National football league Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⡀⢀⡀⠈⢿⡇⠸⣿⣾⣶⣶⣶⣶⣶⣾⣾⣿⠀⣿⠏⠀⠀⢀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠽⠀⠇⠸⠇⠀⠀⠑⠦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠖⠁⠀⠀⠅⠸⠃⠹⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠐⠒⠒⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⣤⢀⣦⣴⠶⢰⢦⣠⣄⡴⡄⢠⡄⢠⡄⡄⠀⡶⢆⡴⣄⣴⢦⠴⡶⢤⢦⡄⣰⡄⣤⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡇⣿⣾⣿⣿⠒⢸⢾⣹⣷⡇⣀⣾⣿⢸⢿⡇⠀⡷⢪⡇⣿⣿⢸⠀⡇⢸⠿⡇⣟⣧⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠛⠋⠋⠛⠙⠛⠘⠘⠛⠃⠛⠋⠋⠙⠙⠈⠃⠀⠃⠀⠛⠃⠙⠋⠀⠃⠘⠛⠉⠃⠙⠙⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢠⡴⢀⣀⣀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡧⠀⠀⠐⠛⠒⠛⠛⢛⠟⠙⠋⠉⠁⠀⠀⠐⢄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠀⣠⣶⠀⢀⣀⠀⠀⠀⠀⠀⢀⢀⡀⠘⣷⣄⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⣼⠁⢀⡇⠐⡟⡇⠈⣿⣇⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠀⠀⠀⠀⠀⣠⣀⣀⡀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠋⠁⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣋⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⢊⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣡⡴⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣦⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠀⠛⠋⠀⠈⠻⣿⠟⢋⣉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢀⣽⣶⣿⠅⡀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠟
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⡄⠀⢀⣴⣾⣿⣿⣿⣿⣏⠈⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠋⠁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠁⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠋⠙⠉⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⢿⡏⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡧⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣶⣶⣶⡇⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠉⠛⠿⠯⠁⠉⠀⠀⠙⠻⢿⡿⠿⠿⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⢰⠉⢶⠉⠍⠹⣍⡼⢰⠀⡃⠀⣾⠀⢓⠀⣆⡬⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠁⠀⠈⠉⠀⠈⠉⠁⠀⠈⠉⠈⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠐⠐⠲⢆⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣠⡤⠤⠶⡶⠒⠶⠶⣤⠙⢦⡀⠀⠉⠂⡠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠀⠀⠀⡄⠀⢀⣠⡤⠶⠺⠛⠉⡋⠩⣽⣦⠒⠤⠛⠈⠡⠘⣿⡧⠶⠶⠷⠠⠄⠴⠼⣶⠾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⢀⣠⡴⠛⠏⠀⠀⢠⡌⠓⠛⠉⠄⠡⠀⠈⡄⠠⠒⠄⠙⠛⠓⡶⠶⣦⠀⡀⠀⠀⠀⢦⡀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢁⠐⠘⠃⢀⣴⣤⠶⠛⣥⠠⡠⠘⠓⡀⠀⠀⠀⠀⢄⡀⠄⠡⠀⠐⢄⠀⠀⠒⢅⡴⢛⡴⠃⠀⡙⡄⠀⠀⠈⢟⢀⠉⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠀⠀⣠⡴⢞⣋⢡⣎⠹⠗⠉⠠⠀⠀⠀⠐⡀⠇⢀⠂⠘⠀⠄⠉⢁⣠⣶⠗⠚⠛⠛⣿⠉⠀⡣⢢⢾⡹⡀⠀⠀⠈⣎⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠚⡀⠈⡐⣠⠟⡥⢐⠇⠘⠋⠀⠀⠀⠀⠀⠆⢀⠀⠀⠑⣀⠀⡀⠢⣠⣤⠞⠋⠁⣴⠀⠀⠂⠀⠺⡧⠀⠵⣿⠾⡑⢳⠀⠀⠀⢙⣿⣿
⠀⠀⠀⠀⠀⢀⠌⠀⠀⢁⢈⢻⡿⢶⣿⢧⢀⠀⠀⠂⠈⢀⡰⠀⠈⣅⣒⣠⣆⡽⡶⠟⠛⡁⢐⣦⢾⠀⠪⠀⣇⢠⢌⠀⠈⣄⠘⡑⡡⢁⢦⣧⠀⠀⠸⣿⡏
⠀⠀⠀⠀⣠⠃⠀⠀⠀⢈⣶⣾⠏⠉⢠⠞⠋⠀⠐⠂⢀⣠⡾⢟⠛⠙⠉⠀⠈⠰⠘⠌⢚⠘⠀⣳⠂⠒⠀⠑⣸⡸⡌⠀⡀⠴⡏⠎⣶⣷⣾⣿⡀⠀⠀⡟
⠀⠀⠀⡴⠃⣆⠀⠀⡤⠞⠙⡆⠀⢰⣏⣬⣛⣾⣮⠶⠋⠁⠁⠀⡎⠀⠀⠤⢀⢆⠄⠀⢀⣠⢀⣴⡑⠂⠨⡤⣑⣱⣁⠘⠟⣂⣰⣿⣿⣿⣿⣿⡇⠀⡼⠁
⠀⠀⡜⠀⢰⣿⡎⠉⠀⠀⠀⢳⠀⣴⡭⠹⣩⠝⠮⢐⢁⣀⠸⡄⠱⠒⠧⢂⣁⡈⡮⠎⠆⡻⡃⠡⠜⢘⢕⣜⡻⢀⢛⣶⣾⣿⣻⣿⣿⣿⣿⣿⣇⡞⠁
⠀⡸⢣⣼⠟⠁⣇⠀⠀⠀⠀⠈⡇⣏⢋⣄⢧⣢⢸⠇⠘⢯⢃⢸⡄⢠⠺⠊⢙⢱⡄⡠⠈⢊⡄⢀⠹⢶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⡿⠋
⢰⣷⠏⠁⠀⠀⠘⡄⠀⠀⠀⠀⠸⡀⣨⣾⣜⣆⣄⣻⢦⢬⢆⠃⡰⠀⡃⣰⣏⠿⣶⡾⡇⣊⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠟⠁
⣿⠁⠀⠀⢠⠠⠀⢻⠀⠀⠀⠀⠀⢱⡌⠱⡿⡹⠶⠈⣟⣵⢾⣲⢗⡐⣸⠔⣸⣦⣷⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⠟⠁
⢧⣰⣌⡶⣤⣰⡁⠀⣧⠀⠀⠀⠀⠀⢻⣃⣀⣳⣆⣰⣿⣿⣾⣿⣻⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣟⣿⡿⠋
⠈⠿⣿⣿⣿⣿⣿⣷⣿⣧⡀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣾⣿⣿⡿⠟⠉
⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣷⡄⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⣀⣀⣀⣀⡀
⠐⠾⣿⣿⣯⡉⠙⠛⢿⣶⣄
⠀⠀⠀⠙⢿⣿⣶⡀⠀⠙⢿⣷⡀⠀⠀⠴⠶⣿⣍⣙⠛⠳⢦⣀
⠀⠀⠀⠀⠈⣿⣿⣿⣄⠀⠀⠻⢿⣦⣀⠀⠀⠀⠙⣿⡻⢦⣀⠙⢷⣄⠀⠀⠀⠲⣾⡷⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠘⣿⣏⠻⢷⣦⣄⣀⣙⣻⣷⣶⣶⣶⣾⣷⡀⠙⠻⢶⣾⣷⣶⣶⣶⡿⠿⠂⠀⠈⠙⢷⣄⣀⡀
⠀⠀⠀⠀⠀⡀⠈⢿⣷⣄⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣦⣄⣀⠀⠈⠉⠁⠀⣀⣀⣤⣤⣤⣤⣤⣭⣛⡻⠷⣦⣄⡀
⠀⠀⠀⠀⠀⣿⣦⡀⠙⠻⠿⣶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣝⠻⣦⣄
⠀⠀⠀⠀⠀⣿⡿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⢛⣿⣿⣿⣿⣿⣿⣷⣌⠻⣧⡀
⠀⠀⠀⠀⠀⠹⣿⣄⠉⠛⠿⠿⣷⣦⣄⡉⠉⠛⠿⣿⣆⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣄⣀⡈⠙⠻⣶⣄⠀⠈⠻⡆⢠⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⡜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢶⣌⣿⣧⠀⠀⠁⣾⣿⣿⣿⣷⠀⠀⠠⠤⠤⢤⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⣿⣿⣿⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣦⣼⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣾⣿⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠉⠀⠀⠀⢿⣿⣿⣿⣟⠿⠋⣿⣿⣿⣿⣇⣿⡇⠀⣼⣿⣿⣿⣿⠙⠟⣿⣿⣿⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠈⢿⣿⣿⣟⣤⡀⣴⣿⣿⣿⣿⣹⡇⢰⣿⣿⣿⣿⣷⡄⣤⣹⣿⣿⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣷⣝⠻⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣮⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⣬⣭⣭⣭⣭⣭⣭⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣾⣿⠿⠿⠿⢿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⠿⢛⣋⢉⣁⣤⣾⡆⢰⣾⠿⢿⣦⢠⣮⠍⠙⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠋⣵⠖⠸⡿⠋⣘⠛⠛⢉⣶⡇⠠⣶⣶⠆⢈⣭⡸⣿⣦⢲⣮⠛⢿⣷⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⠿⠛⣷⣬⣋⣀⢴⣶⣶⡦⠈⠻⣿⡶⠀⢈⣛⠀⣼⣿⡇
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠿⠿⢿⣿⠛⠛⠫⠴⠶⠶⣶⣦⣘⠿⠆⢾⠿⠛⠃⢻⠻⣿⣷⣄
⢠⣿⣿⣟⠛⠛⠛⠛⢛⣻⣿⠟⣩⣶⢀⡐⠿⠿⠇⠀⠒⠒⠾⠿⣿⣿⣿⡿⣿⣷⡦⠤⠄⠀⠀⠀⠀⠹⠉⢿⣷⡀
⠀⠀⣹⣿⠿⢷⣶⣶⠿⠋⠐⠾⠿⠋⠘⣻⣿⣿⣛⡘⠻⠿⠇⠀⠀⠙⠻⠟⠉⠀⠰⠾⠛⠛⠉⠁⠀⠀⠀⢀⣿⢿⣦
⠀⠀⣿⣿⣶⣶⡆⢰⣾⣟⣀⣨⣭⣭⣭⣭⣭⣭⣭⣍⣙⣛⣛⡳⠶⢶⣤⣤⣀⡀⠉⠙⠛⠿⣿⣷⣶⣶⣶⣿⠏⣰⣿
⠀⠀⣿⣿⣤⣶⣿⣿⡿⠿⢿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣷⣶⣌⡉⠛⠿⠖⠀⠀⠀⠉⣻⣿⣿⣥⣴⡿⠃
⠀⠀⠸⣿⣿⣿⣿⣷⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣤⣤⣌⣛⠿⣶⣦⣤⣤⣤⣶⠿⠟⠋⣹⣿⠋
⠀⠀⠀⠹⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠻⣿⡿⣿⣿⣿⣿⣿⣶⡌⠉⣩⣧⣶⡶⣁⣴⠟⠁
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⠀⠹⡇⠀⣿⣿⣿⣿⣯⣤⣾⠿⢻⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣦⡀⠉⠀⠹⣿⣿⣿⠋⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣮⡛⢿⣿⣿⡻⣿⣶⣤⣾⡿⣿⣿⣄⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣌⠻⣷⣌⠛⠟⠋⠀⣋⣋⡛⠛⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡈⠻⣶⣶⡾⠿⠛⠛⢿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣤⣶⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠚⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠒⠲⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢻⠃⠀⠀⠀⣀⣀⡀⠤⠤⠄⢒⡒⠂⢀⣀⠀⠀⣐⡒⠒⠠⡤⠄⠀⢀⣄⢀⣀⠀⠀⠀⢯⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⣸⠠⢔⡊⣡⠢⡄⢴⠉⠤⠤⡇⠱⠀⢾⠀⠇⢸⠀⡧⠀⠏⢸⠤⢤⠉⣆⣀⠔⢌⣒⡄⢸⠀⠀⠀⠙⣶⣄
⠀⠀⠀⠀⠀⣠⠞⢹⠀⠀⠀⠀⣇⠠⠤⢎⣹⠀⡇⡤⢀⠦⠴⠁⠰⠀⠊⠀⠂⢺⠀⠧⠤⠆⢰⠤⢼⠀⡧⢠⠀⣏⡱⠤⢸⡇⠀⠀⠀⢹⠈⠳⣄
⠀⠀⠀⣠⠞⠁⠀⡞⠀⠀⠀⠀⡇⢠⡒⠁⠨⡠⠀⠐⠜⢀⠀⠓⠃⠀⠈⠈⠀⢈⠉⣀⣀⣑⣊⣀⡘⠤⠃⠸⣄⠏⢑⠤⠀⡇⠀⠀⠀⢸⣄⡀⠈⠳⣄
⠀⢀⡼⠁⢀⠤⠊⡇⠀⠀⠀⢸⠇⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⡇⠀⠀⠀⠈⡇⠈⠒⢄⠈⢧⡀
⢀⡞⣡⠔⠁⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⡇⠀⠀⠀⠉⠢⣳
⣼⠞⠁⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙
⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⢻⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⢔
⠈⢳⡀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⢀⠔⢁⡞
⠀⠀⠻⣦⡀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣃⡤⠊⠁⣠⠞
⠀⠀⠀⠈⢧⡑⠄⣹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢸⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠶⣼⡄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢼⠀⠀⠀⠀⣏⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⢸⠑⠒⠂⠤⠄⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠤⠄⠒⠒⠈⠁⠀⠀⡼⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢧⣀⡀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁⠉⠉⠉⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠦⠤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡤⠤⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠒⢲⡖⠒⠒⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠞⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠳⡄
⠀⠀⠀⣰⠃⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠉⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠘⣆
⠀⠀⣰⠃⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠠⣤⣤⣴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣤⣄⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⢧
⠀⢰⠃⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠘⣆
⢀⡏⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⢠⡏⣇⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢹
⢸⠁⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡼⠁⠀⠀⠀⠀⠀⢀⡾⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈
⣿⠀⠀⠀⢸⠁⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢹⣼⠁⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⣇
⡏⠀⠀⠀⣼⠀⠀⠸⣯⣝⢲⡷⣴⣶⣴⣶⣾⣧⣶⣦⣴⣶⣶⣶⠀⠀⣏⣀⣀⡤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⡀⠀⣿
⣇⠀⠀⠀⢿⠀⠀⠰⣦⣿⠻⣷⣿⣽⢿⣯⣿⡿⣿⣿⢽⠇⣿⣿⠀⠀⣏⠙⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠛⠉⣿
⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⡇
⢸⡀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⣇⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀
⠈⣇⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠘⣇⡸⠁⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣸
⠀⠸⡆⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢠⡏
⠀⠀⠹⡄⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠿⣍⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⠶⠶⠾⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⢀⡞
⠀⠀⠀⠹⡄⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠸⣾⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⢤⣤⣤⣧⣤⡤⠤⠶⠖⠚⠉⠁⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⢤⣤⣄⣀⣀⣀⣀⣠⣤⡤⠴⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠾⣻⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠾⠛⠋⠉⠁⠀⢀⠴⠛⠉⠉⠁⠀⠀⠀⠤⠴⠶⢾⣿⣿⣿⣿⣭⣥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⢠⡤⠴⠶⠚⠛⠋⠉⠁⢀⣀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠷⣶⡤
⠀⢀⣀⣠⣤⣴⠾⠷⣤⣀⣀⣀⣀⣤⠾⡛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠷⠶⢶⣤⣤⣀⡀⠀⠀⠀⠀⣀⣴⠞⠋
⢀⣿⠋⠉⠀⠀⠠⠖⠀⠉⠉⠙⢿⣦⣖⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⣶⣤⣾⠋
⣼⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡈⠉⠛⠓⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣷⡤
⢿⡄⠀⠀⢀⡴⠛⠉⠉⠙⢿⡙⢯⠛⢳⡄⠀⣼⠃⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣄⡀⠀⠀⠀⠀⣠⣴⠞⠋⠁
⠈⠻⣦⡀⣿⠀⠀⠀⠀⠀⣸⠃⣼⣠⠞⣡⡾⠛⠒⠶⣦⣍⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣶⠟⠉
