Formal Emojis & Symbols

⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣀⣴⣺⡷⠿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣾⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⠿⣿⣿⣿⣦
⠀⠀⠉⢻⣿⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⢈⣿⢏⠀⠀⠥⡌⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠀⠘⣿⠆
⠀⠀⠀⠐⠃⠶⠀⠀⠶⠈⡅⠀⠀⠀⠀⠀⠀⠀⠙⠛⢴⠀⠀⢰⡆⢀⡿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡊⠘⠂⠀⠘⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠋⠈⠉⠁
⠀⠀⠀⠀⠑⢀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣤⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣤⡖
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⢀⣾⣿⣿⣿⠟⠛⣿⣿⣿⣇⠀⠀⠀⠀⠀⣠⣟⣛⣿⣿⠋⠈⢻⣿⡏
⠀⠈⠋⠉⢻⣿⣾⣿⣿⣿⡿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣇⣿⡆⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣙⣁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢹⣿⣿⠁⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⢹⣿⣿⣿⠀⠀⠀⠀⢀⣴⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣤
⠀⠀⠀⠀⠸⣿⣿⠀⠀⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢻⣿⣿⡀⠀⠀⣶⣿⣿⣿⠁⠀⠀⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠘⣿⣿⡇⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⢻⣿⡍⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⢻⣿⣧⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣸⣿⣿⠀⠙⠛⠿⢿⡇⠀⠀⠀⠀⡀⢀⠀⠀⠸⣿⢿⡿⢿⡿⠿
⠀⠀⠐⠾⠿⠿⠟⠋⠀⠀⠀⣿⣿⡀⠀⠀⠀⣤⣶⣿⣿⠿⠟⠁⣠⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠠⣤⣤⣶⣿⡿⠂⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠛⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⣠⣴⣾⣷⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣶⣦⣤⡀
⠈⠻⣿⠁⠀⠉⠙⣿⠆⠀⠀⠀⠀⠀⢘⣿⠋⠉⠩⣍⢿⣿⣿⣇⠀⠀⠀⠀⠀⠘⢻⡏⠀⠈⠉⢻⡧
⠀⠐⠃⠒⠀⠐⠂⢉⠀⠀⠀⠀⠀⠀⠈⠛⠶⠀⠀⡦⢠⡟⡿⠁⠀⠀⠀⠀⠀⠀⠘⠙⢃⢀⣃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⣈⣴⣶⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇
⢀⣾⣿⣿⡿⠿⣿⣿⡇⠀⠀⠀⠀⠀⣴⠿⣿⣿⠛⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠛⠛⢿⣿⣤⣾⣿⣿⣿⠀⠀⠀⠀⣼⣷⣿⣿⣧⣶⣾⣿⡇
⠀⠀⣼⣿⣿⣿⣿⣧⣿⡄⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠
⠀⠀⢻⣿⣿⢻⣿⣿⣭⣅⠀⠀⠀⠀⠀⣿⣿⣿⡟⢿⣿⣿⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⣿⣿⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⠀⠘⣿⣿⣇⠀⠀⢀⣤⣾⣿⠂⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶
⠀⠀⠈⣿⣿⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠸⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⠁⠀⡀⠀⢰⣿⣿⣿⣿⣇
⠀⠀⠀⢿⣿⠀⠀⢻⣿⠛⠃⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⡄⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⡀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣹⣿⡇⠘⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠲⠾⠿⠛⠃⠀⠀⣿⣷⠀⠀⠀⢠⣶⣿⡿⠿⠃⣀⣴⣿⣿⠟⠀⠀⠀⠀⠀⢤⣤⣶⣿⠿⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠈⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿
⣿⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⡈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣩⡶⠶⣮⣉⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⣿⣿⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠚⠒⠀⠿⠟⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⢸⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢻⣿⣿⣿⣿⣿⡿⢻⣿⣿
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⢸⣿⣿⣿⣿⣿⣇⢸⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⡇⠀⠸⣿⣿⣿⠀⠈⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣷
⠀⠀⠀⢀⣿⣿⡟⠀⣿⣿⠁⠀⠀⠹⣿⣧⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠾⣿⣿⣿⣿⣿⣿⣿⠿⢸⣿⣿
⠀⠀⠀⠈⣿⣿⡇⢰⣿⡟⠀⠀⠀⠀⠻⠿⢷⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣧⣿⣿⡇⠀⠀⠀⠀⠀⢠⣬⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣾⣿⣧
⠀⠀⠀⠀⠘⠛⠉⠻⣿⣷⣶⣤⣤⣤⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣶⣤⣬⣿⣿⠻⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⠀⢻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⢀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⢿⣿⣿⠏⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⣴⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠘⢿⣿⣿⣝⡿⣧⣠⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡇⠀⠀⠀⠘⣿⠻⣿⣷⣾⣿⣿⣿⣶⣤⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠉⠀⠈⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⠀⠀⠿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁

⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠷⠿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣼⣿⣀⠀⠀⠀⠀⣨⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣤⣄⣀⣀⣠⣤⣴⡶⠛⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⠇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⢠⣴⣿⠟⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⡇⠈⢿⠿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠿⠿⠟⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠻⠟⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⢤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣾⣿⣆⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⣤⠀⠀⠀⠀⣰⣿⣷⣦⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢃⡼⢋⠄⠀⣤⣴⣿⣿⢻⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡉⠛⠃⠀⠀⠈⢉⣿⣿⣿⡟⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢤⣑⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⠂⠀⠐⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣶⡗⣿⣿⣿⣿⣿⢺⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡛⠋⢙⠹⣄⣀⣠⢞⡏⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡿⠁⠈⠻⣿⠟⠁⠈⢿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢠⣽⣽⣌⠉⣡⣿⣿⡄⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⠈⠛⣟⠡⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣇⠀⣿⡀⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣧⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡄⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣧⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⡟⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠿⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡆⠀⠠⣿⠅⠀⠀⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⣿⡆⠀⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⣿⡇⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣐⣿⡇⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁⠀⠀⠙⢿⠟⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣬⣿⣭⣄⠀⠀⠀⠀⠀⠀⠀⢀⣬⣝⣫⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⢙⣟⢡⣀⠀⠀⠀⠀⠀⠀⠀⣠⣙⣟⣡⣀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢠⣿⣿⡇⢿⣿⣷⡀⠀⠀⠀⠀⠀⣼⣿⣮⢟⣾⣿⣆⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣾⡿⣿⡇⢸⣿⢿⣇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣇⣿⡇⠀⠀⠀⢸⣿⣼⣿⣾⣿⡇⣿⡇
⠀⠀⠀⠀⠉⢸⣿⡇⣿⣗⠉⠁⠀⠀⠀⠀⠉⠁⣿⡇⣿⣿⠈⠁⠀⠀⠀⠀⠈⠁⣾⣿⢸⣿⡇⠉⠀⠀⠀⠀⠈⠉⢼⣿⢸⣿⡇⠉
⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢼⣿⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠸⣿⠇⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠸⣿⠇


⠀⠀⠀⠀⠀⠀⣴⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦
⠀⠀⠀⠀⠀⠀⠻⢿⠿⢁⣿⠀⠀⠀⠀⠀⠀⠀⠿⢿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠟
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣴⣷⡸⣣⣿⣆⠀⠀⠀⠀⠀⠀⢠⣾⡖⣿⣸⣷⠀⠀⠀⠀⠀⠀⢠⣾⣷⣷⣾⣷⡀
⠀⠀⠀⠀⣼⡟⣿⣿⣿⠻⣷⠀⠀⠀⠀⠀⢰⡿⢻⣿⣿⡟⣿⡄⠀⠀⠀⠀⠀⣿⢻⣿⣽⣿⢹⣇⠀⠀⠀⠀⠀⣾⠛⣿⣿⣿⢻⣧
⠀⠀⠀⠀⣿⢠⣿⣿⣿⡆⣿⡇⠀⠀⠀⠀⣾⡇⣾⣿⣿⣧⢸⡇⠀⠀⠀⠀⢸⡿⢸⣿⣿⣿⡈⣿⠀⠀⠀⠀⢰⣿⢰⣿⣿⣿⡀⣿
⠀⠀⠀⠀⠛⣾⣿⣿⣿⣿⡙⠃⠀⠀⠀⠀⠛⣸⣿⣿⣿⣿⡞⠋⠀⠀⠀⠀⠘⠃⣼⣿⣿⣿⡇⠛⠁⠀⠀⠀⠘⠋⣼⣿⣿⣿⡇⠛
⠀⠀⠀⠀⠘⠿⣿⡿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠿⣿⡿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉


⣿⢿⡿⠿⠿⠻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⡿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡏⠀⠀⠀⢠⡾⠛⠛⠳⠶⠶⢶⣶⣶⣶⣶⣶⠶⠶⠞⠛⠋⠀⢸⡷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣸⠃⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢹⡟⠛⠷⣴⡟⠀⠀⠙⠉⣉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡉⠋⠀⠸⣧⣠⡾⠛⢳⣿
⠀⠀⠀⠀⢸⣧⣀⣴⡿⠁⠀⠀⠀⣿⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠈⠻⣦⣄⣴⡿
⠀⠀⠀⠀⠀⢻⣯⡀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠉⣭⣉⣉⣩⡙⠁⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣀⣀⣀⡀⣉⣀⣀⣀⣠⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠉⠉⠉⠉⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣶⠾⢷⣀⣀⣀⣀⣼⡷⢶⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣤⢴⣿⠉⠉⠉⢻⡅⠀⠀⢰⡟⠛⠛⢿⡇⠀⠀⢨⡟⠉⠉⠉⢿⡶⣤⣄
⠀⠀⠀⠀⠀⢰⡏⠁⠈⣿⠀⠀⠀⠀⠙⠛⠛⢿⣧⣀⣀⣼⣿⠛⠿⠋⠀⠀⠀⠀⣿⠃⠀⢹⡇
⠀⠀⠀⠀⠀⣸⠇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡏⢹⣏⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠈⣧
⠀⠀⠀⠀⢠⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣼⠃⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣿⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⣿
⠀⠀⠀⢀⣿⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡿⣃⠀⠀⠀⢻⡀
⠀⠀⠀⢸⡇⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠘⣧
⠀⠀⠀⣿⠁⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢻⡀⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⢸⠇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⠘⣷⡀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠘⣇⠀⣿⠀⠀⠀⢿⡀
⠀⠀⢰⣧⣀⠀⠀⣿⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⢀⣼⣇
⠀⠀⣿⠋⠉⠛⠛⢿⠀⢿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠃⣿⠛⠛⠋⠉⢿
⠀⠀⣿⣤⣤⣤⣄⣾⠀⢸⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⣿⣠⣤⣤⣤⣾⠃
⠀⢀⡽⠃⠀⠀⢻⡇⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⢸⡟⠀⠀⠈⢿⡄
⢀⣼⠇⠀⠀⠀⠈⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠘⣧
⣼⢷⡀⠀⠀⢰⡄⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⢣⣤⠀⠀⠀⠘
⠻⣇⣸⡆⣰⣿⠛⠛⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠟⣿⡆⢰⣇⣸
⠀⠙⠻⠶⠿⠇⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠹⠿⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣷⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣤⣤⣤⣤⣼⡏⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣼⣧⣄
⠀⠀⠀⠀⣠⣶⠶⠛⠛⠋⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠶⣶⣄
⠀⠀⠀⢼⣯⣤⣤⣀⣠⣤⣤⣤⣤⣤⣤⣤⣾⠀⠀⠀⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣤⣀⣀⣀⣤⣽⡧
⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶
⣿⣿⣿⣿⡟⠋⠹⠟⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣁⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠛⠓⠛⣻⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆⠀⠀⣿⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡀⠀⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⣿⣿⣧⣿⣿⣿⣿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣸⣿⣿⣿⡟⣿⣿⣿⣿⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢁⣝⡻⢿⡟⣴⡜⡿⠿⣛⣅⢹⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⢸⣿⣿⣷⣾⣿⣿⣶⣿⣿⣿⠈⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠁⢻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠟⠋⠀⠀⠀⠙⠻⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡴⢶⠄⠀⠀⠀⠀⠀⣴⣶⠆⠀⠀⠀⠀⠀⠀⣰⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠉⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⢤⡀⠀⢰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠃⠀⠀⠀⠀⠀⣸⣿⣶⣾⡆⠀⠀⠀⠀⢀⣼⣷⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⢘⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⣾⣿⣶⣿⣆⠀⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠘⢻⣿⣿⡆⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣇
⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠘⠛⠛⠛⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⢻⣿⣿⣿⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⣻⣿⣿⣿⡇⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠁
⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠸⡿⣽⣶⣦⡀
⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢀⠙⠛⠛
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣶⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠸⡿⠿⡿⠀⠀⠀⠀⠘⠛⢷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢻⠁⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠼⣴⣾⣷⣶⣶⣾⣶⣶⣶⣶⣶⣶⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣤⣤⣴⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⠤⠤⣤⣤⢤⢤⣤⡤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢶⣦⣭⣿⣶⣿⡿⠿⣳⣄⠀⠀⣿⡏⠉⠛⠳⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠹⡍⠉⠭⠭⠭⠭⠝⣉⣙⠛⣿⣿⣿⢿⡯⠭⠉⢹⡷⢚⡋⢹⣿⣧⠀⣿⡇⠀⠀⠀⠀⠀⠉⣧
