Man with cap Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡛⠉⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣻⣭⣷⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⢞⡿⠿⡳⣮⡙⠛⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡆⠀⠀⠀⠀⡴⠃⣟⠳⠀⠰⡜⡇⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⠆⣸⠀⣤⡛⢳⡀⣸⢷⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡡⠞⢹⡠⡿⠟⠳⣃⡟⡾⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠐⣫⣾⢃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣤⣾⣿⣿⣯⠄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢎⡴⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

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

⣷⣶⣶⣶⣶⡷⢿⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⡾⣶⠿⠶⠶⠶⣶⣶⠷⠶⠷⠶⢶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠉⠉⠙⠛⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢸⣇⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣴⠶⠖⠚⠋⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠹⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀⢿⡛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠋⢁⣴⣤⠉⠛⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣯⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⠶⠶⠶⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠛⠛⠉⠙⠛⠛⠛⠿⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⡰⠋⠁⠈⠙⢆⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⠉⢩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢍⠉⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡔⠁⠀⠀⠀⢼⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⡧⠀⠀⠀⠀⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠋⠀⠀⠀⣴⣦⠀⠀⠀⠙⠁⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠘⢧⣀⠀⠀⣀⡼⠃⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⣀⣀⡀⢀⣀⣀⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⡾⠿⡇⠈⠁⠀⠁⢸⠿⢷⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠁⠀⠀⠀⠀⠰⣀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣾⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣛⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣷⠶⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠉⠀⣾⣿⣿⣿⣎⠿⠉⢉⠉⠙⠛⢿⣿⣿⡉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡀⠈⠙⠃⣠⣾⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⡿⠟⠋⠻⣿⣿⡿⠛⠻⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣫⣥⣤⣤⣄⣉⡉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⣐⣒⣒⣋⠾⠿⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣛⠿⣿⡟⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠰⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣶⣕⡢⣀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡇⣾⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣮⣕⢄⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣽⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣀⣠⣤⣤⡀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣦⡄⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣭⠿⠿⠿⠿⠯⣤⣤⣤⣤⡾⠿⠛⠛⠛⠒⠶⣶⣾⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⠁⠀⠀⠀⠀⠀⠀⣹⠟⢿⡏⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣞⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣀⣤⣴⡟⠀⠈⣧⠀⠀⣀⣠⣤⣶⣾⡟⠸⣯⣻⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣶⣾⣿⣿⣿⡿⠁⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠃⣤⡽⢩⠴⠌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⠀⠉⠉⠁⠀⠀⠀⣿⠃⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣠⣀⣤⣶⣭⣥⣄⡀⠀⠀⠀⠀⠀⣿⠀⠈⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣰⠾⠿⠛⠛⠛⠛⠛⠛⣛⡿⣷⠀⠀⠀⢠⣿⣀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠛⠿⢁⣈⢉⢉⣉⠥⠟⠁⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢤⣄⣠⡤⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⣀⣠⣤⣴⣶⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣛⠛⠛⠶⢴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣷⣾⣿⠇⠀⠀⠆⡀⠉⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣤⣶⡾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣽⣿⣿⣿⣿⡉⠀⣠⣴⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣁⣿⣿⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣷⣌⣿⣿⣿⣿⣷⡈⠉⠛⠿⠶⢶⣤⣤⣤⣤⣤⡴⠶⠾⢷⣄
⠀⠀⠀⢸⣿⣿⡀⠉⠁⠀⢀⣠⣴⣾⣶⣤⣄⣀⣸⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠸⣿⣿⣿⣄⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠿⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⠄⠁⠙⢄⡀
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⣷⣾⣿⣿⡿⠀⠀⠀⠀⡟
⠀⠀⠀⠀⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠟⠛⠋⣠⡾⠃⠀⠀⡇
⠀⠀⠀⣾⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⡗
⠀⠀⣼⣿⣿⣿⣿⣿⠀⢀⣀⣀⡀⠀⢀⣴⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠃⡀
⠀⢠⣿⣿⣿⣿⠋⠉⠀⢈⣿⣿⢿⣿⣿⣿⣿⣿⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡄⠀⠐⠇
⠀⣼⣿⣿⣿⠃⠀⢠⣾⣿⡟⠁⠀⠙⢿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠂⣀⠤⠀⠀⠀⣤⣿⡿⠋⠀⠀⠀⠸⡇
⠀⢹⣿⣿⣿⣦⡀⢸⣿⡟⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣶⣶⣶⣶⣶⡿⠿⣿⣷⡶⠞⠉⠀⣠⠶⢿⣿⣿⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢻⣿⣿⣿⣇⣀⡉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⣿⠛⠛⠛⠉⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠈⠛⣿⣷⡄⠀⠀⠀⠘⠁
⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⡰⠃
⠀⠀⠀⠀⠈⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⢀⠒
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣄⠀⠀⢰⣷⣾⡿⠋⠀⢠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠖⠉
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠏⠀⠀⣰⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣴⠁
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣧⣤⣶⣆⡸⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠟⠁⠈⠁⢙⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣄⣤⣤⣤⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠟⢻⣿⠿⠋⠀⠀⠀⢹⣿⠋⠉⣛⣛⡟⠛⠛⠁⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡿⠁⠀⠚⠁⠀⠀⠀⠀⠀⠘⢻⣶⣷⣿⠛⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠟⠃⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡔⠀⠀⡀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣀⡀⢸⣿⢋⣠⣶⡿⠋⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⣿⠃⢠⣿⣿⡟⠋⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣷⠀⠀⠀⠀⠀⠀⣰⣿⣇⣠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢾⣧
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠸⠟⠉⢻⣿⣿⣿⣄⠀⣠⣤⣤⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠙⢷⣀⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠷⠄⢤⣤⠚⠷⠶⠖⠁⠀⠸⣄
⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣾⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⢀⣿⣿⣿⣿⣿⠛⠯⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⢰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣀⣠⣶⡶⢿⣿⣿⣶⣶⣄⣙⣦
⣤⠚⢋⣭⣿⣿⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠚⢿⣿⣿⡟⠁⠀⣠⣤⣶⢿⣿⣿⣧⡴⠊⠉⣡⣤⠿⠟⠛⠿⢷⣄
⠹⣿⣿⣿⣿⣿⢀⣠⣴⣾⠿⢿⢿⣶⣦⣬⣛⣻⠶⣄⡀⠀⠀⢀⣿⣿⣿⣶⣾⣿⠛⠋⣠⣾⡟⠉⠙⣷⣶⠟⠃⠈⣦⠀⠀⠀⠁⢄⠙⢦⡀
⠀⢻⣿⣿⣿⣿⣿⡿⠋⠁⠀⢀⡴⠋⠀⠉⠉⠙⢦⡈⠻⣦⣄⣹⣿⣿⣿⣿⡿⠁⠀⣰⣿⡟⠀⢀⣾⠟⠁⠀⠀⠀⢡⡄⠀⠀⠀⠀⠑⠀⢻⡄
⠀⠈⢿⣿⣿⣿⡟⠁⠀⠀⢠⠛⣤⠀⠀⠀⠀⠀⠈⢣⠀⠈⠿⣿⣿⣿⣿⣿⠃⠀⣸⣿⡿⠀⠀⣾⡏⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠉⡀⢱⡄
⠀⠀⠀⣼⣿⡟⠀⠀⠀⢰⡇⢰⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠿⣿⣿⣿⠇⠀⢠⣿⣿⠇⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠁⠈⢷
⠀⠀⣰⣿⣿⠁⠀⠀⠀⡆⢀⡌⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⣿⣿⣤⣤⣸⡿⠿⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡸⠧⠀⠀⠀⠀⣀⣀⠠⠄
⠀⢰⣿⣿⡿⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣿⣿⣿⠋⢻⠃⠒⠚⠛⠉⣉⣡⣶⠶⢦⣀⣠⣶⣌⡁⠀⠓⠒⠂⠈⠉⠁
⢸⣿⣿⣼⡿⠠⣄⢀⣘⠿⠟⣇⡀⠀⠀⠀⠀⠀⠀⠼⠆⠀⠀⢙⣿⣿⣿⣤⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤
⢸⣿⣿⡇⠐⠂⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠈⡿⠋⠉⠉⠉⠲⠶⠶⠦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣠⣤⣀⠀⠒⠒⠀⠉⠉
⠀⠙⠻⢷⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⣀⣀⣠⣶⠇
⣤⣤⣤⣤⣬⣭⣥⣴⣶⣶⣶⣶⣦⣬⣭⣽⣿⣿⣿⣿⣭⣉⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⠿⠻⢿⡿⠿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠐⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⣄⠀⠀⠀⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⢁⡠⠒⡉⠀⠈⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠔⠁⠀⡘⠉⠒⠷⢅⡈⠎⠢⡀
⠀⠀⠀⠀⢀⣠⣤⣤⡤⢬⣤⣀⠀⡀⢀⠊⠀⠀⠀⠁⠀⠀⢠⠀⠈⠺⣄⠘⣄
⠀⠀⠀⣴⠋⢁⣡⡶⠋⣉⢙⡷⢔⠠⡉⠀⠐⠀⠀⡀⠀⠀⡝⠀⢐⠀⣲⠀⡄
⠀⠀⠀⠗⣶⡋⢍⣴⠂⠀⠀⠂⠷⠝⢮⡢⡀⠄⠀⠸⢿⣦⡁⠀⠅⠀⢸⠀⡿
⠀⠀⠀⠈⠊⠐⠛⢆⢠⠞⠃⠈⠂⠀⣱⢝⢮⣐⠌⢐⠀⡍⠋⠗⠶⢤⡬⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⡀⣀⠑⠀⢢⠙⠓⠖⠥⢈⠲⢤⠖⠂⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠂⠠⠀⢰⡦⠅⠀⠀⠀⠐⠀⠈⢟⠘⢠⢗⡥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢀⠾⡁⠔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⣅⠀⢙⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠒⠸⠂⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠉⠀⠀⠀⠀⠀⠀⣠⠏⠌⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⣀⡄⢘⠁⡂⠅⠀⠀⠀⡰⠂⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⠈⠀⠈⠈⠤⠐⠂⠁⠀⠀⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢐⢃⣤⠘⠢⠀⣀⡀⠀⠀⣀⡠⠔⠚⠉⠀⠈⡂⠰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢊⢜⡴⠋⠈⠉⡦⠀⠑⠄⠴⠒⠋⠁⠁⠀⠀⡠⢂⡥⠒⠉⡉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠠⢹⠁⣜⠏⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⠒⠁⣠⡢⠀⠁⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠠⠀⠂⠀⠀⠀⠔⠀⠀⡀⠀⡡⠀⢀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠁⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠉⠁⠀⠀⠀⠀⠈⠁⢢⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠤⢰⣿⡟⠿⠶⠶⠶⠖⠛⠋⠙⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⣄⠀⠀⠤⠶⠄⠀⢻⣿⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⡄⠀⠀⢰⣦⠀⠀⠈⢻⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⡄⠀⠀⠀⠀⠀⠀⢸⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⡀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⣉⡉⠁⠀⠀⡐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢀⡠⠊⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⢢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡞⢸⣧⣄⣀⣀⣤⠃⠀⢃⠐⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠃⢰⠀⠘⣿⣿⣿⣿⡏⠀⠀⢈⠄⠀⠰⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡘⠀⠈⢆⠀⣿⣿⣿⣿⠀⠀⡔⠁⠀⠀⢰⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⡁⠀⠀⠀⢢⢸⣿⣿⡇⡠⠊⠀⠀⠀⠀⠸⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡇⠀⠀⠀⠀⠑⣿⣿⠔⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠃⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣭⣷⣶⣶⣶⣶⣬⣍⡛⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠶⠶⠶⠿⠿⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⣴⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣬⣍⣛⡳⠶⣭⣙⠻⢿⣿⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣟⣶⣄⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⢿⣿⣿⣿⣿⡿⢛⣩⣥⣶⣶⣶⣾⣿⣿⣿⣿⣿⣟⣛⠿⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣉⣴⣿⠿⠿⠿⢿⣯⣭⠉⠉⣭⣽⡿⠿⠿⠿⣶⣤⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠚⠻⠟⠛⠋⠀⠀⠹⠛⠿⠟⠓⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⡀⠀⠀⢠⣴⣄⣠⣦⠄⠀⠀⢀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡇⠀⠀⢀⣀⡉⢁⣀⡀⠀⠀⢸⠏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡇⠀⠐⠛⠛⠛⠛⠛⠛⠀⠀⢰⢶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⠀⢸⡶⠶⠶⢶⡇⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⠀⠀⣀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠀⠉⠉⠉⠉⠉⠉⠉⠁⠐⠒⠠⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢀⠠⠤⠀⠒⠒⠀⠀⠉⠀⠀⠐⠒⠒⠠⠄⢄⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⣀⡴⠯⠤⠄⠀⠀⠀⠀⠤⢤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣀
⠈⠁⠒⠂⠤⠀⠀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⡿⠟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⣠⣾⡿⠿⠷⣦⡀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⠿⠷⣦⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠿⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢋⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⣤⣄⠀⠀⢻⡀⠀⠀⠀⠀⠀⣰⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢡⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢿⣿⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⣾⢃⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠻⠋⠀⠀⣼⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡏⠘⠋⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠳⠄⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠉⠒⠤⠬⠁⠉⠩⠭⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣒⣄⣀⣀⣀⣀⣠⣔⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣉⠉⣉⣳⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠟⠋⠉⠀⠀⢩⠟⠛⠉⠁⠙⠢⣍⠉⠁⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡽⠊⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⣠⠊⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢀⡞⠁⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠘⣇
⠀⠀⠀⠀⡼⠁⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻
⠀⠀⠀⢠⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⣀⣠⡤⠤⠶⠶⠶⠒⠒⠛⠒⠒⠒⠲⠾⡄
⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠸⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠴⠶⠚⠛⠛⠛⢋⣉⣉⣩⡽⠭⠥⠬⠍⢹⡀
⠀⠀⠀⠀⢹⣷⣷⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠼⠶⠖⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠈⣿⡿⡿⢿⣿⣿⣿⣭⣓⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⣿⠀⠻⠷⡝⢿⣿⡿⠉⠛⠿⣯⣿⣒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⢸⢸⢳⣿⠿⠈⠙⠁⠀⠀⠀⠘⢿⣿⣿⣶⣭⣕⡢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣄⠀⣈⣙⢢⡀
⠀⠀⠀⠀⠀⠘⣞⣏⠛⣤⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣾⣭⣗⣲⣶⣶⣾⣿⣿⣿⡟⠛⠉⠉⠉⠉⠉⠉⠉⠛⠓⠋
⠀⠀⠀⠀⠀⠀⠘⣎⡀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠿⠿⢿⣟⡋⠀⠀⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠩⢤⣀⠀⠀⣠⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠞⠀⠀⠀⠀⡈⠐⠛⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⣀⣀⣀⣃⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⠉⠉⣉⡉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⢙⣛⠋⠉⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⠰⠿⠛⠁⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣶⣄⣀⣀⣀⣀⣀⣤⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠖⠒⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢋⡉⠁⠀⠀⠀⠙⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡖⠀⠀⠀⣀⣠⣤⣤⣦⣄⢀⡞⠀⠀⠀⠀⢶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡟⠀⠀⠀⠸⣿⣟⣛⣩⣿⣿⠟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆⠀⠀⣠⣴⣷⣄⡀⠀⣠⣾⣷⣦⣄⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡖⠦⡀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⠚⠒⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠉⣀⡀⠀⠀⠀⠀⢀⣉⠐⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢈⣭⡍⡛⠀⠀⢛⢩⣍⡁⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠑⠢⠤⠤⠔⠊⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠗⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⣒⣰⡟⠩⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⢻⣂⣒⠠⠄⢀
⠀⠀⠀⠀⠀⡤⢊⡥⠖⢚⠉⠉⠀⢻⠀⠈⠐⠈⠁⠀⠒⡆⠀⠀⠀⠂⠁⢀⡟⠀⠉⠉⡓⠲⢬⡐⢤
⠀⠀⠀⠀⡜⣰⠋⠀⠀⢀⠀⢐⣨⡴⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢦⣅⡂⠀⠀⠀⠀⠙⣆⢣
⠀⠀⠀⡜⣰⠃⠀⢠⢂⡵⠞⡩⠔⠂⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠐⠠⢍⠳⢮⡐⡀⠀⠘⣆⢧
⠀⠀⡰⢱⠃⠀⡀⣩⣿⡠⠊⠀⢀⣠⡤⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢤⣄⡀⠀⠑⢄⡿⣍⢆⠀⠸⡌⢆
⠀⣰⢡⠏⠀⠀⣹⢧⡜⣧⣤⠞⢋⠡⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠂⠈⠙⠳⣤⣼⢣⡾⣏⠀⠀⠹⡜⡆
⠀⢈⡿⣴⢀⡴⠁⡀⣹⣞⣇⢈⠠⣐⣬⡶⢿⡀⠀⠀⠀⠀⠀⠀⢀⡿⢶⣥⣂⠄⡁⣸⣷⣏⠄⠈⢦⡀⣦⢿⠁
⠀⠸⠡⠏⠛⠆⠐⠴⠋⠘⠿⠶⠟⠋⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠜⠀⠀⠈⠙⠻⠶⠿⠃⠙⠦⠂⠰⠛⠹⠌⠇
⣿⣿⣷⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣛⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣷⣾⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣐⣲⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⢷⡿⣻⣵⣾⣦⣤⣤⣜⠻⣿⣿⣶⣝⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠓⠂⠀⠈⠉⠉⠀⠀⠈⠉⠃⠛⣹⣿⡾⣧⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣖⣸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢧⢻
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣩⣵⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣵⠿⠓⠛⠉⠀⠐⣀⠆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠖⣒⡿⠿⠿⠶⠤⣄
⠀⠀⠀⠀⠀⢀⡞⡵⢲⣝⠇⠀⢀⡶⠂⠛⣿⡿⣉⣩⣭⣿⠷⢦⣀⣀⣀⣤⣄⣿⠉⠈⠀⣀⡀⣀⡠⠬⠷⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢸⣀⠀⠒⠊⠀⠒⠁⠠⠒⠉⠉⠱⠀⠉⢉⠈⣴⡞⠉⠹⠷⢶⢶⠿⠾⣿⡏⠉⠙⣽⠀⠀⢰⠄⠐⠈⠉⠙⢷
⠀⠀⠀⠀⠀⠀⡿⢠⠆⠀⠀⠀⠐⠀⠀⠐⡎⢀⠧⠤⠤⠴⠊⠀⠀⠀⡇⠀⣿⠠⢀⠏⠙⢆⠀⢹⡆⠀⢸⠀⠀⢠⡦⠆⠘⣧
⠀⠀⠀⠀⠀⠀⠻⣦⡈⠀⠠⠀⠀⠀⠀⠈⣧⡸⠀⠀⠀⠀⡄⠀⠀⠀⠙⠒⣿⠒⠋⢢⠀⠀⠙⠻⠱⡀⠘⠀⠀⢾⡃⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢈⣷⡀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠁⠈⢁⠶⠃⢰⠇⠀⠀⠀⠱⡄⠀⠀⠀⠻⡠⠷⠄⢈⡇⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⢷⡀⠀⠀⠀⠀⠀⠀⢹⡄⠤⠄⠀⠠⢈⣀⠀⠀⣼⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠿⣆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠐⠶⠠⠀⢉⣼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⢠⠞⠙⠳⢄⡀⣿⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⢸⣇⣠⣀⡀⢀⣀⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡟⠻⣿⣿⣭⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠈⠙⢦⡄⢠⡇⠀⠀⠀⠀⠀⠀⢳⣼⣿⡟⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠈⠳⣄⠙⢾⠃⠀⠀⠀⠀⠀⠀⢸⣿⢻⡄⠀⠀⢸⢿⣿⣷⣶⣦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠠⢄⠀⠈⠳⣾⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣻⠀⠀⣼⠏⣿⣿⡿⢿⣏⠉⠻⣿⣶⣶⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⡁⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢻⢀⣴⣿⣧⣸⡿⠓⢤⡈⠀⠀⠙⢝⢆⠀⢹⣿⢦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠘⢦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣼⣿⡟⠀⠙⠄⠀⠀⠻⣦⠀⠀⠀⠀⢳⣄⠙⢧⡙⣏⢳⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⢹⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠉⠢⠀⠻⡎⠀⠻⡙⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷⠤
⠀⠀⠀⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠂⠉⠉⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢠⡠⢹⡆⠀⡄⣤⠰⢄⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣰⠎⠫⠄⡾⠃⣼⢛⠆⠋⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠵⠛⠷⠟⣑⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣀⡼⢲⡸⡦⠞⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟
⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⢹⡇⢰⠿⠤⡿⠣⠀⠀⠀⠀⠀⢀⡠⠤⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⢆⣯⠖⣸⢟⢒⡿⢖⣶⡄⠀⠈⠠⠔⠖⠒⠀⠀⠘⡐⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣦⣣⡿⠛⣤⢋⠣⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⡸⣭⣘⣧⡈⢓⡋⠀⠀⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠉⠀⢈⡏⠿⠃⢰⡮⣤⠀⠀⠈⠉⠀⠀⣀⠀⠀⠀⠀⢀⣃⢁⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢐⡘⠟⡁⠟⢺⣩⠏⢿⣆⠀⠒⢈⣡⡄⠀⠀⠀⠀⠀⠸⣟⣙⡀⠀⠀⠀⠀⠀⢸⣿⣟
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠰⠴⠐⠃⡄⠸⠃⡄⢺⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠀⠑⡚⢈⠻⣀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠣⢤⡾⢓⡀⢀⣩⣉⢀⠒⣦⠔⠆⡘⠃⠏⠂⢀⠐⠠⣬⡝⠃⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣠⣈⢧⠘⣁⢯⡅⠭⠂⠋⠉⠁⠋⠁⠀⠀⠀⢨⣤⡒⢉⡕⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠿⠈⠀⠈⠀⠀⠏⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠸⠀⠀⠀⠀⠀⢸⣿⣟
⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⡿⠛⡂⣴⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⠛⠛⠿⠿⠿⠿⠿⠟⠋⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⣤⣄⠀⠀⢀⣀⣤⠴⠦⠄⠀⠙⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⢼⣻⠿⣷⠀⠙⢫⣧⢶⣦⠤⠀⠀⢹⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣯⣤⠤⠤⢤⣟⣤⣀⣠⡤⠤⢤⣄⣀⣴⡏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢧⠀⠹⡇⠀⠀⠀⡀⡋⠀⠈⢁⠀⠀⠀⠀⣿⠛⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠄⠀⠂⠙⠤⣤⡀⠑⠷⠴⠺⠖⢤⣀⢀⠴⠃⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠄⠀⠉⠁⢄⠀⣠⠀⠀⢰⡀⠦⠴⠦⠶⠤⠄⠀⠀⠀⠀⠀⡐⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⡀⡶⠃⠀⠀⠀⢳⡀⠙⠋⠉⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠋⠀⠉⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⢀⠂⠀⠀⡷⣄⡀
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢸⢿⣷⣶⣶⡿⠋⠀⠀⢀⠀⠁⠀⠀⣰⠃⠀⠙⠲⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⣴⡃⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠙⢿⠉⢀⣀⡤⠒⠁⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠈⠙⠒
⠀⠀⠀⠀⢀⡞⠁⠣⠀⠀⠀⠀⡴⠒⠛⠉⣿⣷⡀⠀⣼⣷⣿⣿⡿⠀⠀⠀⠀⣰⠃
⠀⠀⢠⠞⠉⠁⠀⠀⠀⠀⢠⣿⡄⠀⠀⠀⠈⠹⣷⣠⠟⣿⣿⠏⠹⡄⢀⡴⠋⠁
⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⠁⢀⣿⣿⡆⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⢤⣄⡀⠲⠖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣦⣟⡛⠼⠭⠁⠘⠒⠀⢀⠀⢀⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠁⠀⠀⠀⠀⠀⣴⣤⣄⣀⣰⣿⣿⡿⠁⠇⠀⣀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡿⠷⠀⠀⠀⣀⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢛⣉⣉⣠⣤⣤⣶⣾⡇
⠀⠀⠀⠀⢀⡐⠋⢀⡀⠀⢀⣀⣠⣤⣤⣶⠶⠶⢾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢸⣿⣿⣍⠁⠀⠀⠀⠀⠾⢿⣿⡋⢾⣂⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠈⢳⡶⠀⠀⠀⠀⠘⡋⠀⠸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠶⠚⠁⠀⢀⡼⠁⠙⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⢠⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠷⠆⠀⠀⠀⠀⢀⣴⢻⠇⢘⡿⠀⠀⠸⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢀⣠⣴⠟⣁⣿⠇⠈⠀⠀⠀⠀⣹⠈⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡴⢻⣟⠛⣿⡋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⡗⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣯⣾⣿⣿⢆⣇⣸⡄⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢱⠀⠀⠈⠙⠲⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢛⣉⣴⢿⣿⡟⣼⣿⣼⣿⣿⣷⣄⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡙⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⢻⣿⣿⣿⠃⡏⠛⣹⣿⣿⡉⠈⠑⢂⠊⣿⠁⠀⠀⠀⣤⣄⣸⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⢿⣿⣿⣿⣿⢠⡇⢸⣿⠹⡟⢷⡀⠠⠁⣸⡏⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⣰⠃⠀⠀⠐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣧⡟⣸⣏⣿⣿⠋⢸⡇⢸⣿⡄⠃⣾⡷⠁⢠⣿⠁⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡀⢠⡟⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⣿⣧⣿⣿⡿⡼⠀⣾⠃⠀⢻⣿⣤⠟⠁⠀⣼⡏⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⠇⢼⡇⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠇⣿⣿⣟⠇⠀⣿⠀⠀⠠⠿⠁⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢨⠇⣾⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢾⣿⠀⣿⣿⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣯⣼⡇⠐⣿⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⢰⣿⡧⠠⠤⠤⢀⠀⣼⡏⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⡇⠄⣿⣻⠇⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠃⠒⠐⢰⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⣿⡗⡀⣿⣿⡀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⢸⡶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣰⣿⠀⢸⢹⢹⠓⣸⣿⠁⣾⡟⠀⠀⠀⠀⠀⠀⣼⣿⠁⣠⢤⣤⣼⡶⠿⢷⣤⣀⣀⠀⠀⠀⠀⠀⣿⣧⠀⣠⣴⠶⠶⠠⣼⡃
⠀⠀⠀⠀⢀⣟⠹⢠⠘⣰⡀⡀⢙⣿⠀⣿⡇⠀⠀⠀⠀⠀⢰⢿⡏⠠⣇⣀⣀⡀⠀⠀⠀⠀⠀⠈⠳⢄⠀⣀⢀⣿⡟⢈⣉⣥⣤⣀⠀⢸⡇
⠀⠀⠀⠀⡿⢿⠀⣾⡆⡿⣇⠑⢘⣏⠀⣿⠀⠀⠀⠀⠀⠀⣼⢸⣇⠀⠸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣧⠹⣏⠉⠛⠛⠻⠷⠬⣉⠀⢸⡇
⠀⠀⠀⠸⡇⠸⠀⢹⡇⡇⢹⠀⠈⠣⠈⡿⠀⠀⠀⠀⠀⠀⣿⠈⣿⡀⠀⢸⣿⠗⠐⠂⠐⠃⠀⠀⠀⠀⢻⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣼⠁⠀⠀⠈⡟⡦⠈⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢻⠀⠸⡇⠀⠀⣝⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠀⠀⠛⣽⣦⠶⠖⠀⠀⠀⠀⠀⠈⣧⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠸⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣴⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢻⣦⣿⡀⠸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⢦⡙⠇⠀⢣⣄⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠉⠉⠉⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣟⡴⢆⣠⡤⠶⠦⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠟⠀⠙⡁⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠄⠀⠀⠀⠠⠁⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⢶⠀⣠⣄⠠⠀⠀⣀⡤⠚⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣿⣶⡦⠄⠮⠵⠁⠒⠆⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠰⢀⠙⠋⠀⠁⠀⠀⠀⢤⣀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣉⡆⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠁⠀⠀⠃⠀⢈⠃⠀⠀⠀⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢀⡇⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠐⠦⠞⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣇⠀⠀⠀⠀⠒⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⣀⣠⡴⠞⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢀⡀⠸⣆⠀⠀⠀⠀⢀⣎
⠀⠀⠀⠀⠐⡄⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⣰⠋⠀⠀⠏⠓⠠⡤⠰⡞⠀⠀⠈⠀⠂⡀
⠀⠀⠀⠂⠘⢢⡀⠀⠀⠀⢀⡄⠤⠤⠤⠄⠀⠒⠒⠂⠈⠉⠀⠀⠀⠀⢰⠁⠀⠰⠒⢰⡄⠀⠠⣄⠇⠂⠀⠀⠀⠀⠑
⡄⠃⠀⠀⠘⠋⠱⠄⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠇⠀⠀⠀⠀⠸⠀⣼⠁⢸⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠒⠂⠂⠛⡄⠀⠀⠀⡴⠉⣇⠀⠠⠀⢀⣀⣀⡠⠤⠄⠒⠊⠉⠀⡸⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰
⠀⠋⠀⠉⢻⠂⠐⠒⠊⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠃⠀⠀⠀⡆⠀⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣧⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠛⠀⠀⠀⠀⠀⢰⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⡇⢀⠀⢰⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠾⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠤⠚⠒⠒⠒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣀⣀⣀⣀⣀⣸⣀⣀⣤⡄⠀⣠⠐⠉⡀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠚⣛⡇⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠖⠒⠀⠀⠀⠀⣀⣸⠒⣃⠤⠤⠤⢀⠴⠤⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⣀⣀⣀⡀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⣀⣀⣀⣀⣠⡤⠄⠉⠲⢄⡀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠾⣉⣀⣀⣀⣀⣀⡠⠤⠤⠐⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡄⠐⢦⣰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠈⠛⠃⠀⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠸⠛⠛⠋⠉⠉⠙⠛⠻⠿⣿⣿⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⣀⠀⠀⠀⠉⠻⢿⠿⠿⠿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡛⣻⣷⣆⢀⣤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠙⢿⣿⣿⣿⣿⣿⣿⣿⡘⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⢿⣦⣿⣿⣿⣿⠶⠖⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡎⢻⣿⣿⢿⣿⠗⠂⣰⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢻⣿⡸⢋⣤⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⣘⠻⣿⣿⣿⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⣿⣿⣿⣷⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⢻⠛⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢛⡉⠁⠀⠀⠀⠀⠀⠀⠐⠚⠉⠛⠻⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣉⠴⠚⣉⡤⠴⠖⠒⠋⠉⠥⠴⠒⠀⢁⡀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡵⢊⣥⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠦⣀⠀⣀⡿⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣡⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡅⠀⠈⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠐⠒⠒⠲⠶⠤⣤⣤⡤⠤⠶⠒⠒⠛⠛⠂⠀⠀⠀⠀⢠⠸⡆⠙⠄⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠠⠶⠶⠶⠤⣤⣄⣀⣀⣀⣤⣤⠴⠶⠶⠶⠶⠶⠄⠀⠀⠀⠀⣷⠘⠄⠀⢀⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣿⠁⠀⠐⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢠⣼⣿⣶⣧⣄⣄⡀⠀⠨⡄⣄⠀⠀⠀⠀⢤⣄⣴⣤⣤⣀⠸⡄⠀⠹⡀⣼⢿⡿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠛⢛⣛⣛⠛⣛⡻⣿⣧⣀⣀⣙⠂⣠⡿⠿⠛⣛⠛⠛⠿⢿⣁⠉⠀⠀⠘⠁⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡇⢀⣩⡴⠚⢿⣿⡿⢿⣿⡏⠉⠉⢻⡟⢠⣾⠷⣦⣤⣤⣄⡀⢸⣿⣿⡿⢿⠿⣿⡿⠟⠻⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣷⠀⠁⠀⠀⣀⣠⡤⠀⣿⡇⠀⠀⠘⣧⠙⢡⣀⠈⠛⠛⠁⠁⢸⡇⠀⠀⢸⣶⢏⣠⣶⡆⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣠⣿⠷⣦⣄⣀⣀⣀⣤⢀⡿⠀⠀⠀⠀⠹⣦⣀⠉⠉⠉⠁⠀⣠⠟⠀⠀⢀⣾⣿⣿⠇⢀⠃⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠉⠉⠉⠉⠀⣸⠁⠀⠀⠠⣄⠀⠀⠙⠛⠳⠖⠚⠋⠁⠀⠀⣠⣾⡿⠉⣿⢁⡇⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⣴⡟⠉⠀⠀⠀⠀⠉⠀⢳⣦⣄⠀⠀⠀⠀⠀⠀⣼⡿⠁⡀⣰⣿⣾⠇⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⡾⠁⠀⣸⡏⠿⢷⣄⠀⢀⣴⡷⢦⡾⠃⠘⢷⡀⠀⠀⠀⢀⣿⠁⢸⡿⢩⠿⢃⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢀⣴⠟⠀⠀⠀⠙⢻⡿⠋⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⢸⠃⠀⣼⣧⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⢿⡀⠀⣀⣀⣠⣘⣁⣠⣤⣤⣤⣀⣀⠀⠀⣿⠇⠀⠀⠀⢀⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠈⡇⠀⠛⠛⠳⠮⠭⠵⠶⠒⢛⡉⠉⠀⢰⡟⢀⡞⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⡀⠀⠀⠀⠲⠦⣴⠶⠶⠿⠋⢀⠀⠀⣿⠁⣼⠁⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⠀⠀⠘⢦⣤⣤⣤⠀⠤⠶⠋⠀⢀⣿⠀⣿⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣀⣀⣀⣷⣄⣀⣀⣀⣀⣼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠙⠛⠛⠉

⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠿⠛⠁⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡦⣀⠀⠀⠀⠀⠉⠛⠿⠟⢻⣿⣿⣿⡿⠟⠉⠁⢀⣤⣶⢿⣿⣿⣿⣿⡿⠿⠗⠒⣦⣀⣀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⣿⣳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠉⢀⣾⠏⠁⠀⣀⣴⣾⠿⠥⠾⠿⠏⠉⠉⠀⣠⣤⣤⣭⡉⠉⠈⠉⠁
⠱⣦⣭⣿⣻⣦⣀⠀⠀⠀⠀⠀⠀⣠⣆⢄⠀⡀⢠⡀⠀⠀⠀⢀⣾⣿⣟⣡⣶⢶⢶⣿⠿⠿⠿⣿⣿⣿⠿⢿⣿⣿⡄⠀⠀⠀⢀
⠶⢬⣍⣻⣭⢻⣿⣷⣆⠀⠀⣠⡊⠈⢙⠷⣤⣿⣾⣷⣶⣶⣄⣿⣿⣿⣟⣫⡴⣦⣎⣠⣶⣾⣿⣿⣿⣿⣿⣤⣼⣿⣿⡄⠰⣿⣿
⣀⣀⡙⡛⢿⣿⣿⣿⠟⣠⣾⣟⣋⣵⣮⣥⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⢿⣩
⠉⠉⠙⣻⣻⣿⣿⡁⢠⣿⣿⣿⣿⣿⣿⣿⡟⠉⣻⠿⠻⣿⣿⣿⣯⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠺⢿⣿⣿⣿⣿⣿⡄⠸⣿
⠶⠒⠢⠬⠿⣿⠟⠁⣸⣿⣿⣿⣿⣿⣿⣿⣁⡤⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⢿
⠀⠀⠀⠀⠀⣿⡀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣷⣀⠀⣿⣿⣿⣿⣿⣿⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠀⢿⣿⣿⣿⣿⣿⣿⣿⣭⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢝⣻⠿⠟⠂⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⠟⠛⢆⡀⢸⣿⣿
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠈⢋⣤⣶⣟⠋⣉⣉⣉⣙⣿⣷⣄⡴⠂⠀⣀⣿⣿⣿⣷⣤⣤⡀⠻⢸⣿⣧⡀
⢠⡄⣶⢠⣊⢀⠀⠀⠀⣰⣶⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⠁⠀⠀⣿⢿⣿⠩⣿⣿⣿⣿⠆⠘⣿⣟⡇
⡆⢠⣧⣾⣿⠚⡀⠀⠀⠻⣿⣿⣿⣿⣿⣇⠀⠀⠈⠉⢿⡿⢥⣤⡭⠟⠉⠁⠉⡀⠀⠀⢰⠈⠙⠢⢤⠾⠗⠀⠀⠀⡿⢏⠅
⣀⣿⣿⣿⣿⢏⡡⠀⠀⠂⢸⣋⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⣿⣿⣿⠟⣡⠯⠀⠀⠀⠀⢸⣟⢻⣿⣿⣿⣆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡟⠃⠀⠀⠀⠀⠹⡀⠀⠀⠐⠀⠀⣠⣾⡈
⣿⡿⢡⣾⣿⡿⣦⠁⠀⠀⠈⠻⣀⣧⡽⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣷⣶⣶⣦⣦⣴⣿⠃⠀⠀⠀⠀⣿⣿⡿⠃
⠋⣴⣿⣿⣿⣷⠿⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣽⣿⣇⠀⠀⠀⠀⠀⢾⣿⠀⠀⢠⣤⣶⣤
⣾⣿⣿⣿⣿⡿⠀⣀⣀⣀⣀⠀⠀⠀⠀⠛⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠀⣀⣻⢟⣛⡛⠉⠀⣡⠀⠀⣀⣼⡿⠀⠰⠋⠻⠿⢿
⣿⣿⣿⣿⡏⠀⣾⣿⣿⠟⠋⠁⠀⠀⠀⠀⠸⣿⣿⣿⣦⣤⣶⡀⠀⠛⢿⣿⣿⣿⠿⠿⣿⣿⠟⢻⣎⢠⣿⣿⠇⠀⠀⢀⣀⡀⣾
⣿⠿⠿⠋⠀⣀⣿⣿⣿⣧⠔⢺⠆⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣀⣀⠀⠉⢛⣿⣶⣿⣿⣤⣤⡜⠉⣿⣿⠏⠀⠀⡇⠀⠀⠀⣿
⠀⣀⣰⡞⣻⣿⣿⣿⣿⣿⣇⣾⡟⣿⣿⡀⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⠋⠉⠉⠉⠛⠛⠣⣾⡿⠉⡀⠀⠀⡇⠀⠀⠀⣿
⣟⡻⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣶⠆⣀⣠⣤⡾⠋⠀⠀⣻⠀⠀⡇⠀⠀⠀⡟
⣿⣿⣯⣿⡇⢈⣿⣿⣿⣿⣿⣿⣧⣼⡿⣿⣿⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁⠀⠀⢀⠾⣿⡆
⠙⠛⠻⣿⣧⣾⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣀⣷⣶⣿⣿⣶⣄
⣴⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⠀⠀⣴⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣟⣟⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇
⡿⠁⠀⢻⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠙⠁
⠀⠀⠀⠀⠙⠛⠁⣀⡄⠀⠀⠀⠀⠀⠀⠈⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠤⠶⢶⢶⣿⡆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠋⠁⣾⣿⣿⣿⢿⣿⣿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠲⠖⠈⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢀⡄⠀⠀⢠⡀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠊⠀⠀⠀⠀⠓⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣯⣦⣕⢦⡀⠀⠀⠀⢠⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⡄⠀⠀⠀⢀⡴⣪⣴⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡕⢄⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⡠⢪⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣮⠢⡀⡿⣿⣿⣶⣶⡂⠈⠉⠉⠁⢐⣶⣶⣿⣿⢿⢀⠔⣵⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣝⢿⣿⡛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣿⡿⣫⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣍⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣩⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢾⡏⠙⠻⠿⣶⣶⠖⠶⣶⣶⠿⠛⠋⢹⡷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣇⠀⠀⠀⠀⡿⠀⠀⢿⠀⠀⠀⠀⣸⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣦⠀⠀⠀⣥⣤⣴⣤⠀⠀⠀⣴⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣀⣴⣾⣿⣿⣿⣿⣷⣦⣠⣿⣯⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⠿⠛⠉⣉⣉⠉⠛⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣄⡀⠀⠉⠉⠀⢀⣠⣾⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⡿⣿⣻⠷⣶⣶⣞⣋⡵⠋⠀⣸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣿⡛⠂⠉⠁⠀⠀⠀⠀⡀⠀⡼⢻⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⡏⠳⣄⠀⠀⠀⠀⠠⠊⠁⠈⠀⣼⡟⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣿⣇⠀⠈⢅⠀⠀⠀⠀⠀⠀⠀⢘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⣿⢆⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢝⠀⠁⠀⠀⠀⠘⠀⢀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢠⣾⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⢠⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢿⡿⠛⢹⠿⠻⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡛⢶⣿⣦⣀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡛⣿⣿⣿⣶⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣾⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣿⠿⠿⣿⣿⠯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠸⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⢠⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠇⠀⠘⠁⢿⣿⣿⣿⠿⠋⠀⠈⠉⠻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣁⣀⣀⣤⣶⡿⠿⠋⠁⠐⢦⡀⠀⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠛⠋⠙⠉⢡⠌⠀⠀⢀⠀⠀⠠⠄⠀⠀⠀⠀⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡄⠀⠀⠀⠈⢀⠀⢠⣏⣾⡏⠀⠴⠃⠀⠀⠈⠷⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠃⠀⠀⠀⠀⠀⠐⠰⢤⣿⣿⣷⡖⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣠⠴⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡼⢅⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠈⢀⠀⠈⠙⢦⡀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠄⣤⠀⣀⠈⠆⠀⠀⠀⠉⢻⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠺⣿⠀⢀⡀⠀⠀⠀⠘⣇⠈⠉⠀⠀⠀⠀⠀⢄⠀⣠⠁⠰⢤⡎⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⣧⠀⠀⠀⠈⢨⢳⠀⠀⠀⠀⠀⢀⢤⡇⠳⠶⢤⠀⠈⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠸⣦⣀⣀⣠⠀⠀⢠⠎⢰⡟⡀⠀⠀⢠⡥⠎⠀⠀⠀⠀⣰⠋⠀⠙⢦⡀⠀⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠉⠉⠀⢀⡴⠁⢀⣾⠃⡇⠀⠀⢸⠇⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠰⠋⢡⣴⣿⠏⠀⡇⠀⢀⡟⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡀⠀⡰⠁⢀⡾⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠤⣢⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠤⣤⢤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⡠⡄⢠⠤⠤⠤⠔⠒⠒⠉⠁⠀⠙⠦⣄⣀⣀⣀⣠⡤⠖⠋⢣⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠤⠄⣤⡑⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡇⠀⠀⠀⠀⠈⣇⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⢹⡄
⢎⣁⡀⠀⡆⠀⡰⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡒⠓⠀⢹⡀⠀⠀⠀⠀⠘⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠃⠀⣀⡀⠃⠀⠑⠀⢀⣿⠀⣀⣠⣤⠴⠖⠋⠉⣇⠀⠈⢄⠀⠈⢧⠀⠀⠀⠀⠀⢹⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠂⠀⠀⠀⠐⠊⠁⠘⠋⠉⠁⠀⠀⠀⠀⠀⢻⡀⠀⠀⠑⠄⡘⣦⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠙⣧⣀⣀⣠⣤⠼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣀⣤⠶⢏⣁⣀⣀⢀⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣀⣀⣀⣀⣈⣳⡀⠀⠀⠀⠹⣆⣤⠴⠖⠛⠉⠀⠀⠀⠀⠈⠉⠉⢙⡄⠠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⢏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⢱⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢾⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠸⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡖⠚⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⡴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣊⡤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢡⣾⣷⡈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⣠⣖⣭⣭⣭⣈⣉⣉⣉⣉⣉⡉⣉⠉⢉⣉⣉⣉⣉⣉⣁⣥⣬⣭⣭⣭⣶⣄
⠀⠀⠀⠀⠀⣼⢁⣴⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⠉⣧
⠀⠀⠀⠀⠀⢳⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠟⠋⠁⠀⠈⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⡏⡟
⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⣿⣿⡇⣿⢷⣤⣄⡀⢀⣠⣤⠤⠀⣿⣿⣿⣿⣿⣿⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡏⣿⣿⠉⠛⠋⢼⡇⢨⠻⠟⠉⠀⣿⡍⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣎⡏⢀⠀⢀⣴⣇⢀⡀⠀⠀⠠⣏⡼⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢼⣦⣿⣿⣿⣿⣷⣦⣠⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡹⡍⠻⠶⠆⠀⠉⣉⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⣿⣦⣶⡤⠔⠋⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡷⠀⠈⠣⠀⠀⠀⠀⠀⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⣀⠀⣁⣀⡠⣐⡴⡞⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣺⣟⣷⣾⣾⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡤⠞⠋⠉⠉⠓⠦⣄⠀⠀⠀⠀⠀⡴⠋⠙⠚⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢳⣀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⣠⡴⠦⠤⠤⠶⠲⢦⣠⠀⣧⠀⠀⠀⠀⠈⠓⠲⠤⠤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡟⠋⠀⠀⠀⠀⢀⣀⣀⠈⢷⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡾⢡⠶⠶⠀⣀⡀⠋⠀⠈⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣹⠁⠀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⡏⠓⢚⡇
⠀⠀⠀⠀⠀⢰⠦⣤⣶⠀⠀⠀⠀⠛⣇⠀⠀⠙⢦⣄⣠⠴⠋⠀⢀⡴⠃⠀⠀⠀⣀⡤⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠈⠙⠒⠺⠦⢤⣀⣀⠀⠈⠳⢤⣤⣀⣀⣀⣤⠴⣞⣉⣀⣤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢺⡆⠀⢳⠺⢦⡀⢿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⠖⠋⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠭⡿⠀⠀⣿⣫⡽⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠈⠉⠁⣀⡀⠤⠤⠤⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡤⠚⠉⢀⣠⠶⠶⠆⠀⠒⠢⠤⣈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡠⠊⡄⣡⠞⠉⠁⠀⠀⠀⠀⠀⠠⠀⠈⠙⢦⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⢠⠃⠘⢠⠃⠀⠀⠀⠀⠀⠀⠀⣠⠖⠊⠉⠁⠀⠈⠓⢤⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠨⢰⠇⠈⠀⠀⠀⠀⠰⠁⠀⡜⠁⠀⠀⠀⢠⡄⢠⡀⠀⠉⠲⣤⣀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢉⠀⠀⠀⠸⣣⣀⠊⣤⠀⠀⠀⠀⢀⠇⠀⠀⣻⢦⠀⠀⠞⠀⣀⢀⠀⠀⠁⠂⢄⡉⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠑⣤⠁⠈⠉⠉⠉⠀⠁⠂⠀⠰⠷⠚⠂⠀⠈⠀⠀⠀⠀⠈⠉⠀⠈⠀⣀⠀⠀⠉⠓⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⡰⠛⠛⠒⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠤
⠀⠀⠀⠀⠀⠀⠀⢸⡴⠁⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠢⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⡃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣤⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⠿⣿⡆⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⡾⠛⠛⢛⠉⠉⠉⠙⠛⠛⢦⣄⣀⣀⣀⣴⡞⠉⠥⠠⣤⣤⣤⠔⣠⠀⠀⢹⣾⡿⠛⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⣷⣿⠁⠀⠀⠥⠀⠻⠟⠻⠈⢡⡄⣿⡟⠉⠙⣿⠀⠀⠒⠀⠈⠁⠁⠀⠈⠀⠀⠀⡏⠀⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠘⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⠀⠀⠙⠢⠤⠄⠀⠀⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠀⠀⠀⠀⠀⠐⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⢀⣀⣤⣴⣾⣷⣿⣿⣿⣿⣿⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠙⣿⣿⣶⣯⣄⣀
⠀⠀⠈⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢀⡄⠹⣿⣿⣿⣿⣿⡿⢻⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠻⣷⣿⣿⣿⣿⣿⡇⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡄⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡛⠃⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠾⠻⠛⠟⠷⢤⣭⣶⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠁⣾⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠉⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠁⡿⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠀⠘⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠉⠉⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣠⡠⠤⠤⠤⠤⠤⢤⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠃⠤⣤⣤⡄⠀⢤⣤⠤⢹⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢟⠀⠘⠛⠁⠀⢸⠀⠛⠀⢀⡯⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣹⡀⠀⠀⠀⢀⣸⠃⠀⠀⣸⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⣀⡠⠄⣀⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⢧⡀⠈⠛⠟⠁⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⡇⠸⡄⠙⠢⠤⠤⠖⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠀⠀⠀⢳⡀⠙⢦⡀⠀⠀⢀⡞⢸⡦⣀
⠀⠀⠀⠀⠀⠀⢀⡼⠉⢄⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠙⣠⡤⠟⠀⢸⠇⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⢠⠟⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠳⠔⢹⡇⠈⡗⠢⠊⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡇⠀⠀⠀⠀⠀⠀⣸⠀⠘⡇
⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠄⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡇
⠀⠀⣟⡢⣄⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡇
⠀⣸⠀⠙⠲⠤⡒⣢⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣸⣃⣀⠤⢤⠃
⠀⡟⠀⠀⠀⠀⣰⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸
⣸⠁⠀⠀⠀⡴⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⡇
⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⢿⠀⠀⠀⠀⣧
⢸⠀⠀⠀⠀⢻⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣼⠀⠀⠀⠀⢹
⠸⡇⠀⠀⠀⠘⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⢸
⠀⢿⠀⠀⠀⠀⢿⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸
⠀⠈⣇⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣾
⠀⠀⠸⡆⠀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿
⣿⠿⠿⠿⠿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣟⣛⣿⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⣤⡎⠀⢐⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡏⠀⠘⠂⠸⠿⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡛⠉⠁⢀⣿⣿⣿⡿⢂⣀⣤⣤⣤⣶⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠷⣥⡀⠙⢘⣿⣽⣷⣿⣿⠿⠿⠟⠛⢻⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠀⣠⣾⣿⣿⢟⣋⡁⠀⣀⣀⠤⠀⠘⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡐⠿⠛⠁⠀⠸⠯⠝⢿⠃⠛⠚⠛⠀⠙⠏⠒⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢻⡀⣼⠄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⣀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠹⣜⠧⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⢀⣀⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣀⠀⢀⣠⠏⣿⢧⡀
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⡾⠀⢀⡤⠶⠶⢤⣤⡶⠛⡿⠙⠟⠿⡿⠟⢁⠔⢻⠀⠳⡦⣄⣀
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⣣⠞⠁⠀⠀⠀⠀⣴⢣⣔⣾⠀⠀⢀⣷⡎⠁⠀⣾⠀⠀⡆⠀⠈⠙⠓⠦⠤⢤⡀
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⢹⠀⠀⠸⣿⡗⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠹⣄
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠉⠝⠁⢸⠀⠀⣸⣿⡆⠀⠀⡇⠙⢎⠁⠀⠀⠀⢀⡟⠀⠀⠀⢿⡄
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣾⠇⠠⡀⠀⠀⠀⢸⡀⠀⣿⣿⣧⠀⢠⡇⠀⠈⢣⠀⠀⠀⣸⠁⠀⠀⠀⠸⡇
⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⡰⣃⠀⣀⢀⣾⠏⠀⠀⢣⠀⠀⠀⠸⠃⢠⣿⣿⣿⠀⢸⠀⠀⠀⡎⠀⠀⠀⢹⡅⠀⠀⠀⠀⣷
⠀⠀⠀⠘⢧⡀⠀⠀⢀⠾⣪⢕⣋⣵⡋⠀⠀⠀⠀⠀⢣⠀⠀⠀⡇⢸⣿⣿⣿⡀⡿⠀⠀⡘⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠙⠲⠚⠓⠚⠋⠉⠀⠀⣧⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⣾⣿⣿⣿⣇⡇⠀⠠⠁⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⣿⣿⣿⣿⠁⠀⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⡌⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⣾⡖
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⢀⡤⡿⠉⢉⡛⠻⡿⠿
⠀⢀⡏⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣖⠎⠨⣴⠖⣠⣴⠀⢸⡀
⠀⡸⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡈⠀⠀⠀⢸⠀⡈⠓⠂⣈⠧⠀⢰⠗⠃⠀⠀⠀⠢⠄⢁
⢠⠇⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⡀⠈⢎⢉⠞⠀⢀⡜⠀⠀⠀⢀⣠⢦⠐⠂
⣜⣀⡠⢴⠀⡞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣔⡠⠴⠊⠀⡀⠀⠌⠐⠉
⠉⠁⠀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠂⠀⠀⠉
⠀⠀⠀⠘⡇⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠀⡄
⠀⠀⠀⠀⢳⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠
⠀⠀⠀⠀⠘⣇⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠤⠤⠼⡀⠑⠊⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠙⠢⢌⡁⠒⠒⠒⢲⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢰⠃⠀⠀⠀⣀⢴⣁⡓
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠫⢙⣒⣒⡊⢕⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⢆⠀⠐⢈⡤⠚⠀⠈⢂⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠘⠤⠔⠊⠁⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠶⠦⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⢤⣄⣀⡀⠀⠀⣠⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠳⣄⠀⠀⠀⠀⠀⠀⣠⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⠑⠦⣄⣀⠀⠀⡇⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠒⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠶⠤⠤⢄⣀⣀⣀⡠⠤⠖⠒⠉⠉⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣵⣶⣶⣶⡶⠿⠶⠒⠒⠒⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠒⠒⠲⠾⣶⣶⣶⣶⣦⣸
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⣽⣿⣿⡟⠉⠀⠀⠀⠀⢠⠶⠚⠓⠲⡌⠙⢷⡄⠀⠀⢀⡴⢛⡥⠖⠛⠲⢦⠀⠀⠀⠀⠙⠉⢻⣿⣿⣿⠐⢦
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⢣⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀⠀⠀⠀⣿⠟⠛⣿⠁⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⢦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠸⡄⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⣠⡿⠀⠀⢻⣇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⡬⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠙⠒⣷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⣴⠟⠀⠐⠒⠲⣽⣷⣀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟⢰⠧⠊⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢳⢳⠈⠳⢤⣀⣀⠀⠀⢀⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠈⢿⠙⠷⣤⣀⡀⠀⠀⣀⣀⣤⠶⠛⣵⠧⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⢫⠑⠢⠤⠤⠤⠤⠤⠤⠤⠔⠚⢩⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠈⠉⠒⠒⠴⠖⠒⠊⠁⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠒⠒⠒⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠔⠋⠀⠀⠀⠀⠈⠐⢤⡀⠤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠄⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠀⠀⣀⠔⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣠⣞⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⠿⠻⠛⠉⠉⠉⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣾⣦⣀⡀⣄⣀⣠⣴⣾⣦⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⠉⠉⠙⠻⣿⣿⣿⣿⡿⠿⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣽⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⡆⢹⡯⠀⠀⠀⠈⠉⠙⠛⠖⠀⠀⠀⢀⣀⣀⡀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⡀⠈⠁⠀⠀⠀⠞⠻⠓⢦⣤⡀⠀⢀⡞⠉⠉⠙⠃⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢼⡛⠛⠉⠳⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠦⠤⠞⠂⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢀⡤⠶⠶⠶⠶⢦⣀⠀⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠌⠢⡀⠀⠀⠀⠀⠀⠉⠙⠶⠤⠍⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠢⠄⠤⠄⣀⣤⡾⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣠⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠋⠀⠀⠈⠓⠤⡀⠀⠀⣀⣀⡤⠖⠊⠁⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢆⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⠏⠀⠳⠤⢤⣀
⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠘⣄⠀⢠⣿⣿⣿⣿⣿⢷⡀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠈⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡌⠀⢨⣿⣿⡇⠈⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠛⠉⠉⠉⠙⠛⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢀⣤⣀⠀⠀⠀⠀⣀⣀⣀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢳⠴⠒⠨⠿⣄⣀⡼⠟⠒⠢⡼⠟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠸⢸⠀⠲⠗⠀⡎⠉⡇⠚⠓⢠⠇⡶⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡌⠓⠒⠒⠚⠀⠀⡉⠒⠈⠉⣠⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⣠⣶⣿⣿⣷⣤⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣴⣿⡁⢀⣤⣀⣹⣧⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⠻⠿⠿⡿⠿⠿⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣿⣿⡇⠣⣀⠀⠀⠀⢀⣠⠇⢹⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⣹⣶⣫⡀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢀⡞⢿⣿⣿⡟⢆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠘⣿⣿⡇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢀⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢨⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠸⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣁⣈⣉⣉⣁⣙⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⣧⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣰⠿⣄⡀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣻⣷⣶⣤⣄⣀⣙⣿⣦⣄⡙⠓⠶⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⠛⠉⠉⣽⠟⠉⣉⠉⠻⣯⠉⠛⠻⢿⣷⣶⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣏⠀⣾⣿⣵⠀⣿⡇⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣈⠉⢁⣠⡟⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠂⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣇⠀⠀⠀⢀⣼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠆⠀⠀⠀⣠⣾⠏⢹⡀⣿⡀⠀⠀⠉⠁⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⢹⡏⠀⠀⠀⠀⠀⢀⣤⣾⠋⠙⡆⠀⢳⣸⡇⠀⠀⣄⡀⢀⣸⡿⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⠶⠲⡞⠗⠲⣄⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠚⠛⢻⡁⠘⣆⠀⠹⡄⠀⣿⡇⠀⠀⠙⢻⣿⠋⠀⠀⠀⠀⢴⣦⠀⣿⢀⣶
⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠈⢷⡀⠸⣆⠀⢹⣤⡿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣶⣧⠾⣡
⠀⠀⠀⠀⠀⢀⢀⣬⠀⢰⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠻⣦⣽⣤⣶⠟⠁⢀⣴⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣻⣿⠿⠛⠉
⠀⠀⠀⠀⠀⢸⣯⣻⠷⣺⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⢀⣤⡾⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⣀⣴⣿⣿⣻⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠛⠛⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⣠⣾⠋
⠀⠀⡠⠞⠉⠀⠈⠁⠈⠉⠙⠻⡿⠿⠶⢶⣶⣶⡶⣶⡿⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⣠⡾⠟⠁
⢀⡞⠁⠀⠀⠀⢾⣟⡓⠀⠀⠀⢹⡄⠀⠀⠀⠀⢀⣾⠃⢀⡏⡽⣿⠛⠛⣻⣿⠟⢿⠟⠿⣿⠻⠿⠿⠟⠛⠻⣿⠟⠛⠉
⣾⠀⠀⠀⡀⠀⠀⡍⠻⣦⠀⠀⠀⣷⠀⠀⠀⠀⣸⣿⠀⣸⠋⠀⠘⣧⠞⠉⠀⢧⢸⠃⠀⣿⠀⠀⠀⠀⠀⠀⢻⡇
⣇⠀⠀⠘⣧⠀⠀⡀⢀⣼⠀⠀⠀⡟⠀⠀⠀⢀⣿⣿⠀⡇⠀⢠⡾⠹⡆⠀⠀⠘⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿
⢹⡄⠀⠀⠈⠻⠶⠷⠛⠁⠀⠀⣸⠃⠀⠀⠀⢸⡇⣿⢠⠇⣰⡏⠠⡀⢿⠀⠀⠀⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠀⣿⠇⣿⢸⠐⢿⣌⢂⣀⣺⡇⠀⢰⡇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠉⠛⠒⠶⠶⠚⠛⠁⠀⠀⠀⠀⠀⠀⣿⠀⢻⢸⠀⠀⠙⡟⢩⠉⠀⠀⣾⠁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⢾⣾⣦⣤⣴⣿⣤⣤⣤⣼⣯⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠛⡉⠉⠉⢛⠛⣦⠀⠀⣴⠛⠉⠉⠉⢉⠙⢦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣶⣷⣶⣞⣛⣷⣾⣃⣀⣻⣶⣶⣷⣛⣙⣷⣞⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠰⡀⢀⢴⠖⠀⠅⠀⠀⠄⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢨⠃⡌⠄⡐⠁⠃⣠⠀⢀⠄⠒⠒⠶⠒⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠐⢀⡎⠀⡔⠀⠇⠀⠀⠋⠀⠂⠀⡀⠀⢀⣤⣤⣶⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⠀⠃⡇⠀⡇⠀⣼⠀⢠⠘⡇⠀⡼⢀⣴⠿⠋⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡻⣧⡀⠁⠁⡀⠘⣿⣄⠢⠀⢿⣀⡵⣋⢥⣐⣦⡄⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠐⠦⣄⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠝⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢙⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀⠀⢀⣴⠿⠛⠛⠛⠳⠄⠀⠀⠀⠀⡴⠛⠛⠛⠷⡄⠀⢿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⢀⡴⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠤⠠⠴⣄⠀⠀⢸⣿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣄⢟⠃⠀⠀⠀⠀⠀⠒⠀⠀⠑⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠘⡟⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠘⡏⠣⠀⠀⠀⠀⡠⠀⠐⠄⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡾⠀⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⢸⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠹⣷⣤⣤⣤⣤⣤⣤⣾⠏⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⡰⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠱⠀⠀⠈⠉⠉⠁⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⡀⠀⠀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠾⠿⢿⡟⠛⠛⠻⠿⠷⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠋⠉⠀⠀⠀⠀⣀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣦⣄⡀
⠀⠀⠀⠀⣠⡾⢿⡅⠀⠀⠀⣀⣀⡤⠶⠛⠁⢸⠀⠉⠓⠦⣄⣀⡀⠀⠀⠀⠀⠀⢸⠟⠻⣦⣀
⠀⠀⠀⣼⠏⠀⣰⣟⠛⠋⠉⠉⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⢒⣿⡀⠀⠈⠻⣷⡀
⠀⠀⣸⡟⠀⣰⠃⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠹⣄⠀⠀⠘⣷
⠀⢠⡿⠀⣰⠃⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠘⢧⡀⠀⢻⣇
⠀⣾⢇⡼⠁⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠙⣾⡆⣿⡄
⢰⣿⢿⡀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣠⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢀⣾⣿⣷⣼⣇
⢸⡇⣾⣷⣄⠀⠀⠀⠀⠀⣧⡤⠴⠶⠚⠋⠁⣷⠈⠙⠓⠶⠤⠤⠤⠤⣼⡁⠀⠀⠀⠀⣠⣿⣿⣿⣿⣸⣿
⣿⡇⣿⣿⣿⣦⡀⠀⣠⠞⠉⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠇⠙⣆⠀⢀⣼⣿⣿⣿⣿⣿⠁⣿
⣿⡗⣿⣿⣿⣿⣿⣾⠁⠀⠀⠘⣇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⣳⣿⣿⣿⣿⣿⣿⣿⠀⢿
⣿⣷⢸⣿⣿⣿⣿⣿⣷⣄⠀⠀⠘⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡏⠀⠀⣠⣾⣿⣿⣿⣿⡿⣿⣿⣿⠀⢸
⣿⡞⣿⣿⣿⡌⠛⢿⣿⣿⣿⣦⡀⠈⢧⠀⠀⠀⣇⠀⠀⠀⢀⡞⠀⣠⣾⣿⣿⣿⡿⠛⠁⠀⣿⣿⣿⠴⢾
⢹⡇⢻⣿⣿⡇⠀⠀⠙⣿⣿⣿⣿⣦⣌⣷⡴⠖⢿⠒⠒⠶⣾⣠⣴⣿⣿⣿⡿⠋⠀⠀⠀⢸⣿⣿⡇⠀⣿
⢸⣷⢸⣿⣿⣿⡀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡽⣄⠸⡆⠀⡼⣵⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⣿
⠈⣿⠀⡏⢿⣿⣷⡀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣾⣶⠷⣾⣽⣿⣿⡿⠋⠁⠀⠀⠀⠀⢀⣴⣿⣿⣇⡀⢠⣿
⠀⢿⣇⣿⠚⣿⣿⣿⣦⣀⣀⣀⣠⣶⣿⣿⣿⣿⣧⣀⣴⣿⣿⣿⣷⣤⣄⣀⣀⣤⣶⣿⣿⣿⢏⡞⠉⣻⡇
⠀⢸⣿⠈⢧⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢀⡟⠉⢻⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⡞⠀⠀⣿⠃
⠀⠀⢿⡄⠈⢧⠀⠈⠙⠛⢿⡿⢿⡿⠋⠀⠀⣸⠁⠀⠈⣇⠀⠈⠙⢯⣉⡿⠋⠉⠉⠀⢀⡞⠀⠀⣸⡟
⠀⠀⠸⣧⠀⠈⣇⠀⠀⠀⢀⡽⢯⣄⡀⠀⢠⡇⠀⠀⠀⢹⡀⠀⣀⡤⠟⢧⣀⠀⠀⠀⡼⠁⠀⢀⣿⠁
⠀⠀⠀⢿⡆⠀⢸⡄⢀⡴⠋⠀⠀⠈⠙⠢⣼⣀⣀⣀⣀⣀⣷⠞⠁⠀⠀⠀⠈⠳⣄⣰⠃⠀⠀⣼⡏
⠀⠀⠀⠘⣿⡀⠀⣷⠋⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⢀⡴⠛⠲⣄⣰⡿
⠀⠀⠀⠀⠸⣷⡞⠉⠙⢦⡀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⡴⠋⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠙⢦⡀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢠⠞⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠙⢆⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣴⠏⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠈⡟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠙⢻⡀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣄⣀⣀⣤⣤⣴⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⣄⣀⠀⣀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⢀⡄⠀⢀⣠⠋⠁⠰⠀⣴⣾⣧⢀⣼⣇⣿⢀⣿⡆
⠀⠀⠀⠀⢀⡰⠀⢠⡂⡀⠀⠀⡸⢢⢼⣿⣧⣤⣤⣶⡿⣿⡿⠿⠿⠿⠋⣡⣾⣿⡇
⠀⠀⠀⣄⠈⠑⣞⢻⣷⠅⢸⡶⣷⣴⣏⣿⡿⠟⡹⠁⠴⣋⠀⠀⠀⡴⠞⠋⣡⢿⡇
⠀⠀⠀⠘⢷⣦⣌⣹⡿⢖⣿⣥⣿⣏⡠⣻⡷⠈⠀⣦⡀⠟⠀⠀⠉⠐⠋⢉⣡⣿⣁⡴⢃⡤
⠀⠀⠀⢀⠀⣹⣿⣿⠟⣡⣿⣿⣿⡿⣠⡹⢟⣿⣷⣌⢷⣄⠀⠀⠀⠀⠐⢛⠻⠯⠿⠛⠋
⠀⠀⠀⠀⣼⣿⣿⠃⣴⢋⣷⣿⣿⣶⣥⣿⣿⣿⣿⣿⣬⣼⡷⠀⠀⢀⣀⣼⡆
⠀⠀⠀⢠⠼⠟⣁⣼⣿⣿⣿⡿⢃⠚⣉⠉⠉⣙⣿⣿⣿⣿⠶⠀⠄⣸⠟⠁
⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣶⣾⣾⣿⡿⠛⢟⣟⣿⣿⣽⣷⣶⡆⢥⣄⣀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠠⠿⢟⠫⢴⠿⠏⠁⢹⣿⣿⠁⣀⣠⣤⣿⣿⣿⡿⣿⣄⣾⣿⡿⠶⠶⠦⣬⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢤⣀⣰⣿⡿⣹⣍⠉⢉⣭⡿⠟⢀⣿⢿⡿⠙⠁⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⣴⠀⡀⠁⠀⣉⠁⢿⡗⢿⣽⣿⡿⠷⠁⠀⢸⣿⢺⣿⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠄⠈⠐⠁⠀⠀⠠⢀⠈⠇⣼⡇⠀⠁⠉⠀⠀⠀⠀⣸⢧⣿⠏⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⢿⡇⠀⠀⠀⠀⠀⢠⣼⣿⣼⠏⠐⠀⠀⠈⢧⡀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⣀⢤⣹⣿⣼⣿⣧⡄⠀⠀⢀⡀⠑⢬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠻⣶⣤⣘⣠⣽⡿⠿⠛⠋⠀⠂⠀⠈⠁⠤⡄⣁
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⠠⠰⠟⠛
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣀⠒⣠⡄⣴⣾⣷⣿⡾⠄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠉⠈⠁⣸⠃
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡘⠁⠈⠏⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠞⠱⠀⠜⢃⣤⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⡠⢐⣴⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣸⣿⣿⣯⣞⣽⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⠙⠷⠿⠿⠿⠛
⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣀⣠⠄⠐
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢸⣿⣿⣷⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⢍⡀
⠀⠀⠘⠛⠛⠛⠉⠉⠉⠛⠳⠦⣤⣴⣄⣲⡒⡖⠱⣀⡀⣿⣤⡌⠁
⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠻⠻⠿⠿⠛⠉⠁
⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⢉⣙⣿⣦⣴⣴⣧⣴⣿⣿⣿⣽⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⠉⠳⠿⣿⠿⠿⣿⠿⠿⠿⠛⠻⢿⣿⠿⢧⣼⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣤⣶⣤⣀⠀⠀⠀⠀⣤⣴⡾⠿⠿⠶⣆⠀⠀⠙⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⠉⠉⠓⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠀⢀⣾⣿⢃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⡾⣶⣦⠀⡀⠀⡰⢃⡔⢺⣾⣷⡄⠀⠀⠀⠸⣿⣇⡞⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⢛⣫⠀⡇⠀⠁⠀⠀⠬⣥⠄⠀⠀⠀⠀⡀⠀⢈⡇⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⡎⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣤⠔⣿⣇⠀⢠⠆⠛⠐⢤⣀⠀⠀⠀⠀⡖⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢅⡀⠀⠀⠉⠆⠀⠀⠀⢀⣭⠃⠀⠀⠈⢣⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠉⠹⣏⡒⢒⣋⣩⣽⠟⠀⢠⠂⣰⠁⣰⠃
⠀⠀⡖⢒⡦⠤⢀⣀⣀⣀⡀⢾⡉⠳⣄⠀⠀⠀⠀⠀⠀⢣⡂⠀⢌⠻⠟⠛⠛⢃⠆⠀⠃⣸⢃⡼⠃
⠀⠀⠙⠪⢤⣀⣀⣀⠘⠁⠁⠉⠉⠑⠛⠓⢦⠀⠀⠀⠀⠀⠻⡄⠈⠑⠶⠶⠖⠁⠀⠀⢀⣧⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⡟⠉⠉⠉⠙⠓⠲⣦⡙⢦⡀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠒⠒⠒⠢⠤⣤⡀⠀⡀⠉⠲⡀⠀⠀⠈⠑⠒⠶⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣠⠤⢀⡀⠀⠙⠀⠱⡄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⣀⠀⠀⠀⠀⠐⢄⠐⠀⠀⠀⠙⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠿⣿⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⢀⡀⢠⠇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⢏⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠶⠶⠿⢶⣶⣞⣛⠉⣁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠆⠀⢀⣀⣀⣀⠀⢁⣀⣠⡤⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠄⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡟⠀⠀⠀⠈⠙⠛⢛⣿⣿⡟⠁⠀⡀⠀⣀⠀⠀⠀⠀⠀⢳⡄⢠⣄⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠙⡇⠁⠀⠦⣤⣴⣶⡿⣻⣧⠇⠀⣠⣴⣶⣿⣿⡆⠀⠧⠀⣸⣷⣰⣿⣆⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢿⠀⠀⠀⠠⣴⣿⣿⠟⠀⣻⠏⠀⣴⣿⣷⣶⣶⣽⠿⣄⠀⠀⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⡀⠠⣀⠀⠀⠉⠉⢁⣤⣾⡟⠀⢰⣿⣿⣿⣿⣿⣿⣷⣸⢠⡀⣿⣿⣿⣿⣿⣿⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⢻⣿⣎⠻⣦⠸⣷⣤⡶⠶⢟⣻⡿⠁⡄⣾⡟⠉⠉⠉⠙⠛⠿⢿⠇⢡⣿⣿⡿⠟⠛⠡⣄⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣼⣿⣿⣿⣷⣬⡓⢶⣦⣤⣶⣿⣿⣧⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠯⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣿⣿⣿⣯⣿⣿⣿⡶⠈⠛⠛⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣥⣾⣿⣿⠟⡉⠛⠣⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣭⣥⣶⣯⣤⣾⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠠⣾⠿⠿⠿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣀⣴⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⣿⣿⣿⣿⣿⣿⣭⣴⣶⣶⣶⣶⡿⣿⣷⣿⠟⠛⢻⣿⣿⣭⣛⢰⣆⣀⣀⡾⣿⡿⠟⢓⣺⡷⢶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢒⣶⣆⠛⠛⣿⣿⣿⠟⠛⠉⠉⠉⠁⠀⣤⣿⣡⣤⢶⣾⣿⣿⣿⣿⢿⡿⠟⢿⣿⠟⠿⣿⣾⣿⣥⣤⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⢻⣧⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠤⠤⠤⠀⠹⣿⣿⢿⡁⠀⠀⠏⠀⠀⠼⠛⢫⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⡄⣿⣿⡇⢀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠀⣹⢟⠮⣿⡄⠀⠀⠀⠀⠀⠀⢠⡀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠹⣜⢿⣷⡜⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠘⠳⣄⡀⠀⢀⣀⡼⠋⠀⠛⢁⣀⠀⠀⣤⣀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣿⡀⣿⣿⠿⠿⠄⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠁⢠⡆⢠⣴⣮⣝⠛⣓⠈⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣿⡇⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⡉⠉⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣴⠶⠦⠤⠶⠴⠶⣤⣤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠆⠀⢀⣀⣀⣠⣤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣿⣿⣿⣿⣏⠻⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠁⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠈⢻⣿⣿⣿⠀⠘⢿⣿⣿⣷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠃⠀⠀⠙⢿⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣽⠏⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣴⡶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢨⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⢀⣠⡴⠞⢻⡏⠈⣿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠐⠛⠉⠀⠀⠀⠀⠀⠙⡆⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⣿⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⣀⣴⠟⠀⠀⣰⣿⣿⡇⠀⠀⢀⣤⣀
⠀⣰⠄⠀⠀⠀⠀⠀⠉⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢾⡛⠁⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀
⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣟⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⣠⣦⠀⠀⠀⠁⠀⠀⠀⠉⠛⠶⢤⣀
⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠈⠀⣠⣴⣾⣿⣿⠿⠛⠉⢶⠬⠙⠂⠀⠀⣾⣿⣿⣿⣿⢣⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⣠⣴⣿⠿⠿⢻⡍⠛⠶⣤⡀⠀⠠⠀⢷⡄⠀⣿⣿⣿⣿⠏⢸⣿⣿⠇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⣠⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠔⠦⠀⠉⠀⠸⣿⣿⢯⠀⠘⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⠘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⡿⠉⠉⠉⠉⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣥⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⢙⡆⢻⣿⣿⠟⢿⣿⣭⣛⠻⠿⠾⣿⣭⣽⣓⣶⢤⣤⣀⣀⡾⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢁⣼⠇⠀⣿⠋⠀⠀⠈⠻⢿⣿⣶⣤⣀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣷⣦⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣄⡀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⠟⠁⣼⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⢀⡴⣶⠶⣾⠁⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⢀⢀⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⢀⡾⠁⠙⢿⡇⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣀⣼⡇⢩⡟⠻⠿⠃
⠀⠀⠀⢀⡾⠁⠀⠀⠈⢻⣦⡀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⡀⠈⠉⣰⢏⣠
⠀⢀⣀⣾⠁⠀⠀⠀⠀⠀⠙⣷⣄⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⣤⣉⠻⠿⢀⡼⢋⡏⢹⡀⠀⣠⣤
⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠙⣶⣀⠀⠀⠀⠀⠀⠈⢉⣤⠾⠋⠁⢸⡇⢸⡷⢞⣡⡟
⠀⠄⠀⠀⠀⠀⠀⠀⠀⡐⣰⡟⠀⠀⠀⠀⠀⣀⣽⣷⣤⣀⣠⣤⠾⠋⠁⠀⠀⠀⢸⡇⢸⡟⢦⣙⠦⣀
⠀⠀⠀⠀⠀⠈⠀⠀⣰⢱⣿⣷⣤⡀⣠⣴⠟⠋⠀⠹⣧⡉⠉⠀⠀⠀⠀⠀⠀⢀⡾⠁⣽⡇⣾⠉⠳⣌⡳⣤
⠀⠀⠀⠀⠀⠀⠀⢰⢣⣿⣿⣿⣿⣿⣿⠋⠀⠀⢠⡶⠿⣷⣄⣤⠶⠚⠛⢷⡀⣼⠁⢰⡟⣼⠃⠀⠀⠈⠙⠛
⠀⣦⠀⠀⠀⠀⣰⠇⣾⣿⣿⣿⣿⣿⠇⠀⠀⣀⣴⠿⣦⡀⠙⠲⣶⣆⠀⠈⠻⣇⢀⣿⣽⠃
⠀⣟⠀⠀⠀⢰⠏⣼⣿⣿⣿⣿⣿⣿⠀⣠⡾⠋⠁⠀⠙⠛⣦⡀⠀⠙⢷⣄⠀⠈⠁⣽⠇
⠀⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠁⠀⠀⣴⠏
⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢰⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣼⠏
⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⣰⣿
⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠿⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠇⠸⠇⠀⠀⠀⠀⠠⠿⠁⠀⠀⠀⠢⠿⠃⠹⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⠿⠛⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⠀⣀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⡿⢃⣿⣿⠏⢹⣿⣿⠃⢿⣿⣿⣿⣿⣿⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠿⠛⣭⣤⣤⣤⠀⠉⠀⠀⠉⠉⠀⣬⣭⣭⣭⠉⠁⠈⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⠀⠀⠀⠀⡀⠤⣤⡀⠀⠀⠀⠀⠀⠀⢀⠠⢤⣀⠀⠀⠀⠘⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢰⠀⠀⠀⠪⡀⠘⢿⡟⠀⠀⠀⠀⠀⠸⢀⠀⠻⡿⠂⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⣀⣀⠀⠀⢀⣀⠀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⣻⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⢸⡉⠢⠤⢔⣙⠀⠈⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠲⠤⣤⣀⣀⣀⡟⠒⠦⠤⠤⢤⣀⣰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⣀⣀⣀⣀⣀⠤⠴⠶⢄⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣤⣤⣤⠾⠤⠤⠤⢤⣀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠬⣦⣤⣤⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⠏⠀⠀⠀⠐⣋⡑⠈⠱⣤⣤⡴⠋⢚⣉⠂⠀⠀⠈⠉⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⢿⣸⠀⠀⠀⠀⠀⣿⡇⠀⢠⠏⠀⢧⡀⢸⣿⠀⠀⠀⠀⠀⣿⢃⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣼⣿⠧⣀⠀⠀⠀⠈⣀⡴⠋⠀⠈⠛⣷⢤⡁⠀⠀⠀⢀⣠⣿⣾⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⢿⠀⠈⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⡝⠀⠉⠉⠛⠉⠉⢀⡿⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡦⢄⣀⣀⣀⣀⡤⢴⠄⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠑⢤⣀⣀⣀⣀⡤⠊⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⡀⠀⠉⠉⠀⢀⣀⣤⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢶⡶⠋⠁⡼⡏⠉⠉⠉⠉⠉⣷⡀⠉⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⡴⠋⠀⠀⢰⠃⠙⠢⠤⠤⠤⠞⠁⢷⠀⠀⠈⠳⡉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡼⠁⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢱⡀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢀⡴⢁⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠈⡿⡄⠀⠀⠹⡀
⠀⠀⠀⠀⣠⠎⠀⠀⠀⢠⠞⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢻⢹⡀⠀⠀⢳
⠀⠀⠀⡴⠃⠀⠀⠀⣰⠋⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⡇⠀⠀⠘
⠀⢀⠞⠁⠀⠀⠀⡴⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⢠⠏⠀⠀⠀⠀⡜⠁⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢷
⣿⣿⡷⣷⣶⣾⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣾⣿
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⣶⣤⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⣀⣤⠤⠤⠄
⠀⠀⠀⠀⢰⣗⣿⣿⣿⡜⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡾⢋⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠈⠿⣩⣷⣾⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣶⠟⢋⣩⣭⣙⠻⢷⣆
⠀⠀⠀⠀⠀⠈⠙⢻⣿⠁⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀⠀⢿⣿⡏⡔⢡⠤⠤⣌⠑⢌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣀⣠⣤⣤⣤⣬⣽⣿⣷⣬⣿⣿⣷⣕⡊⠑⢢⠳⡈⠳⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠿⠟⠛⠋⠉⠉⠉⢉⣽⠟⠋⠉⠙⠻⣿⡛⠛⠿⣷⣷⣜⣄⠈⢦⡈⢻⣦⡀
⠀⠀⠀⢀⣴⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡻⢸⡇⠀⠀⠀⠈⠻⢿⣵⢦⠙⡆⢻⣷
⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣀⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⠈⢻⣷⡡⡹⣸⣿
⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⢿⡟⢧⡴
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣜⢷⣄⡀
⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠓⠆⠀⠀⠀⣠⣿⣿⣿⣿⣿⡣⠂
⠀⠀⠈⠻⢿⣦⣤⣄⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣤⡀⠀⠀⠀⠀⢀⣼⠃⠸⡇⠀⠀⠀⠘⠋⠀⠈⢿⣿⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⢿⡟⠛⠋⠉⠁⠀⠀⢀⣠⠞⠁⠀⠀⣿⠀⠀⠀⢠⣄⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠰⠴⢶⡶⠚⠋⠁⠀⠀⣠⣾⣿⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢷⣀⣀⣠⣴⡾⠟⢻⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⢁⣠⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⢿⣿⡟⠉⢀⣰⠟⠀⡀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠰⣿⢿⣶⣿⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⣀⣾⠇⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⠿⠛⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⣀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣤⣬⣭⣉⣛⣛⣛⣷⣀⣴⠖⠒⠒⠒⠚⠛⠛⢻⣿⣀⣤⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠓⢿⡟⠉⠉
⠀⠀⣴⠶⣦⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⣸⡇⡀⣸⡇⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⡾⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣤⢿⠆⣸⠃⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⢿⡤⠞⠛⠻⠿⢶⣦⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣍⣉⠉⠉⠉⣿⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡉⢻⡿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣴⠏
⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢶⠐⢏⠀⢸⣀⡠⠠⡀⠀⡀⠀⠜⠠⠂⠄⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠄⢊⢠⠈⢈⣁⠡⠜⢔⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣶⣤⣴⡾⠀⠀⡌⠰⢤⠊⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣄⠀⠘⠻⠛⠁⢀⣜⠀⠀⠀⠀⡀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠈⠀⠩⡖⠒⠒⠾⠁⠘⠉⠁⠒⠲⢼⣄⡀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠘⣄⠀⠈⠂⠀⠀⢸
⠀⠀⠀⠀⠀⣌⡁⠀⡼⠀⠀⠀⠀⠀⠀⡘⠀⠠⠀⠀⠀⠄⠀⠀⠘⣦⣤⣤⠤⠄⠊
⠀⠀⠀⢈⠀⢀⠉⢠⡇⠀⠀⠀⠀⠀⢀⠇⠀⠀⢆⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠂⠀⠈⠇⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢻
⠀⣀⢦⣀⣐⣄⠁⠈⡇⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣼
⡀⠀⠈⠙⠢⢒⡄⠀⠙⢄⠀⠀⠀⠀⠇⠀⠀⠀⢀⠜⠀⠀⠠⠄⡠⠃
⡇⠀⠀⠀⠀⠀⡱⠀⠀⠀⢏⠉⠒⠂⠤⠉⣶⡤⠀⠔⠪⠄⠐⠞
⢇⡀⠀⠀⠀⠐⠹⠀⠀⠀⢘⡄⠀⠀⠀⠀⡀⢄⠀⠀⠀⠀⣌
⠀⠈⠑⠒⠤⠤⠖⠀⠀⠈⠀⠈⠓⠂⠠⡸⠀⠈⢄⠄⠀⠐⠋⠉⠑⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣴⣶⣤⣬⣤⣥⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⣀⣤⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⠛⠉⠀⣾⣤⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡶⠴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠦⠤⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣠⢤⡀⠀⠀⡀⠀⠈⠁⢸⠀⠀⠀⠀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠈⠁⠀⠋⠹⠂⠀⠀⢸⠀⠀⠀⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢀⣠⣶⡄⠀⠘⡉⠓⡄⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠶⣒⡾⢯⡴⠋⠀⠀⢠⡤⠄⢻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⣷⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⠩⠑⠊⠀⠀⠀⠀⢨⣤⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠤⠾⠃⠘⠃⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠆⠀⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠐⠶⠂⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢇⡀⠀⢁⣤⣄⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠛⠁⠀⠀⢹⡶⠋⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⠛⠛⠂⠾⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⠾⠶⠶⠶⠶⠶⠛⠛⠋⠉⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⢀⣴⠿⡀⢀⣠⠶⠛⠋⠉⠉⠉⠉⠛⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠶⠶⠖⠛⠛⠉⠁
⠀⣴⡶⠛⠁⠀⠸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣴⠛⠛⠀⠀⢠⣦⣬⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠻⠖⠻⣆⣼⣄⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡤⠤⠤⠶⠶⠦⠤⠤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠐⣤⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠹⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢠⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢀⣸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠃⠒⢺⣏⣀⣀⣀⣈⣷
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠙⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠦⢤⣒⡶⣦⣄⡀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⠀⠀⠀⠀⠁⠈⠙⠳⢿⣿⣦⡈⠌⡇⠐⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠛⠛⠻⠿⠟⠒⠂⠀⣀⣀⣀⣀⠤⣉⠻⣷⠰⢸⠀⣿⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠒⠀⠀⠀⠀⢠⣴⣶⣿⣿⡿⠿⠛⠛⠂⠑⠈⠃⠈⢀⣿⡀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢛⡿⠿⠶⣶⣿⣿⣿⠿⠛⢁⡀⠄⠀⢀⠀⠀⠀⠠⠄⢸⣷⠃⡀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠒⠶⠶⠿⠛⠛⢉⠠⠔⠊⠁⠀⠀⣀⣤⣴⣾⣿⡿⢷⡿⢃⡼⢋⡄⢠
⠀⠀⠀⠀⠀⠀⠀⣿⣶⣌⡑⠢⠤⢄⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣶⣷⠏
⠀⠀⠀⠀⠀⠀⠀⡬⠽⠿⠷⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠹⠿⢞⣋⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠁⣀⣤⣠⣴⣾⣟⣻⢦⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣦⣀⣰⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⣶⡟⠉⠈⢉⣀⣠⢄⡉⠙⢶⣶⣶⣶⣾⠋⡡⢤⣀⣀⠀⠉⠉⣿⣿⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠠⢤⡌⡙⡇⠀⠀⠘⠁⠀⠀⠊⠀⣹⠋⠀⠙⡇⠀⠀⠀⠊⠛⠀⠀⠀⣿⠁⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠞⡌⣇⢻⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣿⠈⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣻⠈⠓⠶⠤⠤⠤⠤⠖⠋⢀⠀⠀⠀⠀⠙⠲⠤⠤⠤⠤⠖⠋⣸⡚⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠐⢶⣄⣀⣀⣀⣠⣴⠆⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠋⠁⠉⠁⠈⠋⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⣀⣀⠤⠔⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠐⠒⠀⠉⠉⠉⠉⠉⠉⠁⠀⠒⠀⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠘⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀⡀
⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀
⢀⣠⣚⠁⠀⠤⠤⠄⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣷⣶⣶⣶⣤⣤⣤⣄⣀⣀
⠉⠑⠠⠀⠤⠀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀⣤⣤⣶⣾⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠈⢉⣛⡛⠛⠛⠉⠉⠁⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠾⠿⠟⠛⠳⠆⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠷⠄⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢃⣶⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣶⣷⡄⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢃⡿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢿⣿⠇⠀⢸⡇⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⣼⠃⡿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢻⡀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠈⠑⠂⠤⠬⠭⠭⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠋⢼⣦⣾⡏⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣴⠾⠋⠀⠀⠘⢿⣿⡁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠛⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣶⣄⡀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣀⣤⣤⣶⣾⣿⣿⣿⣛⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀⣠⣀⣩⣤⣙⣿⣶⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⢈⣥⣤⣤⣦⣯⣭⣭⣙⡛⠳⠶⣦⡶⠾⠗⠚⣛⣩⣭⣩⣩⣭⣭⣍⣹⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⣾⠋⠀⠀⠀⠀⠀⠀⠈⢿⡆⢠⡶⢶⡄⠠⣿⠋⠉⠈⠀⠀⠀⠈⢹⣿⠛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣿⣀⠀⠀⠀⠀⠀⠀⠀⢘⣁⡞⠀⠀⣷⣠⡟⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠈⠈⠁⢀⠀⠀⠀⠀⠀⣠⡾⢟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠈⠛⠒⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠋⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠙⠿⠳⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢔⠦
⠀⠀⠀⡐⢪⡵⠉⠂
⠀⡆⠐⡉⠣⡴⠬⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡀⠱⡀⠬⠂⠮⢴⠃⡹⡄⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣖⣮⣾⣥⠄
⠘⢄⡈⠂⠀⠀⠈⠓⠓⢌⠢⡀⠀⠀⠀⡠⡼⠍⠉⢿⠍⠉⠫⣀
⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠑⢌⠢⣐⠏⠀⢐⡋⠀⠉⢰⡄⠀⢻⢁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⠀⠀⠈⢉⣀⡴⢻⡿⠀⣸⠀⠀⢿⡠⠶⠄⣀⠀⠀⠀⠀⠁⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢷⣋⡼⡰⠋⡵⣄⠀⠀⠀⠀⠀⠉⠓⠒⠶⠒⠉⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢄⣀⡠⡔⠊⠁⠀⠇⠈⢧⠔⡠⣀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢅⡈⠒⢤⡼⠀⠀⠱⡄⠁⠫⣾⢲⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⡀⠀⠀⠈⢦⠀⠘⠺⡔⢄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡴⠋⢑⡤⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⣉⠷⠌⠀⣉⠀⠪⢦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠲⠧⡘⡄⠑⢕⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣸⠀⠀⠑⢼⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⠮⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⡿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣷⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⣾⣿⣿⣷⣶⣦⣄⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠈⣿⣿⣿⣿⠇
⠀⢀⣤⠶⢾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣤⡀
⢀⡞⠀⣀⣀⣿⡇⠀⢀⣀⣤⣴⡶⠾⠿⠛⠛⠛⠛⠛⠛⠿⠷⠶⣦⣄⠀⢀⣀⠀⣠⣴⠾⠛⠛⠛⠋⠉⠉⠉⠉⠛⠛⠛⠿⣶⣄⣀⡀⢸⣿⣿⣀⠀⢻
⡜⠀⠈⠠⠂⣿⡇⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣶⣶⣄⠀⠀⠈⢻⣿⣿⣿⣿⣿⡃⠀⠀⣠⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢈⣿⠓⠄⠁
⡇⠀⠀⠀⠀⢸⡇⠀⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣸⡏⠁⠀⠉⢿⡆⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠉⢸⡇
⣧⠀⠀⠀⢴⢺⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⣿⠁⠀⠀⠀⠸⣷⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⡗⡆
⠸⡆⠀⠀⠈⣹⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠛⠟⠁⠀⢀⣾⠇⠀⠀⠀⠀⠀⠹⣧⡀⠈⠻⠛⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⢸⣏⠁⠀⠀⣰
⠀⠙⢦⡀⠀⢹⡇⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⣀⣀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⣀⣠⣴⠟⠁⠀⠀⠀⢸⠏⠀⢀⣴⠏
⠀⠀⠀⠙⠛⠛⣇⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⠶⠶⠶⠾⠛⠉⠀⠠⠞⠋⠉⠉⠙⠲⡄⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⣼⠓⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠶⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣤⣄⣤⣠⠶⠖⠛⠉
⣿⣿⣿⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠈⢀⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣄⠀⠀⠀⠀⠛⠿⣷⣶⣤⣑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⡀⠀⠀⠀⠈⣍⣛⣿⣷⡄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣛⣿⣿⣿⠏⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣥⣶⣶⣶⣶⣦⣭⣝⣛⠻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢍⡙⠂⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⢿⣿⣿⣿⣿⣿⠿⠛⠿⣿⣿⣿⣮⡂⠄⠀⠀⠀⠀⢰⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣸⡙⢷⠙⣿⣿⠁⠀⠀⠀⠰⠿⣿⣿⢿⣿⣷⣦⣀⢀⠀⢸⣿⣿⣆⠀⠀⠐⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠙⣿⣌⠀⣿⠋⠀⠀⠀⠀⠐⢾⠿⠋⠻⣿⣿⣿⠻⢿⣿⠶⢿⣿⡿⠦⠤⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⢠⡟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⢰⠟⠿⠃⠀⢻⡀⠈⠀⠀⠋⠀⡉⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣘⠃⢻⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠎⠀⢀⣀⠄⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢾⠀⡌⠀⠰⠋⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢿⡏⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠉⠛⠓⡖⠁⠀⠀⠀⠀⠁⠀⠘⣄⡅⠀⠀⢰⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣯⣷⡀⠀⠀⢹⣿⣷⡀⠀⠀⠀⠀⠈⠀⠈⠉⣉⣙⣓⡾⠁⠀⠀⠀⠀⠀⢹⣧⠀⠸⣥⢆⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣻⣿⣿⣿⣄⡀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢩⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢈⠻⠻⢿⣷⣤⡀⠀⠀⠀⠀⢀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣅⣲⣤⡙⢻⣿⣷⣶⣶⣶⣿⣿⣿⡁⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣛⡿⠆
⠀⠀⠀⠀⣀⣤⣴⠋⢀⣠⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠞⠃⣼⣿⣿⣿⣿⣿⣿⠟⠫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠐⠛⢉⣉⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⣄
⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠿⣿⣿⣿⣿⣿⣯⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣷⣍⠒⢄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⠙⣿⣿⣿⣿⣿⣿⣮⡂⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⡟⢓⣀⣁⣤⡀
⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⡾
⠀⠀⠀⠀⠈⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢹⣿⡿⣿⣿⡏⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠻⣿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⠀⠉⠁⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⡿⠟⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣍⣍⠁⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢇⣞⠀⠀⠀⢊⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣩⣼⣧⠀⠀⢠⣭⣽⣁⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣶⣾⣿⣿⣿⣿⣤⣀⣐⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣟⣯⣭⣉⠉⣉⣭⡝⢿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣿⠀⠐⠂⠙⠀⠈⠒⠒⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠄⠀⠀⠤⠀⢀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⠀⠈⠉⠁⠠⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢀⡔⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣦⣀⣀⣠⠎⠀⢹⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢫⣿⣿⠈⠙⠻⣏⡁⠀⠀⢸⠀⠈⢳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⠿⠃⣿⠀⠀⢠⣿⡇⢀⡴⢻⠐⢶⡋⠀⠉⠲⣤⣀⣄
⠀⠀⠀⠀⠀⣰⠁⣇⠀⠀⠀⠀⠀⠉⠀⣅⣿⡀⠀⢨⣿⣷⠈⠀⣾⠀⠀⠉⢂⡀⠀⠀⠁⠈⢳
⠀⠀⠀⠀⢰⣿⠀⣿⠀⠀⠀⠰⡄⠀⠀⠀⢹⡖⠀⣸⣿⣿⡆⠀⡇⠀⠀⠀⣼⠀⠀⠀⢰⠀⠈⡇
⠀⠀⠀⠀⡾⠇⠀⢹⠀⠀⠀⠀⢳⠀⠀⠀⢸⣷⣄⣿⣿⣿⡧⢀⡇⠀⠀⢠⡇⠀⠀⠀⢸⠀⠀⢻
⠀⠀⠀⢰⠀⠀⠀⠘⣷⡀⠀⠀⢸⡆⠀⠀⠀⣿⣿⢿⣿⣿⢡⣿⠁⠀⠀⣸⠀⠀⠀⠀⣿⡀⠀⢸
⠀⠀⠀⢸⡀⠀⠀⠀⢻⣷⡀⠀⠀⣇⠀⠀⠀⢻⣷⣶⣿⣿⣿⣿⠀⠀⢀⡇⠀⠀⠀⣸⡿⠟⠀⢸⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠿⢷⠀⠀⢸⡄⠀⠀⠸⣿⠿⠗⣿⣿⡇⠀⠀⣼⠁⠀⠀⠀⡟⣃⠀⠀⠘⡇
⠀⠀⠀⢸⣿⡀⠀⠀⠀⣼⣀⠀⠀⠀⢧⠀⠀⠀⢿⣿⠦⣿⣿⠁⠀⢰⠇⠀⠀⠀⢠⣿⣿⠄⠀⠀⡇
⠀⠀⠀⢸⣿⠁⠀⠀⠀⣟⠻⠟⠀⠀⠈⣆⠀⠀⠸⣿⡧⣿⡟⠀⢀⡏⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⡇
⠀⠀⠀⢹⣷⡆⠀⠀⠀⢹⣿⣄⠀⠀⠀⠘⢦⠀⠀⢻⣿⣿⠃⢀⡞⠀⠀⠀⠀⠀⣾⣷⡾⠁⠀⠀⣇
⠀⠀⠀⣼⡿⡁⠀⠀⠀⣸⣿⣟⠓⠀⠀⠀⠀⠀⠀⠈⢿⣿⢀⡞⠀⠀⠀⠀⠀⠀⠏⡏⠀⠀⠀⠀⢿
⠀⠀⠀⢹⣿⠋⠀⠀⠀⢿⣿⡿⡿⠖⠂⠀⠀⠀⠀⠀⠚⠁⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠸⠻⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⡉⠉⢻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣥⣤⣀⣤⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠿⣿⣿⣿⡿⠋⠁⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣉⠉⠛⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⠟⠁⠀⢀⠴⢶⣶⣶⣄⠀⠀⣠⣤⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣤⣶⡾⣤⡾⣿⡟⢿⠛⣧⠠⢿⠏⠾⢿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢿⠛⠻⢿⡟⠁⠀⠀⠘⣇⠀⠀⠀⢸⡏⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡾⣸⠁⠀⠁⠀⠀⠀⠀⠈⠑⠒⠚⠋⢠⣀⣀⣀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡜⠚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣶⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣀⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠷⣦⣤⣀⡀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠂⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣸⣳⠀⢸⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠓⠹⠿⠀⡞⢠⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣷⣶⣶⠏⣠⠎⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠀⠈⠓⠦⢤⣀⠀⠀⠀⠀⠀⣏⠀⠚⠙⠻⡿⣿⣿⡾⠟⢿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⣮⣄⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀⠀⢸⡄⠀⠀⠀⠈⠀⠀⠀⠀⡿
⠀⠀⠀⠀⣠⠴⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣾⣿⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠻⣄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⣀⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⣀⠀⢰⡿⣻⣿⣿⣇⠛⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⢷⡀⠀⠀⠀⠀⢸⡀
⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠁⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⣬⣧⣤⣤⠶⠞⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⣿⠟⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠄⠀⠀⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠖⠠⢄⡙⠿⣥⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⠀⠀⠈⠀⠀⠙⠃⢀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠤⠄⠀⠀⠀⠀⠀⠰⣎⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣿⣟⣛⣛⣋⣀⣀⣀⣀⣀⣀⣱⣦⣤⡙⣃⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⢿⡟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⠫⣥⣶⣤⣽⣓⣾⣿⣿⣿⡿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⠿⠁⠀⠀⠀⢰⣮⣭⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠙⢿⣿⣿⣿⢟⢻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣟⣻⣿⣷⡀⠀⠈⣹⣿⢷⣶⣤⡀⠀⠀⠘⣿⢟⣟⠉⢫⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡜⠋⡇⠀⠀⠀⠱⠮⠋⠁⠀⠀⠀⠀⠀⣘⠉⠀⣼⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠜⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⣦⢔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠠⡵⠤⠴⠶⠖⠂⠀⠀⠀⠀⢀⡼⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠘⠲⠖⠃⠀⠀⠀⠀⣠⣶⠟⠀⠀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢀⣠⣴⣾⠟⠁⠀⠀⢀⡟⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⣠⡿⠀⠀⣠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢏⡍⣿⣿⠿⡅⠀⠀⠀⢀⣼⠁⠀⠐⠋⠙⣻⢰⢿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⢻⣁⣀⠚⠇⢻⣿⠀⠀⢀⣠⣶⣛⠃⠀⠀⠀⠀⢀⣿⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⣿⠟⠷⣦⡦⢻⣤⣾⣭⣷⡿⠛⠓⠢⣄⠀⢀⣾⡇⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠓⠀⠋⠘⠋⠉⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠈⠉⠙⠓⠶⠤⣤⣀⣀⡀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠈⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣀⣀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠷⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣶⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠰⠿⠷⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣩⡤⣽⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢰⠃⢹⡏⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣤⠀⢠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠰⡧⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣄⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣶⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣿⣿⣿⣿⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠸⡍⠛⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣤⣀⣀⣀⣀⣀⣤⠤⠴⠒⠚⠉⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠉⠉⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⣠⣤⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⣠⢄⣤⢠⣄⠠⢠⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⣿⣿⣿⣿⣿⡶⠤⠶⠾⢷⠆⣿⡆⠒⠀⢰⣿⣀⠀⡐⢴⣿⣿⣿⣗⣿⣿⣷⡾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡾⠋⢾⣿⣿⣦⣿⣿⡿⠟⠋⣉⣤⣶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠤⠾⢿⣿⣤⣏⣡⣴⣶⡿⠟⠛⠛⠉⠉⠁⣿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠒⠒⠴⠖⠚⠁⠀⠛⠋⢉⠀⠐⢀⢐⣀⣤⣥⣷⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⣿⠿⣿⣿⣿⠙⣿⣷⣥⡆⢀⣀⣤⣩⣰⣶⣶⣶⡎⢀⣟⣤⣬⣶⣶⣴⣿⡿⢛⡉⢡⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⡿⠀⢿⣿⣿⣍⣻⣛⣛⣿⣿⣯⢸⠇⡅⠠⢿⣿⣾⣿⣶⠶⣿⠛⠀⠚⡉⢙⡿⢻⣿⣿⣿⣿⣿
⠀⠀⢀⣿⣿⣿⡿⠇⢀⣴⣿⣿⣿⡿⣿⣿⣿⣿⠇⣸⠀⠁⠀⢸⣿⣿⠁⢀⣀⢀⣰⠷⠌⣁⣀⡅⠠⣿⣿⣿⣿⣿
⠀⠀⢰⣿⣿⣿⣶⣾⣿⣷⣿⣿⣿⢿⣿⣿⣿⣿⣧⣽⠀⢠⠀⢸⣿⣟⠀⢾⣿⡎⣿⠀⠀⠈⣅⡆⠀⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠒⠶⠠⢿⣿⣿⣿⢸⠀⢸⣿⡏⢀⠈⣩⣤⡄⠀⡠⠄⢸⡇⠀⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⡟⠉⠉⠁⠈⠉⠉⠉⠁⠈⠁⢀⣈⣉⠉⢸⡀⢸⣿⣇⠀⠀⠘⠿⠆⠀⠀⣰⣿⠗⢸⣿⣿⣿⡿⢿
⠀⠀⢸⣿⡿⠋⠀⠀⣤⡄⠀⠀⢀⣀⣀⣀⣠⢸⣿⣿⠀⢸⡁⢸⡗⣿⡀⣴⢆⡀⠀⠂⢀⣿⣿⡆⢸⣿⣿⣿⡇⢀
⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢿⡏⠸⠿⠟⠀⢸⡇⢸⡇⣿⣿⣿⣿⣿⣦⣰⣿⡟⣭⡇⠀⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢺⣇⣨⣇⣿⣿⣟⣻⣟⢀⡩⣹⡇⡝⣧⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣹⣷⠾⣿⣤⣀⡀⠀⠀⠀⢺⣦⣶⣿⣿⠟⠻⠿⢿⣿⣿⣿⣴⣴⢼⡇⡇⣿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⠃⣨⣹⣿⣉⣴⣽⠻⢿⡿⠁⠀⢴⡾⠿⢛⠉⣁⡠⠆⢀⣰⡿⣉⠻⢿⣿⣾⡇⡇⣿⠠⣿⣿⣿⣿⣿
⠀⠀⢠⣴⡇⠠⢴⣿⣿⣫⣭⣷⣶⣭⣷⠀⠀⡾⠖⠋⢉⣀⣤⡶⠶⣻⣿⣾⣿⠿⢿⣿⣿⣤⣇⡯⠀⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣧⣽⣿⣿⣬⣿⣛⣿⣿⡋⠀⢀⣷⣴⣧⣽⣿⠯⠄⢸⠛⠉⠃⣠⠞⢉⣩⠽⣿⣿⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠠⣿⡟⣿⣿⣿⣩⣭⣭⣭⣽⣿⢻⣷⢛⠉⣁⠙⠻⣶⠀⠀⣶⠀⣤⡘⠁⠀⠁⠀⠀⠈⠻⣿⣆⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣏⣷⣾⣿⠻⣿⣿⣿⡌⣿⣿⣿⠈⠿⣿⠆⢀⡼⠒⠄⡀⢰⣿⡇⠀⡄⢠⢲⡀⠠⠄⢹⠛⢻⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⢶⣝⣻⣟⣣⣿⣽⣿⣤⣄⠍⠀⣠⡶⠶⠶⢿⣿⡟⢷⠀⣇⠀⠈⢀⠀⠄⢸⠿⣾⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣽⣿⣜⣛⣛⣛⣛⣛⣹⣿⣿⡏⠱⢿⠋⣖⠆⢆⣓⢹⡆⠀⠀⢸⡄⢸⠈⠀⠆⢸⠀⠈⢿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣦⡀⡼⠓⠤⢠⣝⠻⠿⠾⡄⢸⠧⡀⠀⢄⠀⡟⠀⢤⣬⣿⣿⣿
⠀⠀⠀⣿⣶⠿⡶⠶⣷⣶⣤⣤⣤⡿⢶⢤⣼⣿⣿⣿⢧⡀⠲⣄⡀⠙⠒⢤⡀⠀⠀⠀⠀⡐⠀⢻⠀⢺⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣶⣶⣶⣶⠄⣻⣿⣶⣾⣴⠀⣿⣷⣾⡿⣿⡷⣄⠹⣟⣷⣄⡀⠉⠀⠀⠀⢀⣃⣀⣼⣷⠤⢉⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡁⣼⣿⣿⣿⡿⠀⣿⣿⣯⠈⢹⣿⣿⣆⣈⠟⣿⣿⣗⠶⠶⠾⠟⠻⢋⣿⣯⣴⠃⣍⣙⣿
⠀⠀⢀⡄⢨⡿⠿⠿⠿⠁⠿⠿⠿⠻⠓⠀⣿⡟⢋⣿⣾⣿⣿⣿⠀⠨⣿⣿⣿⡟⣿⡿⢶⣿⢸⣿⣿⣏⡻⢿⣦⣿
⠀⠀⠀⣷⣶⡶⠶⠶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⣜⣿⣿⠿⣷⣿⣤⣀⣿⣟⣿⠃⣥⡇⢸⣿⠀⣿⣿⢿⣿⣿⣷⣿
⠀⠀⠀⣡⣤⣤⣼⡀⠛⣿⣿⣿⣶⠚⠛⣚⣛⣉⣭⣭⣽⠷⠷⠿⢿⣿⣟⣿⣿⢀⣯⡟⠸⣿⡀⠻⣿⣧⢿⣿⡿⢿
⠀⠀⠀⣿⣄⣀⠈⣻⡄⠙⣿⣿⣿⣷⣄⣉⣩⣭⣤⣶⣶⣿⣿⣿⣶⣬⣉⣉⣩⣭⣭⣭⣮⣭⣭⣤⣿⣿⣇⠀⠀⣮
⠀⠀⠀⣿⣾⣿⣿⣄⣙⠢⠾⠿⢷⣿⣿⡿⠿⡾⣿⡿⠿⠿⠟⠟⠛⠛⠿⠛⠛⠛⠛⠛⠛⣿⡛⠛⢛⢿⣿⣦⣴⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣖⣲⣶⣦⣿⣿⣿⡆⣤⣞⣥⣬⣠⣰⣿⣧⣾⡿⠛⣾⢿⡄⠈⣹⣏⣯
⠀⠀⠀⣿⣿⣿⣇⣙⣛⠻⠿⠛⢻⣿⣿⣿⣿⡉⠉⠉⡿⠛⠹⣿⣿⣿⣿⣿⠟⠻⣿⡀⢻⣷⣾⣿⣴⣧⠚⠻⣷⡗
⠀⠀⠀⢶⣍⢿⠿⢿⣿⣿⣿⣤⣿⠿⠾⠿⠿⠧⠴⠒⠷⠀⠰⠿⠿⠿⠿⠿⠶⠂⠻⠗⠼⠿⠿⠿⠿⠿⠷⠶⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣷⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠛⠻⠿⠿⠿⠷⠶⠖⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⣀⣀⠀⠀⢀⣤⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢛⣉⣍⡁⠀⢈⣴⠶⠦⢸⣿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣿⣧⠈⠉⠁⠀⠀⢻⡀⠀⠀⠀⣿⢆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⣿⡇⠀⠀⠀⠀⠀⢀⣽⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢴⣾⣶⣾⣷⡗⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠙⠁⠘⠋⠀⣀⠎⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢢⠤⠤⢴⡉⠀⠀⠀⠀⠀⠘⡅⠀⠀⢥⠒⠊⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠳⠀⠀⠀⢇⠙⠦⣀⡠⠒⠉⠘⢆⠀⠀⣆⡠⠦⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⡴⠁⢠⡇⠀⠀⠀⢀⣷⠀⣨⠃⠀⠀⠀⠀⢃⠀⢸⠀⠀⠈⠢⣄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠁⠐⠃⠒⠒⠒⠒⡺⠓⠢⢸⣿⣷⣶⣾⣿⣿⠊⠁⠀⠀⢀⡠⠔⠈⠙⠁⠀⠀⠀⠀⠀⠙⠢⣀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⢠⠔⠒⠈⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡌⠉⠑⠂⣤⣤⣄
⠀⠀⠀⠀⠀⢠⠁⠀⢠⡤⢴⠀⢀⡄⣠⠤⠄⠀⡆⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⡰⣯⢶⣌
⠀⠀⠀⠀⠀⡆⠀⠀⡆⠀⠀⠀⠛⠁⠀⠀⠀⠀⢁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢻⡝⠳⣦⡌
⠀⠀⠀⠀⢀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠺⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⢸
⠀⠀⠀⠀⠂⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⢸
⠀⠀⠀⢠⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠈⡀
⠀⠀⡐⠃⠀⠈⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣿⣿⣿⣿⡆⠀⠀⠀⡇

⢠⡾⣷⣤⣀⣸⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠃
⠋⣰⠏⢳⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⠟⠛⠋⠉⢳⠀⠀⠀⡜
⠀⠉⠀⠾⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⠀⠘⠀⠀⠀⠀⠈⣀⡠⠊⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠐⠃⠀⠀⠀⠀⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠉⠀⠆⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡛⠂⠤⣀⠀⣀⣀⠀⠀⢰⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⢀⢜⣿⣿⡿⠢⣄⠀⢸⠁⠸⠂⢤⡌⠀⠀⠀⠀⠀⡘⠳⡤⠐⢇⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢘⣾⣿⣿⠃⠀⠀⢹⠃⢀⠃⠀⢀⡇⠀⠀⠀⠀⠀⣇⠀⣀⠀⠈⢆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠃⠀⠀⠀⠈⠣⠼⢆⡀⠀⡇⠀⠀⠀⠀⠰⢯⡀⢸⡄⣀⡈⢢⡀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣦⠀⠀⠀⡠⠃⠀⠀⠉⠀⠀⠀⠈⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠉
⣿⣿⣿⡿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠲⠶⢤⣤⣤⣤⠼⠦⠤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠛⣛⣛⣷⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠶⠾⠛⠛⢿⠛⣶⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠐⠿⠛⢻⣯⣤⡤⠀⠀⠿⣿⣷⣶⡌⠳⣼⢗⢀⣈⣹⣦⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⡏⣠⠤⠀⠀⠀⢰⠦⡄⠀⠀⣻⢡⠎⣤⠈⣿⣦⡀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣤⡤⣿⠀⣿⠀⠀⠀⠘⠀⠀⠀⠈⠛⠋⠐⠁⢠⣿⣿⣿⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣇⣀⡠⠀⠀⢀⡈⡁⠀⠀⠀⣷⣾⣿⡿⠟⠁⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢭⣤⠴⢖⡿⠃⠁⠀⠀⠀⣿⠀⢠⣤⡴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠤⠴⠖⠋⠁⠀⠀⠀⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⣀⣤⡶⠋⠀⣿⣠⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣻⣟⣿⠟⠉⠀⣠⡾⠋⠁⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠉⠁⢻⡄⠀⠀⣰⠏⠀⠀⠀⣰⡿⠿⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢀⡀⢳⡄⢠⣯⡴⣿⡀⢰⡟⠀⠀⠀⠈⠉⠛⠿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⣷⢸⠻⣦⣹⡾⠋⠀⢹⣧⡟⠀⠀⠀⠀⠀⢀⣤⠴⠿⠿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣿⠉⠀⠀⠀⣿⡟⠀⡀⠉⣴⠀⠀⠀⠋⠀⠀⢠⠇⢀⡖⠉⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢹⠀⠀⠀⠀⠙⠀⠘⠟⠀⡇⠀⠀⠀⠀⠀⠀⣾⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢰⠀⡄⠀⠁⠀⠀⠀⠀⠀⣴⡆⢸⠇⠀⠈⡟⠛⠛⢻⡅⣼⢣⠇⠀⣠⠖⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠸⡀⢡⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⡇⠀⠀⣾⣿⢏⡏⠀⡴⠃⠀⢀⣀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⢤⣼⣄⣀⣀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠇⠀⢠⣿⣏⣈⣤⡼⠷⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠲⠶⠶⠶⢶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⡀⠄⡐⠠⡐⠢⡈⠢⡑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠡⠤⣀⣒⠬⡑⢌⠑⡌⣂⣹⠀⢱⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡴⠊⠉⠉⠉⠉⠉⠀⠀⠀⠀⠘⡠⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡼⢦⡶⠲⣶⣶⡶⣤⢤⡤⣶⣾⡷⠶⣤⡖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⡇⠀⠈⠁⢀⠏⠀⢃⠈⠁⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣄⠀⠈⠀⠀⠀⠃⠀⠀⢀⠁⠀⠀⠈⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢄⢀⣉⡁⡀⠀⢀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⡀⠀⠀⠀⠰⠬⠭⠀⠉⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡆⠑⢄⠀⠀⠀⠀⠀⠀⠀⣠⢺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⡶⢻⠀⠀⠹⣄⠀⠑⠦⢤⣄⡤⠔⠊⢡⢿⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠛⠉⠀⢰⠃⠸⡄⠀⠀⠈⢢⡀⠀⠀⠀⠀⢀⡴⠋⢸⠀⠀⠙⢏⠓⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⡤⠒⠋⠁⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⢧⠀⠀⠀⠀⠙⢢⣀⣠⡔⠋⠀⠀⢸⠀⠀⠀⠘⡆⠀⠀⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠘⡄⣀⡤⠖⠛⣏⠉⠉⢹⠟⠦⡀⡾⠀⠀⠀⠺⡁⠀⠀⠀⠀⠀⠀⠀⡘⡆
⠀⠀⠀⢠⠇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⣧⠀⠀⠀⠘⡄⠀⡏⠀⠀⠈⡇⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⣼⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⣠⠋⡏⣇⠀⠀⢰⡇⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠘⡇⠀⢠⠇⠈⠁⠸⡄⠀⢸⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡇
⠀⠀⢸⠁⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢧⠀⣼⠀⠀⠀⠀⣧⠀⣸⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠸⠀⠃⠀⠀⠠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠈⣏⠉⠉⠉⠀⠀⠶⠂⣴⣶⣿⣿⠛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣠⣦⣜⠷⣄⠀⠀⠀⢰⣿⣿⣿⣿⡟⠃⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣹⡿⣿⣤⣄⠀⠉⠁⠀⠀⠙⠋⠉⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣹⣿⣦⣼⣙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡙⠻⠶⣬⣍⣉⣙⣓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣙⣿⣶⣮⣩⣭⣉⣉⣭⣤⣤⡄⢀⣤⣤⣄⣀⣀⡀⠀⢀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣤⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠶⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣤⣤⢠⣤
⠀⠀⠀⠀⠀⠀⠘⢿⡛⠳⣤⡀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣟⢛⣛⢛⠛⠉⠁⠀⠀⠈⠉⠛⠛⠛⠿⠿⣿⣿⣧⣿⡀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠿⡆⠹⢿⣷⣦⣤⡴⠞⠙⡏⠀⣾⣿⣿⣿⣿⣿⣿⠿⢛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣭⣭⣿⣿⣿⡷⠐⢘⣋⣽⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢼⣿⣿⣿⣿⣿⣦⣤⣉⡻⣟⡻⢿⣿⣿⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠟⢛⣉⣥⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⠿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡴⢿⣿⡿⠂⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄⢿⣧⣿⣿⣿⣿⢿⣿⡿⠿⣿⣿⣿⣿⡇⠀⣿⣿⡿⢿⣿⡿⣿⣿⣿⠧⠘⣯⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⢷⡘⣧⡟⠉⠁⠙⣦⣤⠄⠰⣾⡿⢿⡏⠀⠀⣸⣿⣶⣦⡭⠀⠀⠙⠃⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣷⢺⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣿⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⣿⣿⣆⣠⣀⡄⠀⠀⠀⢸⡏⠧⠀⠀⠀⡀⠙⡆⠀⠀⠀⠀⠀⠀⠐⢠⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣿⣿⠏⠀⢀⡀⠀⠙⢶⣶⢦⣤⡼⣿⠿⠁⠀⠂⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣧⠀⣴⠟⠀⠀⠀⠈⠁⠘⠿⠚⠉⠀⠀⠀⢀⠀⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠚⣷⡄⢯⠸⣶⢶⣶⠈⠉⠉⠐⠛⠛⢓⣶⡶⠟⠁⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠻⠄⠈⣿⣿⣷⣄⠘⠉⠈⠀⠉⠛⢉⢁⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⣦⣽⢿⣿⣿⣷⡶⣶⣶⣶⣶⠉⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⠉⢢⡙⣤⠙⠛⠉⠉⠛⠁⠀⠀⠀⠀⣠⣾⡿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⡉⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⡁⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠃⠀⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣙⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡽⠃⠀⠀⢀⣿⠃⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠹⣇⠈⣛⣉⣻⣿⣿⣷⡿⠉⠟⠃⠀⠀⢀⡔⠁⠀⣠⡾⠁⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⢸⡗⠀⠀⠀⠈⠳⡝⠿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠘⣿⠀⠀⠀⠐⠀⠈⠒⢤⢻⣿⣿⣤⠀⠀⠀⠀⣠⣴⠞⠉⠀⠀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⢃⣶⣤⣴⣯⡉⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠛⢓⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡀⠀⣼⣏⣼⢿⣿⣿⣿⣿⣿⠏⣿⡄⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡘⠃⠣⣿⡿⣻⣿⡃⠀⠘⣿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠻⠅⠈⠙⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠿⠛⠋⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠓⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⢠⡟⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢸⠆⠀⠀⠀⣸⡇⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⣰⣿⠀⡀⠀⢠⣿⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣖⠀⢀⣿⣟⠀⣸⣰⣿⢃⣴⣿⠥⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡻⠋⠉⠛⠿⣷⣿⣿⠿⠛⣿⣷⣤⣀⡀⠀⣜⡀⢿⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠈⠙⠃⠀⠐⠙⠿⠿⠟⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣤⣄⣀⣄⠀⠖⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣄⠀⠀⢠⣿⣿⣧⣧⣠⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠳⣦⡟⠛⢹⣿⣿⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⢇⠈⠀⠀⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⠀⠀⠀⠀⢿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⡀⠀⠀⢸⣿⣿⣷⣢⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠂⠀⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢏⠁⠀⢠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠰⢀⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠷⠒⠒⠐⠿⣿⣟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣾⣿⢊⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠛⠻⠷⠿⠿⠷⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠛⠛⠿⠿⠿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡴⢛⡉⣛⠛⠉⠀⠀⠀⠀⠠⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠠⠒⠒⠒⠒⠂⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢸⡷⡿⠿⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠳⣍⣛⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⢀⣰⠂⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⠀⠀⠉⠉⠉⠀⢀⣀⣀⣤⣤⡀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠹⣿⣶⣭⣭⣭⣉⣋⣉⣩⣭⣥⣶⣾⡿⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠙⠿⣍⣀⣉⣉⣀⡽⠟⠁⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣉⣙⡟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠯



⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⣁⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⣠⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣽⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣤⠀⠀⢠⣽⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢛⣭⣿⣿⣿⣵⣶⣦⣄⡀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠠⣿⣿⣯⣿⣿⡿⢟⣛⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣽⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡃⠈⣿⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣩⣭⣭⣵⣶⣶⣶⣶⣶⣶⣿⣭⣭⣭⣙⣛⣛⠛⠻⠿⠀⠈⠃⠀⢀⣀⣀⣀⣤⣴⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⠀⠈⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠟⠛⠛⠉⢀⡴⠃
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠁⠚⠛⠂⣿⣿⣿⢩⣌⠙⢿⣿⣿⠁⠀⠀⠀⠀⠀⣀⣭⠽⢿⣿⣿⣿⠂⣴⠟⢿⠉⠁⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠘⢈⣧⠀⢻⡿⠇⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠁⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⢋⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠘⠀⠐⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠆⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⠛⠁⠀⡄⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⢿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠐⣃⣀⣀⠀⠀⠀⣀⢐⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠸⣿⣷⣄⠀⠀⠃⠀⠀⠀⠀⠠⣁⠍⠙⢁⠴⢿⣷⣼⣿⣿⣿⣻⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣜⢫⣿⣧⠀⠢⡀⠐⣤⡶⠖⠀⢀⡼⢁⡴⠋⠀⢀⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠈⣳⣝⣿⠀⠀⣷⣶⣦⣤⠀⠀⢸⠗⠋⠀⡀⢂⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣆⠀⠀⠀⠀⢤⣌⣘⡏⠀⠀⡿⠋⠛⠋⢀⣤⡀⠀⢠⠾⠒⠋
⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⢻⠇⢀⠜⠁⠀⠀⠀⢺⢯⡋⠀⠉⠀⠀⠀⠀⠆
⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⠏⠀⠘⠀⠀⠀⠐⠀⠀⠐⣿⡆⠀⠴⠾⢏
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⣀⣰⣤⣀⡀
⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠞⠁⣰⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢸⣿⣿⣿⣿⣿⣿⠿⠇⣿⡟
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠀⠀⠀⢀⣤⣴⡛⠉⠀⠀⠘⠻⠿⠛⠋⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠎⠀⠉⠀⠀⠀⠀⠀⠀⠢⠀⣾⣿⣿⡇
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⡿⢿⣻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠗⠛⠉⠉⠉⠁⠀⢙⣻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠙⠛⠟⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣽⣿⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⡺⢭⣭⣭⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠁⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣯⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠂⠀⠈⠃⠀⠩⠐⠉⠀⣀⣤⣶⣶⣶⣤⡀⠀⠀⠀⢹⣿⣿⠟⣛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⣾⠟⣋⣉⣉⠙⢻⣤⠶⠶⠛⠋⠙⠁⣾⢻⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⣛⣿⣦⣤⡶⠶⢶⡃⠸⡉⠛⢛⠉⣸⠇⠀⠀⠀⠀⠀⠀⢈⠙⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⢜⣉⢛⠉⣿⠀⠸⠘⢷⣄⣉⣉⣠⠴⠃⠀⠀⠀⢀⡆⢠⠀⠞⠸⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣇⣈⣥⡴⢻⡄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣾⠀⠈⠇⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠀⠀⡼⠀⠀⠀⠉⡀⠀⢦⣤⣀⠀⠀⠀⠀⠋⠀⠀⣶⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⢰⣿⣤⣀⣴⠟⠁⠀⠀⠉⠙⠳⡄⢸⡇⠀⠀⢀⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠉⢛⣉⣉⣬⣤⠤⠖⣒⠽⠃⠀⠀⠈⣷⠀⠀⢸⠏⢸⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡔⡄⠉⣒⠒⠐⠒⢉⡁⠈⠀⠀⠀⠀⠇⠀⠀⠎⢠⠏⠀⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡄⠉⠛⠲⠛⠋⠀⠀⠀⠀⠀⠀⢀⣼⠀⢠⠏⠀⢸⡇⠀⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⣿⠃⣴⠃⠀⢀⣾⠀⠀⠀⠈⢳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⣦⡀⣀⣀⣀⣠⣼⣿⣿⠟⣡⠞⠁⠀⠀⣼⠏⠀⠀⠀⠀⢸⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡆⠈⠛⣛⣛⣿⣿⠛⣋⠚⠁⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠘⢇⣉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⢛⡒⠒⠲⢦⡀⠀⠀⠀⠀⠀⢀⣠⣤⢶⡿⠁⠀⠀⠀⣿⠀⠀⠙⢿⣛⡽⠛⠁⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠉⠉⠛⠛⠛⠒⠒⢲⣶⡶⠞⠉⠉⠳⣄⠀⠙⣦⡀⣀⣤⡶⠛⠉⣠⠟⠀⠀⠀⠀⠀⣿⠀⠀⣠⣶⣿⡀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⣁⠤⠒⠈⠲⣄⠀⠑⢤⡀⠙⠿⣤⣀⠀⢾⡃⠀⠀⠀⠀⠀⠀⣿⠀⣼⣿⣿⣿⣿⣄⠀⠀⢀⣴⠃⠀⠱⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠚⠉⠀⣀⣤⣀⠀⠀⠉⠂⠀⡈⠁⠀⠀⠙⠳⣌⣷⡀⠀⠀⠀⠀⠀⣿⣼⠋⣿⣿⣿⡿⠋⠳⣤⡾⠁⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⣸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⠒⠋⠁⠀⠈⠓⠀⢀⡀⠀⠀⠈⠀⠒⠀⠀⡼⠉⢙⡶⠦⣄⡀⠀⣿⢁⣾⣿⣿⣿⡄⠀⣠⡟⠀⠀⠀⠀⠀⠀⠘⣦⠀⢀⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣶⠚⠶⢤⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⠃⢀⡞⠀⠀⠀⠉⠛⠿⣾⣿⣿⣿⣿⣧⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⣋⣳⠀⣼⠉⠻⢯⣤⣄⣀⣀⣀⡀⣸⠇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣸⠇⠀⠀⠀⠀⠀⠉⠉⠉⢹⡟⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣄⡀⠀⠠⠞⠉