Sprinkle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠻⣦⡀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢀⣠⣴⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠙⣷⡄⠀⠀⠀⠀⢸⣧⣠⡿⠀⠀⠀⠀⣴⠟⠋⢀⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣼⡟⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠻⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠿⠿⠿⠿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⣤⣴⣶⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠋⠉⣁⣀⣼⡇⠀⠀⠀⠀⢰⣟⠋⠉⠉⠉⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠻⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠿⠟⠛⢛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣶⠶⠿⠛⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣴⡾⠋⢀⣴⡿⠀⠀⢀⣤⣤⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠻⣦⣄⠀⠀⠸⣯⣀⣴⠟⠋⠀⠀⢰⣿⠉⢁⣀⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡈⠛⣷⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣉⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣿⠋⠛⠻⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⣹⣿⠶⠿⠟⠛⠛⠿⢷⣶⠿⠛⣿⡄⠀⠀⠙⠿⢶⣦⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⢀⣴⡾⠛⠁⠀⠀⠀⠀⠀⣴⠟⠁⣠⣾⠟⠻⢷⣤⡀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⡾⠛⠁⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⣤⣤⣴⣦⣄⠀⠀⠐⣿⡀⠙⣷⣄⠀⠀⠀⣾⡋⠉⠛⠻⠷⣦⡀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⢁⣀⣀⣼⡿⠀⠀⠀⠘⢿⣄⠈⢻⡆⠀⠀⠙⠛⠿⠶⣶⣤⡿⠃⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠻⠷⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⢛⡀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣹⡁⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢸⣧⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢙⡃⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠈⣿⡄⠀⠀⠀⠀⣩⡄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⣸⡏⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⢹⣇⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠘⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⠀⠀⠀⠀⢸⡿⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡏⠀⠀⠀⠀⣾⡇⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢸⣷⠀⠀⠀⠀⣿⡆⠀⠀⠀⣿⡇⠀⠀⠀⣼⡇⠀⠀⠀⢠⣿⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⣿⠁⠀⠀⠀⣼⡏⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⢹⣧⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢰⣿⠀⠀⠀⢀⣿⠁⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠈⣿⠀⠀⠀⠀⣿⠀⠀⠀⢿⠇⠀⠀⢸⡏⠀⠀⠀⢸⡟⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠋

⠀⠀⠀⣴⡶⠀⣿⡇⠀⠀⠘⠛⠛⠛⠛⠛⠋⠉⢻⣷⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⣿⡇⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⣿⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⢸⣿⠁⠀⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣾⣟⣋⣀⣠
⠀⠀⢺⡿⠀⠀⠈⠙⢿⣷⣄⠀⣤⣴⣶⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢀⣿⡏⠀⠘⠋⠀⠀⠀⠀⠀⠸⠿⠀⠀⠾⠿⠿⠿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⣾⡟⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠈⠛⠁⠀⠀⠈⠙⠛⠻⠿⣷⣶⣤⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣴⣶⣶⣾⠿⠿⠿⢛⣻⣶⣾⠷⠀⠀⠀⠀⠀⠀⠰⠿⠃⠀⠐⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠃⠀⠀⠀⠀⠀⣠⣴⡦
⠀⢰⣶⣄⡀⠀⠉⠁⠀⠀⠀⣠⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⢀⣠⣶⡿⠟⠉
⠀⠀⠈⠻⢿⣦⣄⠀⠀⠀⠀⠙⠉⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⢾⠿⠛⠁
⠀⠀⠀⠀⠀⠙⠻⣷⡦⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⠘⠿⠿⢿⣷⣶⣦⣤⣝⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⢀⣈⣉⣉⣛⣻⣇⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡄⠀⣠
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠻⠟⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠋⣠⣾⡿
⠀⠘⠛⠿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢀⣴⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠋⠀⠀⠐⠿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⡿⠿⠛⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⠀⠀⠀⠀⠀⢻⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠈⠻⣿⣦⣀⠀⠀⢸⣿⡟⠿⣷
⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠙⠿⣷⣦⡀⢿⡇
⠀⠀⠀⠉⠛⠻⢷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⣤⣤⣈⡙⠿⣷⡄
⣾⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣦⣤
⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠇⠀⠀⠀⠉⠉
⠀⠀⠀⠀⢀⣤⣶⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠇⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣶⣦⠀⠀⢰⣶⡀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⢻⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢻⣧⠀⠀⠀⠀⠙⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠈⢿⣧⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢠⣦
⠀⠈⢿⣧⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⠇⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠻
⠀⠀⠈⠿⠇⠀⠀⣿⡇⠀⠀⠀⣾⡏⠀⠀⠀⠀⠈⠛⠿⢷⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠃⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⡿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⢀⣀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠂
⣿⣿⣿⡿⢿⣿⠿⢿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣦⣽⣤⣧⣤⣿⣬⣼⣯⣼⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧
⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣄⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀
⠟⠁⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠰⣦⡀
⠏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠛⠃
⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢀⣴
⣤⡀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀
⠀⠀⠀⠘⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢴⣆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠴⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣠⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⣤⡀⠀⠀⠀⠀⠐⠻⠿⠇⠀⠀⠀⠻⢶⡄⠀⠀⠉⠉⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⣀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠖⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⢀⣤
⠀⠀⢼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⠀⠁⠀⠀⠀⠐⠋⠀⠀⠈⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠺⠋⠀⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠻⠆⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⢀
⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⣿⣃⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣴⣂⣀⣀⣀⣀⣦⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠙⠛⠁⠀⠀⠀⠀⢼⣿⠃⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⢶⣶⣤⡀
⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠸⠿⠋⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⣴⣄
⣶⡄⠀⠀⠀⠀⠀⠀⢾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠺⠿⠟⠙⠛⠁⠀⢠⣶⡀⠀⠀⠀⠀⣠⡄⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⢰⣶⣤⡀⠀⠀⠙⠿
⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠃⠀⠀⠀⠀⢿⣿⠀⠀⠀⢠⣤⡀⠀⠀⣀⠀⠀⠀⠉⠛⠁
⠀⢀⣤⣶⡦⠀⠀⠀⢰⣶⣄⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⢿⡿⠀⣾⣿⠁⠀⠀⠀⠀⣿⣦
⣄⠈⠛⠉⠀⠀⠀⠀⠀⠙⠻⢿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠀⠀⠀⠙⠛⠀⠀⢠
⠿⠃⠀⠀⣰⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠃⠀⠀⠀⠛⠋⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⢰⣿⠇
⢀⣤⠀⠀⠛⠋⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠄⠀⠀⠈⠛⠃⠀⠀⠙⠋⠀⠀⠀⣴⣤⡀
⣾⡿⠀⠀⠀⠀⠀⠾⡿⠋⠀⠀⠀⠀⠀⣴⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⣶⣦⡀⠀⠀⠈⠛⠟
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣶⡷⠀⠀⠀⣠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠈⠙⣿⣦⠀⠀⠀⠀⢀⡀
⣾⣿⠆⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠛⠋⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠸⢿
⢀⣀⣤⡀⠀⠀⠀⠻⠟⠋⠀⠀⠀⠀⢴⣦⣄⠀⠀⠈⠿⠃⠀⠀⠀⠀⢀⣴⠀⠀⠙⠻⠿⠂⠀⢴⣦⣄⠀⠀⠀⠀⢾⣦⠀⠀⠀⠀⣿⣦
⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⡀
⣼⡇⠀⠀⢀⣀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣴⣿⡿⠂
⠿⠃⠀⠀⠘⠿⣷⠄⠀⠀⠉⠀⠀⠀⢿⣿⡶⠀⠀⢀⣶⡆⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠰⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⢀⣴⣾⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃
⠀⠀⠰⣿⡿⠓⠀⠀⠀⢰⣿⡀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠹⠿⠃⠀⠀⠀⠀⠀⠙⠛⠻⠿⠟⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣤⣶⡷⠀⣴⣦⡀
⣴⣶⠆⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠉⠀⠀⠈⠻⠷
⠉⠁⠀⠀⠀⠀⣾⡆⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⣀⠀⠀⠀⣼⣿⠀⠀⢀⣀⣀
⢀⣀⡀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠉⠁⠀⠀⠈⠻⢿⡇
⠿⠛⠋⠀⠀⠀⠀⠀⠀⣤⣶⡷⠀⠀⠀⠈⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢀⣠⡄⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠛⠁⠀⠀⠀⠉
⠀⠀⠀⠰⠿⠟⠁⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⡀⠀⠀⠙⠛⠛⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢠⣦
⣦⣤⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⢻⣿⠄⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⣶⣤⡀⠀⠀⠀⠀⠰⣿⠟⠀⠀⠀⣰⣦⠀⠀⠸⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⢠⣶⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⡿⢿⣿⣿⠿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠉⠉⠉⠙⣫⣶⣶⡸⢗⣴⡶⢟⣷⣮⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠾⠿⠿⠿⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠄⠀⠀⠀⠀⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠳⣄⠀⠈⠛⠋⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡿⢻⣿⣿⣿⣿⣿⣿⣶⣘⢛⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡟⣛⢻⣿⣿⣶⣿⣿⣿⣶⣿⣿⣿⡿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣭⣼⣿⣿⣛⣿⣿⢋⣭⢻⣿⣿⣷⣍⣿⣿⣭⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⢟⡛⣛⠿⣿⣿⣷⣵⣾⣿⠿⠛⣛⣛⠿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣵⣾⣿⣿⢸⣷⣮⣽⡛⣛⣭⣵⣾⢱⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡎⣿⣿⣿⡇⣿⣿⣿⡟⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⢻⣿⣿⡇⣿⣿⣿⠇⣿⣿⣿⠏

⠀⠙⡆⠀⢀⡀⠀⢸⢸⠀⠀⢠⠃⢀⡏⠀⠊⠀⠀⠀⠀⠁⠀⠀⠊⠀⠀⡇⠀⠘⡆⠀⠀⣇⠀⢀⡀⠀⠀⡔⠁
⠀⢀⠇⠀⠀⠀⠀⢀⣾⠀⠀⢸⠀⢸⣠⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⡀⠀⡽⠀⢰⠁⠀⠀⣏⠑⡄⠀⠀⠀⢧
⠊⠁⠰⠀⠀⠈⢂⣇⡎⠀⠀⢸⠁⠀⢯⢄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⢠⠇⢀⡞⠀⠀⠀⠸⡴⠃⠀⠀⣀⠀⠉
⠀⢠⡀⠀⢀⠖⣫⠎⠀⠀⠀⠀⠳⡀⠈⠣⣕⡦⠤⠀⠀⢀⣀⣀⣀⡴⠃⢰⡜⠀⠀⠀⠀⠀⢇⠱⡀
⠀⠀⣀⡠⠞⠊⠁⠀⠀⠀⠉⠀⠀⠈⠓⠤⣀⣉⡉⢉⠉⠉⠉⠀⢀⣠⠜⠋⠀⠀⠀⠰⠀⠀⠀⠑⠬⣒⠄⠀⢤
⠉⠉⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒
⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠤⠤⣄⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡤⠔⢒⣒⣲⢤⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⣠⠤⠤
⠦⠤⠥⣄⡙⠲⣄⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠉⠀⠀⠀⠉⠉⠙⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⡠⠚⢁⡤⢔⣒
⠀⠀⠐⠂⠈⢆⠘⠳⡀⠀⠀⠀⡜⠁⠀⠀⠃⠀⠀⠘⠁⠀⠀⠈⠀⠀⠀⢳⡹⡄⠀⠀⠀⢀⠾⢁⡔⠁⠊⠁
⡄⠀⠀⡀⠀⢸⡀⠀⢱⠀⠀⠰⡅⠀⠀⠀⠀⠀⡴⠊⠉⠙⢢⡀⠀⠀⠀⠀⢧⢳⠀⠀⠀⡎⠀⡼⠀⡀
⠀⠀⠀⠁⠀⠀⡇⠀⢸⠀⠀⠀⡟⢦⡉⠁⠀⠸⡅⠀⠀⠀⢀⠇⠀⠉⠀⠀⢈⡞⠀⠀⢸⠀⢰⠃⠀⠁
⠀⠀⠀⠐⠄⠀⡇⠀⣸⠀⠀⠀⢧⣠⠃⠀⠀⡀⠈⠲⠤⠖⠋⡄⠀⠀⠰⠀⣏⡇⠀⠀⣿⡀⠸⡰⠀⠀⠀⠁
⠀⠃⠀⠀⠀⡼⠀⢠⠃⠀⠀⠀⠸⡇⢆⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⢀⡴⢊⠞⠀⠀⠀⠸⡀⠀⢳⡢⣀
⠤⠤⠤⠔⠊⢠⡸⠃⠀⠀⣀⠀⠀⠙⠮⣓⢤⡀⢀⣀⠀⠀⠈⠀⣀⡾⠖⠁⠀⠀⠀⠀⠀⠙⢦⡀⠙⠪⢭⣓⣀
⣀⣀⡤⠤⠒⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠒⠲⠬⠶⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠓⠢⠤⣆⣤
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣒⢖⡲⣤⣀⡀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡤⠒⢋⣉⣀⣀⣀⡈⠋⠑⠢⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠔⢒⠩⠭
⠀⠀⢩⠁⠀⠈⡓⢄⠀⠀⠀⠀⠀⣰⠏⣠⠞⠭⠒⠚⠉⠀⠉⢉⠍⠳⡄⢹⢆⠀⠀⠈⠀⠀⡴⠋⠀⡀
⣀⠀⠀⠀⠀⠀⠘⡌⢇⠀⠀⠀⡰⠁⢰⠃⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⠀⠈⢣⠀⠀⠀⡜⠀⠀⠀⠁⠀⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠥⢒⡄⠀⠀⠁⠀⠠⣤⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠙⢀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⣀⣤⡴⠋⠁⠀⠀⡀⠀⠈⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠲⠀⠀⠀⡤⠔⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠤⠀⠀⠘⠠⣄⡸⠁⠀⠁⠀⠀⠀⢀⡠⠊⠑⠦⡄⠀⠀⠀⠐⠂⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠈⠆⠀⣿⠁⠀⠀⠀⠀⣀⠘⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⢀⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠦⢄⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⢠⡤⠃⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄⠀⠀⠀⠀⠀⡢⠈⠀⢆⣀⡔⠉⠓⠀⡄⠀⢀⠤⠤⠔⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⡀⠉⠀⠀⢀⡀⣈⠗⠈⠄⣀⣀⡀⠀⠀⠙⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠘⠂⠀⠀⠀⠁⠀⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀
⠀⠀⠈⠁⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠿⠀⠀⠀⠀⠀⢤
⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠶
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⣴⠄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠈⠃⠀⠀⠀⣠⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⠀⢀⡄⠀⠀⠀⢀
⠀⠀⠈⠁⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠋
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠁⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⡄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠘⠀⠀⠀⠀⠀⢤
⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠾
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⠀⠀⡠
⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠘⠁
⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⢤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠋⠁⠀⠀⣨⠖⠋⠉⠉⠉⠉⠙⠫⣟⡳⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠀⢀⣀⣁⣠⠴⠋⠁⠀⠀⠀⠀⠀⢀⡜⢉⡆⠙⢯⡉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣯⠔⠚⢛⠉⠉⠩⠄⠀⠀⠀⠀⡤⣄⠀⠀⠀⡞⢀⠎⠀⠀⠀⢱⡀⠀⠘⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢱⠃⠀⠀⠘⠶⠶⠖⠚⠁⠀⠀⠀⢃⠘⢦⠀⠀⠈⠁⠀⠀⠀⠀⠀⠑⠦⣄⠀⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⡇⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠀⠀⠀⢀⣠⠴⠒⡦⠀⠈⠑⢄⡀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⢂⡼⠀⠀⠀⠀⠀⠀⡇⠰⠀⠀⠀⣠⡴⠶⠦⣄⣀⠀⠀⠀⠀⠺⡤⠔⠊⠀⠀⠀⠀⢀⣩⡓⠺⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡿⣀⡴⣋⣠⠴⢢⠀⠀⠀⠘⠥⠄⢀⣴⠾⠋⣀⣀⣤⣄⣉⡉⠛⠒⢦⡄⠀⠀⣠⣀⠀⠀⠀⢸⠁⡇⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠸⣍⠤⠒⠊⠀⠀⠀⠀⢀⡴⠛⢁⣤⠞⠋⠉⠉⠉⠉⠛⢷⣄⠀⢹⡀⠀⢧⠘⡆⠀⠀⢎⣠⠇⠀⢰⢿⡆
⠀⠀⠀⠀⠀⠀⢀⡟⢧⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡿⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⢷⡄⠈⢧⡼⠀⡀⠀⠀⠀⡴⠃⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡗⠘⢦⣀⠀⢸⡉⢳⠀⠀⠀⣼⠃⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢷⠀⠀⠀⢸⠉⢃⠀⠀⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⢳⡀⢳⡀⡇⠀⠀⠸⣧⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢸⠀⠀⠀⠘⣇⢸⠀⠀⢹⠀⠃⣿
⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⠈⠈⡇⠀⠉⠁⠀⠀⢀⣿⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣠⠟⠀⠀⠀⠀⠈⠉⠀⠀⢸⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢀⡇⡰⢲⠀⠀⠀⠈⢷⣀⠀⠈⠻⢶⣤⣀⣀⣀⣠⣴⠟⠋⣠⡾⠋⠀⠀⠀⡴⠒⠖⠒⡄⠀⢸⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⡀⣸⠀⡇⢰⠀⠀⠀⠀⠀⠉⠛⠶⠤⣤⣈⠉⠉⠉⠉⣠⠾⠛⠉⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁⠀⡼⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠇⠀⠳⠜⠀⠀⠀⠀⠀⠀⢠⠠⣄⠀⠈⠙⠲⠖⠛⡡⠒⠒⠛⣢⠀⠀⠀⡴⢦⣀⠀⠀⠀⠀⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⣀⣤⣤⣤⡀⠈⢦⡈⢣⡀⠀⠀⠀⠀⠙⠚⠋⠉⠁⠀⠀⠀⠑⢤⣈⢳⠄⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⢄⡀⠘⠧⠤⠤⠴⠃⠀⠀⠉⠚⠀⠀⠀⠀⠀⠀⠀⢠⡖⢲⠀⠀⢀⣤⠤⠬⠟⢋⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣍⡓⠤⡤⠤⠤⠤⢤⣄⡀⠀⡠⠔⠚⣉⡇⠀⠀⢸⠃⣼⠀⣰⡋⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠈⠳⣗⠒⠋⠁⠀⠀⠀⠘⠓⠃⣰⠋⢀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣤⣄⣀⠈⠓⠦⠤⣄⣀⣀⣠⣴⣋⣥⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠒⠛⠛⠋⠉⠈⠉



⣿⠿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣛⣻⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠋⠉⠀⠀⠐⠂⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠠⠆⠀⠀⠐⠂⠀⠀⠀⠀⠙⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢄⡀⠀⠀⢀⡀⠀⠀⠀⠉⠀⠀⠀⠋⠀⠀⠀⠚⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⠲⢤⣉⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡟⡇⠀⣨⡉⠒⠦⣄⡀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠀⡇⠷⢶⠀⢠⣭⠓⠶⢤⣀⠀⠀⠀⠀⠤⠀⠀⠀⠓⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣬⣎⡀⠀⢸⠀⡿⢾⡞⣷⠀⠈⠙⠲⠦⣄⡀⠀⠀⠀⢀⣀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠛⠻⠷⣾⠀⣇⡀⠀⢿⠀⢸⢹⡆⣠⢦⠉⣛⠒⠦⠤⣄⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡁⠀⠀⠈⠛⠛⠻⠿⣿⣀⣧⣘⠷⠃⠈⢋⠸⣇⡔⢲⠀⠀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢦⣄⡀⠀⠀⠀⠈⠉⠛⠿⢷⣶⣤⣄⡈⠀⠘⡆⡔⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⡀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣿⣤⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠴⣤⣀⡀⠀⠀⠀⠀⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣐⣀⡄



