Mirrored Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⢢⡤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⠿⠶⠶⠿⢬⡙⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠔⣮⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣇⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢱⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⢿⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⣽⡘⢿⡀⠀⠀⠀⠀⢀⣠⣔⣤⡾⠋⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢘⢁⣦⡹⠦⣤⣤⡤⠿⠟⣛⣉⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣻⣷⣆⣴⣤⡖⣀⡈⢉⡅⣺⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣤⣴⠾⠁⢸⣏⠍⢉⡼⠋⠐⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣥⣵⠞⠉⠁⠀⣠⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠉⠉⠀⠀⣠⡶⠟⣻⠇⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⣼⠏⡠⠀⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠤⣿⢃⠜⠓⠊⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢰⡏⢨⡤⠖⢋⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣴⣿⣷⣿⠖⠋⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢰⠚⠿⠿⢟⣻⡿⠃
⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠐⠋⣠⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣾⣿⡿⠿⠿⠿⠷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⢋⣩⡼⠟⠋⠉⠀⠉⠉⠛⠷⣦⡀⠀⠀⠀⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⢁⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⡀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢀⣾⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⢀⣾⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣾⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⣼⡏⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⢰⡟⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⢠⡿⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⣾⠃⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⢰⡿⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣀⣤⠶⠶⢶⣤⡾⠃
⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⡾⠋⠀⠀⠀⠀⠙⣷
⣿⠇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠘⢷⣄⣀⠀⠀⠀⣼⡏
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠉⣿⠀⠀⠀⣿⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⣿⡇
⣿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢀⡼⠋⣿⠀⠀⠀⣿⡇
⢸⡇⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⣠⡞⠁⠀⣿⠀⠀⠀⣿⡇
⠘⣿⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⢹⣧⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧
⠀⠀⠹⣷⣄⠀⠹⣆⠀⠀⠙⡆⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠻⣷⣄⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠈⠻⢿⣾⣿⣄⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣶⣾⣿⣋⣉⣀⣀⣠⣤⣤⡶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⠉⠉⠁⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡶⠾⠛⣷⠀⠀⢸⡷⠶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠉⠉⠁⠀⠀⠀⠀⣿⣀⠀⠀⣧⠀⠀⠀⠈⠉⠉⠙⠛⠓⠲⠶⠤⣤⣤⣀⣀⣸⡇⠀⠀⠀⢘⣧⣀⣀⡀
⠀⠀⠀⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣩⠟⠃⠀⠀⠀⣠⠟⠙⠛⠛⠻⠿⢷⣦
⠀⠀⠀⠀⠀⢻⣟⠛⠶⠶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠶⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠈⣿⣄⣀⠀⠀⠀⠉⠉⠛⠻⠷⢶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠙⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣤⣄⣀⡀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠷⠶⢶⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠳⠶⠶⣤⣤⣀⣀⣀⣤⡴⠟⠉⠀⠀⠀⠀⣀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉

⠀⠀⠀⠀⠀⣀⡴⠒⠒⠦⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠠⠐⠀⠂⢄⠀⠀⠀⠀⠀⡶⠒⠀⠀⠀⠐⠲⢶
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠃⠀⠀⠀⠀⠆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠨⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠑⠤⣀⣀⠤⠚⠁⠀⠀⠀⠀⢰⢀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠑⢄⣀⡠⠀⠀⠀⠀⠀⠀⡧⡀⠀⠀⠀⠀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⡀⠀⠠⢅⡄⠠⠄⡀⠀⠀⠈⠃⠀⠀⠒⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠉⠀⠀⢄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠄⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠒⠀⠀⠒⠒⠀⠀⠁⠀⠀⣀⡂⠀⠀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠁⠂⠠⠀⠂⠁
⠀⠀⠀⠀⠀⠰⠆⠳⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠀⠋⠀⠀⠀⠀⠰⠉⠀⠀⠀⠀⠠⠚⠉⠀⠉⠐⢄⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⡏
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠘⢄⠀⢀⠤⠀⠀⠀⠀⢠⢼⠀⠀⠀⠀⠀⢰⣤⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣦⡀⠀⢀⣀⠀⢀⣀⣠⣦
⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠀⠰⠞⠁⠀⠀⠀⠁⠘⠶⠀⠀⠀⠀⠀⠈⠒⠂⠐
⠀⠀⠀⠄⠄⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⣠⢀⣄⣠⡄⠀⠀⠀⠀⠀⠀⠒⠀⢠
⠀⠀⠀⢈⡏⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⠀⠀⠠⠊⠀⠀⠈⠢⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⠀⠀⠀⠀⢺
⠀⠀⠀⡶⠧⠤⠄⠤⠠⠤⠦⠼⣷⠂⠀⠀⠀⢃⠀⠀⠀⠀⢀⠃⠀⠀⠀⢈⣇⢀⡀⣀⢀⣀⣸⡇⠀⠀⠀⠸⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠄⠔⠁⠀⠀⠀⠀⠀⠉⠀⠀⠉⠈⠁⠉⠁⠀⠀⠀⠈⠀⠠⠤⠠⠘
⠀⠀⠀⠀⠀⠀⠐⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠔⠀⠐⠂⠀⠤
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠠⠐⠒⠒⠀⠀⠀⠒⠒⠣⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠈⠋⠓⠒⠤
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠛⠀⠀⠀⠀⠛⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠠⡤⠀⠀⠘⠣⠀⠀⠀⠀⣄⠀⢀⡀⠀⡀⠀⠀⠀⠈⠑⠒⠂⠀⠀⠒⠈

⠀⠀⠀⠀⠀⠀⠙⢦⠑⢵⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⡾⠋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠻⣆⠀⠀⠀⠀⠀⡼⠀⠀⠄⠁⠈⠀⠀⠀⢷⠀⠀⠀⠀⠀⣰⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣧⡀⠀⠀⠀⠇⣠⠂⠀⠀⠀⠀⠀⣤⠸⠀⠀⠀⠀⣼⢋⡴⠋
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⣎⢣⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢀⡜⣡⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠐⢎⣷⣷⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣾⡾⡱⠂⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠓⣽⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⠎⠀⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠌⠈⠻⣿⣧⡀⠂⠀⠀⠀⠀⠐⢀⣼⣿⠟⠁⠡⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠁⠢⣈⢮⡻⣤⡄⢠⡀⢠⣤⠟⡱⣁⠔⠈⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣁⣀⣽⣮⡿⣿⣿⢿⣵⣮⣀⣈⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠⡤⠴⠛⠛⢋⣛⡡⠪⢟⣻⣟⡻⠕⢌⣛⠙⠛⠛⠧⢤⡄⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠉⠽⢶⣶⣦⣤⣜⣪⡷⣾⣷⢶⣔⣣⣤⣴⣶⡶⠏⠉⠁⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⣄⢈⡽⣫⡿⢿⡿⢿⣝⢫⡀⣠⠊⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢀⠄⠊⣠⣣⡾⠙⠁⠀⠀⠈⠉⢷⣝⣀⠑⠠⡀⡰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠡⣠⣾⣿⠏⠀⠂⠀⠀⠀⠀⠐⠀⠹⣿⣷⣄⠌⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⢀⢔⣽⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣳⡀⠀⠈⠐⢄⡀
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠐⣣⢟⡟⠉⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠉⢻⡻⣜⠂⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡎⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠱⣜⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⠏⠀⠀⠀⠀⡇⠉⠂⠀⠀⠀⠀⠀⠉⢸⠀⠀⠀⠀⠹⣦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⡾⠃⠀⠀⠀⠀⠀⢹⠀⠀⠁⡄⢠⠀⠀⠀⡟⠀⠀⠀⠀⠀⠘⢷⡌⠳⡄
⣀⣀⣀⣀⣀⣀⣴⣋⣴⣽⣁⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣈⣿⣦⣙⣦⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠉⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⡈⢣⠊⣸⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠋⢰⣿⣼⣸⣷⠈⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡗⠈⢻⣠⠶⠄⠻⢻⠻⠃⠴⢦⣸⠏⠐⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠰⠷⡸⠀⣼⣉⠶⠶⣛⠛⣛⡳⠶⢎⣹⡄⠸⡰⠷⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⣙⠢⠤⣤⢞⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⢙⢦⡤⠤⢚⣁⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⠝⣱⢞⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⢶⡙⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣌⡼⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⣄⣤⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠠⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠐⣺⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣹⡘⢚⡀
⠀⠀⠀⠀⠀⠀⠀⠰⡖⢁⡀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⢀⠐⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡘⢚⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣜⠚⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢙⣀⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣄⣘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠀⠠⠴⡀⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠁⡰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⡊⢰⡲⣹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣱⣲⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠄⣀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⢀⡀⠔⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡲⠺⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡸⠲⠲⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⠘⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢌⠁⠹⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡼⠡⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣵⠜⠳⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡴⠛⣴⣽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡴⠊⠉⡉⠳⣥⣀⠠⠀⣀⡀⣀⡀⠠⢀⣠⡵⠋⡉⠉⠲⡄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡘⠟⡸⠀⢹⣤⠛⠛⠶⢶⠶⠞⠛⢣⣼⠁⠸⡘⠟⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠠⣿⠈⠛⢁⣾⣾⣾⣆⠙⠋⢸⡦⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⢘⡿⢹⠹⣟⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣤⠎⢢⡸⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠤⠌⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣛⡙⢿⣗⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⣾⣿⣷⣼⣿⠟⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡀⠁⠙⠛⠋⠍⢟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠆⠀⣴⣿⣿⣋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⡿⠃⢀⣴⣿⣇⣯⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠃⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⡯⠀⠀⠀⠀⠀⠀⠴⠁⠀⣸⠟⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⢠⠔⠋⠀⠀⠀⢀⡏⢠⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡁⠀⠀⠀⠀⠠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠁⠀⠀⠀⠀⠀⢸⠇⢘⠃⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠈⠲⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⣿⣿⡏⠀⠀⠀⢀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢀⠤⠤⠾⣄⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢸⣿⡿⠟⡋⠀⠀⠄⠀⠤⢀⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉⠂⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠀⠤⠄⠀⠁⠀⠀⠀⠩⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢐⢄⡂⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⡀⠀⠀⢀⣌⢹⠀⠶⠟⠷⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⡄⢷⣶⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠛⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⠀⢂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢠⠭⠍⠛⣿⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⡇⠀⠀⠀⠂⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢹⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⢙⣻⣿⣯⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡀⠀⠀⠈⠉⠿⠙⢁⠀⠀⠀⠀⠀⣀⣀⡀⠀⠐⢲⡃⠘⠀⠀⢿⣟⡣⢛⣹⣿⠁⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠺⠙⠀⠀⠰⠋⠀⠀⠀⠀⠀⣿⡁⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠌⠀⢀⠀⠀⠀⠀⣀⠐⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣂⠀⠀⠀⠀⠀⠀⣿⣄⡀⣠⢔⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠒⢶⣤⣾⣿⣿⣿⣿⡿⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠔⠀⣤⡀⠀⠀⠀⠀⣔⠁⣾⠇⠘⡦⠈⣄⠀⠀⠀⠀⢠⡆⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠸⠉⠉⠀⠀⠀⠀⠈⠺⣹⠀⠠⣜⠞⠁⠀⠀⠠⠀⠉⢹⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣤⣄⠀⠄⣠⣤⡀⢉⠀⢀⠁⢀⣤⣄⠴⠀⣠⣤⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡁⠀⠙⠆⠘⠃⠀⢀⣯⠀⢀⣿⡀⠁⠛⡀⠸⠋⠁⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⡀⣨⡁⢀⣴⣿⣿⣧⣾⣿⣿⣦⣀⣹⣁⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⡾⢿⡋⠉⢿⣿⣿⡿⠟⢻⡿⢷⣿⠿⠛⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⣿⣿⡂⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡠⢴⣦⣦⣀⣼⡟⢻⡟⢿⣄⢀⣾⣿⡧⣄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣿⡟⠁⡆⠀⠁⠆⠘⢿⣿⣿⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⢀⠆⠀⠀⠀⠀⠢⠈⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠀⠁⠀⠀⠀⠀⠀⠀⢁⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠒⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠠⢾⣷⣄⣀⣀⠀⠀⠀⠀⠀⠈⣉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣏⠑⢤⣸⣼⠿⢋⡿⠀⠀⠀⠀⠀⠉⠉⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣄⣠⣤⣴⡟⠁⢙⣧⣄⣹⣧⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⣬⠇
⠀⠀⠀⠀⠀⠀⢠⡟⠙⣿⣿⣿⣯⣍⣉⣉⣉⣉⣉⣉⣉⣉⡙⠂⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⢀⣼⠃⠀⣿⣿⣿⠿⠛⠟⠛⣿⣿⣿⣿⣿⣿⣧⠀⠀⠹⣶⠿⢿⡀
⠀⠀⠀⠀⣶⣿⡇⠀⠀⣽⣿⣇⡀⠀⠀⣄⣸⣿⣿⣿⣿⣿⣿⣷⣤⣴⣥⣤⣈⠽⣶⡄⡀
⠀⠀⠀⠀⠇⠈⠸⠇⢒⣿⣿⣿⣅⠀⠀⢾⣽⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡅⠀⠰⢿⣿⣿⣿⣿⣿⣿⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠈⢉⣿⡿⠁⠙⢸⣇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠘⣦⣄⣰⠆⠀⣸⣿⣷⡀⠀⠀⠈⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⣿⣿⣿⡶⢐⢻⣿⣿⡇⠀⠀⠀⠀⢦⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢿⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⠻⣿⣿⠛⣸⣿⣿⣿⣧⠀⠀⠀⠀⠸⣷⣄⠀⠀⣰⡿⠻⣿⣿⣶⠿⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⣾⣾⢻⡄⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣷⣟⠁⠀⠀⢣⠘
