New year bunny Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⢩⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡿⠛⢿⣅⠀⠀⠸⠟⠛⠛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⢠⣶⣿⣷⣄⠸⣿⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣻⣿⠛⠙⣿⣅⡀⠀⠀⣿⡟⠁⢹⣿⡦⠁⠀⠀⣼⣿⠃⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⢸⣿⡅⠀⢸⣿⠁⠀⠈⣿⡆⠀⡄⣼⣿⠋⣠⣤⣤⣀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⣼⣿⡿⠀⢸⣿⡇⠀⠄⠋⠁⠈⣾⣿⣿⣿⠿⠿⠟⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⠃⠀⠈⣿⡧⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢁⣠⣤⣶⣿⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠛⠁⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⢀⠀⢀⡀⡀⢀⠀⠀⢀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠈⠀⠀⠈⠋⠀⠙⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠠⠔⠾⠟⠷⠶⢴⠶⡾⠶⣶⠄⠺⣍⠾⠳⠜⠶⠓⠴⠷⠶⠶⠾⠴⠊⠏⠢⠾⠔⠶⠶⠋⢽⠾⠷⠴⠗⠴⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠥⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⠴⠒⠢⡈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢸⠋⠀⣀⠂⢀⣠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢀⠀⠀⠀⠀⢀⣀⣿⠉⢪⣳⡀⠀⠀⠀⠀⠀⠀⡇⢀⠃⠀⢠⡧⠞⠁⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁⠀⣟⠀⠀⠀⠀⠿⠟⢻⡇⠀⢷⢱⡀⠀⠀⠀⣠⡤⢧⠀⠀⢠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣉⠲⣤⠄⢀⣄⠀⢸⡇⠀⠘⣦⠇⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠙⠲⠤⠼⠋⠉⠙⢦⣤⣿⠀⠀⣸⠀⠀⠀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⡠⠤⠦⢤⡀⠠⣄⠀⣦⡀⣠⠃⠀⠀⢸⠀⠰⠾⠏⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣠⣥⠀⠀⠀⣬⠧⢤⡙⡶⠎⣻⠁⠀⠀⠀⢈⢶⣤⠤⠤⠖⠉⠀⣀⣠⠔⣃⡤⠤⡀⢀⣀⡴⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⠛⣷⣒⠦⢼⡗⠚⠁⠀⠀⠀⠀⠈⠙⠓⠶⠒⠒⠚⠋⠁⠀⣎⣁⠴⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢆⠀⠀⢀⡤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠘⡆⣿⠁⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢆⢠⠋⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠛⠛⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠹⣿⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡸⠃⠀⠀⢰⡇⠀⠀⠀⠀⢀⡴⠖⡛⠿⠉⢉⣀⠤⢄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣿⣆⣴⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⡸⠁⠀⠀⣠⡴⠁⠀⠀⠀⠀⠀⠀⠁⠐⢫⠗⠢⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⢹⡿⣿⣯⠃⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠙⡆⢠⡞⠋⢀⣀⡠⠀⠈⠀⠀⠀⠀⠀⠈⣦⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⠤⠌⠛⠁⠈⡇⠀⠀⠀⠀⠀⠀⢸⠀⢀⣠⠦⠞⠀⠀⠀⢀⠆⠈⠉⠉⠉⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⢺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⣿⣽⡟⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢠⠞⠁⣬⠉⠉⢉⡥⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣀⠀⢀⠾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⣟⠀⣾⠝⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡀⠀⠀⠙⢏⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⡀⢸⠀⢰⡇⠀⠀⠀⠙⠲⢻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢣⠀⠈⢳⠇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⢸⢿⠀⠀⢀⡴⠶⡶⠛⢲⣤⣀⠀⠀⠀⠤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠔
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⡜⠀⠀⠘⣇⠀⣹⠠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⠠⢠⣥⣼⡇⠀⠀⠀⠙⢶⢏⣠⣧⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁⠂⠔⠚⠶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠤⠄⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⢀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢶⣿⠀⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡞⣏⡿⠀⣠⣾⣯⣹⢯⡏⢀⣠⣤⣤⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢁⣼⣿⠿⣻⣵⡟⣼⣿⡟⢣⣼⢻⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣟⣩⣴⣿⢟⣵⠿⣟⣯⣾⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⣿⢏⡿⣿⣷⡀
⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣻⣩⢞⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣮⣵⣿⣿⣿⣄
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢩⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣫⣟⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⣾⣿⣿⣿⣿⣧⣾⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣵⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣄
⠀⢰⣿⣿⣿⣷⣾⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣉⣹⣿⣷⣾⣿⣿⣼⣿⣿⣿⣿⣿⣿⢿⡀⣀⡀⣸⢿⣿⣿⡄
⠀⢾⢻⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣯⣠⠿⢿⣀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣟⠁⢀⡽⣿⡿⣅⠀⢹⣿⡇
⠀⠀⠙⢿⣿⠿⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⡿⢋⣾⣿⣿⢻⣿⣿⣿⣿⣿⣿⡏⠈⠉⠁⢈⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣾⣿⣿⣿⣤⣾⣿⣿⣿⣷⡝⢿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⢟⣽⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣮⣿⣿⣿⣿⣿⣿⠋⠈⠉⠉⣩⣭⣭⣭⣭⣥⣍⢿⣿⣿⣿⣝⣻⡿⣻⣷⣿⠟⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⣿⣿⡿⠏⠁⠀⠀⠀⢨⡽⢿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⡧⠘⠻⠶⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡄⡄⠂⡤⠀⠀⠊⠐⠈⠃⠀⠢⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢂⠀⠀⢂⠘⡛⢁⡇⠀⠐⠁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⠞⠐⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⢀⠈⠀⠀⠃⠘⡛⢤⡀⠀⠀⠀⠀⠀⠘⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⡘⠙⣆⣀⢀⡾⡁⠀⢀⡦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣷⣀⠉⠉⠉⢿⣷⣴⣶⠴⠍⡿⢿⡷⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⣼⣽⣿⠟⠀⠀⠀⠈⢿⡉⠉⠉⠙⠛⠔⣛⠛⢿⣿⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠀⣞⡿⠡⡏⠀⠀⠐⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠉⠀⠚⢁⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠐⣿⠟⠻⠟⠁⠀⡇⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣽⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣀⡺⣿⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢝⢺⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠾⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣬⠃⢀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠗⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠲⢌⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡞⠀⠀⠀⠀⢠⣤⡄⠀⠀⢀⠀⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠨⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⣄⣈⠛⠺⠁⠀⣶⣌⡆⡰⠁⠀⠀⠀⠀⠀⢠⠀⠀⠈⠉⣙⣧⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⢶⡄⢄⡀⣄⠀⠀⢀⠀⠀⠀⠈⢧⠰⠀⠀⣽⣿⣿⣠⠀⠀⠀⠀⠀⡐⠁⠀⠀⣴⣤⣼⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣿⣿⣶⠀⢸⣟⣿⣹⣅⣸⣄⢀⡌⠋⠱⠂⢀⠀⠱⣼⠿⣿⣿⣿⡿⡁⠀⠀⡀⠀⠀⠀⠀⢈⣏⢼⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣽⣿⣶⡄⠻⣿⣽⣃⣀⣰⣤⠀⠀⡾⠴⠃⢤⣤⡟⣆⠉⠩⣸⠑⠽⣦⡎⠉⢀⣀⠀⡰⣱⣿⢪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⢄⣀⡀⠻⡏⠁⡼⠧⡤⠀⠁⡀⢴⠿⡏⠉⢈⣧⠀⣻⠀⠤⢿⡖⠋⠉⢁⢿⣵⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠀⠀⠀⠘⠌⠁⠀⢣⣦⣄⠀⠀⠠⡀⠠⣄⣄⠀⠀⠀⠋⠁⡿⠋⠀⡉⢀⡠⣤⣿⣿⣿⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡆⢀⣛⠇⠀⢀⡀⣸⣶⡄⡔⢂⢹⣤⠀⠀⠉⠠⡄⠁⠒⠶⣿⣿⣿⣿⣿⣟⡋⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⠉⡅⠀⠀⠈⠙⠋⠻⠿⢉⣩⣿⡟⠀⠄⣀⡄⠈⠢⢄⠝⠹⠿⠛⠉⠉⠢⠉⠉⠙⣦⣔⡒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠏⠀⠸⠛⠲⠗⠈⢄⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠈⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠠⠤⠶⠶⠶⠶⠶⠤⣤⣀⡈⠃⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⡀⠀⠈⠉⠙⠒⠦⣄⡀⠈⠉⠳⢤⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠈⠲⣌⠢⡀⠀⠀⠈⠙⢦⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢌⡢⣄⠀⠀⠀⠙⢦⡀⠀⠀⠙⠂⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣓⠦⣄⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠭⣉⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠁⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠜⠁⢀⡤⠖⠛⢉⡉⠍⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⡤⠖⠒⠒⠀⠀⠀⠔⠋⠀⣠⠖⠉⠀⣀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⡀⠀⠀⠀⠀⠀⢄⡀⠀⢀⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⠉⠛⠓⠒⠒⠒⠂⠉⢉⡍⠁⠀⠀⢠⣶⣷⠄⠀⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠢⠤⢤⡤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠉
⠀⢰⡟⠉⣻⡉⠉⣉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠏
⠀⢸⡇⣮⣱⡼⣏⣸⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⣩⠝⠋⠁⠀⠀⢀⡴⠋⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⠜⡵⠁
⠀⢸⡇⠩⠟⢰⡟⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠉⡇⠠⠀⢀⣠⠴⠋⠁⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠈⣠⠞
⠀⠘⠷⠤⠤⠼⠷⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰⡅⣀⠄⡹⠉⠉⠀⣠⠞⠉⠓⠀⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠗⠀⠿⠄⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢻⡀⢸⡆⢠⠂⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠙⠆⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠁⠀⠡⡃⠀⠀⣀⡼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⡈⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠘⣶⣾⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣾⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠃⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⢀⢀⠄⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢼⠆⠀⠀⠀⣠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠎⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⢀⣀⣠⡶⠤⠤⠤⠤⠤⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⣀⣤⣤⣶⣶⣿⣿⣿⣶⣶⣶⡶⠶⠶⠾⣿⣶⣾⣇⣀⣁⣀⣉⡀⠐
⠀⠀⠀⠀⠀⢟⣻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⣼⣿⠿⢿⠿⠟⠛⠛⠉⠉⠉⠀⣀⣀⣠⣤⣤⣤⣶⣦⣤⣼⣿⣿⣿⣿⣿⣿⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤

⠀⣆⢀⠛⠀⡵⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠄⠒⠒⢄⡠⠴⠀⠐⠀⠴⠄⡀⡠⠔⠢⢖⠐⢺⡠⡆
⠤⠇⠸⠐⠃⣗⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢘⢇⠀⠀⠀⠱⠋⡄⢡
⠠⡆⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠁⠁⠀⣀⠤⠊⠢⣀⠀⠀⠈⠂⠷⣄⠀⢀⠇⢢⠸
⠀⠃⠓⠒⠁⣁⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡆⢠⢾⡐⢂⢭⢍⠀⠀⢈⢬⢍⠢⡀⣇⠀⡌⣂⡼⠄⠩
⠠⠀⡄⠀⠁⡥⠈⡡⢂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢀⡇⡇⣿⢺⠀⠀⡊⣾⢾⠀⠐⠓⡄⡠⠘⠤⡀⠀⡒
⠐⠄⡧⠐⡢⠯⡙⢄⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠂⠀⡀⢁⡇⠑⠤⠃⠂⠀⣘⠤⠊⠀⠀⡰⠚⠁⠀⠢⣁⠀⠇
⠘⡀⡗⠈⡄⣅⠈⠉⠀⠀⠀⠀⠈⠢⡌⠀⠀⠀⠈⠄⠀⠀⠱⡀⠈⢳⡒⣈⠜⠀⠀⢀⠔⢃⠀⠀⠀⠠⠀⠴⡃
⠀⠶⣉⢓⠀⣃⠀⢀⠆⠀⠀⠀⠀⣖⢎⢶⠄⠀⠀⠈⢒⠁⠀⡘⠒⠦⣤⡄⢤⠶⠚⠁⠖⣂⠙⠀⠀⠀⠀⠀⠙
⠐⡀⣟⠙⠒⡤⠀⡀⠀⠀⠀⠀⠀⠳⡁⠐⠈⠒⠢⠀⡀⠗⠊⡗⠀⠀⣬⣯⡀⠀⠀⢐⠔⠉⢂⠀⠀⢰⠒⠒⢸
⠀⡇⠁⠐⠇⠏⠂⠈⠄⠁⠀⠀⠀⡔⠤⡉⠁⠒⠤⣀⠀⣀⢴⣳⡄⣷⡮⠋⠁⠀⠀⠘⠢⡀⠀⠣⡀⠒⠈⠕⣹
⠠⢇⠤⢠⠀⡇⠀⡀⡀⠀⠀⢠⠄⠁⠒⢌⢂⠀⡀⣀⠀⠀⠈⠙⠏⠈⠉⠀⠀⠀⠀⠀⠀⢉⠢⡀⠘⢄⡀⠁⠠
⠁⠆⢱⠰⡆⣦⠀⠆⠀⠀⠀⠀⠢⢄⡀⠚⠉⣡⣔⠠⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠢⡈⠢⢀⠈
⠀⡐⠘⣀⠄⡥⠘⡠⠀⠒⠐⠀⢀⠎⠀⠀⠀⣟⡟⠀⢈⣾⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢇⠈⠀⠁⠀⠀⠨
⠠⢸⠀⡀⡀⡇⠈⢀⡤⣐⠀⠀⠘⠧⣄⠀⠀⠀⠀⠑⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠸⡀⠀⠰⠓⠜⠮
⠀⠀⠁⠠⠄⡧⠊⠑⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⣃⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢱⠀⠀⢣⠞⠓⠙⠢⠠
⠀⡀⠀⢠⢪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡏⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⢆⠠⢜⡆⠀⡀
⠐⠉⢁⠃⠈⠀⢠⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⣌⡜⠀⠀⠑⢲⠬⠅⡐⣒⠒⠒⢒⢩⠄⠘⠒⠚⡅⠀⠸⠱⠲⠄
⠀⠒⢸⠁⠀⠀⠀⠀⠀⢸⠢⣀⣀⠈⢇⠀⠀⡏⠀⠎⠚⠆⢸⠠⠀⣀⣸⠀⠀⠀⠀⣆⡀⠠⠀⢇
⢣⠀⢸⠣⡀⠀⠀⠀⠀⠘⠀⡃⣠⣼⡆⠀⠀⠃⠀⡠⠚⢄⣈⣉⡠⠒⢋⠀⠀⠀⠀⢱⠒⢖⣈⠁⡨⠁⢢
⠐⠦⠨⠷⢄⡀⠀⢠⣤⠤⠴⠭⣶⡶⣃⠀⠐⡢⡀⠧⠀⢀⠀⡀⠀⠄⡞⠀⠀⠠⡀⢸⡠⠀⠀⠀⠀⠠⠄⡆
⣀⣘⣒⣀⣈⣂⣀⣀⣠⣤⣄⣸⣶⣟⣀⣑⣓⣈⣀⣉⣀⣐⣒⣀⣈⣁⣀⣀⣀⣉⣙⣀⣈⣉⣀⣀⣀⣈⣁⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠅⠂⠁⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠄⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣄⠔⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠠⠀⠠⠚⠉
⠀⠀⠀⠀⠀⢠⣾⠋⠁⣼⣿⠀⠀⢡⠀⠠⢸⢢⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠁⢀⣾⡿⣃⡀⢂⢐⠚⢠⠸⡠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠁⠀⠑⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠘⠛⢋⣁⣀⠀⠀⠀⠀⢀⣤⣦⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⣴⡿⠋⣹⣷⠀⠀⠙⠻⢋⣽⡟⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠉⠀⣰⣿⠏⡀⠀⠀⠰⡿⠛⣿⡄⠀⠀⠀⠀⠀⠙⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⡀⠀⢀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠂⠂⠀⡔⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠘⠿⠛⠉⠀⠀⠈⠛⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠤⠤⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡤⠖⠒⠒⠒⠒⠒⠒⠲⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣷⣴⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣼⣿⢛⣿⣥⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠈⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠠⣾⡿⣶⡄⣀⡀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠰⢲⣟⠹⢟⠀⠀⠀⣠⣶⠂⠀⠀⠀⠀⡀⠀⠀⢀⣤⣤⣰⣶⣯⣿⡏⣼⣻⣿⡆⠀⠀⠀⠀⠘⣄
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⣼⢿⡟⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠘⠛⠛⠋⠛⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣵⠇⣠⣟⡿⣿⣣⣴⡖⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢠⠏⠄⠀⠀⢀⠠⢄⠀⠀⠀⠀⠀⣰⣿⣟⣫⣼⣿⣵⣿⣷⣿⣷⣎⣶⣿⣿⣿⡿⣻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⢸⠀⠀⠀⠀⠈⠡⠜⠁⢀⣤⣶⣿⣿⣿⣿⣭⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣆⠀⠀⠀⠔⡢⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⣿⣿⣿⣿⣿⢿⣿⣷⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⡿⢏⣹⢿⣾⣿⣷⣿⣿⣿⣿⠿⣀⣄⠽⢿⣿⡀⠀⢀⠴⢀⠀⠀⠀⠀⡇
⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠉⢿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣷⣞⣹⣾⢻⣿⣿⣿⣿⣿⣧⡴⠙⠟⢢⣴⣿⠃⠀⠈⠱⠉⠀⠀⠐⣰⠃
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⣿⣿⣿⢿⣿⣿⣿⣿⣿⣵⣿⣿⣷⣿⣿⣿⣿⣿⣶⣴⣾⣟⣽⣷⠀⠀⠀⠀⠀⠁⠀⡸
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠈⢛⣛⣿⣿⣿⡿⠋⠀⠀⣴⣾⣿⣾⣶⣝⣿⣿⣿⣏⢹⣿⣿⡿⠀⠀⠀⠀⠂⠀⣰⠃
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣿⣩⡟⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠋⠉⠉⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢶⡄⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠿⠺⠇⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡀⠀⠀⠀⢀⣤⣶⡶⢯⡽⣛⢿⡦⣶⠶⣮⠖⣒⣿⡶⣶⠦⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡄⠀⠀⠉⠙⠦⠹⣭⣫⣎⣭⡝⣎⢻⣽⣳⣫⡭⠙⠨⠁⠀⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⡀⠀⠀⠈⡉⠉⠀⠉⠉⠉⠉⠁⠀⠀⠀⣀⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠦⠤⠤⠤⠤⠤⠤⠴⠞⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣄⡀⠀⢸⣿⠁⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠉⠻⣆⠸⣿⡀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⠳⣿⡇⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠹⠀⠙⠛⠳⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⡀⢀⣀⣠⡤⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠈⢿⠉⡿⠉⢸⠁⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⢉⠉⣻⣉⣉⢹⠋⣿⠋⠁⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣶⣶⡾⠋⠁⠀⠀⠀⠀⠀⠀⢈⡆⣹⢠⡁⣾⠀⣿⠀⠀⠀⠀⠀⠀⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠠⡿⢀⡽⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠈⠈⣟⠉⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛
⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⣿⠉⠉⢀⠀⠀⠀⠀⡀⠀⠠⠤⠤⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⠉⠉⠉⠁⠀⠀⠀⡇
⠀⣀⣀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣸⠇
⣸⡟⠉⠉⠀⠀⠀⣠⡤⠤⠀⠶⠖⠀⠀⠀⠀⠀⠀⠀⢸⣧⡐⠲⣿⠂⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⣿⣇⠀⠀⠀⠀⠀⣿⠀⠀⣾⠀⠀⣀⠀⠀⠀⠀⠀⠸⢻⡇⢁⢀⣿⡀⣼⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⣿⣿⠀⠀⠀⠀⠀⠉⣠⡀⣿⠀⡀⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠋⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠸⣿⣧⠀⠀⠀⢀⠜⠋⠀⣿⠀⠙⣷⠀⠀⠀⠀⠀⠀⠸⠇⠜⠁⠀⠈⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠹⣿⣧⡀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠙⢿⣿⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠂⠀⢠⡀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠉⠉⠉⠛⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣟⣋⡁⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠿⠿⢿⣿⣷⣄⠀⠈⢻⣦⡀⠀⠀⠀⠀⠻⢿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⡆⠀⠀⠙⢿⣦⣀⣀⣀⣀⣀⣨⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠟⠛⠛⠁
⣿⣿⣿⠟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⣄⠀⢀⣴⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠘⣆⣾⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣸⡏⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⡇⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠁⠀⠀⠀⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⣀⣤⣴⡶⠶⢶⣦⣤⣤⣤⣤⡤⠤⠒⠞⠁⠀⠀⠀⠀⠀⠀⠉⠙⠳⢶⣤⣄⡀⠀⠀⠀⠀⠀⢀⠑⠀⠀⠀⠀⠀⠸
⢀⣴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄
⠈⠻⠷⠶⠶⠖⠛⢦⠀⠀⠀⠀⠀⢰⣍⣎⣸⠈⣁⠀⠀⠀⠀⣠⠀⡀⢀⡀⠀⠀⠀⠀⠀⠙⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⣀⣤⠤⠤⠡⠀⠀⠀⠀⢀⣾⡐⣼⢸⠀⠀⠀⠀⠐⡥⠠⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⠉⠁⠀⠀⠀⠀⣿
⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠃⠞⠘⠁⠸⠀⠀⠀⠀⠐⠛⠐⡇⠀⠃⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠈⡤⢸⠠⣌⠀⠀⠀⠀⣿
⣴⡟⠁⣀⣠⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⣾⠂⢸⠀⣇⠀⠀⠀⠀⠈⠀⠚⠀⠈⠀⠀⠀⢰⣿
⠙⠛⠛⠛⠉⠁⠈⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠟⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁⠀⠐⠦⢤⣀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⣀⠄⠀⠑⠢⢤⣀⣀⣀⣤⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢷⣆⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⢶⣦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣘⣿⠙⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣈⡇⠀⠀⣠⠷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢌⣿⠀⣰⠷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⢿⢠⢗⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡘⡇⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣢⣾⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠄⣻⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠉⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⣉⣰⣏⠀⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣽⣏⠹⡿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣽⡏⠠⠄⠻⣇⣹⡻⣽⣳⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡹⠱⢦⠰⣿⣻⣿⣿⣃⡝⣿⡿⣈⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣜⠇⡑⡌⡟⢩⢳⡍⠅⢻⠿⠛⣷⠎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡻⣷⣳⠀⣟⢶⣚⣓⣲⣗⠀⠂⣀⢷⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⡦⣿⣪⣶⡽⣿⣿⡿⠫⠝⣾⣧⣠⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣇⣟⠲⣿⢸⡕⠀⠀⠀⠀⠘⢻⣟⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣽⣏⢉⠿⣿⣷⡗⠀⠀⠛⠀⠀⣘⣜⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣏⢈⣯⣈⠹⣿⣝⣄⣠⣀⣀⠜⢷⣿⣧⠖⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠾⠕⢨⣧⠠⣹⣉⣻⣻⣿⣿⣧⣤⣴⠿⠓⠺⢀⡁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⢀⣀⡀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⠛⠛⠛⠛⠛⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠛⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠉⠉⠓⠒⣦⡀
⠀⠀⠀⠀⠀⢰⡿⢭⣁⣀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡤⠶⠚⠋⠉⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠉⠉⠓⠲⣯⠀⠀⠀⠀⠀⢀⣀⣤⡶⠛⠉⣸⠋⠉⠉⢻⡈⠙⠒⠦⣤⣀⣀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⡷⢤⣄⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠴⠖⠚⠫⣤⠞⠀⠀⢠⠏⠀⠀⠀⠀⢱⡄⠀⠀⠙⣏⠉⠉⠓⠒⠚⠁⠀⠀⠀⠀⣀⣸⠇⠀⢀⣈⡭⢿
⠀⠀⠀⠀⠀⠈⠳⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢀⡞⠲⠦⣤⠴⠚⠋⢻⡀⠀⠀⠈⢧⡀⠀⠀⠀⣀⣠⠴⠖⢛⣩⡤⠖⠛⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢲⡶⠤⣤⣀⣀⡼⠁⠀⠀⠀⡼⠀⠀⠀⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠳⡴⠚⣉⣡⡴⠖⠛⠉⢸⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢨⣟⣀⣤⠀⠀⣼⣁⡀⠀⠀⡇⠀⠀⣀⣠⣼⣧⠀⠀⣤⣀⠹⡏⠉⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠁⢸⢀⡼⠁⠈⠉⣿⠒⠷⠚⠋⠉⠀⠉⢹⢧⣸⠁⠈⠛⠃⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢻⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣸⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⣀⣤⠼⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣿⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣠⠴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠤⣄⣀⣸⢀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠠⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣦⡄⡀⠀⣠⣤⠄⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠥⠀⠀⢌⠉⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢀⣑⠤⡠⠀⣀⢢⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠓⠀⠀⠂⠀⠀⠀⠐⠠
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡀
⠀⠀⠀⠤⠐⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠂⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠘⢀⠃⠒⡓⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⡀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⡟⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣤⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣉⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⠈⠀⠀⠀⠢
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠂
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡶⠚⠒⠦⣄⠀⠙⢦⠀⠀⢀⡴⠋⢀⡤⠖⠛⠉⠙⢦⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠈⠳⡀⠀⢳⣀⡞⠀⣰⠋⠀⠀⠀⠀⠀⢸⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⢇⠀⠀⠀⠀⠀⠹⡄⠀⠙⠀⡼⠁⠀⠀⠀⠀⢀⡴⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠳⢤⣀⠀⠀⠀⢹⡀⠀⢰⠇⠀⠀⢀⣠⠖⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠉⠙⠓⠲⠶⠇⠀⠼⠴⠒⠋⠉⠀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⣲⠆⠀⠀⣠⣄⠀⠀⢴⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⢠⣾⠀⠈⣧⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢰⠏⣿⠀⠀⡏⢧⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⠏⠀⡟⠀⠀⡇⠈⣇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⢶⣠⠏⠀⠀⡇⠀⠀⡇⠀⠘⣆⡾⢦⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⡇⠀⠀⡇⠀⠀⠹⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⡇⠀⠀⡟⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⠋⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡴⢀⠀⠀⠀⠀⢀⠈⣠⡀⡰⠀⠀⠀⠀⣠⠴⠂⠋⠉⠉⠉⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠌⠀⠀⠀⢀⠀⠀⠀⡠⠘⠀⠀⡠⠀⠈⠁⠐⠐⠠⢄⠀⠀⢸⠁⠁⢨⠀⠀⠀⠀⠀⡆⠈⠄
⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠐⡌⣉⢠⠊⠀⠀⢀⠎⠀⢠⠋⠀⠈⠄⠀⠈⢃⠀⠘⢄⡀⠈⠆⠀⠀⠀⢀⡥⠊
⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⡀⠀⠀⡠⠙⠤⠜⠀⠀⠀⢀⠀⠀⣀⠑⠒⠈⡀⠀⣀⠎⠀⠀⠀⠉⠑⠚⢖⡖⠉⠁
⠀⠀⠀⠀⢈⠠⠤⠤⠤⠤⠬⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⡾⠏⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢏⠀⠘⠀⠀⡽⠀⠀⠀⠀⣰⠚⢉⠉⠁⠑⡄
⠀⠀⠀⠀⠀⡄⡎⠨⠀⠥⠁⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⠠⠠⢀⠉⠁⠀⠀⠈⠂⠀⠀⠐⠀⠀⠀⠙⠛⠠⠀⢁⡸⠀⣠⠂
⠀⠀⠀⠀⠀⡂⡂⢘⠀⢐⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡠⠒⢁⠴⠊⠁
⠀⠀⠀⠀⠀⠅⠅⠰⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠠⠤⠀⠀⠀⠀⠀⠀⠘⠀⠘⢀⠀⠀⠀⠀⠸⠅⠀⠈⠂⠠
⠀⠀⠀⠀⠀⢇⡃⠨⠁⠨⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡉⠀⠀⠀⠁⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠉⢰⠀⢀⠇
⠀⠀⠀⠀⠀⠐⢶⡀⢀⠰⠂⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⠀⢀⠀⡈⠀⠀⠀⠀⡀⠄⠀⠀⠀⢔⡤⢁⡴⠁⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠘⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀⢀⡁⠀⠀⠀⠀⠄⠂⠀⠀⠀⢆⠀⢸⠀⢀⢐⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠄⠠⠀⠀⠀⠀⠀⠒⠘⠂⠀⠀⠈⠒⠠⠨⠝⠊⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣤⣤⣴⣰⣮⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠰⠄⢽⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣾⣿⠛⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢥⣿⢟⡭⣿⣫⣟⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣿⣯⣟⣟⣷⡦⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡿⣿⡟⠛⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⣀⣀⡠⢿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠁⠀⣀⣧⡈⠂⠀⢉⣀⡿⢹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⢨⣷⣾⣾⣿⣿⣿⣿⣿⠛⠛⠃⢠⡏⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣠⣾⣿⣿⣽⣿⣟⣿⣿⣿⣿⣶⣟⣉⡠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣵⢿⠋⠉⠉⢹⢿⣿⣿⣿⡟⣳⡶⠷⣼⣿⢯⠇
⠀⠀⠀⠀⠀⠀⠀⡴⠟⠸⣿⣿⢹⠏⢉⡷⠋⠛⢾⠉⢹⣿⣿⣿⣽⣧⣒⣻⡵⠋
⠀⠀⠀⠀⠀⠀⠈⢷⡀⢁⠻⣿⣿⡌⠘⣧⣀⣀⣾⠈⢰⣿⣿⣿⣿⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢁⣬⣿⣿⣿⣀⠉⢁⣸⣯⡿⣉⠉⠀⠙⠿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠯⠿⠿⠿⠿⠿⠿⢿⣿⠆⠀⠹⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠊⠉⡦⠞⠉⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠢⢼⡀⠀⠳⠦⠀⣗⡶⠚⠁⠀⢈⣦⠚⠉⠙⢦
⠀⠀⠀⠀⠀⣴⡒⠒⠋⢹⢀⠀⠀⠀⣀⣙⣧⣦⡀⣰⣿⢽⠇⠀⠀⣼⣧⣖⣲⣦⡜
⠀⠀⠀⠀⠀⢻⣿⣦⠽⠯⣿⣷⢠⢸⣿⣿⣿⣿⢳⡿⣀⣾⡼⠗⠛⠛⠛⢯⣯⡿⠗⠊⠲⡄
⠀⠀⠀⠀⠀⢠⣟⠁⡠⣠⣾⢿⣿⣮⣯⣿⣿⣻⣾⣮⣿⣿⣿⡆⠀⠀⡀⠈⣟⣭⣁⠀⢠⠃
⠀⠀⠀⠀⠀⢸⣯⣳⣮⠇⠈⣛⣓⣛⣛⣉⣿⣿⣿⣿⣿⣿⣵⣡⣼⣖⣥⣾⣿⣿⣿⣞⣁
⢀⡠⠔⠒⠩⠶⢿⣭⣽⠀⣾⢽⠋⠉⠀⠀⢀⣽⣿⢿⣿⣿⡄⠀⠀⢹⣿⠛⠟⠀⢀⣠⢴⢟⡇
⡎⠉⠑⠒⠤⣄⣀⠭⠛⠷⡶⢾⣇⡆⡄⡀⢸⣭⢠⣿⣿⣿⣵⣵⣴⣼⠤⠉⠓⢿⣍⠴⠣⢺⡇
⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣵⡛⠻⠛⠳⠾⠿⠛⠃⠁⣀⣤⡼⣻⡀⠀⠠⠆⠘⡗⠱⢠⣸⡇
⢧⣀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡇⠈⠉⠒⠢⠤⣠⡴⠒⢻⡿⢋⢬⣺⡇⠀⠣⠢⠀⣿⣵⣿⡏
⠀⠹⠛⢶⣦⣤⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢺⡺⣲⢟⢕⠅⣮⣾⡇⢀⡤⠔⢀⡯⡈⢺⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠓⠂⠀⣧⣀⡀⠀⠀⠀⢸⡾⣀⢀⣶⣾⢞⣿⠇⠈⠀⠀⢸⣟⠶⣹
⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠉⠙⠛⠒⢦⣼⢶⣾⣿⢟⣕⣕⣿⠀⠀⠀⠀⠈⣿⢈⣾⡀
⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣗⡹⡾⣫⢺⢝⢕⣿⠀⠐⠒⡁⠀⣷⢋⣾⣿
⠀⣛⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⢅⣾⡫⡣⣢⡗⣿⠀⠀⢇⠄⠀⣯⣿⣿⠒⠂⠀⠀⠈⢁
⠀⣏⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣿⣕⢿⣪⡪⡫⣪⣿⠀⠀⠁⢁⣠⣿⡋⢠⠄⠉⠁⠀⡠⠄
⠀⠀⠉⠛⠲⢤⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⢕⢕⢟⢝⢋⣾⣿⢀⣤⡶⠛⠉⠉⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠚⠷⣦⣤⢸⡃⠀⠀⠀⠀⣿⣱⢷⢕⣵⠿⠿⠭⠋⠱⠖⠤⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠶⢤⣀⣿⡥⠟⠉⠩⠥⠄⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⢂⡀⠀⠁⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⢢⡡⠺⣍⠄⠢⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙
⠀⡀⠐⢤⠁⠐⠁⠀⠀⠀⠀⢀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠑⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆

⠀⠀⠀⠠⡠⠀⠠⠀⠀⠀⠀⠀⠴⠰⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠸⡃⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠘
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢀

⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⡄⢀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑
⠠⣀⡠⠃⠐⠋⠀⠀⠀⠀⠐⠀⠠⠀⠠⢐⠀⠀⠀⢀⠀⠐⣀⠖⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠊⠤⠖⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡁⣲⠜⡑⡤⠃⠀⠀⠀⠀⠀⡠⠲⠢⠀⠀⠀⠀⠘⠄⠸⠻⠖⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣠⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣀⣰⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣄
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣆⠀⢠⢞⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣆⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣩⠃⠀⠀⠀⠀⠀⠀⣀⣠⠤⠬⢿⡿⠤⠤⣄⡀
⠀⠀⠀⠠⣴⣾⣿⣿⣿⣿⣶⣄⡈⣿⣀⣀⣀⠤⣀⠴⠊⠁⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⠿⣿⣿⣇⠀⠀⡰⠃⠀⠀⠀⠀⠃⢀⢔⡦⡀⠀⢁⠀⠀⠀⠙⡄⠈⠉⠓⠦⣄⣀⡀⠀⠀⣀⣠⣤⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠠⠀⠉⠁⠀⢰⠁⠀⠀⠀⠀⢀⢜⣵⣿⣿⣿⢦⡀⠀⠀⠁⠀⠸⡄⠀⠀⡀⠈⣷⣇⣴⣿⣿⣿⠟⠋
⠀⠀⠀⠀⣀⣠⣴⣶⠯⠤⠤⠤⠤⠥⣀⡀⡮⠀⠀⠀⢀⢔⣵⣿⡍⣷⢠⢺⣷⣟⣦⡀⠀⠀⠀⡇⠀⠀⠀⢰⣿⡿⠿⠿⠿⠶⣦⣄⣀
⠀⣠⣴⣿⣿⠟⠉⠀⠀⠀⠂⠀⠀⠠⣄⣍⠛⢤⡀⠀⠳⣝⢿⣿⣰⡏⠀⢸⣿⡿⡷⠋⠀⠀⢀⠇⢀⡤⠚⠹⠟⠀⠀⠀⢀⠀⠀⠈⠙⢿⣦⣄
⣾⣿⣿⡿⠁⠀⠠⠀⠀⠀⠀⢀⣠⢤⢢⠁⠀⠀⠙⣄⠀⠈⠳⣝⢿⣿⣿⡿⡫⠊⠀⠀⠈⢀⡜⡰⠋⠀⢀⣄⣀⠀⠀⠀⠀⠀⢠⠀⠠⠀⠹⣿⣷
⣷⣝⡻⠁⠉⠀⠀⣀⠤⣖⣮⡵⣾⣿⣧⢧⠀⠀⠀⠘⡆⠀⠀⠈⠳⣝⡫⠊⠀⠀⠀⢁⣠⡊⡼⠁⠀⠀⣜⣾⣿⣵⣲⡤⣀⠀⠀⠀⠈⠀⢀⣩⣾
⡷⣝⠿⣶⣤⣀⡀⠧⢿⣿⡿⢢⡟⡌⣽⣯⢇⠀⠀⠀⣽⢒⣦⡤⠤⠤⠤⠤⠤⠖⠚⠉⠀⠉⡇⠀⠀⣸⣿⠟⢫⠠⠹⣿⡷⠽⢀⣀⣤⣶⡿⣫⢽
⣿⣮⣝⡲⣭⣛⡿⢿⣶⣾⣤⣭⣀⣐⡿⠿⠎⠆⠀⠀⣼⣿⠋⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠇⠀⠠⠻⢿⣛⣈⣥⣤⣾⣶⣿⠿⣛⣫⡵⣚⣵⣿
⣿⣿⣿⣿⣷⣮⣝⣓⠶⠯⣭⣟⣛⣻⠿⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⠿⠿⢿⣛⣛⣭⣭⠶⣞⣛⣭⣶⣿⣿⣿⣿
⣯⡻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣭⣭⣽⣛⣛⣒⣶⡶⠶⠶⠿⠿⠯⠭⠭⠭⠽⠿⠷⠶⠶⢶⣶⣚⣛⣛⣯⣭⣭⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣯⡹⣮⢻⣿⣿⣿⣿⠿⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣻⣭⣭⠷⠶⢾⣭⣽⣻
⡧⠻⠜⣧⠿⣻⣽⠶⣟⣛⣿⣿⣟⣛⡳⢾⣽⡻⢿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣭⠾⣫⣽⠶⣞⣛⣛⣻⠶⣮⣽
⣟⣛⣛⡳⢶⣍⡱⢟⣫⣭⣷⣶⣯⣭⣟⡳⣮⡻⣶⡝⣿⣿⣷⣬⢛⠿⣷⣶⣤⡲⢶⢶⢯⣾⣝⡻⣿⣿⡿⣱⡿⠵⠛⢛⣛⡛⠛⠻⠿⣿⡟⣰⣾
⣿⣯⣭⣝⡳⣮⡛⢷⡝⣿⣿⣿⣿⣿⣿⣿⣮⢻⣮⢿⣎⠿⣫⡾⣛⣵⠾⣙⣻⢿⡌⣿⡿⣶⣭⡻⣮⣻⠟⠉⠀⠀⠀⠉⠉⠀⢀⡠⡀⠈⠙⢯⣾
⢿⣿⣿⣿⣿⣮⡻⣮⢻⣎⢿⣿⣿⣿⡿⢟⣫⣥⡽⠶⠷⠶⣭⣼⣋⣵⠿⠛⠉⠁⠠⠿⠍⠉⠙⠻⢼⡃⠀⠀⠀⠀⠀⠀⠀⡠⣪⣾⣮⡢⡀⠈⢻
⠀⠙⠻⣿⣿⣿⣷⢹⣇⣿⡞⣿⢟⣵⠾⣫⣵⠶⣟⣛⣛⣛⠶⣾⠟⠁⠀⠀⠀⠀⣠⠦⡀⠀⠀⠀⠀⠈⢢⡀⢠⠀⠀⣠⣾⣾⣋⢻⢉⢹⣿⣦⡈
⠀⠀⠀⠀⠉⠙⠻⠾⣿⣼⡇⣡⡿⣱⡿⣫⣶⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⢞⣵⣿⣮⡢⡀⠁⠀⠐⠀⠱⡀⠀⠘⢝⢿⣿⡁⡝⠀⠸⣿⢟⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠚⠿⠿⠿⠿⢿⣿⣿⡿⠀⠀⠀⣠⢞⣵⣏⢛⡏⡩⣿⣮⡢⡀⠀⠀⠀⡇⠀⠀⠀⠱⣝⢿⣿⣶⢞⡕⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⢕⢿⣿⡏⣼⠃⠀⣿⣿⢟⡼⠂⠄⠀⡇⠀⠠⠀⠀⡈⠻⣟⡷⠋⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠑⢝⢿⣿⣶⣶⢟⡵⠋⠀⠀⠃⣰⠓⠤⠤⣤⣀⣤⠤⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠂⠀⠀⠑⢝⢟⡵⠋⠀⠀⠐⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢤⣀⣀⣉⣀⣀⡤⠴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣆⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠯⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣷⣴⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⡿⠿⠛⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣥⣵⡾⠙⣙⣋⡉⠱⠆⣠⡑⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣷⣿⣿⣳⣿⢸⢿⣿⠟⢰⣦⢹⣿⣦⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢿⣿⣿⢳⣿⢿⣿⣦⣦⣾⣇⣿⡇⠻⣿⣷⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡾⣿⣭⣭⡘⢫⣼⠛⢹⡿⠙⢃⡌⠃⣀⡭⠵⠂⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣯⣽⡉⠀⣀⡒⠒⠐⢒⡠⠀⠉⣀⣠⡴⠖⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣯⣿⡛⢻⣛⠛⣿⣯⠛⢛⡚⠉⢩⣜⣿⡤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⡟⢨⣯⣿⣿⣿⣿⣯⣧⠈⠀⣾⣿⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⣼⣿⣿⡿⣿⣿⣿⣿⢀⠀⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⡧⢿⣿⣿⡟⢺⠿⣫⢿⢸⠀⡏⡉⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡿⢿⣿⣿⣷⠈⠉⠀⠀⠁⠁⢸⢸⠀⡇⠀⠏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠁⢀⣿⣽⢿⠠⠤⠀⠀⠤⠔⢸⢸⢠⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⢹⡿⡟⠈⠀⠀⠀⠀⠀⠀⠸⢸⠀⠏⢁⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣭⡺⢟⣷⢀⣤⢢⡀⠀⣠⣢⡌⢼⠀⠒⣈⠴⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣯⣽⣿⢿⢈⣛⣀⣚⣛⣂⣙⠋⠨⠖⣋⣥⠖⢻⣆⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣿⣿⣿⣿⡟⣶⠖⣒⣚⣳⣲⣶⣒⡒⠒⠊⣵⣾⣿⣿⣿⢿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⠷⠾⠿⠟⠛⠛⠛⣟⢿⣶⡾⢟⡍⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢰⣄⡆⠀⡀⡀⠀⡀⠀⡀⠀⠀⠀⢠⡶⢠⢀⠀⠀⠀⠀⠀⣰⢠⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠐⠘⣸⢹⠃⠇⢧⡇⢿⡇⠟⢧⡇⠀⠁⠋⠇⡌⠯⠸⠬⠃⠀⠀⠙⢾⠺⠱⠽⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇⠘⠇⠀⠘⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢇⡴⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⠇⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣴⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣦⣀⣀
⠀⠀⠀⠀⠀⠤⠔⠀⠂⢀⢤⠀⠀⠀⠀⠀⠀⠰⠓⡄⠀⠀⠀⣰⡤⣖⠀⠂⠀⠀⠀⡴⠚⢡⠁⠀⠀⡈⠉⠑⡄
⠀⠀⠀⠀⠂⠀⡠⠌⠁⠁⠁⠀⠀⢀⡰⡄⢀⠔⠉⠀⢀⠐⠁⡄⠠⢄⠀⠁⢂⠀⠸⡀⠀⠸⡀⢸⠀⡁⠰⢀⠆
⠀⠀⠀⠀⠀⠀⢣⠌⠀⠀⠀⠀⠀⠰⢥⡓⡇⠀⠀⢠⢱⠀⠘⠄⠀⠜⠀⠀⢀⠂⠀⠉⠢⠤⢵⡤⡤⠥⠚⠁
⠀⠀⠀⠀⠀⢀⡀⣀⣀⣐⡃⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠐⠚⠂⠀⠒⠂⠒⠁⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣶⠂⠉
⠀⠀⠀⠀⠀⢀⡠⠂⢀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⡃⠀⠂⠀⢸⠃⠀⠄⠀⣠⠒⠛⠁⠑⢢
⠀⠀⠀⠀⢀⣰⠀⡇⠸⢱⠀⠀⠃⠀⠀⠀⠀⠀⠐⠈⠄⢀⢉⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠹⠓⠡⠂⢉⠗⢀⡜
⠀⠀⠀⠀⢀⣐⠀⡇⠈⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⢀⡤⠊⠡⠞⠉
⠀⠀⠀⠀⠀⢒⠀⡂⠀⣒⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠄⣀⠀⠀⢀⠀⠁⠀⢡⠄⠀⠀⠀⠘⠆⣀⠈⠂⢆
⠀⠀⠀⠀⠈⡽⡀⡂⢐⣲⠀⠀⠀⠀⠂⠀⠀⠀⠐⠐⠒⠀⠄⠀⠀⠠⠠⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⡕⢀⠜
⠀⠀⠀⠀⠀⠙⠣⠒⠚⠁⠀⢮⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠤⠠⠄⠂⠀⠀⠖⡒⠀⠀⠠⡱⠋⡴⠃⠀⡀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠒⠂⠄⠀⠒⠐⠀⠀⠀⢀⣉⠠⡀⠀⠀⠡⡀⠓⠤⣦⠆⠏
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

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

⠀⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⢐⠛⢠⠃⢄⣀⢀⡔⢲⠀⠀⢀⠀⢠⣤⣘⣓⣴⡄⠀⠀⠠⠠⠀⢀⠴⠒⢈⡅⢠⣉⠉⠢⣄⠀⠀⠀⠀⠀⠉⢠⡌⠁
⠀⠀⠀⠀⠀⠀⡞⠀⠔⠓⠀⠠⣞⠀⢈⠆⠀⠀⡄⠻⠛⠾⠟⢛⠛⠢⡄⠀⠀⡐⢁⠔⠉⢀⠀⣀⡀⠉⠆⠈⠆⠀⢀⠤⠒⠈⠙⠈⠀⠐⢢⡀
⠀⠀⠀⠀⠀⢐⡉⢀⣍⠒⠊⢤⠊⠀⡹⠀⠀⠄⣀⠀⢸⠀⡀⠘⡀⠴⠘⠆⠀⠀⡎⠀⠀⠀⠶⠄⠇⠀⢘⠀⠠⠀⠺⠦⠤⠠⠄⠀⠒⢀⣩⡄
⠀⠀⠀⠀⠀⠘⢍⠡⡀⡜⠃⡮⢁⡘⠂⠀⣀⠀⢩⠀⡋⠀⡅⠀⠀⠀⡀⢸⠀⢆⠘⢄⠀⠉⠀⠘⠀⠀⠜⢀⠄⠀⣷⣶⣴⣶⣶⣶⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠈⣧⣤⣤⣍⡀⣈⠕⠀⠀⠏⠀⠶⠰⠧⠀⠦⠠⠃⠘⠁⡈⠀⠈⠓⢄⡑⠂⠐⠒⠊⡁⠔⠁⠀⠀⡇⠈⠉⣩⣍⣱⣦
⠀⠀⠀⠀⠈⡝⣉⣀⣠⣤⣴⣾⡿⢦⠀⠀⠀⠳⣟⠒⢢⠀⠀⡜⢂⡃⠊⢀⣒⡀⣀⠀⣀⣈⠉⠀⣄⠀⢤⡦⡀⠀⡁⠀⠀⠘⢿⡿⠃⠀⢀⡅
⠀⠀⠀⠀⡾⠟⠛⠋⠉⠉⡁⠀⠀⠀⡖⠂⠀⠀⠈⢻⣿⣾⣶⣿⠁⠀⠀⢐⣿⠀⠷⠀⣸⡗⠰⠒⣿⠀⣈⣿⡅⠘⡿⢽⣿⣿⣶⣾⣿⡿⠿⠇
⠀⠀⠀⠀⢀⠀⠀⡴⠶⠀⣏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠇⠃⠁⠀⠀⠀⠘⠉⠃⠈⠀⠈⠁⠀⠀⠻⠀⠃⠘⠀⠀⠑⠦⠄⣀⣀⣀⣀⣀⠴
⠀⠀⠀⠀⠈⡄⠀⠟⠆⠤⠉⢀⣀⣠⠇⣠⣤⣄⠀⢀⣀⢀⣀⠀⣠⡀⠀⢀⣀⠀⣀⠀⢀⣀⣀⣀⡀⠀⣠⣤⣀⠀⢀⣤⣤⡀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠹⣤⣴⣶⡾⠿⠿⣿⠋⢰⣿⠋⠻⠆⠈⣿⢸⣿⠀⣻⣷⠀⣿⣿⡀⣿⡇⢸⣿⠉⠉⠁⢸⣿⠋⠛⠀⣾⡏⠙⠁⠀⠐⠁⢰⠴⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⢠⠞⠀⠀⢸⣿⠀⠀⠀⠀⣿⢸⣿⡀⢸⡗⠀⣾⣿⣇⡿⡇⢸⣿⣤⣤⡄⠘⠿⣦⡀⠀⠻⣷⣄⠀⠀⠀⠈⠙⣼⡐⢀
⠀⠀⠀⠀⠀⠰⠖⢣⣀⣀⠔⠀⠀⠀⢺⣿⠀⢠⣤⠀⣿⠿⣿⡄⢸⣏⠀⢻⡇⢿⣿⡇⢸⣿⠉⠉⠁⢀⣄⠈⣿⠀⣀⡈⢻⣷⠀⠈⠢⠀⠀⠒⠈⡀
⠀⠀⠀⠀⠀⣀⠀⠀⠘⠃⠠⣤⣤⡄⠈⢿⣷⣼⡿⠀⣿⡄⢻⡇⢸⣿⠀⢼⡇⠈⠛⠃⢸⣿⣤⣤⡄⢸⣷⣾⠟⠀⢿⣧⣼⠟⠀⣀⣀⣀⣀⠀⠸⠛⠂
⠀⠀⠀⠀⠸⡭⡀⠀⣀⣠⣴⣿⠟⠃⠀⡀⠉⠉⠁⠀⢀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⡀⡀⠀⢀⣎⣠⣤⣶⡈⣿⠀⣾
⠀⠀⠀⠀⠘⠏⠁⠀⠻⠟⠋⠁⠀⠀⠲⣽⠶⠀⠠⡄⠋⣄⠀⣸⣿⣆⣸⡇⠀⣿⡿⠛⠀⠀⣶⠀⣷⠀⣶⠀⠀⠁⡍⠀⢹⣿⠁⠉⢻⣇⢿⡇
⠀⠀⠀⢀⠤⠤⢄⠄⣒⡶⠀⠐⠢⢄⡀⠁⠀⡀⠀⢨⡔⠉⠀⢸⣿⢿⣿⡇⠀⣿⣷⣶⠆⠀⣿⣾⣿⣾⡟⠀⠐⢠⡜⠀⠀⣿⡀⠀⠀⣿⣼⡇
⠀⠀⠀⠧⡀⠐⠠⣘⠠⠇⠀⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⠀⠀⢹⡇⠈⠛⠃⠀⣿⣷⣶⡄⠀⢸⣿⠹⣿⡇⠀⠀⠈⠀⠀⠀⡽⠓⠂⠈⠀⠀⠘⡀
⠀⠀⠀⠀⠈⢂⠀⠀⠉⠀⠐⠒⠂⠈⠁⠀⠈⠀⠀⢠⣤⡀⠀⣠⡀⢀⣄⣤⣤⡀⠀⣀⣀⣄⠀⠀⣀⣤⣤⣄⠀⢀⣤⡄⠀⢡⠀⢄⣢⠠⣤⠀⢁
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢻⣿⣶⡿⠁⢸⣿⡟⠛⠃⠀⣿⡛⣿⡆⠀⣿⡏⠹⢾⡇⠀⣿⡇⠀⠀⠀⠘⣉⡄⡏⡄⠘
⠀⠀⠀⠀⠐⠀⠙⠂⠤⠀⣠⣀⠀⠠⠔⠁⢸⣷⣾⡇⠀⢹⣿⠃⠀⣼⣔⣳⣶⠀⠀⣿⠿⢿⣦⠀⣯⢳⣶⠾⠁⠠⣿⠀⠀⠀⠀⠀⠹⠂⠀⠁
⠀⠀⠀⠀⠁⢀⣠⡴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⢸⣿⠀⠀⣿⣿⣭⣍⠀⢸⣿⠀⢸⣿⡀⣿⡟⢻⣦⡀⢀⣥⡀⠀⢠⡀⠀⠀⠤⠎⠋⢁⢐
⠀⠀⠀⠀⢰⣮⠛⢄⡄⠀⠀⠀⢠⡀⠀⣤⠀⣤⠀⠀⠀⠈⠉⠀⠀⠈⠉⠉⠋⠀⠘⠛⠀⠀⠉⠀⠛⠃⠀⠉⠀⠈⠛⢁⣠⠍⠁⢠⣤⢶⣶⠺⠽⠛⠃
⠀⠀⠀⢠⠟⠐⠒⠀⡰⢆⠀⠀⠈⢃⠲⠈⠋⠰⠚⡃⠀⣀⠔⡋⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠉⠀⠒⢪⣤⣶⣶⣦⡞⠉⠀⡠⠊⡀⠀⣤⠀⠀⠁⠀⢷⣿⡇⠀⡰⠀⠀⢀⠤⠒⠀⠉⠉⠈⢢⣀⡀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣄⠀⡠⠒⠛⠂⠤⣸⡏⣮⣽⣿⣷⡠⠔⠋⠁⠉⡶⢉⡦⠀⠢⢀⠀⢈⠀⣉⠀⢀⠠⡅⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⣿⣿⠚⠋⠁
⠀⠀⠀⠀⠐⢖⢧⠀⠓⣄⠐⠤⢀⡀⠉⢳⡾⡿⠁⠀⠀⠀⣂⡾⡀⠀⠰⢻⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⡀⠠⠊⠉⠁⢢⠀⠀⠀⡀⢀⡷⡇
⠀⠀⠀⢠⡙⣮⠂⢀⠊⣤⣿⣶⡤⢀⣀⡌⠀⣈⣦⣤⣴⣾⠿⣿⡍⠂⠐⢻⠁⣤⣦⣶⣞⣦⠀⠐⠄⠀⠘⣦⣤⣤⡴⣿⢠⡶⢼⣿⡟⠉⠡⠔⡓⠒⢄
⠀⠀⠀⠀⠙⢌⣁⣸⢸⣿⣭⣿⡇⢠⣅⡀⠀⠀⣀⣿⡻⣿⣿⠿⠁⢈⡰⠁⠀⠈⠀⠁⠸⣟⣳⣶⣾⣷⡺⢷⠑⠚⢉⡉⠀⣷⠀⢹⠿⠀⡇⠘⣻⡄⢀
⠀⠀⠀⠀⠀⢰⠁⠀⢷⡭⢭⡭⠔⢋⡽⠻⣤⡼⠛⠳⣝⡶⠶⠲⡟⠙⠄⠀⠀⠀⣀⠀⠀⣀⠀⠀⢹⣯⠀⣻⣃⣌⣀⣈⣶⢻⣷⣷⣄⠤⠌⡒⠚⠂⠁
⠀⠀⠀⠀⠀⠀⠓⢤⣀⡉⠋⠀⠉⠁⢀⣰⣿⣷⣀⠀⠀⠈⠉⠉⠀⢀⠄⠀⠀⠜⢠⣰⡆⠈⠳⠀⡇⠈⢹⣿⣿⣿⢿⠟⠋⣻⣿⣿⣿⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⡰⡈⠙⠤⣠⡃⠈⢹⠉⠉⠉⠉⠃⠒⡆⠠⡤⠰⡇⠀⠀⠀⢄⠀⠼⠍⢂⡴⠀⠜⠃⠘⢿⣿⣿⣿⣄⣤⣿⣿⣿⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠊⠀⡹⠦⣀⠀⠈⠁⠘⠒⠓⠀⠐⠓⠂⠉⠁⠀⡄⠉⢣⠀⠀⢀⡉⢀⠀⠁⠀⠈⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣾⡟⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠈⠒⠈⢇⡀⡾⠉⢰⡆⠠⣄⡀⣀⠀⣀⠤⢤⡔⠈⡶⠐⠊⠀⡀⣘⠉⢸⡇⠀⠀⠰⡀⠀⠀⠀⠠⠾⠋⠽⠛⠉⠉⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠉⠀⠠⠛⢄⠷⡀⠟⢤⡜⠁⠂⠁⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠉⠐⠂⢠⣀⠀⡅⠀⠼⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠰⠿⠋⣁⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⢠⠟⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⣠⣤⡴⢏⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠛⠉⠛⢻⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣴⣦⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣦⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣿⣶⣆⡾⣿⣷⣿⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢿⣉⣿⠀⣿⣎⣽⢇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣚⣉⠛⠛⠛⠛⠁⢘⣯⢍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣉⡾⠁⠀⠀⠀⠀⠀⠀⠙⢦⣙⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⡙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣂⣤⡝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⢦⣶⠶⠶⢶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣲⣞⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠛⣿⡇⠀⢀⣄⢓⣎⣉⣉⣉⣹⣏⣙⣋⡉⢉⠀⣠⣼⡟⠃⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣀⣾⡇⠀⣈⢹⡋⢀⣿⣇⣽⣳⠬⡉⢛⣈⡽⢣⡞⢳⡖⣾⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⢻⠅⠀⠘⠾⠿⠯⣠⡿⣿⣿⠶⣶⠿⠹⣶⣎⣙⠭⠷⠞⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣤⣄⠀⣀⡀⠀⠀⠀⠀⠀⠂⡛⠲⠇⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣷⣉⠀⡄⠀⠀⠀⠀⠈⠁⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣷⣌⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⡇⣉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠛⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⡇⠀⠀⠀⢸⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣿⡿⣤⠚⢣⠀⠀⠀⠀⠀⠀⠀⢿⡇⡇⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣯⣀⢁⡉⢠⡀⠀⠀⠀⠀⠀⠀⣿⡇⡇⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢸⣿⢿⡿⠐⢻⡄⠀⠀⠀⠀⠀⠀⣿⡇⡅⠀⠀⠀⠘⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣄⢸⡟⠚⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⡀⠤⢁⣚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⣍⢿⡇⠀⣀⠀⣀⠀⠀⠀⠛⢛⠛⣀⣛⠃⡴⠈⣹⠶⣆⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡍⢻⣾⣻⡇⠀⠙⠶⣿⢛⣀⢉⣶⢠⠀⠈⢁⡈⢑⡞⠛⣾⣿⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣆⠙⢿⡇⠀⠀⠀⠘⠂⠀⠉⠙⠛⠶⠶⠋⠀⣉⡵⠚⠚⠉⡀⢀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠾⠓⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠆⠀⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢠⠀⣠⡴⣷⣠⡀⠀⠀⣲⣀⡀⠀⠀⢀⣤⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠋⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁⠊⣄⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⢿⡷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⠶⠦⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠾⠆⠀⠀⠀⢿⠿⠻⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠐⠂⠀⠂⠀⠁⣹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠐⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⡀⠀⢿⡿⠃⠀⠀⢀
⠀⠀⠀⠀⡠⠀⠀⠈⢧⠉⠒⠁⠀⠀⠆⠀⠣⡘⢡⡇⠀⢠⡤⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⣌⡱⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠙⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⢠⡂⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠛⢿⣿⡿⠛⠻⣦
⠀⠀⠀⠀⢀⣀⠀⠀⠀⢠⣿⣷⣶⣶⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣔⣃⣼⣿⣧⣉⣤⣿⣷⠀⢀⣀
⠀⠀⠀⠀⣾⡏⠈⣿⣶⣾⣿⣿⣿⡿⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢳⠜⣿⣿⡁⠤⢸⣿⣿⡿⠃⠁⢹⣿
⠀⠀⠀⠀⠟⠁⠸⠿⣿⣿⣿⠿⠋⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠶⠾⠿⠿⠿⠶⠛⠋⠑⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣉⣉⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⠻⣿⠃⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣾⠀⠀⠀⠀⢀⣴⣿⣝⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣸⣷⠛⣿⠀⠀⢀⣴⣯⣍⢿⡿⣱⠃⠀⠀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣷⣾⣿⠏⠀⣴⣿⣟⣟⡿⢟⣵⡏⣴⣾⣿⣿⡋⢹⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣷⡿⠋⢠⣾⣿⡾⢛⣥⣾⣿⡟⣸⣿⣿⣿⣀⣿⣿⣀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣛⣁⣤⣾⣿⡿⣫⣶⠿⢟⣛⣭⣾⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⣍⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣯⡿⠟⢋⡵⢚⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣹⣿⣿⣿⣶⡛⣷⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⢟⣴⣯⣿⣿⣿⣿⣿⣿⣋⣻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⢰⣿⣿⣿⣿⣟⣿⡿⢻⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⣯⣿⣿⡿⣛⣯⣭⣭⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣆
⠀⠀⣸⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣾⣿⣿⣿⣿⣿⡿⠋⠛⣿⣿⣍⡎⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢹⣿⣿⣿⣿⣇
⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢏⣴⣿⢿⣿⣿⣿⣿⣿⡿⠋⢻⡶⣾⠛⠹⣿⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠙⢦⣾⣶⣧⠞⠉⠻⣿⣿⣿⡄
⠀⠘⢏⣻⣿⣿⣿⣯⢻⣿⣿⢿⣫⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⠛⠛⣦⣴⣿⢃⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣠⠞⣻⣿⡟⢦⣀⠀⣸⣿⣿⠇
