Roster Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⣀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠞⣻⢿⣙⣿⣿⡿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣽⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣩⣹⣟⣹⣟⣿⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣳⣏⣼⣿⣫⡛⣟⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣝⣿⣽⣿⣿⣿⣿⣿⣘⣟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣼⣻⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡿⠋⠙⠿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡿⠁⠀⠀⠀⠀⠀⠉⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⠶⣾⣿⠟⠦⠄⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣿⡿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢤⣤⣤⣤⢤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⠛⠛⠛⠛⠙⠛⠛⣛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠉⠛⠛⠛⠿⠿⠿⠿⠿⢿⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣏⣿⣯⣇⣿⣿⣯⣞⣿⣿⣬⣯⣽⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃



⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣏⣹⣷⣦⣤⣶⣦⣦⣤⣶⣦⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡼⢳⠀⠀⣿⠀⡧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢳⠀⢰⠇⢸⠀⢠⡟⢀⡇⠀⣟⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣾⠀⣾⠀⣿⣠⡾⣣⠈⣧⡤⠋⠀⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡇⠀⣿⣠⠏⠀⠘⠿⠖⠁⠀⠀⠀⠀⠀⠻⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠸⣄⡴⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣒⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣯⠷⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣼⣧⡾⢟⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⢀⡾⠹⠟⠁⣁⣀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⢠⠟⠀⢀⣠⡞⣿⡡⠀⠀⠀⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣢⣀⢀⣀⣴⣿⠀⠘⢯⣙⣛⠵⠃⠀⣀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠿⠿⠿⠷⢾⣷⣤⣤⣤⣤⣤⢶⠟⣿⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣤⠶⠞⠛⢉⠝⠒⠒⠦⣄⢀⡟⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡶⠿⠚⠛⣹⡟⢹⠀⠱⣄⠀⠈⢿⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⡼⠋⢠⠏⠀⠀⣿⠀⠀⠈⡟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⢀⡏⠀⠀⢠⠏⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡆⢸⡇⠀⠀⠈⠀⠀⡄⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣀⠱⣤⣀⠀⢀⡼⢁⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⡀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⣿⠋⠉⠁⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⣾⣼⠃⠹⡄⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣰⢿⠀⣰⣿⠀⣰⣿⠃⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣠⢶⣃⡆⢀⡼⠃⣸⡾⠁⣿⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡶⠋⠁⢸⣿⣧⠞⠁⠀⠈⠀⢰⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠚⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠠⠠⠠⠀⠠⠀⠠⠄⠀⠄⠤⠠⡄⠀⡄⠄⡄⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣿⣿⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⣀⣠⣤⣶⣶⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⠻⣿⣿⣿⣿
⠀⠀⢠⣦⡿⠿⣿⠟⠓⠲⠾⣦⣬⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠲⡖⠘⠓⢹⣿⣿⣿
⠀⣠⢿⡾⠂⢀⡴⠶⠒⠒⠲⣤⡙⣯⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⢹⣦⣤⡞⠉⠉⠄
⢀⣿⡝⠃⣰⡿⠁⢀⢀⣤⡀⠀⠙⢿⣿⣟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢸⣿⣿⡇
⣸⣿⡇⢀⡿⠀⣠⣽⣻⣿⣿⣷⣶⣄⢻⣿⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿
⣻⡏⠃⢸⠃⣾⡟⣽⠛⠛⠉⠙⠻⢿⣦⣿⣿⣼⠀⠀⠀⠀⢀⣀⡠⣴⣇⠀⠀⠀⠀⠀⠀⠙⠛⠁
⣿⡁⠀⠿⣻⣿⢠⡇⠀⣀⣤⠶⣦⡾⢿⣿⣍⣸⣤⡤⣶⡿⣻⡄⠠⠘⢻⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠸⡇⠀⠀⡇⠉⢸⣡⠞⢋⣴⣿⢿⣴⣤⠾⠋⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣻⣄⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠁⠀⠸⠇⢠⡾⠏⣰⣿⣿⣷⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣤⡆⠀⠀⠀⠀⣸⠃
⠀⠀⠀⢀⣴⣿⡇⢠⡟⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⣠⠀⢀⣿⣿⣦⡀⣤⢀⣦⠏
⠀⠀⢠⣿⠏⠘⣃⣿⠃⣿⣿⣿⠃⣠⠀⠀⠀⠀⡀⠒⠋⠀⠀⠰⠋⠀⢀⣼⣿⣿⣿⣿⣿⠟
⠀⢰⡟⠉⠀⠈⢻⣿⠀⣿⢸⡿⠀⠈⠐⣐⢀⣈⠥⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⠟⠋
⠀⣿⠀⠀⠀⠀⣸⣿⡄⢹⠈⣧⠀⠀⠆⠈⠟⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠉⢉⠀⢸⠀⢳⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⡿⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠁⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⡤⠀⠀⠀⠀⣸⣤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣯⣽⣿⣯⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠰⡟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢀⢸⡣⠈⣼⠧⠢⣵⣧⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠍⣆⠘⠃⢸⡁⠏⢃⣈⠛⠧⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡋⠁⣡⡾⢛⣟⠻⠿⢿⣶⠽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡿⢾⣿⢤⣿⣋⢱⡶⣶⣬⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢹⣿⣄⣹⣿⣧⢿⣸⡷⡚⠁⣼⢆
⠀⠀⠀⠀⠀⠀⠀⣤⢀⢀⠀⣀⡀⡄⣀⢀⠀⠀⠀⣿⣤⠀⣼⣯⣿⣿⡿⠁⢃⠞⠁⠈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠁⠁⠁⠁⠘⠀⠀⢸⣷⣄⡀⣿⣟⣉⠉⠁⣌⣡⣤⣶⣶⣦⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⡋⢭⣅⡂⠡⠁⠀⣠⠀⠀⣉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣿⡏⡻⠻⡲⠤⢒⠀⢃⢠⠠⣤⠘⢻⠋⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢪⣿⡀⣓⠰⠀⣉⡡⢃⢸⣄⢀⣶⠄⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠠⢴⡾⣹⢣⡀⢠⡠⠶⢀⡿⠉⢙⠀⠃⡀⠀⠃⠀⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⢐⢉⡘⢷⡿⣸⠧⡀⢀⡠⣻⢀⠞⣨⠈⢀⣷⡈⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢆⠛⢗⡒⠘⣧⠋⡤⣳⡶⣔⡛⢃⡰⣁⣀⢄⠁⠀⠄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢨⡵⠇⣿⠿⡶⠅⠀⠥⢿⡀⢤⠲⡅⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠒⠶⠋⠈⠰⠂⠙⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠿⠿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⠀⠀⠀⠐⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠙⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣦⡀⠈⢿⣿⣷⣄
⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠙⣿⣿⣿⣿⣿⣿⣆⠈⢿⣿⣿⣧
⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⢿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣷⠀⠀⣷⣄
⠊⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⣀⣀⡤⠄⠀⠀⠀⠀⣤⠶⢲⣤⣤⣄⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⢠⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⣠⣴⣾⠟⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣟⣿⣦⡀⠈⠛⠛⠛⠿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⣼⡿⠋⠁⠀⠀⠀⠀⣠⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠻⣿⣿⡿⢃⣼⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⢤⣤⣶⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣯⣤⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣌⣻
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⣾⡿⠟⠋⠉⢁⣼⣿⣿⣿⡿⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠘⠁⠀⠀⠀⠀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠸⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⡀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡘⠻⣿⣿⣿⣿⣿⣿⡟⣷⡄⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣆⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠿⠛⠿⣿⠛⢻⠿⢿⡟⡟⢻⣿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣾⣶⣶⣿⣶⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢦⡴⢻⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣀⠀⠈⠁⠳⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠋⠉⢩⠟⢻⣳⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠸⣆⠀⠘⢿⣤⣈⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢸⠃⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿
⠀⠀⠀⢀⣤⣾⠟⠋⠉⠁⠀⠀⠀⠀⠉⠛⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⢀⡠⠀⡀⠀⠀⠀⠀⠀⣠⠎⠀⣿⡇
⠀⠀⣠⣿⠏⠀⠀⣠⡤⠶⠒⠒⠒⠲⠦⣄⣀⠈⠙⠷⣶⣤⣤⣤⣤⠴⠞⠛⠁⠀⠸⣄⣠⡤⠖⠋⠀⠀⠹⣤⣀⣠⡴⠞⠁⠀⠀⣿⡇
⠀⣰⣿⠃⢀⡴⠋⢁⡤⠖⢚⣛⣛⣛⣒⠲⠬⣝⠂⠀⣀⠴⠒⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⣿⠇
⢠⣿⠃⣠⠞⢁⡾⢋⣤⠞⠋⣉⠀⢀⡈⠉⠁⠀⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⣾⡟
⣸⡟⢠⡏⠀⡾⢡⡟⢡⡆⢰⣿⢀⡞⡇⠀⠀⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⠦⢤⣤⣤⣤⠴⠚⠁⠀⠀⠀⣠⣾⠟
⣿⡇⢸⡇⠀⣇⣾⠀⡞⣇⣾⡇⢸⢹⡇⠀⠰⣡⢏⠀⡆⣠⢇⡀⠀⠀⠀⠀⣰⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⣿⡇⢸⢷⣠⡿⡇⢰⠇⠛⢹⡇⢸⢸⡇⠀⠸⠃⠛⢸⢧⠏⢸⣇⠀⡀⡀⢾⢋⡿⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⡿⠋
⢹⣇⠸⡆⠙⠁⣇⣼⠀⠀⠘⣇⢸⡈⣿⠴⡄⢀⣴⣏⣸⠀⢸⡇⣆⣧⢿⢸⡎⠠⠶⠤⠤⠴⠊⠙⢦⣄⢀⣴⠿⠋
⠀⢻⣆⢻⡄⠀⢻⡏⠀⠀⠀⠹⣎⣧⠻⠀⢿⣼⡏⠉⠻⠟⠷⠾⢿⣿⠘⣧⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇
⠀⠀⠉⠛⠿⠦⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠈⠻⣾⠻⣦⣄⠀⠀⠀⠀⢀⣴⠋⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠶⣦⣤⠾⢻⡷⢶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣗⠒⠶⡟⠀⡟⠀⠀⢘⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⡝⠀⢠⣯⣀⠀⠯⣶⣄⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣫⡥⢶⡀⢩⣤⢭⣷⣾⡵⠦⣄⠙⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠉⠀⠈⠛⠶⣭⣝⣮⠁⠀⠀⠘⢦⣿⡓⢮⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣽⠟⣀⣋⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡆⢀⣮⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣶⡿⣋⣵⣿⣯⣾⣷⡟⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢈⣥⣈⡁⡵⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢿⣿⣿⣿⣿⣥⣶⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠝⠉⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣟⣿⡿⠶⠟⠛⠻⢶⣭⣻⣿⣿⣿⣿⣯⡍
⠀⠀⠀⠀⠀⠀⠀⠐⢹⣷⣶⣾⣽⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠶⠦⣤⣦⣤⣤⣤⣌⠙⠻⣝⢿⣿⣯⡽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣿⣿⣿⢝⣷⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⡁⠀⠀⠀⣽⣿⣿⣏⡙⠻⣿⣷⣌⠻⣿⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢿⣼⡿⣌⠻⣍⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡸⢡⠊⠀⠀⠀⣸⣿⣿⣿⣿⣿⣦⡈⢿⣿⣧⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣿⣱⡅⢌⠻⡿⡻⣿⣿⣦⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢸⡇⠃⠀⠀⠀⠀⠀⢿⣝⣿⡌⣿⣿⣿⣜⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡿⣿⢿⢛⣌⡳⣘⣜⣾⣿⣿⣿⡛⠷⣂⢮⣝⣿⣿⣷⠒⠤⠼⣷⣡⣄⣀⢐⠀⠒⠤⢹⣿⣷⡏⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣷⡝⣏⢧⢛⢻⢼⣿⠻⡿⣿⠻⡿⣝⣻⣿⣿⣿⡟⢏⠁⠹⣶⡬⠛⢦⡈⠀⠁⠀⢠⣘⣿⣿⣧⣹⣇⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣼⡓⣝⠂⢏⠋⣧⣵⣬⣿⠮⣪⣻⣿⣻⣦⣼⣈⣷⣤⡈⡈⣿⡎⠛⢆⠀⣸⣿⣿⣷⢸⣿⡿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⢿⡿⡙⡿⣷⣿⣿⢿⡿⢯⣷⡛⢯⣏⣹⡏⢉⣿⣿⣿⣧⢁⠠⠀⡾⣾⣷⣿⣿⣿⣿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡻⣿⣿⣝⣮⠻⣋⣞⠻⣿⣿⡁⢻⡗⢾⣿⣿⣿⣿⡛⢷⣶⣿⣿⣸⣏⠁⢸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣾⣏⣻⣷⣤⣐⣞⣛⣿⣀⣹⣿⣦⣭⡛⠳⠦⣤⣽⣿⣟⣷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣛⠭⣿⣿⠷⣯⢟⣉⣀⠊⣉⣿⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣭⣿⣿⣯⡻⣟⢿⣿⣿⣷⣧⣄⣀⣀⣙⣻⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠿⠛⠑⠚⠛⠛⠋⠈⠉⠛⠿⠿⠿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣛⠣⡄⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣢⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⣾⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠸⠃⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⠢⠆⢙⠛
⠀⠀⠀⠀⢀⣴⣾⡿⢟⣋⣭⣥⣤⣤⣤⣬⣉⡛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⠄⠀⠻⣀⣶⣾⣤⠈
⠀⠀⢀⣴⣿⣟⣵⣾⢿⣻⣭⣶⠶⠶⠶⠶⠤⢍⡓⠌⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠃⠄⠔⡈⢿⣿⠻⠃
⠀⠀⠾⣫⣿⣿⣿⣿⠿⢋⣥⠖⢋⣩⡥⠶⠖⠒⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡴⠃⠞⢈⢄⠔⢀⣤
⠀⣠⣾⣿⣿⣿⠟⣡⣾⢟⡵⢙⣯⣵⣾⣿⠿⣿⣿⡯⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⡴⠠⡊⠜⠉⠘⠘
⣴⡿⢫⣾⡟⢁⣾⣟⣵⢋⣸⡿⠟⠋⢅⡲⣴⠔⣨⢖⡫⢗⡮⣂⢀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠰⠋⠂⢈⠄⠰⠁⠴⠃⡤⡀⠘
⠀⠀⡾⠋⢰⡿⢋⣾⠃⠋⠁⠀⠀⠀⢷⡏⠀⠚⣱⠽⣺⣫⠞⠑⡡⠁⡈⠅⢈⡀⠙⡈⠀⠀⠁⠀⠁⠋⠂⡴⡡⠀⠃⢀⠠⢠⠃
⠀⠀⠀⠀⠈⠀⠸⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠐⡩⢚⠀⣜⢥⠄⠉⢀⠠⠀⠠⠀⠒⠈⠄⠈⠁⠲⢢⢀⠐⢀⢁⢈⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⣦⡇⡇⡀⡀⠘⢔⠁⢠⢢⣤⠄⣤⢠⠄⢂⠀⠀⠂⠈⢐⠂⡁⠈⡀⠈⡀⠋⠈⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠇⣧⠇⢀⡎⠜⠈⠄⠉⠀⡀⡄⠜⡈⠀⡁⠔⢀⠉⠀⠀⣠⠀⠄⢠⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⢔⢤⠀⠆⡧⠇⠃⠘⠗⢈⠀⠀⠊⠀⠀⣰⠃⠃⡀⡦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠤⠄⢀⣒⠀⠀⠁⠀⠂⢀⣴⡾⠋⢘⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠔⠂⢉⡤⠞⠋⠡⠠⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠊⠁⠀⠄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠆⠁⠀⢀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣿⣷⣶⣶⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⣷⡶⣖⡒⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡀⢠⣔⣀⣀⣀⠢⠄⡔⢈⠝⠐⢅⡰⣦⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⢤⡤⠄⠀⡀⠈⠲⡇⠘⠀⠀⠀⠐⠠⢸⣕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢀⠀⠤⠑⠄⠁⢈⠀⠀⠀⠐⠀⠈⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⠠⢀⠒⠬⠁⠐⠄⢀⠔⠈⠁⣴⠟⢠⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⡞⢁⠐⢫⢒⠀⠀⠀⣐⡁⡀⠀⠀⡠⠄⢤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠉⠩⠀⢩⠋⡤⢀⣴⠯⡊⠔⡡⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠑⠃⠲⠙⢿⠖⠭⠒⠁⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡜⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠯⡀⢘⡖⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡥⠈⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠤⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡥⠐⠄⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠈⠀⠀⠀⡀⠀⠀⠀⠀⠁⡀⠀⠀⠙⠇
⠀⠀⠀⢀⠔⣂⣐⠢⡀⠀⡠⠁⠀⠀⢠⣜⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⡆⢀⠛⠀⡀⠀⠀⠀⠘⠀⠨⣥
⠀⠀⠀⡰⢉⠤⠤⠤⡈⠫⠄⠀⣴⠲⠞⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⢀⠀⠀⠀⠀⢭⠀⡚
⠀⠀⠀⡇⠸⠀⡔⠒⠬⡉⠁⠻⡈⠸⠠⠃⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠒⠁⠀⠀⠀⠀⠤⠈
⠀⠀⠀⠀⠀⠠⣉⣀⠤⠂⡰⢆⠱⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠀⠈⢳⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠓⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡼⠟⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣶⣶⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣽⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠴⣊⣤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⣰⣇⣴⣾⣿⣿⣿⣷⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⢀⣦⣀⣻⡦⠐⢀⠀⠠⠄⣾⣿⣾⣿⣾⣍⢉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣴⣿⢟⣠⣤⣶⢿⡋⢷⣐⢘⠂⣰⣿⣿⣿⣾⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⣿⣷⡛⣨⡏⣡⠇⣽⠦⣍⡡⡌⠸⣿⣿⣯⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣡⣾⣿⢧⣼⣧⣶⡟⠲⠦⣱⣿⣶⣿⡿⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣻⣽⣿⡏⠉⠉⢙⢋⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠤⠨⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣦⢠⠆⢲⣀⡆⠰⣄⣆⠐⠂⢰⠀⢰⡆⠐⣶⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣧⢻⡀⢸⠟⢇⠠⠇⠈⠻⠀⢸⠀⢸⡇⠀⣿⢺⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⠂⠁⠁⠀⠀⣠⣤⡴⢶⡄⠀⠀⠀⠁⠀⠛⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠉⠉⠉⠛⠓⠈⠉⠛⠒


