Thinker Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣦⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⢚⡛⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠆⠻⣾⣿⣦⡄⡰⠴⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣸⡄⠸⣧⣄⢄⣰⣶⣿⣤⠙⣛⣛⣉⣛⣓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⢧⣙⣿⣿⣿⣿⣷⣽⣏⣿⠆⣽⡶⣤⣉⣘⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣌⡉⠻⡿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡇⠉⠀⠈⢙⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌⢮⡛⠶⣤⣿⣿⣿⣿⣿⣿⣿⣶⣌⠙⢀⣀⣠⠞⣩⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡙⠛⠪⡙⢷⣮⢯⣽⣿⣿⡿⠋⢿⣿⣿⡇⠸⡿⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠴⢤⣿⣿⡿⠋⣠⣤⣤⣽⣿⣿⣦⣷⣾⣋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣦⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⡿⠟⠙⠻⠷⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠙⣦⣠⡼⢁⡿⠀⣀
⠀⠀⠀⠀⠀⠀⢤⢤⣿⣿⣾⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⠻⣦⣀⣉⣷⣀⣿⢿⣿⣿⣷⣶⠏
⠀⠀⠀⠀⠀⠀⣶⣮⣋⣟⠉⣩⣿⢿⣿⣿⣿⣿⠟⠉⣠⣴⡟⣯⣿⣿⣿⣷⣿⠛⠉⠁
⠀⠀⠀⠀⠀⢀⢉⣼⣿⢽⣿⠄⠐⠂⠉⢛⣿⡿⠀⠈⣠⣿⣿⡿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⢈⡩⠉⠉⢸⠁⢀⠀⣳⣠⣼⣿⠡⠂⣾⣿⣿⢻⣿⠟⠋
⠀⠀⠀⠀⠀⣿⣆⠀⢀⣀⠀⡏⢀⡜⣻⣿⣇⣤⣿⣿⡋⡵⠛⠁
⠀⠀⠀⠀⠰⣯⠉⡁⠈⠿⢐⣵⣿⣷⣶⢟⣿⣿⣿⠛⣷⠃
⠀⠀⠀⠀⠀⣿⠀⠇⠀⠁⢈⣿⣿⣯⣿⢿⡞⠻⠉⣿⣿
⠀⠀⠀⠀⢠⣿⣷⣄⠄⠄⠈⠉⠣⣤⠿⢾⡉⠓⡄⠹⣧⣀⣢
⠀⠀⠀⢠⣾⣿⡿⣿⣿⣧⣦⣴⣷⣤⣄⣤⣿⣤⣦⣤⣼⣉⣁⡀
⠀⠀⠀⠘⠉⠙⠃⠘⠋⠙⠉⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⢟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣬⣀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠋⠁⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠟⠉⢻⣿⣟⣿⠆⠘⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣥⣀⠴⠻⣿⣿⢷⡦⣄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇⣿⣿⣇⠀⠀⠹⠈⣿⡇⢸⠃
⠀⠀⠀⠀⠀⣼⣿⣿⡿⡇⢸⣿⣿⣦⣄⣀⣶⣾⣷⢸
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⣉⠀⠹⡏
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣤⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣶⣶⠆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⡟⣻⣹⣿⡿⣟⣟⣿⠀⢀⣄⡤⣀
⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣿⢟⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⢪⡓⠈⠙⢾⣷⢶⡄
⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣣⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣟⢀⡇⣼⢷⡘⠢⠀⠹⣿⡙⠂
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⠘⣿⣄⠀⠀⠹⢧⠀⠀⠃
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡇⠈⢿⣧⡀⠀⠸⡄
⠀⠀⠀⠿⠿⠛⣛⣛⣛⣛⣛⣿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣀⠀⠙⣿⣦⣴⣗⡀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢿⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⠀⠀⠀⡆
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣿⡟⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⣿⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⠟⠁⠠⢻⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢘⣷
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢰⣿⣿⣤⣤⣤⣄⠙⠻⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠠⡀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣏⠋⢻⣿⣿⣿⣶⣦⣶⡼⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢰⡟⠄⢀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣭⣭⣥⣤⣤⠀⢀⣿⣿⣷⣶⣶⣶⣤⣴⣿⣿⣿⣿⣿⣦⣄⣀⣀⣤⡄
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣤⠿⠿⣿⣿⣿⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠛⠛⠋
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣭⣍⡀⠛⢸⣿⣿⣿⣿⣿⣿⡿⠿⢇
⠀⠀⠀⠀⠙⠛⠛⠛⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓
⡀⡀⢀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⡀⢀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠈⠉⠉⠀⠀⠀⠉⠉⠁⠀⠛⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠀⠀⠀⠈⠙⠢⣄⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠁⠒⠤⠤⢀⣀⣀⠀⠀⠀⠀⣀⣀⣀⣀⠤⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠤⠄⠤⠴⠦⣄⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠳⣌⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠀⠀⠀⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⡸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡘⠋⠛⠉⠙⠓⠲⠶⠦⠴⠶⠖⠚⠋
⠀⠀⠘⠢⣀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠐⢶⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠉⣻⣿⣿⣆
⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠲⣈⣇⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠘⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⡀⣼⡿⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠆⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣼⠁⠀⡠⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⣇⡀⠁⠀⠀⠀⣀⠤⠖⠒⠒⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡀⠉⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⡇⠉⠀⠀⠀⠀⠀⢹⠁⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠒⠁⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠴⠦⠭⢀⣀⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⢀⠀⣀⣀⠀⣀⡀⢦⡀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢰⡄⢠⣼⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠛⠛⠛⠛⠛⠋⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣦
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣀⣠⣶⣀⣀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣸⣿⠟⠛⢻⣿⣁⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢻⣿⣄⣀⣰⣿⠛⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠘⠛⠻⣿⠛⠛⣡⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣿⡋⢙⣿⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠉⠻⠟⠉⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠀⠀⠀⣀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⣒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣩⠴⠂⠀⠈⠁⠀⡰⠛⡍⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡴⠊⠀⠀⠀⠀⣀⡴⠊⠁⠀⠹⡄⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠀⢀⣀⡤⠖⠋⠁⠀⠀⠒⠶⠤⡝⢦⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠴⠒⠛⢛⠛⠛⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠓⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⡟⠀⠀⠠⢼⠞⣷⣶⣄⠀⠀⠀⢠⣶⣾⡳⣯⠄⠀⠀⠉⢻⠴⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢸⢡⢤⣤⡇⠀⠀⠀⢸⡀⠉⠉⣸⠀⠀⠀⢻⡈⠉⠀⡸⠀⠀⠀⠀⢸⣤⡤⡌⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡈⠘⢦⡇⠀⠀⠀⠀⠙⠒⠚⢹⠀⠀⠀⠀⠙⠒⠊⠁⠀⠀⠀⠀⢸⡴⠃⠁⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠚⠛⠛⠉⠉⠉⠉⠉⠉⠙⠒⠒⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠉⠳⡄
⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠛⠊⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠴⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠶⠒⠒⠖⠲⠖⠲⢳⡶⠶⠶⠶⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠀⠀⠀⠀⠻⠿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠶⠶⢶⠖⠒⠒⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⢰⡆⠰⠆⢰⣧⠀⠀⠘⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⠀⠀⠀⠀⠰⠀⠈⠉⠀⠀⢀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⢄⠀⠀⠀⡐⢆⠀⠀⠀⠀⠌⠀⠀⠀⠀⠹⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠁⠀⠀⠀⠀⠀⠀⢻⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⠀⠄⢀⠀⠀⣀⠤⠐⠒⠒⠂⠲⠀⠀⠀⠀⢸⠀⢀⣿⠂
⠀⠀⠀⠀⠀⠀⢀⡾⠁⡧⢠⠔⠈⠀⢀⡀⠤⡧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⡾⠃
⠀⠀⠀⠀⠀⠀⣾⠁⠸⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠻⣤⡖⠀⠀⠀⠀⠀⢉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⡏⠱
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠠⢄⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠁⠑⠠⠤⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣄⣤⣤⣤⠴⠶⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡾⠛⢉⣩⣷⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⢉⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⣄⣀⠀⠈⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⡠⠚⠉⠉⠉⣀⠀⠀⢠⠀⢉⠉⠙⠀⢸⣿⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⠻⠄⠶⡾⠟⠛⣾⡿⠀⠀⠻⠛⠛⠙⠗⠀⠀⣧⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡇⠀⠀⠀⠘⠲⠤⠄⠚⠁⠀⠀⠀⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⣀⣀⣀⣀⣤⠤⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⡼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣆⠀⠀⠀⠀⠀⠀⣰⠟⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣧⣄⣀⣀⣠⠼⠁⠀⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⣆⠙⠻⠍⠁⠀⠀⢀⠄⢀⠇⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⠄⠀⠘⢧⡀⠀⠀⠀⢀⣎⡤⠋⠀⠀⢸⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢨⡀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠀⡞⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠘⢳⡿⠻⡄⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⢀⠎⠀⠀⠃⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠉⠁
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿⣀⣀⠽⠋⠻⠿⠶⣤⣄⣀
⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠀⠀⠁⠀⠀⠀⠉⠉⠀⠒⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡀
⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠛⠛⠻⠶⣄⡈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠒⠒⠛⠉⠻⢶⡶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⣄⠀⠀⠈⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⢒⠒⠤⠄⠂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡸⣿⡶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠈⠉⢉⣻⣿⣷⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⢺⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢿⠋⠀⠀⠀⣠⣴⠾⠟⠛⠿⢿⣿⣿⡿⠿⢷⡄⠀⠀⠀⠀⠘⡃⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠁⡀⠀⢤⣤⡄⠀⠀⠈⠑⠂⠈⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠀⠀⢙⡧
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⠋⠀⠀⢀⠀⠀⠀⠀⠀⢠⡆⣴⢄⠤⢴⣾⡶⠦⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣗⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⢬⣿⡿⠷⠌⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⢠⣷⣶⠚⠷⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠹⢷⡄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠠⣄⠀⣷⡀⠀⠻⡀⠈⠉⠀⠀⢀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠻⣷⣈⠛⠀⣀⣀⡀⠀⠀⠀⢸⣦⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⢀⣀⠀⣸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠘⣷⣄⡀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠰⠀⠀⠀⡀⢠⣄⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀
⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠿⠷⠦⠀⠀⠛⠛⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣬
⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⡆⠀⢀⠀⡀⠀⠀⢠
⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⣀⢠⣶⣶⣤⡀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣆⣾⣇⣤⣤⣎⠠
⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⣄⣀⣾⣤⣾⣿⣿⣿⣷⣴⡶⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣧⣌⣟
⣄⡀⠀⠄⠈⢿⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣦⣬⣿⣿⣿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⠁⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠙⢿⡄⠀⠄⠀⠻⣆⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⣀⣩⣭⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠳⣄⢠⣴⣿⡄⠀⠀⠈⠻⢦⣤⣤⣶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠢⣀⠀⠀⠙⠛⠿⣷⣄⠀⠀⢀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠉⠁⢀⠈⠀⠀⣇⠀⠀⢀⡀⠀⠀⢀⣇⠀⣄⠀⣸⣿⣿⣿⣿⣿⡿⣿⣿⣿
⠀⠀⠈⠁⠀⠀⠀⢀⣀⣉⡓⠤⣸⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⢧⡀⠀⠀⣷⣄⠀⠛⠀⣄⠀⠀⣘⣈⣇⣠⣀⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⡏⠀⢼⣿⣿
⣷⣄⠀⢀⣾⣿⣷⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⠛⢙⣿⢋⠶⠒⣿⣿⣺⣿⣷⣶⣿⣭⣿⣧⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣭⣿⣿⣿⣿⣇⣿
⣿⣿⣷⣼⣿⣾⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣮⡶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠾⣿⣟⣛⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣭⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣷⢏⣼⣿⣿
⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣽⣿⣿⣿⣿⣿⣟⣵⣿⣿⣿⡿
⠿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⣿⣿⡿⣯⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣟⣿⣿⣿⣾⣿⣿⣿⠿⣛⠯⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠶⠶⣶⣦⣄⠤⣄⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠂⠉⠀⠀⠀⠤⠀⠀⠈⠩⠿⣶⡄⢉⠓⣦⡀
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠉⠁⠀⣈⣠⢄⡹⣿⡕⠉⠉⠍⢣⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠐⠂⠈⣙⢛⣛⠋⠤⣫⡟⣽⣷⠦⠤⠂⠀⠻⣦⠀⠀⠀⠀⢀⡴⠃
⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢠⠎⠀⠀⠀⠀⢀⠀⠴⠆⢀⣀⣴⣶⣶⣶⠶⢿⣿⣿⣿⣶⠒⠠⢀⠈⢧⠀⢀⡴⠋
⠀⢀⢂⢈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠋⠀⠀⠀⠀⡈⠤⠄⠒⢻⡿⠿⠛⠉⠁⠀⠀⠀⠀⠉⠻⣿⣯⠀⠀⠀⠀⠿⡁⢀⠀⠀⠐
⠀⠀⢂⠂⠀⠀⠈⠙⠒⠤⢄⣀⣀⠀⠀⠀⠀⠀⣸⠃⣰⠃⠀⠀⡀⠒⢁⣠⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠀⠀⠀⠈⣧
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠌⠄⡤⠤⢤⠀⣰⠇⠐⠀⠀⠀⣴⠦⣢⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⠀⢹⡆
