Black girl Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣄⣿⣿⣿⣧⣤⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣾⣿⢻⣿⣿⢿⣿⣳⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠚⠛⠉⠀⠁⠈⠈⠉⠛⠳⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠔⠛⠓⠶⠄⠀⠀⠴⠖⠛⠓⠂⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⢱⠀⠲⣛⣿⣦⠀⠀⠀⢠⣾⣿⡳⠆⢸⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠠⡀⢀⠀⡀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣤⡀⠀⠀⣀⡤⠂⠲⣄⣀⠀⠀⢠⣄⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠁⢄⣀⣀⠤⠁⠀⣠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠢⣀⠀⠀⠀⠀⡠⢾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠈⠙⠒⠚⠉⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⠛⣿⣿⣼⣿⣿⣿⣿⣄⣀⡀⠀⠀⠀⠀⣀⡀⣰⣿⣿⣿⣿⣷⣿⠟⢛⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡐⡈⢙⣿⣿⣿⣿⡟⠓⢇⣹⣿⡏⣻⣻⣉⠔⠈⢿⣿⣿⣿⣿⠉⡐⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢊⢸⣿⣿⣿⣧⣀⡀⠀⠀⠉⠉⠀⠀⣀⣠⣾⣿⣿⣿⢋⢓⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⣻⡟⢹⠟⠿⣿⡷⢾⡿⠟⣿⠙⢻⣿⣿⣿⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠾⣇⣐⠰⠀⠝⠠⠈⠅⠰⠐⣂⣹⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠛⠋⠉⠁
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠐⠃⠀⠁⠃⠀⢠⡘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⣀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠉⠐⢤⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⠀⢠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢑⣄⠧⠤⠤⠤⣀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠒⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡲⠊⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⢠⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⢀⠛⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⡀⣀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡠⠂⠀⠀⠀⠁⠒⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠸⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⡄⠀⠀⠀⠀⠇⠀⢀⠠⠞⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠀⠀⠀⢔⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠣⠐⢳⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣤⡆⠀⠠⠐⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⣿⣿⣿⣿⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣤⡞⠉⣿⣿⣦⡄⠀⠀⠀⠈⠀⣀⡙⠿⠟⠀⠀⠀⢈⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡷⢀⣘⡻⠿⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⡀⡀⠔⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠓⠢⢤⣀⣀⠤⠚⠁⠰⠀⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠢⠐⡄⠂⠰⠦⣘⣄⠀⠀⠀⠀⠢⠖⠁⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⣾⣴⣬⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢀⠔⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣆⠓⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠃⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⣠⣴⣶⣦⣶⣲⠯⢟⣛⣛⣛⣛⠻⢯⣶⣤⣤⣦⣀⣢⣀
⠀⠀⠀⠀⠀⠐⢺⣿⡾⢿⣫⣿⣶⣾⣿⣿⣿⣷⣮⡙⠿⣿⣿⣦⣌⡻⣿⣿⣿⣿⣿⣿⣿⣿⡛⢛⢒⣤⡀
⠀⠀⠀⠀⠀⠐⣻⣿⡷⠛⠛⣻⣿⡿⣿⣿⣿⣿⣿⣿⣶⡈⠛⢻⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣽⣆
⠀⠀⠀⠀⠀⡸⢿⣏⣤⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⣿⣷⡿⢛⣛⣿⠿⠿⠷⢯⣿⣿⣿⡿⠟⡛⠦⢱
⠀⠀⠀⠀⠀⢁⣿⠿⢺⣿⠟⣿⣿⣿⣿⣿⣿⣯⣭⣍⣛⠻⡿⠿⠿⠛⠉⢁⠴⠚⠋⢉⣛⣯⣿⣯⣗⢾⣿⠷⣾
⠀⠀⠀⠀⠀⠁⠀⢴⣿⣵⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣦⣌⠉⠒⠒⠚⠁⢀⣠⣤⣾⣽⣿⢿⡌⢻⣆⢻⣾⢿
⠀⠀⠀⠀⠀⠀⣨⠟⢛⣭⣵⣶⣶⣾⣭⣍⣉⣛⣿⣿⡿⢿⣿⣷⣶⣤⣒⣶⣷⣴⣿⠋⣿⣧⢸⡷⢸⣿⣾⣿⡏
⠀⠀⠀⠀⠀⠐⣡⠞⠟⣫⣭⣤⣭⣭⣍⣉⣉⡙⠹⢋⣁⣠⣿⣿⣿⣿⣯⣽⣿⣿⡟⠀⠈⢹⠛⠧⣿⣿⢿⣷⠁
⠀⠀⠀⠀⠀⣼⠃⠠⠚⢉⡥⠿⠿⠯⣭⣿⢛⣛⣿⣿⡿⢿⡟⣿⣿⣿⣿⣿⣿⣿⣷⣀⣨⣿⣶⡄⠙⢿⣿⠃
⠀⠀⠀⠀⠀⠇⠀⢠⠞⠁⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠈⠙⠛⠛⠛⠛⠻⡿⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⢻⡿⠛⠉⣩⢿⣯⣿⣿⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡴⠟⠀⢀⣼⣟⣣⠼⣿⡇⠰⢋⠇⠀⠀⠀⠀⠀⠀⢐⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⣀⡈⠁⣀⣠⡴⠟⠈⣡⠤⠶⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⠧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣎⡴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⣾⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢛⣿⣿⣾⣿⣿⣾⣿⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣦⣀⣿⢷⣼⣿⣧⣤⣴⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣨⣿⣿⡿⢧⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣾⣿⡙⣿⣯⣿⣿⣿⣿⣾⣆
⠀⠀⠀⠀⠀⣶⣆⣻⣿⣿⣿⣿⣷⣾⠍⠈⢙⣏⣍⠙⣿⣁⣶⣤⣤⣤⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣻⠿⣿⠝⠛⠉⠉⠙⠿⠟⠿⡆
⠀⠀⠀⠀⠀⠛⢛⣿⡿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⢿⣿⣿⣿⠿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⣠⣤⣾⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⣇⣀⣄⡀⠀⠀⠀⠀⠀⣀⣄⣸
⠀⠀⠀⠀⠘⢻⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⢈⣷⣦⠀⠀⠀⢀⣿⣿⣿⣿⣹⣿⣿⣟⣿⣿⣿⣇⠀⠀⠰⣾⣿⣍⡛
⠀⠀⠀⠀⠀⠘⣿⣿⠿⠀⣀⣀⣤⣀⡀⠀⠀⠀⢸⣿⣿⣿⣿⠾⣿⣿⡇⠀⠀⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⠿⣿⠿⠄⠀⢀⣿⣿⣩⡟
⠀⠀⠀⠀⠀⠀⢈⢿⣦⣾⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⠻⣿⡿⠃⢿⣿⡇⠀⠀⠘⢷⣿⣶⣿⣿⣿⡿⠛⠛⠂⠋⠆⠀⠀⠀⠛⠛⠹⡇
⠀⠀⠀⠀⠀⠀⢺⣿⣿⣟⠿⣋⣿⣿⠻⠄⠀⠀⠀⠉⠀⠀⠀⠐⢸⠏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⣰⣤⣍⡤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡆⠈⠉⠁⠀⠀⣤⣅⣠⠶⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣷⡀⠀⠀⣀⣀⣨⣭⣭⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣄⠀⠀⠀⠀⠈⢉⣡⣤⣤⠞⠂⣰⠃⠀⠀⠀⠀⠀⠀⣿⣿⣏⣿⣿⣿⣿⣷⣄⠈⠀⢹⣿⡽⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣠⠒⠗⠉⢿⣿⠏⢀⡴⠃⠀⣤⣤⣴⣦⠀⠀⣿⣿⡏⣿⣿⢿⠿⠿⣿⣿⣶⣤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣶⣤⣤⣤⣶⣿⡇⠀⠀⠹⢷⣿⠏⠀⢀⣿⢿⣻⣿⣿⠉⠀⠀⡞⢸⣏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇⠈⠙⠿⢿⠙⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠻⠎⠟⠀⠀⠀⠀⢸⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠒⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠶⢾⡶⢦⣄⡀⠌⠩⠘⡋⠀⠀⠀⠀⣾⣷⡇
⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⠙⠉⠀⠀⠀⠀⠀⠀⠪⠊⢁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡅⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠤⡠⠖⠒⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠐⠋⠈⠁⠀⠀⠀⠀⠐⠛⠩⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡠⠔⠒⠀⠐⢄⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⠱⠎⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡆⣀⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠽⠤⠷⠶⠶⠶⠤⢔⣀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⢠⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠞⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⢤⠔⠋
⠀⠀⠀⠀⠀⠀⡹⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⡰⠛⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢀⣀⣀⡸
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⡔⠁⠃⠀⠀⠉⢒⠂⠀⠤⠀⠐⠂⠀⠀⠀⠀⠀⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⡄⠀⠀⠀⠀⡎⠀⠀⡠⢴⠋⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⠀⠀⠀⠀⠇⢄⠁⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠉⠂⠀⠀⢱⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠊⠣⠤⢻⡄⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣤⣀⣠⡤⠀⢡⠤⠺⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠐⠊⠁⠀⠈⠀⠀⠀⠀⠀⠀⢀⡔⢹⣿⣿⣿⣿⡅⠀⠀⠊⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣤⠞⢺⣷⣶⣄⠀⠀⠀⠀⠘⠀⠀⠻⣿⠿⠁⠀⠀⠀⠁⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢹⣄⠈⠻⣿⡿⠀⡆⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠸⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢜⡄⠉⠉⠉⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠄⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠓⠢⢄⣈⠁⠉⠀⢀⡠⠞⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⠢⡑⢤⡙⢽⠉⠁⠀⠀⠀⣇⣀⠤⠒⠉⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠠⠀⠀⠀⡄⢀⡀⢈⡺⠓⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡠⠒⠙⠛⠿⠉⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⢀⠞⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡎⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢸⠎⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡈⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⢻⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣤⣴⣼⣷⣿⣾⣿⣿⣿⣷⣶⣶⣦⣦⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⢀⡾⣾⣷⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡄
⠀⠀⠀⠀⠀⠲⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⡄⣀
⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⠀⡀
⠀⢠⠦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⢛⢀
⠀⢈⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡇
⠀⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⠳
⠈⣵⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣱
⢰⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁⣭⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⣥⣬⣤⣄⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠛⣛⡂⠀⠀⠀⠀⢀⠛⠋⠁⠀⠀⠈⠁⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽
⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⣶⡿⣾⣷⠯⠟⠳⠀⠀⠀⠸⢋⢶⣿⡟⢿⣶⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁⣿
⠐⢿⣿⡏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼
⠀⢾⣿⠁⣿⡟⢿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢾
⠀⠙⠋⠀⠿⣄⢾⣟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣇⣴⣄⣀⡤⣆⡵⠀⠀⠀⠀⠀⠀⠀⢸⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠮
⠀⠀⠀⠀⠀⠋⠲⠿⢩⣿⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡞⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠙⠂⠐⢻⡄⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⢀⣴⣊⣁⣈⣭⣦⣀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢈⣿⣿⣿⣿⣿⣿⣿⣷⡀⢰⣄⢨⣅⠀⠀⠀⠀⢈⡉⠀⢀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣿⡿⣿⣿⣿⣿⣿⣦⡙⣷⣿⢿⣿⣿⠿⠻⠶⠟⠋⢀⣴⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⣿⡷⣿⣿⣿⣿⣿⠛⣿⣌⠻⣁⡀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⣹⣿⡄⢻⣿⣿⣿⣿⡇⠈⢿⣿⣦⣝⣛⣛⣤⣶⣿⣿⣿⠏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠁⠀⢸⣿⣿⣿⣿⡅⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⡤⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣟⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠂⠹⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠟⠋⠉⠀⠀⠀⠀⠀⠈⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⠟⠹⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢

⣤⡀
⣿⣿⣿⣿⣿⣿⣶⣦⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣰⣦⣶⣶⣶⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣦⣤⣤⣤⣤⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠛⠁⣤⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣄⣀⠀⠈⣿⠶⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⡿⠉⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣤⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀⢲⡶⠀⠀⢴⡶⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠨⠤⡢⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠰⠆⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠈⠛⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣦⠀⠀⠀⠀⠀⠀⢀⣴⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡛⢿⣿⣿⣧⡀⠀⠀⣀⣤⣿⣿⣿⣟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠻⣿⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣿⣧⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠉⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋⠹⣿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠿⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⢄⠔⠒⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣁⠤⢒⠂⣹⠖⠨⣅⠺⣾⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣆⠐⢊⡥⠀⠸⠀⠒⠦⣐⡄⢣⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠽⢉⠖⡹⠁⠀⠀⠀⠀⠀⠈⠂⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡄⠌⢰⠃⠀⠀⠀⠀⠀⠀⠀⢎⡒⠃⢏⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡙⠸⡇⠀⢸⠢⣤⡀⠀⠀⣠⡴⠒⠂⠑⡄⠊⠍⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢱⠃⠀⢸⣿⢆⠉⠀⠈⢁⢶⣟⠷⠀⠏⢂⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠌⠂⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠆⠃⢠⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⢰⢈⢈⡌⠟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠁⠀⠞⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠈⠄⣧⢐⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣰⢠⠂⠄⠃⠀⠀⠀⠀⠀⠀⠀⡌⡆⠈⠸⠇⢸⠺⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⠀⡟⢸⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠃⠀⡃⠚⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣸⠀⠈⢸⢀⠀⠐⠀⠀⠀⠀⠀⠀⢘⠈⡈⠐⢠⢲⠆⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠤⠈⠸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⡇⠀⠆⡞⢠⢃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⢧⠃⠞⡸⠀⡌⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠣⢰⠁⢁⡀⠀⠀⠀⠀⠀⠀⠀⢘⡀⡰⡴⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠸⡞⢁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⣴⡦⢤⡦⢠⣶⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⣾⢹⡇⣍⡿⠟⠻⡢⠑⡎⠰⣇⢈⡁⠈⣀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⠊⡁⣘⣿⠱⡈⠕⠻⣐⣇⢈⡙⢺⣿⣹⡷⠈⣿⣿⠛⣧⣿⣰⣆⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣜⡛⠋⢙⡦⣥⠘⣟⡁⣾⣷⡷⢈⣬⣿⢿⡉⠡⠁⢻⣽⠋⣠⢏⣸⢃⣿⣻⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣏⠻⣄⢘⡷⠘⣆⡴⢩⡏⠸⣏⣩⢀⡻⠼⢩⢃⣾⡿⠞⢡⡿⣬⣧⢿⣧⣽⣿⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⣿⣿⣮⣽⣉⡨⢴⡤⢷⡌⢯⡀⢿⡞⣻⣿⠗⣿⠆⣙⢳⢴⡾⣿⡹⣿⣿⢽⣃⡿⢙⠿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⡃⢹⠟⣿⣿⡏⢷⢌⢻⣴⣿⡀⣵⠦⠒⣰⣿⣿⢾⡴⡃⣡⢤⣼⢧⣿⢋⣿⣦⣿⣵⣬⢷⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡟⠛⡆⢦⠉⣻⣧⠴⣾⡷⣿⣭⡟⢷⣤⣿⣿⣿⠉⠈⣀⡟⢽⣼⣯⣼⣿⣿⣻⠟⣝⣻⣿⠾⢫⠶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⠕⠳⡷⢤⡏⢁⠜⠹⣍⡹⢙⠋⡻⠶⠉⡈⠉⠭⣄⣷⢭⡅⢿⣹⣿⣿⣟⣿⣝⣧⣿⡧⢶⢔⣦⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⣀⣤⣇⣼⣉⣁⡀⢁⢮⠿⣾⣿⣗⣷⡤⠱⠃⠀⠛⠉⠛⠿⣿⣿⣿⣿⣍⣻⢾⣧⣯⣿⣦⡜⠻⡟⢻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⣿⣭⠿⣿⢫⣙⠻⢶⠦⣶⣿⣿⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⡟⢙⣿⣿⣟⣿⣛⣽⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢤⣵⣤⣴⣬⣭⡾⡻⢾⠦⣌⣴⠃⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⠸⢟⣏⣭⣿⣯⣿⠃⣿⣟⣻⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⣬⣿⣿⣟⣠⣧⣼⣡⢛⣷⣿⡟⠛⠉⠉⠉⠋⠀⠀⠀⠛⠉⠉⠉⠙⢙⣻⣻⣿⣿⣿⣴⣿⣧⠿⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣿⣿⣿⣴⣿⣿⣿⣿⡿⣿⡄⣤⣶⣶⣦⠀⠀⠀⠀⠀⢰⣶⣶⣤⠘⠿⠙⣿⠻⣿⢿⡷⣿⣛⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡏⣴⣟⣫⣾⣾⣻⣏⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣸⣿⣗⣿⣿⣿⣿⡝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣽⣿⣻⣿⡿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⠟⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⠚⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⣀⣀⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣿⣷⠤⠀⠀⠲⠖⠒⠒⠚⠛⠛⠛⠛⠓⠀⠀⠛⠋⣿⣿⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⣀⣀⣿⣿⣇⣀⣀⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⠛⠛⠛⣿⣿⠙⠋⠙⠛⠛⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉⠁⢹⣿⣏⠈⠉⠉⠉⠁⠀⠀⠀⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣄⡀⢸⣿⣿⣤⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠛⠛⠛⠛⠋⠁⣿⣿⡏⠉⠛⠛⠛⠛⠉⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣿⣿⣧⣤⣤⣤⣤⣤⣤⡤⠄⠀⠶⠶⠶⠶⠶⠶⠾⠚⣿⣿⣷⠒⠲⠶⠶⠶⠶⠦⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣏⣿
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⣤⣤⣤⣤⣤⡤⠀⣿⣿⡧⠤⠶⠶⠶⠶⠖⠂⠐⠒⠒⠒⠛⠛⠋⠋⠉⠁⢸⡏⢻⡏⠙⠛⠛⠛⠓
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠘⣿
⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣦⣤⣶⣿⣷⣿⣾⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡟⠿⠻⠿⠟⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠠⣠⣆⣀⠐⢀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⡖⣚⠉⠍⠠⠤⠤⠀⠤⠤⡀⠐⠇⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡐⡗⣀⡖
⠀⠀⠛⣧⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⢛⣉⠤⠤⠄⠀⠀⠀⠀⠄⢀⡀⠀⠉⢻⡄⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⣹⣶⣮⣥⠤⠦⠦⠦⠤⣤⣤⣀⣀⠀⠈⠑⠦⣼⡅⠀⠀⠀⠒⠰⠤⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢹⠒
⠀⠀⠐⠁⠀⢠⠀⠀⠐⢀⡴⢋⣽⡟⠓⠒⠒⠂⠤⢤⣤⣀⠀⠀⠉⠙⠓⠦⢄⡀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠐⠤⢄⠀⠀⠐⡄⠀⠀⠈⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠎⢠⣿⢿⠏⠑⠒⠠⢄⡀⠀⠀⠀⠈⠉⠐⠒⠢⢄⡀⠈⠲⣄⣈⣻⢿⡿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠐⠀⠀⠀⠀⠀⠀⠘⢠⡟⢁⡾⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠒⠢⠤⢀⡀⠙⢦⡀⠙⠏⡹⠋⠀⡛⠉⠏⠀⢠⣀⠀⠀⢀⡄⠀⠀⢰⡀⠐
⠀⠀⢂⠃⠀⠀⢠⡀⢄⢸⠃⣸⠃⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠒⠂⢀⠀⠙⠦⡀⠙⢾⡞⠀⠀⠀⢰⠀⠘⠀⠈⣏⠂⠀⠈⢻⡀⠀⠸⠁⢐
⠐⡄⠈⠃⠀⠀⠸⠅⢸⠁⢠⣿⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣠⣴⣾⠟⠀⠀⠀⠈⢦⠈⢃⠐⠀⠀⢸⡄⠀⠀⠀⠸⡦⠀⡀⢸⠃⠀⢰⢂
⡀⠘⣔⡆⠀⠀⠀⠀⢸⠅⠸⣿⡆⠀⠀⠀⠀⠀⠉⢀⣾⣿⡿⣿⡏⠀⠀⠀⠀⠀⢸⣷⡀⠡⠂⢠⡏⢡⠀⠀⠀⠀⡇⠀⢻⡾⢀⠀⢐⡉
⠀⠄⠬⠅⠁⠀⠀⠀⠀⠀⢸⠻⣿⣟⠛⣀⣀⠀⠀⠀⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⣼⠀⢹⡀⠺⣿⡇⢸⠀⢀⠂⠀⢹⣀⡼⠃⠛⠁⢸⠄
⠀⠀⡠⠀⠢⡀⠀⠐⠀⠀⠐⡀⢹⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⣧⠸⣿⣇⠘⠁⠈⣱⡦⠞⡋⢡⡆⠀⣰⡇⢘⡞
⠠⣤⣐⡢⡄⠑⠄⠀⠀⠀⠀⠀⠸⡍⢿⣿⢻⡟⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠌⠀⠀⠀⠀⣿⠀⢻⣯⣦⡴⠚⠉⡐⠀⠀⣼⣧⠊⣙⠟⢋
⢤⡈⡉⠗⣧⠂⠀⢀⠀⠀⢀⠀⠀⢳⠸⣿⣾⣿⣶⡾⠋⠉⠀⠂⠀⠐⠂⠀⠀⠀⠀⠀⣰⡿⡇⠀⢿⠏⠀⠀⠈⠁⠀⠘⠛⠁⡄⠀⡀⠋⠉
⢨⠋⠒⠈⠁⠀⢀⠘⡇⣄⠀⠀⢀⠀⠇⢹⣿⣿⣉⡦⠀⣀⣀⡤⠖⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠎⠀⠀⠀⠀⣠⠆⠀⢀⣴⠙⠀⠁
⣀⡀⠒⠄⠀⠀⠀⢠⠇⠈⣰⠀⠀⠀⢻⠘⡏⢿⣿⣿⣾⡟⠁⢀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠉⠀⠀⢴⣶⠻⡟⠻⡿⡏⠘⠅⠠⡔⠋⠰
⠀⠀⠀⠈⠀⠀⠀⠸⠆⠀⡗⠀⠀⠀⡴⠀⡟⢸⣿⢻⡿⣿⣿⠋⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⢳⣧⠀⢇⢁⠀⠁⢀⡴⠀⠐
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠰⡇⢐⠀⠀⡇⢀⡇⢸⣿⡌⣇⢹⣉⣷⣶⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠈⠉⠑⠺⠾⣧⣈⡏⠀⡄
⠀⠈⢦⣤⠴⢺⠁⠀⠀⠀⢸⡟⠀⣰⠀⢸⠁⣸⣿⣇⣿⡼⠷⠚⠋⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠈⠉⠐⠢⠤
⠀⡀⠈⠱⠀⠈⡄⡀⠀⠀⢸⠀⠀⣿⡠⠜⠚⠉⠉⠀⠀⠀⠀⠀⠀⢠⣁⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡂⢀⠇
⠀⠀⠀⠀⡀⠀⢃⠀⠀⠀⠈⢰⣤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠈⠂⢄⣠⠄⠔⠋⠀⢀⠈⠠⠂
⠀⡀⠀⠸⢇⣀⡀⡀⠁⢨⡇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⠀⠀⠀⠀⠀⠀⡀⠆⠁
⠀⠀⠀⠀⠾⡟⢻⡷⠀⠫⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠁⠃⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠉⠙⠀⠀⠀⠸⠜⣄⠐⠄⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤
⠀⠀⠙⠒⠈⠳⡀⠀⠘⢉⣄⠁⠀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠄⠀⡈⠈⠉
⠀⠀⠈⠄⠀⠀⠁⣄⠀⠈⠉⠀⢄⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠀⠀⠄⠀⠀⠀⠀⠐⠒⠂⢤⡠⠁⢀⠄⣀⠈⢂
⠤⠀⠀⢠⡀⠀⡀⠀⠈⢂⠀⠀⢰⡷⣦⡄⠀⠀⠀⠁⠲⢤⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⣀⣈⡙⠀⠀⠀⠀⠀⡄⠉⡄⠈⡿⠚⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣾⣷⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⣀⡀⠁⠈⢛⣉⡛⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⡙⠀⠘⣉⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣄⠀⢀⣄⣤⠀⠀⣸⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⢻⣿⣿⠟⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⠈⡁⠔⡶⣲⡾⠛⠛⠓⠒⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣤⡤⠤⢤⣶⣋⣤⣔⡾⠋⠀⠀⢀⣈⣀⢁⠀⣈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠞⠺⣱⡎⣁⣄⣤⣭⡩⠱⢟⡉⠀⣠⣔⣻⡟⠉⠀⠀⠀⠈⣉⠛⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢨⠇⢿⣧⡯⣶⠏⠉⢉⠉⡉⢉⠡⣠⠝⢿⣍⠁⠀⠀⠀⢀⣼⠏⠀⠤⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡶⠚⠃⠀⡼⠃⠀⠀⠈⡈⠐⣡⣾⣥⣶⡶⢿⡄⡀⣄⢜⣵⢿⣆⣒⠒⢒⢒⠲⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⢱⠁⠀⠀⢸⠁⠀⣠⡠⣾⣑⠯⠿⣿⣯⣙⠛⠛⠛⠻⣿⣿⢿⣷⣶⣤⣀⠀⠀⠑⢤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⣜⠀⠀⠀⣼⣠⣦⢼⠖⠘⣠⢿⠲⣌⣬⠫⠹⣕⡤⣀⠐⠢⠙⢦⡈⢑⡈⢲⡄⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡟⠇⠹⣆⢀⣰⡿⠿⠿⢿⣵⣾⣧⣤⣗⡛⣽⣇⠈⢤⠁⠀⠀⢄⠈⠂⡈⢂⡡⡀⢗⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠅⢇⠀⢘⡟⠉⡴⡶⠿⠋⠁⠀⠀⢈⣻⣭⠟⠲⢆⠀⠑⢀⠀⠀⠣⠀⠘⡆⢁⢵⢸⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡄⠈⠐⢻⠅⠄⠈⠀⠀⠀⢀⣰⡾⢛⠉⠀⠀⠒⠉⢀⠀⠀⢂⠀⠀⠁⡀⠽⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠶⠊⠽⣶⠀⠀⠁⡲⠋⣟⡫⠊⣁⡀⠀⠀⠂⠉⠉⠁⠀⠈⠂⠀⠘⣅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠾⠒⢲⠃⢠⡼⣻⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣴⠁⣠⠛⠛⠳⢦⣄⡀⠀⠀⣠⣴⠖⠛⠓⠘⣧⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡄⠁⣾⣷⠾⣤⠉⠁⠀⠈⠉⣠⢶⢾⣷⠆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠨⠋⠉⠁⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣬⡤⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠉⠀⠀⠀⠀⠀⠠⡄⡀⠀⠀⠀⠀⠈⠈⠁⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠍⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡅⢔⠠⠘⡀⠀⠉⠀⠈⢀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠠⠄⠀⠘⠀⠀⠆⠠⠂⠃⠐⠀⠀⠀⠴⠤⠀⠀⠏⠀⢠⢁⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⢂⠤⠀⠈⠤⠀⠤⠀⠢⠄⠈⡠⠀⠀⠤⠀⠤⠁⠐⠀⡊⢠⠋⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠂⡡⠐⠡⠀⣄⠀⠰⠂⠐⠐⠀⠀⠂⠐⠆⠀⢠⠀⠅⠀⠀⠀⠁⢀⠀⠀⡁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠄⡠⠈⠀⢈⡀⠀⡁⠀⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠀⠈⠁⢀⡁⢀⠐⠄⢀⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠢⢀⠀⠀⠀⠀⡕⠀⢣⠀⠁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠃⢀⠎⠀⠁⠀⢁⠈⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠡⡀⠁⣀⠠⠉⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠌⠀⠂⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠁⠀⣀⢀⣚⡀⣀⠤⠀⠀⠁⠀⠀⢀⠀⡄⠈⠀⠀⠐⠀⠡⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡤⠀⠀⠚⠀⠈⠁⠀⠀⠀⠀⣠⢄⡤⠖⠚⠉⠉⠈⠀⠈⠉⠛⠓⠂⢄⡰⠄⠠⢠⠈⠀⠀⠁⢀⠃⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠁⢶⠀⠌⠁⢌⠀⢀⠠⠐⢀⠞⣁⣄⣰⠠⣀⠀⠀⠀⠀⠀⢠⠄⣤⣠⣌⠲⠀⠀⠀⡄⠀⠡⠈⠁⠀⠁⠀⠖⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢴⠀⡆⠰⠮⠀⠀⠀⢠⠎⠼⠉⠀⠈⠙⠢⠤⠒⠤⠠⠤⠔⠋⠉⠀⠈⠢⠙⣄⠅⠀⠀⠤⠆⠰⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡂⠒⠀⠀⡀⢰⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⡆⢀⠀⠀⡉⠤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⠈⠀⠀⠀⠐⠄⠀⢸⠇⠀⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠀⢸⡉⠀⡀⢀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠠⠀⠄⠰⠀⠆⠈⠃⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠸⠁⡠⠀⠆⠠⠀⠀⠄⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠈⠐⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠐⠂⢁⡌⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠲⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡠⢂⡁⢈⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡄⠢⡀⠂⡘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡼⠇⠐⢀⠀⠀⠀⠄⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠴⠈⠀⡼⢧⠰⡀⠀⠀⠀⠐⠀⠀⠉⠀⠀⠒⠀⠀⠀⢀⠄⡶⢧⡂⠁⠐⠂⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢄⡠⠛⢛⢂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡤⡒⠛⢀⡠⠇⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡁⠠⠺⣆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡚⣰⠗⠄⠈⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠰⣶⠂⠠⠪⠕⠡⠜⠉⠂⣀⠈⠐⠰⠀⠔⠀⠉⢠⠄⠈⢣⠈⠂⠇⠀⠰⣶⠄⠀⠤⠄⠀⠄
⠀⠀⠀⠀⢠⠤⠈⠁⠈⠀⡀⠀⡰⠀⠀⡊⠀⢰⡤⠄⠀⢀⠀⠀⠀⠀⡀⠀⠂⠠⠠⠀⢈⡁⠀⠀⠀⡄⠉⠤⢦⡄⠀⠐⠀⠐⠄⠀⠀⠈⠀⠉⠁⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣁⠀⠀⠀⠁⠄⠁⠎⣐⣾⠀⠈⣇⠀⠀⠀⠁⠀⠀⠤⠄⠀⠈⠀⠀⠀⢐⠃⠀⣶⡁⠳⠘⠠⠈⠀⠀⠀⣈⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠡⢤⠀⣴⡇⠉⠀⠀⠑⡀⠀⠀⠀⠀⡐⡁⢂⠀⠀⠀⠀⢠⠎⠀⠀⠙⡵⣦⠀⡤⠉⢱⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⣇⠀⠀⠀⠀⠙⠲⡄⠀⠀⢀⠀⡄⠀⠀⡀⠀⠊⠀⠀⡀⠀⠧⣧⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢏⠐⡇⠀⠀⠄⠐⠀⠀⠈⠑⠢⠮⠭⠦⠔⠚⠁⠀⠀⠠⠈⠀⠀⢐⠃⡽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⠁⠆⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠈⣮
⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁
⣿⡟⣉⣻⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣠⣤⣤⣄⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⡿⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣫⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠻⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣠⣤⡤⣤⣤⣄⣀⣀⠀⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠓⠦⠀⠀⠀⠉⠉⣉⣥⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⠶⣦⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⡾⠋⠀⡉⠛⠀⣆⠀⠀⠀⠉⠉⠉⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣎⣿⠀⠀⣼⠀⠀⠀⣴⣻⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠁⠀⣴⠏⠀⠀⠀⠛⠛⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠄⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣄⠉⢷⡄⢸⡀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⡄⠀⢈⠟⠀⠀⠀⠀⠀⢧⡉⠁⠀⣠⡗⠈⡇⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠘⠓⠉⠀⢀⣠⡔⠃⢰⡄⠉⠙⠋⠉⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣤⣴⡶⠛⠁⠀⠀⠘⣧⠀⠀⢀⠌⣠⠟⠀⠀⠀⠀⠀⠀⢀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠻⢄⡴⢋⡾⠋⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡴⠋⣰⡟⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠹⢦⡀⠙⠛⠳⢦⡀⠀⠀⠀⠀⠀⠀⣆⣤⠞
⠀⠀⠀⠀⠀⠀⢀⣨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⣿⠀⢀⡴⠞⠁
⠀⠀⢀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠙⢷⣄
⣴⢶⣤⠀⠙⣧
⢿⣌⣛⣀⣠⡿
⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣰⣶⣶⠄⡀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⡇⠀⠀⣤⠐⢀⣶
⠀⠀⠀⠀⠀⠀⣤⠄⢰⣿⠋⣤⣤⣶⣿⡄⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠃⠀⠀⠿⠃⠘⠿⠃⠀⢀
⠀⠀⠀⠠⠀⠄⠀⠀⠀⢉⣼⣿⠟⣋⣉⡄⠀⠀⠀⠀⠀⢂⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠠⠄⠀⠀⠀⠐⠒⢄⠀⠀⠀⠀⠂
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠫⢾⢿⡙⠁⠀⠀⠀⠀⠁⠀⡆⠁⠀⢠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⢀⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⢣⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠀⠣⢀
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⡀⣞⠛⢦⣀⣀⣀⣀⣀⣠⣇⠀⠀⠈⢀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠂⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⠊⡀⢻⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠸⠷⠇⠲⠖⠘⠿⠾
⠀⣠⡄⣰⣦⡠⠀⣠⡅⠈⠀⢀⠀⢠⠖⠂⠀⠀⠀⣠⣀⠈⠀⠰⣾⣿⣯⣯⣿⣿⣯⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠐⢿⣧⡿⣵⣶⡆⠘⣿⡀⠀⠀⠀⠁⠀⢾⠆⠶⡖⠈⠁⠀⠀⠀⢻⣿⣾⣿⣿⣽⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠶⣶⡆⠈⠉⠀⠀⢹⠧⣀⠔⠑⠠⡀⠀⠀⠀⠀⡀⠀⠄⡀⠀⠈⠻⣿⣿⣿⡿⣿⣿⣿⡿⢃⠠⠴⠅⣢⠄⠀⠀⠀⠀⠀⠀⣹⣿⣿⣄
⠀⠰⣤⣤⡀⡠⠔⠀⠁⠀⠈⠐⠄⠀⠀⠄⠀⣤⣀⠀⡜⠀⢀⠱⠖⠂⠘⠛⢏⠀⠘⣿⣿⡌⡁⢀⠀⠒⠌⠀⣀⠀⠀⢀⠀⠚⠛⢿⡿⠋⠁
⠀⠀⢲⡬⠋⠁⠀⠀⠀⠄⠀⠀⠈⠂⠀⠘⣘⠈⢇⢣⠈⠔⠁⠀⠀⠀⠀⠀⠀⠑⣀⣿⣿⡇⠁⠀⠀⠄⠀⢠⠆⠉⠁⠀⠀⠉⠂⢄⠁
⠀⠀⢄⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⣥⡄⠘⡼⡔⠀⠰⣦⠀⣄⡄⢴⡄⠀⠸⠀⠀⠈⠁⠈⢀⣬⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⠅
⠀⠀⠈⠀⠀⠊⠀⠀⠈⠳⠄⣠⣶⣿⣿⡿⣿⣬⡒⠿⠁⠀⠀⠀⠐⠾⠆⠀⠀⠀⠀⠂⠀⣴⡄⠀⠀⠈⡆⠀⠈⠃⠀⢠⠄⠀⢀⡀⠀⠇
⠀⠀⠀⠡⡀⠊⠀⠀⠀⠈⢻⡟⠙⢿⣟⢠⠈⣿⣷⢀⠑⡄⠀⠀⠀⠀⠀⠀⠀⡠⠺⡆⠀⠀⠀⠀⢰⣷⣼⠆⠀⠀⠀⠐⠀⠀⠈⠀⢀⠃
⠘⢷⣀⠢⡌⠈⠒⠀⠀⠀⢻⣧⠉⠸⠋⢂⢀⣿⡿⢠⠀⠐⡌⠀⠀⠀⠀⡀⣯⣀⠀⠰⠄⠀⠀⠀⢸⡿⠟⠣⠁⠀⠀⠢⣠⣤⣤⠔⠁
⠀⠀⠈⢂⢐⠈⢑⣤⣶⣶⣶⣿⠗⠣⠀⠀⠛⠋⠋⠀⠀⠀⡴⢤⠀⠁⠀⠩⠀⠀⠀⠑⠲⠈⠉⠒⠊⠀⠀⠀⠀⠀⠈⠑⢹⣿⣟
⠀⠀⠀⠀⢉⣾⣿⣿⣿⡟⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢋⠀⠀⡆⠀⠀⠀⠀⠀⡀⠐⡠⠏⠀⠀⠀⣈⡿⠂⠀⠀⠈⣏⠁
⠀⠀⠀⣶⣏⠋⡿⠉⠸⠀⢠⣼⣿⣷⣄⠀⠀⣀⠀⠀⠀⣀⠀⠈⠂⠤⠃⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⣀⠀⠈⠛⠀⠀⠀⠀⢐⣅⡔⠄
⠀⠀⠀⣿⣿⣿⣧⡀⠀⠀⣠⣿⣍⣸⣿⠀⠀⠙⠀⠤⠈⠋⠀⠀⡠⠀⠸⠆⠀⠀⠘⠃⠀⣀⡠⠞⠓⠺⣀⡄⢀⣄⣠⠔⠀⢜⠀⣤⠉⢂
⠀⠠⡄⣌⡻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠐⠤⣀⠀⠀⠀⣀⠠⠚⠄⡀⠀⠀⠀⠀⠀⢠⡔⠈⣿⡖⠀⣦⡌⠉⣿⡿⠤⣅⡤⠊⠇⠠⠠⠄⢆
⠀⣃⡉⢞⠀⢀⡈⠉⠉⠉⠉⢉⡁⠀⠟⢻⣶⠀⠀⠀⠀⠑⠄⠀⠀⠀⡘⠉⠉⠉⠇⠀⠂⠀⣿⣧⠠⠋⠉⠒⠉⠉⢼⣿⡇⣸⠀⠀⠀⢀⠦
⠀⠀⠎⠀⡉⠾⢿⣶⣦⣀⣶⣼⣿⣤⣤⡾⠇⠀⠄⠀⡄⠀⠣⠀⢀⠚⠀⢠⢀⠀⢸⣰⠆⠁⢻⣿⠀⡀⠀⠀⠀⠀⣾⣿⢀⣇⣀⠀⠀⠂
⠀⠀⠀⢠⡀⠀⢸⡿⠿⣿⠿⢿⡇⠀⢈⠏⠹⠂⢆⡰⠠⡜⠐⣣⠘⡰⠆⢬⡸⠤⡖⠣⡄⠉⢺⣿⠀⣿⡗⢠⣾⡇⣿⡏⡘⠀⠠⣴⢂
⠀⠀⠀⠀⠈⠐⠚⠒⠔⠳⠂⠚⠳⠄⠋⠀⠐⠀⠊⠁⠀⠓⠤⠚⠄⠐⠤⠜⠣⠴⠓⠀⠃⠀⠀⠉⠉⠉⠛⠚⠛⠒⠛⠃⠃⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣦⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣯⡷⠟⣹⡿⠟⠻⣿⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⣿⡫⣫⡾⡍⠀⠀⠁⠦⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣫⢜⡿⣜⠢⠀⠠⠴⠆⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡜⡽⠉⠁⠀⠀⠉⠁⢸⣿⠀⠀⣴⣾⣿⣿⣿⣶⠀⠀⠀⠀⢀⣾⣿⡿⣿⠿⡿⣿⣶⣽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣹⢳⡇⠀⠀⠀⠀⠀⢠⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣷⣌⡙⠢⢄⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣷⠄⠀⠀⠀⣠⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢺⡿⣟⠿⣿⣿⣶⣷⣽⠈⢻⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀⠀⠀⣰⣿⡟⣿⣿⢹⣿⣿⣿⣿⣿⣿⡟⠿⠛⠀⠀⠈⣿⣿⣿⣪⣙⠛⢿⣿⠇⢘⣗⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⢿⣿⣿⣇⠀⢀⣿⣿⣿⢿⠃⣾⣿⣿⣿⣿⣿⣿⠋⠀⢀⠀⠀⠀⠘⢉⡁⠉⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠟⣿⠿⠀⣼⠋⠙⢛⡎⠀⢸⣿⣿⣿⣿⣿⣿⠀⠐⠚⠀⠀⠀⠀⠰⠦⠕⠀⠀⠀⠺⠥⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣶⣶⣆⠉⢀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢹⠙⢿⣿⣿⠈⠻⣿⣿⡆⠀⠀⠀⢻⣿⣿⣿⣿⣿⠄⠀⠀⠠⢺⣿⠿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⡀⠸⠀⠙⠿⣻⣧⡀⢻⣿⣧⡀⠀⠀⠀⢿⣿⣿⣿⣿⠆⠀⠀⣰⡟⠁⠀⠀⡉⠉⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡽⣷⡀⠁⠀⠐⠻⡙⣷⠀⣻⣿⣿⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⣾⣧⢀⡀⠊⢴⠆⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡌⠳⠟⢢⢦⣄⠀⡐⢺⡇⢸⣿⡗⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣿⣿⠟⠀⢀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⠒⠀⠀⠀⠈⠙⠶⡌⠁⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣀⢸⡵⢶⠀⢠⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡤⣠⣀⣀⣠⡤⠀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⣴⡟⣸⣿⣷⣤⣤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⢀⣠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣹⣿⣿⣿⣿⣍⠉⠈⠉⠐⠙⢄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠁⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠃⠛⠓⠛⠓⠀⠀⠀⠀⠀⠀⠈⠛⠃

⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣴⣆⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡄⠀⠀⠀⣠⡾⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠉⠉⢿⡟⠁⠀⠀⢸⡇⠀⠛⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠸⠛⠉⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠀⠀⣼⡇⠀⠀⠀⢸⠁⠁⠀⠀⢃⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⡟⠀⠀⠀⢀⣾⠂⢠⣿⡇⠀⠀⠀⠀⢀⣀⡀⠀⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠏⠀⠀⠀⢠⡿⠃⠀⣾⣿⡇⠀⠀⠀⠀⢸⠏⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡄⠀⠀⠀⠈⣡⣤⣤⣭⣭⣀⡀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⠁⣹⡄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢠⣾⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠉⠁⢻⣿⡟⠀⢻⣿⡋⣈⠙⢿⣿⣿⣿⣇⠀⠀⣿⣿⣿⠟⠛⠿⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡁⠀⣴⡿⠿⠀⠀⠸⣿⠁⠛⠁⢘⣿⣿⣿⡟⢠⣿⣿⣿⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇⡀⠘⠇⠀⠀⠀⠀⠘⢦⠀⠀⠈⣿⣿⠟⠁⠘⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢿⣿⣿⣿⣦⠀⠀⢀⣾⠋⠛⣿⣷⣦⠀⠀⠀⠀⢀⣠⡄⠿⢿⣶⠄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠰⠀⢿⣿⣿⣿⠀⠀⢸⠃⠄⠀⠼⣿⣿⡇⠀⠀⢀⢿⣿⡇⠠⠈⢻⡆⡄
⠀⠀⠀⠀⠀⠐⣿⣿⣿⡀⠀⣸⣿⣿⣿⡀⠀⣼⣇⠀⠀⠀⣿⣿⡇⠀⠀⢘⠿⣿⠀⠀⠀⢨⣦⣿
⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⣆⣿⣿⣿⠏⠀⠀⢻⣿⣇⠀⠀⢿⣿⣿⣄⠀⠸⡄⠙⠁⠀⢠⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠔⠁⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠⢀⡐⠂⢤⢀⡀⠀⠉⠀⣠⠊⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⡠⠤⠐⠈⠅
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢂⠀⠈⠐⠢⢄⠀⠁⠢⣸⠃⠀⠀⠀⠘⠀⡀⠀⠀⢩⠖⠁⢀
⠀⠀⠀⠀⠀⠀⡆⠑⠠⢀⡀⢃⠀⠀⠀⠀⠉⠢⣄⣿⠀⠀⢀⠀⠀⠘⡗⠀⡰⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠈⢻⡀⠀⠆⠀⠀⠀⠘⣴⡥⠂⠀⠀⡆⠀⠀⢰⣆⠠⠒⠂⢩⠊
⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⠀⠘⣷⠀⡆⠀⠀⠀⠀⠀⢳⣆⡀⠀⠀⠀⢀⣿⠁⠀⠀⠀⡇⠀⠀⡜⠁⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢃⠀⠀⠀⠘⣷⣴⡀⠘⣦⡀⠀⢸⠏⢳⡀⠀⠀⣸⣿⠀⠀⢀⡾⢃⣠⠞⠀⢀⡏⢀⡞⡀⠄
⠀⠀⠀⠠⠀⠀⠌⣇⢸⣆⠀⠀⠀⣿⣿⢷⣦⣄⣽⣷⣿⠀⠀⢹⠶⢶⡿⠿⠶⠶⠟⠛⣿⡥⢴⢂⣼⠣⠊⠁⠀⠆
⠀⠀⠀⠀⢡⠀⠀⠹⣦⡙⣷⣄⣰⣿⠛⠄⠈⠉⠉⠀⠈⢧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⡾⢡⠚⠀⠀⠰
⠀⠀⠀⠀⠀⠣⡈⡉⠻⣿⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣠⡤⠋⠁
⠀⠀⠀⠀⠐⠂⠙⠛⢦⣹⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢿⣾⣿⣥⠴⠋⠊
⠀⠀⠀⠀⠀⠘⡀⠀⠈⣿⡟⠀⠀⠀⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⣀⣀⠀⠀⠀⠀⠘⡟⠉⣠⣀⣐⣀
⠀⠀⣀⡀⠀⢠⣨⣄⣶⣿⠇⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣆⠀⠀⠀⣧⠞⠉⠀⢠⠐⠉⠁
⠀⠀⠀⠀⠑⣄⠀⠈⠻⣿⠀⢠⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢻⣿⣿⣿⣿⡆⠀⠸⠁⠀⠀⢠⠛⢦
⠀⠀⠀⠀⠈⠈⢢⠀⣀⠸⠆⢸⡃⠈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣼⣿⣿⣿⣿⡇⠀⠀⡰⣻⢶⠃
⠀⠀⠀⠀⠀⠀⠈⡳⣝⠂⠀⠘⣷⣾⣿⣿⣿⠃⢀⡀⠀⠀⠀⠀⠀⢀⣀⡈⠻⠿⢿⠿⢟⠕⠀⠀⠀⢀⣃⠜
⠀⠀⠀⠀⠀⠀⠀⢠⡈⣀⠀⠀⠈⠋⠭⠑⠁⠀⢸⣿⣦⣤⣤⣤⣶⣤⣿⠃⠀⠀⠀⢈⠁⠀⠀⠀⢀⡻⡷
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟⠋⠉⠉⡹⠋⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠂⢍⡛⠶⢦⣴⣤⣀⣀⣀⠀⠀⠒⠒⠈⠀⢀⣀⣀⣠⣤⡶⠷⠛⠉
⠀⠀⠀⠀⠀⢀⠤⠠⢷⡄⠀⠀⢱⣀⣠⣾⣿⠿⣿⠿⠿⠿⠿⠿⠿⣿⡿⠿⠿⣿⡷⢦⣔⠁⠀⠂
⠀⠀⠀⠀⠐⠁⣀⠠⠄⣈⡙⢷⡾⠋⠀⠀⣸⣆⠈⠉⠀⠀⠀⠀⠈⠀⠀⠀⠐⣿⡀⠀⠀⠑⣞⢁⠀⠀⠀⡀
⠀⠀⠀⠀⠋⠁⠀⠀⠀⢀⢘⣿⣷⣤⣤⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⢳⣤⣤⡾⣧⠈⠁⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡝⠉⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡈⠉⠙⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠉⢠⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠿⠿⠷⢶⣶⣶⣶⣤⣶⣶⡶⠶⠾⠿⠟⠛⠚⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣶⣤⡀
⠀⠀⠀⢀⣤⣶⣶⡶⣶⡀⠀⠀⠀⠀⠀⢠⣾⣶⢿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠃⣶⣦⠀⠀⠀⠀⠀⠀⣜⣚⢷⣭⠛⠁
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⢈⠷⠿⣿⣿⣿⢾⣿⡄⠀⠀⠀⠀⣠⣿⠆⠀⠀⠀⠙⡷⢯⣦⠀⠀⠀⠀⠌⠙⠃
⠀⠀⠸⣿⣿⡿⢿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠙⢿⡿⣿⣾⣇⠀⠀⠀⢀⣳⢿⡆⠀⠀⠀⠈⣿⢿⣯⡄⠀⠀⠀⢰⡄
⠀⠀⠀⠈⠙⠷⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⣿⣿⠃⠀⠀⠀⣮⢩⡿⠃⠀⠀⣼⣷⣿⣯⠷⡀⠀⠀⠀⡆⠛⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠸⡼⡕⡿⡼⡆⠀⠀⣼⡇⠀⠀⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡅⠀⠀⠀⠀⠀⠈⠆⢡⠃⠡⠀⠀⣼⠇⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⢠⣤⡴⠒⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣒⡖⠚⣛⣽⣷⢶⡤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣦⣤⡀⠀⠀⠠
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢬⡟⣷⢝⣿⡻⢦⣿⡌⣳⠜⡛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠈⠹⠿⣦
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣶⣯⣋⣼⡿⠃⠀⠀⢹⣿⠮⢆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠘⢧⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⣻⣽⣯⣿⡟⠀⠀⠀⠈⠝⣻⣿⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠑⢄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣏⣿⢿⠂⠀⠀⠀⠀⣼⣿⣷⠟⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣿⣿⠃⠀⠀⠀⠸⢿⣿⠺⡊⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢸⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠏⠀⠀⠀⠀⠸⠋⠿⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣮⣳⣴⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⣶⣦⣄
⠀⠀⠀⠀⣀⣀⣾⡷⣦⢞⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡦⠙⠻⢿⣦
⠀⠀⠀⢀⡻⣿⢿⣿⣧⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⠋⢛⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠁⠀⠀⠀⠀⠸⣿⣧
⠀⠀⠀⢻⣿⣷⣾⣼⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠀⢈⣿⣟⣸⣋⣽⣿⡧⠀⠀⠀⣀⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⣠⣾⣿⣧
⠀⠀⠀⠸⣿⡿⡿⠿⢿⠿⡄⠀⠀⠐⣿⣟⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢻⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣤⣴⡳⢤⡉⠉⣍⡁⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠿⢋⡁⢀⣿⣿⣿⡦⣝⠦⢙⣿⣦⣄⡐⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⣻⣿⡿⠃⣠⡞⣧⣻⣿⢧⡉⠀⠈⠀⠀⠁⢿⡻⣷⢉⢿⡟⢯⣛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣯⣿⠏⣄⣽⣾⣿⡸⣿⡿⣮⢳⣄⠀⠀⠀⠈⠘⣇⠈⠃⠘⣇⠈⡁⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⣁⣴⣿⣿⣿⣿⣿⣿⡻⣎⢿⣟⢄⠀⠂⠀⠀⠘⢦⠀⠠⠹⣦⢠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣠⠼⣿⣿⣿⣿⣿⠏⠞⠛⣻⣿⣿⣿⣿⡝⣝⠳⡈⠷⣝⠀⠑⠀⠀⠀⠈⠢⠑⠄⠀⢫⢮⢧⡀⣆
⠀⠀⠀⠀⢀⣾⣵⣿⣿⣿⣿⣿⠇⠀⠀⠀⠉⠙⠿⠿⢿⣿⣮⢢⣁⠀⡜⢧⡀⠀⠘⣄⠀⠑⡄⠲⣄⠀⠉⢷⣳⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⡠⠄⠀⠀⠀⠙⣾⣷⣿⣦⣽⣞⣿⣦⣽⣾⣹⡘⣿⠀⢹⡆⢀⡸⣷⣻⡇
⠀⠀⠀⠀⠸⣿⣏⣾⣿⣿⣿⣿⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⡇⣿⣶⡄⠁⣾⣇⣹⣿⣇
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠿⠃⣄⣠⡾⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⠙⣿⡷⠀⣿⣿⣿⡞⢿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡆⠀⠜⠛⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠝⠣⣸⣯⠇⢠⣿⣿⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣼⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⡀⢀⠀⠀⠀⠀⠀⠰⠾⢏⣼⣿⢿⣿⣿⠋⠉⠛⠿⠟⠻⠿⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⣯
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣧⣠⡴⠦⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⢿⠹⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠃⢸⠀⠉⠻⡿⣧⠙⡀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠒⠒⠚⠉⠀⠀⠙⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠶⠚⠛⠛⠛⠓⠲⠶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⢀⣀⣤⡴⠶⠶⠛⠛⠛⠛⠛⠲⠶⢦⣤⣀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⣀⣤⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⣰⠏⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠈⢻⣆
⠀⠀⠀⠀⣼⠏⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠻⣆
⠀⠀⠀⣼⠏⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢹⣆
⠀⠀⣸⡏⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢻⡄
⠀⢀⡿⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠃⠀⠀⠄⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠈⣷
⠀⣼⠇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⡇
⢀⣿⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣿
⢸⡇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠰⠇⠐⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡄⠀⠀⠀⠀⢸⡇⠀⠀⢻⡄
⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠢⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠈⣷⠀⠀⢸⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⢠⠈⠂⠀⠀⠀⣿⠀⠀⢸⡇
⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⢿⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠁⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡇
⢸⣇⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⣿⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣖⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡟
⠀⢹⡇⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣿⣽⣧⣀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣼⠇
⠀⠀⢿⡀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣿⣿⣿⣟⣷⢶⣤⡀⠀⢠⡟⠀⠀⢰⡟
⠀⠀⠘⣷⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣴⣾⣯⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⢻⣝⢦⡿⠁⠀⢀⣿⠁
⠀⠀⠀⠘⣧⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⡿⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠙⣧⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣟⣽⠟⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣩⡿⠋⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⡿⠏⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠈⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⠿⠋⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠈⠛⠶⣤⣄⣀⡀⠀⠀⠀⣿⠷⠿⢟⣛⣹⣭⡷⠟⠋⠁⢀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⣿⠏⠁⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣶⣶⣶⣦⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣠⠔⠋⠉⠈⠀⠀⠈⠑⠲⣄⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠉⠲⣌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢰⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣤⢶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠁⠚⠻⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⠰⠀⢱⡆⠀⠀⠀⠀⠀⠀⢠⣀⣤⣌⣦⣤⣀⠀⠈⠉⠛⠀⠀⢀⣠⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠘⢷⣅⢈⠀⠀⠀⠀⠀⠒⢠⡀⠉⠛⢛⡿⡿⠪⡃⠀⡀⠀⠀⠀⠀⠀⠀⠈⡙
⠀⠀⠀⠀⠀⠀⠀⠁⠙⣾⠃⣰⢁⢄⠆⠀⠀⠈⠙⠳⡞⠀⠀⠀⠘⠁⠈⠃⠀⠴⣾⡭⡳⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠇⡼⡜⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢠⠞⠲⣶⠁⠀⠘⠟⢣
⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠈⢰⡉⠇⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣠⠴⢁⠀⠀⠈⠀⠀⠀⠈⡵
⠀⠀⠀⠀⠀⠙⢦⣀⣀⠘⣶⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⠁⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⡷⡀⠀⠀⠀⠀⠀⠀⠻⠀⠉⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢸⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⣸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠳⢛⠀⠀⣠⠀⡀⠀⠆⠀⠀⠀⢀⣀⠀⣶⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣟⠁⠀⠀⠄⠀⠀⠀⠀⠀⢄⠀⠻⣤⠊⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢈⡄⠀⠀⠀⠀⡌⠛⢲⡾⠂⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢾⣣⠀⢺⠃⠀⠀⠀⠀⢸⣶⡏⢀⣴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⡸⣏⣠⣾⠀⠀⠀⠀⠀⠀⡉⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠙⠋⠀⢁⠀⠀⠀⢀⠈⠀⠁⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠘⠄⠀⠱⡄⠀⣦⠆⣬⠂⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠿⠀⠟⣰⠃⢀⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⢠⠀⠐⣹⢀⡏⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢠⠿⡄⢸⠀⠀⠻⡈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠶⠶⢦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣟⡻⠶⣖⡒⠬⢍⡛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⢘⣿⣿⣿⣿⡻⢿⡢⣝⠪⣒⠭⣑⠊⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⠁⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣯⡳⣝⠮⣑⠮⢕⡒⠭⣓⢤⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠳⡄⠀⠀⠀⠐⣿⣿⣿⣿⣯⡻⢿⡦⣝⠲⢥⣀⡈⠅⠊⢝⡿⣾⣆⣠⣤⡤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⢡⣄⣀⠀⠈⠀⠀⠀⠀⢻⣿⣿⣯⡛⢿⣷⢍⡺⢕⡢⢄⠉⢛⠶⣤⡙⢾⣿⣿⣷⣢⡥⠤⠤⣌⠢⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⣟⠷⣬⡳⣍⠲⣌⠲⣌⠢⡑⣮⡻⣦⣻⣿⣫⠔⣂⣤⣄⣂⠑⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣮⡻⡌⢧⠘⢷⡜⢆⠘⢮⣻⣾⣿⣿⠃⡠⣶⠿⣻⣿⣷⣽⠇
⠀⠀⠀⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⠿⠿⣿⡷⡜⢎⢣⡀⠹⣦⠈⢦⣻⣿⣿⣿⠜⡜⠁⡄⢸⣿⣿⠟
⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠇⠀⠀⠀⠀⠈⣿⣌⢎⢦⠳⡔⢌⠻⣦⣙⣿⣿⣿⠀⡜⠰⡀⡟⣿⢸
⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢦⠀⢳⣝⢦⣳⣮⣟⣿⣿⣿⡆⡇⣧⢇⡇⡏⡼
⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⢯⢳⡀⠻⣷⣭⣛⣿⣿⣿⣿⣧⣧⣿⠞⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠒⢖⠂⠠⠤⠄⠂⠁⠀⠀⠀⠀⠀⢠⢯⣿⣿⣿⣏⢧⠳⣝⠶⣬⣙⣛⢛⣻⣿⣿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⡝⣮⠳⣌⡛⠶⠶⠖⢛⣵⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣿⡟⣞⢿⣌⡳⠦⣭⣭⡭⠶⠛⣡⠞⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣎⠳⣍⣛⡓⠒⣒⣂⡤⠞⢋⣼⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢝⡲⠬⠭⠭⠭⠔⣂⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣮⣭⣁⣐⣫⣭⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡷⢦⡀⠀⠀⢀⣠⡤⠤⠶⠶⠶⠶⠶⠤⣤⣀⡀⠀⢀⣤⠞⡟⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢦⡉⠓⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠝⠛⠉⣠⠞⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠁⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⡀⠀⠉⠑⠀⠀⠀⠀⠰⠊⠁⠀⡀⢠⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠐⠤⠘⢁⣁⣐⠀⠀⠀⠀⠀⠀⠀⠠⠐⣈⣥⣍⡀⠃⢀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡄⠐⠎⠀⣼⣯⣈⣿⣷⡌⠀⠀⠀⠀⠰⠁⣾⣿⣧⣠⣿⡆⠈⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⡀⠀⢿⣿⣿⣿⣿⠇⠀⣠⠖⢄⠀⠀⢿⣿⣿⣿⣿⠇⠀⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠐⠀⠀⠙⠛⠋⠁⡀⠘⣇⠠⢸⠊⢄⠀⠉⠛⠋⠁⠀⠐⠀⢢⠏
⠀⠀⠀⢀⡞⣹⡄⠀⠀⠀⠀⠀⠈⢧⡑⢄⠀⠐⠀⠀⠀⠀⡠⠞⢻⣀⡞⠑⠦⣀⠂⠀⠀⢐⡠⠔⣡⠏⠀⠀⠀⡜⡆⠀⢀⣤
⠀⢀⠀⠈⢦⢞⠤⢤⡄⠀⠀⠀⠀⠀⠙⣦⢄⡁⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢁⣀⢴⡎⠁⠀⠀⠀⣰⢁⣧⠞⠁⢻
⠀⢿⣍⢣⡈⣿⠶⠚⠀⠀⠀⠀⠀⠀⣼⣁⣀⣈⠉⠒⠲⠦⡄⠤⣤⣄⢤⣤⠤⠆⠖⠈⠉⢣⠠⡹⡄⠀⠀⠀⢳⣼⣏⠄⢀⡟
⠀⠀⠙⢷⡹⢿⡇⠀⠀⠀⠀⠀⣠⠞⢉⠄⠉⡩⠉⠈⢢⠀⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡄⢣⣿⡀⠀⠀⢀⡏⣟⣚⣁⡤⠶
⠀⠀⠀⣀⣻⡄⠳⣄⠀⠀⠀⡼⠁⠠⠁⢀⡜⠀⡀⠀⡘⠢⠤⠊⠢⠴⠉⠲⠚⠙⠂⠊⠀⠂⣃⢸⣟⣷⡴⠚⠉⣀⣤⣤⣤⣤⠞
⠀⢰⠋⠠⢀⡿⣦⠈⣧⠀⢸⠃⢀⠂⢠⡊⠈⠁⠀⠒⡃⢀⢆⡀⡰⢄⢀⠴⣄⣀⠔⢄⡠⠋⠁⢸⠀⣿⠀⢠⣾⣩⠤⠤⣄
⠀⠛⠛⠛⠋⠀⠈⢷⡘⣆⣸⠂⠈⣠⠃⣀⠀⠀⠀⢸⡍⠁⢀⡍⠀⠀⣅⠀⠀⡄⠀⢀⣄⢀⡇⡌⢰⡏⣴⠏⠀⠈⠳⠶⠞⠁
⠀⠀⠀⠀⠀⠀⡀⠈⢷⡈⢿⣀⡞⠁⢀⡀⠀⡈⢠⠇⠈⠒⠊⠈⠒⠊⠀⠑⠊⠈⠁⠁⠈⢠⣿⠀⣾⠿⠁
⠀⠀⠀⠀⠀⣨⠟⢦⡈⣷⣞⠉⢀⣠⠞⡸⢫⢡⡟⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠀⣰⣿⣾⠞⠁
⠀⠀⠀⠀⠀⢿⠀⠄⣷⣷⡉⠉⢉⡰⢏⣠⣧⢿⡀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⣡⢾⡿⠃
⠀⣶⠲⠦⢤⣌⡳⢴⡇⠻⡟⠛⠉⠀⣨⠟⠉⠳⢽⠷⡶⢦⢤⣀⣀⣀⣠⣤⠶⡿⡷⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⡄
⠀⠙⠳⡶⢤⣄⡈⠉⠛⠒⠛⠛⠚⠛⠃⠀⠀⠀⢲⡞⡇⣼⢧⡏⠉⠉⢧⣿⢠⡇⣿⡞⠒⠦⣄⣀⠀⠀⣀⣀⣀⠀⣾⠀⠀⣿
⠀⠀⢠⠷⣄⠈⠉⠛⠶⠶⢶⠄⣀⣤⠴⠶⠶⣤⣤⣤⣬⣡⣬⣄⣄⣤⣤⣥⣥⣼⣤⣤⣀⣀⠀⠉⠉⠉⠉⠍⣫⡝⢿⣤⠖⠃
⠀⠀⠘⣆⣸⠀⠀⢀⠀⣠⢏⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⢀⡤⢛⣉⣤⡴⠶⠶⠶⠶⠒⠶⢶⣄⣉⠓⠦⠄
⠀⠀⠀⠀⠁⢠⠞⡽⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡴⢫⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠦
⠀⠀⠀⠀⣠⣞⡼⣷⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣄⣤⢟⡴⠋
⠀⠀⠀⠀⠈⠁⠀⠈⠓⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢩⢇⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠒⠁⣎⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠄⠠⠄⣀⠀⢄⠈⠀⠀⢀⣀⣤⣀⣀⡈⠉⠉⠀⣀⣀⠀⠀⠀⠀⠄⢀⢤⣀⠀⠡⠀⠈⠀⠁⢀⠀⠀⠀⠈⢇⠀⢀
⠀⠀⢠⠛⠐⠀⡄⠀⢰⠋⠀⡛⠻⠿⠹⠂⠀⢠⠉⠉⠉⡆⠀⠀⠀⡎⠉⠉⡆⠀⠀⠀⢀⣀⢈⣁⡀⠀⠠⡆⠀⠀⢡
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠠⠄⠀⠈⠀⠀⠈⠀⠤⠀⠃⠀⠀⠀⠁⠠⠀⠁⠀⠀⠀⠸⠀⠄⠀⣃⣀⣀⣃⠀⠄
⠀⠀⠀⢠⣤⣤⠀⠀⢠⢀⠸⠀⠈⡆⠀⡠⠀⠀⢀⡀⣀
⠀⢰⠉⠀⠀⠀⠀⠠⡄⣯⡀⠀⠀⠀⣉⠀⠐⠀⠀⠀⠀⠀⠴⠄⣄⠀⠀⠀⠀⡼⠛⢢⠀⠃⠲⠋⣶⢶⣂⡄
⠀⠀⢂⠀⠀⠀⠠⠤⠤⠤⡇⠀⠀⠰⡄⠀⠀⡀⠀⠀⠀⡌⠀⠀⠈⡄⠀⠀⢰⠃⠀⠘⡆⠀⠀⠐⠀⠀⠀⠃
⠀⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠁⠐⠀⠁⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠘⢄⠀⣠⠁⠀⠀⠀⠠⠀⠠
⠀⠀⣀⣀⡀⣀⡀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠥⠀⠇⢀⠀⢀⠐⠃⠀⠘⠂
⠙⠋⠉⣋⠁⣀⡔⠀⠀⠀⠀⠫⠀⠀⢀⠀⠀⠀⠙⠉⢳⠀⠀⠀⠀⠟⠤⠀⣡⠀⠀⡀⠀⠠⠉⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠁⠠⠀⠁⠀⠀⠠⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠁⠀⠊⠀⡀⠀⠀⢢⠁⠀⠀⠃⢀⡀⠰
⠀⠀⠀⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠌⠁⠄
⢤⣄⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⡤⠀⠀⠈⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⡀⡀⠀⢀⠀⢀
⠉⠐⡇⠀⠀⢐⡈⠃⠀⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⣶⠀⣩⡁⠀⠀⠀⣾⡁⢙⣷⠀⣀⣀⢐⠄⠀⠛⡄⠁⠀⠿
⠂⠀⠁⠀⠀⠈⡇⠠⠀⡐⠀⠀⠀⠠⡀⠀⡰⠀⠀⠀⠸⠃⠁⢈⠇⠀⠀⠀⡟⠓⠒⠙⠀⠀⠀⠘⠀⢠⠀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢸⣤⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠸⠂⠀⠀⠀⣟⠈⠁⢈⠀⠀⠀⠀⣀⠀⣀
⡀⠈⠀⠀⠀⠂⡀⠀⠀⠀⡠⣄⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠆⠖⢄⠀⠀⠀⠀⡠⠀⣀⠀⠉⠀⢠⠖⢒⠈
⠁⠀⠀⡀⠀⠀⢠⠀⠀⢶⡁⠀⠀⠉⡀⠀⠀⠀⡄⠀⠀⡰⠀⠀⠀⠸⠀⠀⠸⠈⠇⠀⢠⣦⠀⣫⡀⠀⠸⣧⠁⠈⠉
⠀⠀⠀⠘⡀⠀⣸⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠁⡘⢀⠀⠀⠀⠀⠀⠇⠃⠆⠀⠀⠀⡘⠉⠀⢨⠁⠀⠀⠈⠣⠀⠐
⠀⠀⠀⠂⠀⠈⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠌⠀⢀⠀⢀⠀⠀⣀⠀⠀⣀⠀⠐⠆⠀⠻⠂⠀⠀⠐⠂
⠀⢀⠑⠄⣐⣤⡈⠀⠈⠀⡄⠀⠀⢀⠀⠀⠶⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠈⠈⠀⠠⡆
⠀⠀⢀⠒⠀⠀⠸⡄⣀⠠⠆⠀⠀⠠⡀⠀⠀⡹⠀⠀⠀⠈⠀⡀⠀⡃⠀⠀⠊⠀⠀⠀⠏⠀⠀⠀⡆⠀⠀⡶⠁⠀⠰
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⠀⠀⢀⠀⢐⠀⠄⠀⠀⢢⠐⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣟⣛⣉⣉⡩⠭⢭⣓⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⡄⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠙⡳⣕⠀⢤⡩⣲⣝⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠛⢿⡗⠀⢌⠺⣄⢀⣿⣮⢮⣳⡝⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⡈⡟⢿⡀⢠⣦⣿⣦⡰⣷⣽⣿⣿⣿⣯⣿⣿⣎⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠻⣿⣶⣷⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠈⣿⣿⣿⣿⢿⣿⣿⡿⡟⠙⢆⠀⠙⠻⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠻⣿⣿⣀⢁⡻⢳⢤⣠⣪⡓⠀⠀⠘⠋⠉⠙⠿⠿⠿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣧⢿⣦⣹⣿⣿⣿⣿⣿⣦⣿⣿⢿⡄⠀⠀⠀⢀⠴⢶⡿⠿⣷⡄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠚⠒⠚⠉⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢹⣿⣿⣿⣿⢿⣿⣿⣟⢿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⣿⣿⣿⣿⣷⣙⢿⣿⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣟⠶⠄⠀⢤⠄⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⣿⣿⣿⣷⣿⣿⣿⡏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢠⣿⣿⣿⣷⣿⣿⡏⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣦⡀⠉⠁⠀⠀⠀⣀⣠⠄⠠⠤⠄⠀⠀⠀⠀⠀⠀⣼⡿⣫⣿⣿⣿⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠋⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣵⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠈⢿⣿⣿⣎⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢻⣿⣿⡿⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠊⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡗⠢⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠡⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠄⠒⠒⠈⠀⠀⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⣴⣶⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣤⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢘⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⣠⣾⣿⡿⠋⠉⠙⢿⣿⣿⣄
⢸⣿⣿⣿⠯⣟⢿⣿⣿⣿⣿⣷⠀⢀⣾⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢠⣿⣿⡟⠁⠠⠀⠀⢸⣿⣿⣿⣷
⠸⣿⣿⣏⠉⠀⠀⠙⣿⣿⣿⡇⠀⣿⣿⣿⠠⠆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⣨⣿⡿⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿
⠀⠻⢿⣿⣄⠀⠀⠀⣽⡿⠛⠃⢀⡿⣿⣿⣧⠀⠀⠘⠗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⠀⣠⣼⣿⣿⣿⣿
⠀⠀⡈⠹⠋⠃⠀⣸⣷⣦⣤⣄⡏⣿⣿⣿⣿⣷⡤⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡇⠀⠀⠙⢻⣿⣿⣿⣿
⠀⣾⠃⠀⠀⠀⡔⢉⣿⣿⣷⣿⡇⢹⣿⣿⣿⡀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⠈⢛⣿⡏⠀⠀⠀⠀⠀⠻⣿⣿⣿
⢰⣿⠀⢀⣴⣾⣷⣼⣿⣿⣿⣿⡇⠀⠩⢿⣿⠇⠀⠀⠀⠀⠈⢻⣿⣿⡿⣿⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣤⣿⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠀⢀⣀⣀⠹⣿⠀⠀⠈⠛⠋⠀⠀⣀⣤⣤⣤⣀⠈⠛⠛
⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠁⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣷⣯⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣶⡄
⠀⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⣴⣿⣿⣟⠀⠈⢿⣿⣿⣿⣿⣷
⠀⡞⠴⢸⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣠⣤⣀⡿⣿⣿⣿⠉⠉⠙⠟⠿⠁⠀⣼⣿⣿⣿⡷⠂⠀⠈⢻⣿⣿⣿⣿
⠀⡇⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⣿⣿⡿⢹⡗⠀⠀⠀⢸⠀⠀⢀⣿⣿⣿⣿⣷⡄⠀⠀⢨⣿⣿⣿⣿
⠀⢙⣤⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠀⠠⠴⠀⠀⣄⠀⠀⢹⣿⣿⣿⣿⣿⡖⠐⢻⣿⣿⣿⠇
⠀⠘⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣿⣿⣷⣿⣶⠀⠀⠐⣻⣿⣷⣦⢹⣿⣿⣿⡿⠃⠀⠀⠉⠁⣀
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡟⠯⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠆⣠⣾⣿⣿⣿⣿⠉⠀⠈⢻⠴⠀⠄⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣤⣭⣿⡶⢾⡿⠾⡟⠛⠻⢶⠿⠶⣦⣤⣠⡟⠛⢷⣿⡛⠛⠷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢀⣴⡾⠟⠉⠉⠑⢄⠑⠄⠘⠀⠀⠁⠀⠀⠀⠈⠙⠿⣦⡀⠙⢿⣄⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢻⣆⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠚⠉⠙⠳⢄⠀⢠⠀⢀⠔⠊⠁⠀⠙⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⣠⣄⡀⠈⠑⠎⠟⠁⣠⣤⡄⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡀⠀⠀⠀⠀⠀⢉⣻⠷⠄⠀⢴⠿⠋⠁⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠛⣿⣶⣄⠀⣴⠟⠛⠛⢷⡄⠀⢠⡶⠛⠛⢷⡆⠀⠀⣀⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢛⣿⣿⣯⠙⢿⣏⠀⢰⠄⢸⣧⣤⣿⠀⣤⠀⢈⣿⠶⠟⣽⣿⣼⣟
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⢃⣼⣿⡇⠈⢿⣦⣀⣠⣾⣯⣾⣿⣦⣀⣠⡾⠃⠀⣼⡿⢿⡌⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠺⡿⢈⣿⡀⠀⠉⠉⠉⢻⣦⣴⠟⠉⠉⠉⠀⢀⣼⢏⣿⠄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠛⠙⣷⣄⠀⠀⠐⢿⣿⡿⠷⠀⠀⠀⣠⡾⠻⢿⣥⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣄⣉⣉⣉⣁⣤⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⠿⠿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⣶⠀⢀⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣿⣤⣾⣿⣿⢙⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣤⣴⣿⠋⣿⣿⣿⣧⢿⣿⣿⣿⡟⣏⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠦⠛⠉⠀⠀⢹⣿⣿⣿⣸⣿⣿⡿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣧⡄⠀⠀⠀⠀⠈⣿⠛⠛⡿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡃⠀⠀⠀⠀⣼⠃⠀⣸⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠰⣶⣤⣼⣧⣤⣴⡟⠙⠻⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠒⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢟⣿⠷⢦⣄⣀⣀⣀⣴⣶⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⠟⣿⣿⣿⣿⣦⡈⠙⠛⠛⠋⠩⠭⢭⣬⣙⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢺⠀⠃⠀⠉⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⢿⣆⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⣀⣀⣤⣤⣭⣝⠛⠲⢶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⠈⢛⡻⠿⠟⠛⠉⠉⠉⠙⣿⣿⡇⠀⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡈⠀⠀⢸⡿⣆⠀⠀⢷⡀⠀⠀⠘⣿⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢿⡇⢻⣦⣀⠘⣿⣦⡀⠀⢛⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠙⢿⣿⢮⣟⠿⣷⣦⢷⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⣶⣲⣦⠈⢧⡀⠈⣿⣧⢻⠇⠈⠙⠧⢻⣦⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢀⠈⠉⠋⠁⢀⠘⣿⣤⡿⠁⠘⢀⣠⣶⠇⣼⡟⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠐⠂⠀⠀⣿⣿⠁⢨⣶⣿⡿⠃⣼⠋⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣜⣿⢿⣿⡃⠸⣇⣾⡟⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⣿⠉⠻⣿⣷⣍⠳⡄⠸⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠠⠀⠀⠀⠀⣠⠊⣸⣿⠀⠀⠈⢿⣿⠳⣼⡆⠈⠳⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⣼⣿⡀⢿⣿⣄⠀⠀⣼⠏⠀⣻⣿⢀⠈⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠈⠉⢳⣾⠁⢀⡴⢟⣿⡇⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡄⣰⠿⢋⡞⠉⠀⢸⣿⣿⣦⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡗⠻⣆⣸⠀⠀⠀⠀⠹⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⣧⠀⠀⣿⡀⠀⠀⣇⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠙⢧⠀⢘⣿⡆⠀⠈⠑⢤⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠈⣤⠾⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⡿⠘⢿⣷⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣏⣠⣖⣦⡍⢻⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣤⣤⣤⣴⣟⣛⣛⠛⠛⢛⣳⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠿⢟⣋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣟⣻⣦⠀⠀⠙⢉⣴⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⣷⠀⠀⠈⠛⠋⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⢠⡀⣀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣷⡀⠈⣡⠄⠀⠀⠀⠀⠀⠿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣴⣾⣿⣿⣿⣧⠸⣿⢛⡛⠓⠀⠀⠠⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣽⣿⣿⣿⣿⣿⣧⡘⠉⠁⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡿⠿⠛⠋⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⡿⣧⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣝⢿⣿⣷⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠋⠀⣼⣿⣿⣿⣿⣿⡇⠀⠁⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃⣠⡾⣿⡿⣫⣿⣿⢿⡁⠀⠐⠀⠐⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣵⣿⡁⣾⣿⡟⢡⣼⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠋⣠⣿⣿⡿⠇⠀⠀⠀⠀⢸⠁⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⣾⠿⣿⣟⣈⠀⠀⠀⠀⠀⢼⠆⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣵⡿⢋⣵⣿⣼⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠟⣠⡟⣹⣿⢫⣿⣿⣿⣿⡂⡀⠀⠀⠀⢀⡴⣫⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⢣⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣮⣘⣒⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⢻⣿⣿⣿⣇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠻⣿⡿⣬⣿⣿⣿⣿⣾⣯⠙⢿⣎⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠈⠁⠀⠀⠉⠀⠀⠈⠛⠙⢅⠈⠀⠀⠐⢸⣿⡿⣿⣿⣿⡟⠇⠈⢿⡀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠙⢱⣿⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣷⣦⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⢀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡄⢀⣠⣼⣿⣿⣿⣿⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟⠁⠀⠈⢦⣠⠒⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠁⠂⠀⠀⠈⠀⠀⢏⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⢻⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠉⠃⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⣷⣶⣿⡿⠋⠁⠀⠉⠻⡇⠀⠀⠀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡀⠀⠀⣠⣾⣦⣄⣀⡀⠠⢿⣷⡀
⠀⠀⠀⠀⠀⠁⠬⡀⠀⠈⢰⣿⣿⡿⣇⠀⠀⢰⠂⠀⠀⠀⠀⠀⠛⣠⡞⣠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠀⠘⣿⣿⣿⣿⠃⠀⠀⠈⠻⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠻⠛⠁⠈⠢⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠒⠂⠤⡀⠀⠀⢸⡋⢇⠀⠀⠀⠸⠟⠛⠛⢄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠱⡀⠙⣦⣀⠀⠀⠀⠀⠀⣀⠹⠁⢀⡀⠀⠤⠄⠀⠀⠀⠀⣇⠈⢄⣀⣀⡀⠀⠀⠀⠈⣢⣤⣶⡁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠈⡉⠈⠛⠛⠓⠂⠂⠙⣃⠜⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⢸⣿⣿⣧⣀⠀⢀⣠⣿⣿⡿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣀⣀⡩⠭⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⢶⣶⣶⣦⣄⠀⠀⠀⠀⠙⢽⣻⣿⠉⠀⠀⢀⣨⠿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢒⢾⣿⡁⠀⠀⢄⠀⠊⠀⠀⠀⠀⢀⠔⠁⠀⣠⣾⠟⠋⠉⢉⡉⠣⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣤⣤⣿⣷⣄⣀⣀⡐⠁⠀⢠⣾⠏⣰⣦⠀⠀⠛⠟⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠒⢳⡞⠋⠀⣙⡿⢿⡿⠋⠌⠀⠀⣰⡿⠁⠀⠀⠁⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠐⠁⠊⢡⣦⣽⣿⣿⠀⣁⠀⠀⣠⡾⢿⡀⠀⠀⠀⠀⢀⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣀⣅⡀⠀⡈⠛⢿⣿⡿⠀⠈⠙⢶⡟⢰⣶⠍⣓⡠⠤⠀⠁
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⣙⠛⠉⣻⣖⠂⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠈⣷⡀⠈⠙⣷⣤⣈⣃
⠀⠀⠀⠀⠀⠀⠐⠿⣷⣄⣰⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣀⠀⢀⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣶⣿⣿⣾⣇⣴⣤⠒⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⢿⣿⣿⣿⣿⣿⣿⡟⠈⠁⢻⣿⠏⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣇⠀⠀⢀⣿⣴⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⡆⠀⠀⢻⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣴⠆⠸⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣉⡽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠀⠀⠀⡀⠀⠀⠀⠀⠀⢿⣿⣿⡇⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢠⣿⣿⡄⠀⢰⣿⣿⠀⠀⠀⠰⣿⣿⣿⣾⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢠⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠒⠂⢁⠴⠋⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠠⢂⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⣉⣀⠁⠀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣶⣶⡶⠀⠀⠀⠀⠀⠈⢿⣿⡏⠙⠁⠀⡄⠀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⢠⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠐⠀⠄⠀⢠⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢀⣮⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢱⠄⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠀⢀⠞⠀⠀⠀⠋⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠛⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⢀⡴⠈⠀⠀⣀⡤⠞⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⢰⡼⠀⢸⣿⣶⣦⣤⣤⣄⡴⠁⡇⠀⠀⠛⢧⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣇⢀⠔⠁⡞⠁⠀⠘⣿⣿⣿⣿⠟⠉⠀⠀⠱⡀⠀⠀⠈⣧⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢇⡞⠀⠀⠀⠀⢀⣼⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠸⡛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠉⠉⡀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠃⠙⠲⢄⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠂⠀⠀⠀⠙⡄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣿⣿⣿⣿⣿⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣻⣿⣟⣻⣿⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣽⣷⣌⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽⠛⠂⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣋⠁⠀⠀⠀⢀⣴⠶⠛⠛⢶⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢋⣀⣈⠛⢦⡀⣴⠋⠀⣠⣤⣤⣄⠻⣆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⡟⠉⠉⢳⡄⠻⠁⢀⡾⠉⠀⠀⠹⣆⢻⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡟⠀⢀⣀⠀⢻⡀⠀⡾⠁⣠⣦⡄⠀⢻⡌⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡇⢠⣾⠛⢷⡈⡇⢸⡇⣰⣿⠀⢹⠀⢸⡇⢿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⢸⣿⡀⠘⡇⣷⢸⠃⣿⣿⠀⣼⠀⣸⠁⣼⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡇⢸⣿⣧⣰⡇⣿⢸⡄⣿⣿⣿⡟⢠⡟⢀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢻⡀⢻⣿⣿⣧⣿⣼⣧⣿⣿⡿⢡⡞⢀⡞⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢦⡙⠦⣽⡟⠉⠈⠁⣲⡾⠝⢷⢋⡤⠟⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢹⢇⣟⣀⣤⣴⣞⢹⢣⠶⠋⠈⢳⡀⠀⠀⠈⠙⠻⢶⣦⣄⣀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⠈⠉⠁⠀⢷⡻⣫⡞⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣩⡿⢋⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⣧⡿⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢷⠀⢦⣤⣀⣀⣀⣀⣉⣉⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⠛⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣯⣩⣽⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠯⣝⠲⠤⠀⢀⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠿⠶⣦⣤⣤⣴⣶⡶⠛⠋⠉⠬⠭⣍⣛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣡⠖⠋⡠⠀⢠⡿⠀⠈⠳⣤⡀⠐⠲⢦⣀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣤⣞⠀⣠⡿⠁⠀⠀⠀⠈⠛⠷⣦⣼⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠟⠻⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣭⣤⣤⣭⣷⣄⡀⠀⠀⠙⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈⣿⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠹⣿⡇⠀⣿⠂⣿⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠰⠛⠷⠀⣿⡇⢀⣿⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⣿⠀⠉⣭⠁⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣿⠓⣾⣿⠀⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡏⢰⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣿⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⣻⣴⣿⠇⣾⡏⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⡟⣼⡟⣸⡿⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⢀⡼⢡⣿⢣⣿⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠋⢠⣿⢣⣿⠃⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠿⣶⣶⣶⣶⣿⣿⣿⣠⣿⣷⣿⠃⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠈⠉⠛⠛⠛⣻⣿⣷⡿⠃⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⣿⣿⠁⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟⣿⡇⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⣿⡏⠳⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢻⣧⠀⠸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠻⠗⠀⠈⠿⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⢳⡀
⠀⠀⠀⠀⠀⠀⣴⠋⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠙⢦
⠀⠀⠀⠀⢠⡞⢁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢳⡄
⠀⠀⠀⣰⠋⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠙⣆
⠀⢀⡼⠁⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⢧⡀
⢀⡞⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢳
⣾⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡔⡚⠛⠓⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⡥⠞⢻⣿⡿⢦⣤⡀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠡⠊⠀⣠⢾⡇⣿⡆⢢⠻⣆⠠⡑⣄⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⢀⣠⠞⢁⣾⡇⠹⡿⡈⢇⠙⣧⠱⡈⠳⣤⡈⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣴⣶⠾⠛⣉⣤⡶⠋⠀⢿⡀⠀⠈⢻⣆⠘⣷⡑⠀⠈⠙⠓⠶⠼⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠟⠋⣠⣴⡿⠛⠁⠀⠀⠀⠈⣧⡀⠀⠀⢻⡅⠈⠿⣦⡀⠀⣀⣤⠶⠊⠀⠈⢛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⢠⣾⣿⣿⢄⣀⠀⠀⠀⠀⢀⣈⣷⡀⠀⠀⠙⢦⣐⣬⣉⣉⠥⠐⠁⢀⢄⣴⣿⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡇⢠⣿⣿⣿⣿⣦⣌⠁⠀⠀⠘⢉⣤⣾⣷⣔⣤⣀⣀⣉⣉⣩⣵⣾⣥⣤⣴⣟⢿⣟⢆⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣸⣿⣿⣿⡟⠛⠋⠁⡄⠀⠀⠉⠀⠛⠛⠙⣿⣿⣿⣿⣿⣿⣷⡌⢯⠹⣿⡻⣾⣿⣦⣤⣙⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢹⣿⡿⢻⡇⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⢻⣿⡟⣿⢦⢃⠸⣷⣹⣮⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡟⢁⣿⣷⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⡏⢸⢻⡟⠀⡿⡌⢸⠀⣿⡇⢿⡌⢿⣌⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⣾⣿⣿⣧⠀⣴⣶⡶⣤⣄⠀⠀⠀⠀⠀⡸⢁⡾⠋⠀⡼⢁⢀⣾⢠⣿⣿⠘⣿⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣼⠻⣶⣿⣿⣿⣧⠈⠛⠶⠊⠁⠀⠀⢀⣠⣎⡴⠋⠀⣠⠞⢁⡾⢣⡿⢸⣿⣿⠀⡿⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⠁⢠⣿⢉⣿⣿⣿⣧⡀⠀⠀⢀⣠⣴⣿⡿⠋⢀⡴⠟⣁⠔⠋⠀⣿⡇⢸⣿⣿⡾⢁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡏⠸⣎⠻⣿⣿⣿⣿⣿⢻⣿⡿⠋⣠⡾⢋⡴⠚⣡⠶⢒⡉⠡⣏⡄⢿⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠈⢿⡀⡙⢿⣿⣿⣿⣿⡏⢸⠟⣠⡾⢋⡾⠋⢀⡴⠓⣨⣿⡶⠋⠙⢻⡜⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡼⠙⣆⠈⢷⡀⠘⠈⢿⣿⠟⢠⡟⢰⣿⢡⠏⢠⡞⠁⢀⡾⣱⡿⠁⠀⢀⡾⠛⣼⡟⡷⣍⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⠃⣰⣿⣆⠈⢷⠀⢸⠟⢁⣴⣿⣧⣿⠑⡏⢰⡟⠀⢠⡿⠁⢸⡇⠀⢀⠋⠀⠀⠙⣇⠱⡈⠳⣌⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⢠⡿⢿⣿⠀⠈⡇⢀⣤⣾⡏⣿⣿⣿⡆⢃⣾⡇⠀⣼⡇⠀⣿⠁⠀⠀⠁⠀⠀⠀⠹⡄⠹⣦⠉⠳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡇⠈⣿⠀⣆⣷⠿⢋⢻⠁⣿⣿⣿⣿⡈⣿⠁⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠘⢷⣦⡈⠳⣄⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢧⣠⣿⠞⠋⠁⡠⢃⠇⣸⣿⣿⠙⣿⣧⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⠳⡙⠶⣌⠳⡄⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⢀⡴⠟⠉⠀⢀⡠⠊⣡⠏⣴⣿⣿⣿⢀⣿⣿⣿⠀⠀⣿⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀⠘⢦⣈⠑⢿⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠴⠒⠚⣋⣡⣴⠞⣡⣾⣿⢋⣿⡇⣿⢸⣿⣿⠀⠀⣿⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⡻⣦⡀⠙⠳⣦⣄⣸⠃
⠀⠀⠀⠀⠀⠀⣠⠞⢁⣤⣶⣿⣿⣿⣿⣷⣿⣿⡿⢃⣾⣿⢠⡇⣾⡟⣿⠀⠀⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿⣿⣮⡙⠷⣶⡖⠛⡇
⠀⠀⠀⠀⠀⢰⣷⣾⠟⣻⣭⣥⡴⡶⢤⣤⣞⣋⣡⣾⣿⠃⡜⣴⣿⢡⣿⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿⡿⣏⢻⣆⣸⠇⡰
⠀⠀⠀⠀⣰⠟⢹⠁⣾⣿⣿⢣⢪⡆⢾⣿⠛⣿⣿⠟⢁⣠⡾⣿⠃⣼⡟⠀⢀⡇⣸⡏⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⣴⣾⠋⠘⠟⣼⠟⣡⠎
⠀⠀⠀⠀⢿⡀⠹⣄⢻⣿⣿⢸⢸⣧⡈⠉⣶⣿⣛⣛⣛⣭⡶⠃⡼⣿⠃⠀⣼⡷⠻⢧⣀⣀⣀⣀⠤⠖⠚⠉⢉⠾⠋⠁⢀⠼⣼⡵⠚⠁
⠀⠀⠀⠀⠀⠙⠲⣬⡑⠿⣿⣮⣦⡙⠶⠭⣜⠛⣛⣻⡿⠟⢀⡜⣱⠃⠀⣰⣿⡇⢸⣷⣤⣀⣠⣤⣤⣤⣤⣤⣤⡴⠒⣋⡥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣍⡙⠛⠳⠶⠿⠛⠛⠉⡀⠔⣫⡾⠁⢀⣴⡟⣵⡷⠟⠉⠉⠉⠁⠀⠀⣸⣿⡿⢉⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠷⣬⣕⣒⣤⣄⣀⣀⣠⠴⠞⣉⣤⡾⠛⣿⠀⢿⣀⡀⠔⠊⢉⣉⡭⠭⠍⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣉⣛⣿⣻⠿⠛⠉⠀⠷⢘⣈⣛⣒⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣮⡿⣾⣽⣿⡿⣿⣯⣿⣌⣻⣧⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⢧⣾⣿⣿⣿⣿⣿⣿⣯⣿⣽⠿⣯⢲⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣿⣥⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣾⣿⣿⣿⣧⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣤⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣦⢾⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠩⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢘⢽⣿⣯⣿⣿⣿⣿⣟⣦⡀⠐⠚⢭⣭⡀⠀⢈⣿⢿⣿⣿⣿⣿⡿⢳
⠀⠀⠀⠀⠀⠀⠀⣅⣾⣿⣿⢛⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⢀⣼⡟⣾⣷⣿⣿⣟⢫⡿⣷
⠀⠀⠀⠀⠀⠀⡠⠻⣾⣋⣵⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣾⣟⣼⣿⣿⣿⣿⣿⡤⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣶⣷⣿⣿⣿⣿⣿⣿⣿⣷⠰⠒⠶⠀⠀⠀⣿⣿⣿⢻⣿⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠹⣿⠙⣧⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢀⣀⠜⢹⡉⣧⡾⠋⠻⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠉⢙⣺⣿⣿⠟⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠓⠃⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢋⣵⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦⠀⠀⠀⠀⠀⠀⡀⠀⠱
⠀⠀⠀⠀⠀⠀⣰⣛⠿⣇⣼⣿⣯⣾⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⡇
⠀⠀⠀⠀⠀⠀⢻⣇⣴⠿⣇⣼⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠐⣿⠛⢚⣿⡿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣿⣯⣿⣿⣿⣿⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣼⢿⣿⣿⡿⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⡼⣧⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⢠⣧⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠀⠀⣼⡏⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣾⣿⣿⣿⠟⠁⢀⣼⣿⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⡿⠟⠁⢀⣴⡿⣻⣣⣿⡏⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢁⣀⣴⣾⠟⣫⣾⣿⣿⠟⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠟⠛⢋⣉⣀⣤⣤⣶⣾⠿⠛⠉⣠⣾⣿⣿⡿⠋⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣯⣵⣶⠿⠿⠿⠛⠛⠋⠉⠁⢀⣠⣴⣿⣿⣿⡿⣋⣴⣾⣿⠟⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣴⡾⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣀⣤⡴⠖⠂⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⠁⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣉⡴⣶⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠘⣾⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣡⠞⣡⣾⣿⣿⣿⣿⣿⠟⠋⣴⣾⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢁⣾⣿⣿⣿⣿⣿⡿⠃⢀⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣾⣿⣿⣿⣿⢋⣿⠃⠀⣼⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⣿⡏⢸⣿⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⠀⢿⣇⠀⠘⣿⣿⣿⣿⣀⡶⠈⠢⡀⠀⠙⠻⠿⠟⠋⠀⢀⠄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠈⢿⣦⡀⠘⢿⣿⣿⣿⡇⠀⠀⠑⢄⡀⠀⠀⠀⢀⡴⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⠀⠙⢿⣦⡀⠙⠿⣿⣿⡀⠀⠀⠀⠉⠙⠐⠋⠁⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠳⣄⠀⠙⢿⣦⣄⠈⠛⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢀⠀⠀⠀⠀⠈⠻⢷⣦⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠑⢦⣀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣌⡙⠲⢦⣄⡀⠐⠶⣤⣉⠛⠷⣦⣄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣶⣬⣍⣳⢦⡈⠻⣿⣶⣌⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣦⠙⣿⣿⣿⣆⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⣤⡙⣿⣿⣿⣿⠀⣿⣿⣿⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⠟⣁⣾⣿⣿⡿⢃⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣩⣴⣾⣿⣿⣿⣯⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢀⣀⠤⠭⢉⡙⠳⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⣠⣺⣿⠿⣷⣶⡄⢹⡇⢀⢱⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢰⣿⣿⣯⣽⡿⠿⠿⢿⣷⣼⣷⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠀⠀⠘⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠻⢿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢡⢃⣴⡄⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣹⡟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣿⣿⡇⢀⣇⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⡿⢻⣿⣷⣸⣿⣉⣉⣙⣓⠀⠀⠀⣴⣶⣚⣦⣀⣿⣿⡆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⢴⣿⣿⣿⣿⣿⠙⠋⠙⠋⣿⠛⣿⠁⠉⠾⠂⠈⣿⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠒⠛⠉⠩⢾⠿⣿⣟⠀⠀⠀⣠⠋⠀⠘⣄⠀⠀⠀⣠⣿⡏⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣀⣀⣾⣿⠋⠉⠉⠉⠀⠄⠀⠀⠀⠉⠉⠉⣽⠟⠀⢀⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠂⠀⠀⠈⠉⢉⣿⣿⡀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⣸⡧⠀⠀⣼⡟⠻⢿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⣼⣧⣀⡀⣀⡴⠾⢻⣯⠙⢷⣄⠀⠀⠀⣀⣀⡀⠀⠀⣴⣯⣶⠀⣼⣿⣷⣄⣀⣙⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣾⠉⢿⣿⡻⢦⣄⡀⢈⡇⠀⣷⣿⣷⣄⣀⠀⢀⣀⠀⠀⣿⡿⣿⠰⡏⢿⣿⣦⣄⣈⣩⣿⡟⠃
⠀⠀⠀⢀⣼⣯⠴⠼⣿⣿⣦⢤⣿⣿⡇⠀⠟⣿⡃⠉⠙⠋⠉⠀⠀⠀⢻⡇⠹⣆⠙⠀⠉⠛⠛⠛⠋⢙⣷⣤⣄
⠀⠀⣰⣿⡟⠁⠀⠀⠿⣿⢿⣷⣾⣿⣇⢰⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⢦⡙⠦⣄⣀⠀⣀⣀⣴⢾⣟⣻⣿⣇
⠀⠜⡇⠈⠀⠀⠀⠀⠀⢻⠈⣿⣿⠿⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠤⠴⢚⣷⠀⠙⠷⣶⡿⣿⣭⡵⠶⢿⣭⣭⣿⣿
⠀⢰⠁⣆⡄⣠⣶⣴⣾⣿⣧⡈⠻⢾⣿⡿⠂⠀⠀⠀⠐⠀⠀⠀⠒⠊⠁⠈⠦⡀⠀⠂⠁⠈⠉⠻⣦⣠⣮⣶⡝⣿⡆
⠀⠃⠀⢷⠱⣿⣿⣿⣿⣿⣿⡿⠷⢶⣿⠀⠀⠀⠀⢠⡀⢠⣄⠀⢴⡀⢰⡀⢀⢀⡀⠀⠀⠀⠀⠠⣬⣭⡭⠿⣿⣿⡇
⢸⠀⠀⢸⣿⣿⣟⠁⣸⣿⣿⠴⣾⣿⠇⠠⡆⠹⣧⣸⣧⠘⣿⣆⢸⣵⣨⠷⢸⣆⠹⠒⠤⠀⠀⠀⢿⣷⣶⣤⣿⣿⡇
⡇⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣸⣧⣿⡀⠹⣇⠹⠇⠋⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠘⡟⠈⣿⣿⣿⡇
⡇⠀⠸⣿⠿⣛⣉⠉⠘⠋⠁⣨⠟⠈⠻⣿⡿⡧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⢸⡇⢹⣧⡀
⢁⠀⣼⣷⣿⣿⣿⣦⣶⢞⡟⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⢧⣈⠻⢿⣷
⢸⠀⠙⢿⣿⣿⣿⣿⣏⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢿⡀⠀⠻⣶⣶⣾
⢸⠀⠀⠀⠉⢿⣍⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠈⠛⢦⡀⠀⢿⣿
⢸⠀⠀⠀⠀⢸⣿⣦⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢿⣿⡿⠋
⣾⣶⡶⡶⠶⢾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣾⣿⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⢶⣶⣤⣀⣴⣴⣽⢛⣟⡿⢟⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⣿⣿⣿⣾⣿⣿⣿⣛⣦⣹⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡻⠟⢋⣹⣿⣿⣿⣿⣿⣯⠉⠻⣆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣯⣿⣿⣷⣾⣿⣿⣿⣿⣟⡿⣦⡙⢷⡻⣮⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠻⣿⣿⣇⢯⠳⡝⣿⣽⣿⣋⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⡹⣿⣿⠧⣳⣽⠈⠁⠉⠲⢯⣴⣝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣎⢻⣟⣿⣿⣿⡹⠂⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡹⣿⡻⣿⣷⣜⢦⢻⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢳⡍⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣿⣿⣿⣮⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣻⢻⡟⠿⣯⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣴⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠎⣿⣾⢹⣟⣇⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡘⣿⡄⣿⣽⣄⠀⠐⠛⠉⠀⣠⣄⣠⣴⣾⠿⣿⣿⣿⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⡇⢿⣇⢸⣿⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⡿⣿⣷⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡔⣿⣜⣿⡞⣿⢿⣿⣿⣿⣿⣿⣿⡿⠃⢀⠤⠊⠀⢸⣿⢟⣿⣇⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⠘⢷⡹⣿⣿⡛⠛⠉⠀⠀⠀⠀⠀⠀⣾⣿⡆⢿⣿⣿⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡈⢣⠹⡻⡿⢦⡀⠀⠀⠀⠀⠀⣰⡟⡄⣧⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢿⠻⠛⠂⠖⠁⠉⠚⠛⠀⠀⠀⠀⢀⣿⡇⣷⢻⡈⣿⡾⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⡳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢶⣽⣧⡇⡿⡇⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠑⢤⡖⠀⡀⣤⠀⠀⠈⢿⣿⠂⡷⠃⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠈⠀⢹⡿⢐⢠⣄⣀⡈⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡤⡦⡀⠀⠑⠠⢀⡈⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⢹⣿⠁⠀⠀⢠⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣇⢸⡏⢀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⠇⠀⠀⠀⢠⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀

⠀⢠⡤⠤⠤⢤⣤⠠⠄⠠⠤⠤⠤⠤⠠⠤⠤⠠⠤⠄⣤⡤⡤⡤⠤⠤⠄⠤⠤⠤⡤⡤⢤⡤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⢠⣤⢤⠄⠠⣤
⠀⢸⠁⠀⣰⠞⠉⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠄⠁⠁⠂⡀⢀⡀⠠⠀⠐⠀⠀⠑⢄⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⢧⡀⠀⢿
⠀⢸⠀⠀⢣⠀⢀⠏⠙⠍⠐⠢⠤⣤⣤⠴⠶⠛⠉⠀⠀⢐⡁⠀⠀⠉⠒⠐⠁⠀⠀⠐⣔⠀⠀⠑⠪⠵⣤⣤⣄⣤⡤⠔⠺⠉⠀⠀⢨⠀⠀⢸
⠀⢸⡇⠀⠸⣧⠊⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣄⣾⠀⠀⢸
⠀⢸⣇⡴⠃⠇⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⠔⢁⡤⡠⣏⡒⠄⠀⠀⠀⠀⠀⠀⠀⠰⣉⠇⣶⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⡀⠈⡄⢣⣀⣿
⠀⢸⣉⡱⠞⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡾⠁⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠑⡌⠀⠈⢿⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣏⣹
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠖⠉⣰⠏⠀⠀⢠⠿⠁⠀⡀⢀⡀⡀⠀⠀⠀⠰⡀⠀⠀⠻⣌⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⢠⡾⠁⠀⠀⣠⢟⠇⠀⢀⣀⣸⣇⡇⠀⠀⠀⠀⢣⠀⠀⠀⠙⣷⡀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠀⢀⣴⠋⠀⠀⠀⣰⠏⣾⠴⠚⠌⠁⠀⠀⠘⡄⠀⠀⠀⠀⢇⠀⠀⠀⠈⢻⣆⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⣠⡶⠟⠁⠀⠀⣠⠞⠁⠀⠀⠀⣰⢏⡜⠁⠀⠀⣀⡤⠄⠒⠒⠛⠆⠠⡄⠀⠈⣇⠀⠀⠀⠀⠙⣿⣄⠀⠈⠙⢦⣄⠀⠀⠀⢸
⠀⢸⠀⣠⣴⠟⠉⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⣴⢃⠎⠐⠀⡠⢚⡡⠀⠀⠀⠀⠀⠐⢆⡸⠀⠀⠘⣆⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠉⠻⢦⣄⣸
⠀⢸⠟⠋⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡀⢀⡾⠃⢾⠀⠀⢊⡜⠁⠂⠀⠐⠶⠞⠂⠰⡾⠁⠀⠀⠀⠘⣆⢀⠀⠀⠀⠀⠙⢿⡳⣄⠀⠀⠀⠉⢻
⠀⢸⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢀⡴⢁⡾⠁⠀⠘⢄⢸⣈⠘⠿⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⣆⢧⡀⠀⠀⠀⠀⠱⣌⢳⣄⠀⠀⢸
⠀⢸⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⠟⣠⡟⠁⠀⠀⠀⢀⣿⠹⣄⠁⠀⠀⠐⠆⣄⠾⣿⣧⡆⠀⠀⠀⠀⠀⠘⠈⢷⡀⠀⠀⠀⠀⠈⠢⡙⠳⣄⢸
⠀⢸⣄⣴⡟⠁⠀⠀⠀⢀⡼⠃⣰⠟⠀⠀⠀⠀⠀⣸⠇⠀⠘⡏⠀⢄⡀⠈⠀⠀⠉⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠈⠀⢈⣻
⠀⢸⡅⠀⠈⠉⠒⠆⠴⠿⣄⣾⣏⡀⠀⠀⠀⠀⢠⡿⠀⠀⣠⠏⠠⡀⠈⠙⠋⠐⣖⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⣆⡀⠤⠄⠒⠚⠉⠁⢹
⠀⢸⠉⠒⠂⠠⠽⠥⣀⡀⢀⣤⠀⠈⠉⢋⠉⠤⠛⠓⣶⠞⠁⡄⠀⠀⠁⠠⡀⣠⠊⣣⣄⠄⠲⢖⠖⠒⡂⠉⠉⠛⣤⠁⠈⣂⢀⡀⠿⠤⠒⢺
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠂⠛⠀⣈⠹⠴⠁⠀⠀⠚⡀⢉⠀⠰⡏⢡⠀⠀⡷⣤⠤⠽⠤⠄⠗⠐⠒⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠁⠀⠐⠀⢠⠋⠀⠀⠸⣿⠀⠶⠛⢳⠆⠤⢳⠈⠂⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠱⢄⠀⢀⠌⠀⡌⠀⠀⠀⠀⠘⣆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠓⣠⠆⠰⠁⠘⠀⠀⢠⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡀⠀⠀⠈⡀⠀⠀⠃⠀⠀⠈⡖⠀⠀⠀⡄⠀⠀⣄⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⢿⠠⠀⠀⠀⢱⠀⠀⠸⠀⠀⢠⠄⠀⠀⠀⡇⠀⠀⡗⢠⠇⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣼
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢣⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢰⠃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⢼
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⢆⠀⢠⠀⠀⠀⠀⠀⠀⡇⠀⡜⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⢀⠟⣇⠀⠀⠀⠀⠀⠀⠈⠢⢘⠀⠀⠀⠀⠀⠀⡇⢰⡇⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠠⣠⡟⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⢀⡷⢸⡇⠀⠀⣼⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⡆⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⠀⠀⠀⡇⢸⠃⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠹⡄⢸⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⠀⠀⠀⡇⢸⠂⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢳⡼⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢰⠀⠀⠀⠀⣗⠈⡆⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⣆⠀⠀⠀⠘⡄⢹⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⢀⠸⢆⠀⠀⠀⠀⣠⠎⠀⠀⢀⠄⠉⢃⠀⠀⠀⢡⡞⡄⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠂⠀⠀⠈⢿⡒⠊⠉⠀⠈⠙⢦⡌⠀⠀⠈⡄⡠⠞⠁⢻⡐⡇⠀⠀⠀⡀⢳⡀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠈⠀⠀⠀⠈⢇⣻⠀⠀⠀⢀⠈⢷⡄⠀⠀⠀⠀⠀⢸
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢀⠠⠄⢷⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⡀⠀⠀⠀⠈⢿⡀⢚⡀⠰⠄⠈⡅⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⠿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠻
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠉⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣶⠋⣠⠙⣶
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡉⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦
⠀⠀⠀⠀⢀⣴⣶⣿⣿⣶⣿⣶⣿⣷⣶⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣷⠀⠉⠉⠀⣼⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡁⠀⢨
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡉⠀⢈⡀
⠀⠀⠀⠀⢿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠿⢶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠖⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⢷⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠠⠄⠐⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠅⠈⠉⠅⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⣀⣷⢾⣆⡀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢰⣦⣤⣀⣠⣤⣶⡄⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⣀⣠⣼⠃
⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠸⠟⠿⠿⠿⠟⠻⠃⠀⠀⠀⠀⠀⠘⠀⠘⠀⠘⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⣴⠶⠷⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣖⡀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⣶⡀
⠀⠀⠀⠀⠀⡁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⣿⡆
⠀⠀⠀⢀⣤⣴⠑⡂⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣒⣀⣒⣄⣤⠀⠀⠀⠀⠀⣠⣼⣿⣶⣾⣯⡄
⠀⠀⠀⢸⣿⣿⣾⣧⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣦⣴⣶⠀⠀⠀⠀⠀⠀⠛⢓⣒⣒⣒⣚⠛⠁⠀⠀⠀⠀⡉⣿⣷⣿⣷⡞⠋
⠀⠀⠀⠈⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠴⠖⠒⠦⠤⢤⡀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣴⠶⠯⣭⣤⣤⣄⣀⣀⣠⠴⣲⣿⣛⣋⣉⣀⣀⣀⣀⠉⠉⠉⠉⠉⠛⠒⠒⠲⣤⡄
⠀⠀⠀⠀⡇⡐⠒⠒⠒⠒⠒⠠⡀⡿⢸⡇⠀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⡉⠉⠛⠓⢲⡏⠀⡇
⠀⠀⠀⠀⡇⡨⠭⠭⠭⠭⠉⢉⠀⡇⢸⠀⢀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⡷⠀⢸⠁⢸⠟⡆
⠀⠀⠀⠀⡇⠑⠒⠒⠒⠒⠒⠊⢠⡇⢸⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡾⠀⢸⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡇⠀⡟⢰⠇
⠀⠀⠀⠀⣧⠀⠘⠋⠒⠒⠂⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠃⢀⡇⣸
⠀⠀⠀⠀⢻⠀⠀⠀⣀⠀⠀⠀⡿⠀⢸⡀⠘⠒⠒⠶⠶⠦⠤⠤⠤⠤⠤⣤⡼⠀⣸⠀⣸⠖⠃
⠀⠀⠀⠀⢸⡀⠀⠀⠒⠁⠀⠀⣇⡴⠋⠛⠓⢲⢖⡶⠒⣶⡶⠶⠤⢤⡤⢤⡤⣤⡿⠞⠁
⠀⠀⠀⠀⠀⠉⠉⠙⠛⢛⣛⡛⠉⠀⠀⠀⡊⠁⠸⣧⣤⣀⣉⣉⣉⣉⣤⡾⠀⠈⠒⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡼⡩⢭⢭⠭⡭⢭⢉⡉⣛⣓⡒⠛⠻⠿⠯⠭⣭⣁⣀⠀⠀⠀⠀⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠀⡼⡰⢥⠯⡯⣹⣩⣏⣞⣱⡓⡞⢺⢡⢮⠏⣴⣰⠂⡰⣹⠀⠀⣠⠖⣻⠟⠙⣆
⠀⠀⠀⠀⠀⠀⢸⣥⣍⣉⣚⡒⣓⣒⡒⠀⠧⠼⠥⠃⠎⢞⣲⣲⢃⢜⡴⠁⠀⡼⠉⠉⠉⠒⢤⡟
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠒⠒⠾⠿⠭⠭⢤⣄⣒⣲⡾⠋⠀⠀⠀⢷⣄⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠻⠿⠿⠿⠟⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣀⠀⠠⠀⠙⢿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠃⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠈⠻⡃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢫⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠶⠟⠛⠋⣉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣷⡀⠀⠀⠀⠀⠀⠉⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢠⣾⣿⣿⣿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠞⣻⠟⠉⢡⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣡⣾⠏⠀⠀⣾⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣼⣿⡏⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⡹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣏⠀⣠⣾⣿⣿⣿⡶⠶⠿⠿⢷⣶⣶⣤⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣷⡽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠿⢿⣿⣶⣿⣿⣿⠟⢉⣀⣠⣤⣤⣤⣤⣈⡉⠉⠁⠀⠀⠀⠀⢸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⢀⣴⣿⣿⣿⣿⣿⡿⠘⢿⣿⣿⠟⢻⣿⣿⣿⣷⡄⠂⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠛⠿⠃⠀⠀⠉⠻⢶⣼⣿⣿⣿⣿⠿⠀⠀⠀⠀⠐⣿⣁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠶⠆⠙⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠿⠃⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠝⠻⢿⣿⣷⣆⡀⠀⠀⠀⠀⠲⢶⣶⣶⣶⣶⣦⠄⠀⠀⣰⣿⣿⣿⣿⣿⡿⠟⠁⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⣿⣷⣦⣀⠀⠀⠀⠀⠙⠶⠶⠞⠁⢀⣠⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣄⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣠⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⣴⣿⣿⣿⠿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣻⣿⣿⣿⣠⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣩⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢛⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⡇
⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠿⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⣿⣿⣿⣿⣦⣸⠏⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠉⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿
⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢸⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢷⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠶⠒⠲⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢀⣿⠀⠀⠀⣸⠀⠀⠀⠀⠀⠴⠂
⠀⠀⠀⠀⢀⠄⢥⡀⠀⠀⢀⠄⢀⣤⠀⠀⠀⣀⣀⠀⢀⣠⣤⣤⡀⠀⠀⠀⢠⣾⣿⡿⢿⣦⣤⠴⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠔⠈⠀⠀⠈⠁⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠛⠟⠹⡀⠀⠙⠁⡀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢠⢠⠛⠋⣻⣿⣿⣿⣿⡇⠹⠷⠀⠀⠀⠀⠐⠃⠀⠀⣀⣦⣀
⠀⠀⠀⠀⠀⢰⠇⠀⢠⣾⣿⠿⣿⣿⣾⣦⣤⣴⣾⡟⠿⣿⣿⣇⠀⠀⠀⠀⢀⣀⣀⣀⠀⠺⠟⠛⠛⠻⡄⠀⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠙⠻⣿⣿⣿⣿⣦⣤⡈⠙⠿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⠤
⠀⠀⠀⠘⠦⠀⠀⠀⡿⡇⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣦⣤⡄⠀⠀⠀⠀⡀⠀⠀⠠⣠⣴⣶⣶⣤⣤⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠦⡧⠀⣀⠀⠀⠀⠉⠉⠛⣿⣿⣿⡄⠀⢠⡀⠀⠀⠀⢀⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠉⠉⣵⣶⣿⣦⡀⢠⡿⠁⢀⣶⣶⣤⣤⣼⣿⣿⡇⠀⠀⠁⠀⠀⠀⠐⠀⠐⢿⣿⣿⣿⣿⣷⠛⠛
⠀⠀⠀⣠⣤⠀⠀⠀⣾⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠁⠀⠀⢠⣶⣶⣶⣶⣿⣆⡀⠙⠻⣿⣿⠟
⠀⠀⠀⢻⣿⣧⡀⠰⡟⠋⠁⠀⠀⢀⣀⣀⣀⠀⠀⠀⣤⣶⡀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠘⢿⣿⣿⣷⣤⣀⣀⣠⡄
⠀⠀⠀⠀⠈⠉⠉⠉⣴⠀⣦⣀⣰⣿⣿⡿⠇⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣠⣭⣭⣥⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⡇⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢀⣀⡀⠠⠿⠿⢿⡿⠿⠛⠓⠀⣀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠠⣄⣿⡄
⠀⠀⠀⠀⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠠⢿⣿⣷⡄⠀⠀⠀⢀⡀⠀⠀⠶⢿⡤⠀⠀⠀⠈⠁⣴⣤⡀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠉⠛⠿⠿⠿⢇⡀⠽⣤⣤⣶⣶⣿⠿⠋⠀⠀⠶⣿⣿⣁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣙⣿⣦⠀⢀⣤⠈
⠀⠀⠀⠀⣰⣶⣶⣶⡆⢸⡄⠀⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⢌⠛⢿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡿⠿⠿⠟⠋⢀⣾⡟⠀⣠
⠀⠀⠀⠀⠛⡛⠛⠛⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⡰⠂⠀⠀⣰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣶⣶⣶⣶⣶⣾⡿⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠙⠛⠋⠉⠉⠛⠁⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠚⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠢⠤⠤⢆⡀⠂⠀⠀⠀⠀⠂⠐⠢⠤⠤⠤⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠤⣄⣀⣀⠀⠀⠢⢀⠀⠀⠀⠁⢀⣀⣠⢤⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡴⠊⠀⠀⠀⠀⠢⡈⢂⠄⡀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⢰⠃⠀⠀⢠⣸⣷⡄⢸⡎⢀⣾⡟⡄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⢘⣇⠀⠀⠈⠻⠟⠋⠰⠆⠸⠛⠟⠃⠀⠀⢠⡇⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠲⢤⣀⣀⡠⠖⣇⣠⠲⢤⣀⣈⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠤⣄⠀⠰⠀⠄⠠⠈⠀⠁⠁⠐⠀⠀⠀⠐⢀⠠⠊⠑⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠼⠜⠚⠄⠠⠠⡀⠠⠄⡀⠀⡄⠀⠄⡤⠀⠠⠓⠢⠧⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⡐⠀⡄⣄⢀⡄⠑⡄⠍⢠⠀⠁⠀⠄⠨⠉⢀⠬⣰⣀⠀⢠⣀⢖⠙
⠀⠀⠀⠀⠀⠀⢠⣀⢀⠁⠁⠔⠎⢀⠠⠄⢠⡀⠄⢀⠀⠀⠀⠄⠠⢠⢠⠱⠀⠈⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠓⢀⠀⠀⣀⢀⠔⠀⢀⡀⠡⠄⠁⢀⡆⠀⢆⢸⠀⠀⠀⠾⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⣆⠀⠂⠈⠀⠁⢀⡀⠃⠀⠀⠐⠀⢀⠄⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠃⠀⠐⡀⢐⢀⡙⠀⠑⠢⠘⠚⢑⠚⠁⠒⢠⠀⠀⠘⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠀⡘⢠⡇⠈⡀⠝⢀⡌⢀⡀⢀⣀⢭⢀⠣⢼⣆⠃⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠨⢁⠂⣡⡘⠆⢡⠂⣤⣄⢐⠀⣤⠑⡄⠅⢀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠉⠳⠿⣽⣶⣤⣄⣺⣷⠶⠝⠉⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢲⢹⠙⡇⠆⠉⠉⠰⠈⡏⡟⠖⢠⡀⠀⠀⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠸⠼⠴⠃⠁⠀⠀⠈⠐⠧⠇⠆⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣼⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡟⠃⠀⣶⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⡀⣿⡇
⠀⠀⠀⠀⠀⠘⠛⢻⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⣿⡇⣄⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣥⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠁⢽⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠅⣀⠀⠙⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡺⣿⠇⢀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⣄
⠀⠀⠀⠀⠀⠐⠶⣿⣿⣿⣿⠷⠶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣀⣀⣠⣤⣴⣷⣶⣶⣶⣤⣭⣿⣯⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣛⡦
⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⠙⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⠿⠇⠀⠀⠀⢀⠀⠀⠀⠀⢿⣦⠀⠈⣿⠋⠙⠛⠋
⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⠉⠀⠀⠀⠀⣀⠐⠒⠈⠀⠀⣠⣶⣿⣶⣭⡿⠆⠈⡿
⠀⠀⠀⠀⠀⠬⠿⢿⣿⣿⣿⣿⣿⣍⠋⠀⠠⠀⠀⠀⠀⠈⠂⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⢻⡽⠿⠻⠄⠀⠀⢀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠟⠁⠀⠀⠀⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣠⠀⠀⠀⠁⣀⣴⣿⠟⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠿⣶⣢⣴⣾⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠈⢻⣿⡟⠁⠀⠀⠀⠀⠠⠤⠤⣤⣤⣄⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠉⠉⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠟⠀⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⠀⠀⠀⠀⠀⣠⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⣴⡏⠀⡠⡄⠙⡆⠀⠀⣠⣾⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠉⢧⡜⠈⡇⠀⠳⠤⠴⠷⢶⡾⠁⠈⠳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⡀⠀⠀⢀⣧⠀⠹⡄⠀⠀⠀⣰⠛⣆⠀⠀⣸⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠖⠃⠀⠀⠈⡟⣆⠀⠙⢦⠀⢼⡁⠀⠈⠙⢲⡇⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠹⡌⢧⡀⠈⠙⢦⣉⠓⠦⣤⠞⠀⠀⣀⣤⠾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠐⠀⠘⢦⣉⠳⢤⣀⡈⠉⠙⠓⠛⢛⣉⣡⠤⠖⢻⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢬⣉⣉⣛⣛⡭⠭⠷⠖⠲⠖⠉⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⢰⠞⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡤⠴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣴⢦⠀⠀⠀⠈⠁⠀⠈⠻⣆⠀⠀⠀⠀⢀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠉⠙⠲⢄⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣸⠁⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣀⠔⠒⠋⠙⠛⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⣠⣼⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣠⡴⠞⠁⢸⠀⢰⠇⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⣇⡇⠀⠀⢸⠀⢸⡉⡭⣍⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢷⠀⠀⢸⡄⠈⠳⠶⠛⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠳⢦⣴⣇⡀⠀⠈⢧⡀⠀⢳⡀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠳⣄⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠒⠒⠛⣩⠥⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠓⠦⠤⠤⠀⠀⠀⠀⠀⣼⣀⣀⣀⣀⡃⢋⡷⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠉⠉⠉⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⣀⡀⠀⠀⠀⣀⣠⣞⠳⢤⣤⣀⣀⣀⣠⡤⠴⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠒⠒⠒⠛⠉⢸⡏⠉⠛⣛⠉⠁⠀⠀⠈⠉⡿⡄⠈⠳⢤⣀⡀⠉⠀⠀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢀⡼⠁⠀⠀⢸⡀⠀⠀⠀⠀⢠⠇⠙⢦⣀⠀⠈⠉⠉⠉⠉⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠛⠁⠀⠀⠀⠈⠳⣄⣀⣀⡴⠟⠀⠀⠀⠉⠓⠲⠶⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠰⠷⠂⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⣀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⠾⠶⣶⣤⡀⠀⠀⠀⣠⣤⡶⠶⠶⠖⠶⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠻⣉⢉⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⢉⡹⠟⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠄⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⡇⠀⠈⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢣⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣄⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢺⡿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣄⠀⠀⠈⠉⠀⠀⣀⡴⣿⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠉⠑⠲⢶⣶⡶⠞⠋⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠰⠊⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⢱⠀⠀⠀⠀⡇⠀⠀⢸⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⢀⠀⠀⠀⠀⠀⢀⡀⠈⠋⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠁⠀⠀⢤⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠐⠛⠏⠀⠀⠸⣿⡇
⠀⠀⠀⠀⢠⠀⠀⠀⢀⡀⠀⠀⡠⠀⠰⢀⡈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠆⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠰⢿⡿⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢀⠀⡀⢈⠠⡀⠀⠢⣀⠁⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⢦⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡁⠐⠁⠠⠄⠀⠀⠈⠀⠀⢈⠠⠀⠀⠀⠄
⠂⠀⠀⠀⣤⡄⠠⠄⠀⠀⢢⡐⠀⠁⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡄⠀⠐⠀⠀⠀⣠⣾⣦⠀⠀⠐⠒⠃
⠀⠀⠀⠰⡿⠃⠞⠀⣤⣀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⡿⠀⠀⠀⠔⠑⠂⠈⠛⠋⠀⠀⠂⠀⠠⠀⠆⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⡇⠀⠀⠈⠃
⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠆⠀⠘⠳⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⣶⣄⠀⠀⢀⠐⠋
⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠐⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀⢀⠈⠉⠁⠀⠀⠈⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠐⣄⡀⢄⣤⠞⠀⠀⠀⠀⣀⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠈⠳⡀⠖⢠⣶⣦⣄⣀⣀⣁⣘⠙⡀⣠⠀⠴⠉⠀⠀⠀⠀⠀⢈⢣
⠀⠀⠀⠀⠀⠀⠀⣿⣄⣠⠀⠀⠀⠀⠉⠶⣿⣿⡿⠿⠿⢿⣿⣿⣾⣿⣿⠇⠀⠀⠀⠀⢠⣰⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠋⢽⣿⣶⣦⡤⠤⠄⢠⣿⣡⡀⠀⠠⠾⠿⠟⠛⠋⠀⠀⠀⢠⣩⣿⣿⡿⣿⣿⣏⡀
⠀⠀⠀⠀⠀⢠⣞⠀⣠⣤⣿⣿⣶⠀⠀⣩⣤⣤⣄⠀⠀⠀⠀⠀⠀⣉⣛⡲⢄⠀⢹⣿⣿⣄⣔⣌⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠃⢺⣿⣇⣤⡞⢁⣠⣄⡉⢧⠀⠀⠀⣴⡋⠉⠙⠻⣦⡄⠀⣿⣿⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⢻⡇⢾⢠⣿⣽⣿⣿⠀⠀⠀⢸⣧⣿⣆⠀⠀⢹⣟⠀⣿⠯⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⡰⠁⠸⣌⠘⠿⠿⡻⠀⠀⠀⢸⠿⠟⡌⠀⠀⣼⠋⢰⠃⣴⠊⠆⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢾⡇⠀⠀⠈⠛⠓⠊⠀⠀⠓⠀⠀⠐⠪⢤⡤⠾⠃⠀⠘⣦⣙⠔⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠑⠶⣶⠆⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠉⠙⢓⣲⠶⢤⣤⣤⣤⣤⠤⠤⠤⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠂⣷⢘⠁⣠⡬⠤⡆⠀⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠼⠂⠄⠤⠒⠋⠈⠀⠀⣅⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠤⠄⠀⠀⠀⠤⠐⠂⠈⠁⠀⢸⠋⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠶⡧⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠠⠤⠒⠚⠛⠛⠓⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢦⣄⡞⠉⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠛⠶⣄⡴⠖⠛⠛⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⢸⣧⣤⣀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣴⠋⠀⠀⠈⢷⠒⠺⢧⡀
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣀⣸⣇⣀⡀⣀⣤⠤⣯⣀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠿⣄⠀⠀⣀⡾⠀⠀⠀⢳
⠀⠀⠀⠀⢀⡼⠗⠲⠤⣤⡞⠉⠀⠀⠈⠙⢧⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠈⠙⢿⡉⠀⠀⠀⢀⡾
⠀⠀⠀⢰⠏⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠋⠀⠹⣦⣀⣀⣀⣠⠟⠲⠶⠖⠋
⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡟⠦⣤⣠⡤⣞⠁⠀⠀⠀⠀⣠⠏⠀⠀⣠⠟⠁⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠙⢦⣄⣀⣠⣾⠀⠀⠀⠀⠀⠈⢻⡶⣶⠒⠋⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⣀⣤⡤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⣀⡾⠁⠘⢧⡀⠀⠀⢰⠏⠀⠀⠀⣀⡴⠞⠉⠁⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣛⣛⠛⠉⠀⠀⠀⠈⢷⡀⠀⡿⠀⠀⣠⠞⠉⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⢠⣶⠛⠉⠉⠉⠉⠉⠉⠛⠲⢤⣀⠀⠈⣧⢸⡇⢀⡞⠁⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠈⠳⣤⣸⣾⣱⠏⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⣀⣀⣀⣹⣿⠃⠀⢀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠿⠋⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⢀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⠹⣿⣟⣿⢻⣋⣿⣯⣽⣅⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡯⠁⠁⠉⠈⠉⠙⠋⠉⠻⢿⣿⣇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣮⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣽⢶⣿⣿⣶⣤⣤⣤⣾⣿⣿⣤⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣩⣿⣿⣿⠇⣃⡀⣿⣿⣿⣿⣿⣿⡇⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⡟⠿⠃⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢘⠉⢉⡙⠃⠀⠀⠀⠀⠀⠉⠉⠉⢠⣿⣉⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣿⣆⠀⠀⠀⠀⠸⣷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣶⣤⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢸⣷⣄⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢸⣿⣿⣿⣦⣤⣾⣷⣶⡄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⢀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣭⠖⡂⠥⠤⣤⣞⣽⣭⣙⠛⠓⠶⣦⣉⠳⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡟⡐⡠⢒⢁⣴⡶⠿⠿⢟⣿⣷⣦⣄⡒⢽⣷⡵⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣿⠐⢀⣵⡵⠋⡁⠠⢀⣙⡼⠿⠿⢿⣿⣿⣿⣿⣿⣧⣄⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡵⡿⢹⡀⢸⣯⣖⡥⠶⠛⠋⠉⠁⠀⠀⢻⣿⣿⣿⣿⡿⡿⡿⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⣪⠞⣡⣿⣇⢸⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣯⡻⣍⠪⣀⠈⢪⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⢗⡺⣟⣿⡟⣬⢉⠉⠀⠀⠐⠛⠛⠓⠶⣄⠀⠹⣿⡿⣌⠀⠩⡈⠑⢄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⢱⢋⣾⣿⣿⠞⠃⠘⠟⠀⠀⠀⠀⢰⡶⡦⠀⠀⠀⢹⣿⣌⢷⡐⠙⣗⢌⠢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡸⡘⡌⢿⣿⣿⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⣷⡑⡙⠢⣌⣒⣵⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠱⡜⡌⡇⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡸⣿⠙⣎⠲⣤⣭⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⠇⠇⠇⣧⠀⠄⢀⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣧⡿⣷⣤⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠈⠀⡜⣸⣿⠀⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⣜⣼⢰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⡿⢯⠃⣼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢬⣈⠢⢽⣿⣿⣦⣤⣀⣀⡀⠤⠄⠀⠂⠈⠀⠀⠀⢸⢣⢸⣼⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣺⢿⣿⣷⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢦⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣡⠊⢁⢜⢽⢻⢿⣿⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠹⣦⣁⡈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢃⣴⡵⠋⠀⠀⣪⣿⡇⣿⡿⣿⣿⠀⠀⠀⡠⠐⠀⠀⠀⠀⠀⠘⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢋⡡⢀⡀⠠⢊⡵⢫⡇⢹⢧⡿⢸⡆⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣾⡏⡔⣿⠁⢡⡿⢡⠟⢠⢏⡞⡅⣾⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⢿⣧⡡⡹⣄⢸⡧⠋⡰⢡⢎⡜⣼⣿⡇⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣝⢿⣿⣿⠘⠁⡼⠡⢡⣾⡾⣫⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⢿⣇⢰⢠⡇⡆⡾⢡⢞⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠞⠙⢿⣆⠃⠃⢣⠈⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣷⡀⣆⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠩⠼⠓⡽⠔⠉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢰⠀⠀⠀⢀⢠⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠑⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡼⢃⠀⠀⠀⠊⠻⠶⠦⠤⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠰⣒⢏⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠢⠀⢤⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢀⠚⠑⠠⠀⠠⠖⡑⠉⠁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢐⣡⡄⡀⠂⡐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⡀⠀⠀⠀⠁⡀⢤⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣷⣶⡄⠀⢀⡶⡆⠀⠀⠀⠀⠀⠈⠛⠲⠄⡀⣀⠀⠀⠻⠗⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣤⡔⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⡛⣻⣽⠀⢩⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘
⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠘⠋⠈⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⣀⣦⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠗⠐⠎⠒⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⢀⡀
⠀⠸⠡⠦⠀⢀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣠⡘⠿⠀⠀⠀⠀⠀⠀⠀⡠⠊⢃⡀⠈⠐⠁⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸
⠀⡸⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠰⠤⡆⠀⠀⠀⠀⠀⢈⠀⠂⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣼
⠀⠀⠘⡆⠀⣶⠡⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠔⠁⠀⠀⠀⠀⠀⠀⠁⠢⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠛⠀⠀⣿⣿
⠀⠜⠉⠔⣂⠩⠒⡻⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣀⣠⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⡀⠀⡘⠀⠀⠠⠓⡆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿
⠀⣏⠀⠈⠈⠠⠐⠈⢠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢸⣿⡏
⠀⢸⠱⠊⠀⠀⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡇
⠀⠀⢢⣄⠀⠀⠀⢀⣿⡉⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠈⣿⠇
⠀⠀⠀⠛⠿⣷⣶⣾⣿⣿⣾⣿⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⣿⠋⠒⢁⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠘⠉⢩⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⣀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠉⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣀⡀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠉⢢⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢣⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⢳⠀⠠⠂⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡘⢆⡀⠀⠀⢮⠀⠀⠀⣀⣀⠤⠤⠤⠬⠷⠤⣤⣀⣀⣀⡂⢸⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠱⣀⠈⠉⣢⡀⢳⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⢤⣄⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠖⠓⢌⡶⠊⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠂⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢸⠀⠀⠀⠋⢀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⠗⠀⠀⠛⠋⠁⠀⢠⣶⠀⠀⠀⢀⠊⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠶⠟⠁⠀⠠⠐⣡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡷⣜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡼⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠒⠈⠉⠀⠀⠀⠀⣸⢳⣄⡐⣏⢣⡀⠀⠀⠀⠀⢹⣌⠪⣄⣀
⠀⠀⠀⠀⢀⡠⠄⠒⠒⠒⠒⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠉⠙⡆⢳⠒⠒⢞⠯⡁⠈⠙⢲⡌⡇⠀⠀⡴⢳
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣢⠴⠒⠃⠀⠀⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠧⠞⠀⠀⠈⣗⡼⠀⠀⠀⠉⠀⠀⠸⡀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠤⠷⣄⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡒⠤⣀⠀⢀⠜⠉
⠀⠀⠀⢀⣀⡠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠐⠒⠤⣀⠀⠀⠀⠀⢰⠢⢄⡀⠀⠀⠳⣀⠈⣳⠟⠒⠒⢢⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠈⠉⠙⠂⢲⠉⠉⠁⠁⠒⠤⣀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⢇⠀⠈⠓⢤⡀⠀⡹⠉⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡟⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠈⡆⠀⠀⠈⢦⠀⠀⠀⠙⡼⠓⠀⠈⠁⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡜⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⣰⠋⠓⠤⣀⠀⠙⠒⠒⡺⠱⢄⣀⢀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠀⢧⠀⠀⠀⠙⠢⡀⢰⡁⠤⠤⠤⠤⢤⠄
⠀⠀⠀⠀⠀⠀⢤⣄⡀⢠⠤⠔⠊⠁⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠳⠄⠀⠀⠀⠸⠏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⢀⣤⣶⣿⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣤⣄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⢸⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠿⠇
⠀⠈⠉⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢹⣿⣿⣿⠇
⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⣤⣶⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠸⣿⠟⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⢿⣿⣿⣿⠛⠋⠉
⠀⠀⠀⠀⠹⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⣠⣤⣤⣤⣴⣶⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣆
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠹⡿⣿⣿⣿⣿⠁⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠠⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣛⡛⠛⢻⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢋⣵⣾⣿⣿⣾⣭⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣵⠟⠉⠙⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠞⣳⣶⡤⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠉⠁⠀⣿⠉⠙⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢉⣷⡀⢀⣤⣠⣿⣄⣾⣿⣿⣿⣿⢣⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⢿⣿⣟⣉⣿⣿⣿⣿⣿⣿⡿⠁⢁⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⢋⡻⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣄⠀⢿⣿⣿⣿⣿⣿⣿⢟⡭⣢⣾⣿⣿⣫⣶⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⣿⣧⣻⣿⣟⢿⣿⣿⠁⢁⣼⣿⣿⡿⣿⣿⡿⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡌⢹⣇⣴⠿⢟⠽⠋⠜⠋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣿⣿⣧⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠠⣶⣺⠂⠀⠀⢈⣷⣾⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡄⠀⢰⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠐⠻⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢽⠿⠿⠙⢋⣽⡷⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⣀⡀⣾⡿⢩⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠶⠿⠆⠘⠃⢉⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⢿⠟⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣛⠿⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠈⡇⡲⢴⢰⡇⠀⢸⣷⢰⡇⢠⢶⠠⣄⠆⠛⠐⡖⠀⢰⣙⠆⡮⠀⢰⣿⠀⡆⢠⢶⡰⣋⠀⠶⡀⠀⣶⡀⠀⣦⢦⠆⡦⠤⢰⠘⡇⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⣴⣿⣥⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠹⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠂⠀⠀⠀⠀⠀⠀⢠⣦⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠿⢀⠀⠀⠀⢀⣹⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠛⠛⠛⣿⣿⣧⡀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠃⠙⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡧⡀⠀⠠⡘⢻⣿⣿⡆⠀⠀⢀⣿⣿⣿⣿⠛⠙⠛⠿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡆⠁⠀⠀⠁⢸⣿⣿⡀⠀⠀⠈⢿⣿⣿⠁⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⠄⠤⢠⣿⣿⣿⡇⠀⠀⠀⠈⠛⢿⠀⠈⠁⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠀⠀⠈⠙⠹⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣷⣿⣷⣦⣴⣤⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣶⣤⣄
⠀⠀⠀⠀⠀⠀⣀⣛⣿⣿⠿⢿⠿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⢿⣿⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠉⠀⠈⠻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣽⣿⡏⢀⡀⠀⠀⡀⢻⣿⣯⡅⠀⠀⠀⢀⣴⣿⢿⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠠⠄⠀⠠⠈⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠉⣹⡇⠈⠃⠀⠀⠋⠘⣿⡖⠂⠀⠀⠀⠸⡿⠁⠈⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠈⠀⠀⠀⠁⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⠀⠠⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⢧⠀⠀⠀⠀⠀⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢌⠳⡄⠀⠀⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠘⣇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠠⡀⠠⠀⠀⡠⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢊
⠀⠀⠀⠀⢀⠈⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡊⠀⠀⢅⠠⠀⠀⠀⢀⡒⠁⢠⠐⠈
⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠠⠀⠐⠀⠀⠀⠥⣀⠀⠀⠀⢝⠀⠁⠀⢀⡡⠤⠄⡌⡠⡁⢀⠌
⠀⠀⠀⠠⣢⢖⠔⠀⠀⣀⠀⢀⠀⢀⠐⠷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠤⢀⡀⠀⠑⡂⠁
⠀⠀⠐⠀⠁⠀⠀⠀⠀⠋⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠈⠈⠀⠂
⠀⠀⠀⠀⠂⡊⠀⠀⠀⠀⠠⠄⢰⠀⢂⠀⢁⠀⠁⠀⢀⢀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠂⠴⠀⠁
⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣁⠈⡅⡐⢀⠠⠀⠁⠀⡀⡀
⠀⠀⠀⠀⠀⢠⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠡⠤⠀⠀⠀⠀⡤⣂⡂⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠠⠀⠀⠰⠆⠤⠤⢀⡄⠀⠉⠁⠠⠀⠠⡄
⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⡄⠐⢆⠡⡨⠆⠂⠠⢄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠂⢴⠋⠀⣠⠄⠛⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⢀⠰⠆⠁⠀⠀⠀⠀⡅⠀⠀⠀⠀⢀⡀⠀⠐⣎⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠘
⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⡗⠁⡨⠉⠉⠓⢄⠐⠈⣈⠀⠐⠁⠀⠀⠀⠀⠀⢀⠀⡘⡔⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠔⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣿⣇⣀⣏⡸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢨⠇⠀⠀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣏⣌⠥⠀⠈⠀⠀⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠈⠀⢸⣿⣿⡿⠿⠿⠛⠛⠉⠁⠀⣀⣠⡤⠀⠀⠀⢀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀⣤⣤⣶⣾⣿⣿⡿⢃⣗⠀⠄⢼⡄⢃⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢰⠀⢀⣄⣾⠀⠈⠁⣤⡀⣀⡭⠴⣙⡻⣿⣿⡿⡟⢿⠿⣿⡟⠋⠁⢸⣿⠀⠂⣿⡇⢠⡄⠈⠑
⠀⠀⠀⠀⠀⠀⢀⣿⡀⣼⠿⠓⠁⡠⠟⠪⠴⠀⠩⠈⢁⣄⢂⡡⠈⠉⠉⢠⠀⢀⡄⠁⠊⠲⠄⢄⣠⠈⠙⠛⠻⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠸⡟⠁⠡⠀⠀⢊⠑⢠⡆⡀⠠⣀⠀⠈⠁⠀⠀⡠⠐⠈⠀⡄⢸⣇⠀⠀⠀⠀⠀⠘⠩⢂⠀⠀⡉
⠀⠀⡀⠀⡀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠸⡧⠓⠀⡀⠀⠀⠀⣸⡆⣀⠄⠀⣰⣷⣾⠿⠄⠲⢂⠀⠂⠀⠄⢸⠀⠀⠇
⠀⠀⠃⠠⠀⠀⠤⠈⠐⠒⠀⠾⢿⣿⠿⠆⠀⠈⠉⠀⠚⠃⠾⢧⠈⣉⠈⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠁⠲⠄⠀⢰⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠒⢂⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠈⠀⠀⠀⢰⠀⠀⠀⠀⠈⠄⡂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠊⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⡀⠤⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⠊⠐⠒⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠃⠐⠠⢀
⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣦⣤⣀⠀⠀⠀⠀⠀⡄
⠀⢰⠂⠀⠀⢀⣶⣿⡿⠟⠛⠻⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠛⠛⢿⣿⣷⡄⠀⠀⠀⡗⠂
⠀⠈⡀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⡰⠂
⠀⠀⠀⠀⠁⠻⣿⣷⡀⠀⠀⠀⢀⣼⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⣀⣾⣿⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁




⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⣠⡤⠄⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⡀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠑⢂⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣷⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠋⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠚⠛⠛⠉⠙⠀⠀⢼⠹⣿⣷⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣪⣶⡿⣿⣾⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡀⡀⢸⠀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣿⣿⣿⠹⢭⡈⡛⠷⠜⡿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⢿⣷⣿⡆⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠏⢹⣿⣿⣧⠀⠀⠈⠿⠒⠋⠀⠉⠀⠀⠀⠀⠀⠀⠰⠁⢷⣿⣟⣁⣼⠟⢹⠁⢸⣿⠟⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠃⠀⣼⣿⡏⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⠀⢸⣿⠇⣿⣿⣿⣆
⠀⠀⠀⠀⢠⠞⠁⣴⡿⠀⠀⢀⣿⣿⢣⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡔⢰⣇⠀⣾⡟⡄⣿⣿⣿⣿⡄
⠀⠀⠀⠀⡈⠀⣴⠸⠁⠀⠀⣼⠋⣿⣿⠀⢱⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡟⣸⠂⣿⠁⠃⣿⡟⢿⡏⢷
⠀⠀⠀⠀⠃⣀⠏⠀⠀⠀⠀⣿⠀⣿⣿⡄⠈⠐⣦⡀⠀⠀⠀⠀⢤⣤⣴⡀⠦⣀⠀⠀⠀⠀⠀⡌⢠⢡⣿⣶⡟⠀⢀⣿⡇⠈⣿⠸⡇
⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⢰⣿⠀⢹⣿⡇⠀⠀⢷⣿⣦⡀⠀⠀⠀⠈⠏⠉⠈⠁⠀⠀⢀⣠⣾⢁⢖⣿⣿⡿⠁⠀⠘⣿⠁⠀⣿⠀⡧
⠀⠀⠀⠐⣤⠘⠀⠀⠀⠀⢻⣿⠀⠘⣿⡁⠀⠀⠈⣿⣿⣿⡶⢄⡀⠀⠀⠀⠀⢀⣠⠖⠋⣼⡇⣸⣿⣿⡿⠁⢸⠀⡰⠀⠀⠀⣸⡾⠃
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⣿⣿⡄⠀⢻⡇⠀⠀⠀⣿⣿⣿⠃⠀⠙⠶⠤⠴⠖⠋⠀⠀⠀⣿⡇⣏⣿⣿⢣⠀⡆⢠⠇⠀⡴⢀⣿⠁
⠀⠀⠀⠀⡄⢀⣾⠀⠀⡀⣿⣿⡇⠀⣧⠁⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣿⣿⡇⠀⢃⡿⠀⣴⡇⣼⣿
⠀⠀⠀⠀⢀⡄⠀⡀⠀⡇⣿⣿⡇⠀⡏⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡞⠀⣾⠇⣸⠟⣱⣿⡏
⠀⠀⣠⠖⠛⠉⢿⣿⠀⢇⣿⣿⠇⢸⠁⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣷⣿⣿⡿⣾⡏⢰⣏⣼⣿⡿⠁
⠀⡞⠁⠀⠀⠀⡏⢻⣧⣘⣿⡿⠀⣸⠀⡆⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⣿⣿⣷⣿⠁⡟⣼⣿⡿⠷⢤⡀
⠐⠀⠀⠀⣠⠈⠀⠀⢿⣿⣿⠃⢀⡇⡄⢠⣿⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠸⠻⡷⣼⣿⣿⣿⡇⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣾⠞⣡⡿⠋⠐⠢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢋⠠⠂⠀⠀⢾⡇⣽⣿⣿⣿⡟⠀⡠⠒⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣼⣿⠺⠋⠀⠀⠀⠀⠀⠉⠛⠒⠤⠤⠤⠤⠴⠖⠚⢉⠀⠂⠁⠀⠀⠀⠀⣸⣷⣿⣿⣿⠟⡁⣏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡆⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⢀⠀⣿⡟⣿⣿⣿⣿⡷⣰⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⣿⢻⣿⡿⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡽⣧⣹⣿⣿⡿⠛⠁⠀⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣾⣿⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⢟⡵⡻⠽⣿⣿⣿⣿⣿⣉⠛⢿⡷⣆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢣⢣⢣⠊⣨⡾⣟⣿⣿⣿⣿⣿⣿⣿⣦⡙⡝⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠃⡇⡆⡐⡞⣵⡿⠋⠉⠉⠛⠻⡏⢻⢻⡵⢸⡸⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡘⠰⢰⠰⢸⢱⡿⠀⠀⠀⠀⠀⠀⢑⠘⡏⡇⠂⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡔⣰⢡⠃⣏⡇⡆⣾⠃⠀⠀⠀⠀⠀⠀⠀⠄⢧⢳⠀⢱⢸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣰⢃⠎⡘⣼⢰⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⠘⠀⢸⢸⣵⡸⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡼⢡⢊⢄⡔⠇⠄⣼⠛⠓⠶⡄⠀⠀⢠⡶⠞⠋⢣⠀⠁⠈⠸⣷⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⠟⠀⡡⢢⠎⡔⠀⡼⠛⠙⠛⠀⠀⠀⠀⠀⠀⠘⠛⠙⢣⡀⠀⡀⢻⢣⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣠⠞⡴⣡⠞⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠘⢦⡣⡁⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⠏⣞⡴⢁⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣄⠙⣦⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡞⢸⢏⣴⣿⣿⠟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣿⣿⣷⣬⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⣽⡟⣽⡿⣿⡄⢹⡄⠀⠀⠀⠠⠤⠄⠐⠤⠄⠀⠀⠀⢀⡾⢸⢻⢻⣿⣿⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣂⢻⣇⢻⣷⡠⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢣⢏⡏⣾⢸⣿⣟⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢀⠸⡘⣿⡌⢷⡳⡑⣽⡌⠂⠀⠀⠀⠀⠀⠀⠐⣠⣿⢃⠎⡼⢸⡇⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⡄⠱⡘⠝⣌⢳⡑⡌⢿⣄⠀⠁⠀⠀⠀⢀⡾⣿⠃⢀⣾⢡⠛⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡀⠈⠈⢎⠂⢳⡘⡈⠹⡄⠀⠀⠀⠀⡞⣸⠃⣠⢿⢃⡏⡄⡎⣾⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡇⠇⣿⣌⢆⠈⢇⠀⣧⠁⠀⣷⠀⠀⠀⠀⢁⡏⣴⡏⡞⣸⢰⠀⠃⣿⣷⣜⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠊⣡⠞⡱⠁⣸⣿⣿⣆⢣⢸⠀⢸⡆⠀⡏⠀⠀⠀⠀⢹⣰⣹⠃⠃⡿⣜⡈⡄⣿⢿⡦⣕⠌⠛⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⣡⠞⣠⠞⣻⣿⢱⢿⢸⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⠘⣇⡏⠀⠀⡇⠿⡶⡀⠘⣧⠑⢌⠳⣄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠐⣵⡣⠊⠁⣴⣿⠏⠈⣾⢯⠏⢜⡿⡇⠀⠀⠀⠀⠀⠀⠀⢻⡗⢁⠀⡇⠀⠙⢮⡢⡈⠳⡀⠑⡌⢣⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⢀⣞⡿⠋⢀⡾⣡⠟⣠⠋⡜⡠⠓⠤⢄⣀⣠⠤⠚⠁⢻⡞⢦⡘⣄⠀⠀⠑⡌⠢⣳⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⠡⠒⠉⢀⡤⠊⣀⡬⠊⠀⠀⠀⢀⣀⡀⠠⠐⠊⠀⠈⠈⠅⠊⠳⢄⣀⠀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⠀⠊⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⡇⠈⠉⠉⠉⠉⠙⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⣠⣤⣴⠶⠀⠀⠀⢀⣠⣴⡿⢿⡇⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⡇⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣩⡇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡘⣿⣿⣿⣿⣿⣿⣿⡇⠁⢠⣤⣤⣤⣶⡿⠛⠁⠀⠁⠀⢢⠀⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠹⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⢹⠛⠁⠀⠀⠀⠀⢠⠀⠀⠁⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢷⡹⣿⣿⣿⣿⣿⣿⡷⣤⣀⣿⠀⠀⠀⠀⠀⢀⠋⠀⠀⢀⣠⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣯⢷⡘⣿⣿⡿⣿⣿⣿⡄⠀⠘⢆⠀⠀⢀⠴⠃⢀⣴⡾⣛⠝⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣞⢧⡻⡜⠏⢳⡻⣿⣿⣿⣄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⡳⣝⢮⣂⠱⣌⠿⣯⠻⣷⠠⠄⣀⡀⠀⠀⠀⠀⠀⣀⣤⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣯⣇⠘⢦⡉⠳⠉⠳⣄⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⢿⡎⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⢸⠋⠛⣳⣿⡷⢌⠀⠀⠀⠀⠪⡑⠀⠀⠀⠀⠘⣿⣿⡙⠀⠸⡇⢳⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⢁⡤⠖⠋⠉⠉⠹⣦⠀⠀⠀⠀⠀⠊⠀⠐⢄⠀⠀⢻⣿⣧⠀⠀⠹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠜⠋⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢻⣿⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠒⠂⠀⣉⡥⢟⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⢿⣧⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡠⠴⠶⣚⣫⣽⠞⠋⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⢿⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠡⠔⠒⠋⠉⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠰⣶⣄⢳
⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣾⠟⠻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠛⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋
⠀⠀⠀⠀⣰⣾⣿⣿⣷⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢾⣿⣾⣷⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⢾⣿⣷⣦⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣶⣾⣿⣶⡄
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠾⠿⠿⠿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣴⡿⠿⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⢿⣧
⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⢿⢆⣤⡸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠸⣿⣧
⠀⠀⠀⠀⠀⣴⣶⣶⣿⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⢠⣶⣇⣀⡀⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢶⣀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⢀⣤⣿⣇
⠀⠀⠀⠀⠀⣿⣿⣿⠿⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⣿


⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠉⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠺⣧⠀⠀⣠⡿⠆⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⢰⣆
⠀⠀⠀⠀⢀⣤⣶⣶⣴⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⢀⣽⣃⣀⣀⣙⣅⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣶⣶⣾⣦⡄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⣴⣿⠟
⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠁⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡶⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠉⠻⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⣼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣳⣾⠃
⠀⠀⠀⠀⢀⣤⣴⣽⣿⣧⣦⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⣾⠛⠟⢳⣦⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⢢⣤⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣯⣟⣉⣴⣾⣯⡼⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⠿⢛⣻⣽⣿⣿⣿⣷⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⣀⣸⣿⣿⣿⣟⣿⣿⣿⠿⣿⣟⣿⣿⣽⣿⣿⣿⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣀⣀⣠⣤⣭⡷⠶⣷⣼⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⣘⠃
⠀⠀⠀⠀⡀⠀⢀⣤⣴⣿⣿⣿⣿⣯⣻⣿⣦⡀⠀⠀⣠⠔⣠⣴⣿⣯⣿⣿⣻⣿⢿⣟⣹⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⡇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠰⢋⣥⣾⣿⣿⣿⣿⣽⣾⣿⣿⢟⣯⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠄
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢻⣿⡷⣭⣿⣿⡿⣿⣿⣵⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⣿⣿⣿⣟⠋
⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⣮⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣏⣾⣿⢸⣿⣿⡿⢦⡀
⠀⠀⠀⣐⢿⣏⣿⣿⣾⢿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⡿⣿⣿⢷⡀⠁
⠀⠀⠀⡍⠸⡎⣻⣹⡿⣿⢹⣿⣿⣿⣿⣿⢿⠛⠻⢿⣿⣿⣿⣿⡿⢿⡛⠋⢿⡿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣿⣿⠛⣿⣏⡟⣿⡏
⠀⠀⠀⠁⠀⠸⣿⣿⣿⣽⣿⣿⣯⣿⣛⣿⣿⠟⠛⠛⠈⣩⠟⠹⣄⠀⠙⠦⣈⣏⠂⠀⠹⢿⣿⣿⣿⣻⢟⣿⣿⡇⣼⣿⣿⣧⣿⡇⡆
⠀⠀⠀⠘⠀⠀⠈⠻⢿⣿⡿⡿⠛⣿⣿⠟⢣⣠⢃⡴⣠⠋⠀⠀⠈⠉⠊⠱⠤⢭⣟⣶⣄⣸⣯⠿⠿⠟⣛⢽⡿⢘⣿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢻⣿⣤⡿⣻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠒⢞⣿⣯⣷⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⢹⣿⠋⠈⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠭⠭⠙⠛⣷⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢣⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄⠀⠀⠀⠀⠀⢻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⢉⣀⣤⣤⣤⣦⠖⠀⠀⠀⠀⠀⠀⢄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣶⣦⡄⠀⠀⠀⠀⠀⠈⠉⠁⢀⡶⠻⣿⣿⣫⡝⠁⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡛⣿⣿⡳⠄⠀⠀⠀⠀⠀⠀⠈⠐⠒⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠐⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⠀⢀⠀⠀⣀⠵⠒⠂⠂⠒⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀
⠀⠀⠀⠀⠀⠃⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠠⢀⡀⠀⠀⠀⠀⠀⠔⠈⠀⠀⡀⠀⠘⠁
⠀⠀⠀⠀⠀⠘⢲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠆⠐⠤⢈⢢⠀⠀⡔⠀⠀⠄⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⢀⠃⡌⠀⠀⠀⠀⠀⠀⣠⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠄⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⢠⠌⢰⠀⠀⠀⠀⡠⠔⠈⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠊⠁
⠀⠀⠀⠁⢀⠃⠀⢀⠔⠊⠀⠀⠀⠠⠊⠀⠀⠀⠙⣄⠀⠀⠀⠀⡰⠃⠉⠁⠀⠀⠉⠐⠢⡀
⠀⠀⠀⠀⡜⠀⢠⠂⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠈⠉⢆⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠐⢌⠦⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠑⣌⡄⠀⠀⠀⠀⠀⠀⠨⡤⠓⠊⠉⠊⠀⠀⢱⡉⡀⠀⢀⡭⠚⠲⢤⣄⡀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⡀⡈⠃⠂⢢⠀⠄⠀⠒⢨⠃⠀⠀⠀⠀⠀⠀⠈⠈⠘⠳⣾⠤⢄⡀⠈⡀⠈⠳⡀⠀⠀⠇
⠀⠀⠀⠀⠆⠈⠁⠀⠀⠋⠀⠉⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠑⠒⠂⠀⠀⠀⡟⠩⠤⠀⠁⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⠀⠀⠔⠒⠄⠀⠀⠀⢠⠀⠀⠘⠀⠀⠀⠀⠠⠀⢀⡀⣤⡌⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆⠀⠀⠙⠀⠀⠜⠀⠢⣀⢄⠀⠠⠒⠤⠊⠑⡄⠀⠄⠀⠀⠀⠀⠈⠈⠑
⠀⠀⠀⠀⠀⠀⢔⠐⠐⠀⠀⠀⠀⠀⡒⠒⠉⠀⠀⠀⠁⠒⠒⠶⠀⠀⠀⠀⠐⠈⠄⢀
⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢸⣿⠋⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡀⠈⠿⢿⣷⣤⣴⣶⣾⣿⣿⣀⣁⣴⣻⣧⣀⣀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⣿⣿⠁⠠⠼⢿⣿⣿⣿⣿⣿⣿⠙⠿⣿⠿⠿⣿⣿⣿⡄⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⣿⣯⣄⠀⠀⡿⠋⢻⣿⣿⣿⣿⣇⠀⢱⣶⡆⣠⣤⢹⢿⣿⣿⡏⠤⠠⠄
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣦⣸⠟⠉⢉⣉⣷⣠⣤⣤⣄⠹⠛⠛⠿⠿⡄⢷⣴⢶⣌⣋⣴⣤⣿⣁⣿⣶⠖
⠀⠀⠀⠀⠀⠐⣿⠟⠋⣙⣿⡷⠿⢓⣈⠋⢙⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⡿⠻⠿⣿⣟⣁⣤⣀
⠀⠀⠀⠀⠀⠀⢻⣶⣿⡿⠟⡅⢲⣾⣽⡿⠁⠀⠀⠀⠀⠠⠄⠀⡄⢀⣰⣦⣤⣤⣤⡍⠐⠂⠰⣴⣿⠟⠻⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠣⢞⡟⣵⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⢿⣿⡿⠟⠀⠀⢰⣿⠟⠄⠀⣠⠘
⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠋⠁⠈⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣀⠈⣿⣧⣶⣶⣶⣼⣿
⠀⠀⠀⠀⠀⢸⠟⠛⢻⣿⣿⠇⠀⠀⢀⣿⣗⠀⠀⣠⣶⣶⣦⠀⠀⠓⠀⠀⠀⠛⡏⠉⠛⠛⣻⣦⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠐⠂⠹⡏⠀⣴⣿⣿⡿⠛⢡⣤⡿⠉⠁⠈⢻⣷⣤⣶⣶⣶⣬⣌⣡⣄⢃⡇⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠴⣿⣿⣿⠃⠂⢐⠒⠃⢂⡀⠒⢘⣿⠉⠉⣻⣿⣿⣿⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⣿⣿⠀⢀⣈⣉⣹⣿⣽⣿⢿⣿⣄⠀⠉⣿⡟⠉⠀⠀⠙⠀⠀⠀⠘⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⣠⣤⠤⠤⠈⣉⡀⠻⠿⢿⣿⡂⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⣽⡀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡷⣄⠀⣰⣧⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣼⠟⠁⣾⡶⢻⣯⣴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⣴⠶⢛⣛⡿⢿⣷⣶⣶⣯⣀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⡁⠀⠸⣇⠰⠋⠉⠙⠳⡌⢻⣧⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⣴⠏⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢿⣇⠀⠲⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣏⠸⠘⠦⠀⠀⢰⡏⠀⠳⣤⣤⠞⢁⠀⣼⡿⠀⡀⠀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡌⠻⠶⠂⠀⠠⣄⣈⠻⣿⣶⣿⣶⠞⣋⣼⡿⠃⠀⠈⠛⠿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠻⠒⠀⠀⢀⣀⣀⣭⣭⣽⣭⣿⠶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢻⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣇⡀⠀⣀⣤⡶⢟⠛⠋⠉⠉⠉⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠐⠈⣡⣾⠟⣡⡾⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⣭⣍⡓⠦⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡶⠦⣰⣿⢇⡾⣿⠇⢀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠋⠛⠀⠀⠀⢰⡾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣿⣿⢸⠃⢿⣄⣈⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⣿⣿⡘⠐⣄⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢿⣿⣷⣄⠙⠫⠁⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠘⣿⣏⠛⠿⣶⣾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⣤⠄⠈⢿⣷⣐⢤⣄⣀⣼⠆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⣆⠀⠙⠿⣷⣭⣙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡹⣷⠶⠂⠈⠙⡻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣅⣠⡆⠀⠘⢷⣮⡻⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⠉⠀⠀⠀⠀⢻⣿⣎⣿⣄⡀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣸⡇⢿⣿⠇⠹⣧⡀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⢸⠏⠀⠀⣿⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⢹⡇⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⣋⡵⢸⠃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠛⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠖⠒⠒⠒⠒⠒⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠙⠲⠤⣤⣤⣤⠤⠶⠞⠋⠁⢀⡀⠀⢀⡀⡀⠀⠀⠄⣀⡀⠀⠀⠈⣙⠓⠶⢤⣤⣀⣤⡤⠶⠚⠁⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠢⡀⢀⠔⡡⠐⠈⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⡒⠀⠀⣀⡴⠒⠋⠉⠉⠉⠙⠲⢤⡀⠑⠐⢂⠊⢀⡴⠚⠉⠉⠉⠉⠉⠙⠳⢤⡀⠀⢖⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⠀⠀⠀⠀⣀⣤⠴⠶⣤⣄⠀⠀⠈⢧⣸⠃⠀⠀⠀⢀⣤⡴⠶⢦⣄⡀⠀⠀⠈⣦⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡞⠀⠀⠀⠀⣼⠋⣠⣤⣦⣤⡈⢳⡄⠀⠸⡇⠀⠀⠀⣴⠏⢀⣤⣶⣤⡈⠻⣆⠀⠀⢸⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡇⠀⠀⠀⢸⡇⢰⣿⣿⣿⠛⣷⠀⣿⠀⠀⡇⠀⠀⢠⡏⠀⣿⣿⣿⠛⣻⠀⢿⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⣇⠀⠀⠀⠘⣧⠈⠿⣿⣿⡿⠋⣠⡏⡤⠖⠒⢦⡀⠀⢷⡀⠻⣿⣿⣿⠟⢀⡿⠀⠀⢰⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠹⡄⠀⠀⠀⠈⠳⣤⣄⣠⣤⠾⠋⢸⣁⡀⠀⣀⣻⠀⠈⠻⢦⣤⣀⣤⡴⠛⠀⠀⢀⡼⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣏⢍⠛⣅⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠓⢦⣄⡀⠀⠀⢀⣀⣠⠴⠋⠘⣾⣦⡿⠉⠳⢤⣀⣀⠀⠀⣀⣠⡴⠞⠁⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⡀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠀⠀⠉⠉⣑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠆⠀⠀⠀⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠘⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡿⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢀⣀⣤⠄⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠠⣤⣀⡀⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⣻⣶⠟⠀⠀⢨⠟⣀⣀⠀⠀⠀⣾⣻⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⠀⣀⣈⢻⡅⠀⠈⢻⣾⣏
⠀⠀⠀⣠⡞⠁⣘⣶⠆⡀⠘⢩⡟⢁⡀⠀⢀⡇⡏⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡆⠀⠀⠀⢧⢻⡀⠀⣄⡈⢻⡉⠁⣀⢶⣞⡁⠙⣧
⠀⠀⠰⠿⢶⣿⠋⡵⠋⠀⠀⠈⠋⡿⠁⢀⡞⢰⠃⠀⠀⠀⠀⠈⢆⡐⣁⠆⠀⠀⠀⠀⠀⠑⠦⣉⡠⠞⠁⠀⠀⠀⠘⡎⠳⣄⠈⢯⠉⢁⠀⠀⠳⡌⢻⡿⠾⠇
⠀⠀⠀⢠⡟⠁⠸⣧⣴⠾⠋⠀⢘⠧⣾⠏⢠⡏⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠙⣷⠺⣀⠈⠻⡶⣤⡼⠀⢹⡆
⠀⠀⠀⣾⣁⣴⠟⣿⢡⠃⠀⢀⣿⣰⠏⢠⡟⠁⠀⠀⠀⢧⠀⠀⢠⠂⠀⠸⣌⠢⢊⡼⠀⠀⠀⢠⡀⠄⠄⣸⠀⠀⠀⢂⠹⣄⠘⣧⣿⡀⠀⢱⡹⣿⢶⣤⣿
⠀⠀⠀⠛⠋⠀⢸⡇⠸⣀⣴⡟⢹⣏⣰⡟⡇⠸⡄⠀⠀⠀⠑⠚⠁⠀⠀⠀⠈⠓⠋⢀⠀⠀⠀⠀⠙⠲⠖⠃⠀⠀⣠⣼⠀⣿⣷⣼⡟⢻⣦⣼⠃⢿⡄⠈⠉
⠀⠀⠀⠀⠀⠀⢸⣇⣴⠟⢹⡇⠘⠛⣽⡇⣧⠀⡷⡀⠀⠀⠀⠀⠀⢠⡀⠀⡴⠀⠀⠀⢦⡐⠀⣰⠀⠀⠀⠀⠀⣀⠉⣿⠀⡏⣿⣏⠃⠈⣿⠙⢦⣼⠃
⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⢸⡇⢀⡾⠉⣇⠙⠾⣵⠛⠶⣄⠀⠀⠀⠀⠉⠉⠀⠀⢀⢀⠀⠉⠛⠁⠀⠀⢀⡤⠞⠉⠳⣜⠶⠁⡿⠙⣧⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋⠀⠀⠹⣦⠞⠁⠀⠀⠈⠙⡶⢤⣀⣀⠀⠘⠦⠥⠋⠀⠀⢀⣠⠤⢾⡋⠀⠀⠀⠀⠙⢦⣼⠇⠀⠈⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣓⠼⠉⠙⠛⠓⠒⠚⠛⠋⠉⢧⣊⣤⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠚⠀⠹⠒⠢⡀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠀⠟⠋⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠞⢦⣷⣂⡔⣖⣀⠇⠀⠀⠀⠀⠀⢼⠀⡶⢧⠰⣴⣤⠳⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠿⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⣤⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠶⠂⡟⡟⡷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠄⡟⡿⡴⠾⠸⠑⢉⢿⢿⡏⣩⡄⡀
⠀⠀⠀⠀⠀⠀⣺⡱⣎⢁⣬⠷⣶⠧⣻⣩⠾⢚⢾⣯⣵⣥⡄
⠀⠀⠀⠀⠀⢠⢷⢶⣆⣙⣳⠶⢲⣶⣟⣛⠢⢿⣉⣘⣫⣻⣽⣗⣄
⠀⠀⠀⠀⠀⠈⡏⢪⢢⢿⠛⣻⢧⣏⢏⣟⢻⣾⣿⡥⢱⢿⢿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠁⠮⣟⡾⡟⣘⣩⣤⢷⢿⣇⢴⠆⢾⠏⢺⢳⣿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⡟⣿⣧⠊⣈⣺⣽⣶⣖⡞⡃⡿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢽⣭⣬⣁⠻⣦⡭⠗⡜⠚⡧⢰⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠳⡻⣹⡿⠿⠖⣸⣻⣻⢳⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢴⢣⣿⢧⡚⠃⢟⣏⣱⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠽⢺⡿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⡶⢀⣉⠛⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⠺⣛⣁⠲⠉⢀⡀⠀⠀⠁⠰⣔⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⢋⡙⢢⣌⢉⣉⡛⢦⡀⠈⠳⡠⣝⢦⡈⠓⠮⣝⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠊⡉⠤⠈⠈⠉⠀⠤⡁⢀⠀⢠⡘⠳⣝⢦⠙⣮⡻⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠦⠀⠀⠀⠀⠀⠀⡓⣝⠈⠀⡄⢑⠹⣌⠢⠁⠈⣷⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠍⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡀⡙⢮⠃⢻⡆⠀⣀⠈⢠⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡔⡄⣁⠈⢿⠀⢿⡀⢀⠇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣼⠈⣷⢀⢠⡎⣅⠘⢷⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡄⢀⡀⠀⠀⠀⠀⠀⠀⢤⠖⢺⡶⢀⠀⡀⠉⠀⠉⢸⣃⠀⢻⡄⢶⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠇⠀⠸⠁⠛⠣⠀⠀⠀⠀⠘⠀⠠⠀⠀⠸⡇⠀⡀⡀⠀⢻⠀⠸⣿⣬⠠⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣀⠘⠀⠀⠉⢿⢰⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠴⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⠇⠀⠀⡆⢸⠈⢸⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠀⠀⣿⢢⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⡇⢠⡄⡀⢸⢻⡾⡌⡇⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠁⠸⡷⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡇⣿⢡⡟⠀⢀⣿⠈⠣⢀⠉⠋⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣼⡆⡤⠀⠀⠀⠁⠘⢆⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⢿⢸⡇⣿⠘⣿⢸⠀⢸⣇⠀⡌⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡹⢱⠃⠞⠀⣜⠀⠀⡞⠸⠀⠀⠐⠂⠉⠉⠰⡄⣷⠘⢸⡇⠹⢀⡉⣿⡖⠸⡏⠀⠙⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⡀⠀⣸⠂⡆⠇⠀⠀⠃⠀⠀⠀⠀⠀⢳⡜⣇⠀⢿⡀⠈⠃⡈⠇⢰⡄⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣤⡋⣀⠄⢀⠀⢠⡋⠀⠀⡀⢄⡄⣀⡤⠴⠦⣶⣝⢮⡁⠈⣻⣄⠂⢸⡄⠸⠃⠀⢛⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠟⢀⠈⣄⠈⠃⢸⡇⢰⢺⠀⠈⠁⠀⠀⠀⠀⠀⠉⢳⣍⡢⡌⢙⠻⠎⠣⢠⠸⣧⡀⣎⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢋⡄⠠⣌⢦⠰⢄⠀⡜⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡙⢆⠳⢸⣆⣍⢰⡆⢰⡜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⣷⠀⠘⣼⠃⠀⠀⣶⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢈⠀⡀⡛⡌⡀⣯⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠠⠄⠋⠀⢰⢆⠁⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⡘⠛⡃⠁⠃⠋⣧⠘⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⢠⡀⠀⢀⠸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⢀⠛⢰⡜⢰⠁⢰⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠃⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢈⡳⡜⠃⠘⡷⠈⢋
⠀⠀⠀⠀⠀⠀⠰⠞⣯⠈⢁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⡘⣷⡇⠿⣸⠇⡇⢩⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠉⠰⠏⠀⡇⠀⠉⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣷⢸⠀⠘⣷⠆⠰⢰⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠐⠃⡜⠋⠧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢠⡟⣼⠃⠀⢡⢿⠀⢸⡀⠏
⠀⠈⠀⠀⠀⠀⠀⢤⠀⠀⠀⠆⠀⣼⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢩⡟⢰⡏⣾⠠⢠⠎⢐⠘⠙⡆
⠀⠀⠀⠀⠀⠀⢃⡀⡆⣾⠁⠴⠀⢋⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣰⠏⠘⣰⠀⠀⢸⡆⢠⢆⡈⡈⠇⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⢁⡿⠀⡆⠺⠈⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠰⣸⠇⡀⠋⡀⣶⣿⠇⠄⠸⢉⠃⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣾⣷⣾⣷⣾⣿⣷⣶⣶⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣠⣖⣤⣶⣦⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣾⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⢀⠀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠉
⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠈⣿⣿⠿⠋⠙⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⠀⡀⣀⠄⠀⠀⢀⠀⠁⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠊⠀⠀⠀⠀⠀⣄⣸⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣾⣥⣀⣤⡞⠁⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠂⠀⠀⠀⠀⣠⣴⣿⣿⡿⢿⣿⣿⠟⢻⣿⣿⠉⠙⢿⠛⠻⢿⣿⣿⣿⣿⣷⣿⣴⣮⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⣾⡿⠟⠉⢹⡇⠐⠿⢿⣶⣾⣿⣿⣏⡀⢸⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⠟⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠟⠉⠀⠀⠀⠘⣇⠀⢔⡼⠛⠿⠛⠙⠀⠀⡼⠀⠀⠀⠀⣠⡾⠋⠀⠉⠉⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⠞⢋⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠛⠛⠒⠒⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠀⢒⣚⣥⣴⣞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠖⠶⠐⠶⠖⠋⠉⠛⠯⠉⠃



⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⠐⠒⢀⠠⠀⠀⠀⣀⡀⡙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⣀⣀⣤⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔⠆⠀⠀⠀⠀⠓⠂⠀⢂⢸⣄⠀⠈⢉⡂
⠀⠀⡠⠀⢠⡤⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠐⠀⠙⠁⠀⠡⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⠐⠠⠊⢀
⠀⡈⠀⠀⠀⠀⠈⠐⠀⣿⡿⠛⠉⠁⣀⡄⠘⣧⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠰⡆⣀⠄⠀⠀⠀⠀⠀⠀⢘⠀⠈⠀⠀⡀
⠀⢀⢀⠟⢄⣀⠀⠀⠑⠜⠧⠓⠦⠛⠉⠀⠀⡹⠶⠎⣣⠰⠀⠀⠀⢀⡢⠤⠀⠐⣀⠀⠈⡋⠁⠀⠀⠈⡃⠀⠀⠀⠀⠆
⠀⠚⠅⠀⠀⠊⠀⡀⠀⠀⠀⠁⠖⠐⠤⠀⢎⠀⠀⠀⠘⠂⠁⠀⡀⠌⠀⢀⠐⠄⠠⢰⠊⠁⢠⠐⠀⠐⣥⢮⠇⢀⠃
⠀⠠⡆⠘⡓⢦⡴⠉⠒⡖⠀⠀⠀⠀⡀⠄⠐⢲⠿⠅⠀⡤⠔⢠⡀⠈⡠⠖⠠⠂⠐⠊⠀⠀⣀⠀⠓⠀⢸⡳⠚⠟⢠⣀
⠀⡤⠱⠤⠡⡀⢹⡔⢖⢻⠀⠀⡠⣈⣀⡀⠀⠈⠀⠀⠀⠈⠉⠚⠀⢀⣤⢤⢤⠀⣒⠤⢤⠀⡶⠢⠄⠀⠷⢁⠀⠀⠀⠀⢀
⠠⠥⢀⠜⠢⡑⢄⠡⡈⢮⣆⢠⠀⠸⣏⣹⣿⢳⢀⠀⠦⠌⠙⢶⢚⠋⡗⡃⠠⠒⠓⠢⠶⠃⠀⣀⠀⠀⠀⠀⠈⠒⢷
⠀⠀⠙⢄⠀⠈⠐⢵⣌⣂⡿⢃⠀⠳⣬⣍⣉⡋⡄⢀⣾⡀⣤⣎⢤⣦⠸⡘⠁⠒⠌⡀⢨⠀⠸⣿⠇⠀⠀⣤⣦⡀⠸⠄⢨
⠀⡠⠂⠔⠪⠃⠄⣀⣠⣋⠐⢠⡀⠀⠀⠈⢛⣵⣏⢡⠞⣷⡄⠈⠆⠈⠲⣼⠋⣲⢆⠐⠄⠀⠢⣤⣄⣄⣀⠙⠟⡁⡈⢀⡅
⠀⠣⢄⠀⢀⣁⢹⡙⠉⠀⠠⠄⢡⠤⠂⠀⡿⣯⣼⠈⢷⠛⠀⠀⠄⢀⠀⢈⡒⠃⢀⣤⣀⣀⠀⠏⣉⣝⣿⠁⠈⠠⠀⠀⡁
⠀⠀⠀⢡⣃⠤⠊⠀⢀⡠⠈⡠⠀⠀⠐⠈⡃⠀⠈⠀⠀⢈⣀⣴⠇⠈⡀⣀⣆⣴⢿⣿⣿⣿⣿⣦⠘⠷⣟⠀⠀⣌⠀⠁⠿
⠀⠀⣠⠞⣿⣷⠠⠊⠁⢀⠀⠁⠀⠁⢀⠊⣙⡀⠙⣧⠹⡅⠻⠋⡔⠈⢀⠀⢰⣿⡲⣿⣿⣄⣀⣽⣀⠘⠄⢢⠸⣍⢳⠔⡀
⠴⡾⢿⠾⠋⠓⠁⠀⢈⣤⣠⣄⡜⠒⠁⠙⠀⠙⠜⠌⡖⠁⠐⠀⠈⡘⠁⢠⡄⠈⠙⠓⠶⠿⢿⡟⣄⡀⠁⣠⣀⠡⠋⠀⠐
⠀⠃⡧⡄⠀⣀⡆⣶⣊⡜⠚⠀⡀⠀⢰⠛⠑⠀⠈⠄⢀⠄⠠⠄⠊⠀⣠⠪⡤⠁⠤⠀⠀⡩⢩⣔⢵⡿⢉⣴⡍⣧⡑⠢⠄
⠀⠀⣀⣀⠴⠀⠀⡸⢤⡓⠰⡚⠉⠘⠥⠁⠊⠀⠂⠉⠑⠀⢝⣄⡔⠉⠁⢄⠀⣈⡑⠀⠄⣁⣜⣾⣷⣄⡻⢟⠍⠐⣠
⠀⠀⢄⠀⡀⢠⠈⠀⣀⢁⠡⠈⠂⠀⠔⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⣤⡌⠑⠂⠬⣳⣄⣿⣷⣿⣧⣈⣴⠁⣀⡐
⠀⡕⠲⠀⠀⡔⢋⠀⢀⠘⢆⡱⢂⢈⠢⠀⠀⠀⠀⠀⠀⠀⠀⣱⡔⠊⠒⠚⠮⣀⠇⡐⠛⣣⣿⡿⠡⠋⣉⠩⠟⢲⠠⡀⣁
⢠⠃⣃⣡⠌⠀⢈⠀⢈⠀⡸⠀⠂⠉⠐⣀⠀⠀⠀⠀⡆⠖⠈⠀⠀⠑⡔⠃⠂⡀⠂⠁⢚⡿⠋⢴⢀⠜⠁⠀⢠⠈⠊
⠀⠁⠀⠓⠀⠐⠄⠕⠥⠴⠃⠑⠀⠤⠔⠠⠌⢀⠂⡊⠀⠀⠀⠀⠀⠐⠀⠀⢠⠘⠈⠉⠊⠀⠀⠀⠃⠐⠀⠀⠠⡗⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠘⠠⠀⠠⠒⠀⠁⠁⠂⠒⠈⠀⠠⠒⠲⠈⠄⠂⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠁⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠠⠀⠤⠠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠆⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⠉⠀⢱⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠶⠖⠒⢇⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢄⠄⠈⡇⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⣤⣀⠀⠐⠈⡜⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠉⡇⠀⢀⠀⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣄⡀⢀⢁⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠄⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡈⠀⠀⠠⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠒⠤⠤⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⢢⣤⣀⡀⠀⠗⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⢿⠀⠀⢤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣹⠀⠀⠘⡄⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠃⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠉⠹⠁⠀⠀⠀⠈⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣶⠻⣿⣿⣿⣿⣿⣿⣿⣿⢛⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠀⠉⠉⠉⠛⠛⠉⠙⠁⠘⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠊⢁⣈⡙⠛⠂⠀⠀⠚⠛⢉⣀⠈⠃⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⢽⠀⠛⢟⣿⣟⠦⠀⠀⠀⠀⢞⣿⣯⠿⠃⢸⡯⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⡆⠀⠀⠀⠀⣀⣀⠀⠀⣀⠀⠀⠀⠀⠀⣾⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠈⣀⡈⣁⡈⠀⠀⠀⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⠀⠀⠐⠊⠒⠂⠐⠊⠓⠂⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠓⠒⠒⠋⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠳⢤⣀⠀⢀⣀⡴⠊⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠉⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡶⢾⣿⣿⣿⣿⣿⡿⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⣿⣿⣿⣿⣿⣿⠷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢂⣺⣿⣿⣿⣿⣿⡷⠤⡟⡶⠲⣦⣤⡤⣤⢤⡴⠒⡞⡷⠴⣿⣿⣿⣿⣿⣿⣃⠐⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⣽⣿⣿⣿⣿⣿⣇⠀⠈⠙⠢⠾⠾⣇⡾⠶⠢⠖⠉⠀⠀⣿⣿⣿⣿⣿⣿⣇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀⠀⠀⢀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡏⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⠲⣿⣿⠈⠉⡿⠻⠻⣿⠿⢿⣿⠻⢛⠿⠉⠸⣿⡷⢺⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⣿⣧⣁⡐⠂⠤⢠⡝⢠⠈⢭⠀⠆⠑⣀⣩⣿⡟⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⢹⡇⠈⠉⠙⠓⠒⠲⠶⠶⠒⠒⠛⠋⠉⠀⢻⠃⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⢿⡿⡿⡿⡟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠣⢄⣀⠀⢹⠀⢹⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣇⣀⣠⣤⣘⣄⣸⠀⡼⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆⠘⠧⠤⢤⡈⢇⠉⠙⡏⠓⢯⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠦⠤⠤⠔⣦⣌⠙⠦⣀⠇⢠⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⣞⣥⣖⠒⠚⠛⠛⣋⣉⣉⣱⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣁⡴⣪⢽⠿⠟⠻⠯⡿⣟⢅⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢫⡞⠉⠀⠀⠀⠀⠀⠀⠉⢳⡱⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡇⠐⠒⠲⠤⠀⠀⠤⠖⠒⠂⢸⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡇⠘⠿⠟⠆⠀⠀⠰⠻⠿⠃⢸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠹⡀⠀⠀⠈⠁⠈⠁⠀⠀⢀⠏⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⠈⠉⣀⣀⠉⠁⢀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠢⣀⡀⠀⣀⠔⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣎⣻⢿⣦⣄⡀⠀⠀⢀⣠⣴⡿⣟⣡⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⠻⣿⣜⢯⣻⣑⣌⣉⡚⢻⠟⢓⣉⣡⣊⣟⡵⣣⣿⡟⠛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡠⡸⠿⣷⣕⡦⣜⠢⠿⢿⡿⠿⠔⣣⢴⣪⣾⠿⢇⢌⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠠⠋⠹⠿⢶⣮⣭⣯⣿⣭⣵⡶⢿⠏⠙⠄⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢥⣀⠈⠁⠘⠋⠙⠃⠈⠁⣀⡨⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠀⠀⠒⠉⠉