Dispensaries Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⢧⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣷⠀⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⣭⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⢻⣳⣆⣠⣽⣿⠏⠢⠀⣀⣤⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠾⠿⢻⣿⣿⣁⣽⣷⣶⣴⢞⣡⣀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣄⠀⠀⢀⣾⠟⡁⠀⠀⢹⣿⣿⣿⣟⡒⠚⠁⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡈⠻⣝⠦⣘⠁⢰⣇⣨⣿⣿⠀⢻⡿⠛⠹⢷⣤⡀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⢿⣷⣦⣴⣙⢦⣸⣷⣄⣈⣿⡗⢀⣈⠅⢀⡴⢀⣵⣾⠛⣁⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢤⣦⣬⣝⣻⣿⣿⣿⡿⢿⠋⠛⢧⠾⠂⣠⣿⣱⣿⠿⠟⣉⡥⠴⠒⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢛⣿⣿⣧⣄⢠⣀⣤⡗⠰⣿⣿⣿⣜⣃⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⢿⣿⣭⠿⠿⠟⢻⡯⠀⠐⠿⢿⣯⣍⡉⠉⠉⠉⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣉⡀⠐⣻⣿⣧⡄⢀⣚⣿⠆⠀⢤⠀⣤⣿⠟⣛⠓⠒
⠀⠀⠀⢀⡤⠶⠶⠶⠿⠿⣻⣿⣿⣿⣿⣾⣍⠈⡁⠀⢈⣹⣿⢃⡀⠉⠸⣿⣿⡿⠿⠿⠿⠿⠶⠄
⠀⠀⠀⠀⠀⢀⡠⠔⠚⣛⠭⠭⣽⡿⣿⣿⡛⠁⠀⠙⠛⣿⠁⠉⠁⠀⣸⣿⣿⣿⣶⣆⡀
⠀⠀⠀⣴⠾⠋⠈⠉⠀⢀⣴⡾⠯⢉⣿⡿⠁⢀⡂⠀⣀⣸⢟⡀⣠⣾⣿⣿⣛⣉⠉⠉⠉⠁
⠀⠀⠊⠁⠀⠀⠀⢀⣰⡿⠋⠁⢠⡟⢋⣠⣶⣿⣿⡟⣏⠠⣿⢿⡟⣿⣿⣿⣻⠿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢀⡴⠋⠀⠉⣰⣿⠍⣿⠇⠁⠐⠿⣾⠅⣿⣿⣿⣿⣦⡀⠈⠛⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣠⣾⠏⠉⠀⣿⠀⠠⢤⣤⣽⠀⠚⣿⣿⡜⢿⣿⣄⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⣰⠃⠀⠀⠀⣼⣿⣰⠀⠘⣿⣷⡀⠹⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠸⡏⠀⠀⠘⣿⣷⠀⠈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⣿⣇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⠀⢀⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠈⣿⣿⣿⡿⢀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠰⣤⣤⣄⣀⠀⠈⢻⣿⣿⣿⡆⢿⣿⣿⡇⣼⣿⣿⣿⠟⠀⢀⣀⣠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⣙⢿⣿⣿⣼⣿⣿⢱⣿⣿⡟⣡⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⡿⠟⢹⠛⠻⢿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠁⠀⠀⢸⡆⠀⠀⠈⠙⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣇⠈⣿⣿⣿⣿⣿⣿⡃⣰⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠐⣶⣶⣶⣶⣦⣄⣀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣀⣠⣴⣶⣶⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠘⢿⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⡿⠋⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⠹⣿⣿⣿⣿⣇⢹⣿⣿⡟⢸⣿⣿⣿⣿⠟⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⡙⢿⣿⣿⡀⣿⣿⢃⣿⣿⡿⢋⣵⣾⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣦⡙⢿⣧⢸⡟⣸⡿⢋⣴⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⢦⣝⣾⣿⣯⡴⠿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠿⠛⢋⣽⣿⡿⠃⠸⡇⠘⢿⣿⣯⡙⠻⠿⢿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠏⠀⠀⠀⢷⠀⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠶⠶⠞⠛⠚⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⡤⠐⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣷⡾⠁⠀⠸⠷⠖⠂⠉⠱⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣤⠄⠐⠀⢶⣴⣶⠂⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢈⡹⠋⣤⢠⡶⣻⣿⡿⠧⣀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⠒⠉⠉⠉⠉⠀⣄⡝⠛⠍⠃⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡴⠇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢻⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢐⣾⠋
⠀⠀⠀⠀⠀⠀⠈⢛⢦⡈⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡿⠁⠀⢀⣠⡶⠂⠁
⠀⠐⠒⠒⠶⢤⣠⡈⢨⢻⣮⣆⠙⣗⠀⠰⠀⡀⠀⢀⣀⠴⣿⡟⠀⠠⠒⠹⠂
⠀⠀⠀⠀⠀⠀⠁⢘⢻⣦⣹⣿⡄⠀⠿⠶⣆⣴⢶⣿⣿⣾⡟⣰⢏⠀⠁⣀⣠⣤⣤⠤
⠀⢀⣀⡤⠶⠦⠴⣾⣿⣿⣿⣿⣷⣴⠰⡆⠻⣿⠃⣿⣿⣿⣿⣿⡶⠞⠛⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⢆⡃⠃⠻⣾⠀⢿⣿⣟⠃⠀⠁
⠀⠀⠀⠀⠀⠠⠀⠺⠍⠛⠛⠛⠟⢻⣾⡙⠴⢣⡟⠀⠈⣿⠛⠛⢷⡷⣆⣠⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡀⢿⣇⣀⡞⣀⣤⡤⠄⢀⣠⣄⡈⣉⢈⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⣿⣿⡿⠟⠉⢁⡴⣾⣿⣿⣿⢷⠄⠉⠩⠟⠳⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠉⠉⠀⠀⡴⠋⠠⡝⣃⠽⠉⠛⠋⢃⣢⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡺⠀⠀⢰⢤⠀⠀⠃⠄⠈⠉⠙⢝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⣞⠀⠀⠐⡀⡄⠀⠈⠀⢽⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠙⢢⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣾⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⣿⣿⡾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⢦⣿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣹⣿⡟⣁⣤⣤⡀⠀⠀⠀⠀⣀⣤⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡽⢯⣹⠀⢹⢿⠻⣝⠂⠀⠀⠀⠈⢷⣿⣿⣿⠀⠀⠀⠀⢀⣹⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢤⢘⣷⢠⣾⣿⣥⠤⠀⠀⠐⠚⣾⣿⣿⣧⡄⠀⠀⠀⠀⣻⣶⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⢦⡻⡿⠿⠿⡟⠃⠀⠀⠀⠀⠀⢻⡿⣿⡅⠀⠀⠀⠀⠀⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠄⢻⡦⠀⠀⠀⠀⠀⠀⠀⣀⣀⢽⣿⣿⣧⣤⠀⠀⠀⢙⣻⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣿⡶⣯⡦⠀⡀⠀⠀⠀⠀⠀⠩⠿⠂⣿⠋⠙⠀⠄⢰⣦⠞⢻⡙⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢛⣿⡿⠛⢛⢷⣛⡁⠒⠀⠀⠀⠀⠀⠈⠀⣷⠄⠴⠀⠀⠀⢿⡇⢘⣷⡾⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡾⡏⠙⣦⠀⠀⠀⠀⠀⠀⠀⠈⠐⢶⠒⢴⣾⡾⡶⣿⣧⡔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣷⡜⣨⣧⡗⣤⣬⣷⠴⠶⠆⠀⠀⠀⠰⠀⠀⠃⠈⠑⣿⡿⢽⡏⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⣾⣿⣟⠡⡿⠻⢿⣿⣟⠻⠶⠦⠀⢀⣀⡀⠀⠐⢧⠂⢀⣧⣷⢼⡃⠻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠟⠉⣿⠙⢶⡿⠂⠈⠏⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠠⠞⡈⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣁⠀⠀⣇⣠⣄⣤⣤⣈⡁⠐⢄⣀⡀⠀⠀⠀⠀⠀⠀⢤⣿⣼⠷⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠖⣾⠷⣿⡟⡉⠻⣿⢷⣏⡑⠒⢺⣿⠟⠺⠀⡶⠶⠀⠠⠞⡟⢽⠃⣨⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢠⡞⠁⢰⠟⣿⡌⠀⠹⠆⠈⠙⠀⠃⠁⠀⢶⠶⠀⠀⠂⠀⠀⠣⣼⣿⣿⠯
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⡀⠀⣸⠀⡏⠹⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣤⣧⣄⣀⠀⠀⠉⣸⡏⠉⡃
⠀⠀⠀⠀⠀⠀⠘⣁⣤⡶⣶⣿⣿⣿⣔⣿⣴⣏⠁⣀⠀⠀⠀⠨⠛⢆⠉⠜⡙⠏⠒⣴⣶⡆⠻⠒⡏⠴
⠀⠀⠀⠀⠀⠀⠚⠁⠀⣴⡿⠛⣫⠏⣱⡏⣿⡙⢷⣎⡙⠳⡄⢀⡜⠀⣷⡄⠀⠀⠀⡽⢹⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠟⠀⣿⠀⢹⣦⠀⠙⣿⣆⠑⠚⣷⣴⣿⣇⢰⠖⠀⢠⣾⡗⠛⠛⢧⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⢻⣇⠀⠈⢻⡄⠀⡜⢁⣼⡀⠌⢀⠁⠈⠏⢛⡶⠀⠈⠀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢤⡄⠀⣀⣠⣿⠀⠀⠻⣄⣴⣯⣤⣄⠀⢿⡿⠢⠀⠀⠀⠀⡨⡛⣥⠴⠆⢀⠈⡄⠀⠀⢠⣶⣧⣀
⠀⠀⠀⠀⠀⠠⠔⠒⣻⣿⣾⣿⠟⠉⠀⣏⢠⠞⠋⣼⢻⡛⠷⢩⡆⠈⣴⠄⠀⠀⠀⢊⠀⡧⠹⠆⣀⣀⡀⠀⢀⠈⣾⠀⢳⡄
⠀⠀⠀⠀⠀⢀⡴⠟⠉⣼⠋⣿⠀⠀⣶⢸⡄⠀⢸⡏⠀⢷⠀⠈⠁⠀⣋⠀⡀⠀⢀⡾⠿⠃⠦⠌⠿⠛⠀⠐⡷⠀⢸⡀⠀⢁
⠀⠀⠀⠀⠀⠊⠀⠀⢸⠃⠀⠈⠀⠀⠈⢸⠀⠀⢸⠁⣄⠘⡆⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⣀⠀⠀⠲⡆⢠⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⣀⣸⠀⠄⠀⠀⣹⣶⣷⣶⣶⣦⣿⠆⣀⣈⠑⡶⢠⠄⣿⠀⢀⡾⠁⣼
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⢻⠀⠁⠙⠻⣿⢧⣤⣴⣿⣿⣿⣯⠭⣷⣾⣿⡿⣯⡶⣩⣶⢰⣠⠛⢷⣾⠗⢰⣾⢶⣦⣤⣄
⠀⠀⠀⠀⢠⠂⠹⢧⣵⣿⠛⢷⣤⣦⣤⣴⣒⣋⣉⣁⣶⡶⢶⣶⣾⣷⣿⣿⣻⣷⣿⣿⣚⣥⡺⠛⠿⣇⣹⡿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠙⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢉⣭⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠀⢸⣿⣧⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣬⣽⣛⣿⣯⣵⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣈⡭⠭⠿⠿⠭⢽⣗⡿⢰⣄⠀⠀⠀⠀⠀⠀⢀⢴⡿⡋⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠩⠉⢹⣿⣿⣿⣯⠉⣇⢸⡈⢦⠀⠀⠀⠀⡴⣿⠎⡠⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⠀⠀⣐⣾⣷⣶⣾⡅⠈⣿⣇⠚⡆⠀⢠⡎⡷⣡⠤⡴⠃⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠼⣿⣿⣿⣿⡃⠀⡟⣿⡈⢸⣠⣟⡟⠥⣶⣞⣤⣴⡶⣋⣿⣫⠽⠿⠿⠿⠿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⣹⣿⣿⣿⣿⠅⠀⡇⢻⣧⠘⣿⡟⢒⣵⣿⠷⠟⢛⡉⢡⣄⣰⣦⠽⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⠀⢒⣿⣿⣿⣿⡁⠀⡇⠀⠹⣧⠙⠐⣛⣉⣠⣬⣷⢾⣿⣯⣭⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⠀⠀⢨⣿⣿⣿⣿⠆⠀⡿⠿⢛⣾⠀⢦⣤⣄⣈⠩⡉⣝⢛⡻⠷⢾⣭⡽⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⣺⣿⣿⣿⣿⡇⠀⡇⠀⣼⡟⢈⣤⢌⠻⣿⡿⢶⣾⡤⢽⣀⣶⣀⣯⠉⢛⠺⠿⢶⣤
⠀⠀⠀⠀⠀⠀⠀⣠⢴⣊⣽⠨⠀⠀⠭⣿⣿⣿⣿⠂⠀⣇⣀⣯⡇⢮⡟⠙⢦⡎⢙⠯⣭⠽⠶⣤⡀⠉⠉⠉⠉⠛⠛⠋⠉
⠀⠤⠶⠶⠖⠀⠻⠒⠉⠤⢼⢐⠀⠀⣘⣿⣿⣿⣿⠅⠀⡧⠄⣿⡇⢶⠣⠉⠀⢙⡪⣆⢪⠙⡯⣭⣽⣦⣀
⠀⠀⠐⠒⠒⠒⠒⠲⠶⠶⠞⠮⣀⠀⠰⢿⡿⠿⢿⣣⠜⠱⠶⠛⢧⡞⠒⠂⠀⠀⠀⠶⠯⠓⠾⢄⣩⣛⣻⣷⣄
⠀⠀⠀⠐⠒⠒⠛⠛⠛⠛⠛⠛⠛⠿⠯⠛⠛⠻⠟⠒⠒⠒⠒⠀⠈⠁⠐⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⠔⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣍⠳⣄⣷⣴⣿⡷⡃⠊⣐⠺⠋⠁
⠀⠀⠀⠀⠠⠀⠀⡀⠄⠀⠄⠀⢄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⣨⠷⠨⣿⣻⣯⣿⠉⠡⢔⡢⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⢠⠁⠀⢉⠄⠨⠀⢄⠂⡀⠀⠓⣤⠂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣯⠁⠀⢸⠛⠿⡏⢀⡀⠂⢯⠨⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⢐⠛⠠⠘⡄⠏⠀⢲⣤⣿⡀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠂⡈⠩⠠⠒⠶⠆⠀⣰⣆⣼⣶⡀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⡠⠐⠁⢈⠋⣩⣖⠻⣝⠣⣄⢠⡆⣀⠤⠔⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠊⢱⠄⠁⠟⢋⣺⣧⠢⢦⣶⣄⡀⠀⠀⠒⠰⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢸⠁⠀⢃⠛⠂⠰⠍⠀⠄⡉⠀⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⢈⢄⣴⣡⠄⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠈⠋⢻⠼⢟⣿⣜⣿⠶⠶⠄⡀⠐⠒⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠛⠒⠂⠰⠥⠴⠛⠪⠉⢃⠁⠍⠢⡀⠈⠁⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒⠀⠐⠀⠀⠀⠀⠘⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣖⣏⣀⣀⡀⠀⠀⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⢒⠞⢁⠺⡍⠃⢄⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⣦⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠉⠀⢸⠀⠙⡄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠲⢾⣛⣗⣼⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⠀⠀⠘⣆⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⡶⡏⢿⣿⢥⢹⡛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠚⠉⠘⠉⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣆⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢰⣿⡇⠀⠀⠀⣠
⠀⠀⠀⠀⢃⠢⣄⠀⣿⡇⢰⠀⣠⣴⡞⠀⠀⠀⡀⠀⠀⠀⠀⣿⡇⣿⠀⠀⠀⠀⣀⠀⠀⠀⠸⣵⣄⠀⢸⣷⡇⠀⢠⣞⠏
⠀⠀⠀⠀⠈⢷⣌⠣⣿⡇⢸⣼⡿⡫⠁⠀⠀⠀⢹⣯⡲⣄⠀⣿⡇⡿⠀⣠⣶⢟⠍⠀⣀⣀⡀⠙⣯⢦⢸⣿⡇⣰⢯⠎⠀⣀⣀
⠀⠀⠲⣆⣒⣢⠿⣷⡙⠇⠻⣫⣾⡴⠶⢒⠖⠀⠀⠙⢷⣎⡳⣽⣇⣯⣾⠟⡡⠋⠀⠀⠀⠙⠻⢷⣾⣿⣿⣿⣿⣷⢷⡺⠟⠋⠁
⠀⠀⠀⠀⠉⠙⡻⣻⡴⡌⠔⠳⢖⠂⠉⠀⠀⠀⠀⠀⠀⠙⠳⢿⣿⣿⠕⠋⠀⠀⠀⠀⠀⠠⠤⠶⠾⠿⣟⠉⣹⡿⠷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡗⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣑⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢙⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⡀⣀⢀⡀⡀⡀⡀⠀⠀⠀⠀⠀⠀⣀⢀⢀⡀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠘⠁⠈⠈⠁⠀⠀⠀⠀⠀⠀⠉⠈⠈⠁⠃⠉⠈⠁⠃⠁⠁⠀⠀⠀⠀⠀⠀⠉⠘⠈⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣴⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⡏⢀⣾⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡞⣿⣿⣿⣿⢇⣾⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠲⠶⢿⣿⣿⣷⣶⣶⣶⣤⣦⣤⣀⣙⣿⣿⣿⣿⡘⣿⣿⣟⣾⣿⣿⣿⣿⣿⢋⣠⣴⣴⣶⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢹⡇⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠉⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢉⣀⡤⠴⠒⠛⠙⠋⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⣤⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠈⣇⣼⡯⠖⠒⠚⠋⠙⠛⠓⠛⠙⠚⠒⠲⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⡃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⢏⡉⠙⠛⠒⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢆⠀⠀⠀⠀⠈⠛⠛⠦⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⡍⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⠣⠀⠀⠐⠇⠀⢀⡞
⠀⠀⠀⠀⠀⠿⡄⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⢃⠈⠀⠀⠀⠀⢀⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠠⡂⠀⠀⡁⠀⡼
⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠈⠠⢀⠀⠀⠀⠀⢹⡄⠀⢠⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⠄⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⡀⠀⠀⠸⡄⠀⢠⡘⣦
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠠⣷⠀⠀⠀⣽⠃⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠉⠠⡀⠀⠸⢆⠀⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢆⠀⠀⠀⠘⣌⡄⠠⠀⣿⠀⠀⠀⢠⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⢀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡽⣄⠐⢀⠘⣧⠀⠀⣿⠀⠀⢠⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣳⣬⢷⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣷⣌⠀⠨⡄⠰⡇⠀⠰⣿⠋⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⠦⠿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⡀⠀⠀⠀⠘⢆⡀⢠⢀⠀⣰⠟⠁⢈⢠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠑⢼⣞⣰⢋⣤⠼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⢀⠀⠀⠙⠛⣲⠾⡿⠿⠥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡴⠞⠉⠁⠀⠀⠀⠀⠐⢀⡆⠁⠀⡇⠀⠀⠀⠉⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⠀⢸⠁⠀⠒⠛⠉⠉⠉⠉⠛⠲⠤
⠀⠀⠀⣤⠀⠀⢰⠃⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢴⠶⣦⣄⠠⠤⠤⠤⢀⣀⡀
⠀⠀⠀⢸⠀⠀⡈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⡴⠏⠁⠸⡇⠈⠻⢟⠢⣀⠀⠀⠀⠈⠉⠉⠶
⠀⠀⣄⠀⠣⠀⢀⠈⠀⠀⠀⠀⣀⣄⠂⠴⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠠⣣⠀⠀⠀⠁⠨⠁⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠈⠓⢄⠄⣃⣀⠤⠔⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠰⢹⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⠓⠊⠛⠫⡉⠒⠒⠤⠤⢤⣤⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢉⡇⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⡠⣔⠒⡿⣋⡧⠒⠉⠒⢪⠿⠥⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⢀⠔⣫⠿⠂⠁⠘⢈⠙⣀⠐⠍⠀⠘⠋⠐⡓⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣹⠞⡹⠒⡟⠊⠄⢃⠤⠀⢏⢾⠟⠁⢼⢀⣈⣇⠼⢚⠆⡈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠉⠩⠒⠀⢪⠝⢀⠤⠾⣴⠘⠊⠀⠖⠓⠏⠀⠰⠒⢾⠡⠁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠁⠈⡱⠊⠀⡠⣄⡄⠆⢀⡡⠛⡟⠠⠴⠁⡴⣂⣅⠴⡶⠍⡄⢀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠘⠀⠃⡰⠁⠛⠘⠉⠀⠎⢠⠀⠀⡠⠖⠚⠉⠠⠆⡂⠤⠃⢡⠞
⠀⠀⠀⠀⠀⠀⠀⣀⠘⢅⠐⠀⢠⠘⠀⡀⠀⠐⠐⠛⠦⠀⠊⢀⣡⡄⡠⣭⠘⣲⠌⠂⡩⠜⠃
⠀⠀⠀⠀⠀⠀⠀⡿⢄⣎⠀⡁⠹⢰⢄⠀⢀⠈⠍⣀⠐⣈⠀⠉⢞⠰⢃⡴⠋⠋⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠸⡠⣍⠇⣰⡄⡀⢈⣬⠖⣷⡚⢡⢺⠠⠕⠘⠐⣂⡐⡠⣴⣚⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠋⡆⡀⠋⠆⠀⠘⡁⠤⣄⠌⢓⡲⢄⠨⠉⠩⠙⠉⢻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠇⠀⢬⡬⠥⢡⢉⠊⢁⡴⢲⣭⣀⠖⠓⠒⠉⠉⠁
⠀⠀⠀⠀⠀⣀⡤⠒⡰⠶⠥⠷⠤⠛⠳⢬⣏⡯⠚⠓⠢⠧
⠀⠀⢀⠔⡪⢡⠔⠉
⠀⠀⠘⢌⡶⠁