⠀⠌⠌⠌⠄⠄⠠⠠⠀⠄⠈⠌⠼⡁⠀⠀⣷⡟⠀⠀⠠⠂⣼⣫⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢄⠀⠀⣿⠀⠀⠠
⠈⠈⠌⠈⠀⠀⠁⠀⠀⠀⠈⠌⠌⠉⠁⢩⡿⠄⠀⠀⠛⣼⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣎⢠⢀⢹⡄⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀⠀⠀⢸⡟⠉⠂⡄⢣⣿⡟⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⢤⠀⠀⠀⠀⢸⣿⣿⡄⢾⣟⡇
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⡐⠀⠀⣿⡙⠒⠋⢱⣼⣿⡇⠋⠉⣉⣉⡙⣛⠀⠀⠀⠀⠀⠀⠉⣉⣤⣤⣤⣀⠀⠀⠀⠀⠀⢈⠻⣿⣷⡜⣿⠗
⠀⠠⠡⠁⠀⠀⠀⠀⠀⠀⠠⠡⠁⠁⠀⢉⡇⠀⠀⢸⢿⣿⡇⠀⠾⣟⢛⠛⠳⣄⠀⠀⠀⠀⠀⠰⠿⣭⠁⠉⠛⠛⠂⠀⠀⠀⠈⠀⢹⣿⡀⢻⡷
⠀⠡⠡⠡⠀⠀⠀⠀⠀⠀⠠⠡⠡⠀⠀⠉⣿⠀⠀⠀⢘⣿⡷⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡧⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣿⢀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⣿⣇⢸⡇
⢀⢂⢂⢂⠀⢀⠀⡀⠀⠀⢀⢂⢂⢀⠀⢀⣿⢸⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠉⠉⠉⠹⣷⣶⣴
⠂⢂⢂⠂⠂⠀⠀⠐⠀⠀⠂⢂⣦⣶⣾⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠈⠀⠀⡀⠀⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠻⣿⣿⣿⣦⡄⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠤⠀⠀⠐⡇⠀⣽⣿
⠀⠀⠌⠄⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠈⢿⣿⣿⣿⠀⢿⣿⣿⣧⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠰⠀⢀⠀⠀⠸⠀⠀⠀⠁⠀⣿⡏
⠀⠀⡘⡀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠸⣿⣿⣿⣧⡀⠀⠀⠀⠙⠻⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⣀⠌⠀⠀⠀⠄⠀⠀⠀⣤⣿⡇⣸
⠀⡐⡐⡐⡀⠀⠀⢀⣿⣿⠀⠀⢠⡀⢳⡀⠀⠸⣿⣿⣿⠀⠁⠉⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠈⠀⠀⠀⠀⠨⠀⣠⣴⣿⣿⣡⡿
⠀⡱⡀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢷⡄⢳⡄⠀⢹⣿⣿⣇⠀⠀⠈⢹⢿⣿⣿⣷⡀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠚⠈⣿⣿⣿⣿⡇
⠀⠽⠡⠀⠀⠀⠀⢾⣿⣿⠀⠀⠀⠈⣿⣌⢿⡄⠀⣿⣿⣿⡄⡀⠀⠈⠘⣿⣿⣿⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢺⢻⣿⢃⠀⠀⢹⣿⣿⣿⣧
⠀⢽⠡⠡⠁⠁⠄⣼⣿⣿⠀⢠⣄⠀⠘⣿⣾⣿⡄⢸⣿⣿⣿⣷⠀⠀⠀⠸⡞⣽⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢾⣿⣤⠀⠀⠘⣷⠟⠀⠀⠀⠛⣿⣿⣿
⠀⢩⢁⠀⠀⠀⠀⣿⣿⣿⠀⠸⣿⣷⡄⠹⣿⣿⣿⡘⣿⣿⣿⣿⣧⢊⡀⡀⢹⣵⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠘⣿⣇⣇⡏⠀⢿⡄⠀⠀⠀⠀⠘⣿⣿
⠀⢻⠂⠀⠀⠀⠀⣿⣿⣿⠀⡄⣿⣿⣿⣦⣹⣿⣿⣿⣿⡏⣿⣿⣿⣿⣧⢱⡀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⢃⢸⣷⠀⠀⢀⠀⠀⢹⣿
⠀⢈⠄⠀⠀⠀⠀⣿⣿⣿⠀⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⡟⣿⣿⣿⣿⡎⣷⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣸⣿⣷⡀⣿⡆⡲⢘⣿
⠀⢾⠌⠄⠄⠀⠀⣹⣿⣿⡇⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⢠⡇⠘⣿⠻⣿⣿⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣾⣿
⠈⠜⠌⠈⠈⠀⠀⢹⣿⣿⣧⣿⣇⣿⣿⣿⣿⣿⣿⣿⡿⣸⡇⢠⡇⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠀⠂⠀⠄⢀⠀⠀⠀⠀⠠⠄⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢺⣿⠆⠀⠀⣀⠐⠓⠋⠀⢀⣀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⣤⣌⠁⠂⠄⢀⠀⠀⠀⠁⣴⠏⣷⠀⣴⡄⢠⠀⢠⡲⢶⡿⢹⡇⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡿⠃⠀⠀⢀⠃⠀⡄⠰⣿⣴⠇⣴⣿⡅⠀⣥⠈⢶⣁⣀⣸⠃⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠆⠀⠀⠀⠒⣤⣄⣈⠛⠾⠀⠀⠁⠆⠀⠉⠛⠁⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⢾⠆⠀⣼⣤⢄⡀⢀⡜⠿⠁⠀⠈⠑⠚⠀⠀⠘⡀⢀⠠⣤⣴⠈⠉⠀⠀⠀⠀⡀⠤⠄⠐⠒⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠀⠤⠴⠀⠉⠋⢥⣬⠞⠒⠂⡀⠀⠠⣄⠀⠀⠀⠁⠀⠀⠘⠹⠀⠀⠀⡆⠀⠀⠀⠠⣤⠒⠲⡄⢡
⠀⠀⣀⡀⣀⠀⠀⠠⠤⡀⠀⠐⢐⠢⡀⠀⠀⠙⣍⠠⢀⡀⠀⠈⠠⡈⠆⡀⠀⠀⠠⠀⠔⠁⠀⠀⠀⢰⠀⠄⠰⠆⣀⣰⠒⠻⢧⢄⣀
⠀⠀⠁⠀⣠⠀⣄⠀⠀⠁⠘⡀⢠⣄⡄⠀⠀⠀⠈⠑⠀⣌⢢⠀⠀⠈⠢⠈⢠⠂⣴⠁⠀⠀⠀⠀⠀⣠⡆⠂⠐⠒⠉⠙⠀⡀⠐⠚⢤⢉⠉⠑⠆
⠀⠀⠀⠀⢻⣿⣿⣻⡷⢸⣿⠹⠦⢌⡃⢄⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠃⡞⠛⠡⢀⠂⠤⣂⠅⠛⠙⣄⣀⡀⠤⡆⠀⡇⠰⠶⠄⣗⠀⠘
⠀⠀⠀⠀⠘⠿⠿⠿⠇⢸⠀⠀⠀⠀⠈⣂⠷⠠⡄⣠⠜⢡⡀⠀⠀⠀⠀⢀⡄⠃⣀⠀⠀⠉⠴⡷⣄⠀⠀⠀⠀⠀⠰⠀⠈⢀⣷⣾⠋
⠀⠀⠀⠦⠄⠀⠀⠀⠐⠋⠀⠀⠀⢀⠔⣡⡶⠆⠀⡌⢠⣟⠋⡼⣣⣴⣤⣬⡙⢎⡠⠤⣙⡞⣀⢰⠃⣄⡀⠀⠀⠀⠦⠀⣀⠈⠙⠃⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣽⠀⠋⢁⠔⡙⠇⢸⠛⣦⣌⠃⣬⡗⠡⣼⣦⣢⡼⢉⠇⡀⡾⠀⠛⠫⠐⠤⣀⠠⠐⡻⠏⠉⠐⠒⠒
⠀⠀⠀⠀⣀⠴⠲⡄⠀⠀⢀⠰⠷⠀⣤⣡⣤⡄⢇⣈⣤⡙⠛⠋⣣⠘⠋⠳⣦⣤⡤⢶⣧⡤⢋⣴⣧⢀⡀⠈⠑⠶⠆⠊⠀⢀⢀⣀⡀⠀⠠⠶⠴⡆
⠀⡠⠐⠉⠠⠀⠀⠀⢦⣾⠄⠂⠁⠒⢄⠙⢮⣻⡊⢹⠋⢉⠟⡟⢴⠀⠀⢠⢏⠳⣚⡺⠷⣝⢯⡹⠁⢆⡆⠂⠀⠀⠀⠀⡏⠀⢀⠒⢄⠀⢀⡀⠀⡇
⠻⡀⠀⢠⠊⠈⢂⠠⠈⢃⠀⠀⠀⠀⠀⠉⠐⢶⣶⣡⢞⣉⠁⡾⢃⣜⠳⢭⣒⡷⢈⣄⢀⣍⣷⠑⣶⣯⣴⣷⠐⠒⠤⠿⠇⠀⠙⠂⠋⡠⢒⢉⠴⡇
⠀⠡⡀⠀⡂⠐⠀⠐⠀⢀⠃⠀⠀⠀⣀⠀⢨⣽⣯⣇⣉⣻⠟⣰⣿⣿⡷⣦⠙⣿⣾⡟⢿⣿⣾⢀⣿⣿⣭⠉⢋⠁⠀⠀⢠⠀⠀⠀⠸⡐⠉⠋⠀⠇
⠀⠀⠁⡀⠀⣁⠤⠒⠉⠀⠀⠀⠀⢠⠁⠉⠒⠩⠙⠛⠛⠿⠾⢧⣼⣏⣴⣿⣰⣿⣵⣿⣤⡿⠧⠾⠟⠛⠩⠐⠉⠀⠀⠀⠸⡀⠀⠰⠶⠧⠤⠄⠀⠚
⠀⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⡁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⣀⠣⠤⠤⠤⣈⢢⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠂⠉⣑⠂⢄⠠⡂⢤⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⢾⠀⠀⠾⠁⠀⠀⠀⢠⡀⠙⠃⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠉⠺⠦⠀⣿⢀⣖⠁⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠘⣦⠠⠀⠀⢠⡾⠛⡢⣼⡗⠻⠳⡀⠀⠀⠀⠀⠀⠀⣰⠾⡖⣶⣶⡦⣶⡄⠀⠅⡻⢬⣶⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠋⠀⠀⠀⠀⠑⠢⠤⠐⠊⠠⠐⠁⠀⣄⠀⠀⣦⠀⠁⠦⡀⢋⣛⣂⠤⠃⠀⠀⢵⠠⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣇⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠴⡎⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⡅⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⠀⢦⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⠀⠀⠀⠀⢀⣰⠧⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠶⠶⠟⠛⠐⠃⠀⠀⠀⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⣈⣐⡠⠈⠁⠐⠢⡄⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠾⡯⣭⣄⣤⣀⣀⣈⣉⣉⠐⠚⢷⡄⠀⠀⠀⡀⠂⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠰⣀⠙⠳⢬⣭⣡⣽⡽⠟⠛⠛⠋⠀⠀⠀⠀⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⣀⡄⠈⠀⠀⠀⠀⢐⠑⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢆⣀⡀⠀⠀⠀⠀⠀⠀⠉⠘⠙⠋⠉⠀⠀⠀⠀⠀⠀⣀⢾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠚⠃⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⠣⢄⣀⠀⠀⠀⠀⠀⣠⡀⠄⢠⡟⠀⠄⠀⠀⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠊⠌⡙⠛⠓⠰⠴⠴⠤⠞⠋⠀⠌⠀⠀⢀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠘⠀⠐⠀⠃⠚⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠠⠀⠐⠖⠀⠞⠑⠞⠳⠶⠳⠞⠰⠗⠿⠞⠷⠋⠞⡑⠟⠺⠒⠷⡺⢞⠶⢃⠾⡚⢟⠳⠛⠚⠛⡛⠟⠻⠟⠛⠀⠆⠀⠂⠀⠀⠀⠠⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⠶⢶⣤⣴⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣿⣧⣤⣶⣿⣷⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠉⠉⠉⠛⠛⠛⣻⣿⡿⠟⠋⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠋⠀⡠⠴⠒⠚⢛⣛⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣄⣠⡴⠾⠟⠻⠿⢿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠋⢁⣤⡤⠖⠂⢀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡵⠖⠚⢉⣡⣤⣶⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣶⣶⣶⠿⠛⣻⣭⣽⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⢛⣩⣴⣶⣿⣿⠿⠟⣹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⣯⣿⣽⣿⣿⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢻⣿⣿⠇⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡘⢣⣿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⢠⠞⢋⣤⣿⣿⣿⡟⠀⠱⠄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡘⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠃⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⠙⠋⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣷⣾⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⣦⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣵⣾⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⢿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣏⣿⡿⣟⠗⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠈⠁⠀⢉⣿⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠿⡛⠻⢿⣯⡞⠁⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢺⣿⣍⠉⠻⣶⡀⠀⠘⢿⣖⠁⠈⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣟⠫⡒⢌⠢⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠉⠻⣷⣆⡔⢋⢔⡨⡙⢷⣦⣤⣤⣿⣿⣷⡀⠹⣷⣤⣶⡟⠁⠀⠀⠘⠉⠙⢿
⠀⠀⠀⠀⠀⢀⣼⣿⣿⡟⢭⡑⠬⠑⠈⠂⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣅⣫⣢⣿⣿⣿⡿⠿⠿⢿⣷⣄⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣾⣿⣟⠳⢍⠢⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⢑⡮⢻⣿⣿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣼⣿⡳⣌⠓⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣄⡀⠀⠀⠀⠰⢋⣴⣿⣟⣭⣿⡏⠀⠀⠀⠀⢸⣿⣿⣉⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣸⡿⠁⠙⠦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣄⡀⣰⣿⣿⣟⣡⡴⢾⡇⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣶⣯⢛⠻⢿⣿⣿⣿⣿⣵⣾⣿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠛⠁⠀⠀⠈⢻⣿⣿⠿⠋⢻⡇⠀⠀⠀⢠⣶⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠐⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣁⡴⠞⠛⣧⠀⠀⢠⣾⣿⣿⡿⠿⠿⠿⠷⣶⣦⣤⣤⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠹⡄⢠⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⣶⡶⣾⣷⠶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣴⡆⠀⠀⠄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠶⠻⣿⠿⣿⣿⣷⣤⣤⣶⣶⣶⣤⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠒⠋⠉⠀⠀⣤⠀⠀⠀⣽⠟⠋⠉⠩⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠈⣿⣦⣰⣿⣯⠀⡀⠀⠲⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣧⡜⠤⠤⠌⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢿⣏⣩⡒⠴⠶⠄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢰⣦⣄⣦⠈⣣⣆⠀⠀⢀⢀⠀⠀⢸⣿⣿⡟⠻⠶⣶⣤⣤⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠙⢿⣿⣿⣧⣹⡇⣥⣸⣎⢳⣴⡾⠻⣿⣧⡄⠀⠀⢀⡟⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡴⣷⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣽⣾⣿⡇⠀⠈⢿⣿⡅⢠⣾⠁⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣥⣈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⡄⠀⠘⡿⠋⠁⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠳⣄⣈⡙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⠃⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠀⠀⢈⠛⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣦⣤⠀⢀⣤⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠉⠉⠛⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠆⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠙⠛⠛⠿⢿⣿⣿⡿⠛⠉⠉⢙⠁⣠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢣⡄⠀⠀⠀⠀⠀⣤⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⣀⢀⠀⣦⣼⣿⠂⣄⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠁⡀⡷⠀⠀⠀⠀⣹⣿⣿⣷⣷⣶⣶⣦⡀⢀⡀⠰⠄⢀⣾⠀⠻⣿⣿⣿⣿⣯⣶⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⣤⢀⡶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠛⣭⣻⣶⣾⣿⣷⣶⣦⣾⣿⣿⣿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡡⠂⠀⠀⠈⣷⣿⡿⢛⣩⣍⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢩⣿⣶⢿⣿⣷⡾⣿⣿⣿⣿⣿⣿⣿⣧⠇⣠⣿⠈⣣⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠡⣾⣿⣿⣿⠋⠁⠚⢻⣿⣿⣿⣿⣿⡟⢃⣀⣿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⡟⠛⠿⠦⠀⡼⣿⣿⣿⣿⣿⣥⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⣟⠁⠀⠀⡆⡀⠀⠘⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⡟⠉⠛⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⡈⠀⠀⠘⣋⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡀⠰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣀⣤⠒⠾⠟⣿⣦⡦⠀⠀⢿⣿⠿⣿⣿⣿⣿⣿⣁⠉⠻⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠁⠈⠀⠀⠙⠛⠃⠀⠀⠀⠸⠟⣴⣄⣽⣿⣿⣿⣿⣿⣦⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠄⠀⠀⡠⠀⠀⠀⠀⠀⠀⢿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣮⡀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣟⢳⡀
⠀⠀⠀⠀⢀⡀⣀⣀⣿⣵⢿⠇⠀⢀⣺⣷⠀⠐⡀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠉⠛⣻⣿⣿⣽⠇⠹⣶⣶⣤⣄
⠀⠀⠀⠀⣟⢨⠉⡉⠉⠈⠁⢀⠁⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⢤⡉⠳⣶⣄⣄⢨⣬⠟⠛⠛⠀⠀⠻⢿⣿⣿⣿⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⣿⣟⣀⣠⣐⣀⣁⡻⢷⠤⣶⣤⣤⣤⡀⠀⢤⣤⣶⣶⣿⣆⡈⣉⣙⣋⣁⣀⣤⡀⢀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣷⣿⣷⣬⣿⣿⣿⣯⣍⣽⣻⣭⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⣀⡴⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⠀⠀⣤⢲⢲⡄⠀⠀⢀⠀⠀⠀⢴⣛⣟⣱⠀⠀⢸⡆⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠸⣆⠀⠈⠉⠉⠀⠀⠀⠘⠃⠀⠀⠀⠉⠉⠀⠀⠀⣸⠇⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠳⣤⡀⠀⠀⠀⠈⠐⠚⠁⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠶⠶⠶⠤⠶⠶⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣇⠀⠀⣤⠶⣶⣶⡶⢄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣯⣿⣟⣤⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠦⠭⠯⠴⠚⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣭⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠻⢶⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡟⣰⡃⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⢀⣤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠒⠰⢶⢰⣦⠀⡀⢀⣀⢀⣀⠀⠀⣠⡀⡇⠀⠀⠚⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠁⠈⠉⠈⣻⠀⠀⠉⠁⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣄⣀⣀
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⢠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⣀⡧⣀⠐⢤⣄⣠⠀⢀⡀
⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡑⠀⠋⢳⠀⠈⠘⠓⠺⡠⡄⢐⠏
⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡠⢤⡀⠀⠀⠀⠀⠀⢉⡵⠌⠉⠂⠀⢈⣀⣠⣆⣀⠀⠀⠁⡸⡄
⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡴⠁⠎⠀⡼⠁⡠⢄⠀⠀⠀⠀⠀⠀⡉⠷⠤⠚⠉⠉⠉⠉⣻⣦⡀⠸⢤
⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋⠂⠁⠀⠀⠲⠊⠀⠰⢀⠌⢠⢉⠂⢐⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣍⡁⠰⠥
