Bullet point Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣥⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⢀⣿⣿⠀⠀⠀⣾⣿⣿⣿⡄⠀⢰⣿⣿⣿⣿⡄⠀⠀⣠⣾⣿⣆
⠀⢀⣆⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⡆⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⡆⠀⠀⢠⣾⣷⡀⠀⠀⠀⠀⣤⡀
⠀⢸⣿⠀⠀⢀⣿⣧⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⠀⣾⣿⣿⣧⠀⠀⠀⣴⣭⣥⡄⠀⠀⠀⣠⣄⡀
⠀⣸⣿⡀⠀⢸⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⣷⠀⠀⣼⣿⣿⣷⡀
⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣧
⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿
⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿
⠀⣿⣿⡇⠀⠸⣿⣿⠀⠀⢸⣿⣿⠇⠀⠀⣿⣿⣿⣿⠇⠀⠸⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⡿⠀⠸⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⠀⢈⣍⣉⣉⣉⣍
⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠐⠒⠒⠂⠀⠀⠒⠒⠒⠒⠂⠀⠐⠒⠒⠒⠒⠂⠀⠘⠛⠛⠛⠛⠓⠀⠐⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣧⣤⡀⠀⠹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣏⠉⢻⣿⣿⣿⡿⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⣴⣿⣿⠀⢠⣿⣿⣿⣿⣄⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠙⠛⠁⣠⣾⣿⡌⠻⣿⣿⣷⣄⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣀⡀⠀⠀⠀⠸⣅⣿⣿⠇⠀⢙⣿⣿⣿⣿⡍⠛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠿⡿⠇⠀⠀⠀⠀⠈⠉⠀⠀⠀⣼⡇⠙⣷⣝⢿⣧⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⣀⠀⠀⠀⢠⢾⣿⣷⠀⠈⢿⣷⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⣄⠀⠀⠀⠀⠀⠈⠹⣧⣼⡟⠀⠀⠀⠘⠮⠟⠋⠀⢠⣿⠟⣿⣿⡝⢿⣶⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠸⣿⠟⠀⡀⠀⠀⠀⠀⠀⠺⠻⣿⡆⠀⠀⠀⠀⠀⠀⣤⣾⣿⡄⠈⣿⣟⣼⣿⣿⡄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢶⣿⣠⣤⠄⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⢿⣿⠟⠁⣠⣿⣿⣿⣿⣿⡧
⠀⠀⠀⣰⡟⠁⠀⣶⣷⡆⠀⠀⠀⠀⠈⢿⡿⠻⠿⢻⡄⠀⠀⠀⠘⠛⢛⣾⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⢸⣿⠀⠀⠀⠈⠉⠀⢀⢀⣀⡀⠀⠀⠀⠀⡴⢾⣇⠀⠀⠀⠀⠀⠿⠿⠄⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠈⢻⡄⠀⠀⠀⠀⠐⣏⣸⣿⣇⠀⠀⠀⠀⠀⠛⣻⣶⣤⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⠉⠀⠿⡇⡀⠀⠀⠀⠀⠉⠹⣿⣿⠀⠀⠀⠀⣰⣿⣿⣿⢏⣿⣿⣿⢇⣿⠋
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠘⠛⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣡⣿⣿⣿⢯⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠛⢶⣆⣤⡄⠀⠀⠀⠀⢀⣴⣿⣿⡿⢃⣾⡿⣫⣿⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁⠀⠀⢀⣴⣿⣿⡿⠋⣠⣿⠟⣵⣿⢣⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢋⣾⠟⢁⣴⣿⢫⣾⣿⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⢀⣤⡾⢋⣴⠟⠁⣴⣿⡿⢡⣿⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⣀⣴⣿⢏⣴⡿⠁⣠⣾⡿⠋⣠⡿⢁⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡛⣻⣷⡿⠋⣀⣴⡿⠋⠀⣴⠋⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⣤⣾⡿⠻⢥⣠⠾⠷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⠟⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠙⢛⠛⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡦⣾⡇⢙⡋
⠀⠀⠀⠀⣾⣿⣿⢿⣿⣿⣿⡟⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣯⣷⣶⡾⠃
⠀⠀⠀⠀⢹⣿⣿⣧⡙⢿⠟⣼⣿⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⠛⠉⠀⢀⣀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣄⣼⣿⣿⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀⡀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀⠀⢀
⠀⠀⢸⠉⡇⢹⡇⠀⠀⠀⠀⠀⡏⢹⠈⣩⠇⠀⠀⠀⠀⢸⢨⡇⠩⣿⠀⠀⠀⠀⠀⡇⣿⢀⡿⡄
⠀⠀⠘⠲⠃⠸⠂⠀⠀⠀⠀⠀⠓⠞⠰⠷⠂⠀⠀⠀⠀⠘⠚⠃⠳⠚⠀⠀⠀⠀⠀⠓⠛⠈⠛⠋


⠀⠀⢠⠶⡄⣤⠶⠀⠀⠀⠀⠀⡤⢤⠀⣤⠀⠀⠀⠀⠀⢠⢦⡄⡤⣶⠀⠀⠀⠀⠀⡴⣤⢠⡴⡄
⠀⠀⢸⣀⡇⣙⢹⡆⠀⠀⠀⠀⣇⣸⢰⣟⡇⠀⠀⠀⠀⢸⣘⡇⢰⡏⠀⠀⠀⠀⠀⡇⣿⢸⣛⡧
⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁


⠀⠀⢰⠋⡇⣾⣻⠄⠀⠀⠀⠀⢻⢰⡏⣷⠀⠀⠀⠀⠀⠘⡇⢻⠀⠀⠀⠀⠀⠀⠀⣿⠘⢋⡇
⠀⠀⠸⠤⠏⠨⠏⠀⠀⠀⠀⠀⠸⠸⢧⠿⠀⠀⠀⠀⠀⠀⠇⠸⠂⠀⠀⠀⠀⠀⠀⠿⠰⢯⠄

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣰⢚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⡄
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠁⢀⡀⢀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⣿⠀⠀⣠⠶⡄⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⠀⣇⡀
⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⣡⠎⠁⠂⠀⠀⠀⣿⠀⠀⠘⠚⠁⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢃⣤⡞⠝⠙⠢⣄
⠀⠀⠀⢈⡁⠈⠀⠀⠀⢀⡔⡡⢓⠼⠁⠀⠀⠀⣿⠀⠀⣰⠒⡄⠀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡿⣄⠙⠦⣄⢀⡼
⠀⠀⠀⠈⠁⠀⠀⠀⡰⢊⠄⡠⢚⠀⠀⠀⠀⠀⣿⠀⠀⠈⠛⠁⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⠞⠊⣠⠟⢦⣈⡿
⠀⠀⠀⠀⠀⠀⠀⢘⠢⣡⠊⠁⠁⠀⠀⠀⠀⠀⣿⠀⠀⣴⠒⡆⠀⡞⠛⠛⠛⠛⠛⠛⠛⢛⡿⠕⢁⡜⠁⢀⡜⠉
⠀⠀⠀⠀⠀⠀⠀⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠁⠀⠛⠒⠒⠒⠒⠒⠒⡲⡫⠂⡴⠋⠀⡰⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠍⣠⠞⢀⣮⡞⠐⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠧⣅⡴⠁⢔⡵⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠈⣳⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡯⠟⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⢹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠙⢿⣿⣿⣿⣿⣿⠏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⢻⣿⣿⣿⠏⠀⣼⣿⣿⣿⡟⠉⠉⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⣿⠋⢀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠘⡄⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠊⠙⠛⠿⠿⠿⠿⢿⣿⠛⠛⠋⠁⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠄

⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⢿⣷⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣾⡿⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠈⢹⣿⡇⠉⠻⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠁⠀⠈⢹⠋
⠀⢸⣿⡇⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⢠⡄⣠
⠀⢸⣿⡇⠉⠛⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠍⠩⠩⠍⠩⠉
⠀⢸⣿⡇⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠦⠄⠴⠋
⠀⠚⠛⠃⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒



⣠⡀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠙⠋⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠸⣧⣴⠟⠈⢻⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡇
⢾⡗⠀⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠸⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⢻⠒⠒⢺⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠿⣷⣶⣴
⣤⣄⠀⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⢾⠒⠒⢺⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⢱⣿⣿⣮⣿⣿
⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠾⠤⠤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠫⠿⠿



⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠠⣄⡀
⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⢩⣍⢹⠀⠀⠀⠀⠀⠀⠀⠀⣤⡻⣿⣿⣿⡿⣣⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠟⠀⢾⣿⣦
⠀⠀⠛⠛⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⣄⣤⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢮⡻⢫⢾⣷⢽⡆⠀⠀⠀⠀⠀⠀⢰⡿⠿⠀⠀⠀⠀⢹⣿
⠀⡞⠛⠛⠛⠛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣘⢿⠿⢿⠿⡿⡆⣻⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⢠⣰⡄⣠⣾⠀⠀⠀⠀⠀⠀⠀⠈⢡⣶⣄⡀⢀⡀⠸⠿
⣿⣿⡇⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡦⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⠖



⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⣶⣿⣿⣿⣷⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣿⠉⣿⠉⠉⠉⠉⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣛⣛⣛⣛⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢻⣿⣿⣿⡇
⣿⣿⡿⣭⣭⣝⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣤⣭⣤⣤⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⠿⠿⠿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣾⣿⣿⣿⡇
⣿⡟⣯⣿⣿⣯⣝⢿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⣩⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⣿⣿⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠙⠛⠙⢻⣿⡛⠁⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣉⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇

⠀⠠⠀⠀⠀⣄⡀⠀⠀⠀⣀⠀⠀⣴⡄⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⠤⠤⡄
⠀⢩⣉⣉⣉⣭⡀⠀⠀⢂⣿⡀⢸⣿⣧⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⢲⡆⢰⢶⡶⡆⢈⣙⣒⣒⣒⣒⡃
⠀⣉⣉⣉⣉⣭⠀⠀⠀⠘⣿⡇⢸⣿⣿⢸⣿⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢸⢸⡇⡇⠛⠫⠭⠭⠭⠭⠅
⠀⣉⣉⣉⣉⣥⡀⠀⠀⠀⣿⡇⠀⠛⠃⠘⠛⠃⣰⠀⠀⡐⡀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠰⣟⠘⡇⠸⠸⣷⠅⠴⢟⣛⣛⣛⣛⡃
⠀⣀⣀⣀⣀⣤⠀⠀⠀⠀⣿⡇⠀⠀⠀⣷⠀⠀⣿⡇⠀⡇⡇⠀⢰⠸⡄⠀⢠⣿⠀⠀⠀⠤⠤⢀⠀⠀⠀⠀⣠⣤⠤⠤⠤⠤⡅
⠀⣐⣒⣀⣀⣦⠀⠀⠀⢠⣿⡇⠀⠀⠀⣿⠀⢠⣿⡇⢰⡇⡇⠀⢸⢰⡇⠀⢸⣿⠀⠀⠈⠉⠉⠉⠉⡁⠀⠀⢈⣙⣛⣛⣛⣛⡃
⠀⢠⢀⠀⡄⡄⢠⠀⠀⢸⣿⡇⠀⠀⢸⣿⠂⠀⠉⠁⠀⠁⠁⠀⢨⠸⡇⠀⣙⣿⡇⠀⠉⢹⠀⠀⣿⣿⠀⠀⠛⠫⠭⠭⠭⠭⠇
⠀⢸⡈⡄⡇⢣⢸⠀⠀⠛⠻⠇⠀⡇⢸⣿⠀⡆⡆⠀⠀⣿⡄⠀⣼⡀⣷⠀⡇⣿⠃⠀⣿⢘⡇⢸⢹⣿⠆
⠀⢸⡇⡇⡇⢸⢸⠀⠀⠀⠀⠀⣄⣧⠈⠋⠐⡇⣿⠀⢸⣿⡇⠀⣿⠇⣿⠀⡇⣿⠀⠀⣿⢸⡇⢸⢸⣿⠀⠀⢀⢘
⠀⠸⠳⠇⠇⠾⠸⠀⢠⠀⠀⠀⠿⣿⠀⠀⢰⡇⢹⠀⡾⢸⡷⠀⣿⠀⣿⠀⡇⣿⠀⠀⣿⢸⡇⢸⢸⣿⠀⠀⠀⢸⠀⠀⣸⡄
⠀⢰⡟⢻⣿⢻⠀⠀⣽⡄⠀⠀⡂⣿⡄⠀⢸⡇⢸⠀⡇⣻⡇⠀⣿⡄⣿⠀⡇⣿⠀⠀⣿⢸⡇⢸⢸⣿⠄⠀⠤⠼⠀⠀⣿⡇
⠀⢸⡇⢸⣿⢸⠀⠀⢿⡇⠀⠀⠀⣿⡇⠀⢸⡇⢽⠀⣧⢸⣷⠀⣿⠃⢿⠀⡇⣿⣄⠀⣿⠸⡇⢸⠼⣿⡆⢀⣀⠀⠀⠀⠉⡆
⠀⢸⡇⢸⣿⣼⠀⠠⢸⣇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⡆⠀⢸⠀⡇
⠀⢸⡇⢸⣿⣿⠀⠈⢸⣿⠀⠀⠀⣿⡇⠀⢰⠁⣧⠀⠀⣾⣿⠀⠀⠀⡆⣇⠀⠀⣤⠀⡆⠀⢀⡿⣧⠀⠀⡇⢸⡇⠀⢸⠀⡇
⠀⢸⡇⢸⣿⣿⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⡆⣿⠀⢰⠛⣿⠁⠀⠀⡇⣿⠀⠀⣿⡇⡇⠀⢸⠀⣿⠀⠀⡇⢸⡇⠀⢸⠀⡇
⠀⢸⡧⢸⣿⣿⠀⠀⣸⣿⠀⠀⢰⣿⡇⠀⢸⡇⣿⠀⠰⢀⣿⠀⠀⠀⡇⣿⠀⢐⣿⡇⡇⠀⢸⠀⣿⠀⠀⡇⢸⡇⠀⢸⠀⣇
⠀⠘⠓⠚⠛⠚⠀⠘⠘⠛⠀⠀⠃⠛⠃⠀⠘⠃⠙⠀⠈⠃⠛⠃⠀⠘⠓⠛⠀⠀⠛⠛⠃⠀⠘⠒⠛⠀⠐⠃⠘⠃⠀⠘⠛⠓

⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣠⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⠄⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠛⠃⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀⣴⣤⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⣤⣀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠈⠛⠃⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠈⠛⠟⠀⠀⠀⠀⠘⠛⠁⠀⠀⠻⠿⠟
⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⣤⢤⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠹⠿⠇⠀⠀⠀⠈⠯⠽⠀⠀⠀⠚⠛⠁⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠿⠟⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⡀⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⢀⠤⡀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠙⠛⠁⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠈⠉⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⡠⣠⢄⠀⠀⠀⠀⣠⡄⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⣦⠀⠀⠀⠀⠀⣶⣞⣆⠀⠀⠀⢀⠤⡄⠀⠀⠀⡴⢶⡄⠀⠀⢠⣴⣤
⠀⠀⠀⠀⠓⠓⠋⠀⠀⠀⠀⠛⠃⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠛⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣙⣍⣫⣬⣉⣭⣫⣿⣻⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⠇⣷⢰⡗⢰⡇⢿⣤⣤⣤⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣠⡶⠿⠶⢾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠟⠈⠃⠘⠃⠀⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⢻⡟⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠻⣶⡘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠀⠀⠘⢷⣤⡾⠁⠙⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠀⠀⢸⡇⠀⢀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣤⠶⠶⠶⠶⠶⠶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⡿⠷⠶⠶⠿⣿⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⣀⡀⠀⠀⠀⢀⣤⡶⣿⠋⢿⡀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠸⠿⠟⢀⣤⠾⣯⠁⠀⠸⣧⣘⣧⠀⢀⣴⠿⠷⠶⠶⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⡾⠁⠀⠙⠷⣦⣄⣈⣿⣵⡾⠛⠁
⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⢠⡟⠀⠀⢠⣶⣦⠀⠀⠀⠀⢹⡄
⠀⠀⣾⠁⠀⠀⠈⠛⠃⠀⠀⠀⠀⠘⣷
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠃
⠀⠀⠈⠛⠻⠶⠶⠟⠛⠛⠛⠋⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢻⡟⣻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡜⠳⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡞⠀⠀⠹⡄⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⠎⠀⠀⠀⠀⠘⣆⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢠⠏⠀⣸⠉⠉⡝⠀⠘⢆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣠⠃⠀⠀⡇⠀⣸⠁⠀⠀⠈⢧⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡰⠁⠀⠀⢸⠀⢠⠇⠀⣀⣤⠀⠀⢣⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⡼⠁⠀⠀⠀⡏⠀⣾⠴⠊⠁⡇⠀⠀⠀⠳⡀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⡜⠁⠀⠀⠀⢸⠁⣀⡤⣶⠀⢸⠁⠀⠀⠀⠀⠱⡄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢀⠞⠀⠀⠀⠀⠀⠿⠊⠁⠀⡏⠀⡏⠀⠀⠀⠀⠀⠀⠙⣄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡇⢸⢁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠹⡄
⠀⠀⠀⠀⠀⢠⠎⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠘⣆
⠀⠀⠀⠀⢠⠋⠀⠰⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠷⠀⠈⢆
⠀⠀⠀⢰⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢸⡀⢀⡇⢸⢠⠞⠉⠁⢸⡂⢀⡇⠀⢰⡀⢰⢃⡞⠉⢳⡄⡇⠀⠉⡟⠁⢰⢷⠀⣰⠊⠉⠀⣾⠉⠁⡇
⠀⠀⠀⢸⠉⠉⡇⢸⠘⣆⣈⣻⢸⡍⠉⡇⠀⠀⢧⡏⠘⣇⣀⣸⠃⣇⣀⠀⡇⢠⡟⠚⣇⢳⣀⣉⡇⣿⣉⡁⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠆
⠀⠀⠀⠀⠀⠰⡄⢀⣀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⢀⠤⠶⠿⣧⡄⣤⠿⠗⠦⢄⠀⠀⠀⠀⠀⢲⣾⣯⣿⠻⣿⣦⡆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢨⣿⡄⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣥⣤⣴⠆
⠀⠀⠀⠀⠀⠀⢣⡈⣶⠁⡰⠀⠀⠀⠀⠀⠀⠀⢠⣼⡟⢿⠻⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⠉⢾⣿⣿⣿⡇⠀⠀⠀⠀⠈⠋⠉⠉⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣨⠟⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠈⠀⠀⠀⢸⣷⡾⠁⠀⠀⠀⠀⠈⠻⣿⠏⠀⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⣀⣼⣏
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋


⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀
⠀⠀⠀⠸⠿⣄⣾⣿⣿⣿⣷⡤⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⣘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣤⡆
⠀⠀⠀⢀⣀⡼⣇⣀⡿⣀⣸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠰⣿⡿⠫⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣟
⠀⠀⠀⠘⠟⠀⠀⢿⣷⡿⠁⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠀⠀⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⡼⠙⠯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣶⡦⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⢰⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢣⡆⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠟⣹⠇⠀⠀⠀⠀⢠⡄⢸⣿⣾⣿⢀
⠀⠀⠀⠀⠀⢀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠾⣶⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣽⠋⠀⠀⠀⠀⠀⠀⠀⡀⣻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⡏⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢻⣿⣿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠉⠀⢛⣿⠟
⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠿⠿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣤⣶⣤⡙⠄⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⡴⢃⠄⣀
⠀⠀⠀⢀⣄⠀⠘⢲⣿⣗⠃⠀⢀⡄⠀⠀⠀⠀⠀⢠⣾⠋⢀⣿⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠛⣿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⡈⠀⠴⢊⣡
⠀⠀⠀⠙⠛⠶⣤⣀⠶⢆⣠⠴⠛⠛⠀⠀⠀⠀⢠⡿⠃⠀⢸⣿⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣶⣿⠇⠀⠀⠀⠀⠀⠀⢠⣬⣄⠋⡀⠀⢈⡁
⠀⠀⠀⠀⠀⣠⣠⡽⠛⠯⣄⣠⠀⠀⠀⠀⠀⢰⣿⣁⣀⣀⣈⣉⣀⣀⣀⣻⣧⠀⠀⠀⠀⠀⢀⠀⠉⠳⣿⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠛
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⣶⣯⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣖⣶⣶⣶⣶⣶⣶⣲⣶⡶⠶⠶⡶⡾⠿⣿⣶⣆⣀⣀⡀
⠀⠀⠀⠁⣇⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⡇⡇⠀⠨⡶⠋⠁
⠀⠀⠀⣇⡏⠉⠉⠉⠉⠉⠑⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠓⠒⠒⠀⠰⣇
⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣶⣶⣶⣶⣶⢶⣶⣄⠀⣶⣿⡿⣿⣷⡀⠀⠀⢀⡹⣒⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⡏⠀⠀⢹⡆⠘⣿⠀⢹⣿⣷⣿⣿⣧⠀⡰⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠣⣄⣀⡼⠃⢸⣇⡂⠀⢿⣿⣿⣿⣿⣇⢇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢽⣷⡘⣿⣿⣿⣿⣿⡞⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢻⣿⣿⣿⣿⣿⡽⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡌⣿⣿⣿⣿⣿⣷⢱⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠸⣿⣿⣿⣿⣿⣧⢣⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢻⣿⣷⣼⣿⣿⡎⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣬⠯⠭⠭⠗⠒⠒⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢁⠴⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⡠⠞⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⠢⣈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡴⠋⡠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠳⢄⠉⠢⣀⠀⠀⢀⡴⠋⣠⠖⢁⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠙⠦⣈⣳⠞⢁⡤⠊⡡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡶⠋⣀⠔⢋⡤⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢁⡠⠚⣡⢔⠙⢦⡈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢥⡴⢋⡠⠊⠀⠀⠙⠦⡉⠲⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡩⢊⠁⠀⡱⠃⠀⠀⠀⠀⠀⠀⠈⠲⣌⠗⠁⠬⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠊⠁⠘⠒⣁⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠐⢌⡢⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⡑⢄⢀⡠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡁⠀⠀⢈⢶⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠯⠽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⡶⣴⠋⠂⠣
⠀⠀⠀⠀⠀⠀⣀⡤⢼⢀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠇⠀⢀
⠀⠀⠀⠀⢀⡜⡁⡰⣡⣤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠠⠘⠿⡒⢦
⠀⠀⠀⣠⢊⠎⠰⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠊⠊⠢⠳⡀
⠀⢀⡴⠁⠂⠀⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠄⠑⡙⢄
⡠⠊⡜⠀⠄⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠠⢨⠳⡀
⠑⢄⡁⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠁⠀⠃⢘
⠀⠀⠈⠓⠢⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣀⡠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠒⠚⠙⠉⠙⠙⠒⠲⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠁⠀⠀⠀⢤⡠⠤⠤⡠⠄⠀⠀⠀⠹⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡄
⠀⠀⠀⠀⢠⣶⠟⠇⠀⠐⠊⠑⠒⠢⠀⠀⠀⠀⠀⠔⠒⠒⠉⠒⠀⠘⠟⢷⣄⠀⠀⠀⠀⠀⢰⢻⢸⣷
⠀⠀⢀⡤⣾⠷⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣦⡀⠀⠀⠀⢸⣾⢸⣿⡀
⠀⠀⠸⣿⣿⣄⣀⣀⣸⡄⠀⠀⠀⢸⡇⠀⠀⠀⢀⡇⠀⠀⠀⢀⣾⣀⣀⣠⣾⣷⡟⠀⠀⠀⢸⢿⢸⣿⡇
⠀⠀⠀⠈⢿⣿⢿⣿⣿⣿⣷⣶⣶⣿⣿⣶⣶⣶⣿⣿⣷⣶⣶⣿⣿⣿⣿⢿⣿⠋⠀⢀⣠⣶⣿⢸⣿⠀⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠙⠙⠻⣯⣿⠿⠿⣿⣿⣿⡿⣿⣿⣿⡿⠿⢿⣯⡿⠋⠛⠉⠁⠀⠀⠸⠟⠛⢻⠈⣿⠀⡟⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣤⠖⠛⠛⠛⠛⠛⠉⠉⠉⠹⠛⠛⠛⠛⠳⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⠤⠶⣷⣤⣴⣤⣤⣤⣴⡷⠦⠤⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⣿⡇⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡇⢿⠀⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⢸⡇⢀⣿⡇⢸⣇⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⡏⡟⠏⠉⠉
⠀⠀⠀⠀⠀⢀⡤⠔⠉⡩⡤⠖⠂⢸⡇⢸⣿⣧⢸⣿⠀⠒⢤⢌⡉⠳⠤⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⣀⣤⣤⣿⣇⡾⣿⠿⣌⣿⣤⣤⣄⡀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⣆⢸⠀⢸⢸
⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⡟⠉⠀⠈⠛⠀⠀⠀⠘⠁⠀⠉⢻⣿⣿⣷⣤⡄⠀⠀⠀⠀⠀⠀⢿⠘⣦⠏⣸
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣧⠈⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢶⡤⢷⣤⣴⢦⡴⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⣤⣶⣶⣦⡀⠀⣴⣶⣶⣦⠀⠀⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣅⢈⣿⣿⡄⠀⠿⣿⣿⡿⠁⠀⢿⣿⣿⡿⠀⢀⣿⣿⣏⢙⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡈⠁⠀⠸⠇⠀⠈⢁⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠙⣿⣷⣀⣠⣄⣠⣄⣀⣼⡟⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣤⣿⣿⡟⠁⠀⠀⠉⠛⠿⢿⣿⣿⠿⠟⠋⠀⠀⠀⠹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢤⣶⣶⠤⠀⠀⠀⠀⠀⠠⢴⣶⣦⠤⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣏⠀⠀⠈⠀⢀⣠⣤⣶⣦⣤⣀⠀⠈⠁⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⣤⣺⣟⣟⣻⣿⣿⣿⣭⣭⣽⣿⣟⣋⣀⣀⣀⣀⣐⡷⠶⠿⣗⡤⠤⢴⣻
⠀⠀⠀⢀⣀⣀⣤⣴⣶⡶⠶⠛⢻⣿⡿⣿⣿⣿⣿⣿⣿⡉⠉⠉⠉⠉⠉⠁
⠀⠀⢿⣿⣿⠿⠟⠋⠁⠀⠀⠀⣿⣿⠇⠀⠀⠀⠘⣿⣿⣧⡀
⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠁

⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣄
⠀⠻⠿⠿⠁⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⣸⣿⣿⡦⠀⠸⣿⣿⡄⠀⢀⣠⣤⣶⣶⣶⣶⣤⣤⣀⠀⠘⠿⠿⠋
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠛⠉⠀⠀⠀⠀⠉⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣿⣿⣿⠛⠉⠛⣿⣿⣿⡿⠁⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣷
⠀⠀⠀⢻⣿⡇⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⣿⠁⠀⠙⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠻⣿⣦⣤⣶⣿⢻⡿⣿⣷⣤⣤⣴⣿⡿⠁⠀⠀⠀⠀⠀⠘⣧⣀⣰⡿⣿⢿⣷⣄⣀⣀⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣾⣷⣿⣿⣿⡟⠛⢁⣤⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⠿⠛⡀
⠀⣠⣴⣶⣶⠄⠀⠻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠘⢿⣿⣶⠀⠀⣠⣤⣤⡄⠹⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠘⢿⣶⣦⡄
⠀⠙⠛⣿⠟⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠘⠛⠀⠀⠈⠛⢻⣿⠃⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⠋⠉
⠀⠀⠀⠀⢠⣤
⠀⠀⠠⣾⣿⣿⠀⠀⠀⣀⣀⣠⣄⣀⡀⠀⠀⠀⠀⢠⣶⣀⠀⠀⠀⠀⣀⣴⣶
⠀⠀⠀⠀⢈⣩⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠸⠿⠛⠀⠀⠀⠀⠛⠿⢃⣠⣤⣴⣶⣶⣦⣤⣤⣄⡀⠀⣼⣿⣤
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠛⠋
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢻⣿⣿⣿⣿⣿⠋⠀⠈⢻⣿⣿⣿⡿⠿⣿⣿⣿⡀⠀⠀⠀⠀⢸⣿⡟⠉⠉⠙⣿⣿⣿⡿⠋⠉⠙⢿⣿⡇
⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠸⣿⡇⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣄⠀⢀⣸⣿⣿⣿⣀⣀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣿⣿⢙⣛⣿⣶⣶⣾⣿⠏
⠀⠀⠀⠀⠀⣀⣀⡉⠻⣿⣿⣿⣿⣋⣿⣍⣿⣿⣿⢁⣤⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠋⠁⢰⣶⣦
⠀⠀⠀⠀⠺⠿⣿⡇⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠘⣿⠟⠋⠀⠀⠀⢿⣿⣧⠀⠈⠛⠿⠿⠿⠿⠿⠛⠀⠀⠈⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⣦⣤⣀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣉⣤⡶⠶⠶⢦⣬⡙⠻⣦⡀⡜⠁⠐⣳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣼⠋⢁⡤⠴⠦⢤⡈⠻⣦⡾⢿⡟⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣾⠁⣴⠋⣴⠖⠲⣤⡿⡶⠟⣧⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⢇⡀⢞⡻⢀⡿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢹⣄⠘⠧⣄⣉⣉⣠⠞⠁⣰⠇⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡙⢷⣤⣀⣉⣉⣀⣤⠾⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣬⣉⣉⣉⣉⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⣤⣀⠀⡀⠀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⣀⣄⡀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢧⢀⠆⠀⢁⣾⣿⣿⣿⣶⣤⣀⣴⠆⠀⢰⣤⣀⣤⣾⣿⣿⣿⣷⡘⢀⢰⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣾⢞⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⢀⢤⡘⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⡿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡦⠣⠀⢄⠹⠿⠿⡿⢟⠏⡡⢣⣶⡹⡌⢛⡻⢿⠿⠿⠋⡤⠀⠎⢼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠴⠃⠀⢱⣿⣿⣿⡌⠀⠙⠦⠀⠀⠈⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠶⠶⠂⣀⡀⠤⠀⠀⠀⠭⣿⠍⠀⠀⠀⠄⢠⣀⠐⠶⠶⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⣶⣿⣿⠇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢿⣿⣷⣶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠹⡿⠲⡖⡶⠒⡞⠉⡟⠉⣷⠒⣖⢲⠒⢿⢋⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣷⢸⡦⣧⠤⣧⣀⣧⣀⣷⠤⣷⢼⠇⡟⠘⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢄⡹⠖⢧⣧⠀⡇⠀⣯⠀⣧⠀⣧⠼⠰⢃⠴⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⡄⠀⠀⠉⠈⠒⠉⠂⠁⠋⠀⠀⣰⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢤⣀⣀⣄⣀⣀⠄⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠀⠀⠀⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣀⣀⢀⣀⣀
⠀⠠⠀⠋⠋⠛⠚⠛⠛⠓⠠⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠒⠉⠓⠀⠉⠀⠉⠒⠘⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⣙⠇⠀⢀⠢⡤⠀⠀⠀⠀⠀⠀⡀⠀⠠⠐⢀⡀⣀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠄⠡⠀⠀⠈⠉⠀⠁⠁⣤⠈
⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠠⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢧⡄⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠠⠌⠀⠠⡀⠀⠀⠀⠀⠦
⠀⠀⠀⣤⣀⣤⡀⣀⣀⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⢀⡹⠟⠚⠛⠛⡦⠆⠀⠀⠠⠉⠀⠀⠀⠀⣴⣼⣳⣿⡶⣤⡄⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⢰⠂⠠⠤⠂
⠀⠀⠀⠀⠘⠀⠋⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⢀⡞⠀⠈⠀⠀⠀⣀⠁⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢁⠻⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣄⡀⠀⠀⠀⢀⣀⣠⡤⠶⠋⠀⠀⠀⠆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠓⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠢⢄⠀⠀⠀⠠⣀⢀⡀⡀⠀⢂⠤⣀⣠⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⠻⠟⠶⠓
⠀⠀⠀⠑⠒⠂⠀⠀⠀⠈⠀⠀⠈⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋


⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⢤⣤⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿

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


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

⢀⣏⣑⣲⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡠⠤⠤⠤⠤⢤⠤⡀⠀⠀⡗⠒⢦⡀⢀⡖⠓⢦
⢻⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠿⠀⠀⠀⠀⢸⠀⠈⠀⠀⢳⠀⠀⠉⣉⠔⠉⠙⠃
⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣷⡒⠋⠁
⠀⠀⠀⠘⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢿⡀⠀⡿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠻⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⠁⠀⠀⣀⣿⠓⡄⢠⡴⠖⠒⠋⠁⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣅⡀⠀⠙⣉⡰⢃⣾⣿⡀⠀⠀⢀⠠⠒⠊⠒⠄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠟⠁⣸⡇⠀⠀⡀⠀⠀⠀⠀⢈⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠁⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠐⠤⠤⠤⠤⠤⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⣠⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡿⣤⣀
⠾⣆⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⢀⡀⡀⢀⠀⡀⠳⢿⣬⣗⠲⠒⢢
⠻⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡇⠁⠁⠈⣹⣿⣿⣿⣧⠀⠀⠛⠄⣟⠀⠀⠀⠀⠀⠀⠸⠸⠅⠧⠸⠀⠏⠗⠀⠀⠉⣴
⠀⢸⡛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠹⡟⠹⡿⣿⣿⡟⠛⢛⣿⡟⢋⣛⣛⣛⣛⠉⠉⠉⣿⣀
⠀⠈⠙⠙⠛⠛⠛⠛⠛⠛⠛⠛⡋⠉⠉⠉⠉⠉⢛⣿⢿⣿⣷⡀⠀⠡⠀⢰⣶⣶⣿⣿⣿⣿⣿⣧⠭⠭⠭⠵⠀⢠⣾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣛⠿⠛⠛⠻⢿⣿⣿⣿⣿⣷⡻⡉⠋⠐⠤⠄⠤⠤⠂⠊⢏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⣆⡟⠀⠙⣿⣷⡘⡀⠄⢑⢀⠠⡠⠂⠰⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⣴⠙⠁⠀⢀⣿⣿⡇⢱⠀⡀⠀⠀⠈⠂⠀⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠖⠚⠁⠈⠙⢾⡎⢊⠂⠐⠐⠄⠀⡠⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡌⣆⠀⡀⠠⠀⠈⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠘⡈⠪⠀⠀⠔⠅⠌⠠⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢡⠀⠀⡠⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⡆⠀⠂⠢⠀⠀⠕⠄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⠀⠀⠀⠀⡀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⣨⣀⣁⣀⣀⣊⣂⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠉⠉⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢡⣶⣷⡀⠀⠀⠀⠀⣰⣿⣦⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⣿⣿⣿⣷⡄⠀⠀⣰⣿⣿⣿⣷⡘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣔⠿⠛⠛⠻⢗⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⣿⣿⡏⣴⣿⣷⡌⢿⣿⣿⣿⡇⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⠶⣾⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠹⠿⠿⠃⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠉⢨⡝⠻⣿⠀⠀⢸⡿⠛⣽⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢀⣾⣶⣾⣦⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣷⣄⢹⣿⣿⣿⢁⣼⡟⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⢠⣾⣿⣿⣿⣿⣧⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣺⣿⣿⣿⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣭⣙⣛⣛⣩⣥⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠟⠛⢿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣤⠀⠀⠀⠀⠀⠀⢠⡜⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣷⣿⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣼⢁⡾⠀⣤⣦⡄⠸⣆⢻⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡘⣧⠈⠻⠿⠏⢀⡟⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠷⠈⠀⢠⣆⠀⠈⠰⠟⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⠁⠿⠋⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣸⣿⡀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢿⠀⠀⠀⠟⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠃⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁



⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠬⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠈⠀⠀⠹⡀⠹⠋⣳⡍⣿⡴⢿⣟⠉⡝⠀⠜⡩⠊⠰⢁⠞⠙⠁⡠⠞⠁
⢤⣄⠀⠀⠀⠀⠀⠁⠈⢢⣴⠦⠤⣤⣄⡀⠀⠀⠀⠀⢠⢖⠛⣧⣄⢸⢱⠟⠋⠀⠀⣻⡜⣁⣤⣀⣁⣀⣀⢄⣀⣤⡈⠀⠀⠀⠀⣠⣔
⣒⠠⠀⠒⠄⠀⠀⠀⠀⣟⡀⠠⢀⠤⣼⠋⠗⠓⢀⢴⢟⡀⠀⠈⠀⠚⠓⡶⠀⠀⠈⠋⠉⠉⠀⠁⠠⠏⠀⢀⠃⠀⢻⡆⡠⠐⠉
⡒⠀⠠⠤⣀⠀⠀⠀⠀⣽⠂⠀⡈⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠈⠀⣀⡾⠃⠀⠀⠀⠀⢠
⠉⠛⠂⠀⠀⠀⠀⠀⠀⠈⢲⡆⠁⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠈⠠⢿⠉⠉⡧⠀⠀⠀⢘⡆⠀⠀⠀⠀⠑⠉
⠤⠄⠀⠀⠀⠀⠀⢀⡠⠤⣴⢳⡤⠀⠀⠀⠀⠀⠁⠰⠐⠀⣀⠀⠀⢢⠈⡄⠊⠁⠀⠁⠀⠀⠀⠀⠘⠃⠀⡴⠚⣫⠄
⠴⠒⠦⠤⠀⢀⡜⠉⠛⠁⠘⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠛⠉⠉⠷⣶⢦⡄⢀
⡒⠖⠂⠀⠀⣸⡏⠁⣀⠔⠚⠀⠈⠒⠀⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⣠⠟⣙
⢛⢉⡴⠎⠉⠃⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⡷⠀⠀⠀⠀⠀⠀⠀⡀⠶
⠈⣿⣀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠒⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢤⣤
⡒⡚⠻⠷⢶⣟⠀⠈⠳⢴⣎⡠⠀⠀⠀⠀⠀⠈⠀⢠⠀⠄⠆⠀⠀⠀⢀⠀⠠⠄⠈⠈⠀⠀⠀⠀⠀⠁⠀⠄⢰⡄⠀⠀⣀⣀
⢀⣀⡀⠀⠀⠘⠳⣤⣄⣀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠀⠀⠀⠈⠀⠐⠄⠀⠀⠀⠀⠀⢴⢼⠁⢤⠚⠋⠉⠂⢉⣉
⠿⠁⠀⠀⠀⠀⢀⠀⠍⠁⣻⠀⠀⠀⠀⠈⠙⠉⡉⠀⡀⠀⢀⣀⡁⠀⢆⠀⠀⠈⠁⠀⠈⠉⠀⡠⠶⣤⡤⠞⠋⢀⡖⠀⠈⠐⠒⠄⠠
⠈⠀⠀⠀⡠⠂⠁⠀⢀⣴⠿⣄⣄⣡⠴⡄⠀⣠⡷⢚⠂⠀⠀⣹⡍⠂⢢⣄⣀⣤⠟⠦⠴⢧⣀⣀⣶⠏⠓⠲⠞⠋⠁⠀⠈⠀⢄⠐⠠
⣖⠔⠊⠀⠀⡠⠔⠋⠙⠁⠀⠀⠀⠀⠀⠉⠋⣱⠃⠸⣏⣠⣴⠋⢷⡄⠀⠀⢡⠇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠁⠀⢀⡴⢋⡤⠊⢁⣀⣠⢂⡖⢠⢈⡠⠀⢰⢃⣴⣤⡌⢷⡱⢳⣧⣙⡃⢟⢉⠀⠀⠀⠁⠀⢰⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⣤⡀⠀⢀⣠⣤⡴⠶⢶⡶⡶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡖⠲⠞⠁⡠⠟⠛⢫⣀⡤⠬⠧⠦⠷⠼⢤⣥⣹⢹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⠀⠐⠿⣍⡷⠋⠸⣄⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⢷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣋⣤⣤⠀⠀⣀⠀⢿⠓⠾⠀⠀⠀⢀⣤⣴⠶⠶⠿⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠁⠀⢸⢠⠞⠉⠓⠾⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⢀⣀⣠⣤⡼⠿⠛⠛⠁⠀⠀⠀⠉⠙⠛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⠀⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢁⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠁⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡛⠋⢙⡿⠁⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢀⣴⠋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠛⠉⠀⠀⠉⠛⢷⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢀⡾⠁⠀⣿⡇⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣾⢁⣶⣷⢻⡀⠀⢠⡇⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡏⣼⣿⡟⢈⣿⠞⠁⠉⣏⠀⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣽⡇⢿⠿⢁⣾⠇⠀⠀⠀⢿⠀⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠸⣷⣤⣴⠿⠁⠀⠀⠀⠀⠸⣧⠹⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠾⣫⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣟⡛⠛⠛⠛⠉⠉⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠂⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠈⠉⠉⠉⠛⠳⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠃⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⣀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣄⡀⠀⠈⠛⠿⣿⣿⣿⡟⠛⠛⠛⢻⡍⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣤⣤⣤⣤⣿⡒⠦⣄⣀⢻⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢸⣇⠀⠀⠈⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠛⠛⠒⠾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢀⣴⠶⠾⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⣥⣤⣴⠟⠁
⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⣽⠆⠀⡀⣀⡀⠀⠷⠶⠷⠇⣀⣠⣴⣦⡀⣀⣀⠄⣤⠀⠴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠛⠁⢀⣀⣩⣭⣤⣶⣶⣾⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡄⢀⣼⣿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡟⣋⣉⣩⣭⡀⠀⠈⠉⠁
⠀⠀⠀⠀⣠⣶⠿⠿⣟⣚⣧⣬⣵⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢺⡇
⠀⠀⠀⢰⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣻⣟⣛⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⡿⠿⠿⠟⠚⠁
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠜⠉⠁
⠀⠀⠀⠙⠛⠋⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠁⠀⢹⣿⠀⠀⠀⢠⣖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣯⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠻⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣤⣤⣾⣏⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠈⠙⠛⠛⠛⠋⠉⠉⢀⡀⢀⣠⣤⣄⣰⣶⣶⣾⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣤⣶⣶⣆⣼⣿⠀⣿⡇⣿⡟⢹⣿⠋⢹⣿⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⠁⠿⠇⣿⣿⣸⣿⢷⣿⡇⢸⣿⠀⣾⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠺⠿⢿⣶⡆⣿⡟⢻⣿⢸⣿⠁⣾⡿⠀⣿⡏⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣶⣆⣸⣿⢹⣿⠇⣼⣿⠸⢿⠿⠿⠃⠀⠛⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠛⠛⠛⠋⠈⢉⡀⢠⣤⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣾⠂⠸⠟⠀⠛⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣈⣀⣀⣤⣀⣤⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢆
⠀⠀⠀⠀⠀⠀⢹⡀⡄⠀⠀⠀⠀⠀⢀⣤⣠⣿⣿⣿⣿⣷⣠⣄⠀⠀⠀⠀⣼⠛⠏⠀⠀⠀⠘⠟⢷⡄⠀⠀⠀⠀⠀⠀⡼⠁⠈⢦
⠀⠀⠀⠀⢀⡆⣼⣇⣃⠀⠀⠀⠀⠀⠈⠛⢛⡏⢙⣿⡉⣻⠛⠋⠀⠀⠀⢰⣿⣶⣆⠿⠂⠺⢆⣶⣿⡇⠀⠀⠀⠀⢀⡞⠀⢀⡴⠉⢣
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣤⣿⡿⠾⢿⣯⣄⣀⠀⠀⠀⠀⢿⡉⣩⡐⠶⢆⣌⡉⣻⠇⠀⠀⠀⢀⠞⠀⢠⠎⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠛⠋⠈⠻⠶⠛⠈⠙⠋⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠿⠋⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢠⢃⣀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⡰⠃⢈⣏⠀⠳⡀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠛⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣘⣾⣿⣿⣿⣦
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡰⠁⠻⠏⠈⠿⠃⠱⡀⠀⠀⠀⠀⢠⠎⠘⢿⣅⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⢹⣿⠋⠈⣿⡿
⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠛⠒⠀⠀⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠋⠁

⠀⠀⠀⢀⣴⣶⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣾⣿⠿⣿⡿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⡆⢀⡴⣩⠗⢄⠀⠀⠀⠀⠀⡴⠁⢀⣠⣄⡀⠈⢆
⠀⠀⠀⢸⣿⣤⡞⢳⣤⣿⠁⠀⠀⠀⠀⢀⣾⣦⠈⠟⢁⣾⡆⠀⠀⠀⠀⢠⠇⠀⢨⢁⠋⢀⡥⠞⣉⣇⠀⠀⠀⡜⢡⣦⣿⣿⣿⣿⣴⠌⢧
⠀⠀⠀⠀⢠⣹⡿⢿⢃⡆⠀⠀⠀⠀⠀⠸⣿⡿⠃⡀⠙⣿⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠠⡨⠒⠋⠁⡿⠀⠀⠘⡄⠀⣀⣳⣾⣷⣞⣀⠀⣠⠃
⠀⠀⢀⣦⣈⡛⠿⠾⢟⣁⣴⡀⠀⠀⠀⠀⠙⠁⣼⣿⣆⠈⠁⠀⠀⠀⠀⠀⢣⡰⣿⡿⠀⠉⠨⠉⡹⠃⠀⠀⠀⠘⣌⠛⠉⠳⠞⠉⠛⡰⠃
⠀⠀⢀⣀⣈⡽⠗⠺⠯⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠥⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠈⠦⠤⠤⠤⠤⠔⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⢠⠊⠁⠀⠀⠀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⢀⣧⣴⣶⣶⣶⣶⣶⣆⠀⠘⠀⠀⠀⠀⠀⢀⣶⢸⣿⢀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠀⠄⠀⡴⠉
⠀⠸⠈⠉⠉⠉⠉⠉⢻⣿⣦⠀⠃⠀⠀⠀⣴⣿⣷⣾⣯⡾⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠙⣿⣿⣿⡦⠀⠀⠀⠀⠚⠓⠀⠈⠁⠀⠀⠠⠄
⠀⠀⢣⠀⠀⠀⠀⠀⠈⠛⠛⡿⠀⠀⠀⠀⣿⣿⠿⠻⣿⣿⣆⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠐⠐⠈⡄⠐
⠀⠀⠀⠑⠠⠀⢀⡀⠀⠤⠊⠀⠀⠀⠀⠀⠈⠻⣦⣤⠿⠿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣤⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠈

⡿⠿⠿⠿⠿⢿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣛⣿⣿⣿⢿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣜⣀⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣠⠞⠉⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⠁⠈⠉⠉⠈⠉⠉⠒⠢⢄⡀⠀⠀⠀⠀⣠⡿⠣⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⣀⠤⠒⠊⠉⠉⠉⠉⠉⠉⠒⠢⢄⡀⠈⠓⢄⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⣠⠔⠉⠀⢀⣠⠤⠔⠒⠒⠒⠒⠤⢤⣀⠀⠈⠓⣤⡾⠋⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠞⠁⠀⡠⠚⠉⠀⢀⣀⠤⠤⠤⢄⣀⠀⠈⠑⣦⡾⠋⠙⢆⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣠⠃⠀⣠⠊⠀⢀⠴⠊⢁⣠⡤⠤⠤⣤⣀⠉⣲⣾⠋⠈⢢⡀⠀⢳⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠃⠀⡰⠁⠀⡴⠁⣠⠖⠁⠀⣀⣄⣴⣀⣨⡿⢯⠀⠱⡀⠀⠱⡀⠀⢣⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡟⠀⢠⠇⠀⡸⠀⡰⠁⢀⡔⠉⠀⡰⢁⠿⢿⣄⠀⢳⡀⢹⡀⠀⢧⠀⠘⡇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⠀⡇⠀⡇⠀⡸⠀⠀⣰⣿⠥⠒⠉⠹⠀⠀⡇⠀⡇⠀⢸⠀⠀⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢸⠀⠀⡇⠀⡇⠀⢹⠀⠀⠉⠀⠀⠀⠀⢸⠀⢀⡇⢀⡇⠀⢸⠀⠀⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠧⠀⠘⡆⠀⠱⡀⠸⡄⠀⠳⢄⡀⠀⠀⣀⠴⠃⠀⡼⠀⡸⠁⠀⡎⠀⢠⠇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠘⡄⠀⠘⡄⠀⠑⣄⠈⠢⣄⡀⠈⠉⠉⠀⣀⡤⠊⢀⡜⠁⠀⡼⠀⠀⡜⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠘⣄⠀⠘⢦⡀⠈⠑⠤⣀⡉⠉⠉⠉⠉⣀⡠⠔⠋⠀⣠⠞⠀⢀⠜⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠣⡀⠀⠙⠢⣄⡀⠀⠉⠉⠉⠉⠉⠁⠀⣀⡤⠚⠁⠀⡠⠋⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⠳⢄⡀⠀⠉⠛⠒⠒⠒⠒⠒⠚⠉⠁⠀⣀⠴⠋⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠉⠑⠒⠤⠤⠄⠀⠀⠤⠤⠔⠒⠉⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠤⢀⣀⣀⣀⣀⠠⠤⠴⠒⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⠢⣄⡈⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣀⠤⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠙⠳⡂⢌⡛⢶⠤⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠳⡃⠀⡤⠚⠉⠉⠲⣄⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠈⠲⢌⠉⠓⠊⠙⠳⡴⠚⠓⠶⠉⠈⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⡇⠀⣀⠤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣦⡽⠄⠀⠀⠀⠢⣄⠀⠀⠀⠀⠈⣇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠗⢋⣵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⢿⠄⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡙⠦⡼⠘⢤⠃⡇⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢳⡆⠀⠀⠀⢀⣠⣤⠿⠀⠀⠀⠀⠀⠉⢹⣃⣀⣤⣤⠶⠚⠉⠀⠀⠀⠀⠰⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⢈⣉⡷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣁⣀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠅⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠆⠀⠀⠐⠒⠛⠻⠶⡄⠀⠀⠀⠀⠀⠀⠀⡤⢝⣷⣌⡉⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠥⢄⠀⠀⠀⠀⠀⠀⠱⡄⠀⠈⠉⠓⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠔⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⣴⣴⡀⣶⣾⡁⠚⠆⠀⠀⠀⠀⠀⠀⠀⢺⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠏⠘⠇⢷⡏⠃⠙⠀⠀⠀⠀⡔⠢⠤⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠴⠒⣞⠉⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠙⠢⢤⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⡤⠖⠁⠀⠀⠰⣕⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣖⠚⠁⠀⠀⠀⠀⠀⠀⠁⠠⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⣶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⡏⠹⡄⠀⠀⠀⢀⡄⠀⠀⠀⡇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡘⠀⠀⠉⠓⠒⠚⠁⠘⠦⠤⠞⠁⠀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠰⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠏⠀⠈⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠈⢻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠁⠀⢠⣾⣿⣿⣦⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠙⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡀
⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⡀
⠀⠀⣰⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⡀
⠀⣼⣿⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣻⣿⣿⣿⣿⡄
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡀⠀⠀⣀⣀⣀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠚⠀⠂⠀⠉⠱⠟⠃⠈⠉⠋⠀⠛⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣞⠉⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡔⠒⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠁⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣋⡄⠀⠠⠆⠀⠀⠀⠀⠀⡂⣺⠦⠋⠄⠀⠄⠀⠉⢍⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⢶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢅⣀⣑⠂⠀⠀⠀⠀⠾⠟⠻⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣷⠲⠀⠀⠀⠀⠀⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣀⠂⠐⠒⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠐⠀⠀⠙⠁⡀⡀⠄⣰⠦⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠤⠖⢇⣠⣀⡤⣧⠀⠀⠀⠀⠀⢠⣅⣀⡤⠐⠛⠲⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠑⠊⡀⢿⠃⡀⠀⠀⠀⣿⠉⠑⠋⠉⣉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⣀⣀⡀⠀⠘⠋⠙⠆⢉⠒⠛⠐⠁⢀⡄⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠴⠴⡾⠹⠤⣀⣰⣠⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⣀⠀⠀⠀⣠⠄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠤⢤⡖⠒⠖⡇⠀⢀⠀⣿⠈⠑⠒⡋⠁⠈⠉⡦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠕⠀⡠⠒⠊⠉⠀⠔⠓⡴⣉⡊⠁⠃⠀⠒⠀⠀⠀⠐⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠰⠀⠀⠔⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠴⠋⠁⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠂⠒⠐⠒⠒⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠐⠒⠒⠈⠒⠒⠁

⠀⠀⠀⡭⠭⠭⠭⠭⠭⠭⠭⠭⣽⢃⣀⣀⣀⣀⣀⣀⣀⣄⣏⢧⣿⣤⣤⣤⡤⠿⠛⢛⣛⡃⣀⣀⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣇⣸⣿⡆
⠀⢀⠈⠉⠁⠉⠉⠉⠁⠀⠉⠉⠈⠉⠉⠉⠉⠉⢿⠙⠉⠉⠁⠈⣹⢩⡽⢻⡟⣡⣤⣬⣭⣭⣭⣍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣾⣿⣿⣿⠃
⠀⠈⣳⣄⣀⣀⣴⣷⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣾⠗⠋⣀⡾⠁⣋⣁⣠⣉⠉⢉⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⣿⣿⣿⡟
⠀⠘⣷⣈⡛⢻⡿⣯⡆⣿⣦⠀⠀⠀⡆⠀⠀⠀⠀⣯⠲⠶⠶⠶⠶⣾⠟⠛⠻⣿⡿⣿⣿⣿⡶⠾⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⣿⡷⣿⡍⢱⣧
⠀⠀⣧⠻⢿⣿⡿⣝⣧⢸⣿⣆⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠠⣽⣧⠀⠠⠶⠖⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡿⣧⣿⣿⣧⣿⣧⣼⣹
⠀⠀⢿⣏⡞⣾⣷⣿⡞⢆⠻⣿⡄⢸⣿⠀⠀⢀⣀⡄⡤⣞⣭⣿⡇⠀⠀⠀⠀⢘⣿⣆⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⢀⠤⠒⢏⡁⣿⣿⣿⢸⣿⣻⣿⣿
⠀⠀⠪⣼⢸⣿⠗⣶⢭⡛⡇⢹⣿⣿⣿⣿⣿⣿⣷⡷⣿⡿⠻⣿⡇⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⣻⡆⠀⠀⢧⡀⠀⠈⣷⣿⣿⣿⢸⣿⢼⣿⣿
⠀⠀⢸⢣⣾⣿⡜⣿⣥⣽⢀⠀⣿⣿⢿⡟⠛⠁⢹⡧⢿⠃⠀⢹⡇⠀⠀⠀⠀⠀⠹⠿⣿⡶⣿⣿⣿⣿⢿⠟⠃⠀⠀⡀⠳⠦⣼⡏⣿⣿⣿⢉⢹⣺⣿⢾
⠀⠀⣿⢸⣿⣿⣿⢡⣥⣸⢸⣇⢸⣇⣘⣻⣆⢵⠌⠴⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠠⠟⠁⣿⣹⣿⣎⣘⣿⣿⢿⡀
⠀⠀⣿⡈⣿⣿⣿⡠⣿⣿⡆⣿⣿⣿⢹⠙⢻⣖⡶⠲⠒⠒⠂⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣋⣽⠶⢤⣀⠀⠀⠀⠀⠀⠈⢢⢊⢏⢩⣿⣿⡟⢩⣍⡾⠇
⠀⠀⣿⢰⣿⣿⣿⣷⣜⢶⣄⣿⣿⣿⡉⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣮⡿⡿⣎⡽⢳⡦⣤⣤⡴⣫⣤⣼⣸⣿⣽⣵⣰⡿⠃
⠀⠀⢻⡎⣿⣿⢟⣿⣿⣷⣽⣿⣿⡏⠁⠀⠀⠀⣤⠴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣵⡟⣈⠇⣼⣇⣬⠒⢛⣮⠷⣼⡿⣿⠛⣹⣿⡀
⠀⠀⠋⣿⣹⣷⣏⣴⣿⣿⡿⣿⣿⣶⠀⢠⡎⡘⠀⢛⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢋⢹⡕⢿⠓⣷⣼⢨⡇⣶⣶⡉⢯⣽⣷⠟⠀⢻
⠀⠀⠀⢸⣿⣯⣻⣿⣿⣿⣽⣿⠏⠀⠀⣿⡇⠃⢠⣾⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⡿⣭⠾⠶⠾⠏⠚⠛⠉⠉⠉⠛⠳⠿⠀⠀⠈
⠀⠀⠀⠀⠻⣿⣯⠙⣛⣭⣿⠏⠀⠀⠀⣿⠇⢦⡀⠉⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⢠⣿⣿⡇⢻⣷⣿⡆⠀⠀⠀⢹⡀⠀⣸⡲⡀⠀⢜⢿⡄⠀⠀⠀⠀⠀⢀⢮⠋⠀⢾⠀⠀⠀⢨⡗
⠀⠀⠀⠀⠈⢻⣿⡄⢠⣿⣿⠃⠀⠀⠀⠈⢧⡀⠸⣷⡈⠀⠀⠈⠻⣦⣀⠀⠀⠀⡨⣾⡀⣠⣎⣻⣦⣴⢟⣥⣴⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⢿⣿⣇⢀⣿⣯⠇⠀⠀⠀⠀⠈⠳⣦⣬⣉⠳⠀⢄⠀⠩⠝⠟⡿⢡⣤⣤⢥⡄⢤⣤⣴⡆⢸⣿⣿⣟⣘⣛⣋⣙⣟⣛⣓⣆⣐⣂⣐⣲⣰⡃
⠀⠀⠀⠀⠀⠀⣿⠃⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠰⣦⣔⡠⡄⢠⡇⠷⣿⣽⣯⡟⣾⣷⣿⡇⢿⠿⡟⣟⢹⠋⠉⠉⢉⢉⠙⠋⢉⡙⠋⠛⠛⣿

⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠙⠷⡄⠀⣧⡀⠀⠀⠋⠀⣠⠄
⠀⠀⠀⠀⠀⠀⢻⠻⢦⠴⢲⡟⠁⠀⢀⣤⣤⣤⠄⠀⣀⠄⡖⢰⢂⡐⠀⠤⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠺⣤⣄⣀⠀⠀⠀⣠⠔⠃⣹⠔⠋⠁⠀⠀⠹⡄⠘⡄
⠲⠖⠒⠂⣠⣞⠁⠀⠀⣤⡾⠻⠥⠤⢰⠊⣇⠠⠖⠉⠀⠀⠀⠀⠀⠀⠀⡗⠈⡃
⠀⠀⠴⠚⠋⠉⠉⢻⣴⠋⣯⠘⠃⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣗⣉⣅⣀⣀
⠀⢀⣠⠀⠀⣴⠀⠘⠏⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠿⠛⠛⠋⠉⠉⠉⠙⠛⠛⠻⢷⡆
⠴⠋⠁⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⢤⣤⣼⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⢀⣠⣤⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⡷⠶⠤⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠴⠚⠛⠛⡟⠛⠿⠿⠿⣿⣿⣿⣷⣶⣶⣤⣤⣀⣿⠀⠀⣆⣤⣴⣾⡿⠿⠟⠛⠋⠉⠀⣷⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠘⢽⣿⣿⠏⠉⠙⠛⠛⣿⠀⠀⢻⢿⣿⣯⠁⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⣀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠈⠛⠲⠶⠶⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⣼⡇⠀⠀⠀⠐⠋⣿⠛⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣧⠉⠛⣶⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠴⢶⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⡿⠀⠀⠀⠉⢹⡏⠉⠉⠉⠉⢹⡏⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣷⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⡀⣾⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣽⠞⠹⡆⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢤⣄⣀⠀⠀⠀⣿⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠰⣄⣀⠀⠀⢀⣠⣶⠾⠼⢶⣤⣤⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⢄⣽⣷⣶⣾⣿⣟⣥⣄⠀⠀⠈⠛⠻⢷⣄⡀⢤⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣷⣽⠏⠁⡦⠊⠀⠀⠀⠀⠀⠀⠈⢲⣾⣿⣏⣴⡙⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⡄⠀⠀⠀⡠⠔⠚⠓⠒⠄⠀⠈⠙⠍⠙⢿⣧⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠐⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸⣿⠁⠉⠲⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠻⡀⢠⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡦⠀⠀⠀⠀⠉⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⣦⣼⣧⡑⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣾⣷⣄⢀⡀⢤⣠⢤⡀⠀⠀⠀⡠⠀⠀⠐⢺⣿⠁⠈⠉⠓⠤
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠈⢦⣙⠿⠿⣿⡛⣆⣈⣉⣙⣳⣶⣍⣂⣀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁⢈⠛⠻⠿⠛⡋⠉⠺⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠛⠛⠉⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠋⠀⣀⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⠛⠁⢀⣠⣴⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⠟⠋⠁⣀⣤⣶⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⡿⠛⠉⢀⣠⣴⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⣀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣶⣿⣿⠃⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠀⣿⣿⣿⣿⣷⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠛⢿⣄⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠙⠛⠷⠶⣂⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣀⣀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠻⣿⠇⠀⠀⠀⠸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⣀⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠟⠉

⠀⠀⠀⠀⣀⣀
⠀⠀⠀⣸⣿⠈⠉⠓⠲⢤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⡖⠒⠒⠒⠶⠶⠶⠦⠤⢤⠀⣀⣀⣀
⠀⠀⢀⣿⠿⣀⣛⣛⣋⣓⣒⣐⣀⣀⣂⣀⣀⣀⣂⣀⣀⣀⣀⣠⣦⡼⢧⡤⠘⢿⣛⣛⣻⣟⡷⠈⢛⣡⣶⠿⠃
⠀⢠⣿⢿⡝⡖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣷⣀⣸⣦⣀⣀⣀⣁⡀⡌⢿⣁
⠀⠘⣿⣌⣋⣿⣶⣶⣶⣶⣶⣶⣶⡾⠿⠿⠿⠿⠿⠟⠛⠛⠛⣛⣿⣭⢻⡯⠉⢻⣿⠿⠿⠟⢻⢿⠖⠂⠈⠉⠳⡀
⠀⠀⢨⣿⣿⣜⠛⠒⠉⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣶⣶⣶⣶⣟⣼⠷⠶⣾⣿⣀⣠⣤⣤⡞⠀⠀⠀⣴⣷⡳⡄
⠀⠀⠘⣿⣈⢿⣴⣶⣶⣶⣶⡿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠈⣿⣧⠀⠛⠛⢛⠋⠉⠉⣠⣤⡄⣸⣿⡿⣿⣿⣆
⠀⠀⠀⠈⠙⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣧⣶⣼⣿⣿⣧⢰⣿⠋⣷⣿⣿⣿⢿⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⡌⢿⣟⣼⡇⢠⢿⣿⣿⣿⣗⡾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⠭⠽⠒⠉⢸⣿⡿⣿⣷⢽⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠃⣿⣿⣯⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣿⣿⡿⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢞⣻⣎⡢⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡐⠤⠄⢉⣉⡵⢺⣿⠇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠐⣦⠞⣴⡿⠋⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠛⡻⢃⣾⢏⡁⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⣱⡿⠋⠂⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣨⣿⡿⠥⠶⠀⣌⡀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢰⢿⠏⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣲⡽⠺⠏⡏⠿⢹⠓⢒⡦⢥⢐⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡛⢤⣄⡱⠃
⠀⠀⠀⠀⡰⣠⢾⡇⢿⠀⣀⡤⠶⠞⠒⠒⠷⠾⠴⠦⣿⣁⡎⢥⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⣤⣿⡿⠛
⠀⠀⢀⡾⠕⠁⢸⡇⣼⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠯⣽⣦⡀⠀⠀⠀⠀⡠⠊⢢⣔⣿⠏
⠀⠠⠩⣸⠁⠀⢘⡾⠁⠀⠀⠀⠀⠀⢀⠴⣾⠟⣿⢩⣉⣇⣍⣹⣒⣆⢤⣉⠻⣷⣄⣀⠔⢀⡰⢫⠟⠁
⠀⠀⣄⡏⠀⠁⢸⠁⠀⠀⠀⠀⠀⣰⡋⠀⣚⣠⠟⠉⠁⠀⠀⠀⢁⠉⠙⠛⢿⡦⣹⢃⠄⣫⡝⠁
⡀⢰⣿⢁⡇⡆⣾⠀⠀⠀⠀⠀⢠⢹⠀⠀⣿⠁⠀⠀⣠⢶⠹⣏⣸⠭⢯⣑⣦⡙⢻⣿⣟⠻⣿⣦
⡏⢸⣸⢠⠃⡁⣿⠀⠀⠀⠀⠀⡟⢸⠀⠀⡇⠀⠀⠀⣿⠈⣰⠋⠀⢀⣀⡈⣫⣾⣧⡈⢣⡵⣌⠻⣧⣤
⣿⠙⣇⣾⠀⠃⣽⡀⠀⠀⠀⠀⢿⢸⠀⠀⡇⠀⠀⠐⡏⠂⢹⠀⠀⢿⣿⣿⡀⠹⣿⢳⡀⠹⣼⢦⡈⢻⣦⢀
⢸⡍⢸⡸⣄⠀⡟⢣⠀⠀⠀⠀⢸⣜⠀⠀⡷⡀⠀⠀⠻⡄⠸⡗⣄⠀⠈⠉⠉⢲⣿⡇⣇⠀⠹⣏⢳⡄⠙⢿⣧⡀
⠀⢧⡎⣧⢸⠀⣧⢸⣆⠀⠀⠀⠀⢻⠀⡆⡄⣳⡀⠀⠀⠙⢻⡄⠏⡟⠲⠤⠖⢺⡿⣿⣿⠀⠀⡇⡸⢸⠀⠈⠯⣾⣂
⠀⠘⣷⡜⢆⢿⡇⠸⠟⣆⠀⠀⠀⠀⠻⡄⣇⠺⠸⡲⣄⠀⠀⠈⠳⠷⣴⣀⣆⣼⡧⠟⠑⠒⣸⣧⣅⡌⡄⠀⠀⣧⢷⣃
⠀⠀⠘⣆⠚⣌⢧⡼⠀⡟⢦⠀⠀⠀⠀⠘⢷⡄⠀⢢⠸⡗⢦⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠿⣿⣻⢷⡇⠀⠀⢸⡌⢿⡄
⠀⠀⠀⠙⣮⠈⢎⢿⡀⠁⣾⠑⣄⠀⠀⠀⠀⠈⠳⣌⡀⢸⠰⠆⠛⡶⠦⣤⡄⣀⣤⣤⡞⠹⠀⠁⣽⣹⡧⠤⢤⢸⡏⡸⣶
⠀⠀⠀⠀⠘⢧⡬⢓⣝⢦⣻⠘⡇⢳⣄⠀⠀⠀⠀⠀⠙⠾⢦⣸⡄⡧⢸⠘⡇⠻⢛⡘⢠⠆⠀⢈⡵⠃⠀⠀⠒⢸⣿⣧⣿⡀
⠀⠀⠀⠀⠀⠈⠳⣤⢨⠲⣝⢦⡇⢸⡇⡙⣦⣄⠀⠀⠀⠀⠀⠉⠓⠳⠼⢦⣇⣠⣀⣧⡯⠴⠚⠋⠀⠀⠀⠀⠀⡞⣿⣺⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣔⢩⠗⣿⣼⡃⠃⡗⢹⢱⢢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⠇⠏⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣎⠀⢻⣝⠦⣇⣼⠐⠨⠌⡞⡇⡖⡤⣄⣀⣀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠖⡏⠈⡻⢸⠠⡗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⡎⡑⢮⣛⠦⣄⡄⠇⠁⢠⢑⢳⢸⠉⠉⣿⢻⢻⠋⡏⡏⡼⢸⢸⢡⢡⡆⣎⢟⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣴⡮⠛⣚⢭⣗⢶⡴⠤⢬⣘⣀⣇⠯⢘⢸⠀⠁⡇⡇⡞⣘⣼⠼⢛⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠷⢦⣔⣉⡉⠛⢓⠚⠆⣭⡭⢭⣉⣉⣉⡉⢩⡭⣵⣶⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠈⠏⢴⠂⣀
⠀⠁⠈⠉⠈⠉⠉⠉⠉⠁⢉⣉⡉⢱⠀⠀⠀⠀⠀⠀⠀⠚⠦⡁⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢠⠀⡴⢀⡤⠄⠀⡄⢀⠀⠀⠀⠀⡀
⠂⠀⢀⡖⢠⠀⡇⠀⠀⡞⣻⣱⠷⣺⠆⠀⠀⠀⠠⡄⠀⠒⠆⡅⠠⠲⡀⠒⢠⢰⡖⠀⠺⠙⠶⢂⠂⡀⣛⢹⠀⠀⠀⠀⠀⠀⣸⠀⢠⠥⠆⢤⡄⠉⡄⡐⡠⣱⠂
⠀⡀⣸⡵⢻⡴⠟⠶⠟⢛⡇⣨⢨⣹⠀⠀⠀⠀⠈⠃⠀⢠⠦⢠⠀⠀⢸⠓⢿⠛⠀⢠⠸⠦⠈⠈⠀⠁⠉⠈⠀⠀⠀⠀⠀⠠⠼⢘⢤⣤⣼⣦⣥⣾⣵⣾⡽⠁
⠀⢉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠄⢢⢀⠐⢀⡀⠁⠚⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠉⠀⠉⠉⠁
⠀⠀⠂⣠⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠒⠈⠁⠀⠀⠀⠀⣤⡄⠀⠀⠀⠁⠈⠶⣁⡀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠀⠀⠀⠀⠀⢰⣿⠛⣿⡆⣀⡀⣀⠀⢀⣀⠀⣿⣧⣤⡄⠀⢸⡇⠀⣿⢣⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠆⠀⠀⠀⠀⠁⠀⠀⠀⢠⠶⣄⢸⠇⡟⠀⠀⠀⠀⠀⢸⣷⠀⣿⡇⣿⡇⣿⠀⣿⡋⠁⣿⡏⣿⣇⠀⢸⡟⢘⣟⢸
⠀⠀⣸⣿⡄⣿⣀⢤⡄⢀⡴⣦⢴⡎⢶⢰⡇⣿⠻⠶⠃⠾⠐⠃⠀⠀⠀⠀⠀⠀⠛⠿⠛⠀⠙⠿⠟⢀⠻⠿⠀⠿⠇⠻⠇⣀⣜⣻⣍⣻⠞⠀⠀⣀⠄
⠀⢀⣿⠘⣿⡏⢿⣠⠇⠸⠦⠟⠘⠷⠋⠈⢉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣌⣀⡀⠀⠈⠀⠋⠉⠁⠒⠒⠉⠀⢁⣀⣡⠤⠒⠋
⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠀⠀⠀⠀⠠⠀⠀⠐⠒⠀⠐⠒⠀⠌⠷⣦⡔⠒⠀⠀⠉⠉⠉⠑⠄⢀⠔⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠁⠀⠐⠂⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⠤⢄⠀⠀⠀⡄⠤⣐⡚⢻⢙⣇⠘⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢠⠒⢒⢁⢌⡀⢠⡔⠈⠙⢷⠘⠉⠃⢨⠑⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠀⢻⠈⠓⠈⣇⠘⠃⣸⢧⢮⢳⠎⡱⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣷⡀⢠⠓⣤⢎⣛⣿⣹⣶⣼⣿⡦⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⣀⡂⠠⡄⣸⠆
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠙⠉⠳⣼⣵⣾⣿⣿⠟⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠲⣶⢍⡤⣵⢲⣞⣶⠊⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡥⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢢⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢤⡀⠈⠉⠙⠛⠛⠋⠁⡀⠢⢤⣤⠆
⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⣠⠴⠊⠁
⠀⠀⠄⠀⠔⠴⠽⠦⠮⠷⠴⠾⠗⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠤⠄⠀⠀⠒⠀⠀⠀⠠⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠓⣠⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⠂⠄⠀⢄⣀⡀⠀⠀⠀⣀⠨⠀⠁⢂⠈⠀⢢⡂⠀⡧⠠⠃⠀⡇
⠀⠀⠀⠀⠀⠠⠐⠊⠀⠀⠀⠀⠀⣀⣠⡄⣿⡇⠘⠀⠀⠀⠀⠀⠀⠈⠀⢀⢦⣀⣁⣼⣿⣀⣸⡀⠀⠀⠀⠀⢰⡆⣶⢀⡀⣀⢀⠀⡀
⠀⠀⠀⠠⠊⣤⣆⢠⣄⣰⣦⣾⡟⠿⠟⠃⠉⢡⠂⠀⠀⠀⠀⠀⠀⠀⠈⠞⠿⣿⠿⠿⡿⠛⠛⢉⠀⠀⠀⠀⢸⡿⣿⣻⣷⣿⣼⣷⣿⣿⣾⣷⣿⣾⣇⣤⣤⡄
⠀⠀⠀⠀⠀⣯⠟⠻⠋⠉⠁⠀⠀⢀⡀⠔⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠙⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠁⠠⡀⣀⠀⡀⠐⠐⠃⠂⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣶⣶⣶⣄⠀⠀⣠⣶⣶⣶⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠙⠛⠛⠉⢀⡀⠉⠛⠛⠛⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⣴⣿⣿⡦⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⢢⡀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢀⡔⠁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⢳⡀⡇⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⢀⡞⠁⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠸⠥⣙⡢⠔⠓⠤⠚⠓⠤⠚⠲⠔⣋⠬⠇⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣠⠦⢄⣀⠉⠙⠲⠤⢄⣠⠤⠒⠋⠉⣀⡤⠴⡄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡹⠖⠾⣅⠀⠀⣀⡭⠖⠲⠤⣀⡈⠉⠒⠶⢭⣀⠀⠀⣨⠷⠲⢏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠊⠉⣀⡀⠤⠒⠋⠉⠈⠙⠒⠤⢄⣈⠉⠙⠁⠀⠀⠀⡇⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠈⢳⠀⠀⠀⢰⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡆⠀⠀⠀⡞⠁⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠘⢤⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⡤⠃⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠤⠄⠀⠀⠀⠀⠀⠀⠀⣀⣐⡂⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⠶⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠶⢦⣄⡀⠑⠱
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠙⢷⡀
⠀⠀⠀⠀⢀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⢂⠂⣰⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⢰⣿⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⣾⡏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡩
⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠈
⠀⠌⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣠⠟⠁
⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⣁⡠
⠀⠀⠀⠀⠀⠈⠿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠖⠛⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢍⡉⠛⠛⠻⠷⠶⠶⢶⣶⣶⣦⣤⡤⣄⠀⠀⠀⠸⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠹⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠻⢦⣌⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⡴⠋⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠶⠦⠤⣤⣄⣀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀⠰⠏⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣦⣄⡀⠀⠈⠱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⣠⣾⣿⣿⣿⠿⠛⠛⠛⠻⠿⣿⣿⡿⣢⣄⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⣾⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⢾⣿⣿⣦⠀⠀⢻⣆
⠀⠀⠀⠀⠀⢸⠇⠀⢀⣾⣿⣿⠋⠀⠀⠀⠂⣲⣿⣿⡿⢲⣄⠀⠈⢻⣿⣿⣇⠀⠀⢿⡄
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⡏⠀⠀⣼⣾⡞⠉⠀⠀⠑⣿⣿⡆⠀⠀⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⡇⠀⠀⠫⠍⠄⠀⠀⠁⠀⡸⢿⡇⠀⠀⣿⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⡀⠀⠸⣿⣿⣷⡀⠀⠀⠀⢼⠢⣤⣤⣾⣿⠞⠀⠀⢰⣿⣿⡿⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠹⣿⣿⣷⡀⠀⠀⠉⠒⠚⠛⠛⠁⠀⠀⣰⣿⣿⣿⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠙⢿⣿⣿⣶⣤⣀⣀⡀⢀⣀⣠⣴⣿⣿⣿⠟⠁⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠉⠉⠙⠛⠛⠉⠉⠁⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣤⣄⣀⣠⣤⣤⡴⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⣮⢳⡐⡄⠀⠀⠀⠀⠀⠀⡄⣾⠈⢁⡀
⠀⠀⠀⠀⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⣷⣧⢳⡀⢀⣤⣤⣤⣀⣷⡸⢀⣾⡇⠀⠀⢀⣶⣮
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣸⣿⣟⣷⣿⣿⢟⣛⣿⣿⣷⣼⣿⣷⣶⣷⣿⣿⣃⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢹⣶⣾⣿⣿⣿⡿⢿⣾⣿⣿⣷⣿⣿⣭⣿⡿⣹⣿⡟⠁⠀⢀⠍⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢦⣤⣄⠀⠄⢘⢓⣏⠀⣠⣿⣎⣻⣿⣿⣿⣿⣯⣭⣻⡿⣿⣿⣿⡿⠉⢰⣿⣏⣤⠀⠀⠈⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢿⣷⣷⡦⣈⠁⠀⣾⡟⣛⢛⣿⡿⡿⣿⣟⣭⣽⣷⣶⣾⣿⣡⠀⠈⢿⡿⣿⣾⣶⣶⣀⣀⠈⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠺⡦⠀⠀⠀⠀⠀⠀⠉⠳⢶⣦⣝⣲⣿⣿⣁⣿⣿⣿⣿⣯⣹⣿⣿⣽⠛⠛⢿⣧⣎⣴⣿⣷⣹⣿⣟⣿⣿⣿⣿⣾⣵⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⣿⣿⣿⣿⣽⣿⣿⣭⣿⠏⠀⢿⡛⠩⠀⠀⠘⠃⣸⣷⣍⢻⣟⣮⢻⣼⣿⠿⠛⢋⣉⣤⣴⠂
⠀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠁⣀⣿⣿⣿⣿⣿⣿⠎⠋⢐⡂⠈⠐⢶⣶⠄⠘⣧⣽⣭⠛⢻⡟⢛⣉⠭⠔⢒⡈⠉⣉⠡⠤⠴⠶⡂⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⣽⣉⣿⣿⡿⡿⠆⣐⣿⠀⠀⠀⠀⠁⠀⢀⡿⢿⣿⠀⣠⢫⣧⣶⣾⣿⣯⣭⣶⣶⠖⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠀⣤⣾⢠⣿⢧⢸⣿⣿⣟⢷⣿⠀⠙⠁⠀⠀⢠⣾⣧⡤⠉⣠⣾⣷⣦⣭⡾⢿⡿⣿⣟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠀⠀⠁⢼⣿⣿⣎⣻⣿⣿⣾⣿⣦⡀⣀⠴⠆⣼⣿⣿⡠⠐⠋⠻⠻⢿⣟⣱⡏⣶⡏⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣽⣿⡿⠻⠛⠛⢉⣨⣿⢇⠈⠉⠁⣩⣿⣿⠿⠛⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⣦⣤⣠⣶⡿⣿⣿⣟⡷⣶⣿⣿⣏⡀⠀⠀⠓⠘⢿⣿⡿⠁⣸⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⠟⠉⡈⣩⣴⢿⣿⣧⣽⢷⣼⡿⣿⣿⣤⣬⣽⣻⣿⣎⡓⡄⠀⢈⡻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠀⠉⠁⠀⠉⠹⠿⢷⣽⠶⠾⠿⠿⠿⠿⣿⣿⣿⣿⡿⣆⠢⡹⣿⣧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠟⠀⠀⠀⠀⡈⠀⠀⠀⠙⣧⢻⣎⠻⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⢻⣆⠈⠁⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠋⠉⠉⠛⠛⢿⣿⣿⡿⠛⠋⠉⠉⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⠀⠀⠀⣠⣿⣿⣀⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠛⠉⢻⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⡟⠉⠛⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⣸⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣦⣼⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⣿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠛⠉⠀⠈⠙⠻⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣾⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⣿⠉⠙⠻⣶⠾⢿⣬⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢧⣴⠟⠛⠳⠶⣼⣧⣀⣤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠶⢿⣴⣾⣿⣷⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠉⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⣀⡀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡄⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡘⢿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⡌⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣦⡀⠀⣰⣿⠟⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠛⠁⠀⠀⢠⣴⣶⣿⣿⣿⣿⣶⣶⣤⣀⠀⠀⠉⠻⣿⣿⣿⡦⢛⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠁⠀⣠⣾⣿⣎⢻⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣦⡀⠀⠘⠿⣋⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠏⠀⢀⣼⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠆⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⠀⣾⣿⣿⡿⠁⠀⠀⢀⣠⣶⣶⣦⣄⠀⠀⠀⠹⣋⣵⣿⡆⠀⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⢰⣿⣿⣿⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⢹⣿⣿⣿⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠸⣿⣿⣿⡄⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣸⣿⣿⡿⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠀⠀⢻⣿⣿⣷⡄⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⣰⣿⣿⣿⠃⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆⠀⠀⠻⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠃⠀⢀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡿⠛⠀⠀⣠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⢀⠖⠉⠁⠉⠉⠑⠾⣋⠀⠀⠀⠀⠀⠀⢛⡖⠊⠉⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⢐⠅
⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠀⡸⠀⠀⣎⠀⠀⠀⠀⠀⠀⢀⡤⠃⠀⡠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⠤⠠⠒⠁⠀⠀⠈⠢⢄⡠⠤⠒⠉⠉⢰⠊⠉⠀⠀⠈⠉⠱⢄
⠀⠀⠀⠀⠀⠀⠀⡾⣇⡠⠒⠒⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⣠⠇⠉⠀⠀⠀⠀⠀⡇⠀⠀⢀⡤⠖⠚⠙⠒⠦⡀⠑⡄⠀⠀⠀⠀⡀⠀⣠⠃
⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⡴⠋⠀⠀⠀⠀⠀⠀⢸⠀⠈⠑⠉⠑⢼⣯⡉
⠀⠀⠀⠀⠀⠸⡀⢀⠀⠀⠀⣀⣀⠇⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⡀⣤⣀⣠⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠁⠙⠒⠋⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⣀⠤⠊⠀⢀⡼⠁⠀⠉⠙⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⣇⡜⠁⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠹⢂
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠓⠋⠃⠛⠑⠒⣺⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠚⣆⡀⠀⠀⠀⣘⠄
⠀⠀⠀⠀⠀⠀⠠⣔⠀⠀⠀⠀⠀⠀⠀⢨⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠈⠑⢢⠒⠃
⠀⠀⠀⠀⠀⠀⠈⠁⢳⠤⡤⣠⢤⠴⠲⠉⠁
⠀⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣸⠾⠷⣇⠀⠀⠀⠈⢦⡀⢀⣤⣧⣆
⠀⠀⠀⠀⠀⠀⢈⡆⠙⠒⠒⠤⢄⡀⠀⠲⢤⡤⠞⠈⠀⠀⠈⣟⠀⠀⢀⣨⠎⠉⠀⠀⠙⣆⡀
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣼⠀⠀⠀⠀⢠⡇⠀⠀⢤⡗⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠐⢻⢦⣄⣀⣀⡞⠃⠀⠐⢚⡯⣄⣀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠈⠓⢤⣀⣀⠀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠻⠿⠿⢿⣿⣿⡿⠿⠿⠟⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢸⣿⣷⣄⣀⣀⣠⣾⠟⠻⣷⣄⣀⣀⣠⣾⣿⡇⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣷⣌⠙⠛⠻⣿⣿⣿⣾⣷⣿⣿⣿⠟⠛⠋⣡⣾⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣟⠛⠛⠓⠀⠀⣿⢻⡟⣿⣿⢻⡟⣿⠀⠀⠚⠛⠛⣻⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⠷⣦⣀⠀⠀⠀⠈⠁⠉⠉⠈⠁⠀⠀⠀⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⣉⠳⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⣀⡀⢰⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⡆⠀⣿⣧⠀⢰⣶⡀⣶⡆⢠⣶⣾⣦⢸⣿⠛⠃⢸⡟⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇⣸⣟⣿⡄⢸⡿⣷⣿⡇⣿⡇⣴⣿⢸⣿⠿⠇⢸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠁⠿⠋⠻⠷⠸⠇⠘⠟⠃⠙⠿⠟⠿⠸⠿⠿⠆⠸⠟⠘⠿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣤⠿⣿⠻⡿⠿⢿⣯⡭⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠻⠶⣿⣛⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣮⡳⡀⠀⢸⣭⣭⣭⣭⠷⠶⠋⠁
⠀⠀⠀⠀⠀⠀⣰⣟⣿⣿⣿⠿⣦⡔⠻⡟⠋⠻⡟⠉⠁
⠀⠀⠀⠀⠀⣼⢏⣾⣿⢿⠃⠀⠈⢧⡀⠳⣄⣠⠃
⠀⠀⠀⠀⢰⡟⣾⣿⣿⣿⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠀⠀⣿⣇⠿⠿⠿⣻⡇
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠃




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⠁⠉⠉⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢰⣏⣤⣶⣶⣶⣶⣤⡀⠀⠀⣠⣴⣶⣶⣶⣶⣄⢻⡄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣤⡜⢿⣿⣿⣿⣿⣿⣿⣿⢻⣮⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠘⢿⣝⢿⣿⣿⣿⣿⠿⠋⣼⣿⣿⡌⠛⠿⣿⣿⣿⣿⠟⣸⡿⠃⠀⠀⢀⣠⠤⠶⠦⣄
⠀⠀⠀⢀⣠⣼⡅⠀⠈⠙⢧⡄⠀⠀⢈⣿⣷⣦⣭⡀⠀⠀⢸⣿⡿⣿⡷⠀⠀⠀⢀⣤⣴⣿⣿⠁⠀⢀⡴⠋⠀⠀⠀⣤⣟
⠀⠀⠀⢻⡅⠀⠀⠀⠀⠀⠀⠙⠳⢦⣼⣿⣿⣿⣿⡇⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⡶⠞⠋⠁⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠉⠳⠦⠤⠤⠤⣤⣀⠀⠀⢸⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⠃⢀⣀⡤⠶⠶⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣾⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡶⢤⣀⡈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢀⣀⣤⣾⡋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⠴⠚⠉⠁⠀⠀⣈⣩⠿⠓⠒⠒⠒⠒⠒⠒⠿⢯⣉⠁⠀⠀⠉⠙⠲⢤⣄⡀
⠀⠀⠀⠀⣠⠞⠋⠉⠉⠉⠉⠀⠀⣀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠦⣤⣀⠀⠀⠈⠙⠛⠉⠙⠛⠶⣄
⠀⠀⠀⠀⠻⣤⡄⠀⠀⠀⢠⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡄⠀⠀⠀⠀⣀⣀⣸⠇
⠀⠀⠀⠀⠸⣇⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣼⠉
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉








⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣷⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣷⡿⠿⣿⣿⣿⣿⠿⣻⣿⣯⣵⣾⣿⡆
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣯⣝⣻⠿⠏⠀⠀⠀⠉⠀⠈⠐⣦⠀⠹⣯⣶⠿⢟⣛⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⢲⣟⠿⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢲⣀⣀⣀⣀⣰⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣤⡼⠿⢷⣶⣶⣿⣿⠛⢻⣟⣋⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣾⠿⢻⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠠⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⢛⣛⣛⣛⣛⣿⣿⡇⠀⠀⢰⢸⣿⣿⣿⣿⣿⣛⡛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠚⠃⠀⡀⢸⣾⣿⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣶⣿⣿⣾⣿⣿⣷⣀⣀⣀⡀
⠤⠤⠤⢤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⡤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣷⣀⣀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠿⠿⠁⠙⠛⢛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠉⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣤⣤⠀⣶⡶⠾⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠛⠋⢉⣿⢻⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣺⣿⡿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⢛⣛⣛⣛⣛⠛⠻⠶⣯⣽⣻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣌⡙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣦
⠀⠀⠀⠀⠀⠀⣼⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⣧
⠀⠀⠀⠀⠀⣸⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠹⣧
⠀⠀⠀⠀⢀⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⡆
⠀⠀⠀⠀⢸⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⡇
⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⡇
⠀⠀⠀⠀⢸⣧⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⣿⡀⣿⣧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⠃
⠀⠀⠀⠀⠀⠸⣧⠘⣿⣆⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⡏
⠀⠀⠀⠀⠀⠀⠹⣧⡘⢿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠻⣿⣦⣈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠙⢿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣡⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣌⣙⠛⠿⠿⣿⣿⣿⣿⣿⠿⠿⠛⢋⣡⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⣶⣦⣤⣴⣶⠶⠾⠟⠛⠉

⠀⠀⠀⠀⣠⢤⣀⡀
⠀⠀⠀⣼⣿⣀⣀⣈⣉⣓⠒⢲⣿⡻⠭⠩⣍⣙⣓⣒⣒⣒⠲⠦⠤⠤⣀⣄⣀
⠀⠀⢸⡿⢿⣧⠀⠈⠉⠉⠉⠉⠙⣿⠀⠀⢹⡯⠭⠭⠭⢿⣿⡛⢇⠀⢰⣿⡿⠃
⠀⠀⠈⠻⣶⣿⠛⠛⠻⠿⢿⣿⣿⢿⣶⡶⣾⢿⣷⣤⣤⣤⣤⣥⣤⡄⠀⢧⡀
⠀⠀⠀⠀⣿⣻⣤⣤⣤⣤⣬⣭⣿⣀⣀⣀⣿⣶⠿⠶⠶⠶⠶⢦⣀⠀⡠⠄⠋⠷⠤⢄⡀
⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠈⠹⣿⣏⠉⢹⡿⢿⣆⣀⣀⣀⣀⣈⣉⠟⠈⠁⠄⠀⢀⣤⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠛⢻⣿⣿⣿⣿⡿⠿⠿⠃⠀⠀⠀⠀⠀⣸⣿⣧⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⣤⣤⣤⣤⣤⡴⠦⠤⣶⣶⣶⣤⣴⢆⣿⣿⣿⣎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⣿⣿⠋⠉⢿⣷⠀⣴⣿⡿⣿⣿⢟⣾⣿⡿⢿⣿⣮⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⡆⠀⢸⣿⢀⣿⠿⠇⠀⣬⣾⣿⣿⣶⣶⣿⣿⣧⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠱⣄⢸⣿⣜⡝⠀⣠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠬⠭⠭⠖⠒⠚⠁⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠏⣿⣿⣿⣿⠿⢿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣾⣾⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠹⢿⣿⣿⣿⣿⠿⣟⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣀⣤⡶⠟⠛⠉⠉⠉⠉⠉⠙⠛⠻⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣞⣿⣷⣾⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣠⡾⠛⠁⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⣻⣭⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠘⢷⡀⠀⠀⢀⣴⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⢀⣾⣿⣿⣿⡿⠿⠟⠛⠻⠿⣿⣿⣿⣿⣦⡀⠀⠀⣻⡦⠞⠋⠀⠀⠀⣀⠤⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⡀⢀⡙⢿⣿⣿⣿⡴⠛⠉⢻⡄⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⢀⣤⣤⣤⡠⢊⣁⣤⠿⣿⣿⣿⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣿⣿⠁⠀⠀⢠⣿⣿⣿⣯⠵⣛⣛⣥⡄⣿⣿⣿⡇⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣿⣿⣿⡄⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢿⣿⣿⣷⡀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡧⠀⠀⠈⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣧⠀⠀⠻⣦⠀⠀⠈⠻⣿⣿⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⠏⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠘⢧⠀⠀⠙⣷⡄⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠈⠓⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠉⠉⠉⠋⠉⠁⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⣠⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣿⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⡿⠛⠛⠛⠛⢿⣿⣿⣿⡿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⢀⣀⣴⣿⠟⡟⢿⣷⣄⣀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡏⠀⣷⠀⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣯⠙⣿⣿⣷⣼⣿⣴⣿⣿⣿⠩⣿⣏⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠛⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣠⣤⣤⣤⣀⢠⢠⣖⣸⡭⣭⡥⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠗⣌⠹⣻⠋⢿⢀⡼⢯⡀⠀⠀⠀⠙⢋⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣾⠋⠀⠈⠓⠈⡙⠨⠟⣿⣟⡿⣷⣌⠀⠀⠀⢩⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠀⠀⠀⠒⠙⠁⠁⣼⡄⠀⠀⠀⠀⠀⠀⠐⠖⠀⠈⡈⣉⣡⡔⠀⠀⠀⢋⢁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⣻⡅⠀⠀⠀⠀⠠⣌⡀
⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣴⣿⢇⡀⠀⠀⠀⠀⢢⢥⡀
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠠⢤⣀⡀⠀⢠⣿⣷⠛⠋⠃⢹⣄⠀⠀⠀⠀⠢⡴⢄
⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠈⢧⠀⠀⠀⠀⠈⠲⣛⢢
⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠉
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠄⡏
⠀⠱⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡞
⠀⠀⠹⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡜
⠀⠀⠀⠘⢦⡈⠦⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣠⠋
⠀⠀⠀⠀⠀⠙⢦⡉⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡠⢠⡐⠊⠣⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠆⠝⣘⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣑⡤⣀⣉⠡⠔⠍⠄⣀⡀⣂⠀⠀⠀⢐⡤⠔⠊⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣶⣶⣶⣶⣶⣶⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠢⢄⣠⠃⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘⣆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠹⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄⣀⠀⢹⠀⡜⠛⠉⠉⠉⠉⠉⠙⠛⠛⠿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⠉⠲⡿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⣀⣵⠹⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠈⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⠁⠀⠀⢀⣴⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠈⢻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⣴⣿⣿⣿⣿⡿⠛⢆⢣⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣦⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⠀⠘⡌⢆⣀⣀⣀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⣸⣿⣿⣿⡟⠀⠀⠀⢀⣴⣿⡘⣿⣿⣿⣿⣦⡀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢠⣿⣿⣿⣷⡸⣿⣿⣿⣿⣿⡄⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣧⣹⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⣀⣾⣿⣿⣿⡿⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠻⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⠟⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⢀⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣤⡘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣤⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⠛⣷⡈⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣿⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢠⣾⣿⠀⣿⣷⡌⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠒⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠛⢐⣟⢻⣿⣭⣿⣿⣿⣆⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣭⣭⣭⣭⣍⣭⣭⣭⣭⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀
⠀⡇⠤⠤⠤⠬⠭⠿⠿⠷⠞⡏⠉⠉⢉⣉⣉⣉⣉⣁⣀⣀⣈⣉⠉⠉⢦⣶⠖⠒
⠀⡷⣶⡶⠶⣶⣶⣶⣶⡶⢆⠀⠀⠀⢸⠀⠀⠒⠶⠤⡄⠀⠀⢩⠉⢆⠀⠣⣤⣄
⠀⢸⣀⣀⡗⣀⣀⣀⣀⣀⣘⡿⠿⠿⢿⠀⠀⠀⠀⠈⠁⠀⠀⢸⠀⢸⠀⠀⠀⠉⠒⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸⠀⠈⠙⠛⢲⣞⣿⣿⢞⣠⡟⠠⢂⡔⢄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⠘⠒⠒⠒⠒⠒⠚⠛⠛⠚⠋⠀⠈⠉⠒⠢⢃⣠⣤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠓⠢⡀⡠⠒⠒⠀⠸⣶⡶⠢⡄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡅⠀⠀⠀⠀⣸⠃⠀⢸⢀⣠⣾⣿⣿⣿⣿⣿⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣒⠁⣀⣀⣼⠛⠛⠛⢻⣿⣿⣿⣿⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠐⠈⢣⢀⡨⠀⡀⠐⠈⣀⠅⠠⢇⠀⢀⡢⢸⠀⡀⠁⢀⠃⠄⢁⡊⠀⠀⢹⣿⣯⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠙⠛⠛⠛⠛⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠋⠑⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠘⠄⠁⠀⣄⠀⠀⠈⠄⠀⠀⡀⠀⠀⠀⠀⠠⠘⡀⡀⢀⢀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠑⠀⠀⠀⠀⠰⣿⣦⣄⠀⠀⠀⠂⢡⠠⢄⣾⠃⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢿⠙⠿⣷⣦⣐⣈⣄⣶⣾⠃⠀⢸⣧⡀⠀⠀⠀⢀⣤⣶⣿⡞⠀⠀⡰⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⢿⣿⣿⠿⠃⠀⠀⠀⢻⣿⣷⣶⣿⠿⠛⢁⡞⠀⠀⠈⠀⠀⢀⡄⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣦⣤⣤⣤⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡞⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠷⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣈⣿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡄⠀⠀⠀⠈⠛⢿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠾⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣋⣤⣤⠴⢤⣄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠈⣧⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⣠⠴⠚⠲⢦⣄⡀⠘⣿⡄⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠀⢀⠄⠀⠀⠀⠀⠀⣿⠀⣠⡾⢫⢁⡃⡛⣳⣄⣰⠏⠀⠀⠀⠀⠀⠉⠛⠾⣧⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⡀⠀⠀⠀⢻⡾⠋⡀⠰⢨⠡⠁⠄⠜⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡴⠁⠀⠀⠀⠉⠀⠀⠀⡁⠦⠐⠀⠀⠀⠀⠀⠀⠀⡆⠀⢣⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⡁⠈⠀⠀⠀⠀⠀⠀⠃⠀⠈⠁⠀⠈⠆⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡈⠄⠀⠀⠀⠀⠑




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠺⣄⡙⢷⣦
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠉⠓⠿⣷⣾⣄⣀
⠀⠀⠀⠀⠀⠀⢸⣁⣀⠀⠀⠀⠀⠉⠙⠻⠿⣖⣢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⣰⠁⠉⠛⠶⢤⡀⠀⠀⠀⠀⠀⠉⠛⠶⢭⣑⠲⢤⡀
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢰⠷⠴⣦⣤⣀⠀⠀⠀⠀⠀⠈⠙⠚⠧⣴⣂⢄⣀
⠀⠀⠀⠀⠀⠀⠙⠓⠦⣤⣀⡾⠀⠀⠀⠉⠙⠳⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢽⣍⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⡲⠤⣀⠀⠀⠀⠀⠀⠉⠑⠢⣄⣀⡤⣀⡀⠀⠀⠀⠈⠙⠓⡦⣭⣓⡢⢤⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣭⡓⠦⣤⡀⠀⠀⢼⡁⠀⣽⠶⡭⣕⠲⡄⠀⠀⢠⠁⡆⢈⠉⡷⢬⣝⣲⣴⣊⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡀⢯⣀⠀⠀⠉⠛⠻⠾⣤⣠⠟⢓⡖⠦⣼⠰⠇⢸⠆⡇⡇⢰⠉⡏⠳⠦⢭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢩⡟⠲⠦⢤⣄⣀⠀⣆⠀⠺⣄⠀⢈⡿⢲⡞⠛⡷⡷⣸⢸⡇⣇⠀⠀⣿⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣼⠁⠀⠀⠀⠀⢈⡟⠳⢦⡀⠀⢙⡶⠟⠀⠱⣄⣉⠉⣹⣮⡇⣿⠀⢠⡷⠒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⡇⠀⠀⠀⢀⣠⣞⣀⣠⠼⡇⠀⣸⠁⠀⠀⠀⠀⠀⠉⠁⠀⠙⠳⡏⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⢿⣄⠀⠀⠉⠉⠉⠉⠀⠀⣷⠀⣸⠇⢀⣤⣤⣄⠀⠀⠀⢀⣠⢶⡿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠿⣿⣶⣤⣄⡀⠀⠀⠀⣿⣠⢧⡀⠘⠿⠤⠟⠀⠀⣠⢋⡶⠲⠶⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢽⣟⣲⠾⠃⢻⠞⠁⠀⠀⠀⠀⠀⢠⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠸⣤⠀⠀⠀⠀⠀⠀⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡾⠀⠀⠀⠀⠀⠀⣼⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠁⠀⠀⠀⠀⠀⠀⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡟⠀⠀⡞⣹⢷⡄⠀⢸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⡛⠺⢧⣄⡀⠙⠷⠞⠁⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠶⢭⣽⣓⡲⣆⢀⡾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠛

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

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

⠱⠄⠈⢁⣀⣵⡙⠳⡿⠄⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⢀⠀⡀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠐⢕⣤⠀⠀⣠⠊
⢀⣴⣕⠄⠛⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠉⠀⠛⠀⢀⡀⠀⠀⠀⡔⢠⠀⠀⠀⠘⠁⠀⠜⠁⠸⣆
⣝⠂⠙⢀⣴⠄⠓⠀⠀⠀⣠⠰⠲⠲⠆⠁⠀⠀⠴⠀⠒⠠⠀⠀⠈⠀⠀⠀⠀⠀⠁⠤⡄⠀⠀⠀⠀⠠⠀⠀⠈
⡙⠂⠀⠌⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡚⠶⠄⠀⠀⢀⠥⠒⢄⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠀⠀⠀⠀⠘⠉⠁⢢⠀⠀⠀⠈⠀⠀⢸⠒⠒⢀⡀⠀⠀⠀⠁⠀⠔⠀⠀⠀⡠⠚⠁
⠀⠀⢠⠀⠤⠰⠇⠀⠀⠀⢰⠃⠚⠁⠠⡄⠀⠀⢁⠇⠰⠀⠀⠀⢰⠤⠐⣀⠸⠂⠒⢀⠰⡄⠀⢀⡀⠀⠀⠀⠠
⡰⠈⠙⠂⠀⠈⡱⠔⠶⠤⠌⢠⠆⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠜⠀⠸⡀⠀⠉⣠
⠳⠦⠀⠀⠀⠀⣁⠀⡤⠬⠗⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⠄⠀⠠⠅⠀⠠⠒
⣦⠀⠀⠀⢰⠋⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢈⠂⠀⠀⠉⠉
⠀⠈⡁⠀⠘⠠⠂⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⣤⡤⠤⠄
⣤⡦⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣴⡶⠖⠂⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒
⣼⡷⠖⣋⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⢴⡶⠟⣋⠄⠀⠀⣠⠔⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠢⣌⠙⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋⠄⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡚⣟⣳⠀⠀⠀⠀⢀⡀⢇⣀⡠⠰⣃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣻⠻⢲⠖⣠⢤⣤⡔⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣳⡭⠼⠿⢷⣀⠀⠀⢀⡽⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠃⠈⠉⠁⠰⣄⠀⠅⠘⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⢿⠉⠙⡄⠀⠈⠻⠟⠖⠛⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⢿⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠈⠀⢸⡄⠀⠀⠑⠂⠠⡄⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣄⠀⠀⡄⢸⡇⠀⠀⡇
⠀⠀⢀⣤⣀⡴⠖⠀⡞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢸⣰⠞⠙⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⡀⠀⠀⢿
⠀⠀⢍⢻⡏⠀⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⢸⡿⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⢈⣸⣏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣷⠀⢀⣰⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⠀⠀⠀⣿
⠀⠀⠘⢿⡏⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡁⣾⠿⣷⡾⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡀⡏
⠀⠀⠀⢸⣡⠂⢾⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣼⣿⣿⡟⣷⡾⣿⠿⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⠘⡇⠀⢀⡇
⠀⠀⡇⢸⡏⢀⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠋⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇
⠀⠀⣧⠀⣇⣼⣽⡇⠀⢀⡀⠀⠀⠀⣼⠀⠀⠀⠀⣸⣿⣿⣾⣿⡅⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⡇
⠀⢠⣿⢠⣿⣼⣿⠣⣀⠔⡴⠀⠀⠀⠆⡀⠀⠀⠀⢻⣿⣟⣿⢻⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⢂
⠀⠀⢹⣻⡇⠻⣿⣷⣏⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠛⢡⠸⠙⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄⠀⠘⡄
⠀⠀⢸⢸⡇⠈⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡎⢦⢃⡻⠀⠀⠤⠤⠀⠀⠀⣤⠀⠀⠀⠀⢠⠀⠈⠀⠀⠃
⠀⠀⠀⠀⣹⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢈⣇⣾⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢇
⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⡛⢺⣿⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢰⡅⠀⠀⣄⠂⢸
⠀⠀⠀⠀⢹⡇⢰⡀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⡀⠐⠂⠰⠉⠀⠀⠀⠀⠀⠀⠀⣿⣥⡀⢸⣧⠈⣼
⠀⠀⠀⠀⢀⣿⠀⡇⠀⢰⡇⠀⠀⠀⠈⡆⠀⠀⠀⠀⠸⡿⠃⠈⠻⠆⠀⢈⡵⠀⠀⠀⠀⠀⠀⠀⠀⡟⣼⡇⢰⡿⠀⡏
⠀⠀⠀⠀⢸⡏⢰⠃⠀⠘⣇⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠐⠀⢀⢻
⠀⠐⠀⠀⠈⡇⠘⡄⠀⠀⢹⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠞⠁⠀⠀⠀⠠⠀⠸⣿⡇⠀⠀⢿⣿⢲⡄
⠀⢀⠀⠀⢰⠇⠀⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢘⣿⣜⣡
⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢋⡵⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠈⠀⠀⢀⠈⣿⣿⣋⠄
⣿⣿⠿⠿⡿⢿⠿⡿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