⠀⠀⠀⠀⠀⣴⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠓⠛⠛⠛⠛⠓⠒⠒⠚⠻⠿⣶⣾⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠉⠉⢹⡟⡟⠛⠛⠛⠛⠛⠛⠓⠒⠒⠶⠶⠶⠶⠶⠤⠤⣼⢷⣶⣶⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⠛⠛⠛⠛⠛⠛⠒⠒⠶⠶⠶⠶⠶⠶⠦⠤⠤⢤⣤⣤⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣃⣷⠶⣶⣦⣤⣤⣄⣀⡀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⣼⣯⣭⣭⣭⣭⣉⣉⡙⣿⣿⣿⡿⠿⣛⣷⠀⢠⣶⣻⢷⣶⡄⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⡇⠰⠤⠤⠤⠤⠤⠉⡇⣿⣾⣿⣿⣿⣿⣿⣷⣾⡸⠛⠿⣿⠃⡀⣼⡆⣿⣶⣶⣶⣶⣶⣶⣶⣿⠄
⠀⠀⠀⠀⠀⣶⠷⠾⠿⠿⢷⣶⣶⣤⣧⣿⣿⣟⣛⣛⣛⣛⣻⣷⠇⠴⠶⡖⠓⡟⠻⠃⠿⠛⠛⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣤⣶⣤⣬⣭⣍⠉⠉⠙⠛⠛⠛⠛⠻⡿⠿⠛⠛⣿⠃⣿⠛⠛⠛⠛⠛⠛⠋⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⣿⠇⣿⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠓⠶⠶⠶⠤⣤⣼⣷⣾⣋⣉⡉⠀⣿⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠔⠶⠂⠒⠒⠒⠊⠉⠉⠉⠉⠉⠉⠓⠒⠒⠲⠶⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠥⠤⠤⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠴⠶⠞⠛⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢠⢔⠤⡠⢀⠈⠉⠉⠒⠲⠤⢍⣈⠿⠶⠚⠓⠒⠋⢉⣁⣀⠠⠤⠤⠴⠒⠒⠒⠒⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢱⠅⡨⢊⢕⢍⡮⣪⢔⢵⠦⡠⡇⡤⠖⠒⠚⠉⠉⢁⠤⠤⠐⠒⢲⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢐⠡⡪⢐⠕⡡⡪⡖⢕⡥⡪⢂⡇⠁⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⢸⠀⠀⠀⠀⠀⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⡨⢊⢔⡝⡪⢊⢔⠕⡪⢘⠵⡇⢀⠀⠀⠀⠀⠀⢸⣀⣀⣀⠠⠼⠀⢀⣤⣀⠀⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠨⡪⢒⠕⡪⡢⠕⡡⣪⢔⢕⡉⡇⢸⠀⠀⣀⠀⠀⠀⢀⣀⡀⠤⠄⢒⣚⣛⡣⠇⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣪⢒⢅⡪⢊⢔⡡⡪⢂⢕⡽⡦⡇⢸⠀⠨⡗⠻⠿⠝⠓⠋⠉⠉⣉⡈⠏⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢐⠕⡩⢒⢕⡥⡪⢂⠕⠥⣪⢆⡇⢸⠀⠀⠛⠒⠐⠂⠈⠈⠈⠁⠀⠀⠀⢀⡀⢀⡬⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡡⡪⢂⠕⡡⡪⢒⠕⡡⢊⢔⠅⡇⢸⣀⣀⣀⠠⠤⠄⠒⠒⣒⣂⣉⣉⣉⠭⠭⠤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢊⢔⠕⡩⣪⢔⡥⡪⢊⢕⡵⢋⡇⢈⣩⠤⠤⠒⠒⠊⢉⣉⣉⡠⢤⡄⠀⠀⠀⠀⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢕⡥⡪⢒⢔⡩⡪⢚⠕⡡⢒⠕⡂⢸⠀⠀⠀⠀⠀⠘⢩⠕⠒⠀⢰⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡪⢊⢔⠕⡩⢪⠔⡡⣪⢔⢕⡍⠁⢸⠀⠀⠀⠀⠀⢠⠈⠀⠀⢀⣸⠀⠀⠀⠀⠀⡃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣪⢕⠵⣪⢖⠵⣫⣺⢕⠕⣫⢖⣵⢸⠀⠀⢀⣀⠀⠈⠉⠉⢉⣉⣀⣀⣼⣭⣍⠆⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠕⡡⢊⠔⡡⡪⢋⠔⡡⢊⠕⡡⢺⢸⠀⠀⡿⠳⠾⠯⠭⠭⠬⠟⠛⠚⠏⠀⠁⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣕⣪⢔⠵⡫⢊⠔⡡⢊⠔⡥⢊⠔⡊⢸⠀⠀⠛⠐⠰⠒⠐⠚⠁⠁⠉⠀⠀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠔⡡⢊⠔⡡⢊⠔⡡⢊⠔⡡⢊⢀⠘⠀⠀⢀⣀⣀⠤⠤⠤⠒⠒⠈⣉⣁⣩⣤⣤⣄⢰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣪⢖⡡⣪⢔⡡⣊⢔⡵⢊⠔⡡⢞⠀⠉⣉⣁⣠⣤⡤⢴⣶⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⢤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠕⡫⢊⠔⡩⢚⠵⣩⢔⡥⣊⠔⣹⠀⣿⣟⣝⣿⣟⡷⠠⠤⢤⠄⡿⠿⠟⠛⠫⠭⠭⠶⣶⣶⣿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢊⢔⡡⣊⠔⡡⢊⠕⡣⢚⠵⣫⠴⢀⣿⡿⢿⢿⣿⣿⣶⣶⣷⣿⣴⣶⠾⠒⠒⠂⠈⠀⣁⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡵⣫⢞⡥⢊⠔⡡⢊⠔⡡⢚⠕⣺⣿⡿⣿⣷⣿⣷⣶⣷⣭⣽⣿⣟⣇⠀⠀⠀⠀⠊⡭⠒⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡯⢞⠵⠋⢔⠵⡫⢖⠥⡪⢂⠕⡨⣺⣏⡧⡪⢂⡪⣻⢍⢝⣟⠿⢻⣿⡇⠀⠀⠀⠀⣆⣀⣀⡠⠤⠇⢀⣀⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣒⣂⣒⣒⣒⣿⣩⣊⢔⠥⡪⢔⠕⡪⢚⢕⠨⡪⢚⡷⣯⡊⠊⡪⢐⢕⠩⡠⢖⢕⣗⡇⠀⢀⣠⣀⣀⡀⠠⠤⠄⣔⣛⠯⠼⠃
⢀⣀⠒⡢⠤⠤⠤⠶⠦⠤⠤⢶⣶⣶⠿⠷⢷⣥⣪⢖⠕⡪⣊⢔⠅⡪⢚⢽⡕⣿⣰⢂⢔⠕⡡⣊⢔⠕⠹⣇⡇⠀⢝⣟⢛⠒⡒⡒⠨⠉⠋⠀⢀⣀⡀
⠀⠀⠘⠀⠉⠽⠛⣈⣋⣛⣉⣉⣙⣋⣉⣉⠑⠾⢿⠿⠾⢖⣡⣕⣫⣶⢕⢽⣏⣿⣿⣿⢷⢾⠞⣷⣵⣮⣪⣗⡇⠀⠈⠉⠉⢀⣀⣠⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠩⠐⠒⣒⣚⣛⣉⣙⣒⣀⣈⣘⣛⣉⢭⣽⣿⣷⣾⣗⣓⣛⢭⡭⢭⣿⣛⣉⣿⣷⣖⡒⠻⠭⠍⠩⢗⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠉⠽⠿⠌⢓⣊⣙⠉⠫⠭⠭⠍⠩⣆⣒⣒⠛⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⣊⣉⣍⣍⣭⣭⣙⣒⡰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡛⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢨⣀⣸⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠽⠋⠋⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⡈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠒⠐⠀⡀⢀⠀⠀⠂⢀⠔⠂⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠈⠀⠀⠠⠁⠀⢹⣿⣟⣽⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠰⠁⠀⠀⠔⠁⠄⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢼⢃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠂⠀⠘⡀⠀⢰⣽⠘⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢐⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⢸⣿⢈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡷⠊
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡇⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣟⠃⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⡶⠶⠲⠲⠶⠖⠖⠖⢿⠄⠀⢿⡶⡲⠆⠆⠠⠄⠦⠄⠄⠦⠦⠤⠤⢤⡴⢿⠭⠀⡇⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⠈⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡀⣸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢁⢸⡋⠀⣗⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣆⠀⡇⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠄⠀⠠⢸⡵⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢸⠖⠀⡇⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠨⠈⡧⠀⡇⡔⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠟⠙⠛⠛⠃⠀⠀⠸⡧⠀⣿⢸⣿⡄⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢍⡀⠀⠀⡁⠠⣿⢸⣿⠅⣿⣿⣿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡯⠀⣿⣃⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠈⠉⠁⠀⠒⠂⠠⠤⠤⠄⣀⡀⣀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⣻⡦⢯⠿⠙⣀
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡁⠐⠒⠠⠆⠀⠤⠀⡤⡤⠄⠀⠄⠈⠅⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣉⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣟⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣶⡾⠟⠋⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡍⣿⠋⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⢻⣉⣉⣹⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⣴⣮⡉⠉⣩⣶⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠿⠿⠛⠛⠻⠿⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣷⣿⡶⣶⡶⢶⣖⢲⣆⣰⣶⣶⣖⡒⠒⢃⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠊⠉⠁⠈⠓⠒⠛⠛⠋⢻⣿⢺⣷⢻⣿⠿⢛⡟⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⣼⠞⠋⠁⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠿⣿⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣤⣤⣤⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠸⣧⣤⣼⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⢷⣆⣀⣰⣿⡆⠀⠀⠀⠀⣿⠀⠀⠀⣀⣴⡿⠃⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⠟⠉⠉⠙⠿⠁⠀⠀⠀⠀⣿⣠⣴⠿⠛⠉⠉⣧⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⢿⣶⡶⠶⠶⠶⠶⣶⠶⠿⣿⠁⠀⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⣀⡛⠒⠒⢛⣀⠀⠀⠀⠀⡿⣿⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⣿⣿⣤⣤⣿⣿⠀⠀⠀⠀⡇⣿⠀⠀⠀⣀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠷⠶⠶⠶⠾⠷⠶⠶⠾⠷⠶⠶⠶⠴⢏⣿⠀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠠⠤⠤⠒⠶⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠤⠤⠤⠤⠤⠤⣄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠀⠀⠀⠀⠒⠒⠒⠛⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡇
⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⡔⠐⠒⠀⠒⠒⠒⠒⡄⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⡱⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠰⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠚⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡠⠤⠄⣀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⣉⡠⠒⠕⠂⠱⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠄⠀⠀⠀⠘⠀⠀⠤⠤⠤⢀
⠀⠀⠀⠆⢀⡤⠄⠁⠁⠠⠉⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠻⠿⠛⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰
⠀⠈⠇⠤⠀⠀⠀⢰⠄⠀⠀⢱⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⢸⠀⠀⠉⠀⠀⡡⠅⠈⢸
⡂⢀⡜⠀⠈⡰⠦⠜⠀⠈⡃⠤⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢨⢰⡀⢐⢸⠉⣃⣀⠈⠉⡑⡀⢀⠀⡀
⣇⣎⠃⠀⡰⠀⠀⠀⠀⡴⣁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠘⠀⠉⠁⠀⠭⠤⠤⠬⠭⠠⠈⠀⠀⠃
⠨⡹⠀⠊⠈⠁⠀⠀⠀⢤⠀⢀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣀⠠⠇
⠀⠘⢦⡘⠀⢠⣀⡰⠌⣀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠀⠀⠉
⠀⢀⠐⠀⠉⠙⠃⠙⠉⠁⠂⠁
⠀⠀⠀⠀⠒⠀⠀⠀⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⣂⠠⠤⢀⣄⣂⠀⠀⠉⠐⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⢌⠐⠒⠀⠀⠀⠀⠀⠀⠀⠉⠐⢤⠀⢌⢆⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠈⠈⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⡰⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠹⡄⠀⠀⠀⠀⡃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠐⠀⠀⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠦⠀⠈⠇⠀⠐⡀⠀⠀⠀⡃⠀⠀⠀⠀⡠⢈⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠐⠈⠲⡄⠀⠀⠑⠀⠂⠈⠀⠀⠀⠀⢀⣀⡀⠀⠄⠀⡀⡜⠀⠂⠐⠀⠀⠀⠀⡇⢀⢀⠀⡀⠀⢀⢀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠄⠀⠀⢀⠌⠀⠀⠀⡄⠈⠀⠄⠀⠀⠀⠀⠨⡭⠄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡇⠐⠒⠀⠂⠠⠐⠒⠀⠂⠂
⠀⠀⠀⠀⠀⠀⢠⢎⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠃⠐⢄⠀⠀⢀⠄⠀⠀⠀⠠⠤⠠⢄⠀⠀⢃⠄⠀⠀⠀⠀⠀⡃⠀⠀⠀⠁⠁⠠⠀⠀⠁⠅
⠀⠀⠀⠀⢀⡴⠃⠀⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⠄⠈⠄⠀⠀⠀⠒⠐⠂⠀⠁⠀⠀⠀⠀⠑⠃⠂
⠀⠀⠀⣠⠊⠑⠄⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠌⠁⠁
⠀⢀⠾⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⠤⠤⠤⠄⠐⠊⠁
⠀⠈⠢⡀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⡅⢢⡔⠀⠀⠀⠀⠀⠀⠀⠆⠈⠢⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⢀⠔⣡⠞⢉⠔⠉⡀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠄⢀⠊⠀⠀⡐
⠀⠀⠀⢀⠔⢁⠔⢁⡼⠋⡠⠊⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⡠⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣠⢁⠀⢀⠐
⠀⢀⡐⢡⣔⣁⣴⠋⠀⠊⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠨⠘⢑⠞⠓⠅⠀⠀⡀
⢀⠘⠁⠀⠀⠀⠁⠜⡇⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⡐⠁⠀⠠⢈⠀⠀⠢⠐
⠉⠀⠀⠀⠀⠀⠃⢀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⠉⠀⠀⠀⠀⠀⢕⠄⠀⢁
⠐⠒⠒⠒⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⣇⡶⣼⣠⣴⢴⠢⣶⡖⢲⠂⠀⢴⢲⡦⠶⢰


⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡤⠀⠀⢤⡄⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⡀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄
⠀⠀⠀⠀⠀⠮⡀⣠⠇⠀⠀⠈⣰⣆⠀⠀⠀⠀⠀⡐⢂⡀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠂⡠⠁⠈⠢⡄⠀⠀⢀⠞⢻⠏
⠀⠀⠀⢠⡄⠀⠠⠋⠀⠀⠀⣌⠆⠡⣱⠀⠀⡠⠎⣰⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠆⠰⠀⠀⠚⡩⠀⠀⢏⡠⠂
⠀⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠑⠒⠒⠒⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠂⠐⠊⠀⠀⠈

⠀⠀⠀⢰⠀⠀⠐⠒⡆⠀⠀⠀⢀⠦⡄⠀⢠⠦⠖⠴⠤⠆⠀⠀⡔⠒⠒⠒⠂⠀⠀⠀⣴⣦⠀⠀⢀⠀⠤⢀⡒⠀⢀
⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡒⠇⢴⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⢱⢈⠈⠠⡁⠀⠀⠰⠩⠭⠆⠀⠀⠑⠠⠤⠄⠒⠉⡄
⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⣮⠃⠈⠀⠀⢘⠀⠀⠉⣠⠇⠀⠀⡇⠈⠁⡀⠁⠀⠀⡱⠀⠀⢊⡀⠀⠸⡀⠀⠀⢐⠒⠗
⠀⠀⠀⠈⠓⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢠⡶⠒⢦⡀⠀⢠⠢⠴⢦⡒⡄⠀⠀⠀⢀⠄⠂⠀⠀⠰⠀⡆⠀⠀⠀⠀⢀⡴⣄⠀⠀⠀⠀⡀⠰⣒⢤
⠀⠀⠀⠀⢸⣄⠀⣠⠇⠀⢸⠨⠉⠩⠁⡇⠀⠀⢠⠃⡠⠃⠀⡠⠊⠀⠉⢢⠀⠀⢰⠋⠉⢉⠒⠀⠀⢸⠰⡸⢀⢎⠆
⠀⠀⠀⢀⢎⡟⠉⠁⠀⠀⢸⢀⠀⠈⢀⠃⠀⠊⠉⠒⠁⠀⠀⢏⠀⠑⡈⢩⠃⠀⢸⠐⠈⠐⢸⠀⠀⣨⠳⠤⠢⣟⡀
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠂⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠒⠚⠁⠀⠀⠈⠑⠒⠒⠁⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⢃⡠⠄⠄⢦⣄⠩⢂⠀⠀⠀⠀⠀⢀⣐⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⣦⠈⠆⠀⠦⢀⣶⣿⣿⣿⠿⠿⠿⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠱⠀⠀⠸⠏⢻⠀⠠⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢢⡀⠀⠀⠀⠀⠀⢄⠀⠀⢁⡀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠁⠈⠢⠤⠀⠀⠠⠠⠦⢤⣤⣀⠤⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠠⢄⠀⣻⣣⠄⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠤⠀⢀⠀⠀⠈⢀⡿⠁⠄⠀⠀⡄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⣰⠉⠃⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⢀⠀⠄⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠉⠁⠉⠰⠎⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠰⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢠⠀⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠇⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⡶⠇⢷⣶⣶⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠙⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠋⠉⠉⠉⢉⣉⣉⣉⣉⡉⠉⠉⠉⠉⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⣏⣉⣉⣹⡇⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢀⣈⣉⣉⣉⣉⣁⡀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠘⠋⠉⠉⠉⠉⠙⠃⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣴⠟⠛⠛⠷⣦⣤⣤⣤⣤⣤⣤⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣿⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣿⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡉⠉⠉⢉⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠟⠛⠛⠛⠛⠻⣷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡇⠈⢙⣟⠁⢠⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⣸⣇⠀⣼⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠘⠻⠟⠋

⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⣤⣴⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⠉⠀⠀⠈⠁⠀⠁⠀⠀⠐⡤⠀⢸⣿⣿⣫⡭⣿⣿⣷⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⣧⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⢸⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⠄⠹⣄⠀⢠⡄⠀⠀⠢⡖⠀⠀⢸⣿⣿⢸⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣦⠀⣀⣠⣤⡤⠄⠀⠉⠀⠀⠀⠀⠀⠠⠐⡧⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡁⠀⢀⠀⠀⠀⠀⠀⠠⠖⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⣀⢀⣀⠤⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠑⢑⡤⠁⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⢀⠤⠤⠐⠂⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠑⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⡈⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠠⠄⠀⠄
⠀⠀⠀⠐⠀⠀⠀⢀⠀⠁⠈⠀⣄⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡁
⠀⠐⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠿⠟⢀⣀⣠
⠀⠀⡇⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠹⣄⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢌⠉⠀⣶⠁⠀⠄⠄
⠀⠈⠂⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠀⠀⠀⡀⠀⡂⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡌⡄⠀⢹⣧⣔⡢⠂⠑⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⣃⣉⣀⣵⣘⡦⠀⡀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢻⣷⣿⠀⠈⣿⣷⡀⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣻⠁⠀⠩⠃⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⡆⠌⠙⢠⠦⠤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⡇⠂⠀⠀⠀⠙⠀⠀⠰⢤⡴⠋⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠀⠀⠀⠀⠀⠀⣄⠀⠀⡰⡀⠀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⢐⠀⠐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⢠⡆⠀⠀⠀⠀⠀⠀⠇⠈⢀⠀⠙⠷⠄⠂⡅
⠀⠀⠀⠀⠂⠀⠘⠋⠘⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⠴⠛⠉
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢠⠟⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠒⠋⠀⠐⠦
⠀⠀⠄⡴⠁⠀⢀⣘⠀⠀⠀⡰⠃⠀⡇⠀⠀⠀⠀⠀⠀⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⣠⣴⡾⠛⠉⢁⣀⣀⣤⡤⣐⡴⡀
⠀⠀⠀⠓⠂⠀⠰⠰⠀⢴⡾⠥⠤⠴⠇⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⠀⡀⠀⠁⠀⠁⠀⠀⠀⠚⠒⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠄⠒⠒⢒⠒⣒⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠀⠀⠀⠀⠀⠀⠀⢀⡠⢁⠀⠀⠰⠂⠁⢸⣿⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣷⣗⠀⠀⠀⠀⠀⢀⢘⠚⡉⠉⠁⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⠀⠀⠀⠀⠀⡘⠠⡤⠫⠚⠛⠘⣃⣠⡇⠀⣿⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡯⣿⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢿⡇⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⢸⣇⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡂⢠⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⡀⠸⠀⠀⠀⠈⣿⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠇
⠀⠀⠀⠀⢠⡥⣤⣤⣤⣿⣿⠀⠀⠀⠀⠈⠶⠶⠻⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢐⢘⠂⠀⠀⢀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⢵⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⣍⢹⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠘⠘⠀⠀⠃⡞⠀⠀⠀⠀⠀⢀⠀⠛⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⢸⡍⢹⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⢀⠠⠜
⠀⠀⠀⠀⢸⡓⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣇⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⣴⠀⠀⣀⣀⣘⣉⣀⣘⣻
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⡆⠀⠀⠀⡇⡇⠀⠀⠀⠈⠀⠂⠈⡄⢀⡀⠀⠀⠀⠀⠻⠉⠓⠒⠺⠿⠒⠛⠻⠃
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢲⡀⠔⡜⠟⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢸⡆⢸⠀⠀⠀⠀⢸⢰⡆⠀⠀⠀⡤⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⡇⠇⠈⢆⠀⠣⠀⠀⠀⠀⠀⠞⣰⠃⡨⢄
⠀⠀⠀⠀⢸⡆⢸⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠏⠀⡇⢾
⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠑⠀⠐⠀⠀⠀⠀⠤⠎⠀⠀⠀⡈
⠀⠀⠀⠀⢸⠾⢻⠀⠀⠀⠀⢸⠤⠄⠀⠀⠀⠘⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠿⠿⢽⣿⣿⣿⣿⣿⣿⣿⡿⢿⣛⡛⠻⢿⣏⣉⣉⣽⣿⣍⣉⠉⠉⠉⢻⡟⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⡉⣿⠷⣾⣿⣭⡷⠟⣋⣩⣿⣤⡄⠀⢸⡇⠀⠀⠀⠈⠉⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠠⠤⠤⠤⠤⠤⠬⠭⠍⠉⣿⣿⣿⠅⣿⠒⠆⠀⢸⣟⣛⣉⠁⢸⡿⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠛⢳⡶⢾⣧⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣾⠛⣛⣛⠛⠓⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠛⠻⠾⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠉⢹⡟⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠶⠶⠶⠶⠶⠦⢤⣤⣤⣿⣶⣶⣶⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠈⠀⠉⠉⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠸⠷⠯⢽⣿⣷⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣼⣟⣛⣛⣛⣷⣶⣶⣤⣤⣄⣠⡤⢦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡼⠿⠭⠭⢭⣭⣭⣭⣉⣉⠙⣷⣿⣭⠷⠞⢋⠹⣇⠀⢠⣾⣻⠿⣿⣦⡀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠤⠤⢤⣀⣀⠈⣿⢠⣷⣿⣹⣿⣿⣿⣷⣿⣶⡿⢨⠛⠻⢿⡿⢁⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣴⣷⣤⣶⣶⣶⣤⣤⡤⠀⣻⢸⣿⢿⡿⠷⠿⠿⠿⠿⣿⢣⠇⢐⣒⣾⠿⢿⡗⢿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⢉⣙⣛⣛⠛⠛⠳⠶⠶⠶⠶⠶⣿⣮⣤⣬⣿⣹⣷⠾⠷⢿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⣿⡩⢍⣭⣛⡛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⣾⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠻⠶⠶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣸⡇⣸⡄⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠳⠶⠶⢤⣤⣤⣤⣀⣀⣀⢀⣿⣠⡴⠞⠛⠙⠁⣻⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠓⠶⠶⠶⠶⢾⠿⠋


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣋⣙⣏⣈⣉⣩⣍⣩⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠆⠄⡦⠆⠆⠰⠠⠰⠱⠰⠴⠴⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠘⡯⠍⠙⠟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠭⠍⠻⠟⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠽⠉⠭⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣄⣤⡀
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣿⣿⣧⣄⣀⡀
⠀⠀⠀⠴⠿⠿⠿⠷⠤⠤⠤⠾⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠦

⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢰⣿⡏⠈⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠟⠻⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠛⠿⠻
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠠⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢰⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⣿⣆
⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⣄⣤⢰⢰⣶⡆⡦⣤⣀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠐⠃⠛⠛⠚⠘⠛⠃⠓⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣶⣶⣶⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢸⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿
⠀⠀⠀⢰⣶⣦⣬⣍⣉⣛⠛⠛⠃⣿⣿⡟⠛⠛⣛⣉⣉⣥⣤⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⠉⠙⠛⠶⠶⠦⣤⣤⣼⠀⠀⢸⣤⣤⡤⠶⠶⠞⠛⠋⠉⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣛⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠀⠈⠉⠀⠂⠐⠒⠦⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⣈⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⠈⠉⠉⠉⠛⠛⠒⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠉⠉⠛⠒⠶⠦⢤⣤⣀⣀⠀⠀⠀⠉⠑⢦⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⡤⣀⡀⠀⠈⠉⠁⠲⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠀⠀⠀⠀⡇⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡯⠈⢒⣲⣦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠈⣉⣓⡒⠶⠤⢤⣄⡀⠂⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠢⠬⠝⠛⠂⠀⠀⠀⠁⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏
⠀⠀⠀⠀⡒⠀⠰⠦⠤⣄⣀⡐⠀⠠⢤⣀⡀⠑⠲⢤⣴⣢⣤⡀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡴⠒⣈⡀⠸⢤⣀⣀⠐⠒⠈⠉⠁⠒⠀⠈⠙⠓⠊⠀⠀⠉⠙⡆⠘⡇⠀⡇
⠀⠀⠀⠉⠙⠒⠶⠯⢭⣟⣒⡲⠤⢤⣀⡀⠀⠀⠀⣀⠤⢴⣒⠡⠀⡇⢰⡇
⡆⠀⠀⠀⠀⠀⢤⣄⡲⠦⠈⠉⠙⠒⠂⠀⠉⠀⠀⠒⠉⠉⠀⠀⠀⡇⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡆⠀⠶⠟⠉⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⡇⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁
⢇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⡇⠸⠁⠀⠀⢀⣠⠴⠚⠉
⠀⠈⠉⠒⠦⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⣀⡤⠒⠛⠀⠀⠀⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠤⠤⠀⠀⠘⠀⠤⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣥⠴⠶⠶⠦⣬⡛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣷⣞⣉⣀⣀⣀⣀⣀⣀⣙⣶⣾⣿⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⢸⠀⡇⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣧⣼⣤⣤⣤⣤⣇⣠⣀⣧⣤⣤⣤⣼⣤⣧⣤⣤⣴⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠁⠀⢹⣻⣟⡏⠁⠈⠹⣏⣉⣉⡿⠁⠀⠙⣟⣟⣻⠉⠀⢹⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡟⢻⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠛⣿⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠸⡇⢸⡇⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢿⠀⣿⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠘⢦⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⠞⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠈⣹⡇
⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠛⠂⠠⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣰⡆⠀⠉⠋⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⣹⣿⣿⣿⣿⣿⡀⠀⠀⠀⢰⣿⠁⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇⠀⠀⠀⢸⣷⣶⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⣾⡟⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⢻⣷
⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⡄⠀⠀⣤⡄⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⢿⡇⠀⠈⣿⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢹⡇⠀⠀⢻⡇⠀⠀⠀⠀⢸⣷⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣷⠀⠀⠀⠀⠘⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⠀⠀⠀⠀⠀⣿⠀⠀⢸⣧
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡟⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⠀⣠⣤⣶⣶⡇⠀⠀⢿⡀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠿⠀⠀⠸⠇⠀⠀⠀⠐⠛⠛⠛⠋⠹⠿⠾⠿⠿⠻⠿⠃⠀⠀⠘⠃⠀⠀⠘⠃⠀⠀⠀⠀⠸⠇⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠲⣯⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠄⣤⠀⠀⠀⠀⠀⠀⠀⠈⢦⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣩⡶⣾⠉⠀⢀⡠⠒⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢿⣯⣿⣠⢳⠁⠀⠀⠀⠀⡤⠀⠐⠈⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢲⣾⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⣿⣥⠀⠉⢹⣛⣁⡠⠀⢠⣷⣲⣚⣛⣩⡐⠂⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣫⣶⠀⠀⣾⡈⠁⠀⠐⠋⠉⠉⠉⢉⠉⢁⡀⢀⡀⡠⠟⠀⠀⠀⠈⢰⡆⠀⠀⠀⠀⠀⠈⠋⠙⡛⠉⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠋⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⡿⠷⠠⠤⠤⠤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⣿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠉⠁⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⠀⢒⠆⠀⠐⠂⠀⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠠⠤⠔⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡁⠀⠀⠀⠀⢀⠗⠦⡤⣀⣀
⠫⢉⡁⠒⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠊⠉⡇⠀⢫⠉⣷⣶⠉⡹⠀⠀⡇⠈⡇⠉⠑⠢⢄
