⠀⣠⣾⣿⣇⣿⣿⡇⢠⣿⡇⣸⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⠀⢷⡀⢸⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⡇⢸⣟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢛⠻⠿⠿⣻⣿⣿⣿⠀⣿⣷⣿⣿⡿⠛⣿⠀⣾⠿⣿⢿⡯⠙⠿⢷⡘⣿⣽⣿⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠛⣿⣿⢿⣿⣿⣿⣿⡙⢶⣿⣦⡾⠆⠁⠀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠛⠋⠉⠈⠛⠉⠉⠙⠃⠀⠀⠀⠈⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡄⠀⠙⢿⣷⣦⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠙⠻⣾⠟⠻⡿⢿⣷⡈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣄⡀⢸⠗⠂⠀⣜⠛⢇⣀⠘⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢶⠀⣠⢟⣴⣿⠟⠁⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣵⣿⠋⠁⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠉⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠀⠀⠈⢿⠀⢠⡄⠀⠀⠀⠐⣆⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⠀⠀⢀⠁⣿⣿⡄⠀⠀⠀⣿⣇⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠘⣷⡜⢿⣿⡀⠀⠀⣿⣿⠀⠀⠙⠉⠉⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣭⣝⣈⢿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣿⣿⣿⡜⠇⠀⢰⣿⣿⠐⣿⣿⡇⣶⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠛⠛⠛⢸⣿⣿⣿⡇⣦⠀⠈⠁⠀⡄⣿⣿⣇⣿⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣿⡇⠀⠀⢸⡇⣿⣿⢸⡇⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⢀⣀⠀⠀⢸⣿⣿⣿⢸⡟⠀⠀⢀⣿⡇⣿⣿⣸⡇⠀⠀⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⢸⣿⣷⡦⢸⣿⣿⡇⡾⠁⠀⢀⣾⣿⢧⣿⡇⣿⢁⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⠃⣾⣿⣿⡇⠁⠀⣠⣾⣿⡏⣾⣿⣧⣿⠈⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⠀⢀⣼⣿⣿⠃⠀⢿⣿⣿⣴⣶⠆⣨⣭⣭⣼⣿⣿⢸⡟⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⡿⠁⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠸⣿⣿⡈⠛⠛⠿⠿⣿⣿⠼⠇⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠻⢿⣷⣶⣤⣝⡋⠀⠀⣀⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡄⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⣀⣤⣦⣤⣤⡀
⢰⣶⣶⣶⣶⣶⣶⣶⣿⣋⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠚⠿⠿⣿⣿⣿⣿⣆
⣾⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠛⠙⠛⠛⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⡀
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠙⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠀⠀⠙⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠙⠄
⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⣠⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠀⠙⢿⣿⣿⣿⣷⣦⡀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠻⣿⣿⣿⡿⠃⢹⣿⣿⣿
⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠛⠁⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣶⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠘⠃⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⢀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡅⠐⣿⣷⡀⠹⣿⣿⠏⠀⠀⠘⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡀⠘⣿⣷⣶⣶⣶⣶⣄⠀⢴⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀⠘⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⡀⠘⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡟⠃⠀⣿⣿⣿⣿⠛⠁⢰⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⢿⣿⣿⡿⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⠀⠀⢿⡿⠁⠀⠀⠀⠻⣿⠇

