⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠶⣶⣶⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠁⠀⢂⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡰⣹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⡁⢂⠕⣀⠂⡑⠀⠀⠀⠠⡙⢄⠀⡐⡀⠀⠀⢠⠔⠎⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢏⡅⠉⠄⡡⢌⢤⣁⢌⠢⢑⢤⣤⣌⡤⠟⠾⠷⢤⣀⡰⡌⠠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠓⣤⢪⡝⣦⢦⣀⣈⣀⣨⣷⡶⢿⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⣦⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⣼⢾⡳⢎⠤⣭⣬⡿⣟⠾⡫⢌⠠⡁⢌⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⣐⣏⢻⡴⣶⣶⣾⢫⡽⣯⠂⡔⢪⠒⡅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢈⠶⣻⣞⡷⡟⠾⡱⣞⠷⡁⢌⠡⣉⠄⡠⢀⣠⣤⠴⡶⢖⠲⣒⢞⡿⣧⠀⠀⠀⠀⠀⠀⢸⣟⢍⠻⢦⣄⠀⣾⠏⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡪⠛⣝⢫⡿⣧⡂⠘⣫⡾⣥⣢⡐⣩⡶⠞⠋⢙⣀⣀⣀⡀⠈⢀⣴⡾⣿⠇⠀⠀⠀⠀⠀⠈⠻⣮⣵⣀⠹⣧⣿⣆⠘⣧⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣤⣵⣊⣷⣷⣌⠡⢛⢾⣳⣾⠋⠀⠄⠺⢛⠍⡫⢍⠄⠀⡇⠡⢉⣿⠀⠀⠀⠀⣀⣠⣴⠶⣦⣿⡧⣂⢹⣿⣟⠀⣿⠀⣴⡟⢝⡿⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⣽⡿⡒⠮⣾⣿⣿⠀⡀⠂⠐⢬⣶⡵⠎⠐⠄⠘⢮⠛⢻⡆⠀⢀⣼⠋⣅⢠⡲⡌⠹⡗⡍⢸⡿⣻⠂⣿⣴⠟⠍⣸⡇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠡⢎⢷⠈⣿⣿⠈⠀⠁⢀⠀⠀⠀⠀⢀⢄⡁⠀⡣⢸⣿⢀⡿⠧⣤⡾⠉⢹⠆⠐⢋⠄⡚⠍⠀⣼⠟⠳⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣪⣞⡆⢿⣿⣇⠂⡕⢪⠒⡄⠀⠀⠈⣶⣾⠽⣦⣼⣯⡾⠁⠐⣽⠀⢀⣟⢨⣕⣁⠂⡁⢀⠀⠈⢀⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⡄⠁⠈⠁⠀⠈⠁⢉⣀⣠⣴⠾⠛⠱⡿⠈⣿⠿⣷⠃⣑⢿⡶⢾⠥⣱⢎⠂⠉⠍⠢⠈⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡀⣿⠀⠘⣆⠈⢃⡕⣬⠲⡵⠊⠐⠀⠂⠐⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠇⠀⠠⣳⣄⡁⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⣼⠀⠀⠈⢷⡀⠊⠰⣉⠄⠀⠈⠀⡁⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢯⡟⠀⠂⠐⠨⠊⡝⢲⠶⣤⣂⣐⡀⠂⣔⣪⡾⠋⠀⠀⠀⠀⠙⢶⣄⡁⠂⠐⠀⠀⠐⠻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣟⣷⠉⠄⠁⠈⠀⠁⢌⡡⢉⠤⡣⣭⣿⣛⣫⣭⣤⣶⡶⢶⠶⡶⢶⣤⡘⣷⣠⣈⣀⣀⣀⣠⠷⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⠟⢛⢯⡷⣷⢦⣀⠀⠀⠀⠀⠐⢡⠒⣽⢯⣿⣿⢯⡽⢏⠻⡕⠪⠀⠀⠀⠈⠙⢿⠟⢻⡓⡏⠋⠀⠀⠀⢻⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⠁⠀⠀⠌⢳⣞⢷⠻⠞⠋⠒⠒⡄⠀⡶⡟⠛⢩⣟⠞⡡⠌⠔⠁⠀⠀⠀⠀⠀⠀⠀⢘⡆⠡⣋⠆⠀⠀⢀⣠⠋⡈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠁⠀⠀⠀⠂⠐⠀⠂⠀⢙⢶⡄⠂⠐⠀⠀⢱⢰⠓⡅⠂⣸⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠋⣽⣿⣶⣾⢿⡝⠦⠂⠘⠄⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠄⠠⠀⠄⠁⠀⠠⡁⢌⡡⢉⠍⡳⢬⣠⣀⢬⡯⠘⠔⡋⢉⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⢀⡳⣿⠃⠁⣟⡿⣉⢎⡡⢙⠀⡁⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠂⠐⠀⠂⡐⢮⠒⢵⢪⠐⢠⠂⡕⢪⡷⣽⢫⣕⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢪⡗⣿⠂⠰⣯⡗⡥⠊⠐⠀⠂⡀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢸⣿⠀⠠⠀⠄⡠⢈⠠⡙⢌⡡⢉⠄⡡⢌⠤⡃⢌⡡⣟⢾⡣⢟⠠⠀⠄⠠⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⡳⢞⣿⡀⠘⡡⢛⠶⡁⠈⠠⠀⠄⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠸⣷⣤⣐⡀⠂⠀⢨⠒⡅⢊⠐⣈⠒⡕⢪⠒⡁⢢⡑⣭⢾⡕⢪⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣫⣿⣇⠂⠕⣩⠢⡅⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣷⣿⣶⣷⣾⣶⣷⣮⣥⣭⣤⣥⣬⣶⣿⣮⣽⣬⣴⣷⣾⣶⣶⣶⣶⠶⠶⠿⠿⠿⠿⠛⠛⠛⣿⣇⣈⡦⣉⢆⠀⠀⠀⣀⣀⣠⣴⣾⡿⠛⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