Sincere Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⢤⡀⠀⠀⠀⠀⠀⣴⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠒⠋⠁⣀⣤⣤⢬⠷⣤⣦⣼⣾⠛⢩⠴⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠖⠋⠉⠀⠀⠐⠒⠲⠾⠟⠁⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡾⠛⠈⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣻⠟⠱⠁⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡔⣰⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⣇⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡴⠋⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠘⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠎⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⣂⣿⠀⠀⠀⠀⢋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠃⢸⡴⠒⠲⡆⠀⠀⠀⠀⣶⣶⠶⠒⠒⠒⠀⠸⠀⢷⠀⠉⢿⡀⣶⠀⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⣧⠀⢀⣻⡄⠀⠀⢠⣿⡟⡂⢤⣤⣄⡀⢠⡄⠈⢀⠨⣼⡇⠘⣆⢼⡄⢻⣧⡄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⠛⠀⠈⠁⠙⠀⠀⠉⠙⠋⠀⠈⠁⠈⠈⠉⠀⠀⠀⠀⣿⠃⠈⠹⣆⣧⠘⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⣟⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠹⠀⢿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠘⣿⡌⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠀⠀⠙⠛⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠙⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⣿⠀⠀⠁⢰⠖⠦⠤⠤⠤⠄⠀⡄⠀⠀⢀⣠⣞⠀⠀⣠⢰⠇⢹⡀⠀⠀⠀⠀⠀⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠁⠀⠀⠀⠀⠐⠒⠁⠀⠀⠈⡿⠁⠀⡠⢃⣿⠂⢸⣧⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢾⣿⣦⠀⠀⠐⠒⠒⠦⠒⠀⠀⠀⠀⠀⠀⢀⡀⠊⣤⣾⡟⠀⣸⡿⣇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣶⣿⣿⡿⠁⢰⠋⠀⣽⡆⠀⠀⠀⠥⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢺⣡⡿⣿⣧⣀⠀⠀⠀⠐⣂⣀⣤⠴⠊⠑⢈⣴⣾⠟⠁⠀⠁⠀⢠⣿⣿⡀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠥⣸⡑⢿⣿⣿⣿⣿⡇⠘⣿⡟⠁⠀⠀⢠⠉⠙⠁⠔⠀⠀⠀⢠⣾⡿⠁⡇⠀⠀⠀⠀⣿⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⡀⢻⣿⣿⣿⡇⠀⠛⠀⠀⠀⠀⣼⠀⢀⠔⠀⠀⢀⣰⡿⠛⠀⠀⡇⠀⠀⠀⢀⣿⡏⣦⣟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠀⠀⠀⢠⠛⠿⠛⢉⠃⠀⠀⠀⠀⠀⠸⡏⠀⡎⠀⠀⣠⡾⠉⠀⠀⠀⠀⠇⠀⠀⠀⠈⠉⠸⢋⠟⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⢀⠀⠀⠀⡰⠃⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠀⣰⠟⡁⠀⠀⠀⠀⠄⠀⠀⠀⠀⢼⠀⠀⠀⠈⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡆⠀⠀⢠⠁⠀⠠⠊⠀⠀⠀⡠⠋⠈⠀⠀⠀⠀⠔⠊⠁⠀⣸⠀⠀⠃⠀⠀⢀⡴⠁⠘⠀⠀⠀⠀⠀⡏⣠⠀⡀⠀⡀⣠⡤⠀⠤
⠀⠀⠀⠀⠀⠀⠁⠀⠘⠇⠁⠀⢠⡟⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⠀⠀⠀⢀⡴⠋⠠⠀⠀⠀⠀⢠⠂⢸⣷⢃⠜⢁⠔⣟⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠇⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠁⠀⢀⣔⠉⠀⣤⡟⢠⣿⠇⣠⠟⢠⠾⡥⠁⠀⡼⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠀⠀⠐⠃⣰⠟⣠⡟⠿⡿⠂⠁⠀⠈⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⡀⣠⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠁⣄⣰⣧⣎⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠙⣿⣫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡠⠄⠀⢒⣂⣀⣀⡀⠠⣬⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢠⣶⠶⠀⠉⠁⠀⠉⠀⠀⠼⠀⠀⠀⠀⠀⠀⣆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠘⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠀⡸⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡂⡟⠱⡷⡄⠀⠀⣤⡤⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⣧⠇⠀⠀⠁⠀⠐⠓⠀⢶⣶⣄⡀⠀⢀⣥⠀⢸⡆⡀⠀⠀⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⣩⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⣷⡈⠷⠀⠀⢰⣶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⡇⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠉⠛⡷⠀⣤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡀⠀⠈⠐⠒⠂⠀⠀⠀⠀⠀⠀⣠⡿⣿⠀⠀⠃⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⣵⠾⡃⠀⠀⢀⠀⠀⣃⣄⠠⠀⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠃⢠⣄⡀⢀⣀⠤⠖⠛⠀⡿⠕⠊⣁⠄⢀⡾⠁⡴⢣⣿⠁⣈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠠⢴⣒⢊⣁⣈⠁⠀⠀⠀⠊⠈⣰⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠾⠀⠀⠀⠀⠀⠀⢨⡿⠋⠁⠀⠀⠀⠀⠀⠙⠳⡄⠁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠁⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣴⠆⠰⠟⣶⠒⡠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠠⢢⣠⣸⣧⣾⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⢹⣿⣟⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡠⠤⠤⠌⣉⠐⠀⣀⢺⡿⣻⡗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠊⠁⠁⣴⡶⠶⡶⠶⠲⠤⠄⠸⠿⢡⡿⠋⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⢀⠀⠀⢷⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡊⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠈⢧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⠁⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣵⣾⠀⢸⣏⣑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣇⣸⠃⠸⠿⢼⠀⠀⠀⠰⠶⠒⠒⠂⠂⠀⠀⠀⠀⠀⡆⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢸⣿⢿⣿⣷⠀⠀⠀⠘⠀⠀⠐⠚⠓⠀⣶⡶⣶⡄⢀⠀⠀⠀⣵⡀⠀⢹⠐⡀⠀⠀⠀⠀⠀⢠⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠑⡛⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠀⠀⡾⣾⣷⡄⠸⣷⣀⠀⠀⢢⣤⣦⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣇⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣳⣤⡈⠀⠀⠀⠘⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣿⣿⠀⠀⠀⣈⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⡀⠸⠸⡿⠀⢰⡇⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣆⠀⠀⠀⠉⠉⠓⠒⠶⠂⠄⠀⠀⠀⠀⠀⢀⣾⡿⣿⠃⠀⠀⠃⠀⠀⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣿⣿⣦⠀⠐⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣥⣶⠏⠈⠀⠀⠀⡀⠀⡸⡀⡀⠀⡀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠴⠿⠟⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⣴⡿⢋⡥⠊⠀⠀⠀⣰⡇⠀⢠⡇⣿⡶⠈⠢⢺⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⢨⣶⣶⡖⠒⠖⠒⠛⠉⠀⠀⠡⠖⠊⠁⢀⡤⠊⣀⢴⠟⠀⠰⠟⢰⡯⡧⠐⡦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠅⠀⠂⠀⠀⠀⠀⠀⠶⣦⣶⠛⢃⣀⣈⣁⢀⠁⠄⠀⠐⠋⠸⢀⡼⢃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡿⠛⠉⠀⠀⠀⠠⠀⠀⠈⠙⢾⣍⡁⠁⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠥⠉⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢴⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⢈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⣀⣠⠴⠊⠁⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠒⠈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠄⣣⢤⡀⠈⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⠀⠀⠀⠀⠐⠆⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⡆⠅⡱⡄⠙⠶⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠁⠈⢢⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⣿⣿⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⠛⡟⢟⣻⣿⣻⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠠⠀⠉⠀⠀⢹⡀⠐⡆⢼⠋⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⣠⣸⣧⣶⣶⣶⣷⣶⣶⣷⣾⣗⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣺⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡤⠁⠀⠈⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣛⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠥⠀⢀⠴⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣿⣿⣯⣹⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⠀⠀⠀⠀⠀⢠⣀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣿⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣤⣆⣀⣀⣀⣸⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⡄⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢿⣀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢸⣿⡇⠀⠀⢀⣀⡀⠀⠀⢠⣶⠀⠀⠀⠀⢀⠔⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⢘⣿⡇⠀⣸⣿⡇⠀⣰⣿⢁⣷⣿⠁⣿⡇⢰⣶⠀⡰⠁⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⠃⠸⣧⢻⣿⠚⠘⣿⡇⠀⣿⡇⢈⣿⡏⢠⣿⡇⢰⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⢠⣿⢹⣿⠀⠀⢿⡇⣸⢻⣇⡜⢻⣇⡎⢻⣧⠇⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡉⢉⣼⠏⠀⠀⠈⠳⠃⠀⠉⠁⠀⠉⠀⠀⠀⡼⠸⣿⡇⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⢠⠃⠀⣿⠁⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠈⠐⠠⢠⣔⡉⠀⣰⠞⢳⠀⠘⣄⣰⠏⠀⢰⡟⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠈⢹⡟⠒⢼⠤⢠⣀⣀⣠⣤⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢉⠆⡀⠀⢀⠎⠀⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠜⣸⡇⠀⢎⣤⢄⠀⠤⠀⠹⣧⡸⠀⠀⠀⢀⣶⠟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⣿⡇⣼⣾⣃⠊⣰⡟⠀⠀⢸⢓⣾⠇⠀⣾⡿⠀⣿⡇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠹⣿⣠⢿⣷⢻⣿⠁⢀⣿⡇⠀⢠⠃⢸⣿⠀⢠⣿⡇⢀⣿⡇⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠋⠀⠛⠀⠛⠶⠃⠘⠧⠔⠃⠀⠸⣿⢀⠎⠈⠓⠊⠘⢷⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⢠⢀⢻⣷⡀⠀⣀⢀⡀
⠀⢰⠈⠲⠟⠛⣿⣅⠘⣿⠢⡀
⢀⠋⡷⠶⣦⡀⢸⣿⣧⡀⠀⠿⠦⣄
⠈⣄⡃⢶⣼⢿⣏⠉⠀⠙⠶⣄⡞⠛⠷⢤⣀
⢀⡜⢣⡀⠀⢹⡟⣧⡀⠀⠀⠈⠛⣦⣀⠀⠿⣷⣤⡀
⠀⡃⢾⡷⢦⣌⠻⢿⣷⣤⠀⠀⠀⢠⡿⢷⡄⠡⣿⢿⣆⠀⠀⣠⠊⢉⡼⣷
⠀⠁⢸⡇⠸⠿⣆⠀⠀⠻⣷⣄⠀⠈⢡⡶⢿⣦⡀⠾⣿⣆⠀⡏⠀⢹⣧⡏
⠀⣤⠛⢳⡀⢶⡿⣆⠀⠀⠉⣿⣷⣄⡠⢶⣷⣽⣿⡀⢿⣿⣦⣿⢀⡨⢿⣇
⠀⡏⢀⣼⣧⠀⠀⣽⣶⡀⠀⠰⢟⡿⣷⠀⠻⣿⣿⣷⡄⠙⠻⣿⣄⠿⠿⣿⡄
⢠⠀⣸⣿⣿⣧⠀⠙⣿⣿⡀⠀⠐⣼⣽⣧⠀⠀⠛⢩⣿⣦⡀⠈⣿⣷⣄⠻⣿⣦
⠄⠀⣿⣿⣿⣿⣧⡘⠿⢿⣿⡄⠀⠈⠿⣿⣷⡀⠀⠀⢻⡿⣿⡄⠈⢿⣿⣷⣄⣻⣷⡀
⡄⠀⣻⣿⡿⣿⣿⣧⠘⠛⠀⢹⣥⣄⠘⣿⡼⣿⣶⣿⣈⣷⠹⣿⣷⣼⣿⣿⣿⣿⣿⣿⣦⡀
⡇⠀⢹⣿⡇⢻⣿⣿⣇⠀⢠⣾⣿⡿⢧⠘⢿⣿⣿⣿⡿⣿⣷⣜⣿⣿⣿⣿⣮⢿⡏⠛⣿⣿⣆
⣧⠀⢹⣿⣷⡈⣿⣿⣿⣇⠀⠹⣿⣿⣾⢀⠘⣿⡟⢿⠓⣿⣿⣿⣿⣿⡿⣿⣿⣧⢻⡀⠘⣿⣿⡆
⠘⣧⣌⡙⣿⣷⡌⢿⣿⣿⠀⠀⠘⣿⣿⠈⣇⡉⣀⢾⠇⠀⠙⢿⣿⠻⣷⠀⢹⣷⣾⣧⡀⠈⣿⣿
⠀⠈⢿⣇⣸⣿⣿⣾⣿⣿⡆⠀⠀⠈⣿⣗⣿⠃⠻⣷⣆⣴⣆⠰⣿⡆⣿⣧⠠⡝⣿⣿⣿⣾⣿⣿⡆
⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢿⡿⢻⠂⠀⣿⣿⣯⡁⠀⢹⣇⠸⣿⡇⢻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢿⡟⠻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⣾⣿⣦⢻⣿⡌⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠘⣅⠀⢬⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣿⡿⢫⣾⣿⢻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠸⣧⣤⣹⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢣⡀⠈⠻⢿⣿⣇⡉⣼⣿⡏⢻⣧⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠿⠘⠦⣬⡙⢿⣿⣿⣿⣷⢀⣹⣆⣸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⡀⠀⠀⠹⢿⡇⠀⠉⠻⢻⣷⣿⣿⣿⣇⣿⣿⣿⣿⣷⣤⡴⠚⢉⣉⣉⣻⣷⣦⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠉⠀⠀⠀⠀⠉⠀⠻⢧⣼⣿⣿⠁⠙⠿⣿⣿⣿⡿⠋⠁⠐⣿⣿⣿⣿⣿⣿⣿⠿⠋⣡⣼⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⡀⠹⣿⣿⣿⣿⣿⣦⡔⢶⣼⣶⣶⣶⣶⣿⣿⣿⠿⠆⠀⠀⣼⡿⠋⠀⠀⠀⠐⣿⣿⣿⣿⣿⠟⢁⣴⣿⠟⠋⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡛⣾⣿⣿⣿⣧⣾⣿⣿⣿⡿⣿⣿⣷⣬⣛⠻⢿⣿⣿⣿⣷⠀⠀⠀⢠⡟⠁⠐⣦⣄⠀⠀⣿⣿⣿⡿⠁⣰⣿⡟⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⣿⣿⡟⠿⠿⠿⣿⠿⠟⠋⣩⣿⣿⣿⣿⣦⣍⠙⢿⣿⡳⠀⣠⠏⠀⣠⣾⣿⣿⣷⡄⠸⣿⡿⠁⣴⣿⡟⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡾⢸⣿⣿⣿⠃⠀⠀⠂⢀⡀⣴⣿⣿⣿⣿⡿⢋⣼⣿⣷⣤⡙⢷⡄⠋⠀⠀⠉⠁⠀⠀⠙⢿⡄⠻⢁⣾⣿⣿⠲⠀⠀⣴⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⡏⢻⡄⠀⠀⢀⣀⣴⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⡿⠁⡀⠀⠀⣠⣤⣀⠀⠀⠈⣿⠆⣾⣿⣿⣿⠃⢀⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⣿⡜⢷⣄⡀⢀⣤⣴⣿⠟⠋⢠⣾⣿⣿⣿⣿⣿⣿⣿⠁⠀⣧⢀⣿⣿⣿⡿⠁⠀⢀⡟⣼⣿⣿⣿⡏⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠞⠛⠛⠛⠉⠉⠀⠀⣰⡿⢫⣿⣿⣿⡿⣿⣿⣿⠆⠀⢻⣶⣿⣿⣿⣧⣀⣀⡿⢰⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣶⣶⣤⣶⣦⡒⠃⣀⠸⠋⢡⣿⣿⣿⣫⣾⣿⣿⣿⣧⣶⣂⣿⡿⠿⠟⠛⠉⠀⢧⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣇⣭⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢉⣤⣤⣄⣀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠟⠛⠛⢿⣿⣿⣿⣿⣿⡿⠟⠋⠉⣯⣴⣿⣿⣿⠿⠟⠉⠀⣠⣿⣿⣿⡿⠿⣷⣄⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⢰⣟⠋⠉⠉⠀⣀⣠⣴⣿⣿⣿⠿⠉⠀⠀⠈⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠋⠉⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣷⣠⠀⠀⠀⠀⠀⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⣿⣿⣭⡉⠙⠓⠦⢄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠆⠀⠀⠀⡀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⢀⢰⡷⠀⢹⠀⠀⠀⠤⠤⣤⣤⡄⠀⠀⠀⢠⣤⣤⠃⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣷⡇⡀⢀⠀⢺⡀⠀⠃⢀⠀⢀⣀⣀⠀⢽⡇⠀⠀⠀⢄⡀⣀⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠰⠂⠏⠀⢸⠃⠀⠀⠀⠁⠈⠈⠋⠀⠙⠀⠀⠀⠂⠉⠁⠈⠇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠀⠀⣸⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠀⣸⡇⢸⣇⠀⠀⠀⠆⠀⠀⠘⠀⠀⠄⠀⠄⠀⠀⠀⢠⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠆⢠⠻⠄⠘⣏⢦⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⣾⣧⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠼⠀⠀⢠⣿⠀⠀⠀⠘⣅⡈⠑⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⡜⣿⣇⠀⠂⠀⢀⠋⠀⠀⠀⠈⠓⢦⡤⠀⣤⣤⣴⣿⣯⠳⢿⠉
⠀⠀⠀⠀⠀⠀⠆⠸⣿⡀⠀⠀⠀⡇⠘⢿⣦⡀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠃⢰⡿⣿⣿⡿⡄⠳⣬⡀
⠀⠀⠀⠀⠀⠄⢀⠀⠛⠃⠀⠀⠀⠃⠀⠀⠈⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠹⠿⢧⠑⡀⠀⠲⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠦⣛⣷⠀⢠⡆⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠱⡀⠀⠀⠈⠂
⠀⠀⠒⠩⣦⣠⠀⢄⠠⡈⣦⡄⠀⢀⠐⢌⠢⡀⠀⠘⠁⠀⡇⠀⠀⠐⠀⠀⠀⠀⠀⠣⡀⠀⠈⠲⠄⢀⡀
⢀⡄⠀⠀⠈⢿⣧⡌⠀⠻⣇⢻⣆⠀⣄⠈⣆⠈⠢⡀⠂⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢷
⠈⠀⠀⠀⠀⣦⡙⢻⡆⠀⠙⠃⠘⠳⣿⡰⣿⡳⣄⠈⠂⢄⠠⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠙⠁⠀⠈⠀⠀⠀⠀⠀⠈⠻⠾⠙⠊⠑⠀⠀⠂⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⡀⠠⠤⣰⠦⠀⠀⣝⢣⣶⠯⢠⠶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⠀⠀⠠⣄⢀⠘⣿⣇⣞⡡⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠠⠤⠤⠤⢀⠀⠉⠉⠛⠷⣹⣿⣿⢶⡀⠽⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀⣀⠀⠲⣄⠀⣻⣿⣇⡾⠁⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣀⠀⣀⡠⠤⠤⠄⠠⠤⠄⠈⠑⠀⣼⡟⣿⣾⣷⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣾⡷⠋⠑⢈⣠⣤⣶⣶⣦⣴⣦⠤⣤⣀⡀⠸⠿⣿⣹⣿⡂⠠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⣠⣦⣶⡶⠗⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠘⣿⠀⠈⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠄⠀⠀⠘⣆⣀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⢠⠇⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⡀⠀⠀⠀⠀⠘⠀⠀⠀⠙⠿⠃
⠀⠀⠀⠀⠀⠀⣈⣿⣾⡆⣼⠀⠀⢼⡟⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢨⣶⣿⡟⣰⣿⡆⠀⣾⠋⢩⣶⣅⡄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⠀⠘⣧⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢈⡿⠟⢀⣿⣿⡗⠀⡏⠀⠀⠉⠠⣿⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢮⠀⢠⡿⡄⠀⠀⢸⡄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⡿⢠⡟⠀⠀⠀⠀⠇⠀⠀⠘⠙⠛⠋⠀⠨⣿⠿⣿⣦⣀⣤⠈⠀⢸⡇⣷⡀⠀⠘⣷⣿⢠⠀⠀⠸⠀⠀⠀⠀⣤⢀⡿
⠀⠀⠈⠀⠀⢀⣠⣾⡿⠛⠃⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠀⠀⢸⢰⣿⣷⣄⠐⢋⣯⣌⠂⠀⠀⣴⣤⣔⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⣠⣴⡟⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢋⣦⣀⠙⠋⠀⠀⠀⢻⡟⣯
⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣧⠰⠒⠃⡈⠠⠤⠴⣶⡦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢠⣄⠀⠁⢸⡿⢻⣶⡄⠀⣠⡀⠈⡜⢇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣿⣿⡄⠀⠀⠑⠢⢄⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⢠⣿⣿⣿⠀⣼⠟⢸⡟⠀⠀⣿⣧⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣽⣿⣷⡀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠂⡇⠀⠀⠀⠘⠁⠀⠀⣠⣾⡿⠛⡿⠀⡷⠀⠈⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢒⣿⣿⣿⣷⣄⠀⠰⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠾⠿⣿⣿⣿⠃⢾⣿⢂⣷⠀⣦⠐⢲⣻⣦⣶⠰⣐⠒⠶⢶⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠐⠖⢚⣿⣯⣿⣿⣿⣿⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣠⣤⣤⢀⣀⣠⡿⠁⣸⠇⣸⣿⢠⣼⢠⣿⣿⣴⣎⢣⣀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠛⠓⠀⣀⡀⠀⠦⠀⠀⠀⢀⣀⣤⡶⣿⡿⠿⠿⢀⣡⣤⣀⡼⠃⣀⣤⠁⣠⣤⣄⣸⣧⣾⣿⣿⣾⣟⣧⣹⠛⢿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠿⢿⣿⠿⢛⣩⠀⠀⠀⠀⠛⠛⠛⠋⢁⣠⠞⠉⣀⠜⣛⣫⠼⠉⢉⠋⣹⠁⣿⠋⢩⡟⢀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠄⠀⢸⡏⠀⠀⠛⠀⠀⣀⣠⠄⠀⠀⠈⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠚⠁⢸⠇⣠⡟⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡶⠛⠉⠁⠀⠀⠀⢀⠀⠀⠀⠐⠂⠴⢂⠛⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠔⣴⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⢀⣤⠾⠛⠀⠀⠀⠀⠀⠀⠀⠉⠞⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠋⠀⠀⠀⠀⠀⢀⠞⠑⢀⠀⢁⠤⠄⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⣿⠀⠀⠀⠀⣀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠙⠓⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣭⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢶⣶⢤⡒⠄⠀⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⣬⣿⣿⣿⣧⣁⣶⠏⡄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣶⣟⣏⣡⣹⣿⣿⣇⣭⣶⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠁⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠛⠩⠊⢙⠁⠀⠸⣉⠛⠛⢿⠶⣄⠀⠀⠀⠀⠀⠀⠀⣰⡷⠄⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⢀⠀⢤⣼⡄⠀⠀⢈⢒⠆⠰⠝⣼⡀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡄⠘⢻⠳⠀⠀⡏⠀⠀⠄⠀⠅⢡⠀⠀⠀⠀⠀⣿⠣⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⠏⠀⠈⡳⠀⡠⠉⠀⠀⣈⢀⡰⠛⠃⠀⠀⠀⣨⡟⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⣀⠀⠴⠟⢄⠀⠂⣸⠁⢠⡄⠤⠀⠈⠁⠀⠀⠄⠀⠀⢉⠁⠀⢨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣⠤⣤⠖⠒⠈⠁⠀⠉⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⢆⠀⠁⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣸⣿⣦⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⡄⢷⢒⣿⢴⣾⣿⠿⣿⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠁⠀⡿⠭⠿⠿⣯⡄⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠨⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢰⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⣿⡇⠀⠀⠀⠀⠀⢀⡤⠀⠀⠉⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⡇⠀⣼⠋⣴⡆⣼⡇⢠⡀⠠⠊⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠋⣷⢻⣧⠊⣿⡇⢸⣿⠈⣿⠁⣿⡇⣼⣧⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠀⣹⣿⣿⠀⢻⡇⡼⣿⠸⣿⢠⢻⣧⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠛⣸⠇⠀⠘⠷⠁⠙⠃⠈⠁⠀⢉⡎⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⢨⠀⣿⠁⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⠑⠂⠤⣀⠀⠀⣴⢲⠀⢸⢀⡟⠀⣠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠔⠁⠉⣺⡇⢤⣀⣀⣉⣀⣰⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠄⠀⠀⠌⠀⠀⠀⣿⠀⡈⠀⠀⠉⠉⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⣼⠃⡘⣠⡀⢠⠀⢻⡄⠀⠀⠀⣠⡖⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⢀⣾⡣⢃⣾⠀⠈⠟⣶⠆⢰⣿⢰⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⣠⣿⡞⣿⠁⣸⡏⠀⠐⢰⣿⠀⣸⡏⢠⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠃⠘⠃⠙⠖⠈⠳⠔⠁⠘⣿⢠⠁⠓⠂⠻⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠻⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢰⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠘⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣉⣉⣉⣉⣉⣉⣉⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⠤⢶⣒⣲⣿⠿⣟⣻⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠴⣶⣒⣿⡿⣭⣭⣭⣭⣧⣶⣾⡛⠯⠭⣗⡲⠽⣗⣺⣿⡀
⠀⠀⣤⣴⣶⠾⣟⣻⢭⣗⣾⠭⣗⣚⣯⣿⣿⣿⣿⣿⣿⣿⢿⡗⠬⢷⠶⢟⣛⣯⠵⣾⡀
⠀⣾⣟⣓⠯⢭⣓⣯⣽⠷⠟⣻⡭⠷⣶⣿⣿⣿⣿⣿⠿⡿⣿⢟⣶⣿⣋⣭⡵⣚⣻⠭⣧
⢰⣿⡶⠾⣟⣻⡭⠕⣶⣿⣿⣶⣚⠟⠏⡛⣑⣾⣸⣿⣴⣧⣿⣏⣿⣽⣿⣿⠧⣗⣒⣭⢿⡇
⠀⣿⣿⣭⠵⢖⣛⡯⢵⣊⠻⣿⣷⣾⢻⣟⠉⢯⡛⣿⣸⣿⣿⣿⣿⣿⣿⣟⣿⣯⡕⣀⣬⣿⡄
⠀⣿⣒⠯⠭⣓⣂⣬⣭⣶⣶⣬⣿⠿⠷⠛⡶⠟⠛⡟⠽⡧⢻⣿⣿⣿⢿⣿⡟⢻⣻⡳⠴⣟⣿
⠀⢿⣿⣿⣿⣷⡻⢿⣿⣿⣿⣿⣿⣧⣼⣴⣷⢾⠷⠿⣾⣟⣿⢹⣿⣿⣿⣷⢿⣿⣿⣥⣤⣽⣾⡅
⠀⢿⣿⣷⣿⡛⢩⣠⣾⣿⣿⣿⣿⣉⣀⣶⣿⣼⣷⡧⢿⡿⣿⣿⣿⣿⢿⣿⣿⣿⢉⡿⣿⣿⣿⡏
⠀⣿⣿⣿⣽⣏⣿⣿⡿⣿⣿⣵⣿⣿⣿⠵⢿⢙⣸⣿⣸⣟⣹⣿⣿⣷⣿⣿⣄⢻⣿⣿⣌⢿⠟⢱⡆
⠀⢹⠿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⣿⣿⣿⣿⡟⠙⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣎⡛⣷⡀
⠀⢸⣿⣿⣾⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⡿⣿⡿⢶⡟⣿⢻⠛⢛⠽⢿⣿⣿⣏⢿⣿⣿⣿⡏⣿⣿⣷⡄
⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣅⣇⣹⣷⣦⣷⣽⣾⣿⣾⣿⣿⣧⢻⣿⢿⣿⣿⣿⣺⣿⣿⡿⣷
⠀⢸⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⣿⣿⣿⣿⣿⣿⣾⣿⣿⣻⣿⣿⣿⢻⣿⣯⣷⣦⡾⣇
⠀⠀⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣝⣇⣻⣿⣿⡻⢧⡿⢿⣻⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣨⣿⣟⠝⠿⣇⣹
⠀⠀⣯⣟⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡿⠟⣉⠩⠵⣒⣫⣭⣾
⠀⠀⢿⣿⣿⣿⣬⣻⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢠⣶⣾⠿⠟⣛⣩⡴
⠀⠀⣿⣮⣽⣧⣴⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⠙⣿⣿⣿⣿⣿⣿⠀⣨⣥⠶⢞⣛⣩⣵⡶
⠀⠀⣿⣿⣿⣻⢿⣽⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⢧⣴⠶⠿⢛⣋⣭⣴⣶
⠀⠀⣿⣙⢻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣩⡿⠛⠻⣿⣿⣿⣿⣿⣛⣻⢿⣿⣿⠿⠫⠶⢒⣚⣯⣿⣿⣿⣿⣿
⠀⠀⢸⣏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠴⠶⢒⣊⣉⣻⣿⣿⣿⣽⣶⠿⠿⠾⠿⠿⠟⠛⠋⠉
⠀⠀⢸⣏⣛⣛⣛⣫⣤⢈⣟⣿⣿⣿⣷⠾⠟⠛⠉⠁⠀⠀⠀⠀⠉
⠀⠀⠈⠉⠉⠙⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢛⣛⠻⡿⠟⡛⠛⠻⢿⡿⠟⠛⢛⠻⢿⠟⣛⡛⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⣾⢹⣇⡴⢟⡽⢟⣿⠨⠅⣿⡻⢯⡻⢦⣸⡏⣷⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠘⢫⡾⣫⠾⣫⡿⢠⡄⢿⣝⠷⣝⢷⡝⠃⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡅⠀⠀⠀⠚⡵⢟⡽⠏⣼⣧⠹⢯⡻⢮⠃⠀⠀⠀⢨⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠁⠀⣀⣀⣠⠾⢋⣴⣿⠋⠙⣿⣦⡙⢷⣄⣀⡀⠀⠈⠈⠙⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠉⠁⠰⣿⣿⣿⣶⣶⣿⣿⣿⠆⠈⠉⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡋⡴⣶⣀⣨⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡮⠒⠴⣦⣀⠙⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠓⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠦⢄⡀⠀⠀⠀⢀⠔⠈⠡⣀⡠⠊⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣤⠀⠀⠀⠀⠈⠑⠲⢤⠋⠀⠀⠀⠀⠀⠀⠀⠐⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠄⣾⠟⡀⢱⠤⠤⠄⠀⠀⣀⠆⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠐⠓⢅⠀⠀⠀⠈⠑⢆⢄⡀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⡈⠓⢄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠰⣯⡁⠀⠀⠀⠀⠀⢀⠜⠢⡀⠀⠀⠀⠀⢸⠂⠈⠑⣤⢶⠇⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢳⠀⠀⢘
⠀⠀⠀⠀⠀⠀⢸⡟⢦⣄⠀⠀⠀⠀⠐⡄⠑⠬⠦⡒⠕⢊⣠⣶⣋⠡⢎⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢀⡂⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣄⠀⢦⠀⠈⠢⠊⣀⡤⠖⠋⠁⠀⢀⡤⠈⠑⠦⣄⡀⠀⠀⢀⠀⠀⣠⢃⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠙⣦⡀⣠⣴⠚⠉⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠈⢁⡠⠛⢺⡱⠀⡘⠊⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠋⠁⢸⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⣀⣤⣐⣊⣉⣀⣠⡞⠦
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠲⡞⠉⣿⠀⠀⠉⢹⡇⠸⣧
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆⠀⠀⡘⢇⠀⠀⠀⠘⡀⣿⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠈⢦⠀⠀⠀⢣⣿⠀⠀⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢸⡄⢀⣰⠧⠤⣤⠃⠀⠈⢆⠀⠀⠈⠇⠀⠀⠀⠘⠃⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢸⡏⢉⣀⣀⣤⡼⠀⠀⠀⠈⢆⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠤⠖⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣠⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣁⣤⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣰⣦⡶⣠⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣬⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⢀⣠⡴⢾⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠻⠖⢳⡄
⠀⠀⠀⠀⠀⠀⠈⣥⠞⣿⡿⠋⠀⢀⡀⠀⠀⣀⠀⠚⡉⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣸⠏⠀⠀⠀⠋⡉⣠⡘⠉⠀⣠⠏⠁⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠸⣧⡈⠀⠀⠀⠀⠘⠉⠛⠓⠒⠋⠁⠀⢀⣠⠾⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣅⠀⠀⠀⢀⡴⠋⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣷⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣌⠀⢔⣫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠹⣷⡈⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠳⠶⠞⢩⡟⠀⠀⠀⠀⠀⠀⠀⠸⣷⠘⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⢀⠢⣝⣲⠄⠀⠀⠀⠀⠀⠀⣿⠀⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⠖⠶⢽⠿⠃⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠐⢮⣓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣆⠀⠀⣀⡀⢠⡾⠋⠁⣀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠀⢠⣿⠇⠈⠉⠉⠉⠉⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠿⠿⠶⠶⠶⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⢿⡿⠿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠺⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠘⢿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠙⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⠀⢀⣤⡄⠀⣤⡄⠀⠀⠀⣿⡇⠀⠀⠹⣿⡆⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⢠⣿⠟⠉⠉⠙⢿⣷⠀⠀⠀⢸⣿⠇⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⣿⡇⠀⠀⢀⣿⡇⠀⠀⠀⣾⡿⠋⠉⠉⠻⣿⡄
⠀⠀⠀⢺⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⣿⡗
⠀⠀⠀⢘⣿⣷⣤⣀⣤⣾⣿⠀⠀⠀⢸⣿⡄⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷⣤⣀⣤⣾⣿⡃
⠀⠀⣰⣿⠟⠉⠛⠛⠋⠉⢻⣷⡄⠀⢸⣿⣿⡿⠋⠸⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⣿⡇⠀⢠⣾⡟⠉⠙⠛⠛⠉⠻⣿⣆
⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠘⠿⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦⣈⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠴⢶⠜⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠻⠋⢀⡶⣶⡗⠁⡁⢁⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠾⢂⡉⠀⣋⠸⠓⠓⣿⠉⢃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⠊⠉⠙⠻⠷⣤⣀⡀⢈⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣤⠂⠀⠀⠀⠈⡱⣬⣔⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠖⠀⠀⣠⠇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠗⠲⡈⠀⠀⠀⣸⡧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠉⠙⠛⠶⣤⣀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠣⢤⣠⠞⠁⠀⠙⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⠀⣠⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣄⣠⢞⠇⠀⠀⠀⠂⠀⠀⠑⠀⠀⠀⠀⡠⠖⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠉⠀⢀⡤⣲⡿⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⢸⠂⠀⠀⠄⠀⠂⡀⠁⠀⠀⠀⠘⠀⠄⠀⠂⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⢀⡠⢔⠪⢿⣶⠉⡑⠂⢨⠗⡲⠦⡤⠤⠤⢀⣀⠀⠀⡴⠁⡄⠐⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⣀⠀⠀⡀⠁⠀⠀⢹⣄
⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⢠⡖⡑⠀⠀⠀⠋⠀⠀⠈⠒⢾⣁⡀⠀⠀⠀⠂⠀⠀⠀⠛⠃⢠⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣄⠀⢀⠀⠀⢀⠙⣆
⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠈⠢⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⢅⡓⠦⣄⣀⠀⠀⠠⠀⢠⡎⠀⠂⠂⠀⣀⠀⠀⠂⠀⠀⠄⠀⡶⠀⠈⠳⣌⠀⠀⠀⠀⠘
⢶⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⠙⢦⡀⠀⠀⠀⠀⠀⠀⣨⠗⠀⠀⠉⠳⢥⣤⣶⣿⠁⠀⡈⠠⠀⠀⠀⠀⠈⠀⠂⢀⡼⠁⠀⠀⠀⠈⢦⠀⠀⠁⢀
⠀⢹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡀⠀⠱⣄⠑⣄⠀⠀⣠⠴⠂⣨⠇⠀⠀⣀⣴⠾⠋⢡⠏⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⣾
⠀⢸⣿⠈⠻⣷⣄⠀⠀⠀⢠⡄⡀⠀⠀⠱⡄⠀⠘⠣⠞⠳⣊⡥⠖⠋⢁⣠⣴⣿⣋⣥⣤⠶⢯⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠘⠙⠇⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⢠
⠀⠸⣿⠀⠀⠀⠙⠿⣦⡀⠈⠇⢱⡀⠀⠀⠙⣆⣀⡤⠚⠉⠁⢀⣠⡶⠟⠋⠉⠉⠀⠀⠀⣀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡞⠁⠁⠄⠀⣠⠏
⠀⠀⣿⠀⠀⠀⠀⠀⠈⠻⢦⡀⠈⠷⠀⠀⠀⠀⠁⠀⣀⣤⠾⠛⠉⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠙⠲⠤⣄⣠⠀⠀⢀⣸⠀⢀⡀⠰⠯⢤⡄⡀⣠⡾⠃
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣠⣴⠟⠋⠁⠀⠀⢀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢉⡿⠋⢲⡅⢰⠖⠀⢠⠗⠉⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡶⠟⠉⣿⠀⠀⠀⠀⠰⢤⣷⣶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠉⠀⠀⢀⣿⠉⠀⣠⠃
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣖⣉⣁⣀⣀⣀⣠⣤⣴⣟⠉⠓⠊⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠤⣙⣶⠖⠋⠉⣿⠃⠀⠈⠉⠉⠉⢻⡇⠀⢸⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢻⡄⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡷⠀⢸⣿⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⠻⣆⠀⠀⠀⠀⠀⠀⢻⡀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠹⣆⠀⠀⠀⠀⠀⠈⣧⢸⣿⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⡇
⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠹⣆⠀⠀⠀⠀⠀⠸⣾⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇
⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⣇⣀⣀⣀⣤⡿⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⢸⣇⣤⡶⠾⠛⠉⠉⠁⠙⣧⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⢸⡏⠀⠀⠀⢀⣀⣀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⢸⡷⠿⠛⠛⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠋⠁⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⡾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣯⣤⣤⣤⡴⠞⠁
⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣀⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣀⣤⣤⣤⣄⡈⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣤⣤⣤⣤⣀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣸⣷⡄⠀⠀⠀⠀⢸⣇⠀⠀⣿⠀⠀⠀⠀⠀⣴⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠐⣿⣿⣿⡆⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⠛⠟⠃⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⢀⣠⡾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡁⠀⠀⠀⠉⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁⠀⠀⠀⣹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠶⠶⠂