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢹⣿⠹⡇⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠀⠘⡆
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⡇⠀⢠⡀⠀⠀⠀⠀⠘⣤⡀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣧⡀⠈⣀
⠀⠀⠀⠀⣷⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣀⣹⡼
⠀⠀⠀⠀⠛⠓⣆⠀⠀⠀⢀⡀⣀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣷⣘⡆⠈⠉⠉⠛⠿⠿⠟⠁⠀⢛⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⠀⠂⠒⠐⠓⣻⣿⣿⢿⣏⣉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⠶⠒⠛⠁⢸⣿⡀⠈⠩⣦⣀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠈⠙⠻⠶⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠉⢘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣿⣿⣿⣿⣿⢟⣿⣿⡿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⢫⢟⣾⣿⡿⠁⣿⣿⣿⣿⢹⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣽⣿⣿⣿⣿
⣿⣋⣭⣷⣿⣿⡿⢹⡄⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠻⢿⡟⠟⠻⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣟⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠁⠡⡇⣿⣿⣿⣿⢸⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣇⣸⠇⠀⠀⠮⡷⡹⣿⣿⣿⣿⡏⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⠁⣿⣿⣿⣿⢘⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣿⣶⠀⠀⠀⠀⠀⣺⡻⣿⣿⢯⣼⣸⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣛⣿⣶⣶⣭⡓⠒⠂⠀⠀⠀⠈⣾⣮⡿⣟⣻⣿⣿⣿⣿⣿⣿⣷⣿⣿
⣿⣿⡿⠁⠀⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣵⣿⣶⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠙⣿⣯⡿⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡱⢝⣾⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⡛⠯⡀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢏⣕⠊⠙⣿⣿⣿⠇⡑⣼⠿⠿⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣶⣿⣿⣿⣦⡂⢹⣿⣿⣿⣿⣿⣿⣿⣿
⡅⢂⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠐⠀⠎⣸⣿⣿⡘⠌⢿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠂⢸⣿⣿⣿⣿⣿⡇⠑⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣤⣤⣾⣿⣿⣿⣇⠀⠈⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠻⢿⣿⠿⠛⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠠⠤⢸⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣁⣐⣸⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿
⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⣿⣿⣿⣿⣿⢯⣿⣿
⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠇⠀⣒⢿
⠙⠛⠻⢿⣿⣿⣆⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⠃⠀⣠⣶⣿
⠀⠀⠀⠀⠙⣿⣿⣆⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⠃⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣆⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⠃⠀⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠶⠶⠶⠒⠚⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣶⣦⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⢚⣋⣩⡥⠤⡶⣶⡶⠶⠾⠿⠿⠿⠿⢷⣶⣦⡉⠈⠘⣻⣿⣿⣿⣧⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢚⣋⡭⠶⣚⣋⠭⠕⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⢘⣿⣿⣿⣿⡆⠀⠀⢀⡏
⠀⠀⠀⢀⣠⠶⢛⡩⠔⣊⣭⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⡜⠘⣿⠀⠀⠠⣿⣿⣿⣿⡇⠀⠀⣸⠇
⠀⠀⢀⡾⠁⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⣸⠁⢀⣿⠀⠀⠠⣿⣿⣿⣿⣿⠀⠀⣿
⠀⠀⢸⣇⣰⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⠎⢰⠇⠀⣸⣿⠀⠀⠸⣿⣿⣿⣿⣿⠀⢸⡏
⠀⠀⣿⣿⣯⣀⣀⣀⣀⣗⣃⣀⣀⠀⠀⠀⠀⢰⣲⠀⠀⠀⠀⠀⢠⢟⡏⡎⠀⡟⠀⡄⣿⣿⠀⠀⢼⣿⣿⣿⣿⣿⡀⣼⡇
⠀⠀⣿⣿⡇⠀⠐⠢⠄⠀⠀⠈⠉⠉⠉⠛⠒⠺⠾⢤⣀⣀⠀⢠⠏⠘⣟⣇⢸⠁⢰⢸⣿⡟⠀⠀⣼⣿⣿⣿⣿⣿⡇⣿
⠀⠀⣿⣿⡧⣄⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⢉⣹⣏⠀⠀⠈⠙⠿⣦⣾⣾⣿⡇⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⡇⠈⢦⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢸⣤⣏⠔⠀⠀⠀⠀⠺⢝⣻⣿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣧⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣜⠲⣆⣀⠀⠀⠀⢀⣿⡏⠀⣠⣿⣿⣿⠿⠿⠿⣿⣿⠇
⠀⠀⠈⢿⡈⠳⣟⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣸⠃⡸⠀⢦⡀⢨⠇⠀⢀⣾⡟⣀⣾⠟⠋⢁⣀⡀⠀⠀⠀⣿
⠀⠀⠀⠈⠙⠶⣽⣻⢷⣶⣤⣄⣀⣀⠀⣿⠀⠀⠀⢠⣿⣀⠇⠀⠀⠙⠋⢀⣰⣿⣟⣼⣿⠃⠀⠀⢀⣺⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢯⣽⣛⡛⠿⠿⣿⣷⣶⣾⣿⣿⣿⣷⣶⣶⣶⣿⠿⣫⣾⣿⠏⠀⠀⠀⢈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⢤⣭⣝⣻⣿⣿⣭⣭⣷⣶⣿⣿⣿⣯⣤⣀⣀⡀⠻⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢤⣭⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⠀⢀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢀⠨⢿⡿⠅⠀⠀⡀⠀⡀⠀⠀⠀⢀⡆⠀⠀⣀⢀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣤⢀⡀⠀⡀⢂⠀⠀⠀⠀⠀⠀⡀⠀⡀⠰⣿⡟⠁
⠀⠀⠀⠁⢠⣵⡇⣶⣷⣷⣶⡟⠓⠉⠉⠀⠀⢀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢄⠀⠀⠈⠉⠁⠛⣷⣶⣷⣷⡆⢾⣧⡤
⠀⠀⠀⠀⢨⣭⠀⠀⣿⣿⣿⣷⣤⣠⣤⣴⣖⣶⣤⣶⣶⣤⣾⣿⠿⠾⠿⠒⠚⠿⢾⢿⣿⣦⣴⣶⣶⣤⣶⣶⣤⣤⣤⣴⣾⣿⣿⡏⠀⢈⣭⠉
⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣿⡧⠷⠾⣿⣿⣾⣿⣭⣭⣥⣀⡀⣀⣀⠤⠤⠤⢄⣠⣀⣄⣠⣨⣭⣽⣿⣶⣿⣿⠶⠯⣿⣿⣿⣿⣿⣿⣿⣿⡥
⠀⠀⠀⠀⣺⡿⢿⣿⣿⣿⣿⣿⠙⠺⠾⡷⢒⣂⣿⣿⣿⣡⡶⠴⠖⠀⠀⠀⠀⠀⠀⠶⠶⢦⣸⣿⣿⣿⢖⠒⣿⠶⠞⠻⣿⣿⣿⣿⣿⠿⢿⡄
⠀⠀⠀⢀⡝⠀⠀⣿⢯⡏⣻⡄⠀⠀⢘⡇⢀⠬⣿⡝⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠟⣿⡯⢥⠀⣻⠀⠀⠀⣰⠇⣿⣿⡏⠀⠘⣅
⠀⠀⠈⠡⠛⠀⠀⣿⣼⣧⢾⡧⠴⠲⢾⣿⡇⠀⠻⣿⣿⠁⠂⠀⠀⠀⠲⢦⡶⠆⠀⠀⠠⠆⢩⣿⡿⠇⠀⣹⣿⠶⠲⠤⠾⢦⣿⣼⣇⠀⠐⠛⠈
⠀⠀⠀⢠⠀⢀⢼⣿⣛⣿⣸⢰⡄⡶⠁⠉⠁⢀⣸⣿⣧⠀⠀⠀⠀⠀⠀⣸⣅⠀⠀⠀⠀⠀⠠⣿⣿⣀⠀⠉⠁⠑⡆⢼⢜⢸⣿⣻⣿⢄⠀⢀
⠀⠀⠀⠴⠀⡏⢀⣿⡟⣿⣿⣿⣟⠿⣿⣷⣶⣾⣿⣿⣿⣤⣦⣶⣶⣶⣾⣿⣷⣶⣶⣶⣤⣴⣤⣿⣿⣿⣶⣦⣿⣿⠟⣿⣿⣿⡟⣿⣿⠀⡇⠰⠄
⠀⠀⠀⠀⠀⠀⠘⣿⠃⠻⠛⣿⠟⠓⠟⠛⠉⠙⠙⣿⡏⠉⠉⢉⡉⠉⠉⠉⠉⠉⢉⣉⠉⠉⠉⣿⣻⠙⠋⠉⠛⢳⠛⢿⡟⠻⠃⢻⡟
⠀⠀⠀⠀⠀⠀⣴⣿⠀⢠⢀⡏⠀⠀⠈⠀⠀⠀⢠⣿⡇⢀⡀⠈⠳⠀⠀⠀⠀⠈⠰⠏⠄⣐⠀⣿⣯⠀⠀⠀⠉⠀⠀⠀⣷⢠⠄⢸⣧⣆
⠀⠀⠀⠂⠀⠀⣻⡿⠀⢸⠘⠇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⡛⠂⠀⣠⣄⠀⢀⣤⡄⠀⠚⠛⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠸⡆⠀⣿⡋⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢻⠃⠀⡯⠀⠀⠀⠀⠸⣆⣀⣠⣾⣿⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿⣖⣀⣀⡺⠀⠀⠀⠀⠀⡇⠀⢻⡃
⠀⠀⠀⠀⠀⠀⣿⡆⠀⢏⠀⠀⠀⠀⢸⠏⠉⠉⢻⣿⡇⠀⠀⠀⠀⣴⠀⠀⠀⢀⡄⠀⠀⠀⠀⣿⣿⠋⠉⠉⢹⠀⠀⠀⠀⠀⡇⠀⣼⡇
⠀⠀⠀⠆⠀⠀⣿⣷⠀⢺⢠⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⡶⠂⠀⠉⠁⠀⠀⠉⠁⠀⠰⣴⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⢰⠇⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠻⣿⠀⣸⠼⣧⠀⠀⠀⠄⠀⠀⠈⣿⡇⠀⢀⢀⣞⠀⠀⠀⠀⠀⢐⣦⠀⠀⠀⣿⣟⠀⠀⠀⠤⠠⠀⢀⣿⢰⡇⢸⣿⠇
⠀⠀⠀⠀⠀⡀⢰⣿⡇⣼⣶⣿⣷⡄⣷⣦⣤⣤⣤⣿⣧⣠⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣠⣿⣿⣤⣤⣤⣦⡞⢴⣿⣷⣾⡇⣾⣷
⠀⠀⠀⠛⠀⢧⠈⣿⣷⣿⡿⢿⡷⡿⠿⠏⠛⢻⣿⣿⣿⠛⠋⠛⠛⠛⠻⢿⡿⠛⠛⠛⠋⠛⠛⣿⣿⣿⠓⠋⡿⠿⡷⣿⣿⣿⣧⣿⣿⢀⡇⢘⠃
⠀⠀⠀⠀⠀⠀⢹⣿⣴⣿⢸⠹⠃⠓⣄⣠⡄⠈⠸⣿⡏⠀⠀⠀⠀⠀⠀⢹⡅⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⣀⣀⡀⠀⠨⢩⢸⣿⣼⣿⠉⠀⠈
⠀⠀⠠⢔⡴⠀⠀⣿⢻⡟⣻⡗⠛⠚⢻⡿⢅⠀⢿⡿⡿⣀⠃⠂⠀⠀⠖⠚⠓⠂⠀⠀⠀⠃⣨⠿⣿⡇⢀⠽⣻⠟⠚⠛⣻⢛⣻⣻⡏⠀⠠⣔⢠
⠀⠀⠀⠀⣵⡀⢀⣿⣯⣧⣿⡁⠀⠀⠰⡇⢈⣉⣿⣥⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣽⣯⣉⠀⣼⠀⠀⠀⡘⣧⣿⣽⣇⠀⢠⣏
⠀⠀⠀⠀⢸⣿⣾⣿⣿⣿⣿⣿⡴⠶⠺⣗⢒⣒⣿⣿⣿⣉⣛⠛⣒⠀⠀⠀⠀⠀⢂⡒⢛⣋⣹⣿⣿⣷⣒⢒⡿⠒⠶⣮⣿⣿⣿⣿⣿⣶⣿⠅
⠀⠀⠀⠀⢽⣿⡿⠿⣿⣿⣿⣿⣟⣻⣚⣿⣿⢻⣿⣭⣭⣉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣭⣽⣿⣛⣿⣿⣺⣛⣿⣿⣿⣿⡿⠿⢿⣿⡍
⠀⠀⠀⠀⣬⣭⠀⢀⣿⣿⣿⡟⠋⠉⠉⠙⠓⢟⠛⠛⠛⠛⠻⢿⡿⠿⠿⠶⠶⠿⠿⣿⠿⠟⠙⠛⠛⢛⠟⠛⠛⠉⠉⠛⣿⣿⣿⣇⠀⢀⣫⡤
⢀⠀⠀⡐⠉⡿⠇⠛⠟⡿⠿⢷⠶⢄⣤⠀⠀⢀⢉⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠉⡀⠀⠐⣤⠤⢶⠿⠛⡟⠿⠓⠿⡏⠑⣀⡀
⠀⢰⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⡀
⣀⣈⣁⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣴⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠛⢀⣼⣿⣦⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⡿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣷⣴⣿⣷⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣶⣦⣤⣿⣿⣿⣭⣀⣾⣿⣧⣼⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⢿⣿⡿⠉⠙⣿⣿⣿⣿⣽⣿⠋⠻⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡿⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⣷⣤⡈⢀⣠⡔⠛⠉⠉⠙⢿⣧⣤⣠⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⣿⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠛⢻⡷⠊⠉⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡟⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡍⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣉⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠾⢿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⠁⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢹⡿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠻⢿⡿⣿⡆
⠀⠀⠀⠀⠀⢠⣿⣭⡍⠁⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣜⣷⣬⣅
⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⡟⠻⣷⡀
⠀⠀⠀⠀⠸⣶⠘⠛⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣯⣿⣆⣸⡷
⠀⠀⠀⢠⣤⣼⣿⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣬⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⠃
⠀⠀⠀⠈⠉⣱⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣇
⠀⠀⠀⣿⣏⠙⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⣿⣿⡀
⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⡋
⠀⠐⣩⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⣿⣿⣿⣿⣿⣿⠉⠿⢿⣷
⠀⠀⣿⡅⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢸⡗⠈⣿⡇
⠀⢰⡿⠁⠐⢿⣥⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠈⣃⠀⠸⣿
⠀⠸⣿⠀⣰⡎⠉⠛⠶⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀⣿⣇⢠⣿
⠀⢠⣶⡾⣿⣧⠀⢠⡄⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⢿⡿
⠀⠸⣿⣧⣀⡿⠃⠀⠃⡀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣾⠁
⠀⠀⢻⣿⣿⡧⠀⠀⡶⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏
⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⡿
⠀⠀⢸⣿⣟⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠘⢿⣿⣷⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣰⣦⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠉⠉⠉⠹⡏
⠀⠀⠀⠈⠛⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣾⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣧⣼⠿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⣉⣁⣴⣷⠄
⠀⠀⠀⠀⠀⠀⠻⣿⣾⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⢿⣿⣿⣿⣏⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣾⣿⢇⣀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⢰⣿⠟⠟⠛⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣤⡟⠛⣡⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢣⣴⣿⣤⠀⠀⠀⠀⠀⠸⡿⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⢿⣿⣧⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡄⠀⢀⣶⣦⡀⠀⠀⠀⠀⠀⠸⣿⣠⣿⣿⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣠⣤⣀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣠⣾⣿⣿⣿⣄⣀⠀⠀⢀⣠⣿⡈⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⣿⡏⠙⣷⡾⢇⣴⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣹⠿⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⡈⢤⣾⣟⣿⣿⣿⣇⠀⢈⡙⣻⣿⣿⡄⢉⣁⣤⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠾⠋⠁⣻⣿⣟⠋⠀⠉⢉⡿⣿⣿⠃⢈⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⣿⣿⣿⣿⣷⣤⣿⣷⠘⢿⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠛⠁⠀⠘⠉⠁⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠀⠀⢀⣤⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⣿⣶⣶⣿⣿⣦⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⢿⡿⠉⣠⣿⠉⠛⢧⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡀⠸⣿⣿⣿⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣄⠉⠠⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣦⣤⠈⢻⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⢿⣿⣿⣿⣷⠞⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⣾⢿⣿⣿⣿⣦⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣭⣿⠿⣿⡿⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⢨⣿⠿⣿⣶⣿⣧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⣃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡻⠿⣁⣼⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡟⠉⢰⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠈⠙⣿⣇⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠃⠀⢀⣠⣿⣿⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣾⣿⣰⣿⣧⣾⡀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢸⣿⣿⣿⣿⠁⢿⠇⢸⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢸⣿⣿⣿⣿⣶⣿⡀⣿⡏⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣿⣿⡌⠙⠻⠿⣤⣿⣧⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣶⣶⣾⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⠿⢿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⢹⣿⡏⠛⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣤⣴⣿⣽⣿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠂⠳⣀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⣸⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⡂⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢱⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡖⠉⠁⠀⠀⠈⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠈⠀⠀⠰⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠀⠀⠈⠓⠀⠄⠀⠀⠀⢀⠠⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠇⠀⠀⠈⠲⡀⠀⢀⣠⠎⢀⣤⣴⣆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠘⠓⠙⠛⠳⠟⠛⠀⠀⠀⠀⢰⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠞⡀⠀⠀⠀⠈⠙⠲⠶⠴⠒⠉
⠀⠀⠀⠀⠀⢂⢰⠃⢱⡴⣢⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⠀⡆
⠀⠀⠀⠀⢀⡶⠊⠉⠈⠳⡗⠀⠆⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢙⡧⠠⠄⠤⠀⠠⠄⠀⣼⠃⠀⣨⠀⡃⠀⠀⠀⠀⣼⠉⠉⠉⠉⠉⠉⡉⣷
⠀⠀⢀⠀⣼⡁⠀⠀⠀⠀⡭⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢹⠀⠈⠉⠉⠉⡏⠀⣿
⠀⠀⠀⠀⠠⠷⣄⣀⣀⢴⡃⠠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢸⠀⣻⠀⠀⠀⠀⠀⣇⠀⣿
⠀⠀⠀⠈⢢⡐⡏⡛⣷⣬⠒⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠊⠀⣾⡀⠁⠀⠀⠀⠉⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠀⠀⠀⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⣿⣿⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠁⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⡀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣔⠁⠄⢱⡀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠔⡲⠂⡧⢀⣀⠴⠔⢲⠒⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣀⠀⢀⠄⠂⡑⡬⡀⠀⣃⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠈⣩⠓⠒⢕⢂⠕⠚⢫⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⡌⠢⢄⡸⢃⡠⠔⢁⢇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠅⠛⠫⡁⠒⢄⠈⠓⠥⢈⣀⣀⣉⠨⠗⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢞⡀⣄⣸⡈⠉⠉⣃⣠⠀⡙⡖⠖⠶⡇⠈⠗⣶⠶⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠜⡂⠰⠀⢸⢈⠤⠉⡆⠀⢣⠈⢷⠀⠀⡗⢞⡏⠀⠀⠰⠴⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡄⠑⠢⢠⣊⡠⠈⢜⣈⣦⠸⠊⢸⠀⠀⠈⠉⡇⠀⠀⢠⡼⠒⠀⣿⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⠢⡀⠈⣦⠁⠀⡠⢣⢀⠎⠘⢭⣭⠇⡇⠀⠀⠈⠉⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡢⠬⠟⠀⠓⠥⢤⠧⠊⠀⠐⠍⠉⠇⠧⠤⠤⠤⠤⠤⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡖⠒⠒⠈⠁⣿⡋⣙⢙⣙⡉⣿⠂⠀⠀⠀⠀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠙⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣤⠤⠤⠧⠤⠤⠤⠤⠤⠴⠧⠤⠤⠤⠤⢤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠠⡔⢡⡖⢂⠀⣀
⠀⠸⣧⠀⠐⠋⢻⣆⣤⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡁⠀⠀⠀⠈⠓⠁⠀⠀⠀⠠⠄
⠀⡠⠜⠀⠞⢱⡤⠒⡴⢠⡀⠀⢶⣀⣐⣤⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠓⠶⠃⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠠⡓⢄
⠀⢆⡀⠠⡤⡄⠐⠴⢋⠀⡓⠰⠾⠿⢿⣅⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠔⠀⠁⠀⠀⠀⠀⠀⠐⠉⠀⠒⢄⡀⠈⠢⠁
⠀⠀⠀⢀⠠⣃⢰⠔⡡⠊⡀⠄⢠⣍⠔⠩⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡠⠂⢂⠘⠀⢀⠀⠃
⠀⠐⠉⠻⢤⡤⠖⠠⠀⠂⠀⡈⠉⠀⠀⠁⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⠄⠐⡁⠁⠈⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⢩⠄⣶⡄⠈⡀⠻⢣⢡⠀⠀⠀⠀⠀⠀⡄⠀⠁⠀⠀⠀⠀⡠⠂⠀⠀⡐⠡⠂⠊⠀⠀⠀⠀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡄⡆⠀⠀⠙⢃⣀⣠⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⢀⠌⠀⠀⡠⢈⠄⠔⠁⠀⠀⢀⠀⠸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣇⢀⡀⠀⠈⠉⢁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡠⠁⠀⢀⠔⡀⡡⠂⠀⠀⠀⠀⡄⠀⠃⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡍⠀⢈⠗⠴⠋⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⡠⢁⢊⠄⠀⠀⠀⠀⠀⠠⠀⢠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠘⠀⢀⠐⠁⠀⠠⠊⠐⡡⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⣧⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠉⠹⣦⠠⡀⠀⠀⠀⠀⠁⠀⠏⠀⢀⠔⠠⢂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⢿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⠟⠉⠀⠀⠀⠰⣀⣡⡴⠄⠀⠀⢀⠀⠐⠀⡠⢁⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠚⠳⠏⠉⠃⠈⠀⠀⠀⠀⡀⠀⣠⠈⡀⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⢈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢙⡃⣀⣀⡆⠩⢦⣤⠤⠤⠶⠀⠘⠡⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡘⠀⢄⣀⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣍⠐⢫⠍⠒⠛⠤⠠⠄⠉⢱⠀⠚⡆⢀⣀⡀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⡭⢍⣛⠒⠾⣷⡈
⠀⠀⠀⠀⠀⠀⢼⠁⢔⡁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠁⠈⠀⠈⠈⠁⢀⠁⠀⠉⠂⠜⠠⠔⠀⠀⠀⠉⢰⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠈⠑⠦⢭⣑⣂⠴⠁⠀⡁⣀⠀⡀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢈⣀⡀⠠⠠⢔⣒⣩⠥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠲⠶⠯⠭⠭⠤⠥⠼⣶⣷⣶⣶⣶⣮⣶⠤⠤⠤⠤⠭⠶⠶⠒⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⠛⠛⠛⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⣀⣤⡶⠶⠶⢶⣤⣄⣀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠷⣦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⢹⣆⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⠃⠀⠀⠀⠀⢿⡄⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠸⣷⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⠀⠀⠀⠀⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⠀⠀⠠⠊⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢻⣆⠀⠀⠀⠀⠀⠀⣠⠞⢁⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠹⣦⡀⠀⠀⠀⠈⠁⠴⠁⠀⠀⠀⠀⠀⠀⠀⣸⡏⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠘⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⡈⠙⠛⠷⠶⣶⠶⠟⠋⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣧
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣶⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣿⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣽⣿⠻⠿⢿⠿⣿⠖⠾⢿⣾⠗⢊⣿⣿⠮⠹⠿⢟⠘⢻⠿⠿⠾⠿⠙⠿⠣⠄⢿⣿⡟⠰⠿⣯⣟⡿⠳⢾⣿⠒⢳⠛
⣿⣍⣺⣷⣿⣜⣿⣷⣞⡽⣯⣿⣿⣛⣹⢯⡃⣾⣻⠟⣿⡛⢻⡿⣿⣿⣾⣿⣻⡿⠟⠿⠘⠿⡿⢿⡿⣿⢿⡿⣧⢈⣫
⣿⡏⡉⢿⣁⡀⠀⣿⠏⠀⣀⣀⠀⠻⣿⢶⣽⣾⣿⠿⢯⣿⣉⣁⣻⠿⡳⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣿
⣿⠇⡝⠻⢿⡇⠀⣻⠀⢘⣿⠿⠀⠀⣲⡍⢻⣿⣼⡷⣺⢿⣿⡿⢏⠳⢙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⠹⣿
⣫⣀⠵⣶⡛⠁⠀⣽⠀⠸⢿⣶⠀⢀⣿⢖⣿⣿⣿⡇⠀⠿⣛⢺⣀⢷⣺⢿⡁⠐⢶⣦⣴⣄⣲⣴⣶⣴⣶⣸⣿⡓⣿
⢋⡇⣲⣿⣦⣄⣀⡜⢧⣀⠀⠁⣀⣺⣿⣿⣾⣿⣸⡟⢳⣾⡿⡏⢨⠿⢟⣼⡤⣄⢺⡏⣉⡽⢿⠟⢿⣿⣿⣿⣿⣵⡟
⣹⣟⠿⡦⡀⠬⢓⣤⣜⢛⣎⢿⠉⢉⣿⡿⠿⣿⡿⣷⡻⣷⡙⣿⡦⢻⣼⣿⣷⣼⠿⠛⠹⡖⠾⠷⢻⣿⣯⣫⣿⠻⣿
⣛⠿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣷⡆⠀⠀⠸⠻⠠⠀⠰⠀⡀⠑⠀⠀⠒⠕⠀⠀⠞⠱⠘⠁⠀⠸⠠⠐⠀⠀⠓⠂⠐⠀⠘⠀⠰⠈⠸⠀⠠⠀⠀⢀⣿
⣿⠃⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⣿⡦⠀⠀⠸⠕⠈⠉⠸⠄⠓⠀⠠⠀⠁⠂⠀⠇⠀⠈⠄⠠⠀⠀⠘⠐⠅⠘⠀⠇⠀⠄⠀⠀⠀⠑⠀⠀⠀⠀⠀⢠⣽
⣾⡔⢠⣀⣀⣀⣀⣀⢀⣄⡀⣤⠄⢠⣤⣠⣤⣤⣀⣠⣤⣤⣤⣤⣄⣀⣀⣤⣀⣠⣤⣄⣀⣀⢠⣠⣤⣤⣤⣤⣄⢸⣿
⣿⡧⣿⡿⣏⣿⣛⣛⣟⣿⣙⣛⣃⡜⣿⣏⣿⣸⣟⣻⣁⣈⣃⣟⣛⣓⡹⣿⣿⣩⣿⣿⣿⣼⡿⢻⢯⡙⢻⣻⢟⢛⣿
⣾⢶⣝⣚⣙⣟⠛⣍⣻⢲⡛⢛⣛⣛⣋⣙⣫⣙⣛⡋⣉⣛⠛⣛⣋⢻⣊⣿⣻⣿⣿⣿⣋⣿⣻⣹⣆⣛⣿⣛⣛⣩⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣆⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣵⣦⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣏⣹⢷⣿⣿⣿⣼⣿⣿⡿⣿⣉⣯⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠉⢁⣀⣈⣡⡴⠛⠉⠉⠉⠉⠛⢷⣤⣉⣀⣀⠉⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢿⣿⣷⣿⣿⠟⠉⠀⠀⢀⡄⠀⠀⠀⠀⠈⠙⢿⣿⣷⣿⣿⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⠛⢹⣿⡿⠁⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣏⠘⢳⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣁⣀⣼⠏⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⡿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣜⣻⣿⠃⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣻⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⢀⢴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠓⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠸⣿⡿⠻⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⣻⣿⣷⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣼⠛⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣭⠓⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣿⠙⣯⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡳⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢀⣿⡿⠛⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⣼⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⡄⠁⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⣸⣿⣄⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⡞⢡⠋⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡹⡟⣿⡄⠀⠀⠀⣴⠏⡸⠃⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⣼⡿⠟⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⣧⠀⢀⡾⠁⠆⠁⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⢰⣿⣷⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣽⣿⣆⠀⢠⠊⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣽⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢠⡞⠛⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⣉⣿⣿⣷⡀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣏⣹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣁⣻⣿⣿⣿⣷⣄⣀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣉⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⠏⠀⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠻⣿⣿⣷⣄⠀⠀⢹⣿⣯⣿⠟⠙⠻⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣀⣄⡙⢻⣿⣿⣷⣾⣦⠛⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇⠈⢿⣿⡟⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠿⣿⣿⣻⣿⣿⡅⠀⣌⢣⡄⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⡀⠀⠀⠀⠀⠈⠻⣿⣯⠙⠃⠀⠹⣟⣷⣄⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠰⠿⠆⡀⠀⠀⣸⣿⣿⣏⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠄⠀⠀⠀⠀⠀⢀⣀⣀⠠⣿⣿⣿⣅⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡄⠀⠈⠓⢄⡀⠀⣾⣿⣿⣷⣿⡟⠀⠻⣦⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⣿⣿⣿⣿⣦⡀⠈⠁⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠿⣿⡋⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⢠⣶⣮⣷⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⠲⢦⣀⡀⠘⠿⣿⣿⣿⣷⣄⡀⠀⠠⡾⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⣀⡀⠀⠀⢠⣿⣿⣿⣷⣿⢧⣍⠉⢉⣉⣀⠀⣤⠾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣶⣾⣟⣻⣿⣽⣶⣾⡟⠋⠈⠉⠀⠀⠀⠀⡸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡟⢙⣿⣿⣿⣿⣤⠘⠃⢀⠀⠀⠀⠀⢁⠼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠉⠍⢹⣿⣿⣿⣿⠂⠐⠁⠀⠀⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣷⣾⣿⣿⣿⣏⠠⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⣼⣿⣿⣿⡾⣿⣿⣿⣿⣧⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣷⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠟⢿⣽⠞⣿⣿⣿⣿⣿⣿⠳⣯⠿⡹⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⣡⣦⣴⣴⠞⠉⠀⠀⠀⠈⠙⠳⣦⣶⣤⣈⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⢋⡻⢿⣿⣿⠋⠀⠀⢀⣶⠗⠀⠀⠀⠀⠀⠙⣿⣿⡿⢟⠛⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⠁⣹⠟⠁⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇⠈⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢻⣿⣾⠏⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣿⠃⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⢀⣜⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠐⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢻⣿⢿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣮⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡴⠁⠀⠀⠀⠀⠀⠀⢸⣿⢀⣱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠯⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣟⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣏⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠟⠁⠀⠀⠀⠀⠀⠀⡔⠁⠀⢸⣷⣽⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠙⢻⣧⠀⠀⠀⠀⠀⠀⠀⢀⡴⣳⠎⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣾⡟⢋⣘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣠⠞⡱⠁⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⢠⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣇⠀⠀⠀⠀⣼⢋⠞⠁⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣼⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣿⡄⠀⢠⠞⣡⠎⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⠉⡰⠁⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡀⠁⠀⠀⠀⠀⢠⡞⠏⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⣹⣿⣿⣆⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣍⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣷⣄⠀⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣬⣿⣿⣿⣿⣦⣤⣀⣠⣤⣶⣿⣿⣿⣿⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣟⣽⣿⣿⣿⡿⢿⣿⣿⣿⣏⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⡛⠀⠸⢋⣻⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠻⢿⣿⣷⣆⣠⣾⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡉⢉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠘⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠄⠠⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠹⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⢺⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢋⣀⡄⠀⠀⢀⣀⡹⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⡿⣿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢹⡞⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⣾⡷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢸⣿⣿⢜⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣵⣶⣬⢧⢻⢋⣎⣵⣶⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡄⠑⠦⣼⡿⡼⡼⡘⡞⣿⣤⠔⠁⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢿⣧⡻⠷⠾⣟⣯⡾⠽⠤⠼⢼⣮⣝⡷⠶⠿⣣⡟⠉⣵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⢸⣿⡭⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⡀⢸⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢃⡼⠃⣀⣴⣾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠻⣄⠻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⡠⠴⢛⣡⡶⠋⣠⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡈⠳⣦⣉⠳⢦⣀
⠀⠀⠀⠀⣠⠖⣉⠴⣺⠟⠁⢀⣼⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠹⣯⡲⢌⡙⠢⡀
⠀⠀⠀⡼⠁⠞⣡⣾⡏⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠘⢿⣦⡙⢦⠘⡄
⠀⠀⠀⢣⠀⠾⢉⡟⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠘⣏⠸⠖⢀⠇
⠀⣠⠔⠚⠛⢿⡿⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠘⣷⠾⠛⠲⢄
⣰⠁⣠⣞⠂⣸⠇⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢻⡀⢒⣦⡀⢳
⡇⠀⣿⡟⠿⡿⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠸⣿⠿⣿⡇
⠇⠀⠿⣿⣦⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣧⣴⣿⠇
⢸⡀⠘⡜⢿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣿⡟⡹⠀⢸
⠀⢇⠀⠱⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⣰⠁⢀⡏
⠀⠘⡆⠀⣿⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⣿⠃⠀⡜
⠀⣤⡼⠂⠘⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢠⡏⠀⢺⣡⡄
⠀⢣⣄⣠⠜⠹⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⡼⠙⢄⣀⣠⠏
⣴⡉⠈⠉⠙⠲⣷⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢠⡷⠒⠉⠉⠉⣱
⠀⣿⣿⣿⣷⣦⡈⢳⡀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣰⠋⣠⣾⣿⣿⣿⡏
⠀⠀⠻⠟⠋⠛⢿⡄⠹⡄⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⡼⠁⣼⠟⠛⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⡄⢹⡀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠟⠀⣸⠁⡾⠁
⠀⠀⠀⠀⠀⢠⠞⠲⠇⢠⠓⢄⠀⠛⢁⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢃⠘⢷⠛⢆
⠀⠀⠀⠀⠀⠈⠳⠤⢤⠟⠒⠒⢷⣦⡀⠙⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠒⠚⢦⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⡤⢤⠈⠻⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡿⠋⢀⠤⢄⣀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⠻⡛⢿⣿⠷⠶⣶⡶⠾⢿⣿⠿⡫⠋⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠢⣜⣶⠆⣹⡇⠶⣜⡠⠚⠁⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣠⣾⣿⣿⣦⣀⡀⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣛⠫⣍⠙⣝⢿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠉⠁⠀⠀⠀⠈⠱⣄⢳⠘⡆⢧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⣀⡧⣼⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣡⠚⠁⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠁⠀⠀⢀⣀⣀⣸⣇⣀⡀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣾⠀⠀⢠⡏⠀⠀⠉⠉⠉⠙⠛⡿⠉⠉⢉⣽⡿⣿⣦⠀⣀⣠⣤⣶⣾
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡀⠀⠘⢇⡀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣾⡋⠀⠨⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣾⠿⠯⠭⠭⠭⠭⠭⣓⣒⠢⢄⡀⠀⢸⣾⣿⡇⠀⠀⠀⠉⢉⣙⡶⠶⠶⣼⣇⣀⠀⢻⣧⣤⣷⡿⠺⢿⣿⣿⣿⣿
⠀⠀⠀⣠⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡙⢦⢸⣍⣛⣧⣀⣠⡴⠞⢻⠙⣧⡀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠀⠀⠀⠀⠉⠙⠛
⠀⠀⣾⠋⠉⠀⠉⠓⠒⠦⠤⣄⣀⠀⠀⠀⠀⠙⡌⢻⡏⠉⠉⠉⢹⣧⠀⠈⠳⣌⡛⠷⢦⣤⡀
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⠤⠥⠤⠃⠀⠀⠀⠀⠙⢷⣄⡀⠀⠈⠐⠠⡉⢿⣆
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⢳⣸⡟
⠀⠈⣿⣷⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣼⡿⠃
⠀⠀⠀⠉⠛⠻⠿⠛⠛⠷⢶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠷⣶⣦⣤⣄⣀⣀⢀⣀⣀⣤⣶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣿⣿⡿⠟⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣴⣿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡌⠛⠿⣷⡀⠀⣀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣯⣴⡦⣄⠀⠀⠹⠷⠀⠙⢿⣏⢢⡶⢿⣦⡀⠀⠀⢀⣤⣶⡀⠀⠀⠀⠀⣠⣶⣄
⠀⠀⠀⣴⣴⡶⢿⣿⣿⡿⢿⣿⣿⠓⠀⠙⠀⠀⠀⠀⠀⠀⠙⠳⠄⠀⠀⠘⠶⠶⠿⠟⠋⠻⠗⣶⣤⡴⠋⠉⠙⢗⣀
⠀⠀⠀⠈⠁⢋⣀⣙⠃⢴⣾⣿⡿⠇⠀⠀⠀⢀⣴⣶⣿⣧⡲⣄⡀⠀⠁⠰⣦⣤⣄⣀⣠⠶⠛⠟⡇⠙⠀⡀⠀⠀⠉⠑⠚⠋⠉⠳⠦
⠀⠀⠀⠀⠀⠻⠛⠛⢆⡈⠙⢻⣀⣀⣀⣠⣾⣿⣿⣿⣿⣿⣅⢈⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠟⡿⢡⠊⢉⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣴⣾⡏⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⢀⠀⠉
⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⢀⠃⠀⢈⡴⠟⠋⠙⠁⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⢠⣿⣿⣿⣀⡤⠀⠕
⠀⠀⠀⣿⣿⣧⡤⣄⣀⡀⠀⠀⠀⠀⠀⣄⠀⠀⠈⠠⠄⠚⢀⢀⣿⣶⣾⣷⡄⠀⠀⠀⠀⠀⠻⠿⣟⠂⠀⣸⣿⣿⣿⣏⡀
⠀⠀⠀⠛⠏⠉⢙⣿⣿⣷⣾⣦⣀⣄⢸⣿⡆⢠⣄⠀⠀⠈⠁⠻⢿⣿⡝⠋⠉⠂⠀⠀⠀⠀⠀⠀⠈⢀⣸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⡀⢀⣱⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣆⠀⠀⢠⣶⢀⡀⠉⠛⠧⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣧⣴⣄⣀⣄
⠀⠀⠀⠙⠛⠻⠿⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣷⣠⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁⠙⠋⠙⠻⠙⠛⠿⠛⠛⠛⠿⠿⠿⣿⣤⣼⣷⠀⠀⠀⣠⣄⣸⣿⡿⠿⠿⢿⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠿⠿⠿⠿⠿⠿⠶⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠛⠉⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠙⠻⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⣠⣴⠾⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠳⣦⣄⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⠻⣦⡀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⣠⠞⠻⣦⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠆⠀⢀⡾⠃⠀⠀⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠘⣷⡀⠀⠈⡄
⠀⠀⠀⠀⠀⡄⠀⠀⣼⠃⠀⠀⠛⣡⠞⢁⡤⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⠀⣠⠞⠙⣷⠀⠀⢁⡀
⠀⠀⠀⠀⢠⠀⠀⢸⡏⠀⠀⠀⠀⠁⠰⠛⠁⠀⠀⠀⢀⣴⠟⠀⣠⠟⠁⠀⠀⢀⣤⠞⠁⠀⠀⢻⡆⠀⠘⠇
⠀⠀⠀⠀⣿⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠁⣤⠞⠁⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⢸⡇⠀⠀⢰
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀
⠀⠀⠀⢰⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⣠⠞⠁⣠⠞⠁⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⣠⠞⠁⣀⡾⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡇⣠⠞⠁⢀⡴⠋⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡿⠁⢀⡴⠋⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⣃⡴⠋⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣽⡏⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⣿⡇⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠸⣿⠀⠀⣿⣦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠃
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣿⣅
⠀⠀⢀⣴⣿⠀⠀⢿⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⡟⠁⠹⣷
⠀⢠⡿⠉⣿⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⢠⡞⢷⢻⡇
⠀⢸⡇⠀⣿⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠘⢧⠟⢸⡇
⠀⠸⣧⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⣆⠀⢀⣾⠁
⠀⠀⠙⢷⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣷⡟⢿⡄
⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠸⣧
⠀⠀⠀⢀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡙⠧⣄⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⢿⡄
⠀⠀⠀⣸⣿⠀⠀⣿⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠙⠦⠈⠃⠀⠀⠀⢸⡇⠀⠀⣿⣧⠀⠘⣿
⠀⠀⢀⣿⣿⠀⠀⢿⡇⠀⠀⡴⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡄⠀⢹⡇
⠀⠀⢸⡟⣿⠀⠀⢸⣧⠀⠈⣡⠞⢡⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠘⣷⠀⠘⣷⡀
⠀⠀⣾⠃⢻⣇⠀⠘⣿⡄⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⡏⠀⢹⡇⠀⢹⡇
⠀⠀⣿⠀⢸⣿⡄⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⣾⠁⠀⠘⣷⠀⠈⣿
⠀⢸⡟⠀⣿⠋⢷⡄⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⢀⣾⡇⠀⠀⠀⢿⡄⠀⢻⡆
⠀⣼⡇⠀⣿⠀⠈⢿⣆⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⡞⣿⠁⠀⠀⠀⠸⣷⠀⢸⣷
⠀⣿⠃⢠⡿⠀⠀⠈⣿⣷⣄⠀⠈⠛⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⣠⣾⣿⠀⣿⠁⠀⠀⠀⠀⢿⡀⠀⢿
⢠⣿⠀⢸⡇⠀⠀⠀⢸⣇⠙⢿⣦⣀⠀⠈⠙⠛⠻⠷⠶⠶⠶⠶⠶⠟⠛⠋⠁⠀⣀⣴⠾⠋⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸
⢸⡇⠀⢸⡇⠀⠀⠀⠀⢿⡄⠈⢿⡍⠛⠶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠾⠛⠋⠁⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠈⣿⠀⢸
⢸⡇⠀⣿⠁⠀⠀⠀⠀⠈⣿⡀⠈⢻⡄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⠀⠀⠀⠀⠀⠀⣻⡆
⣸⡇⢸⡟⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⠀⠀⠀⠀⠀⠀⣽⠇⢀
⣿⠃⣼⠇⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⡇⢸⡿⠶⠶⠶⠿⠿⠛⠋⠀⣸
⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠁⢸⣷⣀⣀⣀⣀⣀⣠⣴⠾⠋
⣿⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡏⠀⢸⣿⠉⠉⠉⠉⠉⠉
⢹⣇⠀⠙⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠋⠀⢀⣸⠇
⠀⠙⠷⢶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⢤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠂⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⢀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣼⣷⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⣻⠻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⣿⢿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⠺⣿⢿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⣿⣏⣾⣿⣿⣾⣖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢻⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢻⣿⣟⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠋⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢉⡀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠈⢛⠂⠀⠀⠀⠀⣀⣀⠀⣤⡴⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⠀⠀⢀⠀⠀⠀⣉⡀⠤⠦⠐⠒⠂⠀⠀⠀⠀⣸⣳⣿⢱⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠳⠶⠃⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢣⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣥⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢿⠿⢿⡿⢛⣛⣟⣿⣿⣷⣿⣿⣿⣾⣟⣳⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠚⠛⠻⠛⠛⠛⠛⠛⠋⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣟⡥⢬⣟⣛⠛⣛⠛⡛⠛⡛⠛⢛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⣀⣀⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢴⡻⣏⢹⡿⣿⢿⠉⢉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣶⣶⢲⣿⣿⣦⣄⠙⠋⢀⣼⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⠈⢿⣿⣿⣿⣿⣾⣿⣿⣿⠛⠂⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠈⠻⢿⣿⣿⣿⡿⣿⣳⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠙⣿⣿⣦⣤⣴⡀⣩⣽⡇⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠁⠡⣾⣻⣿⣿⣿⣿⣿⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⢨⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢀⣘⡀⠴⠟⠁⠑⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡈⣸⣿⡶⢶⣿⣷⠠⣄⡀⠀⠀⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠐⠃⠐⠂⠀⠀⠀⠀⠀⠀⠁⢻⣿⣿⣶⣶⣿⣿⣿⣿⢷⣄⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⢻⠛⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⡛⢻⡟⠛⠉⠀⠀⠀⣠⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣀⡈⡀⠀⠀⠀⠀⣀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣾⣿⣟⣉⠉⠀⠉⢙⡖⠁⠀⢩⣿⣾⣿⡄⠢⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠰⣾⣦⣀⠀⠀⠀⣀⠛⠛⠛⠿⠻⠷⠦⠀⠀⠼⠦⠿⠿⠿⢿⣿⣆⢦⡀⠀⠀⠀⢘⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⠀⢠⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠦⠤⠤⠤⠤⠤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠢⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⣴⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣿⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⡀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣈⣷⣦⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠻⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣥⣀⣤⣯⣴⣷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣿⢋⣿⣿⣧⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⣿⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⣿⣿⣾⣿⣿⣿⡟⠃⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢀⣿⣿⡁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢺⡆⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠈⠃⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡏⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠁⠀⡀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢰⣦⠘⢟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠈⠃⠐⣤⡤⠄⠒⠒⠂⠈⠉⣹⢳⣿⠏⣿⣿⣿⣿⣿⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢐⣀⣀⡰⠦⠴⠤⠀⠀⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣺⢏⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⢃⣾⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⣿⣼⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢿⠿⠿⢿⠿⠋⣙⣛⣛⣛⣛⣶⣿⣿⣿⣿⣝⣋⣚⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠌⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⢿⣿⡿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠿⢿⡿⠿⣿⣿⣿⠿⠿⠿⢿⡿⠿⠿⢿
⣿⣶⣶⣶⣷⣶⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠈⠛⠰⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣄⣉⠒⠶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠈⠛⢻⡟⠛⠻⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣀⡀⠀⢻⣟⣿⣿⣿⣟⣿⠿⠋⢹⣿⣿⡇⠉⠿⠀⠀⠀⢠⠅⣀⣼⢣⣴⡶⠶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠉⠳⣼⠟⣿⣿⣿⡟⠙⣷⡀⠘⠀⠈⠀⠀⠀⡀⠀⠀⠙⢾⣁⣿⠸⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣬⣀⡈⢹⡶⢼⣴⣿⡇⢠⡖⠋⠁⠀⠀⠀⠀⣀⣤⡞⠀⡀⠀⠀⠀⠙⠃⠀⠀⠈⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣬⣹⡆⠀⠈⢻⣿⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⡿⠀⠀⠀⢀⠀⠀⢙⣿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡟⡀⠂⢸⡇⠠⣈⣀⣤⣶⣷⣿⣿⣿⣿⣏⠀⠀⠀⣴⣿⢆⣠⣾⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣏⣨⣴⡾⣿⣧⠀⠛⢼⣯⣿⣿⣿⣿⣿⠃⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣞⣿⠀⢀⣀⣙⣦⣚⣛⡛⣉⣀⣠⣴⣾⣿⠿⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⡏⣿⠟⢉⠉⠀⠀⠀⣠⠀⠀⢀⠀⠀⣼⣅⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠁⠀⠀⠀⠐⠿⠶⢃⠎⠁⠀⠹⡿⢷⣶⠟⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⡄⠀⠀⠀⠀⠛⠛⠉⠀⠀⠠⣤⣈⣛⣁⠂⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⡟⠛⠲⢦⣤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⣤⣤⣤⣴⡗⣤⠚⠉⠀⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠵⠿⢀⡇⠀⡆⠀⠀⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠻⠿⣿⣿⣿⠇⣈⠠⠠⡀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡙⠢⡀⠀⠀⣹⠅⣿⠃⣹⡇⣿⡈⢿⣿⡲⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢰⠃⠀⠀⡾⢠⣿⡄⣿⣧⣙⣷⡬⠻⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⣿⡿⢡⡏⠀⠀⣸⠃⣾⣿⣷⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣌⣳⡄⠀⠀⢰⠏⣸⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠇⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣷⡀⠀⠀⠀⠀⠀⠉⠀⠹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣉⣠⣤⣤⣤⣶⣶⣆⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣝⣻⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⣶⣶⠶⢿⣿⣟⠛⠛⠷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⣩⠴⠛⠉⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠂⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣴⠋⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡾⠁⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⡾⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⡾⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣼⠁
⠀⠀⣼⠃⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢠⡟
⠀⣰⠇⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⣼⠃
⢀⡟⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣰⠏
⢸⠃⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡠⠖⠛⠛⢶⡏
⣿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣾⠁⠀⠀⠀⠈⣷
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠙⢦⣀⠀⠀⣼⠃
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⣿⠀⠀⣿
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⡞⣿⠀⠀⣿⡆
⢹⡀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⣠⠏⠀⣿⠀⠀⢸⡇
⠘⣧⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⢀⡾⠁⠀⠀⣿⠀⠀⢸⡇
⠀⠹⣧⡀⠘⣆⠀⠐⡤⠄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠈⠻⣦⣘⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡇
⠀⠀⠀⠀⠈⠙⠿⣷⣤⣀⣤⡴⠶⠛⠉⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢻⡟⠛⠛⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿
⠀⠀⠀⠀⢀⣤⠴⠶⠛⠛⠉⠉⠀⢸⣄⠀⢹⡈⠉⠉⠛⠓⠒⠶⠦⢤⣤⣄⣀⣸⡀⠀⠀⢸⣀⣀
⠀⠀⠀⠰⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡟⡁⠀⣀⡼⠉⠉⠛⠛⠳
⠀⠀⠀⠀⢿⡀⠉⠙⠓⠲⠶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⢀⣠⢼
⠀⠀⠀⠀⠈⠛⠻⠶⢦⣤⣀⡀⠀⠉⠉⠛⠛⠲⠶⠶⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠦⠤⠤⠴⠖⠋⠀⠀⢀⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⢶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣰⡾⠛⣷⡄⠀⠀⠀⠀⠀⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⣴⠟⠛⣷⣿⣏⣩⣿⡁⢾⣿⣿⠏⣠⣦⣄⣰⡿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣽⡇⠀⣀⣼⡇⠐⣿⣯⣤⣾⣿⣽⣷⣤⣤⣿⣷⣿⣏⣿⣿⣵⣏⡀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⡀⣾⣏⢙⣿⣾⡿⠛⠉⠁⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣟⢉⣹⣷⡄⠀⠀⠀⢿⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⣎⢁⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠙⢿⣿⠿⣿⣟⠛⢿⣆⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣬⣿⣻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠙⢷⣌⣻⣿⡿⠿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣾⢣⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠻⣯⣿⠁⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣷⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠉⣩⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢧⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣙⠓⣨⣿
⠀⠀⠀⢠⡶⣿⣦⢻⣧⡉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⢻⡿⠿⣄
⠀⠀⠀⠸⣧⡈⠛⢀⣹⣿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣤⣦⣽⣧⡀
⠀⠀⠀⠀⠈⠛⠛⠛⢻⣿⠙⣿⣷⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡉⣿⣿⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠈⣿⡆⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⣿⠃⢈⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠿⠃⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣧⣿⡉⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣴⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⠈⢁⣰⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢩⣭⡷⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⢿⣿⠿⠛⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣭⣀⣀⣘⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⣧⣴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠉⠛⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢸⣿⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢛⣿⡆⠈⢿⣦⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣾⡟⠘⢻⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠘⢛⣁⣀⣀⡿⣷⣄⠀⠀⠀⢸⣿⡏⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⣸⣿⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣻⡿⢻⣿⣷⣿⣿⣷⣄⠀⢸⣿⡇⢸⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⣸⣿⠁⠙⠿⠟⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣋⣴⠟⠹⣿⣿⠇⠙⢿⣷⣌⠛⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⣰⣿⡿⣾⣿⡶⠾⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋⠀⠀⠀⠀⢀⣀⣀⣀⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⢷⣌⠻⣶⣌⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠹⡿⠃⢀⣹⣿⣿⣷⣦⣤⣀⣀⡀⣀⣀⣤⣶⡿⠿⣷⣿⣤⣾⣿⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣾⣿⢋⣤⣤⠀⢻⣿⠛⠿⣿⠿⠿⣿⣯⣤⡶⢿⣿⣿⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⢸⣇⣉⣠⣼⣿⡇⢰⣿⠃⢀⣼⡿⣿⣧⡀⠻⡛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡅⠈⠛⠛⠉⣽⣿⡇⠈⠻⠿⠿⠟⠁⠸⣿⠛⠻⠿⠶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⣰⠖⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣯⣠⡄⠈⢳⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡙⠛⠁⠀⢸⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⢀⣴⠣⠤⠔⠋⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣋⡴⢛⢲⡄⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠙⠿⢡⣾⣿⣦⡀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠈⢿⣿⣿⣿⡆⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠈⠛⠿⠿⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡛⠶⠶⠟⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣦⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢟⣉⣻⡿⠿⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣾⣍⣽⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠈⠐⠛⠻⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠘⡛⢶⠇⠀⠀⠀⠰⣦⠀⠀⠈⠀⣨⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣷⢸⡇⠀⠀⢾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡘⢦⠀⠀⢹⠀⠀⠀⢀⡟⣸⠇⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠈⠰⠋⠀⠀⢠⣿⣿⣿⣿⠁⠀⠀⢀⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⡾⠁⠀⢸⡗⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⢠⣮⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡾⢤⠂⡄⠀⠀⠀⣤⣿⣿⣿⡿⠋⠀⠀⢰⠆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⠃⠀⠀⠁⢰⣿⣿⣿⣯⣥⣀⣀⠀⠐⢨⠀⣸⣿⣿⣿⣿⣿⠿⡛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠉⠀⠀⠀⠀⢠⡄⢸⡿⠛⣩⣿⣿⣿⣿⣄⠀⠀⠀⡟⠻⣿⣿⣿⣭⠶⠖⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠀⠈⢷⣶⡄⠀⠀⠀⠀⠀⢲⣸⠇⠘⠁⢰⣿⠂⠀⣿⢋⣯⠀⠀⠀⣇⠀⢻⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⡿⠷⢀⡾⠿⠃⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⢸⣿⢿⠛⠶⣌⢟⠀⠀⠀⡿⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡼⢹⡇⢸⡄⠘⣿⡸⡀⢠⣿⣿⡄⠀⠀⡅⠀⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢸⠀⠀⠀⢸⡇⢸⣿⣦⣈⠛⣿⣿⣿⠋⠀⠈⠒⢧⡀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣊⣀⣠⣤⣼⠇⢸⣿⡿⢛⠿⠿⢿⣿⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠒⠀⠈⠹⠿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⣖⠀⠺⣿⣿⣿⣷⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⣷⡁⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠃⠀⠛⠉⢡⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⡿⠛⠉⠀⠀⠀⠀⠀⠀⠨⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣲⣶⣿⣻⣿⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⠛⠻⣿⣿⣿⣿⣿⣟⡻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⣰⣮⡟⠙⠙⠛⠛⠛⠁⠟⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣿⣿⡿⠇⠀⠀⠀⠀⠚⠆⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠲⣄⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢪⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⡰⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⣴⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠾⢃⡰⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠾⠛⠛⠛⠛⠛⠻⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡇
⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢹⡇
⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡇
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡇
⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇
⠀⠀⠀⠀⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡾⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣼⡿⠋⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣤⣤⣤⣤⣤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢛⠟⠉⠉⠉⠙⠛⠉⠉⠉⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⠉⠀⠉⠉⠉⠉⠙⠛⠿⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⣶⣿⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣴⡿⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⢿⣦⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣤⣤⣤⣤⣥⣶⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣯⣤⣤⣤⣬⣭⣤⣤⣤⣾⠟

⡄⢹⠁⠈⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⢀⣀⣀⡀
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⣀⣀
⡇⢸⠀⠀⠀⠛⠒⠒⠒⠒⠲⠶⠶⠤⠤⠤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠓⠒⠒⠒⢶⡄
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣤⣴⣷⡠⣄
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⡞⠉⠀⠀⠈⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣾⢿⣆⡀
⡇⢸⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠘⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠁⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢯⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠈⢫⡀⠀⢠⡤⠀⢀⣠⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣬⣿⣶⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠟⠻⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⡇⢸⠀⠀⠀⠀⠀⠀⠀⢠⣶⣟⠉⠀⣀⣼⣅⣩⣽⣆⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢎⣿⠟⣿⣿⣿⡣⢄
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⣿⠛⠻⣿⡿⣿⡏⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢋⡽⢿⣟⣸⣿⣿⣿⣿⣎⠙⠂
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠀⠀⣿⡇⢻⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿⠟⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣷⠸⣷⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢰⣿⣿⣿⣴⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣷
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠿⠁⢿⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⡇⣼⣿⣿⣿⣿⣿⡄
⡇⢸⠀⠀⠀⠀⠀⠀⠀⣠⠾⠦⠤⠤⠄⠠⠀⢀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣿⣧⣿⣿⣿⣿⣿⣿⡇
⡇⢸⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⢠⡆⠀⠀⠀⢄⠀⢫⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣿⡷⠀⠻⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡇⢸⠀⠀⠀⡰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠀⠀⠙⢀⡐⣤⣤⠤⠴⠚⠛⠒⠈⠉⠉⣿⢡⣿⡟⢹⣿⣿⣿⣿⡟⣿⣿⣿⣿⣧
⡇⢸⠀⠀⠀⣿⣄⠀⢳⠄⠀⠀⣴⣦⡀⢤⠤⠴⠶⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⢏⣿⣿⠳⣼⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣧⡀
⡇⢸⠀⠀⠀⣬⠙⠷⠒⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⣼⣿⡿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⢁⣾⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠃⠶⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⣽⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠿⠟⠿⢿⠟⠋⣀⣉⣛⣩⣙⣿⣻⣿⣿⣿⣿⣷⣼⣛⣛⣛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣒⣚⣠⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣋⣉⣉
⣿⣿⣿⣿⡿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣗⢀⢀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠻⢷⣄⣤⣶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⣤⡀⠀⣿⠋⢀⡀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⣶⠾⠶⠿⣷⣶⣤⣤⡿⠀⠀⠀⠈⠳⠾⠿⠿⠿⠿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⣀⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⡇⠀⠀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠙⣿⡀
⠀⠀⠀⠀⢀⣿⠃⠀⠀⣼⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⣼⠃⠀⠀⢰⠃⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢹⣧
⠀⠀⠀⣸⡏⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢻⡆
⠀⠀⢠⡿⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⣿
⠀⠀⣼⠇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⢿
⠀⢰⡟⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⣼⠃⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⢀⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⣾⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣼
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣿
⢿⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⢸⣇⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⡇
⠈⣿⡀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢰⣿⠁
⠀⠸⣧⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣾⠇
⠀⠀⠹⣧⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⡟
⠀⠀⠀⠹⣧⡀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠙⣷⣄⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⢿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠿⠂⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⣤⣤⣤⣤⣤⣤⣤⡀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡟⠉⢠⣬⠟⠀⣿⠀⠠⠟⠙⣷⡶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄⣀⣀⣀⣴⣿⣤⣄⣀⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠃⠀⠀⠐⠂⠀⠀⠐⠂⠀⠒⠈⠁⠐⠀⠂⠀⠀⠀⠀⠒⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⢀⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣶⣆⣀⣰⣶⣶⣶⡶⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣿⣦⣤⣴⣿⡟⠛⠛⠛⠛⠛⢻⡇⠈⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢠⣝⣛⣿⣿⣿⣿⣿⣟⠀⣀⣀⣤⣴⣾⣿⣿
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠛⣿⣧⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⣿⠟⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠹⣿⣿⠏⠉⠀⠀⠀⠈⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠀⢀⣤⣿⣿⣧⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢻⣏⠉
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣄⢀⣄⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠙⠋⠘⠋⠀⠺⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⢿⡇
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢶⣦⠈⠻⠆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣯⡁⢀⣤⠀⣠⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⣿⣷⣶⣿⣷⣾⣿⠀⠀⠀⠀⡀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡼⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠇⠈⠢⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠢⠜⠉⠉⠙⠛⠛⠻⠿⠿⣿⣧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⣀⣠⣤⣶⣦⣤⡀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⢰⣿⠿⡿⢿⣿⢿⣷⡀⠀⠀⠀⢶⣾⣶⣶⣶⡆
⠀⠀⠀⠀⠀⣿⡿⣿⣿⡿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢈⣏⣿⣹⣉⣏⣿⣹⣹⣏⣏⣽⣹⣏⣏⣿⣹⣉⣏⣿⣹⣹⡇
⠀⠀⠀⠀⠀⢹⠃⠙⠛⠃⠻⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣧⢰⣶⠀⣾⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠨⠿⠿⠽⠿⠄⠀⠀⠀⠨⠯⠽⠽⠭⠯⠿⠽⠽⠯⠯⠽⠽⠯⠯⠽⠽⠭⠯⠿⠽⠽⠇

⣿⣿⣿⡄
⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⣩⡁
⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠹⣿⣿⣿⣿⣿⣿⣧
⣤⡀⠈⠻⣿⣿⣿⣿⣿⣆
⣿⣷⣄⣀⡘⢻⣿⣿⣿⣿⣆
⣿⣿⣿⣿⣿⣦⡽⢿⣿⣿⣿⡆
⣿⣿⣿⣿⣿⣿⣿⠈⠻⣿⣿⣿⡄
⠈⠻⣿⣿⣿⣿⣿⣦⠀⠙⣿⣿⣿⣆⠐⡄
⠀⠀⠈⠻⣿⣿⣿⣿⡗⠀⠈⢿⣿⣿⣷⡄⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄⠀⠀⠹⣎⠻⣿⣦⠁⢀
⠀⠀⠀⠀⠀⠘⢿⣿⣇⠙⢷⡄⠀⠘⢦⣙⢿⣷⣴⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠘⢿⣯⡷⡌⠃⠀⠀⠀⠙⠻⣿⣿⣦⡐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⡄⢿⣿⣿⠇⠀⠓⢢
