Equine Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣆⣬⣶⠶⠶⠶⠶⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⣄⠀⠀⠀⣠⣴⡿⠛⢿⣿⣿⣽⡇⠀⠀⠠⢀⣀⢄⡀⢉⣙⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣷⣿⣿⡿⠋⢰⠀⢸⡏⣿⣿⡇⠀⠀⠀⠲⢿⣿⣿⣷⣿⣿⣦⣝⠻⣶⣄
⠀⠀⠀⠀⢀⡾⣫⠟⠋⣙⣷⣿⣿⣿⣿⠃⠀⢟⢰⠀⠻⣿⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⡙⢿⣯⠛⣷⣜⣿⣷⡀
⠀⠀⠀⠀⠈⠐⠁⣠⡾⠋⠛⠋⣩⣾⣿⡆⠀⠈⢸⣤⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠈⢻⣿⡌⠻⣧⠈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡴⠛⠁⣿⣇⢳⡸⡀⠀⠙⢷⣧⣶⡀⣈⠙⢷⣌⢦⠱⡄⠀⢿⣷⠀⠹⡆⠀⠘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏⠀⠀⠸⣧⠻⣆⠙⢿⣦⡀⠀⠙⠻⣷⡘⢷⣄⠙⢿⣇⢻⡄⠈⣿⡇⠀⠁⠀⠀⠸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡍⠀⠀⠀⠀⢻⠓⢌⡳⣄⡉⢛⡲⢄⣀⠈⠙⠻⣿⣷⡌⣿⡌⣷⠀⢹⣿⠀⣦⠀⢰⡄⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢿⣷⣶⣭⡲⢄⠀⠙⢿⣼⣧⣿⡇⠸⣿⡀⣿⡆⠈⣷⢸⣧⣿⣏⢳
⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣽⢷⡈⠙⢶⡝⣄⠈⢿⣿⣿⡇⠀⣿⡇⣿⣿⠀⣿⡆⣿⣿⣿⡌⡆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠟⠈⣷⡀⠀⠹⣾⡆⠘⣿⣿⡇⠀⣿⣇⣿⣿⠀⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⠀⠀⠃⠀⠀⠀⢠⡾⠀⠀⢸⡇⠀⠀⢿⣷⠀⣿⣿⠇⢠⡿⣿⣿⣿⣸⠏⣿⡇⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⢸⡿⢰⣿⡿⠀⣼⠇⣿⠃⣿⡟⠀⣿⢱⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⣾⣧⣿⣿⠃⢰⡟⠀⠉⠀⣿⢁⡄⢀⣿⣿⣿⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢻⠀⠀⣏⠀⠀⢠⠀⠀⡀⣠⣾⡿⢀⡄⢀⣿⣿⣿⠏⢠⣿⠁⠀⠀⠀⠀⣾⢁⣾⣿⣿⣿⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⠀⠀⣿⠀⠀⣸⠀⢠⡿⠋⣼⠃⣼⠀⠘⣱⣿⡏⢠⣿⠇⣰⠀⣸⠇⣼⣧⡞⣹⣿⣿⡇⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⣿⠀⡞⠁⢰⡟⢀⡟⠀⣰⣿⡟⢠⣿⠏⣰⠇⣰⣿⣼⡿⢋⣼⢿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⠀⠀⣿⡇⠀⢹⣾⠀⠀⣿⠇⢸⡇⢠⣿⡿⢠⣿⠏⣼⢏⣼⣿⡿⠋⣠⡿⢫⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠀⠀⠈⠀⠀⠈⣷⠀⢰⣿⠀⠘⣧⢸⣿⢱⣿⢏⣾⣯⣾⠟⠋⣠⠾⠋⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⠀⠀⠀⠀⢰⣶⡆⢻⠀⢸⣿⠀⣀⣿⣿⣿⣿⣿⣿⠿⠋⠁⠰⠛⠁⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⢷⠀⠀⠀⢎⠀⠁⡇⠀⠘⣿⠀⢻⣿⣿⣿⠿⠋⠁⠀⠀⠀⢀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⢀⣤⡾⠃⠀⠀⢻⡆⣈⣿⣿⠇⠀⣠⠀⠀⢀⡴⢋⡼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⣾⣷⢰⣿⣿⣴⣟⡵⣣⡾⢋⣴⠟⣵⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣧⣿⣿⣿⣿⠞⢁⣴⠟⣡⡞⠁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⠿⣿⣿⣿⣅⣴⠟⢁⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⣀⡈⢻⣿⣿⢀⡴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⠏⣹⣿⡟⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⣿⣰⣿⣿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⣿⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⢻⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⠀⢰⢀⣸⣤⣄⡀
⠀⠀⠀⠀⠀⣠⡟⠛⢿⠿⢿⣿⣿⣷⣶⣊⣀
⠀⠀⠀⠀⡸⠋⠛⠂⠀⠀⠀⠈⠙⢿⣿⣿⣶⠆
⠀⠀⢠⠎⠀⠀⢠⠀⢀⡀⠀⠀⠀⠀⠙⣿⣿⡛
⠀⠀⢷⡃⠠⠆⠊⠁⠘⣷⠀⠀⠀⠀⠀⠈⠿⣿⣧⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠠⠤⠤⣤⡤⠔⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡈⢆⠀⠀⠀⠀⠀⠀⢀⠀⠹⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣻⣯⡀⠀⣠⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢈⠀⠀⠉⠛⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠿⠿⢿⣿⡿⠿⢦⣤⣴⡶⠶⠾⠛⠛⠋⠉⠉⠀⠀⠙⠻⢿⣦⣀⠀⠀⠘⣆⠀⠀⠀⠀⠈⠙⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣦⠀⢸⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⠀⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⣷⣌⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿
⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠼⣄⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣷⣮⡁⣒⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣿⣿⣿⣿⣿⣿⡇⣿⣷⣌⢿⡝⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠛⠛⠻⠿⣿⣶⣸⣿⣿⡝⣿⣏⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⣿⣿⣿⣿⢷⣜⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢟⣵⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣢
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣷⣶⣶⣶⣶⢦⣄⡀⠀⠀⠀⣠⣾⣷⣿⣿⣿⣿⣿⡿⢻⢾⣮⣭⠥⣟⣹⠇
⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠈⠉⠙⠻⣿⣿⣶⣄⣼⡿⣿⡿⠿⢿⣿⣿⣿⣷⣿⣿⡿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡙⠻⣿⠿⢻⣾⣿⣿⣿⣶⣦⣸⣿⣿⠟⢁⣤⣯⣭⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢿⣿⣿⣿⡷⣶⣤⣬⣭⣿⣧⣼⣿⣿⣿⣿⣯⣤⡎⣡⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣽⣛⣿⣥⣻⣽⣾⣿⣿⢟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣛⣿⣿⣿⣿⣿⣿⣿⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⣿⢏⣴⣿⣿⣿⣿⣿⣿⣐⡺⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡁⢠⣾⣿⣿⣿⣿⣿⣿⡇⠈⣻⣝⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⢸⣿⣿⣿⣿⣿⣿⠟⠀⠀⠹⣽⣦⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠈⠻⢿⣿⣿⡿⠋⠀⠀⠀⠐⠹⣿⣰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠘⣿⣿⡿⠁⠀⠀⠀⠀⢰⣧⣿⣿⣿⡿⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⡿⢹⡇⠀⠀⠀⠀⠀⣸⣿⣮⢿⣮⣟⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⡆⠀⠀⠀⣠⣶⣿⣿⣷⣍⣿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠀⠀⢰⡿⠋⠁⠐⠒⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣅⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⢁⣀⠀⢈⡈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⣸⡧⠄⠀⠸⠉⠓⠀⠈⠦⣌⡙⢦⣄
⠀⠀⠀⠀⠀⣰⡏⠀⠑⠀⢀⠀⠀⠀⠀⣄⣠⣄⠉⠳⣌⡃
⠀⠀⠀⠀⠀⣯⠀⢀⢀⡴⠊⠀⠀⣶⠀⠈⣿⣿⣧⠀⠈⠎⢢
⠀⠀⠀⠀⠀⡜⠀⣱⡏⠐⠀⠀⠀⠉⠀⠀⡿⢻⣿⡆⠀⠀⠸⡀
⠀⠀⠀⠀⡜⢀⡀⠹⡀⠀⠀⠀⠀⠐⢀⡾⣇⠘⣿⠃⠀⠀⠀⢹
⠀⠀⠀⠀⢧⢸⡇⠀⠓⠀⠐⠀⡀⠀⣾⠁⢿⠀⠋⠀⠀⠀⢀⠼⢄⡀
⠀⠀⠀⠀⠀⠙⠷⢣⠠⠀⠐⣬⠋⢀⣿⡀⢸⠀⠀⠀⣴⡞⠁⠀⠀⠉⠑⠒⠀⠤⠤⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⣀⣀⡼⠀⠀⠀⠀⢸⣿⣿⠇⠊⣠⠂⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⣼⣩⢤⡤⠄⠀⠀⠀⠀⣿⣿⠏⢠⠎⠀⠀⠀⠀⠉⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⣀⣿⡴⠋⢀⢀⠀⠤⠀⠀⣹⡏⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡀⠀⠀⠀⠀⠘⢻⣇
⠀⠀⢠⣿⢾⢣⡖⣁⠤⠤⠤⠊⢀⣿⡰⣄⠀⠐⣄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠘⢿⡟⠀⠰⣁⠀⠠⠀⠂⠉⣻⣿⣿⣿⣿⣿⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⢸⣿⢦⣄⣀
⠀⠀⠀⠀⢻⡀⣾⠃⢀⠠⠒⠊⠘⠛⠛⠛⠁⡸⠋⠀⠀⢀⣠⣶⣄⣀⣤⣤⣤⣤⣄⣀⣤⢴⣿⣶⣶⣄⠀⠀⠀⠀⠸⡿⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⢀⡷⠇⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣀⡠⠐⠀⠉⠛⠛⠛⠛⠛⠋⠉⠙⣿⣷⣾⣿⣿⣿⠏⢸⠑⣄⠀⠀⠀⠇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠙⠛⠛⠀⢠⠀⠀⠀⠀⠀⠀⠀⣸⡆⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠁⠀⢸⠀⠈⢷⣶⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⡌⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠙⣿⡀⠀⠘⣆⠀⠈⢿⠀⠀⠀⠀⠀⠀⢰⡀⣏
⠀⠀⠀⠀⠀⠀⠀⠙⣏⣼⣷⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠙⣧⠀⠀⠘⣆⠀⢸⡇⠀⢁⠀⠀⠀⠸⡗⢽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠀⠀⣄⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠈⢧⡈⢧⡀⠀⠀⠀⢀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⢉⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠈⢧⢸⠇⡆⠀⠀⢿⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢸⢠⡀⢁⠀⠀⣿⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⢠⠇⠀⠀⠀⠀⠀⠀⡼⠸⡇⢀⡄⢠⡏⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⡜⠛⠀⠀⢀⣀⣠⠴⠚⠁⠀⣿⢋⣤⠟⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠷⠾⠃⣠⣶⠭⠥⠤⢤⣤⣤⠤⢼⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢡⣦⣴⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠻⠻⣇⠡⠞⠛⠛⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢷⣦⣄⡀⠛⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⣿⣿⡿⠁⠀⡄⠻⠂⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣥⣿⣿⣿⣿⠃⠀⠀⡇⠈⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣿⣿⣿⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢛⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣿⣄⣴⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⣄⣠⣀⢸⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⣷⣤⣀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠙⣿⡿⣿⣿⡿⢦⡀
⠀⠀⠀⠀⠀⠘⠛⠛⠛⣋⣽⠿⠟⣻⠿⠟⠁⠀⠀⠀⠀⠀⢶⣆⠀⠀⠘⣿⡟⠻⣄⡠⣀⠀⠀⣿⣆⡀⠀⠀⠀⠀⠀⠊⠀⠀⠙⠻⣦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⢹⠃⠀⠀⠀⠉⠙⣶⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢸⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢳⣄⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢸⡇⠀⠀⠀⣠⣿⣿⣿⡿⠋⠀⠀⠀⠈⠳⣄⠘⢧⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣇⠀⠀⠀⠹⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠈⣷⣬⣿⡄⠀⠀⠀⠀⠀⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠛⠇⠀⠀⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⡄⠐⣷⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠘⣿⣧⣿⣷⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠀⠀⠀⠘⢿⣿⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢐⡀⠀⠀⠀⠀⠈⢳⡀⠛⣦⠀⠀⠀⠷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠘⠿⣿⣿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡄⢀⣿⣇⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⣿⣠⣤⣴⣾⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡟⠻⢿⣇⣽⠉⣛⠿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠛⢳⡶⣬⣿⡧⠈⠃⠈⣿⣿⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣯⣽⣿⠃⣠⡄⢀⡏⢹⣧⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⢃⣼⣿⡿⢀⠀⠀⠛⣤⠀⠀⠀⢸⡿⠙⢿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢀⡌⠉⡡⣰⠆⠀⠀⠀⣿⣦⠀⠀⠈⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠛⡀⢀⡴⠋⠀⠀⠀⢠⡟⢿⣄⠀⠀⠀⠀⠀⣹⣿⣿⣤⣄
⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⣛⠶⠁⢨⣧⡀⠀⢀⣠⡟⠁⠀⠙⢷⡀⠀⠀⢀⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢰⣿⢋⡁⣶⠀⢉⡀⣠⣴⡿⠿⠟⠛⣿⣿⣦⠀⠀⠀⠑⠀⠀⠸⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢼⣿⠘⠋⠉⣤⣼⠟⠛⠉⠀⠀⠀⠀⣿⣿⡇⠀⠀⣄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠘⢿⣤⣤⣃⡼⠁⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⣍⠙⠋⠉⠉⠉⠉⠙⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁⠀⠀⣿⠀⠀⠀⠀⠀⣸⡟⠀⠳⣄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠴⢻⡇⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠁⣿⡀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠉⠛⠀⢻⡇⠀⢠⣿⠀⠀⠀⢀⡾⠁⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⣿⣦⠀⠀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠸⣧⠀⢸⡇⠀⠀⠀⡎⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⣸⣿⠀⠀⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠚⠀⡆⠻⠀⠟⠁⡆⠀⠀⠻⣶⡆⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⣿⠉⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⣰⠇⢠⡶⡀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠐⠃⠀⠁⠀⠀⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⣸⡟⠀⠀⣰⣿⡿⠀⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠁⢠⣄⠀⠀⣀⠀⠀⠀⠀⢀⣴⠟⠉⠓⠀⠀⣠⣿⣅⣤⣾⣿⡿⠁⠀⢀⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⣷⣾⣿⣷⣦⣤⣤⣾⠃⠀⢀⡴⠀⢀⣿⣿⣿⣿⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠶⢻⣿⠏⢹⣿⣿⣿⣿⣿⣿⠀⠔⢋⠀⣠⣾⠏⠉⠈⣿⣿⡏⠀⢀⣴⣿⣿⣿⣿⣿⡟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⣰⡟⠁⠀⣸⣿⣿⣿⣿⡿⠃⠀⣠⣿⠟⠋⠁⠀⠀⠀⣿⣿⠁⢀⣼⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⣼⡟⠁⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⢺⡿⠁⠀⠀⠀⠀⠀⢰⣿⠇⣴⡟⣿⣿⣿⣿⣿⣿⣿⣯⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠘⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⡁⠀⠀⣾⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⠟⠁⠈⠫⠙⢿⣿⣿⣿⣿⣷⡈⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⡿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡀⢰⣿⠃⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠿⣿⠳⡙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⣿⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⢿⠈⣿⠀⠀⠀⠀⠀⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠈⠫⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⢸⡇⣽⡇⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠈⢻⡆⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡎⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠉⢿⣦⣾⣿⣶⢦⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣷⡀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⢻⣿⣿⡟⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠻⣄⣀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠈⢿⣿⣷⣤⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠸⣿⣷⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢋⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⠿⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠛⠿⠟⢿⠛⠿⠟⠘⠿⢻⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻⣛⣛⣛⣙⣻⣿⣿⣟⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⠀⢀⡀⢀⡀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠙⠛⢁⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⠛⠀⠀⠀⠀⢿⠋⠁⠀⠀⠀⠀⠀⠀⠩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠴⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠘⠃⠀⠀⠀⠀⠀⢠⠊⠀⠀⢤⠶⠛⠁⢸⠀⣴⡘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡰⠁⠀⠀⠀⢸⠀⠃⠹⡄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⡎⣆⠀⠀⡞⠀⠀⡴⠀⣀⡀⠀⠀⠀⡟⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢷⠈⠓⠐⠀⠀⠘⠁⢾⢯⡝⡷⠀⠀⠁⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣜⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣤⣀⠀⢀⣀⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠙⠻⠿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⣄⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡄⠀⠀⠀⣾⠿⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠰⣿⣄⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣧⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣦⣭⣿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢻⣮⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⡿⣶⣦⣄⢀⡀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢟⣫⣽⠖⠂⠀⠀⠀⠉⣿⣷⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣫⣶⣿⣿⡇⠀⠀⠀⠀⠀⠠⠿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⣿⡿⢫⣾⣿⡿⠿⠿⠏⠀⠀⣠⠔⠀⠀⢴⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⠿⠛⣰⠟⠋⠀⠀⠀⠀⠀⣠⡾⢋⡞⠀⠀⢲⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⣾⡿⢁⣾⣧⠀⠀⠸⡇⠹⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⣾⣿⣿⡇⠸⣄⠃⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢠⡟⠀⠀⠠⠴⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠈⢢⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠘⠻⢿⣿⣿⣿⣿⠖⢀⡀⠀⠃⢰⣿
⠀⠀⠀⠀⠀⠀⣠⣶⣿⢡⠀⠀⠀⠀⠀⠀⠀⢿⣧⠰⡆⠀⠀⠀⣀⡀⣴⣿⡿⣿⣿⡿⢀⣿⡿⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣴⣿⢻⣿⢸⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣿⣿⣷⢹⣿⠧⢟⣻⣿⠟⠉
⠀⠀⠀⠀⢰⣿⡏⢸⣿⢸⡇⠀⣠⡀⢀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠋⠀⣿⣾⣿⡿⠁
⠀⠀⠀⠀⢰⣿⡇⢸⠇⠀⣿⣼⣿⣿⣿⣿⣷⣶⣿⣿⡟⠛⠉⣁⡀⢀⣰⡿⠋⠉⠁
⠀⠀⠀⠀⣿⣿⡇⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣧⡼⠛⠁
⠀⠀⠀⣼⣿⣿⣷⠘⠀⢠⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉
⠀⠀⣰⣿⣿⣿⠏⠀⢀⣾⡿⠋⠀⠈⢻⣿⣿⣿⣿⠏
⠀⢜⣿⣿⠟⣡⠀⢀⡾⠋⠀⠀⠀⢀⡄⢿⣿⡿⠃
⢠⡿⠋⠀⠀⠁⢠⣿⠁⠀⠀⠀⠀⠟⢀⣼⠟⠁
⡟⠀⠀⠀⠀⠏⣾⠃⠀⠀⠀⠀⢠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⡄⠀⣿⠃⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠘⣷⡾⠏⠀⠀⠀⢰⣇⢸⡟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⡾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⣿⣶⣶⣶⣶⣅⠀⡞⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⣿⣿⠿⠻⡿⣿⣿⣿⣿⣿⣧⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⡈⣱⡯⠀⠀⠙⢮⣟⣻⠿⢿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⣡⡞⠕⠋⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠚⢾⡿⠿⢿⣯⣍⡑
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠟⢡⠄⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⣿⠀⠳⣄⠠⠤⢤⣬⡄⠀⠀⠀⠀⠀⠠⣼⣷
⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⢀⡄⠃⠀⠀⠈⠈⣿⣷⡈⢷⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⢸⡇⠀⠀⠀⠀⣿⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠘⣦⡀⠀⠘⣇
⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⢳⣸⣷⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠘⣷⠀⠀⢹⡀
⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢧
⠀⠀⢰⣿⢹⣿⠁⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⡆
⠀⠀⣾⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⢹⡀
⠀⢀⡿⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⢷⡄⠀⠀⠀⠀⠀⠀⠀⢱⡈⣿⣷⡄
⠀⢸⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⡇⠀⠈⢢⡀⠀⠀⠀⢀⣀⣈⠳⣿⣿⣿⡀
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⡇⠀⠀⠀⠱⡀⣄⠰⢿⣿⣧⡑⢿⣿⡟⠃
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢰⣿⡇⠀⠀⠀⠀⠠⡈⢷⣌⢉⣋⣵⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣸⣿⡇⠀⠀⠀⠀⠀⠙⠦⠿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡿⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢸⠇⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡅⡟⠀⠀⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣼⠁⡎⠀⡿⠛⠟⢓⣒⣒⠯⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠁⢰⡇⢸⠀⢰⡇⡀⠀⠀⠀⠉⢙⡲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⠘⡇⠁⠀⠋⠈⠉⢢⡀⠀⢄⢴⣽⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡠⠋⠀⠀⠀⠀⣀⣀⡸⢃⡴⢸⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠁⢀⠀⣾⠃⠀⠀⠀⠀⣊⡭⠭⣉⠉⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢈⢀⡏⣸⠀⠀⠀⠈⣷⣿⣍⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⡄⡇⣇⠀⠀⠀⠀⠘⠻⠟⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣷⠁⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣠⡇⠀⠀⠀⠙⠂⠀⣄⡀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢀⡔⠉⠀⠀⢀⣀⠱⣄⡤⠖⠒⠲⠤⣤⣀⣠⠽⣡⣢⠟⠀⠀⠀⠀⠀⠀⢸⡇⠙⡖⡀⠀⠀⡴⡆⠘⡆
⠀⠀⠀⠀⡞⠀⠀⠀⡰⠁⢈⡟⠁⠀⠀⠀⠀⠀⠀⠈⠁⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢳⡁⠀⠀⠙⢀⣠⠇
⠀⠀⠀⢸⠃⠀⠀⠃⡇⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠙⠲⠴⠖⠉
⠀⠀⠀⠘⡆⠀⠀⠀⢇⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠓⠦⣀
⠀⠀⠀⠀⢣⠀⠀⠀⢸⡀⢹⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠂⢀⠀⣸⢁⠞⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢸⠂⣠⣀⠤⡄⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⢠⡼⡰⢳⠏⠀⠀⠀⠀⢀⣤⠖⠛⠦⢤⣀⡀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡟⢁⠤⠐⠁⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠿⠋⠀⢸⠀⠀⠀⠀⡞⠁⠹⡀⠀⠀⠘⡇⠉⠓⠒⠶⠖⢻⡀⠀⠀⠀⢳⠘⣆⠀⠈⠢⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡇⠀⠀⠙⡄⠀⠀⠙⡄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⢧⡘⣆⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠄⠤⢼⠀⠀⠀⡗⠐⠒⠀⢻⡀⠀⠀⠀⠀⢹⡄⠠⠤⠤⠈⢧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣨⡇⠀⠀⠷⠤⠤⠤⠼⠇⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣴⣆⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠠⣄⢸⣿⣿⣿⣷⣶⣶⡞⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢫⣾⣿⡿⠿⠛⠛⢿⣷⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠋⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⣿⢇⣴⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠙⢿⣐⡀⠀⠀⠈⣿⣿⣿⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠻⣾⣿⣿⢃⣿⣿⣿⣿⣿⢦⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠉⠙⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⣿⣿⣿⡀⠀⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⠀⢻⣤⣀⣀⣀⣠⠤⠀⢀⣠⣌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣿⣿⡿⢋⡀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⡄
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⢻⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⢠⣿⣄⠀⠀⠀⠀⠀⠀⠀⣸⣆⠀⠀⠀⣼⣿⣿⣿⡇⠘⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣷⣼⣿⣿⣶⡄⠀⠀⠀⠀⠀⢹⣿⣷⣶⣾⣿⣿⣿⣿⡇⠀⠀⠑⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠛⠉⠛⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠻⢝⠻⢿⡿⢿⣿⡿⠿⢿⣿⢿⣟⠒
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧⡀⠀⠀⢀⣿⠀⠀⠀⠘⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠀⢸⣿⣿⣿⡇⠀⠀⣼⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠀⢻⣿⣿⣇⢰⡄⣿⠇⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⠙⢿⣿⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣶⣤⡀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣄⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⢹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⣶⢤⡤⠾⢿⣿⣟⣓⣶⣤⣠⣿⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣬⡶⢶⣿⣷⣤⣰⣾⡿⠟⠛⠛⠛⠛⢿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⣤⣀⣽⣟⣷⣤⣿⡿⠟⡁⠄⠀⣀⣠⣤⠞⢩⠂⢠⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⠬⣿⣷⣯⣵⡿⣻⣿⠏⠐⠈⢀⣴⠾⣫⣿⡏⠀⠀⡿⠛⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢛⣛⠿⣿⣾⠿⠟⠁⠀⠀⣴⡟⢁⣼⣿⡿⢳⣄⢀⣷⣕⠄⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢈⣛⡿⠋⠀⠀⠀⠀⢀⣾⠋⢠⡿⣹⡟⠀⠀⠉⠙⢷⡌⢂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣠⣀⣀⡤⠒⠭⣩⡄⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠺⣶⠿⠃⢠⣿⢃⣿⠁⠀⠀⠀⠀⠀⠙⣆⢰⣶⡆
⠀⠀⠀⠀⠀⠐⢶⣀⣓⣤⣠⣴⠋⣽⣿⡿⠋⠀⠠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣆⠀⠀⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠈⠉⠻⣯⣿⣭⡿⢉⣿⣽⣿⠁⠀⠰⣻⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢦⣄⠀⢰⡄⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠠⠶⣤⣀⠴⣋⣥⣴⣿⣿⣿⠀⠀⣧⢸⣆⢳⣄⠀⠀⠀⠀⠀⠀⢴⣆⣦⡀⠀⠉⠓⠘⣧⢸⣿⢿⣏⣿⡇
⠀⠀⠀⠀⠀⠑⠾⣿⣤⣵⣾⣿⢿⣿⣿⣄⠀⠘⢆⢻⣦⣿⣿⣶⣤⣀⣠⣠⣾⣿⣿⣿⡇⠀⢠⡾⠧⣾⣿⣾⣿⣿⣿⠦⣄
⠀⠀⠀⠀⠀⢠⣤⣶⣤⢞⣋⣑⣿⣿⠟⠙⢷⣦⣤⣀⣉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡴⢏⠀⠠⢄⡙⠻⣿⣿⣿⣷⣶⣯⣖⡶⣄
⠀⠀⠀⠀⠀⠀⠀⣨⠽⣿⠟⠉⠁⠀⠀⠀⠀⠉⠛⢯⣉⠁⢈⠻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠘⠚⠓⠦⠤⠽⠶⢮⣟⣻⢦⡈⠉⠙⢿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣶⣽⣮⡳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣧⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣣⣜⡿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠠⣌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⢙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣚⣳⠋⠀⠀⠀⠙⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣴⣤⣤⣤⢖⣿⢟⠀⠀⠀⠀⡠⢺⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣺⠟⣡⡾⢋⠝⠋⢰⡿⢉⣶⣶⣤⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢣⡾⠋⣐⣵⡏⠀⠘⠁⢸⣿⢿⣿⣿⣧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⢧⣿⠃⣠⣾⡟⠁⠀⠀⣠⢺⢹⢸⣿⡿⢿⡝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⡿⣼⢹⢠⣿⣿⠁⠰⠀⢋⣿⣾⡆⠀⢿⣷⠨⡳⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⡇⣷⣿⡆⢩⠈⠈⠀⠀⠀⢉⣈⣿⣷⠀⣮⢻⠦⡹⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⣾⡇⣧⢻⣿⣿⡄⠀⢀⠀⠀⠀⣽⡿⠋⠉⠃⢹⡀⠁⠘⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠙⡌⣧⢿⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⢃⠀⠸⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣧⠀⢱⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢹⢿⠀⢸⣧⣿⢷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢼⡿⢸⡆⢸⣿⡏⢸⣿⣧⣤⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⡌⡇⠈⠀⣼⣾⠃⢸⣿⠏⣿⢿⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⠀⠄⢹⣿⠀⢸⣿⠀⠘⠈⠻⣌⡃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠀⠈⠀⠀⢿⡇⠀⣸⡿⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⢀⣄⠐⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠇⠀⠸⠁⠀⣾⠃⠀⠀⠀⠀⠀⠀⠈⢀⣧⠀⠀⣿⢿⢧⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠢⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢰⣝⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⢀⣽⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣿⣿⣿⠿⢟⡛⠻⢻⣳⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣾⣿⣿⣿⡿⠋⠁⠈⢁⡸⠀⣀⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣙⣿⣿⣿⣟⡽⠋⠀⠀⠀⢠⣿⠀⠀⢌⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠛⢉⠀⠀⠀⠀⠀⢪⡾⠣⣄⡶⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⠔⠊⠉⠉⠀⠉⠉⠉⡿⠀⠀⡌⣇⠀⢀⡄⠀⣾⡇⠀⠀⠙⣯⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⢠⡆⠀⠀⠀⠀⠀⠀⠃⠀⠀⢳⣈⡀⠈⠀⢠⣿⡇⠀⠀⠀⠙⠛⠃
⠀⠀⠀⣠⣤⣶⣶⣾⣿⣿⢿⡿⢻⠀⠸⣾⠀⠀⠀⠀⠀⠀⢰⢸⠄⠀⠙⠉⢺⡇⣸⠋⣿
⠀⢠⣮⡴⡻⠿⣿⣻⠿⠋⠀⠀⠸⣧⠀⠙⣆⠀⠀⠀⠀⠀⣼⠀⠀⠀⠰⠊⠈⠃⠃⠀⣿
⠀⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠈⣷⣤⣄⣀⣠⣿⠀⠀⣤⡒⠟⠂⠠⠰⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣽⣿⣿⣿⣿⡿⠧⢄⣈⠉⢪⠲⢲⣿⠿⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⢇⠸⣿⡿⠁⠀⠀⠀⠉⠙⣦⣰⠍⠣⣘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⢂⣿⣿⡀⠀⠀⠀⠀⠀⣿⠏⠀⠀⢸⣆⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡙⢿⣧⠀⢀⣤⡀⢤⡏⠀⠀⠀⢘⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠘⣷⣿⠿⠟⠛⣵⣶⣤⢚⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠘⣿⢧⡀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⡈⢻⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣶⣼⡶⠞⢳⣟⣷⣶⣶⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠁⠀⠠⠚⠁⠈⠁⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⠋⠀⠀⢀⠀⠚⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⠃⠀⠀⠀⠏⠀⠀⠛⠉⠀⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⣶⣦⣤⣄⣐⣶⣿⡿⠛⠉⠀⠀⠀⠀⠀⢀⣿⣄⣴⡆
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠳⣦⡸⠄⡀⠘
⠀⠀⠀⠀⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⡇⠀⠈⢡⡞⢷⣶⠇
⠀⠀⠀⢀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⠀⣿⣾⣿⣿⣧⠀⠀⠈⠓⠚⠁
⠀⠀⣰⣿⣿⣿⡏⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⢀⣿⣿⣿⣿⣿⣧
⠀⠈⣿⣿⣿⣿⣷⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⠻⣿⣷⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠙⣿⣿⣿⣿⣆⠀⠈⠻⣤⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠘⣿⣿⣿⣿⣦⠀⠀⠙⣿⣦⣀⠘⢷⣦⣀⠀⠀⣤⣬⣷⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣼⣿⣿⡿⠁⠀⠀⠀⠹⣿⣿⣇⠀⠙⠿⣿⣷⣾⣿⣿⣧⠀⠀⠰⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠐⠶⠿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⢸⣿⡟⠛⠿⠿⠆⣦⠀⠙⢿⣿⡿⠋⠻⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⡆⢸⣿⠇⠀⠀⠀⠀⠈⠀⣰⣾⣿⣷⣄⠀⠀⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣷⣇⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⣝⢦⣄⠀⠈⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣜⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢸⣿⣿⣧⠀⠀⡌⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⠀⠀⠰⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡇⠀⠀⠀⠙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢙⣦⡀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣿⡷⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡜⠉⢳⣄⠀⠀⠀⠀⠀⠓⠛⠛⠁⠀⠀⠀⠀⠀⠀⠘⠟⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡀⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⣿⣦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⡀⠀⡄⠀⠀⠐⠛⠛⡻⠛⠿⡤
⠀⠀⠀⠀⠀⠀⠤⠾⠋⢾⣿⡻⢿⣦⣤⣄⣠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠀⣀⡛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠀⠀⠉⣭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣄⣒⣶⣶
⠀⠀⠀⠀⠀⠈⠛⢷⣔⣿⣷⣴⣾⠟⠟⠏⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣾⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠉⠁⠘⠃⠀⠀⢤⣤⣼⣦⣤⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⢿⠿⣯⣿⣿⣿⣂⣘⣷⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡷⠿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⢀⡶⠟⠉⠁⣠⣶⣚⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡿⣿⠻⠏⠀⠀⠀⠘⢛⣿⣟⣛⣃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡟⠲⠃⠈⣠⠆⠀⠀⠀⠀⠋⠉⠉⠉⠉⢉⣽⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠶⠶⣤⡴⢻⡟⠛⠛⠓⠶⢶⣤⣀⡀⠀⠀⠛⠛⠛⠿⠶⣶⣭⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡁⠀⠙⠁⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⣉⡉⠉⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠐⢤⡉⠻⣷⣄⠀⠀⣰⡾⠿⠶⣽⠄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠖⠙⠋⠀⠀⠀⠀⠸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠹⣦⡈⢿⣧⡀⠀⠉⠙⢶⣬⡀
⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠛⣧⠀⠀⠀⠀⠀⠀⠙⣷⡀⠹⣷⡄⠀⠀⠀⠙⢿⣆
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠘⣧⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⣿⡀⠐⠲⣦⡀⢻⡆
⠀⣴⠟⡁⠀⠀⠀⡤⠖⠛⢻⣧⣀⣀⣀⣀⣠⡾⠿⣿⣷⡀⢻⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣿⡇⠀⠀⢹⣿⣆⣿
⠀⣿⣿⠟⠀⢠⣤⣤⣶⣶⠿⠟⠛⠛⠛⠋⠁⠀⠀⢸⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⣿⠀⠀⠀⣿⣿⡿
⠀⢿⣷⣴⡶⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣿⡇⠀⠀⠀⣿⡿⠃
⠀⠈⠉⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠁⢀⡄⠀⠀⠀⠀⣀⣀⣠⣴⠟⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣾⡀⢧⣀⡀⠀⢙⣮⣭⠀⠀⡀⢀⣿⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠙⢷⣄⠙⠛⠛⠛⠋⠁⣠⣴⣿⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⡾⠋⠁⠀⠀⣀⠀⢠⣾⠋⠘⠉⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡏⠀⢠⣶⠟⠋⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢠⡟⠁⠀⠀⠻⣄⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡁⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣧⡠⢖⣉⣉⣑⡒⠤⢄⣀⣀⣀⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣆⣉⣛⣻⠟⠛⠿⣷⣾⣭⣭⣛⠷⣤⣴⣛⡤⠤⠤⠤⢊⡍
⠀⠀⠀⠀⠀⠀⠀⠀⡹⠯⣿⣿⣿⡷⣾⣿⣿⣿⣿⣿⣿⣶⣍⠻⣷⣶⠦⠙⠻⢄
⠀⠀⠀⠀⠀⠀⠀⡮⣬⣤⡞⣿⣿⣿⣦⡹⣯⢿⣿⣿⣿⣿⣿⣷⣌⠻⣔⠻⣍⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠃⢫⡟⣼⣿⣿⣿⣿⣧⢻⡎⢻⣿⣿⣿⣿⣿⣿⣷⡌⢷⣤⣽⠶⠊⣠⠖⣩⣛⣉
⠀⠀⠀⠀⠀⠀⡏⠀⠛⣠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⣻⣿⣿⣿⣿⣿⣿⣄⠻⣿⣿⡿⠂⢀⣭⠍⣩⠽⠶⠄
⠀⠀⠀⠀⢀⠎⡀⠀⢈⣭⢙⣻⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄⠹⣿⣿⣷⣯⣴⣶⠽⠷⠂
⠀⠀⠀⠀⡜⢸⠀⣰⡿⣫⣾⠟⡿⠛⠉⣸⠏⣿⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣿⣴⣶⠶⢦⣀
⠀⠀⠀⠀⢻⡎⠂⣩⣤⣿⠵⠊⠀⠀⢠⡿⢀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢿⣿⣿⣿⣯⣭⣥⣀⠈⠦⣀⣠⡖
⠀⠀⠀⠀⢿⡁⣰⣯⣟⡇⠀⠀⠀⠀⣼⡇⢸⣿⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⢟⡥⠀⠀⣀⠀⠐⣚⡽⠋
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢠⣿⠁⣾⡏⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢹⣿⣿⣿⢠⣾⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣿⠁⢀⣿⣿⣿⣿⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣼⠇⢀⣼⣿⡿⠟⢋⣡⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⠟⠉⠀⢐⣛⣭⠍⣀⣴⣾⣿⣿⣿⣿⣿⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣥⣴⣶⡿⠛⢉⣴⣾⣿⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠟⠋⠁⢠⣾⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣶⣶⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⡅⡟⠀⠀⣀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣾⠁⡎⠀⡿⠛⠛⠓⣒⣒⠯⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢰⡇⢸⠀⢰⡇⡀⠀⠀⠀⠉⢙⡲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⡃⠁⠀⠋⠈⠙⢢⡀⠀⢄⢴⣽⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡠⠋⠀⠀⠀⠀⣀⣀⡸⢃⡼⢸⢸⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠁⢀⠀⣞⠃⠀⠀⠀⠀⣊⡭⡭⣉⠉⠀⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢈⠀⡏⡸⠀⠀⠀⠈⣷⣿⣍⣿⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠘⡄⡇⣇⠀⠀⠀⠀⠘⠻⠟⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣷⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣠⡇⠀⠀⠀⠉⠂⠀⣄⡀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢀⡔⠉⠀⠀⢀⣀⠹⣄⡤⠖⠒⠲⠤⣤⣀⣠⠽⣡⣢⠟⠀⠀⠀⠀⠀⠀⢸⡇⠙⡖⡀⠀⠀⡴⡆⠘⡆
⠀⠀⠀⠀⡞⠀⠀⠀⡰⠁⢈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢳⡅⠀⠀⠙⢀⣠⠇
⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠙⠲⠶⠖⠉
⠀⠀⠀⠘⡆⠀⠀⠀⢧⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠓⠦⣀
⠀⠀⠀⠀⢣⠀⠀⠀⢸⡀⢹⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⣸⢁⠞⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⠂⣠⣀⠤⡄⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⢠⡼⡰⢳⠏⠀⠀⠀⠀⢀⣠⠖⠛⠦⢤⣀⡀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡟⢁⠤⠐⠁⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠿⠋⠀⢸⠀⠀⠀⠀⡞⠁⠹⡀⠀⠀⠘⡇⠉⠓⠒⠲⠖⢻⡀⠀⠀⠀⢣⠘⣇⠀⠈⠢⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⡇⠀⠀⠙⡄⠀⠀⠘⡄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠈⢧⡘⣆⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠤⢼⠀⠀⠀⡗⠐⠒⠀⢻⠀⠀⠀⠀⠀⢹⡄⠠⠤⠄⠈⢧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣨⡇⠀⠀⠳⠤⠤⠤⠴⠇⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣶⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢸⣿⡆⠀⠀⣿⡇⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠹⣿⣶⣦⣤⣶⣶⣿⣿⣿⣿⠿⠿⠿⠛⣿⣿⠀⢸⣿⣥⠾⠋⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⣀⡈⠛⣿⣿⣿⣿⡿⠛⣁⣤⠶⠋⠁⠘⠋⠁⠀⣠⣔⠶⣤⣶⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣄⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠙⣿⣿⣿⣿⣿⡟⠀⣼⣿⠁⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⠏⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⢀⡴⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⣼⣿⢿⣿⡇
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠀⢸⣿⡀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠙⢿⣿⣿⣿⡟⠈⣿⡇
⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠈⣿⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⣿⡇
⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢹⣷⣄⣠⣿⡇
⠀⠀⢹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇
⠀⠀⠈⣿⠈⢿⣿⣿⣧⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠰⠃⠀⠈⢿⣿⣿⣇⠀⠀⠀⠸⣷⢸⣇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠙⣿⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠋⠀⠀⠀⠀⢻⣇⠘⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠸⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣶⡘⣆⠀⠀⠀⠀⠀⢀⣴⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢹⡄⠀⠀⠀⢰⣿⠁⠀⠀⠀⠠⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢻⡄⠀⠀⣿⡟⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢿⡄⠀⣿⣿⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⢿⣄⣿⣿⡆⠀⠀⠀⠀⠀⢹⣧⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⣨⣿⣿⡟⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⡇⣠⣤⣴⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⡿⣿⡿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠙⠳⣜⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡾⣛⣽⢿⡿⢚⣽⢟⣃⡀⠀⠀⠂⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⢏⡼⠏⢕⡵⠀⠸⠃⣾⣿⣿⣶⣖⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⣿⣯⣿⠁⣸⣿⠇⠀⠀⣀⡟⡇⣿⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡿⣾⢿⣼⣿⡏⠰⠄⢚⣿⣷⠀⢹⣷⠈⡳⡘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡇⣿⣿⡄⠃⠁⠀⠀⠀⣉⣼⣿⡇⢶⠻⡦⢹⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣿⡇⢷⢹⣿⣷⠀⠀⠀⠀⠀⢾⣛⠋⠉⠘⡇⠀⢘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⣆⣧⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⡀⢸⡝⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣿⠿⡇⢸⣿⡏⣷⣦⡀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⢿⠀⠇⣸⣿⠁⣿⣿⢻⣿⡙⡄⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢸⡆⡐⢻⡿⠀⣾⡇⠘⠌⠳⣜⠆⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠈⠑⠀⢿⡇⠀⣿⠇⠀⠀⠀⠈⠳⡀⠀⠀⠀⢀⣄⠰⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅⠀⠛⠀⢘⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⣻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣄⣮⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠖⠛⠛⠓⢦⡀⠀⡾⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡴⠶⠛⠓⠎⢧⣈⠂⣀⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢤⣤⣌⣉⠉⠀⠀⢀⠄⠀⠀⠀⠉⠉⠁⠀⣘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⢈⣹⠛⢛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⠀⠈⢉⡙⠛⠉⣀⡴⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⠈⠉⠛⢛⣋⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠒⠦⣄⡀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠙⢦⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢉⣽⢟⣫⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠘⠲⢤⣤⡴⠟⠉⠀⣴⠟⣡⡞⢹⡟⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠉⠲⢤⣀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡟⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⢈⠛⠲⢦⣤⡾⠋⠀⣰⡟⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⣻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣯⣀
⠀⠀⠀⠈⠙⠲⢤⣄⣀⣤⡾⢋⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⣿⠀⣹⡷⢶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠠⣤⣄⣀⣀⣈⡉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣣⠀⠀⠀⣼⣿⠾⠋⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠻⣍⡉⠀⠀⢮⡉⠉⠛⣻⢆⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢹⡄⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⢷⢀⡴⢫⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣼⡇⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣁⣀⣭⣤⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠹⣏⠛⠶⣦⠀⠀⠀⠀⠀⠀⠀⣼⠁⣀⣙⠛⢉⣁⡠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⣧⠀⡿⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠩⣽⣿⣿⣿⡟⠙⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⠀⠀⠀⣀⣀⣀⣀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠧⠤⠴⣿⣿⣿⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣬⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠋⠛⠉⠉⠉⠉⢿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣷⣄⡿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣭⣿⣿⣿⣿⣧⡍⣋⠠⣶⠆⣹⣿⣿⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⡴⠋⢐⣯⣽⣿⣿⣿⣿⣿⣷⡘⣿⣶⣤⣾⣷⣿⡿⢯⣍⣻
⠀⠀⠀⠀⠀⠀⠀⣰⢫⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣵⡿⠷⠋⣿⣏⠁⠉⣉⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠴⠋⠘⠀⠀⣴⣿⠛⢡⣾⣻⠿⢿⣯⣻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠱⠋⠀⠋⠀⠀⠀⠉⣵⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣿⣿⣗⣲⣤⡀⠀⢀⣦⣴⡄⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⡧⢾⣿⣿⠋⠈⢸⣿⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⢻⣿⣿⣿⣤⣀
⠀⠀⣠⠀⠀⠀⠉⠉⢦⣸⣌⣿⣿⣿⣿⣿⣿⣿⣦⡙⠷⠰⣿⣿⡇⠀⠀⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⣰⠁⣠⣤⠤⠲⠶⢛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶⣽⣿⣷⣶⣴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣽⣿⣿⣿⣿⣦⣾
⠀⣧⠀⣿⢀⣴⣷⣾⣿⣿⣿⡿⢿⢻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢏⣿⣿⣿⣿⣿⣿⣿
⠐⢻⡄⢸⣽⣯⣀⢀⡴⠳⣞⣷⣮⣿⣷⣿⣿⣿⣿⣿⣰⢀⢀⣀⣠⠔⠒⠀⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⣰⣿⣿⣏⣿⣿⠉⢻⣦⣶⣽⣿⠻⣿⣿⣷⣿⣿⣿⣿⠻⣤⣿⡉⣠⠞⠁⢀⣤⠤⣄⡀⠻⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⡭⠉⠀⢿⣿⣿⣿⣿⣿⠁⠀⠀⠓⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣧⠀⠀⠈⠻⣿⣿⣿⣿⣧⡀⠀⠀⢰⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⣿⠍⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⢀⡰⠿⠿⢿⡯⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⣸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣇⠀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⠀⢀⣿⠄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⣛⢿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⣀⣀⠀⣴⣿⣿⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣛⠿⣿⣟⣵⡤⢀⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣟⣿⣿⠟⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣾⡿⢛⣷⣠⣀
⠀⠀⠀⠀⢀⡼⢫⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⢀⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣧⣻⣅⣠⠄⡀
⠀⠀⣠⣾⡿⠁⣼⣸⣿⣿⣿⣿⡏⠉⣿⣆⠀⠀⠀⣿⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣏
⣠⣾⣿⠋⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡆⠀⠀⣿⣿⣿⣰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⢋⣵⣴⡝⢹⣿⣿⣻⣿⣿⣿⣿⣿⣿⡟⠀⠀⣸⡟⢻⡋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣾⣿⣿⣟⠁⢸⣿⣿⣿⣟⣿⢻⡛⣿⠙⡇⠀⠸⣟⣡⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⢋⡍⠁⠀⠰⣾⣿⣿⣾⣧⠀⣢⡘⠢⣆⣴⣾⡿⣻⣿⣶⣿⣿⣿⣿⠏⠉⠀⠀⠙⠻⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⣿⡇⠀⠀⣸⣿⣿⣿⣿⠻⡿⠀⣀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⡿⣯
⠸⣿⣿⡀⢠⣿⣿⣿⡿⠁⠀⠈⢘⣿⣿⣿⡿⢿⡿⠻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣿⣿⠟⠏⠀⠋
⠀⠉⢻⣇⣤⣿⣿⣿⠁⠀⠀⠀⠈⠿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⡿⣰⠆⠐⠞
⠀⠀⠀⠈⠿⠿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⢿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠧⡀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⣀⣠⡤⢤⡈⠑⣄⢸⠀⣠⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣀⣀⣀⣠⠼⠗⠋⠉⠀⠀⠀⠈⠀⠀⢠⠞⠋⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠉⠁⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠤⠤⠖⠒⠉⠉⢀⣀⠤⠖⠒⠂⠀⠀⡠⠚⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⢀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠛⠉⠁⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠐⠒⠶⣶⢦⣀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⢦⣀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣁⣀⡤⠂⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢀⣀⣀⡤⠔⠂⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⠀⢀⠀⠀⡇⠀⠈⣠⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣀⡠⠤⠄⠀⠀⣠⡴⠃⠀⢰⠁⠀⠀⢿⡧⠀⠀⣾⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠎⠁⠀⠀⠀⣠⠞⠋⠀⠀⡄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠉⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡞⠁⠀⠀⣠⠎⠀⠃⠀⠀⠀⠀⠑⠒⠒⠊⠀⠀⠀⡏⢣⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡿⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣰⠻⡆⠀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⠿⢦⣤⣿⡇⣿⣿⠟⣛⣋⣩⡭⠭⠽⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠥⣶⣶⡶⠖⠋⡟⠐⠿⢯⣭⣻⣉⣭⣭⣭⣤⣀⡈⢻⡶⣄⡀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⡧⣾⠟⠉⠀⠀⠀⠁⠀⠀⠀⠀⠉⠛⠿⣿⣷⣦⣈⠻⠳⣿⠀⠙⢶⣄⣀⠀⣀⣴⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣽⠃⠠⣢⣶⠖⣀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠙⠷⣌⠁⠀⠹⠀⠲⢄⣈⣙⣛⡿⠟⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠉⠩⠚⠁⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠠⠤⠒⣫⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣆⠀⠀⠀⠀⠀⠘⢿⣧⣀⠀⠒⠶⠶⠿⣧⣤⡀
⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠹⣿⡄⠀⠀⠀⠀⠀⠈⢻⣝⣻⠤⣤⡄⢰⡄⠈⣿⡆
⠀⠀⠀⡰⢛⣻⡤⠀⠀⠀⠐⠒⠲⢦⣄⣀⣀⣀⣤⣾⣿⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⡿⢛⣫⣽⢖⣯⠵⢾⣏
⠀⠀⠀⢧⠩⠟⠀⠀⢀⠀⣀⣤⣴⡶⠿⠟⠿⠟⠛⠉⣾⠃⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⣷⠻⣿⠸⠋⠐⣤⡄⠈⢳⡄
⠀⠀⠀⢸⣄⣀⣴⠊⣀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⡄⠀⣴⣳⡿⠃⣼⠇
⠀⠀⠀⠀⠈⠉⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡾⠟⣉⡴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣋⠥⢒⣋⣥⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣵⣪⡵⢾⠿⢋⡡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡿⠛⠁⢠⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢖⣺⠟⠉⠈⠀⠉⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣔⣻⡿⠋⠀⡀⠈⠀⢰⡇⠀⠀⠀⠠⢤⡽⠛⠋⠀⠀⠀⠉⠙⠀⠀⢀⣀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠲⠶⠤⠬⠽⠟⠋⠀⠀⠀⢻⣄⠀⠘⣇⠤⠶⠖⠂⠀⠀⠀⠀⠀⣄⠀⠠⠀⠁⠈⠉⠉⠉⠀⠀⠀⠘⠷⠄⠃
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠈⢧⠀⡹⠀⠀⠀⠀⡀⠀⠀⠀⣼⠉⢣⡀⠀⡆⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠙
⠀⠀⠀⣀⣾⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡆⢸⠆⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢰⠃⠀⠀⠹⡄⡙⠀⠀⣠⢆⡶⠋⠉⠧⣄⠙⠀⡆
⢀⡤⢞⣭⢞⣿⡄⠈⢷⣄⠀⠀⠀⠳⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡀⠀⠀⠀⠀⡞⠁⡉⠀⠀⠀⠀⠈⢃⢀⢹
⡞⡟⠉⠐⠉⠀⠉⢀⠀⢹⡟⠲⠤⠠⡄⠀⠀⢠⡆⠀⠐⢦⣠⣤⣀⠀⡄⠀⠀⣦⣬⠞⣙⠒⠺⡇⠀⠀⢀⡄⠠⠾⢧⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣷⡼⠀⡄⠻⣄⠀⣿⠈⠛⠲⠦⡉⠃⠀⠀⠳⣄⠀⣿⠙⢦⡘⢦⠀⢷⠀⢰⠋⠀⠈⠓⠲⡌⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⡇⢸⠁⠀⠙⢦⠘⣆⠀⠀⢰⠁⡇⠀⠀⠀⠈⠃⢸⡆⠀⠈⢈⠁⠘⡆⢸⠀⠀⠠⢤⠜⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷⣈⢣⠀⠀⠈⡇⣼⠀⣀⣨⢀⡇⠀⠀⣀⡔⣠⠏⢀⣀⡄⣸⠀⢰⢃⡞⠀⠀⠈⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⠸⡇⠀⢀⠁⡿⠘⡧⠖⠋⠀⠀⠘⠵⠖⠁⠀⠨⠖⠋⢁⡄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠣⠁⠀⣇⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣤⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣏⣀⡀⣀⣤⣴⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠏⢨⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡻⢿⣿⣿⣿⣇⠀⣼⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⣤⣄⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣆⣿⣿⣿⣿⡶⠛⣿⣿⣿⣿⣿⣿⣿⣿⣆⣸⣿⣿⣿⣶⣶⣶⣿⣶⣾⣿⣧⣤⣾⣿⠿⠿⠛⠛⠛
⣴⣶⣾⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⢹⣿⣿⡹⣿⣿⣿⣿⣧⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠉
⣿⣿⣿⣿⡿⠿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⣀⠹⣿⣿⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡉
⣿⣿⣷⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣝⣷⡌⠛⢷⣬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⣀
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣤⠀⠀⢧⣀⣿⡟⠃⠀⢸⡂⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠙⠛⠓⠛⠛⠛
⡿⠿⠿⠋⠀⠉⠀⠀⠀⠀⠀⠸⠿⠀⠀⠈⣿⣿⠀⢰⣿⡿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡏⠙⢿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿
⣿⣿⣿⣿⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡻⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⣀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣽⣿⣿
⣿⣿⣿⣿⣷⣴⣿⡿⠀⠀⠀⣀⠀⠀⠀⢻⡎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⣶⠗⠄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠿⠇⠀⠀⠈⣿⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣍⡀⠀⠀⢀⣀⣰⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡏⠉⠀⢀⣶⣶⣄⠀⠀⠀⠀⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢿⣧⠀⠀⣸⠿⠋⠀⠄⢠⡀⠀⠀⠀⠀⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠇⠈⠉⠀⠈⠃⠀⠀⠀⠀⠈⣿⡆⣀⡁⠀⠸⣿⣿⣿⣄⢈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠶⠄⠀⠀⠀⢤⡀⢀⡀⠀⠀⠹⣿⡿⠁⠀⢸⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡀⠈⢦⠀⣀⢆⣠⠄⠄⠀⣠⠎⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⣼⡀⠀⡶⣞⢿⡑⣤⡔⡞⠉⠂⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⡀⢸⡿⠀⣀⣷⣿⣿⣿⣿⣿⣆⣐⣦⣜⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠿⠳⣌⡀⢋⠁⢉⣺⣿⠿⠓⠺⢏⠀⠈⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡜⣿⣿⢹⠂⢰⢿⣿⠛⠉⠉⠉⠀⠀⠀⠀⡈⠃⠀⠈⠻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⠹⠸⣿⡀⠣⠂⢨⡏⠀⢀⡠⠃⠀⠀⠀⠔⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣼⡇⢻⢛⢾⣾⣆⢿⠱⡀⢠⠀⠀⠊⣉⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆
⠀⠀⠀⠀⠀⢀⡞⢿⡙⡌⡆⢾⡇⣿⣆⠱⠅⢸⡇⠀⠀⠑⣿⣶⡿⡄⠀⠀⠀⢦⠀⢀⠀⠀⢠⠇
⠀⠀⠀⠀⢠⡟⡸⢸⡝⡞⣸⡘⣇⢹⣿⠑⠄⢸⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠁⠀⢀⠀⣸
⠀⠀⠀⢰⣯⢃⣧⠰⠱⡜⢀⢳⡹⠸⡿⡄⠀⠘⡄⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⢸⡟⠸⡇⠀⢱⣸⡈⣇⡄⢻⡙⠄⠀⢳⣦⣰⡌⢳⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠈⣿
⠀⠀⠀⡠⢊⢄⠀⢳⡇⢀⢧⠁⣿⠻⡀⠳⠀⠀⢻⣿⣿⣷⡀⠳⠤⡀⠀⠀⠠⡀⠀⠀⠀⠘⢮⡆
⠀⠀⠀⡸⢨⠎⠀⠸⢣⡄⠙⠀⢧⡁⠁⠀⠃⠀⠀⢹⣿⣿⡿⢄⣠⡇⠀⠀⠰⡃⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠇⠾⠀⢠⠀⠀⢳⠀⠠⠈⠁⠀⠀⠀⠀⠀⠈⠉⣿⡇⠀⠙⢿⡀⠀⢰⡇⣀⡀⠀⠀⠐⡤⣨⡀
⠀⠀⠀⠀⢠⠀⠸⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⠀⠀⠀⠣⡀⠻⡀⠿⣄⡀⡄⢶⡇⣷
⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠙⠦⣄⠀⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠐⠬⣦⣤⠴⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⢀⣤⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣤⣄⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢠⣿⣿⣿⣿⣿⢿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠛⠉⠁⠀⠀⢹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣰⡀⠀⠀⠀⠀⠀⢠⣦⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠢
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⢿⡄⡄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⢿⣦⣴⣶⣴⡇⣿⣿⣿⣿⣿⣦⡙⠢⠄
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣴⣿⣿⣿⠿⠿⢧⡙⠓⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠻⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⣀⣀⣠⡤⠐⠀⣀⣠⣎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣤⣶⣶⣿⣿⡿⠋⡀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣊⣁⣀⣀⡀
⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠘⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⣾⣿⣿⣿⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣀⣀⠀⣾⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⠀⢀⣼⣿⣿⣿⣿⡇⠀⠙⢌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⡘⠉⠀⠈⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⠙⠿⡟⠛⢿⣯⡛⠛⠿⡿⢿⣯⡝
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣷⡄⠀⠀⠀⢰⡇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠋⠀⢹⣿⣿⣿⣿⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⢠⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠻⣿⣿⣼⣿⡀⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⣤⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⠋⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⡄
⣿⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣦⡀⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋
⣿⣾⣷⣄⠀⠑⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠄

⠀⠀⠀⣄⠀⠀⡇⣤⣤⣤⣄⣀⡀
⠀⠀⠀⢸⣿⣿⣦⠾⠿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠋⠉⠛⠋⠳⠀⠀⠙⠻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠰⠒⠀⠀⠈⢧⠀⠀⠀⠙⢿⣿⣿⣦⡄
⠀⠀⠂⠀⠀⠄⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⠿⣿⣿⡦
⠀⠆⠀⠀⠀⠀⢤⡤⠴⡇⠸⠀⠀⠀⠀⠀⠀⠀⠉⠛⢓⡀
⠄⢤⠀⡀⠀⠔⠉⠀⠀⢹⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⣴⣦⣄⡀
⠀⠀⠠⠵⠀⠀⠀⠀⠀⠀⡇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠘⡏⠙⣿⣿⢿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢹⡀⠘⠟⢿⡙⢿⣿⣿⣿⣯⣽⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣄⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠈⢻⢿⡿⠿⢿⣿⠋
⠀⠀⠀⠀⠀⢀⠠⠒⠂⠈⠙⢦⣀⠙⢄⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠰⠁⢀⠄⠛⠶⠶⠿⠿⢦⣤⣿⣶⠒⠀⠀⠀⣼⣧⣀⡀⢀⣀⣀⣀⣤⣤⣶⣾⣿⣿⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⡴⠚⠻⠿⠿⠟⠛⠛⠛⠉⠉⢿⠉⠛⢿⣿⢧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⡹⡱⡣⣄⠀⠈⢧
⠀⠀⠀⠀⠸⠀⠻⢦⡄⠀⠀⠀⠀⢀⠂⢐⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠘⣎⢣⡀⠘⣷⡄
⠀⠀⠀⠀⠀⠈⢠⣿⡇⠀⠀⠀⠀⢸⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⡰⠃⠀⢣⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠈⠳⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡘⠋⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣰⠋⠀⠀⠀⠀⠘⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⡱⡶⠁⠀⠀⠀⠀⠠⠆⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⣾⣶⣶⠉

⠀⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠴⠶⠖⠒⠶⢶⣶⣦⡲⠶⣦⣀
⠀⠀⠈⠙⢶⣤⣄⣀⣀⣀⣀⣀⣤⣤⣴⣒⠋⠉⣀⣀⣠⣤⣤⣶⣶⣬⣿⣿⣿⣷⣤⣹⣧
⠀⠀⠀⠀⢠⡉⠻⢿⣝⣻⠿⣿⣿⣧⡀⠁⠀⠀⠀⢐⣬⣽⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⡇⠀⡆
⠀⠀⠀⠀⠀⠙⠲⣄⡈⠙⠻⠿⢭⣥⣤⣤⣤⣶⣿⠿⢟⣫⣽⠟⠁⣠⡶⠶⠶⢦⣄⡉⠻⣷⣤⢣⡀
⠀⠠⣄⡀⠀⠀⠁⠚⠿⢷⣶⣶⣿⣿⣿⡿⢟⠋⠁⢪⣿⣿⡟⠀⠘⠁⠀⠀⠀⠀⠀⠉⠳⣌⢿⡆⠻⣄
⠀⠀⠈⠙⠻⢦⡶⠶⠶⠾⠟⣛⣿⠿⠟⠛⠁⢀⣴⠟⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠋⣡⡤⠒⣀⡴⢞⡿⠁⢸⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣠⣿⣇
⠀⠀⠀⠀⠀⠀⢁⡿⠟⣠⠞⢁⡴⢛⡡⠚⠁⠀⠀⡞⡀⣿⠸⡟⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠘⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠊⢀⣷⠃⡐⠁⠀⠈⠀⡴⠀⠀⠈⣰⣿⡏⠀⢻⡀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠉⠁
⠀⠀⠀⠀⠸⢿⡖⠾⢧⡀⠀⠀⠀⠀⡾⠀⠀⢀⣴⣿⣿⡇⠀⢘⣷⡄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⠀⠀⢹⣧⠀⡙⢷⣀⠀⢸⠇⠀⣴⠟⢹⣿⣿⠀⠀⢸⡇⠘⠆⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⣸⡟⣧⡘⣦⠉⢱⡟⠀⡞⠁⠀⣿⣿⡟⠀⠀⠈⠿⠄⠀⠀⠀⠘⣤⡀⠀⠀⠀⠀⠀⢸⡇⠈⢻⣦⡀
⠀⠀⠀⠀⠀⢠⣿⠁⠈⠛⠛⣷⡟⠀⠀⢀⡆⣸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠈⡇⠀⠀⠙⣷⡄
⠀⠀⠀⠀⢀⡞⣿⣤⠀⡄⣰⡟⡰⠀⣠⣾⣾⣿⣿⠏⠀⠀⠀⠀⠈⠳⢤⣄⣀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢻⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠈⠀⢻⣿⢸⣠⣿⣾⢁⣼⣿⣿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡄⠀⠀⠘⢷⡀⠀⠀⠘⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⢸⡏⢺⡿⣿⣇⣾⣿⡿⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠧⢸⡇⢹⣿⣿⠟⢧⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡿⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⣤⣿⠇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇⠀⠀⢾⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠞⣿⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡇⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⢸⡆⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⠀⢿⠀⠀⠀⠀⠀⣥⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠘⠇⠀⠀⢄⠀⠘⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠀⠀⠘⢧⡀⠁⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⠻⣄⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠆⠀⠀⠀⠦⣴⣿⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡄⢠⠀⠀⢠⢸⡟⡵⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⡀⣷⣄⣴⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⢸⣿⡆⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣽⣿⣿⣿⡇⣧⣤⣴⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠿⠿⣿⡉⣿⠛⠻⢿⣿⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠳⣦⣬⣹⣿⡄⠛⠂⠙⣿⣷⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⣿⡿⠀⣠⠉⢠⠛⢿⡇⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠃⣿⣿⡶⠉⠀⠘⠿⠀⠀⠀⠀⣿⣿⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠀⠛⠛⠋⣰⠀⠀⠀⢀⣿⡀⠀⠀⠘⠁⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠰⠟⠀⢈⡴⠃⠀⠀⠀⢠⡿⣧⠀⠀⠀⠀⠀⠀⢹⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⣴⣀⡶⠀⣯⡀⠀⠀⠀⣠⡿⠁⠘⢷⣄⠀⠀⠀⠀⣸⣿⣿⣿⣿⡅
⠀⠀⠀⠀⢀⣴⡿⠏⢠⡀⠰⠛⠀⢀⣹⣷⣦⣴⣾⣿⣷⡄⠀⠀⠙⠧⠀⠀⢴⡿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⠡⣿⠸⠇⠐⣷⣼⡿⠛⠁⠀⠀⠈⣿⣿⡟⠀⠀⡄⠀⠀⠀⠈⠀⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⢹⣿⡀⠂⡠⢛⡿⠉⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⡇⠀⠀⠀⠀⠀⠈⡿⢿⣿⠷⠾⠶⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠉⠛⠿⠗⠚⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣷⠀⣤⡀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⣷⠀⠀⠀⠀⢀⣼⡟⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢻⣿⠀⠀⠀⣿⠀⠀⠀⠀⠈⢸⡇⠀⠀⠀⠇⠀⠀⠀⠀⢠⡄⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⡇⠸⣿⠀⠀⢠⣿⠀⠀⠀⠀⢀⡿⠁⠀⠀⢰⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠁⠀⣿⡄⠀⢸⡏⠀⠀⠀⢀⠏⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠘⣿⡆⠀⠘⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣀⠀⢹⣇⠀⣾⠇⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢠⡿⡇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠈⠃⢸⡆⠛⠀⠁⠀⣦⠀⠀⠙⠿⠟⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⡾⠁⡼⠷⠄⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⣼⡇⠀⠀⠀⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⢠⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⡀⠀⠀⠀⣸⡿⠀⠀⣠⣿⣿⠃⠀⠀⣰⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠈⠀⢠⣄⠀⠀⣄⠀⠀⠀⠀⢀⣾⠋⠉⠉⠁⠀⢠⣿⣧⣴⣾⣿⣿⠏⠀⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣠⣿⣿⣿⣿⣿⣶⣤⣶⣿⠇⠀⣀⡼⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⠀⢠⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠰⠟⣿⡿⠋⢹⣿⣿⣿⣿⣿⣿⠀⠐⢉⡀⣠⣾⠟⠁⠀⢿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⢀⣾⠏⠀⠀⣼⣿⣿⣿⣿⡿⠁⠀⢠⣿⠟⠋⠁⠀⠀⠀⢸⣿⠏⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠃⢰⣿⠋⠁⠀⠀⢠⣿⣿⣿⣿⡏⠀⠀⢺⡿⠁⠀⠀⠀⠀⠀⠀⣿⡟⢠⣾⠿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢻⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠋⠀⠱⠏⠹⣿⣿⣿⣿⣿⣧⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣸⡇⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⡀⢰⣿⠃⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠸⢿⡿⢯⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⢰⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢧⠈⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠈⠟⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣇⢸⡄⣿⡆⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣹⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣾⡇⠈⢻⡆⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢻⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⣷⣄⣼⣿⣤⣄⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠈⢻⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠘⢿⣿⣿⡿⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⢸⣿⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠘⣿⣿⣧⡀⣼⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠉⣿⣷⡄⠀⣰⣿⣿⠇⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣹⣿⣿⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⣾⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉

⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠰⣤⡀⢸⡟⡄⣀⣀⡀⠀⠀⠄
⠀⠀⢠⠜⠟⢿⠋⠛⠿⣿⣿⣶⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣆⠀⠀⠰⠦⣤⣠⣷
⠀⠀⠀⠀⠀⢈⡛⠒⠀⠀⠙⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⣰⣿⣶⠀⣿⠿⣄⡀⣀
⠀⠀⣄⠀⠀⠻⡋⠀⠀⣦⡀⠀⠙⢿⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠛⣇⠛⠦⢁⡀⢽⣿⣥⣀⣀⡤
⠀⠀⠃⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠈⢻⣷⣶⣭⡭⠀⠀⠀⠀⠀⡀⠀⠀⠈⠙⢓⡲⠤⣽⢻⣿⣿⣅
⠀⠀⠀⠀⠀⢀⣾⣦⣶⣿⣿⣿⣿⡄⠀⠻⣕⢦⣀⣀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠹⠟⠠⣿⡇⠹⣿⣿⣿⣦⠄
⣧⣾⣷⣶⠗⠛⠁⠠⣿⣿⣿⣿⣿⠃⠀⠀⠹⣷⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀⢻⣿⣿⣿⡆
⠘⠿⠿⠟⠀⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⢿⣾⣦⣤⡀⠀⠰⠀⠀⠀⢀⠀⣀⣼⣿⣿⠃⠈⢿⣿⣿⣟⡲⠶⠀⠀⢀⣴⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠅⠀⠀⠀⢸⡀⣀⣀⣼⣿⣿⣿⣿⣿⠀⠀⠀⢫⠭⠭⠯⢍⡀⢠⣿⣴⣾⣷⣶⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠇⠀⠀⠀⠈⢻⣿⣿⣿⣿⣝⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣻⣿⠇⡆⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠛⠻⣿⡟⢻⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⡇⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⢸⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⣧⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⠀⠀⠀⣿⢸⢇⡟
⠀⠀⠀⠀⠀⠀⠠⣀⣼⣿⡦⠀⠀⠀⠀⠀⣰⣷⣄⠀⠀⠀⠀⢰⡇⠀⠀⠀⣸⣿⡀⠀⠉⠀⠀⠀⠀⠀⣼⠀⠀⠀⣰⠇⣾⣿⠀⠀⠀⣿⢸⢸⡇
⠀⠀⠀⠐⠁⣶⣾⣿⣿⠟⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣶⣦⣤⠜⠁⢀⣴⣾⣿⣿⡇⠀⠀⠀⠀⠀⣾⢠⣿⣶⣶⣾⢏⣼⣿⡏⠀⠀⠀⡇⠈⠀⠁
⠀⠀⠀⢰⣾⡟⢻⠟⠁⣀⣤⣶⡿⠟⠋⠉⠀⠉⠉⠉⠉⠋⣡⣶⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣼⣇⣾⣿⣿⣟⣵⣿⣿⣿⠁⠀⠀⠀⡇
⠀⠀⠀⠀⢻⡇⠇⣀⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠋⠉⣿⣿⠀⠀⢀⣼⣿⣿⡿⢟⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣾⣱⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⣿⡏⠀⣰⣿⡿⢻⣿⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠠⣠⢹⡿⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⠋⣇⣾⡇⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠃⣸⣇⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⡀⠀⠀⠀⠀⠀⢠⣿⣤⣴⡟⠀⣿⣿⠁⠀⠀⣿⣿⡿
⠀⠀⠀⠀⣼⣏⠀⠙⠲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⣿⠀⠀⠀⠀⠀⠘⢻⣿⣿⠁⢀⣿⣿⠀⠀⣰⣿⣿⠁
⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠒⠃⠀⠀⠀⠀⠀⡄⠈⢹⡇⠀⣼⣿⣿⡄⢰⠇⣼⡇
⠀⠀⠀⠸⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⣼⡇⢰⡇⣿⣿⠁⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢸⠈⣿⠇⠀⠈⠁⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⣿⡄⠀⠀⠀⠀⠀⠇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣶⣧⣿⣿⡏⠀⠀⠀⠀⡤⢴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⣿⣿⠇⠀⠀⠀⢰⣀⣸⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡂
⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⠀⣠⣴⣶⣷⣿⡟⠛⠛⠛⠓⠲⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣄⣤⣴⡿⠋⠁⣿⢻⣿⡇⠀⠀⢀⣢⣶⣦⣰⣬⣙⢷⣦⡀
⠀⠀⠀⠀⣠⢞⡿⠛⣯⣻⣿⣿⡟⠀⢼⠀⠙⣿⣿⡇⠀⠀⠀⠉⠻⣿⣿⢿⣿⢷⣬⣿⣦⡀
⠀⠀⠀⠈⠁⠃⣠⠞⠻⠛⣫⣿⣇⠀⠈⢻⣀⢀⠙⢷⣄⡀⠀⠀⠀⠙⣿⣆⠻⣇⠙⢿⣿⣿⣄
⠀⠀⠀⠀⠀⢰⣿⣠⡴⠛⢡⣿⡘⣄⣄⠈⠻⣾⣶⡄⢈⠛⢦⣣⡰⡄⠘⣿⡄⠹⡄⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⢸⣧⠳⣌⠻⢦⣀⠀⠙⠷⣌⢷⣄⠻⣧⢻⡄⢸⣷⠀⠀⠀⡀⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠁⠙⠚⠷⣤⣿⣽⣒⠤⡈⠙⠿⣷⣻⡞⣧⠀⣿⡀⣧⠀⢷⠸⣿⣿⢿⡀
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣯⠛⠮⣑⣄⠘⢿⣷⣿⠀⣿⡇⣿⡇⢸⡇⣿⣿⣇⢃
⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠻⣿⢿⠈⢧⠀⠈⢿⡆⠸⣿⣿⠀⣽⣇⣿⡇⢸⣷⣿⣿⣿⡎
⠀⠀⠀⠀⠀⠀⠈⡇⠐⡀⠀⠀⠀⠀⠀⠀⢴⠆⠀⢸⡆⠀⠘⣿⠀⣿⣿⠀⣿⢿⣿⣧⡾⢻⡏⣿⣿⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⢸⣿⠇⢰⡏⡼⠁⣿⠃⠸⢹⣿⣿⠸⠃
⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⠀⢀⡀⠀⠀⠀⠀⠀⢀⣿⠀⡀⢸⣿⣿⡟⢀⡿⠀⠀⠀⠋⡆⢠⣿⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣷⣸⠀⢸⡇⠀⢠⠀⢠⡶⢫⡇⣰⠀⢞⣿⡿⢁⣾⠃⡀⢠⠀⣸⣣⢟⣿⣿⠇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⡇⠀⣾⠀⡞⠀⣾⠀⡇⠀⣼⣿⢁⣾⠃⡼⢀⣾⣴⡿⢫⣾⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠀⠈⣷⠀⢸⡜⠀⢸⡇⠀⡇⢸⣿⢣⣾⢏⡾⣡⣿⡿⢋⣴⠟⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣇⠀⣾⡇⠀⣇⣿⣇⣿⣫⣾⣿⠟⢋⡴⠟⢁⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⠀⠀⢀⡿⠷⠏⠀⢿⡇⢁⣿⣿⣿⣿⠿⠋⠁⠀⠉⠀⣠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡘⠀⠀⠘⠀⣠⠆⠀⢸⡇⡘⣿⣿⠋⠀⠀⠀⠀⡠⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠴⠟⠉⠀⠀⢠⣷⣰⣻⣇⣴⢎⢀⡴⢋⣴⢋⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠘⣧⣿⣿⣷⡷⢋⣴⠟⣱⠋⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠜⣿⣿⣯⡴⠛⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣆⣌⡿⣿⣤⡞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣰⣿⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠉⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣐⣦⣤⣤⢴⡶⠺⢟⣴⣷⣾⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣊⣉⣉⣭⣽⡿⠋⠁⢸⣧⡤⠭⠅⡙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠟⢛⣿⣿⣿⡿⠋⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⡟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣫⠴⢻⡟⠁⠀⠀⠀⠀⣸⢷⣇⢳⣶⣆⠓⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡃⢠⡿⠀⠀⠀⠀⡀⠀⢹⠀⢻⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢧⣿⠇⠀⠀⠀⠀⡇⢀⠆⠀⠈⢇⠀⠀⠀⠀⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠻⡄⠀⠀⠀⠀⠰⣼⡀⠀⠀⠈⡄⢦⣤⣴⠞⠛⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣌⡢⡀⠀⠀⠀⠈⠳⠤⣄⠐⣿⣮⢛⠉⠀⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣻⣿⡿⠦⣀⠀⠀⠀⠀⠠⠙⣷⣿⣯⣉⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠧⡀⠈⠑⠄⠀⠀⠀⠱⡉⠛⠛⢿⡟⢇⠀⢰⡇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡤⢀⠀⠀⠀⠀⠀⢱⡀⠀⠀⢗⠘⠄⠉⣡⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⡀⠀⠉⠀⠀⠀⠀⢣⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡦⣄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢧⡄⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆



⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⢛⢻⢛⡟⣻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⣿⣟⡆⠀⠀⠀⣿⣿⢿⡄⠤⠖⠂⠀⠀⠐⠲⠤⠀⡀⠀⠀⢀⣀⡠⠄
⠀⠀⠀⠀⠀⠹⣿⡏⠃⠀⠀⢻⣿⡞⣧⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢹⡟⠄⠀⠀⠀⠙⠷⠏⠀⠀⠉⠳⠦⠤⠤⠛⠓⠒⢤⡀⠀⠀⠀⠀⠠⠀⠡⣀⡤⠖
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⡀⠀⠈⠓⣆⡀⠀⠐⢂⣹⠷⠊⠁
⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠀⠉⠻⣄⠀⠛⠙⠿⠷⠶⠟⠿⣿⣿⣿⣷⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⡳⣄⡀⠀⠐⠤⡀⠀⢄⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡆⢹⠳⣄⡀
⠀⠀⠀⠀⣿⣦⠀⠛⠋⠛⠁⡛⠛⠣⢄⠀⠀⠉⠢⣿⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠸⠃⢸⠓⡄⠀⠈⢄⠀⠀⠐⠂⢤⡀
⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣶⣶⣾⡗⠀⢲⣼⣾⣿⣿⠻⢷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢷⡜⠀⠀⠀⠄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢸⡇⠀⠘⠃⠀⠀⠀⠻⢿⣿⠇⣦⢠⣼⣿⣿⣿⣷⡀⠈⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢸⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠑⡄⠀⢰
⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⣀⡀⢁⢀⠀⢀⠈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⠏⢠⡀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⣿⣿⣿⠻⢿⣷⣡⣄⣴⡈⣰⡏⠀⠀⡠⠀⠀⠀⡠⠂⡴⠁⢠⠀⠃⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠂⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠙⣿⡛⢳⣴⣿⠀⠀⠊⠀⠀⢀⠔⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣆⣴⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠄⠀⠀⠀⠠⠊⠀⠀⠐⠋⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⣾⠟⣩⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⡔⠁⠀⠐⠁⠀⢀⣴⡿⠋⠀⠀⠀⠀⢀
⠀⠀⠁⠀⠀⢤⡄⠀⠀⠀⠞⢋⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠈⠀⠀⠀⠀⣠⣾⡿⠋⣀⠄⠀⠀⢀⠔⠁
⡰⠃⠀⠀⠀⠙⠰⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠎⠀⠀⠀⠀⠀⠀⠀⡤⠚⠋⣁⠴⠊⠁⠀⢀⡴⠃
⠑⡀⠀⡼⣷⣶⡀⡆⣄⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢂⡤⠊⠀⠀⠀⠀⠀⠋
⠀⢷⣿⣆⣿⠿⠿⣧⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠙⢿⣿⣷⣶⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣃⣉⣋⣀⣏⣉⣯⣋⣭⣟⣽⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡠⠂⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣶⠟⣩⣶⣤⣶⣶⣦⣤⣶⡆⠀⠸⡯⠁⣶⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⢶⣶⣶⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣦⣤⣄⣤⣾⣿⡿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣉⣛⣛⣛⣱⣿⣿⣭⣵⣻⣿⣿⣿⠿⠿⠟⠛⠿⠿⠛⠛⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢟⣻⠿⠟⠛⣻⣿⣿⣿⠿⣿⣿⣿⠿⠛⠛⠯⠝⠒⠉⠁⠀⠀⠉⠻⣿⣦⠀⠀⣀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣯⠅⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠈⠻⣿⢤⡿⠧⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣯⣬⣁⣤⣴⣿⣿⣿⣿⣿⣿⣿⡛⠁⠀⠀⠀⢀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣷⣿⣿⣿⣿⣿⠿⠿⣛⣛⣻⡿⠀⠀⠀⠀⢀⣾⣾⣿⣿⡀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣛⣛⣛⣉⣥⣶⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣿⣦⡙⣿⡆⠘⣦⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣟⣋⣭⣿⣿⢟⣫⠷⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⢸⣿⣿⣿⠀⠈⠉⠉⠁⠙⠻⠿⣿⣾⣿⠀⣆⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⡾⠟⣛⡛⠁⠀⠀⠀⠀⠀⠀⢈⣿⡏⠁⠀⢨⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⠌⢳⡜⠦⢉
⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣷⣾⣿⣿⣵⣾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⢨⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠦⠷⠒⠊
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠿⠿⠿⠿⢾⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⢐⣫⣤⣾⢟⣵⣶⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠉⠛⠭⠶⠿⣿⣿⣿⡿⠟⣛⣻⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠹⡄
⠀⠀⠀⢀⣤⣴⣶⣶⣾⣿⢿⣫⣶⣟⣭⣕⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠻⣆
⠀⠀⠀⠀⣦⣶⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣀⢽⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣀⠀⣀⠀⠀⠀⠀⠰⣿⣿⣧⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣦⠀⠀⠈⣻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⠘⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠈⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠻


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏⣿⣤⣤⣤⣤⣤⠊⠁⣴⢫⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣷⣴⡏⠾⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⣿⣻⡿⠋⠘⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢀⣿⢠⡿⠆⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢏⡴⠛⣿⣿⣶⡾⠷⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠿⠿⠿⣷⣬⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁⢀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣇⠀⠀⠀⠈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠀⠀⣾⣿⡿⣄⠀⢀⣀⣀⣘⢆⠀⠀⠀⠀⠀⠀⠀⣄⣹⣆⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⢻⣼⣧⠘⠗⠰⣶⣎⠙⢿⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⢸⡆⢸⡿⣿⡄⢰⡄⠿⠿⣦⠈⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠀⠀⠀⣿⠁⣿⠁⣿⣷⡈⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⠀⠀⣿⠀⣿⠀⢸⣿⣷⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⠀⠀⢀⠀⣿⡇⣿⠀⠀⣿⡹⣧⠀⠹⣆⠀⠀⠀⠀⠘⣷⣄⢀⣀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠘⣆⢹⣧⢻⠀⠀⠸⣧⠘⣧⠀⠈⠣⠀⠀⢀⣤⣾⣿⡿⠿⠟⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠘⣿⣿⣾⠇⠀⠀⢟⣦⠘⢧⠀⢀⣴⣾⣿⠟⡫⠕⣶⠉⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣿⣿⠃⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠘⣾⣧⠈⣷⣿⡿⢋⠑⠉⠀⠀⠘⣧⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣿⣿⠀⠀⠀⠀⠠⢄⣀⣸⣿⣿⣷⣤⡀⢹⣿⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢠⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏⣿⡇⠀⠈⠛⢆⢀⠀⠀⠀⠀⣀⣠⣈⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢣⣿⡇⠀⠀⠀⠈⢻⢳⣄⠀⢾⣿⣿⣄⠓⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢸⣿⡇⠀⠀⠀⠀⠀⢧⡹⣷⣄⠙⠛⠛⣣⣼⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣾⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢸⣿⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠰
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡆⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠷⠒⠉⠉⠉⠀⠐⠒⠤⢄⡀⠀⠔⠁⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠇⠀⢣⡄⠀⠀⠀⠀⠀⠀⠀⠛⠻⢶⡀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⢹⠿⣿⡏⠀⠀⣠⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⢒⠄⠈⠃⠈⠀⠠⠛⠉⠀⠀⢀⡴⢡⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⣡⠚⠁⢀⣴⠿⠁⠀⣠⠂⣠⠟⠁⠀⢆⠰⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⣠⠞⠁⢀⡠⠋⣁⣠⣴⣛⠵⠚⠁⠀⠀⠀⠈⢦⣇⠠⣄⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⣴⠋⠀⠘⠋⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠑⠲⢦⣄⣀⠨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠔⢀⡼⠃⠀⠀⠀⠀⡀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡀⢀⡾⠁⠀⠀⠀⠀⠀⡇⡦⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣴⡷⠋⠀⣾⠃⠀⠀⠀⠀⢀⠀⡇⠹⣟⣉⣹⡟⢆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⢺⠏⠀⠀⣼⠇⠀⠀⠀⠀⢠⠇⢀⠇⠀⠘⢿⡿⠡⠼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⢠⠃⠀⠀⣰⡏⠀⠀⠀⠀⢀⠏⠀⢸⠂⠀⠀⠐⠂⠂
⠀⠀⠀⠀⢠⡟⡀⠀⢠⡿⠀⠀⠀⠀⠀⡞⠀⠀⣸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡼⢋⠄⠀⣾⠃⠀⠀⠀⠀⢸⠁⠀⠀⣿⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡄⠀⣵
⠀⠀⢀⢎⡴⠃⠀⣸⡟⠀⠀⠀⠀⠀⡎⠀⠀⠀⣿⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢻⣴⠏
⠀⠀⣼⢹⠇⡀⢠⣿⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⢦⣀⣠⡄⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠃⣿⡞⠀⣾⡏⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠹⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⡆⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣴⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⢀⣬⣵⡟⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣦⡑⠀⠀⠀⠀⠀⠂⣠⣌⠃⠀⠀⠀⠀⠀⠀⢀⠳
⠀⣠⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⢦⠀⢂⡀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠂
⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣾⠈⠳⣤⠁⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⢀⣾⡇
⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠇⠀⠢⣙⠛⠻⠏⠀⠀⠀⠀⠸⠟⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢣⡀⠱⢤⣉⣙⣛⣃⣁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠛⠚⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣧⣸⣤⠶⠶⠶⠶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣦⡀⠀⠀⠀⢀⣠⣶⡿⢿⣿⣸⣇⣿⡀⠀⠀⠀⠀⠀⣀⠉⢙⠛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣼⣿⣿⣦⣴⣶⣿⠟⠁⠀⠸⣿⢹⣿⣸⠇⠀⠀⠠⢬⣻⣾⣿⣦⣿⣶⣬⡙⢿⣦⣄
⠀⠀⠀⠀⠀⢠⣶⣟⡿⠟⢿⣝⣿⣿⣿⣿⡏⠀⢸⢀⠀⠻⣎⣿⣿⠀⠀⠀⠀⠀⠙⢻⣿⣿⡿⣿⣿⠻⣷⣝⢿⣷⡄
⠀⠀⠀⠀⠐⠏⡴⠋⢀⣤⠾⣿⠿⠿⣻⣿⡇⠀⠘⡸⡇⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠙⣿⣷⡈⢻⣧⠈⠻⣷⣿⣿⣦
⠀⠀⠀⠀⠀⠐⠀⢠⡟⠁⠈⣀⣤⠞⣿⡟⣧⠠⠀⠀⢿⣆⣸⠀⡀⠙⠻⣦⡀⢠⡀⢄⠀⠈⢿⣷⡀⠹⣇⠀⠙⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣬⠾⠋⠀⢠⡟⣷⠘⣦⢣⡀⠀⠙⠿⣷⣷⡀⢶⣈⠙⢦⣿⡈⣧⠀⠘⣿⣇⠀⢹⠀⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⠀⠸⣷⡘⢷⡈⠛⢿⣦⡀⠀⠈⠛⢷⣌⢻⣦⡈⢻⣷⠸⣧⠀⢹⣿⡀⠀⠀⠀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠣⠉⠲⠽⣲⣤⣨⣭⣓⠤⢄⡀⠈⠓⢿⣿⣆⢿⡇⣿⡆⠈⣿⡇⢸⣆⠀⢸⡄⢻⡿⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⢶⣬⡓⢤⡀⠙⢿⣾⣿⣸⣷⠀⢿⣧⠈⣿⡄⠘⣷⢸⣧⣿⣧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡿⣦⠈⠙⢶⡙⣄⠀⢻⣿⣿⣿⠀⢸⣿⢠⣿⣇⠀⣿⡠⣿⢿⣿⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⠁⠸⣧⠀⠀⠹⣽⠆⠈⣿⣿⣿⠀⢸⣿⣸⣿⣿⢠⣿⣿⣿⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠎⠀⠀⣿⡄⠀⠀⢿⣿⠀⣿⣿⡟⠀⣿⣿⣿⣿⣿⣸⠛⣿⡏⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⢰⠀⠀⠀⠀⠀⠉⠀⠀⠀⣽⡇⠀⠀⢸⣿⢀⣿⣿⠃⢠⣿⢀⡿⠹⣿⡟⠀⣿⢱⣿⣿⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠈⣧⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣼⣏⣼⣿⡏⠀⣼⠇⠸⠁⠀⣿⢁⠀⠃⣾⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣿⠀⠀⣿⠀⠀⠀⡀⠀⠀⡀⢀⣾⡟⠀⡄⠀⣿⣿⣿⡿⠀⣸⡿⠀⠀⠀⠀⠁⣼⠀⣼⣿⣿⣿⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣿⠀⠀⣿⠀⠀⢠⠀⠀⢠⣶⠟⣽⠃⣶⠀⢸⢫⣿⣿⠁⣰⣿⠁⡀⠀⣤⠀⣸⣇⣼⢫⣿⣿⡿⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⠀⠀⢸⠀⢀⡿⠁⢰⡏⢰⡇⠀⢀⣿⣿⠃⣰⣿⠃⣼⠃⣰⡟⣰⣿⡟⣡⣿⣿⣿⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⡆⠀⢸⡀⡼⠁⢀⣿⠀⢸⡇⠀⣾⣿⠏⣴⣿⠃⣼⠏⣰⣿⣿⡿⢋⣴⡿⣹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿⠀⠀⡿⠗⠀⠈⣿⠀⠀⢸⡏⠀⢸⡇⢰⣿⡏⣼⣿⢃⣼⢏⣼⣿⡿⠋⣠⡾⠋⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣾⡇⢀⢸⡇⣾⡿⣸⣿⣣⣾⣿⣿⠟⠁⣠⠾⠋⢀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⠀⠀⠀⠀⢰⡶⢶⢸⠇⠀⣿⡇⠘⠘⣿⣿⣷⣿⣿⣿⠟⠋⠀⠀⠛⠁⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡈⠷⠀⠀⠀⢿⠀⠀⡘⠀⠀⢻⡇⠀⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⡀⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⢀⣤⣴⠇⠀⠀⠘⣷⠰⠸⣿⡿⡇⠀⣀⠀⠀⢀⣤⠊⣠⡿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠚⠋⠀⠀⠀⠀⢠⣿⣆⢷⣿⣧⣴⠞⡡⢊⣴⠋⣠⡾⢋⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⢿⣼⣯⣿⣿⣾⡷⠋⣡⡾⠋⣰⠟⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢈⣿⣿⣿⣿⠋⣠⡾⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡟⠈⠻⣿⣿⣿⠋⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸⣹⣶⣇⡾⢿⣿⣶⠏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⢠⣾⣿⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿⣿⣿⣿⣼⣿⡿⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠏⣿⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⢻⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠿⠿⠋
⣦⣤⠀⠀⠀⠀⠀⣤⠾⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣄⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠈⠙⠛⠿⠿⠛⢫⣅⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⣴⣶⢞
⣤⣄⣤⣤⣤⣀⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠊⠀⠀⠀⠀⠶⠿⠿⠿⣿
⠟⠛⣿⣿⣟⣋⣉⣉⣉⣁⣀⣀
⠀⠾⣿⡿⠿⠿⠛⠛⠛⠋⠉⠉⠀⠀⢀⣠⢀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣤⡄⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⠟⠉⠛⣦⣄⣠⣶⠂⡉⣿⡁⡼⠻⣿⣿⣿⣶⣿⡟⠻⣀⣠⣤⣤⣴⣿⣿⣿⣿⡿⠛⠻⠄
⠀⠀⠀⠞⡴⡿⠋⠉⠛⠿⡏⠀⠠⠿⠿⠻⠿⣿⢿⠋⠙⠓⢿⡿⠼⣿⠻⠛⢿⡧⡞⣹⢿⡿⢿⠿⠿⣿⡿⢿⣇
⠀⠀⠀⠠⠂⢰⠀⠀⡜⠒⠁⠠⠀⠀⠀⠀⢲⡇⠀⠀⠀⠀⠀⠇⠀⠈⠂⠀⠀⠁⡜⠁⠾⠀⢹⡄⠀⠱⣼⠈⡆⢷
⠀⠀⠀⠀⠘⠋⠙⠀⠁⠛⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠉⠁
⠂⠀⠀⠀⠚⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣎⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⢠⡀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣷⡌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣧⠉⠀⣀⣀⠤⢀⡤⠐⠻⣿⣿⣿⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠖⢋⠵⣺⠿⠻⢆⣸⡿⣿⣿⣱⣷⡯⣿⣷⣶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⡀⢰⠇⡴⣩⣶⣶⣼⣿⣿⣿⠻⡇⢻⣷⡦⣄⠈⠉⠁⠿⣽⣷⣖⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠜⠁⣻⢸⣿⣿⣿⣿⣿⣿⣿⢉⣷⣄⣼⣿⣇⠀⠀⠀⠀⠀⠀⠉⠉⢆⡹⣿⣷⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⡠⣻⣡⠛⡅⠀⠀⠈⠁⠀⠉⠙⢿⣿⣿⠿⠋⢢⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠙⠪⢝⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣶⡶⠿⠚⡛⠁⠀⠃⠀⠀⠀⠀⣦⠀⠀⠀⠙⢿⠃⢰⠜⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣿⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠉⠀⠀⠌⠀⠀⠀⠰⠀⠀⣀⠀⢫⡷⠀⢿⠀⠘⣷⣄⣠⡘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⡗⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢫⠀⠀⠀⠀⣸⠂⣼⣿⣿⣿⣧⠉⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠻⣵⢀
⠀⠀⠀⠀⠀⠀⠀⠶⢺⣷⣆⠀⠀⠀⠀⠠⠞⠀⠀⠀⠇⠃⣶⣶⣿⣿⣿⣿⣿⣿⠇⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⢾⣿⣿⣿⣿⣿⡏⠀⠰⣿⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠰⠫⣇⡇⠀⠀⠂⠀⠀⢀⡄⠀⠀⠀⠀⠄⠈⠁⢠⣿⣿⡟⠀⠀⠀⣿⣧⡇⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⢰⢀⣿⣿⡟⠀⠀⠀⠀⢸⡇⠀⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠁⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠆⢸⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⢀⣾⣿⡟⠀⠀⠀⠀⠀⢀⣼⢷⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⢀⣼⣳⣾⣿⣷⣶⣶⣤⡀⠀⢀⠄⠚⠹⢉⣾⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢼⠋⢩⠏⠉⢩⠛⢻⣿⣿⣷⣆⡀⠀⢠⣿⣿⡟⠀⠀⠀⠰⠀⢠⣾⡭⣿⢿⠙⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠎⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣿⣿⡟⢀⡀⠀⣀⡤⠚⠘⢿⣿⣾⡧⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠹⣏⣘⣿⣏⢰⣿⣿⡿⠉⠀⠀⠀⠀⢹⡿⠻
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣀⠀⢸⣦⢰⣔⠀⠀⣀⡼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⣶⠐⢧⠘⢀⠋⠁⣠⣾⣿⢰⠃⢠⣾⣿⣧⣿⡿
⠀⠀⠀⠀⢻⣇⣿⣄⢻⠀⣾⠛⠛⢃⡎⣰⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠙⢶⣿⣿⣿⣷⣤⣤⣴⡏⣤⡿⣹⣟⣹⡇
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣼⣾⡿⣫⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠾⠿⠿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⣀⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⣴⣶⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣶⡞⣩⣾⢃⡠⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠡⢞⣿⣿⣿⠟⠉⣀⣤⠴⣶⣶⣾⣿⣿⢏⣾⢿⣿⣿⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⡇⣰⣿⣿⣿⢋⣴⠟⡩⢖⠋⠉⠀⠀⠀⠉⠈⠁⠛⠛⠶⣌⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢻⢻⣿⣿⣿⡿⠁⣨⡾⠃⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠙⣿⠢
⠀⠀⠀⠀⠀⢸⣧⠀⡇⠀⣆⣿⣿⣿⣿⠇⢰⣿⠁⠀⠀⢀⡴⠃⠀⠀⠀⣀⣿⣧⠀⠀⠄⢈
⠀⠀⠀⠀⠀⠀⡏⠻⣿⢸⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢠⣿⣷⠃⠀⢀⣀⠽⠿⠟⢻⡄⠈⢹⡀
⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣿⣿⣿⠋⠹⡇⠈⠃⣼⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢷
⠀⠀⠀⠀⢀⠀⢿⡀⣷⣼⣿⣿⣿⡇⠀⠈⠀⢰⣿⠀⣿⣿⣿⣷⣶⣶⣦⣤⡤⠀⠀⠀⠑⠀⠈⢇
⠀⠀⠀⠀⠘⣷⣼⣧⢸⣿⣿⣿⣿⣿⡀⠀⠀⠘⣿⠀⢻⣿⣿⣿⣿⣏⠀⠈⢿⣦⠀⠀⠠⢄⠈⠘⢦⡀
⠀⠀⠀⠀⠀⠘⣆⠙⠈⣿⣿⡿⢿⣿⣷⡄⠈⢄⠈⠣⠘⣿⣿⣿⣿⣿⣆⠀⠈⣿⣀⠀⠘⣿⠀⠀⠀⡇
⠀⠀⠀⠀⢀⡀⠘⣆⢠⣼⣿⡇⠀⢈⠙⠿⣦⡀⠳⣦⡀⠘⠋⠀⠈⢻⣿⣦⠀⣏⣈⠓⠄⣂⣀⣠⣴⡟
⠀⠀⠀⠀⠀⠹⣦⣘⣦⠻⣿⣿⡄⠈⢿⣶⣤⣍⣑⡈⠻⢷⣄⠀⠀⠀⠹⣿⣷⡈⠙⠷⠶⠤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠘⢿⡻⢷⡿⣿⣿⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⣬⣿⣿⣦⠈⠻⣿⣷⣬⡉⠉⠛⠛⠛⠀⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⣿⣿⣿⠷⣄⠙⢻⣿⣿⣶⣶⣤⣤⡄⠀⠀⣿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⣄⡀⣈⣻⣿⣿⠿⠟⠀⠀⠀⠛⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣶⣤⣤⣀⣠⣤⠆⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣯⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢀⡼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣠⠤⠶⢛⣉⣉⣉⣉⠙⠻⢶⡟⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠦⣄⣀⣀⣀⣤⠽⠶⣀⣠⣤⣶⣿⠿⠿⠛⠛⢯⡹⣶⣾⣥⣽⣿⡿⠛⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⡤⢄⣀⣠⣴⣿⣿⣿⠿⢋⣡⠴⠖⠚⠉⠀⠀⣹⡄⠀⢀⣰⣶⣶⠶⣾⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣤⣈⡉⢿⣿⣿⡿⠋⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⢀⡉⠙⠛⠻⢯⡷⠚⢉⡣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡤⢞⣻⣤⡿⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⣴⠃⣠⠟⠀⠀⠀⠀⢠⠻⣶⣤⣶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⣍⠛⠶⣄⡐⢦⣤⣖⣋⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢰⠇⠀⠀⠀⠀⠀⠀⢣⠻⣟⠻⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠒⠒⠺⠇⠀⠈⡛⢶⣬⣭⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⡞⠀⠀⠀⠀⠀⠀⠀⠘⡆⢹⡄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣙⣷⠌⠓⢶⣤⡠⡙⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⢀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡷⡀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠿⠶⠶⣶⣦⣽⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⣿⣷⣀⠀⠀⠀⢠⠞⠁⢤⡀⠀⠀⠀⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢻⣇⠙⠳⢶⣶⣇⠻⣦⡀⠙⣦⠀⠀⢹⡈⢇
⠀⠀⠀⠀⡀⠀⠀⠀⣠⠞⠉⠀⠒⠛⠿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣿⡄⠀⠀⠈⠻⣿⣦⣕⠀⠛⠃⠀⡼⢡⣜⡄
⠀⢀⣠⣄⣻⡦⠤⠴⠃⠀⠤⠶⠿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⢸⣷⠀⠀⠀⠀⠀⠉⠻⣿⣿⣆⠀⠀⣿⣏⡎
⠀⠀⠀⠙⢦⣙⠂⠀⠀⠀⠀⠲⢶⣤⡙⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣯⣹⣘⣷⣀⢈⡵⠃
⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠿⣷⣦⣻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣿⣇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠛⠻⠿⠿⢿⣶⣦⣤⣤⣤⣤⣤⣀⡀⠈⠙⠿⣶⣤⣀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⡿⠶⢦⣬⣹⣿⣷⣦⡀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣦⣄⡉⠙⠻⠿⣿⣷⡀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣦⣄⠀⠈⠁⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⣄⡀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⣀⣄⣀⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢦⠀⢀⣀⣘⣶⣿⣿⣿⢿⣿⣿⣿⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣭⣽⣿⣿⣿⣿⣷⡿⠿⠿⡻⢻⡷⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣽⣿⣿⣿⣿⠿⠋⢁⠴⠶⢖⡀⠁⠸⢸⣷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⠶⣿⣮⣿⣿⡿⣿⣿⠛⠁⠀⠀⠀⠀⣠⡜⠈⢤⣐⡨⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠪⠿⣿⣿⣾⣿⠟⠫⣾⠟⠁⠀⠀⠀⠀⠀⡼⣿⠀⠀⠀⡉⠃⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣾⣿⣿⣿⣿⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⣸⢟⣿⡀⠀⠀⠘⠂⠘⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣛⣿⡿⠋⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⢣⣾⠏⠙⢦⣴⠿⡂⢓⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣤⣀⡴⠒⠋⠉⠉⠉⠉⠓⠒⠚⠛⣿⡏⠀⠀⢀⠘⣇⠀⠀⠀⠀⡀⠀⠀⣾⣿⠀⠀⠀⠈⠳⡟⢃⠼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣇⠻⣆⠀⢠⡾⠁⠀⢰⣿⡇⠀⠀⠀⠀⠀⢹⣏⡟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠁⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠈⢻⣄⢌⡀⠀⠀⢀⠀⣸⣿⣷⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣾⣿⣿⣿⣿⢿⣿⢿⡇⠀⠘⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⡄⠀⠀⠙⠪⠛⢰⣶⡏⠀⣿⠋⢿⡇
⠀⠀⠀⣠⡴⢛⣻⣿⣿⠿⣿⣿⣿⣿⡿⠣⠛⠈⢸⣷⡀⠀⠹⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠸⠁⠀⠀⠀⠀⣠⠀⢿⡇⣸⠇⠀⢸⡷
⠀⠀⣰⣯⣶⠟⣫⠿⠛⣛⣛⣯⠿⠋⠀⠀⠀⠀⠀⢿⣧⡀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠰⠋⠁⠀⠘⠃⠋⢀⠀⣿⡇
⠀⢠⠟⠝⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡦⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⡠⢴⡟⡴⠀⢠⠀⣸⣼⣿⡃
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⡀⠘⣿⣿⣿⣿⣿⣶⣾⣿⣷⠀⠀⠔⢋⠦⣀⠡⡀⠀⣘⣀⡣⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣿⣿⣿⣾⣿⣿⣿⡟⠛⠳⠾⠀⠲⠦⠌⢳⡝⢿⣛⣿⡻⣿⢮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⠀⢹⣿⣿⣿⠇⠀⠀⠀⠀⠀⠐⠶⠶⠟⠷⢨⡟⠻⣦⢴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠸⣄⠈⢿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⠃⠀⠈⠳⣏⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠉⢦⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⣸⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⡄⠻⢾⣿⣧⠀⠀⠀⠀⡀⠀⢀⣀⣾⠇⠀⠀⠀⠀⢀⢟⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠿⣷⡀⢠⣾⣿⣷⣍⣼⡟⠀⣀⠀⠀⢀⢂⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⣹⣿⣿⣿⠟⠛⠛⠋⣴⣿⣿⣷⡖⢋⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠹⣿⣿⣇⠀⠀⠀⠉⠛⠛⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⣹⡏⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢷⠁⢘⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣍⠋⠻⡈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢉⣁⠀⠀⠀⠉⣼⡛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠈⠀⠀⠀⠀⠀⠛⢆⠀⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠚⠁⠀⠀⠀⠀⠀⠀⠐⡰⠀⠀⠈⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⣴⡖⢠⠶⠶⠲⡶⠞⠁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⡰⠁⠃⠀⠀⠀⠀⠀⠀⠀⢤⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠠⠄⠀⠀⠀⠀⠀⠀⠠⠁⠀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣄
⠀⠀⠀⠀⠀⢀⠤⠤⠤⡀⠀⠀⣠⠞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠴⠀⠈⠉⠑⢢⡀
⠀⠀⠀⠀⡴⠃⣈⣉⡁⠈⠑⠲⡇⠀⠎⡐⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠊⠀⠀⡇
⠀⢀⡤⠊⠀⡼⠁⠀⠙⢄⢀⣠⠆⢸⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠒⠦⣀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⢇
⢠⡏⢀⡖⠞⣡⢊⠉⠒⠢⠷⠿⢀⡎⠀⠀⠀⠀⠀⠀⠔⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠁⡰⠋⡆⠀⠀⠀⠀⠀⡇
⠀⠇⠀⠙⡆⡇⠈⡴⠦⣀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣏⠉⠀⡰⠁⠀⠀⠀⡘⢠⠆
⠀⢧⡀⢠⠃⡀⠈⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠄⢸⠀⠀⡇⠀⠀⠀⠀⣇⢻
⠀⠀⠈⠉⠀⡄⠀⢹⡀⠀⠀⠀⠙⠲⠦⢄⣐⡱⢾⣃⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⡀⠀⠑⣄⢤⡀⢠⠘⣗⠛⠁
⠀⠀⠀⠀⠀⣇⠐⣋⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⠲⠦⢤⣄⣀⣀⡤⠤⢤⣶⣄⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⡇⠀⠀⣈⣿⢧⡿⡇⣸
⠀⠀⠀⠀⠀⢱⠀⠈⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⢵⣌⣉⠑⠋⠀⠀⠀⢘⢇⠀⠀⠈⢀⣋⣠⡷⠃
⠀⠀⠀⠀⠀⠀⠣⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢦⡀⠀⠀⠀⠀⠱⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡓⠶⢤⣄⠀⠈⠑⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠸⡌⠳⡄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡄⢹⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣰⠔⠃⠀⡆⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢰⡃⠀⠀⢰⣧⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⣀⡸⠃⠀⡠⠋⠀⡰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⡜⠁⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠖⠊⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣶⣦⣄⡀⠀⠀⠀⠀⠀⢀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣉⣛⣻⣿⣿⣴⣶⣶⠂⠉⣼⠃
⢀⠒⠦⠶⣾⣿⣿⣿⣿⣿⣿⣭⣽⣿⣾⣯⣭⣥⣤⡾⢏⡀⠀⠀⢄
⠀⢉⣛⣛⠛⣛⣿⣻⣿⣿⣿⣿⣿⡿⣡⠏⠉⢉⡍⠀⠀⠙⠛⢿⣿⡁
⣴⡿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠏⣴⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⢈⡁
⠉⠙⣻⣿⣟⣛⣛⣿⣿⣿⣿⠏⣰⠃⠀⠀⣀⠸⣷⣤⣀⠀⣀⣠⣿⡗⠀⡀⠀⣴⡄
⠶⠿⣟⣻⣿⣿⣿⣿⣿⣿⡟⠀⡃⠀⠀⠀⣿⡆⠹⡇⠉⠉⠉⠉⠉⠻⢮⡱⣄⢭⡇
⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⠀⢇⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉
⣴⣾⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣇⠸
⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢻⡀⢃
⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣶⣶⣶⡶⠆⠀⠈⠻⣶⣦⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⢀⣴⣶⣤⣴⣤⡀
⠛⢻⣿⣿⡿⢿⣿⠋⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠀⠀⠀⠀⠀⢈⠉⠉⠛⠄
⠀⠀⠀⠙⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢀⡴⠚⠛⠛⠻⣶⡄⠀⠸⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠀⠀⠛⠉⢀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠙⢧⣀⡈⠻⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⠀⠀⣠⠾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣮⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡿⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠤⡄⠀⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣭⡙⠿⢿⣦⣿⣯⣻⣿⣶⣤⣤⣄⣤⣤⣶⣦⣤⣤⣀⣴⣿⣿⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⠘⠛⠿⣧⣿⣶⣤⣬⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⣠⣤⣤⠶⠄⠀⠈⠉⠿⣿⣷⣦⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⠖⠈⠉⠀⠘⠒⠲⢶⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠠⢀⣴⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠖⠛⣛⣩⣭⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣪⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷
⠀⠀⣠⣾⣿⣿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⠇
⠀⠀⠹⠛⢉⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡏⠀⢾⣿⣿⡆
⠀⠀⠀⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣷⣄⡄⠀⢿⣿⠀⢇
⠀⠀⠀⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢿⣿⠀⠈⢻⣆⠈⡀
⢀⣤⡾⠛⢙⣡⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⢹⣧⣄⠈⡋⠀⢱
⠀⠁⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢿⣿⣿⣷⣇⡀⢀⢷⡄
⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠙⠻⢿⣿⣿⣿⣿⣿⣆⠹⣿⣿⣏⢻⣷⣄⠾⣿⡀
⠀⢀⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡷⠈⠙⣿⣦⣿⣿⡂⠹⣷
⠀⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣼⣿⠂
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣼⣿⣿⣻⣿⡿⠃
⠀⠀⠀⠀⡟⠻⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠁⠀⠛⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠐⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠘⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣱⡿
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠛⠙⣿⣿⣿⣿⣿⢯⡿⢿⣿⣿⣿⣿⣿⡏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠱⠜⢀⡑⠝⠫⠝⠻⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⣿⣷⣶⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣟⣿⣙⢿⣿⣿⣿⣤⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠹⣿⣷⣿⡟⣩⣾⣿⣟⣉⡀⡤
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣠⣿⡿⠛⠋⠉⢩⡝⠉⠻⢿⣥⣼⡖
⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠋⠉⠀⣿⣿⣿⡿⠀⠠⣤⠖⠸⢷⠀⠀⠀⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠠⣄⡁⢀⣴⡍⠀⠁⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣥⣦⢤⣶⣿⣿⣿⠃⠀⠈⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣶⣿⣿⣿⣿⣷⣄⢻⣿⠋⠁⠀⢀⡀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⣿⣿⣿⣿⠽⢿⡇⠀⢁⣨⣽⣷⠘⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⡷⢾⠿⣷⡸⣷⠇⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣿⣿⡷⣿⣶⣽⣷⡀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣯⣿⠿⠿⠿⠿⢿⣶⣿⣿⣿⣿⣷⣐⣠⣶⠃⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⢳⡟⣸⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣯⠻⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠸⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⢿⢯⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣯⣗⣤⣀⣀⡠⠄⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠋⢹⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠟⢷⡀⠀⠀⠀⠀⠀⠉⢿⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠊⠀⣠⠘⣿⢁⡄⠀⠙⠶⣄⠰⣄⠀⠀⣸⡗⠟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣆⠠⠂⠀⢠⢄⣄⠀⢀⠎⡀⡀⣰⣿⢀⣼⣧⢶⡄⠀⡀⠈⢳⡟⣦⡔⢫⠥⡄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⢀⣀⣈⢀⠱⠆⣤⠀⣸⣿⣠⣴⡊⣡⣿⣷⡟⣿⣾⢿⣯⣾⡷⠊⢤⡄⠀⡇⣭⠛⢁⠀⠊
⠀⠀⠀⠀⠀⡀⢰⣾⣷⣶⣾⣿⣿⣿⣿⣧⢾⠷⣦⣿⣷⣠⡟⣿⣿⣿⣿⣿⣟⣡⣾⡗⢻⣄⣵⣯⡟⢻⢁⡷⠋⠀⡀
⠀⠀⠀⠀⠀⠳⠿⠛⠘⠋⠽⠉⠉⢉⣽⣽⣷⣿⣼⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣧⣾⣿⣿⣡⣯⣾⣡⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠛⠡⢴⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⣿⣿⡿⣿⣯⡝⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠛⠃⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣶⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢸⣿⡆⠀⠀⣿⡇⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠹⣿⣶⣦⣤⣶⣶⣿⣿⣿⣿⠿⠿⠿⠛⣿⣿⠀⢸⣿⣥⠾⠋⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⣀⡈⠛⣿⣿⣿⣿⡿⠛⣁⣤⠶⠋⠁⠘⠋⠁⠀⣠⣔⠶⣤⣶⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣄⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⢁⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠙⣿⣿⣿⣿⣿⡟⠀⣼⣿⠁⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⠏⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⢀⡴⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⣼⣿⢿⣿⡇
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠀⢸⣿⡀⠀⠀⠀⠀⠀⣴⠆⠀⠀⠙⢿⣿⣿⣿⡟⠈⣿⡇
⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠈⣿⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⣿⡇
⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢹⣷⣄⣠⣿⡇
⠀⠀⢹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠇
⠀⠀⠈⣿⠈⢿⣿⣿⣧⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠈⠙⠋
⠀⠀⠰⠃⠀⠈⢿⣿⣿⣇⠀⠀⠀⠸⣷⢸⣇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠙⣿⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠋⠀⠀⠀⠀⢻⣇⠘⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠸⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠋⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣶⡘⣆⠀⠀⠀⠀⠀⢀⣴⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢹⡄⠀⠀⠀⢰⣿⠁⠀⠀⠀⠠⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢻⡄⠀⠀⣿⡟⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⢿⡄⠀⣿⣿⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⢿⣄⣿⣿⡆⠀⠀⠀⠀⠀⢹⣧⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠁⠀⠀⠀⠀⠀⣨⣿⣿⡟⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⡇⣠⣤⣴⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⡿⣿⡿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠙⠳⣜⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣦⣍⣻⡿⣿⣿⣦⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠒⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠉⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣈⣉⣩⣽⣿⣿⣿⠟⠡⣻⣿⣿⣿⡇⠀⢼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⢟⡂⠀⠈⠙⠛⠛⠟⠋⠁⠀⢰⣿⣿⣿⣿⡁⠀⠸⡇⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢋⣴⡟⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠈⣿⣿⣿⣿⣷⠸⣆⠁⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡟⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠈⠱⡀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠸⣿⣿⠁⠀⠀⠀⠁⢸⣾
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⣿⣿⣿⠘⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣰⠿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠸⣿⣦⣤⡄⢠⡟⣿⡷
⠀⢀⣴⠟⠋⣠⣾⢟⣿⣿⣿⡇⠸⣿⣷⠋⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⣼⡿⠟⠋⢠⠟⣰⠿⠁
⠀⠀⠀⢀⡾⠛⠁⣼⣿⣿⠏⣿⠠⠟⠁⠀⠀⠀⣰⠈⠙⠻⠿⠦⠄⠀⢸⠋⠀⠙⢶⣿⡿⡃
⠀⠀⠀⠈⠀⠀⢠⣿⣿⡟⠀⣻⠀⡀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢡⡾⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⣿⢰⠃⠀⢠⣾⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣤⣠⣅
⠀⠀⠀⠀⠀⠀⠀⢿⡻⣟⠾⣿⣿⡄⠀⠈⢻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⡿⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⣬⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠻⣿⣼⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⣦⣸⣄⠀⠀⠀⠀⠀⣀⣤⡒⣀⣠⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⡏⢹⣿⡷⣤⣴⣾⣿⣿⣿⣿⣿⣟⣷⠶⣿⣷⣿⠛⠶⠶⢦⣀⡀
⠀⠀⠀⠀⣠⣤⣷⣾⣏⠀⠸⢿⡟⠿⠿⠶⣦⣭⣙⡻⢿⣷⣮⣝⡻⣶⣤⣀⠀⠉⠲⢄
⠀⠀⢠⣯⢿⣿⣾⣿⠿⠶⠒⠒⠙⠲⢤⣤⣀⣈⠉⠛⠷⣮⣿⣿⣿⣶⣝⢯⡻⢶⣄⡀⠑⢄
⠀⠀⢸⠏⣿⢿⡼⠟⢛⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣟⢿⣷⣽⣶⣝⡻⣶⣄
⠀⠀⣸⣾⡟⠀⠀⠈⠁⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⣛⢿⣎⠙⠮⡢⡍
⠀⠀⣿⣿⠀⡆⢰⣶⡶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾⣿⠶⣬⣮⣂⡀
⠀⠀⢿⡇⢸⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠉⢿⣗
⠀⠀⢸⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠢
⠀⠀⡟⢠⠁⠀⠀⠆⠀⠀⠀⠀⣀⠴⠛⠲⢽⡄
⠀⢸⠁⣸⠂⠀⠠⠂⠀⠀⢉⡟⠁⠀⠀⠀⠀⠙⢆⠘⣄
⢀⠏⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⠣⡀
⣎⣤⣄⡀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠙⢦⡀
⠈⡆⠶⠋⠀⢠⢂⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠙⠄
⠀⠈⠲⠤⠖⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢠⣴⣦⣦⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣼⣿⣿⣯⣙⣽⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣭⣿⡏⢹⣯⣥⠶⠟⣋⡿⠋⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣻⣿⡿⠋⣰⣿⡏⣡⡴⠆⠀⠀⠀⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣩⣽⣿⠈⠀⣴⠛⠛⠃⠸⣯⣿⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢘⣍⢻⣿⣀⠾⠁⠀⣼⡀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠋⠀⠈⣿⣤⠄⠀⢸⣿⡄⠀⠀⣷⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣤⣤⣤⣤⡶⠚⠁⠀⠀⠀⢴⣿⡿⠀⠀⠘⣿⣿⣦⣀⣿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⣤⡴⢿⡟⠁⠀⠀⠀⢿⣿⣿⣯⣍⢧⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠀⠀⠀⠀⠀⢸⣿⡟⠷⢭⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⢸⣿⣧⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣿⣿⡇⠀⠀⠀⠈⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠈⢻⠃⠀⠀⠀⠐⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣻⣿⡟⣿⠀⠀⢿⣿⡿⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡼⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣿⣿⣿⢹⡆⠀⢸⡿⣱⣿⣷⣮⣙⡿⣷⣶⣶⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⣿⣧⣴⡾⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⡿⢸⡇⢠⣿⣷⠟⣱⣿⠟⠋⠀⠀⠉⠛⠛⠻⠛⠋⢀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢞⣵⢏⡿⢁⡾⠁⢿⠉⣿⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠸⣿⣿⡇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠴⣻⢋⡞⠁⠸⣿⣧⢸⠀⠘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠊⠀⠀⠀⠈⣿⠈⡆⠀⠘⣆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⡇⠇⠀⠀⠘⣧⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⠀⠀⠀⠻⣦⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠎⢇⠀⢰⣞⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣀⣴⠿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣮⣂⠘⠿⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⡎⣬⣄⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠿⠿⠋⢹⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢭⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⠀⠀⠀⠀⠀⢀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣷⣦⣄⠘⠁⣿⣿⢶⣤⡀⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⡿⠋⡝⠀⢀⡿⠁⠀⠻⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣩⣿⣿⣿⣿⣿⣿⢿⡿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣠⣤⣶⣿⣿⣿⣿⣿⡿⢋⣵⠟⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⡿⠋⣴⠟⠁⠀⠀⢠⡞⡐⠀⠀⠀⢠⣤⣤⣄⠀⠀⠘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣶⣿⡿⣿⣿⣿⣿⣿⠏⣠⡾⠋⠀⠀⠀⢠⡿⢱⠇⠀⠀⠀⠀⠻⣾⡿⢧⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣴⣶⣿⣿⣿⣿⡿⢁⣴⠟⠀⠀⠀⠀⠀⡿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠴⣶⣿⣿⣿⡟⢠⡾⠃⠀⠀⠀⠀⠀⣼⡇⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⡟⢠⡿⠁⠀⠀⠀⠀⠀⢠⣿⠀⠀⣼⣿⣦⣄⣀⠀⠀⢀⣀⣄⠙⠻⢶⡄⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣾⣿⣿⣿⣿⡟⢠⡿⠁⠀⠀⠀⠀⠀⠀⢸⡏⠀⢠⣿⣿⣿⠟⠛⣿⣿⣿⢿⣿⡇⠀⠀⠹⡄⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣿⣿⣿⠁⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿⣿⡏⠀⠀⠹⣿⣿⣆⠘⢧⠠⣄⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⠿⠟⠃⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡟⠀⠀⠀⠀⠀⠙⢿⣦⡀⠁⠙⣠⣄⠀⠈⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⣷⠸⣆⠻⣿⣧⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠲⣄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡙⢦⡈⢋⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧






⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫⣽⣿⣿⣿⡿⠷⣒⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⣠⡞⢡⣿⠀⠀⠀⠀⠀⠀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣱⣿⡟⣽⡿⢋⣼⣿⣦⣾⠏⢀⣾⣿⣷⡞⠛⠉⠋⠉⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣾⡿⣿⣿⣿⢻⣿⡽⣿⢋⡼⠋⣠⡾⠉⠻⣿⠃⢰⣿⣿⣿⣿⣿⣆⠀⣨⠄
⠀⠀⠀⠀⠀⠀⠀⣼⣼⣿⣿⣿⡟⡿⣸⣿⣿⣿⣿⢻⢰⣿⢳⡿⣽⠇⣾⣿⢳⠃⡼⠁⣴⣿⢿⣦⣀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⢰⢳⣿⣿⣿⣿⢹⡇⣿⣿⣿⣿⡇⠇⢸⡏⢸⠃⡿⠀⡟⠁⢋⡄⢀⣼⢻⠇⠀⠈⠛⠻⢷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⡿⢸⠃⣿⢸⢹⠿⡇⠀⢸⡇⠾⠀⠃⠀⢀⣴⡏⢀⡾⠁⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⡇⢸⠀⣿⠘⠘⠀⡄⠀⠘⠃⠀⠀⢠⡄⣿⣿⠀⣾⠁⢠⣿⠀⠀⢠⡀⠀⠀⠐⢻⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⡇⢸⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣸⠇⠀⠈⣿⠀⠀⢿⣟⣦⠠⠄⠀⢻⡇⣿⣿⠋⡇
⠀⠀⠀⠀⠀⠘⡇⠏⠹⡿⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⢿⡟⠀⠀⠀⣿⠀⠀⠈⠉⠋⠀⠀⠀⠀⠇⣿⡟⡘
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠈⢷⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅⣱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠙⢦⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠑⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣦⣤⣤⣄⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⡉⠙⠛⠻⢷⣶⣶⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡆⡀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢆⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⡟⠁⣢⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⣯⠀⣠⣾⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠳⣿⠀⢿⡿⣆⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠈⠉⠀⠀⠁

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣷⣴⣾⣤⣿⣶⣿⣽⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣽⣿⣷⣛⣀⣀⣀⣠⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠙⠀⠀⠐⣀⣠⣴⣶⣿⣶⣷⣦⣤⣴⣶⣶⣦⣤⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠓⠀⠐⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠶⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⡉⠉⠉⠐⠂⠀⠀⠀⠘⢶⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠫⢿⣿⣿⣿⣿⠿⠛⠛⠉⠀⠀⠀⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣇⠀⠂⠉⠛⠋
⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠈⠈⠉⠀⠀⠈⠙⠛⠙⠻⠿⠟⠛⠿⠟⠁⣠⣶⣾⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣶⣿⠟⢻⣿⠟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡟⠀⠀⠸⣿⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠀⢀⣠⡴⠟⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠈⠉⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠤⣴⣶⣿⣿⣿⣦⣤⣤⡀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡈⣿⣦⡀⠀⣀⣤⣤⣤⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣅⣀⣠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣶⣞⡫⠽⢛⣛⣶⣀
⠀⠀⠀⠀⠀⠀⠀⣼⢵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⣛⣉⠁⠀⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠘⢉⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⡓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠋⢸⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠮⢭⡭⠭⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡤⠭⠲⣬⣓⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡷⢶⣶⣤⡉⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡿⠄⢈⠋⠻⣷⣶⠤⠔
⠀⠀⠀⠀⠀⠀⠀⣨⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡂⠒⠉⠁
⠀⠀⠀⠀⠀⠀⣰⣿⠿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠹⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠬⠍⠉⠉
⠀⠀⠀⠀⠀⢰⣿⣏⣄⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣏⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢋⡴⣻⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠛⠡⠊⠰⠃⣸⣿⣿⣿⣿⣿⢟⣿⣿⣿⡿⣿⢿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣿⣿⣟⣁⣀⣀⣠⣤⣾⣿⣿⣿⡿⣫⣯⣾⣿⡿⢋⡼⢃⠞⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣟⣿⠿⠟⣛⣩⣭⣭⣶⣿⣿⣿⣿⣿⣿⣯⣥⣾⣿⠿⠛⢉⡴⠟⢁⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣥⣴⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣏⣉⣁⣀⣤⠴⠚⠉⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣿⣿⠿⠛⣉⣡⠤⠤⠶⠒⠒⠒⠲⠖⠶⠖⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢞⠽⠿⠛⠁⠈⠉

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣄⣀⣀⡄⠈
⠀⠀⠀⠀⠀⠀⢻⣼⣿⣿⠿⠿⣿⣴⣄
⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⢸⡟⣄⠀⠀⠀⠀⣠⣿⣿⣿⣦
⠀⠀⠀⠀⠀⡿⠘⣿⡀⠀⠀⠀⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠃⠀⠈⠳⣥⣠⣤⣬⢿⣿⣿⣿⣗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⡟⠻⠮⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣿⣧⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣥⣴⣶⣿⠋
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣷⣤⣤⡀
⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠘⣿⣷⡄⠀⠀⣰⡄⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⣿⣿⣷⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠘⣿⣿⣶⣦⣿⣿⣦⡀⠀⠀⠀⠘⣿⣷⣴⣿⣿⣿⣿⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⡋⠁⠙⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠓⠙⠻⠟⠻⢿⠛⠻⠿⠿⠓
⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⢻⣿⣿⣇⠀⠀⣾⠁⠀⠀⠘⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⢿⣿⣿⢠⡀⣿⠀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠹⣿⣼⣧⠃⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⠸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠘⢿⣿⡏⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣤⣀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠙⠛⠉⠁
⣿⢿⡿⠿⢿⠟⡻⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣛⣻⣿⣿⣛⣛⣻⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠟⣤⣤⡼⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⠿⢿⡈⣠⠿⠻⠇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣲⣦⣄⠀⠰⠟⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣶⣿⣿⡏⠀⢀⠰⡦⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣋⣽⣿⣿⣿⡟⠀⠀⢸⠀⠲⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⣿⣿⣿⣿⡟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⢶⣿⣿⣿⣿⠟⠀⠀⠀⠀⢸⣇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣿⣧⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠻⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣤⣄⣄⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠤⣔⣒⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⢶⣴⣶⣦⣾⣿⣇
⠀⠀⠀⠀⠀⠀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠟⠃⠀⠀⠀⢀⠀⠀⠈⢿⣷⣶⣄⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⢙⡿⠛⠿⣿⣟⠳⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠩⠤⠶⠛⠉⠡⠛⠉⠁⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠈⢿⡏⠉⠲⠌⠤⣄⡀⢸⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢸⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡄⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠀⠀⠀⣄⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠹⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⢸⡇⠀⠀⠀⢰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⠳⣄⠘⢧⡀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣧⡀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⣿⡀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠃⠀⠀⠀⠘⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⢀⡀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣴⠀⢈⣷⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣿⣷⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠙⠳⣆⠀⠀⠀⠀⠀⠙⣿⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢘⡄⠀⠀⠀⠀⠀⠹⣆⠈⢳⡄⠀⠀⠘⢶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠀⠀⠻⢿⠿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡀⢻⣧⠀⠰⣦⣤⡀⠀⠀⣴⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣙⠤⠵⠖⠲⠿⠇⠙⠀⣾⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⣠⠶⣛⣧⡄⠀⡀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⡼⠀⠀⠀⢀⡼⠋⠀⡿⠁⠹⠾⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⢃⣤⠀⢀⡾⠁⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⢀⣴⣿⠟⢻⣇⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠈⠀⠀⢀⣿⡿⠁⠀⠤⣄⣠⣤⣤⣀⡙⢦⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⡿⠂
⠀⠀⠀⠀⠀⢀⣰⣿⣥⡞⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠈⠛⣭⣿⣿⠙⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠁
⠀⠐⠲⠶⣶⠿⠿⢻⡟⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⣀⠀⠀⠀⠀⠋⠗⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣾⣇⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⢻⣧⡀
⠀⠀⠀⠀⠀⠀⣿⡏⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠀⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⡇⠀⢠⡀⠀⠀⠀⠈⠻⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⣴⣄⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢹⣿⣿⠀⠘⣧⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠘⢀⡎⠙⠻⡆⠀⣸⣿⠃⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣷⣄⠀⠀⣠⠤⡀⠈⢧⡀⠀⠃⠀⢻⣿⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⠙⠛⠾⣷⣤⠈⠒⢄⠉⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠈⢧⡀⠀⠙⢦⣤⣄⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣷⣌⣷⠀⠀⠀⠀⠀⠙⠦⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠠⡀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠀⡏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⢲⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡆⠀⢠⣶⣄⠀⠸⣿⣿⡀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⣸⡇⠹⣷⣀⣿⣿⡇⠀⠀⠀⢸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⡇⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣲⣤⣴⣦⣤⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⣿⣿⣿⣿⠿⢿⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣿⡿⢛⣿⠏⠀⠀⠀⠀⢿⣿⣷⣾⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠀⠀⠀⠀⠀⠀⠈⢻⣿⡷⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⢻⣿⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣶⣤⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⣿⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣥⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣂⣀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣊⠀⠀⢀⣠⠊⠀⠀⢀⡾⠀⠀⠀⢹⣿⣷⣌⠁⠤⠄⣀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⢶⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣤⣤⡿⢁⡶⣶⡾⠛⠁⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠈⠍⠒⢿⣿⠿⠛⠛⠛⠿⣿⢿⣿⣿⣿⣷⣤⡈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠄⢚⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠈⠀⠹⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⡻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢮⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠂⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⢀⢄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⣿⣿⣿⣿⣿⣦⣤⡐⠀⠀⠀⠀⢀⡴⢀⡀⠀⠀⢀⣀⠤⠒⣹⣿⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠁⢰⣿⡿⠟⠋⠉⠁⠙⢿⣦⣤⣤⣶⠟⣠⣿⡟⠛⠋⠉⠁⢠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⢀⠊⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠟⢉⡴⢾⣿⣿⣷⡀⠀⡀⠀⠀⢀⠟⠁
⠀⠀⠀⠀⠀⢠⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡴⠋⠀⢈⣿⣿⣿⢃⣴⣿⠿⠒⠁
⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⠎⢀⣠⠖⠋⡩⠈⠁⠀⣿⠁
⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⡔⠁⠀⣰⠃⢠⣾⣅⡐⠁⠰⣄⣀⣼⡇
⠀⠀⠀⠀⠻⣿⠀⠀⠀⠀⠀⠀⠘⣷⣶⣾⠏⠀⣼⣿⣿⠀⠺⣤⣤⣿⠟⠁
⠀⠀⠀⡆⢀⣿⠀⠀⠀⠀⠀⠀⣠⣿⣿⠁⠀⠀⣿⡿⠟⢣
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣾⠉⠉⢻⠀⠀⢰⢃⣠⠔⠁
⠀⠀⠀⠋⠁⠈⠓⠀⠀⠀⠀⠹⣄⡤⠋⠀⠀⠈⠉⠁
⠀⠀⠀⠱⠄⠄