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

⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⠀⢀⣤⣴⣶⣶⣤⡀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⣴⡿⠛⣿⣿⣿⠛⣿⣆⠀⣰⣿⠛⣿⣿⣿⠛⢻⣦⠀⢠⣿⠛⢻⣿⣿⠛⢻⣧⠀⢠⣾⠛⢻⣿⣿⡟⢻⣷⡀⢀⣾⡟⢻⣿⣿⡟⠻⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⡆⣿⣿⣷⣿⣿⣿⣷⣿⣿⡇⣾⣿⣿⣿⣿⣿⣷⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡟⣿⡇
⠀⠀⢿⣿⣾⣿⣿⣿⣷⣿⡟⠀⢻⣷⡻⣿⣿⣿⣿⣽⡿⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠸⣿⣟⣿⣿⣿⣝⣿⣿⠃⠘⣿⣿⣫⣿⣿⣽⣿⣿⠇
⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠙⠿⢾⣿⣷⠿⠋⠀⠀⠀⠙⠻⢿⣿⣿⠿⠛⠁⠀⠀⠈⠻⢿⣿⣿⠿⠟⠁⠀⠀⠈⠻⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠀⠀⠠⠤⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⣠⠤⠒⠃⣡⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠒⠂⠠⠄⢠⠊⠀⡀⢀⡴⠋⠁⠀⣀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣼⠏⠀⢀⡤⠞⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠤⠴⠞⠢⠴⠯⠀⣀⣠⡤⠔⠊⠉⠉⠉⠉⠉⠐⠂⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠟⠒⠚⠀⠀⠈⠛⠓⠒⠒⠂⠵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠀⠀⠀⠀⠀⠀⣀⣀⣠⣎⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢀⠀⠀⠄⠀⠀⠀⠠⣤⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢀⡴⠚⠉⣥⣶⠐⠐⠂⣤⡭⢙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡐⡀⠀⠲⣶⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⢋⡖⠁⡀⠐⠀⠁⠚⠉⢮⠙⠂⠀⢱⣬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣹⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⡎⡇⠈⠀⢤⣄⠠⠄⠀⢀⡂⢀⡈⠀⢀⡹⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢈⠀⢉⠰⠆⢀⡀⢁⡀⠐⠊⠁⣼⠨⠀⠉⣀⠈⠳⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢃⡈⣤⠈⠀⠀⠈⠙⠀⢀⠉⠁⠀⠀⠻⠁⢀⣄⠀⢹⡀⠀⠀⠀⠀⢀⣠⣮⡟⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡻⣌⢷⡀⠀⢣⠀⠶⠀⠤⠠⠀⠺⢇⠀⠀⠐⠒⠤⡾⢸⡇⠀⠀⢀⣠⣾⣿⣿⣳⢧⣬⣀⠀⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣹⢦⡙⠦⢠⡀⠀⢰⣇⣀⡈⠁⠈⠓⠀⢀⣀⡔⢁⣾⠀⣠⣠⣿⣿⣿⣻⠟⠃⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡙⠳⣤⣙⠲⠀⠤⢬⣤⣤⣀⠹⠥⠚⣉⣴⣿⣿⣷⣿⣿⠿⠟⠋⣠⣄⠀⠀⠀⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣾⠉⠛⠲⠶⠤⠤⠤⠴⠶⠒⣋⠽⢯⣉⣉⠁⠀⠀⠀⠀⠘⣧⡼⠃⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠽⠿⢿⣶⣤⣮⣤⣤⣤⡴⠞⠋⠁⠀⠀⠈⠉⠛⠛⠃⠐⠚⠛⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣠⣄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⣿⠿⠿⠿⠿⠟⢛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣴⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣄⡀
⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣦⡀
⣿⣿⣿⠁⠀⠀⠀⣠⣾⠟⢩⣶⣶⣾⣿⣿⣿⣷⣦⡀⠀⠀⠀⢀⣴⡿⣻⣿⣿⡟⠁⠘⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡀
⢻⣿⣿⠀⠀⠀⣼⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⣾⡟⢱⣿⣿⡏⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠙⠋⠀⠀⣼⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣼⡿⣠⣿⣿⡟⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣸⣿⢁⣾⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣿⠇⣸⣿⠃⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⠀⣿⣿⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⠀⢻⣿⡀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⠀⠸⣿⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⣿⣇⠀⢻⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠘⣿⣆⠈⣿⣷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⢿⣦⠘⣿⣦⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⢻⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣮⡛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣾⣟⣡⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠉⠉⠉⠑⣆
⠀⠀⡎⠉⠐⢠⠞⠁⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⢀⠀⢣⠀⠀⡆⠀⠀⢆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡧⠀⠀⢀⡤⠔⠀⠀⠲⠤
⡆⠣⣀⡑⠊⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⠀⠀⠀⠀⠀⠀⠄
⠑⠄⡀⠀⠀⠀⠀⠀⠸⢓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠙⠀⠀⠀⠰⡶⠀⠀⢶⡆
⠀⠀⠐⠄⠀⢴⡄⢊⠔⠋⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⡅⠠⠴⣤⠀⠀⠀⠀⠐⠦⠀⠉⠛⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠈⠫⡁⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠳⢄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⣠⠈⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⢀⠀⠁⠒⠠⠄⠀⣀⠀⠀⣀⢠⠔⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠆⠀⠀⡴⠊⠉⠣⠀⠀⠀⡀⠀⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠐⡄⠀⢇⠀⠀⠀⢃⡠⠈⡅⠠⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡄⠀⠜⠁⠀⣆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣇⣌⣀⣤⣃⢀⣄⣁⡴⢤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⠉⠙⡁⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠰⠚⠀⣇⡁⡀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢄⠀⠠⡿⠀⠀⠀⠀⠀⡀⢀⠃⠑⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠷⡀⠀⠀⢀⠔⠁⡘⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣗⠐⠒⠒⠀⠤⠤⠤⠤⠤⠤⠰⠁⠈⠂⠊⠀⠀⠠⣁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣀⡀⢀⣀⠸⠆⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠜⠃⠠⠠⠤⠤⠀⠀⠀⠈⣠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠀⠠⠖⠊⠁⠈⠉⠓⠠⠄⢀⣀⡄⠀⠀⣀⣀⣀⡀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀⠐⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠛⠋⠙⠛⠛⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⢿⣿⠟⠛⢻⣇⣹⠟⢻⢻⣷⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢸⣷⣴⣿⣌⣁⣼⡿⠻⣆⣀⣼⡿⣷⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣾⡏⢠⠘⣿⡿⠿⢯⣥⣿⣿⠿⣦⣾⣿⡄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢠⣿⠓⣶⡾⢿⣔⠂⣸⣧⣼⡀⠃⢸⣿⠿⢿⡄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⢟⡉⢻⣧⣣⣿⣿⡏⣻⡿⢿⡟⢻⣿⣄⣼⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿⢮⣥⡼⠛⢿⠁⢠⢹⣿⣌⣠⣿⢋⡉⢹⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⡾⠛⠳⣥⣾⣷⡶⢿⡟⠛⠻⣿⣧⣥⣾⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⣧⣋⣠⡟⠉⢉⢳⣿⣄⡃⣀⣿⠿⠛⢿⣴⡿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠻⣟⣙⣧⣀⣙⣴⣿⣿⣿⠿⣿⡀⠀⣻⣿⠇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⠻⣿⣿⣿⣏⢤⠀⣻⣶⣿⣿⠿⠛⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠙⢿⣿⣶⣾⡿⠟⠉⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⡀⠀⠀⠉⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠶⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣯⠿⠿⣿⣿⡿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣺⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣆⠀⣼⣿⡄⣠⣾⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣷⣼⡿⠿⠿⢿⣿⠿⠿⢿⣷⣴⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⢟⣫⣵⠶⡟⣻⣏⣿⡛⣟⡷⢶⣭⡻⢷⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣿⢟⣵⣿⢯⣯⠶⠛⠛⠋⠉⠛⠛⠷⣯⡾⣿⣷⣝⢿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡿⢣⣿⣿⣿⠋⠁⠀⣀⠀⠀⣀⠀⠀⡀⠀⠙⢿⣿⣿⣦⢻⣿⣁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢣⣿⣿⡿⠁⣾⡇⢰⠟⣧⢸⠟⡇⣼⢻⢀⡏⠀⠹⣿⣿⣧⢻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣀⣤⣾⡏⣾⣿⣿⠁⠀⢸⡇⢸⠀⣿⣿⠀⣷⢹⣼⣸⢱⢶⠀⢿⣿⣿⡸⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⣿⡇⣿⣤⣿⠀⠀⣸⣇⢸⣦⡟⢹⣤⡏⠀⢀⡏⢹⣸⠀⢸⣦⣼⡇⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣧⢿⣿⣿⡄⢀⡉⡉⣀⣉⡁⡈⡉⢁⣀⡈⣁⣈⠉⠀⣾⣿⣿⢹⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⢛⣿⡜⣿⣿⣷⡈⠷⠛⠀⠫⠷⠃⠷⠢⠀⠀⠣⠏⠃⣼⣿⣿⡏⣼⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡜⢿⣿⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⢟⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣮⡻⢮⣿⢻⡷⣦⣤⣤⣤⣤⣤⡶⢛⣽⣯⠟⣡⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⣹⣿⣶⣝⣛⠶⣧⣧⣽⣎⣿⣤⡷⢞⣫⣵⣿⣿⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⣿⣿⡿⣶⣶⣶⣶⣶⣶⢿⣿⣿⠏⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠸⣿⠋⠈⢿⡟⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢸⣆⣠⣿⣤⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⠔⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⣷⡄
⠀⠀⣼⠋⠃⠀⢸⣤⣾⣦⣶⣿⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⣿⣿⣿⣿⣿⣿⣥⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢄⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠘⠆⠀⣼⣾⢟⡟⠈⣹⠞⡿⢿⣧⡤⠤⢤⣄⡀⠐⣄⠀⣠⣟⢧⣿⡙⣼⣻⣿⣿⡿⡃⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⣾⡄⠀⠀⠀⠀⠀⠸⣿⣤⣤⣾⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣧⣷⠞⡿⢿⠟⠿⣿⣧⣀⡴⠟⠁⠀⠘⠻⢿⣛⢟⣧⣟⣀⣙⣽⣿⣧⡠⡀⠀⠀⠀⠀⠀⣠⢾⠁⠘⡀⢣⡀⠐⣦⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⣶⣂⣹⣿⣿⣼⠅⠛⠖⡉⣨⣷⣶⠀⠀⠀⠀⠀⠀⡄⠐⢻⣿⣾⣯⣼⣿⣿⠿⠿⠀⠙⢮⡆⠀⠀⢀⡾⠁⡇⡆⠀⡇⠱⣿⣆⠈⠳⠦⣤
⠀⠀⠈⠀⠀⢀⣿⣿⣾⣷⣄⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⢠⢀⡟⢻⡿⣿⣿⠛⣿⠇⠀⠀⠀⠈⠁⠀⠐⠋⣀⣼⢀⡇⣸⠇⠀⠙⠻⢷⣦
⠀⠀⠀⠀⢠⣆⢀⠟⠁⠸⡿⠛⡇⠈⠉⠙⠛⠀⠀⠀⠀⠀⢀⣀⠞⠰⢋⡟⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⡾⠁⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠐⠃⠀⡀⢔⣠⣶⣲⣾⠆⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠓⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣄⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠟⠛⣋⣡⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣤⣾⡿
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡣⡄⣦⠋⠀⣀⣰⠃⠀⠀⠀⠀⠀⣀⠀⠀⠙⠿⣿⣉⡁⠔⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢒⡤⢙⣛⣵⢿⣦⣶⡽⠛⣠⡟⠀⠀⠀⠀⠉⠙⡄⠀⠀⠀⢀⡠⡄⠀⠀⠀⠀⠀⢀⣸⣿⣿⣶⣄⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠌⣻⣦⢿⡉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢠⢫⠊⠃⠀⠀⠀⠠⠶⠻⠟⠟⠛⠿⠿⠷⠤⠾⢿⣿⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠱⣠⢤⣄⡀⠀⡀⠀⠀⠀⠀⠙⢼⠃⠹⠀⠀⠀⢠⣶⣦⣄⡀⠀⢸⢸⡀⠀⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⣿⠛⣾⣿⣷⡟⢧⡙⢻⡆
⠀⠀⠀⠀⠀⠀⣧⠀⢹⣿⠛⠁⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⠀⠈⠛⢷⣄⠘⢸⣷⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣼⢰⠿⢹⣹⡌⠙⠳⠿
⠀⠀⠀⠀⠀⠀⠙⡇⢴⣏⠁⠒⢤⣤⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⢾⣿⣿⣧⣄⣀⣀⣀⠀⠀⠀⠀⠀⠘⠊⠀⢰⢃⡟⠀⢸⡏⠇
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⣫⡴⠶⠋⠀⠀⠉⠀⠀⠀⠉⢻⣇⠀⠀⠀⠀⠀⠰⣿⡜⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣶⠛⠛⠛⠛⠓⠒⠶⠶⠶⠶⠴⠿⠭⣿⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⢲⡄
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢛⣉⣩⣭⣭⣭⣍⣉⠉⠓⠲⣄
⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⣖⠆⢠⣞⣁⣠⣤⠶⠛⢉⣠⣤⣤⡹⠋⢿⠟⣿⡿⡿⠆⢀⡼⠃
⠀⠀⠀⠀⠀⠘⣿⡤⠴⠒⢚⣉⣩⡏⠁⠀⠀⠀⣴⣿⠿⢛⡉⠀⠀⠀⢠⣶⣤⣼⡖⠋
⠀⠀⠀⠀⠀⢀⣿⣷⣿⣟⣛⣻⣿⣷⡆⠀⠀⠀⠋⠡⠞⣩⠭⣧⠀⠀⢸⣟⢝⢿⠁
⠀⠀⠀⠀⣼⠋⢉⣍⣉⠛⠻⣷⣾⡿⠀⠀⠀⠀⠀⠀⠚⠛⣀⠀⠀⠀⡰⠿⢦⠈⡆⠀⠀⢀⣤⠶⢓⣛⣶⣄
⠀⠀⠀⠀⠙⢦⣌⠁⠉⢷⡄⠨⣿⣯⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⡖⢊⡀⠀⣸⠇⢸⣤⠞⢉⡤⠞⠋⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠙⡗⠠⣿⣇⠀⠉⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠁⠉⠀⢀⣁⣤⠴⠟⠚⠫⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣉⣉⣀⡧⠀⠀⠀⠀⠀⠀⡞⠓⠒⠒⠒⠒⠛⠉⠉⠀⠀⣀⣀⣀⡐⠂⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⢻⠀⠀⠐⠀⠀⠀⠀⢧⠀⠀⠀⢀⠀⠀⠀⠦⠀⠈⠁⣤⠀⠙⢶⡀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⡇⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠀⠤⠀⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⡄⠀⠀⠀⠀⠐⠂⠀⠄⠀⠐⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⢻⣶⠛⢻⣦⡀⠐⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠈⠀⣀⣡⣴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⣿⠉⣿⣿⠛⢷⣶⠦⠤⣤⣤⣤⣤⣤⡤⠴⢶⣾⣿⠋⢁⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣏⠙⣿⣷⠾⢿⣦⣼⢿⣷⡞⢻⣿⠀⠘⣿⣷⠀⣿⣿⣇⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣽⣿⣦⡬⠿⠿⠼⠿⠷⠼⠿⢧⣀⣿⣟⣀⣿⣿⣿⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢐⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⢲
⢰⠀⠀⢀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⡀⠑⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡞
⠈⢆⠀⠀⠈⢦⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠹⠛⠛⠁
⠀⠀⠑⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬



⠀⠀⠀⡄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣠⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡧⠤⡧⢤⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⠀⠀⠀⢳
⠀⠀⡀⢴⠃⠐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣡⠤⠧⢤⣁⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠒⠚⠦⢊
⠀⠀⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤
⠀⠀⠀⠀⣠⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣄⠀⠀⢀
⠀⠀⠀⠀⠉⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠁
⠀⢀⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡴⠀⠀⠣⣀⠜
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣁⣸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠙⠄⠀⠖⠉⠱
⠰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⠾⠟⣡⡾⠷⠋⣠⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢛⡴⠚⢁⣤⡴⢻⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣿⣿⣷⣾⢟⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⢿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⢠⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⠞⠉⠨⣰⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠰⣧⠀⠀⠙⡂⠀⠛⢏⠑⠛⣠⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⣶⣿⣿⣿⣶⣴⣻⣜⢺⠏⠀⠠⢄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠷⢤⣿⣯⣿⣿⣿⣿⣆⣻⣿⣿⣤⡟⠑⠛⠉⠀⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣄⡀⠛⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⠀⠀⠑⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢈⢣⣶⣿⢿⣿⣿⣿⠟⢿⣧⣀⣀⣴⡿⠏⠉
⠀⠀⠀⠀⠠⠄⠒⠲⠖⠲⢦⠤⣤⣄⠀⠀⠀⠀⣀⡀⣾⢁⡀⢸⡿⠋⠻⣧⠀⠀⠉⢋
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠳⣤⡽⠙⢣⡀⠀⢿⡄⠘⠆⠁⣿⡇⠁⢀⠿⢷⡁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⣹⠟⢦⡈⢿⡄⠀⠙⢸⡇⢀⠈⠀⠀⠀⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣤⣤⣠⣵⠀⠻⣆⠀⢸⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣜⣷⣼⡄⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣉⡉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠟⠛⠛⠋⠛⡻⠟⣿⣿⣿⣍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠀⠀⠀⠀⣴⣶⣿⣿⣾⣿⣿⣿⣿⣼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠃⠀⠀⠀⠀⠀⣴⣷⣾⣾⣿⣿⡿⠈⢿⣿⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⣿⣗⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢻⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣇⠈⠙⢟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠉⠈⠁⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⣅⣠⠄⠀⠀⠀⠠⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢴⠗⠛⠀⠀⠀⣤⡀⣼⠿⠶⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⠠⠠⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠁⠀⠀⠹⠏⡄⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣏⣀⣧⠧⠄⠒⠀⠀⠀⢆⡀⠀⠀⠀⢱⠀⠀⠀⣿⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠜⢹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠥⣄⠀⢸
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⡀⢀⡀⠀⠀⠀⠀⢤⣿⣄⣠⣿⡄⠀⢸⡇⠀⠀⢀⡀⠀⠀⢀⡤⣞⡛⣉⠀⢲⡀⠀⠀⠀⣀⣤⣾⣧⣾
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁⠈⣀⣀⠀⠀⣷⣦⣰⣿⣿⣿⣿⣿⣿⣆⠀⢻⡄⠀⠀⠀⠀⠀⣿⣷⣶⡞⠉⠛⠑⡶⠊⠀⠚⠫⠽⠿⣿⠏
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠘⠁⠀⠀⠀⠙⢿⢃⣀⢹⣿⣿⣿⣿⣿⡆⠀⠻⣄⠀⣀⠄⠀⣿⣿⣿⠁⠀⢀⡜⠁⠀⠴⠒⠉⠁⣴⠋
⢰⣿⣿⣿⣿⣿⢻⣿⠟⣻⣿⣿⣿⣿⣀⡄⠀⠀⠉⠁⠀⠀⠀⠀⢃⣾⣿⣿⣿⣿⣿⣿⡀⠀⠙⢷⣄⠀⢠⣿⣿⠉⠀⠀⠈⠀⠀⠀⠀⢀⣤⠞⠁
⣾⣿⣿⣿⣿⠧⣼⡁⢼⠛⠛⢿⣿⣿⠀⠤⠐⠀⠀⠄⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠛⠾⣿⣇⣀⣀⣀⣀⣀⣠⡤⠞⠋⠁
⣿⣿⣿⣿⣿⡈⠤⠇⠈⠀⣿⣖⢻⣿⡄⠀⠀⡀⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣷⠀⢀⠀⢀⠈⠋⢸⣿⣿⣄⣀⡀⣀⠆⣩⣾⣿⠿⠛⠉⠨⠿⣿⣿⣿⣿⡇
⢹⣿⣿⣿⣿⣿⡄⠀⡱⠁⠀⣠⣿⣿⢿⣿⣿⣦⣤⣾⡚⢋⣰⠆⠀⠀⠴⠒⠚⠛⢿⣿⠇
⠘⣿⣿⣿⣿⡿⠟⠂⠀⠀⠚⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠘⣿⣿⡿⠁⠈⠉⠀⠀⠘⠿⠋⠻⠋⠁⠀⠌⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣦⣤⣤⣤⣤⣤⣴⣶⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⢀⣠⣤⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠈⠀⠀⠀⣠⣯⣯⣿⣫⣽⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣉⡀⠀⠈⠀⠀⠉⢿⣿⠀⠀⠀⠐⢦⡄⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡥⣬⣭⣧⡀⠀⣀⣿⣷⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⡟⢻⣿⣶⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⣼⠀⠉⠉⣉⣁⡀⠀⠛⡛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⢲⡊⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠇⠀⠀⠡⣄⣀⠀⣛⣦⣶⣿⣿⣿⣿⣶⣿⣿⣿⣶⣞⠀⠀⢀⡿⠀⠀⠀⠀⢀⠴⡆⠀⠀⠀⠉⠛⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣏⠙⢿⣍⣉⣉⣉⣽⡟⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⢂⣀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠛⠿⢿⣿⢟⣰⣿⣿⣆⠀⠀⠀⠀⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣦⣦⣾⣿⣿⣿⣿⣿⣿⣷⣀⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣟⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠛⠛⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠿⣿⣿⣿⣿⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣬⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠋⠻⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣀⠀⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⢀⣀⣀⠀⠀⠁⠀⢀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⣈⣉⠀⠀⢠⠈⡈⠉⠙⢄⡈⠀⠰⠐⠷⠠⠐⡔⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠠⠼⠿⠿⠷⢿⣿⣁⡤⠤⣴⡷⣀⠡⠄⠀⠈⠳⢼⡆⣱⠠⠄⠸⡑⠀⢄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀⡐⢠⣶⣴⣾⣿⡿⠦⠞⠛⡀⣄⡀⠤⣤⢞⣶⣌⢻⣦⣶⢞⠃⠀⢳⣄⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣑⣦⣿⣷⣼⠧⣤⣷⠶⢾⣧⣬⣿⣿⣵⣞⣛⠻⣿⣿⡇⠘⣤⠠⠀⠀⢀⡷⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠍⢭⡡⣴⣍⣽⢉⢶⣋⣥⠵⠾⠛⠋⠛⣙⣛⣿⣿⣶⣿⣿⣿⣶⣶⠶⠟⠛⠋⡴⠃⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⢛⣿⣿⠛⢯⠥⠀⠺⠶⠿⠒⠛⠻⠿⣍⣙⠻⣿⡏⢿⣿⠗⠊
⠀⠀⠀⠀⠀⠀⠰⠍⠉⠣⠀⠀⠘⠿⠿⠸⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠤⢤⣾⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⡠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠁⡰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢀⡞⠀⠀⠀⠀⠈⠄⠀⠐⠊⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠒⠒⠶⠦⠤⣄⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠴⠚⠉⣀⣠⠤⠴⠶⠦⠤⢤⣀⣀⠈⠑⢦⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠚⢁⡤⠖⠋⢁⣀⡠⣤⣤⣤⣀⣀⠀⠈⠙⠦⣄⠙⢦⣄
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⠎⡠⠚⢁⣤⡒⡿⣫⠉⡷⢡⡄⣣⡞⢉⠙⡖⢄⡀⠈⠳⣄⠙⢦⡀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠃⡜⠁⣴⢋⡄⣹⠱⠏⠜⡇⠛⣡⡟⢰⡟⣰⠗⠋⡿⢦⡀⠘⢆⠀⢳⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠼⠀⡸⡇⠺⡁⢻⣤⣎⢀⡷⣶⠁⣇⣀⠘⠏⠸⢃⡥⠖⢷⡄⠈⢧⠀⢻
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡃⠀⡄⢰⠇⡷⢶⢀⠎⣤⢸⡎⣴⠌⣧⢊⡉⡿⣇⣠⡞⣰⠆⡸⢹⡄⠘⡆
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠁⢠⠁⢸⡟⣰⡆⡿⠘⠋⣾⠀⠉⢸⡇⠛⣡⠟⠙⣶⡀⢀⣴⡷⢦⡇⠀⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠃⣿⠀⢸⡀⠸⡇⠈⣐⣷⣤⢋⡜⠷⣩⠼⡧⢺⠃⡿⢠⣫⠟⢳⣾⣴⠘⡇⠀⡇
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠁⠀⠀⡴⠁⠀⢸⠀⠘⡇⠀⡟⢺⢋⡄⢸⡎⣴⢀⡇⢠⡘⣾⠀⢀⢆⠏⡶⢸⡇⠈⢸⡇⠀⡇
⠀⠘⢦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⣇⠀⢻⡀⠸⣾⠘⠃⣾⣁⣠⡶⢾⡌⠃⣏⡷⠛⣾⡀⣠⣯⣿⣤⢾⡇⢰⠁⢰
⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠸⣆⠈⢧⡀⢻⡤⢾⠋⣝⢶⡀⣦⠻⠦⠿⢡⡆⣏⡝⣩⠄⡇⢀⡞⢀⠇⠀⣾
⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠳⣄⠙⢦⡱⣍⠈⢿⡌⠀⣦⡆⠀⣠⡟⠀⢀⣞⡴⠋⣠⠏⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⠳⣄⠉⠛⠓⠻⠼⢶⣿⣧⡴⠥⠷⠞⠋⢁⣠⠞⠁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣙⠒⠲⠤⣤⣀⣀⣠⡤⠤⠴⠒⠚⠉⣀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⣤⣤⣤⡤⠴⠖⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⣿⠤
⠀⠀⠀⠀⠀⠀⠀⢰⠚⢦⢰⡖⣲⠀⠰⡇⠐⣇⠀⡇⠸⣛⠆⠸⠻⠆⠀⠀⠐⡟⠀⣦⠖⢠⡦⣆⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠉⠈⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⠴⣽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⠉⠉⠉⠙⠛⠒⠤⣄⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠖⠋⣁⡤⠴⠒⠚⠉⠉⠛⠓⠲⠦⣄⡀⠙⢦⡀
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⡠⠞⣡⠔⠋⣁⣠⡤⢶⣲⠖⢶⠒⡶⢤⣀⠀⠉⠲⣄⠙⢧⡀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡞⢡⠎⢀⡴⠛⣴⢣⡖⣸⠇⡿⢠⡏⣠⡄⣟⣙⣦⡀⠈⠳⡀⠙⣆
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⢠⠃⢠⡏⣼⠢⣏⢈⠔⠹⣆⡠⢻⠀⠛⣴⠋⣤⠘⢙⣦⡀⠙⣄⠘⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡟⠀⡁⢠⡟⣇⡡⠂⠼⠛⣦⡜⣉⢧⢈⠷⢦⣿⠀⢡⠟⣉⢈⢷⠀⠹⡄⠘
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⢿⠇⠘⠃⢸⣰⢉⠹⡜⢰⠇⡾⠸⠏⣼⢣⡞⣠⡽⣞⡏⠘⢋⣜⣀⡇⠀⣇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢠⡏⣼⠀⢸⠀⢸⡇⠛⢸⣇⣀⠎⣧⣠⠎⣫⣀⡴⢣⡆⡸⢧⣴⣩⢋⡈⣿⠀⡟
⠸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢹⠀⢸⡆⠈⡧⢴⠟⡉⢧⡌⡄⢹⡏⠙⢧⡇⠈⢠⢳⢣⡀⣿⠘⠃⡿⠀⡇
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠘⡆⠀⢧⠀⢧⡌⠸⠃⣿⠘⢁⣞⣇⠿⠸⢧⡴⢇⡇⠙⣐⢹⡀⣸⡇⢠⠇⢀
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢹⡄⠘⣄⠘⣧⣀⣼⠛⢶⣹⢠⡽⣦⣸⡏⣠⢸⢳⠞⡋⢹⠉⣱⠃⡼⠀⣸
⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢦⡈⢧⡘⢌⠓⠹⣎⠃⢱⣸⠃⠉⣸⠏⠈⣡⢋⡴⢁⡼⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠙⢄⠉⠒⠷⠦⣏⣦⣤⣿⣀⣜⣹⡤⠾⠋⢁⡠⠋⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣙⠲⠤⢤⣀⣀⣈⡉⣁⣀⡀⠤⠔⠚⠉⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⢤⣬⣭⣍⣀⣀⣤⠤⠶⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡧⠄
⠀⠀⠀⠀⠀⠀⠀⡶⠒⡆⣴⠒⡆⠀⢾⠀⢾⡀⣾⠀⣏⡇⠰⠞⡆⠀⠀⠐⡗⠀⣦⠖⢰⣠⡆⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣟⠉⠁⠈⠉⠁⠀⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣿⣿⣿⣭⣿

⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀⠀⠀⣠⣾⣿⣿⣿⣶⣄⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⢀⣴⣾⣿⣿⣿⣦⡀⠀⠀⣠⣶⣿⣿⣿⣶⣄
⠀⢀⣿⣇⣨⣿⣿⣅⣸⣿⡀⣼⣿⣀⣿⣿⣯⣨⣿⡆⢰⣿⣅⣸⣿⣿⣀⣽⣷⠀⣾⣯⣨⣿⣿⣇⣸⣿⡄⣸⣿⣄⣽⣿⣯⣀⣿⣧
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⣿⢿⣿⡿⣯⣽⡷
⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠈⢿⣮⣿⣿⣿⣿⠟⠁⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠙⢿⣾⣿⣿⣿⣿⠟⠀⠈⠻⣷⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁



⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣾⠆⠀⣶⠂⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠉⠆⠀⠀⠈⢦⠀⠀⠀⠠⢤⡤⠠⠐⠈⠀⢀⠊⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠑⠦⣀⡀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠉⠉⣭⡉⡁⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠌⠐⣿⣿⠈⢢⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⠀⠱⠀⡄⠒⢄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡀⠀⡀⠉⡿⠈⠁⠐⢊⣳⠂⠁⣀⡡⠑⠀⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⠀⠉⠈⠆⠄⠀⠈⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠤⢀⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠉⠒⠪⣑⠬⡒⢄
⠀⠀⠀⡔⠦⠤⠤⠤⠴⠶⠴⠖⠂⠐⠃⠀⠀⠀⠂⠒⡄⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⠀⠀⠑⢜⢆⠱⡀
⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢹⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⡀⠀⠀⠈⢫⡂⠘⡀
⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠠⠄⠒⠒⠃
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠄⠛⠛⠛⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⠤⠄⠀⠀⠢⢄⣀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠒⡦⠀⠀⡐⠲⠭⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⢀⠊⠀⠀⠈⢮⢢⡀⠀⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⡱⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠐⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣦⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⢻⡿⠷⢤⡀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠇⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⠁⠀⠀⠙⣷⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣀⣤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⠀⠀⠀⣹⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⢀⣤⠤⠤⠄⠀⠀⠀⠀⠀⠘⣆⣀⠤⠊⠁⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠳⡄⠀⠀⠀⠀⢀⣼⡋⠀⠀⠀⠀⠀⠁⢤⣤⡴⣾⡿⠇⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠛⠀⠀⢠⣾⣀⣀⣀⣠⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⣿⠰⡏⠀⠀⠀⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡼⠿⣏⠈⢁⣼⠻⢿⣿⣿⠟⠁⠀⠀⠀⣸⠟⠓⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⢀⣿⡷⠞⠛⢷⣄⣀⣀⣀⣀⣠⢴⡟⠁⠀⠀⢸⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠦⠀⠈⠕⢄⡀⠀⠠⠴⠋⠀⠀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠀⠓⠀⠀⠒⢾⣀⣀⣀⣀⣀⣀⣀⣴⣾⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣿⣻⣿⣟⣏⣽⣫⠅⠀⠀⠀⠀⠀⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⠛⠛⠛⠙⠉⠀⠀⠀⣀⠀⣶⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠀⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠲⣦⠄⡤⠀⠀⠛⠀⠀⠀⠀⠀⠀⢀⡔⠚⠁⠀⣀⡈⢇⠀⠀⠀⠀⠀⠀⠔⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⡎⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠐⠃⢙⣁⡀⠑⠢⣄⣀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠃⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠐⠒⠚⠛⠛⠛⠂⠀⠐⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔⠲⡌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡅⢎⡇
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠀⠀⠛⠀⠒⠒⠒⠒⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⢊⠴⠁⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠸⠃⠀⠈⡆⡔
⠀⠠⠀⠀⠀⠀⣰⣁⡍⢀⡴⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨
⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠓⠒⠒⠒⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡱⠀⠀⠀⠀⠀⡠⠂⠉⠉⡇⠢⡀⠀⠀⠀⠀⠀⡇⡤
⠀⠀⠀⠙⠔⠉⠉⠉⠉⠁⡜⢣⠈⠉⠉⠉⠉⠑⠦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢘⠀⠀⢀⡴⢆⠀⢸⠀⠀⠀⠀⢀⠈⡀
⠀⠀⢠⠖⠉⠉⢹⠙⠢⡀⠑⠊⠀⢀⠔⠋⠉⠉⠃⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠙⢦⡀⠀⠈⢹⠈⠁⠀⠀⠸⠛⠀⠀⢀⠔⠁⠀⢈
⠀⢀⠁⠀⠀⢀⠾⡀⠀⢡⠀⠀⢀⠃⠀⠀⢀⠼⣀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠦⣠⡈⢆⡀⠀⣀⠞⢸⡤⠔⠁⠀⢀⠴⠁
⠀⠀⠙⠂⠉⠁⠀⠈⢙⠈⠀⠀⠈⠛⡆⠈⠁⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠀⠉⠈⠉⠀⠁⠀⠀⢀⠔⠁
⠀⠀⠀⢆⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡔⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⡠⠒⠛⠂⠘⠓⠢⡀⠀⠀⠀⡤⠖⠛⠃⠘⠋⠢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣁⣀⣀⣀⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠊⠙⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠈⠈⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⡴⣔⢉⢩⡀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡔⠒⢄⠀⠈⠁⠒⢄
⠀⠀⠰⠢⢄⠀⠀⡰⠛⢤⡣⠄⠃⠈⢣⠀⢀⠀⢀⡆⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠘⠢⡊⠗⢄⠀⠀⠈
⠀⠀⢐⠀⠠⠃⠀⠑⠀⠲⡀⠁⡛⠀⠘⠀⠀⠀⡜⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠙⠦⠴⠃⡀⠜
⢀⠄⣈⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠑⡲⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠲⡀⠑⠄⣀⡠⠎⢠⠒⠄
⠣⠠⠆⠀⠀⠀⠀⠀⡠⠊⠉⠉⠓⢦⠀⠀⠐⠒⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢬⣊⠀⠀⠀⠀⠀⠀⢈⡓⠅⢠⣄⡀
⠀⢠⠄⠤⡀⠀⠀⠠⣁⣀⡠⠚⠤⣀⡇⠀⠀⠀⡠⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢠⠀⠑⢄⠀⠀⠀⡰⠋⠀⠀⢠⠀⠉⢆
⠀⠸⡀⠀⠉⠢⡀⠀⢘⡀⠀⠀⠀⡇⠀⠀⡠⠊⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⠴⠋⠢⣀⣈⠀⠀⠀⣇⣀⣀⠤⠊⠢⢀⣈
⠀⠀⠈⠢⡀⠀⠈⠒⠤⠵⠦⠤⠾⠤⠐⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡁⠀⠀⠀⠀⢨⠈
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣀⣀⣠⠃⠀⠀⠀⠀⡇⠑⣄⣀⣀⣠⠊
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠉⢢⠀⠀⠀⠗⠉⠀⠀⠀⠀⠉⠲⡀
⣀⣀⣀⣀⣀⣀⣀⣘⣤⣄⣀⣤⣤⣤⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣤⣤⣤⣤⣤⣤⣤⣤⣃⣀⣈⣀⣀⣀⣠⣄⣤⣤⣤⣃
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⢦⣼⢳⡀⠀⠀⠀⢀⡴⠋⠁⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡙⣸⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡴⠲⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⡇⣈⠵⠃
⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠈⢧⣡⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠉⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣯⠤⠤⠤⠤⠤⠶⠶⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣺⣿
⠀⠀⠀⠀⠀⠀⢸⠛⢦⣄⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⢀⡀⢀⡀⢒⡒⣒⠂⠀⣀⡴⠚⠁⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠳⢦⡀⠀⠀⣤⣀⡄⠀⠀⠈⠯⠟⠀⠘⢭⢟⡤⠞⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠶⣄⠈⠳⠁⠀⠢⠄⠀⠀⢄⡠⠖⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⣠⠴⠶⢤⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡟⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣙⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣟⣛⣟⣛⣻⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠄
⠀⠀⣠⣾⡿⠋⢁⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠖
⠀⣼⣿⣏⣀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣾⣟⠀⠀⠘⠛⠋⠀⣀⣀⡀⠀⠀⢀⣴⡿⠃
⠀⢈⣛⠻⢿⣿⣷⠀⢀⣴⡾⠿⣿⣿⠏⠘⣿⣿⠟⠛⢃⣾⡿⢃⣴⡿⢛⣿⡿⢀⣠⣾⣟⣀⣤⣾⡟⠁⣰⣿⠏
⣴⣿⣏⣀⣴⣿⣿⢾⣿⣟⢁⣼⣿⣃⣠⣾⣿⣃⣀⣴⣿⡟⣀⣤⠟⢿⣿⣭⡴⣿⣿⣿⠛⢩⣿⣏⣠⣾⡿⠃
⠻⠿⠿⠿⠛⠉⠀⠈⠿⠿⠛⠻⠿⠟⠉⠻⠿⠟⠋⠹⠿⠟⠛⠿⠿⠿⠛⢉⣼⡿⣿⡟⠀⠻⠿⢛⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣰⡿⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣴⠟⠁⠀⠀⠀⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠄⠀⡀⠁⠈⠈⡁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⠀⠀⠀⠘⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣤⠶⣤⣴⣶⣦⣤⣤⡄⣠⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⡀⣀⣀⣤⣤⣶⣫⡟⠉⠁⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⢰⣶⣆⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢟⣤⢶⣥⠿⣾⠿⠟⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡟⠋⢉⣌⡿⠛⣿⣧⡀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣀⣀⠘⠛⠋⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣸⣿⣧⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢹⠃⠀⠻⠟⣧⠀⠀⠀⠀⠀⣿⡄⠀⢸⣿⣿⣷⠀⠀⠀⠀⠒⠀⠘⠀⠀⠀⠀⢰⣿⣝⣻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢁⣀⠀⠀⢀⠘⡄⠀⣠⣄⡸⠀⠀⠀⠀⢠⡿⣿⠇⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠸⡄⠀⠀⢸⣿⡿⠒⢿⣿⡅⠀⠀⠀⢀⡟⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⣀⠀⠀⢰⡿⣺⡶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠀⠀⠀⠉⠀⠀⠀⢻⣷⠀⠀⠀⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠘⠂⠀⠈⣁⡀⠐⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠈⠲⣯⣴⠂⠀⠀⠀⠀⠀⠈⠿⠿⠿⠃⠀⠒⠀⠄⠀⣀⡀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠰⠙⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠺⠿⠆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⡞⢿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠴⡛⠢⣄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⡴⠀⠀⠀⣾⠀⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠓⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠇⠀⢼⣿⠀⠀⠆⢀⣄⠐⡇⣤⣷⣄⢀⡴⠞⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⢀⠀⠀⠀⠈⠉⣩⣥⣄⠀⠀⠙⡀⠀⠀⣿⡿⠛⢇⣸⠿⡻⠎⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⢳⣄⡄⠀⠀⠻⠿⢏⣠⣤⣤⣷⣄⣸⡿⠀⢀⡼⢛⣊⠁⢀⡠⠦⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⢀⣘⡁⢿⡆⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧⣠⢛⣷⢿⡿⠇⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠀⠀⠀⢀⣤⣤⣶⣦⣤⣈⠛⠃⢨⠿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠁⠀⠀⢠⣦⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣆⢀⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣦⣀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠈⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣩⠙⠗⢸⣿⣌⣿⣿⣿⣿⣿⣿⣷⣿⣿⣄⣀⠞⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣾⣏⢲⠿⣤⠛⣹⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣴⡆⠉⢿⣿⣿⣿⣿⣿⣟⣥⡮⣶⣿⣿⣿⠾⣥⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠨⡅⠀⠀⠙⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣦⣾⣿⣿⣿⣿⡟⠀⠀⠁⣰⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠁⣙⡏⠉⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⠇⣿⠀⠀⠀⠀⣴⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⢹⡟⠹⣿⣿⣿⡷⠶⣿⠀⠀⠰⣤⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⢠⣶⣀⣦⣄⡀⠀⠸⡇⠀⠀⠀⠀⢸⣀⣤⡏⠹⣿⡀⠀⠛⠀⠀⠀⠁⠈⣹⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⣻⢿⣩⠿⢃⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣏⣁⡇⠀⠉⠀⠒⠤⣀⣀⠀⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠺⠶⠇⠾⠣⠼⡟⠛⠁⠀⢸⣧⠀⠀⠀⠀⢸⡙⠻⡏⠀⠀⠀⠀⠀⠈⢙⣟⠛⠠⢶⢶⣄⣀⠀⣾⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠐⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠧⠀⡇⠀⠀⣦⠀⠀⠀⠛⠛⠀⠀⣘⣆⡀⠈⠹⠿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣧⠀⠀⠈⢷⡀⠀⠀⠀⠀⠘⠿⣿⠁⠀⠀⠀⢙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡷⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠈⠲⣄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠓⣦⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⡇⠈⠰⡻⣁⡗⠀⠀⠀⠀⠻⡿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⢇⠀⠠⣿⡇⠀⠀⠀⠀⠀⠐⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⠀⢀⢀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡄⢠⡬⠙⠫⣅⣆⡞⣷⣽⠟⣶⢶⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⣁⡘⠖⣣⣞⣽⠿⠛⢟⣧⣿⣿⣷⡵⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⢏⣿⠿⢿⠟⣹⡿⠠⣖⣶⣾⡿⠛⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢿⣿⣿⣿⢿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⣀⠀⠀⠀⠀⠀⣀⣀⢀⡀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⣄⣀⣀⣀⣠⣀⡀⠀⠀⠀⠀⢀⠄⣠⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠑⠲⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠦⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠚⠙⡆
⠀⢀⡤⢄⣾⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣤⠴⠊⠁
⠀⣏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠖⠒⠒⠒⠛⠳⢤⡀
⠀⠘⠦⠤⠤⣴⠇⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠤⠖⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⢀⣠⡤⠤⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⡷
⠀⠀⠀⠀⠀⣀⣉⣳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠃
⠻⣏⠉⠉⠉⠉⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢸
⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢦⣀⣀⡀
⠀⠀⠀⢳⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡠⠚⠉⢧
⠀⠀⠀⢸⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⠊⠀⠀⠀⢸⡀
⠀⢀⣠⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠸⡇
⢰⠋⠁⠈⠁⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠸⣆⠁⠀⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠤⠶⠶⠛⠓⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠶⠒⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣰⡥⠤⠖⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣦⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⢴⠁⠀⠈⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠘⢿⠏⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⢀⠁⠀⠢⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⠑⠒⠀⠘⠋⠀⠀⣀
⠀⠀⠀⠀⠀⣰⣿⣿⣷⣤⣸⣿⣿⣿⠏⠀⠀⠀⠀⠁⠀⠐⠄⠈⠀⢀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⣴⣶⣶
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠈⠁⠀⠀⠼⢀⠀⠀⠀⠀⡐⠁⠀⢀⠀⠀⣀⣤⡀⢰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣄⠐⠀⠐⠁⢀⡀⠑⢄⡀⠄⠙⠄⠀⠀⠀⢰⣿⣿⣿⢀⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⣸⣿⣿⣿⠟⠀⠀⠀⠀⠘⣿⣦⠀⢁⣴⣶⠃⠀⠀⣀⡈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠛⠁⣰⣿⣿⣿⣷⣄⠈⠉⠈⠈⠛⠁⠀⠀⢸⣿⣿⡆⢈⣡⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠛⠉⠀⠢⠀⠀⣿⣿⣿⣿⣿⡿⠀⢀⠀⠀⠀⢠⣿⣿⡆⠻⣿⠇⠛⠿⢿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣤⣄⡑⠀⣿⣿⣿⣿⠟⠁⢀⠂⠀⣠⣶⣄⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⢀⡀⢻⣿⡿⢀⠝⢷⣾⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⡿⢀⠹⣷⣤⣤⣄⠉⠛⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣥⣀⡀⠙⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏

⠀⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡋⠀⠀⠀⣀⡤⠤⠤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⠑⢦⡀⠀⠀⢀⡤⠚⠁⠀⠙⢦⠔⠋⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠳⠔⠋⠀⠀⠀⠀⣰⠃⢠⠴⠒⠲⢄⡤⠖⠒⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠳⡄⠀⠀⠀⠀⠀⠀⣠⠞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣴⢋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⠀⠀⠈⠢⣀⠀⠀⣠⠞⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄⠀⠀⠈⠳⠚⠁⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠤⠤⠴⠖⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣢⣤⣤⣤⣭⣭⣬⣶⣤⣄⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠏⣀⣤⣤⣀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣇⠞⠉⠀⠈⠉⠀⠀⠀⠀⢀⡛⠟⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⣠⠤⠦⣄⠀⠀⢀⣴⣾⡵⡶⢶⣤⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠃⠾⢷⡾⠀⠀⢸⣿⣿⣔⡚⣧⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡍⠈⠁⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⢿⣿⠀⠀⠀⣺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠈⢹⣿⣿⣧⠀⢤⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠶⠖⢒⡚⠿⡿⠂⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢿⣿⣰⡆⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠃⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠀⠀⠀⠈⠛⠛⠛⢹⡟⣸⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡌⣰⣿⣿⣿⣿⡿⡙⣊⣿⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠚⠁⣿⣿⡟⠁⠀⠀⠀⠀⠈⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⡀⠀⠀⠀⠀⢀⣀⡞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⣻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⡿⣷⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠧
⠀⠘⡆
⠀⠀⢃
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠇⠀⠀⠀⠂⠤⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣶⣾⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⣾⣼⣷⣿⣼⣧⣶⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠋⡇⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣄⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠤⡄⡂⠀⢀⢀⣀⢄⡀⡀
⣀⠀⠀⠘⠆⠀⠀⠀⠰⣀⠀⠀⢢⣤⣤⠀⠀⠀⠀⠸⡜⠀⢀⡔⠀⠀⠀⢸⣙⣯⣸⣶⣠⡼⡈⠆⣄
⠈⠁⠀⠀⣀⣢⣄⣠⠁⠉⠤⠄⠀⠉⠁⣀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠉⠘⢳⢸⢟⡹⠿⠂⢫⡷⣄
⠀⠀⠀⠰⣿⢿⡿⣿⡷⠀⠀⠀⠠⠀⠚⠁⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠰⠂⠘⠀⠀⠋⠣⠒⠂⠞⠣⠼⣉⣦
⠒⠒⠈⠙⢻⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⢻⡖⣊⡥⣘⠃⠐⠿⠷⠘⢴⣿⠂
⠀⠀⠀⠀⠐⠛⠛⣧⠀⠒⠲⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠊⠥⠶⡼⣅⠀⢀⠂⣰⣆⣼⠋
⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠈⠁⣿⠧⠼⠛⢙⡹⠃
⠀⠀⠀⠀⠀⡀⠀⢠⣀⡘⠀⠀⠀⠀⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣄⣀⣀⣁⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠜⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠁
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣠⠄⣦⠀⠀⢠⡄⡤⢠⡄⣀⣀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠘⣧⠀⢸⣿⠇⣾⣸⢛⣿⣯⠤⠤⢤⡄⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢸⣇⣼⣿⣤⣿⣯⣾⣯⡶⠛⠛⠿⠃⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣸⣷⡾⠋⠉⠁⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣩⣄⣠⣤⣿⠏⠀⠀⢀⣴⣶⣿⣷⣶⣦⣼⣿⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⢸⣿⣿⠏⠀⠀⢰⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠘⣿⣿⠀⠀⠀⢸⣿⣷⣶⣶⠀⠀⢰⣶⣶⣼⣧⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⣿⣷⢀⡴⠒⡻⢿⣿⣿⡏⠀⣀⣿⣿⣿⣿⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣤⡶⢛⡏⣜⠻⠷⣼⣿⣣⣿⢣⠞⠁⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠋⠁⣴⡿⡀⠀⠈⣿⣿⣿⢹⡄⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢋⠰⠀⢀⣼⡟⠀⠋⠀⢈⠁⠀⢹⣧⠹⢦⣄⣀⡀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⠀⢀⡴⢾⠉⡇⠀⠀⢐⡤⠤⡖⢹⢇⣴⢸⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡤⠎⢀⡟⠀⠃⢀⣀⣉⣀⣠⡴⢋⡾⢻⢸⣄⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠃⢨⠀⠀⣀⣴⠟⢉⢟⣴⢿⠁⡌⡇⠈⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⢀⡔⠛⡾⣱⠋⠀⢏⣴⣥⣿⡍⠂⣿⠇⠤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠻⣿⣿⣦⡀⠀⠀⡞⡰⠋⠀⢀⡾⠃⣼⠘⠃⣾⠟⠶⠀⠀⠀⠀⠉⠙⠲⣤⡀⠀⠀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣊⣀⣤⡤⠶⠚⠋⠁⠀⠀⠶⠿⣿⣿⣿⣶⣾⣾⠁⠀⠴⡏⠀⡰⣇⣠⣾⠋⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠻⢦⣜⣿⣿⣿⣿
⠿⠟⠛⠛⠛⠋⠉⠁⠀⠀⠀⠤⠤⣤⣄⣀⣴⣾⣻⣽⠻⣿⣿⣿⣄⠀⠘⠁⣈⣰⣿⢟⣹⠷⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⢷⡈⠻⣿⣿⣿⣶⣾⠿⠋⢱⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠛⠓⠲⣦⣤⣧⠴⣏⠳⡖⢮⣝⠛⠉⢀⣠⡤⠌⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣉⣀⣀⣉⣉⣁⣈⣉⣩⣭⣽⠷⢮⡳⠬⢭⣭⣭⣭⣍⣀⣀⣤⠤⠶⣂⣤⡴⠾⠛⣛⣉⣭⣭⣿⣿⣶⣄
⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣄⣂⠀⠀⠉⠛⠒⠶⢤⣤⣀⣈⣡⣴⠶⠛⠍⢡⣪⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣷⣦⣤⣄⣠⣠⣌⣭⣭⣠⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⣦⠈⣃⡀⢀⣀⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣉⣀⣀⣀⡁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡤⠃⡞⠁⠉⠋⠀⢱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣋⡐⢿⣿⣿⣿⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠀⠙⠒⢦⡀⣀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⠀⣤⡄⣰⠀⢳⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⡄⠈⠒⢻⠉⠀⠉⢻⠒⠋⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣋⣼⣿⣿⣿⣧⣙⣿⠆
⠀⠀⠀⠀⠀⡜⠀⣠⠴⢒⡋⠀⠀⠀⠘⣒⠶⢤⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⠸⣿⣿⣿⣟⠿⣿⣿⣿⣿⠇⣷⡀
⠀⠀⠀⠀⡼⠁⠀⠳⠤⠥⠭⠷⠶⠶⠭⠬⠤⠜⠁⠀⢱⡀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣤⣤⣶⣶⣶⣦⣤⣶⣾⣿⣷
⠀⠀⠀⠀⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⣀⣀⣀⠀⢀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡈⠉⢹⣶
⣿⠩⣿⡇⠹⢫⣷⠶⠄⠀⠀⠀⠀⢀⠀⠀⣾⣶⡂⠀⠐⠀⠀⠀⠀⠀⠀⠀⢶⣦⢀⣷⡐⠀⣆⣡⣤⣿⣿
⢿⣦⣾⡆⠀⢈⢿⠐⢿⡁⠀⠀⠆⢻⣤⣤⣼⡏⣃⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣟⣱⡈⣷⠀⣿⣿⣶⣿⣿
⢋⣾⡹⣋⣿⣿⣿⠀⣧⣧⠰⠀⣶⢀⣸⠃⢘⣇⡇⠀⠀⢰⠀⠈⠀⢷⣿⢻⣿⡇⢿⣿⣿⠀⠿⡻⢿⣿⣿
⡟⣿⣿⣿⢾⣿⣍⠢⢿⣿⡗⣃⡟⠈⠻⠋⢻⣿⢰⠀⠀⢰⠀⢠⡄⠈⠋⣸⣿⡇⣼⣿⣿⠋⢠⢸⣗⣾⣿
⣾⣿⣿⣿⣼⣿⣿⢹⣿⣿⣿⡇⣩⢻⣷⣼⣾⣷⢺⠀⣷⣾⣇⠈⠇⢠⡀⢾⣿⣟⣿⣿⣿⢃⣽⣰⣼⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢸⣿⣷⣾⡇⣿⢸⣿⣿⣿⡿⢸⠞⣿⣿⣧⣸⡇⣹⣿⡿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿
⣛⣻⣿⣻⣧⣿⣿⣸⢿⣿⣿⡇⣿⣹⣿⣿⣿⣧⣾⣹⣿⣿⡗⢻⣧⣼⣿⣟⣿⡇⣿⣿⣿⢸⣿⢹⣿⣿⣿
⣌⣽⣿⠿⣿⢘⣻⣽⣶⣿⣿⡕⣿⣽⣿⣿⣿⡿⣿⡟⣿⣿⣇⢸⣿⢸⣿⣿⣿⣧⣿⣿⣿⢸⣿⢿⣿⣿⣿
⣦⣿⣿⣶⣿⣾⣿⢹⣿⣿⣿⠀⣿⢻⣿⣿⣿⣇⣿⣇⣟⣿⣷⢸⣯⠸⣿⣿⣾⣿⣿⣿⣿⢿⣿⣼⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⢼⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⢼⣿⠀⣿⣿⣾⡟⣿⣿⣿⣸⣿⢿⣿⣿⣿
⠉⠛⠛⠛⠻⠟⠻⠟⠻⢿⡿⢸⡯⣿⣿⣿⣿⣟⣿⡏⠿⢿⠏⢸⡿⣆⠿⠿⠟⠇⠻⠃⠋⠸⢿⣼⡟⡿⢿
⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠠⠀⠀⠀⠀⢘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣶⣾⣿⣶⣴⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⣤⣤⡬⣅⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣤⣽⣷⣬⢳⡀
⠀⢀⡞⠋⠁⠀⠙⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⣟⡿⡯⣧⡟⣧⠰⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣯⣿⣏⣻⢀⣿
⠀⣾⡇⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣦⣟⣷⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢜⢿⣿⣿⣿⡶⢋⣾⣿⠁
⠀⣿⣯⡀⠀⠀⠀⣠⣾⣿⡟⠀⠀⣼⠷⣭⣓⠶⣤⣥⣀⡀⡀⠠⠘⢮⡷⣿⣽⣯⠏⡘⠀⠀⠀⠀⠀⠀⠀⢬⡳⢿⣯⡵⠞⢋⡽⠋
⠀⠘⢿⣿⣿⣾⣿⣿⣿⠟⠁⠀⠀⢻⠀⠈⠉⢻⡶⣶⣦⣴⡾⠀⠐⠢⠍⠉⠩⠅⠊⢀⠦⠀⠀⠀⣴⠟⢉⣾⠃⠀⠉⠉⠉⠁
⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣵⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⢸⡀⠀⡇⡀⠋⢠⡾⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⣀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠸⢷⣼⡅⠀⣰⣯⠈⠁
⠀⠀⠀⣤⡠⠄⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣆⡀⠀⠀⠶⢋⣾⠃⠀⠀⠈⣿⣿⣿⣿⣁⢁⠀⠀⠀⠐⣲⡶
⠀⠀⠀⢳⠙⢷⡂⠀⠀⠀⣾⠃⠀⠀⢀⡴⠛⣉⠄⠘⠉⠛⢿⣷⣤⡄⣠⡿⠁⠀⠀⠀⠀⠛⠙⣿⣿⣭⣤⣤⠤⠴⠛⠋
⠀⠀⠀⠈⠃⠀⠻⣧⣰⣷⣿⠀⠀⢠⣿⣴⣿⠉⠀⠀⣀⣤⣾⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠁⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢄⣮⣿⣿⡏⠄⠀⠸⡿⠻⣯⣀⣰⣾⣿⠟⡛⡛⠛⠻⢷⣶⣶⢦⣿⡀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⢿⣶⣄
⠀⢠⣤⣤⠤⠔⠒⠛⠋⠉⠂⠀⠀⠀⠹⣦⡈⠛⢿⡿⣇⢺⣽⣻⣿⡟⣦⣬⣫⡀⠀⠀⠀⢀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠛⠻⠿⢇⠘⠷⣍⠻⠷⣿⣿⣻⠇⠀⠀⠀⠀⠈⠁⠸⣿⣿⣿⡄⠀⠀⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠈⢻⣶⣶⣶⣶⡞⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠭⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢴⠦⡀⠀⠀⠀⠀⠀⠀⣀⢴
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠦⠤⠤⠤⠤⡾⠃⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⢀⡀⠀⠀⠀⠀⠀⡼⢄⣀⣀⡠⠤⠔⠒⠋⠉⠉⠉⠉⠑⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠽⣧⢀⠈⡄⢉⠁⢤⡼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⣹⢦⡉⠉⣁⡴⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢣
⠀⠀⠐⢏⠉⠉⠀⠀⠉⢻⠀⠈⠉⠁⠀⠁⠀⢠⠊⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠈⡆⠀⠀⠀⣠⠬⢧⡀⠀⠀⠀⠀⠀⢎⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⢡⠀⠀⠀⣏⡉⠉⠉⠉⠑⠒⠒⠂⠘⢄⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠳⡄
⠀⠀⠀⠀⠀⢣⡀⠀⠉⠓⠉⠉⠀⠀⠈⠉⠉⠉⠙⠓⠤⣀⠀⠉⠑⠒⠒⠀⠀⠀⠀⠀⠀⠀⣀⠚⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠤⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⠠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠤⠄⠒⠒⠒⠂⠤⠤⠄⠀⠠⠤⠤⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣠
⠀⠀⠀⠀⠀⠀⠈⠙⠛⡀⢀⣆⠀⠀⢠⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿
⠀⠀⠀⣴⡿⠃⠀⠀⢸⣷⣾⣿⣆⠀⠀⠙⢿⣟⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠⣿⣿⠏⠃⠀⠀⢰⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⠟⠦⠀⠀⠀⠉⠻⣿⠀⠀⠀⠀⠀⠀⣠⡀⢀⡠⢽⣿⣿⣿⣶⠃⠶⠀⠀⠊⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣼⣿⡿⠀⠀⠀⠈⠻⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠘⣿⡿⠆⢈⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⢀⡠⣴⣾⣿⣿⡆⠀⠀⢀⣴⣶⣾⣀
⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠀⠻⣿⠋⠀⠀⠀⠀⠈⢹⣿⣿⣿⡿⠀⠀⢴⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⠙⣻⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⣿⣏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠟⢻⠿⣇⠀⠀⠀⢹⣿⣿⠿⠏
⠀⠀⠀⠀⠁⠛⠳⢿⡗⣍⠙⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⣾⠇⠉⢀⣀⣐⡤⠀⠀⠀⣶⣶⣧⡀⠀⠀⠀⠀⠈⠑⢦⣠⣴⡟⠤⠀⠤⠤
⠀⠀⠀⢤⡞⣤⣠⣿⡟⣿⠴⠆⠀⠀⠀⠀⠀⠀⣺⡇⠀⠀⠀⢙⣄⠀⣻⣷⣶⡃⠀⠀⠉⣹⠿⣿⡟⠳⠂⠀⠒⠒⠉⠁⠀⠈⢧⡀
⠀⠀⠀⡄⠁⣉⣁⣽⡗⣿⢠⡆⠀⢀⠀⠀⢀⠀⣜⡁⠀⠀⠀⣿⣿⠓⠄⣸⡁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⣀⣤⣤⠀⠉⠒⠦⠤
⠀⠀⠀⠀⠜⠉⢁⣼⡇⠙⠛⠛⠂⠀⠀⠀⢀⡉⣿⣇⠈⠀⠀⠉⠀⠀⠀⠉⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡘⠻⣿⣏
⠀⠀⠀⠀⣶⣀⠈⣿⣿⣷⠶⡇⠀⠀⠀⠀⠀⠀⣼⡏⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡶⢤⠈⠉⠀⠤⢀⠚⠈
⠀⠀⠈⠀⣹⡿⠦⢀⣿⣷⣾⡷⠄⠀⠀⠀⠰⠶⣿⡟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⣤⢀⡰⣤⣽⣿⣿⡦⣤⡤⠄⠀⠀⢠⣦⣹⣿⣁
⠀⠀⠀⠈⠛⠃⠁⣨⣿⡷⢶⡦⠀⠀⠀⣀⠸⣿⣽⣿⣄⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⡀⡄⢷⣠⣄⠈⣀⡀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠊⠿⢡⠄⠙⠀⣻⡾⠏⠉⠀⠀⠀⠀⢺⠉⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⢀⢀⠀⠀⠀⠀⣠⠀⠀⠀⠁⢠⠀⠠⠞⡁⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⣶⣄⠀⠙⠁⠀⢀⡄⣸⡏⠐⣆⡀⢀⡀⢀⢠⡀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣃⣹⡧⠀⠀⡆⠀⣰⡇⠀⠀⢏⠳⠂⠀⠧⠀⠂
⢽⣿⣲⣶⣤⣠⣟⣍⣿⣇⣠⣼⣇⣾⣟⢠⢅⣿⠀⣠⡄⠀⢀⣐⣟⣀⠀⠒⠄⣈⣟⣷⣴⠆⣱⣾⣤⢽⢧⢖⣀⣾⣃⠀⢀⣰⡄⠀⣄⡀
⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡎⢃⣷⡾⣿⣧⡿⢾⣿⣾⣿⣿⣾⣯⣾⣿⣿⣿⣷⣿⣾⣿⣿⡏⣴⣷⣿⣿⣤⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣬⣯⣭⣤⣿⣶⣬⣿⣥⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣷⣾⣿⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠿⣶⣄⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢂⣬⣼⠿⣧⡠⠈⣶⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣺⣿⢟⣽⣿⣾⣿⣇⣼⣶⣌⠳⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣀⣿⠋⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣭⡿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⡅⣿⣿⣿⣇⣶⡿⠛⡙⣗⣸⣿⣿⣿⣿⣿⣇⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠘⣧⣿⣿⣿⡿⢷⣽⡿⣿⣿⣿⣿⣿⣿⣟⣏⣧⡀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⣫⣿⣫⡿⣡⣼⢻⡷⡉⢟⠏⢹⣯⡟⣿⡟⣿⢗⢦⠘⣷⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⡏⣿⣷⣿⡤⠧⠝⢳⣇⠁⠀⣧⡿⣹⣾⡿⣹⠏⢸⣴⠀⣿⡶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣄⡘⣿⣿⣿⣷⠦⣶⢿⣧⡀⢿⣵⣿⡟⢀⣿⢸⢈⢻⡄⣿⢣⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡿⢠⡯⠤⠤⣿⡇⢻⡀⢻⡵⣿⠟⠉⣰⣳⠋⢸⢸⢸⡿⡟⡟⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⣧⣸⠇⠀⠚⠋⠩⢶⣷⠱⣿⡿⠦⠞⠋⢩⠂⣿⡸⢸⡿⣿⠷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⣶⠛⢻⣿⠋⠀⠰⢛⣴⠀⠈⣀⠀⠠⢫⢃⠞⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡸⢠⣿⡧⢀⠄⠜⣼⢃⠰⠋⠡⢐⣥⣵⣽⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠌⢿⣷⣋⡼⠞⠁⣸⣋⣽⣿⣯⣾⡟⠁⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣾⡿⢧⣤⣶⣩⣴⢿⡏⣻⡧⢓⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡔⢿⣯⣇⡎⠄⡿⡟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡉⣿⡂⢹⣿⡙⢶⡇⠀⣿⢀⣀⡀⠀⣠⣄⡠⢤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣿⣷⡈⣻⣧⠀⣿⠻⠀⢹⡄⣾⢻⣿⣤⡙⠀⠀⠀⠀⠀⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣹⢿⡏⢿⢹⣷⢻⠀⠀⠘⢇⡏⠜⢹⣿⡇⠀⠀⠀⡠⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡶⢋⠎⡠⣿⢸⣷⡐⡷⢿⣎⢂⢸⣼⠋⠀⣰⠀⣽⢰⢀⣷⢫⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣅⠃⣸⢳⠛⣧⣸⡇⠳⡟⢿⣾⢺⡇⠐⣡⠋⠀⢈⣧⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡟⢔⣵⡟⠉⠹⡿⣧⠀⠛⣌⢻⡜⣿⠐⡉⠀⣀⣼⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠷⠋⠈⠙⠛⠾⣤⣽⡀⠀⠙⢮⣻⣿⣠⣇⣼⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡴⠶⠛⠛⠛⠋⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠽⠶⠿⠿⢿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡰⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢿⣿⣷⣄⠀⠀⠀⣠⣮⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠳⣿⣿⣿⣦⣴⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡱⡹⣾⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⣤⣹⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢘⡿⢿⣿⢣⠹⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣹⡗⠈⠘⣄⣃⣀⣾⡿⠿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢎⠀⠓⠀⠀⠀⢀⡀⠀⢀⠀⠈⢀⣀⡴⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⢆⡑⡄⣠⢀⠀⠀⠀⠦⠀⢠⣰⠞⠀⠀⠀⢀⡤⠀⠠⣶⠀⠀⠀⡀⠀⣀⠀⠀⠈⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣾⣮⣵⣼⣄⣩⢄⣄⣴⡏⠀⠀⠠⠚⠉⢁⣹⠷⠿⠀⠾⡦⢳⢦⣙⣃⣠⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠹⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠲⠤⠀⡀⠉⠁⢈⣤⡑⣄⠊⣴⠛⣥⡀⢀⠘⡆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡉⠻⣟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢐⣆⠘⢋⣥⡖⠀⢨⣍⠁⢉⡆⣠⠀⠉⠓⠉⠀⠛⠄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⣾⣿⣭⣽⠯⡤⢾⣿⣦⠀⠈⡉⡄⣠⣅⣰⠤⠺⢶⡤⣬⠿⠷⠾⢿⡧⠔⠲⠞⡤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣿⣿⣷⠀⠙⠿⠿⣋⠀⣴⣻⠆⢸⡇⢠⡴⠄⢀⣤⣶⡧⢠⠇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣣⡀⠀⠈⠻⣷⡎⡋⢁⠘⠋⢠⣽⡂⡀⠁⠀⣠⠎⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣢⣀⠀⠀⠁⠉⠛⠷⢦⠀⠈⡀⠀⠀⠀⠉⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡻⣿⣿⣿⣿⣿⣶⣶⣦⡤⢤⠤⠤⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠨⢿⣿⣿⣿⣿⣅⣀⡄⠐⡼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠓⠶⣤⡀⠀⠀⠀⢀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⣀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣸⢃⣠⠖⠶⠶⠶⠖⠛⠉⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠈⠙⢻⣿⠉⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⣴⠚⠛⠛⠛⠛⠛⠚⠛⠛⠛⠓⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠀⠀⠀⠀⠀⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⠀⠀⣿⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣸⠃⠀⠀⣿⠀⠀⢸⣇⣀⡟
⠀⠄⠀⠀⠀⠀⠈⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⣿⠀⠀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣿⣤⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠿⠤⠴⠛