⠀⠀⠀⠀⢸⣿⠻⢿⡷⠀⠉⠑⠈⠷⠰⠄⡀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⢟⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠩⠀⠈⠻⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⣼⣷⠀⢠⣶⣆⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⣿⣿⠇⠸⣿⣿⣆⠀⠈⠙⠻⢦⠀⠀⠀⠀⠀⠀⢀⣤⡞
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡆⢀⣾⡍⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠙⢿⣷⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣿⣡⣶⣼⡷⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣟⡁
⠀⠐⠀⠀⠀⠀⠀⠀⢿⣷⡀⢰⣿⣿⠀⠀⢀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠿⠿⠿⠏⠛
⠀⠀⠀⠀⠀⠀⢷⣄⠸⣿⡇⢼⣿⠏⣠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠽⠦⣄⣀⡀⠀⠀⠀⠀⠀⠙⠉⠛⠿⠷⢀
⠀⠀⠀⠀⠀⠀⠈⠻⣇⠙⠂⠀⠀⢰⡿⠟⣁⣀⠀⠀⠀⠀⠀⠀⠚⠛⣫⣤⠀⠀⠀⠙⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣶⣄⣀⣀⣠⡴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⣤⣄⢠⣧⠸⣷⡀
⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⠉⠀⣼⣿⡏⢸⣿⡇⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀
⡀⢸⣿⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⣿⠈⠻⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣤⡀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠘⠁⠀⠀⠀⣤⠀⢀⡀⠀⠀⠀⢤⣀⣀⡈
⣄⠀⠀⢠⡧⠀⠀⠀⠀⠀⠀⠀⠐⠒⠿⢷⣶⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⡿⣠⡿⠃⠀⠀⠀⠀⠙⠻⠿
⠦⢴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡍⠁⢉⣵⠶⠛⠀⠀⠀⠐⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠶⠙⠉⠋⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠒⠓⠂
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠁⠸⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⡠⠋⠹⠥⣤⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠂⠀⣾⡃⠀⢀⡶⠚⠀⠀⠀⠀⠀⠀⡀⣀⢀⣶⣶⣶⣿⣗⣀
⠀⠀⠀⢲⢶⡙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣿⣇⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣶⠶⠋⠁⠀⠀⠀⠀⢀⡾⠿⠙⢫⣿⣿⢿⣿⡟
⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⢿⣿⠤⠄⠀⠀⠀⢀⠀⠀⠉⠀⠀⢀⣾⡟⠃⠸⣿⡄
⠀⠀⠀⠀⠀⢀⣴⣿⣋⣀⣠⣤⣄⣀⡀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡦⠁⠀⠀⣟⠇
⠀⠀⠀⠀⣄⣿⣿⣿⣿⡟⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣴⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠛
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣷⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠛⠃⠉⠉⠉⠿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠐⢈⢰⡄⠀⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠀⠂⠄⠀⠐⠀⠀⠀⠀⠀⠀⠘⡀⠆⡀⠀⠀⠁⠀⠀⠒⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⢠⣄⢠⠽⠚⠉⠁⢠⠀⠉⠀⠂⢤⠒⠀⣀⠠⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠴⠿⠈⠶⠶⠄⠀⠀⠰⠀⠶⠃⢄⠜⠁⠀⠀⠀⠀⡆⠄⠀⠀⠀⠀⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⣤⠆⢀⠀⠀⠀⠀⢀⠃⠰⠀⠀⠀⠀⢀⠀⠱⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠠⣀⠀⠀⣀⠠⠂⠄⠂⠂⠉⠀⠈⢀⠀⣀⠀⠈⠀⠀⠀⠀⡠⢄⠜⠀⠀⡦⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠠⠍⡠⠛⠋⠀⠠⣬⠁⡀⠠⡄⠀⠱⡀⠁⠲⠀⠀⠖⠁⡠⠃⠀⣠⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⢰⠀⠑⢄⠀⠐⠦⠀⡠⢄⠀⠴⠀⢐⠔⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠄⠀⡀⠀⢨⢠⡀⠀⠈⠁⢲⢂⢸⣆⠰⠂⠈⠀⠀⠀⠞⠀⠙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠈⠑⠆⡀⠀⠋⠁⠀⠈⢈⠓⠀⡀⠀⠁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠠⠐⡒⢠⡤⠀⠀⠂⠃⠀⠀⠀⠀⠠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢀⣀⣀⡀⠈⠀⠈⢀⠀⡀⠠⠆⠀⢀⠀⠀⠲⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠚⠀⡎⠀⠀⠀⠀⢈⠂⠀⡀⠰⠀⠀⠁⠠⠒⠐⢆⠀⠀⠀⠀⢀⠚⠀⠂⠀⠀⠙⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠠⢄⠀⠀⠀⠀⡈⣤⠦⠈⠺⡈⠀⠀⠀⠠⢖⠀⠀⠀⡠⠂⠈⠓⢤⠰⠄⠄⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠖⠈⢠⣀⠂⠠⢄⡆⠀⠀⠀⠈⣧⣀⢸⠲⠞⠂⠀⠎⢁⠀⠀⠀⢘⠤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠋⠈⠉⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠈⠀⡀⠀⠀⢀⡢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡀⠀⠁⡀⠀⠀⠀⠀⠂⠀⢀⡄⠂⠀⠄⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⡇⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣆⡀⠀⠀⠀⠀⢠⡇⠰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢅⠙⣙⣦⠀⠀⠀⢠⣇⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡠⠾⢫⣵⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠘⢧⣷⡄⠀⠰⣿⠆⣿⣿⣿⡄⠀⠀⠀⢀⡔⠉⢀⣴⣿⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⢧⣿⣆⠀⣎⠆⢹⣿⣿⡇⠀⢠⠖⠁⠀⢠⣾⣭⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠡⣼⣦⢸⡂⢸⣿⣿⡇⠼⠁⠀⠀⣠⣿⣿⣿⠋
⠀⠠⢤⣶⣶⣶⠦⢤⣀⠀⠀⠀⠱⡀⠀⠙⣾⢘⠄⢸⡏⣿⠇⠀⠀⢀⣼⡏⣽⠟⠁
⠀⠀⠀⠈⠳⡌⠙⠲⢬⣿⣳⣤⡄⠀⢄⠀⠹⣾⣻⣼⣿⠏⠀⢀⣴⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠈⠻⡤⠀⠈⠛⠲⣽⡶⣄⠳⠄⢹⣨⣻⡟⢀⣴⣿⣿⣿⣯⢶⢒⣿⣿⣭⣿⣿⣿⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠯⣥⣤⠈⠛⢿⣷⣌⠲⠃⣽⣣⡿⢛⣻⣿⣷⣷⣿⠿⢟⠋⢍⣁⣦⡼⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⣓⣮⣽⣿⡁⢠⣿⣾⣿⣯⣉⣭⣤⣤⣶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⠿⠛⠛⢉⡴⡶⠀⡉⠻⠿⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠁⢀⠄⣠⡆⠔⠋⠀⡇⠈⠓⠤⢦⣆⡀⠉⠙⠷⣦⡀
⠀⠀⠀⠀⠀⢀⠾⠡⠾⠐⠂⠐⠛⠉⠉⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠉⠉⠑⠓⢬⡹⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⢦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠧⣤⣤⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠀⠀⢸⣿⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣿⣾⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢻⠀⠀⠀⢿⣿⣿⣿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⢿⢠⠀⠀⢸⣿⣿⣿⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣦⣸⢸⠀⠀⢸⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡬⠋⠉⠱⡴⢄⣀⠀⠀⠀⠀⠀⠀⠀⠳⡀⣹⢸⡄⠀⢸⣿⢽⣿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⡄⠤⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡌⢹⣦⣦⢀⠀⠀⠀⠀⢸⠛⢿⢸⡇⠀⢸⣿⢼⣿⢮⡀⠀⠀⠀⠀⠀⢀⣄⠔⠒⠁⠈⠀⢐⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⠀⠀⠹⢾⣿⣿⣿⣧⣆⠀⠀⠀⢳⣼⢸⡇⠀⢸⣿⢸⣿⠟⠀⠀⠀⡠⡤⠙⠇⠈⢀⣀⡀⠀⢤⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⡀⠀⠙⢿⣿⣿⣿⣿⣧⡄⢸⣾⢸⡇⠀⢸⣿⣼⣿⠂⣀⡌⠛⠀⠁⠀⢠⣴⣾⣭⣄⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡤⠀⠀⠈⠿⣿⣿⣿⣿⣼⣿⣿⡷⠶⢾⣿⢺⣧⡜⠈⠀⠀⢀⣤⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣖⠀⠀⠀⠈⠻⣿⣿⣷⣿⢿⡗⠶⢾⣿⣹⣏⠀⢀⣠⣶⣿⣿⣿⣿⠍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣽⣷⣄⠀⠈⠣⠤⡀⠀⠈⠙⢿⣿⣽⣇⣤⣸⣿⣿⣿⣴⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣾⡿⠉⠛⣦⣤⣤⣄⢀⠙⣒⣶⣀⣿⣿⣯⣉⣹⣿⣿⣿⣟⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⡿⠤⣤⡴⠋⢿⣿⡿⣷⣤⡀⣀⣉⣿⣿⣧⣬⣽⣿⣿⣯⡀⠀⠀⠁⠘⠁⠘⠀⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠁⠉⠀⢀⣀⣠⣽⣖⡺⠿⣀⠀⠙⠿⣿⣿⣿⢿⣧⣤⣼⣿⠿⠟⢿⣿⣷⣶⣤⣤⣀⡀⠀⠈⠁⠓⠀⡄
⠀⠀⠀⠀⠀⠀⠶⠊⠠⣤⡀⣄⢀⣋⣾⣿⣿⣿⣿⣦⣀⠁⠀⠀⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⢀⡙⣿⡿⣿⣿⣿⣿⣿⣶⡶⡄⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠁⠈⠀⠀⠀⠀⠑⣮⣶⣤⠴⠁⠀⠀⠀⠀⠀⡄⣶⣿⣷⣹⡄⠉⠉⠋⠙⠛⠟⠳⠄⢠⠄⠠⠴⠒⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⠉⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⡀⢀⠀⣀⣠⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⣕⣮⣽⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠁

⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⡀⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⢠⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⢿⡿⠟⣛⣉⣉⣛⡛⠣⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣀⢰⣿⣿⣿⣿⣿⠇⣶⣌⠻⣿⣿⣿⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⣰⣿⣧⡹⣿⣿⣿⡏⣼⣿⣿⣧⢹⣿⢏⣾⠟⠁⠀⠈⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣷⡜⢿⡟⣰⣿⣿⣿⣿⠄⣿⢸⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢿⣿⣿⣿⣿⣎⣡⣿⣿⣿⣿⣿⣶⣿⢸⡿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣌⢻⣿⣿⣿⣿⣿⣿⣿⡟⣱⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣌⡛⠿⠿⠿⠿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⡇⣶⠸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⡍⢙⠻⢿⣿⣿⢰⣿⡇⣿⣿⡿⠟⡋⢩⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣎⠻⣶⣌⡻⢸⣿⡇⠟⣡⣶⡟⣡⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣷⡌⠻⣷⣄⣿⣡⣾⠟⢫⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⣸⣿⠋⠩⠴⣶⣶⠾⣿⣿⣿⠷⢶⣶⠦⠍⠙⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⣿⣿⣿⣿⣶⣶⠆⣴⠟⠉⠻⢧⡐⣶⣶⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⣿⣿⣿⣿⣿⣯⣤⣴⣿⠀⣿⣦⣤⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠶⣿⣯⣙⣺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⣾⣿⣭⢏⣯⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣤⡴⣾⣯⢻⣿⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣗⣴⣾⣿⣿⣿⠻⢿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣆⣹⣿⠿⢿⣿⣍⢻⡀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠉⠀⠛⣁⣤⣧⣾⠙⢹⣿⣷⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠚⠛⠛⠛⠛⠛⠻⣿⣷⠀⠲⣿⣿⣿⡏⣰⣿⣝⢿⣮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⠀⢀⣤⡾⠟⠛⢉⡠⢶⣿⣿⡻⣿⣷⣿⣿⡿⠁⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢟⣉⡀⠀⢄⣤⡾⢿⠿⠉⣭⣾⣿⡏⣿⣷⣶⠷⠎⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠳⣿⢿⣷⣞⠫⣁⡀⠀⢀⡙⣿⣿⣿⣿⣟⠿⠿⠭⣍⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⡝⣷⣤⠉⣹⣧⣽⣿⣿⣿⣦⡉⢢⡀⠈⠑⠄⠀⣀⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⠤⠤⠤⠤⠬⠻⣿⡏⣼⣧⣼⣿⣿⣿⣿⣿⣏⠲⣿⣦⣾⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠭⠷⠒⠖⠒⣂⣠⣤⣶⡶⢊⠟⢃⣼⣿⡿⢿⣟⣿⠛⢹⣿⣳⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⠛⠁⢰⣿⡋⢻⣿⣾⣿⣿⣿⡿⡄⣿⡇⠛⠻⢧⣉⠩⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠁⠀⠀⠀⣼⣿⠁⠀⠾⣿⣿⣿⣿⣿⡇⣿⣷⠺⠒⠀⠙⢷⣄⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠁⠀⠀⠀⠀⠀⠀⢿⡇⠀⢀⣬⣽⣿⣿⣿⣿⣿⣿⣿⣷⡍⠉⠀⠁⢹⣧⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠈⣫⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠛⠢⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠰⣿⣿⣿⣿⣿⡀⢻⣿⣿⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⢷⣍⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠃⠈⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣛⣛⣻⣷⣲⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠏⣧⠙⠻⠽⣿⣿⣿⡟⡿⠿⣻⢟⣶⣿⣹⣷⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡽⣮⠙⠿⢷⣼⢽⠂⠐⠀⣾⣿⣿⣧⠠⢎⡛⠋⠌⢈⡹⡛⠻⣿⢿⣦⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣩⠒⠀⠀⠀⠐⠬⠀⠁⠀⣰⣿⣿⣿⣿⡧⠸⣁⠀⠀⠀⠎⠡⠀⠀⠉⢹⣿⣳⣄
⠀⠀⠀⢀⣴⣥⣿⢻⣿⠹⠇⣀⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠛⠸⡽⣿⣦⡀
⠀⠀⢀⡾⢿⠟⠿⢿⠈⠁⡀⠅⠀⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠸⠇⣧⡸⢷⡀
⠀⢀⣿⣷⣿⡛⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠄⠈⠁⠀⢘⣛⣶⣿⡀
⠀⡾⣿⣿⣿⣿⣷⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⡿⣿⣷
⢸⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡏⠸⢿⣿
⣾⣿⣿⡾⠏⡛⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠃⢀⠠⣷⣿
⣿⣯⣴⠡⠄⣀⠈⠽⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢰⣿
⣯⡍⠰⡦⣦⠖⠄⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⡀⢿⣿⣿⣿⣿⡿⢀⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠠⠄⠰⣯
⢿⣱⣿⣻⣙⡐⠀⠀⠀⠀⠈⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠁⠀⠀⠀⠀⠁⠈⠀⠖⡿
⢸⣿⢯⣉⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠉⠀⡆⣸⠇⢙⣽
⠀⢿⣿⡿⠃⠀⠰⠀⠀⡀⠀⣠⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣤⣤⣀⠀⠀⠀⠀⠀⠒⠸⠖⣻⡿
⠀⠈⣿⣀⠹⢢⠇⣠⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⢄⢶⢑⣼⣽⠁
⠀⠀⠈⢿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⣡⣿⣿⠏⠀⠀⠹⣿⣿⣦⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⡿⠁
⠀⠀⠀⠈⠻⣿⣿⣿⣻⠛⠉⠉⠅⡁⢠⠀⣾⣿⣿⠏⠀⠐⠀⠀⠹⣿⣿⣷⡁⠀⠀⢉⠉⠉⠙⡙⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⡟⣺⣃⣼⡗⡄⣸⣿⠟⠁⠀⠀⢐⠆⠀⠀⠉⠻⢿⣇⠀⠠⣌⠠⠙⠘⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⡿⡟⢿⣿⠶⢉⢉⡄⠀⢀⠀⢸⡇⣄⠀⠡⠄⣄⣥⢛⣄⠚⣲⣄⣷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⣿⣟⢲⣮⣶⢠⣦⡲⣞⠻⢉⠇⢂⣿⢧⣽⣿⣿⣯⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⠷⢾⣿⣿⣿⣿⣿⣿⣽⠵⠿⠛⠋⠁
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣟⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠔⡨⠖⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⣤⡾⣄⢰⡯⢦⡴⣢⣤⣆⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣉⣥⠶⡿⣼⡷⣳⣪⣽⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢋⡛⢒⡾⣋⣡⡜⠁⣼⠻⣯⠉⡒⠺⢿⣶⠦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣞⣿⢻⡇⡼⠋⢀⣿⢇⢻⣟⡻⣿⣶⡖⠂⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢙⡶⢻⠟⢙⣦⠴⣧⢿⡼⠿⣧⣿⣏⣙⠲⣝⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⡀⠐⣩⠄⠋⢀⣿⠛⡄⣸⢾⡑⡄⠹⡎⣿⣿⣟⠪⣇⠹⣿⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠊⠀⢈⣰⠞⠁⠀⢠⡾⢋⡴⢿⣼⠞⡧⠸⢄⣱⢋⣿⠍⠃⠈⠳⣌⣓⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⡀⠁⠠⠛⠷⠯⠀⢹⢋⠀⠢⠀⠀⠁⣿⢿⣄⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠡⡄⠂⠀⠋⠀⠐⠒⠃⣰⠋⣴⠀⢈⠈⠊⠀⠀⠀⠐⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⣅⠼⠁⠀⢸⢀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⠸⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠛⠉⠀⠀⠙⠛⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠁⠀⣶⠀⠀⣶⣄⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣄⠀⠙⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⡀⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠛⠻⢿⠀⠀⠿⠀⢀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⣄⣀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣄⣀⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢐⡆⠰⢆⢲⣖⠊⠴⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠐⠀⢨⡏⠂⠀⠐⠀⢹⣨⡷⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢵⡀⠐⢀⠀⢁⠀⡈⠀⢀⡱⠓⠁⠁⠪⢆
⠀⠀⠀⠀⠀⡠⠔⠒⠒⠂⠄⣘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡐⣥⣼⣷⢨⢬⡣⠸⣇⠀⠂⠀⢀⡕⢣
⠀⠀⠀⢠⠊⡠⠂⠉⡀⠀⠀⠈⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⢏⠌⠸⠺⡼⠶⠚⢸⣦⡇⡠⡪⠊⠉⠈
⠀⠀⠀⡆⠐⠀⠀⢏⣈⠂⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠟⠛⠚⠛⠚⠢⣮⢏⢴⠎
⠀⠀⠀⢃⠘⡀⠀⠢⡤⠃⢀⠄⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠁⡠⠊⠉⠉⠉⠢⡈⢻⠁
⠀⠀⠀⠈⠢⡈⠒⠄⠀⠔⢉⢠⠁⠉⠉⠉⡩⠭⠭⠭⠭⡉⠉⢉⢟⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠑⢄
⠀⠀⢰⠉⠅⠭⠩⠩⠉⠉⠉⠉⠉⢢⠤⠤⠭⠬⠭⠬⠭⠥⠤⡎⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠊⠛⠻⠛⠛⠋⠉⠉⠉⠉⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢆
⠀⠀⢱⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠒⠒⠂⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠂⠛⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣽⣉⣉⣋⡦⢰⠒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏⠋⠑⠛⠀⢸⣀⡠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⢀⡠⠄⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡇⠀⠠⠔⠁⠀⠰⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣐⣀⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠥⠚⢧⠀⢀⠗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂⠈⠓⡴⠡⠤⠀⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠠⠀⡀⠀⡀⡀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⣄⢋⣰⠈⢢⠠⠤⠤⠤⠼⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⠨⠈⠂⠀⠀⡀⡀⠀⠈⠁⠆⡈⢸⣀⡀⠀⠀⠀⠀⡎⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡘⠂⠀⠀⠀⠀⠈⠧⡟⢱⠒⡦⢄⡡⠂⠀⠀⠉⠉⠁⠐⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣧⡠⠊⠀⠀⢀
⠀⠀⠀⠀⠀⢀⢤⠖⡫⠉⠉⡐⠠⢍⡁⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⡠⠊
⠀⠀⠀⡠⠚⡱⠃⠀⠀⠀⡠⠃⢠⢄⡌⢝⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠇⠀⢀⣠⠴⡁⠑⡄⢀⡠⠖⠉
⠀⠀⡜⠀⠘⠂⠤⢀⣠⠌⠀⣐⠣⠆⣠⠋⠲⡑⢨⠃⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⢇⠀⠘⣄⠜⠉
⠀⢰⠁⠀⠀⠀⠀⠀⠀⠉⠒⠠⢉⡒⠦⢌⠔⢀⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡆⠀⠀⠀⠑⢄⡇⠀⠀⠃
⠀⠙⠒⠠⠤⠤⠤⠀⠀⠀⣀⣀⡀⠈⠑⠢⣤⠎⠀⠀⠰⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠢⠄⣀⢤⣀⠀⡀⠀⢀⡁⠤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠤⠤⠤⠄⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⡛⡄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⢺⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠸⡓⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡷⢸⣗⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣫⠃
⠀⠀⠀⠀⠀⢳⡘⠪⡙⠢⣄⠀⠀⠀⠀⠀⠀⢸⣿⡪⢸⣿⠀⢱⠀⠀⠀⠀⠀⢀⠔⡯⣆⠎⡎
⠀⠀⠀⠀⠀⠀⢣⣳⡝⢷⡀⠑⢦⡀⠀⠀⠀⡾⣿⢕⢸⣿⡀⢸⡀⠀⠀⢀⣔⢷⠇⣵⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢳⣿⣥⠹⣷⡀⠙⢆⠀⠀⣿⡿⡣⢸⣿⡇⠀⡇⠀⣴⡽⡟⢈⣾⠃⣰⠃
⠀⣄⠀⠀⠀⠀⠀⠀⢻⣿⡧⠌⢻⣤⡀⠳⡀⣿⣟⠎⢸⣿⣗⠀⣇⢾⢾⠃⣡⡿⠁⣰⠃
⠀⠱⡙⢖⡂⠤⣀⠀⠀⠙⣿⣯⠄⠹⣿⣆⠘⡿⡷⠕⢸⣿⣷⠀⣿⡟⠎⣰⡿⠁⡰⠃⠀⠀⢀⡀⠠⠤⠤⣤⠄
⠀⠀⠑⡔⣍⡲⢤⡉⠒⠤⣈⢺⣿⡥⠈⢿⣶⣧⢯⡢⢸⣿⡏⢰⡿⠀⣼⡿⠁⣔⢡⣴⣶⠿⢓⣫⠝⣪⠟⠁
⠀⠀⠀⠈⠪⣧⣤⣉⠻⢦⣌⠑⠻⣿⣧⠄⢻⣿⢑⢅⢸⣿⠇⣸⣇⣼⡟⢡⣾⡗⠋⢁⣥⠶⢋⡡⠚⠁
⠀⠀⠀⠀⠀⠈⠛⢷⢦⣄⡈⠛⢶⣤⡙⠷⣄⠹⣏⡃⢸⣿⢀⣿⣿⠏⣠⠏⣁⣴⠾⠋⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⡈⠙⠷⣬⡳⡹⣮⢸⣇⣼⣿⡥⣚⣵⠞⢋⡠⠖⠉
⠀⠀⠀⠀⠤⣔⠒⢒⢊⣍⣃⣛⣛⣛⣛⣶⠦⠭⣻⣿⣾⣿⣿⣿⣿⣭⣶⣚⣛⣋⣭⠭⠭⠭⣉⡡⠈
⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠦⠶⣽⣽⣭⣭⡭⠟⣛⣿⣟⣛⠫⢛⣻⠯⣷⠶⠶⠶⠛⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠥⢚⣩⣴⠾⠋⠁⣿⡆⠙⠻⢶⣬⣑⠢⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡡⠵⠖⠚⠉⠀⠀⠀⠀⢿⠇⠀⠀⠀⠉⠉⠒⠢⠁⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠇⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡀⠀⡟
⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠘⢮⠦⣠⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⣷⡎⠀⠀⠀⠀⠀⠀⢀⡠⣲⠿⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢗⡘⢧⣆⠀⠀⠀⠀⢘⠆⠀⠀⣿⡖⠀⠀⠀⢀⣠⠒⠁⡡⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠙⢾⣠⠀⠀⠈⡃⠀⠀⣿⠔⣀⠀⣠⠏⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡒⠀⠹⣤⠀⠀⢡⠀⠀⣿⡎⢠⠜⠀⢀⡾⠛⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠈⢳⡄⢢⡀⠀⣿⣷⠟⠀⣴⡏⠁
⠀⠀⠀⢀⣠⣤⣤⣴⠲⠖⠤⠀⠤⢝⣶⡀⠙⣦⡿⡀⣿⡟⢀⣾⣷⠖⣁⡠⠤⢤⠴⠒⠤⠠⠦⠄
⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⡀⠀⠀⠀⠈⠻⠦⣌⢻⣇⣿⣁⣛⡛⢋⣫⣭⡴⠶⠧⣬⠽⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠢⠤⠤⠩⠏⠟⢻⣯⡷⠶⠯⡥⠤⠤⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠂⢀⣀⠄⢸⠂⣉⠳⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠑⠒⠈⠁⠀⠀⢸⠀⠈⠲⣆⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠙⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡤⠤⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠴⠶⠖⠒⠛⠛⠉⠉⠉⠉⠀⠀⠈⢧⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠉⢹⡏⠁⠀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠉⠓⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣠⠟⠀⠀⠀⠀⠀⣰⠏⠀⢀⣼⣷⣤⠀⠈⢳⡄⠀⠀⣠⣤⡀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠀⢀⣀⠀⠀⠀⡏⠀⠀⢿⣯⣤⣀⠀⠀⠀⣷⠀⠐⣇⣀⡽⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡇⠀⠠⡏⠉⣳⠀⠀⣧⠀⠀⠀⠉⠙⣿⡷⠀⠀⣿⠀⠀⠈⠉⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠀⠀⣇⠀⠀⠙⢛⡥⠖⠒⢻⣆⠀⠘⠛⢿⡟⠁⠀⣰⠃⠀⠀⠀⠀⠀⡴⠋⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢿⠤⣤⠞⠉⠀⠀⠀⣸⠋⠛⠦⣤⣤⣤⠴⠚⠁⠀⠀⠀⠀⢀⣸⣇⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⣀⣠⡼⠞⠁⠀⠀⢀⣠⠞⢁⣀⣀⣀⣤⡤⠤⠴⠶⠖⠒⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⢀⡴⠛⠋⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣤⢾⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡴⢦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⣩⣴⡏
⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠳⠶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠘⢧⣤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢏⢄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡞⠰⣟⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢴⡶⣞⣷⢸⣗⡳⢞⣵⣶⣯⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣠⣄⡀⣤⣶⢛⡿⢾⣟⣷⢴⣺⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢼⣩⣟⣷⣿⡏⢀⣿⠈⣷⡎⣯⣙⣻⣟⠓⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣺⣿⣿⣿⢿⡍⢇⡟⣠⢰⣿⡱⣽⣿⣝⡻⣿⣍⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⡡⠟⣹⠋⣠⣿⡘⢻⣹⣿⠙⢿⡿⣷⣷⣝⣦⡻⣿⣤⣄
⠀⠀⠀⠀⠀⠀⢀⠾⠋⠔⣠⡶⠟⡇⢠⡿⢁⣷⢸⣿⡎⢧⠈⢧⡿⡿⢿⠍⢧⡙⢧⡉⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⡩⠀⣰⢿⡵⠛⢹⡿⠁⣟⠈⠑⠎⣾⣿⠀⠀⠀⠙⠷⡆⠀⠂⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠾⢋⠠⠘⡠⠆⢁⣨⡭⢁⡾⢻⠂⠸⠦⡄⠈⠉⠙⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣴⢠⠞⠀⢸⡄⣈⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠈⢿⢹⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⢸⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⢠⡜⢛⠛⠀⠀⠀⠀⠀⢸⡗⣿⢺⡇⠀⠀⠀⠀⠀⠀⢀⡞⠉⢻
⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣟⠀⠀⠀⠀⠀⠀⢀⣿⡧⠄⠀⠀⠀⠀⠀⠀⢘⡋⣿⣙⠃⠀⠀⠀⠀⠀⢀⣿⡗⠓⢺
⠀⠀⠀⣨⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⠀⠀⠀⠀⠀⠈⠱⣿⣿⡇⠀⠀⠀⠀⠀⣨⣿⣷⡶⠚
⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠅⠀⠀⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠿⠄⠀⠀⠀⠀⠀⠙⠻⠟

⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠿⠿⣿⣯⣽⠿⠿⠃⠀⠀⠿⠿⣿⣿⣽⠿⠿⠃⠀⠀⠿⣿⣿⣿⣿⣿⠿⠃⠀⠘⠿⠟⡍⣉⡩⠻⠿⠃⠀⠀⠿⠟⠍⣉⡩⠻⠿⠃
⠀⠀⠀⠀⣿⣿⣿⠀⡶⠀⠀⠀⠀⠀⣿⣿⣿⠀⣦⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣏⠉⢱⠀⡶⠀⠀⠀⠀⠀⣊⠉⢱⢀⣆
⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢄
⠀⠀⠀⠀⢸⣿⡇⢜⢦⠀⠀⠀⢀⠞⠛⢷⣦⣄⠀⠀⠀⠀⠀⢸⣶⣶⣶⡇⠀⠀⠀⠀⠀⢸⠛⠛⠛⡏⠀⠀⠀⠀⠀⠀⣤⣴⣦⣾⣧
⠀⠀⠀⠰⢁⣀⠀⣇⠀⠀⠀⠀⡜⠠⠸⡋⠉⠈⡄⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠘⢤⣀⡠⠃⠀⠀⠀⠰⢶⣤⣠⢣⠀⠀⠸⡀⠀⠀⠀⠘⢻⣿⡿⠃⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⣶⣿⣿⣿⣯⣤⣤⡀
⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠷⠶⠛⠃⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠸⠶⠿⠷⠇⠀⠀⠀⠀⠿⠛⠛⠛⠻⠿⠿

⠀⠀⠀⠀⣤⣿⣤⣀⣤⡀⠀⠀⠀⠀⣤⣿⣦⡀⠀⠀⠀⠀⠀⠀⣤⣿⣤⡀⠀⠀⠀⠀⠀⠀⣤⣿⣤⣀⡀⠀⠀⠀⠀⠀⣤⣿⣤⣠⡀
⠀⠀⠀⠀⣇⣀⣸⠉⠄⢣⠀⠀⠀⠀⣇⣀⣸⣿⣿⡇⠀⠀⠀⠀⣇⣀⣨⣿⣷⠄⠀⠀⠀⠀⣇⣀⡰⠛⡑⢄⠀⠀⠀⠀⣇⣀⣹⣿⣷⡀
⠀⠀⠀⠀⣿⣿⣿⢗⢂⣀⡆⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠍⠉⠀⠀⠀⠀⠀⣿⣿⣿⠆⣤⡄⠀⠀⠀⠀⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠻⠿⠟⠈⠋⠉⠀⠀⠀⠀⠻⠿⠟⠈⠉⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠈⠁⠀⠀⠀⠀⠻⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡾⠹⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⡇⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⡗⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠐⣇⠀⢈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⡇⠀⠈⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠤⡺⠁
⠀⠀⠀⠀⠳⡦⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠈⡇⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⣫⠟⣸⡁
⠀⠀⠀⠀⠀⢙⣦⢍⠋⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢀⠈⡷⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠟⣠⠞⠁⣠⠋
⠀⠀⠀⠀⠀⠀⠙⣶⠷⢤⡀⠘⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠁⠀⠐⣇⠐⠀⡽⠀⠀⠀⠀⠀⢀⡔⠶⠋⠁⣠⠞⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⠙⢦⡄⠀⠀⠳⢦⡀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣟⠀⠸⡧⠀⠀⠀⢀⣴⠎⠀⠀⢂⡞⠁⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠈⠳⣄⠀⠀⠹⠶⡀⠀⠀⠀⠀⢸⡀⡀⠀⢿⠠⢾⡇⠀⠀⣀⠞⠀⠀⢀⡴⠋⠀⠠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄⠀⠈⠛⢦⡐⠀⠹⢶⡀⠀⠀⠘⣷⠈⠀⢼⡀⣾⡁⢀⡜⠉⠀⠀⣠⠏⠀⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡄⠀⠀⠉⠳⣤⡀⠛⣦⣀⠀⠸⣧⠀⢸⡁⡿⠁⣼⠁⠀⢀⠞⠁⢠⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀⠀⠀⠙⠳⣌⠻⢆⡈⣿⡆⢸⢠⣿⣼⠃⢀⡰⢏⣀⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠑⠦⣤⣀⣀⠈⠛⢌⡻⣿⡃⢹⣸⡿⢉⡠⢚⣠⠿⠃⣀⣀⡠⣤⠤⠠⣄⢀⣀⣀⠀⣀
⠀⠀⠀⠤⢤⠠⢟⣉⢉⡉⠉⠉⠀⠉⠉⠙⠋⠲⠖⢤⠬⣭⣿⣷⣄⡑⢜⠷⠉⠞⠥⣎⣽⣯⡥⠚⠉⠈⠁⠀⠀⢀⣀⣀⣀⣠⠬⢥⠼⠋⢟⡀⠴⠆
⠀⠀⠀⠀⠀⠘⠛⠺⠭⣭⠉⡉⠐⠚⠚⠓⠒⠒⠲⠒⠒⠖⠲⠮⠹⠽⡷⠀⠆⠐⣋⢩⡭⢥⠤⠴⠶⠚⠒⠋⠉⠉⠀⡀⠀⣀⣠⠤⠴⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⠤⢤⠀⠠⣤⡠⡤⠴⠖⠟⣳⢾⠋⣰⢠⡌⠻⢾⡛⠛⠛⠛⠛⠛⠛⠛⠚⠋⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⠷⣡⡾⢹⢰⠙⣦⣱⢞⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢂⣹⠟⠁⠀⢸⢸⠀⠀⠘⢿⣁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠞⠁⠀⠀⠀⢸⢸⠀⠀⠀⠀⠉⢶⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⡀⠀⠀⠀⢹⠃⠀⣀⠀⠀⣀⣴⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢦⡀⠀⢸⠀⠸⣍⡷⠀⣿⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣸⣄⣀⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⢁⣠⡤⣤⣀⠉⠓⢯⡀⠀⠀⠀⠀⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣦⣀⣀⣠⠎⠀⠀⣰⣫⠞⠉⠓⢮⣳⡀⠀⠘⢦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣋⣳⣀⠈⠛⠁⠀⢠⠇⠀⠀⠀⡇⡇⠀⠀⠀⠀⡇⡇⠀⠀⠈⢯⠉⠉⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠛⠛⠛⣿⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⢸⡶⠶⠶⠶⠶⢾⣩⠇
⠀⠀⠀⠀⠀⠀⠀⢾⣙⡦⠤⠤⠤⠤⠤⢿⠀⠀⠀⡇⠀⠀⠀⢾⠆⠀⠀⠀⡇⠀⠀⢸⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⣀⣀⣀⣘⣆⠀⠀⡇⠀⠀⠀⠺⠀⠀⠀⠀⡇⠀⢀⡞⠀⢀⣀⡈⠉⠳⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠙⢦⡀⠳⠤⠤⠤⠤⠤⠤⠤⠴⠃⣠⠞⠛⠛⠻⠤⠗⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⣳⠀⠀⠀⠀⠀⠙⡶⢤⣀⣀⠀⢀⣀⣠⣴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⡍⠉⢹⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡞⢦⠀⢸⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢠⡇⠀⠙⠋⢠⠾⣆⠀⠀⠈⢳⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠼⠉⠁⠀⠀⠀⠈⠓⠋⠀⠀⠀⠈⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⣤⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠰⠶⠶⠶⠆⠠⠶⠶⠄⠀⠀⠀⠀⠀⢀⡴⠖⠛⡛⠳⢦⣄⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⣈⣁⡄⠀⠙⣧⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⠤⠤⠤⠴⠤⠦⠤⠤⠴⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⠦⠤⣤⠀⠀⢹⠆⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⠀⢠⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷⠹⣄⠀⠐⠒⡒⠋⠀⢀⡿⠀⠀⠀⢸⡇⠛
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣌⠳⣝⠶⣤⣈⣁⣤⠴⢋⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣇⣙⣛⣛⣋⣉⣉⣉⣉⣉⣙⣙⣛⣋⣸⡇⠀⠀⣤⣤⡄⠀⠀⠈⠳⣬⣛⡲⠶⠶⠶⠚⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⡴⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⢷⡘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣡⠟⣀⣀⣈⠉⠉⣁⣀⡈⠉⠉⣁⣀⡉⠉⠉⣀⣀⡉⠉⢉⣀⣀⡉⠉⢉⣀⣀⠉⠉⢉⣀⣀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠉⠉⢁⣀⣈⣉⣉⣁⣀⣈⠉⠉⠁⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣏⣉⣉⣉⣉⣉⣉⣙⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠖⠀⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠃⠐⠛⠛





