Heat wave Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠉⠉⠉⠁⠀⠉⠉⠉⢙⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢻⣷⠀⣿⠛⠛⠛⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⠀⢀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣦⡀⠀⠀⢿⡄⠀⢸⡇⠀⢰⡟⠀⠀⢀⣴⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣼⠿⠟⠛⠛⠻⠿⣧⣀⣴⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠠⠶⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⣠⡤⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠐⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠶⠶⠶⠶⠀⣿⠀⠀⠀⠀⣿⠀⠀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠈⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠈⠉⠋⠀⠀⠀⣰⡟⠀⠀⣀⣠⣿⣄⡀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢻⣷⢶⣤⣤⡶⣾⡛⠉⠻⣦⡀⠀⠀⠀⣼⠏⠀⢠⡾⠋⠉⠉⠉⠛⢷⡄⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠾⠃⠀⢸⡇⠀⠘⠷⠀⠀⠈⠻⠆⠀⠀⣿⠀⠀⣿⠁⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡄⠀⠀⠀⠀⠀⢠⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠈⠻⣦⣤⣀⣠⣴⠟⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠉⠉⠉⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⠶⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⠀⠀⢸⣿⡆⠀⠀⣠⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⣿⣦⠀⠻⠛⠇⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣤⣀⠈⢉⣤⣶⣿⣿⣿⣶⣤⡉⠀⣠⣴⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠸⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠿⠟⠀⡉⠻⢿⣿⣿⣿⡿⠟⢋⡀⠛⠻⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡷⠀⢠⣤⣤⠀⠺⣿⣧⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⡀⠠⠄⠀⠀⣠⡿⠋⠀⠀⠘⣿⡇⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠑⡇⠀⠀⣀⠄⠊⠀⠀⠀⠀⠹⣤⠊⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠟⠀⠀⠀⠀⠈
⠀⠀⠠⡀⠀⠀⠀⠱⣄⡎⠀⠀⠀⠀⠀⠸⡀⠀⣿⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⢳⣠⠁⠀⠀⠈⠙⠶⣤⣠⠀⠀⠀⠙⣦⡏⠀⠀⠀⢠⠎⠀⢀⡠⠂
⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠹⣦⠀⠀⣠⡟⠀⠀⢀⣠⠞⠉⠉
⠀⡀⠀⠀⠀⣀⡤⠞⠋⠉⠉⠉⠙⢿⣧⣼⣯⣴⠶⠿⠻⠷⢤⣀⠀⠀⠀⠀⣀⠠⠄
⠀⠈⠉⠈⡩⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠙⣏⠛⠛⠉⠉⠂
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⠋⣿⠉⠛⠶⣤⣤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠸⠀⠀⠀⠀⠙⢆⠐⠀⠀⠀⠉⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⢿⣦⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢹⡇⠀⣿⡇⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣴⣶⡀⢸⡇⠀⠈⠁⠀⠰⠿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣾⣿⣶⣤⡀⠀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⣿⡇⠀⠾⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⣿⣿⣿⡿⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠻⠿⠛⠋⠀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣠⡄⠀⠐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⣿⡇⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡇⢸⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⢀⣿⣿⡇⠸⣧⡀⠀⢀⡀⢀⣀⣤⣤⣬⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣠⣾⣿⣿⣷⣆⠙⣿⡀⠀⠙⠛⠛⠛⠁⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⡇⠀⠲⢶⣶⣶⣿⡿⠛⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠘⠿⣿⣿⠿⠋⣰⡿⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣶⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⢀⣈⣋⣀⠀⢀⣴⡿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣾⠟⠛⠛⠛⢿⣦⡉
⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣷⠀⠀⠀⢀⣀⣀⢠⣿⠃⠀⠀⠀⠀⠀⢻⣷⢀⣀⣀⡀
⠀⠀⠸⠿⠀⢠⣶⣶⣶⣶⣶⣦⠀⠀⢠⣶⣶⣶⣶⣶⣦⠀⢸⣿⠀⠀⠀⠈⠉⠉⠘⣿⡆⠀⠀⠀⠀⠀⣸⡿⠈⠙⠉⠁
⠀⠀⢀⣀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣙⢿⣦⣤⣤⣤⣾⠟⣁
⠀⠀⢸⣿⠀⢸⣿⣀⣀⣀⣸⣿⠀⠀⢸⣿⣀⣀⣀⣸⣿⠀⢸⣿⠀⠀⠀⠀⠀⢰⡿⠋⠀⠈⢉⣍⠉⠁⠈⠻⣷
⠀⠀⢸⣿⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⢸⣿
⠀⠀⢸⣿⠀⢸⣿⠉⠉⠉⢹⣿⠀⠀⢸⣿⠉⠉⠉⢹⣿⠀⣼⣿⡀
⠀⠀⢸⣿⠀⢸⣿⣀⣀⣀⣸⣿⠀⠀⢸⣿⣀⣀⣀⣸⣿⣾⡟⠹⣷⡄
⠀⠀⢸⣿⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠘⠛⠛⠛⠛⢛⣿⡏⠀⠀⠹⣿⡄
⠀⠀⢸⣿⠀⢀⣤⣤⣤⣤⣤⣄⠀⠀⢀⣤⣤⣤⣤⣼⣿⡀⠀⠀⠀⣿⡇⠀⠀⢀⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣀
⠀⠀⢸⣿⠀⢸⣿⠉⠉⠉⢹⣿⠀⠀⢸⣿⠉⠉⠉⢹⣿⢿⣶⣶⡾⠟⠀⠀⢀⣾⠏⠀⠀⠀⢀⣾⡟⠀⠀⠀⢀⣾⡏
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⠘⠿⠿⠿⠿⠿⠟⠀⠀⠘⠿⠿⠿⠿⠿⠟⠀⢸⣿⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢻⣧
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⢠⣿⠏⠀⠀⠀⢠⣿⠏
⠀⠀⢸⣿⠀⢰⣿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠸⠿⢿⣷⠀⢸⣿⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡀
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⣻⡧⠀⠀⠀⠀⣹⣷⠀⠀⠀⠀⢹⣷
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⢰⣿⠃⠀⠀⠀⢰⣿⠃
⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠋
⢀⣤⣼⣿⣦⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠹⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⡿

⠀⠀⠀⠀⡆⢀⡇⠀⡆⠀⠀⠀⠀⠀⠀⢀⠄⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⠀⡀
⠀⠀⠀⠈⡇⠀⣇⠀⣇⠀⠀⠀⠀⠀⠀⡏⠀⡏⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡷⡀⠀⠀⠀⠀⠀⠀⠀⣼⠰⣇⢠⡇
⠀⠀⠀⠀⢻⡄⢹⡄⢻⡄⠀⠀⠀⠀⠀⠹⣆⠹⣆⠈⠛⢦⡀⠀⠀⠀⠀⠀⢸⠃⣼⠃⡷⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡆⣷
⠀⠀⠀⠀⠈⣇⠀⣧⠈⣧⠀⠀⠀⠀⠀⠀⠘⠀⠈⢷⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⠀⠀⠀⠀⠀⠀⣾⢡⡟⢠⡏
⠀⠀⠀⠀⠀⡏⠀⠟⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢳⠀⠇



⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣠⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⡞⠀⡏⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢳⡠⡏⠀⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼
⠀⠀⠀⠀⢷⡀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⣧⠈⢷⠈⢷⠀⠀⠀⠀⠀⠀⠀⡼⠁⢹⡄⢯⠀⠀⠀⠀⠀⠀⠀⢷⡀⢻⡄⠻
⠀⠀⠀⠀⠀⢳⠈⢷⠀⠁⠀⠀⠀⠀⠀⠀⠸⣆⠈⣧⠘⠇⠀⠀⠀⠀⠀⠀⠹⢀⡞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠹⣆
⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⡔⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⡼⠁⣼⠀⢻⣄⡀⠀⠀⠀⠀⠀⡇⠀⣷⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⢰⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢳⡀
⠀⠀⠀⢧⡀⠘⠷⣄⠈⠙⠆⠀⠀⠀⠀⢻⡄⠸⣆⠀⡼⠁⠀⠀⠀⠀⠀⠰⡜⣆⠳⣌⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡷
⠀⠀⠀⠈⠻⣄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢷⠀⠹⡆⠁⠀⠀⠀⠀⠀⠀⠀⢳⣌⢷⡙⣆⠇⠀⠀⠀⠀⠀⠀⢸⡀⠙⣆
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⠸⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⢠⠄
⠀⠀⠀⠀⠀⢦⠀⣇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⡃⢸⠀⢻⡀
⠀⠀⠀⠀⠀⠈⢧⠘⣦⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⣧⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠻⣦⠙⢷
⠀⠀⠀⠀⠀⠀⠈⣧⠸⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢹⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⠋⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡟⣸⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⡞⠁⡼⠁⡜⠁
⠀⠀⠀⠀⠀⠀⠸⠁⠇⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠜⠁⠜⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠡⡎⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡇⠀⢆
⠀⠀⠀⠀⠀⢸⡄⢠⡏⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠙⣦⡉⢋⡵⠲⣄⠀⠀⠀⠀⠀⠀⠃⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠁⠀⣼
⠀⠀⠀⠀⠀⠘⣇⠸⡇⢰⡏⠀⠀⠀⠀⠀⠀⣠⠶⣄⠈⢳⠈⢯⠒⠃⠀⠀⠀⠀⠀⠀⢰⡏⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠈⠧⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠹⡄⠳⠘⠁⠀⠀⠀⠀⠀⠀⠙⠾⠈⠇⠈⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⢠⠈⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡄
⠀⠀⠀⠀⠀⡏⠀⣸⠈⣧⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡇⢀⡄
⠀⠀⠀⠀⠀⢻⡄⠁⠀⣸⠀⣹⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⢀⠙⢦⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣴⠏⣠⠞
⠀⠀⠀⠀⠀⠀⢹⠄⣰⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⡟⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢈⡇⠈⡇⠀⠀⠀⠀⠀⠀⡇⠀⡾⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠜⠀⠐⠁⠀⠀⠀⠀⠀⠀⠁⠀⠣⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆⠀⠀⠀⢠⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡍⠀⠀⠀⠈⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠹⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢦⠀⠀⠀⢠⠋⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃⠀⠀⠛⡆⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄⠀⠀⠀⢣⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⠀⠀⣱⢸⠀⠀⠀⠀⣱⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋⠀⠀⢀⠔⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⢼⠀⠀⠀⠀⡎⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦⠀⠀⠀⠘⣜⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⠀⠀⠈⣇⡇⠀⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣙⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣒⣒⣒⣚⣒⣒⣋⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⣀⣀⡀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⡾⠉⠀⠙⣧⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⢸⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠆⠀⢀⣀⣈⣉⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣇⣀⣀⣀⣸⠀⢸⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⢶⣾⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡷⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣾⡿⠇⠀⣿⣿⣿⣿⣿⠀⠸⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠛⠉⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣴⣿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠆⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣼⣿⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠘⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢿⣷⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢿⣷⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⣄⡀⠉⠉⠉⠉⠉⣀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢿⣿⣶⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⣠⣶⡦⠀⠀⠠⠄⠠⠤⠠⠀⠤⠠⠄⠀⠄⠆⠀⠀⠄⠀⢠⣦⡠⢀⣤⣀⣀⡀⡀
⠀⠻⣿⡆⠀⠀⠠⠠⠀⠀⠀⠠⠀⠄⠀⠄⠀⠠⠄⠀⠀⠀⠀⠻⠐⠟⠱⠿⠯⠿⠿⠶⢶⣤⣆⣀
⠀⠀⠈⠀⠀⠈⢰⢲⣠⢦⣀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⢀⠀⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠒⠒⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠆⠈⠀⠀⡀
⠀⠀⠀⠀⢀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⡢
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡀⣆⣰⣀⡀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⣀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⢀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢛⠈⠁⠁⠁⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠟⠻⠘⠃⠂⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠋⢈⡤⠀⠀⠄⠈⠀⠀⠄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⡶⠩⢀⠁⠀⠀⢠⠀⠀⠀⠀⠀⠒⠞⠀⠀⠀⠠⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠋⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠲⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⠗⠒⣀⠀⠀⠀⠀⢀⣾⣄⠄⠀⣀⠀⣴⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠄⢤⡤⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠈⠀⠈⠉⠂⠀⠀⠚