⠀⠀⠀⠻⣿⣿⣿⡿⠉⣩⣶⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⡿⠋⣼⣿⢿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁⠁⠁⠀⣻⣿⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⡏⣠⣾⣿⣿⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⣴⣾⣿⣿⡿⢋⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣟⣽⣻⣿⣿⣿⣿⣿⣿⣶⡿⠻⠾⠿⢿⣿⣿⣿⣿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⢿⣿⠶⠿⣻⣽⣿⣿⣿⣽⡟⠁⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣦⡈⢿⣿⣿⣿⣿⣶⣭⣛⡛⢩⣾⡿⠿⠟⠁⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣾⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢋⣭⣾⠿⣿⣿⣿⣿⣽⣿⣿⣷⣭⣟⣻⣿⣿⣛⣿⡥⠈⠻⠷⠶⠶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠛⠛⠛⠛⠛⠙⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣾⣿⡴⠆⣀⣤⣶⣶⣾⠾⢿⣗⠂⢠⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠱⡿⠋⠀⠚⣿⣛⠃⢍⡀⢐⣦⣩⠿⠙⢄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢅⠠⠞⠚⠛⠈⠰⢀⡀⠀⠸⣿⣷⠖⠛⠇⠀⠀⡔⠛⠶⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁⠀⠀⠀⠀⢀⡤⠞⠉⠙⠢⠸⡁⠀⠀⣀⠰⠀⠀⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⢠⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠞⠋⠳⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠘⠀⠀⢀⠇⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣤⣼⣧⣤⣼⣤⣄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⣧⡝⠛⣯⣝⢁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡔⠀⢀⣿⣤⣬⣿⣟⡈⠈⠱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣛⡟⣛⣿⣿⡿⣧⣈⡿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠏⠉⠉⠙⠋⠉⠁⠛⠉⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣺⠀⢀⠀⠂⠀⠀⠀⢠⠀⠀⠘⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣹⢀⠊⠀⠀⠀⠀⠐⠀⢱⡤⠊⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⠆⠀⠀⠁⠀⠃⠀⠊⠁⠈⠄⠠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢈⣻⠀⣀⠀⠀⠀⠀⠀⠠⠤⠀⠰⠀⢽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⡽⡄⠀⠀⢀⠁⡀⠀⠀⠈⠀⠄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣠⠀⠈⣦⣄⣻⣛⣻⣃⣠⡞⠁⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⡀⢐⣿⣿⣿⣏⡉⠉⠠⣬⠄⠀⢄⡀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠆⠉⠉⠉⠛⠋⠉⣤⣤⣠⡶⠀⠘⢿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣄⣀⣀⣀⣀⣈⣛⣋⣀⣤⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢘⣿⣿⣿⡏⠁⠀⠀⠉⢻⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⠟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠙⠛⠛⠛⠓⠂⠀⠀⠘⠋⢁⣠⡴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣿⣶⣶⣶⣶⣶⣶⣶⡶⢶⣻⣿⠿⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠺⣤⠀⠀⠈⢢⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⣀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⣀⣘⠷⣄⡀⠀⠳⡌⡇⠀⠀⠀⠀⠀⠀⢀⠞⠀⡠⠒⠉⠀⢈⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠀⠉⠙⠻⢿⣦⣀⠙⠻⣀⣀⣀⠀⠀⡠⠋⡠⠊⠀⠀⠀⣀⠞⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⣀⠀⠀⠙⠲⠶⢤⠤⠭⠝⢺⠉⠈⠋⣟⠩⠴⠛⠒⠊⠉⠉⠉⠉⠙⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠉⠀⣀⠈⠀⠐⢢⡤⠤⣤⣤⣤⣼⣿⡀⢸⡇⠉⠒⣌⢛⠛⠛⠛⠛⠃⠐⠀⣸⣀⣩⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠆⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠙⠛⠛⠙⢤⣤⡘⣦⠈⠙⠛⠲⠆⠀⣼⠋⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠤⢄⣀⡀⠀⠀⠀⠀⢀⣨⡇⠸⡇⠉⠉⠓⠒⠢⣼
⠀⠀⠀⠀⠀⠀⠐⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠉⠐⡞⠉⢠⠇⠀⣟⠳⢦⣤⣤⡴⠟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠠⠜⣀⡤⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡞⠀⠀⡿⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡞⠀⠀⠀⡇⠀⠘⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢰⠧⣀⠀⡄⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⣀⣠⡀⠀⣼⠀⠈⠳⡇⠀⠀⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠹⣠⠇⠀⠀⠀⡇⠀⠀⠁⠄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⡇⠀⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⡀⠀⠈⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⢰⠀⠀⠀⡇⠀⠀⠘⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠁⢸⠀⠀⠀⡇⠀⠀⠀⠘⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠜⠀⠀⢸⠀⠀⠀⠇⠀⣀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⣤⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢀⠤⠞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⢤⣀⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠤⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⡀⠀⡇⣀⠤⠒⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢹⣟⡛⣻⣛⡛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠄⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⡄⠐⡀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⡅⠈⡲⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⠀⠅
⠀⠀⠀⠀⡀⠀⠀⢐⠀⠀⢀⠀⠔⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⢤⣠⠔⠀⠀⠀⠀⠑⠀⠈⠁
⠀⠀⠀⠀⢀⣀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠉⠀⡀⠐⠀⠒⠀⢄⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠂⠀⠀⠂⡀⢄⠀⠀⠀⠊⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀
⠀⠀⠀⠀⠀⠠⡀⠈⠀⠊⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠐⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣴⠃⠀⠶⠒
⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠄⠀⠀⡄⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠾⠂⠯⠀⠀⠀⠀⠀⢠⣌⣉⣦⠀⠀⠀⣌⣰⣄⣤⣤⣧⡂
⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠀⠀⠀⠈⠠⢁⣀⣤⣤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠠⠠⠄⣿⣷⣆⠀⠀⠀⠀⠨⠋⠨⡍⠀⠀⠀⠁⠉⠈⠀⢋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡃⠀⠀⢻⣿⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠠⠀⡄
⠀⠀⠀⠀⠈⠂⠀⠀⠈⠄⠀⠀⠀⠀⠄⠀⠀⠐⠄⣠⣾⠿⠃⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠄⠘⠒⢁⡴
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠓⢍⠀⠀⠑⢶⠒⠒⠤⣤⠎⠁⠀⠀⡨⠝⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⣴⠞⠓⠤⣑⠢⢀⠀⠑⢤⡤⠁⠀⢀⠤⣊⠠⠐⠓⣄⡀
⠀⠀⠀⣀⡠⠤⠒⠊⠉⠁⠀⢸⠁⠀⠈⠑⠂⠭⣙⣛⣿⣈⣑⣾⣛⣉⠡⠐⠂⠁⠀⠀⡩⠉⠒⠒⠤⢄⣀
⠀⠀⠀⡏⠉⠒⠢⢤⣀⠀⢀⣀⠬⢒⣂⡤⠝⠛⠭⠑⠋⡹⢌⠙⠊⠉⠛⠫⠍⣛⠪⢍⣀⠀⣀⠤⠒⠊⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢙⡻⠖⠊⠉⠀⠀⠀⢀⡠⠔⠊⠀⠀⠉⠒⠤⡀⠀⠀⠀⠉⠲⣎⡏⠀⠀⠀⠀⢸
⠀⠀⠀⣇⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⣠⠖⠯⢄⣀⠀⠀⠀⠀⢀⣀⠼⠢⡀⠀⠀⠀⠈⠣⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠙⠒⢢⠞⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠈⠉⠒⡖⠊⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⣼⠤⠒⡋
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⣠⣋⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⣢⡀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢀⠜⠁⠈⡏⠒⠤⢄⡀⠀⠀⠀⡇⠀⠀⢀⡠⠔⠚⡇⠀⠐⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠃⠀⠀⠀⡇⠀⠀⠀⠈⠉⠒⠤⡧⠔⠊⠁⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠘⠢⠤⣀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⢀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠢⢄⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⣀⣇⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣿⡫⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠛⠛⠻⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠘⣿⣿⡀⠀⠈⠉⠳⣦⡀⠀⠀⠀⠀⠀⢀⣼⠛⣿⣿⠃⠀⠀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣄⣀⡀⣹⣿⠷⢤⣀⡀⢀⣼⣿⣦⠀⠀⠀⢠⡟⠁⢀⣿⠿⣄⣀⣤⣶⣾⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠉⠉⠉⢛⣻⣧⣀⠀⠀⠙⠻⣿⡿⠉⢷⡀⢠⣿⣷⣴⠋⠀⠀⢸⠿⠛⠋⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢿⣿⣿⣿⡿⠛⠲⢤⣄⣈⢧⡀⠈⢷⡞⢹⡟⣁⡠⢤⣶⣇⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠤⢤⣄⣉⠙⢻⡦⢸⡇⢿⣿⣅⣠⡴⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠿⣿⣿⠉⠋⠉⠉⠉⢻⠋⠉⠉⠉⠛⠛⠓⠒⠒⠲⠶⠶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣠⣼⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⡇⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣴⣶⣶⣶⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠦⣤⣤⣤⣤⡼⠷⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠻⣿⣿⣿⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣧⡘⠃⠈⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠁⠀⠐⢧⡄⠀⢉⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣀⣙⣷⣾⣿⣿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣈⣉⣛⣻⡿⢛⣿⡿⣟⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡉⢉⣿⣧⣤⣤⣀⠀⠀⠈⠀⠀⠈⠓⠽⢤⡠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠑⠀⣀⠀⠀⠈⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠾⣿⣿⣷⣮⢢⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣠⢼⣿⣶⣄⣀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⡹⢻⡏⢧⢸⣷⣿⣇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠘⣻⣭⣭⣭⣉⡋⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡈⠁⠀⠀⠀⠀⠁⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⡼⢣⣿⡿⠹⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠱⣶⠾⡥⠟⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢾⠦⢀⠀⠄⣀⣔⠀⣀⣀⣀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⡌⠳⡀⠁⢄⣠⠞⢫⠋⢉⡝⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡷⠀⣉⢶⡉⠀⠀⢨⣤⡊⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣿⣡⣆⣙⢲⣔⣾⣿⡟⠀⠄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠸⣻⣿⣯⣿⣿⣯⣿⠟⡠⢊⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣰⡀⠈⠛⠛⠻⠟⠃⢈⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣽⡿⠉⠐⣶⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠴⠾⠿⢿⣿⣿⣿⣿⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⡇⠀⠀⠈⣿⣿⣿⣿⣧⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠒⠛⠛⠛⠋⠁


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠁⠀⠀⠀⡄⠀⠀⠂⠀⠀⠀⠐⣀⣤⣶⣿⡿⠛⠻⠟⠿⠿⢿⡿⣿⣿⣿⣿⣄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣵⣶⣄⠀⢀⡿⠲⠎⠀⠀⣰⣶⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠟⠃⠀⠘⡓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⢘⡂
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣰⣾⣿⣿⣿⣿⠀⠛⡃⠠
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣤⡤⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣷⣿⣿⣿⣿⣷⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⡟⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡞⢿⡿⠻⣿⣿⣿⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠂⠠⠦⢠⣾⣿⠟⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠇⠀⠀⠀⠠⠈⠈⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢉⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠳⠀⠀⠀⣠⠀⠀⠀⡀⠀⢉⣽⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⡆⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠓⠲⠁⠀⢀⣿⣿⣿⣯⣿⣿⣿
⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⣿⣄⣰⡿⣶⡟⢿⡿⢻⣿⣿⣿⣿
⠀⠀⢀⣿⣿⣿⢟⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣿⡁⢨⢀⣥⢘⣿⣿⣿
⠀⠀⣼⣿⣯⢻⢿⣟⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠙⠛⣿⣿⣿⣇⣿⡇⢸⠰⠷⢸⣿⣿⣿
⠀⠀⣿⣾⣿⡧⠘⠻⢓⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⢾⣿⣿⡟
⠀⠀⣿⣿⣿⣷⣾⣿⣾⡄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⠀⠦⠀⠀⢸⣿⣿⠇
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠄⠄⠀⠐⠾⢿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⢀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣈⠊⠀⠀⠀⠀⠈⠻⠷⠿⠶⠟⠉⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣶⣶⡶⠞⠋
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠘⠛⠛⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⣠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⢀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⠹⠟⣃⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⠿⠿⣿⣯⣾⣿⣿⣿⣿⣿⣿⢿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣟⣿⣶⣶⣿⠛⢿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣽⣟⣫⣿⣿⣿⣯⣸⢻⣿⣿⣿⣿⣿⣿⡁⠛⣿⣿⣟⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⡿⢿⣾⣿⡿⡿⣿⣿⠏⡛⢻⣿⣯⣛⣽⣻⣿⣿⣿⣿⣿⣿⣿⢿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⢿⣿⣿⣿⣮⣵⣿⢿⣤⣬⣰⣿⣿⣿⣿⣿⣿⣿⣟⠉⢀⠉⢹⡿⡻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠛⠋⠙⢿⡿⠛⢿⡷⢋⣿⣿⠳⣽⣯⣾⣿⣾⡻⢖⡿⠂⠽⠀⢻⣯⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠋⠁⠀⠀⠀⠸⣷⣟⣻⣿⡿⢿⣿⣷⣿⣿⡏⢹⣾⣿⣿⣿⣴⣤⣶⣾⣿⡿⣫⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠀⠀⠉⠻⠿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⢀⣀⣤⣄⣙⣿⣟⣭⣻⣿⣝⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡾⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠀⠀⠈⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⣀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠛⠛⠋⠉⠁⠀⡸⠉⠙⢮⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢣⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⣆⠀⠀⠀⢣⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠢⣄⠈⠳⣄⠈⢣⡀⠀⠈⣿⠀⠀⢀⢀⡠⠔⠋⠁⠀⠀⠀⠈⠉⢉⣹⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣢⡤⠤⠤⣀⣀⡈⠑⠦⣌⡑⠀⠙⢦⡀⢸⡷⡆⠉⡙⢄⣀⣀⣀⣤⠤⠤⠒⠊⠉⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠯⣭⣉⣀⣀⣀⠀⠀⠀⠉⠉⠓⠒⠩⣵⣶⡤⡟⠒⢇⢸⡄⡈⠌⢧⠀⠀⢀⣁⣀⠀⠀⠀⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡖⠓⠛⠛⠩⠭⠥⠤⢄⣀⠤⠭⠙⠛⠛⣛⣋⠭⠟⠒⠺⠧⠖⠻⢄⡇⢠⠀⢨⡭⣙⣒⠒⠒⠒⠶⣾⠭⠭⠭⠭⠚⠛⠒⢲
⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⡎⠉⠉⠉⠙⠒⠒⠒⠒⡶⠒⢺⠁⠘⠀⢸⠉⠉⢹⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡆⠀⠸⠀⠀⢸⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠁⠀⠀⡇⠀⢸⡇⠀⠀⠨⡇⠀⠀⠀⠀⠀⢀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠉⠉⠉⠉⠁⠒⠚⡇⠀⠀⠀⡧⠤⠤⢄⣀⣀⣀⣀⣀⣇⠀⣿⠀⠀⠀⠀⡧⠤⢼⡇⠀⠀⠀⡗⠒⠈⠉⠉⠉⠉⡏
⠀⠀⠀⠀⠀⠀⢀⠜⠁⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⡏⠁⣿⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠐⠛⠒⠒⢻⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⢡⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⢀⡀⠀⢸⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠹⠊⠁⠈⠑⠚⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⠒⠻⠤⣄⣀⠼⠤⠄⣤⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣄⠠⠤⠴⠧⠀⠠⠤⠛⠚⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄
⠀⠀⠀⠀⣠⠤⢥⣄⠀⠀⠀⠐⠀⢠⠿⠟⠿⠿⠿⠿⣧⠀⠀⠀⠢⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⡇⠀⠀⠄⡃
⠀⠀⠀⠸⣿⣀⢈⣿⢆⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⢨⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⢸⣗⣞⡇⠈⠂⠀⠀⠘⠻⠟⠃
⠀⠀⠀⠀⠈⣉⢛⠡⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠌⠀⠀⠀⠀⢽⣷⣿⣳⠀⠀⠀⠀⠀⠠⠄⠀⠀⠄
⠀⠀⠀⣖⣶⣦⠤⠀⠀⠠⠛⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠈⠀⠀⠀⠀⠀⠁⠀⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢤⣤⣄⣉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⠀⠄
⠀⢀⢘⣋⣛⣾⡍⠀⠀⠬⠈⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⢘⣚⠛⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠚⠚⣩⣿⢧⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⣒⠢⠀⠀⠃
⠀⠀⠀⠊⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣰⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠪⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠆⠁⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠰⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⡿⣿⢟⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⢀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⡿⢿⢾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡋⠘⣧⡄⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⢩⡉⠉⠀⡠⠀⢀⡀⠀⠀⠀⠀⠀⠉⠠⠁⠰⠀⠈⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠛⠂
⠀⠀⠀⠀⠀⠀⠀⡠⠄⠈⠁⠈⠀⠉⠀⢰⠁⠉⣰⣁⡔⢠⠃⡴⠀⠆⣖⡂⡔⠂⢠⣒⠂⠀⠀⠀⠀⣀⢠⠶⣄⠀⠀⠀⠀⠀⠑⠐⠈
⠀⠀⠀⠀⠀⠀⡀⠁⠆⠄⠀⠀⠀⠀⠀⠘⠤⠀⠃⠀⠁⠾⠰⠁⠿⠰⠅⠀⡱⠀⣜⠀⠀⠀⠀⠀⢀⡁⠀⠀⢈⡠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⢘⣿⣿⣟⡟⣻⠿⣿⣶⣶⡶⡶⣶⠶⡶⢶⣶⣶⡦⠄⠀⠠⢌⡷⠦⠴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠐⡀⠠⣤⣤⣄⡀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠠⡀⠀⠀⠀⠓⠀⠀⠀⠀⠂⠀⠀⠀⠠⠤
⠀⠀⠀⠰⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠔⠐⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⠄⠚⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀⣿⣧⣿⣇⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⠏⠘⣽⣟⡁⠀⠀⢸⣷⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⣿⣀⣔⣶⠛⢿⣷⣄⣸⣿⣧⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⣿⠋⠻⢷⠀⣹⣿⣿⣿⡿⢡⡿⢶⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣾⣿⠁⠛⠀⠾⠿⠿⣿⣿⣧⣿⠃⠀⠶⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠛⠛⢻⣿⣿⠷⣶⣄⡀⠀⢀⣼⣿⣿⣟⣶⡘⣿⠛⠻⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡀⠙⠻⠶⣦⣭⣁⣀⣀⣉⣻⣶⡿⠛⠁⢸⣿⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⡛⠷⣦⣄⣀⠈⠉⠉⠉⠉⢉⣿⠀⠀⠀⠸⢿⣿⣧⣈⣛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠉⠛⠷⢦⣭⣍⣛⣛⣛⣛⣛⣿⡇⠀⣠⣴⣾⠟⢿⣿⡏⠉⠉⢉⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢉⡿⠰⣿⣿⣿⣤⣤⣤⣭⣤⠶⠞⠋⠁⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⢀⣤⡶⢿⣄⠀⠀⠀⠀⣼⡇⠀⣿⣧⣀⣀⠉⠀⢀⣀⣠⣤⡶⢟⣫⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣴⣾⣿⣿⣿⣿⣿⣆⠀⠀⢠⡿⠀⠀⢹⣿⣏⣛⣛⣛⣛⣩⣭⣴⠾⠛⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠙⣿⣿⣿⣿⣿⡇⣹⣧⠀⣼⠇⠀⠀⠘⣿⣯⠉⠉⠉⣩⡿⢷⣦⣀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠘⢿⣍⣻⣿⠟⠋⠁⢠⡿⠀⠀⠀⠀⣿⣿⠀⢀⣴⣿⣿⣿⣿⡙⢷⣤⠀⢿⡄
⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢸⣿⡇⢾⣿⣿⣿⣿⣿⣿⣾⠋⠀⢸⣇
⠀⠀⠀⠀⠀⢸⣿⡛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠈⣿⡇⠀⠉⠻⣿⣍⣩⡿⢁⣀⣤⢾⣿
⠀⠀⠀⠀⠀⢿⣍⠛⠷⣮⣝⣛⠷⠶⠶⠶⠶⠶⣾⠇⠀⠀⠀⠀⠀⠀⢿⣿⠶⢶⣶⣶⠿⠿⠟⣛⣩⣴⠟⢻⡆
⠀⠀⠀⠀⠀⣤⣿⠷⣦⣄⣉⡙⠛⠛⠛⠟⠛⣻⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠻⠶⠶⠾⠛⠛⠋⠁⠀⣠⣼⠇
⠀⠀⠀⠀⠀⠛⢿⣶⣤⣬⣉⣛⣻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣀⣀⣀⣀⣀⣤⣴⡶⢟⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠙⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠒⣸⣖⠖⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠙⢦⢄⠈⠁⠀⠀⠐⠊⠐⠀⠀⠀⠈⠁⡠⡴⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠡⡑⡄⠈⠀⠀⠀⠀⠀⠀⠁⢠⢊⠌⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡔⢦⠀⠀⡆⠀⠀⠡⠈⢆⠀⠀⠿⡿⠀⠀⡰⠁⠎⠀⠀⢸⠀⠀⡴⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢂⠀⠀⠀⢃⠈⢆⠀⠀⠀⠀⠰⠁⡘⠀⠀⠀⡘⠀⠀⠈⠋
⠀⠀⠀⠀⠰⡲⠀⠀⠀⢀⡀⠈⠢⡀⠀⢸⠀⠸⠀⠰⠆⠀⠃⠀⡇⠀⢀⠔⠁⢀⡀⠀⠀⠀⢖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠈⠢⡸⠀⡀⠧⢀⡀⠼⢀⠀⢇⠔⠁⠀⠀⠻⠟
⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠰⠀⢈⠊⠀⡀⠀⠀⠀⠀⡑⡅⠀⠆⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠠⢀⠠⠀⠀⢀⢃⠈⣆⠀⠈⡃⠀⣠⠃⡘⡀⠀⠀⠄⡀⠄⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢛⠏⠂⠀⠀⢶⣅⠌⠃⠀⠀⠐⠡⣨⡶⠀⠀⠐⠹⡛⠑
⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠠⠃⠈⠛⠐⡀⢀⡄⢀⠃⠛⠁⠘⠄⠀⠀⠁⠀⠀⠀⠀⠻⠂
⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠂⢠⠁⠀⠀⠀⠀⠃⠠⠄⠘⠀⠀⠀⠀⠈⡄⠐⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠘⠶⠃⠠⠄⠠⢼⠄⢀⣀⡀⢀⣁⠀⠀⣘⡀⢀⣀⡀⠠⡧⠠⠤⠄⠘⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠚⣇⡾⠱⠆⠀⠉⢫⡝⠉⠀⠰⠊⢷⣸⠓⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠑⢄⣀⣀⣀⠜⠣⣀⣀⣀⡠⠊⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣤⣴⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⡔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠉⠈⠋⠋⠉⡿⠏⠛⠛⠛⠛⠋⢿⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⢀⣠⣤⣤⣀⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣷⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢹⡿⠋⠀⡰⠛⣿⠟⣿⣿⣿⡿⢀⡖⠒⠋⠙⠛⠒⠒⢺⣿⡆⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢻⡶⠟⠡⣰⡇⣸⡇⡞⣙⡴⠋⠀⠀⠀⠀⠀⠀⠀⡌⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢏⣀⣤⠒⢉⡼⢋⡾⣻⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠉⢉⠐⠉⠉⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠋⢿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣀⣤⡴⣄⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢈⣿⣽⣷⣴⣇⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⣶⣾⣿⣿⣿⡍⠸⣷⣦⡕⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠉⣼⣿⣿⣿⣇⣀⠀⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⣿⣿⣿⠟⠃⢀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢰⣷⣮⣭⣭⣤⣴⣾⠇⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣆⠄⠉⠉⠋⠉⠉⡀⠰⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠐⢀⠀⠀⠀⢀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠠⡀⢠⣼⣶⣶⣶⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⢀⣨⣷⣾⣿⣿⡟⠀⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣛⠁⠀⠀⠴⢮⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣀⣀⡀⠠⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣹⡿⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡋⠉⠉⠙⢢⡄⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢺⠉⢦⡀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⡄⠀⠙⣄⠀⠀⢳⠔⠒⠒⠛⠛⠛⠋⠉⠑⠢⣤⠏⠀⠀⠀⣀⣠⡬⣦⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢷⠚⠉⠉⢦⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⣴⠚⠉⠀⠀⠘⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⠉⠙⣦⡈⢧⡀⠀⠈⢷⡀⠸⡀⠀⠀⠀⠀⠀⢠⠞⠀⢀⡴⠋⠈⢓⣦⡀⠀⠀⢹⡄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢦⡀⠈⢷⠀⢷⣄⠀⠀⠀⣰⠋⣠⠶⢋⣠⠴⠞⠉⠉⠙⢷⣄⠀⣧⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣝⢦⡘⣇⣨⡿⠷⢤⣜⣡⣾⠿⠚⠉⠀⠀⣀⣠⣤⠶⠚⠛⠣⣿⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠘⣎⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠏⠀⠀⣾⡋⠉⢀⣀⣤⡶⢞⣛⣉⣡⣤⣤⣤⡴⠶⢋⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠞⠩⣯⡀⠀⠀⣏⠉⠙⠛⠛⠛⠋⠉⢉⣉⣀⣠⡤⠤⠶⠚⠋⠁⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠑⢦⡀⢀⣠⣤⠴⠖⠛⠉⠀⠀⠀⠀⣈⣽⠿⣟⠉⠉⠉⠉⠉⠻⣍⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⠀⠀⠀⠀⠀⠀⢙⡏⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠀⠀⠀⣀⡤⠴⠚⠋⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡓⠠⣀⠀⠀⠀⢸⡇⠀⠀⠀⣴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⢀⣈⣿⣦⠤⠒⠉⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠈⠑⢤⡀⢸⡇⠀⠀⢀⡟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⣦⡴⠞⠋⠉⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⣷⣄⠀⢸⡇⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⠸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣤⣤⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡏⠈⢷⣼⠇⠀⠀⠀⠀⠈⠢⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣠⣿⠴⠖⠚⠋⠉⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠷⣄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠞⠛⢹⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣸⠀⠀⢀⡿⠀⠀⠑⠢⢄⡀⠀⣿⠀⠀⠀⣀⡠⠴⠚⠉⢹⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡿⠀⠀⣼⠃⠀⠀⠀⠀⠀⠙⢦⣷⠖⠚⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣼⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⡀⠀⣀⣀⣤⡤⠤⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣀⣠⣼⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⡿⠀⠀⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡇⢀⡴⠋⠁
⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⢉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢦⣼⡓⢀⡠⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡤⡀⠒⠑⠴⢤⣆⡀⢠⢶⣀⣄
⠀⠀⠀⠀⠹⠞⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⠁⢰⡷⠀⠀⣉⡙⢧⠋⠉
⠀⠀⢠⣴⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠠⠙⣀⡠⠆⡋⠀⣴⣾⣶⡀
⠀⠀⠿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠾⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣿⠆⠀⠀⠾⠫⢾⢟⡆⠀⠀⠰⠀⠈⠀⠀⠀⠻⣿⡿⠁
⠀⠀⠀⢀⣤⣴⣶⣤⣀⠀⣠⣶⣶⣄⠀⠀⣠⣴⣿⣶⣄⠀⠀⣾⣿⣷⠀⠀⠀⣨⣅⠀⠀⠀⣀⣰⣶⣦⣀⡀⠀⣤⣤⣤⡀⡿
⠀⠀⠀⠟⢻⣿⣿⣿⡏⠀⣿⣿⣿⣿⠀⠀⠙⠿⣿⡿⠋⠁⠀⢈⣭⡅⠀⢸⣿⡿⢿⣻⡇⢼⣿⣿⣿⣿⣿⣿⠄⣿⣿⣿⠀⡇
⠀⠀⠀⣷⠤⠉⠙⠉⠁⠀⣿⣿⣿⣿⢠⣶⣦⠀⠾⠆⠀⠀⠀⣿⣿⣿⠆⠸⣯⣿⣿⣽⠇⠈⠛⠿⣿⣿⠟⠋⠀⣿⣿⣿⢰⠇⢸⡇
⠀⠀⢀⠉⠀⢀⣴⡷⠂⠀⢨⣿⣿⡅⠈⣛⣟⠀⢸⠀⢀⡂⠀⣽⣿⣟⠀⠀⠀⢉⠉⢀⣠⣄⠀⠸⠿⠿⠀⠀⣀⣀⠀⢀⣬⡀⢸⡇
⠀⠀⢻⣿⣷⠙⠿⠳⣿⡷⠈⠙⠉⠀⠠⡿⠿⠇⠸⠀⠈⠁⠀⢻⡟⣿⠇⠀⠀⢸⡇⠸⢿⡟⠇⠀⠠⠀⠀⢘⣿⣿⣅⠻⣥⠟⠈⠁
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠃⣴⢶⠈⢀⣰⣶⡴⠶⣦⣤⣦⡀⢿⣿⣿⣿⠀⢶⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠀⠈⠉⠉⠀⠀⠀⠙⠉⠁⠀⠉⠉⠁⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠛⠙⠛⠙⠋⠙⠉⠙⠋⠛⠋⠛⠛⠋⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢧⣤⢤⡤⣤⡤⢤⡤⠤⠤⣤⡤⢤⣤⢤⡤⣤⡼⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⢁⡴⢋⡴⠋⢠⠎⠀⣸⠃⠀⠀⠘⣇⠀⠳⡄⠙⢦⠛⢦⡈⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠋⢠⡞⠁⢠⠏⠀⠀⡟⠀⠀⠀⠀⢿⠀⠀⢹⡄⠈⢷⡀⠹⣄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⡼⠁⢀⡞⠀⠀⡞⠀⠀⢰⠁⠀⠀⠀⠀⠘⡆⠀⠀⢣⠀⠀⢷⠀⠘⢧⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡸⠁⠀⡸⠀⠀⢠⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡄⠀⠀⡇⠀⠈⣇⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⠇⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠸⡀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡟⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣷⠀⠀⠘⡄⠀⠀⡇⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡿⠀⠀⢀⡇⠀⠀⡇⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠸⡄⠀⠈⡇⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⢰⠃⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢣⠀⠀⢣⠀⠀⠸⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠇⠀⠀⡎⠀⢀⡜⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⢧⡀⠘⣦⠀⠀⣇⠀⠀⢸⠀⠀⠀⠀⠀⢀⠇⠀⠀⡸⠀⠀⣴⠁⢀⡾⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⢳⡀⠘⣆⠀⠸⡄⠀⠈⡆⠀⠀⠀⠀⣸⠀⠀⢰⠇⠀⣼⠃⢀⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠙⢦⡈⢧⡀⠹⡄⠀⢻⡀⠀⠀⢀⡏⠀⣠⠏⢀⡜⢁⡴⠋⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢍⣢⣽⣦⣙⣦⣘⣧⣀⣀⣼⣁⣴⣣⣴⣯⣖⡩⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⡉⠉⠀⠁⠈⠀⠀⠁⠈⠀⠉⠀⡹⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠋⠉⠉⢻⡟⠉⠉⠉⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢸⠇⠀⠀⠀⡧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⢠⡟⢻⡄⠀⠀⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⢻⣛⠀⠀⢠⠟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡟⠀⠀⠀⠘⠃⠀⠀⠈⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⣆⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣯

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡴⠛⠋⠀⠉⠑⠆⠀⠀⢀⡀⠐⠉⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⢠⠁⢰⠀⠀⠀⠀⢀⠀⣰⠃⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢀⠔⠈⡈⠢⡀⠀⠀⠀⠀⡠⠚⢋⠉⡄⢓⠢⡀
⠀⠀⠀⢸⠀⢨⠀⠀⠈⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢾⠀⢠⠀⢀⡴⠊⠈⠢⡈⠂⠀⢀⡞⠰⠁⡈⠀⢠⠀⠇⠑⡄
⠀⠀⠀⠘⡀⠈⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢠⢇⠀⠀⠢⡈⠀⠀⠀⠔⢁⠀⠀⢸⠀⠠⡀⡇⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠀⣀⡇⠀⠀⠑⠦⣀⢀⡀⠔⠃⠁⠀⠀⠀⠈⠀⠈⠃⡐⠁⠀⠀⠈⠢⣘⣄⠰⠀⢀⢀⣃⠔⠁
⠀⠀⠀⠀⠈⠉⠛⡌⠋⠉⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠙⠏⣉⡉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠇
⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⡄⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢄⠀⡀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⠀⠀⠀⡴⡤⠀⠠
⠀⠀⠀⠀⢀⣀⠀⠀⠀⣤⠆⠀⠀⢀⠄⠀⠀⠰⡁⡀⠁⣌⣀⠌⠐⡱⠀⠀⠀⢹⠀⢀⡀⡠⠀⠀⠀⡁⠡⡀⠀⠅
⠀⠀⢀⠎⠁⡠⢔⠃⠐⠂⢄⠁⣉⡀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠉⠑⢤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢣⠀⠈⠺⡄
⠀⠀⠸⡀⡜⠀⠀⠀⠀⠀⠀⢃⠀⠈⢂⠀⠀⡜⠀⢠⠊⠀⡈⠱⡀⠀⠃⠀⠀⢀⡀⡔⠂⠙⠐⢄⠀⢀⠑⠢⠖
⠀⠀⠀⠉⢇⠀⠀⠀⠀⠀⠀⠄⠀⢀⡌⠀⠀⢇⠀⢰⡀⠋⠃⡰⠀⢀⠆⠀⠀⠘⡎⠁⠀⠐⠒⠚⠼⡹
⠀⠀⠀⠀⠈⠢⠄⣀⠀⠄⠚⠀⠂⠉⠀⠀⠀⠈⠢⣀⠉⠉⠉⣀⠠⠊⠀⠀⠀⠀⠈⠢⠄⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣩⣼⣿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⢿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢡⣾⠟⠉⠀⠀⠻⣜⢧⠀⠀⢀⣤⠄⠀⠀⢀⣴⡿⠋⠀⠀⢀⣴⡿⡟⠛⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣰⠟⠁⠀⠀⠀⠀⠀⠹⡌⣆⠀⠘⠿⠃⠀⣠⡾⠋⠀⢀⣠⡶⠟⠁⠀⠉⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⠏⣀⣠⣤⣀⡀⠀⠀⠀⢻⡜⡄⠀⠀⣀⣼⣯⣄⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣴⡿⠟⠛⠛⠛⠿⣷⣄⠀⠀⠹⣿⣴⠿⢛⣛⣩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠸⡿
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡶⢷⠘⠁⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⡼⠱⣿⠿⠿⢿⡏⠀⠀⢠⣤⣤⣄⣀⣠⣤⣤⣤⣤⣴⣿⣤⣀
⠀⠀⠀⢸⠿⢿⣿⣿⣯⣤⣤⣬⣿⣶⣤⣤⣄⣀⣀⣀⣀⣀⣤⣿⣧⣷⣶⣶⢶⣾⠟⠃⢰⣶⣿⣿⠿⢿⣿⣿⡟⢛⣛⣛⣋⣁⣬⣿⣿⣷⣶⣶⡄
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⢛⣿⡿⠛⣫⡿⢻⡿⠟⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡿⠿⠿⠿⠿⠛⠛⠛⠛⢻⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢀⣼⠏⢠⡟⠁⠀⠀⠀⠀⣌⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⢀⡾⠁⢠⡟⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠸⣿⣿⣶⣶⣶⣦⣤⣤⣤⣄⣰⠟⠀⠀⠀⡼⠁⠀⡾⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣤⣤⣤⣶⣶⣿
⠀⠀⠀⣤⠀⠀⣿⠉⠉⠙⠛⠛⠛⣿⠋⠀⢀⠀⠀⠁⠀⣼⣥⣀⣀⣀⠀⠀⠀⠀⢹⠀⠀⠀⢀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⢻⠋⠉
⠀⠀⠀⠿⠀⠀⡏⠀⠀⠀⠀⠀⣼⢁⣠⣾⠿⠀⠀⠀⢰⠏⠙⠛⠿⠿⣿⣿⣶⣦⣼⣤⣶⣿⣿⠿⢧⠀⠀⠀⣤⡀⠀⠀⢠⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⡾⠋⠀⠀⡄⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠋⠉⠁⠀⠀⠸⡀⠀⣼⠋⠻⡄⠀⡾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢳⡀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠇⢰⠇⠀⠀⢻⣾⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠻⠿⢿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣶⣶⣤⣤⣀⣀⠀⣿⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⣿⣿⡿⠿⠟⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠒⠛⠛⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⢀⣀⣤⠤⠶⠶⠶⠦⠤⣤⣀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠘⠄⡠⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⢀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢹⠘⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣼⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠸⡆⣷⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⡶⠋⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢻⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⢀⡀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⢀⠀⢀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⠔⠀⠀⠀⠀⠠⠄⠰⠄⠀⠀⠀⠁⠠⠃⠠⠃⠀⠀⠀⠀⢆⠀⠆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠰⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⡂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠤⠐⠂⠚⠒⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣤⠒⢂⠀⢸⡇⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⢀⡐⠊⠾⡱⠳⡄⢸⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⠘⢨⠀⠀⠀⠰⠤⠀⢰⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⠀⠀⠂⠀⠡⡄⡤⠁⠖⠐⠢⠑⣌⣠⠠⠒⠀⠀⡄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠈⠀⠁⠀⠈⠀⠀⠁⠉⠉⠁⠀⠁⠀⠀⠁⠀⠀⠁⠀⢾⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠐⠀⣤⠀⠀⠀⢠⡀⠀⠂⢠⡄⠐⠀⣤⠀⠀⠀⢠⡄⠀⠂⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⣤⣤⣤⣌⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠈⣤⣤⣤⣤⣥⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡤⠶⠚⠛⠛⠓⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣤⠤⠤⠤⠤⢀⡘⢳⢀⣴⣉⣀⠉⠉⠉⠉⠉⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠈⢣
⠀⠀⠀⢀⡶⠋⠁⢀⡤⠒⠁⠀⣴⣸⣿⠋⠉⠉⠙⠛⠛⠛⠶⠶⠶⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠇
⠀⠀⢀⡟⠀⢀⡴⠋⠀⠀⢀⣴⠋⠈⠁⠈⠛⢶⣦⣄⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⢸⡇⢠⠋⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⣀⣠⠽⠿⣷⡀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠘⠃⣠⣴⠶⠟⠛⠉⠀⠀⠀⢀⡴⠋⢁⡠⠤⠐⢂⡽⢽⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢛⠁⠈⠲⣄⡀⠀⠀⢀⡴⠁
⠀⠀⠀⠋⠁⢸⡀⠀⠀⠀⠀⠀⠠⣡⣀⡊⢁⣀⣠⠾⠋⠀⠀⠀⠀⠈⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠀⠛⠳⢤⣈⠙⠦⣠⠏⢀⣀⣀⣤⣤⡤⣄
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠐⠶⢤⣤⣀⠈⠉⢠⠟⠃⠤⡤⠤⠤⠤⠄⠈
⠀⠀⠀⠀⠀⠀⠘⠻⣦⣀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣀⢠⠉⠠⣤⣀⡀⠈⠉⠛⢲⡿⠟⠀⠈⠉⠙⠛⠛⠒⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠀⠠⣤⣤⠿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⡀⢠⠁⢉⠀⡀⠀⠈⠉⠓⠂⢀⡾⡏⠀⠉⠛⠛⠿⢷⣶⡂⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠦⠀⢀⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠑⢤⡉⠒⠢⠤⣠⡞⠹⠃⠀⠒⠲⣦⣤⢤⣀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⣏⠀⠰⢦⡀⠀⠑⢄⠈⢀⣀⣾⡋⠉⠀⠈⠑⢤⣀⠀⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣤⡀⠈⠳⣄⡀⠙⢦⡀⠀⣀⣿⣏⡭⡀⠀⠓⢦⣀⠀⠈⠛⠆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠐⢤⣀⠉⠛⢦⣄⠀⠙⠆⣠⣾⡾⠿⠂⠈⡀⠘⢦⣀⠀⠙⠷⣦⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠒⠦⣄⠀⠈⠛⠶⣄⠈⠁⣀⣼⠿⠏⠁⠀⢄⠀⠹⣄⠀⠙⢷⣄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⠦⣄⡀⠈⠛⠦⣄⠀⠀⣠⣼⠏⠁⠀⠀⣄⠀⠈⢧⡀⠘⢷⣄⠀⢙⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣙⢩⣀⠀⠀⠙⠓⠤⣀⢀⣱⣾⠟⠠⡀⠀⢧⠀⠘⣧⠀⠈⠷⠀⠀⣣⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠁⠀⠀⠈⠙⠂⠀⢀⣀⣼⠿⠋⠀⣄⠀⠳⡄⠈⢣⠀⠈⠃⣴⣾⠆⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠤⠤⠬⠶⠶⠒⠂⣀⣼⣿⠟⠀⢦⡀⠘⢦⠀⠹⣄⠀⠰⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠤⠤⠤⠄⣠⣴⣿⡉⠁⢳⡀⠀⠳⣄⠈⠳⡀⠈⠂⡴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⡅⠀⡄⢳⠀⠘⡝⢄⠀⠘⠧⠀⣀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣼⠁⢸⠇⠘⣇⠀⠱⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⡟⠀⠀⠈⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢿⠉⠀⣻⠏⠛⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣣⣠⣇⣤⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣢⠔⢊⣩⠟⡇⢸⠙⢦⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣅⣠⡎⠀⠀⡇⢸⠀⠀⠳⣄⡽⠆
⠀⠀⠀⠀⡰⠶⢍⡔⠛⠛⠋⣷⢺⠛⢿⣇⣀⣀⡇⢸⣀⣀⣀⣹⠁
⠀⠀⠀⡞⠁⠘⣤⣇⠀⠀⠀⣿⡞⠀⢀⣿⠁⠈⡇⢸⠈⠉⢹⡇⠀⣀⣤⣤⠴⢦⡀
⠀⠀⠀⠘⢦⡠⠮⠿⢄⣤⠼⠟⢦⣢⣞⣿⠐⠀⡇⢸⠀⠀⢸⣧⣼⠃⠀⠀⠀⠀⣧⠴⠲⢦⣀
⠀⢠⠴⠖⡾⠁⠀⠀⡼⠳⡄⠀⠈⢯⠉⠉⢻⠂⡇⢸⣤⠞⠋⠰⡂⡆⠀⠀⠉⠀⡿⢠⠃⠀⠉⢷
⠀⢸⡀⢰⡷⠺⡆⣼⠁⠀⢷⠀⠀⠘⣆⡀⣼⡉⡇⢸⠾⡄⠀⠀⠱⣽⠀⠀⠀⠀⣿⠃⠀⠀⢀⡟⠜
⠀⢆⢻⠛⠛⠛⠛⣿⠀⠀⠸⡇⢠⣄⣿⢹⡇⠈⡇⢸⠀⢹⣄⣠⠤⠼⢧⣀⡤⠼⠓⢦⣀⣴⣏⣀⣀⣀⡀
⠀⠀⢸⡆⠀⠀⠀⣿⠀⠀⠀⣷⡏⠈⠛⢸⡉⠀⣧⡟⠉⠉⡽⠁⠀⢠⠞⠁⠑⣄⠀⠀⠹⣆⣀⣀⣀⣀⡇
⠀⠀⠸⡇⠀⡀⠀⣿⠀⠀⠀⡏⠀⠀⠀⢸⠀⠀⡏⢳⡶⢶⡷⠲⣄⡞⠛⠛⠛⠛⣆⢠⢤⣸⡏⠉⢹⠋
⠀⠀⠀⣇⠀⠀⠀⢻⠀⠀⠀⡇⡄⠀⠂⣸⠐⠀⡇⣸⠸⡄⠀⠀⠈⠁⠀⠁⠀⠀⠹⠋⢀⢈⣋⣀⡟
⠀⠀⠀⠹⠤⠤⠤⢿⣤⣤⣤⠷⠶⠶⠖⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣼⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣠⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⣽⠏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢘⣣⣴⡯⢶⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⡏⢹⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⣀⣤⣴⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡿⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⣿⣿⣿⣿⣿⢀⣿⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡇⠘⠇⠀⡆⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠹⣿⡶⠿⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⣤⣤⣤⡄⠀⣠⣤⠀⢠⣤⣤⣄⢠⣤⣤⣄⠀⣤⢠⣤⣤⣤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠉⠉
⠀⠀⠀⣿⡷⣾⡃⢰⣯⣽⣆⢸⡷⢺⣯⢸⣿⠺⣿⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠙⠁⠈⠋⠙⠁⠈⠛⠈⠛⠛⠉⠈⠛⠛⠋⠀⠛⠀⠀⠙⠀⠀⠀⢀⣴⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠀⠘⠿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣈⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⣴⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣯⣽⣟⣛⣯⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠄⣠⣾⡿⠉⠈⠉⠀⣼⣷⡒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣼⠋⣹⣧⣀⣀⣀⣀⣠⡍⢻⡎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⠀⢨⣿⣻⡝⠩⢉⣇⠀⢸⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⠀⠈⠙⠛⠛⠋⠋⠹⡄⢸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⣏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡏⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠗⠢⢄⣀⠀⠀⣀⠠⠖⣧⣿⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠇⠀⠀⡈⠙⠋⡀⠀⢰⣿⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣸⢤⡐⣤⡔⣷⡄⠠⡳⣤⡎⣿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣄⣈⣙⣶⡴⣶⡾⣷⣋⣉⣉⡽⣿⡿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢾⣿⣿⠿⣧⠞⢳⣾⠋⠻⢿⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⣤⠤⣿⡶⠾⠃⢤⠔⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣞⠉⠻⣄⠀⠀⠀⠤⠾⠒⣋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠠⠴⠖⠒⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡨⠀⠁⢀⣔⣀⡀⠀⠀⠀⠀⠀⡠⣀⢤⠀⡔⠀⡀⠀⢀⢀⠀⡠⠠⠆⡀⠀⠀⢀⡀⠀⠀⠰⢢⠠⡄⢀⢔⠋⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠂⠀⢈⠀⠙⠀⡃⡁⠀⠀⠀⠀⠀⠢⠊⠀⠀⠀⠀⣴⠁⣤⡄⠁⠀⠉⠀⠀⢁⠟⠁⠀⠀⠀⠂⠊⠀⠈⠁⠀⠀⢀⠂
⠀⠀⠀⠀⠀⡀⠀⠀⢀⡌⠀⠀⢴⠐⠂⠀⠀⡔⣐⢡⢀⠄⠀⢀⠤⠐⠀⠠⢎⡤⠀⠉⠀⠈⢀⠔⢀⠠⠤⠒⠁⠀⠀⠀⠀⠀⠐⠀⠀⢠⣀⠀⠠⠊
⠀⠀⠀⢀⠎⠄⠀⠐⡈⠁⠀⠀⠀⠀⠀⠀⠠⣧⡿⠁⠀⠀⠑⠞⡄⠆⠀⠂⡏⠁⣠⠆⠀⠀⡉⠀⠀⢡⣰⠀⠀⠀⠡⠤⣴⣷⠂⡀⠐⡌⣀
⠀⠀⠀⢈⠀⠀⠀⢀⠄⡑⢠⠀⢈⡇⠀⠀⠀⠈⢳⣤⠂⠂⢠⠎⣀⢆⢀⣌⡀⢰⣨⠂⢊⣸⠧⠄⠀⠸⢯⡀⣀⠀⠈⡠⠟⠒⠦⡀⢸⡆⠃⢠
⠀⠀⠀⠈⠀⣁⡠⠊⢀⡀⠀⢅⣨⣷⠄⠀⠀⠀⠀⠀⠈⠻⣡⠒⡟⠸⠟⠈⢃⡤⢃⡴⠛⠃⠀⠀⠀⠀⠀⠘⠃⠈⡜⣓⡂⣀⡀⡳⡿⢠⣄⡿
⠀⠀⠀⠀⠀⠻⠀⠄⠛⡇⠀⠈⣼⠡⠀⠀⢀⠤⠆⡀⠀⠀⡏⣸⠅⠀⠈⠀⠈⠧⣸⡇⠀⠠⣎⡝⡄⠀⠀⢀⡗⣾⠁⠄⠀⠀⠀⠀⠹⡟⠺⠈⠑
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡰⣉⡙⡆⠐⣈⢠⢈⡿⠁⠀⣸⠯⠀⠀⣠⡤⢤⠀⠀⠈⠳⣰⠽⣁⡼⠁⠀⠀⣼⠗⠁⠀⠀⠄⠂⠀⠀⠀⠻⢂
⠀⠀⠀⠀⠀⢀⡀⠀⢧⡜⠑⣵⡿⢂⢀⡩⣦⢞⠁⠀⣰⠏⠀⠀⡞⠙⠇⣠⢰⡱⡄⠀⢣⣠⠟⠀⠀⢀⡼⠃⣶⠀⠠⠾⠶⢶⠖⠒⠀⠀⠘⠇⢁
⠀⠀⠀⠀⠀⠇⠀⣡⡴⡟⣧⠘⠷⠛⠁⠸⠏⠀⠀⡴⢸⠀⠀⢸⣇⠪⡝⠶⡄⢣⡀⠀⢸⠋⠀⠀⣠⡆⡇⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⡀⠀⡿⢂
⠀⠀⠀⠀⠀⡠⠀⢠⡦⠗⠋⢣⠈⢄⠔⠀⠀⣠⠊⢤⡊⡆⠀⠀⠹⣆⠙⣆⣘⠆⠀⠠⠁⠀⠀⣴⡹⠶⠓⠂⠀⠈⡆⡐⡖⡒⣣⠄⠐⠁⢠⣧⠆
⠀⠀⠀⠀⠀⠈⠀⢫⡀⠀⠀⠀⣱⠌⠀⠀⠐⠉⠉⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠁⠈⠈⠉⠀⠀⠀⣠⠷⡞⡃⠆
⠀⠀⠀⠀⣶⠀⠀⠠⠻⣤⢐⣠⣴⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠣⡀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠠⠶⠎⢉⡨
⠀⠀⠀⠀⠉⠉⢠⠢⠀⡼⢿⡉⠀⢠⠔⠋⠀⠀⠀⠀⡀⢠⠤⡠⣒⡄⢤⠂⠐⠦⡀⢸⠀⢃⡘⡜⠀⢱⠆⠸⠀⡠⢧⢸⢽⠀⠀⠀⡸⠒⠒⠒⢒⠰⠄
⠀⠀⠀⠀⠀⠈⠊⠀⠃⢀⡼⠷⠄⠐⠆⠀⠀⡆⢲⠠⠛⠄⠏⠀⠃⢀⡘⣀⡀⠠⠉⠈⠥⠬⠤⠥⠤⠤⠤⠬⠥⡥⣬⣈⣈⣀⡀⢠⠃⠀⠖⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⣀⣠⣤⢤⣴⢒⣞⠏⠉⠀⠀⣔⣀⣐⠀⠀⢀⠤⣒⣠⣀⡄⢀⠀⠐⣥⡇⢘⣧⡿⢛⢉⣀⣀⣀⣀⣀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢮⣤⣤⠤⠠⣭⠭⣩⢿⠟⡘⢛⠀⠀⠀⠀⠅⠉⠛⠛⣿⠇⠚⠛⠛⣻⠟⠈⠀⠀⡼⡉⡿⠋⡹⠿⡿⣤⠼⣣⢘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⡉⢻⢳⠓⢁⠔⡁⠰⠇⠀⡀⠀⣦⠀⠀⠀⠶⠐⠚⠁⠀⠀⠀⢚⡇⠀⠀⠀⢘⡁⠀⡐⢀⣀⣾⢟⠑⡮⡷⣾⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠖⢋⣃⠄⠀⢧⢰⣇⠰⠡⠐⠆⠀⢲⠜⠀⡄⠴⠇⠀⠀⠀⠀⠀⠀⡨⠶⠀⣠⠀⠈⠠⣃⡒⠉⠤⠛⣿⠞⠀⢹⡯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠻⠆⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠂⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠻⠛⣛⠉⠉⠭⠉⠁⠍⠉⢋⣙⠛⠟⠒⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠏⠁⡈⢁⠂⢀⠅⢄⠀⠃⠤⠴⠀⠤⠴⠐⠀⠔⢈⠐⠠⡡⠈⢩⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠋⠀⠂⠈⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠀⠀⠀⠜⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠀⠀⡀⠆⠀⠀⠀⠀⠀⢀⢀⣤⣤⣀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠑⠀⡊⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢿⣿⣿⣿⣭⡿⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⡠⠚⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠙⢀⠁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⣿⡿⢫⣿⣷⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠎⢷⡀
⠀⠀⠀⠀⠀⣠⠿⠁⠄⠠⠃⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢿⣷⡄⢀⣀⠀⠀⠀⠄⠐⠀⠽⣆
⠀⠀⠀⠀⣰⠏⠀⠀⡀⠀⠀⢠⡶⣾⡟⠛⣿⠆⠀⠀⠀⢀⣤⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⢶⣶⣽⣯⣿⢋⣐⠿⣿⣷⠀⠀⠀⠀⠀⠄⠙⣆
⠀⠀⠀⢰⠏⠁⠀⡄⠀⠀⠀⠀⠀⢿⣿⣶⡶⠃⠀⠀⢠⣿⣻⡇⠀⠀⠀⣠⣴⣶⠗⠀⠀⠀⠚⠉⠙⣿⣿⣿⠿⢿⣿⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⠹⡆
⠀⠀⢀⡟⠂⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣯⢿⡇⠀⣠⣾⣿⣿⡏⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢆⡀⠐⢻⡄
⠀⠀⣼⡁⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡿⢀⣾⣿⠿⣫⡟⣴⣾⢿⣹⢿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⢠⣧
⠀⢀⣯⢀⠉⢂⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⣋⣴⣿⣟⣵⣿⣿⣾⣿⣿⣾⣹⣾⣿⣿⣿⣿⣿⣻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠈⡄⢻⡄
⠀⢸⡇⠈⠠⠀⠀⠀⠛⣠⡺⠃⠀⠀⣠⣴⣾⣿⣿⣿⣿⣛⡽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣾⣵⣿⣿⣧⡀⠀⠀⠀⢀⡰⢆⡀⠀⠀⠀⠀⠄⠀⢺⡇
⠀⢸⡇⠂⠀⠃⠀⠀⠀⠉⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⣾⣿⣿⣿⢟⣿⡿⣿⣟⣻⣿⣿⣿⣿⣿⣷⣄⠀⠀⠈⠹⠏⠁⠀⠀⠀⠸⡁⠘⢸⡇
⠀⢸⡇⠤⢈⡄⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⣿⣾⣿⣿⣿⣻⣿⣿⣿⣿⣿⡿⣿⣿⠿⣿⣿⣏⣾⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⢨⡇
⠀⢸⡇⠀⠐⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⡿⠻⣤⡞⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣤⣶⣤⠛⢿⣿⣇⠀⠀⢠⡘⢃⣄⠀⠈⠂⡀⢼⡇
⠀⠈⣷⠈⣀⡀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡉⣳⣾⢣⣿⣿⢿⣿⣿⣿⣿⣦⡴⠛⠿⠟⢦⣤⣿⡟⠀⠀⠈⠱⡆⠉⠀⢡⡀⠃⢺⠃
⠀⠀⢻⡁⢀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡟⣡⣾⣿⣿⣮⢿⣿⣿⣿⣿⣷⣦⣠⣴⣾⣿⢿⣶⣆⠀⠀⠀⠀⠀⢀⠂⡀⠈⡟
⠀⠀⠈⣧⠄⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠛⢛⣛⣿⣟⣓⣝⠻⣿⣿⣿⣿⠿⠿⣿⣵⣿⣿⣷⠀⠀⠀⠀⠀⠈⢐⠠⣼⠁
⠀⠀⠀⠸⣆⣀⠀⠀⡀⠀⠀⠀⣠⣴⣷⣄⠀⠀⠀⣴⣾⣿⣿⣿⡿⠟⠁⠀⠀⠀⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠛⠿⠿⠟⠁⠀⠀⠀⢀⠘⠀⣀⣰⠏
⠀⠀⠀⠀⠹⣅⠐⠀⠀⠀⠀⠀⢹⣧⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⣸⠏
⠀⠀⠀⠀⠀⠹⣖⠀⠄⠰⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⢾⣛⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣦⣄⠀⠀⠀⠀⢀⠂⠀⠀⣲⠋
⠀⠀⠀⠀⠀⠀⠈⢷⡤⠈⡄⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣾⡏⠀⠀⢀⠄⢀⠁⢆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠈⡀⠀⣀⠀⠀⠀⠀⣀⣀⣠⠶⣶⣶⣶⣄⠀⣀⡀⣠⢞⣛⣳⣦⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠑⣄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣬⠀⠀⠁⠀⠰⣻⣭⣍⡛⢿⣭⢭⣿⢟⣫⣭⣙⠳⣏⣥⣽⡿⣛⣭⣭⡛⢶⡄⢀⡀⠰⠋⡀⠀⢄⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡄⠀⠠⢁⠒⢨⢻⠆⢻⣼⢣⣏⣴⢶⣝⣷⡘⣧⣞⣼⢫⡶⠆⡙⠄⠀⠂⠠⠀⣠⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣇⡀⢀⠀⠄⡁⠔⠉⠉⠕⠲⠁⠄⠰⠨⠉⠁⢐⠈⠠⠠⡐⢀⣰⡦⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⣴⣤⣍⣡⣀⣀⣀⣀⣂⣀⣄⣩⣄⣦⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣮⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣪⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣻⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⣿⣿⣯⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠻⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣿⣿⣿⣿⠄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⣠⣶⣦⣼⣶⣤⠸⣿⣷⣿⣿⣿⣿⣿⣾⣿⣯⣶⡀⠀⠀⠀⠀⠀⠀⠀⣼⣬⣿⣿⣿⠟⠋⠁⠀⢀⠀⠀⠀⣴⣏⣻⣿⣿⣾⣧
⠀⠀⢸⣛⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡇⠀⠀⠀⡀⣀⠀⠀⢹⣿⣯⣿⠁⠀⠀⠀⠀⠀⠀⠤⣀⣿⣟⣿⣿⣿⣿⣿⡆
⠀⠀⠸⣿⣿⣻⣿⣿⣿⣿⠈⠛⠟⠛⠁⠙⢿⣿⣿⣿⡿⠁⢀⣴⣿⣷⣿⣿⣦⡀⠻⣿⣻⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠹⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⣿⡿⠋⠀⠀⢼⣿⣿⣿⣿⣿⣿⣷⠀⠈⠻⠇⠀⠀⠀⠀⠀⣸⠋⠁⠹⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠙⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣿⣿⣽⣿⣿⣿⣿⠀⠀⠀⠀⢀⣀⠀⢀⣴⡇⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣽⣿⣿⡟⠀⠀⠀⡰⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠀⠀⠀⠀⣿⣾⢿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣠⣤⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠃⠀⠀⠀⣿⣽⣿⣿⣿⣿⠟
⠀⠀⠀⢀⣤⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠹⣿⣿⠿⠟⠁
⠀⠀⢰⣏⣻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⣽⣿⣿⣿⣿⣿⡄
⠀⠀⢺⣿⣟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣹⣿⣿⣿⣿⠇
⠀⠀⠘⣿⡿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠘⢿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⠒⠶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣬⣷⡀
⠀⠀⢠⠟⠋⠉⠉⠛⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢋⣩⠤⠶⢦⡈⢳⡀
⠀⠀⡟⠀⢰⡏⠉⠉⠛⠲⠦⢤⣍⡓⠦⣄⡀⠀⠀⠀⠀⣀⡤⠶⠿⠷⠶⠦⣄⠀⠀⠀⢀⣠⠴⠚⣉⣤⠴⠚⠉⠀⠀⠀⣠⠇⠀⡇
⠀⠀⢳⣄⠀⠙⠲⠤⣄⣀⣀⣀⣀⣉⣙⣛⡻⣶⣤⣀⢰⡏⠀⠀⠀⠀⠀⠀⢸⣇⣠⣶⣿⢗⡚⣉⣁⣀⣀⣀⣀⣤⠴⠚⣁⣠⣞⣁⣀⣀⣀
⡟⠛⠛⠉⠛⠲⠦⠤⢤⣤⣭⣭⠭⠭⢭⡿⠛⠛⠋⠉⢉⣻⣶⠦⠤⠤⠶⠖⠛⢿⣉⠛⠛⠛⠛⠛⠛⠿⠿⠟⠗⣶⠛⠋⠉⣻⠉⢻⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠖⠋⠀⢸⡄⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡴⠋⠁⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣿⠀⠀⢀⣿⡀⣸⣀⣀⣀
⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡉⠉⣿⠉⠉⠉⣿⠉⢻⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠁⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠦⣄⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠓⢤⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣸⣀⣠⠤
⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢸⣇⣀⢀⣀⣀⡀⣸⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀⣿⣀⣤⠴⠟⠛⠉⠁

⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⣀⣶⣄⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣰⣷⡀
⠀⠀⠀⢠⠾⣿⠷⡄⠀⠀⠀⣴⣿⣿⣄⠀⠀⠀⣤⣿⣿⣧⡄⠀⠀⢠⡿⠛⢿⡆⠀⠀⠀⣠⣿⣿⡀⠀⠀⠀⣴⡛⠙⣳⡄
⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⠀⡟⠁⠈⣿⠀⠀⠀⡍⡏⠈⡉⡁⠀⠀⢸⡇⢶⢸⡇⠀⠀⠀⠹⠹⠹⠋⠀⠀⠀⣿⡁⠀⣿⡇
⠀⠀⠀⢸⠀⢸⠀⡇⠀⠀⠀⣇⣀⣀⣿⠀⠀⠀⢰⣷⢀⣧⠁⠀⠀⠸⣧⣤⣼⠇⠀⠀⠀⣤⣤⣤⡤⠀⠀⠀⣿⣂⣀⣾⡇
⠀⠀⠀⠘⠛⠛⠛⠓⠀⠀⠀⠋⠉⠉⠙⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠛⠉⠉⠙⠃
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢰⣿⢿⣿⡆⠀⠀⠀⣰⣿⣷⡀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⣿⣿⣟⠀⠀⠀⠀⣠⣾⣦⡀⠀⠀⠀⢾⣿⣿⣿
⠀⠀⠀⢸⡁⠇⢸⡇⠀⠀⠀⢹⠇⢻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⣧⠀⠀⠠⢛⠋⢛⠣⠀⠀⠀⠨⡇⢘⡇
⠀⠀⠀⢸⠁⡄⢹⡇⠀⠀⠀⢸⡇⣸⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⡿⠁⠀⠈⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣇⣨⡇
⠀⠀⠀⢸⠷⠶⠾⡅⠀⠀⠀⠹⣷⡿⠁⠀⠀⠀⠀⠛⡙⠁⠀⠀⠀⣧⣀⣀⣀⣿⠀⠀⠀⠀⢿⠆⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⢛⠻⣛⡛⣻⣟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠐⠀⠀⠐⠀⠀⠀⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣏⣹⣷⣤⣿⣿⡿⠛⣿⣿⣿⣿⣿⣯⣹⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⠈⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⠘⠻⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣧⡀⣠⣿⠿⠉⠁⠿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠿⠿⠿⣿⣿⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣷⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡀⣀⣼⣿⣿⣿⣿⣟⣹⣿⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⣤⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣶⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⠛⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⡿⠃⠀⠀⠀⠀⠈⠛⠛⠛⣿⣿⣿⣿⣿⣶⣿⣷⣶⣶⣶⡾
⠀⠀⠀⠀⠀⡴⠿⠿⢿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⡈⠀⠀⠀⠈⣿⣿⣧⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡟⠁⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣧⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡟
⠀⠀⠀⠀⠀⠻⣶⣀⣀⣠⣴⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣀⣉⠽⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣟⠀⠀⠀⢀⣴⠃⠀⠀⠀⠈⠛⠿⣿⡿⠋⠀⠉⠉⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡆⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠛⠉⠙⢋⣠⣴⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⢀⣠⣴⣾⣿⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⢀⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⢀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢠⣿⣿⠟⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⣠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⡁⠀⢀⣴⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⢀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁⣸⣿⣿⣿⠇⣸⣿⣿⡿⠛⣿⣿⣿⡇⢠⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠉⠀⠀⢠⣾⣿⡿⠛⠁⠀⢿⣿⣿⡇⢀⣿⣿⡿⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠐⠊⣉⡀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⣶⡆⠸⣿⣿⣿⣿⣿⡟⣡⣶⣶⣿⣿⣿⣿⣿⠏⠀⠀⣠⣶⣄
⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⡄⠀⠀⠀⠀⣀⣉⣉⣉⣭⣥⣤⣤⣌⣛⣛⡛⠁⠰⢿⣿⣿⡉⠉⠉⠉⠁⠀⢠⣾⣿⣿⣿⣇
⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢿⣿⣷⣦⣍⡛⠿⡷⠀⠀⢠⡀⠈⣿⣿⣿⣿⡿
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡘⣿⣦⣼⣿⣿⣿⣿⣿⠟⠁
⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⠻⣿⡿⠉⠉⠉
⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⢀⣀⣀⣀⡀
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉⢿⣶⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠈⠙⢿⣿⣿⣦⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠶⢿⣿⠛⠋⠉⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⣀⣸⣿⣤⣤⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡉⢹⣧⡄⢙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⢸⣿⡇⠈⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⣿⠁⠀⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⢿⡄⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠰⠾⠛⠁⠀⠀⠀⠀⠀⠈⠻⣷⣶⣶⣶⣾⣿⣿⡆⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠛⡟⢻⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣍⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡬⡿⢿⠁
⠀⠀⠀⠀⢴⡤⣤⣀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⣀⣠⡤⣴⠄
⠀⠀⠀⠀⠘⣇⠀⠙⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠛⠁⢀⡟
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠙⢿⣳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⠀⠀⢿⣿⡆⠀⠀⠀⢀⣾⣿⠃⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣸⣿⣿⣶⡖⣶⣾⣿⡿⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡬⢏⠯⣽⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⢿⣿⣷⣿⣶⣿⣿⠟⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣿⣽⣿⢿⣿⣿⠟⣩⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣙⠳⠶⠶⠶⢾⣿⣋⣡⣾⠿⣿⣿⣿⣿⣻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠻⢿⣿⣷⣶⣿⣿⡟⢣⠞⣶⣿⡛⡻⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣴⣤⣶⣾⣿⣿⢿⣷⢾⣿⣏⣹⣏⢻⣟⣿⣿⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣮⣍⣉⣩⣽⣻⠟⣶⡳⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣽⡆⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡯⢿⣿⢿⣿⣟⡏⣻⣧⣽⣶⢞⣿⣿⡿⢻⣿⣿⣷⣤⡿⢶⣿⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣷⠏⣹⢻⣹⣿⢿⢯⣿⠿⢾⣧⣸⡷⠿⢿⣿⣧⣼⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⣿⠏⢀⣿⣿⣿⣯⠃⣾⣷⣦⡴⣯⣹⢦⡴⣾⡽⡾⣽⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣍⠀⠻⠏⠀⣾⠌⣩⣿⠃⠀⠹⣿⣟⣀⣼⢧⣀⣹⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⢿⣋⣤⣾⣮⣙⣛⠇⠀⢀⣀⣬⣿⠻⠿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⣿⠿⠿⠟⠁⠀⠰⢿⠿⢿⠿⠿⠶⠾⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠄⡠⢀⠄⢤⠐⠤⠀⡀⢠⠠⠀⠠⡀⡄⠀⢀⠄⠰⢄⠠⠀⡠⡄⠀⠄⢠⠀⡄⢠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⣉⡷⠀⠀⢸⠋⠉⠉⣿⠀⠀⠘⠋⢉⣹⠆⠀⠀⠛⢉⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣞⣥⣤⠀⠀⢸⣤⣤⣤⡿⠀⠀⢠⣾⣫⣤⡄⠀⠀⣤⣤⣭⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⠀⣀⡀⠀⣀⡀⣀⡀⣀⣀⢀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⡭⣽⡟⠛⠛⠛⠛⠛⢛⠻⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡒⠒⠋⠉⠓⢦⣀⣠⣤⣀⣀⠀⠀⢀⡴⠋⠉⠉⠙⠓⠒⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠉⠢⣄⠀⠀⠀⠈⠢⡀⠀⠉⠉⡳⠋⠀⠀⠀⠀⢀⡴⠋⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⢶⣶⠛⠙⠲⢄⣈⠑⠤⣀⠀⠀⠈⢦⣠⡜⠁⠀⠀⣀⡤⠖⣁⡠⠔⠊⢑⢢⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠛⠉⠁⠀⡰⠋⠉⠉⠙⠒⠦⣌⣉⠓⠺⠿⢶⣶⠖⠛⢲⣶⠶⠾⠗⠒⣉⡥⠴⠒⠋⠉⠁⠈⢻⠓⠲⠦⣤⣀⣀
⠀⠀⠀⠀⢰⠾⣍⣁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢀⣀⣀⣀⣀⡀⣀⣭⣵⠶⢖⣻⠦⡄⠾⣛⠲⠶⣮⡭⠤⠤⢄⣀⣀⣀⠠⠴⠋⠀⠀⠀⠀⠀⢉⣉⠷⡆
⠀⠀⠀⠀⢹⠀⠀⠈⠙⠒⠦⢤⣀⡀⠀⣀⡤⠴⠚⣉⣡⡤⠶⠒⠉⠁⠖⠚⠉⠀⡴⠓⣄⠀⠉⠁⠀⠈⠉⠒⠦⢄⣉⠓⠲⢤⣀⠀⣀⡤⠴⠚⠉⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢙⡟⣳⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠉⠳⢴⠚⢻⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠑⣾⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠺⣄⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠱⣄⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⡟⠙⠲⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⢶⠖⠋⠉⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⣎⣠⠴⠚⢻
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⣄⠀⠀⠀⢀⡜⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢱⡀⠀⠀⠀⠀⢀⡴⠳⢤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣸⡤⠞⢣⡀⣠⢾⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠹⣄⠀⠀⣠⠎⠀⠀⠀⡏⠉⠛⠲⢤⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⠴⠞⠋⢹⠀⠀⠀⠙⠁⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢣⡼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠑⠲⠤⣀⣸⣇⣠⠴⠚⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠛⠦⢄⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣬⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣸⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⢸⠀⠀⢀⣠⡴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣼⡴⠖⠋⠁



⣀⣀⣀⢀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠠⠠⠠⠀⠠⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠤⠄⠄
⠀⡀⡃⢈⠁⠹⠽⠸⢠⠄⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⡄⠆⠯⠏⠀⡀⢸
⠀⡀⣆⡀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⢀⢨⢀
⠀⣇⠁⡇⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢰⠊⠙⢳⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠘⢈⡸
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⣠⣜⠊⢀⣼⣇⠀⠁⣄⠀⠀⠀⠀⢀⡀⠀⠀⠰
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⡎⢳⡈⢹⢋⠀⠈⡁⠀⡸⠀⠀⠀⠠⠊⠑⠀⠀⠈⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠊⠉⠉⢹⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠈⠒⠚⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠸⡀⢈⡗⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠑⠄⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠉⡿⠉⠉⠉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠘⠀⠐⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠜⠁⢸⡀⠀⠀⠋⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢀⣄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠊⠁⢇⠀⢘⡖⠒⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⡇⠀⠀⠀⢰⠋⠀⠋⠒⡖⠋⠑⠂⠈⡇⠀⠀⠀⠁⠄⠀⠀⠀⠰⠄⠠⠐⠒⠀⠀⠀⠀⠀⠂⠀⠱⡀⠀⠀⠀⠀⢸⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠉⠀⠂⠐⠃⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠐⠠⠀⢋⠐⠀⠈⠆⠀⢸⠀⠀⠀⠀⢸⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠠⠀⠐⣀⢀⠄⠀⠀⠀⠀⠐⢸
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⠔⠒⢂⠢⠀⠀⠁⠀⠔⠐⠁⠀⠀⠀⠀⠀⠀⠀⡀⠨⠈
⠀⡆⡀⡂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢐⢀⢰
⠀⠁⡇⠁⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠈⠸⠈
⠀⠀⡅⠈⠀⢠⢰⢠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡆⡄⢀⠁⢠⠁
⠀⠀⠀⠀⠀⠐⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠂⠀⠂

⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇⢠⣶⡄⢸⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⠏⠙⣿⣄⣀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠰⣶⣄⠘⡿⢃⣴⣶⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠙⢻⣆⢠⡿⠋⢙⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡶⠛⠛⠛⠒⣾⣿⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⣤⣭⣍⠁⠚⣋⡁
⠀⢸⣿⣿⣿⣿⣿⡟⠋⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡇⠀⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣇⣀⣿⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠿⠿⠿⠿⠿⠀⠸⠿⠿⠿⠇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢎⣠⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⠆
⠀⠀⠀⠀⠀⢠⣬⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⡄
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⢹⢟⠛⠛⡻⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⡿⠿⣿⠿⡆
⠀⠀⠀⠀⢸⡉⣏⠉⢹⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢢⡔⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣇⠀⡇⢸⠁
⠀⠀⠀⠀⠀⣇⢹⠀⢸⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⠂⠐⢄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⢀⡇⡏
⠀⠀⠀⠀⠀⢹⣼⣤⣾⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠶⠶⠾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡙⡆
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣅⡀
⠀⠀⠀⠀⢸⡇⡿⠿⠿⡜⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣼⣿⣇⣧
⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⢿
⠀⠀⠀⠀⢼⡘⣤⣤⣤⡌⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⡏⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⣳⣶⣊⡼
⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣼⣧⣾⣷⠄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⡴⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⡄
⠀⠀⠀⠀⠀⠀⢣⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⡁
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠿⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣦⣀
⠀⠀⠀⠀⢹⢏⠉⠉⠙⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢣⡜⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡛⡟⠛⢿⢻⠋
⠀⠀⠀⠀⠈⡆⠑⣄⠔⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣁⣈⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡟⡼
⠀⠀⠀⠀⠀⢳⡔⠉⢢⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣧⣀⣧⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠂⠐⢀
⠀⠠⠀⠀⠀⠀⠠⠀⠐
⠀⠀⡀⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠈⠀⠀⠀⠂⠀⡐⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠊⠁⡘⡀⠀⠖⡀
⠀⠃⠄⠀⠀⠀⠈⠀⠀⣄⠀⠀⡀⠆⠘⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠔⠡⠀⠄⠪⢈⡀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠠⢂⠡⠀⠁⠄⠚⠦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⠀⠁⣀⠢⠈⠠⠔⠀⠁⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠉⠁⢏⡠⠀⠀⠀⠀⠀⠁⠸⠐⠂⠀⢀⢀⣄⡀⡠⣀⢄⠀⠀⡀⠤⠀⠉⠈⡀⠀⡀⡐⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠁⠘⠐⡰⣀⠒⠀⠨⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠄⠂⠁⠀⠀⡀⠐⠁⠠⠐⠀⠀⠀⠀⠀⠰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⠀⠂⠄⠄⠀⠰⠀⠀⠀⠐⠁⠁⠃⠁⠁⠡⠀⠃⠂⠐⠌⠄⠐⠉⠀⠀⠆⠀⢰⠁⠀⠀⢀⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡂⠠⠀⡤⠃⠀⠂⠸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠨⠀⠓⠀⠀⠀⠀⠀⠌⠀⠀⠠⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠔⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠐⠀⢀⠀⠠⠀⠀⠀⡬⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⡤⠀⠀⠁⠀⠀⠀⠀⠀⠐⠈⠂⠀⠐⠀⠀⠀⠄⠁⠈⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠅⠀⠁⠀⠀⢡⣄⠄⠀⠀⠀⠄⢐⠀⠄⠀⠠⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠠⡀⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⢡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠚⠛⠛⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⠆⢀⡀⠀⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣷⣄⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠸⣿⣿⣿⣷⡄⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠑⠶⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣿⣿⣿⣿⣆⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢀⡄⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣿⣿⣿⣿⣿⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡾⠁⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢻⡜⣿⣿⣿⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣿⣿⢧⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⢿⡜⣿⣟⣿⡆⠀⣿⠀⠀⣀⠀⠀⡀⣿⠀⣸⣿⣿⢏⡞⠳⣄⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠻⣌⢿⡼⡇⠀⣏⣀⡌⠸⠒⣹⣃⣿⢀⣿⣿⣷⠟⠀⢀⣼⠷⣦⣄⡀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢬⣷⡷⠓⠀⠉⠁⠀⠀⠀⠁⠀⠉⠙⠛⢋⣁⣤⠶⠋⠁⠀⠀⠉⠙⠛⠓⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢰⠋⣿⡆⠀⠀⠀⠀⢰⢉⣿⡆⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠉⠀⠀⠈⠉⣶⣿⣭⣭⣿⣦⠉⠁⠀⠈⠉⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⣠⡤⠄⠀⠀⠀⠘⠿⣿⣿⠿⠃⠀⠀⠀⠠⢤⡀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠋⢷⡀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⣠⡞⠹⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠻⠷⣶⠶⣶⠋⢹⡏⠛⣶⢶⣶⠾⠋⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⠀⠀⠀⠀⠈⠛⠒⠒⠛⠀⠀⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠤⠤⠤⠤⠤⠤⠶⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠒⠘⠀⠐⠀⠀⠐⠒⠂⠘⠂⠃⠓⠚⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢲⣤⠖⣦⠴⢢⢰⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠼⠶⠋⠾⠧⠸⠤⠇

⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡰⠚⠋⠁⠉⠑⠄⠀⠀⠀⡠⠐⠉⠑⠠⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢀⠀⠈⠈⠀⡀⠀⠀⠀⠀⣠⠖⠋⠉⡈⢓⢄
⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⢀⡠⠊⠁⠢⡀⠀⠀⠀⡜⢡⠁⠈⠀⢀⠀⠆⠑⡄
⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠢⠈⠀⠀⠀⠐⠁⠀⠀⢸⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡆⠀⠀⠓⢄⡀⠀⣀⠴⠋⠂⠀⠀⠀⠀⢄⠀⠂⡠⠀⠀⠀⠈⠦⡐⡀⠀⠀⠀⢀⣂⠤⠃
⠀⠀⠀⠀⠈⠁⠻⡚⠛⠉⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⠀⠀⠀⠀⠀⠀⠙⠟⠓⠋⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇
⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠰⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⡐⠀⢢⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠂⠀⠀⡠⡄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⡀⠀⠀⢰⠉⡈⠢⡈⢀⠙⠀⠲⠀⠀⠀⢱⠀⢀⠀⢈⠀⠀⠀⠁⢣
⠀⠀⠀⠔⠉⢉⠡⠂⠀⠅⡀⠐⠀⠁⠀⠀⠀⠀⣉⠤⠐⠁⠉⠒⠢⢄⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⢇⠀⠑⢴⡀
⠀⠀⠸⠀⡐⠁⠈⠀⠀⠀⠈⠆⠀⠁⠂⠀⠀⡴⠁⢠⠖⠀⠑⢢⠀⠈⢃⠀⠀⠀⠀⡠⠐⠒⠀⡀⠀⠈⠢⠤⡤⠁
⠀⠀⠀⠑⠇⠀⠀⠀⠀⠀⠀⡄⠀⠀⡘⠀⠀⡇⠀⢺⠀⠐⠀⢀⠀⠀⡌⠀⠀⠸⡝⠃⠀⠀⠀⠸⡐⠻
⠀⠀⠀⠀⠘⢄⡀⠀⠀⡠⠜⠀⠀⠊⠀⠀⠀⠘⢄⠀⠑⠒⠒⠁⣂⠜⠀⠀⠀⠀⠑⢄⡀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠠⣤⣄⡀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠉⠻⢷⣦⣄⢻⣧⣤⣴⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣄⣀⠀⠀⠀⠀⠀⠙⢿⣧⣿⣧⣤⣤⣀⡀⠀⠉⢉⣙⣿⠿⠷⣶⣤⣄⡀
⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⠿⠿⠟⠿⠿⣿⣿⣿⣿⠿⢿⣿⣿⡿⠛⠋⠙⠛⠿⣷⡟⠛⠉⠀⠀⠀⠀⠈⠙⠻⣿⣶⣦⣄⡀
⣶⣿⣿⡿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣿⣷⣦⣽⣿⡇⠀⠀⠀⠀⠀⠘⣿⣦⡶⠶⠶⠶⠶⢶⣶⣿⣿⣍⡈⠙⠛⠿⣶⣤⣀
⣿⡇⠙⠻⣷⣦⣀⠀⠀⠀⣀⣠⣤⣶⡶⠿⠛⠛⠉⠁⠈⠉⣿⣿⣦⣀⠀⠀⠀⣀⣼⣿⣤⣤⣤⣤⣄⡀⠀⠈⠉⠻⢿⣦⡀⢀⣀⣭⣿
⣿⡇⠀⠀⠈⠙⢿⣿⣿⠛⠛⣹⣧⣤⣀⣀⣀⣀⣀⣠⣴⣾⠏⣼⡿⠛⢿⣿⡿⢿⣿⡟⠋⠉⠉⠉⠛⢿⣦⠀⠀⠀⠈⣿⣿⠟⠋⠁⢸
⢿⡇⠀⠀⠀⠀⢸⣟⠻⣿⣦⣿⡟⠙⠛⠛⠻⠛⠛⠛⠉⢀⣾⠋⠀⢀⣾⠟⢦⡈⣿⡇⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢀⣿⡇⠀⠀⠀⢸
⢸⡟⠀⠀⠀⠀⢸⣿⠀⠈⢻⣿⡇⠀⠀⠀⠀⠀⢀⣠⣾⣿⠃⠀⠀⣾⣟⣀⣠⣽⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠘
⢸⣧⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣇⣀⣀⣤⣴⣾⠿⠋⣿⡇⠀⠀⢠⣿⡿⠿⠛⠛⣿⣿⣷⣄⠀⠀⠀⠀⠀⣠⣴⡿⠋
⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⡿⠿⠟⠻⣿⣶⣾⣿⣿⡇⠀⠀⠸⣿⡇⠀⠀⠀⣿⡇⠙⠿⣿⣶⣶⣾⣿⠟⠋
⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠸⣿⠀⠀⠀⠀⢸⣿⠀⠘⣿⡇⠀⠀⠀⠻⣷⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⢻⣿⡄⢀⣀⣠⣼⣷⣦⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⢿⡿⠟⠛⠋⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣧⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠸⣿⡄⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠘⣿⣆⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠈⢿⣦⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠻⣿⣿⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⣿⠁⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣀⣤⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⢸⣿⡇⠀⠀⠀⣀⣤⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣸⣿⣧⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠟⠋⠉
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⢀⠠⠂⢀⠔⠀⢀⢀⣄⡀⢀⣤⡴⠂⢀⣀⣤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⡇⠀⢻⣿⣿⡿⢷⣿⣯⣶⣿⣛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢲⣶⣷⣤⣼⣽⣯⣷⣾⣿⣿⣿⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠙⠻⠿⠷⠿⠟⠋⠛⢿⣿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢩⣥⣭⣤⣴⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣥⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠉⠉⠙⠙⠊⠉⠻⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠐⢲⣼⣶⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⠉⠀⣦⣴⣷⣴⡆⣴⡟⠁⠙⢦⣄⣰⣄⣠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⡿⠿⣉⣸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⣾⣿⣿⣿⣿⣿⣧⣴⣾⣿⣥⡶
⠀⠀⠀⠀⠀⠲⡄⡗⣠⡶⠀⠀⠀⣠⣾⣿⣿⣿⣿⢿⡟⠟⣈⣱⣶⣿⣿⣿⠟⠋⠁⠉⢿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠉⠙⢿⣿⣿⣿⣟⠓
⠀⠀⠀⠀⠴⠖⢻⣿⡁⠀⠀⢀⣰⣿⣿⣿⣿⣿⣽⣶⣾⣷⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⠛⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠟⠈⠻⢶⠶⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⣀⡙⣆⢀⣼⠏
⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⡟⠫⡄
⣿⣿⣿⣟⣤⠶⠶⠦⣄⠀⠀⠀⠀⠀⠀⢶⡄⠀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⠛⠛⠋⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⣛⣷⣿⣾⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣤⣠⣴⣾⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠻⢿⣿⣿⣿⡿⠛⠉⠙⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⣫⣾⣿⣮⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣲⣾⣿⣿⣿⣿⣿⡟⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠛⠓⠛⠛⠛⠛⠛⠻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣶⡶⠶⠶⠶⠶⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⢹⣿⠿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢛⠻⠟⠋⢀⣄⠀⡀⢀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⢀⡄⠉⠛⠟⣛⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢈⠁⠀⠀⠀⠁⠀⠅⠈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠁⠎⠀⠀⠀⠀⠀⠀⢁⣼⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠓⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠐⠁⠀⣀⠄⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠙⢇⠑⢄⠀⡀⢶⡒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣯⣀⢀⣤⣿⠿⡿⠟⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠑⠠⡇⠐⠊⢈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠀⠙⠛⣿⠛⠉⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢨⣯⡀⠤⡞⠩⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠫⠉⢢⣤⣤⣽⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢨⠈⡀⠀⠀⠀⠈⠁⠀⣿⠀⠀⡇⢀⡅⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⢅⠀⢸⣿⣇⣿⠀⠈⠁⠀⠀⠀⢰⠘⡄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠅⠀⠀⠀⠀⠀⠀⣿⠀⢀⣤⠊⢉⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡉⢑⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⡌⠀⢰
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢈⠢⠀⠀⠤⠐⠺⡏⠉⠙⢢⠈⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡿⠟⢉⡩⢛⡲⠦⢤⣀⣀⠔⠀⠄⣪⣇
⠀⠀⠀⠀⠀⠀⠸⢶⣦⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣭⣭⣴⡶⢻⡿⡳⣶⣶⣶⣶⡺⡽⠽⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢸⠬⡡⠿⣿⣿⣿⣧⠗⣡⣓⢯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣭⣭⣷⣿⠿⡿⣿⣷⣿⣷⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣆⠏⣱⣹⣿⣿⣿⢱⠽⠩⣬⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣣⠦⢜⣽⣿⣿⣿⣮⣺⣾⣫⣿⣿⣿⠿⠟⣋⠥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠍⢙⣛⣛⣛⣛⣛⣛⣛⣛⠋⠩⢭⠁⠂⠒⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠒⠒⠒⠖⠚⡖⠒⠒⡖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⡃⠀⠀⠀⠀⠀⣱⣤⣴⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⡀⠀⠀⢀⣾⣿⡿⢿⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠋⠀⠀⠀⠀⠛⠈⠈

⠀⠀⠀⠤⠤⠤⠀⠀⠠⠤⠤⠀⠄⠤⠀⠄⠀⠄⠠⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⡀⢀⡂⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣇⠀⠀⠀⠀⣠⡾⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡆⠀⣠⢾⡟⠁⠀⢸⡇⠀⠀⠠⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⢰⢯⡟⠀⠀⠀⣸⠀⠀⠀⠁⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⣟⣿⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢸⣿⡇⠀⠀⣰⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⣿⣽⡧⢠⠞⠁⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣟⣻⢟⣻⣿⣿⠋⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣽⣼⣿⣿⣾⣿⣮⣿⣷⣄⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢟⠟⠛⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣶⣟⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣼⣿⣶⢿⠿⣿⣷⣤⡀⠙⠱⠛⠿⠿⠟⢛⣭⣷⣿⡿⠟⣻⣿⡏⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⢨⣿⣿⡿⣷⣿⣿⡿⣿⣿⣄⡀⠀⠀⢀⣠⢾⣦⣤⣶⣶⣿⣿⣿⣷⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠻⣿⣾⣿⣍⡹⣤⢿⣿⣿⣿⣿⣿⣯⣯⣽⣿⡿⣿⣿⣷⣿⣿⡟⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⣿⣿⣿⣤⣟⢱⣶⣿⣿⣿⣿⣿⣿⣿⣿⢦⣳⣛⣿⣿⣿⡾⠁⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠈⢿⣧⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣦⣿⣿⣿⠟⢁⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠙⢯⣿⣾⢽⣽⣾⡻⣿⣿⣜⣩⣿⣥⣾⣿⠟⢁⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⣀⠉⠉⠙⢿⣿⢹⣿⣿⠿⠿⠛⣉⣤⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠛⠛⠛⠉⠁

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