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣭⣿⣿⣿⣯⣽⣭

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠟⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⡴⢋⣭⡙⠲⡄⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠻⢿⡟⠀⣏⠀⠻⠷⠃⢀⡌⠀⠀⠀⣿⡿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠿⠿⣿⣦⡄
⠀⠀⠀⠀⠀⢠⣾⣷⠀⠉⠓⠒⠒⠚⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⢠⡄⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⣀⣴⣶⣶⣦⡀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⢠⣀⠀⠀⢴⠚⠋⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⣠⣾⠿⠿⠿⢿⣿⣿⣄⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⠛⠻⠛⠋⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⣠⣾⠟⠁⠀⠀⠀⠀⠉⢿⣿⡄⠀⢹⣿
⠀⠀⠀⠀⠀⠀⢈⣽⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⢀⣴⡟⠁⠀⣠⣴⣶⣦⣄⠀⠈⢿⣿⡀⠈⣿⡆
⠀⠀⠀⠀⠀⢀⣾⣁⠲⠀⣤⠀⢠⡄⣠⣴⠀⠸⢃⣬⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢠⣾⠏⠀⣠⣾⠟⠛⠛⢻⣿⣆⠀⠘⣿⣇⠀⢿⣇
⠀⠀⠀⠀⠀⣼⡿⢃⣴⡇⢀⣴⠀⠸⠛⣡⣶⣠⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀⠀⢀⣾⣿⣱⣿⠃⢠⣾⠏⠁⣀⣀⠀⠀⢻⣿⡄⠀⢻⣿⡀⢸⣿
⠀⠀⠀⠀⢰⡏⠀⠸⠋⠀⣾⠋⣤⢀⣾⣿⣿⠿⠟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠇⣴⡿⠁⣴⣾⣿⣿⣧⠀⠘⣿⣇⠀⢸⣿⡇⢸⣿
⠀⠀⠀⠀⢸⣧⣾⡀⣰⣷⠀⣾⣿⣿⡟⠉⠀⠀⢳⡄⠀⠀⠀⣼⣧⠴⠶⠾⠛⠿⢿⣿⣿⣾⡿⣡⣾⡿⠿⢿⣿⣿⠀⠀⣿⣿⠀⢸⣿⣿⠘⣿
⠀⠀⠀⠀⠈⢿⣿⣷⣿⣿⣿⣿⣿⣿⠀⣀⢘⡷⢋⠀⣷⠀⣘⣯⣀⣀⣀⣠⠤⠤⠶⠿⣿⣿⣿⣿⠟⣁⡀⠀⣿⣿⠀⠀⣿⣿⡄⠀⣿⣿⡄⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢀⡿⠉⡀⣸⠏⢡⣀⣽⡎⢹⣏⠀⠀⣀⣀⣠⣤⠼⣿⣿⣷⣿⣿⡇⠀⣾⣿⠀⠀⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣧⡀⢀⡿⠁⣦⠼⢏⠀⣹⠆⠹⣿⡏⠉⠀⠀⠀⠀⠤⠾⢿⣿⣿⡇⠀⣿⣿⡆⠀⣿⣿⡇⢀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣦⣄⠰⠛⠀⣸⡏⠁⣷⠀⢙⣿⠖⠋⠁⠀⣀⣠⣴⣾⣿⣿⠁⠀⣿⣿⡇⢠⣿⣿⣧⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣁⠀⠈⠁⠀⠈⢻⣶⣶⣾⣿⣿⢻⣿⣿⣿⣿⠀⠀⣿⣿⡇⢸⣿⣿⣿⢸⣿⡏⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⣿⣿⣿⣿⡇⢀⣿⣿⣧⢸⣿⣿⣿⢸⣿⠁⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠘⣿⣿⣿⡇⠨⣿⣿⣿⣸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡏⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠹⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠹⣿⣿⡇⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣶⣾⣿⣿⣿⣷⣾⣿⣶⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠘⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⣭⡿⠿⠛⠉⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⣿⣷⣶⣾⣿⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⣫⣿⠿⠛⠋⠁⠀⠀⠈⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⣒⢦⣀⠀⣜⣙⢦⡀⠀⡔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡻⣿⣿⣷⣾⣼⣿⣿⣾⣿⣿⢀⣤⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠙⢮⣛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣃⡤⢤⣬⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠯⢿⣿⣿⡽⢛⣿⢺⡉⢴⣚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⢻⣭⣧⣄⣤⡞⢗⠲⠾⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣴⠿⣏⢾⡿⢹⢻⣿⣿⣎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢎⣞⣥⢠⢻⣼⠁⠸⡜⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠋⢵⡏⣾⢻⠀⠀⠈⠑⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡆⢶⡄⢶⠸⣇
⠀⠀⡠⢂⡤⢊⣠⣶⣿⣿⣿⠿⠿⠿⠿⣿⣶⣮⣽⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠞⡘⠃⢚⡹⣆
⠀⠜⡴⣡⢶⣿⠟⣋⣭⣴⣶⣶⣶⣶⣶⣶⣤⣉⠛⢿⣷⣝⢦⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⣈⠧⢸⠋⣩⡑⠟⢷⡀
⠀⢊⣴⡵⢋⣴⣾⡿⠟⢛⣉⣉⣉⣙⡛⠿⣟⣿⣷⢄⠹⣟⣯⢷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⣀⢉⢉⡠⣦⡺⡯⣝⣦⡀
⢀⣾⠋⡴⣫⢟⣡⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⢮⢯⣳⡘⣾⡞⡇⠀⠀⠀⠀⠀⢰⣧⠚⢧⡟⢧⡞⡱⢾⣼⣳⣶⣌⠣⡀
⣼⠃⣾⡞⣱⣯⡿⢟⣩⣴⣶⣶⣶⣶⣭⣛⢿⣿⣮⣯⢿⣷⣿⣿⣿⠀⠀⠀⠀⠀⣨⠏⣰⣷⣾⣷⣾⣿⣾⣿⣿⣿⣿⣷⡙⣦⠐⡀⡀
⠃⣸⠏⣰⣿⠟⣴⣿⠿⠛⠉⠉⠉⠛⠿⣿⣷⣿⣿⣿⣾⣯⣿⣿⡿⠀⠀⢀⣠⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣧⢡⠐
⠀⡟⢰⣿⠃⣼⡿⠁⣠⣶⣿⣿⣿⣿⣶⣼⣿⣿⣧⣿⡿⣿⣟⣯⣧⣶⣻⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡆
⠀⠁⢸⡇⢠⡿⠀⣼⣿⠟⠉⠉⠉⠙⠻⣿⣿⣽⣿⣾⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣸⠃
⠀⠀⠸⠇⠘⡇⢸⡿⠁⠀⠀⠀⠀⢀⣴⣾⣿⠛⡩⢋⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣠⡏
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣴⣿⡿⣻⢫⣪⡖⢎⡞⣼⣿⣿⣿⣿⣿⣿⣧⣥⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢰⣧⣿⢫⡇⣼⠧⣌⠻⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣾⣷⣿⣾⣷⡿⠞⣫⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⢸⣿⡏⠸⣿⡇⠀⠈⠙⠮⢝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣭⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠸⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⢯⢿⣿⣿⣻⣿⣿⡏⡿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣇⠀⢸⡇⠀⠀⠀⠀⠀⠈⢯⢿⣧⡏⠳⣝⠗⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠳⣼⠃⠀⠈⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠉⠳⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠐⠉⠳⠶⢾⣿⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⢧⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣲⣶⢿⣦⣶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣦⣄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⡿⣿⡟
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣾⣿⣟⣻⣿⣟⣛⣛⣿⣿⣿⣿⣿⣦⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⢦⣗
⠀⠀⠀⠀⠈⠉⢉⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⢀⣴⣿⠿⣫⣿⡿⢋⣼⣿⡟⠊⣉⣽⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏
⠀⠀⠀⢀⣾⡿⠁⢰⣿⠏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄
⠀⠀⢠⣿⠏⠀⢠⣾⡏⢠⣿⣿⣿⣿⣿⣿⣿⠟⠛⣉⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣦⣰⣶⣟⣸⣧
⠀⠀⢸⡟⠀⠀⢸⣿⠀⣸⣿⠁⢹⣿⣿⣻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣆⠇⢠⣤⣤⣶⣴⣿⣿⡟⣹⣿⣿
⠀⠀⣿⠃⠀⠀⠸⡇⠀⣿⣿⠀⣼⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠈⠉⠙⠏⢱⣿⣟⣿⡿
⠀⠀⣿⠀⠀⠀⠀⣷⠀⢹⣿⢠⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠟⢭⣿⣿⠃
⠀⠀⢹⠀⠀⠀⠀⠈⠀⠈⢿⣜⣿⣿⠃⣼⣿⣿⢿⣿⣿⣷⣶⣿⣿⣿⡟⡅⠀⠀⠀⠀⠀⣰⣦⢀⠀⣄⢀⠀⢠⡴⢓⢠⡘⡺⣽⡽⠃
⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⢠⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠄⡀⠀⢀⢐⣷⣵⡡⢈⣬⢄⡤⠷⡐⠂⠀⣰⢾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⢀⡄⠀⠇⢀⣸⣿⡿⣻⢗⣿⣾⣿⠶⢀⣀⣈⣙⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⣿⡌⠈⢿⣿⣿⢹⣿⣿⣿⣿⣆⢧⣧⢀⣿⣿⣿⣯⣾⣿⣿⡟⠌⣴⢼⣴⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠻⣿⡆⠉⢻⣟⢿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⢿⣷⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠈⢻⡀⠀⠻⡎⠻⠿⣿⣿⣿⣿⢿⣿⣿⣿⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠀⠀⠈⠉⡉⠃⠀⠀⣯⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⠀⠀⠙⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⢦⣀⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠑⡈⢣⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠀⠀⠀⡀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠨⠆⢄⠀⠀⠀⠀⠀⡘⠀⠀⢀⠄⠂⠉⠈⠈⠇⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠀⠂⠀⠠⢄⡀⡠⠐⡢⠤⢄⠀⠀⠬⢄⠀⠀⠀⠰⠀⠀⠠⠃⠀⡀⠄⠀⠀⠬⣄⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠐⠂⡄⠀⠀⠙⣷⣤⠀⠀⢱⠀⠀⢼⠀⠀⠀⠃⠀⢠⠁⡠⠊⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠈⠢⡀⠀⠘⠄⣀⡀⠊⢠⠀⢀⠇⠀⠀⠀⠀⠂⠜⠀⠀⢀⠠⠤⠇⠀⠀⠀⠉⠢⢀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠑⠉⠉⡱⡤⠈⢁⠀⠠⠀⠀⠀⢀⠀⢰⠀⠀⢰⠀⠰⡌⠀⠠⠊⠀⠀⢰⠀⠀⠀⠀⠘⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠊⠀⡘⠂⠄⠀⠀⠄⠠⠐⠉⠉⠁⠀⠀⠈⠀⣸⠀⠐⠁⣀⠄⠀⢸⠀⠀⠀⠀⠀⠃
⠀⢠⠑⠤⡀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⡀⠄⢠⠎⢀⠎⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⢁⠎⠀⠀⠀⠀⢃⠀⠀⠀⠀⢸
⠀⠈⢆⠀⠈⠑⠃⠀⠈⢄⠀⠀⠀⠀⡸⠂⠀⠈⠀⠊⠀⢀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠁⣤⠂⢀⠄⠴⠀⠀⠈⢆⠀⠀⠀⢀
⠀⠀⠀⡇⠂⠄⠡⣀⠀⠀⠁⠢⢤⠎⠑⠂⠠⠀⠂⠠⠄⠊⠁⠈⠉⢡⣌⣁⠀⠀⠀⠀⠀⢠⢰⣇⠄⠀⠀⡄⠀⠀⠀⠀⠁⠂⠤⠜
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠈⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡀⠠⠄⢈⣿⡃⠀⠀⠀⠇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠂⠀⠀⠘⠄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠢⢀⣀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢄⠀⠀⠀⠀⠀⢀⡀⣀⡀⢃⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⡨⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠇⠀⠑⠄⡀⠀⠂⠠⣀⡀⠠⠐⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⣠⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⡀⠀⠐⢄⡀⠀⠀⠀⠀⣀⠠⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠘⠛⣆⠙⢄⡀⠀⠀⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠰⠀⠉⠂⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠄⠀⣀⡠⠁⡆⠀⠀⢀⡠⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡮⠅⠀⠤⠀⠀⠀⠀⠁⡲⠘⡤⠊⡡⠁⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠒⠒⠒⠈⠀⠀⠁⠙⠅⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⢀⣠⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣛⡓⠰⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢗⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⢛⣿⣷⣮⢻⣆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣲⡾⣿⣁⣀⣥⠿⠂⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⣿⣿⠟⠋⠁⠀⠀⠀⠙⠁⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣄⣤⣴⠶⠶⣆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⣿⣭⠉⣉⡛⠿⣦⣶⢦⠈⢧⠀⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣷⡄⠀⠀⠙⢄⠀⣀⠹⠦⠉⣤⡉⠒⠄⠇⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣦⠀⠀⠀⢹⣹⡀⠠⠀⠛⠑⣦⠀⣇⢘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣄⠀⠀⠀⠀⠀⠀⠀⢠⣴⢠⣤⣀⣤⣼⣾⣿⣿⣻⣿⠀⠀⠀⠈⣿⢹⡄⢰⣄⢷⢹⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢷⣦⣤⣀⣀⣀⣠⣿⣿⢯⣩⣽⠻⠿⠟⡟⠶⢋⣽⡀⠀⠀⠀⣿⣿⠁⡌⠻⡦⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⣿⣿⣟⣻⣏⣀⡞⢷⣦⣉⠙⠛⣋⣠⡶⠋⣹⡇⠀⠀⢀⣿⠃⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣯⡿⣽⣿⣦⣿⣟⣻⣿⣿⣿⠳⢾⣿⡇⠀⠀⣼⣿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣾⣷⣧⣾⣿⣿⣟⣛⣻⣿⣿⣿⡇⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⠗⣴⣶⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⣾⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣿⡿⠛⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠏⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⠀⠀⢀⣀⣀⣠⣾⣿⣟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⣤⣤⣤⣴⣾⣿⣿⣶⣶⣤⡘⠟⢻⠟⣿⠿⠿⠛⠛⠉⠁⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⢦⣀⡤⣀⣄
⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠌⠀⠉⠀⠉⠈⢳
⠀⠀⠀⠀⣰⠋⠀⣀⠤⠤⠄⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠀⠀⠴⠄⣠⠀⠘⣧
⠀⠀⠀⠀⡏⢠⡞⠁⣀⣀⡀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⣠⣆⠀⠰⠀⠠⠰⡏
⠀⠀⠀⠀⣇⣿⡴⠋⢁⣠⠼⠓⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠘⠛⠀⠀⠈⢶⡶⣿
⠀⠀⠀⠀⠹⠘⣠⠞⠉⣀⠀⠀⠀⠉⠐⠤⡀⠈⢧⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱
⠀⠀⠀⠀⠀⣰⠃⢀⡞⠁⠀⠀⠀⠀⠁⠂⠈⢢⡘⡆⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⡆
⠀⠀⠀⠀⢠⠇⠀⠌⢀⠀⣀⣀⣀⠀⠀⠐⠀⠀⡀⢹⡀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⢸⠀⠘⢠⣶⡿⠛⠁⠀⠄⠆⠀⢀⠀⠨⠓⠹⡄⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⢸⠀⡶⣻⠋⠀⣠⠶⠶⠂⡀⠀⠴⠂⠀⢀⡄⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠘⠞⢱⠃⢀⡼⠁⠀⠀⡈⠀⠀⠀⢠⡆⠏⠀⠀⢆⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣀⡾⣇⡀⠀⠀⢀⠀⢐⡅⠸⠀⢀⡤⠇⠈⠀⠀⠀⠀⠁⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⡇⢚⠀⢈⠃⠈⡄⠐⡐⡆⠃⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⢈⡔⠡⠏⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡾⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡇⢀⣠⠀⠀⠀⠙⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⠀⠐⠊⠈⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢀⡴⠻⣄⠠⠦⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠦⣠⠋⠀⠀⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⠀⠀⠀⢸⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⣼⠀⣧⠀⠘⢧⣘⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⠀⢹⣤⡤⠤⠼⠀⠀⠛⠛⠛⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼⠇⠘⠛⠋⣩⣷⣶⠢⢤⣐⢲⡖⠦⢷
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣛⣛⣓⣒⣲⣦⣀⣲⣤⣽⣾⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⠻⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠀⣠⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⣿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣿⣿⠀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⣉⣉⣙⣿⣿⠀⣈⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣷⣿⣶⣿⣷⣿⣾⣷⣾⣶⡀⠙⠛⠛⣿⣿⠀⢉⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠋⠉⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠙⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⡆⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠇⠟⢧⠏⠭⠧

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣤⣼⠥⣶⣤⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣋⣴⡛⢗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣯⣿⠋⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡥⣠⣦⠄⢐⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⡎⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡶⠛⠆⢞⠛⠛⠻⣟⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠠⢇⣤⡀⠀⣴⣿⡸⠶⠿⠿⠻⣶⣂⣤⡴⣾⣯⢭⡑⠄⠘⡦⠗⣅⡀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⠀⢸⣿⣇⣤⣿⣹⠇⠦⣀⣖⠳⣨⠙⠦⠠⠡⡴⢤⢺⣳⠕⢄⠀⠈⠻⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣏⡀⠀⢻⡿⣿⡝⠷⢢⠻⣙⠯⣾⢿⣄⣠⡄⢈⠂⢤⡙⣿⠐⣆⣲⣦⠀⣿⢸⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣼⣼⣷⣿⢣⣶⣦⡐⡍⣾⠙⣎⣓⢈⠃⡉⠐⢎⠗⠘⣿⣿⡟⣽⡇⠈⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⡌⡿⢿⠻⣟⠇⡷⠻⠀⣤⡷⠀⠅⠀⠀⠀⣳⠄⠹⣾⣿⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣔⢈⠀⣿⣧⠀⢀⣰⢿⡭⠤⣤⡀⠀⢀⣿⣼⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣼⣳⡶⣯⣶⣏⠁⠈⡇⡀⠀⠀⠀⣼⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣿⣯⠃⢠⠃⠫⢄⢀⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⣿⣮⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⢉⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠈⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠕⠊⠖⠐⢀⡴⡤⠖⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡿⣿⣿⡟⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣷⣿⣷⣴⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠓⠁⠘⢻⠔⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠁⠀⠴⢶⠒⢤⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠀⢁⠈⠆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡇⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣠⣬⣉⡑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⡇⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⡴⠛⣀⣉⣁⣒⠪⢗⣦⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡶⠴⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡘⠠⢊⠴⣋⣩⣤⣤⣍⣻⣗⢝⢮⢢⡙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣰⠁⠀⠵⠊⠁⣀⡒⠦⣌⢝⠿⣦⠱⣷⡙⣞⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢰⣧⢀⢀⣛⠺⠷⢤⣤⡤⠔⠁⠀⠀⠀⠠⠠⢬⣉⠙⠿⡆⠙⢟⣆⠘⢧⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣼⣇⣾⡿⠏⢾⠙⠃⠀⢀⢠⠀⡰⣄⢀⠠⠀⠠⣈⡁⢔⢌⢳⣄⠙⡄⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠟⠘⠏⠀⠀⠈⠁⠀⠀⠀⢢⠹⣷⡝⣮⡇⢡⠑⢬⡸⡵⡝⢷⣯⠳⣼⡀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠻⡟⡇⢷⠀⠃⡀⣟⣮⣿⣾⢿⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢉⣸⣄⢃⣷⡿⣾⠙⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠑⠢⢬⣻⣾⠻⢋⣴⠋⠙⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣀⢤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⣞⣁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⠋⠀⠹⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠃⠀⠀⠀⢃⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢁⣣⡀⢀⣀⣤⣞⣨⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠹⠋⠀⠉⠈⠁⠀⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠠⠀⠐

⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠛⣛⣛⣛⣛⣛⣛⣃⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⣿⡿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⣿⣟⣛⣛⣛⣛⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⡷⠶⠶⠶⠶⢾⣿⡇⠀⠀⠀⠀⠈⣿⣿⠀⣿⣯⣭⣭⣭⣭⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣿⣟⠛⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⠀⠸⣿⠀⣿⣯⣤⣤⣿⣿⢛⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⣿⣟⣛⣛⣻⣿⠿⠿⠇⠀⠀⠀⠀⠀⠀⢻⡀⢿⣿⣿⣿⣿⡿⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⡇⢸⠟⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣶⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡏⣩⠭⠭⠭⠭⠭⠭⠭⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⣀⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢰⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⠀⠀⠀⢠⠖⠚⠁⡇⢈⣉⣉⣉⣉⣁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠐⠒⠒⠒⠒⠂⠀⡇⠀⠀⠀⠀⠈⣇⠀⠀⡇⠠⠤⠤⠤⠤⠤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠒⠒⠒⠒⠒⠒⠀⡇⠀⠀⠀⠀⠀⠸⡄⠀⡇⠀⠖⠒⠀⠀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢸⠀⠘⠙⠛⠋⠛⠃⠀⡇⠀⠀⠀⠀⠀⠀⢳⠀⣇⣀⣀⣀⣀⣀⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠃⣿⢸⠀⠉⠉⠉⠁⠀⡖⣶⠇⠀⠀⠀⠀⠀⠀⠈⣇⠀⣀⣀⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⣾⣿⣛⣛⡿⣗⢲⢶⡶⡒⡶⡶⠖⠲⠶⠶⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⣧⣯⣽⡀⣹⢮⠴⡆⢈⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠍⢉⣽⡭⡝⣹⡛⠯
⠀⠀⠀⣿⣿⢚⢲⡽⡁⢊⣶⣿⣾⣿⣿⣷⡦⠀⠀⢀⠀⣀⠀⡀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠀⠆⣴⣍⢈⣏
⠀⠀⠀⣿⣿⣛⣽⣍⡀⣾⣿⣿⣿⣿⣿⡏⠀⠀⠃⠈⠆⠛⠀⠁⠀⠀⠆⢈⡑⠃⠁⠐⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠑⠸⢡⠒
⠀⠀⠀⣿⣿⢶⡤⠼⠄⢊⣭⣿⣿⣿⠻⠉⢿⡁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠂⠀⠁⢋⣡⣴⣶⣶⣶⣶⣤⡀⠀⠈⣽⢟
⠀⠀⠀⣿⣿⡛⣺⠂⢠⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢒⣶
⠀⠀⠀⣿⣿⣽⣭⠖⠈⠿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣙
⠀⠀⠀⣿⣿⣷⠚⢥⡄⠠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢰⣿⣿⠿⣿⣻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣩⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣻⣿⣭⣿⡿⣿⣧⢻⣿⢿⣿⣯
⠀⠀⠀⣿⣿⡴⡾⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣸⣿⣿
⠀⠀⠀⣿⣿⣗⣖⡫⠀⠀⠀⢰⣤⣀⣧⣻⣀⡶⠬⣶⠚⠂⠙⠓⠀⣂⢶⣾⣿⡿⠿⠍⠓⠠⠈⠻⣏⣿⣿⣿⣿⣿⣿⠇⣸⣿⣿
⠀⠀⠀⣿⣿⣿⣽⡇⠀⢸⠂⣿⣿⣿⢿⣿⢡⠀⠀⠀⠀⠀⠀⢀⡀⢫⣿⣿⣿⣦⡳⡄⠀⠀⢷⠀⢹⣿⣿⢣⣿⣿⣿⠀⣿⣿⠷
⠀⠀⠀⣿⣿⡗⡳⡍⠁⢸⣿⣇⣼⢽⣿⣿⣦⣄⡀⠀⠀⠀⠀⢸⠟⠉⣭⣻⣿⣟⣷⣹⣎⣆⢸⡆⢸⣿⣿⣻⡟⣿⡟⢠⡿⣡⣯
⠀⠀⠀⣿⣿⣟⣽⣿⣢⠸⣿⣿⡿⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣧⡆⠈⢽⣿⡿⠋⣿⣿⣾⣾⡇⠘⣿⡏⣿⠀⣿⠃⠋⠑⠾⠛
⠀⠀⠀⣿⣿⡯⣤⣬⣧⠀⠈⠻⣬⣿⣻⡻⣿⠉⣻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣟⣙⣛⣿⣿⣿⣿⡿⢰⡟⠀⠿⢹⣿⡆⠀⠈⢿⣿
⠀⠀⠀⣿⣿⡟⠟⣟⡣⠀⠀⠀⠈⠛⠫⣤⣤⣅⢼⣿⠛⡿⠿⣿⣿⣿⣿⠿⠿⢿⣿⡿⠟⢹⣿⡇⠈⠄⢰⣟⣼⣮⡃⠀⠀⣞⢧
⠀⠀⠀⣿⣿⣯⣯⣵⠗⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡿⠆⠦⡀⠀⠀⢈⣄⠀⢀⣤⠉⠀⠀⠚⠻⣷⣶⠗⢻⣿⣿⣿⣇⠀⠀⢩⣯
⠀⠀⠀⣿⣿⡿⡾⢷⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠠⢳⣤⣴⣾⣿⣿⡿⠁⠀⠀⠀⠀⠶⠾⠿⢦⡺⠿⠿⠟⠄⠀⠈⢉⢞
⠀⠀⠀⣿⣿⣏⣍⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⡿⠻⣿⡿⠁⠀⠀⠀⠀⠀⢠⣾⡄⢸⣷⢀⣶⣶⡇⠀⠠⢠⠿
⠀⠀⠀⣿⣿⡯⠿⠵⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⠟⠁⠀⢹⡁⠀⠀⠀⠀⠀⠀⣿⢋⠁⠻⣿⣿⣿⣿⡇⠀⠀⢳⡶
⠀⠀⠀⣿⣿⣗⡲⡺⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠀⠀⠀⡜⡅⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠛⢛⠛⣻⡏⠀⠐⡽⣯
⠀⠀⠀⣿⣿⣏⣩⡯⡕⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠡⠀⠲⠂⠡⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⡶⠋⢯⠬⡞⠉⡞
⠀⠀⠀⣿⡟⢳⠷⡎⠙⣦⠃⡄⠐⣢⠄⠀⢈⠉⠀⠉⠀⠈⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠈⠳⠶⢧⠙⣞⡭
⠀⠀⠀⣿⣷⣎⣏⣓⣞⣁⣧⣶⣞⣵⣽⣤⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣴⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⣆⣠⡾⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠟⠁⣿⠀⠰⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⣠⡿⠻⣦⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⣿⣧⣴⡶⠟⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣠⡾⠋⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣠⡾⠋⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣠⡾⠋⠀⣠⡾⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⣿⠀⣿⠀⠀⢈⣉⣉⣉⣉⣉⣉⣉⣉⠁⢀⣿⣻⣦⡾⠋⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⣿⠀⣿⠃⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠘⠛⠋⠉⠀⠀⠀⠀⣿⠛⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣾⠟⠁⣠⣶⣄⠈⠙⢷⡄⠀⢰⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣾⠃⣠⣤⣿⠀⣿⣤⣄⠈⣿⠀⠘⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⠀⠻⣶⣶⠀⣴⣶⠿⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣷⣀⠀⠻⣶⠿⠀⢀⣼⠇⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠋⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⠷⢶⣴⡶⠾⠛⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⣿⣿⠿⠿⡿⢻⠿⡿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠰⠐⠤⢦⣀
⠀⠀⠀⠀⠀⡠⢀⠀⣠⣶⡦⠁
⠀⠀⠀⠀⠠⠽⠁⠘⠂⠂⠑⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣶⡈⠐⠃⡆⠬⡳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣁⣠⣾⣿⣛⠻⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⡀⢄⡘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣵⣾⣿⣟⣭⣥⣤⣤⣄⠨⡹⣿⡆
⠀⠀⠀⠀⠀⠀⡇⠃⡀⠀⣇⢹⡄⠹⡼⣇⠀⠀⠀⠀⠀⠀⠀⣠⢴⣿⣿⣿⣿⣶⣿⣷⣶⣮⣍⠻⣷⣹⡞⣧
⠀⠀⠀⠀⠀⢸⢱⠀⠇⢀⡹⠼⠃⡇⠀⣿⠀⠀⠀⠀⠀⢀⣾⠃⠐⢛⣻⣿⣯⣭⣙⠛⠿⣿⣿⣷⡜⡇⡇
⠀⠀⠀⠀⠀⣿⣸⡇⣰⢫⣇⣠⠠⣷⢆⡏⢣⡀⠀⠀⠀⣼⡿⠂⠴⣛⣻⣿⣛⡿⣿⣷⣄⢨⡻⣿⣧
⠀⠀⠀⠀⠀⠟⠁⣷⠃⠈⠛⠁⠀⠀⠈⠀⢒⠙⠶⣤⠔⢿⣧⠶⠿⠟⠻⠿⠿⣿⣦⡙⢿⡆⣧⣿⡟
⠀⠀⠀⠀⠀⡆⡇⣀⠀⡀⠀⠈⡀⢁⠀⠆⡔⠁⡈⠁⡠⣅⠀⣒⠻⠿⣶⣶⣤⡐⢽⣷⣦⠛⣿⢿⠃
⠀⠀⠀⠀⠀⣇⢀⠁⠐⢀⡰⣠⢠⠀⢴⡄⣧⢲⠘⡆⡀⠘⠀⢬⣙⢷⣦⡙⠿⣿⣎⣿⢹⠀⠛
⠀⠀⠀⠀⠀⠘⣮⢷⡀⠌⠐⣤⠂⠀⠘⡇⡿⢸⣷⠃⣧⠀⣌⣷⣽⢿⣯⠻⡆⠙⣿⠘⠞
⠀⠀⠀⠀⠀⠀⠻⡎⢻⣶⠟⠁⠀⠀⢄⠀⠁⢁⡟⣆⠏⠀⣿⢸⡟⣧⠹⣧⠙⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠙⢤⡝⢦⣄⠀⠀⣿⣗⠄⠈⢰⣏⣰⡀⠛⠀⢷⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠙⠷⣜⢮⡳⠅⡼⠈⢺⢯⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠬⢉⡁⠈⢠⡔⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠻⠛⠊⠉⠊⠃

⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣿⣛⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠋⠉⠀⠀⣉⠉⠉⠉⠛⠛⠛⠛⠛⠳⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⣉⣩⡽⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠛⠋⠉⠀⠀⠀⠀⠠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⣡⡴⠶⠻⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣿⣶⣶⠿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣴⣿⣷⠾⠿⠷⠶⠶⢶⡶⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣻⣿⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⣀⣴⣿⣿⣵⣶⣶⣶⣶⣾⡿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠀⢀⠈⠙⠻⣿⣶⣤⣀⠀⢄⡀⠀⠉⠙⠋⠁⠀⠀⠀⣠⣶⣿⣾⣥⠶⢍⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢶⣿⠛⠛⠛⢛⣛⣛⣛⣛⣛⡛⢿⣟⡶⣌⡁⠀⠀⠀⠀⢀⣠⣾⣿⠿⠛⢉⡰⢜⠃⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠾⣧⣤⣬⣤⣀⣀⣀⠒⠀⠒⠤⠻⠶⣹⣦⣄⢀⣴⡟⠯⢑⢠⡔⠊⠁⠈⠀⣀⡭⣷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣠⣤⣤⠠⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡭⣿⣁⣤⡶⠿⠃⣀⣤⡶⢾⡛⠉⠀⠼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡍⠻⡶⢦⣌⣉⠛⠛⠛⠶⢶⣦⣤⣤⣀⣒⣰⣟⣉⣤⣴⣶⣟⡋⠁⠀⢀⣹⣶⣤⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠶⣿⡀⠈⠄⠈⠉⠛⠷⢦⣤⣀⠀⠈⠙⠛⠿⠿⣿⡿⣿⡿⠛⢉⣠⣴⠾⠟⣛⣫⡭⠴⠶⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢷⡈⢿⡀⠀⠀⠀⠀⠀⣀⣀⠉⠛⠶⣤⣀⠀⠀⠀⢉⣠⣴⣾⣿⣿⣷⠾⣿⣿⣶⣦⣀⠘⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⢳⡈⠛⠛⠛⠛⠛⠙⠛⠛⠛⠷⠶⣤⣍⣛⣶⠶⢛⣽⠟⠋⣉⣤⣤⡒⠀⣀⣤⢼⡏⠙⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣀⣀⡐⠒⠤⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣤⣼⣿⣿⡿⠷⠛⢋⠛⢉⣠⣴⠾⢻⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⣉⣉⠉⠛⠓⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣴⠏⠉⢹⡇⡄⠀⢀⣴⠶⠟⠋⠉⠀⠀⠘⡿⠛⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡈⠉⠙⠓⠶⢶⣭⣍⣛⠶⣦⣄⣴⣿⣧⣴⣶⣾⡧⠶⠚⠉⠀⠒⠒⠒⠀⠀⠀⢠⣷⠀⠀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⠛⠛⠛⠓⠶⠦⣤⣌⣉⡛⠛⠿⢷⣿⡃⢄⡸⢦⣑⣲⠿⠿⡛⠛⠛⠛⠻⠶⣦⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣷⣶⣤⣀⠀⠀⠀⠙⣯⠙⠛⠳⣶⣟⡉⠛⠿⢿⣶⡵⠞⡩⠐⢉⠄⠂⠀⢀⣠⠿⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⣿⣿⡍⠛⣷⣤⣄⠘⠓⠛⠛⠉⠉⠙⠷⣦⣤⣀⠀⠀⠀⠊⣁⣤⣶⣿⣯⣉⣠⠬⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⣠⠟⠙⠛⢷⣤⣾⣿⣿⣶⣤⡀⠀⢀⣤⠾⠿⠬⣿⣷⣶⣶⣿⡿⠟⣋⡩⢬⡐⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣀⣀⣀⠉⠛⠻⠿⣷⣾⣿⢾⣏⣁⣀⣠⣴⡾⠜⠛⣉⠵⢞⣋⣽⣾⣃⡴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠉⣻⣿⢿⣶⡶⢦⣤⣤⡴⠟⠉⠛⣻⣗⠤⠔⠒⢊⣽⠿⢋⣵⣯⣼⠻⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢸⠁⠈⣷⡈⠙⠛⠷⢦⣤⣀⢖⣚⣿⡍⠑⣲⢛⣻⠶⣞⣛⣭⠴⠾⠃⠘⣆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠉⠲⢤⣀⢻⣤⣴⠶⠋⠉⠉⠹⠷⢮⣿⣿⣭⠷⠞⠛⣉⣥⠄⠂⠀⠀⠀⢹⡆⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠐⠦⢄⡈⠛⠿⣤⣀⠀⠀⣰⡷⠲⣤⡉⠉⠀⠀⠐⠉⠁⢀⣀⣤⡶⢾⣛⣛⣡⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣤⣀⠀⠉⠓⠀⠀⠉⠛⣶⣟⠀⠀⡏⢻⣄⠀⣀⣤⡶⠿⠟⠛⠋⠉⠉⠉⢨⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣌⡝⠳⢦⣄⡀⠀⠀⢰⣿⠙⣿⠸⣧⠀⠹⠿⠋⠁⠀⣠⡀⠀⠀⠀⠀⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠈⠱⠤⠉⠛⠷⣦⣼⣿⣿⠃⠀⣹⠆⠀⠀⠈⣉⣡⣤⠴⠖⢊⡽⠋⠁⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣿⠃⠀⠀⠀⠀⠀⣀⡀⠀⢁⡀⠀⠀⣿⡤⠴⣚⠋⠉⠁⢀⣠⠞⠋⠠⣦⡴⠴⣿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣤⣍⣛⠶⢦⣤⣀⠀⠀⠈⠉⠙⢶⢧⣤⣾⢁⠤⠄⠉⠓⠾⠋⠁⠀⠉⠉⢀⣰⣤⣿⠃
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⡿⠿⠿⣶⣮⣭⣻⣶⣄⣀⠀⠉⠑⣺⣏⢀⡀⠒⣀⡠⠤⠔⣒⣠⠶⢟⡛⠉⠁⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣷⣦⣄⡀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣬⣿⣠⣄⣠⣤⠴⠖⠋⠡⠔⠊⣁⡤⠐⠀⠀⢀⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣶⣤⣀⠀⠀⠀⠈⣙⢛⣿⠛⠛⣯⠳⣄⠀⠀⠀⠀⠊⠁⢀⣠⠤⠚⠋⠁
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣈⣉⢻⣀⣤⣬⢷⡌⣳⡀⠀⠀⠔⠚⠉
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠾⠛⠋⠐⠉⠉⠁⠛⠊
⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀⠇⣠
⠀⠀⠀⠀⣀⣤⡶⠿⠞⠛⠻⢷⣷⣶⣤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣉⣀⣀⠀⠄
⠀⠀⠀⠀⠉⠡⠖⠒⠒⠚⠛⠿⠟⠻⠿⣿⣿⣦⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣧⡄
⠀⠀⢀⣤⡶⠾⠟⠛⢛⣿⣿⣿⣷⣮⣷⣦⣝⣿⣿⣌⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢳⡟⢻⣿⣧⠰⠄
⠀⠐⠛⠁⠀⢀⡠⣀⣽⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣿⣤⣄⣹⠄
⠀⠈⠀⠀⢀⣿⠿⠋⠐⠂⣉⣤⣤⣄⣬⣭⣛⠿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡏⠁
⠀⠀⠀⣴⠟⠩⠀⠀⣠⣊⡽⣟⣿⣭⣿⣿⣿⣿⣷⣽⣿⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆
⠀⠀⠀⠁⠀⠀⠀⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣍⠛⠃⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠄⡀⣠⢻⣿⣿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⣀⠀⠀⠀⠒⠖⠒⠒⠒⢨⣄⣠⡤⠀⠀⣖⣵⣿⣾⣿⣿⠂
⠀⠀⠀⠀⠀⠘⢹⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣤⠄⡀⣠⣤⣂⣀⣴⣿⠏⠠⠶⣾⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢿⣿⣧⣿⣿⡿⢻⡿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣋⠀⠀⠠⢢⣣⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠟⠀⠙⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣙⣿⣿⣶⣦⣾⣽⣷⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⢿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣞⡿⠟⠷⣿⣟⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⡿⢿⡿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠿⠛⠁⠂⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠀⠀⠀⠤⣥⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢂⢤⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠐⠰⠖⠴⡀⠀⡀⢰⣿⠶⣽⣛⠢⣟⠂⠀⠀⠀⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠗⠐⠷⠶⠀⠀⠁⠀⠈⠁⠁⠀⠀⠈
⣴⣦⣶⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣦⣤⣶⣶⣶⣶⣶⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣦⣦⣴⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡰⣼⢩⢒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠳⠃⠃⠁⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣰⡾⠧⢾⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⣿⣹⢆⡆⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠖⡥⢋⠜⣃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢡⣷⡿⠃⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⢎⣞⢔⢫⣴⣈⣉⣀⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠋⠁⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣣⣿⣯⣴⣟⠩⣒⡐⠒⣀⠬⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢱⢆⣀⣐⣢⣤⣤⡴⠿⠍⠑⠒⢽⣮⡪⣑⠢⢉⠳⡌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⣠⡼⠀⢤⡄⢾⣛⢁⣲⣚⣋⡒⢄⠼⢷⠙⢎⠿⢧⠣⢘⣰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⢀⠀⣤⡏⡑⣦⠟⢞⢃⣾⢧⡒⢿⠿⠋⠻⣑⡤⢦⠆⢦⡴⠈⠓⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠐⡘⣸⠿⣉⠦⣯⣱⡶⡟⣭⢏⠳⠾⣿⠣⣱⣾⡉⣘⢤⢿⡇⣭⠟⣅⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣿⢍⠒⣍⡗⢪⣈⣤⡟⢃⡸⠷⢤⣌⡳⣌⣬⢦⣿⡿⠿⠣⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠳⣿⡦⣝⠲⡋⢹⣰⡞⣷⢶⢲⣶⡞⢛⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⠧⣼⠇⣚⣹⡱⣌⢈⠈⣀⣶⢶⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢽⣗⢒⣤⣟⣽⣧⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣾⠿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⠛⠒⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢻⣯⠴⠶⢿⡿⠶⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⣴⣶⣶⣶⣶⣶⡆⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠈⣿⠈⠻⣷⣄⠀⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠘⠛⠀⣿⡇⠀⠀⠙⠛⠛⠛⠛⠛⠛⠃⠘⠿⠶⠶⠾⢿⣷⠀⢠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣇⣀⣀⣀⣿⡇⠀⠀⠺⠿⠿⠿⠀⠾⠿⠇⠀⠀⠀⠀⠀⢸⣿⠀⣾⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠶⠀⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⠀⣤⣤⣤⠀⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠁⣀⠈⠉⠻⣷⡄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢰⡿⠛⠛⠃⠀⠘⣿⡄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠘⠿⠶⣶⡄⠀⠀⣹⡇⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⢠⣤⣤⣾⠇⠀⢠⣿⠃⢠⣤⣤⠀⢤⣤⣤⣤⣤⠄⢠⣤⣤⣤⣤⠄⠀⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⣀⠉⣀⣠⣴⡿⠃⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⢸⣿⠀⣿⣇⣸⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⣿⠀⣿⠋⢻⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⣾⠷⣿⡷⠿⠾⣿⠷⠾⠾⠿⠷⣿⡶⠾⠷⠿⠿⣿⠀⠀⢸⣿⠀⣿⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣶⣾⣷⣶⣶⣿⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣿⠀⠀⢸⣿⠀⣿⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣤⣼⣧⣤⣤⣿⣤⣤⣤⣤⣤⣿⣥⣤⣤⣤⣤⣿⠀⠀⢸⣿⠀⣿⡇⢸⣇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣉⣹⣏⣉⣉⣿⣉⣉⣉⣉⣉⣿⣍⣉⣉⣉⣉⣿⠀⠀⢸⣿⠀⣿⡇⣸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⠀⠘⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⠁⠠
⠀⠀⠀⠀⠀⢀⠀⡇⠒⠈⠃⠈⠀⢓
⠀⠀⠀⠀⠀⡄⠘⠁⠀⢀⣠⡖⠊⠁
⠀⠀⠀⠀⠀⡱⠀⢀⠈⣩⡄⠑⠄
⠀⠀⠀⠀⠀⠙⢒⠿⠦⠉⠴⡆⠀⠈⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠐⠂⠀⠐⠂⠤⢀
⠀⠀⠀⠀⠀⠋⡱⠋⢀⠈⠃⠃⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣁⣀⣤⠤⠤⠤⠴⠦⣀⠀⠉⠂⣄
⠀⠀⠀⠀⠀⠀⢇⣂⣀⡀⠆⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠐⡩⠔⡡⠤⠐⠒⠒⠒⠀⠤⡈⠲⣤⡈⠣
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⢜⠕⠉⣀⠄⠒⠐⠊⠓⠦⠀⠈⠑⢯⢳⠀⠡
⠀⠀⠀⠀⠀⠀⢠⡯⠔⡊⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠆⢀⢎⠊⣠⠞⠁⡀⠤⠔⠂⠉⠉⠉⠓⡄⠈⢂⠡⠈
⠀⠀⠀⠀⠀⢀⠏⠀⠀⠷⠰⡄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⠀⢆⠆⡠⣡⡶⠋⠀⠀⠀⠀⣀⡀⠠⠤⢈⣄⠀⠂⠉
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢳⠰⢆⠀⠀⠀⠀⣀⣀⠱⣄⠀⠀⠀⢠⣘⠰⠌⣰⠟⠉⠀⢀⣠⡴⠒⠉⠀⠀⠀⠀⣀⢻⠄⢠
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡆⠀⠀⠑⠈⠣⣐⠁⠢⡄⢫⠉⠊⠑⠤⡀⠀⡏⣷⡶⠗⠊⠉⠘⠫⣁⣀⡤⠖⠊⠉⠀⠀⠀⢘⡂⢸
⠀⠀⠀⠀⠀⢻⡀⠀⠀⡇⣀⠀⠀⠀⠀⠈⠁⠀⡤⣌⣄⠀⠀⠀⢻⡙⢄⠉⡧⣶⠖⣠⣄⠀⠈⢇⢀⣀⣀⡀⠠⢠⡀⠈⠅⠸
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠱⡍⠾⠐⠦⣄⡒⠀⠤⠧⠨⣻⡝⡄⠀⣄⡇⠀⠑⡜⠛⠚⠧⡐⢳⡀⠈⡏⠉⠀⠀⠀⠀⡸⠀⠢⠆
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠘⢆⠀⠀⣀⠀⢀⣀⠐⡶⢇⠑⢽⡄⣸⠃⠀⠀⠸⡽⠷⢆⡘⠏⢹⡇⢸⠑⠄⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠳⣄⠘⢦⣄⢹⡁⠓⢄⡙⢦⡑⡜⠀⠀⠘⣦⡧⠄⣀⣳⠘⣀⡇⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠈⠑⠺⣮⣅⠈⠲⢄⡉⠢⢕⣴⠀⢶⢠⠘⠀⠀⠀⢸⢀⠁⡇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠻⣦⡀⠈⠑⠤⣹⡗⡆⠃⠀⠀⠀⠀⢸⡸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠉⣲⠤⢤⣄⣱⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣆⡀⣀⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠊⢨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠭⠅⢒⠟⡫⠖⠒⠲⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⡉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⣤⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠦⣤⣀
⢠⣾⣿⣿⣿⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣀⣀⣀⣀⡈⠛⢦⡀
⢺⣿⣿⣿⣷⡄⠈⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣴⠞⠛⠉⠉⠉⠉⠉⠙⠒⠿
⠛⠛⣾⣿⣿⡷⠂⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⣰⣯⡴⠶⠞⢛⣛⣓⣶⣦⡀
⠀⠸⣿⣿⣿⠇⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠋⣠⣴⣖⣛⣉⠉⠉⠀⠈⠁
⠀⠀⠈⢹⡅⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠠⠚⠋⣉⣭⡭⠭⠽⠷
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠛⠛⠦⣄⡀
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣤⣠⣤⣤⠾⠯⠄⠀⠀⠀⠀⠀⠀⠳⣦⣌⡹⣆
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠙⣎⠻⡿
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⠱⣼
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠘⣦⡈⣷⡘⡇⠈
⠀⠀⢹⡄⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡿⠀⢸⣷⣸⢳⡇
⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⣀⣀⣀⣀⣤⡴⠞⠋⠀⣸⢾⡇⡿⠘⠃
⠀⠀⠀⠀⠙⠷⣄⡀⠠⡀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣦⣀⠀⠀⠀⠀⣸⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣸⠉⠳⢄⣀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⢏⣿⡀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋⢹⠿⠞⠃⠁⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⡟⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⡿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣸⢹⡄⠀⣿⠀⡇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⡇⢰⡇⢸⠁⢰⡇⢸⡇⢀⡏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⠇⣸⠁⢸⣶⣛⡔⠈⠓⠋⠀⢠⣇⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⣿⠀⠘⠓⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠀⣠⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⠉⠥⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣯⠽⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣎⣸⡿⣭⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⣠⠞⡷⠋⠋⡁⠚⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⡰⠃⠀⣠⣴⢫⣋⠀⠀⠀⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢦⡀⢀⣠⣾⡃⠐⠺⣝⣛⡵⠃⠀⣀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠗⠿⠿⠿⢶⣿⣦⣤⣤⣤⡤⢶⢟⡿⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣠⠴⣒⣋⣉⡱⠋⠛⠲⢤⣀⡏⠈⣿⡀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠛⠉⢉⡿⢉⡇⠈⢷⠀⠀⠹⣧⡀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⡞⠀⠀⢸⠀⠀⠀⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠀⣸⠀⠀⢠⠟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠸⡄⠀⠀⠀⢀⡞⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣽⣶⣦⣴⠿⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡞⢷⡀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢀⣆⠀⣰⡆⠀⣼⡿⠁⠀⢣⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⣠⡄⡀⠀⣰⠏⣿⡴⢻⡇⢰⡟⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡴⠋⠁⣷⣇⡼⠃⠀⠛⠁⢸⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣿⠟⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡖⠉⢳⡀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠙⢾⠀⠀⠀⠟⠁⠀⢸⡆
⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⢳⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠙⠢⣍⠛⠦⣄⡀⠀⠀⠀⡆⠀⠙⣄⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣠⡤⠤⡐⠒⡀⠾⠁
⠀⠀⠀⠈⢢⡀⠀⠙⠲⣄⠀⣿⠀⠀⠈⢣⡀⠀⠀⠀⠀⠸⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠖⠩⣥⣤⣀⣤⣯⣠⣀⣀⣠
⠀⠀⠀⠀⠀⠙⣄⠀⠀⠈⠳⣿⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⡇⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢆⠀⠉⠀⣠⡀⠁⠀⣠⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⡄⠀⠀⢠⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠛⠁⠀⠈⠉⠙⡅⠀⠀⠈⣉⣳
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡼⠀⡘⠀⡷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⡾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠈⣃⡼⠁⠠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡠⠤⠒⠋⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⡰⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠜⠁⠀⠀⠀⢠⠄⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⠁⠀⠀⠀⠀⠀⡠⠊⠀⢀⠔⠁⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠌⠀⢀⠄⠀⣠⠞⠁⠀⠔⠁⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠈⢧⡀⠀⠀⣠⠟⣄⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⡘⠀⠀⠌⠀⣠⠊⢀⡜⠁⠀⠐⠁⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠓⠒⠢⡖⠉
⠀⠀⠀⠀⣰⠁⠀⠀⢀⡔⠁⢠⠋⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⢰⠃⠀⠀⢀⠞⠀⢰⠃⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⢀⠇⠀⠀⠀⠎⠀⠀⠇⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⡜⠀⠀⠀⡘⠀⠀⠸⠀⠀⠀⢸⡇⠀⢀⣠⣤⣤⠴⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢠⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇⢰⡋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠇⠀⠀⠘⠀⠀⠀⠀⠳⣌⠛⠶⣦⡤⠶⠶⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⢀⠀⢸
⠀⢸⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⠁⠀⠀⠀⠀⢀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⢀⡎⠙⠛
⠀⠀⣇⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡴⠛⠋⠁⠀⠀⢀⡠⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠸⣄⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣠⠤⠚⠉⠀⠀⠀⣀⠔⠈⠀⠀⠀⠀⠀⠀⢠⣤⠀⡼⠁
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣀⡀⠀⠀⠀⠀⠀⢴⣁⠀⢀⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠛
⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⢀⣀⣤⠴⠞⠦⣄⣀⣤⣀⣀⣀⣀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⡇⠀⡏⠉⠉⢦⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠃⠀⠀⠀⠙⢦⡙⢦⣤⣴⠛⢷⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠳⣄⢙⣿⠶⣊⡽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢿⠾⣿⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠲⠶⡒⣲⠶⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣌⣹⡆
⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣆⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢠⣤⡤⢤⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⣠⠶⡲⠄⠀⠀⠀⠀⣸⡆⠀⠀⠀⢸⣿⣿⣾⡇⠀⠀⣶⡟⢿⣶⠀⠀⠀⠀⠈⠉⠀⠀⠀⠄⠐⠀⢱⡄
⠀⠈⠉⠃⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠸⠉⠉⠉⠇⠀⠀⠛⠃⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠙⡆⠀⠀⠀⠀⠸⠟⠂⠀⠀⠀⢻⠁⠀⠀⠀⠔⠁⠹⣿⠆
⠀⠰⠾⠶⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠀⣾⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠢⠀⠀⠄

