Sundew Emojis & Symbols

⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣿⣷⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⢸⣿⡯⢞⣿⣿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡻⣿⣿⣯⡻⣿⣿⣿⣯⣿⣿⣧⣿⡿⣟⣋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣝⢿⣿⣿⣿⣻⣿⣿⣷⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⢿⣿⣿⣿⣿⣿⢻⣿⣧⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⣮⣻⣿⣧⣿⣿⢿⣿⣿⣿⣿⣿⡷⢯⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣮⣛⢮⣟⣿⣿⣿⡿⡿⠿⢿⣿⣿⡎⠿⣿⣿⡿⢛⣩⣿⡿⣳⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣟⣛⣛⣛⣺⣿⣦⢻⣇⣤⣿⣿⣷⣿⣿⢿⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠋⡼⣍⣡⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣴⣆⡰⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢀⠖⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⠟⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⡾⠁⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠘⢷⠦⠾⢴⢟⣷⠂⣄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡄⡼⠃⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣰⠟⠓⠒⠒⠤⠥⣄⢻⢛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡠⠐⢦⣤⣀⠐⡔⠂⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡇⠀⠸⣯⡉⢸⠏⠸⠯⣼⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠃⠀⠀⢠⡟⠀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠸⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⡆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠅⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⡃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢾⣶⣾⣶⣄⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣀⣀⡀⠀⠀⠀⠀⠀⣠⡿⠀⣿⢻⣿⣿⣿⣿⠇⠀⣀⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⣸⡟⠁⠼⣿⣿⣦⠀⠀⡀⠀⣿⣧⠀⣿⣻⣿⣿⣿⣿⠀⣰⣯⣿⢿⣿⣿⣿⣱⡆
⠀⠀⠀⠀⠀⠀⠀⠙⣧⣗⣶⣯⣽⣿⣿⣶⣷⣤⣍⣯⡀⣿⣀⣿⠛⣡⡏⢠⡿⠛⣁⣾⣿⣟⣋⡼⠃
⠀⠀⠀⠀⠀⡤⣰⡶⣦⣽⣟⣛⠛⢛⣉⣀⣉⣿⣯⣝⣷⣿⣿⣿⣧⣿⣆⣡⣲⣬⣛⠛⠛⣭⣩⣴⣦⣷⣦⡄
⠀⠀⠀⠀⠀⢃⣿⠀⣻⣿⣿⣿⣿⠟⠛⡿⢭⣥⣤⡽⣿⣿⣿⣿⣟⠛⣿⣿⣿⣿⣿⣇⠲⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣧⣀⠀⣾⣿⣿⣿⡇⢻⡟⠉⢿⣿⣶⣴⣿⡿⠿⣟⣧⣤⣬⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠙⠛⠛⠛⢋⡆⢩⢀⠀⣰⣿⣿⣶⡌⠢⡠⢀⣼⣿⣿⣿⣿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣷⣿⣿⡿⠿⠋⠀⠙⠿⢶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣯⡁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣀⠤⠐⠲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠘⡇⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡘⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠉⠹⠆⠐⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠐⢀⢉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⢸⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⠋⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢂⣿⣿⡾⡆⠀⠀⢀⠀⡄⢸⠠⣶⡟⠉⢙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⣦⣶⣇⠀⠛⠙⡀⢃⢸⠀⠓⡀⢐⢋⣼⣿⣷⣶⣶
⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⠀⠈⠃⡄⠀⠀⢻⠘⣨⠀⢀⡼⠛⠛⢹⣿⣿⣿⣿⠂
⠀⠀⡠⠄⠭⠉⡐⢠⡄⣀⠀⠀⡴⣶⣫⡄⢠⡃⢿⣰⠋⠁⠀⠀⠈⠙⢻⢛⣃⣀⣠⣄⡀
⠀⠸⣅⡀⠀⠀⢀⣴⡒⠙⠒⠀⠁⣌⣉⣈⠈⢳⣾⢷⡀⢀⣴⣓⣤⠒⠚⠛⣿⣿⣿⣿⣿⡅
⠀⠀⠈⠳⠶⠶⣿⣽⣀⠀⠀⠀⢀⠠⣵⡶⣫⡜⣷⡱⢾⡽⣯⣿⣿⣶⡄⠀⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢀⡀⠉⠙⠻⠋⣄⢀⡀⠀⣩⣾⣿⢻⢿⣿⣮⢻⣎⠉⣉⣏⣵⣤⣭⠽⠛⠟⠋⢀⡀⠂
⠀⠀⠀⠀⠨⣇⠀⠀⢠⢸⣿⣿⡟⠛⠻⣵⠏⣘⣿⡄⠚⠂⠛⣷⣬⣾⣿⡯⣿⣶⠤⡄⢠⣮⡥
⠀⠠⣦⠄⠀⠙⣶⣶⣶⣿⣿⠟⣡⢀⠘⡋⠃⠸⠛⠃⠴⠛⡎⢓⠿⡁⢀⣸⣿⠇⠀⠸⢨⣿⣶
⠀⠐⠊⠁⠞⠋⠑⠈⠻⣿⡤⠄⠛⠁⠉⠈⣡⠀⠀⠀⠀⠻⢃⣌⢳⡶⣿⠛⠋⠀⠠⠴⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⣄⡀⠀⠀⣨⣇⠀⠀⠀⣄⠘⠃⠀⣠⡟⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠾⠊⠛⠁⠀⠀⠀⠈⠳⠞⠛⠟⠉⠀⠀⠀⠰⠠
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠉⢉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢢⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣦⢈⠿⠉⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠃⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡴⣤⣌⣀⠂⠀⠀⠀⠀⠀⠀⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡞⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⡏⢾⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠶⣦⠀⠀⠀⠃⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢀⣼⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⢀⢀⣀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⢀⣔⣲⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢃⠀⠀⠀⠘⡀⠀⠀⠀⠀⣿⠀⠀⠀⠙⡀⠀⠀⠀⢘⠇⠀⠀⠀⠀⠀⠀⢀⡟⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣰⡌⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⡄⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠾⡇⢺⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡌⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⢷⡈⠛⣤⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⡇⠀⠀⢰⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡘⢧⡀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⣿⠀⠀⣸⠀⠀⠀⠇⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢦⣀⠆⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠈⣀⠀⣿⠀⠀⠘⠀⣀⣸⣄⣀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠲⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡿⠆⣿⠀⠀⣧⠈⣿⠁⠈⠉⡄⠀⣲⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄⠀⠀⠀⠀⢀⠀⠀⢻⠀⣿⠀⣠⠁⡸⠃⠀⠀⠀⢠⡀⠰⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠑⠀⠘⣆⣿⢀⡃⡘⠁⠀⢀⣠⠼⠛⣷⠀⠛⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠤⣤⡝⣯⣸⣏⣠⣤⠒⠋⠁⠀⠀⢿⣷⣾⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠛⠙⢛⡀⠀⣹⡟⠁⠀⠀⠀⠀⠀⠀⠙⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠙⠏⣹⣾⠇⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢛⣹⡤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⢦⡀


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⣤⠀⠀⠀⢀⣤⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠀⢋⠀⡖⠈⠀⢸⠆⢰⠉⠀⠀⢸⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠙⠀⠈⠀⠙⠀⢀⠀⠡⠀⠀⠸⢀⡈⠀⡛⠀⠃⠀⡦⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠘⠑⠄⢴⠀⠊⠃⠀⠐⠘⡂⠈⠀⠁⠀⠀⡗⠀⡜⢡⠄⠄⠋⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠂⠀⠆⠀⠀⢧⠐⢀⢠⡀⣄⣧⣫⣠⣤⣄⣀⠀⡜⠀⢡⢂⡴⠂⣀⠴⠛⠃
⠀⠀⠀⠀⠀⠉⠑⣤⠀⠠⠄⠀⠑⣤⣤⣾⢻⠒⠛⡀⠬⠟⢓⠛⠿⠯⣵⣹⢀⡞⠠⢋⠤⢊⠵⠆⠀⣀⡀
⠀⠀⠀⢀⡀⠀⠈⠉⠢⠀⠀⠀⣠⣼⠏⢀⡈⠀⠀⣁⠀⠂⢀⡀⢡⠀⠋⢻⡉⣿⡖⠡⠐⠡⠐⠂⠈⠉⠁
⠀⠀⠀⠈⠉⠉⢁⡒⠐⠃⠄⣼⠣⠀⠴⠀⣠⢉⡄⢡⡈⠁⡄⠀⡄⠐⢃⠀⢩⡿⣴⣋⡀⠛⠀⠠⠶⠆
⠀⠀⠀⠀⠀⠀⠈⣉⠈⠉⠀⠵⡖⠓⠀⠂⠠⠄⠲⠀⠄⠋⢀⠰⠈⡁⢉⣁⠠⠦⣟⠵⠶⢶⣖⠒⠒⠒⣶
⠀⠀⠀⠶⠖⠒⠂⠉⠉⠉⠉⣷⠄⠃⠈⠀⠆⣈⢡⠁⠄⠁⠄⢂⠐⠂⠤⠉⠀⢰⢻⠆⣉⡁⢈⡐⠶
⠀⠀⠀⠀⠀⣤⠤⠀⣒⡀⠔⠦⣦⠈⠁⢈⠠⠄⡀⠆⢠⢁⠀⠀⠄⠒⣀⠻⡬⠿⡏⡑⠂⠺⠁⣈⡁⠲⠶
⠀⠀⠀⠀⠀⠀⡀⠤⢌⣠⠐⠀⢁⡶⡀⣉⠐⠂⢁⠶⡀⠒⡘⢣⠜⠁⠬⣰⣶⣿⣑⠨⣐⠂⠠⢈⠁
⠀⠀⠀⠀⠾⠁⠀⠀⡈⠵⡤⠞⡰⢿⡷⣧⣚⣤⡍⠀⣠⢈⠁⣴⣹⣾⣶⡟⢿⠻⣕⠻⡁⠙⠖⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⢙⡡⠂⣠⠞⡴⣣⠐⢡⠏⣸⣧⣖⠃⠼⢠⣧⣟⢻⡏⣻⡈⠃⠈⠳⡈⠷
⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠉⡞⠁⣨⠁⡤⣴⠀⠛⣡⢀⣅⣦⣿⠹⠀⣟⠉⠑⡀⠳⠀⠈⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⢠⡁⠀⡄⢸⡇⢼⢸⣽⡟⠃⣆⠈⣇⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢘⡃⠈⢠⡿⢿⡄⠙⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⠖⠒⢸⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡃⡄⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠻⢀⡛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⡀⢹⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣟⣀⠀⠀⠀⡜⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠆⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⠉⠈⠁⠉⠁⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠉⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠄⠈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊
⠀⠀⠀⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐
⠀⠀⠀⢰⠀⠀⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⡀⡤⠂⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⡂⠀⠃⠀⠀⠠⠈
⠀⠀⠀⢰⠀⠀⠀⢸⡀⠀⠀⠀⡞⠀⠀⣠⠋⠀⠀⠀⡠⠊⡀⢠⠀⠀⠀⠰⠄⠁⠀⠀⢀⠐⠁
⠀⠀⠀⡸⠀⢡⠀⠀⠁⠀⠀⢠⠃⢀⠎⡘⠀⢀⠀⡴⠁⠀⡁⠌⠀⠀⠀⠀⠀⡇⢀⠔⠀⠀⠀⠄
⠀⠀⠀⣇⠀⠀⠢⠀⢣⠀⠀⡞⣠⠃⢠⠃⠀⠈⣶⠁⠀⢠⡇⠀⠀⠀⠀⠀⠀⣴⠃⢀
⠀⠀⠀⡇⠄⠀⢀⣷⠙⡦⢠⡷⠁⠀⡎⠀⠀⡼⠉⢧⡔⢁⡇⣤⡄⠀⢸⠁⣰⢁⡔⠁⠀⢀⡜
⠀⠀⠀⠹⣶⠎⠁⠈⢧⣿⣸⡿⠗⣸⠧⠀⣼⠃⣠⠎⢹⡄⣇⣇⠀⢠⣿⢀⣵⠟⠛⢀⡿⠋
⠀⠀⠀⣰⢻⡜⢢⡀⠘⣿⣿⢸⣄⣏⠤⣾⣣⡞⠁⠀⢀⣻⣿⣼⣦⣾⢸⣿⣟⣠⠾⢋⠀⠂⠆⠀⠀⠠
⠀⠀⠀⣿⠀⠹⣦⠙⣦⣿⣿⠀⣿⣣⡾⣥⡟⣶⣔⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣻⣿⡿⣳⣶⣦⣴⣧⣾⡤
⠀⠀⠀⢻⣀⣠⣽⣷⣿⣻⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣿⣿⣁
⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠅
⠀⠀⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠐⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣍⣹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣹⣿⣉⣀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠻⠿⢿⡿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠟⠉⠛⠻⠛⠟⠻⠛⠿⠿⠿⠟⠿⢿⣿⠟⠿⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣈⠱⠇⠠⠀⠀⠀⠀⣤⣮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢕⣙⠃⠁⠺⠮⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣄⠒⡌⣾⣲⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉⢴⡅⠀⢆⠨⠄⠨⡔⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢐⡀⢂⠀⠀⠀⠀⠀⠈⠍⠉⠈⠀⣷⡃⠀⠀⠢⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠉⠀⠄⣠⡀⠠⢂⣀⡀⠀⠀⠀⠈⠍⡀⣀⣙⡓⠀⠀⠀⠘⠈⣀
⠀⠀⠀⠀⠀⠀⠄⣽⣿⣷⠅⠰⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠰⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⣟⣜⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⢳⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢤⠈⢦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠺⠃⠀⠈⠀⠀⢀⡄⢠⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢲⠷⢀⢀
⠀⠀⠀⠀⠐⠠⠀⢁⡠⠀⠀⠀⠀⠄⠀⠀⠀⠺⠚⠆⢴⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡘⠁⠤⡈⡀
⠀⠀⠀⠐⠢⡁⠀⠈⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⠀⠀⠀⠀⠀⠀⠀⠀⢔⠢⠀⠀⠨⠠⠓
⠀⠀⠀⠀⠀⠀⠀⢈⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠁⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠐⡄⠘⠂
⠀⠀⠀⠀⠀⠈⠀⢂⠼⡄⢘⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠐⠰⢤⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢠⠂⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠔⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣮⡿⠦⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠐⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢻⣴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⢉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⠀⠀⣴⣢⡀⢀⣴⣧⣀⠀⠀⠀⢠⣰⣤⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣧⠀⠀⠻⣿⠃⣾⣷⣽⣿⣆⠴⠂⣾⣟⡿⠿⠟⢻⣥⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢿⡍⠿⠿⢿⣷⠆⠀⠹⡾⢋⠙⠰⣿⣿⠆⠀⣿⣯⣴⠶⠞⣿⣿⠿⣿⢙⡟⢀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⠷⠷⠢⣭⣓⡤⠀⢱⣁⡲⣾⣽⣇⣶⣶⣿⠃⠀⢀⣤⣩⣥⡴⠴⣋⣾⣿⣿⣿⣷⣄
⠀⠀⠀⣴⣿⣿⣿⣷⣦⡀⠀⠀⣀⣠⣿⣷⣿⢿⣟⢧⣹⣿⣋⣿⠿⣿⣶⣾⡷⠞⠛⠀⠀⠀⢸⡿⣏⠙⠛⣿⡿
⠀⠀⠀⢻⣿⣿⣻⣛⣛⣳⡾⣭⠍⣛⣈⡽⢿⣿⣷⠿⣿⢿⣿⣷⣶⣿⣿⣿⣦⣄⣀⣀⣀⣀⣘⣧⣤⣶⣞⠻
⠀⠀⠀⠀⠉⠟⢻⠛⠉⠉⣧⣿⣿⡟⢡⣶⡿⣿⡿⡺⢯⣾⣿⣿⣿⡿⢿⣯⣙⣦⣤⣀⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣤⢾⣿⡿⠻⣿⣷⢟⡼⣿⢹⣿⡻⣿⣿⣿⣾⣍⠉⠀⠀⠀⠀⠀⣤⣀⣄
⠀⠀⠀⠀⢀⣠⣶⣶⣦⣄⣀⠈⠁⣠⣾⡏⠐⣲⣿⣿⢿⡇⢸⣟⣿⣷⣿⣤⣿⢿⣿⣄⡀⠀⣰⣷⣷⣟⣩⣿⡄
⠀⠀⠀⠀⠹⣿⣿⣿⣿⠛⢎⣙⢫⠟⠉⠱⢮⣭⣽⠷⠈⢹⠄⠉⠋⠈⣧⠀⠈⠀⠀⠉⠻⢯⣿⣿⣿⣿⡟⡛⠁
⠀⠀⠀⠀⠀⠙⠛⠿⡶⠾⠛⠉⠀⠀⠀⠒⢻⡇⠀⣀⢤⠇⠀⠀⠀⢀⠛⢤⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠄⠀⠀⠀⠑⠄⠀⠀⠀⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠚⠓⠒⢦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⡰⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣏⠉⡶⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⣸⠇⢈
⠀⠀⠀⠀⠀⠀⢰⣂⠒⠀⡀⢀⣠⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⢰⡅⠂
⠀⠀⠀⠀⢾⣷⠂⢹⣋⠛⠋⠾⢡⢽⠏⠀⣦⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢀⡏
⠀⠀⠀⠀⣴⡖⣛⠜⠉⠉⠘⠒⠤⠔⢦⡀⠏⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜
⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠳⣤⢰⣀⠌⣀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⠻⠀⣿⠃⠴⣋⣠⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢙⣥⠊⠀⠀⠀⠀⠩⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⣀⣘⣀⡀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠒⢨⣦⣿⣿⣿⣿⣆⣸⡇
⠀⠀⠀⠀⠤⣀⡤⣄⣐⣀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠂⠰⡇⢸⣿⣿⣿⣿⣿⠁⠀⠀⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⡀⣾⠋⠁⠴⣿⡿⣧⣄⠀⠀⣀⠀⠀⣿⣧⠄⠰⣗⠩⣿⣿⣿⡏⡿⠀⢠⢿⣴⡿⣿⣿⣿⣿⣏⣳⡄
⠀⠀⠀⠙⣿⣀⡧⣤⣈⡈⢲⣻⣶⣤⣬⣄⡀⠉⢿⠀⢠⣧⠐⣿⢿⠟⢸⡇⠀⣸⡾⠋⢁⣿⣿⣿⠟⢃⡹⠆
⠀⠀⠀⣀⣀⡛⢷⠶⣿⣿⣽⡷⠟⠋⠻⠟⡟⠿⢾⣷⣸⣿⡎⣿⡆⢰⣿⡀⠊⠁⢤⣴⣶⡿⠿⠥⠶⠋⠁⠀⣀⣀
⠀⡞⢰⡟⢉⣻⣾⣶⣦⣤⣤⣤⣤⣦⣶⠾⠿⠿⢷⣾⣿⣿⣿⣿⣿⣿⣷⠶⣶⣶⣶⣴⣦⢀⡀⢘⣶⣿⣿⣿⣿⣿⣾
⠀⠀⣼⣀⣀⣽⣹⣿⣿⣿⠟⠛⠉⢹⠦⢤⣤⣠⡴⢻⣿⣿⣽⣿⣿⡈⠁⠾⣿⣿⣿⣿⣿⡆⠙⢾⣓⣶⣶⣿⣿⣿⡿⠁
⠀⠀⠙⢿⣿⣿⣿⣻⣿⣯⣀⠀⠠⢦⣿⣿⣿⣿⡧⠘⣿⠏⠈⢿⣽⣷⣤⣶⣿⣿⣿⣿⣏⣀⣠⣀⣹⣿⡿⣿⠿⠟⠁
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠉⠙⠲⠾⠿⠿⠿⠋⡤⠙⠁⠀⠀⢈⡽⣿⣶⣦⡉⢦⠀⡀⠈⣹⣯⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣼⣤⣶⣤⣿⣾⣿⣿⣿⡧⠀⠳⣤⣚⣭⣼⣿⣯⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⣿⠿⠛⠋⠁⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢻⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⢿⣾⣿⠘⣷⠀⢠⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡵⣦⣷⣼⡧⣸⡀⢸⠋⣹⣿⠆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡴⠊⠀⠀⠀⠈⠙⢧⡼⠛⣠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⡇⠀⠀⠀⠀⠀⠀⠈⣷⠞⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢀⣿⠀⣴⢿⡶⣠⣤⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣜⠿⣸⣧⠴⠞⠛⠲⢧⣴⠶⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⡛⣤⠛⠀⠀⠀⠀⠀⠀⠸⠷⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣠⣾⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠛⠉⠉⠙⢻⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣰⠋⢸⠁⠀⠀⠀⠀⠀⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠃⢀⡿⠀⠀⠀⠀⢀⣴⢻⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⠀⠀⠾⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢰⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⡇⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢸⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⢀⣿⠀⣼⠇⠀⠀⠀⠀⠀⠀⢁⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⢰⡧⣼⣿⡎⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⣿⣿⣷⡀⠀⠀⠀⠀⢸⡇⣿⠋⠀⣿⢀⣿⠀⠀⠀⠀⢀⣠⠾⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣾⣿⣷⣆⠀⠈⠟⠛⠻⢷⣄⠀⠀⢸⣧⣿⠀⢰⡟⢸⡏⠀⢀⣠⠔⢛⣵⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⣿⡿⢷⣄⠀⠀⠀⠀⠙⢶⣄⠈⢿⣿⣆⣸⠇⣾⢃⡴⠋⢁⣤⠾⠛⡟⠃⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠛⠲⠶⠶⣤⣤⣉⣻⣶⣿⣿⣿⣼⣿⣏⣠⣶⣏⣤⣤⡀⠀⠀⠀⣾⣿⣿⣿⡧
⠀⠀⠀⠀⠠⣠⣤⣤⣤⣀⠀⣀⣀⣀⣀⣀⣤⡴⠶⠊⠙⣉⣭⠿⠟⠛⢋⣹⠿⣿⣿⣿⠿⣿⣿⡿⠛⢿⣍⢻⣏⠛⠛⠙⠿⠿⠟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠋⠉⠉⠉⠉⠀⠀⠀⣠⡴⠛⠁⢀⡠⠒⠈⠁⠀⣶⢛⣿⣿⠰⢦⣤⠀⡷⠀⠈⠳⡘⢦⡀
⠀⠀⠀⠀⠀⠠⠝⠉⠁⠀⠀⣀⠀⣀⠀⣀⣴⠞⠁⠀⢀⡴⠁⠀⠀⠒⠋⣼⠏⠘⣿⣿⠀⠄⣀⣀⡧⠀⠀⠀⢱⠈⠻⣶⣶⣶⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⣿⣿⡏⠀⠀⠀⣠⡞⠁⡀⢀⣤⣾⣿⣿⣖⠀⣿⡽⢣⣾⣿⣿⣷⣄⠀⠀⢸⡄⠀⠈⢻⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠛⡿⠛⠉⢈⣱⣿⣿⣿⣷⣈⠍⠘⣿⣿⣿⣿⡿⢦⡽⠀⠩⠿⣿⣿⣿⡿⠔⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠿⠁⠈⠂⠈⠉⠉⠛⠉⠁⠀⠀⠈⠀⠀⢠⠅⠘⠓⠺⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡟⢛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢃⠴⠱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢨⡄⠘⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⢠⡄⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡻⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡰⣦⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡛⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡶⢿⣷⣦⡀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢁⡼⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⡿⠋⠀⠀⠀⠀⠹⠿⣿⣿⡀⠀⠀⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣠⣤⠋⠀⠒⠘⠻⠀⠀⢤⣤⣀⡙⠃⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠋⠄⠀⠀⠀⠀⠀⠃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠑⠊⠀⠀⠐⠶⠾⠛⠓⢦⡀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡾⠓⠀⠀⠀⠀⢀⠀⣤⠾⢆⣙⣿⣦⣶⣤⡎⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢈⣷⠀⠀⠀⠀⢀⡀⡾⠃⠀⠀⠀⠀⠀⢶⣴⣶⣼⡷⠻⣿⣩⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⣿⣄⣶⣀⣴⣯⠀⠀⠀⠀⠀⠐⠀⣤⡄⢸⣿⣼⣿⡾⣿⣿⣿⣿⣿⣿⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣤⣴⣶⣼⣿⣿⣿⠻⠉⠀⠆⠀⠀⠀⠀⠀⠈⠙⠛⠟⠉⠉⠉⠉⣷⣝⣿⣿⣿⡅
⠀⠀⠈⣤⠀⡰⣇⠀⠁⠴⠤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⡿⠟⢿⠈⠉⠀⠀⠁⠒⠒⠢⠦⢤⠤⣤⡀⠀⠀⠀⠀⠀⠛⠜⠉⠻⠋⠛⠁⠒
⠀⠀⠀⡈⠈⠀⢸⣠⣦⣤⣤⣴⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⡜⠀⣼⡇⠱⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠱⠶⠶⠲⡦⠠⠶⠆⠙⠁
⠀⠀⠀⣠⡄⢀⣸⣿⣍⣼⣿⣿⠟⢁⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⣾⠃⠀⡳⠀⠀⢠⣄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢞⠀⠈⣾⣿⣿⣿⣿⣿⣿⣿⣶⠟⢀⡀⠰⠃⠀⠄⠀⢡⣄⢰⣾⠁⠀⢸⣟⠀⠀⠁⠀⠀⠀⢙⠛⢻⣷⡶⠶⣄⠀⡀⠲⡴⢋⡷⠂
⠁⢘⠃⠀⠹⢿⠿⣿⣿⠤⠄⢀⠀⠀⡘⠃⠀⠀⣁⣴⣶⣿⣿⠻⣿⡀⠀⠸⡟⠀⠀⢀⠄⢀⣴⣿⣷⣼⡿⠻⢿⣅⠀⠀⠤⠀⢼⣿⣶⣿⣆⠉
⠀⡈⠳⡀⠀⠀⠿⠏⠁⣴⡄⠁⠀⠀⠀⠀⢤⣖⣙⣿⣿⣿⣯⣿⣿⡇⠀⢠⡇⠀⠀⣯⠐⢀⢿⣿⣿⣿⣧⡤⠈⢹⡇⠙⠀⠀⠠⠁⡭⠿⣿⣇⡘⢓⡄
⠀⠃⢰⠀⠈⠋⠀⠀⠈⠀⠀⠀⠠⣎⠀⠀⠀⢻⣿⣿⣾⣿⣿⣿⣿⠁⠀⠀⠁⠀⠠⡏⠁⠈⣤⡙⣿⣿⣿⣧⡀⢻⣟⠻⠆⠀⠀⠈⠀⠰⠏⣿⣷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣴⣆⣼⣿⣿⣏⣿⣿⣾⠟⠀⠀⠀⠀⠀⠀⠸⠇⠀⠻⣿⣿⣿⣿⣿⢷⣾⠿⣿⣄⡀⠀⠀⠀⠀⠀⠙⢇⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠻⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡶⠀⣍⠹⠋⣏⣥⠾⠉⠂⠈⠀⠉⠂⠶⢶⠆⠀⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡁⠈⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡻⠅⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠛⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢠⠇⡄⠀⠀⠀⠀⠀⣴⡾⠿⢶⣄
⠀⠀⠀⠀⠀⠀⢰⡎⠉⠈⠙⢿⡦⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠀⠀⢿⡄⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠸⢇⡀⠀⢀⣼⡯⠀⠀⠀⠀⠀⠆⠀⣲⣶⠟⠻⣧⠀⢘⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠻⠟⠃⢀⣴⠶⣤⡀⢰⠀⣽⢿⣤⣠⠟⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢳⣼⣇⠀⠈⡷⠘⡀⢹⢸⡏⠁⠀⠀⠀⠀⠀⠀⣠⣴⣴⣦⡄
⠀⣀⣤⣤⣤⣄⠀⠰⡏⠀⠈⢓⡀⠀⠙⠻⣿⠟⠀⡇⢸⡾⠀⢀⠎⠀⠀⠀⠀⢸⡏⠁⣀⣼⠇
⠸⣿⡿⠿⠟⠫⢤⡀⠙⠦⢤⣌⠷⠀⠀⠀⠀⠀⠀⠣⢸⠃⡠⠃⠀⠀⠀⠀⠀⡼⠛⠛⠉⠁⠀⠀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀⠀⠀⠈⠳⡄⠀⠀⢠⡐⣾⣤⣴⠃⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⣀⡠⠌⠻⠷⢦⣍⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⢄⡈⠐⢄⣀⣘⡿⣻⣇⣴⣯⣤⣤⠀⠐⠒⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⢀⡼⠛⠛⣻⡶⠒⠒⠂⠈⠁⠉⠉⠙⠻⣷⣿⣿⠟⠁⠀⠀⠈⠑⢢⣴⠶⢿⣶⣤
⠀⠀⠀⠀⢺⣁⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⣴⣤⣼⡉⠛⠐⠲⠀⠀⠀⠀⠸⣟⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⢀⣀
⠀⠀⠄⠂⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡴⠶⠶⠶⣤⠛⠛⠛⠛⢷⠶⢶⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠞⠋⠉⠉⡀⠀⠀⠀⣧⠀⠀⠀⠀⡀⠀⠀⢨⡏⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠙⠆⠀⠶⠊⠱⣆⢀⠤⠙⢤⣀⣀⣰⣏⠀⠀⣠⠀⠀⠀⠸⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⠀⢀⠉⠁⠘⢣⣤⡟⠀⠀⢀⡾⠀⠀⣽⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠉⠳⣄⣀⡴⠶⠹⢦⣀⣠⠾⠷⣄⣀⡠⠾⠀⣀⣼⠃⠀⠀⣀⣠⠟⠀⠀⣴⠃⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⢀⠉⠀⠀⠀⠘⠁⠀⠀⠀⠈⠏⠀⠀⡸⠉⠉⠓⣤⠚⠈⢙⣤⣤⠞⠛⣆⣠⡞⢿⡄
⠀⠀⠀⠀⠀⢀⣾⠀⠙⠶⣶⠖⠋⠲⣄⣠⠔⢿⠀⢀⣤⣶⡎⠀⠙⠛⢱⣶⣦⣄⡀⠄⢠⡞⠀⢠⡄⠀⢙⠃⠀⠀⣿
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠠⠀⠀⣰⠏⠁⠀⣰⠗⣈⠉⠺⣦⠀⠼⠀⣾⠟⢨⠉⣡⣄⠀⠙⢶⠀⢰⠟⠛⢷⡞⠛⠻⣧
⠀⠀⠀⠀⠀⡟⠀⠀⠀⣠⠆⣀⢸⠇⠀⠀⣞⣁⡾⠉⣁⣀⣭⠀⠴⠀⣼⡇⣀⡀⣹⣼⣷⣄⡸⠶⢶⡄⠀⣼⠁⠀⠀⣹⡄
⠀⠀⠀⠀⣰⠟⠒⠠⡼⠃⠀⠈⡇⠀⢠⡼⠇⣠⡀⡀⠉⣽⣧⠀⣤⠄⢉⡉⠛⠋⠉⣧⣤⣿⣄⠀⠈⣻⠆⠙⢧⣴⠟⠛⢷⡀
⠀⠀⠀⢀⣿⠀⠀⠀⠁⠀⠀⢨⠀⢀⡜⠂⠀⣨⠁⣈⣉⣹⣯⠀⠀⠀⣿⠷⢶⣿⠀⢿⡀⢶⣿⡄⠀⣿⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⢸⠟⠀⠀⠀⣀⠛⠛⡇⠀⡘⠣⠄⠘⡧⢈⠛⠁⠹⡇⠠⠆⠀⢨⣶⣤⣀⠀⢾⡟⠁⢼⣿⢀⣟⠀⣴⠎⣶⡀⠀⣸⣿
⠀⠀⠀⢸⡦⢤⣀⡴⠋⠀⠀⡐⠶⡉⢠⡄⠰⣆⣈⠉⠑⠲⡶⠀⠐⠀⢺⣄⡀⠛⠀⢸⡟⠳⣤⣹⡆⠉⢿⠁⠀⠈⣿⠛⠙⣿⡀
⠀⠀⠀⣿⠀⠀⠀⠱⡄⠀⠀⡷⠀⢟⡠⠤⠀⢹⡉⠉⠁⠘⣿⠀⠛⠀⢻⣤⠁⠾⠇⢸⡟⠓⣤⣾⡇⠀⠹⣄⢀⣾⠋⠀⠀⣻⣷
⠀⠀⠀⣿⠀⠀⠀⣠⠋⠀⠉⣄⠀⢈⣠⠄⠀⢠⣝⠉⠁⠀⣿⡃⠙⠀⠙⠉⠛⠶⠆⠸⣿⣷⣤⣿⡀⠀⠀⣿⠉⢻⡄⠀⠀⣿⣿
⠀⠀⠀⣶⠀⠀⣠⠟⠀⠀⠀⠈⡄⠘⢧⡤⠂⠸⣟⢳⣄⠁⠸⢟⣐⣦⠰⢿⡦⣤⣶⣰⣷⣌⠙⣿⡇⠀⠀⡧⠀⠀⣷⣄⣠⣿⠃
⠀⠀⠀⡹⠒⠂⠛⢤⠀⠀⠀⣤⠘⠂⠈⣀⠖⠲⣉⠞⠛⢿⣶⣟⠉⢀⠔⠺⣧⠀⢠⣿⣯⡙⠷⣼⡇⠀⠀⢳⣀⠾⠁⠀⠉⣿⡇
⠀⠀⠀⣷⠀⠀⠀⠀⡷⠤⢤⠋⠀⠸⠄⠽⠀⢸⠃⢰⣦⠀⢠⣷⡄⠠⠄⠘⡄⣰⣸⣿⠋⣡⣶⣽⣿⠀⣰⠃⢸⡀⠀⠀⠀⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⣃⠀⠀⢠⠀⠿⠿⠂⢠⣧⡀⠀⣩⡷⠀⠒⠈⡿⠻⠻⣿⣉⡙⢿⣏⣿⠷⣏⠀⢸⡇⠀⠀⢠⣿⠟
⠀⠀⠀⠙⠀⠤⠤⠴⢇⡀⠀⠛⠀⠀⢸⠀⠀⣦⡀⠀⣘⣃⡀⠀⣡⠤⠄⢸⣿⠉⠀⣽⣉⣉⣧⣈⣿⠀⣿⠀⢸⡷⠶⠦⢾⣏
⠀⠀⠀⠀⡇⠀⠀⠀⠈⠱⣦⠈⠣⡀⢈⠀⠀⢹⣄⠀⠉⢙⡃⠀⠻⡆⠄⢸⣿⢉⠀⣹⣟⠉⣉⣹⡇⠀⠻⣀⡸⠁⠀⠀⢸⣯
⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡇⠀⠀⠱⡞⠀⠀⠈⣯⠀⠀⣾⠛⠀⢸⡧⠄⢸⡿⠋⠀⢸⡟⢀⡀⣸⡇⠀⣤⠉⢰⡄⠀⠀⣸⠏
⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⠀⠀⠀⣼⠀⠀⠀⢻⣄⠀⢠⡿⠂⠘⠣⠄⢺⡷⠂⠀⣸⣷⠌⢸⣿⠁⢠⡟⠀⢸⣧⣄⣸⡇
⠀⠀⠀⠀⠀⢱⡌⠛⠉⠻⣄⠀⠀⠀⢈⡄⠀⠀⠈⣥⠀⠀⣿⡇⠘⣇⡀⠹⣷⡄⠀⣛⡁⣸⣸⡟⠀⣸⡀⢠⡞⠁⠈⣿
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠘⢧⠀⣰⠎⠙⣇⠀⠀⣿⠀⠐⣿⣄⣤⣿⡛⠻⣇⡀⠐⣿⣁⢸⣿⠀⢠⡍⠻⡞⠁⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⣀⣀⠛⠃⠀⠀⠹⡇⠀⠸⠇⠀⠻⠋⠠⣞⡁⢰⣏⠀⠰⣿⣿⣶⡿⠀⡸⠃⠀⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⡶⠊⠉⠁⠀⢱⡄⠀⠀⢳⠀⢰⡆⠀⢶⡟⠁⣬⢤⣼⣿⠃⢠⣿⠿⣿⠇⠸⠇⠀⣤⠛⠂⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠈⢷⡀⠀⢸⠀⢨⡄⠀⣾⡷⢰⡇⠠⣾⡻⠆⢹⣿⠀⢸⣶⠶⠦⡾⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠈⠷⢠⡎⠀⣼⠀⠀⢹⣶⣞⣥⠞⣟⠁⠄⣿⠋⠀⣸⡅⠀⠰⠇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⣿⠀⢠⡍⠀⠀⣿⡷⠟⠱⣾⡿⠳⣶⡿⠀⠀⡻⠁⣀⡐⢇⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠿⠀⠰⣿⠁⠀⢠⣿⡄⢳⡿⣧⣴⣛⡿⠀⠀⢈⡷⠟⠉⠀⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠻⣧⣴⣿⣷⣿⠟⣹⡿⢿⡟⠁⠀⠀⣼⠃⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣯⠀⢸⠇⠀⣷⢹⣧⢸⡇⠀⠀⣘⠃⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⢀⠟⠀⢸⠇⠀⣫⣾⣿⠾⣷⠶⠟⠙⣧⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠘⢧⣀⡜⠀⠀⣟⣹⣿⣾⡇⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠇⠀⠠⣿⣿⣟⣹⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣟⣻⠏⣿⠷⣦⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠐⣏⣹⣼⡿⠀⠄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠃⠀⢘⣏⣹⣿⡇⢰⡆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠰⢾⣉⠿⣏⠁⢰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇⠀⣼⣿⡿⢿⠀⢸⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡇⠀⢹⢹⣧⣾⠀⠸⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⢻⡀⠀⣿⠾⣧⠾⢀⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠈⠀⠀⣿⠀⠠⡖⠈⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⣿⣼⣦⡟⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢧⣠⠀⢹⣿⠃⣤⠀⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣠⡇⠀⠀⣿⠘⣿⣧⠀⣳⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠉⡇⠀⠀⣿⣦⣄⣿⠐⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠀⣿⠀⠀⣿⣟⣁⣶⠀⡇⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⡿⠀⠀⣻⣼⣏⣿⠐⡇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢻⠀⠀⠙⢾⣏⣵⣾⡇⠀⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃

⠀⠀⠀⢶⡲⣢⣶⣢⢖⡀⣤⣤⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠴⢒⠙⠛⠶⣤⡀
⠀⠀⠀⠀⠁⠁⢞⣟⣾⠷⠊⢇⠠⣦⡼⣟⠋⣀⠀⠀⠀⢀⡤⠤⠄⣀⣀⢆⠘⠁⠀⠀⡐⠁⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⢷⡦⠙⢶⢟⡇⢈⢟⡄⠀⢠⡏⠀⠀⠀⠀⠀⠀⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⣏⣽⢺⠃⠋⠳⢺⡿⢠⡄⣼⠀⠀⠀⠊⠀⢠⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⢰⣣⠶⢛⠶⡁⢻⠀⠀⠀⠀⢀⠃⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣾⡧⠘⢦⠀⠰⠀⡆⠈⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠃⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢫⣥⣤⣄⠈⠀⠀⠄⠀⠀⠀⠘⣿⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣫⡭⢤⣀⣯⡽⣶⠄⢠⡞⣿⡚⠓⢢
⠀⠀⠀⠀⠀⠀⠀⢴⡋⠀⠀⠀⠙⠶⣍⠲⢄⣀⣄⠀⠀⠀⠹⣶⠠⠁⠀⠀⠀⠀⠀⢀⠞⣥⢖⡫⠕⠊⢉⣀⠔⠂⠘⠻⣧⣿⡻⠉⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠘⠷⡶⠾⠃⣦⠀⠈⠛⠶⣍⡳⢿⡶⣄⠀⡏⡃⠀⠀⠀⠀⠀⠀⣼⢐⡵⡯⠔⠚⠋⠉⠀⠀⠀⠀⠀⠙⢾⠅⠀⠰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠈⠉⠉⠻⢶⡻⢷⣇⠀⠀⠀⠀⢠⠞⢡⢈⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⡜⡀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣽⡆⠀⠀⡰⡃⠡⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠀⡥⢔⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⡇⠃⡀⠀⢿
⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠼⣛⠋⠙⢛⣻⣷⣄⠢⢄⣤⣀⡀⠀⣿⡀⠀⠠⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢷⡴⠃⠀⡼
⠀⠀⠀⠀⢀⡶⠔⢋⣩⣤⠤⠤⠖⠒⠉⠀⠀⠈⠳⣄⡙⠋⠻⠃⢿⣠⠔⣡⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⣀⠜⠁
⠀⠀⠀⠀⡾⠡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠠⣄⠀⠻⡏⠀⣼
⠀⠀⠀⠀⣇⠀⢀⣴⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⢣⢠⠃
⠀⠀⠀⠀⢹⡄⠀⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠸⢸
⠀⠀⠀⠀⠈⢷⢸⣏⡈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠘⡆
⠀⠀⠀⠀⠀⢸⢸⣿⡅⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣿⠀⠀⣠⠔⡋⠽⠦⠈⠙⢿⡉⢙⠻⢖⠦⣄⡀
⠀⠀⠀⠀⣠⠏⢼⣿⠁⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣸⠀⡴⢡⠊⠐⠉⠑⠀⠀⠀⠘⢦⡀⠀⠙⢆⠙⠲⡀
⠀⠀⠀⠀⠙⢻⣾⡞⠒⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡔⣁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠑⢄⠘⣄
⠀⠀⠀⠀⠀⡄⣿⣇⣼⡵⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠃⠈⠇
⠀⠀⠀⠀⠀⠣⢽⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⢋⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⠀⠁⢸⠛⡌⠤⠰⣖⡒⠒⠲⢀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡐⠈⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠆⣜⠇⠀⠀⠀⠀⢣⡻⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⡀⠄⠘⣧⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⠌⠀⠀⠈⠻⢆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠤⠶⠤⢄⠢⠄
⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠹⠐⠀⠁⠀⠀⠙⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠊⠁⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠑⠿⠷⠔⠀⠀⠀⠀⠀⠀⠀⠐⡸⠟⢛⣁⠀⢀⠀⢀⡀⠂⠀⠀⠀⢰⣒⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠻⠛⠛⠿⠛⠛⠛⢛⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠙⢷⠶⠦⠰⠶⠾⠿⠿⢿⣏⠉⠉
⠀⠀⠀⠀⠀⠀⠁⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⣶⣴⢶⣾⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠀⠀⠈⠁⠒⠒⠤⣄⣀⣴⣦⣀⡤⠔⠒⠂⠉⠀⠀⠠⠀⠀⠀⣀⡀
⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⡠⠄⠰⠒⢽⣿⣿⣿⣿⠕⠒⠀⠤⡀⠀⠀⠀⠀⠀⢰⣿⣿⣄
⠀⠀⠀⠈⠻⣿⣿⣷⠀⠀⠀⣷⡆⠀⠀⠐⠀⡼⣿⣿⣿⡿⡆⠐⠀⠀⠀⢶⡇⠀⠀⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⢿⣧⡀⠀⠀⠀⢠⣶⢠⠎⠀⢿⣿⣿⡇⠘⢦⢰⣦⠀⠀⠀⠀⣠⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠈⠀⠈⣠⠋⠀⠀⢸⣿⣿⠐⠀⠈⢣⡈⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠶⡤⠊⣌⠔⠃⠈⢸⣿⣿⠘⠀⠓⢌⡌⠢⡴⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠁⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠉⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⢀⠀⠀⠀⢀⠆⢆⠀⠀⠀⢀⠀⠈⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠁⠀⠀⠁⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣀⡤⣴⡦⢾⣿⣷⠾⠿⢿⡻⣿⣿⣿⣶⡲⢶⢦⣄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠳⠿⣿⡷⣾⣿⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⡀⣀⣰⣶⡶⣷⣦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠶⠶⠟⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⠀⢀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⠉⠙⠻⢿⣿⣿⣿⣿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣾⢟⠉⢀⣠⠴⢒⣈⣭⣤⣶⣶⣶⣶⣶⣦⣬⣉⡛⠿⢷⣦⣀⣽⣿⣿⣿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⣀⣼⣿⣿⣧⣠⡾⣋⡴⠞⢹⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢷⣦⣉⠻⣿⣿⣿⣿⣾⣫⠚⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢂⢁⣴⣿⣿⣽⣿⠟⣡⣾⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⠀⢸⢈⠀⠒⠉⠻⣷⣌⠻⡿⠿⠏⠂⠁⠀⣁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⣿⣿⠿⣳⣾⣷⠈⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢑⠀⢀⠖⠀⠀⠀⠠⠀⢉⡿⢳⡌⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠣⣝⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠘⠛⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠂⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⣠⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠿⢋⣴⣷⠈⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⢀⣾⣧⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣳⠖⠿⠵⠿⠀⣿⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣽⠉⠀⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⡃⢀⠀⠀⠀⠀⠉⣠⣴⣶⣤⣄⣀⠀⠀⠀⠀⢀⡟⠀⠀⢀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⠙⠿⠉⡀⣀⠴⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⣾⠇⠀⢀⣾⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⡟⠀⣠⣿⣿⣿⣿⠿⠛⠋⠉⠀⠰⣿⣷⣶⣿⣿⣿⣿⣦⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣝⣾⣿⡟⠋⢠⣾⣿⡿⣻⣽⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠋⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⡠⠒⠺⠿⠃⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣒⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠙⠛⠿⠿⠿⠇⠀⣧⣄⠈⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢹⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉⠙⢷⡀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⡄⠀⠀⠀⠀⣷⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⡇⠀⠀⠀⢠⡏⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⠁⠀⠀⣠⠟⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⠀⠀⣰⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⠀⢠⡇⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⢧⣦⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣏⠀⠀⠚⠉⠙⠛⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⢀⣠⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⡀⠀⣴⡆⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⢿⣿⣿⠇⠀⠸⠿

⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣶⢶⣦⡀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡶⠶⣤⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⣾⣿⣿⣿⣹⢿⣿⡆⠀⠀⠀⠀⢀⣿⣿⣆⠁⣀⢈⣷⡄⠀⠀⠐⢽⣷⣦⣴⣤⣴⣴⠃
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢿⣟⣿⣿⣿⣶⡋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡦⢿⡇⠀⠀⠀⠀⢹⣿⣿⠟⠉
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠘⠟⠓⠀⠁⠹⣷⡀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡃⠀⣸⡇⠀⠀⢰⣶⣿⣿⣿⣦⡀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢀⠙⢷⡀⠀⠚⠋⣽⠃⠀⢰⣿⡇⢸⣿⠁⠀⠀⠸⣿⣿⣿⣫⣿⠃
⠀⠀⢰⠁⠀⠀⠐⠀⠀⠀⠀⠀⠉⠙⢻⣧⡀⠳⠀⠀⠀⠁⠀⠀⠀⣿⣧⣾⡏⠀⠀⠀⢀⡱⢺⣿⣿⡏
⠀⠀⠸⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢹⡽⡿⠀⠀⠀⠀⠀⡇⣻⣿⣿⠁
⠀⠀⡇⠀⠀⠉⠳⣦⣤⡤⠤⠀⠀⠀⠚⣿⠿⣿⡄⣆⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢸⣷⣿⣿⡏
⠀⠀⡇⠀⢀⣀⢰⣄⣿⣧⣄⣠⣄⡀⠀⠈⢧⢹⣧⠘⢆⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠸⢻⣿⣿⠁
⠀⠀⠃⢠⡞⣠⣿⣿⣿⣿⢿⢭⣬⣛⠓⢄⠈⢆⢻⡄⠈⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⡆⣾⣿⠇⠀⢀⣠⣄⡀
⠀⠀⢸⣸⣅⢻⣿⣿⡋⠀⠀⠀⠙⠻⠇⢸⡆⠀⡄⢿⡀⢀⠀⠀⠀⠀⠀⢿⡆⠀⠀⢠⣰⣿⡟⠀⠀⣾⡟⢿⠃
⠀⠀⠈⣟⠻⣤⣨⢿⠿⢂⢀⡀⠀⣀⣤⣾⡇⠀⠠⠸⣷⣾⠀⠀⠀⠀⠀⠘⣿⡀⢀⣾⣿⡿⠀⠀⠀⣼⠇⠘⠂
⠀⠀⠀⣽⡀⠀⠉⠛⠒⢶⣾⣿⠆⢹⡋⢹⡇⠀⠀⠂⣿⣿⡀⠀⠀⠀⠀⠀⠸⣷⣾⣿⡟⠀⠀⠀⣾⠏
⠀⠀⠀⢯⢳⠀⠀⠀⠀⠸⣿⡇⠀⢸⣇⣿⡇⠀⠀⢸⣸⣿⡇⠀⠀⠀⠀⠀⠀⡻⣿⡟⠀⠀⠀⢠⡇
⠀⠀⠀⠈⡌⡆⠀⠀⠀⢸⣿⡇⠀⢸⣿⡿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢀⣶⣿⣷⣄⠀⠀⣇⣷⠀⠆⠁
⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⣿⠇⠀⣼⣿⠃⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⣰⣿⠃⠘⣿⣿⡀⢰⣿⣿⢰
⠀⠀⠀⠀⠀⠹⣇⠀⢀⡀⢸⠀⣸⣿⠏⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⢠⣿⠃⠀⠀⢹⣿⡇⢸⣿⡏⠚⢀
⠀⠀⠀⠀⠀⠀⢻⣷⣼⡇⣸⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⣾⠇⠀⠀⠀⢸⣿⣧⣿⠟⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠀⠀⠀⠀⠘⠛⠿⠇⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⣀⠀⠀⢀⠤⢤⠀⠀⡤⠤⡀⠀⠀⡀⠛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⣹⡇⣿⢎⠀⠀⣱⣿⢸⣏⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣄⠀⣑⣦⣀⣼⡏⣬⠿⣂⣒⠿⣥⢻⣧⣀⣴⣊⠀⣠⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣾⢙⢓⣻⠛⠳⣿⣿⣿⣿⣿⣿⠞⠛⣟⡚⡋⣷⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣴⡿⠘⣝⡧⠬⢼⡐⣿⣿⣿⣿⣋⡧⠧⢼⣫⠃⢿⣦⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣨⣿⣿⢸⡿⠿⣶⠀⣉⢷⣿⣿⡿⣉⠀⣶⠿⢿⡇⣿⣿⣍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⢿⣯⡴⢉⡝⢣⡼⢾⣤⡌⢡⣤⡷⠧⡘⢫⡉⢦⣽⡿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠇⢾⢲⡓⠞⠲⣌⠛⠛⣿⡅⢈⣿⠛⠛⣡⠖⠲⠚⡖⡷⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⠷⠆⠀⣑⠉⢿⣾⣿⠁⠈⣿⣷⡿⠉⣋⠀⠸⠾⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⣿⣵⣬⣉⠛⠀⠀⠛⣉⣥⣮⣿⣿⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠉⣿⢋⣽⣥⣮⣯⠙⣿⠉⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠿⠾⠗⠓⠘⠺⠇⠿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡴⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⢢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠀⠀⠙⠃⠀⠀⠘⠋⠀⠀⠸⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⢠⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠒⢐⣶⣶⡿⣷⣒⡆⠐⠂⠀⠀⠐⠀⠀⠐⠒⠂
⠀⠀⠑⢠⡄⠀⠼⡇⢸⡷⠀⠀⡄⠈⠀⠀⢀⠀⡼⠀⠀⠀⠀⠀⠐⠀⠘⠉⢿⠈⠉⠀⠂⠀⠀⠀⠀⠀⡸⠦⣤⠴⢇
⠀⠀⠀⠀⠐⠒⠓⠉⠋⠚⠃⠂⠀⠀⠀⠀⠘⣄⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠈⠀⠐⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠀⢀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠐⣀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠁⢂⣰⠉⢧⣀⠈⠀⠀⠀⠀⠐⢄⡁⠶⢈⡠⠄⠀⠀⠠⠀⠀⠀⠀⢸⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⢠⠘⠧⡙⠂⢛⡠⠃⡠⠀⠀⠀⠀⡀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠰⣞⣆⡈⢿⠁⣰⣳⠆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠙⠒⠀⠈⠀⠐⠊⠁⠀⠀⠀⠀⠀⠑⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠂⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⣀⡀⢀⢠⢠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢠⠠⢤
⠀⠀⠀⠀⠈⠀⡿⠩⡇⠈⠀⠀⠀⠀⠀⠀⠐⢀⡠⠧⠀⠆⠀⠀⠀⠀⠈⡇⠀⠀⣀⣀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡆⢸
⠀⠀⠀⠀⠀⢠⡚⠚⢢⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠬⠁⠀⠀⠀⠀⠀⢸⡄⣐⣿⣿⢂⢀⡟⠀⠀⠀⠀⠀⢸⡧⡞
⠀⠀⠀⠀⠀⠘⠄⠀⠜⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⢂⠀⠀⠀⠈⠸⣇⠈⠛⠟⠁⣰⠗⠃⠀⠀⠀⠀⢸⡓⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⡇⠀⠀⠀⠀⠈⠀⠘⠂⠀⠀⠐⠃⠀⠁⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂
⠀⠀⠀⠀⠢⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠄⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⢀⠯⠼⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠒⣋⠉⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠊⠇⠀⠀⠀⢤⡀⠐⠒⠀⣠⠄
⠀⠀⠀⠀⠎⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠠⠀⢀⠘⠶⠃⡀⠀⡄⠀⠀⠀⠀⠀⠚⠑⠂⠱⠂⠀⠀⠐⠐⠀⠋⢈⠁⠀⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⣈⠁⢁⠈⢠⠇⠀⠀⠀⠀⠀⢠⠀⠀⠰⠈⠀⠀⠀⡀⢂⠀⠊⠀⠂
⠀⠀⠀⠀⠀⠦⢤⠤⢚⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠰⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡁
⠀⠀⠀⠀⠀⠈⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⢴⡗⣚⣟⠒⡿⢰⠯⣄⡖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢸⣪⢃⣡⣿⣅⣼⣷⡯⣰⠾⢉⣱⢮⣲⢌⡀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣟⣤⣾⠟⣿⡾⣦⣤⣊⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⣾⣿⣿⡿⠿⣛⣿⣿⣿⣿⣿⡛⠛⢿⣿⣿⣿⣿⣶⣿⣷⠿⣓⣠⢴⣋⡡⢎⠍⡠⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣵⣿⣿⣿⡟⠕⣫⣾⣿⠟⢛⣋⣛⠻⢿⣦⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣋⣴⢗⡯⣠⠎
⠀⠀⠀⠀⢀⡴⡴⣿⣿⣿⣿⡟⠀⢰⣿⡿⢁⣾⢛⡭⣍⠳⡌⢻⣷⠀⢹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⠾⠛⠋
⠀⠀⠀⠀⠀⢪⡴⣿⠟⣿⣿⡇⠀⣿⣿⡇⢸⡇⢿⠁⡠⢇⡟⢸⣿⠀⢸⣿⣿⡇⢸⣿⣏⣙⣻⠍
⠀⠀⠀⢠⣾⠿⢓⢹⣄⣹⣿⡇⠀⢻⣿⣧⡈⢿⣮⣙⣛⣋⣴⡿⠋⢀⣾⣿⣿⡷⣿⣵⡒⠦⡁⠁
⠀⠀⢆⡀⠙⠋⠉⠛⠿⢿⣯⣍⡀⠚⠿⢿⣷⣤⣉⠛⠛⠛⢉⣠⣴⣿⡿⢟⣭⣖⠶⡶⢌⢳⠌⠂
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠊⡁⠀⠚⠀⢀⢀⢈⢍⢉⡉⣩⡭⣍⢭⢤⣴⡛⢛⠻⡳⡌⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠘⠀⠈⠁⠙⢆⠙⠈⠘⠀⢫⠙⠆⠳⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣭⣬⣭⣯⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠦⠀⠀⠀⠀⠄⢤⡄⠀⠀⠀⠀⠀⣴⡒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠈⢁⠀⠄⡹⢃⠀⡀⠀⠈⢻⠁⠀⣶⡾⠛⠀⠺⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⡴⠒⠀⠠⡟⠀⠀⠁⢀⣂⣡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣿⣿⣻⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣦⡶⣔⣿⠋
⠀⠀⠀⠀⠀⠀⠐⠂⢉⠿⢛⠙⡟⠉⠁⠀⠀⠀⠀⠀⠀⢠⢶⣾⣿⣿⢿⠺⠃⡿⣽⠜⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⠩⡅⠁⡶⢃⡜⡀⢓⢱⡾⣻⠂⠀⠀⠀⠀⢀⣤⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡀⠀⠀⠀⢻⣌⢳⡇⠠⣠⣎⣄⣁⣾⣿⣿⠏⠀⠀⠀⣠⣾⣽⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠈⠀⠀⡀⠀⠙⢧⣾⠟⢻⡿⠛⢿⣟⡙⣿⠀⠀⠀⠀⠉⠛⠿⠙⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⢲⠃⠀⠀⠂⠀⠙⠲⡀⠰⠹⣧⢼⡈⡓⡯⢌⣿⣶
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⢘⢷⡄⠙⡀⢠⢠⠀⠀⠂⣸⠏⠀⡇⢻⣦⡟⢾⡛⣩⠁⣀⣤⠴⣶⡄⣀⣠⡤⣤
⠀⠀⠀⠀⠀⠀⠴⣾⠏⠍⡳⠥⣀⠐⢆⡋⠘⣞⠀⣠⠞⠉⠀⠀⢻⠈⣿⡷⣶⣿⣷⣾⣿⢎⡇⠚⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢱⢒⠁⠐⣯⠃⠘⠁⠂⠁⣰⣯⡀⠀⠀⠀⢸⡀⠸⠄⢿⣿⣿⠟⡽⠇⠑⠀⣻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠦⠂⡀⡘⢤⠇⣱⠀⡄⡿⣿⣄⡀⢀⣀⠘⡇⠀⣧⢸⣿⢵⣠⣀⣐⢂⣰⠿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠠⡜⡀⡏⠀⡇⢱⡛⢿⢿⣟⡯⣙⠿⡄⢻⠸⡿⣦⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠖⣿⣿⣆⡇⢈⣷⠎⣟⣹⣖⢿⣿⣿⢸⡘⠀⠉⠏⢻⣿⡿⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢶⣸⣼⣿⣿⣿⣯⣿⣿⡿⠘⠀⠀⠀⠀⠀⢻⣿⢹⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⣘⣿⣿⣿⣿⣿⣿⣿⠿⣧⠀⠀⠀⠀⢶⠀⢸⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⡟⠋⠀⢹⠀⠀⠀⠐⠘⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠩⢻⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⡟⡿⢽⡷⣿⣿⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿
⠀⠀⠀⠈⣳⣾⣿⣿⣿⡗⠀⠀⠀⠀⠀⣿⣿⢋⣷⠚⣧⡏⣿⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢠⡀⠉⣿⣿⡄
⠀⠀⠀⢸⡿⣿⣿⠟⠻⠆⠀⠀⠀⠀⠀⣿⡞⣾⣹⠖⣿⣿⣿⡀⠀⠀⠀⣨⣿⣿⣿⠿⠿⣧⠀⢰⣿⠟⠁
⠀⠀⢠⢏⠾⡏⠈⠟⠀⠀⠀⠀⠀⠀⠀⢹⣧⣿⡟⣧⠗⢸⣿⡇⠀⠀⠀⡿⣯⣿⡇⠀⢠⣿⣾⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣦⠤⡶⠒⢳⠀⡗⢀⠂⠐⢒⡾⢦⡦⢀⣤⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⡞⠛⠉⢦⡀⠀⢦⣸⣄⣸⣧⣷⣞⣰⣥⡾⢡⡞⣠⣞⠁⢀⣩⠞⣳⣎⠕⠀⡠⠔⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢶⡛⠏⢤⡈⠢⣌⣲⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣿⣿⣿⣶⣟⣋⣥⠟⣻⣤⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠐⠀⠑⢾⣦⣻⣿⠟⠻⠛⣿⣿⣿⣿⣿⣇⣀⣉⣉⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣽⣫⠭⣷⣎⣡⠴⠊
⠀⠀⠀⠀⠀⢀⣐⣤⣶⡿⠛⠋⠀⠤⠀⠨⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⠻⠈⠽⠻⣿⣿⣿⣷⣶⣏⣉⣉⡤⠖⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⣀⣴⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⢿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣹⡇⠀⠀⠀⠀⢉⣽⠟⠷⠿⣟⣍⡉⣉⣩⡴⠬⠴⠚⠋
⠀⠀⢀⠸⠟⠋⠉⠄⢀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣀⠈⠻⢿⣿⣿⣿⣿⡋⢀⣾⡿⠁⠀⠀⣀⡼⠟⣋⣰⣾⣧⠤⢤⠭⢿⡋⠛⠋⠉
⠀⠀⠈⠲⠤⡤⠤⠀⠤⡴⠂⢀⠀⡀⠀⠀⠀⠈⠙⠧⣤⠀⠈⡁⣁⣴⣶⠾⣋⣠⠤⠒⣟⣩⣤⣾⠬⣭⣭⠉⠙⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠝⠺⠇⣰⢴⠸⣃⣄⣈⢹⡋⣉⢩⣭⣭⣤⠴⣔⣋⠛⢝⠻⣓⠦⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠁⠘⠸⠋⠎⠉⠸⡿⠎⡞⣯⠳⡌⠳⡸⠈⠳⡀⠳⠈⠓⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠙⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠙⣆⡀⠀⠀⠀⣀⣤⡤⠶⠶⠶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠲⠶⣯⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⠀⢀⣠⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠛⠛⠓⠛⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⢀⡟⠻⢷⣤⣄⣠⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡀⢸⠇⠀⣀⣬⣍⣉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠸⡌⢧⣿⢠⡞⠉⠀⠀⠈⠁⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠛⠷⠶⣶⠶⢿⡏⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⣹⣦⣿⠞⣋⡁⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⠴⠞⠛⠉⠁⠐⠒⠲⢦⡀⠀⢸⡇⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠸⢁⡟⠁⠉⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⣧⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⣾⢁⡼⠃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠓⠲⢤⣀⠀⠀⠀⠈⣧⠀⠀⢘⡇⠀⠀⠀⣀⣨⣿⣷⣤⣾⣶⡆⣠⡏⣿⠀⣀⢀⠀⠀⠀⠀⣀⣀⣈⠺⣵⡟⣋⡴⠃⣟
⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠛⠁⡀⠀⠀⠀⠀⠤⣍⡉⠒⢤⡀⠘⢷⣴⠋⠀⠀⠀⠸⠣⣄⣾⣻⡿⢿⣿⠟⠋⠀⠹⣾⣿⣿⣧⣤⣶⠿⠛⠉⠉⠑⢹⡟⣯⠀⠀⢿
⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⣿⠃⠀⣀⣀⣀⠀⠀⠈⣷⠀⠀⠱⡀⠨⡇⢀⡠⠄⠀⠀⠀⠈⠉⠛⠒⠋⢹⡆⠀⠀⢀⡾⣿⠛⣿⣻⠍⠓⣄⠀⠀⠀⢸⠇⠸⠀⠀⣾
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣰⢏⠅⠒⠈⠁⠀⢢⣸⠀⠀⠀⠈⢢⣿⡏⠀⠀⠀⠀⠀⢀⣀⣀⡙⢶⣼⠃⠀⠀⣼⢁⣀⡙⠛⠛⠒⠋⠋⠀⠀⠀⢯⡀⠀⣠⡼⠃
⠀⢀⠔⠋⠣⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠱⣹⣦⠀⠀⠀⠚⡉⢰⠋⠉⠀⣿⠀⠀⢠⡿⠋⢥⣄⡀⠀⠀⠀⠠⢄⡀⠀⠈⣷⠟⠉
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣰⠇⠀⡠⠀⠀⠀⠀⢻⣇⣀⣠⣤⣾⣷⡦⢼⡳⣼⡇⠀⠀⢸⠃⣀⠀⡏⡙⢦⠀⠀⠀⠀⠈⣦⡼⠃
⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣤⣶⣏⣠⠜⠁⠀⠀⠀⠀⠀⠹⡟⠛⢻⡻⣿⣝⢿⣿⣿⠀⠀⠀⣿⣯⣽⣖⠤⣽⡀⠁⠀⢀⣤⡶⣻⡇
⣸⠉⠑⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠙⢷⣀⢠⠆⠀⠀⠀⠀⠀⡇⠀⠀⢡⠹⣏⠿⣿⣿⣀⠀⣰⣿⣿⣋⣭⣿⢷⣿⣦⣤⣹⣟⣴⡿
⡇⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⣿⡏⣀⡤⠄⠀⠀⠀⣇⠀⢀⠀⣇⢻⡂⠄⠁⠙⠋⡛⢙⢹⣽⠟⠁⠃⠀⠉⢹⡿⠛⠉
⠰⡄⠀⠀⠠⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠛⠒⠀⠀⠀⠀⢻⡄⢸⠀⠘⣧⠻⣤⣀⡈⢀⣁⣤⢞⣡⠊⠀⠀⠀⢠⡟⠁
⠀⠙⢆⠀⠀⠹⡄⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⢿⡌⠂⠀⠈⠛⠦⣭⣉⣉⣭⡴⠛⠁⢠⠇⢀⡴⠋
⠀⠀⠈⠳⣄⠀⠹⣄⠀⠀⠑⠦⣤⣤⣂⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢷⡀⠀⠀⠀⠐⠒⠤⠀⠀⠀⠀⠰⢋⡴⠋
⠀⠀⠀⠀⠈⢧⡀⠘⢦⡀⠀⠀⠀⣀⡟⠀⡎⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠚⠁⠀⠘⣧⠀⣴⣋⣽⣿⡝⣦⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⠦⣄⣽⢛⣳⠞⠁⠀⠀⡇⠀⠀⠀⠀⠴⠓⠉⠀⠀⠀⠀⠀⠀⠀⠘⢷⣏⣿⣿⣿⣿⣿⣀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠙⡍⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⢰⡄⢀⢉⠛⠋⣉⡀⢰⠆⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⡄⠀⠀⠀⠀⠀⠑⢤⣆⠀⠀⣰⡤⠊⠀⠀⠀⠀⠀⢠⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⡸⡯⣴⣦⡽⢇⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⣠⢶⢟⣶⣾⡿⣿⣶⣾⡳⡦⣄⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⡁⢰⡿⣷⣿⡇⢸⣿⡾⢿⡅⢸⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡇⢸⠁⣿⣿⡇⢸⣿⡿⢈⡅⢸⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢉⣩⣿⣽⣇⠸⣿⡇⣾⣿⠇⣸⣷⣿⣉⡙⢷⣄
⠀⠀⠀⠀⠀⣀⣀⡠⠤⠴⠾⠟⠁⢠⣯⣤⡄⠉⠙⢿⡦⡤⢤⢴⠿⠋⠁⢠⣬⣽⠀⠙⠿⠷⠦⠤⠄⣄⡀
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⣷⢴⣾⡦⡿⣱⣮⢞⣰⠷⡖⢿⡿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢘⡯⣯⡽⣷⣌⢡⡆⢰⣾⢞⣳⣿⠇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢚⠛⣛⣿⣽⣯⢸⡆⣿⣗⣿⢹⠧⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣀⣃⣻⢤⣻⣹⡷⣏⡖⣟⣭⣤⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⣙⣶⡢⡿⢸⡇⢺⢳⡇⡅⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡉⡍⠫⠃⢸⡇⠙⠫⣵⢛⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠙⠛⠓⠒⠛⠓⠒⠚⠛⠋⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠟⠛⠛⠛⠛⠻⠿⠿⣿⣿⣷⡄⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⡿
⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⢀⣈⠉⠁
⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣠⡾⢿⡟
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⢹⣷⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠱
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣧⣼⣿⡇⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠿⠿⠿⠟⣿⡿⣿⣿⣿⣿⡀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢠⠂⣨⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⣿⣿⣿⣽⣷⣾⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢀⣤⣤⣿⡟⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⠀⠀⠀⢨⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⠿⣿⡿⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠐⢀⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⣄⠀⠉⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢫⠀⡤⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⡄⠶⠇⢀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠋⠚⡃⢀⡀⠈⠃⠀⠀⡀⠀⠐⠃⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠙⢎⠁⣿⠁⡀⠀⠁⠉⠀⠀⡄⠉⣭⡀⠀⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠷⠰⠋⠑⢧⡀⠋⠀⠀⠀⠀⢠⣄⢰⣶⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠄⠂⠀⡀⠀⢀⠄⠀⠘⠢⠤⠀⠁⢦⠀⠀⠀⠐⣯⡉⣿⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠦⠈⡃⠀⠸⠃⢠⡓⠀⠀⠆⠀⢀⡃⠀⣻⡄⠈⣧⠉⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢠⡤⠁⠀⠈⣁⠀⡟⠆⠀⠛⠀⠀⢻⠃⠈⠛⢿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣙⠀⠀⠰⠎⠁⠘⠋⠁⠴⢀⡙⠀⠶⠀⢻⠤⣀⡀⢠⡤⠴⠆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠀⠠⠠⠆⠐⢀⠀⠠⠙⠰⠆⣀⣼⠀⠉⠷⠀⠀⠀⠀⡠⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⣀⠰⣌⣀⣧⡄⠀⠀⠀⠁⠈⠱⠋⠀⠀⠀⠈⢻⣆⡀⠶⠛⠚
⠀⠀⠀⠀⠀⠀⠈⠁⠋⠙⠿⢁⣈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡏⠠⠆⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠒⣰⠃⣠⣄⠀⠀⠀⠀⠐⡃⠀⠀⢸⣬⡅⠈⠓⢀⠀⢰⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠛⢀⡀⠶⣧⣄⣻⠀⠀⢠⡶⠀⠈⠁⠀⠀⣸⠉⠀⢮⠁⣄⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⡏⠶⠚⡻⠀⠀⠀⠀⠀⢠⡜⠀⠀⠀⠁⠸⠐⢶⣋
⠀⠀⠀⠀⠀⠀⠀⠈⠰⠁⠶⢄⠶⣀⠤⢨⠗⠀⠀⠀⡄⠀⠀⣿⠀⠀⠀⠀⠀⣿⠛⢁⡀
⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠃⠒⠀⠀⠟⠀⠀⠀⠹⠥⡶⠄⢀⣖⣠⠟⠛⠁
⠀⠀⠰⣿⡃⠀⠀⠀⠀⠀⠀⢒⡀⠠⡐⠆⠀⡀⣾⠃⠀⠠⠀⠲⠀⠀⠠⠈⢙⣁⡀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠘⢣⡄⠀⠀⠀⠀⠀⠀⠀⢂⣤⠘⠢⣤⠀⠀⠀⠄⠀⢀⠀⠀⠲⡶⠆⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃
⠀⠀⠀⢀⣄⠙⠀⠀⠀⠀⠀⠀⠐⣿⣿⢀⣤⠘⣀⠰⠂⡄⠀⣼⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠐⢶⠀⢠⣦⠀⠠⣌⠛⣤⡿⣀⡽⢃⡀⠀⢰⠇⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣅⠀⠀⠙⠻⣿⣾⡏⠀⠁⢄⠀⡜⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⠀⣴⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣤⣠⡀⠘⡿⠀⠈⠀⠀⠰⡇⢀⠃⠀⠀⠀⣀⣦⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢺⣷⠀⠀⠀⠀⢰⣦⠀⠠⠐⠚⠛⠛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣴⠀⠀⠀⢈⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠘⠻⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠈⢰⡞⠀⠀⠀⠐⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢄⢸⡇⢀⣾⠇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⡆⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠈⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠤⠈⠂⠀⢠⠀⠀⠀⠀⠑⢢⡀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡓⠂⢬⣉⠀⠀⣉⡚⠢⢄⣀⢗⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠈⠀⠀⠀⠀⠀⣀⡀⠀⠀⢤⣈⠙⠀⠀⠀⠀⠀⠀⠀⠀⢑⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⢐⡦⠲⠤⣤⠀⣿⢁⠀⠀⠀⠒⠁⡀⡰⠀⠀⠀⠂⠀⢠⠊⣡⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠠⡌⠀⢀⡴⠁⠀⣿⡀⠧⠀⠀⢀⣠⠤⠶⠶⠶⠶⠶⠶⢶⣦⣼⠂⠀⠛⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠙⡄⠀⢼⠁⠀⢀⣯⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠉⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⣇⠈⡆⣧⠀⢀⠀⢽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠠⢤⡋⢦⡀
⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⡀⠀⠀⠀⠨⢆⡘⢻⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠒⢤⡇⠀⠳⡄
⠀⠀⠀⠀⠈⠀⣀⠴⠀⠀⡏⠀⠀⡀⠀⠈⠷⠤⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠠⠔⠁⢈⠐⠈⠹⢆⠀⢄⠀⠀⡂⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⢴⣧
⠀⠀⠀⠀⠀⠀⠐⠂⠒⠁⠀⠀⠑⠀⠀⠚⠑⠀⠀⣷⠀⠀⠀⠠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⠌⠃⢸⡀⠀⠀⢰⡿⣿⠿⡖⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣡⡞⠀⡆⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡥⠖⠂⢹⠃⠀⠀⠀⠱⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢡⣿⣿⡇⢠⣧⠃
⠀⠀⠀⠀⠀⠀⠤⠀⠈⢀⣠⠤⣧⣀⡔⠋⠀⠀⠀⡟⠀⠀⠀⠀⢀⠈⢀⠀⠀⣀⣠⣤⣄⡀⠀⢀⠀⣰⠰⠁⣿⣿⡟⢁⣾⡁
⠀⠀⠀⠀⠀⠀⠀⢰⣒⣁⣠⠟⠀⡽⠀⠀⠀⡠⢞⣻⣾⣷⣶⡶⠣⠀⠀⠉⠉⢿⣛⡽⠄⠉⠓⠁⠠⠃⠀⠈⡋⠁⢰⡿⠯⡻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣷⡠⢠⡞⡴⠉⢀⡞⠛⠛⠋⠀⠃⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠈⠂⠀⠀⠈⠀⠘⡇⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣿⠘⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⣾⠳⣄⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⢄⠀⠀⠀⠀⣀⡀⠀⢀⣤⣤⡼⠁⠀⠀⠀⠀⠀⠀⢸⠆⠀⠈⢣⠑⠌⢷⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⡆⠀⡤⠤⡈⠙⠲⠛⠁⠀⠀⠀⠀⠴⠒⣦⣤⡔⠃⠀⠀⡀⢀⡇⠀⠈⣷⢸⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣦⡐⢄⠀⠀⣧⡼⠓⣦⣀⣀⣤⣴⠒⣶⣿⠀⢸⣿⠃⠀⠀⡆⠀⣷⣾⣅⢠⡄⣿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠙⣶⢵⣦⣼⣁⣴⣿⣿⣿⣿⣿⣼⣿⡿⠀⠿⠃⢀⠀⢰⢀⢀⡿⢳⡙⣿⠁⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⠀⠀⠀⣨⡛⠛⠻⠍⡻⡏⢹⣿⣿⡿⠁⠀⠀⠀⡀⢸⣸⠿⠟⠁⢰⠃⢸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⡔⠀⠀⠈⠙⢛⠓⢦⡈⠣⠈⠿⠏⣄⣀⡀⠀⠀⣷⠿⠃⢀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⠰⡇⠀⠀⠀⠀⡀⠀⠁⠀⠰⠒⠛⠉⢁⠄⣿⣿⣁⡤⠖⠁⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢧⡆⠀⠀⠈⠳⣀⠀⠀⠀⢀⠔⠚⠙⢲⣷⡶⣶⣶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⡄⠀⠈⠀⠀⣠⠀⠀⠀⠀⠀⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠘⠛⠁⠀⠀⠀⠀⠀⠟

⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⢰⣿⡆⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢶⣢⣶⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣖⣶⠄⡀
⠀⠠⢀⡐⣒⠚⠍⠁⠀⠁⢳⣿⡞⠀⠀⠉⠹⠓⣀⢀⡀⠄⠀⠈⢧⣕⣿⣯⣃⣹⢁⣸⠗⠠⢶⠂⠚⣏⡀⣏⣘⣽⣿⣚⣵
⠀⠀⣀⣀⠈⠁⠐⠄⠀⠂⢾⢛⡇⠐⠀⠠⠂⠉⠁⣀⡀⠀⠀⠀⠀⠀⣿⣿⣖⡠⣶⣪⣿⣷⣈⣼⣿⣯⣲⢄⣰⣿⣿⠃
⠘⠓⠒⠂⠀⠀⠀⣀⣁⡤⠜⣍⠢⢤⣈⣀⠀⠀⠀⠐⠲⠚⠀⠀⠀⠀⠉⠻⢿⢷⠏⢡⡟⣯⣟⣯⡟⡄⠙⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡄⠑⠊⢀⡟⡄⠙⠒⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢶⢄⠃⠀⠀⠈⠻⡶⣶⠞⠁⠻⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠤⠄⢀⣑⣾⣭⣿⣊⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢹⢿⣤⡿⠏⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠇⢨⣿⡇⠘⣾⢃⠀⠀⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⢠⠀⡏⠀⠀⠀⢸⠀⣆⠀⠀⠀⠀⠀⠀⠁⠃⠃⢸⣿⣇⠈⠈⠈⠀⢸⠀⠀⢠⣄⠀⡇⠀⣰⠏⠙⠒⠂⢄
⠀⠀⠀⢀⣠⡤⠑⢮⡀⣶⣀⠥⠊⠠⣄⡀⠀⠀⠀⠀⠀⢀⡜⢻⣿⡟⢢⡀⠀⠀⠸⡗⠀⣼⣿⣿⣷⣶⣇⣠⠴⠓⠒⢤⡀
⠀⠀⠀⠀⠤⡤⠀⣀⣻⠭⣿⣁⠀⠤⠤⠀⠀⠀⠀⠈⠓⠋⠀⠀⠉⠀⠀⠙⠚⠁⠀⣈⡛⢻⣿⣟⢟⣿⣽⡷⠚⢧⡀⠀⠈
⠀⠀⠀⠀⢠⠘⠋⣩⣤⠛⣡⣍⠙⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠈⢉⣻⡿⣿⠿⠇⠀⠀⢱
⠀⠀⠀⠀⠘⠰⡄⠙⢿⣶⠿⠋⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⣰⡏⠘⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⣿⣛⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣻⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠖⠉⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣠⣀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠠⡇⠈⢳⡀⠀⠀⠀⢰⡇⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣧⠀⢸⡇⠀⠀⢀⡾⠀⠀⠀⠀⣼⠁
⠀⠀⣀⠴⠖⠒⢒⣚⣛⣛⣉⣩⠤⠤⠤⢀⠉⠳⣄⠀⠀⠀⠀⠀⠸⡆⢹⡄⣸⠁⠀⢀⡾⠁⠀⠀⠀⣼⠃
⢀⡼⠁⢀⠀⠀⠤⠤⠤⠤⠴⠶⠒⠒⢒⣂⠀⠀⠘⢦⡀⠀⠀⠀⠀⢻⡈⣷⠃⠀⣠⠟⠀⠀⠀⢀⡼⠃
⢎⠀⠀⠀⠀⠀⠤⠒⠒⠢⠔⠒⣊⣩⡤⠤⠤⠤⣄⡀⠙⢆⠀⠀⠀⣨⣧⠸⣆⣴⠋⠀⠀⠀⢠⠞⠁
⠈⠛⠶⢤⣤⣀⣀⣤⡤⠶⠚⠋⠉⠀⢀⣀⣠⡤⠬⠤⠀⠈⠳⡀⣰⠋⠸⡆⢿⠁⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣞⠯⠛⠀⠀⠀⠀⠀⢀⡈⢿⠁⠀⣠⡇⢸⡇⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⡊⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣸⠂⣰⠃⡇⢸⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢚
⠀⠀⠀⠀⠀⠀⠀⢠⢞⡵⠋⠨⢊⣥⡦⠂⠀⠀⢀⡔⠃⠀⣠⡟⢰⠇⣰⠃⣼⠀⡟⠀⠀⠀⠀⠀⠀⢀⡴⢞⠁⢠⠟
⠀⠀⠀⠀⠀⠀⣰⢏⡞⠀⠀⠔⠟⠁⠀⠀⢀⡴⠋⠐⣀⡾⢫⠇⡟⣰⠃⣰⠇⠈⣧⡀⠀⠀⢀⣠⠖⢋⠔⠁⢠⠏
⠀⠀⠀⠀⠀⠀⡟⢸⡄⠀⠀⠀⠀⣀⡤⠖⠉⠀⣡⠾⢫⠃⡼⢀⡟⠁⣰⠋⠀⠀⠈⠙⠛⠛⠉⢀⡴⠋⠀⣠⠏
⠀⠀⠀⠀⠀⠀⢷⡀⠍⠓⠒⠛⠉⠁⢀⣠⡴⠛⡏⠆⢸⢀⡷⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠉⠢⡄⠀⠀⠐⣶⡉⠁⡄⢸⠃⠀⣼⢸⣀⡴⠋⠳⣤⣀⣀⣠⡤⠶⠛⠁⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠆⠈⢽⠣⠀⠃⢸⠀⠀⣿⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⡆⠀⠀⢻⠀⠀⢻⡘⡇⠙⠢⣄⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢹⠀⠀⠀⡇⠀⠀⢸⡀⡀⢸⡃⢻⡀⠀⠀⠉⠓⠒⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⢾⢀⠀⡄⣷⠀⠀⠘⡇⠧⠈⡇⠀⢧
⠀⠀⠀⠀⠀⠀⢰⠇⠤⠀⡘⠈⠀⠁⣿⠀⡄⠀⢷⠈⠀⢱⠀⠈⢇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⢳⡇⠀⡀⠀⡿⠀⠇⠀⢸⠀⠀⠀⡆⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠳⣄⠃⢰⠇⠀⠀⠀⢸⡇⠀⠀⢸⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣶⠈⠙⠾⣄⡀⠀⠀⢀⡇⠀⠀⠈⣧⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠷⣄⣀⡈⠻⠲⠀⢸⠇⠀⠀⠀⣿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢠⠟⢻⣄⠀⠀⡀⡸⣇⠀⢀⠀⣟⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠟⡰⠁⠅⣀⣨⡼⠋

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢡⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⢾⣋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⣲⢤⡀⠹⣿⣧⣿⣿⢃⣤⡖⣲⣶⣔⣦⣄⡀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣾⣷⣤⣿⣆⣿⡦⡈⠻⠛⢡⣾⣻⣷⣿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⠁⣸⣿⣧⣤⣴⣶⣒⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠓⢦⣤⣝⠙⡻⣿⣿⣷⠀⠂⢸⣾⡟⣡⠇⠀⠀⠀⠀⠀⠀⠐⡄⢀⡆⢾⣧⣴⠛⠋⠀⠙⠻⠿⠿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠷⠤⠽⢿⠀⡔⠐⡯⠖⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⡴⠏⠓⠀⠀⠀⠀⠀⢀⣠⠆⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁⠀⢀⣀⠀⠀⠀⣀⢈⣡⣶⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢠⣄⣀⣀⡀⠀⢄⠀⠀⢀⠂⠀⠀⢀⣸⢉⣉⣠⣼⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤⣲⣦⣽⣦⡐⣤⣄⣠⣤⣄⠀⠀⠀⠈⣷⡿⣿⣿⣿⣷⡶⠦⢱⣦⣼⣿⣍⣶⡾⢿⣿⣿⣿⣗⣾⠗⠲⣤⡀
⠀⠀⠀⠀⠀⢀⢐⣿⠟⠩⠤⠤⠈⣉⡛⢿⣼⣿⣿⣿⣿⣿⣆⡀⠀⣼⣿⠟⠻⣿⣿⣧⡀⡟⠓⡛⢉⣠⣤⣴⣿⣿⣿⣿⣿⣿⣾⣿⣾⣿⣦⡤⠄
⠀⠀⠀⠠⠖⠋⠉⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡉⠋⢀⣰⣾⣼⠏⠀⠀⠉⢹⣿⣾⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠑⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠛⠉⠉⠁⠀⠀⠈⢹⣽⠏⠁⠀⠀⠀⠀⠀⢻⡿⠟⠋⠋⠩⢹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠷⣄⠀⠀⠢⢤⣬⣵⣏⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣄⣀⠉⠉⢉⢉⣁⣀⣹⣧⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⠿⠛⠛⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠙⠿⡿⢻⡿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣼⣿⣿⣿⣿⣿⠁⠀⠀⣠⣿⣿⡇⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢸⡷⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣱⣶⣿⣿⣿⣿⡇⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣽⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡠⠀⣀⢀⣀⠀⣀⣠⠔⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠛⠿⣿⣿⣿⣿⣇⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⣿⣾⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⡿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠶⠶⠖⢀⠀⠀⢸⡆⠀⠀⠰⠀⠀⠄⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⠀⢰⡇⠀⠚⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⢀⠀⢐⡀⠀⢱⠼⣄⠗⠀⡌⢀⡀⣁⠀⢆⣨⣔⠁⡀⠀⠀⣊⣀⡔⢉⣩⠝⠛⢳⢶⡄
⠀⠀⠀⠀⠴⢛⠉⣉⢉⡶⣤⣾⡿⠃⠀⠒⠀⠙⣦⠈⠄⢨⡷⠀⠀⣸⠁⠀⠠⠃⠀⢠⠛⡆⡙⢩⣄⠀⣬⠾⠏⠉
⠀⠀⠀⠀⠀⠻⠲⢪⠄⢀⠄⠀⠊⢳⢀⡇⠀⠛⠻⢷⠐⣿⣟⠀⢸⠟⠓⠐⠀⠛⠷⢊⢀⡇⠘⠂⡁⠀⠀⢠
⠀⠀⠀⠀⠀⠞⢆⠁⢸⠏⠲⠇⢀⣴⡟⡤⠀⠀⠀⠈⠣⣘⣍⡲⠿⡀⠂⠀⣄⡀⢆⡇⠈⠃⡀⠾⠃⡆⠈⠀⠀⠀⣠⡆
⠀⠀⠀⣧⣠⠀⡺⢀⠌⠀⠙⠆⠈⢀⣦⢭⡀⡆⠀⠆⠀⢸⠎⠉⢳⠂⠀⠀⠠⣢⣯⣀⠘⠟⣡⠄⠳⡁⠀⢱⠀⣾⠟⠁
⠀⠀⠀⠛⠃⡜⣷⣻⡆⠀⠘⢾⣟⠉⠈⢀⣘⣥⣤⠞⠋⢻⠖⠦⠼⢦⡶⣤⠂⡹⠎⠉⠀⣤⣤⣾⠧⡀⢀⡈⢆⢁⣴⠇
⠀⠀⠀⠀⣴⠀⢀⠿⠃⣄⠴⠿⢭⣝⢛⣿⠏⣽⡏⢀⣼⣫⣷⣤⣤⣾⣧⠈⢷⣾⣷⠻⠟⢫⣦⣀⠀⠟⠛⠁⠪⡘⠋
⠀⠀⠀⠀⠘⠃⣠⠞⢻⠛⠻⢶⣶⣌⣿⡏⠠⣿⡇⠀⠁⣰⡇⢀⠀⠉⣿⡇⠘⣧⢁⡴⠟⠛⠛⠛⠛⢣⣄⣠⠀⠇⠀⢀
⠀⠀⠀⠀⢇⣴⣤⡾⠛⠻⠛⣶⣦⣽⡿⡇⢀⣿⣿⠀⢤⣼⢧⡀⢀⣠⣿⡇⠸⣷⣋⣤⡶⠿⠋⠛⠚⠻⢿⣏⣤⠸⣾⠏
⠀⠀⠀⣦⡾⠛⣥⣴⣿⣿⠗⢶⣽⣿⡿⢷⠀⣹⣿⣷⣾⠷⣞⣿⡀⢹⣟⠃⣼⢻⣷⢀⣤⣶⣾⣿⣶⣄⡀⠙⣆⠁⡏
⠀⠀⠀⣾⢁⣾⣿⣿⣿⠁⢘⣾⢩⣷⡄⠸⢧⠸⠟⠁⠻⣶⢛⣿⠣⠈⢻⡝⢩⣟⣿⣿⠸⠃⣸⣷⣿⣿⣷⠄⢸⡄⡏
⠀⠀⠀⡇⢸⣿⣻⣿⣿⠀⠙⢹⣿⣿⡇⡀⠀⠀⢀⣤⣴⣟⣸⠁⠄⠤⠆⣰⡾⣿⣿⠟⣩⠿⢻⣿⣿⣿⣿⠇⣼⡇⣜⢠
⠀⠀⠀⣧⠈⣿⣿⣿⣿⣤⡆⠀⣿⠟⣡⣽⣾⡿⣿⣿⡿⠛⠁⢠⣤⣶⣶⣶⣦⡝⠿⣄⣩⡞⢁⣿⣾⣿⡏⠠⣿⠁⡈⢱
⠀⠀⠀⢻⣂⣹⣿⣿⣿⣏⠀⠀⣌⠠⣿⣿⡿⢻⣿⣿⡗⢀⡄⣿⣿⣿⣾⣿⣿⡏⠈⢮⣟⠡⢼⣿⣿⣿⣶⣾⣯⣀⡼⠆
⠀⠀⠀⢿⣿⣀⡀⠙⠛⢻⣄⠰⡙⠀⠙⠓⠞⠛⢋⣍⢰⣿⣷⠈⠉⣛⠿⠟⠛⠁⠁⣾⡝⢃⣾⡿⠟⠛⠛⡹⢿⠁⣴⣆
⠀⠀⠀⣿⠘⣿⣿⣶⣄⠀⠙⢿⡵⢄⣉⣠⡘⠶⠀⢱⣾⣿⣿⡄⠀⠆⠐⢉⣀⣀⣴⡟⣠⠾⠏⠀⡄⣰⣾⢓⣾⢿⢿⡌⠁
⠀⠀⠀⢹⢧⢛⣽⣿⣿⣶⣀⠈⠹⡄⢭⠟⣧⠀⠉⠸⣿⡟⠻⠃⠀⣀⢆⣻⡷⢡⢎⡾⠻⠆⠀⢴⡞⢛⠁⠈⠉⠿⡁⠁
⠀⠀⠀⢠⡈⠛⢛⣿⢻⣿⣆⠀⠂⣷⠀⣺⠸⣷⣶⣤⣦⣶⡄⣄⢢⣽⣾⣧⠁⠰⡿⠓⠀⠀⣵⠏⡹⠋⠀⢰⡀⣦⠄
⠀⠀⠀⢈⣄⠀⠘⢛⣷⣿⣿⡴⣇⢸⡀⢸⠇⣿⣍⣼⢩⣿⣿⡿⠛⣯⡿⢱⠀⡘⣟⠊⠁⣰⡏⣰⡇⠀⠀⣼⡟⢁
⠀⠀⠀⢸⡟⢦⡴⠛⣛⠟⢻⣿⣷⠾⠁⣻⡇⠃⠻⢿⣿⣈⠛⣷⠷⣿⠡⠸⡙⠻⠻⠶⣤⣹⡿⠋⢣⣔⠂⣝⣠⠙⠳⠄
⠀⠀⠀⠉⠀⠸⠀⣸⡉⢆⢠⠹⠿⠒⢡⣿⠋⠲⠀⢄⣈⡉⢁⡀⠀⠀⢠⠔⣱⡶⠛⠛⣯⡁⢀⡁⠻⠞⠓⢧⠄
⠀⠀⠀⠀⠀⡽⢳⢿⠃⢈⣿⣄⠟⠛⠛⢿⣏⠋⠑⠀⠤⠴⠂⠁⠠⠘⢱⣿⠟⠒⢳⡿⡤⠟⡁⠐⠀⠂⠀⠐
⠀⠀⠀⠀⠀⢦⠛⣠⠆⡘⢿⡎⠐⢀⠂⡀⠬⠅⠠⠆⡀⢠⡄⠠⠄⢀⠉⣬⠂⣱⣿⡷⠙⡤⠁⠤⡀⠔⢷⠶⠂
⠀⠀⠀⠀⠀⢴⣗⠣⠀⠀⠈⢷⠀⠀⠲⣂⣀⣠⡁⠀⠀⣾⣽⠂⠂⢈⣡⣀⡚⠙⠋⡀⠀⠀⠈⠰⣿⣿⠞
⠀⠀⠀⢀⠀⢰⠋⣅⣿⡄⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⢻⡆⢳⣶⣿⣿⣿⣿⣿⣏⡉⠀⠀⢰⣶⠻⣷⣤
⠀⠀⠀⠀⠀⠈⢀⠉⠙⣧⣠⠀⠣⠘⠋⠙⣿⣿⣿⣿⣿⣿⡶⣾⣿⣿⣿⣿⣿⠟⠛⠔⠁⣠⣾⢟⠴⠐⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠸⣀⠁⢈⢭⠀⡁⡘⠡⠀⠩⡩⢉⠛⣹⣟⣃⣿⢋⠈⢅⠀⠉⢉⠛⠤⠀⠩⢁⠄⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠰⢀⠀⠠⢤⡄⠀⠀⠀⠀⠀⠳⡙⢿⣿⢿⡍⠀⢀⡀⠀⢠⠄⠒⠂⢠⠀⠒⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠃⠀⢠⡄⠘⠀⠂⠀⠀⢀⡀⢀⠐⣈⠁⠠⠄⣤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⡲⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⢷⠀⣉⣉⣛⡛⠻⢿⡟⠉⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠛⠟⠛⢿⣿⣿⣿⣿⣻⣿⡿⢻⡿⠟⣀⣴⣾⣻⣶⣄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠⣤⣶⣿⣿⣿⡿⣿⣿⠟⠀⠀⠀⣸⡉⠙⣙⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠙⠛⠋⠉⡆⢸⣧⢸⣿⡇⢀⣿⡧⣀⣺⡟⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣄⠀⠀⢧⣿⣿⠳⢿⣿⢸⠏⠀⣿⡿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⣿⣿⣿⡄⠘⢹⡟⢷⣀⣠⣝⠻⣤⣄⡸⣧⣏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣷⣄⠈⣷⣔⣟⢿⣿⡷⠀⣟⡃⢿⢻⣷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⢿⣿⣿⡿⠿⣿⣿⠉⠋⣿⡇⠀⠁⠉⠘⡆⢽⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠰⣿⡿⢹⠇⠀⠸⣿⡅⠀⢻⡇⠀⠈⠀⠀⠹⣞⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢰⡇⠀⢺⠀⠀⠀⣿⣧⣠⣿⠇⠀⠀⠣⠀⠰⠜⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠁⠀⣸⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣤⣬⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⡿⠀⠀⣤⣿⣿⣿⠁⠀⠀⣀⠀⣅⢿⣿⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠠⣜⠂⠀⣇⠀⠀⣹⣿⣿⠇⠀⠀⢸⣿⣶⣿⣇⣨⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣿⣾⡂⠻⣶⣾⣿⣿⠃⠀⠀⠛⠛⣿⠛⣿⣇⣠⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⢠⣄⠏⠉⠀⠀⡀⠀⠀⠀⠁⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠎⠀⠀⠈⠇⣠⡀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠀⠛⠀⠈⠀⠉⠀⠀⢀⡀⠀⠀⠀⠀⠠⣤⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡀⢹⡶⢀⡤⠀⠂⠀⠀⣰⣿⣿⡆⠀⠠⠤⣴⠟⢹⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠟⡀⢸⣟⣥⠀⠀⠀⠀⠈⣿⠿⣗⠀⠀⠼⢏⣰⣾⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣗⣉⢿⣟⣶⣶⣴⡆⢘⣿⣿⡷⣠⣴⣶⣯⠿⠿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣦⣿⠿⣿⣿⣷⣪⣽⣿⡻⣿⣯⠽⢿⣯⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣈⣿⣏⣳⣿⣿⣿⣿⣾⣵⣾⣿⣿⣴⣿⣯⣥⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠛⠛⠛⠛⠿⠋

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

⣀⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣀⡀⠴
⢩⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣫⣁
⠈⠉⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⣿⠞⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡙⠈⠉⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⢀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⣿⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡿⠇
⠀⢀⠠⠒⣀⡰⡖⠀⠀⠘⡀⠀⠰⢦⣀⠐⠢⠀⠀⠀⠀⠀⠊⣌⠋⣬⠁⠀⠀⠀⢀⠠⠒⣀⡐⡖⠀⠀⠈⠀⠀⠲⢦
⠀⠁⠀⠀⠀⠀⠀⠐⡇⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⠠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⣀⣤⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣄⡀⠀⢀⠀⡀⠀⠀⣀⣤⣤⣦⡄
⣩⡽⣻⣿⠟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡽⣦⣍⡓⢄⢤⡤⣒⣩⡽⣻⣿⠟⠀⠀⠀⠀⡀⠄⢀
⣦⣼⡿⠃⠀⠀⠀⠀⠀⣤⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣶⠛⣩⢤⡙⠳⣦⣼⡿⠃⠀⠀⠀⠀⠀⣤⡀⢠⣄
⣤⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡞⠹⡜⡌⣦⣮⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃⠸⣟
⠾⠋⠀⠀⠀⠀⠀⠀⠀⠠⠃⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠩⠃⠘⠏

⣀⡀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⡀⠘
⢵⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣰⢵⣥
⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠿⠋⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡌⠀⠈⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠢⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢤⣦⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇⠀⢇⠀⢸⡇⢠⡏⠀⣠⠞⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⣄⠀⢸⣄⠼⡦⠜⠧⠾⢇⣴⡃⠀⡴⠋⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⣄⡸⢉⡁⠤⠀⠒⠀⠓⠂⠠⠏⣼⢏⣡⢴⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠤⠋⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣖⢛⡋⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠢⣌⣢⣄⣶⣄⣄⠀⠀⠀⠀⠀⠐⢶⣁⠀⠀⠀⠀⠀⠀⢀⡄⣀⣀⠀⠀⠀⠀⠀⢸⣩⡛⠁⠀⠀⠰⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠉⣷⣴⣿⣿⣿⠛⢉⡄⢭⡀⠀⠀⠀⠀⠘⡍⢀⠀⠀⣠⡠⢯⣜⣳⣧⣿⣝⣆⠠⠀⠀⣩⠉⠁⠀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠉⠛⢲⣴⣿⣿⣟⠿⠀⢰⡏⢠⡌⢻⡄⠀⠀⠀⠀⠈⠳⡅⠈⣻⣿⣿⣿⣿⠿⠿⡿⣿⣷⣖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠦⣄⣴⣿⣿⣿⣟⠁⣠⣾⠟⠉⠀⢰⠀⠀⠀⠀⠀⠀⢀⡠⠾⣭⠋⢧⡀⠘⡇⠀⢸⠏⠀⣿⠛⣀⡄
⠀⠀⠀⠀⣀⣽⣿⣿⣿⡉⠀⣾⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠙⣆⠘⣇⢈⣇⣠⣧⣀⣿⣄⣴⡇⣰⠏⣡⠞⣠
⠀⠀⠀⠀⠈⢻⣿⣿⣯⡁⠀⢻⠟⠁⠀⠀⠀⢀⡀⠀⠀⠀⠠⣽⣶⣾⠷⠿⠛⠛⣿⣿⠋⠉⠉⠙⠛⠿⠾⢧⠴⠃⢀
⠀⠀⠀⠀⠁⠛⣿⣿⣿⡄⠀⡘⢠⠀⠀⠀⠀⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠠⣶⣟⣵⠗
⠀⠀⠀⠀⠈⠉⠛⠿⣿⣧⡀⠁⡴⢧⣤⣴⡟⠻⣼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⠆⠀⠀⠀⠀⠀⢀⠤⡀⠀⠈⣱⣾⡔⣡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠂⠭⠿⠟⠉⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣬⠄⠀⠀⠀⠀⠀⠀⠌⠙⠻⠿⠆⠀⣾⣿⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣀⠀⠰⣿⣯⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣿⣿⠀⢰⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⠀⠀⣰⢻⣄⡀⠀⣀⣿⠏⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⣘⡃⠀⠀⠀⠀⢠⡟⠉⠈⠐⠢⠍⣿⠦⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠀⠀⠀⠀⣋⡃⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠃⠂⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣧⠀⠀⢸⠠⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠈⠓⠀⠀⡰⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠁⠀⠘⠊

⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⡅⢶⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⣇⠀⠀⠟⠁⣸⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣿⣿⣿⠀⣿⠀⣆⣀⡤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⢿⣿⣧⢰⠦⠟⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣶⠆⢸⣿⣧⠈⣠⠚⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⡟⣋⣨⡽⠿⠛⣿⣿⠗⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣏⣻⠟⣿⠀⠶⠂⠁⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠈⢿⣷⡶⡄⣄⠒⣘⣋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣸⣏⣤⣤⢰⣂⣭⣿⣿⣿⣷⠶
⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠏⠀⠈⠙⠋⠛⠛⠋⠉⠁
⢀⠙⠉⣠⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⡀⠀⣿⣰⣿⠄⠀⠀⢀
⠀⢱⡶⢼⡧⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣦⢙⣟⣁⡤⠂⣠⡎
⠀⠈⠃⡤⠛⠀⠈⠁⢠⣤⠀⠀⠀⠀⠀⠀⠀⢀⡘⢿⣿⣿⡿⠛⠁⡤⣨⠏
⠀⠀⠀⠀⠀⣤⡄⢳⢀⡟⠀⠀⠀⠀⠀⠀⢠⡌⢻⣄⠙⣿⣿⣧⡾⠟⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡈⠃⠎⠀⢠⠀⠀⠀⠀⠀⣿⣿⣄⢿⠀⡞⠉⠀⣀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⡞⠀⠀⠀⠀⠘⣿⣿⣿⣧⣷⢁⣶⠈⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠁⠀⢀⡰⣤⡀⣿⡿⠙⢿⡿⠈⣳⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡄⣠⠛⢱⡍⢿⣾⠃⡀⠀⢀⡾⣥⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡆⣿⣀
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣯⣄⠀⣤⡈⣿⡀⢲⠀⢸⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⢿⡍⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⣉⢈⡉⠀⠘⠿⠻⣷⢸⣦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⢿⣿⣦⡀⠀⢿⣿⠋⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣧⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡇⢸⠟⣿⣿⠇⠀⡏⣀⢃⡼⠃⠀⠀⠀⠠⣀⠀⠀⠀⠐⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠊⠀⠈⠻⠀⠆⠘⣿⠚⠁⠀⠀⠀⠀⠀⠙⢷⡌⠀⠰⡀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⢠⡇⠀⢺⣇⠀⠉⠀⠀⠀⠀⠀⠀⢀⣿⣤⣹⣦⠷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠁⠀⠀⣿⡄⠄⠀⠀⠀⠀⠀⠀⠈⢻⡉⣿⣿⣀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢰⠀⢠⡆⢸⣇⠀⠀⠀⠀⠀⢀⡴⠀⢾⣇⣡⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢲⣤⣄⡀⠀⠙⠷⣦⠸⡇⠘⡟⠀⠀⢀⡦⢀⠈⠁⠀⣬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣌⡳⢦⡄⠀⠘⠇⠀⠀⠀⢐⣈⢰⣷⠐⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠻⢿⣿⣷⡆⢰⡇⠀⠀⣴⠿⣿⠀⢻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠙⢿⣇⣘⡁⢉⡼⢁⣤⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⢀⠙⢿⠁⢩⡄⢸⣏⢉⢡⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠈⡀⣼⡇⢸⣿⡤⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⢘⣷⠘⡇⢠⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⣿⣿⢀⡟⠀⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⡆⠘⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢎⣿⡄⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠦⠤⢄⡀⠀⠤⠤⣀⣀⠀⠀⠀⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠠⠭⠅⠈⡑⠤⠵⣾⣷⣿⣴⣦⡽⣷⣮⣒⢤⣁⠲⣤⣠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠑⠄⠈⠛⠿⠝⠻⣿⣿⣿⣿⣷⣮⣳⣇⡹⢿⣽⣿⣷⡼⣦⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠟⢮⠻⡝⢷⡿⣌⣿⣿⣿⣿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠛⠙⠒⠝⢷⣿⣧⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢂⣆⣠⣤⢰⣤⣠⣬⣬⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⢋⣤⣌⣴⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡰⣽⣷⣿⣾⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⢉⢼⣟⣿⣿⣿⡇⠉⣹⡇⠀⠀⠉⠙⠛⢿⣶⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣛⣿⣿⣿⣿⣿⡿⠟⠁⢸⡄⠀⠀⠀⠙⠻⢿⠿⠃⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠩⠍⠭⠍⢀⡀⠀⠠⢻⣄⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡍⠀⠄⡠⠄⠀⠉⠑⠒⠶⠤⠤⠾⠥⠤⠴⠖⠒⠚⠋⠉⠉⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡆⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢻⡄⢜⡵⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣤⣄⣈⠳⣯⣬⣿⠀⢧⠀⣺⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢧⣾⣿⣿⣿⣿⣿⡏⣿⡿⠃⢠⣼⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⢇⣿⡗⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢸⢹⣿⣿⡿⣼⣿⠃⠀⢰⣽⣿⣿⣿⡿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⡀⠀⠀⠙⠄⠀⠀⠀⡜⣼⡿⢫⣾⣿⣿⠀⠀⣿⣿⣾⣿⣿⣿⣟⢿⣿⣿⣯⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠛⠀⢀⡰⣾⠀⢀⡼⠰⠯⠴⠛⠉⠀⠈⢀⣠⣿⣿⣿⡟⣿⣿⣿⣯⣻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠦⡄⠀⠘⢣⡌⠉⠙⠁⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣷⣻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡦⢄⢀⣏⠀⠀⠀⠐⠂⠀⠈⠉⠀⠀⠙⠿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⠠⠤⠤⠤⠄⠀⢀⣠⠔⠺⣿⣷⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⣉⠛⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢀⣀⠀⢀⣚⢁⡠⠋⠐⠒⠠⠤⣤⣀⣀⡀⠈⠁⢀⣀⠼⠟⠉⠀⠈⢫⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣬⡿⣿⠛⠋⣿⠿⣿⣿⣿⣿⣿⡇
⠩⠟⠓⠉⣠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⡜⠡⠾⣿⣿⣯⣿⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣶⣦⣤⣴⣶⣾⣿⣿⣿⣷⣤⡎⠀⢀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠚⠿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠠⠤⠴⠚⠛⣯⣿⣿⣿⣿⣿⠿⢿⡗⠙⢿⣿⣿⣷⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠠⣴⠶⠒⠀⠀⠀⡀⠤⠚⠉⠁⢈⣿⠉⡿⣅⠀⠀⠸⡀⠀⢻⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠲⠒⠒⠚⠛⠓⠂⠂⠀⠒⠛⠁⠀⠐⠓⢺⡀⠀⢀⢱⡀⠘⠿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠘⠓⡄⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⣀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⡤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠛⢛⣩⡽⠿⠟⠛⠳⠶⠦⣤⣉⠉⠉⠙⠛⠻⠷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⡄⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣤⣄⣤⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣷
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠙⠳⣤⣀⣀⠀⠀⠀⣀⣀⣤⠶⠛⢁⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⢤⣤⣬⣭⣭⣿⣯⠭⠽⠶⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣭⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠟⠷⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⠶⠶⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣤⣤⣶⣴⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠗⠲⠶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣁⣤⣤⣤⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠓⠶⢶⣬⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠁⢀⣀⣀⣿⣿⣦⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠛⠳⣦⣌⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⠇
⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⢻⣿⣷⡾⠋⢉⣁⣤⣿⣿⣧⣀⣴⣾⣿⣿⠿⠿⠿⠿⠿⣿⣶⣤⡀⠀⠀⣠⣿⣿⡛⠛⠶⣄⠘⣿⣿⡇⠀⠀⢀⣴⣿⠟⢹⣿
⠀⠀⠀⠀⣸⡿⣿⣧⠀⠀⠀⠈⣿⣿⣷⡿⠛⣉⣠⣤⣿⣿⡿⠟⠉⠀⠀⠀⠀⣀⣤⣤⣾⣿⣿⣷⣴⣾⣿⣿⣅⠙⠻⣦⣈⣿⣿⡿⠀⢀⣴⣿⡿⠋⠀⢸⣿
⠀⠀⠀⢀⣿⡇⠸⣿⣧⠀⠀⠀⠘⣿⣿⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⠉⠉⣹⣿⡿⠁⠉⠙⢿⣦⣈⣿⣿⣿⣷⣾⣿⡿⠋⠀⠀⠀⢸⣿
⠀⠀⠀⣼⣿⠁⠀⢹⣿⣦⠀⣠⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⣄⡀⠀⠉⢿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⣸⣿⠇⠀⣤⡀⢻⣿⣿⣿⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠛⠉⠉⠉⠉⠙⣛⣿⣦⡀⠀⠉⠈⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⣰⣿⡏⠀⠀⢹⣷⠀⢻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠛⠛⠛⠛⠀⠀⠀⠀⠙⢿⣿⡄⠀⠀⠀⠀⠀⢸⣿⡆
⢠⣿⡿⠀⠀⠀⣼⣿⣇⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⠀⠀⠘⣿⣷
⣼⣿⠃⠀⠀⢠⣿⣿⣿⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⠿⢿⣿⣿⣦⣄⠀⠀⢠⣿⣿
⣿⣿⡀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠈⠻⣿⣿⣆⣠⣾⣿⡿
⢿⣿⡇⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⢀⣼⡿⠟⠛⠛⠛⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣀⣠⣀⡀⠀⠀⠀⢹⣿⣿⣿⣿⠟
⢸⣿⣧⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠻⣿⣧⠀⠀⠸⣿⣿⡿⠋
⠈⣿⣿⡄⠸⣿⡟⠉⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢹⣿⡄⠀⠀⣿⣿⠁
⠀⠘⣿⣿⡀⠉⠁⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀⣀⣴⠇⠀⠀⠀⠀⢸⣧⣴⣶⣶⡄⠈⣿⡇⠀⢀⣿⣿
⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠈⠛⠗⠀⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣷⣶⣦⣄⠉⠛⠛⠛⠉⠀⢠⡆⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⠃⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⣠⣾⣿⡿⠿⠛⠛⠛⠛⠻⣿⣿⣦⡀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⣼⣿⠀⠀⣼⣿⠁
⠀⠀⠀⠀⣿⣿⡟⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠛⣶⣾⡿⠟⠋⢁⣀⣀⣤⣤⣶⡾⠿⠿⠿⠿⠷⠀⠀⠀⢻⣧⠀⢀⣼⡿⠿⠿⠿⣷⣿⡏⠀⣰⣿⠃
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠁⠰⠶⠛⢻⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠈⠀⠀⠀⠀⠀⠀⠉⠀⣴⣿⣧⡀
⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣴⣶⡄⠀⣠⣤⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠃⠀⠀⠀⠀⠀⠸⣿⡄⠀⠿⠿⠰⣿⠇⠀⣼⡿⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠛⠳⢶⣤⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣾⣿⠿⣆⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⡿⣿⣿⠃⠀⠿⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡻⣿⠿⣿⡿⢿⡿⠟⣿⠉⣿⣴⣿⠏⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⢻⣿⣿⣶⣿⣦⣼⣷⣤⣿⣶⣿⡿⠃⠀⢠⡀⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣴⣿⣧⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⡿⠿⠋⠀⠀⢠⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⢰⡆⠀⠀⠀⠀⣠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⢸⣷⡀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣄⣿⣷⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿

⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⢀⣿⡄⠀⠀⠐⣄
⠀⠀⠀⠄⡀⠀⠀⠂⣤⡀⠘⣿⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠫⢝⣶⣄⠀⠀⠀⠀⠀⡖⠀⢹⣿⠀⢠⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⢀⡀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠙⣆⠀⢀⠀⣱⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠖⠀⠀⣠⠀⠀⠀⠀⠁⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⠀⠀⠀⠀⠙⠇⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠋⠁⠀⣀⣤⣤⣀⡀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠟⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⣴⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡆⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠘⡆⠸⡄⠸⡇⢰⠃⢠⠞⠁⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢦⡤⠷⠄⢱⠊⣕⠙⢒⠿⣤⣼⠁⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠈⡷⣊⠐⠀⠁⠀⠀⠀⠀⠉⠑⠋⠶⣋⣥⠴⠒⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢄⠠⣄⢠⡄⠀⠀⠀⠀⠀⠀⢤⠇⠀⠁⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⢹⠶⠶⠒⠂⠀⣤⣴⡊⠈
⠀⠀⠀⠀⠈⠑⣦⣬⣿⣿⡿⠛⠉⢐⡀⠀⠀⠀⠈⢛⡂⠀⠀⢀⢜⡌⣿⣿⣿⣇⣀⠄⠀⢘⠛⠁⠀⠀⠀⠙⣿⠇
⠀⠀⠈⠒⠲⣶⣿⣿⣿⠏⢀⠆⠀⣈⠕⠀⠀⠀⠀⠈⠪⣀⣌⣿⣼⣿⢿⠿⡿⣿⣿⣤⡀⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠐⠦⣴⣿⣿⣿⡍⠀⣼⠂⠁⠀⢠⠀⠀⠀⠀⠀⢀⡄⠻⣝⠹⡆⠈⡆⠀⣿⠁⢸⠟⢠⡄
⠀⠀⠀⠤⣼⣿⣿⣿⠀⢴⣷⡏⠀⠀⠀⠀⠁⠀⠀⠒⢄⠙⣆⣸⣄⣷⣤⣷⣤⣿⣤⣿⣰⢋⡰⢂⡄
⠀⠀⠀⠠⣼⣿⣿⡷⠀⠀⠛⠁⠀⠀⠀⢸⠀⠀⠀⠈⠚⠻⠛⠋⠉⠁⢹⣿⠀⠀⠀⠉⠙⠛⠓⣋⡤⢄⡀
⠀⠀⠀⠀⠬⠽⣿⣿⡂⠠⡉⣢⠀⢀⣤⣟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠀⠀⠀⠀⠀⠀⠈⠛⠻⣯⡕⡤⢀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠤⢘⣾⣿⠟⠁⠈⢻⡷⠀⠀⠀⠀⠀⠀⠀⢠⡙⠁⠀⠀⠀⠀⢀⠿⢷⣤⡀⢛⣿⣎⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠁⠀⠀⣻⡆⠈⣽⣶⣂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⡔⠀⠀⠀⣿⣇⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⣞⣷⡤⠀⣠⣿⠉⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⠀⠀⢠⡅⠀⠀⠀⠀⢼⠁⠀⠀⠑⠐⠻⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡂⠀⠀⠸⠁⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⡼⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢄⠀⠳⠄⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠈⠀⠰⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⢀⢨⢿⠠⠶⣾⣁⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣼⣿⡎⠀⠀⠀⠐⣿⣿⡄⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢆⠀⣿⣿⣷⢀⢀⠀⣨⣿⣿⡇⠀⠀⢀⡴⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⡆⠈⢛⣾⣾⣾⣷⣿⣭⣤⣁⡀⢀⢾⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠟⠙⠛⠉⠁⠉⠉⠙⠹⣿⣻⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢊⣶⠔⢀⡀⠀⠀⠀⠀⠀⣀⢠⣿⣤⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⣤⡴⠊⡐⡂⣺⢿⣿⣾⣤⣴⣆⡄⣄⣮⣷⣿⣭⠭⢳⠀⠈⠳⣤⣄⣄⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠚⠓⢚⠍⠁⠀⡰⠃⠉⢉⣹⣿⠻⠿⡏⡿⣟⠿⢻⣙⣓⠚⠃⢣⠀⠀⠈⠛⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⢁⣞⣯⣽⡃⠀⠀⠁⠀⠈⠁⣀⣠⡏⠿⣭⣛⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠈⢠⡙⣆⠋⠹⠓⠶⠠⠖⠒⠚⠛⠉⠁⠘⠄⢀⡀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢠⠁⠀⠈⣇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠇⠀⠰⠀⠘⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠞⣽⠟⠒⠀⠄⠄⠀⠀⠤⠠⠐⠊⠈⠙⠀⢲⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⢇⡾⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣧⠀⠀⠻⣝⡂⠃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠿⠾⠧⡀⠀⠀⠀⠀⣀⣀⡥⠘⠏⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⢀⡰⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⠶⠤⢄⡀⣄⠤⡤⠾⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣦⣶⡶⣒⣶⡶⣿⣿⣿⣷⣶⣦⣔⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⣻⡿⠿⢫⡽⠟⠛⠛⠛⠛⠛⠿⠿⣻⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣶⣶⢾⣿⣿⣻⣿⣿⣿⠿⠿⠻⠟⠱⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡫⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣿⣿⣿⡿⣫⣿⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡿⣿⠟⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢀⠀⠀⢀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠿⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣷⠶⢿⣿⣿⣿⡖⢶⣿⣤⣶⢁⡞⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢿⣏⢹⣆⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣵⣯⣋⣴⡿⣁⣰⠆⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢿⣇⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣶⣶⣮⣭⣛⣿⣯⣴⣾⣿⣾⡿⣃⡤⠎⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⠶⣵⣬⣿⡿⢿⣿⠟⠛⢿⣿⣿⡉⠙⠛⢻⣿⡇⠉⠛⠷⣮⣝⣿⣿⣿⣯⣠⣴⠟⣣⡤⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⣶⣾⡿⠛⠀⠘⣿⣶⣶⣿⣿⣿⣿⣿⡿⣦⣙⣷⠀⠀⠀⠀⠉⠛⣼⣿⣿⣿⣿⣟⣩⡤⢚⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⠿⠁⠀⠀⠀⢻⣷⡞⣿⣿⣿⣿⡿⣿⣿⣿⠃⠀⠀⠀⠀⢀⠔⠀⢿⡙⠻⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⠀⠀⣀⣤⣾⠏⠀⠀⠀⠀⠀⠀⠈⢿⣾⡿⣭⣿⣯⣽⣿⠟⠃⠀⠀⠀⡠⠔⠁⠀⣔⡛⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⡙⢆⠀⠀⢀⣀⡀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠀⠀⠀⠠⠐⢀⣤⣾⡿⢿⡝⢝⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⣹⡼⢋⡴⢬⣥⢑⠖⡒⠀⣀⠀⠀⠀⠀⢀⢠⣠⣤⣿⡿⠿⣧⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠏⠐⠹⡇⠋⡞⢡⣷⢟⡇⢿⡆⢷⡟⡆⠳⡙⣿⡇⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠘⠛⠀⠇⢻⠆⠸⠇⠃⠀⠁






⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⠲⢦⣄⣀⡀⠀⠀⣀⣠⡴⠶⠛⠻⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⢀⠀⠁⠀⠀⠀⠀⠉⠉⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣠⡤⣀⠈⠶⢤⣀⡀⠀⠀⠀⢀⣀⣤⡶⠂⠶⠶⢁⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣠⠖⠋⠘⠃⠀⠀⡀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠈⢦⠊⠀⢢⡝⠢⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⣠⣾⣿⣦⣤⣦⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷⣄⣨⣦⡀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣤⣤⣤⣤⣴⣦⣤⣤⣤⣴⣾⣶⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢻⣿⣿⡿⠋⠉⠀⠀⠀⠉⠉⠉⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠻⣿⣿⡟⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠸⡏⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢨⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠶⠤⠤⠤⠶⠖⠚⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠉⠉⠉⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢰⣶⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⣼⣿⣀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠋⠛⠃⠀⠀⠀⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠀⠐⠁⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⢀⣠⣴⠿⠛⠛⠛⠛⠛⠛⣿⠟⠀⠀⠀⡶⠻⣿⣶⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠛⠛⠛⠻⠻⢶⣤⡀
⣤⣶⠟⠉⠀⠀⠀⠀⠀⣤⣤⣤⡏⠀⠀⠀⠀⠖⠀⠈⣿⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⡄⠀⠀⠀⠈⠙⢷⣦
⠈⠛⠷⣦⣀⠀⠀⠀⠀⠈⠉⢹⠇⠀⠀⠀⠀⠈⣄⠀⠀⠀⠉⠉⠉⠀⠉⠙⢶⡄⠀⠀⠀⠀⢫⠉⠉⠁⢀⠀⢀⣠⡶⠟⠋
⠀⠀⠀⠀⠙⠻⠿⠶⠶⠶⠶⣿⠀⠀⠀⠀⢠⠰⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢈⢷⠾⠶⠾⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠠⠄⠀⡠⠂⠀⣀⣬⡆⠀⠀⠀⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠞⢹⣷⣄⣀⠀⠀⠀⠀⠀⣠⡾⡏⡸⣿⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣄⠀⢊⢠⣿⠻⣿⠆⠀⣠⣿⣻⠀⢠⠇⠙⠀⠀⠀⠀⢆⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠘⣆⠃⠀⠈⠒⠐⠊⠏⢀⠀⡙⠁⠉⠁⠞⠀⠀⠀⠀⠀⠀⠸⡆⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡇⠀⠀⠀⠘⣆⠁⠀⠀⠀⢀⠀⠀⠀⢣⢀⡀⠀⢃⡇⠀⠀⠀⠀⠀⠀⣯⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠇⠀⠀⠀⠀⣿⡄⡀⣀⣠⣼⣦⡀⠀⢸⣎⣁⣠⢸⡇⠀⠀⠀⠀⠀⠀⣶⠀⡄⢸⡏⠢⡀
⠀⠀⠀⢀⡤⠂⠁⡌⠙⠀⠀⠀⠀⠀⠀⣰⠟⣹⠇⠈⢷⣮⣭⡿⣶⣾⣫⡟⠉⡃⠃⠀⠀⠀⠀⠀⠀⡏⠀⡀⢸⡇⠀⠘⡆
⠀⠀⠀⡎⢸⠀⠀⢣⠀⠀⠀⠀⠀⠀⣨⣴⣾⣿⠀⠀⠀⠙⠿⢿⣿⠿⠛⡀⣰⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⣷⣿⡁⠀⠇⡇
⠀⠀⠀⠘⢆⠀⠀⠀⢣⡀⠐⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠲⠤⠤⠴⠊⠀⡿⠀⢠⠁⠀⠀⠀⢠⠏⠀⢀⣿⡿⠁⠐⣰⠃
⠀⠀⠀⠀⠀⢧⠀⠀⠀⢷⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣿⣿⣦⣾⠁⠀⡎⠀⠀⠀⢠⠏⠀⠀⢸⠟⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⣇⠀⢠⣻⡛⠿⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⡰⠋⠀⠀⡰⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠘⠳⣦⣄⣽⠀⠀⠹⣿⣦⣀⡀⠀⠉⠉⠁⠀⠘⢿⣿⣿⣷⠀⢸⣷⣤⡞⠁⠀⡠⠊⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⣤⣀⣼⣿⣅⠀⠀⠢⢤⣀⠀⠀⠀⠈⠉⠹⠀⢸⣿⡟⣠⠖⢊⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠯⠭⠑⠊⠛⠉⢻⠶⠔⢺⣦⣄⣤⣠⡶⠀⣾⣿⡧⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⣿⠈⠻⣭⡥⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡠⠀⠀⠀⠀⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢣⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠰⡀⠀⠀⠀⠀⠀⣠⢎⡕⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠀⠀⠳⡀⠀⠀⠀⡴⢡⠎⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡈⠻⡲⢄⠀⠀⠀⡠⠜⠀⡿⠀⠀⠀⢱⠖⠑⠋⣰⠏⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠈⠢⡱⣄⠀⡇⠀⠀⣇⠀⠀⠀⢘⠀⠀⣰⠏⢀⣄⠀⣙⠀⠀⠀⠀⣀⣤⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠙⢮⡑⠋⠳⠀⣿⠀⠀⣀⠜⠀⢀⡟⡰⡟⢠⠀⢧⠀⢀⢔⡫⠋⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠳⣄⠀⢇⢹⡀⢸⠀⠀⠀⣸⠏⡜⢀⡜⠀⡰⢟⡵⠋⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⢦⣀⠀⠙⣆⠈⠛⠧⡀⣇⠀⢀⡏⡼⠀⡜⡔⠉⣴⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠻⡳⡄⠘⣧⠀⠀⢃⡸⠓⠊⡼⠁⠀⢧⣹⡾⠁⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣄⡀⠀⠀⠀⠘⢆⠘⢮⢢⣘⣧⠀⢘⡂⠀⣸⠃⠀⠀⣸⡟⠁⠀⢀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⠛⢕⠢⣀⠀⣸⠀⠈⢳⡄⠈⣧⣿⠇⢀⡏⠀⠀⠀⢻⠀⠀⠰⣋⡠⢄⡶⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠙⢮⡉⠉⢄⠀⠀⠹⡄⠸⡉⠀⣸⠀⠀⠀⢀⡼⠄⠒⣋⡵⠚⢁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠙⢦⡎⠀⠀⠀⠹⡄⠈⠱⡇⠀⠀⣰⠛⢁⡴⠞⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⢻⡖⢤⡀⠀⠹⡄⠀⡄⠀⠀⢱⣴⠋⠀⠀⠀⠀⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠆⠀⠀⠹⣤⡃⠀⠀⢳⣸⠃⢀⣴⠟⠁⠀⠀⡔⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠖⠒⠀⠛⠘⠒⠀⠘⠛⠚⠃⠛⠂⠐⠒⢖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⡀⠀⠀⠀⢀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⢰⠀⠰⢶⠄⠀⠀⠀⠀⡁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛⠆⠠⣾⠇⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⠀⠀⠀⠆⣰⡿⠀⠀⠀⡀⠀⠀⠀⠀⠀⣟⠀⠀⠀⣴⡀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉⠁⢀⣴⣿⣷⣄⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠐⠲⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡹⠆⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣰⣏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⣸⣷⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾⡇⠀⠀⠠⣶⣧⠀⠸⡇⠀⠈⢱⡄⢨⠁⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⣼⣿⣦⣎⣿⣄⠀⠀⠀⣀⡚⠂⠀⠀⠀⠈⠁⠀⠁⢰⣿⠁⠀⠀⠀⢀⠀⣈⠀⠊⠉⣿⡷⠆⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠐⢿⣿⡟⣿⣿⣯⣿⠀⠀⠈⢻⣅⠀⠀⠀⠁⠰⠦⣤⡿⢘⣿⡷⣄⠀⠀⢠⣓⣻⣟⣷⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠐⠿⡿⢿⣿⡿⣰⡿⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠇⠈⠀⠀⢿⠁⢈⣿⣿⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠠⢤⠃⠀⠀⠑⠀⠀⡀⡅⡟⠀⠀⢠⡇⠀⣺⡇⠘⠁⠀⣘⢁⡀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢶⡟⠀⠀⠀⢧⡄⠀⠀⢸⡇⢰⠁⢘⠇⠀⠀⠀⠈⣿⡄⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠈⠀⠔⡖⠀⠚⡁⠠⠀⢸⡇⣿⠦⣼⠆⢀⠀⠎⠀⣽⠇⠀⠀⣷⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡄⠠⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣧⠀⠀⠘⠳⠿⠀⠻⠀⠀⠀⢀⡾⠁⠀⠀⠀⠋⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣦⣄⡀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇⠀⢸⠀⠀⠀⠀⢠⠏⠀⠀⡼⠁⠀⠀⠀⠀⢶⣿⠊⠆
⠀⠀⠀⠀⠀⢀⠀⢠⣌⣦⣼⣡⣄⣡⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⠀⠀⠀⠠⠍⠀⠀⢀⡟⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠞⠛⢛⠷⠂⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠁⠀⢠⠀⢸⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣄⠀⠻⡀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠁⢠⠟⠀⠀⠀⠀⠀⠘⠃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢹⣷⡄⠀⠀⡆⠀⠀⣿⠀⠀⠆⠀⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠏⣠⣠⡀⠀⣻⡅⠀⠀⠸⣴⡤⠁⠀⢠⡄⢀⠇⢀⡼⠁⠀⠀⠀⠀⠀⠀⠠⣆⣀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣴⢿⣝⣿⣾⣿⡇⠀⠀⢀⡍⠀⠀⠀⣸⡟⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠳⠀⢠⡞⢥⠈⣻⣿⢫⡛⠄⠈⠹⣿⣶⡤⢠⠀⠀⠀⣰⣿⣁⡼⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⣤⠈⠉⠀⣀⠀⢠⠤⣶⡄⠀⢈⣿⠻⡦⡹⢷⣶⣿⣿⡏⠀⣾⣿⣤⣶⣿⣍⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠰⡄⠀⠀⢈⡟⠻⠈⠳⡉⠀⢰⡏⣴⣆⣶⣿⣿⣅⣤⣼⣟⢳⣄⡙⢃⣠⣌⣛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠠⠿⢤⡔⠲⠤⠒⢲
⠀⠀⠀⢠⣶⣠⡄⢈⣥⣤⣤⡴⠛⢾⠛⣷⠀⠉⠉⡩⢤⣤⡀⠓⣫⡛⢟⣁⣙⠛⢿⣧⠀⠀⠀⠀⠀⢀⡀⠀⠰⡄⢀⣉⣻⠒⠚⠟⠀⠀⣀⠠⠂
⠀⠀⠀⠸⣷⠞⠾⠋⠀⠈⠹⣧⠀⠚⢣⢾⡿⡶⠾⣤⠼⡿⠙⠿⠿⣥⠿⠼⠿⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠒⠲⠀⠐⠀⠁⠀⠀⠂⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠉⠛⠦⣤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢺⠗⠛⠲⣄⠀⠀⣀⣀⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⢀⣠⡴⠚⠉⢰⠏⠀⡀⠀⠘⢿⡏⠁⠀⠀⠀⠀⢀⡏⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠳⢤⣄⣀⡟⠀⠀⢻⠀⠀⠘⢿⣶⣤⣤⣄⣀⣼⡁⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠸⡆⠀⠈⠻⣇⠀⠀⠈⢧⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣿⣿⣄⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠈⠳⢦⣄⡈⢧⡀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠿⠿⣿⣿⣿⣦⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠈⠙⠳⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⣄⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣹⣿⡿⠿⠒⠒⣺⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠉⠙⠓⠲⠦⠤⠴⠞⠋⠉⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⢶⡶⠶⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣤⡖⠋⠉⠙⡇⠀⠀⠀⠘⡇⠀⠀⠀⢸⠋⠉⠙⢲⣤⣀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣴⠋⠁⠀⣇⠀⠀⠀⣧⣀⣀⣀⣴⣧⣀⣀⣀⣼⠀⠀⠀⡸⠀⠈⠙⣦⡀⠀⠀⠹⡄
⠀⠀⠀⠀⢠⠏⠀⢀⣴⠋⠈⡄⠀⣠⣾⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣧⣀⠀⢠⠁⠙⣦⡀⠀⠹⡄
⠀⠀⠀⠀⡾⠀⢠⢏⠀⢆⣠⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣷⣄⡰⠀⡿⡄⠀⢷
⠀⠀⠀⠀⡇⢠⣏⢦⣶⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣄⣹⡄⢸
⠀⠀⠀⠀⣇⠈⠻⣿⠻⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⡟⣿⠟⠀⣸
⠀⠀⠀⠀⢹⡀⠀⠈⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠛⠁⠀⢠⡇
⠀⠀⠀⠀⠀⠳⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠦⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠁⠀⠀⠀⠀⠎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠹⣄⠡⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡈⠑⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠔⠁
⠠⡄⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣶⠞⠁
⡆⢰⠀⠀⠀⠀⠀⠀⠀⣸⠉⠛⠿⠷⠶⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠑⠈⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠘⣜⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡁
⠀⠀⠘⣮⠢⡀⠀⠀⠀⢿⡀⠀⠀⠀⢠⡥⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠎
⠀⠀⠀⠈⢷⡄⠀⠀⠀⢸⡇⡄⠀⠀⡟⠀⠀⢀⣠⣴⠶⠶⠶⣶⣤⡀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣠⠞⢡⡾
⠀⠀⠀⠀⠀⠘⢧⡀⠀⢘⣇⢳⡄⢸⠃⠀⠔⠋⠁⠀⠀⠀⠀⠀⠈⠙⣦⡀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠙⡇⠀⠠⠈⣡⠶⠩
⠀⠀⠀⠀⠈⠀⠘⡇⠀⠀⢻⡄⠹⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢻⣿⣶⡆⠁⠀⠀⠀⠀⠀⡎⠀⠀⠀⣀⣠⡄⠀⠷⠀⢀⣼⠃
⠀⠀⠀⠀⠀⡀⠀⠁⠀⢹⣮⣷⠀⣿⠀⠀⠀⢀⣠⠔⢋⡀⣀⣀⣀⠉⢢⡀⠉⢿⠁⠀⠀⣀⣀⣀⣀⠀⣴⡶⠟⡉⢀⠀⠀⢰⡄⡎⠀⠀⡰
⠀⠀⠀⠀⠀⠱⡀⠀⠀⢸⣿⠏⢻⡏⠀⠀⠀⠀⠈⠃⠠⠄⢻⣶⣋⣸⡁⠲⠀⠀⠁⠒⠉⠀⠀⠈⢻⠞⢛⢊⡥⣤⠀⠙⠀⣼⢣⠀⠀⣎
⠀⠀⠀⠀⠀⠀⠙⢄⠀⠈⣿⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠔⠀⠀⠀⠀⠀⠀⠀⢀⡞⠐⠀⠤⠙⠋⠤⠀⢀⡉⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⠓⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⠄⠀⠓⠤⠤⠤⠴⠂⠀⣷⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⢀⣸⣇⣠⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⢦⠀⠀⠀⠀⠀⠾⠶⣤⡀⠀⠀⠀⠀⠀⠀⢸⠆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠚⠁⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠀⠀⢀⡆⠐⡀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢹⣯⣐⠂⠀⠀⠤⠴⠒⠒⠉⠉⠉⠓⢤⣤⡀⠀⠀⠀⠀⢰⣨⠘⠁⣀⣵⣤⣄⣴⡟⠀⠀⡁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣿⠉⠛⠟⠛⠛⠛⢻⣿⡛⠻⣶⣤⣤⣿⣷⣄⣀⣀⣠⡾⢳⣶⣾⡟⠋⠠⠏⠙⠛⠳⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠀⣄⠉⠐⠎⢛⡋⠈⣿⣿⣿⣿⠋⠀⡀⠙⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄⡈⠳⣄⠀⠀⠀⠙⣿⣿⣿⠏⠈⠁⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠂⠀⠀⠙⠲⣬⡳⢄⡀⠀⠙⠿⠋⠀⢀⠔⠁⠀⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠀⠀⠀⢦⡀⠀⠙⠲⡍⠛⠀⠀⠃⠀⢀⠌⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠰⢄⡈⠒⠦⠴⠆⠘⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⣀⣤⣴⣾⣿⣿⣷⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣁⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⡅⠀⠀⢀⡗⠀⠀⠀⣠⡤⢠⢀⢄⣠⣀⢄⢠⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⢢⠀⢻⢛⡛⢿⠃⡀⠀⠀⠙⢦⡀⢢⣾⡷⣿⣦⠂⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠣⢼⣯⣽⡯⠔⠁⠀⠀⠀⠀⠹⣦⣀⣹⣋⣠⡼⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢎⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⢀⣺⣿⣷⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠸⣶⣧⡶⠄⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡠⣼⠿⢿⢤⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⣯⡧⠀⠀⠀⠀⠀⠈⡜⢹⣄⣸⢱⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠈⠁⠈⠻⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣽⣷⣇⠀⠀⠀⠀⠀⢐⣛⡂
⠀⠀⠸⣿⠇⠀⠀⠀⢀⣿⣷⣾⣧⠀⠀⢀⡜⠻⡿⠛⡀⠀⠀⠀⣿⣽⣿⣯⡟⠀⠀⠀⠀⠹⣿⣿⡐⠄
⠀⠀⠀⣉⠀⠀⠀⠀⢸⢀⢠⢇⣭⡇⠀⠀⣵⣦⣶⣶⡉⠀⠀⠪⡇⠈⠛⢀⣶⡅⠀⠀⠈⠁⢸⡏
⠀⠀⠀⠉⠀⠀⠀⠀⠰⠻⠾⠾⠣⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠰⡹⣴⣶⣿⢿⠃⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣟⣛⣛⣛⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠐⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢛⣉⣤⣤⣤⣴⣶⣶⣶⣤⣤⣬⣉⡛⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣍⣛⠛⠶⣭⣛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠿⢟⣛⠋⠉⠙⣛⣛⣛⠿⠿⣿⣿⣿⣶⣤⡉⠛⢮⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡿⣟⣯⣷⣾⣿⣿⣿⣷⣶⣾⣿⠿⢿⣿⣿⣶⣮⣝⡻⣿⣿⣷⣤⡈⠓⠍⠳⠄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⢞⡽⢟⣵⣿⣿⣿⣾⣿⣿⠟⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠘⣿⣟⢿⣿⣿⣿⣿⣿⣿⣟⢦⣄
⠀⠀⠀⠀⠀⠈⠉⠵⠚⠉⠐⠉⣻⣿⣿⣿⣿⠟⠁⣸⣿⣿⠿⣿⣿⣿⣿⣄⡀⢀⣠⣿⣿⡆⠙⢿⣿⣿⣿⣿⡋⠑⠈⠓⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠇⢿⣿⠏⠀⠀⢿⣿⡿⠀⣿⣿⣿⣿⣿⣿⡟⢩⣿⣿⡇⠀⠀⢻⣿⡇⡿⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⣷⣄⡈⢿⠀⠀⠀⠘⣿⣿⡄⠈⠻⠿⠿⠿⠋⢀⣽⣿⣿⠁⠀⠀⢸⠟⣀⣴⡿⣟⣫⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠙⢿⣶⣥⡀⠀⠀⠘⢿⣿⣦⣄⣤⣠⣠⣤⣾⣿⡟⠁⠀⠀⣀⣴⣾⣿⡿⣻⡽⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣐⢄⡙⠻⣿⣷⣶⣤⣄⣉⡛⠻⠿⠿⠿⠟⣛⣁⣤⣴⣶⣿⣿⡿⣿⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣭⣒⠤⣉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⡿⢟⠽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢮⣥⣄⣀⣀⡉⠉⠉⠉⢁⣠⡶⠟⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⢠⡄⡀⠀⡄⠀⡀⠀⡆⡄⠀⠀⠀⢦⢀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢄⢣⠹⡘⡄⠌⡆⡇⠇⠇⡃⠃⡇⡇⡇⡇⡂⠀⢈⡙⡌⢀⠆⢂⢆⢎⢄⠔⢗⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡳⡙⢄⢆⢣⠡⠱⡱⠀⠀⢠⠀⠀⠇⠄⠇⡇⠁⡇⠃⠈⣠⠳⠀⣀⠂⠂⠂⠢⡂⡠⠋⡠⠂⡪⡫⠂
⠀⠀⠀⠀⠀⠀⠀⡠⡔⢦⡀⣔⠀⠈⠪⠢⠣⠱⡁⠡⠀⠀⠀⠀⠀⢀⠅⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠔⠜⢐⣌⢌⢈⢜⡔⡡⡢⢂⢖⡀
⠀⠀⠀⠀⠀⠀⠀⢩⡪⡑⢍⠪⡑⢄⠀⠀⢔⠐⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣴⣶⣶⣶⣽⣿⣷⣶⣖⡒⢬⣠⡀⠐⡑⡵⠱⡫⠊⢎⢔⡵⡪⢆⠀⣀
⠀⠀⠀⠀⢠⣙⠦⡐⢌⡢⣑⢕⠪⡂⠑⠄⠀⠁⢀⡀⣠⣴⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣶⣦⣿⣄⡬⣘⠔⢑⣡⡫⢞⢵⡺⢽⠂
⠀⠀⠀⡀⠀⡪⢕⠪⡑⢬⡂⢕⠀⠊⠀⢀⡤⠖⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣯⡪⠒⠉⡪⢝⡠⢄⡀
⠀⠀⠲⠭⣓⠩⢒⠉⠢⠔⡈⠓⠀⣠⣘⣫⣷⣾⣿⣿⣿⠿⠿⠿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⡐⠯⢗⠺⠍⣓
⠀⢠⡀⠬⣒⠩⢐⠠⢀⠀⠂⣠⣾⣿⣿⠿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡓⢒⣍⢉⡅
⠀⢖⣢⠭⠐⡀⠀⠈⢀⠈⣴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣦⣄⣶⣿⣮⣿⣿⣿⣾⣿⣿⣿⢿⣿⣿⣿⡿⠇⠯⡐⠒⠊⠉⣿
⢀⣀⡒⠊⠤⠄⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡾⠿⠿⠿⠛⠛⠛⠚⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣾⣾⡿⣿⡷⠠⠤⠖⠌⠛⢲⡒
⢐⣒⣒⠒⠈⠁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠋⠁⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣒⣒⣒⣒
⠀⠀⢒⣒⣒⣒⡒⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠛⠉⢀⣤⣶⠾⢛⣿⣿⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣿⣧⠉⠙⠻⣶⣤⡉⠻⢿⣿⣿⣿⡿⠁⠀⠤⠀⠐⠒⠒⠒
⢐⡒⠒⠒⠒⠀⠀⠀⠀⠰⠶⠶⠒⠛⠉⠁⢀⣠⠴⠿⠛⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠘⣿⣿⣦⡀⠉⠛⠛⠁⠀⠀⠀⠀⠬⠨⠤⠤
⠐⠒⣲⣤⠤⠬⠥⠶⠀⠀⠀⠀⠤⠴⠶⣿⡟⠁⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢹⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⠔⢀⡰⠭
⠀⢠⠯⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣅⣺⣿⣿⠀⠤⢀⡀⠀⢀⡲⡄
⠀⠘⠋⠉⠄⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠘⣸⡿⠟⠛⠟⠛⠋⠀⠀⠀⠀⠡⢀⠂⠭
⠀⠀⢀⣨⠅⢐⠨⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣄⣈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣀⣀⡤⠖⠋⢁⣀⣴⣼⡄⠀⠀⠀⠀⠀⠐⡠⢔⡂
⠀⠀⠀⠶⠫⢕⡨⣒⠍⡠⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠋⠉⣠⣤⣴⣶⣿⠿⠋⠃⠀⠀⠀⠐⠄⠠⢔⠪⡐⠎⠁
⠀⠀⠀⠀⠀⢴⡪⢖⡭⡊⢔⠑⠀⡨⠂⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⣀⢀⣀⣴⣠⣤⣴⣶⣴⣷⢿⣿⡿⠿⠛⠉⠀⠠⠐⢔⠠⡢⢐⠄⡠⣐⢄
⠀⠀⠀⠀⠀⠈⠙⠉⡨⢖⠕⡡⢊⢄⠔⠁⡐⡁⡡⠀⠀⠀⠀⠀⠁⠀⠐⠀⠐⠍⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠠⠀⠀⠈⢊⡢⡪⣑⢕⠨⡢⠑⠎⠲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠵⠊⠔⡵⡣⡢⢊⠌⠔⡐⡀⡀⠀⠀⢀⠆⠀⠐⠀⠀⡂⠀⡄⠀⠐⢰⠂⠈⠀⢂⠈⠄⡐⡐⡐⡑⡕⢝⢌⠪⡲⣕⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣔⡵⠃⢊⢎⡔⡜⡐⡐⡘⡜⡘⡂⢀⠀⠘⠇⡄⡇⠀⣾⢀⢰⠀⢃⡄⡖⡜⡜⡜⡜⡜⣎⢪⡁⠁⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢡⢎⡜⠀⠑⣡⢱⢱⠱⢰⠀⠈⡀⠄⡇⡇⠀⡇⠌⠘⡰⢨⠰⠹⢹⡸⡜⠀⠈⠈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠋⠀⠀⠁⠂⠁⠀⠁⠃⠃⠇⠀⠀⠃⠚⠃⠀⠀⠁⠁

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠘⣻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣯⣤⣸⣆⠨⢯⠹⡯⢫⡖⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣁⡄⠰⠂⢉⢡⡀⢑⣎⣙⣿⣷⣅⡸⡅⣿⣽⣋⣌⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⣤⣾⣿⣦⠄⠀⣴⣦⣇⣬⡷⣼⣿⣷⣿⣤⣯⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢢⠤⠒⠒⠒⠂⢒⠖⠁⣇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠞⠀⠀⠀⠀⠀⢇⣀⡔⠀⠑⢄⠀⠀⠐⢁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⠦⡀⠀⠀⠀⠀⠀⠀⢃⠀⠐⢺⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣉⠁⢀⠤⡒⠄⠀⠀⠀⠀⠄⡰⠂⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠟⠀⠎⠀⢻⡇⠀⠀⠀⠀⠀⠁⠀⠀⡤⠑⢁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡩⠦⠴⢀⠰⢠⡄⠤⠁⠀⠀⠀⠀⢀⠃⠀⡠⡅⠀⡐⠀⢸⠂⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠚⣏⠀⠀⠀⣠⡔⠁⠀⠀⠀⠀⠀⠄⠔⠁⠀⡰⠯⠧⢏⢀⡼⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡑⡀⢌⠒⠀⠚⠻⣇⠀⠀⠀⠀⠀⢠⠖⡦⢠⡌⠀⠀⠀⠀⠁⠀⢀⢂⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡔⠄⠳⠄⠀⠤⠎⠁⠂⠀⠀⢀⠃⠎⡠⠢⠊⠀⠀⠀⠀⢀⡰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠐⢳⣶⣶⣦⠀⢀⣠⡤⠐⠁⠜⢰⠚⡄⠀⠀⣠⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠰⣿⡿⣋⠀⠀⠘⡞⠀⠀⠀⢀⡜⠀⠙⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠚⠁⠀⠈⠓⠒⠤⣘⣶⡒⠈⠒⠀⠀⠀⣛⣁⢀⠀⠠⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⡀⠠⣀⢀⡀⠀⠀⠀⠀⠒⠒⣒⣛⣆⠀⠀⠀⠀⡵⠐⠒⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣀⠀⣀⣁⠸⡤⠒⠒⠂⠈⠉⠉⠁⠀⠀⠀⠉⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠠⢘⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠄⠀⠀⠀⣤⡤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠤⣶⠄⠀⠀⢀⠊
⠀⠀⠘⡂⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⣀⠀⠀⠀⠀⣠⡴⣦⡄⠀⠀⠀⣀⡤⠔⠂⢀⣯⡄⡀⠀⠀⠠⠦⠀⠀⠀⡠⠃
⠀⠀⠀⠈⢆⡀⠀⠀⠐⠀⠀⠻⠱⡂⣄⡀⠂⠨⢉⣓⣼⣏⣥⣼⣧⣉⣾⣴⡫⢛⠀⠀⠰⠬⠛⠊⠀⠀⠠⠀⠁⠀⠠⠔
⠀⠀⠀⠀⠀⠱⠄⢆⠀⠀⠀⠀⣠⣀⣀⣀⣀⣈⡠⠾⣤⣘⣿⣷⣿⣟⣃⣽⠷⣤⢤⢄⣠⣤⣄⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠙⢢⣿⣿⣿⡛⢻⣿⣿⣿⠂⠁⠉⠁⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⡿⣿⣿⣟⢿⢿⡋⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢣⣿⠿⣿⡿⢿⡗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⠛⠂⣽⡀⠙⡑⠏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⡿⠿⢿⣋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠑⣒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠿⠶⢶⣤⣤⣀⣠⣤⣴⣶⠶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠍⠉⠉⠉⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠁⠀⠀⠀⠀⠀⢀⣀⣤⢶⡿⠛⠻⠶⢶⣦⣤⣤⣤⡶⠶⠶⠶⣶⢶⣤⡀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣁⣐⣀⣀⣤⣤⣴⣶⣿⣯⣄⣀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣯⣀⣈⣉⣳⣦⣤⣀⡀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⣠⣿⣤⣀⣀⣀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠘⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⡿⠋⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠘⣿⣿⣿⣿⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠁⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⢀⣠⣤⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠿⢶⣦⣄⠀⠀⠀⠀⠀⢹⣇⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠟⠛⠛⠛⠛⠛⠛⠛⠻⣶
⣤⣾⠋⢡⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠠⠀⠀⠈⠙⢿⣦⡀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⢀⣀⣤⣶⡶⠶⠞⣩⡿⠋
⠈⠉⠐⢦⡀⠀⠀⠀⠀⠉⠉⠉⠉⠻⠶⣶⣄⡀⠀⠀⡙⢿⣄⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⣰⣤⣴⣾⠿⠉⠉⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣤⣀⠀⢻⣆⠀⠀⠀⢿⣷⠀⠀⠀⣰⡿⠋⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⣶⣄⢻⡄⠀⠀⢸⣿⠀⠀⣼⠟⠀⠀⣠⣾⢛⣁⣀⡤⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠀⢸⣿⠀⣼⠏⠀⣰⣿⠟⠓⠉⠉⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠺⢿⣦⡀⢸⡇⢠⡟⢠⣾⠟⠁⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⡇⠘⣷⣿⣃⣤⣴⣶⡶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣰⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣶⡶⠶⠶⢾⣿⣿⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠟⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢶⣦⣤⣀
⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⡀
⠀⠀⢸⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⢻⡇
⠀⠀⢸⡇⠉⠛⠻⠷⢶⣶⣤⣤⣤⣤⣴⣶⣾⣛⣛⣛⠛⠉⠉⠉⠉⠉⠉⢹⣏⣿⠉⠉⠉⠉⢛⣛⣛⣛⣻⣷⣶⣦⣤⣤⣶⣶⠾⠟⠋⠀⠀⢸⡇
⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠻⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠏
⠀⠀⠀⠀⠀⠉⠉⣿⠳⠶⢶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠶⠶⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠊⠉⠉⠙⠒⠤⣀⠀⢀⡠⠔⠒⠉⠉⠑⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀
⣠⣖⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠔⠒⠉⠀⢸⠀⠉⠑⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣈⣑⣦
⠘⣟⢿⣿⣿⣿⣿⣭⣭⣉⣹⠉⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣸⣭⣭⣽⣿⣿⣿⣿⠟⣡⠋
⠀⠈⢆⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣶⣶⣤⣤⣤⣤⣼⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡴⠁
⠀⠀⠈⢆⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢀⠞
⠀⠀⠀⠈⢧⠀⠀⠈⠻⣿⣿⣿⠿⠛⠋⠉⠉⠙⠻⣿⠿⠛⠋⠉⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⡿⠋⠀⠀⣠⠋
⠀⠀⠀⠀⠈⢣⠀⠀⠀⢈⠟⠁⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣽⠟⠁⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⢣⡀⢠⠏⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠉⢁⠔⠁⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠳⡎⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡴⠋⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠋⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⢀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠔⠚⠉
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠘⠦⣀⡀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋
⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣶⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⢛⠋⠉⡉⠉⠉⠉⢙⠛⠻⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣷⣿⠛⢹⣄⣸⣤⣼⣦⣴⣷⣤⣶⣤⣏⡀⠀⡔⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣴⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡄⠈⢿⣿⣄
⠀⡀⠀⠀⠀⠠⢄⡀⢈⣲⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣤⣿⣿⣦
⠀⠙⠻⢶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢁⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠉⠙⠻⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠚⠛⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢸⠏⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠉⢉⣉⣹⣿⣿⣿⣿⡁⠀⠀⠀⠀⢸⣧⡀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢉⡭⠶⠛⣛⣿⣿⣷⡀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢰⣿⣏⣀⣘⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⣉⣭⠿⣿⣿⣦⣄⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⠛⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡴⠚⠉⣹⠿⣿⣿⣶⣦⣤⣤⣄⣈⣉⣉⣁⣀⣤⣤⣤⣴⣶⣿⣿⠿⢿⠻⢏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠞⢡⠞⣡⢟⡿⠛⣿⠿⣿⣿⣿⣿⢿⣿⣿⠿⡟⠿⡝⠳⣌⠓⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⢸⠃⢰⠇⠀⢸⠇⠀⢿⠈⠇⠙⠄⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠈⠻⣦⠀⣠⡶⠛⠉⠁⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⣠⡴⠞⠛⠛⠳⢦⣀⠀⠀⢀⣠⣤⠶⠞⠛⠶⢦⣀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⣀⣴⣿⣏⣀⣠⡄⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⣀⡴⠟⢁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣠⣶⣄⠀⠀⠀⢀⣴⠶⣤⣀⣹⣆⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠈⠻⣄⠀⢺⣿⣿⣿⣿⣿⠿⠋⠁⠀⠈⠉⢉⣿⠟⠋⠉⠛⠛⠛⠋⠁⠀⠀⠉⠉⠙⠳⢦⣤⡄⠀⠀⠀⣹⡗
⠀⠀⠀⠀⠀⠙⢧⡀⠹⣿⣿⠟⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢷⣤⠟⠁⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⣰⠏
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣰⠏
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⣰⠏
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣼⠋
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⣠⡞⠁
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣀⣴⠞⠁
⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⠚⠉
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⡀⠀⠀⠀⢀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⠤⡤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⣀⣀⡴⣠⠽⠳⠂⠒⠛⠒⠂⠀⠒⠊⠐⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢀⣜⣷⣀⣴⣷⣾⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⢀⡀⣤⣤⣾⣿⢿⢿⣿⣿⣶⣖⣶⡂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣶⢿⣻⣧⣤⣶⣷⣶⣷⣶⣿⣿⣿⣿⠿⣾⣿⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢤⣻⣿⣿⣵⣾⣿⣿⡿⠟⢻⣿⣽⣿⣿⣛⢿⣿⣯⣿⣯⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣻⣹⣿⠟⢹⣿⣿⣡⣿⣾⣿⣿⣿⠀⠀⢸⣧⠙⢿⣿⣿⣿⣾⣿⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⣾⣏⣼⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠤⣿⣿⠃⡿⠈⣿⣿⣿⣿⣿⡕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⣿⣿⣿⣇⠘⡀⢻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣶⣿⡟⢠⡟⠀⢸⣿⣿⣿⠿⠛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡖⠃⠈⢿⣿⣮⣿⣿⣿⣾⣿⣿⣿⣟⣡⡿⢁⣴⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣄⡙⠻⣿⣿⣯⣿⣿⣿⣾⣿⣿⣾⢿⡛⠛⠙⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠟⠿⠛⠻⣿⠿⠛⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡴⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⢳⡀⠀⢀⣤⣴⠞⠖⠋⠉⠉⠀⢀⡴⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⡀⣷⠐⠋⠀⠀⠀⢀⡖⠀⢀⣴⠟⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡄⣸⠀⢰⡇⠀⠀⢸⣇⡴⠋⠉⠉⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⢚⣱⠏⠀⢸⡇⢀⣠⣞⣋⣀⣀⡀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⡋⠁⠀⢀⣸⠷⠋⠉⠉⠉⠉⠉⠁⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡤⠖⠻⠧⣤⣄⣀⣀⡀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢚⣫⠝⢷⣄⠀⠀⠀⠀⠀⢀⣀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡴⠞⠋⠀⠀⠀⠈⠙⠛⠓⠒⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣾⣷⣄⣽⣤⠤⢀
⠀⠀⠀⠀⠀⢠⠂⠐⢢⡸⣿⣿⣿⣷⡿⢿⣿⣷⡄⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣷⠮⠿⠿⠿⠋⢿⣷⣤⣄⠑
⠀⠀⠀⠀⠀⠈⠙⠢⡀⢹⠈⠛⠛⠋⢸⠘⣿⣿⡇⠀⠏⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢆⠉⠉⢡⡿⢿⠁⠀⠀⡘⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠈⠚⡀⠀⠀⠀⠸⣦⢻⡏⢀⠞⠀⢀⡔⣃⣀⡀⡄⠀⠀⠀⠀⠀⠀⣀⢠⣀⣼⣶⠿⠏⠐⢻⡀⣰⣾⣅⡀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢀⣁⣤⣉⡄⠙⠆⠀⠀⡇⣿⣄⡤⢁⠤⠞⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡄⠀⢤⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄⢄⢀⣏⣿⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠠⢿⠉⢡⠀⠈⢿⣿⣿⠿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢷⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢾⡳⠾⣿⠃⠀⠳⠆⢼⣿⣇⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣇⣀⠄⠀⠀⣏⠙⠻⠏⠉⠉⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢀⣀⣹⡇⠀⣀⣀⠏⠀⠀⡆⠀⠀⠛⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃⠈⠛⠋⠀⠀⠈⠁⢠⣀⣠⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣀⣤⣤⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠗⢿⣿⣿⠂⢼⣭⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠁⣤⣼⠾⠷⠤⢠⡿⣃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⢋⠳⣴⡏⢛⠁⣰⣶⣽⠗⢰⣶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠸⠋⢉⡓⢰⣇⣸⡏⢉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠠⢀⣈⡀⠀⣱⣼⣀⠀⠉⠡⠾⠨⠥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡇⠀⣾⡇⢸⠙⣿⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣀⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣷⠀⣿⡇⢸⢀⡏⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠛⠛⠁⠀⠀⠀⢶⣦⡄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⢻⡇⠸⢸⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢇⣀⣀⠀⠀⠀⠈⠓⠺⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⢆⣤⣐⢒⠂⣠⣺⣟⢙⣷⣷⣶⢶⣤⠀⠀⠀⡀⣤⣴⣶⣞⡟⡟⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠲⠋⣻⡉⠁⠚⢀⠛⠀⠀⢦⣁⠛⢓⣼⣿⣯⣿⣷⡿⢿⣻⡍⠂⠀⣰⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠐⠛⢻⠶⣯⣯⡀⣼⡤⣶⣽⡿⣿⣷⡝⠻⣉⣵⣿⣟⢹⣧⡗⢡⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠃⠈⠓⠛⠛⠛⠀⢒⣾⠓⠖⠚⢿⠝⣪⠟⠛⠘⠷⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⠄⠂⠛⠀⠀⠀⠀⠀⠀⢀⡴⠋⣇⠀⠀⠠⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠈⠙⠆⠘⠉⢿⣿⠚⣖⣖
⠀⠀⠀⠀⠀⠀⠀⢉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⢠⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢶⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠹⢄⠀⠀⠀⠀⠀⠀⣤⣤⡤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠩⠖⠂⠀⠀⣚⡧⠄⠀⠀⠀⠀⠈⠀⠲⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣶⠦⢴⠀⠀⠀⠀⠀⠀⠈⢨⣖⡀⠀⠀⠠⠄⢀⠀⠤
⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⡀⠈⠡⠀⠀⠀⠀⠀⠈⠀⠀⡄⡄⣀⡠
⠀⠀⢧⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠈⢿⠀⢄⢢⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠁⠀⠀⠀⠤⡀⠀⠀⠒⢝⠄⠁
⠀⠀⠀⠸⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢠⠀⠈⡀⠤⠀⡐⠀⠀⢀⠀⠈⡀⠈⢀⠀⡀⠀⠀⠀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠈⢀⢂⠀⠃
⠀⠀⠀⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠶⠀⣦⠀⠀⠀⠀⠀⠀⠾⣸⡜⣯⣗⢯⠵⡆⠀⠀⠀⠀⠠⠒⠤⡀⠀⠀⠀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⢠⡄⣤⠡⣶⣦⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠼⣠⡎⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⢿⣿⣧⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⡁⢹⡂⡈⢀⠎⠀⠀⡁⠀⠂⠀⠐⠀⠀⠐⡘⡿⢰⠀⢀⠂⠀⠀⡄
⠀⠐⠲⣄⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠆⠀⡄⠀⠰⠁⠀⠀⠐⠀⠀⠀⢀⡙⢦⣾⡣⠋⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢈⢳⡁⠀⠀⠀⠀⢈⡗⠈
⠀⠀⠀⠴⠆⠋⠀⠀⠀⠀⠀⠀⠄⠠⠀⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠡⠒⠻⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠉⠀⠉⠓⢤⡴⠋⠁⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣠⡤⠤⣄⣀⣀⣤⠤⣤⣀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣠⠴⠋⢹⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡏⠓⢦⣀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⢀⣠⣴⣋⣡⣤⣶⣾⣯⣀⣀⣀⣼⡀⠀⠀⠀⣸⡇⠀⠀⢀⣀⣧⣀⣀⣈⣽⣶⣦⣤⣈⣓⣤⣀⡀
⠀⠀⠀⠀⠀⠻⣍⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⣽⡟
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠙⠿⣿⣿⠿⠟⠛⠛⠛⠛⠻⢿⡿⠛⠛⠛⠛⠿⣿⣿⣿⣿⡿⠋⠁⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢿⠉⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣶⣶⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀⡀⠀⠀⢹⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠉⠉⠁⠀⠀⠒⢤⡀⠘⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢀⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢰⡟⠁⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⣿⢳⡟⣿⡄
⠀⠀⠀⠀⣰⡝⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠠⢟⠂⣀⡉⣿⡀
⠀⠀⠀⢀⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣏⠀⠟⠉⠀⠃
⠀⠀⠀⡸⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡗⠀⠀⣿⠀⢀⠀⠀⠀⠇
⠀⠀⠀⡇⢸⣿⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣠⠀⢾⠛⠁⢶⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠃⠀⢀⠀⠀⠀⢀
⠀⠀⠀⢿⣿⣿⢻⣤⡀⢀⣠⣤⣤⣤⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣴⢇⡀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⡟⠉⠉⣙⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠠⠘⠃⠀⠀⠎⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⡰⠃⠀⢀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢐⡇⠀⠀⠀⣂⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⣾⠉⠀⠀⢀⡉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢠⠀⠀⠀⠀⠀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠠⠄⠠⠄⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠄⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠱⠄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⠀⠀⠀⠀⠀⠀⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠠⠀⠀⠀⢠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⡇⠀⠀⠁⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⢀⡌⠀⠋⠀⠀⠀⠈⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠙⣆⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠃⠔⠀⢀⣸⠟⠀⠀⠠⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⠇⢰⡄⠀⢀⢀⣼⣿⣿⡏⠀⣀⠀⢸⣿⠆⠀⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢶⣾⣿⣶⣿⣿⣿⣿⠋⡀⢀⠀⠀⣾⣿⠖⡎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣀⣴⣾⣿⣿⣿⣿⣿⠟⢡⣌⡃⠈⠀⢰⣿⠄⣴⠁⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⢉⡑⠖⠀⣸⡿⠶⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠁⢶⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⣶⣿⣯⠑⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⣿⣧⠟⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⡅⢸⣿⢻⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡜⠛⣿⡟⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠿⢸⣿⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡆⢾⣿⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣿⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠐⢒⣲⣾⣦⣀⠀⠛⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣇⣿⣿⡆⢰
⠀⠀⠀⡄⠒⢀⡴⠋⠉⠈⠉⠙⢷⣄⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⢠⣇⣿⣿⣿⢹⠇
⠀⠈⠁⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⢻⣇⠀⠀⠀⠀⠀⠀⢸⣿⣹⣿⣿⠷
⠠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠻⣧⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠙⣧⡀⠀⠀⠀⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠘⣿⡄⠀⣼⡟⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⢿⡶⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠘⠃⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠸⣿

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

⠀⣀⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤
⠀⢣⠀⠀⠀⠀⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⡀
⠀⠀⢑⡆⠀⠀⠀⠠⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⢐⡊
⠀⠀⠈⢷⡀⠀⠉⠀⠈⠦⣀⣙⢦⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣶⠆⠀⠀⠀⠀⠰⣾⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⡴⣋⢀⠴⠁⠀⠉⠀⢀⡾⠁
⠀⠀⠀⠈⢿⠤⠀⠲⠦⣄⣘⠛⢣⡷⣄⠀⠀⠀⢠⠞⠀⠈⠉⠑⣤⣶⣶⣶⣶⡴⠊⠉⠁⠀⠳⡄⠀⠀⠀⣠⢾⡜⠻⣓⣠⠴⠖⠀⠠⡿⠁
⠀⠀⠀⠀⠘⣀⣀⠀⠀⠀⠀⠤⡆⠐⢌⢣⡀⠀⠸⣄⡀⠀⠀⣠⡿⣿⣿⣿⣿⢿⣄⠀⠀⢀⣠⠇⠀⢀⠜⡩⠂⢰⠤⠀⠀⠀⠀⣀⣄⠃
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠋⠃⠀⠀⡀⠙⢦⡀⢸⣿⢀⣾⢽⣾⣿⣿⣾⣿⣷⡯⣷⡀⣿⡇⢀⡴⠋⢀⠀⠀⠈⠙⠂⠀⠀⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠙⣦⡹⢿⣿⣤⣿⣿⣿⣿⣿⣿⣤⣿⡿⣏⣴⠋⣠⠖⠁⠀⠁⠀⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠀⠴⠶⠿⢿⣷⣾⣿⡛⡿⠟⢛⣿⣿⣿⣿⡛⠻⠿⢛⣿⣷⣾⡿⠟⠶⠦⠀⠀⠀⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠔⠉⠛⣿⣿⡿⡀⠀⢘⣷⣿⣿⣾⡃⠀⢀⢿⣿⣿⠛⠉⠢⠄⠀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⢀⠈⣸⣿⡇⡀⠀⡚⣿⢿⡷⣿⢃⠀⢈⢸⣿⣎⠁⡀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠈⣸⢻⡇⠀⠠⠋⢻⣼⣇⡗⠙⠄⠀⢸⡟⣇⠑⢀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⣼⣿⣿⡇⠀⠀⠼⢻⣻⣟⡿⠇⠀⠀⢸⣿⣿⣧⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣷⡇⣿⣇⠂⢀⠀⡐⣿⣿⢂⠀⠀⠈⣸⣿⢸⣾⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢻⣿⣾⣄⠓⢿⣿⣿⡿⠚⣠⣷⣿⡟⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠉⠻⠶⠶⠟⠉⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⣲⣀⡀⠀⠀⠀⠀⠀⠀⣀⣌⣹⡟⢛⣲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⢸⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠹⣿⡇⠻⣿⣿⣶⣍⡓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⣿⣁⡾⢻⠀⠀⢀⠴⠒⠊⠙⠒⢄⠀⠀⢹⡎⡻⣮⡻⣿⣿⣿⣷⣮⣓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣳⣿⣿⣿⣟⠘⠀⢠⣿⠄⠀⠀⠀⠀⠀⣷⠀⢸⠏⣿⣿⣿⣌⢿⣿⣿⣿⣿⣻⣿⡢⡀
⠀⠀⠀⠀⠀⠀⢨⣿⣿⢣⣿⣿⣿⠀⣦⣼⣿⠀⠀⠀⠀⠀⠀⢸⠀⡘⢀⣿⣧⢿⣿⣦⢻⣿⣿⣿⣿⣿⣿⣮⡢⡀
⠀⠀⠀⠀⠀⢠⣿⣿⡟⣾⣿⣿⣿⣇⠸⣦⡉⠀⠀⣠⣀⣄⣀⣰⡴⢁⣾⣿⣿⣾⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿⣮⣦⡀
⠀⠀⠀⠀⣌⣿⣿⣿⢣⣿⣿⠟⠛⠛⠀⢿⣿⠆⠈⠈⢋⠈⠁⠟⠀⠛⠛⠻⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠰⣼⣿⣿⣿⣿⣿⠃⠀⠈⠻⠄⣿⣿⣗⠲⡌⢻⣤⠛⢀⣠⠀⠀⠀⠘⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣆⠈⠉⠁⠀⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡷⡀
⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⡄⠢⣀⠀⠀⠀⠈⠿⠿⠿⠀⠀⠀⠀⢸⡀⠀⠀⢠⠋⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢄
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡀
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡀⠀⠈⢻⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡔⡄
⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠸⣿⣿⣿⣿⣿⡞⣿⣿⢷⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⡀
⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠛⠀⠀⠀⠆⠀⠤⠤⠀⣿⣿⣿⣿⣇⣿⣿⣾⢣⡟⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣷⠁⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣽⣷⡀⠀⠀⠀⡄⠀⠀⠀⣒⡛⠛⣶⣶⣿⢸⣿⣿⣿⣿⣿⣿⣼⢳⡏⣿⣼⢿⡏⣽⡿⣿⣿⣿⣿⣇
⡟⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⣟⠦⣄⡀⡇⠀⠀⢀⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⢻⣿⡟⣿⣼⣇⣿⣿⣾⣻⣏⣿⠏⣿⣿⣿⣿⡄⢇
⡇⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣟⣷⣿⣿⡇⠀⠀⠸⠛⠿⣿⣿⣿⠿⣿⣿⣿⣿⢸⣿⣇⣿⣿⣼⣿⣿⣿⣿⣼⣯⣼⢻⣿⢻⣿⣷⢸
⡇⣿⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣷⠀⠀⠸⣿⠙⠉⠀⠀⢿⡇⠀⠀⠃⠀⠀⡠⠈⠁⠆⠘⣦⢹⡿⢸⡿⠋⠉⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⣿⣾⣿⣿⣾
⡇⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣷⠀⠀⠿⣀⠀⠀⠀⣼⠁⠀⢀⡤⠐⠁⠀⢀⣣⢼⠄⢆⣠⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡯
⠇⣿⣿⣿⣿⣿⣿⣿⣿⣌⣷⠿⠟⠉⢻⣿⣴⡆⠀⠀⠐⠀⢠⡝⡀⠀⡉⠁⢠⣀⠔⠋⢙⠀⠀⠈⣧⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢀⣿⣿⣿⣿⣿⠟⠋⠁⠉⠋⠾⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠑⠄⠁⠘⣥⠃⠀⠀⠈⠒⠒⠋⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢰⣿⣿⣿⣿⠋⠀⠀⠀⠀⢀⡆⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡟
⠘⢹⣿⣹⡏⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢨⡟⢻
⠀⡼⣿⣿⡃⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⡎⠀⣿⣤
⠀⠠⢿⣿⡇⠀⠀⠀⠀⠀⠀⡻⠀⠠⢆⢳⠀⠈⠛
⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢷
⠀⠀⠈⢫⣿⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠸⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⡤
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰
⠀⠀⠀⠀⢰⡀⠃⣴⠀⠀⠀⠀⠀⠀⠏⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣥⠁⠀⠀⠀⠀⠀⠈⠓⠺⠃
⠀⠀⠀⠀⢸⢻⡾⠸⡠⢲⠃⠀⠀⣀⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣿⣿⠓
⠀⢀⠀⠀⠸⠂⡟⡺⠁⡏⠀⠀⠀⡆⠉⠱⡇⠀⢸⠀⡀⢰⠀⠀⠀⠀⠀⢸⣏⡙⢹⡆
⠀⠀⣷⢄⠀⢠⡇⠃⠞⠀⠀⠀⠀⠙⠀⠚⠀⠀⠀⣼⡇⣾⣧⠀⠀⠀⠀⠸⣏⠀⣨⠃
⠀⠀⠸⣌⢳⡀⢡⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡗⣿⣼⡶⢠
⠀⠀⠀⠈⠑⠀⡄⢰⣦⣀⡈⢇⠀⠀⠀⠀⠀⠀⠓⣆⣿⣾⣿⢇⢈
⠀⠀⡠⠒⢀⠐⡇⠁⠀⡨⠙⠾⠀⠀⠀⠀⠀⠀⠸⣯⣿⡻⣿⣷⡏⠀⠀⢀
⠀⠴⠁⣴⠏⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⣓⡻⣿⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠻⡜⠁⠀⠈⣧⠏⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣧⢸⣅⡿⠀⠀⠸⣄⢀⠇⠀⠀⠀⠀⠀⠀⣆⣁
⠀⠀⠁⠀⣤⣴⡿⠦⢤⠀⠀⠀⠀⢂⡀⠐⠀⠀⠀⢀⢽⣿⣿⢧⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⢠⡄⣾⣿⣤⡀⠀⢀
⠀⠀⠀⠀⢿⠋⠀⠂⠒⠀⠀⠀⢠⠸⠷⢤⣄⠀⠀⠀⠀⠉⠉⢁⠀⠀⣠⣧⣤⡀⠀⠀⠀⠀⢸⡃⠀⠀⠈⠇⠀⠈⠃
⠀⠀⠀⠀⠸⣧⡀⣦⠆⠀⠀⠀⠸⣄⠀⣀⠌⠀⠀⠀⠀⠀⠀⠜⠀⠀⣿⣸⣽⠇⠀⠀⠀⠀⠈⢧⣀⣀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣶⣷⠈⠋⣉⠈⠛⠿⠿⣿⣿⣿⡿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⡜⠀⠟⢀⣉⡛⣧⣶⢶⡶⠇⣨⣽⣦⣈⣄⣹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠆⣀⣄⣤⣴⣿⣿⡿⣿⡿⠁⠀⢀⡎⠙⣻⣾⣿⠛⣿⠉⠝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠾⠿⣿⣟⠈⠉⠀⡆⣽⣆⣿⡇⢸⠿⣤⣿⡍⢿⠀⠻⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠘⣿⣿⣿⡆⠀⢾⣿⡏⠙⢷⣌⠀⢹⣿⡇⢸⠀⠀⣈⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⣿⣿⣿⣀⠘⣇⣿⢾⣷⡌⢻⠟⣿⣷⡌⡀⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠉⣿⣿⡿⠻⣿⡏⠋⣿⠀⠈⠀⠸⡬⡃⢇⣄⣿⡘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢐⡿⢋⡇⠀⢸⣧⠀⣿⠀⠀⡆⠀⠹⣅⠘⢿⣿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣿⣿⡿⠀⠘⠁⠈⡇⠀⢸⣿⣾⡿⠀⠀⠈⠀⢉⣙⣏⠛⢿⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠿⣉⠀⠄⠀⠀⢰⠃⠀⣸⣿⣿⠃⠀⢀⠀⡔⣿⣿⣿⣤⢲⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣯⠐⡆⣬⡀⢸⡀⢀⣽⣿⠏⠀⣀⣿⣷⣿⠄⠏⣉⡹⣄⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⡷⠛⠛⠈⠛⠿⠿⠇⠀⠀⠈⠼⠉⢿⣶⣾⣯⣽⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣥⢄⠃⠀⠀⠀⠄⠀⠉⠇⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⢂⠀⡀⠀⠀⠀⠂⠈⠀⠉⠀⢀⡀⠀⠀⠀⠠⣄⢡⡨⢹⡛⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠋⣄⢀⣧⠀⢇⡴⠀⠀⠀⠠⣿⣿⡄⠐⢒⡾⣋⡟⢡⣗⢻⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣷⡈⠺⣯⡄⢸⣷⣆⣀⣀⠀⣾⣿⡇⣀⣨⡵⣿⣿⡷⢷⣦⠾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢠⣭⡟⠹⡿⣿⣧⣹⡿⣿⣿⣖⣻⣿⢾⣯⣽⣿⣳⣿⣿⣿⣭⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣼⣅⣽⡖⢿⣦⣨⣿⣱⣎⣻⣿⣾⣵⣿⣿⣶⣿⣯⣽⣶⣿⣈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠁