Rope Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣴⢧⢾⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠈⠑⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⢲⣄⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡇⠀⢸⣆⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠹⣀⡸⠃⠀⠀⠀⣠⢤⡶⠋⣩⠟⠒⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢷⡀⠀⡏⠀⠀⠀⠀⡜⠁⣾⠀⢀⡇⠀⢀⣀⣤⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⢦⡀⠉⢫⠀⠀⠀⡠⠻⣆⠀⠘⡶⠊⠙⠒⡋⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠙⠶⡌⠀⠀⢀⣧⠀⠈⠓⡞⠀⠀⠀⠀⢙⠛⠛⢦⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢦⣀⠀⢸⠀⠀⠌⠈⠛⠶⣴⠃⠀⠀⠀⠀⠸⣀⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠛⠻⡀⠀⡱⢦⣄⣀⣸⠀⠀⠀⠀⠀⠰⠉⠛⢦⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠦⣤⣤⡇⠀⢣⡀⠈⠉⢹⠀⠀⠀⠀⠀⢸⠦⣤⡀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠙⡄⢸⠉⠛⠛⠛⡀⠀⠀⠀⠀⢊⡀⠀⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠛⠛⢳⡀⢳⣤⣤⣤⡇⠀⠀⠀⠀⣎⣉⠳⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠦⠤⢼⡃⠘⣦⣤⣀⣱⠀⠀⠀⢠⠉⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠀⠀⡷⠀⢣⠤⠠⠙⡆⠀⠀⠈⠟⠲⢤⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠓⢣⠀⠀⡆⠀⠀⡇⠀⠀⠀⢇⣀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢦⣀⠀⡸⠀⠀⡧⠤⠤⢧⠀⠀⠀⠜⠉⠙⢦⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⠙⡇⠀⠀⢇⡀⠀⡶⠇⠀⠀⡜⠷⣄⡈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡋⠛⠦⣤⠇⠀⠐⠎⠙⠛⠁⠀⠀⢀⢧⡀⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠙⣦⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠈⠛⠲⡇⠀⠀⠀⠀⠀⠀⠀⣠⠻⣆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⣠⣧⡀⠀⠀⠀⠀⠀⡠⣧⠀⠘⣧⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠉⠀⠀⣸⡤⣄⡤⠒⢺⠀⢸⡇⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠶⠛⠉⢀⡿⠀⢀⡟⠀⡸⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⠠⢾⣁⡠⠛⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⢒⣊⣁⣌⣩⡡⣉⣈⢂⠲⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢊⣁⠤⠒⢩⠁⠤⠀⠀⠀⠀⠀⠀⠄⢉⠓⢤⣉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠔⣃⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠲⡌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⣿⣾⢧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠘⡆⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠔⣃⠌⠀⣄⣌⣌⠛⠤⣠⣤⡴⡄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⠠
⠀⢠⢀⣀⠀⠀⠀⣾⡜⠁⠀⣀⢽⠃⣿⣵⣶⡈⠹⢿⣟⡖⠦⡙⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠸⢂
⢌⡃⠆⠧⠥⢩⣉⣌⣩⡣⡵⠇⢸⢀⡟⠂⠋⠁⠀⠈⠹⢦⣀⠉⣇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠏⡰
⠀⠀⠀⠈⠀⠀⠂⢤⢴⠀⠈⠩⢾⣎⣷⠀⠀⣀⣀⣀⣀⣙⣫⠅⣟⠘⣤⣀⣀⣀⣀⡀⡠⠔⢊⡥⠊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠕⠲⠲⡚⠛⢋⣉⡉⠉⠀⠀⠛⠓⡿⠀⡗⣨⣷⣷⠧⠤⠤⠴⠜⠊⠅⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣟⡓⠶⠖⠒⢃⡤⠛⡰⠋⠉⠉⠁⠁⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢉⣙⣙⣉⡡⠀⠊

⠀⠀⠀⠀⢰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠂
⠀⠀⢠⣀⡠⠼⠛⠶⣤⣀⡀⠀⠀⣠⡴⢚⡏⠉⠉⠉⠉⠉⢉⣹⠷⠶⣿⣉⡽⠁⠀⠀⠀⢀⡤⠞⠛⠳⢦⣀⣀⣤⢶⠏⠁⢀⣠⠄
⠀⠀⠀⠙⠳⢦⣄⠀⠀⢉⣙⠷⠾⢧⣤⣼⣄⣴⣶⣶⠶⢶⣯⣀⢀⠀⠀⠉⢛⣳⠿⢿⣿⣿⣿⢷⣦⡀⠀⠙⢯⡴⣃⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠺⢭⣄⣀⡀⠀⣀⠔⠉⠀⠀⢀⡴⢋⣠⡽⠟⠛⠳⠶⢯⣄⣀⢀⡀⢀⡴⢋⣀⣹⣯⡉⠒⢿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡉⠉⣿⠛⠚⠓⠒⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠘⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠹⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣰⠏⢱⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⡤⠤⠿⠶⠶⣤⡤⢤⣤⣀⡀⠀⠀⠀⠀⢀⣠⠶⠛⡹⠋⠀⠀⢀⣤⠟⠁⠀⢸⣾⣅
⠀⠀⠀⣀⣤⠶⠞⢋⠝⠉⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠉⣉⡻⠶⣤⡴⠋⠁⠀⣰⣷⣶⣾⠿⠋⡏⠀⢀⣴⡏⠀⢈⣙⣦⣄
⠀⠀⠘⠉⠀⠀⡴⢁⣀⣤⡤⠶⠶⣟⠛⠛⠛⠳⢶⣶⣭⣥⠴⢞⡏⠀⣀⣤⡾⠟⠋⠉⠁⢀⡜⣀⡴⠛⠉⠙⠻⣭⣁⠀⠉⠓
⠀⠀⠀⠀⠠⠾⠛⠋⠁⠀⠀⠀⠀⠈⠳⢶⣖⡊⠉⠀⠀⠀⢀⣾⡴⣾⣟⣁⣠⣀⣀⣤⡴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁

⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠶⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠶⠶⠶⠶⠶⠶⠶⠶⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶

⠤⣤⢤⡤⢤⠤⣤⢤⡤⢤⡤⣤⢤⡤⢤⡤⣤⠤⣤⢤⡤⣤⠤⣤⢤⡤⢤⠤⣤⢤⡤⢤⡤⣤⢤⡤⢤⡤⣤⠤⣤⢤⡤⢤⠤⣤⢤

⣠⣀⣄⣠⣀⣠⣀⣄⣀⣄⣠⣀⣄⣀⣄⣠⣀⣄⣀⣄⣠⣀⣠⣀⣄⣠⣄⣠⣀⣄⣀⣄⣠⣀⣄⣀⣄⣠⣀⣠⣀⣄⣠⣀⣠⣀⣄
⠁⠀⠁⠈⠀⠁⠀⠁⠈⠀⠈⠀⠁⠈⠀⠈⠀⠁⠈⠁⠈⠀⠁⠀⠁⠈⠀⠉⠀⠁⠈⠀⠈⠀⠁⠈⠁⠈⠀⠁⠀⠁⠈⠀⠉⠀⠁
⢀⡀⢀⡀⣀⠀⣀⢀⡀⣀⠀⣀⢀⡀⢀⠀⣀⢀⡀⢀⡀⣀⠀⡀⢀⡀⣀⠀⣀⢀⡀⢀⠀⣀⢀⡀⢀⡀⣀⠀⡀⢀⡀⣀⠀⣀⢀
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠚⠛⠛⠛⠛⠚⠛⠛⠓⠛⠛⠛

⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶

⠀⠄⠄⠄⠠⠠⠠⠠⠀⠄⠄⠄⠤⠠⠠⠠⠠⠄⠄⠄⠄⠠⠠⠠⠠⠀⠄⠄⠄⠄⠠⠠⠠⠠⠄⠄⠄⠄⠠⠠⠠⠠⠀⠄⠄
⠀⠈⠈⠁⠁⠁⠁⠉⠈⠈⠈⠈⠁⠁⠁⠁⠉⠈⠈⠈⠁⠁⠁⠁⠉⠈⠈⠈⠉⠁⠁⠁⠉⠈⠈⠈⠈⠁⠁⠁⠁⠉⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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



⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣐⠢⠄⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣫⠀⠀⠀⠀⢠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠄⠀⠁⠀⠄⠤⠄⠀⠉⠀⠀⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀
⠀⠀⠐⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠄⠀⠀⠀⠈⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠈⠊⡢⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠅⠈⠀⠀⠐⠈⠀⠀⠀⠈⠀⠠⠀⠀⠀⠊⠂⠀⡀⠀⠀⠀⠀⠀⠀⡀⢔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⣞⡯⠝⢿⡷⣌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣾⣿⢿⡿⠋⠁⠉⠀⠀⠪⢦⣑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡊⣠⠞⡟⠠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⡞⢁⠞⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡜⠀⣾⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣴⠇⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⠓⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⢹⠀⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⠀⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⣻⠞⣿⠁⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⡿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⢸⡿⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠂⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢯⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢿⡈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢃⣾⣿⣷⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠈⢧⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⢞⣿⣻⣧⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢕⢾⣧⠙⣦⡀⠀⠀⢀⡀⠴⠋⣠⠃⣸⢫⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠙⠷⣷⣌⠛⠮⣄⣀⡤⠔⢊⣡⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠈⠉⠹⢦⣤⣀⣴⡶⠛⢛⣷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣤⣤⣴⠿⠛⠋⢀⡆⣿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⢀⡴⣤⠠⢖⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢤⡰⣄⣶⠰⠦⠲⠴⠤⠦⣶⣐⣄⣤⢠⡀⡀
⠀⠀⢸⠅⢸⢆⢺⡇⠀⢀⣠⣴⡲⠟⠋⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠺⢴⣄⡄
⠀⠀⠘⣧⠎⡐⠈⣷⡖⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢴⡀
⠀⣠⣴⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤
⠰⢯⠁⠀⣼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡀
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠓
⠀⠀⠀⠀⡇⠐⢛⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡺⠉
⠀⠀⠀⠀⡇⠀⠀⠐⢪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⡗⠊
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠈⠑⠲⢤⣀⢀⠀⢀⣠⣄⢦⣄⣲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⣦⠼⠗⠉⠁
⠀⠀⠀⠀⠑⢦⣠⣀⣄⢄⣠⣄⣤⠾⠿⠓⠊⠯⢽⣿⣯⠯⠟⠲⠴⠤⠦⠴⠦⠷⠚⠓⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠘⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡞⢡⡖⠒⠒⠲⣤⡙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡴⠚⠋⠙⠳⣌⢻⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠉⠙⠶⢶⣲⠦⣌⣻⡌⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢠⣏⠄⢀⠀⠀⠛⢁⣀⣀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⢰⣏⣉⡙⠲⢦⣌⣉⠛⠛⠿⣆⠻⣆⠀⠀⠀⣀⠀⠀⠀⠀⣰⠋⣰⣛⣁⣀⣀⡤⢖⣊⣉⣤⡽⠂⠀⡽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢶⣽⠛⠿⣭⣉⡛⠛⠻⠿⠛⣻⣷⣝⢷⡾⢻⡝⣷⡶⢶⠞⣡⣾⣿⣟⡿⠿⠯⣍⣩⡭⠒⠉⣀⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠋⢉⣉⣉⣉⣀⠭⠤⠒⠒⢛⣉⣛⠛⠿⣿⡇⢸⡇⣿⣿⢸⡿⠟⢉⣉⡭⢭⣭⣓⣒⡶⠬⠭⣭⣭⣈⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢰⡏⢁⣠⠄⠀⠠⠤⢤⣤⣤⣤⣭⣽⣶⣼⡇⠘⡇⣿⣿⢸⣷⣾⣯⣶⣶⣶⣶⣶⣈⡹⠿⠶⠦⢄⣹⣇⡀⢨⡇
⠀⠀⠀⠀⠀⠀⢀⡜⣋⣤⣬⣶⡶⠶⠶⠚⠛⠛⠉⠍⠛⠂⣩⣭⣉⣻⡇⠀⡇⣿⣸⢸⣿⣿⣿⣿⣛⠛⠿⢿⣟⡿⠿⣟⣻⠶⢶⣶⣮⣍⣳⡆
⠀⠀⠀⠀⠀⠀⢸⣏⠋⠉⣁⢠⡖⠿⠄⠀⣂⣡⠴⠖⣛⠭⠟⠛⠛⠛⣷⢰⣷⣿⣿⢸⣟⣛⣛⡛⠾⢭⣛⠲⢮⣝⡳⠶⠟⢻⡶⣤⣈⡉⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠉⣩⠟⢻⡼⢧⡤⠖⠛⣉⣤⠶⠛⢒⣈⣩⡭⠭⣭⣿⢸⣿⣿⣇⣼⣧⡶⠶⠦⢭⢛⠛⠻⢶⣬⣙⠳⠶⠾⣳⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣟⢸⣇⠙⠓⣢⠖⠋⣁⠤⣒⡼⢛⣉⠤⠒⠈⠉⠙⠛⠉⠀⠉⠀⠈⠉⠛⠶⠤⠐⠶⣶⠲⠬⢙⠻⠶⠛⠁⠸⠯⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠭⠭⢥⡔⠲⠷⠖⢚⡡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢮⣝⡛⠛⠛⢻⡿⠺⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡄⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠂⠙⢇⠀⠈⢇⠀⠈⡇⠈⢱⠖⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡋⠀⠈⡢⠤⠚⣉⣉⢉⣉⢀⡙⠢⠼⡀⠀⡟⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢉⡖⢉⣤⣾⣿⠟⠻⠛⠻⠟⢿⢶⣶⣈⡙⢇⠀⡸⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⡹⠉⣴⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣷⣌⠹⡀⢀⡵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠒⢾⢡⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⠑⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⢤⠃⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢣⠖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠀⣾⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡃⢸⣀⡤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠊⠉⡆⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⡜⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣠⠜⢣⠈⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢠⠛⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣸⢧⡉⢿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠟⣡⠗⠢⢴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡁⠀⡱⢤⡘⠿⣾⣦⣤⣄⣀⣀⣤⣤⣶⣿⠟⣉⠴⠧⣄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⡁⠀⡱⠒⢌⣉⡙⠛⠙⠛⢋⣛⣁⠔⠒⠣⣀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢧⡀⢨⡃⠈⢻⠉⠉⢯⠀⠈⢳⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠓⠶⠚⠋⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣿⡿⠛⠙⠛⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣿⣿⣧⣄⣤⣤⣤⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣷⣿⣷⣶⣶⣶⣶⣤⣾⣿⣿⣯⣤⣤⣠⣿⣿⠋⠉⢉⡛⢛⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⢿⣿⠿⠋⠀⠉⠉⠉⠉⠀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠒⠢⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣙⣹⣓⡀⠀⢠⣐⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠊⠉⠉⠀⠀⢈⠉⠉⠝⠙⣿⢸⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠖⠋⠁⢀⣠⡔⠚⠛⠛⠛⢻⠛⣻⠒⡶⠛⢆⠰⣧⠊⠈⠟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢴⠞⢋⠟⠃⠈⣠⣶⡿⠛⠛⠉⠙⠙⢋⠏⢉⠉⠉⢉⡟⢛⣷⠻⢻⣿⣿⠖⠚⢶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠍⢠⣁⣠⡃⠀⣠⣾⣥⣬⡤⠤⠴⠶⠶⠶⠶⠶⠶⠶⢲⡾⠓⡿⠞⢷⣶⡛⢑⠲⠓⠾⢆⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠥⠖⠈⣽⡹⠐⢇⡾⠋⡈⠁⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢀⣯⠄⠘⢁⠀⣾⣍⡷⢤⣌⣚⠂⢠⡄⠸⢶⢦⡀
⠀⠀⠀⠀⠀⠀⡴⣻⠁⢐⣀⡀⣿⡀⢧⡿⠶⡶⠖⠚⡶⠒⠳⠒⠀⠚⠋⠉⠛⠉⢩⠟⠋⠉⠛⠛⠛⡛⠶⢺⡿⠿⢿⠿⠷⢦⣄⡊⣷
⠀⠀⠀⠀⠀⠀⣁⣧⣦⠾⠛⣛⠇⢠⡏⠀⠠⠃⠀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠡⠄⠀⠎⠁⠀⠾⠏⣠⡻⣿⠀⠦⠈⡛⡟
⠀⠀⠀⠀⡠⠞⡛⠁⠀⠀⠀⡿⢀⣾⣧⣄⡠⠠⠤⠤⠶⠶⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠓⠛⠒⠒⠒⠿⠶⠿⠶⠶⠟⠒⠁
⠀⠀⠀⠰⡇⢐⣠⣠⡤⠖⡾⠁⡼⠉⠁
⠀⠀⠀⠀⠉⠉⠉⠀⣠⣾⢙⡾⠁
⠀⠀⠀⠀⠀⢀⣠⠚⠙⣌⠟
⠀⠀⠀⠀⠀⡼⢀⠇⡴⠃
⠀⠀⠀⠀⠀⠙⠋

⣤⣤⣄⣤⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣠⣤⣤⣄⣠⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢊⡅⠀⠀⠀⠀⣡⠔⢉⠀⠀⠀⠀⢈⡡⠂⡑⠀⠀⠀⠀⢈⠴⢊⡁⠀⠀⠀⠀⣡⠔⢉⠀⠀⠀⠀⢈⡡⠂⡁⠀⠀⠀⠀⢨
⢁⡅⠀⠀⠀⠀⡥⠊⣠⠀⠀⠀⠀⢈⡔⢊⡕⠀⠀⠀⠀⢨⠖⢁⡄⠀⠀⠀⠀⡥⠊⣠⠀⠀⠀⠀⢨⡔⠊⡄⠀⠀⠀⠀⢨
⡠⠆⠀⠀⠀⠀⡖⢋⠠⠀⠀⠀⠀⢠⠖⣁⠌⠀⠀⠀⠀⠰⠊⡡⠆⠀⠀⠀⠀⡖⢋⠰⠀⠀⠀⠀⠠⠖⣁⠄⠀⠀⠀⠀⠰
⠠⠇⠀⠀⠀⠀⠖⣁⠴⠀⠀⠀⠀⠰⢊⡠⢆⠀⠀⠀⠀⠰⢉⠠⠆⠀⠀⠀⠀⠖⣁⠴⠀⠀⠀⠀⠰⢊⡠⠆⠀⠀⠀⠀⠰
⠔⡃⠀⠀⠀⠀⢋⡠⠚⠀⠀⠀⠀⠐⢋⠄⠢⠀⠀⠀⠀⠘⣡⠔⡃⠀⠀⠀⠀⢋⡠⠚⠀⠀⠀⠀⠐⢋⠄⠂⠀⠀⠀⠀⠘
⠚⡇⠀⠀⠀⠀⣋⠔⢊⠀⠀⠀⠀⠘⡡⠔⡣⠀⠀⠀⠀⢘⡠⠚⡃⠀⠀⠀⠀⣋⠄⢚⠀⠀⠀⠀⢘⡡⠔⡃⠀⠀⠀⠀⢘
⢊⡅⠀⠀⠀⠀⣥⠖⢉⠀⠀⠀⠀⢈⡥⠂⡑⠀⠀⠀⠀⢨⠔⢈⡅⠀⠀⠀⠀⣥⠔⢩⠀⠀⠀⠀⢈⡥⠂⡁⠀⠀⠀⠀⢨
⢁⡆⠀⠀⠀⠀⡥⠊⣠⠀⠀⠀⠀⢨⠔⢉⡄⠀⠀⠀⠀⢠⠖⢁⡄⠀⠀⠀⠀⡥⠊⣠⠀⠀⠀⠀⢠⠔⢈⡄⠀⠀⠀⠀⢨
⡠⠆⠀⠀⠀⠀⠖⢉⠴⠀⠀⠀⠀⠠⠞⡁⢌⠀⠀⠀⠀⠰⢊⡠⠆⠀⠀⠀⠀⠖⢉⠴⠀⠀⠀⠀⠠⠞⡁⠄⠀⠀⠀⠀⠰
⠤⠇⠀⠀⠀⠀⠞⡁⠔⠀⠀⠀⠀⠰⢊⡠⠆⠀⠀⠀⠀⠰⢉⠴⠂⠀⠀⠀⠀⠞⡁⠴⠀⠀⠀⠀⠰⢊⡠⠂⠀⠀⠀⠀⠰
⠐⡃⠀⠀⠀⠀⣋⡤⠚⠀⠀⠀⠀⠐⢃⠄⠢⠀⠀⠀⠀⢘⡡⠐⡃⠀⠀⠀⠀⢋⡠⠚⠀⠀⠀⠀⠘⣃⠄⠣⠀⠀⠀⠀⢘
⠊⡅⠀⠀⠀⠀⣋⠔⢈⠀⠀⠀⠀⢈⡡⠐⡁⠀⠀⠀⠀⢈⡤⠊⡁⠀⠀⠀⠀⣋⠔⢈⠀⠀⠀⠀⢘⡡⠔⡁⠀⠀⠀⠀⢘
⢈⡅⠀⠀⠀⠀⡥⠒⣉⠀⠀⠀⠀⢈⡤⠊⡑⠀⠀⠀⠀⢨⠔⢈⡅⠀⠀⠀⠀⡥⠖⣩⠀⠀⠀⠀⢈⡤⠊⡑⠀⠀⠀⠀⢨
⣡⡅⠀⠀⠀⠀⡴⠊⡠⠀⠀⠀⠀⢠⠔⢁⡌⠀⠀⠀⠀⢠⠒⣡⡆⠀⠀⠀⠀⡤⠊⣠⠀⠀⠀⠀⢠⠔⢁⡄⠀⠀⠀⠀⢠
⠠⡆⠀⠀⠀⠀⠖⢁⠴⠀⠀⠀⠀⠠⠎⡁⢌⠀⠀⠀⠀⠰⢊⡠⠆⠀⠀⠀⠀⠖⢁⠴⠀⠀⠀⠀⠰⠎⡁⠆⠀⠀⠀⠀⠰
⠔⠇⠀⠀⠀⠀⠞⡡⠐⠀⠀⠀⠀⠰⢊⡠⠦⠀⠀⠀⠀⠘⣁⠔⠃⠀⠀⠀⠀⠞⡡⠐⠀⠀⠀⠀⠰⢋⡠⠂⠀⠀⠀⠀⠸
⠐⡃⠀⠀⠀⠀⣋⠤⠚⠀⠀⠀⠀⠐⣃⠔⠣⠀⠀⠀⠀⢘⡡⠐⡃⠀⠀⠀⠀⣋⠤⢚⠀⠀⠀⠀⠘⣃⠔⠃⠀⠀⠀⠀⢘
⠊⡅⠀⠀⠀⠀⣃⠔⢈⠀⠀⠀⠀⢈⡡⠂⡑⠀⠀⠀⠀⢈⠤⠊⡁⠀⠀⠀⠀⣃⠔⢉⠀⠀⠀⠀⢈⡡⠒⡁⠀⠀⠀⠀⢈
⢉⡅⠀⠀⠀⠀⡥⠒⣩⠀⠀⠀⠀⢈⡴⠊⡑⠀⠀⠀⠀⢨⠔⢈⡅⠀⠀⠀⠀⡥⠒⣩⠀⠀⠀⠀⢈⡴⠊⡁⠀⠀⠀⠀⢨
⡡⠆⠀⠀⠀⠀⡴⢊⠠⠀⠀⠀⠀⢠⠔⢁⡌⠀⠀⠀⠀⠰⠊⡡⠆⠀⠀⠀⠀⡴⢊⠠⠀⠀⠀⠀⢠⠔⢁⠄⠀⠀⠀⠀⢰
⠠⠆⠀⠀⠀⠀⠖⣁⠴⠀⠀⠀⠀⠠⠎⡡⢆⠀⠀⠀⠀⠰⢋⠠⠆⠀⠀⠀⠀⠖⣁⠴⠀⠀⠀⠀⠰⠎⡡⠆⠀⠀⠀⠀⠰
⠔⠇⠀⠀⠀⠀⠏⡡⠒⠀⠀⠀⠀⠐⢋⠤⠂⠀⠀⠀⠀⠘⣁⠔⠃⠀⠀⠀⠀⠏⡡⠚⠀⠀⠀⠀⠐⢋⠤⠂⠀⠀⠀⠀⠘
⠐⡇⠀⠀⠀⠀⣋⠄⢚⠀⠀⠀⠀⠘⣁⠔⠣⠀⠀⠀⠀⢘⡠⠐⠃⠀⠀⠀⠀⣋⠤⢚⠀⠀⠀⠀⠘⣡⠔⠃⠀⠀⠀⠀⢘

⠀⠀⠤⠒⠐⠐⠂⠂⠂⠂⠐⠐⠐⠐⠂⠂⠂⠂⠐⠐⠐⠐⠄⢀⠀⠀⠀⠀⠀⠀⡀⠄⠢⠒⠑⠘⠐⠐⠠
⠀⠬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀
⠀⡪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠀⡨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡪
⠀⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⠈⠐⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠌
⠀⠈⠆⢄⢄⢄⢠⠠⡠⡠⡄⢄⢄⢄⠠⠠⡠⡠⡀⢄⢄⢄⠠⠐⠀⠀⠀⠀⠀⠀⠃⠆⠄⢄⢄⢀⠠⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠈⠈⠈⠁⠁⠁⠁⠈⠈⠈⠈⠀⠁⠁⠁⠉⠈⠈⠈⠈⠁⠁⠁⠁⠈⠈⠈⠈⠀⠁⠁⠁⠉⠈⠈⠈⠈⠁⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠘⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢄⢄⢤⢠⢠⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠬⠁⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠃⠆⣄⣀
⠀⠀⢀⢀⡠⡀⠤⠁⠀⠀⠀⠡⢢⡠⡀⡀⡀⡀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⣄
⠒⢌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⢑⠃⠀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋
⠀⠈⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠍⠀⠀⠀⢐⠁
⠀⠀⠀⠈⣂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒
⠀⠀⠀⠠⠂⠀⠠⡐⠌⠂⠆⡄⡀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂
⠀⠀⠀⠈⠄⠂⠁⠀⠀⠀⠀⠀⠁⠃⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡄⡄⡀⡀⣀⢀⠠⡠⠔⠌⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⠀⣤⣤⣤⣄⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣬⠻⣿⣿⣿⣷⠸⣿⣿⣿⣷⢸⣿⣿⣷⡄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡙⢿⣿⣿⣿⣧⠙⠛⠿⠟⠃⠙⠛⠿⠿⠘⢿⣿⣿⡇⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⡶⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢻⣿⣿⡟⣸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣍⡛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⣿⣿⣿⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠾⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⣡⣾⣷⡄
⠀⠀⠀⠀⠀⣠⣶⣿⣷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⢰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢛⣥⣿⣆
⠀⠀⠀⠀⣠⣶⣦⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠟⣣⣄
⠀⠀⠀⠛⣛⣛⣩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿
⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟
⠀⠀⠀⣿⣿⠿⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⣴⣾
⠀⠀⠀⢉⣶⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣛⡁
⠀⠀⠀⠈⠿⢋⣥⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠹⣿⡟⣣⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⠂
⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⣡⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢿⣿⣿⣿⢰⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣮⡹⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇⢾⣿⣿⣿⡆⣿⣷⣦⡄⢠⣤⣤⣄⡠⣶⣶⣶⣶⡍⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡄⢿⣿⣿⣿⠸⣿⣿⣿⣷⡸⣿⣿⣿⣿⠶⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠃⠙⠛⠛⠛⠃⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠀⠀⢀⣀⣤⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⣀⣤⡾⠟⠋⢻⣿⣶⡾⠟⠋⠉⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⣿⣿⡾⠛⠉⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣴⡿⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣏⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣤⡀
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⣴⡆⣠⣾⠿⠿⠿⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⢸⣿⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠘⢿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣶⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠤
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠉⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣰⣟⡹⠂⠃
⠀⠀⢀⣤⣀⠀⣤⣤⠶⠀⡀⢠⣄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⣼⣿⣭⠐⠀⠀⠸⣿
⠀⠚⢛⣡⡥⣾⢤⠷⠄⠚⠑⠲⠍⠗⠀⠀⠀⠀⠀⠀⣠⡄⣴⡿⣡⣾⣿⣿⠅⣉⠀⠀⠀⠈⠃⢀⣀⣴⠖
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠋⢀⣾⡿⠋⠉⠁⢺⠇⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣴⡁⣰⣯⠖⣀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⢠⡞⠿⣵⣿⡿⢧⡌⠀⠀⠘⢧⠀⠘⠆⠀⣹⣄⣧⡀
⠀⡤⠀⠾⠔⢙⣷⣿⠿⡿⠃⢠⠀⠀⢀⡆⢀⢲⠦⣰⣤⡿⢻⣁⡀⠀⠀⠀⠀⢀⠔⢂
⠀⣄⢰⡄⠀⠾⠋⡀⠈⢁⡀⣾⣆⣾⣼⣷⣿⡿⠶⠟⣛⢳⡾⠋⠁⠀⠀⠀⠠⠁⠀⠀⣄
⢠⣿⣾⣷⡀⡴⠴⠷⠷⢿⣿⠿⠋⠙⠛⠉⠀⠀⠀⠚⠋⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠉⠐⠶⣦⣌⣰⣆⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠃⠆⠀⠀⠀⠀⠀⠈⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠁⢂⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠁⣠⡟⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢹⡌⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⢀⣼⠋⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠈⣧⠀⠹⣦⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⢠⡞⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣿⠀⠀⠘⣧⡀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢸⡄⠀⠀⠘⣧⠀⠀⣼⠃⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡆⢸⡇⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡀⠀⠀⠀⣿⠀⡿⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠶⠶⠶⠶⠶⢦⣄⡘⢧⡀⠀⢸⡇⠀⠀⢰⡏⢸⡇⠀⢀⣼⠋⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠹⣿⠶⣤⣀⠀⠀⠀⠀⠀⠈⠛⢮⡻⣄⠘⣇⠀⠀⣿⠀⣼⠁⣠⡞⠁⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠙⠛⠶⣤⣄⣀⠀⠀⠈⢻⣝⢷⣿⡄⢸⡇⢠⣟⣴⠋⢀⣤⠶⠛⣉⣤⡶⠶⠛⠛⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠈⠉⠛⠶⢦⣄⡹⣦⠻⣷⣾⢀⣾⣿⣥⠞⠋⣡⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣤⣤⣤⣤⣬⣿⣿⣷⣿⣿⣾⣿⣋⣤⣶⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢛⣳⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⢛⣛⣻⣶⣶⣾⣿⣿⣿⣟⠻⠿⣭⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⢛⣩⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⡛⠛⠉⢉⣩⡴⢛⡽⣻⣿⡝⣿⣷⣦⣤⣍⣙⣳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠋⠉⢠⡟⢡⡟⣿⢿⣸⡇⠙⢷⣌⠉⠉⠛⢿⣍⠛⠛⠶⠶⠶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣿⠀⣿⠘⣿⣿⠀⠀⠉⠻⣦⡀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⡄⣿⢀⡿⠸⣆⠀⠀⠀⠀⠙⢷⣄⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣯⡾⠁⠀⠙⢷⣄⡀⠀⠀⠀⠈⠻⣦⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⠶⠶⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡤⠀⠠⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠈⠀⠀⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠤⠄⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢡⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣠⠄⠀⠀⣀⣠⣠⣤⡶⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣴⣤⣤⣄⣤⣶⣶⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠉⠉⠭⠭⠩⠭⠍⠍⣉⠉⣉⠐⣯⣮⣉⣙⣿⣾⠆⢉⠻⣎⠓⡻⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠊⠉⣉⣉⣉⣩⣥⣶⣤⠌⠙⠿⠛⠛⠿⠒⠈⠀⠈⠉⠉
⠀⠀⠀⠑⠒⠒⠒⠀⠀⠀⠸⠍⢀⣀⣈⡉⠉⠉⠁⠉⠙⠉⠀⠉⠀⠀⠐⠒⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢠⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡾⠀⡇⢠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⢧⠀⣷⠋⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⣸⠀⠈⢳⠏⠀⣇⡴⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠹⣄⢀⡏⠀⢀⡟⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢿⡀⢀⡞⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢳⡞⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣸⠁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡤⣯⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⣏⠀⣸⢧⡉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⠏⠸⣴⠏⠀⣻⣇⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⢳⡀⣰⠛⢦⣰⠋⠈⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⡿⡅⢀⡼⠻⣄⣰⠃⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣶⠋⠀⡽⢯⡀⢀⡼⠓⢦⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⣷⡞⠀⢀⡽⢯⡀⣠⠏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⠀⢙⣶⡋⠀⣠⠟⠳⣄⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡋⠀⣹⠾⣅⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣥⣀⣼⡿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠹⠒⠉⠉⠑⠔⠒⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠚⠒⠒⠒⠲⢄⡀⠀⢀⡠⢘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡞⠉⠉⠉⠓⣆⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠼⡤⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢼⣤⢀⣀⠀⠀⣗⣒⡄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣛⣻⠷⣾⣛⣛⣭⣿⣷⡾⡧⣤⣴⢾⣛⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⢘⡇⠉⠉⠛⠻⠿⣿⠿⠛⠉⠁⣿⠻⠷⢿⠍⠛⠋⣉⣸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣶⣾⠿⠿⠛⢻⡇⠷⢀⡀⢠⠄⢸⠀⡀⠀⠀⣹⣤⣤⣼⣶⠟⣛⡥⠖⠛⡇⠀⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣽⣿⣷⢶⣿⣷⣾⣷⣷⣌⣁⣼⡦⠶⢚⣯⠿⠁⣁⡤⠞⠋⠁⢀⠰⠆⡿⢹⡽⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣮⣯⠐⡆⡭⢙⡛⣿⠉⣻⠾⠗⣏⣡⡷⠞⠋⠁⢀⠄⠠⠅⠈⣀⣴⠗⣫⢶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣧⢼⠿⡟⠷⣷⣬⡄⣿⣁⣴⣶⠞⠛⢉⠀⡆⣂⠀⠈⢁⣠⣶⡿⢋⡵⣊⠁⣘⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⠴⡞⠛⡇⣿⠴⠛⣧⣌⣠⠋⡛⡟⣿⠁⢸⠀⠘⠘⠀⢏⣸⠴⣚⡽⠛⣡⠞⡉⠂⢐⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⣉⠑⣢⠴⠛⢳⡿⠔⠊⣀⠤⠈⣙⠷⣾⢻⣄⣸⡇⢀⡤⠖⣫⠴⠚⣁⡴⠛⠑⡠⢊⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⡁⠂⣠⠴⠋⣁⠤⠂⢁⠠⠒⢉⠠⠐⡉⠄⣂⠽⣸⡏⠝⣛⡥⠖⠋⢀⡤⠚⡁⠀⠀⢊⡴⠋⡠⠃
⠀⠀⠀⠀⢤⢀⠀⣀⡴⠚⢁⠁⠂⣤⠴⠋⡡⠔⠈⣀⠄⠊⡡⠔⠊⠠⠀⢀⣔⣚⣧⠴⠟⠛⠉⠁⣀⡤⠞⢉⠀⢀⠔⢁⡴⠿⠖⠋
⠀⠀⠀⠀⠘⣧⣞⣁⣈⣒⣢⣔⣋⣀⣔⣈⣀⣤⣊⣠⣴⣥⡤⠴⠶⠖⠚⠋⠉⠁⠀⠀⢀⣠⠴⠚⠉⠰⠀⡠⢀⣤⠞⡛⠠
⠀⠀⠀⠀⠀⠙⢢⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠉⠀⠒⠀⠀⡠⢊⣠⢟⡹⠊
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⡠⠐⢁⡴⠿⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠈⠉⠉⠉⠀⠀⡔⠉⢻⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣠⣴⠏⢔⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠈⢧⣶⣾⠎⠀⠀⠀⠀⠀⠐⠈⠠⠔⣲⡞⠑⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡶⣼⣿⡷⠶⠬⣄⣀⡠⣴⣬⣴⢊⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣿⡿⠿⡒⠾⠛⠙⠋⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⢠⠃⠀⠀⠀⠀⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣿⣷⠶⠸⣿⠷⠁⠀⠀⠀⠀⢠⡟⢻⣿⣿⣷⣤⣀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⡿⢋⠤⠊⠁⠀⠀⠀⠀⠀⠰⠋⠀⢸⣿⣿⣿⣿⣿⣷⡀⠀⠙⠷⣦⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣶⣦⡀⠈⠙⢷⣮⠙⢦⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠈⠓⠀⠈⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣀⠀⣀⡀⣀⣀⣀⢀⠀⣀⣀⢀⣀⠀⢀⣀⡀⡀⠀⠀⡀⠀⢀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀
⠀⠀⠀⢛⡛⠛⡿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠂
⠀⠀⠀⣾⣿⣼⣿⡆
⠀⠘⠛⠛⠻⠿⠿⠷⣶⣶⣶⣶⠶⠶⢶⣶⣦⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣈⣶⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠇⠀⢏⡉⠛⠂⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠑⢄⠀⠀⣸⠀⠀⠈⢳⣄⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡳⠀⠀⠀⠑⢀⡟⠀⠀⠀⣸⠛⠣⡀⠀⡿⠀⠒⡶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⡀⠀⣴⠛⠡⢤⠐⡄⠀⠀⠈⠓⡀⢀⡴⠣⠀⠀⠈⠲⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣽⠁⢄⠀⠈⠙⢃⢠⣤⣴⣟⠋⠶⠖⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣾⠀⠀⠙⠢⡀⠀⢩⣯⠻⣿⣿⣶⡆⠠⢑⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣿⢠⠀⠀⠀⠈⠱⣾⢀⠷⠈⠻⣿⡧⠀⠀⢹⡦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠐⢤⢆⡜⢤⡀⠀⠀⠀⠀⣬⣿⢸⡆⠀⠀⠑⢄⠀⣾⠁⠀⠐⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣀⣀⠀⠂⣿⠃⣼⡁⡀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠛⢿⠁⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠢⢄⡀⢰⡿⠀⠓⢈⣡⣗⣤⣿⢏⢬⡀⠀⢀⡀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⢤⣀⣤⠀⢭⡼⠾⠑⠍⢉⢜⣷⠒⡄⣠⡄⠀⠀⠀⠀⠉⠢⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣄⡙⢷⣤⣿⡿⣟⡑⢶⡶⢾⣭⣀⠁⢡⠇⢢⡄⣰⠄⠀⡄⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢡⠀⠈⠙⠿⢾⣿⣄⠀⠉⠳⠶⣄⣻⡟⠈⠉⢢⡄⠁⠉⣿⠀⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⠀⠀⠀⠀⠆⠙⣯⠀⠀⠀⠀⠀⠉⠳⠢⢄⣿⠀⠈⠐⣜⠃⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠀⣼⣷⣠⠀⠀⠀⠈⡎⣿⢀⡖⠁⠀⠀⠀⠀⠀⠀⠉⢲⢶⣀⠘⢗⠢⢽⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡘⠸⠎⣰⣻⣿⠹⢆⡆⢠⡞⢰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣿⣿⣁⠀⠈⠒⠠⣀⠀⢀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠔⠀⠨⢻⣿⣽⣏⡋⣶⢿⡟⡡⣿⣿⡇⠀⠘⡆⠈⡀⠀⠀⢀⠀⠠⣴⢀⡝⣿⢿⣆⡄⣳⡀⠀⠉⠛⢅⡀
⠀⠀⠀⠀⠀⠃⠐⠒⠒⠻⡿⣿⡿⢝⣿⡿⠁⣸⣿⣿⣇⣿⡀⣇⢸⣿⢠⠆⡈⠀⠀⣿⣸⠀⣾⣿⡟⠃⡧⢠⢀⣦⠀⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⢀⢀⣀⢳⡾⢀⣓⣈⣭⠥⣽⣯⣟⣻⢻⡏⠉⢱⠈⠋⠈⣀⣋⣠⣼⣽⣧⣼⣯⣭⣡⣤⡅⠐⣻⢱⠀⡸⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡿⢿⢿⣿⣿⣽⣿⣄⣀⣠⣌⡛⡈⣝⡤⢶⣶⣿⠿⢛⣩⣭⣤⡭⠽⢿⡿⠿⣿⣿⣿⣿⡀⠀⠀⠀⠢⠰⣴⡿⠁
⠀⠀⠀⠀⠀⣸⠇⠰⣚⠞⠈⠙⢹⣿⣿⣿⣯⡤⡾⢿⡖⣋⠣⠞⠛⠙⠁⠠⠤⠤⠤⠤⣤⣷⣿⣾⠇⣿⣿⡄⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⢿⠀⢒⠃⠈⢥⣀⠒⠠⠽⠛⣽⣯⣿⣿⢿⠿⠂⠀⠀⠀⠀⠒⣀⣀⣀⣀⣒⣻⣿⢏⣾⡉⡽⠓⠀⠀⠂
⠀⠀⠀⠀⠀⢸⡄⠀⠉⣀⠒⠢⠄⠤⠄⢨⢏⣉⢺⠙⣦⣀⠈⠀⠀⠀⠀⠁⠤⠤⣤⣭⣿⡿⣵⡟⢣⠋
⠀⠀⠀⠀⠀⡸⣿⡍⢐⡀⠤⢍⣁⠐⠂⣚⢀⠈⣩⡾⢷⡤⠄⠒⣂⠀⠤⠐⢒⣠⣶⣾⢟⣾⢇⡘⠁
⠀⠀⠀⠀⠀⠷⠞⡛⠘⣿⣷⣒⣲⣒⣀⠩⣇⣀⡴⠾⠛⠓⠛⢟⣣⣴⣶⣭⠴⢶⣿⣟⣾⢁⣮⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠈⠉⠉⠙⠁⠈⠉⠀⠀⠀⠉⠉⠉⠉⠛⠛⠓⠂⠀⠙⠻⠿⠷⠾⠖⠋⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢻⣿⣿⡳⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣀⠀⠀⣼⡇⢀
⠀⠀⠀⠀⠀⢰⡄⠀⠀⢀⣴⣿⣿⣿⡛⠛⢿⣿⣲⣄⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡀⣀⣴⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣸⡄⡀
⠀⠀⠀⠀⣦⣼⣇⣰⢴⣿⣿⢻⣻⣻⣷⣦⣀⡀⣄⣽⣷⣆⣾⣿⣼⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⠾⣿⣯⣛⡛⠛⠻⠿⠏⠻⣿⣿⣿⣧⣧
⠀⣀⣀⣀⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠈⠙⠛⠿⠿⣿⣟⣺⣿⣿⣷⣤⢀⡀⠀⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣦⣤⣉⣉⣉⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠀⠀⠀⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⢶⣿⣿⣦⡀
⠀⠀⠀⢀⠀⠀⢀⣴⣀⡀⠀⣀⣤⣾⣫⣾⣿⣿⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣴⣿⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣤⣄⣤⡀
⠀⠠⠴⢾⣷⢶⠿⠿⢿⣿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⣤⣶⣾⣷⣄⣀⠀⠀⠀⠀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠼⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠙⠋⠉⠉⠀⠀⠀⠐⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡠⢲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⣠⣶⣯⣴⣮⣿⣿⣷⣄⠀⠀⠀⠀⠀⣠⠾⣦⡞⠀⠈⠋⠑⠄⠀⠀⠀⠀⠀⢀⠴⠞⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⡰⠂⢀
⠀⠴⠿⠿⠿⠿⠿⠿⠿⠷⠶⠾⠷⠄⠀⠀⠐⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠀⠠⣾⠀⠀⣠⠂
⠀⠀⠛⠀⠀⠊⠁⠀⠈⠛⠒⠀⢢⡄⠀⠀⢀⣤⣀⣤⣴⣿⣄⣠⣾⣤⣤⣤⣶⣦⣀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢠⠃⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣀⡀⠠⡯⠣⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⠀⠀⠀⢸⠀⠀⠠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣼⣏⣩⣖⢦⣤⣄⣀⣀⠀⢈⡇⠀⠀⠁⡀⢀⣆⣩⣤⡶⠃
⠀⠀⠀⢀⣀⣊⣛⣿⣷⣴⣶⣶⣦⣼⣿⣿⣿⣿⣓⡻⠶⠾⠶⠾⢒⡚⢳⣿⣿⣭⣉⣙⠝⢁
⠀⠒⠀⠀⠿⡉⠹⠿⠶⠟⡉⠽⢯⣭⠋⡐⠿⠿⠧⠽⠿⠟⢀⣽⣿⡋⠙⣛⡂⠘⠛⠯⠐⠛⠀⠀⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠀⠖⠐⠛⠛⠧⠄⠀⠛⠂⠰⠦⠉⠉⠡⠄⠀⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⢀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠄⡀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⡀⠀⠀⡠⠤⠤⠤⠤⠤⢄⠀⠀⢀⠤⠤⠤⠤⢄⠀⠀⢀⠤⠤⠤⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⡇
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⠀⠀⠀⠀⠀⠀⠠⡄⠀⢠⠀⠀⠀⠀⠀⡆⠀⢠⠀⠀⠤⡄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢰⠀⢰⠄⠀⠀⠀⠀⠀⠀⠤⠄⢴⠀⢰⠀⠀⠀⠀⠀⠀⠤⡆⠀⢠⠀⠀⠀⠀⠄⡆⠀⢠⠤⠠⠤⡄
⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠀⢈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⠀⢈⣀⣀⣀⣀⣀⣀⣀⡁⠀⢀⣀⣀⣀⣀⣀⡁⠀⢀⣀⣀⣀⡁
⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠈⠀⢸⠀⠈⠀⠀⠀⠁⠀⡇⠀⠈⠈⠀⠀⠘⠀⠁⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠇⠀⠀⡀⠀⠀⠀⠀⢘⠀⠀⢀⠀⠀⠀⠀⡃⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢀⠀⠀⠀⡀⠀⡇⠀⠀⢀⠀⠀⢀⠀⠇⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠘⡖⠈⠉⠀⠀⠀⠐⠉⠑⢰⠃⠀⠀⢳⠂⠉⠂⠂⠀⠈⠑⢲⠃⠀⠀⡖⠉⠀⠂⠀⠑⠲⠀⠀⠸⠊⠑⠒⠈⠲⠀⠀⠰⠋⠒⠑⠇
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⡈⠀⠀⠀⡄⠀⠰
⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠡⡀⣀⣀⣀⠌⠀⠀⠀⠀⠀⠀⠀⠰⡀⣀⣀⡐⠀⠀⠀⠀⠀⠀⠘⣀⣀⣀⠁⠀⠀⠀⠀⠀⣄⣀⡘⠀⠀⠀⠀⢠⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠐⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠐⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠤⠀⠴⠺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⠀⡀⠁⢀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠠⡄⠸⠷⠀⠸⠖⣄⣴
⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠈⠁⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠁
⠀⠀⠀⠀⠀⠰⣁⢀⠀⢀⠀⠀⠀⠀⣠⠔⠉⠑⠀⠤⠀⡀⠀⠀⠀⠀⠡⠂⠀⡀⠀⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⡤⠐⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⡁⠤⠀⢀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠠⠆⠀⠈⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠠⠀⠀⢃⢀⠀⠀⠡⡄⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠢⢲⣤⠉⠿⠗⠛⢃⡔⠅
⠀⠀⢰⠀⠈⠀⠠⠀⠀⠀⠀⣀⠠⠵⢄⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢈⡼⠀⠐⠀⡐
⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠈⢂⠀⠀⢱⠄⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠇⠂⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠉⠁⠐⠂⢄⡀⠈⠳⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣤⣀⠤⣀⠀⠙⢄⠀⠘⡄
⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡗⡶⠈⠑⣴⡌⠲⣼⣀⠀⣧
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠈⠀⣁⡯⠚⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⣿⠈⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢒⣛⡓⣒⠒⠒⠛⠒⠒⠶⠶⠾⠶⠶⠶⠶⠶⡀⠔⠊⠀⣷⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣙⣿⣿⣿⣿⣿⣿⣿⡟⠿⡷⠶⣶⣾⣷⣶⣶⣶⣶⠿⠶⠶⠶⠼⠶⠶⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠿⠿⠿⡿⠿⠿⣿⡿⢋⣴⣿⣿⣿⡿⠿⠿⠛⢻⡿⠿⠿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠒⣒⣚⣛⣛⣒⣒⣒⣒⣒⣻⣟⠒⠒⢲⣶⠿⠛⠛⠛⠒⠒⠒⠒⠀⣼⡇⠀⠀⠀⠈⠙⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⢛⣛⣓⣒⣒⣒⣒⣒⣒⣲⠈⠓⢲⡶⡶⠶⠶⠶⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠛⢛⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⠶⣛⢅⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡏⠀⠀⠀⣠⠾⣫⣵⡿⢣⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣻⡇⠀⠀⠈⠛⢛⣻⣿⣿⣿⣿⣿⡇⣿⣿⣿⡟⠀⢀⣴⢟⣵⣾⣿⡟⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠯⢿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⠟⢀⡴⢋⣵⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⢋⡴⣋⣴⠿⠿⠿⠿⠋⢰⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢳⣶⠀⠀⢀⣛⣻⣿⣿⣿⣿⣿⣿⡿⢡⣟⡵⢋⣡⠥⢶⣒⡚⣛⠃⠀⠀⣹⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⣽⡇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⠿⢛⣫⣵⣶⡿⠟⣋⣱⠃⢠⣆⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣛⣿⣥⠾⠟⠋⠉⠁⢸⡟⢀⡤⢖⣫⣵⣾⡿⣟⣫⣵⣾⣿⢋⡟⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣾⡟⠀⠀⠀⠀⠀⢀⡿⣛⣥⣾⣿⣿⣿⣷⣿⣿⣿⣿⣶⡟⣾⠁⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣭⠿⠧⠤⠤⠤⠤⠤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⠇⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣭⣴⣶⣿⠛⣛⡛⢛⣟⣳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠏⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⣿⣧⣼⢟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⠏⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣳⠏⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡽⠁⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓⠋⠠⠴⠿⠿⠟⠛⠛⠛⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠽⠿⠿⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣾⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠑⠀⢠⣞
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠏⢠⠀⠀⠀⠀⣠⠾⠀⠀⠀⠀⠀⠀⠿⢇⠀⠀⣠⡆⠀⠸⣋⠀⠀⠀⠀⡄⢠⠋
⠀⠀⠀⣠⡦⠀⠀⠾⡇⠀⠀⠀⠁⠀⠀⠼⢯⡄⠀⠈⠓⢀⣄⠀⠀⠀⠀⠀⠛⠣⠄⠀⢉⣀⠀⠀⠙⠃⠀⠁
⠀⠀⠘⠛⠇⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠈⠙⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⣧⡀⠀⠀⠄⠀⢀⣤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢨⣆⠙⠦⠀⠀⠀⣿⠏⠀⠁⠀⠀⠀⠀⢀⡴
⠀⠀⠀⣠⠀⠀⢀⣾⣦⠀⠀⠀⠛⢻⡄⠦⣄⡀⠘⠇⠀⠀⠀⠀⠀⠀⠀⣾⣥⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⡀
⠀⠀⣴⣿⡆⠀⠀⠈⠻⣧⡀⠤⣄⡀⠙⠆⠀⢀⣴⡇⠀⠀⠀⣠⡾⠀⠀⠀⠈⠃⠀⠀⣰⠿⢋⡁⠀⠀⠀⢾⣇
⠀⠀⢺⣿⡀⠀⢀⠀⠀⠈⠛⠃⠀⠠⣶⡀⠀⠘⠛⠛⠂⠀⠈⠛⠻⠆⢀⣴⡆⠀⠀⠰⣿⣧⡀⠑⢄⠀⠀⠀⠉⠁
⠀⠀⠈⠙⠛⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠂⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠙⠻⠿⢿⣿⠿⠟⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠉⠛⠿⠿⠿⠟⠋⠀⠀⢀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠿⠛⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠻⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⣿⣿⣿⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰
⠀⠀⠀⠀⠀⡀⠀⠘⢂⠐⠁
⠀⠀⠀⠀⠀⡄⠀⠻⢧⠀⣀⠀⠉⠉⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠒⠒⠒⠊⠉⡽
⠀⠀⠀⠀⠀⡀⠀⠉⠁⠀⠠⠘⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠠⠀⠀⠈⡠⠔⠚⠉⠀⠂⠐⠠⡀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠡⠂⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠄⠒⠉
⠀⠄⢊⡔⠀⠀⠀⠀⠀⠀⣴⣛⠀⠀⡂⢀⡇⠀⠀⢀⡠⠄⠒⠉⠙⢇⠀⠀⠀⠀⢠⠀⠀⠉
⠀⠀⠸⠀⠀⣤⣤⣀⡀⠸⠛⣿⠆⠀⢰⣨⠧⠴⠖⠋⠀⠀⠈⢄⢂⢢⠛⠖⠂⠀⠊
⠀⠈⠀⠒⠘⢣⡈⠻⠃⠀⠀⠀⠀⢠⠸⠁⠀⠀⣀⣤⣖⠉
⠀⠀⢖⠢⣄⠀⢙⠤⣀⣀⣀⡤⠞⠋⠀⠀⠀⠈⠁⠙⢮⡳⡀
⠀⠀⠈⢇⠀⠉⠒⠂⠤⠤⠬⡀⠀⡶⢄⠀⠀⠀⠀⠀⠀⠱⣌⠢⡀
⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠱⡜⠀⠀⠳⡀⠀⠀⠀⠀⠀⠘⣆⠙⡆
⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢹⡤⠤⠄⠀⠀⠀⠘⡆⠁
⠀⠀⠀⠀⠀⠀⠣⡀⠁⠀⠀⠀⠈⡇⢠⠊⠁⠀⠀⠈⣷⣦⣀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣧⠏⠀⢀⠖⠛⠉⠙⢯⡀⠀⠘⠦⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⣰⠏⠀⢀⡞⠀⠀⠀⠀⠀⠙⠤⠤⠤⠠⣄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠐⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⣴⡖⢤⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣦⡀⠀⠀⠀⢸⡆⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡷⠘⣏⠑⢄⡀⢸⡇⠀⠀⠀⣿⡿⠁⠢⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠘⠀⠘⡀⠀⠙⢾⣧⠀⠀⠀⡏⠁⠀⠀⠈⢂⣄⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡏⠀⠀⢀⠄⠀⠀⠀⠀⠀⢉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡆⠀⢸⠀⠀⠀⠀⠀⠀⠋⢹⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢼⡇⢀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢹⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣾⢷⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⡿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠃⠀⠀⠀⠀⠀⠁⢈⣷⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢻⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⡛⠠⠁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⢿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣼⠀⢣⠀⠀⠀⠀⠀⠀⠈⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⠀⠀⠀⠒⠻⣿⠟⣋⠴⣶⠶⠦⠄⠀⠀⠀⠀⠀⠾⢶⡁⠀⠀⠀⠐⠂⢀⣠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⣟⣀⠠⠘⣁⡀⣀⣀⣀⠤⣈⣿⣿⡿⢿⣿⠀⠒⠒⠛⠀⠀⠀⠀⠿⠿⠿⠶⢀⣠⣴⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠐⠛⠐⠊⠤⠤⠤⠄⠀⠀⠀⠀⠉⣻⣻⡿⣾⣿⡆⠀⠀⠀⠀⠀⢠⡤⣟⣭⣴⣾⣿⣿⣿⣿⣿⣿⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡃⠀⣸⣿⠏⢀⣀⣤⣶⣾⣿⡿⢿⣿⠿⠛⠃⠀⠈⠙⠃⡠⠎
⣀⣀⣀⣀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣬⣭⣶⡿⠛⠀⣴⡆⠀⣺⣿⠂⢀⣄
⣿⣿⣿⣿⣿⣖⠀⠑⠪⠄⡀⢀⣤⣀⠀⠛⢿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⣟⣻⣿⣭⠉⠁⠉⠉⣴⣶⣾⣿⣿⡇⢸⣿⡿⠗⢸⣿⡷⣄⣀⡀⠀⢀⡠⠀⡀
⠛⠛⠛⠳⠆⡀⠀⠀⠀⢰⣬⣀⠉⠙⠛⢶⣶⣭⣭⣭⣭⣭⣿⣿⣭⣤⣼⣿⣿⣿⣿⣷⣦⣿⢿⣿⣿⣿⣿⣿⡟⠘⢿⠀⠀⢾⣿⣿⣏⠻⢿⣿⣷⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣄⠈⠉⠻⠿⠟⠻⣯⣝⣛⡛⠛⠋⠉⠀⠀⠈⠉⠙⠷⠀⠉⠉⠉⠀⠀⠀⠀⠀⠆⠸⣿⣿⣛⠂⠀⢨⣝⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⣄⡀⠀⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣴⢿⠁⠀⠀⢀⠀⠀⠀⢀⣤⢶⣄⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⠒⣀⣶⡈⣷⠄⠀⠜⠡⢇⣴⣷⣶⠸⣿⣄⠀⣾⣿⣿⣿⢉⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠚⣳⣄⢺⣷⠞⠀⡀⢸⣯⣧⠈⠀⠿⠀⣸⠋⢡⠇⣿⡶⣟⡙⡾⢉⠏⠁⠙⠟⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⣿⣿⠧⣾⣧⣼⣿⣿⣇⠀⢸⠁⠀⢀⡞⣼⡿⠽⡽⠿⢷⣊⠀⠀⠄⠠⠃
⠀⠀⠀⠀⠀⢀⣀⢀⠈⠀⢨⣥⣴⣾⢿⣷⣿⣿⣿⣦⢀⡀⢀⢰⣿⡿⠟⠒⣾⣶⣾⣥⡴⢋⣄⣰⣳⣤⣀⡀
⠀⠀⠀⠀⣴⠏⠹⠯⡅⠇⢨⣿⣿⠭⠛⠶⡤⣝⠿⢯⣇⣿⡿⡟⣡⡴⢞⣯⣯⣭⣯⣭⣷⢞⡋⢋⡽⠿⠿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠑⠈⠠⣂⣐⠒⠀⣤⠰⠶⢲⡶⠞⠉⣿⣿⣹⣷⣧⣾⡷⠖⡰⠻⣿⢿⠉⠁⠀⠀⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⣰⡞⠛⠉⠁⠉⣼⣒⣠⡴⢻⣿⡁⣰⠆⠻⣷⣿⣹⡡⢤⡾⢁⢾⡉⠳⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣆⣦⣊⣽⣿⣿⡕⠀⣼⣾⣅⠀⢶⡻⣯⣿⠕⠈⠀⢸⣿⣌⠂⢸⣟
⣶⣶⣶⣶⣶⣤⣄⡀⢠⣿⣛⣽⣻⣿⣷⣿⠁⢠⡟⠛⣿⣧⠸⣷⣿⠋⠀⠀⠀⠀⠻⣿⡆⠀⣿
⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⡿⠋⣿⡿⠀⠀⠀⠀⣿⣿⣷⣶⣿⠀⠀⠀⣀⣀⣀⢹⡇⢠⠞
⢸⣏⣿⣿⢿⣿⣿⣿⣿⡿⣎⢻⠀⢰⢛⣛⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣶⣄
⢿⣿⣿⣿⢿⣿⣿⣿⣿⣽⠟⠋⣂⡸⣸⣿⠀⠀⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⢿⢻⡷⢸⣿⠀⡜⣱⡿⠁⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣿⣿⣿⣿⠿⠟⠛⠉⠀⠀⠀⠘⡦⣷⢸⣿⠐⢸⣿⠁⠀⠀⣹⡁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⣿⠀⣿⠇⠀⠀⠀⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣅⢸⡿⢰⣿⡄⠀⠀⠀⢹⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⣘⡁⣼⣿⣧⠀⠀⠀⠈⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡀⠹⣥⣴⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⢠⣾⣿⣿⣿⠇⠀⠀⠀⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢼⣻⣿⣿⣿⡇⠀⠀⠀⡟⣿⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⡷⢲⣿⣿⣿⣿⣇⠀⠀⠀⣧⣿⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠶⠒⠆⢻⣿⣿⣿⣿⡆⠀⠀⣿⡿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠒⠛⠑⠴⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡷⠛⠀⠰⢿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣴⡀⠀⠂⣦⣨⣭⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢀⣠⣤⣷⣽⣿⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠟⠃
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⢸⠉⢉⣍⠉⣩⣭⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⢤⡀⠀⠀⠀⣀⣀⠀⠀⠀⣠⠖⢻⣇⣀⣀⠀⠀⠀⢀⣤⣤⡤⠤⣤⡀⠀⣿
⠀⠀⠀⢸⡇⢸⠉⠟⢁⣿⣀⣠⡤⠖⠛⠉⣻⣭⣟⠲⠤⣄⡾⠁⠘⠳⣶⠋⠙⢧⣌⣽⣶⡤⢷⡄⢀⣤⠼⢧⣀⣠⡾⠋⣾⣵⠀⢶⡿⣁⠀⢻
⠀⠀⠀⢸⡇⢈⡷⠶⠖⣫⡀⠀⠀⠀⠀⣞⣥⠀⡾⠃⠀⠀⢷⣀⣸⡟⠉⠀⢠⡏⠀⠀⢬⡗⠀⠙⠛⠃⠀⠀⣠⡶⠛⢆⠀⠈⠳⣼⠃⠘⡆⢸⡇
⠀⠀⠀⢸⡇⣼⠓⣶⠊⠠⣯⣄⠀⠀⠀⠉⠘⠶⠏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢳⡞⠦⣬⡇⠀⠀⠀⠀⠀⠀⢻⠃⣴⡼⠀⠀⠀⣀⡤⠾⣤
⠀⠀⠀⢸⡇⠸⣄⢻⣤⠷⢦⡼⠇⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⢰⣧⠀⢠⣬⡷
⠀⠀⠀⢸⡇⠀⣽⣷⣤⡀⠀⣀⠀⣴⢶⡇⠘⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡼⢹
⠀⠀⠀⢸⡇⣼⠿⢿⠝⣹⠊⠈⠳⣬⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡟⠰⡆
⠀⠀⠀⢸⡇⠘⢀⡀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⠒⢤⣷⡀⡇
⠀⠀⠀⢸⡇⠀⣸⠹⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢡⣄⣿⢣
⠀⠀⠀⢸⡇⠸⣯⣤⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⢁⡈⠁⡼
⠀⠀⠀⢸⡇⠀⠈⠉⠀⣹⠦⣤⠀⣠⣄⢠⢶⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢛⠊⠀⡄
⠀⠀⠀⠘⣧⠀⠀⡴⠛⠉⠀⢸⣤⠏⠹⣼⠿⠟⠁⠀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢸⣀⡀⠇
⠀⠀⠀⠀⣿⠀⣈⣧⣤⣤⣤⣤⣄⣀⠀⠀⠀⢀⣴⢟⣽⠟⣿⢟⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠃
⠀⠀⠀⠀⣿⠚⠋⠉⠀⠀⠈⠉⠉⠛⢷⡀⣴⠋⠀⠞⠁⠞⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠶⣤⣄⢠⣄⣤⠀⠿⢁⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢧⠄⢰⡆
⠀⠀⠀⠀⣿⠀⠀⣠⡴⠛⠉⠁⠀⠀⣀⣤⣶⠿⠶⠿⣧⣄⠶⣤⡀⠉⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠭⠉⡷⠀⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⣠⠞⠉⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠈⢻⣆⠈⠻⣦⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⡴⠋⣷⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢻⣇⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣀⣄⣠⡶⣆⠘⢧⣠⠾⠃⠐⢯⡀⠀⣸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠾⣍⣀⡉⠀⣼⣄⡀⢻⠳⠶⣤⡴⠟⠀⣿⠁
⠀⠀⠀⠀⢸⡇⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣿⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠉⢿⣤⡏⠀⠙⣾⠃⣠⠴⠶⢤⠀⣿
⠀⠀⠀⠀⢸⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢰⣿⠀⢀⣿⠓⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⢃⣠⣌⠉⠀⣿
⠀⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⣼⡏⡰⣿⠃⠀⠀⠘⢧⠀⠀⠀⢠⠞⢳⡀⠀⠀⢀⡾⣴⠏⠀⢿⣄⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⣼⡟⠀⠀⠘⣧⡠⣦⣀⣘⣧⣀⣀⡇⠀⠘⠛⢛⣶⡍⠀⣏⣰⣄⡀⠘⡆⣿
⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣈⣛⣃⠈⠁⠀⠈⠉⠘⢦⣀⡘⢿⡁⣀⣀⣙⣁⣈⣉⣛⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠿⠉⠛⠻⡿⠉⠿⣦
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠀⠁
⠀⠀⠀⢀⣴⣿⢿⣿⣿⣿⡟⠁
⠀⢀⣴⣿⣿⣿⣷⣤⡀⠉⠀⠀⠀⠀⠀⠀⠀⠒⠶⠦⠀⠀⠀⠀⠀⠀⣀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⢰⣄
⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠈⣿⣦
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⣻⣿⣷⡀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠈⠿⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠹⠏⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⢿⣿⣦⣀⣀⣀⣠⣾⡿⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣼⡶⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠰⢾⣼⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⣿⠉⠙⠉⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣴⣶⣿⣿⣿⠿⠿⠿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠛⠿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠟⠋⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⣶⣦⣴⠁⣠⣎⠉⢉⣦⠀⢳⣤⣶⡾⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣶⢿⡔⠛⠛⠓⠛⠛⢃⣾⢷⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢉⣴⡾⢻⡷⠀⣾⡄⠶⣿⠻⣦⣌⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⢠⣿⠀⡘⢿⠟⡀⢸⣧⠘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢼⠅⢻⠊⣷⢠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⢾⠗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠿⣿⢤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡖⢻⡆⠠⠈
⠀⠀⠀⠀⠀⠀⠙⢗⠒⠒⠶⠼⣧⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠁⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠉⠁⣿⡿
⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣿⣴⣿⣛⣻⣿⡿⣿⡾⣦⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡵⢌⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠲⠤⠶⣼⠇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡀
⠀⠀⠀⠀⠀⠰⠧⠐⠒⠲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⢐⣃⣹⣿⣠⡤⠬⣽⡿⠶⠖⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⢀⣿⣄⣠⣴⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣻⡷⣧⣙⠤⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠾⢿⣛⣿⣿⣿⣿⣍⡉⣴⣜⣯⡾⠞⠓⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⠀⢻⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⠛⠻⣿⣍⢙⡿⢿⣿⡇⢠⣿⡿⣼⢀
⠀⠀⠀⠀⠀⠀⣨⠿⣿⣿⣿⣯⣮⡀⠛⢛⣤⢀⠈⠛⠲⢿⡟⠓⢿⣇⠳⣄⡀
⠀⠀⠀⠀⢠⣶⣿⠆⠠⠀⠘⠉⠉⠉⠒⠀⠀⠈⠠⠂⠀⠠⠄⠀⢘⠻⠿⢮⣉⠁⢤⣴⣶⣤⡤
⠀⠀⠀⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣤⣽⣄⣈⠙⠳⠦⠀⠀⢀⣠⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠋⠉⠐⠀⠈⠙⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣧⣸⣯⣝⣿⣭⣉⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⣴⠂⠀⠀⠀⠀⠀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠈⠛⠀⠂⠛⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣆⠀⠀⢀⠴⣿⡠⠀⢸⢟⡀⠀⢹⣿⣿⣦⣀⡀⠈⠀⠲⠀⠀⠀⣰⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣿⡷⠶⠀⠀⡀⠀⠠⢶⣿⡁⠀⠀⠈⡈⠛⠿⠛⠛⠂⠀⠀⢀⣾⣿⣿⣿⣷⡶⠗
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⠰⠀⠀⠈⢾⣿⣁⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠈⠋⠀⢀⣴⣏⠀⠀⠐⢿⣿⣿⣿⣿⣟⠁⠀⠀⠀⡄
⠀⠀⠀⣀⣀⣤⣾⣿⣿⣿⣶⣤⠁⠀⠟⠿⠿⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠉⠉⠀⠀⠀⣼⣿⣿⣿⠟⠉⠀⠀⠀⠈
⠀⠘⠛⠋⠽⠟⠿⠛⠻⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠻⠿⣷⣦⣀⠀⠀⠀⠿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄⠀⢀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣇⢰⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠈⣆⠃⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡆
⠀⠀⢀⣠⣤⡀⠘⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⠇⠀⣠⣄⡀⠀⠀⠀⠀⡠⠤⣄⠀⣇⠀⠀⠀⢀⠇⡆⠀⠀⠀⢰⠁⣀⠤⣄
⠀⢀⣾⣿⣿⣷⡀⠈⠛⠿⠟⠁⠀⠛⠿⠟⠋⠀⣼⣿⣿⣿⡄⠀⠀⡞⠀⠀⠈⢣⠈⠢⠤⠤⠞⠀⠱⠤⠤⠔⠁⡰⠁⠀⠀⢳
⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⢀⣀⣄⣀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⢰⡃⠀⠀⠀⠀⢧⠀⠀⢀⣀⠤⢄⣀⠀⠀⡰⠁⠀⠀⠀⠀⡇
⠀⠈⢻⣿⣿⣿⡿⣁⣴⣾⣿⣿⣿⣿⣿⣦⣄⢿⣿⣿⣿⡿⠃⠀⠀⢣⡀⠀⠀⢀⣜⡤⠚⠁⠀⠀⠀⠈⠑⠦⣣⡀⠀⠀⠀⡼
⠀⠀⠀⠈⠉⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠙⠉⠀⠀⠀⠀⠀⠉⠒⢚⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡒⠒⠋
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠻⢿⣿⣿⣿⡿⠿⠛⠛⠿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⣀⣀⠤⠤⢄⣀⠀⠀⢀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡟⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠉⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠆⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠚⠈⠀⠈⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣀⣀⣄⠀⠀⢄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡴⣿⢻⣏⢙⣁⠀⠈⠈⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠵⠓⠛⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡠⠀⣸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠈⣤⡀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣀⣴⣺⡥⢿⣤⣉⣰⣄⣠⣨⣦⣼⣅⣀⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⡴⠞⣛⣩⡤⠷⠞⢻⠚⠛⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⢿⠚⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣃⣢⠘⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡄⣘⡿⣿⣿⣿⣿⣯⣝⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⠞⠁⣹⣁⡟⣻⣔⣻⣿⣿⣿⣷⣮⣟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⣦⣤⢿⣻⣮⣷⣿⣿⣿⣿⣿⣿⣿⡏⠈
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠀⠛⠉⠉⠙⢿⡿⠋⠄
⠀⡂⠀⠀⠀⠀⢤⠤⣤⣤⠤⠶⠦⢤⣤⣬⣿⢿⠿⢁⣤⡀⠀⢀⢢⣼⡇⠀⡄⠖⠠⣤⠄⠀⠀⠀⠠⠶⠦⠠
⠀⠂⠀⠀⠀⠀⠀⠀⠡⢶⣾⢂⣶⢶⣦⣤⡤⠀⠐⢺⣽⡿⠈⣽⣾⣿⣇⣀⠻⢿⣿⣃⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢶⣶⡬⢙⣠⣴⠚⠿⠃⠀⠈⣛⣯⡌⠻⣿⣦⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢮⡟⣤⡀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠁⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⣱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠂⣨⠀⠀⠀⠀⢺⡞⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⢀⡲⢨⣶⡀⠀⠀⢀⠀⠀⠈⠘⠛⠿⠉⠉⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠉⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠶⠶⠶⠶⣶⣤⡄⠉⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣿⣿⣡⣿⡟⠒⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⣠⣴⣶⣤⣾⠾⣟⡿⣷⣦⣍⡑⠲⣤⣄⣀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣯⣟⣿⢷⣦⢾⣏⠡⣿⡤⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣶⠀⣰⣿⣶⣾⣭⣟⣻⢿⣿⣾⣿⣿⣾⣕⢻⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢹⣿⣿⣿⣿⠆⣿⣏⣹⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡷⣿⠿⣟⢿⣆
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠿⠿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣏⠄⢹⣷⣽
⠀⠀⠀⠀⠻⢮⣝⣿⣿⣿⡯⠿⠚⠛⠉⠉⠙⠓⠮⢽⣛⣿⢿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣷⣄⠿⠋
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⠯⠭⠥⠾⠋⠀⠈⠻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠇⠀⠀⢰⣶⠆⠀⠀⣤⣄⠀⠀⢀⣀⠀⠀⢀⣠⠀⠀⠀⣴⡦⠀⠀⠰⡾⠆⠀⠀⢲⣶⠄
⠀⠀⠀⠀⠀⠀⠾⡿⠀⢤⣄⠀⠀⢼⡦⠀⠀⢠⣦⠀⠉⠁⣀⡀⠘⠛⢀⣀⠈⠛⠁⣤⣄⠈⠀⢰⡶⠀⠀⠀⣷⠆⠀⢀⣄⠀⢀⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠘⠛⠃⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠑⠁
⠀⠀⠀⢰⣿⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⣀⡀
⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃
⠀⠀⠀⣤⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠉⠁⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇
⠀⠀⠀⣀⡀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆
⠀⠀⠀⠛⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡄⠉
⠀⠀⠀⢿⠧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀
⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠘⠟
⠀⠀⠀⠀⠀⢴⣦⠀⢠⣶⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⣢⡄⠀⠀⢲⣶⠀⠀⠀⣴⡆⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⢠⣤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣀⠀⠙⠇⢀⣀⠙⠛⢀⣤⡀⠉⠀⣴⣦⠀⠀⢰⣶⠀⠀⠀⣤⡄⠈⠋⢀⣀⠘⠻⠃⢀⡀⠉⠀⠘⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠛⠛⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠚⠛⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⣴⠆⠰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⣾⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢲⡯⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠋⠉⠀⠂⡀⠉⠀⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠠⠄⢀⣔⡏⠀⢰⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢿⣇⣀⢸⡇⢰⡶⣄⣀⠐⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠈⠙⡛⣸⠃⠀⠉⠙⠻⡾⠧⣤⣛⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⣤⠄⠀⠀⢨⣵⡏⠀⠀⠀⠀⠀⢹⡄⠀⠈⢿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠚⠂⠲⠀⡜⣨⡿⠀⣀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠛⠛⠿⠿⣿⣿⣾⣶⣿⣥⣶⣿⣿⡛⠛⠻⠿⣿⣆⡉⢰⡦⢿⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢩⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠯⠿⣿⠃⠀⠀⠀⠀⠀⠉⠙⠳⢾⡧⣤⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠚⣇⠀⠀⠀⠀⠀⠀⡠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣠⣾⠟⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠈⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣰⣃⢋⢌⣼⣦⣗⣴⣷⣧⣾⣿⠋⠀⠐⠇⠴⠃⠀⠀⠀⠀⠀⢰⣽⠃⠀⣀⣀⠐⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣤⣬⣅⣟⣻⣟⣿⡿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣶⣤⡀⠀⡀⣀⢀⣼⡿⢀⣴⠟⠉⠓⢢⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣉⡉⠛⢻⡿⣿⣿⣷⣮⣿⠋⠰⢟⡿⠿⡿⣿⡿⣿⣷⣦⣤⣤⣽⣿⣿⣷⣿⣧⣦⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⠷⢤⢄⣉⠛⠃⠀⠀⢈⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⠆
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣠⣞⣛⢻⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠏⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⣀⣸⠃⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⡅⡟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣭⡤⣄⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⢂⠀⠀⠀⢴⢞⣛⣩⣥⣶⣤⣿⣿⣿⣿⣿⡿⣿⢟⣩⣾⣏⠀⠀⠉⠿⢿⢿⡿⠿⠿⢿⡅⠾⢿⠀⠀⠂⠀⢀⣤⡞
⠀⠀⠀⠀⠀⠈⠳⣤⣀⣐⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣟⣛⠛⠿⣿⡅⠀⠀⠀⠀⢠⠷⣍⠢⣶⣿⣧⣀⢤⣀⣀⣰⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠐⠓⠶⢚⣋⣩⣭⣭⣭⣏⣉⡛⠿⣦⣝⠂⠀⢐⣕⣾⣶⣮⣷⣮⣽⣿⣿⣿⣿⣿⣟⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⣀⣁⣐⣨⣽⣿⣿⣿⣿⣿⣿⣿⣄⠈⠛⢷⣄⠀⠉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣭⣭⣤⣶⣾⣿⣿⣿⣿⢿⡏⠁⢾⣆⠙⠶⣤⣈⣛⠿⠿⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢈⠙⢿⣿⣧⡍⠙⠀⠶⢿⣷⣴⣤⣉⣉⣛⡛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠻⠶⠦⠉⠀⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃

⠀⠀⠀⠀⠀⠀⣠⣀⢀⣀⠀⡀⠀⠀⣤⠄⢠⡀⢀⡀⡀⠀⠀⢀⢀⢀⠖⠀⠀⠄⢠⣤⠀⢀⠀⢀
⠀⠀⠀⠖⠘⢻⠆⣀⠈⠋⢴⠋⣠⠀⣰⠚⢋⣤⡿⠟⢠⣄⠉⠑⡀⠈⢠⠴⠖⠦⠙⠸⠗⠘⡛⠿⠗⣦⡀
⠀⠋⠓⠉⠁⠀⠉⠒⠒⠁⠀⠠⠤⠀⠳⠿⠶⠲⠖⠚⠉⠁⠈⠉⠙⠁⠉⠉⠂⠀⠚⠚⠓⠀⠈⠐⠢⠐⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣀⣴⣶⣀⣄⣤⠟⠁⠀⢀⠀⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠛⠛⠉⠻⣿⣿⡃⠄⠀⠀⠘⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⢚⣿⣿⡿⣿⣧⣶⠷⠰⠰⠆⢉⣂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢰⣷⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣠⡟⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣣⣴⠿⠿⠿⠿⠟⠛⠋⠻⣿⣟⣷⣦⣁⣴⡆⡀
⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠐⣷⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣯⠽⠛⠋⠁
⠀⠀⠀⠀⣠⣤⣽⡟⣣⣠⣿⠿⠛⠛⠁⠀⠀⠀⠀⠀⣀⣴⣾⠿⠟⠁
⠚⠛⠓⠚⠋⠀⠀⢰⠈⠉⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠋⢻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⣠⠎⠀⠀⣀⣀⣀⣤⡎⢿⣿⣿⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⣾⡧⠀⠀⠀⠘⢧⣴⣾⣿⣿⣿⣿⣿⣎⣄⣠⣀⢻⣶⣤⡶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣤⠟⠁⠀⠀⠈⠛⢿⣿⣿⣿⣟⣿⣿⣿⣯⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢛⣁⠀⠀⡀⠄⠀⢀⣾⣿⣿⡟⡋⠋⠉⠉⠁⠁
⠀⠀⠆⠀⠀⢤⠤⡶⢲⣶⣶⣬⣶⣿⣇⣰⣿⣿⣟⣛⣷⠒⢸⣿⡿⠁⠂
⡠⠄⠀⠈⠀⠀⠶⠀⠈⢛⣿⣿⣭⣵⣾⣿⣿⣭⣭⣒⣐⣠⣯⣭⡅⠀⣀⢠⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠿⠿⠿⠟⠛⠟⠛⠿⠿⠛⠛⠛⠛⠛⠻⠛⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⣸⣬⡱⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢃⣘⠋⣴⡿⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠉⠉⣷⣠⣤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣶⣙⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠟⠙⠋⠀⠀⠀⠀⡿⡘⢃⣤⠀⠀⠀⠀⠙⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡸⠆⡶⠀⠀⠀⣶⠸⠇⣤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⣿⣿⣧⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣷⡶⠇⠀⠰⣾⣿⣆⣥⠀⠀⠀⠀⢨⣧⣅⣃⠀⠀⠀⠈⠛⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠙⠁⠀⠀⠀⠈⠛⠁⠀⣀⢀⡄⠀⠀⠙⠋⠁⠀⠀⠀⢀⣤⣺⠟⢴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣌⠱⠶⠀⠀⢠⠴⢆⡀⠀⠈⣿⣷⡗⠰⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣀⠀⠸⡿⣘⡋⢾⠿⠀⠀⢠⠄⠶⣀⠘⢻⣿⠷⠞⠀⠀⠾⣿⣮⠄⠀⠀⠻⠝⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⠄⠀⢴⣿⣿⣿⣦⣶⡆⠀⢾⣿⣎⣍⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡿⠃⠁⠀⠀⠀⠉⠀⠀⠀⡄⢦
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠆⢆⣀⠀⠀⠀⠠⡿⢰⠖⠀⠀⠀⠀⠀⠛⠓
⠀⠀⠀⠀⢤⡜⣃⣿⡂⠺⣿⡦⠅⠀⠀⠸⡏⠙⡆⠚⠛
⠀⠀⠀⠀⠰⠶⣿⠿⠋⢀⠀⣀⠀⠀⠀⠀⠘⠞⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣙⢫⣤
⠀⠀⠠⣔⡰⡀⠀⠀⠘⢿⣿⢶⡦
⠀⠀⠀⠛⠛⠁⢀⠀⠀⠀⠁
⠀⠀⠠⢀⠀⠈⣷⡬
⠀⠐⠻⠗






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠚⠋⠁⠀⠀⠀⠄⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠖⠚⠉⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢳⡄⠢⡈⠢⠀⠱⡀⢀⠈⢄⠀⠀⠀⠄⠈⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠷⢦⣀⠀⠉⢻⡄⢱⡀⢣⠀⠘⠀⢆⠈⡆⠀⠀⢘⠀⠀⠀⠀⣾⢀⣀
⠀⠀⠀⠀⠀⣰⡾⠯⣤⣀⠀⠈⠳⣄⠀⢿⡀⠇⠀⠀⠀⠀⢸⠀⢸⠀⠀⢜⣀⠥⠒⠋⠉⠉⠉⠉⠉⠉⠙⠓⠒⠶⠤⣄⡀
⠀⢀⣤⣶⡛⠉⠛⠲⠶⣬⣙⠲⣄⡈⢧⠈⣇⠸⠀⠇⠀⡇⠸⢀⡠⠔⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣤⠞⠓
⠀⠀⠉⠉⠉⠙⠛⠲⠶⢤⣬⣙⠲⢽⣦⣣⣹⠀⠀⢃⠠⠔⣊⠁⡀⠀⠀⠀⠀⠀⠤⠤⣀⠀⠀⠀⠀⠈⣀⡤⠖⣋⡥⢶⣯⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣽⣿⠿⢶⣿⣷⣖⣒⠒⠶⠶⠶⠶⠶⠤⢤⣄⣀⠀⠁⣀⡤⠖⣋⣥⣶⡯⠵⠚⠋⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠉⠓⠦⢭⣙⡒⠶⠤⣍⡀⠀⠀⠉⣙⡿⢵⣲⠿⠛⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠀⠀⠀⠀⠐⠢⠄⠀⠉⠛⠲⠦⠬⣝⣚⡩⠵⠚⠉⠀⢀⣠⠴⠖⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠔⠒⠉⠀⠀⢀⣠⠤⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠠⣄⡀⠉⠓⠶⢤⣀⡀⠀⢤⣇⠀⠀⠀⠀⠠⠖⠛⠉⠀⠀⠀⠀⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠂⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠈⠀⢀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠠⠐⠉⠀⠀⠀⢀⣀⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠈⠀⢠⡅⢀⣠⠔⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠒⠦⣄⡀⠀⠀⠈⠙⠓⠦⠄⠀⠀⢰⡖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⠴⠒⣁⡤⠖⠋⠁⠀⠀⢀⣼⢛⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢦⡀⠑⠲⢤⣄⡀⠀⠀⠒⠦⢄⠀⢸⠃⠀⠐⠉⠀⠀⠈⠀⠀⠤⠒⣁⣴⠞⣫⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢮⣷⣤⣀⠈⠙⠳⠦⠀⠀⠀⢄⣸⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢞⣫⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣟⡶⣤⣀⠀⠀⠀⢀⢸⡇⡀⠀⠀⢀⣠⡶⢟⣩⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣽⣷⢶⣤⣀⢸⡇⢀⣠⡶⢛⣥⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢮⣝⠻⠟⣋⡵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⡀⠀⠀⠠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠄
⠀⠀⠠⠶⠀⠑⡀⠀⠀⠀⠀⠛⠙⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠋⠁⠀⠀⠀⠀⠀⠀⡠⠖
⠀⠀⠀⠀⢠⠀⣹⠆⣠⣴⣿⠏⠀⠀⠀⠀⡀⠀⢀⣤⣿⠟⠀⠀⠰⠂⠀⠀⠀⣤⠒
⠀⠀⢀⡄⣀⣽⡿⢺⠛⡏⠹⣦⠀⢀⣤⣦⣽⣾⣿⣿⣿⡄⢀⣴⠀⣀⠀⠀⠜
⠀⠛⠸⣾⣿⡁⠸⣶⣀⣻⣶⣿⣶⣿⣿⣿⣛⣿⣿⣿⣿⣷⣿⠇⡈⣷⣷⠏
⠀⠀⠺⠃⠙⠿⣶⠟⠙⠯⠁⣿⡟⠻⣿⡿⣯⣍⡉⠉⠙⠿⣿⣷⣫⣧⡟
⠀⠀⠀⠀⠐⠶⠉⠀⠀⠀⠀⢻⣿⣄⠀⠉⠂⠉⠉⠛⠶⢶⣾⣿⣿⣿⡁⠀⠀⠀⠀⠀⣰⡯⢗⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⢷⣬⣶⣼⣶⣤⣤⣤⣄⣽⣿⣟⣷⣞⡁⣶⠶⠀⡻⠚⢿⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⢿⣤⣀⠉⠙⣿⣿⣿⡟⣻⣿⣟⣴⣿⣿⣟⣾⠁⠀⠈⣤⠎⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣏⠿⠓⠀⠉⠀⠈⠀⢿⣟⠻⠿⢿⣿⣿⠇⠀⣠⡾⠋⣀⠀⠙⠛⠛⠓⢒⡶⠶⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡀⠀⣀⠀⠀⠀⢀⡌⠻⠷⢤⣿⣿⡛⠀⣰⣿⣤⠾⠛⠀⠀⠀⣠⣶⡏⠀⠀⠀⢰⣿⣿⡇⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠰⣿⠋⠀⠀⣸⣿⣟⠟⢰⣿⣿⣍⡀⣀⡄⣠⠞⠁⣾⡇⠀⢀⣴⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡞⣿⡀⢠⡾⢿⡏⣿⣶⠟⣻⢟⣿⣿⣿⣧⣿⡀⠸⡿⠋⠁⠩⣿⣹⣽⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⣾⠑⠋⢰⡾⢻⡟⠁⠸⠁⠸⢓⣿⠏⠛⠟⠃⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⡿⡉⢡⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡴⠸⣿⢟⣀⢀⠀⡰⠀⡸⠈⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣮⣁⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢷⣾⣿⣿⣿⣿⣾⠧⠂⣀⡤⠻⣷⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⣿⣿⠿⡣⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣽⣷⣶⣒⣉⣀⣠⡾⠛⠧⣌⡁⠀⠀⠀⠀⠀⠀⠐⢦⡴⢿⣽⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠋⣹⣟⠛⠛⠋⠀⠀⢄⠈⠿⠀⠀⠀⠀⠀⠀⠀⢸⡦⡀⠹⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣯⣈⠉⠁⠀⠀⠀⠀⠘⢘⡃⠀⠀⠀⠀⠀⠀⠀⠀⢊⣼⠋⠸⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠀⠈⠻⣿⡛⠒⠀⠀⠀⠀⠀⢸⡿⠦⠲⠶⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⢛⣷⣦⣄⡀⠈⠛⠳⠖⠒⠀⠀⠀⠸⡇⠀⢀⡴⠁⠀⢀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⣽⡓⠦⠄⠀⠠⠤⠄⠤⠀⣠⣴⠏⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢟⣡⣾⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⢬⣿⣿⣿⣿⣿⣦⣤⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠⡀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣧⣴⣠⣅⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣍⣑⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠋

⠀⠀⢾⣿⣿⡇⠀⠛⢿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣾⣿⣿⣿⡄
⠀⠀⢿⣿⣿⣇⠀⣠⣾⠟⠛⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣹⣿⣿⡟⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⣿⣿⣾⠟⠁⠀⠀⠀⠈⠙⠿⣿⡇⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣳⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠄
⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣤⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠰⠆
⠀⠀⣰⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠇⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣾⣷
⠀⠀⣿⠋⠈⠻⣿⣿⣷⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠁⠀⠀⠀⠙⢿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠉⠉⠁⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⣿⣿⣿⣿⣿⣷⣶⣯⣿⢡⡿⢹⣷⡨⠀⠀⠀⠀⠀⠀⠹⣿⣧⣨⣿⣿⣶⣿⣿⣿⣿⣿⣿⡟⡁⠀⠀⠀⠀⠀⡄⠀⠀⠁⣀⣠⣿⣶⠾⠿⠛⠛⠉⠉
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠘⣿⡇⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⣹⣿⡿⠁⠀⠀⠀⠀⢀⣤⣴⡾⠟⠛⠛⠡⠄⠀⠀⣀⣤⣶⡿
⠀⠀⠀⠈⣿⡟⠻⠿⣿⣿⣿⣿⡇⠀⢻⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡛⠉⠙⣷⡀⢀⣼⣿⡿⠁⠀⠀⠀⠀⠀⢸⣉⣀⡀⠀⠀⠀⠀⣀⣴⣾⡿⢟⣿
⠀⠀⠀⠀⣿⡇⠀⠀⢰⣏⣉⣿⣶⣾⣿⣿⡇⠀⠀⠀⠀⠘⢛⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣾⣿⠟⠁⠀⠀⠀⠆
⠀⠀⣤⣴⣿⣷⣶⣾⣿⣿⣿⣿⡿⠿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣟⠛⠉⠁⠀⠀⠀⠀⠘⠛⠛⢿⣿⣿⣿⣿⣿⣦⣼⡂⠀⢀⠎
⠀⠀⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⢻⣿⣄⣼⣿⡿⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠻⢿⣿⣿⣿⣿⣦⠋
⠀⠀⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠘⣿⣿⣿⡿⠁⠀⠈⢻⣿⡆⠀⠀⠀⠀⢠⣾⣿⡿⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⣠⣤⣤⣤⣤⣤⣤⣠⣄⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⢀⣤⣤⣤⣀⣤⡀
⠀⠀⣿⣿⣿⣿⣿⣿⡟⢿⠇⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣴⣶⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣧⣀⣀⣠⣼⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠿⣿⣿⣿⣿⣿⠀⡜⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡿⠿⠛⠛⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛
⠀⠀⢀⣿⣿⣿⣿⡇⠀⣰⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢻⡏⠁⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⢸⡟⣻⡿⠶⠶⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠈⣿⣿⣿⣿⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣬⣇⣀⡀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⢸⡇⢹⡇⠀⠀⢸⣿⣿⣿⣽⣿⣿⣿⣿
⠀⠀⣸⣿⣿⢹⣿⣴⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡄⠀⠀⠀⠀⢸⡇⣿⣧⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣧⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠙⠃⠀⠀⠀⠀⠀⠀⠀⣶⢶⡿⠟⠛⠛⢿⠟⠉⠉⠉⠛⠟⢿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣭⣥⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡶⢲⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⠉⠁⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠴⠂⢠⡤⢐⡺⡭⣭⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠸⠓⠲⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢠⣤⣤⢶⣶⢄⠀⠀⠀⣤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⡾⠛⣿⣏⠳⣾⡥⠤⠾⠛⠓⣿⡶⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⠃⢀⣠⠿⠒⠚⠉⠉⢻⠏⠀⠀⠀⠀⠀⠀⠀⣰⣦⡤⢶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡉⠀⡀⠛⡾⠋⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⣀⠀⠀⠀⣿⣿⠁⠀⢸⡇
⠀⠀⠀⡀⠀⠀⠀⠀⣾⣏⡉⣉⣤⡟⠀⢠⣦⠴⠶⢳⣶⣶⣜⣩⣭⣭⣿⣿⣶⣤⣹⣿⡄⠀⣼⡀
⠀⠀⠀⠈⠲⢤⡀⠀⢿⣿⣾⣿⠿⢇⣴⠯⠤⠴⠾⠛⠛⢿⣾⣿⠿⠟⠛⠛⠻⠿⣿⣿⡇⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠙⠢⣌⠻⡏⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⢧⣠⣦⣶⣦⣤⣀⡀⠀⣿⣷⣶⣦⣄⠀⠀⠁⣿⣿⣿⣶⣿⣿⣷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣽⣁⣀⣄⡀⣿⡿⣦⡉⠀⠈⢙⣿⠷⣦⡀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⢿⣿⣿⣿⣿⣧⣬⡙⠦⡄⠀⣿⠀⠹⣿⣷⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣯⢷⡿⢟⡛⠙⣿⣷⣯⣟⣿⢻⠶⠿⡷⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⣍⣻⣶⣿⣿⣿⣌⡿⢉⣿⡟⣷⣶⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢬⡘⢪⣿⡏⢉⠛⠻⡟⢻⣿⡿⢿⣿⣿⡿⢻⣿⠿⠟⢻⢿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠈⠉⠁⠀⠀⠐⠃⠀⠉⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣥⣤⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡀⠈⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇⠀⠈⠻⣿⣿⣿⣿⠇⠀⢠⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠉⠉⠁⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⣠⣶⣿⣷⣶⣄⡀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣴⣶⣶⣶⣶⣄⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠙⢿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡄⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡧⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⠠⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠠⠒⠈⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣀⢴⡤⠴⠓⠀⠀⠀⠀⢸⠀⠀⣀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⠐⠛⠉⠀⠀⠀⢀⡀⡀⠀⣈⣺⠼⠓⠚⠉⠙⣦⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠀⠀⠀⠀⠀⠀⢀⡰⠍⠋⠁⠀⢈⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⠉⠀⠀⠀⠀⠀⠀⢺⡇⠀⢈⣤⣤⣶⠶⠓⠒⠚⡂⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠿⠑⠛⠉⠁⠀⠀⠀⠀⠠⡄⣀⣠⣿⠇⠀⠋⠁⠀⠀⠀⠀⠀⠀⢰⣄⠀⠛⢧⣀⣀⣀⡀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣇⡯⠴⠾⣟⡋⠉⠩⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠈⠉⠋⠉⢁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡬⠉⢒⠈⠁⣥⠟⠷⠛⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⠾⡾⠿⢿⣀⣀⣀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠲⠶⠖⢒⡫⠈⠀⠀⠀⢸⡆⠀⡀⡴⢀⡄⠴⠒⠋⢹⡅⠹⣿⣤⣶⣾⣿⣿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠻⠈⠀⠀⢤⠀⠘⠀⠀⠀⠀⠀⠀⣷⠀⠀⡴⠛⠀⠀⠀⠀⣠⣷⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣫⣠⢴⣶⣶⣿⣿⣿⡿⢻⠿⠿⣿⣿⣿⣵⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣯⣼⣯⣤⣥⣾⣷⣖⡚⠻⢉⣑⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣭⣤⣤⣬⡁⠈⣿⣿⣿⣿⣦⡄
⠀⠀⠀⣶⣶⣤⣀⡀⠀⠀⠙⣶⣶⠒⢀⡤⣤⣶⣶⣶⣿⣿⣟⣛⣛⣛⠻⠿⠿⠟⠻⣂⣽⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣏
⠀⠀⠀⠈⠽⠟⠛⠿⢷⣶⣤⣬⣭⣶⠿⠿⠖⢸⡋⠋⠀⡉⣈⡉⠉⢉⣩⣄⣀⣴⣿⣿⣿⡿⣿⣿⣿⣿⠃⠟⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠁⠀⠀⠀⠓⠂⠈⠉⠁⠀⠈⠉⠉⠉⠉⠉⠈⠻⠛⠿⠛⢫⡿⠋

⠀⠀⠀⠀⠀⠀⢰⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠈⣿⡿⠃
⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀
⠀⠀⠀⠀⣠⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣴⣿⣿⠋⠀⠈⠉⢿⣿⣿⣿⣿⡇
⠀⢀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠉⠉⠻⣿⣿⣆
⢠⣿⣿⠟⢁⣠⣴⣿⣿⣶⣄⠀⠀⠀⠘⣿⣿⣧
⣿⣿⣿⣶⣿⣿⠿⠛⠻⣿⣿⣆⠀⠀⠀⠈⢿⣿⣷⡀
⠘⠻⠿⠟⠋⠁⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠈⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⠀⠀⠀⠀⠹⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⡀⢸⣿⣿⠉⠙⠛⣿⣿⣧⠀⠀⠀⡄⠀⠀⠀⢿⣿⣿⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⢸⣿⣿⡄⠀⠀⠘⣿⣿⣆⠀⠀⢿⠀⠀⠀⠸⣿⣿⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⠀⣿⣿⡇⠀⠀⠀⠸⣿⣿⡆⠀⣿⣧⠀⠀⢰⣿⣿⡄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣼⣿⡆⣿⣿⡇⠀⠀⠀⠀⠹⣿⣿⡄⣿⣿⣧⡀⢸⣿⣿⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣷⣌⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⢹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢠⣿⣿⣿⣿⡏⠀⠀⠀⠸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇⣽⣿⣿⣿⣿⠀⣰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿
⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣠⣄⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣠⡴⢶⣶⣖⡶⣒⠂⠤⣀
⠀⠀⣴⡋⠁⠀⠀⠀⠀⠈⠙⢾⣷⣄⠀⠙⠶⠒⠶⠒⠈⠉⠉⠑⠲⢄⡀
⠀⠀⠩⠀⡠⠀⠀⢀⡀⠀⠀⠀⢙⣿⡦⣄⠀⠀⠀⠀⠀⢠⣠⠦⠒⠒⠛⠆
⠀⢀⡆⢰⡇⣀⣠⣾⣿⡛⢶⡀⠋⠐⣰⠁⠀⠀⠀⠀⣠⠏⠁
⠀⠘⠓⢾⣿⠋⠉⠹⡿⠧⠖⢱⠀⠲⢿⣄⠀⢀⡴⠞⠁
⠀⠀⠀⠘⠛⠀⠀⡀⠀⠄⠀⢈⣷⣄⣘⣷⡾⣭⣤⣀⣀⡀
⠀⠀⠀⠀⠑⢄⣀⣁⣀⡠⢴⠟⢙⡏⢻⠁⠀⠀⠀⠀⠀⠈⠓⠦⠤⣄⣀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠙⠛⠃⢠⡞⣆⣸⠀⠀⠙⠒⣶⠀⠈⠉⠓⠢⣀⠀⡼⢀⡽⢋⣽⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠈⠁⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⠉⠁⠠⠃⣾⣿⡛⠯⢭⣛⣓⣶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢣⡀⠀⢀⣠⣶⣋⣿⡿⠦⣄⠀⠀⠀⠀⠀⠙⠋⢀⠉⠒⢄⠹⡁⠄⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣿⣴⣧⣿⡿⠋⠁⠀⠀⠈⠑⠒⣤⠀⣀⡠⠞⠻⣝⢦⡀⠈⠑⠒⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠉⢠⡾⠋⠈⡟⠁⣀⡴⠆⠀⠀⢠⣶⡟⣡⢮⡀⠉⢦⠀⣨⢆⢳⡀
⠀⠀⠀⠀⠀⠀⣼⣷⣻⣷⠾⠀⠀⠀⢇⡴⠃⠀⠀⢀⠔⠁⠛⠉⠀⠀⠱⣠⣜⡠⠧⠤⠧⠃
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⠀⠀⠀⢠⠋⠀⠀⣀⠔⠛⠢⠄⠤⠄⠀⠊⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠛⢦⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣷⣄⣼
⣿⣿⣿⣷⣶⣄⡀⠀⣾⣿⣿⣿⣿⣷⣿⠋
⣿⣿⣿⣿⣿⣿⣿⡄⠘⠻⠿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣧⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟
⠀⢀⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟
⠀⠠⢾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⣠⣾⣿⠀⣼⣧⡀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⣸⣿⣿⣯⣀⠀⠀⠀⠀⣀⣤⡶⢶⣿⠃⠀⠀⠀⠀⠀⠈⠉⠙⠀⠙⠉⢸⠀⠀⠀⢀⣴⠟⠋
⡤⢶⣿⣿⣿⠷⠖⠒⠒⠛⣟⠋⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠉⠀⠁
⢶⣿⣿⣿⣿⡿⠦⠀⠐⠈⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀
⣻⣿⣿⣿⣿⣿⣶⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⣿⣷
⣿⡛⢻⣟⣿⣿⡿⠷⠶⠤⠄⠄⠤⠤⠤⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣠⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣠⣄⣤⣤⣀⣀⣠⣤⣤⣤⣬⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣏⢹⣟⠿⣿⠟⢟⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣯⣝⡛⠻⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣍⠒⢭⣛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⡈⠙⠮⣝⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⠈⠛⢮⣻⣷⢶⣶⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣴⠟⡡⠀⠙⢷⣌⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠡⠚⠁⠀⠀⢀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⣦⡀⠀⢀⠔⣡⣾⡟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⣴⠟⠈⢻⣦⣠⣾⣏⠀⠹⣮⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣣⠞⠁⣀⣴⠟⠉⠉⠀⠙⢷⣄⠈⢷⡹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠻⣧⡀⢳⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠱⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡁⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡌⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⠞⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣫⠞⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣴⡾⣫⠞⠁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣌⠁⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣼⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢛⣛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣖⣯⠭⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⢲⡄⣿⠁⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣖⣾⣿⣧⡿⠀⠀⠀⢀⣀⣸⣧⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⢀⣀⣿⡷⠶⢚⣛⣉⣉⣀⣠⣤⣤⣤⣤⣤⣤⣤⣽⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣰⡏⠉⣟⣋⣉⣭⣭⣥⢤⢤⢶⢶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣬⣭⣉⠙⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⠶⠖⠚⡛⠉⠩⢹⡇⠀⡿⠷⠷⠓⣛⣛⣛⣉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣉⣛⣛⣤⠇
⠀⠀⠀⠀⠀⠀⢸⣐⣀⣁⣩⣬⠤⠴⢖⣒⣿⢀⠉⡟⢛⣛⣉⣉⣩⣭⣤⢤⡤⣤⢤⡤⡤⡤⣤⢤⢤⣤⣤⣬⣽⡇
⠀⠀⠀⠀⠀⣯⣭⣵⣶⠶⠶⠶⠿⢟⣛⣛⡿⠀⠤⣿⢹⡇⡇⢸⣀⣇⣸⣸⣇⣿⣼⣥⣧⣧⣿⣸⣼⣿⣷⠿⠿⠗⠚⠲⣦⡀
⠀⢀⣤⣴⠚⡓⢒⠒⡉⣉⢉⡉⣉⠁⠄⠠⠀⠆⢀⡟⠛⠛⠋⠉⢉⣁⣀⣤⣤⠤⠶⠶⠚⠛⠋⠉⠉⠀⠀⠀⢀⣀⣀⣀⣟⣿⠆
⠀⠘⠦⣤⠤⠥⠬⠤⠤⠤⠶⠶⠶⠖⠒⠒⠛⠉⠉⠙⠒⠚⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠒⢚⠋⠉⠉⣁⣸⡿⠃
⠀⠀⠀⣇⣐⣀⠀⠀⠀⠀⠰⠐⠐⠂⠂⠂⢀⣀⣁⣈⣀⣠⡤⠤⠤⠶⠖⠒⢛⣉⣉⣡⣤⠤⠴⠶⠒⢾⣿⡟⠉⠉⠉⡿⠁
⠀⣀⣀⢸⡟⠛⠛⠿⠿⠿⠿⠭⠭⠭⠭⠭⠭⠭⠴⠶⠖⠒⠒⠒⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠳⣌⣯⣤⡄⠀⢸⠇
⠈⠉⠉⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡿
⠚⣉⣱⣢⣿⣽⠷⠲⢶⣦⠤⢤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡷⢤⣤⠖⠲
⠚⠁⠀⠈⠑⠦⣀⣀⡞⠙⠻⢷⡲⠶⠋⢈⡽⠿⠭⣭⣽⠛⠻⢷⣒⠒⢒⡖⠶⣖⠒⠲⢲⡶⢶⡒⠒⢒⣿⡿⠭⣿⠛⠳⣤⣀⣀
⠀⠀⣠⢤⣀⡀⠀⠀⠀⠀⠀⠀⣈⡛⠒⠋⠀⠀⠀⠀⢈⣉⡓⠒⠚⠛⠋⢀⣀⠈⠙⠲⠶⠃⠀⠉⠳⢮⣍⡽⠂⠈⡷⠦⢤⡉⠁
⠉⠉⠁⠀⠀⢈⣉⣉⣉⡒⠒⠚⠉⠉⠙⠒⠦⠤⣄⡤⠞⠉⠉⠛⠲⠦⠤⠜⠉⠙⠒⠦⣄⣀⣠⡟⠙⠲⠤⣄⣠⠴⠃⠀⠀⠙⢦
⠀⣀⣠⡴⠞⠋⠁⠀⠀⠉⠙⠒⠤⣄⣀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠉⠒⠒⠤⢤⣀⡀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠉⠓⠒⠤⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⣴⣶⣄⠑⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⣧⡀⢿⣿⣿⡇⢀⣤⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠈⠻⣿⠇⠸⣿⣿⡆⢱
⠀⠀⠀⠀⠀⠀⢠⣶⣄⠛⠿⠃⣴⣶⣤⡀⠀⠹⠿⠇⡸
⠀⠀⠀⠀⠀⠁⠸⣿⣿⡆⠀⢰⣿⣿⣿⣿⣷⣦⣀⠰⡁
⠀⠀⠀⠀⠀⠈⢄⠙⠿⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⢸⣿⣿⡿⠟⠋⣉⡉⡠⠞⠀⠀⠀⠀⠀⠀⡠⠀⠠⠁⣴⣿⣿⠀⠉⣠⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣉⣉⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⣾⡆⢰⣿⣿⡿⢀⣾⣿⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⣿⣿⠇⠈⠛⢛⠀⢸⣿⣿⡟⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠛⠃⢀⣠⣶⣿⣿⡄⠙⠉⣴⣿⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⣾⣿⣿⣿⣿⣿⣧⠀⢰⣿⣿⠇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠻⣿⣿⣿⣿⣿⣿⣿⡆⠈⠛⣁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⡉⠻⢿⡿⠇⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠢⠄⠂⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⣿⣿⣿⡿⠋⠀⢠⣿⣿⣿⣿⠟⠁⠠⠀⢀⣤⣶⣿
⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠆⣀⡠⠀⠀⠀⠀⠀⠀⣞⣡⣾⣿⣿⣿⡟⠁⠀⢀⣾⣿⣿⡿⠁⢀⠀⢀⣼⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣾⣟⣉⣠⣤⠴⠂⢀⣠⣾⣿⣿⣿⣿⡿⠋⠀⠀⢠⣾⣿⣿⠟⠀⡠⠂⣼⣿⣿⣿⣿⣿⠟
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⣀⣴⣿⣿⣿⣿⣿⣿⠟⠡⠀⠀⢀⣿⣿⣿⠏⠀⠀⢠⣾⣿⣿⣿⣿⡿⠃
⠀⣀⣴⣿⣿⣿⣿⠿⣛⡋⠉⠁⠀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣠⣾⣿⡟⠁⠀⠀⣰⣿⣿⣿⣿⣿⡟
⣼⣿⣿⣿⡿⠋⣡⠟⠁⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⣠⣼⣿⠟⠋⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠀⠀⠉
⣿⣿⡿⠋⢠⣾⠟⠀⢠⣴⣿⣿⣿⣿⡿⠿⠿⠛⢛⡿⠉⠀⠀⠀⠀⣀⣴⣿⣿⠟⠁⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⠟⠀⠀⠠⠀⣠
⣿⡟⠁⣰⠟⠁⠀⣰⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⠿⠛⠉⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣴⣿⣿
⠋⠀⣸⠃⠀⣠⣾⣿⣿⡿⠋⠑⠂⠀⠀⣤⠶⠾⠿⠛⠛⠋⠉⣁⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⣿⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟
⢀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠄⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⠉⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠋
⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠋⠁⠀⠀⢀⣠
⠋⠁⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⡿⠿⠛⠉⣀⣠⣤⣶⣾⣿⡿
⣀⣀⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⠟⠋⠀⠀⠘⠛⠛⠛⠉⠙⠉⠁⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣶⣾⣿⠿⠛⠋⠁⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣭⣤⣤⢤⡀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠉⠉
⢀⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠉⠀⠀⢀⣀⣀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶
⣉⣭⣥⣤⣶⣶⣶⣶⣶⣶⣶⣶⣿⠿⠿⠛⠛⠉⠀⣠⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠟⠛⠉⠉⠉
⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⠉⠀⣀⣠⣴⣶⣿⣿
⣿⣿⡿⠷⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢀⡤⠂⣠⣾⣿⣿⡿⠋⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣴⠋⢀⣾⣿⣿⣿⠏⠀⢀⣾⣿⣿⣿⣿⣿⡿⠿⠛⠉
⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⣼⠇⢀⣾⣿⣿⣿⡟⠀⢠⣿⣿⣿⣿⣿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡀⠀⠀⢠⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⠁⠀⠀⢀⣔⡀⣀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⢘⠀⠀⠀⠀⠈⠀⠈⡁⠀⠀⠀⠀⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡀⠀⣺⠞⠂⠀⠀⠁⣀⠀⠀⠀⠀⠈⠀⠀⡠⠂⠀⠀⠘⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠄⠀⠘⠓⠀⠘⠛⡀⠀⠀⠀⠈⠰⠄⢀⠔⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⢠⠐⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠒⠀⠤⡀⠀⠀⠀⢀⡄⠀⠎⡄⠀⡴⠀⠀⠘⠠⠀⠀⠀⠀⠀⠈⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⠄⠀⠀⠀⠀⠀⠀⠞⡀⠀⠀⠀⠀⡠⣢⠄⠠⠀⠀⠀⠆
⠀⠀⠀⠀⢠⡄⠀⠀⠀⣤⠀⠓⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡀⣶⡤⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠋⠀⠀⠀⠀⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠈⠁⠘⠂⠀⠀⢀⣤⠂⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠚⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⣠⠴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⡴⠋⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⣼⠁⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⡇⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡇
⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⠴⠟⣒⢉
⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠾⢟⣛⣯⠭⠗⣒⣋⠭⠤⢰⡆
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⢀⣠⣤⣶⣿⣿⡯⢵⣒⣻⠭⠖⠚⢋⣉⠥⠖⠒⢉⠈⡇
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠈⣣⣴⣾⣟⣋⠭⠥⠖⠒⠋⣉⣠⠤⠖⢒⣋⡭⠤⠖⣚⣉⠥⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⣠⣾⣷⡫⠤⠤⠤⠶⠖⢚⣋⣩⠥⠖⠒⣋⡩⠤⣒⣚⠭⠕⣒⣀⡄⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣴⡿⠁⠤⠤⠒⠒⠒⢚⣉⣉⡤⠴⣒⣫⠭⠗⢚⣉⣥⡴⢾⣛⣯⣵⡦⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣿⠇⠒⢒⣚⣋⣉⡭⠭⠥⠶⣒⣩⣥⣴⡾⢟⣫⣭⣶⠾⠋⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣯⣭⣀⣀⠤⠖⠒⢚⣉⡭⢵⣚⣭⣷⠾⠟⠛⠉

⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉
⣏⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠻⠟⠿⢻⣿⡿⠛⠛⠛⠛⠛⠛⢛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⢿⣿⣿⣤⣶⣤⣦⣤⣤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠁⠀⠰⣷⠀⠀⠀⠀⠀⢐⡈⠙⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢰⣶⣿⣦⣀⠀⠀⠉⣉⣋⣀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⠛⢚⣳⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⣀⠴⠒⠉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢾⠘⣿⠳⣄⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠠⠀⠃⠹⣧⠈⢃⠄⠀⠀⠁⠀⠀⠀⠀⠀⢠⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣼⡏⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠁⠐⠀⠀⢻⣆⠀⠀⣱⣇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣶⣶⣦⣰⣤⡠⢤⣤⣌⣿⣿⣷⡄
⠀⠀⠀⠀⡘⠀⠘⡄⢀⣀⣻⣄⣴⠟⣨⠓⢄⠀⠀⠀⠿⠿⢻⣟⣿⣿⢹⣿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣠⣇⠀⠰⢃⣼⣿⣿⣿⣿⣧⢹⣧⣌⠳⣄⠈⠼⡀⣾⣿⣿⣿⣸⣿⡇⠀⠈⠙⠻⠝⡻⢿⣿⣿⣿⣿⣧⠀⠀⠄
⠀⢀⣴⡿⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⣳⣆⣥⣿⣿⣿⣿⣿⣿⣷⣶⣴⣦⢤⣀⣉⡳⠿⠿⣿⣿⣿⣧⠀⠰⡄
⠀⣸⡟⠀⠀⠀⣾⠟⢩⣿⠟⠉⠹⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⢸⣿⠀⠈⠉⠙⠛⠿⢾⣿⣿⣿⣧⢰⣿⣦
⠀⣿⠁⢴⠀⣠⣸⣇⣛⣀⣀⣀⣀⣹⣆⣏⣛⣤⣿⣿⣿⣿⣯⣥⣼⣧⣤⣤⣄⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣈⣙⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠙⢉⣿⣿⣿⡿⠿⠻⠏⢿⣿⡏⠉⠉⠉⠉⡻⠏⠉⡟⠻⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⢷⡀
⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⡴⠁⠀⠺⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⡏⠋⠉⢉⣉
⣧⣴⣦⣤⣤⣤⣼⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⢦⡤⠤⠴⡦⠶⠤⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣌⣾⣇⠀⠀⠀⠁
⢥⠀⠀⠀⠀⠀⢰⡇⢶⡀⠀⠀⠀⠀⠀⠀⢿⣿⡄⢀⠶⣤⡄⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠈⣿⣿⡏
⠀⠃⠀⠀⠀⠀⠈⣿⣶⣾⢶⣤⠀⣠⣦⣤⣾⣻⣿⣟⣴⣯⣤⣤⣴⣤⣶⣶⣶⣶⣷⣶⣦⣤⣤⣀⣴⣷⣶⣤⡜⣧⣿⣿⣿⠁
⠀⠀⠀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠹⣿⣿⠿⠟⠉⣿⡟⠻⠿⣿⠏⢙⣿⣿⣿⠿⠻⢿⣿⣿⠿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⢹⠗⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣀⣿⣧⡄⠀⢀⡀⢸⣿⠘⡀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢀⣶⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣧⣾⣿⣤⣦⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡖⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣄⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠶⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡉⠉⠛⠛⠟⠛⠉⠛⠢⠦⣤⣀⡀
⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣤⣤⣄⣤⣄
⠀⠀⠀⠀⠈⠉⣛⠿⠛⠛⣉⣉⣀⣀⡈⠋⠙⠛⠛⠛⠛⠛⣻⣿⣿⠛⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠻⣿⣋⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠟⠋⠉⠛⠻⠷⢶⣦⣴⡶⠾⠿⠛⠿⠟⠛⠛⠿⠿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠙⠿⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⣰⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣆⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣠⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⢿⣿⣿⣿⣿⣿⣿⠏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠙⠿⣿⡿⠿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠀⠈⠻⢿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣿⣷⣶⣤⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀⠹⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠋⠙⠋⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢴⣤⠤⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠰⠐⢀⠄⣄⠄⠀⠘⡂⢻⠐⠀⠀⢸⡟⡗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠒⢢⠇⢀⠡⠀⡖⠀⠀⠐⣦⠀⢧⠀⠀⠀⠀⢹⡄⠛⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡿⣾⣿⣆⢸⣿⡂⠀⠀⡞⠀⢸⡆⠀⠀⠀⠈⢻⡀⠈⢂⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⡇⠀⠀⠀⠀⠘⢧⠀⠑⣷⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⠙⠛⠛⢛⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡩⠀⣶⡶⠆⠀⢸⡆⠀⠀⢻⡄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠿⡇⠀⠀⠉⠉⠉⠉⠉⢼⡀⠀⠀⢿⡄⠀⠀⠈⢻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣻⡇⠀⠀⢀⠀⠀⡀⠀⢀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠘⠻⠄⣤⣤⡤⢅⣙⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⢀⣼⡿⠀⡀⢀⠀⠄⠠⠣⢐⣾⡻⡏⠀⢤⡀⠀⠀⠀⠀⠀⠻⡦⠤⢤⠾⠬⠭⣽⣿⣦⡀⣀⡭⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡡⠤⠞⣿⢃⣤⠴⣶⠖⣲⠀⠐⠚⠛⢿⠃⣈⠶⠑⣀⣀⣀⡤⡀⢶⡇⡰⠃⠀⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣄⣀⣀⣀⣿⡁⢀⠐⠼⠀⢸⠀⠀⠁⢀⡎⠠⠔⣞⡏⣿⠼⣻⣀⣀⣿⣾⣤⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄⠛⣛⠻⠛⠟⠻⠛⠳⠤⠾⠶⠶⠶⠿⠷⠶⠺⣛⣛⠛⠭⢽⡿⠿⠿⠿⢟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣒⣂⣈⠀⠉⠈⠁⠉⠈⠁⠉⠈⠁⠀⠈⠉⡁⡀⠠⣤⣴⣷⣾⣿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠈⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠾⠿⠿⠷⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣋⣁⣀⣀⡀⠀⠀⠀⠀⠈⠉⠛⠿⣶⣄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠿⠛⠛⠛⠛⠛⠛⠿⢷⣶⣤⣀⠀⠀⠀⠀⢙⣿⡿⢿⣦⡀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄⣴⡿⠋⠀⠀⠙⠿⠟⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢠⣾⢿⣦⡀⣠⣾⠟⠁⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢀⣴⡟⠁⠀⠙⠿⠿⣷⡄⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢸⡿
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣿⠇
⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⠋
⠀⠀⣼⡟⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁
⠀⠀⢻⣧⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠙⠿⣶⣶⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠴⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⢀⣰⣾⣿⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠁⢀⣐⡞⣿⡿⢿⡄⠀⢽⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠛⠀⢨⣿⣿⡇⠈⠀⠀⠘⢻⣿⠀⠄⠀⠈⢱⣿⣄
⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡄⠰⣿⠿⠛⠁⠀⠀⠎⣸⣿⡄⠀⠀⠀⠀⠀⠻⠷⠄⠀⠀⢿⣿⣿⣷⡄⠀⢀⣴⠈⠀⠆
⣿⣿⣶⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀⡀⠀⢀⣀⣤⣨⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⣿⣷⡄⠉⠁⠀⠀⠀⠃
⣿⣿⣿⣿⣷⣀⢀⣶⣿⣿⣿⡿⢻⣿⣿⣿⣶⣀⡉⠉⠸⢿⣿⣿⣿⡇⢠⡀⠀⠀⠀⠈⠆⠀⠀⠈⣿⣿⣿⣿⣆⠀⠀⠀⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣯⣟⢻⣹⠿⠿⠿⠗⠀⠀⠻⣿⣷⣿⣷⣄⠀⠀⠀⠀⠳⢦⢀⠛⠻⣿⣿⣿⠹⡀⢠⡀⠀⡅
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⡛⠛⠛⢉⣭⣤⣿⣶⣶⣿⣿⣿⣿⣿⠿⣿⣿⡿⢿⣿⣄⠙⢶⣄⠀⠈⠑⠄⠈⣿⣿⣦⡡⡀⠛⠒⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠙⠿⠎⠻⡿⠷⠀⠀⣡⣄⠀⠀⠀⠈⠉⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣤⠀⢀⠀⠀⡀⡶⠿⣃⡑⠀⠀⠀⠀⠀⠈⠁⢀⣤⣶⣤⣄⣀⣤⣦⣴⣶⣿⣿⣿⣿⣿
⣬⣿⣿⣿⣭⣮⡙⠛⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡷⠸⣿⣧⣀⠀⠀⠁⠀⠀⠀⠀⠙⠒⠀⣀⣤⣴⣶⡿⣿⣿⣿⡿⣋⣍⣽⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⣿⣿⣿⣿⣴⣯⣿⣿⣿⣿⣿⣿⣿⣇⣽⣿⣿⣿⣿⣿⣿⣧⣤⣦⣄⠀⢤⣿⠿⣿⣿⣷⣶⣦⣴⣾⠿⠿⣿⣿⡿⠟⣛⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠊⠉⠉⠉⠐⠀⢀⡈⠦⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠈⠉⠁⠒⠂⠀⡠⠔⠚⠁⠀⠀⠀⠀⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠠⠤⠤⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣿⠀⡜⠀⠀⠀⠠⠀⢰⡇⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⡎⢸⠀⠀⠀⠀⠀⠀⠀⡟⢀⠀⠀⠀⠀⠀⠀⠙⢻⡖⠦⠤⠤⠄⣀⡴⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⡾⠀⠀⠀⠇⠀⠀⠀⡷⠀⠳⢄⠀⡀⠀⠀⠀⠀⢻⠀⢀⡴⢛⡝⢁⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠉⡇⠀⠀⠀⢸⡀⢀⡜⠁⢀⡴⠞⠻⢟⠐⣄⠀⠀⣸⡤⠋⢠⠟⠀⣾⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⠁⠀⠀⠀⣠⡃⠚⠡⡶⢻⡿⣟⣿⣿⠁⠀⢃⢰⠏⠑⢄⠃⠀⣼⠃⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢒⣬⡤⣴⣿⡇⠀⠀⠀⠘⠀⠘⠃⠁⠀⠸⢰⡞⠀⢀⠀⠳⣴⠁⠀⠀⢀⣘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠚⠋⠙⢷⣿⠻⡿⠷⠄⠀⠀⠀⠀⠀⠀⢀⠇⢈⢹⡆⠀⢓⣤⣹⡀⢀⣴⠋⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡾⢀⠀⠀⡮⣟⡿⠿⠏⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⡟⢻⣆⠀⠈⠁⠀⠀⣀⠀⠀⠀⣠⡟⠀⠄⠀⠀⣧⡿⣷⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⢀⣿⡽⠋⠳⢦⡀⠉⢠⠀⠀⢠⢞⡉⠔⠀⢀⣠⣞⣁⣁⠹⣆⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡊⠠⢤⣀⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠉⠓⣤⣤⣚⣋⠁⢀⡠⠶⠋⠁⠉⠍⡉⠛⠛⠦⢤⣀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠘⠋⠀⠀⠀⠀⠀⠀⠀⠠⡾⠋⣉⣠⠶⠊⠉⠉⠓⠒⠢⢤⡀⠀⠈⠁⠐⠰⢄⢙⣿⡦⠴⣿⠀⠘⠂
⠤⠔⠀⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣣⠿⠋⣀⠔⠂⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⣠⡤⠴⣶⣿⡟⢻⣟⣢⡀
⠤⠀⠀⠸⣄⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⢠⣧⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢾⡀⣧⣤⣄⡈⠱⢶⠀⢸⠟⠲⣄
⠀⠀⠀⠀⢻⡄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⢻⡋⠀⢫⡈⠳⣦⣰⡟⣆⠀⠘⡄
⠀⠀⠃⠀⠀⢻⡄⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣿⡀⠈⣧⠀⠈⠧⠀⠹⡄⠀⢷⣄
⠄⠀⣠⡀⠀⠀⢻⣄⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠸⣷⣀⣘⣆⣀⢀⣼⣀⣻⣄⠈⠏⠢
⠀⣼⡟⢣⠀⠀⠀⠹⣦⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢻⡿⡏⢿⣏⣩⣍⣭⣭⣿⡇
⣼⠟⠀⠀⢳⡀⠀⠀⠈⢷⡄⠹⣆⠀⠀⠀⠀⠀⢀⣼⢁⠀⠄⠀⣂⠀⠐⠚⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡄⠈⢿⡘⣀⣸⣀⣀⣀⠀⡠⡇⠄⣠
⠏⠀⠀⠀⠀⢻⣄⠀⠀⠀⠙⣦⡈⠻⣄⠀⠀⣠⠾⠛⣠⡿⠛⠉⠉⠉⠓⠢⣤⡀⠀⠀⠀⠀⠀⠀⠘⣠⠋⠀⠀⠀⠀⠈⢻⡇⠹⡌⠉⠙⠛⠀⢱⣾⠃
⠰⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠈⠻⣦⡈⠻⣦⣿⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⡄⢠⠂⠀⣼⠏⠀⠀⠀⠀⠀⠀⠈⣿⡄⢳⠀⠀⠀⠀⠈⢧
⠀⢡⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠈⠻⢦⡈⠻⣿⣿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⢀⠉⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣇⠀⠀⡄⠠⡘⣇
⡆⠈⡄⠀⠀⠀⣠⠽⠛⠀⠀⡴⠂⡄⠀⠀⢙⣶⣄⡀⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣎⠳⡄⢰⡇⠀⠀⠀⠀⠀⠀⠀⡴⠁⢻⣧⠘⡇⠀⠁⢀⠀⠸
⢁⡶⠃⢠⡠⠀⢹⡄⡀⣀⠸⣷⣀⣹⡄⠀⠀⠀⣿⠉⠙⠾⢹⡆⠀⠀⠀⠘⠀⠀⣴⡛⠹⣦⢹⣾⠃⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣻⡶⠓⠒⢢⠈⡄⡀
⠀⠀⠀⠘⣧⢧⠀⢹⡉⠀⠀⠈⠛⠛⠀⠀⠀⢰⠏⠀⠀⣀⣾⠁⠀⠀⠀⠀⢠⣾⡷⡁⠰⣽⣿⡏⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣿⡴⡀⠀⢺⠀⢸⣰
⠀⠀⠀⢠⡀⠈⣇⠀⢷⣶⣶⡖⠀⠀⠀⠀⣰⠏⠀⠀⢰⣿⠃⠀⠀⠀⣀⡴⢋⣜⣫⣴⡾⠻⠊⠀⠀⠂⠀⣠⠞⠁⠀⠀⠀⢠⣿⠉⢷⣷⠀⠈⣧⢀⠃
⠀⠀⠀⣼⠁⠀⡝⠀⠈⣿⡟⠉⠀⠀⠀⣰⠏⠐⢀⣠⣿⠁⠀⣀⣴⠞⠋⠙⠛⠛⠋⠁⢀⣆⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⢀⣿⠃⠀⠘⣿⣧⠀⢹⡏⣰
⠀⠀⡴⠋⠄⢰⠁⠀⣰⡿⠁⠀⠀⠀⣼⠋⠀⠀⢀⣾⣋⣶⠾⠋⠁⠀⠀⠀⢀⣀⣤⡶⠞⠃⣀⣠⠎⠑⠁⠀⡀⠀⢀⡴⠿⠁⠀⠀⠀⢹⣿⡆⠀⣷⡏
⠀⣨⠥⠴⠄⠁⢀⣼⡿⠁⠀⠀⢠⣾⠣⠀⠰⣤⣿⠟⡋⠀⠀⠒⠄⠀⠐⠂⣹⣉⣠⣤⠶⠟⠉⠀⠀⠀⠀⢀⣠⡴⠟⣠⡏⠀⠀⠀⠀⣰⠛⣿⡶⢿⡄
⠀⠀⠀⠀⠈⢠⡟⠁⠀⠀⠀⢠⡟⠁⠀⠀⢰⠏⠉⢁⡙⢀⡀⠀⣀⣀⣤⡴⠿⠟⠉⠁⠀⠀⠀⠀⣀⡤⠾⠛⠉⣠⠞⠃⢱⡆⢀⣴⣦⣷⣀⠸⣷⣀⣡
⠀⠀⠀⠀⠀⢨⠋⠀⠀⢀⣾⠟⠂⠀⣤⣼⠏⢀⣴⡟⠻⠶⢿⡛⠛⠉⠁⠀⠀⠀⠀⣠⣤⠴⠞⠋⠁⢀⣠⡴⠛⠁⢀⣴⢋⣲⠿⣁⣾⡧⠽⢾⣿⣧⡄

⠀⠀⠀⠀⠀⠀⠀⠠⣦⡠⢸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣦⣄
⠀⠀⠀⠀⢀⡠⢀⡤⣿⣆⣿⣿⡿⣪⣷⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣼⣿⣿⣷⡦⢐⣦⡀
⠀⠀⠠⢖⣻⠛⣿⣾⣿⣿⣿⣿⣭⣾⣷⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣶⣾⣴⣿⣿⣿⠟⡉⣄⣸⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠉⢹⠋⣯⢿⡎⣿⣿⣙⡟⢿⢽⡟⣿⠷⠚⠭⠅⠀⠀⠀⠀⠀⣠⣼⣷⣿⣾⣿⣿⣿⣿⣯⣭⣼⣿⣿⣿⣥⣿⣿⣿⣿⣦⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⢀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⣴⣥⠑⢄⠀⠀⣀
⠀⠀⠀⠀⠀⢀⢴⡝⠀⣹⣿⠟⢠⡶⠈⠳⣮⡆⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠏⠀⣤⢀⡀
⠀⠀⠀⢀⢴⣷⡟⢃⣾⣿⣿⣤⡟⠀⠀⠀⠈⠣⢀⠀⠑⠢⠀⠀⠀⠀⣠⣺⡿⠃⢼⢦⣴⣤⠀⠤⢴⡟⡟⣇⡴⠚⠃⠘⡝⢶⣄
⠀⠀⠀⠷⠋⢙⠵⢽⡿⠋⠛⠿⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠶⠟⠛⠛⠀⠀⠈⢢⡉⠙⠃⠀⠈⠀⡼⠋⠀⠀⠀⠀⠈⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠁⠀⠋⠀⠀⠀⠀⢀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣦⣀⠔⠁⣼⣿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢉⠓⣄⣀⢄
⠀⠀⠀⢀⣀⣴⣯⣥⣾⣻⣷⡶⣾⣿⣿⢻⣻⢻⣿⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠺⣍⠉⣠⡼⣡⡾⠀⠳⣄⣀⡀
⠀⠤⠈⣛⡛⠿⢽⣟⣿⡿⠗⢛⡩⠭⠄⠠⠤⠬⣉⣓⠛⠿⠦⠤⠄⠀⠀⢤⣴⣏⠉⠱⣮⡉⠀⠘⠭⠴⠛⢁⠔⠪⠿⢇⡬⠓⠦⠄⡀
⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢲⡴⢫⣿⣦
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡔⠁⣀⠀⠀⠰⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠎⣴⡟⢁⣼⣿⣿⣷⣄⣠⡀
⠀⠀⠀⠀⠀⣠⠊⠀⣩⣾⣾⣣⣾⣴⣮⣀⣤⡀⣀⢀⡀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣷⢄⡤⢸⣾⡄⣠⣿⣿⣯⡍⣩⣾⣿⣦⢤⣄
⠀⠀⢀⣴⣭⡵⠾⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠂⠀⠀⠀⣠⣶⣿⣿⣶⣿⣿⣧⣾⣿⣿⣿⣿⣯⣭⣼⣿⣯⣿⣷⣿⣿⣷⣤⣀
⠀⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢀⠤⣀⣴⣱⣦⣽⣦⣾⡓⢦⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⣎⢢⣄⣠⣿
⠀⠀⠀⢀⣴⣿⣤⣾⣿⣿⣿⢿⢿⣿⣷⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣰⣿⣿⣾⣿⢦⣍⣻⣿⣷⣴⣦⣀⣤
⠀⠀⣰⠿⣫⣭⡉⠉⢉⣡⣶⣄⠘⢁⣔⣌⡙⢛⣭⡻⠦⠀⠀⠀⠀⠀⣰⣒⣿⣿⣵⣶⡾⣻⣿⣬⡻⠃⠺⣭⣿⣾⣷⠙⢯⣿⣋⠑⡀
⠀⠈⣠⠾⠿⠿⢿⣷⡿⠿⠟⠿⠟⠛⠛⠛⠿⠿⠛⠿⠆⠀⠀⠀⡠⠘⢉⣿⣉⣩⡵⠷⣜⢿⣿⠷⢦⣀⣤⣬⠽⠛⠛⠛⢢⣣⣙⣗⢺⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⣖⠈⠭⠍⣑⣲⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠐⣒⣉⠭⠤⠐⠒⠈⠀⠒⠂⠤⣄⠀⠀⠈⠉⠁⠒⠂⠤⠤⣀⣀
⠀⠀⠀⠀⢀⡤⢤⡒⠊⢉⡠⠤⠒⠊⠉⠀⠀⠀⠀⠀⣀⡠⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠲⣃⠒⠢⢤⡀
⠀⠀⠀⢰⠃⠀⢠⢌⡓⠢⢌⡓⠢⠄⣀⠤⠄⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢀⣀⠴⠃
⠀⠀⠀⠈⠢⣀⠣⢄⡈⠑⠢⢌⣑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠊⠉⠈⠀⠘⡦⢤
⠀⠀⠀⠀⠀⠀⠉⠲⢌⡑⠢⢄⡈⠙⠂⠤⣉⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠊⠉⠀⠀⠀⠀⠀⣀⠤⣒⠭⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣌⡑⠢⢄⡀⠉⠒⠤⣈⠑⠢⢄⣀⣀⡠⠔⠒⠉⠁⠀⠀⣀⠠⠄⣀⡠⢔⡪⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣈⠑⠢⢄⡀⡑⠀⡰⢻⠀⠀⠀⠀⠀⠴⡀⠀⠉⣶⠭⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⡇⢸⠀⠀⢀⣠⣴⠾⠻⠖⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠷⠧⠶⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⠀⢠⣾⡿⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣵⣤⣿⡟⢁⢈⡤⢶⣶⣔⡲⠶⠒⠒⠢⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠒⠂⠈⠉⡉⠿⠿⠿⠿⣿⣾⣿⣿⣶⡶⣿⡟⢛⡳⢤⡀⢄⣀⠀⠈⠙⠢⣄
⠀⠀⠀⢀⠀⠀⠀⠠⢐⠋⠀⠀⢠⣤⣴⣶⣶⣿⠀⢠⣦⡀⠀⠈⠻⣿⣷⣿⣿⣿⣖⣵⡆⢸⡷⡈⣎⢿⣶⣄⠀⠈⠳⡀
⠀⠀⠀⢸⡷⢤⠶⠏⠉⠛⣷⣴⣿⢿⣿⣿⣷⡌⠳⣼⠃⠙⢦⡀⠀⣿⣿⢿⠛⠛⠻⢿⣇⠈⡅⠐⠘⣿⣿⣿⣇⠀⠀⠘⡄
⠀⠀⠀⣼⣇⢠⣶⡀⠀⠀⣿⡿⠁⣿⣿⣿⣿⡇⠀⠘⠓⠒⠉⠁⠀⣿⡿⢿⣆⠀⠀⠀⢸⡀⢷⣄⠀⢼⣿⣿⣿⡄⠀⠀⠸⠦⠤⢄⣀
⠀⠀⢀⣿⣿⣿⡻⣧⠀⠀⢨⠀⠀⢿⣯⣥⢞⣡⣿⡶⠀⠀⠀⠀⢠⣿⣿⡀⠀⣀⣀⣀⣘⣷⣌⢻⡟⢷⣽⣿⣿⡇⠀⠀⠀⢀⣀⣀⣀⣈⣁⠂⣄
⠀⠀⢸⣿⡿⠿⠿⣿⣀⠠⠦⠤⠀⣌⠛⣶⣿⡿⣿⣿⣷⣶⣶⣴⡿⠃⣸⠃⠀⠉⢻⣿⣿⡿⣻⣿⣿⣾⣿⣿⣿⣿⣄⣤⣾⣿⣿⣿⠿⠛⠛⠛⠛
⠀⠀⣿⣿⣷⡀⠀⠀⢸⣆⠐⢶⠀⠀⠀⢸⣯⡇⡿⠻⣿⢟⣿⣿⣧⣤⣤⣀⠀⠀⠀⠀⣀⣯⡃⠻⣿⡟⢿⡛⠉⠛⠿⠿⠛⠉⠁
⠰⠋⠙⢿⣿⣿⣶⣶⠏⠈⠣⡀⠂⠀⠀⠘⣿⣿⣿⣷⣤⡼⢩⣄⠉⣿⣦⣭⣤⣤⣶⣾⣿⣿⣷⣷⣾⣿⣦⣵⡄
⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠈⢄⡄⠀⠀⠀⢹⣿⣿⠋⠄⣾⣿⣷⣿⡿⠛⢋⣽⣷⣶⣿⣇⣿⡏⠉⢹⡟⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⢸⣿⣿⠀⢣⣹⣿⣿⣿⡟⠉⠉⠉⠁⠀⠀⠈⣿⣧⠄⢚⣿⣿⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⣿⣟⠉⠁⢀⣠⣴⣶⠀⠀⡠⢠⣿⣿⠿⢿⢿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣷⣄⣀⣼⣶⣾⣿⣿⣿⣿⣴⣾⠇⢊⣿⣏⣀⡀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡿⣯⣿⣿⣿⣿⣷⠟⢿⡿⠿⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⣿⣿⣿⣿⡿⠙⠛⠁⠀⠀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠁⠀⠸⣿⣿⣷⡾⢿⣄⠀⠀⠀⠀⠉⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠉⠹⣷⣄⠀⠹⣿⣄⡀⠀⠈⠙⠶⣶⣤⣤⣤⣤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⣻⣿⣷⣤⣘⣿⣿⣿⣶⣦⣤⣬⣿⣿⡟⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠇⠀⣴⡟⠛⣿⣿⣿⡟⠉⠉⠛⠛⠉⠻⣻⢿⣿⠀⠘⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡷⣶⣿⣿⣶⠟⢻⡟⠁⠀⠀⠀⠀⠀⠀⠈⠰⢿⣧⣀⡉⠙⡌⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠁⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣀⣥⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⣤⡼⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣶⣆⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⣄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠁

⠀⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦
⠀⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⠀⠀⠀⠀⢀⠐⠒⢖⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⡀⠀⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⢷⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣤⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠈⠉⠙⠛⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣮⣧⣤⡄⠀⠀⠘⠉⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠋⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠘⠻⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠋⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢴⢿⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣾⡿⠁⠘⣧⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⠀⠀⣪⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡆⠇⢘⣷⣾⠀⠠⠀⢄⠙⠦⠜⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⢏⣆⠀⢀⣹⠂⠁⠀⠀⠻⡇⠀⠀⠀⠀⠀⠈⣀⡼⠃⡲⠚⡯⡉⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣗⡀⠈⣹⡇⠀⠈⠀⠀⠀⠀⠀⠙⢦⡀⠀⢴⡶⣋⣿⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢌⣿⣿⠅⠋⠁⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠁⣯⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⡨⣧⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠠⣝⣻⣶⡧⠀⠀⠀⠈⠛⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣾⣿⡂
⠀⠀⠀⠀⣀⠀⢀⡲⢾⣿⣶⣰⣤⣧⣴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⣦⠀⡠⠀⣾⣧⠐⢤⣙⣿⡥⠀⡄
⠀⠀⠀⠀⠿⠶⢴⣞⣿⣿⣭⠟⠿⠿⠿⠛⠛⠛⠐⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠤⠤⣬⠴⢿⡴⣳⠿⠀⠘⠃⣘⣿⣿⣄⣒⠶⣿⣏⠁⢡⣤⣦⡤
⠀⠀⠀⠀⠀⢙⣲⣚⣟⣛⣛⡡⡦⠓⢺⠿⠿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢰⣶⠾⠿⢷⢬⠭⣿⣿⡀
⠀⠀⠀⠀⠀⢴⣤⣽⣿⣷⣷⡶⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠀⠀⠀⠲⠯⣶⠟⢻⡟⠀⠀⠙⠁
⠀⠀⠀⠀⠠⠶⠦⢤⡀⠀⠈⠉⠐⠿⠆⠀⠀⠀⡀⢀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠓⠶⠆⠀⠀⠒⠂⠉⠀⠈⠁⠤⠼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠃⠀⠻⠛⠛⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠦⠤⠦⠠⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠚⠛⠛⠿⠟⠛⠀⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠟⠓⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣕⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠷⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⣯⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣮⡟⡅⠈⠳⡀
⠀⠀⠀⠀⠀⠀⡴⢧⣴⠊⠈⢂⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢶⣤⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⠒⠢⠭⢄⡩⣄⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠱⢾⣏⣿⣌⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢵⠋⠢⣀⠀⠀⠀⠀⢀⡴⠃⢀⡴⠚⠁⠀⠙⢷⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⢄⠀⣠⠎⠀⣠⠊⠀⠀⠀⠀⠀⠀⠳⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡟⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣴⠛⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡴⠋⠈⠑⢦⣈⣗⣤⡀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⣠⠞⠀⠀⠀⠀⠈⠣⣝⢮⡻⢶⣶⡿⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢢⠉⠂⠙⢧⡀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠳⠄⠀⠙⠆⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⢦⡙⢢⡄⡘⣆⠈⡃⣀⢁⡛⡀⡀⢂⣙⢢⡀⡀⣀⢠⢀⡀⢀⢀⣄⢀⡁⡀⠊⠀⠀⣀⣠⡚⠁⠠⠀⠁⠠⠋⠀⢀⣴⠒⠀⠰⠋
⢾⠢⠄⠀⠉⠲⢤⠁⠈⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⢈⡀⢀⣀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠋⠀⠀⠀⢀⣴
⠢⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠁⠀⠄⠀⠈⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠢⡀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠁
⣄⣀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⡀⣠⢎⣠⣄⡀⠐⠀⠀⢠⣄⠀⠀⠀⢻⠀⠀⣀⣠⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠀⠉⠂⠤⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⡍⢳⡾⡏⠀⣠⡆⢤⠀⢀⢡⡀⠋⣦⣼⠖⠛⣿⡛⡆⠀⠛⢤⠀⠀⠀⠀⠀⠀⢀
⠇⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⡙⠓⢶⣶⣿⠻⣾⣿⢠⠐⡁⢻⣎⢳⣬⣇⠣⣠⡸⢿⣶⠀⠘⢷⣦⣌⢲⡾⢷⣄⠀⠀⠀⠀⠀⠐
⢀⠀⠀⠀⠠⠄⠀⠀⠀⢄⠀⡈⠉⠁⠀⡾⠀⠙⣿⣸⣶⣿⣦⠙⠷⠙⢿⣔⠈⠻⣿⣿⣧⣼⣿⠯⣙⣷⣦⣤⣌⠗⢾⣷⣥
⠑⣦⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠀⢎⠁⢀⣼⣏⣿⡟⠉⠋⠀⢠⠴⢿⠿⢳⡶⣬⠉⣿⡁⣀⠠⠴⠿⣻⡿⣷⣄⠀⣹⡏⠀⠐
⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢰⣦⣶⣿⠀⢿⡿⡀⠀⠠⠀⡉⠀⠀⣴⣿⣿⣋⠴⠋⠈⠁⠛⠛⢇⣹⣦⠀⢿⣟⣿⠆
⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡰⠟⠛⢉⣍⡳⣌⠳⠿⣄⠐⠜⢀⡠⢞⣿⢾⡟⠁⠀⠀⠀⠀⡞⣻⣫⣌⣡⡙⠾⣛⠁
⠛⠂⠀⠠⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠂⠙⢎⢳⡄⠙⣍⣋⠽⠋⠘⣧⣈⢨⠀⠀⠀⣠⠾⠟⠈⠛⡎⣻⠉⣧⠸⠀⠀⠀⠈
⠀⠔⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠄⡀⢼⣿⣀⢙⡅⠴⠖⠶⣿⢻⡜⢀⢔⡥⠂⠀⠀⢰⠟⠉⠁⠀⠣⣶
⠀⠀⠂⡀⠀⠀⠀⠀⢸⡇⠀⣈⠤⠉⠁⠀⠀⢠⡈⢦⠃⣿⡿⢡⡞⠀⢆⣘⣿⢟⠕⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠉
⣡⠀⠈⠀⠀⠀⠀⠀⠰⡍⢺⡅⠀⠀⠀⣀⣀⣤⡿⠾⠻⠏⠁⡁⠃⢰⡿⣫⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠉⠀⡀⠀⠀⠀⠤⠀⠀⣹⣿⣴⠶⠖⠛⠉⠁⠀⢀⡠⠴⣭⠟⢷⠀⣿⡾⠀⠀⠀⠀⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠄⠁⠀⠀⠀⡠⡔⠚⠉⠉⠀⠀⣀⠀⠤⠐⠂⠉⠀⠀⠀⠻⢤⣶⣾⢻⣃⠀⠀⠀⣀⠞⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢤
⠀⠂⠀⠀⠀⡇⠀⣰⣶⠀⠸⡦⠀⢀⡤⢒⠾⠟⠋⠀⠀⠀⠀⣻⡿⠍⠉⠐⠊⠉⠻⣄⠀⠈⠲⣄
⣠⠄⠀⠀⠀⡇⠀⢈⣉⣠⡞⢁⡴⡩⠊⠁⠀⠀⣠⡚⠛⠃⢰⡿⠴⣶⡄⠀⠹⡷⡄⠈⠳⣄⠀⠈⠳⢄⡀⠀⠀⠀⠈⠀⠀⠀⢀
⠃⠀⠀⠀⠀⠇⠀⢹⠉⠉⢠⠏⡜⠁⢀⣀⣀⣠⢟⡽⠀⣠⣾⠃⢀⣴⡇⠀⠀⠹⡜⡄⠀⠈⢳⡄⠀⠈⠻⣦⡀⠀⠀⠈⠢⣀⡀
⡔⠁⠀⠀⠀⡄⠀⠸⠀⠀⠻⢦⣍⠳⢻⡯⠭⠵⠋⢀⣴⣍⣹⣇⠈⠎⡿⣀⠀⠀⢱⡘⡄⠐⡏⢀⣾⠷⠄⠀⢻⠆⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠃⠀⠈⡄⠀⠀⠀⠈⠑⠋⠀⣀⢀⣴⣏⠈⢦⠋⢹⡆⠀⣿⣍⠻⣆⣸⡇⢠⠀⢿⣀⡑⠙⠁⢀⡟
⠸⠀⡀⠀⠀⢰⠀⠀⡇⠀⠘⡛⠶⣄⡀⢾⡿⠋⠉⠉⢛⡛⠓⠶⣿⣦⣄⠈⠧⣉⠉⣥⡼⠀⠈⢻⠋⠀⢀⡾⠀⠀⠀⠀⠀⠀⠄
⣤⠊⠀⠀⠀⠸⡀⠀⠸⡀⠀⠘⡳⢦⣍⡒⢘⠁⢀⡀⠉⢧⠀⠀⣄⠻⠟⠀⠀⢀⣀⣀⣠⠄⢀⠆⠀⢀⡾⠁
⠁⠀⠠⠀⠀⠀⠇⠀⠀⠳⠐⢶⡙⢠⢬⡅⠻⠀⠈⠳⣠⠞⠁⠀⣙⠀⢀⣈⣉⣉⡁⠄⠃⢀⠎⠀⠀⡼⠁
⠄⠀⠀⠀⣀⠀⠐⡀⠀⠀⠀⢈⣧⠈⢉⣓⣠⠜⢀⢤⣇⠀⡤⠚⠁⣤⡭⣲⡶⠖⠀⠀⢠⠏⠀⠀⣼⡁⠀⠀⠀⠀⠀⠈⠁⠀⢘
⡖⢋⣠⠎⠁⢀⡴⢷⠀⠀⢠⡞⠁⠀⠀⠈⠀⡐⢯⣿⣉⣧⠈⢨⣷⠤⠛⠉⣠⣤⣀⡰⠋⠀⢠⡞⠉⠻⣦⠀⠀⠀⠀⠀⠀⠠⣄
⣾⠏⠁⠀⣠⡏⠀⠈⣆⠀⠈⣇⢰⡀⢸⡶⢎⡇⠘⠡⢁⣦⡀⢀⠀⠀⠀⡞⠀⠀⠙⠁⢀⣴⠋⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈
⠿⡀⠀⢼⣁⡈⠟⠓⢼⡆⠀⠸⡎⠣⢀⣄⣈⠀⢢⠄⢈⣛⠛⠋⣠⠆⠀⢸⡆⠀⢀⣴⡿⠁⠀⢀⡠⡴⠂⠘⢻⡄⠀⠀⠀⡄⣀
⠀⡤⠁⠘⡿⠋⠲⢦⣦⡘⢀⠀⠘⢼⠖⢁⡼⠤⠌⣀⡙⢿⡓⠹⠋⣠⠴⠉⠀⣠⣾⠏⠀⠔⠈⠁⠀⣠⠴⠺⡅⠀⠀⠀⠀⠐⣌
⠎⡴⠀⡔⠀⠀⠀⠀⠈⠛⠚⢦⡀⠀⢳⠏⠀⣀⠀⠰⣿⢈⣡⠴⠚⠁⠀⣠⣴⣟⣤⣤⡤⠴⠒⠋⠁⠀⠀⠀⣽⡄⠀⠀⠀⣳⢌
⣿⢄⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠹⣿⡇⠀⠙⠉⠀⠀⢀⢴⡞⠉⠀⠈⢙⢷⢦⡀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢰⣄⢿⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡗⠙⢲⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⠇⢀⣼⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣴⠦⢄⣀⣀⣀⣀⣀⡀⠀⠀⣀⣴⣿⣿⣴⠟⢠⡎⣿⣸⡌⣷⣤⠴⡄⠀⢀⣠⡤⠤⢤⣤⡾⠏⣉⠓⠲⠶⠶⣿
⣤⡖⢒⣶⣾⣿⡟⣦⡀⢈⠙⠣⠌⠷⣐⣠⣼⣿⣿⠟⣿⣯⡀⠈⢀⠋⡇⢃⣿⠋⡀⠹⡛⠻⠏⣠⣴⣿⢿⡛⢀⣈⣛⡄⢄⣀⣤
⣿⣾⣿⣿⠟⣿⣧⡈⠙⠦⢹⡦⡀⢂⣼⣿⣿⡿⢫⡶⣇⣿⡟⢦⣀⠀⣀⣼⡟⢿⢷⡀⡈⠒⠾⣏⠸⠃⢸⣼⣏⡀⢠⠀⠀⣼⣿
⣿⣿⣟⡵⢻⡇⠀⠛⠦⠀⠘⣡⣴⣿⣿⢛⡿⢧⣾⣿⠿⡏⠃⢀⣽⣶⠛⢻⡤⣈⠃⠷⣜⣳⡆⠀⣉⣰⣿⡿⣏⢳⡘⠂⡾⢋⡏
⢿⣿⡿⠁⠋⣷⣄⠀⢀⣠⣾⣿⣿⣿⣡⣿⣇⠀⠻⣆⣀⣥⣾⣿⣿⠟⠳⣄⠉⠀⠉⠒⢤⣴⣿⣿⣿⡿⣿⣷⢤⣀⠀⠚⠃⡼⠉
⣩⠞⠁⠀⠜⣁⣤⡴⣿⣿⢿⣟⣵⡿⡿⣁⣩⣵⣶⣿⣿⣿⣿⣿⢇⣾⣄⠉⢦⠈⢶⣤⡀⠀⣉⡛⠯⠄⣿⣏⡀⠻⠑⠄⠀⣠⣶
⢀⣀⡤⢴⡾⠋⣩⢾⣿⡷⢛⣽⣿⢖⡻⠟⣋⡽⠛⣱⠋⣹⣿⣵⠟⠛⢿⠷⣤⣀⠸⠀⠙⢶⢤⣉⠁⠀⠀⠀⠙⠲⠤⠀⠈⠊⣈
⠉⠀⠚⠁⠒⢉⠵⠛⠉⠀⠉⠀⠀⠀⠐⠋⠁⠀⠠⠐⠚⢉⠜⠁⠀⠀⠈⠀⠈⠒⠁⠂⠄⠈⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⡼⡇⣞⠉⢱⢸⠀⡇⣿⢦⡇⢹⠁⣼⣳⡀⡇⡿⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⠁⡁⠨⠉⠁⡈⠉⠁⠉⠀⠁⠈⠀⠁⠈⢁⡁⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣶⣤⡀⢀⣤⣶⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠉⠛⢿⣿⣷⣾⣿⡿⠋⠉⠛⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠉⠉⠙⣿⣿⣦⡀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⣴⣿⣿⠋⠉⠉⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⡀⠀⠀⠀⣠⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄⠀⠀⠀⢀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣶⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣶⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣄⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⣠⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⣖⡛⠙⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣶⣀⠈⠉⠛⠲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠉⠛⠲⣤⣀⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⡉⠓⢤⡉⣿⠣⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡆⠁⠘⠷⣤⣼⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄⠀⠀⢐⣶⣾⢟⠦⣌⡙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠂⠀⢠⣿⣿⡿⢿⣥⣀⠉⠓⠈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣴⣿⣿⠟⠀⠀⠀⠉⠙⠳⢦⣄⡀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣬⣪⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⣼⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢹⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢨⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠐⠒⠶⠖⠲⠆⠀⠀⠰⠒⠦⠶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⡍⣷⠀⠀⢀⡄⣶⣤⣠⣴⠀⣶⣄⡰⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣇⠀⠀⣽⠀⠀⠘⠃⠁⡉⡉⠀⢀⡀⠁⠀⠀⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠐⠋⢀⡇⠸⠀⡖⠀⠏⡖⡸⠂⣾⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠛⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰⠒⠛
⠀⠀⠀⠀⠀⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⡠⠒⡰⠄⢢⣎⣁⠀⠐⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠶⠒⣒⡒⠲⢤⡀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣠⠖⠻⡄
⡟⠓⠺⣄⠀⢀⣇⣀⣹⡄⠀⢸⣄⠀⠀⠂⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⠀⢀⠀⠈⢑⡦⢷⡀⣌⠐⢣⡄⢀⣀⡰⠟⢫⣾⠟⠁
⠇⠀⠀⠀⠉⠉⠀⠀⠀⣹⢿⣧⡎⠀⠀⠀⠀⠀⠀⠀⢸⡂⡈⠀⠀⡐⢀⡀⣈⢀⣀⣤⣾⠁⠉⠀⠛⠓⢀⣤⠶⠛⠉
⢰⠀⠀⠀⠄⠀⠀⠀⡐⠉⠀⢧⠀⠀⢐⠀⠀⠀⠀⢴⠉⠻⠽⠿⣭⠿⣯⠽⠛⣛⡻⢟⡟⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠸⠀⠀⠀⠀⠀⠀⠐⢀⣠⠴⠚⠀⠀⠀⠉⠉⠁⠀⠈⠲⢍⣀⣒⣐⣒⣀⡨⠭⠄⠒⠉⠀⠀⠰⠀⠀⠀⠀⠈⠀⠈⠁⡆⠀⠀⡀⠀⠀⠨⢆
⠀⠉⠙⠛⠛⠒⠚⠞⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢀⡀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠀⠀⠱⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡨⠓⠀⠀⠀⡈⠓⢦⡀⠀⠀⠘⣆⠀⠀⠀⠀⣼⠃⠀⠀⠀⣤⣶⠶⣤⣤
⠀⠀⠀⡀⠀⢀⡄⠀⠀⠀⠀⠰⢄⣨⠁⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠈⠢⡀⠙⢆⠀⠀⠁⣆⠀⠀⠀⡇⠀⠀⠀⠀⠁⢀⠠⠐⡄
⠀⠀⠀⠀⠐⠳⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠤⠀⠀⠀⢀⠎⠀⠀⠐⠦⠀⠀⡀⢀⣨⡦⠠⠇⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⡁⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠄⠀⠀⢀⠀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡀⢂⣤⠾⠛⠁⠐⠃⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⡆
⠀⠀⠀⠀⠀⠀⠄⢙⣣⡄⠀⠀⠈⠀⠉⠀⠀⠀⠀⢸⣀⣴⠟⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⠶⠤⠀⠀⠀⠒⠃⠀⠀⠀⠀⠀⠘
⠀⢀⠄⠒⠢⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⡤⠒⠛⠛⠉⠉⠉⠓⠦⡀⠀⠀⠀⠀⡤⢀⢄⡀
⠀⡀⢰⠀⠀⢧⠀⠀⠀⠀⣠⣄⣴⣄⠐⢃⣀⡀⠀⠀⠀⢠⠏⠀⠀⠈⠀⠀⠀⠀⢀⣠⢼⠀⠀⠤⣜⠷⠻⣇⢱⠈⡁⠀⠀⠀⠐⠀⠀⠈⠠
⠀⠄⣸⠀⠀⢸⠀⠀⢠⣾⠏⡠⠊⠀⠀⠀⠀⠉⢶⠀⠀⡟⠀⠤⠀⠀⠀⣠⣤⣶⠿⢳⣻⡀⠀⠀⣿⠄⠀⠟⣸⣄⡿⠃
⠀⡇⣇⠀⠀⣼⠀⠀⢼⡫⠜⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠳⣤⡤⠶⠊⣙⢋⠽⣷⢦⣎⡥⢴⠋⠀⡋⠘⠚⠷⠛⢛⡇⠀⠀⠀⠀⠀⡀⠠⠤
⠀⢶⢸⠀⠀⡇⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠤⠀⠠⠃⠀⠀⠈⣼⣄⣸⣭⣴⡶⠿⠋⠃⣱⠎⠀⠀⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠃⠀⠄
⠀⢸⡌⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⢀⣀⣈⣉⣀⠤⠒⠊⠁⠀⠀⠀⠀⠃⠀⠀⠠⠓⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠳⣤⠟⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠂⠐⠚⠀⠀⠀⢀⠀⠐⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠟⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣠⡌⠢⡀
⠀⠀⠀⢸⠀⠀⠀⠐⠂⠀⠀⠐⠖⠒⠲⡤⠠⢤⡶⠀⠀⢀⠤⣠⠶⠶⠚⠋⠉⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⡆⢀⣬⡦
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠂⠀⠀⠁⢨⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠘⠦⠂⣀⣒⠀⠉⠝⠃⠚⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣵⣤⣤⣤⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⣽⣋⣛⣛⣛⣛⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