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠜⠁⣀⠀⠀⠑⠀⠀⠙⠒⠀⠀⠦⠤⠀⣦⠀⢠⠇⠀⠘⠁⠀⠀⠀⢦⡀⠐⠒⠒⠀⡴⠃⠀⠙⠆⠀⡏⠀⠈⠓⠦⠀⠰⠃⠀⡴
⠖⠂⠈⠉⠃⠀⠠⠖⠂⠀⢀⠴⠀⠀⠀⠘⠀⠀⣀⠤⠂⠀⠉⠲⠀⠀⠉⠀⠀⣀⠀⠀⠀⠲⢤⠀⢀⡀⠀⠤⣤⠀⠀⢀⣀
⠀⢠⠀⠐⢄⠀⠀⠤⣀⡀⠁⠀⠀⣦⠀⢀⡖⠀⠀⠀⢠⠀⠠⠤⠤⠀⠀⡇⠀⣿⠀⣠⠞⠀⠀⠀⠀⠳⠄⠀⠀⡀⠀⠈⠉⠁
⠀⠾⠀⠀⠀⢀⣀⠀⠀⠀⠀⡆⠀⠉⠀⠈⠀⠀⢳⠀⠸⠀⢀⠀⢀⡀⠀⠁⠀⠀⡀⠀⢰⡄⠀⠓⠦⠀⠀⠰⠋⠀⢠⠆
⠀⢀⡴⠀⠀⠉⠀⠀⢰⠀⠀⠁⠀⢀⠀⠐⢤⡀⠀⢀⠀⠀⠏⠀⠀⠙⠆⠀⠰⠊⠁⠀⠀⠃⠀⢀⣀⡀⠀⢠⠀⠀⠉⠀⠈⠙⠒
⣆⠀⠠⠤⠴⠀⢳⡀⠈⠀⣠⠖⠀⠈⡇⠀⢀⠀⠰⠋⠀⠀⠀⠀⡆⠀⢀⡀⠀⠰⡄⠈⠳⠤⠀⠀⠀⠀⠀⠈⠇⠀⠐⠒⠃⠀⣀
⠉⠀⢀⡄⠀⠀⠀⠀⡄⠀⠀⠀⠤⣄⡀⠀⢸⠀⠀⣀⠀⢸⠀⠀⠃⠀⠀⠙⠂⠀⠙⠀⠀⠀⠠⠞⠁⠀⢧⠀⢀⡄⠀⡀⠀⠞⠁
⡀⠀⠋⠀⠀⢱⡀⠀⠇⠀⠲⡄⠀⠀⠀⢀⠀⠀⠴⠁⠀⠀⢀⡴⠀⢀⡴⠀⠀⢠⡆⠀⡞⠀⠀⠀⠲⣄⠈⠀⠘⠀⠀⠙⠆
⠈⢀⡴⠂⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠋⠉⠀⠤⠤⠄⠀⠀⠉⠀⠀⠈⠀⡀⠀⠈⠁⠀⠀⠠⠞⠁⠀⠈⠀⢰⡀⠐⠦⣄⠀⠀⡤
⣀⠈⠀⠀⠒⠒⠂⠀⠞⠀⠀⠀⠉⠒⠀⠀⣀⣀⡀⠀⢰⠀⠀⡇⠀⠀⠀⠹⠄⠀⠖⠒⠂⠀⢰⠂⠀⠳⣄⠀⠃⠀⣀⡀⠀⠘⠁
⠈⠉⠀⣠⠄⠀⡀⠀⢀⠀⠘⠛⠁⢀⡄⠀⣀⠀⠀⠀⠉⠀⠀⠁⢠⠞⠀⢀⠀⠀⣤⣀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠉⠉⡀⠀⢤
⣠⠀⠈⠀⡀⠀⠳⠀⢸⠀⠀⡀⠘⠁⠀⠀⠈⠦⠀⠀⠒⠚⠃⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠤⠖⠂⠀⠈⠉⠁⠀⠀⠀⡇⠀⠘⠂
⠃⢀⣠⠄⠙⠦⠀⠀⠀⠀⠀⠙⠆⠀⠈⠲⡄⠀⣠⠆⠀⣠⠞⠀⣸⠀⡀⠀⠀⠴⠚⠀⠀⢠⠀⠀⡤⠖⠂⠀⠙⠦⠀⠀⣠⠆
⡀⠈⠀⠀⢄⠀⠀⢳⠀⠉⠲⠄⠀⠀⣠⠄⠀⠀⠁⢀⠀⠀⣀⣀⠀⠀⠉⠓⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⣠⠃⠀⡀⠀⠈⠁
⠉⠁⠀⡄⠀⠑⠀⠀⠀⢠⠀⠀⠀⠉⠀⠀⠀⡇⠀⠈⠳⠀⠀⠈⠁⠀⠴⠖⠂⠀⠈⠦⠀⠀⠓⠦⠀⢀⠀⢀⠀⠙⠆⠀⠘⢦
⠶⠆⠠⠇⠀⠤⠖⠂⠀⠼⠀⠘⢦⠀⠀⢤⣀⠀⠀⠲⢄⠀⠀⣇⠀⢀⠀⠠⠴⠂⠀⠀⠤⠤⠄⠀⠀⡾⠀⠈⠧⠀⠀⠠⣄⡀
⡄⠀⠀⠀⡄⠀⠀⣠⠄⠀⣠⠀⠀⢀⣀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠈⢧⠀⠀⢠⠆⠀⢄⡀⠀⢀⣀⣀⠀⢀⠀⠀⢀⣠⠀⠀⢀
⠀⠀⡇⠀⠹⠀⠈⠁⠀⠘⠁⠀⠈⠉⠀⠀⡆⠀⠸⠁⢀⡀⠉⠙⠃⠀⣀⠀⠀⠉⠀⢀⡀⠉⠁⠀⣀⠀⠀⠸⠀⠀⠁⠀⡀⠀⠉
⢦⠀⠀⠀⠀⠀⠲⠦⠄⠀⠶⠒⠂⢸⡀⠘⠃⢀⡤⠀⠀⠉⠃⠀⠀⠘⠁⠀⢶⠀⠈⠉⠀⠀⠸⡀⠈⠙⠂⠀⣠⠖⠀⠀⠉⠓
⠀⠀⠐⠚⠋⠀⣠⠀⠀⡰⠂⠀⡄⠀⠁⡀⠀⠉⠀⢠⠀⠐⠛⠉⠀⢀⡞⠀⠘⠀⠀⠒⠒⠀⠀⠁⠀⠦⣄⠀⠁⠀⠰⠤⠄
⠉⠀⡀⠀⠀⠊⠁⠀⡀⠁⠀⠀⠻⠀⠰⠇⠀⢦⠀⠈⠇⠀⡴⠀⠀⠈⣀⠴⠀⠈⢧⠀⢠⠀⢀⠞⠀⠀⠀⠀⠰⡄⠀⠀⣀⡄
⠀⠀⡇⠀⠴⠒⠃⠀⠹⠀⠀⠦⣄⠀⢰⡀⠀⠀⠁⠀⣀⠈⠁⠀⢦⠀⠁⠀⣀⣀⠈⠀⠚⠀⠀⠀⠀⠉⠉⠀⠀⠃⠀⠈⠁⠀⠈
⣀⠀⢀⡤⠄⠀⢰⠃⠀⣠⠄⠀⠀⠀⠀⠁⠀⠀⠦⡀⠈⠙⠂⠀⠈⠁⠀⣀⡀⠀⠀⢀⣀⠀⠐⢦⡀⠠⠴⠒⠀⡴⠀⢠⠔⠃⢀
⠀⡀⠀⠀⢠⡀⠈⠀⠀⠁⠀⠴⠚⠀⠐⠒⠒⠀⠀⠀⠀⡄⠀⠒⠛⠁⠀⠀⠉⠀⠀⠉⠀⠀⡀⠀⠁⣀⠀⠀⠈⠁⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠
⠀⠀⠀⢀⣤⣤⠖⠠⠀⠀⠀⠐⠁⠀⡠⠀⠀⠀⢀⣀⢤⠀⡀
⠀⠀⢠⣿⣿⡿⠦⠀⣤⡖⠞⠁⠀⠒⠀⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⢈⣿⣿⡂⠆⠘⠁⠈⠃⢀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠂
⠀⢀⣼⣿⡿⢁⠀⠀⠀⠀⠀⠸⡆⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢀⣼⣿⡟⠀⠘⠀⢀⡐⠂⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⠐⠀⢀⠠⠐⠒⠉⠉⣀⢠⡆
⢰⣿⡏⠀⠀⠀⠀⠀⠑⠤⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣻⠃
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠠⠄⠀⠉⠀⠂⠀⠠⡀⠀⠀⠀⣸⣿⣿
⠀⠙⢺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⡄⠒⠆⠁⠀⠀⠀⢉⢽⣿
⠀⠀⠈⠀⢶⠦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠘⠒⠒⠒⠒⠲⠶⠶⢶⠦
⠀⠀⠀⠀⣁⣤⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⣼⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠁⠀⠀⠀⡀
⠀⠀⠀⠸⡻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢠⢞⢉⣎⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠈⠁⠀⠘⠉⠀⠀⠲⠄⠄⠀⠀⠀⠀⠀⠐⠄⠴⢧
⠀⠀⠀⠀⠀⠁⣧⡠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠀⡄⠀⢀⠀⠀⢀⠀⢠⢀⣦⡀⡁⠀⠀⡔⠁⠀⠀⠀⠈⢑⠄⠠
⠀⠀⠀⠀⠀⠀⡧⢌⡉⠁⠀⠒⠒⣊⣉⣹⠤⡀⠀⠀⠀⠀⢢⣀⠀⠀⠂⠈⠀⠀⠘⠐⢀⣀⣤⣞⠀⠀⠀⢸⡶⣐⢢⣤⣒⠋⠄⠂⠄⠀⢤⡀
⠀⠀⠀⢐⠀⣾⡠⣤⣤⣤⣤⣤⣭⡭⣅⣋⡰⠈⡇⠀⠀⢀⢰⠓⠬⢁⠀⠀⠐⠒⣒⣉⡭⠴⢛⠆⡂⠀⠀⠀⢧⣵⣸⣸⣸⣤⠤⢄⣤⢴⣼⠁
⠀⠀⠀⠀⠑⠦⠭⣖⣀⡀⠀⠀⠲⣒⣋⠩⠀⠈⠀⠀⠀⠈⠓⠭⣄⣀⠀⠀⠀⠀⠀⣀⣀⣠⠥⠚⠁⠀⠀⠀⠀⠀⠉⠁⠀⣻⣹⢹⣹⣫⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠔⢡⡀⢀⠠⠀⠒⠒⠀⠀⠒⠒⠬⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠰⠄⡐⠅⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⡠⠖⠩⠙⢥⠒⣀⣤⣀⢀
⠀⠀⠀⠐⠎⠑⠚⠶⠤⠤⠤⠤⠤⠤⠀⠘⠒⠙⡆⠀⠀⠀⠀⠀⠀⠰⠁⡀⠀⠀⠐⠐⠈⠄⡸⠁⠀⠀⠀⠀⢠⠔⠋⠀⠀⢨⠁⣀⠰⠄⡤⠀⢀
⠀⠀⢠⠀⠶⣤⣤⣀⣀⣀⠀⠀⢀⣈⣀⣠⣤⡾⠀⠀⠀⠀⢠⣔⡊⠀⠡⠀⠀⠀⠡⠀⢐⡔⠁⠀⠀⠀⡠⠚⡀⠀⠐⡀⠉⠁⢎⡀⣀⠆⠀⡠⠁
⠀⠀⠸⠀⠀⠈⣉⠉⠛⠛⠛⠛⠋⠙⠛⢉⡼⠁⠀⠀⠀⠀⠈⢋⠻⢔⡂⠁⠀⠐⣀⠜⠉⠀⠀⠀⠈⠣⢕⡀⠁⠀⠂⠡⠊⠁⠐⡄⠀⠀⠄
⠀⠀⠀⠑⠲⠖⠁⢀⡂⠀⠤⠤⠴⠶⢚⠁⠀⠀⠐⠃⠀⠀⠀⠀⠑⠌⣳⣴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡀⠑⠒⠊⡠⢀⠂
⠀⠀⠀⠀⠀⠀⣟⠉⢁⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠬⠭⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⠀⢀⠵⠀⠀⠀⠀⠀⠉⠀⠱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⢀⣁⠀⠤⠀⠀⠐⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⠎⡥⠤⠀⠴⠖⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⡐⠐⠀⠃⢀⣠⡶⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⣧⠀⢀⣼⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⣀⣴⠶⠛⠁⠐⠣⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠈⠀⢹⣷⣿⠏⠀⡇
⠀⠀⠀⠀⠀⢐⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠁⠀⠀⠀⠀⠀⠘⡮⣧⢦⣄⠀⠀⠀⠀⠀⠀⠼⠀⠀⠰⠦⣬⣉⣠⢠⡇
⠀⠀⠀⢴⢈⠜⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⡳⡌⠇⠀⠀⠀⠀⠘⢀⢠⡄⠰⠘⠛⣙⣉⡼
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂⢂⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠞⡠⠂⠾⠰⡄⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⠒⢉⠀⠄⠀⠀⢑⣤⡚⠳
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⢀⢴⣾⣙⡏⠀⠓⡄
⢂⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⢴⣿⣭⠞⠒⡛⠉⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀
⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠓⠉⠉⠙⠋⠁⠀⠀⡴⠀⡇⠀⠁⠠⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠐
⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠈⠀⠀⠐⠀⠀⡂⠙⠧⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢂
⣖⠂⠤⠄⣀⣀⣀⠀⠀⠀⠀⢀⡌⠘⣥⠀⠀⢀⢄⠀⣀⣀⠂⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠻⡄⠀⣠⡀⠀⠀⠀⠀⠀⠈
⡇⠈⢩⣥⠴⠒⠒⠉⠁⠀⠐⠁⢴⡄⢈⣀⠮⠁⢈⢂⠐⠛⣦⣀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠄⠙⠁⠀⣐⣋⣂⣀⣁⣀
⣟⣋⣒⠀⠤⠤⠄⣀⡀⠑⢬⠕⠀⢥⠀⣃⠘⡇⠘⠗⡀⠀⠠⠉⠀⠀⠀⠀⠀⡰⠒⠒⠒⠛⠢⠀⡄⠄⠊⠉⠁
⣓⣒⣢⠤⠕⢛⠭⠀⠀⠀⢐⠈⠢⣰⡦⢴⡀⠛⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠄⠀⠀⠀⠐⡄⢀
⡋⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⡴
⠃⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⡄
⡁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠄
⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠐⠠⠀⢠⠄⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⡽⠷⣤⣄⡀⠀⠈⢉⣉⠉⠛⠳⣯⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡶⠾⣏⠀⠀⠀⠙⠶⣤⣌⣿⠀⠀⢸⡉⣇⠀⠀⠸⣆⣹⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣿⣄⠙⢷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⣧⣹⠀⠀⠀⠈⠉⠀⠀⠙⣷⠟⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⡏⠉⠳⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡴⢤⡀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢦⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠻⣦⡙⢦⠀⣾⣷⣿⣿⣿⣿⣶⣦⣀⠀⠀⢸⡇⣿⠀⠀⠀⣼⠿⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣈⠛⠾⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⠷⠋⠀⠀⠀⠈⠛⠳⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⢀⡔⠛⣉⣥⠿⠂⠀⠀⠸⣟⠉⠉⠙⠛⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠛⠛⠉⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡄⠀⠀⢀⣤⠶⣦⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⡴⢻⡆⠀⠘⢷⡶⣄⠀⠀⠀⠙⣿⣿⣿⣿⠀⢰⣟⣡⠞⢋⣄⣀⠀⠹⣆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡄⠀⠀⠀⢠⡟⢀⡾⠁⠀⠀⢸⡄⢹⡄⠀⠀⠀⢹⣿⣿⣿⣇⡀⠉⠁⠀⠘⠦⣭⣙⣶⢻⡆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠸⣷⠟⠁⠀⠀⠀⠈⢷⠀⣿⠳⣦⣤⣨⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⣷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⠀⠀⢸⡏⢷⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣶⡴⠶⠚⠛⣷⠀⠀⠀⠀⠀⢠⠏⢸⠇⠀⠀⠀⠀⠀⠀⢻⡌⣷⠀⠀⠀⣼⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⣶⣾⣟⡋⠀⠀⠀⠀⠀⣿⣠⡟⠀⠀⠀⢀⣄⠀⠀⠀⠻⠟⠀⠀⣼⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠈⠉⠀⠀⠀⠀⣿⠈⣇⠀⠀⠀⣀⣤⣾⡟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣄⣿⠀⣿⣴⣶⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠒⠂⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠦⠤⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠑⠄⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠉⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠁⠀⠄⠀⠀⠊⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠰⢷⠀⠀⠀⠀⠀⠀⠙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⡂⠀⠡⠂⠀⠄⠀⠀⠀⠀⠠⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⡀⠀⠄⠞⠂⡑⢄⠀⠀⠀⠀⠔⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠊⠈⡀⢀⠀⠀⠀⠀⠘⢄⠁⠀⠀⠀⠈⠢⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠄⠰⡄⠀⠄⠈⠕⠀⠀⠀⠀⠀⣬⡤⠴⠓⠀⡐⠀⠀⠠⠑⠀⠀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠢⡑⢄⠂⠀⠀⢈⠢⡀⠢⡑⣌⠢⣀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣦⠀⠊⠁⢈⣣⡚⠀⠀⢂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠁⠁⢀⠀⠋⠜⠂⠈⠑⠶⠈⢢⡄⠀⡀⠤⣖⡱⠐⠀⠀⢀⠀⠰⡣⠀⠀⠨⠓⠁⠀⢀⢘⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⢌⠢⢃⠂⢄⠈⠀⠀⠒⠲⠦⣤⣤⠖⠋⠁⠰⠂⠁⠘⠛⠛⠃⠁⠀⠀⠁⠘⠁⡤⣌⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢡⣌⠓⡀⠄⠐⣊⣤⠗⠋⡙⠦⢰⢤⠂⡿⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⢀⡆⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⠜⢋⡡⠶⡛⠁⠀⠀⣚⣅⢀⡉⡆⠰⡏⠀⠀⠀⠉⠉⠉⠉⠙⢻⠀⢸⠀⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢁⠡⠂⠀⠀⠀⠢⡙⢯⠀⠈⠁⠀⠀⠀⢠⣘⣇⠀⠀⣠⣤⡀⠀⠀⠀⠌⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⣑⠄⠀⠀⠀⠰⡏⠀⠀⣹⣿⠁⠀⢙⣷⣤⣤⣤⣤⡞⢠⠍⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠊⠁⠈⠑⠒⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⠈⠀⠀⢄⠀⠀⠀⢀⣠⡂⣶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣁
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⣀⠀⠀⠀⣁⡀⠀⠀⠀⣄⠀⢀⣤⡶⠋⠀⠀⢔⠃⠀⠈⠀⠘⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⣷⠉⠉⠀⢙⡟⠁⠀⣼⠁⠀⠀⠀⡿⠀⠠⠤⠄⠈⣻⣧⠀⠀⠆⠀⠀⠀⠢⠀⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢀⠀⠀⣠⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣥⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣇⠀⠀⢰⠀⠀⠀⡄⠀⣠⡗⠀⠀⠀⠀⠀⡀⠀⠂⠀⠃⢀⣀⡀⠀⠀⠀⠈⠰⠰⠄⠠
⠀⠀⠀⠀⢰⠏⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⣠⣿⡀⢀⣰⣇⣴⠟⠁⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠈⠉⡅
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⣴⠀⡀⣠⠀⠀⢀⠠⡂⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠄⠀⠀⠈⠃⡏⠀⠅⠹⠆⠀⠈⣳⠀⠀⢤⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⡄⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣾⠂⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠄⡐⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠃⠀⡠⠄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠆⠀⠀⠀⣄⣠⣧⠈⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐
⠀⠀⠀⠀⢀⡀⢀⠀⣀⡌⠀⠀⠀⠀⠨⠟⠉⠊⠋⠉⢙⠈⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠘⠀⠀⠀⠀⢺⠁⠀⠁⠀⠀⠀⠀⠀⢀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠩⠀⢀⡀⠀⡀⠀⡠⢀⣠⠆⠀⠀⢠⠐⠄⢀⡠⠄⠀⡀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠠⡄⠀⠀⠀⠀⡄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠂⠀⠋⠀⠀⠀⠀⠉⠒⠦⠴⠾⠓⠁
⠀⠀⠀⠀⠀⠀⢀⡄⣂⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⠂⠀⠀⠻⠛⠃⠀⠠⠶⠶⠥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠅⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠉⠉⠱⠆⠀⠰⠄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠘⠶⠶⠦⠤⠀⠀⠀⠀⠂⠐⠂⠒⠂
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣒⠠⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡂
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠁⠀⠀⣠⠀⡅
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢠⡄⠀⠺⠷⠂⠀⠀⠀⠀⠀⠀⠀⠯⢀⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡀⠀⠀⠀⠀⢈⠴
⠀⠀⠀⢿⣿⣛⡛⡟⣟⣿⢻⢛⣿⢻⣛⣟⣿⣿⠀⠀⠈⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠍⠙⠉⠑⠛⠛⠻⠼⠇
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⠒⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠚⠋⠁

⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣷⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠦⠀⠄⠂⠀⠀⠀⠁⠀⠔⠃⠀⠀⠀⠀⠀⡤⠂⠤⡄
⠀⠀⠀⢀⣯⣤⣶⣶⣄⣄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢈⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠆⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠘⠛⠟⠁⠀⠸⡆⠀⢀⡤⢀⣀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⣉⠥⠐⠛⠁⠀⠈⠈⠩⠉⠀⠂⠀⢼⣆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⠁⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⣠⠶⠦
⠀⢀⠈⡠⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⠀⠀⠀⠄⠀⠀⡀⢀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠘⠁⠔⠀⠀⠀⠀⠀⣸⠏⠀⠀⢼⣒⠲⠞⢒⣠⣈⡀⣸⡏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠓⠀⠀⡆⡂
⠀⠀⠀⠰⠖⠀⠒⠀⠀⠞⠁⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠶⠞⠁⠀⠀⠚⠓⠚⠓⠀⠐⠐⠛⠓⠛⠂⠀⠀⠀⠀⠀⠄⠀⢀⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢡⠋⠔⠁⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⣠⢀⡀⢀⣠⣤⣤⣶⣶⡆⢾⠁
⠀⠀⠠⢀⠀⠀⠀⠀⠀⠄⠀⠠⢁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡟⠋⡽⠋⠹⠈⠈⢣⡼⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⣀⡀⠂⠀⠀⠤⣤⣤⣤⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠊⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠦⠄⠀⠀⠀⡀⠠⠂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⢀⠴⠶⢾⠲⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠀⠄⠈⠆
⠀⠀⠀⠀⠠⠀⠀⠀⠘⠄⠐⠓⠀⠂⠐⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠤⠤⠖⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡅
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠀⠠⠤⡄⣤⣤⣤⣠⣀⣤⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⢉⠽⠋⠀⠀⠀⠀⠈⠙⠮⣉⠓⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠁⠰⠒⠊⢁⣀⢤⡀⠀⠀⣀⡀⠀⠀⠈⠉⠒⠒⠲⠾⠷⠤⣀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢴⣉⣡⠤⠔⠃⠀⠀⡇⢸⠀⠀⠀⢀⡤⠚⣱⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⢠⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⢰⣉⡤⠊⠁⠀⡤⣄⠀⠸⡙⣄
⠀⠀⠀⢠⠃⣸⠀⠀⠀⡠⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆⠀⡇⠘⡆
⠀⠀⢠⠇⣠⠃⠀⠀⠀⠑⠒⠒⠒⠊⠁⠀⢠⡶⠶⠤⠤⠴⠶⢶⡀⠀⠀⠀⠀⠀⠀⢣⣈⠆⠈⠢⣸⡀
⠀⠀⣸⡔⠁⢰⠒⣄⠀⠀⠀⠀⠀⣀⡤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⢀⡏⠀⠀⠈⢣⠈⢆⠀⠀⠀⢘⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠀⠀⠀⢀⠴⠋⡹⠀⠀⢸⡆
⠀⠘⡇⠀⠀⠀⠀⠳⠜⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢰⣁⠴⠋⠀⠀⠀⣼
⠀⠀⡷⡄⠀⠀⠀⠀⣠⢄⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡆⠀⠀⠀⠀⠀⠀⢀⠎⢸
⠀⠀⢱⠘⡆⠀⠀⡰⠁⡜⠀⠀⢸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠵⠃⠀⠀⠀⢠⠎⡱⢸⠀⡎
⠀⠀⠸⡀⢸⠀⢰⣁⠞⠀⠀⠀⠀⠉⠓⠢⡀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣀⠀⠀⠀⢀⠎⢠⠃⠘⣄⠇
⠀⠀⠀⢣⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣤⠔⠒⠤⣔⣊⠏⠀⠘⢄⠙⠦⡀⠘⠶⠃⠀⠀⡿
⠀⠀⠀⠈⢿⠀⢀⣀⣀⠀⢀⣀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⡤⢄⠀⠀⠑⠦⡼⠂⠀⢀⣠⡴⠁
⠀⠀⠀⠀⠈⢇⠘⠤⠤⠤⠤⠚⠀⢀⡤⠊⡠⠊⠀⠀⠀⠀⠀⢸⡇⣸⠀⠀⠀⠀⠀⢀⠖⢉⡜⠁
⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⠀⠀⠘⠤⠊⠀⠀⡠⣄⡀⠀⠀⠸⡀⡿⠀⠀⠀⠀⠀⡞⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣉⠉⠒⢤⡀⠀⠈⠳⢤⣉⠓⢦⠀⠈⣀⣀⣀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠬⣓⣤⣀⠀⠈⠉⢁⡴⣋⣀⠤⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠙⠛⠛⠛⠶⠶⠒⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠀⠀⣰⠖⠀⠀⠀⢀⣀⠀⠀⠀⠀⠨⠭⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠠⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡀⠀⠀⠀⠀⠀⠀⡴⠞⠉⠉⠉⡉⠉⠉⠙⠲⣄⠈⠙⠁⠀⠀⡀⠙⠃⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠠⡇⠸⠃⠀⠐⠛⠁⠀⠸⣧⠖⠋⠉⠉⠉⠉⠉⠐⠢⣼⡇⠀⠀⠀⠐⠁⠀⠀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⣀⣀⣀⣠⡤⠴⠛⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⠀⠀⠞⠃⠀⠈⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠶⠃⠀⢀⡄⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⢻⡀⠀⠀⠀⠀⠀⠀⡴⠆⠀⠀⠙⠃⠀⢀⡀⠀⠀⠀⠘⠃⠀⠀⣤⠖⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⡇⣼⠿⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢀⠈⠀⡤⠴⡄⢸⡇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠉⠁⠀⠀⠈⠉⠹⣄⠀⡴⣦⠀⡴⠶⠛⠒⠋⢷⢰⡇⠀⠘⠾⢁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠒⠦⢤⣀⣀⠈⠛⠁⢸⢀⡇⠀⠀⠀⠀⠘⠛⠀⠠⠔⠊⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣄⣀⠉⠉⠉⠀⠘⠿⠃⠀⠈⠉⠁⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠶⠶⠶⠶⠶⠖⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠚⠛⠿⠿⢿⣶⠶⠶⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡀⠈⢻⣶⣄
⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⡄
⠀⠀⠀⣰⠏⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⣼⡟⠀⠀⠉⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⠀⢧
⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⢠⡟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣸⠀⠀⠘⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦
⣿
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰
⢸⣿⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿
⠘⡏⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⢳⣄⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠐⠁⠀⠀⡔
⠀⠀⠀⠹⣷⡄⠀⠉⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⢀⣤⠎
⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠁⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣦⣤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠦⢤⣤⣤⣤⣴⠾⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠀⠀⠀⠀⣀⠄⠤⠐⠊⠀⠀⠀⠀⠀⡘⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢹⠄⠀⠀⠀⠁⠀⢂⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠀⠀⠈⠁⠀⢀⡼⠙⢦⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠙⠦⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠑⠄
⠀⠀⠀⠀⠀⠠⠀⠀⠈⠃⠀⠀⠀⠀⠁⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⡀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠄⠀⣀⠀⠀⠁⠀⠀⠀⠁⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠄
⠀⠀⠀⠀⠘⠒⠒⠒⠋⠉⠁⠈⠀⠀⠀⠢⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢰⡠⠀⠀⠀⠠⠐⠀⠉⠀⠀⠀⠀⠑⠄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠹⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦
⠀⢀⠠⠂⠁⠀⠀⠀⢠⠁⠀⠀⠀⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠉⠉⠒⢤⡀⠀⠀⠈⣆⡀⠀⢠⣇⠀⠀⠀⠀⠲⢌⡀⠀⠀⠀⠄⠀⠀⠀⠀⠃⢀⠏⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣤⡦
⠀⠠⠤⠤⠤⣌⣷⣀⣠⣤⣤⣤⣤⠿⠤⠤⠤⠤⠤⠤⠐⡀⠀⠀⠠⡐⠀⠀⠠⠤⠘⣄⢀⠀⠄⠀⠊⠁⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠁⠀⠐⠒⠒
⠀⠀⠀⠀⠀⠀⠀⢠⣀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠓⠠⠀⠀⠀⣁⠀⠀⠀⠒⠀⣀⣀⣤⣴⣾⡿⠟⠉⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠐⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠈⠲⡄⠈⠱⠚⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠠⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠤⠄⠀⠒⠤⠤⠤⠖⠀⠘⣄⠀⠈⠑⠲⠄⠀⠀⠀⠀⢀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⢀⢀⣀⣀⡤⠀⠲⠤⠤⠤⠵⠆⠈⠑⠮⠥⢄⠀⠀⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⠀⠀⠀⠀⡀⠀⢆⣀⣀⣀⣀⣀⠀⠰⣈⣄⣀⣠⡄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣉⣭⣥⠀⠀⠈⠉⢩⣭⣤⣤⢀⠀⠶⠶⠶⡶⠶⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠤⠠⠊⠀⠀⠈⠀⠐⠀⠀⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠻⣦⠀⠐⢦⣀⠀⢸⠆⠀⢦⡀⠀⠈⠃⠀⠰⣦⡀⠀⠿⠀⠀⠻⠄⢸⠇⢀⣠⡄⠈⠙⠃⠀⠙⠂⠀⡀⠐⢦⣀⠀⠼⠁⠀⠙⢧⠠⡟⢀⣠⣤⠀⠉⠋⠀⠈⠃
⡀⠀⠀⢤⡀⠀⠉⠀⠈⡀⠀⠀⡉⠀⠀⠈⢳⠄⢈⡁⠀⠰⣦⡀⢸⠆⠰⣦⡈⢤⡀⠀⠀⠀⡿⢀⡾⠁⠹⡆⠀⣉⠀⠐⢷⡄⢸⡇⠀⢶⡀⢠⣄⠀⠀⣼⠃⣸
⠙⠶⠀⢈⠓⠀⢦⡀⠀⠙⠦⠠⡇⠀⠀⢠⡖⠀⠀⠻⠄⠀⠈⠁⠙⠀⠀⠈⣃⠀⠙⢂⣤⣤⠁⣀⠀⢀⡶⠀⠀⠙⠦⠀⠀⠉⠘⠃⠀⠀⢉⠀⠉⠓⠀⣀⣀⣉
⠀⠀⢠⠟⠀⠀⠀⠙⠀⡀⠀⠀⢠⡦⠀⠋⠀⠀⢰⡄⠀⢰⠀⠙⢦⡄⠀⢰⠋⠐⢷⡀⠀⠀⢴⠏⠀⠘⠁⠀⢠⡆⠀⢰⡆⠈⠳⡄⠀⠠⠟⠀⠲⣄⠈⠉⠁⠀⣰
⣼⠃⣀⠀⣀⢀⡇⠀⢸⡇⡀⠀⠟⠁⢠⡇⣰⠇⠸⠀⠀⠛⠰⣦⠀⠀⠀⠀⠠⣤⣤⡉⠀⡦⠀⠀⠀⣾⢀⡆⠸⠇⠀⠘⠃⢤⡀⠶⠦⠤⠀⣤⣤⣌⠀⣤⠀⠀⠁
⠁⠀⡟⣰⠏⠀⠁⠀⠀⠀⠹⣦⠀⢀⡈⠀⠋⠀⠰⠶⠶⠄⠀⠈⠃⠀⢸⡆⢠⡄⠀⣄⠘⠃⠀⠀⡈⠃⠈⠀⠀⠶⠶⠆⠀⠀⠃⠀⠀⣦⠀⣦⠀⢀⠀⠋⠘⠓⠶
⠀⣤⠀⠀⣤⣀⠉⠉⠉⠀⠀⣬⠀⠘⣧⡀⠀⠓⠶⢀⣀⡤⠀⠀⢷⡀⠀⠛⠀⠗⢸⣇⠀⠈⢷⠀⢻⡄⠀⠛⠲⠄⣀⣤⠀⠀⢻⡄⠀⠈⠀⠹⠀⡿⠀⠀⠀⠰⣦
⠀⠘⠃⠀⢀⡍⢣⠴⠞⠃⠀⠘⠇⢠⣄⠀⢀⡾⠀⠈⠁⢤⠀⠀⣤⣀⠀⠴⣤⣄⡀⠈⠛⠛⠃⠀⣤⡀⠀⣸⠇⠈⠉⢰⡄⠀⣤⣄⠀⠠⢤⣄⡀⠀⠀⠛⠒⠀⠙
⠀⠰⣦⡀⠾⠁⣀⠀⠘⣇⠀⠛⠶⡤⠉⠳⠊⠁⠀⣆⠀⠘⠃⠀⡀⠉⠁⢷⡀⠈⠁⠛⠓⠶⢀⡿⠈⠛⠂⠁⠀⢶⡀⠀⠗⠀⣀⠈⠁⢸⡆⠈⠁⠘⠛⠶⠂⣠⣤
⢰⡄⠀⢀⣀⠀⢹⡆⠀⠡⣤⡤⠀⠀⠀⠤⠶⠛⠀⠛⠀⠀⠉⢩⡅⠰⣦⡈⠁⠒⠲⠶⠀⠀⣈⣁⡀⢠⠶⠞⠃⠘⠃⠀⠈⠉⣍⠀⢦⣀⠉⠒⠲⠶⠆⠀⣀⣀⣀
⠘⠃⠈⠋⠁⠀⠈⠁⠀⠀⠀⣷⠀⠀⠀⠤⣤⡀⠀⠀⠰⢦⣄⠀⠷⠀⠈⠙⠐⠶⢦⡤⢤⡀⠉⠉⠁⠤⢤⣄⠀⠀⠀⠶⣤⡀⠻⠀⠀⠙⠁⠲⢦⣤⠤⣄⠉⠉⠉
⣤⡄⠙⠛⠒⢀⣀⣀⡙⠳⠄⢘⣀⣀⢀⣤⣄⡀⠶⠒⠒⠀⠈⢀⠘⠲⠶⢤⡀⠀⠀⣰⠀⠙⠂⠀⠀⣤⣄⣀⠰⠖⠒⠂⠈⠉⡀⠓⠶⠀⡀⠀⠀⢠⡇⠙⠃
⠀⠤⣤⣄⠀⠈⠉⠉⠀⠀⢀⡄⠈⠉⣁⠀⠀⡁⠀⢤⣤⣤⠀⣼⠀⢀⣀⡈⢿⠀⣀⣉⡀⠀⠐⠓⠒⠒⠀⢉⠀⢠⣤⣤⡄⣸⠇⢠⣀⣀⡹⠀⢀⣈⡀⠀⠀⠘⠒
⠀⠲⣄⠀⠤⣤⣀⠛⠓⠂⠘⠃⢠⡀⠙⠦⠀⠉⠛⠆⠀⣀⣀⣀⠀⣀⡈⠉⢤⡈⠉⠉⠁⠸⣧⠀⣸⠇⣀⠈⠙⠷⠀⠰⣆⠀⠀⢀⡈⠉⠡⣄⠈⠉⠁⠈⢷⡀
⠳⣄⠙⢀⡀⠀⠉⠀⠰⠤⠴⠄⠀⠙⠗⠀⠀⠛⠶⠆⠀⣉⠉⢉⠀⠉⢉⡋⠀⠙⠀⡀⠀⠀⢈⡀⠉⢠⡏⠀⢰⡄⠀⢀⡙⠂⡀⠈⠉⣉⠀⠘⠃⢀⠀⠀⠈⣁
⣀⡀⠀⠈⠉⠛⠃⠀⢰⡄⠀⣤⡈⠛⠶⠂⠀⠠⣄⠀⠀⡿⠀⠈⢳⡄⢸⠇⠀⣀⡈⠉⠛⠂⠈⠻⠆⠀⠀⠠⣤⡉⠀⢸⠃⠀⠹⣦⢀⡿⠀⣀⣀⡉⠛⠂⠀⠙⠦
⠉⠉⠁⣄⠀⠻⣤⠀⠘⠁⠀⠈⡓⠀⠀⠀⢷⡀⠈⠃⠀⠰⣄⠀⢰⡄⠠⣄⠈⢉⡁⠀⠀⠀⣿⠠⣄⠀⠳⣄⠈⠛⠀⠀⢦⡀⠀⡆⠀⣤⡈⢩⡉⠀⠀⠀⢸⠆
⠛⠛⠁⠈⠷⠀⣤⡀⠈⠻⣦⢀⡟⢰⡆⢀⡦⠀⠈⢳⡄⠀⠘⠃⠘⠀⠀⠙⠓⠈⠻⢂⣀⣀⡋⠀⠈⠓⣤⠀⠀⠹⣦⠀⠈⠓⠘⠃⠀⠈⠛⠀⠙⠖⣀⣀⡉
⡶⠀⢠⡿⠀⠀⠈⠳⠀⠀⢀⡀⠁⠟⠀⠟⠁⠀⢠⡄⠀⢠⠀⠳⣦⡀⠀⣠⡟⠠⣦⡉⠉⠉⢠⡟⠀⠘⠃⠀⠀⡄⠀⢠⡄⠐⢦⡀⠀⢀⡾⠀⢠⡄⠉⠉⢀⡾⠁
⠀⠀⠉⠀⠀⠀⡆⠀⢀⡇⠀⠙⠦⠀⢀⡶⢠⡆⢼⠇⠀⠸⢠⣄⠈⠁⠀⠀⢀⣀⣈⡛⠀⣄⠀⠀⠀⣴⠀⡤⠸⠇⠀⠸⠃⣤⡀⠁⢴⠀⠀⣀⣀⣙⠀⢠
⢦⡀⡾⢠⡟⠈⠃⠀⠈⠁⠳⣄⠀⠀⠸⠃⠛⠀⠠⣤⣤⡄⠀⠙⠇⠀⠠⡄⢠⡌⠉⡁⠰⠇⠀⠀⠀⠋⠘⠁⠀⢤⣤⡤⠀⠈⠳⠀⠘⠃⡀⠀⢉⡉⠀⠟⠀⠙⢦
⠀⣄⠀⠈⣀⠀⠘⠓⠂⠀⠀⣉⠀⠘⣧⠀⠀⠶⣤⠀⢀⣀⠀⠘⣷⠀⠀⠛⠀⠷⢰⡏⠀⠘⣧⠀⢿⡀⠀⠰⢦⡄⠀⣀⠀⠀⢲⡀⠀⠀⢻⡀⣾⠁⠀⠀⢳⡄⢰
⠀⠹⠦⠀⠉⡙⢂⣤⠴⠀⠀⠘⠦⠀⡈⠀⠀⣴⠀⠘⠋⢡⠀⠀⣈⠁⠀⢀⣄⡀⠈⠘⠳⠶⠎⠀⢀⠁⠀⣰⡇⠀⠋⢩⡀⠀⣀⡁⠀⢀⣌⣀⠀⠘⠛⠶⠊⠁⠿
⣤⠀⣦⡀⢼⠃⠀⠀⠘⣧⠀⠰⢤⡀⠙⠶⠈⠁⠀⣄⠀⠸⠇⠀⠉⠙⠂⢶⠈⠉⠀⠐⠶⠤⠀⠀⠈⠳⠄⠋⠀⣤⠀⠀⠗⠀⠈⠙⠓⢰⡄⠉⠁⠐⠶⠦⠄⣸⣆
⠀⠀⠈⠙⢂⠀⢸⡄⠀⠛⠀⠀⠀⠀⠀⢠⡤⠆⠀⠿⠀⠀⠛⢛⡛⠀⣄⠈⠃⠰⠦⡄⠀⠀⠀⠀⠀⢀⣤⠴⠂⠸⠇⠀⠘⠛⣛⠁⣤⡀⠛⠰⠶⢤⡄⠀⠀⠉
⣠⡴⠀⠚⠋⠀⠈⠁⠈⠛⠛⠁⢲⡀⠀⣀⣀⠀⠀⠀⠀⣤⡀⠈⣧⠀⠈⠓⠠⢤⣄⣀⢀⡀⠛⠉⠁⣠⣄⣀⠀⠀⠀⢤⣀⠀⢻⠀⢈⣛⠂⠠⣤⣀⢠⡄⠛⠋⠉
⠀⠀⠒⠶⠶⠄⠀⠀⠚⢶⡄⠀⡘⠃⠀⣀⡈⠁⡤⠶⠶⠀⠙⠀⠰⠦⡤⠀⢠⡀⠀⢰⡆⠓⠀⠀⠀⢀⣈⠉⠠⠴⠶⠀⠉⠃⠀⠀⢶⡉⠙⢡⡄⠀⠀⠙⠆⣀
⠀⠈⣧⠀⣀⣘⡋⠉⠁⠀⠀⣄⠉⠛⠀⠉⠉⠁⠀⢀⣀⣀⠀⣼⠃⢤⣤⣀⡀⠓⠀⠛⠀⣄⠀⠶⠶⠮⠉⠉⠀⢀⣀⣀⡀⢰⡟⣀⣈⠓⠀⠀⠻⠆⠐⣦⠀⠉⠉
⠛⠁⠈⠀⢩⣉⡁⠶⠶⠄⢸⠇⡀⠴⠶⠶⠂⠛⠳⡤⠈⢩⡄⠉⢀⣀⣀⢨⡀⠘⠛⠉⢰⡏⠸⣆⠀⠀⠀⠘⠳⢦⠀⣉⠁⠉⠀⠈⠉⠉⢡⡄⢠⣤⠤⠌⠀⣸
⠰⠒⠒⠀⠀⠈⠙⠀⠀⠰⠖⠛⠁⠰⣆⠀⠓⠶⠆⠀⠀⣤⠛⢀⡀⠉⢩⡍⠻⠦⢀⣀⡀⠀⢠⡄⠀⠉⠛⠛⠀⠀⠀⢩⡙⢋⣄⠈⠛⣗⠀⠙⠦⣀⣀⠀⠀⣅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣐⡄
⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠋⠄⠀⠀⠀⠐⠒⠀⠀⠱⠀⢀⠀⠸⣹⣿⢳⡆⠀⣾⣹⣿⡿⢿⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⣤⣦⣤⣦
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣀⣶⠀⠀⠀⢀⣀⣸⣀⣀⠀⠘⠤⠄⢻⠀⠀⠀⠀⠿⠯⠥⠥⠼⠃⠀⠀⢻⡯⣝⠛⠀⠀⠀⠷⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⣿⣦⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⢠⡶⢶⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠉⣿⣿⣿⣶⡆⠀⠠⣿⢟⣷⡿⠃⠀⠠⠒⠶⠶⠖⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠿⠦⠤⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠙⠟⠁
⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣿⣿⣿⡿⠀⠀⠀⠰⠶⣾⣶⣶⠀⠀⠰⠶⣾⣂⣀⠀⠀⠀⡀⢸⣦⠀⠀⠀⢀⣶⣿⡏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⢺⡿⢷⡦⠀⠀⠀⠰⠴⠿⠿⠿⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⢙⣍⣙⠃⠀⠀⠠⠤⠤⠤⠿⠧⠀⣰⣷⣾⣧⣤⡀⠀⢾⣦⡎⠱⠿⠆
⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠋
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢾⡥⠙⠀⠀⠀⠀⣴⣶⣶⡄⠀⠀⢰⣦⣀⣠⣞⠀⠀⢠⣀⣂⡄⠀⠀⠀⠀⣠⢀⣀⣀⠀⠀⠀⣬⠀⢀⡃⠀⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠈⠁⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⠙⠈⠁

⠀⠀⠀⢠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⠀⠀⠀⠀⢀⡄⢀⣀⣀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢀⣶⣠⡄⠀⠀⠀⡶⢶⠶⢤⡀
⠀⠀⠤⠀⠀⠙⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠘⠛⠛⠉⠛⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠛⠟⠀⠀⠀⢀⣷⣾⡶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠙⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠿⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⢋⣴⡄⠀⠀⢀⠀⠀⢸⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠀⡇⠘⠙⠀⠀⢰⠿⠇⣠⠏⢸⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠋⡞⠀⠀⠘⠤⣀⣀⣀⡠⠴⠊⠁⠀⡼⢀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⡇⠀⢧⠀⠀⣴⡦⠀⠀⠀⠀⠀⠀⢀⠜⢱⢿⠃⢠⠃⠑⡄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠳⡀⠀⠑⠦⣅⣁⡀⢀⣀⣀⠤⠚⠁⠀⠀⢀⡴⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢰⣶⠈⠲⠤⣀⣀⠀⠉⠉⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⢸⢹⡦⣄⡈⠀⠀⠀⠀⠈⠉⠉⠛⠒⢚⣻⠉⠉⠀⠀⠀⢀⣤⠞⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠈⡆⡇⢻⠙⡗⢲⢤⡤⣀⣀⣀⣀⣀⣀⣋⣠⠤⡴⢲⠋⡇⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⢱⢸⠀⡇⢸⠈⡇⢸⠀⡇⢸⢸⠀⡇⢸⠀⡇⣸⢸⠁⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡀⡇⢧⠘⡆⡇⢸⠀⡇⢸⢸⠀⡇⢸⢠⡇⡇⢸⢠⠇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⢇⢸⠀⡇⣇⢸⡄⡇⢸⢸⠀⡇⡏⢸⠀⡇⡏⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢹⢸⢸⠀⡇⢸⠘⡇⡇⢸⢸⢀⡇⡇⢸⢸⢁⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⡼⡌⡇⢹⢸⠀⡇⡇⣼⢸⢸⠀⡇⡞⢸⢸⢠⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣇⡇⡇⢸⠸⡀⡇⡇⡏⢸⢸⢰⠇⡇⡞⣼⣸⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠶⠧⠧⣧⣧⣼⣼⠼⠤⠷⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠿⡻⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⠉⣨⣴⣿⣿⣿⣿⣿⡟⢿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣷⣶⣼⣾⣷⡟⢩⣿⣿⣿⣛⢛⣭⣿⠎⣉⡠⢧⡃⠉⢻⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⣟⣟⣭⣭⣽⢻⣬⣷⣾⣿⣿⣿⣿⢿⣿⣿⣿⣭⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠞⠁⢁⣬⣶⣿⣿⣿⣿⣿⢿⣯⢿⣿⢿⣿⣿⣦⡥⢿⣿⢿⣿⣿⣿⡿⠒⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⡟⣼⣿⣿⢿⣹⣿⣟⣻⣿⢹⢿⣿⣿⣿⣿⢸⢻⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠷⣼⣿⣿⣿⣆⣻⣿⣿⣖⢿⡜⠘⠏⢹⣿⣿⣌⡀⠀⠹⠛
⠀⠀⠀⠀⠀⣼⠿⠟⠏⠈⢠⣿⣿⡝⠯⡧⢹⣿⣯⡓⠸⣷⠀⠀⠏⢿⢿⡓⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡿⡝⠆⠀⢸⣿⣦⡉⠀⣿⢀⣤⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠑⠈⠀⠀⠀⠻⠇⣩⣶⣿⡝⢿⣿⡿⣶⣤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⡿⣫⠵⢿⣹⡇⠀⠀⠹⡄⢹⠷⠄⠀⣀⣙⠿⣔⣲⣤⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠛⠛⣿⣷⣾⣖⣲⣤⣄⣿⡗⠛⠛⠓⠂⢲⣶⡼⠶⢦⣸⣷⣶⡿⠿⠿⠿⠛⠛⠛⠻
⠀⠀⠀⠀⠛⠉⠛⠒⠒⠒⠚⣁⣤⣤⠤⠤⠌⠈⠉⠒⠒⣿⡇⠉⢁⣷⡆⢠⠈⣡⠈⣋⣉⣣⠤⠴⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡾⢸⢻⢃⣮⢞⢥⠀⠠⠤⠭⣭⣿⣿⣯⣈⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣬⡏⢀⡟⣯⣈⣬⡃⢻⣾⣯⣀⠀⠐⣾⠿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠒⠊⠉⢛⡽⠋⠋⠀⠸⠘⠹⠉⠳⠈⠙⠛⠿⠿⢽⣿⠿⠿⠓⠒⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⣤⣴⣦⣤⡄⠀⠀⢐⠒⠒⠂⣀⡀⠀⠀⠀⢀⣰⣾⣾⡆⠀⠀⠀⠠⢤⠤⠀⠀⠀⠀⠀⣠⣾⣧⣠⠂
⠀⠘⠿⠿⠿⠿⠇⠀⠀⠘⠀⠀⠣⠀⠀⠀⠀⠻⡿⠛⠻⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣆⠀⠀⠀⠀⠄⣦⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣾
⠀⠀⢸⣿⣤⣄⠀⠀⠀⠀⠸⠿⠿⠿⠄⠀⠀⠀⠀⠚⠁⠟⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⣾⣶⣾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣾⣿⣄⠀⠀⠀⠀⠠⢨⡂⠀⠀⠀⠀⢸⡇⣶⣿⣿⠀⠀⠀⠀⠀⣠⡺⠟⠁
⠀⠀⢹⣿⣾⡿⠀⠀⠀⠀⢠⢏⠷⡄⠀⠀⠀⠈⣇⣻⣿⡟⠀⠀⠀⣠⣦⠟⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢠⣤⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣶⡦
⠀⠀⣾⣿⣿⣿⡄⠀⠀⠈⠉⠉⠉⠈⠁⠀⠀⠀⠀⡇⢥⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠖⠋⠉⢉⡩⠚⠉⠉⠉⠙⢻⣓⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣋⣀⣀⠤⠤⠒⠉⠀⠀⠀⠀⠀⡔⢉⠃⠙⢌⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⡟⠁⠀⣆⣀⣀⡆⠀⠀⢰⠳⡄⠀⠘⠄⠃⠀⠀⠈⢦⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⡼⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠡⠨⠀⠀⠀⠀⠀⣠⠤⡄⠈⠳⢄⠈⠻⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠁⣠⠃⠀⠀⠀⠀⡇⠀⠀⠀⣠⠶⠦⢤⣀⡀⠀⠀⠫⠤⠚⠁⠀⠀⢀⣽⡒⢿⡆
⠀⠀⠀⠀⠀⠀⣾⡤⢪⠵⠒⡰⠀⠀⠈⠒⢀⡴⠛⣁⡤⠴⠶⢤⣌⡉⠓⣄⠀⣴⢦⠀⠀⢸⠁⡇⠈⣿⡀
⠀⠀⠀⠀⠀⢠⣿⠀⠈⠊⠉⠀⠀⠀⠀⣼⠉⣠⠞⠁⠀⠀⠀⠀⠈⠻⣆⠹⡄⠘⣆⣃⠀⠘⠚⠀⡴⠉⣧
⠀⠀⠀⠀⠀⢾⠘⢦⡀⠀⡞⢢⠀⠀⢰⠇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢹⡀⠈⠁⡖⢢⠀⢸⠀⠀⣟
⠀⠀⠀⠀⠀⢸⡆⠀⠙⢆⠹⣄⠆⠀⠘⣧⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⡇⠀⠀⠹⣈⠀⠈⣇⠀⡿
⠀⠀⠀⠀⠀⢸⡅⠀⠀⢸⠀⡀⠀⠀⠀⣏⠀⠙⢦⡀⠀⠀⠀⠀⣀⣴⠏⣠⠞⠀⠀⢀⣀⣀⡀⠀⢇⢸⡇
⠀⠀⠀⠀⠀⠈⢷⡀⠀⡎⢸⢈⠀⠀⠀⠙⠲⢤⣀⣙⠛⠶⠶⠛⣩⡴⠞⠁⠀⠀⠀⠥⠤⠤⠃⢀⡞⣼⠁
⠀⠀⠀⠀⠀⠀⠘⣧⢸⠁⠸⠜⠀⠀⠀⠀⠀⡤⢄⠈⠙⠢⠴⠚⡥⠀⢒⡀⠀⠀⡤⢤⡀⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⢠⠤⠤⢦⡄⠑⢄⣣⠀⠀⠀⠀⠉⠉⢉⠀⠀⠀⠑⠲⣽⣆⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣗⠦⣈⠂⠐⠋⠀⠀⠀⠀⢀⣀⠤⡀⠀⢀⡇⡵⠀⡠⠒⠒⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣍⠉⠉⠉⠉⠑⢴⡥⠴⠚⠁⠀⠘⠧⠇⡰⠃⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣤⣄⣀⡙⠢⠤⣄⣀⣠⣴⣊⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⢀⡀⡠⠐⣲⣷⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠭⠁
⠀⠀⠀⠀⡴⠛⣻⣟⣿⣡⢾⠏⠚⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣷⠏⣯⡉⠀⠀⣠⣶⡒⣂⣤⣤⡀
⠀⠀⠀⠀⠀⢸⠟⠹⠓⡹⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⡙⢀⠀⠀⠋⠘⢷⠈⠋⠗
⠀⠀⠀⠀⢀⠎⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⡀⠀⠑⠠⡀
⠀⠀⠀⡠⠃⠀⡠⠋⠀⠤⠀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠑⠀⡀
⠀⠰⠞⢁⣴⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠗
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⢿⠿⡿⡿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⡀⠠⠤⠀⠀⠀⠠⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠈⠀⢀⡠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠉⠘⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⢀⣀⡤⠇⠀⠐⠀⠀⠐⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠐⠧⣄⣀
⢀⡴⠚⠉⠀⠀⠀⠂⣀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠑⠐⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠲⣄
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⡇⠀⠂⠀⠀⠀⢀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⡀⠀⠠⠂⠁⠀⠀⠀⠀⠀⣼
⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠉⠉⠛⠓⠶⠶⢤⣤⣤⣤⣀⣁⣂⣒⣀⣀⣒⣒⣀⣐⣒⣈⣉⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⢿⠏⣭⡙⠟⣭⡹⣿⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⢘⣏⠛⠖⣀⣉⣉⣙⡉⣣⣀⣨⣽⣎⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣵⡉⢉⠁⠉⣿⣯⣯⣽⣤⣝⡆⢠⣄⢠⠞⢹⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠧⢛⠈⢉⡤⠸⣿⠛⠛⠻⣿⠇⢀⡁⢬⠛⠻⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢱⠁⣤⠴⣶⡄⠬⠉⠉⣛⠉⢘⣊⠋⡠⡾⢇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣞⠦⡥⡀⢈⡰⠄⠺⠶⠈⢁⡘⠛⣡⣋⡖⣻⣵⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⢿⣾⠇⣿⡵⣜⠗⣰⡆⣿⣷⢐⡾⠿⣼⠟⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣭⣓⠻⢿⣮⣿⣵⡛⣿⣿⡼⢟⣻⡵⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣤⣿⣿⣿⠷⠞⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠊⠉⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠛⡿⡿⠿⣇⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠱⣴⣿⠏⡆⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢻⣶⢞⣽⡀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢷⣤⠦⢷⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠘⣜⣺⣧⣧⢀⣀⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⣿⣾⣿⣿⣿⣧⠀⠀⠀⠀⢀⣀⡠⠼⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠘⣿⣿⣿⣿⣿⠦⠔⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆⠀⠀⠀⢶⡆⠘⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠄⠀⠀⠠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠃⠀⣀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠛⠃⠀⣀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡀⠀⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠻⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣘⣷⣤⣧⣾⡻⠋⠉⠙⠛⠲⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣣⠅⠼⠘⠞⠒⠝⠸⠞⢀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣬⣭⣻⣿⣿⠵⠾⠿⠿⠒⠶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⢾⡿⢛⡿⣯⣛⠟⠻⣶⣤⡈⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠞⣵⣿⡆⣇⠙⣦⡀⠈⠙⠆⠙⠅⠀⠀⠀⠀⣠⣼⣷⡛⣚⠷⠦⠀⢀⣳⣠⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠋⡄⣤⢻⣿⡄⠹⢶⣽⣿⣄⠀⠀⠀⠐⡀⠀⠐⣣⢞⢿⣷⣬⠈⢳⢤⣴⠞⣻⣿⠛⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢡⠻⠘⣿⡇⠘⠀⢹⡷⠙⠐⠆⠀⠀⠀⠀⣸⠧⠈⠈⣿⣟⢷⡀⢸⡏⢺⡏⢌⠓⣆⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⠀⠀⢠⡀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠸⣿⡁⠀⠀⠀⠁⠂⠰⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠦⣄⡀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠈⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠠⡀⠀⠈⠙⢦⣀⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠤⠬⠯⠿⠿⠤⠴⠖⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣆⣤⡀⠀⠀⠹⣿⡄⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢈⣉⡀⣠⣤⠴⠚⠋⠀⠀⢠⢴⠖⠈⠀⠀⠤⠐⠂⠙⢿⣯⢦⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⡤⠶⠶⠶⣶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠈⠐⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣯⣷⡀⠀⠈⠳
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡤⠤⢤⠤⣤⡤⠂⢀⡤⠖⠲⠿⠟⠛⠋⢀⡾⠀⠀⠀⠀⠀⢀⣤⠖⠒⠈⠁⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠶⢞⣋⣩⣥⣤⡤⣶⢶⡶⠶⢶⣦⣤⣄⣀⣀⣀⣠⣤⡤⡶⠚⠋⠀⠀⣀⣀⠤⠚⢉⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠠⠤⠤⠤⠤⠴⠶⠖⠒⠒⠂⠉⠉⠉⠉⠀⣀⣀⡤⠨⠀⠀⠴⠒⠊⠁⢀⡀⠄⠚⠁⠈⠀⠀⢀⣜⠛⣋⣲⣆⣤⡤⠟⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢉⣉⠥⠀⠀⠀⠀⣀⠠⠄⠀⠲⠉⠀⠀⠀⠀⠀⣠⡖⣥⣦⡾⢸⡞⠈⢙⡓⠢⠄⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠒⠒⠒⠒⠚⠉⠀⣠⠴⠒⠋⠉⠀⠐⠂⠀⠠⠂⠀⢀⠤⠐⠛⠱⠋⠽⠇⠀⠙⠋⠙⠛⠑⠃⠀⠀⠀⠀⠈⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⢀⠊⠀⠐⠦⡀⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣈⠉⠙⠓⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠤⠀⠀⠛⡀⢠⠇⠀⠀⠂⠂⠼⠇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠁⠀⠈⠉⠙⠓⠲⠶⢤⣄⣀⡉⠉⠛⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠤⠤
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡠⠁⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠱⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⢤
⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢸⡤⣤⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⡈⠛⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠋⠔⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠈⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠐⠀⢹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂⠀⠀⠀⠀⠠⠀⡀⠀⠀⠓⠐⠁⠀⠀⠀⠀⠀⠒⡌⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣴⠟⢻⣧⢀⣼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠙⠷⠾⣿⡾⠃⢠⡞⢻⡗⣴⠟⠷⣰⠟⣿⠇⣺⠟⢃⡾⢠⡞⢻⡆⣸⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⠟⠁⠀⠘⠛⠛⠓⠙⠓⠒⠛⠛⠛⠒⠛⠓⠘⠓⠚⠛⠛⠑⠋⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠄⠠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡂⠀⠀⣀⠊⠀⠈⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠁⠀⠁⠀⠀⠇⠓⠠⠄⠠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠦⠀⠀⠰⠀⠀⠀⠒⠀⢤⡄
⠀⠀⠀⠀⠀⣀⣉⡂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠟⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠈⠇
⠀⠀⠀⠘⠿⠟⠛⠃⠐⠿⡷⠆⠀⠀⠁⠀⠀⡀⠀⠀⠀⢀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⣴⣶⣄⡀⠀⠀⢀⣠⣶⣦⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⢀
⠀⠀⠈⠛⠿⠋⠀⠀⠈⠿⠟⠁⠠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠸⣿⠟⠃⣠⣈
⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠲⢠⡶⠀⡰⠠⡀⠀⠀⠠⠼⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡀⢸⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠉⠀⠐⠚⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⠃⠀⠠⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠚⠉⠒⡶⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠊⠲⢶⣦⣤⣤⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⣾⣷⡿⠂⠀⠀⠀⠉⠋⠛⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣠⡴⣦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢸⡿⠾⠛⠀⠀⠐⠛⠛⠆⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠘⠀⠀⠀⠘⢋⡥⠀⠀⡤⠤⢬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⢀⡓⠂⠀⠀⠀⠀⠀⠀⠀⠇⠛⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡰⢊⡴⠋⢋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣕⡀⠙⣿⣷⡀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⠀⠀⠀⡠⠋⠄⠚⠡⠀⡀⠀⠀⠘⣰⠋⠀⣀⠤⠛⠖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣴⣄⠈⣽⠆⠰⠋⠰⣄⠀⠀⠀⠀⢾⣿⠀⣼⠀⠀⠀⠀⡀⠀⢀⢈⠂⠀⠉⠜⣁⠠⠳⠁⠐⡛⢁⣠⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣛⣿⡏⢻⣷⣾⡇⠀⠙⠳⢦⣀⠀⠈⢁⣾⠟⣄⠀⠀⡰⠐⢀⠏⢂⠀⣰⣿⣿⣶⣶⣤⣄⣀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⣰⣿⣿⣾⣿⣿⣾⣡⠇⡰⠁⡠⢎⢀⠂⣼⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠛⠻⣁⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣷⣿⠟⡀⡀⡰⠁⢀⡆⣼⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⠴⠋⣷⡄⠀⠈⠲⣄⠀⠀⠀⠀⠅⠀⠀⡘⢿⠉⠁⡘⡀⡐⠁⢀⡼⣽⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⢸⡇⠉⠾⠋⢄⠁⡀⠈⢆⠈⢆⠀⠀⠀⠉⠒⣤⣄⣀⡀⢸⡆⢀⢀⡜⠐⠀⡰⠀⢜⣿⣿⣿⣿⣿⡟⠳⣤⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣼⣿⠿⣷⣤⠺⢗⠈⠀⠈⢢⢛⠳⣤⣀⣴⠂⠀⢠⣽⠟⠎⠀⢾⠞⠀⠀⠀⢀⢎⣾⣿⣿⣿⣟⠃⠑⣄⢹⣿⡿⠟
⠀⠀⠀⠀⠀⢹⣤⣟⢾⣿⡄⠀⠱⣿⡄⠀⢡⢢⣌⠉⠑⢀⣤⡀⠉⣤⣶⣄⠸⠙⠓⢶⡀⢮⣿⣿⣿⣿⣿⡏⠰⢂⣇⠜⠉
⠀⠀⠀⠀⠀⣼⢿⣿⠀⠈⢳⣶⠄⢨⠙⣆⠀⢳⡉⣠⠖⡋⠉⠁⠀⠐⠀⢤⣅⠄⣄⡸⠎⢟⡛⠛⣻⠛⡛⠃⠞⢈⠃⠀⠀⠀⠀⠀⣤⣠
⠀⠀⠀⠀⠀⢿⣮⡙⢠⣶⠁⠁⠪⣾⣄⠙⡆⠀⠣⠉⠀⠀⠀⠀⢀⡴⠋⠀⢱⣶⠀⡇⠀⠘⠿⣴⡏⠠⠀⠀⠀⠎⠀⠀⠀⠀⢀⣼⣫⡕⣡⠄
⠀⠀⠀⠀⠀⠀⠻⣿⠆⠉⠂⣠⣬⠟⠣⣠⣧⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠤⠄⠊⡉⢿⣂⣀⣽⠁⠸⠇⠀⡜⠀⠀⠀⠀⠀⠙⠻⢛⠄⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡖⠒⠴⣍⢻⣦⣄⠉⢻⠀⢀⡠⠖⠉⠁⠀⢀⣴⡧⠒⢲⠀⡀⢰⡏⠁⠀⢠⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⢋⡀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⣿⣫⠴⠚⠋⠁⠀⠀⣠⠞⠉⠉⠁⠀⠀⠘⣶⡇⢸⣇⢀⡴⠋⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⢂⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢠⠐⡇⢸⠉⢻⣬⣤⣷⠄⢠⡏⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠚⣡⠖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢠⠃⠀⢣⢬⠔⠁⢃⠀⠁⢠⠞⠀⠀⠀⠀⠀⢿⣤⡼⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢀⣀⣤⠶⣿⠀⠀⣴⠃⠀⠀⠀⠀⠀⠸⡄⠀⠸⣀⡀⠀⣼⣀⣴⠋⠀⠀⠀⠀⠀⠀⡼⠀⢠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠋⣉⠀⠀⢿⡗⠐⠛⠓⠒⠲⠦⠂⠀⠂⠙⡶⠦⢭⡦⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⠆⢸⣷⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡇⠀⣿⡻⡁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠓⠊⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢧⣤⡴⠛⠁⢀⡼⠃⠀⠀⠀⠀⠉⠉⠉⠁⡀⢀⠐⢀⣇⣗⣱⡀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣶⣆⣀⣉⣀⡈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠸⣧⣄⠀⣤⣤⡄⠤⠤⣤⣄⣀⢤⡤⠤⠤⠤⠤⢀⠆⠀⣿⡇⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠉⠉⠉⠒⠋⠀⠀⠀⠀⠀⠁⠀⠀⠰⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⡆⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⠀⠀⣿⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⣿⣄⠈⠻⠆⠀⠀⠀⠀⠀⠀⣠⡎⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠈⢻⡗⣲⣄⠒⠒⠒⠒⠲⣟⠋⠀⠀⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠰⠂⢘⠛⠉⣩⠿⠿⢶⣶⣦⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣚⠁⠀⢀⡴⠖⠪⠤⠊⠁⠀⠀⠀⠈⢧⠈⠓⢮⡑⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣰⠋⠀⢩⠛⠋⠀⠀⠀⢄⠀⠀⠀⠀⢸⠀⠘⢥⣀⣀⣙⢦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠂⠀⠀⠈⡆⠹⣆⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀⠋⠉⠀⠀⡀⠂⠀⠀⠘⠂⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⣠⠤⣀⠙⣆⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢰⠁⢳⠀⡀⠀⠀⠁⠀⠀⠀⢀⠴⠫⠗⢠⢤⡠⡀⠀⠀⢀⡠⠀⠀⠈⠀⠈⢳⣸⡆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠇⠀⡟⠀⠀⠤⠂⠀⠀⢀⡔⠃⠀⠀⠀⢸⢰⠀⢱⡅⠀⠀⠀⠀⠀⢀⠀⠀⠀⡟⢷⠀⠃
⠀⠀⠀⠀⠀⠀⢠⢠⡌⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⠀⠀⠀⠘⠈⡆⢨⠈⢦⣀⠀⠀⠀⠔⠁⣠⠞⠁⢶⡀⠠
⠀⠀⠀⠀⠀⠀⢸⠸⡄⠀⠀⠑⠀⠀⠀⠀⣏⠀⠈⠒⣠⠀⠀⠀⠬⣸⡆⣰⠋⠀⠀⠀⠀⡞⠁⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡆⠸⠆⠙⠈⠀⠀⢀⠀⠑⡀⠀⠠⠌⠀⠠⠤⠀⠀⣽⠅⠀⡖⠀⠀⠀⠙⢦⠀⢀⣿⡄⡄
⠀⠀⠀⠀⠀⠀⠀⢱⠈⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠤⣀⣀⣠⣠⠤⡼⠃⠀⠀⠀⠀⠀⠀⠄⠀⠁⣘⡟⡘
⠀⠀⠀⠀⠀⠀⠀⠈⢆⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠒⠈⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⠔⡹⠝⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⠳⣦⠀⠀⠀⡀⠀⠘⠀⠀⠀⠄⠁⠀⠀⢀⡠⣀⠀⢀⡞⠩⣇⠜⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⢆⠐⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠑⢾⠀⠐⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⡤⠒⢦⡀⠀⠀⠀⠂⣀⡤⠔⠷⣄⣀⢀⢀⣈⣈⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠍⠳⠶⠶⠊⣤⡤⠤⠶⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠳⢸⢆⡇⡖⠀⡈⠸⠀⠀⡇⡷⢰⢸⢘⢐⠆⠀⡁⠀⡁⢸⢸⠐⡆⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠄⠸⠀⠀⠀⠇⠐⠢⠄⠄⠇⠘⠸⠨⠸⠀⠆⠄⠀⠄⠠⠌⠐⠃


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠶⠶⠾⠿⠿⠛⠛⠛⠻⠿⠿⠶⠶⢶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠟⠛⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠠⣦⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠈⠻⢷⣄
⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⣦⣄⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣰⡿⠹⣷⡀
⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⣶⠀⠀⠈⠛⠀⠀⠀⠀⠀⣼⠋⠀⠀⠘⣿⡄
⢰⡿⢻⣦⣀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠈⢿⣮⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠸⣷
⣾⡇⠀⠈⠙⠛⠿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⠿⠛⠁⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⢸
⢻⡇⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠛⠁⠀⠀⠀⠀⣿
⠘⣷⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠹⣧⡀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠙⢷⣄⠀⠀⠀⠀⠙⢷⣦⣄⡀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠁⠀⠈⠉⠛⠷⣶⣤⣄⣀⣀⡀⠀⠀⢀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠾⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣟⣛⣛⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⢤⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⣀⣠⣄⣀⣀⡀⠈⠙⣦⠡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⡀⠀⢂⣠⣠⣤⣐⣲⣤⣤⣤⣤⣀⣀⣘⣷⣿⢉⣀⠀⠈⠩⠙⣳⣶⠋⠄⢂⣠⣤⠒⣠⣐⡠⠠⠀⠄⣒⣄⣂⢠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢐⠪⣢⠶⠶⠶⠿⠛⣫⣩⠟⠛⣲⣶⡿⠟⠉⠉⠭⠿⠿⠍⠩⠭⠭⠿⠿⠿⠿⣿⡿⠟⠓⣲⡮⠽⠟⢛⣛⣛⣛⠛⠫⢉⢳⡄
⠀⠀⠀⠀⠀⠀⢈⠰⣿⡀⡀⢀⡠⠞⠁⠀⠀⢸⡏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡏⠀⠀⢎⠀⠀⠀⡀⠈⡆⣯⠀⡗⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠁⣿⠀⡴⠁⠀⠀⠀⠇⠀⠘⢣⠀⠀⠀⢈⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⡇⠀⢀⠀⠙⢆⡒⠋⠀⢷⡇⢰⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠰⡅⢻⠜⠁⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⠀⢀⠈⠁⠁⡰⠖⠀⢀⠀⠀⠀⠀⠀⠟⠉⠉⢺⡇⠀⠀⠀⠀⠀⠙⢄⣀⡜⣇⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠐⠠⢺⠀⡀⠂⠁⠂⠘⡀⠀⡠⠇⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠘⢄⠀⡠⠛⠑⠀⠀⠀⠀⠠⡐⠦⠀⠇⢻⠀⣿⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠁⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠐⡝⠀⠀⠀⢸⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⢠⡀⣿⠀⠀⠁⣤⠆⠀⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠒⠚⠈⠀⠀⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣿⠀⠀⠈⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⡀⠀⠀⣿⠠⠁
⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠄⠠⠤⠄⡀⢨⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢨⡀⠙⠊⠀⠂⠄⡀⡇⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⣿⠀⠁⠀⠀⠂⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠅⣁⣀⠌⠙⢺⠃⠃⡆⠀⡟⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣄⠀⠀⠀⠠⡀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠆⠀⠀⠀⠰⠇⠀⠀⠓⠠⠀⠈⠁⠒⢿⠆⢀⠀⡇⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠁⣯⠀⠣⡀⠀⣠⠶⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠠⠀⠀⠀⡀⠀⠀⡀⠀⢡⠀⠀⠀⠀⢀⢸⠀⡇⠈⡇⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡀⣿⢀⣰⠈⢦⡀⠀⠸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣠⡄⢣⡘⠆⠄⠈⡇⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠻⠶⠶⠶⣶⡿⠶⠶⣤⢤⣼⠷⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⠀⠀⡠⠂⢐⡦⠀⣠⣿⡶⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠻⠁⠦⠄⠐⠒⠂⠉⠀⠘⠆⠠⠁⠀⠈⠉⠉⠉⠩⠉⠙⠛⠛⠛⠛⠙⠉⠝⠓⠲⠶⠟⠛⠓⠿⠤⣤⣤⣤⡼⠋⠂⠈⠗⠀⡇
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠤⠀⠀⠠⠬⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⡿⠛⠉⠀⠀⠀⠉⠛⢿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠿⠿⠿⠛⢉⣀⣤⡀⠀⠀⣀⡀⠀⠀⠈⠛⠛⠿⠿⠿⠷⢤⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢴⣿⣿⠿⠟⠃⠀⠀⣿⣿⠀⠀⠀⢀⣤⣾⣷⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢰⣿⠿⠋⠁⠀⣤⣄⠀⢸⣿⣄
⠀⠀⠀⢠⣿⣿⠀⠀⠀⣠⣤⣤⣤⣶⡄⠀⠀⢀⠀⠀⠛⠃⢀⣀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠘⣿⣿⡆
⠀⠀⢠⣿⣿⠏⠀⠀⠀⠙⠛⠛⠛⠉⠁⠀⣾⠿⠟⠓⠒⠚⠿⢿⣦⠀⠀⠀⠀⠀⠀⢻⣿⠆⠙⢿⣿⡀
⠀⠀⣸⡿⠃⢰⣶⣄⠀⠀⠀⠀⣠⣴⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⡟⠀⠀⠈⢻⣿⣆⠀⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⢀⣴⣿⡿⠀⠀⢸
⠀⠀⣇⠀⠀⠀⠀⠛⠟⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣿⠟⠋⠀⠀⢀⣾
⠀⠀⣿⣆⠀⠀⠀⠀⣠⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⢹⣿⣦⠀⠀⣰⣿⡟⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠁⠀⠀⢠⣿⡷⣿⣿⡏
⠀⠀⠸⣿⣿⡆⢸⣿⠟⠀⠀⠀⠙⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⣀⡀⠀⠀⢀⣾⣿⠃⢸⣿⠇
⠀⠀⠀⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣤⣤⠤⣤⣴⣾⣿⠁⠘⢿⣿⣦⡀⠘⠛⠃⠀⠀⡟
⠀⠀⠀⠀⢻⡄⢀⣀⣀⣀⣤⣀⠀⠀⠀⠀⣨⣭⠉⠀⠀⠀⠉⠉⣵⣄⠀⠀⠙⠻⠿⠀⠀⣀⣤⡞⠁
⠀⠀⠀⠀⠀⢳⡘⠿⠿⠿⠛⠛⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⢠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠙⠢⣤⣄⣀⣀⡀⠘⠿⠋⠀⠀⣤⣄⡀⠀⠀⠸⣿⡇⠀⠀⠀⠀⢀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣦⣄⠀⠀⠛⠿⣿⣷⣦⠀⢀⣠⣤⣀⣀⣀⠼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣦⣄⣀⡀⠉⣡⣴⣿⡿⠿⠛⠋⠁
⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠁⢒⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⠀⢠⣎⣀⠀⠀⠀⠀⢀⠔⠉⠀⢛⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⠀⢝⠂⠀⠀⠈⠙⢲⡀⢠⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡉⠉⠋⠀⠀⠀⠀⠀⠀⢱⣇⣀⡀⠀⠀⠀⡇⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⡀⡰⣖⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠀⠀⠀⠀⠆⡄⠀⠀⠀⠀⠀⣸⠋⠑⢄⠀⡇⢀⠀⡾⠀⠀⠀⠀⠀⠀⠠⣀⡳⠜⠀⡏⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠈⢀⡀⠀⠀⠀⣰⠃⠀⠀⠈⠷⠋⡠⠈⠴⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣴⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⢤⠠⠒⣓⡄⢠⢀⠤⢡⠒⠢⢴⠃⠀⡠⠀⢀⡠⠊⡱⠷⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠈⠖⢥⠀⠀⢠⠃⠀⠀⠀⠑⡏⣀⠴⢋⠔⠙⠦⠤⠊⢀⡀⠀⠀⠀⢀⠔⢲⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢽⡖⠨⡷⠔⠂⠄⠀⠀⡾⠀⠀⠀⢀⣠⢼⡥⢚⡁⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⡠⠎⢳⣀⣠⠗⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠃⠠⡀⠔⠀⠐⢦⣀⣇⠀⡠⠞⠉⠀⢀⡟⡉⢁⡀⠀⠀⠒⠀⠀⠀⠢⡀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⡿⠀⠄⠉⠋⠀⠑⠀⡇⠀⣏⠀⠀⠀⠀⣼⠁⡀⠄⠩⡒⠀⠒⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠒
⠀⢀⡗⡁⠀⠀⠀⣀⣂⣀⣃⣤⣼⢥⣿⣿⣱⠁⣠⣨⣁⡖⡀⣉⣞⣀⣑⣤⡠⠾⠷⠏⠘⠚⣢⠈⠂⠀⢄⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣳⡜⢻⡄
⠀⡌⠁⠁⢠⠀⣀⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠙⡖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠉⠂
⠀⡃⠀⠘⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠇
⠀⠿⠲⢶⠃⠀⠀⣠⠔⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏
⡴⠛⠈⠚⠲⡀⢸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣠⣇⣀
⡇⠀⠀⠀⠀⡇⠀⠀⢠⠁⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⡆⡴⠁⠀⠀⠈⡷⠀⠀⠀⠀⠀⢈⠀⠀⠃
⠘⠠⠄⣀⠴⣧⠃⠀⠘⢀⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⡠⠀⢁⢧⠀⠀⠀⡰⠃⠰⠄⠉⢹⡂⠘⠆⠁⢀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠙⡤⠤⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠈⡏⠍⠉⣉⢱⠀⡔⠀⠀⠠⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠈⠀⠀⠈⠁

⠀⠑⡄⠀⣀⠀⠀⡇⡇⠀⠀⡞⠀⡼⠀⠘⠀⠀⠀⠀⠁⠀⠀⠃⠀⠸⡄⠀⢹⠀⠀⢸⡀⠀⡀⠀⠀⡴⠉
⠀⣠⠇⠀⠀⠀⠀⡰⡇⠀⢠⡇⠀⡇⡄⠀⠀⠲⠀⠀⠀⠀⠀⢠⡀⠀⡇⠀⡞⠀⠀⢸⠉⢢⠀⠀⠀⢣⡀
⠉⠁⠂⠀⠀⠈⣸⣳⠁⠀⠀⣏⠀⠹⡠⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡼⠀⣰⠃⠀⠀⠀⢧⢋⠀⠀⠠⠀⠈
⠀⠠⠀⠀⣰⢋⠔⠁⠀⢀⡀⠈⢦⡀⠙⠮⣓⣒⣀⣀⣀⣀⣀⠤⠞⣁⡳⠃⠀⠀⠀⠀⠸⣜⢆
⣀⣀⡠⠔⠋⠁⠀⠀⠀⠈⠁⠀⠀⠈⠒⠤⢄⣄⣠⣀⣀⣀⣀⡠⠔⠋⠀⠀⠀⠀⠂⠀⠀⠈⠓⠭⣒⣀⠲
⠀⢀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁
⠉⠑⠖⠤⣄⠀⠀⠀⠀⠂⠀⠀⠀⢀⡤⠒⠩⠉⠟⠯⠟⢗⣲⣤⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡤⠖⠒⠋
⠉⠑⢒⠒⢄⠙⣦⡀⠀⠀⠀⠀⡔⠁⠀⡄⠀⠀⢀⠄⠀⠀⠄⠀⠀⠈⡳⡄⠀⠀⠀⠀⢀⣴⢁⡴⢚⠭⠟
⠀⠀⠁⠀⠀⣇⠀⠱⡀⠀⠀⡼⠀⠀⠀⠀⠀⢀⡠⠤⢤⡀⠀⠀⠀⠀⢱⡸⡀⠀⠀⢀⠎⢀⠎
⠃⠀⠀⠂⠀⢸⡀⠀⢣⠀⠀⢱⣄⠠⠆⠀⠀⡏⠀⠀⠀⠘⡄⠀⠤⠀⠀⢇⡇⠀⠀⡞⠀⡼⠀⠒
⠀⠀⠀⢀⠀⠀⡇⠀⡆⠀⠀⢸⠈⡧⠀⠀⠀⠳⣄⣀⣀⠜⡁⠀⠀⣀⢠⢾⠀⠀⢰⠃⠀⡇⡀⠀⠀⠦
⠐⠆⠀⠈⠁⡸⠀⣰⠃⠀⠀⠈⡟⢅⠀⠀⠒⠀⠀⠀⠀⠀⠁⠀⠀⣀⢜⡞⠀⠀⠘⡏⠀⢳⢡
⣀⣀⣀⣠⠜⡁⣲⠃⠀⠀⠀⠀⠑⢌⠢⣀⠀⠀⠀⠀⠀⠐⠀⢀⣞⡴⠊⠀⠀⠀⠀⠙⢄⠀⠳⢝⡢⠤
⠀⠀⣀⣀⠤⠛⠁⠀⠀⠐⠃⠀⠀⠀⠉⠒⠥⢤⣓⣠⠤⠤⠒⠋⠁⠀⠀⠀⠀⢠⠀⠀⠀⠑⠢⢄⣈⡍⢉
⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣲⣶⠤⣄⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⠴⠒⢉⣉⣉⣉⠉⠗⠒⢤⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡤⢖⡒⡿
⠈⠈⢉⠉⠉⠙⠢⡀⠀⠀⠀⠀⢀⠾⢁⠔⠫⠕⠒⠚⠉⠉⣉⠑⢢⡈⣳⡀⠀⠀⠁⠀⢠⠞⠁⢀
⣀⠀⠈⠀⠀⠀⢱⡹⡄⠀⠀⢠⠏⢀⠎⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢣⠀⠘⡄⠀⠀⢠⠃⠀⠀⠈⠀⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠉⠈⠛⠁⠉⠋⢹⠙⠋⠈⠙⠉⠉⠁⠀⠀⢸⣿⣿⣿⣷⣾⣷⣴⣿⣶⣴⣶⣦⣾⣷⣦⣾⣷⣿⣿⣿
⠀⠀⠀⠀⢠⣦⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠙⢿⣧⣿⢿⣦⣄⣬⡴⠶⢦⣄⣀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡉⠉⢠⣈⠉⠻⠟⣋⣉⣛⠿⢿⣿⣿⣿⣿
⠀⠀⢀⣀⣞⣀⣉⣛⣾⣭⣵⣞⣛⣛⣾⣯⡀⠀⠀⠀⢸⣿⣿⣿⠿⢱⠿⡶⣬⠑⠒⠋⣡⢶⣬⡉⠚⢿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⢶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢴⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⢶⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢀⠀⠀⠀⠀⠀⠀⢨⡍⠛⠿⠿⠿⠻⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡿⢂⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠅⢀⡇⠀⠀⠀⠀⢀⡠⢌⣇⣈⣉⣹⡿⡞⠛⠛⠉⠉⠉⠉⣉⣉⣃⣪⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣼⠃⢀⣠⠴⣚⣭⣶⡽⠿⣤⣼⣿⡟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢼⣯⣥⣴⣿⠿⠛⠁⠀⢬⠟⠰⠿⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠙⠛⠉⠀⠀⠀⠀⠀⢲⣦⢀⢺⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⢸⡇⢠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣷⢨⣽⠟⠩⣙⠿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢔⣒⠭⠺⣿⡆⠸⣿⢸⠏⠀⣈⣡⣬⠿⠞⠛⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣺⠭⢒⠉⢀⠀⠀⠛⠣⠄⣻⣤⡶⠟⠋⠉⢀⡠⠤⠔⢢⣤⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠿⠿⠿⣿⣋⣴⡋⡤⢒⠭⣂⡥⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠈⠀⠀⠀⢀⣤⣽⣿⡿⠿⣿⣶⣦⣤⣀
⠀⠀⠀⣠⣶⠿⠛⣩⣴⣶⠿⠿⠿⠱⣿⣿⣿⣧⣞⡵⣊⣡⣴⠿⠋⢁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠖⠈⠀⠀⠉⠛⠿⣷⠶⣭⣿⣿⣿⣿⣷⣦⣄⡀
⠀⣠⣾⠟⠁⡴⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⣿⣏⡀⢀⣠⣄⠀⠀⠀⠀⢠⣴⣦⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠬⣙⠛⢿⣿⢷⣮⡙⠳⣄⡀
⢰⡿⠁⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⢰⣯⡻⣿⣷⣶⣶⡀⢚⣽⠏⣹⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠬⣙⠿⣿⣶⡈⠙⣆
⠈⠳⢾⣋⠻⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⢹⣿⣿⡿⣷⣌⠛⢛⡡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠒⠚
⠀⠀⠀⠈⠓⢦⣉⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠛⢦⣍⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣍⡻⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⡿⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣭⡻⢶⣄⡀⠀⠀⠀⢸⡇⣿⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣽⡻⣶⣄⣀⣾⣇⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡉⣿⢹⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠌⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣴⣿⣿⣿⣟⣿⣿⣆
⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢼⣿⣷⡏⠁⠈⠹⣿⣿⡄
⠀⠀⠀⠀⡀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⢿⣿⣧⣄⣠⣼⣿⣿⠁
⠀⠀⠀⠀⠁⠀⠀⠤⣀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⠀⣀⣀⠄⠀⠀⠀⠀⠀⠿⢷⡿⡿⠟⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⠄⠔⠒⠑⢀⡄⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⢿⣿⣶⣄
⠀⠀⠰⠡⠃⠀⠀⠀⠀⠁⠠⢄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⣿⣿⣿⠟⠛⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣧⡀⠀⢀⢿⣿⡟⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠠⠊⠀⠀⢈⢀⠀⠀⠀⠀⠀⠀⠐⢀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠙⠋
⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣦⣤
⠀⠀⠀⣴⣿⣿⠿⠻⢿⣿⣷⡀⠀⠀⣴⣿⣿⣿⠿⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠿⠿⢿⣿⣷
⠀⠀⠘⣿⣿⠀⠀⠀⠀⣽⣿⡇⠀⠀⢿⣿⣿⠀⠀⠀⣹⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠈⣿⣿⠗
⠀⠀⠀⠻⣿⣷⣤⣤⣾⣿⣿⠁⠀⠀⠘⢿⣿⣷⣶⣾⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⣶⣾⣿⣿⠂
⠀⠀⠀⠀⠈⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⠈⠉⠿⠛⠿⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠏⠛⠋⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣤⠴⣤⡁⠂⢄⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⡤⢾⣦⡰⡟⠻⡉⠉⠙⢦⢌⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡕⠁⠈⠁⠈⠛⠒⣻⢏⠎⣂⢰⠀⠀⣼⠀⠑⡈⢂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠒⡄⠀⠀⠀⠀⠎⢀⠇⠯⡀⠘⠔⠁⠀⠀⠀⠈⡄⢂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡔⠀⠀⠀⠀⠀⠉⠀⠖⠺⠀⠀⠀⠀⠀⠸⣮⠿⡀⡀⢠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⢀⠀⠀⢄⠀⠀⢀⣀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢐⠵⠾⡇⠁
⠀⠀⠀⠀⡀⠨⠰⠉⠉⠉⣷⠨⠀⡀⠈⠓⢄⣇⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡄⡀⢀
⠀⠀⠀⠎⠇⠀⠀⠀⠀⠀⠀⠀⠀⡉⡄⠀⠀⠀⠙⠶⠒⠂⠀⢀⠀⠀⠀⠀⠀⠑⠂⢲⢁⠁⠈
⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⡀⢀⠀⠁⠀⠀⠀⠀⠠⡀⠀⠀⠀⡄⠵⠤⠤⠤⠤⠤⡧⠾⠴⠓⠲⢟⣚⡛⢍⠉⠙⠛⠛⢋⠆
⠀⠀⢰⠀⠘⠠⡷⠀⠀⠀⠰⡧⠈⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡏⠈⢻⡆⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠘⣄⡀⠀⠀⠀⠀⠀⠄⠁⠰⢢⢀⠀⠀⠀⠀⠰⠀⠀⡠⢄⠙⠒⠚⠁⠃⠀⠀⠀⠀⡔
⠀⠀⡄⠀⡠⠐⠒⠒⠒⠒⠒⠒⠠⠀⠀⡷⢤⡁⡐⠂⠀⠒⠀⠒⣘⡜⡠⠀⠡⠄⠀⠬⠅⠀⠀⠈⢆⠀⠀⠠⠀⠀⠀⠀⠀⡠⠂
⣆⠀⠀⠀⠃⠤⠤⠤⠤⠤⠤⠤⠄⡆⠀⠂⢀⠧⠀⢀⡀⠀⠀⡀⠠⠖⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⢀⠦⠀⡖⠐⠀⠂⠈⠀⠠
⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠰⠠⠀⣠⠑⠶⣷⣴⡀⠀⠀⠀⠀⠀⡔
⠣⣘⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡠⠂⠐⠒⠓⠒⠒⠓⠒⠀⢀⠀⠀⠀⠀⠀⢀⡤⠊⠈⠁⠐⠃⠀⡀
⣀⣈⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣌⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣎⣶⣶⣶⣦⣬⣤⣦⣤⣖⣒⣒⣚⣉⣉⣉⣉⣉⣉⣉⣉
⣿⣿⣋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠁⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡰⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⠤⠖⠉⠀⠠⠜⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⣅⠈⠁⢠⠀⠈⠁⣰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣄⠀⠠⠄⠀⣠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠀⢗⠂⠐⡿⠀⠙⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠄⣡⣌⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⡴⣹⡠⠊⠉⠉⠙⠉⠉⠉⠑⠄⠏⢆⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠂⢄⠀⢀⡼⠥⠛⠂⠄⡀⠀⠀⠀⠀⢀⠠⠒⠚⠬⢧⡀⠀⠠⠐⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⢢⡠⠒⠁⠀⠀⠀⠀⠀⠀⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠒⢄⡔⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢰⡑⡐⣠⢠⠁⠈⠒⠒⠒⠒⠒⠒⠒⠂⢤⠄⠒⠒⠂⠒⠒⠒⠒⠒⠁⠈⡄⣄⢂⢪⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠖⠂⠁⠢⢄⠀⠀⠀⠀⠘⠁⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣈⣐⣤⣀⣀⠤⠔⠁