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣯⠊⢻⣿⣿⣿⠸⠀⡇⢹⢸⣿⣿⣿⠟⠀⢰⢿⣿⣿⠟⣴⡿⣫⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡙⢿⣷⡀⠹⣟⣿⠀⠁⠁⢸⢸⣿⣿⡏⠀⢠⠏⣾⡿⢣⡾⢋⣾⢟⣫⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣆⠙⢿⣄⠙⡞⣇⠀⠀⠀⢸⣿⣿⠀⠀⠈⢠⠋⡴⢋⣵⢟⣵⢟⢽⡯
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣮⣙⢷⣄⠙⢦⡈⠘⠄⠀⠀⠘⢻⠇⠀⠀⠀⠀⠊⡠⢋⠴⠋⠀⠐⠝⣋
⠀⠀⠀⠀⠀⠀⠀⢨⣁⠉⠛⠿⠿⣇⡈⠳⢄⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣬⠷
⠀⠀⠀⠀⠀⠀⠀⠠⣬⣍⣒⠀⠀⠀⠈⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢉⣁⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠨⡥⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⠠⣦⠀⠀⠀⠀⠀⠐⠒⠭⠽⢷⣞⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠶⠶⠶⠤⠀⠀⠀⠀⠈⠟⠋⠙⣈⠀⣡⠀⠀⠀⠀⠀⠀⠨⠭⠽⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠯⠉⠉⠀⠀⠀⠀⠀⠀⠀⢰⠸⠟⠟⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠭⣙⡓⠶⠿
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡙⢦⣈⢷⣦⣄⡀⠉⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠐⣋⢀⣠⡶⣀⠔⡀⢀⡴⠁⠀⠀⠀⡄⠀⠀⠀⠀⢀⠀⠹⣄⠙⢷⣽⣿⣿⣷⣦⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⢟⣭⢞⣵⣞⣵⠟⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠘⣎⣆⠘⢷⣄⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⡷⣛⣵⣿⣿⡿⠃⣴⠃⡼⠀⠀⡿⠀⣾⢠⠀⡄⠀⢹⣞⣧⠈⢻⣷⣌⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⡟⢁⣼⣟⣼⢁⡾⢸⡇⣿⣿⢸⠀⣷⢀⢸⣿⣿⣷⡀⡹⣿⣿⣿⣿⣿

⣿⣭⣭⣍⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⠿
⣿⣿⣿⣿⣿⣿⠷⠶⠶⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣶⣶⣶
⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⣭⣭⣭⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣽⣯⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿
⣯⣭⣭⣭⣭⣭⣍⣉⣉⣉⣉⣉⡉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠋⠛⠛⠛⠛⠉⠙⠉⠙⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐
⣟⡛⠛⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣒⣒⣲⣶⣾
⣿⣿⣿⣛⣛⣒⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣙⣻⣿⣿⣿
⣛⣛⣉⣉⣉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⠿⠿⠿⣿
⣿⣿⣿⣿⣿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶
⣿⣿⣿⣛⣛⣛⣛⣉⡉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣚⣛⣿⣿⣿⣿
⣷⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠛
⣿⣟⣋⣉⣉⣉⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣉⣩⣽⣿⣿
⣿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣩
⣿⣿⣿⡷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶
⣿⣿⣿⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿
⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿
⣿⣛⣛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠻⠿⠿⠿
⣿⣯⣭⣭⣭⣭⣭⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣬⣭
⣓⣒⣒⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠶⠾⠿
⣿⣿⣿⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣼
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣽⣿⣷⣤⡎⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣏⢹⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣴⣿⣿⣿⣿⠟⠛⣸⡿⠛⠙⠻⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠶⠶⢤⣀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⡿⠛⠁⣤⣶⡉⣶⣄⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡡⠀⠀⡰⠋⠀⠀⠀⣈⣭⠓⢒⣾⠏⢉⡿⠛⡋⠁⠀⢠⣾⣿⡇⠉⣦⣀⣱⣀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⡈⠀⠀⣰⡔⠀⠀⠀⠀⠀⠁⠐⠋⠁⠀⠜⠀⡞⠀⠀⠔⢋⡀⢸⣇⣴⣯⡿⣏⠃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢸⡄⢀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⡟⠀⣿⡟⠛⢦⣏⠈⢀
⠀⠀⠀⠀⠀⠒⠶⣾⣿⣿⣿⣿⠀⣇⢸⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠁⠀⣼⡟⢀⣼⠏⠀⠀⠀⠙⣦⣛⣷⡞
⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⣿⣧⡈⠀⣿⣧⠈⠻⢦⡐⢤⣄⣠⣾⠃⠀⣧⠀⠀⠀⣼⡿⢡⡟⢁⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠁⠁⠘⣧⢠⠾⠿⢷⣄⠀⠈⠑⠈⠉⣿⠀⣠⡞⠳⣤⡀⠹⠵⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣱⣶⠇⠈⢳⣶⣶⣤⣤⣿⣾⢿⡇⣰⡿⠧⢀⣐⡺⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⣿⠋⠀⠀⡼⡿⢻⠋⠙⢻⠁⠀⠹⡿⠷⠻⢿⣟⠓⢤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣻⡇⠀⠀⠀⠹⣼⢿⡀⠀⠘⣧⡄⠐⡇⠀⠀⠀⠉⠻⣆⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠈⠻⡹⣄⡀⠘⣿⡄⢃⠀⠀⠀⠀⠀⠘⢿⡦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⣆⠀⠀⠀⠀⠀⠙⣿⣷⡀⢸⡅⠈⡆⠀⠀⠀⠀⠀⠀⠹⡜⣦
⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠀⠀⠀⠀⠀⢷⣿⣇⠀⠀⠀⠀⠀⢹⣿⠛⣸⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⢻⡈⣷
⠀⠀⠀⠀⠈⠙⠓⠀⣈⣛⠛⠛⣓⠒⠒⣾⡟⠛⡆⠀⠀⠀⠀⢠⡾⣏⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠙⠊⠉⠛⠁⠀⠀⠀⠀⢸⣿⠟⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠘⠶⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠙⠛⠛⠓⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣒⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠛⠻⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠉⠛⣳⣦⣤⣄⠀⠀⠀⣀⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠇⠀⠀⠐⠿⠿⣿⣿⣿⣿⠿⠟⣰⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⢿⠉⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣌⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣄⣠⣿⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣬⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠱⢶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣇⠀⠀⠈⢿⣿⣶⣤⣀⡀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢻⡄⠀⠀⣰⠟⠈⠻⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠃⠀⢠⡟⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠛⠶⣤⣀
⠀⠀⠀⢀⣿⣿⠿⠿⠛⠻⠿⠁⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠤⠄⠀⠀⠐⠒⠒⠛⠛⠛⠂⠉⠉
⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠚⠓⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡜⢣⣤⡄⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱⡎⠉⠁⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠶⠶⣉⡱⠶⠆
⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠾⠿⣉⡹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣀⣀⣤⣴⡿⠋
⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⢀⣀⣀⣤⣴⡶⠿⢿⣿⠉⠙⠛⠉⣿⡿⠿⢶⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠛⠋⠉⠀⠀⠀⠈⠻⣷⣦⣴⣾⠟⠁⠀⠀⠀⠉⠙⠛⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⢠⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⡄
⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠿⠟⠛⠛⠻⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⢼⣿⡶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣿⣟⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣻⣿⣶⡶⠶⠶⠶⠶⠶⠶⠶⢶⣿
⠀⣿⡀⢠⣤⠀⣤⡄⢠⣤⠀⢹⡟⠛⣿⠛⢻⣿⠛⣿⡟⢻⣿⠛⣿⡟⠛⣿⠛⢻⡏⠀⣤⡄⢠⣤⠀⣤⡄⢸⣿
⠀⣿⡇⢀⣁⠀⣈⡀⢀⣀⠀⢸⡇⠀⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⠀⢸⡇⠀⣀⡀⢀⣁⠀⣈⡀⢸⣿
⠀⣿⡇⠘⠋⠀⠙⠁⠈⠋⠀⣸⡇⠀⣿⠀⢸⣿⠀⣿⡇⢸⣿⠀⣿⡇⠀⣿⠀⢸⣇⠀⠙⠁⠈⠋⠀⠙⠃⢸⣿
⠀⣿⡇⠀⠀⢸⣿⠿⠿⠿⢿⣿⣷⣶⣿⣶⣾⣿⣶⣿⣷⣾⣿⣶⣿⣷⣶⣿⣶⣾⣿⡿⠿⠿⠿⣿⡇⠀⠀⢸⣿
⣾⡿⠿⠿⠿⢿⣿⠀⠀⢠⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⡄⠀⠀⣿⡿⠿⠿⠿⢿
⣿⣧⣤⣤⣤⣼⣿⠀⠀⣸⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻⣇⠀⠀⣿⣧⣤⣤⣤⣼
⣬⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭
⣿⣯⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠟⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣀⣤⣶⣿⣿⠟⠁⠀⠈⠻⣿⣿⣦⣄⡀⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⠀⣰⣶⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣶⣦⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⡄⢹⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣧⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣦⣶⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠛⠛⠛⠛⠋⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣴⣾⣿⣿⣷⡀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣤⣤⣤⣤⣤⣄⣀⠀⠀⢰⣿⣿⡋⠙⢿⣿⣷⣄
⠀⠀⠀⣠⣴⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠉⠹⣿⣿⣆⠀⢠⣾⣿⡟⠉⠙⠛⠛⠛⠛⠻⠿⠿⣿⣿⣶⣴⣿⣿⣿⣆⡀⠙⢿⣿⣷⡄
⠀⢀⣼⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠟⠉⠻⣿⣷⣤⡀⠙⣿⣿⠄
⢀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⠿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⣨⣿⣿⣿⣿⣿⠟
⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠹⣿⣿⡀⠀⠀⣀⣤⣄⣴⣿⣿⠟⠁⠀⠀⠀⣠⣾⣿⣿⠋⠉⠉⠁
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⢻⣿⣧⢀⣾⣿⡿⣿⣿⣿⡁⠀⠀⠀⣠⣾⣿⡿⢿⣿⣷⣄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⣄⠈⠻⣿⣿⣦⣠⣾⣿⣿⣿⣄⡀⠙⢿⣿⣷⣄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡙⢿⣿⣷⣄⠈⠻⣿⣿⣿⠋⠈⣿⣿⣿⣦⡀⠙⢿⣿⣷⣄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠙⢿⣿⣷⣄⣼⣿⣿⠀⠀⣿⣿⡿⣿⣿⣦⡀⠙⢿⣿⣷⣄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠙⠻⠿⠿⠟⠁⠀⠀⣿⣿⡇⠈⠻⣿⣿⣦⡀⠙⠻⣿⣷⣄
⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠈⠻⣿⣿⣦⡀⠈⢻⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠻⢿⣿⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣥⢲⠴⠶⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠾⢻⢾⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠦⠤⠴⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⠾⢶⠲⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣦⢰⢢⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠠⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣒⠆⠀⠀⠀⢑⢒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠄⠀⠀⠀⠀⠀⠢⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢈⢡⢸⠖⣓⣶⡈⢁⡀⡀⢀⡤⡞⠻⡦⠲⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠠⠸⡿⢿⠹⢿⠻⠹⠿⠿⠿⠿⡟⢿⠇⠠⢒⡔⠀⡰⠹⠷⡿⠿⠾⠾⢾⡷⠇⠠⠤⠄
⠀⠀⠀⠀⠀⠀⢠⣶⣺⣶⣶⢲⢶⣶⣷⣶⣶⣿⢸⢼⡷⣇⡇⣟⢻⢼⠼⢿⡿⢧⢤⠤⣦⢤⣦⡄⣬⡥⠐⠒⠒⠂
⠀⠀⠀⠀⠒⠒⠒⠶⠀⠂⠶⠆⠲⠒⠶⠶⠶⠠⠶⠖⠶⠶⠦⠆⠶⠶⠲⠶⠒⠈⠰⠶⠖⠰⠶⠠⠶⠖⠶⠆

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⣿⣿⣿⣿⣿⣿⣿⣷⡝⣆⠹⣿⣿⣿⣿⣿⣿⠀⣷⠀⢻⡆⢸⣿⢸⣿⣿⣿⣿⣿⣿⣼⠁⠏⢠⣿⡟⣾⣿⣿⣿⠟⢀⣾⣿⡿⢋⣾⣿
⣿⣿⣿⣿⠋⠻⣿⣿⣿⣌⠣⠘⣿⣿⢿⣿⡿⡆⠘⡆⠘⡇⠀⣿⢸⣿⣿⣿⣿⣿⠇⠇⠈⠀⣾⠏⣼⣿⣿⣿⠋⣰⣿⣿⢟⣵⣿⣿⠿
⣿⣿⣿⣿⣷⣄⠈⠻⣿⣿⣦⠀⠈⢿⣯⢻⣷⠑⠀⢁⠀⠣⠀⢿⠸⣿⣿⣿⣿⡟⠈⠀⠀⣸⠏⡼⣻⣿⡿⢁⣾⣿⠟⣡⣿⣿⠟⣡⣾
⣿⣿⣿⣿⣿⣿⣷⡄⠈⢿⣿⣷⡀⠀⠻⣆⢻⣇⠀⠀⠀⠀⠀⢸⠀⣿⢻⣿⣿⠁⠀⠀⢠⠏⠘⣱⡟⠁⣠⣿⠟⣡⣾⡿⠋⣡⣾⠿⣫
⣿⣿⣿⣷⣝⠻⣿⣿⣦⡀⠙⢿⣿⣄⠀⠙⣆⢻⡆⠀⠀⠀⠀⠘⠀⣿⢸⣿⡏⠀⠀⠀⠈⠀⣰⠋⢀⡼⠋⢀⣼⠿⢋⣴⡿⢛⡵⢚⣥
⣿⣿⣿⣿⣿⣷⣮⡙⢿⣿⣦⡀⠙⢿⣦⡀⠘⢆⢻⡀⠀⠀⠀⠀⠀⣿⣾⣿⠀⠀⠀⠀⠀⠐⠁⡠⠋⢀⣴⠟⢁⣴⠟⠩⢒⡡⢞⣫⠴
⢽⣛⠿⣿⣿⣿⣿⣿⣦⡈⠻⢷⣄⠀⠙⢷⣄⠀⠀⠳⠀⠀⠀⠀⠀⡇⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⡠⠊⢁⠴⠋⠀⠀⠐⠩⠐⠉⣀⡤
⣤⣈⠉⠀⠉⠛⠿⣿⢷⣽⡢⠀⠙⠳⣄⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢒⣩⣵⣾
⣉⠙⠛⠶⢤⣀⡀⠀⠀⠈⠉⠓⠄⠀⠈⠑⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠛⠉⠁
⣻⠿⠷⣶⣤⣄⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣠⣤⣶⡾⣿
⣉⣉⡁⠀⠀⠈⠉⠙⠓⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠄⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⣚⣋⣩⣭⣴⣶⠿⠿