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠛⢉⣙⣛⡒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋⠁⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢠⡀⢀⣴⣿⠟⠁⣀⣴⣿⣿⡿⠟⢛⣉⣡⣤⠤⠤⠀⠈⠉⠛⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠘⣷⣿⡿⠁⢠⣾⣿⣿⠟⣉⣴⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢰⡀⢹⡟⠀⣰⣿⣿⠟⣡⣾⣿⣿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡆⠘⣧⠀⠀⣰⣿⣿⠋⣰⣿⡿⠿⠿⠿⠿⠿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢸⣿⠀⣿⠃⢰⣿⣿⠇⠀⠉⣀⣀⣤⣤⣤⣤⣀⣀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡞⣿⠀⠀⠀⡔⠀⠀⠀⢠⡀
⠀⠀⠀⣼⡟⢠⣿⠀⣼⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣷⣦⡀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⣾⣿⠸⣰⣆⢸⣇⠀⠀⢠⣿⣷
⠀⠀⠀⣿⡇⢸⣿⠀⣿⣿⡇⠀⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⡖⢀⣴⣿⣿⣿⠀⣿⣿⣦⣿⡀⣶⠘⡿⠁
⠀⠀⠀⣿⡇⢸⣿⠀⣿⣿⡇⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⣾⣷⣿⣿⣿⣿⣿⡆⣽⣿⣿⣿⣷⣿⡇⠀⠀⡄
⠀⠀⠀⢿⣿⠈⣿⡀⢸⣿⣿⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⣿⣿⣿⣼⣴⣿
⠀⠀⠀⢸⣿⡄⢻⣇⠀⢿⣿⡇⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢼⡔⢋⡡⠖⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⠘⣿⣿⣿⣿⣿⣿⡿⠀⠘⢻⣿⣿⣿⣿
⠀⠀⠀⠈⣿⣧⠘⣿⡌⠘⣿⣿⡄⠹⣿⣆⢤⣤⡤⠶⠶⠚⣛⣩⡥⠎⣷⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠈⠉⠿⣿⠿⠋⠀⠀⠀⣸⣿⣿⡿⠋
⠀⠀⠀⠀⠹⣿⣆⠘⣿⣆⠘⢿⣿⣄⠙⣿⣧⡐⠒⠚⠛⠋⠉⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣭⣤⣤
⠀⠀⠀⠀⠀⢻⣿⣆⠘⢿⣦⡈⠻⣿⣧⡈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⢀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⠤⠤⠤⠤⠀⠈⣉⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠹⣿⣧⡈⠻⣿⣦⡈⠻⣿⣦⣈⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⣛⠿⢷⣭⣛⡦⠄⠀⠀⠙⠻⢦⣤⡀⠀⠀⣀⣴⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣌⠙⢿⣦⣈⠛⠿⣷⣦⣄⡉⠛⠻⠷⣶⣤⣤⣄⣛⣛⣶⣾⣯⣭⣛⣛⣛⡤⠤⠐⢀⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣌⡙⠛⠦⣌⣙⠛⠿⢷⣶⣤⣤⣀⣈⣉⣉⣉⠉⠉⠉⠉⠉⢁⣀⣤⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣉⣉⣉⣉⣉⣡⣤⣤⣶⣶⡿⠿⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠢⠀⠠⠤⣄⡀⠀⠀⢀⣀⠀⠀⠀⣄⣀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠐⢅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡂
⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠒⠢⢄⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠘⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠸⠀⢀⢳⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠰
⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣨⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠠
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠎⢀⠀⠀⠀⢰⠀⢈⠐⢳⠂
⠀⠀⠀⠀⠐⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡠⠎⠀⠀⠀⠀⠀⠀⠓⠊⠀⢢
⠀⠀⠀⠀⠀⠀⢄⢨⣇⡓⠀⣀⠀⢀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⡄⡀⠄⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠐⠀⠀⢏⠈⡄⡂⡹⢦⢩⣷⢀⠀⠐⡆⠘⡀⢢⣄⢀⣆⣰⡀⣤⣀⠠⣃⠀⢄⡀⡀⠀⠈⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⠀⠉⠀⢸⠉⠀⠀⠰⢷⠀⠁⠣⠁⡁⠇⣼⠟⢉⠁⠀⡜⠇⠙⣷⢾⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡇⠠⠼⣧⢸⡯⣄⡆⠈⣿⡂⠠⣀⡄⡇⠂⣡⠀⢸⣂⠀⡆⠀⡀⠠⢀⠉⠃⣽
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⣃⡏⣹⣨⣍⢬⣣⠀⢠⡇⠀⠐⠃⡇⢪⢽⡰⠨⠦⠀⢧⠃⠹⠤⢘⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⢛⣙⡋⡧⠽⠚⡀⠘⡧⠀⠸⠙⢳⣠⢸⣠⠀⡆⠤⢸⣤⢰⡀⠌⣎⡁

⠀⠀⠈⠐⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠀⠀⡤⠊⠀⠀⠠⠤⡐⠀⠾⠓⠒⠒⠈⠀⠀⢀
⠁⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈
⢤⠖⠀⠀⠀⢠⣀⠄⠀⠀⠀⠌⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⣄⠀⠀⠀⢶⠄
⠀⠀⠀⠀⠀⠋⠀⠀⠔⠛⠀⠀⠀⠀⠀⠢⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠱⡁⠀⠀⠈⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⡄⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢆⢤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂
⠀⠀⠀⠀⠀⡈⠄⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⡎⠂⠀⠀⠀⠀⠀⠀⢄⡇⡀
⠀⠀⠀⢆⠀⠁⣦⠀⠀⠀⠀⠀⠀⠀⠳⡇⠀⠀⠇⠀⠀⠀⡇⠀⢀⠀⠈⠃⡃⠀⠀⠀⢀
⣄⠀⠐⢾⡔⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢣⠃⠀⡠⠂⠁⠀⠀⠀⠀⠀⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⢀⡴⠀⠀⢄⡀⠀⠈⠀⣠⡆
⠀⠀⠘⠀⠘⠀⠀⠀⠀⣠⠀⠀⠀⠇⠀⠀⠀⢀⠀⠊⠙⠀⠀⡠⢺⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡊⠔⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠝⠋⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡠⠊⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠞⠋⠀⠀⠰⠀⠀⠀⠘⠄
⠐⠠⢀⣀⠀⠀⠀⠈⠉⠉⠁⠌⢠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠀⠀⢀⡀⠀⠀⡀⠀⠀⡀⠈⠚
⠀⠀⠀⠀⠀⠠⠄⡐⢀⣀⡀⠀⠀⠑⠀⠀⢠⣧⠀⠀⠀⠀⠀⠐⠢⢤⣤⠀⡠⠋⠀⡠⠊⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠑⠹⠯⡁⠀⠀⠀⠀⠀⠈⠃⢀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠈⣢⣀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠗⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢀⡀
⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣐⣛⣛⣉⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢾⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣼⣇⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⣸⠀⠸⣧⣸⣦⡀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣠⠀⣿⣿⣿⡇⢰⡀⣷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣼⣿⢰⡄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⣷⣾⡟⣼⠁⣿⣿⣿⡧⣸⣴⠀⢀⣼⣿⠃⢠⡆⣿⣤⣿⣿⣿⣀⣾⡷⣸⣷⠀⣦⠀⠀⢀⠀⣼⣿⣿⣿⣿⣎⣿⣾⣿⣦⠄
⠀⠀⠀⠀⣄⠀⣴⣿⡇⣿⣿⣿⣿⣿⣤⣿⡿⣹⣿⣿⣿⣷⣿⣿⣿⣴⣿⣃⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣄⢷⣌⣷⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⣀⣦
⠀⠀⠀⠰⣿⣿⣿⢻⣷⣟⠹⡙⢿⣿⣿⣿⠰⣿⣿⠟⣿⣧⣿⣿⠙⢿⢻⣿⣿⡏⣿⠃⢿⠙⣿⡟⠠⣿⣿⢹⣿⡿⣿⣿⠻⣿⣿⠐⣿⣿⡿⠃⠟⢛⣿⣷⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠉⠁⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢂⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠠⠂
⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠀⠀⠄⠀⠀⠀⠀⠘⡀⠀⠀⠄⠀⠀⠀⠀⠆⢀⠠⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣤⠀⠀⠀⠒⠂⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⡄⠂⡄⠀⠀⠂⠀⠄⢦⡀⡀⠄⠄
⠀⠀⠀⠡⡳⠀⠀⠠⠐⠩⠀⠀⠀⠀⠀⢃⠀⡀⣀⡈⠎⠀⠀⠀⠈⠒⠜⠄⠤⠊⠀⠀⠀⠐⠆⢠⠃⠠⠂⠠


⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⠀⠁⠸⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⠄⠐⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠩⠤⠊⠄⠈⠀⠀⠀⠈⠂⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠔⠤⠀⠂⠁⠂⠀⠀⠀⠀⠀⠂⠁⠐