⠀⢰⣶⣶⣶⠀⠀⠀⠂⢀⣀⡀⠀⠀⢀⣶⠶⣦⠀⠀⠀⠤⠄⠄⢤⠀⠀⠀⠠⣆⣤
⠀⢸⣿⣻⣿⠀⠀⠀⠀⣘⡟⠃⠀⠀⠀⣿⣴⡿⠀⠀⠀⠀⠄⠀⣸⠀⠀⠠⣾⡏⠁⠀⠀⠀⠀⠠⡄
⠀⠈⠛⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁
⠀⢀⣀⣀⣀⠀⠀⠀⣀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⠀⠠⢄⡀⠀⠀⠀⢠⣤
⠀⢸⣿⠉⢙⡇⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣿⣿⠿⣿⠀⠀⠀⣾⣿⣿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠉⠉⠙⠛⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠻⠟⠛⠟⠀⠀⠀⠿⠿⠿⠀⠀⠀⠻⠿⠿⠿⠃
⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠞⠉⠆⠀⠀⠀⢸⠱⢜⡆⠀⠀⠀⣯⣭⣽⠀⠀⠀⠉⠉⢠⢹⠀⠀⠀⢸⣈⣸⠀⠀⠀⠀⣰⣿⡷
⠀⣠⣤⣤⣤⡀⠀⠀⢠⣽⣯⡄⠀⠀⠀⣿⣿⣿⠀⠀⠀⠇⠁⠈⠰⠀⠀⠀⢸⠉⣹⠀⠀⠀⠰⢿⡟

⠀⢠⣠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣶
⠀⣼⣼⣅⢨⡄⠀⠀⠘⠀⠈⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⠈⠳⠚⠁⠀⠀⠰⠶⠾⠶⠆⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠙⠻⡿⠿⣿⢿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⡄⠨⠄⠀⠿⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣆⣀⣀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠿⠛⠿⠿⣿⡿⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠿⠀⢀⠀⠀⠀⠀⠂⠘⠿⠿⣿⣿⢿⠶⠎⠙⠿⠿⠹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠭⠉⠋⠽⣶⡖⠒⠞⠠⡈⠀⠀⠀⠀⣀⡀⣀⣀⠐⠤⠤⠴⠲⠤⠄⠀⠠⠼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠔⠒⠒⠉⠉⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠚⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠜⠘⠞⠀⠂⠰⠆
⠀⠀⠀⠀⠀⠠⠃⠼⠃⠀⠀⠀⣀⠄⠄⠂
⠀⠀⠀⠀⠀⠸⡄⠀⠀⢠⠴⠚⠷⡀
⠀⠀⠀⠀⠀⠰⣅⣀⣤⠠⠾⠅⡀⠈⠢⡀
⠀⠀⠀⠀⠀⢀⣈⠭⡉⠓⠂⡘⢷⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠈⠉⠉⠉⠉⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠈⢀⠎⠀⠸⠀⢹⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣁⠠⣤⠴⠄⠒⠒⠒⠓⠦⢄⡀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠸⣆⣀⣀⡄⠌⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢊⠥⢊⠤⠐⠀⠁⢀⠈⠁⠒⠤⡈⠳⣴⡀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⡠⢊⠂⠁⣀⠤⠐⠒⠒⠚⠲⢄⡀⠈⠒⣥⠻⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣏⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡐⡡⠁⣤⠞⠁⢀⣀⠤⠐⠒⠀⠀⠛⠦⡀⠈⢆⠡⡀⡄
⠀⠀⠀⠀⠀⠀⢠⠋⠁⠀⣿⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⡇⡐⠰⠁⡘⣁⣤⠖⠉⠀⠀⠀⠀⠀⠀⣀⣀⠑⡀⠀⠆⠑
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠁⠀⢰⠀⣠⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠁⢡⠁⣴⠴⠋⠀⠀⠀⣀⣠⠔⠂⠉⠁⠀⠀⠈⣿⡄⠘
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⡄⠀⠈⠣⡍⢦⡀⢰⠠⡀⠟⢶⠦⢽⠢⣀⠀⠀⠘⡶⣤⡇⣼⣫⠤⠤⢤⡴⠚⠉⠀⢀⣀⡀⠤⠀⠀⠚⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠸⠀⠈⠃⠈⡆⠀⠀⠀⠑⣦⣀⠇⠽⡟⠁⠀⠀⠀⠀⠈⠲⡞⠛⠉⠀⠀⠀⠀⠀⠀⢠⠆⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡇⡴⡀⡀⠀⠀⠀⠀⠀⠀⡟⠲⣾⠤⡀⠀⠀⠻⡌⠳⣄⠷⣾⡏⣩⠗⢦⡀⠀⠱⣤⠤⠴⠦⠀⠐⠒⢄⠀⡀⢠⠁
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠘⢧⠘⠈⠙⠲⠼⠉⠁⠒⠒⠀⢮⡷⢹⡄⠘⣦⡇⠀⠈⢞⡉⠈⠉⠲⣉⣱⣤⠀⢡⡀⠀⠀⠀⠀⢀⠜⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠳⡀⠀⢠⡀⠀⢠⣄⠈⠟⠻⣀⠢⣵⡀⣝⠁⠀⠀⠈⡷⠟⠓⢄⡘⠅⠹⡇⠈⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠙⢦⡀⠙⣦⡀⢹⡁⠑⠤⡈⠳⣌⠳⡙⠀⠀⠀⢧⡴⠤⣀⡈⣷⢘⡀⡇⢠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠉⠒⠴⣬⡁⠉⠳⢄⡈⠑⢬⡣⣅⠀⢰⢀⠈⠃⠀⠀⠀⢹⢸⡇⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⣀⠀⠈⠛⢷⣄⠀⠈⠓⠤⣈⢻⢧⡈⠛⠀⠀⠀⠀⠀⠸⢰⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡗⠀⠀⠀⠈⠑⠢⣄⣀⡀⠙⢇⠁⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡾⠃⠈⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⠒⡉⠠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠧⠴⠒⡉⠟⡩⠖⠛⠛⠭⠕⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⠶⠶⣶⣶⡶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣼⣷⣿⣿⣷⣏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠹⣷⣶⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⣀⣠⣀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢢⣠⣤⣤⣼⣿⣿⣿⣿⣿⣿⣆⢱⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠐⠋⠉⠻⣾⣯⣿⣿⣿⣿⣿⣿⣿⣧⣿⠀⠀⠀⢀⣼⣿⣿⣿⢿⣿⢿⣷⡄
⠀⠀⠀⠀⠈⠓⠶⢶⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⣿⣿⣿⣿⣟⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣴⣿⡿⢛⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⣠⣾⡿⠋⡴⠟⣡⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠈⠁⠀⡾⠁⣰⣿⢣⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣟⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢸⠁⢰⣿⡏⢸⡿⢸⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢸⡇⢸⣿⡟⠤⠾⠋⡽⢿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠃⠈⣿⠁⠀⠀⠀⠀⠀⣿⡇⠻⠿⠖⠛⠋⡹⠷⢦⠄⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠧⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠖⠲⣤⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠛⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠠⠒⠖⢴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⡐⠔⠇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢸⣤⠴⠏⢸⠌⠝⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⡆⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢤⠔⠁⠀⠀⠀⢸⡆⠀⠀⠀⢀⡴⠋⣁⠀⠁⠀⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣰⠃⠀⠀⠁⠠⠀⠀⠀⠀⢑⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⡷⢰⠴⠣⠠⠄⠀⠀⠀⣿⠄⠀⣏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠑⢸⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⣹⠀⠀⠀⠈⠉⠑⢄⠀⠐⡄⠀⢀⢳
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠈⠙⠂⢄⡄⠀⠀⠀⠀⢱⡀⢱⠈⢢⢻⠂
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠫⡴⠉⠁⣾⠴⠦⣬⣧⢸⠀⠈⣾
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠀⠀⢀⠞⠀⠀⠀⠀⢹⣾⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠈⢸⣾⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⠀⢀⣀⣀⣀⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣞⠁⠀⠀⠙⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡇⠀⠀⠀⠀⠸⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⠞⠹⢥⡀⢀⣤⣤⡴⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡶⣖⡴⠋⠳⠷⣟⡭⣶⣤⠷⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠉⠳⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⣦⣶⣶⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡀⢀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⣷⣤⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠢⡀⡔⠄⢌⢂⢀⠀⠄⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣻⣿⣿⢿⢏⢞⡟⡱⡿⣫⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠄⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡏⠀⠀⠀⠉⠊⠈⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢹⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠤⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠄⢠⠟⠆



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠀⢀⣠⠔⠙⠓⠛⠀⠀⠀⠰⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠄⠀⠀⠀⣀⣴⣋⠀⠈
⠀⠀⠀⠀⠀⠂⠈⠀⠀⠈⠁⠒⠀⢤⢀⢀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠈⠀⠀⡠⢔⡾⠋⠁⡠⢔⡫
⠀⠀⠀⠀⠀⠀⡀⠄⠒⠤⢄⣀⠀⠀⠀⠀⠁⠉⠉⠋⡉⠀⠠⠀⠀⠀⣀⠠⠂⣩⠔⢁⡠⢒⠍⢚⠵⣺
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣄⢀⢀⠉⠒⠠⠤⣀⡀⠀⠀⠀⠀⣀⡠⠔⢊⡠⠖⡁⢔⡪⠔⠉⢀⡤⠖⠋⠩
⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⠀⠈⠈⠙⠒⠒⠤⠤⣍⣑⣒⡋⢁⡠⢖⡡⣐⠥⠒⠁⣀⠴⠚⠉⠀⣠⡠⡇
⠀⠀⠀⠀⠀⠀⠗⠀⠒⠂⠈⠭⠅⣒⣀⠤⠄⣀⠀⠀⠈⢈⠤⣒⠅⣊⢄⣨⠵⠊⣁⣠⢤⡺⠟⣋⣶⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠈⠩⠀⠚⠩⠭⠕⠊⣔⡯⠑⠉⢥⢴⣛⡽⠚⡹⢖⡯⠗⣻⠃
⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁⢀⣨⠴⢞⠍⠀⣀⠥⠚⡡⠔⡋⢸⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢺⣷⡾⣫⠕⠋⠄⠒⠉⣠⡔⠬⠐⠁⢠⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁⠐⠆⣭⠝⠊⠁⠂⢀⣠⠶⠛⠁⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠦⣄⣀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠄⠂⠀⢀⠤⠚⠉⡀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠢⠄⣀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠠⣐⡼⠞⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠄⣰⠿⠭⠄⠀⠀⠀⠀⠒⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠵⠋⠀⠒⠒⠒⠒⠦⠄⣄