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⢻⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⠏
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣄⠰⣿⣿⣿⣿⣿⣿⣿⡷⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣦⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣤⣴⣤⣴⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠀⠈⠛⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠻⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠳⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠉⠙⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⣰⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⣆⠀⠀⢀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣏⣀⣛⣛⠁⣿⠀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⣿⠈⣛⣛⣈⣹⠉
⠀⠀⠀⠀⢠⣴⣄⣀⣉⣉⣉⣉⡁⣿⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⣿⢈⣉⣉⣉⣉⣀⣠⣶⡄
⠀⠀⠀⠀⠈⠛⠋⠀⣭⣥⣭⣬⡁⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⢠⣭⣤⣭⣤⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣧⡀⣶⣤⠄⣿⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⣿⠠⠤⣶⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠙⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠋⠀⠀⠀⠈⠻⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠂⠀⠀⠀⢒⠤⢹⡶⡖⠒⠒⠒⠀⠂⠀⠀⠐⠒⢒⡊⢳
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸⡦⡸⢹⡄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠇⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⠉⠁⠀⠘⣡⣿⡞⠃⠀⠀⠀⠀⠀⠈⠀⠈⢉⡀⠀⠀⠇⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⣽⡏⢡⡀⠀⠀⠀⠀⠀⡤⠀⠀⠻⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⣀⠀⡈⠀⠀⠻⢣⣸⡇⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣯⠘⠋⠁⠀⠀⠀⢀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⢘⡫⠀⠀⠀⠀⠐⠂⠎⠸⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣯⠀⡀⠈⢉⣭⣿⣶⠆⡖⠰⣦⠴⠆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⣿⣯⣿⣟⢚⠸⡱⠿⢿⣻⣖⠀⠀⠀⠀⡀⠀⠀⠐⠛⠆⣳⣾⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣯⣭⣭⣟⣻⣿⠏⠁⣀⡀⠙⠛⠳⠶⠶⣤⣤⣄⣀⣀⠚⠛⠉⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠤⠤⣈⣀⣐⠶⠯⢥⡀⠉⢟⣛⠿⢿⣶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠙⠛⠛⠙⠒⠚⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡶
⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠹⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣾⠏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣦⠀⠀⠈⣿⣿⣿⣿⠅⠀⠀⣴⣿⣿⣿⣿⣿⡿⠇
⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣧⡀⠀⢹⣿⣿⡟⠀⢀⣼⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠉⠛⢿⣿⣿⣿⣾⣷⣴⣦⣠⡀⠀⠉⠻⣿⣿⣿⣧⡀⠘⣿⣿⠃⢀⣼⣿⣿⣿⠟⠉⠀⢀⣄⣤⣦⣶⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠙⠿⣿⣧⠀⢹⡏⠀⣼⣿⠿⠋⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⠙⢧⣸⣇⡼⠋⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠙⠛⠛⠿⠿⢛⣛⣛⣓⣿⣿⣒⣛⣛⡛⠿⠿⠛⠛⠋⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠿⠃⢸⡇⠘⠿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⡿⢿⠟⠛⠁⠀⠀⢸⡇⠀⠀⠈⠛⠻⡿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠛⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣿⣿⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣿⣿⣿⣿⣏⣤
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣻⣿⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣴⣇⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣾⣧⠀⠀⠀⠀⠀⣤⣻⣿⣿⣿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣿⣿⣿⣿⣿⣛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣾⡇⢀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣼⣿⣿⣿⣿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣷⣿⡆⠀⣶⣽⣿⣿⣿⣿⣿⣿⣥⡄⠀⠀⠀⢀⡀⣼⣧⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣧⣶⠈⢻⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⠇⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⣀⡀⡀⢀⣀⠠⣄⢀⣀⢀⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢁⣈⣿⣿⣿⣏⣁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠉
⠀⠀⠀⠀⠈⠙⢿⣿⣷⣿⣿⣿⣿⣿⣾⣷⣴⣦⣴⣬⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡛⠹⣦⣼⣿⣿⣿⣿⣿⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡟⠛⠿⠟⠻⣷⣿⣿⣿⡛⢿⣿⢈⣩⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⢛⣿⣯⣹⣿⣾⣿⣿⣿⣿⣷⣿⣿⣾⣷⣿⣿⣶⣇⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠻⠿⠻⠿⠋⡿⠻⠋⣀⣸⠷⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⡗⠀⣾⣿⣾⣤⡓⠘⠿⠻⡿⠿⣿⠻⠿⠛⠿⠙⠛⠙⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠀⡀⢸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠃⠀⠀⡇⠀⠉⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠁⠀⠀⠀⢀⠁⠀⠀⠈⠘⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠚⣩⡤⠶⠦⣬⡙⠂⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⢀⡀⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠁⠴⢚⠲⠄⠁⠀⠀⠀⠀⠀⢀⠂⠀⠋⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣇⣀⣠⣿⡇⠀⠀⠀⢀⠀⠀⠤⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠽⠿⠿⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠇⠀⠀⠽⠿⠿⠀⠒⠐⠃⠐⠂⠿⠿⠿


⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣒⣒⠴⠶⣂⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⣲⠁⢈⠆⢘⣒⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣍⣉⣁⣀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢔⣀⡢
⠀⠀⠀⠼⠷⠤⢤⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣧⣤⡆⠀⠀⠀⠀⠀⢸⣿⣟⣉⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠪⠭⣕
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠟⠿⠛⠛⠃⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠀⠀⣶⣤⠚⣿⠳⣴⣶⠀⠀⠀⠀⣀⣤⣶⣶⣄⣀⠀⣤⣴⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⣶⠖⣿⣿⡿⢶⡶⠀⠀⠀⠘⠿⠿⢻⠟⠿⠟⠈⠛⠛⢻⡟⠛⠛⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠸⠛⣿⣧⣠⣄⣀
⠀⠀⠀⠀⢠⣤⣭⣄⡄⡀⠀⠀⠀⠀⠀⢀⣐⢴⠀⠀⠀⠀⠀⣀⣲⠀⠀⠀⠀⠀⢸⣦⣍⣉⣵⣿⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⢿⣿
⠀⠀⠀⠄⢤⢤⡤⡤⡌⠄⠀⠀⠀⠀⠀⠀⠀⡭⠀⠀⠀⠀⠀⠁⠭⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⡩⡀⠀⠀⠀⠀⠀⠾⡷⠤⠤⢬⣷⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⣄⡟⠙⢂⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣙⢒⠂⣾⣿⠈⢳⠀⠀⠀⠀⠀⣖⢂⣐⠁⢤⠤⠐⢂⡁⠇⠀⠀⠀⠀⢸⠐⠩⠃⡇
⠀⠀⢀⣴⣆⣿⣿⣿⣿⣴⣦⠀⠀⠀⠀⠀⠀⠲⣒⣆⢇⠀⣿⡟⠀⠀⠀⠀⠀⢄⠀⡶⠔⠒⠀⢀⠁⡦⠆⠀⠀⠀⠀⣸⢖⠤⡖⠇
⠀⠀⠀⠙⠁⠈⠙⠋⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠓⠒⠁⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠈⠒⠚
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀⠀⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⣄⣀⣄⣀⢀⣀⣀⣀⣠⡀⣼⠀⡓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢸⠀⡇⠀⡉⢉⠉⢉⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣀⠀⠁⠀⠀⠀⢸⠀⠱⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠈⠑⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠳⠚⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⢤⡀⠀⠵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡷⠀⠄⠘⠦⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⢺⠀⠀⠀⠀⠀⠀⠸⣆⠀⠘⠒⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⢒⠒⠒⣒⠒⣐⢒⠒⡒⡒⠒⡒⠒⢲⠀⠀⠀⡠⠶⠉⠋⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠋⠀⠀⢠⠟⣡⠌⠻⡄⠀⠀⠙⡆⢸⠀⠀⣜⠃⠀⢰⠛⠄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⢸⡀⢈⡦⢀⡇⠀⠀⠀⡇⢸⠀⠀⢏⡀⠀⢀⣙⠆⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠳⠤⠤⠤⠷⠦⠴⠾⠤⠤⠤⠖⠁⢸⠀⠀⠈⠣⣤⠀⡁⢀⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡰⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⢸⡇⢱
⠀⠀⠀⠀⢸⡳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡷⢸⣇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣫⠃
⠀⠀⠀⠀⠀⢳⠙⠮⡑⠢⣄⠀⠀⠀⠀⠀⠀⢸⣿⢮⢸⣷⠀⢱⠀⠀⠀⠀⠀⢀⠐⡌⢆⠎⡎
⠀⠀⠀⠀⠀⠀⢧⣷⠝⢶⡈⠑⢄⠀⠀⠀⠀⡾⣷⡕⢸⣿⡀⢸⡀⠀⠀⢀⣄⣷⠇⣵⠃⣰⠁
⠀⠀⠀⠀⠀⠀⠈⢯⣯⡤⠻⣷⡀⠑⢄⠀⠀⣿⣿⡢⢸⣿⡆⠐⡇⠀⣰⡿⡿⢈⣼⠋⣰⠃
⠀⣀⠀⠀⠀⠀⠀⠀⢻⣿⡥⠌⢷⡥⡈⠳⡀⣿⣗⠍⢸⣿⣗⠀⣇⢜⣾⠇⢡⣾⠃⢠⠃
⠀⠻⡁⢖⠠⢄⣀⠀⠀⠹⡿⣍⠀⠹⣿⡄⠹⣿⡯⡁⢸⣿⣗⠀⣿⡿⠈⣰⡿⠁⡰⠃⠀⠀⠀⣀⠀⠠⠤⢤⠄
⠀⠀⠱⡰⢝⠢⣌⡙⠢⢄⡈⢾⣿⡄⠘⢿⣦⣧⢿⠪⢸⣿⡏⢰⡿⠁⣴⡿⠃⡴⣁⣤⣶⣯⠿⣛⡭⢂⠔⠁
⠀⠀⠀⠈⠪⣶⣦⡙⠲⣦⡈⠓⠿⣿⣤⠈⢻⣿⢳⠕⢸⣿⡂⣸⡇⣼⡯⢀⣼⡿⠟⢁⣡⡶⠛⣡⠔⠁
⠀⠀⠀⠀⠀⠉⠻⣲⣤⣄⠙⠻⢦⣌⠙⢧⡀⢻⣏⡢⢸⣿⠀⣿⣼⡟⣡⠞⢃⣤⡶⠋⣁⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⢷⢦⣀⠈⠛⠶⣌⣦⡹⣍⢸⣗⣼⣿⣫⢚⣥⡾⠟⡁⠔⠊
⠀⠀⠀⠀⢤⣴⠖⢒⡒⢒⣒⣚⡛⠻⠶⠦⠬⢙⡻⣿⣼⣿⣿⣿⣖⣫⡥⠖⣛⣒⣒⣀⡀⠤⢠⣀⠜
⠀⠀⠀⠀⠀⠈⠑⠒⠦⠴⠶⣭⣭⣭⣭⣭⣥⣶⠿⢟⣿⣟⠿⠟⣛⠯⣭⣭⣭⡥⠴⠶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡫⢝⣋⣥⡶⠟⠁⣿⡄⠙⠷⣦⣭⣓⢬⡲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢓⠩⠤⠚⠋⠁⠀⠀⠀⢿⡇⠀⠀⠀⠉⠛⠣⠬⢒⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⡿⠿⠿⠻⠿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⣠⣴⣶⣶⣦⠀⠐⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⠋
⢰⣮⣿⣿⣿⣿⣿⣿⠿⡳⣬⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡆
⣸⣿⣿⢋⣿⢿⣿⣿⣇⠀⢡⠹⣿⡕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⠁
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣾⣖⣿⣷⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠤⢿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿
⢿⡿⢿⣿⢦⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠄
⣙⠛⣻⣿⠿⣿⣋⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⣿⣿⣿⣧⣄⣀⣈⢻⣿⣿⣿⣿⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠈⠛⣷⣉⠙⠒⠯⣭⣿⣿⣿⣿⣆⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠋
⠀⠀⠈⠻⣦⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠈⠛⣆⡀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣄⣀⣠⣼⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠨⠴⢿⣿⣿⣿⡏⣿⣿⡿⠀⠀⠀⠀⣠⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⢿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣿⣿⣿⣿⣿⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⡟⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⠫⣍⣁⣀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡏⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠆⠀⣠⣴⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢠⡶⢛⣽⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠋⣵⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣀⡄⠀⡄⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡏
⢀⣾⠿⠿⣧⠀⠇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⡷
⣾⣥⠔⠚⡇⠘⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⣿⡀⠀⢠⠃⠀⠁⠀⠀⠀⣾⣆⢹⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⡄
⢿⣿⣾⣿⠀⠀⢀⣄⠐⣶⣿⣿⣿⣿⣿⣿⡟⠉⠛⠿⣶⣤⣄⣀⣀⠀⠀⠀⢀⣀⣼⣿⣿⡿⠟⠃
⣿⣿⣿⣿⣦⣬⣿⣿⣦⣬⣿⣿⣿⣿⣿⣿⣧⣀⣄⣀⣈⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⠀⡀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⢀⣉⣍⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠨⢉⢉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢘⠆⠀⠀⠆⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠉⠀⣀⠒⠈⢌⡐⠂⠤⢀⣀⠠⣤⠼⠉⠠⠶⠏⣤⣀⡀⢐⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠉⠉⠅⢘⡂⠸⠁⠀⠈⠉⠁⠀⠀⠀⠀⢀⣀⣤⣠⠀⠐⢊⠓⠟⠇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠂⠀⠐⠢⠖⠚⡇⠀⠀⠀⠀⠀⠀⢀⠂⣅⡉⠉⢙⠃⠀⠀⠐⠛⠛⠉⠲⠦⠤⠀⡠⠀⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠖⠒⠒⠠⠴⠀⡄⠸⠂⠀⠀⠀⠰⠤⠜⢃⣀⡀⠄⠤⣄⢰⢒⣋⡕⣒⠷⠶⣣⣅⣒⡦⠤⡁⠀⠉⠓⢈⡉⢄⠔⢒⠉⠉⡅
⠀⠀⠀⠀⠀⠃⠤⠀⠠⠼⠁⡅⠰⠦⣀⣀⣀⣀⣀⡠⢞⠈⠉⢈⠢⣿⠸⠀⠒⠨⠍⠛⠁⠀⠬⣁⡈⠉⠀⠍⠐⠽⢒⠉⢉⠩⢀⠀⠀⡇⡀⣠⢀
⠀⠀⠀⠀⠐⠣⠤⢤⡗⠟⠋⠭⣝⠄⠀⠀⠀⠀⠀⢀⡼⢄⠀⠀⡥⣴⠶⢄⠤⣠⣤⣀⣤⣄⣤⣄⡀⠀⠀⢀⣤⠤⠶⠎⣒⣩⡅⠖⠓⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣐⣀⣠⣆⣀⠀⠀⠀⣙⣰⠞⠓⠲⠒⢚⢭⢆⢲⠂⠀⠀⠀⠀⠈⠙⠙⠻⠽⠀⡠⠤⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁⢪⡁⠒⠀⠐⡒⠂⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠐⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣇⡀⠀⣽⣿⣿⣿⣿⣿⣿⡅⠀⣀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣧⡀⢸⣿⣿⣿⣿⣿⣿⠀⣠⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣷⡈⣿⣿⣿⣿⣿⡏⣰⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣶⣤⣤⣀⡀⠀⠀⠙⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟⠁⠀⠀⣀⣠⣤⣴⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣶⣤⣀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣠⣴⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣍⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⣠⣾⡿⠟⠁⢸⠀⠙⠿⣿⣦⡀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢠⡞⢳⡄⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⠀⢿⣴⣇⠀⠀⢹⡏⠀⠀⣸⣦⡿⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡶⠃⠀⠀⠀⠈⣷⠀⢸⡇⠀⣾⠁⠀⠀⠀⠘⢧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣉⡿⠶⠶⣄⠈⠳⣄⠀⢀⣀⣿⠤⠼⠧⠤⣿⣀⠀⠀⢀⡴⠋⢀⡤⠤⢴⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠳⣄⣨⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣏⢀⡴⠋⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⣾⢛⣷⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⣾⡛⣷
⠀⠀⠀⠀⠀⠈⠉⠙⢦⣤⣤⣤⣤⣼⠁⠀⠀⠀⠀⢸⡏⠀⠀⢹⡇⠀⠀⠀⠀⠈⣧⣤⣤⣤⣤⡴⠋⠉⠁
⠀⠀⠀⠀⣠⠶⣄⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⡴⠞⠓⠒⠒⠚⠳⢦⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣠⠶⣄
⠀⠀⠀⠀⠙⠶⠋⠉⠉⠉⠉⠉⠉⡇⠀⠀⠀⠀⡇⠀⠀⢺⡗⠀⠀⢸⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠙⠶⠋
⠀⠀⠀⠀⠀⢀⣀⣠⠞⠛⠛⠛⠛⢻⡀⠀⠀⠀⣇⡀⠀⠘⠃⠀⢀⣸⠀⠀⠀⢀⡟⠛⠛⠛⠛⠳⣄⣀⡀
⠀⠀⠀⠀⠀⢿⣬⡿⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⢿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠞⠁⣹⠷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠙⢦⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣯⣸⠗⠒⠚⠁⣠⠞⠁⠀⠀⠉⣿⠒⢲⡖⠒⣿⠉⠀⠀⠙⢦⡀⠙⠶⠶⣾⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⡄⠀⠀⠀⢀⡿⠀⢸⡇⠀⢿⡀⠀⠀⠀⢠⠾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣾⠻⡏⠀⠀⣸⣇⠀⠀⢹⠟⣷⠀⠈⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠘⢧⡼⠃⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣿⣿⣧⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣿⣿⣿⣿⣏⣤
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣻⣿⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣴⣇⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣾⣧⠀⠀⠀⠀⠀⣤⣻⣿⣿⣿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣿⣿⣿⣿⣿⣛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣾⡇⢀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣼⣿⣿⣿⣿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣷⣿⡆⠀⣶⣽⣿⣿⣿⣿⣿⣿⣥⡄⠀⠀⠀⢀⡀⣼⣧⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣧⣶⠈⢻⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⠇⢀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⣀⡀⡀⢀⣀⠠⣄⢀⣀⢀⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢁⣈⣿⣿⣿⣏⣁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠉
⠀⠀⠀⠀⠈⠙⢿⣿⣷⣿⣿⣿⣿⣿⣾⣷⣴⣦⣴⣬⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⡛⠹⣦⣼⣿⣿⣿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡟⠛⠿⠟⠻⣷⣿⣿⣿⡛⠿⣿⢈⣩⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠑⠀⠀⠀⠀⢛⣿⣯⣹⣿⣾⣿⣿⣿⣿⣷⣿⣿⣾⣷⣿⣿⣶⣇⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠻⠿⠻⠿⠋⡿⠻⠋⣀⣸⡷⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⣿⣿⡷⠀⣾⣿⣾⣤⡃⠘⠿⠻⡿⠿⣿⠻⠿⠛⠿⠙⠛⠙⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠀⡀⢸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠃⠀⠀⡇⠀⠉⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠁⠀⠀⠀⢀⠁⠀⠀⠈⠘⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢠⠰⣾⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣰⣾⡿⠷⣂⠀⠀⠀⠙⠁⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣶⣭⣿⣿⣿⣏⣅⠀⠈⣛⣻⣽⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠆⢡⢼⣿⡿⢛⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣤⣤⣀⣀⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢠⣤⣬⣿⣿⣿⣿⡿⠟⠛⠙⠫⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⡉⠉⠀⠀⠐⠒⢧⣸⣄⠤⢤⣄⣠⠄⠀⠀⢀⣤⣴⣤⣦⣶⣾⠿⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠿⡻
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠹⠟⠛⢠⣀⡘⣇⠤⠴⣾⠏⠀⠉⠉⠁⢈⣁⠀⠈⣻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠎
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⢿⣮⣤⣄⣀⢙⣿⡆⠈⡀⢠⣾⣿⣿⣭⣋⣁⡴⠃
⠀⠀⢠⠾⠁⠀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠈⠙⠻⠿⠿⣶⣿⡾⠛⢀⡉⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈⠛⠻⡋⠁⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣒⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡫⠭⠭⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠉⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢁⣴⣶⣶⣄⠁⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣀⣀⣀⡀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣋⣹⣟⣁⣀⣀⣀⣀⣻⣟⣘⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠵⠧⠤⠾⠿⠿⠿⠿⠦⠤⠷⠝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡿⠿⢿⣿⣿⣿⣗⣺⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠀⠀⠀⢀⣠⣀⠀⠀⠀⠬⠅⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣺⣿⣿⣿⡅⠀⢀⣿⣿⣿⣗⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠠⠭⠤⠄⠀⠸⣿⣿⠂⠀⠠⠤⠥⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⡿⠀⢹⣿⣿⣿⣿⣺⠅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⠩⠭⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠭⠁⠀⠀⠈⠛⠋⠀⠀⠀⠨⠍⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣵⣿⣿⣿⣿⢿⣾⣙⣻⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣽⣿⣿⣿⢿⣵⣿⣿⣾⣿⣿⣿⣭⣭⣝⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⠀⠀⠀⠈⠿⠋⠈⠋⠁⠉⠉⠛⠛⠋⠉⠉⠉⠛⠛⠻⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣏⠀⠀⣀⣠⣤⣶⠿⠛⣉⣠⣤⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣭⣴⣶⡿⠟⢋⣁⣤⣴⡶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣶⡿⠿⠛⠋⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⣏⣿⣿⣿⣿⣿⣏⣀⣤⣤⣶⡶⠶⠂
⠀⠀⠀⠀⠨⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⢻⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠐⢶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⠖
⠀⠀⠀⠙⣿⣿⣿⣿⣷⣦⣦⣀⣀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⡀⣠⣦⣶⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢹⣿⣿⣿⣿⣿⣿⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⡟⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣷⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣿⡿⢿⠿⣿⣿⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠛⠛⠋⠉⠀⣴⣿⣿⠟⠀⢸⠀⠘⢿⣿⣷⡄⠈⠉⠛⠛⠻⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⢸⠀⠀⠀⠈⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣠⣄⠀⠀⠀⠀⠀⠀⡀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣯⣦⣶⡆⣠⠆⢀⣽⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⠛⣿⣿⣿⣿⣿⣿⣏⣹⣿⣷⣏⣿⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⠓⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⣿⣿⣿⠻⠿⠛⣿⡟⣹⣻⠉⢀⣀⣴⠁⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡘⢛⣯⣿⣷⠀⠀⠀⢸⣟⣿⣿⣀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⢴⠄⠀⠈⠙⣿⣿⣿⠗⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣠⣤⣫⣿⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠀⠀⣾⣦⡀⢹⡄⣿⣿⣿⣿⣿⣿⣯⣿⣍⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣮⣿⡿⡛⠉⠙⢿⣿⣿⣗⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⠀⠀⣀⣀⣠⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡟⠛⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⢿⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣽⣿⣿⣿⣿⣿⣿⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣶⠀⣼⣿⣿⣿⣿⣿⣿⠃⢀⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⡟⢀⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠲⢶⣶⣤⣦⣦⣤⣤⣄⣀⣀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⠏⠁⢀⣀⣀⣠⣤⣤⣤⣴⣤⣶⡶⠒
⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣘⠻⣿⣿⣿⣿⣿⡜⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⢟⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣹⣿⣿⣿⣧⠹⣿⣿⢇⣿⣿⣿⡿⣋⣵⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣟⣿⣿⡎⣿⡟⢸⣿⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⣿⣿⣷⣿⣷⣼⣇⣿⣯⣾⣿⡿⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣭⣿⣿⡟⢻⣿⣭⣭⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠻⠿⠿⠿⠷⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠠⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⠛⠋⠛⠃⠓⠘⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠓⠇⠀⠈⠶⠖⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠏⠏⠀⠀⢰⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠀⡀⠀⢀⠤⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠀⠠⠒⠀⠀⢨⠐⣀⡠⠄⠣⣼⠀⠀⠀⠀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⣠⠀⠈⠃⢀⢆⢀⠀⣈⠖⢁⠋⠈⠁⠀⠀⠈⠃⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠙⠂⠄⡀⠠⠞⠀⠀⡰⠀⠀⠸⢀⡀⠀⠀⠀⠇⠉⣦⠤⠚⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠰⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠊⠠⡠⠛⡀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤⠀⣠⢎⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⡧⠰⣄⣀⣀⠄⢀⣠⠖⠃⠲⠒⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣀⠀⠀⠰⠀⡾⢁⠀⣠⣍⠓⠒⠋⠀⣠⢄⣀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⣰⠞⠁⠠⠈⢀⠀⠀⠀⠈⠉⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠴⠋⠉⠙⠦⠞⠀⠐⠉⠀⢀⣠⠂⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠆⢀⣀⡼⠁⠀⠈


⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣄⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡄⠀⣿⣿⣿⣿⡀⠀⠀⢠⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⣿⣿⣿⣿⡃⠀⣰⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⣀⣀⣤⣴⠖
⠀⠀⠀⠤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠘⢿⣿⣿⣷⣿⣿⣿⣿⠀⣼⣿⣿⣿⡿⠃⠀⠀⣠⣴⣶⣾⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣷⣤⡈⠻⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⠏⠀⣀⣴⣾⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣦⡹⣿⣿⣿⣿⣻⣿⣿⠟⠁⣠⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣌⢻⣿⣿⣿⠟⣡⣴⣿⣿⣿⠿⠛⢉⣀⣠⣤⣤⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣙⠳⣿⣿⡷⣞⣛⣋⣉⣡⣤⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⠿⠛⠛⣩⡿⢿⣿⡿⠿⠿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠛⠛⠁⠀⠀⠀⣼⡿⠁⣾⠻⣿⣶⣄⡀
⠀⠀⠀⠀⠀⣰⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⢰⡟⠀⠈⠛⠿⣿⣶
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢨⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣑⢼⠈⢧
⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢎⢸⡂⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠸⡙⣢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⡣⢸⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣥⠇
⠀⠀⠀⠀⠀⠀⢳⡈⠳⣉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡳⡅⢸⣿⠀⢸⡀⠀⠀⠀⠀⠀⠀⡠⢺⠱⣠⠋⡞
⠀⠀⠀⠀⠀⠀⠀⢷⡩⣌⢺⣤⡈⠳⣄⠀⠀⠀⠀⠀⣼⣿⣝⠂⢸⣿⠄⠘⡇⠀⠀⠀⠀⣤⢫⢸⢘⣴⠃⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣳⣒⡙⢧⣆⠈⠳⣄⠀⠀⠀⣧⣿⡪⡁⢸⣿⢕⠀⡇⠀⠀⣠⣪⡾⡜⢀⣾⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⣮⠌⠻⣾⡄⡈⠳⡄⠀⣿⡷⡑⠄⢸⣿⣕⠀⣿⢀⡜⡽⡇⡇⣠⡿⠃⢠⠏
⠀⢶⢤⣀⠀⠀⠀⠀⠀⠀⠻⡿⣭⣔⠘⢿⣵⡀⠙⢦⡇⣟⢌⠂⢸⣿⡗⠀⣿⣞⣾⠇⠁⣴⡿⠁⢠⠏
⠀⠈⢦⠙⠍⡑⠲⢤⣀⠀⠀⠙⣟⢷⣒⡀⠻⣷⣧⠈⣧⢯⠢⡁⢸⣿⡯⠀⣿⣿⠹⢀⣼⡟⠁⣠⠋⠀⠀⠀⣀⡤⡀⡂⢒⡒⡢⠒
⠀⠀⠈⠢⠱⣈⡳⢤⣈⠙⠲⢤⡈⢻⣻⣯⠄⠙⣿⣧⣷⠳⡕⠄⢸⣿⡯⠀⡿⡇⢁⣾⡟⠁⣰⣃⢄⣶⢿⠷⠇⣃⡥⠓⣡⠊
⠀⠀⠀⠀⠑⢞⣶⣤⡉⠳⢧⣠⡈⠓⠿⣿⣯⣄⠈⢿⣿⡜⢎⠂⢸⣿⡂⢨⣿⢠⣾⡟⢁⣴⣿⠺⠙⢘⣨⡶⠛⢁⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠑⢽⠯⣤⣀⠉⠛⢷⣤⣈⠙⢿⣆⠈⢻⣯⠳⡁⢸⡿⠂⣾⣿⣿⠟⣡⡞⠉⢁⣤⡶⠛⠁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢝⣶⣦⡀⠈⠙⠷⣦⣌⠳⣄⢻⣞⠆⢸⡏⢰⣿⡿⢫⡴⢋⣤⡾⠟⢉⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣙⣻⣷⣦⣄⣀⠙⠳⢮⣷⣽⣆⢸⣶⣿⣿⣞⣥⡾⠛⣉⣴⣒⣉⣀⣀⢀⠀⠤⠄⠀⡄
⠀⠀⠀⠀⠀⠒⠯⣉⡀⠐⠴⠦⠤⠤⢤⣬⣭⣭⣝⣓⣒⣪⣿⣿⣾⣿⣿⣿⣿⣶⣾⣯⣭⡤⠤⠴⠶⡒⠒⣒⣤⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⠚⠷⠧⠶⢶⣶⣶⣶⣶⠶⠖⢛⡽⣿⢯⣛⠋⠩⢙⣒⠶⣶⠶⠿⠗⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠴⢚⣩⣴⡾⠟⠉⠀⣿⠀⠙⠫⣗⢦⣌⡓⠬⣑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣊⣤⠤⠟⠋⠁⠀⠀⠀⠰⡗⡇⠀⠀⠀⠉⠛⠻⠤⣈⣓⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⢤⢤⠤⠤⣄⢀⡤⠤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠚⠚⠚⠒⠒⣿⢸⣛⣛⣛⣛⣛⣛⣛⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣄⠀⠀⣿⢸⢐⠶⡶⡶⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠘⠧⣌⠠⠄⣹⠘⠒⠖⠒⠶⢒⣲⣒⠲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠶⠟⠀⠀⣿⠀⠀⠀⢀⡼⠋⣭⡙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣠⣀⣄⣿⠀⠀⠀⠸⣇⠀⣽⡆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠤⠤⠿⠧⠤⠤⠟⠀⠀⠀⠀⠙⠳⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡿⠿⣿⣿⣿⡿⠿⢿⣛⡇⠀⣰⠞⣛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣏⢺⡇⡷⠀⠀⣿⡇⢸⡇⠀⣻⡅⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣙⣶⣶⣾⣿⣿⣷⣶⣾⣉⡇⠀⠻⣦⣭⣥⠾⠁





⠀⠀⠀⠀⠀⠀⠀⢰⣖⡆⣴⡄⢦⡴⢰⣄⣶⢰⣒⢰⣆⣴⢲⡖⠀⣦⣠⡆⣶⡒⢲⡖⢰⣀⣆⡔⢲⣰⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠘⠉⠙⠘⠁⠘⠉⠘⠘⠒⠘⠈⠋⠈⠁⠀⠃⠁⠃⠛⠒⠈⠃⠘⠀⠃⠑⠚⠘⠓⠋

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

⠀⠀⠀⠀⠀⡰⠫⠍⡒⠤⣀
⠀⠀⠀⠀⡰⠑⠀⢠⣀⠁⠀⠉⠒⠤⣀
⠀⠀⠀⡰⠁⠀⢠⡏⠻⠿⣶⣤⣀⠀⠀⠉⠒⠤⣀
⠀⠀⡰⠑⠀⢠⣿⡃⠀⠀⠈⠉⢿⣓⡄⠀⠀⠁⠂⠍⡒⠤⣀
⠀⠠⠀⠀⢠⡿⠷⠹⠦⣄⡀⠀⣼⡟⠃⠀⠀⠀⢀⣤⣀⠀⠀⠁⠀⠄
⠰⠀⠀⢀⡻⣏⠀⠀⠀⠈⠐⠂⠚⠃⠀⠀⣀⣄⢻⢿⣿⡇⠀⠀⠀⠁⠂⠅⡢⢄⡀
⠡⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⡿⠛⠁⢸⣿⡇⠀⠀⠀⠀⠀⣄⣀⠁⠀⠕⡢⢄⡀
⠀⠀⠁⠒⠤⡀⠀⠀⠀⠀⠀⢀⣴⣶⡍⠿⢳⣦⣀⣨⣿⡇⠀⠀⠀⠀⣼⡿⠏⠀⠀⣤⣀⠁⠊⠕⡢⢄⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⢻⣿⡇⠀⠀⠀⢸⣾⠏⠀⠀⠰⣿⡿⢿⣶⣤⡀⠁⠊⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⣀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠤⣄⠉⠀⠀⠐⣿⡟⠀⠀⠈⠙⣿⣷⡀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢐⣄⡟⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⠂⠀⠀⠈⠀⠀⢀⠂⣌⡀⠀⠀⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠙⠳⢗⣦⣠⣶⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠈⠉⠉⠀⠀⠀⠐⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣶⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⣶⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠘⣿⣿⣿⣿⣿⣿⠃⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣷⡈⣿⣿⣿⡿⢁⣾⣿⣿⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣤⣿⣷⣾⣿⣷⣿⣷⣾⣿⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣿⣷⣾⣿⣾⣿⣷⣾⣿⣤⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣫⣿⣿⡿⢹⡏⢿⣿⣿⣝⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⠻⠿⠟⢿⠿⢻⠿⠟⠋⠙⢉⣴⣿⣿⣿⠿⠁⢸⡇⠈⠿⣿⣿⣿⣦⡉⠋⠙⠻⠿⡟⠿⠿⠻⠿⠟⠻⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⢸⡇⠀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠐⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠈⠀⠀⣿⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⡇⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⠜⢆⣱⠀⡀⠀⠀⠀⠀⠀⠀⠀⢘⣧⠀⠀⠀⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⣾⠻⠋⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠄⠹⣧⠀⡀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⢸⣿⣤⠃⠀⠀⠀⠀⠀⢀⡤⠴⢉⠋⡸⠁⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠦⠀⠻⣷⢠⡀⠀⠀⠀⠀⢰⣍⠀⠀⠀⢸⡿⢃⠄⠀⠀⠀⢠⡖⠞⠀⠀⣠⡾⠀⢘⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠈⠻⣇⢆⠀⠀⠀⠀⣉⠂⠀⠀⢸⣿⢏⣀⡄⢀⡤⡟⠀⠀⣠⣾⠟⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡖⠤⠄⡘⢿⡀⡄⠀⠀⠱⡄⠀⠀⢸⣷⣎⠀⣠⡼⠀⠁⢀⣴⠿⠇⢒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠀⠀⠙⣧⣤⠀⢀⠃⠀⠀⢸⣿⣯⣼⡇⠀⠀⣰⠿⠇⠀⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢶⡄⠀⠈⢻⣾⠘⢷⣁⠀⢸⣿⣿⠏⠁⢠⣾⠗⠀⡐⠈
⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣶⡒⠶⠑⠶⡒⠢⢤⢤⣙⣲⡂⠀⠙⣿⣬⢿⡀⢸⣿⡟⠀⣰⣿⣿⡵⠋⣀⣀⡤⢤⠴⣶⠶⠒⠦⠖⠒⠶⠦
⠀⠀⠀⠀⠀⠉⠉⠉⠒⠳⢦⣭⣄⠀⠀⠈⠂⠀⠀⠈⠙⠿⣧⣄⠈⠻⣼⣷⢸⡷⢃⣾⣿⣟⡿⡾⠋⠉⢁⣀⣠⡌⠀⠐⠒⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠣⠤⣄⣀⣀⠀⠀⠀⠠⠉⠷⣦⣹⣇⣾⣡⣾⡿⡛⣉⣤⢤⣞⠛⠋⠀⠤⡤⠩⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠒⠲⠤⠐⠘⠂⠝⠉⣿⣷⡿⠿⠿⠯⢄⡲⠤⠾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠉⠀⠀⣀⡠⠘⡗⠀⠙⠻⡥⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢋⣤⣠⡤⠐⠛⠉⠀⠰⡇⠀⠺⣄⢠⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠈⠙⢧⣀⠈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠳⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊⠻⢿⣭⡿⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⣟⢻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣸⡿⠀⣼⢋⢻⣤⣿⣿⣿⠸⣛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠃⣰⢿⠛⣼⣿⣿⡯⣿⡀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⢛⣿⣿⣿⣿⣿⣇⠱⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠏⠸⠿⣿⢹⣿⡋⠉⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⣹⡟⢃⠙⠇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠈⠀⠠⣼⠇⠈⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢰⣠⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣈⡉⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣺⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢸⣿⣿⣾⣿⠇⠀⠀⠀⠀⠀⠀⣾⣄⣼⣿⠛⠈⠛⠂
⠀⠀⠀⠀⠀⢰⡾⠅⠀⠀⠀⠀⡀⠘⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡆⠀⠀⣰⣶⣴⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣄⠽⢀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠸⣿⣿⣞⢸⣿⠙⣿⣿⣿⣇⡀

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


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠃



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡏⠛⠀⠀⠀⠀⠀⠘⠃⡍⠒⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⡠⢠⠞⠙⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⣤⣌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡷⠀⠐⠒⢃⡤⢶⠦⠀⠀⣀⣀⣦⣠⣰⣎⣙⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠠⠖⣪⣿⣿⣋⣩⣷⣦⠀⠀⠀⠛⣾⣇⠀⠈⠛⣟⠻⣿⣿⣿⠉⢉⣻⣿⣯⡿⠂
⠀⠀⠀⠀⢠⣤⣽⣿⡿⣿⣿⣿⣿⣷⣤⣀⢀⣼⣿⣿⡄⠀⠀⢀⣤⣿⣿⣿⣿⣶⣿⣿⣿⣥⣿⠿⠗
⠀⠀⠠⡄⠘⢷⠃⠀⠒⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⡟⣿⣷⣌⠷⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢰⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠛⠿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣣⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡝⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠘⢳⠊⠀⠀⠀⠀⣴⠛⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⡧⠀⠀⠀⠀⠀⢫⠂⠀⢨⠀⠀⠀⢸⠀⠀⢰⡆⠀⠈⡟⠁⠀⢱⠁⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⢸⠀⠀⢐⠀⠀⠀⣇⠀⢰⠁⠀⠀⠀⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⣁
⠀⠀⠀⠀⠀⠀⡴⠲⣄⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣤⣼⣿⣤⣤⣤⣼⣷⣤⡄⠀⠀⠀⠀⢸⣏⣉⣉⣛⣀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⡄⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡤⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠁⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠴⠋⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⢸⡗⠒⠒⠪⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⠔⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠶⢶⡶⠶⣶⠶⠶⢶⡶⢦⠶⢶⡶⠶⣶⠶⢶⡶⠦⠼⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠀⢸⠀⢸⠀⢸⡄⠀⣷⠀⠘⢄⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣹⠦⠤⠤⠴⠃⠀⠀⠿⠂⢰⠁⠀⢸⠀⠈⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠸⣅⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⢠⠞⢦⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣻⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢽⣿⡟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣽⡿⢧⣤⣤⡀⠀⠀⠀⣤⣼⣿⣧⠀⠀⠀⠀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠟⠣⣧⠘⢻⣽⠍⠀⠀⠀⣈⣻⣿⡿⠀⠀⠀⠀⢼⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡲⣼⣷⣿⡿⠏⠁⠀⠀⠈⣽⣿⡟⠀⠀⠀⠀⠺⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠉⣭⠀⠈⠁⠀⠀⠀⡀⢀⣿⣾⣇⡀⠀⠀⠀⠶⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣿⢶⣧⠄⠀⠀⠀⠀⠀⠘⠟⠸⡿⠋⠁⠄⢰⡜⠳⡞⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠿⠟⢋⠳⣾⣅⠀⠀⠀⠀⠀⠀⠐⠆⠰⠂⠀⡸⡷⢀⡽⢖⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⢼⣿⠈⢢⡀⠀⠀⠀⠀⠀⡀⠑⡍⠲⢿⣟⣻⣷⡎⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣑⣿⡧⣻⣿⣾⣾⣿⣭⣉⠁⠀⠀⠇⠀⣁⠀⠸⣯⣙⠇⣤⠴⠄
⠀⠀⠀⠀⠀⠀⠘⠉⡴⠛⡿⠧⣼⠄⠙⠟⢷⡄⠉⠁⠀⠈⠁⠠⠱⠑⢈⠟⢹⠖
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡃⠀⢸⢀⣄⣀⣀⡁⠐⢦⣀⡴⠆⠀⠀⠀⠀⣼⣄⡶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⡠⠖⢒⡿⢷⣿⢍⠙⣿⠿⣍⠒⠰⡿⠃⠃⠸⠆⠀⠀⢻⠻⣃⣜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣟⢹⢧⠀⠙⠀⠀⠀⠁⠀⠘⡓⢁⠀⠀⠀⢈⢾⣻⢿⠂
⠀⠀⠀⠀⠀⡠⠖⣒⣃⣶⣤⡏⣼⢆⡤⠀⠀⠀⠀⠀⣤⠠⠟⠷⡿⠂⣀⣄⢼⡧⠤⡁
⠀⠀⠀⠀⠀⠐⠊⢉⣽⠿⣟⠛⣻⢻⠳⣶⡒⠦⡄⢀⠀⢰⡄⠀⠀⠀⡟⠟⢦⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠏⠈⡇⠸⣦⠈⠳⣦⠀⠞⣧⣴⡧⣀⠆⠀⢠⣷⠾⠳⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠹⣆⠀⢈⡇⠀⠏⣠⡇⠀⠀⠀⠀⢉⣴⠀⠀⠠⡤⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⣤⣤⣰⣦⣠⠤⢺⠀⢀⡹⣦⣿⣖⡦⠀⠓⠃⠂⠀⠀⠠⢪⢰⠞⠃⠀⠐⠀⠀⣸⠻⡶
⠀⠀⠀⠀⠀⢈⡴⢞⡿⣿⠁⢀⡨⡄⠁⢰⡏⢣⠉⠙⠃⠀⡇⠀⠀⠀⣶⣼⢈⠃⡶⢆⠀⣰⠄⢻⠀⢱
⠀⠀⠀⠀⠔⠁⠀⡾⠁⠘⠀⠀⠓⠏⠀⠸⢀⠀⡇⠀⠀⠀⡆⠀⠀⠘⠘⠀⢈⠁⠈⢄⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡇⠀⠀⢈⣦⣶⣶⢦⢤⡧⣀⣀⠰⠤⡀⢺⠀⢀⠎⢠⡇
⠀⠀⠀⠀⠀⢀⣤⠞⢙⡀⠀⠉⠿⠧⣶⡶⠾⠿⠿⠤⠷⢾⡾⣛⡗⣰⡆⣃⢘⢛⣾⠃⣾⣗⣢⣄
⠀⠀⠀⠠⠬⠉⠚⠛⠉⠛⠾⣷⣾⡯⠥⠤⠤⠚⠛⠛⠿⣿⣿⣿⣿⣿⣿⣽⣭⡍⠻⠵⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣹⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⣿⠿⣿⢶⣆⠁⠀⠀⠀⣀⣴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢓⣿⣛⣶⡿⢻⣿⠉⢀⣰⣿⠟⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣤⣀⠀⣾⣿⣿⣿⣿⣿⡴⠆⡤⣹⡏⠀⢀⡴⠮⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⣿⣿⣿⣿⠿⣶⣶⠶⣻⣷⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⢹⣿⢿⣹⣿⢆⣤⣶⣤⣤⣶⠶⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣶⣾⣿⣿⣯⣽⣟⣻⣿⡷⣮⣽⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠛⠋⠁⢻⣾⣿⣿⣿⣧⣶⢾⣿⣷⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⡿⡿⢻⡿⣿⢿⡟⣿⣿⣿⣿⣿⣟⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⢟⣹⣟⡲⢡⡉⠐⣧⡎⣸⡿⣙⢿⣿⡿⣟⣿⣦⡈⠛⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⣰⣿⣿⡯⢁⣾⣏⡇⢼⣷⠈⣿⡌⢦⢿⣷⡌⠳⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣤⣄⡀⠀⡻⠟⠀⣠⣿⣿⡥⡁⣼⣿⣧⢿⣿⡈⠾⢿⣷⠀⠈⠛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣤⣤⣯⣽⣛⣾⡧⠀⣴⣿⣛⣿⣤⠁⣾⣿⣟⣿⣿⡇⠈⣾⣿⡆⠀⠀⠸⢿⣧
⠀⠀⠀⠀⠀⠀⢀⡤⠶⠟⠛⠋⠛⠛⠙⠛⢿⡿⣇⣴⣿⣿⣾⣯⣷⣼⠀⣾⣯⣿⣿⣿⣇⠆⠈⣿⣧⠀⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣏⣾⣿⣿⣿⡿⢿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣄⡀⠀⠰⣿⡆⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⣿⠿⠟⠉⢩⣿⣿⡿⢡⣾⠿⠿⢿⠄⣿⣿⡟⠙⢉⡉⣁⣀⣀⣼⣷⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠟⠀⠁⠀⠀⣸⣿⢟⡏⠀⣿⡟⣻⣀⣀⣸⣿⣿⡷⢀⣾⠿⠋⣹⣿⣿⣿⣿⠻⣿⣾⣿⣻⣿⣻⠿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠁⢀⣤⣴⣶⣿⣿⣿⠾⣿⣿⣿⡇⣹⡟⠿⣮⣻⣿⣷⠏⠀⢠⣾⣿⣿⣷⡏⢻⣷⡌⢋⡻⢿⣿⣿⣷⣄⠀⠉⠙⠛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠞⢁⣴⠾⠟⠛⠉⢡⣿⣿⣿⣿⣿⣿⣿⡃⣿⢠⠀⠙⠹⣿⠛⠀⣠⣾⡟⣻⣿⣤⠋⠈⢻⣿⣧⠸⢆⠉⠻⣿⣿⣷⣄⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣞⡃⠀⢾⣿⠇⢰⣿⡟⢀⣿⣯⠙⠀⠀⠀⠙⣿⣷⡆⣷⠀⠀⠙⠻⣿⣇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⢹⣿⠏⢀⣿⠟⣵⢏⣡⣀⣼⣟⣠⣿⠟⠁⠘⣿⡧⡟⠀⠀⠀⠀⠙⢿⣿⡄⣦⠀⠀⠀⠈⠻⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⡟⠃⠠⣿⠻⠂⠀⣽⢋⡷⠍⣻⠿⣿⣿⣿⢷⣤⡀⢸⣿⡇⡅⠀⠀⠀⠀⠀⠈⢻⣿⡦⣄⠀⠀⠀⠀⠀⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠃⠀⢀⣾⡏⠖⢀⣤⣿⣿⣿⣤⣧⣾⣿⣟⣿⣷⣍⠃⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⣸⣿⢧⣶⣾⣿⠗⠈⣿⣨⣿⣿⣿⡿⡇⢻⣿⣧⣼⣿⡟⢿⣿⣦⡀⠀⠀⠀⠀⠀⠹⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠀⠀⠀⠀⠀⣿⡟⣴⡿⠟⠀⣠⣾⣧⣸⠇⣸⣿⡃⣄⠈⠙⣿⣦⣿⡍⠀⠘⠿⣿⣆⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣿⣷⠏⠁⠀⣠⣿⣿⣍⡃⠀⣿⣿⣉⠊⠀⠀⠙⣿⣟⣧⠀⠀⠀⠘⠻⣧⡀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡥⠀⠀⣠⣿⣿⡿⢻⠀⠀⣿⣿⠆⡀⠀⠀⠀⠹⣿⣷⣆⠀⠀⠀⠀⠈⠻⠄⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢸⠁⠀⢠⣿⣿⡿⠀⠈⠀⠀⢿⣿⢛⠁⠀⠀⠀⠀⠹⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⣼⣧⠗⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡉⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⢺⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠋⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⣉⡭⠷⡄⠀⠀⠀⠀⢠⡏⠗⡦⣄⣀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣉⡡⠴⢚⡉⠀⠀⠀⢹⠀⠀⠀⠀⡼⠛⠲⢤⣈⠈⠟⢲⢤⣀⡀⠈⠙⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣉⡤⠖⢋⡥⠖⣚⣨⠵⠚⠛⣆⢸⠀⠀⠀⢠⠇⢀⠀⠀⠈⠙⠓⠦⣌⡁⠿⢳⠦⣄⣀⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⡞⢀⡜⠁⢀⣠⠤⠒⠋⠁⠀⠀⠀⠀⣿⢸⠀⠀⠀⢾⡀⠈⠁⠲⠄⣀⠀⠀⠀⠉⠛⠲⢤⣁⠛⠹⠖⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠉⠓⠦⢄⣀⠈⠁⠲⠄⣀⠀⠀⠀⠈⠉⠓⠦⣼
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⡇⠀⠀⣠⠖⢋⣉⣉⠙⠲⠄⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣨⠀⡀⢠⠞⣡⠚⣉⣤⣍⠙⠂⠀⢺⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡇⡏⡼⠁⡴⠁⡠⠄⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⡋
⠀⠀⠀⠀⠀⠀⠀⣧⣽⠘⡇⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢼⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⣚⣩⠷⠒⣿⡉⣿
⠀⠀⠀⠀⠀⠀⠀⡿⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⣀⣠⠤⠖⠚⣉⣡⠤⠖⠋⠁⠀⢀⣀⣻⡧⠿⢤
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⢀⣠⠴⢺⡇⢸⢸⠀⠀⣴⣋⣁⣠⣤⣖⣋⣭⡤⠤⠴⠖⠚⠋⠉⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡆⡇⠀⢀⣠⢴⢚⣭⠰⣼⠀⡇⢸⢸⠀⠀⡇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⢚⣹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣾⢉⣿⣯⡇⢹⢸⡇⢸⢸⠀⠀⡗⠶⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⢹
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⣿⣿⠉⠸⠇⢃⣨⠼⠃⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⢺
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⡏⠋⣀⠴⠚⠉⠀⠀⠀⣼⢸⠀⠀⡷⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⢾
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⠀⠛⠉⠁⠀⠀⠀⠀⣠⠞⠁⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡃⡇⠀⠀⠀⢀⣠⠤⠖⠋⠀⢀⡴⠋⠀⠀⣟⠒⠒⠚⠉⠉⠀⠀⠀⠀⣀⣀⣠⠤⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠙⠒⠚⠋⠉⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠙⠶⠶⠒⠒⠒⠚⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠾⣇⡀⣀⣀⡤⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢰⠿⡿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢎⣶⣶⣦⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⣠⣄⣠⣤⡀
⠀⠀⠀⠀⢸⣷⣿⡿⢿⣿⣶⣶⡆⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⢀⣤⣜⢟⣩⣛⢿⣿⣿⣿⣷⣦⣄
⠀⠀⢠⣴⣦⡼⠿⣷⡄⠻⣿⡿⢷⠄⠀⠀⠀⠀⢿⣿⣿⣿⣉⣥⣬⣿⣶⣼⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠘⢿⣿⡗⢀⣬⠀⠸⣿⣇⣤⣤⡀⠀⠀⠀⠀⠉⣽⢿⠿⣿⡿⢿⣿⢿⡿⠟⠀⠀⠀⠀⣿⡍⠛⠛⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠰⣿⣿⣿⣿⣿⣷⣿⣿⡏⠉⠈⠃⠀⠀⠀⠀⠈⡛⣛⠋⣭⣵⣌⢻⣦⡀⠀⠀⠀⠀⠀⠛⣷⣶⣾⣿⣯⣝⣿⠾⣿⣿⣿⠆
⠀⠀⠀⠈⠉⠀⠻⠟⠿⣿⣿⣧⣾⡆⠀⠀⠀⠀⠀⠘⣿⡛⢶⣿⡏⠛⠿⠟⠁⠀⠀⠀⠀⢰⣷⣾⣿⣙⡿⣿⡿⣿⣦⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠑⠂⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠈⠛⠋⠉
⠀⠀⠀⠀⢀⣤⣤⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣤
⠀⠀⠀⠀⢿⠍⠻⡿⠿⣿⣿⣿⠳⠀⠀⠀⠀⠀⣀⣼⣿⣿⡷⣴⡦⢴⡶⠄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣶⣿⣿⡧
⠀⠀⢰⣿⣷⣶⣼⣿⣦⡾⠿⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣹⣿⣰⣿⡦⣀⠀⠀⠀⠀⡀⠐⢿⡿⠿⢟⣿⣿⣷⡄
⠀⠀⠀⣿⣿⡷⠈⣥⣼⣦⣶⣤⣄⠀⠀⠀⠀⣠⣾⣿⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣶⣦⣀⠾⣛⢿⣿⣿⣿⡆
⠀⠀⠀⠈⢩⣿⣮⣵⣶⣿⡿⠇⢋⣀⠀⠀⠀⣴⣶⣿⣿⣿⣷⢌⣙⣿⣿⣿⡿⠇⠀⠠⣿⣿⣿⣤⣉⣻⣿⣾⣿⣿⣯⣤⡀
⠀⠀⠀⠀⣻⣿⡿⢏⣉⣵⣶⣿⣿⢿⡗⠀⠀⠙⠛⠛⣫⣽⣧⣿⣿⣿⣿⠿⠁⠀⠀⠀⠸⣿⣯⣽⣛⣛⣋⣤⣀⣸⣿⣿⡇
⠀⠀⠀⠀⠸⠿⠿⠟⠛⣻⣵⡿⠻⠞⠃⠀⠀⠀⠀⠀⠻⠋⠉⠛⠛⠻⠟⠛⠂⠀⠀⠀⠀⠀⠙⠛⣿⠛⠽⠛⠋⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⣠⣤⣴⣿⣿⡷⢂⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⢸⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣿⣦⡄⣀⡀
⠀⠀⠀⣠⡿⢋⣭⣝⣿⣷⣿⣿⣀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣦⢿⣿⣿⣷⣿⣿⣦⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⣽⣿⣿⣦
⠀⠀⠈⣱⣶⠿⠿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠀⣠⣿⣿⡿⠿⣿⣷⣻⡿⣿⣿⣿⡆
⠀⠀⣠⣽⣥⣿⣿⣼⡟⣿⣿⣿⣿⠇⠀⠀⠀⠀⠙⣿⣿⣿⣿⣯⣩⣶⣵⣿⠿⠃⠀⠀⠀⠙⠻⣟⢡⣿⣶⡦⣿⣿⣿⣿⣿⠟
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠸⡟⣽⡿⠻⣿⠗⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⣘⢛⣿⣿⣾⣿⣿⠿⠛
⠀⠀⠉⠛⠉⠙⢹⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡞⣻⡆⠀⠀⠀⠀⠀⠀⠀⠢⣄⣹⣿⣿⠿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⢀⣠⣤⣤⡀
⢀⣴⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⠀⢠⣿⠋⠉⠙⣿⡆
⣾⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠘⣿⣦⣀⣠⣿⠃
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠛⠋⠁⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣠⣶⣶⣶⣶⣾⣿⠁⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⠉⠉⠻⣷⣤⣶⡿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣤⣶⣶⣄
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡏⠀⠈⢻⣷
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣷⣤⣤⣾⠏
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⢿⣷⣄⣀⣀
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⣿⡆
⠙⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⣿⣇⠀⢀⣼⡟
⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣈⣻⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠠⣷⢠⡄⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⠿⣧⠀⣠⣼⣟⣿⣿⣿⡿⠟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣟⠀⠀⢉⣤⣿⣿⣿⣏⠻⣿⠀⢰⣾⡇⠘⠛⠂⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠏⣀⢳⠀⣾⣿⣿⣄⠀⠀⣧⡀⢀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⢻⡿⣿⣿⣿⣿⣾⣿⣾⣾⣿⣿⣿⠿⡷⠾⠏⠀⠶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣾⠟⠁⠀⣠⣿⣿⣿⡿⠁⣿⣼⣿⣿⣿⠿⠃⠐⡇⠀⠀⣴⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠘⠋⣉⠉⠀⣀⡰⠞⠋⢹⡿⣿⣇⠠⣿⣿⡇⢽⣿⠇⠀⠀⠃⠀⢠⠏⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣴⣤⣹⣦⣤⣶⣶⣄⡤⠒⠂⠀⠉⠁⠈⠀⠀⢀⡜⣷⣾⠟⣁⣼⣿⠃⢸⣿⠆⠀⠀⠴⢢⣼⡀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠛⠁⠀⠀⢀⠀⠀⠀⠀⣿⠀⡠⣿⣿⣿⠏⠋⠀⣸⡇⠀⠀⠀⠀⢨⣿⡿⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠙⠁⠀⠀⠀⠂⠀⠘⠁⠀⠀⠀⠀⣿⠿⣷⡿⠛⠋⣀⠀⣸⣿⠁⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠉⣌⣉⡙⣿⡟⢦⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣞⣿⣿⣿⡶⠛⠄⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢠⠀⢀⣤⠀⠀⠀⠀⠀⠀⠠⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⢀⠄⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⠋⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠜⠁⢠⣶⠚⠁⣦⣈⣦⣤⣀⠀⠀⠀⠀⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠞⠁⠀⠀⠀⣹⠶⡂⠉⠉⠈⠉⠙⠳⣄⠀⠐⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⣀⣀⣀⠀⠀⠘⠆⠙⣿⣷⠄⠴⠦⡀⠘⣷⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⣿⣿⣿⣿⣆⡀⠀⠈⣤⡟⠀⠀⠀⣀⠀⣢⣽⣧⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⣇⣉⣿⣿⣿⣿⣾⣶⣷⣧⣀⠀⢠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣤⡶⣦⡀⠀⠀⠀⠀⣈⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⠿⢿⡃⣠⡦⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⢀⣿⡏⠀⠌⣿⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡛⣻⣦⣄⣠⣿⣿⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣾⠟⠀⠀⠀⠀⠛⠛⠹⠿⣿⣿⣿⣿⣽⣿⠋⠉⢻⣿⣿⣿⣿⢿⣄⣤⣤⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠀⠀⣹⣿⣿⣿⣇⣸⣧⡌⢠⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡶⠒⣀⣴⣶⣦⣤⡀⠀⠀⠀⣸⣿⣿⣿⣄⠀⣼⣿⣿⠟⠁⠀⢈⡿⠿⢿⡿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⣿⣤⣼⣿⣿⠁⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠠⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠇⠈⢿⣿⣿⣿⣿⠟⠁⠁
⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠈⢯⣉⠉⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣁⠉⠻⣧⣀⣄⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢀⠘⠋⠉⠹⣿⣿⡿⣿⣿⣿⣿⣋⣉⣉⣉⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⡿⠿⣿⣿⣟⠛⠋⠉⣹⣿⣿⣿⣿⣾⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠐⠏⠉⠻⢿⣶⡾⠛⠿⣟⢿⡟⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠈⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣾⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣿⣯⣿⣿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣬⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢴⣿⣳⡗⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⣶⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣻⣿⣷⡾⢹⠏⢲⣿⣟⣛⢦⡀⠀⠀⠀⠀⠲⣿⣻⣿⣿⣧⠀⠀⠀⠀⠀⠘⢿⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⢸⠉⠣⣼⣄⠀⠟⣧⣨⡷⠈⠀⠀⠀⠀⢀⠻⣿⣿⣿⡟⠀⠀⠀⠀⠀⠠⢶⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢤⣈⣿⣆⣾⣿⣯⣶⠖⠂⠀⠀⠀⠉⠻⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⣸⣿⣿⠯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣷⣂⡽⣿⠿⠿⣿⣟⠛⠀⠀⠀⠀⠀⠀⠘⡟⢻⣿⡄⠀⠀⠀⠀⠀⠀⠉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣡⠈⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⣿⣿⣆⡠⠀⠀⠀⠀⡙⢿⣿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣨⣷⣤⠾⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏⣹⣿⡿⢉⠉⡀⠀⣠⣀⡾⢿⡢⡿⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣉⣿⡟⢡⣔⡂⠀⠀⠀⠀⠀⠀⠚⠙⠀⢸⠟⠀⠀⠀⠈⠀⢻⣿⠀⠈⢧⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠜⠛⠋⠉⣸⢻⡮⣮⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠛⠀⠀⢀⡘⢿⠃⣠⡶⠻⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⣾⣿⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢦⠙⠦⣾⣿⡿⡟⣾⣿⣦⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣘⣦⡄⢀⣿⣹⢀⣤⣈⣧⣤⣤⡤⠤⠀⠀⠀⠀⡷⠀⠀⠇⠀⠉⠙⣷⡿⣼⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣬⣽⣿⡷⢋⣿⣿⣿⣿⣿⡿⢷⣶⣄⡀⡀⠀⠀⠀⠇⠀⢤⣠⠄⠀⠘⡧⣶⣈⡇⠸⡶⠷⠗⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠁⣴⠟⢻⡿⣷⢀⣿⡄⠈⢻⡟⢷⣤⠀⠈⠉⠀⠀⠀⠉⠃⠀⣐⠡⡜⠆⠚⡿⠛⣩⡶⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⠇⠈⠘⣟⠁⠀⠀⠁⠀⠙⠇⢀⣀⠀⠀⠀⢠⠀⠀⠈⠀⠁⠀⠘⠁⣄⠸⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠼⣀⠀⢀⣿⣀⣶⣤⣶⣴⣤⣄⠀⠙⠳⣄⣰⠛⡇⠀⠀⠀⠀⠀⠀⢶⣿⣴⡟⠛⣋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠞⠓⣿⠿⣿⣋⣉⡉⢿⣿⢷⣿⣉⠑⠚⣸⣿⡟⠉⠛⠠⠿⠖⠁⠀⠠⢻⡟⢹⡟⢀⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢀⣴⠏⠁⣼⠸⣿⣄⠃⠀⢻⡄⠈⠙⠂⢠⠁⠉⠀⢤⣦⡼⠀⠀⠒⠀⠀⠀⠧⣤⣿⣿⡿⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣿⠀⣿⠛⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣆⣀⠆⣀⠀⠀⠀⠀⠖⣿⣤⠛⠷
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⣙⣉⣶⣶⣴⡟⢀⣿⡶⣤⡴⠆⠈⠁⠀⠀⠀⠀⣴⡄⢠⡟⢻⣿⠿⠟⠲⢀⣄⠀⣾⣏⣦⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⣠⠶⠿⠟⣻⣿⣿⣿⣷⣮⣿⣶⣿⣤⣀⠤⣤⣀⠀⡀⠚⢉⠁⢀⠀⠀⠄⠃⠀⠹⣿⠿⣇⣈⠁⡴⠆⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣼⠟⠉⢛⡥⠒⣴⡏⢸⡆⡙⢿⣆⠄⠙⢷⠀⢤⡌⠀⢸⣷⠀⠀⠀⠀⠰⠃⣘⠀⢹⡃⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⢸⠃⠀⣿⠁⠘⣿⡅⠀⠙⣿⣧⡀⠙⠛⢿⣴⣿⡿⢀⣸⠗⠀⠀⣤⣿⡿⠻⠟⢿⡌⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⢻⣧⠀⠹⣿⡆⠀⠈⠻⣧⠀⠈⣽⠃⣠⡇⢀⠉⠄⠁⠀⠘⠟⠷⣰⡆⠀⢉⢤⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣿⠀⠀⠸⣷⡀⢀⡴⢛⡀⠀⠃⢰⢻⣟⠀⠀⠀⠂⠀⠀⢀⡾⠟⠀⣠⠀⠈⠛⡋⠀⠀⠀⢸⣠⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠒⢒⣒⣶⣄⣹⣿⣆⣴⠶⠚⢛⠀⢀⣠⠼⢷⣾⣿⣯⣛⠷⠀⢘⠈⠁⠘⠀⠀⠀⠀⠔⠁⡄⢻⡟⠉⠀⠀⠀⠃⠀⠀⡀⣺⠛⢿⡛⠆
⠀⠀⠀⠀⠀⠀⠐⠈⠉⣩⣽⠟⣿⣿⠛⠁⠀⠀⢸⡀⠞⠁⢠⡾⠹⣆⠉⠛⢲⡇⠈⢠⡟⠀⠀⠀⠀⠈⣦⣀⡇⠈⠇⣠⣴⣤⠀⠀⢠⡔⠠⢿⠀⠈⢧
⠀⢰⠀⠀⠀⠀⢀⣴⠟⠉⢀⣾⠃⢹⡆⠀⠠⢻⢸⣷⠀⠀⢸⠇⡀⠸⡄⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⡼⢻⡟⠅⠸⠤⠋⡏⠙⠀⠀⢋⠋⠀⢻⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣼⠃⠀⠀⠃⠀⠀⠈⢸⠇⠀⠀⠘⠀⣅⠀⢷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠈⠁⠀⢰⠀⠀⢀⠄⡔⢀⡜⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠀⢹⣄⣼⣦⣤⣤⣀⣠⣧⠀⠀⠀⠂⢠⡄⠀⠀⣾⡂⠀⢀⡼⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⢶⠀⠀⠀⢴⠾⣺⡇⠀⠀⣀⣡⣶⣿⡛⠛⠛⢭⣉⣹⣭⣴⣿⣛⡂⢌⡡⢉⠀⢿⣧⡄⣾⣥⢀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⡏⢀⣜⣤⠀⠀⠀⠈⠛⠛⠛⣿⡿⠛⠛⠛⠛⠛⠛⢻⣾⣻⣿⣋⣧⢟⣱⣿⣇⣨⡄⢠⣉⣿⣟⣁⣸⣿⣧⣭⣿⣷⠾
⠀⠠⠀⠀⠀⢴⣍⡉⠙⠾⠿⠿⠉⠙⢷⣿⣷⣶⣾⣯⣥⣤⣤⣄⠈⠛⠛⠛⠿⢿⣿⣿⣿⣛⣽⣿⣿⣿⣿⣴⣮⣉⣉⡙⢻⣥⡴⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣾⡆⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⠀⠀⢰⣷⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣟⣿⣿⣿⣦⠀⢘⣿⣿⣿⣿⣿⣿⡃⠀⣴⣿⣿⡿⣻⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠙⢦⣤⣄⡀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡻⣿⣿⣇⡘⣿⣿⣿⣿⣿⣿⢃⣸⣿⣿⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣶⣦⣄⣄⡀⠀⠀⠙⢿⣿⣿⣟⢿⣿⣧⢸⣿⡏⣹⣿⡇⣼⣿⡿⣳⣿⣿⡿⠋⠀⠀⢀⣠⣠⣴⣶⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣾⣷⣤⣄⠘⢿⣿⣷⡛⣿⣇⢻⣇⣼⡟⣸⣿⢟⣾⣿⡿⢃⣠⣠⣾⣷⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣻⢿⣮⣿⠀⠛⠋⠀⣿⣱⡿⣟⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣷⣿⣿⣟⣻⠷⣌⠉⠠⡀⠀⠄⠉⣡⡾⣟⣻⣿⣿⣿⣿⣿⣿⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢛⣛⣿⣛⣯⣁⠀⢻⡟⢀⣈⣹⣛⣿⣛⡛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⠟⠀⢸⡇⠀⠻⣿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⢸⡇⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠛⠛⠛⠋⠋⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠙⠙⠛⠛⠛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⡞⠀⣤⠲⠌⢻⡄⠀⠀⠀⢀⣴⣾⡿⠟⠀⠀⠀⠀⠀⣠⣾⣿⣶⣀⠙⠛⢣⡀⠀⠀⣴⢴⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠉⠀⠨⣟⣓⠛⠁⠀⠀⠀⠀⠻⠿⣤⣤⣀⠀⠀⠀⢾⠟⠋⠀⢀⣽⣇⠀⠀⠀⠀⠘⢿⣯⣻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠠⢶⣿⠿⠿⠀⠀⠀⠈⠀⠀⠘⠙⠋⠙⠛⠆⠀⠀⠀⠀⠉⠙⠳⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⢠⣴⣿⣿⣷⣷⣤⡀⠀⠀⠀⠀⣴⣾⣿⣿⣿⡷⠀⠀⠀⠀⢠⢠⣿⣧⣄⠀⠀⠀⠀⠀⢀⣴⣾⣛⣖
⠀⠀⠀⠘⠯⠭⣟⣛⣻⡿⠃⠀⠀⠀⠸⣏⣴⡧⣿⣿⡆⠀⠀⠀⠀⠈⠡⢩⡭⢽⡄⠦⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠐⠶⢭⡤⠏⠀⠀⠀⠀⠀⠉⣫⣿⣮⣫⠟⠀⠀⠀⠀⠀⢠⡤⠭⠮⠜⠀⠀⠀⠀⠀⠈⠉⣹⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⣀⣤⣤⣤⠤⠠⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣄⣀⡀⠀⠀⠀⠀⣠⣶⣿⣿⣷⠀⠀⠀⠀⢀⣴⡿⢷⡶⢶⣦
⠀⠀⠀⠴⠿⢿⣿⠒⠴⣞⣉⠉⠀⠀⠀⣀⣼⣿⣿⣿⣿⡿⠀⠀⠀⠀⠿⢿⣿⣿⡟⠀⠀⠀⠀⠈⣿⠀⡀⡀⠀⣿
⠀⠀⠀⠀⠠⣴⠟⢫⡖⠋⠉⠀⠀⠀⢰⣿⣿⣿⡿⣻⡿⠀⠀⠀⠀⠀⠀⢀⣿⠿⠃⠀⠀⠀⠀⠀⠈⠀⢧⠃⠈⠉
⠀⠀⠀⠀⠀⠈⠓⠄⠝⠀⠀⠀⠀⠀⠀⠈⠉⠁⠎⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⣀⣶⣶⣶⣶⣄⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢤⣀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣄
⠀⠀⠀⢠⣾⣿⣿⣿⣿⡿⠿⠀⠀⠀⣾⣟⣉⣛⣉⣛⣿⠆⠀⠀⠀⣶⣏⣀⢀⣤⣏⣀⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠘⣿⣿⣿⣍⠉⠀⠀⠀⠀⠀⠀⠈⣉⣩⣉⣉⠀⠀⠀⠀⠸⠿⠤⣤⠤⢀⣀⡼⠃⠀⠀⠀⠈⠹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⢻⣿⠗⠀⠀⠀⠀⠀⠀⠀⠙⢒⠒⠋⠀⠀⠀⠀⠀⠀⠘⠛⠳⢿⣧⠀⠀⠀⠀⠀⠀⠀⠉⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢢⣀⠀⢀⣿⡆⠀⢀⡴⠀⠀⠀⠀⣦⡀⠀⣾⣿⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀
⠀⠀⠀⠀⠀⠈⢿⣧⣸⣿⣇⣴⣿⠃⠀⠀⠀⠀⠘⣿⣦⣿⣿⣴⣿⠃⠀⠀⠀⢤⣀⠀⠀⢸⣿⡇⠀⠀⣀⡤
⠀⠀⠀⠀⢀⣀⣈⣻⣿⣿⣿⣿⣋⣀⣀⠀⠀⠠⣤⣬⣿⣿⣿⣿⣥⣤⠄⠀⠀⠈⢿⣷⣆⠸⣿⢁⣴⣿⠟
⠀⠀⠀⠀⠀⠉⠛⢻⠿⠛⠿⣟⠛⠋⠁⠀⠀⠀⠀⠀⠘⠉⠉⠃⠀⠀⠀⠀⠀⢀⣀⣈⣛⣷⢿⣟⣋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠈⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⣸⣇⢀⣾⠀⠀⠀⠀⠀⣦⡀⠀⣸⣿⡆⠀⣠⡄
⠀⠀⠀⠀⠀⠱⣦⣄⠘⣿⡇⣀⣴⡞⠀⠀⠀⢀⣀⠹⣷⣿⣿⣾⠇⣀⡀⠀⠀⠀⠸⣿⣦⣿⣿⣧⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣴⠟⠋⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⠿⠋⠁⠀⠀⣀⣀⣘⣿⣿⣿⣿⣟⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠉⠙⠉⠛⠉⠁⠀⠀⠀⠀⠀⠐⠚⠋⠙⠓⠂⠀⠀⠀⠀⠀⠉⠻⠿⠟⠛⠿⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠈⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠈⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⠲⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡴⠈⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⢀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠄⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠠⢆⠀⠀⠀⠀⠀⡰⠇⠀⠀⠀⠀⠀⢀⠴⠂⠀⠀⠀⠄⠀⠀⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⠠⠀⠀⠀⠓⡀⠀⠀⠀⠈⡀⠀⠀⠈⠀⠀⢀⠂⠀⠀⠀⠀⠔⠃⠀⠀⠀⠈⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠰⠀⠀⠘⠡⡀⠀⠀⡞⠄⠀⠀⠀⠀⢒⠂⠀⠀⡐⠙⠀⠀⠠⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⠀⠀⠀⠂⢀⠀⠙⢀⠀⠐⡀⠀⠠⠀⢀⠌⠀⢀⠛⠀⢀⠐⠀⠀⠀⢀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⠀⠀⢀⠈⢦⡀⠞⠀⠀⠀⠺⠂⣤⠃⢀⠀⠀⠀⢀⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠆⢀⡀⠀⠠⠐⢨⠄⠈⢀⣄⠜⠠⠀⢀⣀⠐⠦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⡀⢉⠉⢖⣁⡂⢀⣠⠀⠘⣈⣰⠂⢉⢀⠀⣀⢀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠂⠉⠘⠀⠈⠁⢀⡈⠀⡈⠹⠀⢀⣾⠯⢬⡍⣥⠨⣿⡀⠀⠞⠁⡀⠉⡀⠀⠁⠀⠛⠈⠀⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠠⠀⠀⠀⠀⠀⠈⠀⢀⠀⡀⢒⡴⠏⠄⠀⠌⣷⠑⡀⠠⠽⢧⡔⢄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠒⠒⠃⠀⠲⠇⠐⠇⠐⠊⠈⠁⡰⠁⠄⠀⠰⠂⠀⣿⠀⠐⢆⠀⠠⠈⢤⠈⠁⠁⠂⠸⠖⠀⠇⠂⠘⠓⠒⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢠⡔⠁⠀⠀⣿⠀⠀⠀⠂⣤⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⢿⣶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡗⢼⡇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢀⣾⢯⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⢿⣶⡍⠁⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠔⣿⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠘⠓⢾⣿⣿⡦⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣙⣛⣿⣿⣤⣤⣾⣿⡿⠈⠳⠀⣀⡴⢶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣿⠻⣿⣿⣿⣁⣸⣿⣷⣦⣴⢟⣫⣐⣁⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⡈⠀⠀⠀⠀⣴⣿⡟⠃⠈⠉⠹⣿⣿⣿⣾⣿⠭⠽⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⡷⣤⡀⢸⠟⠁⣸⡆⠱⣦⣶⡟⠻⣿⣽⣿⡶⣦⣄⠀⠀⠀⠀⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠶⢶⣶⣶⣶⣤⣄⡙⠤⡙⢦⡀⠀⣿⠳⠾⢿⡿⣧⣀⠚⠛⣄⠉⠉⠙⠿⠦⢀⣠⣶⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣶⣷⣬⢿⣦⣿⣷⣲⣄⣹⣿⠋⢠⡤⠖⠀⣤⡞⣀⣴⣿⡟⣁⣤⣤⡤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⢶⣶⣦⣬⣝⣿⣿⣿⣿⣿⡿⢿⠋⠉⠻⣴⠿⠓⢠⣾⣏⣼⣿⣿⡿⢏⣉⡤⠴⠖⠚⠉⠉⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠙⠛⠛⠿⠿⣿⣿⣿⣧⡄⢀⠀⢀⣸⡆⢀⣿⣿⣾⣿⡽⠗⡈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣟⣛⣿⣷⣽⣿⣿⡏⠁⠀⣼⣿⣿⣟⠿⠟⠿⠻⠿⠿⠛⠿⠷⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡴⠟⠛⢉⣽⣿⣿⠛⠉⠉⣡⣼⡟⠉⠀⠀⢈⠙⠻⠿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣛⣶⣶⣤⡀⢉⣼⠿⣿⡷⠀⢤⣤⣿⣟⠂⠀⣺⠀⣱⣿⣿⣋⣭⣍⣉⣉⣀
⠀⠀⠀⠀⢠⡴⠶⠾⠻⠿⠿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣅⠀⢉⡀⠀⢠⣴⣿⡏⢁⠀⠉⠐⢿⣿⣿⡿⠿⠟⠻⠛⠻⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠗⣛⣛⢩⣽⣿⣿⣿⣿⠛⠃⠈⠉⠛⠟⣷⡏⠈⠉⠁⠀⢐⡿⣿⣿⣷⣶⣶⡀
⠀⠀⠀⠀⣀⣴⣾⠋⠷⠒⠋⠉⢁⣤⣾⣟⠿⣻⣿⢿⠃⢐⠠⠀⠀⠀⢹⣷⠖⠀⠀⣰⣾⣿⡿⠿⠿⠟⠛⠛⠓⠦
⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⣰⣾⠿⠋⠁⣠⣿⡿⠃⣠⣼⣶⣦⣭⣶⠈⣯⣿⣦⣿⣿⣿⣿⣿⣿⣷⣠⣄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠀⠀⣴⠿⢉⡤⠞⣿⣿⣿⣿⡟⣧⠒⣾⡟⣿⡏⣿⣿⣿⣯⣯⡋⠻⢿⣿⣕⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⣠⠞⠉⠀⠀⣀⣾⣿⠖⢹⡿⠃⡁⠀⠛⢷⣿⠄⣿⣿⣿⣿⣿⣿⣆⡀⠉⠙⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠉⠀⢸⡇⠀⢠⣄⣠⣬⣿⢀⡜⢿⣾⣷⠘⢿⣿⣧⡀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠁⠀⠀⢠⡾⢁⠀⠀⠀⢨⣷⣿⢰⠆⠈⣿⣾⣧⠈⠻⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠞⠁⠌⠀⠀⠀⠘⠋⣿⡾⠀⠀⠈⢿⣿⣇⠀⠈⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠈⣿⣿⡆⠀⠀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⢿⣿⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣀⣀⣀⣀⡐⠂
⠀⠀⠀⠀⡠⣤⡤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣬⣈⠬⠄⠁⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠉⠉⠉⠉⠈⡇
⠀⠀⠀⠀⡇⣿⡇⠀⣠⣄⡋⠀⢂⠀⠀⠀⠀⠀⠀⠀⢰⣀⣂⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⢨⢐⢒⢒⢒⢒⢊⠠⠀⠀⠀⠀⢀⣈⣀⡠⠀⠀⠀⣠⠀⠀⠇
⠀⠀⠀⠀⡇⣿⣿⡈⠷⠊⠑⠄⠀⠣⠀⠀⠀⠀⠀⠀⠰⠒⠒⠒⠂⠐⠲⢰⡆⠀⠀⠀⠀⢸⠨⢎⡨⢌⡨⢌⠀⠀⠀⠀⠀⡆⠀⢾⣵⠔⣁⣈⡀⠇
⠀⠀⠀⠀⡇⣿⣿⡇⡄⢑⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⢸⡇⠀⠀⠀⠀⢸⢨⣣⢠⡣⢘⡣⢐⠀⠀⠀⢀⣥⣤⣤⣶⣾⣿⣿⣶⣧⣤⡀
⠀⠀⠀⠀⡇⣿⣿⠇⠃⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣐⠆⣂⠆⢆⡒⢸⡇⠀⠀⠀⠀⢸⠐⡱⠔⡁⠄⠡⠈⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠤⠒⡠⡃⠢⠔⢸⠇⠀⠀⠀⠀⠘⢴⢲⣶⠒⠒⡦⠄⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠊⡒⠉⠒⠉⠑⡀⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠭⠤⠂⠀⠀⠀⠀⠀⠀⢸⣭⣭⣭⣭⣭⣷
⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⢁⣀⣸⣀⡀
⠀⠀⠀⠀⠀⣾⣒⣛⠃⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠓⠠⣤⠄⠃
⠀⠀⠀⠀⢨⣝⠩⣫⢡⡢⢙⣥⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣄⣰⣖⣲⢰⡦⣑⡀⠀⠀⠀⠀⠀⠀⠠⡏⠉⠉⡹⠋⣹⡂
⠀⠀⠀⠀⠀⣆⣀⣀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠄⠀⢽⢹⣿⡆⠉⠀⣯⠀⠠⠀⠀⠀⠁⠀⢀⣠⡀⡠⢊⣼⡶⠀⠀⠀⠀⠀⠀⣇⠠⠊⡠⠊⢸⡁
⠀⠀⠀⠀⠀⠥⠤⠤⠤⠥⠤⠄⡠⣪⣾⠗⢄⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡹⢊⣴⣿⠋⠀⠀⠠⠀⠀⠀⠀⡇⡠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⡴⠀⡠⣪⣾⣿⠁⠀⡠⠃⠀⠀⠀⠀⢸⣿⠿⠇⠀⡇⠀⠀⠀⠀⠀⠖⠖⣪⣴⡿⣫⠇⠀⢀⠔⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠰⢀⣀⣀⣁⣀⣘⠿⡻⠁⡠⠊⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠐⠀⠛⢌⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠇⠀⢀⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠃⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣄⡀⠀⢀⡀⠀⢻⠁⠀⠀⠀⣼⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢸⡇⠰⣏⡽⠀⢸⠀⠀⣠⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣇⣀⣼⣀⡀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠆⠀⠀⠀⠀⣸⠗⠋⠁⣀⣀⣀⡀⠉⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠞⠀⠀⢠⡾⠋⠉⠉⠻⣦⠀⠀⠈⢦⣀⣀⣠⡖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠉⢩⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢧⠀⠀⠙⠋⢀⣰⣋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠶⠶⠶⠶⠶⣾⠀⠀⠀⢰⣿⣷⣶⣶⣶⣶⣿⣷⡄⠀⠀⢸⡟⠛⠛⠛⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⠀⠀⠀⢸⣿⣿⣿⡁⣹⣿⣿⣿⡇⠀⠀⢸⠧⠤⠤⠤⠤⠤⣞⣹⠄
⠀⠀⠀⠀⠀⠀⠀⣴⠓⡶⠋⠉⣀⣀⠀⠘⣆⠀⠀⢸⣿⣿⣿⣇⣿⣿⣿⣿⡆⠀⢀⣞⣀⣀⣀⣀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠧⠼⠛⠛⠛⢦⡀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠁⣠⠞⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣤⣀⣀⠀⢀⣀⣠⠴⡞⠁⠀⠀⠀⠀⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠉⠉⣽⠀⠀⡗⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢸⠀⢠⠞⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢲⠋⠀⠀⢀⡾⢦⠈⠛⠁⠀⡇⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠉⠹⠤⠏