⣿⣭⣭⣭⣍⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡿⢻⡽⣿⢂⢿⣂⢸⡼⠃⠀⠀⠀⠀⠀⠀⠐⣀⣉⣉⣩⣥⣤⣴⣶⣶⣾⣿
⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠖⠒⠒⠀⠀⠀⠀⠀⠀⠹⠟⠘⠀⠈⢁⣈⡅⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠲⠿⠿⣿
⣿⣿⣿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣷⣿⣾⢻⣂⢙⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⣓⣶⣶⣿
⡿⠶⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠙⠉⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⣀⡀⠀⠀⢈⣉
⣤⣴⣶⠆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠐⠒⠤⣄⣉⠙⠛⠷⢶⣶
⠛⠉⢀⣀⣤⡴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠙⢦⡀⠀⠐⢤⣀⠀⠀⠀⠈⠙⠓⠶⣤⣄
⣶⡿⠟⣋⠥⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠙⢷⣌⡓⣮⣛⢶⣦⣤⣀⠀⠀⣀⠉
⡵⠖⠋⣀⠄⢀⡤⣒⠅⢀⠴⠊⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢷⣄⠀⠻⣿⣮⣝⢿⣿⣿⣿⣷⣦⣄⣉
⣠⠴⣊⣵⣞⡽⢊⣡⠞⢁⣴⡄⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⠀⠀⠀⠀⠀⠈⢧⠰⡄⠀⠻⣷⣄⠈⠻⣿⣷⣿⡻⣿⣿⣿⣿⣿
⣵⣿⣿⠞⣡⣴⠟⣡⣴⡿⣫⣾⡿⠃⠀⠀⠀⢠⠀⠀⠀⢸⡇⠀⢸⠀⡆⠀⡀⠀⠀⠘⣧⠹⣆⠀⠙⣿⣷⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿
⡿⣋⣴⣿⠟⣡⣾⣿⢟⣵⣿⡟⠁⣠⠆⠀⣰⠇⠀⠀⢀⣿⠀⡄⣼⡀⡇⠀⣤⠀⠀⠀⢹⣧⡹⣦⠀⠈⢿⣿⣦⡐⢌⠻⣿⣿⣿⣿⣿
⣾⡿⢋⣵⣾⣿⢟⣵⣿⣿⠏⠀⣰⣯⠎⢰⡏⠀⠀⡆⣸⡟⢠⡀⣿⡇⣧⠀⢹⡆⠀⠀⢀⢿⣷⣹⣷⡀⠀⠻⣿⣿⣦⡳⣝⢿⣿⣿⣿
⣫⣴⣿⣿⣿⣿⣿⣿⡿⠃⢀⣼⣿⡟⣰⡿⠀⣸⡿⢀⣿⡇⢸⣿⣿⡇⣿⠀⢸⣷⠀⠄⠘⣾⣿⣷⣿⣿⣄⢢⡙⣿⣿⣿⣮⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⢠⣾⣿⡿⣱⣿⠇⣰⣿⠇⣸⣿⠁⣿⣿⣿⡇⣿⠀⠘⣿⣇⢸⡀⣿⣿⣿⣿⣿⣿⣦⠹⣌⢿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⢒⣟⠮⠛⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢀⣊⠬⠚⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠂⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠚⠉⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢤⣴⠿⡞⡃⠀⠀⠀⠀⠀⠈⢪⡀⣈⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠤⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡦⢻⣽⣬⣿⣆⠘⡜⡄⠀⠀⠀⠀⠀⠀⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣒⣤⠹⣯⠤⡄⢫⢛⠻⣝⣆⠘⠛⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⣐⠀⡰⠀⢀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⣐⡌⢞⢄⣥⣦⠹⡩⣤⣤⠱⠃⠈⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠈⠑⠳⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢆⢹⢡⠹⡱⡾⣶⡌⢼⢬⣿⠇⠙⠉⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡀⠀⠀⠀⠀⠀⠀⠈⢮⡆⠱⣣⠹⣷⡘⠾⠀⠉⠀⠀⠀⠀⠀⠀⠄⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡀⠀⠀⠀⠈⠜⠰⠟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⢀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⢠⠜⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠤⠀⠀⠉⠓⠶⠶⢦⣤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠿⢿⣷⣶⣄⣀⠀⠀⠀⢠⠈⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠻⢗⢒⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠘⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠅⠓⠁
⠀⠀⠀⠀⠀⠀⠠⠚⡄⠀⠀⠀⠀⠀⢀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠠⠀⠂⠂⠘⠈⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠠⠊⠁⠀⠄⠀⠀⠀⠈⢀⠀⠄⠠⠀⠰⠀⠂⠐⡀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠅⠰⠂⠀⠐⠈⠀⠉⠀⠀⠀⠀⠀⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠂
⠠⠐⠀⠂⡠⠈⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠒⠻⣿⠃
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⣤⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡇⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⣰⢄⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡶⠂⠀⢸⡇