⠀⠀⠈⠉⢐⡤⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⢠⠃⠀⠈⡄⢹⡏⢀⠇⠀⠀⡇⠀⢱⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢣⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢸⠀⠀⠀⠸⢄⣀⡀⢱⣼⣷⡸⠀⣀⣀⠅⠀⠸⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠑⣄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠁⠈⣿⣿⠇⠀⢣⠀⠀⠀⠀⢃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⢦⡠⠊⠁⠀⠀⠀⠀⢀⡤⢺⠀⠀⠀⠀⠀⠈⢢⡀⢸⡿⠀⡠⠊⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠳⡀⠀⠀⢀⡤⠚⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠦⠧⠎⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣱⡠⠖⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠞⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢃
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡴⠒⠋⠙⠛⠒⠲⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣔⡴⠶⣶⠴⢦⣠⣴⣲⠶⠤⡤⣿⣦⡤⠤⠤⢤⣤⣤⣴⣧⣤⣤⡴⣆⣀⣀⢶⣶⡤⠤⢤⣤⣤⡤⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⣠⠞⠛⠁⠠⠾⠋⠉⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠉⠁⢠⡾⠋⠡⡄⠀⠉⠁⠐⠢⡀⡆⢸⡆
⠀⠀⠀⠀⠀⢸⠑⢊⠔⠋⠁⠀⣇⠀⣀⣧⠀⠀⠀⠠⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⠀⢸⡇⠀⠀⠙⢦⡀⣈⡦⠀⣧⡇⢸⡇
⠀⠀⠀⠀⠀⢸⣀⠏⠀⠀⠀⠀⢧⠤⠤⣤⠀⠀⠁⠀⠓⢄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢤⡇⠀⠀⠀⠀⠙⢦⡀⠀⡿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠠⠀⠀⠀⣿⠀⠀⠀⠑⢀⠈⠐⠉⠀⠀⠂⠂⠀⠀⠀⠀⡃⠀⠀⡼⠇⠀⠀⠀⠀⢀⢠⣙⠐⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠀⠑⠔⠉⠀⠀⠀⠀⠀⠐⣠⢦⠀⠀⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠠⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠰⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⢀⡀⡀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠀⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⠀⢫⡀⠀⢀⠀⠀⠅⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⡈⠃⠀⠀⠀⢰⠀⠈⠁⠂⠀⠤⠤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⡳⠠⠀⠀⠤⣤⠆⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢡⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⡆⡀⠈⠑⠪⡥⢤⡌⠀⠀⠃⢸⠃
⠀⠀⠀⠀⠀⢠⠋⢢⠀⠀⠀⣡⡠⠀⠀⢸⠀⠀⠐⠀⠀⠀⠉⡀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠁⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠼⡄
⠀⠀⠀⠀⠀⢸⡀⢀⠑⣄⠉⠀⢸⠀⠀⢸⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⢀⠀⠂⡘⡄⡀⠑⡇⠀⠀⡆⠀⡘⠀⠀⠀⠀⠂⠈⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠸⣅⣨⣤⣈⣷⠤⢬⣀⡀⣸⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠁⠁⠀⠀⡇⠀⠀⡗⠈⠀⠀⢀⠀⠁⡀⠀⢟⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠧⠤⠀⠷⠦⠠⠴⣁⣀⣐⣁⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⡤⢤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⢧⠀⠀⠀⠀⢰⠁⢸⠁⠀⠀⣰⠋⠑⠢⣄
⠀⠀⠀⠀⡔⢋⣉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⡆⠀⠀⠀⡏⠀⠸⠀⠀⡴⠃⠀⠀⠀⡜
⠀⠀⠀⠀⡄⠸⡀⠱⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠲⢄⡀⠀⡆⠀⠸⡀⠀⣸⠀⠀⡇⠀⡼⠁⠀⠀⢀⠞⠀⣠⠔⢾
⠀⠀⠀⠀⠘⢄⠑⢄⢱⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢶⢫⡱⡀⠹⣤⠶⡇⠀⣷⠦⣤⢁⡞⠀⠀⠀⢀⠎⢀⡜⡝⡳⡼
⠀⠀⠀⠀⠀⠈⠳⣄⠙⢧⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠳⡱⡘⣄⢹⡀⢻⢀⡏⢠⣇⠞⠀⠀⠀⢠⠋⢀⢎⠜⡜⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢴⠉⠙⡦⣤⠤⢤⡀⢀⣀⣀⣀⣀⣳⣱⣜⣌⣆⣷⣀⣿⣁⣼⣏⣀⣀⣀⣠⠇⠠⠯⠮⠾⠼⢤⣐⣒⠢⠭⢽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠊⠁⠉⠉⠉⠉⠉⠉⠉⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠉⠉⠉⠉⠉⠉⠁⢀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠧⠝⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠓⠚⠀⠀⠀⠀⠀⢴⣰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡠⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⣔⢲⠀⠀⠀⠀⠓⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣠⣄⠀⠑⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠓⠚⠀⠀⠀⠀⠀⠀⠺⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⠂⠀⠀⠀⢯⡹⠀⠀⠀⠀⠰⠽⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠤⠄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠤⠤⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⣠⡮⡃⠿⠃⠀⣴⡄⠀⠀⠀⠀⠀⠉⠑⠒⠦⢤⣀⡀⠀⠀⠀⠀⢐⠾⠟⠷⠾⠿⢦⡀⢠⠦⣤⠄⡀⢀⡀⠐⠀⠀⠀⠐⠶⠀⠁
⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⣘⡛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⣠⠤⢤⡀⢠⡐⠟⣀⡀⠃⠀⠢⠹⠶⠋⡈⢿⣀⠗
⠀⠀⠀⡠⠋⠀⠙⢠⣄⠀⠐⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡕⢀⠀⠈⡀⠀⢀⠀⠰⠂⡀⣀⡈⠈⠁⣀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⡐⠀⠀⠈⣿⠃⠀⠀⠿⠃⠀⠓⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠙⡆⢸⠀⢸⡇⠸⣍⡉⠀⠂⡏⠁⠙⡆⡮⠤⠽⠸⣍⡉⠸⣅⡉
⠀⢤⣈⠠⠃⠀⠀⠀⡀⣠⣤⠄⠀⠀⠖⠀⠀⠀⠈⡉⠛⠓⠲⠤⡄⠀⠀⠀⣇⣀⣀⠆⢸⣀⣠⣇⢠⣀⡸⢸⠀⡇⠀⢰⡏⢧⣀⡰⠠⣄⡸⠠⣄⡸
⠀⠀⠈⠙⠈⢀⠉⠀⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡠⢠⣤⣤⣤⣤⡤⠤⢤
⠀⠈⠒⠃⠀⠈⣀⢹⣿⣯⠉⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⡇⠺⣿⠿⣿⡟⠡⠶
⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣯⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⣴⣿⣽⣿⠙⣉⣤⣠⠀⠀⣧⣤⣤⣤⣤⣼⣦⣤⣤
⠀⠀⠀⠠⠀⠀⠂⢀⠀⣄⠀⠀⠀⠀⠀⣀⠀⠄⠀⠠⠄⠀⠀⢄⠀⠀⣿⣯⣿⣽⣯⣿⡇⠻⣿⣿⠟⣼⣻⡞⣞⣿⠀⠙⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⢀⠓⠒⢶⠎⠀⢿⢌⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢱⠀⠿⠿⠿⠿⠿⠿⠛⠃⠀⣀⣀⠘⣾⣭⡵⠏
⠀⠀⡐⠁⠀⠀⠈⠇⠀⢾⠀⠁⠀⠀⢀⣠⠤⠒⠒⠲⡄⠀⢸⠀⡌⠀⠀⠀⠀⠒⠂⠀⠀⢀⣴⣿⡿⠀⠀⠀⠀⢀
⠀⠠⡁⠀⠀⠀⠀⢹⠀⠘⡀⠤⠶⠶⢶⠶⠶⢦⣤⣠⣿⣁⣿⡖⢃⠀⠀⠀⠀⢀⠁⠀⠀⣼⣿⡾⠆⠀⠀⠀⠀⠘⢛⠛⠃⠀⠀⠀⠀⠠
⠀⠀⠈⠂⠄⠤⠐⠊⠀⠀⡇⠀⠀⡞⠃⠀⢄⡀⠀⠉⠉⠀⠈⣧⣰⢄⠀⠀⠀⠀⠀⠀⠘⠟⠛⠀⠀⠜⠦⠀⠀⠠⠍⠁⠀⠀⠀⠀⡀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⢠⣧⠀⠀⠺⠂⠀⠀⠰⠆⠀⢸⢧⡈⠂⣄⡀⠈⠁⠀⠀⠈⠀⠘⠀⠂⠀⠀⠅⠀⠀⠀⣀⣴⣤⣤⣤⣤⠀⠠⣴⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠰⠙⠂⠀⠀⠀⠀⡆⠀⠀⠀⣼⠀⡝⡹⠁⠀⠑⠤⡀⠀⠀⠀⠢⠀⠀⠀⢠⠀⢀⠀⡄⠈⠉⠉⠉⢉⡛⠂⠰⠶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡏⠀⠑⡷⠋⠁⠀⠀⠰⠀⠀⠀⠄⠀⠀⣼⠈⠀⢸⢰⡇⣿⠀⠀⠀⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠓⢦⣰⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠃⠇⠀⠀⠘⢻⣿⡉
⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⢹⠣⡀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣀⣧⣄⣀⣸⠀⠀⢀⣀⣀⡀⠀⠀⠐⠶⠶⠶⠶⠲⡀⠀⠀⣿⣇
⠀⠀⠀⣀⢠⢀⠀⠀⠀⠀⠀⠀⣸⡆⠉⠲⢤⡤⢤⡴⠋⢀⢦⡀⠀⡠⢠⣧⣤⣄⣀⣀⡀⣔⡵⠒⠚⠪⠣⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢻⣿⣾⣷
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡠⠊⠸⣇⠀⠀⠀⠀⠀⡓⠄⢘⢷⠄⢝⡅⣼⣿⣿⣿⣿⣿⠋⢸⠁⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠈⠁⠀⠀⢘⠛⠙⣿⡆
⠀⠀⠀⠨⠀⠀⢀⣀⣤⡞⠁⠀⠀⠈⠓⠦⣤⣠⠔⠁⢰⣌⡑⠀⠚⠀⣿⣿⣿⣿⣿⣿⠀⠳⢄⡀⢀⣨⡜⠀⠀⠀⠀⢀⣀⠤⠴⢦⡀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⠃⠀⢀⣠⠴⠺⣿⣿⣿⡟⢦⣌⣿⣿⣶⡄⢸⣿⣿⣿⣿⣿⡇⠀⠀⢀⠀⠙⡄⢳⡀⠀⠀⠘⠧⠔⠒⠈⠀⠈⠈⠀⠀⠈⣿⣧
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣶⣿⣏⠀⠀⠀⠘⣿⣿⠀⢸⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⠃⠀⠀⠒⠀⠀⢡⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⠿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢣⠹⡧⠀⠂⠀⠀⠀⠀⠀⠀⢀⣠
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠈⡀⠠⠔⠒⠈⠁⠈
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⠁⠀⠀⠀⠂⢄⡀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠖⠚⠛⠛⠛⠙⠋⠋⠙⠛⠛⠓⠒⠒⠒⠒⠲⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠸⠆⠀⠐⠒⠲⠒⠖⠒⠒⠐⠒⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠄⡤⠠⢀⠄⠀⣤⣠⣤⣤⠄⠤⠀⠤⡤⠤⠤⠤⣤⣤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣤⣀⣀⠀⠠⢢⠀⠀⠀⡄⠀⠒⠒⠀⡠⠒⠀⠄⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠰⡄⠀⢈⠂⠰⡨⠤⠒⡄⡄⠀⠀⠀⡆⢀⡀⠀⡠⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠟⠀⠉⠉⠁⠀⠀⠈⠁⠉⠀⠈⠉⠁⠀⠀⠀⠈⠁⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠳⠤⠤⠶⠒⠖⠠⠲⠒⢶⠒⡲⠶⠶⠶⠲⠶⠦⠦⠤⠠⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣤⢤⡄⠀⢠⠞⠉⠉⠉⠙⢳⠀⠀⠀⣠⡄⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⣀⠈⠉⠀⠀⠈⠳⠒⠶⠒⠚⠋⠀⠀⠀⠀⠀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠚⠒⠓⠚⠛⠛⠛⠛⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⠿⠟⠛⠋⠁⠀⠀⠈⠙⠛⠿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣶⣤⣀
⠀⠀⠀⠀⣀⣤⣴⣶⣾⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣦⣄⡀
⣤⣶⣿⣿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣿⣷⡀
⣿⣿⠛⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠟⢻⣿⠁
⣿⣿⠀⠀⠀⠈⠙⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡾⠿⠛⠉⠁⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣤⣀⡀⠀⠀⣀⣠⣤⣶⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣷⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⣿⠀⣀⣤⣤⡀
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠸⢿⣦⣄⢀⣀⣠⣴⣶⡿⠿⠟⠋⣀⣠⣤⣾⡿⠿⣿⣿⠟⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠛⢉⣁⣤⣴⣾⠿⠟⠛⣉⣁⣤⣴⣿⣿⠿⢇⣀⣠⣤⣴
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⡿⠿⢛⣉⣩⣤⣶⣾⡿⠿⠛⣋⣭⣵⣶⣶⡿⠟⢻⣿⡿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢁⣴⡿⠿⠛⣋⣉⣵⣶⣾⡿⠿⠟⠛⠉⠁⠀⢀⣴⡿⠋
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣰⣿⠟⠁⢠⣾⡟⢁⣾⡿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣤⣀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣀⣠⣤⣾⡟⠁⠀⣰⣿⠋⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⡿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡛⠉⠉⠀⠀⣾⡿⠁⣰⣿⠏⠀⠀⠀⣀⣠⣤⣶⣶⡿⠿⠛⠛⠉⠁⠀⠀⣼⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠙⠛⠿⢿⣷⣶⣤⣀⣀⣴⣿⣧⣴⣶⡿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠉⠙⠻⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣆⠀⠀⠀⠀⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⣠⣤⣶⡿⠟⠃⠀⠀⠀⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⢿⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠃
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠿⠛⠋⠉
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣤⣄⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣠⣤⣶⣾⠿⠿⣿⡏⠁
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠉⠙⠻⠿⢿⣶⣦⣼⣿⣤⣴⣶⡿⠿⣿⣏⡁⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⢀⣤⡉⠛⠛⠛⢉⣀⣠⣴⣾⡿⠟⠂⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⢿⣦⣴⣾⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠿⠛⠋⠉⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⣠⣴⣶⡾⠿⠟⠋⠉⠀⠀⠀⠀⠀⠰⣶⡀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠛⠋⠁⣀⣀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡿⠿⠃⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠀⠀⠀⠉⠻⣷⣤⣶⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠟⢻⣿⡇
⣿⣿⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠟⠋⠉⠁⢸⣿⠀⢸⣿⡇
⣿⡇⢸⣿⠉⠙⠛⠻⠿⠿⣷⣶⣶⣤⣄⣀⡀⠀⠀⠀⢸⣿⠀⠀⣀⣀⣤⣶⡾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⡃⢸⣿⡇
⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣿⣶⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇
⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋
⣿⣧⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠇
⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⡁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡤⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⣀⣤⡴⠟⠋⠀⠰⠁⠀⠀⠀⣀⡴⠚⠉⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠀⠀⠀⠠⠄⣀⣘⣦⡶⠟⠛⢡⡶⠂⠀⡆⢀⠀⣠⡴⠛⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⣴⠾⠛⢰⣿⡀⢠⠇⡎⠚⠁⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⡀⢀⡀⠀⠤⠒⠂⠀⠀⣀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠉⠁⠐⠒⠒⠒⠒⠀⢠⠿⠞⠋⢁⣴⣾⣞⡇⢿⢸⢠⣱⠀⠀⠀⠀⢀⣀⢀⠀⠀⠀⡷⠛⠃⠉⠀⠀⠀⠀⠀⢀⣀⣀⠟
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣇⢀⣴⣾⢿⠋⢘⠈⠁⠛⠛⠘⢻⡶⠛⠛⠉⠁⠀⢠⠀⠀⢀⣇⣤⣤⡤⠤⠶⠶⠟⠛⠋⠁⠒⠒⠒⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠚⡿⠋⢁⣼⣿⣿⠀⠀⠀⠀⠀⣾⣧⣄⣠⣤⣤⠶⠃⠀⠀⣼⣿⣿⢂⣀⣀⣀⣀⣀⣘⣂⣀⣀⣐⣐⣺⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠄⠠⠄⠀⠀⣀⠀⠀⢧⣴⠿⠋⣠⠴⠀⠀⠀⠀⠀⣿⠀⠉⠉⠁⠀⠀⠀⣠⠾⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⣬⡆⠀⠀⠀⢀⣿⠤⠤⠀⠠⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡇⢀⣼⠿⠋⠁⠀⠀⠀⠾⠋
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣧⠈⠙⢿⣿⣿⣿⠋⢠⡀
⠴⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⣸⣏⠀⠀⠈⣿⣦⣤⣀
⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⣿⣏⠀⠀⠀⢹⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣾⣿⣦⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣿⣿⣿⣿⣷⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡿⠟⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡉⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⢀⣼⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣿⣿⣿⣿⡟⠙⠿⣿⣿⣿⣿⣿⣶⣦⡀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⠈⠉⠛⠻⠿⠿⠟⠁⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣴⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⡉⢹⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠘⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠈⢿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠿⠉⠉⠉⠉⠙⢿⣿⡿⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢻⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣿⣿⡿⠿⠿⠛⠛⠛⠛⠻⠿⠿⣿⣿⢷⣶
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣶⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣾
⠀⠀⠀⠀⠀⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣉⣉⣉⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣡⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠛⠛⠛⠛⠃⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠚⠇⠄⠛⢒⣶⣌⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⠂⠀⠀⠀⠀⠀⠉⠉⠛⢆⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢎⢔⡞⠁⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣷⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢿⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡰⢛⡍⣼⠇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣷⣴⣿⣿⠏⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⡀⠀⠀⢀⣐⣶⣿⡿⠟⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣠⡾⠒⠀⠀⠀⠀⠀⠈⠀⣲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢽⠻⠻⠻⠟⠛⢙⣋⡴⠊
⠀⠀⠰⣤⣤⣤⣤⣦⣤⣰⣼⢀⡾⣋⠤⠤⠄⠀⠀⠀⠀⠀⣠⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠤⠓⠂
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠹⣿⣿⣿⣷⠆⢀⣐⣠⣤⣤⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠘⣷⣤⣄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠛⠛⠉⠉⠉⠉⠁

⣿⠿⠿⠿⡿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠿⠆⡾⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢴⡤⢾⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣤⣑⣂⣾⣇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠶⣶⡙⡞⢦⣈⣿⡟⣿⠖⠊⠉⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃⠀⠀⢀⢹⣧⣀⡀⠙⠢⣿⣷⣄⠀⠀⠀⠈⠦⠒⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣓⢎⠂⠢⣼⣿⣿⠘⠧⣀⡀⠈⠳⣻⣦⠀⠀⢸⠄⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠈⠫⠀⣄⢹⠛⣯⣵⡴⠠⠭⠝⠲⡿⠧⢷⣤⣤⣝⡶⠶⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣦⣴⣶⣶⣼⣾⣤⡿⠋⣠⣴⣴⣶⡶⢶⣶⣬⣤⣁⢫⡉⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠐⡒⢤⡀⣾⢶⠋⠁⠐⠋⠁⠈⠁⢸⡗⠛⡉⠍⠁⢀⣠⣶⣿⣅⡸⣿⡟⣧⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠀⣼⡌⢏⠛⢌⡾⣿⣷⡆⠀⠀⠀⢸⡶⢻⣿⣷⣷⣾⣿⣿⣿⣿⣿⣏⠻⣿⡷⠶⠋
⠀⠀⠀⠀⠀⠀⠀⣏⠻⣌⣦⡎⠆⢿⣦⣷⠀⠀⠀⣸⢿⣿⣿⣿⠙⠻⠟⠋⠘⡽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢰⣦⣬⣭⣍⣈⣉⣉⣉⣁⣾⡕⢻⣿⣿⣿⠁⢰⠀⠀⠀⠁⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠈⠉⠉⠙⠛⠛⠛⠋⠈⠣⡘⠋⣿⣷⡦⠖⠀⠀⠀⠘⠶⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠋⢹⡄⠀⣀⡤⠖⠋⠉⣉⠛⠷⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⣀⣀⠀⢷⣉⠀⣀⠴⠉⠊⠍⠀⠁⠀⠀⢀⣠⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣰⠞⠉⠀⢘⣧⠤⠾⠋⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠇⠀⢹⡠⠴⠚⠉⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⣸⠤⠴⠒⠋⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⣉⡤⠒⠺⢉⡍⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⣶⣏⣁⣀⡀⠼⠂⠡⠐⠊⠁⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠉⣟⠙⠛⠓⠶⢤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠲⢿⣄⣀⠀⠀⣼⠀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡆⠈⠙⠿⣿⣿⠿⠋⠁⢰⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⣴⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣦⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣹⣏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢰⣿⣿⡆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣼⣿⣿⣧⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠁⠿⠛⠉⠀⠀⠀⠀⠀⣨⡀⠈⠀⠀⠀⡀⠀⠰⣔⠒⠼⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠩⣝⠢⣄⡀⡲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠠⠚⡃⠀⠀⠀⠀⠦⠀⠰⣘⢫⡟⢨⡓⣾⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⣑⣱⠏⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣖⣤⠆⠀⢠
⣀⣤⣆⣼⣴⣄⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⡁⠀⠀⢸⡀⠀⣀
⠉⠙⠊⠉⠀⠈⠉⠀⠈⠀⠀⠀⣠⠔⠛⠁⠀⠐⠃⠀⠀⠀⠀⠀⠘⠉⣉⠉⠉⠉⢉⡛⠀⠠⠈⠀⠄⠀⠀⠀⠏⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠤⠀⠀⠤⠀⢢⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⢀⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀
⡈⠨⣮⡏⣹⠋⠀⠀⠀⠁⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠄⢜⠲⢏⠅⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀
⣷⢮⣸⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⡇⠀⠀⡎⠉⡖⠉⠉⣧⠀⠀⠈⠈⠉⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣦⣥⣽⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⡷⣟⣾⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢄⠀⠁⠀⠀⠠⡦⠀⠀⡀
⠔⠀⠀⠀⠀⠀⢼⡤⢀⡀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⢂⠓⡀⢠⠂⢀⠀⢠⢺⠿⠟⠛
⡾⠇⠀⠀⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣇⣀⡾⢠⠏⠀⠸⢶⣶⠶⠶
⠐⠛⠄⠀⠀⠀⠀⡇⠀⡇⠀⠀⢠⣤⡄⠀⠀⠐⠋⠀⠀⠀⢰⣿⠛⣧⠀⢢⠀⠀⠈⣧⠘⣆⣷⠃⠀⠀⢸⠷⠁⠚⠛
⠀⠀⠀⠀⠀⠀⠀⣇⣠⠁⠀⠀⠈⡈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠈⠀⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂
⢐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤
⠘⠀⠀⠀⠀⡀⢀⣀⡀⠀⢠⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠸⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢚⠥⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠛⠛⠲⠶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⣀⠀⢁⠂⡁⠢⡈⢢⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠠⢤⣅⣂⣍⣢⣑⠌⠤⠅⠀⡇⢀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠌⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⡊⠀⢀⠀⢀⠀⠀⠀⠀⠀⣀⡀⠀⢓⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⡏⠉⠻⠿⠍⡶⢲⠚⠿⠿⠏⠙⣿⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠀⠣⠀⣀⣀⡐⠁⠀⠣⢀⣀⣀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠐⠐⠒⠢⠦⠀⠔⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡏⢢⡀⠀⠀⠈⠉⠉⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠃⠀⢳⡀⠑⢤⣀⡀⠀⢀⣀⡴⢋⣿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠋⡽⠈⡇⠀⠀⠱⣄⠀⠈⠉⠋⠉⠀⣠⠎⣼⠀⠙⢶⠦⢄⣀⡀
⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⡼⠁⠀⢣⠀⠀⠀⠈⠳⣄⠀⠀⣠⠞⠁⠀⡇⠀⠀⠈⢇⠀⠀⠉⠙⠒⠦⢤⡀
⠀⠀⠀⠀⡼⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠀⠘⡆⠀⢀⣠⢴⡾⠟⠿⢷⠦⣀⠀⡇⠀⠀⠠⡞⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢰⠇⠀⠈⡀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠙⣞⠁⠀⠀⢱⡀⠀⡎⠀⠈⢳⡇⠀⠀⠀⠙⡦⠀⠀⠀⠀⠀⠀⡄⢹
⠀⠀⠀⡼⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⠀⠀⠀⢠⠗⡞⡅⠀⠀⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠇⢸⡆
⠀⠀⢀⡇⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⡇⠀⢀⡏⠘⠃⢹⠀⠀⣼⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢳⠀⣸⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣧
⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠋⠀⠀⠀⠀⠳⠀⠃⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠛⠛⠛⠀⡄⡏⠉⠉⢹⢠⠀⠛⠛⠛⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣤⣤⣤⠀⠃⣇⣀⣀⣸⠘⠀⣤⣤⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠉⠉⠀⠀⠀⠐⠒⠀⠀⠀⠉⠉⠉⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠾⠃

⠀⠀⠀⠀⠄⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⣠⡀⣀⠀⡀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣄⣀⡑⠑⠁⠀⡇⠀⠀⠀⢁⠀⠀⠁⠈⠉⠐⠀⡡⠁⣠⡀⡈⠉⢉⡁⠀⠀⢀⡐⠆⡇
⠀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠐⠀⠙⡄⠀⢃⠀⠀⠀⢈⠀⠐⠀⠀⠠⠆⠀⡇⠈⡄⠀⢁⠀⠸⠤⠤⠤⠤⠃⠀⠃
⠀⡇⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⡇⠐⠌⠀⠀⠙⠀⢰⠂⠀⠀⠉⠇⠀⠇⠄⠉⠉⡁⠴⠦⡀⢸⠀⠉⢱⢸
⠀⠘⠢⠄⠇⢰⡖⠀⠀⠀⠀⠒⠄⠂⠀⠀⠀⠤⠤⠅⠸⡀⣉⠁⠀⠔⠀⡀⣀⠀⠀⠳⠤⠤⠃⠸⠀⠀⠌⠘
⠀⣀⡀⢠⠀⠀⢀⠄⠀⠀⠀⠀⠈⢄⡆⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⠓⠒⠒⠒⠒⠒⠒⠂⠀⠒⠁
⠀⠱⠖⢃⠠⠔⢊⠓⠀⠀⡀⠀⠐⠉⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⡀⠤⠤⠒⢂⠰
⠀⠀⣮⠁⠀⠀⠀⠡⠀⠘⠂⡔⠀⠀⠆⠀⡠⠀⠀⠀⠀⠘⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠰⠠⠤⣀⠠⡀⠀⠈⡀
⠀⠘⢿⣧⠀⠀⠀⠀⠡⠀⠸⠁⠀⢀⠀⠈⠈⠀⠀⠐⠀⠀⠀⠐⢄⠆⠀⠀⢀⠌⠀⠀⡖⠀⠀⠀⠀⠘⡄⠀⢡
⠀⠀⠈⣿⣧⠀⠀⣀⠼⠀⠀⡆⠀⢪⠀⠀⢼⠐⠆⡀⠀⡀⠀⠀⠀⠁⠀⠈⠀⠀⠄⢠⠘⡄⠀⠀⠁⠀⠸⡤⠚
⠀⢠⠄⠈⠻⠛⠉⠀⢄⠀⢰⢰⠀⠀⠀⠀⢐⠀⠀⠀⢡⠖⠉⠁⠀⠀⠀⠀⠀⠁⠀⡈⡄⢱⠀⠀⠀⠀⠀⣱
⠀⠠⠼⡀⠀⠦⡀⠀⠈⠀⠀⡘⡀⠀⠀⠀⠈⣄⠀⢀⡌⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠉⠀⠚⢧⠠⠐⠂⠁⡄
⠀⠈⠀⠀⠀⢆⡀⠁⢉⣛⠄⠁⠃⠀⠐⠀⠀⠘⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⠀⢤⡀⠀⠀⠠⠤⠤⠄
⠀⠀⠠⠀⠀⠈⠓⠃⠍⠀⠀⠈⢁⡤⠖⣀⣀⣠⡘⡎⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⡀⠘⡀⠀⠀⠀⠀⠀⡄
⠀⠐⢒⣽⠧⣲⣋⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠛⠉⠉⠉⠉⡀⠀⠄⡲⠃⢀⣉⣉⣀⣀⣀⡀
⠀⠀⠀⢀⠤⠤⢤⠀⠀⠀⠀⠒⠀⠀⠤⠀⠀⢀⡀⠀⠀⠀⠀⢀⣀⠀⠩⠄⠀⠂⢉⡀⠀⢫⠀⠀⠀⠀⠀⡇
⠀⢠⠲⡉⠉⠉⠉⠉⠉⠳⠀⠀⢀⠀⠄⠒⢆⠠⠄⡄⠀⡀⡰⠀⢀⣀⣠⣠⣤⡄⠈⠁⡠⠚⢀⠀⠀⠀⠸⠳⡄
⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⡇⡄⡄⠀⠀⠀⠘⡄⠀⡇⠀⠈⠁⠸⠏⠁⠀⠀⠂⠘⡄⠰⡥⠀⠀⠀⠤⠦⠀⠀⢼
⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⡀⣇⢱⠀⡔⠒⠂⠉⣉⣉⡩⢇⠀⠄⠈⡀⠀⠀⠀⠀⠰⠀⢹⠐⡒⠒⠒⢒⣲⢲⠁
⠀⠀⠀⠉⠒⠀⠀⠈⠉⢉⠅⠾⠀⠄⢧⠀⣀⣀⠀⠀⠀⠜⠀⠂⠀⠳⠤⠀⡀⠂⣤⠁⠸⡀⠛⠀⠃⠋⠛⠈
⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠂⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠙⡆⠀⠀⠀⠀⢀⡴⠖⠒⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢒⡒⠶⣇⠀⠀⠀⢠⢿⣀⣀⣄⣀⣸⠀⠀⠀⠀⠀⢘⠀⢸⠋⠉⠉⠉⠉⡏⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⢒⣒⣋⣿⠀⠀⠀⢸⡌⣿⠀⠀⠈⡏⠀⠀⠀⠀⠀⢘⠀⢸⠀⠀⠀⠀⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠈⡇⠀⠀⠀⣿⡇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣿⡇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡿⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⠀⢸⣿⡇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡗⠀⢸⠀⠀⠀⠀⠀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠸⡞⡇⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠱⠅⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⠀⠀⠀⣷⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠁⠈⠹⡆⠀⢻⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⣹⠀⠀⠀⠀⠀⡧⠀⢸⡀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⣴⠃⠈⠲⢄⡀⢀⡇⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢉⡞⠁⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼⠋⠀⠀⢹⡄⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⢀⡾⢀⠀⠀⠀⠀⣠⠏⠀⠀⠀⢸⡆⠀⠀⢸⠀⠀⠀⠀⠀⡟⠀⠀⠀⠘⡇⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⣾⠀⠀⠙⠢⢄⡴⠋⠀⠀⠀⠀⢸⣇⡤⠤⢼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⢸⡇⠀⠀⠀⠀⡇⠀⢸⡆
⠀⠀⠀⠹⣦⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣾⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⡗⠀⠀⡇⢀⣀⣀⣀⣇⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⡇⠀⠀⠀⠀⠀⢸⣀⣀⣀⣠⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⣇⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢉⠡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠰⢖⣒⡢⠤⣠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⣥⣶⠯⢃⣁⣄⠳⡄⠀⠀⠀⠀⠀⠀⢀⠄⠐⠒⠒⠒⠈⠀⠀⠉⠀⠐⠒⠒⠂⢄
⠀⠀⠀⣀⠞⠙⠲⠖⠢⡀⠀⠒⠨⡳⡀⠄⠠⠀⠀⠀⠀⠀⠀⠱⠽⠿⠟⠒⠉⠉⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⣠⢀⡀⣠⠠⠀⡀⠠⢀⣀⡄⣀⠉⠘
⠀⠀⠄⠘⣁⠦⡀⠀⠀⢹⠂⢀⡠⠘⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⡄⠃⠀⠃⣇⢀⡀⠁⢀⢠⡿⠀⠘⠀⠀⡄
⠀⢨⠂⣠⠃⠀⢀⠛⠶⢎⠀⠘⡁⠦⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⡇⢧⣀⡇⣇⣠⠧⠄⢀⣀⠿⣄⣸⠰⠀⡇
⠀⠸⣄⣍⠐⢀⣆⠀⠀⠀⢀⠞⡤⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢀⠃
⠀⠀⠣⡻⠉⠁⠈⠀⠀⠀⣀⠦⡀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢀⣀⠀⠤⠘⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⡹⢤⡉⠂⢴⣈⠒⢡⣀⠴⢊⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⣠⠁⠀⠡⠍⠉⠩⠍⠐⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠀⣢⠤⠤⠄⣦⣸⠀⡐⠁⠉⢢⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠤⠄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣌⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠉⠲⡔⠢⢱⡀⠀⠀⠀⠀⠀⠁⡖⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢼⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡧⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠈⠂⠀⠋⠀⠀⢠⠄⠀⠀⠀⠀⠀⠔⠄⠀⠛⠀⠐⡁⠀⠀⠀⠀⠀⠉⠡⠈⠀⠁⠉⠩⢸⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠐⠊⠂⡀⠀⠀⠊⠀⠀⠀⠀⠀⡀⠀⠀⡀⣀⠄⠀⠀⢀⢤⠀⠀⠀⢡⠀⠀⠀⠀⠀⡀⢰⢐⠀⠀⠂⢐⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠓⢄⠀⠀⢀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠒⠂⠐⠉⠀⠐⠰⢠⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠁⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢄⠂⠀⢀⡠⠄⠀⠀⠀⠀⠀⠤⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠄⠀⢠⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠁⠀⡠⠊⠀⠀⠀⠀⠀⠀⠜⠤⢰⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⡅⠤⢀⠂⠀⠀⠀⠆⠓⠊⠈⠒⠀⠑⠚⠈⠒⠂⠆
⠀⠀⠀⠀⠀⡰⠋⠢⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡠⠋⠈
⠀⠀⠀⡠⢎⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠄⠀⠀⠀⠀⠤⠕⠊⠁
⠀⠀⠀⠳⢄⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣏⠲⣈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠄⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⠼⢻⡽⠋⡉⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈⠂⠀⢠⠊⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠔⢁⡴⢙⡵⠋⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⢂⢔⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⣱⡁⠈⠀⠔