⠀⠀⠀⠀⠀⠀⠠⡌⢻⣿⣿⣄⠀⠀⠀⠀⠀⠈⠛⢿⣿⣎⢧⣀⠀⣄⠀⣠⣶⠀⠀⠀⢀⣠⡤⠤⠈⣷⣿⣿⣿⠂⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⢻⣄⢿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠙⠿⣷⣝⣦⣿⣶⣿⡟⠀⠀⠀⡬⠋⠀⠀⠀⠈⢻⣿⣿⡀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠻⣿⣿⣿⣿⣷⣄⠀⠀⡀⠀⠀⠈⠻⣿⡝⢿⣿⣄⠀⠀⠀⠃⠀⠀⠀⠀⢠⣾⣿⣿⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡹⣿⣿⣿⣿⣿⡀⢸⡇⠀⠀⠀⠀⠈⠻⣶⣝⢿⣆⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⣿⣿⣿⣿⣿⣯⣷⠖⠀⠀⠀⠀⠀⠘⣿⣷⣻⢄⠀⠀⠀⠀⢰⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡄⢀⠀⠀⠀⠀⠀⠙⠻⣿⣦⡁⠀⢀⣠⣾⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡦⣆⠀⠀⠀⠀⠀⠈⠉⠻⢦⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣶⣤⣤⣾⣶⣤⣾⣿⣿⣿⣿⣿⣿⡟⠲⢖⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⡿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠻⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠀⡀⢀

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡲⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⡀⢀⣠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⢤⡂⠀⠀⣄⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⡄⢤⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡤⡻⣄⠐⠈⢷⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣔⣀⣤⠬⠶⠒⠃⣁⠆⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡛⢶⣄⡀⡉⠙⠛⠓⣾⣾⠾⢲⠾⠿⠿⠿⠟⠋⠉⠉⠉⠉⠁⠁⠀⢀⡠⠟⠀⡈⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡖⢊⠛⣷⡦⡤⣄⡉⡀⡀⢀⡠⠀⠀⠀⢀⡀⢀⣀⣀⣤⢴⠴⠋⠁⠀⢰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣵⣀⡷⠀⠀⠀⠀⠀⠁⠀⠀⠐⠐⠛⠛⠉⠉⠉⢉⠀⢠⣀⣤⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠈⠹⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡘⠑⠚⠉⠁⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣫⣄⠈⠙⠛⠓⠗⠒⠳⠒⢸⣀⣣⠴⠋⠁⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣈⡛⠞⠄⠉⠉⠐⠛⠙⠋⠉⠁⠀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣮⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡶⠒⠛⠉⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣴⣳⡔⣄⡠⢠⣩⣈⣤⣰⣏⣠⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⡛⠛⠛⠛⠛⡯⡼⢟⣉⠙⢉⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡁⠀⠀⠀⣼⣿⢏⠚⠛⣂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢋⠿⠵⢶⣶⣿⣿⡷⡷⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⣼⣿⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣇⠀⢿⣿⠀⠀⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢟⡨⣻⠀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣄⢹⡆⠿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠁⢽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡌⢟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⣼⣷⡤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣾⠁


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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡕⡗⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⣄⣤⠔⠙⢦⣄⣤⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⣁⣹⣶⣿⡿⠿⠿⠿⣿⣿⣦⣋⡩⢝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢊⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣮⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⢟⠱⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠡⠈⢙⠿⣿⣶⣦⣤⣤⣶⣾⠿⠟⢉⣊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠛⣛⢳⣿⡞⣋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠁⠀⠁⠀⠉⠈⠉⠁⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⣻⣿⣭⣟⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣵⣾⠟⢻⢿⡿⡟⠻⢷⣮⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⢄⠀⠀⠀⢀⣀⣀⣰⣋⣾⠏⠀⠀⢈⣲⣖⡁⠀⠀⠹⣷⣹⣆⣀⣀⡀⠀⠀⠀⡠⣖⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⣦⣶⡿⠟⠛⡛⠛⠛⠿⣷⣤⠺⡱⡟⢻⢎⠗⣤⣾⠿⠛⠛⢛⠛⠻⢷⣶⣴⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢻⣶⡢⡀⢃⢸⠀⠀⢸⣳⠏⠀⢎⡱⠀⠹⣞⡇⠀⠀⡆⡘⢀⢔⣶⡟⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠈⠪⣿⣿⡘⡈⣠⣜⣰⣯⡴⠞⠛⠛⠳⢦⣽⣆⣣⣀⢁⢇⣿⣿⠕⠁⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠈⣁⣐⡒⠪⡽⣻⠥⢭⣿⢏⣤⠲⣛⣛⠶⣤⡹⣿⡭⠬⣝⣿⠅⢒⣒⣉⠁⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣫⣽⣿⡄⠀⠀⣀⢾⡅⣇⣯⢟⡇⠿⠱⢏⡡⢌⡹⠎⠿⢸⡻⢿⣸⢨⡷⣀⠀⠀⢠⣿⣯⣝⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⣿⠋⠁⠙⣿⢶⣲⡴⣾⡿⢟⣩⡍⡆⠉⡉⢸⠀⠀⡇⢉⠉⢠⢩⣭⡻⢿⣶⢦⣖⡶⣿⠋⠀⠙⣿⣎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣳⣿⣁⡀⢀⢞⡢⠋⠁⣰⠏⢠⣟⡵⢦⣃⣈⣮⣿⡦⢴⣿⣷⡁⣘⡰⢮⣛⡄⠹⣆⠈⠙⢔⡳⡀⢀⣈⣿⣞⣧
⠀⠀⠀⠀⠶⡷⡷⠶⡷⣾⣿⡷⣸⢾⣿⣉⢎⡱⣿⠀⣿⢺⡁⢎⠀⠀⢈⣏⢾⡷⣹⡁⠀⠀⡱⢈⡇⣿⠀⣿⢆⡱⣋⣷⡷⡏⢾⣿⣿⢾⠶⢾⠾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⣿⡉⠁⠈⢮⠕⣄⡀⠹⣆⠘⣭⡳⠎⡍⣉⢿⣿⠗⠺⣿⡿⡉⢩⠳⢞⣭⠃⣰⠟⢀⣠⠮⡵⠁⠈⢉⣿⢯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡹⣿⣄⡀⣠⣿⠾⠽⢳⢿⣷⣮⣛⣃⠃⣀⣁⢸⠀⠀⣇⣈⣀⡘⣸⣛⣵⣾⠿⠞⠯⠷⣿⣄⠀⣠⣾⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣝⣻⣿⠃⠀⠄⠉⢾⡃⡏⣟⣎⡇⣶⡰⣎⡑⢊⣱⢆⣶⢸⣱⣻⢹⢘⡷⠉⠀⠀⠘⣿⣟⢫⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢀⣉⠩⠥⢔⣷⣽⡒⣚⣿⣎⠛⠼⣭⣭⠧⠛⣱⣿⣓⢒⣫⣾⡢⠬⠍⢉⡀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⢀⢔⣿⣿⢱⢁⠙⢫⠸⣟⠳⢦⣤⣤⣴⠞⣻⠏⡝⠋⡈⡎⣿⣿⡢⡀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣼⠿⠕⠁⡎⠸⠀⠀⢸⡽⣆⠀⢌⡱⠀⣰⢯⡇⠀⠀⡇⢁⠈⠪⠿⣧⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⠟⠿⢷⣦⣤⣥⣤⣤⣶⠿⠛⢴⡱⣧⣽⢎⡦⠛⢿⣶⣤⣤⣬⣤⣴⡾⠿⠻⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠊⠀⠀⠀⠈⠉⠉⠹⣍⢿⣄⠀⠀⢈⡽⠯⡁⠀⠀⣠⡿⣹⠏⠉⠉⠁⠀⠀⠀⠑⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡻⢷⣦⣼⣾⣷⣧⣴⡾⢟⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣽⣹⣟⣯⠷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣠⣾⣾⣿⣯⣿⣿⣷⣷⣤⣤⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠿⠋⢉⣙⣿⣿⣟⣿⣿⣿⣋⣉⠛⠿⢿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⢉⣤⣤⣴⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣦⣤⣤⡉⠛⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⢉⣥⣶⣶⣿⣋⣿⣿⣿⠟⠛⢿⡿⣿⣿⢿⡿⠛⠻⣿⣿⣿⣹⣿⣶⣶⣬⡙⢻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⢻⢗⡛⣷⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣛⢺⡟⣿⣷⣶
⠀⠀⠀⠀⠀⠘⢣⣴⣾⣿⣿⡿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡞⠃
⠀⠀⠀⠀⢰⣷⡀⠸⣿⣿⣿⣷⣿⣻⣿⣫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣜⣷⣟⣷⣿⣿⣿⣿⡇⢀⣾⡄
⠀⢀⣀⡀⢾⣿⡗⢲⡝⡿⢿⣿⣷⣿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⣿⣿⣿⣿⢿⢻⡶⢻⣿⡗⢀⣀⡀
⠀⠘⠋⢻⣤⣿⣿⢽⣷⣿⣶⣽⢿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⡿⣯⡶⣿⣾⡽⣿⣿⣤⡟⠙⠃
⠀⣀⣤⣸⣿⣽⣿⣟⣿⣿⣿⣽⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣯⣿⣿⣿⣻⣿⣯⣿⣇⣤⣀
⠀⠙⣿⣿⣿⣿⣿⣟⢹⣿⣻⣿⡟⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢻⣾⣿⣿⡏⣻⣿⣿⣿⣿⣿⠉
⠀⢠⣿⣧⢟⣿⣿⣛⣾⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣶⣛⣿⣿⡿⣿⣿⡄
⠀⣾⣿⣿⣟⣿⣟⣻⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣻⣿⣿⣿⣿⣷
⢠⣿⣿⡏⢹⣿⣏⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣩⣿⡏⢹⣿⣿
⢸⣿⣿⠠⢾⣿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⣿⣿⡷⠀⣿⣿
⣾⣿⡏⢀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡀⢹⣿
⣿⣿⡇⠘⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⢸⣿
⣿⣿⡅⣾⣿⡿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⣿⡗⢨⣿
⣿⣿⡁⠻⣿⣾⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⣿⠗⢈⣿
⣿⣿⡇⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⢸⣿
⢹⣿⣧⠀⢠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⣼⣿
⢸⣿⣿⠐⢻⣿⣿⡄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢠⣿⣿⡟⠂⣿⣿
⠀⣿⣿⣤⡿⣿⣯⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣽⣿⢯⣤⣿⣿
⠀⢻⣿⣿⣿⣿⣿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣯⣿⣿⡏
⠀⠘⣿⣯⣸⣿⣿⣷⢯⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡿⣾⣿⣯⣦⣿⣿⠁
⠀⣴⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣧⣿⣿⣿⣿⣿⣦
⠀⠀⠙⢸⣯⣿⣟⢫⣿⡟⣷⣉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣉⣿⣿⣿⣝⣻⣿⢽⡇⠋
⠀⢠⣤⡼⢃⣿⡿⢱⣿⠿⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⠟⣿⡎⣿⣿⡈⢧⣴⠄
⠀⠀⠈⠰⢾⣿⡟⡿⢷⣿⣼⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣧⣿⡾⣿⢺⣿⡷⠂⠁
⠀⠀⠀⠀⠸⠟⢀⢸⣿⣿⣿⣿⣯⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣽⣿⣿⣿⣿⣇⡀⠻⠇
⠀⠀⠀⠀⠀⢰⡛⠙⢿⣧⣿⣧⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⢿⣿⣼⣿⣾⠟⠋⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣯⣼⡶⣥⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⢀⠀⣀⣀⢀⡀⠀⠀⢀⣀⣤⣶⣿⣿⣿⢿⣯⢶⣧⣽⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⣈⡛⠛⠿⢻⣟⣿⣿⣿⣷⣶⣿⣶⣿⣿⣾⣿⣷⣿⣿⣿⣿⣹⡟⠛⠛⢋⣥⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⣀⡙⠛⠻⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠟⠛⢋⣠⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣦⣄⣉⣿⣿⣿⣿⣿⣿⣍⣠⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠛⣿⠿⣿⣯⣽⡿⠿⣿⠛⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⢿⡿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢒⠲⣄⠀⠀⣠⢖⡒⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠉⠉⣽⡌⣇⡼⣱⣯⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡔⠳⠨⡅⠰⠋⣿⠀⠄⣽⠛⠇⢩⠅⠞⢢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡉⡀⠁⡱⠚⢀⣀⣳⣎⣀⡀⠱⢏⠈⢀⢉⣉⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡸⢦⣏⡀⢨⣾⡥⢗⠫⠁⠒⠒⠒⠒⠊⢝⡺⢽⡶⠅⣀⣹⡴⢇⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⠬⢅⠰⠒⠞⡴⡫⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢝⢆⠳⠒⠄⣨⠥⣶⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⢰⠳⡦⠀⣵⢀⡞⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢳⡀⣮⠀⠴⠞⡄⢛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠛⢄⣉⡁⢀⡎⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢱⡈⢈⣉⡠⠛⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⡈⠞⠱⠛⢸⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡇⠛⠮⠿⢁⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣡⡞⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⢳⣜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣥⠲⠄⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠖⣬⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣛⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢛⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣄⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡹⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠚⢟⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡫⣰⣄⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⣠⣆⢹⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣝⡴⣺⠟⢌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠛⣓⢦⣹⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡿⡄⣠⠀⠀⠙⠪⣔⣢⠤⠀⠀⠤⣔⡪⠕⠋⠀⠀⣄⢠⣿⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⢰⢾⢿⡤⠄⡴⠒⢫⡝⢆⡠⢫⡝⠒⢄⠠⢤⣿⡷⡟⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⡘⢄⠐⠃⠰⣦⠇⠼⠧⠸⣴⠂⠘⠂⡠⣃⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⡽⠦⣥⣄⣪⣵⣀⣎⣱⣀⣬⣕⣠⣬⠴⢧⣬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡄⢝⡏⢹⡫⠠⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣶⣶⣶⢶⣷⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣷⡞⠛⠛⠛⠛⠛⠛⠉⠉⠉⠛⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣉⡀⠀⢀⣀⣀⣀⣀⣀⠀⢀⣴⣟⣿⠂
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣿⠋⠉⠉⠉⠙⠛⠛⠛⠛⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠁⠀⠁⠉⠉⠉⠉⢻⣿⣿⡇⣼⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠁⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠿⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⢈⣿⡆⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢻⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠂⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⣇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿⡀⠐⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⣯⠀⢀⡟⠁⠀⠒⠂⠀⠛⠳⠟⠓⠛⠛⠓⠒⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠚⣿⣷⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡤⢋⡀⠀⠀⣀⣴⣶⠀⣠⠒⠀⣀⣠⢠⡔⣀⣤⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣠⣶⣺⣠⣶⣲⡄⠀⠀⢰⣶⣼⣿⣽⣷⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠛⠉⠉⠉⠉⠉⠉

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣌⣤⣬⣯⣠⣤⣠⣸⣆⣡⣶⣏⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠲⠒⠒⠂⣤⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⣀⣀⣠⣍⣌⣆⡄⡤⠐⡀⣄⠀⠐⠂⠒⠀⠀⠀⠁
⠀⠀⠀⠀⠠⣤⣤⣆⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠀⣀⡀
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣵⣶⣶⣷⣿⣿⣷⣶⣦⣄⡀⠈⠻⣿⣷⣀
⠀⠀⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⠷⣤⡈⠻⣿⣶⣄⠀⠀⢀⢀⣠⡀
⠀⠀⠀⠀⠀⠈⠛⡛⠉⢁⣬⣩⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣿⣧⣥⣼⠃⠈⠻⠦⣌⠻⠿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠸⠿⠛⠻⣿⣟⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⡛⠶⢤⣙⣻⢿⢿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣷⣶⣤⡤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠿⡍⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣄

⠀⠀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⢀⣾⣿⣿⡀⠀⠀⣼⣿⣛⣿⡗⠀⠀⣴⣿⠛⠻⠋⠀⢀⣿⡟⠻⣿⡆⠀⠀⣿⣿⣛⠛⠀⠀⢸⣿⠛⠛⠃
⠀⢠⣿⣿⣾⣿⡇⠀⠀⣿⡟⠛⣿⣦⠀⢰⣿⣇⠀⣀⠀⠀⢸⣿⠃⢠⣿⡇⠀⢰⣿⡟⠛⠃⠀⠀⣿⡿⠿⠟
⠠⠿⠏⠀⠈⠿⠟⠀⠘⠿⠿⠿⠟⠁⠀⠀⠻⠿⠿⠟⠃⠀⠾⠿⠿⠟⠋⠀⠀⠸⠿⠿⠿⠇⠀⠸⠿⠇
⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣀⡀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⣀⡀
⠀⢀⣾⡿⠛⠿⠃⠀⠀⣸⣿⡃⣀⣿⡟⠀⠀⠚⣿⡿⠛⠀⠀⠀⢸⣿⠇⠀⠀⣿⣿⣰⣿⠗⠀⠀⢸⣿⡇
⠀⢸⣿⡁⢿⣿⡇⠀⠀⣿⣿⠿⢿⣿⠇⠀⠀⢠⣿⡇⠀⠀⠀⠀⣾⣿⠀⠀⢠⣿⣿⣿⣅⠀⠀⠀⣼⣿
⠀⠈⠻⠿⠿⠟⠁⠀⠀⠿⠏⠀⠾⠿⠀⠀⠸⠿⠿⠿⠁⠀⠾⠿⠿⠃⠀⠀⠸⠿⠁⠻⠿⠆⠀⠀⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣼⣿⣧⢀⣼⣿⡇⠀⠀⢠⣿⣿⡄⣾⣿⠀⠀⢀⣶⡿⠻⣿⣧⠀⠀⠀⣿⡿⠿⣿⡆⠀⠀⣠⣾⠟⢿⣿⡄
⠀⢰⣿⢻⣿⣾⢿⣿⡇⠀⠀⢸⣿⢿⣿⣿⡏⠀⠀⢸⣿⡃⠀⣿⡿⠀⠀⢸⣿⣷⣶⠿⠃⠀⠀⣿⣿⣄⣸⣿⠃
⠀⠿⠏⠈⠛⠋⠸⠿⠇⠀⠀⠼⠿⠀⠿⠿⠃⠀⠀⠘⠿⠿⠿⠟⠁⠀⠀⠾⠿⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⣿⡿⠿⣿⡦⠀⠀⠀⣠⣾⡿⠿⠗⠀⠀⠀⠿⢿⣿⡿⠿⠀⠀⠀⢸⣿⡇⢰⣿⡇⠀⠀⠀⢻⣿⡆⢠⣿⡗
⠀⢸⣿⣷⣾⡟⠃⠀⠀⠀⠙⠿⢿⣶⡄⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⣼⣿⠀⢸⣿⠃⠀⠀⠀⠸⣿⣧⣿⡟
⠀⠼⠿⠈⢿⠿⠄⠀⠀⠐⢿⣶⡿⠟⠁⠀⠀⠀⠀⠿⡿⠀⠀⠀⠀⠀⠹⠿⣷⠿⠋⠀⠀⠀⠀⠀⠿⠿⠏

⠀⢰⣿⡇⢠⣤⡄⢰⣿⠇⠀⠀⠀⠀⠀⢻⣿⣆⣴⡿⠀⠀⠀⠀⠀⠘⣿⣇⣠⣿⠇⠀⠀⠀⠀⠀⠿⢿⣿⣿⠇
⠀⠈⣿⣷⣾⣿⣷⣿⡟⠀⠀⠀⠀⠀⠀⢠⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠁⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⢿⣿⠏⠸⣿⠟⠀⠀⠀⠀⠀⠀⠰⡿⠋⢻⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣧⣀⠀⢉⣣⢳⣜⡓⠚⣧⠆⠍⣳⠽⢿⣽⣿⣯⣿⣿⣎⣻⣯⢭⡝⠁⠀⢹⠋⢰⠀⠹⡇⠈⠻⠈⠛⢿⣵⣆⡼⠃⠈⡾⠃⠀⣛
⣿⣿⣿⣿⣿⣿⣤⣄⣛⠃⠀⠘⠲⡄⠘⡷⣼⠟⠾⠷⣿⣿⢿⣻⣭⣟⣧⡖⠙⢐⡗⡀⠐⠠⡄⠀⠀⠀⠀⠀⣿⠃⠀⠰⠄⡑⣀⡤⠻
⣿⣿⣿⣿⣿⣍⣻⣷⡄⣿⣆⣤⡄⠀⠌⠀⣴⣼⡛⠻⢿⠟⠾⢳⢯⣎⠐⠁⣰⠼⣿⣿⣿⡇⠀⢠⡄⠀⠀⣄⠘⠀⠠⠂⠀⢻⣁⣠⣷
⣿⣿⣥⣄⣤⡟⢹⣿⣡⣯⡴⠁⣀⠄⠀⣀⡻⠄⠉⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⣂⢀⡟⡜⠘⠁⠖⠐⠃⠄⠀⡁⠁⠀⠐⢺⡟⠉⠈