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⣶⠖⠒⢲⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⣋⠥⠔⣀⣀⣀⣈⢀⣀⣀⣀⡀⠉⠉⠙⠛⢲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⠅⢀⣨⣶⣿⣧⣼⣿⣿⣿⣟⣿⣿⣟⣹⣿⣿⣶⣦⣄⣀⢛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢅⣴⣿⣿⣿⣿⣟⣿⣿⣟⢻⢿⣿⣿⠿⣿⣿⣿⡙⣿⣿⣾⣗⣭⡢⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢁⣴⣷⣭⣿⣿⢋⣿⣿⣿⣋⣿⣿⣿⣿⣿⣷⣿⣛⣿⣿⣿⣿⣊⣿⣿⣿⣦⡣⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣰⣿⣷⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⠛⠛⠛⠛⠛⠻⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⡄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⠀⣽⣙⣿⣿⣧⣾⣿⣿⣯⠞⣁⡴⠞⠛⠉⠉⠛⠒⢤⡉⠸⣿⣿⣩⣿⣿⣿⣿⣿⣨⣿⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⣿⣿⣿⣿⣯⣿⣿⠟⣁⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠉⢿⣿⣯⣿⣿⣿⣭⣿⣿⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠀⣼⣟⢻⣿⣿⣿⣿⡏⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⡖⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⢰⣿⣿⣿⣿⠛⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢸⣿⠟⣿⣿⢿⣿⣷⣾⣿⡿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⣿⣿⣏⣻⣿⣿⣿⠠⠼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⣿⢹⣿⣿⣿⣿⣿⣿⡇⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠊⡎⣿⣿⣿⣿⣿⣟⣿⣤⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠴⣿⣿⣿⣿⣿⣿⣷⣼⣿⣳⠃⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠹⣹⣟⣋⣿⣿⣿⣿⣿⣷⣄⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣴⣿⣿⣿⠿⣿⣿⣿⣿⣯⠎⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⣿⣿⣿⣿⣿⣴⣿⣿⣿⣦⡀⠉⠙⠲⠶⠶⠶⠚⢉⣠⣾⣾⣿⣿⣿⣿⣶⣿⣟⣿⣿⠟⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡆⠘⢿⣿⣏⣹⣟⢻⣷⣿⣿⣿⣧⣤⣒⣲⣤⣶⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣈⣿⣿⠿⣿⣿⣿⣶⣿⣿⣷⣶⣿⡿⠋⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡈⠻⣿⣿⣿⡟⢿⣻⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣤⣿⣿⣿⠟⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠙⠫⠻⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣯⠿⠛⠉⠰⢒⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣶⣀⠀⠈⠉⠙⠛⠛⠻⠟⠉⠓⠊⠉⣤⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠤⠤⠴⠖⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣸⡇⣾⠀⣾⣦⣸⣿⣼⢻⣄⣼⣷⡾⣷⣸⠀⣸⡇⠀⠀⣰⠟⢹⡿⣰⠏⢳⣄⣦⡟⣿⢰⡇⢰⣷⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⢿⡇⣿⣠⢿⡿⣿⢹⡏⢸⢻⡟⢿⠇⣿⢿⣠⢿⡇⠀⠀⢿⣠⡼⡇⢻⣄⣼⠿⣿⠃⢿⠸⣇⡾⢿⢸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⠁⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⠁⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⢉⡀⢀⡀⠀⢉⠀⠉⠀⠀⠀⠁⠀⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠄⠀⢀⡀⠀⠘⣀⢀⠀⣀⣠⡀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣍⠀⠀⠀⠀⠀⠉⣤⠀⣀⠀⠀⠄⡄⠉⠀⠉⠉⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⣀⠀⠀⡀⠀⢠⡄⠀⠀⠁⡀⠀⣰⠣⠀⠀⠀⠤⠀⠀⡠⠔⠁⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠉⠀⠀⠀⣀⠔⠀⣀⠾⠭⠄⠁⠁⠉⠈⠠⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠓⠀⠀⠠⠀⠀⠂⠁⠀⠀⠚⢅⡀⠁⠠⡀⢀⠄⠀⢀⡀⠀⠐⠂⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠂⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠒⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢
⠀⠀⠀⠀⢘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠪⣷⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠤⣄⠀⠀⠀⢀⡀
⠀⠘⠷⣄⠂⠈⢽⣇⡕⠏⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠳⠲⠄
⠀⢠⣄⠙⢻⡀⠱⢃⣴⣤⣧⣦⣶⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀⣀⡀
⠀⠀⠨⢧⠈⢥⣠⢀⣌⡀⠀⠈⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠑⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈
⠀⠀⠀⠈⣣⠜⣉⡁⠙⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡜⢲⠟⠛⠷⢧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⡀⡆⠃⢀⢠⠇⡄⢠⠃⡄⢀⠄⠀⠀⠀⠀⠂⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡰⣌⢆⠣⢱⣠⣘⣘⡰⠡⢃⠎⡰⢋⠀⠀⣀⠀⠀⠀⢀⠀⡀⠀⢀
⠀⠀⠀⠠⠔⠀⢀⠀⠈⣇⠀⠀⠀⠐⠦⣄⡐⠦⣀⠦⡑⢌⠾⠋⠉⠀⠀⠀⠉⠓⢮⢊⠴⣋⠴⠊⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⢈⠳⠀⡿⠒⠒⠒⠠⠤⣀⣉⠓⠬⡑⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠵⢚⣁
⠀⠀⠀⠀⠁⠀⠈⠂⢧⢈⡉⠓⠶⠆⠀⠠⠬⢭⡑⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠭⠤⠒⠒⠉⠁
⠀⠀⠀⠛⠀⠀⠀⠨⠶⢦⡉⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈
⠀⠀⠐⠂⠀⠈

⠀⠀⡀⡀
⣤⣤⣉⣄⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀⣀⡀

⡀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠉⠀⠀⠀⠀⢈⠂⠀⠀⠀⢠⡀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⢈⠂
⠉⠂⠀⠀⣀⡄⠀⠀⠀⢀⡔⠀⠀⡄⠀⠈⠉⠒⠀⠀⠀⠉⠂⠀⠀⣀⡄⠀⠀⠀⢀⡔⠀⠀⡄⠀⠈⠉⠒
⠀⠀⠀⠊⠁⠀⠀⠀⠀⠊⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠊⠀⠀⠀⠱
⠀⠀⠀⠠⣄⠀⠀⠐⠲⢤⠀⠀⠀⢀⠀⠀⠀⠓⠒⠂⠀⠀⠀⠀⠠⣄⠀⠀⠐⠲⢤⠀⠀⠀⢀⠀⠀⠈⠑⠒⠂
⣄⣀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠠⠔⠚⠀⠠⡄⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠔⠚⠀⠠⡄⠀⠀⠀⠀⣠
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠠⠊
⠴⠂⠀⠀⠀⡆⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠂⠀⠀⠀⡆⠀⠀⢀⠔⠂
⠀⠀⠀⠀⠀⠓⠀⠀⠁⠀⠀⠀⠉⠉⠉⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠁⠀⠀⠀⠉⠉⠉⠀⠀⢢
⠈⠑⠲⠄⠀⠀⢀⠀⠀⠀⠀⡤⠀⠀⠀⡄⠀⠸⠀⠀⠀⠈⠑⠲⠄⠀⠀⢀⠀⠀⠀⠀⡤⠀⠀⠀⡀⠀⠸
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠼⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠾⠁⠀⠀⠀⠹⡄
⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠐⠤⣄⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠐⠤⣄
⠀⠀⠀⠤⠚⠁⠀⠀⠀⠴⠁⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⠴⠁⠀⠀⢧
⠀⠀⠀⢀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠈⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⢀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠈⠀⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠉⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠉⠀⠀⠀⠘⢆
⠉⠒⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠈⠁
⠀⠀⠀⠒⠦⠄⠀⠀⠈⠀⠀⠀⠀⢣⡀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠒⠢⠄⠀⠀⠈⠀⠀⠀⠀⢣⡀⠀⢀⡤⠊
⣀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠄⠀⠀⠀⠁
⠁⠀⠀⠀⠀⣇⠀⠀⠚⠁⠀⠀⠤⠤⠤⠀⠀⣀⠀⠐⠊⠁⠀⠀⠀⠀⣇⠀⠀⠚⠁⠀⠀⠤⠤⠤⠀⠀⣀⠀⠐
⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠁⠀⠀⠴⠀⠀⠀⠰⠃⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠀⠀⠀⠠⠃⠀⠀⠀⠣

⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠈⣙⢿⣾⣿⣿⣿⣶
⣿⣟⣴⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣿⣿⣿⣿⣿⣿⢿
⣿⣿⣿⡷⣿⡿⢶⣿⣷⣿⣿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠿⠀⠁⡀⠰⣿⣍⢛⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⡿⢿⣿⣿⣿⣿⣿⣿
⣿⣯⠀⠀⠀⠈⠆⠙⠿⠿⠤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡠⠿⠛⠛⢻⡿⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⡄⢠⣄⠀⡴⢠⡄⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠃⠀⠀⠒⠂⠚⠉⠁⠂⢀⣿⣿⡆⠀⡏⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠠⠤⠶⠒⢂⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⣀⣠⠤⠤⠤⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠚⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠈⢀⠀⡠⠤⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣀⣀⠈⣭⣿⣥⣤⠀⣀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠍⠉⠛⠂⠙⠉⠉⠀⠠⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⢤⣤⣰⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⡠⠀⠀⠐⠄⠀⠀⠀⠀⠂⠀⠀⠂
⠀⡤⢊⠄⠀⠀⠀⡀⠠⠀⠀⠀⡡
⠀⠄⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠰⡄⠀⡆⠄
⠀⠠⠀⠀⠀⠀⠀⠈⠘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢰⣿⣿⣿⣿⣿⡇⡄⠀⢀⣶⣿⣿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣿⣿⣷⠁⠀⠀⢿⣿⣿⣿⣿⣿⡾
⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡾⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠟⠃⠢⡤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀⠀⠀⠀⠀⠠⣤⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⡆⠀⠀⠀⠀⠀⢀⠾⢑⠶⠆⠀⠀⠀⠀⣠⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠙⠏⠉⠀⠀⠀⠀⣴⡉⠙⠶⠁⠀⠀⠀⠀⠀⠁⠈⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⣂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠰⣭⠂⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡾⠃⠀⠀⠀⠀⠀⠀⠀⣴⡇⣌⡳⠀⠀⠀⠀⡎⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠑⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢾⠛⠁⠀⢰⠂⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣢⠂⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡷⠆⠀⠀⠀⠀⢠⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠈⠁⠁⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠒⠤⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠒⣀⡀⠀⠀⠀⢠⡄⠂⠙⣲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠈⠒⣤⠄⢤⡀⠀⠀⠀⠀⠀⠀⣶⠱⡈⣇⠟⠃⠀⠀⠀⠀⠈⣇⠔⠼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠉⢦⠀⠀⢇⠀⢀⡇⠀⠀⠀⣴⠲⠀⢈⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⡤⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠑⣄⠀⠈⠁⠑⢤⡖⢢⣙⣢⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⠒⢖⣾⡏⠈⠺⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢀⣼⣿⣷⣄⠀⠀⢀⣙⣸⠛⠁⠀⠀⠀⠔⠈⠒⡄⠀⠀⠀⢀⣀⣾⡇⠑⠾⠟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⣴⣿⣿⣿⣿⡿⠓⢄⠙⠋⠈⠣⡀⠀⠀⠀⠳⠤⠔⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⣠⣾⣿⣿⣿⠟⠉⠀⠀⠀⢉⣦⣄⠀⠑⡀
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣾⣿⣿⡿⠋⠁⠀⢀⣠⣴⣾⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⣸⡟⢀⣴⣿⣿⠟⠉⠀⣀⣴⣶⣿⡿⠿⠛⠉
⠀⠀⠀⠀⢰⡟⣠⣾⡿⠋⣁⣤⣶⡿⠿⠛⠉⠁
⠀⠀⠀⢠⣏⣾⢟⣡⡴⠾⠛⠋⠁
⠀⠀⢀⣿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠀⠠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠈⢦⠁⢈⣀⠀⠄⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠁⠈⠀⠀⠀⠂⢨⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣬⠐⠀⠐⠀⠠⣢⢠⢁⠀⠀⠢⢠⡾⠀⠉⠠⠿⠇⠄⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠾⢮⡁⠁⠈⠉⠹⣘⠉⣀⡄⠀⠀⠆⠐⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠐⠁⠐⠋⠓⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⡇⢀⠶⣾⣤⣤⡿⡦⠀⡀⣾⣷⣴
⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠉⠉⠀⠀⠁⠘⠚⠀⠀⠀⠀⠙⠀⠈⠁⠈⠋
⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠂
⠀⠀⠀⠂
⣀⣀⣀⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣐⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠾⠁⠀⠀⠚⠀⠀⠋⠉⠭⠉⡟⢉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠠⠀⠀⣽⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠁⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⠴⠿⢿⣿⣿⡗⠀⣀⠀⠀⢠⡀⠀⢠⠂⢰⠋⠀⠀⠀⠈⠀⣀⠞⢋⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⣉⠀⢀⣍⠉⢩⠉⠀⣀⠀⠐⠘⣿⣿⣤⣼⣶⣶⠾⠋⠉⠉⠛⢷⡀⡀⠀⠀⠀⠉⠁⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠂⠘⠿⠇⠀⢥⠀⡉⠀⠀⠉⠀⢠⡀⠸⣇⣙⣿⣿⡏⠀⠀⠀⠀⠀⠀⢳⣄⡴⠀⠀⠀⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠀⣠⡄⠀⠃⣴⡄⠀⣶⡄⠈⢠⣤⡀⠀⢤⠄⠈⢩⡉⠣⡀⠀⠀⠀⠀⠀⣸⠟⣡⠀⠀⠀⠀⠀⠀⣬⠇
⠀⠀⠀⠀⠀⢠⢫⣄⡘⠘⠿⠛⣠⣄⠈⠀⠀⠉⠀⠄⠘⠛⢃⡀⢠⡄⠐⠀⠀⠀⠹⣄⡀⢀⣀⡔⠁⠉⠀⠠⠀⠀⠀⠀⠀⢠⡼
⠀⠀⠀⠀⠀⣎⡸⠿⠇⠀⢀⣄⠈⠁⢦⣤⣶⠶⠾⣶⣦⠀⠙⠃⠈⠁⠀⣤⠄⢀⣀⢸⣿⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠁
⠀⠀⠀⠀⢸⣧⣸⣷⠆⠀⠸⠟⠉⢠⡿⠋⠀⠀⠀⢸⡁⢰⣦⠀⠀⠛⠀⠉⠀⢺⡿⠄⣿⢱⡆⡄⠀⠀⠀⠀⠀⠀⢀⣴⡾⠃
⠀⠀⠀⠀⢸⣿⠉⠁⠀⡀⠀⠀⠂⠘⣇⠀⠀⠀⠀⠀⠹⡀⢀⣴⡌⢠⣤⡀⠀⣴⢀⡀⣿⠗⣿⠀⢄⠀⢀⡄⢀⢞⣿⠏
⠀⠀⠀⠀⠘⡏⠀⣶⠆⠀⠀⣀⡀⢼⡛⢦⣀⠀⠀⢀⡰⠀⠈⠁⠀⠈⠛⠋⠀⠀⠀⢀⣿⣷⣝⣂⣼⣦⣾⣷⡿⠋⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⣀⣀⠛⠋⠈⠙⠛⠛⠛⠛⢉⣤⣄⠀⠆⠀⠀⢀⣀⠀⠀⠀⡼⢿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠘⣄⠀⠀⠿⠷⠃⠤⠀⠀⠀⠃⠀⠀⠘⠿⠛⠀⠐⠀⠀⠸⠿⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⣤⠀⠂⠐⠷⠀⠀⠀⠀⠀⢀⡀⢠⢀⡛⡃⢰⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⡙⢳⣄⠻⣿⠆⢤⣤⡀⠀⠈⣁⠀⠘⠻⣣⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠿⣷⣦⣤⣈⣹⣂⣀⣨⣯⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉



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

⠞⠀⣀⠀⠈⠃⠀⠈⠓⠀⠐⠦⠤⠀⡆⠀⡞⠀⠀⠃⢀⠀⠀⢦⡀⠒⠒⠂⢠⠏⠀⠘⠆⠀⠇⠀⠈⠒⠀⠀⠏⠀⡴
⠚⠁⠈⠉⠀⠠⠖⠃⠀⢠⠖⠀⠀⠀⠁⠀⠀⠴⠂⠀⠈⠑⠀⠀⠁⠀⠀⡀⠀⠀⠐⠢⠄⢀⡀⠐⠦⠄⠀⠀⣀⣀
⠀⣤⠀⠑⠤⠀⠐⠦⡄⠀⠀⠀⣇⠀⣰⠁⢀⠀⢠⡄⠀⠒⠒⠀⠸⡄⠀⠇⠠⠎⠀⠀⡀⠀⠙⠀⠀⣠
⠀⠃⢀⠀⠠⠤⠆⠀⡀⠀⡇⠀⠀⠀⡀⠀⠈⠆⠀⠁⢠⠀⠠⣄⠀⠀⢀⡤⠀⠀⣇⠀⠈⠑⠀⠀⠋⠀⢀⠞⠀⣄⡀
⡀⠰⠋⠀⠀⢠⡀⢸⠇⠀⣀⠀⢰⡀⠉⠲⠀⣠⠂⠀⠃⠀⠀⠈⠁⠀⠉⡀⠀⣀⠀⠀⠰⠶⠆⠀⢢⠀⠀⣀⣀⠀⠈
⠧⠀⠈⡉⠁⠀⠃⠀⠀⠚⠁⠀⠀⠃⠀⡄⠀⠁⠀⢰⠀⠀⡇⠀⠲⢄⠀⠹⠄⠀⠙⠀⣠⠄⠠⡄⠀⠁⠀⠀⠀⢀⠔
⠀⠀⠞⠀⠠⡄⠀⣸⠀⢠⡀⠈⠓⠀⠀⠃⢀⡴⠀⠈⠃⢀⠀⠀⣀⠀⠀⢀⠀⢠⠄⠈⠁⣀⠀⠓⠀⡞⠀⠰⣄
⠓⠀⣠⠀⠀⠙⠀⠀⠀⠀⠓⠀⠠⠖⠒⠀⣀⣀⠀⠀⠚⠁⠀⠚⠁⠀⠀⠟⠀⠚⠀⡤⠆⠀⠑⠀⣄⠀⢤⡀⠈⠀⣀
⠀⠈⠁⠀⠦⠤⠀⢠⠃⠀⠀⠒⠤⠀⠀⣀⠀⠀⢀⡄⠀⡆⠀⠀⠈⢦⠀⠠⠤⠄⠀⢠⡄⠀⢦⡀⠘⠀⠀⠈⠀⠠⠃
⠉⠒⠀⣠⠀⢀⠀⠀⠀⠐⠒⠂⢀⡀⠀⠉⠉⠀⠘⠀⠀⠁⢀⠖⠀⡀⠀⢀⣀⡀⠀⠘⠀⠀⠀⠁⠀⠐⠋⠁⠀⢠
⣀⠀⠉⢀⠀⠘⠄⠸⡇⠀⡀⠐⠉⠀⠀⠱⠄⠀⠐⠒⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠴⠂⠀⠈⠉⠁⠀⠀⢸⠀⠈⠃
⠃⣀⡤⠀⠑⠄⢀⠀⠀⠀⠙⠆⠀⠘⠦⠀⢀⡞⠀⢠⠞⠀⣸⠀⡀⠀⠠⠖⠃⠀⢀⠄⠀⠤⠖⠀⠀⠑⠄⠀⣠⠔
⣀⠀⠀⠠⣄⠀⠘⡄⠈⠑⠆⠀⣀⡤⠀⢀⠀⠀⡄⠀⠠⣄⡀⠀⠈⠑⠀⢀⡀⠀⠋⠀⠀⠀⠀⡴⠁⢀⡀⠀⢁
⠀⠀⢀⡆⠈⠁⠀⠀⢰⠀⢀⠀⠀⠀⠀⠸⠀⠀⠘⠂⠀⡀⠀⠀⠒⠋⠀⠀⠙⠂⠀⠈⠓⠀⣀⠀⢤⠀⠱⠀⠀⠳
⠛⠁⠘⠀⠀⠚⠁⠀⠘⠀⠈⠓⠀⠀⠲⠄⠀⠈⠓⠄⠀⠷⠀⢠⠀⠒⠋⢀⠀⠐⠒⠃⠀⠀⠏⠀⠈⠃⠀⠐⠲⠄
⠂⢠⡀⠰⡄⠀⡴⠃⢀⡔⠀⠠⠤⠆⠀⠀⠀⡆⠀⠀⢠⣄⡀⠀⠃⠀⠠⠞⠀⠐⠦⠄⠐⠒⠂⠀⣴⠀⠠⠖⠂⠀⠠
⡀⠀⠃⠀⠁⢀⣀⠀⠈⣀⣀⠀⢤⠀⢸⠀⠀⠁⠐⠦⠄⠀⠀⡠⠖⠀⡀⠀⠤⠖⠀⢠⠀⠲⢤⠀⠁⢀⡀⠀⠠⣄
⠙⠀⢠⠤⠄⠀⠈⠁⢀⡄⠀⠀⠘⠀⠀⠠⠚⠁⠀⠀⠤⠤⠀⠀⡄⠀⠇⠀⠠⣤⠄⠈⠃⠀⣀⠀⠐⠋⠀⣀⡀
⠂⠀⠀⠀⠀⠞⠁⠀⠊⠀⠀⠧⠀⢸⠁⢠⡀⠀⣧⠀⢠⠀⠀⠘⢀⡀⠀⢠⠀⢀⡀⠀⡠⠀⠈⠉⠀⣄⠀⠀⠉
⠀⠀⡇⠀⡤⠔⠀⠸⡄⠀⣄⡀⠀⣄⠀⠀⠑⠀⠀⠀⠃⠀⣄⠀⠉⠀⠀⠈⠃⠸⠀⠈⠁⠀⠒⠒⠀⠘⠀⠐⠊⠁
⡀⠀⢀⣠⠀⠀⡴⠀⢀⡤⠀⠉⠀⠈⠂⠀⠠⣄⠈⠓⠄⠀⠈⠀⠀⡈⠉⠁⠀⢀⡀⠐⢄⠀⢀⠤⠆⢀⡄⠀⡠⠆
⠁⡀⠀⠀⣄⠈⠁⠀⠉⠀⠤⠒⠀⠐⠒⠂⠀⠈⠀⡤⠀⠐⠚⠀⠀⠉⠓⠀⠈⠉⠀⡀⠈⠁⡀⠀⠀⠊⠀⠀⢀
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣻⣻⣻⣿⣛⣿

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⣤⠐⠀⠂⡉⠀⠀⠤⠀⠀⠀⠀⠐⠁⠈⠉⠁⠄⠤⠀⡀⠀⠀⠀⠀⡎⠀⠀⠐⢀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⡀⠧⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⢂⡀⠀⠄⠀⠀⠀⠆⠀⠀⠀⣿⣻⡟⣿⣿⣿⡇⠀⠀⠀⡀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣿⣫⣵⣽⣿⣿⠇⠀⢠⠀⠀⡠⠀⠃⠀⠰⠀⠈⢀⡀⠀⠀⠠⠄⠀⠀⠀⠀⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠄⡀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠛⠋⢉⠀⠀⠘⠉⠀⠉⠀⠘⠅⠀⢀⠄⠀⢁⡀⠰⡂⠀⢠⠀⢈⡠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠁⠀⠀⠀⠀⠰⠞⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⡄⠀⠀⠀⠀⠀⠘⠀⡀⣰⠂⠉⠠⠀⡇
⠀⠀⠀⠄⠀⠀⠀⡾⠃⢈⣢⡤⠄⠂⠨⠁⠠⠐⢧⡠⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⢀⠜⣠⣇⠀⠀⠀⠀⠀⠉⠂⠠⠤⢀⠀⠀⠀⠀⠠⣴⠄
⠀⢆⠀⠀⠀⠀⠀⣄⠁⠈⠁⠀⡋⠈⠀⠀⠐⠀⠁⠀⠀⠀⠎⠀⡀⠀⠀⠀⡀⠰⡆⠀⠈⢲⠓⠑⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⢁⠀⠀⠂⠀⠀⠀⠀⠀⠃⠈⢁⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣀⠀⠀⠀⠀⠠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠱⢀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⡀⠂⢀⠀⠁⠀⡀⠈⠀⠀⠀⠠⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠠⠠⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠀⠁⠀⠀⣀⣀⣤⠠⡀
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⢀⠀⠤⠃⠀⠀⠀⠀⢀⣲⡄⢛⢄⠣⡜⢡⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠒⠃⡤⠀⠀⠀⠀⠀⠀⠀⣼⡿⠏⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⠈⢣⠌⢱⡜⢣⠄
⠀⢀⣀⣠⠀⠀⠀⠀⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠊⠤⠐⠀⠃⠈⠀⠀⡀⠀⠀⠀⠀⢠⠤⡄⠀⠀⠀⠐⢀⡢⢠⡀⠛⣠⠚⢡⠘⠻
⠀⠀⠈⠀⠀⠀⠀⢹⢸⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠍⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⣄⠰⢆⡀⠦⣀⠶⣀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⢀⢱⡇⠀⠀⠀⠠⠶⠤⠀⠀⠀⠤⠀⡀⢀⠔⠊⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠤⠈⢡⡔⢁⠔⠉⡦
⠀⠀⠀⠇⠀⠀⠐⠺⣙⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠁⠀⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠹⣄⠘⣀⠜⢃⡐⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⢂⣀⣤⣄⡀⠂⡠⠀⠀⠀⢀⣄⡀⠀⠀⢈⡄⠀⠀⠀⢨⠉⠀⠀⠀⠀⡙⣈⠖⣀⠲⠀⠠⢴
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⠁⠀⠀⠀⢐⡀⠃⠀⠀⠀⠂⠀⠠⠩⠀⠈⠁⣸⡧⠄
⠀⠀⠀⣀⠤⣒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣹⣯⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣠⡗⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋⠉⠀⠀⠀⠀⠀⣇⠐⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⡀
⢰⢎⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⢲⠀⠀⠀⠈⠀⢀⠀⠀⠀⡀⠌⠀⠄
⢸⠀⠀⠰⠄⢀⡀⠠⢆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠂⢀⠠⠖⠁⠘
⢸⠀⠀⠀⠀⠈⡠⠑⠣⠀⠀⠀⠀⢨⠁⠀⣀⣀⣠⣤⣤⣤⣤⣴⡄⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠀⡀⠀⠈⠀⠀⠀⢰
⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⣿⣟⣿⣿⣟⣻⢿⣽⠻⣿⣧⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡄⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠉⢀⠀⠀⠀⠀⠀⣿⣧⣿⣝⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠰⠀⠀⠄⠀⡤⠤⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⠸⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⡇⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠈
⠀⢇⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⢀⠀⠀⠂⢨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠑⢄⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡇⠈⠀⠐

⠀⠀⠀⢠⡞⠲⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠚⠉⠉⠙⠓⠦⣤⠴⠒⣦
⠀⠀⡴⠚⠳⠤⠇⠀⠀⠀⡈⠉⠙⠒⠶⣴⠊⢉⣧⣤⣤⠤⠤⠶⠒⠛⠉⢁⡤⢤⠀⠀⠀⠀⠐⠀⠀⠘⠶⠚⠷⠤⢤⣄⣀⣀⣀⠀⠀⣠⣄
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢧⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢧⡈⣧⡀
⠀⡇⠀⠀⠂⠀⢀⡴⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢲⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⢦⡀
⠀⠹⡄⠀⠀⠀⠘⠦⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠧⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢳
⠀⠀⠙⠦⣀⣀⢀⣀⣀⣠⠤⠴⠖⠒⠚⠒⠒⠶⠤⣄⡀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⢹⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⣠⠤⠖⠋⠉⠀⠀⠉⠙⠲⢦⣤⡤⠖⠁
⠀⠀⠀⠀⠘⣧⠤⠤⠤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣤⠤⠤⠤⠤⢤⡞
⠀⠀⠀⠀⠀⢷⠀⡴⠚⠁⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠉⠳⣄⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠸⣾⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣤⣤⣴⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣤⣄⣤⣶⣉⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣤⡏
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠤⡤⠒⠒⠒⠒⠂⠐⡖⠐⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠖⠂⢩⠶⠿⠙⠉⠀⠀⠘⠟⠛⠙⠉⠀⠒⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⢂⡀⡄⠈⠀⠤⠤⢤⡤⠤⠤⣤⢤⡀⡆⠈⠀⣉⠁⠐⠂⠳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⡄⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠰⢀⠀⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠀⡠⠀⠀⠀⠃⠈⠀⠉⠈⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠊⠀⠁⠀⠀⢀⠠⠓⠖⠁⡀⠉⠀⠀⠀⠀⠀⢀⡀⠤⢀⡤⠶⡶⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠊⠀⠀⠀⠀⢀⠄⠈⠀⣠⠞⠁⠀⠂⠀⠀⠀⠙⢦⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠃⠀⠀⠀⠀⡰⠁⠀⠀⡜⠁⠀⠀⠌⠀⠀⠀⠈⠀⠀⢣⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠂⠀⡀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠑⠂⠀⠘⠀⠀⠀⠰⠄⢀⣀⡀⣀⠀⠀⠀⡇⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠰⢷⡴⠻⡿⠿⠦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠐⠦⠤⠀⣀⣀⠁⠀⠀⠈⠁⠐⠃⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢳⠼⣦⡀⣿⠉⢹⣿⠗⠀⠢⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠏⠘⣡⠆⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣈⠙⠉⠘⣿⣷⣼⡋⠈⠋⢀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡤⠀⠀⠀⠀⠀⢀⠇⠀⡾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠰⠀⠙⠶⠶⠶⠛⠿⠿⠷⠦⢀⠀⠁⡀⠀⠀⣤⠀⠑⠀⠞⡈⠀⠀⠀⡀⠀⢘⡄⣪⠃⡜⠁⢙⠃⢆⠅
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⣌⣉⡀⢈⠀⠀⠈⠀⠀⠀⠀⠑⡄⢋⠜⢁⡞⠁⡠⢞⡴⠁
⠀⠀⠀⠀⠀⠀⢀⠂⠐⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠁⠀⠀⠀⠀⠐⠶⣀⠀⢞⣐⡭⠞⢟⣤⠞⠁⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⠂⠀⠀⠀⠀⡀⠠⢼⢝⠳⢯⡐⠦⣞⣪⠦
⠀⠀⠀⠀⠀⠀⠐⢀⡃⠐⠀⢲⡎⠉⠉⣶⡎⠉⠙⣦⣀⣤⣤⣤⣤⣤⠧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡬⠑⠂⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠙⠒⠊⠉⠙⠒⠺⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠉⣩⠝⠛⠛⠛⠛⠿⣷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⢀⣀⣀⡠⠴⠋⠁⠀⠀⠀⠀⣠⠊⡏⠙⢏⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⠗⠉⢩⠉⠁⠀⡄⠀⠀⠠⢦⡀⠀⠰⣁⠜⠀⠀⠈⢇⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢁⡎⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⢄⠑⡀⠀⠀⠀⠀⠀⣀⣀⠙⠢⣄⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⡸⠀⠀⠀⠀⢰⠁⡅⠀⠀⢀⣀⣀⡉⠀⠀⠀⠀⢰⣚⡡⠞⠀⠀⠈⠓⡤⣼⣧
⠀⠀⠀⠀⠀⠀⢰⠏⣠⢞⣡⠤⡄⠀⠀⠸⡀⠀⣠⡴⠋⣀⣀⣉⣛⠒⠒⠤⡀⠁⢀⣀⠀⠀⠀⣾⢹⠀⢹⣆
⠀⠀⠀⠀⠀⠀⣿⠋⠐⠯⠄⠒⠁⠀⠀⠀⡠⠞⢁⡴⠞⠉⠉⠉⠉⠛⢦⡀⢹⡄⠸⣌⢣⠀⠀⢇⡜⠀⣰⢿⡄
⠀⠀⠀⠀⠀⢰⡋⡆⠀⠀⢀⣄⠀⠀⠀⣰⠇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢻⡄⠘⠶⢀⢄⠀⠀⡔⠁⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠈⠢⣄⠸⡄⢃⠀⠀⢿⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣧⠀⠀⢸⡀⠇⠀⢳⠀⠈⡇
⠀⠀⠀⠀⠀⢸⡗⠀⠀⠘⡆⠙⠚⠀⠀⢸⡇⠘⣧⡀⠀⠀⠀⠀⠀⠀⢀⣾⠃⢠⡏⠀⠀⠀⠓⠂⠀⢸⡆⢸⡇
⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⡇⡄⠄⠀⠀⠸⣇⡀⠈⠳⣤⣄⣀⣀⣠⡴⠟⢁⡴⠋⠀⠀⡠⠤⠶⢦⠀⠀⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠸⣇⠀⡼⢸⡇⠀⠀⠀⠀⠈⠙⠓⠦⠤⣍⠉⠉⢁⡴⠛⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⢸⢠⠏
⠀⠀⠀⠀⠀⠀⠀⢻⣄⡇⠀⠑⠁⠀⠀⠀⠀⠠⡒⠢⡀⠀⠉⠉⠁⣎⣉⣩⠵⠀⠀⢰⡓⠦⣀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⣖⠚⢛⣳⠀⠈⠢⠵⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠉⢲⣾⠤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⡢⢌⣁⣉⣀⣀⣀⠀⠀⡠⠤⠒⡄⠀⠀⡇⢸⠀⣠⠋⠉⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀⠀⠀⠀⠙⠾⡖⠚⠉⠀⠀⠀⠓⠋⡰⠃⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⣤⣄⣈⣒⣒⣤⣤⣤⡶⣯⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡔⢦⠀⡠⠤⠤⡀⢰⠒⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡠⡀⠑⠉⢜⣀⣀⣀⣘⡆⠉⠁⡤⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⡔⢤⢸⠁⠀⠀⢸⢀⠔⢦⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⡞⠓⠒⠒⠚⢦⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠓⠒⠒⠒⠒⠚⠒⠒⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣰⠦⣖⡄⣴⢤⢠⡄⡆⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣄⣆⣴⡾⠷⠢⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣬⣝⣿⣿⡷⡶⠶⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⢏⡽⡛⣝⠙⠷⣦⠠⡀⠀⠀⠀⠀⢠⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠀⣿⡇⢱⣘⣷⡀⠀⠁⠀⠀⠀⠠⣸⣿⣏⠀⢍⠁⣠⢾⣶⠧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠇⢹⡗⠀⠈⣿⠙⠠⠀⠀⠀⠀⡾⠉⠘⣿⣧⡈⢸⠳⡟⢹⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⡈⠣⣄⠀⠀⠀⠀⠀⠀⠀⠉⢿⡅⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢄⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠓⡂⣂⣀⣲⢄⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⡀⠀⠀⠈⠑⢦⡀⠀⠀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢻⠛⠚⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠈⠳⣄⡀⠀⠘⢦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⡀⡀⠀⠙⣶⠄⢴⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠲⣀⠹⣷⣄⢀⡈⠈⣲⣄⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠆⠀⠀⠀⠀⢀⣠⡴⠞⠊⠉⢉⡽⠥⣑⣌⠿⣹⡽⣟⣠⣶⠉⣿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣤⣴⠾⠉⠀⡄⠀⣴⠖⡀⡄⢠⣆⠹⣿⣇⣤⡀⠘⢙⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢋⣭⢏⣴⣾⢐⡋⢡⢰⣷⡷⡿⠿⢿⠿⠿⠿⠿⢷⣄⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⠛⠿⠿⠷⠿⠷⠿⠋⠃⠀⠀⠀⠀⠀⠑⠀⠐⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⡆⣾⣿⣴⣿⢀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡀⣿⣦⣿⣷⢸⡷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣷⡹⠟⣼⣫⠟⠁⠀⠀⠈⠻⣿⣯⣽⣿⠟⠀⠀⠀⠈⢿⣝⣧⠻⢏⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⢀⣠⣤⣤⣤⣭⣭⣤⣤⣤⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⣠⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣄⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⣾⣿⣿⠗⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠸⣿⣿⣶⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠟⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠻⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣦⣌⣛⠿⢿⣿⣿⡿⠿⣛⣡⣴⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣥⣚⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⢻⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣝⡻⠿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠒⠉⠙⠛⠿⠿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠈⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⡅⣙⠿⢷⡦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡀⠘⠿⢻⡿⠁⠀⠻⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣿⣇⣠⣾⣿⣿⡰⠶⠺⠾⢿⣿⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⢿⡟⢛⣶⡦⠤⠛⠁⠀⣪⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠈⠁⠀⢀⣠⣶⣾⣿⣿⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠿⠿⠟⠛⠉⠉⠉⠉⠉⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠛⠶⠤⠤⣤⣠⢠⣷⠾⠿⠛⠉⠀⠀⠀⠀⠀⠈⢻⢂⢠⣿⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⣾⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠠⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⢶⡚⣻⠻⢿⣼⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡉⢋⣉⢽⣬⡀⢠⠐⠂⣤⢙⣿⣃⢿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠐⣻⢒⡼⢖⠦⣰⡤⣄⣘⣥⡀⢠⣿⣾⣿⠿⢧⣿⣶⡏⠺⢿⣿⡆⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣯⠁⠀⠈⠛⠁⡰⠼⠋⣸⠴⠋⠀⠻⠿⡄⠀⠙⢻⣿⠀⠀⣾⠇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣻⠀⠀⡄⠀⠀⠀⠀⠀⣿⢀⡀⠀⠀⣤⣧⠀⠀⣾⣿⡀⢸⣿⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣿⠀⠀⢶⠀⠀⣿⢸⣧⡇⠀⣿⡿⡀⢀⣿⡿⠀⣸⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢹⡇⢀⣸⠀⢀⣿⣿⣷⡇⠀⣿⣿⡃⢸⣿⣿⠨⣿⢿⣿⣷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣾⣧⢘⣿⣿⢸⣿⢼⣿⣇⢸⣿⣿⠀⣼⣿⣧⢸⡿⣾⣿⣯⣔⣢⣶⡧⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⣿⣿⢩⣿⣿⡅⣿⣾⣻⣷⣾⣿⣻⠀⣿⣿⣟⢸⣾⣿⣿⣗⣮⣥⢴⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠤⠿⣿⣅⣉⣽⣿⣋⣹⣷⣿⣥⣿⡯⢽⣿⣿⡿⢿⣿⣿⢿⣧⠴⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⣿⠿⠷⠛⠓⠛⠛⠁⠤



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⡛⢫⣝⢛⣛⣋⣙⡻⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣾⡟⣿⣬⣭⣾⢟⣩⣾⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠙⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⢀⡀⠀⠀⢠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⠻⡄⠀⠀⠙⠉⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⡿⣻⣿⣿⣿⣿⣿⣿⣶⣆⠻⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢛⠻⣿⣿⣯⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣛⣣⣿⣿⣿⢿⣿⣿⠟⣛⢿⣿⣿⣷⣌⢻⣿⣿⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣌⣛⣼⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢛⣫⣴⣾⡆⣶⣍⡻⠿⢿⡿⠿⢛⣩⡖⣼⣿⣶⣬⣝⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣧⢻⣿⣿⣿⡆⣶⣿⣿⣿⢇⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠸⣿⣿⣿⡇⣿⣿⣿⣿⢸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠛⠛⠛⠃⠛⠛⠛⠃⠚⠛⠛⠛