⠀⠀⣾⣧⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣴⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣴⣿⠇⠀⠀⠀⠀⠀⢸⣷⣼⣿⠐

⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢗⣄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢨⠀⣠
⠀⠀⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣷⣤⣾⣶⡆⠀⠀⠀⠀⠀⠀⣾⣤⣿


⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠥⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡉⣦⠀⡎⢩⠀⠀⠀⠀⠀⠀⠀⢸⣯⠃
⠀⠀⠀⠺⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠾⠿⠀⠀⠀⠀⠀⠀⠸⠦⠿⠴⠿⠼⠇⠀⠀⠀⠀⠀⠀⠟⠘⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣴⠒⢦⢞⢙⠆⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⡀
⠀⠀⠙⢲⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠂⢻⡇⠀⠀⠀⠀⠀⠀⢰⣶⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⢸⣶⣇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠾⠶⢘⣿⣀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⡠⣦⢀⣸⣿⣧⡀⠀⠀⢀⡴⠾⠗⠒⢳⣦
⠀⠀⠀⠐⢶⡄⠀⢸⣿⣿⣧⡀⠀⢠⣤⣤⣿⣿⣿⠶⣤⠀⠀⢠⡞⢋⣿⠉⢷⡄⠀⠀⢸⣿⣿⣾⣿⠛⢿⡶⠄⠀⢈⡿⣰⣺⡆⢸⡇
⠀⠀⠀⠀⣾⣵⠃⣿⢯⣅⡀⠀⠀⠀⠀⢘⢿⣿⣿⣀⡀⠀⠀⠈⣷⠟⠁⢀⣾⠇⠀⠀⣼⠿⠿⢸⣿⣿⣿⣧⠀⠀⢸⡇⠻⠿⠇⢸⣿
⠀⠀⠀⠀⠻⠛⠛⠁⠀⠙⠿⠆⠀⠀⠀⠿⢷⣭⣽⡯⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠛⠁⠀⣿⠀⠀⠀⠸⠇⠀⠀⠰⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠒⠒⠒⠛⠃⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠈⠙⡇⢀⣠⣤⣤⣤⣤⣴⡶⠶⢶⣦⠄⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣶⣶⠾⠿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠋⠁⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⢆⣶⡄⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⠴⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠀⠀⠀⠀⠀⠀⠘⠃⢸⡆⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡏⠀⣤⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠛⠉⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⣧⣀⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡶⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣇⣼⠏⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠘⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠓⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⢦⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣧⡀⢀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠙⠻⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⣀⣀⣀⣤⣴⣾⣿⣿⡿⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣀⣠⣀⠀⢿⡛⢿⣿⣿⡿⣿⣿⣿⢿⣿⣿⠿⠿⠛⠛⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡞⠉⠀⠼⡇⠀⠙⠿⣧⣴⣟⡩⠴⠶⠚⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⣈⣷⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠈⠳⣤⣤⣤⣿⡇⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡄⠀⠀⢸⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⢹⣿⠟⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣆⠰⠚⢧⡍⠁⢙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠀⠀⣙⡋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣩⣆⣤⣾⡡⠖⣩⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠙⠛⠿⣿⣿⣶⣶⣶⣶⣶⠶⠿⠛⢻⣿⣿⣿⡿⠃⠈⠋⣠⠴⠋⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧⣤⣾⣿⣿⠿⠿⠿⣿⣿⣷⣄⡤⠿⣿⣿⣿⣧⣀⡴⠚⠁⠀⢀⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠴⠿⠛⠁⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⣿⡿⠟⠁⠙⢦⣄⣀⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⠇⢀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⠿⠿⠛⢀⣾⣿⣿⣿⣿⣿⣿⣿⡟⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠈⠙⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣤⡀⠀⢠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡟⢀⣾⡿⠃⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣾⠏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣭⠍⠉⣭⣭⠉⠩⣭⣭⠉⠉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⣿⣿⠀⠻⢿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⡟⣡⣴⣦⢰⡆⣰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣯⣭⣥⣬⣭⣭⣥⣬⠀⣿⣿⣿⡞⢧⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠀⠂⠀⠂⠀⡆⠀⠂⢀⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⡈⠂⠀⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⢀⡇⠀⢀⢀⠀⡐⠁⠀⢬⢪⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢸⡇⠀⠀⠀⡴
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠒⠀⡠⠐⠒⠒⠒⠒⠒⠒⠂⢄⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⢀⣠⣤⣤⣤⣤⣤⣤⣄⡀⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⢪⠔⠋⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢡⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣧⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡠⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⣭⣿⡟⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡯⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠽⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡺⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠤⠴⣶⣶⣶⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⢿⣿⣍⠁⠀⠀⠀⠀⠀⠐⠒⠲⠶⠿⠷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣥⣿⠖⠂⠀⠀⠀⠀⠀⠀⠚⣓⣐⣪⣬⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⠆⠀⠀⠀⠀⠀⠠⠤⠶⠶⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣆⠀⠀⠀⠀⠀⠀⢐⣺⣿⣿⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⢙⣻⣿⣿⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣟⣁⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠴⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣮⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣿⣿⠒⠀⠀⠀⠀⠀⠀⡀⢺⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⣿⣿⣿⡽⢂⣀⠀⠀⠀⠀⠘⣫⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠿⠟⠛⠉⠀⠀⠀⠐⡚⢿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣙⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⣷⣶⣤⣄⡀⠀⠀⠀⣀⣠⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⡏⠙⢿⣆⢀⡴⠋⠁⠀⠈⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠛⠛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⠾⠋⠉⠉⠻⢦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄
⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠿⠶⣤⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣀⡀⠀⣀⣴⣿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⠦⣤⣄
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢠⣿⣧⣄
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠁⠀⡹
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢹⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢸
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⢰⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⣿
