False Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⢠⣾⡿⠿⠟⣻⣿⠿⢻⣿⣇⣿⣷⣿⣷⣦⣼⣿⣀⣀⣤⣾⣿⠿⢿⣿⣶⡀⠀⠀⣿⡿
⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⣡⣤⣼⣿⣿⣿⢿⣷⣬⣙⣿⣿⣿⣿⣿⣏⣀⣀⣀⣼⣿⠃⠀⢀⣿⡇
⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⠟⢹⣿⣿⡟⢀⣩⡝⠻⠿⣿⣿⣿⡟⠻⠿⠿⠿⠟⠁⢀⡀⢸⣿⠁
⠀⠀⠀⠀⠉⠉⠻⣿⠉⢿⣏⣀⣼⣿⣿⣇⣸⣿⣧⣄⣀⣈⣹⣿⣿⣄⡀⠀⠀⢀⣠⣿⢃⣈⡉
⠀⠀⠀⠀⠀⠀⠀⠹⠇⠈⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠿⠛⠁⠙⠻⠿⣿⣿⠿⠟⠁⠸⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠦⠤⡤⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⡛⡫⠅⠒⠊⠉⠉⠉⠉⠁⠒⠪⠭⣛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⠕⠊⠁⠀⢀⣀⡠⠤⠤⠤⠤⠤⣀⣀⠀⠀⠉⠠⣝⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⠔⠁⠀⢀⠔⣂⣬⡶⠞⠛⠛⠛⠛⠻⠶⣮⣕⠢⠀⠀⠀⠑⣝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡱⠃⠀⢀⠖⣵⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡑⣄⠀⠈⢪⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡁⠁⠀⣰⢣⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣌⢆⠀⠀⢣⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢩⠉⠉⠉⠉⠍⠛⠩⠉⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⡙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⣤⣤⣤⠄⠀⢠⣤⡀⠀⠀⠄⠀⠀⠀⣠⣴⣤⡀⢠⣤⣤⡤⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⣿⣧⣤⠀⢀⣾⠻⣧⠀⠀⣾⡇⠀⠀⢿⣦⣉⠁⢸⣯⣀⡄⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⣿⡏⠉⠀⣼⠟⠛⢻⣆⠀⣿⣇⣀⠀⣤⣈⣹⡇⢸⣿⣉⠁⠀⠀⠀⠀⠀⠃⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠉⠁⠀⠈⠋⠀⠀⠀⠙⠀⠙⠉⠉⠀⠈⠛⠋⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠿⣿⡷⡿⠶⠷⣿⢯⣾⠷⠿⠿⠿⠿⠿⠿⠿⠿⠶⠿⠷⠿⠿⢾⣿⢶⠽⠾⢾⡿⣷⠿⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡑⡄⠀⠈⠢⡙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⡵⠃⠀⢀⢞⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠎⠢⠀⠀⠈⠒⠍⠛⠷⠶⢤⣤⣤⣤⡶⠾⣛⡩⠔⠉⠀⢀⡴⣫⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⡲⢄⡀⠀⠀⠀⠀⠀⠐⠂⠀⠊⠉⠁⠀⢀⣠⢔⣩⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣮⣕⣒⠤⠤⢀⣀⣀⡀⠠⠤⣐⣪⣭⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠶⠖⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⠶⠶⠶⠿⠟⠛⠋⠉⠙⠛⢷⡆
⠀⠀⠀⣼⠏⠉⠉⠉⠙⠛⠶⣦⣤⣤⣤⣴⣶⠶⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⡄⠀⠀⠀⢸⣿
⠀⠀⠀⠙⠷⣦⠀⠀⠀⠀⠛⠶⠶⣤⣤⣤⣤⣀⣀⣀⠀⣀⢀⠀⠀⠟⠙⠂⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣀⣀⣀⣤⡤⠴⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⢉⣉⣀⣀⠀⠀⢀⣀⣀⣀⣤⡀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⣿⠃⠀⠀⣶⠶⠶⠶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⠶⠶⠶⢶⡞⠋⠉⠉⠉⢿⡏⠉⠉⠀⠀⠘⣧⣶⠿⠋
⠀⠀⠀⠀⠀⠻⣦⣤⣤⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣿⠀⠀⠉⠆⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢸⣄⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣤⣤⣤⣤⣿⣧⣀⠀⠀⠀⣀⣼⣧⡀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠀⣀⣿⣿⣦⣄⣠⣴⣿⣿⠟⠛⠛⠉⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠈⢿⡟⠙⠛⠛⠛⠉⠙⣿⡿⠿⠻⠿⠟⣿⠟⠛⠛⠛⠋⠉⠈⣿⠉⠉⠉⠁⠀⢿⠠⠄⠀⠀⢀⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡇⠀⠤⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢻⡁⠀⠀⠀⠀⢸⠀⠀⢀⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡆⠶⠀⠀⠘⣿⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⢰⡟⠋⠉⣧⡀⠀⠈⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠀⠀⣸⣇⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⠟⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠈⠉⠛⠛⠛⠛⠛⠓⠲⠶⠶⠛⠛⠉⠉⠙⠶⠶⠶⠶⠚⠛⠳⢤⣤⡴⠶⠛⠉⠈⠙⠓⠒⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⢀⣀⣀⣀⣴⠞⠋⠀⢀⣴⡿
⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠳⠶⢤⣤⣤⣴⠞⠛⠳⠀⡀⢀⡀⠀⠀⠀⠀⠉⠁⠈⢷⣤⣤⠶⠋⠉⠉⠉⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠶⠶⠶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠳⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠿⢶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣤⣤⣤⣤⣤⣴⣶⡶⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉⠀⢀⣀⣠⣤⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠁⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⠿⠿⠛⠋⠉⢻⣿⣧
⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⡿⠿⠟⠛⠋⠉⠀⠀⠀⢀⣀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⢿⣿⣿⡿⠿⠛⠛⠉⠉⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⣀⠀⣤⣤⣶⣶⣿⣿⣿⠄⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⣀⣠⡄⠀⣠⣾⣿⠏⠀⢻⣿⣿⣉⣁⣤⣤⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⢿⣿⣷⣰⣿⣿⠃⠀⠀⠘⣿⣿⣿⠿⠟⠛⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⣿⣿⡏⠁⠀⠀⠀⣀⣠⣤⣴⣶⣾⣧⠀⢀⣿⣿⣷⣄⠀⠀⠸⣿⣿⣿⣿⣿⣷⣄⠀⠀⢻⣿⣿⣤⣶⣶⣿⣷⠀⠀⠀⣿⣿⡆
⠀⠀⠀⢻⣿⣧⠀⠀⠀⠈⣿⣿⣿⠛⠋⣉⡁⠀⢸⣿⡟⢿⣿⣷⡀⠀⢻⣿⣿⠁⠙⢿⣿⣷⣄⠈⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⣸⣿⡗
⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⢸⣿⣿⣿⣿⡿⠷⠀⣿⣿⣧⣶⣿⣿⣿⣦⡘⣿⣿⠇⠀⠀⠉⠉⠁⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⠿⠁
⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⣿⣿⣯⠀⠀⠀⢰⣿⣿⠟⠉⠉⠙⠛⠉⠁⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⡿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠸⣿⣿⠄⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⣀⣤⣤⣶⣾⣿⣿⣿⠿⠿⠛⠛⠉⠁⠀⠀⣀⣠⣤⣴⣶⣶⣦
⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⠿⠟⠛⠋⠉⠁⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⣆⣀⣀⣤⣤⣶⣶⣿⣿⣿⠿⠿⠛⠛⠉⠉⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠿⠿⠛⠋⠉⠁⠀⢀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡶⠟⠛⠛⠚⠷⢻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣧⣤⣶⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠶⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠻⠟⠉⠀⠀⠀⠀⠀⠙⠻⣷⣶⣄
⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣿⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⣰⡞⠋⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢻⣧
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⣿⠀⣿⠃⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠘⢦⡀⠀⢀⡔⠀⠀⠀⠀⠀⢸⡇⢠⣿
⠀⠀⠀⢠⠏⠀⣤⡀⠀⠀⠀⠀⢀⣠⠃⠀⠀⠀⠀⠀⠀⡏⣠⣿⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠙⣦⡞⠀⠀⠀⠀⠀⠀⢸⠇⢼⡧
⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⡴⠋⢳⡀⠀⠀⠀⠀⠀⣼⣼⣿⠁
⠀⠀⠀⢻⡆⠀⠀⠽⣄⣠⣼⠛⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠹⡄⠀⠀⢀⣴⢿⠟⠁
⠀⠀⠀⠸⣷⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⣀⣤⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠉
⠀⠀⠀⠀⢹⣷⣠⣀⣀⣠⣀⣀⣤⣶⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⠿⠉
⠀⠀⠀⠀⠀⠹⠝⠾⠭⡿⢿⠹⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣟⣗⣻⣿⠿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢩⠽⠅⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠶⠶⠶⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠴⠤⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣤⣤⣤⠀⢠⣤⣄⣀⠀⠀⠀⣠⣤⣄⠀⣠⣤⣤⠀⣠⣤⣄⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡽⣿⣿⣿⠀⣾⣿⡿⣿⣷⡀⠀⣿⣿⣿⠀⣿⣿⡟⠀⣿⣿⣿⢿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⢹⣿⡇⠀⣿⣿⣿⢰⣿⣿⠃⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣷⣶⣶⠀⠀⣿⣿⠁⢸⣿⡇⠀⣿⣿⣿⣾⣿⣿⠀⠀⣿⣿⣿⣶⣶⠂
⠀⠀⠀⠀⠀⠀⣲⣿⠀⠀⢸⣿⣿⡞⠛⠛⠀⢠⣿⣿⣷⢸⣿⣿⠀⣿⣿⣿⢹⣿⣿⡀⠀⣿⣿⣿⠙⠛⠃
⠀⠀⠀⠀⠀⠰⢾⢿⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⠛⢸⣿⣿⠀⣿⣿⣿⠀⣿⣿⣇⠀⣿⣿⣿⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃⠀⠀⠀⢸⣿⣿⠀⢸⡿⠻⠀⢸⣿⣿⠀⢻⣿⣿⡀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠀⠈⠉⠉⠀⠈⠉⠉⠁⠈⠉⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠟⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠻⠻⠿⠛⠛⢿⣿⡿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣩⡁⣸⢰⣿⢯⣓⡦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢸⡑⣻⣣⡿⠃⠀⠀⡴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡞⣡⠶⠯⠿⠟⠁⠀⠀⠀⢸⣹⠁⠀⠀⠀⣠⣤⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠳⣿⠋⠁⠀⠀⠀⠀⠀⣀⣀⣀⣈⣥⣤⣤⣤⣞⣽⣿⣿⡿⣿⣟⠿⢯⣍⡓⠂
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⠏⠀⠀⣠⡤⠶⠚⠛⠉⠉⠠⠚⠉⠿⡦⠀⠀⢉⣭⠭⢟⡓⠶⢿⣷⣤⣉⠑⠂
⠀⠀⠀⢀⣼⣿⣿⣿⣿⡿⠃⣀⡴⠛⠁⠀⠀⠀⠀⣤⠶⠖⠒⠒⠒⠒⠀⠀⠀⠀⠀⠘⢻⠀⠀⠈⠙⠻⢿⣦⣄
⠀⠀⣰⣿⣿⣿⣿⣿⡿⠁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠙⠻⣷⡀
⠀⠘⢿⢿⣿⣿⣿⡿⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠈⢻⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠶⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣠⠴⠚⣋⣭⣟⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣳⠦⠈⠙⠚⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣽⣂⡀
⠀⠀⠀⠀⠀⠘⠚⠉⡽⠁⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠙⣆⠀⢻⠉⠳⠬⣷⠲⢤⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣸⠀⠀⠀⠀⠠⢔⣀⡉⠙⠒⠤⠤⣘⣯⡉⠙⢲⠒⠦⠬⣷⣦⡀⠈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡟⠀⠀⠀⢀⡞⠉⠈⠉⠓⢄⠀⠀⠀⣀⣰⣤⠈⢧
⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⡏⡼⠁⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢷⣰⠋⠀⠀⠈⡆⠈⢧⣀⡀
⠀⠀⠀⠀⡞⢉⣀⣉⠑⢾⡼⠁⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣷⠀⠳⠆⠺⡄
⠀⠀⠀⠀⢇⠀⠈⠳⠻⠄⠁⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⣠⣀⣸⢠⢴⣄⠀⢀⡇⠀⠘⢯⡍⡣
⠀⠀⠀⠀⠀⠑⠲⠤⠴⢶⡀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣳⣿⠋⠻⠿⠯⠤⠚⣠⠖⠒⡄⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠒⠲⠖⠛⠁⠀⠀⠀⠀⠀⠀⠁⢀⡴⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⢀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠄⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⡠⢀⡴⠒⣒⡧⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡸⢀⠆⠀⠀⠀⢸⠉⣉⣹⠭⠼⠧⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⢱⠓⢺⠀⠀⠀⠀⢸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢖⡓⠄⠛⠀⡞⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⠤⠤⠴⠃⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⡖⠒⠒⠒⢻⢹⡉⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠼⢾⠷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⣀⣠⣤⢾⠒⠲⣍⣹⡆⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣢⠤⠤⠾⠷⢓⣚⡪⠽⠛⠉⠀⠀⠀⢀⣠⣤⣤⣤⠄⠍⠉⠉⠉⠛⠛⠳⠶⠶⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠶⠿⠯⠭⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠟⠻⠎⠙⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠠⠐⠀⠁⠀⠀⠀⠈⠉⠒⠦⢀⣠⠞⠀⠀⠀⠀⠀⠒⠈⢷⣄⠀⠀⠀⠀⠀⢼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠳⢤⣀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⢀⣀⣠⡤⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠀⠀⠀⠈⣇⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⢉⡥⠤⠄⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣀⡀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⡇⠀⠀⠀⠀⠀⢰⠋⠀⣠⣴⡆⠈⢳⠀⠀⠀⠀⡴⢉⣤⣤⠀⠈⢢⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⡶⢸⡇⠀⠀⢰⠀⣿⣿⣷⣶⠀⠀⡇⠀⠀⠀⠀⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⠀⠀⠘⠲⠤⢌⢛⠋⣁⠜⠀⠀⠀⠀⠣⠙⠛⣛⢋⢀⡼⠁⠀⠀⠀⢀⡇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛
⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢀⣄⡀
⠀⠀⠀⠀⠈⢧⡈⠙⠦⣄⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⡤⠖⠋⣠⠟
⠀⣠⠤⠤⠶⠶⠞⠓⠀⠈⠙⣆⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠙⠂⠀⠀⠀⠀⣠⢞⡟⠉⠀⡀⠛⠷⠶⠤⢤⢤⡀
⣾⣃⣈⠭⠅⠀⠀⠈⠉⠙⡆⣜⠀⠀⠉⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠁⢘⡇⡔⠉⠉⠀⠀⠀⢭⣭⣁⣙
⢯⠁⣀⣀⣀⣀⣀⣠⣤⠴⠃⢹⡀⠀⠀⠀⠀⡽⠓⣶⢤⣤⣄⣀⣀⣀⣤⣤⡴⠶⣏⠁⠀⠀⠀⣸⠃⠳⢤⣤⣄⣀⣀⣀⣀⡀⢨
⠈⠉⠉⠀⠀⠀⠀⠀⠸⡄⠀⠀⢳⣀⣀⣠⠞⠁⠀⣿⢢⡀⠴⣿⣿⡷⢄⡌⣿⠀⠙⢦⣄⣀⣠⠏⠀⠀⣼⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠉⠀⠀⣠⠞⣯⠀⠀⠀⢨⣽⡀⠀⠀⢸⣦⡀⠀⠀⠉⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⠤⠖⠚⠁⠀⡃⠀⠀⠀⠈⠛⣇⠀⠀⠈⡇⠉⠓⠦⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⣿⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⣿⡿⠃⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠒⢲⡆⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⢰⠀⢸⠃⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⡴⠋⠀⠘⢧⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⢯⡀⠀⠀⠀⠀⡹⠂⠀⠀⣽
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣧⣦⣤⣤⣷⣄⣀⣠⣾⣥⣤⣤⣴⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣸⡿⠃⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⢀⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⣴⣶⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⢾⠏⠉⢻⣷⡄
⠀⠀⠀⠀⣿⡁⠀⠙⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⡇
⠀⠀⠀⠀⠙⠋⠀⢼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⠑⠦⣄⠀⠀⠈⢯⠳⢄⡀⠀⢀⡀⠀⠀⠀⠀⣀⡛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣴⣶⠀⠀⠀⠠⣤⣤⣤⠼⠀⠀⠀⠉⠒⠒⠋⠀⠀⠙⠦⠼⠏⠲⣄⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣈⣢⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⡤⠤⢤⣀⠀⠀⠀⠉⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡏⠀⠀⠀⠈⢳⡀⢠⠒⠒⢢⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⡇⠀⠀⢸⡀⠀⠀⠀⠀⣸⡇⢻⠉⠀⠀⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣥⣬⡓⠋⠀⠀⠀⠈⢇⠀⠀⠀⠀⣹⠁⠈⠳⠤⠞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠤⠖⣆⠀⠀⠀⠀⠈⠣⣄⣀⡴⠃⠀⠀⠀⠐⢦⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠒⠐⠋⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠦⣄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⠀⠈⠙⠒⣦⢤⣄⣀⣁⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠦⡴⠃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⠺⠟⠀⢰⠁⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠭⠀⣀⣯⣀⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠁⠀⢨⡭⠉⣹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠉⠙⡟⢒⣶⣶⡾⠖⢺⠟⠛⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⠭⠭⠟⠋⠉⠑⠛⠭⠿⠯⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡀⠀⠀⣠⠄⠀⣀⠤⣦⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡠⠀⢼⣤⣾⢟⡦⠄⠤⠜⠻⢷⠛⣷⣶⣄⠀⢲⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠲⡯⠐⡚⠛⣻⢖⠈⠁⠂⠒⠦⠐⣎⡁⠾⡖⢩⠲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⠶⢗⠀⠠⡎⠉⠀⢀⣀⣤⣄⣀⠀⠁⠠⠓⣷⣀⡹⠶⢒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠏⠀⣑⠀⢀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠋⡰⠂⠀⣀⣁⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⢄⡶⣿⣼⠃⠀⠀⠸⠿⠿⠟⠀⠀⢹⣿⣿⠀⠸⣀⠄⠸⡄⠈⠉⡗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣞⡿⣃⡀⠀⢤⠀⠀⠤⢀⡔⠀⣠⣿⣿⡟⠀⠐⠋⢓⠸⣀⣀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⡅⠀⣦⠀⡰⠋⠠⡟⠍⢀⣾⣿⣿⠟⠀⠀⡀⢠⡎⠈⢉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⢁⣀⡭⣦⣳⠸⣏⠀⠀⣾⣿⣿⠋⠀⠀⠰⠛⢻⠷⡄⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣿⠂⠀⠻⣼⣋⣀⠀⠉⠉⢉⠤⣠⡒⠒⢢⠁⠹⡑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⡄⠊⣱⣿⣿⡁⠀⣿⣿⣿⠀⠈⣕⣦⠬⠀⠀⠈⠉⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⢀⡧⢰⡟⣷⡿⢷⠀⠉⠉⢉⣠⢼⡥⢤⠵⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠷⠞⠛⠀⠤⠉⠌⢉⡀⠀⠀⠰⠄⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠋⠉⠛⠛⠛⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠤⠄⠀⡤⠤⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠿⠂⠀⠀⠷⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣠⣴⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢲⢰⠃⡖⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⢵⣏⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⡀⣤⣶⣦⡄⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⡟⠋⠉⠻⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⣾⡿⠋⢿⣿⣿⠏⠻⣿⣦⣄⠀⠀⠀⠀⣿⣿⣿⣿⠟⠋⢀⣴⣄⠀⠀⢀⣦⡀⠈⠻⣿⣿⣿