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⡀⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⣷⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⢠⣿⠁⠀⠀⢀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⢰⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⢀⣾⠃⡀⠀⣴⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡏⠀⠀⢠⡇⠀⠀⠀⣰⣿⣿⠏⠀⢠⠀⣼⡟⢀⠃⣼⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠀⣠⣿⠀⠀⢀⣼⣿⣿⡟⠀⠀⡄⢰⣿⡇⢸⠠⠏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⣿⣿⣿⠀⢀⣼⣿⠃⠀⣴⣿⣿⣿⡟⠀⠀⢰⡇⢸⣿⠇⣾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⣿⣿⣿⡇⢠⣾⣿⠏⢠⣾⣿⣿⣿⠏⠀⠀⠀⣾⠃⢸⠟⣰⢇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢰⠃⠀⠀⠀⣰⣿⠟⣿⣿⣇⡞⣿⡟⣴⣿⣿⣿⡿⠁⣰⢣⠆⣼⠟⠀⠈⣰⢏⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢺⠀⠀⠀⢠⣿⠃⠀⣿⠟⣿⢀⣿⣿⣿⠋⣼⡟⠀⢰⣷⣿⣾⠏⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⠀⣻⡄⠀⠀⣼⡇⠀⠀⡇⠀⡏⢸⣿⡿⠃⣼⣿⢡⢇⣿⣿⣿⠏⢀⣾⣿⠟⠁⠀⣠⡾⠀⠀⠀⢸⡇
⠀⠀⢸⠀⢿⣿⠀⠀⣿⠃⢰⣴⠇⠀⠀⢸⣿⠁⢰⣿⡏⣾⢸⣿⣿⣿⢠⣿⡿⠋⢀⣴⣿⡿⠃⠀⠀⠀⢸⡇
⠀⠀⢸⡄⠈⢿⡇⠀⣿⠀⣼⣿⠀⠀⠀⣼⡇⠀⣼⣿⢧⣿⢸⣿⣿⣿⣾⣿⠃⢠⣿⡫⠋⣠⡜⣡⠆⠀⢸⡇
⠀⠀⢸⣇⢠⣞⣿⠀⠁⢀⣿⡟⠀⠀⠀⣿⠀⠀⣿⢻⢸⣿⣸⣿⣿⣿⣿⡇⢠⡿⠁⣤⣾⡿⠛⠁⠀⠀⢸⡇
⠀⠀⠸⣿⣆⣿⣿⠀⠀⣸⣿⠃⠀⠀⠀⠉⠀⢀⣯⡏⣿⣿⢹⣿⡇⣿⣿⡇⢼⠁⣾⡿⡋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢻⣿⣿⣿⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⢱⣿⡟⣼⣿⡇⣿⣿⡇⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⣿⣿⢠⣾⠏⠀⠀⡆⠀⠀⠀⣰⣿⢇⠏⡿⣹⣿⣿⣾⣿⡿⠁⣼⡿⠋⠀⠀⠀⠀⣠⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣿⡇⣼⠏⢠⠃⣴⡇⠀⠀⢠⣿⡏⢀⠀⣽⢿⣿⣿⠿⠋⡀⠞⠉⠀⠀⢀⣀⣴⡾⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠋⣼⠏⢠⠇⣰⣿⠃⠀⢀⣾⠏⢀⣾⡼⣿⣿⠟⠁⢀⡠⠔⠀⣠⣴⠾⠛⢡⡞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡏⢀⣾⣴⣿⠏⠀⢀⣾⠃⠀⣼⡇⣿⡿⠁⢀⣴⠋⠀⣠⡾⠋⠀⣰⣾⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡿⠀⣸⣿⣿⠏⠀⠀⡾⠁⠀⣰⣿⣧⢹⠇⢠⣿⡏⠀⠞⠁⠀⣠⣾⣿⣿⠃⣸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠘⠁⠀⣰⢿⣿⡏⣻⢠⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⠏⢀⢻⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡴⢋⣾⣿⢳⣿⣾⣿⠃⠀⠀⢀⣾⡿⣭⣿⠏⠀⠎⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡜⣡⣿⣿⣢⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⡿⠃⠀⠜⢀⡾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢈⣼⣿⣿⣿⠿⠋⠁⠀⠀⠀⣰⣿⣿⠟⠋⠀⣠⣎⣤⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⠀⢀⠞⣵⠿⠋⠁⠀⣀⡀⠀⣠⠾⠟⠋⢁⣀⣤⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣶⠀⠀⢂⠞⠁⠀⣠⣶⠿⠋⠛⢋⣠⣤⣶⣾⣿⣿⣿⣿⣿⣻⠟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⢀⠀⣠⠞⢉⠀⠀⡠⢚⣩⣭⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⡿⠃⠀⢀⡴⠁⠔⣡⡖⠁⠀⢀⣴⡿⠿⠿⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣿⠋⢀⣤⣶⠟⠁⣠⣾⠋⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠜⢁⣴⡿⠟⠁⠠⠾⠋⠁⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠄⠀⠠⠤⠠⠄⠼⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⡌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿⣿⢛⣉⡻⣿⠟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⢸⣿⣿⣿⡿⢸⣿⣿⣡⡆⣮⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢇⣬⣙⡛⠿⠿⠿⠟⣛⣩⣴⣿⣿⢰⣿⣿⣿⣿⢸⣿⡇⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠛⣰⡜⣿⣿⣿⣿⣿⣷⣬⣛⠻⠿⠿⢋⣾⣿⣿⣿⣿⢸⣿⡇⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⢸⡕⠿⠟⣸⣿⣿⣿⣿⣿⣿⣿⡇⢾⣿⣦⢹⣿⣿⡿⢋⣼⣿⣧⡻⢧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠻⠿⢏⣼⣿⡟⣰⣿⢟⣛⡻⣿⣎⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡇⢿⣿⡸⠿⢃⣿⡿⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢸⣿⣿⣿⣎⡻⢿⣿⡿⢟⣱⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣛⣛⣛⣉⠁⠀⠀⠀⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣷⣮⡻⣿⣿⣿⣿⣿⢟⣡⣾⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣍⣛⣩⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢠⡄⠀⠻⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⠋⠉⠁⠀⠰⢶⡄⣠⡶⠟⠛⠛⠷⣦⡀⣴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⣯⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⣀⢰⡟⣴⠟⠀⠀⠀⠘⣷⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠉⠉⠀⠈⠙⣷⠀⠀⠀⠈⠋⢸⣇⠀⠀⠀⠀⠀⢀⣿⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⢠⣴⠖⠹⢷⣤⣀⣠⣴⠟⠁⠷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠀⠀⠀⠀⠀⣠⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠈⣩⡍⠀⠿
⠀⠀⠀⠀⠀⠀⠀⣾⠃⠙⢷⣄⠀⠀⠀⠹⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⢀⣀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠉⠻⢶⡄⠀⠀⠉⠙⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠙⠻⠾⠟⢹⡇
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠹⢷⣤⡀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⠀⢀⣤⣶⠶⠿⠁⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢸⡏⠁⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢘⣷⠀⠀⠀⠀⠀⢺⡏⠀⠀⠀⠀⢀⣤⣴⠾⠟⢛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⠇⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣀⣼⠏⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⣀⣀⣀⣈⣁⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠉⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣴⠟⠛⠉⠉⠉⠙⠛⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⠀⠀⠀⠘⣷⠀⣀⣀⣤⡴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢰⣿⡇
⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⣸⣿⣧
⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⣿⠄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⣿⡿⠁⠀⠀⣸⣄⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⣿⣿⡟⠇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣠⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢠⡀⠹⣿⣿⣾⣿⣆⠀⠀⠀⢀⠀⠀⠀⣼⣇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⠏⠀⠀⠀⣄
⠀⠀⢀⡀⠀⠀⣸⡗⠀⠙⢿⣿⣿⣿⡆⠀⠀⠘⣧⠀⢸⣿⣿⣶⣤⣴⣿⣿⣿⠀⠀⢻⣦⡀⠀⠀⠙⢿⣿⣿⣿⠟⢰⣿⣿⣿⣿⣿⣆⡀⠀⣰⣿
⠀⠀⠀⢻⣷⣼⡟⠀⢸⣆⣨⣿⣿⣿⡟⠀⢀⣼⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣼⣿⣿⡆⠀⠀⠀⠙⠿⢣⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⡆⠀⠀⣿⡿⠀⠀⠈⢿⣿⣿⣿⣿⢃⣴⣿⡟⠁⢀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣾⣿⣶⣾⣿⡇⠈⣷⡄⠀⠛⢿⡿⢡⣿⣿⣿⣧⣤⣼⡧⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠙⣶⡘⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀⢸⣇⢿⣿⣿⣿⣿⣿⢃⣶⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠈⠹⣿⣿⣿⣿⣿⠏⠀⠀⢸⣯⣿⣿⣿⣿⣿⡿⠁⣾⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣤⣾⣿⣿⣿⠏⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠙⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠈⢄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠈⠤⠈⠀⠀⠀⠀⣀⣀⣀⣀⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢁⡾⠤⠒⠒⠚⠛⠛⠓⠒⠒⠤⢄⡀⠉
⠀⠀⠠⢤⡀⠀⠀⠀⠐⠢⠄⣸⢀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⡐
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠉⠒⠤⠤⠒
⠀⠀⠀⠀⠀⢱⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠠⠐⠈⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣠⠒⠁
⠀⠐⠓⠲⠄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠙⢦⡀
⠀⠀⠀⠀⠈⠢⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⢤⣀⣀⡄
⠀⠀⠀⠀⡠⠔⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠜
⠀⠀⢠⠊⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣀⠀⠄⠂⠁
⠀⢠⠇⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⡄
⠀⠸⠈⠁⠉⠐⠠⢀⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠔⠂⢙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡈⠀⠈⠙⠲⠦⢤⣀⣀⣀⣀⣀⣀⣤⡤⠖⠋⠁⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⣀⠤⠚⠩⡀⠀⠀⢀⠞⠉⠢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡴⠋⠁⠀⠀⠀⠀⠑⢄⡀⠏⠀⠀⠀⠈⠑⠒⠠⠵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⣀⣠⣤⣬⣥⣤⣄⣀⠀⢴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡦⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⢴⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠾⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠈⠉⠛⠛⠻⠟⠛⠛⠉⠁⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⢠
⠀⠐⠈⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣦⣥⣽⣥⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⠉⠉⠙⢿⣷⡀⠀⣿⡇⠀⠀⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠛⠃⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣶⣶⡄⠀⣿⡇⠀⣤⣤⣤⡀⠀⠀⠀⣀⣤⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⣿⣿⣿⣿⣷⡀⠀⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⠀⠀⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⣿⣿⣿⣿⣿⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⣿⡿⠿⠛⠁⠀⠀⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⣀⠀⠀⠀⣶⣄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⣿⡇⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⣿⠇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⣀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⣸⣿⣿⣧⡀⠻⣷⣄⠀⠀⠀⣼⡇⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢀⣾⣿⣿⣿⣿⣿⡄⠘⣿⡆⠀⠀⢿⣿⡄⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣿⣿⣿⣿⡿⠀⣿⡇⠀⠀⠈⢿⣿⡀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠻⣿⣿⣿⣿⡿⠃⣰⣿⠃⠀⠀⠀⠸⣿⠇⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⣉⣉⣁⣤⣾⡿⠁⠀⠀⠀⠀⠸⠋⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠁⠈

⠀⠀⠀⠀⠀⢀⣴⡞
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄
⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⢸⣿⡟⣧⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⠀⠀⢀⠀⢸⡇⠀⢸⠀⢱⢰⣿⣇
⠀⠀⠀⠀⠀⡜⠉⢠⣸⣇⠀⢀⣼⣇⠀⠀⢀⣿⣿⡀⠀⡘⢦⣈⣿⡄⢸⣧⣾⣿⣯⠻⡄⠀⠀⢱
⠀⠀⠀⠀⠈⠀⠀⠸⣿⠙⠳⡾⠉⣿⣦⣴⠟⢻⡿⢻⣦⡇⢸⡟⢹⢇⡞⢿⣯⡾⠋⠀⢿⣾⠀⠚⠀⢠⡀
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠰⡀⢻⡇⠉⠀⠘⠇⠀⠀⠁⡞⠀⠀⠊⠀⠊⠀⠀⠀⠀⠈⠟⠁⠀⠀⢼⠁
⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢳⡀⠀⠀⡇⠀⢸⠃⠀⢀⠞⠁⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠐⢄⠈⢢⠀⡀⢣⠰⠀⡇⠆⢏⠄⣠⠋⠀⣠⠖⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣀⠀⡑⢄⠑⠌⠀⠂⠀⡇⠘⠌⡔⡡⡢⠞⢁⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀⠁⡂⠀⠀⠀⠀⣀⠀⠁⠀⠪⡈⢀⠠⢚⡡⠀⣀⡤⠴⠂
⠀⠀⠀⠀⠀⠰⠶⠦⠤⢤⣀⣀⡉⠑⠀⠀⡐⢡⠊⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢉⣀⣠⣄⣠⣤⡴⠒⠦⠤⠴⠒⠶⠄
⠀⣀⣀⣀⣀⡤⠴⠤⠤⠤⢄⡀⠀⠄⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣁⠀⣀⠴⠉⠉
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⢄⣀⣀⠨⠤⠤⠤⠤⠔⠊⠉⠀⠀⣀⠜⢁⡤⠂⠀⠐⠀⢀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠁⠐⠄⠄⠀⣀⠐⠄⠀⠀⠵⢦⣒⠂⠀⠀⠐⠒⡴⠊⢉⠁⠂⠉⠀⠶⠀⣒⠍⣁⣀
⠀⠀⠒⠐⠒⠐⠀⠐⠂⠐⠒⠀⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⣰⠚⣡⣀⣄⣖⡰⠀⠀⠂⠁⠚⠀⠘⠲⠗⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⢤⣭⠴⠞⠟⠋⣁⣉⠁⣀⣈⠛⠛⠛⠋⠉⠉⠓⠒⠛⠛⠚⠓
⠀⠀⠀⠠⠶⠒⠒⠛⠒⠛⠓⢾⣍⠉⠁⠀⠩⠤⠬⢒⣐⠈⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠰⠤⠤⠄

