.
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢐⠩⠽⠯⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡂⠚⠁⠁⠀⠀⠀⠀⠀⢥⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⢏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡑⡝⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡆⡀⢀⢀⢰⡀⢀⠀⢘⣦⢠⡔⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣇⡿⡸⣀⢳⢷⣦⣿⣦⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣴⠾⢛⠻⣳⠻⢳⣽⣷⢟⡟⠛⠀⠹⡅⡿⣱⡄⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣴⡴⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⣤⣤⡤⢶⡀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠭⠴⡺⠞⣡⡽⡳⣾⢼⣷⡀⠀⡀⣀⣭⢢⣻⣇⣿⣿⣷⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⠁⣀⣣⣽⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢷⠷⣻⣟⣛⣚⣓⣓⣶⣶⠖⣒⣓⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⠂⢸⠀⣿⡟⡱⠿⠿⠟⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⣡⠤⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠈⡉⠉⠻⣹⡻⣿⣛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⢹⠛⠀⠉⠘⠔⡩⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⠄⠐⣿⣿⣿⣿⣿⡿⣏⠀⠀⠤⠄⠤⠠⣵⣿⣿⣡⣵⠼⠦⡀⠀⠀⠀⠀⠀⢀⠠⠙⠲⠮⠶⡳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⠖⢂⡉⣿⣿⣿⠿⣇⠙⠘⢗⡆⠀⣐⣤⣽⡿⠿⡿⡛⣄⠀⠈⠪⣢⡀⠀⠀⠀⢈⠦⣀⣀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⣔⡭⣒⠉⣛⠮⠛⠉⠀⠠⢻⣄⠀⡺⣟⣻⠉⣫⠩⠕⣗⠲⡊⢊⢂⢀⣄⣾⠧⡀⠀⡀⠁⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⠋⠈⡡⠊⠀⠀⠀⠀⠀⢀⡠⠻⠓⢹⢁⡇⠀⢘⠣⠄⠘⡄⢱⠈⢜⠝⠉⠀⠀⠈⠎⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡎⡰⡟⠀⠀⠀⠀⠀⢀⣴⠟⢆⢇⠀⢆⡸⠁⢀⣸⡇⠀⠀⠛⠚⢚⠝⠤⡀⠀⠀⠐⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢇⠀⣣⠀⠀⠀⢀⣴⠏⢓⢧⣀⠉⠉⠉⠀⢀⣡⡽⣈⠢⣤⡤⠔⠁⠀⠀⡈⠑⢤⡀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠑⢌⠛⠷⢯⡩⠋⠀⢸⣹⠓⠂⠀⠚⠍⣛⣙⡏⣠⣶⠉⠀⢀⡠⢔⣬⢾⡻⠝⠹⠻⠲⠓⠁⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣠⣗⣯⣹⣳⣭⣲⣭⢷⣽⣷⣧⡿⠶⢛⡩⢔⣩⠴⠋⠀⠀⠰⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣏⣿⡿⡿⣼⢿⡛⠿⣜⠬⣪⣴⠟⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢡⠦⠐⡢⠙⢉⡔⢱⠟⢋⠡⠐⡩⡴⡹⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡩⠦⢐⠨⢔⣊⠥⠦⣠⢷⠫⣐⣵⣾⠋⠀⠀⠴⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⣀⠀⠀⠰⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⢁⣒⠒⣮⠭⠽⠒⠈⡉⢄⣪⢷⣿⡏⠻⡇⠀⠀⠀⠈⢄⢄⠀⠀⠀⡠⠒⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠀⠕⢀⡀⠈⡂⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⡀⠀⠀⠀⠀⠀⠀⢌⡏⠋⠉⠁⠀⡀⠄⣂⠥⠚⠋⠀⠀⡹⡇⠀⣏⠂⠀⠀⠀⠀⠉⠚⡤⠊⠀⠀⠑⣄⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠊⣀⠈⠆⠠⠜⢀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⢳⠀⠀⢠⠀⢀⢮⣿⣷⣴⠦⢃⣡⠴⠻⣥⠀⠀⠀⠀⠀⠡⣧⣠⡅⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠈⡷⠊⠀⠀⠀⠈⢲⠀⠀⠀⠀⠘⠂⠈⡀⢠⠘⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⡐⡈⣃⠆⡸⠁⢪⢻⡅⣧⠴⠚⠉⠀⠀⠀⠀⢡⠀⠀⠀⢀⣬⣿⠧⡷⠤⠄⠒⠈⠀⠀⠀⠀⣨⠀⢠⠂⢸⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠘⠣⠶⡎⢀⢈⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⢥⠐⢙⠀⠙⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⣀⠴⡋⢹⡇⠀⠀⡀⠠⠀⠀⢀⠀⢀⠀⡟⠇⡃⠀⠀⢶⠦⠀⠀⢀⡀⣤⢣⠀⠀⠄⡀⡀⡤⠒⡃⡘⠀⠀⠀⠀⠀⠔⣋⠄⠀⣀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀
⠈⡂⢺⢠⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣞⡀⠀⠘⣸⠃⠀⢰⠠⠁⢀⢘⡡⠠⠒⠐⡁⠠⠌⠀⡀⠀⢡⣠⡾⠍⠃⠘⡀⠠⠅⠀⠁⠆⠈⢂⣜⠑⡆⢰⡀⢀⠂⡡⡁⠤⠞⢁⡨⣡⠠⢀⠜⢡⠔⢀⡀⡌⢁⠤⠀⠀
⢗⢱⡌⠗⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠈⡄⠀⠠⡉⠀⠦⡕⣀⠀⡡⠴⢢⠃⠡⠎⠊⢁⣈⡀⣇⡀⠀⠉⢳⡀⠀⠀⢡⠀⠢⢰⠄⠀⠀⠌⠠⠀⠁⠀⠀⠀⠁⠀⢄⠀⠀⠲⠊⠁⡀⠀⠀⠁⠀⠁⣨⠑⠁⠀⠂⠀
⢦⡘⣞⢀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⣠⣾⣧⠊⠝⠱⠈⠒⠵⠂⠀⠀⢠⣬⠐⠀⠉⠊⠛⢒⣊⡤⣤⣬⠥⠘⠀⠀⠀⢀⢆⠈⢢⣇⠀⠀⠀⠀⢀⠀⠠⠁⢀⠞⠘⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠄⢒⠀
⠾⠧⠋⢸⠀⠀⠀⠀⠀⠐⣰⠋⠀⠀⠀⡠⠪⠣⠒⠒⠉⠀⣄⣤⡤⠤⠀⠠⠀⠉⠛⠿⠟⠒⠀⠀⠀⠉⠉⠉⣁⡀⠼⠔⠄⠉⠒⠒⠣⡀⠀⠀⠀⡘⠀⡌⡐⠠⡂⠃⠀⢆⡀⢀⠤⣂⠰⡋⠀⠀⡁⡈⠄⣀⠠⠆⡁⠀⠁⠀
⣶⣿⣟⢸⠀⠀⠀⠀⠀⠊⠈⠀⠀⣠⣮⣽⣧⣄⡠⣄⠦⠿⢭⠉⠉⠍⠀⠶⠄⠀⢀⡤⢄⠀⠀⠀⢀⡀⠦⣥⡦⡚⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠂⠐⢰⠒⡡⡪⠀⠀⠃⠈⠁⠀⠁⠀⠀⠃⠈⠈⠀⠀⠀⡠⠂⠈⠉⠉⠁⠀
⠀⠀⠀⢸⠤⠀⠀⠀⠀⠀⣀⣤⠪⢵⠘⠒⠂⠙⠛⠒⠈⠉⠀⠀⠁⠀⠀⠀⠀⠈⠈⠑⠚⠛⠒⠀⠤⠤⣤⣤⣤⣅⠒⠂⠠⣭⣦⢄⠀⠀⠀⣀⣉⠲⢤⡈⠨⡤⠄⠀⣀⣐⣒⠀⠝⠷⠖⠠⠆⣀⠀⠀⠁⢀⡤⠤⠤⠶⠤⠂
⠀⠈⠛⠉⠉⠛⠛⠋⠉⠛⠓⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠐⠒⠛⠛⠑⠒⠒⠒⠒⠂⠛⠂⠁⠀⠀⠀⠈⠁⠀⠀⠀