⣿⣿⣿⣿⣿⣷⣿⣟⣭⣥⣾⡿⣛⢢⡬⠽⠅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠃⠀⠀⠀⠀⠛⠂⣶⣀⢀⠁⣿⠂⠐⠀⠘⠀⠀⠈⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢲⠛⠈⠀⠀⠀⠉⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⡦⢽⣔⡏⠀⠀⠀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣀⣾⣰⣶⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⢺⢻⣥⡿⡽⡽⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⣴⡆⠀⠀⠀⠀⢚⠉⠉⠀⠀⠀⢀⠀⠈⠃⢀⠤⠚⠋⢸⡅⠊⢱⠛⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⢠⡀⠈⠃⠀⠀⡈⠥⠖⡿⠀⠀⠁⠀⠀⠀⠛⣷⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠾⢛⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠗⠙⢤⠀⠀⠀⡀⠀⠀⠀⠀⠘⢉⣱
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡾⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙⠁⠛⠀⣀⠀⢤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣾⡅⠀⠀⠀⠀⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⡆⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⡨⣾⣯⣀⣣
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⢁⣹⠋⣠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣶⣄⠀⠀⠁⠀⠀⠀⢀⢘⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠳⠀⢱⡶⣂⣂⠀⠀⢺⠹⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢎⢡⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠺⠟⣒⢰⡇⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢁⣽
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡜⢁⡦⣤⠀⠘⠇⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢂⣈⣉⣁⣧⣿⢙⢾⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢥⣠⣭⢿⠛⣚⡿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡄⠀⠀⠀⠀⠀⠀⠠⢾⠈⠹⠧⣽⡟⠉⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠠⢞⡒⠂⠈⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⣤⣄
⣿⢸⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣴⣾⣿⣿⣿⣿⡄
⠉⠈⠉⠉⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣶⡀
⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⣀⡄⠹⠿⠿⣿⣿⡿⠿⠟⠉⠀⢠⡀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣟⣴⣿⣿⡇⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠘⠟⠁⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠚⣿⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣼⣿
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⣿⣷⣤⣤⣤⠀⠀⠈⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿
⠀⠀⠀⢀⣾⠏⠻⢿⣿⣿⣿⣿⣿⣿⣦⠀⠆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣰⡿⠃⠀⠀⠈⢿⡿⢿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠿⠿⠏⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⣼⠟⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢸⣿⣿⣿⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿
⡿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣼⣾⣿⣿⣿⠀⠀⠀⠀⣴⣤⡄⠀⠀⠀⠀⠀⢠⣄⣤⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠙⠟⠛⠀⠀⠀⠀⠀⠺⠛⠿⠇⠀⠀⣸⣿⣿⣿⣤⣤⣴⠦⠾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠸⣷⣾⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⡏⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣤⣤⣼⣿⣿⣿⣿⣿⣿⠾⣷⣤⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣶⣟⢿⣿⣿⣿⣿⣿⠿⠋⢀⡀
⣷⡄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢻⣿⣶⣤⣤⣄⡀⠀⠀⢀⣀⣀⣤⣴⡿⢏⠀⢺⡿⣿⣾⣿⣿⣶⠤⠘
⣿⣿⣦⡐⣿⣿⣿⣿⣿⣿⣿⠋⠻⣿⣿⣿⠀⠈⢿⣿⣿⡟⠻⣿⣿⣿⣿⣿⠟⢻⣿⡗⡦⢀⣿⣷⡟⠀⠙⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶
⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⢲⣄⣹⣿⣿⣦⣄⡈⢻⣿⠀⠀⠀⠀⠉⠀⠀⠀⠘⣿⢛⣁⣿⣟⣿⣉⠒⠘⣿⣿⣿⣿⣿⣟⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠿⣿⣿⣿⣟⢃⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⣽⣿⣿⣿⠋⠀⠀⢻⣿⣿⣿⣿⣿
⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⢿⣿⣿⣿⢾⣿⣤⣴⣶⣧⣤⣿⣶⣶⣤⣿⣼⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶
⠀⠀⣛⣛⣛⣛⠛⠉⠉⠉⠃⠀⠀⠀⠀⠘⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣿⣟⢛⡿⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠁⠀⠀⡤⠠⣦⠌⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⣧⣏⠙⣿⣿⣟⡿⠿⣿⣿⣿⣷⠀⠀⢀⣾⠂⠀⠀⠀⠀⠻⣿⣿⣿⣿
⢻⠉⠉⠁⠀⣀⣀⠀⠀⠀⠀⠀⢷⣀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣀⢹⣿⣿⣿⠀⣠⣾⡅⠀⠀⠀⠘⠂⢰⣿⣿⣿⣿
⢸⠀⠀⠀⠢⣽⡟⠂⠀⠀⠀⠀⠀⠌⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⠍⠉⢀⡶⠀⠀⠀⠀⣸⣿⣿⣿⣿
⣼⣤⣤⣤⣤⣿⣿⠆⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡿⣥⡠⣶⣴⣿⠕⣶⣤⣤⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣬⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⢻⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⠿⠟⠻
⣶⣾⣿⣶⣶⣿⣶⣿⣧⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⡗⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢸⣿⣿⣿⣿⡗⢦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣷⣻⣿⣿⣿⣿⣟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠛⠛⠛⠙⣿⣿⣛⣛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣙⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣽⣿⣿⣿⣿⣿⣟⣛⣛⠁⠀⠀⠀⠀⠈⣉⠉⣉⣉
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣵⣾⠆⢱⠈⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⢻⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠇⠀⢀⠐⠒⠒⠿⠿⠿⠿
⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠀⠈⡆⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⣛⣩⡭⠥⠶⠒⠒⠛⠛⢛⣛⣛⣋⣉⣉⣉⣉⣉⣉⣛⡛⠲⠿⢿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡴⣞⡭⠶⠚⣉⣁⣠⣤⣤⣶⣒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠛⠓⠶⣌⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢒⣛⣩⡭⠭⠶⠶⣶⢫⡞⣡⣴⣾⣿⣿⣿⣿⣛⣛⣻⠿⠽⣿⣓⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠴⢚⣩⠴⢖⣛⣭⠵⠶⠒⠛⠛⠋⣿⢸⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⢭⣻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⡽⣻
⠀⠀⠀⠀⣰⢟⡥⠞⣉⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⡞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⣧⠀⠀⠀⠀⢀⣠⠴⢚⣭⠞⢉⡴⠋
⠀⠀⠀⡼⢡⣾⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡙⢮⡳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣀⡤⠶⢛⣩⡴⠚⠉⣠⠴⠋
⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠲⢽⣷⣶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⠴⠖⣛⣩⡥⠶⠚⠉⢀⣠⠴⠛⠁
⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣈⠉⠙⠛⠓⠶⠶⠶⠶⠶⠒⠚⠋⠉⠉⣀⣠⡴⠖⠚⠉
⠀⠀⠀⣿⡈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠶⠶⠶⠶⠖⣶⢞⡿⣿⠉⠁
⠀⠀⠀⠸⣧⡘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣡⡞⣵⠃
⠀⠀⠀⠀⠹⣷⣌⡓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢛⣩⠴⢚⣡⠞⠁
⠀⠀⠀⠀⠀⠘⢿⣝⠲⢤⣉⡛⠲⠦⣤⣄⣀⣀⣀⣀⣤⣤⠤⠤⠶⠖⢚⣛⣩⡭⠴⠖⣛⣩⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢬⣍⣙⣓⣒⠒⠒⠒⠒⠒⠒⠒⠒⣚⣛⣫⣭⠥⠶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⢬⣛⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⢥⡸⠁⢀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⣀⡋⡀⠽⠉⢹⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠇⠀⠀⠀⠁⣼⠩⣼⠩⣖⡀⠐⢶⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠀⠀⠀⠀⠀⠀⢤⠀⠀⢸⡤⠀⠛⣹⣾⢽⣃⣄⣀⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⠀⠀⡀⣄⡰⣦⣴⣬⣿⣿⣿⣿⣇⡘⡻⣧⡴⠶⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡆⠀⠀⠀⠀⠀⠀⢐⠄⠀⠀⢨⠃⣀⣳⡿⢿⣿⣽⢻⣿⣿⣻⣯⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣃⢄⡄⢈⡿⣅⣿⠀⠀⢻⣿⣥⣿⣿⣿⣿⣍⣶⣋⢰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢢⡅⠂⣤⢀⡿⣽⣿⣶⡀⠈⠹⣿⣽⣷⣯⣿⣞⣷⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⢡⠘⠁⠘⠀⡄⠀⡐⠛⣟⣹⣶⣾⣷⣿⡿⠃⠀⠀⢀⠙⢾⣿⣿⣿⣿⣿⣞⠀⠠⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡄⠀⠙⡆⠀⢆⣠⠡⠘⣻⠇⠻⣾⣿⣿⡿⠟⠃⠀⢀⠔⠀⠄⡌⢿⣿⣿⣿⣷⣆⣾⠇⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠐⢰⣆⣰⣙⣻⣧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⣀⠐⠆⢸⣿⣿⣿⣿⣹⢧⡈⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣱⣿⣇⡤⣙⣷⣸⡿⠁⠈⠀⠀⠀⠀⠀⠀⠀⢀⠠⡂⠀⠠⢺⢠⡀⣼⣿⣿⣿⣿⣯⣯
⠀⠀⠀⠀⠀⠀⠀⢶⡔⢅⣹⣿⣛⣿⣧⣼⡟⠁⠀⠀⠀⣦⠐⠒⠂⠀⠉⠈⢠⠁⠀⠀⣰⣯⢄⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡳⣷⣿⣿⡿⢿⣟⣻⣿⠃⠀⢀⠀⣀⠠⠀⠀⠄⢂⢀⠀⠀⠀⣠⠀⢩⣿⣻⣟⣻⣟⠙⢻⣾⡇
⠀⠀⠀⠀⠀⠐⡄⢹⣿⣿⣿⣷⣾⡿⠟⠁⢀⣴⣿⠷⡀⢢⠐⠀⠈⠄⢐⢀⠄⠀⣘⣥⣄⡿⢳⣿⣛⢩⣷⣴⡻⡷
⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⡿⣿⠟⠃⠀⠀⣾⣿⣿⢿⣇⠀⠈⠡⢄⣀⠀⠐⠠⢀⢪⣸⡿⣐⣿⣿⣿⣿⣿⣶⣿⣧⣀
⠀⠀⠀⠀⠀⠈⠟⢸⣤⠿⠷⣾⣾⣿⡿⣿⡿⢿⣿⣟⣿⣻⣿⣄⣈⣀⣈⣀⣯⣬⣿⣿⣿⣿⣿⣿⣿⡙⢿⢢⡿⠚⠂⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠥⠌⢣⣸⡗⠞⢿⣤⣿⡿⣿⣿⣿⣿⠿⣿⣿⣿⠿⠛⠻⠉⠛⢧⣼⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠞⠉⢻⡗⠬⢫⣿⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⣭⡴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠏⠀⠒⡀⠀⠀⠀⠀⠀⠀⠈⠣⠡⠈⠡⠴⠦⠄⠀⠀⠀⠀⠀⠺⣷⣶⣤⢤⣤⣤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⣖⠀⠀⡀⠀⠖⠢⠀⡗⠀⢀⡀⠀⠠⠐⢨⡄⠛⡙⠂⠀⠀⠈⠐⠤⠦⠖⠒⠖⢿⣿⣿⣿⢷⣼⠿⢿⣶⣶⣞⠀⡄
⠀⠀⠀⠀⠀⠀⠙⡀⠙⠀⢰⣅⠢⣀⢀⣀⡝⠃⠁⠰⠉⢠⡈⠚⠇⣘⠀⠈⠐⠊⡀⠟⠀⢀⡢⠀⠀⠀⢹⡙⠿⣿⣿⣻⣾⣭⣹⣿⡿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠉⣦⣾⣿⡿⣷⣄⠘⠋⣀⠀⠀⠀⠔⣼⡿⠀⣀⣝⣷⣦⠙⣻⣃⠉⢻⣿⠟⠀⠀⢰⣼⣭⡲⣼⣿⠿⣟⣿⠁⣤⢀⠀⡔
⠀⠀⠀⠀⠀⠀⡀⠄⢈⠃⢾⠹⠙⠛⠀⠀⣿⠀⠴⠂⠀⠀⠶⠟⠁⣻⣩⡻⢄⠉⡝⢈⠾⠁⢀⠀⠀⠘⣿⣿⣿⣮⣿⡝⠛⢅⠶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠑⣈⠃⡰⠮⠄⠙⠿⠤⢄⠽⣗⡀⠀⠀⠀⠰⣶⣿⣿⡟⠉⠒⠋⠀⠋⠁⡀⠀⡀⠢⠀⣿⠃⢿⣿⢾⢿⡄⢺⡁⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⢀⣾⣿⡿⣧⣴⣶⣶⠞⠋⠀⠉⡁⣤⠈⠀⠀⠀⠀⠏⢠⡄⣰⣌⣡⣄⡈⠟⢠⡀⢈⡛⢁⣰⠀⠀⣀
⠀⠀⠀⠀⠀⣈⠑⠦⢦⣴⣦⣤⣿⠿⠛⠛⠿⢧⣌⠛⣿⡷⠶⠖⠓⢪⢧⡀⠈⠀⢀⣀⠀⡁⢰⡴⡏⠉⠀⠀⠐⣦⢼⣿⣿⡇⠈⠋⠂⢆⠁
⠀⠀⠀⠀⠠⠀⡀⠀⠘⣻⣿⢋⣤⡶⠛⠛⠓⢾⣭⠛⣿⠶⠀⠀⠐⢤⣿⣯⣭⣄⣛⣩⠔⠁⢸⡇⠤⠶⢀⣀⠞⢱⣼⠿⣏⠀⠀⠤⠠⠄⠁
⠀⠀⠂⠀⠓⠀⠀⠙⢼⣿⠁⡿⠁⠀⢠⠤⠀⠀⠹⢗⢘⢦⡐⠠⠂⣸⡷⠉⠈⣛⠋⢁⣼⡁⣽⢧⠠⠀⠁⠀⠀⠸⡆⠠⣏⡖⠛⠛⠲⠆⠀⡀
⠀⠀⠀⠀⠀⣄⣴⠁⠈⢿⣾⠃⢀⠀⠧⠀⠈⠀⡀⢰⠾⠏⠓⣤⠞⣷⠀⠈⢃⡌⢃⣴⣿⣃⡷⣆⢐⡀⢀⣀⣤⠞⢈⠥⢞⡃⠠⠭⣀
⠀⠀⠀⠀⠀⠚⠉⠁⠢⣼⣏⠠⠈⠀⠀⠀⠀⠘⠻⣪⣄⣠⣄⣠⣾⠻⣃⡬⣤⢾⡿⠿⠟⠛⢚⡍⠸⠟⠫⠊⢙⣣⣔⠚⠿⠃⠀⠞⠁⠀⢀⣄⡘
⠀⠀⠀⠀⣀⠀⠀⡀⢀⠘⠉⡙⠀⠀⠰⠘⠁⠀⡴⢛⡘⠛⢿⡝⢋⠁⠀⠀⣁⠬⠀⠀⠀⠀⡠⢸⡟⠿⠶⠀⠈⠀⠀⠀⠀⡀⣀⡀⠤⠀⢨⡟⠁
⠀⠀⢀⠠⠀⠀⠳⣯⡌⣃⡗⠍⢄⣠⣀⣸⠿⡀⢷⡆⠀⠀⢀⠹⡇⠀⠀⠀⠀⣠⡶⠃⠀⠈⠀⢀⣅⡀⠀⣦⣤⠀⠀⠀⠀⠈⢡⠀⠾⠆⢀
⠀⠀⠀⢀⠀⡈⠀⠁⠶⠄⠀⠂⠀⠀⠈⠻⠰⢮⣺⡇⠀⠰⡿⣀⠀⠤⠔⠖⠋⢉⡔⠃⠀⣤⣴⡿⠀⣀⠀⠙⣿⠿⢀⣤⠴⠲⡀⠐⣀⠀⢠⠀⠐⠄
⠀⠀⠄⠈⠀⢀⠀⠀⠂⠘⠃⣐⠘⠀⠀⠀⡢⠤⡄⠻⢾⣀⠐⠏⠁⡠⠄⠶⣴⡏⠀⠀⠀⢸⡇⠀⣠⡉⠡⢄⠈⠛⠉⠀⠀⢠⣟⠤⡈⠁⡨⠄⠤⢄⡀
⠀⠀⠀⠠⢢⣄⠀⠀⠀⠁⢠⠙⠂⢤⠤⢀⣉⣵⣏⠀⠀⣉⡀⠐⠩⣀⣼⠳⢿⡇⠀⠀⠈⢩⡖⣿⣿⣶⣤⣠⣍⠘⣒⣟⠒⠦⠣⡄⢈⣾⣶⣦⠒⠤⠃
⠀⠈⠀⠀⠁⠈⠀⡀⠀⠀⠄⣀⡀⢀⡀⠘⠉⣠⡽⠓⠀⠙⠣⣀⠀⠹⠇⠀⠘⣷⠟⠙⠀⠈⠁⠿⠈⠛⠋⠉⠉⠈⠉⠉⠙⠚⠁⡄⠈⠉⢠⣶⠄⢠
⠀⠀⠀⠀⠀⠀⠀⣷⣤⡴⠾⣿⢿⣿⠁⢠⠞⣽⣷⣄⡀⢀⣼⣏⣷⡄⠀⠆⠸⣇⠀⠀⠀⣀⡄⠀⠀⢀⣀⢤⣾⣁⠈⠁⠚⢀⡄⠑⣀⠉⠈⠹⠇⠋
⠀⠀⠀⠀⢀⣴⡾⠏⠉⣰⠛⠈⢀⢷⣶⣤⣤⡌⠉⠉⢁⠿⠿⣋⡉⣝⠋⠀⠀⠉⢉⠖⠀⠉⠋⠍⠉⠟⢅⠒⢫⡥⠆⣠⡄⠂⠀⠀⠀⠎⠙⢲⠒⠆
⠀⠀⣠⠀⠚⠋⠉⣁⠃⠁⠀⠀⣸⢸⣿⡿⠛⠓⢁⠀⠆⣀⡄⠁⠄⠈⢻⠹⣁⣠⡆⢂⡀⠐⢆⠀⠂⠀⠘⠻⠄⠄⣼⣿⣇⠀⠀⠇⣞⠻⠧
⠀⠀⠟⠀⣤⠀⠀⠀⠠⠆⡶⣈⢻⢬⣿⣤⣶⠊⢣⠤⠀⠙⡧⢠⠀⠒⠀⠐⠋⢰⣆⣠⡚⠻⡶⠄⠐⠀⢀⡈⢻⠖⠉⠉⠉⠘⣤⠀⢀⠰⠀⠀⠀⠈
⠀⠀⠀⢰⠛⢂⠀⠀⠀⠠⡀⢉⠐⠀⢣⡍⠭⢰⣆⡄⡄⠐⠲⠿⡏⣬⠉⠉⠉⠃⠀⠉⢹⠆⢧⣀⣤⣴⡌⢦⡻⠀⠀⠀⠀⠀⢸⠆⠛⢦⠀⠀⠀⠁
⠀⠀⠠⣼⠄⠼⠇⣠⠀⠌⡾⠵⡂⠀⢶⡉⢡⠤⠀⠀⠉⠳⠛⢐⠺⠇⠨⡶⡼⡷⠾⠶⡶⠡⣤⠀⠀⠈⢱⡜⢿⢆⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠜
⠀⠀⠀⠀⢸⡗⣠⣴⣗⠀⠂⠴⡇⢠⡥⠉⠀⠐⠶⠌⠀⠃⡔⡆⠐⠆⠖⠐⠀⠀⡄⠀⣿⡀⠛⢃⡀⠰⠛⠛⠢⣤⠙⣦⠀⠀⠀⠀⢀⣌
⠀⠀⠀⠀⠀⠘⠋⠈⠈⠣⢄⣀⠀⠛⠃⢀⡁⢰⠒⢆⠜⠂⠀⠠⠀⠈⠠⠬⠕⠀⠙⣄⠘⠖⠛⠚⢁⡴⠚⠋⠀⣼⡓⠈⠲⠄⠀⠐⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠒⠉⠁⠀⠀⠀⠈⠈⠁⠀⠀⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣶⣿⣿⠿⠿⠿⠿⢷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢛⣩⣽⠿⠛⠋⠁⠉⠉⠙⠻⢶⣄⠀⠀⠀⠀⠉⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⢀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢰⡟⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⢠⡿⠁⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⢀⡿⠁⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⣾⠇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⣸⡏⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⢰⡟⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⣿⠃⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡀⠀⠀⣰⡿
⢸⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡴⠛⠛⠙⠛⢶⣿⠁
⣸⡇⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣰⠏⠀⠀⠀⠀⠀⠈⣿⡄
⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⣿⠃
⣿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠘⠷⣦⣄⠀⠀⠀⣾⡏
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⣿⡇
⣿⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⣿⡇
⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣠⡞⠁⣿⠀⠀⠀⣿⡇
⢸⣇⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣿⠀⠀⠀⢻⡇
⠈⣿⡀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠹⣧⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠹⣷⣄⠀⠘⣷⡀⠀⠈⢧⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠻⣷⣄⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠻⣷⣮⣿⣦⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠋⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣦⣤⣶⣿⠛⠋⠉⡀⣀⣀⣀⣤⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⢿⡟⠛⠛⠋⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣇⠀⠀⠸⣧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡶⠾⠟⠛⠋⠉⠀⠀⣿⠀⠀⠀⣏⠉⠉⠛⠛⠳⠶⠶⢦⣤⣤⣄⣀⣀⡀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣦⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⢶⣿⠇⠀⠀⠀⠀⣿⠶⣶⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⢻⣿⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣁⣀⣀⣀⣤⡾⠁⠀⠀⠀⠀⠈⠉⢻
⠀⠀⠀⠀⠀⠘⣿⡀⠈⠉⠙⠛⠻⠶⢶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿
⠀⠀⠀⠀⠀⠀⠹⠷⣶⣤⣄⣀⡀⠀⠀⠀⠉⠉⠛⠛⠿⠶⢶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠷⢶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⠶⠶⠶⠾⠛⠉⠀⠀⠀⠀⣀⣴⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⢷⣄⣴⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠴⣄⠈⢿⠁⣤⠀⢹⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠟⠛⠛⠛⠷⠶⠶⠟⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⢀⡴⠒⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣄⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣰⡏⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠘⢿⡄
⠀⠀⠀⠀⢠⡟⠀⠀⢠⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⣷⡀
⠀⠀⠀⢀⡿⠁⠀⢀⡏⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⣧
⠀⠀⠀⣾⠃⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢹⣇
⠀⠀⢸⡏⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⡿⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸
⠀⢸⠇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘
⠀⡿⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⢰⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣸⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸
⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣾
⢿⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢠⣿
⢸⡇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣸⡏
⠀⣿⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣿⠁
⠀⠸⣧⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣸⡏
⠀⠀⠹⣧⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣰⡟
⠀⠀⠀⠹⣧⡀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣴⡟
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠰⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡏⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣤⣤⣤⣤⣤⣄⣀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠋⠠⠤⠟⠀⣿⠀⠸⠏⢻⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣀⣤⣴⠿⣦⣤⣤⣼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⢻⡏⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡋⠀⠀⢸⡇⠀⢀⣈⣿⣶⠖⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⠳⣿⣿⡞⠉⣼⢿⣿⢦⣀⡜⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⣠⠏⠉⠳⣄⢿⣾⡟⠺⡿⠒⠤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠙⢶⣧⣤⣤⣴⣾⣿⣿⣤⣤⣧⣄⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⣄⣸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣦⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣻⣿⡿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠿⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠈⠻⠟⠃⠀⣀⣘⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⢻⣟⣛⣿⣛⣻⣿⣛⣛⣛⣛⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠿⠶⠶⠶⠶⠶⣿⣿⢛⣿⠛⠛⣿⠉⣿⣿⣏⠉⠉⠉⠉⢻⣿⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣾⠃⠀⠀⣿⠀⣿⣿⢿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠉⠙⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣿⣿⣀⠀⠀⠀⠀⢀⣿⣿⣏⣀⣀⣤⣿⣤⣿⠿⠾⣷⣤⣤⣤⣼⣿⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠋⢹⡟⠉⠉⠁⠀⣿⠀⠀⠀⠀⠸⣧⠀⠀⠈⠉⠙⠛⠶⢦⣄⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⠾⠿⠿⠿⠿⠿⠿⠷⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠟⠉⠀⠀⣠⣴⡶⠿⠛⠉⠉⠉⠉⠉⠉⠙⠻⠶⣦⣄⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣦⡀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⢀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡼⠻⣦⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⡌⠁⠀⣰⡟⠁⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠀⠀⠈⢿⡄⠀⠀⢃
⠀⠀⠀⠀⠀⢠⠀⠀⢠⡿⠀⠀⠰⠞⣩⡶⠀⡀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢀⡴⠋⠀⠀⠀⠀⣤⠞⢿⡄⠀⠘⢁
⠀⠀⠀⠀⠀⠘⠀⠀⣿⠁⠀⠀⠀⠘⠋⣠⠟⠁⠀⠀⠀⠀⣀⡾⠃⢀⡴⠋⠀⠀⠀⢀⣴⠟⠁⠀⠸⣧⠀⠀⢸
⠀⠀⠀⠀⢸⡆⠀⢰⡿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡼⠋⢀⡴⠋⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣿⡀⠀⠀⡆
⠀⠀⠀⠀⣸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⢀⣿⠃⠀⢸⡇⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠏⠁⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿
⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⢀⡴⠋⠀⣰⠟⠁⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿
⠀⠀⠀⢸⣿⠀⠀⢸⡟⠀⢀⡴⠋⠀⣠⠞⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⡄⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⢸⣧⡴⠋⠀⣠⠞⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⢸⡟⠁⣠⠞⠁⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⣿⡇⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⢸⣷⡞⠃⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠸⣿⠀⠀⢸⡏⠀⠀⣀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⢸⡗⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⢸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⠀⠀⠀⠀⣸⡇⠀⢸⡇⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⣠⣿⣀
⠀⠀⠀⣠⣿⡃⠀⢸⣷⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣼⠏⠈⢹⣧
⠀⢀⣾⠋⣿⡇⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡏⢠⠲⡆⣿
⠀⢸⡇⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠸⣤⠇⣿
⠀⢸⣇⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⡀⠀⢠⡿
⠀⠀⠻⣶⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠷⣶⠿⣧
⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⠀⢿⡆
⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠱⢦⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⠀⠈⣿
⠀⠀⠀⢰⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠳⣤⠙⠦⠀⠀⠀⠀⣿⡇⠀⠀⣿⡄⠀⢹⡇
⠀⠀⠀⣼⣿⠀⠀⢸⣧⠀⠀⠀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠈⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣧⠀⠈⢿⡄
⠀⠀⢀⣿⢿⡇⠀⠸⣿⠀⠀⠾⣡⠞⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢰⣿⢻⡇⠀⢸⣇
⠀⠀⣸⡏⢸⡇⠀⠀⣿⡆⠀⠐⠋⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣸⡇⠈⣿⠀⠀⣿⡀
⠀⠀⣿⠃⢈⣿⡀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⢀⣿⠁⠀⢻⣇⠀⢸⣧
⠀⢠⣿⠀⢸⡿⣷⡀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⢀⣼⠇⠀⠀⠈⣿⠀⠀⣿⡀
⠀⢸⡏⠀⢸⡇⠘⣷⡀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢀⣾⣿⠀⠀⠀⠀⢻⡆⠀⢸⣇
⠀⣼⡇⠀⣿⠇⠀⠈⣿⣆⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⣠⡿⢉⣿⠀⠀⠀⠀⢸⣧⠀⠘⣿
⠀⣿⠁⢠⣿⠀⠀⠀⢻⣟⠳⣦⡀⠀⠙⠻⠶⣦⣄⣀⣀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠁⠀⢀⣤⠾⢿⡇⢸⣿⠀⠀⠀⠀⠀⢿⡆⠀⢻
⢰⡿⠀⢸⡏⠀⠀⠀⠈⢿⡄⠈⢿⣷⣤⣀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⢀⣀⣴⡾⠛⠁⠀⢸⡇⢸⣟⠀⠀⠀⠀⠀⢸⣇⠀⢸
⢸⡇⠀⢸⡇⠀⠀⠀⠀⠘⣧⠀⠘⢷⡌⠉⠻⠷⢶⣤⣤⣤⣤⣤⣤⣤⣴⠶⠿⠛⠉⠁⠀⠀⠀⠀⢸⡇⢸⡏⠀⠀⠀⠀⠀⠈⣿⠀⠘
⢸⡇⠀⣿⠁⠀⠀⠀⠀⠀⠹⣧⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⡇⠀⠀⠀⠀⠀⠀⣿⡄
⣸⡇⢠⡿⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣸⡇⠀⠀⠀⠀⣀⣀⣿⠀⢀
⣿⠃⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⠙⠛⠻⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠀⣿⡟⠛⠛⠛⠛⠛⠉⠁⢀⣼
⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⢻⣧⣤⣤⣤⣤⣤⣤⡶⠟⠁
⣿⡀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⣈⣿⠁⠀⣹⡏⠈⠉⠉
⠹⣇⠀⠙⠛⠿⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠾⠿⠿⠿⠿⠛⠛⠋⠁⠀⣠⡿
⠀⠙⠳⢶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣾⣱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⢣⡿⣿⣿⣿⣽⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⢀⣿⣿⣇⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣸⣿⠘⣿⣿⣿⣿⣿⣿⠏⣿⡆⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⣋⡙⢿⡀⠘⠻⣿⣿⠟⠋⢀⣿⠛⠛⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣼⣿⡾⣯⣥⣾⣿⣧⣽⠎⠱⣽⣽⣿⣷⣼⡹⣾⣿⣅⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡿⠿⣿⣶⣿⡌⠙⠋⠉⠀⡀⠉⠛⢉⣥⣿⣷⣿⢿⣿⣿⣧⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⠋⣼⠧⣄⣠⣾⣿⣿⣶⠾⠿⠿⠿⠿⢷⣿⣿⣿⣭⡁⣀⣿⡏⢿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣌⣹⡿⠟⢡⣶⡿⢿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣮⠻⢾⣿⠿⠁
⠀⠀⣠⣤⣤⣤⠀⣿⣿⣿⠿⢶⣤⣿⢏⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣠⣽⣷⣾⣿⠇⠠⣄⣤⣤⡀
⠀⠀⣟⡻⣿⣧⣲⣿⠋⠀⣴⣾⡿⣣⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣯⣤⡌⠙⣿⣷⣞⣿⡿⢛⡇
⠀⠀⣨⣹⣿⣿⣿⢛⠛⢦⣽⣿⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣳⡶⠟⢿⣿⣿⣿⡉
⠀⣾⡟⣩⣹⣿⣿⣬⡱⡄⣿⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⣠⢚⣽⣿⣿⣫⡙⣦⡀
⠀⢸⣷⣿⣿⣿⡏⠀⣀⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠉⠻⣿⣿⣿⣷⣿
⢠⡾⢿⣿⣧⣾⣿⡾⠛⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⢦⣼⣿⣿⣿⣿⠿⣦
⠘⠶⣿⣿⡷⢿⣿⠀⣀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⡀⠈⣿⣿⠷⣿⣿⣶⠞
⠀⣿⣿⣿⣷⣾⣿⣀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣏⢀⣿⣿⣶⣿⣿⣿⡂
⠀⢻⣿⣿⢻⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⢽⣿⢿⣿⣿⠃
⠀⠀⠈⣹⣿⣿⣠⣾⣿⡌⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⢋⣿⣯⡅⣸⣿⣿⡋⠁
⠀⢀⡞⣻⣿⣿⠻⣿⣿⣿⣾⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣧⣿⣿⣿⠷⢿⣿⣿⠻⣆
⠀⠈⠻⠷⠟⠋⠀⢙⠿⣍⣹⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣛⠹⡟⢿⣀⠘⠛⠿⠿⠏
⠀⠀⠀⠀⠀⠀⡴⡾⣠⣿⣿⣿⣿⡿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠿⢻⣮⠋⡧
⠀⠀⠀⠀⠀⠀⠷⡼⠟⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣿⣿⣿⣿⣍⡛⠟⠛⠛⠻⠟⠛⠛⠉⠛⢛⣩⣿⣿⣿⣿⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠻⣿⠟⢉⣿⣶⡺⣷⢤⣤⠶⡾⢓⣶⣟⠙⢿⣿⠛⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣎⣿⡿⢻⣿⠁⠀⣶⣿⡖⠀⢹⣿⢿⣿⣚⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢰⣿⣷⣶⣿⣿⣷⣶⣾⣧⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢰⡿⣿⣷⠋⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢴⣿⠷⠀⣤⣀⠀⠀⠀⠀⠀⢠⣶⣶⠶⢤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⣶⣶⣶⣻⣶⣆⢀⠀⠀⢻
⠀⢠⣾⣷⣀⡀⣴⣴⣤⢷⠄⠀⠀⠀⠀⢸⣿⣿⠰⡧⣯⢠⣿⡏⠘⡏⢰⣾⠀⠘⢱⡟⠛⣿⢰⣿⣿⣿⡟⠿⠿⠟⠗⠈⣸
⠤⡶⢿⡟⠉⣶⣤⣿⠟⠀⠀⠀⠀⠀⠀⢸⠿⣿⢤⠥⠦⠤⢿⣭⠉⢷⣜⣻⢀⣠⠀⣷⣧⡿⢸⣿⣿⠉⠀⠀⠀⠀⠀⠀⢸
⣾⣿⣿⣷⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣾⣄⣥⣼⡁⡖⢰⡇⡿⠀⡆⠙⠁⡆⠙⠀⢰⠙⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘
⢉⠿⣿⣇⡖⠃⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣴⣤⣧⣤⣄⣻⣄⣃⣴⣀⣈⣐⡇⢸⣿⣿
⣾⡿⣿⢩⡧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡀
⠘⣧⡷⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣻⢿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⢿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⡿⣻⣶⣿⠃⢸⣿⣿⣷
⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣿⣿⣿⣿⠾⠟⠀⠿⣯⣿⣿⣿⣿⣿⣿⣏⠀⢸⣿⣿⣿
⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⣿⣿⣿⠀⣿⣶⣆⠀⣿⣿⣿⣿⢓⣺⣾⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⢸⣿⡿⣻⣿⣿⣿⣤⣿⣿⣿⣰⣿⣿⣿⣧⣻⣿⠟⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣸⣿⣟⣉⢋⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⢀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⡄
⡄⢸⣿⣿⡄⠀⠀⡀⠀⠀⠀⢻⣿⣿⣿⣿⣟⡽⠝⢽⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⡀⢠⠀⠀⢸⡇
⡇⣼⣿⣿⣧⢠⣴⠃⢀⣀⣀⣾⣿⣿⣿⣿⣷⣶⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇
⡇⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠻⠿⠿⠿⣿⣿⣿⣿⣿⢛⣛⢻⡄⠀⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇
⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠁⠉⠉⠀⠐⠚⠇⠀⣿⣿⣿⣿⣇⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⣄⡀

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⡶⢦⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣻⣿⣷⣿⣿⣿⡿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⢿⠏⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣭⣶⡿⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢶⣤⡀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣏⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠙⣿⡿⠉⠂⠀⠀⠀⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠝⠋⠙⠃
⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⣠⣾⣿⣶⣤⣀
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣦⣴⡴⣿⢲⣢⣀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢹⣷⣿⣽⣆
⠀⠘⠊⢙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣟⡏⠿⠢
⣀⣀⣀⣀⣀⣄⣀⣹⣿⣟⣿⣿⣯⣻⣁⣀⣙⣅⣠⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠘⣿⣦
⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⡶⢶⣴⣾⠛⣿⠃⣶⣿⣶⢂⣀⡀⢰⡟⣽
⠀⠀⠀⢸⢏⣿⠀⢀⣰⡏⠰⣾⣣⣴⣿⣽⣷⣬⣽⣷⣼⣏⣿⣿⣵⣃⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠘⢧⣁⠰⣿⠉⣿⣾⠟⠋⠉⠀⠀⠀⠀⠀⠉⠙⠻⢯⣟⢩⡿⣧⢀⣀⡀⠿⠿⡷
⠀⠀⠀⠀⣤⠾⢿⣀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠿⣿⠻⣿⣉⣻⣦⡾⠃
⠀⠀⠀⠈⠿⠖⣾⢯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠙⢷⣾⣿⠟⠛⣧
⠀⠀⠀⢀⣠⣼⣣⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠹⣿⡄⠻⢟⣀⡀
⠀⠀⠀⣾⢏⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⢿⠍⣻⡄
⣠⣶⣦⢿⣎⠛⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⣾⠿⣤
⢿⣄⣛⣀⣻⣷⣾⣧⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣴⣸⣧⡀
⠀⠉⠉⠉⠻⠏⠹⣿⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⡄⠘⣯⣹⠿⣼⠟
⠀⠀⠀⠀⣠⣶⣦⢸⣷⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⣠⣤⣤⡀
⠀⠀⠀⠀⢿⡌⠁⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⡻⠇⢈⣇
⠀⠀⠀⠀⢨⡟⢛⣿⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⣧⣶⠿⢻⡇
⠀⠀⠀⠀⠘⢷⣝⣉⣀⣘⣧⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⣥⣶⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠉⠛⣷⡀⠀⠀⠀⠀⠀⣸⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣿⠟⢻⣧
⠀⠀⠀⠀⠀⠀⢰⠟⢉⣿⡆⠈⢿⣆⠀⠀⠀⠀⣿⣿⠇⣾⠃⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢸⣿⠈⣻⣧⣄
⠀⠀⠀⠀⠀⠀⠘⣦⣤⣭⣤⣤⣴⣻⣷⡀⠀⠀⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⡏⢼⣯⡍⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⡽⣿⣿⣿⠻⣶⡄⢻⡟⠀⡿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣰⣿⣥⣬⣙⣡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠋⠀⠉⠛⠉⠀⠈⢻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⣿⣯⣭⡛⢯⣉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠻⣿⠀⢙⣿⣿⣶⣤⣀⡀⠀⠀⢀⣀⣴⣾⣿⣽⣏⣼⣿⠀⠉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣾⡿⣩⣤⡈⠻⣿⠻⠿⡿⠿⣿⣿⣠⡾⣿⣿⡻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⣄⣤⣼⣿⡇⣾⣟⣀⣼⡿⣿⣧⡈⣯⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠈⠉⠁⣾⣿⠃⠈⠛⠛⠋⠀⠸⣯⠉⠛⢓⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⢻⡆⣰⠋⠉⠻⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠶⠇⠀⡷⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⣠⢏⣠⠴⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣜⡵⡶⢦⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠛⠛⣴⣿⣶⣄⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠹⣿⣿⣿⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣌⣙⣛⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣍⣉⢁⣰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠖⠒⠊⠉⠉⠁
⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠒⠚⠋⠉⠉
⠀⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⢀⣀⣀⠤⠤⠴⠒⠒⠋⠉⠉
⠀⠀⠀⣼⣿⣯⠤⠒⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠶⠖⠒⠛⠉⠁
⠀⠀⢰⣿⣿⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠤⠐⠒⠒⠈⠉⠉⠁
⠀⠀⣿⣿⠇⠀⠀⢈⣢⣀⣀⠤⠤⠔⠒⠒⠒⠋⠉⠉
⠀⢀⣿⣿⠛⠫⢍⡀⠀⠙⢄
⠀⢸⣿⣿⠀⠀⠀⠈⠁⠂⠈⠢⡀⠀⢀⡀
⠀⢸⣿⣿⠉⠉⠉⠉⠉⠉⡉⢀⠟⠹⠇⠉⠉⠉⠉⠉⢱⡯⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠘⣿⡿⠀⣀⠠⠐⠚⠁⡠⠊
⠀⠀⣿⣷⠈⠉⠀⠒⣰⠞⠤⢄⣀⣀
⠀⠀⢹⡿⡇⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⠦⠤⠄⣀⣀
⠀⠀⠈⣿⢿⡴⠡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠂⠤⠤⠄⣀⣀
⠀⠀⠀⠹⣾⣷⠀⠀⠀⠉⠑⠒⠢⠤⢤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠲⠦
⠀⠀⠀⠀⢻⣮⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠻⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠹⣿⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠦⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