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⢠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⢸⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⡰⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⣼⡿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⢀⠃⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⣼⡿⠁⡀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⣠⡞⠀⠀⠀⠀⣴⣿⣿⡿⠁⠀⢀⠀⣸⣿⡃⣸⠁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⠀⢀⣴⣿⠁⠀⠀⣠⣾⣿⣿⡿⠁⠀⢠⡄⢰⣿⣿⠃⡿⢰⠋
⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠁⠀⣰⣿⣿⠃⠀⣠⣾⣿⣿⣿⡿⢁⠀⠀⣸⡇⢸⣿⡿⢀⠳
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⢀⣼⣿⣿⠃⣠⣾⣿⣿⣿⣿⠋⢀⠂⠀⢠⣿⠁⢨⡿⢁⡿⣈
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣼⣿⡿⠙⡿⣿⠇⣼⢡⣿⢇⣴⣿⡿⣿⣿⠟⠁⣰⢋⡞⣠⣾⠋⠀⠈⣠⣟⣼⠃
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢰⣿⠏⠀⢰⡇⣿⢸⠇⣸⣿⣾⣿⠋⣴⣿⠏⡌⣰⣧⣞⣴⡿⠁⠀⣠⣾⣿⡞⠁
⠀⠀⠀⡄⠀⣿⣆⠀⠀⠀⣿⡏⠀⠀⢸⠃⠋⢸⠀⣿⣿⣿⠁⣸⣿⡏⣼⢠⣿⣿⣿⡿⠁⣠⣾⣿⡟⠉⠀⢀⣤⡾
⠀⠀⠀⡇⠀⢿⣿⡄⠀⠀⣿⠁⢠⡇⣾⠀⠀⠈⢀⣿⡟⠀⢠⣿⣿⢰⡏⢸⣿⣿⣿⠁⣼⣿⡿⠋⢀⣠⣶⣿⡟⠁⠠
⠀⠀⠀⣧⠀⡘⣿⣷⠀⠀⡿⠀⣸⣿⡟⠀⠀⠀⢸⣿⠀⠀⣼⣿⡇⣾⣇⢸⣿⣿⡿⣸⣿⡿⠁⢀⣾⣿⠟⠋⣠⠶⣡⠔⠁
⠀⠀⠀⣿⡄⠱⣍⢿⡆⠀⡇⢀⣿⣿⠇⠀⠀⠀⢸⡇⠀⠀⣿⢩⢣⣿⣿⢸⣿⡟⣿⣿⣿⠁⢠⡿⠋⣠⣴⣿⡷⠞⠁
⠀⠀⠀⢿⣿⡄⣿⣿⣷⠀⠃⣸⣿⡿⠀⡀⠀⠀⢸⠀⠀⢸⡇⡾⣸⣿⣿⢸⣿⣧⢹⣿⣿⠀⣿⢀⣼⣿⢟⠁
⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⢠⣿⡿⠁⠀⡇⠀⠀⠸⠀⢸⣼⣸⢣⣿⣿⢇⣾⣿⡇⣼⣿⣿⠀⠁⣼⣿⡿⠋
⠀⠀⠀⠀⠘⣿⢿⣿⡇⢠⣿⡿⠁⠀⢀⡇⠀⠀⠀⠀⣾⣿⢇⡞⢹⢟⣾⣿⣟⣴⣿⣿⠟⢀⣴⣿⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠸⣿⣿⢁⣿⠟⢀⡞⢀⣼⡇⠀⠀⠀⣸⣿⠏⢌⢀⢫⣾⣿⣿⣿⢿⠿⠋⣠⠞⠋⠀⠀⠀⠀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠘⠇⣼⠏⢀⡞⢀⣼⣿⠁⠀⠀⣰⣿⠋⠀⣼⣠⣟⣵⣿⠿⠋⠀⢀⣈⠄⠀⢀⣤⣴⠾⠛⣩⡟
⠀⠀⠀⠀⠀⠀⡈⣾⠃⢀⡾⣠⣾⣿⠇⠀⠀⣼⡿⠁⠀⣼⡇⣏⣿⡟⠁⢀⣠⠞⠋⠀⣠⡾⠋⠉⢠⣴⣾⡟
⠀⠀⠀⠀⠀⠠⢰⠏⠀⣼⣿⣿⣿⠋⠀⠀⣼⠟⠀⠀⣼⣿⣇⢻⡟⠀⣠⣾⡏⠀⣠⠞⠉⠀⣠⣶⣿⣿⡿⠀⣾
⠀⠀⠀⠀⠀⠀⡎⠀⢠⣿⣿⡟⠁⠀⠀⢼⠋⠀⠀⣰⣿⡿⢻⢸⠃⣰⣿⡿⠀⠀⠀⠀⢀⣼⣿⣟⣾⡿⠁⢠⢻
⠀⠀⠀⠀⠀⢠⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⢀⡼⠋⣾⡷⣄⣾⣰⣿⣿⠃⠀⠀⠀⢠⣿⣿⢻⣾⡿⠁⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢠⠏⣰⣿⡟⢇⣿⣿⣿⡿⠁⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⢠⠊⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠰⢣⣾⣿⣿⣿⣾⡿⠟⠋⠀⠀⠀⢀⣾⣿⣽⡿⠛⠁⢀⡴⠁⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣶⠀⠀⠀⡴⢋⣽⣿⠿⠋⠁⠀⠀⠀⠀⢀⣴⣿⠿⠟⠉⢀⣠⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠠⢸⣇⢰⡇⠀⢀⠞⣰⠟⠋⠀⢀⣀⣤⡶⣮⡤⠞⠋⣁⣀⣤⣴⣾⣿⣿⣿⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡟⠀⠀⢀⠞⠁⠀⣠⡶⠟⠋⠁⠀⣀⡴⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⢠⣡⣿⡟⠀⠐⠀⣠⠄⢀⡾⢋⡠⠆⠀⠀⢊⣥⣶⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⢀⣠⡾⠁⠐⣡⡴⠋⠀⠀⣠⣴⠿⠛⠋⠉⠉⠁
⠀⠀⠀⠀⢰⡿⠋⢀⣤⣶⣿⠋⠀⣠⣾⠟⠁⠀⣠⠾⠋⠁
⠀⠀⠀⢀⠟⢠⣾⠿⠟⠋⠀⠐⠚⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⢠⣾⣿⡟⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢿⠀⢠⣿⣿⡿⠀⠀⢠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⠏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠈⣿⡀⣸⠏⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⣦⠀⠀⠸⡇⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠹⣷⡈⢿⡆⠈⣿⣿⣿⣿⣿⣷⡀⠀⠁⠀⠀⠀⠀⣾⠏⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠈⠁⠈⠿⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⣤⠀⠀⠏⠀⠀⠀⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⢀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⣾⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠙⣿⣦⠀⠸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣧⠀⠀⠀⠸⢏⣼⠟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠀⢻⣿⡇⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡆⠀⢀⣀⠸⠁⠀⠀⣠⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣸⣿⣧⠀⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠁⢰⣿⠃⠀⢠⣶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣤⠀⠀⣿⣿⣿⣷⣿⣿⣿⠀⠀⠹⣿⣿⣿⣧⣠⣾⣿⡄⢰⠟⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡙⣷⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⠀⠴⢋⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠘⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠒⠫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⢹⣿⣿⡇⠙⢿⣿⡀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠉⠙⠀⠀⠀⢀⣿⡿⠃⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠘⠋⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣦⣀⠀⠀⠀⠀⣀⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢀⡴⠋⠉⠁⠈⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠉⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠈⠉⠳⢤⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠁⠉⠉⠙⢦⡀
⠀⢠⡏⠀⠀⢂⡀⢀⡀⠀⠀⢻⡀⠀⠀⠀⠀⡼⠁⠀⠀⢂⣂⠀⠀⠀⠘⡆⠀⠀⠀⠀⢰⠃⠀⠀⠀⣀⠀⢀⠀⠈⢧⠀⠀⠀⠀⢀⡞⠀⠀⣀⠀⣘⣿⠃⠀⢹⡄
⠀⢸⡁⠀⡁⠞⢠⣄⠑⡁⠀⢈⡇⠀⠀⠀⠀⡇⠀⠀⣼⠇⠀⠹⠆⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⢈⢸⣿⠀⢒⠀⠀⢸⠀⠀⠀⠀⢸⡁⠀⣾⠏⠀⢈⣿⡀⠀⢈⡇
⠀⠘⣧⠀⠀⠄⠀⠉⠀⠀⠀⡼⠁⠀⠀⠀⠀⢳⡀⠀⢉⠐⢿⣆⠄⠀⣠⠇⠀⠀⠀⠀⠸⣄⠀⠀⠀⠉⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⢧⠀⢉⠐⠿⣎⡉⠀⠀⣼⠃
⠀⠀⠈⠳⣤⣀⡀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⠀⣀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠘⠶⣄⣀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⢁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⣀⡤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣀
⠀⠀⣠⠏⠁⢀⡀⢀⡀⠈⠳⣄⠀⠀⠀⠀⠀⢠⡞⠉⠀⢠⣤⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣶⡄⠀⠉⢳⡄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⣄
⠀⢰⠇⠀⣴⡟⠀⠀⠹⣦⠀⢸⡆⠀⠀⠀⠀⡏⠀⠀⠀⢸⣿⠀⠀⠀⠈⣇⠀⠀⠀⠀⣼⠁⠀⠀⣾⣿⣿⠀⠀⠀⢹⠀⠀⠀⠀⢠⡇⠀⢠⣤⠀⠀⣰⡆⠀⠸⡆
⠀⢸⡄⠀⢟⠀⢶⡆⠀⠘⠀⢰⡇⠀⠀⠀⠀⡇⠀⠀⠀⣼⣿⣄⠀⠀⠀⡿⠀⠀⠀⠀⢿⠀⠀⠀⣏⣹⣙⠀⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⢸⣿⣤⣾⣿⡇⠀⢠⡇
⠀⠀⢳⡀⠈⠑⠠⠀⠛⠀⢀⡞⠀⠀⠀⠀⠀⠹⣄⠀⠈⢿⣿⠟⠀⢀⡼⠃⠀⠀⠀⠀⠘⢧⡀⠀⢿⣿⣿⠀⠀⣠⠏⠀⠀⠀⠀⠀⢳⡄⠈⠋⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠙⠲⠤⣤⣤⠤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣤⣤⠤⠞⠋


⠀⠀⠀⣠⠴⠒⠛⠛⠒⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⠛⠛⠒⠦⣄
⠀⠀⡼⠁⣀⣀⣀⣀⣀⣀⠘⢧⠀⠀⠀⠀⠀⣰⠋⠀⢠⡖⠀⠲⡀⠈⢳⡄⠀⠀⠀⠀⢠⡞⠁⠀⠀⣰⡆⠀⠀⠙⣆⠀⠀⠀⠀⠀⡴⠃⠀⡆⠀⠐⠀⠀⠈⢧
⠀⢸⠃⠀⢿⣿⣿⣿⡿⠿⠀⠘⡇⠀⠀⠀⠀⡇⠀⠀⠈⣀⣷⡆⠁⠀⠀⣷⠀⠀⠀⠀⣾⠀⠀⠁⠸⢛⠿⠠⡄⠀⢸⡄⠀⠀⠀⢸⡇⠀⣠⡶⠈⠱⣴⡆⠀⢸⡇
⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣧⠀⠀⠀⠸⡿⠁⠀⠀⢀⡏⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠸⣇⠀⣿⠧⣶⣤⣼⣷⠀⣸⡇
⠀⠀⠹⣦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠳⣄⠐⠛⠛⠓⠀⣀⡞⠃⠀⠀⠀⠀⠀⠹⣦⣈⠳⠾⠟⠛⢛⣴⠟
⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠚⠛⠉⠁

⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⡴⠚⠉⠁⠈⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠀⠉⠙⠶⡄⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠈⠉⠛⢦⡀
⠀⢠⡟⠀⢀⣾⣿⣿⣤⣄⠀⢳⡀⠀⠀⠀⠀⡼⠁⠀⠀⣀⣀⡠⠀⠀⠙⡆⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⢰⣿⡆⠈⢧⠀⠀⠀⠀⢀⡞⠀⢀⣠⠄⣤⣾⣀⠀⢻⡄
⠀⢸⡅⠀⣿⣿⣿⣿⣿⣿⠀⢨⡇⠀⠀⠀⠀⡇⠀⢀⠰⣿⣿⣿⢈⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⡴⠋⠉⠀⠀⢸⠀⠀⠀⠀⢸⡁⠀⣾⠃⣀⡟⠉⠿⠀⠈⡇
⠀⠘⣇⠀⠙⠛⣿⣿⡿⠀⠀⣼⠁⠀⠀⠀⠀⢳⡀⠀⠠⠉⠋⠡⠀⠀⣠⠇⠀⠀⠀⠀⠸⡄⠀⠙⠯⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠈⣧⠀⠡⢈⠛⣧⠄⠀⠀⣸⠃
⠀⠀⠈⠳⣄⣀⠀⢀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⡀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⢀⣀⡤⠏⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⡀⢀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣄⣀⣤⣤⢠⣤⠀⣠⣄⣠⣀⣤⢤⣤⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣇⠄⠀⠀⠀⠀⠟⠟⠛⠳⠸⠚⠀⠳⠐⠛⠛⠆⠘⠘⠟⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠶
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⡀⠀⠀⠸⡆⢸⢨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠨⠀⠀⠀⠛⠿⠘⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠴⢠⡄⢿
⠀⠀⠀⠀⠇⢀⠰⠀⠀⢀⣸⠋⠀⠀⠀⠀⠀⢰⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠆⠀⢐⠀⣾⠀⢾⡇⠀⡇⢲
⠀⠀⠀⠀⠛⠈⠓⢀⠚⠻⠿⠀⠀⠀⠄⠀⠂⠉⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠘⠓⣾⡿⢹⢲⣼⡇⡆⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠁⠸⣼⢸⣿⢹⣡⡇⡄
⠀⠀⠀⠀⠀⠀⣀⡀⣀⡲⡄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣀⠠⠴⠀⠄⠠⠐⠚⠂⠀⠀⠀⢀⡀⠀⢀⡹⠎⠋⠈⠋⠾⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠁⠈⠉⠉⠠⠀⠖⢊⠁⠄

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

⠀⠀⠀⢠⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦
⠀⠀⣀⢀⣴⠂⠈⢲⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⣶⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠒⠲⣤⣿⠃
⠀⠀⡀⠘⠃⢠⣤⡄⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⠀⠉⠤⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⢠⣿⣄
⠀⠀⠀⢀⠀⠀⠙⠃⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⢴⣶⣦⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠂⢰⣶⣦⡜⢛⠁
⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢀⡉⠻⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣉⣹⠇⠊


⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡶
⠀⠀⣰⣿⠇⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⣴⡄
⠀⢸⣿⠋⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣠⣿⡁
⠀⠸⡌⠁⠸⣿⡃⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⢾⢀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣼⣿⣿⣿⣇
⠀⠀⠘⠢⢄⣀⡀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⣀⡀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢉⣾⣯⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡆⠡⢰
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠨⠁⠰⣶⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡾⠿⣿⣀⣀⡂⠀⠀⠀⠀⠀⠀⠀⣠⡶⠉⠑⢦⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣶⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑⠀⣶⣶⣦⣰⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠾⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠉⠷⠙⠿⠋


⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣾⣿⣿⡟⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣥⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⢠⣤⣿⣧⡀
⠀⢰⣶⣬⣛⣿⣿⡿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠎⠋⠉⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢸⡟⠞⡛⣿
⠀⠘⣿⣿⣿⡟⣿⣿⣶⡉⠃⠀⠀⠀⠀⠀⠀⠐⠀⠻⣿⣿⡿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠠⣤⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠓⠠⣶⣦⡇⠀⠇⠁
⠀⠀⠈⠉⠉⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡈⢹⠧⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢿⠛⡿⢻⡻⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⡟⣧⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⣀⡴⠞⡻⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡇⣿⠀⠀⠀⠀⠀⠀⣿⢸⣿⣇⠀⠀⠀⢠⡾⢋⡴⢋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢻⣄⠀⠀⠀⠀⠀⣿⢸⡇⣿⡀⠀⠀⣾⠀⡾⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡀⠀⠀⠀⠀⢹⣄⠻⣄⠉⠛⢶⣤⡀⢸⡇⢸⡇⠸⣇⠀⠀⢻⡄⣷⠈⣷⠀⠀⠀⠀⠀⢀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠲⣤⡀⠀⠈⠻⢦⣈⠙⠲⣄⠈⢿⣾⠁⢸⡇⠀⢿⡄⢀⣼⠃⣿⠀⢹⡆⠀⠀⣀⡴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡑⢤⡙⠳⢤⣀⠀⠙⣦⠀⢻⣆⣸⣿⣤⣼⣧⣤⣼⣿⡟⠉⣠⠟⢀⣾⣃⣤⠞⢩⡖⣿⠋
⢀⣤⣴⣦⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄⠙⢦⡀⠉⠓⢶⣿⡶⠟⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠻⢿⣤⣾⠟⠋⣠⡴⢋⡼⠃⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣶
⠾⠛⠳⢦⡈⣷⣄⠀⠀⠀⠀⢀⣀⣹⣆⠀⠻⣦⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡾⠋⢠⡞⠁⠀⢀⣴⠞⢛⣉⣉⣉⠙⠛⣩⡿
⠀⠀⠀⠈⢷⡹⣍⡛⠿⠿⠛⠋⠉⠙⠻⢷⣴⠟⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣴⣿⣀⣀⣴⠟⣡⠞⢩⡴⠶⠶⠶⠟⠋
⠀⠀⠀⠀⠈⠻⣦⣉⡛⠚⠛⠛⠛⠲⣦⣼⠏⠀⣠⠶⠚⠋⠻⠿⢷⣴⣿⠀⠀⣾⣆⣤⣶⣟⣿⠓⠶⣦⣄⠹⣏⠉⠉⣁⡴⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⣦⣼⠇⠀⣀⠁⢠⣶⣿⣿⣿⣿⣿⣿⠂⠀⢻⣿⣿⣿⣿⢿⣿⣦⠈⠙⣆⢻⡟⠛⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠛⣿⠀⠀⠻⠷⠿⢿⣷⣯⣻⡼⠟⢿⠀⠀⢸⡏⠿⣿⣧⣽⣽⠿⠳⠞⠃⠈⣿⠻⠿⢧⣤⣀⣀
⢀⣀⣠⣤⣶⣒⣚⣛⣉⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⢸⣷⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣈⣉⣙⣛⣿⣶⣦⣤⣤
⠈⠙⠛⠛⠻⠿⠧⣤⣤⣀⣀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣏⠀⠀⠀⣿⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⣀⣀⣤⡤⠴⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡽⠿⠛⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⠟⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣠⡶⠚⠛⠛⣷⡀⠀⠀⠀⠀⠀⣠⣀⣀⣠⣴⣤⣤⢶⣤⣄⣤⣄⠀⠀⠀⠀⠀⢠⣿⣄⠀⠻⢤⣤⣤⣤⣄⣀
⠀⠀⠀⣀⣀⣀⣠⣼⠇⣰⠏⣠⡤⠴⣾⠟⠳⣄⠀⠀⠀⠀⠟⠛⠛⢻⣿⠿⠷⠛⢻⡟⠛⠛⠀⠀⠀⠀⣠⣿⡀⠉⠳⠶⠤⠶⠶⠶⢬⣙⠳⣄
⠀⣴⢟⡭⠥⠤⡤⠴⠞⢁⡾⠃⠀⣰⠏⠀⢀⣿⢦⣄⠀⠀⠀⠀⠀⠀⠙⠷⣶⠶⠟⠁⠀⠀⠀⠀⣠⣾⡃⠈⢿⣦⣤⣠⣤⠴⠶⠾⠶⣌⢣⣹⡆
⢸⣷⡶⠟⠛⠓⠶⠶⠟⠋⠀⠀⣰⠏⠀⣴⠟⠁⣀⡽⠷⣤⣀⡀⣂⣤⡀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣅⠈⠙⢦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢦⣽⣟⣳⣶
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠞⢁⣴⢾⡟⠀⢠⡟⠉⣙⣿⡿⠿⣶⡶⠶⢾⠟⢻⡇⢀⡟⠉⠻⣦⣀⠙⢦⡘⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣼⣥⠞⠋⠀⠘⣷⠀⣾⠀⣸⠏⠈⣿⠀⢹⠀⠀⣿⡄⠈⢧⠈⢷⣄⠀⠀⠙⠳⢦⣉⢮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠀⠀⢸⡇⢸⡆⣿⠀⠀⢹⡆⢸⠀⢰⡟⢳⣄⠈⠳⢤⡉⢷⡄⠀⠀⠀⠈⠛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⢀⣼⠃⡼⢀⡿⠀⠀⠘⣧⢸⠀⣾⠁⠀⠉⠛⠷⣦⠹⡄⢿⠀⠀⠀⠀⠀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⡜⢁⡾⠃⠀⠀⠀⢻⣼⢰⡇⠀⠀⠀⠀⠀⢸⡆⠇⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢰⣯⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⢸⣰⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠈⢿⣗⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠦⠄⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⢋⣠⣤⣤⣀⠀⠀⠰⠶⠂⠀⠀⠀⠀⢰⠟⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠶⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠠⡀⣤
⠀⠀⣀⠀⠤⠀⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⢀⣤⠀⠤⠆⠀⡀
⠒⠁⠀⠀⠑⠈⠩⠖⣨⡥⠔⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠁
⠀⠐⠊⠁⠈⠔⣊⠁⢀⡴⠖⢉⡤⠄
⠆⡠⠖⣂⡴⠏⣡⡶⠋⣴⠞⢉⣴⠞⢋⣡⣤⠀⢀
⣉⡤⠞⠡⠤⠖⠋⢀⡼⠁⠔⢉⡴⠚⣉⠥⠴⠚⢋⣠⠴⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠟⠀⣁⠖⢚⣩⠶⠋⣠⠶⠒⣩⡤⠶⠒⣠⡤⠞⠛⢁⣴⡶⠗⠋⣉⣤⠔⢂⠄⢀⣀
⣤⡿⢃⡴⢋⣤⠶⠛⣉⡤⠞⣋⠴⣠⡿⢋⣤⠾⠁⣸⠟⠀⣠⡾⠋⣠⡾⠋⣠⡾⠋⢠⣼⠃⣠⡤⢀⣠
⣩⣔⣯⣖⣋⣴⣞⣉⣡⣶⣛⣡⣞⣩⣴⣛⣈⣴⣚⣁⣤⣞⣉⣤⣾⣋⣠⣾⣛⣡⣴⣛⣀⣜⣡⣾⣋⣡⣤⣟⣃⣤⣶⣂⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⢀⡀⠀⡏⠈⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠙⡆⠀⠀⠀⢨⢷⡀⠰⢈⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡟⠀⠀⠀⠘⣕⠙⠢⡀⠁⠪⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡀⠀⠀⠀⠀⠀⠱⡨⢀⠌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠊⣣⠀⣴⠀⠀⠀⠀⠘⢤⡀⠀⠂⠑⣄⠀⠀⠀⠀⢠⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠌⡠⢁⡿⢸⠹⣄⢀⡀⠀⠀⠀⠙⣬⠀⢑⠙⡆⠀⠀⡔⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⠅⡢⠰⡇⠀⠑⠬⣢⠹⡳⣄⠀⠀⠸⡆⠐⠂⢻⠀⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠐⠁⣄⠀⢷⡀⠀⠀⠈⠀⠇⡈⣷⠀⡸⠂⡷⠀⢹⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⢸⡨⠀⡄⠣⡀⠑⢄⠀⢀⡬⠁⠀⣾⡼⠃⡐⢨⠀⡵⠀⢸⡇⣷
⠀⠀⠀⠀⠀⠘⡈⠑⢢⡀⠀⠀⢣⠀⠸⡄⠈⠢⡈⢳⡞⢀⠞⠀⠍⢀⠞⠀⠎⢰⠃⠀⡸⡁⢸⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠂⠀⣙⡀⠀⠀⠑⣄⠐⡄⠀⠐⡀⢠⠃⠀⣇⠔⠁⠀⠌⢠⠏⣠⡞⠃⠀⠀⡇⠀⠀⠀⣠⢷
⠀⠀⠀⠀⠀⠀⠈⠹⢖⠤⣁⠀⠀⠀⠘⡆⠈⠀⠀⢱⡎⠀⠀⠋⠀⠀⡜⢠⠿⠚⠉⣀⡤⠀⢰⠇⠀⢀⠌⣁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⢦⠀⠀⢿⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠁⢀⠄⠒⠉⡔⠀⢰⠏⠀⠀⠧⢊⡁⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢦⡀⢳⢀⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⢀⠎⢀⡴⠃⢀⡀⢀⡔⢉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⡀⠰⡈⡟⢁⠌⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⡞⢡⣴⡏⢰⠋⠈⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠃⠀⠆⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠀⣷⠟⠲⣇⡟⡰⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡀⠇⠸⠀⠟⠱⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢴⠀⠱⣄⡃⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠔⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠈⠀⠀⠤⠀⠀⢀⣀⣀⣀⡀⠀⠔⠈⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠈⠀⠈⠐⠠⠐⠨⠀⡂⠄⡀⠄⠠⠀⡠⢀⠐⠀⠠⢐⠀⢀⠀⡠⢔⠢⢀⡐⢀⠔⡢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⣠⠔⠈⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⢂⠠⠤⠈⠀⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠒⡀⠊⠀⠁⠀⠃⢀⡴⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠀⠀⢀⠘⢤⡀⠀⠢⠀⠈⢠⠀⠀⠀⠀⠰⡃⣀⠤⣤⣤⣶⡎⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠧⣀⡅⠀⢀⣅⠐⠠⠒⠣⠤⡰⠀⠀⠀⠞⢡⠿⠆⠛⣘⢋⣅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠠⠁⠐⢀⣀⠀⡀⢺⢄⣤⡶⠃⠠⢀⠀⣀⣀⡀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢦⠀⠢⠄⠐⠒⠒⠛⠫⠛⠁⠀⠈⣉⡉⡡⠔⢁⠴⣿⣦⡥⠤⠅⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤
⠀⠴⠒⠀⠂⠂⠐⠀⠀⣀⠀⠀⠀⠴⠤⠀⣐⠒⠀⠰⠦⠶⠶⡴⠖⢚⠋⠁⠀⠀⠞⠂⠁⠙⠙⠃⠀⠀⠀⠐⠀⠐⠒⠐⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠠⠂⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠖⠉⣭⠿⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣎⣀⣼⡠⠃⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣛⠯⣭⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⢆⠀⠀⠄⠐⢄⠹⡂⢠⠈⢖⣠⠂⢠⠀⡀⣀⡀⠤⢀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠑⠦⣸⡄⢄⠻⣇⡜⢣⣸⢁⢂⣾⠟⠋⠁⣀⠔⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣧⡀⠰⠈⢻⣬⣶⣿⠞⠳⣿⣿⡏⠀⠰⢀⣼⡉⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⠛⠻⣦⣴⣯⠝⠉⠀⠀⠀⠀⠉⢋⣷⡶⣋⠥⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢈⣑⠶⠔⣿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣥⠶⠿⠒⠉⠉⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠠⠐⠉⠀⠔⠚⢅⣻⣿⠀⠙⠒⠂⠀⠀⠑⠚⠋⢈⣿⢌⡵⠒⢀⣤⠄⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣀⣠⠔⢲⡿⢋⡹⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣒⠻⢍⠑⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠖⠁⢂⣡⡞⠟⢭⣷⣤⣀⣀⣀⣠⣴⣯⠟⠉⠻⣦⡅⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⢋⠘⠂⣡⣿⣿⡟⠷⢻⡟⢯⠻⣅⡘⠀⠘⣷⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣀⠄⠉⣀⣠⣴⢾⠏⠌⣸⡏⢶⠋⣿⣄⠣⠹⡏⢦⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠉⠁⠀⠀⠐⠀⡔⠁⢧⠀⠀⠸⣄⠑⠄⢩⠀⠌⠢⠑⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⡶⢯⠀⠀⠀⠁⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢐⠙⠃⠀⡄⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢾⣀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠙⠶⠄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢰⡿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢠⢿⡅⠀⠀⠀⠀⠀⠀⠀⠸⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢻⡆⠀⣼⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⡟⢸⡇⠀⣿⠈⣇⠀⠀⠀⠀⠀⢀⠀⠀⢹⡄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣇
⠀⠀⠀⠀⠀⠀⣸⠃⢸⠃⠀⢹⠀⢻⡀⠀⠀⠀⠀⠀⢷⡄⠀⢳⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⠇⠀⣼⠀⠀⢸⡇⠈⣇⠀⠀⠀⠀⠀⢸⢹⡀⠈⢧⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⡞⠀⠀⡏⠀⠀⠀⣇⠀⢸⡄⠀⠀⠀⠀⣸⠈⡇⠀⠈⣇⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⡿⡇⠀⢸⡆⢻⡀⠀⠀⠀⠀⠀⠀⠀⡏⣿
⠀⠀⠀⠀⢰⠇⠀⠀⡇⠀⠀⠀⣿⠀⠀⣧⠀⠀⠀⠀⡟⠀⡇⠀⠀⠸⡆⠀⠀⢻⡀⠀⠀⠀⠀⢰⠃⡇⠀⠀⣧⠈⣇⠀⠀⠀⠀⠀⠀⠀⣇⢹⡀
⠀⠀⠀⠀⢸⠀⠀⠀⢿⣤⣀⣴⠏⠀⠀⢻⢠⣇⠀⠀⡇⠀⣇⠀⠀⢀⣇⠀⠀⠀⣇⠀⠀⠀⠀⡟⠀⡇⠀⠀⢹⠀⢸⡄⠀⠀⠀⠀⠀⠀⢻⠈⣧
⠀⠀⣄⠀⢸⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⡿⢸⢹⡄⠀⣧⠀⢻⡶⠶⠾⡟⠀⠀⠀⡿⢠⡄⠀⢠⡇⠀⣧⣀⣀⡼⠀⠀⣧⠀⠀⡄⠀⠀⠀⠈⣇⠘⢧
⠀⠀⣿⡇⠸⡆⠀⢀⠀⠀⠀⠀⠀⠀⢠⠇⠘⡆⢳⡀⠸⡄⠀⠙⠀⠀⠁⠀⠀⢀⡇⣿⡇⠀⠸⡇⠀⠘⠿⠋⠁⠀⠀⢹⠀⠀⢹⣆⠀⠀⠀⢹⡄⠈⢳⡄
⠀⢀⡇⣿⠀⢧⠀⡾⣆⠀⠀⠀⠀⢀⡟⠀⠀⢻⡈⢧⠀⠹⣆⠀⠀⠀⠀⠀⠀⣼⠁⢷⢻⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡟⡆⠀⠀⠀⢷⠀⠀⠙⣆
⡀⣸⠁⣿⠀⠸⣶⡇⢸⡄⠀⠀⠀⡾⠀⠀⠀⠀⣧⠈⣧⠀⠘⢦⠀⠀⠀⠀⢠⡇⠀⢸⡆⢳⠀⢹⡀⠀⠀⠀⣶⠷⣦⣸⠀⠀⠸⡇⢹⡀⠀⠀⠘⡆⠀⠀⠘
⡇⡏⠀⣿⠀⠀⢹⡇⠀⡇⠀⠀⢰⡇⠀⠀⠀⠀⣿⠀⢸⡄⠀⠈⢷⡀⠀⠀⢸⣇⠀⣸⡇⠘⡇⠀⢷⠀⠀⠀⡇⠀⢸⡏⠀⠀⢸⡇⠈⡇⠀⠀⠀⢻
⣿⣷⠀⢻⣄⠀⠀⠀⢠⡇⠀⠀⠸⣧⠀⠀⠀⣠⣿⠀⢘⡇⠀⠀⠘⣧⠀⠀⠘⣿⠛⠉⠁⢠⡇⠀⠈⢧⠀⠀⢳⡀⠀⠀⠀⠀⣸⠁⠀⣷⠀⠀⠀⢸⡆
⠹⣿⡄⠀⠻⣷⠶⠾⣿⠃⠀⠀⠀⢻⣿⠷⠿⠻⠃⠀⢸⠃⠀⠀⠀⣿⡆⠀⠀⠈⠀⠀⠀⢸⠁⠀⠀⠈⢧⠀⠀⠻⢶⣤⣴⡾⠋⠀⠀⡟⠀⠀⠀⣸⡇
⠀⠈⠻⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣦⣤⣤⣴⣿⡇⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⣀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⣼⠻⠿⠛⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠈⠉⠉⠉⣿⠃⠀⠀⠀⠀⠀⠀⠘⢿⠛⠛⠋⠁⠈⣇⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠣⠀⠀⠀⠃⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠈⣀⠀⠀⠀⠀⡆⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⡇⠀⠃⠀⠁⠀⠀⠈⠀⠀⠀⢀⠀⠈⠁
⠀⠀⠀⢀⠀⡏⢱⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡜⣙⣎⣸⢇⡇⡀⠀⡀⠆⠀⡔⠀⠀⠀⠈⠁⠀⠀⠀⣀⡀⠀⢠⠖⢦⠀⠀⣀
⠀⠀⠀⡋⣇⡇⢸⣠⢏⡇⠀⠀⠀⢠⠶⡄⠀⡇⠉⢩⠦⢊⡀⠘⢆⣡⡠⠜⣀⡔⠉⢀⠀⠠⠖⠀⠀⠁⢀⠀⢸⠁⠸⠀⡰⠁⠱⡀⠀⠠⡄⢀⡀
⠀⠀⠀⠳⠤⡄⢰⠓⠊⠉⠉⠉⠓⢣⠀⡧⢬⡇⠦⣿⢰⢳⠈⢲⠊⠀⠀⠀⠀⠙⣏⠥⠒⠐⠒⡀⠀⠉⠁⠀⠸⡄⠆⢧⣇⠀⠀⡇⠀⡦⡇⢨⠀⢶
⠀⠀⠀⠀⠀⠁⢸⡄⠀⠀⠀⠀⠀⠈⣆⠉⠒⠁⠀⠻⠞⣸⢅⣣⠀⢀⣀⠤⠔⠒⠛⠂⠉⠉⠀⠀⣫⡄⡿⠋⠉⠙⠤⢠⡀⠀⠀⠳⢖⣡⢇⣈⡄⡇
⠀⠀⠀⠀⠀⠘⠈⠀⠀⠰⠷⠆⠀⠀⠈⠑⠒⡄⠈⡴⠉⠁⠀⠉⠙⠢⠤⢀⣀⣀⣀⣀⢀⠀⠀⠀⠀⠈⠁⠠⠄⠀⠀⠀⢣⠀⠀⡏⠉⠀⠀⠈⠇⠀⢤⣀
⠀⠀⢀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⣶⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠁⠒⠀⠒⠀⢠⠀⢠⢸⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⠈⠁
⠘⠚⠚⠓⠚⠒⠒⠛⠛⠛⠚⠒⠒⠚⠒⠚⠒⠓⠂⠛⠓⠚⠚⠚⠓⠓⠚⠓⠒⠚⠒⠒⠓⠒⠒⠒⠒⠒⠚⠒⠛⠚⠒⠓⠛⠒⠒⠓⠛⠒⠒⠒⠒⠚⠛⠚⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂
⠀⠀⠀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠴⠀⠀⠀⠀⣀⠠⢀⣀⠀⣀⡀
⠀⠀⠠⡀⠀⠈⢄⠨⡒⢄⡠⢀⠃⢱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠲⣀⣀⠤⠒⠩⠔⠁⠀⠂
⡤⠀⠀⠀⠀⢀⠈⢂⠑⠒⠠⠤⠐⠂⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠐⠒⠄⠃⢄⡀⢀⣀⣀⠄⠀⠀⠀⣀
⠀⠠⡌⢁⣀⡡⠤⠤⠗⠒⠒⠦⣄⣀⡳⠚⠃⢄⣄⡠⡆⢧⠀⠀⡀⠠⠠⢤⣀⡠⢀⢀⣭⡤⠄⠈⠙⠀⠐
⠀⠀⠈⠠⢅⣀⣀⠀⠀⠀⠀⠈⠀⠀⡀⠤⢴⣆⣀⠔⡁⠀⠀⠰⠔⠒⠀⠐⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠐⠒⠒⠁⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⢄⠈⢷⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣸⠃⢻⡀⠀⢀⣾⠀⠀⢹⡆⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣸⠇⢰⠀⠀⠀⠀⠀⢠⠈⢻⡄⠀⡄⠀⢠⣿⠀⠈⣷⠀⢸⣷⡀⠀⣸⡇⠀⠀⠀⢠⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢀⠀⣿⠀⠘⣧⠀⠀⠀⠀⣼⠀⠀⣿⠀⣧⠀⠀⢿⣷⣶⡿⠀⠸⣿⣿⣿⡿⠀⣧⠀⠀⣾⠀⠀⠀⣿⠀⣇
⠀⠀⠀⠀⢸⡄⠀⢀⣾⠇⢾⡀⣿⡄⠀⢸⣧⠀⠀⣼⣇⠀⣠⡿⠀⢻⣧⠀⠀⢻⣿⡇⠀⠀⢻⣿⣿⠁⠀⠹⣆⠀⢹⣷⣤⣶⡏⠀⢹⡄
⠀⠀⢀⣴⡿⠁⠀⣾⡟⠀⠈⣷⠘⣿⣷⣿⠇⡄⠀⢿⣿⣿⣿⠁⠀⢀⣿⠀⠀⠈⣿⣧⡀⢀⣾⣿⣿⡀⠀⠀⣹⡆⠀⢻⣿⣿⡀⠀⢀⣿
⠀⢀⣾⡟⠀⠀⠀⣿⣷⣄⣤⡟⠀⠘⣿⡟⠀⢳⡀⠈⢻⣿⣿⣄⣠⣼⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⡿⠁⠀⠀⣿⣿⣷⣶⣿⣿
⠀⠈⣿⣇⠀⠀⢀⣿⣿⣿⣟⠀⠀⢀⣿⡀⠀⢘⣇⠀⠀⢹⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⣿⣦⣤⣾⣿⣷⣶⣾⣿⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⢠⣄⣰⣆⣠⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣆⡠⠀⠀⠀⠀⠀⡄⣐⣰⣂⡀
⠀⠀⠀⣤⣿⣿⣿⣿⣤⠀⠀⠀⠠⣿⣿⣿⡧⠀⠀⠀⠀⠬⣿⣿⣿⡎⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⣽⣿⣿⣿⡧⠀⠀⠀⠐⣾⣿⣿⣿⡒
⠀⠀⠀⢠⠿⢿⡿⢯⠀⠀⠀⠀⠀⠹⠿⠟⠃⠀⠀⠀⠀⠈⠽⠿⠿⠓⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠐⢚⠿⡿⢟⠓⠀⠀⠀⠈⠝⢿⠿⠛⠌
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⣄⣂⣠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⡄⣂⡐⢀⠄
⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢾⣿⣿⣿⣿⡇⠀⠀⢀⢹⣿⣿⣿⡧⠄⠀⠀⠀⠠⣿⣿⡯⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⢒⢾⣿⣿⢀⡒
⠀⠀⠀⠻⣿⣿⣿⣿⠏⠀⠀⠀⠹⣿⣿⣿⡿⠁⠀⠀⠀⠚⡻⢿⠿⡓⠂⠀⠀⠀⠀⠋⠛⠑⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠁⠆⡉⢁⠲⡌⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⢤⣰⣦⡆⡀⠀⠀⠀⡀⢦⣴⣤⠆⡀⠀⠀⠀⡀⢆⣶⣠⢄⠀⠀⠀⢀⡄⣴⣇⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣅⡂⠄
⠀⠀⠀⣽⣿⣿⣿⣿⡥⠀⠀⠠⣽⣿⣿⣿⣿⠥⠀⠀⢠⢽⣿⣿⣿⡯⡄⠀⠀⣀⣻⠁⠀⠙⡟⠆⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠅⣾⣿⣿⣿⡎⠄
⠀⠀⠀⠚⣻⢿⣿⡟⠂⠀⠀⠀⠊⣻⢿⢿⡛⠂⠀⠀⠀⠊⡻⠿⢿⡓⠂⠀⠀⠈⣙⣷⢤⡞⠻⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠂⠝⠿⠿⠟⡔
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁
⠀⠀⠀⠠⡰⠘⠰⢢⠀⠀⠀⠀⢀⣠⣤⣆⡀⠀⠀⠀⠀⣠⣢⣖⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣷⣶⣄⠀⠀⠀⢀⠰⣀⣆⡐⢀
⠀⠀⠠⠅⢰⣿⣷⡄⠩⠀⠀⠀⢺⡏⠀⠈⣿⡂⠀⠀⢸⣹⠁⠀⠈⣳⡅⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠌⣿⣿⣿⣿⡅
⠀⠀⠀⢫⢌⠙⢋⠠⠑⠀⠀⠀⠙⠷⡦⣶⠟⠀⠀⠀⠘⢭⡷⢤⡴⡿⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⠀⠀⠀⠐⢙⠿⠿⢟⠐
⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣵⣭⣿⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣷⣿⣿⣾⣽⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠘⢷⣄⠀⢀⣠⣽⣧⣀⡀⢀⣴⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣋⣙⠻⣷⡄⠀⠀⠀⠀⠀⢹⣿⠿⠛⠉⠉⠛⢿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⢸⣷⠀⢀⣀⣀⣠⣿⠏⠀⠀⠀⠀⠀⠀⢹⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⢸⣿⠀⠛⠛⠛⠻⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⢸⣿⠀⠀⠀⢀⣴⡿⠛⠻⠿⣷⣾⠿⠛⠛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⢸⣿⠀⠀⠀⠈⠋⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢸⣿⣿⡇⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⢀⣼⣿⣿⣧⡀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⣸⡏⠀⢀⣿⣿⣿⣿⣿⣷⡀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⣿⣿⣿⣿⣿⣿⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠛⠻⠟⠛⠁⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣤⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠠⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠚⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠁⠀⠅⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⣀⠀⣤⠦⠐⣄⠤⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠌⠳⡄⡀⠀⠀⡕⠊⠀⠀⠀⠀⠀⠀⠀⠐⡢⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠘⠃⠀⢀⠀⠀⠈⠛⡠⠀⠀⠀⢀⠄⠤⢈⠴⠫⢀⠔⠢⠤⠁⢀⠀⠀⠀⠰⠆⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⡀⠀⠀⠀⢠⠣⡈⠢⢀⢠⠐⠁⠀⠀⠀⠤⡀⠐⠃⠀⠲⢿⠁⠂⠄⡌⠀⠠⠒⡂⠑⠃⠀⡘⠤⠃⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠉⠌⢊⠢⡀⠀⠀⠀⠑⢄⢡⡀⡀⢀⡞⢞⣳⠀⠀⠀⠐⢄⠱⡀⡆⠨⢰⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠁⠜⢀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡀⠄⠀⠀⠂⠀⠀⠺⠓⠂⠉⠀⠢⡠⠴⠀⠀⡃⢀⠃⡝⢑⠂⡰⠈⠀⠐⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⢀⠰⣀⡐⠁⠀⡠⢠⡟⡏⠅⠀⠀⠠⢈⡤⡁⠂⠄⠀⠰⠖⡐⠋⠀⡀⠔⣡⡮⠀⡑⠀⠀⠀⠠⠄⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡣⠄⠠⡄⠕⠐⣬⡂⡊⡤⠋⢈⠄⠨⡐⠪⡴⠛⣁⣤⣄⡀⢤⢲⠿⠤⠤⢤⣆⣁⣼⣀⢨⣥⠀⣉⠉⠠⠔⠄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢜⢯⡑⠞⠫⠒⠀⠐⢁⣭⡈⢀⡙⢹⡆⠀⠈⠓⢓⠈⡀⡀⠀⠠⢌⠀⢨⡼⠀⠈⣣⣰⢁⣾⡷⢎⣡⠁⡰⠀⢂
⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠑⠩⠑⢤⣀⠀⢠⡟⠄⡽⠿⣉⡉⠀⠀⣠⡤⠗⠛⠲⣤⡤⠋⠷⠔⡉⢳⠀⠉⠈⠡⠈⡋⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⣤⠀⣎⠆⠀⠆⠐⠂⠀⠰⡙⡖⠫⠉⢉⣤⠴⠃⠂⠐⠑⢀⡤⣐⡒⠖⠠⠤⠐⠒⠀⠂⠉⣀⠀⢄⠁⠉⠠⡩⠂⠀⠛⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣀⡆⢠⠒⠀⠀⠀⠀⠀⠀⠀⢰⡈⠀⠘⠚⠑⠲⠓⠉⠙⣻⣮⢔⣲⡭⠁⢠⠒⢒⡂⢁⠀⠂⠐⠀⠡⠈⠀⠀⠂
⠀⠀⢀⠀⠀⠀⡄⣡⠴⢄⠀⠀⠂⡠⢀⠆⠀⠂⠀⠀⢀⡄⠰⢆⢬⡁⠉⠀⠠⢀⠀⢗⠀⡆⢀⢀⠀⠀⠀⠀⠐⠀⠀⠔
⠀⠀⠀⠁⡄⢠⣅⡨⡗⠀⠦⡭⣤⠴⠻⠄⠀⠀⠀⣄⠴⢀⡾⠞⠛⠁⠒⠒⠂⠁⡀⠀⢇⡱⠈⠊⠀⣤
⠀⠀⠀⠂⠁⠪⡉⢫⡀⠀⠀⠀⠠⢄⡁⠀⠀⠀⣊⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠜⣆⠈⠀⠀⠑⠈⡠⠔⠂⠉⠀⠀⠀⠘⠀⣀
⠀⠀⠠⠀⠀⠀⠀⠠⣨⢗⡂⠀⠀⠀⡁⠀⠀⠀⠣⠃⠀⠀⢀⡀
⠀⠀⠀⠑⠀⠀⡠⠚⠉⠋⠐⠀⡀⠀⠐⠂⠠⠤⠎⠬⠄⢒⠄
⠀⠀⢀⠔⠰⠶⡇⠀⠒⠀⠒⠂⠀⠂⠠⠆⠒⠨⠄⠒⠠⢣
⠀⠀⠈⠉⠀⠀⠐⠈⠀⠀⠘⠄⠐⠐⠠⠄⠉⢀⣘⡄

