Dumpling Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢖⡦⢟⣛⣋⡓⢒⡦⢉⡉⠛⠛⠛⠶⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣠⡾⢋⡴⢉⠖⠉⠀⢀⡕⢪⠭⠥⢬⠋⡵⠂⢀⡀⠀⠉⠙⠻⢶⣤⣀
⠀⠀⠀⠀⠀⢠⣿⠀⡼⠀⡗⠀⢀⡴⢣⠞⠁⠀⢀⣠⢞⣫⠭⠭⠍⣩⣗⣀⢄⣀⡈⠙⠷⣦⣀
⠀⠀⠀⠀⠀⢸⣸⠀⣧⠄⡃⠠⣼⠂⣏⠀⠀⢠⠚⡰⠋⠀⠀⣠⠞⡫⠴⠒⠒⠉⣼⢥⡤⢌⣛⣳⣄
⠀⠀⠀⠀⠀⠀⢯⢆⠙⢪⡙⠋⢹⠂⣃⠀⣀⡞⠀⡇⠀⠀⡞⢣⠎⠁⠀⢀⡴⢏⠵⠲⠶⠒⠠⡎⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠣⡳⣄⠙⠦⣈⠉⠉⠉⠁⣟⣉⣤⣀⣼⠁⢸⠀⠀⢠⡋⡱⠁⠀⠀⠀⠀⣶⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣕⠦⣀⠀⠢⢤⣀⠉⠁⠀⠀⢸⣋⣥⣄⣺⣏⣀⡗⠀⠀⢀⣠⡺⠋⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢝⣲⢤⣈⡉⠓⠲⠤⢬⣁⣀⣀⡀⢻⡡⠦⠤⠾⠟⠋⣀⡴⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠯⢿⣲⣶⣤⣤⣤⣄⣀⣈⣠⣤⡤⣴⣖⡯⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠒⠀⠀⣴⢃⡄⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢀⠔⠂
⠀⠀⠀⠀⠀⠀⠖⠀⠐⠀⠀⠀⠠⠎⠁⠼⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢚⡡⠚⠁⠀⣀⠔
⠀⠀⠀⠀⣰⠀⠐⠐⠋⠀⠀⠀⠀⠀⢀⠛⠁⡀⠀⠀⠒⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⠔⠋⠀⣠⠔⢋⡤⠒⠁
⠀⠀⠀⢰⡟⢀⣀⠀⠐⠂⠀⠀⠀⠀⠾⠀⠀⠑⢀⠀⠀⠀⢄⠀⢀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡴⠋⠁⡠⠐⣉⠴⠊⠁
⠀⠀⠀⠀⣷⢾⣟⡁⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀⠷⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡪⠞⠁⡠⠐⣁⠴⠊⠁
⠀⠀⠀⠀⠘⢯⣻⣿⣿⣶⡦⠄⣀⣴⣤⡤⢶⣦⠀⡠⣤⢿⣡⣶⠋⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⡀⠐⣩⠔⠋
⠀⠀⠀⠀⠀⠀⠓⢿⡁⠸⣷⡩⠗⢿⡇⠀⣽⣿⣋⣼⣟⣿⠟⠉⠀⠀⠀⣤⡤⠤⡾⠛⡡⢐⡡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⠤⠽⠧⠤⠨⠿⠵⠟⠉⠀⠀⢠⣖⡀⠄⠈⢑⣠⢔⡳⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⠀⠀⠀⢰⣿⠟⡉⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠄⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⡔⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡁
⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠘⣷⡖⢠⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠁
⠀⠀⠀⣸⣧⢸⡆⠿⠿⠿⡿⠟⢘⡿⢿⢿⣿⣕⡀⠀⠘⣿⣹⣿⡦⣀⣀⢀⣠⣄⣀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⢶⣿⣜⣵⣤⣤⣤⢀⣴⣥⣤⣄⡀⣿⣿⡀⠀⠀⠈⠉⠻⢯⡿⠿⢽⠟⠵⣿⣿⣷⣿⡿⠃
⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠊⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⠀⠀⡀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⠔⠿⠶⠻⠳⣄⠀⠀⠀⣞⢿⡿⣣⣼⢿⡿⣱⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⠀⢠⠞⠋⠓⠄⠀⠈⠇
⠀⠀⠀⢠⡏⣴⡄⣀⡠⣶⠘⡆⠀⠀⡏⠓⡢⠽⠯⠶⠚⢹⠀⠀⣰⠚⣡⡀⠀⢠⡝⠳⣄⠀⠘⢄⡀⡠⠮⠲⡄⠸⡀
⠀⠀⠀⠈⢇⡋⠀⠹⠃⠘⡰⠃⠀⠀⢧⣠⠋⠶⡶⠛⢄⡼⠀⢸⡅⠀⠋⠰⠏⠉⠃⠀⢸⠂⠀⠀⠀⣇⠐⠂⡹⠀⠰
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠋⠁⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠀⠀⠈⠓⠒⠛⠛⠙⠛⠓⠒⠊⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡴⣄
⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣎⣹⣤⡀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⣀⡶⣟⣿⢟⣻⣦⢤⣤⡀⠀⠀⣰⠋⠈⠈⠀⠀⠙⣆⠀⠀⠀⢀⠞⠓⠒⠋⠢⡀⠀⠀⣠⢒⡩⠭⡝⠲⣀
⠀⠀⢿⣿⢟⡵⢋⣄⠀⠀⣤⡉⠧⡀⢧⠞⠉⠴⣶⠉⠹⣾⠀⠀⠀⣿⡎⢂⣀⡐⢳⡷⠀⠀⣇⣈⡀⡠⢺⡉⠈⢩⡳⣄
⠀⠀⠀⠀⢸⡇⠀⣋⣐⣂⣈⠃⢐⡇⠈⠳⢤⣄⣀⡤⠖⠁⠀⠀⠀⠘⠦⣄⣈⣠⠴⠃⠀⠀⠀⠀⠀⢇⣀⡠⠭⢀⣁⡸
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠀⠀⢀⠴⠒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠧⡸⣄⠀⠀⠀⠀⠀⢰⠁⠼⠢⡔⠧⠈⡆⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣀⣁⣈⡀⠀⠀⠀⠀⠈⠓⢪⣩⣭⡅⠚⠁⠀⠀⠀⠀⠀⠈⣊⠓⠀⠀⠀⠀⠀⡠⠚⠘⠓⢄⣀
⠀⠀⠀⣠⡖⣋⠈⠀⠉⣙⠳⣄⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⢀⡴⢋⡉⠉⢉⡓⢦⡀⠀⢸⠀⠀⠀⡴⠛⠂⠓⢄
⠀⠀⠐⡇⠀⡜⠃⠤⠜⢣⡀⢨⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡏⠀⠟⠱⡖⠘⠧⠀⣹⠀⠀⠑⠦⠸⡜⠡⣶⠈⢺⠇
⠀⠀⠀⠳⠖⠒⠒⠒⠒⠒⠲⠞⠁⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠓⠶⠒⠒⠒⠒⠒⠶⠚⠀⠀⠀⠀⠀⠑⠂⠄⠐⠋
⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀
⠀⠀⠀⢰⡋⡀⠠⠖⢄⢀⠙⡆⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⠞⠽⠖⠛⠲⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⢸⠙⢲⠦⠤⠴⡖⠋⡇⠀⠀⣰⢚⣅⠁⠈⣨⡓⣆⡀⠀⢰⢃⣤⠀⠀⢀⣤⢸⡄⠀⢀⣴⢚⡅⠁⠈⣨⠓⣆
⠀⠀⠀⠘⣄⡝⠃⠤⠈⠻⣠⠇⠀⢸⠄⠘⠈⠾⠓⠁⠃⠠⡇⠀⠘⣶⠋⠐⠴⠀⠙⡼⠃⠀⢺⠀⠘⠁⠛⠗⠉⠂⢨⡇
⠀⠀⠀⠀⠀⠉⠙⠒⠋⠉⠀⠀⠀⠈⠓⠒⠛⠉⠉⠛⠓⠚⠁⠀⠀⠀⠙⠓⠒⠚⠉⠀⠀⠀⠈⠓⠚⠛⠉⠉⠛⠓⠚⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⣡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣠⠞⠉⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⣁⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡴⠚⠁⠀⠀⠀⠀⣀⡤⠖⠋⢉⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠚⠚⠛⠦⣴⠋⠀⠀⠀⣀⡤⠖⠋⢁⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣟⠁⠄⠀⠀⠁⠀⠐⠀⠌⣳⡤⠖⠋⣁⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠶⣌⠀⠀⣀⡤⠶⠋⣁⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢚⣩⠷⠚⣉⣤⠶⢋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⡤⠖⠛⠳⣄⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠖⠛⠛⠋⠉⠉⠉⣉⣉⣉⠉⣙⣛⣛⠒⠶⠦⣤⣀⣀⠀⠹⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⣉⣁⠀⠀⠀⢀⡴⠒⠋⠉⠉⠉⢁⡴⠛⠉⠉⠈⠉⠍⠙⠓⠦⣤⡉⠙⠒⠦⣄⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢀⡴⠚⠉⠉⠳⣄⢀⡞⠠⠀⠐⠁⠀⢠⠞⠀⠀⠀⠀⠀⠤⠄⠀⠅⠀⠀⣙⠳⣄⠀⠀⠉⠓⢦⣄
⠀⠀⠀⢀⣴⡾⠁⠀⠀⣰⠋⠀⠐⢀⠓⠄⠈⢻⡁⠀⠀⠄⠀⡴⠃⠀⠀⣪⣄⢀⣠⡤⠴⠶⠶⠚⠛⠉⠉⠉⠛⠲⢤⣀⠀⠈⠻⣦⡀
⠀⠀⣰⡿⠋⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠄⠀⠰⣇⣤⡞⠛⠁⠸⣯⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⡈⠁⠀⠈⢧⠀⠀⠘⢿⣆
⠀⡼⡿⠁⠀⠀⢀⡴⠛⠙⠧⣄⣀⣀⣀⣀⣈⣀⡘⢆⠀⣠⡤⠿⠧⣄⣘⣁⣐⣿⠄⢀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣈⣁⡀⢈⡇⠀⠀⠘⣿⣦
⢸⢳⡇⠀⠀⣰⠋⠀⠘⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠳⣾⠁⠤⠒⢹⡏⢍⡉⠩⣿⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠉⢉⠙⠺⣇⠀⠀⠀⢹⡿⡆
⡟⢸⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠐⠂⠀⠂⠀⢀⠀⠀⣴⣏⣤⣀⣠⣾⣃⠀⠀⠀⣿⠓⠖⠒⠲⠶⠤⣤⣈⠒⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⡇⢷
⡇⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⢀⣠⠞⠁⢸⡏⠀⠉⠉⠻⠓⠲⠻⣌⠂⠀⠐⢄⠀⠀⠈⠳⡄⠀⠈⠀⠀⠀⢸⡄⠀⢀⡇⢸
⢷⠈⡇⠀⠀⠘⣆⠀⠀⢺⡙⠷⠦⠤⠤⠶⠒⠚⠉⠀⠀⠈⠀⣇⠀⠀⠀⡀⠠⠂⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠂⢸⠃⠀⣸⠁⡞
⠸⡆⠹⡄⠀⠀⠈⠓⠶⢼⡇⠀⢀⠀⠠⠀⠀⠀⢀⠀⠃⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠰⠀⠀⠀⠀⠘⣇⠁⠈⠀⣰⠃⠀⣰⠃⣸⠃
⠀⠻⡄⠹⣄⠀⠀⠀⠀⠈⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢘⡇⠐⠀⠂⠀⠀⡀⠀⠀⢸⡄⠀⠀⡀⠀⢀⠀⡏⠙⠻⣼⠃⢀⡼⠁⡴⠃
⠀⠀⠹⣄⠘⢧⡀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⣼⡇⠀⠀⠐⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⠀⢀⡼⠁⠀⠀⣀⡴⠋⣠⠞⠁
⠀⠀⠀⠈⠳⣄⠉⠳⣄⡀⠀⠀⠙⠲⠤⣄⣀⣀⣀⣤⠴⠞⣩⠏⠀⠀⠂⠂⠀⠀⠀⢀⡴⠃⣏⠀⣠⡴⠚⠉⠀⣀⡤⡞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠳⢦⣀⠉⠓⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠻⠤⠴⠦⠤⠤⠴⠶⠚⠉⠀⠀⠙⢛⣁⣤⠴⠖⠋⣁⣬⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢤⣀⣈⡉⠙⠛⠒⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠒⠚⠉⢉⣁⣠⡤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠶⠐⠒⠉⠉⠀⠀⠀⢀⠀⠒⠤⣌⡉⠙⠛⠲⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡠⠤⣀⡀⠀⠈⠀⠀⠈⠁⠀⠈⠱⠀⠀⠀⠈⠙⢫⣁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠂⠈⠁⠀⠌⢉⠓⢄⠀⣀⠄⠈⠒⠤⢄⡀⠀⠀⠀⠀⠈⠃⠀⠀⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠢⣈⠑⠲⠤⣄⣀⡀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⣙⡤⠲⠊⠀⠀⠀⠘⢗⡲⠤⢤⣬⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠲⢍⣒⠤⠤⢄⣀⣉⣉⣉⢀⠀⠀⠀⢉⣉⡁⠤⠔⢂⠅⢀⡀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⠤⠬⠭⠭⠠⠶⠂⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠠⡀⠀⠑⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢤⣈⣑⠂⠤⢀⣈⠉⠑⠒⠂⠤⠤⠤⠄⠀⠀⠀⠠⠤⠤⠄⠒⠂⠁⢀⡀⠠⠀⠂⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠒⠲⠬⠽⢶⣖⣒⣢⣤⡤⠤⠤⠤⠤⠀⣀⣐⠀⠬⠡⡤⠴⠲⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣾⣿⣏⣽⣿⣿⠂⣤⣦⣄⡀
⠀⠀⠀⠀⡀⠀⠐⠙⣿⣷⣬⣛⠛⠛⣋⡵⠾⠛⠋⠛⠻⠶⣤⣀
⠀⠀⠰⣿⡇⠀⠀⠀⠈⢛⠿⣿⣿⠟⠁⠚⢀⡠⠐⠂⠂⠉⠛⢿⣷⣦⣀
⠀⠀⠀⠈⠳⣾⣧⣀⠀⣸⣿⠏⠀⠀⠀⣨⠏⠀⡀⠀⠀⢤⣀⠴⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⠀⠀⢠⡇⠀⠀⠃⢐⠞⢣⠤⠀⠀⠀⢈⠩⢿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣶⣿⡇⠀⠀⣼⠃⠀⠚⠀⢀⡶⢊⠁⠀⠀⠄⠀⢩⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣠⣾⡟⠀⠀⠀⣼⡟⢀⠀⠀⢀⡴⠓⠀⠉⠉⠙⠻⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣧⣀⣴⣾⠛⠀⠀⢠⡰⠋⠀⠀⠀⣤⠊⢀⣠⣶⡿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣧⠀⠀⡨⠇⠀⠀⠁⠀⣨⣴⣾⠿⣯⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣿⠇⠀⣀⣤⣾⣿⡿⢛⣣⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣾⣿⣿⣿⣟⣣⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠤⠤⢄⡀⠀⠀⠀⠀⣀⡤⠶⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠢⠄⠒⢢⠤⡀⠀⠀⠀⢀⣾⠏⠁⠀⠀⠀⠈⢣⠀⠀⢸⠋⠀⠄⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⣀⣤⠔⠀⠡⣤
⠀⠀⢰⡇⠂⠀⠀⡈⠷⡀⠢⡄⠀⠀⠀⡌⣿⠀⠀⠀⠐⠀⣀⣸⠇⠀⢸⣦⡠⠂⠀⠀⠀⢀⡿⠀⠀⠀⠀⡴⠯⠁⣀⣠⡴⠋⠈⢦
⠀⠀⢸⣇⠆⠀⠈⠀⠀⢻⡀⠰⢄⡀⠀⢷⠘⢣⣀⠀⠂⣟⡵⡏⠀⠀⠸⣟⠛⢧⣀⣭⠴⠋⣴⠃⠀⠀⢰⡇⠀⠀⠀⠀⡼⠛⠒⡾
⠀⠀⠀⢹⡀⠆⢀⠀⠔⠉⣗⠀⠀⠁⠀⠈⠳⢤⣌⣉⣡⠤⠚⠀⠀⠀⠀⠈⠙⠲⠤⣤⠴⠛⠁⠀⠀⠀⠈⠻⢦⣄⣠⣾⡤⠔⠚⠁
⠀⠀⠀⠀⠳⣶⡃⠀⠀⠀⢘⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠺⡷⡀⠀⢀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢵⡄⠀⠀⠀⠠⠐⠁⠀⠀⠀⢀⠈⠂⢀
⠀⠀⠀⠀⠀⠀⠦⠤⠤⠂⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⡀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⠈⢢⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿
⠀⠀⠀⠀⠠⢦⣄⠀⠀⢰⠀⠀⠀⠀⠀⠙⢄⣀⠀⠑⠲⡏⡠⠁⠀⠀⠀⠹⡀⠀⠀⠀⢀⡆⠀⠠⠂⣼⠀⠀⠘⣄⠀⠀⠀⠂⠀⠈⢀⣴⠇
⠀⠀⠀⠀⠀⠀⡙⢇⠀⣼⡎⠀⠀⠀⠀⠀⠀⠈⠉⠛⠂⠂⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣀⣀⣠⠤⠞⠁⠀⠀⠀⠈⠉⠒⠲⠴⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠠⡙⣦⡀⢙⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠤⠀⢀⡀⠀⠀⠀⠀⢀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⡹⢻⡗⢿⠃⠀⠀⠀⠀⣄⢠⣀⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢰⠶⠒⠒⠺⠯⠣⠄⠀⠀⠈⣿⠶⣯⡳⣄
⠀⠀⠀⠀⠀⢰⠃⠈⡇⠀⡇⠣⠀⠀⠀⡅⠈⠉⠑⠄⢀⠀⠁⢠⠀⠀⠀⠀⠀⢠⣀⣶⠏⠉⠁⠀⠀⠀⠈⢹⠈⠀⠀⣠⠏⠀⠈⢻⠉
⠀⠀⢠⠀⢀⡇⠀⢰⠃⠀⣿⠀⢸⡆⠈⢧⡈⠀⠸⡏⠁⠉⠛⠻⣀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠐⠀⠀⡀⠔⠚⠀⢀⠴⠛⠦⠀⠀⠀⡇⠃
⠀⠀⠘⢦⡀⠀⢀⠈⠀⠀⢰⣆⡼⠁⠀⠈⠛⠷⠿⠇⣠⡶⢯⠉⠙⠿⣦⡄⠀⠰⡾⣦⣤⡤⠀⠂⠉⠀⠀⢷⣞⠁⠀⠀⠀⠀⢀⠜⢁
⠀⠀⠀⠀⠉⠓⠟⠳⠦⠤⠽⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠑⠒⠄⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⠴⠖⡉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡉⠳⢤⡀⠀⣼⡉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠤⣤⠿⢽⣲⢤⣉⠳⢿⣯⣗⣦⣍⡳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠒⠛⠉⠉⠁⣠⣶⣾⣶⡀⣿⠀⠀⠈⠛⠾⣽⣶⣍⣻⢿⣷⢿⣶⣭⣓⠦⣄⡀
⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⠀⠀⠀⠀⣠⡾⣿⢿⢿⢻⠷⣿⣀⣀⣀⣀⢀⣾⣿⣿⣿⣷⣬⣙⠮⣝⡺⢽⣦⣝⡳⢦⣄⡀
⠀⢀⡴⠋⠀⠀⢠⣾⣿⣿⢦⠴⠒⠚⣯⣝⠁⡏⢸⣌⣃⣻⠉⠉⠉⢉⣿⣿⣿⣿⢿⣿⣿⡿⢿⣶⣝⡲⢽⣿⣿⣷⡮⣝⡦⣄⡀
⢠⠏⠀⠀⢀⣠⣾⢿⣿⢿⠧⣝⢦⠀⠻⣌⠛⠀⢀⣾⢿⣿⣷⢤⡀⣿⣉⠹⠏⡏⠼⠈⠀⡟⠀⠈⠙⣿⣷⣮⣝⠦⣟⣷⢯⣷⢿⣷⣤⣀
⢸⠀⢀⡴⢻⠏⠇⡞⣸⠀⢧⠘⠊⡇⠀⠈⠉⢉⣿⢋⡞⣿⠉⢧⠹⡜⠾⣝⣦⣀⣠⣿⣿⣿⣆⠀⠀⣇⠸⢻⠛⠿⣿⣿⣿⣬⣿⡾⢽⣻⣿⣲⢤⣀
⣾⣇⡞⠀⠘⣆⠀⠐⠿⠖⠒⠀⣰⠃⠀⠀⠀⡾⠙⠚⠀⠿⠀⠉⢀⡇⠀⠀⠀⢡⣾⣿⣿⢿⢮⡓⢦⠈⠳⠦⣤⡤⠤⠟⢿⣿⣿⣿⣶⣬⡙⠻⢷⣮⣛⠶⣄
⣿⡘⢧⡀⠀⠈⠙⠒⠒⠒⠶⠚⣡⣤⣴⣤⡄⠹⣄⠀⠀⠀⣠⣶⣾⣇⠀⠀⢰⠋⡇⡏⣿⠘⣇⠹⠘⡆⠀⣠⣦⣤⡀⠀⢀⡿⣹⣿⠻⣿⣿⣷⣦⡌⠛⠿⣾
⣏⠳⣄⡙⠶⣄⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⠦⣌⠙⠛⣻⣿⢿⢿⣿⣷⣄⠸⣄⠀⠱⠷⠶⠖⠀⣰⠃⡼⡿⣿⣿⣿⣦⠞⣰⠃⣿⠀⠀⠉⠛⠿⠟
⡟⠀⠀⠙⠳⢤⣉⠙⠲⢤⣠⣿⡿⡿⣿⢻⡎⡇⢸⠀⢰⠃⠟⣼⠘⣇⠓⠛⡇⠈⠓⠒⠲⠶⠶⠚⠁⣼⣱⣧⠷⠟⣋⡤⠞⠁⠀⢿
⡇⠀⠀⠀⠀⠀⠈⠙⠒⠦⢬⣉⣉⠛⠓⠾⢶⣇⣾⣀⣘⣆⣀⣀⣀⣀⣀⣰⣃⣀⣀⣤⠤⠤⠖⠒⣋⣉⣥⠤⠖⠋⠉⠀⠀⠀⠀⣸
⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠶⠤⠬⢭⣭⣭⣭⣩⣿⣯⣥⣤⠤⠤⠴⠶⠒⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣀⣀⣤⡤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠐⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠀⠀⠉⣦⡀
⠀⠀⠀⠀⠉⠐⠂⠀⣀⣀⡀⣠⡄⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⢢⢸⠄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⢠⡄⠘⠁⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠤⠀⠰⠀⡄⠀⠀⠀⠰⡀⠀⡀⠀⠄⠀⠘⠀⠀⠀⠔⠊⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠘⠛⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠒⠁⠉⠈⠉⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠
⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣮⣷⣶⣤⣾⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡿⢿⡿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣧⠀⢀⠔⠁⠀⠁⠀⠓⢄⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠘⣿⡿⣫⣾⣿⣾⣿⣬⣻⣿⣿⣿
⠀⠀⠀⠀⠀⣿⡿⠿⣿⡇⠹⣴⠀⣶⣄⣠⣦⠊⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⣀⣀⡀⢹⣟⠛⣿⠛⠻⠟⠛⣽⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣾⣀⣿⣿⣿⣿⠻⡆⠀⠀⠀⣿⣿⣿⣿⣿⠐⣿⣿⣷⢸⣿⡇⠿⠀⠀⠀⠀⣤⢹⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣿⣠⡟⠀⠀⠀⣿⣿⣿⣿⣿⠠⣿⣿⡿⢸⣿⡇⠀⠀⠀⠀⠀⠟⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⢿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠈⠉⠀⢸⣿⡇⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿



⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀
⠀⠀⠀⢠⣾
⠀⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠊⠚⠒⠀⠀⠀⠀⠊⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡆⠀⠀⠀⠶⡄⠄⠈⠉⠙⠗⢲
⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣿⣿⡷⠀⠀⣾⣿⣿⠀⠀⠨⠀⠸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢤⣴⠀⠀⢀⣿⣷⣗⠀⠀⣹⣿⣿⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⡻⠀⣄⣽⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⠼⣿⣿⣟⠀⠈⣿⣿⣿⡂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⣿⠀⠀⢰⣿⣿⣯⠀⠀⣹⣿⣿⡅⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣿⠏⠀⠀⢾⣿⣯⠁⠀⠀⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠆⠀⢸⣿⣿⡏⠀⣀⣺⣿⣿⡂⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⢸⣿⡿⡇⠀⠀⢸⣿⣿⠀⠀⠀⢿⡄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠠⠺⣷⠶⠺⠛⢹⡟⡿⠚⠋⠛⣿⣿⡃⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⣺⠚⠈⡷⠶⡶⡶⡯⡉⣵⠀⠀⠈⠁⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠈⠶⢭⣀⣀⣀⣀⠀⡀⠀⡀⠀⡀⢀⣀⣀⣀⢤⠵⠂⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠲⠾⢿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠴⠖⠚⣷⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⠀⣆⠀⣿⢋⠕⠛⠓⢯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡷⠼⠀⠀⠀⢈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢰⡇⠀⠀⢀⡠⢎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣾⡴⣒⣫⣥⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣩⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⠉⠉⢳⠀⠀⠀⣶⣀⠀⠀⠀⢀⣆⣸⣁⣉⣽⠿⠛⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠉⠁⠈⠣⢤⣄⣙⣒⣒⣪⣭⣭⠷⠛⠈⠁⠈⠉⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢶⣖⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⢖⣲⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠒⠒



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠂⢊⠀⢫⣟⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⣿⣷⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣀⣲⡶⠒⠒⠒⠒⠀⠙⢻⠇⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠹⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣼⠀⡸⢠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠨⠛⣹⣿⡆⡄⠀⠁⠇⠀⣰⣿⠃⣀⣀⡀⠀⠇⢸⠇⠀⢰⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠘⣿⣆⠐⣯⡃⠀⢠⣤⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⡅⠤⠈⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⣷⣬⠀⢀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠬⠄⢀⡀⠀⠀⠀⠔⠀⢀⣿⣿⢰⠃⠏⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⡀⠠⢤⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠃⡸⠀⠀⡀⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢋⠿⠇⠀⠀⠀⠀⠀⢀⣠⣴⠾⠿⠋⠠⠞⣁⠀⣰⠁⢀⠄⠁⠂⠀⠸⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡟⢻⡟⠫⠆⠀⢀⠉⠀⠀⠊⠁⠊⠑⠀⠈⠜⡸⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠁⣰⣶⠖⠂⠉⢀⡀⠀⠀⠀⢅⠋⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⠋⠀⠀⠀⠀⢀⡠⠋⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠔⠋⠀⠀⠀⡀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠉⠁⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢰⠏⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠘⢷⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠸⠇⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡶⠛⠉⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠙⠳⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣶⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣻⣭⣭⠹⣦⠀⠙
⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠈⣿⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⣿⠿⠿⠛⠛⠛⠋⠉⠁⣸⠏⠁⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣤⠶⣿⢦⣄⠀⠀⠀⠀⠀⢠⣿⣠⡴⠟⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣆⠄⠀⠀⠀⠀⠀⣿⠀⠋⠀⣿⠀⠀⠀⠀⣠⣿⣩⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣙⣧⡀⠀⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⢀⣴⣟⡉⠁⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠉⠉⠙⠷⢦⣤⣤⣬⣥⣭⣭⣤⡤⠶⠋⠉⠉⠙⠛⣦⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡟⠛⠛⠂
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⣠⣄⠀⠀⡾⠁⠀⠀⠀⡇⢀⣤⡄⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⢸⡇⢿⠀⢠⡇⠀⠀⠀⢿⢈⡇⠰⡷⠄⠀⠀⠀⡇⠸⣷⣃⡸⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠶⠛⠀⠻⠞⠁⠸⡗⠋⠀⠀⠀⠀⠈⠋⠀⠀⣷⠀⠀⠀⠀⠃⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢁⠟⣦
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣠⣀⡄⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡀⠘⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡆⢨⣿⣶⣷⢊⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡀⣀⠟⠉⠉⢹⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢃⣿⣿⠝⠹⡍⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⣸⣽⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢰⣿⣿⣑⣀⣈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢴⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⢠⠏
⠀⠀⠀⠀⠀⣤⠉⠛⠛⠛⠛⡋⣵⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣏⠀⠀⠀⢸⣄⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠒⡏
⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣀⣤⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡛⠉⠑⠚⠋⠉⠩⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢐⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠙⢂⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⢸⠁⠈⢿⣿⡿⠏⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡤⠛⠋⢦⣤⡉⠓⠁⢀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠸⠀⠀⠈⠉⠁⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⣬⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠰⠶⠶⠤⠤⠤⢤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢠⣴⣶⠀⠀⠀⠀⠀⢠⣿⣶⣦⣶⣶⣶⣴⣶⣶⣶⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣭⣿⠎⠉⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢠⡀⡀⢠⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡷⠠⠀⡠⠀⠀⠀⠀⠀⠉⠛⠛⠛⣛⡛⠋⠉
⠀⠀⠀⠀⠀⠀⢸⣧⡶⠂⢁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢨⣿⣿⣷⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡴⠆⡀
⠀⠀⠀⠀⠀⠀⢼⣏⣯⣭⢍⡉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠶⠀⠉⠀⠰⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⢃⣤⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⡴⢁⠤⠂⠀⠀⠀⠀⠀⠀⠀⢀⢸⠅⠘⡻⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⡔⠁⣴⠋⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠂⠀⠝⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⢣⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠐⣠⠞⠡⠂⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⢠⠨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠐⠢⠜⠋
⠀⠀⠀⠀⣠⣾⡫⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣞⡛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⠄⠊⣀⡀
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠘⠁⣟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠀⠀⢿⣷⡴
⠀⠀⠀⠈⣆⡀⠸⢰⠄⢸⠂⣃⡈⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡀⣠⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠿⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠑⠀⠍⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⠁⣟⡄⠀⠀⠀⠀⠀⠀⠰⡇⠠⠈⠓⢬⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⢤⠀⠒⢀⡧⠄⠀⠀⠀⠀⠀⠀⠀⠁⢢⣠⠇⠐⣾⠿⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⣀⡠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⢠⠀⠀⠀⠐⠄
⠀⠀⠀⠠⡄⠄⠽⠞⠛⠒⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣷⢉⣹⣿⣄⣌⢠⣤⡇⠀⠀⠀⠀⠀⣴⠁⠀⠄⠂⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠁⢠⡖⣤⣀⢢⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⣟⠃⠀⢽⡄⠀⠈⠁⠀⠀⠀⠀⠀⠁⢴⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠆⠈⠈⢙⣏⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠔⠐⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢤⡄⣁⡀⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡧⠤⢤⠝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣤
⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⣂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⣾⡅
⠀⠀⠈⠀⠀⠸⡇⠄⢀⠰⣷⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣆⢰⡌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⠿⢿⣤
⠀⠀⠀⠀⠀⢠⡇⠀⠨⠄⢻⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⡸⠛⢠⠈⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⠁⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠠⢥⣝⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢰⣶⣛⣛⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⡤⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠄⢀
⠀⠀⠀⠀⡠⠈⠁⠁⣀⣠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠐⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡄⠀⠀⠐⢢
⠀⠀⠀⠀⣇⣈⡤⠞⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⣾⣿⣦⣤⣦⣼⣤⣶⢴⡆
⠀⠀⠀⠀⣿⠃⠀⠐⠀⠠⣶⡏⠀⠀⠀⠀⠀⠀⠀⢈⠛⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠤⠲⣦⣀⡟⠒⠒⠿⣶⣦⠉
⠀⠀⠀⠀⠈⠀⢀⠀⠀⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⠆⠀⠐⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣸⡏⢹⡛⠛⠻⡟⠶⠶⢶⠶⠶⠶⡶⠶⠶⣶⠶⠶⢲⡖⠞⠻⡟⠋⢻⢉⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⣇⣀⣇⠀⢘⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⠀⢸⡇⠀⢘⡇⠀⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠈⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣸⠇⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠟⠀⠀⠀⠀⢸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⡀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢿⡍⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⡟⠀⢸⡏⠀⠀⠀⢸⡀⠀⠀⠈⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟⠀⠀⠸⠷⣤⣤⠶⠾⠷⠶⠶⠶⠋
⠀⠀⠀⠀⢠⡴⠶⠶⠶⣶⠶⣤⣤⣈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣴⠶⠲⢶⣄⡀
⠀⠀⠀⠀⣿⠀⠀⠀⢰⠃⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠈⣿⣄⠀⣠⠞⠛⢶⣄
⠀⠀⠀⠀⠙⢷⣄⣀⣸⣄⠀⠀⠀⣀⡾⠃⢀⡀⠀⢰⡟⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠉⠳⣄⣴⠟⠙⢷⣇⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣛⣯⡶⠛⠋⠙⢷⡈⠻⣄⠀⣠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢸⡅⠀⠀⢀⣴⠟⠁⠀⠀⠀⠙⠷⣤⣀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠳⡄⠀⢀⣼⠇⠀⠈⠻⣧⡀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠙⠶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⣀⣿⠶⠟⠁⠀⠀⠀⠀⠈⠻⣦⣀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⢀⣀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠏⣻⠇⠀⢀⠃⠀⠀⠀⠀⣰⣿⣿⠟⠛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡠⠄⠒⠙⠠⠤⠐⠁⠀⠀⠀⠀⠀⢿⣿⣯⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠘⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠐⠒⣸⣿⣯⣐⠒⠂⢰⠦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣤⣀⠔⠊⠁⠀⠀⠀⠀⣸⣿⣿⡿⠁⠀⢀⠆⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⡟⠀⠀⠀⠈⢛⠿⣿⣿⣿⣶⣄⠀⠀⠀⠀⣿⣿⠏⠀⠀⡠⠃⠀⠀⢠⣶⣄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠠⠾⠋⠀⠀⠀⠀⡠⠁⠀⠈⠙⢿⣿⣿⣷⡀⠀⠀⠙⠿⠄⠄⠊⠀⠀⠀⠀⢺⡟⠁⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠉⠐⠢⢄⣴⣿⣿⣷⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢀⡤⠖⢐⣾⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠑⢄⡀⠀⠀⢀⣠⣿⠟⢡⣿⠁⠀⣾⣿⣿⡟⠀⠈⠁⠂⠤⣀⠀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⢰⣿⣿⡿⠁⠑⠢⢤⣤⡾⠆⠀⠈⠉⠉⠉⠉⠀⠀⠘⠛⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣈⠁⠀⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⢦⠀⠀⠀⣿⣿⣿⠁⠀⢠⣶⣶⠀⠀⠙⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢀⣴⣾⡿⠷⣆⠀⠀⣾⣇⠀⠀⢠⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⡆⠀⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⢠⣿⡟⠁⠀⠀⣰⣶⣤⡹⣿⣦⣠⠃⠀⠀⠀⣿⣿⠁⠀⠀⡆⠻⣿⣀⡠⠀⠀⠀⢘⣿⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢸⣿⠀⠀⢀⣴⣿⣿⠿⠃⠙⣿⣿⣦⣄⠀⠀⢻⣿⠀⠀⠀⠃⠀⠈⠉⠀⠀⡠⠒⠁⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⠃⠈⠉⠁⣼⣿⠟⠁⠀⢀⠌⠀⠛⢿⣿⣷⣄⠈⢿⣇⠀⠘⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢀⠂⠀⠀⠀⢰⣿⠃⠀⣀⠔⠁⠀⠀⠀⠈⠻⣿⣿⣷⡀⠙⠒⠁⠀⠀⡠⠃⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠐⠁⠀⠀⠀⠀⠈⠃⠄⠊⠀⢀⠤⢤⣄⡀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢹⣿⡄⠀⠀⢹⣿⣿⣧⠀⠀⠀⢁⠀⠀⠀⠀⠀⣠⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠈⠋⠉⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠁⠂⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⣀⣀⣠⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣰⣷⣬⣭⡙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⣿⣿⣿⣿⣿⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⣤⣭⣤⠄⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠸⣿⡟⢸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⢹⣿⡆⢿⢃⣿⣿⣿⡇⠙⢋⣵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⣿⣿⣬⣼⣿⣿⣿⡇⠀⠀⠹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣌⠻⣿⣿⣿⣿⡇⢀⡄⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠻⢿⣿⠌⣿⣿⣿⠇⡞⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣥⣾⣿⣿⡏⣼⣷⣬⣉⣴⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠟⣠⣍⡛⠿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣷⣦⡀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣶⣿⣿⣿⡟⠀⠀⠹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡀⢀⣀⣛⣛⣛⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠸⣿⢡⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣌⢻⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠘⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠸⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⢀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⣀⠀⠐⠈⠉⠁⠀⠒⠒⠒⠛⠓⠲⠶⠀⠉⣉⣀⠴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠀⠀⠐⠐⠒⠒⠛⠋⠉⠉⠉⠁⠀⠀⣸⡥⠔⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⢠⡏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠉⠀⠀⢀⡾⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⡖⠁⠀⠈⠀⢹⡆⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠳⣄⠀⠀⡠⠞⠁⠀⠀⢀⣾⠡⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣄⣀⣀⣀⣀⣀⡤⠘⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡾⠋⠉⠉⠉⠉⠻⣿⣿⣿⣇⠀⠀⠀⠀⣾⡟⠋⠉⠉⠉⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⢿⣿⠟⠛⠻⠿⣿⠀⠀⠀⠀⠀⠠⠴⠄⠠⠦⠤⠀⠸⣿⣿⣿⣧⠀⠀⠀⠻⣤⣤⠀⣤⣤⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠃⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠘⠃⠀⠘⠃⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣆⠀⠀⠀⠀⢸⣿⠿⠛⠛⠛⠛⠛⣿⣿⣷⠀⠀⠀⠀⢈⣿⡿⠟⠛⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣯⣤⡄⠀⣤⣄⠀⢻⣿⣿⠀⠀⠀⠀⠈⢃⣀⣀⠀⣀⣀⠀⢻⣿⣿⠀⠀⠀⠀⠸⣿⠤⠴⠄⠠⠤⡤⠀⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⣀⣀⡴⠶⢶⣄⣀⣀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠉⠓⠒⠒⠛⠛⠲⢦⡀