⠀⠀⠀⠻⠿⣿⣷⣶⣤⣭⣥⣴⣶⣿⡿⠿⠃⠀⠀⠀⣿⣿⣿⣧⣀⡀⠈⠉⠛⠓⠚⠛⠋⠁⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⡀⢉⣿⣿⣿⣿⣿⣏⠁⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡼⠉⠉⠀⠀⢸⠀⠀⠈⠉⠹⡄⠀⠀⠀⠀⣿⣿⣿⣿⡏⣴⣶⣿⣿⣿⣾⣿⣿⣿⣷⡹⣿⣿⣿
⠀⠀⠀⣼⣧⣤⣒⣒⣒⣼⣔⣒⣒⣢⣤⣿⡄⠀⠀⠀⣿⣿⣿⡟⠸⠿⠿⠭⠭⠟⠹⠭⠭⠿⠿⡇⢹⣿⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿

⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡼⣻⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⣈⠀⠈⠘⣿⣿⡇⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⢉⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⡌⢈⡉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⢻⣿⣿⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠓⠚⠋⢸⣿
⠀⠀⡠⠖⢒⣛⣛⣛⠒⠲⡄⠀⠀⠀⣾⣿⡿⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⡦⢐⣷⡆
⠀⠀⡓⠶⠤⠭⠭⢭⢤⠔⣻⠀⠀⠈⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⡂⠀⠐⠹⠆⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⢏⠒⠦⠤⠤⠤⠀⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠗⠷⠚⠛⠒⣄⠀⠀⠀⠀⠀⠀⠈⠲⡕⠄⠀⠀⠀⢸⣿⣶⣿⡿
⠀⠀⠀⠁⠒⠀⠂⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠸⡇⢠⠤⠔⠒⢲⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠸⡜⢋⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢹⠀⠀⠀⠀⠀⠀⢻⡈⠉⠁⠀⠈⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠄⠄
⠀⠀⠀⠀⠀⠰⡶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣀⣰⢶⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠖⠚⠁⠀⠀⠀⠀⠀⠈⠻⣏⣀⡇
⢀⠤⠚⠆⠀⠀⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⢠⠑⠁⠀⠈⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠰⡒
⠀⠀⠄⠀⢃⠀⠀⠀⠀⠀⠈⠉⠛⢦⣘⣷⠀⡀⣠⣛⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠰⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⢾⣿⣿⣿⣦⠀⠀⠀⠘
⢂⠉⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂⠁⠀⠀⠀⠀⠀⠀⠀⢷⣺⣿⣿⣿⡿
⠀⢣⠄⢀⠈⢸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⠛⠛
⠀⠀⠳⡤⢤⠊⡧⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⢀⠀⠰⠷⣶⣶⡶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⡝⢶⡆⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠊⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠤⠒⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡉⠢⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠑⠒⠀⠉
⠀⢰⠀⠀⠀⠀⠀⠤⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⢀⠀⠼⠂⣈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡾⠑⠒⠓⠒⠬⣉⠹
⠀⠀⠓⠀⠀⠀⠀⡀⠙⢄⡠⠀⠀⠀⠀⠀⠀⡔⠁⢠⠀⠈⡀⢀⢠⠀⢑⡬⠂⠀⠀⠀⠀⠀⠀⠁⣀⣠⣶⣒⣓⣓⣶⣀
⠀⠀⠀⠀⢀⡴⠛⠷⡤⠊⠀⠀⠀⠀⠀⢠⠎⠠⠀⠀⠠⠄⠀⢠⡤⠒⠁⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠢⡀⠀⠒⢀⡤⠚⠁⠀⠀⠀⠀⢀⣸⡍⢆⠀⠀⠀⠘⣿⣿⣉⣉⡉⠛⠛⣿⣿⡇⠀⠀⠀⢠⣤⡴⢦
⠀⠀⠘⢄⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡒⢈⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⢧⣠⠏
⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣁⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⢄⠙⠦⢄⣀⠀⠀⢀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢳⠀⠀⠀⠀⠉⠉⢀⠴⠒⠚⠛⠒⠲⢤⡀⠀⠀⠀⡞⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡇⠀⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠰⠀⠀⣰⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠻⣄⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⣶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠉⣻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡇⠀⠛⠦⠴⠶⠖⠩⠩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡇⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⢠⣇⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⣾⢿⡐⣆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⢸⠃⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡞⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣄⠀⠷⣚⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⡻⣧⠤⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠑⠐⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠰⠍⠉⠉⠋⠙⠭⡆⠀⠀⡠⠤⠤⣤⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⣰⠂⠀⠰⠂⠀⠀⠀⠀⠋⠁
⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢲⡐⣀⣿⠀⠀⠈⢹⠁⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠄⠁⠀⣠⢀⡀⠀⠀⠈⠀⠁⠁⠀⠀⠀⢸
⠀⢀⡴⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠈⠉⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠠⠆⠀⠀⠀⢸⠀⠀⠀⠁⠀⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠰⠃⠀⠀⠐⠀⠀⠀⠠⣂⠄⠀⠀⢰⠀⠀⠃⠀⠀⠰⠘⡄⠀⠀⣗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⡠⠄⠈⠁⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠹⠁⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⡀⠀⠀⠀⠀⢀⡠⠒⠉⣠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⡀⠀⠁⠀⠌⠂⠀⠀⠀⣀⠔⠉⢀⡴⠊⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⠄⠂
⠀⠀⠀⠄⠈⠀⠠⠀⠀⠀⠀⢀⠤⠊⢀⡠⠞⠁⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⣀⡠⡠⠁
⠤⠈⠀⡀⠄⠀⠀⠀⠀⡠⠖⠁⣀⠔⠋⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠕⠂⠀⠀⠀⠒⠀⠀⠀⣴⣾⣧
⠘⠁⠀⠀⠀⠀⣀⠔⠊⢀⡤⠚⠁⠀⠀⠂⠀⠀⢀⠔⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠀⠸⣿⣿⡡
⠀⠀⠀⠀⡠⠞⣡⢀⠖⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠰⠋⠁⠀⣠⠺⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢈⠀⣀⠔⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⣀⡀⡄⠀⣀⠈
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣤⣤⠤⠔⠒⠈⠉⠁⠀⠀⠀⠀⠀⠁

⢠⣴⣶⣿⣷⣶⣾⣷⣶⣤⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣷⣦⣶⣶⣆⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄
⠀⠻⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣴⣾⣿⣿⣟⣛⡉⠙⣋⣻⡇⠀⠀⠀⠀⠀⣿⣿⣿⣭⣭⣽⣿⣿
⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢀⣠⣀⣀⣀⡀
⢀⣴⡿⠛⠛⠛⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣴⣴⣤⣤
⢾⣿⠀⠀⠙⠿⠿⢿⣿⣷⠀⠀⠀⠀⠀⣰⣿⣿⣯⣁⣉⣀⣽⣿⣿⣦⠀⠀⠀⠀⠻⣿⣶⣤⣭⣥⣶⣾⡿
⠘⢿⣷⣤⣤⣤⡤⣾⣿⠃⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣿⡿⠋⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟
⠀⠀⠙⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣠⣄
⢠⣶⠿⠛⠻⠿⠛⠻⢷⣦⡀⠀⠀⠀⠀⠰⣿⣿⣛⣛⣛⣛⣛⣿⣿⡆⠀⠀⠀⠀⣴⣿⣟⡙⠛⠛⣛⣿⣷
⠀⠉⠻⣾⣶⣶⣶⣿⣿⡟⠁⠀⠀⠀⠀⠀⢻⣿⣟⠛⠛⠙⢛⣿⡟⠀⠀⠀⠀⠘⣿⣿⣿⣭⣉⣩⣽⣿⣿
⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣴⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀
⠠⣾⣿⣯⣿⣿⣽⣿⣿⡦⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣷⣄⣀⠀⠀⠀⠠⣶⣿⣿⣛⣿⣟⣿⣿⣷
⠀⢹⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠞⠉⠉⠉⠉⠛⠛⠒⠶⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣄⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠂⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⣿⣦⠀⠀⠀⠀⠀⠹⠿⠿⠛⠻⠻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⣠⡤⠴⠶⠶⠦⠤⠤⠄⣀⣀⣀⡀⠀⠀⠀⢉⣠⠾⠁⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠁⠙⣦⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⠒⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠴⣞⣽⣃⣶⠀⠀⠀⠀⠰⣟⣛⣻⣿⣿⡑⡄⠀⠀⠀⣿⣉⠀⢊⠁⣿⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠊⠉⠉⠉⠈⠀⠙⡄⠀⠀⠉⠉⠀⠀⠉⠉⠁⠀⠀⠀⣿⣀⠄⠉⣰⣇⡀⠀⠀⠀⢰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣀⠠⣤⣤⣄⠀⢸⠀⠀⠀⡀⠀⣤⣤⡀⠀⠀⠀⠀⠈⢷⡀⣸⠋⠈⢳⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠂⠀⠀⠀⠈⠉⠉⠀⡌⠀⠀⠀⠑⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⣧⠃⡴⠀⢘⠀⠀⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⡀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠐⠀⡟⠀⠀⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠃⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠋⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠠⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⢰⡿⠶⠶⠀⠀⠀⠒⠒⠻⢧⡀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠸⠤⠔⠒⠒⠒⠒⠂⢐⠂⠘⠁⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢰⠲⣄⠀⠀⠀⠀⠀⠈⢶⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⣠⣤⣤⣤⡬⣷⡌⠳⣄⠀⠀⠀⠀⠀⠈⠓⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡼⠁⢀⠀⡀
⠀⠀⣾⠿⠶⠾⠀⠀⠀⠴⢤⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢻⣾⠛⢛⠛⢿⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣀⡙⠁⠉⠿⠽⣟⣷⡀
⠀⠀⠳⠤⠤⢤⣤⣀⣀⣀⣠⠷⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠏⣠⣎⠁⠀⢀⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⣘⠁⠀⠀⠀⢀⣁⣀⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡅⠀⢀⡽⣄⠀⠀⠀⠀⢀⡴⢿⠀⠃⠘⢻⣿⡿⠆⠴⠋⠀⠀⠉⡿⢦⠀⠀⠀⠀⠀⠀⣠⠋⠀⠉⠳⠚⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡁⠀⠈⠳⣄⠀⣠⠞⠁⢹⠀⡀⢠⣾⣿⡇⠔⠀⠀⠀⠀⣸⠃⠈⢳⡄⠀⠀⠀⠴⠉⠑⢢⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢹⠋⠀⠀⢰⢀⠀⢸⣿⣿⠃⠀⠀⠀⠀⣰⢦⡀⠀⠀⠙⢦⡞⠁⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣾⠀⣀⡴⢻⡆⠄⢸⣿⠃⠀⠀⠀⠀⢰⠏⠀⠙⢦⡀⠀⢸⡀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠚⠉⠀⠘⡇⠀⢠⠁⡠⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠦⣌⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢠⠀⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠈⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢠⡅⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣾⠻⣄⣀⣀⣀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠲⡍⠉⠉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡦⠄⠘⣷⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢤⣀⠀⣷⣶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠚⠛⠁⠀⣀⡈⣻⠋⠠⠀⢹⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠐⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠘⠂⠇⠀⠀⠈⠈⡃⠀⠀⠀⠃⠀⠀⠀⢀⠄⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠀⠄⠂⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣤
⠀⠀⠴⠒⠒⠒⠒⠲⠒⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⢱⡆⠀⠀⠀⠀⠀⠄
⠀⠀⡇⢀⠖⠉⠁⠉⠢⡀⢰⠉⠉⡆⡊⠀⠙⠄⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠞⠃⠀⢀⠀⠀⠀⢀
⠀⠀⡇⡌⠀⠀⠀⠀⠀⠱⠈⡲⣒⠁⢙⡶⡊⠀⠀⠀⠀⠀⠀⠀⣀⠀⣃⠀⠀⠠⢀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠛⠄
⠀⠀⡇⠡⡀⠀⠀⠀⠀⡔⢸⠀⠀⡇⣇⠀⢈⠀⠀⠀⠀⠀⠀⠀⡅⠀⠬⠀⠀⢀⣀⣀⡀⠀⠠⠀⠰⠀⠀⠀⠀⠁⠤⠀⠄
⠀⠀⣇⠀⠑⠢⡤⠄⠊⠀⠀⠉⠉⠀⠈⠉⠁⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⠀⠤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⠌⡆⠀⠀⠀⡇⠘⠀⠀⠀⠀⠅⡀⠀⠀⠀⠀⡈⠂⠀⠰⠰⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠁⠀⠀⠁⠀⠀⢮⡰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡾⢩⠏⢿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢤⢀⠠⢄⡀
⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠄⠀⠀⠀⡅⡆⠀⠹⢍⡘⢹⠀⢀⠔⠛⠋⠲⡄⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠀⠀⢻⡀⢀⢈⡬⢷⡾⠀⢸⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⠀⢠⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠢⡉⠓⠛⠛⠉⠀⠀⠀⠓⠤⠤⠖⠁⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⢢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⢄⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠒⠀⠘⠀⠀⠀⠀⠸⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠄⠀⠀⠀⠀⡀⠀⠀⠡⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠘⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠲⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⣀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⠉⡿⢷⣤⣶⡞⠛⠀⠀⠀⠀⠠⠿⣦⣤⣤⡴⣞⠃⠀⡀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⠀⠀⠀⠀⡅⠠⠀⡀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣀⡀⠀⠀⠀⠀⠀⠒⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣇⠀⠀⠀⠀⢠⠚⣥⠄⠡⠷⡄⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠀⠻⣦⣀
⠀⣠⡾⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠁⠒⠤⠜⠃⠀⠀⠀⣀⡴⣖⠀⠀⠀⠀⢀⡀⠀⠀⢈⡽
⢸⡏⠃⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⣻⠏⠀⠈⠳⡾⠛⠶⠿⠉⠛⠛⠋
⣟⡡⠀⠃⢸⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⢿⡇⠀⢀⡟
⠘⣧⠀⡀⠀⢰⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣿⠀⠀⣼⣇
⠀⠸⣆⡀⠂⠀⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠛⠆⣈⣁⠈⣀⠸⢧⣀⡀
⠀⠀⠈⠙⢳⡆⠀⠀⠀⠈⠙⠛⠓⠶⠶⠚⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠙⠛⠳⢦⣀
⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣄⣀⣀⣀⣀⣠⣤⣤⡊⠀⠀⠀⠀⠀⠀⢀⣶⠛⠛⠳⢦⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠈⣻⠀⠀⠀⠀⠐⠂⢸⣷⠀⠀⠀⠀⢹⡄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢀⡼⠃⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⠖⠛⠋⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⣦⣀⣤⣤⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠉⠀⠐⠀⠀⠀⠀⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠃⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠃⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⡀⠠⣦⡀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠀⠀⢸⢛⡇⠐⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠁⢠⣦⢸⣿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠛⠛⢾⣧⣴⡿⣷⣇
⣿⣿⣿⡿⠿⡿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⡀⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠉⠀⠐⠉⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⣀⡤⠤⢀⡀⠄⢠⠄⣀⡀⠀⠈⢤⡀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠈⠻⣔⠒⠒⠚⠋⠤⠄⡀⠀⠀⠀⢀⣀⠄⠃⠚⠓⠂⠹⠋⠀⠀⠀⠻⠟⠒⠒⠒⠐⠂⠈⠛⠃⠀⠀⠬⠓⠉⠉⠀⠒⠀⢒⠓⠃
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⡠⠊⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠀⠀⠈⠂⠀⢀⡠⠄⠠⡀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠉⠀⠈⠁⠊⠁⠀⠈⠲⣄