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣉⣸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢀⡐⠈⠏⠀⠀⠀⠀⠀⡀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠋⡰⠁⠀⠀⠀⢀⠘⢝⣂⢴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣶⠂⠍⠀⣵⣀⠀⠀⠀⠈⠿⠁⠀⠂⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⠚⢀⣼⣿⣿⣿⣦⣄⣠⠂⠣⡈⡴⠑⢄⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⣠⣿⣿⣿⣿⣿⣿⣿⠀⡀⢄⠔⠥⠂⠰⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠉⠑⠢⡀⣴⣿⡿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡿⢠⠀⠀⠂⠀⠀⠹⠤⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠺⠟⣿⡀⠘⡟⡿⠁⢀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⣐⠀⠈⠶⢤⠢⡀⠀⠆⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⢸⠇⠤⡉⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠉⠓⢁⣀⣘⡆⠀⠘⢀⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⠀⠀⠉⠚⠠⠟⠀⠀⠟⠛⠉⠘⣿⣿⣿⣿⣿⡷⠀⠀⠀⠎⠉⢹⣿⡖⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⢂⠀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠉⢁⠀⠀⠆⠀⢠⣤⠀⠀⡻⢃⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠄⠁⠀⠀⠀⠈⠀⠀⢢⠀⡇⠀⣠⠀⠀⢀⠠⡀⠈⡄⠀⢦⠀⠀⠐⠶⠔⠁⠸⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⢊⠠⠀⠀⠂⢠⠀⠀⠀⠀⠀⢈⢢⠁⠞⠁⠀⢀⠌⠀⢳⡀⠀⠀⠀⠇⠀⡌⢀⠐⠀⢀⡄⠀⢠⣃⣠
⠀⠀⣀⣤⣤⣤⠃⠀⢠⣼⣀⠇⠀⠀⠀⢀⠀⣮⡜⠀⠀⠀⠀⡌⠀⠀⠀⠱⡄⢀⡀⣀⠐⠀⣰⠀⠀⢀⢙⣈⢉⣴⠟
⠀⠘⠻⠿⠿⣧⣤⣶⣿⣿⣿⠀⠒⣻⣿⠁⢀⣿⡇⠀⠀⠀⣼⣿⣷⣦⡀⠀⣈⠃⡈⠀⡤⠊⠑⢄⣀⣤⣾⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⢠⣿⣷⠀⢾⣿⡇⠀⠀⢰⣿⣿⣿⣟⡁⠀⠀⠀⠀⠂⠋⡆⢠⠞⠛⢿⣿⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠻⢷⣾⠛⠇⠀⢀⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠘⠴⠃⠘⠀⣠⠂⠈⡁⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠢⣄⠒⣾⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠐⠁⢀⠀⠈⠲⢤⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠥⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⡁⠰⠃⡡⠄⠀⠀⠀⠀⠑⣄⡈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣱⠀⠤⠄⠁⠀⠀⠀⠀⠀⠏⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠰⡀⠀⠀⠁⠈⠁⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣄⣠⣶⣶⣷⣶⣶⣦⡀⠀⠀⠙⠒⠀⠂⠁⠤⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣴⡿⠃⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠋⠀⠀⢿⠆⣤⢸⣿⣿⡿⣿⣿⣿⡟⠀⠀⠀⠀⣠⣰⣿⣤⡄⠀⠀⠀⠀⠘⣿⣿⣷⣶⡀
⢠⡶⠂⠀⠀⠀⠀⠀⠻⠃⡠⠀⢀⢀⣤⣤⣀⠀⢠⣿⣿⣵⣶⣾⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⣣⠀⠀⠀⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡁⠉⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢛⠉⠉⢰⣶⣶⣴⣶⣤⠀⣉⠋⠀⠀⠀⠀⠀⣤⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣴⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⢿⣿⣿⣿⡦⡤⢤⣀⡀⠀⠀⠀⢀⣸⣿⣶⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠒⠒⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠏⠉⠉⠉⠉⠉⠁⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠻⣿⣿⣇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣸⠿⠋⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠁

⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣤⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⣾⡿⣿⠿⠷⢬⡀⠤⣽⡿⠿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⢸⢃⠀⠈⠃⢰⠋⢠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠁⠀⠀⠀⢇⠀⠀⠈⠀⢸⠀⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⢿⡋⠒⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⢡⡀⠀⠀⠀⠀⠀⢳⠀⠀⣀⣿⣄⣦⣤⣀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣤⣙⣶⣆⣶⣶⣶⣼⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠿⠟⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣛⡛⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠄⠀⠀⠀⢀⣀⡄⢀⡉⠉⠑⠒⢒⣒⣂⠠⠄
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⢒⣒⠒⠀⠠⣤⣄⠀⣀⣀⣀⢉⣉⣉⣉⣀⣀⣤⣤⡶⠞⠋⣉⠁⠀⠀⢐⢒⡟⠉
⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣐⣠⣬⡥⠶⠶⠶⣾⣻⠻⠿⠟⠛⠛⠋⠉⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠈⠁⢲⣶⡤⣄⠀⠁⢠
⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⣶⣶⣶⠿⠟⠛⠉⠀⡾⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠛⢛⣉⣉⣉⣥⣤⣤⣤⣤⣤⣤⡀⡀⠀⠈⠀⠀⠀⢀⣀⣠⣤⠶⢻⡅⠀⠀⠀⢀⡠⠔⢋
⠀⠀⣶⠀⠐⠒⢀⣉⣉⣙⣋⣉⣭⠭⠤⠶⠞⠛⠛⠋⠉⠁⠛⠙⣋⣁⣀⣉⣉⡽⠟⢛⠛⠛⢛⡭⠍⠑⠊⠉⠒⠺⠿⠖⣒⠉⠀⠀⣀⣈⡦
⠀⠀⠛⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠖⠒⠋⠩⠍⠐⠊⠉⠀⠂⣉⡠⠖⠋⠀⠀⠀⠀⠀⠀⡀⢀⡀⠐⢺⣷⡞⣶⣯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠀⠀⣀⣠⡤⠖⠒⠉⠔⠈⢂⠀⡀⠀⠀⠀⢀⡤⢺⣼⣮⡿⠯⣵⣿⠁⣤⣴⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠋⠉⠉⠁⠀⠠⠴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠈⠉⢀⡛⠴⠞⠃⠀⠀⠉⠀⠉⡁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡦⣤⣤⡤⢤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣺⢯⢛⡿⣞⠛⣠⣍⠚⢩⣟⣻⣻⣏⡓⢿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡋⠠⣮⠞⢹⡵⠊⢡⢤⣬⣮⣧⢠⣄⢀⣔⡵⠃⠀⠁⠯⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠓⠋⣤⡤⣴⣿⣶⢤⣣⢫⣷⣥⠿⢾⣻⡈⠋⠀⠛⣭⠭⠟⣈⠛⢹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⡴⣽⣀⣐⠯⠯⢽⡯⢷⣧⢬⣉⣡⣋⣛⣥⣤⣤⣙⡛⠳⣮⢝⠚⢳⢾⡛⢿⣮⢑⣿⡦
⠀⠀⠀⠀⠀⠀⣸⠞⠈⠙⣻⡽⢀⠼⡾⣭⡇⣠⠟⠋⠁⠀⠀⠀⠀⠈⠙⠷⣼⣿⢳⠀⢷⡇⣤⣒⠭⢻⣇
⠀⠀⠀⠀⠀⢠⣏⡀⠀⡜⡝⣁⠈⠉⣩⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢾⢖⠭⢛⣄⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠸⡟⢯⣢⡛⠈⠻⣕⣿⡶⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⡪⠟⢠⢧⠣⣖⢤⡈⢻⡇
⠀⠀⠀⠀⠀⠀⢷⡀⢩⢖⠯⢟⣈⡁⠉⣪⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⠝⢀⠈⠋⠀⠀⠱⠗⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣮⡁⣔⡽⢻⣻⣸⡶⣿⣿⡛⢶⡶⣦⣤⢤⣴⠯⣟⣧⢳⡧⡄⢯⣇⠻⢽⡢⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠘⣎⢻⡍⠀⣀⠙⢃⠬⣿⠍⠀⣮⣮⢔⠎⢉⠺⠗⠀⢹⡱⠊⠁⠸⢏⣠⡤⠽⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠳⠞⠉⢷⣗⡉⠀⠸⣷⣔⠷⠁⡼⡝⡴⡆⠾⠭⠭⢇⣀⣠⠞⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠈⠉⠛⣆⠈⠾⣀⣀⡳⠁⣟⡇⢀⠀⢠⡏⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠙⠶⠚⠁⠈⠙⠓⠾⠞⠉⠓⣋⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠒⠒⠒⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣤⠀⠀⠀⠀⡄⠀⠀⠀⢠⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠾⠁⠀⠀⠀⠟⠀⠀⠀⠰⠏⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⢠⡆⠀⠀⠀⢰⠆⠀⠀⠀⣴⠂⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡄⠀⠀⠀⢀⡄⠀⠀⠀⢠⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠃⠀⠀⠀⠈⠃⠀⠀⠀⠘⠁⠀⠀⠀⠘

⠀⠀⠀⠀⠀⢀⡠⠔⠒⠀⠀⠐⠒⠤⣀
⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⢀⠎⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡄⠀⠀⠦⣀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀
⠀⠀⡘⠀⠀⠐⢤⢀⠆⠀⠀⣴⢀⠀⢤⢀⢀⠘⠀⠀⠀⠀⠀⢼⠃⢀⣀⡀⠻⣷⣄⠀⠀⠀⣠⡀⠘⣯⠀⠀⠀⠀⠀⠀⠀⠈⢀⡏⢮⡁⠄⢰⣣⡀⠰⣀⣠
⠀⠀⢃⠀⠀⢀⡎⢈⡶⠈⠆⠚⠒⠀⠈⢩⠃⢀⠀⠀⠀⠀⠀⢿⡀⠘⠛⠀⢀⠈⠻⣷⣄⠀⠛⠁⠀⡯⠀⠀⠀⠀⠀⠀⠀⠤⠎⠐⠊⠉⠉⠁⠉⠀⠀⠀⠃
⠀⠀⠘⡄⠘⠊⠀⠁⠀⠀⠠⠐⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠹⢇⠀⠀⠔⠃⠀⠀⠈⠙⠦⠀⠀⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠦
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠑⠦⢄⣀⠀⠀⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⣠⣾⠟
⠀⠀⠀⠀⡀⠀⠀⠈⢿⣦⡀⣠⣾⠇⠀⠀⣖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣛⣸⣷
⠀⠀⠀⢨⡇⠀⠀⠀⢠⣿⣑⣿⡇⠀⠀⠀⠘⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣷⣾⣿⣧
⠀⠀⠀⢾⡆⠀⠀⠠⢾⣿⣷⣿⣷⠄⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⢘⣧⠀⠀⢀⣾⣿⣿⣿⣷⡄⠀⠀⢬⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠿⣷⣄⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⢠⡤⣄⠸⠿⢿⠏⢹⡿⠿⠀⡀⢤
⠀⠀⠀⠀⠘⠁⠀⠘⠿⣿⠛⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠻⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⣻⢳⢳⢻⡟⡟⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠿⠴⢦⢾⡼⠷⠷⠿⠿⠈⠁


⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢰⠂⢠⡄⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠘⠒⠚⠓⠸⠇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⡌⠀⠀⢠⣶⣿⣿⣷⣿⣿⣷⣦⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣀⡀⠀⣀⢀⡄⣀⠀⠀⡀⠀⢀⡀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⢿⡀⠀⠈⠉⠛⠛⠛⠛⠛⠋⡉⠀⠀⢠⠇⠀⠀⠀⠀⣤⣴⣶⣦⣤⣾⡟⣿⡇⠛⢸⡇⡆⣿⠈⢡⣿⠘⢻⣿⠸⢃⣿⡘⣸⣿⢛⣷⣦⣤⣶⣶⣤
⠀⠀⠀⠀⠈⠁⠀⠀⠃⠃⠁⠨⠬⠘⠇⠧⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠙⠿⠿⠇⠛⠸⠇⠶⠸⠸⠧⠻⠜⠋⠿⠰⠌⠿⠇⠿⠿⠿⠏⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠐⢧⣰⠒⣶⣲⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⡆
⠈⠠⠡⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠂⠀⠀⠀⠀⠐⠀⣀
⠀⠀⠐⠀⠀⠀⠀⠀⢠⠠⡅⠇
⠀⠀⠀⠂⠀⠀⠀⠀⠀⢘⢰⡆⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠲⣾⠗⡛
⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠈⠉⡽⠙⠆
⠀⠀⠀⢀⡄
⠀⠀⠂⠠⠇⠀⠀⡔⠀⠀⢀
⠀⠀⠈⠀⢈⠀⠐⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢂⠀⠈
⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣀⣤⣤⠄⠤⠀⠀⠐⠠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣤⠨⠁⠀⠀⠀⠀⠀⠀⠨⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠐⠀⠀⠀⠀⠀⠀⠀⠈⢉⠦⠀⢠⠀⠀⡄⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⠊⢹⣿⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⠀⠀⠀⠀⠈⠋⠉⠉⠘⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⡀⡀⡀⢠⢀⠠⡀⠠⠀⠠⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠄⠄⢠⠄⡄⠀⡄⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠚⠂⠀⡄⢇⠀⠸⣀⠀⠀⢀⡸⠊⣄⡤⠼⠀⠠⠀⠀⡀⠈⠁⣃⡠⣀⠇⣀⠇⠰⣀
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄⠀⠠⠀⠀⠰⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠦⠀⣀⠀⠀⠀⠀⠀⠂⠀⠀⢀
⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⣀⡀⠉⡍⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠆⠃⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢁
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⢃⠄⠃⢀⣀⠔⠊⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⣸⠄⠶⠀⠀⠀⠀⠀⠀⡀⠀⠠⢤⣾⣦⣤⠔⣂⡀⠒⠀⠀⢀⢀⡄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⢀⡠⠀⠀⠀⡀⠀⢀⠖⠂⠀⠀⠀⠀⠐⠀⢁⢟⡌⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠀⠀⠐⠓⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠙⢗⠂⠀⠀⠀⢐⠀⠀⠀⠂⢿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⢫⡨⠀⠀⠀⢥⡀⢡⣪⠞
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⠂⠃⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⣊⠉⠉⠉⠉⢓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡡⠔⢁⣠⠤⠤⠤⠤⣈⡀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠰⠋⢀⡴⢻⢆⡠⠤⠤⢤⡀⠉⠲⣄⢀⡜⢦
⠀⠀⠀⠀⠀⢰⠃⠀⠤⣠⠋⠀⡰⡋⣤⠀⣀⣄⠈⠳⡀⠈⢦⠀⠀⢧
⠀⠀⠀⠀⠀⡏⡄⠀⡰⠁⠀⢸⢡⠀⠀⣼⠀⠀⡇⠀⡳⠰⢸⡄⠀⢸
⠀⠀⠀⠀⢠⠓⠀⠀⢇⠰⠂⢻⠀⡄⠀⠘⠮⣂⣢⠔⠁⠀⡼⠀⠀⢸⠀⣀⡤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠘⣄⠀⠀⠳⣄⠀⠀⠀⢠⡄⠀⣠⠞⠡⡄⢀⣯⠊⠁⠀⠀⠘⠃⠀⠀⢉⠓⢦⡀
⠀⠀⠀⠀⠈⡇⠀⡀⠀⠘⢦⡀⠠⡄⠉⠒⠢⠤⠔⠊⠁⠀⠀⢀⡞⠁⠐⢒⡤⠖⡒⠒⠒⢤⡈⠁⠀⠈⢢⡀
⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⠃⡙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠔⡡⡿⠑⠀⡴⡁⠂⡡⠴⠒⠲⢄⠈⠣⡀⠐⠂⠱⡀
⠀⠀⠀⠀⠀⠀⢣⡀⠠⡀⠀⠈⠐⠈⠉⢒⠢⢤⣍⣤⠴⠊⢵⠃⠀⡞⠀⠀⡼⠁⡔⠒⢦⠀⢳⡀⠙⡄⠀⡀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠑⡄⠀⠀⠚⠀⠀⠉⠐⠀⠀⠀⠂⡞⢠⠀⡇⠀⡄⢣⡠⡀⠀⣨⠇⠀⡇⠀⢹⠀⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠐⠄⠀⠀⠀⠀⠀⠛⠀⠀⢄⢳⡀⠀⠹⡄⠀⠀⠙⠒⠊⣡⠀⡸⠁⢉⠏⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⠳⣄⠀⡌⠢⣔⣀⣀⣀⡤⠚⠁⣠⠎⠀⠀⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠁⠀⠀⠀⢀⡈⠑⠦⣄⡀⠀⣁⡠⠄⣠⠞⡁⠀⡐⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠉⠙⠒⠒⠉⠁⠀⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡂⠀⠤⠤⠄⠀⠀⢀⠀⠄⠂⢠⣊⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⠤⠤⠤⠔⠒⠊⠉

⠀⠀⠀⠀⣀⠤⠔⠒⢒⣒⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⡤⠤⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢐⣒⡒⠢⠤⣀
⠀⠀⡠⢊⡖⣉⡒⡝⢓⠀⠑⠤⢈⡳⡄⠀⠀⠀⢀⠔⢉⡠⠒⢉⠴⠊⠀⣀⠬⠲⣄⠀⠀⠀⠀⢀⡴⠅⠒⠊⠁⠀⠀⠉⠉⠉⠑⢄
⠀⡜⡡⠎⠡⠖⡲⠔⠒⠠⢍⢊⠐⠍⡎⣆⠀⢀⡮⠚⠁⡠⢒⠥⠀⠴⢍⢀⠤⠊⠈⢆⠀⠀⢠⡏⠀⠀⠀⢀⠤⠀⠠⢄⠀⠀⠀⢸⢣
⠀⢣⠃⡑⠈⡜⠀⠀⠀⠀⠈⡆⠄⡀⠧⣸⠀⠈⠀⡠⠊⢠⠃⠀⠀⠀⠀⢡⢀⡠⠔⢺⠀⠀⡇⢸⠀⠀⢠⠃⠀⠀⠀⠀⢱⠀⠀⠈⢆
⠀⣧⠀⡁⠔⠰⣄⠀⠀⠀⡠⢃⡂⠀⢒⣿⠀⢀⡜⢁⠤⠊⢆⡀⠀⠀⣀⠞⠉⢀⠔⢻⠀⠀⠇⡌⠀⠀⠈⢆⠀⠀⠀⢀⠜⠀⠀⢠⠃
⠀⠱⡳⣔⠀⠡⠉⠭⠩⠙⠠⠡⠁⡥⢪⠃⠀⠈⢖⠁⢀⡔⠁⠈⡩⠛⠁⢀⠔⠁⢀⠎⠀⠀⠘⡇⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⣸⡜
⠀⠀⠑⢾⡤⠭⣈⣀⡁⢊⣐⠭⢼⠗⠁⠀⠀⠀⠈⠳⢎⢀⠴⠊⠀⠀⢠⠊⢀⠴⠃⠀⠀⠀⠀⠈⠢⣤⢄⠀⠀⠀⢀⣀⣀⣀⡠⠋
⠀⠀⠀⠀⠈⠑⠒⠠⠭⠛⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠛⠢⠭⠥⠔⠒⠉
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣤⣄⣀⠄⢀⣤⣀⣠⠄
⠀⠀⠀⠀⠀⠒⢻⠃⠀⠘⠋⣹⠂⠀⠀⠀⣽⠉⠉⣹⠆⣠⢦⣤⣤⣄⣤⠀⢀⡄⠀⡄⠀⣠⣧⣤⡆⠀⠈⠈⠈⠁⠈⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣤⠾⠶⠂⡴⠾⠥⠤⠀⠀⠀⠿⠶⠚⠁⠘⠧⠞⠀⠸⠋⠘⠶⠛⠧⠞⠛⠚⢹⠁⠈⣳⠀⢀⣀⣀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⡉⠋⢀⣓⡋⠘⠃
⠀⠀⠀⢀⠤⣒⣈⣉⣭⠭⣥⡒⢄⡀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠈⠑⠭⡑⠢⡀⠀⠀⠀⠀⠀⣠⢒⣉⠤⡒⢩⠒⠠⠬⠷⢦⡀
⠀⠀⡔⢡⠞⣊⠩⠤⣐⡒⠩⠵⢶⢌⣆⠀⠀⠀⢠⠊⡅⠀⠀⠀⠀⡀⠀⠀⠈⠢⢌⢢⠀⠀⢀⡞⠙⠀⡀⠀⢐⣄⣀⠉⠈⠐⠂⠹⣆
⠀⣼⡔⣋⠭⢒⠞⠉⠉⠉⠲⡒⢉⡠⣏⢆⠀⠀⢇⠞⠀⠀⠠⠊⠁⠀⠉⠢⡀⠀⠀⢡⢃⠀⡎⢆⢀⢀⢄⠞⠁⠀⠀⠑⢆⠉⠈⠐⡌
⠀⣧⢋⡠⠔⢊⠀⠀⠀⠀⠀⣏⡠⠔⠛⣼⠀⠸⡎⠀⠀⠀⢇⠀⠀⠀⠀⠀⠇⠀⠀⠈⢿⠀⡃⡎⠀⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠒⡸
⠀⢿⢧⠤⠐⠊⣢⡄⣀⠠⢞⡡⠄⢒⡼⡟⠀⠀⣷⡀⠀⠀⠈⠢⠤⠤⠤⠊⠀⠀⠀⠀⢘⠀⢧⠇⠉⠀⠀⡗⠤⠄⠤⢔⠁⠈⠀⠳⢁
⠀⠈⢎⢳⡔⢋⡀⠄⠒⢉⡠⠔⠊⣝⡜⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠈⢇⠈⠀⠂⠀⠐⠀⡀⠈⠀⣡⠀⣸⠜
⠀⠀⠀⠙⠫⢝⡒⠢⠴⢡⣔⡪⠝⠉⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣀⣠⣤⡶⠟⠁⠀⠀⠀⠀⠀⠙⠯⣲⣌⣀⠀⠀⢰⡀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡡⠓⠁