⠀⠀⠀⠀⢿⡂⠀⠀⠀⢠⣾⣷⣦⣤⣄⣀⣀⣀⡇⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⡟⢹⣶⡀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⢰⡇⣀⣛⠚⡋⣀⣀⣀⣀⣀⣀⡀⣀⡀⡇⠀⠀⠀⠀⡇⣀⣀⠀⡀⢀⣀⣀⣀⣀⣀⡀⣀⡀⣷
⠀⠀⠀⢸⡇⣿⣿⢰⡇⣿⣿⣿⣿⣿⣿⡇⣿⡇⡇⠀⠀⠀⠀⡇⣿⣿⢠⡇⣾⣿⣿⣿⣿⣿⡇⣿⡇⣿
⠀⠀⠀⠀⡇⣿⣿⢸⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⣇⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⡇⡏
⠀⠀⠀⠀⢻⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠃⠀⠀⠀⠀⢻⢸⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠇
⠀⠀⠀⠀⠸⡎⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⡇⡾⠀⠀⠀⠀⠀⠘⡆⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾
⠀⠀⠀⠀⠀⢷⢸⣷⢹⣿⣿⣿⣿⣿⣿⣿⢰⠇⠀⠀⠀⠀⠀⠀⢳⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⢡⡇
⠀⠀⠀⠀⠀⢸⡌⣿⣿⣿⣿⣻⣿⣿⣿⡏⣾⠀⠀⠀⠀⠀⠀⠀⠸⡆⣿⣿⣿⣿⣻⣿⣿⣿⡟⣸
⠀⠀⠀⠀⠀⠈⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⡟
⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⢸⠇⠻⠿⣿⣿⣿⣿⡿⠿⠁⢷⠀⠀⠀⠀⠀⠀⠀⢰⠇⠙⠿⣿⣿⣿⣿⡿⠿⠃⢻
⠀⠀⠀⠀⠀⢿⣀⣀⡐⠒⠒⠒⠒⠒⣀⣀⣼⠃⠀⠀⠀⠀⠀⠀⢺⣄⣀⡐⠒⠒⠒⠒⠒⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣖⢲⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⡐⠟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣭⡛⠫⠭⠙⠉⠒⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠹⣆⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠉⠛⠢⢄⠠⣴⣵⡄
⠀⠀⠀⠀⠀⢸⡷⠲⠶⢶⠒⣿⠀⠀⠀⣀⣹⣄⠀⠀⠘⡿⠿⠿⠿⠶⣶⡶⠶⣤⣧⣀⡏⠉⠁⠒⢲⠀⠈⡆⣿⣿⡇
⠀⠀⠀⠀⠀⣸⣧⣀⣠⣸⠀⢋⡄⠀⣾⡟⠙⠻⣇⣀⣴⠗⣤⠀⠀⢀⣀⣈⣲⡀⠈⣿⣿⠀⠀⠀⢸⠀⠀⡇⣿⣿⡇
⠀⠀⠀⢠⡞⠉⠛⠾⠭⠭⠷⣿⠙⠓⠛⢻⡷⢾⠿⠿⣥⣾⣷⣤⡤⠴⠿⠿⠿⣿⣿⡿⠋⠀⠀⠀⢸⠀⠀⣟⣿⣿⡇
⠀⠀⠀⢸⠉⠓⠒⠒⠀⠐⢺⣿⣿⠛⠉⢹⡇⣧⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇⢀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇
⠀⠀⠀⢸⠀⡖⠲⠤⠤⠤⠞⣿⣻⡶⠶⣾⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠈⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣤⣿⡛⣉⣃⡨⡳⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣃⡉⠉⠀⢀⣲⡄⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⣼⣽⣿⣷⣿⣿⣯⠨⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠁⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡟⠻⠿⠿⠿⠿⠿⠂⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢯⣉⡓⣒⣒⣛⣩⡵⡀⠀⠀⢸⢿⣿⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠰⢤⡰⡶⠶⠀⢸⣷⢄⣀⠀⠀⠀⠀⠀⢀⣷⢤⣄⣈⣉⣉⣁⣤⢴⡟⠋⠉⠙⠛⠻⠷⢤⣄⣀⡀
⠀⠀⠀⢸⠀⠧⢄⣀⣀⡇⠀⣴⠋⠀⠈⢲⠄⠀⣿⣿⠟⠻⠿⠿⠿⠿⠓⡏⠙⠓⠒⠶⠒⠒⠊⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢓
⠀⠀⠀⠘⠦⣄⣀⣀⣀⡇⠀⣈⡳⣤⡔⣁⠀⠀⣿⡟⠀⠠⡤⠒⠚⢿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠈⠉⠉⠉⠉⢀⣠⡟⠧⢤⣀⣷⣷⡶⡟⠟⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣤⡴⠊⠁⠀⢀⡤
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿⣿⠿⠿⣿⣿⣿⠇⢿⠇⡀⠀⣠⣯⣧⠤⠂⡼⣿⠢⢤⣄⣀⣀⣠⠤⠖⢻⠿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠉
⠀⠀⠀⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⠊⠁⠘⠿⠾⠉⠉⡟⠒⠚⠋⢻⡀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠉⠉⠛⠻⠖⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧⠼⣧⣤⠴⠠⡄⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠚⠋⠁
⠀⠀⠀⣿⠤⠤⠤⠀⠐⠒⠒⠒⠒⠂⠉⢻⠀⠘⠶⠒⠊⠉⠁
⠀⠀⠀⠹⠴⠶⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢠⡿⠁⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠘⢷⣄⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⢻⡆⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠟⠁⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⣠⣶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⠀⣸⣷⣤⡶⠶⠶⣶⣄⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⡶⠶⠿⣶⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⡄⠀⠀⠀⢻⣇⠀⠀⠈⠉⠙⠛⠛⠛⠛⠻⠿⠿⢿⡿⠿⠛⠛⠛⠛⠋⠉⠉⠀⠀⢃⣿⠃⠀⠀⢀⣾⠇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⢄⣼⡏⣀⣤⡶⠟⣁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠸⡗⠈⣿⡇⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⣉⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠄⠀⠀⠀⠙⣷⣎⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⣿⡷⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣦⡀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣿⡇⠀⠀⢀⣠⣾⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠉⠉⠉⠻⢷⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⠋⠉⠉⠙⠛⠷⣦⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣄⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⠿⠿⠷⠶⠶⠶⠾⠿⠿⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡶⠀⠀⠀⠀⢸⡇⠀⢀⣀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡄⠀⣴⠀⣾⠛⠛⢻⡇⠀⠀⠀⠀⣼⠛⢷⡀⠀⣿⠀⠀⠀⠀⠀⢸⡇⢀⡟⣹⠆⢀⡾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⣀⠈⣧⢀⣿⡀⢻⡄⢀⣼⠃⠀⠀⠀⠀⢸⣆⣼⠃⠘⣿⠛⠀⠀⠀⠀⢸⡇⠈⢿⣯⣤⠌⠷⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡶⠶⠛⠁⠀⠻⠟⠉⠁⠘⣿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢹⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡤⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠉⠃⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⡄⡨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣦⡌⠀⠸⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠚⠈⠀⠈⠂⠐⠂⠈⠑⠀⠀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣐⣻⠿⠟⠓⠁⢻⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠊⠀⠀⠀⡀⠀⠀⣀⣀⠘⠉⢑⠠⠠⠄⣄⣀⣰⡂⠉⠁⠀⠀⠀⠁⠈⠀⠀⢀⡀⠀⣄⣤⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣷⣶⣷⡎⠸⢦⣶⣾⣷⣶⣷⣶⣷⣤⣿⣯⣭⣄⡨⣉⠀⢂⡰⢦⠒⣊⣥⣾⣾⣿⣿⣿⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣿⣿⣄⡼⠿⠛⠉⠉⠉⠉⠛⠿⣿⣿⣿⣿⣧⣿⣤⣸⣶⣿⠿⠿⠛⠋⠉⠀⠀⢀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠉⠉⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⡠⠀⠀⠘⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⣀⠄⠀⠖⠒⢀⡉⠙⠤⠤⠥⣱⠀⠀⣀⠈⠐⠒⣒⠂⠀⣀⣊⠈⢧⡈⢢⣽⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢀⣀⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⠀⠀⠂⠀⠀⠀⠀⠈⠄⢀⡬⠩⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠁⢰⠁⢠⡄⣀⡤⠂⠒⠀⠤⢄⠀⠀⠀⠀⠈⠃⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠥⠔⠋⠁⠀⠀⠀⠀⠀⢀⠱⠢⢀⠀⠀⠀⠀⠈⢢⠄⠀⠀⠀⢀⡠⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⡀⠱⠒⠤⠤⠤⠤⠄⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⢾⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⢀⠇⠀⠀⠀⠐⠶⡒⢯⡞⠁⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⡀⠀⠀⠀⠈⠢⠀⠐⡄⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⢀⠀⠀⠳⣄⡀⠀⠘⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡐⠀⠀⠀⠀⠀⡄⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⢲⡀⠈⠀⠀⠀⠀⠠⢌⠹⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢌⠄⠀⢻⣄⠜⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢗⡀⠀⠀⠀⠀⠢⡕⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠳⡀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠤⠨⠥⠌⠘



⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣿⣵⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⠉⠀⠀⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠀⠀⠀⠉⠐⢳⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠰⠁⠀⠀⢈⠂⠀⠀⠀⠀⠴⠒⠀⠀⠀⠒⠀⢳⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠰⠆⠠⠀⠂⠒⠉⢸⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡠⢊⡠⠊⠁⠒⠤⠤⠤⠤⠤⠤⠤⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠆⠀⠀⠀⠀⠀⠐⠀⠠⠁⠀⡇⠀⠄⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠬⠙⡆⠀⠀⠀⠠⣤⣤⠤⠴⠋⡼⢄⡀⠀⡀⣀
⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠰⠀⠆⢀⠏⠀⠈⠓⠄
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠕⡦⢀⠀⢀⣠⠆⠀⠀⢸⠀⠀⠀⡰⠈
⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡌⠀⣠⡈⠓⠢⢸⡀⠀⠰⠁⢱
⠀⠀⠌⠀⠀⠑⡄⠀⠀⠀⠜⠀⠀⠀⠀⠠⣀⠘⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⡸⠀⠀⠀⢦⡇⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠑⠢⢄⢰⡀⣀⠄⠀⣸
⠀⠁⠀⠀⠀⠤⣙⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢦⠀⠀⣀⣀⡴⠃⠀⢠⠃
⡄⠀⠀⠀⠀⢀⠘⠐⠂⠤⠀⠀⠀⠀⠀⠉⠁⢠⠀⠘⡀⠀⠀⠀⠀⢸⠀⢸
⢷⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⢀⡇⠀⢸
⠀⠳⡇⠀⠀⢸⡤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⡾⠀⠀⢸
⠀⠀⠹⠄⠠⠒⠀⠐⡄⠀⠀⠈⠑⠢⠤⠤⠤⠜⠀⠦⡀⠀⠀⢠⠇⠤⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠁
⠀⣁⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠁⠒⠤⢤⡀⠀⣀⣀⣰⡿⢯⡙⠃⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠶⠞⠀⠀⠀⠁⠀⡀⠀⠈⠜⢇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠂⠉⡆⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⢀⠃⠀⢠⡇⡟⠁⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡴⠫⣤⣤⡀⠀⠀⠀⡎⠀⠀⣸⠁⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠩⠀⠀⠈⠁⠂⢀⣀⣼⡄⠀⠒⠋⠀⠀⠈⠁⡄⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠑⠀⠀⠀⠁⠀⢀⡰⣄⠀⣀⡔⣇⠀⠀⠀⠈⠀⡐
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠈⠁⢏⠲⢶⡶⠸⢇⠀⣀⡀⠀⠸⠋⠙⠉⠀⡞⠓⠀⠣⠔⠊
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡁⠀⢀⣨⡷⡾⠀⠀⠀⢢⡿⠿⠶⣾⣦⣶⠖⠛⠛⠂⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠢⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠧⣤⠼⠷⣶⣿⡏⠀⠈⢤⠞
⠀⠀⠀⡀⣠⠇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠀⠀⢀⣾⣶⣿⢶⡄⠀⠈⠓⡄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⢻⠀⠒⠀⠄⠤⠄⠑⠒⠊⠘⠉⡉⠹⣭⣠⡦⠴⠉⠒⠈⠓⠀⠈⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡨⠀⣠⡆⠀⠀⠂⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠖
⠀⠀⠀⡠⠄⠀⠀⠐⢾⣆⠀⠀⠀⠀⠀⠀⡘⠁⠀⠃⠁⠀⠀⠈⣀⠄⠀⠀⠠⠄⠀⠀⠀⠀⠠⠊⠀⢄⠀⠈
⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠜⠋⠔⠀⠀⡀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣸⠶⠞⠋⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢧⣸⠃

⠀⠴⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡰⢄⡀⠀⠀⠀⠀⡐⢤⠐⣆⢆⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⡠⠀⠠⠄⠀⠀⠴⠒⠂⠀⢶⡖⠦⣠⠶⠄
⠀⠔⣺⠟⠂⢔⠲⠔⠂⠰⣄⢀⣀⠀⠙⢀⣉⡉⢠⣀⣀⠙⣙⡶⣬⠎⡧⠂⣠⠖⠋⣤⡶⠷⢟⠉⠓⠂⠀⠀⠀⠀⠀⠈⠢⣤⣴⡀⣤⠁
⠀⠀⠙⠰⡤⢀⡑⠀⠀⠀⠈⠋⠉⠀⠀⠘⠓⡚⠃⠩⡉⠉⠉⠁⠈⠀⠁⠬⠀⠀⠀⠉⠣⢄⠀⠀⣂⣀⡠⢸⠦⣄⣀⠀⠈⡉⠽⠛⠁
⠀⠀⠀⠀⠉⠰⢄⠀⠀⠀⠀⠀⠀⢀⣬⣧⡞⠏⠀⠑⢉⠀⢀⡀⣔⠃⢀⡔⠂⣈⠀⠄⠐⠪⠤⢤⠈⠃⠀⠐⢤⣈⣀⣙⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠴⠆⠀⠉⠀⢀⠀⠀⠀⠀⠀⠙⢿⣄⣈⡆⠸⣤⣾⣁⣀⣀⣀⣀⠀⢤⠒⢤⠰⣄⠀⠙⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠘⠒⠦⣄⠀⠀⠀⠀⠹⡫⠀⠸⠛⠀⠉⠛⠤⢄⠀⠷⢴⠛⠿⣄⣈⡉⠳⡚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠀⠈⣉⠀⠀⠀⠙⣄⠀⢈⣶⠦⠼⠤⠤⠤⣁⠄⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠙⠒⠋⠁⠉⠒⠛⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁⠀⠀⠀⠀⡄⣄⢄
⠀⠀⠀⠀⠀⢀⣄⡀⡀⠀⠀⣴⡤⣆⠀⠀⢰⡖⠀⠀⠀⠃⠀⡀⢸⠀⡄⠀⠀⠀⠀⢁⢀⠀⠀⠀⠠⡀⣁⡀⠀⠀⠀⠀⡀⠀⠀⣄⢀
⠀⠀⠀⠀⠀⢆⢐⣝⠓⠀⢀⠛⠀⢁⠁⠀⡛⠠⡆⠀⠺⣸⡟⢥⣸⠀⡇⠀⠀⠔⠁⠀⠀⠀⠀⠀⢾⢧⠈⠙⠆⡐⠀⠘⠋⠈⡰⠆⢤⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⢸⠀⡀⢸⡀⠀⠀⠀⠀⠀⠠⣴⠅⠠⢿⠀⡇⠀⠈⢀⡆⠀⠀⠀⠀⢀⣈⣇⢀⠂⠯⠡⠀⠐⣍⡚⡿⠀⠸⣄
⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠋⠉⠉⠈⠰⠀⠀⣧⠀⠀⠀⠀⠸⠀⠈⠀⠇⠀⠉⠋⢠⠀⠀⠀⠀⠋⣧⠈⠌⢴⠀⠀⠀⠀⠅⠀⢹⠒⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠘⡆⠀⡸⠤⠄⠀⠀⣇⠈⣎⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠃⢸⣄⡀⠀⢀⣀⠀⣧⢰⠁⡄⠀⠀⠀⠈⠀⠛⠃⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠈⠀⢫⣤⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡐⠉⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⣤⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠐⢷⡇⠈⠙⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠟⢠⠂
⠀⢠⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣇⠀⡀⠹⢦⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠈⠁⠉⠀⠀⠀⣦⣤⡄⡀
⠀⣏⠀⢱⣶⠀⣧⠀⢸⠁⣄⠀⠀⠀⠲⢿⣷⣶⣾⣿⣿⣛⣛⣛⣻⣛⣿⣿⣷⣶⡾⠟⣃⠸
⠀⠹⣦⡄⢹⣶⡻⠀⡖⣠⡘⢆⠀⢴⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣻⣿⠁⣆⣲⣦⣄⢲⡀
⣀⣀⣈⠳⣄⣹⡇⢰⠃⣿⡇⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢸⠋⠁⠀⡼⣸⠇
⠘⡍⢶⣄⣩⣭⣽⣿⣦⣿⣧⠟⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠇⣎⣠⠴⢊⣼⠟
⠀⠈⠢⣀⠜⠋⠹⠛⣿⠿⣁⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⣸⢋⣤⠾⠛⠁
⠀⠀⠀⠈⠐⠒⠒⠁⠀⠀⠙⠷⠆⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠏⡰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢾⣒⠁⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣫⣞⠓⢤⣤⣐⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠀⠉⠙⠋⣆⠰⢍⡛⠷⠶⠶⠶⠶⠾⡿⢿⣿⠿⠋⠻⢂⠽⠂⢉⣆⡕⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⡴⣞⡋⠠⡆⠀⢀⠀⢀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣻⣲⣿⣻⣿⣻⣻⣿⣷⣿⣿⣿⠿⣿⠛⠋⠉⠀⠀⠉⠙⠣⡀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣏⠙⠶⣤⣤⠤⠴⠒⠚⣷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣿⡀⠀⠀⠀⢀⣽⣬⠿⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠮⣿⣷⣠⠤⠚⠋⠁
⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠏⠐⠂⠒⢄⡀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⢢⡀⢀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⣀⡀⠂⢀⠀⠈⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⢀⠀⠀⡫⠵⠀⢤⡔⠌⠀⠀⢱⣦⡈⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢼⠡⢠⡶⠋⠀⢀⣀⠀⡀⠀⠁⠀⠀⠀⣸⠃⣝⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢄⠑⠃⠈⠃⠠⠄⣠⡖⢀⡴⣻⠟⠂⠀⣀⣀⣼⣿⠎⣩⠇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠈⠁⡤⠛⠀⢘⣇⣴⣀⣤⣾⣿⣿⣿⠃⠀⡟⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠘⠹⠿⢿⠿⠟⠛⠉⠀⠀⠀⣇⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠀⠀⢀⠰⡀⠀⠀⠀⠀⠀⠀⡏⠀⢢⠈⠉⡇⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⡠⠴⠄⡁⠀⠀⢀⠀⠀⠀⡠⠀⢸⠀⢆⠸⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣂⣤⠖⠁⠀⠀⠀⠁⠀⠀⠶⠀⠊⠀⠈⠢⡁⠈⠂⠀⠀⠸⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣘⠁⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣀⣀⠀⢀⠀⢀⢱⣀⣀⣀⠀⠁⡀⠁
⠀⠀⠀⠀⠀⠀⣠⠃⢃⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⡀⠀⠀⠀⢔⠛⣻⣿⠋⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⡴⠁⠀⢸⠀⠀⠀⡐⠸⠀⠀⢀⡝⢶⣄⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢒⣒⠒⠒⠒⠀⠀⠀⠀⠒⡒⠞⠋⠉
⠀⠀⠀⠀⡜⠀⠀⠀⠸⠀⠀⡐⠀⡀⡧⠀⠀⠑⣄⠙⠿⣦⣄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠃⡉⠛⡋⠉⢩⡍⠀⠡
⠀⠀⠀⠰⠁⠀⠀⠀⠀⡆⠰⠁⠀⡇⠱⣄⠐⢄⣀⠁⠀⠈⠛⢿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢒⠃⠠⢾⡇⠀⠀⢣
⠀⠀⠀⡧⠀⠀⠀⠀⠀⠱⠃⠀⠸⠁⠀⣨⠗⠸⠁⢀⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢦⣀⠠⡄⠀⠀⠀⠄
⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⡠⠖⣿⠐⢉⡤⠛⣧⠀⠀⠙⠳⣦⡀⠀⠉⠒⢄⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠘
⠀⠀⣜⣀⡀⠤⠤⢴⠖⠉⠀⠘⠁⡠⠋⠀⠰⠁⠀⠸⡀⠀⠈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⢤⢚⠀⠀⠀⢀⠆
⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠁⠈⠁⠀⠈
⠀⠀⠀⠀⠈⠒⠦⠠⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠳⠎⠁⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣵⣶⣶⣭⣵⣿⣷⡌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣫⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⡻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⢤⡏⡇⠀⡟⠉⠙⢦⣄⡤⠋⢹⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣮⡛⢰⢸⣿⢰⣶⣦⣙⠿⢟⣵⡆⣡⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡇⢠⠀⡄⠀⠀⢠⠘⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⢸⣿⢸⣿⣿⣿⣿⣿⣿⣧⣭⣭⣛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢇⣀⡇⠐⠀⡇⠀⠀⠐⢸⠋⠙⡆⢹⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⡸⡿⢸⣿⣿⣿⣿⣿⣿⡏⣶⣶⢹⡎⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠀⠀⠀⡇⠀⠀⠐⢸⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⢸⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠠⠀⡇⠀⠀⢘⢸⣄⡤⠇⣸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠻⢟⣸⢇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡂⠀⠀⠰⠀⡇⠀⠀⢘⢰⠶⠖⠚⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣩⣭⣵⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠈⠀⠈⠀⠁⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⠤⠤⠤⠤⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣬⣛⣛⣛⣛⣛⣛⣛⣛⣛⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛








⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣽⣿⣿⣷⣿⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⢀⣀⠀⢀⡀⠀⢀⣀
⠀⠠⠁⠀⠲⡄⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠈⠉⠀⢫⡀⠀⢀⡈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫
⢀⠀⠀⠀⢀⠀⠉⠀⠂⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙
⡎⡀⠀⢀⠂⡀⠀⠀⠀⡌⠀⢸⡀⢠⠤⡴⡂⠉⠀⠀⠂⠀⠒⠀⢒⣀⣾⣷⡶⢠⣐⡀⠀⠀⠀⠀⠀⠂⠐⢻⡂⠀⠀⠀⠀⢧⠙⡄
⢠⠁⠠⠃⢠⠁⠀⠀⢀⠆⠀⠈⠀⣼⢀⠀⢉⠄⣀⠀⠀⣀⡀⣺⡟⡉⢡⢔⠀⠀⠈⡕⠒⢤⡆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠐
⠇⡰⠃⢠⡇⠀⠊⡁⠀⠀⠀⠀⠀⡄⠘⢀⢸⠂⣿⠴⢋⢹⢭⠉⠉⣡⠱⠛⠓⢤⣤⣄⠠⣠⣶⠀⠃⠀⠀⣴⡈⡀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢡⢀
⠐⠋⣥⡄⠀⠰⠍⠠⢄⠠⠄⠠⢈⣧⠟⣬⡇⠨⠄⠠⢘⣼⠖⣏⠚⣧⡟⢿⡉⢸⡆⡿⠒⢘⣿⣄⠀⡴⠀⠄⡡⠶⠀⠔⠀⡐⠀⠀⠀⠀⠀⡀
⠢⢄⠈⠑⠢⣄⠀⠑⠠⡐⠀⠑⢼⠉⡖⢦⢄⡑⠦⣀⢸⣛⠲⠦⣌⣧⠥⡤⠦⢤⠭⡿⠆⠹⣿⣧⣾⡢⠄⢠⠀⠀⢂⢀⣌⠄⠊⡀⠄⠂⡀
⢸⠀⠹⢢⣄⡀⠙⠢⣀⠈⠐⠄⡆⠀⡇⠸⠨⠘⡑⠠⢙⣎⣂⣁⣚⣛⡛⣶⣯⡴⣊⢀⡤⠤⣭⠿⡏⠀⠰⠈⡄⡈⠌⢋⡠⠐⢀⠠⠂⠁⣀⠴⠞
⠇⢀⠇⡸⢘⡏⢰⠤⠀⠙⠢⢀⠃⢀⠇⡆⠈⠦⠥⠀⠀⠉⠁⠦⠄⣀⠀⠀⠀⠞⣁⣩⠜⠋⠀⠀⠀⠀⠀⠀⠉⢡⠈⢌⡄⠊⢀⣠⠔⠊⢅⠘⡄
⠀⡼⢠⠇⢸⠁⢸⡂⠀⠁⠂⣼⣄⠈⠠⠄⠀⠀⠔⠀⠀⢄⠀⠠⠀⠈⠙⠒⠶⣶⠿⠀⢀⣤⣄⣠⡄⢀⣠⡄⢂⡔⠀⣪⡼⡖⠉⠀⠀⠀⠈⢢⠐
⠀⣇⣞⠀⠛⠀⠈⠁⠐⠂⠈⠉⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⣴⡿⢶⠐⢂⡁⠲⠀⠀⠀⠀⠀⣠⡴⠚⠉⠒⠒⠶⠶⠶⠶⠶⠶⠷
⣸⣿⣟⣉⣙⣑⣀⡒⠒⠂⠒⠒⠖⠖⠀⠈⠳⢦⣄⠀⠀⠀⠀⠀⠠⢀⠑⠦⠖⣊⠴⠊⠁⠀⠀⠀⢀⣠⡖⠛⠁
⣍⣁⣀⣀⣀⣉⣉⣉⣉⣙⣛⣛⠛⠒⠒⠒⠒⠖⠚⠻⠦⣄⡀⠁⠂⢀⠉⠒⠋⣀⡤⠊⠁⣀⣴⠾⠉⢁⣀⣀⣀⣀⣀⣀⡀⠀⠈⠁
⣿⣽⣯⣭⣭⣭⣭⣭⣭⣭⣙⣛⣛⣛⣛⡛⠛⠛⠓⠶⠶⠾⠿⢶⣄⡀⠑⠲⠟⠁⢀⣴⠾⠯⠤⠤⠤⠤⠤⠤⠄⠀⠠⠤⠤⠤⠤⠤⠄⠤
⣿⣭⣽⣯⣭⣯⣭⣭⣭⣩⣍⣩⣉⣉⣉⣛⣛⣛⣛⣛⣓⣶⣶⣶⣮⣿⣶⣤⣤⣾⡛⡒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠄
⣿⣯⡛⣿⣯⣝⣯⣿⢿⣭⡋⠩⠻⣿⣍⠉⠙⠻⣯⣍⡉⠛⢿⣍⠉⠉⠉⠙⠛⠉⠉⠉⣉⡿⠛⠉⠙⣻⡿⠛⠛⠛⣓⡿⠛⠒⢒⡶⠛⠒⠒⣲⡶
⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣖⢹⣿⢷⣌⡠⠐⠉⠻⣶⣄⡙⢷⣄⠀⠀⠀⢀⣤⠾⠋⠀⣀⣴⠟⠉⠀⢀⣤⠞⠉⠀⣠⠶⠋⠀⣀⡴⠛⠁
⣿⣿⣿⣽⣿⣿⣾⣿⣿⣿⣽⡯⣿⢷⣧⡶⣉⠻⣦⣔⠉⡈⠙⢿⣦⡙⢷⣶⡾⠛⠁⢀⣤⠞⠋⠀⢀⣤⠾⠋⠁⣀⣴⠟⠁⢀⣤⠞⠉⠀⠀⣠⠴
⣿⣻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⡻⣿⣿⣦⣗⡬⣻⢷⣤⡀⠀⠙⢻⣦⡉⠀⣀⣴⠟⠁⢀⣤⠾⠋⠁⣠⡴⠟⠉⢀⣠⡶⠋⠁⠀⣠⠴⠋⢁⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣮⣿⣷⣤⡷⣝⡿⣦⣄⠀⡈⢿⣿⡟⠁⢀⣴⠟⠁⣀⣴⠟⠉⠀⢀⣴⠟⠁⠀⢀⡴⠋⠁⣠⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⢩⣽⣻⠿⠇⠀⠀⠀⢰⣿⣿⣿⢰⣶⡆⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⡇⠀⠀⠀⣿⣿⢰⣶⣿⣿⣿⣶⡆⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⡇⠀⠀⠀⢻⣿⣬⣭⢻⣿⡏⣭⣥⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢸⣿⣿⣿⡇⠀⠀⠀⠈⠻⣿⣿⣬⣭⣥⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢸⣿⠿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣡⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣫⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠉⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢖⣶⣀⣤⠴⠶⠶⠒⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢚⣩⢴⣾⣿⣿⣿⣷⠶⣿⠛⠲⠶⠚⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣛⣩⣶⢾⡽⠖⣿⠛⠛⠛⠉⢀⡾⠁⠀⠀⢸⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠿⠛⠛⠛⠛⠉⣿⢠⠀⠀⢹⡜⣇⠀⠀⢸⡅⠀⠀⠀⢼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠿⠛⠛⠛⢷⢰⡄⠀⠀⢠⡿⣎⠷⠀⠀⢳⡜⢦⡀⠀⠳⣄⢠⡆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣉⠰⣄⠀⠀⠀⠀⢸⡇⣷⠀⠀⣸⣧⠘⢦⡓⠀⠀⠙⢦⣀⠲⠀⠘⣧⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⣾⣿⣟⡛⠛⠛⠻⠿⢷⣤⡄⠀⠀⠀⡇⠋⠀⠀⣿⠁⠀⠀⠙⠳⣤⡀⠀⠙⠳⣄⠀⢸⡏⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡞⢉⡀⠀⠈⢳⠸⡆⠀⠀⠀⣇⣷⠀⠀⠀⡇⡶⠀⢠⡟⠀⠀⠀⠀⠀⠀⣩⣷⣾⣦⡼⠶⠾⠶⠖⠛⢉⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠻⣄⠀⠈⣧⠛⠀⠀⠀⣿⣉⠀⠀⢀⡇⢀⣠⣴⡗⢚⣻⣷⡶⠖⣻⡿⠟⠋⠁⣄⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠀⠀⠸⡎⠃⠀⠀⣿⣈⡤⠶⠚⠛⠛⠉⠉⣿⣾⠛⠉⠀⠀⠘⣦⠀⠀⠀⠙⣆⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⢻⠀⢀⣴⣏⣥⣤⣤⣤⡄⠀⠀⠀⣷⢹⡄⠀⠀⠀⠀⠘⣆⠀⠀⠀⠹⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣷⣚⡛⠛⠛⠛⠉⢉⡇⠀⠀⠀⢻⠀⢻⡄⠀⠀⠀⠀⠹⣆⠀⠀⠀⠂⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠛⠛⠛⠛⠛⠛⠛⡆⠀⢶⠀⢸⡇⠀⠀⠀⢸⡆⠀⠹⣄⠀⠀⠀⠹⣌⠳⣄⠹⣆⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⢡⡞⠁⠀⠻⣆⠀⠀⠀⠀⠀⡇⠀⢸⠇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠈⢷⣄⠀⠀⠈⠳⢬⡳⣌⢳⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡴⠋⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣇⠀⣾⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢷⠀⠉⠳⣄⠀⠀⠀⢹⡈⠷⠿⡆⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣅⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⣿⠀⠀⠀⣸⢷⡀⠀⠈⢧⠀⠀⠈⢧⠀⠀⢸⠃⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡇⠀⠇⠀⢀⡟⠀⠀⠀⠉⠀⢳⡀⠀⠀⠀⠘⡄⢸⡇⠀⣿⣠⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠆⠀⠀⠀⠀⣼⠀⠸⠇⠀⣸⠃⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢸⠇⠀⠀⢀⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠋⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠓⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠳⠶⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⠤⠤⠤⠤⠴⠶⠖⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⡀
⠀⠀⠀⠉⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠑⠄⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠂⠀⠀⠁⠀⠘⠐⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑
⠖⢺⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠠⢀⠆⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠐⠅⠀⡄⠐⠈⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⡍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⠒⠛⢻⠛⠛⣿⠛⠛⠛⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣄⣀⣀⣠⣤⡤⠤⠤⠤⠤⠴⠦⠤⠤⠤⢤⣤⣄⣀⡀⣙⣿⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠳⣶⠒⠒⠛⠋⠉⠁⣿⠶⢦⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡧⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣄⠀⠀⠀⠀⠀⢰⣇⣀⡤⠔⢁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣴⡉⣿⠀⠀⠀⢠⠿⠤⠶⠖⠚⠉
⠀⠀⠀⠀⠀⢀⣀⣠⣤⡼⠿⣦⡀⠀⠀⠀⠀⠀⠀⣿⢾⡓⠀⢸⡄⢀⡴⠓⠶⠤⢤⣀⡀
⠀⠀⠀⠀⣾⠋⠉⠀⠀⠀⠀⢨⣿⣶⣤⣀⠀⠀⠀⠻⠾⠶⢚⣫⣴⣿⡄⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⣶⣶⣿⣿⠿⠟⠛⠁⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⣀⠀⠀⢀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠐⠀⠀⠀⠀⠀⣾⣿⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡋⠛⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠙⠛⣹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⡀⠀⠐⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢿⣿⠿⠿⠆⠀⠘⠿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠄⠄⠂⠀⠀⠈⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⣀⣠⣤⡀⠀⠀⠀⠀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢿⣿⡿⣻⣿⣿⡿⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣴⠀⠀⠀⠀⢢⠈⣿⡥⠽⠿⠋⡄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠰⣅⠈⠀⠀⠀⠀⠀⣐⠕⠀⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⠀⠠⠤⠄⠤⢠⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣨⣾⡇⠀⠀⠀⠀⠀⠀⠰⣶⣾⣾⣶⣶⣤⣌⣀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⠠⣐⣈⣴⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⠛⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⠉⣁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠈⠁⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠀⢄⠔⠒⠢⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠈⠉⠂⢀⠀⠀⠈⠂⠀⣀⡀⣀⠤⠄⠊⠀⠀⠀⠀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄
⣤⡒⠁⠁⠀⠈⠈⠝⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠚⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡎
⠑⠈⠦⣄⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠠⠁⠀⠈⠀⠀⡤⠄⠚⢡
⠀⠀⠀⠈⠢⡀⠉⡀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠈⠁⠙⡄⠀⠀⠙⠄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠄⠠⠀⠰⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠈⠐⢲⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠁⠒⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⢄⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠁⠺⠀⠀⠱⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠑⢄⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⣀⣄⠐⠂⠀⠠⢄⠀⢤⠃⠀⠀⠀⠀⠠⠄⠒⠂⠒⡠⠴⢤⡀⠀⠀⠂⠀⠀⠁⠢⡀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢏⠙⢳⣻⠁⠀⠀⠀⠀⠀⠀⣀⡘⠀⠀⠀⠀⠁⠀⠀⠀⡠⢡⡀⠤⢈⠀⠀⠀⠰⡀⠀⠀⠈⠑⠠⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡋⠀⠄⡀⢐⣠⢜⠤⠉⠀⠀⠀⢄⠀⢠⡄⠀⠀⠁⠀⣀⣀⠚⠁⠀⠀⠀⠈⠐⠢⢀⡀⠀⠀⠈⢛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠍⢐⡀⠀⠀⢄⠀⠈⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠁⠁⠀⠄⢛⣒⣐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⢀⢀⡀⠀⠀⠘⠁
