⠀⢴⣶⣶⣿⣶⣶⣶⣶⣶⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠍⡄⢀⠄⣰⠀⣄⢠⣄⠀⠀⣠⣀⡀⠄⠠⡀⠀⣄⡀⠀⠀⠀⠸⡀⠀
⣽⣿⣿⣿⣿⣿⣿⡟⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠰⠀⡞⠀⡅⠀⠛⣿⣿⡆⠀⡿⠛⡇⢰⠀⣿⠀⠉⠱⡄⠰⠀⠀⠇⠀
⣿⣿⣿⣿⣿⣿⣿⣛⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈⠀⠀⡏⠀⠀⠈⠈⠃⠀⠁⠉⠀⠀⠛⠘⠀⠘⠀⠉⠀⠀⠀⠸⠀
⣾⣿⣿⣿⣿⣿⣿⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣿⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠰⠿⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⡆⠠⠀⠇⠀⠀⠀⡄⠀⠀⢀⠀
⣿⣿⣿⣿⡿⡝⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⡈⠀⢠⣶⣶⣶⣾⣶⣷⣶⣶⣶⣶⣶⣤⣶⣿⣶⣶⣶⣷⣷⣶⣿⡿⠿⠶⠷⠦⣄⡀⠀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠄⡇⢀⠀⡇⠀⢸⠿⠿⠛⠛⠛⠛⠿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⣀⣀⣀⣠⣄⣀⠙⠀
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⡇⠀⣤⣤⣤⣤⣤⣤⣤⣄⣀⠈⢸⣿⣿⣿⣿⣿⣿⣿⣦⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀
⣿⣿⣿⡏⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠙⠛⠿⠿⠀
⣿⣿⣿⠑⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠇⠀⣦⠀⠐⠪⠭⢍⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⠀⠀⢠⡀⠀⠀⡀⠀
⣿⣿⡿⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⣤⡀⠤⢤⣤⣤⣤⠄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣇⡐⣛⣈⡒⠁⠛⠁⢀⣴⡇⠀
⣿⣿⢣⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣽⣿⣿⣷⣶⣿⣿⠀⠀
⣿⢏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀
⡟⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀
⢱⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⣠⣶⣿⣖⣄⠀⠀⢙⣛⡛⢿⣿⣿⣿⣿⣿⣧⣌⣐⣢⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡐⣩⣴⡻⠿⠿⠇⠀⠴⢶⣿⣿⣿⡿⣆⠻⣿⣿⣿⣿⣿⡟⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀
⣿⣷⣴⡃⠀⠀⠀⠀⢀⣼⠀⡿⣿⣿⣧⠄⠀⠸⣿⣿⣿⡄⠙⢃⣿⣆⠙⢧⣍⣁⣠⠄⣁⣀⣄⣈⠙⡛⣿⣿⣿⣿⠏⠠⠂⠀⠀⠀⠀
⣿⣿⡙⣀⠀⠀⠀⠀⢸⣿⣶⣌⠻⢿⣿⣿⣦⡀⢻⣿⣿⣿⠀⠈⠪⠽⢆⣸⣿⣿⣶⠷⠶⠶⢿⣿⣿⣿⣿⣿⡿⣡⠌⠀⠀⠀⠀⠀⠀
⣿⣿⣇⢻⣆⠀⠀⠀⣠⣈⠻⢿⣷⡈⠻⣿⣿⣷⡘⣿⣿⣿⣾⣧⣄⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⡽⠋⢀⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣆⢻⡄⠀⠀⠈⠻⢷⣤⡙⠿⣦⡈⢿⣿⣧⢸⣿⣿⠿⣿⣿⣿⣷⣦⣭⣝⠛⠿⢿⣿⣿⣿⠿⠛⢁⡤⣲⡇⠀⠀⠀⠀⠀⠀⠀
⣿⣿⣿⣿⣧⣯⣢⣤⣄⠀⠀⠙⠻⢷⣌⠻⢆⠙⢿⣌⢛⡛⢿⣯⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠋⢁⣠⡴⢫⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀
⣾⣿⣿⣿⣿⣿⣿⣿⣷⣷⣤⣄⠀⠀⠉⠣⢘⡓⠲⣦⣵⣮⣵⣮⣙⣿⣯⣟⣻⣿⣿⣿⣿⣿⣆⠀⠈⣴⣿⣿⣿⡇⣶⣄⣀⠀⠀⠀⠀
⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡄⣾⣿⣿⣿⣿⣧⢹⣿⡿⠉⠀⠀⠀
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠄⢀⡄⢀⡤⢀⠄⢀⡤⡠⡠⠀⣠⠄⢀⠄⠀⣠⠄⣀⢄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡪⣴⡫⡺⡫⡴⢟⡔⠫⠪⡪⣫⢞⣵⢕⢕⡴⣫⣣⠞⣵⢗⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣞⢕⢞⣡⠭⣺⡪⡪⡢⡀⡾⡡⡾⣡⢾⢟⡴⠿⣣⠞⣱⢿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢪⠯⡡⢑⠞⢩⣈⡈⠯⠎⠬⠮⠎⠄⣞⣋⣞⣫⢪⠞⣡⡗⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠬⣟⣪⠕⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⣭⡓⢮⣺⣩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⡔⢁⢠⣰⡤⣾⢗⢔⠲⡢⣫⢗⣀⣦⢠⡀⡻⠻⠌⢀⣰⣕⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢵⣿⢟⠯⡫⡲⢕⢕⠨⢘⠕⣊⣄⡀⡟⢹⣷⣯⠠⡶⣋⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢟⠕⠁⠊⠈⠀⠁⠨⣐⠨⠤⢼⣿⡇⠸⢿⣿⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡳⠁⠀⠀⠀⠀⠀⠀⠈⢂⠥⣯⣿⡿⣷⣀⠀⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⣐⣽⣿⣿⢿⣬⣍⢧⣤⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⡠⡢⣪⣪⣮⣿⣿⣿⣯⣯⣙⣦⡉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡇⠀⠀⠀⠐⢁⠎⣔⣼⣿⣿⣿⣿⣿⣿⣿⣿⣽⣮⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⠀⠀⠀⠀⠱⢾⣿⣿⣿⣿⡟⠏⢟⡯⢙⣫⡛⠦⢤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⡆⠀⠀⢀⣴⣾⣟⣙⣋⢃⣮⣭⡓⣖⢍⡅⣿⣕⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⣇⠀⠀⢤⣤⣿⣿⡲⠴⣾⡄⠹⡧⣿⡌⡇⢯⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠿⡀⠈⠂⢜⢻⣇⠦⠀⣿⣷⠀⠹⡎⣷⠠⠘⢜⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⢿⣷⡀⠀⠀⢣⢿⡭⣧⠉⣿⡇⠀⠻⣼⡇⠀⠀⢪⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠶⣿⣷⠖⢄⠡⡻⡇⠛⠀⢽⠇⠀⢠⣙⡷⠤⠤⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠶⣾⣿⣏⣒⠀⣿⠇⠀⢀⣸⣀⡀⣀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠶⣿⡟⢿⡦⠐⠁⠀⢀⣸⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⢻⣿⡇⠈⠳⢤⣤⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢤⣭⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