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣟⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣷⣄⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⣠⣾⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣿⣿⣦⠀⠀⢹⣿⣿⣿⣿⡟⠀⠀⣴⣿⣿⣻⣿⡿⠃
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡀⠘⣿⣿⣿⣿⠃⢀⣾⣿⣟⣽⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠙⢷⣶⣤⣄⣀⠀⠀⠀⠀⠛⣿⣷⣿⣿⣿⡀⢻⣿⣿⡿⢀⣾⣿⣿⣾⣿⠛⠀⠀⠀⠀⣀⣠⣤⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⣄⣀⠈⠹⣿⣿⣿⣧⠸⣿⣿⠏⣼⣿⣿⣿⠟⠁⣀⣠⣤⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣽⣿⣿⣷⣦⣌⠻⢿⣿⣆⢻⡟⣰⣿⡿⠟⣡⣴⣾⡿⣛⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣦⣙⢿⡈⢃⡿⣋⣵⣿⣿⣿⣷⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣛⣻⣿⣷⢿⡯⣾⣿⣟⣋⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⠿⠿⠛⠉⢸⡇⠈⠛⠿⠿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇


⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⡀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡀⠀⣿⣿⣿⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡄⢿⣿⣿⣿⣿⡿⢠⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣶⣤⣄⡀⠀⠀⠻⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣇⣾⣿⣿⣿⣿⠟⠀⠀⢀⣠⣤⣶⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣦⣄⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣠⣴⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠋⠉⢸⡇⠉⠙⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⢀⣠⢴⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡨⡀⢀⢁⠀⣠
⠀⠀⠀⠀⠀⣴⣯⣿⣟⣻⣷⣯⣷⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠁⣡⠞⣋⣑⠷⡁⠈
⠀⠀⠀⠀⠀⢸⣿⣿⠈⠅⢹⣿⡟⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢧⢾⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣌⠻⢟⣴
⠀⠀⠀⠀⠀⠀⠀⠙⠯⡽⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠊⠙⠉⠑⠁⠾


⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⡇⠀⠀⠀⡔⠲⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⢠⣤⣤⣤⣤⣤⣤⡄⢸
⠀⠀⠀⠀⡇⠀⠀⠸⣿⣿⠇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠠⠤⠤⠤⠤⠄⠀⢸⠀⠀⠀⠀⡇⠀⠈⠉⠀⠈⠉⠁⠀⢸
⠀⠀⠀⠀⠷⠤⠤⠤⣤⣤⡦⠤⠤⠼⠀⠀⠀⠀⠷⠤⠤⢤⣤⣤⡦⠤⠤⠾⠀⠀⠀⠀⠷⠤⠤⢤⣤⣴⡤⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⣶⣶⣶⣶⢲⡆
⠀⠀⠀⠀⢰⣶⣶⣶⣆⣀⣀⣀⣀⡀⠀⠀⠀⠀⢰⣾⣗⣶⣶⣶⣶⣺⣇⡀⠀⠀⠀⠀⢰⣶⣶⣶⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣟⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⠉⠁⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⠉⠉⣿⣿⣿⡇⠀⠀⠀⠀⢸⣷⣤⣥⣴⣦⣬⣤⣾⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢨⠂⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠓⣄⡀⠀⠀⠀⠀⢼⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⡱⣄⠀⠀⠀⢼⡀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⠴⢛⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⡀⠙⣜⢶⠀⠀⢸⡷⠈⣿⣿⣷⠀⠀⠀⢀⡤⠊⠁⣠⣿⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⢤⡿⡄⠰⡕⠀⣿⣿⣿⠀⠀⠴⠈⠀⠀⣴⣯⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢡⣹⡆⢆⠀⣿⣿⣿⠠⠋⠀⠀⢀⣾⣿⣿⠟
⠀⠠⢤⢶⣶⡶⠦⣤⣀⠀⠀⠈⢶⠀⠀⢻⡷⡾⡄⣿⣿⡿⠁⠀⠀⣠⣿⣿⡿⠋
⠀⠀⠀⠈⠳⡌⠙⠲⢭⣿⣳⣤⡀⠱⢄⠀⢳⣟⣣⣿⣿⠁⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠘⢣⡀⠀⠈⠛⢦⣿⢦⡀⠢⠈⣧⢹⣿⠃⢠⣼⣿⣿⣿⡵⠖⣻⣿⣯⣽⣿⣿⡷⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠭⣤⣤⠈⠳⢿⣦⡙⠞⢈⣿⣴⠿⣟⣿⣿⣾⣾⡿⠿⠛⢉⣉⣠⡤⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣛⣂⣭⣿⣇⠀⣼⣵⣿⣿⣍⣩⣥⣤⣴⡶⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠿⠿⠛⠋⢩⢶⠆⠈⠙⠿⠿⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⢠⢂⣠⡂⠖⠁⢸⠀⠙⠢⡤⣴⡀⠉⠙⠻⢦⡀
⠀⠀⠀⠀⠀⢠⠾⠡⠦⠐⠂⠒⠋⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠉⠉⠚⠦⣝⢦
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⢀⣀⡀
⢠⣾⢟⣩⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⡂⠀⠀⠀⠀⣀
⠿⠣⠯⡞⣣⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠛⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠑⠋⡶⠶⠶⠂⠀⠹⡆
⠀⠀⠀⠀⣿⠀⣀⣒⣶⣾⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⣶⣦⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣻⡜⣷⡄
⠀⠀⠀⠀⣿⠀⣿⣾⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠻⠾⡚⢹⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⣗⣖⣖⣖⣇⢹⣿⣿⣿⣷
⠀⠀⠀⠀⣿⠀⠈⠉⠉⣽⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣆⡀⣿⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣷⣿⡆⢿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣏⣹⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠈⢿⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠀⠀⠀⠀⠀⢸⡀⢸⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⠸⠗⢄⠀⠀⠀⠀⢸⠀⠨⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢞⣽⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠂⠐⢾⢣⡀⠀⠀⢸⣷⠀⣿⣿⣿⠀⠀⠀⠀⠀⢀⠤⠊⢁⣴⣟⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢷⠹⣤⠀⢘⠯⠀⣿⣿⣿⠆⠀⠀⢀⠴⠋⠀⢠⠾⠿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠳⠏⢇⡈⡇⠀⢺⣿⣿⡇⢀⠀⠁⠀⠀⣰⣟⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠹⣮⠃⢱⠀⢸⡿⣿⡇⠉⠀⠀⢀⣴⣯⣽⡿⠃
⠀⠉⠙⢿⡻⠿⣟⡓⠶⣤⣀⡀⠀⠫⠀⠀⠘⡿⣍⣃⣼⣇⡿⠣⠀⠀⣰⣿⣿⡾⠏⠁
⠀⠀⠀⠀⠈⠢⡀⠈⠓⠶⣯⡻⢶⡄⠘⠆⠀⠹⡇⣽⣿⣿⠁⠀⣠⣾⣿⣿⠟⠁⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠙⢤⣀⠀⠈⠙⠷⣟⢳⣄⠑⣀⢻⠨⣿⠋⣠⣾⣿⣿⡿⠿⢛⢉⣽⣽⣷⣾⠿⠿⠿⠛⣶⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⣀⠈⠛⠿⣷⣌⠛⡀⡿⣴⢟⣻⣫⣿⣶⣿⠿⠟⠛⣋⠡⣤⣔⡶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣙⣒⣶⢿⣿⠁⠸⣷⣻⣯⣭⣭⣤⣤⣤⠽⠷⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠿⠟⠛⠋⢩⢴⡆⠈⠉⠛⠿⠟⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠋⠉⠀⠠⢂⡠⡀⠔⠁⠈⡃⠈⠳⡤⢤⣀⠀⠉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠴⠤⠰⠄⠐⠓⠉⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢄⡙⢦
⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡖⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢈⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠐⠁⠈⢠⠀⠀⠀⠀⠓⠈⠂⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠀⠠⠈⠀⠀⠀⠀⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡌⠃⠀⠀⠀⠀⠀⡮⡄⠆⠀⠀⠀⢠⠄⠂⣠⠀⠀⠀⢀⡄⠀⠂⡄⠀⠀⠀⠐⢣⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠙⢁⠆⠀⠀⠐⠆⠠⠀⠆⠀⠀⠀⠀⠊⠀⠂

⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⢀⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠃⠘⠀⠌⠀⠀⠀⢈⡠⠔⢙⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠁⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠘⣿⣿⣿⣿⣿⣿⠁⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣦⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⣴⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣧⡀⢸⣿⣿⣿⣿⡃⢀⣼⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⠉⠻⢿⣿⣿⣧⡀⢿⣿⣿⣿⢀⣼⣿⣿⡿⠟⠉⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣉⡻⣿⣷⠸⣿⣿⢇⣾⣿⢟⣉⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠙⠛⠙⣛⣻⣿⣿⡿⢻⠛⢿⣿⣿⣟⣛⠋⠛⠋⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⡿⠋⠀⢸⠀⠀⠙⢿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠟⠛⠙⠁⠀⠀⠀⠀⡇⠀⠀⠀⠈⠋⠛⠻⠿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠠⠀⠀⠀⠒⠒⠒⠂⢠⡍⠉⢠⣉⣉⣝⣍⣭⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠏⠠⠤⠶⠖⠀⣒⠒⠒⠀⠒⠒⠀⣹⣿⣉⣈⡋⡆⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢡⠰⠀⠓⠀⠀⠈⠓⠀⠉⠉⠁⢸⠆⠂⠑⠍⠁⢧⠸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⡾⡘⠄⡐⠀⢠⠃⠈⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢃⢠⠀⠀⠀⠀⠀⠀⠀⠀⣸⣃⣼⣰⢡⢡⢃⠂⢠⢿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⢀⠔⠊⣁⡤⢤⣤⣉⠙⢦⡾⡞⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⡄⠀⠀⠀⢠⣁⠔⡺⢱⡏⡱⠀⢪⣳⠂⠀⣟⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠐⢠⠀⠀⣠⠤⠬⣤⠳⢸⢰⠡⢡⢣⢉⣏⣿⡝⢰⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡄⠀⠐⠀⡀⠛⠧⣧⣣⣆⣞⡧⠖⠉⢀⢱⡣⢊⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢠⠀⠀⠃⢨⢓⣤⣄⣈⣉⣀⣀⣤⠶⡻⡳⡡⣦⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⣃⣭⡧⠤⠤⠭⠽⠟⠒⠛⠛⠓⠓⣧⢩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⡀⠀⣾⠋⡥⢠⡀⡔⢰⣴⣾⣇⠲⢼⠹⠀⣽⡆⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢁⢠⢼⡀⠁⠈⠁⠀⣀⣀⣠⣤⡤⢤⢒⠎⠘⡅⢺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⡏⢈⢏⠍⡝⡙⡙⢩⣳⢃⣿⢣⢪⢊⠎⢐⠄⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣥⣰⣾⣷⣲⠾⢖⠲⠖⠷⠂⠥⣷⡾⠧⠀⣼⣻⢟⡿⡗⣿⢿⠿⣶⣶⡤⠀⢤⣀⣀
⠀⠀⠀⠀⠠⣶⠢⢀⠀⢐⣶⡶⢖⡛⠟⡻⢛⠟⠹⣏⣉⣉⠙⣷⢶⣤⣀⡀⠀⠀⠀⠉⠒⠲⢾⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⣿⢙⣶⣌⣜⣋⠙⠳⠥⣚⠛⣷⠦⡿⣾⣟⣿⣰⣷⣿⣿⣿⣷⠀⡖⡀⡀⣀⣱⢸⣿⣭⣏⣽⣿⣧⣷⡿⡿⣶⣿⣾⣿⡅
⠀⠀⠀⠀⠀⣿⣖⢤⣭⣉⣛⠯⣒⠲⠦⢐⣻⣿⣽⣿⣽⣿⣿⣿⣿⣿⡿⠿⣋⣀⠙⠛⠛⢻⡟⣿⣿⣱⣷⣿⣷⣿⣟⣿⣿⣿⣿⣿⣧⣤⡀⢀
⠀⠀⠀⠀⠀⣿⣧⣬⠍⢛⡛⠻⣿⣛⠶⠀⣞⣿⣿⣶⣿⣿⡿⣿⣿⣿⣧⣀⡀⠀⢈⣹⢙⣏⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⠉⢀⠋
⠀⠀⠀⠀⠀⠙⠲⢭⣝⡓⠦⢽⣟⣳⠿⣞⣻⣽⣿⡯⠝⣠⣄⠀⠀⠈⠈⣻⣱⣄⡉⠹⢉⡏⣍⣼⢏⡳⣞⣲⠊⠉⠘⠛⠒⢻⣻⣐⠤⠔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠮⢷⣽⠺⣻⣿⣿⡿⣀⣀⠀⢈⣙⢸⣉⣘⣼⡏⠑⢺⠛⢛⡋⣽⢃⠘⣇⠀⠈⠉⠋⠩⠉⠋⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠑⠘⣧⡀⠀⠀⠀⠉⠙⢹⠉⣿⡿⠷⠚⡛⠻⣯⠟⠒⠀⠈⠉⠓⠒⠛⠚⠛⠋⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠚⠛⠛⠛⠛⠛⠋