⠀⠀⠀⠀⠔⢁⡴⡙⡴⠊⢀⠀⠁⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠊⠐⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡑⠞⢳⠷⢖⣁⣀⡀
⠀⠀⢀⠞⠉⠉⠉⠉⠁⠶⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠠⠀⠐⡆⡰⠀⠐⠀⠔⠀⠀⠍⠢⡀
⠀⠰⠃⠀⠀⠀⠀⠐⡆⢀⣁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⡀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⣀⢼⠍⠀⠀⠀⠈⠺⡄⡀⢀⠀⠅
⠀⠀⠤⠤⠤⠤⠤⠤⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⠤⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠒⠁⠀⠀⠀⠀⠀⠀⠉⠂⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⣷⡀⠀⠀⠀⠰⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠸⢟⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣠⡄⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠀⡴⣦⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠼⢄⣠⡆⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⡧⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡇⠀⠀⠀⠀⠏⠀⢿⣿⠃⠀⠀⣶⡄⠃⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠏⠛⠋⠀⠀⠀⠀⠀⠀⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡴⠟⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠀⠀⠀⠀⡰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡄⠀⠀⠀⠈⠿⠾⠋⠀⠀⠀⣤⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣶⠄⣀⣀⣀⣶⠤⠔⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⠈⠉⠉⠀⠀⠀⠀⣀⠀⠐⠶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠙⠛⣶⣶⡞⡀⠀⠀⢠⡞⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣤⠾⣄⢨⡟⠋⠳⣴⠆⠀⢰⡏⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⠇⠀⢩⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠀⢸⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⢸⠇⠀⣾⠀⠀⠀⠀⠇⢠⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⡄⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡸⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢹⠆⠀⠙⢻⣇
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⢏⡀⠀⠀⣿⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠟⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⣴⠏⠀⢀⠇⠀⠀⠀⠀⢣⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠋⠀⠀⠀⢠⣤⡄⠙⢿⣦⡀
⠀⠀⠀⠀⢀⣾⠏⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠙⢿⣦⣀
⠀⠀⢀⣴⠟⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠈⠀⠀⠀⠀⠈⣦⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢿⣆⠀⠀⠙⠻⣷⣦⣀
⣀⣴⠟⠁⠀⣤⣴⣞⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢆⠀⠀⠀⠀⣰⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠙⣷⣄⠀⣀⠀⠈⠉⢱
⠟⣁⡀⠀⠀⠈⠉⠛⠛⠿⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠓⢤⣄⡾⠋⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢻⣿⣿⠟⠀⠀⠈
⠀⠙⠳⣦⡀⠀⠀⠀⢀⠰⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠠⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣠⡶⠿⠋⠀⠀⠀⣀⣀
⠀⠀⠀⠈⠻⣦⡀⢀⡞⠀⢱⡄⠀⠀⠀⠀⢀⡏⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⣀⣤⡶⠟⠉⠁
⠀⠀⠀⠀⠀⠈⢻⣾⠀⠀⠸⠏⠀⠀⠀⠀⣾⡇⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⠤⣤⡟⠉⢿⡄⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠈⠿⠋⠁
⠀⠀⠀⠀⠀⠤⠀⠙⠇⠀⠀⠀⣴⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠈⠲⠶⣄⣠⣤⡀⠀⠰⠋⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠑⠄⠀⠀⢶⣄⠹⣛⠿⠆
⠀⠀⠀⠀⢠⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⢶⡆⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣛⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⢼⣇⡀⠀⠀⠀⠀⠀⠚⠋⠀⣀⣀⣠⣤⣴⣶⣾⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡻⠛⠉⠛⠋⠛⠉⠁⣀⠀⠀⢈⣧⠀⠙⠻⠶⠶⠶⠶⢶⣶⡿⠟⠛⠛⠋⠉⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⣠⣤⡴⠞⠈⠻⠗⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢤⣤⣀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⣠⣄⣁⡠⠤⠶⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣄⡀
⣤⣤⣤⣬⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⠿⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡂⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣫⣴⠾⠛⠛⠻⣿⣿⣤⠤⠴⠶⠤⢃⠈⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠟⠁⠀⠀⠀⠀⠈⣏⢹⠦⣀⢀⠀⠠⠀⢙⣢⠀⠰⠊⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠋⠀⠀⠀⠀⠀⠀⠠⢈⣁⡠⢤⠳⠒⠃⠉⠻⣿⠠⠰⠰⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠤⠈⠁⠀⠀⠀⠀⣀⠁⠀⢐⣀⣠⣱⡶⢻⠛⢹⠸⠃⠀⠀⠀⠀⢀⠴⣿⠀⠀⢀⠴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⢀⠀⠄⢐⣢⣄⠀⢱⣶⢿⡛⠋⠉⣈⠡⠀⠀⠘⠀⠀⠀⣤⠀⠀⠈⢦⣻⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⢀⠔⢈⣥⣶⣿⣿⡿⡿⠐⠸⣏⠐⠁⠄⢈⣴⡶⠈⠀⠀⠀⠀⠀⠂⠑⠄⠀⠀⢹⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⠀⠀⠀⠎⢠⣿⣿⡿⣯⣿⣧⣿⠠⠀⡏⢀⠀⣰⣾⡿⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠸⢸⣿⣿⣟⡿⠋⢿⣷⠶⠚⡇⠀⠀⠟⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠸⠀⣿⡿⡿⠇⠀⠈⢿⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⣇⣟⡠⠒⠦⢼⠆⠰⠃⠀⠀⣟⣃⡒⠤⢀⠀⠀⠀⠉⡅⠀⠀⠀⡀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⣿⣿⡜⢱⡁⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠑⠃⠀⠀⠓⠀⠢⢄⠘⠆⢸⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⣿⢻⡆⠮⡁⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⢸⠀⠀⠀⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⢻⣨⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢈⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡁⢸⡇⡀⡀⠀⠀⠠⠀⠀⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣴⣿⣿⣟⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⢸⣿⢁⡇⠀⠀⣰⡆⠀⢺⠉⠢⡄⠠⠐⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠟⠏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⡀⠀⠀⢸⠈⣿⣿⣿⡶⠀⡿⡇⠀⢸⡄⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣶⣤⣀⠸⠅⣿⣻⣿⣇⡆⡇⣿⠀⢸⣷⣀⣠⣴⣶⣿⣿⣯⡛⠲⢭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠛⠿⢿⣦⣿⣿⣿⣩⣛⣧⣽⣤⣾⣿⣿⢿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠦⢶⠝⠻⠿⣿⣿⣿⣿⣿⣿⡿⢿⡻⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⢁⠀⠀⠀⠉⠋⠛⠙⠢⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⡤⠒⣀⠐⠂⠤⢤⡤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠈⣀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⢰⣦⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠳⠦⠀⠀⠀⠀⢄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⢰⠀⠘⣄⠀⢀⡱⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠠⣀⠀⠉⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠁⠴⠂⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⢀⣘⣀⣀
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢉⠓⢒⡶⡒⢺⠉⠀⠀⠀⠑⠤⡀
⠀⡀⠀⠀⠀⢠⡀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠤⠂⢁⠎⢀⠎⢸⠈⣶⠉⡆⠈⠠⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠐⣩⠀⠈⠀⠀⡏⠑⠒⠒⠶⠒⠉⠀⠀⢀⡜⢐⠉⠀⠸⢀⡟⠀⡆⡰⠁⠀⠀⠀⡀⠀⠀⠀⠉⠒⢤⡀
⠀⠂⠀⢠⠆⣤⣀⠰⡇⠀⠀⠀⠀⠀⠀⣀⡴⢻⠅⠀⢂⠀⠀⢠⠁⢀⡰⠃⠀⠀⠀⠀⠍⠒⢄⡀⠀⠀⠀⠈⠁⠂⠄⣀
⠀⠀⠂⠠⠔⠀⠀⠀⠉⠉⠀⠀⠀⠒⠈⠁⠀⣌⠀⠀⠈⡄⠇⡌⠀⣸⠁⠀⠀⠀⠀⠀⢰⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠉⡶⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⣄⠃⣧⠏⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠙⠤⢿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠂⠄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠘⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⡀⢀⣰⠀⢸⠀⢀⣀⣀⠀⠀⠀⡀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠍⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⠂⠠⢤⣤⣴⠦⠤⠤⠤⠐⣒⡶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠒⠂⠉⠱⠮⠤⠤⠔⠒⠚⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⣿⠀⠀⠘⠀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⡾⠛⠉⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡏⠁⠀⣠⣤⣭⣭⣭⣭⣭⣭⣭⣥⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡿⣿⣻⡿⠉⠉⣿⣿⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡇⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣿⣇⣀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⢍⠁⠀⠈⠉⠀⠁⠀⠀⠘⡤⠀⢸⣿⣿⣫⡭⣿⣿⣿⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⢷⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⣿⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⢄⠹⣄⠀⢠⡄⠀⠀⠣⠖⠀⠀⢸⣿⣿⢹⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢐⣟⣛⣛⣛⣛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⠤⠤⠤⠤⠤⢌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⠿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣟⣛⣛⣛⣛⣻⠂⠀⠀⢸⣿⣿⡇⡀⠀⠀⠀⠀⢀⣿⣿⠠⠤⠤⠤⠤⠄⣿⣿⣿
⠀⠀⢸⣿⣿⠿⠿⣿⣿⡇⢸⣿⣿⠿⠿⣿⣿⡇⠀⠀⢸⣿⣿⡇⠀⢀⣀⣀⠀⠀⣿⡇⠀⠀⣀⣀⠀⠀⢹⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡇⡀⠀⠀⠀⠀⢀⣿⡇⡀⠀⠀⠀⠀⢀⣽⣿⣿
⠀⠀⢸⣿⣿⢿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⡇⠀⠀⣀⣀⠀⠀⣿⡇⠀⠀⡀⣀⠀⠀⢻⣿⣿
⠀⠀⢸⣿⣿⣾⣿⣿⣿⡆⢸⣿⣿⣷⣶⣿⣿⡇⠀⠀⢸⣿⣿⡇⡀⠀⠉⠉⠀⢀⣿⡇⡀⠀⠉⠉⠀⢀⣺⣿⣿
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⢸⣿⣿⣧⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠄⠀⠀⠠⠴⠿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣟⠙⢦⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⢸⡇⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣄⣾⣿⣿⣿⣿⡿⠿⠿⠷⠖⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣛⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠿⠿⠟⠛⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⣯⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⣲⡿⣿⠻⣶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠇⠀⠉⠀⠀⠹⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠛⠉⠉⠉⠉⠉⠓⠒⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⣠⠴⠚⠛⠉⠉⠉⠛⠓⠲⠶⠦⢤⣤⣤⣤⠴⣎⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠞⠁⠀⢀⡀⠀⠀⠀⠀⣠⢶⣄⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣀⡴⢏⠗⠀⠀⠀⠀⠈⠻⢮⣿⡶⢦⠄⠀⠀⢸⡗⠀⠠⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠛⠁⢛⣁⣀⡀⢢⠀⠀⠀⠀⠀⢈⣁⠀⠀⠀⠀⢸⠃⡀⠚⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠊⠀⢴⣦⡟⠀⡁⠀⠀⠹⠀⢠⣤⡑⡄⠀⠀⠈⢻⡅⢱⠟⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠖⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠃⠀⠀⠀⠀⢷⡋⡀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⣠⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⢀⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⢰⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡠⠄⠂⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠨⠍⠉⢉⣁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⢠⡖⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠒⢶⡶⠶⠲⠾⣏⠉
⠀⠀⠀⠀⠀⠀⢀⣹⠶⢀⣀⡈⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡏⣠⣆⠀⠀⣹⣤⣀
⠀⠀⠀⢀⣴⣞⡋⠀⠀⠀⠀⣹⡜⠳⡄⠀⠀⠀⠀⠀⢀⣾⡃⠀⠐⠛⣿⡿⠢⠞⠁⠀⠈⣹⡆
⠀⠀⠀⠸⠖⠋⠁⡀⠤⠐⠋⢡⠀⠀⠘⠆⡀⠀⠀⢠⠏⢸⡇⠤⠀⣾⠻⣥⠀⠀⠀⠀⡼⠁⢷
⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠉⢧⡀⠀⠙⠳⣴⠃⠀⠘⡿⠀⢨⣧⡿⠁⠀⠠⠀⣰⠃⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⣹⠀⣀⠞⣷⠀⠈⡿⠁⠀⠀⠀⢀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣥⠞⠁⠀⢹⠀⠐⠀⠀⠀⠀⠀⢸⠙⣦⠔⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⠃⠀⠀⠀⠀⣹⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠇⠀⠀⠀⠀⢿⠀⢳⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⠀⢘⠀⢸⡀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣸⠳⣄⣀⣀⣀⣼⠆⡏⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠛⢻⣍⠉⠁⢹⣠⡟⠛⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⡟⣆⠀⠈⡟⠧⣤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣧⠸⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⠀⢷⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢷⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡀⢸⡀⠀⠸⣆⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣅⡀⠈⣇⠀⠀⣧⣴⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠶⠚⠋⠀⣀⡉⣿⠁⠀⣠⠏⠁⠀⣇

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⡀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠚⣿⠃⢀⡴⠾⢉⣋⣉⡻⠲⣄⠀⣀⣄⠰⣿⠆⠀⣼⠁⠀⠲⡌⢳⡄⣴⣶⣾⡓⣶⣤⡼⢧⡼⠃⠀⣠⡴⠒⠚⠳⢬⣍⠳⣦
⠀⠀⠀⠀⠀⣰⠏⣣⠞⠉⠡⠈⠙⢳⡌⠛⢉⢹⡄⢀⣤⠀⢿⠀⠀⠀⢡⢨⡇⠈⠉⠛⠛⢋⡉⢻⡛⣷⢠⡞⠁⠛⠋⢻⠀⠀⣸⢷⡈⢷
⠀⠀⠀⠐⠀⣿⠃⣏⠀⠉⣠⡴⠚⢉⣀⣦⠞⠋⢷⡛⠈⣳⠌⢧⡀⠀⠀⡼⠁⣀⠀⢰⣆⢸⡿⣮⡉⠁⣾⡴⠶⠶⠦⠾⠃⢸⡅⠈⡇⢸⡇
⠀⠀⠀⢾⣷⠘⣏⠹⠶⠋⣁⣰⡶⠚⠉⠰⣶⠆⠸⠟⠳⠧⢀⣀⠙⠳⣿⡇⠀⣿⢷⣿⣩⣿⡷⣟⢧⠀⢹⡷⠛⣩⠷⣴⡄⠘⢓⣻⠇⣸⠁
⠀⠀⠀⠠⠀⣶⢋⡀⣰⠶⢋⣡⣦⢀⣤⠶⠶⣤⡴⠖⠚⢋⠉⠙⣶⢶⡀⢻⠀⣿⢻⡹⡜⣎⢧⢹⡌⢧⠀⠳⣄⣹⡄⠙⠁⣰⡿⢋⡴⠃
⠀⠀⠀⢶⣶⠹⠶⠋⠁⣴⡋⣹⢃⣞⣠⣤⣿⡎⡷⣖⢛⣙⡇⠀⢹⡈⣧⠸⣆⠸⡎⢧⢻⡸⡜⣇⢳⡘⣧⠠⠀⠉⣻⣿⣯⡭⠖⠋⠀⣀
⠀⠀⠀⠀⠀⠠⠀⢀⡞⣡⣟⡽⠃⣯⡛⠉⣀⡼⠃⠉⣉⣡⡤⠀⠈⣧⢹⡄⠘⡇⢹⡜⣇⢷⣳⡘⣆⣷⠾⣆⢺⣉⠟⠓⣆⠸⡯⣽⠆⣭
⠀⠀⠀⠠⣶⡄⠀⡼⠛⠁⣠⠞⢙⡯⠉⢻⡉⠤⠀⠛⠉⣀⣠⠀⠀⠸⡆⢳⡀⢧⡀⢳⣸⡾⣿⠋⢹⣄⡤⣿⠀⠉⠙⣉⡁⠀⡇⣿⠰⠦
⠀⠀⠀⢀⣨⠴⠶⡆⢰⡞⢁⡴⠟⠀⠀⠈⣇⠀⠀⠒⠋⢉⣀⣤⠀⠀⢻⠈⢧⠈⣷⠈⢧⣠⢾⡿⠽⠋⠉⢡⣦⠀⣾⠽⢣⠞⠁⠘⢧⡀
