RIP Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠒⠒⠒⠒⠒⠒⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⢉⣤⣴⠶⠟⠛⠛⠛⠛⠛⠛⠻⠶⢦⣄⡉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⠟⠁⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⠀⢀⣀⣀⡀⠀⠀⠀⠈⠻⣦⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⣾⡏⠀⠀⠀⠀⠀⣿⣏⣿⠆⠀⢸⣏⠀⠈⣿⣉⣿⠆⠀⠀⠀⠀⢹⣷⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢻⣇⠀⠀⠀⠀⠀⣿⡏⢿⡆⠀⢸⣏⠀⠀⣿⠉⠁⠀⠀⠀⠀⠀⣼⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⣿⡀⠀⠀⠀⠀⠛⠃⠻⠛⠛⠛⠛⠛⠛⠻⠃⠀⠘⠃⠀⠀⢀⣿⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⣿⣦⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢖⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣻⣥⣾⣿⣿⣯⡏⢻⠉⢻⠿⡇⠀⠘⣿⡟⡿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠟⡇⢸⠀⠰⠀⡇⢸⠐⢹⣴⣦⣷⣿⢿⣿⣿⣷⡌⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⠛⣻⢡⣿⣿⡇⠂⠂⠈⠀⠀⠀⡇⢸⠀⢠⠬⣑⡯⣿⣾⣿⣿⣿⣇⣀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢟⣤⣬⡷⠲⠷⠶⠶⠶⢺⠶⠷⠾⠶⠶⢾⠾⡿⣿⣿⣿⣟⠙⠛⢿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡎⠁⠀⠀⠀⠄⠀⠀⠀⢸⠀⠆⢰⠀⠀⠰⠀⢇⡿⣿⣿⣿⡿⠶⣾⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢻⣿⡿⢷⣶⣦⡀⢺⣶⣶⡟⢶⣶⣶⣶⣾⣧⣿⣿⣿⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣸⣿⡇⠀⣹⣿⡇⢸⣿⣿⠃⢠⣿⣿⠀⢈⣿⣿⣿⣿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢹⣿⣧⣶⣿⣟⠃⠘⣿⣿⠰⠘⣿⣿⣤⣾⡿⠟⢻⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⣿⡇⠈⢿⡹⣆⠀⣿⣿⠆⠐⣿⣿⠀⠝⠀⠀⢻⣿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢼⠿⠷⠄⠘⢷⣿⣄⣿⣿⣇⣠⣷⣿⣦⢀⠀⠀⣿⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠁⠀⠈⠉⢹⠰⠀⠀⣿⡏⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⣤⣦⣴⣴⡂⠒⠂⠒⠐⣦⣶⣤⣾⡶⠀⠀⣹⡇⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠈⡝⢁⠀⠀⠈⢯⠀⠀⢸⢸⠀⡀⣾⣿⠛⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⢠⢰⣞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣄⣰⠷⠾⠓⡂⣸⠘⠀⣷⣿⣿⣦⣶⣿⡇⣰⣦⡄⣰⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⣄⣦⣘⣿⣻⣃⣠⣤⣀⣀⣀⣠⢆⣀⣀⣀⣉⡁⢈⣀⢀⣆⣿⡃⣿⡘⠰⣿⣿⢛⠉⠥⠿⣻⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠀⠿⢟⢻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠈⠁⠈⠉⠉⠉⠉⠛⡙⠿⡻⣟⢿⣧⡀⠀⠀⣀⣴⢈⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡚⢦⣠⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣽⣿⣿⣦⡾⠁⠿⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠑⠀⢙⠃⢀⠸⣿⡿⣿⣿⣿⣿⢿⡽⡟⠏⠟⠻⠻⢯⡻⠉⠉⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠠⡴⢋⡾⢠⠍⣐⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠚⠻⠮⢸⡈⠇⠉⠀⠊⠀⠊⠄⠀⠘⠰⠀⠀⠀⠀⠀⠄⠠⠄⠑⠐⠻⠛⠙⠛⢿⢿⠿⣿⣿⡿⢿⡿⠃⠈⠐⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡶⣶⣶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⢳⣿⢽⣷⠛⠿⣿⡫⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣮⣶⢣⠟⠋⠉⠉⠙⠢⡽⢯⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣏⡴⠋⠀⠀⠀⠀⠀⠤⠛⢻⡿⣧⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢻⠁⠀⠀⠄⡀⡀⡤⡀⠀⠀⡟⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⣿⠈⡇⠀⠀⠟⠄⠂⠋⠀⠀⢰⢐⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⡛⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠘⠀⠀⠀⠀⠀⠀⠀⢀⢇⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠄⠀⠀⠀⠀⠀⠀⣘⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⢷⢸⣀⡀⠀⠀⠀⢀⡇⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣫⡦⢷⣧⠤⠤⡤⠼⠉⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⣷⣶⣶⣶⢦⣤⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⡿⢿⣿⡿⠿⢿⠿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠶⠿⠿⠿⠿⠿⠶⠿⠿⠶⠶⠾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣫⡥⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⠋⡴⠋⠀⠀⣀⠀⠀⠙⣆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⢸⠁⠀⠀⢬⡿⡇⠀⠀⢻⡀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠈⠀⠀⠀⠀⠐⣋⠄⠀⠈⠉⢿⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠂⠀⠀⠀⠀⠙⢾⠁⠀⠀⠀⠀⠈⢷⣝⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣷⠛⡥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢻⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢶⠖⠲⣦⡄⠀⢻⡇⠀⠀⢹⣯⠉⢿⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⢹⡆⢀⡼⠃⠀⢸⡇⠀⠀⢸⣿⢀⡾⠏⢨⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡄⠀⠀⢹⡇⠙⢿⣆⠀⠸⣧⠀⠀⢰⣿⠀⠀⠀⢨⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠈⣿⡇⠀⠀⠸⠧⠀⠈⠻⠦⠸⠻⠄⠀⠀⠛⠃⠀⠀⠀⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠯⡇⠀⠀⠰⠒⠒⠒⠒⠒⠒⠛⠛⠛⠋⠉⠙⠛⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠶⢿⠄⠠⠄⠀⠒⠒⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠉⠉⠉⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣤⢀⣄⡀
⠀⠀⠀⠀⠀⡀⣀⣀⣄⣀⣸⣿⡟⢟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠋⠑⠚⠻⠗⠾⠗⠓⠚⠓⠃
⠀⠰⠶⠃⠿⠻⡿⠿⠟⠻⠫⢫⠞⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠹⡄⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠳⡀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣠⠔⢢⣹⡤⡆⠀⠀⠉⠑⠢⣄⠀⠀⢀⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⡞⠁⠀⠀⠙⢻⣇⠀⠀⠀⠀⠀⠈⢳⡀⠈⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⡭⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠠⡤⢤⡤⢤⢠⣤⠤⣤⡤⢤⣤⢤⡀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇⢸⣽⣠⠇⢸⠀⡇⡇⢸⣘⠆⣷⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇⢸⡍⢧⠀⢸⠀⡇⡇⢸⠿⠔⠃⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇⢸⢳⠸⠂⢸⠀⡇⡇⢸⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠛⠋⠈⠳⠇⠘⠚⠃⠑⠚⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠀⢸⠀⠀⠀⠀⠀⣀⣠⡤⠀⠀⠀⠀⠀⡀⠀⡞⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣚⠷⠴⣺⠟⠋⠁⠀⠀⠀⠀⠀⠀⠙⢏⡁⠀⠘⠦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣭⣽⣭⣤⣤⣤⣤⣤⣤⣤⣤⡼⡏⠉⠉⠁⠠⠤⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠽⠤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠛⠛⠛⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢰⣿⡿⣷⣦⠀⠀⢰⣶⡄⠀⠀⣿⡿⢿⣶⡀⠀⠀⠀⣿⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣤⣾⠟⠀⠀⠸⣿⡇⠀⠀⣿⣇⣠⣿⡇⠀⠀⠀⣿⢤⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢸⣿⠛⢿⣧⠀⠀⠀⣿⡇⠀⠀⣿⡟⠛⠉⠀⠀⠀⠀⣿⠠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠘⠿⠀⠘⠿⠃⠀⠀⠿⠇⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⣿⢠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠀⠀⡿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣻⡵⠂⠀⢰⡟⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⠛⠁⠀⠀⠀⢸⡟⠳⣿
⠀⠀⠀⠀⠀⠀⢸⣷⣀⣿⠀⣰⡆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠶⡿
⠀⠀⠀⠀⠀⠘⢿⡿⠙⣿⣠⡿⣧⣶⡿⣡⠾⠃⠀⠀⠀⣤⡀⢸⡆⠀⠀⢠⣤⣿⣦⣀⣼⣉⣿⣧⣤⠄
⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠉⠀⠁⢨⠿⠿⠶⠶⠶⠶⠖⢿⠿⡮⠿⠒⠒⠚⢿⡙⠟⠙⠛⠋⠛⠻⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⣖⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠹⣶⣽⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠦⣭⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⣀⡤⠴⠒⠒⠒⠒⠲⢤⡀⠀⠀⠈⢳⣄⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⠉⣀⣀⡀⢀⣤⠀⡤⢤⡀⠹⡄⠀⠀⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠠⠆⢇⢩⣠⡇⠀⡇⠀⡷⠖⠁⢰⠀⠀⢰⢟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⠈⠇⠙⠂⠃⠀⠃⠀⠀⡘⠈⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡆⠀⣀⣀⣀⣀⣀⠀⠀⣧⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⢰⠀⢠⣀⣤⣤⣤⠀⢰⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠉⠈⡆⠀⠤⠄⠤⠄⠀⢸⠀⢸⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⠒⠒⠓⠒⠒⠒⠲⠦⠤⠿⠤⣾⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠦⠧⠴⠦⠤⠤⠤⠿⠼⠦⠧⠼⠿⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢚⣩⠥⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣴⠏⣴⠟⠁⠀⠀⠀⠀⠙⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⢸⠃⠀⠀⢀⣿⡗⠀⠀⢸⡇⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡏⢁⠘⠀⠀⠀⠈⢡⠷⡁⠀⠘⠻⢶⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠁⠄⠀⠀⠀⠀⠲⣄⠞⠁⠀⠀⠀⠈⢳⡜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣯⠿⠃⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠳⡌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠐⠁⢀⣀⣀⣀⣀⠀⠀⢴⡶⠀⠀⢰⣶⠶⢶⣄⠀⡗⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⣿⠀⠀⢻⡇⠀⢨⣷⠀⠀⠈⣿⡁⢀⣿⠆⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢿⡆⠴⣯⡀⠀⠈⣿⠀⠀⠀⣻⡆⠋⠉⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢻⣹⠀⠀⠀⢹⡇⠀⠸⣷⡀⠀⣿⡀⠀⠀⣿⣇⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠸⣿⡇⠀⠀⢚⣋⣀⣀⣈⣉⣈⣉⣁⣤⣤⣤⣬⣤⣤⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠻⡇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣃⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢒⡛⣿⠀⠚⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣶⣠⣤⠄⣀⡀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⣠⣤⣴⣤⣤⣿⣿⢛⣻⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠘⠉⠙⠛⠿⠛⠛⠉⠛⠉
⠀⠐⠚⠋⠹⠓⠽⠛⠟⠛⠋⠟⠵⠛⠋⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⡷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⣉⣭⠭⠭⠭⠭⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣼⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡏⠀⠀⠀⠀⢰⡿⠲⣆⠀⣶⠀⢸⠗⢶⡄⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣧⠀⠀⠀⠀⢸⣇⣠⡟⠀⣿⠀⢸⣥⣼⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢻⠀⠀⠀⠀⢸⡏⠉⣷⠀⣿⠀⢸⡁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⡆⠀⠀⠀⠘⠃⠀⠛⠀⠛⠀⠘⠂⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢷⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣳⡀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠳⣄⡀⠀⠈⠳⢤⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠦⣄⠀⠀⠉⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠾⠿⠿⠿⠿⠿⠷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⢻⣿⡛⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠋⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠟⠁⠀⠀⠙⠻⣶⣄⠉⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠰⠋⠁⠀⠀⣴⣿⠏⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⣀⣤⣀⣀⣀⣴⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠈⠉⠀⣩⠟⠉⠁⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣿⣆
⠀⠀⠀⠀⠀⣠⣶⡶⠿⠆⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠙⢷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⣿⡆
⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⣠⣄⣀⣤⡆⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⣠⣶⣤⣴⣶⣄⠀⠀⠀⠀⠀⢻⣧⠀⠀⠸⣷
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠠⢿⣿⡏⠙⣿⣿⣆⠀⠀⠀⠀⠰⠞⣿⣿⠁⠀⠀⠀⠀⠰⠞⢻⣿⡇⠘⢿⣿⣦⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣇⢀⣸⣿⠟⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⣺⣿⣇⣀⣸⣿⠟⠀⠀⠀⠀⠘⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⢻⣿⡟⠙⢻⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠛⣹⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⣴⠿⣿⣿⠆⠀⢿⣿⡞⠁⠀⢠⡾⠿⢿⣿⡗⠃⠀⠀⠀⣴⡛⠛⠿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠷⠂⠀⠁⠀⠀⠀⠉⠁⠀⠀⠹⠞⠀⠀⠉⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⣤⠤⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠂⠀⣸⡿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⣿⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠈⣿⡆
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣇
⠀⣤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣷⠀⣴
⠀⠸⣿⣷⣤⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣷⣿
⠠⣄⣹⣿⣿⣇⠀⣼⣧⣤⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⣴⣦⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢠⣿⣿⣿⣿
⢀⡘⢿⣿⣿⣯⣰⣿⣿⣿⣀⣀⣀⣀⣀⣀⣸⣿⣷⣀⣀⣀⣀⣸⣿⣀⣀⣀⢀⡀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣾⣿⣿⣧⣴⣿⣿⣿⣿⡟
⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠟⠿⢿⣿⡿⠋⠉⠉⠛⠿⣿⣿⡿⠛⠛⠻⢿⣿⣿⣿⠟⣿⣿⣿⣿⡟
⣀⣀⣈⣛⣛⣛⣛⣋⣁⣀⣀⣉⣻⣿⣿⣟⣋⣙⣛⣃⣀⣈⣛⣿⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣈⣛⣛⣛⣁⣀
⣿⣿⣿⠟⠻⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠈⠑⢮⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣾⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⢐⠨⡇⡇⣂⣽⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⡯⣗⣾⣺⠽⠂⡇⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣷⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠁⠈⠈⠈⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣟⣻⣟⣿⣶⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠋⠒⠒⠒⠒⠦⠤⢤⣷⠖⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⣠⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣯⣷⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠑⡉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⠿⣿⢿⠿⢿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢀⣀⠀⡀⣀⡀⢰⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣇⣈⢸⢸⢈⣴⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⢸⠿⡇⣿⡷⠃⢸⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣯⣯⣽⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡍⠀⠉⠉⠉⠉⠛⠒⢺⠋⠉⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣄⣀⣀⠀⠀⠀⠀⠘⠁⣠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠿⠿⠷⠶⣶⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⡀⠀⠀⠀⠀⠐⠒⠒⠢⡄⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⢰⡀⡆⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠶⠶⠶⣦⣤⣀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⡇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢴⣉⡻⢦⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⣿⣿⣿⣿⠿⠷⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⠀⠄⠀⠀⢰⠀⠀⠀⠉⠁⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣿⣧⣾⠟⠀⠀⠀⠀⠀⠉⠙⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠘⠀⠀⠀⠀⠀⠘⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⡴⠎⠉⠁⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢰⠀⠀⣶⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠘⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⡀⠒⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣶⣦⣤⡀⠀⢀⣼⠿⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⠃⠘⣿⣿⣿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣼⡇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠐⠀⢰⡀⠀⠈⣻⣿⣍⠁⠀⠂⠲⣶⣶⣶⣄⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠻⠁⠀⠀⠀⠀⠀⠀⢄⠀⢸⡁⠀⢺⣯⡈⠻⠷⠶⠶⣤⣬⣿⣿⣿⣦⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣼⠃⠄⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠟⠀⠀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⠆⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠶⡀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠻⠷⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⡀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢋⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠞⠁⠶⠼⠃⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠎⢀⡠⣾⠀⠀⠀⢸⣿⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⡾⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⣴⠶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⣤⠟⠀⠀⡀⠀⠀⠀⢸⠋⠱⣄⠀⠀⠀⡀⠀⠀⠀⢻⠀⠃⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⣿⠿⣅⠀⠀⠀⡇⠀⠀⢀⡿⣀⡠⠟⠀⠀⠀⠁⠀⠀⢠⠀⡐⠀⣾⠇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠙⠃⠀⢠⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢸⣿⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠙⠀⠠⠀⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠃⡀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣶⠀⠀⢰⣿⠃⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠑⠀⠀⣼⠇⡆⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠈⠀⠀⢠⡟⠀⠃⢠⣍⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠂⣰⣦⣼⡇⠀⠀⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢂⣼⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⠴⢼⡟⢀⣿⢷⡆⠀⣲⡄⠂⠟
⠀⠀⠄⠀⠀⠀⠀⠀⠘⠋⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠔⠒⠃⠀⠀⠀⠈⠠⠊⠀⡘⠋⠀⠊
⠀⠀⣤⡀⠀⠀⠀⢰⡄⠀⡿⠀⠀⠀⠈⣡⠀⠀⠀⠀⠀⢠⠖⠀⠐⠀⢀⣀⣠⡟⠹⠇⠀⠀⠈⠉⢂⠀⠀⠀⢀⡀⠉⢡⡴⡁
⠀⠀⠛⠍⠁⢠⡄⠃⠀⠀⠃⠀⣠⠞⠙⠋⠙⠢⠀⠴⠖⠋⠀⠀⠀⠀⠀⠀⠈⠙⢤⡀⠀⠀⠀⡀⠈⠁⣀⢠⠀⠰⠀⠠⠄
⠀⠀⢳⣦⠶⡈⠑⠀⠂⢛⣶⠃⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⢀⠀⠹⠃⠀⣸⠀⠈⠃⠀⠀⠀⠀⢠⣴⠆
⠀⠀⣠⡤⠶⠀⠠⣤⠔⠋⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠘⠛⣷⠂⠀⠀⠠⠐⠣⠀⠀⠀⠀⠙⢣⡤⡀
⠀⠀⠙⠠⡀⠈⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠚⠀⠈⠀⠀⠀⠀⠰⠀⠀⢀⠀⠘⠀⡁
⠀⠀⡀⠀⢀⡴⠃⠀⢀⡀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⠀⠉⠀⢀⡀⠀⠀⢀⡀⠀⠦⡀⠀⢣⡀
⠀⠀⡄⠰⠞⠀⠠⣤⡦⢶⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⠀⠀⠴⢧⠴⠄⠀⠈⠁⠀⠀⠀⠀⠈⠶⠆⠀⠀⠈⠣
⠀⠀⠀⢀⣠⣄⡛⢁⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠆⠛⡁⠀⠁⠀⠄⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣤⠤⠤⠄⠀⠐⠛⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂⠀⠀⠀⣀⡐⠂⣀⣀⡀⠀⠙⠂⠀⣀⣠⣤⠄⠂⠉⠈⠀⠀⠈⠉⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢉⡉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣯⣷⣿⣿⡟⡟⠹⠿⡏⠉⢿⡿⢿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⠟⠅⡇⢰⠀⠇⡀⠙⢷⣾⣿⣻⣿⣿⣧⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢪⣹⠘⢿⣟⣀⣀⣀⣀⣀⣄⣃⣀⣢⣸⣿⣿⣿⡿⣿⣤⣤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⠛⠋⠁⠁⠉⠉⠈⠉⠁⡍⠉⠉⠉⢹⣿⣿⣷⣤⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⣶⣶⢶⣦⣄⠰⣶⣶⠷⣶⣶⣦⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⣿⠀⣽⣿⠀⣿⣿⠀⣿⣿⠀⣹⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⡿⠻⣿⣏⠀⣿⣿⠁⣿⣿⠳⡟⠉⢽⡏⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣷⡀⢹⣼⡆⣿⣿⢀⣿⣿⡀⠀⠀⣼⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠋⠉⠀⠀⣿⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠰⠶⠶⠶⣶⣐⠂⠂⣲⡶⠶⡶⠆⠀⣼⣇⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢰⣤⡄⢠⣜⣆⠀⣥⠅⡄⣾⣏⢁⣸⡆⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡇⢀⣤⠰⣾⣦⣀⣀⠀⠀⢀⠀⠀⠈⠳⠏⠁⠁⡄⣿⢰⠓⢹⣿⡿⠛⣃⡿⡇⣾⡏
⠀⠀⠀⠀⠀⠀⠠⣮⠰⠿⠻⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠻⢺⡻⣯⡀⠀⠈⣰⢉⠉
⠀⠀⠀⠀⡀⢰⣿⠏⢦⣾⣷⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣿⣿⣿⣿⣿⣾⣶⣶⣶⣶⣶⣿⣿⣶⡏⠘⣿⢀
⠀⠀⠀⠀⠀⠀⣥⡠⡝⢿⠿⠟⠛⠛⢟⠋⡅⡉⠉⠱⠁⠀⠉⠋⠙⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠄⠘⠕⠋⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠉⠋⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠒⠀⠀⠂⠒⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠁⠀⠂⠑⢀
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⢐⡤⠓
⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⠀⠀⠀⠀⠀⣊⣀⠁⠀⢂⣀⣀⡈⢀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⠙⠁⠀⡀⠀⡏⠘⠀⠀⢸⠀⠉⢉⢂⢀⠄⠡
⠀⠀⠀⠀⠀⡀⠀⠀⢸⢰⢞⠄⠀⠀⠀⠍⠀⠀⠀⢸⢀⠶⠦⠀⠀⢈⣰
⠀⠀⠀⠀⡀⠀⠀⠀⣸⢀⠀⡧⡀⣀⠀⡆⢀⢀⡀⣸⢀⡀⣀⠘⠀⢀⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠈⠁⠀⠀⢀⢊⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠉⠄⠈⠲⡇
⠀⠀⠀⠀⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢢
⠀⠀⠀⠀⡃⢈⠀⠀⠀⠀⠁⠊⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠌⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠧
⠀⠀⠀⠀⠀⠐⠀⢀⠀⣀⡂⡀⢎⣤⢤⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡒⣒
⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⢁⠁⠀⠀⠀⠘⠇⢈⡙⠭⣷⠖⠴⣤⣀⠈⠈⠏
⠀⠀⠄⠠⠐⠂⠄⠀⠀⠀⠉⠁⠀⠀⠈⡁⠠⠀⠃⠢⠵⠄⠙⠨⢒⣳⣠⣿⢀
⠘⠲⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠛⢉⡠⢴⠀⠚⡂⠀⡩⣼⡟⠹⣶⣤⡀⠀⠀⠀⢀⡀
⠀⠀⠈⢄⠀⠀⠀⠀⠀⢀⡄⠐⠀⠀⠀⢀⡐⠈⠁⠀⠀⠀⠀⠟⢛⣨⠁⡜⠛⠻⠐⠉⡹⣶⠤⡊⠁⠔⠊⠀⢀
⠀⠀⠀⠀⠙⠢⠤⡀⣀⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠝⢞⢁⡠⠖⢒⡿⠋⠈⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠒⠲⠒⠲⠖⠖⠚⠲⠶⠲⠶⠖⠛⠓⠒⠚⠋⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠟⠛⠛⠛⠻⢶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠻⣿⣙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⢠⠖⠛⠛⢛⣿⠿⠿⠛⠛⠛⠉⠀⠀⠀⠈⠻⣷⣄⠈⠛⢧⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠙⢷⣦
⠀⠀⠀⠀⠀⠀⢾⠁⠀⠀⢀⣤⣄⣀⣠⡞⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣀⣸⡆⠀⠀⠀⠀⡴⠉⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠹⣧⡀
⠀⠀⠀⠀⢾⠛⠛⠁⠀⠀⠀⠜⠀⠀⠀⠀⠀⠈⠻⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠈⣷
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣀⠀⣀⡄⠀⠀⠀⠀⠀⠁⣤⡀⠀⠀⠀⠀⠀⣠⣄⣠⣤⠀⠀⠀⠀⠘⣷⡀⠀⢹⡆
⠀⠀⠀⢰⡇⠀⠀⠀⠠⢾⣿⠙⢿⣷⡄⠀⠀⠀⠰⢿⣿⡏⠀⠀⠀⠠⠞⣿⣿⠙⢿⣷⡄⠀⠀⠀⢹⡇⠀⠈⣷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⣿⣀⣼⣿⡃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣀⣿⣿⣠⣼⠿⠃⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠸⡇⠀⠀⠀⢈⣽⣿⣀⠙⣿⣆⡄⠀⠀⣠⣼⣿⣇⠀⠀⠀⢀⣥⣿⣿⡤⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢀⡄⠀⠀⠀⣯⠉⠛⠁⠀⠙⠛⠀⠀⢸⡥⠉⠛⠁⠀⠀⠀⠻⠆⠀⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠘⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢰⠖⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣼⠇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠻⣆
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡆⢀
⠀⢿⣦⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣧⣸
⢠⣈⣿⣿⡇⠀⣾⣠⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⣴⣿⡇⣠⣆⠀⠀⠀⠀⠀⣼⣿⠀⣨⣿⣿⣿
⢠⣙⣿⣿⣧⣼⣿⣿⣤⣤⣤⣤⣤⣼⣿⣧⣀⣀⣀⣿⣇⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣴⣿⣿⣿⡟
⠀⠹⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⡿⣿⣿⠛⠻⣿⣿⡿⠋⠙⠻⠟⠁⠀⠀⠙⠻⠿⠋⠉⠉⠛⠿⠿⠋⠻⣿⣿⡿
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠋⠉⠉⠉⠉⠉⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⣀⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢻⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣿⡟⢻⣿⠀⠀⢸⣿⡆⠀⠸⣿⡟⢻⣷⡀⠀⠀⢸⡗⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⣿⣏⠀⠀⠀⣿⡇⠀⠀⣿⣷⣾⠟⠀⠀⠀⢸⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣿⡇⠘⣿⡆⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣇⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣯⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣋⠁⠀⣸⣦⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⠿⠋⠁⠀⠀⣿⢦⣽⡇
⠀⠀⠀⠀⠀⠀⢶⣄⢸⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⢦⣼⠁
⠀⠀⠀⠀⠀⠰⣾⡟⢿⡇⣸⣿⣠⣶⢇⣴⠆⠀⠀⠀⣀⠀⢰⡀⠀⠀⣠⣀⣦⣄⠀⡟⣳⣿⣀⣄
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠙⠃⠙⠁⣿⠿⠷⠶⠶⠶⠶⢿⡷⣼⣷⠶⠶⠾⣿⠻⠏⠻⠟⠹⠟⠿⠖⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⢚⣉⣭⣭⣭⣭⣭⣍⣙⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢉⣠⠶⠛⠋⠉⠀⠀⠀⠀⠀⠈⠉⠙⠳⢦⣈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣰⡟⠀⠀⠀⠀⢸⣟⣻⡆⠀⢻⡇⠀⢻⡟⣷⠀⠀⠀⠀⢹⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣷⠀⠀⠀⠀⢸⡟⢻⡇⠀⢸⡇⠀⢹⡟⠉⠀⠀⠀⠀⢸⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢻⡆⠀⠀⠀⠘⠛⠘⠛⠛⠛⠛⠛⠛⠃⠀⠘⠃⠀⠀⣾⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⣷⣄⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣸⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠉⠉⠉⠉⠉⠉⠉⠐⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠦⢤⡀⠀⠀⠀⡄⠀⠀⢦⠤⣄⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⢹⠀⠀⠀⡇⠀⠀⣭⠀⠀⣹⠂⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⣖⠉⠁⠀⠀⠀⡇⠀⠀⣿⠋⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠈⠳⣄⠀⣀⠀⡇⢠⠀⡟⠀⣀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠐⢄⠠⠀⠀⠀⠀⠇
⠀⠀⢀⠀⠀⠀⣤⣤⣤⣤⣤⣤⠤⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢢
⠀⢠⡌⠓⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢨⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡀⠀⡤⠒
⠄⠀⠛⢄⢃⠈⠰⣃⠀⠱⡀⡀⠐⠓⠦⠀⣤⠈⠂⠳⠀⠀⠘⠂⠐⠡⢀⠔⠁⠰⡐⠃⠀⠠⠃⠀⠤
⣀⣀⣈⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣐⣀⣀⣀⣀
⣿⣏⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢏⠁⠀⠀⠈⠉⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠑⢀⢠⠤⠤⠤⠤⣀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⣠⠞⠁⠻⢿⡇⠀⠀⠀⠙⢦⠀⠐⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢰⠃⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢳⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠩⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡟⡿⡑⡔⢿⢹⢳⢸⢷⢢⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⣿⢯⠁⢸⢸⢸⢸⣮⠜⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⡟⡄⠇⢸⢸⢨⢸⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⡇⠀⠉⠁⠈⠃⠈⠉⠈⠉⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠀⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢳⣲⢤⡶⠞⠋⠁⠀⠀⠀⠐⢶⡃⠀⠢⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠻⠽⠭⠤⠤⠤⠤⠤⠤⠼⠏⠉⠁⠐⠤⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠄⠂⠐⠢⠤⠄⠤⠤⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⡑⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠀⠀⠀⠀⠈⠊⠔⠕⡥⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢊⠔⣡⠘⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠒⠀⠀⡀⠀⠀⠀⠀⡝⡡⠪⠘⡀
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⣤⠤⠤⣀⠀⠀⠀⠀⠀⣤⡤⢤⡄⠀⠀⢠⡤⠤⠤⠤⣄⠔⢅⢅⠱⡄
⠀⠀⠀⠀⠀⠀⢣⠀⠂⠀⠀⠈⣷⢸⠉⢪⢱⠀⢀⡀⠀⠸⡇⢸⠀⠀⠀⠀⣿⠀⠉⠑⡌⢗⡡⣢⠂⡜⠆
⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⣿⢸⣠⠞⡸⠀⠈⠀⠀⠈⡇⠸⠀⠀⠀⠀⡏⠰⣦⣤⢃⠎⠀⢑⠕⠁⣄
⠀⠀⠀⠀⠀⠀⡃⠸⡀⠀⠀⠀⣿⢰⠀⢢⣡⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⢰⠉⠉⠁⢀⢔⠅⠸⠟⡿
⠀⠀⠀⠀⠀⢀⠇⢨⠀⠀⠀⢀⣼⢸⡄⠘⡞⢤⢀⣤⠀⣨⡇⢤⡀⣠⡀⣀⡇⢨⡀⢀⢄⠁⡡⢂⢈⡪⣺
⠀⠀⠀⠀⠀⠸⠀⢨⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠀⠉⠀⠉⠉⠉⠁⠉⠀⠉⠁⠉⠁⠈⢈⠐⠔⢡⠏⠜⢴
⠀⠀⠀⠀⠀⠀⠄⢨⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⢀⠀⠠⠬⠂⢃⡼⣂⣹⡄
⠀⠀⠀⠀⠀⢀⠆⠐⡨⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠕⡡⠀⠀⠀⠀⡠⠂⢀⢀⠂⠜⡀⠠⠨⠋⡇
⠀⠀⠀⠀⠀⠘⠆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠞⠀⢀⢀⠌⢀⠔⡐⡁⡀⠀⠜⡀⠐⢋⠔⣧
⠀⠀⠀⠀⠀⢸⠀⢠⠆⠀⠀⠀⠀⢀⠜⠀⡴⠃⠤⠖⠁⢀⢀⠔⡐⡡⠀⠀⢀⢈⠀⠔⢀⠂⢈⠎⡸⠃⢈⢜⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠟⠀⠠⠀⢀⠄⡀⠀⠀⠀⠔⢁⡴⡡⡢⢊⢌⠄⢐⡔⡠⡡⠀⠨⠁⠀⠠⠊⠁⠡⡠⢊⡇
⠀⠀⠀⠀⠀⠀⡆⠀⠕⠠⠂⠀⠁⣊⠅⡀⡨⢊⣰⢊⠈⠐⠕⠑⡡⡪⢈⢄⠄⠔⠐⠐⠁⠐⡐⠑⠀⡐⡰⠱
⠀⠀⠀⠀⠀⠀⡇⠀⠪⢊⠔⡡⢊⠔⡨⢊⢔⠄⠕⡁⡀⡡⡪⣊⢈⠔⠡⠁⠈⠂⢂⢀⢈⢌⢀⠈⡈⣐⡴⡇
⠀⠀⠀⠀⠀⠀⠘⠀⢄⠀⡡⢒⠕⡠⢒⠅⡠⣂⡴⡪⠊⢊⠄⠐⠡⢊⠔⠔⠐⡁⠠⡠⠀⠁⢀⢊⣏⢉⢈⡍
⠀⠀⠀⠀⠀⠀⢀⠀⡃⠪⠐⡡⢊⣔⡱⢨⡊⠎⣴⣴⠳⡇⢺⣔⡡⠡⠊⠈⠔⠀⠜⡔⠐⠀⠁⡰⣧⡉⣉⣉
⠀⠀⠀⠀⠀⠀⠸⠀⢆⡵⠈⠁⠈⠀⠈⠈⠁⠀⠀⠀⠀⠀⠺⣄⠀⠋⠑⣎⡻⣦⡔⢆⣬⣄⢐⠌⠔⠔⠁⡅
⠀⠀⠀⠀⠀⠀⣠⣠⠁⠀⠀⠀⠀⠎⠱⢷⠂⠀⠀⠐⠤⠖⠀⠀⠀⢻⡆⢀⠼⡑⠂⠰⢥⠙⢟⣫⡌⠪⠀⣶
⠀⠰⠒⠉⠉⠊⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⣰⠀⠐⠂⠀⠉⣀⠛⠉⠒⠂⢀⢍⡉⣶⡝⢷⣵⣯⣂⣔⡠⠑⡊⠄⢂
⠈⠛⢷⣃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠠⡬⠑⠠⠛⠉⣉⠰⣨⣆⠑⠾⢣⠤⡰⡸⣤⣺⣽⠇⢯⣹⣖⡠⢒⠥⢐⠠⢀⠠
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⢠⣀⠛⠂⡀⠉⠁⠈⠑⠨⣞⢯⠇⠐⢛⠄⡿⡞⣶⡶⠬⠙⠟⢳⣊⠄⠀⡡⠚⠉⣃⡥⠒⡂
⠀⠀⠀⠀⠙⢢⣄⡄⠀⡀⠀⠀⠀⠀⠏⠁⠀⣠⣄⡀⠀⠀⠰⠶⠀⠀⠆⢁⠀⢁⠀⠃⠄⢃⠀⣶⣾⢸⡰⣃⡀⢠⠉⠀⢰⢠⣇⣿⠟⣩⠦⡀⠈⠨⢐⠅⠂⠴⠂
⠀⠀⠀⠀⠀⠀⠉⠛⠴⢤⣄⣰⡀⠀⠀⢀⠀⢀⠀⠀⠀⠀⡀⠀⠆⠀⡀⡈⡈⡀⡀⠀⠀⢸⠀⠀⠁⠠⠹⡇⡇⣙⡰⠖⠋⣸⡽⡃⠋⠁⡠⢀⠁⡨⢀⢁⡬⠢⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠓⢬⠶⢮⡮⣼⠴⡴⣥⡥⢤⡥⠷⢴⣠⢡⣤⣧⢇⡬⢶⠷⠶⢤⡴⠒⢖⠉⡀⢐⠔⠙⠀⠊⠀⠁⠊⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠙⠛⠛⠛⠳⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⡀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠒⠒⠒⠲⠶⢤⣤⡀⢻⣆⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢀⠀⠀⠣⠝⣳⡿⠀⠀⠀⠀⠀⠀⢿⣹⣿⢛⣿⠛⠛⠳⠶⠶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠁⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⢁⡠⠒⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡄⠀⡆⣹
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠘⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠤⠤⠤⠴⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⣦⣄⠀⢀⡽⠟⠀⢰⠀⠀⠀⠀⠀⠘⠉⢿⣿⣇⢀⡀⠀⠀⠈⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢾⠇⠀⣀⠄⠀⠀⠀⠀⠈⠀⣤⠀⣈⡿⢧⣀⣀⠘⠻⣶⣦⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠈⠀⠀⠀⠀⠀⠸⠀⡷⠀⠘⢷⡌⠉⠉⠙⠛⠛⠛⠷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢄⠀⠀⠐⠒⠒⠒⠒⠖⠶⠦⠤⠄⣀⠀⠀⠀⠀⠀⣾⣷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡄⠈⣦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⡌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡤⠖⠛⠀⠀⣸⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⢠⡴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⢸⣇⡼⠁⠀⡀⠀⠀⣾⠉⢢⠀⠀⠀⠀⠀⠸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⢸⡟⢦⠀⠀⠇⠀⢠⡇⠒⠉⠀⠀⠀⠀⠀⡆⠀⢸⡇⠄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠘⠁⠀⠀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡆⠀⢸⡟⠐⠒
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⡾⢸⠀⠿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣠⣠⡇⠀⠔⠳
⠀⠀⠀⠀⠀⠈⢀⣾⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠴⢼⠇⡿⢷⠀⢰⡄⠲⠂
⠀⠐⠀⠀⠀⢀⠈⢸⠀⠀⠀⠀⠘⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣠⠳⠊⠉⠀⠀⠀⠀⠀⡤⠉⠀⠁
⠀⠸⡖⠄⣀⠜⠁⠸⠀⢀⠤⠶⠢⠀⢀⡠⠞⠀⠀⠀⠐⠘⢇⡈⠀⠀⠀⠐⠆⠀⠀⢂⠀⡉⠃
⠀⠐⣤⠦⠙⠀⠐⢳⠆⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⠠⠤⠤⠀⠘⠇⠀⡎⠐⠂⠁⠀⠀⢤⡆
⠀⢠⠖⠒⠀⠲⠊⠁⠀⠀⠀⠀⠒⠢⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⢻⠁⠀⠀⠈⠃⠀⠀⠀⢱⠖
⠀⠀⠀⠂⡠⠁⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⡆
⠀⠀⠠⠞⠀⣤⡬⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠼⡤⠄⠀⠊⠁⠀⠀⠁⠤⠀⠀⠙⠄
⠀⢀⠀⠤⠌⠁⠄⠤⠤⠤⡀⠀⠀⠀⢀⠈⠉⠃⠉⣀⠈⠀⠂⠀⠀⣄⠠⠠⠤⠤⠴⠒⠒⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠔⠂⠀⠀⠀⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠒⠈⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢹⠉⠙⠲⡄⠀⠀⠀⣾⠀⠀⠀⡿⠉⠓⠦⣄⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠀⠀⣀⠇⠀⠀⠀⣿⠀⠀⠀⣇⠀⣀⣠⠟⠀⠀⠀⠈⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⣯⡉⠁⠀⠀⠀⠀⣿⠀⠀⠀⡟⠉⠁⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣸⠈⠻⢦⡀⠀⡀⠀⣿⢀⡀⠀⡇⠀⢀⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠁⠀⠁⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠳⡄⠰⡀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠀⠁⣠⣤⠤⣬⣤⣤⣤⠤⠬⠭⠤⠠⠤⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠒⡄⠀⠀⠀⠀⠂
⠀⢀⡀⠋⠆⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢙⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠐⠀⢁⡤⠖
⠀⠀⢻⢆⠀⡇⠸⠀⡠⠀⠀⣆⠀⠀⠀⣠⢄⡂⡀⠈⠀⢧⡩⡔⠀⠀⠀⢤⠀⢀⠄⡀⠀⡠⠒⢁⠀⢆⠀⠀⢘⠄⡪⠊
⠁⠀⠈⡎⠏⠐⠀⠨⠐⠆⠀⠊⠣⠒⢀⠀⠀⠉⠀⠻⠐⠃⠀⠋⠈⠀⠀⠀⠀⠑⠀⠀⠊⠐⠀⠘⠊⠘⠀⠔⡁⢀⠐⠉⠁⠀⢤
⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣡⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⡿⠿⠿⠿⠿⠿⢿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠟⣫⣭⣤⣄⣀⣀⣀⣀⣠⣤⡾⠏⠀⠈⠙⠻⣯⡙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⢀⡾⠋⠁⢈⣹⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⣰⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣠⡤⠶⣶⣶⠾⠋⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠙⣿
⠀⠀⠀⠀⢀⣠⣤⣷⠀⠀⠀⠀⣼⠋⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠙⣷⡀
⠀⠀⠀⠀⢹⠛⠉⠈⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠸⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⢹⣧
⠀⠀⠀⢀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠈⣿⡆
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣤⠴⠤⣄⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢤⡼⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⣀⣽⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠃⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇
⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇
⠀⠀⠀⢰⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⢀⣾⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢻⣷
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡄
⠀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⡇⢠
⠀⢻⣶⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣾
⠐⢦⣿⣿⡇⠀⢠⣴⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⢰⣸⣷⣤⣾⣿⣿⣿
⠐⢦⣽⣿⣷⣶⣿⣿⣦⣤⣄⣠⣤⣤⣶⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠰⠾⢿⣶⣶⣦⣴⠶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠙⠻⠿⠿⠛⠋⠉⠙⠿⣿⡿⠟⠻⠿⠟⠉⠛⠿⠿⠋⠁⠀⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠋⠀⠙⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⠿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⢛⣋⣉⡉⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏⠈⠉⠛⢿⣿⡛⠿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⢀⣾⠟⠛⠛⠛⣿⣿⠟⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⠻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠟⠁⠀⠀⣤⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠈⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⠀⠴⠶⠛⣿⡿⠿⠋⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠹⣿⡀
⠀⠀⠀⠀⠀⣀⣤⣤⣾⡀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆⠀⠹⣿⡄
⠀⠀⠀⠀⠀⣿⠟⠋⠉⠁⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢹⣷⡀
⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⣿⣧
⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⢸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠉⠙⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⠉⠉⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⢾⣏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠐⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣴⠶⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⣸⡟
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠐⣿⣦
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣧
⠀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⠀⢸
⠀⢹⣷⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣷⣼
⠠⣄⣻⣿⣿⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣧⣀⣤⣿⣿⣿⣿
⠀⣈⠻⣿⣿⣀⣀⣼⣿⣃⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣄⣀⣀⣀⣀⣀⣀⣤⣧⣽⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠿⣷⣶⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠟⠻⠿⣿⠟⠋⠈⠉⠛⠻⠿⣿⠟⠛⠙⠛⠿⢿⣿⡿⠛⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡵⣕⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣪⠟⠁⠀⠈⠳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠚⠁⣀⣤⣤⣤⣀⡈⠳⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⢰⠋⢀⣤⣷⣤⣈⠹⡆⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⠓⣶⣄⢈⡇⢸⣰⣿⣿⣿⣷⣿⣧⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢹⣧⣤⡟⡄⠀⣿⣿⢻⡇⢸⡟⣿⣿⣿⣿⣿⣿⣿⢸⡧⠦⣴⡏⠙⠛⣦⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠾⠖⠉⠉⣤⣷⡧⢤⡿⣿⣸⡇⢸⣧⣿⣧⣿⣿⣿⣿⣿⠨⡇⠀⢸⣇⠀⠀⣿⠒⠒⠺⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣠⢴⣦⣀⠀⠀⠚⠳⠚⠓⠒⣒⣚⣓⣒⡒⠚⠛⠙⠓⠘⠉⠉⠉⠉⠉⠀⠀⠀⠈⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠰⡆⠈⠻⣧⡀⠀⠀⠀⢠⡏⠁⢠⣄⠀⠹⡆⠀⠀⢰⡞⡏⠉⢩⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠁⠀⠀⣿⡇⠴⠶⠆⢸⠁⠸⢯⡽⠗⠀⡇⠈⠁⢸⡄⡇⠀⢹⠉⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠛⠀⢸⢃⡀⠀⠀⠀⠘⣿⡇⠀⠀⠀⢸⣲⣤⠸⡷⠀⠀⡇⠀⠀⢸⡇⡇⠀⢸⠀⠀⡇⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠷⣶⡶⣦⣦⣾⡿⠷⠆⠀⠀⣾⣷⠿⠶⢶⣶⣶⠷⠀⠀⠀⠷⠷⣴⣶⣾⠶⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠉⠁⠀⠀⠰⠤⢤⣀⢀⣹⠂⠀⢠⣾⡥⠀⠀⠀⠀⠀⢈⣧⡛⠋⠀⠰⣆⠀⠀⠀⠀⠴⠶⠿⠿⣗⢶
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⠟⠁⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣌⢻⡟⠋⠀⠀⠟⠆⠀⠀⠐⠖⠀⠒⠒⠛⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠦⠤⠦⠶⠤⡴⠦⠞⠁⠀⠀⠀⢸⠃⣀⠀⠀⠀⠀⢀⣀⡀⠻⡇⠀⢠⡔⢦⣶⣀⠀⠀⠀⡀⢀⡀⣀⣠⠞⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠋⠙⠓⠺⠤⠟⠛⠀⠀⠀⠉⠉⠉⠉⠛⠉⠉⠉⠁⠀⠀⠁

⢰⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⣰⠀⠀⠀⠀⠀⢀
⢸⣿⣿⣿⣾⣿⣾⣿⣿⢿⠹⡿⠿⢿⣻⣯⣥⣦⣤⡀⠀⠀⣿⣯⣼⣇⠀⣰⣿⣧⠀⠀⠀⠀⠀⢀⠄
⢸⣿⣿⣿⠿⣈⣍⡛⢿⣿⣿⣿⣷⣿⡿⣿⣷⣿⣿⣿⣿⣖⣿⣿⣧⣿⢸⣿⣏⣿⠀⣾⡄⠀⠀⠁
⢸⣿⣫⣶⣯⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣷⣿⣟⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡿⢿⣾⣿⡿
⢸⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣟⣿⣟⡏⠙⢫⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣇
⢸⣿⣭⣬⣍⠻⣿⣋⣾⣿⣿⣿⣿⡟⠁⠀⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽
⢸⣿⣿⣿⣿⣷⡽⣿⣿⣿⣿⣿⡟⡀⡄⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣽⣿⣟⣿⡿
⢸⣿⣿⣻⣿⣿⣇⢸⣷⣿⣿⣿⣿⠁⠀⠀⠀⠀⢷⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣟⣿⣷⣏⣄⡀
⢸⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⡀⣤⢠⡆⠀⠸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡇
⢸⣿⣻⣿⣿⣿⣭⣬⡻⠟⠗⠤⠀⢀⢠⢦⡤⣀⣠⣤⣭⣽⡠⠽⣏⡙⠻⣿⣿⣿⣿⣿⣿⣾⣿⣶⡄⠀⠀⢀⣀⣀⡀
⢸⣿⣹⣿⣿⣿⣿⣿⣿⣿⣤⣌⣃⡙⠂⠉⠉⠉⠭⣍⢁⠶⠋⣁⣁⣰⣄⣿⣀⢹⣿⣿⠿⠿⠿⢿⣿⣾⣿⣿⣿⣿⣿
⢹⣿⣿⣿⣿⣯⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡄⠀⢀⠀⠀⠀⠀⠁⠏⠀⠈⣠⣤⣿⡇⢀⣀⠀⠀⠙⠛⠛⣺⢟⠻⢿
⣿⣿⣿⣿⣟⣛⣿⢿⣿⣿⣿⣿⡋⣿⣍⣿⣿⣿⣔⣛⡛⠭⣥⠀⣀⣈⣉⣁⡀⣈⣉⣁⣀⣀⣀⣠⣤⡄⠀⢀⡀⠉⠭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠃⠘⠈⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠒⠂⢄⠀⠀⡍⠂⣈⠀⠐⠁⠀⠐⣆⠀⡀⠤⠐⠂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠓⠀⣋⣀⣈⣀⣈⣈⣀⣈⣁⠈⠂⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣗⠐⢉⣃⢀⣠⠂⣸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣰⠁⠀⠼⠿⠿⠿⠿⠿⠿⠿⠦⠈⢳⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⠏⠓⠲⠸⠸⠄⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⡟⠠⢉⡅⠂⡀⣀⢄⠴⣡⠂⢀⡆⠸⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠰⠆⠒⠒⠐⠂⠲⠒⠂⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠃⠀⠤⠌⠁⠁⠀⠊⠀⠀⠠⠢⠉⠀⠙⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡦⢎⣡⢰⢹⢼⠙⡶⣂⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠛⠻⠘⠀⠸⠉⠠⣂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠠⠄⠰⡄⠤⠂⣿⠀⡀⠀⠀⠀⢀⣔⠄⠀⡀⠀⡤
⠀⠀⠀⠀⠀⠀⣀⠠⠦⠄⠀⠀⠀⠙⠉⠆⠨⡍⠁⡰⢀⠈⡏⠐⢄⠀⠀⠀⠀⠀⠀⢸⡇⠈⠩⠭⣭⣥⠀⠃⡤⠤⠃⠊⣉⡉⢩⣀⠀⣈⢀⣀⠂⡀
⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠉⠀⢨⠤⡊⡡⡄⢘⢑⠯⠀⡇⠀⢠⠀⠀⠀⠀⠀⠀⠘⡇⠈⠉⠑⡚⢻⠀⢠⡇⢀⡙⠙⠙⠧⠟⠪⠜⠓⠙⠚⠖⠁
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠰⠒⠃⡈⠁⠘⢈⠃⡐⠃⠒⢁⠀⠀⠀⠀⠀⠀⠀⢒⣀⣀⡀⠠⣸⡄⠃⠈⡁⠀⠰⠦⠀⠀⠂⠐⠆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢩⠁⢀⡔⢻⠀⠰⠶⠀⠈⠁⡛⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠈⠀⠺⠀⠀⠀⡇⢠⢶⡤⢂⡐⢶⠓⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⢸⠀⢾⠗⠺⡔⠻⣋⠴⢊⣉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠘⠀⠀⠀⣅⡠⠥⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⣀⡀⠀⠠⠀⣄⢄⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠦⠀⠤⡅⡴⠣⡖⠓⢞⠈⢢⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⣶⠀⠂⢠⢀⡄⣴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠄⠀⠉⠃⠂⠁⠀⠀⠈⠂⠣⠂⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡏⠀⠃⠘⣾⢁⣟⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⠂⢁⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠀⣙⠉⠁⠀⡀⢀⠉⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠐⢸⠀⣿⢸⣹⠆⣇⡾⢸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢠⠀⠀⡁⠌⠉⠀⠐⡂⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠄⢸⠀⠷⠤⠅⠀⠘⠃⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠑⠄⣀⠀⡀⠠⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⡀⢰⣛⡆⠶⢳⡄⣇⡼⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⢸⡋⣳⠰⣿⡇⡏⢧⢸⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠹⢀⡠⣤⠤⠤⠤⢢⡀⠶⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢠⠁⠈⠉⠁⠀⠈⠁⠁⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢰⣴⣶⣤⣀⣀⣤⣀⣀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠘⣿⠀⢀⣀⣀⡈⢻⣿⠀⢿⣿⡆⠀⣌⣉⡳⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⢰⠟⠀⢻⣿⡿⠋⣠⡟⠁⢀⣿⣿⠀⣀⠀⠀⠈⠉⢦⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠙⠁⠀⠻⣿⢠⣾⣿⣿⣿⣿⠋⢸⣷⣤⣀⡀⠈⠐⡀⠈⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⢀⣿⠁⢾⣿⣿⣦⠀⠈⠛⢿⣧⠁⠙⡆⠀⠹⣿⣿⣿⡷⠂⠩⠄⠈⠉⢳⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠛⠋⠛⠃⠀⢸⣿⣤⣶⡄⠈⠁⢀⣶⠀⢸⣿⠃⠀⠁⠀⠀⣿⣿⣿⣷⣤⣤⠀⠀⠀⠈⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁⠀⠀⠘⠢⡀⠀⠀⠀⠘⠛⠻⠋⢵⣦⣠⣾⣿⣷⣌⠁⠀⡆⠀⠀⠀⠛⠁⠙⠛⠛⠉⢀⠀⠺⢗⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣠⡄⠀⠀⢀⣤⡤⠴⠏⠉⠙⠛⠉⠙⠳⢦⣴⣶⣤⠄⠀⠀⡀⢀⡀⢀⣠⣶⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⠀⠸⠿⠁⢠⣾⣿⣶⡀⠀⡇⠀⢠⣿⡟⠋⠀⠀⠀⢀⠋⠈⣿⣿⡿⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⢿⠀⠀⠀⣤⠀⠸⣿⡟⠿⠿⠾⢻⡆⠈⣩⣥⡀⠘⠃⢀⣤⠀⠸⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⡄⠀⠻⣷⠖⠀⠀⢸⡇⠀⠘⢿⣿⡄⠈⠻⣿⠀⠀⣿⣿⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣍⣹⠇⠀⠀⠀⠴⠁⠀⠀⠀⢸⣿⣄⡀⠀⠀⣀⡴⠚⠃⠀⠰⢂⣿⡇⠀⠀⢨⡗⢤⠀⠀⢤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠛⠻⣷⡔⢀⠀⣾⠀⠒⠀⠀⢸⣿⡿⠿⠛⠛⢿⡄⢠⠀⠀⢰⣿⡿⠀⠀⠀⠀⢱⣤⠴⠛⢳⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡏⠀⠈⠀⠟⠀⠀⠀⣠⣿⡟⠁⠀⠀⣀⡄⠀⢀⣴⣠⡞⠁⠀⠀⠀⢠⡄⠿⠁⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠇⡜⠀⠀⠀⢰⣶⣿⣿⢿⣇⠀⣠⡾⠟⠁⢀⡾⢻⡟⠀⠀⠀⠀⢁⡘⠇⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠈⢷⣷⡄⠀⠐⣿⣿⣿⢏⡼⠃⢸⣿⣷⣄⠀⠙⡇⢸⡧⠀⠀⠀⠀⠀⠀⠛⣶⣤⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣿⣿⠉⠀⠀⠈⣿⣧⠂⠠⠐⠊⠉⠁⠛⠂⠀⠈⡌⠳⠆⣤⡄⢀⠀⣤⣤⣿⣿⠀⢀⣀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢶⡀⠙⠿⡧⣠⠄⣤⠀⠈⢉⢵⣶⣶⣤⣤⠀⠀⠀⠀⣤⡿⠀⣤⣵⣶⣿⣿⡿⠋⠉⠉⠀⠀⣿⣿⣿⡶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠸⣿⣤⡀⠀⠀⡀⣿⠀⠀⢠⠀⠹⣿⣿⣿⠀⠀⠀⠰⠀⠀⣀⣀⠉⢿⣿⡁⠰⣿⣿⡆⢀⡖⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠈⢄⠀⠙⣅⠀⠀⠀⠀⠀⢻⣿⣿⠗⠀⠀⠀⠐⣿⡿⢿⣧⠘⠿⠷⠀⠘⠋⢠⡎⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡼⠀⠀⠀⠑⢄⠿⠑⢤⡀⠀⠀⢈⢿⣥⡶⠒⠀⠀⢴⠏⢀⠀⠛⠀⠂⠀⠀⠀⣤⡈⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⡀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡾⠆⠀⠋⠀⠀⣴⣾⣿⠁⠀⢈⣃⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠊⠀⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠶⠀⠀⠀⣀⣀⠀⠀⠀⠀⠻⣿⠛⣿⠃⠀⢼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢰⣿⠙⢮⣯⠐⣆⡰⢶⠶⠛⠛⠁⠀⠀⠀⠀⢈⣤⡿⠃⠀⠀⠀⠀⢴⡏⠀⠘⠇⠀⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠛⣤⠍⠁⠀⠀⢀⣆⠀⠀⣀⡀⠀⠈⠻⣇⠀⢠⣾⡆⠀⠘⠁⠀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡀⠀⠀⠀⠒⢦⠀⠀⢸⡟⠀⣼⣿⣿⠀⣤⣾⠟⡠⣿⣀⠀⠀⠀⠀⢠⡉⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢛⡃⠀⡼⢃⣴⣧⠈⠉⠄⣴⣆⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⡀⠀⠀⠐⣤⣤⠀⠀⠀⠓⡈⠉⠳⠄⣶⣶⡯⠈⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⠉⠀⠀⠀⠈⠁⢠⣦⡀⠀⠱⠆⠀⢸⠟⢫⣴⣿⠟⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡇⣿⣶⡖⠀⠀⢴⣤⣦⣠⡄⠀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⣯⠀⠈⣿⡇⠀⠀⣴⠃⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⢹⠄⠀⣯⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⣿⡿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣷⣶⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠏⠁⣤⣈⡙⢆⣈⣙⣿⠘⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢁⣰⣾⣿⡿⠟⠂⠉⢿⣿⡏⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢻⣫⣾⠟⣋⣠⣴⣿⣦⡀⣹⡟⢑⣥⣼⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢷⣨⣿⣿⠋⠉⠙⠚⠿⠕⠺⠿⢷⣦⢭⣵⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣽⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⣿⡇⠰⠛⣛⡓⠀⠀⢀⢘⣯⣉⡓⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣙⣿⠀⠀⠺⠻⠟⠊⠀⢸⠛⠛⠋⠁⠸⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⣿⡄⠀⠀⠀⠀⠁⣀⢸⣄⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢷⠀⢰⠐⠀⣀⡍⢭⣀⠌⠀⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠀⠈⢍⡭⠇⠀⢀⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⡉⠳⢦⣀⡀⣀⣀⣴⣿⠇⢹⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⣾⠉⠛⠈⠷⣤⡀⠉⠙⠛⢉⡽⠋⠀⠸⠿⣿⠉⣿⣷⢦⣄⣀
⠀⠀⠀⠀⢀⣀⡴⣿⡉⡇⣧⠸⣆⠀⠀⠀⠈⠙⢳⢖⠶⣾⠁⠀⠀⠀⠀⣿⠀⢳⣿⣸⣿⣿⣭⠷⣄
⠀⠀⢠⣾⣋⣵⣶⡏⡇⢿⢹⡀⢹⣆⠀⠀⢀⣤⣾⣜⣄⣵⣷⣦⣄⡀⢀⡟⠀⢸⢸⣿⣿⣿⣿⣿⡟⣷⡀
⠀⣰⣿⡄⡇⣿⣿⣿⢡⠀⠀⠃⠀⢹⣷⣾⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠘⣿⣿⣿⣿⣿⡇⣿⣿
⠀⣿⣿⣷⢹⢹⣿⣿⡞⡆⠀⠀⠀⠀⠹⣿⣿⣿⠉⣿⣿⡟⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣻
⠀⣹⣿⣿⣏⣏⣿⣿⣷⢳⠀⠀⠀⠀⠀⠙⣿⣿⠀⣿⣿⡇⣿⣿⣿⠟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿
⢠⡿⠸⣿⣿⣸⡼⣿⣿⣏⣇⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣿⢧⡿⢻
⢸⡿⠶⣦⣿⡇⡇⣿⣿⣿⡜⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣿⣿⣧⣿⠼⠁⢿
⣿⣷⣌⣶⣿⣿⢱⢹⣿⣿⣿⣝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⣿⣿⡇⣿⣿⢀⣴⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡤⠄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣦⢀⡶
⠀⠀⠀⠀⠀⡀⢀⠀⢸⡇⠀⠀⡘⠇⠐⠥⢠⡟⠀⠀⠀⠀⠉⠼⠶⠂
⠀⠀⠀⠀⠀⠙⣊⢃⢸⣧⡀⠀⠀⠀⣠⡄⢸⣇⠀⢨⡅⢀⡤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠋⠀⠀⢿⣆⠈⣷⣀⣿⠀⠸⣇⠀⢸⡇⢿⡂⠀⣿⠀⠸⣠⡶⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣿⠁⠉⠉⠀⠙⠂⠀⠁⠈⠁⠀⠸⡧⠀⢿⡁⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠠⠤⠶⠀⠀⠀⠴⠋⠀⠀⠀⠀⠦⠞⠣⠄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢢⡉⢤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠻⠀⣀⡀⠀⣦⠀⢀⡀⠀⠀⠀⠰⡀⠀⢠⠠⠀⠀⠀⣾⠀⣠⠄⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⢹⡆⣼⠦⢤⠀⣷⠀⢠⡇⠻⣦⣠⡾⠀⣿⢰⡇
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⡇⠳⠀⠘⠃⠻⠂⢸⡅⠀⠈⠃⠳⠄⢻⣸⣧
⠀⠀⠀⠀⠀⣤⠍⣁⠀⠀⠀⠀⣼⠃⠠⢀⣀⠀⣠⠄⣀⠀⠀⠀⠀⣀⠐⡀⠈⢧⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠶⠞⠁⠀⡶⡍⠁⠀⢱⠇⠙⠃⠀⠀⠀⢫⡌⠳⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠚⠛⠛⠉⠛⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣤⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠉⠉⠁⠒⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⠀⠈⠉⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⢉⣒⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣀⠀⠀⠀⠈⠓⠲⠒⠚⠉⠉⠀⠉⠳⠶⢾⢋⡉⠉⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠒⠂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⢛⣲⣀⣀⣀⣠⡀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠿⠶⣄⡀⠀⠀⠉⠁⠀⢠⠞⠯⠛⠛⠿⣾⣶⣶⢚⣻⡶⠖⠒⡶⠛⠚⠒⠦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⠁⠀⠀⠀⠀⠈⠋⢀⡠⠒⢶⠒⠒⣦⢄⣀⡀⠀⠀⣴⡿⡾⠋⡈⠂⣼⡁⠠⢀⠈⠀⠈⠢⡅⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠛⠙⠛⠙⠀⠐⠢⢀⣶⡴⢋⣱⣤⣶⣷⣾⣭⣦⠀⡷⣀⣾⡩⡎⠀⠀⠀⢸⠇⠀⠠⠂⠀⠊⢂⠈⣾⠂⡙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⠁⠀⠀⠤⡄⠀⠀⣰⢼⠋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣻⡟⠀⠀⠀⢀⣴⠋⠀⠀⠁⠀⡁⡠⣣⠾⠡⠌⢀⢿⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣷⡄⠀⠀⠀⠀⠀⢻⡋⣻⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡾⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠙⠊⢑⠢⣰⢻⣼⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣰⠟⢑⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣛⠲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⠏⠀⠀⠀⠀⢠⠎⠀⠀⢚⢺⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣤⡋⢚⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⠂⠀⠈⠁⠀⠀⠀⠂⣼⢛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⢾⡿⣬⣷⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣎⣼⣿⣿⣿⣿⣿⠏⠛⠻⠿⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡢⡈⢫⣾⠟⡿⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⠀⠀⠀⠀⠀⣀⠀⠀⢀⡿⠯⠍⠯⠉⠿⠉⠽⠉⠸⠋⠹⠅⢹⣿⣀⣀⣤⣝⣼⣷⣿⣢⣄⣾⣏⣞⣵⣿⣦⣼⣷⡅
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⣀⣤⣀⡀⠀⣤⠞⠉⠛⢷⣟⣛⣻⣛⣓⣶⣒⣒⣖⠒⣖⣋⣸⡚⣻⣟⣻⣿⣯⢿⣿⣿⣿⣿⣿⣵⣮⣿⠿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠉⠈⠉⢷⡞⠋⠀⠀⠀⣸⣿⣤⣤⣤⣤⣬⢥⣤⣤⣤⣤⣤⣴⣦⣤⣤⣀⣠⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠏⠐
⠀⠀⠀⠀⠀⠀⠺⣯⣤⣄⣀⣀⣀⣼⠿⠷⣶⠶⠳⠿⠛⠛⠛⠛⠛⠛⢋⢹⣟⣋⣋⣁⣀⡀⠀⠈⣿⠋⠉⠙⠷⢭⣯⡏⠩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠒⠒⠒⠚⠋⠉⢉⠉⢉⠉⠉⠉⠉⠛⠶⠶⢶⠶⣶⣟⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣽⣯⣽⣽⣮⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣶⣶⣿⣷⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣒⣠⣼⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣭⣶⣿⣿⡿⠿⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠉⣲⡀⠀⠀⠀⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠃⠀⠛⠁⢐⠺⠏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⣠⠄⣈⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣈⠛⣋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠴⠲⡎⠀⢀⠀⠉⠉⡋⠰⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣄⠀⠀⠀⠘⢀⣀⠈⠄⣀⠀⠇⢀⢙⡗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⠀⠀⠈⠛⢋⡉⠢⡀⠀⠈⡀⠞⣠⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢾⡆⠀⠀⠀⠀⠙⢶⣄⡷⣈⡴⠞⠁⠀⣾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢠⡞⣸⠇⡀⠀⠀⠀⠀⠀⣹⡿⣉⣴⣶⣦⣸⡷⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢁⡖⠀⣿⠀⠛⢶⣦⣤⠤⠀⠈⣶⠉⠉⠉⢩⡏⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠟⠀⠲⢸⡀⠀⠀⠀⠀⢀⣀⠸⠆⠀⢀⣴⠟⡷⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢁⣀⡸⠇⠘⣧⠀⠀⠀⠀⠀⢉⠆⠀⣠⠞⠁⠀⢸⡆⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠧⢌⠋⠀⢀⣿⣠⡤⠤⣄⣠⠟⠀⣰⠏⠀⠀⠀⠀⣷⠤⢤⠜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡘⠀⣴⠛⠉⠉⠛⠂⠈⡙⠀⠀⢻⡄⠀⠀⠀⠀⠈⡄⠐⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⠁⡾⠁⠀⠀⠀⠀⠀⠀⢽⠀⣀⣌⣷⣄⡀⠀⠀⠀⢱⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡛⣸⣶⣶⣾⣿⣾⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⣷⠀⠀⢩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⡌⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⣆⣠⣾⣧
⠀⠀⠀⠀⠀⣀⣀⣠⢿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠹⣿⣿⡀⠀⠀⠀⠘⣿⣿⣿⠆
⠀⠀⠀⢠⡟⣡⣄⡀⢠⠼⣥⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢀⣿⣿⣧⢤⡤⢤⠤⠽⠋⣄⠀⢀⣀⡉⠜⢢
⠀⠀⠀⠘⠧⠚⠊⠉⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⡟⠀⠀⠀⠀⠀⠆⢀⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⡘⢠⣼⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⠞⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠏⠀⢀⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣐⠃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡼⡇⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠘⠀⣷⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢹⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣀⡒⠒⢾⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⠉⠉⠀⢸⠀⠠⢶⡧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡶⣶⣴⣤⣤⣤⣤⣾⣶⢿⢿⣿⣷⣶⣶⣧⣄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⣥⠷⠾⠛⠋⠉⠁⠈⠀⠀⠀⠀⠀⠐⠒⠉⠙⠻⠿⢽⣭⣵⡤⠴⡶⢶⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⡀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠓⠲⠤⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣯⣭⣉⠛⠃⠤⠶⣶⣦⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠠⢶⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⣉⡉⠧⣤⣉⣍⠉⠉⣱⣾⣿⣷⣾⣶⣾⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢐⣶⠿⠛⠿⠋⠹⣻⣷⣾⣝⡛⢿⣿⣿⣿⣶⣯⣽⣽⣿⣿⣿⣋⡉⠹⣷⡛⠛⢿⣿⢿⣿⣿⣿⣿⣷⣀
⠀⠀⢠⡞⠙⠧⠀⠀⠀⠀⢚⣿⣿⣿⣷⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢽⣿⣆⠈⠻⣦⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠲⠖⣤
⠀⠀⣼⠡⢂⠀⠀⠠⠀⠀⠀⢹⣿⣿⣷⣷⡼⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣤⠀⢻⣦⠙⢹⣿⣿⣿⣿⣿⣿⣛⢿⣦⡈⠀⠠⠆
⠀⠀⣯⡠⠀⠂⠀⠀⠈⠀⠀⠘⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣆⢸⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢻⠟⢛⡙⠉⠉⠈⢠⣿⣅⣿⣿⣽⣿⣿⣿⣿⣷⡎⠓
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⢚⣿⣿⣻⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣽⢻⡷⠟⡛⠻⠬⣄⡀⢿⣿⡇⣿⡿⣿⣯⣿⣿⣛⣫⣝⣿⡇
⠀⠀⠉⠉⢹⣟⣲⣞⣀⣤⣮⣽⣿⣛⣥⣾⣿⣿⡿⢿⣿⠿⡿⠿⢿⣛⣋⣳⡿⡳⠒⠦⠦⠬⠦⠵⠫⠏⠿⣿⣿⠿⠿⣻⢩⡿⠛⠁
⠀⠀⠀⠀⠤⠽⠿⠿⠿⠿⠿⠿⠿⠛⠋⠩⠤⠶⠶⠷⠶⠶⠿⣦⣼⣿⣿⣿⢷⢶⠦⠤⠤⠤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣁⣴⣶⣿⣿⣿⡇⠈⡆⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⣤⣾⡿⢿⠋⢹⣿⣿⠃⣸⠀⠀⠀⠀⠄⠤⠉⠀⠀⠙⢂⠒⢤⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢋⣠⡞⠉⠛⡀⠀⡝⠐⣿⡟⢠⠇⠀⠀⣠⠀⢀⡀⠀⠀⠀⠐⢿⣷⣿⡇⠀⠀⢠⠊⢀⡄⢄⠠⡀
⠀⠀⠀⠀⠀⠀⣠⠎⣀⠀⠉⠀⠀⠀⢄⢀⡉⢡⣿⠂⡞⠀⠀⠀⣿⣴⠋⠀⢶⠂⢀⣀⠸⡆⣿⡁⠀⠀⠆⢠⠎⡁⢸⠆⢱
⠀⠀⠀⠀⠀⡔⠁⠈⠣⠄⠀⠀⢀⠀⡀⠀⡘⠺⣿⠀⡇⠀⠀⠐⢠⣾⠀⠀⠀⠀⠉⠟⢛⣿⣿⠁⠀⡘⠀⡎⡄⢹⠊⠤⠜⡆
⠀⠀⠀⠀⠈⠃⢤⠀⠀⠈⠀⠁⠀⡄⡈⠈⢳⠞⣿⠀⣇⠀⠀⠰⣼⠿⣄⠀⠶⠁⢀⣰⣿⠟⠁⠀⡰⢁⣶⡁⠟⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠈⢀⠀⣄⠀⡀⢉⣽⣻⣇⠘⢦⡀⠀⠠⠂⠘⢶⣶⣶⣿⡿⠿⠁⣀⠜⣰⡷⡟⠉⠈⠀⣀⠸⠁⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠘⠂⠀⠀⡀⢈⡀⠁⠄⡰⢷⣮⡦⠴⠀⠀⠀⠀⠀⠀⣩⣿⣀⣠⣤⣾⣥⢾⠟⡉⠰⠁⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⡀⠈⠄⠀⠐⠉⠁⠀⠀⠀⠠⠂⠀⠘⠺⠟⡿⢿⣿⣿⣿⣿⢁⠰⠁⠀⠔⠀⠀⠠⠁⡜
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠔⠒⠶⠦⣔⠀⣈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠈⢻⣿⣿⡟⠀⠀⠘⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢳⠇⠀⠀⠀⣀⠤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢩⡄⡀⡁⠀⠀⠀⠀⠤⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣠⠖⠁⠀⢖⡠⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠁⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡤⡾⠁⠀⢀⣾⠿⠂⠀⠀⠀⠀⠀⠀⠀⣼⠹⠀⠀⠀⠀⠉⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠘⠁⢸⠁⠀⢀⣼⠿⠮⠀⠀⠀⠀⠀⠀⣀⡴⢻⠀⡇⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⢀⣾⣿⠋⠀⠀⠀⠀⣠⠔⠉⠁⠀⢸⠀⢍⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣴⣿⣿⠛⠂⠤⢀⣀⣘⣁⠀⠁⠀⠀⠘⣄⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣠⣤⠟⠛⠁⠀⠀⠉⠉⠙⠂⣰⠃⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠾⠋⠁
⠀⠀⠛⠉⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣴⣿⣟⢛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⢻⣅⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣟⠹⠫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠉⠀⠀⠀⠀⢠⡄⣾⣿⣆⠀⠀⠀⠀⠀⠠⣿⣿⡿⠟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡀⠀⠛⢿⣷⣾⣿⣿⢤⠀⠀⠀⠀⣠⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠱⣄⠀⠀⠁⠀⠤⠙⢶⣆⣴⣶⣾⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⡏⠻⡏⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄⠀⠀⢰⣄⠔⠈⢱⣤⣄⣸⣏⠉⠀⠀⠰⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⢧⣀⠀⠈⠙⠀⣠⠏⠹⣿⠏⠙⠀⢀⣀⣠⣤⡶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠈⠙⣶⡒⠺⢁⣀⡼⠧⠴⠶⢾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⢀⡀⣼⣷⣤⠿⠟⠁⠀⠀⠀⠀⠈⠻⡏⠛⠻⣤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠀⠀⢰⣾⣷⣿⣿⣶⣿⣟⠁⠐⠆⠐⠀⠀⢀⡀⠀⢠⣬⣆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⢿⠟⠉⠙⠛⠋⠩⡄⠀⠀⠀⠀⣀⠀⢀⣸⣴⣿⣿⣿⣦⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⣀⡙⢻⣿⣿⣿⣿⣿⣿⢿⣿⣯⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠺⣿⣿⣿⣿⣿⣿⣁⠀⠀⠋⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⢿⣦⡀⠀⠈⢻⡿⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣧⡙⠁⠐⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠇⠀⣽⣿⣿⣿⣿⣿⡎⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿

⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢐
⢠⣴⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⠧⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢈⠀⠀⠉⠉⠚⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣶⡄⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣴⡖⠊⠀⠄⠀⠀⠀⡤⡀
⢸⠈⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠙⠤⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣸
⢸⣷⡀⠀⠀⠀⠙⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠛⠇⠀⠀⠀⢠⠀⠀⠀⠳⠦⠀⠀⠒⢆⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠘⠀⠀⠀⠀⣰⠣⠀⠀⠀⠒⠎⠀⠀⠀⠀⢠⣄⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣠⣴⣶⠷⠾⠄⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠁⠈⠉⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠏⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⠸⠦⠀⡀⡀⠀⢀⣤⡀⠀⠀⢈
⢸⠒⣠⡾⠋⠁⠀⠀⠀⠀⠐⠀⢀⣦⣀⡠⠀⠀⠐⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡗⠿⠶⠀⠀⠀⢸
⢸⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡶⠉⠀⠁⠀⡄⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⠶⠀⠄⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠈⣤⢆⡐⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⢀⠰⠀⠀⠀⠈⠙⢄⠀⠀⠀⠀⠀⠀⠉⠋⠠⡀⢨
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡀⠀⠀⠀⠀⠀⠀⠠⢾
⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⡆⠒⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠐⠓⣼
⢸⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⣀⠀⠓⡀⠀⢀⠀⠀⢰⡯⠽⠀⡀⠠⠀⠀⠠⠀⢹
⢸⣿⣿⣷⠂⢀⣀⣶⡀⠀⠂⠀⠀⠀⢀⡜⠛⢛⣩⡟⠁⠀⠀⠀⢐⠻⣳⠞⣋⣰⠖⠂⠁⠀⠀⠀⢀⡴⠺
⢸⡟⠛⠛⠀⠉⠙⠋⢽⡷⠦⠀⠀⠺⠟⠛⠉⠉⠉⠁⠀⠀⠀⠴⠞⣚⠁⣈⡉⠁⠀⠀⠀⠀⠀⠀⠸⠦⢴
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠚⠂⠀⠈⠙⠛⠿⣿⣷⢤⣶⣶⣶⣶⣶⣤⣸
⢸⡄⠀⠀⠠⠄⢤⣤⡤⢤⠁⠐⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠉⠉⠈⠒⢂⠀⠀⠈⠉⢹
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⠀⠀⠀⠈⠁⠈⠁⡘⠟⠀⠀⠀⠐⣿⣄⢀⠀⠀⢺
⠸⠦⠤⠤⠀⠤⠤⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠷⠶⠶⠷⠶⠴⠶⠶⠽⠷⠮⠡⠤⠤⠤⠤⠤⠠⠤⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡔⠒⠒⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⢀⢀⠀⣀⠈⠙⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡘⠀⠀⠀⠀⢘⡈⠉⣋⠊⡀⠀⢹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣇⢰⢀⠐⢀⠙⢄⣷⠸⣗⣇⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢻⣻⠆⣳⣼⣤⡀⠠⣿⢀⠘⣿⠁⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣼⣿⣧⣿⡿⢫⡙⠀⠀⢹⣶⠟⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡘⢻⣻⣽⣵⢀⡆⠀⠁⠟⠀⠀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠸⣿⠨⣿⣿⡟⣬⣇⢰⣇⠀⠀⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡿⠀⣿⡇⣇⣼⡧⣸⣾⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡰⢃⢠⣿⡟⢹⣿⣇⣰⡇⠀⠂⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡇⢸⣿⣿⡇⢀⣸⣿⣯⠀⠠⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣾⡿⣿⣿⣿⣿⣿⡿⠀⠰⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⢁⣿⣿⣟⣯⣿⠃⡰⠄⠠⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⡿⣸⣿⣿⡿⣿⠃⢐⡇⠀⡖⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣴⡷⣿⣿⣷⣿⣿⣿⣿⣿⣷⢸⢿⠀⠀⢸⣿⣿⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣦⡀⡄⢸⣿⣿⣿⣿⣠⡞⣠⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠛⣉⣥⣾⡿⣟⠉⠙⠻⢿⣿⣿⣿⣿⣿⣟⣧⢸⣿⣿⣿⣿⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⣟⣛⣿⣿⣿⣿⣏⣾⣿⣷⣶⣖⣀⣬⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠿⠿⠿⠿⠟⠛⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⢀⠂⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⢀⢂⡴⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⢀⠀⢸⣿⠀⠀⠀⡠⢡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣼⠟⠀⣿⡇⠀⠀⠈⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠻⡀⠀⣿⡅⠊⠀⡄⢸⡟⠀⣴⡟⠇⠀⠀⠀⣨⠀⠀⠀⣀⠻⣧
⠀⠀⣀⠄⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣿⡄⠀⣾⠇⣾⡇⢰⡿⠀⠀⠀⠀⢠⣿⠀⠈⡀⢹⡆⢹⡇
⢀⡞⠁⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⢹⣇⢀⣿⢀⣿⠁⢾⡇⠀⠀⡄⠀⠈⠧⢸⠂⠀⢼⠇⢸⡇⠀⠀⠀⠀⢀⡤⠄
⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠜⠘⠚⣿⠃⠘⣧⠀⡰⠀⠀⠀⠀⠀⠂⠀⠀⢀⠜⠁⠀⠀⣠⠞⠁
⡇⠀⠀⠀⠀⣠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⡠
⢣⠀⠀⢀⡾⠁⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠂⠀⢹⡗⠄⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠈⢆⠀⣼⠃⠀⠀⠆⢀⡴⠃⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣈⠄⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⢱⣿⠀⠀⠌⠀⢸⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⢸⡟⠀⡘⠁⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⣿⠃⡔⠁⠀⠀⠀⠀⢰⡟
⠀⡆⢸⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠠⡄⢰⣿⠀⠀⣠⠀⢀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢠⠤⡀⢀
⠀⠁⠸⣿⠀⠀⠀⠀⣀⣄⢀⡤⢄⣤⠀⣠⡆⣰⡇⠀⣿⠁⠀⣾⠃⢸⡟⠀⣼⡏⢠⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠙⠂⠘
⠀⠀⠀⣿⡄⠘⣶⣰⣿⠄⣾⠇⢸⡟⢀⣿⠀⣿⠁⢠⣿⠀⢸⣿⠀⢾⡇⠀⢿⠁⢸⡏⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢹⡇⠠⣿⠘⣧⡠⠻⡤⠙⢧⠂⠛⠈⠹⣄⠊⣿⠀⠜⠻⠈⢸⡷⠀⠈⠂⠀⠓⠌⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠃⠀⢀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⡂⢀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⢃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⢀⡀⠀⢀⠄⠀⠰⢤⠀⠔⠃⠀⠀⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢀⡀⠀⠤⣬⠝⢫⣴⠊⢉⢰⠀⠴⠒⠚⠍⣤⠄⠀⠝⠀⠀⠠⠄⠀⠀⢤⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠃⡤⠄⠀⠀⠐⠂⠠⠉⠨⠀⠀⠀⠀⢬⠴⠀⠄⠘⠈⠏⢨⡁⠀⠐⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢀⣴⠶⠷⠐⠀⡖⠒⣆⠴⠀⠉⠊⠚⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢜⢢⡄⠀⠀⠀⠀⠀⣿⠷⡄
⠀⠀⠀⠰⠏⠀⡤⢤⡀⠀⣁⡴⢉⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⢀⠀⠘⠐⣀⣀⠀⠄⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡿⠀⠀⣐⠀⠀⠂⠨⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠩⡄⢍⣷⡁⢾⠉⠀⠃
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⠃⣰⠋⠹⢏⣤⣒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠅⣸⠀⢀⣌⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢋⠅⢠⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⡁⡟⢴⡄⠛⠅
⠀⠀⠀⢀⣀⠀⡀⢠⡞⢶⣇⣴⣵⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢸⠻⡎⢷⠂
⠀⠀⠀⠀⡀⠀⣠⡏⣵⣙⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠣⣾⠀⠀⣸
⠀⠀⠀⠀⠳⣴⣿⣶⣝⣿⣿⣿⠿⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠁⠀⢀⠀⠀⠐⠀⣠⡊⠈⠓⠚⠃
⠀⠀⠀⠀⣠⣿⣿⣾⡿⢿⠟⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⢺⡄⠰⢆⠀⢠⡜⠃
⠀⠀⢀⡄⣙⡿⠛⣩⣛⠿⠆⠀⠲⡀⠑⡀⠀⢀⡀⢠⠀⡀⠀⠀⠀⠠⡀⠉⠡⣁⠈⠁⢞⡁⠠
⠀⠀⠐⡻⠋⠐⡹⣵⠿⠇⠀⠀⠀⣅⣤⢈⣀⠄⠉⢀⡀⣀⡀⣀⡘⠀⠀⢳⡀⠈⠽⠯⡁⠁⢦⠰⡿⠈
⠀⠀⠈⣄⢨⠙⣠⣿⣠⡆⠀⠀⠀⠸⠓⠸⠋⣢⢀⡁⡀⠉⠁⠀⠈⠉⠒⠈⢤⠠⠀⠈⢳⡤⠈⠇⠄
⠀⢀⣀⡀⠠⣼⠋⢉⣉⡀⠀⠀⠀⠀⠀⣰⣞⡻⠯⠉⠀⠒⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⡄⠀⠁
⠀⠈⠄⣠⠹⡏⡇⢸⢶⡧⠀⢀⠀⢰⡷⡿⢿⡅⠢⠀⠀⠀⠀⠀⣤⠀⠠⢠⡂⡀⠀⠀⠀⠀⢀⢠⡶⢀
⠀⠀⢠⠃⠀⣱⣿⣶⣻⣿⣄⡏⣄⢾⣿⠇⠈⡠⢀⠀⠀⠀⠀⡀⠀⠁⡰⠁⠈⠀⠂⠀⠀⠀⠠⠎⠀⠀⠀⣠
⠀⠀⠰⠆⠀⠒⠒⢒⣾⣩⡏⠀⠿⠓⠋⢀⠈⠀⠈⠁⠀⢀⠈⣀⠀⡗⠁⡈⠁⡀⢀⣨⣀⣶⡅⠡⠆⣤⠊⣉⠅
⠀⠀⠀⠓⠀⡀⢀⣿⣰⢿⠫⡆⡀⣑⡴⠈⢠⡄⢤⡥⠀⢬⣤⡴⢸⣧⡀⠀⢒⢻⣄⣠⡹⣽⠾⣱⢦⣼⣇⡀
⠀⠀⡃⠘⠃⣤⢤⡈⠋⠸⡟⠗⠃⢘⢣⣌⣿⣉⢳⣶⣖⣆⡽⡃⠾⠀⣝⠶⣯⣤⣼⠚⠏⢹⡳⣴⣿⣿⡘⠳
⠀⠀⢃⣐⢤⡍⠀⠀⠀⠐⡇⡦⠐⠺⠺⠙⠟⠹⠛⠉⢉⠁⠀⠉⠉⠋⠛⠷⢭⣦⡿⠀⠀⠀⠀⠎⢼⠋⣃
⠀⠄⠀⠿⠛⢿⠀⠀⠀⢀⣹⣄⠀⠀⠈⢀⠈⠤⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⢀⣌⠈⣰⠓⢽⠂
⠀⡂⠀⢠⠐⡎⢃⠀⣰⣾⡛⠋⠃⠀⠀⠀⠁⠀⠤⠀⠠⢀⠀⠀⠁⠀⠀⠀⠛⠿⣴⣆⡄⠟⠛⣇⣹⠉⠫⠁
⠀⠀⠀⠤⠄⠼⠈⣦⠋⠨⡅⠀⡄⠀⠀⠀⣰⣳⢘⡞⢓⣞⢻⡦⠄⠀⠀⡀⠘⡄⠸⣿⡗⣼⠋⢀⣛⠠
⠀⡆⠀⠀⠻⠶⠓⠚⢂⣠⢦⠀⠤⡐⠉⣡⠺⠷⢭⠞⠛⠟⠫⡔⠻⢊⣿⣲⣞⣹⣅⣾⡱⢦⢮⣟⠉
⠀⡄⠀⠀⢀⠠⡌⠉⣂⣋⢈⠀⡘⠛⠀⠩⠋⢙⠷⠉⠀⠀⠠⠷⠷⠟⠶⠿⡯⡏⠉⢶⠆⠀⠈
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣶⣿⠋⢹⠋⢻⠿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠏⠁⠹⠁⠹⠉⢻⠶⡄⠀⠈⢀⠋⣿⡄
⠀⠀⠀⠀⠀⣸⣿⠛⠀⠀⠀⠀⠀⠀⠀⠆⠀⠇⠀⠀⠀⠀⠀⠀⠁⡿⢦⣴⠶⣤⣄
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠹⠀⠋⣿⣷⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⡖⠖⢲⡶⣤⣄⠀⠀⠀⠈⢁⣿⣿⣆
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⠉⠓⣀⣤⣤⣤⣈⡐⠩⠟⣷⣄⠀⠀⠀⣠⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣼⣿⡉⠓⢀⣴⣿⡿⠿⠿⣿⣿⣿⣦⡈⡩⢿⣿⠿⠟⣫⣽⠟⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⣼⡟⠄⢁⣶⣿⣿⣿⠖⠩⢅⢾⢻⣿⣿⣿⡶⣋⣉⣲⡾⠋⠀⠀⠀⠀⠀⣿⣷
⠀⠀⢠⣶⡿⣿⣿⡄⠀⠀⡯⠁⣰⣿⣿⣿⣿⣿⣟⣭⣶⣣⡏⣿⣿⡟⠠⢤⣼⠟⠁⠀⠀⠀⠀⠀⠐⣸⣿
⠀⢰⡟⠘⠡⢃⡌⡇⠀⢰⡇⢰⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣇⣿⡟⠀⢙⣿⠏⠀⠀⠀⠀⠀⠀⠰⡀⣸⣿
⢀⣿⠁⠀⠀⠀⠀⠁⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡇⣬⣽⡏⠀⠀⠀⠀⠀⠀⠐⡰⡑⣼⡏
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡗⠒⣿⡀⠀⠀⠀⠀⠀⢀⠀⠐⢐⣿⠇
⢸⣇⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣹⣇⠀⠀⠀⠀⠀⠀⢣⠀⣼⡿
⢸⣿⠀⢀⠀⢠⠀⢢⠘⡈⣇⠀⠀⠛⠿⠿⣿⣿⣿⡿⠿⠟⠛⠙⢿⣦⡤⠿⣦⡀⠘⢄⠂⠐⣄⣵⣿⠃
⠈⣿⣦⡘⡆⠈⣆⠈⢧⡙⣿⣆⣆⠀⣄⢀⡀⢠⡀⢠⠀⢦⡀⣧⣈⣿⣿⡿⢾⣿⣶⣶⣷⣷⣿⡿⠏
⠀⢻⣇⣧⣽⣦⣼⣷⠾⠿⠿⠿⣿⣷⣼⣦⣷⣼⣷⣾⣷⠿⠿⠟⠛⠋⠁⠀⠀⠀⠉⠉⠙⠋⠉
⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠃⠒⠚⠒⠒⠐⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇
⠀⠀⠀⢠⠀⢀⠀⠄⠀⠀⠈⠁⠀⠀⠈⠀⠈⣧⠀⠀⠀⠠⠤⠄⠤⠤⠤⠤⠔⢀⡀⢿⣿⡇⠀⠀⣀⠔⠁
⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⠀⢈⡤⣤⣓⣎⣴⣮⣼⢇⠴⠋
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢠⡶⠒⠂⠐⠒⠀⠊⠈⠉⠈⠿⠁⠿⠥⠤⠼⠀⠂⠐⠀⠐⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢠⠀⠰⠀⢠⡀⠀⠀⠀⠀⠀⣏⡆⣿⣿⣿⣿⣿⣾⣷⢸
⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⣶⣿⣄⡀⠀⠐⡱⡀⠀⠀⢻⡄⠀⠀⢰⠀⣩⡆⠋⡿⠿⠿⢻⢸⢿
⠀⠀⠀⠀⢸⡄⠐⠀⠂⢀⣩⣭⡛⡟⣟⡿⣿⠿⠿⣿⣿⠦⡀⠐⡙⡄⠀⠀⠹⣄⠀⢸⡀⣀⣀⣠⡤⠤⠤⠼⠤⡼⢀
⠀⠀⠀⠀⠈⠄⡈⠄⠀⠈⠘⡎⠃⠁⠁⠡⠀⠈⠀⠀⠀⠀⠉⠢⡈⢾⣆⣲⣿⣽⡅⡀⠉⢁⣈⣉⣀⢀⡀⠲⠾⠿⠟⠟⠟⠛⠃
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠠⠄⡀⢀⣐⣤⣤⡤⠶⠶⠛⠋⠀⡉⠛⠛⢚⣡⣄⢀⡥⣌⣭⡭⣭⣍⢅⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠪⢠⣄⣤⡴⠶⠶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⡤⠀⡀⠀⠨⠐⠀⠈⠀⠉⢓⢈⠂⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣐⠚⢉⠄⠀⠀⢠⣼⠅⠠⠀⢀⠠⣠⣴⡽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⣄⠠⠉⠀⢀⠐⣻⣯⣵⢾⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⠀⣀⣀⢉⣥⠴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⣦⣤⣄⠀⠀⢀⣤⣀⢠⠖⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⣇⢹⣷⡴⠁⠈⣿⠏⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡟⠈⣿⡇⠀⠀⣿⡇⠀⢸⡇⠀⠀⢀⡤⠄⠀⡟⡐⢀⣴⠂⢼⣡⣠⡿⠁⠀⣰⡶⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⡇⠀⠀⣿⣇⠂⠀⡹⠓⣿⠃⠀⠈⢡⡿⠁⠠⢺⡿⠁⡰⣿⠇
⠀⠀⠀⠀⠀⠀⣠⡶⠛⢳⡀⠀⢰⡖⢆⠀⠀⣿⡇⠀⠀⢹⡇⠀⢸⡇⠀⠀⢻⣧⣠⠔⠁⠸⣏⡠⠊⠀⠸⡧⠔⠀⢿⡥⠊⣼⠏⠀⠀⢀⡷
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⡇⠀⢻⣇⢸⠀⠀⣿⡇⠀⠀⠉⠁⠀⠀⠻⠦⠴⠃⠀⠠⠤⠤⣤⣤⣴⣆⣀⣠⡤⠂⠀⠀⢀⣼⠯⠤⠤⠶⠟⠁
⠀⠀⠀⠀⠀⢸⣿⠈⠷⠞⠁⠀⠸⣿⡸⠀⠀⠀⠀⢀⣀⡀⢠⣾⠀⠀⠰⠆⠀⢰⡏⡁⠀⠀⣸⡟⠉⠉⠀⢠⠊⣀⣤⠞⠁
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⣾⠁⣇⢼⣿⠀⠀⣶⡆⠀⢹⣯⠀⠀⢀⣿⠁⠀⣠⠀⣈⠉⢉⠁⠀⠀⢀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠈⣿⣆⠐⠙⣷⡘⠷⣟⢸⣿⠀⠀⣿⠀⠀⠌⢿⣇⠀⢼⡇⠀⢀⣿⠁⣸⠆⢸⡇⠀⣴⠏⢠⣿⠁⢸⣇⠆⠀⢦⡀
⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢸⣇⠀⢸⠈⣿⣀⠜⣿⣆⠌⣀⠼⣿⡄⢺⡇⣠⢾⡇⠀⡿⠀⣾⠁⢸⡿⠀⣼⡇⠀⠸⣿⡀⢀⡾
⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⣿⣧⠀⠈⣿⡀⡔⠀⠀⠀⠀⢈⠽⠊⠁⠀⣿⡇⠈⠉⠀⠈⠀⠈⠃⠀⢿⠔⠻⠷⠊⠻⠤⢊⡀⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⡏⢿⠇⠀⠈⠋⠁⠀⠀⠀⡴⠁⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⢌⠱⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢀⠎⠈⠁⠈⣦⠀⠀⠀⠀⡿⣿⡄⢀⠁⠀⣀⣀⣤⣤⣄⣀⣀
⠀⣦⢶⠛⡿⠛⣦⣤⡀⠀⠀⠀⠀⠀⠈⠣⣄⢰⣃⡔⢀⢀⣀⣯⣽⣄⠀⣼⣽⠟⠀⠀⢀⡔⠁⠘⢯⣽⣉⣿⡿
⠀⠈⠺⠖⠛⠛⠁⠀⠙⢄⡀⠀⠂⠀⠀⠀⠈⠁⣾⠿⠄⢸⣿⣿⣿⡟⠃⣠⣿⣤⠤⣶⣿⡇⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⠤⢶⡶⣢⠟⢆⠀⠉⢁⡼⠏⠀⣠⣿⣧⢰⠐⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⠿⠿⠀⠀⠁⠀⠀⠀⠐⣡⡻⡿⢿⠟⠀⠹⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣤⣔⣠⡴⠁⢀⣾⣿⡠⠑⠖⠀⡐⠀⢠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⣻⡿⣚⢁⣴⣿⣿⡟⠛⠀⠀⠈⣠⡶⢻⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣧⢾⣿⢿⠿⠉⡇⣿⠉⣿⡄⠀⠀⣻⣾⣇⣘⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢨⠵⡿⠁⠀⡀⠀⠀⠀⡀⢿⣿⡀⠀⢹⠿⠉⡺⣿⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⢀⠀⠀⡇⢸⣿⣧⠀⠀⠶⠚⠏⣷⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⡀⢸⡇⢀⣿⡿⣀⢠⡦⠀⠹⢸⣿⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠞⢑⠀⠀⠀⠀⠀⢇⡾⡗⢸⣿⣿⡿⠀⠀⠀⠀⠀⣿⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡄⠀⠀⠤⠤⠖⠀⣀⣀⣠⣟⣡⠔⣫⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⠀⠀⠈⢻⡏⣧⢐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠖⠀⠀⢠⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠴⠀⠀⠀⢊⣾⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠀⢰⠟⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⡀
⠀⠀⠀⠈⠃⠂⠀⡀⠀⢀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡢⠀⠀⢂⣤⠔⠛⠛⠁
⠀⠀⠀⠀⢒⣌⡳⣾⣄⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⡄
⠀⠀⠀⠀⠈⢉⣿⣿⣷⣮⣶⠁⠂⠀⠀⠈⠀⠀⠀⠀⠀⠐⠊⠀⠑⠀⣀⠄⠀⠀⠀⠀⠀⠈⡠⣦⡀⠶⡤⠄
⠀⠀⠀⠀⠀⠰⢻⠛⢳⡦⣿⢀⡀⢈⢤⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠘⠉⠀⠀⠀⠀⠀⠐⠚⠁⣄⡁⠀⢀
⠀⠀⠀⠀⠀⠀⢈⣹⣭⣽⣆⠄⠀⠀⠎⠠⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠂⠀⠀⠀⠀⠀⣠⣤⠀⠰⠷⠅⠊⠐
⠀⠀⠀⠀⠐⠀⠀⠙⢿⣧⣟⡊⠀⢲⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠩⣶⠥⠀⢉⣯⠐⡀⠀⠀⠀⠀⠐⢠⢁⡔⠀⠀⠀⠀⠐⠀⠀⠈⠀⡼⢂⠐⠀⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠈⠽⠃⠐⠀⠀⡆⠀⠀⠈⠀⠉⢀⠀⠀⠀⠀⠀⢀⡀⠌⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠑⣤⣤⡤⠤⠤⠂⠁⠀⠄⡘⣀⡃⢼⠆⠈⠑⢏⠁⠂⠠⡬⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠠⡄⠀⠀⠀⠀⠓⠒⠀⢀⡠⠀⠋⢰⣋⣠⣉⡁⠘⢄⠠⣀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⢀⠔⠊⠁⠀⠀⠀⠁⡠⢉⡃⢴⡄⠈⢃⠀⠑⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⣛⡚⢛⢹⡇⠀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠲⡿⠥⣈⠛⠏⠩⠀⢠⠎⠀⠀⣤⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠚⠷⢶⠒⠘⠆⠘⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡤⠤⠤⠔⢛⣂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠆⠀⢱⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠘⢠⡟⠲⣄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⡤⠀⠀⢹⡀⠀⠻⠇⢠⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡤⠀⠀⠸⡆⠀⠐⡀⠀⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⡇⠀⠀⢸⣥⠀⠀⣿⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠠⠤⠀⠠⠄⠶⠂⠾⢀⠇⠀⠀⠸⡇⠀⠀⠈⡷⠀⠀⢻⠀⠠⠄⠺⠇⠠⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⡀⢀⣰⡆⠀⠀⡠⠟⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡡⠁⠀⠀⢠⠀⢆⠀⠀⠀⠀⣀⡄⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠀⡁⠂⠔⠁⠁⡂⠀⠂⠉⠁
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⢢⠀⠀⠀⠀⠀⠀⠊⠀⠀⠆⠀⠀⠀⠄⠘⠀⠈⠡⠀⠋⠀⡀⠀⠀⠄⠄
⠀⠄⠀⠀⠀⣠⠀⠀⠀⣀⠈⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⣤
⠀⢀⠀⠀⠀⠐⠁⠀⠈⣿⠀⢠⡄⠀⠈⠁⠁⠒⠈⠉⠃⠄⡈⠡⠀⠠⠤⠀⠊⠑⠦⠀⠀⠀⢠⣄⠀⢸⡏⠀⠀⠑
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣀⡀⠶⢶⠀⠀⠀⠀⠈⠀⠐⠀⠁⠀⠁⠀⠀⢰⠶⠆⣀⡀⠹
⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠇
⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠉⠁⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢳⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⡴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⢂
⠀⠀⠀⠀⣄⠀⡇⣴⠉⠀⠀⠑⡄⠙⠢⠀⠀⠀⠀⠀⠀⣼⣄⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢾⠀⠱⠟⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢤⣄⡘⠙⠁⠀⠀⠀⠀⢸⠀⠈⠈⠩⠁
⠀⠀⠀⠀⢨⣇⢀⡷⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠸⣾⠁⠀⣠⠤⠤⢄⡀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠒⢿⠀⢠⣇⠄⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠘⣧⠞⠁⢀⣀⡄⠀⠠⠀⡀⠀⠀⢻⣿⡇⡇⠀⠀⠀⠀⠀⠚⢰⢱
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠙⣏⠀⠀⠀⣄⣷⣆⢠⣿⣿⡿⣇⢀⣀⣀⣄⠀⠄⠚⠋⠀⢰⡀⠀⢀⡀⠀⢀⣄⣰
⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⣿⣤⣴⣤⣾⣿⣿⣶⣬⡛⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠸⠿⠳⣿⣤⠀⠀⢉⡿⢤
⠀⠀⠀⠀⠀⢀⣴⣿⣯⢠⡾⣛⡭⠂⠀⠙⠛⠿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⡘⠒⠀⠻⣦⣄⡼⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣿⣟⠏⡞⢁⣠⡶⡠⠚⠉⠀⠀⠉⠉⣂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣷⣾⣿⣷⣡⡶⡋
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⣦⣡⣿⣾⡄⢰⣌⣵⣧⢊⣿⢔⣫⣼⣻⢿⣄⣔⣀⠀⠀⠀⠀⢀⠀⠈⡻⣿⣿⣿⣿⣿⣷⣶⡀
⠀⠀⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣾⣷⣿⣿⣿⣽⣿⣿⣟⣴⣠⢒⣦⣶⣾⣿⣿⣷⣿⣿⣿⣿⣯⠉⠉
⠀⠀⠀⠀⠀⠰⠾⠿⠛⠛⣛⣛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠋⠉⠉⠁⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⡟⣿⣛⣋⠉⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⢀⣾⣿⠏⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⢠⣿⣿⠏⢀⣴⣿⠿⣿⡇⢀⣶⣿⡿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡄⢠⣿⣿⠏⣰⣿⣿⢣⣾⡿⢣⣿⣿⠋⣠⡾⠁⣠⣄
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⢿⣷⡄⠀⠀⠀⠀⠀⣿⣿⠏⢰⣿⣿⠃⠘⠿⠇⣿⣿⡧⠾⠋⠀⢀⣿⡿
⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⣠⣿⣿⢃⣴⣶⣦⡀⠀⣿⣿⡀⠘⣿⣿⣄⣀⣤⠞⠻⣿⣄⠀⢀⣠⣾⠟⠁
⠀⠀⠀⣼⣿⣿⡿⠁⢠⣾⣿⡟⠁⣼⣿⡅⣸⠇⠀⠈⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⣀⣀⡀
⠀⠀⣸⣿⣿⡿⠁⠀⠸⣿⡏⠀⠀⠘⢿⣿⣿⣤⣤⠀⢀⣤⣶⣿⣿⣦⠀⢀⣴⣾⣿⠿⣆⣾⣿⠆⢀⣿⣿⡟⡴⢿⣿⠇⣴⢟⣿⣷
⠀⢀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣹⣿⡟⢀⣾⣿⡟⠁⣼⠏⢀⣾⣿⠟⠁⢰⣿⣿⠏⠀⣼⣿⣿⡟⣱⣿⢏⡾⢁⣾⣿⠃
⠀⠸⣿⣿⡏⠀⠀⠀⣸⣷⠀⠀⣠⣾⡿⣱⣿⡿⠁⢸⣿⣿⡴⠞⠋⢀⣾⣿⡏⠀⣠⣿⣿⠏⠀⣼⣿⣿⠏⢰⣿⣿⡟⢀⣾⣿⠃
⠀⠀⢿⣿⣇⠀⠀⣠⣿⡿⠐⠿⠿⠛⠀⣿⣿⣁⣴⠟⣿⣧⣀⣠⡴⠛⢿⣿⣷⡾⢻⣿⣯⣴⠞⣿⣿⠏⠀⣿⣿⡟⠀⣾⣿⠃
⠀⠀⠀⠙⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠉⢉⣥⡀⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⢀⣉⣀⣤⣬⣽⣶⣾⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⡿⠿⠛⠛⠉⢻⣿⠁⠈⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠿⣷⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⢿⠿⣿⢿⣿⣿⡶⠀⢿⣿⡟⡟⠃⠙⠙⣿⡟⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠈⠀⠀⠸⠈⢿⠀⠛⡟⠀⢄⣿⣹⡿⢿⣿
⠂⠀⠁⢻⡿⠃⠰⠖⠷⣿⣿⣿⡂⣞⡀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⢿⠛⠁⠘⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⢺⠁⢉⠀⠁⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢸⠇⠀⠀⠀⠀⡾⠀⠀⠀⣴⠀⡄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣾⣂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣴⣶⣄⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⠟⠳⢾⢳⠀⢠⣾⣿⣿⣿⣶⠀⢀⣤⣵⢦⣽⣅⠀⠀⠀⣴⣿⡄⠀⠀⢺⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠂⢹⠀⢴⢲⠶⠓⢶⠞⠀⢸⡟⠛⠛⢻⣿⠀⣰⣾⣿⣿⣻⣶⠀⠻⠛⠉⠉⠹⠛⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⢘⢿⢀⣷⣿⠀⠀⢨⡆⠀⠀⢠⡿⠀⢸⣽⠁⠀⠙⣷⠀⢠⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣂⣐⣤⣹⠀⢸⣿⣿⣭⣿⠷⠀⢘⣻⣤⣤⣭⣿⠀⢘⣽⣤⣤⣴⣏⠀⢸⠷⠤⠤⠼⠿⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠾⠇⠨⠿⠿⠟⠿⠾⠀⠽⠷⠧⠼⠿⠿⠀⠸⠿⠿⠿⠿⠿⠀⠀⠶⠶⠶⠶⠯⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠐⠀⠀⠀⠀⠒⠲⠖⠒⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣲⡾⢰
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⣿⠇⢻
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡦⢿
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⣠⣿⡇⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠾⠛⠛⠛⠛⠲⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⣀⣤⣤⠶⠶⠶⠶⢦⣤⡀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⣠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠸⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⢛⣉⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣌⡉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡴⢋⣥⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⠈⠻⣆
⠀⠀⠀⠀⠀⢠⡟⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠹⣧
⠀⠀⠀⠀⠀⣿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⢹⡆
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠘⣧
⠀⠀⠀⠀⢸⡇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢰⠛⣩⣥⡤⣤⣬⣍⡛⢶⣄⠀⠘⠛⣷⠀⣴⠟⠁⢰⡟⢉⣩⣭⣬⣭⣉⠛⢷⡄⠀⠀⡟⠀⣿
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⣼⠀⣿⠀⠀⠀⠀⠈⢻⡄⠹⣧⠀⠀⣿⠀⣿⠀⠀⢸⡇⢸⡏⠀⠀⠀⠉⢿⡄⢻⡄⢸⡇⢀⡿
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢸⡆⣿⠀⠀⠀⠀⠀⣸⡇⠀⣿⠀⠀⣿⠀⣿⠀⠀⠈⡇⢸⡇⠀⠀⠀⠀⢸⡗⣸⠇⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡇⢻⠀⠀⠀⢸⡇⣿⡆⠀⠀⣀⣴⠟⣠⡾⠃⠀⠀⣿⠀⣿⠀⠀⠀⣿⢸⡇⠀⠀⠀⣠⣾⢳⡟⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠸⡇⠘⠛⠛⢛⣭⠶⠛⠉⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⡿⠈⠻⣶⣶⣛⣩⣴⠟⠀⠀⣸⠃⣾⠁
⠀⠀⠀⠀⠈⡇⢸⡇⠀⠀⠀⡇⢠⣶⣤⣌⡛⢦⣀⠀⠀⠀⠀⠀⢻⡆⣿⠀⠀⢀⡇⢀⡟⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⡇⢸⡇⠀⠉⠻⣆⠙⣷⡀⠀⠀⠀⣿⡇⣿⠀⠀⢸⡇⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⢀⣷⢸⡇⠀⠀⠀⢘⣷⣈⣷⣄⡀⠀⢙⠁⣿⡀⠀⢸⡇⢠⣇⠀⠀⠀⠀⠀⠀⠀⢀⡟⢠⡏
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠾⠿⠿⠾⠿⠿⠂⠀⠉⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣾⠁
⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⣴⢿⡞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣧
⠀⠀⠀⢀⣼⠃⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠘⣧⡀
⠀⠀⢀⡾⠁⠀⢸⣇⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠛⠃⠀⣿⠀⠈⢷⡀
⠀⣠⡟⠁⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠛⠛⠋⠀⠀⠈⢷⡄
⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⣿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠛⠛⠛⠛⢻
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠙⠡⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣾⣿⣻⣽⡛⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤
⠀⠀⠀⠀⠀⠉⠉⠙⢻⣿⣿⣏⠙⠛⠃⣠⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠉⠛⠒⠛⠻⠿⠧⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠦⠬⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠉⠀⠠⠴⠶⡶⣤⣄⠀⠀⠀⠈⠑⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⢹⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠉⣠⡀⠢⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣶⣄⣤⣤⣤⣤⡀⡘⠢⡀⢀⡙⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣝⡻⣿⣷⣦⣈⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠊⠁⢬⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠋⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣷⣭⣫⣿⣿⣿⣷⣿⣿⠛⠳⠄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣨⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣭⣿⣗⡀⠀⠀⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⣿⠿⠟⠛⠁⠀⠈⢙⡻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⣠⠴⠒⠛⠁⠀⠁⠀⠀⢀⣠⣰⠾⠛⠁⠀⠈⠙⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡌⠀⢈⠉⠁⠀⠀⠀⠀⡀⢠⣤⡶⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠆⠀⠀⠘⠀⠀⡠⠒⠀⠀⠀⠀⢢⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠀⡈⠰⠰⠁⠀⠀⠀⠀⢂⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠈⠁⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠀⠀⠄⠒⠀⠀⠈⠃⠐
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠈⠀⢀⢀⣀⠈⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠄⠀⢸⠞⠉⢆⠀⠀⠀⢀⠉⠀⢀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⡀⢀⠈⠙⠖⡋⠘⠂⠂⠀⠚⢂⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠄⠉⠐⡀⠚⡅⠛⠐⠧⡤⠀⢶⠀⠆⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠸⠖⠌⠀⠸⠧⡇⠀⡀⠠⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢀⠷⡀⡁⠞⠀⠀⠀⠀⢀⠀⠆⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠐⡄⠀⡐⠂⠄⠈⠙⢱⠾⠶⣁⠀⠀⠒⣈⠦⡲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠘⠛⠲⣄⢸⡆⢠⡄⣵⢀⡔⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠈⠀⠀⠀⠀⠄⢸⠀⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠠⠀⠀⠀⠀⢀⠊⠘⡁⡀⠀⠀⢀⣸⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠸⠷⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣠⠞⠀⠀⠀⢱⠈⠑⠔⠀⡆⢥⡀⠀⠀⠀⠻⣄⠀⠀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠙⢄⠁⠀⠀⠀⠄⠀⠀⠀⠀⠁⢀⢧⠀⠀⠀⠀⠈⠀⠀⠙⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠟⠴⡆⣿⢳⡄⠧⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⢀⣬⠀⠀⠀⢿⠠⠃⢰⠀⢀⠀⠀⠀⠐⠶⠄⠀⠠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⣁⠛⢢⡀⠀⠸⣧⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣍⡄⠀⠳⠀⠀⡿⣇⢰⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣧⡟⢰⡗⠀⣰⠇⣇⣿⠀⢀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠁⠀⠀⠈⠀⢀⣤⡠⠞⡩⠀⡞⠀⡐⠛⣠⣿⠟⠀⠸⣁⠉⣠⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠂⠂⠀⠴⠦⡈⠀⠄⠀⠊⢀⣼⠁⠐⣇⠀⢼⡸⣟⡄⠁⠀⠀⢨⣄⠠⠀⠀⢤⡀
⠀⡀⠀⠀⠀⠀⠀⠘⣃⡀⠀⢄⡑⠆⠀⠑⠲⡆⠰⠶⠋⠉⠃⢰⡿⠄⠀⠓⠿⠄⠒⠀⡴⠕⠘⠧⠀⣀⠀⠀⠀⢀⡀
⠀⠅⠀⠀⠀⡠⠄⠀⠈⠀⠘⠉⣔⠊⡇⠀⠀⠱⠀⢢⡀⠁⠸⠸⠫⢶⡀⢢⣴⠲⠶⢏⡁⠀⠈⠦⠘⠋⡈⢰⠐⠊
⠀⠂⠀⠀⠀⠀⠤⠀⠀⠠⠀⠀⠈⡠⠇⣘⠂⢀⡲⡦⢄⢠⡴⢐⠁⢺⠀⣠⠀⠀⠐⢀⠱⠀⠁⠀⠆⠀⠐⢣⣄⠀⡁
⠀⠀⠀⠀⠀⠀⠘⡂⠀⠘⠑⠄⠘⣠⠠⣀⡼⠏⠀⠀⠀⡈⢤⢀⣀⠈⠻⠑⡀⢠⠀⢀⠠⠃⠂⠉⠤⡷⠄⠀⠀⠈⠉
⠀⠃⠀⠀⠀⠈⠒⠓⠖⣠⡄⠐⠀⡠⠏⠈⠀⠚⠀⠀⠀⠈⠠⠠⠈⡼⠀⢈⠲⢈⡕⠠⠆⠠⡄⠹⠏⠀⠉⠁⠠⠐⠐⠒
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠃⠀⠘⠂⠀⠀⠀⠀⠀⠚⠒⠈⠁⠀⠁⠀⠉
⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠋⠉⠉⠉⠹⣿⣿⡷⢠⡾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣁⣠⣤⡀⠀⠀⣀⣰⣿⣿⢧⣿⠁⢀⡞⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣠⡾⠿⠿⣿⡿⠀⠀⠙⠛⠋⠁⣾⡏⢀⡾⠁⠀⢿⠿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡀⠀⠞⠁⠀⠀⠀⠉⠁⠀⢀⣴⡞⠀⢠⣿⣣⠞⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⢀⡇⠀⠀⠀⠀⠀⣀⠀⡀⠀⣾⣿⣦⣀⣼⣿⠋⢀⣀⠀⢸⣷⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⡼⢁⣠⡀⢤⢨⠁⠓⠁⠁⠀⠙⠿⣿⠛⢻⣿⠀⠈⣿⠀⠀⣿⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⡼⠃⠀⠃⠀⠀⠀⠀⠀⣀⣀⣴⡦⠀⣿⡇⢸⣿⠀⠐⣿⡄⣸⣿⣧⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⢀⣴⡾⠁⠀⣿⣿⡾⠁⠀⠀⣠⣤⡄⠐⣿⡼⣿⡟⢸⡇⠀⣿⡇⠈⢿⣷⡄⠙⠛⢁⣿⣿⠉⠙⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣠⣿⡿⠀⠀⢀⣾⠏⢀⣥⠀⣿⠃⣿⡇⢸⣷⡄⠻⠿⠂⠀⠀⠀⠀⢠⠟⣿⡇⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠘⠻⠿⠛⣉⣿⣇⡀⣀⣼⣿⠀⢸⣿⠀⣿⡇⠙⠗⠀⠈⠀⢀⢠⡀⢐⠂⠀⢠⡟⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠛⢧⠘⣿⣦⠏⠛⠋⠀⠁⠀⠀⠀⠀⠡⠌⠘⠋⢈⡄⠀⣼⠀⢨⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⡇⠈⠁⠀⠀⠀⠀⠀⠀⢠⣾⣇⣀⣀⣤⡶⠋⡠⠀⡇⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣿⠀⣤⣾⣿⡿⠿⠀⠀⠸⢿⣿⠿⠟⢁⣠⠞⠀⠀⠓⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣤⠇⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣷⣶⣤⡶⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡤⠖⠛⠉⠉⠉⢁⠜⠙⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⢏⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⡀⠰⡿⠁⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠘⢸⡂⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢾⠇⠀⢀⣀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⢹⡶
⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠐⠒⣿⠀⠀⡸⠀⡷⠀⠀⠀⡄⠀⠀⠀⡖⠲⡀⠀⢰⡇⠀⠑⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⠀⠀⠀⠀⢠⡀⢀⡏⠀⠀⡧⣖⠁⠀⠀⢠⡇⠀⠀⢸⡃⣠⠇⠀⢸⠀⠀⠀⢸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠇⠀⢸⠁⠈⡆⠀⠀⢸⠀⠀⠀⡞⠉⠁⠀⠀⢸⠀⠀⠀⢸⡥⠂
⠀⠀⠀⠀⠀⢣⡀⠠⠀⣄⠀⠀⠀⠀⠀⣀⣾⠀⠀⠼⠀⠰⠃⠠⠀⠃⠀⡠⠀⠇⠀⡀⠂⠀⢸⠀⢀⡴⢹⡇
⠀⠀⠀⠀⠀⠀⠙⠒⠁⠉⠀⠀⣠⢤⡾⢁⠩⢹⠖⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⣸⠔⠉⠀⢸⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢰⡇⣀⣂⠗⠻⠁⠠⣄⡽⡤⢤⡀⠀⠀⠀⠀⠀⠀⠠⢶⠀⠐⣟⠀⠀⠀⢸⠏⠑
⠀⠀⠀⠀⡌⢄⠄⠀⠀⠀⠀⡤⠷⣄⠘⠢⠀⠀⠀⣧⣀⣳⡄⠙⡆⠀⠀⠀⡠⠊⣰⠋⠀⠀⠇⠈⣁⠀⢸⠂⢄
⠀⠀⠠⠄⠈⠈⢱⣄⠀⠀⢸⡀⠀⠰⠀⠀⡠⠀⠀⠀⠉⠉⣀⡴⠃⠀⢀⠘⣰⠊⠁⠀⠀⢀⡤⠴⠟⠀⢸⠩⡄⢑⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⣾⠅⠀⢠⠔⠈⡸⠀⠄⠰⢦⠺⠀⠱⢋⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠦⣀⡀⠀⠠⢾⠉⢁⡐⠒⠉⡁⠀⠊⠑⣚⠉⠫⠀⣀⠀⠀⠈⠁⠀⠀⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠶⠁⠀⠑⠂⠀⢹⡄⠀⠀⢸⠤⢀⣀⢀⣀⡀⠀⠀⠈⠀⠀⠀⠙⢤⡀⠁⠀⠀⠑⠀⠑⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡨⠒⣀⡀⢀⠐⠄⢠⢹⡇⠀⠀⠸⣇⣳⠴⠋⠀⢀⣀⠀⠀⠀⢀⡀⠀⠠⠕⡤⠤⠘⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⢄⣐⠒⠤⠈⢿⡇⠀⢀⠀⢿⠀⡀⠤⠒⠀⡄⠀⠎⠐⠡⠓⢀⠐⠘⡢⠂⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⣀⣈⣁⠀⢐⣻⣇⣀⣠⣼⣿⣿⣾⡥⣀⠈⠃⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠫⠛⠛⠭⣉⠉⠭⠉⠉⠀⠒⠤⠭⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣀⣶⠶⠺⡏⠉⠈⡉⠁⠠⠀⠀⣠⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⡿⢷⣶⣩⡅⣲⠔⡪⢙⢛⣷⡒⠠⣠⠀⣰⣿⣿⣿⣿⡆⢄
⠀⠀⠀⠀⠀⢀⠀⣰⠲⠿⣒⠩⠉⣈⠤⢒⡺⠷⡺⠝⣻⠗⠾⣲⢷⡿⣫⢿⢷⣰⠀⢹⣭⣿⣭⣭⠾⠋
⠀⠀⠀⠀⠈⠎⢉⣀⠂⠅⣔⠒⠉⠀⠘⠉⠤⠬⢀⠊⠡⠨⠌⡵⠊⠻⠴⣫⠌⠹⣋⣬⢿⠿⣿⠃⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡘⡺⣫⣿⢿⣿⡷⠇⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠚⠴⣭⣡⠎⢀⠻⣠⠄⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠈⢑⠲⠲⠕⠀⠉⠈⣿⣟⠷⡷⣶⠲⠎⣩⣴⡛⠋⢹⣂⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠍⠀⠀⠀⠀⢀⣠⡤⠜⠱⢀⢀⡁⠀⠰⢛⡦⢤⠵⠒⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣯⡟⢁⠔⣠⣆⡌⠊⠙⣀⡹⡡⠀⠊⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⠀⡓⠈⠀⢻⠀⠈⠑⠋⠁⢀⢪⠕⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣡⠊⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡢⠢⡦⢁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⣀⠀⠂⠤⣀⣠⠀⠀⠘⠋⢀⠠⠰⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁⠀⠀⠐⠂⠲⠄⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠡⠠⡄⠒⠶⠒⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⠀⠒⠀⢁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄

⠀⠀⠀⠀⠀⠤⠄⠤⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠹⣿⣿⣷⣦⣤⣀⠀⠀⠀⣀⣠⡤⣤⣄⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣢⣤⣤⣭⣻⣿⣿⣿⣾⢝⣧⣶⣿⣷⡜⢿⣿⣿⣿⠁⣹⡿⠟⢿⣿⣷⣶⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⣛⣿⣿⣿⡟⢋⣤⣾⣿⣿⣿⣿⡇⣼⣻⣿⣿⠀⢿⠸⠯⢸⠟⠉⢁⢹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠐⣒⣛⣛⣿⣿⠟⢉⣼⣿⣿⣿⣿⣿⣿⡿⠥⣛⡿⠉⠸⢀⠈⠛⠚⠁⠀⠀⠞⠧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠭⠛⠛⣫⣿⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⡈⠙⢠⡈⠀⠀⠀⠑⢢⠀⠀⠀⣠⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠤⠖⣺⣿⣿⠏⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡀⠀⠀⠀⠈⣄⠀⠀⠡⣠⠔⠁
⠀⠀⠀⠀⠀⣀⣤⣴⣿⢃⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡀⠀⠀⠠⠴⠐⠦⣌⣙⡲⠞⠁
⠀⠀⠀⠀⠈⠁⠘⣽⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠀⠿⠳⢤⡀⡈⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠴⣿⡟⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣧⣀⣀⠀⠁
⠀⠀⠀⠀⠀⣠⣾⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠈⢙⣯⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣀⣀⣴⣿⣿⠋
⠀⠀⠀⠀⠠⢻⣿⠨⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⢾⣿⣻⠿⡫⠃⣰⣿⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⢠⢿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠭⣛⡈⢯⣳⣉⣑⢶⡿⠏⠉
⠀⠀⠀⠀⢀⠁⢸⡆⢿⣿⣿⣿⣿⣿⣿⣿⡓⠄⡁⠀⠉⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡄⠙⢣⠸⣿⣿⣿⣿⣿⣿⣿⢃⠢⠄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣳⢾⡶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠡⣀⠙⣁⢻⣿⣿⣿⣿⣿⣿⣧⡟⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠁⠀⠀⠈⠉⠙⢳⣝⢮⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⢿⣤⠑⣤⢿⣿⣿⣿⣿⣿⣟⡰⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠃⠙⡞⡶⡄⠀⠀⠀⠀⠀⠀⢸⣿⡁
⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⣌⣿⣿⣿⣿⣿⣿⣽⢳⡂⠀⠀⢀⣴⣶⡶⠦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠃⣳⣧⠀⠀⠀⠀⠀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡡⣿⡆⣙⢿⣿⣿⣿⣿⣿⣽⣄⠘⠿⣯⠭⠴⢦⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⣿⠀⠀⣠⣀⣴⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠻⢄⢻⣆⠛⣿⣿⣿⣿⣷⣯⣦⠀⠀⠂⠀⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣮⣿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢘⠏⡛⢶⣼⡻⢿⣿⣿⣿⣯⣴⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠙⠺⣏⢀⡀⠈⠙⠻⢿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣴⣭⣤⣄⣴⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠦⢭⣭⣭⣽⣿⣿⣿⣽⡿⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠶⠶⠶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣍⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⣦⡈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠂⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠈⣷⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣀⣀⡀⠀⠀⣀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢹⡄⠂⠸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡅⣾⠉⠈⣿⢰⠏⠉⢹⠄⣸⠃⠀⠀⠀⠀⠀⠈⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢳⣌⠛⠛⠁⠘⠳⠶⣋⡴⠋⠀⠄⠀⠀⠀⠀⠀⡇⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⢹⡇⢠⡶⡄⠀⣸⠋⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣧⣬⣭⣅⣤⠟⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣶⠟⠛⣮⠛⠛⠛⠋⠁⠀⠀⣿⠉⣧⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠛⠳⣞⢧⡀⠀⠀⠀⣠⠾⣣⡤⠞⠁⠀⠀⠀⠀⢺⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣦⡴⢛⣵⠞⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣡⣶⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⣤⣤⢾⣫⠶⠋⠁⠙⢿⡟⠦⣤⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠻⣄⣀⡾⠁⠀⠀⠀⠀⠀⢿⡼⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠂⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣠⡾⠋
⠀⡀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠤⠤⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣿⠞⠋
⠀⡀
⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢔⢶⣪⣧⣵⣥⣭⣏⣜⡰⠢⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⣤⣲⡽⠞⠛⠉⣽⣿⣿⣿⡟⣿⣿⡉⠙⠳⢭⣗⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⡋⠁⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⡿⡇⠀⠀⠀⠈⠙⢿⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠨⡑⠦⣄⣀⠈⢿⣿⣿⣿⣿⣟⣹⠃⠀⢀⣀⡤⢖⡫⠒⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠴⢂⢌⣉⡛⠛⠿⠿⠿⠿⠗⠒⢋⡩⡵⢮⠋⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠜⠙⠑⢰⠐⠺⡙⠌⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠶⠶⠲⠒⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⡠⠔⠋⠉⠀⠀⢀⣀⣀⣀⣀⠀⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠶⢷⣶⣶⣶⣶⡯⠶⢤⡀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢷⣶⣶⣶⣶⡶⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠼⣿⣿⣿⡿⠁⠀⠀⠁⠀⠀⠈⡄⠀⠀⠀⠆⠀⠀⠈⠀⠀⠌⣿⣿⣿⣿⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠠⠦⠤⠐⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⠀⠀⠄⢀⣀⠄⠀⠀⠀⠀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⢀⣀⣀⣀⠀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠈⠙⠢


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢢⣄⣤⣾⣦⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⣿⡟⠁⠉⣿⡆⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠐⢀⠜⠀⠀⠀⠀⠀⠀⢈⣷⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠀⠁⢉⠀⠀⢠⠀⣷⡄⠀⢈⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⠄⠀⠀⠀⠀⢻⣆⠀⣴⣀⣿⡇⠀⠸⠀⠀⣼⢿⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠸⣿⣿⣇⠿⠼⠃⠀⠀⠀⠀⣿⠀⢹⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠑⢤⠤⠤⠤⠴⣶⣿⠓⣿⣆⠀⠀⠀⢀⠀⠀⣿⠀⣈⠓⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣙⠁⠐⠀⡀⠀⣀⣀⣀⣀⡼⠀⢽⣿⠄⢆⣀⣸⣀⣀⣽⢼⡏⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣹⠀⠀⠀⣬⣭⣭⣭⣭⣵⣶⠁⣾⣿⣀⣸⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⠏⠚⠛⢻⠃⢿⣿⣻⣿⢿⣿⣿⣿⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⣾⠀⢸⣿⣿⡏⠸⣿⣿⣿⠇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⠀⣿⢻⡀⠀⢸⡇⠀⢸⣿⣿⠃⠀⣿⣿⣿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⡄⣿⢈⣇⠀⣼⡄⠀⢸⣿⣿⡆⠀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠿⢷⣿⠀⢻⠄⣿⠀⠀⢸⣿⣿⡇⠀⣿⣿⣿⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠈⣿⠀⠀⠀⣿⠀⠀⢸⣿⣿⠁⠀⣿⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠑⢠⣿⠀⡄⠀⡅⠀⠀⠛⢻⣿⠀⠀⣿⡿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠓⠚⠁⠀⠛⠒⠃⠀⠙⠛⠋⠉⠀⠀⠠⠼⠇⠀⠀⠄

⠀⠀⠀⠀⢀⡀⠀⢀⡀⢀⠀⠀⠀⢀⡀⢀⡀⠀⠀⢀⡀⣀⠀⢀⡀⡀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡘⢧⠀⢸⡇⢺⡀⢾⠀⠀⠁⣸⠀⢸⡇⢸⠶⠈⠀⣿⣴⠉⢼⠁⠀⣈⠀⠀⢀⠀⠀⡔⠀⡄⠄⠰⢀⡄⣽⠀⡄⠆⢰⡇⠠⢀⠆
⠀⠀⠀⠐⠀⠚⠃⠀⠓⠚⠁⠈⠒⠚⠁⠛⠂⠉⠀⠘⠂⠚⠀⠘⠂⠀⠈⠁⠐⠃⠀⠀⠘⠘⠂⠃⠀⠁⠘⠒⠀⠁⠘⠀⠑⠂⠘⠃⢐⢲⡄

⠀⠀⠀⠀⣯⠀⡟⠀⢸⠀⠀⣾⠀⢰⣥⡊⠁⢠⡇⠀⠀⠚⢧⡀⢻⠁⠐⠓⣌⡇⠀⠀⠠⢠⠀⢨⠀⡉⢀⣀⠄⠀⡃⠀⡤⢤⢤⠀⠠⢠⡀
⠀⠀⠀⠀⠧⠀⠷⠀⠸⠄⠂⠟⠠⠸⠄⠱⠄⠀⠆⠴⠂⠤⠈⠁⠸⠆⠀⠀⠸⠇⠀⠀⠠⠀⠄⠈⠀⡇⠘⠍⠦⠀⠇⠠⠄⠸⠸⠄⠠⠔⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢠⠂⠐⣦⠀⠒⢀⠄⣠⠀⠀⡄⠀⡆⢐⡆⠠⣀⠲⠀⠀⡂⠒⠀⡆⠀⠆⠀⠀⠀⢀⠀⢀⣀⠀⢀⣀⠀⢀⣀⡀⢀⡀⢠⡀
⠀⠀⠀⠈⠄⠀⠟⠀⣻⠀⠀⠙⠀⢀⠇⠀⣄⠘⡄⢠⡀⡳⠀⠀⣇⠀⠈⣇⣀⠄⠀⠀⠀⢸⠀⣃⣨⠇⢀⣠⡄⢸⡀⢀⣱⠄⢰⡀⢢⣰⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠠⠀⡠⠀⢠⡄⢠⡄⢠⠀⠠⡀⠄⠀⢠⠀⡠⢀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠈⡶⠀⠀⠀⢳⠎⢱⠃⠀⠀⠛⡄⠀⠀⣺⠁⠀⠠⠂⢀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⢲⠜⡦⠁⠘⡇⠀⢱⡌⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠉⠁⠀⠉⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈⠠⠜⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⡀⢱⠀⠀⠠⠀⠀⠀⠀⡇⠀⠀⢀⡇⠀⠀⠀⡄⠀⠀⢤⡋⠀⠀⡄⡀⠀⠀⠀⡟⠀⠀⢃⡴⠀⠠⡀⢲⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠇⠘⠀⠀⢠⣄⠀⠀⢾⣤⠀⠀⣀⠛⠀⠀⠒⡷⠀⢀⣀⠸⠀⠀⢇⡽⠀⠀⢰⠁⠀⠀⣀⡽⠀⢀⡱⠋⠀⠀⠆⠉

⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠐⠸⠃⠀⡇⠀⣀⣆⣀⠀⠀⠀⠀⢰⡜⢀⠀⢀⠊⡠⢤⠑⡀⠘⢿⡓⠀⢲⡗⡠⠃⠀⠀⢰⢷⠂⠘⡆
⠀⠀⠀⠀⠘⡇⠀⢤⠀⢠⠀⠀⠆⠀⠀⠆⠀⠀⠇⠀⠈⠉⠁⠸⠤⠽⠧⠄⠸⡐⠷⠯⠜⡀⠠⣼⠟⠀⠀⡔⠁⠺⡗⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⢿⡿⢿⢹⣟⡿⠟⠛⠿⠻⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⡿⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿
⠟⠻⠈⠀⠈⠀⠀⠀⠈⠁⠀⠀⠈⠁⠈⠀⠉⠙⠟⠛⠛⠟⠛⠛⠛⠛⠋⠙⠸⠻⠙⠁⠋⠁⠀⠀⠋⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠻


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢻⣯⣽⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⣭⣴⣾⣿⣿⣿⣿⣿⣷⣦⣶⣾⣋⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⢹⣿⣿⣿⣿⣿⠉⠉⠙⠉⠙⠲⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠻⠿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣴⡿⠋⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⠀⠀⠀⠀⠘⡏⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢿⣿⠀⠀⠀⠀⠁⠀⡀⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠁⠉⢱⣶⣦⡄⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠂⣿⣿⠁⠀⠀⠀⠀⠀⠁⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⣐⣒⣀⣤⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⢹⡟⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡿⣤⣋⣤⡀⠀⠀⠀⣤⣴⣮⣿⠿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁




⠀⠀⢠⡀
⠀⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠠⠄⠒⠐⠂⠤⠤⠤⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⡑⡒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠀⠀⠀⠀⠈⠂⠔⠕⡠⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢂⠔⣤⠘⠆
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠓⠀⠀⡀⠀⠀⠀⠀⠜⡀⢢⠘⡀
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢀⡤⠤⠄⣀⠀⠀⠀⠀⠀⣤⡤⢤⡄⠀⠀⢠⡤⠤⠤⢤⣄⠔⠄⢀⠱⡄
⠀⠀⠀⠀⠀⠀⢣⠀⠅⠀⠀⠈⣷⢸⠉⢪⢱⠀⢀⡀⠀⢸⡇⢸⠀⠀⠀⠀⣿⠀⠉⠑⡌⢗⡡⣢⠂⡘⠆
⠀⠀⠀⠀⠀⠀⡆⢐⠀⠀⠀⠀⣿⢸⣠⠾⡸⠀⠈⠀⠀⠐⡇⠸⠀⠀⠀⠀⡟⠰⣦⣴⢃⠎⠀⢑⠕⠁⣄
⠀⠀⠀⠀⠀⠀⡃⠸⡀⠀⠀⠀⣿⢰⠁⢢⢁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⢰⠉⠋⠁⢀⢔⠁⠸⠟⡻
⠀⠀⠀⠀⠀⢀⠇⢨⠀⠀⠀⢀⣼⢸⡀⠘⡖⢤⠀⣤⠀⣸⡇⠦⡀⣠⡀⣠⡇⢨⡀⢀⢄⠁⡡⢊⢈⡪⣺
⠀⠀⠀⠀⠀⠸⠀⢠⡀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠀⠁⠀⠉⠉⠉⠁⠉⠀⠉⠀⠉⠁⠈⢀⠐⡐⢡⠊⠜⢴
⠀⠀⠀⠀⠀⠀⠀⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⢀⠀⠠⠢⠀⢂⡴⣊⣸⡄
⠀⠀⠀⠀⠀⢀⠄⠀⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠕⡡⠀⠀⠀⠀⡠⠂⢀⠀⠌⠜⡠⠠⠨⠫⡇
⠀⠀⠀⠀⠀⠘⠀⠀⡎⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠄⠀⠞⠀⢀⢀⠄⠀⠔⡀⡡⡐⠀⠜⡀⠰⢁⠔⣧
⠀⠀⠀⠀⠀⠸⠀⢠⠆⠀⠀⠀⠀⠀⠜⠀⡴⠂⠰⠖⠁⢀⢀⠔⡀⠠⠀⠀⠀⢈⠀⠔⢀⠈⢈⠌⡰⢃⢈⢜⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠞⠀⠀⠀⢀⠄⠀⢀⠀⠀⠔⠀⡤⠁⡠⢈⠈⠄⠐⡐⡡⠡⠂⠠⠁⠀⠀⠈⠁⠠⡠⢈⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠔⠠⡀⢀⠁⢊⠔⡀⡠⢊⢰⢊⠔⠐⠕⡁⡠⠪⠀⢄⠄⠐⠁⠀⠁⠀⡰⠑⠀⡔⡠⠱
⠀⠀⠀⠀⠀⠀⡇⠀⠪⢊⠐⡡⢊⠄⡨⢊⠔⠑⠔⡁⡠⡠⡪⢊⢄⠔⡡⠁⠊⠀⢈⢀⢌⠄⠄⠀⡀⣐⡱⡃
⠀⠀⠀⠀⠀⠀⡘⠀⢄⠀⡩⢒⠅⡨⢔⠅⡠⣀⡾⡪⢢⢂⠄⠐⠡⢊⠔⡑⠐⠁⠀⡠⠀⠀⢀⢸⠏⠋⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⡃⠈⠐⣡⢊⣔⡱⢪⡊⠎⡰⣵⢳⡗⢺⣔⡡⠠⢊⠀⠔⡀⠜⠔⠐⠀⠀⡰⣁⡐⠃⣃
⠀⠀⠀⠀⠀⠀⢸⠀⢄⡴⠈⠁⠉⠀⠈⠊⠁⠀⠀⠀⠀⠀⠹⣄⠀⠋⠑⣎⣻⣦⡴⢄⣤⣄⣐⠄⠔⠐⢁⡇
⠀⠀⠀⠀⠀⠀⢨⣠⠉⠀⠀⠀⠀⠎⠩⢶⠆⠀⠀⠀⠆⢶⠀⠀⠀⢻⡆⢀⠬⡓⠂⠠⢥⠛⢟⣪⣤⠪⠀⣧
⠀⠰⠒⠊⠉⠊⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠖⢀⠂⠐⠢⠀⠉⣈⠚⠃⠒⠀⢈⢉⡋⣦⡽⢷⣵⣿⣀⡅⡀⠁⡂⠄
⠈⠻⢷⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⡤⠐⠀⠛⠉⣋⠰⣸⣇⠑⠾⢣⠤⡰⠰⣤⣾⣿⠎⢯⣻⣖⡠⢐⠠⢐⠠⠀⡠
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⢀⣀⠳⠂⠀⠉⠁⠈⠑⠨⣞⣮⠧⠐⢟⠈⣵⡞⣶⡶⠩⡛⡟⢷⣂⠄⠀⡡⠞⠉⡃⣡⠖⡒
⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠎⠁⠁⢀⡀⡀⠀⠀⠰⠦⠀⠀⡆⠁⢠⠁⠂⠂⠀⢁⠀⣶⣺⢸⡰⣁⡀⢠⠉⠀⢐⢠⣎⣿⠗⣉⠄⡐⠔⡈⢐⠉⠂⠤⠂
⠀⠀⠀⠀⠀⠀⠉⠛⠴⢤⣄⣠⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡆⠀⡀⡄⡄⡀⡀⠈⠀⢘⠀⠀⡁⠠⢹⡇⡇⣙⡰⠖⠋⣸⡽⡃⠉⠀⡨⢐⠁⡀⢀⢀⡨⢂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⢬⠶⢬⡬⣴⡴⢴⣥⡤⢥⡥⢷⢥⣠⣡⣥⣦⣇⣬⠶⠷⠶⢤⡤⠖⠖⠉⠀⢀⠔⠽⠈⠊⠀⠉⠊⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠆⠀⠀⠀⠉⠀⠒⠶⠶⢦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡁⠉⠴⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣀⠨⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣤⣀⠊⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢧⡀⠀⠀⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡌⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢇⢰⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠈⠻⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠾⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢀⡇⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠃⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡦⡄⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠄⠡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡀⠃⣭⣽
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠓⡟⢿
⠀⠀⠀⠀⠀⠀⠀⠱⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣿⠀⠀⡇⣇⠰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⡄⢰⡇⣿⠋⣠
⠀⠀⠀⠀⠀⠀⠐⣄⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⣧⠀⠀⢁⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⣟⠀⢠⡇⣿⣇
⠀⠀⠀⢀⠀⠀⢨⣼⣿⣶⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣸⣿⣧⣀⣿⣿⡷
⠀⠀⠀⠀⠹⡆⣾⣿⢿⡙⣱⣂⠀⡀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠄⠀⣀⠂⣸⣿⣿⣯⡷⣿⣿⡧⣰⡟
⠀⠀⣄⣀⣀⣸⣿⣿⠜⠃⢿⣿⣅⣧⣴⠀⣠⢀⣸⠷⠂⢀⣠⣶⢮⡆⡇⢀⠀⢠⣤⡄⡰⢀⠀⢄⠘⠾⡦⠈⣸⡧⡟⣷⠔⢡⣴⣿⢸⣿⣿⣿⣿⣿⣮⣛
⠐⠋⠋⠝⠻⠟⠛⠩⠄⡆⠀⡁⠈⣙⠋⠐⡿⠐⠧⠀⢀⠂⠈⠻⢾⣁⠙⠚⠰⢏⢉⠁⠐⣝⠒⠏⠀⢰⣃⣰⣛⠛⠛⡫⠀⢘⠋⠉⠺⢏⣡⡇⠩⠍⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠙⠁⠂⠀⠛⠘⠳⠀⠀⠀⠒⠂⢀⠀⠀⡀⠀⠑⠀⠁⠀⠀⠄⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣻⣿⣿⣿⣿⢿⣿⣿⣿⡟⢛⠻⣿⠟⠚⠿⠿⢿⠚⠛⠻⡾⠛⠿⣾⠟⠿⣿⣿⡿⠿⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣬⣽⣿⣿⣟⣀⣻⣻⣱⣀⣀⣤⣆⡿⣆⣀⣀⣘⡆⢀⣉⡳⡄⠀⡘⢿⣄⠀⠈⠛⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠙⢌⠙⢍⠙⠻⣍⠉⠉⠛⢻⡟⠉⠉⠛⢿⡻⠛⠏⢻⠛⠛⡟⠛⡟⡟⠛⠛⠛⣿⠟⠙⠛⡿⡿⠛⣻⠿⣿⠿⠛⠛⡿⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠑⢄⠈⢧⠂⢄⣠⣙⣄⠀⠀⠈⢇⠀⠀⢈⠀⠀⡇⠸⠁⡏⠀⠊⡰⠁⠀⣠⡜⣫⠔⣋⠴⠊⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄⠐⢄⠉⠚⠻⣦⡀⠀⢨⡃⠀⠘⡀⠀⣇⠀⠠⠀⠀⢪⣞⠅⣪⣝⡩⠔⠋⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡑⢄⠠⡀⠀⠈⠛⠲⣄⢱⠀⠐⡁⠀⣿⡀⣸⢀⣴⣿⢑⢯⠞⠁⠀⠀⠀⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢀⣷⣀⣿⣴⣿⡿⠛⠋⠀⠊⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⢙⡻⣿⡟⠁⠀⠀⢀⠎⠀⠀⡠⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡁⠀⠀⡀⢀⣼⣠⣧⣿⡄⠀⢠⠃⠄⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠱⣸⣇⣸⣬⡻⣷⣤⣣⡔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣽⣿⣿⣾⣷⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⣶⡿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⡭⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⠶⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣯⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠁⠀⠐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⢇⡴⣤⢴⠬⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⢤⣶⣾⡶⢠⠀⠀⠀⠀⠀⣹⣦⢤⠀⠠⣖⠀⣿⡆⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠡⢻⣿⣿⡗⠉⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⠋⢠⣿⡧⠀⠀⠀⠀⠀⠀⠙⣿⣛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠏⠀⣿⣯⡋⢀⠀⠀⠀⠀⢪⡉⠹⠿⠆⠋⢀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢥⡂⣻⣿⣟⠂⠈⢧⠐⢄⠄⠀⠀⠀⠈⢦⡀⢫⡚⡻⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣻⢃⣾⣙⢿⣾⣯⠆⢀⣹⣧⡀⠀⠀⠀⠀⠀⠐⢔⠀⢳⠞⢷⠀⠀⡀⠀⠀⠀⠀⠀⣾⣯⡦
⠀⠀⠀⠀⠀⠀⣴⠟⣯⣾⣏⢁⣾⣿⡟⠋⣀⢈⡿⡇⠢⣀⠀⠀⣴⠀⠀⠑⢮⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⢷⣀
⠀⠀⠀⠀⠀⢰⡟⠀⠐⢷⠿⡿⣈⣿⠀⣹⢻⣾⣵⣾⡀⠈⠑⠄⠋⠀⠀⠀⠀⠀⠀⢠⣀⡠⠀⡀⠀⠀⠀⠈⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣘⠅⢁⢈⢹⠛⠛⣸⣼⣿⣬⡀⠀⠀⠀⢠⢠⣄⠻⠀⠀⣼⡿⠁⠀⠀⢆⠀⣠⣠⣤⣤⡿
⠀⠀⠀⠀⠀⠐⠃⠀⠈⠀⢹⠂⠈⠐⣿⣶⣧⣿⣿⣿⣟⠀⡠⠤⢄⣾⡿⠇⠀⣠⣿⣿⣟⣓⢀⠀⠻⠋⣽⠟⠿⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈⠀⠀⢠⣼⡼⣿⣿⣹⠷⣿⠆⣤⡀⠀⠀⠈⠓⠿⢄⡿⣿⣿⡟⠁⠀⠀⣼⠏⣀⠂⠐⡦
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡆⢠⢇⠀⠈⢡⠟⠙⢿⣿⣿⣿⣿⣶⢤⡄⠀⠈⠙⠻⠏⢴⣦⢢⡆⡿⠀⠄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣴⡀⠀⢸⣿⣿⣿⣿⣧⣿⣻⣟⣠⣶⣧⣀⡀⠀⢨⣶⢀⡘⠐⠃⠁⠑⢶
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠹⠼⠁⠀⢰⠀⠙⠄⠀⠹⣞⠛⡛⢻⠀⣬⡿⠟⠛⠉⠙⢏⣤⣤⣉⣁⣡⣿⣾⣮⣌⢛⡀
⠀⠀⠀⢀⠀⢀⠠⠀⠀⠀⠀⡰⠁⠀⠀⠀⢘⠀⠀⠀⠀⠀⠻⡛⣾⣠⡄⠉⠀⠀⢠⣤⠀⣴⡿⡿⣿⣿⠋⠀⠀⠙⠃
⠀⠀⠀⠘⣷⢡⢂⠂⡌⢠⠛⠁⠀⠀⠀⠀⣸⡷⠂⠀⠀⠀⠀⠀⠙⣿⣿⣠⡾⣷⣿⠋⠘⠋⠁⠙⣞⠁⠀⣤⡀
⠀⠀⠀⢠⠤⢃⠎⡸⢠⠇⠀⠀⠀⠀⠀⠀⠸⢿⡄⠀⠀⠀⠀⠀⠀⠘⢿⣫⡞⠉⠁⠀⠀⢠⠀⣰⠟⠀⢠⡿⠀⢠⣤⣴
⠀⠀⠀⠘⠖⠋⡰⠳⠋⠀⠀⠀⠀⠀⠀⠀⢰⣶⡍⠁⠀⠀⠀⠀⠀⠀⠠⣤⠁⠀⠀⠀⢒⡃⢀⣿⠁⣰⠈⠁⢀⣼⣿⣿
⠀⠀⠀⠘⠧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⣿⠀⠁⠀⠀⠀⠀⠀⢤⠆⠀⠀⠀⣠⡼⠁⠀⠀⠀⠁⢀⣴⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⡓⣒⡀⢀⢀⠀⢀⠎⠀⢀⠂⣠⡿⠁⣼⠀⠀⠀⣠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣄⠀⠀⠈⠀⠂⠀⢀⠎⣸⡿⢁⣴⠟⢀⣠⣾⠏⣪⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⡌⠈⠁⠈⢁⣄⡤⠀⣠⠔⣡⠞⢋⣴⣿⣣⣴⣿⣿⣿⣾⡏⣻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠌⡱⣌⠂⠠⢿⠿⠟⢡⣼⣯⡤⠤⠶⠿⠟⠙⣋⡩⠉⠉⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣠⣴⣿⣤⢤⣭⣭⠿⠶⠦⠒⠊⠠⠀⠀⡐⠀⠄⡔⣟⣀⡡⡤⣤⣤⢄⠠⠴⠒
⠀⠀⠀⠀⠀⠀⣀⠿⠶⠛⠛⠛⠉⠉⠠⠐⠀⣀⣀⣀⡠⠬⠄⠢⠶⡒⢑⠝⠉⠽⠟⠐⠛⠈⠉⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠶⠶⠖⠚⠛⠩⠉⠁⠀⠀⠀⠚⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠐⠤⣼⣧⠤⠔⠀⠀⠀⣸⣧⡀⠀⣤⡀⠀⠀⢰⣦⠀⠀⣸⣇⡀
⠀⠀⠀⠀⠀⢀⡀⠀⣰⣆⠄⢸⣿⠛⠀⣸⣿⠀⠀⠀⠀⢰⣿⣿⣿⢀⣿⣏⠀⢠⣾⣿⣦⣾⣿⣿⣿⡄
⠀⠀⠀⠀⣠⣼⡇⠀⣸⢧⠀⠈⡿⡁⠀⢸⡇⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⡭⠂⣼⣿⣿⡌⣿⣿⣿⣿⠁
⠀⠀⠀⠈⠙⠁⠈⠀⠃⠘⠀⠘⠁⠃⠀⠘⠁⠀⠀⠀⠀⠈⣿⣿⡏⢸⣿⡇⠀⠻⣿⣿⠃⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⡿⡇⢸⡿⣆⠀⠀⡿⢻⠀⠀⣿⢹⡇
⠀⠀⣀⠀⢰⠀⣰⣶⡀⠀⠀⢀⣿⣆⠀⣬⣿⣄⠀⠀⠀⠀⠼⠃⡧⠼⠀⠘⡄⢀⡇⢸⠄⠀⡿⠘⣇
⠀⠀⠸⣼⣾⢴⣿⣿⡿⠷⠀⣿⣿⡿⣶⣿⣿⣿⠆
⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⢿⣿⣿⠷⣿⣿⣿⠀⠀⢠⣤⠀⠀⢠⣶⠀⠀⠀⠀⢀⣄⠀⠀⢠⣤
⠀⠀⠀⣿⣿⡀⠘⣿⠇⠀⠀⠈⡿⡏⠀⠈⣿⡇⠀⠀⣼⣿⣦⢠⣾⣿⣷⡀⠀⢀⣼⣿⡄⢰⣿⣿⣷⡄
⠀⠀⢀⠏⠘⡇⠀⣿⠀⠀⠀⢀⡇⠧⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡟⣠⡾⣿⣿⡷⢿⣹⣿⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣦⠀⠀⠀⠀⠘⣿⣿⡇⠹⣿⣿⣿⠀⠀⠀⣿⣿⡇⢸⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣂⠀⠀⢠⣶⣿⣦⠀⠀⠀⠀⣿⣿⡇⠀⡿⠘⣿⠀⠀⠀⡟⢹⠀⠈⢸⠹⣿
⠀⠀⠀⠀⠀⠀⢶⢿⣿⣿⡷⢺⠏⣿⣿⡟⢇⠀⠀⠀⣿⢻⠃⢠⡇⠀⢽⠀⠀⢠⠁⣏⠀⠀⣼⠀⢿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⣿⣿⡇⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠙⠙

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠈⢉⡉⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⣨⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣿⡿⠋⠉⢠⣭⣽⣿⣿⡿⠛⠋⠁⠀⠈⠙⠻⠿⠿⠟⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣤⡴⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠋⣠⠀⠀⠀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣯⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⣩⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⢶⣴⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣻⣿⣧⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡷⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⡄⢠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠉⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠂⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⡄⢻⣿⣷⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠂⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⡿⠀⠉⣧⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠈⠂⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠛⠋⢤⣀⢠⣿⣿⣿⡆⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⢼⣿⣇⠙⡿⠿⡧⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠛⢿⣿⣿⣿⣿⣿⣿⠛⠻⢿⣿
⣿⣿⣿⣿⣿⣿⠃⠀⠀⢤⣁⠀⠀⠀⠀⡀⠈⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⡟⢫⣿⡿⠄⠀⢸⣿
⣿⣿⣿⣿⣿⣿⡃⠀⠠⣾⣿⠹⣿⣶⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠘⠋⠀⠀⠁⢀⣀⣰⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣶⠈⠉⠁⠈⠀⠹⣿⠖⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⠀⠆⠈⠀⠀⠀⠀⠀⠀⠺⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⡼⠿⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢀⣘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣤⣁⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⠻⣿⡉⠀⠀⢴⣄⠀⠀⢹⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠈⠛⠀⣴⣿⡿⠂⠀⠀⠂⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣿⣶⣤⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⣄⠀⠠⠀⣄⣀⡀⠀⠀⢠⣴⣴⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣷⣦⣴⣿⣦⣤⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿
⠀⠀⣠⣤⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠓⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣧⣾⣿⣿⣿
⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢀⣠⣶⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⡿⢇⡀⠠⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣿⣦⣄⣴⡶⠀⠀⠀⠀⠀⠀⠂⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⢾⣿⣿⣄⣶⣀⡀⠀⠀⢠⠂⠄⠀⢲⣧⣦⣶⡤⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣄⠀⣬⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠘⢿⣿⣿⣿⣟⠷⠀⡀⢸⣶⣿⣿⣦⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠝⠛⠛
⠀⠀⠀⢿⣿⣿⣏⣹⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⡿⠁
⠀⠀⠀⠀⣉⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠛⢿⠿⠈⠀⢠⠀⠰⠄⠈
⠀⢸⣇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡏⣿⣿⣿⣿⡇⢻⠘⠻⡿⣿⣿⠛⠚⠀⠀⠀⠀⢼⠀⠀⠀⠈⠀⠀⢠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠈⠻⠇⣿⣿⣿⣿⡇⢻⠀⠀⠀⢸⡿⠀⠀⠀⠀⠠⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡛⡻⡟⠁⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⣿⣿⣿⠿⣿⣿⣿⣿⣧⠾⣿⡿⠛⢹⠁⠋⢙⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⡏
⠛⢿⣿⣿⣿⣿⣿⣿⠟⠀⣿⠛⡀⠉⠀⠀⠸⠛⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣼⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠶⠦⢤⡀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠳⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠉⠑⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⠀⠀⠀⠀⠀⠉⠢⡄⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⡴⠂⠉⠀⠀⠀⠀⠀⠈⠉⠓⢄⠀⠀⠀⠀⠈⣣⠤⣲⣶⡟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⡖⠢⠀⣤⡄⠀⠰⠒⠒⠄⠈⡇⡀⠀⠀⢠⡶⢚⣿⡧⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⡰⠀⠀⠀⠀⠀⠤⠔⠁⢰⠀⠉⠀⠀⣘⡠⣾⠏⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠏⠑⠆⠰⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠃⣾⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠸⢾⣿⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⣗⣿⣟⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠴⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠀⢀⠇⠀⠀⠀⣆⣽⡴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣿⣯⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠣⠤⠄⠠⠤⠅⠀⣀⣀⣀⣀⠀⠀⠀⠘⠀⠀⠀⢀⠫⡶⠞⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣠⣤⣠⣠⠠⢠⠀⠄⡄⡀⡈⠉⠉⠉⠉⠉⠉⠙⢻⣡⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠂⠛⠀⠰⠺⠿⠾⠴⢤⡼⣴⣤⣼⣯⠜⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⠿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣠⣴⠟⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠷⣶⣬⡟⠁⠚⠋⠠⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣏⣽⣿⠀⠀⠀⠨⡇⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⣿⣿⣷⣤⡤⠔⠁⠀⣋⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣆⣴⣈⣷⣾⠋⠀⠀⠀⠀⠸⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⠿⠛⠛⠛⠒⠀⠀⠀⠀⠀⣱⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⣻⡏⠀⠀⠀⠀⠐⠀⠢⢀⣰⠍⠢⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣾⣿⣿⣦⣀⠘⠋⢁⡀⠀⠀⡁⢠⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠏⣿⣿⣿⣷⣾⣟⣠⠀⠘⢱⠀⠙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⣾⠃⠸⠟⠛⢻⣿⣿⣿⣿⣶⣷⣄⣠⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⠇⠀⠀⠀⠀⠸⣿⣿⡟⠀⠉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣾⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣠⣯⣤⣶⣿⣿⣤⣶⣦⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⢹⡟⠻⣿⠿⡳⠍⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢡⣿⣿⣿⡆⠀⢓⠀⠈⠳⣌⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⣷⡄⢮⣶⣤⣤⣌⠳⣄⡀⢄⠀⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⣿⣿⣿⠿⠿⣾⠷⣿⣿⡟⠛⡟⠛⠙⠛⠀⠀⠠⠀⡈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡼⠋⠀⠀⢹⡇⠀⢀⣤⡝⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢷
⠀⠀⠀⠀⡠⠞⠋⠈⣀⣀⣠⣤⣴⣾⣧⣄⣿⣿⣇⣸⣇⡀⡀⣦⡀⠀⣴⣀⣀⣀⣰⣤⡞
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠸⠶⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⢷⠃⡏⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⢢⠄⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⢦⠀⠁⣤⠀⠀⣤⣠⠄⠀⠰⠬⠀⠀⠄⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⠄⡟⠀⠀⠀⡄⠀⠀⠀⠀⠠⡄⢛⠂⠘⠴⢆⠙⠆⡀⠀⠁⠀⠀⠀⠇⠀⠀⠘⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⡀⠀⠀⠀⠀⠘⠀⡒⠙⠂⠀⠀⠀⠄⠠⡙⠛⠀⠀⠀⠀⠀⠀⠐⡀⠀⠰⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠀⠀⠀⠀⠀⠀⠆⠡⠀⠀⡄⠘⠀⠄⡀⠀⠻⣨⠆⠀⠴⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠁⢀⡄⠀⠀⢂⠀⠀⠀⠁⠂⡀⠀⢈⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣘⡣⠀⠠⣅⠀⠒⠀⠂⢰⡫⠀⠀⢈⡅⠌⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⢂⠈⠁⢀⠀⠑⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⠀⠀⠈⠀⠈⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠄⠀⠂⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣩⠀⠀⡄⠀⡀⠀⠞⠀⠀⠰⠀⠀⠁⠀⠂⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠒⠀⠀⠰⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠠⠠⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⢀⠀⠀⠀⠀⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣈⠀⠶⠚⠚⠀⡂⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⠀⡘⠁⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⣤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢠⠁⠀⠀⠀⠀⠂⠀⠀⠐⠰⠄⠀⠀⠀⡀⠄⠀⠀⠈⠀⠀⢀⡠⠁⠂⠀⠀⠀⢰⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠀⠀⡠⠀⠀⠀⠀⠂⠠⠈⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠙⠃⡄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢄⡀⠀⡐⠀⠀⠄⣀⠴⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⡠⠀⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡘⠀⢀⠈⠘⠧⣄⠀⠘⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⢀⡽⣀⠀⠀⠀⠀⣠⡴⠂⣀⠄⠁⠀⢩⠄⢀⡀⠁⠄⠀⠈⠰⣄⡈⢷⡄⠀⠀⠀⣰⠠⢀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢠⣦⠀⠀⠀⣿⠠⡈⢿⣄⡀⠤⠀⠀⠀⠁⠀⠀⠀⠒⣤⠋⠁⠨⣇⠀⠀⠠⠔⠋⠀⠈⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢃⠀⠀⢹⡀⠈⠀⣈⡗⢂⠀⠀⠀⠀⠀⠀⠠⡀⠒⠖⠀⢸⠇⢀⣴⠎⠁⠀⠀⠀⣀
⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡶⠀⠀⠃⢀⣀⡬⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠑⣀⡔⠋⣁⠐⠀⠀⠀⠀⠀⠀⠁
⠀⡘⠀⠀⠀⠰⠁⠀⠀⠀⠀⡘⠀⠀⠁⢐⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢚⠣⠀⠴⠫⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠒⠂⠀⠀⠀⠂⠀⢀⠀⣀⠀⠀⠂⠢⠀⠀⠀⠠⠀⢀⠀⠀⠀⢀⠀⠀⠈⠀⣀
⠀⢨⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠂⡃⠀⠀⠀⠀⠀⢀⡅⠂⡱⠀⠄⠀⠀⠐⠑⠀⠀⢀⡀⠈⠑
⠀⢉⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠲⠁⠈⠀⠠⠉⢀⠀⠀⠸⠀⠀⠀⠁
⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠛

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣶⣤⣤⠤⣄⣀⣀⣀
⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡿⣿⣿⣷
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣟⣭⣿⠿⢿⣿⣿⠿⠏⠙⠁
⠀⠀⣾⣿⣿⣿⣿⣿⣛⣻⣿⣿⠯⢒⡶⣟⣛⠛⠀⠀⠆
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⣿⣿⣿⡃⠀⠀⠀⠸⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣾⣭⣶⠋⠀⠀⣠⠶⢶⠇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⡁⡉⢿⡿⠀⠀⠀⠀⠔
⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⢆⢀⡀⠀⠀⠀⠀⠀⠀⠃⠀⡄
⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡸⠂
⠀⠀⢸⣿⣿⣿⣿⠟⠀⠀⠀⠛⠦⠂⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠈⠋⠉⠀⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠚⣢⠄
⠀⠀⠀⠁⠠⣄⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⢧
⠀⠀⢀⡀⠀⠈⠑⢦⡀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠴⠃
⠀⠀⠀⠙⢦⡀⠀⠀⠙⢦⠀⠀⣰⠉
⠀⠀⠀⠀⠀⠈⠳⡄⠀⡄⠑⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⢰⣧⣤⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣿⡷⣿⠁⠀⠀⠀⠀⠀⠀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣀⠀⢀⢔⡿⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⣅⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢐⡵⠋⠠⢀⡀
⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡠⠒⠢⢄⣀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⢀⡴⡡⠋⠀⠀⠀⠀⠈⠑⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠈⢳⡄⢀⡤⠂⠀⠚⢆⠀⠀⠀⣄⠀⠀⠀⠀⠈⠁⠒⠤⢀⡀⠀⠀⠀⢀⠒⢤⣠⡞⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡼⠁⠀⠀⠀⠀⠀⡄⠀⠀⠙⠒⠤⣀⠀⠀⠀⠀⠀⠈⠑⠢⢀⡌⠁⣾⠀⢁⠀⠀⠀⠉⠐⠒⠤⣄⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠴⣫⠏⠰⣤⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠉⡲⠆⠀⠀⠀⠀⠀⡟⠀⢸⡇⠀⣾⡀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⣰⠏⢈⡆⠀⠈⠻⢦⡀⠀⠀⠸⡇⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⡔⢁⠞⠁⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠊⢁⠄⢸⡄⠀⠀⠀⠀⠈⠁⠀⠀⢞⠀⠀⠀⠰⢅⡀⠀⠀⣠⠊⢀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠈⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⢀⡔⠁⠀⠀⢿⡇⠀⠀⠀⢀⣴⠊⠀⠛⠀⠀⠀⠀⠀⠈⠑⠒⠁⠀⢸⠀⠀⠀⠀⠀⠄⣠⠊⠀⠀⡰⠁⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠈⠀⠈⠉⠐⢿⡀⠀⠀⠀⠚⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠞⠁⠀⠀⠜⠁⠀⠀⠀⠀⠰⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠀⠘⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣤⣤⣤⣀⣀⡙⣶⠀⢰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠁⠀⠉⢉⢋⠟⠂⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠒⠀⠴⠀⢀⡀⢀⡀⣥⠁⠀⠀⡀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠁⠀⠀⠀⠠⠊⠀⡠⠋⠀⠑⣤⠊⡷⣤⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠁⠲⠀⠄⠤⠀⢀⠴⡁⠀⢀⡴⠓⢒⡓⠛⢦⡀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠢⠄⢠⠊⠀⠀⠀⠀⡀⠁⠀⠈⡢⠎⠀⠀⠻⠀⠀⠈⠿⣤⠒⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠒⢦⣄⡀⠀⠂⠠⠌⠉⠢⣀⠴⠥⠓⠀⠢⡖⠒⠒⠊⠉⠙⠷⠤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠘⠂⠄⠉⣰⠲⠶⠖⠒⠃⠀⠀⢀⣴⣾⣿⣷⣄⠀⠀⠀⡄⠤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⢍⡂⠀⠀⢲⣀⡀⢠⣾⣿⣿⣿⣿⣿⠀⠤⠄⡆⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠠⠄⡀⠀⠠⡃⠀⠀⠰⠈⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⡀⠄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠁⠒⡟⠲⠒⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠤⠿⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠂⠤⢀⡱⣇⠀⠀⡐⡞⠉⣿⣿⣿⣿⣿⣿⣿⡄⠀⠐⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠯⣖⠤⠄⠀⠈⡇⠙⠉⢃⡇⠀⣿⣿⣿⣿⣿⡿⠿⢆⡤⠘⠂⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠐⠤⢙⠂⢄⡀⠧⢀⡀⡷⠧⠀⢛⠉⠉⢀⠀⠀⠀⠀⠛⠓⣄⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡀⠀⠈⠐⢬⣁⠀⢒⠒⠋⢽⠊⠉⢉⠄⠀⠀⠀⣀⠠⠄⠘⢤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢦⣄⠀⠀⠀⠀⠀⠀⢘⠠⠤⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡤⠤⠖⠚⠦⣀⠀⠀⠉⢂⣀⣀⡤⠤⠖⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠳⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠏⡇⠀⠀⠀⠀⢀⣾⠋⠁⠀⠀⣀⡀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⢺⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⡄⠀⠀⠀⠀⢀⣿⠃⠀⠀⢐⣽⠇⠑⢷⣄⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⢳⡻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠃⠼⣧⠀⠀⠀⠀⢸⣿⠀⠀⡆⠈⠀⠀⢀⠘⢿⣤⠀⢸⣿⠀⠀⠀⠀⠀⣠⣿⠃⠹⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⢛⣧⠀⠀⠀⠘⣿⡄⠠⠘⡆⠀⠀⠀⠀⠀⢳⡄⣸⡿⠀⠀⠀⠀⣴⣿⡟⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⢻⣿⣷⣄⡀⠀⠹⣿⣤⠀⠙⠤⠀⠠⠖⠀⠀⣷⡿⠁⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣦⣬⣿⣴⡁⠆⠀⠀⠁⠠⡆⠈⢠⣤⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⡘⣷⡄⣀⣀⣀⠀⣄⣀⠀⠩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⢰⠾⣇⠀⢀⠄⠀⠀⢠⠈⠙⣷⡅⠀⠀⠀⣿⠈⣿⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠈⠀⢾⣦⠋⠀⠀⣼⡇⠀⠀⢁⠐⢀⡿⢀⢹⠀⣷⠈⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⣿⡇⠀⠀⠌⠴⢋⠄⣨⠀⠀⣿⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⡟⠃⠀⣿⡇⠰⢚⣡⠞⣡⡾⢋⡀⢸⠻⣧⠑⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⢶⡀⠐⢁⡾⣀⠙⢒⡾⢋⣴⠟⢈⣅⣤⣤⢸⠀⣏⠑⢦⣱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡿⠗⠲⠿⡇⢻⡿⣫⠴⢛⣡⠾⠏⡇⢰⢹⣇⣇⢸⡄⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⢡⡟⢀⣤⠖⢛⠀⠈⠀⡇⢸⡹⣟⠎⠀⢛⣠⠊⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣾⠉⣿⠁⢸⠀⠘⣶⠄⢀⡇⢸⡇⢿⠈⠄⠈⢿⡁⢠⣈⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢛⠀⣿⠀⢸⠀⠀⠛⠀⣼⡇⢸⢿⠈⡆⠀⠀⠀⠙⠂⡽⢿⡄⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣼⠁⠀⠀⠈⠀⣿⠀⢸⠀⠀⠀⢰⣿⡃⢸⡌⠃⠸⡄⣰⣄⠀⣠⡆⠀⠀⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡸⠉⠀⠀⠀⡇⢸⣿⠀⢸⠀⠀⢠⡟⢸⠀⢸⢷⠀⠀⠹⡸⣿⣿⣿⣓⣷⣤⣿⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⢸⣿⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⡇⢸⣿⠀⢸⠀⠀⢻⡆⢸⠀⢸⠈⠀⠀⣶⢳⡙⠋⡁⢀⣤⠀⢸⡆⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⢰⠁⠀⠀⠀⠀⡇⣿⣿⠀⠸⡀⠀⠸⠃⣸⠀⢸⠀⠀⠀⠸⡄⢻⡾⣰⠏⢻⡆⠈⣷⢠⠀⠀⠀⢸⣷
⠀⠀⠀⠀⣾⡿⠀⠀⢀⣧⠎⠀⠀⠀⠀⠀⡄⠟⢸⠀⠀⡇⠀⠀⠀⣿⠀⢸⡆⠀⠀⠀⢹⢨⣤⠃⡔⢸⡇⡀⠘⢧⡄⠀⠀⠸⣿
⠀⠀⠀⢰⣿⡇⠀⠀⠸⠈⠀⠀⠀⠀⠀⠀⠙⣆⢸⠀⠀⣷⠀⠀⠀⣿⠀⢸⡇⠀⠀⢠⣾⠸⡏⣸⠀⠀⣷⠀⠀⠘⣇⠀⠀⠀⣿⡇
⠀⠀⢀⣿⡟⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⠀⢻⠀⠀⠀⢻⡀⠸⣷⠀⣦⠘⣿⡆⣿⠃⠀⠀⢿⢠⠀⠀⢻⠀⠀⠀⠸⣿
⠀⠀⣼⡟⠀⠀⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⢸⡆⠀⠀⢸⡇⠀⡿⡆⢸⡆⢻⡇⠇⠀⠀⠀⢸⡏⠀⠀⠈⡇⢀⠀⠀⢻⣇
⠀⣸⡟⠀⡄⡸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡧⠀⠀⣸⡇⠀⡇⠈⠀⢷⣼⣷⠀⠀⠀⠀⠘⣇⡇⠀⠀⢸⡘⡄⣄⠈⣿
⢰⣯⣤⡞⠰⢁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⣧⠀⠀⢩⡇⠀⡇⢇⠠⡘⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢃⢻⡼⡷⣼
⠈⠁⡀⢠⠳⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⡷⣄⠸⡇⠀⢧⠸⡀⢳⢹⣿⠀⠀⠀⠀⠀⢹⣼⠀⠀⠀⠈⠾⢷⣷⡄
⢀⡞⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠶⠶⣶⣿⠳⡈⠁⠀⠀⢸⡀⢳⡄⢸⣿⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠀⠀⠳⡙
⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠖⢻⠈⡟⠀⠀⠀⠀⠐⣌⡻⠄⠹⣿⠿⡄⠀⠀⠀⠀⠀⣷⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⠀⢸⠀⡇⠀⢸⡇⠲⣤⡈⠻⣶⠒⢶⣾⡇⠀⠀⠀⠀⠀⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⠀⢸⠀⡇⠀⢸⡇⠀⡟⠀⢲⣦⠅⣈⣩⠂⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡀⢸⠀⡇⠀⢸⡇⠀⡇⠀⠸⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⠉⠛⠖⠚⠓⠒⠧⠤⠞⠛⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄⠀⠀⠀⢰⡀⣠⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠛⢻⡾⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠋⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣟⣻⣷⠀⢹⣿⠀⠀⣿⡟⢛⣷⠀⠀⠀⢠⡗⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿⣿⣿⡁⠀⣾⣿⠀⠀⣿⡟⠛⠁⠀⠀⠀⢸⣧⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠿⠃⠻⠗⠀⠿⠿⠀⠀⠿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣻⣦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣴⠂⠴⢺⠟⠙⡟⢻⠟⠩⠿⠛⠽⠟⢻⡿⢛⣿⠷⣺⣿⠶⣶⣾⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠉⠘⠉⠀⠉⠙⣛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠒⠒⠶⠶⠔⠒⠒⠒⠦⠤⠄⠒⠶⠤⠤⠤⠤⠤⠴⠖⠀⠤⠴⠖⠲⠴⠶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⢿⣿⣿⡍⠛⠛⠻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣶⣬⣽⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡻⣿⡏⠁⠠⣶⡔⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡇⠉⠡⠀⠀⠑⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡄⠀⠀⠀⠾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠌⠙⠲⠦⣤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⣿⠛⠛⢲⡶⠶⢦⣄⡀
⠀⠀⠀⠀⢀⣴⠟⠁⠈⠳⠂⠀⠐⠒⠶⠁⠊⠀⠀⠰⣤⡀⠈⢻⣦
⠀⠀⠀⠀⣼⠇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠈⠙⣦⠀⢿⣷⡄
⠀⠀⠀⠀⣿⠀⠙⣷⣤⠄⠀⢀⣀⠀⠀⠀⠀⠀⠹⣇⠰⣦⡿⠀⠀⠈⠙⠛⠛⠛⠷⣦⣀
⠀⠀⠀⣼⣧⠀⠀⠀⠀⠀⢰⡟⠑⠶⣤⣤⣤⣤⠄⠀⢀⠘⣿⡶⠂⢠⢸⡆⠀⠈⠙⠾⠛⠷⡄⣀⣠⣤⢶⣟⣷⣆
⠀⠀⢸⡗⣿⠀⠀⠀⢠⡆⣸⠇⠀⠀⠀⣤⣤⣄⣀⠀⢸⡇⢸⣷⣤⣼⣆⠻⣤⡴⠶⢀⣀⣤⣴⡾⠟⠛⢻⡟⢛⣿
⠀⠀⢸⡇⠀⠴⠴⠖⢋⣰⠏⠀⠀⡐⠶⠶⠤⠉⠀⣠⡟⣠⡿⠁⠀⠀⢙⣷⣶⣶⡾⠿⠛⠉⠀⠂⠀⠀⣽⡇⢸⣿
⠀⠀⣼⠃⠀⠲⠶⠒⠻⠷⣤⣄⠀⠉⠉⠀⠀⠰⠚⢉⣰⣟⣀⣤⣶⣾⣿⡿⠏⢷⡄⠀⠀⠀⠀⠀⠀⠀⣿⣥⡿⠃
⠀⢸⡏⢀⣤⣄⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⡶⠚⢻⣿⣽⣿⠿⠟⠟⠉⠀⠀⠀⠀⠁⠀⠀⠀⣠⣴⣶⠶⠛⠋⠁
⠀⠈⢷⣤⣬⡻⠷⣦⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⢉⣩⡅⣀⢠⡄⠠⡄⡀⠀⠠⢠⣴⠴⠟⠋⠁
⠀⠀⠀⠀⠈⠛⠛⠳⠶⠖⠚⠻⠶⢶⣤⣄⠀⠀⠀⠀⢼⡇⠉⠈⠀⠀⣀⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧⣄⣀⣸⣿⣁⣠⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠿⠛⠉

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡕⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣿⠿⣿⠿⣿⣷⣿⣿⣿⣿⣍⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣯⣿⣟⣿⣟⣿⣿⣿⣿⠿⢿⡿⢿⣫⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣇⠻⣷⣟⡁⠉⡛⢻⣿⣶⡽⣏⣀⠀⠛⢿⠛⠋⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡟⢛⣽⠿⠈⢹⡓⣽⡇⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢈⣿⣤⠻⠿⢻⠟⢧⡀⢳⣤⣆⢺⡿⠀⠀⢸⣿⣴⣿⠟⠛⠛⠿⡏⢻⣿⣿⣿⣿⣿⠿⡏⢻⢀⡞⢛⠀⠴⢼⡟⣿⣳⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣟⠈⡙⠃⠘⢀⡿⢨⡿⠿⣶⣥⣼⣇⢸⣷⣿⣿⡶⢃⣔⣠⣶⣿⡷⠿⣥⣬⠘⢐⣧⠀⢸⠋⣾⢀⡀⠈⡴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣀⣬⣿⠇⣿⠀⠗⢬⣿⣿⣿⣿⣿⡿⣁⣤⣤⣸⣿⣿⣤⠄⠀⢀⣤⡶⠟⢻⣠⢬⡁⣿⣘⠖⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⣯⢴⣦⣜⣯⣻⣿⢟⣿⡿⠿⠿⣿⡽⠟⠋⣀⡠⣶⠋⠀⣴⣾⣟⣋⠼⠚⠛⠻⠗⣸⣿⣿⣿⢙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣭⡟⠛⣷⣼⣿⣿⣷⣿⣶⣄⠀⠀⣠⣬⣀⣧⠠⣿⣿⣿⣿⣿⣷⡤⣿⣿⣧⡈⠟⣋⡡⠚⢿⣽⣸⢩⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣯⡚⠳⠈⠻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣻⣷⠛⣿⣿⣿⠏⠁⠀⠀⣤⡿⠆⢸⠛⠇⠀⣺⣿⣿⡹⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⡀⢠⣼⣿⣿⣮⡛⢿⡆⣿⣿⣿⣿⣯⣿⣞⠊⣿⢃⣼⡀⠀⠈⡟⠃⢠⣾⣷⣆⣴⡛⠙⣿⣷⣾⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣇⣨⣾⣿⣿⣿⣿⣿⣿⣄⠛⠻⢿⣿⠛⠻⢿⣿⣧⣿⣿⡟⠀⠀⢀⣿⠀⢘⣿⣵⡿⠿⣿⣣⣿⡿⢻⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣄⠉⢿⣿⣿⣿⣿⣷⣦⣀⡟⣘⡓⢢⡟⣹⣿⣿⣿⡟⠀⣾⣿⠀⣼⣿⡽⢇⣤⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠻⣿⡟⠉⠉⠙⣻⣥⣄⣃⣩⣌⠻⣿⣿⡿⢃⣼⣿⣿⣧⣽⣿⢿⢿⣾⢿⣿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⠿⠶⠋⡈⠙⠉⠉⠀⠀⠀⠀⠀⠛⠛⠀⢉⣴⡿⠿⠿⣿⣿⣿⣧⡐⠿⠛⣋⣉⡀⠀⢼⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣟⢛⣷⣿⣿⣿⣿⣿⣿⣽⣯⢠⣴⣦⣴⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⢀⣴⣶⣾⣿⣿⣿⣿⣖⣿⣿⣾⣿⠟⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣇⡼⣿⣿⣿⣿⣿⣁⣠⣖⣒⠀⠀⡀⠀⣀⣀⣤⣴⣾⣶⣶⣾⣿⣿⣿⣿⣿⣿⣇⠻⣭⣽⣧⣀⣴⣶⣿⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠽⠿⠿⠻⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠛⠉⠉⢿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠿⠶⢶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠚⡏⢹⠶⢬⣉⠛⠳⣄⠀⠰⡖⢒⡆⠀⠀⣠⣤⣄⣀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⢠⠀⠀⠙⣧⠀⢹⡆⠀⡿⢸⠀⠀⠀⠁⣴⠶⣍⠻⣆⢸⣧
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⡇⢸⡆⢀⣠⠟⠀⣸⠁⠀⣇⣿⠀⠀⠀⡆⣿⠀⠸⡆⢹⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⢸⣏⣉⡥⠴⠚⠁⠀⠀⣿⢻⠀⠀⠀⣷⣿⠀⢰⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢀⣤⡾⢃⡤⠖⢻⣦⠴⠚⢹⡇⠀⠀⢸⠀⡟⣎⢷⡄⠀⠀⠀⠀⣿⢸⠀⠀⠀⡟⠻⠖⣋⣴⠟⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡤⠞⠉⠈⠛⠉⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢸⠀⣧⠘⣆⠹⣆⠀⣀⠀⣿⣸⡤⢤⠐⡇⣶⠋⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢴⣿⣤⡽⠂⠹⣦⡼⠟⠛⠁⠉⠉⠙⢫⣾⣧⣽⠀⠾⠗⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⠚⠓⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢀⡴⢻⢃⡴⠋⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢀⡴⠋⠀⠷⠚⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣼⠏⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠦⣤⣀⠀⢸⣇⣠⢶⡄⠀⢠⡾⣇⠀⠀⣠⡼⣷⠀⠀⣀⣀⠀⠀⠀⣀⣤⠾⠋⣡⣿⣵⡿⡾⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠉⠛⠾⠿⠁⠀⢷⣀⡟⠀⢿⣄⡾⢹⡇⢿⣤⡾⠋⣿⣠⡴⠞⠋⠀⠀⠀⣿⠟⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠈⠓⠛⠋⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡾⣡⠞⠉⠉⠈⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢱⠇⠀⠀⣺⡗⠀⠀⣷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠁⠈⠀⠀⠀⠀⣛⠅⠀⠙⠛⡆⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⡔⠀⠀⠀⠈⠱⠋⠀⠀⠀⠀⠙⢮⣒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠱⠊⠀⠀⠀⠀⠀⢠⣄⠀⠀⣤⡤⣄⡀⢹⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⡏⠉⣻⠆⠈⣿⠀⠀⢻⡇⢨⣿⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⡇⠰⣯⡀⠀⣿⠀⠀⢸⡇⠉⠁⠰⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⡀⠀⢸⣷⠀⠘⠧⠄⠿⠄⠀⠸⠿⠀⠀⠀⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢿⡇⠀⠰⠒⠒⠒⠒⠒⠒⠛⠛⠛⠉⠛⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⣿⠀⡀⢀⣀⡀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠠⠴⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠛⢋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⡀⣀
⠀⠀⠀⠀⢀⣀⣀⣄⣀⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠻⠓⠾⢶⠶⠶⠲⠖⠂
⠀⠖⠚⠸⠻⠿⠿⠛⠛⠟⠝⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠊⠂⠉⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠠⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠁⢂
⠀⠀⠀⣤⢄⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢄⡐⢀⠄⢀⠗⠁⠀⠀⠀⢀⠁⠀⠀⠈⢄⡀⠀⢣
⠀⠀⠀⠮⠣⠠⠜⠀⠀⠀⠀⠀⠀⠀⣡⠄⠁⠀⢤⣆⠀⠀⠀⠀⠀⠂⣀⣇⠥⣀⡨⢍⣦⡀⢁
⠀⠀⠀⠐⡈⠢⠀⠂⠀⠀⢀⠤⠐⠘⣿⣷⣄⣴⣿⣿⡃⡂⠄⠀⠀⠢⠐⠀⠰⣍⡅⠀⡘⠰
⠀⠀⠀⠀⠈⢰⣶⣿⡄⢠⣿⠀⠀⠀⠈⠻⣿⣿⠟⡹⢋⠃⣀⣿⡀⠀⠈⠒⡒⠂⠒⡖⠈
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣿⡄⠀⠀⠀⠀⠈⠁⠀⠀⢠⣀⣼⣿⡇⣠⣾⡟⠊⠴⠌⠐⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠃⠀⠀⠀⠀⠛⠉⠉⠉⠃

⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⣿⡇⠸⠁⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠋⠀⠒⠀⠉⢸⣿⣿
⣿⣿⣿⣿⣿⢣⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿
⣿⡟⠉⢻⠟⠀⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿
⠟⠁⠀⠀⠈⠀⢙⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣀⣀⣤⣤⣶⣄⣤⣾⣿⣿⠁⣴⣷⠀⡀⠰⠀⠀⣸⣿⣿⣿
⠈⠈⠃⠀⠀⠀⠈⢿⣿⣿⣿⣟⣿⣆⠀⠀⢸⣿⡏⣶⠿⣿⣿⣿⣿⣿⠏⠸⣿⣿⡀⡿⠀⠀⠀⣹⣿⣿⣿
⠀⠀⠀⠀⡆⠀⠀⠈⣿⣿⣿⣿⣿⣿⠗⠀⣤⠟⣴⣿⣷⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⡇⢸⣤⠀⠈⠿⣿⣿
⠀⠀⠀⠐⠇⠀⠀⠀⠙⢝⡛⠿⠝⠀⠀⠀⠀⠀⠻⠿⣿⣟⢻⣿⣿⡟⠀⠸⢯⣿⣿⡗⢸⠋⠀⠀⠐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡟⠂⣀⣦⠄⠀⢀⠀⠒⠒⢾⣿⣿⠋⠀⠀⠀⠀⣿⣿⣿⡿⡆⠘⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡋⠈⣻⣿⠰⠞⠋⠰⠃⣴⣾⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⠇⢸⡇⢀⣾⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠚⠻⡟⠀⣀⣾⣧⣼⣿⣿⣿⣿⡀⠀⠀⠀⠘⠛⣿⣿⣦⢸⣿⣿⣿⣿⣿
⡗⢀⣶⣶⡀⣠⢀⣀⣠⣆⠸⣇⢠⣤⣤⣤⣿⣿⣿⣿⣟⣿⣿⣿⣅⠀⠀⠀⠀⢈⡿⠛⢀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⢈⣼⣿⣿⣿⣧⢀⣾⣿⣿⠟⢛⣟⢫⣭⣿⣸⣿⣿⣿⣦⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⡟⢿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣴⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣻⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣼⣿⣿⣿⣹⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⡿⣷⣿⣿⣿⣿⡿⠭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠘⠿⠏⢹⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⡀⠀⠀⣼⣾⢧⡠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡔⠀⠀⠀⠀⢠⠆⠀⡖⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⡀⠀⠀⠀⠀⢀⡤⠂
⠐⣿⠀⠀⣆⡄⠀⢸⠃⠀⣠⠀⠀⢣⠀⠀⠀⠠⢽⡀⠀⣯⢶⣶⠀⠀⣦⣾⡇⠀⠀⡞⠀⢰⣿⣒⢾⡏⠀⠀⣦⣄⣀
⠘⣿⠀⠀⠉⠀⠀⣼⠀⠀⣿⡆⠀⢸⠀⠀⠀⠠⢽⡇⠀⠉⠘⣿⠀⠀⠁⣸⡇⠀⠀⡇⠀⠘⠀⠈⠈⢧⣀⡀⠈⠉⠛
⢺⣿⠀⠀⠦⠼⠿⣟⠀⠀⣿⠃⠀⢸⠀⠀⠀⢀⣸⡇⠀⠀⣠⣟⠀⠀⠀⠶⠷⠀⠀⡧⠀⢠⠀⣀⣀⣠⠟⢶⣤⣦
⢘⣿⠀⠀⠀⠀⠀⣿⢄⠀⠀⠀⢠⣞⠀⠀⠀⠀⠈⡇⠀⠀⣘⣯⠀⠀⣇⣒⡒⠀⠀⡗⠀⠈⠛⠁⢠⣿⣄⠀⠙⠋⢀
⢨⣿⣽⠋⠀⠀⠀⠘⠳⠭⣴⡞⠁⠿⠧⠤⠴⣀⣼⣽⣶⠀⢀⣿⣗⠖⠁⣬⣟⣡⡰⣯⣍⣉⣒⣶⠛⠛⠬⣕⣀⣴⠏

⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⣀
⠀⠀⠀⠀⠀⠀⠠⠬⣉⣘⣻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠂⠤⠤⢬⣽⣿⣿⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡤⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠉⢀⣀⠠⠈⢛⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣷⡾⠇⢂⠠⠟⠀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢀⡤⢅⣈⣻⡿⣿⢿⣿⣆⠀⠀⠀⠀⢠⣴⣿⡇⠀⠈⡾⢏⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⢿⣭⡉⠁⡌⠁⠀⢀⡀⡀⠀⠈⢻⠀⢀⣠⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠭⠿⠠⠄⠀⠀⣲⠶⠜⠒⠀⣴⡤⢨⡄⠈⢂⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣤⡶⠀⠄⠀⠀⢺⣷⡟⠈⢃⣿⡟⠘⠀⠀⠲⠾⠷⣤⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠐⠀⢀⣀⣠⣼⣿⣖⢀⣴⣶⣶⣶⣶⢃⣼⣿⡆⠈⠏⣿⢿⣿⡛⣿⣿⢯⠄⣀
⠀⠀⠀⠶⠀⠀⣀⣽⣿⣿⣿⡿⣿⣼⣿⣟⠻⠍⠛⠻⠅⠔⠈⠀⠀⠠⣾⣷⣟⣛⣟⡿⢧⡄⠤⠐
⠀⠀⠀⠀⠀⠈⢉⣰⣟⣿⣿⣼⣿⣿⢙⣲⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⡀⠈⠉⠛⣿⣷⣦⢞⠷
⠀⠀⠀⠀⠀⠀⢹⣄⣸⣿⣭⣿⣿⣿⣿⣿⣿⡟⠹⡴⠐⣧⡀⠀⠀⠺⣿⣿⣧⣴⣶⣾⣿⠿⠆⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠭⣿⣛⣻⣿⣿⣿⣆⠀⠀⠙⢿⣄⠀⠀⠙⢹⣿⣿⣽⣭⣬⣭⠀⠀⣀
⠀⠀⠀⠀⠀⡄⠀⠲⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⢨⠿⣄⠄⠐⢀⠙⢿⣿⣿⣿⡿⠠⠶⠁
⠀⢀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠻⡆⠀⠡⣦⠿⣇⢂⠱⡔⠘⠻⣿⣿⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠘⢿⣿⣿⠟⢳⢿⣿⣿⠟⠀⢀⣼⡀⣤⣿⣼⣿⣾⣆⢷⣴⣞⢈⠋⠀⠀⡌
⠀⠀⠀⠀⢈⠀⠀⠀⠠⠌⢻⣯⣾⣿⣽⡿⠀⡀⠀⣼⣿⣷⣬⣩⣏⣿⣟⣹⣿⣿⡟⠻⠃⠀⠀⠁
⠀⠀⠀⢀⠘⠀⠀⠀⠀⠐⠛⢫⠟⢿⣟⡔⣸⠁⣼⠿⣽⡿⡇⠠⢭⣿⣿⡷⠙⠁⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠐⣹⡽⠟⣸⡇⣧⢣⢻⣹⠀⠞⢯⠁⢄⠀⠀⠀⠠⠀⠀⢰⡷⠋
⠀⠀⠀⠀⠀⠀⢶⡦⠀⢡⠀⠀⣤⣼⠐⢃⠀⢻⢰⣿⣦⢳⣿⠄⡌⣦⠷⡀⠱⢪⠅
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⢀⡂⠀⣿⠀⢸⠀⢠⡼⣿⣿⣧⠻⠇⠀⣸⡑⠈⢢⣴⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣤⣾⢿⠇⠘⠀⠻⣦⣼⣿⢿⣷⡄⣤⠷⡿⠲⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠓⠚⠦⣔⣰⣶⡿⣹⡄⢿⣷⣟⢀⠘⢆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠜⠳⣌⣿⣽⣴⣏⣇⣈⣯⣭⣥⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣯⡿⢿⣷⡌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠋⠃⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⠤⠄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⠶⢧⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣤⡀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠶⣴⣿⣷⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠋⠛⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠠⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⠀⠀⢸⣶⣞⢬⣷⣶⣾⠠⠀⡐⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢿⣡⠄⡀⢠⣼⣏⣷⣝⣿⣤⠀⡀⢤⣹⢿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠁⠀⠉⢲⣾⠟⣿⡟⢻⣶⠊⠁⠀⣽⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢻⡇⠀⠀⠘⡇⠄⠈⠀⢄⠏⠀⠀⠀⡿⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⠂⠃⠀⠀⠀⠘⡂⢈⠀⡊⠀⠀⠀⠀⠃⢊⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠰⢁⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠋⠁⠀⠀⡀⢀⠀⢈⢋⡃⠀⠀⢀⡀⠀⠀⠉⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡄⠀⠀⠀⠀⠿⡀⡄⢬⡁⣩⢄⡄⣸⠃⠀⠀⠀⠠⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢜⡴⠛⠁⠀⠠⣤⡐⠂⠄⠮⢮⣟⣧⠾⢆⡠⠚⣠⣤⠀⠀⠙⠳⣜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠁⢀⡘⠃⠚⠉⡄⢶⡀⡀⢴⡶⣿⡷⣶⢄⡀⣱⠖⡌⠙⠀⠋⣀⠁⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠒⢲⠏⠀⠐⢶⣆⠀⠘⣧⠀⢠⠴⠻⠳⢄⠀⢸⡟⠄⢀⣶⠦⠀⠈⢶⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⣀⠀⠈⠋⠁⣀⣩⣱⢂⡗⢰⡀⣰⢐⣇⢳⡩⣁⡀⠙⠁⠀⢀⣨⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡠⠋⡩⠛⠻⢂⣶⣀⣷⡏⠘⠋⠚⠷⠟⠎⠛⠈⣷⣆⣘⣆⠚⠛⠫⡈⠣⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠂⠂⠓⠁⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠙⠂⠂⠒⠈⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢰⠀⠀⣤⠀⢀⠀⠀⢰⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⠀⠁⠒⠣⠀⠓⠀⠉⠘⠁⠘⠚⠀⠀⠈⠉⢨⠀⠃⠀⠈⠁⠓⡃

⣀⣉⣉⣙⠙⠻⣿⣿⣿⣿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⡟⠀⢁⣏⣈⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⣿⠋⠉⡀⠀⠈⠻⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⣷⣄⣀⣤⡀⢀⣠⣤⣄⠀⠀⠰⠶⠶⣎⣀⣠⡜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠂⠀⠀⠀⢰⣶⣦⠀⢠⣿
⣿⣿⡟⠀⠀⠀⢠⣽⣿⣿⠇⠀⠀⠀⠀⠙⠛⠛⠶⠦⣄⣉⣉⣿⣿⣽⣿⣿⣿⠃⢠⣄⠀⠀⠀⠻⠟⠛⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣧⠀⠀⠀⢀⣙⣛⠶⢷⠀⠀⢀⣹⣿⢿⣿⣿⠇⠈⠁⠈⢙⠀⠈⠛⠲⣤⣴⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⡏⠀⠀⠀⢻⣿⣿⠂⢼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣰⢿⡦⣀⣤⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⣰⡏⠉⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡌⠷⠘⠳⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠘⣿⣿⣾⡿⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠿⢿
⣿⣿⠟⠛⠛⠉⠉⠉⡽⠀⠀⠀⠀⣿⣿⣿⣧⢀⠀⠉⠉⠛⠋⠘⠓⠈
⠋⠉⠁⠀⠀⠀⠀⠈⠁⠀⢴⣴⣶⣿⣿⣷⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣚⣿⣻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⡿⢿⣿⣿⢿⠿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣾⣶⣤⣄⣀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⡿⢿⣿⣿⢿⣿⠿⠿⠷⠶⠿⣦⣾⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣿⣿⣿⣿⣿⣶⣦⣦⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣛⣻⣟⣛⢛⣟⣛⣿⣛⣛⣛⡛⠀⠀⠶⢴⡆⠀⠜⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣤⣿⣿⣏⣽⣭⣩⣭⣽⣿⣉⣉⣉⣉⣁⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡄⠀⠀⠀⡴⠋⢄⡅⠈⠙⢻⣿⣿⣿⣿⣿
⠀⠀⢉⣿⣿⣿⣿⣿⣿⣻⣿⠛⠛⠛⠋⠋⠉⠉⠉⠋⠉⠉⠀⠀⠀⠁⠀⠁⠉⡁⢀⡼⢷⢶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠶⠾⠏⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⢧⡀⠀⠀⠀⠀⠀⠀⢠⡿⢾⠾⣿⣿⣿⣫⣻⣿⡉⠉
⣿⠧⢴⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⠷⠀⠀⠀⠀⠀⠀⣈⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠁⠀⠈⠙⠫⣵⡴⡭⢿⠿⡿⣿⠟⠃⠉⠉⠻⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⢙⣹⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣄⣰⣶⣶⣾⣷⣾⣿⣷⣷⣶⣶⣶⣶⣶⣶⣶⡶⠆⢰⣿⠿⠒⠀⠀⡀⠀⠉⠘⢧⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡄⣶⣿⣿⣼⣿⣶⣿⠀⠀⣀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡃⠹⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠞⠛⠛⠛⠛⢛⠷⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠁⢀⡴⠉⠀⠀⠈⠙⠻⢿⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠐⡆⢷⣀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⠀⢸⡆⠑⠤⡈⢻⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠢⠒⡀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⢸⠋⠹⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢸⡇⠠⠐⠔⠅⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡟⠀⢀⡇⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣼⠁⠈⣷⠀⠀⠀⢸⠅⠀⡀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⣧⣴⠋⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢀⡟⠀⢀⡿⠀⠀⠀⢸⠀⠀⡀⠁⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣾⠀⠈⢷⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢸⠟⠛⠛⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠤⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⠇⠀⠀⠾⠀⠀⣶⠀⠼⠀⠀⢰⠄⢀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⡄⠀⢸⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠠⢀⠀⣿⠀⣀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⢧⠘⡶⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡇⠈⠀⠀⠀⠠⣿⠞⠁⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠀⣥⣡⣿⢰⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠑⠢⠥⣀⠀⣿⢠⠂⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠰⡒⢾⡀⠇⢹⡇⠾⠀⠀⠀⠀⠀⢠⣾⠶⠤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠆⡤⢀⠀⡇⢀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠶⠒⠺⢦⡴⢃⣼⠇⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡀⠈⠄⠀⡿⠉⣅⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡀⠼⠂⠀⠅⣠⠼⢤⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠘⣄⡀⣠⣇⠈⠸⢤⡼⠋⡹
⠀⠀⠀⠀⠀⠀⠱⣆⡀⠀⣸⠓⠲⠄⠀⠈⢳⣤⡟⠀⠀⠀⢀⡀⠀⠀⠀⢀⣀⡤⠶⠒⢤⠀⠀⠀⣀⡔⠊⢈⠁⠐⠒⠔⠉⢻⡁⠻⠀⠀⠊⠤⣮
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⠛⠒⠒⠒⠲⠤⢤⣤⡭⠭⢉⣉⠁⠈⠉⠐⠛⠉⠁⠀⠀⠐⠋⠉⠣⠞⠁⠈⢦⡀⢀⡀⠀⠀⠀⠀⠘⡀⠀⡀⠂⠉⡹⠂
⠀⠀⠀⠀⠀⣠⠴⠛⠛⠓⠀⠀⠀⠀⠒⠀⠀⠀⠄⠀⠀⠀⠈⠉⠉⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⢵⡎⠉⠃⠒⢄⡀⠁⠀⢦⡀⠀⠧⣀
⠀⠀⣠⠖⠒⠛⠒⠀⠀⠀⠀⠶⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠦⠤⣤⣀⣀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢀⠀⠀⠁⠀⠀⣈⠃
⠀⠀⠈⠙⠒⠦⢤⣄⣀⣀⡀⠀⠀⠠⠄⠀⠀⠀⠠⠤⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠐⠂⠀⠀⠀⠢⠤⠌⢢⣀⣠⠄⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⢒⣒⠒⠢⠤⠤⠤⣤⣄⣀⣀⡀⠀⢀⣀⣀⠀⠀⣠⠄⣀⠴⢻⣄⠤⢀⣠⠤⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠍⠉⠛⠫⠩⠍⢉⠀⠙⠂⠀⠐⠁⠉⠰⠄⠀⢄⡀⠜⠁⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠐⠢⠤⣀⡀⠀⠉⠉⠓⠚⠉⠉⠀⠀⠀⠈⠁⠠⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⠈⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⣿⣿⣯⣉⣽⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠙⠉⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡐⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣯⣯⣽⣽⣿⣿⣯⣿⣄⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠕⢌
⠀⠀⠀⠀⠀⠀⠀⣿⢼⣿⣿⣿⣿⠛⢸⣿⣿⡟⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠙⣿⣶⣿⣿⣿⣿⣶⣿⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠙⣿⣿⣿⣿⣿⣿⡏⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠙⣿⣿⣿⣿⡟⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⢘⢿⣯⣻⣿⡁⠀⠀⠀⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣿⣿⣾⣿⣻⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⣿⣿⣿⣿⣿⣿⢤⡀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣴⡶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢹⣿⣿⣿⣿⣿⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⣿⣿⣿⡿⣻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⢼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡬⢀⣄⠀⣿⣿⣿⠉⠙⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣿⣿⡟⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠳⠦⠄⠀⠀⠀⠀⠀⠙⠛⠛⠛⠳⠦⠄⠄

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣤⡀⠀⠀⠀⠀⠀⢠⡼⠆⠰⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣀
⠀⠀⠀⣠⠞⠳⣄⠀⠀⣴⣯⣷⡄⠀⠀⠀⠀⠀⠉⠉⠀⣠⡤⠀⠀⠀⠀⠙⣥⠀⠙⣥⠀⠀⠀⠀⡶⠛⠛⢿⡿⠛⠛⢶⠀⠀⠀⠀⢀⡀⠈⠋⡀⠀⠈⢛⣥
⠀⠀⠀⠁⢀⡀⠈⠀⠀⠀⣈⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⢇⡀⠀⠀⢠⣿⠇⣠⣿⠆⠀⠀⠀⡇⠀⠀⢈⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⢠⣿⠇
⠀⠀⢸⡇⠈⠁⢸⡇⣿⠂⠉⠁⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⠀⠀⢹⡏⠀⢹⡇⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠸⠧⠤⠶⠾⠇⠿⠦⠴⠶⠾⠇⠀⠀⠀⠀⠀⠀⠀⠸⠛⠻⠷⠀⠀⠘⠇⠀⠘⠇⠀⠀⠀⠀⠧⠤⠤⠼⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢒⡲⣄
⠀⠀⠀⠀⣠⡆⢀⣠⠐⣶⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⣦⠀⠀⠀⢸⡀⠈⠁⡸⠀⢠⣦⡀⠀⠀⠀⠀⠈⠁⠀⢀⣤⡘⣿⣶⡄⠀⠀⠀⡴⢦⠀⣠⡄
⠀⠀⠀⢸⣿⣿⣤⣽⣾⣿⣿⡤⠀⠀⠀⢀⣿⣷⡄⠀⠈⠛⠀⠀⠀⠀⠉⠒⣋⡡⠄⠈⠉⠁⠀⠀⠀⣀⡠⠀⢺⣄⣽⣷⡻⣿⣿⠀⠀⠀⠙⠚⢠⣾⣷⡄
⠀⠀⠀⠘⠁⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡅⠰⠈⠀⠀⠀⠀⠀⣻⡅⠀⠀⢨⣼⣿⡇⠀⠙⠀⠀⢀⠀⠀⣾⣿⣿⡧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⠀⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠻⠛⠟⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠼⠿⠇⠀⠀⠀⠀⠀⠀⠾⠿⠷⠀⠀⠘⠟⠗⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠿⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢸⡇⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⢀⠀⢸⡇⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠉⠂⢘⣇⠀⠏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣤⣴⣶⣿⣿⣿⣿⣷⣶⣤⣜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⠁⠀⠸⠏⠀⣿⠀⠹⠇⠀⠈⠃⠀⠐
⠀⠀⠀⠀⠀⣠⡶⣿⣯⣽⣿⣿⢿⣿⣏⣵⣿⣿⣟⡿⣿⣿⣟⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠆⠀⠀⢀⡀⠀⣿⠀⠀⡀⠀⠀⠰⠀⠀⣂⣀⡀
⠀⠀⠀⠀⣠⢻⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⣿⣿⣶⣶⣿⣿⣷⣦⣄⠀⠀⠀⠀⠰⠿⣿⣿⣿⣶⣶⣤⣠⣇⣑⠀⣿⠀⢀⣉⣄⣤⣴⣷⣿⣿⣿⡿⠦
⠀⠀⠠⠾⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⡿⠿⣮⠝⠛⠉⣉⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣷⣿⣾⣿⣿⣿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠋⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠄⠀⠈⠙⢦⡀⠀⡴⣶⡀⢀⡴⠋⠁⠀⠠⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣿⣧⠏⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣣⣼⣠⣆⣶⣠⣄⣀⠴⠋⢀⡉⠙⠦⣀⣠⣄⣶⣰⣄⣷⣜⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣱⠟⣹⠟⢹⠏⠟⢁⢝⠁⠀⣠⡀⢀⣅⠀⠈⡫⡈⠻⠹⡏⠻⣏⠻⣌⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠁⠀⠠⠔⢚⠽⣻⣞⠴⠋⢰⣦⠙⠦⣳⣟⠯⡓⠢⠄⠀⠈⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠻⣦⠀⠀⢈⡡⠀⠀⠀⢸⣿⢸⡇⠀⠈⠁⣀⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⢀⣀⣀⡠⠄
⠀⠀⠀⠀⠀⠤⢄⣀⣀⣐⣒⣒⣉⣩⣤⣤⠶⠶⣦⡞⠃⣰⣞⣥⠤⡄⠑⠀⠀⢻⣼⣇⠀⠀⡾⢉⣤⠄⠀⠈⠉⠉⠉⣉⢉⠩⠝⠓⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠼⠇⠸⠿⣿⢣⡎⠀⠈⡾⠄⠠⡙⢿⣆⠀⠃⠸⣇⡀⠀⠀⣀⣀⠤⠀⠒⢲⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⢲⡌⢾⡀⠀⡴⢚⠉⠐⠜⢆⠙⢧⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠛⠁⠘⢿⡾⢁⡞⣴⢇⢸⡌⢧⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠞⣡⣾⢠⡇⢸⡆⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣼⠫⠋⢼⣷⠾⢡⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠘⣧⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢱⣤⣿⣇⣼⣀⡔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⠀⠀⢤⣹⣿⣿⣿⢥⣬⣽⣿⣾⣥⠤⠀⠀⢰⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣟⠃⠐⠺⢿⣯⣝⣻⠾⠿⣋⣠⣿⠷⠶⠄⢉⡿⢯⡁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣌⣷⡶⠀⠈⠀⠈⠀⠀⠚⢙⣽⣿⢿⣯⣽⣿⣿⠻⢦⠀⠀⠀⠀⠀⠀⠐⣲⣮⠤
⠀⠀⠀⠀⠀⠀⠀⠰⠏⠙⠀⠀⠀⠀⠀⠀⠂⠈⠹⠾⡛⠹⠛⠿⠙⠀⠀⡠⠊⠀⠀⠀⠀⠈⠁⠙⠆⢀⡀⢀⡀
⠀⠀⠀⣈⣣⣞⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⡄⠘⣴⣾⣷⡄⢀⠁⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢿⡯⡤
⠀⠀⠀⠀⠸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠎⢈⣿⣆⣾⣧⣿⣔⠍⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣿⣿⣿⣿⣿⣿⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢀⡀
⠀⠀⠀⠀⣑⣾⡴⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠧
⠀⠀⠀⠀⠀⠘⠇⠀⡀⠀⠀⢀⠔⠒⡤⡴⣾⣿⣝⣛⣛⣻⣏⣿⣿⣿⠟⢿⠒⢀⠔⣦⡀⢠⡀⡤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⢽⣾⣃⠀⠛⠶⠶⣤⣤⣬⢟⣩⡍⣿⣍⣽⣡⠰⠄⠄⢦⡽⠚⠛⠁⠶⢿⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠙⣿⣽⠉⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