⠀⠀⠈⠙⠛⠓⠀⠀⢠⠞⠡⣞⣋⣵⠞⠋⠀⠀⠀⠀⠀⠙⣷⡀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣀⣀⣤⡬⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣤⣤⣬⡿⢷⣤⠶⠶⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠘⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡙⠛⠿⠿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣷⣤⣀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣍⡙⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠻⣿⣷⣦⣀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⠀⢹⣿⣿⣿⣷⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣌⡛⠻⢛⣽⣿⠟⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣭⣤⣀⣀⡀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣏⠉⠉⠉⠉⠛⣓⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⣠⣴⡾⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣦⣤⣀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣶⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡀⠀⢰⡶⠶⣶⠶⡄⠀⣶⠖⣶⢰⡶⣶⠶⠶⡆⣠⠶⣶⠶⣄⢠⡶⠶⣆⣶⠶⣶⠀⠀⣴⠶⣆
⠀⠀⠀⠀⠀⠀⠀⡾⣸⡎⣧⠀⠈⡇⠸⠽⢇⡿⠀⢹⢸⡇⠈⢉⡽⢋⡾⠁⣿⢸⠉⡇⢸⠀⣿⢰⡘⣿⣿⡏⠀⢰⢧⣷⢹⡀
⠀⠀⠀⠀⠀⠀⣸⢣⡞⣷⠸⣄⢀⡇⢸⣳⡘⣆⠀⣼⢸⣇⢠⠞⣴⣋⡶⡆⣿⢸⣀⡇⣸⠀⣿⢸⡹⡜⢻⡇⢀⡟⣼⢻⡄⢧⡀
⠀⠀⠀⠀⠀⠀⠳⠶⠃⠻⠖⠟⠘⠷⠾⠛⠷⠞⠃⠓⠒⠛⠘⠶⠶⠖⠶⠃⠘⠳⠖⠖⠋⠘⠳⠲⠃⠙⠾⠃⠸⠶⠛⠘⠷⠾⠃
⣠⠶⠶⢶⡀⠀⠀⡶⢶⡄⠀⢰⡶⠶⠶⠶⣄⠀⣶⠶⠶⠶⣦⠀⣶⠶⢶⠀⣶⠶⢦⢰⠶⢶⠀⠀⢰⠶⣶⠀⠀⣶⠶⢶⡄⠀⠀⣴⠶⠶⢦
⡇⢸⠛⣧⡇⠀⢸⢣⡇⣧⠀⠈⣿⢸⣟⡇⣿⠀⢹⡆⡟⢻⢸⡇⢹⡇⡟⠀⢹⢠⡘⣏⡇⣿⠀⠀⡟⣼⢸⡄⠀⢹⡇⣿⠁⠀⢸⡇⣾⣹⣾
⡇⢸⠀⣀⡀⠀⡟⠘⠛⢸⡄⠀⣿⢨⣭⠱⡏⠀⢸⡇⡇⢸⢸⡇⢸⡇⡇⠀⢸⢸⣷⡘⡇⣿⠀⢸⠃⠛⠃⣧⠀⢸⡇⣿⢀⣀⡈⠳⠤⢤⠙
⣧⠸⠾⢋⡇⣼⠣⣯⢹⠆⢷⢠⡿⠸⣮⡇⠻⡄⣼⠃⠷⠾⣸⠇⣼⠃⢷⠀⣾⠸⣇⢷⠀⣿⢠⡟⢼⡍⡿⠸⡆⣼⠇⠿⠾⢹⡄⣏⠻⠾⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣸⣿⣿⣿⣿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⡿⠋⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠻⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠈⠙⠛⠉
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⡿⠛⠋⠀⠀⠙⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⠶⠶⣶⣾⠿⢿⣿⠿⣿⡿⠿⢿⣿⠿⠿⣿⠿⠿⣷⣶⠶⢶⣶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⡏⢀⠘⣿⠀⣿⡇⠒⢀⣿⠀⠼⣿⠀⡁⢾⡷⠄⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣣⡉⢿⣿⣿⡿⠉⠀⢸⣿⡇⠀⠈⢻⣿⣿⡿⢋⣼⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠻⠷⠄⠉⢾⠃⠀⠀⢸⣿⡇⠀⠀⠈⢷⠍⢠⣾⠟⣸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣧⠊⠀⢀⠀⠘⠿⢇⡀⠀⠀⢈⣠⣿⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠀⢠⣤⣤⣤⠐⠩⠤⡌⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡆⣄⠛⠛⠛⠀⠀⠈⠁⡆⢀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣸⡄⠠⠀⠀⠀⠀⠀⢁⣼⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠣⣟⡀⠀⠀⠀⣠⠟⡜⠛⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣭⣤⣤⣶⣿⠻⣿⡿⠞⡣⣶⣽⣶⣤⣬⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣤⣾⣷⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⠿⠋⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⡼⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⢰⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⡎⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⡀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢿⣿
⠀⠀⢸⡇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸
⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⡆
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⠿⠛⠋⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠒⠛⠛⠛⠛⠛⠛⠳⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢁⡀⠠⠀⠀⠀⠀⣀⠀⠀⣀⣀⣈⠛⣛⣛⣫⣟⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⢀⠴⠊⠁⠀⠀⢤⣤⣽⣶⡿⠿⡏⠁⠀⠀⠀⠈⠻⣿⣿⣧⡀⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⣀⠴⠋⢠⣀⣨⣷⣿⠿⠹⡏⠀⠃⣀⠉⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⠔⠊⠁⠀⢐⢦⣶⡯⢿⡉⠘⠃⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⢀⡴⠃⠀⠰⠁⠀⠀⣈⣷⣿⠗⠻⠀⢈⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧
⠀⠀⠀⢠⣾⣄⡀⠀⠀⠀⢈⠳⣟⠋⠙⠂⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆
⠀⠀⢠⢿⣿⣿⣿⣦⣀⠔⠁⠀⠙⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢣
⠀⢠⠏⠀⠈⠻⣿⢟⣵⣆⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⡎
⢀⠏⠀⠀⠀⡴⠋⢾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡞
⡸⠀⠀⢀⠎⠀⠀⠀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠇⠀⣠⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡴⠋
⠀⢤⠃⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢋⣠⠞⠁
⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⣀⡴⠛⠁
⠀⠐⢤⡉⠙⠓⠲⠶⠤⣤⣬⣭⣛⣛⣃⣀⣤⣤⣤⡤⠤⠶⠖⠚⠋⠉⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠲⣤⡀⠀⠀⠀⠀⠀⢀⣠⣤⡶⢦⣄⠒⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣴⣿⠟⣋⣴⣾⣿⣦⡈⢿⠀⠀⠀⢠⣴⣿⠟⢉⣤⣾⣿⣷⣄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣾⡿⠋⣱⣾⣿⣿⣿⣿⣿⣷⡄⠀⠀⣴⣿⠟⢁⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠈⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⠻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⣾⣦⠀⠹⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣷⡀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠸⣀⠀⢀⠀⣀⣀⡠⠔⠊⠁⠀⠀⠀⠀⢿⣿⣷⡀⠈⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⢻⣿⣦⡀⠙⢿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡶⠞⠛⠶⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⠀⠀⠀⠀⠈⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⣇⠀⡀⠀⠀⠀⠀⢀⣼⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣃⣼⣿⣿⣀⣀⠀⣀⣀⣿⣿⣧⣸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⢿⣿⣭⣭⣶⣭⣽⣿⡿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣤⣀⣙⡛⢸⣿⣿⣿⣿⣿⣿⣿⠣⢛⣋⣀⣤⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⣿⠿⠿⠻⠟⠟⠻⠛⢻⣿⣷⡶⣾⣿⠉⣿⣶⢶⣿⣿⡟⠻⠟⠛⠿⠛⠿⢿⣟⢦
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⠃⡀⠀⠀⠀⠀⠀⠈⠳⡻⢧⣿⣿⢀⣿⡟⡿⣏⠟⠀⠀⠀⠀⠀⠀⠀⠘⢷⡈⡄
⠀⠀⠀⠀⠀⠀⠀⣇⢻⣷⠇⣽⣷⣶⣶⣶⣶⡶⠬⠿⠙⠳⣾⣶⣷⠟⠈⠿⠴⢶⡶⠶⣶⣶⣾⣇⢲⣿⡏⣸
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣇⣿⣥⣷⣿⣿⡿⠁⠀⠈⢷⢰⣍⠋⣥⢠⡿⠁⠀⠈⢿⣿⣿⣶⣸⣷⣸⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠳⢼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣳⠏⠈⠻⡞⠁⠀⠀⠀⢈⣿⣿⣿⣿⣿⠵⠚⢇
⠀⠀⠀⠀⠀⢀⣴⡵⠃⠀⠀⠈⢿⣿⣿⣷⣤⠞⢀⢔⣚⣋⠄⠀⠠⣙⣒⠠⡀⢷⣴⣾⣿⣿⡟⠁⠀⠀⠘⢾⣦⡀
⠀⠀⠀⢀⠔⠋⠡⣤⣴⣶⣶⠟⠁⠏⠹⣤⡄⢈⣨⠭⠥⠤⠔⠐⠤⠤⠭⢭⣉⡀⢠⡤⠏⠇⠈⢿⣶⣶⣦⣤⠈⠛⠢⡀
⠀⠀⣰⡿⠆⠀⠀⠀⠉⠛⠃⠀⠀⠀⠦⢼⡷⠐⠲⠶⠟⠒⠀⠀⠀⠚⠳⠾⠒⠂⢿⠥⠆⠀⠀⠀⠙⠋⠁⠀⠀⠀⠰⢷⡄
⠀⠐⣧⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢤⣿⣿⣿⡷⠂⠀⠀⠀⠒⢾⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣼
⠀⠀⠿⠿⢿⣿⣿⣿⡿⠿⣶⡶⠶⠾⠿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠿⠷⠶⣶⣾⠿⣿⣿⣿⣿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠾⣿⣿⣿⣿⣤⠀⠀⠀⣼⣿⣿⣿⣿⠶⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣶⣾⣿⣿⣿⣦⣀⣾⣿⣿⣿⣷⣶⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠢⡀⠀⠠⠚⡄⠀⢀⡠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠢⠀⠃⠀⠈⠒⠁⠀⠘⠒⠁⠀⠀⡆⢀⠤⠒⡄
⠀⢠⠤⠤⠤⠤⡄⠀⢀⣧⣔⠢⠤⢀⡠⠖⠦⡄⢀⡤⠤⠤⠤⠤⡤⠒⠒⠤⡉⠀⠀⡴⠓⠲⢀⡰⠐⢲⠀⠠⡖⠲
⢠⢸⠀⠀⠀⠀⢠⡴⠁⠀⠀⢣⠀⡞⠀⢀⡀⠘⡎⣄⣀⠀⢀⣠⣷⠀⢠⡀⠘⡄⡘⡇⠀⠀⠉⡫⠀⢸⠀⠼⡆⢸
⢸⢸⠀⢰⠯⠟⠛⡇⠠⣝⣆⢸⣟⠇⠀⢏⢹⠀⠛⠘⡾⠀⢘⠋⢽⠀⢰⡵⠀⡇⡃⡇⢰⠀⠳⡆⠀⢸⡄⢈⡇⢸
⢸⢸⠀⠘⠶⠶⡆⡇⠰⠨⣾⢸⣿⠀⠀⠈⢖⠀⢸⠀⣾⠀⢸⠀⣩⠀⠼⠃⢠⠇⡆⡅⢸⠆⢀⡇⠀⠸⠁⠒⡇⠘
⢸⠰⠀⢠⠤⢠⠇⠃⠀⢸⢧⢸⢹⠀⢠⠀⣽⠀⢸⠀⣾⠀⢸⠀⢚⠀⠀⠀⢏⠀⣅⡃⠘⠃⠙⠇⠀⠠⠤⡥⠃
⠀⢘⠀⢸⠛⠁⢨⠀⢠⢨⡇⢸⣸⠀⢸⠨⡿⠀⠸⠀⣿⠀⠠⠀⢸⠀⣔⣢⠀⢳⢱⠀⢰⡄⢸⠅⠀⠀⢀⣭
⠀⣫⠀⢸⡀⠀⠐⡄⠈⠻⠃⢸⠸⡀⠈⠓⠃⠀⡇⠀⣿⠀⠘⠚⠉⠉⠉⠛⠠⠙⠺⠀⢸⡇⢸⡯⠀⠘⠓⣾⠀⠐⠐
⠀⡟⠤⠼⠊⠀⠣⡱⢄⣠⣤⠞⢤⡳⢤⣀⣠⠞⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⡂⠹⣧⣨⣷⠄⢤⣤⣿⡤⠤⢤
⠀⠙⠯⢀⡀⠀⠀⠈⠉⣉⣀⣀⣀⣈⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠒⠈⠁⠉⠉⠉⢟⠉⠈⠈⠁
⠀⠀⠀⠀⡠⠀⠀⢠⠏⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣀⣤⡶⠀⠀⠀⠀⠙⢦
⠀⠀⢠⠎⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡞⠻⠁⢨⣧⠀⠀⣠⠔⠊
⠀⠀⠀⠀⠀⢉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠻⡄⠀⠀⣠⠖⢾⣿⠟⠋⠑⠂⠐⠂⣟⣶⣶⣦⡀⡄
⠀⠀⠀⠀⠠⠅⠸⡟⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢻⡀⢰⠗⢦⠘⣧⠀⢌⠀⠀⠀⡰⠉⢻⣭⢟⣯
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣧⣏⡀⠈⠳⠼⣧⠤⠶⠶⠛⣛⣋⣩⣷⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠏⢧⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠛⠒⠒⠻⡟⠉⠉⠉⢹⠁⠀⣿⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣀⠀⠀⡽⠯⣥⣂⣎⣀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣂⠀⠀⠀⠀⠀⠀⠀⠀⣸⢄⠀⢀⠁⠑⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⢤⣤⣤⡴⠃⠀⠱⠆

⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⣤⣅⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡛⠉⠀⠀⠈⠉⠀⠀⢀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⡀⠉⠀⡀⠀⠀⠀⠀⠸⣿⣧⣤⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣿⡿⣿⣷⣀⣀⣤⣣⣴⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢽⣿⣇⣈⣿⣿⣿⣿⣿⣿⣿⢏⣫⠏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣠⣼⡟⢹⣿⢿⣿⣿⡿⢿⡿⠋⣈⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣛⢻⣿⣿⣿⣿⣿⣿⣷⣦⣿⢎⠝⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢶⣿⣿⣿⡟⠉⢛⢻⣿⣿⡖⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠻⣿⣿⠰⣶⣶⣦⣾⡿⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⣧⠹⡟⢠⣿⣿⣿⣿⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⢸⣿⣿⣿⡏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⡮⠀⣼⣿⣿⣿⣤⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⢰⣿⣿⣿⣿⢸⣿⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠨⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⣾⣿⣿⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠟⠙⣻⣿⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣋⠁⠤⠀⣴⣿⣫⣾⣛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠎⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⡾⠿⢾⡷⣿⣇⣤⣽⡿⢷⡄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣾⣶⣿⣯⣿⣾⣷⣿⣷⣦⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⠿⣿⣿⣿⣟⣿⣿⣿⣿⡿⠿⠯⠁⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠐⠒⠂⢈⣉⣈⣁⣀⣒⣂⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠁⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡉⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠑⠠⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠑⠠⡈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⣠⣤⣤⣤⣤⣤⣤⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⣀⠔⠉⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡠⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠀⠂⠀⠀⠀⡀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠈⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠀⠈⠻⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠉⠉⠉⠉⠉⠁⠀⠀⠱⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⣀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⢀⣴⣿⣿⣿⣷⣶⣤⣤⣀⣀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠛⠉⣉⣈⠉⠀⠤⠜⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣖⠂⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⢻⣵⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠛⢁⡿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣒⠄⠉⣼⠿⠋⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠈⣿⣿⣿⣿⣦⡶⣡⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⣿⠛⠿⠿⣿⣭⠟⠋⢇⡿⠉⠙⢆
⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⣹⣷⡿⢡⠁⠃⠀⠉⢀⣤⠁⢸⡇⠀⠀⢠
⠀⠀⠀⠀⠀⡼⠁⠀⣤⡞⠷⣿⣿⣷⣤⡐⣠⣦⡟⠗⠀⣰⡏⠀⠀⣿
⠀⠀⠀⠀⡼⠉⢩⣷⠠⠀⠀⢿⡍⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⣿
⠀⠀⠀⠀⠘⠻⢾⡽⣇⠀⢸⣿⣅⢀⣤⣀⣀⡉⠉⠁⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣆⣼⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⣟⣸⣿⣿⣿⡿⢻⣿⡿⢃⣼⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠁⠋⣿⣿⣿⠛⣉⣼⣿⣷⢶⣽⣧⡀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠤⢛⣩⣶⣿⣿⢹⡇⣱⠞⠛⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣻⣿⣋⠉⣿⣿⣿⣏⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠉⢹⣿⣿⡿⢿⠿⣷⡄⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡴⠋⠁⠐⢼⣇⣹⣷⡀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠈⠻⣿⣿⣿⣆⠀⢀⣈⡑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠹⣿⣯⠙⢟⢿⣿⣷⣿⣇⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⣀⠿⣿⡄⠀⣢⠙⢿⣿⣿⣷⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠙⢷⣶⠟⠀⠈⠙⠻⣿⣿⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡉⢹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠴⠶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⠀⠉⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠰⡄⡸⠑⣖⢠⡇⠐⠊⠀⠀⠀⠀⠀⠀⠀⠧⠀⢤⣣
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠨⠱⣶⡏⢭⠴⠃⠀⠀⠀⢠⡲⣶⡖⣛⣲⣾⣴⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⣼⣗⡀⠀⠀⠀⢠⣿⠗⢺⣶⠒⡿⡏⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠈⠊⠁⠀⠀⢠⣿⡟⠀⠀⢻⣇⠐⣿⣦⣽⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠐⡂⢢⡀⠀⠀⠈⢻⣿⢦⡀⠈⢿⡄⢈⣖⣼⣧⣤⣶⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠑⠀⠃⠀⠀⠀⠀⢙⠻⠻⣟⣛⣶⠙⠻⠿⠿⢤⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢀⠀⠀⠀⠀⠀⠀⣾⣻⢿⠟⠛⢻⠹⣇⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠤⣄⣙⣿⣶⣶⣎⣁⣀⣽⣄⣀⣤⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⢰⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣧⠀⠈⠻⣿⣿⣿⠛⠀⠙⢿⣿⣿⣿⠛⠛⢿⣿⣿⠀⠘⢻⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠈⠛⠛⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣤⣴⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣆⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠙⠛⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡀⠀⢹⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⣠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡆⠀⢿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⠀⠘⢿⣿⣿⠀⠀⣴⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠛⠿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⣿⣿⣽⡇⠄⠀⠀⠀⠀⠀⢾⣝⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣀⣀⣀⠤⣠⠖⠀⠀⠀⠀⠀⠸⣥⣍⣉⣭⢽⡿⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⠶⠛⠁⣠⠞⠀⠀⠀⠀⠀⣦⣤⡤⠖⣫⢔⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⢔⣪⠞⠀⠀⠀⠀⠀⠀⠀⣠⢖⣫⢗⡯⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⡉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡬⠽⠷⠘⠀⠀⠀⠀⠀⠀⠀⢀⣠⢬⡿⢷⡾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡴⠶⢶⢶⣶⡆⠀⠀⠀⠀⠀⣿⣷⡞⡾⢷⢳⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠋⠛⠛⠚⠛⠃⠀⠀⠀⠀⠀⠛⠙⠓⠛⠛⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠶⠶⠖⢲⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠉⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⡓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠛⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆
⠀⠀⠀⠀⡾⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⢰⠃⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⣸⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠻⣿⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠹⡇
⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠁
⠀⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠸
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⣠⣾⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣟⣛⣛⣃⣀⣠⡤⠤⠀⠀⠈⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠒⠒⠲⢒⠂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠁⠇⠀⠀⠈⡜⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣗⡮⢽⣎⣘⣑⣛⣓⣀⣙⣷⣭⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣭⣭⣥⣴⣶⣶⣿⣫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣦⣤⣴⣿⣏⢽⣟⣛⡛⣛⣛⣿⢹⣾⣿⣴⣾⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣩⣿⣿⣿⣿⣿⣿⣿⡸⣜⣶⣾⡆⣽⣶⡖⣾⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣮⡘⠿⠇⠿⠿⣈⣵⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠹⣯⣿⣿⣿⣿⣿⣿⣿⢿⣾⢽⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣦⣿⣸⣿⣿⡿⠿⢛⡇⠀⠈⠘⢧⡙⠻⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⠟⡠⠾⠟⠀⣠⣠⢤⣐⡛⠣⡈⢻⣿⡏⠹⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠉⠻⣿⡿⣻⣿⠋⢈⠴⠞⢛⣛⣛⣻⣦⣭⣙⠳⠄⠀⠻⣿⣿⣿⣿⣿⡿⠛⠁⠈⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⠉⠒⠲⠴⢤⣿⣵⠿⠃⠀⠀⠒⠛⠋⠉⣿⢽⡀⠤⠘⢛⡃⠀⠀⠉⣻⢿⣽⣿⣤⣤⣤⢤⣄⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⡄⠀⣀⣴⣿⠋⠉⢿⣻⠄⠙⠛⣿⣧⣀⡀⢊⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢨⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣾⣿⣿⣿⣿⡀⠀⠀⠙⠀⠀⣠⣿⣿⣿⣿⣿⣷⣶⢔⡀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠛⠓⠒⠚⠛⠛⠙⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠀⠈⠑⠒⠦⠤⠤⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠻⠛⠛⠁⠀⠉⠙⠋⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣀⣠⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⠿⠷⣄⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣃⠀⢈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡾⠟⢿⡈⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠸⣥⡀⠀⠀⠀⠀⠀⢀⣴⡞⣠⠀⠀⠈⣿⣿⣿⣿⣗⠲⣿⣏⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠉⠛⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⢸⣿⠷⣄⣾⠆⠀⣿⣿⣿⣿⣿⣿⣻⡻⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠂⠀⠀⠀⠠⠤⠤⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⣈⠛⢿⡉⠿⠟⣠⣼⣿⣿⣿⣿⣿⣿⣍⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢡⣀⣴⡆⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡙⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⠈⠙⠋⣀⣠⣶⣿⣿⣾⣿⣿⣦⠀⣤⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠘⢁⣿⣿⣿⣿⢿⣿⣿⣀⠀⠟⠘⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⢾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡃⠀⣿⡟⠉⠁⠀⠀⠉⠛⠋⠉⠛⠿⠿⠳⢢⠀⡀⠈⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢆⡠⢬⣭⠙⡋⢉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⢿⣥⣘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣾⣿⣿⣿⣶⢉⣶⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢠⣿⣿⣍⡠⠴⣾⣷⣤⣤⡀⠀⠀⠀⣴⡞⣋⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⡿⠿⢿⣿⣿⣹⣷⣭⣾⡟⢿⣿⠿⠿⠛⢿⣿⣿⣿⣿⣿⣎⣻⣿⣿⣷⡔⠁⠈⠛⠻⠄⠀⠀⠞⠛⢿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣤⡀⠿⠟⠋⠀⣿⣿⠷⠀⠀⣶⢾⣷⢠⣿⣿⣿⣿⣿⣿⡿⠛⠛⢻⣇⠀⠀⠀⠀⠀⠀⢀⣀⣴⣦⣤⣜⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣶⣶⡆⢀⠀⣀⣀⣀⠀⠀⢠⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣧⣛⠃⢸⡟⢺⣄⠀⢀⣀⣰⠿⠛⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣼⣿⣠⣿⢦⡌⠸⣿⣷⠀⣨⣷⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣾⣷⡀⣿⣿⣿⣿⣷⣦⡀⠈⠀⢉⡛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⠿⣷⣶⣄⣸⣏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⠟⣳⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣶⣶⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣦⣿⣾⣿⡀⢻⣿⠹⠛⠛⠛⣿⣿⣿⣿⣿⠶⣿⣿⣿⣿⣿⡟⢻⣿⣿⣇⠀⢿⣿⣿⣿⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠟⠀⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⠞⢛⣛⣋⢻⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⣷⣾⣿⣷⣿⣷⣾⣿⣿⠟⠉
⠀⠀⠀⠀⢀⠤⠔⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣿⡈⢿⣿⣿⠛⠿⠟⢋⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠿⣿⣿⡇
⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⣿⣿⣇⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⢹⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠉⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣷⣾⣿⣿⣿⣧⣬⣿⣿⣿⣿⣿⣗⠀⠹⣿⣿⣿⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡍⢿⣿⣿⣿⣿⡟⠁⠀⠿⣿⡇⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠇⠘⣿⣿⣿⠿⠿⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢻⣿⣿⣿⣿⣿⣿⡿⠿⣆⡶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⠴⠖⠛⠻⣿⣿⣿⣿⠟⣁⣱⣶⡶⢸⣿⡿⠟⠋⠁⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⢯⡑⠞⠛⠉⠀⠀⠈⠉⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢲⠶⡀
⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠇⠀⠀⠃⠈⠢⡀
⠀⠀⠀⠀⣠⠞⠉⣠⣰⠲⣤⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠒⠢
⠀⠀⢀⡾⠁⠀⠀⡇⢹⣳⠶⡃⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢂⠀⠄⠤⠄⠀⠀⠀⠀⠙⢃
⠀⠀⡾⠁⠀⠀⠀⠛⢻⢀⡒⠁⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠰⠀⠠⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀⠠⠄
⠀⣸⠇⠀⠀⠀⠀⠘⡖⡽⠇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐
⠀⣿⠀⠀⠀⠀⠀⢠⡷⠾⢀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⠄⠐⠂⠀⠀⠀⠚⢁⡀⠀⠈⠥⠀⠀⠈⡹⠃⠀⠀⠀⠀⣀
⢀⣿⠀⠀⠀⠀⠀⡈⠑⡚⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⡄⣀⢀⡀⠀⠀⠀⣤⣀⠄⠠⠀⠀⠀⣈⠴⠀⠀⠀⢀⠀⠤⢉⣭⣀
⠐⣿⠀⠀⠠⠆⠀⠄⠤⠤⠀⠆⠰⠤⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣬⣥⣄⡀⠀⠀⡆⠀⠀⠀⡀⠈⠁⠀⠛⠒⠋⠀⠀⠀⢀⠔⠘
⠀⢿⣷⣶⣶⣶⣶⠶⣶⢶⣶⢶⣶⣶⣶⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡿⠃⠀⢠⠇⠀⠐⠈⠀⠀⠀⠀⠀⠀⢠⡄⠐⠂⠁
⠀⠀⠻⢿⣿⣟⣿⣻⣻⣿⣻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠁⠈⠀⡶⠚⠛⢀⢀⠀⠀⠀⠀⢀⡀⠠⠄⠚⠁
⠀⠀⠀⠀⠈⠙⠿⣿⣟⢙⣿⡿⠛⠉⠀⠀⠀⢀⡀⠀⠣⠀⠀⢀⠀⠀⠀⠄⢀⠒⠤⠔⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠠⠊⠀⠀⠀⢠⣠⡀⠐⠓⠒⠀⣠⡄⡁⢘⠚⠶⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢐⡁⠀⠁⠀⠀⠀⠀⡀⠀⠁⠂⠿⣼⣸⠡⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠋⠁⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠀⠀⠀⠘⠀⠐⠒⠐⠒⠒⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⠈⢀⠀⠀⡀⠐⠂⠀⢤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠒⠲⠬⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢡⠀⠆⠀⠒⠠⡀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠑⣆⠀⠀⠀⡖⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡶⠉⠉⠁⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠐⠀⠈⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠞⣁⠀⠀⢀⡌⢲⠇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠭⠬⠥⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠈⠀⠀⣀⣠⣤⣾⣦⣄⡀⠈⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠉⠉⠓⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⡀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣛⡒⠒⠸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠿⢿⣿⣿⣿⣿⣿⠟⠁⠊⢿⣿⡿⣿⣞⢥⠄⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠄⠀⠀⠀⠨⢽⣿⣿⣷⣖⡄⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢩⣿⣿⣿⣿⣕⢔⠈⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢀⣠⣴⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢀⢕⣿⣿⣿⣿⣿⢕⣴⠇⠉⢣
⠀⠀⠀⠀⠀⠀⠁⠈⣦⣶⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣶⣿⣽⣿⣿⣿⣿⣿⡥⢋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⡊⠀⡃
⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠀⠀⠅
⠀⠀⠀⠀⠀⠀⡤⠀⠁⠉⠛⠻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣕⡁
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠨⡻⣮⡀⢄⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢏⡄⠉⢢⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢿⢿⣿⣶⣕⣷⡶⡡⣢⣆⢁⢄⣴⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⡊⠀⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡤⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⠂⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⡀⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠈⠉⠀⠀⠠⢬⡉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠥⠤⠒⠂⠤⣀⠀⠀⠀⠉⠲⣄⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠙⢧⡀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠠⠀⠀⢰⠇⠀⠀⠀⠐⠙⠁⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠉⠙⠢⣄⠀⠀⠀⢸⣿⣯⣿⣤⣀⠀⠀⢀⣤⠐⡋⠀⢀⣀⣀⢀⣄⣀⣀⣀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⡀⢸⣿⣿⣿⠄⠙⠛⠈⠉⠁⠀⢠⣾⣿⡿⣱⣿⣿⣿⣿⡛⣻⣿⡷⢶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠻⣿⣟⠂⠀⠀⠀⠀⠀⠀⠘⠿⢏⣾⣿⣿⡋⠻⣿⢿⣿⡏⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣽⣿⣄⠀⠀⠀⠀⠤⣀⠀⠿⢷⣶⣶⣮⣥⠀⣘⡻⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⡐⢦⣬⠘⠂⠀⣸⣷⣶⣯⣽⠃⠰⢯⣟⣋⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣿⣿⣿⣿⣿⣿⣶⣍⣥⣄⡺⣿⣿⣿⣿⠃⢠⣿⣷⣈⡁⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⠻⠏⠀⢻⣧⣉⣩⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣤⠞⠁⠀⠀⠀⠀⠀⠀⡼⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣦⣄⡐⠿⠉⢹⠆
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠋⢀⣠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠸⣎⢿⡿⠃⠁⠀⠀⣈⣠⣿⣿⠛⠢⢤⣍⠉
⠀⠀⠀⠀⣀⣴⣟⣩⣤⡶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡠⣤⡀⢰⠟⢛⣿⣿⠏⠀⠀⠀⠈⠳
⠀⠀⠀⢀⣿⠟⠛⣉⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣮⣿⣶⢾⣿⣿⠏
⠀⠀⠀⣾⠷⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠻⠯⠍⠉⠠⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢻⣦⠀⣰⣦⡀⠈⡁⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡔⠒⠀⠐⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣦⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠘⣷⣶⣯⡻⢏⢽⣿⣿⡻⣶⢕⠉⢿⣦⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠸⣿⣿⡿⣾⡜⠟⠻⢧⢾⢋⣼⡷⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣿⣿⣧⠈⣧⠀⢖⠙⠀⠘⢋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⢿⠿⠙⠀⠠⣀⣴⠙⣿⣿⣷⣿⣷⠄⠀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠆⠀⠀⠀⠀⠉⠛⠓⢿⣿⣶⣿⣿⡿⣆⣌⠻⠛⠟⠋⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣛⣛⠴⠉⠉⠁⢂⣀⣠⣷⣶⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣍⠋⠉⠻⠂⠀⠀⠐⠠⠚⢛⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣎⠻⢶⠆⠀⢠⣴⡟⠊⠀⠐⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠐⠂⠀⠙⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣠⣿⣿⣾⡶⠂⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠐⣿⠃⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠏⠀⢸⣿⣿⡿⠋⢠⣾⣿⠏⣶⠦⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⡄⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣶⡿⣿⠇⢠⣾⣿⣿⠏⠀⠴⠤⢤⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣍⡓⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡏⠀⠈⠶⡞⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠛⡿⣿⣿⣿⣷⣶⣄⣐⠂⠀⣀⠀⠀⠀⠀⠉⠉⠠⠤⠄⠾⠅⢀⠀⣀⠀⠞
⠀⠀⠀⠀⠀⡔⢠⣾⢋⠋⢿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠐⠒⢃⣀⣄⠉⠉⠆
⠀⠀⠀⠀⢀⠀⣾⣿⡇⠀⠀⠙⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠻⣗⠒⠀⢹⣿⣻⠿⠀⠐
⠀⠀⠀⠀⢸⠐⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠘⠊⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠁
⠀⠀⠀⠀⠀⢡⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠈⡌⣿⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⣇⠘⣿⡆⠀⠀⠀⠀⠀⢸⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠳⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⠅⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠁⣾⡿⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠛⠃⠀⠀⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣷⣿⣽⣦⣰⣿⣦⡐⣦⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣧⡀⠀⠀⠀⠠⣤⣤⡽⣿⠻⣿⣿⣿⣟⠧⠿⣷⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡀⠀⠀⠸⣿⣿⣦⣇⢿⠿⣧⣹⠷⣰⣍⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣿⣿⡎⢹⡀⢠⠴⠍⠰⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠂⠀⢄⣠⠙⣿⣿⣶⣿⡄⠀⠠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠉⠛⠺⣶⣷⣾⣿⡿⣆⣈⠛⠛⠛⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣴⣾⣿⣿⣯⣭⠐⠉⠉⠁⣠⣤⣾⢷⣦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⢀⣴⣿⣿⡟⣿⡍⢈⠙⠃⠀⢀⡈⠂⢉⣵⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⣀⠀⠀⣴⣿⣿⣿⣿⣿⣮⠙⠋⠀⣰⣾⣋⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠃⠀⣴⣿⣿⠟⠋⣠⣶⠆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⡦⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣯⣄⣰⣿⡟⢁⣠⣼⣿⠁⠈⠉⢈⡁
⠀⠀⠀⠀⠀⠀⠀⣰⣦⡓⠤⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡾⠁⠘⣭⡍⠁⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠁⣀⡀⣼⠁
⠀⠀⠀⠀⢀⠀⣴⣏⠚⢿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠐⠒⢂⣀⡀⠉⠑
⠀⠀⠀⠀⡄⣼⣿⡇⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠙⠻⠓⠀⢸⡟⡩⠟⠠
⠀⠀⠀⠀⠇⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣄⡠⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠹⣿⣧⠀⠀⠀⠀⠀⠀⢰⡿⠋
⠀⠀⠀⠀⠀⡆⣿⣿⠃⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⢠⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⢸⠘⣿⡄⠀⠀⠀⠀⢀⠏⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠸⠀⢹⣧⠀⠀⠀⠀⢸⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⡇⠸⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⣀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⣶⣶⡶⠶⠶⢶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⣍⡐⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⡁⠀⠉⠛⠛⠶⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢿⣟⠙⠀⠠⠄⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣛⢦⣙⣾⣏⣀⠀⠰⣤⣤⣀⢬⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⢀⣠⡼⡶⠾⠿⠛⠉⠹⣷⣬⣿⣿⣿⣷⣦⣙⠛⣿⣶⡁⠀⠀⠀⠀⢠⠴⣿⣿⣿⡏⣙⢧
⠀⠀⠀⠀⣀⣾⣦⣶⢂⣀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⡟⢹⣷⡾⢾⡙⠿⡆⠀⠀⠀⠀⢻⣿⠹⠿⠱⢋⠘⡇
⠀⠀⠀⠀⠈⠉⠹⠟⠷⠶⠒⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣾⣬⣘⣀⣿⣷⣔⣿⡄⠀⠀⣼⣿⠇⠰⡶⠷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⡿⠀⣼⣟⣀⡉⢙⣿⣿⡿⣿⡂⠀⣐⠛⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠃⣰⣿⣿⠟⢩⢿⡿⠋⠀⠈⠿⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣀⡀⡿⠋⣡⣴⡿⠛⠃⢀⠀⠀⣠⣶⡽⠋
⠀⠀⠀⣶⣶⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣶⠯⠋⠀⠀⠀⠀⠈⠁⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⡾⠛⠋⠛⢢⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠿⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡈⠂⠐⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⣑⣶⠁⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠜⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠈⠁⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⣶⣤⣧⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡏⠈⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡤⠀⣀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣶⡆⢙⠙⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆⣩⠐⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠋⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠂⠀⠁⠂⠄⠐⠰⠶⠰⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠴⠧⠓⠶⠘⠾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⡿⠋⣩⠔⠉⣿⣿⣷⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢁⢩⣿⣯⢁⢩⣿⠏⡠⢝⠊⠀⠀⣿⡍⡈⣽⣿⡏⠈⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⡿⣿⣿⢀⠏⠁⠀⠀⣰⣿⣿⢿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣸⣿⣿⣀⣸⣿⡜⣀⣀⣤⣾⣿⣿⣆⣀⣾⣿⣇⣀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⣿⡄⠸⣿⣿⠇⢿⣿⣿⣿⣿⡇⢾⣿⣿⠆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣷⠀⣿⣿⠀⢸⣿⡇⠀⠙⠃⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣇⣿⣿⠀⢸⣿⣧⣴⣶⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⠸⣿⣿⣿⠀⢸⣿⡏⠙⠛⠀⠀⣿⡇⠀⣀⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣿⣿⠀⢻⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⣿⣇⣀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡘⠛⠿⠗⠈⣿⣿⠀⣸⣿⡇⠀⠀⠀⣾⣿⠿⠿⠛⢃⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣤⣤⣄⣁⠀⠙⠻⠿⠆⠀⣀⣠⣤⣤⣶⡾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣶⣄⣠⣶⠿⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡇⢰⡆⣄
⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⡇⢸⡇⣿⢠⣤⣤⣤⣤⣤⣤⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠲⢶⣖⣛⣺⣿⣿⡟⠻⢿⣿⣿⡇⢸⡇⠟⢸⣿⣿⣿⣛⣳⣶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠗⣺⣿⣿⠟⢀⡀⣉⠛⠇⢈⠀⣀⢸⢿⣿⣟⡲⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣂⠀⢸⠃⣿⢠⡆⢸⡇⠛⠐⢐⣊⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⢷⠀⠀⢹⠸⡇⢸⠁⠀⢸⢿⣾⣟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡇⢸⠀⠀⢸⠀⡇⢸⠀⠀⡏⠀⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⢻⣆⣊⣀⣤⣤⡄⣥⣬⢠⡄⣑⠂⣟⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠘⣿⠋⢸⡇⢹⡏⠁⣿⣷⢸⣿⣿⠸⣯⡇⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠺⠇⡘⠗⠸⠟⠿⠚⡏⠻⠚⠷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠙⠛⠗⠟⠺⠚⠛⠋⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠒⠛⠛⠉⠉⠉⠉⠉⠙⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⣁⠔⡀⡖⡤⡔⠦⠤⢳⠤⠖⡤⠴⢢⢤⢲⢀⠖⣀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⣀⡴⠋⠁⡦⠖⠚⠒⣤⠒⠦⠋⠣⠋⠉⠌⠉⠧⠉⠩⠌⠙⠜⠘⡴⠚⠙⠒⠆⠀⠉⠲⣄
⠀⠀⠀⣠⢞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢙⢦⡀
⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣄
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠱⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠦⣄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠞⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣓⣒⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣖⣒⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣙⣛⣟⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠃⠀⠀⠀⠀⠀⠀⢀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣼⣿⣿⣿⠸⠃⣟⣤⠑⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠴⠮⠹⠛⢣⠜⠈⠁⠁⣠⣿⡟⠉⠉⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⢀⡿⣶⣟⢟⣦⣤⡶⠟⠉⠁⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⡀⠘⢈⡉⣿⣽⣾⣿⡟⠁⠀⠀⠀⠀⠀⡠⠀⣦⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣝⣿⣟⣿⣿⣿⣿⢟⠓⠀⠀⠀⠀⢀⡉⣡⣤⣿⣿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠘⣟⢿⣿⣿⣟⠻⣧⣾⣶⣮⣶⡦⢊⠁⠀⢻⣷⣞⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠙⣿⡿⠟⠋⠙⢶⡄⢿⣧⠙⠿⣷⣿⣧⣶⣾⣿⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣤⣶⣿⠿⠗⠁⠀⠀⠀⠀⠀⠉⠀⠛⢿⣷⣤⣤⣤⣤⣾⣿⣿⣿⢙⣇
⠀⠀⠀⠀⠀⠀⠠⢀⠀⠄⠐⢀⣰⡄⠝⠋⠉⠀⠀⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⠛⠛⠿⠿⣿⡟⣿⣿⢀
⠀⠀⠀⠀⠀⠐⠀⠂⠀⢄⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢷⣄⣀⣀⣴⡿⠟⠛⠛⠁⠉⢹⡟⠂⣼⡿⣿⣿⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣅⡀⣀⣠⣤⣴⣆⢀⣠⣦⣿⡿⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠶⠶⣿⠿⢿⣿⣿⣿⣿⣯⣿⣿⣿⣶⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠏⢡⣾⣗⣒⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢈⣧⣼⣧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢠⣲⡿⠋⠁⢺⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⣠⣞⠁⠀⠀⠀⢸⣿⣷⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⡹⠊⠁⠈⠓⣀⠀⠀⢟⣻⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⠀⠀⣠⠒⣻⣷⡾⢟⣿⡃⣤⣀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡤⣌⣠⣾⣿⣟⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡠⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢬⣯⣾⣿⣿⠟⠛⠋⠙⠉⠙⠛⠋⠿⢿⣿⣿⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠃⣨⣦⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠍⠙⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣷⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⢀⣤⡈⣿⣿⠿⠋
⠀⠀⠀⣄⠙⣶⣽⣿⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠉⠻⠿⠿⠿⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠻⠯⠍⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⣦⠀⣰⣦⡀⢀⡁⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣦⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⡄⠀⠀⠀⠈⣷⣶⣯⡻⢉⢽⣿⣿⡹⣶⣅⠉⠻⣦⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡦⡀⠀⠀⠸⣿⣿⡿⣾⡘⠟⠻⠧⡾⢋⣼⡷⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢿⡟⠁⠀⠀⠀⣿⣿⣧⠈⢧⠀⠒⠀⠀⠘⢋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⢿⡳⠙⠀⠠⣀⣤⡙⢿⣿⣷⣿⣧⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠉⠛⠛⠿⣿⣶⣿⣿⡿⢀⠈⠛⠛⠛⠋⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣛⣋⠐⠉⠉⠁⠀⢀⢠⣶⣶⣄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡍⠋⠉⠻⠂⠀⠀⠐⠈⠊⢙⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣎⠻⢤⠂⠀⢠⣴⡯⠀⠀⠀⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⣠⣿⣿⣶⡶⠂⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠐⢟⠃⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠇⠀⢸⣿⣿⠿⠃⢠⣼⡿⠋⠀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣯⣷⣶⡿⣿⠇⢠⣶⣿⠿⠁⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣌⡀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⠏⠀⠈⠴⡖⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⠀⠄⠾⠀⠀⠀⣀⠀⠒
⠀⠀⠀⠀⠀⡀⢠⣾⠫⠊⢻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠐⠀⢃⣀⡄⠉⢉⠂
⠀⠀⠀⠀⢠⠀⣾⣿⡀⠀⠀⠙⠛⠿⠛⠋⠛⠛⠛⠛⠛⠋⠻⣗⠒⠀⢸⡟⣙⠟
⠀⠀⠀⠀⢸⢀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠈⡌⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⠞⠁
⠀⠀⠀⠀⠀⢠⡘⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠂⣿⡿⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡝⢸⣿⠀⠀⠀⠀⠀⠀⢸⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⣧⠘⣿⡆⠀⠀⠀⠀⠀⢸⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢹⠀⢸⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠈⡄⢸⣿⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣄⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⣿⡿⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢟⣵⣿⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡞⢁⠀⠀⢰⣴⣴⡀⠀⢠⠈⢦⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⣱⣿⣿⣿⠟⣛⡛⣿⣿⣿⣧⠻⣿⣿⣿⣿
⠀⠀⠀⢀⡞⠀⡆⠀⠀⠐⣻⡗⠂⠀⠀⢇⠈⢧⠀⠀⠀⢸⣿⣿⣿⡿⣰⡿⣾⣿⣿⣭⠁⣭⣿⣿⣷⢻⣧⢹⣿⣿⣿
⠀⠀⠀⣸⠁⢸⠁⠀⠀⢉⣿⣏⡁⠀⠀⢸⠀⠸⡆⠀⠀⢸⣿⣿⣿⢣⣿⡇⣿⣿⣿⠶⠀⠶⣿⣿⣿⡾⣿⡇⣿⣿⣿
⠀⠀⠀⡿⠀⢸⠀⠀⠀⢠⣼⣥⠀⠀⠀⢸⠀⠀⣧⠀⠀⢸⣿⣿⣿⢸⣿⡿⣿⣿⣿⣟⠀⣛⣿⣿⣿⡇⣿⣿⢸⣿⣿
⠀⠀⠀⣧⢸⢸⡄⠀⠀⠰⣾⡷⠄⡀⠀⣸⢰⠀⡿⠀⠀⢸⣿⣿⣿⢸⡻⡇⣿⣿⣿⣛⠀⣛⣿⣿⣿⢱⣟⣿⢸⣿⣿
⠀⠀⠀⢿⠰⣄⣇⠀⡅⢘⣿⣟⠂⡇⢀⣏⣴⢠⡇⠀⠀⢸⣿⣿⣿⡸⡟⢧⢻⣿⣿⡭⠀⠨⣿⣿⡟⡼⢻⡏⣾⣿⣿
⠀⠀⠀⠘⣇⢫⠻⠄⠰⣈⣿⣯⡰⠁⠼⢫⠇⣼⠁⠀⠀⢸⣿⣿⣿⣇⢻⣰⣌⣿⣞⠧⣴⠰⣫⣿⣣⡆⣾⢱⣿⣿⣿
⠀⠀⠀⠀⠹⡌⢇⠀⠀⠀⢸⠈⠀⠀⢀⠎⣸⠃⠀⠀⠀⢸⣿⣿⣿⣿⣎⢷⣻⣿⣿⣿⣿⣿⣿⣿⡿⣼⢣⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢎⠂⠀⠀⠸⠀⠀⠀⢊⡼⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣮⠻⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣝⡿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⢸⣿⡇⠀⢀⡆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⢠⣾⠋⠀⠀⠀⢠⣿⣿⡅⣠⡿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣄⢰⣿⡃⠀⠀⠀⣠⣿⠿⣫⣾⡿⠃⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⣤⣄⠀⣤⣭⣶⣿⣿⠟⠁⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⢾⣝⠻⢿⣿⠀⣿⡿⣟⡯⢀⡔⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡿⣮⡄⢸⣿⠀⣿⣾⢫⣵⠋⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢇⣿⠃⢸⣿⠀⣿⡿⣿⣿⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⢸⣿⠀⣾⣿⠀⣿⡇⢹⣿⡄⣾⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⣾⣿⠀⣿⣿⠀⣿⣧⠀⣿⣇⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠐⣿⣿⠀⣿⣿⠀⣿⣿⠀⣿⡿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠘⠇⢠⣿⣿⠀⣿⣿⠀⠻⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡺⣿⣿⠀⣿⣿⢧⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠊⠻⠀⠟⢡⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⢀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⣡⡄⢤⡤⠄⣤⣤⠀⣤⣤⣄⣤⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢀⣿⠁⣾⡇⠀⣿⣿⡄⣿⡝⣿⡈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⡏⢀⣿⠁⢰⣿⣽⣇⢻⣇⢹⣧⢀⡙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠠⠿⠃⠸⠟⠀⢘⣋⣙⣛⠘⠛⠘⠿⠘⢷⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠠⣤⣶⣶⠛⠛⠛⣛⣛⡓⠶⢦⣄⡀
⠀⠀⠀⠉⠛⢷⣄⠀⠙⢿⣷⣄⠈⠙⢷⣶⠶⠛⠻⣯⣟⣛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠙⢷⡀⢀⠻⣿⣧⡀⠀⠙⢿⣷⣶⣄⠸⣝⠷⣦⡙⢷⣶⣿⣟⠛⠛⠛⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢰⣾⣷⡈⢧⣉⠻⢿⣦⣄⡀⠉⠛⠛⠀⠘⣷⣬⣉⠃⠈⠉⠉⠀⣀⣿⣿⣦⣌⠙⠷⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠸⡇⠻⣷⣄⠙⢿⣶⣶⣶⣶⣶⣶⣶⣤⣄⡈⠛⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣶⣦⣌⡙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡉⠻⢷⣶⡄⠉⠉⠛⠻⠿⣿⣿⣿⣿⣦⣤⣤⣾⡿⠟⠉⠀⠀⡀⣰⣆⣀⠀⠀⠉⠻⢿⣦⣌⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡻⣦⣄⣀⠠⢤⣤⣤⣄⣀⠈⠙⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢙⣿⣿⡉⠀⠀⠀⠀⠀⠙⢿⣦⡘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣮⣙⠻⠿⢿⣶⣭⡛⢿⣷⣄⠘⣿⣿⡟⠁⣀⣀⣀⣀⣀⣀⣈⣀⣈⣁⣀⣀⣀⣀⣀⡀⠈⢿⣷⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⡉⠻⠄⠹⣿⣧⣿⣿⠀⠀⢻⣶⣶⣶⣶⣶⠶⠂⣶⣶⣶⣶⣶⣶⣶⠏⠀⠈⣿⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣷⣄⠀⢹⣿⣿⡇⠀⠀⠀⠟⠀⠀⠀⠠⠀⠀⣿⣿⠃⠀⠀⠈⠛⠀⠀⠀⢹⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠁⠀⢀⣿⣿⣇⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⣿⡿⠀⠀⠀⣀⠀⠀⠀⠀⣼⣿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡘⢿⣿⣿⣿⣿⡀⠀⢻⣾⣧⣀⠀⠀⢠⠀⣿⠇⠀⠀⣠⣽⣷⡟⠀⢀⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣉⠛⠿⢿⣷⡐⠉⠹⡇⠀⠀⠀⠀⠄⣿⠀⠀⠀⠀⢸⠏⠉⢀⣾⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⡄⠻⣿⣆⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣴⣿⠏⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠻⢿⣦⣄⣀⠀⠀⠈⠀⠀⠀⣀⣤⣶⡿⠟⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣉⠛⠻⠿⠿⣿⣿⠿⠿⠟⠛⣉⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠉⢹⢸⣿⠓⢾⡇⠀⣶⠀⢠⡄⢀⣄⡄⢠⣄⡄⢠⡄⠀⣤⠀⢠⡄⠀⡄⠀⣴⠀⠀⡏⣶⠏⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠀⠀⢤⣤⣤⠄⣈⢀⢀⣀⣁⣈⠀⢁⠈⢁⣀⡉⠀⢈⣁⣀⠁⢀⣤⡤⠀⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠀⠀⢸⠀⡇⠀⠟⠉⡇⣿⠙⠿⠀⠁⣿⠙⡏⡏⠀⠀⡇⣿⣆⢸⢸⡇⠀⠑⠿⣵⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠀⠀⢸⠀⡇⠀⠀⠀⡇⣿⠀⠀⠀⠃⣿⠾⡅⡇⠀⠀⡇⣿⢞⢾⢸⡇⠀⠀⠀⠀⢹⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠀⠀⢸⠀⣇⠀⠀⠀⡇⣿⠀⠀⠀⠀⡇⠀⡇⡇⠀⠀⡇⣿⠈⢊⢸⡇⠀⣦⡀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣼⣿⣤⠀⠼⠶⠿⠂⠀⠐⠛⠛⠂⠀⠘⠛⠃⠀⠛⠛⠂⠘⠓⠛⠀⠘⠿⠧⠀⠿⢿⣦⣚⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⣠⣼⣧⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠒⠂⢤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢟⣿⣖⠖⠀⠤⠄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠑⢿⢟⣵⣷⢶⢐⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠀⠀⠀⠀⠉⠝⢹⣷⣗⣵⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⣿⣿⣿⣿⢟⢄⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⣀⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⢄⢿⣿⣿⣿⣷⣔⠀⣸⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠈⠄⣀⣠⣴⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⣄⣕⢝⢵⣿⣿⣿⣵⡾⢋⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⠕
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠿⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠙⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⠢⠄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⢗⣵⣖⠄⡀⢆⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣿⣿⣶⣾⣷⣉⣊⣀⣀⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⡤⠀⠀⠂⠈



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣶⣄⡀⠈⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⢿⣿⣷⣄⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠉⢠⣶⣤⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣤⠉⠀⣶⣤⣽⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⠈⠁⣴⣦⣼⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⣿⣿⣧⣈⠁⢴⣦⣬⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠈⠻⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀⠈⠙⢿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣶⣄⠀⠈⠛⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠛⠻⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⡀⠀⠀⠛⠷⣄⡀⠀⠀⠈⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⢀⣀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⢠⣿⠇⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣸⠟⠀⠠⣾⣧⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠋⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢹
⠀⣿⠃⠀⠀⢘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⣿⡄⠀⢀⣼⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀
⠀⣿⠁⠀⠈⠹⣿⣷⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢀⣤⣄⣀⣀⣀⡴⠖
⠀⣿⠀⠀⠀⢸⣿⣿⣿⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣤⠜⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴
⠀⣿⠀⠀⠀⠘⣿⣿⣿⣯⣉⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠘⠋⠉⠁
⠸⣿⣦⠀⠀⠀⢸⣿⣿⣿⣿⣷⣀⡶⠀⣾⠟⠁⡀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡆⠀⠀⠀⠀⣠⣤⣤⣤⠀⣴
⣶⡟⠻⣦⡀⠀⢸⣿⣿⣿⣯⠉⢻⡀⠀⢿⣤⠈⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡰⣿⣅⠀⠻⣷⣿
⣿⡇⠀⢹⣿⠀⠀⠈⣿⣿⣿⣦⡜⣿⣷⣄⠀⣦⠀⠀⣸⣿⣠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢄⣀⠇⢀⣾⣿⡟⢱⡿⠋⠀⠀⣿⡇
⢿⣿⣄⠀⢻⣄⠀⠀⠘⠿⣿⣿⣧⣿⣿⡿⠀⢹⣧⠀⣿⣿⣞⣿⣿⣶⣤⣤⡄⣴⣿⠀⣴⣿⣦⠘⣿⣼⢟⣠⡿⢸⣿⢠⡎⠁⠀⠠⣾⣿⠃
⠀⠹⣿⡀⠸⣿⡀⠀⠀⠀⠉⢿⣿⣿⣿⡆⠀⠀⠛⠀⢸⣿⣟⣿⣿⣿⣿⣿⣇⣿⣇⠈⣾⣟⠛⣷⣼⣿⣼⣿⣧⣼⠟⠈⠀⠀⠀⠀⣻⣿
⠀⠀⢈⡇⠀⠹⣷⡀⠀⠀⠀⣀⠉⠻⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⣧⣽⣁⣿⠿⣶⢈⣿⣿⢏⣽⠟⢁⡴⠇⠀⠀⠀⠀⣿⡟
⠀⠀⢨⣷⠀⠀⠘⢿⣦⠀⠀⠉⠷⠀⣸⡃⠀⠀⠀⡀⠸⣿⡿⣿⣧⠀⣿⣿⣿⣿⣿⣯⣤⣶⣼⣿⣿⣽⠿⢁⣴⠟⠁⠀⠀⠀⢀⠀⣿⡇
⠀⠀⠘⣿⠀⠀⠀⢀⡹⢿⣦⡄⠀⠀⣿⡇⠀⠀⠀⣷⠀⢹⣿⠈⢫⣼⣿⣿⠟⣿⡿⠿⠟⠉⢁⣴⠟⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⢿⡄⠀⠀⡼⠿⢦⡉⠻⢷⣶⡍⠀⠀⠀⢠⣿⠀⠈⢿⣷⣦⣁⡀⠁⣀⣀⣠⣤⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢰⣶⠾⢿⣿⡟
⠀⠀⠀⢸⣷⠀⢠⡄⠀⠈⠛⣦⣄⣻⠃⠀⠀⠀⢸⣿⣦⠀⠈⢿⡉⠛⠿⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠈⢿⡄⠈⠃⠀⠀⢀⣈⣿⡇⠀⠀⠀⠀⣸⡟⠀⣶⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠘⢷⠀⠀⠙⠃⠈⢿⣿⠃⠀⠀⠀⠀⣿⡇⠀⢹⣧⠀⠸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠸⠆⢸⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣆⠀⠸⣇⠀⢠⣌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⣼⡇⠀⠀⠀⠀⣰⠛⣇⠀⠀⠀⢻⡄⠀⢸⣷⡎⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⣿⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⣷⠀⠀⢹⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣿⠁⣰⠂⠀⠀⡿⠀⣿⠀⠀⠀⠀⢻⡇⠀⠀⢻⣦⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⢱⣾⠀⠀⣤⠀⣿⠀⠀⣠⠀⢀⣿⠀⠀⠀⠀⠈⢁⣾⠇⠀⠻⣇⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠹⡄⠈⠀⠀⣿⠀⢸⡿⠀⡄⠀⠀⠐⠻⢏⣾⣧⠀⠻⣷⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣸⡄⠀⠀⠀⠀⢸⡇⢀⣧⠀⠀⠀⠀⣾⡿⣡⠛⢂⠈⢿⣯⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⣿⡇⠀⠀⣠⠀⢸⡇⢸⣿⠀⠀⢀⡄⢹⣤⡿⢦⣿⡀⠈⢿⣇⡀⠀⡤⠀⣀⠀⢸⡟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣿⠁⠀⠀⠉⠀⢸⡇⠘⣿⠀⠀⠸⡇⢸⣿⣅⣀⣿⠿⣤⡀⠛⣇⠀⣀⡶⠏⢰⡾⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠀⠀⠀⠟⠀⢸⡇⠀⢹⣄⠀⠀⢸⡟⣿⡟⠙⣿⣄⢿⣿⠀⠙⣿⡏⠀⠰⡿⠁⢠⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⢱⣿⠃⠀⣿⡟⢿⣿⣧⠀⠈⢿⣄⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⠂⠀⢨⣇⠀⠈⠋⠀⠀⠀⠘⢻⣾⠀⠈⣿⠏⢻⡟⠀⠀⠈⢿⣀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣀⠀⢨⡇⠀⠀⠈⡇⠀⠀⠀⢸⡇⠀⠀⢹⣶⣿⣠⣴⠀⠀⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠀⢸⡗⠀⠀⠀⢇⡀⠀⠀⢿⣿⠀⠀⠈⠉⣻⣿⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠟⠀⠸⣧⠀⠀⠀⢸⡇⠀⠀⠘⠿⣷⡀⠀⢺⣿⠃⣶⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⣦⠀⢰⣿⠀⠀⠀⠈⣧⠀⠀⢀⡀⢻⣅⠀⠀⣿⡆⣿⣿⡀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⣶⡄⢠⡏⠀⠀⣿⠀⠀⠀⠀⠙⠀⠀⠘⢃⣸⣿⠀⠀⠙⠓⢉⣿⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡟⢧⡌⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢼⡿⢉⠛⣦⠀⠀⠀⠸⠿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⣿⢷⣿⣦⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣾⠿⣿⡄⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⣾⠀⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢸⣦⡈⣷⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠘⣿⡟⢾⠛⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠷⠟⠁⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠠⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣾⠟⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⢀⣤⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠿⣦⣄⡀⠀⠀⠀⠀⢀⣦⢀⣼⠏⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⢘⣿⣿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣼⣷⡀⠀⠈⠙⠻⠿⠿⣦⣤⣾⣧⣸⣿⡀⠀⠀⠀⠀⠀⠸⣿⣤⣤⡾⠿⠿⠛⠉⠁⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⣸⠇⠈⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠙⢧⡀
⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠙⠛⠢⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣤⣤⣤⠄⠛⠛⠁⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⡜⠻⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢻⣆
⠀⠀⠀⢠⣧⠀⠀⠈⠱⠶⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠠⠶⠋⠁⠀⢀⣿⡇
⠀⠀⠀⠈⠿⠷⣄⡀⠀⠀⠈⠉⠉⠉⠒⠒⠶⠶⠶⠶⠶⠆⠀⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠚⠋⠉⠉⠉⠀⠀⠀⣀⡴⠟⡙⠃
⠀⠀⠀⠀⠀⠐⢀⠸⠶⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠶⠾⣍⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⢉⣤⣶⡖⠛⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠻⠿⢧⣤⣬⠉⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠃⡄⠀⠀⢅⠞⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠈⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⠀⠀⣀⣀⠀⣀⠀⢀⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣚⠯⠭⢭⣭⠥⠭⠽⣖⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣿⣿⣿⣿⣿⡿⣿⣽⢁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⣿⣿⣿⣿⣿⡟⡌⣟⣒⣉⣓⣺⣧⡟⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢿⣿⣿⣿⣿⣿⣿⣜⠻⣿⠿⣿⠏⢛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡧⣻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⡿⢻⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡽⣿⣿⡿⠟⣫⠇⠘⠀⠹⢬⡙⢿⣿⡿⡿⣶⣿⣿⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠻⠿⢿⣿⣿⣷⣿⠟⠁⣉⣩⣤⣴⣔⠲⢦⣉⠂⢹⣷⣥⣿⣿⣿⣿⡿⠿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠤⢤⣀⣀⢻⣟⣿⠏⠀⠈⣭⡴⠶⣶⣿⢓⡲⢦⠄⠀⠹⢿⣿⣿⡿⠋⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠈⠉⠉⣇⠀⠀⢀⣴⡖⠒⣿⣟⠰⠶⣶⣆⠀⠠⡾⠉⠉⠉⠉⠉⠉⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣵⣾⣿⣿⡀⠀⠙⠇⠀⢀⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠒⠛⠛⠉⢿⣿⣿⣿⣿⣿⣿⣦⣄⣠⣴⣿⣿⣿⣿⣿⣿⡿⠀⠉⠐⠲⠦⠤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠁⠀⢀⡠⠖⠛⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣿⣿⣿⣦⣤⣤⣀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣯⣽⣿⣿⣯⣙⣿⣿⣧⢴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣈⢛⡿⠷⣿⠿⢛⣏⣽⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠴⠐⢛⣙⣽⠖⢠⠻⣄⠀⣀⡐⠤⠖⠂⠄⠴⠤⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣘⣿⣛⡓⣛⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⢀⣠⡤⠤⠲⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢾⣿⣿⣿⣿⣿⣿⢿⡿⢛⡿⠟⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠈⠙⢿⠿⢿⠿⣿⣄⣴⠋⠀⠀⠀⠀⢀⣿⠇⠀⣠⣶⣿⣷⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠿⠋⣰⣿⠇⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⢀⢀⣿⣿⠀⠀⠀⠀⠻⣿⣷⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⢀⡀⣴⣿⢆⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣄⠀⠈⣿⣿⣇⠀⢸⣿⣴⣿⡿⠀⠀⠽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠎⢩⣟⣿⣿⣿⣷⡄⠀⠸⣿⣿⠉⠈⠹⣿⣿⡄⠀⠈⢹⡿⢀⣤⣽⣿⣷⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠿⢿⣿⣿⣿⣟⠷⡆⢀⠉⠉⠀⠀⢀⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⠟⠀⠀⠀⠺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣧⣾⡿⣿⣮⣍⠹⣾⣎⠳⡀⠀⣴⣿⣿⠋⠀⠀⠀⠀⣹⡿⣿⣿⣧⠀⠀⠀⢈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠊⢇⣭⣷⡧⣤⢤⣼⣿⣿⣦⣸⠂⠛⠻⢿⣿⡆⠀⠀⠀⠉⢻⣿⣿⣿⣧⡀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢨⣽⠭⣟⠻⣾⣟⡿⢿⡿⠋⠀⠀⠀⣿⣿⢻⠒⠒⠒⠐⠘⠛⠻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡟⡻⣿⡟⠸⠛⡌⠀⠀⠙⣶⣤⣠⣾⣿⣿⣿⣤⣤⣤⢄⡄⠀⠀⠨⡿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣌⢻⣿⣁⠀⠁⠀⠀⠀⠛⢿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣷⣷⣾⣷⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠃⠉⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣯⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⣿⠛⠿⠿⢿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣤⣤⣤⣤⣤⣤⣤⣄⣠⣄⣠⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠬⣉⠛⠿⢿⣿⣿⠯⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠩⠲⣶⣶⣠⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠀⠐⠒⠒⠒⠒⠂⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⢠⠀⠀⠀⠀⠀⠀⢤⠀⡔⢂⣀⠒⣀⡞⢢⣰⢱⣀⡎⣆⡔⢢⣠⢲⢀⠤⠀⠀⠀⠀⠀⠐⡀⠈⠲⣄
⠀⠀⠀⠀⠀⡠⠊⢹⠠⠀⡆⠀⠀⢀⣠⣤⣶⣀⣿⣧⣸⣿⣤⣿⣧⣼⣿⣼⣿⣇⣿⣷⣼⣿⣸⣿⣀⣷⡦⣄⣀⣀⣀⡃⠀⠀⡏⠓⢄
⠀⠀⠀⣠⠊⠀⣰⠃⠀⠀⡿⠟⠋⠉⠉⠉⠛⠿⠏⠛⠟⠉⠛⠁⠙⠋⠈⠛⠁⠙⠉⠙⠋⠙⠟⠛⠿⠋⠀⠀⠉⠉⠛⢷⠀⠀⠸⣦⡀⠱⡄
⠀⠀⠰⠁⣠⣺⣑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡙⠐⠻⢤⡈⡆
⠀⠀⣟⠉⠀⠉⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠙⢁
⠀⠀⠘⣷⣤⣤⣠⢼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡇⣀⣀⣠⣶⠋
⠀⠀⠀⠈⢿⣿⣿⢸⣤⣆⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⣾⢿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠙⠿⣾⣿⣿⣿⢻⣀⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣄⣴⣴⣧⣿⣿⣿⣼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣏⠿⣿⣿⣿⣧⣷⣶⣴⣰⣠⣤⡄⡀⡄⣄⣀⢀⢠⢠⢀⣠⣦⣦⣷⣾⣿⣿⣿⣿⠿⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣭⣛⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣻⣭⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢻⢶⠾⣭⣭⣝⣛⣛⣛⣛⣛⣛⣛⣛⣩⣭⣭⠶⠶⠛⠻⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠉⠩⠭⠭⠭⠭⠍⠙⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⢀⡠⢤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠛⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡄⠀⠉⠓⢤⠀⠀⠉⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠄⠀⠀⠘⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢺⣷⠆⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠀⢻⣿⣵⣦⡀⠀⠀⠀⠀⠀⠀⠀⢨⣆⠀⠀⠀⢀⣀⣀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠛⠉⠀⠉⠁⠀⠀⠀⣀⠀⠀⠀⢀⣤⡶⠆⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⣿⣿⣶⡆⣀⠀⣤⠖⠒⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⠀⠀⢀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⣄⠀⢸⣿⣿⡏⢻⡆⣀⠫⡄⠃⠈⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⢀⣼⣟⠙⢷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢻⠀⠀⠹⣿⣿⡟⢻⣿⡄⢡⡀⢠⣿⣴⣷⡀⠀⢀⡀⢀⠀⢀⣀⡀⣦⡼⢇⣰⣿⠇⡽⠋⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠈⣷⠀⠀⠙⠿⣿⣿⣿⠃⠘⢧⠘⣿⣿⣿⣿⣶⣿⣿⡟⢶⣿⠟⢷⣞⢳⣾⣿⣿⠀⠁⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢹⣦⠀⠀⠀⠈⠻⣿⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣝⣷⡼⢿⡾⣺⣿⣿⡿⠋⡤⠂⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠘⢷⣄⠀⠸⠂⣽⠀⠀⠀⠀⢻⣿⢿⡆⣿⣿⣿⣿⣷⣾⠿⢿⡿⠏⢰⠎⠁⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⣴⣽⠷⣦⣄⡟⠀⠀⢸⡄⠘⣿⣄⠻⠛⠋⠙⠋⣉⣤⡴⠋⠀⠀⠀⠀⠀⣠⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢘⡇⠈⠲⣄⣻⠃⠀⠀⢸⣷⡀⠸⣟⠛⢷⣶⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢠⡀⢠⣽⡏⠀⠀⠀⣿⡇⢳⡀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⣿⠇⠀⠀⠀⣿⠁⠈⢣⡀⠻⡃⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢀⣆⠀⠀⠀⠀⣿⠀⠀⠈⣇⠀⢻⣶⢫⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢸⡿⠀⠀⠀⠀⣿⠀⠀⠀⢿⡇⠀⢻⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⡇⢸⠆⠀⡄⢸⡇⠀⠀⠀⠻⢆⡈⠻⡏⠁⠀⢀⠀⠀⠀⠀⠀⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠐⠇⠘⠀⠸⠃⢸⠃⣀⠀⠀⠺⣾⣷⣀⠻⣦⣾⡿⠃⠀⠀⠀⠀⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢰⡆⠀⠀⠀⢸⠀⣿⠀⠀⠀⢾⣽⠛⣦⠙⣿⡄⠀⠀⠀⠀⢰⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⡇⠀⠀⠃⢸⠀⢿⠀⠀⣴⣿⣧⣘⣷⣆⠈⢷⡄⠈⣠⠆⣠⠁⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡇⠀⠐⠀⢸⠆⢸⡄⠀⠙⣸⡿⠉⣿⡘⣷⡀⢻⣾⠁⢰⠟⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠇⠀⣀⠀⢸⠃⠘⡟⠀⠀⠻⣷⡄⢻⣟⣿⣇⠀⠹⡆⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⠀⢛⠀⠈⣯⣤⡏⠀⠀⢰⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⠃⠀⠈⠃⠀⠀⣿⡇⠀⠘⢻⣟⠄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠈⠀⢸⡇⠀⠀⢸⡀⠀⠸⠿⠀⠀⣿⠛⣷⠆⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⠀⣿⠀⢸⠃⠀⠀⠘⠃⠀⠀⣾⣷⠀⠸⠟⣿⠇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠻⣄⠀⠀⠻⠇⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡆⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢉⡇⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⢈⣼⣧⡀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡿⠋⠋⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⣴⠟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⡞⠁⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⢀⣴⠋⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠉⠻⢶⣤⣤⣤⣼⠀⣾⠁⠀⠀⠀⠀⣾⣇⣠⣤⣤⠾⠛⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠑⠲⠂⠀⢤⠀⣠⣤⣤⣀⣤⣤⠤⠄⠴⠶⠂⠉⠉⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠐⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠒⠉⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡦⢀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠀⠀⠀⣀⡠⢜⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢡⡴⠶⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡐⠒⠶⣦⡝⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣦⣤⣾⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡷⢾⣿⣿⣿⣿⣯⡻⣿⣏⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠉⠃⠈⢙⣿⣿⣿⡟⢻⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⢹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠇⠀⠀⠀⣰⣾⣿⣿⢿⣿⠏⠀⠘⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣿⣿⣿⣍⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣻⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⣾⣿⠿⣫⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⡿⢛⣡⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠉⣶⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣁⣀⣀⣀⣀⣀⣀⣙⣁⣀⣀⣀⣀⣀



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣧⣈⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⣉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⠉⠉⠙⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣠⣾⢟⠻⢿⣿⣣⠴⠃
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⠀⣾⣿⣏⠘⣆⠈⢿⣇
⠀⢸⡿⠛⠿⠷⢶⣶⣶⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠛⠿⠟⠛⢻⡄⠹⣦⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣶⡾⠿⠿⠿⢷⣶⡄
⠀⣾⡇⣾⣿⣿⣷⣶⣶⣮⣭⡍⠋⣝⣛⣛⡛⠻⠿⠿⠿⠿⠿⠿⠿⠷⠾⡿⠿⠿⠷⠾⠿⣶⣶⣾⠿⠿⠿⠿⠿⣿⣿⣿⠛⣛⣫⣍⢩⡿⢋⣵⣾⣿⣿⣿⡇⣿⠃
⢠⣿⠀⣋⣭⢹⣿⣿⡿⠛⠻⢡⡆⣿⣿⣿⠇⡇⣾⡿⢿⣿⣿⣿⣿⣿⡇⣿⡦⢸⣿⣿⣿⡜⣿⡿⣷⡘⣿⣿⣿⣎⠻⣿⣷⢸⣿⡇⣿⢡⣿⣿⡟⢥⣶⣬⣁⣿⡃
⣸⣿⠿⢻⣿⢸⣿⣿⡇⢹⣧⣼⡇⣿⣿⣿⡇⣤⣴⣶⢸⣿⣿⣿⢰⣤⣸⣿⢃⣿⡟⣿⣿⣷⡹⣷⣿⡇⣿⣿⣿⣿⣷⡝⢿⢸⣿⡇⣿⠸⣿⣿⣿⣦⣝⠻⣿⣿⡇
⠋⠁⠀⢸⣿⢸⣿⣿⡇⢸⡟⣿⡇⣿⣿⣿⡇⣿⢹⣿⢸⣿⣿⣿⢸⣿⣿⠇⣾⣿⠀⢸⣿⣿⣧⢻⣿⡇⣿⣿⠙⣿⣿⣿⣦⢸⣿⡇⣿⣦⡙⢿⣿⣿⣿⣿⣮⢻⣧
⠀⠀⠀⢸⣿⢸⣿⣿⡇⢸⡇⣿⡇⣿⣿⣿⡇⣿⢸⣿⢸⣿⣿⣿⢸⣿⡏⣼⣿⠟⠛⠛⢿⣿⣿⣇⢿⡇⣿⣿⢰⣌⠻⣿⣿⣿⣿⡇⣿⣿⣿⣦⣌⠛⣿⣿⣿⡇⣿
⠀⠀⠀⢸⣿⢸⣿⣿⣇⢸⡇⣿⡇⣿⣿⣿⡇⣿⣸⣿⢸⣿⣿⣿⢸⡟⣰⣿⡇⣾⣿⣿⠄⣿⣿⣿⣜⢃⣿⣿⢸⣿⣦⡙⢿⣿⣿⡇⣯⢰⣭⡛⠿⠗⣸⣿⣿⡇⣿
⠀⠀⠀⢸⡿⢸⣿⣿⡿⠸⣷⣿⡁⠛⣛⣛⣃⣙⣿⣷⣬⣭⣭⣭⣤⣤⣭⣥⣥⣿⣿⣥⣤⣭⣭⣭⣭⣬⣭⣭⡼⠁⠙⢿⣦⣙⠻⠇⣿⡀⠻⢿⣿⣿⣿⣿⠟⣱⡿
⠀⠀⠀⠀⠳⣤⣤⣶⣶⠶⠿⠿⠿⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⢹⣏⠉⠉⢻⡏⠉⠙⣿⡁⠀⠀⠀⠉⠛⠿⠷⣿⠿⠿⣶⣤⣬⣭⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⢿⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠘⡇⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢿⣿⣿⠃⠀⠈⣿⣿⡿⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡌⠻⣿⣷⣤⣶⣿⠿⢁⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡛⣿⡇⠀⠀⠉⠉⠉⠁⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⢿⣿⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣸⣿⡄⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠃⠀⠀⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⠀⠀⢰⣿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⣿⣿⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠻⠋⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⣠⠖⠋⠁⠀⠀⠀⠈⠉⠙⠛⢿⣿⡿⠟⠋⠉⠀⠀⠀⠈⠑⠢⣄⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⣀⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣮⣧⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠒⠶⠤⣤⣀⣀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⢀⣀⣠⡤⠴⠖⠿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣸⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡄
⠀⠀⠀⡟⢿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⡇⠈⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃
⠀⠀⠀⣧⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢻⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠸⡇⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⢻⡀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠁
⠀⠀⠀⠀⠈⢷⡀⠀⢹⣤⣀⣀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⣀⣠⣴⡃⠀⢠⠃
⠀⠀⠀⠀⠀⠈⢷⡀⢸⣿⣿⣿⣿⣿⣶⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣶⣾⣿⣿⣿⣿⡇⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠻⣌⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⠟⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠙⠿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠤⣤⣀⣀⣀⡀⠀⠀⢀⣀⣀⣠⡤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⣶⣖⢳⡶⢶⢶⠶⢶⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣿⣛⣛⣛⠛⢛⣲⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠟⠛⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⠟⠁⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣷⣶⣶⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⡀⠀⠀⠀⢀⣠⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄⠀⠀⡰⣿
⠀⠀⠀⠀⣿⣷⣀⡠⠖⠉⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⠚⠁⣿
⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⡟
⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣸⠇
⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢠⡟
⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠈⢿⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠁⠁⠀⠀⠸⡇⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⠳⢄⡀⠀⠀⠀⠀⢀⣠⣾⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢰⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠲⢤⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣭⣭⣥⣤⡽⠿⠿⠟⠛⠉⠁


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠛⠉⠉⠉⠉⠙⠓⣲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣧⣄⣀⣀⣤⣤⣤⣤⣄⣴⣾⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠘⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉⠉⠉⠻⣄⠘⣇
⠀⠀⠀⠀⠀⠀⠀⣼⢀⣿⠛⠉⠁⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠹⣦⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡏⣸⠇⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣯⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣤⣴⠶⠞⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠟⠋⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⣠⣴⣿⣤⣤⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣴⢖⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡾⠿⠟⠛⠛⠿⠿⣿⣽⡿⠥⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢼⡶⠖⠋⣉⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠉⠲⢬⡙⠻⣧⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⣾⣿⡿⣿⣻⠉⣹⡿⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⠹⢇⣂⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣛⡽⠾⠿⠍⣾⠷⠟⠁⠀⣀⠔⠀⠀⠀⠀⠀⠀⠋⠀⠀⠉⠻⠏⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠕⠊⢉⣠⡔⠚⠊⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⠴⣣⡶⢊⣽⢾⣡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣩⠞⠁⠀⠊⠁⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⣴⠾⠻⠛⠛⠛⢻⡟⣧⠰⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⢸⣿⣇⠐⠂⢂⡙⠀⠀⢸⠑⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠁⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⢊⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⠁⠀⠀⣸⣿⡀⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢤⠀⠀⠀⢸⣿⡇⢸⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⣿⡇⢸⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠓⠀⠀⠀⠀⠀⠀⢀⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡁⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣮⣽⣓⠂⢀⣄⣀⣀⣀⣈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠁⠘⢟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⢲⡦⣀⠀⠀⢙⣿⣿⣿⣝⣿⣿⣿⣿⣩⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣼⡟⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⢹⠏⠀⠉⠛⠿⡿⣿⣿⡿⣿⡿⠟⣿⣿⣿⣿⡆⢰⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣷⣾⣟⡹⠏⢉⣩⡤⣩⠍⠷⠲⠤⠀⠀⠉⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⣙⠻⠿⠿⠿⠿⢿⣿⣿⣷⣾
⠀⠀⠀⣰⣼⠶⠋⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣀⡀⠀⠀⠀⠀⣿⣿⣿⢦⠀⠀⠀⠀⠀⠀⠙⠻⠿⣆⡀
⠀⠀⠐⠿⠃⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⣄⠀⢻⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⠈⢿⣿⣾⣵⠀⠀⠀⠀⢀⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠐⠐⠚⠓⠒⠢⠏⠁⢹⣦⣾⠗⠓⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠰⠿⠿⠟⠀⠀⠀⠀⠘⣿⣯⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⣀⡄⣠⣤⣄⡀
⠀⢀⣼⣍⠈⢿⣯⣟⣿⡄
⠀⣿⡿⣿⡆⣶⡍⠙⠋⣵
⠀⠹⣿⡿⠁⣡⣶⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⣦⣶⣰⡆⢦⣦⣦⢲⣴⣦⡆⢰⢢⢠⡒⣆⣴⡶
⠀⠀⢲⣾⡷⣿⣿⣿⡿⠁⠀⠀⠀⢀⣀⣀⡀⢀⠀⠀⡀⠀⣁⣁⠁⣁⣈⡈⢀⠉⠀⢈⡉⠉⠁⢀⣉⡈⠁⢀⣀⡀⠀⡀⠀⣀⣀
⠀⠀⠸⢿⣿⣝⠃⠛⠀⠀⠀⠀⠀⣿⢀⣨⠟⣼⠀⠀⣷⠸⢦⣄⡀⠷⣤⣀⢸⡇⢠⠎⢻⡀⠀⠋⢀⣿⢰⡏⠀⢻⡄⣿⢘⡧⢼
⠀⠀⠀⠈⣷⡝⣧⡀⠀⠀⠀⠀⠀⠻⠉⠙⠆⠙⠦⠴⠋⠠⠤⠼⠃⠤⠤⠟⠸⠇⠿⠖⠊⠷⠀⠴⠯⠤⠈⠳⠤⠞⠁⠟⠘⠦⠼
⠀⠀⠀⠀⠿⣾⠘⣷
⠀⠀⠀⣇⣀⡴⣵⣿⡇
⠀⠀⠀⠿⣵⠾⣣⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣤⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⠙⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⢀⣠⠴⠚⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠃⠂⠤⣀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⣤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠉⠛⠒⠶⢦⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣤⡴⠶⠚⠛⠉⠉⠈⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷
⠀⠀⠀⢸⡛⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿
⠀⠀⠀⢸⠀⠈⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠏
⠀⠀⠀⢸⡆⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠸⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠸⡆⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⡀
⠀⠀⠀⠀⠀⢻⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⣷⣦⣤⣄⣀⡀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣀⣠⣤⣶⣿⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⣿⣿⣿⣿⣿⣿⡇⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠦⢤⣤⣤⣤⣤⣤⡤⠴⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⠤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠒⠋⠉

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

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣦⣄⠠⢤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⣄⣈⣿⣿⣿⣷⣤⣀⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣉⣉⣉⣉⣙⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⣼⣿⣿⣿⣿⣿⣿⣿⠟⣡⣶⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⠋⣼⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣧⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⡟⢰⣿⣧⣠⣤⣤⡀⠀⠀⣤⣤⣄⣸⣿⡇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣿⣿⣇⢸⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⡇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⡈⢿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⡿⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣌⠻⣿⣿⣿⣿⣇⣿⣿⣿⣿⠟⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣉⡛⠛⠿⠟⠛⣋⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⠟⠉⠉⠐⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⣿⡿⠟⠛⠋⠉⠂⢄⠀⠀⠀⠀⠀⠉⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠑⣦⣤⣤⣤⣀⡜⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⣴⣷⣄⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⡮⢄⣰⣿⣿⣿⣿⣿⣦⠤⠒⠂⠉⠁⠈⠻⣿⣿⣿⣿⡿⠟⠆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠋⠀⠀⠀⠐⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠹⠛⠛⠛⠛⠦⡀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡇⠀⠀⠀⠀⠀⠈⠢⠀⠀⣀⣠⣐⣾⣦⡀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⢘⣟⣛⣿⣟⣛⣟⣛⠀⠚⣩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⣀⡀⢼⣿⣿⣿⣿⣿⠿⣋⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⠤⠬⠭⠛⠓⠉

⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠉⠉⠉⠈⠉⠉⠉⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠈⠉⠀⠉⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠈
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢆⢀
⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣴⣿⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣢⠀⠛⡿⠋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢨⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠻⠿⣤⣤⣗⣫⡜
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⡿⣶⢂⡤⡀⣤⡶⣖⣦⠀⡀⠀⠀⠀⠀⠀⠛⠶⣿⡏⠹⠁
⠀⠀⠀⠀⠀⢀⣾⣏⢀⢲⣄⣠⣴⠀⠀⠀⠀⠀⠀⠉⠞⠁⣼⣷⠿⢾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐⣀⠘⡀⡀
⡆⠀⠀⠀⢰⡟⣿⣿⣿⡿⠿⠛⠃⠀⠀⠀⠀⠀⠀⢐⣫⠟⣇⡈⠉⢀⠙⣟⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⠖⢇⡃
⡇⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠙⣷⣻⣇⣴⣿⣫⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⢟⡀
⠇⠀⠀⠀⢠⠏⠀⠉⢻⣳⡀⠀⠀⠀⠀⠀⠀⢰⠋⠓⠚⣿⣿⠃⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠐⣄⣀⠀⠀⠱⣷⡀⠀⠀⠀⠀⠀⢈⠀⣾⡄⢸⡇⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠐⢁⠻⡶⣦
⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⢣⢳⠀⠀⠀⠀⠀⠀⢘⣧⣀⣸⠃⠀⠀⠀⠀⠀⠈⢷⣤⠀⠀⠀⠀⠀⢈⣿⠋⠙⠳
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⠚⣷⡇⠀⠀⠀⠀⠀⠈⢿⣀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠁⠠⠤
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠅⠈⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡀⠀⠀⠑⠆⠀⣀
⡆⠀⠀⠀⠀⠀⠀⢿⣿⠐⢦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⣄⣀⠀⠦⣄⠀⢀⣤⣦⣄⠀⠀⠀⠙⠂⠀⠀⠀⠀⠈⠙⠳⣦
⠃⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣑⣂⡿⣷⣾⣿⢹⣿⡀⠀⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⡀⠓⣯⢦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠆⠈⢻⡛⢟⠣⢿⣭⢬⣼⠛⠶⠄
⠀⠀⠀⠀⠀⠿⣤⡀⠱⡝⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⠆⠠⠿⢟⣳⠄⡻⣛⡏⢰⣤⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠹⣷⡄⢓⢷⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣿⣀⠀⡀⣸⣇⣀⡶⠋⠑⠈⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣧⣍⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠙⠋⠁⡄⢠⣄⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣿⡿⠟⠁
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⣿⣿⢗⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠑⠀⢻⡟⠀⠊⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠈⢿⣿⣿⣿⣿⣿⣿⡿⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠈⠋⠀⣿⡿⠀⠙⠁⠀⠌⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⣻⣇⠀⠀⠀⠀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⡀⠈⣿⣿⠁⢀⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣿⢹⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣿⠀⣿⠀⠈⢧⠿⣼⠃⠀⠀⠹⠎⠀⠿⠨⠇⢸⡧⣟⠀⡿⢯
⡖⢒⣒⠒⠒⢒⡒⠒⢰⠒⠒⢲⡖⠒⣒⠒⠒⢒⡒⠒⢢⡖⠒⠒⣒⣒⠒⠒⢲⣴⠒⠒⢲⡀⢰⠒⠒⡆⡖⠒⠒⣒⣒⠒⠒
⠓⠋⢹⠀⠀⣾⠙⠒⢸⠀⠀⢸⠓⠚⢹⠀⠀⢸⠙⠓⢨⡇⠀⢸⡉⠉⡁⠀⢸⣿⠀⠀⠀⠱⣾⠀⠀⡇⡇⠀⠘⠧⣬⣶⣒
⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⢸⡇⠀⢀⣉⣉⡀⠀⢸⣿⠀⠀⢸⡄⠉⠀⠀⡇⣈⣐⣒⠀⢤⡀⠉
⠀⠀⢸⣀⣀⣻⠀⠀⢸⣀⣀⣸⠀⠀⢸⣀⣀⣸⠀⠀⢸⣇⣀⣸⠀⠀⣇⣀⣸⣿⣀⣀⣺⠘⣆⣀⣀⡇⡇⠀⠈⠓⠚⠁
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠉⠉⠁⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡜⣿⣯⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⢿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⣦⠂⠆⣦⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠉⠀⠀⠀⠰⠿⢿⣿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⢀⣤⣾⣿⣿⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣭⠛⠋⠁⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠿⠿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠶⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠏⢰⣿⡿⡿⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣌⠉⠀⠘⣿⣿⣷⠈⢰⠃⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⠀⠈⠻⣿⣿⣯⣶⣿⣿⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣾⣿⣿⡇⠀⢶⣄⡀⠉⠛⠛⠛⠛⠋⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡋⢹⣿⣿⡇⠀⢸⡿⠛⠃⠀⠀⠀⠀⠛⢥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠁⠀⠉⢀⣠⣶⣿⣿⣿⣿⣶⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁⠀⠀⣰⣿⣿⣟⣿⠟⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣤⠶⡇⢰⣿⣿⣿⣏⣽⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠹⣿⣿⠉⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⣹⣿⣿⣦⡀⠉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢋⣀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠙⣷⡈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢻⣿⠇⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠜⠁⡄⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⣤⡾⢁⡆⠈⡯⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣾⡿⠋⣠⠾⡇⢠⡷⢾⡇⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣠⣴⠚⣯⠟⢁⣼⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠁⠐⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⡀⣀⠀⣀⢀⡀⡀⣀⣀⢀⣀⣀⡀⢀⣀⣀⠀⣀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢸⢸⡗⣸⣿⡆⢹⡿⣷⠿⣇⡽⢿⢺⣽⣇⣸⣇⡽⠀⢯⣀⢿⣨⠿⡗⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⡇⠉⠉⣻⡆⢸⡇⠀⠀⣿⠀⢾⡉⠀⠉⠀⢰⣉⠀⠉⠀⠨⡇⠀⠀⣠⠛⣷⠀⠀⠀⠰⠉⠈⢻⡆⢠⡾⠁⠈⢳⡄⠐⣿⠀⢸⠁⠈⣷
⠀⠀⢸⡇⡤⣾⠋⠀⣿⡁⠀⠀⢸⡆⠈⠙⠻⣶⡄⠀⠙⠻⢶⣄⠸⡇⠀⣼⠁⠀⠸⣆⠀⠀⠀⢀⣠⠟⠁⢸⡇⠀⠀⢸⡇⠀⣿⠀⣤⠃⠘⣧⡀
⠀⠀⠸⠇⠀⠈⠳⠄⠘⠣⠤⠤⠛⠀⠠⠤⠤⠾⠃⠠⠤⠤⠼⠃⠐⠇⠐⠿⠶⠊⠀⠹⠆⠀⠴⠿⠤⠤⠄⠈⠳⠤⠤⠞⠁⠀⠟⠀⠻⠤⠤⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣻⡏⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⣦⣤⣄⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣿⣿⠿⣿⣿⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢺⢿⣯⣿⣷⡻⡿⠣⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠄⠿⣚⡳⠀⢿⡁⢀⡠⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣽⣿⣉⣿⣿⣿⣿⡿⠀⠀⠀⣠⡴⠖⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠻⢿⣿⣿⣿⡟⢟⡼⠋⠀⠀⢠⡾⠋⢀⣠⣄⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠉⠉⢉⠹⠋⠀⠀⠀⠀⣾⠇⠀⠿⢋⣿⠇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢠⣿⡄⠀⢀⠘⢿⣇⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠤⣰⣿⣧⡀⠀⢠⣾⡿⠿⣿⣷⡄⠀⠿⣷⠀⠻⣷⣿⣋⣀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⣾⣿⣿⣿⣿⣄⠈⠟⠃⠀⢈⣿⣇⠀⠀⣿⣶⣿⡿⠛⠋⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣤⣿⣿⣭⣙⣾⣌⢆⠀⣤⣾⣟⠁⠀⠀⣹⣿⣿⣷⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⢣⣿⣷⣶⢲⣿⣿⣧⠇⠉⠛⣿⣇⡀⠀⠀⢻⣿⣿⣷⣄⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡾⣧⣯⠩⣿⠋⠙⢯⣀⠀⣸⣿⣦⡀⣀⠀⡀⠈⠙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣧⡻⣷⡂⠐⠀⠀⠘⢿⣿⣿⣿⠿⣿⣿⣷⣷⣶⣤⣯⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠟⠈⢿⠀⠀⠀⠀⠀⠀⠀⠉⠡⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡿⠉⠙⠛⠻⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠈⡛⠿⣿⣿⣿⡿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠲⠤⣍⣉⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢋⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣿⡟⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢻⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢳⣶⣾⣶⡿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⢠⣤⣤⣤⣤⣿⠀⠀⠀⣨⣭⠄⠈⢻⣤⣤⣤⣤⠀⣰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⢻⣧⣬⣅⣙⠳⢦⣄⣿⣡⡴⠞⣋⣡⣤⣼⠏⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣹⡿⡿⢿⢷⣦⡾⢿⡵⢶⢿⢿⡿⣿⢃⣾⠏
⠀⠀⠀⠀⠀⠀⠀⣿⠳⠶⠶⣦⣤⣤⣬⣷⣇⣷⣻⣸⣿⣡⣌⣧⣿⣻⣠⣜⣧⣾⣥⣤⣤⣴⠶⠶⢾⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣶⡇⠀⣶⣼⡇⠀⣿⣩⡍⠉⣭⣹⣿⠏⡉⠹⣿⣿⡍⠉⠹⣷⠀⣿⠀⠺⠷⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠀⣿⣿⡇⠀⣿⣿⡇⠀⣿⣿⡟⢠⣿⡀⠹⣿⡇⣿⣆⠙⢀⣿⠻⣶⠆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠶⠾⠿⠷⠶⠿⠿⠷⠶⣿⡿⣶⣾⣿⣷⡶⠿⠶⠾⠿⠶⠾⠿⠿⢶⠶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡜⠿⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡾⠛⣦⣤⣄⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠛⠉⠉⠁⠀⠈⠉⠙⠛⠛⠻⠂