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠊⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⠔⠋⠀⠀⠀⠀⠀⣀⡤⠖⢋⣁⡴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣔⢉⡤⠞⠁⠀⠀⠀⣀⡤⠖⠋⣁⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠁⠀⠀⠀⠈⠙⢦⡀⣀⡤⠖⠋⣁⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⢦⣅⠀⠀⠀⠀⣀⡤⠖⢋⣁⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠼⣳⡤⠞⢋⣡⡴⠖⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣚⣩⠴⠺⣏⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠳⣤⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠒⠛⠛⠛⠛⢋⣉⣛⠛⢛⣓⡒⠶⠦⣤⣄⣀⠀⠘⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣋⣁⠀⠀⠀⢀⠴⠒⠛⠉⠙⠉⣹⠟⢋⠉⠉⠉⠙⠛⡒⢦⣍⠉⠓⠦⣤⣀
⠀⠀⠀⠀⠀⢀⡴⠞⠉⢀⡴⠚⠉⠈⠳⣄⢠⠏⠀⠀⠈⠄⢀⡞⠁⠀⠀⠀⠀⠒⠂⠈⠁⠀⢈⣛⣦⡀⠀⠈⠙⠶⣄⡀
⠀⠀⠀⣠⣾⠋⠀⠀⢠⠏⠀⠀⠀⠙⠀⠈⢻⠀⠀⠐⠀⢠⠋⠐⣀⣬⣶⣠⡤⠶⠖⠒⠚⠋⠉⠉⠉⠉⡓⠢⣄⡀⠈⠹⣦⡀
⠀⢀⣾⠟⠁⠀⠀⢀⣸⡀⠀⠀⠀⠈⠀⢀⠘⣇⠐⠂⠀⢿⣴⠞⠉⡁⠘⣿⠀⠀⠀⠐⠈⠀⠚⠀⠀⢂⢘⡀⠀⢳⡀⠀⠘⣿⣆
⢀⣟⡏⠀⠀⢀⡔⠋⡁⠙⠦⣬⣤⣤⣤⣦⣤⣜⣦⡠⡶⠚⢛⢶⣤⠤⢤⣿⣂⡤⠴⠖⠶⠶⠲⠶⠶⠦⠤⢤⣄⣠⡇⠀⠀⠘⣿⣆
⣼⢸⠀⠀⢠⡏⠀⠀⠁⠄⠀⠀⡀⠀⠀⠀⠀⠀⠉⣻⡇⠐⠈⢀⡯⠒⠤⠘⣿⣀⣐⣃⣀⠀⠀⠠⣀⠀⠀⠐⠀⠙⢧⡀⠀⠀⢹⢹⡄
⡇⢸⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⢐⡿⠖⣶⠶⠻⠷⣤⣄⣀⣟⠈⡉⠉⠉⠉⠙⠲⢦⡀⠀⠠⠀⠐⠀⢷⠀⠀⢸⠇⡇
⣇⢸⡀⠀⠈⣇⠀⠀⢠⡤⣄⣀⣀⣀⣀⣀⡤⠴⠋⠄⠀⢻⠀⢀⠀⠈⠁⡈⠉⠳⡄⠀⠀⠂⠀⠀⠀⢳⡀⠄⠂⠀⠀⢸⠀⠀⣸⢀⡇
⢹⡀⢧⠀⠀⠈⠳⣄⣈⡷⠈⠈⠉⣉⠉⠀⠀⠀⠠⠄⡀⠸⡆⠀⠀⠃⠀⠀⠀⠀⠹⡄⠀⡄⠀⠀⠀⠈⡇⠀⠠⠀⢀⠞⠀⣠⠃⣼
⠈⢧⠘⢧⠀⠀⠀⠀⠉⣧⠀⠉⠁⠀⠀⠀⠈⠉⢀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⣹⠲⢦⣰⠏⠀⡴⠃⣰⠃
⠀⠈⢧⡈⠳⣄⠀⠀⠀⠘⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠘⠀⠀⢠⠏⠀⠀⠉⣠⠞⢁⡜⠁
⠀⠀⠀⠻⣄⠈⠳⣄⡀⠀⠈⠓⠦⣄⣀⠀⠀⣀⣀⡤⠖⡽⠃⠀⠤⠀⠀⠀⠀⢀⡴⢋⡇⠀⣠⡴⠞⠉⠀⣠⡴⠋⣀⡴⠋
⠀⠀⠀⠀⠈⠛⢮⣌⠙⠒⠦⢄⣀⡀⠈⠉⠉⠉⠀⠀⠘⠧⢤⠤⢤⣤⠤⠴⠖⠋⠀⠀⠓⢛⣁⣠⠴⠖⠋⣑⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⣀⣉⠉⠛⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⠚⠋⣉⣁⣤⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡆
⣠⡶⠶⠶⠶⠶⠶⠶⠶⣿⡇⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⢸⡷⠶⠶⠶⠶⠶⠶⠶⣦⡀
⣿⠀⠀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⡀⠀⢸⡇
⣿⠀⠀⣿⠉⠉⠉⠉⠉⢻⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠏⠉⠉⠉⠉⢹⡗⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠰⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠠⠶⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠠⣴⣶⣶⣶⣦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⢀⣾⠃⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⢠⣾⠟⠋⠉⢻⣆⠀⢀⣾⠃⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⢷⡄⠀⣠⡿⠁⠀⠀⠀⠀⠙⣦⣸⠏⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠈⢻⣶⠟⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⠀⠀⠀⣤⣄⠀⠀⣼⡏⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢸⡇
⣿⠀⠀⣿⠀⠀⢠⡾⠋⠙⠻⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⣿⠀⠀⣿⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⢸⡇
⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇
⠙⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣤⣀⣤⡄⣠⠀⣠⣄⠀⢀⢀⠤⢀⠠⠀⢤⠄⠠⠄⡀⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣧⣸⣟⣁⣽⣇⣸⣀⣧⣟⣀⣆⣸⣁⣐⣀⣀⣀⣀⣀⣀⣃⣓
⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⢄⠀⡀⢲⣾⣻⣍⣯⣿⣦⣾⣤⣌⡓⣒⣠⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⣰⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣺⡿⠻⠩⠿⣯⡿⠛⠀⠀⠀⠠⠤⢀⣤⣦⣲⠀⠀⠉⠙⠻⢯⠿⠷⠶⠿⢿⣶⣤⡀⡞⡵⠚⠉⠉⢝⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣧⣶⠐⠗⠃⠀⠀⠀⠀⣤⣄⢸⣎⡙⣣⠜⣻⠠⣴⠒⡦⣀⠀⠀⠀⠀⠀⢠⡄⢙⣿⡿⣇⢿⡀⠀⠀⠀⠀⢹⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡯⡋⡷⢻⣿⣦⠀⠀⠐⣠⣦⡽⣿⣷⡿⢿⣧⠲⣌⠶⢏⠁⠀⠀⠀⠀⠀⢀⣴⢎⡴⢣⣿⡷⡁⢸⣷⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡻⡃⠑⡘⢛⣩⢈⠀⠀⠀⠸⠾⠃⠛⣀⠨⣅⡐⠛⠭⠈⠁⠀⠀⠀⠀⠀⠀⠙⠉⢀⡠⡾⣔⣻⠿⠶⠀⠀⢀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢶⣥⣬⣉⣊⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠆⠀⠀⠀⠀⡠⢤⣲⣖⣠⣥⠶⠿⠋⠀⠖⡦⢧⠀⠧⠄⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠟⠃⢀⠀⠈⠉⠙⠛⠛⠓⠿⠿⠛⠛⠖⠲⠟⠷⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠐⡦⠔⠀⣁⡤⠐⢫⡀
⠀⠀⠀⠀⣠⣮⠀⠀⢀⠰⡇⣀⣂⠀⠄⠀⠀⠀⠄⡀⠀⢠⣄⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣄⣦⡀⠀⠀⠀⠀⠠⢋⠴⢂⣉⣉⠀⠀⠀⢻⣚⣀
⠀⠀⠀⢠⣿⡇⠀⠀⠋⡐⢛⠉⠼⣣⠀⠀⠀⠀⣼⣇⡀⢈⣁⡜⣠⠤⣶⠚⣯⢣⢄⠀⡀⠀⠛⢾⠏⡀⠀⠀⠀⠀⢀⠊⠈⣋⢬⠏⠉⠀⠀⢀⠉⢹⣷
⠀⠀⠀⢾⡏⡀⠀⠀⠒⣤⣿⡛⣾⣀⢀⠁⠀⠘⠛⠻⠛⢀⣍⣤⢧⣉⡰⢇⡿⡞⠂⡞⣉⡶⠗⠛⠛⠀⠀⠀⠀⢾⣋⠈⠁⠉⠅⠀⠀⣂⢢⢾⣴⢎⡿
⠀⠀⠀⠘⢿⣿⠿⠦⠀⠈⠓⠉⠀⠁⢴⠆⠑⡀⠀⠀⠒⠁⠠⠀⠸⣿⠊⠁⠀⠉⣡⠟⠛⠇⠀⠀⠀⠀⠀⡀⠐⠯⣭⠔⠀⠀⠀⠀⠀⣤⣴⣿⠔⠋
⠀⠀⠀⠀⠈⠙⢶⣢⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⢠⢀⡤⠀⠀⣀⣈⣤⠼⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠾⣦⢶⣤⣀⣀⡀⠀⠀⠀⢫⡑⠂⠤⢤⣤⣠⣤⣤⣔⣶⣴⡬⠴⠖⢋⣝⣀⣤⣬⣭⠬⠖⢛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⡀⠉⠉⠓⠒⠒⠛⠒⠒⠶⠦⠴⠶⠒⠒⠒⠒⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⢶⣶⣤⣦⣤⣤⣤⣴⣰⣶⠤⠤⠆⣲⠖⠲⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⢀⣿⣿⣿⣿⡀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣾⣿⣿⣿⣷⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋⠀⠀⠙⠛⠛⠛⠛⠛⠛⠋⠀⠀⠙⠛⠛⠛⠟⠻⠛⠛⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠁⠀⠈⠛⠿⠛⠛⠛⠛⠛⠋

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

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣭⣭⣭⣭⣬⣍⣿⣩⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣿⣿⣿⣧⣤⣬⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡀⠄⠀⠶⠂⠖⠒⠒⠒⣶⣶⣦⣤⣄⣀⡜⠲⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡤⠀⠀⠀⡀⠠⣍⣀⣀⣤⠈⠉⠉⠂⠛⢛⡛⢿⣷⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠰⣇⠀⠀⠀⠀⣡⠀⢩⡀⠈⠹⠷⣤⣈⣁⣼⡿⢃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣒⡐⠀⠀⠀⠀⣠⣶⡀⢼⠭⢽⢟⠉⢰⣏⣹⡷⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠴⠖⠀⠉⠙⠀⠋⡿⣿⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⠶⠒⠋
⠀⠀⠀⠀⠀⢀⡔⠟⠛⠉⢛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⣶⣤⣀
⠀⠀⠀⠀⠐⠈⠀⠀⠀⣰⠻⡿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⣻⣿⠵⣺⡀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣛⣷⠋⡷⢲⣦⣧⣀⡀⠀⡀⠀⢀⢃⣠⣶⣟⢓⣏⣹⣿⣧⠆⠀⠀⣘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢋⠥⢠⠈⢉⡛⠛⠛⠿⠛⠿⠛⠻⣿⣯⣯⣯⠋⠁⠀⢀⣠⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠛⠁⠀⣀⣀⠀⠀⠀⣀⠀⠈⢀⣤⠄⢰⠿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠉⠉⠁⠀⠓⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠖⠒⠛⠛⠛⠉⠉⠙⠛⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢁⠠⢀⡂⠀⢀⠀⢀⠠⠀⠀⠐⡀⠄⠀⡀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠫⢁⠻⡄⠀⠀⣴⠷⡄⠀⢹⣎⠔⠈⡀⠤⠀⠒⠁⠈⠀⠒⠂⠀⠠⠆⠀⠀⠑⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⢁⠂⠈⠀⠹⣄⡼⠛⠀⢿⣀⣸⡏⠳⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠖⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢀⡠⠐⢀⠀⠁⢰⡴⠋⠀⠀⡐⠀⢯⠉⢷⣀⢀⣰⠿⠙⣿⠛⠛⠛⠛⠛⢻⡟⠙⣧⠄⠆⡾
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠐⢁⠔⠁⠀⣢⠞⠋⠔⠉⡠⠈⠀⡄⠘⡇⣘⡿⠞⢁⠁⠀⠈⢧⠀⠀⢀⡴⠋⠀⢠⠹⣎⡼⠁
⠀⠀⠀⠀⢀⡴⢾⡏⠀⠀⢀⠀⡘⠀⣠⠞⠁⠀⠜⠀⠄⠀⢄⠀⣨⠶⠛⠉⢀⠄⠀⡠⠀⢄⠘⣧⡴⢋⠤⠊⡠⠂⠀⢻⡛⠶⣄
⠀⠀⣠⡾⡫⠀⣼⠀⠀⠀⢊⠔⠁⣸⠃⠀⠊⢀⠀⠊⠀⠀⢠⠞⠁⠀⡀⢊⠄⠂⠀⡀⠀⣠⠟⠁⠠⠀⠠⠊⠀⠀⢂⠈⡧⢬⠊⢳⡄
⠀⣴⠋⢛⡥⢺⡇⠀⠃⠐⠁⠀⢠⡇⠀⠠⠀⠆⠀⠀⢄⢠⠏⠀⠀⢀⠀⠄⠀⠀⠀⢠⡟⠁⡠⠂⠁⡠⠁⠀⠁⡀⠒⠁⣿⠀⠑⠄⢿
⢰⠇⡰⠋⠀⢸⠃⠀⠀⡀⠀⠀⢸⠁⠔⠁⠀⠀⠀⠀⢀⡟⠀⠀⡠⢂⠔⠀⠠⣀⢠⡟⠀⢀⠀⡐⠀⡀⠀⠀⠐⠔⠀⠀⣸⠀⠀⠈⣾
⣿⣿⠀⠀⢀⡟⡠⢊⠔⠀⠀⠀⡟⡀⠠⠀⠀⠀⠀⠀⢼⠇⠠⠊⡔⠁⠀⠀⠀⠀⢸⠃⢀⠈⠀⠁⠀⠀⠑⠄⢐⠀⠀⠀⡟⠀⠀⢰⢿
⣿⠹⣄⠀⣾⠃⢐⣁⣀⣭⣍⡼⠡⠁⡀⠀⠀⠀⠀⠒⣾⠀⢀⠀⠀⠀⠐⠀⠀⠕⡾⠐⠁⠔⠁⠀⠀⠀⠀⡀⠀⠀⠀⣸⠃⠀⣰⠏⢸
⢹⣆⠘⠷⣝⡛⠋⠁⠀⠀⣸⠃⡀⠐⣀⣂⣈⣁⣈⣠⡇⢀⠄⡐⠀⠀⢀⠀⠀⢰⠇⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⢀⡼⠃⢀⡴⠋⢠⡏
⠀⠹⣦⠀⠈⠛⠶⣤⣀⠀⠹⠴⠟⠋⠉⠉⠉⠉⢀⡾⠂⡐⠈⠐⠒⠒⠒⣁⣰⡏⠈⢠⠂⡀⠀⠄⠀⠀⠀⣀⡴⠋⣀⡴⠋⠀⣰⠏
⠀⠀⠈⠳⣤⡀⢀⠠⣉⢛⠳⢦⣤⣀⠀⠀⠀⠀⠈⠳⠶⠛⠋⠉⠉⠉⠉⠉⣿⢀⣢⡤⠤⠶⠶⠶⠒⣚⣫⣥⠶⠛⠉⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠙⠲⢵⣦⡍⡁⠀⠀⠉⠛⠳⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣈⣭⣥⡤⠴⠶⠖⢛⡛⠉⠁⢀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣒⣬⡵⠷⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠒⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠈⠉⣹⣋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡟⣀⠔⠊⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢒⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢐⣀⣩⣤⠤⠄⣤⣀⣀⠀⠠⡄⠤⢤⣀⡙⢅⠢⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣀⡤⠖⠋⢩⠏⠁⠀⠀⠀⠀⠀⠈⠉⠒⢼⡳⡄⠈⠉⠓⠧⠀⢠⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢨⠁⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢸⡄⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⠀⠀⠀⠈⠀⠀⠙⡆⠀⠀⠀⠀⠀⢸⠋⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠙⠦⡤⠖⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣄⡀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠈⠑⠲⠦⠴⠖⠋⡁⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠄⣤⣠⡤⡤⠼⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠒⡈⠃⠛⢐⠒⠂⠀⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡀⠀⠀⠻⣄⡔⠁⠀⠀⠀⡄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⣀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⣇⣀⡴⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⡔⠀⢨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢧⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡉⠂⢀⠠⠒⠈⠁⡴⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠊⢁⡠⠤⠗
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⡀⡄⠤⠴⠒⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢊⣡⢄
⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣎⡵⠊
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠤⠤⠤⠤⠶⠶⠒⢻⡉
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⣢⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠉⠙⠒⠐⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣠⣤⠤⠼⢰⣤⣤⡤⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠀⣸⠀⢸⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠁⠀⠀⠘⠢⠚⠡⠀⠀⡘⠀⠙⠋⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⣀⠀⠀⣀⣀⣀⣤⠴⠖⠒⠃⠀⠑⠦⣤⣀⡀⠀⠀⠀⠈⢢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⣛⣛⣒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⢉⣀⣈⣀⣀⡉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠭⣭⣭⣬⣭⢭⣭⣬⣭⠭⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⠈⡍⠩⢭⠉⢩⢯⠉⣩⠭⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢠⠊⠀⠀⠀⠀⠈⠿⢫⡀⠀⠀⠀⠀⠀⠀⢰⠰⠈⡇⢸⢸⠀⡇⡄⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠘⢿⢿
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢸⢸⠀⡇⡇⡇⠀⠀⠀⠀⢠⡦⠤⣤⠀⠀⡜⠀⠀⣀⣾⣿⣀⡀⠈⢧
⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⢸⢰⠀⡇⢸⢸⠀⡇⠇⢇⠀⠀⠀⠀⠀⠷⢎⢧⠀⠀⢇⠀⠸⠿⣿⣿⠿⠇⠀⢸
⠀⠀⠈⢣⡀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⡏⡜⢸⠀⢸⢸⠀⣇⢰⢸⠀⡆⠀⠀⠀⠀⠈⣸⠀⠀⠘⢆⠀⠀⠛⠛⠀⠀⣠⠏
⠀⠀⠀⠀⠈⠓⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⢸⢣⠃⡜⠀⡞⢸⡄⢸⠈⡌⡇⠙⠦⣄⣀⡤⠔⠁⠀⠀⠀⠈⠑⠦⠤⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣌⠚⠁⠀⢧⣼⠃⠀⠓⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠓⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠖⠒⠒⠒⠒⠒⠒⠒⢤⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠠⣤⣤⡀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠀⠀⠀⠀⠀⢀⡼⠒⠒⠒⠲⡄⠀⠀⠀⠀⣧⠇
⠀⠀⠀⡜⠁⠀⠀⣠⣤⠜⠛⣦⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠀⠀⠀⡖⠦⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠳⠴⠶⠶⠼⣟
⠀⠀⢸⠀⠀⠀⢀⣽⣷⣴⠃⠸⡆⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠘⡆⢠⣾⠟⠁⠀⠀⠀⢸⠃⠀⠀⠀⠀⢸⠀⠀⠀⠛⠦⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠙⢦⡀⠀⠀⠀⣀⡴⠋⠀⠀⢀⡠⠄⠊⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠂⠤⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠇
⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠎⣁⣀⠀⠤⠤⠤⠤⠄⠀⠀⠠⠤⠤⠤⠤⠀⣀⣀⡹⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⠰⡁⢠⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⢁⡤⠤⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠢⠀⠀⠀⠀⠄⡀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠊⠉⣀⠘⣢⠲⣐⠀⠀⠈⠀⠀⢀⡇⠈⠀⠉⡇⠳⠤⣀⠔⢺⠉⠡⠀⣀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠗⠒⠒⠲⡟⠀⠀⠀⠀⠀⢸⡇⢠⠀⠀⠇⠀⠀⠀⠀⢸⡄⡄⡞⠉⠉⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠯⣝⢉⣁⣹⠀⠠⠀⠠⠄⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⡇
⠀⠀⠀⠀⠀⣀⠀⠀⠙⡂⠀⠇⡄⣀⠀⠀⠀⠀⢠⠇⡇⠀⢀⣤⡀⠆⠙⠀⠀⠁⠀⢱⠤
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠰⡓⢬⣀⡀⠀⠰⣊⣰⠁⠀⠈⡈⢳⡄⠀⠀⠠⠄⣱⠈⡱⠢⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠌⠭⢭⣭⠏⠜⠀⠀⠀⡏⢶⢶⠶⠶⢚⡋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠁⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠠⠤⡀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⢸⠁⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠈⠓⠄⡀⠀⢧⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⢀⡈⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⠟⠃⠀⠰⠟⠛⠛⠦⡈⠐⣄⠃⠀⠀⠂⠠⠆⠀⠀⠀⠀⠀⠞⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠽⠀⠀⠻⠥⠀⠀⠀⠀⢹⡆⠀⠀⠀⡨⢄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⢀⠁⠄⠀⢀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⠀⢠⠀⣀⣀⣤⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡄⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢊⠄⢁⠀⠀⠸⠀⠀⠐⠿⢿⣛⣛⢻⡷⠖⠀⠀⠀⢸⣷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡀⠀⠀⣠⠜⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⡠⠚⠀⢨⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⢦⣤⡤⠔⠊⠀⠀⡠⠋⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠐⣾⡇⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⠙⠲⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠐⢊⠁⡘⠃⠀⠉⠿⠦⣤⣀⣀⢐⣐⠧⠀⢀⡴⠊⣴⠁⠀⠀⢉⢑⠖⠦⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⢋⡜⠁⠀⠀⣠⠀⠀⠀⠀⠀⠉⣉⠉⠉⠆⠁⣸⠏⠀⡆⢰⠃⠏⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠐⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⣦⡿⠀⠈⢀⡟⣀⣔⢀⡀⠀⢰⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⣸⣿⢣⡇⢠⢾⠇⣿⠟⠀⠀⢠⣤⠑⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠰⠙⣧⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢡⠏⠞⢸⣠⠇⡜⠸⢻⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⠀⢸⠁⠀⠀⢸⡟⠀⠃⠀⢿⠀⣠⠄⢠⣤⠀⠈⠢⠀⠠⠂⠈⣁
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠃⠐⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⣸⠃⠀⠀⠀⠈⠣⡄⢰⡿⠋⡁⠀⣤⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢰⢸⠇⡀⠳⡄⠘⠧⡀⠀⠀⡈
⠀⢀⡆⠀⠀⠀⠀⠀⠀⢰⠀⠘⡷⠸⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⢀⡄⠀⡇⠀⠀⠘⠀⣰⠀⠀⠀⠀⢠⠂⠀⠁⠷⠄⢁⡀⡀
⠀⡇⠠⠀⠀⠀⠀⢀⠔⠠⠁⡤⠃⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⢐⡅
⠀⢱⠐⠀⠀⠀⡰⠃⠀⡴⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢼⡇⠀⠀⠀⠘
⠀⠀⢣⣄⠀⠈⠀⣠⡌⠔⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠁
⠀⠀⠀⠐⠢⠔⠊⠁⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠰⠆⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⣄⠀⠀⠀⠀⠀⣠⡖⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠁⠉⠉⠉⠉⠉⠊⠀⢀⡇⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⣄⠀⠤⠄⠀⣤⡀⠀⠀⡼⠑⠒⠤⠖⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢍⡓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⣳⡄⢆⣰⣆⡠⠀⣒⡾⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡌⢢⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣠⠓⢤⣀⣀⣠⠴⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢱⡀
⠀⠀⠀⠫⡉⠈⠀⠀⠀⠀⢹⡀⠀⠈⠉⠀⠀⠀⠀⠀⡔⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣠⣷⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠈⡇⠀⠀⠀⣞⠒⠒⠛⠒⠒⠲⠤⠤⠤⠈⢧⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠳⣄
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠻⠦⠴⠒⠒⠒⠒⠒⠒⠒⠒⠮⠷⣄⡀⠀⠉⠒⠦⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠄⠐⠒⠒⠀⠠⠤⠤⠤⠤⠄⠐⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠔⠒⠊⠉⠉⠉⠉⠒⠒⠲⠤⠤⠤⠖⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠛⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣄⠀⠀⠀⣀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⠀⠉⢦⠀⡼⠉⣧⠀⢿⢮⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⢀⠀⢀⣨⠟⠀⠀⠘⡗⠻⡄⠈⢉⡶⢶⡶⠦⠤⠤⠶⣶⠛⣟⡉⢄⡟
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠠⠈⠀⠀⢠⡴⠋⠑⠀⠀⠀⠀⢹⡄⣻⠶⠋⠀⠂⢳⡀⠀⢀⡴⠋⠀⠹⣆⡞
⠀⠀⠀⢀⡴⢺⠇⠀⠀⠀⡀⢀⡴⠋⠀⠈⠀⠀⠀⠀⣨⠞⠋⠁⠀⠀⠀⠀⠀⣳⠔⠫⠀⠀⠀⠀⢻⠛⡦⣄
⠀⢀⣾⡿⢀⡟⠀⠀⡀⠀⠀⡞⠀⠀⠀⠀⠁⠀⢠⠞⠀⠀⠊⠐⠀⠀⠀⣠⠞⠁⠐⠀⠀⠀⠀⠀⠈⡗⠊⠘⣦
⢠⠏⡠⠊⢰⠇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢢⡏⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠊⠀⠀⢀⠀⠀⣧⠀⠀⢸
⣾⡔⠁⠀⡸⢀⠄⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⣿
⣿⢧⡀⢰⠇⢀⣀⣀⣀⣰⠣⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣼⠐⠁⠀⠀⠀⠀⢀⠀⠀⠀⣰⠃⢀⡼⢱
⢹⡄⠙⢮⣛⠉⠁⠀⢰⠇⠀⣈⣀⣀⣀⣠⣼⢃⠀⠀⠀⠀⢀⠀⢠⢇⠀⠂⠀⠀⠀⠀⠀⠀⢀⡴⠃⣠⠞⢁⡞
⠀⠻⣆⠀⠉⠓⠦⣤⣈⠓⠋⠁⠀⠀⠀⢠⡟⢀⣀⣠⣤⣤⣤⢤⡏⠀⠐⠀⠀⠀⠀⣀⣠⠴⣋⡤⠞⠁⣠⠏
⠀⠀⠈⠙⢦⣤⣐⠠⠏⠙⠳⠦⣤⣀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠘⠤⠖⠋⣉⣉⣩⡭⠵⠖⠋⠉⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠈⠙⠻⠶⣄⣀⡀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠡⢐⣒⣦⠴⠖⠋⠁

⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⡇⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠐⣿⣛⣛⠛⠛⠛⠛⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠠⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠉⢶⣄
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄
⠀⠀⠀⢸⡷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡿⠖⠒⠀⠀⠀⠀⢶⣤⡀⠀⠀⠸⢻⡄
⠀⠀⠀⢸⠀⠈⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠁⠀⣿⣿⠀⠈⢹⠿⠷⢶⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢹⡀
⠀⠀⠀⠀⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⡄⠀⠀⢸⡇
⠀⠀⠀⠀⣷⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠈⠛⠿⠁⠙⠟⠙⠥⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡇⠀⠀⠘⡇
⠀⠀⠀⠀⣿⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⠀⠀⡇
⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣧⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣭⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣄⣀⣄⣀⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡎⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡛⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⢸
⠀⠀⠀⠀⠁⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⣸
⠀⠀⠀⠀⣶⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠘⣫⠙⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⣈
⠀⠀⠀⠀⣿⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⢢⠟⣻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣷⠀⠀⢀⡏
⠀⠀⠀⠀⣿⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⢐⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠇⡀⠀⠘⡇
⠀⠀⠀⠀⣿⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠁⡀⣾⠃
⠀⠀⠀⠀⠛⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⢻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣇⣸⣿⠰⣶⣶⣶⣏⣁⣠⠖⠀⠀⠀⠀⣀⡜⠋
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣿⡀⣿⡿⠟⠋⠁⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣾⣏⣿⡇⣿⠃⠀⠀⠀⢀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠰⡿⣿⣻⣇⣿⣤⣠⣤⠞⠋
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⢼⣦⣿⣿⣿⢀⠟⠁
⠀⠀⠀⠀⢸⣾⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢄⣀⣠⣿⣿⣿⣿⠈
⠀⠀⠀⢀⣌⠿⣿⣿⣧⣔⣀⣀⠠⣄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣤⣄⣱⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠙⠻⠶⣭⣝⠛⠿⢿⣿⣿⣶⣾⣿⣿⣷⣦⢠⣦⣤⣴⣦⣤⣴⣿⣤⣦⣼⣿⠿⠟⢛⣩⣶⣿⣿⠁
⠀⠀⠀⠀⢶⣤⠀⠀⠈⠉⠛⣶⣦⣤⣭⣍⣉⣉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⣿⡧
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠏⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠀⠀⠀⠀⠀⠉⢁⣀⣀⠠⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠉⠉⠛⢶⣄⡀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢻⡇
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢳⡀⠀⠀⠀⠀⠀⠉⠳⢌⠀⣠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠹⠃
⡼⠁⠀⠈⢆⠀⠀⣀⡀⠀⠀⠀⢶⠘⠻⣆⠀⠀⢀⡤⠂⠀⠀⠙⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠂⠀⠀⠀⠁
⢷⡀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⣤⣀⣄⣦⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒
⠈⢷⡀⢦⠀⣈⣆⠀⠀⠀⠀⠀⢰⣲⡀⠀⠀⢠⣤⣄⠀⠀⢄⡀⠀⠀⢘⠟⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣠⣤⡶⠏⠀⠀⠀⠀⠀⣄
⠀⠈⣷⣾⣇⠹⣿⣧⠀⠀⠀⢀⣿⢿⠆⠔⠛⠃⠊⠚⠻⣶⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⠛⢉⣡⣴⠀⠀⠀⣀⡤⠞⠃
⠀⠀⠘⣿⣿⣆⠹⣿⣆⠀⠒⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠉⠉⠉⠫⠶⡀⠀⠀⠀⠀⠀⢷⣠⣶⣾⣿⣿⣿⠶⠚⠋⠁
⠀⠀⠀⠘⣿⣿⣦⡟⡿⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠛⠛⠃⠀⠸⣍⠀⠀⠀⠠⣙⣦⣄⠀⠀⠀⢀⣀⡀⢠⠤⠖⠁⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠳⢦⣄⣠⣜⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠹⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣤⡀⠀⠀⠀⠀⢀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠙⠻⣿⣶⣌⠙⠿⣿⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢣⠀⠈⠻⢿⣷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠹⢦⣄⣠⣀⡤⡤⢒⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠙⠾⣿⣷⣴⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⠀⠀⠠⠠⠤⠤⠄⠤⠤⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠂⠈⠌⡁⠀⠰⠀⠀⠐⠋⠉⠈⠀⠀⡤⣀⣄⡀⢔⡂⠚⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠠⠀⠋⠀⠀⠀⢄⠀⠘⠛⠀⢀⠀⣐⡷⠘⠋⠀⠀⠄⣈⣄⡈⠉⠁⠀⡀⠀⠀⢠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠲⠀⠀⠀⠀⠀⠈⠀⠀⢀⣤⠀⡀⠌⣈⢡⢶⠅⠘⠟⠛⠁⠘⢶⠂⠑⠉⠀⠖
⠀⠀⠀⠀⣴⡄⠀⠀⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢰⠈⢋⣁⣀⣰⣚⡉⠛⠁⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠠⠀⠁⠀⠁⠀⠐⠾⢃⡁⠒⠃⠀⢄⠀⠀⠀⠀⠀⠊⠁⠀⠈⠁⠀⠀⠀⢉⠀⢀⣀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠁⠠⠁⠐⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠈⠁⠀⠀⠈⠀⠠⢄⠄⠀⠐⠁⢈
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢅⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡏⠉⠉⠛⠛⠲⠶⠶⠶⣶⣤⣤⣤⣤⣤⠄⠀⠰⠀⠦⠀⠐⠶⢶⣶⣶⠒⠒⠲⣶⣶⣶⢶⣶⣶⣶⣶⣴⣤⣦⡴⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠤⡁⣀⠀⣈⣩⣨⡶⠿⠛⠛⠉⠏⠀⠀⠠⠀⠀⠀⢚⠁⢘⣄⣄⣀⢠⣌⡇⢀⣀⢠⣿⣍⡉⠉⠉⠉⠈⠉⠉⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠉⢛⠙⠀⠀⠁⠐⣤⣤⣄⠀⠆⠐⠙⢄⣩⡤⢯⡟⢢⠉⣉⣉⣉⣿⣿⣓⠒⠒⠤⠤⠤⠤⠄⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⣴⡧⡖⠀⠂⠃⠀⡀⠀⠰⠚⢿⣿⣿⡷⠊⠀⡀⠀⣤⣬⣏⣆⣩⣈⣁⡀⠈⠉⠉⠉⠁⠉⠀⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠈⠀⢁⠉⠀⠀⠠⣴⣿⣈⠁⠐⠊⠥⢷⢳⡞⢛⣻⣿⣿⣿⣧⣾⣇⣀⣼⣆⣭⣏⣛⣛⣻⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠈⡀⠀⠈⠁⠀⠠⢟⣳⣌⣹⡿⠏⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠋⠀⠉⠉⠉⠀⠀⣀⠵⢿⠫⠹⣟⣿⢿⠶⠒⠂⠀⠐⢉⠃
⠀⠐⠿⠏⠰⡀⠀⠢⣈⡇⢫⣫⠀⠀⠁⢀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢺⡄⠀⠃⠀⣞⠿⡉⠀⠀⠀⠀⢀⠀⠈⠠⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠻⢣⠍⠏⠐⠀⢀⣰⣶⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠏⠡⡶⠅⠐⠀⠄⠱⢂⡀⡀⠀⠐⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⢁⣀⡅⠀⣯⢹⠻⡟⢦⡀⠀⠀⠀⠀⡠⠎⠁⠀⠉⣒⣀⣀⡀⠲⡆⠀⢨⠶⠤⠟⢀⣘⣚⡳⠆
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈⠛⠿⠷⠜⠁⠀⠀⠆⠑⢄⠀⠀⠀⢴⠄⠑⠋⠀⠀⠀⠨⣉⠀⠀⠀⠀⠀⠀⠀⠀⢉⠙⠂⠀⠀⠠⠁
⠀⠀⠀⠀⠄⠀⠀⣰⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⢔⣤⣶⣶⣶⣶⠲⠴⠖⠒⢲⢶⣦⣤⣠⡄⠤⠐⠠⡆⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠐⠈
⠀⠀⠀⠀⠀⠈⠒⠼⢤⣀⡀⣀⣀⡠⣄⠀⠀⣀⠉⢚⣛⣓⠛⠒⠒⠒⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⠀⣰⢀⣀⡀⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠒⠒⠛⠁⠁⠒⠒⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉

⣿⣿⣿⡯⢘⠀⡏⠉⠀⣈⣇⠉⡎⣿⢑⣸
⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠀⠁⠀⠁⠈⠉⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠎⠿⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠈⡙⠒⠂⠀⠠⡄⠀⠀⠀⠀⠀⠜⠀⠀⠂⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠈⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⢄⠘⠗⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠉⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠐⠀⠀⠀⠀⠀⠈⠦⠠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡄⠀⠀⡀⠐⠊⠀⠀⢡⡀⠀⠀⡀⠀⠆⠀⠀⠄⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠢⣄⠈⣟⠡⠎⠀⠀⢸⠀⠀⠁⠀⠀⠀⡧⠀⠀⠘⠀⠀⠘⠃⠀⠰⠂⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢁⠕⠀⠀⠈⠓⣂⣄⠠⠄⠀⠂⠀⠀⠀⢄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⣀⣀⡀⠒⠀⠀⠀⠀⠀⠈⠉⠒
⠀⠀⠀⠀⠀⠀⠀⢹⠒⢌⠀⠀⠀⠀⠀⠀⠀⣠⠤⠈⡁⠀⠀⠀⠀⠀⠀⠀⢐⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢣⡄⠔⣹⠢⠀⠀⠀⠀⣀⠠⠊⠀⢠⠾⠒⢳⡀⠀⠀⢀⡴⠈⠈⡦⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠉⠰⠛⠓⠉⠀⢄⠀⠀⡟⢁⡄⠀⠀⠀⠀⢚⠳⠀⠀⡁⠠⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠊⡠⠀⢀⡇⠀⠈⠃⠀⡇⠘⠿⠝⠀⠀⠀⠔⠀⢁⠜⠀⠀⠀⠀⠸⠀⢀
⠀⠀⠀⠀⢈⡀⠀⠁⠀⠀⠀⢀⡀⠃⠀⠃⠒⠀⠛⠋⠉⠀⢁⠀⠀⠀⠀⠀⠠⠀⠈⠀⠘⠃
⠀⠀⠀⠀⠁⠉⠀⠤⠀⠠⠜⠈⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⡀⠀⣀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠋⠀⠈⠃⠀⠀⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠘⡄⠀⠀⠀⣀⣀⣀⠀⠂⠀⡰⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈⠢⣄⡀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠉⣭⡉⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⡟⠈⢆
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⠀⠀⠀⢤⡀⢀⠀⢨⡄⠀⠀⢂⢀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡇⠀⠆⢸⠁⠀⠀⠰⣿⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠐⠒⠤⣐⢕⠠⡀
⠀⠀⢠⠠⠤⠤⠤⠤⠤⠄⠀⠐⠒⠀⠒⠒⠒⡄⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⡀⠈⠑⢕⡌⢂
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠁⠀⠀⠈⢫⠀⠃
⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢁⠀⠀⢀⣀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠈⠐⠢⠤⣀⡀
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⠜⠀⠀⣖⡄⠁⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠠⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⢀⡀⢀⣀⡀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠈⢎⠆⠀⠀⠀⠉⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⡒⡳⠓⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠅⠚⠙⠣⠴⣶⠪⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢗⠉⠑⠿⠤⣄⢈⢀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠸⠛⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠂⢀⡔⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⡠⠞⠉⡅⠀⣠⠞⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⠀⠀⠀⠈⠙⠒⠤⣀⣀⡾⠀⠀⠀⠈⠙⠁⠀⠀⡠⠕⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⣠⣶⡟⠓⢄⡀⠀⠀⠀⠀⠀⠀⡍⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⢄
⠀⠀⠀⠀⣀⣤⠶⠋⠁⢀⢤⠂⠀⠙⠋⠐⠠⡎⠈⠉⠉⠁⠀⠀⠾⢁⠃⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠈⢦
⢀⣠⠴⠚⠉⠀⠀⠀⠀⠈⢈⠢⡀⠀⠀⠀⠀⠈⢣⡒⠤⣀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠑⢌
⠻⣧⡀⠀⠀⠀⠀⠀⠀⠠⡊⠑⢌⠢⡀⠀⠀⡀⢸⡁⠀⠈⣡⠾⢻⢀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡇
⠀⣿⠙⠷⣄⠀⠀⢠⠀⠀⠘⣄⠈⠢⠟⣖⠭⠒⢉⣠⣶⣋⡥⠴⡇⠀⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠀⡸⠁
⠀⣿⠀⠀⠈⠻⢦⡈⠘⡄⠀⠈⠢⠖⠉⣀⡤⠞⠋⠁⠀⠀⢀⡄⠈⠒⠤⣀⠀⠀⠀⠀⢀⠀⠀⢀⠞⠁⠐⢀⠞
⠀⣿⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠀⣰⠟⢴⢡⠄⠈⠧⠚⠁
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠶⠋⢹⡆⠀⠀⠐⢼⡶⠋⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠋⠁⢀⡸⣃⡠⠎
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣸⠀⠀⠀⢀⣤⡤⠖⣾⠛⠒⠒⠒⢶⠚⢻⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⢹⡀⠀⠀⠀⠙⡆⠀⣿⠀⠀⠀⠀⢸⡇⠸⣿
⠀⢹⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⡇⠀⠀⠀⡏⢳⡀⠀⠀⠀⠹⡀⣿⠀⠀⠀⠀⢸⡆⠀⣿
⠀⠘⢧⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⢳⡀⠀⠀⠀⢳⡟⠀⠀⠀⠀⢸⠁⠀⣿
⠀⠀⠀⠻⣄⠀⠀⠀⠀⢸⠀⣀⣠⡷⠤⠴⡖⠃⠀⠀⢳⡀⠀⠀⠈⡇⠀⠀⠀⠀⠚⠂⠀⢹⡄
⠀⠀⠀⠀⠈⢷⡀⠀⠀⢸⠋⠉⠀⣀⣀⣀⡗⠀⠀⠀⠀⢳⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠸⠷
⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⠤⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣶⣶⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣠⣤⡤⣤⣄⣀⣤⢤⡤⢶⣤⣀⣀⣀⣤⠤⠤⡶⢤⣤⣄⣀⣠⣤⣴⠶⠦⣤⣀
⠀⠀⠀⢀⡴⠋⠀⠀⠳⠤⣤⣤⠤⠞⠀⠘⢆⡀⠀⣀⣀⡤⠞⠁⠘⢦⣀⡀⣀⣠⠜⠀⡤⡀⠈⠳⣄
⠀⠀⠀⣼⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠀⠀⠀⠈⠉⠉⢀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⣴⠛⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⢲⠲⠒⠒⠒⢒⡒⠒⠛⣉⣭⣍⡉⢉⣭⣍⣉⣉⣉⣀⣠⣤⣉⡗
⠀⠀⠈⣛⣿⣙⣲⣒⣉⣶⣖⣟⣶⣖⣉⣉⣓⣶⣫⣭⢯⣉⣭⡤⠤⠤⠽⠧⠤⠤⠤⠤⠤⠧⠤⠖⡟⣶
⠀⠀⠀⣏⠓⠒⠊⠓⠤⠴⠒⠲⠒⠊⠓⠒⠒⠛⠒⠒⠛⠉⠉⠈⠙⠋⠙⠋⠉⠙⠋⠉⠑⠒⠊⠉⢀⡏
⠀⠀⠀⠙⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠞
⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣴⡶⠿⠿⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⢯⣾⡇⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣭⣾⣿⡇⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣕⣿⣿⣬⣺⣿⠿⠿⢿⣿⡿⠿⢿⣿⣿⣿⡇⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⣛⣭⣭⣭⣟⠛⠟⠿⠿⠿⠿⠿⠿⠻⠟⠺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⣿⣿⣿⣷⡄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢾⡷⣹⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣾⣿⣿⣿⣿⠃⠉⠉⠙⠿⠁⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠁⠒⢀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⠠⠀⠀⠀⠀⠑⢀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠄⢀⠄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠄⡀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠠⠤⠄⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡠⠡⠂⠁⠠⠂⠀⠀⠀⢀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠎⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠄⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠑⠠⢀⠀⠀⠀⠀⢀⣀⠠⠄⠂⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⠀⠤⠄⠀⠐⠈⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣲⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠩⠽⠔⠒⢒⣒⣒⣒⣒⣂⣉⠩⠤⠐⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⣶⡒⡶⣲
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢻⡶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⣼⣇
⠀⠀⠀⠀⠀⠀⣠⢟⠁⣿⣷⡙⢿⣄
⠀⠀⠀⠀⠀⡼⢡⠁⠈⣿⣟⢱⠈⢻⣦
⠀⠀⠀⠀⠀⡇⢸⠀⣷⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠉⠉⠉⠙⠧⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠂⠀⠀⠈⣀
⠀⠀⠀⠀⢸⡧⠀⠀⠀⠠⠀⠀⠀⠖⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠰⣿⣄⣀⡀
⠀⠀⠀⠀⢸⠁⠀⠈⠁⠀⠀⠈⠀⣤⢹⡇⠀⠀⠀⠀⠀⣿⠀⣶⡖⣲⣶⠀⣶⣶⣿⠿⠛⠙⢹⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠀⠀⠀⠉⢾⡇⠀⠀⠀⠀⠀⣿⠀⣿⠁⠸⣿⢠⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⠀⠀⠀⠁⠔⠂⠀⠀⠠⢸⡇⠀⠀⠀⠀⠀⣿⠀⠃⠀⠀⠀⠸⣿⣯⣿⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢅⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣉⣽⠇⠀⠀⠀⠀⠀⣷⠀⠀⠆⠀⣁⢰⣤⢿⣿⠿⠇
⠀⠀⠀⠀⠀⡏⠀⠀⣶⣤⣤⣴⡆⣸⣿⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠺⢺⣽⣿⣿
⠀⠀⠀⠀⠀⣧⣤⢀⢿⣿⣿⣾⣧⣿⡿⠀⠀⠀⠀⠀⣰⡿⠛⠓⠒⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠈⠉⠓⠀⠀⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⣉⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⢛⣡⡴⠞⠋⣁⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⡴⠞⣋⣴⢞⣋⣥⠶⠞⢛⣉⣤⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠛⠛⠛⠉⠙⠳⠶⣦⡶⢛⣭⣶⣿⠿⢛⣋⣭⡴⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣠⡄⠀⠀⠀⢀⣀⣤⡼⠟⣛⣩⣥⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢸⣏⣠⣤⠶⣞⣫⣭⡶⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣷⣶⣿⣿⠿⢛⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⠉⣁⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠶⠶⠶⠒⠛⠛⠛⠛⠒⠶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣍⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⣩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⢉⣭⣭⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠉⢹⣿⣿⠉⠉⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⡀⠀⠀⠑⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡄⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢀⣀⣀⢨⡤⣀⣀⣖⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⡀⠀⠀⠀⠳⠇⠀⠈⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠘⠋⠋⠈⢀⠲⠘⠚⠁⠈⠙⠃⠀⢠⠈⠀⣤⣤⠀⠓⠦⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠤⣤⠄⠤⡤⠤⠰⠦⠄⠀⢀⢀⣤⠀⠙⢀⢷⠀⠰⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠙⢱⠀⠀⠈⠀⠄⠀⢀⠀⠀⠸⠀⠀⠀⠀⢉⢠⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠯⠉⠉⠉⠉⠋⠉⡋⠁⠀⢀⠀⠲⠟⠀⠸⠆⠰⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣼⣤⣀⣀⡀⣀⣀⣩⡀⠀⠘⠓⠒⠂⠚⠁⠀⠀⠀⠄⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠁⡀⢀⠀⠀⠀⠀⠀⢀⣠⡀⠠⡜⠋⠀⠠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠴⠶⠤⠤⠤⠮⠧⠤⢒⠄⠀⢈⠉⠀⣀⡉⠈⡧⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠞⠀⡀⡀⠀⠠⠖⠀⠀⠀⡐⠬⠀⠈⠉⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠐⠂⠉⠁⠀⠀⠐⠒⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁





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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠠⢌⡑⠂⢄⠀⠀⠀⣀⣤⣤⣶⣿⣿
⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⣂⡐⠲⠿⣓⡀⢄⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⣾⡿⣻⣡⣿⣾⣿⣿⣿⣿⣟⠿⣏⠟⠻⠛⠛
⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠂⠀⠐⠀⠀⣠⠀⠀⠀⠀⠀⠀⢺⢷⡶⠿⠻⢛⠟⣹⣟⢿⡻⣌⢣⠙⣄
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠅⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢠⣾⣿⣿⢦⠳⡈⢀⢡⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠠⠀⠀⠀⠰⣟⡻⠯⡹⠽⠖⢱⡀⠀⠃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣐⠀⢀⡀⠄⠀⠀⠀⠂⢀⠀⠀⠀⠀⠒⠉⠉⠁⠂⠀⠀⠀⠀⠀⠚⢸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣨⣷⣄⡀⢤⡁⠀⠁⠀⠀⢈⢲⣴⣋⠀⣀⠀⢀⠀⣀⠀⣀⣀⣀⣀⡀⡉⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠠⠾⠻⠙⠋⠦⠈⠑⠈⠙⠭⡙⠽⡿⣿⢯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠈⠑⠀⠀⠀⠀⢸⢦⡀⠀⠑⠤⠑⢌⠛⡾⠁⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⡀⠠⠄⡒⠈⠁⠀⠀⡀⠀⠀⠀⠀⠏⡙⢦⣀⠠⠈⠢⠈⠃⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠌⠷⣶⡾⠁⠀⠀⠀⠔⠀⢞⠀⠀⠀⠀⠁⠈⢻⣷⣄⠐⠄⠐⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠋⠀⠀⠀⢀⠄⡠⢁⣾⠀⠀⠀⢸⠀⠀⣛⣷⡍⠳⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢁⡴⠠⣿⣿⠁⠀⠀⠈⠀⠛⡡⢋⠔⢁⠄⠉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⠄⠀⠒⠀⠈⠒⠀⠉⠑⠀⠀⠀⠀⠘⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠃⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⡀⣠⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠄⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡹⡆⠀⠀⠀⠀⢠⡀⠀⣠⠄⠶⣇⠤⡨⠂⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢀⠴⠚⠋⢙⠀⠀⠀⠈⠀⠀⢗⣆⠀⣨⠅⣀⠄⠀⠀⠀⠀⠀⢠⠏⠸⠄⠈⢊⠀⣄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⢀⠠⠴⠚⠋⠀⠀⠀⠀⠢⠀⠈⠛⠫⠀⢂⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢠⣎⣠⠄⢋⣘⠁⣄
⠀⠀⠀⠀⠀⠀⣀⣠⠄⠴⠋⠀⡀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣀⠀⠀⡤⢯⡇⠈⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠰⠛⣉⠔⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠦⠰⠚⠁⠸⣏⠲⠀⠀⠀⢘⡇
⠀⠀⠀⠀⣰⣎⣁⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⣒⠂⠀⠀⠈⠀⠀⠀⠐⠒⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡔⢢⣤⠛
⠀⠀⠀⠀⠙⢿⡧⠊⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⠭⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠳⣄⡀⢀⠠⢄⡤⠤⣄⣠⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⣀⣴⡯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠘⡆⠑⡶⠉⠑⠂⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠍⠚⡅⢰⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠙⠷⠚
⠀⠀⠀⢠⣍⣤⠞⡙⠠⡤⡌⠄⠀⠄⠀⠀⢱⠘⡧⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⢣⠏⡠⠊⠙⡆⢳⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠚⢣⡼⠲⠔⠼⠀⠀⠀⠀⠀⢃⢧⠈⠑⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⡀⠤⠤⠒⠊⠉⢀⣮⠏⠉⠁⠀⠜⢁⠌⠀⠀⠀⣠⠘⢃⠀⠙
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣟⡠⠐⣀⠄⠂⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⣡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣠⣞⠋⠉⠉⠉⠉⢉⣁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣟⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢹
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⣀⣈⣉⣢⣶⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣫⡄⢸
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠛⠋⢉⡴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⢐⡇⢸⡆
⠀⠀⠀⠀⢀⡃⣀⣀⣀⡰⠂⠀⢠⠎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⢻⣟⠁⡈⠀⠈⡇⢸
⠀⠀⠀⠀⠈⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣿⣷⠿⣡⣶⡽⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢠⠈⠁⠱⣈⠳⡴⢁⡎
⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡹⢉⣺⣷⣤⣄⠀⠀⠀⠀⠴⣒⠉⠡⣀⣆⠀⣬⣀⠀⣲⠿⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢸⠅⠒⠀⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⣵⠿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣉⣉⣀⣉⣉⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⠻⠯⠤⠤⠤⠤⠭⠭⠭⠽⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣭⣙⣒⣒⣒⣒⣒⣒⣒⣒⣚⣩⣬⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠁⡏⢹⠉⡏⢹⠉⡏⢹⠉⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣷⠾⣦⣧⣼⣤⣧⣼⣤⣧⣼⣤⠷⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠶⠶⢶⡶⠶⠶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⠻⠶⠶⠶⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠀⠀⠀⠀⢀⡴⡖⢶⡄
⠀⠀⠀⠀⠀⢀⡴⠶⣤⣤⣄⡀⠀⠀⣿⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⣿⡀⠀⠀⣴⠟⢮⣁⣼⡇
⠀⠀⠀⠀⠀⢸⣄⣀⣸⠐⠚⣻⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⠆⠘⣧⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠋⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⢶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣉⣸⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⠿⢿⣆⠀⣀⣰⣿⠿⢷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠘⠿⠟⠛⠿⠃⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⢀⣴⡿⠃⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠘⢿⣦⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⠿⠋⠀⠀⣴⡿⠟⠋⠉⠉⠙⠛⢿⣶⣄⠀⠙⠿⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠃⠀⣾⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠘⠿⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⣿⡿⣿⣍⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣴⣶⣄⠀⠈⠻⣷⣦⣤⣤⣤⣤⣴⡾⠟⠁⠀⣠⣶⣦⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠻⣿⡆⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢰⣾⠟⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⠀⠀⢠⣶⣶⣶⣶⡄⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣿⠏⠀⠀⠹⣿⣶⡿⠟⠋⠁




⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⣀⠀⢀⠀⢀⡀⠀⣀⠀⢀⠀⢀⡀⠀⣀⠀⣀⠀⢀⡀⠀⡀⠀⠀⠀⢀⠀⠀⡀
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⡾⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⠶⠾⠟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⠛⠋⠉⠀⠀⠀⠀⠀⣙⣿⣶⣶⡾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠘⠻⠷⠄⠀⠰⡶⠿⠛⠉⠉⠙⢿⣄⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣄⡀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⡿⠟⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⡙⠻⣦⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠿⣷⡄⠀⠀⠀⠀⣠⣴⣦⠈⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⣀⡀⠀⠈⢿⡆⠀⢀⣼⠟⠁⠹⣷⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⠀⠀⠙⠿⠆⠀⢸⣿⣶⣿⠋⢀⠀⠀⠹⣷⣤⣤⣀⣼⠿⢷⣄
⠀⠀⠀⣠⣾⣿⣿⣿⠋⠀⢀⣀⡀⠀⠀⠀⢀⣿⠟⠁⠀⠸⡧⠀⠀⠹⣿⣿⣿⠏⣀⠈⠻⣷⣄⣀
⠀⢀⣼⣿⡿⢻⣿⠃⠀⣀⠈⠉⠉⠀⠀⢠⣿⠃⠰⣷⡀⠀⠀⠀⠀⠀⣿⡿⠃⠀⢿⣆⠀⠘⣿⣿⣿⣷⣄
⢠⣿⣿⠋⠀⣾⠇⠠⣶⡛⠿⠂⠀⠀⢠⣿⠃⢀⡀⠙⠃⠀⠀⠀⠀⣼⡟⠀⠺⣦⡈⠋⠀⠀⠸⣿⣿⣿⣿⡆
⣿⣿⣏⠀⢸⣿⠀⠀⠉⠁⠀⠀⠀⠀⣿⠃⠀⠈⠻⢷⡄⠀⠀⠀⣼⡟⠀⠀⣀⠈⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⣿⣿⣿⣄⠘⣿⡄⢀⣴⡶⣶⣶⣶⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣀⠀⠛⠃⠀⠀⠀⠀⢀⣿⠇⣿⣿⣿⡇
⢻⣿⣿⣿⣦⠈⠻⠟⠁⠀⠀⠀⠀⢸⣿⠀⣤⣶⠶⠿⠷⠶⣶⣾⡇⠀⠙⠻⠆⠀⠀⠀⠀⣠⡾⠋⠀⣿⣿⣿⡇
⠀⠻⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠈⢿⣾⠏⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣀⣀⣠⣀⣤⡾⠛⠀⣀⣴⣿⣿⣿⡇
⠀⠀⠉⠻⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣠⣴⠿⠛⠉⠉⢉⣁⣤⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡅⣦⣖⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣯⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣷⠀⠀⠀⢀⡴⠚⠓⠶⠋⢳⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡸⣿⣿⣿⣿⢫⣵⣬⣭⣶⣎⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠘⣆⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢇⣾⣿⣿⣧⠹⣿⢋⣶⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⠀⠳⣴⣦⢰⠒⠶⣄⠴⢲⠞⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⡇⣿⣜⠋⢹⡏⣭⣝⠻⣫⡍⣱⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠰⢸⠀⠀⢸⣿⢸⠰⠀⠠⠀⢨⣿⡳⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⡇⢸⡇⣿⣿⣿⣿⡷⢚⢭⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⢸⠀⠀⣾⠙⠋⠸⠀⠐⠀⢸⠀⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⡇⣧⣵⣿⣿⣿⣿⡇⣿⢸⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⢸⠀⠀⢿⠀⠀⢀⠀⠀⠀⠘⣶⣫⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⡇⣿⣿⣿⣿⣿⣿⣧⢉⠜⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⢿⠠⠄⠸⠄⠐⠄⢸⠉⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡇⣿⣿⡇⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠚⠀⠀⠘⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣭⣭⣭⣭⣭⣥⣿⣿⣧⣭⣭⣭⣭⣭⣭⣵⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿








⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣋⡥⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢲⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠛⠁⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠾⣇⢀⣀⣤⣤⡿⠓⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⢀⡶⠛⠛⠳⢦⡀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⢶
⢸⠁⠀⠀⠀⠀⣙⣃⠀⢀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠹⡀
⢸⠀⠀⢀⡴⠋⠁⠀⠉⠃⠀⠀⠀⠱⡀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢣⡤⠒⠠⣀⠀⠀⠀⠀⡠⠞⠙⡆⠀⠀⢀
⠈⣇⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠆⠀⢠⠞⠀⠀⠀⠇⠀⣠⡟
⣠⠟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠞⢠⣧
⢇⠀⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠈⢻⠆⠈⠉⠐⠒⠤⠤⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠖⠊⠉⠀⢴⠞
⠀⢸⡆⢠⠒⠂⠤⠄⣀⣀⣿⠀⠀⣼⠁⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⣷⠀⢰⣏⡠⠤⠔⠲⡄⢸
⠀⠈⡇⠘⡀⠀⠀⠀⠀⠀⠉⠲⠖⠋⠀⠒⠲⠶⠖⠒⠖⠒⠒⠒⠖⠒⠒⠛⠛⠉⣿⠀⣼⠁⠀⠀⠀⢰⠁⢸
⠀⠀⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⣸⠀⡏
⠀⠀⠸⡆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⣹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡏⢰⠃
⠀⠀⠀⢻⡀⢷⠀⠀⣠⠔⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠹⠆⠀⠀⠀⣼⢁⡟
⠀⠀⠀⠈⢷⡈⢧⠀⠘⣄⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢀⢰⠇⠀⠀⣰⢃⡾⠁
⠀⠀⠀⠀⠈⢷⡈⢳⡀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⣰⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠻⣆⠙⣄⠀⠀⠀⠀⠀⠀⠀⢠⠉⠉⠉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣄⠀⠀⠀⠀⠀⠈⠳⠤⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣌⡑⠢⢤⣀⡀⠀⠀⠀⠀⢀⣀⡤⢖⣩⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣷⠀⠀⠉⠉⠉⠉⠉⠁⠠⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣍⣒⣠⡤⠦⢤⣤⠤⠤⢤⡤⠤⠤⠤⠶⢶⣒⣫⡽⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠁⠉⠉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣀⡀⠀⠀⠠⠂⠈⠀⠀⡄⣤⣥⡀⠀⣀⢀⣀⡁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⣼⣿⣿⡇⠀⣾⣾⣿⣿⡆⠀⢷⣿⣿⡇⠀⣿⣿⣿⡇⠀⣀⠆⠈⠀⢸⡠⡀
⠀⠀⠀⣼⣷⣿⣿⡇⠀⢿⣿⣿⠃⠀⢻⣿⣿⣿⠇⠀⢸⣿⣿⠁⠀⢹⣿⣿⠃⠀⣰⢸⣶⣿⣾
⠀⠀⠀⠻⣿⣿⣿⠇⠀⠸⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠹⡏⠀⠀⣿⣼⣿⣿⣿⡄⢀⠆
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠿⠿⠀⠀⠀⢿⣿⡇⠀⠀⠀⠉⢋⠀⠀⠀⠘⡀⠀⠀⠉⠛⠛⠛⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠀⠀⠉⠉⠉⠉⠀⠀⠐⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡨⠐⠂⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⡘⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠠⢐⠡⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡨⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠢⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠐⠁
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣾⣷⣿⣾⣥⣾⣿⣷⣞⣛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⣿⣷⣾⣿⣿⡿⠿⠟⠛⠛⠛⠻⠿⣿⣯⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢟⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠻⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣥⡾⣻⣿⠟⠁⠀⠀⡀⠀⠀⠀⡀⠐⠋⠉⠀⠀⠀⠀⠀⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡿⠃⠀⠀⠐⠋⡉⢁⡀⠋⠉⠀⢠⠟⠁⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠁⠀⠀⠀⠀⠠⠛⠾⠿⠤⠴⠞⠉⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡟⠉⠀⠀⠀⠀⣠⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⢶⣦⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣄⡀⠰⠋⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⡄⠫⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣴⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠒⠚⠋⢰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⠀⠀⠦⣝⣳⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⠶⠶⣬⣓⣶⠋⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⢌⡲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠙⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠿⠆⠀⠀⣶⡇⠀⢿⣧⡤⠴⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣁⠀⠀⠀⠀⢀⣸⣿⡇⠀⠀⠀⢀⣀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠙⠿⠿⠟⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠚⠛⠋⠉⠉⠉⠛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢰⠋⡁⠀⠀⠀⠀⢀⠀⡠⠀⠀⠐⠀⠀⠄⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡉⠀⠳⡀⠀⣰⡛⣦⠀⢸⣧⡂⠀⠄⠐⠂⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⢁⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠐⠀⠀⡀⢹⡴⠃⠀⠘⣦⣼⡇⠙⠓⠦⣤⣄⣀⣀⣀⣀⣀⣀⣤⢤⡴⠒⠋⣼
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⡠⢀⠁⠄⠂⣠⡴⢋⠄⠀⠐⠀⢹⡄⢹⣤⣠⠞⠁⡘⢧⠀⠀⠁⠀⣰⠫⠀⢷⡢⣱⠇
⠀⠀⠀⠀⠀⢀⣤⠎⠀⠀⠈⠀⠁⠀⣠⠟⠁⡀⠀⠐⠀⠀⠀⣀⣧⠴⠛⠁⠈⠀⠀⠈⣇⠀⣠⠞⠁⠀⡐⠘⣷⣯⡀
⠀⠀⠀⣠⠞⢋⡟⠀⠀⠠⢂⠄⢠⡞⠁⠀⠀⢀⠌⠀⠈⢠⡴⠋⠀⠠⠀⠀⠄⠀⠀⢢⡼⠋⠁⠈⡀⠈⠀⡀⢹⡄⣝⣳⡄
⠀⢠⠞⠵⡁⢼⠃⠠⠊⠔⠀⠀⡞⠀⠀⠀⠀⠠⠀⠀⣰⠋⠀⠀⠂⠐⠀⠀⠀⢀⡴⠋⠀⠔⠀⠀⠀⠀⠀⢈⠀⣇⠑⠄⢹
⢠⠏⣠⠊⠀⡾⠀⠀⠀⠀⠀⢸⠇⡀⠀⠐⠀⠀⠀⢱⠃⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠄⠈⠀⠀⠀⣡⠀⠀⣿⠀⠀⢂
⣿⡔⠁⠀⢠⠇⡠⢂⠄⠀⠀⣼⠀⢀⠊⠀⠀⠀⢀⡾⠀⡄⠀⠀⠀⠀⠀⠁⣾⠀⠀⠀⠈⠀⠈⠂⠀⠠⠀⠀⠀⡏⠀⠀⣸
⣿⢳⡀⢀⡿⠀⠔⣁⣀⣀⣰⢃⠂⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠄⢀⠀⡟⠐⢀⠄⠀⠀⠀⠀⡀⠁⠀⠀⣸⠃⠀⣰⠃
⢻⡄⠙⢦⣛⠚⠋⠉⠀⢸⠇⠀⠐⠁⢀⣀⣀⣀⣸⠁⡠⠠⠀⠀⢀⠀⠀⣰⢃⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡼⠃⣸
⠀⠻⣄⠀⠉⠳⢦⣄⡀⠘⠦⠗⠋⠉⠉⠉⠉⢰⡗⠀⠂⢀⣒⣒⣐⣁⣰⠃⠀⡠⠀⠀⠠⠠⠄⠂⣀⡴⠋⣠⡴⠋⢀⡼⠃
⠀⠀⠈⠳⣄⡀⡀⠬⣉⡛⠶⢤⣄⡀⠀⠀⠀⠘⠓⠚⠋⠉⠉⠉⠉⠁⣿⣠⡤⠴⠶⠒⠒⢒⣚⣩⡵⠖⠛⠁⣠⡴⠋
⠀⠀⠀⠀⠀⠙⠺⢷⣄⡅⠀⠀⠈⠉⠛⠒⠶⠦⠤⣤⣤⣤⣤⣤⣤⣤⡬⠥⠴⠶⠒⠚⠛⣭⠉⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣬⠽⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠋⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠈⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣼⠧⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⡞⠁
⠀⢀⣤⠶⠶⠶⠶⠶⣤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟
⢠⡟⠉⠢⡀⠈⠠⡀⠀⠈⠙⠶⣄⣿⠛⠒⠒⠒⠓⠛⠛⠋⠉⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣿⢧⠀⠀⠈⠳⣄⠈⢦⡀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⣿⠘⣆⠀⠀⠀⠈⠳⣄⠙⢦⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠸⣆⠈⢢⡀⠀⠄⣀⠈⠳⢄⠙⢦⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠁⠀⢀⡾⠁
⠀⠘⢦⡀⠙⢦⡀⠀⠀⠀⠀⠑⢄⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣸⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠙⢦⣀⠈⠓⠶⢤⣤⣀⣤⢽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣤⣤⠶⠟⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠉⢉⣽⠟⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⣠⠾⠋⠁⠠⠄⠀⢀⡤⠞⠉⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⠀⢀⣀⣠⠴⠛⠁⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⡠⠚⠁⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⠦⠤⢤⣄⣀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⣠⠖⠁⠀⣠⠖⠉⠀⠐⠀⠀⣠⠎
⠀⠀⠀⠀⣠⣴⡛⠛⠲⢤⣀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢴⣟⡁⢀⣠⠞⠁⠀⣀⣀⡤⠖⠋⢁⡼
⠀⠀⠀⣰⠏⠈⠙⠲⢦⣀⡈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠈⠉⠉⠉⠉⠉⢉⣀⣠⡤⠶⠋
⠀⠀⠀⣿⣄⠀⠀⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢤⣀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⢹⣏⠑⠦⣄⡀⠀⠂⠀⠀⠈⠙⠲⢤⣀⠈⠙⠲⢦⣤⢿⡆
⠀⠀⠀⠀⠻⣆⠀⠀⠉⠓⠒⠤⠤⣀⣀⣀⣀⣬⠽⠶⠒⠉⠀⢀⡿
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⠀⠀⣀⣀⣀⣤⡴⠞⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠂⠉⠰⠖⠚⠉⠑⠲⠒⠒⢤⢄⡀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⢀⡤⣄⠀⠀⠀⠀⠀⢠⠤⠖⠀⠒⠒⠓⠒⠊⢳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⡼⢥⠈⢁⠴⢾⡃⠀⠀⠀⣠⠀⠀⠀⠀⠀⡀⠀⡄⠀⡀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠁⠀⠀⠘⠀⠘⣆⠀⠀⠌⠁⡀⡇⢠⠀⠀⠸⠀⠀⠀⢀⢡⣼⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡏⠀⠀⠀⠀⠀⠀⣼⠀⢀⠀⠀⠀⠃⢸⠀⠀⠐⠀⡀⠀⠈⢸⠐⠀⠒⠒⠀⠈⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣱⣯⠀⢤⡀⢀⡀⢀⣿⠀⡎⠀⠀⠀⣀⢸⠀⠀⠀⡀⣇⠀⠠⢼⣀⡤⠞⠛⠛⠒⢄⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠁⡇⠀⠀⢸⠋⠀⠀⠀⠀⠀⠈⠇⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⢀⠀⠀⠀⠀⢻⡴⠀⠀⠀⢰⠀⢠⠀⠀⠀⡆⠀⠀⢀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⡇⠀⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⡇⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠀⢸⠀⠀⠀⠃⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇⠀⢰⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠘⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡇⠀⢸⠀⡄⠀⢸⡃⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⢸⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⢀⡅⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⡼⠀⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡇⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡁⡏⠓⠲⠤⠀⠊⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢨⡇⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⠀⣿⣀⠤⠤⣄⣢⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⡘⠀⡅⠀⠀⡇⢰⠀⠀⠀⠀⢀⠀⠇⠀⠀⢸⠀⢇⠀⠀⢻⡴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡈⡀⠁⠀⠈⠂⠚⠁⠈⢦⣀⠀⠀⠰⠀⠈⠂⠀⠈⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠶⢭⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠉⠉⠓⠒⡂⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠉⠁⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⠤⣤⣥⣄⣤⣤⣤⡤⠤⠶⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⢻⢯⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠘⡅⡰⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣷⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠒⠒⠒⠶⠶⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⣛⣉⣉⣭⣭⣭⣭⣭⣉⣉⣙⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣼⣿⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠘⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃⠀⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢹⡷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣠⡾⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⢁⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺
⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐


⠀⠀⠀⠀⠰⣮⣶⡄
⠀⠀⠀⠀⠘⣻⣿⣷
⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠐⡻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