⠀⠀⠀⠘⢶⡖⢾⠁⠀⠙⠋⠀⠀⡿⣆⠀⢹⡀⠒⠀⠘⠉⠁⣀⡄⠀⠈⣧⠘⡆⠘⠂⠈⠙⠉⢀⡴⢚⡦⢈⡵⣤⠀⣰⢿⣋⡽⠟⠉⠳⡄
⠀⠀⠀⠀⢈⡇⠘⣧⣀⠀⠈⠀⢸⣇⠹⡄⠈⣧⠀⢁⠀⠚⠋⣉⣀⣤⠤⢾⣠⡽⢀⣤⡶⢺⠃⠙⠋⠉⠀⡼⠧⣇⠸⠧⠤⠤⠤⠶⠶⠶⠟
⠀⠀⠀⢠⡟⢁⡀⣠⠼⣷⠀⠀⣾⡀⠀⢻⡄⠸⣆⣤⠴⠖⣋⣩⡤⠖⢛⡉⠀⢀⡟⠈⣹⠏⠀⡟⠛⣛⣿⣓⠒⠛⠲⠶⡄⠀⡶⢻⡛⠓⣆
⠀⠀⠀⠘⣷⠋⠉⠛⠚⣻⠀⢠⣏⠀⣤⠀⢻⡄⠀⠙⠖⠛⣉⡀⠦⡿⠿⢃⡴⠋⣹⠋⣡⡴⢠⣇⣰⡿⠒⢾⣇⣤⡀⠀⡇⠀⡇⢸⣥⣄⣿
⠀⠀⠀⠀⠈⠳⠶⠶⠚⠁⠀⣼⡍⢰⡟⣇⠀⢷⡀⢠⡴⣿⣿⣷⠀⢀⡴⢯⣠⠞⠁⢀⣷⠀⢸⣏⣁⣀⣀⠀⠙⠀⠙⢺⡇⠀⡇⢺⡉⠋⣿
⠀⠀⠀⠈⢀⣾⣆⣴⡟⠀⢠⣯⠀⣼⠀⠸⡆⠈⢷⠀⠙⢯⡿⠁⣰⠋⢀⡴⠃⣠⡞⣿⠀⠀⣼⠉⠉⣉⡽⠏⢿⠙⠛⣻⠀⠀⡇⢼⠃⠀⣿
⠀⠀⠀⠀⢸⡇⠋⣾⠀⠀⣾⠅⠀⠧⣤⣤⣿⠀⠘⣧⠀⠄⢀⡞⣡⠖⢋⡴⠛⢯⣿⠁⣀⢀⣿⠖⠋⠁⠀⣠⠞⠀⠀⡿⠀⠀⡇⣸⠇⠀⡏
⠀⠀⠀⠀⠘⢧⠀⢿⠀⠸⣧⣤⣤⣄⣀⣀⣀⡀⠀⢹⡆⠀⠚⠋⠁⣠⠞⣵⠛⢶⡟⢸⣋⠉⠉⠛⠛⠓⠒⠻⠶⠦⠤⣧⣄⠀⣷⡛⠒⢺⡇
⠀⠀⠀⠀⣴⠚⠓⠚⠳⠦⣤⡀⠠⠀⢈⣉⣉⣉⡉⠛⠋⠀⣤⠀⡴⢯⣄⡉⣳⠏⢀⣤⡉⢩⠟⢛⡟⠓⠒⣶⠶⣤⢤⣤⡟⠀⠌⠉⠉⠉
⠀⠀⠀⠀⡟⠐⠒⠛⠓⠶⣤⠙⡶⣋⣭⣭⣭⣍⣉⠓⣦⠀⡟⢦⡙⠲⣬⣙⣏⡷⠈⠋⢠⡟⢠⠟⠀⠒⡀⢿⠀⡇⠀⣤⢶⣶⣶⣒⣲⡶⣦
⠀⠀⠀⠀⡇⠐⠋⠉⠛⠲⠆⢸⢳⣇⣀⣀⣀⣠⠟⢰⠇⠀⠳⣄⡷⢀⡤⡍⣭⣤⣄⢠⡟⢠⠏⠀⣴⣀⡀⢸⠀⡇⠀⣿⣿⡿⢏⣉⣉⡇⣿
⠀⠀⠀⠀⡇⠐⠋⠉⠙⠳⠀⡟⢨⠭⠤⠤⣭⡉⢀⡟⢀⢠⡶⠶⠦⡄⠙⠁⣿⠒⠛⠘⠓⢋⣀⣠⡟⠛⠁⢸⣀⡇⠀⣿⢨⢉⡉⡍⢥⠀⡏
⠀⠀⠀⠘⢧⡵⠶⠶⢤⣄⢸⣇⣀⣠⣄⣀⠀⠀⡾⠀⢈⣸⡏⠉⢛⡇⢠⠶⡏⢀⣴⡆⣴⠋⠉⠙⠛⢳⠀⣬⣭⡁⠀⢻⠰⠀⠂⢲⡟⡇⡇
⠀⠀⠀⠠⠀⠛⠛⠛⠛⠛⠻⠛⠓⠶⢤⣨⠟⠾⠃⠀⢯⡽⠀⢾⣹⠇⠈⠛⠁⣾⢹⠇⠻⣄⠤⢀⣠⠞⣴⣌⢧⣙⡇⢸⡈⠁⠁⠁⠙⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈⠉⠉⠁⠀⠉⠁⠀⠁⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠖⠒⠚⠛⠉⠉⢉⣨⠷⠖⠒⠒⠒⣦⡤⠶⢛⣹⡷⠞⠁
⠀⠰⡖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠴⢛⣩⠴⢾⡉⠁
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣀⣤⠖⠛⣉⣤⣶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣺⠵⣦⣶⣻⣻⡿⠃⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢠⣾⣭⠶⠻⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⢻⡙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⡀⠠⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣂⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⠲⠶⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠤⡠⠉⠉⠐⠀⠒⠠⢌⣐⢂⣦⠤⠶⠒⠒⠒⠊⣉⣉⣀⠤⠤⠤⠔⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠨⠀⠀⠀⢕⢥⢖⠤⣢⠤⡠⡇⡤⠔⠒⠒⠀⢉⡩⠤⠀⠐⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠊⠠⠂⠄⢴⠚⠉⡪⢂⡅⡇⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⢠⡠⠀⠐⢡⠞⡠⢈⢴⡇⡇⠀⠀⠀⠀⠸⣄⠄⠤⠤⠂⠀⣤⣄⡀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠊⡴⠘⠁⠪⡢⣨⠀⡇⠁⠀⢠⣄⣀⣀⣤⣤⣴⣖⠂⢭⠭⠕⠋⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠁⠢⢊⠔⠄⠑⠈⡪⠪⠂⡇⠀⠀⢸⣉⠉⠉⠉⠀⠠⠤⠒⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⠠⠈⠐⠕⠀⡪⠊⠔⡵⡇⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⠤⠄⠀⠐⠒⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡢⠢⢀⠕⠑⡨⠢⠐⢕⠕⡇⠐⠒⣂⣉⠭⠤⠔⠖⠒⠒⡒⠈⠉⠉⠉⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡊⢀⢔⠕⠁⡪⡨⢀⠔⡡⡊⡇⡎⠉⠀⠀⠀⢠⢒⠠⠉⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠑⠕⠀⠈⠈⢂⠕⣈⡪⡊⡅⡇⠀⠀⠀⠀⢸⠘⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢐⠁⠀⠂⠔⣁⠂⠀⠊⠔⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⢀⣠⣯⣍⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠠⠪⢀⢴⠀⡡⠂⢈⠔⠁⠀⠀⠀⢸⠓⠺⠿⠾⠗⠀⠀⠋⠛⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⡈⢀⠄⡠⡀⠈⢀⠄⡠⢂⡄⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠤⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢊⠔⠑⡡⢊⢔⠐⡡⢊⠔⡡⢀⢰⣀⠠⠤⠄⠐⠐⢂⣉⣁⣤⣤⣴⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠔⡡⠊⢀⠔⡡⡪⢖⠴⡪⢂⠁⠀⣤⡤⢶⣶⣾⠿⠟⡛⡿⢽⣳⡾⠿⠛⠟⢛⣛⡉⠩⣇⣤⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠔⠑⡪⢊⠔⡡⣊⢔⠥⣚⠐⣿⣿⣿⣧⣮⣠⣂⣘⡞⣉⣠⢔⡒⠨⠭⠀⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⠔⡡⢊⠔⠁⡪⢚⠕⠡⠊⢴⣼⣿⣴⣽⣿⣽⣿⣻⡿⣿⣿⠁⠀⠀⠀⠔⠒⠊⠉⢹⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠥⡪⢈⠕⠡⡊⢔⠅⠨⡪⢐⢽⣿⡏⠉⢩⠻⡟⠻⡿⡿⣿⣿⠃⠀⠀⠀⣆⠁⠀⣀⣸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠤⣿⢊⣔⢅⠠⠪⠐⢑⠅⡪⠪⡚⢽⡿⣯⠸⠏⠁⠠⠺⡋⢀⢼⣯⠀⢀⣀⣀⣈⣉⠁⠤⢤⣾⡿⠵⠂⡇
⠀⠀⠀⣀⠴⠤⠬⠤⠤⠤⣤⣤⣶⣾⣖⣚⣳⣷⣥⣪⢊⢔⢕⣡⠊⡪⢐⢽⣼⣿⡆⡠⡄⠨⠊⠐⠑⢽⣿⠀⠘⣟⡛⡒⡒⠒⠈⠙⠁⢀⣀⠠⠇
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠹⠖⢒⣒⣒⣒⣓⣓⡒⠶⠽⠵⠷⢦⣭⣪⣶⣻⢻⡧⣿⣿⢿⣞⣛⣷⣷⣵⣽⣿⠂⠀⠀⣁⣀⣤⡤⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠨⠭⠭⠭⠭⠭⠭⠥⠤⢤⣭⣝⣛⣳⡿⣯⢭⣭⣖⣒⣮⠭⠼⠟⠷⠍⠉⠛⣒⣒⠠⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠋⠉⠭⠑⢒⣒⣒⣀⡀⢠⠭⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠤⠤⠴⠒⠒⠒⠒⠒⠲⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢩⠤⠤⠴⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠉⠛⠒⠒⢢⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⡆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣄⠧⠤⠤⠤⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠚⠋⠉⠉⠉⢹⠉⡇
⠀⠀⠀⠀⡏⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹
⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡸⡇
⠀⠀⠀⢀⡿⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⢠⠇⣧
⠀⠀⠀⢸⡄⡇⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⢿⣿⣀⣀⣀⣤⠤⠴⠒⠚⠉⣁⡤⠤⠚⠁⠀⢸
⠀⠀⠀⢸⡇⠙⢤⣀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⡇⠀⠀⣿⣇⣀⡤⠤⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⡇⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⣷⣤⣤⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⠏
⠀⠀⠀⠀⢻⣻⡟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠖⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠙⠳⠿⠷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠰⠒⠒⠒⠊⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⢻⣿⣛⣛⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠙⠛⢿⡿⠿⠿⠛⢫⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⡒⠲⣄⠀⢠⣤⣤⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢿⠀⠉⠁⠹⠀⠒⠒⠀⡄⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⠀⢠⠘⠃⠐⢀⠀⢠⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠈⣥⠐⢀⠔⡡⡛⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡿⡉⠿⣦⣀⡀⢀⣸⣌⢰⣡⣆⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⠿⢟⠿⣿⠁⡆⠐⢌⠉⠉⢱⡏⠓⠋⠙⠁⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⢛⠉⠉⠙⠁⠀⢀⡎⠀⠏⠈⡇⠀⢀⣷⣄⠙⣿⡄⠸⠀⠀⠟⠛⠻⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⠀⠈⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⢷⢠⠋⣿⡟⠂⢸⣿⣄⠀⠀⡈⠧⡀⠀⠀⠈⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢸⠗⠀⢳⠀⠀⠤⠌⠢⢤⣄⠀⠀⢸⠁⣾⣿⣿⡆⣼⣿⣿⠔⠉⡴⠁⠱⡀⠀⡜⠀⣸
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠑⠈⠙⢦⣸⡀⢸⡆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⣿⡖⠀⠚⢇⡀⠀⠘⢦⣠⡾⣿
⠀⠀⠀⠀⠀⢰⣯⠀⠁⢒⣠⠤⢤⣿⣧⠈⢿⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠏⢻⣿⣿⡄⠀⠈⡆⠄⠀⠀⠹⡔⠛⠆
⠀⠀⠀⠀⠀⣾⡇⠣⢄⡀⠉⠙⠎⣻⡏⠀⠀⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳⣄⢤⡀
⠀⠀⠀⢀⠼⣋⣤⣶⠶⢮⣍⢵⣾⣿⡇⠀⢸⡟⠹⡀⠀⠀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠀⠰⡌⢳⢱⡀
⠀⠀⢰⣷⣿⡖⠋⠿⢲⣄⠀⣼⣿⡻⠇⠀⣼⠃⠀⢱⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢹⣿⢻⡙⢦⠀⠀⠀⠀⢣⠀⣸⢣
⠀⠀⢹⠟⡕⠁⠀⣀⡀⠈⠻⢿⣿⡿⢖⢤⣿⠀⠀⠀⢳⡀⠀⠀⠀⢹⡿⠀⠀⠀⠀⢀⣾⣿⠞⢻⣆⡀⠀⠀⠀⠀⢢⢏⡞
⣶⣶⣾⡔⠀⣠⠞⠁⢀⡠⢂⣴⠋⢤⡀⠱⢴⠗⣤⣶⣤⣷⣶⣶⣶⣼⣷⣤⣶⣶⣶⣾⣿⣿⣷⣦⣿⡿⢦⣄⠀⠀⣠⡾⠶⠶⠤⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠻⣿⣧⣤⣄⣧⢰⠋⣿⠐⢦⡄⠀⠄⡑⢿⣿⡿⠿⠟⠛⠛⠛⠋⠉⠉⠉⠉⠉⠙⠢⣤⣀⠉⠙⠛⠛⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠉⠙⣻⣧⠕⠻⢎⠲⡀⠁⢲⠤⠝⠇⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⡀
⠀⠀⠀⠀⠀⠰⣯⡉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⠀⠀⠀⠀⢀⣀⠠⣄⠀⣀⡤⠤⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣱⠦⠴⠚⠒⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀⠀⢀⣀⡤⡴⠶⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⣉⣉⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢰⠀⠁⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠉⠁⣰⣾⡿⢿⣶⣦⠀⠀⠀⣆⢆⠠⠤⠀⠀⡀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠀⠈⡟⠀⠀⠀⠻⣦⡑⠲⠖⢚⣡⡿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⠏⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠙⣟⠀⠈⠓⠒⠛⢤⡀
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢹⡞⡗⢸⠀⠀⠀⠙⠦⣄⣀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⠀⠀⢠⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠸⣷⠀⠀⢀⡾⠁⠀⠀⣠⠎⠀⠀⠀⠀⢸⣿⡏⠀⠀⣿⠳⢤⣄⠀⠈⢿
⠀⠀⠀⢠⡶⠿⠟⠀⠀⣾⡇⠀⠀⣰⠟⡆⠀⠀⠀⠀⠘⣿⠃⠀⢸⡏⠀⠀⠹⣇⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡇⠀⠀⢻⡄⢳⠀⠀⠀⠀⠀⠉⡄⠀⣿⠁⠀⠀⠀⣿⣤⣼
⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠻⣆⠀⠀⠹⣄⢀⣀⣀⣠⣤⣤⣤⣴⣿⣤⣤⣤⣴⠏⢸⡯⠴⠶⠶⠶⠶⠶⠤⡄
⠀⠀⠀⠀⠘⣷⣀⣠⣤⠤⠶⠾⠓⢦⢀⣼⢿⣉⣹⡇⠀⠐⠲⠤⣄⣀⠀⠼⠱⡤⠠⢹⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⣀⣤⡴⠶⠛⠋⠉⠀⠀⠀⠀⢠⠄⠀⠿⢧⡀⠰⣟⡗⠀⠀⠀⠀⠀⠈⠙⠲⢤⡃⠀⠃⠀⠀⠀⠀⣠⠖⠲⠦⢀⡙⢷⣄
⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠹⠔⠾⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠙⠷⠀⢀⡠⠴⠒⠫⠓⠒⠲⡄⠉⢳⠙⢷⣄
⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣝⠲⢤⡀⠀⠀⠀⢀⣠⡴⠚⣏⠁⠀⠀⠀⠈⡳⣄⠀⣀⣨⠴⠂⢻⣤⡼⠳⡄⠙⠓
⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠙⢷⣠⠖⠋⠁⢀⣀⠼⠃⠀⠀⠀⠀⠹⣌⠉⠉⣀⣠⠶⠋⠁
⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣉⣁⣠⣤⣤⣤⣤⣤⣶⣦
⠀⠀⠀⢻⡹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⣀⣀⣀⣤⣤⣤⣤⣶⣶⠶⠿⠿⠛⠛⠛⠋⠉⠉⠉⠉⠀⢸⠁
⠀⠀⠀⢸⡇⠹⡄⠀⠀⢀⣀⣀⣀⣠⣤⣤⣴⣶⡶⠿⠿⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣇⠀⠙⠛⢻⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣦⠀⠀⠀⣿
⠀⠀⠀⠀⠉⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⡶⠾⠿⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⢀⢤⠄⠀⠀⠀⢀
⠀⢈⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠤⠤⠤⠴⠦⠤⠦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣤⣭⣤⣤⣬⣼⠀⠀⠠⠤⠶⠶⠶⠶⠶⠶⠄
⠀⢀⠀⠀⠀⠀⠀⣀⣀⣆⣀⠀⠀⢠⠀⠀⢀⣏⡛⠨⡿⠿⠫⢦⠀⠀⠀⠀⣀⣴⡶⠀⠀⠀⠀⠀⠑
⠀⠘⡀⠜⠛⠛⠛⠋⠛⠋⠉⠒⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁
⠀⠀⠀⠀⠆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀
⠀⠀⢀⠰⠄⠂⡆⡰⠂⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⠈⠀⠀⠀⠀⠐⠩⠂⠁
⠀⠀⠀⢢⠸⣽⣷⠁⠀⠀⠀⡄⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⠀⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⡀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣴⠇⠀⠀⢀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠁⠀⠀⠇⢀⡇⠠⢸⠂
⠀⠀⠀⠀⡨⠐⠀⠀⠀⠀⠀⠀⢨⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠠⠰⠇⠀⠀⠂⠀⣸⣻⣿
⠀⠀⠀⣠⢦⡀⠀⠀⢀⡤⠀⠐⠀⠀⠐⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣤⠄⠀⠀⠂⠀⠺⠹⠚⡇
⠀⠀⣰⢇⡲⣌⠓⢤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢰⠀⠀⠁
⠀⠓⠭⡺⢴⢃⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠀⠀⢠⣀⠀⠤⠀⠠⠄⠐⡄⠀⠀⢸
⠀⠀⠀⠈⠐⠶⠃⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠋⠀⠀⠀⠀⢻⠇⡩⠠⡪⢋⡻⠁⠀⠇⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢠⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠂⢘⠶⡣⠀⠀⠀⠓⠃
⠀⠀⣈⠗⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢢⣾⠅
⠀⠀⠀⠀⠊⠄⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⣠⣴⣄⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⢿⣿⣿⠆⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢨⠾⠛⠉⠀⠀⠀⠀⠿⣤⡿⠁⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠠⠤⠤⠀⠐⠐⠒⠒⠒⠒⠒⠒⠠⢤⡀
⠀⠀⠀⠀⠠⠔⠀⠀⠚⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⠿⠿⠿⠟⠛⠋⠀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⡄
⠀⠀⢹⠀⠀⣀⣀⣤⣤⣤⣤⣴⣦⣤⣤⣴⣶⣶⣶⣾⣿⡿⠿⢿⡿⠛⠟⢛⡝⠉⠉⢹⢋⣿⡇
⠀⠀⣸⡆⠀⣿⣿⣿⣿⣿⠟⠻⠛⠛⠙⠉⢻⣅⣸⡿⢋⣰⢀⡠⡀⠀⡠⡎⠀⠀⠐⠀⠀⢿⣳
⠀⢠⣿⡇⠀⢿⣿⣿⡿⡃⠀⠀⠀⠀⢀⡴⠃⠛⠋⡴⢿⣧⣟⣼⠟⣠⣿⠀⢀⠀⠀⠀⠀⢸⣿
⠀⠸⣿⡇⠀⢸⡿⢋⠜⠁⠀⢀⠀⣠⣪⠀⠀⠀⠀⠀⠀⢸⠟⠁⢀⡽⢫⢦⠀⡀⠀⠀⢠⣨⣿⡆
⠀⠀⣿⣿⠀⢠⠔⢡⢀⡄⠀⠀⠘⠋⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢞⡄⠁⠀⡀⠓⣿⡇
⠀⠀⢙⣿⠀⢸⠖⣱⣯⠀⠀⠂⠀⠘⣯⠪⠆⣠⢃⠔⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⣖⡴⠁⣤⣿⣿
⠀⠀⢸⣿⡇⢸⡾⡈⠃⠀⠀⠀⠀⠀⠈⠐⢻⣷⡋⠀⣠⣂⠀⠀⠀⠀⠀⠀⠀⠉⣾⠇⠀⠀⢸⣿
⠀⠀⠈⣿⡇⢸⠷⢣⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠙⢁⡴⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠰⣺⢸⡆
⠀⠀⠀⣿⡇⢸⠠⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢿⢿⡇
⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⢀⡄⠀⠀⠀⠀⠀⠟⠀⠀⢸⣿⡇
⠀⠀⠀⠘⣿⢸⠀⣣⡎⢘⡶⠐⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠁⠀⠀⠀⠀⢠⠞⠂⠀⠘⣷⣧
⠀⠀⠀⠀⣿⢰⡿⡹⢡⠆⡀⡀⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡀⠀⠀⠀⠞⠀⠀⠀⠀⡿⣸
⠀⠀⠀⠀⢹⣮⡇⠑⠁⠈⠘⠀⠀⠈⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢞⡴⠄⡀⠀⠀⠀⠀⠀⠀⠀⣴⢿
⠀⠀⠀⠀⢸⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⡵⢀⠈⠀⠀⠀⠀⠀⣻⣿⠄
⠀⠀⠀⠀⠀⣿⡥⠂⠀⠀⠀⠀⠀⠀⠀⠠⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠂⢀⠀⠀⠀⠀⢼⣿⡄
⠀⠀⠀⠀⠀⣿⡷⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡔⠀⠀⠀⠀⠀⠀⠀⠀⠰⡡⣠⠀⠀⠀⠠⠀⣸⣿⡇
⠀⠀⠀⠀⠀⣿⢷⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡴⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⢹⢸⢊⡴⠃⠀⠀⠀⠀⠀⠀⠀⡠⠈⠁⡼⡁⢀⠀⠀⠀⢀⣾⠃⢀⡔⡄⠀⣠⡾⣿⣿⣧
⠀⠀⠀⠀⠀⢸⠀⣿⡠⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠰⢛⣴⠀⠀⠈⡡⢢⢾⡿⠀⣼⣟⣽⣿⣿⣿⠄
⠀⠀⠀⠀⠀⢸⡄⢻⡅⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡠⡾⣁⠂⢀⣜⣰⣏⣜⡵⣺⣿⡿⠿⠿⠋⢹⡇
⠀⠀⠀⠀⠀⠘⡇⠘⣿⣶⣀⣊⣀⣠⣿⣠⣦⣴⣶⣿⡿⢷⠟⣻⣟⠛⠛⠛⠋⠉⠁⠉⠁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣧⠀⠈⠙⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⡴⠉⠈⠁⠈⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠈⠉⠁⠁
⣿⣿⡿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣾⡿⠟⠛⠛⠉⠉⠉⠉⠉⠁⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣷
⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠋⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⡇⠀⠀⠀⢰⠛⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠖⠚⢓⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠈⠑⠒⠂⠤⡤⠒⢂⣉⡩⠤⠤⠔⠒⠒⠆⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡁⠀⢀⡀⣀⣀⠦⠘⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡄⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⣀⡠⠤⠤⠒⠒⠋⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠀⣀⣀⣤⣴⣶⣷⢸⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣶⣾⣿⣿⣿⣿⣿⡿⠟⠚⠃⠉⠉⠉⢱⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡆⠀⠀⠉⠛⠿⠿⡇⠀⠀⢄⡠⠴⠀⢸⡄⠀⠀⠀⢘
⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢯⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⢖⡦⢄⡀⢸⠀⠀⠀⣀⡠⠤⠞⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠢⠤⠼⣄⡠⠴⠒⠉⠀⠀⠈⠸⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣤⣤⣴⣾⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⢿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣿⡟⣻⣽⣾⣿⣷⣽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣾⣿⣷⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢉⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⣿⡆⢀⣿⣆⠀⠀⠀⠰⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⣿⡇⣸⣿⣿⣷⡄⠀⠨⠭⠭⠭⠭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡆⠙⠁⣿⣿⣿⣿⣿⣶⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢀⣾⠿⠛⢉⣿⣿⣿⣧⠀⢰⣿⣿⣿⡏⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡍⠀⠀⠀⢸⣿⣿⣿⣿⡀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⠙⣿⣿⣿⣷⣿⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠃⠿⠿⠶




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠁⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣿⡔⣫⡤⠒⠐⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠉⠀⠀⢄⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠜⠁⡀⠃⠀⠀⢘⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⣿⠀⣿⣥⣦⡀⣎⠛⠀⢀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠙⡀⠙⠋⠁⠀⠀⠀⢀⣀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢄⡎⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠋⠀⠀⠐⠀⣀⠆⠀⠀⢰⡄⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡆⠀⢀⠀⢸⠇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠁⣁⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⠀⠀⠀⠀⠀⠀⣀⡴⠋⠘⡄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠉⠁⠀⠀⠀⣘⣄⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠈⠄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⡴⠁⠀⡜⠧⢤⡤⢤⠏⠉⠁⠐
⠀⠀⠀⠀⠒⡄⠀⠁⢀⣀⠤⠤⠤⠀⠀⠐⠒⠂⠈⠉⠀⠀⠀⡰⠀⠀⠒⢰⡤⠀⠠⣄⠐⢀⠀⠀⠀⠡
⣄⠉⠀⢈⡛⠘⠄⠀⢀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢴⠃⠀⠀⡄⠸⠀⡾⠀⠋
⠠⡁⠀⣈⡅⠀⠀⣠⠏⠸⡧⠆⠤⣤⠤⠐⠶⠂⠉⠁⠀⠀⡈⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡆
⠀⠁⠀⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠇⠀⠀⣠⠃⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸⠀⠀⠰⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⡤⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⣀⠄⠚⠒⠒⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⡤⠤⠤⠤⠦⠤⠴⠤⠤⣔⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⡡⣿⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡌⠀⣀⣀⡀⣘⢒⣉⠓⣋⡁⠉⠖⠉⠽⡈⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠚⠠⠄⠀⠀⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠤⠤⠤⠤⠤⣴⠖⠂⠁⠈⠃⠀⠀⠀⣀⡴⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣶⣶⣶⣶⣶⣶⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠛⠎⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⡛⠍⠀⢠⣬⡷⣿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⢿⣿⣶⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠉⠛⠛⠉⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣋⠓⢆⠀⠀⣤⡶⣤⠀⠙⠛⠁⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡇⠉⠛⠋⠀⠐⢤⣤⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣄⢀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢴⣶⡦⠄⠀⠀⠀⣠⠎⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣆⠀⠀⢀⣀⡴⠞⠁⠻⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠺⠭⠀⢾⣧⡻⢿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⡟⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⡆⠀⠀⢸⠉⠁⢀⡹⠀⠀⠀⢀⣀⠀⠀⠀⢠⠄⠀⠈⠙⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠈⠀⣴⠏⠀⠀⣀⠀⠈⠿⢆⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠃⣀⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡄⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠞⢤⣤⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⠣⠀⠂⢤⡶⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠶⣬⣿⡟⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣇⡠⠒⠒⠂⠠⢍⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣿⣿⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⣀⣄⠀⠀⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠐⡄⠂⠐⠂⢄⠀⠘⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣄⠀⠀⠀⢠⡎⠠⢀⠒⢣⠀⠇⠀⠚⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣇⡀⠰⣿⠀⠀⠀⠀⠰⡘⠀⠁⠶⡦⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⠘⠛⠙⢄⠀⠀⡀⣴⣄⠀⠈⠈⣷⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡄⠀⠁⢨⣤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⢡⡀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⣠⠶⣋⡉⠁⢠⣤⣬⣉⣉⣹⣿⣿⣋⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⢿⡷⠶⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢢⠄⢠⣀⣀⣀⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⠛⠛⠋
⠀⠘⠀⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠉⠉⠉⠁
⠀⢠
⠀⠘

⠀⣀⣤⠶⠿⠛⠛⠛⠛⠓⠒⠒⠲⠶⠶⠒⠒⠒⠒⠒⠒⠒⢲⡆⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⡟⣯⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠉⣈⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡌⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⡟⣷⣿⣽⣿⣿⣠⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡄⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢹⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠛⠟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠿⣿⣿⣿⢿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣯⣿⣾⣽
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠐⠃⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠛⠛⠿⢾⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⡀⣀⣀⡀⣀⠀⣸⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⣀⣀⣀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣴⠟⠉⠉⠉⠛⣦⡀⠹⡆⠀⠀⠀⠀⢠⡞⠛⠛⡶⠶⠦⣤⣄⡀⠀⢀⡴⢋⡠⠊⢠
⠀⠀⢀⣤⠶⠛⠋⠉⠙⠳⢦⡀⠀⠀⢸⠁⠀⣼⠃⠀⠀⠀⠀⠀⠈⣧⠀⢿⠀⠀⠀⠀⣸⠁⠀⠀⠇⠀⠀⠀⠏⠙⣷⢋⡴⠋⢠⢀⡟
⠀⣴⠋⠀⣀⡤⠶⠶⢦⣄⠀⠙⣦⠀⢸⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⡇⠀⠀⢠⠇⠀⠀⢸⠀⠀⠀⢸⠀⠀⡿⠊⡠⠔⣡⠟
⣼⠁⠀⡼⠋⠀⠀⠀⠀⠈⢷⡀⠘⣧⠸⡇⠀⣇⣀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⡞⠀⠀⠀⡏⠀⠀⠀⡏⠀⣰⠷⡄⠀⠊⠁
⣿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢻⡄⠘⣧⣷⠸⡋⠙⡇⠀⠀⠀⢀⣀⣸⣇⣠⡇⠀⢰⠃⠀⠀⣸⠁⠀⠀⢸⢃⡼⠁⢀⠃
⣿⠀⢸⡄⠀⠀⠀⠀⣶⣤⡀⠀⢻⡄⠈⢿⡆⢃⠀⢱⠀⠀⠀⡏⠁⠀⠀⠀⢹⠀⡯⠀⠀⢀⡟⠀⠀⠀⣼⡾⠁⠀⡜
⢹⡄⠀⢷⡀⠀⠀⠀⣷⣈⢷⣄⠀⢻⡀⠘⣷⠾⠒⠛⢋⣻⡄⣃⣀⣀⣀⣀⣀⣼⠃⠀⠀⣸⠁⠀⠀⢀⡟⠀⠀⣰⠁
⠀⠹⣆⠈⠳⣄⠀⠀⢹⡀⠀⠙⢦⡈⡇⠀⣻⡶⠞⠋⠉⠉⢿⡧⣤⣤⡤⠤⠤⢿⠀⠀⢀⡟⠀⠀⢀⡾⠀⠀⢰⠇
⠀⠀⠙⢷⣄⠈⠳⢄⡈⣧⠀⠀⠀⣻⣇⢺⠉⣷⡀⠀⠀⠀⠘⣷⠒⠲⠖⠶⠒⡎⠀⠀⣸⠃⠀⢀⣾⠁⠀⢀⡿
⠀⠀⠀⠀⠙⠷⣄⡀⠉⠹⣖⠒⠋⠀⠹⣎⠋⢽⡇⠀⠀⠀⠀⢿⠀⢀⠀⠀⢸⡇⠀⠀⡟⠀⠀⡼⠁⠉⠂⣼⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠻⣆⠀⠀⠀⠹⣶⡸⣿⡀⠀⠀⠀⣿⠀⣸⠀⢸⢸⡇⠀⢸⠇⠀⣼⠃⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠹⣷⣿⣧⠀⠀⠀⣿⠀⣿⠀⣿⠀⡇⠀⡿⠀⣼⠃⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠹⣿⣿⡄⠀⠀⡿⠀⣿⠀⣿⠀⡇⢰⠇⣰⠇⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠹⣏⢳⠀⠀⡇⠀⣿⠀⡿⠀⡇⡞⢰⠏⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠹⡎⡇⠀⡇⠀⠹⠀⡇⠀⣇⢡⠏⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠼⠦⠤⠤⠤⠿⠾⠤⠷⠤⠶⠶⠷⠶⠷⠾⠶⠶⠶⠶⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⢤⣀⢀⣴⣖⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⢄⣀⠈⢻⠻⣛⣀⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⢰⠒⡒⠋⠽⠁⠱⡜⢢⠀⢟⠁⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠘⠦⣇⠈⣥⠀⠈⠃⠀⠀⡜⠀⢈⣷⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡍⢿⢻⡄⠀⠘⣄⠀⠰⡞⢉⡇⣰⠥⡔⠁⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⢾⡄⠀⣠⢼⢒⡤⣼⣭⡜⡹⣆⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠗⠉⣀⡤⣼⣾⠟⣶⠿⣥⠼⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡳⣄⣀⣀⣠⡵⠤⣠⣤⡾⣃⠴⠮⡀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠉⠉⠉⢻⠉⠁⠀⣰⡛⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⢀⡀⠀⠀⠀⠈⢇⠀⡸⠅⠁⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡍⠑⣢⣄⣀⣀⠉⠧⣀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡺⠋⣀⡔⠁⠙⢄⠀⠉⠙⠒⠒⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⡾⠋⠀⠀⠀⣨⠷⢄⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢻⠀⠀⠙⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⣇⣠⠤⠤⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣃⣤⡄⠀⠀⠀⢈⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡿⡟⢻⣷⠿⠛⠋⠙⠛⠛⠻⠿⠻⣿⢽⣏⣱⡟⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠡⢿⠇⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠰⠤⣼⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠉⠛⠛⠓⠂⠂⠠⠤⠤⠤⠤⢤⣤⢀⡀⢸⣿⣿⣿⣷⠶⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣤⣻⣾⣿⣿⣍⣉⣓⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣷⡅⡄⠀⠀⠀⠀⣀⣼⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠂⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠒⠀⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠠⠵⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠦⠶⠴⠖⠒⠓⠒⠒⠋⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⢴⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⡀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⢘⠑⡆⠀⢀⣀⣠⠤⠔⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣷⡸⠿⣭⣭⣭⣽⣿⣿⣷⣶⣶⣶⠦⠤⣤⠀⢈⠄⡏⠉⢉⠀⠀⡀⣀⣠⠤⠔⠒⢲
⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣤⣤⣶⣟⣋⣉⣁⣈⣩⣭⣿⣾⣿⣿⣿⣿⡿⠿⠛⠋⣉⣩⢹⣦⣾⢹⠀⠈⠀⡥⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⠒⠲⠦⠾⢿⣿⣿⣿⣿⣿⡿⠿⡿⣍⣩⣍⣤⣶⣷⣿⣏⣿⣾⣿⣿⢸⡀⠀⠀⡇⠀⠢⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⣢⠀⠀⠀⠀⠀⠀⠀⣽⣿⣦⠞⣿⣟⢸⣿⣿⣿⡟⣽⣿⡿⠿⠟⠛⠉⠉⠀⠀⠠⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⢘⣾⢌⣿⡿⠾⠛⠃⠉⠁⠀⠀⠀⠀⠀⠐⠀⠀⢀⡇⣄⡁⠠⠀⠀⠀⣀⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠴⠞⠛⠉⠁⠀⠀⠀⠀⠠⢤⡀⠈⠀⢀⡠⠼⠀⢸⠴⣄⢘⣿⡿⠶⠚⠁⠀⢀⣀⠤⢶
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠶⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣀⡤⠟⠓⠉⠁⠀⠀⠀⢠⡀⡟⠉⠄⣀⣠⠤⠒⠋⠁⠀⠀⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠁⠢⠂⢀⣤⡄⢰⠋⠁⠀⢀⣀⡤⠒⠈⠀⠀⠰⡈⡧⠞⠋⠉⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠁⠒⠀⠠⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢀⠂⠤⠁⣢⡤⣶⢧⡤⠖⠟⠉⠃⠀⠀⠀⠀⠀⢰⠘⡇⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠤⠄⣀⠀⣸⠪⠄⠡⠀⠀⠀⠀⠀⠰⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠒⡀⠀⠀⠀⠀⠀⣀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠹⡇⠀⠁⠀⠀⣠⠴⠊⠉⠀⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡁⣤⣴⠖⠋⠡⢀⠠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠘⡟⠁⠀⡠⠴⠂⠁⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⣙⡂⠀⠀⠀⠀⠀⠀⠀⠸⠀⡧⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢠⠀⠀⠄⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⡝⠻⣿⡿⠿⣿⣷⡀
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣟⣻⣿⠙⣧⠀⣂⣶⠀⠀⠀⠈⣽⠿⠆⠛⠀⠀⠙⡿⣿⣆
⠀⠀⠀⠀⢸⣿⣇⠀⠀⣢⠱⡀⠀⣠⡀⡀⠀⠀⡀⣍⣿⣹⣿⣿⣿⢿⣿⣷⠾⣇⠀⠀⢀⠁⠀⠀⣰⠀⠀⠀⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠸⣿⣿⡄⡀⠁⠀⢷⡈⠉⠁⠛⠒⠂⣷⣃⣻⣿⣿⣿⡿⠾⣏⣟⣻⡿⠂⠀⠘⠀⠀⠀⠋⠀⠀⡀⢻⡿⠟⠁⠀⠀⠀⠀⠀⢀⡄⢫⡀
⠀⠀⠀⠀⠐⠹⣿⣿⡔⡀⠀⠀⡿⠄⠀⠀⠀⠀⢿⣟⣉⣉⠛⠃⠄⠀⠀⠉⠀⠀⠀⢶⠿⡀⠀⡌⠀⠀⠈⢁⡿⠁⡆⠀⠀⠀⣠⣂⣼⡟⠀⣾⡗
⠀⠀⠀⠀⠀⠆⠹⣿⣿⡀⠈⣴⠁⠿⣄⣤⣶⣶⣞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠁⠉⡥⠴⠒⢃⠸⡀⠀⠀⢀⠄⠀⠀⢀⠀⠀⣀⣴⣿⢟⡄⢠⡿⠁
⠀⠀⠀⠀⠀⠒⠊⠻⣿⣿⡔⠍⣀⡀⠘⢎⠋⢉⠛⠁⠁⠀⠀⠀⠠⣤⣾⠀⠀⠀⠁⠀⠀⠀⠀⠈⣄⣠⠎⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠃⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⢨⣥⣄⢤⡄⠂⢿⣷⣄⣾⣷⣿⣿⡿⠤⠀⠀⠠⠤⠄⣀⡀⠀⠠⠏⠀⠄⠀⠀⡠⠒⡀⣀⣿⣿⠀⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠀⠀⠈⠉⠛⢻⣇⢸⡟⠛⣿⠿⣿⡿⠃⠘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⢾⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⡛⢟⣙⠛⣻⢛⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