Hose Emojis & Symbols

⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠶⠖⠒⠒⠚⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠓⠒⠒⠲⠶⠶⠶⠤⣤⣄⣀⣀
⠀⢀⡴⠞⠋⠉⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⣀⣀⡀⠀⠀⠀⠈⠉⠙⠳⢦⣀
⣴⠋⠀⠀⠀⣠⠖⠛⠉⠁⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⣄⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⠀⠀⠀⠀⠳⣄⠀⣠⠴⠒⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠓⠲⠤⣄⣀⠀⠀⠈⡷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⡉⢧
⠙⣦⡀⠀⠀⠀⠈⠛⠧⣄⡀⠀⢀⣠⡤⠶⠖⠒⠛⠉⠉⠉⠉⠉⠉⠉⠛⠒⠲⢤⣀⠀⢀⣀⣨⠷⠖⠋⠁⠀⠀⣀⡴⠻⣦⠀⠀⠀⠀⠀⢠⠞⠻⡞⠁⠀⢙⡼⠞
⠰⣇⠉⠓⢦⣀⡀⠀⠀⠀⠉⠙⠛⠓⠶⠤⠤⣤⣤⣤⣤⣤⠤⠤⠤⠴⠶⠶⠒⠒⠛⠉⠉⠁⠀⠀⣀⣠⠴⠖⠋⠁⠀⢀⡾⠀⠀⠀⠀⢀⣼⡀⠀⠙⣦⠔⠋
⠀⢹⠶⢤⣀⠀⠉⠓⠲⠦⠤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠴⠒⠚⠉⠁⠀⠀⠀⣀⣠⠴⠛⣷⠀⠀⣠⠔⠋⠈⣻⠶⠤⠾⠂
⠀⣿⠀⠀⠀⠉⠛⠒⠦⠤⣤⣀⣀⣀⣀⣀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣀⣀⣀⣀⣤⡤⠴⠖⠚⠋⠉⠀⢀⣠⣿⡴⠋⠀⢀⣠⠞⠁
⠀⢸⡷⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣀⣀⣀⠄⠘⠋⢉⣿⠀⣀⣴⠟⠁
⠀⢸⡀⠀⠀⠉⠙⠓⠲⠶⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠶⠒⠒⠛⠋⠉⠁⠀⠀⣀⣠⣬⣡⠜⠋
⠀⠀⢻⡷⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠖⠛⢉⣼⠏
⠀⠀⠈⢷⣄⠀⠀⠀⠉⠉⠉⠙⠓⠒⠒⠒⠒⠒⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠉⠉⠉⠁⣀⣀⡤⠶⠚⠉
⠀⠀⠀⠀⠈⠙⠓⠶⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠶⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣤
⠀⠀⠀⠀⢄⡀⠀⠈⢠⡀⠀⠦⠀⠄
⠀⠀⠀⠀⠀⠉⠐⠦⠀⡙⢸⣰⢞⡔
⠀⠁⠒⠲⠠⠤⠄⢀⣀⣙⠲⢅⣾⡤⣤⣀
⠀⣀⣀⣀⣀⡀⣀⣀⡀⣉⡉⣸⡏⢰⡇⢺⣷⣀⡀⠀⣠⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢿⡇⢸⠀⢀⣄⡿⢿⢿⠑⣾⡆
⡀⡀⠤⠶⠒⠚⠃⠈⠉⠀⣀⢸⣤⢸⡄⢞⣽⣷⠖⣿⣆⢡⡇
⠁⠀⠀⠀⠀⠀⣀⡤⠐⠋⣉⣄⢻⠼⠗⠋⠘⣿⡍⣿⡏⢸
⠀⠀⠀⠔⠚⠀⠀⠀⣠⢌⡥⢩⢫⡷⠀⠀⠀⣿⠇⢹⠻⣌⣧⡀
⠀⠈⠀⠀⠀⢀⡴⠊⢁⡄⢠⠏⠦⠀⠁⠀⠀⢿⣒⣸⠇⠈⠁
⠀⠀⠀⡀⠂⠀⠀⠀⠈⠀⠂⠀⠆⠀⠀⠀⠀⢸⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠙⠦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣭⣿⣿⣶⣶⣦⣬⣭⣩⣭⣗⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⣻⠿⢟⣛⣋⣭⣭⡭⠽⠿⠿⠿⠿⠿⠿⣷⣶⣤⣉⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣩⣿⢟⣛⡭⠭⢶⣒⣒⣻⣯⠭⠭⠭⠭⢭⣭⣉⣻⣿⣿⣿⣿⡜⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣿⣽⡭⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡗⢬⣛⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠶⢚⣩⣿⢟⣧⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣷⡮⣝⣒⡦⠭⣭⣍⣙⣛⣛⣛⣋⣉⣭⣭⠥⠴⣖⣲⣿⣿⣿⣥⣮⣿⣬⣿⣿⣿⣽⣯⣯⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣯⣷⣶⣦⣭⣭⣓⣒⣒⣒⣶⠶⠶⠒⢒⣒⣋⣉⣭⠿⠟⣋⣹⠿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣛⣒⣶⣦⣤⣭⣭⣭⣭⣭⣭⣭⣷⣒⣒⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠂⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠉⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡄⠀⠀⡀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⠢⡀⠘⣄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢀⡤⠖⠉⠉⠉⠉⠙⠂⠀⠀⠀⠄⠀⢱⡀⠘⡄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡞⠀⢀⠔⠂⠀⠐⢦⡀⠈⠆⠀⠀⡀⠀⠀⠀⢻⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠰⠆⠀⠀⠀⠸⠀⠀⢇⠀⠀⠀⠀⢀⡄⠀⢀⠆⠀⠀⠀⠸⠀⠀⡄⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠱⠀⢐⠀⠀⡄⠀⢠⣀⠀⠐⠀⠒⠋⠀⠀⣠⠎⠀⢠⠀⠀⡄⠀⠠⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠣⠀⣸⠖⠁⠀⠀⠀⠀⠀⠐⠂⠈⠁⢀⡴⠃⢀⠞⠀⢠⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⡹⠁⠀⠀⣴⠉⠓⠶⠶⠤⠀⠀⠈⠁⣀⠀⠉⠀⡴⠃⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠁⠀⠀⢠⡀⠉⠓⠂⠀⠀⠤⠤⠒⠊⠁⢀⡴⠊⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠤⣈⠁⠐⠂⠠⠤⠤⠄⠀⠀⠈⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠀⠐⠷⠤⢉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⡃⠃⠀⠈⠘⢸⠁⠀⠀⢸⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⣀⣀⢶⡀⠀⠀⠉⠉⠀⠀⠒⠒⠀⠀⠀⠀⠄⠀⠀⠲⠚⠶⠤⠤⠧⣤⣠⣠⣰⣘⣄⣀⠠⠼⣬⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣹⠈⠉⠁⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢇⣨⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁



⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣒⣯⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⠎⠀⢀⠱⢀⠄⢠⠄⡠⠄⢠⡠⠤⠤⠤⠄⢡⠎⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠂⠀⠀⠁⠠⠁⠐⠁⡐⠀⢀⠀⠀⠌⠀⠌⢠⠃⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠌⠀⢀⠀⠠⠁⠀⠀⠰⠀⠀⠀⠀⡘⠀⠰⠀⡎⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⡇⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢸⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠈⠀⢠⢱⢣⠐⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⢨⠡⢤⠠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢁⡸⠀⢀⢇⠀⠀⠀⠀⠀⠐⠀⠘⡄⠀⡀⠰⠀⢀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠁⣌⡇⠠⠉⠀⢀⠎⠈⣄⠀⢀⠀⡀⠀⠂⠀⢱⠀⢠⠀⠆⢸⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠃⢀⠿⠵⠦⠔⠀⠁⠈⠑⠚⠀⠀⠦⠄⠱⠄⠄⠐⢸⢀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢀⢄⡠⠀⠁⠀⠠⠀⠂⠀⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢼⢘⠀⡀
⠀⠀⠀⠀⠀⠀⠸⠼⡣⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠤⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡤⠴⠶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠤⣾⠀⠀⢸⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⡤⠭⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⠹⡙⣶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⡻⣸⠶⣟⡢⢀⣀⣀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⡉⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠧⠔⣒⣳⣬⣭⠭⠭⢯⣅⣬⣿⣻⣽⡏⣉⠁⠀⣀⠁
⠀⠀⠀⠀⠀⠀⢀⡠⢒⡩⢴⡖⠉⠁⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠒⠢⢤⣀⡈⠀⠀⠐⠂
⠀⠀⠀⠀⢀⠔⡩⠖⠁⠀⢺⣧⣤⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⡀
⠀⢀⡠⢊⡴⠋⠀⠀⠀⠀⣼⢻⡄⠀⠘⡆
⠐⡡⠚⠁⠀⠀⠀⠀⠀⢠⡇⠀⢹⣆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡶⠃⠘⡟⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⡜⠀⠀⠀⢧⢸
⠀⠀⠀⠀⠀⠀⠀⠰⣏⡹⣄⡀⠀⠀⡼⠼⣆⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠂⡈⠆⠀⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠄⠃⢠⢃⡎⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣁⠡⠉⢀⡎⡼⠀⠈⠲⢭⠓⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠸⢴⣁⠤⠤⣤⣀⡀⠁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢹⠳⡄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣅⣀⣸⣀⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠒⢲⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠣⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠉⣍⣉⣉⣉⡍⠉⠉⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠭⠭⠭⠭⠭⠭⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⣀⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⡀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⢀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢁⣠⠤⠖⠚⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠢⠤⣄⣀⠇⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⠞⢁⠤⠐⠂⣉⢉⣉⠉⣀⣀⣈⣉⣉⠉⠉⠐⠒⢠⠎⢀⠾⣄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡠⣆⠀⠋⣀⣴⣾⣷⣿⣭⣍⠉⠓⠠⠤⢌⡉⠉⠉⢉⠖⠁⡠⡎⠑⠜⡅⠈⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠜⣦⣀⠈⠑⠂⠤⢀⣀⠀⠉⠑⠒⢤⣤⡤⠕⠊⣀⣴⠊⠀⣸⠀⠸⠰⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠣⢜⡄⠻⣶⣶⢤⣤⣤⣀⣀⣀⡠⣤⣀⡤⠴⠺⢅⡀⠵⠊⠁⣠⡢⠃⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠄⣈⠛⠶⢄⡈⠁⠒⠀⠉⠭⠭⣭⣥⣤⣐⣂⡉⠁⠤⠴⠚⠋⠁⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠊⠳⠶⠦⠤⠤⠄⢀⣀⣀⣀⣀⠠⠤⠤⠄⠐⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠉⠉⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⢤⣀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⢸⣧⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⡸⠿⠿⠿⠿⠛⠛⠻⠿⠻⠿⠿⠿⠿⢶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⢀⣠⡤⠖⠚⠉⠁⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠛⠻⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⢀⡠⠒⠉⠁⢀⣤⣴⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠶⣦⣤⣀⡀⠀⠉⠙⠿⣿⣦⣄
⠀⠀⠀⠀⠀⣴⣿⠋⠀⢀⡶⠃⠀⢀⣴⣾⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⡀⠀⠙⣟⢻⣷⡀
⠀⠀⠀⢀⣾⢿⠁⠀⢀⡟⠀⠀⢠⣿⡿⠛⠁⠀⠀⠀⣀⣠⠤⠶⠖⠒⠒⠛⠛⠛⠛⠒⠲⠦⣤⣄⡀⠀⠘⡏⢦⠀⠈⣇⠙⣿⣆
⠀⠀⢠⡿⠁⣿⠀⠀⢸⠀⠀⠀⢸⡟⠀⠀⣀⡴⠟⣋⣡⣤⠤⠶⠶⠶⠶⠶⠤⠤⣤⣀⣀⠀⠀⠈⠙⠳⣶⠃⠀⡇⠀⢸⠀⢹⣿⣦
⠀⠀⣿⠇⠀⣿⠀⠀⠸⣇⠀⠀⠈⣷⢤⣾⡯⠖⠛⠉⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠈⠉⠛⠶⣄⣀⡠⠋⠀⠀⡇⡰⠃⠀⢠⠏⣿⣆
⠀⠀⣿⠀⠀⠸⣄⠀⠀⠈⠳⣄⠀⠘⢦⡈⠓⠦⢴⣚⠋⠉⠁⠀⠀⠀⠉⠉⠛⠛⢿⣿⣶⡦⠶⠛⠉⠀⠀⢀⣼⠞⠁⠀⢀⡞⠀⡏⣿⡄
⠀⠀⠸⣆⠀⠀⠙⢦⡀⠀⠀⠈⠓⠦⣀⠉⠓⠤⣀⡈⠉⠛⠒⠲⠶⠶⠖⠒⠚⠋⠉⠁⠀⠀⠀⢀⣀⣤⠖⠋⠀⠀⠀⣠⠞⠀⢠⠇⣿⡇
⠀⠀⠀⣿⡷⣄⠀⠀⠉⠲⣄⠀⠀⠀⠀⠉⠓⠶⢤⣍⣓⣲⣤⣤⣄⣀⣀⣀⣠⣤⡤⠤⠴⠖⠛⠋⠉⠀⠀⠀⢀⡤⠚⠁⠀⣰⠋⣰⣿⠃
⠀⠀⠀⢹⡇⠈⠑⢄⠀⠀⠀⠉⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠉⠀⠀⣠⠞⢁⣼⠟⣿⡀
⠀⠀⠀⠘⣷⠀⠀⠀⠙⢶⡤⣀⠀⠀⠀⠉⠛⠒⠶⠦⠤⠤⠤⣤⠄⢀⡀⠠⠤⠤⠤⠴⠒⠚⠉⠉⠀⠀⠀⠀⣀⠴⠊⣁⠔⢩⠋⠀⢹⡇
⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠙⠶⣍⡐⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠒⢋⣡⠴⠚⠁⣰⡇⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠈⠙⠲⠤⣤⣉⣙⡓⠒⠒⠤⠤⠤⠤⠖⠒⠒⠒⠒⠉⢉⣁⣀⣤⠤⠖⠚⠉⠀⠀⣠⣾⣿⣇⡀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠒⠒⠒⠒⠒⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⣀⣤⠾⠋⣼⡯⠤⣬⡳⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⠾⠟⠋⠁⠀⢰⡟⣵⣶⠦⢱⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠶⠶⠶⠶⠶⠾⠿⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⣧⡈⢉⣴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠺⠤⠙⣧⣼⠫⠬⠇
⠀⠀⠀⠀⢀⡀⠀⢾⠯⠽⡇
⠀⠀⠀⢰⠁⣿⠂⠀⠀⠀⠑⠲⠶⢤⡀
⠀⠀⠀⠈⠦⠿⠉⠓⠶⠖⠒⠉⢳⣄⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡦⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡿⡇⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠿⡿⣟⣟⣻⣯⣭⣭⣽⣭⣿⣭⣭⣭⣟⣛⣻⡶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⢟⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣕⣳⡏
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢾⣯⣷⣿⣛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣭⣟⣻⣿⣿⣿⣿⡿⣯⡏
⠀⠀⠀⠀⠀⠀⠀⣀⣟⣿⣿⣿⣿⣫⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣽⣻⣮⣷⣿⢏⡷
⠀⠀⠀⢀⣠⢶⣟⣻⡿⣿⣿⣻⡿⣿⣻⣿⣿⠿⠿⣿⡿⢿⡿⠿⣿⣿⠿⣟⣿⣻⣟⣝⣓⣽⡿⡟⣷
⠀⢀⡴⣿⣻⡵⠛⠁⠙⢧⣮⣿⣿⣿⣿⠿⣷⣶⣤⣽⣯⣽⣿⣿⣿⣿⣾⠿⠿⢿⣟⡿⢿⣁⡤⠞⠁
⢠⣿⣞⡾⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠾⠿⠭⢿⣿⣿⣾⣿⣿⣿⣶⣿⠿⠭⠿⠾⠛⠋⠉⠁
⠸⣷⣾⣧⣀
⠀⠙⢯⣚⢿⣿⣳⡶⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡔⢚⠋⠶⠤⢀⣀⣀⡀⠀⢀⣀⡀
⠀⠀⠀⠈⠙⠻⠾⢯⣏⣀⣘⣿⠿⢿⣿⡿⡽⢯⣭⣿⣿⣿⣿⣯⣭⠉⠈⣿⠀⡎⠀⠀⠀⠀⠀⠀⠀⠉⡏⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠻⠶⠶⠷⠶⠶⠾⠶⠾⠾⣆⣸⣂⣤⠤⠶⠖⠒⠒⠒⠓⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⢒⣋⣭⣭⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⠉⠉⠙⠓⠲⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣡⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣶⣶⣤⡲⢶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⣵⢫⣿⣿⠿⠛⠛⠉⣉⣀⣤⣤⣴⣶⣶⣶⣿⣯⣶⣶⣶⣶⣶⣤⣤⣤⣤⣰⣤⣤⣉⡉⠛⠻⢿⣏⣻⣿⣿⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣇⣠⣤⣶⣾⣿⡿⠿⠿⠛⢛⣛⣋⣉⣉⡩⢭⡍⠉⠉⠉⠛⠛⣛⠛⠿⠿⢿⣿⣿⣷⠶⣄⡈⠙⣿⣿⡀⣎⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⢿⣿⣷⣽⣿⣿⣏⣩⣤⣶⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣾⣿⣿⣷⣶⣤⣈⠙⠳⣦⣹⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣝⢿⣿⣿⣿⣯⣟⡿⢿⣿⣏⣉⣩⣤⣤⣤⣶⣶⣶⣶⣶⣤⣤⣤⣤⣀⠉⠉⠙⣻⣿⣿⣿⢾⣿⣿⣟⣿⣿⣿⡏⣽
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⣿⣮⣝⡻⢿⣿⣿⣷⣾⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡿⠛⠉⠈⠉⠻⠀⢿⣿⣾⣿⢧⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡛⢿⣿⣿⣷⣾⣽⣛⣿⣿⣿⣬⡝⢡⣾⣯⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⢀⣠⣴⠾⠋⣣⢿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣮⣿⣿⣿⣿⣿⣿⣿⣿⣽⣁⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⣶⣶⣤⣆⣤⣴⣶⣿⠿⠟⠁⢀⣸⡵⠋⢟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣛⢿⠻⠿⣿⣛⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢀⣠⣤⣶⠿⠋⢀⣴⣿⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣶⣤⣄⣉⡙⠛⠲⠶⠶⠭⣭⣭⣭⣿⣿⠿⠿⠿⠟⠛⠛⠋⢉⣉⣁⣤⣤⣴⣶⣿⠿⠛⠉⣀⣴⡾⢋⡽⣣⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣮⠍⠛⠿⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿⠿⠿⠟⠛⠋⠉⣁⣠⣤⣶⠿⠛⠁⢀⡠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⢿⣿⣶⣦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣿⠿⠟⠋⠀⣀⣤⡶⠛⣠⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢤⣉⡛⠛⠿⠿⣿⣟⣻⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠟⠛⠉⢁⣀⣤⣴⣶⡿⠛⢉⡠⠀⢁⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠤⢤⣤⣈⣉⣉⣉⣉⣉⣉⣉⣩⠭⠥⠴⢒⣒⣻⣽⣿⡿⠟⣋⣥⡶⠛⣁⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣯⣥⠶⢟⣋⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⢻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠿⠿⢟⣛⡯⠥⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢻⣤⣼⡽⠿⠿⠿⠿⢿⣿⣟⣛⣛⣯⠭⠭⠖⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠒⡦⠀⠐⠒⠒⠒⢒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠉⠁⠀⢸⡀⢣⠐⢒⣂⡩⠭⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⢀⣀⠤⠔⠒⢇⠼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣀⣀⣠⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⢀⣀⣘⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣎⣀⣀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⢀⡠⠬⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠧⠤⡀
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠈⠢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠒⠉⠉⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⣀⠠⠔⠓⠄⠠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡅⢄⠐⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠒⠂⠛⠒⠤⣀⠀⠉⠑⠒⠂⠤⢄⣀
⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⢀⣀⣠⣤⣤⣤⣶⠶⢶⠶⡾⣷⢶⣤⣤⣀⣀⠀⠉⠑⠢⢄
⠀⠀⠀⢀⡔⠁⢀⣤⠶⣛⣩⣷⣿⣿⣿⣶⣷⣿⣷⣿⠶⣾⣿⣿⣿⣯⣟⡽⣓⢦⣄⠀⠑⢄
⠀⠀⢠⢻⠀⠀⣾⢿⡟⣯⣹⣋⣴⣾⣿⣿⣿⣿⣾⡿⠷⠿⢿⣾⣧⣽⣿⣻⢿⣾⡞⡇⠀⢸
⠀⠀⢨⠀⢣⡀⠈⠳⢾⣟⣻⣫⣵⣶⣷⣾⣿⣿⣿⣷⣦⣶⣮⣽⣿⣟⠽⣛⣻⠾⠋⠀⣠⠺⡀
⠀⠀⠈⣦⡀⠈⠒⠤⣀⠈⠉⠙⠛⠒⠶⠶⠿⠿⠿⠿⠷⠶⠶⠶⠛⠛⠋⠉⢀⣀⠤⣪⠃⠀⡇
⠀⠀⣸⡀⠙⢦⠤⣀⡀⠈⠁⠒⠂⠤⠄⠤⠀⠀⠀⠀⠀⠠⠤⠤⠄⠒⠂⠉⣁⡠⠚⠁⢀⡜
⠀⠜⠰⠣⣄⠀⠙⠢⢌⡉⠒⠂⠤⠤⠄⠠⠤⣤⠤⠄⠠⠤⠤⠤⠐⠒⠊⠉⠀⣀⡤⢒⠝⠇
⠰⠂⠀⢣⡈⠓⠤⣀⠀⠀⠉⠐⠂⠠⠤⠤⠤⠄⠀⠀⠀⠀⠤⠤⠤⠄⠒⠊⢉⡀⠔⢉⠜
⠀⠳⡀⠀⠙⠢⢄⡀⠉⠐⠂⠤⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⣀⠠⠤⠤⠒⠊⢁⣠⠔⠁
⠀⠀⠈⠒⠤⣀⠀⠈⠑⠒⠆⠤⠤⠀⣀⣀⣀⣀⣀⣀⣀⠀⠤⠤⠄⢖⡰⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠒⠒⠂⠉


⣿⠿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡛⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠉⣹⣦⠴⠖⢛⣷⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠴⠞⠉⠀⠀⠐⠛⠋⣉⣉⠉⠉⢉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠛⣻⡉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⣤⠴⠖⢛⣛⣋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠳⣾⢋⣀⡴⠞⣋⣥⣶⣞⣛⣉⣉⣉⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠉⢷⣙⡯⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠴⠚⠉⠀⠀⣠⡴⠋⠉⠁
⠀⠀⠀⠀⠀⣮⠙⢯⠉⢾⡋⠁⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠸⣍⢷⡈⢳⡈⢳⡄⣠⡴⠋
⠀⠀⣀⡴⠛⠙⣦⠻⣄⠻⣄⢹⡅
⠀⡴⠋⠀⠀⢀⣨⢷⣙⣦⣙⡏
⣸⠃⠀⠀⣰⠏⠀⠀⠉⠉⠉
⣟⠀⠀⢰⡇
⢻⠀⠀⠸⡇
⠸⡆⠀⠀⢻⡄
⠀⠹⣄⠀⠀⠹⣆
⠀⠀⠙⢦⡀⠀⠈⢳⡄
⠀⠀⠀⠈⢳⡀⠀⠀⢻
⠀⠀⠀⢀⡾⠀⠀⠀⣸
⠀⢀⡴⠋⠀⠀⢀⣴⠋
⣀⣈⣀⣀⣀⣠⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⠿⠛⢉⣀⣤⣤⣤⣤⣤⣄⣈⠙⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⠟⠁⣠⣾⣿⣿⡿⠿⠿⠿⢿⣿⣿⣷⣦⡀⠙⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁
⣿⣿⣿⠋⢠⣾⣿⠿⣿⣿⡯⠀⣤⠀⢸⣿⣿⠿⢿⣿⣦⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠠
⣿⣿⠇⢠⣿⣿⠋⠀⠀⠉⠃⣴⡟⠓⠘⠋⠁⠀⠈⢿⣿⣧⠈⣿⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠐
⣿⣿⠀⣼⣿⣧⣄⡀⠀⠀⠀⠈⢟⡆⠀⠀⠀⠀⣀⣴⣿⣿⡀⢸⣿⡇⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢈
⣿⣿⠀⢿⣿⣿⡿⠟⠃⠀⠀⢰⡏⠀⠀⠀⠀⠻⢿⣿⣿⣿⠇⢸⣿⡇⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⡄⠸⣿⣯⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⣨⣿⡿⠀⣼⣿⡇⠀⠀⠀⠠
⣿⣿⣷⡀⠻⣿⣧⣀⣠⣴⡇⠀⠀⠀⢸⣶⣤⡀⣰⣿⡿⠃⣰⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⢧⠀⠠
⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣇⠀⠀⠀⢸⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⡇⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⡀⠀⠐
⣿⣿⣿⣿⣿⣷⣄⡈⠛⠿⢿⣿⣿⣿⣿⠿⠟⠋⣀⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠢⠈⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡟⠛⠛⠛⠛⢻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⡉⡉⡉⡉⠙⠛⠿⣶⣄
⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⡇⠀⣴⡄⠀⢸⣿⣿⡿⢿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⠀⠀⢸⣿⠿⣿⣿⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⣼⣿⠏⠀⠀⠉⠛⠇⣰⢾⠲⠆⠸⠛⠁⠀⠀⠹⣿⣧⠀⠀⠀⠀⢀⣾⠏⢀⣾⣷⣦⣀⢸⠟⢐⡛⣿⢀⣤⣶⣷⡄⠈⢿⣆
⠀⠀⣸⣿⣏⠀⠀⠀⠀⠀⠀⠙⢾⣤⠀⠀⠀⠀⠀⠀⠀⣹⣿⡇⠀⠀⠀⣾⡏⢀⣾⣿⣿⣿⣿⣿⣌⠸⢿⣿⣿⣿⣿⣿⣿⡄⠘⣿⡄
⠀⠀⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⣸⠞⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⠀⠀⠀⣿⠃⠀⠈⠙⠿⣿⣿⣿⡿⢛⣸⣿⣿⣿⡿⠛⠉⠀⠀⣿⡇
⠀⠀⣿⣿⣿⠟⠋⠁⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠈⠙⠻⣿⣿⡿⠀⠀⠀⣿⡆⠀⣠⣴⣾⣿⣿⣿⣧⡸⣿⣿⣿⣿⣿⣦⣄⡀⠀⣿⡇
⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⢻⣧⠈⢻⣿⣿⣿⣿⣿⣿⣱⣿⣿⣿⣿⣿⣿⡿⠃⢰⣿⠁
⠀⠀⠀⠹⣿⣧⡀⢀⣤⣶⡇⠀⠈⠀⠀⢸⣶⣤⡀⢀⣼⣿⠏⠀⠀⠀⠀⠀⢿⣧⡀⠻⠟⠋⠁⢸⣿⣿⣿⣿⠀⠙⠻⠿⠁⣠⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣶⣶⣶⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣤⣀⣠⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣴⡦⢠⣤⣔⣒⣲⣦⣴⣯⣭⣤⣤⣤⣤⡟⠙⣧⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢩⢤⢷⠶⢿⣿⣿⣯⡿⠟⠛⠙⣉⡉⠩⠭⠭⠭⠇⣸⠧⠤⣬⣅⣀⠀⢀⣁⡛⣷⣶⣤⣀⠀⣀
⠀⠀⠀⠀⠐⠸⢴⣊⢙⣩⢗⣛⠛⣋⣩⠤⠒⠚⣉⡩⠤⠄⠔⢒⣒⡂⣺⡁⠀⠈⠒⠒⠤⠭⣙⡒⠤⣝⠩⢝⡻⣭⣛⠍⠒⢀
⠀⠀⠀⠀⣠⠞⠃⠐⡐⡀⠀⢀⡊⠋⣀⡴⠞⠉⠁⢒⡈⠉⣉⣉⣉⣁⠃⢉⠉⠉⠑⠒⠢⢤⣄⠈⠁⠂⠄⠀⠈⠑⠉⠑⠢⣀
⠀⣰⢃⡞⠁⠀⠄⠀⠀⢀⡀⠀⢀⡾⠉⠀⡠⠀⠀⣴⣶⣾⠿⡿⠕⠁⠀⠀⠽⣿⣷⣶⡦⢄⡀⠓⢤⣠⣀⠀⠀⠀⠀⠐⡆⠈⠂⠀⠀⠀⠠⣄
⠰⠃⣸⠐⠀⠘⠀⡀⠀⡏⠈⠋⠿⠀⡂⢸⠀⠀⣶⢸⠇⠁⠀⠀⠀⣀⣀⣴⣮⣮⣿⠈⠂⠀⠀⠀⠀⣈⣿⣷⣤⠀⠀⠀⠈⢣⠀⠀⠀⠀⠛⡌⢦
⠘⠀⢀⢰⢀⢰⢡⢡⠀⣇⢀⣀⡀⠀⢇⠀⠀⠀⠙⢇⡀⠀⠀⠀⠀⠉⠛⠛⠛⠉⢁⡠⠆⠀⠈⠀⠀⢨⣽⢿⣛⠄⠀⠀⢀⠈⡷⠀⠀⠀⠀⢸⠈⣇
⠀⠀⠀⠈⢎⢧⢻⢧⠱⡈⢮⡻⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠁⠀⠈⠉⠁⠀⣀⣀⡤⠤⠶⠀⢀⡾⠉⠀⣠⣾⠏⠀⠀⢀⡄⠀⠐⢸⢄⣷
⠀⠀⠀⠀⠈⢦⠳⣙⣴⣌⠂⣙⠪⣝⡷⣦⣀⠀⠀⠐⠢⠄⠀⡀⠀⢈⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⢠⠴⠋⢀⣤⠾⠋⣀⠴⢋⠤⠋⠀⠀⢀⠎⣼⠃
⠀⠀⠀⠀⠣⣀⠑⢌⠳⣗⠑⠢⣔⠠⣙⠻⢭⣟⡶⢤⣤⡀⠀⠀⠀⠀⠬⡭⠭⠭⠄⠀⣀⣀⣤⠤⠶⠚⣋⣉⠤⢖⣫⠵⢚⣡⠔⢠⠄⠀⠂⠚⠁
⠀⠀⠀⠀⠀⠀⠙⠦⠈⠀⠑⠂⠈⠙⠲⢭⣒⡬⢽⡓⠲⠬⣭⣽⣛⠓⠚⠛⠛⣛⣛⣩⠭⠥⠖⣒⣛⡩⠥⠖⣊⡡⠄⠚⠁⣀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡁⠀⠀⠈⠻⢡⡀⠀⡄⠀⠀⠒⠒⢈⠉⠩⠥⠴⡒⠊⠉⠀⠀⠤⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⣀⡀⠀⠀⠈⠑⠈⠑⠊⠀⠀⠉⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⠀⠀⠒⠂⠀⠀⠰⠀⠀⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣦⣤⣦⣠⣄⢠⣄⣤⣔⢠⡆⠀⠀⠙⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣠⣶⣶⣶⣶⣶⡆⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣡⣾⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠰⡇⣀⡴⠛⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⢀⣼⣧⣤⣬⣤⣬⣭⣭⣭⣭⡍⣉⣉⢨⣤⣤⣤⣤⣭⣭⣬⣭⣭⣬⣥⣭⣭⣤⣭⣭⣭⢬⣭⣥⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣻⣿⣷⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⣿⣮⣻⣧⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣿⣟⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⠤⠤⠦⠀⠒⠿⠻⣿⡏⠙⣿⣿⣓⠚⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡶⠦⠴⠤⠤⠄
⠀⠀⠀⠀⠀⠠⠔⠒⠒⠚⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⠰⢦⣴⠿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡦⠤⣤⣤⣿⣏⣀⣀⡀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⡤⣤⣤⣽⡟⣻⣛⡛⠛⠲⠶⣤⣄⣀⣀⣀⡀⠀⢀⣾⡏⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⡿⠐⠀⡀⣀⣀⠀⢨⣡⣭⠙⣋⣻⣇⣏⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠔⠀⣿⣿⣿⣿⣿⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠁⣀⣬⡤⠴⠒⠒⢛⣧⣘⣋⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠠⠟⠿⠝⠉⠹⠤⠀⠀⠠⢤⣤⠀⠒⣦⣤⡤⠃⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣉⠉⠛⠛⠻⠿⠛⠓⢲⣶⠟⠻⠿⠁⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣠⡤⡄⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢴⠆⠀⠸⢃⠈⡅⠻⣿⡿⣿⣿⣿⣿⣷⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣭⣭⣭⣉⣽⣧⢤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠗⠚⡼⠃⠀⣛⣿⡞⣿⣿⣿⣿⣿⣿⣷⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⣿⣿⣿⠻⠿⠿⡾⠟⠛⡐⣾⣶⡆⠀⠀⠀⠃⢀⠞⡟⠛⠁⠀⣤⣿⣿⣤⣍⣉⠉⠉⠉⢉⣩⣭⣭⣤⣤⣄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣷⢠⣤⣤⣤⣤⡆⣴⣿⣿⡿⠀⠀⠁⣀⠘⠀⠁⠀⠀⠀⠋⠰⡆⠀⣀⣉⠁⢰⣷⣾⠩⢿⣿⣿⣿⠟⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⢿⣿⣿⣯⣽⣿⣇⣈⣈⣩⣥⡤⢴⣿⣻⠏⠁⠀⢰⣾⣿⢿⣰⣷⣾⣶⡶⢦⣤⣤⣼⣿⣿⣷⣘⣛⣛⣂⣉⣭⣽⣿⣦⣥
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠻⠷⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡿⠋⠙⠛⠓⠚⠛⠛⠿⣏⠪⣿⣿⠿⠿⠿⠿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢠⣀⠠⠄⠤⠤⢤⣤⣠⣤⣤⡄⠠⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠃⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠗⠛⣾⣷⣘⣈⣿⣞⣭⣤⣤⣶⣶⣶⣶⣶⣴⣶⣤⡬⠉⠁⠀⠈⣴⣦⣴⣦⡔⢻⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⠀⠀⢸⣿⣿⣿⣿⠸⣿⡎⣹⣿⣯⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⣿⣦⠀⢸⣿⣿⣿⣿⣴⣾⡷⠒⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣭⣭⣭⣭⣬⣭⣭⣭⣭⣭⣬⣽⣷⣦⣼⣿⣤⣿⣤⣤⣤⣦⣭⣤⢺⣿⣏⠀⣿⣿⣿⣛⣻⣟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣾⣿⣯⠀⣿⢇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣼⣷⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⣿⡿⠿⠿⣿⣿⣿⡇⣏⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣉⣉⣹⡋⣉⣩⡏⢾⡿⢻⣿⣿⣶⣾⣿⣿⣿⣤⠿⢿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣾⣤⣤⣤⣄⣉⣀⣨⣭⣍⡉⠉⠉⣩⡟⣩⣮⣿⣿⣿⣵⣿⡇⠰⣿⡟⠉⠉⠛⠃⠀⠉⠙⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⣿⣿⣿⠟⠋⠉⠉⠀⠀⠘⣿⣿⣷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠶⠶⠷⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣾⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠋⠉⠉⠉⠉⠉⠁⠀⢹⡇⠀⠀⠀⠀⠀⠛⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠶⠶⣶⣶⣶⣦⣾⡇⠀⠀⠀⠀⠀⣿⠿⠿⢶⣦⡀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⣤⣤⣴⡿⠀⠀⠀⠈⢻⣦⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠃⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⢀⣴⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠟⠁
⠀⠀⠀⠀⢠⣶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⣄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⢠⣶⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⣶⣄
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠙⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢀⡈⣠⡀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢾⡠⠄⠊⣡⠞⠁⠀⠈⠀⠀⠀⠀⠀⢧⡄⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣔⠀⠏⠀⢈⡶⠁⠀⠀⠀⠀⠀⠀⠐⠒⢦⠀⠀⢳
⠛⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣿⠁⠀⡸⠂⡼⣁⡀⠀⠀⠀⣴⠂⠀⠆⠀⢄⣠⡄⠸
⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⠉⠁⠀⠀⢡⣴⣿⣿⠗⢀⠙⡄⠊⠉⠀⠀⠀⠀⠀⠀⠁
⣷⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⡂⠀⢫⣿⣷⠛⣣⡛⠈⠀⠀⠀⠀⠀⢀⣀⣆⠀⠈
⢼⠗⢄⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⡾⠟⠀⠀⠀⠈⠁⠀⣿⣿⣶⢾⢽⣷⡇⠀⠀⠀⣀⣠⡟⠉⡈⠀⢠
⣾⣷⣾⣷⣤⣤⣄⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡹⠃⠀⡰⠂⢠⠀⠀⠀⢠⣿⡿⢡⡟⣼⣽⡇⠀⠀⠀⠀⠹⣧⢽⢣
⣿⣿⣿⣋⣿⡿⠿⠿⣧⣾⣿⣿⡗⠠⠀⠀⠀⠀⢀⣀⠀⠀⣠⣤⣤⠟⠀⠀⠀⠀⢰⣯⡇⠀⠀⠀⢹⣿⣿⣆⣽⡿⣿⡁⠀⠀⠀⠀⠀⠘⠶⢧⡀⡀
⣿⡿⠛⠉⠀⠀⢲⣶⣿⣿⣿⣿⡇⣀⡨⣿⢿⣶⣬⣯⢶⣾⡿⠋⢋⣀⣀⣀⣾⢟⣟⣮⡖⠀⠀⠀⠈⢿⣿⣿⣿⣧⡈⠷⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⢿⣿⣿⣾⣿⣇⣀⡉⠻⠿⢿⡦⠰⢯⣉⡿⠿⡿⢋⣾⣿⣿⡇⠀⠀⠀⠀⠈⢻⣏⣻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣆⣿⣿⣿⣿⡿⣿⣇⡈⠁⠀⠤⣀⠒⠿⣞⠛⣾⠿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣧⣿⣿⣏⡳⢶⣶⠮⣯⣷⡴⠶⠏⣲⣿⠿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠙⠯⣭⣀⠈⠻⢷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡒⠶⠿⣿⣿⣿⣶⣤⣄⣀⣀⣀⣤⢞⣛⣡⣿⡳⠻⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣶⣷⣷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⢿⣿⣿⠿⠛⠫⠭⢭⣿⣿⣟⠚⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢿⣿⠟⠒⢀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⡳⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠻⠿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣄⣀⣠⣾⣇⠀⢀⣠⡎
⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⢠⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣤⣴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣯⣿⣶⣿⣿⣿⣿⣿⣶⣿⣽⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣵⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉
⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡇
⠀⠀⠀⠀⠈⠉⠻⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠲⠀⠈⢻⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣷⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠹⠦⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡄⠀⠀⠀⣤⣴⣾⣏⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣯⠉⠁⠀⠀⠀⢻⣿⠿⢿⣷⣶⠄⡼⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠻⣦⣼⣯⣽⡗⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⡟⠀⣴⡌⠀⢻⣆⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠙⠚⠆⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢳⣷⣧⡄⣍⡆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠈⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣾⣬⣏⠓⠛⠂⠀⠀⠀⠀⠀⢀⡜⠁⠿⠀⠀⠀⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⠖⠀⠀⠀⠀⠀⠀⠰⠋⠀⠆⠀⠀⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⣀⠀⢨⣯⣤⣄⣀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⣈⡟⠉⠛⠉⠀⠀⠀⠀⡽⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠁⠈⠀⠀⠀⠌⠀⠀⠘⢶⣶⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⣷⡄⢭⡉⠈⢀⣠⣤⣀⣀⣀⣀⣀⣀⣀⣤⣶⡄⠀⠀⢻⣿⣇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⣻⣿⡄⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠈⢿⣿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⡿⣟⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣏⢾⢿⣿⡄⠀⠀⠈⢿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣿⠻⣧⢫⣿⠟⠀⠀⠀⣿⣿⡝⢇⣿⣿⠋⠏⣾⡋⢘⢟⢀⡄⠀⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⡾⣃⣼⡟⠀⠀⠀⢀⣿⣿⣿⣾⡟⣦⡊⣿⢈⣿⣆⢸⣆⣃⣤⣄⣹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠸⠿⠿⠀⠀⠀⠀⠹⠿⠿⠥⠿⠽⠿⠷⠞⠛⠉⠻⠿⠛⠛⠛⠛⠋

⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀
⠀⢀⣤⣿⣿⣤⡀⠀⠀⣶⡶⢲⣶⠂⣀⠀⠀⢸⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠇
⠀⠈⠛⣿⣿⠛⠁⠀⠀⢿⡿⠾⠿⢿⡿⠀⠀⠀⠹⣿⣿⠏⠀⠀⠀⠈⠐⠊⠀⠐⠆⠀⠀⠀⢠⣀⠎
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⢸⠓⠚⠇⠀⠀⠀⠀⢸⠓⠚⡇⠀⠀⠀⠀⠀⣶⣿⡿⠀⠀⠀⠀⢸⢀⣁⠀⠀⠀⠀⠀⠀⣰⣦⡖
⠀⠀⢘⢢⠴⠂⠀⠀⠀⠀⢘⢢⡴⠃⠀⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⢨⠈⠛⠀⠀⠀⠀⠀⣸⣿⠏
⠀⠀⠘⠐⠒⠂⠀⠀⠀⠀⠘⠀⠀⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⢀⡴⢊⡑⢦⡀⠀⠀⠠⠔⢒⡒⠢⠄⠀⠀⢠⣤⣯⣽⣦⡄⠀⠀⠀⠀⣰⠉⠇⠀⠀⠀⠀⠘⡛
⠀⠢⠸⠤⠤⠇⠜⠀⠀⠱⠀⠘⠃⠠⠌⠀⠀⠸⣿⣆⣴⣿⠇⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠸⣿⡿⡻⣿⠇⠀⠀⠀⠀⡅⢨⠀⠀⠀⠀⠀⢰⠈⠁⡆⠀⠀⠀⢸⡏⢉⡉⢸⡆⠀⠀⢸⠢⣤⣤⣤⡆
⠀⠰⣿⣿⣾⣿⠆⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠘⢳⡞⠃⠀⠀⠀⢸⡆⠻⠟⢸⡇⠀⠀⢬⠉⠉⠉⠉⡥
⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⠈⠓⠒⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠚⠆⠄⠀⣀⠐⠂⢄⠠⠄⠀⠀⠀⠀⠌⠂⠀⠀⣢⣠⡤⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⣴⣯⣶⡄⣷⣸⢿⣄⣤⣴⣷⠆⠊⣛⣍⠱⣿⣶⡘⡒⢢⢤⣄⡆⡤⠀⡀⢁⢺⣁⣴⣹⡇
⠀⠀⠀⠀⠀⠀⢠⣴⣦⣊⣶⣷⣻⣷⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣶⣿⣽⣿⣿⣿⣿⣯⣦⣤
⠀⠀⠀⠉⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠙⠋⠉⠁⠁
⠀⠀⠀⠀⠐⠀⠘⢁⣠⣴⣖⣀⣠⠀⣦⠀⠀⢠⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣧⣦⠈⣳⣴⡆⣯⣦⣶⢸⣯⡀⣤⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⠿⠿⣿⣟⣛⣙⣿⣿⣿⠀⢀⣰⣦⣆⢀⣾⢿⡇⢰⢼⡄⠀⠀⠸⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠿⠉⢉⡛⠻⠆⠀⠀⠏⣿⣿⣿⣿⣷⡠⠼⠇⠀⠀⣆⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠂⠀⠀⠀⠀⢳⡶⢖⠪⡿⠿⠶⠶⣿⣁⣈⣛⣻⣦⣤⠤⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣄⢄⡀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠁⠉⠉⠉⠙⠛⠒⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⡜⠿⢷⠦⠴⣠⣴⣦⣶⣾⣿⣿⣦⠀⠈⠉⠉⠙⠛⠓⠒⠒⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠀⠄⠹⣿⣿⣿⣿⣿⠿⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠥⢄⡉⠂⡇⣑⣒⣆⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠹⠃⠳⠆⡿⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠂⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⠦⠤⢤⣀⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠦⢤⣄⣀⡀
⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⢤⡀⠘⠀⠀⡰⠀⠀⠸⠁⠐⠒⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⢤⣄⣈⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠙⠀⠈⠀⠀⠀⣁⡀⠠⠤⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⣤⣄⣀
⠀⠀⠀⠈⠉⣁⡤⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤
⠀⠀⡀⣸⢈⡿⠁⠑⠠⢀⡈⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠀⡆
⠀⠰⠿⠿⣿⠃⠀⠀⠀⠀⠈⠐⠢⣀⠈⠑⠢⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠩⠳⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⡇
⠠⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⡀⠀⠉⠐⠄⡀⠈⠀⠀⠀⠀⠉⢐⣲⡭⢅⡂⢽⡀⠀⠀⠀⠀⠁⠀⠀⠀⡇⡇
⢦⣶⣄⢀⠈⠒⠄⡑⠠⠀⠀⠀⢽⡲⠄⠀⣀⠀⠁⠀⠄⠀⢀⣈⣁⠉⣴⣶⣿⢦⢻⡐⠲⢤⣀⡀⠀⠀⠀⡇⡇
⢸⣿⣿⣏⣢⣄⠀⠈⠐⠀⠑⠠⠀⠈⠁⣹⣿⡶⠀⠀⠀⢞⣾⣿⠂⠀⣿⣿⡟⢸⣜⡇⠀⠀⠀⠉⠲⢤⣄⡇⠇
⠀⠉⠉⠲⣭⡺⡝⠢⣄⠀⠀⠀⠐⠤⡀⠀⠛⠋⠀⠀⢸⣿⠿⢟⠁⣤⢻⠓⠉⢀⡿⣷⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⣄
⠀⠀⠀⠀⠈⠙⢧⣀⣧⡁⠀⣀⡀⠀⠁⠀⠂⢀⣀⣀⣾⠛⠉⡩⠀⠖⠉⠁⡀⠀⠇⣿⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣦⡀⠀⠉⠘⠋⢹⠀⡀⣴⠤⠖⢚⡡⠔⠊⣁⡏⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣇⠀⠀⠀⠀⢸⣀⡀⠄⢒⡩⣦⣤⣶⡯⣀⡼⠃⠀⠐⠠⡈⠙⠦⣄⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣷⣤⣤⠀⠉⡀⠤⠔⢾⡿⢟⣫⠵⠊⠁⠠⣀⠀⠀⠀⠈⠑⢤⡈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢀⠀⠀⢀⡤⠔⠚⠉⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠈⠲⣄⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢤⡀⠑⢤⡀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠀⠑⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡉⠲⣄⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀

⠘⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠇⠀⠀⢸⠀⠀⠀⠀⢸
⠀⢷⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡾⠀⠀⠀⡟⠀⠀⠀⠀⡟
⠀⢸⡀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢀⡞⠀⠀⠀⠀⢰⠇⠀⠀⢠⡇⠀⠀⠀⢰⡇
⠀⠈⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠹⡆⠀⢠⠞⣡⣄⠀⣀⢠⣄⢸⢧⠘⡆⠀⣠⠏⠀⠀⠀⠀⠀⣾⠀⠀⠀⣸⠀⠀⠀⠀⢸
⠀⠀⢷⠀⠀⠀⠀⠀⠀⣧⠀⢀⡼⠁⠀⠀⠀⠀⢀⣀⠠⣦⣹⠀⠈⠋⠀⡟⢸⢿⣿⢸⣼⠈⠷⠇⣰⠃⡀⠀⠀⠀⠀⠀⠹⣄⠀⠀⡏⠀⠀⠀⠀⡟
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡀⠸⠶⠚⠉⢧⠀⣷⣼⡝⡆⣯⠻⠃⠀⠀⢰⣇⡞⢸⣿⠘⠿⠀⠀⠠⣧⢞⡟⣴⡆⣠⠀⡶⠲⢬⣷⢸⠃⠀⠀⠀⢠⡇
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡀⡇⡇⣧⢳⡼⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⡴⣿⣰⢻⢆⡇⠀⠀⠀⡾⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠳⣤⣄⣀⡈⠳⠃⠻⢏⣠⠤⠖⠒⠚⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠶⠦⢤⣀⡙⠃⠘⠞⠀⠀⠀⢠⡇⠀⠀⠀⠀⡿
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠴⠇⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⢀⠀⠀⠀⠀⢀⠀⢳⠀⠀⠀⠀⡜⠁⠀⢀⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠀⠀⠀⣰⠃⠀⠀⠀⠀⠰⡇
⠀⠀⢰⡇⡿⡆⢰⣿⠀⡟⣇⠈⡇⠀⠀⡼⠀⠀⠀⣾⠤⢼⢁⣶⠂⣼⢰⡴⢲⠀⠀⢰⡰⣆⡴⡀⣴⠖⠠⠁⠀⣼⠀⢠⠞⣁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠸⣿⠃⡇⠘⣿⡇⣧⠹⡄⢻⠀⠀⣧⠀⠀⠀⠃⠀⠘⠈⠓⠃⠛⣸⠛⠁⠀⠀⠘⠁⠉⠀⠃⠓⠚⠐⠀⣰⠃⠰⣧⠞⢸⠀⣤⠀⡀⢀⡀⢻
⠀⠀⠀⠀⠀⠹⡴⠃⢧⣸⠀⠙⠋⠀⠀⠈⠧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⢸⢸⣿⢰⣿⢸⡇⢸
⠀⠀⠀⠀⢀⣀⠀⠀⠈⢁⣤⡀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⠤⣤⣤⣤⣤⠤⣄⠀⢠⠴⠶⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣞⣸⣿⣸⣿⡾⢷⡼
⠀⠀⠀⠀⣏⡈⠳⡄⠀⠻⣄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⢦⡾⠀⠀⠀⢠⣤⢠⣶⣠⡄⠀⠀⠀⠀⠉⠁⠘⠃⠈⠁
⠀⠀⠀⠀⠀⠉⠙⠛⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⡀⡞⣧⣿⣷⢿⠄⠀⠀⠀⠘⠃⠀⠀⠀⢸⡘⣾⢸⡇⣧⣾⡆⠀⠀⠀⣠⠖⢦⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣞⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⣿⢿⡏⣼⠀⠀⠀⠀⠀⠀⠀⢶⡒⠾⠃⠀⠈⠁⠋⣸⠁⠀⠀⣞⣡⠴⠋⣰⢊⣁⣨⠇
⠀⠀⠀⠀⢠⠶⣄⠈⠓⠲⠶⠷⠀⣠⢤⣄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⠛⣻⠇⠀⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⢠⠇⠀⠀⠈⠁⣀⣀⠘⠉⠉⠁
⠀⠀⠀⠀⠘⠳⠾⠷⠀⠀⣀⡀⠀⢧⣀⣈⣳⡀⠀⠘⡇⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡾⠀⠀⠀⢀⠞⢁⣨⠇⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢭⣽⡦⠀⠀⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠛⠋⠉⠀⠠⠾⠧⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠛⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠛⠋⠁⠀⣀⣀⣤⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⣏⢏⢷⣀⣤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠙⢿⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀⠀⠀⡾⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠞⣛⣫⣭⡭⠭⠥⠆⠀⠀⠀⠀⠀⠚⠛⠛⠛⠿⠿⠿⠿⠿⠛⣿⠃⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣯⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠲⠾⠿⠭⠭⠭⠭⠍⠙⠛⠛⠉⠩⠭⣭⣭⣭⣭⡭⠭⢩⡏⠀⢀⡟⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡾⠀⠀⣼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣤⠶⠖⠚⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⣾⠃⠀⢰⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡟⢛⣶⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠲⠶⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡤⠶⡾⠛⠋⢩⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣽⠁⠀⣸⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢠⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢰⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⡾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠞⠋⠁⠀⣸⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠶⣭⣳⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⢖⣚⣩⡵⠞⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠲⠶⠾⠿⠿⠿⠿⠿⠿⠷⠒⠛⠛⠉⠉
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠚⢠⠔⠁
⠀⠀⠀⠀⠀⡠⠚⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠄⠀⢀
⠀⠀⠀⡠⠺⣰⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⡭⠖⠚⠉⣉⡉⠙⠒⠮⣕⠠⡀
⠀⡠⠾⣀⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠁⠀⠀⣶⣦⡟⣧⣶⡄⠀⠀⠙⢮⡢⡀
⣮⣄⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡱⠁⠒⠀⠐⠀⣴⣿⣯⣿⣷⡄⠐⠀⠀⠂⠱⡑⡀
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱⠃⠀⠀⠻⠀⠀⠈⠀⠿⠇⠈⠀⠀⠸⠃⠀⠀⢳⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⠒⠒⠂⠉⠱⠖⠚⢓⠒⢤⡄⠀⠒⠛⠒⢤⣀⣸⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⠋⡢⣞⠑⠀⠀⠀⠀⠀⠀⢸⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣆⠠⠄⠄⠠⠄⠀⠨⠀⠈⠀⠤⠀⠠⠤⠀⢲⡎⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⢦⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢝⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⠵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠭⠓⣒⣒⣒⣒⠫⠅⠂⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣯⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣿⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⡖⠒⠛⠓⠲⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣻⣻⣻⣻⣿⣻⣟⣟⡛⣿⢿⣿⣿⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣼⣿⣇⣀⣀⣀⣀⣀⣀⣹⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⠿⢿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣵⡿⠿⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣿⣿⢯⣶⡿⢿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⢻⣿⡀⠀⣸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢿⣯⠀⢀⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁

⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⡶⢶⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⢢⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠈⣧⡠⣂⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⣤⣬⣀⣄⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡄⢹⣄⣈⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠦
⠀⠀⠀⡧⠤⠤⠄⠄⠤⠄⠤⠀⠀⠀⠀⠀⠀⠀⠒⠂⠒⠂⠒⢂⣇⡗⠒⢷⠛⠓⠒⠒⠒⠋⠂⠀⠛⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⠀⠀⢱
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⡤⠈⠃⠆⠀⠀⠈⠃⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⢀⠠⠐⠁⢀⡀⠀⠈⢡⣀⣀⠠⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⣤⣾⣽⠇⠀⠤⣤⠠⢀⡀⠀⠽⣀⠀⣼⡟⠉⢰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠉⠋⠁⠀⡠⢰⡅⠀⠘⠓⠢⠀⠓⠺⣿⡷⠴⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣤⣀⡶⠁⠀⠀⠀⠃⠰⠀⠀⠀⠈⠥⡠⠊
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃
⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠠⡀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀
⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠁⠄⠀⠀⣀⡀⣠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠔⠀⠈⠃⠀⠾⢶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢁⡙⡇⠀⠀⠀⠀⠀⠀⠲⣿⣿⣯⡆⢀⣀⣤⣴⡶⠖⠚⠉⠉⠉⠉⠉⠙⠓⠤⣄⡠⡄⣀⠀⠀⡐
⠀⠀⠀⠄⠀⠤⡀⠐⠊⣿⣽⣧⠿⠃⠀⠀⠀⠀⠀⠀⠙⠿⣉⠛⠋⠉⠟⠉⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⢋⠀⠀⠀⠀⠦⣀⡀⠀⠀⠄⣀⠀⠁⠀⠁⠀⠀⠻⠉⠉⠉⠁⠀⠉⠁⠀⠠⠉⠙⠛⠉⠥
⠀⠀⠀⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠲⠂⠀⠈⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠖⠐⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠐⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠛⠛⠉⣉⣁⣤⣽⣿⣷⣶⠿⠿⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⣷⣶⡾⠿⠿⠛⠛⠋⢹⣿⣧⣤⣤⠶⠶⠞⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⠿⠿⠟⠛⠛⣿⣿⣁⣤⣤⡴⠶⠖⠛⠛⠉⢛⣁⣀⣤⣤⣴⣶⣾⠿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⢀⣠⣤⠶⠶⠚⠛⠛⠛⣉⣀⣠⣤⣴⣶⣶⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⠟⠁⢀⣠⣴⣶⣾⠿⠿⠛⠛⠋⠉⠉⠀⣀⣀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡏⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⣁⣀⠈⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢸⡟⠉⣿⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣸⡇⠀⢿⠶⠶⢾⡆⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⠇⢸⡟⠉⠉⠁⠀⢀⣀⣤⣼⠇⣸⠃⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡄⠸⠷⠶⢻⡇⠀⣿⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⣇⣠⣿⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣉⣁⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣾⣇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⢀⣀⣀⣤⣤⣶⣶⡾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⠿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣶⣶⡾⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠁
⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡈⢀⠄⠀⠀⠈⡀⠀⡀⠀⠀⠀⢀⡀⠐⠂
⠀⠀⠀⠈⠂⠑⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡐⠃⠀⢀⡀⡀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠠⠄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠘⠀⠀⠀⠐⠀⠈⠁⢀⣸⠏⠹⡃⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⡀⠂⡀⢇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠄⠂⠁⠀⠐⠃⡂⢀⠐⢂⠐⠊⠠⠄⠀⠒⠀⠊⠀⠀⠐⠒⠈⠀⠀⠒⠀⠠⠀⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡠⠀⠀⠀⠀⠀⠗⢪⠔⢈⡀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⡄⢀⡠⠒⠢⠐⠂⠈⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠺⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡖⠷⢤⠤⣎⠓⠆⠀⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈⠀⢱⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⢉⡙⠚⠤⠉⠀⠐⠉⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠉⠀⠀⠀⣠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠊⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠈⠀⠠⠀⠐⠂⠈⠀⠀⠁⠈⡉⠉⠀⠀⠠⡄
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠂⠀⠀⠀⠀⠚⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⡀⢷⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⡀
⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠁⡮⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠃⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣠⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠹⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠈⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⡤⠂⠠⠁⠀⠀⠠⠄⢂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠎⠀⠀⠀⠀⠐⠈⠂⠐⠤⠈⢂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠜⡀⠄⠀⠄⠠⠘⠀⠀⠀⠀⠐⡌⠀⠀⠀⠀⠀⠀⢀⠁⠀⠂⢀⠦⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢤⣉⡤⢇⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠈⠂⠰⣄⣀⣀⣀⣨⠄⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠠⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠬⢩⣤⠝⠰⠇⠀⡀
⠀⠈⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠰⣀⡀⠙⡑⣠⡔
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠒⢃⠀⢠⠀⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠠⠀⠁⠈⠂⠀⠀⢀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠰⠈⠀⠈⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣀⡠⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠂⠊⠠⣤⠀⠀⡈
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢆⠀⠒⠀⠀⠃⠘⠀⠛⠀⠐⠃⠀⠀⠤⢀⡀⣀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠰⠀⠰⠠⠤⠴
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠶⠆⠀⠇⢠⠀⠀⢀⠐⠈⠀⠀⠔⠁⠀⠀⠀⢀⡅⡍⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠉⠐⡋⠀⠙⠁⠀⠆
⠀⠀⠀⢀⣠⠔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⡛⣛⣛⣛⡘⣛⣻⢛⢛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣽⣯⣭⣿⣿⣿⣿⣿⣿⣉⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠓⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠂⠈⠁⠀⠉⠁⣀⡀⠀⡁⠀⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠤⠴⠀⠀⡀⣀⠹⠿⠄⠈⢀⣠⠀⣤⠤⠁⡙⠁⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠢⢆⣀⡀⠀⢀⣤⠖⠀⠙⢚⣙⠀⠀⠄⠀⠠⣤⣀⡁⠀⣀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⣠⢤⣾⠁⠀⠁⠯⡿⠀⠀⠀⠀⠀⠲⢂⣾⣶⣤⠄⢠⣽⡀⠃⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠀⠀⠀⢐⠘⠃⣹⢷⡄⠀⠀⠀⣻⣧⠀⡀⠀⠂⠉⠱⡀
⠀⠀⠀⡀⠀⠀⠀⣠⡀⡸⠮⠲⡇⠒⠼⣷⣿⠏⠀⠉⣰⣦⣺⢿⣯⢏⠛⠰⠆⠤⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣠⡀
⠀⠀⠠⠀⠀⠀⠀⡙⠈⢡⠄⠈⠈⡒⠄⢈⣻⠇⠀⠀⠊⠉⢼⣾⠟⠉⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠆⡈⢹⢿⢿⡗
⠀⠀⠀⣀⠀⠚⠁⠙⠚⠣⠇⠀⡀⠷⠌⠋⢉⣇⠀⠀⠀⠀⡼⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⠀⠀⣤⠄⣀⡀⡀⠠
⠀⢀⣍⣻⣿⡏⢀⣓⣶⢷⣶⡏⣯⣤⣤⣶⠂⣤⢶⣾⣷⣾⣶⣀⣔⡀⠒⠀⣀⣀⣀⡀⠀⣀⣀⠀⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠈⠉⠁⠈
⠀⠸⡷⢶⠄⠻⠻⣿⣿⠏⡟⠿⠐⢚⡷⠶⣿⣟⠙⠛⠀⠛⠻⣟⣩⣐⠂⠀⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠠⡀⠤⢄⣀
⠀⠀⠤⠤⠤⠥⠤⠭⠝⠛⠛⣁⣘⣥⠖⣖⣵⠀⠁⠀⠁⠀⢀⡘⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢄⣀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠁⠉
⠀⠀⠤⠤⠦⡄⢀⠀⠀⠀⠊⠁⠀⠀⠛⠠⠉⠁⠂⠀⠀⠀⠈⠃
⠀⠀⣀⡠⠎⠁⠀⢉⣤⡍⠀⠤⠢⣀⣀
⠀⢈⣍⠁⠚⠀⠀⠘⠓⢢⣤⡴⢠⣬⡍⢉⡂⠀⠄⠀⠀⠀⠈⠉⠁⢀⠄
⠀⠈⠁⠰⠄⠀⠢⠀⠈⠁⠀⠂⠀⠒⠘⠉⠀⠉⠁⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⣸⡇⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⠀⠀⣈⣁⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠿⠗⢰⣿⡏⠀⠀⢹⣿⡆⠺⠿⠿⠿⠿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣼⣿⣧⣤⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠿⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⢁⣠⣴⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠛⠉⠁⣀⣀⣀⠀⠈⠙⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢀⣠⣴⣿⡿⠁⠐⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠈⠉⠉⠉⢹⣿⣿⣿⣿⠋⠉⠉⠉⠁⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠈⠛⠛⠛⠉⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣶⣶⣶⣾⠿⠟⠉⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠉⠉⠙⠻⣿⣆⠀⠀⠀⣿
⢻⣿⣄⣀⣀⣀⣀⣠⣿⠋⠀⣴⣿⣿⣷⣄⠀⢻⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⠁⢠⣾⣿⣿⣶⡄⠘⣿⣇⣀⣠⣿
⠀⠙⠻⠿⠿⠿⠿⢿⣿⡀⠸⣿⣿⣿⣿⡿⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠀⢸⣿⣿⣿⣿⡇⠀⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠉⠛⠛⠛⠁⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠙⠛⠛⠋⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣴⣶⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣦⣶⣶⡿⠟⠁
⣶⣶⡶⢶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡎⢀⢤⡤⠔⠂⠀⠀⢔⣶⠂
⠀⠀⠀⠀⠀⣶⣾⣖⣀⣀⣀⣀⡀⢠⡤⢾⡷⠤⡀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⣶⣾⣧⣴⢶⠟⠋
⠀⠀⠀⠀⢰⠟⠛⠛⠛⠛⠛⠛⠛⣿⠀⠨⣧⣤⣥⣀⣀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢸⣏⡿⢿⣿⠿⢿⡿⣿⡿⠿⠿⠿⢿⣿⣿⢿⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠁⢯⣸⠗⢏⡹⠋⠉⠉⠉⠉⠉⢿⣉⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣴⣿⡟⠉⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⣤⠀⠀⠀⠀⣯⣿⡇⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⡆⠀⣠⠞⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠿⣶⣶⣶⣾⣿⣿⣦⣤⣤⣤⣴⡇
⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⡛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠿⣫⣽⣿⣿⣿⠋⡃⢐⢰⣦⣶⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠐⠒⠓⠒⠚⠒
⠀⠀⠀⠀⠀⠀⠳⠜⠈⠉⠁⠀⠀⠀⠀⠘⠦⠞⠳⠜⠁⠀⠐⠀⢀⣀⠀⢠⣤⣄⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣚⣀⣸⣿⣟⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣍⡭⣭⣭⣭⣭⣭⣭⣩⢭⣭⡅⠀⣠⣶⣷⣦
⠀⠀⠀⠀⠀⡌⠀⠀⢸⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠋⠉⠉⠉⠉⠉⠓⠞⠉⠀⣼⠏⠀⠀⠙⣷
⠀⠀⠀⠠⣶⠿⢿⡄⠈⠀⠀⣼⠿⣶⣆⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣀⣀⣀⣀⣸⡄
⠀⠀⠀⠰⢷⡞⢳⡷⠶⠦⠾⣷⠛⣿⠿⠀⠀⠀⠀⠀⠶⣷⣦⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⡄⠂⠀⠀⣿⣿⡇⠀⠀⠀⣤⣼⡃⠀⠀⠀⠀⠸⠿⠶⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⣒⣒⣒⣛⣛⣛⣒⣒⡲⠍⠀⠉⠀⠀⠀⠀⠈⣷⣄⣀⣀⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠹⠏⠉⠿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⡴⢄⠀⠀⠀⠀⠀⠀⠔⠀⢸⠟⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠈⢾⡁⠀⠀⠀⠀⠀⠀⠀⠸⠄⠤⠷
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢈⠀⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠖⠂⠀⠀⠀⠏⠃⡇⠀⠈⠀⠀⠀⠀⠀⠀⠿⠛⠀⠀⠀⠆⠄⠁⠀⠀⠀⠀⠀⠀⠀⠠⡐⠀⠀⠀⠀⠀⠂
⠀⠈⠈⠹⠛⠋⣧⡍⠀⠀⠀⠀⠀⠀⠀⠢⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⡀⣴⡆⣰⣶⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠠⠤⠤⠆⠀⠀⠀⠀⣰⡄⠀⠷⣾⡇⣿⡇⢹⡿⠀⠀⠀⠐⢦⢄⣠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠃⣣⠀⠾⠟⠀⠿⠇⠼⠧⠀⠀⠀⠀⠠⠂⠙⠄
⠀⠠⡀⠀⠂⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠊⠄⠀⠀⠀⠀⠈⠃⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠑⠂⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣄⠀⢀⣤⣤⣤⡀⠀⠀⠰⠤⠤⠄⠀⠶⠶⠂
⠀⠠⠀⠀⠄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠂⠀⠀⠀⠀⠀⠀⠀⣠⡀⠘⠁⠘⠛⠒⠘⠘⠋⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⡬⢰⢤⡟⢀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⣢⠀⠈⠀⠠⡆
⠀⠀⠀⠀⠸⢿⣿⠃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠠⠀⠀⠀⠁⠀⠀⡄⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠎⠀⢀⣱⡄
⠀⠀⠈⠐⣀⡘⠀⠃⠈⠁⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢇⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠄⠭⠡⡄
⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⠄⠀⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠆⠀⠁⠀⠃⡀⠻
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣠⣴⠞⠁⠀⢀⣀⣀⠀⠈⣟⣾⠂⠂⠒⠡⠤⣤⣤⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⠌⠳⠿⠶⠦⠭⠭⠭⠤⠤⠭⠽⠛⠄⠀⠀⢀⠀⠀⠀⠀⠀⠈⢲⡆
⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠁⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣶⣶⠿⢷⣮⡉⣡⡶⠿⢦⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⡶⠶⣦⡵⣄⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠦⠦⣿⡃⠰⠀⣿⡿⣿⡄⠶⠀⣿⠘⠓⠶⠆⠀⠀⣄⣀⣠⠒⠻⡟⡄⠆⠈⣿⠒⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠾⠋⠀⠙⠳⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⣤⡴⠋

⠀⣀⣀⣀⣀⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠰⡶⡶⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⢠⣿⣿⣿⠋⠙⣿⣿⠉⠁⢄⠀⠀⠀⠀⢀⣴⣯⡀⢀⣀⣀⣉⣀⣀⡀⠀⠀⡿⠿⡿⡛⣿⠿⠿⣷⠟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠙⠚⠀⠀⣼⠃
⢸⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣷⣶⡄⠀⢸⣿⡄⠛⠘⠭⠟⠛⢿⠿⠇⠀⠀⣿⣿⣿⣿⣬⣿⣿⣿⣷⣶⢿⣶⡄⠀⠀⣿⡹⠞⠿⠿⠿⠿⠾⣿⡀
⠛⠛⢿⡽⠛⠛⠛⠛⠛⠛⢻⣭⠟⠃⠠⡞⠀⢿⠀⠠⠴⠞⠉⠑⠦⠄⠀⠘⠛⢻⣽⠟⠛⠛⠛⠛⠛⠛⢯⡽⠛⠀⢰⠏⡧⠀⠀⠀⠀⠀⡼⠉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⣤⣀
⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣢⠀⠀⠀⠀⣾⣿⣷⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⢸⠂
⢠⣦⣶⣶⣷⣶⣶⣶⣷⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣵⡏⠀⠉⠀⠀⠀⠀⣿⣿⡿⢀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣠⠞
⢸⣿⣿⣿⢫⣴⢶⣌⢻⣿⣿⣿⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⡌⢀⡠⠊⠁⣴⠒⠚⠒⣦⠀⠀⠀⠀⢀⡤⢄⡤⣌⢳⡞⠁
⢸⣿⣿⣧⢻⣦⣠⣼⢢⣿⣿⣿⠀⠀⠀⠀⠀⢯⣾⢟⡿⠋⠀⠀⠀⠀⠀⠀⢸⢀⠊⠀⠀⠀⣾⣫⠉⣝⣿⠀⠀⠀⠀⡏⠀⠈⠻⠋⢰⠆
⢸⣿⣿⣿⣷⣬⣭⣴⣿⣿⣿⣿⠀⠀⠀⠘⢷⡇⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣦⣤⡤⠤⣿⣽⣻⣫⣿⠀⠀⠀⠀⠳⣄⡀⣀⣠⠞
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣰⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣖⣲⣦⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆
⢰⡿⠙⢃⣀⡀⢀⡀⣀⠀⠀⠀⠀⠀⢻⡛⡛⣛⢛⡛⡛⣛⢻⡟⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⢀⢀⢸⡇⠀⠀⠀⠘⢿⣿⣿⣟⡶⣾⢶⣺⣿⣿⡷
⢸⡇⠀⢾⣅⣿⢸⡇⣿⠀⠀⠀⠀⠀⢸⠭⡭⠭⠭⠭⡭⠭⢽⡇⠀⠀⠀⠀⠀⠀⢸⡇⠯⣼⢾⣼⢸⡇⠀⠀⠀⠀⠀⠈⠁⠣⣹⠤⠯⠜⠈⠁
⢸⡇⠀⠰⣭⡿⢸⡇⡿⠀⠀⠀⠀⠀⢸⣓⣓⣚⣚⣓⣓⣛⣺⡇⡴⠲⣦⠀⠀⠀⢸⡇⠙⠋⠘⠙⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣡⠶⣚
⠈⣿⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣧⡴⡴⣦⢦⣴⣼⡇⠻⡶⠋⠀⠀⠀⢸⣿⣛⡗⢺⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢚⠅
⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠃⠉⠘⠛⠛⠓⠚⠛⠂⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡖⡒⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡆
⠀⠘⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣓⣚⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣟⠃⠀⣠⣦
⢰⠉⠀⠀⠈⢱⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢀⣀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠻⣿⣿⣿⣮⠉
⢸⡇⢀⠸⢿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠈⡿⠿⠿⠿⠿⢾⡫⢝⡇⠀⠀⠀⠀⠀⣈⡛⠟⣁⡀⠀⠀⠀⠀⠀⠀⠀⣤⣽⣿⣵⣿⣿⣾⡿⠁
⢸⠿⠿⠀⠻⢿⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⣾⣷⢠⣾⠿⠂⠀⠀⠙⠋⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠿⠋
⠸⣀⣀⣀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠛⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣛⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣥⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠬⢭⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣛⠳⠄⠀⠀⣤⣤⣾⣯⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠒⠒⠒⠒⠒⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⡖⠀⢰⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⠀⠀⢀⣭⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡾⠁⠀⠋⢠⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠆⠸⣇⠀⠶⠀⠏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡤⢿⡆⠀⠀⠘⠈⠃⠀⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢞⣯⠉⠁⠀⠀⠀⣰⠃⠀⡆⠀⢀⣴⠀⠀⠀⠀⠀⢉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣌⣋⣀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠁⠀⠀⢀⣀⣠⣤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⡀⠀⠉⣉⣉⠐⠒⠓⠒⠒⠒⠒⠓⠛⠛⠉⠉⠉⠀⠀⠀⢸⡿⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⣀⣈⣻⠛⠃⠀⢰⡇⠠⡄⢠⣆⠀⠀⠀⠀⠼⢫⡿⠀⢸⡇⢠⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⠏⠉⠙⢢⣤⠤⡄⠹⡄⢀⡄⠉⡇⠀⣠⠶⢶⡄⠀⠀⢸⣷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣨⠴⠶⣄⠀⠀⠐⠃⣠⠇⠸⠁⣼⢁⡾⠀⢈⣸⠇⠀⠀⢸⣿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠺⣧⡶⠂⠘⣷⠀⠀⠐⠋⠀⠀⠀⣉⡴⠃⠰⠟⠁⠀⣰⠆⢸⡏⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⣸⠏⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠰⠖⠛⠁⠀⢸⣇⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⠆⠀⠀⠀⠀⣠⡾⠏⠀⠀⠈⠉⠁⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⣸⡆⢸⣿⡛⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⢠⡾⠋⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⠛⠳⣦
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⢧
⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣦⣤⣴⠏
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⢸⣿⠁⠀⠘⡄
⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠤⠦⠞
⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢸⡇
⠴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠈⠀⠒⠚⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣁⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠉⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠀⠉⠙⠲⠶⢤⣤⣍⣉⣙⣛⣛⣀⣀⣬⣤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⡿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠜⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠈⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢀⣀⣿⡟⢻⣿⣀⡀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⡏⠀⢸⣏⠉⠁⠈⠉⣹⡇⠀⢹⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⡀⠘⠛⣿⣇⣸⣿⠛⠁⢀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢷⣄⠀⠉⠉⠉⠁⠀⣠⡾⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠻⠷⠶⠶⠶⠛⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⡀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠛⠛⠁⠀⠘⠛⠛⠃⠀⠈⠛⠛⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⠄⠀⢠⣤⣤⡄⠀⠠⣤⣤⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⠀⠘⠛⠛⠂⠀⠘⠟⠛⠂⠀⠐⠛⠛⠃⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⡀⠀⢀⣀⣀⡀⠀⢀⣀⣀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠦⠀⠰⠶⠶⠆⠀⣿⠀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠁⠀⠀⣿⠀⠰⠶⠶⠆⠀⠴⠶⠶⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠦⠀⠰⠶⠶⠆⠀⣿⠀⢸⣿⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣈⣉⣿⡇⠀⣿⠀⠰⠶⠶⠆⠀⠴⠶⠶⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⣿⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⣿⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢶⣶⡦⠀⠰⣶⣶⠆⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠰⣶⣶⠆⠀⢴⣶⡶⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓

⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲
⡰⠒⠢⡄⠀⠀⣀⠀⠀⠀⠀⠸⢱⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⠒⡄⠧⠴
⠑⠤⠠⠎⠀⡘⠂⠃⠀⠀⠀⠀⠀⣤⡀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⣩⠆⠰⠅⠜⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⢎⣠⠇
⠀⠀⠀⢠⡏⠉⠑⢦⡤⡀⠀⠀⠀⠂⠣⡀⠀⠀⠘⢍⢧⠀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⢀⠴⠤⢔⡺⠛⢉⡭⠝⡩⠿⠽⢦⠀⡠⢤
⠀⢀⠀⠀⠙⠛⠛⠲⣄⠙⣄⣀⣀⡼⠆⣈⡉⡒⠤⠒⠃⠉⠁⠀⠈⠉⠉⠢⠴⠊⠑⢒⣁⡴⣤⠤⡝⣯⡛⠒⢒⣉⣰⡴⠋⠀⠀⡿⢀⡾⠔⠊⠉⠁⠀⠈⠉
⠀⠈⠀⠀⠀⢀⡴⠒⠈⢳⣉⠲⢷⡖⠲⣆⣏⣁⣠⠦⣀⣀⡤⠀⠀⠒⠲⠴⠂⠀⠀⠀⠠⠀⠀⡜⢵⠋⠙⠒⠋⠀⡜⠀⠀⠀⣠⠇⢸⠁
⢀⣀⠀⢠⠒⠉⡠⢄⠤⢌⡳⣧⡀⠉⠓⠏⠙⠿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠉⠢⠤⠦⠶⠤⠀⠀⣸⠁⠀⢸
⠣⠔⠁⠈⠁⠋⠀⠠⠀⠀⠳⢌⣻⣦⡀⠀⠀⠀⠀⠙⠓⢦⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠦⣶⡦⣄⠀⠀⠀⠀⠙⠦⣤⡤⠤⣄⣀⣄⣠⠖⠶⠃⠀⠀⠀⠀⣀⡀⠀⠀⣸⡇⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⠀⠁⢿⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣖⠈⠓⠶⠛⠛⠋⢩⠉⠛⠉⠅⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠙⠶⠤⣤⠴⠋⠳⠦⠴⠒⠐⠦⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠊⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡆⠘⢆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠳⣄⠀⠀⠁⠒⠂⠤⠤⢤⡀⠀⠀⠀⠀⠀⠤⠤⠒⠒⠉⠀⠀⠀⠀⠀⠀⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠂⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠂⠤⠤⠤⠤⠤⠤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⠶⠿⠿⠿⠿⠿⠿⠿⠷⠶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣒⣒⠻⢧⣄⣹⡷⣄⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣿⣿⣶⣬⡛⢷⡝⣆⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⣼⠟⣡⠶⠚⣛⣳⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠧⠄⣠⣤⠙⠿⣦⣽⣿⡀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⢀⣾⢯⡞⢁⣴⣿⣿⣯⣿⣿⣻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⢐⡓⠄⠀⠀⢰⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⣼⠏⣿⠀⣿⣿⣿⣿⠏⣿⣿⡇⣿⣿⣶⣄⡀⠀⠀⣠⡴⠶⠦⣤⣀⣀⣀⣀⣀⡀⣀⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⣼⡟⠀⣿⠀⣿⣿⣿⣫⡼⣿⣿⢡⣿⣿⣿⣏⠻⢶⣤⣏⣰⣶⢶⣄⠩⠝⠻⢯⣝⠛⠹⣿⠿⠷⠶⠆⢰⣾⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⢠⣿⠁⠀⠘⣷⣝⣿⣿⣿⣷⠟⣡⣾⣿⣿⣿⡟⠀⢸⣿⠿⢿⣧⣄⠈⠓⣒⡶⠄⢈⢻⡆⣿⡇⠀⠀⢀⣿⠇⠀⢹⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⢸⡏⠀⠀⠀⠈⠙⠫⠭⣭⣴⣾⣿⣿⣿⣿⠟⠁⠀⣸⣟⣴⠞⠋⠙⠛⣋⠥⣖⣛⣭⣾⣧⠸⡟⣶⠶⢛⡁⠀⠀⢸⠈⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣯⣤⣀⣠⣾⠟⢸⡇⠀⣾⣷⢀⡾⠛⣿⣄⡁⠀⠈⢸⠏⠔⠛⠛⠻⢷⣦⡾⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⡽⠿⣿⣷⣦⣬⣷⣤⣨⣥⠞⠃⢰⡟⠙⣿⣄⠀⡞⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠘⠉⢀⡴⢻⣿⠁⠙⠿⢿⣿⣷⠶⢶⣏⣁⣴⠇⠈⠳⣇⠀⣠⣤⣀⣀⡀⠀⠀⢠⡏⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣿
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠘⠀⣾⣿⠷⢄⣞⠉⠉⠀⠀⠀⢻⣿⣧⣄⠀⠀⠈⢧⡙⢿⣿⣿⡿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣿⣿⣄⠀⠉⠛⢛⡿⠀⠀⣸⠇⠈⠻⣦⡀⠀⠀⠳⡄⠀⠀⠀⠀⢰⣧⣀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢠⣿
⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣿⣷⣀⠉⠉⠉⢙⡇⢀⡴⠋⠀⠀⠀⢨⠿⣆⠀⠀⠹⡄⠀⠀⣰⡟⠉⠛⢷⠶⠚⠃⠀⠀⠀⢻⠀⣼⡏
⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣈⠓⠚⣉⣿⡟⠋⠀⠀⠀⠀⢀⡿⠀⠘⣧⠀⠀⠙⣄⣼⠏⠀⠀⢠⣿⣤⣤⡀⠀⠀⠀⣾⢠⡿⠁
⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡋⠉⠉⣿⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⢷⡀⠀⠸⡇⠀⠀⢠⣿⣉⠛⠿⣿⣦⠀⣰⢧⣿⠃
⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠁⢠⣿⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣈⣷⡀⠀⢻⡶⠶⠾⠉⠙⢷⣄⠈⢿⣷⢯⣿⠃
⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⣤⣶⣿⣿⣿⣶⣶⣦⣤⣿⣷⣶⣶⣾⣿⣿⣿⣧⡀⣸⠇⠀⠀⠀⠀⠈⣿⣄⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠋⠁⣀⠀⠀⠀⠀⡼⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡅⣉⡤⠀⠀⢀⡜⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠔⢂⣤⣾⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠘⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢠⠇⠀⠀⢸⠁⠀⠀⠀⢸
⠀⢷⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡼⠀⠀⠀⡿⠀⠀⠀⠀⡾
⠀⢸⡀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⠇⠀⠀⢠⡇⠀⠀⠀⢠⡇
⠀⠈⡇⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠘⣆⠀⢠⢞⣡⣄⠀⣀⣠⣤⢸⢳⡘⡇⠀⣠⠏⠀⠀⠀⠀⠀⣼⠀⠀⠀⣸⠀⠀⠀⠀⢸⠁
⠀⠀⢿⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠁⠀⠀⠀⠀⢀⣄⠀⣦⣹⡀⠈⠉⠀⡟⢸⢿⣿⢸⣾⠀⠳⠇⣰⠃⣀⠀⠀⠀⠀⠀⠙⢄⡀⠀⡟⠀⠀⠀⠀⡿
⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡄⠘⠒⠚⠉⢳⠀⣿⣾⡌⡇⢿⠛⠃⠀⠀⠰⣇⡞⢸⣿⠘⠟⠀⠀⠀⣧⢎⡟⣼⡇⣠⠀⡾⠓⠦⡵⢸⠃⠀⠀⠀⢠⡇
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢸⡄⡇⡇⣧⠻⡼⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠴⣿⣰⢻⢇⡇⠀⠀⠀⡾⠀⠀⠀⠀⢸⠁
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⣤⡀⠓⠃⠙⣋⡤⠶⠒⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠲⠦⣤⣙⠁⠈⠛⠀⠀⠀⢠⡇⠀⠀⠀⠀⡾
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠒⠃⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⣠⠀⠀⣀⠀⣀⡀⢳⡀⠀⠀⢀⡞⠁⠀⢠⠀⢀⡄⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⡇
⠀⠀⢰⣧⡟⡇⢸⣿⡀⡏⣇⠈⣇⠀⠀⣾⠀⠀⠀⡾⠶⢾⢡⡾⠃⣾⢸⡖⣹⠀⠀⢸⡴⣧⠾⡆⣼⠟⠐⠃⠀⡼⠀⣠⢎⣤⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠘⡿⠁⣇⠈⡟⡇⢷⠘⣆⣹⠀⠀⢷⠀⠀⠀⠁⠀⠈⠈⠙⠁⠉⣼⠉⠁⠀⠀⠈⠀⠈⠀⠉⠙⠉⠈⢀⡴⠁⠐⠧⠋⢸⠇⣶⡀⣄⢠⡄⢸
⠀⠀⠀⠀⠀⠙⠾⠃⢷⡼⠀⠈⠁⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⢸⢸⣿⢡⣿⢸⣿⢸
⠀⠀⠀⠀⣠⢤⡀⠀⠀⣠⠦⡄⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠶⠶⠶⠶⠶⠶⢦⠀⢰⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⢿⣸⢿⡞⠻⠞
⠀⠀⠀⠀⠧⣄⣙⣆⠀⠙⢄⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⡀⢀⠀⠀⠀⠀⢳⡼⠀⠀⠀⢠⢶⢰⢿⣤⣆⢀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⢉⣀⣀⠀⠙⠃⠀⠀⠀⠀⠀⠀⢀⣄⡏⣷⣿⣿⢻⠆⠀⠀⠀⠀⠁⠀⠀⠀⢸⡎⠿⢸⡇⣧⢿⡇⠀⠀⢀⡴⠋⣱⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢮⡀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⢸⠻⠇⠿⠛⠏⢸⣀⣀⡀⠀⠀⠀⠀⠻⢍⡛⠁⠀⠀⠀⠀⣼⠀⠀⠀⡾⠴⠚⢁⣴⣩⡤⠴⠃
⠀⠀⠀⠀⢰⡛⠢⡀⠉⠉⠛⠛⠀⡴⠒⢦⡀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣉⡼⠃⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢰⠃⠀⠀⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠉⠛⠛⠀⢀⡤⣄⠀⠓⠦⠤⡽⠆⠀⠈⡇⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡟⠀⠀⠀⣠⣋⣠⠴⠃⢀⡤⠴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⠓⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⠉⠁⠀⠀⠀⠛⠛⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⡭⢉⣳⣤⡀
⣿⠻⠿⣟⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⠛⠛⠛⠛⠛⣋⣾⣿⣿⡿⢻⠛⠛⢳
⣿⢰⠖⠤⠤⣈⣉⡓⢲⡄⠀⠀⢀⡴⢚⠉⠈⠻⡍⠙⠲⢦⣤⣴⠙⠛⠀⠈⠑⣿⠀⠀⢸
⣿⢸⠀⠀⠀⠀⠈⣿⢸⡇⢀⡴⢋⡶⣯⣳⠀⠀⠀⠀⠀⠀⢹⠈⡆⢸⣧⣀⡼⠋⠀⠀⢸
⣿⢸⠀⠀⠀⠀⠀⣿⢸⣷⡋⠀⠘⠓⠛⠁⠀⠀⠀⠀⠀⠀⠈⣧⢿⣿⣧⡈⠀⠀⠀⠀⢸
⣿⢸⠀⠀⠀⠀⠀⣿⡼⡿⠁⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡝⣄⠀⠀⠀⢸
⣿⠘⠛⠿⠶⢶⣴⣿⣽⠃⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⡯⢗⣂⠈⣧⠀⠀⢸
⣿⠀⠀⣀⡤⣴⢲⢷⣿⣀⡀⠀⣾⡿⡸⢀⡀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣭⣴⣶⢸⠀⠀⢸
⣿⠀⠀⣒⣿⠏⣼⢸⣿⠖⠂⠀⣿⢣⡇⣸⠀⠀⠀⠀⠀⠀⣼⡿⠟⠉⠉⠛⠛⠾⢦⣀⣸
⣿⠀⠀⠀⠀⠀⠸⡎⣿⠀⠀⠀⣿⢸⠇⡟⢸⠀⠀⠀⢀⡼⠋⠀⠀⠀⣠⠒⠒⠦⡀⠀⠉⠑⠢⢄⡀
⣿⣿⢰⠞⠓⢠⡶⢿⡸⣆⠀⠀⢹⣼⡴⢗⣯⠟⢝⡤⠊⠀⠀⠀⣠⠾⠁⠘⠀⢠⣾⡆⠀⠀⠀⠀⠈⠑⠢⣀
⣿⠿⠘⢦⣀⢸⣟⡛⣧⣻⣷⣴⣾⣵⣾⣷⣿⣠⢎⣠⡖⠚⠓⠲⣽⣄⣀⠀⠐⣟⡿⠁⠀⠀⠀⠀⠀⠀⣠⡞
⡿⠀⠀⠀⠀⢀⣩⣵⣺⣵⣾⡇⠀⣿⣿⣿⣿⠗⣻⠿⢤⣄⡀⠐⢌⣉⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⠏⢁
⡗⠀⣀⢤⡺⠟⠊⠁⢸⣿⣟⡥⠤⢿⣟⣡⠾⠋⠁⠀⠀⠈⠙⢦⡀⠀⠉⠉⠳⣄⠀⠀⠀⠀⠀⣴⡿⠁⠀⡼⢄
⣧⢾⠟⠁⠀⢴⣒⡟⣸⣿⣿⣆⠀⣰⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⣼⠃⠀⠀⣠⣾⠉⠀⢠⠞⣠⣾
⡇⠃⠀⠀⠨⠽⢞⡣⠟⣫⣾⣝⠋⠁⠀⠉⠓⢦⡀⠀⠀⠀⢀⡴⣿⣲⡄⢀⣴⡃⠀⣠⣾⠏⠁⢀⡴⣣⣾⠟⠁
⡇⠀⠀⢀⡠⠖⢉⡴⠛⠉⠀⠉⠙⠶⣄⠀⠀⠀⢳⡤⢔⣾⠗⢮⣁⠈⣾⡏⠀⣹⣾⠿⠉⠀⡰⢋⣴⡟⠁
⣇⡤⠖⢉⡠⢚⣡⠔⠉⠀⠀⠀⠀⠀⢈⣷⢴⣾⣵⣾⣟⣁⣤⣾⣯⢿⣿⣠⣾⢿⡇⠀⡠⢊⣴⡿⠋
⡏⣠⢖⣭⠖⣫⠔⢊⠀⠀⠀⣀⢴⣺⡿⠞⠋⠁⣈⣩⣽⣿⡿⠟⢁⣴⣿⡟⠋⢀⡷⠊⣠⡿⠋
⣿⣿⣫⠶⣋⠴⢊⠡⢀⢔⠽⣪⠕⠉⣠⡤⠶⠟⠻⣏⢉⣠⣴⣿⠻⠻⣿⣀⡴⠋⣠⣾⢿⠁
⣿⣫⡵⣋⡵⠚⠁⢰⡷⠁⣼⣿⣶⠞⠀⠀⢀⣠⣴⢿⠟⠃⠃⠀⣀⣴⠟⠉⢀⣾⠟⠁⢸
⣿⡵⢚⡥⢒⠁⠀⣸⡷⠊⠛⠛⠁⣠⣴⡞⠟⠋⠉⢸⣀⡤⠖⠋⠁⣀⣬⣿⠿⠃⣀⡴⢿
⣿⣚⡵⢚⡡⠔⠋⢹⣇⣀⡴⡶⠟⠙⠘⠀⢀⡠⠔⠋⠁⢀⣤⣶⡿⠛⠉⣠⡴⠛⠉⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠈⠀⣠⠔⠈⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⢂⠠⠤⠈⠀⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠒⡀⠊⠀⠁⠀⠃⢀⡴⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⠀⠀⢀⠘⢤⡀⠀⠢⠀⠈⢠⠀⠀⠀⠀⠰⡃⣀⠤⣤⣤⣶⡎⡁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠧⣀⡅⠀⢀⣅⠐⠠⠒⠣⠤⡰⠀⠀⠀⠞⢡⠿⠆⠛⣘⢋⣅⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠠⠁⠐⢀⣀⠀⡀⢺⢄⣤⡶⠃⠠⢀⠀⣀⣀⡀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢦⠀⠢⠄⠐⠒⠒⠛⠫⠛⠁⠀⠈⣉⡉⡡⠔⢁⠴⣿⣦⡥⠤⠅⠈⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤
⠀⠴⠒⠀⠂⠂⠐⠀⠀⣀⠀⠀⠀⠴⠤⠀⣐⠒⠀⠰⠦⠶⠶⡴⠖⢚⠋⠁⠀⠀⠞⠂⠁⠙⠙⠃⠀⠀⠀⠐⠀⠐⠒⠐⠐⠒⠒⠀⠀⠁
⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠠⠂⠀⠀⠀⠀⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠶⠶⠒⠚⢿⣿⣿⣉⣿⣿⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣒⣛⡋⠉⣉⣀⣤⡤⠴⠶⠒⠛⠋⠉⠉⠁⠀⠀⠉⠳⡄⠙⣦⡤⠶⠶⠒⠒⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠖⡽⠟⠛⠋⠉⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣏⣁⡀⠀⠀⠀⠀⠀⠙⠲⣌⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠇⢧⡀⠀⢀⣠⠶⠚⠋⠉⠉⠉⠙⠛⠶⢶⣞⠛⠉⠿⠿⢿⣷⣶⣬⣉⠛⢷⣶⣦⣤⣀⠀⠀⠙⢦⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢳⣴⣋⣀⡀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠈⠛⢯⡛⢯⡛⠲⢬⡛⢉⣓⢦⡙⢦⡙⢯⡻⣶⣄⠀⠙⢦⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠞⠙⢦⠻⡿⣄⢸⡆⠀⠀⡇⠀⠀⣰⠆⠀⠀⠙⢦⢹⡷⡄⠘⢷⣬⡓⠙⠢⣝⣆⠙⣮⠻⣷⣄⠀⠹⣌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠈⠳⡝⢾⣯⡁⠀⠀⠁⠀⢠⡏⠀⠀⢀⡀⠀⠻⣷⣝⢦⢠⡻⡝⠳⣄⣨⠿⠓⢾⣧⠘⣎⢧⡀⠹⠌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆⠀⠀⠀⠀⠙⢮⠳⣿⣄⠀⠀⠀⠈⠀⠀⢠⠞⠀⠀⠀⠹⣟⣎⢧⢳⡹⡆⢸⠁⠀⣀⡤⡽⣧⠘⣎⢧⠀⢦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⡆⠀⠀⠀⠀⠈⠳⡝⢿⣧⡀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠸⡞⣎⢧⢳⠹⡌⢧⡴⠋⠀⠙⢎⢧⣹⡌⡇⠘⡆⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣆⠀⠀⠀⠀⠀⠘⢦⡡⡽⣄⠀⠀⠀⠀⠀⣀⡴⠒⠀⠀⢹⡼⡾⡎⣇⢧⠀⠹⣄⠀⢀⡤⠶⠿⣇⢸⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⣆⣠⣴⡖⠦⣄⠀⠳⣜⠋⢧⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣷⢁⢿⢸⢸⡀⡆⣟⢶⡋⡘⢦⠳⡜⢾⡅⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢤⣤⢠⡟⣡⢤⡙⠶⣌⠳⣀⠘⡆⠘⣇⠀⠀⠀⠀⣀⣀⡀⠀⠀⢸⣸⣿⠘⠋⡇⣇⣿⢈⢳⡝⢮⠳⡜⢮⢳⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡇⣿⠀⣻⠀⠈⠳⣍⢲⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡮⡇⡇⣿⣿⢸⢸⠙⣎⠳⢍⡤⠖⢿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣷⣌⠙⠁⠀⠀⠀⠈⠳⣝⢾⡃⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡇⢸⡇⠁⡇⣿⣿⣾⣼⣶⢸⢷⣏⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡼⣁⠈⠳⢦⡀⠀⠀⠀⠀⠙⢦⡙⢦⡀⠀⠀⠙⠳⢤⠀⠀⢸⡇⢸⠁⢸⢇⣉⡇⠏⡷⠟⡟⢸⢛⣦⠀⠀⠴⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡞⣱⠏⠀⠀⠀⠉⢳⣤⣀⣤⠤⠤⣝⣆⠳⡄⡀⠀⠀⠀⠀⠀⣸⠃⣼⠀⣸⢸⣽⣿⢲⠷⣾⢃⣿⠟⠈⣳⣄⣠⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣰⠃⠀⠀⠀⢰⠋⢻⡅⠀⠀⠀⠀⢸⣿⣆⡇⠙⢦⣀⠀⠀⢀⡟⢠⡏⢀⡏⢀⡟⣠⣯⠴⠟⢋⣠⢆⣴⡃⠈⠓⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣰⠃⠀⠀⠀⠀⠘⠦⠾⠷⠶⣶⢶⣶⡛⠞⠛⠀⠀⠀⠉⠀⢀⡾⢁⣾⣠⣾⡴⠛⢿⡅⠤⠤⠀⣉⢴⡏⣹⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠏⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡡⡝⣆⠀⠀⠀⠀⠀⢀⡾⠓⠋⠉⠁⠉⠙⠓⠒⠒⠒⢶⣉⠀⣼⢿⡁⠈⠓⢦⣄
⠀⠀⠀⠀⠀⠀⢠⠏⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣜⠎⣳⡄⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠚⠉⠳⢤⡙⣎⢳
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢉⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢶⣚⠑⠋⡼
⠀⠀⠀⠀⠀⠀⠧⣄⣀⠀⠀⣀⡤⠖⠒⠛⠓⠲⠦⣤⣀⠀⠀⢠⣿⡞⣙⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠒⠋⠉⠁⢀⣀⣭⠤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠾⠶⠷⢬⣓⢦⣉⠛⠒⠶⠶⠒⣚⣛⡉⠉⠀⠀⣀⣠⡤⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⢭⣉⣛⣀⣉⣉⡥⠴⠒⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡤⠂
⠀⠀⠀⠀⡃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠐⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠠⠴⠀⠀⠀⠀⣀⠠⢀⣀⠀⣀⡀
⠀⠀⠠⡀⠀⠈⢄⠨⡒⢄⡠⢀⠃⢱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠲⣀⣀⠤⠒⠩⠔⠁⠀⠂
⡤⠀⠀⠀⠀⢀⠈⢂⠑⠒⠠⠤⠐⠂⠀⢈⠀⠀⠀⠅⠀⡀⠀⠀⠀⠐⠒⠄⠃⢄⡀⢀⣀⣀⠄⠀⠀⠀⣀
⠀⠠⡌⢁⣀⡡⠤⠤⠗⠒⠒⠦⣄⣀⡳⠚⠃⢄⣄⡠⡆⢧⠀⠀⡀⠠⠠⢤⣀⡠⢀⢀⣭⡤⠄⠈⠙⠀⠐
⠀⠀⠈⠠⢅⣀⣀⠀⠀⠀⠀⠈⠀⠀⡀⠤⢴⣆⣀⠔⡁⠀⠀⠰⠔⠒⠀⠐⠂⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀⠐⠒⠒⠁⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠛⠉⠉⢻⡏⢹⡏⠉⠉⢻⡇⠰⠶⠶⠶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣠⣤⣴⠶⠶⠶⠿⠷⠾⠷⠶⠶⠾⠃⠘⠛⠷⠶⣤⣄⡈⠉⠛⠶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠘⢿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠲⣦⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⠀⠉⠙⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣤⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⣼⣇⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠞⠛⠋⠁⠀⠀⣠⡾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠷⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠋⠁⠀⠀⠀⣀⣀⣤⡶⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣤⣀⠀⠀⠀⠈⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠘⠛⠿⠷⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠾⠛⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠙⠛⠶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠶⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠲⠶⠶⠶⠶⠶⠖⠛⠛⠛⠛⠋⠉⠉⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠈⠉⠛⠛⠳⠶⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⠛⠛⠉⠉⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⠦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡴⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⡉⠠⠤⠀⠀⠀⠀⠠⠤⠈⡉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⠴⠊⠁⣀⡠⠤⠔⢒⣒⡒⠒⠠⢤⣀⠁⠢⢌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⠞⠁⢀⠴⣋⠥⠒⠉⠉⣈⣉⡁⠈⠉⠒⠬⣙⠢⡀⠑⢄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡰⠃⢀⠜⡡⠊⢁⡤⠖⠋⠉⠀⠀⠀⠈⠉⠒⢄⡈⠑⢌⢢⡈⢣⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⠁⢠⢏⠜⠀⡴⠋⠀⢀⡠⠔⠒⠒⠒⠢⢄⠀⠀⠙⢆⠈⢣⠱⡀⢣⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⡇⠀⠏⡜⠀⡞⠀⠀⡴⠋⣠⠴⠒⠒⠒⢤⡀⠱⡄⠀⠈⠆⠀⡇⢣⠈⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⢰⠁⢸⠀⠀⡸⠀⡸⠁⠀⠀⠀⠀⠀⢹⠀⢹⠀⠀⢸⠀⢸⢸⠀⣧⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢀⢸⠀⢸⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⢸⠀⢸⢠⠀⡟⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡀⠸⠘⡀⠸⡀⠀⠸⡄⠘⠦⣀⠀⢀⣠⣔⣁⣠⠃⠀⢠⠇⢀⡇⡸⢀⠇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢇⠀⢇⠡⠀⠱⡀⠀⠈⠢⢄⣀⣀⣀⣀⣀⡀⠉⠉⠓⠯⣀⠞⣰⠃⡜⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡈⢦⠈⢦⠱⣄⠙⠢⣄⡀⠀⠀⠀⠀⠀⢀⡩⠷⠄⠀⠀⠈⡞⢁⠞⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠳⣄⠑⢬⡓⠤⣄⣉⠉⠁⠀⠉⢉⣁⡤⠔⣺⠀⠀⠀⡖⢁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠓⠤⣈⡑⠲⠦⠬⠭⠭⠥⠄⠒⢒⣉⣽⠀⠀⠀⡶⠋
⠀⠀⠀⠀⠀⠀⡀⢠⠤⠶⡦⠖⠀⢤⠀⠤⣤⠠⠤⠤⠤⠤⠤⠙⠲⠤⣍⣉⣁⡀⣀⣀⣈⣉⢨⠤⠴⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠘⠀⡇⣇⢰⠀⢸⠀⠀⢹⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⡟⠙⠛⠉⡇
⠀⠀⠀⠀⠀⠀⢿⡰⢄⣠⣿⠼⠴⠾⠶⠒⠚⠓⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣵⣲⣬⠿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠒⡄⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠓⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⠀⡀⣀⠀⣀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠶⠾⠛⠛⠛⣿⠀⣿⠿⠶⢶⣤⣤⣀⣴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠈⢙⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣿⠀⣿⣀⣤⣤⡶⠶⠿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⣠⡶⠟⠛⠉⠉⠉⠉⠉⠻⠶⠟⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠁⠀⠀⠀⣠⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣯⣤⣤⣀
⠀⠀⠀⠀⠀⢠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣧
⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⣠⡶⠶⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠷⠶⣦⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠻⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣼⠏
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣰⡎⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠠⢄⣤⣤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡄⣀⣀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠀⠠⠤⠤⠤⠀⠀⣀⠀⠈⠀⠋⠉⠉⠙⠛⠋⠛⠂⠙⠒⠒⠲⠶⠶⢶⣮⣉⣉⠉⠉
⣥⣤⣤⣄⣷⣤⣤⣶⣿⣿⣿⡿⠿⣛⣙⠃⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠋⠍⠉
⠭⠭⣽⣯⣭⡉⠿⠿⠿⡿⠯⠽⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣄⣀⣀⣀⣀⢀⡀⠀⣀⣀⣸⣶⣷⣶⣶⣶⣶⣶
⣶⣶⣾⣿⠿⢭⣭⣭⣭⣥⣀⣀⠀⠤⠤⠤⠤⠶⠶⠿⠛⠛⠓⠒⠚⠛⠛⠛⢽⠽⠿⠿⠷⠖⠲⠶⠶⣾⣿⣏⣉⡉⠡⠤⠤
⠛⠉⠓⠂⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣤⣤⣶⣖⣒⣉⣭⣤⣄⣠⣤⣴⣶⣶⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣀⡀⠀⠉⠭⣽⣿⣿⣟⣛⣛⣉⣉⣀⣀⣁⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⢛⡛⠿⠻
⠛⠛⠩⠭⠭⠥⣤⣤⡄⠀⠶⠶⠾⠿⠿⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠤⣼⣤⠿⠭⠵⠶⠶
⢖⣶⣶⣿⣿⣿⡶⣶⠶⠶⠶⠒⠲⠶⠶⠶⢶⡖⢂⡀⠐⠂⠀⠀⣀⣀⣠⣤⡄⠁⠀⢀⣀⣉⣉⣉⣉⣙⡛⠓⠒⣒⣲⣶⣶
⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣶⠶⠶⠶⢾⣿⣿⣛⡛⠿⠿⣿⣿⣭⣍⣀⣀⠀⣀⣀⣀⣀⣈⣉⣉⣩⣤⣴⣶⣲⣶⣶⡶⠶
⣤⣤⡤⠤⠤⣴⣦⣤⣤⣤⠴⠶⠒⠒⠶⠶⠤⠀⠀⠐⠒⠒⠲⠶⠖⢺⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣭⣭⣭⣤⣤⣤⣶⣶
⠉⠁⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⢿⣷⠶⠶⠶⠶⠾⠛⠟⠛⠾⠿⠗⠒⠂⠀⠐⣚⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿
⠥⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠿⢟⣛⣛⣛⣛⣛⣿⣿⣿⡿⠯⠭⠽⠟⠛⠛⠛⠛⠉⠉⣉⣛⣉⣉⣀⣈⣉⣉⣀
⠿⠿⠿⠿⠿⠿⠾⠿⠿⠶⠾⠛⠛⠛⠋⠉⠉⢉⢁⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉
⣒⣒⣂⣠⡤⣄⣠⣤⡤⠤⠤⢴⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠻⠲⠒⠒⠒⠒⠂⠛⠋⠀⠒⠀⠈⠙⠛⠓⠂
⣿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣤⣤⣤⣤⣤⣤⣀⣀
⣀⣀⣀⣤⣤⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣭⣭⣽⣿⣿⣭⣥⣬
⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⡀⠀⠈⠉⢉⣩⣭⣭⠭⠭⠴⠶⠚⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠛⠛⠛⠋⠉⠉⠙⠉⠛⠛⢛⣛⣛⣛⣛⡛⠻⠿⠷⢶⣶⣶⡶⠶⠶⠶⠶⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣛⣻⣿⣿⣿⣭⣭⣍⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⠿⠟⠛⠛
⣀⣀⣀⣲⣀⣈⣭⣭⣭⣿⣿⣿⣤⣬⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⣯⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⣀⠀⠒
⠀⠀⠀⠀⠢⠐⠇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡰⠈⠀⠠⢀⡀⠸⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠀⠀⠁⢂⣻⣟⡺⢇⠆⠂
⠀⠁⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⡱⠄⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⡀⠀⠀⠁⠸⠥⠊⡀⠰⠄⠀⠔⢊⡁⠀⠶⠄⠐⠀⠀⠀⠶⠤⠊⠀⠀⠤⠂⠀⠀⢀⡀⠠⠄
⠠⠄⠀⠤⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠀⠀⠀⠀⠧⢜⡠⠜⠀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠐⠀⠀⣠⠤⣤⡀⠀⠀⠤⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠈⠁⠀⠰⠂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢁⣈⣇⢀⠠⣤⢤⡄⠚⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⢠⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠗⠯⢽⣉⡿⠀⠡⠤⠶⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠠⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠰⠀⢈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠀⠘⠀⠀⢠⠇⠐⠋⠐⠄⠑⠀⠐⠂⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⣏⡀⢃⠀⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⣤⠀⠀⡄⠄⠀⠀⠁
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡄⢠⠀⡄⠰⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢶⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠃⠀⠀⠈⡇⠀⠆⡿⠀⢷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠓⢃⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⠁⠀⠘⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢺⠉⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠄⠁⠈⠐⢈⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠂⠀⠖⢁⡀⠓⠀⠀⠀⠚⡁⠰⢀⠀⣡⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⢢⠀⠀⢠⡀⠀⠀⠀⠀⠁⡘⠃⠀⠡⠀⠀⠀⢀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡶⠘⢀⠓⠀⡃⠐⠀⠀⠀⠀⠀⠀⠑⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢡⠄⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢮⠴⠖⢚⡰⠀⡁⠊⠀⠀⠀⠀⠀⠀⠀⠈⢠⡛⡤⡤⣤⠤⣜⠂⠀⠁⠔⠈⠁
⠐⠀⠀⡀⠀⠀⠔⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠆⠉⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠗⠠⠶⠔⠋⠐⠁⣈⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠲⡀⠂⢱⡉⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⣀⠈⠃⣀⣰⠆⠀⠠⠄⠠⠀⠄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠋⠉⠀⠁⠀⠀⠄⠔⠀⠀⠐⠒⠘⠇⠘⠀⠀⠀⠀⢸⠀⠀⠁⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠰⠀⠀⠠⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠶⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⠤⠄⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⡰⠤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠆⠀⠀⠀⠀⠀⠂⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⠃⠐⠒⠒⠀⠉⠑⢤⡄⠀⢰
⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠊
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠐⢆⠐⠒⠂⠀⠒⠘⠂⠘⠂⠒⠓⠀⠀⠐⡄⣀⣀⡀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠦⠤⠤⠄
⠀⠠⠐⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠐⠈⠲⠶⠦⠆⠀⢰⠀⡆⠀⠀⠀⠒⠁⠀⠀⠌⠁⠀⠀⢀⠈⠉⡋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠉⠀⠳⠖⠀⠀⢑⠛⠐⠋⠁⠀⡄
⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⠠⠄
⠀⠀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠠⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⠀⠈⠁⠀⠚⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠁⠀⠅⠘⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⡀⣀⠀⣤⠦⠐⣄⠤⢀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠌⠳⡄⡀⠀⠀⡕⠊⠀⠀⠀⠀⠀⠀⠀⠐⡢⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠘⠃⠀⢀⠀⠀⠈⠛⡠⠀⠀⠀⢀⠄⠤⢈⠴⠫⢀⠔⠢⠤⠁⢀⠀⠀⠀⠰⠆⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⠀⠀⡀⠀⠀⠀⢠⠣⡈⠢⢀⢠⠐⠁⠀⠀⠀⠤⡀⠐⠃⠀⠲⢿⠁⠂⠄⡌⠀⠠⠒⡂⠑⠃⠀⡘⠤⠃⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠠⢠⠀⠀⠀⠀⠀⠀⠉⠌⢊⠢⡀⠀⠀⠀⠑⢄⢡⡀⡀⢀⡞⢞⣳⠀⠀⠀⠐⢄⠱⡀⡆⠨⢰⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠁⠜⢀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⢸⡀⠄⠀⠀⠂⠀⠀⠺⠓⠂⠉⠀⠢⡠⠴⠀⠀⡃⢀⠃⡝⢑⠂⡰⠈⠀⠐⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⢀⠰⣀⡐⠁⠀⡠⢠⡟⡏⠅⠀⠀⠠⢈⡤⡁⠂⠄⠀⠰⠖⡐⠋⠀⡀⠔⣡⡮⠀⡑⠀⠀⠀⠠⠄⠠⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡣⠄⠠⡄⠕⠐⣬⡂⡊⡤⠋⢈⠄⠨⡐⠪⡴⠛⣁⣤⣄⡀⢤⢲⠿⠤⠤⢤⣆⣁⣼⣀⢨⣥⠀⣉⠉⠠⠔⠄⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢜⢯⡑⠞⠫⠒⠀⠐⢁⣭⡈⢀⡙⢹⡆⠀⠈⠓⢓⠈⡀⡀⠀⠠⢌⠀⢨⡼⠀⠈⣣⣰⢁⣾⡷⢎⣡⠁⡰⠀⢂
⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠑⠩⠑⢤⣀⠀⢠⡟⠄⡽⠿⣉⡉⠀⠀⣠⡤⠗⠛⠲⣤⡤⠋⠷⠔⡉⢳⠀⠉⠈⠡⠈⡋⠀⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⣤⠀⣎⠆⠀⠆⠐⠂⠀⠰⡙⡖⠫⠉⢉⣤⠴⠃⠂⠐⠑⢀⡤⣐⡒⠖⠠⠤⠐⠒⠀⠂⠉⣀⠀⢄⠁⠉⠠⡩⠂⠀⠛⠀⠀⠀⠀⠀⠀⣀
⡀⠀⠀⠀⣀⡆⢠⠒⠀⠀⠀⠀⠀⠀⠀⢰⡈⠀⠘⠚⠑⠲⠓⠉⠙⣻⣮⢔⣲⡭⠁⢠⠒⢒⡂⢁⠀⠂⠐⠀⠡⠈⠀⠀⠂
⠀⠀⢀⠀⠀⠀⡄⣡⠴⢄⠀⠀⠂⡠⢀⠆⠀⠂⠀⠀⢀⡄⠰⢆⢬⡁⠉⠀⠠⢀⠀⢗⠀⡆⢀⢀⠀⠀⠀⠀⠐⠀⠀⠔
⠀⠀⠀⠁⡄⢠⣅⡨⡗⠀⠦⡭⣤⠴⠻⠄⠀⠀⠀⣄⠴⢀⡾⠞⠛⠁⠒⠒⠂⠁⡀⠀⢇⡱⠈⠊⠀⣤
⠀⠀⠀⠂⠁⠪⡉⢫⡀⠀⠀⠀⠠⢄⡁⠀⠀⠀⣊⠡⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠈⠜⣆⠈⠀⠀⠑⠈⡠⠔⠂⠉⠀⠀⠀⠘⠀⣀
⠀⠀⠠⠀⠀⠀⠀⠠⣨⢗⡂⠀⠀⠀⡁⠀⠀⠀⠣⠃⠀⠀⢀⡀
⠀⠀⠀⠑⠀⠀⡠⠚⠉⠋⠐⠀⡀⠀⠐⠂⠠⠤⠎⠬⠄⢒⠄
⠀⠀⢀⠔⠰⠶⡇⠀⠒⠀⠒⠂⠀⠂⠠⠆⠒⠨⠄⠒⠠⢣
⠀⠀⠈⠉⠀⠀⠐⠈⠀⠀⠘⠄⠐⠐⠠⠄⠉⢀⣘⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠀⠀⠀⠀⠐⡨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢓⠁⠀⠀⠀⠀⠀⠐⠰⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⢶⣒⣒⣛⣛⣿⣿⣿⣾⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⣶⣻⠁⠀⣤⡏⠀⠀⡀⣴⢰⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠒⠶⢬⡟⡉⠀⠀⣹⣦⣀⠀⢀⣠⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⣷⣄⡀⣸⠏⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠱⠈⠀⡤⢿⣟⢦⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⢇⣤⡾⠁⠀⠙⢷⣹⡇⠈⠙⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠳⠈⣟⠁⠀⠀⠀⢠⡈⣿⡇⠀⢠⠈⠉⢂⢺⣿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣰⠃⠁⠀⣏⡔⠀⠀⠀⠀⢻⣿⡇⠀⢸⠀⠀⠈⣸⡟⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢀⣏⠆⠀⠀⣿⠁⠀⠀⠀⠀⠹⣿⠁⠀⡘⠀⠀⣠⡿⣿⠄⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣛⠃⠀⠀⣇⠘⠀⣀⣤⣤⣤⣼⠃⢀⣁⣴⡾⠋⡁⣿⠇⠀⠐⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠠⠄⠐⠁⡁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⠿⠛⠛⡩⠿⡟⢿⠿⣥⠤⠤⣤⣿⡥⠀⢀⣤⢠⣿⣄⣠⠴⠒⠋⠙⢾⣿⣿⣿⡄⠀⠀⠐⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠈⣿⣦⠴⣶⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢻⣽⠛⠋⠁⠀⣴⣾⣿⣿⡆⠈⠀⠀⢨⣿⣿⣿⡇⠀⠈⢁⠀⠁⠀⢀⢐
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⢻⠧⣥⡤⠤⠀⠀⢀⡀⠀⠀⠀⡀⠀⣌⣿⠀⠀⠀⠀⠈⠻⠿⡿⠋⠀⠀⣠⡼⠿⠿⠟⠃⠄⠠⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡤⢿⠈⠇⠑⠈⠉⠛⠉⠛⠛⠛⠷⠶⣶⠦⣼⣿⣿⣿⣷⡤⠀⣤⡰⠾⡷⢶⡖⠋⠁
⠀⠀⠀⠀⠀⢀⣾⣅⠸⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢸⣧⣾⣿⣿⠿⢹⠏⠛⠶⠖⠒⠛⠉⠉
⠀⠀⠀⠀⣰⠿⣭⣛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡿⠋⠀⢸⣿⢷⡄
⠀⠀⠀⠘⢷⡋⠀⠉⠛⠷⢯⣯⣿⣷⣶⣶⣶⣦⣾⣿⣿⣿⣧⣶⣚⣯⠷⠛⢝⣦
⠀⠀⠀⠀⠀⠙⣶⢒⢶⣒⣀⣀⣈⡉⠉⢉⣽⣿⣿⠟⠛⢻⣿⡟⡉⣀⡤⠴⣟⢳⡀
⠀⠀⠀⠀⠀⢀⡇⣾⣿⠦⠀⠈⠉⣳⣶⣿⣿⣿⣥⣤⠖⡚⠁⠓⠚⠁⠀⠘⠉⣣⢻⡄
⣀⣀⡀⠀⠀⣼⢡⣯⣷⣴⣶⣾⣿⣿⣿⣿⣣⠋⠙⢦⡻⣼⠮⡂⣀⢀⠀⠀⠀⠛⠚⡷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⡛⠭⡶⢃⡏⠀⠀⠀⠙⠲⣝⡿⣯⣧⣶⡏⠀⠀⠈⣳⡸⡄
⣿⠋⠉⢩⡟⣽⡿⢭⡝⠂⠀⢀⣾⣯⡖⠋⠀⠀⠀⠀⠀⠀⠈⠹⣷⣮⣽⣴⠤⠂⠀⢚⡷⡹⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠶⢦⡀
⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⣿⠉⠛⠳⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠶⠶⠶⠞⠛⠛⠉⠁⣿
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⢸⡇⣀⣠⡴⠶⠞⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣤⣀⢸⡇
⠀⠀⠀⠀⠀⢸⣿⣍⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⢿⡇
⠀⠀⠀⠀⠀⠘⡇⠈⠙⠳⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⠛⠶⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠞⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠚⠛⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣽⣥⣬⣩⣯⣥⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠚⣩⢞⣋⠁
⠀⠀⠀⠀⠀⠀⠀⣰⢃⣄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⠑⢈⡿⣿⠾⠁
⠀⠀⠀⠀⠀⠀⣰⢃⠎⡍⢆⠳⡀⠀⠀⠀⠀⠀⠀⣰⠣⠀⢸⠁⢸⡉⠮⡳⣄
⠀⠀⠀⠀⠀⡰⢣⠂⡐⠑⡀⢢⠻⡄⠀⠀⠀⠀⢀⡇⠇⡔⠁⠀⠀⠉⢧⠘⠪⠇⠀⠀⣀⣠⡤⠶⢛⡏⠙⠒⠒⠒⠒⠛⠉⡟⠳⠦⣄⣀⡀
⠀⠀⠀⠀⡼⠡⠃⡜⠀⠀⠘⡄⠡⠹⡄⠀⠀⠀⢸⢰⢐⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣯⠥⠤⠒⠉⢁⠀⠀⠀⠀⠀⠀⠰⠉⠓⠀⠤⢼⠃
⠀⠀⠀⡼⢡⡇⣊⡉⣉⣉⣉⣉⣀⣥⠙⡄⠀⠀⢸⢸⢸⣦⣤⣤⣤⣤⣾⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣼
⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠀⠀⢸⢸⠘⡖⠂⠀⠀⠂⢺⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⡄⠀⠀⠀⠀⢼⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣹⠛⠲⠶⠖⠚⢻⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢀⣠⠤⠤⢤⡤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠈⠓⠦⠤⠤⠶⠊⠀⠀⠀⠀⡷⠒⠒⠒⠒⠲⠢⠒⠒⢒⡗⠒⠒⠒⠢⠂⠲⠒⠒⢲⡇
⠀⠀⢸⠁⣀⣀⣀⣽⣯⠀⣀⡆⠀⠀⠀⠻⠚⠁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢸⢠⡛⠛⠛⠛⠋⡀⡝⡇⠸⠀⠸⠘⠓⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠒⠀⠠⠀⠠⠄⠠⠒⠚⡗⠀⠠⠠⠄⠠⠀⠀⠒⢺⠁
⠀⠀⢹⢠⠀⠀⠀⠀⠀⠀⡔⠋⠰⠤⢤⠀⢄⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡸⠀⠀⠀⠀⠀⠀⠉⠃⡈⠉⠙⠐⠉⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠙⡖⠆⣖⡀⠒⣀⢛⣟⣹⠿⣿⣶⢬⡉⣉⣭⣷⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⡷⠤⡤⠀⣀⣀⡀⢀⡀⢀⣇⡀⡀⠀⣀⣀⡀⠤⡤⢴⡇
⠀⠀⠀⠀⡇⠀⠀⠉⠻⠛⠻⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠧⠼⠀⢠⡇⠀⠀⠀⠀⠀⠓⠤⣤⣄⣀⣈⢉⣈⣁⣨⣇⣈⣉⣀⢁⣀⣀⣄⠤⠾⠃
⠀⠀⠀⠀⠙⠢⢄⡀⠠⠀⠀⠁⠁⠀⠀⠀⣀⣀⣀⣀⣀⣠⡤⠤⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠓⠋⠉⠉⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣟⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⠢⣀⠠⣀⠀⠀⠀⢸⣾⡠⠄⢰
⠀⠀⠀⠀⣶⣿⢾⠞⠡⡸⠊⠀⠀⠀⠀⠀⠈⠀⠀⠘⣮⣤⠀⠉
⠀⠀⠀⠘⠃⠈⢳⡠⢈⣣⡄⠀⠀⠀⢀⢦⣄⣆⠀⠰⠋
⠀⠀⡁⠁⠀⠀⠰⣏⣿⣿⣷⡀⠀⠀⡀⠀⠉⣿⠃⠘
⠀⠀⠀⠀⠀⠀⠸⢗⣺⣿⣿⣿⣶⠀⠉⠁⠉⠋⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠐⢼⣧⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⣠⣄⠀⠀⢹⣵⠙⠙⠿⡿⣿⡻⠇⠀⢏⡢⠀⢀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐
⠀⠀⠐⠛⠺⣿⣦⠰⠁⠀⠀⠀⠣⠌⢮⡀⠀⠼⠥⠀⠷⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠇
⠀⠀⠀⠒⠟⠛⠛⠒⢶⣶⣄⠀⢀⡀⠀⠁⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⣭⣢⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢟⠂⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠈⢻⡛⠳⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠶⢞⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣼⠀⠀⢳⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢦⣸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡀⠀⠀⠀⠀⠀⠸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢰⡆⠒⣲⣟⠙⣷⣖⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⡿⠿⣧⡀⠉⠁
⠀⠀⠀⠀⠀⡠⣤⡄⠀⢳⠓⠒⢳⡇
⠀⠀⠀⠀⢸⠁⣸⣏⠒⠉⠀⠀⠀⠙⠒⠒⠶⣦⡀
⠀⠀⠀⠀⠘⣆⣸⡿⠙⠶⣦⣤⠤⠖⠒⠲⣴⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⣉⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡧⣷⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⢷⢿⡿⠟⢋⣽⣭⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣟⡙⠿⡟⢳⡶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡻⢭⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣭⣿⣽⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣜⣿⣽⢿⣛⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣫⡧⣿⣿⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡩⣻⡿⣶⣯⣟⣯⣟⣾⡿⠿⢿⣯⣽⣿⣿⣿⣯⣭⣭⣿⣽⡿⢿⣷⣶⣿⣯⣯⣽⡷⢟⣻⢝⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⣾⣽⣻⣾⢯⣭⣿⣛⣿⣻⡿⠿⠿⠿⠿⠶⠾⡿⠿⠿⠿⣿⣛⣿⣻⣯⣭⢿⣿⣻⣿⣿⣟⣯⣿
⠀⠀⠀⠀⠀⠀⣀⣤⢶⢛⣿⣯⣣⣝⣽⣛⠿⠷⣶⣾⣯⣭⣯⣿⣿⣿⣿⣿⣟⣻⣿⣭⣭⣭⣽⣿⣷⡾⠿⣿⢟⣯⡽⣾⣿⡾⣧
⠀⠀⠀⢀⡴⣞⢯⣷⣭⠾⢿⣯⣿⣿⣷⣯⣿⣟⣟⣷⣾⠿⡿⢿⣯⣭⣭⣽⣯⣽⣿⣿⠿⢿⣿⣶⣾⣿⣽⣿⣷⡾⣿⣯⠽⣪⠟
⠀⢀⣴⣫⠎⣠⠾⠋⠀⠀⠀⠙⠷⣽⣷⣻⣯⠭⣟⡛⠛⣛⡻⣿⡿⣿⠿⠿⠿⣿⣿⠿⣿⣛⣛⣻⡛⣩⡽⢿⣞⣋⣭⠶⠛⠁
⢀⣾⠳⣅⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠦⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠭⠴⠶⠞⠛⠛⠉⠁
⠸⣇⢧⣸⣇
⠀⠹⣾⣿⣯⣟⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⢤⡀
⠀⠀⠈⠛⢯⣾⣻⠷⣯⣭⣯⣹⣟⣛⣒⡶⠶⠶⠶⠶⠶⣶⣶⣶⣶⡶⠶⠶⠶⡶⠶⠶⣶⢶⢶⡾⠁⢎⠖⠀⠙⠒⠒⠒⠲⠶⠦⠤⢤⠔⢲⣦
⠀⠀⠀⠀⠀⠈⠉⠛⠳⠾⢿⣭⣽⣟⣻⣟⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷⠾⠶⢿⡄⠫⠢⠤⣤⠤⠴⠶⠶⠒⠒⠒⠛⠶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢧⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⢀⣤⠶⠛⠛⠙⢛⡓⠶⣤⡜⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣄⠀⢀⣴⠛⢁⠔⠁⠀⠀⠀⠀⠑⠈⠻⣆⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣾⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⢹⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡯⣀⠃⠀⠀⣰⣾⢿⢧⠀⠀⠀⢠⣒⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⠈⠣⠀⢰⣿⠃⠀⠈⡇⠀⠀⠽⣫⣽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢭⠀⣿⡟⢀⢄⡀⡄⠀⠀⣧⡱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠿⠀⣿⣿⡇⠀⡱⡇⠀⠀⣷⡐⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢗⡀⠸⣿⣿⢉⣧⠇⠀⠀⢳⡲⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄⡀⠙⢒⣍⠹⣿⡾⡟⢁⠁⢠⣯⣷⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀⠀⡹⠋⠙⠟⠛⠒⠒⠚⠉⠉⢫⡏⠙⠛⢟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢩⠟⠛⠛⠻⠶⠾⠶⠚⠒⠒⢺⣏⣉⣥⣹⣽⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠧⣘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⠕⣢⣽⣟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣖⠟⠛⠻⠶⠲⠶⠶⠶⠶⠖⠒⠒⢚⡫⢥⣀⠤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣥⣄⣤⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⢼⣚⣩⣤⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠷⠿⠾⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠿⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢹⣷⣦⠿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠙⠊⠹⠟⠁⣄⠁⠀⠀⠷⠳⣄⡆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡤⢀⣀⣸⣶⣶⣾⣁⣀⢠⣤⠘⠓⠮⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⣤⢶⣶⠿⠟⡻⠛⣏⢹⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠙⠀⢈⣤⡄⠀⢀⣿⣼⠈⡃⠀⠀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⣿⠁⠀⠙⢿⣧⣤⣈⢻⡟⢠⡇⠀⠠⠄⠀⠀⠀⠀⠀⠰⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡇⠁⡏⠀⠀⢢⡠⠄⠠⠈⠘⣧⠾⠁⠀⠀⠐⠒⠐⠒⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⡀⡇⢣⠀⠀⠠⣶⣶⣶⣶⡌⠁⢀⡀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⠃⢸⡆⢶⣾⢁⡿⠾⠿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⣦⣉⣈⣧⠈⣘⣉⣡⣀⣄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠸⡟⢷⣤⠤⠤⢽⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⡿⣛⣏⠙⠛⠋⠀⠶⢸⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⠉⠂
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠴⢬⣄⡿⢿⠎⡀⢀⡿⠂⠴⢸⡄⣿⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡸⢸⣤⡞⢻⣿⢆⠙⠛⠁⢁⠀⣸⠠⡇⠀⠀⠀⠠⡇⠸⡆⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠠⠠⢄⡀⢈⠃⣸⣿⠁⣼⣿⠀⠀⠀⠀⣸⠀⠋⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⢢⣤⡀⠈⠣⣤⣾⡏⠀⣸⡿⠃⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠀⠀⠰⠀⠇⠀⡀⠀⠀⠁
⠀⠀⠀⠀⣰⠶⠶⠈⠙⢂⠘⠟⠁⢰⣿⠇⠘⠳⠋⠉⠶⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠸⠃⠀⣾⣿⣄⠀⠀⠀⠰⣤⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⣿⠛⠻⠷⣦⡀⠀⢰⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿
⠀⠀⠀⢰⠖⠒⠓⠲⡆⠀⠀⢀⣽⡿⠁⠀⠀⠀⠉⠙⢻⡄⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⣠⠀⡀⠀⠐⠀⠀⠀⠿⢻⣇⠀⠀⠀⠀⠀⠞⠉⠂⠀⠀⠀⠀⠀⠘⠋⠁⠸⠛⠁⠀⠀⣄⠀⠑⠀⠣
⠀⠀⣰⣋⠞⠀⠂⢱⡄⠁⡀⠀⡸⢁⠀⡐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡅⠀⠀⠘⠀⠀⢁⠀⠱⡀
⠀⢨⣽⣏⣀⠀⠀⠀⠀⢀⠄⢠⠀⢠⠀⠙⠋⠀⠀⠀⠤⠬⠀⠀⠀⠀⢠⠂⠘⠂⠀⠀⠁⠀⠀⠀⠀⠀⠈⡄⠀⠠
⠀⢘⣽⣯⡀⠆⠀⣀⠔⠚⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠉⢽⣿⡤⢘⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⡄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠄⠀⠀⠀⠀⠀⠡⡀⠈⠠⠀⡀⠀⠀⠠⠞⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⢀⣀⡴⠀⠒⠀⠀⠀⠀⢠⠀⠀⢨⣿⣶⡶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡈⢁⣀⡀⠀⠀⠀⠀⠴⡔⠁⠦⣘⣿⣿⣿⡤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠿⠉⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⠤⢤⣀⣀⣀⣠⡤⡤⠤⣤⡴⠒⠉⠒⠒⠒⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣡⡴⠷⠶⠾⠴⣦⡯⠶⠾⠚⠒⡟⠓⠒⠲⠶⢴⣥⣦⢀⡗⣦⠤⢤⡤⠔⢲⠖⠂⢠⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢇⣴⠋⠀⢀⠤⠒⠉⠁⠀⠀⢀⢀⢰⠁⠀⠀⠀⠀⠀⠀⠉⡽⢉⠛⠲⢾⣶⠶⠷⠶⠖⠳⠒⠶⣍⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣴⠞⠁⡠⠊⠁⠀⠀⢤⠀⡀⡆⢸⠘⡄⢘⠰⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠈⠑⠦⣄⠀⠀⠀⠀⠈⢫⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⡾⠁⢀⡜⠀⠀⠠⡈⢢⡘⠆⢃⢸⠘⡀⡇⢀⢠⠀⢀⠂⢄⠀⡸⠠⢀⠆⢀⠀⠀⠀⠁⢊⡓⢤⡀⠀⠀⠈⣇⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡇⢀⠎⠀⠀⣀⠳⡌⢦⢳⡘⡌⣆⢣⣣⡿⠤⠤⠌⣀⣈⡈⠈⢱⠀⡘⢀⢊⡔⢠⠄⠀⠀⠐⣀⠙⢦⡀⠀⢷⣀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⡇⠂⠀⠀⡐⢮⡳⡌⢦⢳⢣⡵⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢧⣌⡌⡐⢡⠞⡀⠀⠀⢒⠀⠨⢷⡄⠈⣇⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⡵⠔⠒⠒⠚⠒⠓⠺⢤⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣥⢃⠞⡡⠀⠀⢐⠂⢈⣁⢻⠀⢸⡉⢻⡇
⠀⠀⠀⠀⠀⠀⢰⠁⠀⢈⠃⠁⠀⠀⠀⢀⡐⠊⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢊⠔⠀⠀⠀⠀⠀⠖⠬⡇⣎⢙⣾
⠀⠀⠀⠀⠀⠀⡇⠀⡀⠘⠀⠀⠀⠀⠀⠀⣀⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠬⠖⠒⠒⠚⠚⠛⠒⢟⣯⣱⡇
⠀⠀⠀⠀⠀⠸⡀⠠⣂⣇⣀⣀⣤⣀⣀⣐⡂⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠹⡀
⠀⠀⠀⠀⠀⠀⢹⢿⡇⡇⠀⠀⠀⠈⣓⠢⣍⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡈⢸⡆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡸⢜⡇⢋⠀⠀⠀⠀⠀⠉⣙⣛⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⣀⣐⠀⢨⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣷⢬⡇⠸⣡⠆⠀⠐⢄⠀⠀⠐⠊⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠉⠩⢉⣑⡒⠭⣛⢠⡓⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢰⠘⣇⠀⢣⡌⡰⢀⠀⠢⢀⠀⠀⠀⠨⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠰⣜⠒⢍⠳⣤⢋⢿⡷⠋
⠀⠀⠀⠀⠀⠀⠈⢯⡉⣳⣄⠳⡀⠇⠎⠰⢀⠂⠀⠀⠀⠀⢠⠛⠒⠤⣄⣀⣀⣀⣀⣀⣀⣠⠤⠖⠉⠀⠀⠀⡐⢮⡙⢦⡙⢆⣱⢃⠞⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣘⣧⠙⢾⣀⠃⡅⡜⢠⠀⡄⡠⠆⠀⠀⠀⠀⠀⠀⠀⢨⠁⠂⠀⠀⠀⠀⠀⢀⠃⣈⠳⡙⢂⢓⡜⣡⠿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡼⡄⠀⠙⢶⣅⡂⡆⡘⠰⠱⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⢠⠠⡐⡄⢦⠹⡌⢡⡈⡄⢹⡼⣫⠞⣯⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣌⡳⠦⣤⣈⣓⠧⣄⡇⢀⢘⢰⠀⡄⠀⡄⣀⠀⢠⢸⢸⠀⡃⢱⠈⡇⢡⣀⡧⠞⣡⠞⣇⡾⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠴⢩⣭⣭⣟⣛⣿⡦⢬⣄⣀⡀⠃⠁⠀⢸⡸⡄⣆⣹⣀⡧⠜⠊⠉⠀⣰⠋⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢦⣀⠁⠊⠊⠩⠉⠏⣟⢛⡿⠛⠷⣤⠤⠤⣤⡤⠚⢇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠋⠉⠙⠛⠚⠲⠶⠴⠵⠶⠾⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⡠⠔⡠⠔⣀⠔⣀⠔⣀⠄⡠⢀⡠⢂⡠⠀⡠⢄⡠⠀⡠⠄⡀⠄⡠⠄⣀⠀⡠⢀⡠⠂⡠⠀⡀⠔⡠⢀⠤⢀⠔⣠⠔⡠⢂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠐⠋⠴⠊⠤⠊⡤⢊⠔⣡⠔⢡⠔⢁⠔⢉⠔⢉⠔⢋⠔⢊⠴⢊⡤⢊⠕⣪⠔⢉⠔⢉⠄⢊⠴⣋⠔⣡⠖⡡⠊⡴⠊⠔⠡⠊⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠀⠁⠀⠁⠀⠁⠀⠉⠀⠀⠀⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠲⢄⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡴⢶⠢⢄⡀⠀⠤⠄⢤⣀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠁⠀⠠⠈⠀⠀⠀⠈⡸⠻⠿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⡀⠀⠀⠀⠀⠠⠉⠐⠴⢦⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠠⠂⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⢷⡀⠀⡰⠁⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡷⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡐⠀⠀⠛⡲⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠂⠉⠉⠀⠀⠸⡀⠁⠀⠀⠀⠊⠂⢀⡀⠀⠐⠆⠄
⢀⠔⠂⠀⠈⠀⢊⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣂⠀⢘⠀⡈⠀⠢⠜⠁
⠃⠤⠚⠁⠀⠀⠂⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⡏⠰⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠙⠂⠦⠈⡃⠒⠆⢠⠤⠤⠤⠠⠀⠒⢈⡅⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠦⢀⡀⠈⡁⢘⢀⡇⡇⢸⠀⣉⢠⠄⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⢦⣄⣄⠈⠁⠈⠘⠃⠣⡶⠴⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠀⠀⠀⠀⠀⢠⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⡀⢀⢠⠂⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠆⠈⠳⠀⠢⢀⠀⠀⠠⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⢀⣀⡀⠅

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⣴⣶⣦⣤⣤⣤⣿⣿⣿⣷⣤⣴⡶⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢈⣉⣭⣭⣁⣉⣁⣌⣤⣀⡀⢀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⠋⠉⢉⣿⠛⢻⣿⡉⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡿⠿⠿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢿⡛⠿⠿⠿⠟⠛⠛⠻⢶⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣴⠾⠛⠛⢷⣄⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡉⢻⠉⠉⠉⠉⠉⠉⠉⠉⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠈⢻⣇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠘⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⣼⣶⣿⣡⡟⠀⠀⠀⠀⠀⠀⢀⣴⡾⠷⠾⠿⠶⠶⠶⠶⠶⠶⠾⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠀⣿⣍⡙⠛⠛⠛⠛⠛⠛⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣼⣿⡉⢿⣦⣀⢀⣀⣀⣀⣿⠟⣿⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠙⠶⢭⣙⣛⣛⣛⣩⡥⠞⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣦⣤⣬⣭⣭⣥⣤⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⣦⠈⠉⠉⠉⠉⠉⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⠀⠀⡄⠀⠀⠀⣿⡇⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡧⠀⠀⠀⣿⡇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣿⠀⢰⡇⠀⠀⠀⢸⣿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣿⠀⢸⡇⠀⢠⡄⢸⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⡇⠀⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⠃⠀⢸⠇⢸⡇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⠀⠀⢠⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡟⠀⣀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⡏⢸⠀⠀⠀⢸⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⠀⡇⠘⠀⠀⠀⢸⣇⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢸⡟⢠⠇⠀⢀⡀⠀⢸⣿⠀⠀⢰⠞⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣀⡀⠀⢸⡇⠀⠀⠀⢸⠃⠀⢸⡇⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⣴⡟⠛⠻⠇⠸⡿⠀⠀⠀⠘⠀⠀⠘⣧⠀⠘⠋⠉⣠⣄⠀⠀⠘⠋⠉⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⠇⠀⢀⣤⡄⠀⠀⠀⠀⢀⡀⠀⠞⠁⠀⠲⣦⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠸⣤⠀⠀⢰⡟⠁⠈⠙⠃⠠⣇⠀⠀⠀⠀⠐⠛⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⣠⡄⠀⠈⠀⣀⠀⠀⣀⠀⠈⢉⣤⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠿⠛⢿⣤⣀⣠⡿⠃⠀⠻⣶⣶⠾⠛⠿⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡦⠚⠓⠒⠒⠲⠤⠤⠤⢄⣀⣀
⠀⠀⢀⡠⠐⣂⡬⣉⣀⣀⣤⣤⡤⠴⠶⠶⠶⠶⠒⠚⢛⣛⣛⣛⣛⣛⠃⡟⠒⠶⠶⠶⠶⣶⡶⢶⡶⠶⣲⣾⠓⢀⣈⣀⠀⠒⠤⣀
⠀⢰⡁⠰⣿⠷⢛⣋⣩⣽⠶⠶⠶⠒⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⢙⠉⠉⠛⠛⠛⠛⠻⡿⠇⣿⣿⣿⣿⣯⣿⣿⣿⣭⣗⡄⠈⠳
⣰⡻⣮⣀⡉⢛⡿⠿⢷⣾⣿⣶⣭⡴⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠛⠸⠛⠒⠶⠶⠶⠶⣮⣽⣤⣻⣿⣭⡽⢾⣾⣇⡠⠿⠿⣿
⡹⣿⡀⠙⣱⠿⢶⣤⣤⣀⣈⡉⠉⠉⠙⠛⠛⠒⠂⠒⠶⠶⠶⠶⠶⠶⠾⠶⠶⠶⠶⠶⠖⠚⠛⠛⠛⠛⠉⠉⢺⢃⣀⣠⣴⣷⡿⠀⣰
⢷⡈⠙⠇⣿⣤⣀⣀⠈⠉⠉⠙⠛⠛⠛⠲⠶⠶⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠶⠾⠛⠛⠛⣏⡛⣛⡿⠛⠡⣪⣿
⢾⡙⠻⠇⣧⣀⡈⠉⠉⠛⠛⠓⠲⠶⠶⠶⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣶⣯⠛⠉⣀⣴⣾⠟⠛
⠈⣿⢷⣤⢻⡉⠉⠛⠛⠲⠶⠶⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣀⣤⣤⣤⠤⠶⠖⠛⠋⠉⣁⣠⣿⣶⣿⠿⠛⢁⣐⣽
⠀⠘⠷⣤⣄⢿⡛⠻⠶⠶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣠⣤⣤⡶⠶⠾⠟⠛⢛⣇⣁⣠⣤⣾⠿⠛⠁
⠀⠀⠀⠀⠉⠑⠻⠶⠶⢦⣤⣤⣤⣄⣉⣉⣉⣉⣉⠉⠉⠉⠙⠛⠛⠛⠛⠋⣉⣉⣉⣉⣉⣤⣤⣤⣴⣶⣶⡾⠟⠼⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⢰⣾⣶⠆⠀⠀⠀⠀⢰⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢰⣿⣶⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⠄
⠀⠀⠀⠀⢀⣤⣝⣥⣄⠀⠀⠀⢀⣬⣝⣥⣄⠀⠀⠀⠀⣠⣤⠀⠻⡿⢇⣬⣝⣥⣄⠿⡿⠃⢠⣤⡀⠀⠀⠀⠀⠀⢠⣶⣾⣛⣧⣄⢀⣤⣄⠁⠂
⠀⠀⠀⠀⣾⣿⣿⣿⢿⡆⠀⠀⣿⣿⣿⣿⢿⡆⠀⠀⢀⣝⣫⣾⣿⣾⣿⣿⣿⣿⣿⣿⣾⣷⣌⣻⣅⠀⠀⠀⠀⠀⠸⢻⣿⣿⡿⢻⣿⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣿⢸⡇⠀⢠⣧⣿⣿⣿⢸⡇⠀⠀⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⡇⣸⣿⣿⡟⢿⡄
⠀⠀⠀⠀⠁⣿⣿⣿⠎⠁⢀⡾⠛⣿⣿⣿⠈⠁⠀⠀⠟⣿⣿⡃⣿⣿⡏⣿⣿⣿⠇⣿⣿⡇⣻⣿⡟⠇⠀⠀⠀⠀⠀⣼⡟⢸⣷⠹⣿⣿⣷⣜⠗
⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠛⠚⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⣿⣿⣿⡅⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⡀⣸⡿⣸⡿
⠀⠀⠀⠀⠀⠻⠏⠿⠀⠀⠀⠀⠀⠿⠏⠿⠀⠀⠀⠀⠀⠿⠻⠁⠿⠻⠇⠹⠏⠿⠀⠿⠻⠇⠻⠻⠇⠀⠀⠀⠀⠀⠀⠿⠁⠀⠿⠇⠿⠃⠿⠃

⠀⠀⠀⣀⠀⠀⠀⠀⠐⠒⠒⠛⠛⠛⠛⠛⠛⠓⠒⠲⡖⠲⠶⠶⠶⠶⠶⠶⠶⠤⠤⠄⠀⢰⣿⣷⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠒
⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢠⣶⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡍⠁
⠀⠀⠀⠀⣸⣿⠓⠾⠿⣿⣿⣿⣿⣟⠩⢹⣿⣿⣿⡇⢠⡆⣿⣿⡀⠠⠙⢦⣄⠀⠀⠀⣿⣿⠃⢀⣤⣤⣶⣴⣶⣶⣶⣶⣶⣶⣦⣴⣷⡆⣾⣿⠃
⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠉⠉⠙⠓⠛⠿⠿⣿⣧⣬⣥⣿⣿⣿⣷⣶⣾⠿⠷⠀⠠⢿⣿⡧⢾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣾⣿⡷⢿⣿⣦⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣷⣀⣀⣈⣷⣀⣀⡠⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠛⠀⠀⠀⠋⠀⠀⠛⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣴⡿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡆⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⣭⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⣈⣿⣷⣴⣶⠶⠶
⠀⠀⠀⠀⠀⠀⢠⣤⣄⣼⣁⣀⣹⣁⣹⣧⡤⣤⣤⠀⠀⠀⠀⠀⠀⠛⢹⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠭
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⡷⠀⠺⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⣼⡇⠸⣿⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢂⠀⠀⠀⠈⠀⣴⣷⡦⢾⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠻⠃⠀⠿⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣥⡴⠏
⠀⠀⣶⣶⣶⣶⣶⣶⣶⣞⣲⣒⣞⣲⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⣴⣿⣷⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿
⠀⠀⣿⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣴⠿⠷⣄⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠉⣵⣤⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢸⡇⣿⣿⣿
⠀⠀⣿⡀⠛⢀⣿⣿⣿⣿⣿⣿⣿⣿⣻⣶⣴⣾⣷⣤⡀⠀⠀⢩⣷⡆⣠⣤⠀⠀⣤⢶⣾⣻⣿⣷⠀⠀⠀⠀⣼⣿⣉⢿⣍⠻⣿⠀⠁⣿⣿⣿
⠀⠀⣿⣿⣿⢟⣿⡻⣿⣿⣿⣿⣿⣿⢸⣿⣿⣻⣟⣿⣇⠀⠀⢸⣿⣿⡿⢻⣧⠀⣤⠬⣿⢻⣿⡿⠀⠀⠀⣸⠃⣿⡙⢷⣝⢷⣿⠀⠀⣿⣿⣿
⠀⠀⠉⠉⠉⠻⣤⠟⠉⠉⠉⠉⠉⠉⠈⠉⠸⢤⡼⠈⠉⠀⠀⠘⠿⠟⠁⠈⠻⠃⠿⠤⠿⠿⠿⠿⠿⠃⠰⠏⠀⠿⠿⠶⠿⠷⠿⠀⠀⠿⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣾⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠄⠀⠛⠁⠀⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⢶⣤⡀⠀⠀⣀⣤⣄⡀⠀⠀⣠⣴⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠈⠀⠀⣾⣿⣿⣿⣿⡆⠀⠈⠀⠀⠀⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠇⢻⣴⠶⠶⠖⠀⠀⠛⠛⠛⠛⠛⠃⠀⠰⠶⠶⢶⣼⠋⠷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⡇⣶⣶⣶⠈⣿⣿⣿⣿⣿⣿⣿⣿⢰⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣼⣿⣿⣿⣿⠿⠿⠿⠇⣿⣿⣿⡀⠿⠿⠿⣿⣿⣿⣿⣿⢸⣷⣼⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⢸⣿⡿⠿⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠠⣶⣶⣾⣿⣿⣿⣇⠸⠿⠿⠿⣿⣿⣿⡿⠿⠿⠟⣸⣿⣿⣿⣿⢸⣿⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣛⣛⣠⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣋⣭⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠟⣋⣭⣙⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠋⢸⣿⣿⣿⣧⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠋⣼⣿⣿⣿⡧⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⢀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⠟⠃⢸⣷⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⣠⣴⡿⠈⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⡿⠋⠀⠀⠀⠘⠛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⢀⣴⣾⣿⣿⣷⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⣤⣤⣄⠀⢸⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠟⡛⠛⢿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⡟⢷⣴⠟⣷⠈⢿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⣿⢠⡌⠛⣡⡄⣿⣄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⣿⣿⣿
⠀⠀⠀⠘⢷⣏⣙⣷⠟⢀⣀⣙⠛⠿⠿⠿⠿⠿⠿⠿⣿⣛⣛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣌⠡⠶⠌⣡⣿⠿⠷⣤⣄⣀⣀⣀⣀⣀⣀⣀⠤⠤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢒⣛⣛⡁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣯⣭⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⠶⠤⠤⠤⠤⡤⠤⠤⠤⠴⠶⠲⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠠⡀⢀⠀⠀⠀⠀⠺⣷⡿⠿⣷⡿⠂⠀⠀⢀⣤⣤⣤⣤⠄⡀
⠀⠀⢿⠛⢻⣿⣿⡟⠛⠖⠀⠀⠘⢧⠀⠀⠄⠀⠀⠀⠀⠀⠀⢿⣧⣥⣿⢿⠀⠀⠀⢸⣿⣯⣀⣽⣦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠗⠀⠀⠀⠀⠀⠀⠀⠀⠯⠽⠗⠊⠀⠀⠀⠈⠹⠏⠙⠋⠻⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⡤⠔⠒⠁⠀⡀⠀⠀⠀⠀⠤⢀⣃⠀⠄⠀⠀⠀⠀⠀⠀⣴⡿⢿⣦⠀⠀⠀⠀⣠⣾⢿⡿⣿⣆
⠀⠀⣬⣿⣿⣯⣿⣧⣤⠀⠀⠀⠤⢠⣿⣿⣷⠠⠄⠀⠀⠀⣰⠀⢿⣧⣴⡿⠁⠀⠀⠀⣿⣿⢨⡅⣿⣿
⠀⠀⠛⠻⠟⠛⠛⠿⠛⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠡⠠⠃⠉⠉⠀⠀⠀⠀⠀⠘⠿⣾⣿⡿⠋


⠀⠀⣤⣴⡆⣰⣦⣤⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣷⣄
⠀⠀⠙⣡⠽⢯⡘⠁⠀⠀⠀⠀⣼⣿⠻⣿⣾⡆⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⣿⣏⡁⢈⣙⣿
⠀⠀⡾⠁⠀⠀⠙⠦⠀⠀⠀⠀⠘⠇⠀⠀⠹⠇⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠙⢿⣷⣼⡿⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣂⡀⠀⠀⠀⠀⠠⠠⢰⣆⠀⠀⠀⠀⠀⠀⢀⣶⣶⡀
⠀⠀⢀⣐⣻⣳⠖⠒⠐⠇⠀⠀⢀⣼⣅⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣾⡇⢸⣷⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢳⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡸⣧⠻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⣀⣤⣴⣶⣶⣶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠲⣶⣤⡀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⢀⣴⡿⠛⠉⣁⣀⣤⣀⣀⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⢹⣿⣿⣦⠀⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢠⣿⠋⢀⣴⡿⠛⠋⠉⠙⠛⢿⣦⡀⠙⢿⣆⠀⠀⠀⠀⠀⢿⣿⣿⣆⠈⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣿⠇⢠⣿⠋⢀⣴⡾⠿⠿⣶⣄⠙⢿⣆⠈⣿⡆⠀⠀⠀⠀⠘⣿⣿⣿⣆⠈⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢸⣿⠀⢸⡏⠀⢸⣿⠀⠀⠀⠘⣿⡆⠘⣿⠀⢸⣿⠀⠀⠀⣷⡄⠘⢿⣿⣿⣧⡀⠙⠻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠘⣿⡀⠸⣿⡀⠈⢋⣾⠆⠀⢀⣿⡇⢠⣿⠁⢸⣿⠀⠀⠀⢻⣿⣦⣄⠙⠿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠹⣷⡀⠙⢿⣦⣄⣉⣀⣤⣾⠟⢀⣾⡟⠀⣾⡏⠀⠀⠀⠈⢿⣿⣿⣷⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠙⢿⣦⣄⡉⠙⠛⠛⠉⣀⣴⡿⠋⢀⣼⡟⠀⠀⢀⣀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠉⠛⠿⠿⠿⠿⠿⠛⠉⣀⣴⡿⠋⠀⠐⠻⢿⣿⣿⣷⣦⣈⠙⠿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣤⣤⣴⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄⡀⠈⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⣀⠉⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠁

⠀⣩⣷⣶⣦⣤⣬⣉⠉⠋⠉⠛⠛⠛⠀⣀⢙⣃⠈⠉⢁⣀⣄⢾⣷⣌⣳⣤⠬⣭⣉⡉⡁⢀⡴⣢⡶⣴⣾⠳⣿⠒⣭⣿⣵⣶⣤⣒⣀
⠲⠶⠶⠄⠉⢉⣉⡀⠀⠘⠛⢀⣤⠶⠛⠉⠉⠉⣍⠛⠛⣿⠿⠿⠟⢿⣛⣿⣿⣿⣿⣿⣿⣯⣝⣳⣶⣯⠙⠛⠻⠷⠶⠮⢉⣭⣭⡟⢛
⠿⠿⠿⠿⠿⠿⠟⠛⠋⢠⣶⣿⣥⣤⠤⠤⠤⠶⠖⣶⣶⣻⣷⣤⣠⣽⢽⣿⠛⠠⢿⠏⠙⠂⠉⠠⠀⠀⠀⠀⠀⠀⠛⠛⡷⢠⡜
⣋⣹⡿⠟⣛⣋⣠⡤⢶⣖⣺⣿⡟⠛⠻⣶⣆⣧⠀⠉⢭⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠠⠂⠀⣼⣧⣴⠼⣧⣴⡶⠛⠛⣿⣿⣦⣠⣭⡥
⣈⣁⣀⣬⣿⣿⣶⣶⡲⠛⠛⠳⠖⠺⠿⠿⣟⠁⠀⠀⠈⠝⣿⣷⣶⣄⠀⠀⠀⠀⢒⣀⣀⡀⠠⠦⠄⣀⣉⡹⠿⠯⢭⣉⣛⡻⢅⡀⠲
⠙⣿⣿⡏⠁⠀⠀⠀⠀⠙⠿⠦⠤⠤⣤⣤⡄⠀⠤⠶⠶⠿⠛⠛⠋⠉⠁⠀⠐⠿⠿⢿⣿⣀⣀⣴⣶⠿⠟⡀⣀⣤⣶⠾⠟⣁⣄⠙⣩
⠀⠻⣿⣏⠀⠀⠀⠀⣤⣤⣤⣤⣶⣶⣶⣶⣶⠶⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠚⠃⠀⠸⢿⣿⣯⣥⡶⠶⠛⣛⣩⣴⠶⠛⠛⣑⣊⣥
⠀⠀⠉⠻⣷⣤⡀⠀⠈⠛⠛⢛⣛⣭⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⠛⠛⣉⣛⡿⠟⣠⡶⠒⣦⣤⣤⣦⣍⠩⢛⣭⣀⣀⣀⣈⣩⣭⣥
⠫⢵⡂⠤⣬⣿⡿⠃⠀⠀⠾⣿⣿⣭⣀⣀⣀⣀⣉⣹⣟⣿⣿⡿⠛⢋⣭⣬⡰⠶⠟⠋⠉⠛⣁⣉⣠⣴⣶⠿⣿⣿⣿⡿⠉⠉⠁⠚⠛
⣤⣤⣬⣿⡦⠀⢰⣶⠶⠒⠲⠶⠶⠤⢤⠭⢭⣭⣉⣉⣍⣉⡁⠀⣀⡬⠕⠒⠚⠻⠓⠒⠛⠛⠛⢿⣷⣶⣿⣿⣿⣿⣿⣷⣿⣿⠟⠛⢹
⠀⠈⠉⠉⠁⣀⣀⣉⡓⠂⠤⠀⠿⣦⣥⣤⠤⠤⠀⠐⢋⣡⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⣛⠉⠡⠀⠐⠁
⠀⠀⠀⣠⣾⣿⣿⣿⠛⠛⠛⠛⠲⠶⠶⣶⣶⣿⣿⣿⣿⣿⠟⠛⠛⢻⣿⣶⡄⠀⠀⠀⠀⠀⠐⠶⠶⠶⠦⠀⠰⠄⠀⠹⡗⠀⠀⠴⠆
⡀⠀⠀⠙⠻⢿⣿⣷⣤⣄⣀⣀⣀⣤⣴⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠉⢛⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀
⠛⠟⠲⠀⠀⠀⠀⣈⣩⡿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣤⣤⣤⣭⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣁⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣧⣀⣀⣤⣤⣶⡶⠿⠛⠉⠉⠉⠉⠉⠙⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⢛⡛⣿⣿⣿⡿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣏⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠒⠀⠀⢲
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠿⠿⠷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⣿⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⠸
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣹⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠴⠦⠤⠤⠤⠤⠤⠀⠤⠤⠔⠒⠺⠿⠿⢟⣿⣯⣥
⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⢿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢀⣀⣀⣴⣾⡿⠟⠛⠛
⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⣰⣾⡟⣹⣿⣿⣿⡏
⠀⠀⠈⠉⠙⠛⠻⠷⢶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠿⠿⣷⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢰⣿⡟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠛⠻⢿⣿⣿⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣠⣴⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣩⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⣶⣶⣶⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠶⠿⠿⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠛⠛⠿⠷⠶⠶⠶⠾⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀

⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠑⠢⢀⣤⣤
⠀⠀⠈⠙⠋⠽⠿⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⡷⢀⣴⣶⣶⣦⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⣾⠿⢛⣭⣤⣤⣤⣶⣾⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠛⠛⠋⠉⣉⣉⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣯⣽⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠷⢿⣿⣿⣿⣇⠀⢠⠀⠀⠀⣠⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⡀⣾⣷⡤⣤⣿⡿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⠈⢿⣧⡈⣿⣿⣣⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢃⣘⣿⢧⣾⣿⠿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠹⠿⠇⠉⠙⠧⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀

⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠈⢐⡽⢮⡆⠨
⠀⠀⠀⠀⠀⠀⠀⣹⣏⣼⣇⡈⠁⠀⠤⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡠⠂⠨⠤⠠⠠⠄⠤⠲⠶⠐⠶⠶⠖⠒⠒⠶⠖⠶⠦⠶⠦⠤⠤⠑⣄⠀⠀⠉⠉⠉⠁
⠀⠀⠀⡈⣹⢷⠉⠙⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⢰⣖⣶⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⡸⠈⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⣠⣤⡸⣯⣿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⢀⡀
⠀⠀⠄⢀⠟⡁⠀⠙⠀⠴⠁⠀⠀⠀⠀⠀⠹⠴⢾⢀⣴⢯⣿⠀⠀⠀⠀⠀⠀⠀⢸⣼⠀⠠⠤⠄⠠
⠀⠀⠀⠀⣀⣇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢰⠩⠤⠬⠭⠿⠤⠤⠨⠭⠤⠤⠤⠤⠤⠄⠠⢤⠤⠤⠤⡤⢤⠤⠤⣄⣀⣀⣤⣀⣈⣻
⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⣿⠇⢠⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡄⠀⠠⠀⠂⣨⠃⠀⠀⠀⠀⡀
⠘⢤⢿⣿⣩⣟⡿⣦⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣸⣿⡛⣿⣿⠀⢀⡠⠞⠁⠐⠀⠀⠀⠀⠐⠂⠀⠀⠈
⠀⢀⠀⣉⣛⣋⣀⣀⢀⣀⣀⢀⣀⠀⣀⠤⣀⠠⢄⠠⠄⡠⢙⣿⡟⣁⣉⡅⣀⢀⣀⡀⣀⣀⣀⡀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠉⠉⢹⡇⢸⠉⠉⠉⠛⠛⠓⠒⠶⠶⠦⢴⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠾⣇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⣀⣤⠴⠶⠒⠚⠛⠛⠛⠻⠷⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⣀⣠⣤⠤⠤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣠⠞⠁⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⣰⠏⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢠⡏⠀⠀⠀⢰⠏⠀⠀⠀⢀⡴⠞⠉⠁⠀⠀⠉⠙⠳⣤⡀⠀⠀⠀⠹⣆⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣾⠀⠀⠀⢠⡏⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠹⡆⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢠⡞⠉⠙⠳⡄⠀⠀⠀⢹⡄⠀⠀⠀⢿⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⠀⢿⠀⠀⠀⠘⠳⠂⠀⠀⣿⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢻⠀⠀⠀⠈⣧⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢸⠇⠀⠀⠀⣼⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⣧⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⠳⠦⠤⠶⠚⠉⠀⠀⠀⢠⡟⠀⠀⠀⢠⡏⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⣦⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡞⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⢷⡀⠀⠀⠀⠈⠙⠲⢤⣤⣀⣀⣠⣤⠴⠞⠋⠀⠀⠀⠀⣠⠟⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠛⠶⢤⣄⣀⡀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠶⠶⠖⠒⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠩⣍⠛⣶⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⣛⡿⠇⣀⠀⠀⡼⠓⠉⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⣡⣴⣶⣤⡈⠙⡿⣿⢷⣯⣀⡤⠤⠤⠧⣟⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠎⠀⠈⣿⣿⣿⣾⣧⣿⢸⠉⠹⠍⢛⡟⢶⠋⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⡠⠃⣠⠶⠛⠻⠧⣍⠻⣿⣻⡘⡄⠠⠄⣸⠐⡼⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡔⠁⡴⠁⠀⠀⠀⠀⠺⣿⣮⣿⣧⣹⣺⣧⣥⣥⣷⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠎⠀⡼⠀⢀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣯⡽⣿⣿⠀⠈⣇⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⡇⠀⣼⠁⢠⣿⠀⠀⠀⠀⠀⢤⣀⣿⣿⣭⣾⣿⡟⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⣿⣾⡇⠀⣾⣿⣆⠀⠀⢦⣀⡀⠙⣿⣿⣿⣿⣿⣷⡀⠀⠈⢳⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⣿⢿⣿⠃⢰⣿⣿⡇⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣁⡀⠈⢿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⠃⢸⣿⠀⠀⣿⣿⣧⠀⠀⠀⣤⣄⣀⣹⣾⣿⡟⢸⣿⠋⠀⠀⢸⡆⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⣧⣿⣿⣷⣾⣿⣿⣿⣿⡀⠀⠀⠀⠉⠉⣙⣿⡀⣸⣧⣤⣤⣤⣸⠃⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠉⠉⠉⠙⢻⣿⣿⠏⢹⡏⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣾⣫⠁⡟⠉⠈⠁⣀⣁⣀⣦⣾⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⢸⣇⢇⠀⠀⠀⠀⠀⠻⣿⣿⣯⢷⣾⠋⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣫⣿⣿⡦⢤⣀⣀⣈⠻⣿⣿⣿⣄⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡤⠖⠋⠀⠀⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⢠⣤⣴⢦⣤⠤⠄⠄⢀⠠⢤⡴⢀⡠⠦⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⡄
⠀⠚⠒⠛⠛⠗⠂⠀⠀⠸⠏⠉⠁⠉⠁⠉⠀⠀⠀⠀⠀⠈⣧⠀⣶⢀⡇⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠢⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣯⡸⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠋⠁⢀⡀⠀⠈⠉⠙⠃⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⠻⠿⢻⡿⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣿⣿⡁⢰⣦⣶⡀⠀⠀⠀⠉⠁⠒⢤⣀⡀
⠀⢸⣿⠏⣐⢀⣀⢀⣼⡖⠚⣭⠿⣌⠛⠣⡀⢀⡤⠄⠀⠀⢸⡆⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠛⠋⠀⠘⠛⠿⣿⣿⣶⡆⠀⣀⠀⠀⠀⠉
⠀⢸⢿⣿⠟⠦⠉⣹⠿⢈⠿⠏⢰⣿⣁⣀⠑⢀⠰⠆⣀⠀⠛⠃⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠙⠂⠨⣿⣿⣶⣄⣄
⠀⠘⠛⠋⠛⠛⠛⠛⠓⠛⠓⠓⠛⢺⣿⣿⡖⠿⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣴⣾⣿⠀⢰⣿⣶⠶⠄⡀⠀⠀⠀⠉⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠘⠓⠘⠷⠀⠤⣶⣆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢈⡿⠂⠀⠀⠀⣸⠿⠛⠋⠀⠘⢻⡇⠀⠀⣙⡃⢰⣶⣤⣀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠀⠶⢠⣅⠀⠖⠀⠀⠀⣀⣠⡄⠀⢀⠀⠷⠛⠛⠙⠁⠘⠿⠿⣿⣿⣿
⠘⠀⠀⠉⠀⠀⢀⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⡺⠿⣀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⣠⡈⠙⠛⠐⠀⠻⣿⣿⡇⠀⢸⣿⣿⠳⠄⠜⢀⣀⡀⠀⠀⠀⠉
⠡⡀⠀⠀⠀⣠⠘⣧⠀⠀⠀⠈⢳⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⢀⠀⠘⠉⠙⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⢘⠋⠉⢤⡀⠆⢈⡏⢻⣷⣶⣦⣤
⠀⢧⣐⣢⣴⣿⠄⠘⣆⠀⠀⠀⠈⠿⠿⠯⣴⠀⢀⠦⠀⠀⠀⠀⠀⠀⣿⣿⢷⣿⠋⠀⠀⢀⣤⢀⣄⡠⣄⡀⠀⡌⠉⠀⠀⣈⠉⠀⠈⣁⣸⠉⠙⠿⠏
⢀⠜⣿⣿⣿⣿⣧⠄⣘⣤⣴⡶⠾⠿⣿⣿⣾⡇⢈⠀⠀⠀⠀⠀⠀⢠⣿⣿⠸⠃⡰⠀⠀⢠⣿⢸⣿⣿⣿⣿⣿⠻⣆⠀⠀⡛⠓⠈⠉⠐⠛
⠀⢀⠿⣿⡏⢋⣵⠟⣋⣉⠀⠰⠿⠃⠸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠙⢣⡄⠀⢠⡈⠻⠎⠀⠀⠀⠊⢳⠠⣿⡀⠀⡽⠻⠻⠿⣿⠅⣸⣯⠒⢶
⠠⠃⠀⣿⣷⣿⣇⠘⠋⠉⠀⠀⠀⠀⠀⣼⣿⢰⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣧⠀⢸⡇⢠⣤⣶⣼⣶⣶⣶⣶⠝⠁⠀⠃⠀⠀⠀⣿⢀⣿⡟⢀⣿
⠀⠀⠀⠸⣿⣷⣝⢧⡀⠀⠐⣒⣥⣀⡴⣿⣯⠞⠀⠀⠀⠀⠀⠀⣼⣿⡗⠀⣶⣾⡕⠒⣾⣿⠚⠁⠙⠟⢻⣿⠏⣸⡁⠀⠀⠀⠀⢲⡆⣁⡀⢘⣿⡟⠉
⠀⠀⠀⠀⠈⠛⣿⡏⠛⠀⠛⠛⠋⣩⡆⠹⠥⠆⠀⠀⠀⠀⠀⢀⣿⣿⡃⢰⣿⣿⠀⠀⢸⡿⠀⠀⠀⠀⡤⠀⠿⡟⠀⠀⢠⡀⠀⣨⠀⣿⣻⣾⣿⡟⠋
⠀⠀⠀⠀⠀⠉⠻⢿⣤⠀⠀⠀⠀⡯⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠁⠹⠛⠛⠂⠀⠈⠓⠀⠀⠔⠀⠀⠀⠀⠈⠀⢀⡾⠁⠀⢹⠀⣬⡉⢀⣁⢀⡀
⠀⡄⠀⠀⠀⠀⠀⠀⢻⣦⣀⣠⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡃⠀⠀⣾⢠⣎⡉⣠⣿⠤⣧
⣰⡔⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⢃⠀⠀⠀⠀⠀⠘⢧⣤⡀⠀⢰⠃⠀⠀⠀⠀⠈⣿⡇⠀⠀⡏⣼⡏⢹⣏⣠⠶⠹
⣿⣿⡴⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡷⠀⡄⢠⡀⠀⠀⠀⠀⠀⠈⢻⣶⠁⠀⠀⠀⠀⠀⠀⣿⡃⠀⢰⠗⢿⣯⡼⢿⠁⢟⣄
⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣁⠈⠓⠄⠀⠀⠀⠀⠀⢈⣿⠆⠀⠀⠀⠀⠀⠀⣿⡏⠀⠹⣦⣀⣉⣻⡶⠥⠎⢁
⢹⣿⡿⠀⠀⠉⠀⠀⠀⠀⠀⠼⠇⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣿⣷⡀⠠⣇⠄⠀⠀⣀⣠⣠⣿⣇⡄⢐⣀⣠⣄⢀⣿⣷⣄⣠⣿⣿⣿⣿⡡⢤⣀⣨
⢿⣿⡗⠀⠀⠄⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⡀⣸⣟⣿⠋⣄⣹⣿⣿⣷⣾⣿⣿⣿⡞⣿⣷⣿⡿⢯⣌⠀⠛⠳⢾⣬⣓
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠌⠉⠀⠀⠀⠀⠈⠀⠉⠼⣿
⢸⡿⠁⠐⠀⠀⠀⣀⠀⠀⠀⣾⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⢋⠃⠀⠀⣤⠦⡤⢀⣲⠐⢂⡿
⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣽⠿⡷⣾⣻⣿⣂⣿⣿⣿⣿⣿⣿⣿⣿⠛⢋⡃⠴⠆⠀⠤⠟⣽⡿⠛⠃
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣼⢠⠾⢧⠛⠉⢡⣿⣿⣿⡿⢿⣿⣿⣿⣶⣾⣷⡀⡰⢢⣶⣦⣬⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠄⠈⡄⠀⠀⠀⠀⣀⠠⣾⣿⣿⣿⣿⠶⢀⠋⠀⠀⢻⣿⣿⣿⠃⢸⣿⣿⣿⣿⡋⠉⠉⣹⠷⢤⠘⢿⡿⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⠞⠁⠀⣿⣿⣿⣿⣽⡿⢿⡟⠿⣷⣻⣿⣿⣿⠀⠀⣿⣿⣿⣿⡆⡀⣙⣷⣦⣤⡀⠀⠀⠚⡋
⠀⠖⠂⠤⠤⠤⠀⣀⣀⡸⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣋⠠⠀⠀⠂⣉⠀⠀⢹⣿⣿⣿⠐⠀⠹⣿⣿⣿⣿⠷⣀⠀⠉⠁⠉⠓⠀⠊
⠀⣦⠀⠀⠀⣀⣀⣠⣭⣴⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣥⠇⠀⠁⠄⡀⠀⣀⡀⠀⠀⣺⣿⣿⡿⠀⠀⠀⣿⣿⣿⣟⠀⠈
⠀⠰⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠀⠉⠀⠑⠈⠉⠁⠀⢀⣿⣿⣿⡇⢀⠀⠰⣿⣿⣿⣿⠄⠀⠂
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡄⢠⣿⣿⣿⣿⡷⢺⣤⣀⣀⡙⠛⠛⣀⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠰⢦⣴⣶⣦⣬⣦⣈⣙⢿⣿⡁⢀⣭⣭⣽⠛⠿⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣷⣭⣽⣯⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⠛⣛⣟⣛⣙⡛⣿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣭⣽⣯⣽⣿⣿⣿⣟⣻⣛⣛⣿