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

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣤⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⠊⠁⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣤⣆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⡟⠉⠉⠉⠹⣿⣬⣿⠀⣏⠂⠄⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣯⡴⠛⠢⣴⡆⣤⡀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣅⠀⠀⠀⠀⣈⣿⣿⣿⡶⠀⢻⡗⠛⠀⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⣿⣿⣿⣿⣿⣧⣴⣏⠀⣦⣤⡼⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠀⠀⠹⠿⠛⠿⣿⣿⣿⣿⣶⣿⣯⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣃⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡁⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣿⡏⢻⡿⠗⠂⠀⠀⢀⣾⠛⢿⣿⣿⣾⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⣉⣀⢨⣿⣷⣶⠄⠀⠈⠁⠀⣸⣿⣿⡿⣿⣿⡇⢠⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢀⡙⣿⡟⢻⡍⠁⠀⢀⣠⣴⣿⠛⣿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣶⡆⢸⣿⠻⡯⠈⢻⣿⣿⣿⣿⣿⠟⠀⣿⣽⣿⣏⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡗⣀⡹⠶⠀⠋⢹⣿⣿⡟⠛⠁⠀⠀⠹⣿⣿⣟⢿⣭⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣿⣿⣿⡄⢀⣿⣿⠏⠁⠀⠀⠀⠀⡀⠀⢿⣿⣶⠟⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⠟⠃⠀⠀⠀⡀⠀⣉⣴⣿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣝⣿⣿⡟⠉⠆⠀⠀⠀⠈⢀⣺⣿⠙⢿⣏⢰⠄⠀⠈⢷⠄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⡟⢿⣿⣿⡏⢷⣼⡌⡇⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣀⢐⣄⣀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⡟⠀⠞⠻⡇⢁⡀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠀⠸⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡶⠾⠿⣿⣿⣿⣿⣿⣿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠭⣤⣤⣶⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣋⡉⠁⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟⣛⣛⡛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣤⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠛⠉⢙⣿⠋⠟⣻⣿⣿⣿⣿⡿⢿⣿⣿⣻⠿⢭⣭⣍⣙⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣤⠴⠂⠀⡞⢈⣿⡿⠿⢏⠿⠇⠀⠊⠀⣀⣀⠉⣻⣿⣿⣿⣍⠿⣟⠻⢽⣛⠲⢦⣬⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠁⡾⢠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠈⠁⣀⣀⠉⣿⣿⣿⣿⣧⡈⠳⠦⠈⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣵⣾⣷⡾⢀⣿⣿⣿⣧⠤⠶⠲⠋⠉⠙⠛⡀⠀⢩⣽⣿⣿⣋⢐⣻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃⣼⣿⣿⣿⣿⡀⠱⠈⠀⠀⠀⠀⠟⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢸⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠰⠃⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⣆⠈⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣥⣤⣾⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣉⣀⣁⣈⣈⣁⣉⣉⡉⣉⢉⣉⣉⣉⡉⣉⢉⡙⣉⣉⣉⣉⣉⣉⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣿⣿⣿⣿⣿⣯⣽⣯⣽⣯⣿⣿⣿⣿⣿⣿⣯⣭⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣭⣿⣯⣯⣭⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠏⠙⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢊⣿⣶⣄⠀⠀⠀⠀⠀⠀⠸⣂⣸⣅⣤⣄⡴⠀⠀⡄⢻⣋⣭⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡏⠀⢀⡜⢧⣬⡍⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⡇⢠⠘⠳⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠈⠁⠙⠋⠀⠀⠀⠀⠈⠉⠻⠿⠿⠿⠿⣿⠃⢸⠅⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠠⠆⠱⡆⢸⣿⠆⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠑⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠉⠉⠛⠤⠀⠀⣀⡀⢀⠀⠀⠀⠠⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣚⣓⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⢸⣿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⣿⣄⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⣹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣶⣤⡄⠀⠀⣿⡿⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢥⠀⠍⢻⣧⠀⠀⣤⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠙⠋
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠐⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠠⢀⠀⠀⡀⠢⠁⠀⠀⠀⠀⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⠢⠄⠀⢀⠄⠂⠀⠀⠈⠉⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠐⠀⡀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⡀⠀⠠⠀⠣⡀⠀⠀⠈⠀⠀⢀⠌⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠂⠈⠉⠁⠠⠀⠀⠑⠄⡈⠐⠂⠤⠀⢒⡡⠊⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠀⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⡀⠔⢨⠂⠀⠀⠆
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠄⡐⠀⠀⠀⠐⠒⠁⡀
⠀⠀⠀⠀⠀⠐⢄⢀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠎⠀⠀⠀⠀⠀⢑⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠂⠀⠀⢀⡀⠀⢀⡐⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⢉⠁⡀⠈⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⡋⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠠⠼⢬⣼⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣶⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⣻⡿⠟⠛⠛⠛⠿⣿⣿⠿⠿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⣤⣴⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⣾⡟⠀⠀⣀⣀⣠⣤⣬⣿⡇⠀⠀⠀⠈⠉⠙⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠚⢟⣻⣿⡇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠉⠉⠀⢀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣷⣶⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⢿⣷⡄⠀⠀⠀⠠⣶⣾⠿⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠘⠛⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣾⡿⠃⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣷⣶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠉⢹⣿⡟⠻⠿⠿⠿⠿⠿⣿⡿⠿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⠆⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣇
⠀⠀⠀⠀⠀⠀⣠⣴⡿⣿⣿⠉⠉⠉⠉⠉⢩⠟⠙⣿⡛⠛⠛⣿⠟⠛⠛⡟⠛⢻⡏⠉⠉⢹⡏⢹⣿⣶⣄⡀
⠀⠀⣀⣤⣶⠿⠋⠁⢀⣿⡏⠀⠀⠀⠀⣠⠏⠀⠀⣿⢳⡀⢸⠇⠀⠀⠀⡇⠀⠈⣷⠀⢠⠿⡇⠀⢿⣿⡉⠛⠿⣶⣤⣀
⣴⣿⡛⠉⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⢠⡏⠀⠀⠀⣿⠀⠹⣾⠀⠀⠀⠀⡇⠀⠀⢹⡶⠋⢘⡇⠀⠘⣿⣇⠀⠀⠀⠉⠛⣿⣦⡀
⠈⠙⠻⢷⣦⣀⠀⠀⣿⡏⠀⠀⠀⢠⣾⠗⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢨⡇⠈⢙⣿⣿⡀⠀⣠⣶⠟⠛⠉
⠀⠀⠀⠀⠈⢻⣧⢸⣿⠃⠀⠀⠀⠀⠻⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⣼⠛⣿⣧⣿⠋
⠀⢠⣦⣴⣠⣾⠏⣼⡿⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃⣼⠃⠀⢿⣿⣿⣄⣠⣄⣤
⠀⠲⣽⡿⠿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣬⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⢸⣿⡌⠻⢿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣿⣤⣤⣤⣤⣤⣤⣧⣠⣤⣤⣤⣤⣿⣤⣤⣤⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣟⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⢻⣿⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⡶⠶⠲⠶⠾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⠶⠖⠛⡻⠛⠛⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⣃⠀⠀⣀⣠⡀⠹⣦⠀⠀⠀⠀⠀⠀⢠⡾⠉⢠⣄⣀⣀⢀⣇⢀⣀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠉⢹⣧⣤⠿⠇⠀⠀⠀⠀⠀⠘⠷⠶⠿⠉⠙⠛⠛⠛⠛⠋⠉
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⡄⢀⡄⠀⠀⣠⣀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣾⡇⠀⠛⢿⣿⣅⡀⠀⠀⠹⠋⢀⣈⠁⠀⠐⠖⠀⠀⡀
⠀⠀⢠⡄⣠⡶⢀⣀⠉⠛⠛⠛⠛⠛⠀⠀⠀⠀⠻⣶⠶⠀⠈⠀⠀⠀⠐⠀⠀⠐⠁
⠀⠀⢼⣿⣿⣃⣈⠙⢿⣿⣿⠶⠲⠖⣠⣀⣤⡄⠀⠀⠀⠈⠚⠁⠀⢀⣀⠀⠂⠀⠀⡘⠀⠂
⠀⠀⠀⠉⠉⠉⠛⠂⠈⢹⡟⠀⠀⠀⠀⠉⠉⠀⠂⠠⠀⠀⢠⡶⢊⣥⠶⠒⠚⠁⠐⠲⠄⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⣦⠈⣥⠉⠀⠀⠀⢠⡏⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⢀⣿⢠⣷⣰⠐⠀⣄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠘⠀⠂⠀⠀⠀⠀⣿⠈⣿⣇⠁⢸⡿⠿⣆⠈⠀⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠄⠀⠀⠀⠀⠀⠰⣿⠀⠙⠟⠀⠀⢀⣼⠛⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠶⠀⠀⠀⠀⢿⣦⡀⠀⡞⠈⢸⠇⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣅⠀⡜⠀⠀⠠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⢰⠀⡄⣿⡱⣼⡿⣦⠈⠙⠷⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣚⡀⠀⠉⠁⠐⣿⡿⢨⣷⣄⣾⡙⠇⠁⢚⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⡏⠀⠀⢠⠣⢻⡷⡿⢟⣻⡉⡇⠀⠘⠁⠙⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣇⣄⠀⢈⡀⣾⠅⠠⠤⠤⠐⠐⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄⢻⣧⠇⠐⠐⢲⡖⠀⣼⡟⠀⣶⡄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣻⣿⣿⣿⢀⣾⠏⠀⠀⢠⡿⠁⣼⠟⠁⠀⢻⣇⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢺⣿⣿⡿⣿⣿⡟⠀⠀⢀⣆⢛⡼⠃⠀⢀⣀⠘⢹⡄⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣀⣤⣀⣘⠀⠀⠈⡿⠉⢰⣿⣿⠃⠀⢠⣼⡿⠈⠁⠀⢨⡉⢡⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠛⠿⠻⠏⠀⠀⠀⢀⡠⣾⠉⠻⡄⠀⣿⣿⡆⠀⡀⠀⢸⢇⡸⠌
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣴⠇⠀⠀⡠⣂⠔⣉⣧⢀⣤⣾⣿⣛⣤⣶⣥⡴⠚⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣀⠙⣉⣡⣿⣯⣀⠈⣁⣭⣤⣄⣀⠙⠿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠤⠾⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⣟⣃⠀⠀⠀⢀⡀⠖⠢⠆⠀⢀⡄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠿⠿⠿⠶⠦⢤⣤⣤⣤⣼⡿⠍⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠒⠀⠀⣠⣴⣾⠿⠿⠿⣶⡄⠀⠐⠀⠀⠀⣤⠀⠀⣠⡶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⡀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠈⣿⠠⠶⠶⠶⢶⡿⠾⠛⢫⣶⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⣿⣿⣿⣿⣿⣧⠀⠀⣿⡇⠀⠀⠀⢀⡀⢠⡏⢠⣾⠀⢀⣿⠃⠀⠀⣀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠀⠀⠻⣿⣄⠀⠀⠸⠷⠏⠀⣤⠁⠀⣼⡟⠀⠀⣼⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⣡⢀⣠⣤⣀⠈⠻⣷⣄⠀⠀⠀⣸⡏⠀⢰⣿⠃⠀⣼⣿⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⢠⡟⣵⡟⠉⠀⣹⠀⣀⡈⢻⣧⠀⢰⡿⠁⡰⢹⡟⠀⣰⣿⡇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠰⡟⠀⠀⣠⠏⣾⠋⠀⠀⣿⠀⢸⣇⡴⠁⠸⣧⠞⠁⢿⣇⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠙⠷⣤⡤⠞⠁⠀⠙⢧⣤⠾⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠆⠀⣀⣤⣤⣀⠀⠀⠀⢀⣤⡀⢀⠀⣦⣰⣦⣿⣠⡷⢀⡆⠀⠀⠉⢀⣾⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢶⡀⢠⣾⠟⠉⠉⠙⣷⡀⠀⢈⡛⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⡇⠀⣠⡴⠋⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⣇⠀⠀⠀⠀⢹⡇⢀⣾⡇⠀⠛⢻⣿⣿⣿⣿⠿⠏⠀⠙⠛⠁⠀⢠⣿⡇⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠻⣦⣀⠀⢀⣸⡗⣹⣿⠀⠀⢠⠀⣿⣿⣿⣿⠀⠀⣤⡬⠊⠻⠿⢿⣿⣷⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣿⡇⠀⠀⠉⠉⠁⣿⠀⣿⡇⠀⢠⠃⠀⣿⣿⣿⡟⠀⢰⣧⠀⠀⠀⠀⣴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣸⠇⠀⢿⣇⡴⠁⠀⠀⣿⣿⣿⡇⠀⠈⢿⣄⣀⣀⣴⡿⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢿⡇⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⣀⠀⠉⠉⠉⠀⠀⢠⠀⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⢿⣦⡴⠋⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄

⢿⢿⣷⣄⣀⠀⠀⠀⠢⡀⠀⠀⠀⢀⣶⣶⣶⣿⠿⢻⣿⣿⢿⣿⡟⢻⣿⣿⣿⣶⣤⣐⠀⠀⢰⠟⠀⠀⠀⠔
⢶⣅⠀⠀⠈⠙⠦⣀⠀⠀⣄⣴⣾⣿⣿⣿⡿⠛⠉⠀⢠⡄⠀⠀⠈⠙⠻⣿⣿⣝⠿⣯⣷⣦⡏⠀⠀⠀⠈⠀⠀⠀⣠
⠀⠈⠀⠀⠄⠀⠀⠈⣳⣾⣿⢿⣻⣿⡿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⡈⢻⣿⠎⠻⣾⠿⣦⡄⠀⠀⠀⢀⡔⠉
⣄⡀⠀⠀⠀⠀⢀⣼⣿⡾⠛⠻⣿⣿⣀⡀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠼⠃⣼⠟⠀⣶⣮⢻⣏⣻⣦⠀⠒⠁
⠉⠙⠒⠦⣄⢀⣾⣿⢏⣴⣦⡀⠈⠛⢷⣿⠂⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠚⢁⠐⠚⠻⣿⣷⠹⣯⣽⣦
⠂⠂⠀⡀⠀⣾⣿⡏⣾⣿⣿⠟⢛⣓⣀⡅⠈⠀⠀⠠⢐⣒⣤⣐⡒⠦⣀⠀⠀⠉⢀⠉⡀⠈⠀⠀⠘⣧⣼⣇
⠀⠀⢀⣀⣀⣿⣿⠀⠈⠻⠃⡀⠀⠐⠛⠿⠷⠀⣠⠊⠁⣾⣾⡋⠀⠑⢢⡄⢤⣶⠿⠿⡿⠂⠀⠙⣷⢹⣦⣿⠀⠀⡀
⠛⠋⠉⠁⠀⣿⣿⣿⣷⠄⢀⣷⠂⢀⡤⣶⢲⡖⣿⡥⣴⡆⢩⡄⢠⣶⣠⡶⢾⡶⣄⡀⠀⣀⠂⢠⣿⣸⣧⣿
⠄⠀⠂⠀⠀⢻⣿⣿⢱⠈⠉⠇⣰⣿⡳⢟⣿⠛⠛⠓⠻⣏⣯⡽⢯⡽⣯⣿⣞⣳⣟⣷⡀⠁⠀⠀⠛⢹⣧⣿
⡀⠀⠀⠀⠀⢸⣯⣿⠂⠣⣀⢠⡿⡞⣱⠋⣤⠦⠀⠀⠀⠀⠙⣻⠏⠔⢒⠦⠬⣍⡻⣴⡇⠀⠀⠀⢁⣾⣤⡏
⠉⠉⠉⠋⠉⠉⢹⣿⢧⡀⠈⢸⣻⣇⡏⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠠⠘⡩⡏⠀⠀⠀⢠⢷⡟
⢃⣀⣀⣀⣀⣀⣀⡙⣿⣷⣤⢸⣟⣿⡧⠀⠁⠠⠀⠀⠶⠶⠀⠀⠀⠀⠀⠄⠤⠐⠁⠀⠃⠀⢠⣶⣿⠟⠁⠀⠀⠀⠈
⡋⠉⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⢙⠻⠇⠀⠀⠀⠈⢠⣤⣤⣄⠀⠀⢠⣤⣤⣤⡄⠠⠁⠀⠀⠛⠉
⡷⠠⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠈⠖⠀⣶⡞⡡⠀⠀⠀⠀⠉⠀⠀⠀⢻⠿⢋⠐⠁⠀⠀⠀⠀⠀⠀⠀⠂
⡗⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠔⠲⠈⡀⠘⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡟⠒⠒⠖⠒⢳⡀⠀⠀⡋⠀⠀⠀⣿⡄⠂⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⠜⡷⠓⠉⠁⠒⡀
⣦⠒⠈⢠⠞⠀⠳⡄⠀⠧⠀⠀⠀⡿⢿⣆⠉⡄⠑⠀⠀⠀⠀⠀⠀⢀⡴⣫⠋⢘⠀⠀⠀⠀⠀⢸⣦⡀
⠁⠀⢀⠎⠀⠀⠀⢱⠀⠘⠀⠀⣼⠃⠈⣿⡇⠀⠂⡀⢰⣶⠀⠀⣞⠯⠛⠁⠀⡆⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣤⣤
⠁⠀⡜⠀⠀⠀⠐⢁⡀⠀⣠⡾⢃⣼⡇⢸⣇⢰⣄⡀⠸⠿⢀⣈⣴⠂⣴⠀⢰⡇⠀⠀⠀⠀⠀⣸⣿⠳⣬⣙⣛⣿⡿
⠶⣶⡇⠀⠀⠀⢀⣁⣴⠞⣋⣴⣛⡱⠃⣸⣿⡼⢦⡿⠀⠹⣯⠥⢻⡀⠈⠰⣌⠻⣄⠀⠀⢀⣼⣿⠿⢇⠒⣹⡿⠋⡀
⢳⣿⣡⣤⣶⣿⣿⣍⣀⠀⠀⠉⠁⣀⣴⣿⢧⣟⡾⠃⢠⡀⢌⠾⣋⣧⠀⢴⣿⠳⠈⠻⣿⡟⢿⣙⠚⢢⡾⢋⡴⢿⠁
⣾⣿⣿⣫⣵⠾⠛⢛⢛⠟⠿⠿⠛⣛⣉⣴⠿⠋⠀⣠⣬⣥⣤⡐⠈⠛⠷⣄⠉⠀⢠⣦⡄⠙⢦⣷⣵⣿⠶⣬⡙⢿
⣽⣛⣻⡿⢁⣶⡿⣛⣛⣟⣷⡄⠓⢿⠅⠀⠶⡆⣾⠛⣧⣤⡞⢻⡄⣤⡄⡠⢄⡄⠀⠁⠀⢀⣨⣥⣤⣤⣀⠀⠉⢧⣀
⣙⣛⣿⢱⣿⡏⢀⣤⣶⣭⡙⢷⣄⠀⢰⡷⡀⠀⠻⣟⣻⠶⣏⣹⢃⢕⠊⢠⠶⠆⡖⠀⠰⠋⣀⣤⣤⣬⠙⢿⡄⠀⣿
⣛⣛⡧⢸⣿⡀⣿⡁⢶⢈⡿⠀⣿⡆⢠⣼⡏⠀⠀⢈⣙⣒⠛⡱⡀⠀⢠⠧⣶⠀⡥⠀⣀⣼⡏⡴⣻⡻⣧⢸⡇⠀⢻
⣛⣻⡇⠘⣿⣷⣜⠻⠿⢟⣁⣾⡿⢁⢺⣋⣴⢠⠀⠀⢿⠏⢜⣬⠃⠀⡘⡦⣴⠀⠰⡆⠓⠹⣧⡳⠬⠷⢋⣼⠇⠀⢸
⣿⣻⡧⠀⠈⠛⠿⢿⣿⡿⠿⡋⢤⣼⢧⡜⡇⣸⢃⠀⢈⠄⣾⠙⣂⠀⣯⣅⣩⣡⣄⡀⣤⢠⡈⠻⠿⠿⠟⠁⠀⠀⢸
⣿⢿⡗⠂⣴⡿⠇⣤⣄⢻⣶⠣⣾⢮⣷⣿⢡⣯⣼⠀⠛⠀⢯⣩⠗⣦⠸⣎⣉⣵⣯⢴⣿⢋⣤⡆⠀⣾⡉⣻⠄⠀⣾
⣿⣿⣧⠁⢻⡇⠀⢄⡽⢋⣭⣼⢳⡞⡾⠁⠨⢶⠜⢀⣆⠀⠙⠶⣾⡷⠂⠙⠶⣇⣰⠶⣇⣸⡷⠃⠀⠈⢠⡟⠀⢀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢠⣶⡶⠤⠤⠤⢤⣄⣀⡀⠀⠀⣠⣤⡀⠀⠀⠀⣿⣿⣿⠛⠛⣛⣛⣛⣻⣿⠿⠿⣿⣿⡿⠛⠻⣿⣿
⠀⠀⠘⠻⢿⠀⠀⠀⠀⠀⠀⠈⡆⠀⣿⣿⣷⠀⠀⠀⣿⣿⣿⣤⣀⣿⣿⣿⣿⣿⣿⣿⡜⣿⡇⠀⠀⢿⣿
⠀⠀⠀⠀⡾⠀⢴⣶⣦⣶⣦⠀⡇⠀⢹⣿⣿⠀⠀⠀⣿⣿⣿⣿⡏⣿⡟⠉⠙⠋⠉⣿⡇⣿⣷⠀⠀⢸⣿
⠀⠀⠀⢠⡇⠀⠈⠻⣿⠿⠃⢰⠃⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⢳⣿⣿⣦⡀⢀⣴⣿⢸⣿⣿⠀⠀⣾⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⠇⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⣠⣿⣿
⠀⠀⠀⠸⡄⠀⢠⣤⣤⣤⣶⣾⣷⡆⠸⠃⠀⠀⠀⠀⣿⣿⣿⣿⡬⣷⣿⠿⠛⠛⠛⠋⠈⢙⣿⣴⣿⣿⣿
⠀⠀⠀⠀⠙⠤⠴⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣝⣛⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠀⠀⠉⠉⠉⠉⠉⠉⠐⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠤⠒⠂⠁⠀⠀⠐⠀⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠡⡍⠀⠀⠀⠀⠀⡰⠋⢁⠀⠀⠀⠉⢛⠂⠸⡄⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠄⠀⠀⠀⠀⢵⡆⠠⠀⠐⠰⠆⠉⢨⠃⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡠⠊⢀⡔⠒⠤⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠄⠀⣷⠀⠀⠀⠠⢊⡠⠖⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⡌⠁⠀⢀⠂⡀⠀⠀⠀⠁⢀⠈⣄⢀⡟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠴⠀⠸⡀⠀⠈⠀⠀⠲⢤⠤⠴⠶⠞⠁⡼⡧⢄⣀⢀⡤⢄⡀
⠀⠀⠀⠀⠉⠉⠙⠋⠀⠀⠀⠀⢁⠀⠳⣄⡀⠀⠀⠀⠀⠒⠒⠈⠁⠰⠁⠰⠀⢠⠋⠀⠀⠛⠦⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡲⢤⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡌⠀⠀⠀⠀⡤⠙
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢔⠢⠔⠒⠋⠰⣤⡀⠀⡇⠀⠀⠀⡼⠁⣤
⠀⠀⠸⡄⠀⠀⢆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⢳⡀⡠⠧⠄⠁⠈⠑⢳⠄⠀⢠⠁⠀⡟
⠀⠀⡐⠀⠀⠀⠀⢣⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠠⢄⡀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡎⠀⠀⡇
⠀⢠⠃⠀⠀⠀⢴⣼⠀⠀⠀⠀⠀⠀⠀⠠⠀⠢⠀⠀⠈⠉⠢⣤⡀⢸⡀⠀⡠⠂⠀⢰⠇⠀⠀⡇
⠀⠘⠀⠀⠀⠀⣀⡹⠇⠀⠀⠀⠀⠀⠀⢀⠃⠈⠀⠀⠀⠀⠀⠰⡌⠀⡉⠈⠀⠀⣠⠏⠀⠀⢰⠃
⢠⠀⠀⠀⠀⠀⠀⠙⡏⢀⡀⠀⠀⡀⠔⠁⠀⠀⠠⠤⠄⢀⠀⠀⢣⠀⠈⠐⠒⠋⠁⢠⠀⠀⡟
⡇⠀⢀⠀⠀⠀⠀⠆⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⡄⠀⠀⠀⠀⠀⣸⠀⠀⢻
⢻⣇⢸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⡇⠀⠀⠀⠀⢠⠇⠀⠀⢎
⠀⠹⣾⠀⠀⠀⠀⣇⠠⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⢀⡾⠀⠀⠀⡂
⠀⠀⠘⢇⠀⢀⡤⠂⠀⢠⡀⠀⠀⠈⠑⠢⠄⣀⣀⣀⣀⣸⠀⠀⡀⠀⠀⠀⠀⢸⣇⣀⡢⢾⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⣾⠀⠀⠀⠀⡆
⠀⡠⠄⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁
⠀⢄⣀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡞⡄⠀⡰⡄⠀⣴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⡀⢸⠈⢦⡜⠀⠸⡔⠁⠳⠜⠰⣆⠌⣽⠀⣠⢶⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠢⣄⣸⠀⠈⠃⠀⢀⡀⣀⣤⢤⠤⡄⣤⣀⣉⠰⠸⢞⢵⢙⣀⢔⣹⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢤⢄⣀⢘⡀⠀⠉⣀⢤⣲⠬⠗⠓⠛⠊⠉⠉⠉⠉⠛⠛⠿⢵⣳⢥⣁⠕⡍⣺⠤⡪⣭⠀⢀⡀
⠀⠀⠀⠀⠀⣀⡀⠈⣆⠀⠉⠁⣠⣺⠝⠋⢀⠀⠀⠴⠒⠀⠰⡀⢰⠙⠂⠒⡦⠄⠈⠙⠺⢽⣲⢬⣎⠆⢷⢊⢵⠃⠀⡀
⠀⠀⠀⠀⠀⠩⡈⠉⠉⠀⣠⣺⠞⠁⢠⡀⠛⠤⢄⠘⡎⠀⠏⠙⠘⣤⠄⠀⡁⠀⢀⠁⡠⣄⠈⠑⢮⣗⢭⡑⡿⡒⡹⠃
⠀⠀⠐⢖⢖⠒⠚⠀⢀⣞⠞⠁⡠⡂⠀⠑⢄⠀⠚⠀⣐⣠⡤⡤⡴⡶⡤⣤⣀⠀⠁⠸⡀⠎⣠⡆⠀⠈⠻⣟⣮⢚⢗⠒⣨⠃
⠀⣀⣀⣠⣱⠄⠀⢠⣻⠊⠠⣄⣀⡈⠳⠄⢠⢠⢵⣻⠝⣚⢛⢉⢉⢍⢉⢉⠛⠿⣶⣄⡀⠐⢀⡧⠔⠁⠀⢊⢾⢧⡣⠺⠧⠤⢤⠄
⠀⠉⠢⡂⡀⠀⢠⣷⠃⡠⠀⠘⢮⡉⠁⢀⡶⡧⠋⠀⠀⠀⠈⠀⠈⠈⠀⠈⠈⠑⢌⡙⢽⣦⠀⠀⠀⠀⠀⠀⠈⢯⢧⢕⢵⣟⣁⡀
⢀⡤⠄⠞⠂⠀⣞⠃⠀⠁⠈⠀⢹⠀⠨⣸⡿⠁⢀⣀⠀⠀⣠⠤⣀⠀⠀⡴⡖⣄⠀⠀⠈⢻⣷⠀⠀⠀⡶⡀⠀⠼⡝⡆⢺⢀⡶⠞⠁
⠀⠑⠪⣀⠄⢰⢾⠀⠀⠀⠀⠀⠀⠀⢀⢯⠃⡴⢩⢤⡇⢸⢳⣶⢎⢧⠸⢉⡾⡮⣇⠀⣀⠀⢻⡇⠻⡉⢠⣿⠝⠁⣯⢵⢐⢝⣑⠢⣤
⠰⢖⣊⡉⠀⠸⠿⠀⠀⠀⢠⠀⠀⠐⢸⢿⠀⠓⢳⡇⡇⢸⢺⡇⢳⣻⡀⡞⡇⢸⣽⠘⢾⠃⢸⢿⠀⣯⠝⢾⡄⠀⢸⢽⠀⡻⣟⡋⠁
⠀⢀⣉⡦⠀⢸⣷⠀⣀⡠⠃⠣⢤⡄⠺⣿⠀⠀⢀⡣⣛⠸⡮⣧⣸⡺⡇⢡⢽⣆⢼⡇⡜⠯⢺⣿⠀⠉⠀⠀⠀⠀⢸⢽⠎⣴⢧⣮⠵
⠛⠫⢔⣒⠀⢸⣻⠀⠉⡆⢰⡿⡏⠀⠁⢯⣆⠀⠈⠚⣽⡀⠻⣕⣫⡾⠁⠈⠣⠥⠟⠁⠀⡀⣾⠇⠀⣠⠤⣄⠀⠂⣎⡿⢺⢪⠫⣀
⠀⢀⡤⠊⠁⠀⠿⡆⠀⠷⠁⠉⠓⠀⠈⢌⣿⣆⠀⠓⠋⠁⠀⠀⠀⠀⠀⠂⢀⠠⢀⠀⢂⣼⡟⠀⡀⠑⠢⡬⢀⢵⣶⡵⣡⡥⠷⠿⠓
⠀⠉⠀⠒⡲⠁⠘⡽⣄⠀⠀⠀⠀⠀⠀⠀⠩⠱⣷⣤⣀⡀⠀⠀⠄⠌⠄⠌⠆⠂⣀⢴⡿⠋⠀⠐⢧⡆⣠⠀⢊⢯⡇⣴⣑⣍⢦⡀
⠀⠀⢠⣎⣀⣴⡀⠘⢯⢆⠀⠀⠀⠀⠔⠉⠃⡀⠀⠉⠛⠻⠽⣿⣖⣖⣶⣲⣾⠯⠟⠉⠀⠀⢮⣠⠀⡈⢁⣀⣯⣏⣬⣌⣯⡉⠉⠁
⠀⠀⠀⠀⢀⠞⡀⠀⠈⠻⡵⢄⠀⠸⠔⠃⡜⢀⠎⢀⡆⠀⡤⢄⠀⢀⡀⠀⡀⠀⠀⠐⠤⠀⠀⠐⠊⣰⣺⢯⢟⣾⢷⠻⠿⠟
⠀⠀⠀⠀⠚⠒⠓⣹⠃⢀⠈⠫⢗⡦⣀⠀⠓⠃⡴⠛⡆⢀⠓⡎⠀⣨⢷⠀⠋⠲⣧⠀⠈⠃⠀⠠⣫⢾⢿⢇⢾⣝⣏⣇
⠀⠀⠀⠀⠀⠀⠰⠧⠐⠊⡇⠀⠀⠉⠫⠽⣲⣤⣀⡀⠀⠀⠀⠁⠀⠃⠈⠂⠘⠀⡈⠀⢐⣀⢴⡻⢾⢡⣎⡻⠌⡆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⠔⢻⠀⣠⣀⠀⢝⣞⠻⠯⠧⣖⣶⣤⢤⢤⡤⣶⣲⡖⠿⠽⠺⢁⢂⣥⡑⢹⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠉⢸⠐⡣⢶⡨⢊⣢⣖⢔⢄⡌⡘⣙⡱⡰⣡⣴⡱⣱⠳⡗⠃⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠎⠁⢸⡀⠏⠁⢳⢃⠎⢫⠊⡘⠱⡜⡰⠈⢣⡝⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠋⠀⠀⠛⠀⠀⠘⠃⠀⠀⠁
⠀⠀⠀⠀⢤⢔⠔⡠⢂⠔⡠⢄⡠⢂⠔⢂⠄⡠⠀⡠⢀⠀⢀⠄⣀⠀⡀⢀⠀⡠⠀⡠⠀⡀⡠⢀⠄⡠⢀⠄⡠⢀⠀⡀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠈⠈⠁⠊⠐⠁⠒⠁⠊⠡⠊⡠⢊⠔⡡⢊⡡⢊⡴⣈⢔⡡⢊⡠⢊⠔⡨⠊⠔⠡⢊⠔⡡⢊⠔⡡⢊⠞⠕⠱⠡⠊⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⡆⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⢠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢰⣦⠀⠙⢠⣾⡇⠀⠉⢀⣠⠄⠀⠀⠠⠦⠀⡧⠸⡇⠓⢸⡿⢋⡻⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⢸⡟⣷⣠⣿⢹⣇⣠⠶⣿⠋⢠⣦⠸⡃⠘⠂⣴⠾⣶⡛⢸⣷⠉
⠀⠀⠀⠀⠀⢀⣄⠈⢿⡷⣦⣼⠿⠈⢉⠀⠚⢉⣿⣾⣃⣀⣀⣀⣀⣫⣘⢸⡏⠾⠋⠁⠈⢿⣧⡀⢀⡀⣶⡰⣆
⠀⠀⠀⠀⢀⣀⣀⣀⡈⢻⡆⠡⠀⠀⡄⠀⠀⠾⢛⣉⣭⣩⡿⠟⢁⣀⠉⢸⣿⡀⠀⠀⣀⠀⠻⢷⣮⣴⣿⡁⢹⡆
⠀⠀⠀⢀⡀⠙⢿⣍⠛⠋⠆⠀⠀⠀⣀⠀⣀⠸⣦⣭⠛⢿⣤⣤⣈⣉⠀⠀⠹⣷⣼⣧⣿⠀⠛⡃⠈⠶⠈⢙⠀⣿⡀
⠀⠀⠀⠸⠛⠂⠀⢙⣶⠀⠀⠆⠀⠀⠈⣠⣿⠆⢛⣥⠞⣣⣶⠿⠛⠩⠀⠀⠀⢉⣽⠿⣟⣿⠂⠛⣀⡐⢆⠈⣠⡿⠁
⠀⠀⠀⢠⣤⣶⡾⠟⠃⠒⠀⠀⠀⠐⠀⣈⠰⣾⠈⢡⣾⡿⢦⣄⡀⠛⠀⠀⣴⡿⠃⠀⠀⢈⣴⠾⣏⣻⢇⣴⠟⠁
⠀⠀⠀⣀⣈⠉⠛⠛⣷⠀⠀⠁⠀⡀⢾⣿⠗⣀⠷⠘⣿⠟⠛⠛⠛⠀⠀⠰⣿⡷⠆⠀⣴⠟⠁⢀⣠⠿⠋⠁⠄⠀⡀
⠀⠀⠀⠿⠋⢀⣠⣾⠟⠀⣀⣤⠀⠻⡦⠩⣾⣿⣿⡶⠙⢷⡄⠰⠇⠀⠀⠀⠀⠀⠀⠈⣻⣷⡾⠋⠁
⠀⠀⠀⠀⣴⣿⣯⣴⣾⣾⠻⢿⠀⢣⣌⠳⠌⠛⣽⡛⠻⠿⠿⢀⣠⠴⠒⠋⠛⢻⣶⣾⠛⠁
⠀⠀⠀⠀⢠⣤⣤⠄⢀⣴⡿⠆⠀⣋⠡⣶⣬⣿⣙⣷⠀⣶⣾⠻⠅⠀⠀⣠⣼⠟⣯⠙⣦
⠀⠀⠀⠀⠀⠼⠃⢀⣾⠟⠰⠚⢂⣿⣷⢆⣿⡉⠉⢙⣻⡏⠋⢀⠠⢀⣶⡿⠋⠢⠐⡂⣿⡆
⠀⠀⠀⠀⠀⠀⢠⡾⠿⠾⠿⠿⢻⣿⠷⠶⣼⣇⠼⠻⣷⣻⣆⠂⠁⢼⣿⢁⠀⠀⢟⢶⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢠⣾⢏⡀⢠⡄⠉⠀⠁⢻⣇⠙⠳⣤⣸⣯⢀⠀⢀⣠⡾⡃⢙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣿⡷⠾⢻⣯⠔⠀⠀⠠⠈⠻⣇⡀⠀⠉⠙⠛⠛⠋⠉⠀⠃⢘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡤⠀⣰⡿⢫⠂⠁⠀⠀⠂⠀⠙⢻⣦⢠⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠷⠾⠿⣆⠂⠐⠀⠀⠂⠀⠀⠙⠫⣶⣦⣤⣄⣀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⢀⣾⠟⠀⡀⠀⡀⠀⠄⠀⠀⠀⠀⢨⢉⡙⠛⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣬⣦⣈⠠⢀⠐⠠⠀⡀⠂⠠⠠⠄⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠁⢀⣼⢆⠑⢀⠀⠂⡀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⣟⣑⣊⡐⠢⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠉⢉⣹⣇⠂⠠⢁⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣟⣛⡉⢘⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠉⢙⣻⣿⠪⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠛⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠠⠐⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⢀
⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠑⠂⠰⠀⠐⠒⠉⢀⠀⠀⠀⠀⠀⠀⢀⠀⠒⠀⠀⣀⣀⣄⣀⣠⣀⠀⠐⠂⠂⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠉⠀⠢⡀⠀⠀⢈⠔⠒⠃⠈
⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⣠⡆⣀⣒⢀⠄⠠⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠀⠀⠀⡀⠠⡐⠀⠀⠀⠈⠄⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠂⠀⢈⢠⠟⠀⠀⠉⠉⠓⠺⡄⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⡀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠔⠅⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠼⠀⢀⡀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠐⠀⡼⠴⠀⠠⢬⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⠀⠉⠀⠀⠈⠉⠁⡤⠃⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠴⠦⠆⠀⠔⠋⠁⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⠠⠄⠀⡆⠠⢸⡀⠀⠄⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠐⢄⡀⠀⠀⡈⠀⡠⠂⢀⡀⠀⠀⠀⠀⠀⠙⢄⡁⠂⠀⣀⡄⢁⠀⡀⠳⣄⠈⠀⠁⡠⡞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠢⢄⣀⢥⠚⡇⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠆⠙⠒⠊⣩⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣁⠀⠁⠂⣤⠐⠈⢀⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣤⠒⠓⢤⡀⠀⡠⠓⣥⠤⣇⠸
⠀⠀⠀⠀⠀⡄⠒⠉⠀⠀⠀⠙⠒⠚⠑⠙⠒⠉⠀⠀⠀⠉⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠁⠂⠤⠤⠅⠀⠈⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⡇⠀⠒⠀⢰⠀⠀⡖⡆⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⡀⡠⠂⠔⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠁⠊⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡾⠋⠉⠉⠉⠉⠻⣿⣿⣿⣇⠀⠀⠀⠀⣾⡟⠋⠉⠉⠉⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⢿⣿⠟⠛⠻⠿⣿⠀⠀⠀⠀⠀⠠⠴⠄⠠⠦⠤⠀⠸⣿⣿⣿⣧⠀⠀⠀⠻⣤⣤⠀⣤⣤⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠃⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠘⠃⠀⠘⠃⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣆⠀⠀⠀⠀⢸⣿⠿⠛⠛⠛⠛⠛⣿⣿⣷⠀⠀⠀⠀⢈⣿⡿⠟⠛⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣯⣤⡄⠀⣤⣄⠀⢻⣿⣿⠀⠀⠀⠀⠈⢃⣀⣀⠀⣀⣀⠀⢻⣿⣿⠀⠀⠀⠀⠸⣿⠤⠴⠄⠠⠤⡤⠀⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⣀⣀⡴⠶⢶⣄⣀⣀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣀⣤⡀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⠀⠀⠀⢲⡄⢸⡄⠀⠀⡟⢸⡇⠀⢸⠃⡀⠀⠀⢳⠀⢸⠃⣿⠀⢀⡄⠀⣿⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⣷⠀⣾⠁⢠⡇⠘⣧⡀⠸⡇⢸⡇⠀⣼⠀⣷⠀⠀⣿⠀⢿⠀⣿⠀⠀⢧⠀⢹⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⣿⠀⠰⠃⠀⠈⠀⠉⠀⠀⠹⡇⠀⠉⠀⠑⠀⠈⠁⠘⢦⠼⠁⠀⠈⠁⠈⠂⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⡇⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠑⠒⠠⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠴⠒⠒⠋⠉⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠓⠶⣄⡀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⣠⣀⣀⣀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⡤⠖⠛⠛⠲⣼⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠟⠛⠻⢿⣷⡖⠋⠉⠙⠳⣄
⠀⠀⣾⠁⠀⠀⠀⠀⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢸⡆
⠀⠀⢿⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⠳⢤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠞⠁
⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⠀⢠⠄
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸
⡇⠀⢸⠀⠀⢠⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢠⠀⠀⢸
⠳⣤⣾⠀⠀⢸⠀⠀⠀⣧⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⡇⠀⠀⢸⠀⠀⢸⡦⠴
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⣿⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠹⣄⠀⢸⡀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣼⣀⣀⡼⠁
⣶⣶⠶⠾⠿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠚⠋⣙⠎⠱⢾⡀⢯⢫⢰⠂⣀⠐⠿⡿⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⡓⠦⠉⠸⢸⠞⠚⢉⣑⣤⠃⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢉⡭⡿⠀⡃⠈⠉⢡⢀⡀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⢤⣶⣥⢮⣹⢗⡆⠀⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠛⢶⣅⠁⢫⢥⠲⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣦⡤⢍⠉⣁⣀⡀⠃⢢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠡⠌⡾⣸⢠⢈⡑⢄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢢⣄⠀⣠⠤⣄⠀⢀⡴⠀⡄⠀⠀⢀⠀⠄⠀⠘⢣⢇⠻⣀⠱
⠀⠀⠀⠀⠀⠀⣤⡄⠀⠠⠀⠀⠀⠀⠀⠃⡑⠀⠀⠀⠀⠙⣦⢉⣦⡉⣤⠞⢁⠀⠠⠀⠠⡁⠂⠀⠀⠀⠀⠤⠳⠼
⠀⠀⠀⠀⠀⠀⡠⣄⠆⠀⠀⠀⠀⠀⠀⠀⠼⠒⢗⠺⠛⢖⣻⣜⠿⢃⣿⡦⠶⢒⠒⠈⠂⠀⠀⠀⠀⠀⠀⠀⠸⠊⠐
⠀⠀⠀⠀⠀⣄⠑⣤⢠⠆⡤⠀⠀⠀⠀⠀⠒⠀⠠⠐⢂⢼⠿⢥⣤⡤⠼⣷⡈⠂⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠰⣌⠃⠙⢸⢰⢡⠒⡀⠀⢦⡄⠀⠀⠀⠐⣡⠊⣓⠒⠒⠒⣚⡟⢫⡀
⠀⠀⠀⠀⠰⠬⠿⡀⢀⠛⠙⣁⣈⡉⠙⠻⠦⠀⠀⠀⠀⠀⠙⣉⣉⣉⡍⠁
⠀⠀⠀⠀⠀⠄⠀⡢⠩⡴⠓⣡⢐⢹⢳⡆⠀⠰⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠂⡑⠀⢴⣱⣏⣳⢊⠟⠛⢄⠀⢸⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠑⠃⣤⠄⠰⠂⠹⠓⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠠⡘⠣⣁⡤⢶⢳⢻⡰⣌⠙⠊⡰⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡤⠀⠁⠸⠪⠧⠗⠐⡏⠀⢘⠂⠉⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣯⣿⣯⣽⣯⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣭⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⣏⠛⠀⢀⣄⡣⠤⣄⠘⠂⠀⣠⠘⠿⠷⠄⠒⠉⢉⡉⠂⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣥⣂⠘⡟⠶⠿⠈⠓⡦⣄⠑⢠⣾⣟⢛⠀⠀⠐⢄⣠⣼⠁⠀⣼⠁⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠙⢿⡜⠈⠀⠀⠀⠀⠑⠠⣱⣾⣿⣿⠏⠀⠀⠀⠈⣿⠘⢶⣶⠙⠶⠸⣆
⠀⠀⠀⠀⠀⠀⣸⣿⣥⡀⢀⣼⡇⠀⠀⠳⣆⠀⠀⡠⠀⠀⠙⠋⠒⠲⠄⠀⠠⠄⠀⠀⢸⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋⠀⠠⡄⠀⠀⠀⠻⠀⠀⠀⠲⠄⠀⠠⢄⢀⡀⠀⠀⠀⠀⢻⠛⢏⣇⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢀⡝⠀⠀⠘⠂⢀⣀⣀⣀⡟⠀⠈⡏
⠀⠀⠀⠀⠀⡀⢿⣿⣿⣿⣷⠤⠄⠺⡶⠆⠈⠉⠿⠀⢰⠀⠀⠈⠁⠈⢠⣀⣀⣀⣁⡜⠂⠠⢰⡞⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⠀⠀⢀⣸⣄⠀⢀⠐⠒⠚⠆⠀⠀⠀⢠⢋⠏⠈⠁⠀⠀⠖⠁⣼⠏⠈⠁
⠀⠀⠀⠀⢀⠀⠀⠘⣿⣿⣿⣷⣿⡿⠋⠙⣏⠁⠀⠀⠀⠀⢆⡀⠠⠛⠾⠀⠀⠀⠀⣴⣄⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣄⠀⣨⣿⠒⠒⠒⣿⡁⠉⠁⠀⣠⣴⣄⣠⡾⢛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠻⣿⣦⣄⠀⢸⡦⣤⠄⠈⢁⡴⢁⣯⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣾⣽⣿⣿⣓⣻⣽⣿⣆⣴⣿⠿⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣍⠉⢇⣾⣿⣟⡟⠋⠀⠀⢠⡡⠄
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣼⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⠀⠀⠈⡂⠠⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠄⢀⠉⠀⠀⢀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠠⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⠄⠐⠀⢀⠃
⠀⠀⢀⣀⠀⠀⡁⢄⠀⠓⠀⠀⠀⠀⠘⢄⣴⣾⣿⠿⣿⠋⢹⣿⣿⡏⠉⡙⠻⣿⣿⣿⣷⣦⡠⠃⠄⠀⠀⠀⠚⠀⡠⢈⠀⠀⣀⡀
⠀⠐⠉⠚⢝⡂⣬⡐⠀⠈⠀⠀⠀⠰⠶⢿⣟⠉⠀⠀⢸⡇⠸⣿⣿⣿⣿⠃⢸⡇⠀⠉⠛⣿⡿⠷⠆⠀⠀⠀⠁⠠⢂⣥⢐⡫⠃⠉⠂
⠀⠀⠀⠀⠀⠑⢭⠤⢦⡄⢈⣀⠴⣷⠶⠉⣘⣿⣦⣄⡀⠻⣆⡈⠉⠉⢁⣠⠟⢀⣠⣴⣾⣋⢉⠶⣾⠦⣀⣀⢀⡄⠤⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠔⠋⠤⠚⢓⡒⠒⠞⣻⡿⠛⢻⣿⢿⣿⣿⣿⣿⡿⣿⡟⠛⢿⣟⠳⠒⠒⡚⠓⠤⠉⠢⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡀⠐⠉⠈⠀⠐⠀⡼⠁⢠⢻⠟⠰⡠⢸⡇⢀⠆⠹⡗⡄⠈⢧⠀⠀⠀⠁⠉⠂⢀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⣀⣀⡀⡴⠁⢀⢆⡞⠰⠀⠀⠸⡇⠀⠀⠀⢳⡸⡀⠈⢦⢀⣀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠁⠠⢃⠎⠊⠀⣠⠂⠀⠃⠑⣄⠀⠑⠱⡘⠄⠈⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣁⠤⠔⠊⠀⠀⠀⠀⠀⠀⠑⠢⠤⢈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣤⣤⣤⣀⡲⠟⠀⣠⡶⣶⡄⠀⠀⣠⣶⡄⣾⣠⡾⠃
⠀⠀⠀⠀⣺⣼⠇⣶⣿⠟⠛⠉⣉⣀⣠⣬⣭⣉⣙⣛⡻⠿⣶⡿⠀⣿⠷⣶⣾⡟⣿⡇⠐⠫⠶⠿
⠀⠀⠀⠀⠉⠉⣴⣿⣥⣴⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠟⢳⡆⢸⣟⣆⣛⣭⣾⣟⢷⣤⣴⡶⢿⣧
⠀⠀⠀⠀⠀⢸⣿⡇⠈⠿⠛⢿⣿⣿⠿⡻⡟⠰⣣⣿⡿⠓⢻⠁⣾⡏⢈⣿⠉⣿⣽⢘⡛⠋⣀⣢⣿⣤⣄
⠀⠀⣶⣄⢿⡘⣿⠇⢠⣦⡀⠘⢟⡅⠠⠿⣁⠀⢿⣿⡟⠀⣼⢰⡿⠀⣼⡟⠀⣿⢯⡿⠀⣰⣿⣿⠟⣩⣿⡇
⠀⠀⢬⣽⠓⠀⣿⠀⢸⣿⡷⠀⣿⠁⣾⣿⣿⣦⠀⢿⠃⢰⡏⢸⠃⢠⣟⣄⠀⢏⣾⡇⢠⣿⢟⣥⣾⡿⠋
⠀⠀⠀⠀⢀⣾⡏⠀⢸⣿⡷⢀⣿⠀⣿⢧⣿⢿⠆⣸⣀⣻⣾⣤⣷⣤⣾⣿⣷⡟⢻⣷⣶⣿⠿⠛⣡⡄
⠀⠀⠀⠀⣸⣿⢷⠀⣾⠻⢃⣼⣿⣦⣽⣿⢿⣷⡟⢻⠛⣿⠁⣿⣿⠿⢿⣿⣿⠀⣼⣿⣯⡀⠀⠸⠷⠶⠶
⠀⠀⠀⠀⣿⣿⣾⠀⣿⣿⣿⣿⣿⢿⣍⠃⠀⠛⠀⠞⣒⠁⢠⣶⣾⠿⣿⡿⠋⢰⣿⠋⡉⣿⣤⣤⡙⠷
⠀⠀⠀⠀⣿⣿⣿⢠⣿⣿⡿⠋⡀⢸⣿⠁⣾⠃⣼⠏⣹⠀⡏⢠⡟⢰⠏⢠⠂⣸⠇⣰⣾⡿⣡⣿⣿⠂
⠀⠀⠀⠀⣿⣿⠟⢸⣿⡿⠀⣼⢇⣿⡏⢰⡏⢰⠏⢠⡏⢸⠁⢸⠇⡟⠀⢊⡆⢫⣄⣻⣯⣾⣿⠿⠃
⠀⠀⠀⠀⣿⣿⡄⢸⣿⡇⠸⢏⢸⡿⣶⣘⣦⣈⡆⠸⣤⣨⣷⣶⣿⣾⡷⣿⣿⠟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠸⣿⣷⣿⠋⡙⡲⢿⣿⠞⠉⠻⠛⠻⢿⡿⠛⠛⠋⠀⠉⠉⠁⠴⠟⣱⣷⡆
⠀⠀⠀⠀⠀⠀⠉⢡⡆⣿⡝⠇⠀⠀⠀⠀⠀⠀⠴⠟⣱⣷⡆⠀⠀⠀⠀⠀⠐⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠁

⠀⠀⠀⣀⣀⣀⣀
⠀⠀⢸⡟⠛⠛⢻⡇⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⡇⠀⠀⠀⣠⠴⣖⣷⠦⣀⠀⠀⠀⠀⠀⢤⠀⢀⣀⣀
⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⢻⢫⠝⡟⢹⠀⠀⠀⠀⠀⢿⠀⢀⣿⣽⡗
⠀⠀⠀⣧⣤⣤⣼⡇⠀⠀⠀⠘⠛⢛⣿⣿⡛⠛⠃⠀⠀⠀⠈⠛⢼⡤⠗⠊⠀⠀⠀⠀⠀⠉⠀⠀⠈⠗
⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⣠⣾⣫⣟⣷⣄⠀⠀⠀⠀⠤⢒⣉⣉⡒⠤⠀⠀⠀⠀⠀⢘⣧⡴⢒⣆⠀⠀⠀⢰⠒⠒⡒⠒⣶⣶⡆
⠀⠀⡧⡧⠴⡧⢾⢾⠂⠀⠀⠀⠀⠡⢒⣒⠌⠁⠀⠀⠀⠀⢰⣫⣤⣿⣿⠏⠀⠀⠀⢸⠹⡇⡇⢦⣬⠍⡇
⠀⠀⠙⢿⣾⣷⣻⠟⠀⠀⠀⠀⠀⠀⢱⡦⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⡀⠀⠀⠀⠸⠤⠤⠥⠥⠶⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠀⠀⡆⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠠⢲⢶⣀⣶⡲⡄⠀⠀⢠⠒⠉⠀⠉⠉⠐⡄
⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⢠⢀⠵⠤⠤⢎⠙⠀⠀⠀⢰⣟⠋⣾⣿⡦⢏⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⡏⣿⣿⡿⠁⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠀⠀⠀⠈⠹⡶⠾⠿⠷⡾⠀⠀⠀⠀⠉⠒⢦⠔⠀⠉
⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠠⡀⢀⢄
⠀⡔⠀⣲⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢸⠄⢠⠀⠀⠀⠀⣦⣼⣿⣿⣿⣼⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠑⠠⠿⢁⠔⡠⠤⣄⠀⠀⠀⡄⠀⠉⠉⠀⢸⠀⠀⠀⠀⢿⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠮⠥
⠀⠀⠀⠀⠃⠚⢇⡀⠿⠀⠀⠀⡇⠀⠀⠀⢠⠞⠀⠀⠀⠀⣼⣛⣿⣟⣟⡿⠀⠀⠀⠀⠀⢰⢲⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠘⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠒⠒⠒⠒⠷⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⡴⠋⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢠⠞⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢰⠃⠀⠀⠀⠐⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠦⠤⠤⣀⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋⣉⣠⠤⠔⠒⠲⠦⠤⢄⣀⠉⠳⣄
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠔⢋⣁⣠⣤⢰⠶⣴⢤⣤⣀⠈⠑⠢⣌⠳⣆
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠎⡴⠋⣠⠶⢭⢯⡆⣸⢣⡖⣸⢃⣤⢸⣛⣦⡀⠈⢢⡈⠳⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡜⠀⣼⢱⡆⣾⢈⠐⢹⣀⡴⡏⠘⣣⡟⣡⢨⢉⣦⡀⠳⡄⠹⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⠀⡼⢣⣠⠊⡜⠛⢦⢞⡙⣆⡹⠺⣴⡀⢁⠞⣡⢈⢷⠀⢱⠀⢹
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⠀⠸⠀⣧⢋⡌⣶⠰⢇⡟⠘⢃⡟⠸⢃⡯⢿⣝⠈⢁⣮⣌⡇⠀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣾⠀⢂⠀⣿⠘⢁⣷⡤⢊⠷⣔⣡⣧⢴⠏⡶⢸⣹⠶⣥⢣⡌⡇⠀⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣠⠃⠀⢻⠀⢸⡀⢸⠓⢏⡤⢱⢧⡆⣸⠁⡌⣾⠀⢠⢻⢣⡆⣿⠈⢡⡇⢠⠃⢰
⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⣇⠀⢇⠈⣾⠈⢃⣾⣀⢴⠳⣧⠁⢸⡟⠛⣾⣀⡼⢼⢦⢾⠇⡸⠀⣸
⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠹⣆⠘⣆⠘⣗⠻⡁⣽⢾⡸⡞⡗⢻⠰⠇⣧⠋⠖⡸⢀⠞⣰⠃⢠⡏
⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣤⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠳⣌⠳⢜⣦⣈⡷⡀⢸⡇⢀⡼⣇⣠⡾⠗⢁⡴⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⢤⣀⣉⠉⠙⠛⠛⠋⠉⢉⣁⡤⠖⠉⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢬⣉⣙⣒⣋⣉⣉⣁⣤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⢰⠒⣦⢰⣒⡆⠀⣮⠀⣮⢠⠆⢾⡷⠀⠟⡆⠀⠀⢺⡗⢰⡴⠂⡆⡆⠐⡆⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠐⢍⡉⠓⠤⠬⠿⣄⣀
⠀⠀⠀⢖⠒⠢⠴⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠒⣄⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠉⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠑⠒⢤⡎⠀⠀⠈⠉⠒⢦⡀
⠀⠀⠀⠀⠠⣔⡒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⣟⠒⠤⠤⠖⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠉⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢰⡓⠤⡤⠤⠖⠉⠉⠉⠑⠦⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠘⣷⣝⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⠦⣄⠀⠉⠀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢤⣀⢀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⡿⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⡄
⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡈⠛⠋⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦
⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⣀⡀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧
⠀⠀⠸⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠈⠁⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⣠⣤⡀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⣿⣿⡟⠉⠉⣹⣿⣿⠉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⣿⣿⡇⠀⢀⣴⣶⣦⡈⠛⠁⣤⣶⣦⡄⠀⠀⣿⣿⡇⠀⠀⣿⣿⣷⣶⣶⣾⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⣿⡇⠀⠘⠻⠿⠛⢁⣀⡈⠻⠿⠿⠃⠀⠀⣿⣿⡇⠀⠀⠙⠿⠿⠿⠿⠿⠿⠋⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷
⢻⣿⣿⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆
⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡉⠉⠉⠉⠁
⠀⠀⠀⠀⠙⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠆⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢰⣶⣦⠈⠉⠉⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⢠⣤⣤⣤⠈⠛⢁⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠘⠿⠿⠟⢀⣀⡈⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢠⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠉⠉⠉⠉⠉⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣶⣶⣦⣶⣶⣶⣼⣿⣧⣴⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣿⣿⣴⣶⣶⣶⣶⣶⣶⣤⣶⣶⣦⣴⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⡴⠛⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⣷⣦⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡇⠀⠸⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⡏⠀⠘⢿⣿⡇⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⡇⠀⢰⣿⣿⠀⠀⠀⢀⣸⠀⠀⠀⡗⠀⢰⣿⣿⡇⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠼⠷⠶⠾⠿⠿⠆⠀⢰⡋⠁⠀⠀⠰⠷⠶⠾⠿⠿⠷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⡤⢤⣤⡤⠤⣤⡤⡤⠤⢤⣤⣤⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣇⣀⣀⣸⣧⣴⣶⣿⡿⠃⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠛⠟⠛⠛⣿⡿⠋⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠛⠻⠿⠧⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉




⣶⡶⢶⠶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⠶⢶⣶⣶⠶⣶⡶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈⠉⠉⠉⠉⠛⣷⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡤⢻⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠄⣠⠀⠀⠀⠀⠀⡀⠠⡌⠟⡻⠁⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣨⣿⣧⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣷⣖⣀⡀⠈⣀⣀⣄⣜⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⣻⣿⠿⢋⣠⠤⠒⠒⠀⠀⠙⠓⠲⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠻⢯⣽⣟⣿⣛⣥⠆⠙⠉⠉⠀⣴⠋⠀⠠⠖⣤⡗⢠⣠⡀⢤⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⣀⡈⠃⢠⣄⡥⠽⢣⣍⡐⡄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠷⠈⡚⣀⢰⣀⣲⡦⣤⡄⠀⣍⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⠛⣋⡠⣶⢸⡅⠔⠀⠤⠶⡸⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⠈⠛⠰⠦⢄⠛⠀⠂⠘⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣦⣄⣀⢀⡀⠀⣀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣋⡉⠱⠟⠋


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

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


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⢿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⠿⠿⠿⠿⣿⣷⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠉⠀⠀⣀⣤⣴⣶⣿⣿⡿⠿⠿⠿⠿⠿⣿⣷⣶⣤⣄⡀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⢀⣤⣶⡿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣷⣦⣄⠀⠈⠛⢿⣶⣄
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⣠⣶⡿⠟⢉⣼⡿⠋⠀⠀⣀⣴⣶⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⡀⠀⠙⢿⣦⡀
⠀⠀⠀⢀⣼⡿⠋⠀⢠⣾⡿⠋⠀⠀⣾⡿⠁⠀⠀⣴⣿⠛⠁⢀⣤⣤⡈⠙⢿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠈⠻⣿⣄
⠀⠀⢀⣾⡟⠁⠀⣴⣿⠏⠀⠀⠀⢸⣿⠃⠀⠀⢸⣿⠁⢀⡀⠀⠉⠻⣿⣆⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠙⣿⣆
⠀⢀⣾⡟⠀⠀⣼⡿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⢻⣿⡀⠀⢀⣿⡟⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄⠀⠘⣿⣆
⠀⣼⡿⠁⠀⣼⣿⠁⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠘⣿⣧⡀⠛⠿⠿⠿⠛⢁⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠹⣿⡄
⢰⣿⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠈⠻⢿⣶⣤⣤⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⣿⣷
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⢸⣿
⣿⡟⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿
⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠿⠿⢿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿
⣿⣧⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿
⢹⣿⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠘⠟
⠘⣿⡇⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠋⠉⠙⠛⢿⣷⣄⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⢹⣿⡄⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⢀⣴⣿⠿⢿⣶⡀⠹⣿⡆⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⢀
⠀⠀⢻⣷⡀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⢾⡿⠁⠀⠀⣿⣷⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣼⡿⠁⠀⣰⣿⠃
⠀⠀⠈⢻⣷⡄⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⢀⣤⣶⡿⠃⢠⣿⡇⠀⠀⠀⣿⡿⠀⠀⠀⣠⣾⡟⠁⠀⣴⣿⠃
⠀⠀⠀⠀⠹⣿⣦⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣄⣀⠉⣉⣠⣴⣿⠏⠀⠀⠀⣼⣿⠃⠀⣠⣾⡿⠋⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠈⠻⢿⣶⣄⡀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠁⠀⠀⣠⣾⡿⣃⣴⣾⠿⠋⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣤⡀⠀⠉⠻⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⠿⠋⠁⠀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣀⠀⠈⠉⠛⠛⠿⠿⠿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠁⠀⢀⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⣾⠿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡴⢞⣩⡤⠶⠶⠶⠿⢤⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠋⠀⠀⠀⠀⣀⠀⠀⠈⠻⣆⠀⣀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡶⢀⣟⣈⡟⠀⢀⣤⡀⠀⠀⠛⠁⠀⠀⠀⠘⣆⣹⣎⣁⣦⡾⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣬⠟⠋⠉⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠲⠂⠀⠀⠉⠀⠀⠀⠉⠉⠲⢴⡻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣢⠞⠀⢀⣀⠀⠀⠛⠁⠀⠶⠀⠀⠉⠀⠀⠶⠀⠀⠀⠶⠀⠀⣤⣤⠀⠀⠀⠹⡍⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠲⠀⡟⠀⠀⠀⠋⠀⠀⠲⠂⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠶⠀⠀⠀⠈⠁⠀⠀⠛⠀⢹⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣧⠀⠀⣤⡄⠀⠀⢄⠀⠈⠃⢰⡀⠀⠀⠀⠀⠻⠃⠀⢠⠀⠘⠟⠀⢀⠀⠀⠀⣸⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠹⣆⠀⠈⠀⠀⠚⠀⠓⢤⣀⡀⢧⠀⠘⠋⠀⣤⡄⠀⡞⠀⣀⠤⠚⣁⣀⠀⣠⢏⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣈⢦⡀⠰⠆⠀⠰⠆⠀⠈⠉⠛⢧⡀⠀⠀⠈⠀⣰⡷⠋⠁⠀⠀⣉⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠉⢳⠦⠤⢄⣀⣀⣀⣋⣀⡈⢷⡀⠀⠀⢰⡏⣀⡤⠖⠚⢹⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣷⡄⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣦⣀⡀

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