⠀⠀⠀⠈⠒⠦⠤⣤⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⢠⢋⠀⠀⠠⢤⡀⠀⠀⠀⣿⡏⠉⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢰⠳⠤⢠⡌⠀⣠⣴⠅⣀⠼⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠈⠑⠒⠊⠀⢼⡵⢒⠲⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠆⠀⠀⠀⠀⢀⣀⡸⠰⠹⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢄⠀⠀⠱⡀⠀⠉⠉⠀⠐⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠁⠒⠒⠃⠀⣇⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠉⠉⠓⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⠄⠈⠫⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢣⡆⠀⢀⡴⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠁⠒⠋⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠠⠖⠋
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⡀⠄⠐⠒⠒⠒⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠄⠐⠄⠄⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⡰⠁⠀⠀⠀⠀⠸⡄⢀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡖⠀⠈⠢⠀⠀⠹⠀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⣶⢟⣿⣉⣩⣟⣝⣉⢝⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⢸⠀⠀⠁⠀⠀⢀⣗⡖⠲⢶⠶⠶⣶⠦⣤
⠀⠀⠀⠀⠀⠀⢠⣿⡿⡟⠀⠀⠈⠉⠉⠉⢹⠀⠀⠀⢀⠃⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠉⠉⠛⠛⠳⡖⣻
⠀⠀⠀⠀⠀⠀⢸⣿⡕⡇⠀⠀⠐⠂⠉⠀⠀⡆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠒⠀⠀⠀⠀⣇⣼
⠀⠀⠀⠀⠀⠀⠈⣿⡆⣿⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⣇⣽
⠀⠀⠀⠀⠀⠀⠀⣿⣄⣾⠀⠀⠀⠀⠀⣀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠠⢀⠀⠀⠀⠀⢸⡏⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣕⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣯⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠑⡖⠒⠒⠒⠒⡎⠉⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣻⡇⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠷⠤⠤⠶⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡧⢋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠽⠯⠍⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠋⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⢃⣿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⡼⢇⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠇
⠀⠀⠀⣀⣀⣀⣀⣀⣿⣾⣿⣣⣟⣝⣡⣜⣽⣛⣩⣋⣛⣻⣫⣋⣛⣽⡫⣩⣝⣁⣽⣟⢭⡽⡫⠋⣩⢯⢝⣽⡿⢋⠝⠉⣽
⠀⠀⢸⡍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⢳⡶
⠀⠀⠀⠉⠉⠙⠛⠛⠓⠚⠛⠛⠒⠛⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠓⠓⠒⠚⠒⠛⠒⠒⠚⠚⠓⠒⠛⠒⠒⠒⠶⠶⠶⠶⠦⠴⠦⠴⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠲⠚⡓⠒⠶⠤⠤⣄⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠠⠒⢀⠀⠀⠑⠀⠀⠘⠃⠀⠐⢦⠈⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠠⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠰⠶⠄⠀⠀⢸⠇⠀⠀⠀⠀⡔⠊⢉⡛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣦⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣦⠀⠀⠀⠀⠷⡴⠋⠙⢳⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⠀⠉⠁⠀⠀⢸⡶⢻⠀⠀⠀⠀⠀⣀⣀⣠⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⣦⠾⠀⠀⠀⠀⣎⣠⡤⠴⠖⠒⠉
⢀⣠⠴⠶⠦⣄⡀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠼⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⡄
⡎⠠⡟⠉⢳⡀⡳⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠓⠃
⠛⠚⢋⡴⢋⡴⠃⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠤⠔⢚⣉⠍⠁⠀⠀⢀⡞⠁
⠀⠀⠼⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⠀⠀⢠⠒⢢⠀⠀⢀⡴⠋
⠀⠀⠀⠾⡽⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠉⡹⢻⠒⠢⢤⠀⠀⣤⢾⠋
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢧⡹⡀⡠⠬⠀⠀⡏⢹⠧⣄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠁⠀⠀⠀⡇⡟⠀⠈⠛⠦⣄
⠀⠀⠀⢀⡞⠁⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠁⢹⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⡾⠁⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠸⣧⢄⣀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢠⡇⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⢹
⠀⠀⣸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢗⠒⠒⠒⠒⢲⡶⢾⡇
⠀⠀⣿⠀⠀⠀⠀⠉⠛⠛⠲⠶⢦⣤⣀⣀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠘⣆⠀⠀⠀⢸⠟⠉
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠶⢿⡀⠀⠀⠀⠀⠘⣏⠭⢷⣼
⠀⠀⠈⢳⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⢱⡈⢩⣿
⠀⠀⠀⠀⠈⠉⠉⣿⠋⠛⠛⠓⠒⠒⠒⠒⠒⠲⠶⠦⠤⢿⣇⠀⠀⠀⠀⠈⣧⠴⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠷⠀⡟
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡗⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣏⠀⣠⡀⠀⣀⡉⠛⠾⠿⠛⠛⠛⠻⢝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠄⢿⣿⣿⣷⣟⡿⢠⡀⠐⡞⠀⠀⠀⠀⠀⠁⢉⠜⠃⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⣿⣿⣿⣾⣿⣶⣿⣧⣄⣠⠄⣤⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⢩⣝⣿⣿⣿⡿⠏⠉⠁⢸⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⡙⣿⢿⣿⣿⠀⠀⠀⢹⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⢸⡿⠋⠀⠀⢀⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⢀⣀⣴⣤⡤⠼⣿⣿⣿⣿⡿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣟⡁⠀⠙⢿⡿⠛⠟⠛⠛⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣦⡀⡈⢿⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣯⡈⣿⣿⣿⣧⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣎⢻⣿⠻⢿⣿⣿⣷⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⡿⣿⣿⣿⣇⠿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⣿⣷⣦⣼⣿⠟⠃⢉⣿⡿⠋⠙⠢⠀⠀⠑⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣵⢟⠛⠁⢤⡤⣤⣀⠀⣣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⠀⠉⠁⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠁⠈⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⠉⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⣠⠤⣀⠀⠈⢧⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⣰⢚⠉⠀⠀⠈⢦⠀⠘⡆⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⠖⠁⠉⠀⠀⠀⠈⠉⡆⠀⢻⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⠧⡀⠀⠉⠀⠰⠀⠉⢀⡇⠀⢸⠀⠀⠀⠀⠀⡼⠀⠀⣀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠹⣄⠀⠺⡭⠗⢀⡜⠀⢀⠏⠀⠀⠀⠀⢠⡧⠖⠋⠁⠀⠙⢦⡀⠀⠀⠀⠀⢀⣠⠎⠀⠉⠛⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⡀⠀⢀⡼⠓⠒⠒⢶⡉⠀⣀⣫⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠉⠙⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡭⠻⡟⠁⠀⠀⠀⠈⠛⣫⠤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣠⢾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠒⠤⠤⠤⠴⠚⠁⠀⠀⠙⢦⠀⠀⠀⠀⢀⡇⠋⠀⢙⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⣄⣀⣄⡀⠀⠀⠀⠀⠀⢳⡀⠀⢀⣼⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡀⠀⢀⠔⠱⠫⢕⡥⡿⠀⠀⢀⠀⠀⠀⠱⣄⠎⠀⠉⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⣠⣼⠀⠀⠀⢘⡾⠃⠀⠀⢸⠃⠀⠀⢠⠏⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡠⠋⠁⠀⠙⠲⣶⠉⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠴⠋⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⢸⠈⠳⣄⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢤⠤⢤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠙⠛⠛⠻⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⠁⠀⢀⣤⡀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠛⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣧⣾⣿⠏⠹⣿⣦⠀⠀⠀⢀⣠⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣶⣾⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⡿⠟⢻⣿⣿⣿⣿⣿⠟⣿⣷⣼⣿⣿⣤⣿⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣍⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠣⠀⠄⠀⡀⠈⠀⠀⠀⠐⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣦⡁⠐⠉⠂⠀⠀⠀⠄⠀⠀⠀⢀⣴⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣆⠀⠠⠀⠀⠀⡈⠀⢀⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⠀⡄⠰⠀⡄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⣿⠛⡇⢸⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡔⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠻⣤⣤⠀⣿⡇⢠⣤⡴⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠅⣠⡝⠀⠹⠀⠘⣥⡀⠭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠈⠠⠠⠀⠀⠀⢠⠀⠈⠠⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠤⠤⠀⠈⠛⠀⠀⠀⠘⠃⠀⠠⠤⠴⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡀⠈⠀⠀⠁⢠⣤⣀⣠⣤⠂⠁⠀⠈⠀⡀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢃⠀⠈⢂⠀⠀⠙⢻⠛⠁⠀⢀⠊⠀⢀⠃⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠦⣀⠀⠐⣤⡴⢾⠶⣤⡔⠀⢀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢈⠒⠹⡄⠀⠀⡼⠑⢊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠓⠀⢰⣄⠀⠀⠀⢀⣴⠀⡐⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠸⡿⠟⠋⠛⠿⡿⠀⢀⣀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠙⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣠⡀⠔⢮⢴⠠⡰⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢽⣹⠞⡁⢸⡘⣿⠚⡵⢟⣿⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣟⣓⣇⠈⠸⣦⠀⢰⡄⢤⠈⠀⣍⠉⠭⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡀⠄⠀⡉⠖⠁⠀⠘⠈⠈⢁⠘⣆⠁⠸⣀⣋⠤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢈⠄⡐⡘⠂⠉⠀⣘⠈⠀⠃⠈⠀⠉⠀⠀⠀⡐⡂⠀⣈⢇⡇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢏⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠌⠠⢀⠆⡥⠑⠀⠁⢲⣽⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠘⣫⠀⡀⠀⠀⠁⠸⠀⠊⠈⠈⠈⣑⢠⡔⣆⠘⣼⢻⣶⡂⠗⣭⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠷⠌⠉⢀⡠⣤⠠⣴⠞⣰⢿⢶⡾⠲⠎⠀⢩⠟⠝⣧⡑⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⣞⢂⡇⠧⢪⣀⣾⣾⣼⣪⣷⠟⣦⠀⢩⠆⠲⣆⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⠆⡳⣯⡯⣞⢿⣻⠽⡖⠟⢏⠱⢧⣘⣧⣼⡛⠓⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⠀⠀⠐⠳⠛⣯⢎⠰⢇⢃⡴⡴⣚⣷⣯⠌⡁⠀⠀⠀⠀⠑⠪⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢯⡍⢁⣏⣈⢡⣞⣿⡵⣶⣮⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣭⢭⣽⠙⣝⠌⡶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⢏⢴⡥⡔⢦⣴⣚⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣚⣻⡯⠄⠈⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠉⣉⣁⣤⠴⠖⢋⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣉⣁⣤⠤⠔⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⠀⠀⠀⠀⣰⣿⣇
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣦⣶⣾⣿⣿⣿⣿⣿⣦⡄⡠⣀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢰⣿⣿⡿⠛⠉⠉⠛⠻⣿⡿⠋⠉⠈⢣⢡⠀⢣⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠿⠟⠻⠿⣿⣿⠷⣄⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⢰⣶⣦⠘⠁⢠⣶⣦⡈⡀⠂⠀⡇⠀⢠⣾⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠈⡇⢀⡘⣿⣟⣯⣿⣷⡄
⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⣹⣿⣿⠀⠀⢸⣿⣿⣧⠁⠀⠀⡆⣰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣷⠀⠀⣿⣿⢿⣏⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣷⠀⡠⡤⣁⡀⠉⠀⠀⠉⠉⢉⣟⣥⣭⣲⠀⠘⠻⣿⡿⣿⣿⣿⣷⡀⠀⠀⠀⠙⠋⢀⠶⣘⠋⡞⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣆⠢⡁⡘⢸⢸⡸⠀⢀⡴⠋⢸⣿⣿⡇⠀⠒⠻⠿⣿⣿⣷⠋⡉⠛⠷⠖⠒⠒⠈⠁⠀⠸⣿⠿⢼⣷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡒⣣⡭⣭⡒⠚⠁⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠉⠻⢆⡑⠤⣀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⣣⣄⣠⡋⠉⠒⣶⣶⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣠⣭⣶⣦⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣾⣿⣿⣿⣿⣿⣷⠞⠛⠛⠛⠋⠉⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⢀⣴⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠻⣿⣿⣶⣿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣯⣝⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢻⣿⢟⠠⢙⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣅⠀⠖⠙⣸⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⡿⣙⣭⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⣷⣿⣯⠻⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠟⠛⠋⠉⢠⣿⣿⣿⡏⠙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣠⣿⣿⠏⠀⠀⠀⠀⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⢻⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣞⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠳⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠛⠁⠀⠈⠙⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢢⠀⠀⠀⢀⡤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⢦⡘⠁⠀⢧⡠⠒⠁⠀⣏⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠒⠦⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠩⠿⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠰⢤⠤⠼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⣀⣠⡌⠑⠚⠉⢉⠙⠲⢭⣑⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⣀⣀⣈⡉⢀⠀⠔⣛⡋⠁⠀⠀⠈⠳⣜⠉⠉⡹⠋⠀⠀⢀⠤⢄⠀⡀⠀⢙⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡖⠊⠁⠀⠀⠈⠲⠊⠉⠀⠀⠈⠑⢤⡀⠀⢈⢦⠀⠈⠓⣀⠄⠳⣄⠀⠙⠑⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡟⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⣼⠊⡇⢠⠊⠁⠠⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠁⠀⣨⠟⠉⠉⠀⠀⠀⠰⠶⣴⠄⠀⠀⠀⢀⠘⠀⢱⠀⢉⣦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡸⠒⣲⣾⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠈⠀⠀⠂⠀⠀⠀⠤⠀⠀⠀⡇⠀⠀⡰⢁⠞⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⡀⠀⠀⠀⠀⠀⠀⡔⠄⠀⠀⢀⡴⠃⠀⠀⣇⡼⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⢀⠊⠈⢇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡠⢈⡁⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠀⠁⠀⠀⠈⠳⣄⠀⠀⠀⠳⢖⠶⠆⠀⢀⠀⢀⠔⠁⠀⠑⠶⠊⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠒⠒⡡⠔⠁⡀⠀⠙⠒⢦⠄⠀⢀⠔⠉⠀⠧⣎⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣀⠀⠀⣀⡤⠖⠒⠒⡛⠀⠀⢊⡀⠈⠀⠀⠀⢙⣦⡀⣀⡤⣪⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⡠⠺⣅⠀⠀⠀⠉⣿⠶⠖⢉⣉⡘⠀⣦⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠈⠑⠒⠐⠚⠷⠖⠀⠀⠀⠉⠉⢻⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢚⡃⠀⠀⠀⡀⠘⡇⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡍⡁⠀⠀⠘⣇⠀⢳⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⡉⠀⠀⠀⠸⡄⠎⠧⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⢠⣇⣀⣀⣀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⡟⠂⠀⠀⠀⠀⠀⠀⠘⢟⣀⠀⢰⠉⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⢀⠸⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⡂⢸⣹⡽⡸⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⣂⡐⠒⠀⠲⠆⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢊⡡⠔⠢⠀⣀⣀⠀⠀⠈⠘⠰⢲
⠀⠀⠀⢀⠴⠊⠉⠁⠀⠀⠈⠙⠒⠛⠓⠶⣄⠀⠀⠀⠀⠀⠠⢿⡆⢨⢀⠤⠄⢉⡀⠀⠢⢀⠀⠀⠀⢈⣧
⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⣤⠖⠢⢠⢤⠙⠀⠀⣨⠀⠀⠀⣈⡆
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⠀⠀⠀⣦⠋⢸⡀⠀⠀⢀⠃⠀⠀⢱⠀⠀⠀⡅⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡇⢆⠀⠫⠁⠀⠈⠀⢀⡀⢌⠀⢈⠙⡄
⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣛⠶⠆⠀⢳⡀⠑⠒⠠⣴⣶⡞⠁⠀⣠⠼⠤⠴⠃
⠀⠀⠙⣦⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣙⢦⣤⣀⣉⣡⠴⠒⠯⣁⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⠞⠁⠀⢀⡤⢎⠉⢠⣧⠌⣀⠁⠀⠀⠀⣠⠞⠫⣆
⠀⠀⠀⠘⠷⣄⣀⠀⠀⣀⡴⠣⣄⣀⣤⠞⠁⠀⠀⢡⡀⢀⠘⣷⠫⠥⣄⣀⡅⠀⡀⡜⡇⠑⣄⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⡿⠁⠚⠙⠣⠈⠀⢀⡭⡗⠿⢟⡡⠛⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣸⠁⠀⠀⠀⠀⠀⢀⠜⠀⠘⠋⠁⢀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⣹⣄⣀⠀⠀⠀⠀⢣⣤⢆⠀⠚⡯⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣁⠀⠈⠉⢉⣉⠉⠁⠀⠀⠉⠒⠒⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠦⣀⡰⠉⢆⣀⡤⠴⡒⠺⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡮⡩⡇⠀⠀⠈⠁⠀⠀⠑⣄⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠉⠙⠻⣟⣯⡤⡺⠀⠀⠀⠀⠀⠀⠀⡸⢒⠚⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢍⣒⡛⠽⠋⠉⠀⠀⠀⠀⠀⠀⢀⡾⠓⠚⠻⠦⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢍⣒⠐⠒⢒⣊⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠀⢀⣀⠀⣀⢀⡀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠄⠈⡐⣽⣻⣿⣿⣿⣿⣿⣿⡇⠈⠛⠷⣤⡙⢦⠽⠞⠃⠉⠉⠙⠓⠦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰⣑⣿⣿⡿⠿⠛⠛⢻⣧⠘⠿⠶⣦⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠈⣿⡄⠻⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡓⠶⣤⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠚⠘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠔⠀⢉⠀⠀⢀⠹⢷⡶⣲⢲⢺⡆⠀⠀⣬⣙⠳⣦⣌⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡴⠇⠺⠘⠁⣴⠻⣦⠙⢷⣄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣇⢈⣁⡉⠁⠀⠉⢄⣠⢤⠀⢀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⢿⠔⠉⠈⣃⠀⠀⡀⠉⠈⠀⠀⠀⠀⠀⠁⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⡔⠈⠀⠀⠉⠙⠶⠚⡄⠀⠀⠀⠀⠀⠠⣿⣿⣿⢾⠖⢤⠸⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢾⡟⣿⣛⡃⢈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣼⣟⣿⣉⡇⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⢀⠀⠀⢠⠜⠋⢁⠐⢠⡊⠉⠉⣟⠋⠛⠀⠃⠀⠦⠄⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠘⠃
⠀⠀⠀⢀⡴⠀⠰⡀⠀⠛⡤⠶⣀⠶⠴⠤⠜⠁⠆⠀⢀⠀⡤⠃⠀⠀⢾⠉⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠈⢷⠀⠀⠈⢢⠀⠈⠂⠀⠤⠔⠒⠉⠙⠢⣄⡀⠂⠰⠤⠤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⢀⣤
⠀⠀⠀⠀⠈⠒⠀⠐⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⠄⢀⠀⠀⠀⠀⠀⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠠⠀⡄
⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠛⠓⠆⠘⠃⠁⠀⠀⠀⢀⡀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠈⢀⡀⢀⣤⣀⣥⢰⡄⣿⢰⡟⣰⠗⣠⣶⠖
⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣄⢠⣉⡀⠤⣤⠀⠉⠀⠈⠉⠀⠈⢂⡀⠀⠀⠀⠀⠘⠻⠶⠋⠁⠀⠘⠛⠓⠾⣼⣫⠀⢋⣡⣶⠇
⠀⢠⠈⠓⣀⠀⠀⠀⠀⠀⠀⢀⠀⠃⣀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⢁⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢛⣉⣥⣤⡆
⠀⢰⠀⠂⠀⠈⠓⠦⠤⠦⠔⠠⠶⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡐⡂⠘⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠭⠤⢤⣤⡄
⠀⢸⣀⣇⣨⣨⡁⣏⢨⣁⡇⣸⢀⣙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⢼⡏⡁⠀⠀⢉⣷⣦⣄⣀⣤⣄⡀⣤⣴⣯⣽⠗⢶⣤⣤
⠀⠈⠀⠉⣿⡾⠖⠻⣼⡶⠗⢺⠊⠉⠀⠘⣧⡀⠀⢀⣤⡀⠀⠀⠺⠇⠀⠀⠀⠀⠈⠁⠠⠷⡀⠀⠰⢟⠹⠾⣻⠟⣿⣿⣿⡿⣿⢷⣌⠲⣷⣬⣍
⠀⠀⠀⠀⠀⠠⠀⠀⠈⠑⠀⠈⠀⠀⢠⠀⠀⠙⠋⠉⠀⠈⠉⠋⠀⠒⠓⠚⠛⠛⠂⠀⠀⠀⠙⠰⠾⠿⠏⠾⠿⠂⣰⡿⠿⠻⠟⠃⣛⢋⡙⢫⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠄⣤⠤⠤⠦⠴⠖⠆⠐⠒⠠⠀⠖⠛⠈⠁⠍⠻⠄⠀⠀⠀⢥⠼⠷⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡀⠀⢀⣀⡄⠀⠀⠀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢲⣿⣧⣴⣤⣶⣴⣷⣶⡶⢶⣴⣆⠀⢘⣷⣼⣯⣿⣯⣿⣿⣴⣷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⣿⠁⠀⠀⠈⢿⠿⠛⠿⣿⣿⣿⣿⣿⣇⣀⣸⣿⡇
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠈⢐⣄⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡷⠄
⠀⠀⠈⠉⠑⠢⠖⠚⠓⠂⠀⠀⠀⠀⠀⢣⡀⠀⠀⣀⠀⢀⣀⣀⣨⠁⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⣀⠀⡀⠀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠡⠄⠀⠉⠙⠉⠉⠉⠁⠀⠀⠁⠀⠀⠀⠀⠀⠁⠛⠋⠉⠛⢙⡓⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠊⠀⠄⠀⠀⠀⠀⢰⠀⠀⡆⠀⣶⠀⢸⢣⣞⢆⢲⠀⡶⠀⠀⠀⠀⢀⢀⠀⢀⠀⢀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠖⠀⠀⠀⠀⠀⠀⢸⠖⠒⡇⢰⠧⣇⢸⠎⢸⠎⠈⡿⠁⠀⠀⠀⠠⠁⠀⠙⠁⠀⠀⠃⠁
⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⣸⠀⠸⣸⡄⢸⠀⢰⠃⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠁⠀⠀⠉⠁⠈⠀⠉⠀⠀⠀⠀⠀⠐⠈⠑⠄⢀⠔⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠏⡡⢈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡤⠀⠀⠀⢠⠀⢠⡄⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠄⡇⢀⡀⠀⢸⠀⠈⡇⠀⣀⣀⠀⠀⠀⣇⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⣸⠀⠀⡇⠸⡇⠀⢾⠀⠨⣧⠀⢧⣠⠂⠀⠀⢹⡀⢰⠆⢠⠇
⠀⠂⠀⡀⠀⠠⢀⠀⠀⠀⠀⢠⡙⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋
⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠰⢮⣙⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠳⢤⡍⢀⠀⠃⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⢄⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠖⠰⢈⠀⠁⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠄⡀⠀⠀⠀⠀⠀⠈⠒⠈⠉⠀⠁⠈⠛
⠀⠀⢠⠄⠀⠒⠲⠒⠒⠒⠐⠂⠀⡸⠃⠀⠤⡈⢃⣀⢠⠤⠄⠒⠒⠒⠂⠀⠀⠒⠲⠤⢄⣠⠃⠔⠊⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠨⠀⠀⠰⠂⢠⣄⢤⠆⠀⠘⠢⣀⠀⢠⠌⠈⠍⠟⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠓⠂⠤⠀⣃⠀⠀⠀⠀⢀⠔
⠀⠀⠘⠂⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡤⡀⣤⠄⣤⡄⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣂⠒⠀⢠⠀⠀⠀⠀⠀⠀⢸⡗⢸⢿⣹⡧⢸⡧⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠁⡀⠔⠁⠀⠀⠀⠀⠀⠀⠘⠃⠘⠈⠛⠓⠘⠗
⠀⠀⠀⠀⢠⡄⣤⠀⠀⠀⠀⠀⠀⠄⠉⠁⠀⢀⠄⠠⡀⠂⠈⠐⠄⠀⠈⠢⡀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢸⡟⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠘⡇⣰
⠀⠀⠀⠀⢸⡇⠿⠸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠸⠠⠤⠶⠀⠀⠖⠆⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⢃⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠉⠀⢀⣀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣡⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⣀⠄⠂⠄⠄⠁⠀⠀⡀⠐⠈⠁⠛⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢚⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⠀⠀⠀⠀⠀⠘⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒
⠀⣀⣀⣀⣀⣀⣀⠀⠀⣿⠀⠐⣈⡙⢳⡄⢸⡇⢠⣄⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⠃⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢲⡈⠷⠁⣼⡇⠘⠳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠋⠻⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠈⠻⠛⠃⠈⠛⠟⠁⣽⣿⣿⣿⡇⠀⣠⣶⣤⠀⠨⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠁⠀⢀⣀⣀⡀⠀⠈⠻⠿⢿⣿⡆⠀⠻⣿⡿⠀⠠⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣴⣾⣿⣷⣆⣠⡀⣀⣰⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⣷⡄⠀⠀⠉⠉⠀⠀⢠⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⣶⣶⡄⢠⣴⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠃

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠉⠉⠉⠉⠓⠲⢄⡀
⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠲⢤⡀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠉⠓⠒⠒⠒⢲⠀⢀⡖⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡆⠀⡼⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠴⠊
⣠⠖⠒⠒⠛⠛⠛⠒⠒⠒⠒⠦⠤⣄⣀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⢤⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠤⠤⠤⠤⢤⣤⣀⣀
⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⣀
⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⠓⠦⠤⠤⠀⠠⠤⡄⠀⣀⣀⣀⣀⣀⣤⠤⠴⠚
⠀⠀⠙⢦⣄⣀⠤⠤⠤⠤⡀⠀⢠⠖⠒⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣍⠙⠳⣦⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⣴⠞⠋⣹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⢷⡀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢀⡾⠋⢸⡇⠀⢖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⠀⠈⢿⡄⠈⢿⡀⠀⠀⠀⠀⠀⠀⢀⡿⠁⢠⡿⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⢿⡄⠘⣧⠀⠀⠀⠀⠀⠀⣼⠃⢠⡿⠁⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠈⣷⠀⢻⣄⣀⣀⣀⣀⣠⡿⠀⣾⠁⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⣸⡷⠞⠛⠉⠉⠉⠉⠙⠛⢶⣇⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠻⣤⡿⠀⠀⠀⠀⠻⣧⡿⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠘⢯⣤⣤⣽⠇⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⢀⣀⠀⣸⣯⠀⢀⡄⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠙⢻⡏⢹⡟⠛⠁⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠶⠰⠶⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠶⠶⠶⠶⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠶⠆⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠤⠤⠤⠤⣤⣤⠤⠤⠤⠤⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠛⠒⠒⠂


⡶⣶⢶⠶⠶⡶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⢶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠾⠿⠷⠶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠿⠦⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⢀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣆
⠀⠀⠀⣸⡟⠀⣀⡀⠀⠀⢀⣴⠟⠁⣿⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⢀⣼⣏⣴⣾⠏⠀⠀⢀⡾⠃⠀⠀⠸⣇⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣿⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠰⠿⠿⠋⢹⡟⠀⠀⢀⡿⠁⠀⠀⠀⠀⠹⣦⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⣼⡇⣀⠀⣾⠃⠀⠀⠀⣤⠾⠷⠙⢷⣄⠀⠘⢷⣄⠐⠛⢿⣷⡀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⣿⡿⢿⣷⡏⠀⠀⠀⠈⠋⠀⠀⠀⠀⠉⠻⠶⣦⣽⣿⣶⣾⡟⠁⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠻⠇⢸⣿⡇⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⢻⣷⠟⢻⣆⣰⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠈⠻⣦⣀⠀⠀⠀⠀⠀⢤⠤⣤⣤⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠙⠛⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⢿⡁⠀⠀⠀⣀⣀⣤⣶⠶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⢹⡟⠳⠶⠶⠾⠛⠋⠁⠀⠀⠘⠷⣶⡾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠘⠷⣦⣤⣤⣤⣤⣤⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠉⠻⣶⡄⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⣤⣤⣤⣤⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠁⠀⠈⠉⠉⣹⡟⠉⠉
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣼⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⢀⣼⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡿⢷⣤⣤⣾⡃⠀⠀⢀⣿⠛⠷⠶⠶⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡃⠀⠀⠉⠉⠻⣷⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣻⣶⣶⣶⣟⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠀⠒⠂⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠉⠀⠒⠒⠢⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠖⠛⠛⠳⠦⠤⣤⡀⠀⠀⠀⠀⠀⠈⠉⡝⢀⠔⠒⠒⠒⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠳⡦⠤⠶⣤⡄⠀⠙⡆⠀⠀⠀⠀⠀⡜⡰⠋
⠀⠀⠀⠀⢸⡏⠀⠀⠀⢂⢀⡇⠀⣰⠟⠛⠢⡀⢳⡀⠀⠀⢀⣼⠛⠁
⠀⠀⠀⢀⣼⠇⠀⣰⠋⠉⠉⠀⢀⡏⠀⠀⠀⢻⡄⡇⠀⠀⠺⠁
⠀⠀⢠⡟⠁⠀⠀⠣⣴⡟⠉⠛⠉⠀⠀⠀⣀⢼⡼⠁
⠀⠀⢸⡇⠀⠀⠀⣠⢾⡇⢈⣭⣭⠁⠀⠠⣶⣾⠻⣦⡀
⠀⠀⠀⠳⣄⠀⠀⢻⡈⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⣸⣿
⠀⠀⠀⠀⠀⢱⡆⠈⣹⣦⠀⠀⠀⣀⢀⣀⠀⣾⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠘⣧⣤⣿⣿⠀⠀⠀⠙⠛⠋⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⡀⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡜
⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⠈⠁⠒⠢⣄⣈⣳⣄⡀⠀⠀⠀⠀⠀⠀⢠⠈⠋⠀⠘⢳
⠀⠀⠀⠀⣀⡤⠊⠀⠀⠉⠂⢄⡴⠂⠉⢿⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡆
⠀⠀⠀⡾⠃⠐⠂⠀⢄⡀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠙⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠑⡄⠀⠐⠂⠀⠀⢀⠔⠁
⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⠄⠁⠀⠀⠔
⠀⠀⠀⠈⠷⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠃⠀⠔⠁⠀⠀⠀⠌
⠀⠀⠀⠀⢱⡙⣆⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⢀⡾⠛⢣⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⢳⡈⢢⡀⠀⠀⠀⠀⠀⠘⣯⠒⠉⠱⡀⠀⢆⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠳⡀⠙⢄⠀⠀⠀⠀⠐⠿⠁⠀⠀⢁⢘⢪⣳⠁
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⢢⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠈⠑⢤⡀⠀⢀⡠⠖⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡆⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣄⢰⣂⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠚⠋⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣰⣶⣦⣌⠁⢏⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀⠀⢻⡏⠟⠻⣷⣦⣀⠢⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠱⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢿⣄⠐⠻⣿⣿⣿⣷⣄⡀⢁⡶
⠀⠀⠀⠀⠀⠀⠀⢐⣿⠉⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠦⢤⣭⣿⣿⣿⣿⣿⡉⠀⠠⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠿⠀⠀⢩⡏⠀⠸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢰⢿⣤⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⢷⣶⡎⠀⠀⠀⢏
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢻⣷⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣼⣿⣾⣿⡁⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⢨⣉⣽⣿⣯⣿⣿⣔⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⡿⠊⠀⠀⣮⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠐⢼⣿⣿⡿⠟⠋⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⢷⣿⣴⡗⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⣾⢋⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡆⠉⠀⠘⠇⠀⠀⠈⢷⣹⡇⠀⠀⠀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠺⣄⣀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⢠⣤⡄⠀⠀⠀⠀⠈⣿⡧⠀⠐⠀⠛⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⣾⠘⠀⠀⠀⠀⠉⠉⠀⠁⠀⠀⠀⠀⠐⠄⠀⠀⠐⠁⠀⠀⠈⠃⡟⠂⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆⣀⢀⣄⠀⡀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⡛⠈⠁⣀⡀⠀⠀⠀⠇⠳⣞⣇
⠀⠀⠀⠀⠀⠀⣧⡧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⣿⣿⣿⣿⡇⠀⠀⣇⠉
⠀⠀⠀⠀⠀⢀⡟⡇⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠉
⠀⠀⠀⠀⠀⢸⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⣿⡿
⠀⠀⠀⠀⠀⢸⡇⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠁⢳⣷⣤⣄⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣼⣽⠿⢿⣟⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠶⠶⣶⣦⣄⠤⣄⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠂⠉⠀⠀⠀⠀⠀⠀⠈⠩⠿⣶⡄⢉⠓⣦⡀
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢄⡹⣿⡗⠈⠁⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⣰
⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠈⠀⠀⠀⠠⠔⠂⢈⣙⣛⣛⠉⠥⢫⡟⣽⣷⠦⠤⠀⠀⠻⣦⠀⠀⠀⠀⢀⡴⠃
⠀⢈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣠⠞⠀⠀⠐⠀⢀⠀⠴⠄⢀⣠⣴⣶⣶⣶⠶⠿⣿⣿⣿⣷⠒⠠⠀⠈⢧⡀⢀⡴⠋
⠀⢀⢂⢈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣰⠋⠀⠀⠀⠀⣀⠤⠄⢒⢻⡿⠿⠛⠉⠁⠀⠀⠀⠀⠉⠻⣿⣯⠀⠀⠀⠀⢿⡁⢀
⠀⠀⢂⠂⠀⠀⠈⠙⠒⠠⢄⣀⣀⢀⠀⠀⠀⠀⣸⠃⣸⠃⠀⠀⡠⠒⢁⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠀⠀⠀⠈⣧⠂
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠄⡠⠤⢤⠀⣰⠇⠀⠀⠀⠀⣴⠦⣪⡵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠀⠀⢹⡆
⠀⠌⠌⠌⠄⠄⠠⠀⠀⠄⠌⠌⠼⡁⠀⠀⣷⡟⠀⠀⠠⠆⣼⢫⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠄⠀⠀⣿⠀⠀⠠
⠀⠈⠌⠈⠀⠀⠁⠀⠀⠀⠈⠈⠌⠉⠉⢩⡿⠄⠀⠀⡛⣼⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⢢⢊⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢸⡟⠉⠀⡄⢣⣿⡟⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⠶⢤⠀⠀⠀⠀⢸⣿⣿⡄⢻⣾⡇
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠀⠀⣯⡙⠒⠋⢱⣼⣿⡇⠋⠉⣉⣉⡙⣛⠀⠀⠀⠀⠀⠀⠉⣩⣤⣤⣤⣀⡀⠀⠀⠀⠀⢈⠿⣿⣿⡜⣿⠷
⠀⠠⠡⠡⠀⠀⠀⠀⠀⠀⠀⠡⠡⠁⠀⢉⡇⠀⠀⠸⢿⣿⡇⠀⠾⣟⢛⠛⠳⣄⠀⠀⠀⠀⠀⠰⠻⣭⠁⠉⠛⠛⠂⠀⠀⠀⠀⠀⢹⣿⡀⢻⡷
⠠⠡⠡⠡⠀⠀⠀⠀⠀⠀⠡⠡⠡⠁⠀⠉⣿⠀⠀⠀⢸⣿⡷⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡷⠀⠈
⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢁⠁⠀⠀⠉⣿⢀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⣿⣏⢸⡇
⢀⢀⢂⢂⠀⠀⠀⡀⠀⠀⢀⢂⢂⢀⠀⣀⣿⢸⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠛⠉⠉⠉⠹⣷⣶⣴
⠀⢂⢂⠂⠂⠂⠐⠐⠀⠀⢂⢂⣦⣶⣾⣿⣿⣿⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠈⠁⢀⡄⠀⣿⣿⠿
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠛⠻⣿⣿⣿⣦⡄⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠤⠀⠀⢈⡇⠀⣽⣿
⠀⠀⠌⠄⠀⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠈⢿⣿⣿⣿⠀⢿⣿⣿⣧⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠰⠀⢀⠀⠀⠸⠀⠀⠀⠁⠀⣿⡏
⠀⠀⡘⡀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠘⣿⣿⣿⣧⡀⠀⠀⠀⠙⠻⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⣀⠌⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⣸
⠀⡐⡐⡐⡐⡀⠀⢀⣿⣿⠀⠀⢠⡀⢳⡀⠀⠸⣿⣿⣿⠀⠁⠉⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠈⠀⠀⠀⠀⢨⠀⣤⣴⣿⣿⣡⡿
⠀⠐⠐⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢷⡄⢳⡄⠀⢹⣿⣿⡇⠀⠀⠀⢽⢿⣿⣿⣶⡀⠀⡀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⡶⠚⠈⣿⣿⣿⣿⡇
⠀⠠⠠⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠈⣿⣌⢿⡄⠀⣿⣿⣿⣄⡀⠀⠀⠙⣿⢿⣿⣿⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⢺⢻⣿⢃⠀⠀⠹⣿⣿⣿⣷
⠀⠡⠡⠡⠁⠀⠀⣼⣿⣿⠀⢠⣄⠀⠘⣿⣾⣿⡄⢸⣿⣿⣿⣷⠀⠀⠀⠸⣟⣽⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢾⣿⣄⠀⠀⠘⣷⠞⠀⠀⠀⠙⣿⣿⣿
⠀⠀⢡⠀⠀⠀⠀⣿⣿⣿⠀⠸⣿⣷⡄⠹⣿⣿⣿⡘⣿⣿⣿⣿⣧⢢⡀⡀⢹⣽⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠘⣿⣏⣇⡞⡀⢿⡄⠀⠀⠀⠀⠘⣿⣿
⠀⠀⢂⠀⠀⠀⠀⣿⣿⣿⠀⡄⣿⣿⣿⣦⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣧⢱⡀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⢃⢸⣷⠀⠀⢀⠀⠀⢹⣿
⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⡈⣇⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⡟⣿⣿⣿⣿⡎⣷⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣻⣿⣷⡀⣿⡆⣾⢸⣿
⠀⠌⠌⠄⠄⠄⠀⣻⣿⣿⡇⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⢠⡇⠘⣿⠻⣿⣿⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿
⠈⠈⠌⠈⠀⠀⠀⢹⣿⣿⣇⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣸⡇⢠⡏⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⣽⣆⢸⡟⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠙⠂⠀⠀⠀⠀⣀⠀⣠⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⢼⡇⢸⣧⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⠄⠀⠸⠇⠈⠓⠛⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢄⡀⡀⣠⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⠾⠗⠀⠿⠟⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣤⠀⠀⢀⠤⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡨⢿⣿⢿⣶⣿⣷⣴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣷⣭⣭⣭⣥⣾⣷⣤⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⣿⡿⢾⣿⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣿⣿⣿⣿⣿⡿⣿⣿⡿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⢟⣋⣁⣼⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠿⢿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠓⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡘⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⡁⡀⠠⠀⠐⡄⠀⠠⠀⠉⠀⠀⠀⠀⢁⠀⠀⠀⣤
⠀⠀⠀⠐⢀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠆⠀⠨⠂⠫⠀⠀⠉⠈⠉⠂⠚⡇⠀⠀⠀⠀⠐⠀⠀⠂⢀⠃
⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠄⠘⠂⠀⠢⠐⠐⠊⠀⡄⠀⠀⠀⠀⢀⣀⠀⠈⠈⠁⠈⠀⠐⠀⠀⠀⠈⠂⢀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣸
⠈⠀⢻⠃⠈⠀⠁⠘⡟⠀⠉⠈⠀⢻⠃⠈⠉⠀⠘⡟⠀⠀⠈⠁⠙⠋⠈⠉⠉⠈⠛⠉⠈⠈⠁⠈⠛⠁⠉⠁⠉⠙⠋⠈⠉⠈⠀⢻⠃⠉⠈
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣸
⠀⠀⢻⠃⠀⠀⠀⠘⡟⠀⠀⠀⠀⢻⠃⠀⠀⠀⠘⡟⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢻⠃
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣸⡀⠀⠀⠀⢀⣇⠀⠀⠀⠀⣸⡀⠀⠀⠀⢀⣇⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣄⡀⠀⠀⡀⢀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣸⡀
⠀⠀⢹⠁⠀⠀⠁⠈⡏⠀⠉⠀⠀⢹⠁⠈⠁⠉⠈⡏⠀⠀⠈⠀⠙⠉⠀⠁⠀⠈⠋⠁⠀⠈⠁⠈⠉⠁⠈⠀⠈⠉⠉⠀⠁⠀⠀⢹⠁
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣼⣄⣠⣀⣄⢠⣧⣀⣀⣠⡀⣼⡄⢀⣤⣤⣤⣧⣤⣄⣀⣀⣤⣄⣤⣀⣤⣤⣤⣄⣤⣀⣀⣤⣶⣀⣠⣀⣠⣤⣤⣀⣄⣀⡀⣼⣄⣀⣀
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠤⠤⢾⠦⠴⠤⠦⠴⡷⠤⠶⠶⠤⢾⠆⠴⠦⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠤⠴⠤⠴⠶⠶⠤⠦⠤⠤⢾⠦⠤⠤
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠘⠛⠿⠛⠛⠛⠛⠻⠿⣿⣿⠟⠛⠿⠛⠻⠛⠛⠟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⡗⠀⠀⠀⣠⣴⡄⠉⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠒⠒⢻⠓⠚⠒⠓⠚⡟⠛⠐⠛⣀⣿⣿⣿⡟⠒⠚⡿⠓⠛⠺⣿⣿⣿⣿⣿⠐⠚⠿⠓⠛⢓⡒⠚⠛⠒⠚⠒⠚⠻⠟⠒⠓⠚⠒⢻⠓⠓⠒
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⣾⣿⣿⣿⣿⡷⠀⠀⡇⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⣤⠀⣼⣿⣿⣷⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣸⡀⠀⠀⠀⢀⣃⠀⠀⡈⠛⣻⣿⣿⠃⠀⢀⣇⠀⢀⠀⠀⣸⡟⢁⡀⠀⢀⣀⡀⢹⣿⣿⣿⣿⡟⠀⠀⠀⢀⡀⠀⠀⠀⠀⣸⡀
⠉⠁⢹⠁⠉⠉⠉⠈⡏⠈⠉⠉⠉⢹⠁⠀⠉⠁⢸⣟⠉⠉⠉⠉⢉⠉⢉⣅⠈⠉⠛⠉⠈⠻⠟⠛⠋⠁⠉⠉⠉⠉⠉⠉⠉⠉⠀⢹⠁⠉⠉
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣿⣦⣀⠀⠀⣈⣀⣼⣿⣧⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⠀⣼⡄⣀⠀⣀⢠⣧⠀⣀⣀⠀⣼⡄⢀⣀⠀⣼⣿⣿⣿⡿⠛⠉⣿⣿⠻⣿⡿⢿⣄⠀⣀⡀⢀⣤⡀⣀⣀⣀⣠⣄⢀⣀⢀⠀⣼⡄⣀⣀
⠀⠀⢸⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠁⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⢿⣿⡆⠈⠷⠸⣿⡷⣀⠀⠀⠉⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⢹
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠉⠁⠀⠀⡄⠉⠁⠈⡆⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⣼⡄⠀⠀⠀⢀⣧⠀⠀⠀⠀⣸⡆⠀⢠⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣀⣀⣰⣆⢸⡄⠀⠀⠀⣠⣤⣤⣄⠀⠀⣾⡀
⠀⠀⢸⠀⠀⠀⠀⠀⣥⣾⣿⣧⠀⢸⠁⢀⣼⣿⣿⠟⠛⠛⢧⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⢹
⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⣸⠂⠘⣿⣿⡇⠰⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠇⠠⠀⠀⣠⣴⣾⣿⣿⣿⠁⠀⢹
⠀⠀⢾⡆⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣄⠈⠹⢷⠀⠹⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⠃⢀⡶⠀⣳⣤⣾⣿⣿⣿⣿⣿⣿⣶⣤⣽⠄
⠀⠀⢸⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣷⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⣋⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⢸⠘⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣌⣛⡛⠿⠛⢛⣁⣴⣾⣿⣿⣿⣿⣿⣿⠟⠋⠈⢿⣿⣿⣿⣿⣿⣿⣿⠃
⠶⠒⢾⠶⢿⣿⣿⣿⣿⡿⠿⠛⠂⢶⠆⠚⠛⠛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠿⠂⠐⠶⠂⠶⠟⠛⠻⠿⠿⢿⠟⠓⠒
⠀⠀⢸⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡆⠀⠀⠈⢉⣽⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⣠⣴⣶⣿⣷⣤⣄⠀⠀⡗⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠒⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣸
⠉⠁⢻⠃⠉⠉⠉⠙⡛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠉⢻⠉⠉⠉⠛⠛⠋⠙⠋⠉⠁⢻⠃⠉⠉
⠀⠀⢸⠀⠀⠀⠀⠀⡟⢿⣿⣿⣿⠿⠋⠙⠛⠿⠿⠿⠿⠛⢹⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣄⠀⠀⢀⣤⣾⣿⣶⣤⡀⠀⢸
⣀⡀⣸⡀⣀⣀⣀⢈⣇⡈⢛⣛⣁⣸⣀⣀⣄⣀⣀⣇⣠⣤⣿⣿⣿⣿⣿⠀⣀⣀⣉⡹⣿⣿⣿⣿⣿⣷⣠⣿⣿⣿⣿⣿⠿⢿⣀⣹⣀⣀⢀
⠀⠀⢹⠁⠀⠀⠁⠈⡏⠀⠈⠉⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠉⠈⠉⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠠⠇⢹⠁⠁
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠟⡁⠀⣠⣀⠷⠀⢸
⠤⠤⣼⡤⠤⠤⡤⢤⣧⠤⠤⠤⠄⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠁⣤⡤⠤⠤⠤⢤⣦⡤⠤⣤⣤⠼⣿⣿⣿⡏⠀⠀⠀⢠⡿⠦⠄⣼⡤⠤⠤
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡇⠀⢀⡀⠃⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⠉⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠈⠁⠀⠀⠀⠀⢸
⠀⠀⢾⠆⠀⠀⠀⠰⡷⠀⠀⠀⠀⢾⠆⠀⠀⠀⠸⡧⠠⠆⠀⠀⢼⠆⠀⠀⠀⠰⠶⠆⠀⠀⠀⠀⠶⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⢾⠆
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡅⢀⣴⣄⠀⣀⣀⣀⢀⣀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠙⠛⠿⡿⠟⠻⠿⠛⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢻⠃⠀⠀⠀⠘⡟⠀⠀⠀⠀⢻⠃⠀⠀⠀⠘⡿⠀⠀⠀⠀⠻⠓⠀⠁⠒⠘⠟⠃⠀⠀⠀⠘⠛⠃⠀⠀⠀⠛⠃⠀⠀⠀⠀⢻⠃
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠐⠀⢻⠃⠀⠀⠂⠘⡟⠀⠂⠐⠀⢻⠃⠐⠂⠒⠐⠟⠐⠂⠐⠂⠺⠓⠀⠂⠀⠐⠟⠂⠐⠐⠂⠐⠻⠂⠐⠀⠐⠚⠗⠀⠂⠐⠀⢿⠂⠐⠐
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣸⡀⠀⠀⠀⢀⣇⠀⠀⠀⠀⣸⡀⠀⠀⠀⢀⣇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣸⡀
⠉⠀⢹⠁⠈⠈⠉⠈⡏⠀⠉⠈⠁⢹⠁⠈⠉⠉⠈⡟⠈⠉⠉⠁⠙⠉⠈⠉⠉⠈⠛⠉⠉⠉⠉⠉⠉⠁⠉⠁⠉⠉⠋⠈⠉⠉⠁⢹⠁⠉⠉
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣼⣄⣀⣀⣀⣠⣧⣀⣀⣀⣀⣼⣄⣀⣀⣀⣠⣧⣀⣀⣀⣀⣤⣄⣀⣀⣀⣠⣤⣄⣀⣀⣀⣠⣤⣄⣀⣀⣀⣤⣤⣀⣀⣀⣀⣼⣄⣀⣀
⠀⠀⢸⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⡍⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢹
⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡱
⠀⠀⠀⠀⠀⣟⣶⡓⢀⣤⠀⢀⡀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢥⡀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠸⣯⡙⢦⠉⠂⠀⠀⠀⣦⡘⡖⡇⠀⠀⠀⠀⠈⠙⠰⢦⠄
⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⡖⠙⠂⢳⣴⠈⠦⠀⠉⠀⠀⠀⠀⠀⢀⠐⢀
⠀⠀⠀⠀⠀⢿⣠⠇⣄⠑⠆⠁⠀⠀⠈⢉⣠⣤⡀⢠⣄⣀⠀⠀⠀⢾⡹⣇⠄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⡀⠰⠆⣀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⢀⠤⠂⠈⠉⠀⠀⠱⡄⠙⡌⠙⠲⣄⠀⠀⠘⠂⠘⠟⢠⣁⠂⠀⠀⠀⡴⠉⣀⣀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣄⠐⣤⠿⡠⠂⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⠀⠀⠈⢳⡄⠀⢰⣆⠀⠀⠁⠀⠀⠀⡜⢀⣤⡄⠈⠛⠁⠀⠈⢷⡀⠀⠀⠀⠀⠀⢻
⡀⢠⠀⢀⠀⠀⠻⠃⠀⣠⠃⠀⠀⠂⠈⠀⠀⠂⡂⠀⠁⠀⠀⠀⠀⠀⢹⡄⣾⠛⠦⠤⠾⠴⣄⢆⠁⠈⠛⠁⠀⠀⠀⠀⠀⠀⢣
⠱⣞⢳⠈⠛⠀⠀⠀⣆⡹⠀⠀⠀⠰⠿⠂⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⡇⠀⠀⠄⢀⣀⠘⠦⠄⢀⣀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠉⠀⣀⡔⢦⠀⠀⠈⡇⠀⠀⠈⠓⠒⠀⠀⠑⠂⠉⠀⠀⠀⠀⠀⠀⠈⠶⠲⡁⢠⡤⡄⠀⠀⢀⡀⠀⠀⡄⢉⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠰⢄⣟⠳⠄⠀⠘⢿⡀⠀⢠⡆⢚⠛⣿⠚⢻⡂⣻⣁⠗⠀⠀⠀⢰⠃⠀⢣⠘⣟⠳⣧⣀⠼⠷⠘⠛⢃⠂⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠙⢦⠀⣠⣅⠀⠳⣄⠀⠳⠚⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⣠⢇⡀⠀⠘⠆⠀⠀⠀⠀⠄⠀⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⢨⠇⣴⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢆⡀⠛⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⡹⠟⠢⠤⣄⡤⡀⠤⢼⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⣤⡂⢦⣼⡟⠀⠀⠀⠈⣯⡕⠲⠖⠒⠈⠉⡔⣤⠘⢁⣼⡀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢿⠞⠁⠀⠀⠀⢸⡀⠀⠀⢸⠀⠀⠀⡌
⠀⠆⢀⠰⢀⠀⢫⠁⠀⠀⠀⠀⠀⠀⣐⣳⠀⠸⠀⠀⠀⠙⠟⠀⠀⠈⠋⠙⠢⠾⠂⡿⣀⠆⠰⢶⡟⠂⠀⠀⠀⠀⠀⣸⡇⠀⠀⠸⡇⠀⠀⡇
⠀⠘⠄⠀⠉⠀⡎⠀⠀⠀⠀⠀⠀⣀⡙⠇⠐⠉⢟⠲⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠙⣆⠀⢠⡄
⠀⠀⠀⠀⠀⣴⡀⠤⠄⠒⠂⠉⣡⣌⡒⠃⠀⠀⠸⡆⠀⠀⠈⠙⠒⠢⠤⢄⣀⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⢏⠀⠀⠀⠀⠈⠙⠮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡙⢯⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⡨⠄⠰⠀⠘⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⡄⠀⣀⣴⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠰⢣⡞⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠇⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠴⡄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣀⠀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠊⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⠇⠀⠀⠀⠀⢸⣅⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣚⣂⣐⢒⣉⣀⣀⣀⣀⣀⣈⢳⣆⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣂⣀⣺⣅⣙⣂⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠂⡱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣷⣟⣿⣿⣷⣦⣄
⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣶⣶⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠘⠗⠀⠈⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⠛⣿⣇⣿⣿⣞⣏⠉⣉⣛⠁⣝⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⡙⠈⠉⣾⡁⠁⢹⣷⣿⣿⣿⡯⠿⠛⠛⠲⠟⠁
⠀⠀⠀⠀⢀⡴⢀⣿⣿⣿⣿⣿⣿⣿⠿⠿⢾⣭⣙⡿⢿⣿⢯⡾⠛⠂⣁⣤⡼⣿⣧⣈⣴⣿⣿⡿⠃
⠀⠀⠀⠀⢸⣶⣿⣿⠿⠿⠿⠿⠟⠉⠀⠀⠀⠀⠈⠉⠒⠀⠀⠁⠳⠚⠛⡻⠻⣿⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⢈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢮⣟⣿⣿⣦⣀
⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣟⣷⣦
⠀⠀⠀⠀⠀⠈⠿⢿⣷⣶⣲⣶⣶⣶⣶⣶⣶⣶⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣝⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠐⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠲⣄⣀⣀⣤⠂⠀⠀⠀⣆⡀⣀⠠⠀⢠⣤⠀⠀⠐
⠀⠀⠀⠀⠸⣧⣀⡘⢠⣤⢠⣤⢠⣤⣠⠄⣤⢀⣤⢠⣄⣰⣇⠀⠀⠀⡏⢸⠀⢻⠀⣀⢀⡀⢹⠁⣿⢀⣀⡀⡇
⡀⠀⠀⠀⠠⡈⠻⠻⣦⢻⠀⢻⠠⠁⢹⡀⠿⠀⣇⠠⠇⢸⡇⠀⠀⠘⠀⢸⠀⠈⠀⡅⣨⡇⢸⠀⣿⢠⠋⠀⡇
⠀⠀⠀⠀⠠⠷⡄⠠⠋⠸⠧⠸⠧⠀⠿⠅⠠⠐⠧⠀⠀⠸⠧⠀⠀⠆⠠⣾⣦⡀⠜⣇⡸⣧⣼⣄⣿⡀⢶⠰⡦⠀⠀⠀⢀
⠀⠡⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠺⡄⠀⠀⠀⢀⠂
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂
⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠂⠄⠀⠀⠀⡀⢀⢀⠀⡀⢀⠠⠀⠄⠀⠐⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠒⠊⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠉⠙⠲⠶⢤⡀⠀⠀⠀⠀⠀⠀⠃⡔⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⣰⠟⠉⠉⠉⠻⠂⢛⣳⣖⡀⠙⡄⠀⠀⠀⢀⢣⠊
⠀⠀⠀⠀⣯⠀⢀⡤⠤⠾⠀⢸⠉⠀⠑⣄⢳⠀⠀⢠⠟⠁
⠀⠀⢠⡞⠁⠀⢾⢠⡴⠦⠤⠋⠀⠀⠀⣓⡼
⠀⠀⣿⠀⠀⠀⢀⣿⠁⢐⣒⡂⠀⢀⣩⡿⣧⡀
⠀⠀⠘⢦⡀⠀⢿⠘⠀⠉⠉⠀⠀⠀⠉⠁⢈⣿⠆
⠀⠀⠀⠀⠈⡦⠈⣻⡆⠀⠀⢀⣀⡀⢀⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠈⠷⣴⣿⣇⠀⠀⠈⠛⠃⢾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠌
⠀⠀⠀⠀⠀⠀⢀⠜⠣⢄⡀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢁⡈
⠀⠀⠀⠀⠀⣀⠞⠠⠄⠀⠀⢉⠶⣤⠛⠰⢄⠀⠀⠀⠀⠀⠎⠀⠀⠀⠸⡀
⠀⠀⢀⣴⢊⣀⡀⠀⠀⠈⠚⠀⠀⠙⡀⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⢸⠁⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠈⠢⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠙⠀⠀⢁⡠⠀⠀⢀
⠀⠀⠀⢹⢄⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⡰⠀⠀⠉⠀⠀⠠
⠀⠀⠀⠈⣎⢧⠀⠀⠀⠀⠀⠀⢰⡀⠀⣠⠏⠓⠀⠀⠀⢠
⠀⠀⠀⠀⠘⣆⠳⣄⠀⠀⠀⠀⠀⣯⠉⠀⠢⠀⡡⡀⡄
⠀⠀⠀⠀⠀⠈⣆⠈⠢⣀⠀⠀⠈⠈⠀⠀⠀⠀⠁⡽
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠑⢤⡀⠀⠀⠀⢀⡔⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠈⠁⠂⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡄
⣶⣶⣶⢶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⡀⠀⠀⣢⠠⢄⡀⠀⠀⠠⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠠⢠⣀⠀⢀⣤⢠⡄⠆⢄⡀⡨⠃⢠⡰⠌⠖⠆⢀⠀⢀⡃⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⡀⠀⠈⠐⠀⡠⢊⣭⡭⣒⠏⣤⣤⢓⣬⡆⣷⡂⣦⠄⡠⣸⠤⠼⡂⠈⠈⢊⠀⢀⡀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⣀⣦⣯⡷⡱⣧⠾⣭⣓⡻⠎⣓⣛⡵⣧⢞⡕⣹⢫⡞⣅⢢⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠙⠛⠛⠙⠓⠚⠛⠲⠻⠻⠙⠛⠚⠛⠳⠿⣯⣶⠿⠿⠟⠓⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠠⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⢂⠀⠀⠀⠀⠂⢀⠁⠀⢄⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠄⠀⠐
⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⡆⠀⠈⠁⠀⠀⢀⠎⠀⠀⠀⡀⠊⠁
⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⡅⠦⠠⡀⢠⠄⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠛⠒⠶⠒⢊⡁⠴⡀⠀⠀⠙⠂⠦⠈⠀⠀⠀⠀⡀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙⠀⣤⠄⠐⠠⠊⠀⠸⠴⠦⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠂⠐⠂⠁⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⡴⠚⠋⠀⠀⠀⠀⠛⠠⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠈⠉⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠤⠖⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⡀⢀⡀⠀⠀⠀⠀⢀⠀⠐⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⡿⠿⠿⠿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠋⠉⠉⠁⠀⠀⠀⠀⠀⣀⡙⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣶⣤⣤⣶⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡏⠁⠈⠉⠙⠿⣿⣿⣿⣿⣿⣿⠿⠟⠋⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡉⢻⣿⣿⠏⠀⠀⠀⢀⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣷⡄⢹⡿⠀⠀⠀⠀⠈⣀⡀⠉⠛⠋⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠃⠈⠁⠀⠀⠀⢠⠟⠻⠟⠳⣦⣦⡄⠀⠀⣰⡿⠉⠉⠉⠛⠂⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⣿⠻⠿⠟⢳⡄⠀⣨⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠂⢀⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣇⣀⠀⠀⠀⣠⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠓⠤⠞⠋⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⡴⠶⠶⠶⠶⢦⣄⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣤⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣨⠛⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠈⠑⠶⠶⠶⠶⢤⣶⡿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡾⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⣀⡠⠴⠞⠋⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⠁⠀⢻⡀⠀⠀⠀⠀⢉⣷⣶⣶⣿⡇⠉⠁⠀⠀⠀⠀⢀⡿⠉⢻⣄⡀
⠀⠀⠀⢀⣤⠖⠀⠁⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⠁⠉⠻⠶⣤⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢠⡟⠛⢿⣿⣿⣿⠟⢿⡄⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⠃⠀⣸⣿⣿⣿⠀⠀⠠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄⠻⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⠈⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣫⣿⠀⢹⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⠙⡏⠀⠸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠛⠁⣿⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡴⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⡉⠉⠉⠙⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠽⡿⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠟⠙⠋⡿⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠙⣿⣷⡆⣀⣠⠄⢀⣨⣭⡉⠙⠻⣿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣴⣿⣿⣿⣿⣷⣿⣿⣳⣿⣿⣿⣿⣷⣌⢻⣷⡈⢻⣿⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡹⣦⣠⣿⣿⣿⣿⠋⢿⢿⣿⣿⣿⢷⣿⣷⢸⣧⠀⢈⣷⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣿⡗⠘⣿⣮⣄⠉⣿⣿⣿⢀⡀⢈⠻⠛⢫⣾⣟⠁⣈⣿⣶⡾⠿⠿⠛⢛⠻⢿⢿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡤⡉⢻⣦⣬⡛⠛⠛⣫⣿⡿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣟⠛⣷⣶⡄⢶⣷⡄⠈⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠻⢿⣷⡘⣿⣷⣶⣆⠀⠈⣰⡟⠉⠉⠉⠁⠹⣿⣿⣛⢿⣿⣿⡿⠈⢿⡇⠈⣿⣿⢀⡿⣿⣏⠻⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣮⢻⣿⡿⠿⣿⣿⡾⠿⠋⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⠙⢿⣿⣂⡾⣰⣾⣿⠏⠀⢸⣿⣿⡆⠉⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⢿⣾⣿⣟⣥⡄⠠⢻⣿⣿⣇⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠲⣾⣿⣿⠟⠀⢀⣨⣿⠟⠃⠞⣹⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠆⠀⠀⠀⠰⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠘⣿⣷⣾⣴⣟⣋⣁⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠶⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠉⠙⠳⠄⠀⠀⠀⠀⠀⠘⣿⡄⠙⢿⣿⣿⣿⠋⠀⠀⣩⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡤⠀⠀⠀⠀⢀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢶⣬⣿⣶⣦⣀⣠⣴⣾⠟⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢠⠁⠀⠀⠈⠛⠿⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠻⡿⠟⣿⣿⣟⠁⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠶⣄⠸⠿⣿⣿⡁⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⡄⠈⢳⣾⣿⡏⠀⠀⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠓⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⠀⠘⣿⣿⣍⠳⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠢⣆⠀⠀⠇⠀⠀⠟⢿⢿⣿⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢢⣀⣀⣀⣀⡀⠀⠀⠀⠠⣄⠀⠀⢀⠄⠀⠀⠀⠉⠀⠂⢀⠈⢂⠀⢰⠀⢀⠀⠀⠈⠿⠿⢟⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢢⠀⣄⢿⣾⣷⣾⣷⣶⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠁⣇⣼⣿⣿⣿⠙⠏⠻⡿⠏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡝⣆⣠⣤⣶⣾⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠢⠳⢄⠀⠀⢠⣄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⡿⠿⠿⢁⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢶⡀⠁⠈⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢿⡎⠉⣀⣼⣿⣿⣿⣿⡝⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⣀⠅⠀⠀⠀⢣⠀⠀⢀⠀⡜⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠓⣿⡿⣿⠛⠻⣋⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⡸⠀⠀⠀⠀⠈⢇⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢿⣧⣀⣰⣶⣿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣈⣃⣀⣀⣀⣀⣀⣈⣆⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣐⣄⣀⣀⣀⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢱⡀⠀⠀⠀⣼⣆⡀⠀⢀⡜⠛⢠⡀⠀⢀⣠⡏⠀⠀⠀⠀⣿⠁⠈⣿⠀⠀⠀⢱⡀⠀⠀⢀⣼⣄⡀⠀⢀⡜⠛⢠⡀⠀⢀⣠⡏⠀⠀⠀⠀⣿⠁⠈⣿
⣤⣴⠋⠉⢲⡶⠁⠀⠀⠙⣿⠿⠷⠤⣾⡿⠋⠉⠉⢻⣤⣀⣠⣴⠟⠒⠾⣿⣦⣤⣴⠋⠉⢲⡶⠁⠀⠀⠙⣿⠿⠷⠤⣾⡿⠋⠉⠉⢻⣤⣀⣠⣴⠟⠒⠶⣿
⠀⢹⣄⡀⠼⢧⡀⠀⠀⢠⡇⠀⠀⠀⠈⣇⠀⠀⠀⢀⠟⠉⠹⡇⠀⠀⠀⣻⠁⠀⢹⣤⡀⠼⢧⡀⠀⠀⢀⠇⠀⠀⠀⠈⣇⠀⠀⠀⢠⠟⠉⠹⡇⠀⠀⠀⣹
⠲⢿⣟⠀⠀⠀⡿⠒⠈⠛⢧⡀⠀⢀⣠⡟⠲⣶⣶⣿⣄⣀⣠⠿⠦⣤⣴⡿⠶⠲⢿⣟⠀⠀⠀⡿⠒⠘⠛⢧⡀⠀⢀⣰⡟⠳⣶⣶⣿⣄⣀⣠⠿⠦⣤⣴⡿
⠀⠀⣿⠶⠤⣶⠀⠀⠀⠀⠀⢹⠟⠉⠉⠉⠻⡿⠁⠀⠈⡟⠁⠀⠀⠈⢿⠁⠀⠀⠀⣿⠶⠤⣶⠀⠀⠀⠀⠀⢹⠟⠉⠉⠉⠻⡟⠁⠀⠈⡟⠁⠀⠀⠈⢿⠁
⣀⣴⡁⠀⠀⢨⣄⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⣿⣄⣀⣠⣷⡀⠀⠀⢀⣾⣦⣀⣀⣴⡁⠀⠀⢨⣄⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⣿⣄⣀⣠⣷⡀⠀⠀⢀⣾⣦
⠉⠉⠓⣤⡤⠞⠛⠓⢶⡞⠉⠉⠣⣄⣀⣀⡔⠁⠀⠉⣿⠋⠉⠐⣾⡏⠀⢸⠟⠉⠉⠳⣤⡴⠚⠛⠓⢶⡞⠉⠉⠣⣄⣀⣀⡔⠁⠀⠉⣿⠋⠉⠑⢾⡇⠀⢸
⠀⠀⠀⣾⠀⠀⠀⠀⠀⣄⠀⠀⣠⠋⠉⠙⣧⡀⠀⣀⣶⡀⠀⢀⣼⠿⠶⣿⡀⠀⠀⠀⣼⠀⠀⠀⠀⠀⣄⠀⠀⣠⠋⠉⠙⣧⡀⠀⣀⣿⡀⠀⢀⣼⠿⠶⣿
⣦⠄⠚⠻⣄⠀⠀⠀⣠⠟⠉⠙⢿⣄⡀⣀⡞⠋⠙⢿⡀⣨⣿⠏⠀⠀⠀⠈⢳⣦⠄⠚⠻⣄⠀⠀⠀⣠⠟⠉⠙⢿⣄⠀⣀⡞⠋⠙⢿⡀⣨⣿⠋⠀⠀⠀⠈
⡃⠀⠀⠀⠈⠟⠁⠈⠙⢄⡀⣀⠞⠛⠛⠻⣆⠀⠀⣸⠟⠉⠛⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⠟⠁⠈⠙⢆⡀⣀⠞⠛⠙⠻⣆⠀⠀⣸⠟⠉⠛
⢷⣄⣀⣠⣮⠀⠀⠀⠀⢈⡋⡋⠀⠀⠀⠀⢸⠟⠋⠙⢢⣀⣤⡿⠦⣤⣤⡔⠛⢷⣄⣀⣠⣶⠀⠀⠀⠀⢈⡋⡋⠀⠀⠀⠀⠸⠟⠋⠙⢢⣀⣠⡿⠦⣤⣤⡔
⢾⡟⠁⠀⠙⣦⣀⣀⣠⠞⠉⠻⣄⡀⣀⣠⡆⠀⠀⠀⠀⣿⡏⠀⠀⠀⢻⣷⠤⢾⡟⠁⠀⠙⣦⣀⣀⣠⠞⠉⠻⣄⡀⣀⣠⡆⠀⠀⠀⠀⣿⡏⠀⠀⠀⢿⡷
⠀⠡⣄⣀⣴⡟⠀⠀⢹⣄⣀⣠⡁⠀⢹⠿⠿⠦⣤⣤⡜⠋⠓⣄⣀⣤⡟⠀⠀⠀⠡⣄⣀⣴⡟⠁⠀⢹⣄⣀⣠⡁⠀⢹⠿⠿⠦⣤⣤⡜⠋⠓⣄⣀⣤⡟
⣀⣼⣿⠟⠋⠙⠢⣤⠎⠁⠀⠉⢻⣶⠁⠀⠀⠀⠀⢻⣄⠀⢀⡏⠀⠀⢳⣀⠀⣀⣼⣿⠟⠋⠙⠢⣤⠎⠁⠀⠉⢻⣶⠀⠀⠀⠀⠀⢻⣄⠀⢀⡏⠀⠀⣳⣄
⠉⠙⢏⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⢿⠀⠀⠀⠀⠀⡼⠉⠀⠀⠙⣶⠖⠻⣿⠋⠉⠙⢏⠀⠀⠀⠀⢸⡀⠀⠀⠀⢨⠿⠀⠀⠀⠀⠀⡼⠉⠀⠀⠙⣶⠖⠻⣿
⠀⢀⡼⠦⣀⣀⠐⠛⠓⢦⣤⣶⣇⠀⣸⣶⡤⠴⠾⣧⠀⠀⠀⢀⣿⣀⣠⣿⡀⠀⢀⡼⠦⣀⣀⠔⠛⠓⢦⣤⣴⣇⠀⣰⣶⡤⠴⠾⣧⠀⠀⠀⢀⣿⣀⣠⣿
⠛⢿⠀⠀⢸⡏⠀⠀⠀⠀⡟⠉⠉⠉⠻⡏⠀⠀⠀⠘⣿⠶⣶⡟⠁⠀⠉⣿⠟⠛⢿⠀⠀⢸⡏⠀⠀⠀⠀⡟⠉⠉⠉⠻⡏⠀⠀⠀⠘⣿⠶⣾⡟⠁⠀⠉⢿
⣀⣼⣷⠂⠉⠙⣦⣀⣠⣼⡅⠀⠀⠀⠀⣷⣀⡀⣀⣴⠁⠀⠈⣧⡀⠀⢀⣾⣀⣀⣼⣷⠂⠉⠙⣦⣀⢠⣴⡅⠀⠀⠀⠀⣷⣀⡀⣀⣼⠁⠀⠈⣧⡀⠀⢀⣾
⠉⠙⣷⣀⠀⣀⠋⠀⠀⠀⠙⢦⣤⠴⠾⠧⣬⣿⠟⠛⠳⣶⠞⠉⠙⠻⣿⠏⠉⠉⠙⣷⣀⠀⣀⠋⠀⠀⠀⠙⢦⣤⠴⠾⠧⣬⣿⠟⠛⠳⣶⠞⠉⠙⠻⣿⠟
⠀⢀⠟⠉⠉⠛⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⣯⠀⠀⠀⣇⠀⠀⠀⠀⣹⡀⠀⠀⢀⠟⠉⠉⠻⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⣯⠀⠀⠀⣍⠀⠀⠀⠀⣹⡀
⠶⠿⣆⠀⠀⣸⣷⣄⣀⣠⠴⠿⡀⠀⠀⠀⢀⠟⠓⠶⣾⡿⠦⢄⣠⡴⠛⢻⣶⠶⠿⣄⠀⠀⣸⣷⣄⣀⣠⠴⠿⡄⠀⠀⠀⢀⠟⠓⠶⣾⡿⠦⢄⣠⡴⠛⢻
⠀⠀⠀⣿⠋⠁⠀⠀⠙⠃⠀⠀⠈⡷⠶⢶⡇⠀⠀⠀⣛⠀⠀⠀⢹⣧⣀⣼⠁⠀⠀⠈⢻⠋⠁⠀⠀⠙⠃⠀⠀⠈⡷⠶⢾⡇⠀⠀⠀⣿⠀⠀⠀⢹⣧⣀⣼
⡀⠀⣠⣿⠀⠀⠀⠀⠀⣷⠤⢤⣾⠀⠀⠀⣻⠦⢤⣶⠛⠢⣤⡴⠛⠉⠉⠻⣆⡀⠀⣠⣿⠀⠀⠀⠀⠀⣷⠤⢤⣾⠀⠀⠀⣻⡦⢤⣶⠛⠢⣤⡴⠛⠉⠉⠻
⡟⠁⠀⠈⠳⣤⡄⠠⢾⠁⠀⠀⣹⡷⣤⣶⡃⠀⠀⢹⣦⠾⣿⠀⠀⠀⠀⠀⢸⡟⠁⠀⠈⢳⣤⠄⠠⢾⠁⠀⠀⣹⣷⣤⣶⡃⠀⠀⢹⣦⠾⢿⠁
⣆⠀⠀⠀⡀⠁⠀⠀⠀⠱⣦⡶⠁⠀⠀⠈⢳⣤⡤⣾⡁⠀⠀⣤⡀⠀⠀⢀⣼⣇⠀⠀⠀⣀⠁⠀⠀⠀⠱⣦⡶⠁⠀⠀⠈⢳⣤⡤⣾⡁⠀⠀⣤⡀⠀⠀⢀
⣹⣷⠖⠚⢿⡀⠀⠀⠀⣰⠦⣦⠀⠀⠀⠀⡘⠁⠀⠀⠘⣶⡾⠋⠉⠛⣿⣏⠀⣹⣷⠖⠚⢿⡀⠀⠀⠀⣰⠦⣦⠀⠀⠀⠀⡈⠁⠀⠀⠘⣶⡾⠋⠉⠛⣿⣇
⠙⠇⠀⠀⢀⣿⠒⠐⢾⠁⠀⠈⠗⠂⢶⣾⣧⡀⠀⠀⢠⡿⣇⠀⠀⠀⣼⠛⠉⠙⠇⠀⠀⢀⣿⠒⠒⢾⠁⠀⠈⠗⠂⢴⣾⣧⡀⠀⠀⢠⡿⣇⠀⠀⠀⣽⠛
⠀⢨⣷⣶⡿⢧⡀⠀⡸⠗⠒⠾⣦⣀⠘⠉⠀⠉⠳⣾⠃⠀⠀⡷⠒⠛⣇⠀⠀⠀⢨⣷⣶⡿⢧⡀⠀⣸⠗⠒⠾⣆⣀⠘⠉⠁⠉⠓⣾⡃⠀⠀⡷⠒⠛⣇
⠶⢿⡿⠁⠀⠀⠈⢻⠁⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⠷⠀⠐⢧⣀⣀⣼⣷⡤⠶⢿⡿⠁⠀⠀⠈⢻⠁⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢸⠷⠀⠐⢧⣀⣀⣼⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠉⠀⠀⠈⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⣤⠀⠀⣤⠀⢴⠆⢠⣤⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣿⣤⣤⣿⠀⢰⡆⢸⡏⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⣿⠀⠀⣿⠀⢸⡇⢠⡄⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⡤⠤⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠎⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣦⡀⠀⠀⠀⣰⣾⣧⡀⠀⠁⠀⢀⣾⣿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣈⣛⠻⢾⡏⠁⠈⣿⡏⠀⢉⣿⡾⠟⣋⣀⠄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⣤⠄⢀⣀⡙⠀⠀⢻⠃⠀⠘⣁⣀⠀⢤⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠆⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠛⠛⠀⠀⠸⢉⠱⡀⠀⠀⠀⠀⠰⣉⠙⠀⠀⠘⠛⠛⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⣀⣀⣀⡀⠀⠘⠾⠟⠀⠀⠴⠆⠀⠐⠿⠞⠀⠀⣀⣀⣀⣠⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠉⠉⣉⠭⠭⠤⠤⣀⠀⡎⠀⠀⠀⠀⠀⠈⡆⢀⡠⠤⠬⠭⢍⡉⠉⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡤⢠⠎⠁⠀⠀⠀⠀⠌⠙⣶⣤⣤⣤⣤⣤⣴⡞⠁⠀⠀⠀⠀⠀⠉⢦⠠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠑⢦⠀⠀⠀⡘⠻⢿⣿⣿⠿⠛⠀⠀⠀⢢⠖⠁⠀⠀⢸⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⢦⠀⠀⠀⠈⢎⠀⠀⢀⠀⠀⣸⡀⠀⠀⠀⠠⢀⠎⠀⠀⠀⢠⠏⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢕⣄⠀⠀⠈⠓⣦⣤⣤⣶⣿⣷⣦⣤⣤⡖⠋⠀⠀⢀⣔⡥⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⡏⣿⠋⠀⢩⠁⠈⢹⡏⣇⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠧⣀⣈⣀⡠⠟⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣤⣤⣤⣤⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠟⠋⠀⠈⠛⠿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠟⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠁⠀⠀⣀⣴⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⢁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠀⠀⢠⣾⣿⣿⡿⠟⠋⠁⠀⣴⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⣰⣿⡿⠋⠁⢀⡀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⣀⣀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣤⣤⣀⣠⣶⠀⣿⣯⣤⣤⣴⣿⡏⠀⠀⠀⠀⠀⡀⠀⠀⢨⣟⠿⢿⣿⣿⡿⠟⠻⣿⡷⠦⡈
⠀⠀⠀⠀⠀⠀⢀⠔⠻⣿⡿⠿⠿⢿⣿⣿⣿⣶⣿⣿⠟⠛⠛⢿⣿⠗⣴⣿⣷⣶⠶⣶⣾⣿⣿⢷⣿⣿⡏⠀⠀⠀⠈⠃⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⠀⠀⠀⠀⠃⠀⠈⣿⠋⠀⠀⠀⠹⠏⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⣰⣾⠆
⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⢂⠰⠷⠀⠀⠀⠶⠾⣿⣿⣿⣿⣿⠶⠄⠀⠀⠀⠀⢠⣾⣇⠀⠀⠀⠀⠀⠀⠀⢀⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⡸⠿⠿⠛⢥⡀⠀⠀⠀⠀⡀⠔⠁⠙⠢⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⣤⣤⣤⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣾⣿⠀⠀⠀⠀⠀⣾⣷⣶⣾⣿⣷⡀⠀⠀⢠⣶⣭⣿⣴⣦⡀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⡟⠋⠛⢿⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣠⢔⣲⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆⠘⣿⣷⣿⣿⣷⣾⣿⠀⢀⣤⢿⢿⣿⣿⡏⢹⣏⣹⣷⣀⠀⠀⡰⢡⣿⣵⢿⡗⣼
⠀⠀⠀⢶⣶⣄⠀⢹⣿⣿⣿⣿⣿⣿⣧⣀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣶⣾⠁⠀⠀⠉⠿⣿⣿⣿⣿⣿⣿⣿⣷⢏⣿⣹⣿⣿⣯⡟
⠀⠀⠀⡈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣟⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠛⠿⠿⢀⡶⠂⠀⡀⠋⠉⠛⠛⠛⠛⠿⠷⡀⣿⣫⣿⡫⠋
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠋⠉⠛⠉⣡⣿⣿⣿⣿⠟⠙⠛⠉⠋⠉⠛⠻⠿⣿⡿⠿⠟⠁
⠀⠀⠀⠀⠈⠀⠀⢻⡿⠋⠀⠀⠀⠰⣿⣿⠿⠋

⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣷⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⡤⠀⣤⣤⣤⣤⣤
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣦⣴⣆⢸⣿⣿⣧⣴⣿⣿⣿⣿⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⠉⣿⣿⣿⣿⣯⣿⣿⣿⣿
⠀⢀⡇⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡟⠁⠺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⡿⣿⣿⣿⠿⠟⢿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣮⣷⠀⠀⠀⠀⠀⣴⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣿⡟⠀⠀⠀⢸
⠀⢸⡇⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡉⠀⠀⠀⠀⠀⢿⣦⠀⠀⢿⣆⠀⠀⠀⠀⡴⠁⠈⠀⠀⠄⠀⠀⢀⣠⣤⣀⣰⠀⢙⣿⣿⣿⣷⡀
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠈⠻⢿⡶⣦⣼⣧⣤⡀⠀⠀⣸⠀⣴⠛⠁⠈⢙⣿⣷⠈⢿⣿⣿⠿⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⠀⠀⢀⣤⣤⣶⣦⡈⡓⠠⠄⠀⢀⣴⣿⣿⣿⣿⣿⣦⠴⠃⠀⡇⠀⠀⠀⠀⣻⣿⠆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢟⡕⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⠀⠀⠀⠻⣄⡀⢀⣼⣿⠿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠋⠀⠀⠀⠀⢀⣘⣿⣿⣿⣟⢻⣿⡿⠀⠀⢀⣀⠬⢙⣿⣉
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⣉⠹⠆⣾⣿⡈⠳⣄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⠛⠃⠀⢀⠈⣿⣇⠀⢽⠄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢻⠿⣦⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣶⠀⠺⢷⣿⣿⣷⣌⡀
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠔⠀⢸⣿⣿⡀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⠀⠀⢸⠟⠙⠋⠙⡃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣠⣿⣿⣿⠃⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢳⣍⠀⠀⠀⠀⠀⠋⠙⠁⠀⢀⣤⣤⣤
⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠈⠛⠓⠂⠀⠚⠀⠀⠀⠀⠘⣿⣿⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛
⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣿⣧⣤⣤⣀⢀⣀⣠⡾
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠿⠛
⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠳⠦⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢢⠘⣀⣀
⠀⠀⠀⠀⠀⡠⠖⠐⠢⡀⠀⠀⠀⠀⣨⠄⠈⢅⠜⠁⠀⠡⠀⠀⠀⢀⡤⠒⠠⡮⠂⠀⠀⠀⠀⠘⡠⠖⠂⢤⡄⠀⠀⠀⠀⣤⡀⠮⠅⣀⡀
⠀⠀⠀⠀⠀⡅⠀⢀⡤⠢⡀⠀⠀⠰⠀⠀⠀⠀⠈⡒⢲⠁⠀⠀⠀⠀⠀⠀⣠⠴⡀⠀⠀⠀⠀⢸⡀⠀⠶⢈⠀⠀⠀⠀⠀⠬⠰⠒⡂⣙⠃
⠀⠀⠀⠀⠀⠈⠐⠚⠤⠠⠂⠀⠀⠀⠑⠺⡉⠋⠺⠩⠂⠀⠀⠀⠀⠾⠒⠐⠇⠀⠄⠀⠀⠀⠀⠃⠚⠂⠒⠦⠀⠀⠀⠀⠀⠀⠈⠉⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⢀⡤⠄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⢈⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣸⡅⠼⡀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠄⠀⠀⠀⠀⠀⠀⡤⠖⠒⢌⢁⢄
⠀⠀⠀⠀⠈⢲⠒⠲⠡⠖⠢⠀⠀⠀⠀⣞⡝⠋⠁⠎⠀⠀⠀⠀⠈⠀⠀⠀⣱⡀⠀⠀⠀⠀⠀⣀⡞⠉⣠⡘⡂⠀⠀⠀⠸⠀⠀⠀⣠⠤⡈
⠀⠀⠀⠀⠀⠀⢡⠥⠈⡒⠊⠀⠀⠀⠀⠰⠑⠤⠤⠃⠀⠀⠀⠀⠈⠢⠤⢜⠁⢸⠀⠀⠀⠀⠀⠘⠤⣻⢁⠜⠀⠀⠀⠀⠀⠉⢒⡐⠃⣀⠜
⠀⠀⠀⠀⠀⠀⠈⠉⠈⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡰⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠜⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⢠⠖⡄⢀⡀⠀⠀⠀⠀⠀⢀⡠⠤⢄⠓⣈⠀⠀⠀⠀⠀⢀⣀⠖⠂⠀⠀⠀⠀⠀⡜⠉⠀⠈⡄⠀⠀⠀⠀⠀⠰⣀⠀⡉⠈⣁
⠀⠀⠀⠀⠀⣈⢬⠛⠁⠀⠱⠀⠀⠀⠀⡎⠀⠀⠀⣆⡒⠁⠀⠀⠀⢀⡬⠤⡹⠅⠀⠀⠀⠀⠀⠣⣀⠀⣰⠓⡄⠀⠀⠀⠸⡏⠁⠀⠈⣊⠒
⠀⠀⠀⠀⠐⢅⢀⢇⠀⢀⡴⠀⠀⠀⠞⡹⠦⢤⢼⠁⢈⠀⠀⠀⠀⠘⠄⠠⢃⡀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢡⡄⠀⠀⠀⠀⠱⢄⣃⡋⠁⡃
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠤⠀⠀⠀⠀⠀⢀⡃⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⢠⢢⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⡟⠀⠀⢖⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣆⠹⣿⣿⣷
⠀⠀⠀⠀⣴⣿⣿⣿⣿⡟⠀⡰⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣇⠀⠙⣿⠇
⠀⠀⠀⠀⠻⣿⣿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣼⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠈⠉⠳⣄⢀⣤⣶⣶⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⢀⣤⣄⣀⣤⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠙⢾⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠛⢿⣷⠋⠉⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⢳⣀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⣀⣤⣤⣬⣽⣯⣭⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠬⠭⠭⠤⠤⠤⣄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡀⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠠⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡄⠀⠠⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠇⠀⢰⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠙⠚⡇⠀⠀⡇⠀⠀⡗⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠀⠀⢸⠀⠀⢸⠛⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⡇⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⢸⡀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠴⣆⣀⡴⠚⢷⡄⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠻⠄⠀⣠⠞⠁⠀⠉⠁⠀⠀⠈⠛⠁⠀⠻⡤⠖⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⠓⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⡠⠒⠉⠉⠒⠤⠄⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⠟⠊⠉⠑⠀⠀⠀⠀⠀⠠⠒⠒⠒⠪⡲⡄⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⢾⠗⠊⠐⡄⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀⠀⠀⢰⠀⣰⠃⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢼⠀⠀⠀⠈⠀⠠⠂⠀⠀⠐⠋⠀⠀⠈⠇⠀⣼⠖⢫⡄⢀⣤⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣿⡻⠁⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡏⠉⠑⡆⠀⠀⠀⠀⠀⠀⢀⢀⣠⠟⣿⡃⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⢦⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⣰⠋⠁⠀⢀⣼⠳⢤⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣇⢹⢸⡤⢤⣀⠀⠙⠳⢤⣀⠀⠀⢇⠀⢠⠁⠀⠀⠀⢀⡴⠃⠀⠀⢀⣞⡃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠀⢀⡴⢢⣹⠆⠀⠀⢀⣀⠈⢙⣿⠉⠁⣴⡶⢒⣛⡉⠀⠀⣴⠶⠾⢥⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠈⠣⠜⢹⠆⢀⡴⠋⠉⠉⠙⠏⠒⢚⠞⠉⠉⠉⠙⢳⣼⡏⠀⠀⣸⡗⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡆⠀⢲⡚⢉⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡤⣾⠏⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢷⢹⡮⡒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡤⣪⠟⣦⠃⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠸⠏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣔⠁⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈⠳⣤⣀⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠞⠁⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⡀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣶⣾⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣧⣴⣿⣿⡿⠀⠀⣿⣿⣿⣦⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⠟⠉⠀⢶⡶⠀⠙⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⠸⣄⣀⡼⢆⣀⡰⠀⢹⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⣄⡀⠈⠓⠊⠀⢀⣤⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠉⠉⠀⠀⠉⠉⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡄⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠘⠁⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠃


⠀⠀⡀
⠀⠁⠀⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⣢⠀⠤⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠈⠁⠢⢄⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣰⠀⠀⠀⠀⢸⠀⠀⠅⠠⠃⢰⠁⣰⣀⡆⣶⠆⡤⠀⣠⠀⠀⣀⠀⠑⠀⠈⠂
⠀⠀⠀⠄⠀⠀⠀⠀⣤⡗⡼⠞⠉⠠⠂⠀⠀⠐⠀⠀⠀⡐⠀⠙⢄⠉⠈⠙⠓⠚⠷⠶⠧⠤⣾⣽⢖⠃⠀⢀
⠀⠀⠀⠀⠄⠀⣦⡷⠏⠉⠀⡠⠂⠀⠀⠀⠀⠁⠁⠂⠁⠀⠀⠈⠢⢕⠂⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠄⠀⠀⢀⠀⢀⡠⠐⠂⠀⠀⠀⡀⠀⠀⠈⠀⠐⠂⠠⠍⠩⠩⠅⠀⠂⠀⠑⢌⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⢤⠠⣄⢲⡀⣖⡆⢹⠀⠀⡁⠀⠀⠀⠠⡰⡄⣖⠰⡖⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠸⠸⠀⠏⠏⠃⠛⠁⠐⠀⠀⡂⠀⠀⠀⠐⠂⠃⠙⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠐⠀⡆⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠠⠀⠀⠀⠀⠀⠂⠀⠃⢀⠀⠀⠀⠀⠀⠄⡂
⠀⠀⢎⠀⢄⠀⠠⠀⠀⠀⠐⠀⠀⠉⠉⠀⠀⠒⠀⠐⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠀⠐⠀⡘⢸⢡⢲
⠀⠀⠄⠀⡀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠂⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠴⠰⠖⠶⠆⠆⠰⠶⠷⠆⠀⠧⠇⠒⠀⠂⢀⡄⠀⠐⠑⠠
⠀⠀⠀⠀⠀⠂⠀⠀⠑⠀⠒⠂⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠒⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⢀⠠⠀⠒⠒⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠢⠄⠄⠄
⠀⠀⠀⠈⠔⠁⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⡐⢀⢄⡠⡠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢰⠀⠀⠚⠘⠀⠹⠀⠀⠘⢢⠀⠀⠀⠀⠉⠁⣁⠍⡘⢡⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠃⠀⢰⠏⠇⢶⢶⣏⠀⠀⢀⠀⠀⠸⠀⠆⠳⠉⠸⠃⠃⠠⠀⠀⠀⢀⠀⠀⠀⠀⠐⠈⠀⠀⠐⠁⠀⢀
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠈⡁⣀⡂⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠂
⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⠁⠀⠠

⠀⠀⣀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣤⡀⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠉⢪⡁⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⣀⡀⠀⠀⢱⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢼⣿⣿⡇⠀⠀⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠈⠙⠋⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠰⠏⠛⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣤⣤⣀
⠁⠀⠘⠃⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣶⣄⠀⠀⠀⠀⠀⠀⣀⣼⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣴⣶⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠟⠻⠀⠀⠀⠀⠀⠀⠿⠃⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⣍⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢿⣿⣿⣧⣶⣶⣶⣤⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣷⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣶⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣿⣿⡿⠇⠀⠀⣠⣄⢰⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠛⠿⠿⠿⠟⠁⠀⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠁⠀⠉⠀⠀⠀⠘⢿⠿⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠰⢿⠆⠀⠀⠀⠀⠀⠹⠿⠟⠀⠀⠈⠉⠛⠛⠛⠛⠩⣿⣿
⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⢀⣤⠖⠛⠋⠉⠉⠈⠉⠀⠈⠉⠉⠉⠙⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⣠⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠶⢦⣄⡀⠀⠀⠀⠙⣦⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠁⠀⠀⠀⠈⠛⢦⣀⠀⠀⠸⡆⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⣿⠁⠀⠀⠀⠀⠀⠀⠀⣴⠟⢀⣴⠟⠀⠀⠀⠀⠀⠀⢀⡤⠤⣝⢦⡀⠀⣿⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⠀⠀⠀⠀⠀⠀⢀⣼⠏⣰⠟⠙⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢳⣤⣼⡄⠀⠈⠉⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⢻⠀⠀⠀⣰⠆⠀⣾⡏⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠙⣧⠀⠀⠀⠀⠙⢷⣤⣀⣀⠀⠀⣀⣴⡾⠋
⢸⡄⠀⠀⣿⠀⣼⢿⣿⠇⠐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣧⠀⢸⡆⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠁
⢸⡇⠀⠀⣿⢰⡏⠸⠏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⢀⡾⠁
⠀⣷⠀⢠⣾⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⠀⠀⠀⠀⠀⠀⣿⠻⡇⠀⠀⠀⠀⠀⠀⠀⣰⠶⣄⣤⠀⣀
⠀⢿⠀⢸⡞⣇⠈⠳⣤⡀⠀⠀⠀⠀⠀⢸⣍⠉⢁⣼⠃⠀⠀⠀⠀⣠⡟⠀⣇⠀⠀⠀⠀⢠⣤⡀⡟⠀⡟⢸⡿⢹⡇
⠀⢸⡇⠈⢷⠈⠃⠀⠈⠙⠳⣄⠀⠀⠀⠀⠉⠉⠋⠁⠀⠀⣀⣤⠾⠋⠀⠀⣿⠀⠀⠀⠀⢸⠀⠻⠇⠀⠀⠈⠀⣼⢧⡄
⠀⢸⡇⢀⣈⣳⣆⣀⣀⠀⠀⠈⣇⠀⠀⠀⠀⠀⣤⡶⠶⠛⠋⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢁⡾⠁
⠀⠈⣧⣿⠁⠀⠈⠉⠉⠙⠛⠲⡏⠀⠀⠀⠀⠀⠿⢦⣤⣄⣀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠷⢦⣤⣀⣀⠀⢠⡞⠁
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⠷⠤⢤⡤⠤⠶⠚⢻⠇⠀⠈⠉⠻⣦⡀⠀⢸⡇⠀⠀⣰⠇⠀⠀⠀⠈⠉⠛⢻⠆
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠙⢶⣼⡇⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣾
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠛⠶⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠹⣄⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⢶⣤⣀⣀⡀⠀⠀⢀⣼⠏
⠀⠀⠉⠛⠶⢶⡶⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠉⠉⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⣶⣶⠶⠶⠶⢾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡇⠀⠀⠀⠀⠀⠀⣰⠫⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⡰⡳⠃⠀⠀⠀⠀⠀⠀⠳⣀⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣰⡳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⢀⡨⠟⠁⢀⣀⣄⣀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⢀⡪⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⣴⠋⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠿⣷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠉⠉⠙⠲⢤⠀⠀⠀⠉⠓⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⢀⡴⠖⠛⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⢀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢙⢹
⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⢀⢜⠟⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢌⢆⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⡼⡜
⠀⠀⠀⣀⣀⡀⠀⠈⠙⠲⠤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠠⣪⠟⠁
⠀⢠⠊⠁⠈⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢀⢞⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠋
⠀⢸⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠤⣄⡀⠀⠀⠀⠙⠦⣄⣀⠀⢀⣰⠵⠋⠀⠀⠀⠀⢀⡔⠻⢷⣦
⠀⠀⠙⠲⢤⢶⣷⣤⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⣻
⠀⠀⠀⠀⣏⠀⠀⢹⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠋⠆⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠲⠦⣄⡀⠈⠑⠊⠁⠀⠀⠀⠀⠀⢠⢴⣶⣤
⠀⠀⠀⠀⠈⠒⠴⠚⠁⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢀⣏⠃⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⣐⠵⠃⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⣀⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⣸⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠛⢻⣧⠀⠈⠳⣄⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣠⠏⠀⠀⠀⠈⠙⠒⠶⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠹⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠈⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠸⣿⣿⣿⣦⣶⣶⣶⣶⣶⣿⣿⣿⠇⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⡉⢻⣿⣿⣿⣿⡿⢋⡙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣈⢋⣰⣿⣿⣿⣿⣧⡈⢁⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣇⠐⠻⠿⠃⣨⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⢿⣿⡆⠰⣿⣿⠿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣄⡁⢠⡆⠈⣡⣴⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣾⣿⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢅⠠⠟⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠲⠂⠀⠀⠘⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠠⡄⠀⣤⠀⠡⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⡀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠴⣒⡶⠑⢄⠀⠀⠁⢀⣀⠠⢞⢷⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠀⠀⠀⠹⣀⠀⠀⢙⠋⠉⠀⢀⣴⣷⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠹⣷⡀⠀⠀⠈⠉⠐⢸⠀⠀⢸⠁⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣀⠀⠀⠀⡟⢷⡄⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉⡇⠈⢻⣆⠀⠀⠀⢸⠀⠀⠀⠘⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠲⠤⣀⠴⢛⠳⢤⡀⢸⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠸⡀⠀⠀⠀⠀⢀⢐⠢⠭⣾⡇⠘⡄⠀⠀⠀⡸
⠀⠀⠀⡎⣉⠭⠛⠉⠉⠉⠀⠀⠉⠉⠙⠻⣒⠒⠒⠒⠚⠪⠽⠿⠋⠻⠉⠉⠙⠒⢞⡓⠒⠐⠒⠒⠀⠀⠂⠀⠀⠀⠲⢄⡀
⠀⠀⢸⠈⠁⠒⠢⠤⠐⠒⠈⠉⠉⠀⠒⠒⠢⠷⠤⠤⠐⠒⠒⠒⠂⠀⠉⠉⠉⠉⠒⠚⠓⠀⠀⠀⣀⡠⢤⡢⠀⠀⠀⠀⠙⠢⣀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣒⠭⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣢⣄
⠀⠫⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠛⢦⢰⡏⠈⠁⢹⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣵⣍⢻⢣⣿⣿⣿⡜⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠹⣄⣠⠞⠀⠳⢤⡤⠞⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣙⠿⢟⣼⣧⣛⣛⣋⣼⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠞⣋⣩⣍⡛⢶⣄⢠⡞⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢋⡥⠶⠶⠶⣌⡻⣿⣏⣴⡎⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⢁⠞⠁⠀⠀⠈⠀⠹⡆⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢰⣯⣶⣿⣿⣿⣿⣷⡘⣷⣭⣵⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠻⢿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣼⠛⠂⠈⠹⡄⠀⢀⣀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣱⣿⣿⣷⡙⣿⣿⡿⠿⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣄⠀⢀⣰⠳⢶⠋⠉⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⡹⣿⣿⡿⣡⣍⢡⣶⣶⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠈⠳⠴⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣮⣝⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣄⡀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢏⡀⠀⠈⠛⢿⣿⣿⡇⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠟⠁⢠⣦⠀⠀⠀⢠⡄⠙⣣⠀⢀⠜⠀⠀⢀⠀⠀⠀⠤⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⡇⠘⠓⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠸⣷⣶⣶⠟⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⣀⣀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⣧⣀⢫⠰⢄⠀⠀⠀⠀⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠘⠠⠍⣃⠰⠾⠠⠤⡔⠈⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⡍⠗⣾⣄⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣤⣤⣤⠈⠁⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠛⡟⠛⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡐⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣴⡀⠀⠀⢸⣄⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠀⠀⠀⠀⠀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠀⠀⢀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣠⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠒⠒⠒⠒⠂⠲⣶⣶⣶⣶⠖⠒⠒⠒⠒⠂⠀⠠⣾⣿⣿⣿⠿⠟⠛⠛⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣀⣀⡀⠀⢀⠛⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀
⠀⢠⣾⣿⣿⣿⡄⢨⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀
⠀⢸⣿⣿⣿⣿⣿⣾⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄
⠀⠀⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠑⠢⠄⣀⢀⣀⣀⣀⠤⠖⠉⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠈⠙⠻⠿⠿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠒⠒⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠶⠂⠀⠉⠉⠉⠉⠉⠉⠉⠰⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠀⠀⠐⠒⠂⠀⠤⠄⠀⠀⠀⠀⠀⠀⠈⢖⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢄⠀⠈⠑⡀⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⡠⠊⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠱⠀⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠴⠀⠁⡀⢁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⡇⠀⢠⠀⢠⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⡀⠀⠀⡆⠈⡀⠀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠄⢰⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⡈⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⡆⠀⠀⠀⠁⢆⡹⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡠⠂⢠⠀⠀⢰⠂⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⣄⠀⠀⢁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⠐⠢⡔⠂⠁⠀⠀⠌⠀⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠹⠖⠀⣽⡄⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⣠⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡴⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢶⡀⠀⠀⠀⠀⠈⠀⠋⠀⢸⠵
⠀⠀⠀⠀⠀⠀⠀⢠⠙⢄⠀⠈⠘⠃⠢⠄⢰⠀⠀⡔⠀⠉⠐⠟⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⢁⠐⠀⠐⠢⠀⠀⣀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠘⠢⠄⢀⡇⠀⠀⠄⠀⢃⡀⠀⡀⠂⠀⠀⠀⠀⠀⠐⠢⠤⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠀⢀⠃⠀⠘⡀
⠠⠦⠤⠀⠀⢀⣤⠁⠀⠀⠀⠀⡠⠋⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠈⠀⠀⠀⠐⢄
⡀⠶⠆⠀⠉⣹⠋⠐⢄⠀⣠⠊⠀⢀⠆⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠰⠄⡀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠳⡀
⠁⠒⠂⠀⡝⠀⠈⠐⢠⡜⠉⠀⠦⣿⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⢈⠀⠀⠀⢀⡄⠊⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⡄
⠀⠀⢠⠂⠀⠀⠀⠀⣶⠀⠠⠊⠀⠀⠈⠁⠒⠤⠀⠀⠀⡀⣽⠲⡤⠐⢂⣉⣀⠙⠂⣠⣤⣀⡐⠢⠤⠐⠚⠡⠄⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠃⠀⠀⠀⠀⣌⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠾⠛⣀⣡⠐⠂⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⢝⡳⠦⢀⡀⠀⠀⢸⡇⠀⡠⠃
⠀⠀⠳⢄⣀⡀⠜⠈⢢⡘⠢⢄⡀⠀⠀⠀⠀⠀⠐⢁⠔⠃⠉⠠⢃⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣐⡢⠔⠂⠀⠀⠀⠁⠘⠂⠺⢧⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠼⣷⠤⢀⡀⠀⠀⠄⠀⠀⠀⠀⠏⠀⠁⠤⠊⣉⣁⣀⣀⣉⠁⠠⠄⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠁⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⠐⠶⠄⠳⠦⠾⠻⠶⠶⠆⠶⠂⠶⠂⠶⠰⠖⠂⠤⠠⠤⠤⠤⠤⠐⠶⠲⠶⠰⠟⠶⠶⠂⠶⠂⠰⢦⠀⠀⠀⠀⠀⠷⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠆⢠⠔⠂⣤⠀⠀⣴⡀⢰⣀⠀⡖⠂⠀⡤⠒⡄⣶⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡆⠈⠃⠂⠘⠀⠘⠃⠃⠐⠙⠂⠊⠃⠀⠑⠀⠃⠛⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠇⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⢐⣒⠒⠒⠒⠒⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠈⠀⡔⠂⠀⠀⠈⣼⣰⡀⠀⠀⠙⣀⠐⠂⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⡀⠠⡄⠀⡹⠀⠀⠀⠠⢄⡀⠈⡤⠚⢫⡑⠄⢀⡀⠀⠈⠢⡀
⠀⠀⠀⠀⢠⠊⠀⢀⡀⢇⠒⠂⠈⡾⠒⠒⡆⢸⠀⠀⠀⠀⠀⠂⠒⢄⠔⢅⡀⠀⠀⠀⠈⢢
⠀⠀⠀⠠⠃⠀⠀⠓⡆⢀⢑⡃⠤⠤⠄⠀⠀⠀⡮⡭⢄⠀⡀⠀⢸⡌⠀⠄⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠆⠀⠀⠈⠒⠁⠇⢠⡒⣒⡨⡀⡄⠘⠚⠉⠀⡛⠂⠥⠀⠀⡟⠀⠒⢄⠀⠀⣀⠀⠀⡠⢣
⠀⠀⠰⠀⠀⠀⠀⡙⢀⡄⠑⢟⡺⡏⠆⠏⠝⠘⠠⠊⠿⢤⠀⡔⠐⠣⠄⠀⡠⠻⠈⠀⠀⢈⠴⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠠
⠀⠀⠀⠀⠀⠉⠉⢛⠀⣀⠀⣈⠛⣁⣀⡼⠀⠂⠔⠆⣂⣖⠣⠂⠠⣴⠢⢥⠀⠀⠃⠀⠀⠀⠀⠀⢃
⠀⠀⠀⡀⠀⠀⠀⢈⡀⠦⠿⠀⢀⠁⣄⠙⡀⢳⣦⠤⣉⣛⡧⠄⠊⡀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠁⠀⠀⢀⡀⣡⡄⣂⠁⠈⠀⠐⣶⢤⠜⡁⠐⡀⢰⣴⣾⣅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠈⡄⠄⠀⠀⠀⠑⠄⣀⣈⠓⠚⠛⢢⠰⠀⠀⠇⠚⣈⡏⢩⠐⠀⠀⠀⠀⠀⠀⠀⠠⣀⠠⠀⡠⠃
⠀⠀⠀⠀⠰⡀⠄⠀⠀⠀⠀⠀⠀⣈⠐⠖⢁⠠⠲⢀⠘⠤⠯⠹⠍⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠱⡐⠢⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠆⠀⠁⠀⠀⡰⠒⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠘⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⠀⢂⢄⠠⢀⠄⠀⠀⡀⠀⠀⢀⡀⡈
⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⢀⢀⠔⡠⢀⠡⠊⠔⡡⢊⠔⢁⢔⠅⡡⠪⠊⠐⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠐⡡⢊⠄⢊⠔⡡⢐⠕⣁⠬⠐⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠔⢀⠔⡁⠐⡡⢊⢔⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠠⠀⢐⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⡀⠀⣀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠐⠀⠐⠈⠐⠀⠀⠀⠂⠈

⠀⠠⠤⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⠸⠞⠃
⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡐⢤⠀⠀⢀
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠘⡏⠋⣻⣶⣄⡒⠠⢀
⠀⠀⠀⠀⠀⠀⣷⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡈⠻⣿⣿⣿⣦⣤⠜
⠀⠀⠀⠀⠀⠀⡟⠀⣠⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠙⢿⠟⢻⣿⡇⠀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢼⣣⣤⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣤⠋⠀⠀⡏⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿⣿⣿⠀⡀⢀⠇
⠀⠀⠀⠀⠀⣷⣾⣿⣿⣿⣷⣀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠸⣿⣟⡀⢀⣹⣾⣷⡀
⠀⠀⠀⠀⠀⢨⣿⣿⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠻⠉⢻⠋⠉⠻⡿⣿
⠀⠀⠀⠀⠀⣾⠳⠁⠀⢦⣀⣨⣄⠀⠀⠀⠀⠀⠀⠀⡼⠟⢀⣤⡀⠀⠀⠀⠹⣿⠀⠔⠒⠒⠛⠛⠃
⠀⠀⠀⠀⠀⠿⠃⠀⠀⠈⠉⠁⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠑⠑⠀⠀⠀⠀⢻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⣲⣍⠀⠠⠶⠾⠧⡼⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠉⠁⠀⣀⢀⠀⠀⠀⠚⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣤⣀⣀⠀⠸⡆⢀⢀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⠹⢿⣿⣿⡄⣬⠏⠘⡿⢶⡌⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣥⡟⠀⠀⠀⢦⡿⠁⠈⡟⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠠⡿⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠶⣌⡉⠀⢻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣶⣤⣾⣷⡦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠃⠀⠀⠀⠉⠃⢨⡏⠃⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢻⡅⠀⠀⠀⠀⠀⠀⢐⠙⢀⠃⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠀⢠⠀⠀⢀⡼⠀⢸⣆⠀⢸⡆⠙⠆⡀⠀⠀⠀⠀⠘⢀⡟⢇⡘⡇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢀⣿⠆⣴⡿⠁⠀⢸⣿⠀⠀⢿⣤⣤⣭⣓⣦⣀⣨⣃⣾⣥⣬⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠙⠛⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣮⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