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


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

⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡰⠀⠀⠦⢔⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠞⠀⠘⡄
⠀⢰⠃⠀⠀⢀⡾⢡⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡘⣧⠀⠀⠀⢹⡄
⠀⣿⠀⠀⠀⠘⠷⢋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠳⠟⠀⠀⠀⢸⡇
⠀⠹⣆⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⢀⡼⠁
⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⣠⣾⠟⠋⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⣼⡿⠋⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠰⡜⢿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣿⣇⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠘⡎⢿⣧⠀⠀⠀⠀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿⡀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⣠⣾⠟⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠻⣷⣾⣟⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⡿⠀⣼⡿⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠈⣿⡆⠀⠉⠛⠛⠉⠀⠀⠀⢀⡼⡇⠀⠀⠀⢰⣿⠇⣼⡿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢠⠏⠀⢧⠀⠀⠀⢸⣿⣾⡿⠁⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⡀⠀⠀⠹⣿⡀⠀⠀⠀⠀⣰⠃⠀⠀⠘⡆⠀⠀⠀⠉⠉⠀⠀⠀⡄⣼⠃⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣷⡀⠀⠀⣿⡇⠀⠀⠀⢰⠇⠀⠀⠀⠀⢹⡀⠀⠀⢀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢀⠞⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡘⢿⣦⣤⣿⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣇⠀⢀⠜⠀⠀⠀⠀⢣⡹⢦⣤⠴⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⠓⠮⠭⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠉⢂⠱⡀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠑⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⢿⠢⠞⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠐⢂⡿
⠀⢶⣶⡌⢷⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⡾⣡⣾⠗
⠀⠀⣿⠙⠟⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠛⢋⡿
⠀⠀⠘⢧⣀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣠⡾⠁
⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠄⠄⡀⠀⠀⡠⠀⠀⢀
⠀⠀⠀⠀⠄⠀⢆⠊⠼⠛⢩⠇
⠀⣤⠀⠀⠠⠀⢨⠴⠀⠀⣀⡄⠒
⠀⠀⠐⠃⠀⠂⣴⠀⠀⣄⣀⠀⢤⡀⠀⢀⢠⡄
⠀⠀⠀⠀⠄⠋⢿⡡⠀⠽⠉⠉⠂⠐⣔⠡⠛⣀⠆
⠈⠉⠀⠀⠀⠀⠀⠘⠂⢠⢤⡤⠂⣠⡏⠐⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠛⠱⠀⠘⠁⠁⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⢐⠀⠀⠀⠁⠀⠀⠀⠀⢠⡼⢏⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠦⠀⠀⠀⠀⠀⣠⣴⣗
⠀⠘⠧⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠟⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠏⠀⠉⠐⠃⡒⠰⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⢛⡁⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠄⡃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠖⠒⠀⠤⠂⠀⠀⠀⠐⠂
⠀⠁⠀⠀⠀⠒⠃⠀⠀⠀⠨⠥⠄⠀⠀⠀⠠⠀⠁⠒
⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⠤⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢀⠀⠀⣀⠀⡉⢀⠁⠀⠀⠀⢀⠀⠁⠀⠐⣾⣣⣠⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⡀⠈⠉⢀⡀⢠⣠⣀⢀⣀⣹⣄
⠋⠡⠀⣀⠀⢀⣠⣷⣿⡾⠵⠞⠛⠓⠒⠛⠉⠉⠉⠀⠀⠀⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠶⠶⠤⠶⠾⠻⠿⠛⠛⠛⠟
⠛⠛⠛⠛⠋⠉⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⢸⣿⡀⢸⣦⡀⣿⡏⠀⣾⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠹⣶⣦⣿⣷⣤⣿⣿⣿⣷⣼⣿⣧⣾⡟⠀⣤⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣼⣿⡿⠟⠛⠋⠉⠉⠙⠛⠿⣿⣿⣧⣾⣷⣀⣴⣾⠟⠁
⠀⠀⠀⠒⠶⢶⣤⣤⣀⠀⠙⠾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣾⡿⠀⢀⣀⣀⣀⣤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⣩⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⠤
⠀⠀⠒⠛⠛⠛⠛⠛⠛⢛⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠿⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡛⠛⠛⠻⠿⠷⠶⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠓⠂
⠀⠀⠀⠀⠀⣠⣶⠿⠟⠋⠉⢸⣿⠟⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⠻⢿⣦⣄
⠀⠀⠀⠠⠚⠉⠀⠀⠀⠀⠀⠋⢁⣴⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⢦⠀⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⢻⡟⠀⣿⡿⣿⣿⠻⣿⡿⣿⣧⠈⢻⣷⡀⠀⠀⠀⠀⠀⠈⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠈⠀⣸⣿⠁⠉⡏⠀⣿⣇⠀⠘⠂⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠹⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⠄⠀⣀⣠⣤⣀⡀⠀⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠛⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⢸⣿⠃⠀⠀⠀⠀⠈⣿⣧⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⢸⣿⣄⠀⠀⠀⠀⢠⣿⡏⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢿⣷⣦⣴⣶⣿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠈⢉⡉⠁⠀⠀⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠀⠀⠉⠻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠁⠀⣠⣶⣷⣦⡀⠀⠙⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠀⣀⣴⣿⠿⠋⠉⠻⢿⣷⣄⠀⠈⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠁⢀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠙⠻⣷⣦⣀⠀⠉⠻⣿⣶⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁⠀⣠⣶⣿⠟⠋⠀⠀⣠⣶⣾⣿⣶⣦⡀⠀⠈⠛⢿⣷⣤⡀⠈⠙⢿⣷⣦⡀
⠀⠀⠀⠀⢸⣿⣏⠀⢀⣴⣾⡿⠋⠀⠀⠀⠀⣸⣿⠋⠀⠀⠈⢿⣿⠀⠀⠀⠀⠉⠻⣿⣶⣄⡀⢈⣿⣿
⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠟⠁⠀⢠⣤⡀⠀⢀⣤⡄⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣶⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣰⣿⠟⠀⣠⣿⡟⠀⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠻⣿⣄⠀⠹⣿⣆⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣽⣿⠂⠀⣽⣿⠃⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢾⣿⡁⠀⢾⣿⡃⠀⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⢿⣷⡄⠈⢿⣷⡄⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣰⣿⠟⠀⣠⣿⠟⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣸⣿⣇⣀⣀⣀⣀⣀⣀⣿⣿⠀⠀⠀⠀⠹⣿⣆⠀⠹⣿⣦⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠘⠟⠀⠀⠘⠟⠃⠀⠘⠿⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⣀⣠⢄⣀⣀⡀⢀⡀⠀⣀⡀
⠀⠀⢰⡇⠀⠀⠀⠀⣴⣿⣿⣆⢠⣿⣿⡇⠀⠀⢠⡄⣄⡀⠀⢘⣿⣿⣿⣿⣿⣾⣿⣿⠇⣠⡼⣠⣶⢫⡄
⠀⣸⣿⢷⣶⠀⠀⢰⣿⣿⣿⣿⡄⠁⠛⠀⣠⢬⣴⣞⣿⣿⣦⣂⠉⠉⣩⣉⣭⣭⣭⣥⠀⣹⣷⡽⢛⣏⣿
⠰⣿⠃⠘⣿⡇⠀⠘⠛⠛⠛⠟⠁⠀⢠⣾⡿⣿⣿⣿⡿⡿⢿⡿⣷⠜⣿⣮⣭⠯⠄⠀⠀⠉⠋⠿⠿⠋⠁
⠀⠈⠀⠈⠉⠀⠀⢀⢀⢠⢀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠠⠀⠀⣾⣿⣿⠁⡀⠀⠀⣀⠐
⠀⠀⢀⠀⠀⣠⣶⣶⣿⣽⣪⢢⣴⡆⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠘⡋⠁⢢⣬⠄⢠⣽⣶⣧⡀⠀⠠⣀
⠀⡀⣼⠀⠀⠳⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⣠⣾⣿⣿⣿⡀⠀⠀⣰⢤⡤⠈⠉⠀⠻⣿⣿⠊⠀⡤⡟⣯
⠘⣇⢣⣮⡀⠀⢀⢾⣤⡀⠀⠀⢦⣦⠂⣀⠀⠿⣿⣿⣿⣿⡟⢠⣼⣷⢳⣷⡄⠀⠚⠛⠛⠛⠓⠂⢉⣿⡟
⢀⣿⣸⣿⡃⢠⡿⠻⠿⣷⠀⢜⣿⣻⣗⣼⢃⠀⣿⣿⣿⡿⠁⢹⡿⣿⣿⣿⣧⠀⠀⠀⢂⠀⠀⠀⠻⢯⣄
⠈⠻⠿⠟⠁⠈⠓⠀⠐⠋⠀⠘⠟⠛⠛⠛⠟⠀⠈⠇⠹⠋⠀⠸⠿⠿⠿⠿⠿⠀⠶⠿⠿⠷⠶⠆⠰⠾⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡀⠀⠐⠀⠀⠀⠀⠀⣀⡀⠘⢃⣀⣉⠀⠰⠇⠀⠀⠀⠀⠀⠀⣄⠹⣇⠹⠀⠀⡜⢀⡀
⠀⠀⠀⠀⠀⡀⣄⣴⠟⢯⡓⣄⠀⠀⠀⠀⠀⠀⠙⠁⡄⠀⡀⠀⡀⠀⠀⠶⠂⠀⠀⠀⠲⠀⠁⢈⣤⣤⣄⠐⣡⡴⡆
⠀⠀⠀⠀⠀⢀⠸⡧⣸⣅⣼⣇⠀⠀⠀⠀⠀⠛⠃⠰⠀⠢⣀⣀⡡⠄⠁⠠⡤⠀⠀⠨⠭⠄⠀⣿⣿⣿⣿⡇⠀⠰⠶
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠛⠏⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠶⠒⣡⠌⠛⠛⠋⠀⢌⡙⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⣤⠀⠲⠄⠀⠀⠀⠀⠀⠀⠺⠃⠀⡆⢰⠀⠘⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠲⠄⠸⡄⠀⢠⡟⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢃⣀⠃⢠⠆
⠀⠀⠀⠐⠶⣄⠢⠄⠐⠈⡔⢀⣤⡄⠀⠀⠀⠀⠀⢠⠿⣟⣛⡛⠿⣷⡀⠀⠀⠀⠀⠀⠐⠄⢠⣾⣿⣿⣿⣦⡀⠤
⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠠⠄⠀⢤⠄⠀⠀⠀⠀⢿⣜⢾⣜⣟⡶⣿⠁⠀⠀⠀⠀⠀⠐⠂⣿⣿⣿⣿⣿⣿⡇⠤⠄
⠀⠀⠀⠙⠋⠁⠔⡀⠄⣀⠢⠘⠲⠄⠀⠀⠀⠀⠀⠈⠮⣯⣿⣿⡵⠃⠀⠀⠀⠁⠀⠀⠐⠂⡘⠿⣿⣿⣯⠟⠡⢄
⠀⠀⠀⠀⠚⠁⡸⠁⠀⠘⡄⠐⠦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⢀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢀⡀⠀⢠⡄⠀⣠⠆
⠀⠀⠀⠀⢠⣀⢻⠢⡸⠣⠞⣏⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣤⣀⣹⠟⠀⠀⠀⠀⠑⠧⢤⣤⣤⠀⠀⠀⠀⠀⢀⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠍⠫⣝⢦
⠀⠀⠀⠀⠈⣲⠄⠀⠀⠀⠀⠀⠺⣍⣀⠀⠀⠀⠀⠀⢰⠡⠄⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢆⢞⣻⣾⣿
⠀⠀⠀⠒⠛⣡⣟⣀⠀⠀⠀⠀⣏⠈⠁⠀⠀⠀⠀⠀⠈⠢⣈⣋⠠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠧⣂⠥⣕⣵⠏
⠀⠀⠀⠀⠘⠉⠀⡶⠉⣦⠲⡏⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠀⠀⠀⠀⠀⠀⠀⡆⠉⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⡿⡖⣄⠀⠀⠀⠀⠀⣇⢀⣀⠀⠀⡄⡾⠀⠀⠀⠀⠀⠀⢀⣀⣠⠒⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠟⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⣿⣆⠀⠀⡀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⣄⢠⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡄⡀⠆⠘⠀⣳⠀⠀⠀⠃⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠈⠁⠀⣄⣃⣌⠏⣄⡈⠀⠱⠄
⠀⠀⠀⣰⡀⢸⡽⠁⠈⠦⣽⡉⣿⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠐⠶⠇⠐⡆⠸⣄⣸⠖⠃⣦⠇
⠀⠀⠀⠙⣍⡩⢀⡀⢤⠀⣼⠀⡆⠀⢀⡀⠀⠀⠀⢀⢤⡀⠀⠀⡤⣤⡄⠀⠀⠀⠀⣟⣉⣀⣇⠖⠈⠩⣤⣴⠁
⠀⠀⠀⠀⠀⣇⠀⠈⡿⢺⣼⣿⠗⠀⠈⠁⠀⠀⠁⠋⠉⢳⠄⠁⣄⠀⡇⡰⠙⠂⠀⠈
⠀⠀⠀⠀⠐⠚⠛⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠼⠤⠤⠞⠚⠁⠓⠀⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠖⠶⠶⠤⢤⣀
⠀⠀⠀⠀⠀⠙⠛⠦⣄⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢱⡀⠀⠀⠀⠐⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣿
⠀⠀⠀⠻⠿⠛⠛⠀⠀⠀⠀⣇⠀⠀⠀⢺⣄⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡟⠀⠀⠀⠀⢀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠈⢦⣉⠓⠒⠶⠶⢒⡿⠋⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠈⠙⠒⠒⠋⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠖⠚⠉⠁⠀⠀⠀⠀⠀⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣤⡀⢬⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠘⣿⣿⣦⠹⣧⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢠⠀⠀⠀⠀⢹⣿⣿⡇⠟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠲⣦⡀⠀⠀⠀⠘⢠⣿⣇⠀⠀⠀⢸⣿⣿⡇⠀⡀⠀⠀⠸⣷⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠉⠹⠀⠀⠀⠀⣼⣿⣿⣶⣶⣶⣿⣿⣿⠀⠀⢻⣷⣄⠀⠈⢿⡿⠄⢹⣧
⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢸⣿⡇⠀⠀⠀⠀⢳⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣠⣼⣿⣿⣇⠀⠘⡏⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣾⣿⠃⠘⣶⣄⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢰⡄⣰⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣦⣴⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⢃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠐⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣶⣶⣶⣶⣶⠀⣿⠀⠰⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⠰⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⢠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⢀⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⢈⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⢈⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⠀⣿⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⢋⣴⣿⣿⣿⣿⣿⣤⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣍⣉⣉⣉⣭⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣼⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⣿⡏⠸⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣶⠄⣉⣤⣭⣥⣌⡁⢴⡞⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡌⢛⡷⠞⠛⠛⠻⢾⣟⢉⣾⣿⣿⣿
⠀⠀⠀⠀⠦⢀⣾⣿⣿⣿⣿⣿⣿⣆⠠⠆⠀⠀⠀⠀⣿⣿⣿⣿⣟⣻⠏⠀⠀⠀⠀⠀⠀⠙⣿⣻⣿⣿⣿
⠀⠀⠠⣴⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣶⡤⠀⠀⠀⣿⣿⣿⣛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⢛⣻⣿
⠀⠀⠀⠀⣤⠘⣿⣿⣿⣿⣿⣿⣿⡟⢠⡄⠀⠀⠀⠀⣿⣿⣿⣿⣟⣻⡄⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⣿⣿
⠀⠀⠀⠀⢀⣴⡌⠛⠿⠿⠿⠟⠋⣴⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⠇⢹⡦⠄⠀⣀⣠⣴⣟⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠁⠀⠚⠀⢶⠆⠘⠃⠀⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣯⣾⡏⢩⣿⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿

⣤⣶⣶⣴⣶⡶⠶⠶⠶⠶⠒⠒⢂⣀⡒⣒⣂⣰⣆⣠⣤⣤⡤⠤⠤⠤⠤⠤⣤⣄⡠⠄⠀⢀⣀⣀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣷⣶⣤⣤⣤⣤⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣯⣽⠛⠛⠻⠏⠡⢤⡤⠄⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣶⣤⣤⣀⣀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢤⢢⡰⡀⣠⢠⡄⣄⢠⡀⡀⣄⣀⣀⣀⣀⣀⣤⣀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣏⣎⣷⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣴⣶⣦⣤⣤⣤⣿⣿⣿⣷⣤⣴⡶⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢈⣉⣭⣭⣁⣉⣁⣌⣤⣀⡀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠋⠉⢉⣿⠛⢻⣿⡉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠿⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢿⡛⠿⠿⠿⠟⠛⠛⠻⢶⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣴⠾⠛⠛⢷⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡉⢻⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠈⢻⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⣼⣶⣿⣡⡟⠀⠀⠀⠀⠀⠀⢀⣴⡾⠷⠾⠿⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠀⣿⣍⡙⠛⠛⠛⠛⠛⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⡉⢿⣦⣀⢀⣀⣀⣀⣿⠟⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠙⠶⢭⣙⣛⣛⣛⣩⡥⠞⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣦⣤⣬⣭⣭⣥⣤⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣦⠈⠉⠉⠉⠉⠉⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⠀⠀⡄⠀⠀⠀⣿⡇⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡧⠀⠀⠀⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⠀⢰⡇⠀⠀⠀⢸⣿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣿⠀⢸⡇⠀⢠⡄⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⠃⠀⢸⠇⢸⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⠀⠀⢠⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡟⠀⣀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⡏⢸⠀⠀⠀⢸⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⠀⡇⠘⠀⠀⠀⢸⣇⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢸⡟⢠⠇⠀⢀⡀⠀⢸⣿⠀⠀⢰⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣀⡀⠀⢸⡇⠀⠀⠀⢸⠃⠀⢸⡇⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣴⡟⠛⠻⠇⠸⡿⠀⠀⠀⠘⠀⠀⠘⣧⠀⠘⠋⠉⣠⣄⠀⠀⠘⠋⠉⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠇⠀⢀⣤⡄⠀⠀⠀⠀⢀⡀⠀⠞⠁⠀⠲⣦⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠸⣤⠀⠀⢰⡟⠁⠈⠙⠃⠠⣇⠀⠀⠀⠀⠐⠛⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⣠⡄⠀⠈⠀⣀⠀⠀⣀⠀⠈⢉⣤⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠿⠛⢿⣤⣀⣠⡿⠃⠀⠻⣶⣶⠾⠛⠿⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