⠀⠀⠀⠀⠀⡀⠐⠁⣀⡄⠐⡐⠂⠒⡴⠀⢩⠔⠈⢰⠑⢄⠀⠄⠀⠀⠀⠀⠀⢀⠄⠊⢀⣀⡄⠰⠒⠒⢰⠆⠒⠰⡆⠄⡀⠑⢤⡀
⠀⠀⠀⠐⠚⠁⠲⢭⣜⢆⣀⡶⡄⠀⠇⠀⣸⠷⠂⠈⠀⠈⠉⡩⠒⠀⠀⠀⠻⣋⠚⠈⠁⠀⣁⣉⣀⠒⠊⡁⣀⣈⣁⣀⠉⠐⠒⠚⡻
⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠢⣖⠀⠁⠀⠀⠉⠐⠛⠒⠈⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠠⠤⣀⡠⠔⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠘⠈⠀⡄⠀⠒⠢⠆⡀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⡰⠃⠀⣀⣀⢠⠂⠁⠀⠀⠀⠀⣓⣦⣀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⣠⠤⠈⡇⠀⢸⠀⠀⠀⡇⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠶⡄⠀⠈⠁⠀⠀⠀⠑⠂⠂⠐⠁⠀⠀⠀⠈⠁⠀⢒⡳⠆⠀⠀⠈⠀⠀⠀⣀⠈⠀⠉⠀⠂⠐⢣⠀⢀⠁⠒⡈⠀⡜⠂
⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠀⠠⠼⠀⢀⡥⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠤⠀⠀⠠⠤⠤⢀⡀⠠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂


⡿⠿⠿⢿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠀⣠⠔⠊⠉⠁⠀⠀⠀⠉⠉⠒⠤⣀⠘⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠊⠀⣠⠤⠒⠈⠉⠉⠉⠑⠲⢤⣀⠈⠳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣾⠁⢠⠞⠁⢀⣀⠀⠀⠀⠀⢀⣀⠀⠈⢣⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⢸⠀⠀⠃⠈⠀⠀⠀⠀⠀⠈⠁⠀⠈⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠸⡄⠀⠀⠀⠰⢄⣀⠴⠀⠀⠀⠀⣸⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠳⡀⠘⠢⢤⣤⡤⠤⠤⠤⠤⡤⠴⠚⠁⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⣿⣿⡯⢭⡭⣭⣭⣭⣭⣭⣿⣿⡇⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⢀⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⡇⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⡏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣀⡼⠁⠳⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠲⣶⣖⣒⣖⣶⣒⡶⠒⢦⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⠀⢀⡀⣀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⡲⠒⠆⣀⣀⣀⣀⣀⡼⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⢰⠁⠀⠈⡇⠀⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⠀⠈⠳⠤⠔⡇⠀⡇⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣀⠤⢄⠀⢠⡇⠀⡇⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣩⠃
⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡠⡏⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣀⣀⠀⠀⣀⣀⣠⠤⠴⠼⠮⣍⠒⠃⠋⠁⠚⠐⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠒⠼⠦⠬⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣛⠂
⠀⠀⠀⠀⠀⠀⠶⣶⠦⠴⠞⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣀⣤⣤⣤⣤⣀⠈⠙⣆
⠀⠀⠀⠀⣠⠤⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⡘⢾⡜⣡⠖⠋⠁⠀⠀⠀⠀⠈⠳⡄⠀⠑⢄
⠀⠀⠀⠙⠓⠒⣦⠖⠁⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠙⢾⡟⠁⢀⣄⣀⡀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢳⡀
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⣙⠳⣤⠀⠀⠀⠀⡇⠀⠀⢻⠁
⠀⠀⠀⢠⣣⢴⠆⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠀⠀⠉⠳⣎⠁⠀⠀⠀⣿⠀⠀⢸⡄
⠀⠀⠀⠋⠁⢸⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣿⣄⣀⡀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠞⡋⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢍⡁⢀⡇
⠀⠀⠀⠀⠀⢸⠈⠩⠭⣷⠀⠒⢶⠶⠶⠾⠟⠛⠋⠉⠀⠘⠿⠂⠀⠘⠗⠀⠀⠀⠀⠀⢠⡿⠀⢠⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠈⡇⠀⠀⢹⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣾⠀⠀⠀⡾
⠀⠀⠀⠀⢀⣀⣸⡄⠀⠈⡆⠀⠀⠈⠛⢦⣤⣀⣀⣤⣤⡶⠂⠀⠀⣀⠀⠀⠀⠴⢾⡯⠀⠀⠀⣇⠀⣀⠴⠗⢒⢶⡄
⠀⠀⠀⣴⠋⠉⢀⠈⠓⠊⠁⠀⠀⠀⠀⠀⠀⠠⠴⠿⠋⠀⠀⠀⠘⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣴⢾⡉⠀⢹⡄
⠀⠀⠀⣇⠀⠞⢩⡿⣆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢻⠇⠀⢸⡇
⠀⠀⠀⠹⣄⠀⠘⠀⠹⠀⣧⠀⠀⠀⠀⢀⣤⣖⡋⠉⠉⠉⠉⠀⠀⠈⠉⠉⢙⣶⣄⠀⠀⠀⠀⢀⣷⣄⠀⠉⠀⣀⡾⠁
⠀⠀⠀⠀⠈⠳⠦⣤⡤⠞⠹⡆⠀⠀⠀⣾⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣶⣶⣿⣿⣿⣇⠀⠀⠀⣼⠃⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠙⢟⣿⠿⠿⠭⢌⣙⣉⣩⠿⠿⠟⠻⣿⣿⠏⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠉⠒⠦⠤⠤⣄⣀⣠⣤⣤⠤⠖⠋⠁⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁
⣿⣿⣿⣿⣻⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣻⣿⣛⣻

⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⣄⣠⣀⣀⣀⣀⣀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠆
⠀⠀⣉⠙⠛⠉⠛⠋⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣤⡄
⠀⠀⠗⠹⠱⠮⠾⠱⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠹⠄⢀⡀⣀⡀
⠀⠀⡴⣤⡆⣶⡤⡤⣴⡦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣧⣿⣿⡄
⠀⠀⣤⣤⡀⣀⣀⣄⡀⣀⣅⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⠀⣿⣿⠗
⠀⠀⠉⠉⠙⠉⠋⠋⠈⠋⠁⠉⠊⠁⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⣿⠇⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠀⣼⣿⣿⠀⠀⠈⠻⠿⠋⠁
⠀⠀⠸⠿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠠⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⢻⠿⠋⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠂⠀⡠⠶⠖⠀⠖⠀⠆⠀⠆⠀⠛⠒⠠⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⣀⣈⠁⠀⢠⠂⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣀⠀⢐⠀⠀⠁⠀⠀⠄⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⡯⠖⠀⠀⢹⠁
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⠉⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⢒⠒⠐⠂⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠓⠋⠉
⠀⠀⠀⠀⠠⡀⢰⠀⠀⡆⡆⠘⠀⠀⡇⠀⠀⡏⠀⠀⡔⢸⣽⢲⡄⢁⣈⠀⠀⢀⡀⠐⠃⠀⢀⠉⠀⠀⠘⡁⠀⠒⠒⢀⡀
⠀⠀⠀⠀⠀⠁⠈⢀⠟⠛⢻⡁⠀⢠⡇⠄⢠⠃⠀⠸⣚⣚⣗⣰⡇⢀⠀⠀⠀⠈⠀⠀⠀⣰⠛⢿⡄⣴⠀⠀⠀⠀⣄⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⡇⠀⠈⠁⠀⠀⠙⢋⠏⠃⠄⢀⠁⠀⡀⢠⠀⠀⠀⠘⠳⠾⠃⢸⠀⠀⠀⢰⡟⠊⠃⠀⢠
⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⠀⣿⠀⠘⠓⠀⠀⢸⢠⡖⠀⠸⠀⢰⠀⢸⠀⢀⣇⢀⠀⠀⢰⠛⠉⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠈
⠀⠀⠊⠀⠀⠀⠀⠈⠆⠀⢰⠏⣀⠯⠀⠀⠀⠈⢸⡇⠀⠀⠀⢸⠀⠈⠀⠐⠶⢺⡀⢀⡀⠧⠔⠰⠶⠆⠀⠀⠐⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢒⡋⣀⢹⠀⠀⡄⠀⠀⠠⠁⠆⡏⠀⢸⠀⠀⢠⡆⠀⠀⠀⠈⢱⢸⠀⠀⢆⠉⠀⠀⠀⢰⡇⠀⠁
⠀⠀⠛⠒⠋⠁⠀⠀⠀⠸⡝⠹⢸⡀⠀⠳⠀⠀⡄⠀⠀⠁⠴⠞⠀⠤⢹⠀⠘⠌⠀⠀⠈⢘⠀⠀⢸
⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠑⠃⠀⢡⣤⣤⡄⢰⠁⠀⠀⠀⢠⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠇⠀⠀⠀⣀
⠐⠒⠒⠒⠒⠒⠒⠀⠒⠒⠓⠒⢢⣤⣤⣤⡄⢸⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⡄⠀⢀⡃⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠘⠀⠀⠂⠀⠀⠏⠀⠀⠀⠀⠀⠋⠀⠀⠄⠀⠃⠀⠀⠳⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢤⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⣴⠋⠙⢒⡞⠁⠀⠀⠀⠙⠛⠤⢤⠀⠀⠀⠀⠀⠀⠈⢦⣉⡓⠦⣄⡀⢀⡠⠖⠉⠉⠙⠛⠒⠶⢤⡀
⠀⠙⠦⣄⠈⠳⣄⠀⠀⢀⠜⣿⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⢦⡞⠀⠀⠀⠀⠀⠀⠀⣼⣿⠉⠓⠦⣍⠛⢷⣦⣄⡀⠀⠀⠀⠀⢸⡇
⠀⣀⡀⠈⠓⠆⠈⠁⢠⢎⠜⣸⣧⡀⠀⠈⠻⢶⣶⣴⠾⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠻⣹⣄⡠⠀⠀⠙⠲⢌⡻⢿⣶⣤⡀⠀⢸⡇
⠀⠈⠙⠲⢤⠀⠀⣰⣧⠎⡴⣿⣬⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠐⠁⠚⠉⠛⡆⠙⢿⣿⣦⣾
⠀⠀⠀⠀⠀⠀⠰⠋⠏⡼⠁⣿⢿⡛⠈⠳⢤⣀⣠⠤⠤⠒⣮⢥⠀⠀⠀⠀⠀⠀⠀⢸⢠⣤⡀⠀⠀⢠⠔⠔⠀⣻⢀⡀⠙⠫⣿⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⡠⢈⡓⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠋⠀⠀⠀⠀⠀⠀⠀⠘⡆⣠⠁⠀⠀⠀⢠⢵⣷⣏⢸⣷⡄⠈⣷⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠃⠀⠀⠀⠬⠥⠀⠀⡄⠨⠥⣄⠀⠀⠀⠀⠀⠀⠀⣠⠟⠈⡤⠆⠀⠀⠀⠀⣧⠀⡇⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠒⠀⠀⠀⠀⠀⠀⠒⠛⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠙⠂⠀⢀⣀⠀⠀⠀⠠⠤⠎⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⠟⢹⣤⠋⢳⠇⠲⡖⠒⢒⠖⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢩⠏⠀⠀⠀⢀⣤⣤⣤⠀⠀⠐⣿⡇
⠀⠀⠀⣠⣶⣿⣿⣿⣀⣅⣰⡾⢡⢠⠁⠜⠁⠉⠛⠋⡰⠞⠁⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⣿⠋⠀⣀⣄⠀⠀⠈⢴⡟⣦
⠀⠀⠀⢿⣿⣿⣿⢿⣿⣿⣿⡇⠂⠂⢀⠎⠙⢲⠖⠊⠀⠀⠀⠀⠀⠀⠀⠱⣕⢄⠀⠀⠀⠀⠘⠦⠴⠚⢧⠈⠙⠋⣿⠋⣠⣿⠀⢸⣷⡟
⠀⠀⠀⠀⠙⣻⣿⣿⡿⠟⠉⢧⠀⠀⠈⣆⠀⡸⣤⠤⠤⣄⠀⠀⣠⡴⠀⢠⢬⣧⠑⡔⠳⡄⠀⠀⠀⠀⢸⣀⣠⠤⢝⣫⣽⠮⣷⡿⠏
⠀⠀⢠⡎⠉⠉⠉⠉⢦⠀⠀⠈⢧⠀⠀⠘⡶⢥⡹⢂⡖⠉⠐⠻⣏⠀⠀⢌⡒⠆⠺⠞⠁⣸⠀⠀⠀⣠⠞⣩⠴⡚⠉⢈⣿⣷⡀
⠀⠀⠀⣿⣶⣔⡲⠦⡈⣆⠀⠀⢈⡧⠀⠀⡷⠀⢰⡦⠬⠂⠀⠀⠹⡄⠀⢺⣉⣛⠊⠀⢠⡏⠀⠀⡴⣡⣞⣥⡴⠾⣿⡋⠈⣝⣧
⠀⠀⠀⠘⣷⡉⠻⣦⡀⢹⡀⣠⠊⠓⠀⢛⠉⠒⠒⠀⠀⠀⠀⠀⠀⠓⠀⠀⠙⠮⣅⡀⢠⠟⠆⣰⣟⠘⣻⡗⢼⣦⠸⣷⣄⢿⣿⣇
⠀⠀⠀⠀⠹⣧⠀⠀⠀⣀⠔⠙⢷⣄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠋⣠⢴⢁⣯⢲⡿⣅⣼⠳⠁⢿⡷⡠⣳⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠙⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠃⢹⢏⡿⡓⠸⠁⠉⠀⠀⠀⠳⠑⠑⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠉⠉⠳⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠃⠀⣀⠜⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡬⠂⠀⠀⠀⠀⠀⠀⠀⡃⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠉⠁⠒⠒⡆⠀⠀⠀⠀⠀⡝⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡨⠀⠀⠀⠀⢀⣤⣄⡅⠀⠀⠀⠀⠀⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠹⠀⠀⠀⢀⠨⣀⠨⡄⠀⠀⠀⠀⠀⡀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠢⡀⠀⠀⠀⢀⣀⢀⡞⠉⡗⠀⠀⡁⠀⠀⠀⠀⠸⡶⠅⢪
⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠠⠉⠐⠐⠶⣚⠁⠀⠀⠀⠈⠉⠉⠀⠈⠛⢻⣻⠃⠀⠀⠀⠀⠀⣫⣁⡯⣀
⠀⠀⠀⠀⠀⠀⠀⠅⡠⡍⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠣⠄⠀⠠⠔⣊⢙⣯⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⠒⠲⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠉⠘⣼⣻⠀⠀⣷⠘
⠀⠀⠀⠀⠀⠀⠀⡘⠶⣤⡞⢉⡱⠈⠉⣽⡄⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠜⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠘⢄⢠⠄⠀⠈⠓⠒⠦⠏⠢⡜⠗⠊⠙⡀⠀⡔⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣀⣀⣀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⠚⠈⠉⠀⠀⢁⠀⢠⠐⠉⠀⠀⠀⠀⠈⢱⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢉⠉⠉⣤⣹
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠⠖⠛⠀⠈⢄⡤⠶⡄⢀⣀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡄⠀⡟⠋
⠀⠀⠀⠎⠉⢒⣁⠀⠀⣠⣬⡓⠈⢁⠀⠈⢡⠐⠁⢺⠀⣥⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⠀⡅
⠀⠀⠀⠣⠤⠏⠉⠀⠘⣇⡀⣹⣤⠼⠀⠀⠈⢄⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣘⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠀⠒⠒⠒⠂⠐⠚⠛⠛⠓⠂⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⣉⠯⠍⢙⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡴⢿⣿⣿⣿⣷⣄⠀⠂⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣍⡟⡛⠻⣿⣿⣿⣿⡇⠀⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡷⢦⡈⠻⣿⣿⣶⠧⣤⣀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡇⠶⣿⣷⢦⣥⣙⠒⠠⠤⠿⠊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⣿⡿⠇⠹⡞⠛⠂⠯⣿⣿⠓⢳⣎⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠝⠀⠀⠀⠈⠁⠀⠀⣀⠀⠈⠁⢸⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀⠀⠀⢒⠶⡶⠍⡹⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠉⠈⠳⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⠀⠀⠀⢰⣆⠀⠓⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⠒⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣷⠈⠣⡀⠀⠘⣿⣿⠷⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢤⣴⣶⣾⣿⣿⣷⣻⣿⣿⡄⠀⠘⠒⠦⠄⠀⠘⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣠⢞⠉⠂⠀⣿⣿⣿⣔⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡿⠯⣼⣿⣲⡁⠈⠉⠀⢠⢯⢿⣿⣿⣷⣾⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⡔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢟⢿⡁⠤⠜⢫⣿⣽⣦⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣻⠀⠀⠀⠀⠀⠀⠀⠀⡜⢿⠞⣁⣦⣤⡤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣻⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⡋⣽⣷⣶⣶⣮⡳⣾⣴⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣧⠇⠀⠁⠈⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⢟⣻⠃⣤⣿⣿⣭⢻⡿⣻⣶⣿⢿⣯⣿⣿⢿⣽⡿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⣿⣶⣿⣿⣿⠏⣿⣿⣿⡽⣿⣿⢦⣿⢟⢿⡟⣟⢹⡟⣿⣹⣿⣾⣿⣿⣿⠛⠈⣿⣿⣿⣵⠀⠀⠀⠀⢀⡼⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣭⣻⣟⢷⣼⣿⣿⣷⣟⣯⣝⡿⠏⢋⣺⢍⡂⣿⡝⢋⢮⣿⣿⣻⠿⣿⢼⣿⣿⣿⡿⡄⠀⠀⡄⢻⣄⠃⢦⡤⠀⠰⠳
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣭⣿⢷⠿⡬⣹⡵⣸⣿⣿⣟⣛⣽⣿⣷⣶⢿⣿⣿⣿⣅⠘⢿⡄⠘⣻⢯⡿⢽⡅⢸⣿⣿⣿⣧⠋⢀⡴⠃⠀⠉⠓⠢⠄⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⣾⡿⣵⣷⡷⠯⣽⡿⠿⢴⣿⣿⣻⣽⣿⣟⣿⣿⣿⡧⠴⢀⠈⢠⣅⠀⣧⠘⠿⠧⠺⡟⠁⣸⣿⣿⢿⠗⢀⣎⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⣴⢋⣤⡽⡯⢋⣻⣛⣍⣀⢘⣻⡏⠋⢬⣿⣿⣿⣾⣶⠿⠩⠀⠀⠘⠋⢀⡏⠂⠀⠀⡀⡡⠀⢹⣿⢃⡾⣾⠋⠃⠀⠀⠀⠀⠄⠀⢀⡞
⠀⠀⠀⠀⠀⢀⢟⣽⠟⢂⡤⠤⠀⠀⠀⠘⠤⣭⣇⠉⠛⠻⠧⠽⠿⠿⠧⠀⠀⠀⠀⠀⡸⠀⠀⡀⠠⠜⠃⠀⣸⣿⣋⡴⠁⠀⠀⠀⠀⠀⡀⠀⠀⡼
⠀⠀⠀⠀⠠⠀⠋⠀⠀⠁⠀⡴⠀⠀⠀⠀⠀⠀⠉⠃⠀⠘⠒⠈⠀⣀⠠⠲⠀⠀⢂⠜⠁⠀⠀⢡⠀⠀⢈⠀⡇⡍⣿⡀⠀⠀⠀⠀⠀⠈⠀⢀⠜
⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠰⠁⠀⡐⠂⡀⠀⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⡑⠌⠈⠰⡀⠳⡟⠷⡆⠀⠀⠀⠀⣀⠄⠋
⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠄⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢀⡀⠀⠈⠉⠁⠠⠔⠀⠀⠀⠈⠁⠈⠀⠀⠤⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠐⠂⠠⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⢀⡶⠀⠢⠀⠀⠁⣐⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠂⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠘⣛⣛⣛⣛⣛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⠤⡀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠛⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠟⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣄⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠰⠈⠓⡦⠤⠼⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠛⠄⠘⢏⢢⣀⠀⠀⠀⠀⠀⠀⢀⠜⠒⠈⠁⠀⠉⠁⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠪⠉⠂⢌⠢⡀⢼⠀⢂⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠣⡀⠀⠁⠚⢼⡆⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠈⠢⡀⠀⠀⠀⣀⣴⡄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠢⢀⠀⠈⠀⠈⠀⢀⠰⠛⡐⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠑⠠⠄⠂⠁⢸⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢰⠀⠀⠁⠒⠒⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣶⣶⣶⣾⣿⣿⡆⠀⠀⠀⠀⠀⢰⣶⣤⣤⣤⣤⣴⣶⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⡥⣄⠀⠀⢄⠀⢀⣠⠀⠀⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠁⠀⠀⢦⠀⠀⠱⠂⠀⢀⣀⣀⣨⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠱⠶⠖⠁⠀⠙⠛⠁⠀⢀⣀⠁⠊⡠⠂⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⣸⡇⠈⠀⠀⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠉⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣼⡇⠀⠀⠐⠒⠉⡄⠀⠸⣠⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠸⠿⠇⠀⠀⠀⠀⠀⠸⠆⠀⠛⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⠀⠁⠀⠀⠀⠡⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠛⠃⠀⠀⠀⠈⠀⠀⣀⡜⠁⣀⠆⠀⠀⠀⠀⠀⠘⣄⡀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⠆⠘⠀⠀⠀⠀⠀⢄⡀⠀⠌⠀⣶⡄⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡎⠰⠈⠀⠀⠀⠀⠀⢘⠤⠒⠉⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⠀⠀⢀⠋⡔⠂⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢠⠊⢁⠀⠀⠀⠀⠑⢦⣼⡀⢀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⡞⠑⠤⣀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⢀⡀⠠⠞⣀⡀
⠀⠀⠀⠀⣀⣣⢀⣀⠀⠀⠀⠀⢀⠴⠒⠓⠒⠲⢏⡉⠁⡈⢆⠀⠆⠆⠀⠀⠀⡔⠉⠀⠀⠀⠀⠈⠳⡄⠠⣤⡀
⠀⢀⠔⠉⠀⠀⠀⠀⠉⢢⡤⡴⢁⠤⠀⠀⠀⠀⠀⠑⡄⠱⡸⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⡎⠐⠉⠁⠀⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇⠀⠀⠀⠀⠀⠘⢖⠒⠂⠉⡆⠀⠀⢀⠎
⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⠃⠈⢦⡂⠀⠀⠀⠀⣀⠜⡰⠧⣈⡆⠀⠀⠀⠰⣲⡲⠷⠶⠯⠄⠒⢖⠁
⠀⠀⠈⠒⠤⣤⠤⢤⠒⡳⠀⢀⠾⣌⠏⠀⢈⠙⡍⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠂⠀⠀⡇⠀⠰⠄
⠀⢠⠒⠒⠊⣁⠤⢸⠾⠥⠴⠛⠲⠛⠀⠀⠈⢆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀
⠀⠀⠉⢉⠃⠀⠀⢸⠀⠀⠀⠀⠀⠸⡄⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⠊⠒⠤⠤⠒⠇
⠀⠀⣀⠎⢠⠒⠢⣀⠑⡆⠀⠀⢀⡴⠁⢀⠼⣁⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠜
⠀⠀⠈⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠑⠊⠁⠀⠈⠒⠁

⠀⠀⠀⣠⣤⣠⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢻⡿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⣿⣿⡇⠀⢰⡇
⠀⠀⠀⢸⣷⣶⣶⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠅⠀⠀⠀⣿⣿⣷⠀⠸⠇⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣤⣾⠀⠀⣶⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡏
⠀⠀⠀⠀⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡞⠂⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡷⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡇
⠀⠀⠀⠀⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣴⣧⠀⣿⡇⠀⠀⠀⠀⣿⣇⡀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣷⣿⢰⣾⣷⡆⠀⠀⢀⣘⣿⣧⣇⡀
⠀⠀⠀⠀⢿⣿⣿⠃⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⢠⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⢸⣿⢈⡇⠀⠀⠘⣿⠂⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠘⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠈⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣠
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⣀⣤⣄⣀
⠀⠀⠀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢹⣶⣾⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢀⡾⠋⠀⠀⠙⢷⡄⠀⠀⠀⠀⣀⣙⣛⣛⣛⣛⣛⣋⣠⠀⠀⠀⠈⠻⠟⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣾⠁⠀⠀⠀⠀⠘⣷
⠀⠀⠀⣿⣿⢶⣶⣶⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠉⠉⠉⠉⠋
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢹⡄⠀⠀⠀⢀⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠻⠲⠒⠓⠓⠚
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠙⢶⣤⣾⡷⠏⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣿⣿⢚⣛⣾⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣈⡇⠀⠀⠀⠀⠀⣾⠉⠙⠻⣿⣿⠟⠋⠉⣷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⡆⠀⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠦⣄⣰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⣀⣴⡾⠃
⠀⠀⠀⠿⠿⠛⠛⠛⠁⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣄⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣌⡛⠿⣿⣿⠟⢃⣤⣴⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣧⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⣿⣏⣿⡟⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣿⣽⣿⣰⣿⣿⢛⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣜⣣⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠇⠀⠀⠀⣠⣄⠀⠀⣧⣍⠀⠀⠀⠀⠀⠛⡀⠀⠀⠘⢽⠂⠀⢀⠈⣾⣀⠀⢰⠆⠀⠀⠀⠀⡀⡇⠀⡴⢤
⠀⠀⠀⠸⣿⡿⠀⣾⣿⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠈⠱⡿⣿⣛⣏⠀⠀⣀⠀⠉⠁⡏⠀⣸⣿⣧⡆⢠⡆
⠁⠂⠀⣸⣿⣿⠀⢿⣿⣿⣷⠌⡇⣆⣠⠞⣻⣿⠀⣠⣤⣼⣶⠀⠈⣷⢿⣿⣽⠀⢀⠀⠀⠐⠀⡇⢀⣯⣿⣿⣧⣿⡗
⣟⣉⣿⣷⡟⢿⣿⣉⣻⣹⣉⣶⡇⣇⣻⣏⣙⣿⣼⣿⣟⣸⣿⢐⠀⠀⠃⢿⣷⢀⣻⣿⣼⣿⡆⣷⣶⠊⢁⣠⣷⣿⣷


⠸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⣠⠚
⣶⣯⠹⡗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣜⣣⣤⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⣨⣿⣿⣻⣖
⣝⣷⣸⣍⣤⣤⣶⡄⠀⠀⠀⠀⠀⣸⡻⣒⡶⠶⠶⠿⠽⠿⢿⡿⠷⣶⠶⣶⣶⢟⣇⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿
⣫⢹⣿⣿⣿⣿⣿⠣⠀⠀⠀⠀⢸⣸⡷⠾⠶⠿⠷⡾⠶⠶⠶⠿⢷⡿⡷⠶⠶⢿⡇⡇⠀⠀⠠⣾⣶⣾⣿⣾⡟⣻⣻
⣹⡏⢿⣿⡟⣋⣟⡆⠀⠀⠀⠀⢸⢸⣷⣶⣶⣶⣶⣿⣶⣿⣷⣶⣶⣶⣶⣶⣶⣾⡇⡇⠀⠀⠀⠈⣿⣿⣿⣿⠇⣿⣿
⠙⠛⠟⠻⠛⠋⠙⠁⠀⠀⠀⠀⢸⠾⠷⠷⠾⠺⠛⠗⠛⠟⠛⠾⠿⠓⠳⠻⠟⢾⢷⡇⠀⠀⠀⠒⠟⠿⠿⡿⠻⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣈⡀⠀⢠⣶⣦⡀⠀⡀⠀⠸⣛⢀⠀⠄⠀⠀⠀⠀⠀⢠⠀⠠⡄⠀⠀⠀⢀⣿⣀⡀⠁⣸⣯⣷⣄⢹
⠀⠀⠊⡁⠀⢀⠄⠁⡀⠘⢁⣦⣥⣼⠀⣤⠶⣶⡯⠂⣆⠀⠀⠀⣤⣤⣄⠺⠇⠛⠁⠀⣠⠟⣿⣿⣷⢺⣿⣿⡿⣿
⣤⡀⠍⢥⣶⣈⠳⡒⢂⠀⣾⣷⡖⣀⡵⣷⢟⡟⣷⣞⢋⠇⠀⢼⡟⢿⣿⣇⢰⣿⡎⠉⣤⣶⣿⡿⣿⣿⣿⣿⣧⣦⠔
⠭⠿⠿⠿⠿⠿⠧⠀⠿⠠⠿⠿⠿⠿⠇⠾⠋⠑⠹⠿⠇⠀⠀⠸⠿⠿⠿⠿⠿⠿⠇⠄⠹⠾⠿⠿⠿⠿⠿⠿⠽⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⢲⣦⠀⠀⠰⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⣶⢼⣗⢶⣾⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⡽⣿⣿⠋⠻⠷⣹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣄⣹⣿⣷⣶⣶⣤⡶⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⢞⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠧⢤⣤⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠈⠃⠛⠛⠉⠉⠉⠛⠛⠋⠉⣁⣀⡀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠓⠚⠙⢛⠋⠉⠉⠉⠉⠉⢹⣿⠋⠀⠈⠙⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠔⠛⠀⣀⣀⣙⣿⣶⣸⣿⣀⣀⣀⣀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣭⡍⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣟⠒⠂⠚⠂⢀⣀⢀⣀⣀⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠉⠉⠉⠉⠉⢸⣿⣿⣿⣿⣿⣿⣇⠀⠒⠒⠒⠒⠂⠉⠁⠉⢹⡾⣿⣿⣿⣿⣿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⠀⠀⠘⠻⠿⠿⠿⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠛⠃⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠂⠐⠐⠐⠀⠀⠀⠀⠈⠒⠀⠐⠀⠂⠐⠀⠐⠐⠐⠀⠒⠂⠒⠒⠀⠀⠀⠈⠐⠀⠐⠂⠀⠀⠂⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⡀⡀⠀⠀⢠⠠⢀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠤⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⢀⡠⠐⠈⠉⠁⠀⠉⠉⠛⠢⣀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⢻⣁⣠⣔⣉⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⠶⠛⠉⠉⠀⠀⠀⠈⠉⠉⠓⠦⣀⠀⠀⠀⢸⠄⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⣼⡄⢠⡀⣾⣦⣿⡘⢧⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣃⡀⠴⢋⣧⠞⢳⡟⠀⠈⢳⠘⡆⠀⠀⠀⠀⢀⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠞⠀⠀⠀⠠⣦⣤⣀⡈⠁⠀⠀⠀⠀⠀⠈⡇⢹⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⠘⠛⣛⣛⣉⡉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠃⠀⠀⠀⣷⢸⡆⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⡴⠋⠀⠀⠀⠀⡴⠋⠁⣀⣀⡉⠳⡄⠀⠀⠀⠀⢀⠔⠊⡉⠉⠓⢦⡀⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⣾⠁⣿⠀⠀⠀⠀⠀⢸⠁⠀⣼⣿⣿⣤⡄⢹⠀⠀⠀⠀⡏⣴⣿⣿⣀⡀⠀⢳⠀⠀⠀⠀⢹⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣄⣿⠀⠀⠀⠀⠀⠘⣀⠀⠘⠿⣿⠿⠁⣘⠀⠀⠀⠀⣇⠻⣿⣿⡿⠃⠀⡸⠀⠀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡄⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠶⠞⠁⠀⠀⠀⠀⠘⠦⠤⠶⠶⠒⠚⠁⠀⠀⠀⢀⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢠⠴⢄⡀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⢦⣀⠙⠲⣄⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠒⠒⠲⠦⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣀⡤⠖⠋⣙⡆
⢀⣤⣀⣒⣒⠂⠒⠉⢁⣀⠀⠘⣆⠈⠱⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢀⡔⠉⠁⠀⠴⣮⣉⣀
⣾⣩⡭⠭⠤⠀⠀⠀⠈⠉⢹⡄⡟⠀⠀⠀⠙⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠋⠁⠀⠀⠀⢸⢄⡶⠛⠛⠀⠀⠀⠩⠭⣽⣶⡄
⢻⣀⣀⣀⣀⣀⣀⣀⣠⡤⠾⠁⢿⠀⠀⠀⠀⠀⣠⠏⡿⠙⠛⢶⣖⠒⠒⣒⡿⠉⠹⡎⣧⠀⠀⠀⠀⢀⡟⠘⣧⣀⠀⠀⠀⠀⠘⠓⠶⢬⣿
⠈⠉⠉⠀⠀⠀⠀⠀⠀⣧⠀⠀⠘⣆⠀⠀⣀⡴⠃⢀⡇⠀⠀⠀⠈⣉⣉⡉⠀⠀⠀⢷⠈⠳⣄⣀⣠⠞⠀⠀⢀⡟⠉⠛⠓⠒⠶⢤⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠙⠋⠁⠀⠀⣼⠃⠀⠀⠀⠈⠛⢻⣿⡆⠀⠀⢸⡆⠀⠀⠉⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣤⠴⠋⢹⠀⠀⠀⠀⢿⣿⣿⣿⠃⠀⠀⠀⣿⠲⠤⣤⣤⡤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠤⠤⠤⠶⠶⠶⠶⠶⠒⠒⠒⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠰⣶⠂⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠛⠐⠒⠂⠂⠀⣻⠒⠲⠖⠦⠴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⡀⠀⣿⣀⣀⣀⣀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠼⣯⡉⠉⠉⠉⠙⣧⡽⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢻⠀⠀⠀⣾⠁⠀⠀⠙⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⣿⢸⣿⡇⢀⣠⣤⣄⠀⣶⣶⢰⣶⠀⢠⣤⣠⣤⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⢸⣿⣸⣿⡇⣼⣿⣻⣿⡇⣿⣿⣾⣿⡄⢸⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⢸⣿⣿⣿⡇⣿⣿⢹⣿⡇⢹⣿⣿⣿⡇⠀⣿⣿⡏⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣾⡂⣿⣿⡿⠟⢸⣿⣿⣿⡇⣿⣿⣼⣿⡇⢸⣿⢿⣿⡇⠀⣿⣿⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⣿⡇⣿⣿⠀⠀⢸⣿⣿⣿⡇⣿⣿⣾⣿⠀⢸⣿⡘⣿⣧⠀⣿⣿⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣤⡘⠛
⠀⠀⠀⠀⢻⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠻⢿⣿⠟⠀⠘⠛⠃⠛⠋⠀⠿⣿⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠈⢉⣤⣤⣶⢾⣿⣶⣦⡀⠀⠀⢀⣀⣀⣤⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡀
⠀⠀⠀⠀⢸⡇⠀⣿⡷⢷⢂⢷⣿⣿⣿⡇⠀⠀⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡭⢙⣋⣻⣿⡇
⠀⠀⠀⠀⢸⡇⣰⣿⣷⣷⡾⠿⠿⢿⣿⣷⡴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠛⠭⢺⠀⢻⡇
⠀⠀⠀⠀⢸⡧⣿⣿⣿⣿⣷⡀⢀⣾⣿⣿⡧⠀⣿⣿⣿⡿⢿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⡄⠃⠀⠈⠀⠀⠁
⠀⠀⠀⠀⢸⡇⢿⡛⠉⠛⠛⠛⢸⡟⠻⣿⣷⣄⣿⡝⣿⣯⣈⢹⣠⣼⣿⣿⡿⣿⣿⣿⣿⣿⡎⠀⠀⢀⡀
⠀⠀⠀⠀⢸⡇⢸⣷⣄⠀⣤⣤⡄⣧⠶⣿⣿⡿⠋⠀⠈⠛⠉⠃⠋⠉⠀⣿⣵⣿⣿⣿⣿⣿⣿⣦⡛⠉⠀⠀⣰⢀⡀
⠀⠀⠀⠀⢸⣇⠘⣿⣇⠀⠀⠀⠀⠀⣼⡿⠟⠀⠀⠀⣀⣀⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣆⣶⠊⢡⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠉⠁⠀⠈⠓⠲⣾⣿⣷⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷⣿⣿⡷
⠀⠀⠀⠀⠸⣿⠀⠀⢠⣀⠤⢀⣼⣽⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣗⣴⣿⣿⣿⣿⣿⣿⣿⣧⣿⡿⢻⣿⣿⠟
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠀⠸⢿⣿⡿⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣷⣾⣿⠏⢸⡆
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⠏⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣉⣀⣀⣼⣿⣿⣷⣶⣶⡿⠋⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⡟⢁⡀⣀⢂⡠⣠⠄
⠀⠀⠀⠀⠀⣿⡄⢠⣿⣿⡿⢿⣿⣷⡀⢀⣀⣠⣤⣼⣿⣿⠋⡽⢋⡩⠋⠀⠀⣀⡔⡁⡠⢠⡒⡜⣡⠋⠹⠱⠏⠐⠽
⠀⠀⠀⠀⠀⠉⠁⢸⠟⠉⠀⠀⠀⠉⠓⠀⠉⠉⠉⠉⠉⢀⠀⢉⡜⡱⢫⠣⡎⠾⠾⠘⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡅⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⡒⠒⠒⡒⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣷⣶⣦⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⢠⣴⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠸⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡟⠛⠁⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢿⠿⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠘⠖⠂⠀⠀⠀⠀⠀⡠⢪⡽⠂⡉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠦⠀⠀⠀⠀⡀⠀⢸⠏⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌⠀⠄⠀⠰⡀⠀⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠈⢀⠐⠘⠟⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀
⠀⠀⢀⡠⠔⠒⣉⣙⣒⣊⣉⣉⡒⠂⠄⠠⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠉⠉⠀⠙⢄
⠀⠀⠀⡑⠤⢤⣍⣀⣀⣀⣀⣤⠤⠔⠁⠀⠀⠀⠤⠴⠮⢍⣁⣒⣈⣩⣴⣦⠤⠄⠀⠀⠀⠀⠀⣠⡔⠐⠂⠒⠒⢤⡀
⠀⠀⠀⠘⠢⣄⡀⠀⠉⠁⠀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠢⢄⠀⠀⠈⠓⠒⠁⠀⡄⠀⠀⠀⠐⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠠

⠀⠀⠀⠠⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠄⡀
⠀⠀⠀⠐⠒⠤⢬⣥⣄⣀⣉⣑⣢⣤⠤⠀⠀⠀⠈⠀⣀⣀⣀⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠿⠿⠶⣭⣤⠤⠀⠀⠀⠀⠀⠖⠚⠡⠾⠯⠋⠆⠑⠢⠄
⠀⠀⠀⠀⠀⠉⠓⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⡀⠀⠀⣀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⡠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣀⣀⡤⣄⡀
⠀⠀⠄⢤⣀⣀⣈⠡⠤⠴⠒⠒⠒⢰⠟⠀⠀⠀⠀⡠⠊⠀⣀⣀⣀⣀⠀⠑⡄⠀⠀⠀⢠⣐⣊⣁⣀⣄⣀⣀⣤⣀⣉⣒⣠⡄
⠀⠀⠀⠀⠈⠑⠒⠲⠶⠶⠶⠖⠛⢁⠄⠀⠀⠀⢰⣁⣴⢮⣤⣤⣤⣤⣽⣦⣈⡄⠀⠀⠀⢍⠛⠦⢤⣄⣀⣀⣤⠤⠖⢛⠅
⠀⠀⠀⠀⠀⠀⠂⠄⣀⣀⣀⡀⠄⠊⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠓⢤⣀⠀⠀⠀⢀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠔⠊⠉⠉⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠲⠶⠖⠒⠦⣄
⠀⠀⠀⠀⢀⣴⣋⣉⣉⣉⣉⣭⣦⣄⠀⠀⠀⠀⠢⣤⣄⠠⠥⠤⠒⠒⠒⢀⡾⠃⠀⠀⠀⠀⠁⢀⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠙⠻⠿⣾⣿⣿⡿⠟⠡⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀⡀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⢀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠍⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠛⠋⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⣶⣀⣄
⠀⠀⠀⠀⠀⠀⠀⢿⡿⠏⠁⠀⡁⡀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⡉⠘⠃⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⠄⡛⢍⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢀⣉⠀⠀⠀⣰⣿⣿⣿⣿⣿⡛⠀⠀⠀⠀⢔⠙⢊⣴⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠁⠈⢘⣿⣧⡀
⠀⠀⠀⠀⢀⠤⠒⠀⠀⠐⢠⣄⠀⠀⠀⠀⢻⣿⣿⣿⣽⣿⠁⠇⠀⠀⠀⢨⢩⣍⠝⡏⠁⠈⠰⠀⠀⠀⠀⠀⠀⠀⠘⠞⢀⠀⠈⢧⡽⠃
⠀⠀⠀⢀⣀⡐⡀⡄⠀⠠⠸⠋⠉⢛⠉⡀⠸⢿⣿⡿⠿⠟⠀⣦⢀⠀⠀⠰⢰⣧⠘⠁⠀⢠⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠐⠠⡀
⠀⠀⠰⠀⠁⠘⣶⠘⠄⠀⠀⣀⣤⣿⠯⠀⠀⠈⠉⠙⢶⠏⠈⣮⠄⢤⡆⠀⢿⠏⢠⣀⠤⣼⠃⠘⠡⣀⠠⣀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠄
⠀⣤⣴⠈⢀⡤⡿⠂⠀⠀⢈⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣦⠋⠁⣠⠛⡄⠴⡦⡄⣀⣠⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸⣶⡖⠒
⠀⢸⣿⣇⣡⠊⠀⠀⣰⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠱⣶⠾⢏⠀⠉⠁⠀⡉⠀⠉⠀⠙⠊⡏⠉⢀⠈⣦⠠⡀⠀⠀⢀⣀⠀⠀⣿⣿⠃
⠀⠈⣿⡟⠃⠀⠀⠀⠉⠛⠻⠇⠀⣼⣄⣀⣀⣀⠀⡀⠀⠀⣀⠀⠀⠙⢄⠀⢶⠴⡶⠀⠀⢰⠓⣤⣤⣬⣤⡴⠀⠻⠋⠑⠕⢄⠐⠄⠀⠀⠀⣿⣿
⠀⠀⣿⠄⠈⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⢉⣿⡂⠐⠁⠀⠀⠀⠀⠢⠀⠚⠛⠂⠉⠀⠘⠀⢀⠀⠀⢰⣿⣿⢠⠁
⠀⠀⢸⠀⠀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠐⠒⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣾⡟⣿⡄
⠀⠀⠘⣆⠀⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠋⠀⣿⠃
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠦⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠐⠀⠑⢀⠀⠂⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⠄⠐⠢⠔⠉⠑⢆⠀⠀⠀⠠⢀
⡄⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈
⠣⡀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠐⠁
⠀⢀⠀⠀⠀⠀⠀⠙⢦⡀⢀⠎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠑⣄⢠⠊
⠀⠈⠢⠄⠀⡀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⡀⠀⠀⠀⢀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠈⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠠⢤⢤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠒⠒⠤⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢤⠀⣤⠉⠀⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠐⡀⣀⡀⣀⣀⣒⣀⣀⣒⣒⠂⡀⠠⢄⠀⠀⠀⠀⠀⠀⢠⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣀⣰⣀⣂⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠉⠒⠐⠀⠒⠁⣘⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠊⠀⠀⠀⠀⠰⠒⠂⠂⠀⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠈⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠉⠈⠉⠉⠉⠀⠁⠀⠂⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣂⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣠⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⣀⠄⠒⠒⠒⠒⠚⡗⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⣄⣀⣀⡀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡀⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣀⠤⠒⠉⠉⠹⡀⠀⠈⠑⣄⣎⡠⠤⢄⣀⣀⣠⡤⣖⠉⠉⡇⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠊⠁⠀⠀⠀⠀⠀⠙⠢⢤⡶⡿⠛⠈⠉⠀⡠⠊⠁⠉⠉⠳⢤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⢀⠜⡇⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡏⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⢹⠱⡀
⠀⠀⠀⠀⠀⢀⡎⠀⢰⠀⠀⠀⠀⠀⢀⣾⠋⠀⢠⠎⠀⠀⠀⠀⠘⢄⠀⠀⠀⢀⠼⠁⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘
⠀⠀⠀⠀⠀⡘⠀⠀⠀⢣⠀⠀⠀⢀⡿⠁⠀⣠⠧⠴⠒⠒⠒⠒⠢⣄⠉⠒⠊⢁⣸⠴⠆⠦⣖⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠣⡀⠀⢸⢃⣴⠟⠁⠀⠀⠀⢀⠔⠉⠁⢸⠉⢢⡔⢩⠃⠀⠀⠀⡼⢢⢀⠤⠒⠉⠉⠑⢾⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠢⣾⠉⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡸⢀⠏⣳⠃⣀⠤⠒⠢⡆⡼⠁⠀⠀⠀⠀⠀⡇⠙⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠏⠑⠢⢄⣀⠀⠀⢸⠀⠀⠀⢠⢃⣸⢖⡷⠊⠀⠀⠀⣸⣿⡿⠀⠤⢄⠀⠀⢠⠃⠀⢰
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡌⡄⠀⠀⠀⠀⠉⠁⠚⡖⠒⢲⠋⣡⠼⣡⠃⠀⠀⠀⡰⠋⡿⢆⠀⠀⠀⡇⢀⡇⠀⠀⠰
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠃⢡⠀⠀⠀⠀⠀⠀⠀⠈⡲⣁⠞⠁⢠⢇⠀⠀⡠⡾⢁⢼⠃⠈⠳⢄⠀⠇⡜⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰⠀⠀⢢⠀⠀⠀⠀⠀⣠⠎⡰⠋⠁⡲⠃⠜⢖⠈⠀⡧⠋⠸⠀⠀⠀⠀⢹⠟⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠸⠀⠀⠀⠱⣄⠀⡠⠚⠁⡐⠁⢀⠔⠁⠀⠀⢀⡱⢞⡇⠀⡇⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⡆⢀⣀⠤⠚⠛⢄⡀⣰⡡⠔⠁⠀⢀⡠⠖⠉⠀⠀⠟⠲⠂⠒⠒⠉⡸⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣭⠤⠄⠒⠒⠃⠉⠉⡛⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⠀⠈⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠈⠢⡀⢀⠆⠀⠀⠀⢀⡠⠊
⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢈⠟⠒⠒⠒⠊⠁
⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠎
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣈⣢⣀⡠⠤⠒⠁⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⣀⣀⣀⡠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠐⠂⠈⠉⠁⠉⠉⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠈⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⢀⠀⠀⠀⠀⢀⡑⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡠⠊⠀⠀⠀⠀⠀⠉⠐⠂⠠⢄⠀⠀⡤⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⢀⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠇
⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⢀⡀⠤⠂⠀⠀⠀⠢
⠀⡠⢁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠉⠁
⠀⠘⠀⠀⠀⠈⠐⠠⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠄⠀⠤⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠂⠈
⢠⠖⡂⠀⠀⡔⢀⠀⠀⠠⠐⠄⠀⠀⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢱⡄⠀⢠⠀⠀⠀⠀⠂⠂⠀⠀⡼⠐
⢇⠀⠇⠀⠀⠡⠄⠀⠀⠐⠀⠃⠀⠀⡄⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢘⠀⠀⠈⠔⠀⠀⠀⠢⠂⠀⠀⡅⡀
⠀⠑⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠹⠀⠀⠀⢠⡆⣁⣦⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣤⣤⣦⠀⠀⠀⢀
⠀⠀⢠⡄⠀⠀⠀⠈⠉⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢁⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⡇⠈⠄⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠣⡀⠀⠀⠀⠀⠀⠀⡐⠈⡄
⠀⢠⠀⠀⠀⠁⠂⠠⡤⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠂⠤⠤⠤⠐⠀⠀⠀⢁
⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⡤⠤⠤⢤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠊⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⡄
⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠂⠤⠔⠊⠀⠀⠀⠀⠘
⣧⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⢀⣀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢳⣹⣼⠺⡎⣏⣱⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡺⡶⢤⡐⢾⣾⡟⣼⣈⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡎⢅⠢⡻⣮⡂⢳⣿⡿⠟⣩⣾⣿⣝⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠉⣿⣮⣡⣄⣘⣿⣿⠟⣡⣾⣿⣿⣿⡛⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣾⣿⣿⣭⠿⣛⣥⣾⣿⣿⣿⣟⣎⣳⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣟⢷⣽⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⢿⣿⣿⣿⣿⣿⣶⣿⡿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣾⣛⣯⠮⠟⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⣸⣿⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣿⣦
⠀⠀⠀⠀⠀⠀⠙⣻⠖⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡛⣿⣿⡧
⠀⠀⠀⠀⠀⢸⣿⣄⣠⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⡇⣸⣷⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣾⣦⣿⣿⣽⣿⣦
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠸⣿⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⣿⣿⡳⠀⠀⠀⠀⠀⠀⣿⣿⣧⣽⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠇⠻⠂⠀⠀⠀⠀⠀⠀⠃⢼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⣠⡶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⠀⠀⣼⡛⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠊⠁
⠀⠀⠀⠀⠀⢸⣟⣿⡞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⢰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣂⣀
⠀⠀⠀⠀⠀⣻⣿⣿⡄⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢺⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣽⠿⠼⡇
⠀⠀⠀⠀⣴⣿⣿⣿⣷⡇⠁⠀⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣘⣽⡇⠀⠀⡇
⠀⠀⠀⠀⠈⠉⠉⣟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠿⣿⣿⡿⣿⣧⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠂⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡠⠀⠐⠀⢀⠐⢀⠀⠒⠈⠉⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⢇⠒⠠⠓⠀⠀⠐⠁⠉⠀⠀⠀⢂⡀⠈⠀⠐⠦⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡤⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⢶
⠀⠀⠀⠀⠀⡀⠀⢘⠀⠀⠀⠀⠀⠀⠊⠀⠁⠀⡀⠀⡌⢀⠀⠀⠀⠀⠂⢀⠠⠴⠀⠀⠀⠀⠀⠐⢠⠄⠀⠖⠀⠒⠀⠀⡀⠀⠱⣂
⠀⠀⠀⠀⠀⠀⠅⠈⢀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠁⠀⠀⠀⠄⠑⠁⠀⠀⠀⠀⠀⠀⢐⠀⠤⠀⡀⠠⠐⠉⠂⠈⠑⠀⠄⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠀⠀⠂⠸⠄⠐⠀⠀⠀⠀⠂⡀⠄⠐⡀⠀⠀⠚⢁⠄⡘⠀⠀⠀⠂⠀⠈⠁⣄⠈⠐⢄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⣀⠐⢀⠂⠌⠐⠀⠉⢡⠀⡄⠀⠠⠉⢩⠀⠉⠉⠁⠀⠉⠀⠈⠁⠀⠈⠀⠀⠀⠀⢀⠀⡄⠀⠀⢸⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣀⠠⠲⠐⠈⠁⠄⢀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠀⠈⠀⠀⠀⠠⠠⢠⡶⡦⣦⢴⣤⡔⠁⠐⠂⠁⠠⠀⠀⠀⠐⠀⠀⢁
⠀⠀⠀⠀⠀⡀⠀⠀⠁⠤⠀⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠷⠆⢳⣗⣮⠀⣙⣣⡀⠀⢈⠐⢀⡠⡡
⠀⠀⠀⠀⠐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠉⠉⠁⠀⠐⠰⢦⢯⠈⢐⢚⠉⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢥⡄⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠊⣽⠂⠀⠀⠀⠀⠀⠀⠀⠘⢘⠀⠀⠀⠀⢀⡠⠐⠠
⠀⠀⠀⠀⠀⠀⠁⠂⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣄⠀⠀⢀⡄⠀⠀⠈⠉⠀⠁⠈⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡆⠈⠁⠈⠁⠠⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣿⢧⡀⠀⠀⠀⢰⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣧⢿⣶⣤⣤⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⣱⣿⣻⣿⣷⠩⣽⡟⢩⣿⡇⠀⠀⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡻⣿⣽⠯⠶⣟⣿⣿⣿⡇⠀⠀⢸⣿⣿⣷⡀

⠀⠀⢷⠀⢸⠀⢠⠃⢠⠔⠉⢀⠁⠠⠀⠀⠀⠄⢀⠁⠀⡩⠔⣂⠭⠴⠒⠒⠒⠒⠀⠀⢄⠀⠀⠙⡇⠀⢸⠀⠄⠀⡼
⢂⠀⠸⡄⠈⡀⢀⠰⠁⠂⠔⠁⠀⣤⠀⠀⠀⠠⠀⠈⠢⠔⠊⠀⠀⠀⠀⠀⢀⠐⠀⠈⠂⠁⡀⠀⠅⢀⢃⠌⢀⠞⠁
⠀⠀⠀⠱⡀⠀⠘⡈⡄⠄⠄⠀⠀⡇⠀⣠⢀⠐⣴⢃⠀⠀⠈⠁⠰⠤⠋⠱⠴⠂⠴⠄⠠⠄⡇⣤⣇⡴⢁⠄⠁⣀⠤
⠀⠀⠀⠀⣴⣤⣖⡖⠲⠺⣞⠖⠲⠛⠛⣲⣤⡤⠴⢖⡒⠤⣴⡷⠒⠒⠒⠒⠒⠶⠒⣒⡚⡛⢻⣿⣟⠈⠄⣂⡀
⠀⠀⠀⢘⣿⠁⠄⠄⠐⠖⢏⠐⠒⢲⠒⢰⢊⠴⠚⠉⠉⠙⢲⣅⡎⠀⠀⠀⠈⠀⠈⠀⠀⠁⢸⣿⡿⠦⠄⠀⠀⠀⠐
⣀⠄⢀⠘⣿⠀⠀⠀⠘⠀⠘⠃⠀⡆⠸⠀⠃⠀⠀⣤⣤⡀⠀⢹⣇⣀⣀⣀⠀⠀⣄⠄⠄⠖⠸⣿⣷⣄⠐⠤⢄⣀⠉
⢀⡴⠃⠀⠡⠀⠀⠀⠂⠀⠀⠄⢰⠀⢄⠀⠀⠀⠀⠙⠓⠄⡀⢸⠁⠀⡀⠀⠀⠀⡆⠀⣶⠄⠀⠈⠉⠀⠀⢀⠀⠠⠤
⠁⢀⠎⢰⢿⡇⡄⡄⠀⠸⡄⠀⡞⠀⣼⠈⡄⠀⠀⠀⠁⠀⣥⢋⣾⣟⡇⠀⠀⠀⡇⡜⡏⠜⠀⠉⠀⠑⢌⠂⡈⠐⠠
⡴⠃⢀⡏⣸⡇⠰⠧⠦⠖⣀⠈⣉⣉⣁⠶⣌⣃⣒⡐⣒⣋⣴⣿⢗⣺⣧⣜⣭⣹⣥⣿⡇⠀⢀⢠⠀⠀⠈⠂⠈⠒⠄
⠀⢀⡞⠀⠹⠿⠶⠿⡿⢯⡿⡟⠶⢖⠰⠾⢛⠻⠿⣟⠿⠭⠩⠤⢽⡟⠹⠉⠉⠉⠁⠉⠀⢠⢸⠀⣆⠀⠀⠀⠀⠑⢄
⣠⠏⠠⠃⢀⠄⠀⠀⠘⡄⠁⠈⠢⡀⢸⢀⠅⠚⠀⠴⢄⡂⠀⣶⣞⡃⣤⠀⠀⡆⠀⠂⠀⢸⠀⣇⠘⣄⠐⢄⠢⣀
⠁⡰⠁⠀⠊⠀⠀⠀⠀⡇⠀⠀⠀⠈⢈⢸⢦⠸⠏⣤⡚⣃⡀⠉⠭⠁⡇⠄⠀⠁⠀⡇⠀⠀⢇⠘⣆⠈⢂⠈⠢⡈⠑
⠌⠀⠒⢀⣐⣐⣀⣀⠢⠇⠀⠊⣀⣃⣘⡈⠀⠀⣐⢶⣈⣉⣍⣉⣉⣙⡃⠞⢚⣒⢒⣓⣚⣒⣊⣀⢈⡂⠠⢥⡀⠈⠣
⡟⠛⠛⠉⠉⠉⠉⡿⠛⠋⠉⠛⣿⠉⠉⢻⡏⠉⡿⠛⠉⠉⠙⢟⠉⠉⠙⢷⣿⡏⠉⠉⠉⢹⠁⠈⠉⠙⢯⣢⡘⡦⢀
⣇⣀⣀⠀⠀⣀⡀⠇⢀⣤⣤⣤⣿⠀⠀⠘⠇⠀⡀⠀⣠⣄⠀⠀⡄⢶⠆⢀⣿⡇⠀⣶⣶⣿⠀⢰⣶⡄⠀⢻⡃⠐
⢉⣉⣿⠀⠀⣿⠀⠀⣾⠛⠛⠛⣿⠀⠀⠀⠀⠀⡇⠰⣿⣾⠀⠀⡆⠀⢰⣿⣿⡇⠀⣉⣿⣿⠀⢸⣿⠇⠀⣘⡇⠀⠓
⡟⠉⠉⠀⠀⠉⠉⠁⠙⠟⠃⠀⢛⠀⠃⠀⠀⠀⠃⠀⠉⠁⠀⠀⠀⠀⠀⠻⠻⠃⠀⠉⠙⡿⠀⠈⠉⠀⡰⡋⢀⠬⡀
⡇⠀⠀⣀⣀⣀⣀⣤⣀⣀⣠⣴⣬⣀⣬⣄⣀⣠⣥⣄⣀⣀⣤⣤⣤⣶⣤⣤⣴⣦⣤⣤⣤⣧⣤⢤⠖⠞⣩⠔⣛⢲⠆
⣉⣉⣭⣭⢭⣉⡠⠍⣙⣛⠻⠉⠉⢭⠭⠉⠁⠠⠌⠄⠀⠠⠉⢤⠤⠩⠅⠐⠤⠐⠒⠒⠲⠆⢶⠶⣾⣹⣼⢂⠀⣎
⢉⣾⡟⣿⣧⠙⠀⠂⣶⢶⣦⣤⣄⡀⠄⠂⠀⠀⣠⣤⠶⣦⣄⠐⠄⣤⣤⣰⢶⠊⣤⣤⣴⠐⢸⣯⣳⣹⢇⠂⠰⠈⠳
⠟⠉⠇⠸⢻⠸⡄⠀⣿⣾⣿⣿⣿⣿⣦⠘⣠⡾⢦⠛⣷⣿⠶⣷⡀⣿⣿⢿⣾⣷⣷⢷⢿⠀⠀⣿⣳⣻⠀⠀⠆⠘⠆
⡀⡰⠀⠀⠸⡇⠃⠀⣿⡧⣿⠙⣿⣻⣿⡇⣟⡷⣻⡟⠉⢻⣛⣟⡇⣿⣜⠶⠷⢷⢿⡫⢾⠀⠀⣷⣶⣿⡀⢰⠀⠢
⡠⠁⢀⠀⠀⡇⢰⠀⣿⠯⢿⣶⣿⣯⣻⠇⣿⡷⣼⠷⣶⢛⣿⣻⠁⡧⠀⣿⣤⠀⠈⠁⢼⡄⠀⠿⠽⠾⢇⢸⡀⠀⠐
⠁⠀⡌⠸⠀⣿⢸⢀⣿⣴⣳⡯⠼⠟⠁⠂⡘⢷⣮⣛⣿⣏⡿⢃⠀⡇⣀⣏⢠⣀⣁⣀⣸⣇⠀⣴⣾⣶⠀⠀⠧⡀⠁
⠀⣰⠀⠀⢀⠇⢸⠀⠀⠀⢀⡀⢀⠠⠰⣄⠀⠀⠈⠉⠉⠁⣀⠀⠄⠁⠀⠘⠂⠀⠀⠀⠀⠀⠄⠙⠛⠁⠀⠀⡄⠐⢄
⣰⠃⢀⠃⠸⠐⠈⠋⠉⠉⠀⠀⡀⢣⡱⡄⠳⣄⠁⠀⠱⡅⢺⡄⠲⠂⠉⠛⣅⡠⠤⠀⠘⠳⠖⠖⢤⣠⡤⠞
⠃⠀⡜⢀⡇⢀⠀⠀⠀⠀⠀⠀⢁⠀⢳⠹⡄⠈⢧⡈⢆⠘⡀⠐⣀⡰⠖⠉⠠⢐⠤⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠠
⠀⣰⠃⡸⠀⡜⠀⢰⠀⢀⠀⠀⠀⡄⠈⡄⢻⠀⠀⢳⡌⡆⢡⠀⠉⠀⠀⢁⠔⠁⡠⠊⠀⢠⠋⠀⢠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠻⠶⠟⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢀⣀⣤⠾⢿⣄⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠖⠒⠛⠉⠉⠀⠀⣀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣦⣀⣤⣤⡶⠾⠛⢋⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢀⣤⣶⣿⠿⠟⠛⠛⠛⣻⣿⠟⠛⠛⢛⣿⣿⣿⠿⠿⢿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⢸⣿⣿⡎⢻⣇⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠉⠉⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⡀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⣷⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⣤⣤⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⣹⡇⠀⠀⠀⠀⢰⣿⡇⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠙⠛⠛⠀⠀⠀⠀⠀⢸⣿⠀⠺⣿⣽⣻⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⠦⠀⠀⠀⠀⠀⠀⢸⣿⠀⠰⢾⣯⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⢰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡆⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⡿⢿⣿⣷⣄⡀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⢀⣠⣴⡾⠟⢉⣿⡇⠀⠀⠀⠀⣰⠏⠀⢸⡟⡀⠀⣼⢿⡄⠀⠀⠀⡿⣧⠀⠀⣼⡇⠘⣿⣇⠙⢿⣦⣄⣀⠀⠀⢹⣷
⢀⣴⡾⠟⠋⠁⠀⠀⣸⣿⠀⠀⠀⠀⢠⡏⠀⠀⢸⡇⠳⣴⡇⠈⢿⣄⠀⣼⠇⢸⣂⡴⢉⡇⠀⢹⣿⡄⠀⠈⠙⠻⢿⣶⣾⠟
⠛⠿⣶⣤⣀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣾⡀⠀⠀⢸⡇⠀⠙⠁⠀⠀⣹⣿⣇⠀⠈⠋⠀⢸⡇⠠⢴⣿⣧
⠀⠀⠀⠉⠻⣷⡄⢸⣿⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⣼⠏⠀⠙⣦⠀⠀⠀⢸⡇⠀⢰⢿⣿⡀
⠀⢀⡀⡀⠀⣼⡏⣼⡟⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⣿⠀⠀⠀⣼⠏⠀⠀⠀⠘⣷⡀⠀⢸⡇⢠⠟⠈⣿⡇
⠀⣼⣿⣿⣿⠏⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⣿⠀⠀⣸⡟⠀⠀⠀⠀⠀⠘⣷⠀⢸⣷⠏⠀⠀⢻⣿
⠀⠀⠉⠉⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⢿⣦⣤⣀⣀⣀⣠⣴⠿⠃⢸⠃⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠸⠿⣿⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣸⣆⣀⣀⣀⣉⠙⠛⠛⢉⣀⣀⣀⣿⣀⣀⣀⣀⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⢻⣿⠟⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⢤⣤⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⡶⠶⢶⠶⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⡆⠀⠀⢀⣀⠙⣷⡀⠀⠀⠀⠀⠀⠀⣴⠟⢉⣁⡀⠀⠀⠆⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠷⠾⠛⠋⣿⣄⣼⡷⠀⠀⠀⠀⠀⠘⢷⣤⣾⠏⠛⠛⠛⠛⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢸⣿⡆⠀⠀⠈⣹⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⡄⠀⢸⡇⠀⡀⠀⢿⣿⠂⠀⣴⣶⣶⣶⣦⣴⣶⣶⣶⡄
⠀⠀⠀⣼⣿⣧⡀⠀⣾⣿⣿⣆⠀⠘⣿⣿⣿⣿⡇⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣶⠀⠀⡇⠀⢸⡇⢰⡇⠀⢸⡟⠀⠀⡿⠉⠙⡟⠉⠙⡟⠉⠙⡇
⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⢘⣿⡀⠀⠀⠀⠀⠀⢾⣿⠀⠀⣿⡇⠀⣿⠀⠀⡇⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠻⠦⠶⠿⠦⠾⠷⠦⠾⠃
⠀⠀⠘⣿⣿⣿⠃⠀⣿⣿⣿⣿⠀⠀⢿⣿⣿⡿⠀⠀⢸⣿⡇⠀⢸⣿⡆⠀⢺⣿⠀⠀⣿⡇⠀⣿⠀⢠⣷⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠐⠀⠐⠒⠒⠒⠒
⠀⠀⠀⢻⣿⡿⠀⠀⢿⣿⣿⡟⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⠇⠀⢸⣿⠇⠀⢸⣿⠀⠀⣿⠇⠀⣿⠀⠐⣿⠀⢸⡇⠸⡇⠀⠀⡇⠀⠀⠚⠂⠚⠛⠛⠛⠒⠒⠒⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣾⣯⡀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⠀⠀⢀⣿⣿⡟⠃⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣿⡟⡇⠀⠀⢠⣤⡹⢿⠃⠀⢰⣿⣦⣤⣤⣴⣾⣷⣶⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡄⠀⠀⠀⠀⣬⣬⡉⠀⠀⠀⣿⣿⣿⣿⠿⢿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⡇⠀⠀⠼⠿⠇⠀⠃⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣷⠀⠘⢠⣶⣿⣿⣷⣤⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣿⡏
⠀⠀⢸⣿⣿⡗⠀⢁⠰⠶⠀⡇⠈⠀⠀⠈⠻⢿⡿⣿⣿⣿⣿⣿⠀⠀⣼⣇⠀⠀⣼⣷⠀⠈⡏⠀⢠⣿⡈⠄⠀⢸⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⠿⠿⠇⠀⠸⠦⠤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠁⠀⠸⠿⠿⠃⠘⠿⠿⠃⠀⠁⠀⠻⠿⠗⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⢿⠿⠿⠿⠷⡿⠿⠿⢿⡇
⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢀⣄⣄⣄⣤⣠⣀⣠⡀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⢹⣟⢸⠠⠀⠆⠀⠀⠀⣿⣿⣧⣤⣼⣿⢿⡃⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣴⡿⢻⡟⡝⠙⢻⣷⡀⠀⠀⠀⠀⣼⣿⣄⠀⠀⠀⠀⠿⠶⠶⠶⠶⠷⠶⠶⠾⠇
⠀⠀⢸⡇⢸⢐⠀⡇⠀⠀⢸⡏⠈⡉⠀⠉⠁⢹⡇⠀⠀⠀⣿⣿⣄⠀⠀⠀⢸⣿⢠⡿⡱⠀⢀⡿⢻⡷⠀⠀⠀⢀⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣄⡀
⠀⠀⢸⡇⢸⢈⠀⡇⠀⠀⠘⡅⠀⠁⠀⠀⠀⢨⡇⠀⠀⡆⢡⣠⡌⢰⠀⠀⠘⣿⣿⡱⠁⢀⣾⣣⣿⡏⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣞⣩⣶⠟⠋⡲
⠀⠀⢸⡇⢸⠘⠀⠇⠀⠀⠀⡟⠈⠉⣿⣿⣿⣿⠁⠀⠀⣇⠈⠉⠀⣾⠀⠀⢀⠈⢻⣿⣶⣾⣿⣿⠋⣀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡿⠛⠁⢀⣤⡇
⠀⠀⠈⠃⠘⠈⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠒⠛⠋⠀⠀⠈⠛⠛⠓⠒⠒⠚⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⣀⠴⠚⠳⠦⣴⡋⣸⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢷⣦⣄⠀⣀⡴⠛⠁
⠀⠀⢈⣩⠿⠛⠻⢯⣉⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⢠⡟⠁⠀⠀⠀⠀⠈⢧⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢰⣿⢿⢏⣤⡄⠀⠀⠀⢸⣿⣿⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣷⣶⡆
⠀⢾⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣰⠋⢋⣼⣿⣧⡀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠘⣧⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣿⣿⣦⣤⣄⣀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠔⠁⠀⠘⣿⣿⣿⠆⠀⠀⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠙⠲⠦⠶⠚⠋⠀⠀⠀⠀⠻⠿⠛⠤⠬⠿⠀⠀⠀⠀⠘⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠿⠿⠿⠃⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡍⣿⠛⠿⠿⠿⠻⠟⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠉⠬⠈⡀
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠘⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠘⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢤⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢁⠂⠤⣤⠠⡀⡱⡀⡀
⠀⠀⠀⠀⣄⣀⣄⣀⠀⢀⢸⢁⡢⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⠀⢀⣀⣳⣁⣢⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠴⠤⠤⠤⡑⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⢈⠀⠀⠀⢀⠈⠀⠒
⠀⠀⠀⠠⣀⣀⣀⣀⢥⠀⠀⠀⠀⠠⠉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⢯⠀⠀⠀⠌⣀⣀⣀⣀⣀
⠀⠀⠀⣠⣄⣭⣥⣤⣤⠹⠤⠭⠉⣥⡄⠀⣀⣤⣀⣤⣤⣄⣀⣤⣀⣀⡀⠈⠀⠀⠀⠀⣤⣤⣨⣅⣀⡄
⠀⠀⠄⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀
⠀⠀⠀⠘⠇⠀⠀⢻⣤⠀⣤⠀⢤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣸
⠀⠀⠀⠀⠃⠀⠀⢠⡀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡤⠤⠤⠤⢥⡆⠀⠀⠻
⠀⠀⢀⠀⠀⠀⠀⠰⠄⠘⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠘⠨⠁⠀⠀⠀⠠
⠀⠀⠀⠘⠂⠀⠀⠀⠀⢠⠀⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⣇⡀

⠀⠀⢀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠢⢀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠤⣀⣀⡀⠠⠈
⠀⠀⠀⠀⠈⠉⠉⡁⠀⢀⠂⠀⠀⣀⠈⠀⠀⠄⠀⢀⠄⠂⠚⠒⠒⠓⠀⠉⠉⠉⠀⢅⠈⠁⠀⡀⠈⠀⠀⠀⡀⠀⠈⠠⡀⠀⠐
⠀⠀⠀⠀⠁⠀⢐⠁⠀⠀⣸⠇⠒⠶⠴⢥⣀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠠⡀⠀⠀⠐⢀⠀⠉
⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡇⠀⠀⠀⠀⠀⠘⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⢀⣄⡀⠀⣀⡘⠆⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠐⠃⠀⠀⠒⠀⡙⡇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⣄⡄⠀⡠⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⢀⢇⠀⢀⠀⠀⡀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠁⢄⠀⠀⡀⡘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⢏⠙⠄⠀⢀⡀⠃⠀⡘⢆⠀⠐⠐⠀⣴⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠣⣀⣁⣠⠜⡇⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠈⠀⠸⠙⠒⡶⢋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠁⠀⣤⠀⠊⢁⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡥⠒⠢⡄⠀⢀⠿⢤⡤⣥⠈
⠀⠀⠀⠀⢀⠔⠂⠁⠀⠀⠉⠒⠚⠑⠙⠒⠁⠀⠀⠀⠉⠐⠄⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠄⠬⠀⠀⠁⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⡀⡃⠀⠂⠀⢰⠀⢰⢲⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⡀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣤⡀⠀⠘⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⣶⣶⣄⠀⠈⢻⣿⣷⡀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⢻⣿⣷⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⣿⣿⡆⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢹⣿⡇⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣼⣿⡇⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⢀⣿⣿⠇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⣼⣿⡟⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠀⢀⣾⣿⡟⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠞⠛⠛⠛⠛⠙⠲⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⡅⠐⠀⠀⠤⠤⠤⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢠⠳⣶⣀⠀⢀⣤⣶⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⣀⣾⠀⠘⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⠋⢻⠟⠻⠇⠀⠀⠀⠀⠀⠀⠈⢹⠗⠾⠏⠙⠀⠀⠰⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⡀⠐⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⡂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠔⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠶⢉⠶⠒⠒⠙⡆⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠙⢧⣴
⠀⣠⡞⠟⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠁⠀⠉⠐⠚⠁⠀⠀⢀⣠⡴⠻⣄⠀⣀⡀⠀⣤⣀⣀⣀⣠⠞
⢸⡇⠁⡀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⢸⡏⠀⠀⠙⠳⠏⠙⠻⠏⠈⠉⠉⠁
⢯⡑⠈⠁⠸⡦⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠶⠟⢻⡏⠀⠀⣾
⠈⣷⠀⠀⠀⠠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢾⠁⠀⣰⣧⡀
⠀⠘⣇⡈⠐⠀⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⠉⠆⠀⢉⠀⢁⠸⢷⣄⣀
⠀⠀⠈⠙⢳⡶⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⠉⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣀⠀⠀⠀⢀⣀⣀⣤⠆⠀⠀⠀⠀⢀⠀⠀⣴⠶⠒⠶⢦⣄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⣷⠀⠀⠀⠀⠰⠄⠀⣿⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⢀⡿⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⣄⢤⠶⠟⠁⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⢀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⡼⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⢤⣤⣀⣀⣀⣀⠀⠀⢠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠠⠶⣬⣩⠉⠉⠉⢹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠁⠈⠀⠀⠀⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⣡⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠁⠀⢹⠟⣷⠐⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⣀⢸⣶⠋⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⣤⣤⡾⢿⣼⣟⢀⣴⣦⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⡉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠛⠋⠀⠀⠈⠦⢀⠈⡆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢉⡴⣺⣁⠀⠀⠀⠀⠀⠀⢈⡦⣌⠰⡀
⡔⠄⠒⣆⠀⠀⠀⠠⣾⢷⠗⢢⠑⢄⠀⠀⠀⡖⡁⠀⠀⠙⢗⠒⠦⣀⠀⡠⠖⠒⢄
⡩⠃⠀⠤⠇⠀⠀⠘⠾⠘⠀⠀⠳⡈⣉⣂⡄⠊⢈⡢⠔⣒⠀⠱⡀⠀⠑⢇⠀⠀⢸⡀
⢇⣀⠴⠀⠀⠀⠀⠀⠀⠀⢃⠀⡖⠁⠀⠀⡙⡶⠋⠀⠀⢸⢆⠀⠘⡄⠀⠈⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤⠳⠤⢀⣀⢽⠥⠀⠀⢀⠏⠰⡗⣀⣘⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢍⠁⠀⠀⠀⠀⠓⠈⠉⣉⣤⡄⠉⠡⠠⣹⠀⠤⠄⠒⠒⠀⠀⠀⠁⠀⣼⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠐⠒⡝⠔⣉⣴⠾⠋⠙⡅⠀⣚⣓⠒⠠⠭⠅⣀⡈⠉⠉⠁
⠀⠀⠀⠀⠈⡘⣋⡆⠀⠀⢈⡱⡶⠤⣤⣤⣤⣔⡒⠬⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⣀⣤⡿⠖
⠀⠀⠀⠀⠀⢰⠹⠖⢲⠒⠛⠠⣀⡀⢠⠋⢠⣄⠀⠆⢰⠂⠃⠀⠀⠸⠀⠀⣀⢀⡀⠀⠀⠀⠙⠀⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠔⠒⠒⠚⠓⠠⢀⠀⠈⢯⡽⠚⠤⠤⡤⠔⠁⠀⠐⢄⠀⠨⡬⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠊⠡⠤⢠⣼⠥⢄⠀⠀⠀⠈⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⡀⡸⢶⡹⣧⠀⠀⠀⡊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠗⠂⠀⠐

⠀⠀⠀⢀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⣯⢺⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠘⣎⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠙⡆⠀⠀⠀⠐⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣮⣿⢫⣿⣿⣦⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡃⠀⠀⠀⢸⣷⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢮⡙⢿⣿⣿⣷⣄⠀⠀⠀⢀⣧⣤⣤⣤⣤⣤⣧⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠻⣿⣿⣿⣦⡀⠀⢰⣿⡀⢀⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡜⠁⢺⣿⣿⡄⠘⣛⣣⣬⣭⣭⣭⣭⣄⣀⠀⢸⣿⣿⠀⠀⠀⠀⣠⡾⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⢀⣀⣤⣶⣦⡀⠀⠀⠀⠻⣄⠈⢛⣩⣶⣿⠿⠛⠋⠉⠉⠉⠉⠛⠻⢿⣷⣌⡻⠀⠀⠀⣰⡟⢀⡄⠈⢙⣿⣿⣿⡿⠃⠀⣠⣶⣦⣄⡀
⢾⣧⡤⢿⣿⣿⡟⡆⠀⠀⠀⠈⣱⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⣰⡟⢠⣾⠁⢠⣾⣿⣿⡿⠀⠀⠀⡇⠀⠈⠙⢿⣦
⢺⣿⣷⠀⣿⣿⣷⢹⡄⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣟⣠⣿⠃⢠⣿⣿⣿⡿⠁⠀⠀⡜⣰⡀⠀⠀⠀⢹⠂
⠈⢿⣿⣧⢻⣿⣿⣧⢻⡀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⠟⢤⣿⣿⣿⡿⠁⠀⢀⣼⣿⡟⢡⠆⣠⠎⠁
⠀⠘⣿⣿⣇⢻⣿⣿⣯⢷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢀⣾⣿⣿⡿⠁⠀⣠⣾⡿⠋⣰⣿⡿⠁
⠀⠀⠸⣿⣿⡆⠿⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⣿⣿⠙⠺⣿⣿⡿⢁⣤⡾⠋⠎⢀⡼⣿⡿⠀⠀⠀⢀⣠⡾
⠀⠀⠀⠹⣿⣿⣷⠹⣿⣿⣿⣷⠀⠀⠀⣠⣴⣾⡿⢿⣛⡿⠭⠭⠭⠭⠭⠽⣟⣿⢿⣿⣧⣄⠀⣼⡿⢡⣾⣿⣦⣀⠀⠞⢱⣿⠁⠀⢠⠂⢿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣆⢻⣿⣿⣿⣧⣴⣿⣿⡯⠗⠋⠁⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠉⠺⢻⣿⣿⣿⢡⣿⣿⠛⣿⣿⣷⣤⣼⣿⠀⣰⠟⣡⣿⡿
⠀⠀⠀⠀⠀⢿⣿⠈⣦⡹⣿⣿⣿⣿⡿⠋⢀⡠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⡀⢹⣿⣿⣿⣿⠧⢄⣿⣿⣿⣿⣿⠃⣴⠃⣴⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣦⠛⠛⣾⣿⣿⠏⢀⠔⠉⠀⠀⢀⣄⣀⢸⣧⣀⡤⢖⣀⠀⠀⠀⠀⠈⢠⣿⣿⣿⣧⢀⣼⣿⣿⣿⣿⠃⠚⢅⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠘⣿⣦⣄⣸⣿⣿⡀⠁⠀⠀⠀⠀⣀⡤⠟⠋⢻⣥⠶⣈⣭⠉⢀⠀⠀⢀⣼⣿⣿⣿⣿⣸⣿⣿⣿⣿⠇⠀⠀⠀⣰⣹
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⣻⣿⣧⡳⣄⠀⠀⠐⠿⢥⣭⣶⣒⣻⡧⠞⣋⡤⠖⠁⣀⣴⣿⣿⣿⣿⣿⣏⣝⣻⣿⡿⣵⡆⠀⠀⡴⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠙⣿⣿⣷⣬⡑⣢⣄⣀⣀⠀⠀⠉⠉⣉⣉⣡⣤⣶⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⡿⠛⣽⣽⡇⢀⣼⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠉⠀⠀⠀⠑⢦⡙⠻⢟⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣬⡛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⣉⣠⣽⣿⠟⠀⠀⠀⠀⠀⠀⢰⢸⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣯⣿⣿⣿⣿⣿⣥⣴⣶⣶⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡜⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇⠀⠀⠀⠀⠀⣠⣾⣾⣿⡟⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⢶⣤⣤⣤⣿⣾⣿⡿⢋⣤⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⠾⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠈⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⣀⡦⢄⡠⣄⢀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡆⠠⢀⠀⠀⡐⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠛⠀⠘⠃⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠀⠀⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠳⢄⡐⠒⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠳⢄⡐⠂⠀⠀⠀⡀⠸⡀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⡯⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠸⠐⠦⠈⠉⣿⠩⡐⠤⠠⠑⠌⠆
⠀⠀⠀⠀⠀⠠⠤⢤⣦⠤⠤⠀⠠⠄⡇⠤⠕⠤⡀⠤⠤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⣀⣀⣋⣃⣈⣳⢄⢠⣤⣤⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠈⡤⠤⠤⠥⠤⠤⠤⠬⠢⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠰⡀⠀⠇⠀⠠⠀⠀⠀⠀⠁⠐⠒
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢓⡇⠀⠀⠀⠀⠀⠀⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡃⠀⠀⠀⠀⡜⠆⠀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠒⠒⠖⡖⠒⠊⠑⠀⠀⠄⠄⠤⠄⠃⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣒⣒⣒⣒⣒⣒⢹⠀⠀⠀⠀⠓⠒⠒⠒⡖⠖⠒⢂
⠀⠀⠀⠤⠤⠤⠤⠶⠤⠤⠤⠤⠐⠖⠧⠌⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠈⠂⠀⠐⠒⢂⠤⠤⠤⠴⠦⠤⠤⠤
⠀⠀⢐⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠸⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢨⢀⣀⣀⣀⣀⣀⣁⣀⡈
⠀⠀⠀⠀⢰⢰⠀⠀⠀⢺⡄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⢰⡇⠀⠀⠀⢸
⠀⠀⠀⠀⢸⢸⠀⠀⠀⠈⠛⠋⠀⠘⠃⡆⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠐⠀⡀⠀⠀⠀⠀⠀⠀⠇⠛⠁⠀⠀⢰⠐
⠀⠀⠀⠀⢸⠈⠀⠀⠀⠈⢰⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠖⠒⠲⠒⠒⠂⣾⠀⠀⠀⠸⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠀⢘⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠘⠂⣉⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠉⢰⢀⠀⠀⠀⠀⠀⠀⠨⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠸⠂⠸⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠀⠘⠴⠄⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠺⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⣠⣤⠀⠘
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⣾⡟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠾⠟⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⢰⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⣿⠗⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡆
⠀⠀⠀⠀⢈⠁⠀⣉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣭⣮⣫⣴⣍⣽⣯⣾⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣴⣆⢀⠠⠀⠑⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡉⠋⣹⢭⣿⣤⣦⠴⣦⡤⠬⢧⣄⣠⠄⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⠊⠀⠼⣦⣶⣿⣋⣿⣾⠭⠶⣿⣹⠲⢊⠛⠩⣹⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠻⠌⠄⢰⠿⠟⢢⣶⣿⣿⣷⣽⣿⢷⣿⣿⡿⢷⣿⠛⣤⣤⡘⣿⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣰⣾⣿⣉⢉⣽⣿⣿⣾⣚⣾⡿⣷⣶⣍⣯⠉⣼⣿⣯⡰⣤⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣸⡥⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠹⠭⣛⡛⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢃⠛⠀⣿⣥⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⢿⣶⠛⡟⣍⢻⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⢴⣿⣿⣿⠟⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢶⣿⣤⠇⣙⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⠶⠀⠳⡌⢧⡹⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⣿⣿⣁⡨⡝⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⢁⠀⠃⢈⣷⡘⠳⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠤⠤⢧⡿⣍⡴⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⢐⡊⠙⣿⣇⣲⣮⡿⣿⣿⣿⣿⣻⣯⣿⠟⠉⣱⣸⠿⠘⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⢷⣄⡈⣹⠟⠓⠂⠻⠿⠟⠛⠻⠛⣧⢚⣥⡿⠉⠐⡀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠛⢭⡋⠑⠒⢻⣥⢤⡤⠉⠀⠀⣨⠿⡨⠧⠤⣤⣥⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠾⠀⡾⢯⠋⠛⠯⠀⠬⠁⠆⠀⡛⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⡀⠀⢀⠓⠀⠀⠀⠄⠛⠳⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠆⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⠤⠤⠤⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠁⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠖⠊⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠧⣀⡀⣀⡀⡀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⠀⠘⠞⣻⡟⢭⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠸⣤⣄⢀⡛⡁⠀⢸⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⡃⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀⠀⠠⠖⠀⠀⠘⠦⣀⣀⣀⣀⠀⠀⠠⠤⠤⠄⠠⠀⠀⠀⢀⣀⡀⠀⠀⠠⠤⠤


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀⡤⠴⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠊⠉⠒⠒⠊⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠲⠤⠔⠓⠢⡀
⠀⠀⠀⠲⡌⠭⠤⠤⠤⠴⠶⠶⠤⠤⠤⠤⠭⡕⠂⠁⠀⠀⠀⠈⠀⣀⠠⣀⣀⣀⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⠀⠀⠀⡀⠀⠈⢆
⠀⠀⠀⠠⡈⠓⠲⠦⠤⣤⣤⣤⣤⠤⠶⠒⢋⡀⠀⠀⠀⠐⠒⠚⠛⠲⠤⠤⠭⢥⣤⣴⠿⣷⠖⠒⠂⠀⠀⠀⠀⠀⣀⣴⣊⣀⣉⣉⣉⣉⣓⣤⡀
⠀⠀⠀⠀⠈⠲⣄⣀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠐⠢⣀⠀⠀⠀⠈⠓⠒⠁⠀⠠⠀⠀⠀⠀⠀⠊⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⡀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⡈⠉⠓⠦⠤⠤⠴⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠑⠒⠖⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠢⠤⠖⠒⠢⡀
⠀⠀⠀⠀⡈⠉⠓⠶⠶⠶⠤⠤⠤⢤⠭⠵⠶⠦⠂⠀⠀⠀⠈⠀⢤⣄⣠⣤⣀⣀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⣀⠀⠀⠀⣀⠀⠈⠢
⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠀⠀⠀⠀⠀⡀⠀⠉⠑⠲⠶⠿⠷⢶⣭⣤⡤⠄⠀⠀⠀⠀⠀⠰⠒⠛⠳⠶⠯⠭⠵⠶⠛⠲⠦⠄
⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠶⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠒⠒⠒⠂⠀⣀⡀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠙⠒⠋⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠲⠶⠖⠒⠲⢤⡀
⠀⠀⠰⠂⢤⣤⠤⠬⠤⠴⠖⠒⠛⠋⠉⠉⣩⠟⠀⠀⠀⠀⠀⣠⠎⠀⢀⣤⣀⣀⣠⣄⠀⠈⢦⠀⠀⠀⠀⠠⣤⣉⣠⣤⡤⠤⣤⣤⣤⠤⢤⣤⣌⣁⣤⠤
⠀⠀⠀⠀⠀⠀⠉⠒⠲⠶⠶⠶⠶⠶⠞⠛⠁⡐⠀⠀⠀⠀⢰⢁⣠⣶⣧⣤⣤⣤⣤⣤⣽⣦⣀⢣⠀⠀⠀⠀⢈⠙⠳⠦⣤⣀⣀⣀⣀⣠⣤⠴⠞⢋⠅
⠀⠀⠀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠈⠉⠁⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⣀⣀⣀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠶⠶⠖⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠉⠁⢀⡀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠚⠛⠒⠒⠛⠛⠓⠦⡀
⠀⠁⠀⠀⠀⢀⣤⣞⣉⣉⣉⣉⣉⣉⣱⣦⡀⠀⠀⠀⠀⠀⠦⣤⣤⠄⠪⣥⠤⠴⠶⠒⠒⠒⣸⡟⠀⠀⠀⠀⠀⠈⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠈
⠀⠀⠀⠀⠐⠛⠛⠻⠿⢿⣿⣿⡿⠿⠛⠛⠛⠂⠀⠀⠀⠀⠀⠈⡙⠿⣶⣶⣶⣶⣶⣶⣶⡿⠋⡄⠀⠀⠀⠀⠐⠒⠚⠛⠻⠿⠿⡿⠿⠟⠛⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⠉⠉⠛⠉⠉⢀⠜⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣤⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠄⠤⠤⠤⠤⠤⠤⠤⠄⣀⣀⣀⣀⣀
⠀⣠⣶⣿⣿⡆⠀⢀⠏⠀⠖⠀⢢⡀⠀⢀⣶⣶⢦⡶⠶⢴⣤⣄⠀⢀⡈⣀⣀⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡄
⢰⣿⣿⣿⣿⠇⠀⣸⠀⠀⠦⣠⠜⠁⠀⢸⡏⣿⢸⡇⠀⣼⢻⣿⡄⢸⡇⣿⠉⠉⠉⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⣿⣿⣿⣿⠀⠀⡇⢀⣶⢶⣤⡄⣾⠃⣿⢡⡿⢼⣥⣀⣿⢸⣿⣇⢸⡇⣿⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠉⢁⣀⠀⠀⠀⣀
⠀⣿⣿⣿⣿⠀⢐⠇⣼⣇⠀⠀⢰⡟⢰⡟⢸⡇⠀⢈⣿⣿⣸⡇⣿⣾⠇⣿⠛⠃⠈⠀⠀⠀⠀⠀⢀⣴⣿⠇⢀⣴⣿⡿⢁⣴⣿⣿
⠀⢹⣿⣿⣿⠀⠈⠓⠬⣙⠛⠶⣬⣝⡻⠃⢾⠃⠀⢸⡏⣿⣿⡇⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⣴⣿⣿⠟⣡⣿⣿⣿⠃
⠀⠀⢿⣿⣿⡄⠀⠀⠀⠈⠙⠲⠦⣭⡁⣘⡛⠷⠶⠾⠃⠿⠿⠁⠀⠿⠀⠿⠶⠂⡇⠀⠀⠀⣼⣿⠟⢀⣾⣿⡿⠃⣰⣿⣿⠟⠁
⠀⠀⠈⣿⣿⣿⣄⠀⠀⠀⣴⣶⣄⠀⠀⠉⠉⠉⠛⠛⠛⠛⠓⠓⠒⠒⠛⠛⠃⠉⠀⠀⣠⢾⡿⠋⣠⣿⣿⠋⠀⣰⣿⠟⠁
⠀⠀⠀⠘⣿⣿⣿⡇⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⢀⠔⠉⣴⠟⠁⡴⠻⡟⠁⠀⡰⣿⠿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠘⢿⣿⣧⡀⠈⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⢠⠃⢀⣾⠁⠀⡼⢁⡞⠀⠀⡔⡱⠁⠀⠀⣠⠖⠉⠘⡇
⠀⢰⣶⣶⣄⠈⢿⣿⢿⡀⠸⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⠇⢠⠾⠀⠀⡜⢡⠏⠀⢀⡞⡼⠁⢀⡴⠊⠀⠀⠀⠀⡇
⠀⣼⣿⣿⣿⡀⠀⠱⡈⢳⡀⠹⣿⣿⣿⣄⣠⣤⡀⠀⠀⢸⠋⠛⢣⡀⠀⣼⡴⠚⢀⣴⣋⢠⠏⠀⢠⢎⡜⠀⡰⢫⣄⡀⠀⠀⠀⠀⡇
⠀⣿⣿⣿⣿⠀⠀⠀⠙⡄⠹⡄⠹⣿⡿⢋⡴⣋⡁⢀⡀⠀⠇⠀⠘⡅⠀⡏⠀⣠⡻⠊⡜⡇⠀⣠⢋⡞⢀⡜⢁⡳⣮⠟⠀⠀⠀⠀⡇
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠘⢆⠙⣆⢘⣥⣏⠋⠀⠀⠀⠈⠱⠸⠀⠀⡇⠀⠀⣰⠟⠁⣰⡞⠀⢠⢃⠎⠀⠎⣰⠒⣬⠀⠀⠀⠀⠀⢀
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢆⠘⢦⡀⠸⠀⠀⠎⠑⠀⠀⠀⢣⣰⡄⠀⣼⠃⠀⣰⠟⠀⢠⢃⠎⠀⡜⢰⡎⢉⠁⠀⠀⠀⠀⠀⢸
⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢳⡈⢳⡄⠣⢀⠀⢀⣠⠞⠀⢀⡙⠀⠘⠁⠀⡼⠃⠀⢠⡵⠊⠀⢀⠃⠀⠑⠊⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⣄⠙⣦⠀⠀⠀⢀⣤⣾⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠻⠿⠋⠑⢄⣀⣀⣀⣀⣀⣀⣀⠈⠳⡈⢧⠀⠀⣿⣿⣽⣿⣽⣿⣿⡇⢀⣠⣶⡻⣿⡄⠀⠀⠄⣐⡐⠂⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⢒⣂⢨⡀⠀⠈⠻⣧⠀⠘⠿⣿⣿⡿⠿⠋⣴⣿⣿⠝⠁⣹⣿⠀⠀⡁⠤⠿⠅⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢘⠚⣈⢸⠃⠀⠀⠀⠈⣀⣀⣀⠀⢀⣤⠀⣾⣽⠛⠁⣀⣼⣿⠃⠀⠀⠡⠪⣧⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢘⣚⡉⣀⠀⠀⠀⣠⠊⣸⡟⢁⡔⢫⠃⣀⠈⠻⣿⣿⠿⠛⢿⣷⣄⠀⢾⢸⣻⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣸⠤⢁⡏⠀⢀⠞⢁⣾⠏⢠⠊⠀⢸⠃⢹⠓⡀⠀⢀⣀⠀⠀⠻⣿⣷⡌⠂⠀⠀⠦⠤⢄⡀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⡌⠀⡴⠃⣰⡟⠁⡰⠃⠀⠀⡆⠀⡟⠀⢃⠀⣨⠉⢢⡀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⢱⣶⣶⣶⡇
⠀⢸⠀⠀⠀⠀⠀⣠⣄⡢⠄⡰⢡⠞⢀⣴⠏⠀⡜⠁⠀⠀⢰⠁⠀⡇⠀⢰⠀⢻⠀⠀⠙⢦⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠈⠀⠀⠀⠀⠀⠙⠻⢛⡔⡱⠃⢀⣾⠃⠀⡸⠁⠀⠀⠀⡎⠀⠀⡇⠀⢸⠀⠘⢆⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⡆⠀⠀⠀⠀⠀⣠⢊⠜⠁⢶⡿⠁⠀⡴⠁⠀⠀⠀⢰⠁⠀⠀⣇⠀⢸⠀⠀⠈⢶⡀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⡇⠀⠀⠀⢀⠜⢡⠊⠀⣶⠏⠀⠀⣰⠃⠀⠀⠀⠀⡎⠀⠀⠀⠉⣷⣿⠀⠀⠀⠀⠳⣄⠀⠀⠀⠘⢦⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⣇⠀⣠⠞⠁⡰⠃⢠⣾⠃⠀⠀⢰⠃⠀⠀⡠⠤⠴⠁⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠱⣄⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠷⠊⠀⠀⣼⣷⣤⡿⠁⣠⠀⢠⣿⣷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠘⠿⠿⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⡤⢤⣈⣧⡀⠘⣿⣿⣿⠇
⠀⠀⠀⠀⢀⣼⣿⠋⠉⠀⣼⡟⠀⣾⣿⣿⠃⠀⢠⠖⠒⠒⠒⠒⠒⠒⠖⠲⠒⠶⠶⠲⠖⠶⠒⠀⠤⣄⣀⠀⠈⠻⣿⣆⠈⠛⠉
⠀⠀⠀⢠⣾⣿⠁⠀⢀⣞⡞⠁⣸⣿⣿⣿⠀⠀⡌⠀⢠⡴⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⠙⢦⡀⠿⣿⣷⡄
⠀⠀⢠⣿⣿⠁⠀⢠⣾⡿⠟⠀⣿⣿⣿⠁⠀⠀⡇⠀⣸⣤⡴⠋⠀⣾⢠⡞⠻⡆⣶⣄⠀⡟⢶⠀⠀⣠⠄⠀⠀⠙⣄⠀⢻⣿⣦
⠀⠀⢿⣿⡷⠀⢰⣿⡟⠁⠀⠀⣿⣿⣿⠀⠀⠀⡇⠀⡟⠀⣾⠶⠖⣿⢸⡄⢠⡇⡿⠈⠻⡇⢸⡇⣴⠋⠀⠀⠀⠀⠈⣆⠸⣿⣿⣇
⠀⠀⠀⠈⣠⣤⣤⣤⣀⡀⠀⠀⢿⣿⠟⠀⠀⣠⣧⡀⠁⠀⣿⠀⠀⠋⠀⠉⠋⠐⠇⠀⠀⠀⢀⡿⠁⢀⠔⠢⡀⠀⠀⢸⡀⠈⠻⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣶⣾⣿⣿⣷⠀⠀⠀⠀⠘⠳⣤⣀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠸⣀⣀⠇⠀⠀⠀⠃
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠈⠁⢀⣀⣀⠤⠆
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣤⣄⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⢠⣿⠛⢦⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡀⠀⠀⠈⣿⡀⢸⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢀⣴⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⠹⣧⠀⠀⠀⢿⡇⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣿⡄⠀⠀⢸⣷⠀⢹⠀⠀⠀⢀⣿⣿⡇⠀⠉⢹⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣹⣷⠀⢠⣾⣿⡖⠘⣿⡇⠀⢸⣿⣿⡇⠀⠀⡞⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⣹⠿⢇⠈⣿⣿⣷⠆⢿⣿⠀⢸⣿⣿⠃⠀⢰⠇⠀⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠸⡀⣿⣿⣿⡂⠸⣿⡇⢸⣿⡿⠲⠤⡜⠀⡼⢁⣼⣿⠏⠙⠻⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣷⣾⣿⣿⡇⠀⢸⣇⣿⣿⡇⠀⣾⠃⡼⢡⣾⣿⡟⠀⠀⠀⢈⡿⠛⠋⠉
⠀⠀⠀⠀⠀⣠⣴⣾⠿⠿⠟⠋⠉⠉⠉⠙⠻⣿⣿⣷⡀⠸⣿⣿⣿⡯⢉⣿⡼⣡⣿⣿⣿⣄⡀⠀⢠⠎
⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⣿⣿⣿⠙⢺⡿⣵⣿⣿⣿⣯⣀⣼⣷⠋⠀⠀⢀⠀⠀⢐
⠀⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⠁⠀⣼⣿⣿⣿⣿⣟⣵⣿⣿⣷⡶⣶⡆⢸⣷⣴⡟
⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⢹⣿⡀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⡇⣿⡇⢸⠛⠛⡇
⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢰⣿⣿⣿⣷⡼⢤⣤⠧⣄⠠
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣽⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣿⠀⣸⣿⠀⢸
⢻⣿⣿⣦⣀⡀⠀⠀⢀⣀⣤⣶⢿⣿⣿⡿⠛⠉⠉⢙⣿⣿⣾⣿⣿⣿⣿⣿⣀⢐⣗⣀⣽⠀⢸⣿⣿⣿⣿⠁⣿⣿⠃⢹
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠋⠉⠻⣦⣤⠞⠋⠀⢈⣹⢿⣿⣿⣿⣿⣮⣽⣿⣿⣿⠀⢸⣿⣿⣿⣿⠀⣿⣿⡇⢹
⠀⠀⠉⠛⠛⣻⣿⡿⠋⠉⠉⠻⣦⣴⠞⠋⠉⢙⣷⣶⡞⠉⠀⢀⣉⣿⣿⣿⠫⣭⣿⣯⣿⠀⢸⣿⣿⣿⣿⠀⣿⣿⡇⢸
⠀⠀⠀⠀⢸⣿⣿⣷⣤⣤⠶⠋⠉⠙⣷⣶⠞⠋⠁⢀⣙⣷⣾⣿⣿⣿⡿⢿⠀⢿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⣿⣿⡇⢸
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣶⣤⠖⠋⠁⠉⣻⣶⣾⣿⣿⣿⣿⠿⠛⠉⠀⢸⡇⢸⣿⣿⣿⠀⠘⠻⠿⠿⠟⠓⠛⠛⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⣦⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠉⠻⢶⣤⡤⠞⠉⠁⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠁⣀⡀⠀⠀⢠⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⠃⡀⠀⣿⡇⠀⠀⠀⣷⡀⡀⣀⣿⢻⡆⠀⠀⣰⣷⢷⠃⡀⠀⢀⠀⢀⡀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⢀⣴⣿⣧⡿⢃⢃⠠⣸⣿⣦⢰⡐⡑⣷⢷⢻⣿⣿⣿⣦⣿⣿⣟⡞⡸⢡⢧⣿⢇⣾⡖⠀⣀⡄⠸⠻⣦
⠀⠀⠀⠀⣠⣾⡿⣿⣏⣷⠘⢏⣿⣽⣝⣿⣦⣣⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣷⣯⡟⣠⣾⢯⣿⢃⣷⢹⣧
⠀⠀⢀⣼⣿⣿⣿⣿⣯⣝⣷⣶⣿⣿⢿⡿⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⠿⠿⣿⡟⠛⠟⠛⢏⠻⣿⢿⣷⣿⣵⣷⣷⣧⣞⣷⡀
⠀⠀⠈⢻⣿⣿⡿⣿⣿⣿⣿⣯⣻⠃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠘⣿⢻⢿⣿⢿⡿⣿⠇
⠀⠀⠀⠀⠻⣟⣼⡿⣹⣿⣻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣮⣿⡆⡏⡏⣿⠋
⠀⠀⠀⠀⠀⠙⢎⠱⠻⣧⣿⣿⡏⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣼⣸⣿⣿⣇⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠏⣿⢠⣿⣧⡞⢠⣄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠰⢀⠀⡆⢀⢠⣿⣧⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠉⢻⣿⣿⣿⣿⡏⡞⣰⢀⢠⡇⣀⠀⡀⠀⢀⢰⣼⣶⣷⣼⣿⣿⡟⢻⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣿⣿⣿⣿⡇⠇⡇⠆⠈⠁⣿⡿⣿⣿⣿⢸⣿⣿⣿⠿⣛⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣤⣅⣀⣀⣀⣃⡀⢿⣾⣈⣈⣤⣼⣷⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠿⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⣀⣀⣀⡠⠶⠦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠉⠉⠰⠶⠆⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠠⣾⡟⠉⠛⠙⠛⢧⣄⠀⠀⠀⠉⠙⠛⢶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠤⣤⣤⡤⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡾⠛⠃⠀⠉⠁⠀⠀⠀⠀⠀⠘⢷⣤⡀⠀⠀⠀⣀⣀⠽⣷⣤⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠟⢠⣄
⠠⣾⣿⠛⠶⠶⠀⠒⠀⠀⣀⣉⣀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠰⠆⠈⠙⢿⡀⠀⠀⠀⠀⠀⠈⠛⣏⠀⠀⠀⠺⠷⣀⠀⠀⣀⡀⠰⠂⠀⠀⠀⠀⠁⠰⠆
⠀⠈⠉⠀⠀⠀⠠⠄⠀⠈⠙⠛⠉⠁⢰⡿⢷⠦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠀⠀⠀⠀⠀⠀⠙⣀⡀⠀⠀⠀⠿⣿⡆⠉⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⢦⣄⠀⠀⠀⠀⠀⠀⠈⣶⣄⣠⣤⣤⠀⠘⢿⠟⠙⠁⠀⠀⠀⣼⡆
⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢠⣴⠃⠀⠀⠀⠈⠃⠀⠀⠉⢷⣤⣀⠐⠛⠓⠂⠈⠙⢧⡀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⢣⣄⠀⠀⠀⠀⠀⠀⢾⡏⠀⠀⠀⣰⡶⠶⠖⠀⣶⠟⠻⣿⣷⠶⢄⣀⠀⠀⠈⠹⢶⡀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠈⣳⡤⠈⠉⠛⠁⠀⠀⣤⠀⠀⠀⠘⢃⣀⠈⠻⣷⣤⠠⣤⡴⠿⣶⣤⣀⣹⡿⠂⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⣧⠀⠸⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⡿⠃⠀⠉⠁⠀⠀⠀⠀⣀⠛⠛⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣽⡇⠀⠀⠀⠀⠀⢿⣄⠀⠐⠀⠈⠁⠀⠀⢀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢉⣹⠶⠶⠖⠂⠀⠀⣠⡄⠐⠒⠖⠊⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠉⢿⠀⠀⠀⠀⠀⠀⠹⣿⣆⣀⡀⢀⣶⡆⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠁⠀⢶⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⡀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⡄
⠀⢀⣴⠀⠀⠚⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠿⠿⠲⠂⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⢀⣀⣀⡠⠶⠶⠶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢷⣦⠄⠠⠾⣷⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣦⣤⣤⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡄⠀⠀⠀⠀⢀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠶⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠂⣀⣤⣀⡀⠀⠀⢀⣀⣀⡀⠘⠛⠃⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢻⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢄⣀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⠦⢀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣤⡄⠀⠀⠈⠃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣀⠈⠋⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠾⣿⣄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⢀⠀⠀⠐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣀⣠⣤⣄⣠⣤⣀⣤⣤⣄⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣷⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⡄
⠀⠀⢾⣿⣟⡁⢙⣿⣿⣟⠁⣨⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣉
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⣶⣤⣤⣶⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡟⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣾⣿⡇⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠙⠻⢿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠛⠛⠛⠛⠛⠛⢻⣧⠀⠀⠀⣠⣶⠶⠷⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⠀⢀⡀⠀⣀⡀⢸⣿⠀⠀⣾⠟⠁⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠀⠈⠁⠀⠈⠁⢸⣿⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣶⣶⣶⣶⣶⣄⣠⣾⠏⠀⠀⠹⣷⣄⡀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠇⠀⢀⣀⣀⣈⣙⠛⠛⢛⣉⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠉⠛⠻⠿⠟⠋⠉⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣴⡆⠀⠀⠀⠀⠀⠀⠀⣶⡄⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢠⣿⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⢻⣧⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣇⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⣾⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢿⡆⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⣼⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⣄⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⢸⡇⠀⠀⢀⣀⠀⠀⠀⣿⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⢻⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣾⠏⠈⢿⣦⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠉⢁⣠⣾⣿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣋⣿⣿⣿⣿⣿⣿⡿⢤⡄⠛⠉⠛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣅⠀⠀⠀⠀⢀⣠⣴⣿⣽⣿⣿⠘⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠛⠛⠛⠁⠀⠋⠀⠞⢠⣏⢹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢿⣿⣿⣷⣄⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⡉⠋⠙⠻⣿⣿⣿⣿⣷⡀⠙⠀⠀⠀⠀⠀⠀⠘⢃⡾⣿⣿⣿
⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⣴⣿⣾⣿⣿⣿
⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⢀⡀⠀⠀⠄⠀⠀⠀⡀⢸⣿⣿⣾⣿⡆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⣶⣾⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣤⡆⠀⠀⠐⠀⢀⣀⣤⠖⠄⢸⣿⣿⣡⣾⣿⣆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠉⠳⢤⣿⢿⣿⣿⣿⡿⢛⣿⡆⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠰⠿⠿⠿⠿⣯⠉⠛⠻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣾⠏⣼⣿⣿⣫⣴⣿⣿⣿⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣠⣴⡏⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠤⢭⡟⠻⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣶⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣶⠀⠀⠈⢿⣿⣿⣿
⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠟⢶⡉⠙⣶⣟⣿⣿⣿⣿⣏⠁⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠈⢿⡛⣿
⣿⣿⣿⣿⣶⠀⢄⣀⣀⠀⠀⠘⠃⠉⠑⣏⠻⣿⣿⣿⣿⣷⠙⢀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣾⣿⡟⠉⠉⠻⠿⢿⣿⣿⣿⣿⣟⡾⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⡏⠁⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣗⡦⣀⢼⢟⠛⠉⣸⣿⣿⣿⣿⣿⠃⠀⠀⠀⣴⣿⣿⠋⠉⠻⣿⡇⠀⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣷⡀⠀⠀⠸⡏⠀⠀⠀⠀⠀⢤⠀⠀⠀⠘⢿⣿⣿⡷⢢⠦⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢰⣿⣿⠃⠀⠀⠀⢶⡇⠀⠀⠀⢀⣿⣿
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⣄⠀⠀⢹⣿⣿⢆⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⢾⣇⠀⣤⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠒⠒⠒⠒⠒⠂⠒⠆⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠄⠀⢀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⡔⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡚⠀⣀⣀⢘⢲⡀⠐⠀⠀⠀⢰⠒⢢⠤⠄⠀⠀⠠⠄⠀⠀⠀⠀⢃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢸⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡔⠃⠊⠙⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠘⢀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠂⠘⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠖⠲⠖⠒⠒⠒⠂⠒⠂⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⡠⠒⠂⠐⠒⢤⠀⠀⡴⠂⠑⢢⠀⠖⠉⠙⢆⢸⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠇
⠀⠀⠀⡇⡘⠀⠀⠀⠀⠀⠀⢱⡀⢣⡀⢀⡼⠀⢦⣀⣀⠎⢸⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⡃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⣠⠤⢤⠄
⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⠆⠠⢄⠀⡴⠒⠢⡄⢸⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠧⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⡇⠱⡀⠀⠀⠀⠀⠀⣰⠁⣇⠀⠀⣸⠀⣄⠀⢀⡄⢸⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠰⠄⠀⠀⠠⠄⠄⠤⠀⠀⠐⠀⠀⠰⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡇⠀⠈⠢⠤⠀⠤⠞⠁⠀⠈⠉⠈⠁⠀⠈⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⡀⠀⢀⣄
