Obese Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣝⡛⠿⢿⡿⣿⠿⠿⢟⣫⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣾⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣷⡀
⠀⣼⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣧
⢠⣿⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿
⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿
⠈⠙⠿⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠛⠿⠿⠛
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣈⣉⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⣉⣁⣤⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⠀⠀⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⣩⣉⣉⣉⣉⣉⣩⣍⣍⣩⣉⣍⣉⣉⣩⣍⠸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⢽⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠑⠒⠐⠂⠀⠀⢤⡆⠀⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⢶⣙⠉⢛⣴⠚⠤⡐⢄⡃⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠩⠒⠁⣹⢦⠀⠋⠀⠁⡀⠀⠀⠀⢂⡸⢊⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡓⢒⣙⣩⢵⠞⠐⠀⠠⣄⠔⠂⠇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠌⢨⠃⠀⠛⠀⠀⢠⣦⣄⠁⠀⢀⠀⡀⢠⠄⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠰⠋⠀⠃⠀⠀⠠⠔⠀⡘⠀⠒⠀⣈⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡤⠒⠀⠂⠀⠀⠀⠀⠂⠤⡇⠀⢦⠠⢉⠆⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢂⡀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠤⠤⠤⠄⠰⠀⠀⠁⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠒⠀⠳⢤⠀⣾⡄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⢀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡠⠆⠀⠀⠀⡋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⢲⣠⠏⠀⠀⢣⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉⠀⠀⠀⠬⠬⠄⠁⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠠⠃⠀⠀⠀⠸⠄⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⢿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠃⠴⢶⡬⠀⢨⡽⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⣿⠀⠀⠰⠶⠀⠀⢰⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠛⠐⠀⡀⣀⣀⡀⢘⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⠀⠀⠙⠻⠻⣛⡻⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⢿⠀⠱⣄⠀⠀⠈⣠⣾⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠁⠀⣿⡌⠀⠀⠈⠛⠿⠿⠟⣿⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⡉⠑⠀⡁⠀⠀⠘⠿⣢⣄⣀⣀⣠⣴⡾⠋⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠆⠂⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⡐⢢⠂⢀⡄⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣆⢀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣠⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠠⠔⠤⣦⠀⠀⣾⠀⠊⠁⢂⣩⡤⠾⠒⠒⠂⠀⠀⠀⠤⢤⣿⡀⠀⢹⣇⠀⠀⠀⠀⠀⠀⢀⣠⣼⠋⠀⢰⣏⠀⢠⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠛⠀⠀⠀⢀⡀⠐⠬⠃⠄⠀⠀⠀⠀⠴⠋⠑⠈⣀⠤⢀⣄⣠⡀⠀⠀⠀⠈⠙⢶⣞⠙⢳⣤⣤⡶⠶⠟⠋⠉⠁⠀⠀⠸⣯⠀⠄⣿
⠀⠀⠀⠀⢀⣴⣿⠤⠤⠤⠤⠤⣄⣈⡙⣡⣾⠗⠀⠤⠐⠀⠀⠈⠁⠈⠀⠐⠉⠁⢀⡀⠀⠀⢀⣀⠀⠀⠘⢧⣼⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡟⠶⢲⡟
⠀⠀⠀⠀⣾⡃⠈⢉⡀⡀⠐⠒⠦⠬⣙⡏⢸⡗⠀⠦⠂⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⡀⠉⠉⠀⠀⣈⠀⠤⠄⢻⡀⠀⠀⠀⠀⣀⣤⠾⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⣿⠡⡶⠟⠋⠉⠉⠉⠉⠛⢻⡇⣼⠁⠀⠤⢒⣡⠴⡚⠉⣀⣠⣴⣾⠿⠷⠞⠛⠛⠛⢛⠛⠛⠷⢦⣘⣧⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠻⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣠⡴⠚⠭⣒⣽⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣿⠋⠁
⠀⠀⠀⠀⠀⠉⠛⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠹⣦⣴⠞⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠈⢻⣄⡠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⡏⠡⠀⠀⠀⢿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡹⠇⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡶⢿⣒⡒⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣬⣄⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⠿⠿⣃⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⢀⣶⡆⠀⠀⠀⠀⠠⣿⡦⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⣿⣟
⠀⠀⣠⣾⣿⣿⣿⡄⢀⣾⣿⣿⣿⡄⠀⠀⠀⣼⣿⣇⠀⠀⠀⣠⣾⣿⣷⣦⡀⠀⠀⣸⣿⣷⡄⠀⠀⢠⣶⣾⣿⣦⠀⠀⣴⣿⣿⣿⣿⣆
⠀⠀⢿⣿⣿⣿⣿⡟⠈⢿⣿⣿⣿⡷⢦⠀⠀⣿⣿⣿⡇⠀⠐⣿⣿⣿⣿⣿⡗⠀⢀⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⢸⡿⣿⣿⣿⣿⢿⡃
⠀⠀⢘⣿⣿⣿⣿⠀⠀⠸⣿⣿⡿⠃⠀⠀⣾⣿⣿⣿⣧⡀⠀⢻⣿⣿⣿⣿⡇⠀⣼⣿⣿⣿⡇⠀⠀⣷⣿⣿⣿⣿⡇⢸⡇⣿⣿⣿⣿⣸⠅
⠀⠀⠀⣿⣿⣿⡟⠀⠀⢀⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠃⠀⠙⢹⣿⡿⠃⠀⠀⢸⣿⣿⣿⡿⠁⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⢻⣿⣿⠀⠀⠀⣾⠏⢹⣯⠀⠀⠀⣸⡟⠁⣿⡇⠀⠀⠀⢻⡿⣿⠏⠀⠀⠀⢸⡟⢿⣆⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢿⣿⢻⣿⠇
⠀⠀⠀⠈⣿⡟⠀⠀⣠⣏⠀⠈⢿⣄⠀⠀⣻⠃⠀⠘⢿⣀⡀⠀⢸⡇⣿⠀⠀⠀⣀⣿⠁⠀⢹⣦⠀⠀⠘⡟⢸⡄⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠉⠀⠘⠛⠉⠀⠉⠁⠀⠀⠈⠉⠀⠀⠈⠁⠛⠀⠀⠈⠉⠉⠀⠀⠉⠁⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠛⠀⠉
⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣾⣿⣿⡅⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠐⣿⣿⡀⠀⠀⠀⣠⣴⣿⡏⠀⠀⠀⠀⢻⣿⡇⠀⠀⢠⣤⣿⣿⣷⡄⠀⠀⢸⣿⡇
⠀⠀⣿⣿⣿⣿⡄⠀⠀⢰⣿⣿⣿⡄⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⣼⢛⣋⣽⣃⠀⠘⠛⣿⣿⣿⡇⠀⠀⣿⣿⣷⡄
⠀⢸⣿⣿⣿⣿⡇⠀⢀⣾⣿⣿⣿⡄⠀⠀⣠⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣧⠀⢀⣿⣽⣿⣯⡿⡄⠀⢸⣿⣿⣿⡇⠀⢀⣿⣿⣿⣿⣄
⠀⠀⢻⣿⣿⡟⠀⠀⠉⠘⣿⣿⡿⠁⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⡇⠀⢸⢿⣿⣿⣿⡇⠻⠀⠈⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⠈
⠀⠀⠀⣿⣿⡿⠀⠀⠀⢸⣿⢿⣇⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⢀⣿⣿⣿⡅⠀⠀⠈⣿⣿⡟⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⣿⡿⢿⣿
⠀⠀⠀⣿⣿⣇⠀⠀⠀⢸⡟⠈⣿⡀⠀⠀⢰⣿⠃⠹⣿⡄⠀⢠⣾⠛⠿⣿⠃⠀⠀⠀⣿⣹⡇⠀⠀⠀⠀⢀⡟⢿⠀⠀⢸⣿⠁⢸⣿
⠀⠀⠀⠿⠿⣿⣤⠀⠘⠛⠃⠤⠿⠇⠀⠒⠻⠏⠀⠠⠿⠇⠀⠈⠙⠂⠀⠿⠷⠄⠀⠐⠋⠸⠆⠀⠀⠀⠀⠺⠧⠾⠇⠀⠺⠿⠀⠘⠿⠦

⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠷⠿⠿⠷⠶⠶⠷⠷⠿⠶⠾⠶⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⠀⢴⣿⡆
⠀⠀⠀⠀⠀⢠⣾⣿⣶⡀⠀⠀⣰⣶⣿⣶⡄⠀⠀⢠⣶⣾⣷⣶⡄⠀⠀⢀⣴⣾⣿⣶⣆⠀⠀⠀⢀⣴⣶⣿⣶⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠀⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠛⣿⣿⡟⠃⠀⠀⠛⣿⣿⣿⠛⠀⠀⠻⢿⣿⣿⡿⠟⠀⠀⠛⢻⣿⣿⣿⠛⠃⠀⠘⠟⢻⣿⣿⣿⡏⠛
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⢿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣿⢿⡇⠀⠀⠀⠀⢿⠿⡿⠀⠀⠀⠀⠸⡿⢿⡇⠀⠀⠀⠀⠸⣿⢸⡿⠀⠀⠀⠀⠀⠈⣿⠻⡿

⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣦⣰⣶⣶⣶⣶⣶⣶⡴⣶⣶⣶⣶⣶⣶⣦⡰⣶⣶⣶⣶⣶⣶⣶⣦⢶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠑⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠋⠚⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠿⣿⡀⠀⠀⠀⣹⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠈⠁⠀⠀⠛⠋⠀⠉⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⢀⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⢻⣿⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠻⣷⣴⠏⠁⠀⠀⠀⠀⢀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⣠⠶⣤⠀⢀⡤⠈⠛⠿⣶⣶⣶⣶⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⣧⠀⠈⠀⠋⣴⠟⠁⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⡾⠋⠀⠀⠀⠀⢡⡀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠹⢦⣤⠴⢦⣠⡼⠃⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠂⠀⠀⠀⢀⣰⡿⠃⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢀⣿⠟⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⣶⣶⠿⠋⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠉⢙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡟⠻⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣶⣦⠀⠀⠀⠀⠀⠀⢸⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠿⠿⢻⣿⣿⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⡇⣿⡟⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡿⢚⣛⠛⠳⣤⣿⢟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⣿⡆
⠀⠀⠀⠀⠀⠀⢀⡾⠋⣄⠻⢭⣭⣤⡴⠋⢿⣯⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⡿⠟⠁⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣾⣅⣈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⢉⣉⣀⣀⣀⣼⢃⡿
⠀⠀⠀⠀⠀⠀⣿⡉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⣿⣼⠃
⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠶⠶⠶⠶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢖⣒⣛⣭⣭⣭⣽⣶⣶⣶⢶⣍⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡞⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠘⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⣿⣿⣿⣽⣧⣯⣾⣿⣿⠋⠀⠀⠀⠀⣹⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⣿⣿⣿⣿⠿⠻⠋⣸⠷⢚⡀⠀⠀⢀⠈⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡼⣛⠛⢿⠏⠀⠀⠈⠀⠚⠧⠀⠀⠀⠻⠄⢷⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⣏⠀⠑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⣧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡝⣷⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠐⠀⠀⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢇⣿⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢷⣽⣲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠋⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠙⠲⠮⢭⣓⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀⠀⠀⠈⠙⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣄⠀⠀⠀⠀⢻⡹⡄
⠀⠀⠀⠀⠀⠀⠀⡴⢫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⣄⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣧⠀⠀⠀⢸⡇⣧
⠀⠀⠀⠀⠀⢠⢞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⡇⡇
⠀⠀⠀⠀⣰⢯⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣆⠀⣼⢻⠇
⠀⠀⠀⠀⡏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⡟
⠀⠀⠀⠀⢿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢚⣏⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣆
⠀⠀⠀⠀⠈⢧⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⣸⡟⣿⡀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣆
⠀⠀⠀⠀⠀⠈⢳⡹⣦⠀⠀⠀⠀⠀⠀⠀⣇⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⢦⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⣩⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡍⠾⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠶⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣭⣛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣻⡵⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠾⠭⣭⣭⣽⣿⣭⣭⠭⠵⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⡄⠀⠀⠀⠀⠰⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠿⣶⣦⣄⣀⣽⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠛⠛⠋⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⣿⣄⠀⠀⠀⠀⣠⣿⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠁⠀⠀⠈⠛⠿⠶⠶⠿⠛⠁⠀⠀⠈⠛⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠙⠻⣶⣄⠀⠀⠀⠀⣠⣶⠿⠋⠀⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠿⣦⣄⡀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢀⣠⣴⠿⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠈⠙⠛⠿⠿⠛⠀⠀⠀⠀⠛⠿⠿⠛⠋⠁⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⢿⣦⣤⣀⣀⣀⣀⣀⣀⣤⣴⡿⠛⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠉⠛⢻⣿⣿⡟⠛⠉⠁⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢠⣶⠿⠃⠘⠿⣶⡄⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣾⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣾⣿⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣿⣷⣶⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠀⠀⠈⠉⠹⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣀⡴⠞⠛⠫⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠑⣔⠢⡀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⢀⡜⠁⠀⠟⠀⢀⡀⠀⠰⠇⠈⠳⡐⢄⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⠀⠀⠹⣌⢧⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢰⠧⡀⠀⠀⠀⠀⠀⠀⠀⠲⠋⠀⠀⢀⣼⡎⣇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠸⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⢹⡸⡆⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣴⣶⣾⠛⠁⠀⠀⠀⠈⡇⢻⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡟⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠠⠀⠀⠀⠀⠀⠀⣠⠼⠁⠀⠀⠀⠀⠈⠙⠷⣤⡤⠤⢤⣄⣀⡀⠆⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⣀⡴⢿⠀⠀⠈⠀⢀⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠀⠈⠙⠳⢦⣀⣼⠏
⠀⠀⠀⢀⣴⠞⠉⠀⢈⣧⠶⣟⠋⠉⠁⠀⢀⣀⣀⣀⡀⠀⢀⣤⠤⠤⣄⡀⠀⠀⢀⣼⣧⠀⠀⠀⠀⠈⠳⣄
⠀⠀⢠⡟⠁⠀⠀⢀⡞⠁⠀⠈⠲⣄⡀⠀⠀⠀⠀⠈⢻⡶⠁⠀⠀⠀⢀⣀⣤⠾⠛⠁⣾⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⣾⠀⢀⡤⠚⢻⡅⠀⠀⠀⠀⠈⠙⠳⠶⢦⣤⣤⣼⣥⣤⡴⠶⠞⠋⠉⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⡏⢀⠞⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⣇⠘⠀⠀⢀⢀⣾⠻⢦⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠐⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⢯⣀
⠀⠀⢻⡈⢉⣁⡄⠈⢻⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⣀⣙⠲⠾⡀⠀⠀⡰⠿⠟⠒⠒⠒⠒⠒⠒⠛⠛⠉⠁⠀⠀⠈⠳⣄
⠀⠀⠀⣩⠟⠁⠘⠦⣹⡀⠀⠀⢀⡠⠤⡴⠚⠋⠉⠉⠉⠛⠶⣌⠢⠜⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⠀⢸⡇⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠈⣷⠀⠀⠀⢠⠼⢅⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠘⣧⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⠦⢤⣤⣤⣤⣈⠳⣄⣀⣀⠀⠐⢠⣤⣦⣤⣶⠶⠟⠋⠁
⠀⠀⠀⠈⠛⠶⢤⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⢷⡄⠙⠉⠉⠀⠀⣠⡈⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠖⠛⠉⠁⠀⠀⠹⣆⠈⠢⡘⠢⠤⣄⡀⣀⣀⣘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣈⣦⡀⠀⠙⣇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣾⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⣷
⠀⠀⢰⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠸⣿⣿⣿⣿⣿⡆
⠀⠀⣼⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⡇⢀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡀⠸⣿⣿⣿⣿⡿
⠀⠀⠀⠙⠛⠛⠉⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⣶⡶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⡄⠀⠀⠁⢀⡆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣤⡤⠴⠞⠋⢹⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠃⠀⠀⠃⠈⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠲⠦⠴⠂⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠠⠤⠤⠘⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣠⠄⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⢤⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⠳⣬⡀⠀⠀⠀⠀⣨⡴⠃⠈⣧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⣧⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣠⣿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠘⡿⠙⠦⢤⣄⣀⣀⣀⣀⣤⠴⠞⠹⡟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠛⠛⠛⣧⡟⠛⠛⠛⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢸⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠿⠶⠶⠶⠃

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

⠀⠀⠀⠀⣼⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢿⣿⡿⣀⡀
⠀⠀⢀⣀⣼⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣾⣶⣄⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣿⣿⣷⣀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣶⣿⣿⣟⣦
⠀⠀⣿⣿⣿⣿⣿⡟⣿⠀⠀⠀⠀⠀⠀⣼⣟⣿⣿⣿⣿⡟⣷⠀⠀⠀⠀⠀⣼⣟⣽⣛⣿⣿⣻⣽⣿⡆⠀⠀⠀⣾⡟⣾⣿⣽⣿⣿⣯⣦⢿
⠀⠀⣿⣿⣿⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⡄⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⣿⢾⣷⣿⣿⣿⣿⣿⣾⣏
⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠛⠋⠉⠉⠈⠀⠀⠀⠀⠀⠈⠈⠉⠋⠛⠛⠋⠉⠈⠀⠀⠀⠀⠈⠀⠉⠙⠛⠛⠛⠋⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⣠⣴⣶⣶⡶⠶⢶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⣰⠛⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠿⠟
⣿⣿⣿⡏⠀⠰⠀⣹⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣆⡿⣯⣳⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡏⣿⡀
⢹⣿⠟⠉⢻⣿⡟⠉⠻⣿⡇⠀⠀⠀⠀⢀⣤⣮⣟⣻⣵⣦⣄⠀⠀⠀⠀⠀⣟⣿⣾⣿⠷⠀⠰⠀⠀⡂⠀⠀⠀⠀⠀⢸⣷⣝⣿⠇
⢸⣿⡄⠀⣾⣿⣧⠀⢠⣿⡁⠀⠀⠀⣰⣿⣿⣉⣿⣿⣧⣼⢻⣧⠀⠀⠀⢀⣽⣮⣿⣟⣿⡆⢀⡂⠰⠀⠀⠀⠀⠀⠀⠀⢻⣽⠃
⢸⣿⣇⠀⣹⣿⣏⠀⣸⣿⠇⠀⠀⢀⣿⢧⣮⣿⣽⣯⣟⣿⡜⣿⡇⠀⠀⠀⡕⠳⡽⣿⣜⣼⢻⠛⡆⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⢀⣀
⠀⠉⠋⠛⠉⠙⠉⠛⠋⠉⠀⠀⠀⠈⠋⠉⠙⠉⠉⠋⠋⠙⠙⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⠋⠉
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢰⣟⢿⣓⣀⢀⣀⡀⠀⠀⠀⠀⠀⣦⣿⣹⣆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⠊⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣷⠎⠀⠈⠉⠁⠀⠀⠀⠀⠀⢿⣿⣝⠪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠐⠀⠀⠀⠀⠰⣮⣽⣟⣻⣛⣛⣿⣯⣷
⠀⠀⣴⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⢻⣆⠀⢀⣤⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣾⣿⣶⡄⠀⠀⠀⠀⢰⣿⣾⣽⣻⠿⣟⣭⣶⣿
⢀⣀⣁⣀⣙⣁⣀⣤⣶⣾⡇⠀⠀⠀⠀⢸⡿⠀⠀⣫⣼⣿⣿⠀⠀⠀⠀⠀⣼⣟⣝⣛⣿⣟⣟⣅⣿⡄⠀⠀⠀⠘⣿⣿⣿⣿⠘⣿⣿⣿⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣀⣬⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠒⠒⠒⠒⢀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⢿⠇
⠀⠀⣴⢋⣿⣿⣿⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣷⣶⣶⣶⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⠀⠀⠀⠀⣰⣿⣿⣦⠀⣴⣿⣿⣿⢷
⠀⠀⢡⣼⣩⣩⣝⣧⡀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣯⢻⣿⣿⡿⠀⠀⠀⢠⣶⣦⣄⣴⣶⣼⣿⣿⣧⡀⠀⠀⠀⠇⣿⣿⠸⠘⢡⣿⣿⣿⣧
⠀⠀⠀⠘⣿⠈⣿⠀⠀⠀⠀⠀⠀⢈⣭⣭⣭⣭⣭⣭⡸⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⡟⣿⠇
⠀⠀⠀⠀⣿⠀⡟⠀⠀⠀⠀⠀⠀⠈⣯⣭⣭⣭⣭⡽⣹⣿⣿⠃⠀⠀⠀⠀⠻⣿⣿⣿⠟⣽⡿⣿⣿⠟⠀⠀⠀⠀⠟⢹⠀⠀⠀⢸⡇⣿
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡿⠋
⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣾⡟⣿⣿⣿⣿⣿⣿⡏⢿⣧⠀⠀⠀⣼⡟⢸⣿⣿⣿⣿⣿⠈⣿⡇⠀⠀⠀⣾⡟⣿⣿⣿⣿⢹⣿
⠀⠀⠀⠀⠛⠁⢿⣿⣿⢹⣿⣿⡇⠘⠋⠀⠀⠀⠙⠁⠘⣿⣿⢹⣿⣿⠀⠙⠃⠀⠀⠀⠙⠃⣿⣿⢹⣿⠘⠛
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⢲⡄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡖⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣙⣟⣋⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣿⣋⣸⡇⠀⠀⠀⠀⠀⠀⠀⣈⣻⣟⣡⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⣰⡿⣿⣿⣿⣿⣿⣿⡿⢿⡄⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⣿⢿⣧⠀⠀⠀⠀⣼⡏⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠰⡿⠁⣿⣿⣿⣿⣿⣿⣿⠘⢿⠀⠀⠀⢸⡟⢸⣿⣿⣿⣿⣿⡆⢿⡆⠀⠀⠀⣿⢹⣿⣿⣿⣿⡼⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⣾⣿⣿⣿⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠈⣿⣿⢻⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢻⣿⡏⠁⠀⠀⠀⠀⠀⠀⠉⣿⡟⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡿⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⠇⠀⡨⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢁⠀⠀⠀⢠⠏⠠⠞⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠈⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠰⠶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⢢⡒⢢⣀⠤
⠀⠀⠀⠀⠁⠀⠁⠠⠈⠉⠉⠋⠥⠯⡐⠲⢂⣲⠐⠊⠆
⠀⠀⠀⠀⠀⠀⠀⠢⢤⡁⢀⠉⢉⠉⠉⠀⠄⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠠
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⢄⡀⠄⠀⠂⠂⠤⢀⡀⠠⠀⢠⠄⠀⠀⠄⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠓⠀⠀⢀⠀⢀⣈⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡐⢠⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⠄⠄⡉⠀⠚⠋⡀⠀⠀⠀⠀⢱⠀⠀⠀⢀⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠙⠂⢶⠌⢢⠤⠠⠂⠀⠀⠀⢀⠜⠤⣀⠀⠈⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠑⢄⠀⠀⠀⠁⠁⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠙⢴⠁⠀⠀⠀⠀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠰⠀⢀⠁
⠀⠀⠀⠀⠀⠀⢀⠞⠀⠠⣪⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⢸⡀⠸⡀
⠀⠀⠀⠀⠀⢠⠋⠀⡐⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠑⢦⡈⠲⣄
⣀⠖⢢⢀⠊⡏⠀⢀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⢢⠈
⠑⢢⡊⢷⠸⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡞⡠
⠀⠀⠡⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡜⢰⠁
⠀⠀⠀⢂⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡠⢤⡙⢄⢣
⠀⠀⠀⠀⢊⠆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠠⣀⡷⢈⡆
⠀⠀⠀⠀⠸⠴⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⡠⠶⠯⠤⠎⡸
⠀⠀⠀⠀⠀⠀⠈⠓⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⡠⠚⠉⠐⠓⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠉⠓⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣖⣒⣫⣭⣭⣭⣷⣶⣶⣶⠶⣮⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⡞⣻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⣎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣿⣿⣿⣾⣧⣯⣾⣿⣿⠋⠀⠀⠀⠀⣹⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⠟⠛⠋⣸⠷⢊⡀⠀⠀⢀⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡞⣟⠛⠿⠃⠀⠀⠀⠀⠚⠣⠀⠀⠀⠙⠆⢳⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⣇⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣷⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡝⣷⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⣿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣿⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢷⣽⣳⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠉⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡈⠙⠓⠮⣭⣟⡲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⠀⠀⠈⠙⢦⡹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⠀⠀⠀⠀⢻⡜⡆
⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣤⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣷⠀⠀⠀⠈⡇⣿
⠀⠀⠀⠀⠀⣠⢞⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⡇⡿
⠀⠀⠀⠀⣰⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⠀⣸⢧⡇
⠀⠀⠀⠀⡏⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⣞
⠀⠀⠀⠀⢷⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢚⣏⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣦
⠀⠀⠀⠀⠘⢯⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡏⣿⡀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣧
⠀⠀⠀⠀⠀⠈⢳⡹⣆⠀⠀⠀⠀⠀⠀⠈⣇⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⡳⢦⣀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡶⣩⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡍⠼⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣝⠶⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⣽⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣛⡭⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠯⠭⣭⣭⣿⣿⣯⣭⡭⠽⠖⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⣉⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢀⣾⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣾⣿⣿⣿⣿⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣷⡀
⠀⠀⣸⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣧
⠀⢀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⡀
⠀⢸⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⡟⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸⣿⣿⣿⣿⡇
⠀⠈⠻⣿⣿⠿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡾⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡟⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠨⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⢽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡟⠛⣁⣀
⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⢠⣌⣵⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢀⠀⠀⠀⠀⠀⣤⣿⣖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡄
⠀⠀⠀⠀⠀⠻⢀⡤⠛⠦⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⣠⡀⠊⠀⠀⠀⠀⠀⠈⢿⣧⡾⠗⠤⠂⠀⠀⠀⠀⢀⡟⠀⠀⣀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⣾⣶⣿⠁⠀⠀⠿⠳⠶⠆⠀⠂⠀⠀⠀⠀⠈⠻⠎⠀⠀⠀⠀⢰⣶⣾⡟⠀⢘⡽⠟⠶⠶⠆⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡽⠀⠀⠈⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⠇⠈⠈⠁⠀⠀⠀⢀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠖⠐⠲⠶⠶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡷⠖⠶⠶⠶⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡀⠀⠀⣀⣠⣴⣶⠟⠛⠻⢗⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣤⣶⣾⠿⠻⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⠝⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠛⠻⣿⣿⡿⠛⠁⠀⠀⠀⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢀⣿⣿⡿⣦⡐⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⢿⣾⡷⡤⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⡿⣿⠃⠘⡎⠀⠉⢰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⢻⡇⠈⠁⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢉⣇⠀⠀⢹⡄⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⢻⡀⠀⠈⣷⡀⢄⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣁⠀⠀⠀⠸⣝⠆⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣋⣄⡇⠀⠀⢯⡻⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⠳⠆⠀⢰⣝⢶⣄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡛⠶⠀⠀⢮⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣡⣤⣤⠭⠏⠀⠀⡴⢭⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣤⣬⠿⠿⠁⠀⡠⢬⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠦⢤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⡼⠟⠁


⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⡿⠻⡿⠛⠿⢿⡿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢀⣿⣷⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⣷⠄⠀⠀⠀⣀⣿⣇
⠀⠀⣰⣿⣿⣿⣷⡀⣰⣿⣿⣿⣇⠀⠀⠀⣼⣿⣧⠀⠀⠀⣴⣿⣿⣿⣦⠀⠀⢠⣿⣿⣦⠀⠀⢠⣶⣾⣿⣦⡀⢀⣾⣿⣿⣿⣿⣄
⠀⠀⢿⣿⣿⣿⣿⠃⠸⣿⣿⣿⣿⠳⡄⢀⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⡇⢸⣟⣿⣿⣿⡿⣿
⠀⠀⢸⣿⣿⣿⡿⠀⠀⢻⣿⣿⠃⠀⠀⢸⣿⣿⣿⡷⠄⠀⢻⣿⣿⣿⣿⠀⢰⣿⣿⣿⡏⠀⠀⢿⣿⣿⣿⣿⡇⠸⢿⣿⣿⣿⣧⠿
⠀⠀⠈⣿⣿⣿⠁⠀⠀⣼⣿⣿⡄⠀⠀⠸⣿⡿⣿⡇⠀⠀⠀⣿⣿⣿⡟⠀⠀⠉⣿⣿⡉⠀⠀⠘⢿⣿⣿⡟⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⢻⣿⡧⠀⠀⢸⡟⠀⣿⡄⠀⠀⢸⡟⠀⢹⣇⠀⠀⠀⢻⡿⣿⠁⠀⠀⢀⡿⠙⢷⡀⠀⠀⠸⣿⢻⠇⠀⠀⠀⢿⡿⢹⠟
⠀⠀⠀⠀⣿⡇⠀⠀⢾⣦⡀⠘⣧⣤⡄⢸⣇⠀⠀⢻⡤⠀⠀⢸⠄⣇⠀⠀⢠⣼⡇⠀⢠⠿⠆⠀⠀⡿⠘⠆⠀⠀⠀⢀⣧⠿⡄
⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣼⣿⠃⠀⠀⠀⢠⣿⣦⠀⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⠀⣰⣿⡆⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣻⣿⣄⠀⠀⠀⢰⣶⡄
⠀⢰⣿⣿⣿⣆⠀⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⣻⣿⣄⠀⠀⠀⣾⣿⣿⣆⠀⠀⠀⣠⣿⣿⣄⠀⢀⣿⣿⣿⣿⡇⠀⠀⣼⣿⡅
⠀⢨⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⠀⠀⠠⣿⣿⣿⣿⡄⠀⢸⣿⣿⣿⣿⡄⠀⢹⣿⣿⣿⡇⠀⠀⣿⣿⣿⡄
⠀⠘⣿⣿⣿⡟⠀⠠⠞⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⣻⣿⣿⣿⠃⠀⣼⣿⣿⣿⡏⢳⠀⠘⣿⣿⣿⠇⠀⠰⣿⣿⣿⡟⠂
⠀⠀⠸⣿⣿⡇⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠈⣿⣿⣿⠀⠀⠋⢿⣿⣿⠇⠀⠀⠀⢻⣿⡟⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⣿⣿⡇⠀⠀⠀⣿⠏⢿⡆⠀⠀⠀⣿⡟⢻⣷⠀⠀⢀⣼⢿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠈⣿⣷⠀⠀⢰⡿⠁⣿⡏
⠀⠀⠸⣿⣿⣧⡀⠀⠴⠿⠀⣸⣇⠀⠠⢼⡿⠀⢀⣿⡇⠀⠚⠷⠀⢈⣷⣄⠀⠀⢼⢻⣏⠀⠀⠀⠀⢰⣇⣹⡆⠀⣾⣇⠀⢺⣧⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⣴⣿⠟⢠⡄⠻⣿⣦⣀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠻⠿⠻⣿⣅⡀⢸⡇⢀⣨⣿⠟⠿⠟⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⠺⣷⣄⣤⡜⢿⣿⢸⡇⣿⡿⢣⣤⣠⣾⠗⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⣬⣿⣿⣇⣼⣿⣾⣷⣿⣧⣸⣿⣿⣥⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣶⣌⣉⣽⣿⣿⡀⢸⡇⢀⣿⣿⣯⣉⣡⣶⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⣉⡀⠙⣿⣾⣷⣿⠋⣀⣉⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠿⠿⠿⠿⠿⠿⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡾⠿⠛⠛⠛⠛⠉⠉⠛⠛⠛⠻⠿⢶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⠀⠀⠀⢀⣴⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⡤⡤⠄
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠁⢀⣄⡾⠂⠀⠀⠀⠀⠀⢠⡀⠙⢦⡀
⠀⠀⠀⢀⣴⠟⠡⠒⠀⠀⠀⠀⠀⠘⠫⠁⠀⠀⠀⠀⢀⠀⠀⠡⠀⠈⠙⣆
⠀⠀⣠⡟⡡⠀⠀⠀⠀⠀⠀⠀⣴⠇⡀⠀⠀⠀⠀⠰⠏⠷⡀⠀⢢⡄⠀⠙⣦
⠀⢠⡿⠠⠁⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠑⡆⠀⢸⡄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⡭⠼⢧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⠀⠇
⠀⣿⠀⠀⠀⢀⣸⡖⠉⠀⠀⠀⠀⢌⢢⡀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⣿⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⢧⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⡀
⠀⣿⠀⠀⠀⢸⠇⠦⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⠀⡀
⠀⣿⠀⠀⠀⢸⢀⡄⠀⠄⠠⠂⠈⠀⠀⠈⠂⢀⡤⠒⠗⠈⢁⠀⡤⠀⠨⢉⣉⢛⣆⣼⡛⠂⠀⠑⠆⠀⠀⡞⠋⠃
⠀⢿⡄⠀⠀⠙⠀⡠⠀⠀⠤⣆⠀⡀⠀⠀⢸⡿⠉⠐⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠸⣙⡀⠀⠀⠀⢶⣀⣚⠀⠠⡄
⠀⢸⠏⠀⠠⣸⡄⠃⠀⠀⠀⡈⠠⢠⠄⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠈⠁⠈⠁⠀⠩⠉⠀⢲⠁
⠀⠀⠀⠀⠀⣿⣿⣷⢀⠂⠀⡓⡋⠀⢚⠛⣈⠡⡏⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⢡⡀⠉⠀⠀⠀⠀⠀⠀⡎
⠀⢸⡄⠀⠀⠀⢉⠇⠘⠂⠀⠉⢉⡉⠥⠀⠈⠁⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⢀⣀⣴⠀⠀⠀⠧
⠀⠀⠻⣄⢠⠀⠀⢀⣀⣀⣀⣘⣁⣤⣤⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠞⢋⡉⠉⠛⣻⡆⠀⠀⡀
⠀⠀⠀⠈⢻⡄⠀⠾⠋⠉⠀⠀⠀⠈⠀⠀⠀⢻⡃⢠⣀⢀⣀⣤⣴⠚⢿⣥⣼⠟⣡⡾⠛⠈⠉⠁⠀⢹⡄⠅⠃⠀⠰
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢶⠀⡅⠀⢸⣷⡾⠛⠉⠉⠉⠉⠉⠉⠛⢿⡞⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇
⠀⠀⠀⠀⠀⢰⣿⣧⡄⠀⠀⠀⠀⠉⠉⠉⠀⣸⠏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇
⠀⠀⠀⠀⠀⠀⣷⡈⢙⡷⠒⡏⠀⠀⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⢿⣷⣀⠁⠀⠇⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⣸⠇⢠⣦⡄
⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣿⣿⡄⠀⠀⠠⠀⠴⠻⠶⢀⠇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣸⡄⠀⢀⣀⠀⣼⣿⣿⣄⠀⠀⠀⠀⢀⣀⡤⢤⣼⣥⡾⣇⣀⠀⠀⠀⠀⠀⠐⠊⢧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠧⠈⠉⣀⣠⠞⠻⢧⡤⣧⣠⠶⠋⠉⠀⠀⠀⠀⠀⠙⡾⠛⢀⠀⠀⠀⠀⠀⠀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⠢⠤⠤⢤⣤⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠛⠢⠤⠖⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠦⣤⣤⣤⣤⣬⣦⣀⠀⠀⠀⣀⣀⡀⠔⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣥⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⠖⠛⠳⠾⠳⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠯⠄⠀⠀⠰⠦⡄⠀⠀⠀⠀⢈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣼⠦⢶⡀⠀⢀⡖⠆⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠻⣄⠀⣀⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⣚⣉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠋⠙⠛⠛⠛⠒⠒⠛⠒⠀⠀⠀⠀⠀⠀⠉⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠻⡄⠀⠀⠈⠙⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⠙⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣠⠆⢠⡍⠓⠐⣦⡄⠀⣀⣀⣀⣀⣀⣀⣀⠠⠴⠚⠋⠉⠀⢀⠀⠀⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣆⠀⣰⠃⠀⢸⣧⣴⣶⣷⡀⠀⢀⣉⣭⠽⠛⠉⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⣉⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡏⠉⣰⠇⠀⠀⣀⣴⣿⡏⠁⠙⣾⠿⠋⠁⣤⣄⣀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⣴⣋⣥⠄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢸⠀⢠⡏⠀⠀⣾⣿⣿⡿⢁⡴⠋⠁⠀⠀⠀⣸⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠀⠁⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⠀⡾⠀⠀⢸⣿⣿⣟⡴⠋⠀⣀⣤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⢸⠇⠀⠀⣼⣿⢿⡏⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣾⠀⠀⢀⣿⠏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣴⡆⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⡏⠀⠀⢸⡏⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠘⠁⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠞⠠⠤⠤⠤⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠋⠀⠀⠀⠀⠀⠀⣇⢀⣀⡤⠖⠂⠀⠀⠉⠉⠛⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⡀⠙⠦⣄⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠿⢥⣤⠈⣵⣀⠀⠛⢯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠒⠒⠚⠉⠛⠓⠶⠶⠶⠶⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣰⣿⡟⣽⣿⣿⣿⣿⣿⣿⡏⢿⣿⡄⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⣿⣿⢿⣿⡄
⠀⠀⠀⠀⠀⢰⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⠀⠀⠀⠀⢀⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣷⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠸⠿⠃⢸⣿⣿⣿⡟⣿⣿⣿⣿⠇⠸⠿⠃⠀⠀⠀⠘⠿⠇⠀⣿⣿⣿⡟⣿⣿⣿⡇⠀⠻⠿⠀⠀⠀⠀⠀⢿⡿⢸⣿⣿⢻⣿⣿⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠸⣿⡿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⢈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣛⣛⣁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣛⣛⣁⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣛⣛⣁⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⠒
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⣸⣿⣻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢠⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣷⠀⠀⠀⠀⠀⢰⣿⡟⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⡄⠀⠀⠀⠀⢀⣿⡏⣾⣿⣿⣿⣿⣧⢻⣿
⠀⠀⠀⠀⠀⣿⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⡇⠀⠀⠀⠀⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⡧⠀⠀⠀⠀⢸⣿⢡⣿⣿⣿⣿⣿⣿⡘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⠁⠀⠀⠀⠀⠈⠋⣸⣿⣿⣿⣿⣿⣿⣇⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠀⠚⠀⠂⢣
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠘⡸
⠀⠀⠀⠀⠀⠀⢀⣀⡆⠀⠀⠀⢨⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡅⠀⠀⠀⢸⣀
⠀⠀⠀⠀⡤⢺⠁⠱⡌⠀⠀⠀⡱⠃⢸⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢺⡁⠹⡈⠀⠀⠀⣡⠃⢹⠒⣄
⠀⠀⢀⡞⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢈⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⠃⠀⠈⠀⠀⠀⠁⠀⢸⠀⠈⢆
⠀⠀⡾⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⣆⠀⠀⠀⠀⠀⠀⡸⠁⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠘⡆
⠀⢸⠃⠀⣸⠀⠀⠐⠚⠉⠉⠙⠒⠀⠀⠸⡁⠀⢹⡀⠀⠀⠀⠀⢠⠇⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⢱
⠀⡜⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠘⣆⠀⠀⠀⠀⡘⠀⠀⡽⠀⠀⠐⠒⠉⠉⠉⠑⠂⠀⠐⢇⠀⠘⠄
⢸⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠃⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢰
⠸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠆⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠸
⡜⢀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠈⡆⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠣⠸⡎⠀⠉⠒⠤⣤⣀⣀⣀⣀⣠⡤⠔⠊⠁⠀⣾⠠⠃⠀⠀⠐⠸⡆⠀⠉⠲⠤⣄⣀⣀⣀⣀⣀⡤⠴⠚⠁⠀⣾
⠀⠀⡇⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⡏⠀⠀⠀⠀⠀⠀⣇⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⡏
⠀⠀⢹⠀⠀⠀⠉⠓⠲⠶⡴⠖⠚⠉⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠓⠲⠶⡴⠖⠚⠉⠀⠀⠀⢀⠇
⠀⠀⠈⠳⢤⣤⣀⣀⣀⣀⣇⣀⣀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣤⣄⣀⣀⣀⣧⣀⣀⣀⣠⣤⠴⠚
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣧⠀⠀⠀⠀⢫
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⢿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⢿⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠰⡀⠀⠀⡞⠈⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡜⠀⢇⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⢰⠁⠀⢸⠀⠀⠆
⠀⠀⠀⠀⠀⡐⠀⠀⠘⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠈⡆⠀⠈⠂
⣀⣀⣀⣀⣀⣑⣠⣤⣆⣀⣀⣀⣦⣤⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣠⣤⣌⣀⣀⣀⣡⣤⣐⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣭⣿⣯⣽⣽⣯⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠒⠋⠁⠀⠈⡇⠀⠀⠀⠐⠂
⠀⠀⣠⠤⠖⠚⠉⠉⠀⠀⠀⠀⣠⢧⡀⢻⣤⠖⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠛⠶⠤⣤⣤⣤⣤⠴⠶⠛⠁⠀⠳⣼⠃⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠙⢦
⠀⠀⠀⠀⣠⠛⢦⣄⠀⠀⠀⠀⠠⢤⡀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⡴⣾⠋⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢠⡏⠀⠀⠈⠙⠲⢤⣀⠀⣀⡤⠽⣤⣿⠆⠀⠀⣰⡇⠀⠀⣀⣠⠴⠋⠁⣼⠃⣴⠛⡆⢀⡔⢦⢹⡆
⠀⠀⢀⡟⠀⠀⠀⢨⡟⠐⠲⠬⢽⠁⠀⠀⣠⣻⣄⣀⡴⠟⠒⠒⠋⠉⠀⠀⠀⢰⠇⢸⠛⡇⣿⣸⢻⢸⣼⡇
⠀⢀⡾⠀⠀⠀⢀⡾⠀⠀⠀⠀⢸⣆⠀⠀⣉⣹⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⢸⠇⣿⡇⣸⢸⢿⡇
⠀⡼⠁⠀⠀⢠⡞⠁⠀⠀⠀⠀⢸⠉⠻⠊⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡼⢀⡇⣧⡏⣸⢸⡇
⡼⠁⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⡿⠀⣠⡶⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠹⢥⠞⠀⠙⠒⠃⣸⠃
⠙⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⠋⠀⢀⡟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠠⠻⡄⢠⠏⠀⠀⡼⠓⢦⣀⡀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠙⣾⠀⠀⣸⠃⠀⢀⡟⠉⠓⣶⠤⣤⣀⣀⣀⣹⣦⣄⣤⡴⠛⠀⠀⠀⠀⣠⠶⠶⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠠⡀⠀⠀⠀⠈⠳⣴⠇⠀⠀⣸⠁⠀⢀⡏⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣤⣴⣱⠀⢠⡇⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠑⠢⣀⠀⠀⠈⠓⠦⣤⣇⣀⣀⣼⣥⣤⡀⠀⠀⠀⢸⣇⣀⡀⠀⠀⣀⣀⣧⠀⠹⢤⠞⢧⣠⢿⡆⢠⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⣫⡉⠉⠉⠉⠉⠁⢹⠠⣶⣤⢦⠀⣠⠎⣰⡓⡽⢻
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣼⡙⠉⠉⠉⠛⠛⠓⣿⠀⠈⠁⠀⣰⠋⣴⢥⡞
⠀⠀⠀⠀⠀⣀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠈⠀⠠⢀⠀⠀⡖⢲⢠⠟⠀⠀⣴⣿⣑⣶⢶⠧⠀⣀⣀⣰⣁⡞⣢⠟
⠀⠀⠀⣠⢫⣭⢇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣀⡈⠑⠢⢄⣀⠀⢰⠋⠉⢁⠘⢿⡀⣀⣀⣧⠀⠈⠉⠙⠛⠳⠶⠶⠭⠽⠕⠁
⠀⠀⠀⠹⢬⣟⣛⣚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄⣈⠉⠛⠢⠾⣩⡟⠶⠋⢧⠀⠿⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⡿⠉⠉⢧⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⡦⣤⣀⡈⠉⠒⠒⠷⠀⠀⠀⠈⠉⠉⣩⠇⠀⠀⠀⢧⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣇⡀⢰⣤⣀⣀⠀⠀⠀⣀⣀⣠⡤⠖⠋⠑⡆⢀⣉⡟⠒⠦⠤⡄⠀⠀⠀⠒⠛⣓⡦⠀⠀⠀⢰⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠂⠀⢿⡷⢾⣄⠙⡏⡳⣮⢉⣩⡿⠭⣭⡿⢤⡀⠀⠀⡸⠥⡾⠀⠀⠀⠀⠀⠉⠓⠒⠒⠛⠛⠓⠒⠢⠤⠤⠤⠶⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡼⠆⠀⢿⣦⠼⢿⡁⠘⣯⠳⣦⡀⣱⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠈⣧⡾⠲⢬⣙⠗⠻⣮⣛⣧⣀⠛⠿⢭⣭⣓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠈⠉⠒⠒⠒⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠳⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢀⣴⠞⠋⠉⠁⠰⢆⡴⠞⠓⠉⠉⠉⠉⠹⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠈⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⣧⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣄⠀⠀⢀⡤⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⡀⠈⢣⡞⠁⣠⣴⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠙⠉⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢾⣡⣾⠟⣛⠛⢮⡇⠀⠀⠀⠀⣰⣞⣛⡳⣄⠀⠀⠐⠦⣿⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠛⠛⠛⠳⣾⣯⢁⣘⣃⠀⠀⠀⢰⡟⢩⡮⠙⣿⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣯⣽⠂⠀⣠⡧⠾⢿⣂⣋⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣤⣶⠶⠶⠶⣾⣿⣿⣯⣅⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣶⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠛⠶⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠉⠉⠉⠙⠛⠻⣷⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⠀⠀⣶⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣶⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠖⠲⠦⣄⣀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⣧
⠀⠀⠀⡟⠁⠹⣤⠀⠀⢠⠷⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠃⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢠⠖⠚⣧⠀⠀⠛⠀⠁⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⡏⠀⠀⠀⠀⠠⡄⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⣿⣶⡶⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⢧⠀⠀⠀⠀⠀⢹⡄⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠲⣶⣤⣤⣴⡿⠁⠀⠀⠙⠻⣷⣦⣤⣠⣤⣤⣶⠋
⢸⡄⠀⠀⡆⠀⠀⣷⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⡏⠀⠀⠼⠉⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢰⡏
⠀⢷⠀⠀⢿⠀⠀⣿⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠘⣧⠀⠘⢧⣼⠏⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⠶⣦⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⣀⣀⡤⠔⠚⣋⣉⣩⣍⡉⠙⢦
⠀⠀⠘⢧⡀⠀⠀⠀⣠⠿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠃⠀⠘⣇⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⣠⠞⠙⠦⣭⠭⠵⠶⠖⠛⠋⠉⠁⠀⠈⢻⡄
⠀⠀⠀⠀⠙⠓⠖⠛⠁⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⢿⣇⠘⡄⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠒⠛⠛⢹⡄⠀⠀⠀⣿⠀⠀⢠⡦⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠐⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠤⠴⠒⠚⠉⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠚⠁⠀⠀⢀⣀⣤⡴⠖⠚⠛⠋⠉⢉⣉⣁⣀⣀⣀⡀⠤⠤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⣁⠠⠄⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⡌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⠀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡿⢀⠚⣽⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣏⣀⡀⠀⠀⠀⣠⣷⢄⠀⠀⠀⠀⠀⢀⣴⠉⠃⠀⠀⠈⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⡛⢿⣿⣯⣉⣩⣼⣿⠞⠀⠀⠀⠀⢀⣿⣿⠂⠀⠀⣀⣼⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣤⣴⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⣠⣴⣶⣟⣛⣒⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣀⠴⠶
⠀⠀⠀⣿⣿⠽⠛⠉⠉⠈⣿⣦⡀⠀⠀⠀⠀⠐⠿⢭⣉⣡⠬⠿⢟⠄⠀⠀⠀⠀⢀⡿⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⣾⣿⣽⣷⣤⠤⠤⠞⢻⣿⡟⠀⠀⢠⣴⡀⠀⠀⠀⠀⠀⠀⠈⢸⢆⡀⠀⡄⣽⠓⠤⠄⠀⣤⣴⣾⣷⣽
⠀⠀⠻⢿⣧⣤⣤⡤⠤⠶⠚⠋⠀⠀⠀⢻⣿⣿⣶⣶⠤⣤⣤⣰⣾⣿⡿⠃⠀⠣⣉⡓⠲⠶⠶⢟⣻⡿⠟⠋
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣦⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠗⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠻⠀⠈⠻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⣀⡀⢀⠀⠀⠀⣤⣤⣤⣤⣿⠿⠃⠀⣼⡇⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠿⠿⠟⠁⠀⢀⣤⠟⠁⠀⠀⠀⢿⡇⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡞⠋⣤⠟⠃⠀⣴⣦⡀⠻⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣠⡾⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠘⣿
⣀⡶⠇⠀⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠁⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣄
⣿⠀⠀⠀⠀⠸⠷⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠋⠀⠀⣠⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⢀⣀⠈⠉⠉⠲⣶⠆⢀⣀⠀⠀⠀⠀⠀⠈⣹⣶⣀⣀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡿
⠋⢠⣤⣤⣤⡾⠛⠙⠓⣤⣤⠀⠀⠙⠛⠃⢤⣤⡄⣤⡾⠋⠁⡙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⢀⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⣤⣄⠀⠀⠀⠀⠀⢀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠙⡟⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢢⡤⠚⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠰⠶⠆⠉⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⠴⠶⠞⠉⠉⣀⣀⡠⠶⠶⠞⠁⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠉⠙⠳⠶⠎⠉⢁⣀⠠⣴⡶⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠃⠀⢹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⣁⡀⠀⠻⠿⠿⠏⣩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠷⠚⠉⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠶⠶⠦⣄⣀⣙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⢁⣀⠴⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠃⢶⠶⠛⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠿⠿⠿⢷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠙⠂⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡤⠖⠒⠂⠀⠀⠐⠒⠦⣄⠀⠀⠀⠈⣿⣷⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢀⡤⠄⠄⠀⠀⣠⠔⠒⢤⠈⠁⠀⠀⠐⠉⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢺⣀⣿⡓⠀⠀⠋⢰⣟⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⣸⣀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠘⣧⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠰⠋⣉⠻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠙⠳⢶⣤⣤⣴⠾⠃⠀⢠⣤⡀⠀⠀⠀⠀⠁⠀⡅⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠛⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠠⣄⡀⠀⠀⢀⣀⡴⠃⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⡏⠙⠛⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡠⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡰⠊⠀⠀⠈⠻⣷⣄⡀
⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⡘⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⢠⠞⣀⠀⠠⠿⠏⢻⡄⢤⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠸⡄⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠞⠛⠁⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⡰⠂⠙⠂⠠⢤⣄⠀⠀⠀⢠⠀⠙⠞⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⢻⡄⠀⣀⣀⣠⡿⠀⠀⠀⢸⡇⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⢯⠀⠀⠉⠉⠉⣹⡆⠀⠀⢸⣧⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⢸⡇⠀⠀⠚⠛⢻⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⢻⠃⠀⠠⠄⢾⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠙⠲⠶⠶⠿⠶⠖⢛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⣀⡴⠛⣷⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡦⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠠⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⣴⣶⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠶⠶⣛⠵⣿⠀⢠⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⣉⡑⠂⠈⠃⠀⢶⡟⢀⣾⣇⣸⡧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⠿⣿⡁⠀⠁⠀⢀⠔⠀⢡⠀⠘⠛⠛⢁⣉⠅⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣰⠀⠀⠀⠐⠑⠂⢐⡠⠤⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⢧⣧⣈⣈⣠⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⡠⠀⠆⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣬⣿⣄⡀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠋⠉⠉⠙⠛⠿⣷⣤⣤⣀⣀⣠⣬⣷⣤⣄⡀⠀⠀⠀⡇⢰⠂⠁⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡿⠟⠉⠉⠉⠉⠉⠻⣶⣄⣀⣳⣎⣀⣁⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡉⠉⠉⢉⣉⣠⣞⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠶⠶⠶⠦⠴⠶⠶⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠛⣋⣭⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣶⣤⣤⣤⣤⣤⣤⣶⠶⠶⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣤⣭⣥⣬⣤⣥⣬⣧⣤⣬⣭⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⣷⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠂⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⡠⠀⢻⠲⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠉⠢⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⢁⠂⢰⠀⠈⠻⢆
⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠆⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⢀⡄⠀⠰⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠊⠁⠀⠀⠀⠖⠲⡄⠠⢤⡄
⠀⠀⠀⠘⠇⢀⠃⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠧⢨⡗
⠀⠀⠀⣀⠘⠁⠀⢠⠃⠀⢠⠆⠀⢡⠀⠀⢈⠀⠠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⢛⠂⠀⠀⠀⠀⡰⠋⠀⡴⠀⠡⡀⠈⠑⣤⠥⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠈⠑⠠⠤⠤⠄⠐⠂⠁⠀⠀⠀⠈⠢⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠒⣀⠠⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⡄⠀⠑⠢⠄⣀⡀⠀⠀⠀⠀⠀⠓⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢀⡠⠤⠌⡉⠈⠈⠉⠀⠈⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⡠⠄⢼⠀⠀⠀⢀⠚⠁⠀⠀⠀⠈⠠⡀⠀⠈⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⢤⡄⢱⠀⠸⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠖⠋⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠓⠛⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢄⠡⡀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠈⠢⡀⠀⠀⠀⣀⠀⠈⠁⠒⠠⠤⠤⠄⠠⠄⠀⠐⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠢⣀⠜⠁⠒⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠉⠉⠉⢀⡼⠃⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠙⡆
⠀⠀⠀⢀⡤⠤⠤⠄⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⡐⣿⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⡄⠀⠠⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠰⣀⡿⢆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠈⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠈⠢⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⠒⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣠⣄⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⠴⠟⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡛⢀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠏⠀⠀⠋⠂⠐⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠲⠆⠀⠀⢲⠖⠀⢸⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁⠀⠀⣀⠀⠀⠀⠀⠸⣏⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣴⡞⢉⣟⣷⡠⠤⠄⠀⠄⢀⡠⠞⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⠀⠀⠱⠌⠛⠳⠒⠀⠀⡉⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣿⠀⢲⣾⠏⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠒⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠇⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⣏⠜⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⠀⠀⠀⢇
⠀⠀⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⢰
⠀⠀⠀⠀⡝⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⠀⠀⢈
⠀⠀⠀⠀⣷⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⣨⠋⠙⠋⠉
⠀⠀⠀⠀⢿⣠⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠯⠁⠀⠀⠤⢄⡀⢉
⠀⠀⠀⠀⠸⣿⡀⠘⠳⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠈⠀⠂⠈⡑⠂⡂⠐⠀⠐⢠⡄⠄⠐⢲⡌⠀⠀⡀⠂
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠠⠊
⠀⠀⠀⠀⠀⠈⣿⡀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣼⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠃⠀⠀⠀⣠⣴⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⣴⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣯⠁⠀⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⣰⡿⠇⠀⠀⠀⢀⠀⣽⣿⡟
⠀⠀⣀⣤⣼⣿⡷⠤⠄⣤⣤⣾⣿⣿⣿⡄
⢠⣾⣿⡿⠛⠋⠀⠲⣶⣦⣿⣿⡿⠛⠛⣱
⢻⡟⠋⠀⠀⢀⡀⣄⣰⣿⣿⡿⠁⠈
⠀⠉⠚⠛⠛⠈⠡⣿⣿⡯⠁⠀⠀⠀⡀
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣒⡒⠐⠒⠺⠿⣒⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢽⣟⣒⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣶⠤⣀⡀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠙⡦⠀⠠⠶⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠏⡠⠊⠀⠀⠐⢦⠺⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣉⣀⣉⣧⣀⠤⠅⢀⣀⡠⠚⢁⣴⣧⠞⠑⠟⢦⠜⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣬⡟⠁⠀⠀⠀⡄⢀⣠⠟⢀⣾⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠁⣠⠆⠳⡖⠲⠖⠒⠋⠉⠀⠀⠼⢊⡥⡦⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⢀⣠⠴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢡⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠁⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⠀⠀⠈⠃⠀⠀⠀⠀⢸⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣀⣀⡤⢤⡀⠀⠀⠀⠀⠇⣾⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡚⠂⠘⢦⠂⡠⠃⠀⠀⠀⠀⣿⠋⠀⠀⠀⢀⡤⢞⠟⠁⡀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡙⢲⡷⢾⣎⠓⣄⠀⠀⠀⢰⠉⠑⠒⠒⠊⠁⢠⠎⠀⠀⡇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣑⠉⢀⡵⠦⠓⠒⠚⠛⠑⠢⠀⠀⠀⠀⠈⠒⠒⢺⠃⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣤⡇⠰⡖⠈⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⣀⣀⣠⣤⠴⣖⠛⠉⣛⠁⢘⣻⡀⠀⠙⣾⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⠘⢿⣧⣼⡿⠇⠸⢿⡇⢀⣀⡦⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠿⣿⣼⣿⣂⣸⣿⣦⣼⣷⠴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡛⡄⠀⠀⠹⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⢦⡠⠤⠒⠒⠐⠃⠈⠦⡀⠀⢣⠘⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡖⠈⠀⣀⠔⠒⠉⠉⠉⠒⠒⠃⠐⠯⢤⣄⣀⣈⡑⠦⡤⠴⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣄⠀⠐⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣤⣼⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠉⠉⠉⠁⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣋⠀⠀⠀⢀⣀⣠⣤⣄⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⡿⡟⠛⣷⣀⣰⣿⠟⣍⠉⢹⣧⣤⣬⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⠀⣆⣹⣿⡟⢻⣇⣃⡜⠀⣸⡏⠀⠀⢿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡏⠛⠛⠛⠛⣿⣃⣬⠟⠻⠿⠟⠋⠁⠀⠀⣺⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠙⠋⣁⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠲⣦⣀⠉⠁⠀⢀⣶⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠟⢻⣧⠀⠀⠀⠀⠈⠛⠿⠶⠿⠟⠁⠀⠀⠀⣼⠟⠀⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠸⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⢀⣀⣤⣶⠟⠋⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣰⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣰⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⣿⡟⠿⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡏⠉⠋⠉⣿⠁
⠀⠀⠘⣿⡄⠈⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣴⡶⣿⣿⣿⣿⣿⡄⠀⣸⡏
⠀⠀⠀⠈⠛⠛⢿⣿⣿⣿⣿⠉⠛⠻⠿⠷⠶⠶⠶⠶⢾⣾⠿⣿⡿⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁⠀⠀⠸⣿⣿⣿⣿⣷⡾⠏
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠸⣧⠀⠀⠀⢀⣀⣠⣴⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠘⢻⡶⢿⣷⠿⠶⠿⠛⠛⠋⠉⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠘⠃⣸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⣤⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠁⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⢰⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⣀⣸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⣨⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠙⣉⣻⣏⢈⡝⢛⣯⠛⢹⡽⢋⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⠿⠟⠉⠋⠀⠙⠉⠹⠿⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⠀⠒⠛⠂⠀⠀⠀⠀⠀⠐⠛⠂⠀⠙⠿⣿⣿⣿⣿⡟
⠀⠀⠀⢾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠂⠀⠰⣿⡇⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠸⣿⣿⡿⠋
⠀⠀⠀⠈⣿⣿⠛⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡁⠀⠀⠀⠀⠀⣀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠈⠙⢧⡀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠘⣷⠶⣶⣶⣶⣿⡿⠟⠛⠻⠿⣿⣷⣶⣶⠶⣳⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠉⠛⠛⠶⠶⠶⢼⠷⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠙⠳⡶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡶⠶⢦⣤⣀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠉⠳⢦⣈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢹⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠈⠿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢀⣴⣶⣦⡀⠀⣴⣦⡀⠀⠀⢀⣠⣤⡀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣿⣿⠁⢰⣿⣿⣇⣴⣾⣿⡿⣿⣿⡆⢰⣿⠃
⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣿⡅⢀⣿⣿⠀⣼⣿⣿⣿⠟⠋⠁⠀⣿⣿⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣄⠀⠘⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠘⣷⣸⣿⡇⠀⣿⣿⡇⢈⣴⣶⣶⣾⣿⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣽⣿⣿⣿⣷⣶⡆⠀⠀⣿⣿⠀⠀⢹⣿⣿⠃⠀⣿⣿⣧⣿⣿⠛⢉⣿⣿⣿⣿⣷⣶⣶⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⠿⠟⠟⠛⠁⠀⢀⣿⣿⣤⣤⣾⣿⣿⣶⡶⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠜⠻⣿⡿⠛⠋⠁⢻⡇⠀⢸⣿⣿⣿⣅⣺⣻⣛⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠛⠛⠈⠉⠉⠙⠟⢹⣿⣁⠀⢀⣀⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠤⠀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠄⣴⢠⠀⠀⣘⣀⣀⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠁⠀⠀⢱⠈⠁⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⢀⡤⢄⡀⠀⢻⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠘⣇⡀⠤⣲⠶⠿⠿⢏⣉⣻⣞⢦⡤⣴⠲⣄⢰⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢄⣪⠔⢊⣧⡘⣷⠶⠖⣚⣛⠟⣋⣥⣀⠚⣼⡷⠄⣛⣛⠯⣳⣖⢤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⢟⣦⣝⡛⠿⣫⣿⢾⣟⣛⣫⡶⣋⠝⣫⣿⢿⣿⠯⣍⣒⣖⢚⣭⣝⣒⢭⡓⠻⣿⣿⣷⠂
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣭⣒⡲⢶⣶⡒⠮⣽⣿⣿⠾⣓⠉⢃⣰⣦⣤⣿⣷⣿⣿⡭⣾⣿⠭⣼⣤⣽⡯⣓⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣲⠦⣭⠯⠭⠭⠾⠧⠤⠴⠶⠶⠶⠮⠛⠛⠋⠀⠉⣩⠾⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠘⠖⠴⠇⢐⠖⠴⠃⠘⠲⠶⠆⠀⠒⠲⠖⠶⠢⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⠤⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⠤⠄⠀⠔⠋⠁⠀⠀⠀⠀⣠⠇⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⠳⢤⡀⠈⠒⠤⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣾⣙⠦⣄⡀⠈⣉⠗⠲⠤⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢘⢣⣄⠀⠈⣛⠋⠑⠀⠀⠀⡀⠉⠓⠦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠬⠏⠀⢸⣦⠀⠀⠀⠀⠀⢇⢀⠴⠒⢦⠑⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠾⡅⢀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢀⣀⣀⡤⠜⡉⠀⠀⠀⠀⠀⡀⣀⡥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣘⣦⡀⠀⠀⠀⣗⢛⡹⠁⠀⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠒⠉⡱⠊⠉⠀⣤⡉⠳⣄⡀⠈⢉⣀⣀⡤⠴⠚⠛⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠘⠁⠀⠀⠀⠤⣬⠂⠰⣌⡭⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠏⠁⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⡤⠒⠈⠉⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠤⠤⠚⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣡⠔⡀⠀⡀⣀⡄⠀⠤⠤⠴⢶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠛⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢖⠒⠦⠤⠤⢤⢤⡤⠤⠤⠴⠖⠚⠋⠉⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⣘⡇⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠚⠁⠈⠉⠉⠁⢙⠶⠤⠤⠴⠶⠚⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠶⠶⠰⡇⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠚⠛⠁
⡶⠶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⢶⡶⢶⣶

⠀⠀⠀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡄
⠀⠀⠀⢘⣟⡁⠀⠀⠀⠀⠀⠀⠀⠀⣙⣟⡁⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣥⡀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣥⣄⡀
⠀⢰⣶⣿⣿⣿⣷⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣻⣆⠀⠀⠀⣾⣟⣿⣿⣿⣿⣛⣷
⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⣿⣾⣿⣿⣿⣿⣷⣿
⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠉⠁⠁⠀⠀⠀⠈⠈⠉⠉⠉⠉⠁⠁⠀⠀⠀⠁⠈⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄
⣾⣿⡿⠋⢉⠙⣿⣿⡇⠀⠀⠀⠀⢰⣧⣿⣷⠀⠀⠀⠀⠀⣀⣴⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡍
⣿⡿⠛⢲⣶⠞⠻⣿⡇⠀⠀⠀⢀⣤⣿⣿⣧⣄⠀⠀⠀⠀⢿⣷⣿⣿⠁⠠⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⢸⣧⠀⣾⣿⡄⢀⣾⡁⠀⠀⣰⡿⣏⣹⣿⣧⡟⣷⡀⠀⠀⣼⣿⣿⣿⡆⣀⡀⠂⠀⠀⠀⠀⠀⢹⣿
⠸⣿⣤⣼⣿⣥⣼⡿⠁⠀⠀⡿⣿⣿⣿⣿⣿⣷⡿⡇⠀⠀⠸⣩⠛⠛⠛⠻⣩⠂⠀⠀⠀⠀⢀⣬⣯⣤⣤⠄

⠀⠀⢀⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⡺⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣦
⠀⠀⣿⣻⣧⡄⣀⣄⠀⠀⠀⠀⣾⣿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣅⡀⠀⠀⠀⠀⣻⣿⢿⣿⣿⡿⣿⣟
⠀⠐⠋⠹⠀⠀⣀⣤⡀⠀⠀⠀⢸⡟⠈⠷⣠⣼⡇⠀⠀⠀⢠⣿⢿⣿⣿⣿⢻⣆⠀⠀⠀⢿⣿⣿⣿⢽⣿⣿⡿
⠘⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠈⠁⠀⠙⠛⠛⠁⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠋⠀⠀⠀⠘⠛⠛⠃⠈⠛⠛⠃
⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀
⠀⠀⣠⣶⣿⡶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣉⣁⣁⣈⡀⠀⠀⠀⠀⠀⠈⠛⣃⡀⠀⠀⠀⠀⣀⣏⡀⠀⣀⣽⣧⡀
⠀⠐⢡⢿⠿⠷⠙⠀⠀⠀⠀⢀⣤⣤⣬⣿⣿⣿⣿⠁⠀⠀⣀⣀⠀⣀⣀⣿⣿⡀⠀⠀⢠⢿⣿⢳⠰⢻⣿⣿⡹
⠀⠀⠉⣿⢹⡏⠉⠀⠀⠀⠀⣛⣛⣛⣛⣛⣹⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⠀⠀⢹⡿⣿⠃
⣀⣀⣀⣻⣘⣃⣀⣀⣀⣀⣈⣿⣿⣿⣿⣿⣹⣿⣇⣀⣀⣀⣙⣿⣿⣿⣻⣿⣿⣟⣁⣀⣀⣘⣙⣀⣀⣈⣇⣿⣀
⣿⣟⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠈⢣⠀⢀⣀⣀⡀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡴⠛⠉⠉⠀⠐⢢⠶⠛⠉⠁⠀⠀⠉⠃⠀⠀⠀⠀⢀⣤⠴⠦⢤⣄⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢻⡄⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣠⠴⠂⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⢹⡏⢠⣴⣛⣛⢶⡄⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢿⡆⣀⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣷⡟⣉⣤⠌⢻⡇⠀⠀⠀⢠⣿⠭⢭⣻⡆⠀⠀⠙⠻⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠀⠀⠀⠉⠻⣷⣤⣾⡦⣄⠀⠀⢸⡇⣾⣃⢹⠇⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠟⠀⠀⠚⠛⠛⠛⠲⢦⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⣀⣀⣀⣀⣠⣤⣤⡴⠞⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠛⠋⠉⠉⠉⠛⠛⠛⠷⢿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⠖⠻⠿⠿⠷⣶⣤⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⢀⣀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣄⣸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⡾⠛⢦⣀⠀⠀⣰⣤⡾⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠦⠀⠀⠀⠀⠐⠂⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⡀
⢀⡤⢶⡇⠀⠈⠏⠁⠲⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⡞⠀⠀⠁⠀⢀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣤⣤⣾⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⣇⠀⠀⢰⠀⠈⢷⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣤⣀⣀⣰⡿⠋⠀⠈⠻⢷⣤⣀⣀⣀⣀⣤⠖⠋
⢸⡀⠀⠸⡄⠀⠸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⡟⠀⠀⢸⠋⠉⢻⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣽
⠀⣧⠀⠀⣧⠀⢀⡇⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠁⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠸⣆⠀⠹⣦⣼⠃⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⡤⣤⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠀⢀⣀⣀⠤⠒⢚⣉⣉⣉⡉⠙⢦
⠀⠀⠘⢦⡀⠀⠀⢀⡴⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡏⠀⠘⣧⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⣴⠞⠻⢬⣭⠭⠷⠶⠚⠛⠉⠉⠀⠈⠹⡆
⠀⠀⠀⠀⠙⠓⠚⠋⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⢸⣷⠨⠇⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠙⣇⠀⠀⠀⣿⠀⠀⢰⡖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠐⠋⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠤⠖⠒⠋⠉⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠋⠁⠀⠀⣀⣤⠴⠖⠚⢋⣉⡉⠀⠀⠀⠠⠤⠤⠤⠤⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢉⡤⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⣴⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣷⣤⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠉⠀⠀⠀⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠟⠰⠦⠄⢀⣤⡤⠄⠋⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡇⠩⠅⠀⠈⠆⠍⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡿⣇⠀⣈⣓⣒⣊⡀⢀⣿⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠚⠁⠀⠀⣧⣼⡦⣍⠒⠒⢚⣥⣞⣸⠇⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠹⣟⠉⠀⠉⠉⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⣀⣀⡀⣀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢘⡇⠻⠆⠀⠀⠀⠀⠀⠀⣸⠟⠃⠀⠀⢭⠿⠛⢹⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠰⠶⠶⠐⣿⠆⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⠴⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠳⣤⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⢱⡆⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠉⢙⡿⠦⠶⠶⠖⠀⠶⠶⣤⠸⣤⡟⠁⣀⣤⣤⣤⡀⠸⡏⣦⠀⠀⠀⠀⢡⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠒⠢⠤⢄⡀⠱⡀⠀⠀⠀⠈⢳⡀⠀⠀⠶⠛⠛⠛⠓⢠⣿⠄⠀⠼⠶⠶⢤⣄⠀⠀⣽⠃⠀⠀⠀⢸⠀⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣠⠞⠋⠉⠓⠦⣄⠀⠀⠀⠉⠀⣼⠁⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠹⡆⠀⢠⠞⠁⠀⠀⠤⠀⠀⠈⠳⡄⠀⢀⡼⠁⠀⠀⢀⠜⠁⠀⠀⣀⡀⠤⠤⠤⠤⠄
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢘⣷⣄⠀⠀⠙⢞⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡞⠀⠀⣀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣳⡄⠀⠀⠈⠓⢄⠐⠦⠤⠤⢆⡴⠒⠉⠁⠀⣠⣾⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⢿⡀⠀⠀⢤⣤⣕⣶⡒⣲⣾⣴⣿⡄⠀⣴⣻⣿⣿⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠂⠀⢿⣿⣿⣿⣿⣶⣷⣾⣿⣿⢿⣿⣿⣿⣿⣭⣴⣾⣿⣿⣿⣿⡇⠀⠀⠈⠉⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠛⠛⠛⠛⠛⠋⠋⠃⠘⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡼⠁⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⡇⣠⣠⡶⠋⠉⠉⠉⠒⠢⠤⠥⠔⠁⢀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠟⠉⠁⠉⠁⠐⠀⠀⠀⠀⣀⣀⣐⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣲⠶⠶⠖⠲⣦⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠂⠁⠀⠀⠈⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⡀⠤⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣤⣄⣀⣀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣬⣵⣦⣤⣤⣀⣀⣀⣠⣤⣴⣖⣭⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣴⠶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣾⡿⠿⣟⣯⣥⣆⢰⣭⣵⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠉⣱⣿⡿⣋⣵⣼⣿⣿⣿⣿⡷⡙⣿⣻⣛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣿⣾⣿⠛⣻⣿⣿⣿⣿⡝⠻⠛⣻⣿⠿⠟⣿⣿⣿⣮⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣿⣿⣿⣟⣵⣿⣿⢿⣿⣿⣿⣿⠄⢀⣴⣶⣿⣿⣿⣿⣿⡿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣛⣵⣷⣿⢟⣻⣿⣿⣿⣿⣿⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣿⣿⣿⣿⡸⠟⢻⣿⣥⣻⡿⣻⣽⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣵⣷⣿⣿⢟⣵⣿⣿⣿⣿⣿⡿⠟⠋⠁⡀⠀⢀⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣻⣿⣿⡿⣋⣶⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⢠⡀⣼⠀⣆⢐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣴⣿⣿⣯⣾⣿⣿⣿⣿⣿⡿⣠⣴⣾⣿⣿⣿⣷⣶⣽⣯⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠈⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣠⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣗⠉⠙⠛⠿⠵⠀⠀⢻⣿⣿⣿⣿⠿⠟⢁⡄⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⣤⣶⣄⠀⠀⠀⠀⠀⠀⢴⡒⠋⠙⠉⠄⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠿⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠘⠖⡄⣏⡄⡀⢃⣀⡈⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠉⠑⠶⠴⠌⠘⢠⡶⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠲⠤⢤⣤⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠟⠻⠟⠻⠿⠿⠿⠿⠇

⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣊⣁⠀⠀⠀⣤⡀
⠀⠀⠀⢀⣸⣿⣧⣤⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣤⣼⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣧⣄⣰⠏⠹⣆⣠⣼⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢰⣿⣿⣿⣯⠉⠀⠀⠉⣽⣿⣿⣿⡆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠈⢻⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡟⠁
⠀⠀⠀⠸⣿⣿⣿⢹⣿⣿⡇⠀⠀⠀⠘⢿⣿⡿⢿⣿⣿⡏⣿⣿⡿⣿⣿⣷⠀⠀⠀⠈⣿⡟⢿⡇⠀⠀⣼⣿⢻⣿⠁
⠀⠀⠀⠀⣿⣿⡟⠈⣿⣿⠀⠀⠀⠀⠀⢸⣿⠇⢸⣿⣿⠃⢹⣿⡇⢸⣿⠇⠀⠀⠀⢠⣿⠃⢸⡇⠀⠀⢸⡇⠘⣿⡄
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠈⠁⠀⠈⠁

⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⣤⣄⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶
⠀⠀⣠⣿⣿⣄⢐⣶⣶⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣠⣿⣿⣄⢘⣶⣖⠀⣨⣿⣧⠀⠀⠀⠀⠀⠀⣾⣷⠘⢶⣾⣿⣆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡄⠀⠀⠀⢨⣷⣶⡿⠋⢻⡟
⠀⠸⢿⣿⣿⡿⣿⣿⣿⡿⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⢿⣿⣷⠀⠀⠀⣸⣿⣿⣆⠀⠈⠳
⠀⠀⢸⣿⣿⡇⣿⣿⣿⣿⠀⠀⠀⢻⣿⡟⢹⣿⣿⡟⢸⣿⣿⢣⣿⣿⣿⣇⢹⣿⣿⠸⣿⣿⠂⠀⠀⠹⣿⣿⣟
⠀⠀⠈⠿⠿⠀⠹⠏⠹⠇⠀⠀⠀⠘⠿⠇⠈⠿⠿⠇⠘⠿⠿⠀⠹⠏⠽⠁⠸⠿⠏⠀⠿⠇⠀⠀⠀⠀⠈⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⣠⣼⣿⣧⣄⠀⠀⠀⠀⣤⣿⣿⣦⡀⠀⠀⠀⠀⢀⣿⣽⡀⠀⠀⠀⠀⠀⢠⣿⣭⡀
⠀⢰⣿⣿⣿⣿⣿⣶⡀⠀⣸⣿⣿⣿⣿⢷⣶⡄⠀⠀⣾⣿⣿⣿⣶⣄⠀⠀⠀⣾⣿⣿⣿⣦⡄⠀⠀⠀⣴⡆⠀⠀⣠⡀
⠀⠸⠿⣿⣿⣿⣿⣿⣏⠀⠻⢿⣿⣿⣿⣨⣿⡆⠀⠀⢿⣿⣿⣿⣿⣿⡀⠀⠀⢿⣿⣿⣯⣿⣇⠀⠀⣴⣿⣷⡆⢠⣿⣧
⠀⠀⠘⣿⣿⣿⣿⣿⠛⠀⠀⢸⣿⣿⡟⢿⣿⡇⠀⠀⠘⣿⣿⣿⣿⡟⠀⠀⠀⠸⣿⣿⢿⣿⡿⠀⠀⠘⣿⣿⠃⢸⣿⡿
⠀⠀⠀⢿⠿⠏⠻⠿⠀⠀⠀⠈⠿⠿⠇⠸⠿⠁⠀⠀⠀⠿⠿⠸⠿⠇⠀⠀⠀⠀⠿⠿⠘⠿⠇⠀⠀⠀⠿⠟⠀⠈⠿⠇

⣿⣿⣿⣿⣛⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡿⠿⣿⣷⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠀⠠⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⠀⠀⠀⠀⠉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⣀⣤⣶⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠖⠶⠶⣦⣤⣤⣤⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠸⣿⡀⠠⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠉⠙⢿⡟⠛⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⣿⡏⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠋⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣧⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⣤⣤⠴⠶⠶⠾⠿⠿⠿⠛⠛⠉⠉⠉⠛⣻⣷⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣀⣀⣀⣴⣿⠁⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣦⡤⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⡿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⣰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣿⢿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⣿⠿⠿⠿⠿⢿⣿
⣿⣦⣶⣦⣤⣶⣶⣷⣴⣷⣴⣦⣾⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣾⣷⣿⣷⣿⣷⣾⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡹⠿⠋⣾⣿⣓⣸⣿⣷⡄
⠀⠀⠀⠀⠀⢠⣴⣿⣿⠟⠁⠀⠀⠙⠻⣭⣽⡿⣛⢧
⠀⠀⠀⠀⠀⢿⢉⣿⠋⠀⠀⠀⠀⠀⠀⠹⠿⠀⠿⣸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⠾⠃⠀⠀⠀⠀⠀⣘⠹⣧
⠀⠀⠀⠀⠀⠀⠰⠿⢳⣦⡀⠚⠀⠀⠀⠀⠀⠀⣿⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣫⣤⠳⠀⠀⠀⢰⠏⠀⠀⠉⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡉⠴⠶⠀⢀⣠⠀⣀⣀⣀⣀⠀⠀⠉⠻⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠛⠛⠛⠛⠋⠁⠋⠙⠛⠋⠉⠀⢀⣤⠀⠀⠙⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⡀
⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⢀⣴⠟⣿⠇⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⢀⣾⠃⠀⣿⠀⠀⠀⢀⡟⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠍⠉⠉⠹⣿⡿⣷⣤⣀⣀⣤⣤⠀⠀⠀⣿
⠀⢸⣿⠀⠀⢿⣦⣤⣴⡿⠁⠈⢿⣦⣀⣀⣀⣀⣒⡒⣀⣤⡿⠛⠁⠀⠀⠀⠀⢻⣷⠀⠉⣹⡟⠉⠀⢠⣤⢠⣿
⠀⠀⢻⣄⠀⣾⠏⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⢠⣿⠃⢀⣴⣿⣣⣾⠃
⠀⠀⠀⢻⣿⠏⣰⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠃⠀⠘⣫⣿⡿⠃
⠀⠀⠀⣼⡏⠀⣿⡟⠛⠟⠀⣰⣶⠀⢠⣶⣶⣶⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⣠⣼⠟⠁
⠀⠀⣸⡿⠀⢸⣿⣇⠀⠀⢰⣿⣿⡄⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⢀⣾⠟⠁
⠀⢀⣿⡇⠀⣾⡿⠙⠃⢠⣿⡏⣿⡇⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣾⣵⣆⡀⠀⠀⣰⣿⡇
⠀⠘⣿⡇⠀⠙⠃⠀⢀⣾⡿⢿⣿⣇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⡷⢿⢞⣻⣿⠇
⠀⠀⠻⣿⣄⡀⠀⠀⠈⠙⠁⠀⠻⠿⠀⠀⠻⠏⠀⠀⠀⣀⣤⣶⣀⣀⣠⣤⣶⣶⠈⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⠛⣿⣿⣿⣿⣿⡿⠿⠀⠉⠹⣿⡀
⠀⠀⠀⢠⣿⠃⠈⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦
⠀⠀⠀⠀⢘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆
⠀⠀⣰⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠘⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡏
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢸⡿⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠏
⠀⠈⠋⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠁⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇
⠀⢸⣿⢿⣿⣿⢿⣿⡿⠂⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⣿⠛⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⢸⣿⠸⣿⡏⢸⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡀
⠀⠀⠀⠀⢻⡇⢸⡟⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⢿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠘⠿⠌⠙⠳⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠿⠿⠿⠂⠀⠀⠈⠛⠛⠛⠋⠹⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⣀⣤⣾⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣷⣤⣄
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⠿⢛⣻⢿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡏⣿⣿⣿⣿⣿⣿⣿⢹⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⣞⣿⣿⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⣿⣿⣿⣿⣿⣿⣿⢸⡟⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠻⣿⣿⣿⣿⣿⡟⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏
⢿⣷⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣿⣿⡏⠀⠀⠀⣴⡏⣹⣿⣿⣿⣿⣿⣿⡿
⠈⠁⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢿⣿⣿⢻⣿⣿⠇⠀⠀⠀⠀⠀⠙⢿⣿⣿⠇⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⢹⣿⣿
⠀⠀⠈⣿⡏⠈⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿
⠀⠀⠀⢸⢷⢰⣿⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠸⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣶⣶⡴⢦⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠋⠀⠀⠀⠈⠁⠟⠈⠇⢹⠆⠙⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢷⡄⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⡄⠀⠀⠉⠙⠿⠀⠀⠀⢀⡤⣤⣰⢾⣦⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠷⠟⠀⠀⠺⠃⠀⠀⠀⠀⠀⢈⡄⠈⢿⠾⠛⠟⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢀⡾⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠰⣆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣟⠉⠀⣿⡀⠀⠀⠀⠀⠀⠀⢠⠾⠋⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⢠⣄⠙⠆⠀⠃⠈⠀⠀⠀⠀⠀⠀⢸⡇⢹⡏⠉⠉⠻⢶⣤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠘⠲⠆⠁⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠰⣄⠀⢀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⡶⠾⠟⣿⣧⡀
⠀⠀⣤⠀⠀⢀⣴⠏⢸⠇⣤⣶⣶⣦⣄⠀⠀⠈⠻⣤⡈⠉⣁⣤⣴⠶⠟⠋⢉⣠⣴⠿⠛⣧⣠⡔⠚⠉⠹⣿⡄
⠀⠀⢠⡄⠀⣿⠁⣠⢿⣴⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⡿⠛⠉⠀⠀⠀⢀⣴⡿⢿⠀⣠⣴⣿⠿⠛⠛⣧⡀⢿⣧⡀
⠀⠀⠀⢀⡄⠙⠋⣴⣾⣿⣿⣿⣿⣿⠟⢉⣿⠿⠋⠁⠀⠀⠀⠀⠀⣠⡾⠋⢀⣸⣿⣿⠟⠁⢦⣠⠞⠉⢻⣄⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⣾⠟⠿⠛⠃⢀⡴⠎⠻⣄⠀⢀⡾⣇⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡶⠃⢹⣧⠀⠀⠰⢏⠀⠀⠀⣹⣶⠉⠀⠘⢧⣿⡇
⠀⠀⠀⠀⠀⠀⢀⠀⢀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢻⣤⡾⠛⠉⠀⡀⠀⠀⢳⣴⠎⠁⠀⢣⡀⣠⠞⠻⣿⡀
⠀⠀⠀⠀⠀⣴⣟⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢡⣾⡏⠀⢳⣤⠚⣿⡀⠈⠁⠙⢦⠀⢀⡴⠻⣅⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⢉⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⡴⠋⢣⡄⠘⣿⣄⡀⢀⡤⠿⣏⠀⠀⠘⣧⣀⣤⠾⠿⣷⣄
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⠿⢯⡀⠀⢀⣵⡞⠁⠙⠿⣿⣶⡄⢹⣦⡔⠚⠉⠻⠀⠀⠐⢿⣿⣦⡀
⠀⠀⣾⣆⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⣳⣶⠋⠀⠘⢦⣠⡔⢙⣿⣧⣤⣼⡷⠖⠶⢾⣶⣶⠖⣾⠋⠻⣿⡄
⠀⠘⠟⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣍⣀⠙⢦⣀⡴⠚⣻⣴⢾⣇⠈⢙⠿⢤⣤⣶⣿⡿⠋⠀⡟⠀⢀⣾⠟
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣭⡿⠟⠻⣿⣤⣾⡏⠀⠀⣨⢷⣏⠀⢀⣼⠛⠉⠑⢶⠾⢳⣴⡿⠋
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣶⣶⣶⠶⠿⠟⠛⠉⠉⠀⠀⠀⣿⢻⣯⠀⢹⣦⡞⠉⠀⣈⣷⣏⠀⢀⣴⠞⣀⣴⡿⠋
⠀⠀⠀⣿⣧⣴⣶⣾⣿⡿⠿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢻⣷⡊⠉⠓⣤⡞⠁⠀⢈⣽⣿⣶⠿⣿⡁
⠀⠀⠀⠙⢿⡇⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡆⠀⠙⢿⣦⡞⠁⠉⢳⣴⡿⠛⢉⣧⢀⣽⣧
⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠄⠀⣿⣿⣦⣴⠿⢿⡗⠊⠉⡟⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣴⡾⢯⠀⣿⡇⠀⢘⣇⣠⣤⣴⡖⠾⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣆⠀⠀⣿⡟⠉⠉⢋⠀⠀⢸⡇⠀⣻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢿⣷⣶⣦⣾⣶⣶⣾⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣾⠟⠋⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣩⣿⢧⡄⠾⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠛⠋⠉⠁⠸⣿⡄⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣿⣿⠀⠙⣷⣤⡀
⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣀⣀⣨⣿⠟⠛⠻⠿⠿⣶⣶⣶⣶⣶⡿⠿⠿⠿⠛⠛⠉⠙⢿⣶⣿⣿⣿⣇
⠀⠀⢠⣾⣿⣿⣿⠿⣿⣿⣿⣯⣋⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⠀⢠⣤⣴⣶⡶⣶⣾⣿⣿⣿⣿⣿⠇
⠀⠀⢿⣿⣿⣿⣿⣶⣤⣽⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣼⣿⣿⣷⣶⣾⣿⣿⣿⣿⣼⣿
⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⠉⠉⢻⣧⣤⣤⣤⣤⣤⣴⡿⠋
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠠⣿⣷⠀⠀⠀⠀⠀⣤⣤
⠀⠀⢠⣤⣭⣥⣄⠀⠀⠀⠀⢀⣤⣬⣥⣤⡀⠀⠀⢀⣻⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⢰⣿⣿⣿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⣴⣿⣿⣿⣿⣆⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⡄
⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⠋⣿⠏⣿⣿⣿⣿⣿⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠈⠸⣿⣿⣿⡇⠁⠀⠀⠀⠁⣿⣿⣿⣿⠀⠈⢸⣿⣿⣿⣿⡏⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⣤⣤⣴⣿⣿⣿⣿⣿⣦⣤⣄
⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⢻⣿⣿⡏⠀⠀⠿⢿⣿⣿⡿⠿⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⣿⡇⢻⣿⡄⠀⢠⣿⡏⢸⣿
⠀⠀⠀⠙⠋⠛⠀⠀⠀⠀⠀⠀⠘⠋⠙⠃⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠛⠁⠈⠛⠁⠀⠘⠛⠁⠘⠛

⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⢠⡇
⠀⠀⢀⣴⣶⣬⠛⠋⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣃⣀⠀⠀⠀⠀⢀⣙⣛⠁⠀⠀⠀⠀⢸
⠀⠀⣾⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣀⣼⡿⠇⠀⣀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⡆⠀⢀⣿⣿⣿⣧⠀⠀⠀⠀⣿
⠀⠘⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢻⢸⣿⣿⣿⣿⣁⣠⣶⣆⣻⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢀⣿
⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣯⠉⢿⡟⠉⠁⠀⠀⠀⢹⣿⣿⣿⡟⠀⠀⠀⠈⢿⣿⣿⡟⠁⠀⠀⢸⢻
⠀⠀⠸⣿⣿⠀⠀⠀⠂⠀⠀⠀⢸⠶⠿⣯⠉⣿⡟⠀⢸⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⡾⢸⡆
⠀⠀⠤⠭⠥⠤⠴⠦⠀⠀⠀⠀⠾⠀⠀⠿⠐⠿⠇⠀⠼⠤⠀⠀⠀⠀⠀⠀⠿⠻⠿⠀⠀⠀⠀⠀⠈⠿⠟⠀⠀⠀⠀⠠⠼⠧⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⠁⢾⣿⡇⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣇⠀⣠⠋⠀⠀⠀⠀⠀⠀⢻⣷⣶⣶⣶⣾⠟⠀⠀⠀⠀⠀⢠⣤⠀⠀⢀⠔⠋⠀⢀⣀⣀⡀
⠀⠀⣴⠒⠚⣿⣿⣿⣿⡟⠚⢻⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⢸⣿⢠⣴⣄⣠⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣿⠀⢀⣿⣿⠻⣿⣧⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣘⣻⣋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⣿⠀⣿⡿⠃⠀⢿⣿⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣶⣿⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣿⣷⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⢻⣿
⣶⣾⠷⡶⢶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿
⣿⣿⣤⣦⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠉⠉⠁⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠿⠿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢤⡀⠀⠺⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠙⠲⣄⡀⠈⠙⠛⠿⣷⣶⣤⣄⣘⠛⠛⢛⣛⣛⣛⣛⣛⣛⣛⣻⣿⣶⣝⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠳⢤⡀⠀⠉⠓⠦⣤⡀⠀⠈⠉⠛⠻⠿⢿⣾⣯⣭⣭⣭⣉⣉⣉⣉⣉⣉⣿⠀⠹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠳⢤⣀⠀⠈⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⡀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣿⠀⠉⠳⠤⠀⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠻⣦⣄⠀⠀⠀⣿⠀⠙⠻⠶⣦⣄⣀⣀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠉⠛⠶⣦⡇⠀⠀⠀⠀⠀⡏⢹⣟⠛⠛⠛⠛⠋⠉⡉⠀⠀⠀⢰⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⣄⠀⢹⠈⣿⠀⠀⠀⠀⠀⢰⢀⣀⣠⠴⠚⢉
⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢸⡛⠻⠶⢶⣤⣤⣤⡄⠀⠀⡟⠉⠁⠀⠀⠀⡌
⠀⠀⡴⠿⠿⠿⢿⣿⣷⣦⣄⡙⣷⣤⣀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡇
⠀⣼⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣇⠈⠙⠳⢦⣄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⠀⠀⠀⢰⡁
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠉⠳⢤⡀⠀⠈⠙⠀⠀⠀⠰⠶⣦⣤⣀⣀⠀⠀⢿⠀⠀⡟⠀⠀⠀⢀⣠⠜⠁
⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠉⠓⣤⣄⡀⠀⠀⠀⠀⠀⠈⠉⠙⡟⢻⡟⠀⠀⡷⠶⠒⠋⠉⣿⣶⣶⣶⣶⣶⣶⣤⡀
⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡉⠛⠻⢶⣦⣤⣀⣀⠀⠀⢳⠘⣷⠀⢀⡇⠀⠀⠀⠀⣿⠉⠀⠀⠀⠉⠙⢻⣿
⠀⠀⠀⠹⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣤⣈⣉⣛⠛⠛⠟⠓⠛⠛⢋⣀⣤⡤⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠙⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣶⣶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠰⡲⡔⢁⠔⠉⣱⡄⠐⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠕⠻⣠⢎⠘⡖⠉⠉⠓⢦⡜⠁⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠛⢷⡘⣄⠀⠀⠛⣹⣴⠀⠀⢀⣜⢀⣀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠒⠬⠿⠈⠑⠲⠶⠏⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢢⣉⣚⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠤⣀⢀⠀⡀⠀⣰⠃⠀⠀⠀⢠⣔⡦⠯⢈⣯⡶⠖⠫⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⢮⠃⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢔⣄⡀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢳⣠⡀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⢳⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⢱⣀⠀⠀⠀⠀⠀⣣⠔⣇⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠚⠉⠁⠀⠉⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡗⠭⣉⢻⠋⠉⠀⢀⠼⡄⡠⠔⠁
⠀⠀⠀⠀⠀⢀⠔⠋⠐⠢⢤⡀⠀⠀⠀⠀⠀⠈⠻⡒⠲⢤⡤⠤⠤⠒⠒⠉⢁⡰⠁⠀⠀⠉⠲⢤⣀⡜
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠈⢳⣤⠄⢀⣀⣀⡀⡸⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠘⢄⡀⠀⠀⠀⠀⠺⠊⠩⣏⡀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠐⢲⡄⠠⠤⢤⡷⠙⠤⠴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⢲⠖⢆
⠀⠀⠀⠀⣀⣤⣴⠖⣶⠈⢷⣄⠀⠀⠓⠒⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⠃⠀⢸⠀⡠⠆⣗
⡆⠀⢴⡾⢋⣯⢼⠉⠧⡤⠚⠀⠉⠒⠢⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠒⢤⣤⢤⣤⣤⣤⣤⠤⠶⠚⠉⣀⡠⠴⠦⢄⠀⢞⣡⢟⠏
⠀⠀⠀⠉⠁⠀⠈⠢⣐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀⣀⣀⣤⠴⠖⠚⠉⠀⠀⠀⠀⢈⢒⢟⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠤⠤⠤⠤⠤⠰⠖⠒⠒⠒⠒⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠤⠀⢤⣄⠤⢤⣄⣀⠤⣤⡤⠤⠤⠴⢦⡴⠶⠖⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢀⡤⠏⠀⠙⠦⣄⣀⣀⣀⣤⠴⠂⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡶⠋
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠸⡆⠀⢾⣽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣿
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⠖⠀⡴⠋⠀⠸⣆
⠀⠀⠀⣷⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠉⢀⡞⠁⠀⠀⠀⠹⣆
⠀⠀⢀⡇⠀⠀⠀⠉⠉⠉⠛⠛⠛⠓⠛⠛⠛⠛⠉⢉⡟⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⠈⠉⠉⠉⠉⠐⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠙⢦⢀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠒⠄⡐⠀⠀⣀⡀⠀⠀⠀⢀⠀⠐⠒⠚⠛⠛⠒⠒⠭⣕⡢⣌⠳⠀⠁⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠈⠑⢾⠋⡠⢊⡠⠊⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠙⠮⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠤⠤⣄⠀⢀⠤⠤⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠙⡓⠌⠀⠤⠂⠈⢧⠂⠚⣻⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⣤⠆⠀⠀⠀⢐⡆⢠⠏⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣒⠂⠀⢀⣀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⠀⠀⠀⠀⠬⠁⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠒⠚⠁⠀⠔⠒⠒⠶⢦⢤
⡦⢕⠻⣶⠄⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⡀⡂⢐
⠹⣖⣙⣈⣁⣠⣤⠤⠤⠶⠾⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠲⠶⠶⠶⠶⠤⠤⠤⠤⠌
⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣤⠤⣀⣀⣀⣤⢤⣠⡤⠶⣿⣿⣿⣿⣿⣿⣿⠟⠭⣑⣲⠶⠦⠰⠒⠖⣫⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢭⡀⠑⣄⣰⠀⠀⠀⠉⢀⡄⡿⠋⠁⠀⠀⠙⣿⡀⠁⠀⠀⠀⠀⣦⡴⠊⠠⢈⡵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⠞⠁⠀⠀⠀⠀⠐⣡⠞⠁⠀⠀⠀⠀⠀⠈⢮⣂⣀⠀⠀⠀⠀⠀⠘⠶⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠤⠤⠠⠀⢀⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣤⡀⠀⠤⠠⢐⠵⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠋⠉⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠿⡍⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠈⠒⢤⡂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠛⣄⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⠋⠀⠀⣤⡀⠀⠀⠛⠃⠀⠈⢳⡈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢻⡄⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⢹⣆⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡇⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢻⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠋⠁⠀⠀⢷⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠉⠛⠷⣶⣦⣴⣶⣾⡿⠛⠁⠀⠀⠀⠀⠀⠸⡄⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠇⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣤⡤⠤⢤⣤⣄⣀⢸⠸⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠸⠀⠇⠀⠀⠀⢀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠈⠉⠛⠻⢦⣄⡀⠀⣸⡿⠁
⠀⠀⠀⠀⢀⣴⠾⠋⠀⢻⡄⢀⣌⣤⣤⡶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠈⠙⢾⣟⠁
⠀⠀⣠⡾⠋⠁⠀⠀⠀⣸⡿⠛⢻⣅⠀⠀⠀⠀⢀⠤⠤⠶⠶⣶⣤⡀⢀⣠⠶⠚⠛⠛⠲⠦⠀⠀⠀⣠⣴⠟⢻⡄⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⣰⡟⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠊⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠛⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⢀⡿⠀⠀⠀⣠⠴⢾⡏⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣀⣀⣀⣰⣿⣀⣀⣤⣤⣴⣶⠿⠛⠉⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⡇⠀⣠⠞⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢸⡇⢰⠃⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⢸⡇⠨⠀⠀⣀⢀⡀⣸⡏⠙⠷⣦⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠛⢦⣀
⠘⣧⠘⠂⠈⢉⡀⠉⠻⡇⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠙⠲⠤⣮⠀⠀⠀⠀⢀⣾⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠋⠁⠀⠀⠀⠙⠳⣄
⠀⠙⠷⣶⡾⠋⢳⡘⢆⡃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠷⢶⣦⣄⠳⡄⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⢀⡾⠋⠀⠀⠀⠙⠢⡽⠀⠀⠀⠀⣠⠞⠉⠛⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⡉⠂⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡆⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⢹⣇⠀⠀⠀⠀⠠⠞⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠏
⠀⢻⣆⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣦⣤⣤⣤⣤⣤⣀⣀⡀⠙⢦⣀⠀⠀⢀⠠⠄⠐⣒⣂⣲⣤⣤⣴⣶⡾⠟⠛⠁
⠀⠀⠙⢷⣤⣀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⢳⣄⠉⠻⡞⠣⠴⠦⠀⣈⠉⠛⠛⠻⢦⣀
⠀⠀⠀⠀⠈⠉⠛⠛⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠾⠛⠙⢷⡀⠈⢢⡀⢠⣆⠀⠚⠃⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⣄⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠈⢿⡄⠀⠙⢦⡉⠓⠲⠤⣄⣀⣠⣤⣤⣌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠹⣄⠀⠀⠈⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠳⠶⣤⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡾
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⣀⣿⡀⠀⠀⢀⣾⣆⠀⠀⢀⣾⣆⣠⠈⠲⣤⣷⡤⠞⠁⣀⣾⣤⠤⠖⢀⣴⣆⠀⠀⣀⣿⡀⠀⠀⠀⣀⠀⠀⢯⣼⣧⠽
⠀⠀⣿⣿⣿⠀⠀⣿⣿⣿⡆⠀⣿⣿⣿⠁⠀⠀⣿⣿⡇⠀⠀⣿⣿⡧⠀⠀⢸⣿⣿⡇⠀⢻⣿⡟⠀⢠⣾⣿⣷⡄⠰⣿⣿⠆
⠀⠀⢸⣿⡇⠀⠀⠘⣿⡇⠀⠀⠘⣿⡏⠀⠀⠀⣿⠙⣇⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⢸⡟⠙⣿⠁⠀⡟⢿
⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠀⠀⠀⠁⠀⠉⠀⠀⠁⠈
⠀⠀⣰⣶⣦⠀⢀⢀⣶⣄⢀⠀⠀⣴⡄⠀⢀⣤⣄⠀⠀⣰⢀⣄⢰⠀⠀⣰⣆⠀⠀⠀⣰⣆⠀⠠⣤⣶⣶⠄⠀⠀⣶⣆⡤⠤
⠀⢰⣿⣿⡇⠀⠈⢻⣿⣿⠁⠀⢸⣿⣿⠀⢸⣿⣿⠀⠀⠈⣿⣿⡏⠀⢸⣿⣿⡷⠀⢾⣿⣿⣧⠀⢸⣿⣿⠀⠀⢾⣿⣿
⠀⠀⢹⣿⠃⠀⠀⠸⣿⡇⠀⠀⠈⣿⡏⠀⠈⣿⡟⠀⠀⣰⠟⠛⢷⠀⠈⣿⣿⠁⠀⠈⣿⣿⠇⠀⠀⢹⡿⠀⠀⠈⣿⡿
⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠉⠁⠀⠀⠉⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁⠀⠀⠀⠉⠁
⠀⠀⢀⡶⠀⠀⠀⣾⢂⡄⠀⠀⠀⠀⠀⠐⡷⠀⠀⠀⢀⡾⠂⠀⠀⢀⣀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣄
⠀⠀⢸⣷⡄⠀⢸⣿⡟⠀⠀⣸⡟⠀⠀⢸⣿⣆⠀⠀⢻⣿⡆⠀⠀⣹⣿⡤⠀⠀⠀⢐⣿⣶⡖⠗⠀⠀⠀⠀⠀⠀⣾⣿⠛⠃
⠀⠀⠈⣿⠁⠀⠈⣿⠁⠀⠀⣿⣷⡀⠀⠈⣿⣷⠀⠀⢨⡿⣧⠀⢀⣼⠿⣧⠀⠀⠀⣠⡿⠯⠉⠂⠀⠀⠀⠀⠘⠳⠟⠉⠻⡄
⠀⠀⠀⠉⠀⠀⠀⠙⠀⠀⠀⠛⠛⠃⠀⠈⠁⠘⠁⠀⠉⠀⠙⠀⠀⠀⠀⠙⠀⠀⠘⠉
⣶⣶⡶⣶⠶⣶⣶⡶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⢠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄
⠀⠀⠈⢻⣍⠉⠛⠛⠲⠶⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣧⣀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠶⠶⠚⠛⠋⢉⣽⠋
⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡟⠉⠙⠛⠛⠛⠛⠉⠙⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⢘⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠋⢹⡅⠈⣷⡀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣿⠉⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟
⢀⣀⣤⠶⠞⠋⠉⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⢸⡇⠀⠈⠳⣦⣀⣀⣀⣠⡾⠋⠀⠀⣿⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣄⣀
⠻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡟
⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠈⠳⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠋⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⠁⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⢻⡀⠀⢀⡀
⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⣿⠀⠀⠀⢀⣿⡇⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⣼⣇⠀⠀⠀⢸⠇⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠁⣷⠀⠀⠀⠀⠈⠙⠓⠶⠶⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⢸⡇⢹⡄⠀⠀⣿⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢿⠀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢸⠇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢰⣷⣼⡇⢸⡏⠙⠳⠶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠾⠛⠉⣿⠀⣿⣼⣷⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣧⠉⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠉⠉⣩⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠈⢡⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠛⠛⠳⠶⠶⠶⣦⡾⠁⠹⣦⡶⠶⠶⠶⠛⠛⠋⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣂⣀⣀⣀⣀⣀⣰⣏⣀⣀⣀⣈⣷⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠟⠛⠛⠉⠉⠉⠙⠛⠛⠻⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠛⠉⣀⣤⣴⣶⣶⣿⣿⣿⣿⣷⣶⣦⣤⣀⠉⠛⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⣠⣶⣿⣿⣿⣿⣿⠟⠛⠛⠿⠛⠙⠻⢿⣿⣿⣿⣶⣄⠈⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⣰⣾⣿⣿⣿⡟⠁⠀⠈⠀⠀⠀⠀⠀⢀⠞⠉⠉⠉⢻⣿⣿⣷⣄⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⢠⣾⣿⣿⣿⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠿⣿⣿⣿⣷⡀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣾⣿⠃⢠⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡄⠘⣿⣷
⠀⠀⠀⠀⠀⣸⣿⠇⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡀⠸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⡏⠀⣿⣿⣿⡄⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⢹⣿
⠀⠀⠀⠀⠈⣿⣇⠀⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣸⣿
⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⢹⣿⡆⠈⣿⣿⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⡄⠘⣿⣟⠀⣆⠀⠀⠀⠀⠒⠄⠀⠀⠀⢠⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠘⢿⣿⣿⣦⣤⡄⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⢀⣠⣾⡿⠁⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠹⢿⣿⣿⣷⣤⠀⠀⠟⠀⠈⠀⠀⠀⠈⣁⠀⢀⣠⣤⣴⣿⡿⠋⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠙⠿⣧⣀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠉⣽⠿⠋⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣄⣀⠉⠛⠲⠶⠾⣿⣿⣿⣷⠤⠄⠀⠀⠈⣀⣤⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣦⣤⣤⣄⣀⣀⣠⣤⣤⣴⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠈⠉⠀⠈⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⠁⠀⠲⠂⠀⠐⠒⢢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⢠⡤⠀⣿⢦⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠁⠀⠀⠀⠈⠀⠀⠛⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠈⠀⠀⠀⠀⠀⢰⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⢶⡾⡆⢀⡀⠀⠁⡀⠀⠀⣠⠿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠟⠁⠘⢆⠻⣦⣤⣀⠀⠀⠴⣟⣁⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⢠⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡅⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⣿⣿⡷⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢻⡃⠈⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣷⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⣺⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠘⡇
⠀⠀⠀⠀⢠⣞⠋⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⣧
⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣄
⠀⠀⠀⠀⢸⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⡜⠉⠓⠋
⠀⠀⠀⠀⢻⣀⡻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠌⠀⠀⠠⠤⣀⠈⢁
⠀⠀⠀⠀⠀⣿⡇⠈⠳⢆⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠅⠀⠀⠀⠀⠀⠀⠈⠳⠟
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠐⠀⠛⡦⠀⠀⠀⠀⠀⢠⡄⠠⠠⠠⡌⠁
⠀⠀⠀⠀⠀⢸⣇⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⣷⠀⠔⠁
⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⢀⣿⣿⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠷⢤⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠈⠀⠀⠀⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠰⣿⣿⡿⠃
⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⣦⣸⣿⡿⠁
⠀⢀⣤⣶⣾⣿⡟⠓⠂⠰⢶⣾⣿⣿⣿⣿
⣰⣿⣿⠟⠋⠉⠀⠈⠻⢿⣾⣿⣿⠋⠉⠡⠇
⠹⣏⠁⣀⣀⠀⣀⢴⣤⣿⣿⠿⠋
⠀⠀⠉⠉⠉⠁⠀⠾⣿⡿⠆⠀⠀⠀⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⣖⡒⠒⠒⢲⡿⠤⠤⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⡒⠒⠒⢺⡿⠤⠤⣤⠆
⠀⠀⠀⠀⠀⠈⠉⠀⠰⠀⢸⡇⠀⠀⣦⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠈⢉⣀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠁⠀⢠⠁⠀⢿⡀⠋⣠⠄⠀⣿⡏⣿⠀⢰⡟⢩⣧⠀⣿⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⢿⡇⢠⠏⠀⠀⠈⣻⡞⠁⠀⠀⢿⢠⡟⣠⠻⣇⣈⡿⠊⣿⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠋⠀⠀⠀⠀⠛⠁⠀⠀⠀⢀⣀⣛⠁⠀⠈⠁⠀⠀⠘⢗⠃
⠀⠀⠀⠀⠀⠀⣠⢖⡚⠋⠛⠛⣶⡄⢸⠀⣷⣏⣟⢹⠆⢠⣿⠁⢈⡇⢠⡴⠲⣄⠀⣾⣿⠀⣦⣄
⠀⠀⠀⠀⠀⢸⣁⣸⡇⠀⠀⢀⣾⠇⠼⠆⡿⢧⢯⠼⠄⣾⡇⠀⣸⢡⣿⠁⢀⡟⠀⠙⠿⢮⣿⠏
⠀⠀⠀⠀⠀⠀⠉⢹⡇⢠⣴⣞⣁⡀⠀⠀⠀⣀⣀⡀⠀⣿⢃⡴⠃⣸⡏⢀⡞⠀⣠⡀⢀⣬⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠻⣧⠀⣼⠏⣠⠇⢀⣿⠋⠀⠀⣿⡷⠋⠀⢠⣿⠀⣸⣿⠀⠀⡄
⠀⠀⠀⠀⠀⢠⠖⣻⡟⢲⡀⠀⠀⢀⣿⠁⣿⠉⠁⣠⠛⣿⠀⠀⣰⣿⡇⠀⢀⡾⣿⡰⠃⣿⣀⠞⠁
⠀⠀⠀⠀⠀⢸⡀⠙⠁⠘⠁⣀⣤⠾⠋⠀⠙⠓⠋⠁⠀⢿⣆⡴⠃⢹⣇⠀⡼⠀⠀⢀⣰⣿⠁
⠀⠀⠀⠀⠀⢀⣉⣓⣒⣛⠋⠉⠀⠀⠀⣄⣠⣄⠀⠀⠀⠈⠉⠀⠀⢈⡻⠞⠁⢀⡴⠋⣸⡏
⠀⠀⠀⠀⠀⢀⣝⣻⣺⣺⠗⠒⠒⠒⠒⠻⣿⠟⠒⠒⠒⠒⠒⠒⠒⢺⣏⠥⠆⢺⡀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⣴⣒⣒⣒⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⣶⣿⣿⣿⣿⣿⡟⠉⠙⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣸⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠉⠡⢟⡩⠀⠀⢠⡘⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣞⠗⠄⡉⠀⠀⠀⠀⠁⠀⠀⠀⠁⢱⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⣦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡯⠟⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢓⠫⢷⣲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡵⠋⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠉⠓⢯⡲⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⠀⠀⢀⣿⣿⣦⡀⠀⠀⢹⣽⡀
⠀⠀⠀⠀⢀⡴⡻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⣿⣷⡀⠀⠀⡇⡇
⠀⠀⠀⢠⢮⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠀⢰⢷⠃
⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⡞
⠀⠀⠀⢳⢧⠀⠀⠀⠀⠀⠀⠀⣒⣭⡷⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⣄
⠀⠀⠀⠈⢫⣣⡀⠀⠀⠀⠀⠀⢯⢏⣷⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⡆
⠀⠀⠀⠀⠀⠱⣝⢦⡀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡀
⠀⠀⠀⠀⠀⠀⠈⠳⢝⠦⣀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡼⠇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⢧⣅⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠻⠭⠽⠿⠿⠿⠿⠭⠿⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣴⣤⣄⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠻⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⣿⣇⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡓⠀⢙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠿⠿⠿⠏⠴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣦⣀⣀⣀⠀⠈⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡟⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣛⣿⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡏⠈⠁⣿⡒⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡻⡄⠀⠸⣧⣽⣎⡙⠛⠛⢛⠋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣽⣿⣿⣼⣦⠀⠈⠻⠿⠛⠃⠀⢈⡳⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣿⣿⣿⣿⡄⠀⣴⠟⣛⣙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣽⣿⣿⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠉⠁⠀⠀⠀⠙⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⣤⢤⠤⡤⠔⢂⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠨⠤⢦⠦⠒⣀⡷⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡶⠴⠾⢆⣹⣍⣴⠾⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡉⠻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣷⠘⡻⣿⢟⣵⣶⠾⣟⣿⣿⣿⣿⣿⣿⡍⢿⣿⣿⣿⣦⡙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠇⡏⠀⠀⠐⠋⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣿⣿⣿⣿⣬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣦⣧⣀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⠿⠛⡿⢻⡿⢻⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⢱⠀⠀⠀⣠⠄⠉⣽⠋⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣇⣀⢰⡇⠀⠘⢤⣀⡴⠛⠢⠶⣀⡀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡉⣝⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠄⠆⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⣉⠉⠉⠉⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠟⠢⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠀⠀⠀⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠱⡄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡴⠁⠀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡐⠀⠀⠀⠘⡄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡠⠋⡠⠊⠀⠀⠀⠈⢁⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⢹⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⡇⠀⠀⡄⠀⠘⡇⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⡮⣠⠀⠀⠀⠀⠀⢀⠼⢄⡀⠀⢈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢸⠁⠀⠀⡇⠀⢸⠃⠀⠀⠀⠀⠀⠉⠲⠤⣀⣀⡰⠂⠘⢇⠀⠀⠀⠀⡁⢀⣀⡏⠀⢇⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢸⠀⠀⠀⡧⠐⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠤⠀⠀⣀⠀⢇⣀⠤⠖⠃⢸⡄⣀⣀⣘⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠒⢾⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⡜⠀⠀⠀⣸⣉⠉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡓⠠⠟⠉⠉⠉⠀⠙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢘⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠘⠤⠔⠊⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⢀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠈⠉⠑⠒⠒⢲⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⣦⣤⡤⠤⠤⠤⠬⠯⣭⣅⣈⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠎⠈⡇⠀⠀⠀⠀⠀⡸⠉⣇⣀⣀⣀⣀⠠⣀⣀⣀
⠀⠀⠀⠀⠀⢠⣞⣫⣷⣫⠼⣋⠀⠀⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⣉⡱⢦⡹⠎⢵⠦⡀
⠀⠀⠀⠀⠀⢸⢌⡇⢀⣀⠑⣆⠙⡦⠤⠤⠤⠠⠤⠤⠚⡄⠀⠸⡣⠤⠤⠔⠤⠤⠎⢡⡞⠁⢀⡘⣚⠴⣧
⠀⠀⠀⠀⠀⠈⠓⠭⠬⢀⠠⠤⠄⠐⠂⠀⠐⠒⠒⠒⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠈⠛⠋⠤⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⣦⠀⣀⠤⠆⠄⢄⡠⣂⡠⠤⠆⠀⢀⡾⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠈⠀⠀⠀⠒⠛⠃⠈⠁⠈⠀⠐⠒⠀⠀⠉⠀⠈⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⢠⠖⠉⡑⢀⠀⢠⢖⠀⠒⠄⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⢃⣀⣀⣀⣘⠁⠀⠸⠀⠀⠀⠀⠀⠀⠀⢈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠊⠉⠉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠊⠑⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⢀⣠⡶⠿⠿⢧
⠀⠀⠀⠀⠀⠀⠀⣠⠛⠻⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⣀⣤⣶⣶⣾⣧
⠀⠀⠀⠀⠀⠀⢀⠟⠛⠷⠶⣌⡙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⠋⠁⣀⣀⣠⣼⡄
⠀⠀⠀⠀⠀⠀⣬⣤⠤⠄⠀⢀⣹⡓⠤⠉⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠋⠁⠀⣬⠄⠉⠉⠉⠉⠉⣇
⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠐⠀⠀⠈⢦⡀⠀⠀⠀⠈⠉⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠐⠒⠊⠉⠀⠀⠀⠀⠀⠎⠀⠀⠀⢤⣤⡴⠶⢿
⠀⠀⠀⠀⠀⠸⡷⠶⠆⠀⠀⠂⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⣦⣀⣀⠀⠀⠀⠤⣀⠀⣀⣺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣥⣀⢀⡴⠣⠤⠴⠶⠿⠟⡏
⠀⠀⠀⠀⠀⠀⢻⠛⠋⠉⠉⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⢀⣸⠃
⠀⠀⠀⠀⠀⠀⠈⣦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡶⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⢀⣬⣭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣛⣛⣲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢂⠉⢈⡉⠙⢻⣦⣧⣶⣄⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣾⣿⠛⠙⠉⠉⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠉⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠉⠉⠉⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⢀⡀⣠⠀⢀⣤⠀⢠⡆⢀⡀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠻⣤⠎⠿⠉⠶⠋⠰⠾⠁⢸⠞⢧⡴⢿⣠⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠉⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⡌⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣾⠃⠀⢀⡴⠖⠲⢶⣄⠀⠀⢀⣤⠶⠶⢦⣄⠀⢸⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣠⠏⠀⠰⠏⠀⠀⠀⠀⢹⡄⢠⡎⠀⠀⠀⠀⠈⢷⠘⢿⡛⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢠⡆⠀⢠⣄⠀⢸⠇⠈⢧⠀⢠⡄⠀⠀⡾⠀⠀⠹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠙⠶⢤⣤⠖⠋⠀⣀⠈⠳⠾⠷⠶⠋⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⡠⠴⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠚⠁⠀⠀⠀⠀⠀⢀⡴⠛⠲⠤⡤⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠧⠖⢶⠟⣀⡙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠋⠀⠈⠻⣖⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠓⢪⢋⣀⠀⠀⣈⠉⠛⢛⣓⣒⢲⣖⣶⣿⣛⠉⠁⢀⡿⠤⣬⣻⣶⣟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠋⠁⠀⠀⢹⡔⠋⠉⠀⠈⢻⡟⠉⠀⠀⠙⣦⠞⠀⠀⠈⠙⢷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠓⠶⢤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠤⠒⠋⠙⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⣀⣤⡤⢤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣖⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⢠⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⢠⠶⣄⡀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠙⢷⣤⡟⠀⠀⠉⠓⠶⣤⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⣶⠶⠤⢤⣤⣤⣤⣠⣤⣤⣤⣤⣶⡶⠟⠙⢇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⠉⠓⠲⠖⠺⢧⣤⣤⡤⠟⠋⠁⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⠿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⠉⠉⠛⠻⣿⡆⠀⠀⠈⠉⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣤⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣰⣿⠏⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠛⣿⣧⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣤⣤⣤⣶⣶⣾⡿⠿⠛⠉⠀⠀⠀⠸⣿⣆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠋⠉⠉⣉⡉⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⢰⣶⣄⡀⠀⣠⣶⣶⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⠀⠀⠙⠻⠿⠿⠿⠋⠀⠀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠋⠉⠉⠉⠀⠲⣶⣶⣶⡶⠆⠀⠉⠉⠉⠙⠛⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣀
⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⡄
⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⣰⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⣿⣆
⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⣾⣿⠀⠀⠀⠀⠀⣴⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⠀⠀⠀⠀⠀⣿⣧
⢠⣿⡇⠀⠀⠀⠀⣼⣿⠃⠀⢠⣤⡀⠀⢾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⠀⢀⣤⡄⠀⠹⣿⣇⠀⠀⠀⠀⢸⣿
⣸⣿⠁⠀⠀⠀⢰⣿⡏⠀⠀⠈⠻⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⠟⠁⠀⠀⢹⣿⡆⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠈⠙⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠋⠁⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⢹
⣿⣇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣾
⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿
⢹⣿⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⢀⣿
⠘⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⢸⣿
⠀⢿⣷⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠁⠀⠀⣾⡟
⠀⠸⣿⣆⠀⠀⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠀⠀⣰⣿⠃
⠀⠀⢹⣿⡄⠀⢿⣿⣿⠛⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⣿⣿⡿⠀⢠⣿⠏
⠀⠀⢠⣿⠏⠀⠘⣿⣿⠀⠀⠈⠙⠻⠿⢿⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡿⠿⠛⠉⠀⠀⠀⣿⣿⠁⠀⢸⣿⡄
⠀⠀⢸⣿⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢰⣿⡇
⠀⠀⠈⢿⣷⣄⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣠⣾⡿
⠀⠀⠀⠀⠙⠿⠿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡄⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣾⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠋⠀⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⠤⢀⡀⣀⠀⠤⠤⠤⠤⢄⡀⠀⡀⠀⠠⢠⠀⠀⠀⢀⠊⠋⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⠆⠀⠙⠀⠀⠀⢀⠂⠀⠀⠈⠉⠀⠀⢀⠎⠀⠀⠀⠓⡄⢡⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠶⠐⢰⣷⠂⠀⠀⠀⠀⠀⠱⡅⠀⠀⠀⠀⢰⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⡉⠈⡳⢄⠐⠀⠀⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢼⡀⠀⠀⠈⠙⢟⠋⡱⠒⠒⠉⠀⠀⠀⠀⢀⡼⠉⠒⢤⣀⠈⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠈⠲⡏⠀⠀⠀⠀⠀⢀⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠶⠶⠶⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⢠⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢀⠔⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠸⡄⠀⢇
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠠⢣⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠱⣄⠘⠳⢄
⠀⠀⣀⠀⠀⣠⣸⠁⠀⠰⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⡀⠙
⡴⠙⠀⢦⠰⢠⠇⠀⠀⠄⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢹
⠙⠒⣍⠈⡇⡸⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡞⢠
⠀⠀⠈⡄⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡞⠀⡇
⠀⠀⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣀⣀⠳⡀⢧
⠀⠀⠀⠀⠱⡸⡀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⡀⠉⢧⠘⡄
⠀⠀⠀⠀⠀⠱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⣀⣑⠵⠃⢠⠇
⠀⠀⠀⠀⠀⠈⠚⠳⣄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢁⣠⣖⢚⠓⠒⠚⢁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣢⡔⠀⢀⣠⠔⠉⠁⠈⠉⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋⠁⠙⠒⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣒⣲⣶⣶⣶⣦⣴⣶⣶⣖⣒⣋⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠻⣿⠲⠶⠦⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠶⠶⢺⡿⠃
⠀⠀⠀⠙⣧⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣍⣉⣉⣥⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⣾⣆⢻⣿⣿⣿⣿⣿⣿⢃⣾⡆⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏
⣀⣠⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠆⣿⣿⣦⡙⠻⠿⠿⠛⣡⣾⣿⡇⢸⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣀
⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣍⡛⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣋⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⡇⠀⠀⢀⡀
⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⣿⣿⣿⠁⠘⣿⣿⣿⣟⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣿⣿⣿⡟⠀⢹⣿⣿⡇⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢻⣿⣿⣿⣿⣶⣦⣤⣬⣭⣭⣤⣤⣶⣾⣿⣿⣿⣿⠃⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢧⠀⠀⢈⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣋⠀⠀⠀⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⢸⣿⣷⣶⣤⣬⣭⣍⣉⣉⣉⣉⣩⣭⣥⣴⣶⣿⣿⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣉⣉⣛⡛⠛⠛⠃⠀⠀⠛⠛⠛⣛⣋⣉⣩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⣴⢶⡄⣴⠶⢠⡶⠆⢰⡔⢶⡶⠰⣄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣰⠇⣿⣹⡆⣿⣉⠀⣙⡷⢸⠇⢸⡇⠀⢹⠁
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢛⣋⣡⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢷⣦⣤⣀⣀⡠⢿⣿⣿⣿⣿⡿⠿⠿⢛⣋⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⠿⠻⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣿⣿⣭⣿⣷⣟⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⠀⢘⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣷⡀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⠟⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⣿⣿⣿⣯⣅⣀⠀⠀⡠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣻⡿⢯⣽⣿⣿⣯⣭⣭⣽⡷⠛⠉⣀⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣏⣙⣛⠺⣿⡿⣿⡿⢋⣁⣤⡤⠤⢤⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠳⣿⣿⠿⢶⣿⢹⠈⡿⠽⠶⣶⣤⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠶⣶⣶⣿⣿⣾⣴⣿⣄⠲⠤⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣟⢉⠅⠉⡻⣿⣿⣿⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⡀⠀⠀⢀⠸⢤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⣀⡱⠀⠰⣉⣠⠫⠤⢃⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠩⠍⠃⠀⠀⠀⡀⠀⠀⠀⠉⠁⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠈⠢⢄⣀⠀⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢂⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢣⠈⠢⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠤⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠹⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠢⡀⠀⠀⠀⠈⠑⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢰⠀⠀⢧⠐⠁⠀⠀⠀⠀⠀⢈⣉⣈⣁⠀⠰⡀⡀⠴⠀⡃⠁⢀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠑⢺⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣀⠠⣁⠀⣀⡅⡁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠊⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠴⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠐⠢⠤⠀⣀⣀⡀⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠤⣤⣀⢀⣀⣛⣗⣒⣒⠀⠀⢀⣨
⠀⠀⠀⠀⠀⣀⢀⡀⣀⡀⠤⠤⠔⠋⢇⠀⠀⠀⢀⠉⠫⠀⠀⣀⡤⣤⣄⡀
⠀⠀⠀⠀⡞⢿⠛⠾⡃⢄⠀⠀⠀⠀⠀⣁⠀⠀⡆⠀⢀⠀⢀⠤⡪⠗⠃⡼⢢
⠀⠀⠀⠀⠡⠔⠒⠀⠈⠀⠉⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠈⠀⠑⠶⠂⠬⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠸⣆⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣆⡿⠀⠀⠀⠀⠀⠀⠀⢻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣟⣼⡷⠲⣤⡀⠀⠋⢉⠉⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⡿⠟⠋⢹⡇⠀⠈⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠮⣅⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠁⠀⠀⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠀⠹⠻⣿⣿⣿⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣀
⠀⠀⠀⠀⠀⢀⡴⠚⠟⠋⠁⠀⠀⢠⣽⠿⠿⡷⠿⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢠⡴⠋⠀⢀⣤⣴⠮⡍⠳⣆⣠⡶⠆⠀⠀⠀⠘⠀⠂⢀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣺⡇⢀⣔⣻⣛⣿⣿⣋⣀⡋⠀⠀⠠⠄⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⣜⣤⠀⠀⠀⠀⠀⡰⠋⠀⢈⣼⣯⣽⢭⠭⠉⠙⠃⠀⠀⠢⠀⣠⠴⣄⡀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⣰⠋⡅⠀⠀⠀⠀⠠⢃⠀⣀⣼⣷⣶⠶⠒⠒⠚⠁⠀⠀⠀⡗⠀⢻⡆⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠈⢢
⠀⢰⠇⠀⠀⠀⠀⣠⠔⢠⣿⣆⠛⠛⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⡙⠛⠶⠄⠀⢠⡋⠛⠐⠒⠀⠤⢄⠀⠀⢃
⠀⣞⠀⠀⡠⠔⢋⠡⣶⣮⡟⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠓⠒⠦⠀⢠⢎
⠀⣏⡜⠁⠀⠀⣸⣾⡿⠟⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠶⠶⠴⣶⣤⣤⣥⣄⡀
⠀⡟⠀⠀⠀⣤⣤⡹⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⣀
⠀⣇⣠⣼⣿⣯⡟⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⡇⠁⠀⠠⠴⠊
⠀⠛⠯⣛⠻⠟⢋⡠⢾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠁⠈⠁⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡧⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠁⠀⠀⠀⣿⣿⣿⡇
⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇
⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷
⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⠟⠙⣿⣿⣿⡆
⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⢻⣿⣿⣿⡄
⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡀
⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣧
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡀
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄
⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣧
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢸⣿⣿
⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠘⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢀⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠃
⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠟⠁
⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠋
⠀⢸⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠋
⠀⠸⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⡿⠟⠁
⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⣿⣿⣿⣧⠀⠈⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿
⠀⠀⢹⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⢸⣿⣿⣿
⠀⠀⠘⠻⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⡋⠉⠻⠿⠟⠉⠙⠿⠿⠋⠻⠿⠿⢿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⢿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠛⣊⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣟⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠠⣼⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡛⠫⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⡄⠲⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠠⡐⠂⢀⠀⠀⠀⠀⠀⢙⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣙⣻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣧⣬⣠⣦⣀⡱⡀⠃⠀⢃⣤⣤⣤⣬⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⡓⢤⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢡⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⣀⠴⣊⣭⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣭⡂⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣶⣼⣿⣿⣿⣿⣿⣿⠿⢟⣪⣵⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣶⣭⣭⣟⣛⣻⣿⣟⠛⠛⣛⣛⣛⣛⣻⣭⣭⣵⣶⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠉⠙⣟⠛⠋⠙⠛⠛⠛⠛⡙⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⢿⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⣸⣡⣾⣿⣿⣿⣿⣶⣦⣄⠀⣠⣴⣾⣿⣿⣿⣿⣶⣽⣄⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⢃⣼⣿⣿⣿⣿⣿⣻⣿⣧⣤⣾⣿⣹⣿⣿⣿⣟⣿⣄⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠁⠈⠙⠛⠉⠉⠉⠩⢽⣿⣿⣿⣿⣿⠍⠉⠉⠉⠉⠛⠁⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢴⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⢷⣾⣿⣷⡦⠀⠀⠉⠉⠉⠉⠁⠀⠠⢾⣿⣿⣿⡿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣃⠀⠀⠀⠺⠷⢢⠀⠀⠀⣘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠉⠙⠉⠙⠓⠐⠘⠙⠉⠉⢉⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⣓⠦⠤⠄⣄⣀⣠⠤⠤⠦⣻⠿⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠳⠬⣉⠒⠦⠤⠤⠤⠴⠚⣋⠵⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⣀⠀⠀⠀⠀⠀⢀⠤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⠀⠀⠰⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢳⠀⡀⠀⠀⠀⠤⠤⠤⠤⢀⠀⢷⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠘⠋⠀⠀⠐⡋⢀⣀⣀⠀⠀⠙⡏⣰⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣸⠀⠀⠀⢠⠁⡀⢣⠋⡉⠙⠂⠀⠀⣿⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢠⡏⠀⠀⠀⢊⠘⢿⠿⡘⠿⠀⠀⠀⠀⢯⠳⣄⡀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⣠⡿⠀⠀⠀⠀⡈⠓⠉⠀⣙⠒⠊⠀⠀⠀⠀⢳⡀⠉⠙⠓⠲⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⡰⠁⢀⠔⠊⡥⠂⠈⠉⠉⠉⠛⠶⢄⡀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡔⠁⠀⠘⣄⡀⠀⠀⠀⢀⣀⡶⠀⠉⠢⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠎⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠈⠲⡀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⢢⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠉⠈⣧⠀⠀⠱⣼⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡒⠉⠁⢀⣠⡴⠾⣿⠀⠀⠀⢹⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠻⣿⣶⣿⣿⣏⡤⠖⡾⠀⠀⠀⢸⡇⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠘⢦⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠡⠔⠚⠁⠀⠀⠀⣼⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠙⠲⢤⣌⡓⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡤⠞⠁⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠒⠲⠶⠤⠤⠤⠔⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢀⡴⠖⢴⡏⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⣴⠋⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠙⠓⢦⡀
⠀⣸⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠹⡄
⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢻
⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢿⡀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠈⢳⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠙⣄⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⣀⣀⡈⠓⣄⡀⠘⢆⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢠⠞
⠀⡏⢀⡀⠀⠀⠈⠉⠀⠀⠑⠠⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⠉⠉⠉⢦⡀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠉⠓⢶⠋
⠀⣇⣿⠀⠀⡏⠀⠀⢀⣀⣀⣀⣀⣀⣠⡴⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⣘⡷⠋⠀⢀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠙⠦⠤⠿⣤⣤⡼⠁⠈⠉⠉⠉⠀⠀⠘⢧⣀⠀⣆⠀⢠⠀⠀⣶⢀⡏⠈⢧⡀⣠⠏⠀⠀⡴⠀⠀⣼⠋⠋⠉⠉⢧⣀⡇⠀⢀⠂⠀⣆⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠳⠤⠷⠴⠋⠀⠀⠀⠀⠙⢿⣀⣀⣼⣥⣴⠞⠁⠀⠀⠀⠀⠀⠈⠙⠲⠾⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠈⠀⠀⠉⠐⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠆⠀⠶⠦⠀⠐⠂⠤⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡧⣀⡀⠀⠀⢀⣀⠀⣴⣧⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠘⠁⠀⠀⠘⠉⠀⠸⠋⣹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠐⠤⠒⠀⠀⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡅⠀⠐⠀⠒⠀⠀⠀⠀⣹⡇⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠺⡟⠁⢼⡭⣧⣄⡒⠒⠀⠀⣀⣤⠚⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠐⠌⡉⠛⠋⠀⢀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⣿⣦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡄⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⢸⡿⠀⢀⣼⡿⠟⠛⠃⠀⠀⠀⠀⠀⠠⠤⠤⠤⣤⡀⠂⠀⠀⠀⠀⠀⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣻⣿⡀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⢸⡟⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣨⡷⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢠⣾⡍⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢘⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣶⣄⠀⢀⡀
⠀⠀⠀⠀⢸⡖⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⠟⠈⠉⠉⠀⢀⡀
⠀⠀⠀⠀⢸⣆⡸⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠯⠀⠀⠒⠠⢄⣀⠀⠉
⠀⠀⠀⠀⠀⢿⣇⠀⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⠹⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠁⠋⠐⢲⠆⢠⡀⠀⠀⠀⣀⣀⡀⠀⠀⣄⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⡀⠠⠊
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡧
⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⣨⣿⣶⠲⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠛⠳⠀⠀⠀⢀⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣼⣿⠄⠀⠀⠀⢀⠀⢹⣿⣿⠏
⠀⠀⠀⠀⣀⣾⣿⣄⡀⠀⠀⢀⣼⣶⣾⣿⡏
⠀⣰⣾⣿⣿⡿⠏⠀⣈⣈⡙⣿⣿⣿⣿⣿⠿⡀
⣾⣿⡟⠋⠁⠀⠀⠀⡈⢫⣿⣿⣿⡏⠀⠀⠈⠇
⠘⠧⣀⣠⣄⡀⠤⢖⣽⣿⡿⠟⠋⠀⠀⠀⡠⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⣀⡀⢀⣀⠠