⢹⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣧⡈⠙⢿⣿⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⠿⠻⠿⠁⠀⠀⣿
⠈⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣍⣷⠶⠀⠀⠀⠀⢠⣿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⣿⡌⠻⣿⡙⠛⠛⠿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢿⡟⣿⡉⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠒⠓⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠓⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠏⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡉⠃⠀⠉⠿⠁⢀⡞⠉⠉⣹⡅⠛⢐⣧⡿⢋⡿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⡄
⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡐⢢⢄⢠⡔⠂⣠⣠⣄
⠀⠀⠀⠀⢠⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⡀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⠆
⠀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠇
⠀⠀⠀⠐⣴⣂⣆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⢀⠤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣿⣷⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣿⣿⡿⣿⢿⡿⣿⣻⣿⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣛⣧⣿⣤⣟⣿⣾⣧⣿⣽⣭⣿⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⢀⡄⣠⡤⢠⡀⢀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠋⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⢧⠤⡄
⠀⠀⠀⠀⠀⠸⡄⠹⠋⢠⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠙⡞⠒⠃
⠀⠀⠀⠀⠀⠦⡤⣤⢴⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⣄⡨⡍⣄⣄⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⡤⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⣰⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢙⡈⠁⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠠⠞⠸⠀⠁

⠀⠀⠀⠀⠀⠀⠨⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠃⠀⠀⠀⠀⠀⠐⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢤⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⠀⠀⠀⠠⠴⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠉

⠀⠀⠀⠀⠀⠠⠤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡷⠀⠀⠀⠀⠀⠈⢹⣿⡟⠉⠀⠀⠀⠀⠀⢀⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠒⢲⣶⣦⣤⣤
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉

⠀⠀⠀⠀⠈⠐⢤⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣇⠀⠀⠀⠀⠀⠀⠠⠴⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠁⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠭⠿⠿⠿⠒⠂⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠃⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠴⠂⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⠀⠀⠀⣀⠤⠴⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠺⠿⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⣿⠒⠀⠀⠀⠀⠀⢀⣠⣶⡾⠋⠀⠀⠀⠀⢀⣺⠒⠒⠂⣀⣠⡕⠒⠀⠀⠀⠀⠀⢨⡁⠐⢾⠖⠲
⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠠⢬⣭⠙⡒⠂⠀⠀⠀⠀⠀⠀⡀⣭⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡆
⠀⠀⠀⠀⠀⠒⠚⠨⠭⠥⠀⠀⠀⠀⠀⠀⠀⠬⡝⠓⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠀⣿⣿⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠀⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣐⣶⡦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⡆
⠀⠀⠀⠀⠀⠠⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⢿⣓⣒⣀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⡟⠛