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠳⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠉⠒⢄⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠢⢀⣀⣀⣀⣀⡀⠠⠊⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣅⠼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠴⠊⢠⠘⠅⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠀⡀⠤⢀
⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⠄⡀⠀⠀⠀⠈⠀⠄⠀⠀⠤⠂⠀⠘⡆⠀⢀⡔⠚⠒⠒⠢⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⠀⠀⠀⢨⣀⠠⠀⠠⠐⠨⠀⠐⡇⠀⡎⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⢀⡠⠀⠀⠀⠀⠑⢤⡐⣀⢶⠡⡈⠎⠀⠀⠡⡀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠔⠉⠑⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠈⠛⠂⠑⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⢐⠀⠀⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡦⠀⠀⠠⠄⠀⢠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠜⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠢⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠊⠀⠁⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡴⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶
⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⢀⣾⠏⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢀⣿⠇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡀⣀⣀⣀⣀⣤⣄⣀⣀⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⠀⠀⠀⠀⠀⠀⠀⢷⣶⣶⣶⡶⢿⣶⣶⠶⠶⠾⠖⠿⢿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠋⢹⡇
⣿⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⣤⣤⣴⡿⠇⠀⠀⠀⠀⠀⠀⠻⠄⠸⢦⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀
⣿⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⢸⣧⠀⠀⠁
⣿⠀⠀⠀⠀⠀⢠⡆⠀⠀⢱⣤⣤⣤⡟⢣⠀⠀⠀⠀⠘⢿⣤⣤⣴⡿⠁⠀⠀⠀⠀⠀⢸⣧⠀⡄
⣿⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠉⠋⠉⠳⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠈
⣿⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣾⣿⠶⣿⡿⠿⠖⢀
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡸⠃
⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡎⠁
⠀⢀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⡟⠋⠁⠀⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠁
⠀⣾⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠛
⠀⢻⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⣦⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⣿⡀⠀⠀⠀⠀⠈⠙⠶⠄⣀⣀⣀⡀⢀⣀⣀⣀⣤⠀⠴⠀⠀⠀⠀⠴⠆⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣤⣀⣀⣀⣀⣀⣈⠋
⠀⠀⠸⣷⠾⢷⣶⣴⣶⣄⣀⣀⣉⠉⠉⢉⣉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣤⣤⠴⠶⢶⣶⣶⡿⢷⣶⠟⠛⠻⠛⠁⠉⠛⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠿⠛⠛⠻⠟⢿⣷⠀⠀⠀⠀⠀⠀⣿⡋⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣼⣧⡀⣠⣤⣤⣤⣴⠟⠛⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠃⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⢾⡆
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠐⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡇⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠆⢀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⣶⣄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠿⠟⠛⠛⠛⠛⠻⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⣶⠶⠒⠶⠖⠲⣶⣿⠃⠀⠀⠀⢸⣿⣁⠀⠀⠀⠀⠀⠀⣀⣼⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠻⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣤⡆⠘⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⣿⣦⣶⣶⡤⠀⠀⠀⠀⠀⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣷⣦⣀⣾⣿⣿⣾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠹⡿⠿⠀⠛⠃⠉⠀⠈⠁⠈⠁⠿⣿⣿⣿⢹⣿⣿⡿⠛⠋⠀⠉⠉⠀⠙⠛⠛⠛⠿⠟⠛⠿⠿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⣬⣍⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣤⣶⣾⣿⣦⣴⣶⣶⣴⣶⣶⣶⣦⢀⡀⠀⣀⠀⠀⠀⠀⢠⣄⢀⣤⣴⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⡿⢿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⡟⢿⡿⣿⣿⣿⣿⣿⣿⣿⠏⠿⠟⠛⠃







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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣤⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣷⣦⡄⠀⠀⠀⠀⠀⠘⢡⣿⠆⠀⢀⣀⣀⡤⠄⠀⠀⠀⠀⢀⣻⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⢀⣿⣿⣀⣠⠤⠆⠀⣀⡤⠶⠚⠋⠉⣀⣤⠤⠶⠒⠛⠉⠉⠉⠙⠋⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠋⢁⣀⠤⠒⠉⣀⣀⣤⣶⣚⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣩⣭⡠⠖⢋⣠⣴⣾⣿⣿⣿⣿⣿⣿⡟⣰⣾⣶⣦⡀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠋⢀⡠⠟⠛⣡⣴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠸⣿⣿⣿⣿⣿⣦⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⡴⠋⢀⣤⣾⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⣀⣀⠀⠹⣏⡙⢿⣿⣿⡄⢋⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⠉⢸⣿⡿⠟⣿⣿⡟⠁⠀⣀⣉⡙⠛⣩⡤⠂⠀⠉⠁⠀⣿⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡠⠚⣁⣀⠀⠘⣿⡅⢠⣿⠏⠀⠀⠉⢁⣈⠉⠛⠁⢀⣴⣶⡄⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠞⠀⠘⢛⣿⡷⠀⢻⡇⠘⠿⠀⠀⣴⣾⠿⠻⣿⠀⢰⠟⠋⠹⠇⠀⢸⣏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⣀⠀⠀⢨⣥⠀⠀⠘⣷⠀⠀⠀⠸⠋⠁⠀⠀⠁⢠⡀⠀⢀⡀⠀⠀⢸⣿⣭⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠙⠀⠀⠀⣠⠶⠦⣿⡄⠀⠀⠀⠀⢠⣶⡆⠀⠀⢷⠀⠻⡿⠀⠀⣾⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡀⠀⢰⣿⠀⠛⢻⣷⠀⠀⠀⠀⠀⠙⢡⠆⠀⠸⣇⠐⢦⡀⠀⣿⣿⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⣿⣇⠀⠀⠀⠀⠴⠋⠀⣀⠀⢻⡀⠀⠁⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣀⣹⡟⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⣁⣤⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠺⢿⡟⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠠⠿⠶⣦⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣼⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣀⣀⣴⡧⠾⠋⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣦⣤⣍⣩⣤⣤⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠛⠻⠿⠟⠛⠻⠿⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣶⣿⣿⣿⣿⣿⣟⡳⣦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡧⢹⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡌⢿⣿⣿⣿⣿⣿⡿⢣⡟⣼⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢣⣦⣙⠿⣿⣿⣿⠟⣫⣬⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢣⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⡇⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣌⢻⣿⣿⣿⣿⣿⣿⡿⢋⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠷⠝⠻⢿⣿⠿⠋⠴⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⡶⢆⣼⣿⣿⣿⣿⣷⡐⢶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡅⢶⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆⣼⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡌⢿⣿⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⡟⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣎⢿⣿⣿⣦⠻⣿⣿⣿⣿⡿⢣⣾⣿⣿⡟⣰⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣎⢿⣿⣿⣷⡹⣿⣿⡿⣡⣿⣿⣿⡟⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣷⡜⠟⣰⣿⣿⣿⠟⣼⣿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⣿⣿⡙⣎⢻⣿⣿⠏⣼⣿⣿⣿⠏⣼⣿⡟⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⢸⣿⣿⡜⠆⠻⠋⠾⠿⠿⠿⠏⣼⣿⣿⡇⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣭⣭⣥⢸⣿⣿⣿⣿⣿⣿⡇⣬⣭⣭⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⠿⠿⠿⢟⣛⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠸⣿⡿⣿⣿⣿⣿⠇⠛⠛⠉⠉⠉



⣿⣿⡿⠿⣿⠿⢿⠿⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠃⠀⠀⠦⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⢀⠠⠀⠐⠂⠁⠀⠠⢼⠄
⠀⠀⠀⠀⠀⠀⢀⠐⠈⠁⠀⠀⠀⠁⠀⠰⠄⠃⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣈⠀⠀⠀⢁⡁⠀⠀⠀⠀⠴⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡀⠀⠀⠂⠐⠂⠀⠐⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠘⠀⠀⠈⠛⠀⢀⠀⠙⠃⠀⠀⡂⠀⡄⡀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⣰⣦⡎⠇⠀⠀⠸⠇⠀⠀⠀⠿⠀⠀⢨⢱⣴⣦
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠃⠀⠀⠀⢠⠉⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠁⠑⠼⡀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠰⠊⠈⠀⢣
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠒⠀⠀⠀⡀⠄⠃⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠐⠀⠀⠀⠐⠒⠂⠀⡀⠔⠁⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⠀⣰⣀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⣎⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣈⣷⣁⡉⢣⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⣀⠀⠀⠀⠀⠜⡋⣱⣞⢙⠓⡄⠀⠀⠀⢀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠞⠃⠀⠀⣿⡇⠀⠀⡏⠒⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡚⠀⠀⢸⣯⠀⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢠⢸⠀⠀⢰⡇⡇⠀⠀⢇⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣾⣻⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣸⠀⠀⠸⡟⡿⢀⠤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣻⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠛⠋⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠘⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⣤⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣤⣤⣤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠃⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠗⠸⠿⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⣀⣀⢀⢀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠛⠛⠛⠋⠛⠐⠚⠂⠘⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠈⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⣿⣿⣿⠿⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⡆⢀⡤⠤⠤⠤⠴⠃⠙⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⢀⡼⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋⠈⠈⠙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠉⠉⠉⠉⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⣀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠐⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⡧⡀⢀⣰⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣗⠀⢘⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣼⣷⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠈⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣾⠿⠿⢟⣛⠉⠉⠉⠛⠛⠛⠉⠉⠉⣛⡻⠿⠿⣷⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⣀⣀⣠⠤⣶⣶⣶⣦⣤⣤⣴⣶⣶⡦⢤⣄⣀⡈⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠟⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣭⡁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣭⣥⣦⣤⣀⣀⣀⠀⢀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠘⢿⣿⡋⠉⠈⡇⢰⣿⣿⣷⡄⠀⠘⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⣿⠁
⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣆⠀⠀⠘⠛⣷⡀⠸⣿⣷⡀⠀⣷⢸⣿⣿⣿⣇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⡰⢃⣼⣿⠃
⠀⠑⠤⡀⢀⣠⣴⣄⠀⠀⠀⠀⠹⣉⠀⠀⠀⢸⣿⣧⠀⢿⣿⣷⡀⢹⠀⣿⣿⣿⣿⠀⢸⠀⠀⡿⠀⠀⠀⠀⢠⣮⣀⣿⣿⠃
⠀⠀⠀⠈⢻⣿⣿⣿⣦⡀⠀⠀⠀⢻⣧⢣⠀⠈⡿⣿⣆⠘⣿⢿⣿⣹⡇⢻⣿⣿⣿⡆⢸⠐⠒⡇⠀⠀⠀⣰⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡄⠀⠀⠀⢻⣧⢣⡀⢹⣿⣿⡄⢹⣽⣿⣀⣷⢸⣟⣿⣿⣇⢸⠈⠉⡇⠀⠀⣰⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣀⣨⣿⣿⣿⣿⣶⣦⣤⣌⣿⣧⢳⡀⢿⣿⣷⠀⢿⣿⣿⣿⡀⣿⣿⣿⣿⡆⠀⢸⡇⠀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡳⠘⣿⣿⣧⠸⣿⣿⣿⡇⢹⢹⣿⣿⡟⠀⣼⣿⡏⠉⠛⠿⣿⣃
⠀⠀⣴⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣸⡟⠙⡆⢻⣿⣿⣿⠘⡟⣿⣿⡇⢸⣿⣿⣿⣿⣶⣦⣄⡙⠳⣦⡀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⠀⢹⡈⣿⣿⣿⡄⣧⢿⣿⡇⠀⣿⣿⣿⡿⠟⢿⣿⣿⣶⣄⡙⢦⡀
⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠻⢇⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢧⢹⣿⣿⣇⢸⣸⣿⡇⢰⣿⢿⣷⣍⠀⠀⠉⠛⢿⣿⣿⣦⣽⡄
⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⣠⣔⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⡆⢿⣿⣿⣼⣷⣿⣦⣼⣯⣾⣿⣿⣷⣦⣀⠀⣰⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣅⣀⣠⣴⣿⠾⢋⣽⣿⣿⣿⣿⣿⢯⣿⣿⣿⣿⠸⣆⣽⣾⣿⠿⣛⠛⠉⠙⢛⡻⢿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⡃
⢻⣿⣿⣿⣿⣿⣿⣿⣯⢁⣤⣾⣿⣿⣿⣿⣿⣫⣾⣿⣿⣿⡿⣰⣿⡿⣫⣖⣭⡶⠾⠟⠛⠛⠻⢶⣌⠻⣿⣿⣆⠉⠛⠻⠿⠿⠟⠁
⠘⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠉⠀⠀⠀⠀⠀⠀⠐⠂⠈⠳⡜⣇⠘⣇
⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⡿⠿⠟⠛⠛⢿⣿⣿⣿⠃⠀⠠⠀⠀⠀⠀⠸⠗⠀⠀⠀⢀⢹⣿⡄⣿⡀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⡁⠀⠤⢤⣤⡀⢸⣿⣿⡏⠀⠰⣦⣶⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟⠁⠀⠈⠳⣌⣧⣾⣀⣀⣸⣿⣿⣷⡀⠀⢨⠙⠓⠀⠀⠀⠀⠀⠆⠀⠀⣸⣿⢡⣿⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⡏⠀⣤⣀⣀⣤⢀⣀⠈⠓⢦⠜⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢯⣾⣟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠉⠹⣿⠌⠉⠑⠂⢁⣤⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣤⣤⣤⣤⣤⣶⣾⣿⢟⣫⠶⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠆⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠈⠀⣰⡿⠛⠙⠃⣾⠟⢷⣆⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠁⠀⠀⠀⢹⣇⠀⢀⠀⣿⣷⡿⠃⢰⡿⣷⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠳⠿⠙⣿⣀⣿⡏⠹⣧⡀⢁⣾⢆⡀
⠀⠀⠀⠀⠶⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠲⣦⣼⣷⡾⢻⡿⠻⣦
⠀⠠⠀⠀⠀⢀⠀⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣅⣘⣷⠀⠁
⠀⠀⠁⣴⣦⠈⡛⠀⠀⠀⠒⠺⠆⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠉⠛⠋
⠀⠀⠀⠘⠉⠛⣀⠀⠀⠀⠀⠀⠀⢤⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠎⠠⠀⠀⣀⣀⣠⠦⠀⢘⣤⠀⠀⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣏⠴⠆⢰⡀⠀⡘⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦⡐⣺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣷⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠠⠀⡄⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣙⣦⣤⣀⣀⣤⣬⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⣶⣤⣀
⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡶⣶⣤⣄⣀
⠀⢀⡞⠉⠉⠉⠉⠉⠉⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠉⠙⠷⣤⡀
⠀⣸⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣼⣯⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣞⣣⠀⡄⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⢹⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⢹⣿⣇⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠹⢦⣤⣤⣤⣦⣴⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣶⠶⠶⢤⣤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣾⠿⠟⠛⠋⠁