⠀⠀⣠⡤⠚⠚⠦⣄
⠀⡼⡥⠚⠿⣡⣾⠟
⠤⢤⠀⡘⢃⣴⠉⣲⣄⡀
⠀⢀⢸⣹⣶⣍⠶⣌⠳⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣄⣀⡀
⠀⠀⡰⠡⠏⠈⠳⠹⡃⠀⢫⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡔⡩⠀⠀⢀⡠⠤⠐⠒⣂⠠⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠉⡆⠀⡍⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡡⠊⠈⠀⣠⣞⡁⡤⠶⢭⣭⣉⡃⠀⠈
⠀⡅⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⢈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⢀⠇⣠⣚⣥⣤⡴⠄⠶⠤⠤⣄⡙⡳⣄
⠰⡁⠀⠀⢀⡆⠉⢀⡆⠹⡄⠀⢈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢸⢁⡿⠋⠉⠈⠀⠀⠀⠈⠁⠀⠀⠀⠙⠍⠙⢧⡀
⠀⡇⢰⣿⠆⣧⠈⢈⡆⠀⠹⢦⣤⡽⢇⠀⠀⠀⠀⠀⠀⢠⢟⠉⠠⠀⠠⡤⠀⠀⢠⢤⡄⠁⠀⠄⠀⠃⠀⠈⢧
⠘⠗⠚⠉⠀⠘⠀⠁⠸⠄⠀⠀⠐⠲⠿⢷⢦⣀⢀⢠⡄⣥⣠⠤⢄⣉⠀⠀⠲⠆⢄⠈⠀⠘⢦⠀⠈⠶⠀⠀⠸
⣸⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠊⢧⠀⠌⠻⣧⠀⣨⣵⠆⠀⠀⠀⠀⠀⠀⠀⠂⠘⠃⣀⠈⢠⠷⡆⡔
⣯⡰⠀⠢⡀⠀⠀⠤⠀⠀⠀⠀⠘⠈⠄⠘⡄⢸⢧⡮⣅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢢⡈⣷⡌⡄⢹⡾
⢸⡄⡘⡆⠝⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢹⡀⠀⡀⠀⡀⠀⠀⠀⠑⢄⠀⠀⠈⠁⣵⠏⠈⡇
⠀⢻⡧⡴⢸⣄⠉⢢⠀⠡⠀⠀⠀⠀⠀⠠⠁⠈⡇⠀⠰⠀⠑⠀⠀⢠⠀⢀⠑⢀⠀⠀⠿
⠀⠀⠻⣃⡄⢵⠆⠀⠫⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈⣷⡄⣯⠈⢦⡴⠁
⠀⠀⠀⠙⢶⣈⡔⡛⠃⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠹⠀⠙⠾⣏⠀⠃
⠀⠀⠀⠀⠈⠋⣴⣕⢂⠀⠑⡂⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣⣴⣀⡀⠂⢀⠀⠀⠀⠀⠰⡄⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣙⠷⣌⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣤⠘⠣⣴⡦⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⠆⠀⠀⢀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠈⠒⠂⠀⢸⡇⢸⠤
⠀⠀⠀⠀⠀⠠⢤⡤⣶⣂⣬⡩⠼⠦⠶⠂⠀⠸⠁⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣀⢀⠰⢖⣖⣦⠄⠈⠋⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣙⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⡇⢸⣿⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣿⡄⠸⣿⡈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠛⠛⠛⣷⣦
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠈⢿⣿⣄⠹⣧⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠙⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠈⠻⣿⣦⠹⣷⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠄⠁⣄⠀⠀⠀⠀⠀⠀⠴⠛⠁⠀⣀⣤⡶⢿⣿⡇
⠀⠀⠀⠀⠰⣦⡀⠙⣿⣷⣤⡀⠙⢿⣷⣽⣿⣿⣿⣦⠀⠀⠀⡠⠛⠁⠄⢰⡆⠀⠀⡰⠟⣹⣏⡀⠀⠀⠀⠀⠀⣠⣴⠟⢋⡥⢞⣻⣿⡇
⠀⠀⠀⠀⠀⠙⣿⣦⡈⠻⣿⣿⣦⣄⠙⢻⣿⣿⣿⣿⣷⡄⠞⠀⢠⣷⠀⠀⣴⣄⠀⠀⠀⠀⠀⢀⠀⠀⢶⣤⠾⠋⣠⡾⢋⣴⣿⣿⣿
⠀⠀⠀⢀⡀⠀⠈⠛⢿⣷⣤⣙⠻⢿⣿⣶⣬⣿⣿⣾⣿⠁⠀⢀⣹⡟⠒⠻⣿⠛⠁⠀⠀⠀⠀⠈⠁⠀⠟⠁⣠⡾⢋⣴⣿⣻⣿⣿⣿⡀
⠀⠀⠀⠀⠙⠿⣶⣦⣤⣈⡙⠻⠿⣶⣬⣭⣿⣿⣿⣿⣿⠃⠀⢠⣾⣇⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⣤⡀⢠⣾⠏⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣶⣶⣿⣿⣿⣿⣿⣿⣧⠀⠸⣷⣭⡞⢠⣼⠏⠀⠐⠀⠀⠀⠀⣈⣹⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠛⠷⣶⣶⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣦⣾⣿⡅⣶⣾⣿⣶⣄⠀⠀⠀⠄⠂⣼⠿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⣤⣬⣿⠿⠿⣛⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⡜⢣⣴⣟⣻⣿⣿⡿⣻⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⣶⠾⠟⣩⣴⢿⣷⣿⣩⣶⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⠀⠀⠰⣿⣾⣿⣿⣿⣿⡟⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢡⣤⡤⠶⢛⣭⡞⣫⣿⣿⣏⣽⣿⣿⠿⠀⣿⣿⣿⣾⣿⢿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣷⣿⣿⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢋⡵⠞⡹⢟⡑⢫⣾⠏⠟⠀⠈⢷⣽⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠵⠊⣭⠟⠁⠐⠀⠀⠀⢠⣿⣿⣿⣯⣿⣿⣼⣿⡿⢻⡟⢻⣿⣿⠁⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⡿⠁⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣻⣿⣿⣿⣿⣯⣵⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⣧⣼⣿⠿⠿⠿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⣩⣾⣿⣿⡟⣡⣿⠟⣱⠇⣴⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣼⣟⣛⣛⣿⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣾⣯⣴⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣋⡭⠐⠒⠛⢋⣩⣵⣿⣛⣛⡻⢉⢱⣟⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣀⣤⡾⠛⣹⣿⣿⣿⡟⣡⣿⢸⣿⣿⣿⡿⢻⡏
⠀⠀⠀⠀⠀⠠⠟⠁⢀⣴⠾⠛⣁⣴⠿⣿⣿⣿⣿⣾⣿⣿⣸⣿⡏⣼⣇⡾⠁
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⡾⠛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣿⡿⠁
⠀⠀⠀⠀⠀⢀⣿⠀⠀⢰⠏⣠⡾⠛⢉⣾⡟⠸⣿⢇⣿⣷⣿⣿⢸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⡿⠁⠀⣼⣿⡅⠀⠉⣾⡟⢱⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠃⠀⠀⢀⣿⠃⠀⢰⡟⣿⠇⠀⠀⣿⣷⣿⡟⢿⡼⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣾⡧⣿⢀⢰⡀⢫⣿⣿⣇⢸⣿⣿⣷⣹⣇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⠀⣿⠀⢼⣿⡼⣿⡿⡇⢸⣿⣿⣿⣿⣿⡀⠀⠀⣠
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⣿⡄⠀⠙⠃⣿⡇⢷⢸⣿⢿⣿⣏⠻⣿⣶⣶⡇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⡀⠀⢹⣧⠀⢹⣿⠈⢿⣿⡄⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣿⠀⠀⠻⣦⡈⣿⡆⠈⠻⡿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡀⠀⠈⠁⠸⣿⡄⠀⠀⠀⢉⡁⢒⣠⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠙⢿⣦⣀⠀⠈⠉⠉⠉⠉
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠄
⠀⠀⠆

⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⣿⡇⠀⢀⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⢿⣦⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠉⣿⡏⢹⣿⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⠄⢸⣿⣤⣤⣽⣿⣦⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢠⣤⣤⣤⡀⢀⣤⣤⡀⠈⠉⠉⠉⠉⠙⣿⠀⢠⣿⠻⣷
⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣶⣶⣶⣾⣿⠀⠀⠀⠈⠉⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⣿⠀⢸⣿⠀⣿⡇
⠀⠀⢀⣠⣴⣶⣶⣶⣦⣤⡀⠀⠀⠰⣶⣶⣶⣶⣶⠀⢰⣶⣶⣶⣶⡆⠀⣶⣶⣶⠆⠀⠀⣿⠀⢸⣿⠀⣿⡇
⠀⣴⡿⠋⠁⢀⣤⣀⡈⠙⢿⣦⡀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⣿⠀⢸⣿⠀⣿⡇
⣼⡟⠀⠀⣰⡿⠛⠛⠛⠀⠀⢹⣷⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⣿⡇
⣿⡇⠀⠀⠘⠿⠶⣶⣄⠀⠀⢸⣿⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⣿⠀⢸⣿⠀⣿⡇
⢻⣧⠀⠀⣤⣤⣤⣼⡟⠀⠀⣸⡿⠀⢠⣤⣤⠀⢠⣤⣤⣤⣤⡄⠀⣤⣤⣤⣤⣤⡀⠀⠀⣿⠀⢸⣿⠀⣿⡇
⠀⠻⣷⣄⡈⠉⠛⠁⢀⣠⣾⠟⠁⠀⢈⣉⣉⣀⣈⣉⣉⣉⣉⣁⣀⣉⣉⣉⠉⠉⠀⠀⠀⣿⠀⢸⣿⠀⣿⡇
⠀⠀⠈⠙⠻⠿⠿⠿⠟⠛⠁⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⣿⠀⢸⣿⠿⣿⡇⢸
⠀⠀⠀⠀⠀⣤⡄⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣿⠀⢸⣿⠀⣿⡇⢸
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣏⣩⣿⣉⣉⣹⣿⣉⣉⣉⣉⣁⣽⣏⣉⣉⣉⣉⣉⣿⡇⠀⠀⣿⠀⢸⣿⠀⣿⡇⢸
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡟⢻⣿⠛⠛⢻⣿⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⣿⡇⠀⠀⣿⠀⢸⣿⠀⣿⡇⢸
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡿⢿⣿⠿⠿⢿⣿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⣿⡇⠀⠀⣿⠀⢸⣿⠀⣿⣧⣾
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣷⣶⣿⣶⣶⣾⣿⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣿⡇⠀⠀⣿⠀⠸⣿⣤⣿⠋⠁
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡌⢧⡀⠀⠀⠀⠀⠀⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣍⣓⠒⢤⡀⠀⢸⣯⡰⢄⠀⠀⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⠳⠶⠶⠛⠷⣄⠀⠀⠈⠻⣿⣿⣶⡄⠀⠀⠀⢻⣿⡌⢣⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣾⣿⣧⡄⠁⡀⠀⠀⣸⣿⣿⡖⠄⠈⠲⣾⣿⡅⢸⠀⢸⣿⣿⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢻⣿⣧⣷⣀⠀⠐⢾⣿⣿⣿⠓⠀⠀⠀⢸⠟⠁⠸⣆⢈⣿⣇⠠⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣅⠄⠀⠀⠀⠉⠉⠷⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⠀⢸⣿⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠹⡧⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣴⣦⣀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡻⢿⣿⠷⢤⡀⠠⠄⠀⠢⡄⣄⠀⠀⠀⠀⠀⠀⠀⠻⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⢿⣿⣷⣦⡍⠈⠀⠀⠀⠀⠀⠈⠛⠂⠀⢄⠀⠀⠀⠀⠀⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣩⣴⠶⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⣾⠿⢛⢳⣥⣶⡟⠀⠀⠀⠸⠛⣿⣿⣦⡀⠀⠀⠀⠈⢷⣶⣴⣄⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠤⠒⣋⣫⣿⠿⠃⠀⠀⠀⠀⢘⡃⠾⠿⣿⣦⣤⣶⣿⣿⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⣠⠴⠛⠉⣱⡀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢾⣿⣿⣿⡟⠛⠉⠉⠉⠛⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢁⣴⠞⠁⣠⣶⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣾⣿⣿⣿⣶⣿⣿⡿⠷⣶⣶⣤⣄⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⣴⠟⠃⣴⠟⡻⢏⣾⣿⡆⠀⡀⠀⠀⠀⣀⡀⠀⢠⣿⣿⣿⣿⣿⢉⣿⡗⠀⠉⠉⠉⠛⠋⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⡟⠀⠀⣡⡞⠁⡾⣿⢿⣿⣶⣧⢀⢰⣿⠟⠀⠀⠀⠀⠀⠹⢟⣷⠸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⣼⡏⢀⣾⣶⠏⢀⣾⠐⠃⢀⠰⣿⣿⣿⣯⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⣠⡟⣿⠋⣰⡿⠁⠀⣰⡇⠀⣹⡿⡿⢿⢿⢀⣧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⣰⡿⠀⠀⠿⢹⡇⠀⣴⣿⠃⠀⣿⡇⢠⡏⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣿⠃⣰⠀⢀⣴⢠⡇⠘⠃⠀⣸⢿⣇⠈⣷⠘⣿⣿⣿⣆⡂⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⠇⢰⣿⢀⣾⣇⣾⡇⠀⣀⠀⡄⠈⣿⡀⣿⡀⠈⢻⣿⣿⣿⣿⣦⣤⣆⣀⡀⠀⠙⣷⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣠⡟⣻⣿⣿⠇⠈⠳⣰⣿⡀⣇⠀⢻⠃⡽⣷⡀⣾⠙⣿⣻⣿⣿⣿⣿⣿⣗⢠⣤⡟⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣸⣿⣠⡿⠛⣿⠀⠀⠀⢻⠇⠀⣿⡀⢀⠀⠀⣼⣿⣾⣷⡍⢹⣿⡛⠻⣿⣿⣿⣿⣟⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢸⣿⣗⠀⣸⡆⢰⠀⢀⡀⠀⡿⢳⣾⣧⠀⠈⠿⠉⠻⠗⣿⣿⠛⣇⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠋⣻⣦⣿⠙⢻⣆⠘⣧⠀⣧⠀⢻⣿⣧⣠⡀⠈⣦⡀⢹⡏⠮⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⣯⡄⢸⣿⣦⣿⣷⣿⣦⠈⣇⠀⢻⣷⡀⠸⣷⣼⣷⠀⠉⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠻⠁⢸⣿⠛⢿⣯⠋⢿⣷⠿⣦⠀⣏⣿⣦⡟⢿⡿⣧⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠘⣿⡄⢸⣷⠀⢹⣷⣿⣄⠈⢷⡀⠁⢹⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⡿⢳⣼⢿⡇⢸⣯⠁⢻⡄⢸⣷⡀⠸⣿⡏⣦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⡆⢸⣷⡀⣿⣧⢰⣿⡇⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠸⡟⠃⢸⣿⠗⢻⡏⣿⣿⣥⣼⣿⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠃⠈⠛⠙⢻⡟⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⢀⠈⠃⠈⠿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣤⣄⣀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⠺⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠇⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠿⠿⢿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⠀⠻⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⡀⠀⠀⠠⢿⣿⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⡄⠀⠀⠈⠃⠀⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣬⣽⠿⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠿⠛⠻⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣆⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣿⣿⡿⠛⠛⠿⢯⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⣿⣿⣷⣄⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣄⣤⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠛⠛⠁

⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠒⠀⠀⠀⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢉⡀⢉⣉⣁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠞⡶⢄⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡤⢏⠉⣠⠖⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢠⡤⢉⠤⡴⡩⠏⣀⠠⡾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡕⢠⠤⠤⢠⠄⢐⣈⠃⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣄⡈⠋⠴⠨⠛⠐⠒⠏⠠⣄⢎⡟⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡶⣧⡤⠃⡒⢠⠔⢲⣠⡦⠜⠒⡒⠲⠎⠁⠀⠀⠀⠀⠀⠀⠀⡖⠤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⠁⠀⠀⠉⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠠⣤⣄⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣦⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣴⣿⡞⠃⠀⠀⠀⠀⠀⢸⠸⠿⠿⠿⠿⠇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠛⠋⠀⠀⠀⠀⠀⠀⢀⣸⢸⡟⣟⣻⢻⡇⣇⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣼⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⣿⣿⢸⡇⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠷⠿⠿⠾⠃⠇⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿


⠀⠀⠀⠀⠀⠀⠀⢀⡄⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢐⢒⣒⣀⡀
⠀⠀⠀⠀⠀⠀⣐⣛⣣⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⣶⠶⠶⢶⡆⠀⠀⠀⠀⠀⠀⣭⣭⣬⣬⣬⡌
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡧⣿⠭⠭⢽⡇⠀⠀⠀⠀⠀⠀⣿⢹⡏⣿⢹⡇
⠀⠀⠀⠀⠀⠀⣿⣿⡅⠇⠀⣸⡇⠀⠀⠀⠀⠀⠸⠾⠯⠟⠇⣿⣭⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠶⠶⠯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠿⠾⠿⠿⠾⠇


⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣠⣦⣤⣤⣤⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢂⣴⣶⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⢀⣤⣀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⡿⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣗⣒⣒⣒⣒⣺⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡞⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣴⣝⠿⠋
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠈⠞⠋


⣿⣿⠟⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢉⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠪⡐⠃⢀⠺⠀⡄
⠀⡠⠏⠙⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠁⠖⠉⠂⠈
⢀⠧⣀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⡀
⡸⠦⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⢱
⢣⣀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠆⠀⠁⠀⠀⠈⠆
⠈⠳⡄⠀⠀⠀⠀⠀⢀⠄⠄⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀
⠀⠀⠙⠖⠦⠀⠠⢄⠈⠂⠄⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡔⠊⠉⠓⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠂⠉⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠑⣤⠄⡶⠛⠈⠉⣁⡀⠤⠍⢀⠾⠍⠩⠩⠋⢉⠭⣋⠀⠉⠀⠀⠀⠀⢆⠀⠄⠈⠀⠀⠀⠆⠒⢰
⠀⠀⠀⠀⠀⠘⠧⠀⠣⠠⠐⠉⢉⡴⠀⠀⠈⠚⢠⠄⠔⢋⠜⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠑⠤⠤⠒⠀⡐⠈⠢⠤⢊⠠⢚⢀⡎⠠⠈⢐⢀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠊⠀⡀⠀⡠⢀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⠐⢧⣈⠑⠠⠄⠀⠀⠀⠀⠘⠀⠘⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢀⠀⠀⠆⠄⠈⠈⠁⠈⠈⡀⠠⡀⠙⣈⡁⣈⡡⠃⠀⣡⠖⠀⡀⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠈⠀⠀⠀⠀⠀⠀⠈⠠⠤⠀⠋⡀⣀⠀⢁⠉⡀⠂⠀⠀⠀⠀⠀⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠢⡤⠔⠂⠀⠀⠀⠠⠀⢀⠀⠀⠀⠁⠀⣢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⣀⠐⠠⢬⠄⠀⠐⠂⠠⠊⠀⡀⢀⠀⡠⠀⠀⢐⡀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠘⠀⠁⢠⠀⠀⠀⡐⠂⡀⠀⡘⠀⢡⠜⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠢⠤⠀⠢⢤⡠⠆⠀⠀⠫⢀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡙⢤⡀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢄⡌⠁⠐⢢⡀

⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣠⣽⣿⣿⣷⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣸⣷⣾⣇⣆
⠀⠀⠀⠀⠀⠁⠀⠀⠐⠻⢿⣿⣆⠁⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣬⣿⣿⣷⡌⢠⡀⢀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣻⡿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⣀⣼⣾⣶⣿⣿⣿⠛⣿⣹⣾⣷⣦⡛⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠻⣿⣿⣿⣿⡟⠛⠂
⠀⠀⠀⠀⠔⠙⠛⣻⣿⣿⣿⣿⢷⣿⣿⢿⡿⢿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣐⠀⠀⠀⣤⣹⣿⣿⣻⣇
⠀⠀⠀⠀⠀⠀⠀⣨⣽⣯⡿⣿⣿⠿⣿⣷⣶⣼⣿⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⣙⡗⣄⢘⡿⢿⢽⣿⣿⡿
⠀⠀⠀⠀⠀⢀⣼⣿⣿⢋⣼⢯⣯⣾⣿⣿⣿⣿⣿⣷⣾⣿⠇⠀⠀⠀⠀⣠⣿⠁⠀⠈⠠⠛⠌⠀⠈⣿⣇
⠀⠀⠀⠀⠀⣾⣿⡯⡇⠸⣿⣿⡏⡓⢾⣿⣮⣿⣿⣿⣶⣯⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⢠⣷⢀⡟⢿
⠀⠀⠀⠀⢀⣿⣿⠀⠁⢠⣿⣿⣇⣙⠻⠟⠉⠙⠛⠙⠛⠿⢿⡶⣤⣦⠟⠀⠀⠀⠰⠆⠃⣠⣾⣿⣿⣿⣽⡇
⠀⠀⠀⠀⠈⠋⠙⠀⢰⣻⣿⣷⣾⣷⣶⡄⠀⠀⢀⣠⣴⡭⠄⠐⣻⡼⠆⠀⡀⠄⠀⣀⣾⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⣏⡨⣱⣖⣶⠛⠛⢉⠀⡀⠠⣾⣟⣼⣲⡄⣳⢧⣸⣇⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⡿⠿⢇⡆⠀⣡⠔⠀⠀⠉⠀⠘⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⡏⣸⣿⣯⣜⣛⡿⣿⠟⢁⣰⠇⡀⠀⠀⠀⢀⣽⣿⣿⣿⡿⠿⢋⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇⣿⣿⣿⢿⢑⣿⡀⣼⣿⣾⣷⣾⢀⣾⣿⣿⣿⣿⣏⠀⣤⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⣾⣤⣡⡼⠉⢻⡽⣿⣿⡿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣟⣷⣾⣻⡿⣿⣿⣷⣶⣯⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠉⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⢿⣿⣿⠟⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⣾⣿⠋⠀⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠟⠋⠀⠀⠹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠘⣿⠆
⠀⠀⠀⠀⠠⠤⣦⣤⣯⡶⣾⣭⣿⣿⣿⣿⣷⣀⣀⡀⣸⡟⠀⠒⠒⠀⠀⠀⠀⢹⣆⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠙⠊⠋⠻⢿⣿⣿⣿⣿⣿⣿⠀⣤⣤⣶⣶⣶⣶⣾⡷⢿⠸⣿⣿⣷⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⢿⡀⣛⠿⠛⠉⠙⠛⢲⡾⣋⣰⠯⡙⣏⣥⣤⠄
⠀⠀⠀⠠⠀⢠⡤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠴⠎⠑⠛⠓⠮⣛⣖⠂⠀⠀⠉⠉⠉⠉⠓⠬⣿⠛⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠉⢻⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢸⡇⠀⠀⠀⡟⠛⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⣀⡀⢸⢸⡇⣀⣀⡀⡇⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣬⣭⡅⢸⢸⡇⣭⣭⡅⡇⠀⢀⣠⣄⡠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⡇⠴⠶⠆⢸⢸⡇⠶⠶⠄⡇⠀⣸⣁⣈⣷⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡇⠒⠒⠂⢸⢸⡇⠒⠒⠂⡇⢰⡏⢠⡄⠉⣧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠻⠶⠶⠶⠞⠀⠳⠶⠶⠞⠃⢸⡇⠘⠃⠀⣿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣻⡿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠰⠒⠆⠰⠰⠘⠰⠸⠐⠄⠸⠇⠆⠂⠖⢰⡰⠰⠂⠂⣞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠿⢿⣿⣶⡄
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⣴⡏
⠀⠀⠀⠀⠉⢉⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⢀⣴⡿⢛⣽⡿⢋⣼⣿⣋⣉⣽⣯⣷⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⢠⣾⠟⠀⣼⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡾⣷
⠀⠀⣾⠇⠀⢸⡿⠀⣿⡟⢻⣿⣿⣟⣫⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⡂⠀⡀⣀⣀⣿⣾⡿⣷⣿⡆
⠀⢠⡿⠀⠀⢸⡇⠰⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠙⠁⠘⠉⠛⠛⠿⢻⣶⣿⣿⠃
⠀⠀⡇⠀⠀⠀⣇⠀⣿⡀⣿⣿⡿⣿⣿⣷⣿⣿⡿⢿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢽⣿⡏
⠀⠀⡇⠀⠀⠀⠈⠀⠸⣧⣿⣿⢁⣿⣿⢿⣿⣿⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⡴⠆⠄⢤⡀⠀⡤⠚⠂⠌⠻⣿⠝
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⡿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⠇⠐⢠⠀⠀⣨⣿⣶⣢⣶⣢⡚⠣⠉⢀⢴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⢸⡿⠇⣿⣿⣿⣿⣿⣿⣿⡐⣀⡜⢠⣾⣿⣿⣫⣿⣿⡿⠉⢀⣲⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⡀⠘⣿⣿⠘⢻⣿⣿⣳⣿⣷⣿⣿⣿⣿⣿⣿⢃⣴⣯⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠈⠙⣇⠀⠻⡌⠿⢿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠀⠀⠉⢉⠋⠀⠈⡏⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡇⠀⠈⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢶⡤⠄⣈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⡘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⠿⠟⠀⠀⠘⠛⠛⠛⠛⢛⣷⠟⠀⠀⠙⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡇⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⣻⡟⠁⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠲⠒⠒⠒⠒⠒⠶⠂⠀⣿⡻⣦⣀⣾⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠶⠶⠶⠶⠶⠶⠶⠆⠘⠿⠟⠛⠋⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣾⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡏⠉⢹⣿⣿⠉⠉⣿⣿⡏⠈⢹⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃


⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⢀⠄
⠀⠀⠀⠀⠀⠀⡗⢎⢾⠙⢲⡁⢘⢉⠆⡇⣎⢹⠀⢸⠻⡀⡏⠃⣎⠓⣘⢙⢶⢰⡏⡇⣎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠊

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣠⣿⣿⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠟⠻⠟⠛⠛⠛⠛⠻⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣤⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠛⠟⠿⠿⠿⠿⠿⠿⠿⠟⠟⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣶⣶⣶⣶⣶⣾⣿⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢹⣿⡇⠀⠀⠙⠛⠛⠛⠛⠛⠛⠁⠀⠀⢠⣤⣤⣤⡀⠀⠀⢸⣿⡇⠀⠀⣠⣤⣤⣤⣤⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⢸⣿⡇⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⢛⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣦⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⢠⣄⠀⣷⢀⣷⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣻⣿⣿⣿⣿⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣦⡀
⢀⣤⣨⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⢰⣿⣿⣿⣿⠟⢛⢩⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣿⣿⣿⣿⡷⣺⣋⣭⣽⣭⣫⣟⣻⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣯⣾⣿⣿⠏⠃⠈⠙⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⡌⣿
⢈⢹⣿⣿⣿⣿⢃⠆⢠⠦⡀⠀⠀⠑⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⠿⣿⣿⣿⣽⣿⣿⠿⣇⠀⢿⣇⠘
⠀⠸⣿⡇⢻⣿⣌⠴⢋⠀⡀⠀⠀⠀⠈⢿⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠋⢁⣈⣉⣙⣻⣿⣿⣯⣽⣻⣿⣷⣽⣿⣿⣿⣿⡄⠀⠀⠸⣿
⠀⠀⠹⣏⣶⣿⣿⠀⠉⠀⠘⠀⠀⠀⠀⠀⠐⢽⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⡸⢁⠀⣉⣹⣻⣛⢿⣿⣿⣻⣿⣿⣿⡝⢿⣿⣿⣾⣿⣷⠀⠀⠀⣿
⠀⠀⠀⠈⠟⠛⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠱⣄⢹⣿⣿⢦⣀⠀⠀⢀⣤⣤⣶⠦⣷⣼⠋⣠⣤⣀⠂⠉⠙⠻⣮⣭⢿⣿⣽⣿⣿⣿⣶⣿⣿⣿⡿⣿⠀⠀⠀⣿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⣤⢸⠀⣄⡀⢾⠈⠑⠈⢿⣿⣻⣿⠟⠛⠁⠀⠐⠤⡅⠠⠒⠫⡘⠉⠛⠛⣿⡿⣿⣟⡆⢻⣿⣿⣿⣿⣿⣾⣿⣿⡇⠏⠀⠀⠀⣽
⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⡇⡿⠈⢀⣻⡇⢸⡇⢿⢷⠿⠘⠛⠉⠀⡈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⡻⣿⣿⠀⠀⠀⠀⢀⡇
⠀⠀⠀⢸⣆⡂⡆⠀⣠⠠⠀⡁⡄⣬⡉⠀⣧⠓⠃⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠛⣿⣿⣿⠻⣿⣾⣿⣿⢿⣿⣳⣿⣿⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⢿⠸⢁⣴⣧⣆⡾⣿⢻⣿⣿⡆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠄⢀⠀⠀⠀⠀⠰⣷⡀⣿⣿⣿⣷⣹⣿⣿⣿⣼⡇⢸⣿⢿⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⢠⣾⠁⠨⣾⢁⡟⢮⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⣀⢳⣄⠈⡇⢄⠀⡄⢦⠘⣷⣿⣿⣿⢻⡏⢿⠿⣿⠋⠀⣾⠇⣾
⠀⠀⠀⠀⠀⠀⠙⣦⣀⣧⡞⣠⠎⣼⡿⣏⣿⣄⠀⠀⣤⠀⠈⠀⠘⡌⢶⠀⣽⣾⡟⢶⡅⢈⡆⢷⠘⡇⢰⢹⣿⡿⠘⡇⠸⠁⡿⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣏⣼⣿⣷⣿⣿⣿⣼⣦⣿⣾⣖⣇⠀⠙⠀⢻⣽⠎⠻⣫⢿⣄⠇⣸⡇⣷⢸⡿⣿⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣟⡟⣿⣻⣿⣿⣷⣿⣿⣿⡿⢳⡀⠀⣄⣿⡎⠃⣀⠙⢧⣿⣧⢻⣷⡟⠉⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢵⡙⣿⣟⣿⣿⣿⣿⣿⣿⡇⢾⡿⣿⣷⡆⠘⣿⣾⣽⣿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣟⣿⣿⣿⣿⣿⣁⠈⠻⢮⣽⢷⣄⢿⣿⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⡶⢦⣳⣮⣷⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃⠀⠈⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⣀⣤⡤⠶⠚⠋⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠉⠋⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⡡⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⢀⣀⡠⠤⣤⣀⣒⣢⣤⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠞⢻⡞⠀⠀⠀⠀⠀⠀⠀⠀⠴⠦⡶⠄⣠⡾⢹⣿⣷⣿⠿⢶⠶⠟⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠶⠞⠛⠛⠓⢀⡿⠅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣻⣟⣿⣯⣝⣻⣟⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣿⣿⣿⣭⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⣦⣀⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡄⣴⡿⠋⠙⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⣩⡿⠋⠀⠀⠀⠀⠙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣾⠟⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠛⣿⡇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢰⣿⣧⡀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣼⣧⣈⠻⣦⣄⣴⢿⣷⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⣿⣿⡷⠿⠛⠁⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣷⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠹⣿⣿⣿⡟⠉⠉⢻⣿⣿⣿⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⢀⣠⣿⣿⣿⣦⣀⣀⣼⣿⣿⣿⣄⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⡷⢀⡤⣄⠀⣠⠤⢀⡤⢤⡀⣀⡤⢀⡤⣼⠀⠀⢸⣠⠞⠀⡠⢤⡀⣠⠤⡀⢠⡠⢤⠀⣉⢠⡤⢤⡀⢠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣇⢸⡓⠚⠸⣅⠀⢸⡄⢀⡇⣿⠀⢿⡀⣸⠀⠀⢸⠙⢦⠸⣗⠚⠃⣟⠒⠋⢸⡀⣨⠇⣿⢸⡇⢸⡇⢯⡀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠉⠁⠸⠉⠁⠀⠀⠀⠀⠀⠀⠠⡥⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠘⠀⠀⠀⠐⠐⠂⠘⠀⠀⠀⠀⠃⠐⠀⠀⠠⠈⠃⠀⠀⠀⠄⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⢠⡐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣀⣨⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠔⡟⠉⣀⣀⡈⠱⣄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⠖
⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⣯⠿⠭⣁⣈⠳⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡞⣽⣿⣷⣿⣀⣀⡙⢦⡹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠸⢠⣿⢻⣿⣿⣷⣶⣭⣷⣿⡻⠧⠀⠀⠀⢀⣄⢀⣀⠀⢀⣠⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠏⣿⣿⣯⢾⣿⣶⣶⡄⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢙⣿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠸⢸⡻⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⣾⠀⠀⠀⠀⠔⣴⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⠀⠀⠀⣷⣆⣜⣈⣷⠶⠿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠛⣿⠻⠀⣤⣼⠉⠉⠉⠁⠀⠀⠀⢱⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⢿⡁⡤⠀⠀⠀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡷⠞⠁⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⣿⣿⣷⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢳⣰⡏⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠋⠄⠸⠿⠿⠿⠛⠙⠛⠛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠘⠿⡃⢸⣿⣿⣿⠿⠿⠿⠿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡈⠈⣤⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⣴⡇⢠⣤⣤⣤⣤⣤⣤⣤⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠂⠁⢂⠘⠛⠛⠛⠉⠉⠉⠉⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢧⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢜⣛⠅⢸⣻⣿⣿⠛⠛⠛⠛⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⣶⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠋⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⠉⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠀⠀⠀⣀⡀⠀⠐⠁⣀⣈⡠⡤⠶⠥⠚⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢉⣹⣋⣀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢰⡏⡠⠊⠉⠀⡼⠁⠀⢈⠆⡷⠖⠒⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠧⡠⢤⠞⠙⠓⠒⠛⠚⠧⢤⡀⠒⠠⠀⠒⠠⡁⢶⣀⡂⠤⠬⠍⢰⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠒⢏⣀⡀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⣄⢀⠀⠀⡀⠍⠀⠀⡀⠀⠀⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣀⠀⠀⠀⠰⠁⠀⠀⠀⠃⠀⠉⣹⡖⢒⡒⢠⣖⢚⡛⠲⠦⢄⣂⢭⣀⣒⠢⠭⡅⠹⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⢰⠦⠬⠃⠀⢀⠇⠀⠀⠀⠈⠀⢀⡴⢉⠁⣿⡏⠹⠼⠁⢹⣶⣤⣤⡈⠁⠀⠀⠀⠀⠈⡦⠤⠄⣀⡀
⢹⡿⣭⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⢢⣤⠀⠀⠈⠒⠒⠣⢄⡜⠀⠀⠀⠀⡷⠖⠉⠀⢸⠀⣿⡇⠀⠀⠀⣸⣿⠀⢸⡿⠀⠀⠀⣤⣀⠀⠇⠀⠀⠀⠙⡇
⠈⣿⠷⣿⣷⣦⢄⡀⣀⣀⣠⢤⣴⠶⢮⣛⠯⣻⠆⠀⠀⠀⠀⠀⢰⠁⢀⠠⡴⠶⠁⠀⠀⢠⡎⢸⣿⣧⣤⣶⣶⣿⣿⠀⣼⠇⡄⢸⠙⠛⠈⣠⠀⠀⠀⠀⢀⠁
⠀⢻⢔⠽⣿⣝⠑⢫⣉⡢⠼⠓⠂⠉⠀⠀⠀⢘⡇⠀⠀⠀⠀⢀⠃⠀⢀⠜⠁⠀⠀⠀⠶⣡⠁⣾⣿⣿⣿⣿⣿⣿⣿⢀⣿⢠⢠⠺⡄⠀⠀⢙⣤⠀⠀⠀⡜
⠀⢸⡗⠦⣦⢝⣳⠜⢢⡄⠀⠠⢄⡀⠠⠤⣀⣻⠀⠀⠀⢀⣀⡎⢀⠔⠁⠀⠀⠀⠀⠀⣠⡞⢸⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠆⡌⠀⠓⢒⣋⡅⠀⠀⠀⢠⠃
⠀⠀⣇⡈⠀⠝⢿⣛⡴⢶⠛⠻⢟⣛⠛⠋⢭⡿⠀⠀⠀⢹⠟⠴⣿⠄⠀⠀⠀⠀⠘⠊⣰⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⠀⣇⣀⣀⣈⡅⢿⠀⠀⣠⠃
⠀⠀⣯⢻⢮⣔⣽⣯⠑⠒⠛⠉⠀⠀⠈⠉⢻⡇⠀⢀⠤⢿⠂⢴⠁⢀⡀⠀⠀⠀⠀⢠⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛⠁
⠀⠀⣟⠡⡳⢊⣛⡲⠤⠄⠒⠢⠤⠬⣤⣀⣿⡇⢀⡟⡧⡇⠀⣿⠀⡸⠀⠉⠒⠂⠤⠧⢈⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⣉⣉⠁⠈⠁
⠀⢈⡷⣮⣮⢥⡿⠿⠛⠛⡛⠛⠒⠒⠶⠴⣿⠀⡸⣤⡽⣤⡤⢭⡶⠥⠤⠤⠔⠊⠙⢦⣀⠙⠶⢦⣬⣉⣙⡛⣿⡿⠿⠾⢄⠀⡠⠊⠀⠀⠉⠒⠠⠄⣀
⠀⢸⢟⠗⡻⣷⠮⣍⣓⠒⠤⠉⠉⣉⣈⠀⣿⡀⠁⣈⣵⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣲⣆⡀⢋⠉⠁⠀⠀⠀⠀⠀⠫⡀⠀⢀⡀⣀⠔⠓⠂⠬⣀
⠀⣿⡭⡫⢺⠶⢿⡶⠾⠟⠛⠓⣂⣀⠌⠭⢿⣇⠼⠿⣦⣞⠳⣄⣀⣀⣀⡤⠤⠴⠂⣺⡋⠠⢚⡅⠈⠉⢣⡀⠀⠀⠀⠀⠀⠀⠙⢶⡁⠀⠉⠑⠒⠦⠤⣄⣉⠒⢠
⠀⠀⠀⠀⠻⠛⠲⠜⠛⢛⣛⡉⠥⠖⠓⠚⠋⠉⠑⢤⡀⠙⠯⣢⣄⣉⣑⣦⣤⣀⡴⣍⠂⠉⠉⠉⢁⠟⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⣠⡚⠕⠊
⠀⠀⠀⠀⠀⠀⠈⢫⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠀⠉⠁⠀⠀⠀⠀⠀⣉⡵⢂⡴⠁⠀⠀⠰⠀⠑⢦⣀⣀⣠⡤⠀⠂⠂⠉⡀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣀⠀⠀⠀⣀⣀⡠⠴⢒⠩⠒⠙⠢⣀⣠⣤⠴⢖⡒⠉⢀⠔⠉⠉⠙⠲⣖⠃⠀⠀⢀⡨⣖⠉⠈⠁⠐⠛⠁⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⠙⢏⠉⢉⡡⠐⠛⠀⡀⠀⠀⠀⠈⢁⡠⠔⠚⠉⠛⠁⠀⠀⠀⣠⠊⠈⠲⡤⠚⠁⠀⠀⠈⣁⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠳⣤⠔⠊⠉⠁⠒⠒⠤⠄⣀⠀⠀⡔⠣⡀⠀⠀⢀⣀⠠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠙⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⢟⡯⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣖⣭⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢻⠍⣐⣭⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣽⣿⣿⣿⣿⣿⣷⣿⡿⣟⠏⢙⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣴⣾⣿⠿⠻⣿⣿⣿⣿⣿⣿⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⣠⣤⣶⣿⡿⢟⠻⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣯⢁⣢⣴⣾⣿⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢽⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⠉⠻⠻⠿⣿⣿⣿⣿⣿⣿⡿⠛⠉⣀⣬⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡟⠁⢀⣀⣤⣤⣤⢾⣾⣿⣿⣿⣿⣿⣿⣶⣿⡿⠟⡏⢻⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⢃⣴⣾⣿⣿⣿⣿⣾⣽⣿⢿⣿⣿⣿⣿⣿⣇⣠⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⢿⣿⣿⣿⣿⡻⣯⡙⢿⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣄⠙⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⠀⠀⠀⠀⠀⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠃⢄⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇⠀⠀⠑⣦⡀⢸⣿⣿⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⣿⣿⣿⠀⠀⠀⠈⢻⣖⣲⣿⡟⠉⠙⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣿⣿⠀⠀⠀⠀⠀⠈⠻⠿⣿⣷⣶⣦⣤⡄⠉⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⢿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⠋⠀⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠁⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣀⣀⣤⡿⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣠⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠁⢉⣈⠳⠂
⠀⠀⠀⠀⠀⠀⢼⣠⠔⣠⠞⠻⣟⠃
⠀⠀⠀⠀⠀⠰⠶⠅⠀⡙⠁⡑⠈⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠶⠦⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢳⡇⡈⠐⠃⢶⠰⢌⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠋⢀⣀⣿⣽⣛⡛⠻⢿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡽⡁⠀⡄⠈⠀⢀⡁⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢋⣴⣿⣿⡿⣛⣉⣭⣭⣀⡀⢨⡙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⢱⡇⠀⠀⢠⠘⣦⠙⢷⡸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣛⣭⣭⣭⣭⣍⡛⠿⣿⣦⡹⣦⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠈⢀⡆⠀⢸⡆⣼⡆⡄⠇⢿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢻⣷⢹⡇⠟
⠀⠀⠀⠀⠀⠀⢸⡇⣆⠸⠃⡄⣆⠃⠛⠀⡇⠀⣬⣿⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢉⣩⣵⣾⣿⣿⣶⣤⡉⠙⠻⣿⣿⣿⣇⢻⠸⠃
⠀⠀⠀⠀⠀⠀⢸⣇⣾⡆⣼⠃⣿⣀⡆⢰⢣⠇⣼⠈⢦⡀⠀⠀⠀⠀⣾⣿⠋⠴⢛⣛⣭⣭⣭⣛⠿⣿⣿⣧⡀⢬⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠸⠏⠸⡷⠃⠜⠻⠛⠀⠀⠀⠀⠃⠀⠢⡙⢶⣤⣠⠞⢿⣇⣴⠾⠿⠿⠿⠿⢿⣿⣿⣦⡙⠻⣷⣸⣇⣿⣿
⠀⠀⠀⠀⠀⠀⡠⣴⠄⠁⠀⠀⢀⡄⠒⠐⠃⠀⠀⢀⢫⡆⠠⠖⠂⣀⠋⠉⠤⢤⣶⣶⣤⣤⣀⠀⢍⣻⣷⢰⠙⡇⣿⢹⡏
⠀⠀⠀⠀⠀⠀⣿⠘⠐⠂⣀⠚⢀⡀⠐⠃⠸⢀⣇⢺⡇⣇⠹⡌⠘⠎⢷⠀⠑⠶⣤⣍⡛⠿⣿⣿⣦⡹⣿⢾⣧⠀⡿⠈
⠀⠀⠀⠀⠀⠀⠻⡄⢧⠠⠄⠢⡎⠑⠇⡑⠀⠸⣿⠘⡇⢹⡄⣿⢸⡀⢈⠠⣉⠳⣮⣛⢿⣦⡈⠻⣿⣷⢻⡎⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡎⢷⣄⠆⠀⣣⡞⠁⠀⠀⢻⢠⡇⣸⣻⠏⢸⡇⠈⣧⢹⣶⣝⢿⣷⡙⢷⡀⠘⣿⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⢿⣷⡿⠋⠀⠀⠀⢢⡀⠈⠀⢃⡿⣷⣸⠃⠀⢹⡟⣿⡙⣧⠙⣿⡄⠃⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⡝⢦⣄⠀⠀⠀⡳⣝⢦⠀⠘⠁⣿⣁⣰⠀⢸⠀⠹⡇⠻⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠙⢷⣦⡲⣝⢮⡓⠀⣰⠠⢃⣿⢯⣦⡠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡉⠻⢮⣷⡍⠰⠃⠀⣰⠯⠞⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠀⢦⡀⡄⢒⡏⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣾⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣴⣄⡞⡿⣠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉

⠀⠀⠀⠀⣤⠀⢀⣤
⠀⠀⣼⣾⣿⣶⣿⣿⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣦⣤⣀
⠀⠐⠻⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠚⣻⣽⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣯⣭⣍⠙⠻⣿⣷⡄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢹⡟⣿⡆
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣮⣝⠻⡝⣿⡄⣧⠘⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣦⡈⠇⠁⠀⢿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡟⢷⣍⢷⡄⠀⠀⡞
⠀⠀⠀⠀⠻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠙⢷⡀⠉⠂⠙
⠀⠀⠀⠀⠀⠈⣿⣭⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣌⣙⣻⣿⣿⣿⣿⣿⡿⣻⣿⣿⡏⣿⣿⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣭⣙⣛⣛⣛⣯⣾⡿⠋⠈⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠞⠛⠻⠧⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⣧⣽⣦⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡛⠛⠹⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡿⢓⠊⠉⠲⣆⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⣭⣿⣿⣿⠷⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣠⣤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁⠈
⠀⠀⠀⠀⠀⢀⣦⣴⣾⣿⣷⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠘⣿⣻⣿⣿⣿⡁
⠀⠀⠤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠂⠀⠀⠀⠀⠀⠀⠈⢻⣍⣉⡾⠁
⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡉⠿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⢠⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣶⣾⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠆⠀⠀⠁
⣾⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⠝⣈⢉⣈⢉⣽⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⠂
⢸⣿⣿⣿⡇⠀⢺⣿⣿⣿⣿⣿⣿⣿⡿⢻⣵⣞⣧⣯⡿⣡⣶⣦⠀⠤⠀⠀⠀⠠⠄⠀⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠈⣿⣿⣿⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⢂⡾⣻⣏⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠛⠏⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡿⣿⡿⣿⡿⡿⠁⠀⠃⠉⠤⠁⠀⠀⠠⢀⡀⠀⠠⠀⠀⠀⠀⢠⠀⠀⠀⠀⠴⠗⣄⣸⣿
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢋⣵⣯⣾⠁⣀⡂⠀⠀⠀⢀⣴⡆⠀⠀⢀⣀⠤⡵⢤⡀⣄⠀⠀⠀⢀⠀⢠⠘⢃⠿⠋
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢠⣿⣧⠀⣶⢲⣾⣿⡇⠀⣷⠚⡷⢰⣿⣶⣌⡿⢣⣴⣦⡾⠃⣠⣤⣾⠘
⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡗⠘⠁⡙⣇⣿⣿⣿⠛⣟⡃⢹⣷⣯⣬⣽⣿⡶⠛⢯⡄⣡⡼⠛⣡⡶⠿
⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠈⣧⠀⠀⣽⣿⡿⢿⣿⣿⣿⡋⣽⢡⣾⣿⣿⣿⡏⠍⠀⠙⣷⣤⣛⣛⢿⣿⣿⡷⡷⠆⠁⢀⡠⠰⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⣧⠀⠈⠉⠀⠘⠿⢿⣟⣿⣷⣼⣿⣿⣿⣿⣤⠂⠰⢊⡸⠉⠛⠿⢶⣮⣤⣤⣤⠀⣶⠈⣥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠿⣿⣿⣿⣿⣿⡿⠟⣡⣴⠛⣡⣄⠆⠀⣀⣾⡛⣏⠀⣠⢉⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠷⢿⣛⡭⠴⢟⣫⣬⣶⣿⣿⣿⣿⣿⡏⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⢯⣴⣶⣾⣿⣷⣿⣿⣧⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⣿⣿⣿⣿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⡿⣦⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠋⠉⠀⣸⢻⡄⣃⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⡆⠀⠈⠛⣿⠀⢀⠀⠛⣼⣯⢻⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣷⣤⣤⣒⣀⣠⡟⠁⠀⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⡠⠀⠂⠉⠐⠐⠀⠉⠛⠻⠿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠀⠂⢐⣶⡶⠶⣤⣤⣄⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢬⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⢀⡖⠈⠉⣍⢳⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣄⢁⡀⠤⡀
⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠺⠶⠶⠀⠠⠘⠀⠈⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⣄⡀⠀⠀⠀⠠⠆⠠⠀⠄⣀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣓⢾⡧⠀⠀⠀⠀⣺⣿⣿⣿⡿⢿⣿⣿⡀⠀⠀⠀⡀⣀⣀⣭⣭⡏
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⢠⠤⠤⠤⡇⠀⠀⠀⠁⠭⢾⣷⠶⠷⠛⣛⢹⠶⠠⠀⠉⠉⢛⣭⡶⠯⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢔⣀⡀⠈⠧⠀⠢⠤⠀⠀⠀⠤⢄⣀⣶⣶⣿⡃⠀⢈⠀⠤⠒⣒⠚⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠍⠉⢅⣲⣇⠀⠀⠀⠀⠩⣔⣟⣻⣯⠽⣾⣿⣤⠀⣪⢠⡀⢠⠄⠬⠔⠓⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠒⠛⠛⠛⠛⠛⠚⠰⠆⢰⣿⣿⣿⣛⠛⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⢠⡤⣤⢀⡄
⠀⠀⠀⠀⠀⢈⣉⣉⢘⡇
⠀⠀⠀⠀⠀⠘⠓⠛⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻
⠀⠀⠀⠀⠀⢠⡤⣤⢸⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠶⠶⢦
⠀⠀⠀⠀⠀⠈⠉⠉⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠸⠯⠿⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣀⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠘⠓⠛⢨⡟⠛⠉⠉⠙⠉⠛⠛⠈⣷⠙⠉⠋
⠀⠀⠀⠀⠀⢰⣖⣶⢈⡷⠶⠶⠶⠶⠶⠶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠘⠿⠟⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⢠⡤⣤⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠈⠉⠉⢨⡏⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠁⠀⠉⠑⠲⢄⡀
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⣴⠃⠀⡀⢀⣴⠞⣛⣉⣉⣍⣉⡙⠲⢦⣀⣠⡜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢛⡶⠀⣀
⠀⢰⡏⠀⠀⣱⣿⠕⠁⠀⣀⡉⠙⠛⣿⣿⣦⡙⣿⣿⣎⢳⡀⠀⠀⠀⠀⠀⣠⣞⠙⢿⣅⠠⢀⠖⠈⠒
⠀⢸⡟⠀⣾⣿⠁⢀⣤⣤⣤⣭⣿⣿⣿⣿⡻⣿⣮⢿⣿⡌⢷⠀⠀⠀⠀⢰⠛⠀⠀⠘⢻⣿⡀
⠀⠸⣷⠀⣿⡟⢰⡿⠋⠁⠀⠀⢈⣿⣿⣿⣿⡜⣿⣿⣿⣿⣸⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠻⡆⠙⢣⡿⠁⢀⣤⠖⠛⠛⣛⡛⠻⢿⣿⣸⣿⣿⣿⣿⡇⠀⠀⠤⠀⠀⠀⠰⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⣸⠇⣰⣟⡵⠀⠀⣀⣈⣽⣿⣷⣮⡛⢿⣿⣿⣿⡇⠀⢠⠇⠐⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠐⠋⠀⣿⡜⢀⡴⠋⠉⣁⣀⣬⣉⡛⢿⣦⣻⣿⣿⡇⠀⡼⢰⢡⠀⡆⣸⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⣿⠀⣴⢟⣭⡷⣿⣿⣿⣿⣿⣷⣿⣿⠀⣸⣿⣿⣿⣾⣿⣯⣴⣧⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⣿⣵⢟⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣣⣿⣻⣿⣿⣿⣿⡾⢟⣽⣿⡟⠛⠛⠛⠛⠻⢿⡟⠉⠳⣬⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢾⣼⣿⣿⡿⠋⠁⠀⣼⣿⣿⣗⣤⡀⠀⠀⠀⠈⣿⣤⠶⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣦⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣷⡷⠀⢀⣼⣟⣴⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⢀⡆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⣙⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⣿⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢻⣿⣿⡟⠀⣀⣀⣈⣽⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣥⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣯⣸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣘⣷⣀⣄⡀⢀⣠⣤⣴⠶⠀⠁
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⡉⠉⠉⠁⠀⠀⠀⠀⠈⠙⠛⠛⠛⣛⡟⢿⣌⣙⣉⣩⣍⡉⠁⠀⣤⣶⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣴⣤⣴⣦⣀⣀⣉⣉⣉⡉⠛⠿⣿⣿⠷⣦⣤⣾⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠈⠉⠙⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⠷⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣿⡅⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠚⢳⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢛⢏⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⣀⣤⣤⣤⣈⡙⠷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣧⠀⣀⣷⣄⠀⠀⠀⠀⠀⠀⣿⣿⣧⣾⣿⠿⠛⠋⠛⠛⠦⡉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⣸⣿⣿⣿⣷⣄⠀⠀⠀⠀⢻⣿⣿⣿⢃⣴⣾⣿⣿⣿⣟⡶⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠐⠮⣷⣾⣿⣿⣿⣿⣿⣿⣿⣷⣭⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⣿⣿⣿⣿⡿⢻⡏⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⡹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠈⠉⠟⠀⠀⠃⠀⠀⠐⢄⡀⠀⠀⠀⡹⣿⡇⠿⡟⢿⠇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣷⣤⣄⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣽⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⣷⣰⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣄⢀⠀⠀⢹⡿⡟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠮⢹⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠦⡀⢸⡗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠉⠓⣿⣦⠀⢹⣴⠇⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⢦⡈⣿⣧⣼⡟⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉⠉⠉⠛⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣤⣶⣄⠀⠙⣏⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡉⠓⢦⡀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠸⣿⣿⡿⠀⠀⠘⣦⡤⢼⡿⠂
⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠤⣄⠉⠳⣄⠳⡄⠀⠀⠀⠀⠀⣸⠃⢀⠀⠀⠀⠉⠀⠀⠀⠀⢹⡶⠋
⠀⠀⠀⠀⠀⣴⣯⠴⠶⠤⣄⠈⠳⡄⠘⡆⢹⠀⠀⠀⠀⢀⡟⣠⠟⡇⠀⠀⣶⠀⢀⣾⡆⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠹⡄⣿⢸⠆⠀⠀⠀⣼⠟⠁⠀⡇⢀⠞⢹⠀⡞⠸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠲⣄⣷⠀⣧⡟⣸⠀⣀⣤⠞⠁⣠⠀⢰⡷⠋⠀⠘⣷⠃⠀⢷⣰⡇
⠀⠀⠀⠀⠀⠀⣰⠃⣠⡤⢤⣈⠙⣿⠉⠙⠛⣉⡥⠴⠖⠋⡵⠂⠈⠀⠀⠀⠀⠉⠀⠀⠀⣽⠁
⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⣠⡼⠟⣿⠀⠀⠐⢧⡀⢀⣀⠀⠳⢾⣃⣤⢀⠀⡦⠀⠀⠀⣰⠇
⠀⠀⠀⠀⢀⡏⣰⠃⢠⡞⠁⣠⠞⠻⣆⠀⠀⠀⠉⣿⠁⠀⢀⡈⠉⠛⠋⣰⠃⠀⠀⣰⠏
⠀⠀⠀⠀⠸⡿⠃⠀⣾⠀⢸⠃⠀⠀⠘⢦⡀⠀⠀⠈⠓⠒⢿⣁⣀⣠⠾⠃⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠀⠀⠀⠀⠀⠙⢶⣤⣀⣀⡀⠀⠉⢉⣀⣀⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡇⠀⠀⠀⠀⠀⠀⠸⣧⣸⡏⠉⠉⠉⢻⣉⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡞⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠺⠟⠲⠀⠀⠀⠀⠴⠻⠟

⠀⠀⠀⠀⠀⢀⣀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⢐⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⣤⣤⣄⣀⣀⡐⣾⣿⣿⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣛⣛⣻⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⣤⠋⠙⠻⢏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢿⣿⣿⣿⣿⣶⣦⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠅⠀⣶⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣼⡟⠁⠀⠀⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣠⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⢀⣼⣿⣿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢉⠉⢧
⠀⠀⠀⠀⠀⠀⠀⢠⣶⢖⣶⡶⠶⠶⠋⠰⢯⠗⠈⠳⢶⣶⣖⣶⣲⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣹⣿⡽⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣫⣿⠁⣠⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢯⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢟⣽⠀⣿⢼⣹⠆⡏⠀⠀⠉⠉⠉⠛⠀⠀⢸⡵⡳⡧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⡟⠊⠉⠓⡇⠀⠘⠋⠉⠉⠁⠀⠀⢸⣞⢟⡷
⠀⠀⠀⠀⠀⠀⠀⣼⣣⣿⠀⠙⠛⠋⠉⢁⡀⠐⠛⠛⠛⠓⠀⠀⢸⣫⢞⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡫⣿⠀⣰⢶⣲⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⡿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢟⡿⠀⠙⠚⠋⠐⠒⠒⠒⠒⠒⠒⠒⠒⠀⢸⢟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⡷⣫⡇⠀⢸⠋⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢟⡇⠀⠈⠉⢩⠄⠛⠛⠛⠛⠛⠒⠂⠀⠀⣿⣿⢿⠇⠀⠀⠀⠀⢀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⢰⡿⢻⡇⠀⣾⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⠀⠀⠀⣀⡴⠋⣠⡏
⠀⠀⠀⠀⠀⠀⢸⡷⣻⠁⠀⢉⡉⠁⢉⡿⠟⠓⠒⠒⠒⠒⠀⠀⣿⡫⢿⢀⣠⠞⢁⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⢸⡟⣽⠀⠀⣿⣽⡶⣋⣀⣀⣀⣀⣀⣀⠀⠀⠀⣟⣴⡿⠋⣠⠶⠟⠋
⠀⠀⠀⠀⠀⠀⢸⣟⢾⣤⣤⣬⣭⣤⣄⣉⣉⣉⣉⣉⣉⣀⣀⣠⠟⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢸⣧⣫⢎⣞⡵⣱⣫⣫⢞⡝⡵⣱⣣⣫⣯⠞⣁⣴⣋⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⡧⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣰⣂⣤⣀⣠⠶⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⠯⣩⠝⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠈⣼⡿⣷⡿⠃⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⡌⢀⢴⡚⣉⣀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⠁⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣠⣳⡷⣡⣿⣾⣭⣭⣥⣁⢷⣄
⠀⠀⠀⠀⠀⠀⢠⣴⠃⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⡾⢟⠷⡚⣯⣟⡙⠻⣯⡙⠋
⠀⠀⠀⠀⠀⠀⢈⡏⠀⣀⠀⠀⠀⢀⣼⡎⢧⣴⣶⣶⠶⣤⢄⣠⣼⣾⣿⣭⡛⢮⠈⣿⠿⣧⣽⢹⡷⣌⠉⠂
⠀⠀⠀⠀⠀⠀⢾⡟⠀⠉⣶⣼⠀⣸⡟⣷⠄⣀⠀⡔⢦⢟⠓⠯⢉⣂⣬⣤⣽⣿⣿⠧⡄⠀⠙⢿⡷⣌⠦⠴
⠀⠀⠀⠀⠀⠀⢸⢠⠀⠀⠻⣿⣶⣟⠛⡁⠄⣔⠻⣟⡶⣷⠀⣀⡀⢀⠢⢄⡝⢿⡯⢲⣼⣶⣦⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣠⣄⠀⣸⣸⣿⢂⣅⣈⠀⡘⢰⠙⢎⠮⠙⠛⠘⠃⢦⢻⠮⢿⣿⣿⢿⣿⡇⡟⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⣺⠻⣿⢿⣾⣿⢟⣢⣧⡘⠃⣈⠉⣂⠑⠈⡄⠲⣷⣜⣶⣿⣿⣿⡿⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣻⡆⠉⠃⠙⣲⡌⠉⠀⠀⣽⣓⠀⠀⠈⠁⠀⢸⣧⣻⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣁⠀⠠⡀⠚⡟⣀⣠⠾⠋⢽⠉⠉⠉⠀⠀⣱⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢷⣯⣜⣹⣿⠦⠀⠘⡐⡆⠀⠀⣺⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⣿⣧⣤⣮⢀⣍⣳⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠮⠅⠄⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣘⣀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠁⠀⠐⠛⠖⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠶⠶⣶⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⣄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠻⠤⠤⠤⠬⢷⡄⠀⠀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⠄⠤⠤⠄⠤⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠖⠲⠶⢤⣀⠀⠀⠀⠀⠒⠒⠀⠒⠒⠀⠒⠒⠐⠒⠒⠐⠒⠒⠐⠒⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡀⠀⠀⠉⢳⡀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⡞⠉⠉⠉⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠑⠶⠶⣄⠀⠀⠀⢸⡇⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⣀⣀⣠⠟⠀⠀⠀⣼⠁⠀⢀⡀⣀⣀⡀⣀⣀⡀⣀⣀⡀⣀⣀⡀⣀⣀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠁⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠦⠴⠶⠚⠉⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⢸⡇⠀⢸⠗⠒⣿⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡏⠉⢹⡏⠉⠉⠉⡏⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⢹⡇⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡟⠒⢺⡗⠒⠒⠚⡗⠒⠒⠒⠒⠒⢺⡗⠒⠒⠒⠒⠒⢺⡅⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⣤⣼⣧⣤⣤⣤⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣼⡇⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⢸⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⠀⠀⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⡇⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣛⣾⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣯⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⢄⡀⠘⢦⣄⠀⢄⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⢸⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⡝⢦⠈⠳⣄⠀⠀⠀⠀⡐⢱⣿⣧⣼⣷⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⢿⠈⣧⠀⠹⠀⠀⠀⣰⠋⣼⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⣿⣿⠇⣿⠀⠘⠀⠀⣰⠇⣰⣧⣿⣯⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣉⣤⣶⣶⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⡉⣹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢋⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⠻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡏⠀⠀⠛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡠⠄⠰⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠖⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠶⠤⠤⠤⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣲⠶⠛⠉⠉⣉⣈⠭⢟⣠⡀
⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⡤⠖⠋⠀⢰⡇⢀⡠⠔⠊⣡⠴⢒⣉⣼⠃
⠀⠀⠀⠖⢻⡷⠶⠶⠶⠶⠶⠶⠶⢶⣶⣦⣤⡤⠤⣤⣤⣤⣤⣤⣤⣤⠴⠯⠬⠷⣖⣋⣻⡏⠉⣉⣉⠽⠋⡡⠴⠚⣉⠴⠊⠁⡗⡏
⠀⠀⢰⠶⠟⠒⠒⠒⠚⠋⠉⠍⠍⠀⠀⠀⣀⠀⣐⣂⣀⣀⣀⡀⢒⣀⣀⣀⣐⣒⡒⠒⠀⠤⠭⠭⠭⣶⡡⠔⢂⠩⢛⠨⢙⡩⠷⢇⣀⣀
⠀⠀⢸⡄⠀⢀⢀⣀⣀⠤⠿⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠬⠭⠭⢉⣉⣙⣻⡂⢩⣰⠼⠒⠊⠁⠀⠀⠀⠀⠈⣉⣳⡦
⠀⠀⣼⠀⣀⣉⣁⣤⠤⠤⠤⠶⠶⠒⠒⠒⢒⣛⠛⣛⣛⣒⣒⠒⠒⠒⠒⠒⠲⠦⠤⠤⠤⠤⠤⠴⠶⠾⠗⠉⠀⠀⠀⠀⣀⣤⣤⠒⠚⠉⠁
⠀⠀⠉⡠⠋⠁⣀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠬⠥⠭⠭⠽⢶⣶⣶⣦⣤⢤⣄⣀⣀⡀⠀⠀⠀⠀⣀⠤⠖⠩⢑⡒⠊⣻
⠀⠀⠚⠉⠉⣩⠉⠀⠀⠀⠀⠀⠐⠐⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠐⠒⠒⠒⠒⠒⠛⠂⠀⠀⠀⠉⠉⢹⡲⠭⢀⣒⠤⠍⠀⣒⣀⡽
⠀⠀⠀⠒⠂⡇⠀⣀⣀⣀⣀⠀⠈⠉⠉⠉⠉⣉⣁⣒⣛⣛⣒⣒⣒⣒⣒⣒⣶⠦⣤⣤⣤⣭⠍⢉⡀⣼⠐⣊⡡⠴⠒⠊⡉⠉⠀⠠
⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠭⠛⠉⠉⠉⠩⠭⠍⠁⠈⢨⠔⠂⠀⣈⠠⠀⠀⠀⡇
⠀⡰⠁⠀⠤⠤⠐⠒⠒⠒⢈⣉⡉⣉⣉⣉⣉⠉⠉⠭⠷⠶⠶⠶⠶⠶⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⡀⠄⠒⠀⠠⢀⣰⣇
⠀⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⠤⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒⢸⠀⠀⠐⠚⠣⠤⠤⣼⣄⣱⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠒⢺⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠙⠛⠛⠋⠙⠋⠡⠤⠤⠤⠤⢀⣀⣀⡀⠀⠈⠉⠉⠉⡝⠀⢸⠀⢈⡀⠄⠀⠂⢁⡏
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠬⠭⢍⣉⣙⣛⣁⣀⣼⢈⠠⠀⠀⠈⢁⣰⠇
⠀⠀⠀⠀⠄⠚⠑⠠⡤⠤⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣈⣙⣛⣛⣁⣒⠒⠒⠒⠢⠤⠤⠄⣇⢀⣀⠤⣔⡻⠿⡟
⠀⠀⠀⠀⠀⠀⢹⢎⡀⢤⡤⠄⠒⠒⠒⣊⣩⠭⠽⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⡉⢉⣙⠛⣋⠥⢒⣋⡁⠄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠉⠓⠀⠈⠁⠀⠀⠀⠀⠀⢠⣤⡤⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⣈⡥⠤⠖⣂⢹⠇
⠀⠀⠀⠀⠀⠀⠀⡸⠤⠤⠤⠤⠄⠰⠶⠶⠶⠶⠶⠒⠐⠲⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠁⠀⠐⠘⠋⠉⡟⠀⠀⠀⡀
⠀⠀⠀⢀⡠⠀⠀⢳⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⣶⡶⠶⠶⢶⣶⣤⣤⣄⣀⡀⠀⣽⠀⣀⣡⣤⣤⣶⡧⢀⡴⠊⡀
⠀⠀⣉⠤⢒⣩⠔⢊⡹⠗⢒⡲⠖⠒⣶⠶⠶⢶⡶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⠦⠶⠤⣤⠤⢤⡿⠭⢗⠟⢉⠵⣪⠔⣉⠴⢊
⠀⢀⠥⠒⣉⠤⠚⢡⠔⠊⠉⠀⠔⠉⠀⠔⠊⠁⠀⠂⠁⠠⠐⠉⠀⠠⠐⠁⠠⠒⠁⣀⠔⠊⣠⠖⠉⠠⠊⠡⠚⠵⠊⡤⠊⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⠞⡧⠚⢱⡧⠇⢹
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠏⠀⠁⠀⠉⡀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠟⠂⢠⣦⣴⣷⣞⡻⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⡸⢶⣟⣁⠈⣫⣿⢿⡓⠈⢦⡀⠀⠀⠀⠀⠀⠀⣰⡟⢀⡤⢞⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⢿⡟⣉⡾⡏⠧⠀⠀⢧⠀⠀⠀⠀⠀⣼⣿⣡⢏⣴⣏⡤⠖⣒⣫⣿⡿⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⢸⡿⢹⠃⠇⠀⠀⠀⠸⡆⠀⠀⠀⢸⢹⣿⢡⣿⢛⣡⣶⣿⣿⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢹⣆⡴⢻⠃⠃⠀⠀⠀⠀⠀⠀⣇⠀⠀⡀⢸⢸⣿⣿⣷⣿⣿⣭⣥⣤⣤⣄⡙⢷
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⢰⣿⠀⠀⣷⣼⣼⣿⣿⡿⠿⠿⠿⠿⣿⣿⡝⢿⣿⡇
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡆⣸⡷⢿⣿⣦⡀⣿⣿⣿⠟⠁⠀⢀⣠⣤⣤⣤⣽⣿⡄⢻⡇
⠀⠀⠀⠀⢰⣇⡔⠀⠀⢀⠄⢀⡄⢀⣾⠗⠋⠀⠸⣿⠉⠻⢾⣿⣿⠀⠀⠘⠉⠀⠀⠈⠻⣿⣿⣇⢸⡇
⠀⠀⠀⠀⠈⣿⠀⠀⣴⣋⣴⣾⣴⠋⠀⠀⠀⠀⠀⢻⣇⠀⠀⠉⠻⣆⠀⠠⠴⠶⣶⣦⣄⢹⣿⢿⡘⡇
⠀⠀⠀⠀⠀⢿⢠⣾⡟⠋⠀⣿⡇⠀⠀⠀⠀⠀⠀⠠⣝⣦⡀⠀⠢⣜⢧⢀⣤⣤⣬⣿⣿⣿⣿⢸⡷⣧
⠀⠀⠀⠀⠀⠈⠻⣿⡀⢀⠘⢹⣇⠀⠀⠀⠀⠀⠀⠠⡈⢻⡅⠀⢦⡘⣿⡟⠉⢻⣿⣿⣿⢻⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⡀⠈⠿⣦⠀⠀⠀⠀⠀⠀⣈⠻⣿⠀⠀⢿⣼⡇⠀⣼⠽⣿⣿⢸⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢈⠳⣤⡠⣄⠐⢦⡈⢳⣿⠀⣧⣸⠟⠀⠀⠀⢠⣿⣣⠞⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣷⣤⣝⠻⣷⣄⡹⣶⣿⣾⠿⠃⠀⠀⠀⠠⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡟⠉⠙⠀⠉⠛⣾⡟⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⣋⠋⠀⠀⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠚⠿⠛⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣾⣷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⣿⣯⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣽⣿⣷⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠛⠻⢿⣿⣿⣿⣿⠉⠙⢿⣟⠛⠻⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⣿⡄⠀⠀⠀⠀⢰⡟⣡⠚⣩⠴⠶⣶⣾⣿⣿⣟⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣞⣿⣿⣇⠀⠀⠀⠀⢸⣴⡗⣾⢿⣏⢅⡀⠉⠹⣿⡟⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣿⣿⣟⣿⣦⣄⣀⣠⣾⡯⣿⡆⢸⣽⠪⠙⢳⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣻⠯⢝⣿⣿⣿⣿⣿⣿⣷⣯⣍⣬⣝⣻⣷⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣟⡽⢻⣿⣿⣿⣿⣿⣻⣿⣿⣿⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣾⣟⣓⣤⣿⣯⣿⣿⣿⣷⣿⣿⣾⠻⣉⣀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⣀⠀⣠⢀⣀⣔⣀⠀⠂⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠈⠉⠛⣿⣯⣈⣘⣸⣟⣳
⠀⠀⠀⠀⠀⠋⠘⠋⢾⣿⠿⠿⠷⠦⠀⠀⠀⠈⠻⣟⢿⢿⣿⣿⣿⣿⡿⠛⠛⠋⠛⠒⠁⠀⠀⣀⠀⡙⠉⠉⠙⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⠞⠳⣄⡏⠉⠉⠁⠀⠀⠀⢀⡀⠀⠰⣤⣬⣿⣷⣴⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⠀⠀⠘⡇⠀⠀⠀⡰⢤⣬⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⡀⠀⠀⢀⡤⠶⠖⠒⢂⣷⠗⢛⣻⣿⣿⣿⣿⠿⠟⠉⠀⠀⠈⢧⠤
⠀⠀⠀⠀⠀⠀⠒⠾⢿⣿⣿⣿⣿⣿⣶⠶⠄⠀⠀⢀⣀⣼⣴⠶⠛⠋⠉⠉⠉⠃⠀⠒⠒⠂⠀⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣶⣤⠤⠤⣐⠂⠁⠀⠀⠀⠀⢀⣀⠊⠁⠀⠀⠀⠼⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠙⠓⠂⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠘⠀⢸⡇⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠿⠤⠤⠬⠷⡄⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢘⡇⠀⠀⠀⠴⠄⠤⠤⠠⠤⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠠⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⡇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⣀⣀⢀⣀⡀⣀⣀⢀⣀⠀⠀⠀⡇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⠈⠙⢢⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣴⠚⠓⠂⠀⠀⠹⡄⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⠲⠦⡄⠀⠀⠀⡇⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⠀⠀⡇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠⢤⡤⠃⠀⠀⣰⠃⠀⠠⠀⠤⠄⠠⠤⠄⠤⠤⠠⠤⠄⠤⠤⠀⠀⠀⡇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡀⠀⠀⢀⣠⠞⠁⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⡇⠀⢸⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠉⢹⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⠒⠒⡖⠒⠒⢲⠒⠒⠒⠒⠒⢲⠒⠒⠒⠒⠒⢲⠀⠀⠀⡇⠀⢸⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠶⠶⡷⠴⠶⢾⠦⠶⠶⠶⠶⢾⠶⠶⠶⠶⠶⢾⠀⠀⠀⡇⠀⢸⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡤⢤⣧⠤⡤⣼⡤⠤⠤⠤⠤⣼⡤⠤⠤⠤⠤⣼⠀⠀⠀⡇⠀⢸⠀⢸⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣄⣀⣇⣀⣀⣸⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣸⠀⠀⠀⡇⠀⢸⡀⢸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠒⢠⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⢄⣀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢋⡖⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠘⣆⡆⠀⡜⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡬⠿⠿⢭⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠡⠈⠻⣄⣈⡼
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣅⣠⣭⣥⣭⣭⣛⣦⢳⣀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢠⠠⠀⢈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣴⡿⡟⣿⠶⣋⣅⣀⣁⡉⠓⢽⣆⠀⠀⠀⠀⢀⡞⠠⡂⠀⡀⠠⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠸⠿⡟⡼⣿⣷⠟⣻⡭⢭⡉⢻⣆⢸⣿⡆⠀⣀⣴⢫⣆⠀⡁⡄⠀⠁⡈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⢹⣽⣯⣿⣿⣯⣍⠲⣝⣼⣿⣼⣿⣧⡾⠛⠙⠻⠛⢦⡁⢰⡄⠤⣽⡾⣿
⠀⠀⠀⠀⠀⠀⠀⣏⢛⣿⣿⣹⣿⣯⡙⢦⡉⠻⠿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠘⢛⠑⠢⠀⠄⢹
⠀⠀⠀⠀⠀⠀⠀⢿⠞⢹⠃⠁⣾⠅⠀⠒⠀⠀⠀⠀⢀⠄⡸⠂⠀⣤⠊⠁⠀⣾⡀⠀⢀⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢛⣦⣤⣤⠖⣀⣴⠟⠉⠀⢠⣯⠇⠀⠀⠀⠹⠃⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣵⣴⣿⣯⣞⣠⢴⡟⠁⠀⠀⣠⣦⣪⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣃⠞⢀⣠⣶⡿⠟⡛⢩⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀⠐⢰⡋⠉⠁⢂⣰⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣶⣠⣤⣤⣴⡿⡋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣏⢁⣐⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡿⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠸⡇⠘⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢒⡛⠡⠄⠉⠋⠨⣗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⡞⠙⠢⣀⠀⠀⢠⠏⠱⣤⠎⠈⠙⠲⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠑⢦⣸⠁⠀⠀⠈⠳⢤⡏⠀⠀⠈⠑⠦⣀⠀⠀⠀⠈⣉⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠂⠠⡉⠳⣄⡀⠀⠀⠀⠉⣻⠂⠀⠀⠀⣨⠗⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠢⣈⠳⢄⡉⢓⡆⠀⢰⡟⠀⠀⠀⢰⠃⠀⠀⡾⣄⣀⠀⢀⣀⣀⡤⠔⣄
⠀⠀⠀⠀⠀⢠⣖⠒⠒⠒⠳⣄⠐⢄⡑⠢⡀⢸⡶⠀⠈⣧⠀⠀⠀⣿⠀⠀⢸⣇⠀⠉⠉⠉⠁⠀⠀⠀⠈⠑⠢⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⣶⣄⣀⠈⠓⢤⣈⠓⠄⠸⣧⡅⠀⠈⠳⣄⠀⢻⡀⠀⠈⣏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠙⠻⢮⣟⣿⣿⣦⣄⠈⠙⠲⣄⡈⠳⣄⣀⠀⠙⢦⣀⠱⣄⡀⠘⢎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣯⣤⣶⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⢝⣿⡿⣿⣦⣄⠀⠉⠙⠚⠻⠿⣶⣤⣬⣟⣶⣿⣶⣦⣽⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⡶⠶⠾⢿⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⡺⣷⣝⢷⣤⣖⠭⢭⣿⣿⣿⡿⣿⡿⢿⣿⣿⡿⠷⠶⢟⣛⡶⠾⠿⠛⠛⠚⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⢵⣛⣚⢩⣭⣦⡶⠮⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠋⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣮⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣷⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠉⠁⠐
⠀⠀⠀⠀⢠⠂⠁⠤⠀⠀⠈⠉⠉⠒⠂⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⣀
⠀⠀⠀⠠⠃⠘⠁⠀⢀⡀⠠⠀⢀⣀⡀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠈⠄⠘⠀⠐
⠀⠀⠀⠀⠀⣀⠔⠊⠁⠐⠛⠀⠀⠈⠁⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠈⠀⠀⠀⠙⣅
⢠⠞⠀⡠⠊⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢰
⣄⣤⠊⣠⣠⠔⠁⢀⡀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠈
⠁⠈⠉⠉⡷⢀⠄⡴⠋⠀⠀⠀⠘⠤⠤⠀⠉⠀⠈⠃⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠜⠁⠉
⠀⠀⠀⠸⢠⠂⠸⣤⠄⠋⠀⠀⠀⠀⢠⠚⠃⠀⣼⢍⠁⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⢀
⠀⠀⠀⠐⢣⠄⠊⡠⠒⢈⣀⣄⡀⠀⠀⢉⡴⠊⠉⠀⠤⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡃⠡⡀⢹
⠀⠀⢀⠔⢁⠴⠎⡠⢾⠋⡡⠁⡠⠔⠉⠀⠀⠀⣇⡀⠀⠠⠀⢀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣸
⠀⠀⢮⠶⠁⣸⠔⢱⠁⠸⢲⣯⠔⠈⣀⣤⠴⠖⠳⠆⠀⡀⢠⡏⠀⠀⠀⠀⢀⠰⠂⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠖⠉⠁
⠀⠀⠀⣆⡔⠀⡠⣯⠆⢀⡞⢁⣴⣏⠍⠁⢠⡄⠀⠀⠈⠐⠇⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢩⠁⡘⢠⠃⢀⡜⣠⠋⠈⡍⢀⡌⠀⣶⠔⣡⢀⠋⠀⠀⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢩⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⢠⠁⡎⢰⠏⣷⡇⢀⠀⠁⣼⠁⡇⣼⡔⢡⠟⠁⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠘⠾⠀⢧⢸⠀⢸⡇⢸⠀⠀⣿⠀⡏⢸⢠⢟⠀⠀⡀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⡼⠁
⠀⠀⠀⠀⠀⠀⠘⠋⡆⢸⡇⣸⠸⣤⢻⡤⠃⠸⡾⠁⠀⠐⡀⠤⠀⡀⠁⢂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠛⠃⠀⢰⡄⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢧⡏⢺⢨⣞⠔⣁⠔⠋⠀⢓⣰⠁⡄⠒⠡⢀⠀⠀⡅⠀⠀⠀⠀⠀⠈⠳⣤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢠⣯⠜⢁⡠⠔⠚⠁⠈⠙⠂⠰⡠⣄⡈⠈⠲⣀⢠⣀⣄⣀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣋⣁⡠⠤⠀⠀⠐⠀⠶⠿⠓⠸⠶⠅⠂⠒⢒⡞⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡧⠤⠀⠀⠉⠢⢠⡀⠐⠧⠀⠀⢀⠃⢴⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⠀⠁⢶⡀⢀⠐⠀⠀⠀⣠⠌⢁⠒⠠⠄⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠃⢄⠀⠣⡀⠀⡉⠂⠈⠀⠐⡓⠂⠄⢆⡙⠒⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣄⡀⠐⠪⢖⠺⠒⠀⠂⠠⠈⠣⢔⠁⣀⢠⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢉⡵⠰⣄⢀⠀⠈⡁⠀⠀⠀⢀⣍⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡚⠀⠀⠀⠉⠐⠰⣋⣉⣓⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠞⠀⡴⠁⠀⠀⠀⠀⠀⠀⠨⠁⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣉⠀⠀⢇⣀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢇⠀⠀⠀⠀⠀⠠⠀⠀⠐⠾⢭⡆⠀⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠑⠀⠀⠀⠀⠀⠀⠣⢀⠤⢄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⠂⠀⠧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⡭⣟⣻⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠽⠥⢬⣼⣡⡻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣀⠀⠀⠈⠙⠛⠶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣀⠉⠙⠳⠦⣤⣀⠀⠀⠈⠉⠛⠷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣛⠛⠛⣛⠛⢛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢀⠉⠙⠓⠦⣤⣀⡈⠙⠓⠶⣤⣀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⠉⠙⠳⠦⣤⣀⠈⠙⠓⠶⣤⣀⠈⠀⣸⠏⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠈⠙⠓⠆⠀⠀⠉⠙⠓⠦⣤⣀⡈⠁⢠⡟⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠁⠉⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣄⠈⠓⠶⠄⠀⠈⠁⠀⠿⠁⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣠⣴⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣴⣤⣤⣦⣦⡄⠀⠀⠀⠈⣹⠆⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡶⠒⠺⠷⣾⣷⣶
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢀⠿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠀⠀⠀⠉⢹⣷⣿⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠾⠷⠶⠶⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⢶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⠙⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⠶⠶⠶⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣿⣿⣿⠿⠃
⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢸⠃⠹⣿⣿⣯⠁
⠀⠀⢀⣼⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠐⠚⣿⣿⣿⡇
⠀⠀⣼⣿⣿⠋⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠘⣿⡟⠁
⠀⣼⣿⣿⡇⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠙⡄
⠀⣿⡇⢿⡇⠀⢻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠘⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣼⣿⣿⡆⠀⠀⠀⠀⠀⣠⡿⢆
⠀⢸⣷⠀⠈⠀⣿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣀⡠⡶⠞⢻⣿⣿⣿⠃⠀⠀⣀⣤⡾⠋⠀⠨⡄
⠀⠀⠻⠀⠀⠀⣿⠘⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠏⠀⣠⠀⠀⠉⠀⠄⠀⠀⣴⠟⠉⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠃⠈⣿⢹⠟⣿⣿⣿⣧⠊⠉⠉⠀⢠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠈⢿⣽⢀⣀⠀⢴⣿⠁⢀⠀⠠⢀⣰⣦⣈⡼⠀⠀⠀⢀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣮⣽⠟⣹⣿⣾⣷⣿⣿⣿⣿⠋⣀⣀⢄⣦⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢰⣬⣟⣽⣯⣽⡿⣿⠇⠠⢵⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣽⣿⡯⠙⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢠⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠐⡀⠀⠀⠀⠠⠤⠾⣿⣿⣿⣿⣿
⢀⠞⠓⢷⣤⣵⣆⣰⣮⠐⡣⣀⣀⣙⠻⢿⣿⣿⣿⣷⣆⡀⠀⢴⠇⠀⠀⠀⠀⡇⠀⠐⠒⠂⠄⠀⠀⢉⠝⠛⠛
⢉⣐⢿⣿⣿⣿⣿⣿⣿⣿⣶⣽⢛⢿⣿⣿⣿⣿⣿⣿⣻⣿⢿⣿⡷⠆⣀⠀⠀⣱
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠎⠉⠛⠻⠿⠿⠿⡿⢿⣶⣷⣶⣤⣤⣤⣾⣦⢖⡀
⠀⠐⠾⠿⠿⠻⠟⠛⢻⡙⠿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⣉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⡠⠀⠀⠀⡠⠒⠉⠙⠛⠉⠙⠒⢦⡀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠉⠐⠂⠶⠈⣀⠀⠀⠀⠀⠤⠐⠀⠀⠀⠀⠈⠉⠐⠲⠤⢀
⠀⠀⠀⡠⠚⠠⢐⣀⡞⠁⠀⡀⠤⠊⢡⡄⠀⠀⢹⡿⡎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⠴⢛⠢⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⣴⣶⡂
⠀⢀⣼⠀⠀⢀⠚⠋⠐⠒⠀⠀⠀⡀⢉⣦⠀⠀⣸⣿⡧⢀⢀⠠⣂⣵⣶⣾⡄⣀⢔⠩⣶⠿⠟⣁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣿⣿⣿⣯⡥⠔
⠈⠀⢬⣑⠈⠁⠀⠀⠀⠙⠲⠒⠋⠁⠈⠉⠂⠰⢿⣿⣷⣏⣵⣾⣿⢿⣿⣿⢻⣫⠵⣾⣶⣿⣟⡍⢿⣦⣤⣴⣶⣿⣿⣿⣿⣟⣽⣿⣿⣿⠗⠊
⠀⠀⢚⠾⠷⣦⣄⣀⠀⠀⠀⠀⣀⣠⢴⣶⠶⠴⠒⣘⡿⣿⣿⢿⣧⠸⡿⠿⠾⣛⣲⣋⣫⣭⡖⠌⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠻⠟⠛⠛⠛
⠀⠀⠀⠠⣒⣲⣦⣉⠝⠒⠦⣝⡫⠶⡅⣴⣢⠾⢛⣋⣐⣼⣤⠬⠽⠿⠷⠖⠛⠛⠛⠛⠉
⠀⠀⠀⠸⠶⠾⢭⣟⣢⠤⣄⠀⠚⣳⢞⡩⢒⡽⠿⣿⠿⠛
⠀⠀⠠⣠⡍⢙⡛⠒⠚⠉⠓⠦⠄⠖⠁⢈⣡⡾⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠉⠁