Downhill Emojis & Symbols

⠑⠦⣴⣛⡧
⠀⠀⠿⠛⠣⢤⣀
⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⣷⣶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⣠⣾⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠛⠻⠿⣿⣿⣿⣿⣿⣋⣉⣉⣍⡙⡛⠉⠉⠙⢿⣿⢿⣿⣿⢿⡻⣿⣿⡆
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣞⣳⠄⠀⠈⠹⣿⣿⡪⣝⣿⣷⣿⣾⣽⣶⣄⣠⣿⣾⡿⠗⠈⠉⠿⣿⠃
⠀⠉⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣶⢤⣀⡉⠛⢷⣮⡭⡟⠿⢿⣿⣿⠛⠁⠉⠓⠢⣄⣠⠶⠋
⠀⠀⠀⠀⠈⠙⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣝⢷⣤⣉⢻⣷⣷⣦⣈⠛⠳⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣶⣿⠿⡝⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣾⣿⣿⣿⣻⣿⡿⠋⠀⠀⠉⠛⢦⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⣠⡴⣫⣭⣙⢻⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢾⣿⣾⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡉⠓⠶⣤⣀⡀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠘⢶⡆⠀⠀⠀⠀⡀⠠⣤⣤⣶⣆⢈⡁⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢃⣤⣾⣿⣀⡀⢀⣾⣿⣄⣽⣿⣿⣿⡾⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣯⣾⣿⡞⣻⣿⣿⣿⠋⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⢙⣿⣏⣿⣿⣿⣿⠋⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢀⣾⣿⣿⣿⣿⣋⣝⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣼⣿⡿⣿⣿⣯⡼⣽⣿⣿⣧⡤⠀⢴⣄⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢩⣿⡿⡏⣿⣿⣿⣷⣹⣿⣿⣿⣿⣿⡿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⡻⡇⣿⣿⣿⣿⡻⣿⣿⣿⣥⢋⣠⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⣿⣿⡇⠇⢻⣿⣿⢿⣿⣿⣿⣿⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⣿⣿⣧⣥⣿⣿⢹⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠉⣄⠀⡼⢛⣏⣿⡿⣼⣿⠟⠃⠙⠻⣻⣿⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⢙⣷⣧⣾⣛⣭⣾⣿⡷⠐⠀⢄⠴⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢿⣿⣾⣷⣿⣿⣿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠿⠋⠛⠻⠿⢿⠿⠿⠿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⢀⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣠⠴⣿⠟⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣭⣿⣿⣿⣛⠛⣿⣿⣿⣀⣠⠞⠁⣼⠃⠀⠀⣘⣷⣾⡷⠶⠶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠉⠻⣿⣿⣿⣧⡽⢡⣤⣼⠇⠀⠀⢠⣿⠛⢿⡄⠀⠀⠀⠉⢻⣦
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⣠⣴⣄⠀⣀⣨⣿⣿⣿⣷⣾⣿⡏⠀⠀⠀⣿⠃⠀⠈⢿⣄⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⢿⠟⠓⠒⣿⣿⣿⣿⡯⠟⠀⠀⠀⠀⣿⡀⠀⠀⠈⠻⠃⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠈⠉⠙⠆⠀⣠⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣀⣀⣤⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣤⣤⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠢⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣷⣌⡀⠀⣤⣿⣦⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣀⡤⠖⠚⠛⠛⠻⣿⣭⣿⠷⠎⠛⠉⠙⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣠⠔⠊⠉⠁⠀⠀⠀⠀⠀⠀⠙⢿⣻⡀⢷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⣞⠁⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⣀⣀⣳⡙⠞⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢸⡿⠿⣷⣶⣶⣾⣿⣿⠿⠛⠋⠙⠻⣿⡛⠉⠻⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢻⣀⣀⡤⠀⠉⠙⠛⠳⣦⣄⡀⠀⠀⠈⠙⠦⣄⣈⠙⠳⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠿⣧⡀⠀⠀⠀⠀⠈⠿⢿⣶⣄⠀⠀⠀⠀⠉⠙⠂⠀⢹⣿⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⡳⠶⢶⣶⣤⣤⣬⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡟⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⢟⣩⣭⣭⣭⣭⡍⣛⠿⡿⣢⣾⠟⠉⢹⣿⣿⡇⠀⠀⠀⢀⣀⣠⣀⣼⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣠⣾⠟⠋⠁⠀⠀⠀⠈⠛⢷⣾⠟⠋⣀⣴⣿⣿⣿⡶⠶⠒⠛⠉⣡⠞⠁⠈⠻⣆
⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⡟⠁⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⣠⡌⠁⣟⢻⣿⣿⣤⡤⢴⣾⠋⠀⠀⠀⠀⠀⢿
⣿⣿⣿⣿⣿⣿⣿⣿⢸⣟⠀⠀⠀⠀⣠⣶⣶⣶⣼⣿⣾⣷⣽⣷⠾⣿⠿⣿⣿⣅⣴⠟⠁⠀⠀⠀⠀⠀⢀⣼⣷⣶⣶⣶⢦⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⠀⠀⠀⡀⣿⣿⣿⣿⣟⠻⣿⣿⣾⣿⠀⣿⣰⣿⣿⣿⡁⠀⠀⠀⠀⠀⣠⣾⠟⠋⢻⠀⠀⠀⠈⠑⠪⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⡀⠀⠀⠐⠈⢿⠛⠉⠉⠉⠉⠙⣿⡿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⣰⡿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢻⣦⡄
⣿⣿⣿⣿⣿⣿⣿⣯⡀⠙⣷⣄⠀⠀⠰⠇⠀⠀⠀⠀⢀⣴⡟⣀⣬⣭⣵⣶⣦⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠠⢠⣿⣿⣷⡄⠀⠀⠄⠈⣿⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⡻⢷⣦⣤⣤⣤⣄⣤⡾⠟⣩⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠸⣿⠀⠀⠀⢀⢸⣿⣿⣿⡇⠀⠀⠀⠁⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⣭⣭⣭⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢄⣀⢻⣆⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⢰⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠙⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠙⠛⠿⣵⣦⣤⣤⣤⣴⡾⠟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡜⠃⡌⡇⠃⠏⡇⠉⣍⠸⢸⢸⡏⣿⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠙⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣘⣣⣄⢃⣧⣀⣀⣇⣇⣛⣸⣸⣸⣇⣿⣘⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠿⢩⡿⠙⣿⢩⡝⡏⡻⣹⠉⣽⡉⢽⡉⢽⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣷⣷⣾⣾⣾⣧⣷⣾⣶⣶⣮⣾⣮⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣾⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⠿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣥⠴⠶⠻⢋⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠩⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡿⣿⠶⣴⣿⣿⣿⣿⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡇⡼⠀⠹⣧⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⢻⡿⣿⡇⠀⠀⠹⣧⠘⣿⣿⣧⣤⣴⠿⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⢀⡾⣰⠟⣿⡀⠀⠀⠹⣇⣹⣿⣿⣿⣟⠦⣄⣀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⢠⣾⣥⡟⠀⢹⡇⠀⠀⠀⢻⣿⣿⣿⡷⠬⣷⣿⠿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠨⠙⠉⠁⠀⣾⡇⠀⠀⠀⠀⠙⠋⠹⣷⠀⠏⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠻⢿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⣄⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠿⣯⡙⠛⠶⢤⣤⣀⣸⣿⣿⣿⣿⣷⡀⢀⣠⣤⣴⣶⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⠻⣦⣴⣶⣆⡙⠻⢻⣿⠛⣿⣿⣷⣿⠟⠉⠉⠉⠉⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣤⣤⣰⣾⠃⠀⠀⠀⠀⠀⠻⣧⡈⠙⠻⠷⣿⣿⣄⡸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠛⠉⠉⠙⣻⣿⣷⣄⠀⠀⠀⠀⠀⠙⣿⣦⣀⡸⣿⣿⡽⣿⣿⣿⠛⢶⣴⣿⣿⣷⡄⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⢠⡿⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⠷⠾⠖⠾⢿⣿⣿⡇⠀⠀⠀⢸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⣠⢒⣿⣿⣧⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⠿⠿⣿⣆⠀⠀⠀⠼⠏⠉⠁⠀⠀⢀⣾⡟⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢧⡾⣿⣿⠏⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣀⠀⠀⠀⠀⣀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⢷⢶⡶⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⡀⠀⠀⠀⠀⠀⣠⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣶⣶⣾⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⡄⢠⡶⡄⠠⡶⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⢿⠹⡄⠸⡇⢸⡗⣧⠀⡇⢠⡇⣾⠀⠀⢰⢶⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣆⠘⣎⢷⡘⣇⠻⠀⠛⠈⠃⠉⠘⠛⠘⠑⠃⠀⢠⣏⡿⢀⡼⠋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⡀⠳⣌⣷⠘⠛⠁⠀⣀⣠⣤⣴⣶⣶⣶⣶⠆⠀⠀⠀⠀⠉⠁⠾⢁⡼⢳⠞⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡳⣍⠲⠈⠁⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠿⢶⣦⠀⢦⣄⠀⠈⠰⢋⡴⠻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠙⠊⠀⠀⠀⢘⣉⡩⣤⣤⠀⠉⠻⣿⣿⣿⣿⡶⠖⢀⣽⣧⠈⢿⣷⣄⠀⠈⠓⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣉⠀⠛⣯⠀⣴⣶⣾⣿⡿⠟⢉⣤⣾⣿⣿⣿⣥⠈⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣆⠹⣆⡉⠉⣉⣠⣴⡾⠟⢛⢟⣿⠃⣄⠹⣧⡈⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⡿⣾⣶⣤⣀⠙⢿⡄⢻⣿⣿⠿⠫⣵⣦⡤⣪⡾⢁⣼⣿⠆⠘⣷⣴⣶⣾⠿⢿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⣩⠴⠖⠒⠂⠈⠑⠮⡳⣄⠙⠀⢿⣁⣤⡾⡿⢛⣽⠋⣠⣾⣿⠋⣠⢞⡽⣿⡥⠖⠒⠒⠀⠉⠪⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢡⡞⠁⣀⠤⠀⠀⠀⠀⠀⣱⣿⣦⠶⢿⡿⢟⣪⡾⠋⣠⣾⣿⣿⠃⣰⣿⢋⡾⠙⣿⣤⠀⠀⠀⠀⠀⠈⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡇⡟⠀⠘⢱⣿⣿⣷⡶⠚⢛⣉⣹⣿⣀⣼⣿⣿⡅⢀⣈⣉⠉⣿⣿⠀⣿⡏⣼⠁⢠⢙⣿⣿⣿⡆⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠇⠀⠀⠘⢿⣿⡿⠛⠛⠛⠛⣻⣿⠙⣿⣯⣽⣿⠋⢡⣤⣤⣿⣿⠀⣿⣇⢹⠀⠀⠈⠿⣿⡿⠃⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣁⣴⠖⢒⣒⠲⣤⡀⠔⠒⠉⣹⣿⠃⢀⡈⠙⠋⢁⣤⣿⣿⣿⣿⣿⡆⠸⠟⣠⡶⣄⠀⠀⢀⡀⠀⣠⡶⡄⢠⡴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⣿⠃⠀⢸⣿⠀⠤⠞⠛⠁⡰⠛⢿⠟⠻⠿⠛⢿⠟⠿⠟⠛⢿⠆⣼⡟⣰⠃⠀⠀⠛⠃⣰⡟⡰⢱⡿⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣾⠇⠀⠀⣸⣿⢀⣴⠟⢻⡆⢠⣿⠆⣰⡷⠀⣿⠀⣴⡶⢚⣿⠆⢰⣿⠟⣻⡷⠀⣾⡇⢰⣿⠗⢁⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣛⣻⣾⣯⣀⣀⣴⡿⠃⢸⣿⣠⣾⣧⣼⣏⣠⣿⣇⣼⣏⣤⣿⠁⢸⣿⣀⣾⡏⠠⣿⣃⣼⣿⣀⣼⣿⣀⣼⣿⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠉⠀⠀⠉⠁⠈⠁⠀⠈⠉⠀⠉⠉⠀⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⣴⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠶⠶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣭⡉⠛⠻⠿⢿⣿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⣹⣿⣿⣿⣿⣿⣿⣿⠛⠿⢿⠿⣶⣶⢦⣶⣾⣿⡍⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⢴⣿⣿⣿⣿⣿⣿⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⡾⢻⣿⠻⣯⡻⣦⡀⣀⣤⣴⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣇⠀⣠⠟⠀⣿⠇⠀⠈⠻⣾⣿⣿⠋⠉⠉⠉⠙⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⡾⠁⠀⢸⡟⠀⠀⠀⢠⣿⣿⣝⢷⣄⡀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣾⣿⣿⣶⣦⢀⣿⠃⠀⠀⠀⢸⣿⠀⠙⣷⣿⣿⣷⡄⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠸⣿⡆⠈⠛⠿⡟⡻⠃⠀⠀⢐⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⢠⣾⡟⢹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣿⡇⠀⠀⣾⣿⣿⣿⣠⣼⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⣿⡇⠀⠀⢽⣿⣿⡏⠁⠘⢿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠉⠈⠿⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⣀⣀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢿⣿⣿⣿⣿⠿⡇⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠻⣶⣿⣿⣿⣿⣿⠿⡞⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡄⠀⠀⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣼⣿⡀⠀⠛⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠺⠷⢶⣤⡀⠞⢁⣼⡿⠁⠀⠀⠙⢦⡢⡄
⠀⠀⠀⠀⡀⠀⠀⠀⢀⣾⠋⠀⣀⣀⣀⣹⠟⡀⣼⣿⣀⣀⣀⣀⣀⣸⡿⠗
⠀⠀⠈⠂⠈⠲⣄⠀⠸⡿⠀⠐⣿⣿⣧⡛⢸⣿⣯⣄⠀⢀⣉⡭⠿⢻⡇
⠀⠀⠀⠀⢀⣑⠈⠳⡄⠻⣆⡀⠀⠰⣬⡛⠿⣷⣿⡿⠛⠉⠁⢀⣀⣸⣅⣀
⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣬⡛⠛⢛⣛⣉⠁⠛⠛⠁⠀⢀⣴⠛⠉⢸⠈⠛⠶⡀
⠀⠀⠀⠀⢸⣿⣿⣿⡏⠙⣿⣿⡆⣿⣿⠛⢿⡄⠀⠀⠀⢸⡇⠀⠖⣽⠑⠄⠀⣷
⠀⠀⠀⠀⢸⣿⣿⣿⣇⣠⣿⣿⡇⢿⣿⣤⣼⡷⠀⠀⠀⠹⣧⠀⠈⠒⠊⠀⣠⡟
⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠋⣀⣀⠙⣻⣿⣁⠀⣤⣤⡄⠙⠷⣦⣧⣤⠶⠛
⠀⠀⠀⠀⢰⣶⣶⣠⣾⣿⣧⣼⣿⡿⠀⣿⣿⣿⢷⣿⣿⣿⣦
⠀⠀⠀⠀⢸⣿⣿⣿⠏⢹⣿⣿⣿⠇⠀⢸⣿⣿⣧⠈⠻⠿⠿⠷
⠀⠀⠀⠀⠘⠛⠛⠋⣀⠀⢉⣩⡍⢠⣤⣴⡄⠀⠀⢾⣿⣿⣦⠀⣀
⠀⠀⠀⠀⢸⣿⣇⣸⣿⡇⢸⣿⣿⠀⣿⣿⣿⡀⠀⠈⢿⣿⣿⣧⠙⣥⣤
⠀⠀⠀⠀⢸⣿⡿⢿⣿⣷⠘⣿⣿⡇⠸⣿⣿⣧⣤⣤⡈⢿⣿⣿⣷⣭⣭⣤⡀
⠀⠀⠀⠀⠸⠿⠇⠨⠿⠿⠀⠿⠿⠿⠀⠻⠿⠿⠿⠿⠿⠘⠿⠿⠿⠿⠿⠿⠿⠄⠖⠦
⠀⠀⠀⠀⠰⠖⠀⠶⠀⠲⠀⠰⠂⠐⠂⠰⠆⠰⠀⠲⠀⠀⠀⠶⠀⠀⠰⠆⠐⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⠉⠉⠉⠉⠁⠀⠙⠿⣿⣷⣄⡀⠉⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⡀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡙⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⣿⣿⣿⣿⣿⣿⡙⠻⣿⠛⠉⠉⠀⠀⠀⠈⠉⠉⠛⠛⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣶⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣆⢹⡄⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠋⠃
⠀⠀⠀⠀⢀⣼⡿⠋⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡏⠘⣷⣄⣀⣠⡴⠞⠛⢉⣼⡿⠋⠙⣿⡄
⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⡀⣀⠀⠀⣀⣸⣿⣿⣿⡁⢀⣽⡿⠋⠁⠀⠀⣠⣿⠏⠀⠀⠀⢻⣧
⠀⠀⠀⢸⣿⠀⠀⢀⣰⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣾⣿⣤⣤⠀⢀⣼⡿⠁⠀⠀⠀⠀⠸⣿⣀
⠀⠀⠀⢻⣿⠀⠀⠘⢿⣿⡛⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⡷⠾⣷⣿⠋⠀⠀⠀⠀⠉⢛⣻⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠸⣿⡄⠀⠀⠻⠇⠈⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣴⡾⠟⠁⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⠉⠉⠉⠉⠙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⢀⣠⣼⡿⠻⣿⣿⡿⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⢠⣿⣿⣄⡄⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⡿⢿⣿⠿⢃⡀⠀⠀⠀⠀⠀⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣶⣦⣴⣶⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣤⡀⠀⣶⠀⠀⡦⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⣶⠀⢸⣧⠀⢹⡇⠀⣿⠀⢸⡇⠀⣸⠃⠀⣾⠁⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢻⣧⠀⢹⣇⠀⢿⡄⠸⣇⠀⣿⠀⢸⡇⠀⡿⠀⣰⠇⠀⣼⠃⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠻⠷⠀⠹⠷⠀⠻⠆⠘⠇⠀⠿⠀⠻⠀⢸⠁⢸⠇⢠⡟⠀⣼⠃⢀⡼⠁⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⣴⣿⣿⣷⠌⠀⣾⠀⡼⠁⠰⢃⣀⣚⡀⣠⠞⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⣉⣉⣉⣛⣛⣛⣛⠛⠛⢃⣀⣸⣿⢿⣿⠄⡀⠋⣀⣥⣶⠞⢻⣫⠟⡁⠁⢀⠴⠉⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠬⠭⠭⢭⣭⣭⡍⢉⣩⣶⣿⣿⣿⣿⡛⠻⠿⠿⠿⠟⠛⠋⠁⠴⢘⣯⡀⠀⢐⣁⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⢀⣯⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢷⣶⠟⠋⠉⠉⠉⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⢀⣼⣿⣿⣿⣿⣟⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⡾⣫⡿⢱⡿⠙⣦⡀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣉⣉⣉⣉⣉⣉⣉⡁⠟⣻⣽⣷⣾⣿⣿⣿⣿⣿⡿⣿⣿⡀⠀⣠⡾⠋⣰⡿⠁⣿⡇⠀⠈⠻⣶⣿⣷⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣉⣉⣁⣈⠻⠿⠿⠿⠻⠛⢻⣿⠃⠀⠘⢿⣷⡜⠋⠀⣼⡟⠁⠀⢻⣧⠀⠀⠀⠙⠛⠋⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⠶⠶⠶⠶⢶⣤⣤⡄⠀⠀⠀⣀⣤⣤⣤⣤⣜⡟⠀⢀⣴⣦⠙⠻⣦⠾⠟⣀⣤⠄⠀⠻⣷⣄⠀⠀⠀⠀⠀⣀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣶⡶⠶⠶⠶⠆⠀⣠⡾⠛⠉⠉⠀⠉⢹⣿⣷⡟⠋⠙⣷⣠⡬⢿⣿⣿⡧⠀⠀⠀⠈⠛⠿⠷⣶⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣴⠶⠶⠆⣼⡟⠀⠀⠀⠀⡀⣀⣼⣿⣿⣿⣿⣶⣿⣿⠟⠋⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⢁⣀⣠⣤⠶⠖⠂⣿⠅⠀⠀⠠⣿⣿⣿⡿⠿⠛⢻⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠛⠋⠉⣀⣠⡴⠞⠃⢿⣇⠀⠀⠀⠉⠛⠉⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠉⢀⣤⠶⠛⢈⢻⣦⣀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⡴⠞⠋⢀⣠⠞⠋⣠⠌⠛⠿⠷⣶⠾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣠⠶⠋⠁⣠⠞⠁⣠⠋⢠⠖⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⢀⣴⠞⠁⢠⠞⠁⣰⠋⢀⠎⢀⠎⠀⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠏⠀⠀⠈⠻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⢀⡴⠃⠀⡴⠁⢀⠎⠀⡸⠀⢰⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣳⣤⠞⠁⠀⣀⣤⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⡞⠀⢀⡎⠀⢠⠃⠀⠆⠀⠀⢀⠀⣰⣄⠀⢀⣤⣶⣿⣿⡟⠁⣠⣷⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣬⣤⣤⣤⣤⣤⣴⣤⣾⣶⣿⣿⣿⣶⣿⣿⡏⣿⣷⣶⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⡿⣿⢻⢿⡿⣿⣿⣿⡿⣿⣿⣶⣿⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠈⠳⠟⠛⠉⠉⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⢹⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠙⢿⣿⣿⡿⢿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠈⠙⠻⢷⣭⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣤⣄⡀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢈⣿⡟⠿
⠀⠀⠀⠀⠀⢀⣴⣿⠟⠛⠉⠉⠀⠉⠉⠙⠻⣿⣦⡀⠀⠀⠀⣿⣏⣽⣿⣿⣿⣿⣤⡤⠴⠶⠒⢛⣿⣿⣿⣧
⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⣠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⣠⣴⡿⠋⠁⠈⣿
⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⣀⢤⣶⣤⡀⠀⢀⣀⣻⣿⣿⣿⣿⣿⠟⣿⣿⠃⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⢹⡇
⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⣌⣼⣿⣷⢿⠛⠛⠙⢿⣿⣿⡇⠀⣿⠁⣸⣿⡇⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⣸⣿⣧⣴⣶⣶⣤⣀⡀
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠙⠿⣿⣿⠛⠳⠶⢦⣤⣽⣿⣿⣴⣿⣾⣿⣿⣷⣟⠁⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⣿⡏⠉⠉⠙⠛⠿⣿⣶⣄
⠀⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠠⣾⡇⠀⠀⠀⠀⣠⣿⡟⠻⠿⡿⣿⣿⣟⣿⠿⠿⠦⠀⠀⠀⠀⣴⣿⠟⠁⠀⢹⣇⠀⠀⠀⠀⠀⠈⠹⣿⣦
⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⢸⣿⡀⣀⡀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣶⣶⣶⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⣽⣿⣷⣾⣴⡆⠀⠀⠀⠀⢻⣿
⠀⠀⣀⣀⣀⣀⣤⣤⣤⣄⣈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⢻⣟⣛⡵⠀⠀⠀⠀⠀⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣦⣤⣤⣴⣶⣿⡿⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⠀⠉⠉⠛⠛⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢀⡀⠀⣸⠃⣰⠞⠀⠀⠀⡠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤⠀⠀⠀⠀⠀⢀⣴⡯⣰⣿⠁⡴⢡⡞⠁⠀⠀⠀⢠⢆⡴⠃
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⠇⣤⠀⢀⡀⢠⣾⡟⣿⡿⠋⠸⠘⠋⠀⠀⠀⡠⢂⡾⣫⣶⠏
⠀⠀⠀⠀⣾⣽⡿⢀⣤⢀⡀⠀⠀⢀⠀⠀⠀⡞⢰⠏⡼⣣⢰⡿⣿⣿⣯⣾⣿⣣⡼⠁⠀⢀⠀⣠⢞⣴⣿⡾⠟⠀⣤⡤
⠀⠀⠀⠀⢻⣷⣶⠆⢹⣿⣧⠀⠀⠀⠈⢓⠖⠱⢇⣾⢳⣷⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣠⢆⣵⣿⣿⣛⣦⣮⡾⠏
⠀⠀⠀⠀⠀⠈⣷⣦⣾⣿⣿⡇⠀⡄⠀⠀⢀⠀⠀⢻⣷⣿⣿⣿⣿⣿⣿⣿⣛⢁⣠⡴⠊⣰⣿⣿⣋⢉⣠⣿⣡⠞⠁
⠀⠀⠀⠀⠀⢈⣿⡿⠛⠛⠻⡇⠀⣵⣶⣖⣿⡄⠀⢀⣿⣿⣿⣷⣿⣿⣿⣿⣶⡟⢠⣶⣾⡿⢿⡟⣽⢟⡵⠛⣁⡀
⠀⠀⠀⠀⠀⢸⠋⠁⠀⠀⠰⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣟⣿⣿⣉⣽⣿⣿⣥⣾⡿⣫⣷⣭⠞⠛⠀⢀⣀
⠀⠀⠀⠀⠀⡇⠀⠀⢀⢠⡁⠀⠀⣿⣿⡿⡿⠀⠀⣿⣿⣿⣿⣿⣟⣿⣵⣿⣿⣿⣡⣾⠿⢿⡾⠏⠉⠀⣠⣴⡿⠛⣉⠠
⠀⠀⠀⠀⠀⣇⠀⢀⡼⠎⠁⠀⠠⣿⣿⣿⣯⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣟⣵⣿⡿⠿⠔⠋⢀⢀⣴⣾⣧⣴⣾⠭
⠀⠀⠀⠀⠀⣃⡴⢿⡆⠀⠀⢠⣾⣿⣿⣿⣿⣦⡀⠘⣿⠟⣽⣿⡿⡿⢿⣿⣿⡃⠋⠀⣠⣶⣿⣿⣿⣿⡿⠟⠋⣶⠖
⠀⠀⠀⠀⠀⠉⠁⣿⡒⢦⣤⣆⣽⣿⠿⠝⡿⣿⣇⠀⢸⣼⣿⣿⢟⣿⡿⢑⣋⣤⣾⣿⣿⣿⣿⠿⢋⣡⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡏⢹⣿⡙⣿⡿⢦⣄⣿⣳⣿⡄⠀⠄⣨⡙⢿⣿⣷⣿⢿⡿⣿⣿⣿⣿⣯⣶⡉⠁⠀⠀⣤⣆
⠀⠀⢀⣠⣤⠶⠾⣿⣧⣈⠤⠛⣯⡇⠀⣿⣿⣿⣳⣧⡀⠸⣿⢿⣌⢿⣿⡷⣛⣿⣿⣿⣿⣟⣉⣽⣿⡿⠿⠟⠋⢀⡀
⢀⣴⠟⠉⠀⠀⢸⣿⠈⠙⢷⣼⣿⠟⢦⡟⣻⣿⠿⣿⠇⢰⣿⡏⢥⠘⣷⢿⣿⣿⣿⠿⣿⡿⡿⢿⣃⣀⣤⣖⠚⠉
⣼⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠹⣷⣶⣶⣿⣾⣿⣿⣉⡴⠻⠙⣷⣿⢀⡿⢿⣿⣿⣗⠋⠛⠙⠛⠉⠉⠁
⣿⡀⠀⠀⠀⠀⣾⠻⠀⠀⠀⠀⣿⣿⣿⣿⣿⣙⣿⣮⡲⣶⣷⣾⢏⣼⣷⠿⠟⠋
⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣻⣿⣿⠟⠷⠶⠶⠛⠉
⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠿⣿⡿⠿⠛⠁
⠀⠀⠈⠻⠷⣶⣤⣤⣴⠾⠟⠁⠀⠈⠁
⣿⣟⣻⠿⠿⠿⠻⢻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡄⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⠿⠿⠛⠛⣹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⣠⣾⡿⠋⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⣀⠀⠀⣠⠟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠹⣧⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⣠⣴⣿⣿⠿⠛⠛⠙⠛⠛⠿⣿⣿⣿⡃⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢀⣴⣾⣿⣿⠿⠛⠛⠛⠛⠿⢿⣿⣦⣄
⠀⠀⢀⣼⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⢿⣿⣦⠀⢻⣧⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠀⣴⣿⡿⠋⠘⣿⡆⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⡀
⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠹⣿⣧⣬⣿⡄⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⢸⣿⡏⠀⠀⠀⠀⠀⢀⣴⣶⣴⡟⠀⢀⠀⠀⠀⣻⣿⣝⣿⣷⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⢹⣷⣀⡀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⢸⣿⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⡷⠤⢤⣤⣤⣤⣼⣿⣇⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⡷⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣧⠀⠀⠀⠀⠀⠈⠛⣿⡏⠀⠀⠀⠀⠀⠀⣾⣿⠋⢻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠤⣼⣿⠟⠀⠀⠉⠉⠉⢳⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠻⣿⣷⣄⡀⠀⠀⠉⠁⠀⠀⠀⢀⣤⣾⣿⠋⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟
⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣤⣤⣤⣶⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣶⣦⣤⣤⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠀⠀⢄⠀⠀⡇⠀⡇⠀⡇⠀⢰⠀⠀⡞⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠐⣆⠈⢆⠀⠸⡀⠀⡇⠀⡇⢀⡇⠀⡇⠀⡼⠀⢠⠂⠀⡠⠋
⠀⠀⠀⠀⠠⡀⠀⠀⣄⠈⢣⡀⠈⢆⠘⡆⠀⢇⠀⢧⠀⡇⢸⠀⢸⠀⣰⠁⣰⠃⢀⡜⠁⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠈⠣⡀⠑⣄⠈⢆⠘⡄⠸⡀⢸⠀⡇⣸⠀⡇⢠⠃⡰⠃⣠⠊⢀⠔⠁⢀⡴⠊
⠀⢀⡀⠀⠀⠤⡀⠈⠳⢄⠈⢢⡈⢢⡈⢧⠹⡀⢇⢸⠀⡇⡇⢸⢀⠏⡰⠁⡴⢁⡴⠋⣀⠔⠉⢀⡤⠂
⠀⠀⠉⠒⠤⣀⠈⠑⢤⡀⠑⢤⡙⢦⠑⣌⢧⠱⠘⠈⠀⠁⠃⠃⠞⡰⢡⠎⡠⠋⡠⠞⢁⡤⠚⠁⣀⠤⠒⠉
⠀⠀⠠⠤⣀⡀⠉⠒⠤⣈⠓⢤⡙⠦⡑⠈⣠⠔⠚⠉⠉⠉⠉⠑⠲⢄⡁⠊⡴⢊⡤⠚⣁⡤⠖⠋⢁⣀⡤⠄
⠤⢤⣀⣀⠀⠉⠙⠒⠤⢄⣙⠲⢬⠃⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠘⡡⠖⣋⡡⠤⠒⠊⢉⣀⣀⠤⠤
⠀⢀⣀⣀⣉⠉⠑⠒⠒⠤⠬⢍⠃⢰⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢐⡩⠥⠴⠒⠚⠉⠉⢀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠉⢉⣉⣑⣒⣲⣤⣿⣤⣤⣤⣿⣿⣿⣾⣶⣷⣴⣶⣤⣶⣦⣼⣿⣒⣒⡚⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡷⡛⠉⠙⢧⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢁⠀⠀⢘⢃⣿⢷⣦⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠿⢸⣿⡃⢿⣿⣿⣿⡻⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣧⣤⣬⣿⣟⠧⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣷⠆⠹⣿⣿⣷⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣹⣿⡀⢠⠘⣿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡌⠛⢿⣿⣿⡟⠅⠀⠀⠙⠉⡛⢶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡧⠀⠀⠈⠟⢀⣠⣴⡾⢿⣿⠂⢿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠁⠀⠀⣠⠖⣋⣥⡶⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⣴⣿⡇⠴⠶⢊⣵⣿⣿⣿⠁⠀⣰⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣽⠶⣶⣿⡏⠀⠀⠃⠀⣠⣾⣿⣿⡿⠃⠀⠸⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⠀⠀⢀⣴⣞⢿⣿⠟⠁⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣿⠻⣦⣀⣸⣧⣯⡷⠝⢂⣤⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⠀⠀⢠⡶⣿⡿⠉⢷⡘⡿⢿⣿⣿⣿⡶⠹⡿⠋
⠀⠀⠀⢀⣴⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣾⢿⣽⠏⠀⠀⠀⠻⡜⣆⠻⣿⡇⠀⠀⢠⡀⠀⠰⣴⣄⡀
⠀⢀⣴⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠈⣿⣿⡜⣿⣄⠀⠀⠀⠀⠹⣼⣆⢹⣟⢆⠀⠈⣿⣆⠀⠀⢻⣿⣦⣀⡀
⠀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡟⠘⣿⣿⣦⠀⠀⠀⠀⠹⣿⣎⣿⡏⠃⠀⣙⢿⣧⠀⢸⠟⠻⠿⠛
⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠹⣿⣿⣿⠒⠀⢻⣷⣮⣧⡆⢀⣠⣴⣶⣿⣿⣿⣿⣿⣶⣦⣄
⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⣺⣿⠋⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠘⣿⡿⢿⣯⡌⠿⢉⣿⣷⣝⢿⠿⠛⠋⠉⠉⠉⠙⠛⢿⣿⣿⣦⡀
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠸⠎⠙⠋⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣀⣨⠵⠚⠛⠀⠳⢸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠈⢉⣒⡒⠒⠒⠒⠒⠒⠲⣿⣿⣿⢿⠛⣿⣅⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆
⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠀⠈⠛⠛⠛⣿⠟⠛⠸⣿⣿⣧⣦⣬⣬⣽⣿⣷⡄⢦⡀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⢻⣿⣿⠙⠶⠶⠶⠾⡿⢿⣷⣾⣗⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠙⢿⣿⣿⣷⣶⣤⣤⣤⣴⣶⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠘⠠⠤⠴⠁⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣶⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⡀⠀⠀⠿⣿⣻⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣤⣾⣿⣶⣌⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣷⣶⣶⡶⠂⠉⢩⡟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣼⣿⣮⣻⠋⠀⠀⣠⣿⣷⠀⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣏⠃⠀⠀⠀⠘⣷⡀⠀⠈⠙⠢⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠘⣿⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠙⢶⣤⣾⡙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠻⢿⣿⣷⡀⠀⠀⠀⠀⠀⢠⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⣷⢶
⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⣦⣄⡀⠀⢠⣿⡿⠟⠁⠀⠀⠀⠀⢀⠌⠀⠀⢘⣼⣆⣶⡶⣦⣤⣀
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣽⣻⣶⡛⠹⡄⠀⣿⠀⠀⠀⡐⠁⠀⢀⣴⠟⢱⣿⡏⠀⠀⠀⠙⠳⣄
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡀⢷⠀⢿⠃⡠⠊⠀⠀⠀⣼⠃⠀⠀⢻⣻⡄⠀⠀⠀⠀⠹⣧
⠀⣼⡇⠀⠀⠀⠀⢠⣾⣰⡆⠀⠀⢻⣿⣿⡇⢰⣶⠆⠀⣀⠀⠀⠀⠠⣿⠀⠀⠀⠘⣷⣷⡄⠀⠀⠀⠀⢹⡇
⠀⢿⡇⠀⠀⠀⠀⠘⠋⢭⣥⠀⠀⠀⠀⢰⡞⢿⣿⣷⣾⣿⣿⣿⡶⠆⣿⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠘⣷⡀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⣠⡿⠁⠀⠙⠛⠉⠀⠉⠁⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠈⢿⣄⡀⠀⠀⠀⠀⠉⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣀⠀⠀⠀⠀⢀⣤⡾⠁
⠀⠀⠀⠀⠈⠛⠷⠶⣶⡶⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠉⠙⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⢿⣿⣿⣿⣿⣆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⠋⠉⠀⣠⣿⡿⠋⠉⠻⢿⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⢿⣾⣠⣴⡶⠿⠛⠉⠁⠀⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⠀⠀⢹⣿⣀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⣴⠟⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣴⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠈⠉⠛⠻⣿⣿⣿⣧⡀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⢀⣴⣿⣿⣿⢿⣿⣏⠁⠀⠀⠀⠈
⠀⢠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠙⢿⣿⣿⣄⢻⣧⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⣠⣿⣿⡿⠋⠁⠀⢿⣿⡄
⢠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠈⢿⣿⣿⡎⣿⡆⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠘⣿⣷
⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠸⣿⡄⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⢹⣿⣧⣤⣀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣶⡀⠂⠀⠀⠀⠀⠀⢺⣿⣿⡆⢻⣷⣤⣿⠏⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⡌⣷
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠙⠛⠛⠻⠷⠶⢶⣾⣿⣿⣷⣿⣿⣿⣿⣧⣀⣀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠻⣬⣛⣥⠟
⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠙⢿⣿⣿⡿⠋⠀⠈⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣾⠁⢀⣀⣀⣤⡤⠤⢤⣾⣿⣿⠋⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄
⠀⠈⢻⣿⣿⣦⣄⠀⠀⠀⠀⠀⠛⠋⠉⠉⠀⠀⢀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡀
⠀⠀⠀⠙⢿⣿⣿⣷⣶⣤⣀⣀⣀⣀⣀⣤⣶⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⠐⢉⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⣀⠤⢀⢰⡐⣨⣤⠶⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⣤⣙⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠉⠀⠄⠀⠀⢚⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠨⠀⠀⠈⠀⠁⠀⠒⠈⠀⠂⠀⡳⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠏⢉⠏⡜⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⡈⡉⢽⡀⣈⣠⠭⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢨⣑⠒⠂⠀⠀⡀⠀⠈⠠⣀⠕⢦⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢆⢀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⡀⢀⡏⠀⠀⠲⢷⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠀⠀⠐⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣏⢁⣴⣦⣄⡙⢿⡁⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣛⣛⣛⣛⡛⠿⠿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢛⣯⣝⢿⣿⡿⠋⠀⠀⣰⣇⠙⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣾⣿⣿⣷⣧⠀⠀⠀⠺⢿⣟⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣌⡁⠀⠀⠀⠀⢻⣄⡀⠀⠀⠈⠳⣄⣠⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠙⠷⣄⠀⠀⠀⢘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠄⠀⠀⠀⠀⠈⠳⣶⣴⣯⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡛⠿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⡌⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⣴⣻⡿⠇⠀⠐⠀⠈⠀⠀⡀⢺⣦⣶
⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣀⠀⠀⠀⣹⣿⣿⠿⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢹⣸⡠⣤⣤⣤⣄⡀
⠀⠀⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠈⢙⠷⣦⡾⠟⣯⠀⠀⣶⠀⠀⠀⢀⠜⠁⠀⢀⣴⠿⣯⣿⡟⠀⠈⠉⠻⠷⣄
⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⡄⠸⡆⠀⣿⠄⠀⡠⠁⠀⠀⢀⣾⠋⠀⠘⣿⢷⡀⠀⠀⠀⠈⠙⢷⡀
⠀⢰⡟⠀⠀⠀⠀⠀⢀⣠⢀⡀⠀⠈⢿⣿⣿⣿⡀⣣⣀⠉⠀⠊⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⣞⣧⠀⠀⠀⠀⠀⠙⣷
⠀⢸⣇⠀⠀⠀⠀⠀⢾⡿⣾⡇⠀⠀⠀⠛⠟⢻⣦⣾⣿⣄⣠⣶⣦⣤⣀⣀⢻⡇⠀⠀⠀⠀⠻⠿⠷⠀⠀⠀⠀⠀⢻⡆
⠀⢸⣧⠀⠀⠀⠀⠀⠈⠀⠛⡷⠀⠀⠀⠀⢠⣿⠉⢿⣿⣿⠿⠿⠿⠿⠟⠉⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⢴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⡀⠀⢀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⠶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⢰⣿⡄⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⣿⣿⣶⠶⠿⠿⠿⠶⣾⣿⣿⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡾⠛⣁⣤⣴⣶⣶⣶⣦⣤⡉⠙⢶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⣶⣿⣿⣿⣤⣽⣿⣿⣿⣿⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀⣛⣛⣛⣂⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠾⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣉⣉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠾⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⠍⠉⠉⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣯⣭⣭⣽⣯⣬⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣯⣿⣿⣭⣿⣯⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡛⠁⠀⠘⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⣄⠀⠈⠛⠿⣿⣻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⡀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠉⢻⡿⠧
⠀⠀⠀⠀⣴⣾⠟⠋⠉⠉⠉⠙⠻⣷⣄⠀⠀⢿⣯⣿⣿⣿⣧⠤⠶⠒⢒⣿⣿⣧
⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⡀⠀⠀⠈⢿⣧⣴⣿⣿⣿⣿⠿⠁⠀⣠⡾⠟⠁⠀⢹⡀
⠀⠀⢸⣿⠀⠀⠀⢀⣾⣾⣽⡷⠶⢶⣾⣿⠋⢻⠋⣼⡟⢀⣠⠾⠋⠀⠀⠀⠀⢸⣷⣄⣄⣀⣀
⠀⠀⠸⣿⡀⠀⠀⠈⠻⣿⡟⠒⠶⢤⣽⣿⣧⣿⣾⣿⣷⣟⠁⠀⠀⠀⠀⣠⣾⠿⣿⠛⠉⠛⠻⢿⣦⡀
⠀⠀⠀⠻⣷⡀⠀⠀⠀⠻⠀⠀⠀⣠⣿⠏⢛⡟⠿⠿⠟⠛⠛⠀⠀⠀⣼⡟⠁⠀⢹⡆⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠙⠿⣷⣦⣤⣤⣤⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⣷⣶⣢⠀⠀⠀⢸⣿
⣀⣀⣀⣀⣠⣤⣤⣉⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⢿⡛⠯⠀⠀⠀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣦⣄⣀⣀⣀⣤⣶⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣀⡀⠀⠉⠙⠛⠛⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠈⠹⣿⣷⣦⣤⣄⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣷⣄⠀⢀⡤⠿⢿⠋⣧
⠀⠀⠀⠀⠀⠀⣤⣴⡖⠛⠀⠀⠈⠻⢿⣿⣿⣆⣘⣧⣠⠞⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⣀⣠⡤⠞⣿⣿⡏⣸⢷⣤⣠⣴⠶⠶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣏⠉⠀⣴⣾⣿⣿⢡⠏⣨⡿⣿⣄⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠘⢦⣰⣿⠟⢿⣿⡟⢠⡿⠀⠈⠻⢧⣀⡀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⣀⣴⠶⠛⠛⣿⠷⣦⣾⠿⡅⠀⠸⣿⠁⢸⣇⠀⠀⠀⠀⠛⠃⠀⠀⠀⢰⡟
⠀⠀⠀⣴⠟⠁⠀⠀⢠⡏⠀⠙⣿⣦⣽⣿⣽⣷⣄⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⢀⣀⣤⣴⠆
⠀⠀⢸⡏⠀⠀⠀⠀⣸⣁⣠⣤⠾⢿⡿⣿⣻⠟⠉⠀⠀⠙⢷⣤⣀⣀⣀⣤⣴⣿⣥⣴⣾⣿⣿⣿⣿⣿⡟
⠀⠀⢸⡇⠀⠀⠀⠀⠙⠉⡤⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣀⣠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⡟⠁
⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣴⣾⣯⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⡿
⠀⠀⠩⠿⠿⠿⠿⠿⠿⠟⠛⠉⠙⠓⠻⠿⠿⠟⠟⠿⠛⠛⠛⠉⠉⠉⠀⠐⠒⠛⠛⠛⠛⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣤⣮⢾⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡽⠃⣴⣤⣿⣯⣾⣷⣶⣤⡀⠀⢀⣠⣤⣴⣿⣿⡿⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣟⡅⢰⣿⣤⣿⣿⣿⡿⠏⣹⣷⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢗⣫⡞⠀⠉⠘⠟⠋⠃⠁⠀⡀⣽⠟⣻⣿⣿⡿⠟⠁
⠀⠀⢀⠀⠀⣀⣠⣶⣶⣟⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⠻⠟⠿⠟⠇
⠀⠀⠀⠻⠻⣿⣿⣿⣿⠽⠛⠉
⠀⠀⠀⠀⠀⠘⠛⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠈⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠀⠀⢠⣹⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠘⠃⠈⠳⢤⣈⢿⣿⣷⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠛⠛⠉⠉⠉⠙⠻⠆⠀⢀⠲⣤⣍⣛⣻⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⠟⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡿⠀⠈⠻⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠈⠀⠀⠀⠀⠀⣧⠀⢀⠄⠀⠀⠀⣿⡀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢐⣦⠀⠠⣀⣷⣿⡼⠋⠀⠀⠀⠀⠘⠿⠖⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⣲⣾⡟⠀⠀⠘⡟⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠰⣿⣟⡇⠀⠀⢠⣿⣾⣷⣾⣾⣷⣿⣾⠇⠙⣧⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠂⠳⣿⢸⣄⢀⣤⣽⣿⣿⠿⣿⡹⣿⣿⡇⣤⡀⠈⣀⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠀⢄⣻⣷⡗⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠳⣬⡻⣆⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⡀⠈⠻⠀⠀⠉⠉⣭⣝⢿⣿⠦⠀⠠⠴⢾⡿⢷⣤⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻⠇⢀⡄⣀⣀⣠⡤⢤⣴⣶⣒⠀⠀⣰⣶⡟⠙⠢⣮⠻⣶⣄⣤⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⢀⡾⠶⠟⠛⠛⠋⠉⠉⣀⠀⢀⣾⣿⡏⠁⠀⠀⠃⢀⣼⣏⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠂⠀⣴⡿⠁⠉⢀⣴⠋⠀⠀⠀⠀⠀⠀⣴⠟⢁⣴⣿⣿⠹⠿⠀⠀⠀⠰⣿⣿⣿⢷⡌⠿⣡
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠁⠀⠀⣸⡟⠀⠀⢰⡟⠁⠀⠀⠀⠀⢀⣠⠞⢁⣴⢿⣿⡟⠟⠀⠘⠿⣶⣦⡀⠈⠉⢹⣿⡥⠺⠟
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⢠⡟⠀⠀⣠⢋⣾⠀⠀⠀⣀⡴⠋⣡⡾⠟⠁⣼⡿⠀⡀⠀⠀⠀⠙⣿⣿⡆⠀⠀⠉
⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⣠⠔⠀⡾⠀⠀⠐⢱⣿⣯⡾⠋⠉⢀⣴⠾⡋⠀⠀⢠⣿⠃⣾⡇⠀⠀⠀⠀⢻⣿⣷
⠀⠀⣠⣿⣿⠋⠀⣀⡴⠋⠁⠀⠀⠁⠀⠀⠀⣿⣿⠁⠙⣠⣴⠟⢁⣾⠁⠀⠀⣸⡟⠀⣿⠇⠀⠀⠀⠀⣸⣿⡟
⠀⣰⣿⡿⠁⢀⣾⣿⣿⣧⡀⡄⠀⠀⣠⣶⣿⣿⣿⣴⣾⠟⠁⢠⣿⠇⠀⠀⢀⣿⠁⠀⣿⠀⠀⠀⠀⠀⣿⣿⠇
⢠⣿⣿⠁⠀⢋⣽⠟⠉⠀⣼⠇⠀⢰⣿⣿⣿⣿⣿⠟⠀⠀⢠⣿⡟⠀⠀⠀⢺⣿⠀⠀⡟⠀⠀⠀⠀⣼⣿⡟
⣼⣿⡇⠀⠀⠻⠇⠀⢀⣼⣟⠂⠀⢸⣿⣿⣿⣿⠋⠀⠀⢀⣿⡿⠁⠀⠀⠀⠀⠀⠲⠴⠇⠀⠀⠀⣰⣿⡿⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣤⣀⠀⣈⠛⠋⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⡈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠁
⠸⣿⣿⣷⣦⣤⣤⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋
⠀⠈⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣀⠀⢀⣀⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠖⠀⢀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⡶⠞⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠮⠐⠀⢀⠴⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠂⠀⠀⠈⠒⠁⠀⠒⠒⠀⠀⠀⠀⡩⠋⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣀⢀⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠒⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠂⠀⣠⣭⣤⠄

⣀⣀⣀⣀⣠⣠⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣄⣀⣠⣀⣤
⣿⣽⣿⣽⣭⣽⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣯⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⢿⢿⣷⡀⣉⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⠋⠉⠹⠟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠛⠛⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⡿⣿⣿⣿⣿⣆⢻⡇⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠻⣿⣿⣿⣿⣿⣿⣿⢻⣄⣿⡀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⢹⣿⣿⣹⣿⣿⣿⣼⣿⣾⣷⠄⠀⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣾⣹⣿⣿⣿⣿⣿⣿⡟⠻⠿⠋⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡙⠋⠹⣿⣿⡿⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⣿⡇⠀⣀⣙⣿⣿⣿⡆⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣾⣿⣾⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣿⣦⣈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣆⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⠀⢠⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢤⡧
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠠⢾⣿⣯⡀⠀⢸⣟⡁⠀⡀⠀⠀⠀⠀⠀⢀⣿⠙⠣⡀
⠀⠀⣄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠐⠀⠀⠙⠙⠛⠀⢠⡌⠓⠀⠀⠀⡀⠀⠀⢠⣼⣿⠄⠐⡋
⠞⠛⠋⠙⢷⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢸⡀⠒⠂⠀⠀⠀⠀⢀⣴⣿⣿⣤⣾⠇⠀⠀⠀⠁⠀⠀⠿⠿⣷⣄⠀⢼⡧⠀⠀⣤⣿⣿⣦⡄⠡
⠀⠀⠀⠀⠀⠻⣾⡁⠀⠈⠀⣰⠧⠀⠐⠀⠀⠀⢀⡜⠀⠈⢦⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣷⣦⠄⠀⠀⠐⠀⠀⠀⠈⠉⢀⣿⠇⠢⠐⣾⣿⣃⡘⠟⠟
⣅⣰⢄⡀⠀⠀⢻⣶⣤⣠⣚⣀⠿⢿⣦⠀⠄⠀⣾⣄⢀⣄⣈⡆⠀⠀⣀⠀⠀⠘⠻⠛⣿⣿⡤⠀⠀⠀⠀⠡⡀⠀⠀⠀⣬⡟⡓⢡⣴⣟⣿⢿⣧⢺⣦
⢿⢿⡿⠧⠘⠂⠀⠉⠙⠗⠉⠈⣢⠀⢰⡄⢀⠈⡃⠉⢻⠟⢻⠟⠒⠀⠉⠡⠀⠀⠀⢀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡿⠤⠤⢹⡏⣴⠆⢁⣆⢡
⣼⣮⡀⠀⠀⠀⠘⠲⠂⡀⠀⢺⣿⠀⠈⠰⠀⠀⠰⡀⠀⢀⡞⠀⠁⠀⠀⠀⠀⠤⠄⠈⠽⠿⣶⣀⣀⠀⠀⠒⠀⣄⣤⣿⢟⡉⠘⠻⣀⣾⣷⣴⣮⣽⡷
⢿⣿⡇⠆⠆⢘⣁⠠⠀⢆⠀⠀⠀⢀⡞⠁⠀⢀⡤⠴⠒⠁⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⢤⣀⡀⠀⣀⣄⢀⠀⠀⠀⠴⣿⣷⣶⣄⠀⠘⠛⠙⠛⠛⠛⠉
⠀⠈⠉⠁⠀⠘⠁⠀⠀⣈⣀⣀⡠⠊⠀⠀⠀⠈⠐⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⡀⠀⠀⠉⠀⠠⢿⣿⣿⡇⠀⠀⠀⠈⠉⢹⣤⠀⠀⠀⠀⣀⡴⠆⡀
⠸⠗⠛⠉⠀⠀⠙⠠⠀⠠⠜⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠈⢉⡀⠀⠀⠈⠉⠀⠐⣤⠤⠄⠀⠈⠉⠙⠀⠠⢤⣦⣄⠙⠧⠔⠀⠀⣠⣴⣷⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣓⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣧⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⣼⣿⣿⣷⠰⡞⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣏⣇⣀⡴⠋⠉⠈⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠻⢽⣿⣏⣿⣿⡗⠒⠶⣦⠀⠀⢸⡆⠀⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠲⢾⣯⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⣿⠶⣄⠀⠀⠈⢿⣿⡞⣧⡀⠀⠙⠀⢀⡞⣠⣾⣿⣿
⠀⢀⣀⢠⡇⡄⢆⣺⣿⣿⣦⣄⣴⡄⠀⠀⣿⠄⠀⣰⣾⣆⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⡿⠀⠈⢷⠀⠰⠛⠋⠁⠈⠙⠒⠖⢚⣿⣿⣿⣿⣿⣿
⣾⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣧⣾⣿⣿⣿⣿⣧⣰⣦⣤⣴⣠⡄⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠀⠀⣼⠀⠀⠀⢀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠻⢦⣄⣀⣤⣾⣣⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣆⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠂⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠶⠊⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⣴⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠠⠤⠀⢀⣼⠃⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣦⣄⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣄⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠁⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠒⠎⠀⠀⡀⠐⠀⣁⠠⠤⠤⢀⡉⠑⢄⡀⠀⢀⠌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢄⡤⠄⢀⡠⠒⠁⠀⠀⠀⠀⡊⠀⠐⢄⠑⡄⡌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠆⠁⠀⠀⠀⠀⡄⠂⠁⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠈⠆⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⣠⣄⢤⢠⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠉⠀⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠓⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡛⠀⢀⡀⠀⢈⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⢀⠠⠀⡏⠀⠈⠁⢀⠒⠀⡏⠀⢀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢻⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠅⠀⠀⠀⠠⠂⠀⣁⠀⠀⢢⡤⠖⠚⠒⠒⠒⠒⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠄⡄⠀⠀⠄⢀⠀⠀⢠⢹⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢠⠉⡇⠀⠀⠀⠀⠀⠀⡇⢠⡇⠰⠋⠀⠀⠀⢀⣮⠚⡀⠀⠀⣠⠾⡟⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢡⠀⠀⡀⠤⢄⠀⡇⠈⠁⠀⠐⡀⠀⠀⡜⠁⠀⡇⠀⠟⠁⢰⠇⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠰⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⣷⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⡀⠀⣧⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⡾⠀⠀⠀⡞⠙⡅⠰⢠⡆⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⢄⣸⠀⠀⠀⠀⠀⢀⣇⣀⡴⠀⢰⡇⠀⠀⢰⡁⠀⢻⠐⠃⢇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⢹⡀⠤⠔⠂⠈⠁⠀⠐⡄⠁⢸⠀⠀⠀⣸⠅⢀⠸⡆⠐⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣿⠀⠀⠀⣿⠀⠀⠀⠩⡄⢤⡄⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡀⠠⣤⠒⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢏⣀⣀⠜⠀⠀⠀⢸⡀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠳⣄⡀⠀⠀⠀⣀⡴⠞⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣀⣸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⣽⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢰⠙⠲⠟⠛⣧⣄⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⣀⣠⡿⢷⡘⠀⠀⠀⡀⡿⠁⠀⣧⠀⠀⠀⠀⢀⡔⡀⠂⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⢠⠔⠋
⠀⠀⠀⠀⠀⢀⣤⡾⠃⠀⠀⠉⠀⠀⠀⣿⡇⠀⠀⣿⣧⡀⢠⡴⠋⠀⠀⠀⠀⡾⠃⠀⠀⠈⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⢀⣾⠞⠁⠀⠀⠀⠀⠀⠀⠐⣾⡇⠀⣴⣿⣋⢇⡞⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣰⠞⠋⠀⢀⡤⠂
⠀⠀⠀⣰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡟⠀⣿⣟⣳⠾⠀⢀⠁⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⢠⡤⠞⠁
⠰⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡴⠊⠁⠀⢰⣿⣿⡿⠃⡠⠛⠀⠀⡠⠀⢀⡴⠋⠁⠀⠀⠀⣁
⠀⠀⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠋⢀⣠⡆⢀⣾⠟⠁⠀⠈⠀⠀⠀⢀⠀⠘⠁⠀⠀⢠⣤⠴⠛⠁⠂⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣬⣹⣿⡉⢄⣨⠆⠀⠀⠀⠀⠀⣀⠔⠃⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠠⠛⢁⠀⠀⠀⠄⠀⠉
⠀⠀⠀⠀⠘⢆⡀⡀⠀⡰⠋⣶⣿⡿⣿⣿⣇⠀⠀⠀⣠⠴⠛⠁⡀⠀⠐⠐⠋⠁⠀⡠⠅⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⠃⠈⠀⢰⣿⣿⣇⣻⣿⡿⣤⣴⡎⠁⢀⠐⠋⣠⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣿⣉⠛⠓⢲⣿⣿⣿⣼⠁⣾⠏⣊⢻⡆⠀⡘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣴⣿⣇⡙⠲⣄⣼⣿⣿⣿⣿⡶⢫⡎⠁⢸⣿⡟⠃
⠀⠀⠀⣼⡿⢛⡏⢀⡾⠛⡻⣿⣿⣷⣶⣶⡾⢷⣄⣼⠃
⠀⠀⣼⣿⢁⣼⡷⠿⠤⣼⢷⡾⢿⣿⡿⠚⠀⠈⣰⠃
⠑⣺⣿⣧⣶⡿⣿⣿⢛⢧⣦⣶⡿⢿⣦⣬⣤⠞⠁
⢀⣿⡏⡿⢩⣿⣿⣵⠾⠋⠉⠁⠀⠀⠉⠉
⢸⣿⠏⣡⡿⠿⠋⠑
⣀⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣷⣸⣯⣛⣿⣭⣫⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠠⡄⠀⠤⠤⢀
⠀⠀⠀⡠⠁⠀⠀⠀⠀⠙⡤⠐⠈⠀⠈⠢⠀⢀⠤⠐⠒⠒⠒⠀⠠⢀⡀
⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡍⢀⠠⠄⠀⠀⠀⠀⠀⠈⠳⢀
⠀⠀⠀⠈⠑⠒⠒⠂⠀⠈⠅⡀⠤⠜⠀⢠⠊⠁⠀⢀⠤⠒⠂⠀⠀⠀⠒⠢⢀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡰⠁⢀⠔⠊⠀⠀⠀⠀⠀⢀⡀⠒⣄⠀⢙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠩⠀⢍⡲⠇⠀⠀⠀⠀⠀⡀⠔⢈⡇⠀⠈⢦⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠉⠈⢉⡱⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠂⠂⠐⠂⠉⠀⠀⠈⠀⠀⠀⠀⢢⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠔⢉⡀⢉⡉⠉⠀⠀⠀⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⡔⠀⠀⠀⠀⣀⣤⠂⠀⢀⣀⣤⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡂⠀⠀⠈⠁⠐⠒⠂⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢒⡐⠦⠑⡒⠰⡊⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠙⠁⢠⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⡉⢙⡢⣄⡀⠀⠀⠀⠀⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢀⡀⠠⠤⠀⠐⠲⢤⡤⠤⢀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠳⣄⢀⣈⠀⠒⢒⠖⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⡴⠪⢄⡀⠔⠁⠀⠀⠀⠀⠑⠒⠀⠐⠓⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⡗⢎⠉⣠⠜⠁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡄⠀⠀⠀⠀⠀⠀⣮⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣰⠉⠀⠀⠛⠀⠐⣀⠐⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡨⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⠿⠀⠀⠀⡀⠄⠊⠉⢙⠉⠁⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠂⠀⡤⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢸⣦⣇⢴⠊⠀⠀⡟⠁⢈⠇⣀⠼⠇⠁⠰⠀⠀⠀⠀⡎⢉⠉⠘⠋⠉⠀⠀⠀⠎⠉⠒⠤⢀⣀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⡞⢈⠈⠙⠳⣦⢰⠁⠀⡜⠈⠀⠀⢐⡀⠀⡷⡄⢀⠴⠋⠈⠉⠀⠒⢠⡀⠀⠀⠘⢌⠓⠒⡰⠉⠉⠂⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠡⡂⢄⡌⡏⠓⢴⡞⠦⠖⠮⢑⠺⣉⣥⠂⢁⠜⡁⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠈⡉⢀⢀⣀⡀⠀⠈⠓⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢑⠠⢤⢀⡰⠒⡶⠿⠛⠉⠓⠈⠁⠐⠁⠀⠐⢅⠀⠀⠀⠀⠀⠀⢘⢝⠔⠁⠀⠀⠀⠉⠘⠚⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠨⠀⠔⠁⠀⠀⠀⠀⠀⢠⠊⠈⠁⠀⠉⠢⢀⠀⠀⠀⠀⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢀⣀⡀⠠⡍⢒⠒⠳⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠀⣈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡀⠔⠊⠁⠀⠠⠀⠙⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠎⢀⠠⢀⠂⠤⠀⣠⢠⠈⡀⠀⠀⠠⢄⠀⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⠀⠀⠘⠭⠄⠀⠀⠀⠈⣮⠘⠀⠀⣠⢄⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠾⠀⠀⠀⠳⣧⠈⢀⠀⠀⠀⠀⢀⢄⠀⡅⠊⣠⠊⠉⠠⠅
⠀⠀⠀⠀⠀⠀⠀⠁⢤⣀⣀⠜⠳⠔⠠⣄⣓⠤⠀⠀⠀⠨⠥⠊⢱⠚⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣈⡦⠀⠈⡐⠁⡈⠷⠄⠀⠀⠠⠐⢈⠌⠀⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⣀⡠⢤⠚⠁⠸⠀⠀⠀⠁⠒⠐⠚⡅⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠠⠔⠀⠀⠣⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⡔⠀⠀⢀⠂⢠⠁⢀⡀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⢀⣣⡘⠁⡠⠴⢲⡦⠜⠉⠁⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠐⠈⣉⠀⠳⡔⠒⠊⠁⠀⠈⠁⢰⣠⠔⠢⠀⠀⡱⠶⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⢳⠂⠬⣑⠓⠴⢒⣢⠀⠀⠀⠀⣆⠀⠀⠙⠫⣀⣤⣔⠡⠒⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠄⠀⠲⢌⣀⠀⠀⠉⠑⠂⢬⣅⠀⠀⢀⣜⠀⠀⠀⠀⣠⡊⣡⡴⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢗⣒⠤⢀⣄⡈⠲⡒⠈⠙⢒⣤⣀⢰⠗⣯⠶⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠊⠵⠦⣴⣓⡂⠉⡀⠔⢋⠜⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠈⠁


⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡏⠉⠉⠀⠀⢀⣀⣤⣴⡶⠿⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⠶⠿⠛⠋⠉⠀⠀⣴⡿⠋⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣤⣀⢀⣴⠛⣧⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⢀⣹⣷⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉⠁⠀⠀⠉⠙⣿⣿⣅⠀⢹⡄⠀⠀⠀⢀⣴⡿⠃⠀⢀⣴⡿⠟⢿⣇⠀⠀⠉⠙⠻⣿⣦
⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⣠⡾⠁⠙⣿⣧⠈⣷⣤⠄⣠⣾⠋⠀⠀⢠⣿⡟⠁⠀⠘⣿⡆⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⣿⡟⠀⠀⠀⢀⣶⣿⣿⣏⠀⠀⠀⢸⣿⡆⠸⣷⣴⡟⠁⠀⠀⠀⣾⣿⠀⠀⠀⠀⣼⣿⣶⡄⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠘⢿⣿⣿⠟⠛⠛⠛⢻⣿⠿⠿⣿⠃⠇⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⢿⣿⣿⠏⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠉⢼⡇⠀⠀⢠⣾⡿⠀⠐⣻⣉⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠙⣿⣶⣄⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⣀⣤⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣷⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⠶⢶⣤⣤⣶⣭⣻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⡛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣙⢻⣿⣿⡟⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⠉⠙⢳⣽⣿⣿⣿⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⠉⠛⠳⣦⡀⢧⠈⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡏⢿⣿⣿⣿⣛⣋⠀⠀⠀⠀⠀⠙⢢⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢻⣿⠿⠟⠛⠆⢀⡤⠋⠀⣠⠞⠁⢳⣤⣀⣀⣈⠁⠐⠒⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠶⢿⡶⠶⠶⣶⠋⠀⣠⣾⣷⣶⣶⡿⠿⠿⠛⠛⢻⣿⠓⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⠀⠀⠀⢿⡄⠀⢻⡀⣼⠃⠀⠀⠀⠀⠀⢹⣿⣷⡀⣀⣉⣩⣍⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠬⣍⣛⠷⢄⣼⣿⣀⣴⣿⣷⠄⢸⠀⠀⢀⣴⣿⣿⠿⣷⡉⠁⠀⠀⠬⣍⣛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⣿⠻⣿⣿⣿⣿⣿⠀⠏⠀⣠⣿⢿⡟⠁⠀⠙⣷⡀⠀⠀⠀⠀⠉⠳⣌⠻⣄
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠈⢧⠹⣿⡟⠻⣿⣿⣤⣾⠟⢡⡟⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠈⢧⠹⡆
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⣶⣶⣞⠁⠀⠀⠀⠬⡤⣽⣯⣿⣶⣿⠟⠁⠀⣼⠁⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⠈⠃⣿
⠀⠀⠀⠀⠀⠀⢿⡀⡀⠀⠀⠀⢠⡸⢿⣿⠟⠀⠀⠀⠠⠤⢤⣿⣿⣿⣿⣿⡆⠀⠀⢿⠀⡀⠀⠀⠀⠀⠰⠝⠛⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⣇⢳⡀⠀⠀⠀⠙⢲⣆⣀⣀⣀⣀⡀⠀⣰⣿⣿⣿⣿⣿⠇⠀⠀⠸⣇⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢩⣿⣿⣿⣷⣤⠄⠀⠀⠙⣦⡹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣝⡓⠦⠀⠀⠀⠀⣀⣴⠞⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠻⢦⣙⡓⠶⠀⠀⠀⠀⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⠿⠿⠷⠾⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠾⠿⠿⠿⠶⠶⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠠⡄⠀⠀⠀⠀⡏⠀⠀⢠⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖
⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⢻⡄⠀⠀⠀⡇⠀⠀⣾⠁⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠘⢦⡀⠈⢧⡀⢀⡀⢳⡀⢶⠀⡇⠺⢰⠇⡔⠀⡴⠃⠀⠀⣠⠞⠃⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠙⢦⡀⡑⣄⠱⡄⢣⠸⠀⡇⠆⡌⡼⢁⡞⢠⠜⣠⠞⠁⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠢⡙⠢⡈⠣⡀⠀⢀⡀⠇⡀⠈⣠⠋⠔⡡⠊⢁⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⢦⣤⣀⠀⢬⡑⠬⡂⠀⠠⠂⣠⠤⠄⠢⠄⡁⠐⠄⢀⡠⠖⣉⠤⠒⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠶⠶⠦⣤⣤⣀⣀⣀⠀⠈⢉⡒⠈⠑⢈⠟⢠⠊⠀⠀⠀⠀⠀⠈⠳⡔⢅⠀⠪⠄⠒⠊⠁⢀⣀⣠⡀⣀⣀⣀⣴⠖⠶⠤⠤⠤⠶⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠈⠉⠉⠉⠑⠒⠒⠤⠌⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠂⠀⠐⠒⠊⠉⢉⡝⠒⠛⠁⠈⠉
⠠⡤⠶⠒⠒⠒⠚⠋⠉⠀⠀⠀⠉⠉⠳⢄⣀⡉⠉⠉⠀⠐⠀⢇⠀⠀⠀⠀⠀⠀⢀⣀⡧⠬⠥⠤⠬⠭⣱⠖⠋⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠤⣄⣠⠤⠸⠓⠒⠒⠒⠒⠚⠉⠁⠀⠀⠀⢀⣠⠼⠃⣀⠖⠋⠀⠀⠈⠉⢁⣠⠤⠤⠀⠀⠒⠀⠉
⠀⠀⠀⠀⠀⠐⠂⠁⠁⠐⢦⡀⣀⡀⠀⠀⠀⠤⡀⠀⡀⢠⣳⣖⡒⠒⠠⠤⠤⠤⠤⠤⣴⠞⠉⢉⡀⠐⠬⠈⠀⠴⡆⡀⠾⣉⣉⠁
⠀⠀⢀⣄⠀⣤⠄⢤⠄⠀⣠⠀⠤⠤⠬⠭⠥⠴⠦⠤⠤⠤⠴⠶⠿⠖⠂⠀⠀⠀⣤⠼⢃⢰⣀⠉⡤⡀⣀⠠⠀⣠⠔⢁⡽⠄⠈⢫⣉⣙⡤⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢠⡼⣃⣤⠾⠷⡼⠖⠛⠶⢅⣀⣀⣀⣜⣠⣤⣅⣐⣀⣈⣉⣄⣁⢇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡖⠒⣒⠲⠤⠶⠻⢛⣛⣋⢀⣤⣄⠤⠤⠤⠤⠤⠶⠤⠭⠭⠁⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣀⣴⠶⠒⠒⠚⠛⠚⠛⠛⠒⣾⣉⡉⠀⠀⠀⠈⣡⣤⣵⣐⠒⣒⣒⣃⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⠤⠤⠔⠀⠀⠦⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠬⠤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢘⠄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣸⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡄⠀⡤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠀⠘⢄
⠀⠀⠀⠀⢀⡠⠤⠒⠒⠒⠒⠤⣴⠁⡈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠓⣌⠣⣀⠤⠔⠒⠒⠒⠢⠤⣀
⠀⠀⢀⠔⢉⡤⠒⠊⠉⠉⠑⠲⠄⠈⠪⡢⡈⢢⡀⠀⠀⠀⠀⠀⢰⠁⡎⠀⡨⠃⡀⢔⠂⠉⠉⠉⠒⠢⣌⠑⢄
⠀⢠⠃⡰⠃⠀⠀⠀⠀⠀⢀⡜⣠⠛⢆⠑⡜⢆⠙⢄⠀⠀⠀⢀⠇⡸⠀⡜⢠⠊⠙⢌⠣⡀⠀⠀⠀⠀⠀⠳⡀⢣
⠀⡇⢰⠁⠀⠀⠀⠀⢀⡠⢊⠔⠁⠀⠈⡆⢱⠀⠑⢄⠑⣤⣴⡞⢠⠃⢸⠀⡎⠀⠀⠈⠣⡙⢄⠀⠀⠀⠀⠀⢱⠀⡇
⠀⡅⢸⠀⠀⠀⠀⠀⠧⠴⠁⠀⠀⠀⠀⡇⢸⠀⠀⠀⠳⡈⠁⠁⣞⣀⡸⠀⠧⠐⠒⢒⣊⣁⠀⠇⠀⠀⠀⠀⢸⠀⡇
⠀⢇⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡸⠀⠀⠀⠀⠈⣄⠀⡠⠤⠰⡀⢲⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠇
⠀⠈⢆⠙⢄⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⡰⠁⠀⠀⠀⠀⠀⠈⣟⡄⠀⠀⠱⡀⠣⣀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⠎
⠀⠀⠀⠑⢄⡉⠒⠤⠤⠤⠤⠒⢉⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠀⠀⠀⠈⠢⣈⠑⠢⠤⠤⠤⠔⠊⣁⠔⠃
⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣧⣀⣀⣀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⣿⣏⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡄⠀⠀⠛⠁
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣷⣶⣦⣤⣾⣿⡏⠉⣿⣿⣿⢸⣿⣿⣷⡿⣿⣷⣶⣾⣿⣶⣶⣤⣀
⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⣽⡿⣿⣿⣷⣼⣿⣿⠇⢸⣿⣿⣿⣿⠟⣿⡆⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⢠⣾⡟⠁⠈⢿⣿⣿⣿⠏⣴⣿⣿⣿⡿⠁⠀⠸⣿⣄⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⣿⡇⠀⠀⠀⠀⣴⣿⣿⣏⣀⠀⠀⢸⣿⣿⣯⣾⣿⣿⣿⣿⣧⣀⠀⠀⠘⣿⣿⣷⡄⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠻⠿⠿⠛⠛⠿⣿⣿⣿⣿⣿⡧⠀⠀⠸⣿⡍⠉⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⣸⣿
⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⣿⣿⣿⣟⡁⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠘⢿⣦⣄⡀⠀⠀⠀⠀⣀⣴⡿⠏⠀⠈⠉⠀⠀⠀⠀⠀⠹⢿⣦⣀⠀⠀⠀⠀⢀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠉⠒⠀⠴⠒⠚⠉⠑⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢀⠔⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠁⠀⠀⠑⢬⣲⣄⡢⢀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠁⠐⠀⠀⠦⠀⠙⢍⡀⠀⠀⡀⢀⡀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠓⠀⠀⠆⠀⠀⠀⠀⠈⠉⠁⠤⠀⠙⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣀⣀⠀⠀⠀⠀⠰⠄⠀⠀⠀⣀⣀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠉⠀⠀⠘⠈⠁⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠓⠢⡄⠀⠀⠠⠤⣲⠋⡅⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⢤⡁⠉⠉⡏⢁⠰⠖⡇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⡲⢶⣤⣤⣀⣭⣀⣤⡷⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠟⡴⡻⠋⠀⠉⠉⠉⠀⠀⠀⢣
⠀⠀⠀⠀⠀⢠⠈⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢡⠎⡜⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢠⠀⠀⡌⠉⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡊⢣⣤⠁⣰⢣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠈⡀⠀⠃⠀⡀⠈⢗⢒⢒⠒⡖⠀⠢⠀⠒⠒⠀⠓⢆⠀⠸⠃⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡇⠀⡃⠀⡀⠀⠘⡄⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠙⠁⡆⠤⠀⣀⠔⠀⠀⠀⠀⠀⡀⠀⠀⣰⠉⠈⠉⠉⠁⠒⠂⠤⠤⣀
⠀⠀⣀⡠⠴⠒⡧⠀⡇⠀⠀⠀⠀⠁⢹⠉⡍⣹⠀⠀⠃⠀⠀⠉⠁⠀⠀⠈⢩⡗⢧⡀⠀⠀⠀⣠⠎⠥⠨⠽⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡔⠋⠀⠀⠀⠀⢑⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠤⠄⠀⠈⠑⢝⢢⡤⠊⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⡠⠖
⠑⢄⡀⠀⠀⠀⢸⠀⢺⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡄⠑⠁⠀⠀⠀⠀⠀⠀⠚⠀⠀⣀⠠⠤⠒⢨⠃
⠀⠀⠈⢿⠒⠢⠤⢀⣀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢉⠂⠀⠀⠀⠀⠀⡇⣀⣀⣀⡀⠠⠤⠐⠒⠚⠈⠁⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠉⠉⠉⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢦⣀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠊⠉

⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠈⠟⢳⡄⠀⠀⢀⣠⡤⠀⢴⠖⠀⢄
⠀⠀⠀⠀⠀⠀⡰⠁⢀⣠⡀⠀⡀⢹⣄⢴⠗⠀⠀⠁⢛⣢⢀⣜
⠀⠀⠀⠀⠀⡼⠀⢰⣿⢋⠏⠆⠇⡘⠡⠄⠄⠀⣤⡬⠚⢿⣏⠙⡖⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡌⠀⢰⡷⠤⢷⡴⠄⠁⠐⠀⠈⠣⣠⡴⠋⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠟⢶⡴⡏⠁⠀⣠⢂⠀⠀⢀⡿⠋⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡴⣏⡼⢇⢳⠚⠙⢢⡾⠅⡾⠓⢢⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠶⣿⠋⠒⠌⢁⢄⢀⡸⠇⠀⣧⡄⢀⠃⠠⢰⠁
⠀⠀⠀⠀⠀⠀⢠⣀⣼⡟⢀⡡⠜⡆⢐⡵⠻⠐⠦⣬⠓⠄⣙⡑⢿⡅⡠⠊⢠⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠁⠀⢸⡅⠀⡸⠇⢱⣌⠃⠰⢦⡉⣱⣯⠳⣄⣀⣵⢻⢭⡓⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡚⣛⡛⠓⢷⡄⢳⣟⠆⠘⠻⣿⣿⣷⠿⡉⠘⢷⡂⢹⣦⣾⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢿⠿⠚⡛⠿⣶⣼⡘⣦⡑⠛⢶⣌⠻⣭⣧⡴⢷⣌⡛⣦⡶⠏⣤⣽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢰⣏⠀⢌⢻⣹⡄⢯⠷⠸⣽⣆⡀⠈⠳⣌⡑⠲⣅⣼⣻⢿⢶⣿⣾⠳⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⠐⠠⡽⣿⣟⠘⢳⣶⣿⣿⣷⡄⠀⣼⠛⡴⠯⠳⡝⠒⠈⠫⢫⠑⠴⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⡄⠀⠀⢻⡿⣅⠛⡃⣿⠋⣷⡈⢄⠙⠀⠻⠖⠀⠙⢦⡁⠈⡄⠀⠁⡆⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⣆⠀⢨⣡⡜⠶⢔⡹⢆⠘⢷⡌⠆⠀⠀⠀⠀⠀⠀⠈⠳⢮⣓⣲⣅⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠘⠣⡀⠀⠃⠀⢘⠦⡀⠆⠸⡟⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⡘⠢⡄⠀⠀⠁⠀⠄⢲⡇⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⣀⡑⠲⠤⠤⢔⣋⠄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠷⡅⠤⠴⠒⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⢿⣿⡛⠻⠿⣶⣦⣤⣀⡀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠙⢿⣦⡀⠀⠉⠙⠛⠿⢷⣶⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣷⣿⣿⣿⣿⣷⣤⣀⡀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⡀⣠⣴⣶⣾⣿⣿⣾⣷⣶⣤⡄
⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⠿⠛⠛⠛⣻⡿⠿⣿⣿⣷⣤⠀⠈⢿⣷⡀⠀⠀⠀⠀⣾⡟⠈⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⢀⣿⡇⠀⠈⠹⣿⣿⣷⠀⠀⠙⢿⣦⡀⠀⢸⣿⠁⣶⣿⣿⠿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⢘⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠈⠻⣷⣶⣿⣷⣸⣿⣿⠃⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⡃
⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⣿⣿⣯⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡧⠀⠀⠉⠉⠿⣦⣤⡾⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠛⣿⣿⣷⣤⣀⣀⣀⣀⣀⣤⣾⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣤⣀⣀⣀⣀⣀⣤⣾⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠹⠻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⡿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠑⠒⠤⢄⣀⠀⠀⠀⠀⡖⠲⣶⡖⠛⠻⡿⠿⠇⠻⣷⠀⠀⠀⠀⢀⢀
⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⣶⡶⣿⣆⡸⢌⠒⢠⣄⠀⡈⠱⠉⠀⡀⠀⡠⢃⢪⢊⠜⠔
⠀⠀⠙⢖⠢⠤⡀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡀⢨⡙⠦⠔⡼⠁⠁⠀⠀⢀⡜⡐⡱⡡⡳⡡⢊⠊⡰
⠀⠀⠀⣀⡀⠀⠈⠒⢠⠀⠸⠟⠢⢄⡀⠀⠀⢸⣿⣿⣿⣿⡿⠓⢢⣤⣊⠱⠃⠀⠀⠀⢀⠊⠜⠔⡔⠑⡐⡡⠃⡔⠡
⠀⠀⠐⠀⠉⠂⢄⡀⠆⢕⠀⠀⠀⠀⠈⠑⠲⣾⣿⣿⣿⣿⣷⣤⣿⢦⡋⠀⠀⠀⠀⠀⢃⢈⠌⢎⠎⠔⠐⢁⠌⡰⡱
⠀⠀⠀⠈⠑⠦⣤⡀⠀⠀⣀⠉⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡇⠀⠀⠀⠀⠀⠃⠂⢀⡠⠴⠦⢄⠈⡴⡝⡰
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠢⢈⠀⠰⡄⢦⣶⣿⣿⣿⣿⠃⠈⣹⣿⣿⣿⣿⠀⠤⠤⠖⠒⠒⠚⠉⠀⠀⠀⠀⠉⠒⠲⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠐⠉⠉⠩⠉⡀⣠⣾⣿⣿⠿⠛⢯⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠌⡀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⠓⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡙⠢⢄⡀⠈⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⢍⠓⢦⣬⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠨⡉⠻⠟⢋⡀⠀⠀⠀⠈⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠲⢄⠑⠹⠳⠦⠤⠀⠀⢠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠈⠓⢄⡀⠀⠉⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠉⠲⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠀⠀⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣀⣀⠀⠀⠊⡘⢛⣻⠛⠓⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠘⣿⣿⠟⣛⡛⠿⠿⣷⣾⠟⠓⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡟⠛⠁⠢⠴⠿⠛⡩⠋⠙⢾⣿⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠌⠀⠀⠀⠘⠙⣿⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣷⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⢀⠀⢠⣿⡙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⢨⣀⣼⣀⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠘⢊⣉⠉⠉⠹⢇⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠃⠀⠀⡉⢉⣶⠈⢆⠀⠈⠳⣄⠀⠀⠀⠀⠂⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠂⠀⢀⣶⠀⠁⠀⢹⠆⠀⠆⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠹⠁⡄⠁⠀⠈⠀⠘⠁⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⠿⠛⠻⢿⣿⣿⣿⡇⠀⠙⠀⠀⠀⠐⠀⠊⠀⠀⠀⠀⠘⣇⣴⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣠⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⢠⠁⠙⢿⡇⠀⢤⡀⠀⠀⠀⢡⠘⣇⠀⠀⠀⠘⣿⣿⣶⣾⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠆⠹⡄⠀⠀⢀⣿⣿⣿⣿⣿⠿⠁⠈⠻⣿⣿⡆
⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣷⠀⠀⠀⠀⡠⢳⠀⠀⠸⠏⢈⣿⣿⠃⠀⠀⠀⠀⠹⣿⣿⡄
⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠰⢛⣧⠀⣰⣦⣾⣿⣏⠀⠀⠀⠀⠀⠀⢹⣿⣧
⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠋⠙⢧⡦⢣⣻⣿⢻⢀⣤⣠⣶⣷⡄⠀⣿⣿
⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡉⢀⢋⡀⠀⢠⠀⠀⢸⣿⣿⡆⠀⠀⠰⢊⣵⣀⣿⡾⠿⣿⢄⣿⠁⠀⠉⠈⢡⡄⣸⣿
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣼⣃⣹⣾⠟⢑⠀⡇⠀⠀⢸⣿⣿⡇⠀⠀⣿⣾⣿⣫⣿⠟⡿⢃⣶⣭⣤⣬⡔⣬⡶⠖⢻⣿
⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇⣿⢿⣿⣆⠈⠲⠇⠀⠀⢸⣿⣿⡇⢀⣄⣿⣻⢿⡟⠛⢠⣿⣿⠁⣀⣈⠟⠀⢀⠉⠁⢸⣿
⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡜⠻⣏⠀⠟⠀⠀⢀⣼⢿⣿⣷⡾⠋⠀⠛⠋⠁⠀⠀⣿⣿⡄⠉⠹⠀⠀⠀⠁⠀⣾⣿
⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠋⠀⠀⠀⠘⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⣠⡆⠀⠀⠀⢀⣿⡟
⠀⢹⣿⣟⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠈⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣄⣀⣴⣾⣿⠏
⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠹⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣤⣾⣷⣿⣿⠋
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⣉⣿⣉⣙⣛⣛⡿⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢶⡛⣉⣩⡤⠖⢺⡭⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠈⠉⠉⠛⢒⣲⡄⠀⠀⠀⠀⠓⠒⢛⣉⣥⠴⠖⢚⣉⣧⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⡞⢻⠋⠉⠁⠀⢀⣀⡤⠴⠒⠋⣉⣡⠴⠖⠚⣩⠟⢙⡦⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡤⠴⠒⠋⣉⣠⠤⠖⠚⠉⠁⠀⠀⣠⠞⢁⡴⠋⢳⣤⣷
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣠⠞⡇⢸⡤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁⡴⠋⠀⠀⠀⢧⠸⣆⣀⣀⣀⣀
⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠉⠙⠲⢤⣠⠞⢁⡴⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠊⠀⠀⠀⣀⠴⠚⡆⢻⠀⠀⠀⠉⠉⠓⠦⣀
⠀⢀⡴⠋⠀⣠⠴⣞⠋⢹⡏⠙⢳⠦⣤⠞⢁⣴⡋⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠊⠀⠀⠀⢠⠞⠁⢀⣠⢿⡈⣏⣿⠙⠓⡶⢄⡀⠈⠳⡄
⢀⡞⠀⢠⠞⠁⠀⠹⡄⢸⡇⣠⣯⠞⢁⡴⡏⠀⢳⡀⡇⢸⡇⠀⠀⠀⡀⣠⠞⠁⡠⠊⠀⠀⠀⠀⣰⠃⠀⡰⠋⠈⢣⣧⠸⣿⠀⡼⠁⠀⠙⣆⠀⠘⣆
⣼⠀⢠⡏⠓⠦⣄⠀⢳⣸⣷⠟⢁⣴⣫⡤⣿⣀⣠⠧⠷⠾⠗⠒⡖⢉⣍⠑⣤⠊⠀⠀⠀⠀⠀⢠⡇⠀⣸⠓⠦⣄⣀⠹⣆⣿⡼⢁⣀⡤⠴⠚⣇⠀⢸
⡇⠀⢸⠤⠤⠤⠬⠽⣾⠉⢹⡶⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⡘⣎⢣⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⣏⣀⣀⣤⡬⢽⣟⠉⣻⣭⣥⣤⡤⠤⣽
⣧⠀⠸⡆⠀⢀⣠⠖⢫⢿⢿⡛⢿⣛⠓⠒⢲⠖⠶⡶⠶⠶⠶⠶⠒⠛⣻⢶⣻⠷⠀⠀⠀⠀⠀⢸⡄⠀⢿⠀⢀⡤⠖⢫⠟⡟⣏⠉⠲⢤⣀⠀⡾⠀⢠
⠸⡄⠀⠹⡔⠋⠀⢠⠏⢸⡄⢳⡀⠈⠙⢦⠏⠀⢰⠇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢟⠉⠀⢠⠏⢀⡇⠸⡄⠀⠀⠈⡽⠃⠀⡼
⠀⠙⣆⠀⠙⠢⣄⡞⠀⠈⡇⠀⢹⣤⠔⠁⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠈⠳⢴⡃⠀⢸⠇⠀⢻⣀⡤⠞⠁⢀⡜⠁
⠀⠀⠈⠓⢤⣀⠀⠉⠙⠛⠛⠋⠉⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠉⠙⠛⠓⠚⠋⠉⠀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⢠⢠⡀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⡾⠁⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠠⣴⡶⡀⠙⡄⠂
⠀⠀⠀⠀⠀⠀⠊⠁⠀⠷⠏⠀⠁⠀⠀⢀⣨⣴⢙⣿⡿⣿⣟⣿⣾⢿⣿⡇⠀⠑⠳⠽⠇⢲⡿⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠙⢯⠠⠑⢶⣶⣬⠭⣤⣥⣭⣤⣼⣿⣿⣣⣬⣛⢿⣿⣇⣐⣀⣖⣚⣿⣷⣾⣷⣿⣿⣦⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠶⣛⣽⣒⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⡭⢙⣮⣡⣌⣿⣿⣿⣿⡯⠿⢿⣏⡷⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠁⠀⠈⠉⠋⠉⠋⠉⠚⠋⠉⠙⠛⠋⠋⠉⠁⠉⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠙⢒⣤⣤⣤⣤⣤⣤⣄⣰⣿⠛⠀⠀⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣖⣠⠖⢀⣴⣾⠟⠁⠀⠀⠀⠀⣈⣽⣿⣿⣿⣦⡈⢢⡀⠢⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣽⣿⣿⣧⣤⣤⣤⣀⣀⣾⣿⣿⣇⢻⡙⣿⣷⡄⠙⡆⠙⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣻⣆⠙⠿⣆⢘⣄⠴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣿⣿⣿⣿⠷⢬⣟⣿⣿⣿⣿⣿⣿⠟⠛⠛⠿⣿⣿⣿⣿⠑⠀⢈⣤⣿⣿⠟⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣻⣿⣿⣿⣇⠀⠀⠙⢿⣿⣿⣿⣿⡷⠀⠀⠀⡿⠛⠋⠉⠁⠠⠜⠛⠿⣿⣷⡀⠈⠄
⠀⠀⢀⣴⣾⠿⠿⠿⢿⣶⣌⣻⣿⣿⣿⣿⣷⡀⠀⠀⠙⣿⣿⣿⡇⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⠘⡀
⠀⣰⣿⠋⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣆⣀⣀⣀⣸⣿⣿⣏⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢧⠀⠐
⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⠟⢻⡁⠀⠀⠙⢿⣷⡌⠉⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣿⣆⠘⢷⡀⠁⠀⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡤⢄⣀
⣿⡇⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⠀⠸⣿⡄⠀⠙⣦⠀⠀⠄⠀⡿⣟⠁⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟⢻⣿⣶⡄
⣿⣇⠀⠀⠀⠀⣴⣾⣿⣿⡆⠀⠀⠀⠀⢿⣧⠀⣸⣿⠂⠀⠀⡸⠁⠈⢦⡀⠀⠀⠈⢿⣶⣄⣠⣾⣶⣦⣤⣴⣾⠟⠋⠀⠀⠀⣹⣿⣿
⢸⣿⠀⠀⠀⠀⢿⣿⠿⢿⣍⠲⢄⡀⠀⢸⣿⠀⣿⡟⠀⠀⡰⠁⠀⠀⠀⠀⠂⣀⠀⠀⠙⡛⢿⣷⣦⣾⣿⠟⠁⠀⠀⠀⠠⡾⠛⠎⠛
⠀⢿⣇⠀⠀⠀⠀⢷⣄⠀⠙⠳⣦⣙⠢⣼⣿⢰⣿⠇⠀⡼⠁⠀⢀⣠⣤⣤⣄⠈⠻⢶⣄⠘⣤⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠘⣿⣆⠀⠀⠀⠀⠈⠓⢄⡀⠈⠙⢷⣼⣿⣾⣟⣒⣾⣷⣶⣿⣿⣿⣿⣿⣯⣤⣴⡾⢿⣆⠉⠈⠙⣿⣿⣇
⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠙⠢⡀⢠⣿⣿⣿⠍⢹⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⢙⣆⠀⠀⢻⣿⡿⠿⣷⣦⡀
⠀⠀⠀⠀⠙⢿⣦⣄⡀⠀⠀⠀⣀⣼⣿⢏⣿⣿⣄⠈⢻⣿⡟⠉⠀⠀⠀⠀⠀⠀⣶⣄⣠⣾⣧⣤⣴⣿⣿⡇⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠋⠀⠀⢹⣿⣷⣆⠀⠻⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⡏⢸⣿⢿⣧⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠻⢷⣆⣠⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠁⠈⣿⠚⣿⡀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⢻⣧⢻⣧⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠈⢿⣦⣿⣷⡄⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⣠⣤⣾⠿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣭⣥⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠉⠉⠉⠙⠛⠛⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⣀⡀⣾⣿⣤⡦⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢀⣀⣾⣶⢶⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⡀⠄⠀⣾⣿⣿⣇⣰⣿⢉⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠻⣆
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠤⠄⢿⣿⣿⣯⠹⡷⠋⠈⠻⣧⠀⠀⠀⠰⠄⠀⠀⡩⠀⢹⣆
⠀⠀⠀⠀⠀⢠⣿⣾⡆⠀⠀⠀⠀⢰⣿⡿⢿⣷⣅⣀⣀⣀⣈⣣⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣯⡀⠀⠀⠀⠈⢿⣷⡌⢻⡏⠉⣸⡋⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣶⣄⠀⠀⠀⠈⢻⣿⣿⣧⣴⣿⠓⣆⠀⢀⣀⠀⠀⠀⢠⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣀⣠⣿⣀⣿⡿⣾⢃⣷⣿⣿⣿⣆⠠⢂⣠⣾⣿⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣧⠙⠛⢻⣿⣿⢿⣿⣿⣷⣿⣇⣸⡆⢈⣿⣦⣽⣿⣿⣯⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡝⣶⠀⢸⣿⣿⣾⣿⡟⢹⣿⠻⣿⣿⡿⣻⣿⣵⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣻⣾⣿⣿⣉⣿⣿⢷⣶⣿⡿⠿⠿⣸⣿⣿⣿⡿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣉⠛⢿⠿⣶⣟⣑⣾⣷⣾⣾⣿⣿⣃⣰⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣧⣤⣈⣙⣻⣿⣿⣿⣿⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠋⠉⠉

⠀⠀⠀⣠⢄⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⣇⠘⠃⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠉⠉⠀⠁⠈⠀⠀⠀⠈⠀⠉⠀⠀⠀⢀⡿⣔⣦⡦⡆⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡏⣷⠘⡏⠃⠁⣀⠀⠀⠀⣴⠒⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠙⠊⠀⠁⠀⠀⠀⠈⠉⠙⣮⠈⠉⠀⣥⣶⡤⠤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⣈⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡃⠀⠀⡄⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣶⣼⡦⠀⠀⠀⠀⢀⣏⡼⠃⠀⠉⠀⠉⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⣤⣶⣿⡿⣻⣿⣷⣶⣷⠀⠀⠈⠉⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠰⡎⠛⠿⢻⣦⣭⣾⠟⠫⡜⠯⠛⠀⣨⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠦⢰⡛⣿⣩⠀⠀⣜⣵⠖⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⣡⠵⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠀⠀⡀⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠞⢆⠠⢀⠀⠀⠀⣠⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣾⣿⡟⣹⣿⡀⠀⠘⠀⠀⠐⠺⠿⡏⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠉⠀⢈⣿⠃⣿⣿⣧⣤⣴⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡀⠀⠀⠈⢻⠃⠉⢈⣿⣯⣽⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠙⢻⣿⡁⠁⠀⠀⠀⠀⠀⠂⠘⣣⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠉⡏⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣟⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⣴⢻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡷⣴⣿⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⣿⣿⣿⡿⠛⠿⣷⣿⣿⣿⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⠀⠀⠀⣿⣇⡀⠀⠀⠈⠉⠛⠛⠛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣽⣶⣶⣦⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⣿⣿⣯⣿⣯⣷⡄⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠻⢿⣿⣿⣷⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠀⠀⠀⠀⠀⢹⣿⣯⣿⣾⣽⣇⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⢸⣿⣿⣿⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣤⣀⣿⣿⣿⣏⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣵⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣶⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠋⠙⠛⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡟⠋⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⢛⣿⡿⠛⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⣀⣀⣀⣀⣤⣾⠿⠋⠁⢠⣾⠏⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣤⣤⣄⡀⠀⠀⠀⠀⠈⣿⡉⣩⣽⡿⠟⠋⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⢻⣄⣠⣤⣤⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠿⠛⠛⠛⠛⠻⢿⣿⣷⣤⡀⢀⡀⢸⣿⣛⣭⡄⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠛⠛⠛⠛⠿⢿⣿⣷⣄
⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⠛⠛⣿⠉⣿⡇⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⣰⣿⡿⠋⠙⣿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠉⢿⣿⣆⠀⢹⡆⣿⠁⠀⢀⣾⠟⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆
⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⢀⣴⣶⣶⣿⡛⠁⠀⢀⣀⣈⣿⣿⣄⣸⣧⣿⣤⣴⡿⠋⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⣻⣷⠒⠲⣄⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠐⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣉⣉⣉⣿⣿⣿⣋⣿⣿⣿⣋⣀⣰⠇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⡁⠛⢿⡆⢸⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠈⢿⠿⣿⠟⠁⠀⠀⠀⠀⢈⣿⣿⠻⣍⣉⣩⡟⠉⢩⠏⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠑⠤⠤⠤⠋⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠇
⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠉⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣤⣤⣤⣤⣴⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣤⣤⣤⣤⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⣤⣤⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠻⠶⠮⣽⣿⣟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡆⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⣏⣿⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢷⣤⣤⣤⣀⡀⠀⣰⠇⡾⠙⠾⠋⠛⠓⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣟⣀⠀⠈⣿⣛⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡟⠋⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⠦⣤⣄⣀⠀⠀⠀⠀⢿⣿⣦⠤⣄⣀⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣿⡓⣶⠶⢭⣽⣛⡲⠦⣤⣄⣉⣿⢳⡶⠦⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠓⣺⠃⢳⡌⢷⡀⠀⠉⠉⠛⠲⠦⣽⢃⣾⣅⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠛⢻⡇⠀⣿⣄⡀⠻⡄⢳⡄⠀⠀⠀⠀⢰⡏⣼⠳⢮⡳⢦⣠⡴⠛⣉⣤⡴⠶⠶⡦⣤⣈⠙⠷⣄
⠀⠀⠀⠀⣠⠞⠋⣀⣤⠴⡶⣾⠀⢠⣇⡈⠙⢶⣽⣆⠻⣄⠀⠀⢀⡟⣰⠇⠀⠀⢹⣦⡙⢶⡟⣍⠀⢇⠀⢸⠁⢠⠏⠛⢦⡈⠻⣄
⠀⠀⠀⡾⠁⢠⡞⠉⢣⠀⢳⡏⠀⣸⣫⠛⣦⡀⠹⣿⣆⠙⣦⠀⣾⢡⡏⠀⠀⢠⡟⢠⣿⣦⣝⠿⣦⢸⡀⢸⠀⡞⠀⢀⡴⠻⡄⠘⣧
⠀⠀⣼⠁⢰⡏⠙⠢⣄⠳⣼⠃⠀⣿⡡⢊⡬⢷⡄⠸⣿⣦⣘⡿⠧⣾⡦⠤⠤⠾⣤⣿⣤⣬⣿⣷⣬⣿⣧⣟⣼⠀⡰⠋⠀⣀⢿⡄⠘⡇
⠀⠀⡏⠀⣾⠉⠙⠒⠢⣽⣿⢠⢸⡿⠿⠗⠒⠊⢿⠀⣿⢸⢯⢖⣒⢦⢻⡦⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣀⣈⠉⠻⣦⠒⠉⠀⠘⡇⠀⡇
⠀⠸⡇⠀⢿⠊⠉⢉⡩⠖⡿⡶⡟⢿⠫⢍⡉⠉⢹⠆⣿⢸⡌⣾⠿⠼⣰⠷⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠽⠃⢀⣿⡓⠒⠲⢴⡇⢀⡇
⠀⠀⢻⡄⠘⣷⠚⠉⢠⠞⢀⡇⢧⠀⠱⣄⠈⢑⡾⢠⡟⠀⣹⠟⠲⠾⠿⠶⠶⠶⣶⠶⣶⠶⡶⠶⢶⠶⢶⠶⢶⠛⣅⠙⠢⣤⡞⠀⣼⠃
⠀⠀⠀⠹⣆⠘⠷⣴⠃⠀⢸⠀⢸⡀⠀⢈⣳⠟⢁⡾⠉⠛⠷⠄⠀⠀⠀⠀⠀⠀⠹⣆⠙⢿⣄⡠⠋⠀⡼⠀⠸⡄⠈⣦⡴⠋⢀⡼⠃
⠀⠀⠀⠀⠘⠷⣄⡈⠛⠶⠷⣤⣤⠷⠖⠋⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡈⠙⠲⠦⠧⠤⠴⠷⠛⠉⣀⡴⠛
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⣤⣤⡤⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡾⠛⠉⠉⠉⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠤⣾⣏⣠⣤⣤⡄⠀⠀⠀⣀⠹⣷⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣽⣿⣷⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⡿⠟⠶⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠿⣿⡟⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⠿⠟⠉⠉⠉⠉⠛⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣄⠀⠀⠀⠀⠀⠀⣠⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠦⠴⠶⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⠀⠉⠉⠉⠙⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡂⠀⢐⣾⠂⠀⠀⠀⠀⢀⠀⠀⠠⠆⠀⢀⣼⣻⡀⠟⣃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣷⣼⣯⣿⣷⣶⣼⣦⣤⣶⣶⣴⣧⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣤⣤⣤⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠀⠁⠈⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⣀⣠⠊⠙⢂⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣀⣀⣀⡀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠀⠠⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣉⣷⣶⣄⣀⠀⠀⠀⠀⠴⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⢻⣿⣿⣿⣶⡶⢤⡤⠤⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠚⠿⠿⠿⠿⠟⠷


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⣿⣿⣿⣿⣿⣇⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⡿⠿⠛⠛⠛⠛⢿⣿⣿⣧
⣿⣿⣿⢿⣿⣿⣆⠀⣼⣿⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣤⣴⣿⡿⠏
⣿⣿⣿⠈⢿⣿⣿⣾⣿⣿⡿⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠛⠛⠛⠻⣿⣿⣦
⣿⣿⣿⠀⠀⢻⣿⣿⣿⡟⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣴⣶⣿⣿⡿
⠙⠛⠛⠀⠀⠀⠛⠛⠋⠁⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⣠⣤⣤⣀⣰⢟⣿⣟⠷⣆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢲⣤⣤⣬⣯⠁
⠀⠈⠉⠉⢻⣿⠿⣾⣦⣷⢿⠆⠀⠀⠀⠀⠀⣴⠿⠿⣶⣟⠿⣿⣿⠇⠀⠀⠀⢠⠖⢛⡿⣽⠄⣰⡿⣟⠳⣄
⠀⠀⠀⠀⠘⣯⡳⠯⠽⣟⡿⢷⣶⣶⡄⠀⠘⠁⠀⠀⣩⣷⣿⡻⢿⠿⣦⡀⠀⢻⡀⠛⢓⡿⢿⠻⡄⠙⠀⡼
⠀⠀⠀⠀⠀⠈⠙⠓⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠁⠁⠀⠀⠀⠉⠀⠀⠙⠛⠋⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⢒⣊⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠭⢿⣭⣭⠭⠟⣛⣯⡥⣴⠲⣏⢻⣎⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠦⠤⠤⣀⣀⠀⠀⠀⢰⣊⡽⠤⢒⣊⡭⢿⠻⡷⠽⠷⣛⣫⣭⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⢤⠴⡖⠒⠉⠀⢀⣀⠤⠔⢒⣋⡭⠴⢒⠟⣦⢷⣏⢿⡌⣷⡸⣧⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠖⢒⣉⡥⠴⠒⠋⠉⠀⣠⠖⢁⠔⠹⡤⣿⠬⣗⣚⠯⠝⠒⠉
⠀⠀⠀⠀⣀⡤⠤⠴⠤⠤⢄⡀⠀⢀⠔⢹⠀⡗⠋⠉⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠀⣱⠸⡭⠤⠤⠤⣀⡀
⠀⢀⡴⠋⢀⣠⠤⢶⠦⠤⣄⢈⠗⢁⠔⢹⠀⡇⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⢀⡴⠋⢈⣇⢳⢤⠤⢤⣀⠉⠲⣄
⢠⠎⢀⡔⠁⠘⡆⢸⠀⣼⡵⢁⢔⠉⢣⢸⠀⡇⠀⠀⢰⣶⢾⣁⠔⠁⠀⠀⢠⠋⢀⠔⠙⢾⡜⣿⢀⡜⠈⠳⣄⠈⢆
⡎⠀⡞⠙⠢⢄⣸⣼⡾⢃⣴⣥⣾⠧⠤⠿⠒⠓⢲⠏⡹⡹⡟⠁⠀⠀⠀⠀⡇⠀⡞⠒⠤⣌⣳⣹⣞⣠⠤⠖⠛⡄⠘
⡇⠀⡟⠒⠒⢒⡿⣄⣽⣭⢥⣀⣀⣀⣀⣀⣀⣀⣈⣳⡽⠵⠃⠀⠀⠀⠀⠀⡇⠐⡷⠒⠒⣲⢿⣤⡽⢶⡒⠒⠒⡇
⢳⠀⠱⡤⠖⠁⡼⢹⠱⡌⠑⠦⣴⠃⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢣⠖⠉⢠⠋⡇⢳⠀⠈⠑⡶⠁⢰
⠀⠳⡀⠙⠦⣴⡁⢸⡄⢘⣤⠖⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠑⢴⣃⠀⡇⠀⣧⡠⠞⠁⡠⠃
⠀⠀⠈⠲⢤⣀⡉⠉⠉⢁⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡈⠉⠉⠉⢁⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⣈⣉⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠇⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⡠⠐⠒⠤⣌⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣧⡀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡠⠞⠉⠙⢦⡈⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠐⢤⡊⠀⠀⠀⠀⠀⠉⠓⠦⠈⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣠⠝⢦⡀⠀⠀⠀⠀⠀⠀⠀⢘⣆⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠇⠀⠈⡇⠀⠀⠀⠀⠀⣠⡴⠋⣽⣞⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⡄⠀⣼⣇⠀⣀⣠⣴⠞⠁⣠⠞⠁⠘⣆⢀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⠖⣫⠍⠁⠈⠉⠙⠳⣦⡼⠁⡼⢻⠉⠙⢿⠀⣿⣠⡾⠃⠀⢀⡴⣻⣯⠉⠀⠉⠉⠛⢶⣄
⠀⠀⠀⡴⣡⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⠴⠋⠀⠀⢸⣧⡿⠋⠀⠀⢠⢋⠞⠁⠘⠄⠀⠀⠀⠀⠀⠹⣦
⠀⠀⢰⢣⡇⠀⠀⢰⠋⠀⠀⠔⠚⠻⡉⣿⢀⡤⠀⣠⡼⠋⠀⠀⠀⠀⡏⡞⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠘⠸⡇⠀⠀⠘⢦⣀⡴⠃⠀⠒⠚⣿⠺⣖⣛⡁⠀⠀⠀⠀⠀⠀⠇⣧⠀⠀⠀⠳⣄⣤⠆⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⡀⣰⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⡠⢠⡟
⠀⠀⠀⠀⠀⠉⠳⠦⢤⣤⠤⢖⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⣤⠤⠴⣊⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠋⠉⠁


⡟⠉⣿⣛⣻⣛⣻⣛⣟⣛⡟⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⣟⣻⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢶⣶⣶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠉⠛⠛⠿⠿⢿⣶⣦⣤⣄⣀⣾⠃
⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣤⣤⣠⣾⡇⠙⢿⣷⣄⠀⠀⠀⠀⠀⠈⠉⠙⣿⡿⢷⣄⠀⠀⠀⣀⣤⣤⣶⣶⣤⣤⣀⡀
⠀⠀⣀⣴⣿⠿⠛⠋⠉⠉⠙⢻⣿⣿⣧⡀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠙⣷⣴⣿⠿⠛⠋⠉⠉⠙⠛⠿⣿⣦⣀
⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⣾⣿⠉⢻⣿⣆⠀⠈⠻⣿⣦⠀⣀⡀⢠⣿⠃⠀⠀⣰⣿⡿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⢹⣿⡆⠀⠀⠈⠻⣷⣽⣇⣾⡏⠀⠀⢰⣿⡏⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆
⣾⣿⠀⠀⠀⠀⠀⢠⣾⣿⣿⠏⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⢹⣿⣿⣿⣷⣀⣀⣾⣿⣥⣤⣤⣤⣤⣽⣿⣷⡄⠀⠀⠀⠀⠀⣿⣷
⢿⣿⡄⠀⠀⠀⠀⠀⠙⠋⠈⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠈⢿⣿⣿⠟⠉⠉⢹⣿⡍⠉⠉⠉⠀⠀⠙⠋⠀⠀⠀⠀⠀⢠⣿⡿
⠈⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁
⠀⠘⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠃
⠀⠀⠀⠙⠻⠿⣿⣷⣶⣶⣾⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣷⣶⣶⣾⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠭⠀⠒⠘⠂⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡀⢀⡀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⠀⠀⠀⠀⡀⣀⣠⣀⣠⣀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣠⣴⣶⣿⠿⠿⠿⠛⠛⠉⠡⡄
⠀⠀⠀⠀⠀⠀⠀⠐⣀⣤⣾⣿⡏⠉⠀⠀⠀⢀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠛⠻⠏⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠂⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣗⠂⠀⠀⠀⢰⡒⠚⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠲⠂⠄⣸⠁⡀⠠⡴⠐⣇⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠸⠁⡀⠀⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠿⣤⣤⣥⣠⣤⣴⡖⢞⠉⢻⠒⠂⠀⠀⠒⠂⢳⠀⠀⢰⠺⠁⠂⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠉⠁⠀⠈⢏⢢⠻⡀⢣⠀⠀⠀⠀⠀⢸⡒⠀⢸⢸⠀⠀⠀⠀⡜⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢣⠰⡎⠳⡄⠤⠤⠐⠒⠓⠁⢸⠀⠑⠦⠒⠊⠀⠀⣠⠊⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⣆⠈⣤⠈⢑⠆⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⡠⠊⠉⢠⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠄⢀⡰⠛⠒⠒⠂⠀⡖⠉⢲⠐⡐⡆⡾⠁⢀⠀⠸⠁⢀⠁
⠀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣆⡀⠀⢀⠀⠀⠀⠀⠀⠢⣄⠠⠄⢰⠈⢁⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⠇⠀⠀⠀⠘⠀⢰
⡎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠉⠍⠨⠱⢄⠀⠀⠀⢀⡼⢺⡍⠀⠀⠀⠀⠉⠉⠁⠘⠀⠀⠍⠹⠉⠏⠉⠀⠀⠠⠀⢨⠀⢼⣶⣤⣄⣀
⠈⠒⠌⡙⠛⠿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠑⢤⡔⠅⠊⠀⠀⠂⠀⢄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠘⡅⠒⠦⠤⣈⠉⠙⠓⠀⠀⠀⠀⠀⠀⠈⠂⢠⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡗⠀⠿⠿⠿⠛⢉⠤
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠈⠉⠐⠂⠒⠠⠤⠄⢀⣀⡀⢸⠀⠀⠀⠀⠀⠐⡉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠐⠒⡞⠁
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⣀⠴⠓⠂⠀⠐⠒⠒⠀⠐⠒⠒⠂⠀⠉⠉⠁⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣻⣿⣿⣿⣿⣿
⠀⠀⠛⠒⠤⢤⣀⣀⣠⣤⣴⣶⣾⣿⣷⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣥⣽⣟⣻⠿⠿⠿⠛⠛⠉⠉⠀⠙⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣱⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠏⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⣤⣀⡀⠈⠉⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠻⠿⣿⣷⣶⠂⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠺⠯⣽⣾⣿⢿⣅⣀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣬⣭⣒⡢⠁⢀⣀⠺⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠺⠭⣿⡯⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣍⣒⡪⢐⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠤⠙⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠫⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢾⡀⣀⢋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠠⡀⠀⠉⠁⣬⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠒⠳⣄⡀⠀⠀⠀⣾⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠈⢡⠌⠀⣴⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣺⣷⠿⣫⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐⠯⣅⡀⠀⢀⣀⠤⢤⣤⣾⠎⡰⢋⣡⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠁⠀⠀⠘⠋⠁⠀⠀⠀⣙⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠤⠤⣤⣀⣀⣁⠀⠺⠟⠙⠀⠀⠀⡠⣖⢒⡒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⣄⠀⠀⠀⢈⠉⣀⠀⠈⢿⡷⣄⣿⠟⠉⠉⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⢛⠟⣦⣈⢣⡀⠀⠈⠀⠀⡀⠀⠀⣽⣞⢇⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢸⢀⡇⠙⣧⠙⢦⡀⠁⠀⠁⠀⣀⣻⡏⠁⠁⣤⡠⠄⠀⢸⡇
⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⡆⢸⠁⠀⢹⡇⠀⠙⢦⡀⡀⠀⣹⣿⠇⠀⣬⣿⠿⡜⠀⢸⡇
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣴⢁⡏⠀⠀⢸⡧⠀⠀⣸⣿⣟⡧⠿⢿⡂⠀⠵⠾⠒⠁⠀⣾
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⢿⠟⠀⠀⠀⢸⡇⠀⢀⣿⣿⠟⠁⠀⠘⣇⠀⠀⠀⠀⠀⣼⢃⣀⠤⠤⠒⠂⠁⠁
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠑⠀⠀⠀⠀⡼⠁⠰⠿⠛⠁⠀⠀⠀⠀⠘⠷⣤⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠚⠛⠉

⠀⠠⠤⠤⠤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣄⣾⣿⣿⣇⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠿⠛⡿⠿⠿⠿⠿⠟⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠺⠀⠐⠉⠘⠛⣿⣿⣿⡟⠙⠁⠀⠠⠀⠀⠀⠀⠀⠀⡹⠯⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠒⠀⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⠒⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⢤⡈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠹⡄
⠀⠀⠀⣠⠴⠖⠒⠓⠒⠲⣴⣇⠀⠀⠉⠲⣄⠈⠙⠲⠤⣄⣀⣀⣤⠔⡿⠀⠀⠙⢦⣠⠤⠴⠶⠦⢤⣀
⠀⣠⠊⠁⠀⠀⠀⠀⠀⣰⠃⠈⠳⣄⠀⠀⠀⠙⠦⣀⠀⢀⡄⠀⠀⣸⠁⠀⢀⡴⠋⠑⣄⠀⠀⠀⠀⠈⠙⢦⡀
⣸⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠓⢬⣳⣤⣤⡇⠀⢠⠏⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢳
⡇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢹⠋⣻⠓⠒⠦⢤⣀⡀⢳⠀⠀⠀⠀⠀⠈
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⢯⡀⢸⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⢠
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⢀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠁⠒⠚⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⣀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠻⠿⣿⣿⣿⣿⣷⣦⣤⣄⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠊⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘⠛⠉⠹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠰⠄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠰⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠚⠋⠛⠉⠛⠋⠁⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣢⠄⣀⣀⣀⠀⠀⠀⡀⡤⠚⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⡘⠃⠤⠤⠤⠤⠐⢋⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡈⠛⠀⠀⠀⢶⠂⠀⢙⣍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢀⡨⠛⣹⡶⢶⣤⣼⣤⣶⠾⠛⠙⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠘⢱⢋⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢡⡞⣠⠛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠐⡀⠈⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠚⢃⣀⠊⢀⡆⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠿⠀⠀⠀⠱⡄⠤⠤⠤⢤⠄⠠⣄⠀⠀⠀⠠⠴⢦⡈⠁⠀⡞⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠈⠃⠀⡦⠀⠐⠀⠀⢰⠘⠀⠂⠘⠀⠀⢨⡀⠀⠀⠀⠀⠀⠑⠦⠾⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡼⠀⠤⠤⠤⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡃⠀⣦⠀⠐⠀⠀⠀⠆⢀⡀⡤⠠⡤⠼⠦⣀⠀⠀⠀⠀⠀⠀⠐⠃⠈⣁⣐⠖⠁⠀⠀⠀⠀⠀⡠⢀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⢢
⠀⢀⡠⠔⠒⠋⠉⡓⠀⢦⠀⠀⠂⠀⠀⠀⠘⠂⠐⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠝⠢⣀⠀⠀⠀⡠⠚⠀⠂⠀⠘⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡔⠉⠀⠀⠀⠀⠀⢘⠀⢰⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡤⠒⠐⠁⠀⠀⠁⠈⠳⡔⠋⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠑⠤⣀⠀⠀⠀⠀⢀⠀⢘⡄⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡄⠐⠉⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢀⣀⠠⠄⠒⢉⡏
⠀⠀⠀⠻⡒⠢⠤⣀⣀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢃⠂⠀⠀⠀⠀⠀⠨⡇⠀⠀⠀⢀⣀⡀⠠⠤⠔⠒⠂⠉⠁⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠙⢆⡀⠀⠀⠈⠉⠁⠒⠒⠒⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⡀⠀⠀⠀⠀⣠⠞⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠃
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠖⠚⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣥⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣬⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠂⠀⠐⢲⣦⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣤⡖⠂⠀⠑⣲⣆⣀⢀⣀⣀⣀⡀
⠀⠀⣯⣉⣉⣛⠛⠿⡿⠿⢛⣛⣛⣛⣻⣯⣟⣻⣛⣯⣿⡷⠀⢠⣯⣉⣉⣛⠛⠿⡿⠿⣛⣟⣛⣛⣻⣭⣟⣛⣻⣽⣿⡇
⠀⠀⣭⣭⣭⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢠⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠛⠛⠋⠉⠉⠉⢛⠛⠛⠛⠙⠛⢛⡉⠛⠛⠛⠛⠛⠃⠀⠈⠛⠛⠉⠉⠉⠙⣟⠛⠛⠋⠉⠉⢙⠉⠛⠛⠙⠛⠛⠃
⠀⠀⠀⠀⠀⣦⠀⣠⣾⠀⠀⠀⠀⠀⢸⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣿⠀⠀⠀⠀⠀⣿⣄⢰⡇
⠀⠀⠀⠀⠀⣹⡞⣫⣾⣇⠠⠤⢤⡀⢸⣯⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢞⣡⣿⣄⠤⠤⣤⡄⢿⣯⣻⣧
⠀⠀⠀⢀⠔⢻⣾⣿⠧⠀⢴⡶⠾⠯⠤⢿⣿⣿⠳⣄⠀⠀⠀⠀⠀⡠⠞⣿⣾⡿⠅⠀⠤⡴⠯⠭⠌⢿⣿⣏⠳⣄
⠀⠀⠖⠁⣴⣿⡿⠁⠀⠀⠀⠠⠤⠤⠾⠯⣿⣿⣷⡎⠳⡄⠀⠠⠊⢠⣶⣿⡿⠁⠀⠀⠀⠀⠀⠀⠻⢯⣿⣿⣷⡌⢳⡄
⠀⠀⣴⣿⡿⣿⡉⠉⠉⣉⢉⡉⢉⢉⠉⣉⣛⣻⣿⣿⣇⡈⠀⠀⣶⣿⣟⣟⡉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣻⣿⣿⣧⡁
⠀⠀⣿⣟⣼⠩⠤⠤⠄⠀⢀⣀⣀⣀⣀⡈⣈⣸⣿⣿⣿⣷⠀⠀⣿⢏⡾⠥⢤⡤⠤⠄⢀⣀⡀⣤⡀⠀⣤⣼⡿⣿⣿⡇
⠀⠀⣟⡾⣉⣀⣀⣀⣀⠀⠀⠐⠀⠀⢀⣤⣄⡍⢭⣿⣿⡿⠀⠠⢏⡟⠁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⠭⢭⣽⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡄⠀⠀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣴⣿⣄⡀⣼⣿⣇⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢻⣿⣿⡿⣻⣿⣿⣯⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣛⣋⣠⣿⣿⣿⡟⠀⠀⣤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡿⠿⠟⠛⠛⣦⣴⣿⡟⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⣤⣭⣥⣤⠴⠶⣶⣾⣿⣿⣿⣿⣧⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡿⠛⠉⠉⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⣄⣤⠆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣤⣴⡾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⣿⣿⣿⣯⣤⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠟⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣽⠿⠛⠛⠋⠉⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⣤⣴⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⢉⡩⠿⢍⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢗⡟⣿⢻⣄⠢⡀⠀⠀⠀⠀⠪⣄⢠⣀⠀⢉⣤⣤⣾⣧⣀⣠⣇⣽⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⡞⠀⠘⣇⢹⡑⢌⠢⡀⠀⢠⣤⣿⣿⡿⣷⢾⣿⣿⡿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⢶⣤⣤⣶⣿⠀⡞⠀⠀⠀⠘⢆⢳⡀⠙⢮⣢⡀⢿⠐⣿⣷⡗⠀⢴⣿⠁
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠈⢹⣿⡏⣴⡁⠀⠀⠀⠀⠈⢧⠙⡄⠀⠙⢮⡻⢦⣼⡝⠟⠉⢹⡿
⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡽⣿⣿⡄⠀⠀⠀⠀⠈⢳⡈⢦⡀⢀⠙⣶⣴⡁⠀⠀⡿⠁⢀⣠⣤⣶⣶⣶⣶⠂
⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⡁⡸⠁⠘⣿⣿⠀⠀⠀⠀⠀⠈⢷⣀⣹⡻⢿⣿⣿⡟⠀⢠⣧⣴⡻⣿⠟⠛⠋⠉⠉⠉⠓⠶⣄
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⣿⣼⣴⡅⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠈⣿⣶⣭⣀⢻⠻⡿⣤⣿⣿⣯⡙⠃⠀⢀⠀⠀⠀⠀⠀⠀⠈⠱⣆
⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⠀⠀⢈⡗⠀⠀⠀⠸⡇⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⣧⣬⣿⣇⡜⢠⣑⢤⡀⠑⠠⢤⣄⠀⠀⠀⠀⠈⢧
⠀⠰⡀⢫⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣟⠓⠿⣟⣿⣶⣾⠆⢙⡀⠀⠀⠀⠀⠈⠇
⠀⠀⠱⡄⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠉⢻⡏⢿⠛⠓⠿⢧⢭⣽⣖⠚⠃
⠀⠀⠀⠙⢦⡑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⠂⠀⠀⢀⡞⠛⠉⠛
⠀⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⠀⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣉⠂⠀⠀⠈⠁⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣷⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡿⠿⠛⠿⠟⠋⠛⠛⠛⠙⠛⠿⢿⣿⡷
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡒⠄⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡷⠟⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⣸⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠤⠀⢀⣀⣀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠉⠉⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡩⠒⢷⡀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢄⠀⠀⠀⠔⠀⠀⠀⠛⠶⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠑⠒⠦⢤⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣤⢤⡶⠶⠤⠤⢤⣤⣀⡀⠀⠀⠀⠈⠑⢄⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢠⡞⠕⠀⠀⠳⣄⠀⠀⠀⠀⠀⠉⠙⠲⢄⠀⠀⠀⠑⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣤⣒⣚⢒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣇⠐⢲⡄⠐⢉⣷⣤⡀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡅⠀⠀⢀⣀⡀⠀⠀⠀⢀⡀⠀⠤⠐⠊⠁⠀⠈⢧⡔⣭⣛⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡦⠹⡃⠐⠬⠝⠁⢨⡑⠒⠤⠤⢄⣀⠀⠈⢤⠜⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⣀⣀⣀⣀⣙⣏⣭⣼⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠙⣄⠈⠀⠀⠠⠔⣡⠛⠀⠀⠀⠀⢈⣷⡀⠀⠙⠒⠤⠤⠤⠤⠄⠀⠀⠀⣀⣤⠴⠚⠋⠉⠉⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠳⢄⡉⠐⠛⠛⠁⠀⠀⢀⣤⠞⠁⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠤⣀⠀⠀⠀⠀⡠⠐⢉⡀⠤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣈⣿⠑⠒⠖⠒⠛⠉⠀⠀⠀⢀⣨⠶⢄⠀⠀⠀⢠⠤⠤⠤⠤⢄⣀⠀⠀⠉⠑⠒⠉⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⣀⠤⠖⠋⠉⠀⠀⠀⠈⠑⠢⣤⠤⠤⢀⣀⠀⠈⠉⠑⠒⠂⠤⣴⠶⡅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠘⣶⡀⠤⣀⠑⠄⣀⡠⠔⠀⠀⣉⣠⢘⢸⠔⢀⡄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣱⡄⠀⠀⠘⣟⢶⡂⠩⠥⠰⢖⡉⠈⠀⠈⠒⠶⠊⠀⠁⣄⠑⡌⣖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡠⠂⢐⡀⠀⢀⠜⠒⠄⠀⣰⠊⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⣽⣀⣁⡤⠤⠤⣤⠑⠒⠆⣀⣀⠀⠄⠈⠇⠈⢁⢟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡴⠂⠓⠒⠉⢲⡒⠧⠔⣺⠀⠙⡀⢄⡀⠀⠀⠀⠀⣀⣠⠤⠒⠋⠁⠀⠀⠸⡄⠀⠀⠈⠀⠐⠒⠒⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⣃⣁⡤⠤⠔⠊⠁⠀⠀⡘⠊⠙⢋⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣭⠤⠓⢤⣤⣤⠤⠖⠋⠉⢂⣠⠴⠒⠒⠒⠲⠶⠦⠤⠤⣤⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠲⣂⠤⢤⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⢸⡙⠒⠀⢀⡱⠀⠀⠀⢀⣠⡤⠴⠖⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡠⠂⠈⠫⡒⠒⠋⠀⠀⠀⠉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠈⠒⠤⣀⣀⠠⠤⠖⠛⠢⣀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠈⢁⡈⠁⠶⠐⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠈⠐⢄⢀⡠⠀⠀⠈⠳⡄⠄⠂⢈⡀⠤⠄⠒⠈⠁⠀⢀⣀⡤⠴⠒⠊⠁
⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠉⠈⡁⠤⠈⡂⢀⡉⠀⠀⠀⠀⠀⣘⠖⠉⠀⣀⣀⣤⠴⠖⠚⠉⠁
⠙⢦⡉⠑⠒⠒⠂⠀⠀⠀⠀⠐⠒⠒⠒⣈⣉⣉⢥⠽⠶⢶⣎⣀⣀⣉⣁⣀⣤⠤⢴⣒⣲⡭⠴⠖⠚⠉⠉
⠀⣦⢉⡛⠓⠒⠒⠒⠒⠒⠚⣛⣉⣉⡭⠥⠤⠐⠒⠈⠉⣀⣀⣭⠭⠥⠖⠒⠋⠉⠉
⠀⠈⠳⣌⣉⠉⠉⠉⠉⣉⣉⣁⣀⣤⡤⠤⠶⠒⠚⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣶⣤⣤⣀⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⢻⣿⠟⠉⢻⣯⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣼⡇⠀⠀⠀⢻⡟⢻⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⡄⠀⠀⠸⣿⣜⣿⣿⣽⠏⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⡀⠀⢿⣿⣷⡿⠁⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣤⣤⣶⣶⣶⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⣜⠀⠀⢀⣠⢤⡻⣿⠷⣼⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⡠⢂⣥⣾⡿⠛⠋⠉⠉⠀⠀⠀⠉⠙⠻⢿⣿⣦⣄⠀⠙⠳⣿⣿⣿⢺⡟⠅⠀⠐⠨⣷⣄⣀⠀⠀⠀⠀⡀⡀⠀⢻⡆
⠀⠀⡠⢊⣴⡿⠛⠅⠈⣀⠀⠀⠀⠀⠀⠀⣀⡤⢒⡠⢻⣿⣏⣧⠇⠀⢮⣿⡿⣸⣧⢰⣄⠀⢀⠰⢾⡻⢦⠙⣷⣶⡝⠛⠂⠈⠉
⠀⡔⣰⣿⢋⠊⠀⠀⠀⢻⣷⡀⣀⡤⣖⡯⠗⠊⠁⠀⠔⣋⣾⠀⠐⣆⠀⢸⢇⣿⣿⡄⣿⣷⣬⣷⣌⠀⠆⠞⠙⠛⢡⡄⢀⡀
⠘⣸⣿⠣⠁⠀⠀⠀⠀⢀⡽⣿⣿⣛⣧⣠⠄⣂⡥⠖⢋⣿⡟⠀⠀⠘⢆⠘⠸⠻⠟⠇⠸⠿⠿⠿⠗⢄⡀⠀⠀⢰⣿⣷⠘⠃
⣇⣿⡇⡀⠀⠀⠀⠀⠀⢾⡃⠿⣿⣯⣵⡖⠋⠁⠀⢀⣼⡿⠁⠀⠀⢠⢂⡅⠀⠰⡄⠀⠀⠀⠀⠀⠀⠈⢉⡀⢈⢱⣿⡟
⢿⣿⡇⠁⠀⠀⠀⠀⠀⠈⠓⠒⠚⠿⣯⠁⠀⠀⢠⣾⡟⠁⠀⠀⠀⡄⠨⠴⡖⣿⣽⣬⣌⠀⠀⠀⠀⠐⠚⠻⠿⣷⡝⠃
⠘⣿⣿⣎⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠋⠀⠀⠀⠀⢀⠇⠀⡼⣿⣶⣦⣍⣿⣿⡷⣶⣤⣀⢀⣤⡶⠿⠁
⠀⠈⠻⣿⣷⣦⣤⣉⣁⣀⣁⣠⣴⣶⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⡘⠀⠠⣷⣟⡿⠉⣉⣽⣿⠿⠿⠛⢿⣾⣿⠁
⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠇⢀⣄⣾⣿⣷⠿⢛⣿⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡞⠀⣼⠿⢟⣩⣴⣾⣿⣿⣿⡿⠿⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡋⢀⣼⣿⣶⣿⡿⠟⠉⢉⣿⢸⡇⠀⠀⠀⠉⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⣴⣶⣴⣄⣿⣿⡿⢿⣦⡀⠈⢪⣿⢸⡇⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⣭⣟⣛⣻⡟⠀⠀⠙⢿⣷⣤⣿⣼⣧⣆⠀⠀⠀⠀⠠⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠈⢀⣀⣀⣠⣽⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠛⠋⠙⠻⣅⢻⢿⣷⡜⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⡢⢄⡀⠀⠀⠀⠀⠀⠠⠐⣡⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣮⣭⣭⣭⣥⣴⣾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠻⠿⠛⠋⢹⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⣸⣿⣿⡄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀⣀⣀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⠟⠿⣿⣿⣯⣭⣽⣿⡼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡀⣀⣸⣿⣿⣿⣿⡇⠀⠀⠈⠙⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⣩⣿⣿⠟⢱⣿⣿⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⣶⣿⣿⣿⣵⣿⡟⠁⢠⣿⣿⣿⣿⡟⠛⠛⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣠⣿⣿⣿⣿⣿⣿⡄⢠⣿⠟⣼⣿⠈⢿⡀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡰⠁⢿⣿⣿⢿⣌⣿⣿⣿⠏⣸⡏⢹⣇⠘⣧⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⣷⣀⣼⣿⣏⣸⣿⣧⣿⠏⠀⣿⠁⠀⢿⡀⠹⡄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢺⡟⠷⢽⣿⣿⣿⣿⣿⠏⠀⠀⣿⠀⠀⠘⢧⣤⠷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠇⠀⣸⣿⣿⣿⣿⠏⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠻⣿⣿⣿⡋⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⠾⠛⠿⠟⠛⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣄⣠⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⢹⣏⣛⣛⣙⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠈⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⢀⠐⢲⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⡼⣟⡶⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠭⠎⠃⠀⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣠⠾⡀⠀⠀⠀⡰⢿⠢⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠜⠁⠀⢱⡀⠀⠈⠐⣦⡄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⡜⠰⣈⣦⣀⣡⠤⠷⠶⠛⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⢼⠖⠉⠙⠿⣧⣹⡤⠤⠤⠄⠐⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⡿⣶⣏⢳⡀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣗⣧⣿⡜⠻⣿⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⠔⢻⣿⣿⣿⡿⣄⠘⡅⠀⢿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠸⣿⣹⢸⢿⣞⣆⢸⣦⡈⠳⡄⠚⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡀⠀⣷⠹⣾⢿⡞⡞⣦⣿⠓⣤⠽⣄⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⣹⣘⣿⣸⣿⠘⡠⡿⣁⡾⠀⢈⡆⠘⣿⡄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢹⡀⠉⠀⠀⠀⡟⡗⢧⣣⠈⡃⢲⣭⡈⡇⢩⣧⠀⡀⠀⠀⢄⠜⠀⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢣⠀⠀⠀⠀⡗⡇⠀⠀⠀⠸⡘⡌⠉⠀⢸⣿⠀⠇⡀⠀⢀⢁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡳⣄⠀⡰⣣⠇⠀⠀⠀⠀⠳⡘⠄⠠⢾⠏⠐⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠈⠲⠤⠞⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠿⣷⣦⣄⣀⣴⣾⣿⣷⡄⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠻⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣶⣤⣄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⣀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⢷⣶⣶⣾⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠻⠿⢿⣶⣶⣶⡾⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣾⣿⣷⣶⣶⣦⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠟⢻⠋⣏⠉⢩⡍⠉⣠⠈⠁⠉⠉⠉⠙⠏⠛⢻⠿⢿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢻⡏⠀⡇⠀⠀⣾⠀⡏⠀⢸⠳⡄⢹⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢰⠀⠀⡝⠛⢻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⢀⣴⡟⠋⠀⠇⢸⠇⢀⡇⠀⠀⡷⢀⡇⠀⣾⠀⢀⡿⣀⣀⡀⠀⠀⠐⡄⠀⠇⠀⡺⠀⢰⠀⠀⡘⠀⢈⠙⠿⣿⣿⣦⣀
⠀⠀⠀⣠⠟⠁⢸⠀⢰⠀⢸⣀⣰⣧⠴⢶⡗⢿⠷⣾⡟⠛⣻⠟⠉⠙⠛⠟⠛⢻⠛⢻⠻⢶⡶⣦⣌⣀⣠⠃⠀⡧⠀⢠⠋⠙⣿⣿⣷⡀
⢠⠀⣼⠏⠀⠀⣼⣤⡼⠶⣏⠁⣸⠁⢀⡼⢀⠟⢀⡼⢣⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡌⠉⠉⠿⠿⠶⣦⣥⠂⠀⢠⠏⠙⣿⣿⣦
⡜⢰⡏⢀⠤⢞⠇⢈⠗⣴⢃⣰⢃⣐⣊⣡⣭⣤⣽⣥⣤⣴⣴⣦⣴⣦⣤⣤⣤⣤⣀⣐⣀⣀⠒⠀⣀⠀⠀⡄⠀⣠⠛⠿⣶⣧⣀⡜⠈⢻⣿⣷⡀
⡇⢸⡷⠇⢀⣄⣠⣤⣽⣷⣾⠿⠿⠟⠟⠛⠛⠛⠛⠛⠉⠉⠙⢻⡉⠛⣟⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣶⣶⣴⣴⣁⣀⠐⡇⢙⠿⣶⣤⠞⣿⣿⣷
⢷⢸⡇⣴⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⠻⠿⣷⣾⣧⣤⠞⣋⡴⢿⣿⣿⡆
⠘⣆⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠻⠿⣷⡶⢻⣿⣿⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡌⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⢀⣀⣀
⠀⠉⠛⢍⡛⠳⠶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠁⠒⠤⠈⠉⠙⠛⠿⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠙⠂⠀⠀⠀⠀⠀⠐⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠛⠳⠿⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⠄⣀⢀⡀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠢⠤⠄⣉⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⣀⡀⠀⣠⣤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣶⣾⣤⡴⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⠄⠀⠀⠀⠀⣾⡟⢁⡦⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡬⠿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠯⡆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣊⣤⡴⠶⠶⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣦⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣤⡴⠶⠖⣻⣿⡯⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣦⣭⣭⡍⠉⠛⠛⢻⣿⠛⠉⠁⠀⠀⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠃⠀⠀⠀⡠⠀⠀⢴⠃⠉⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⢀⡠⠷⠛⠛⠓⠒⠤⣀⠀⠀⠀⢰⣿⠀⠀⢈⡴⠂⠀⠈⠘⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⢁⠞⢁⣤⣖⣿⣭⣿⣿⡶⣤⡑⢄⠀⠈⣿⡆⠖⠁⠀⠀⠀⠀⣆⢀⣀⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠈⣬⠟⠠⠟⠉⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀⠀⢻⣿⡀⠀⡀⠀⠀⢠⣿⣿⡉⢹⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠉⢬⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡸⣟⢋⣴⡾⣿⠆⠀⡂⠈⡹⠛⠋⠁⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⡟⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠿⣿⡟⠸⢿⣯⢿⡿⠀⣴⣷⣶⣴⣦⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⠇⠀⠀⠀⠀⠀⣀⠀⠀⣄⡔⠋⠁⠀⡟⣱⣿⣸⣿⣿⡎⢹⡏⠉⢙⣿⣿⠀⠄⠀⠸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣼⠀⠀⠀⠀⣠⣿⣿⣶⡿⣿⣷⡄⠀⡔⣡⣿⣿⣿⣿⣿⣴⣿⠇⠀⠈⣿⣿⣦⣦⣵⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⡏⠀⠀⠀⢴⢉⢟⣿⣿⣿⣿⣿⡿⢢⣾⠿⣿⣿⡿⢿⣷⣿⣟⠀⠀⠀⢿⣿⡍⠁⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡇⡷⠀⠀⠀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠛⢻⣻⠇⠀⠀⠀⢸⣿⡇⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⡅⣷⠀⠀⠀⠸⣆⣻⣿⣿⣿⣿⣿⣿⣷⣄⠸⣿⡟⡏⠀⡠⠉⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢿⠀⠀⠀⠀⠈⠩⠛⠛⠛⣿⣷⣾⡿⣿⡇⢸⢹⠇⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⠸⡆⠀⠀⠀⠀⢀⠀⠠⡑⢿⣿⣿⣭⣿⢃⠾⣾⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⢻⡄⠀⠀⠀⠠⠀⠀⣀⠀⣽⣯⢠⢧⡞⣰⠧⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄⠳⣄⠀⠀⠀⠀⠀⠁⠀⢾⡿⣠⠟⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦⡙⢷⣄⡀⠀⠀⠀⠀⣶⠞⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣬⣙⡓⠒⢒⣋⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⡀⣀⡠⠄⠠⠄⠀⠀⠀⠀⠀⠒⠈⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈⠁⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠄⢀⡀⠀⢀⠀⡆⠀⠀⠀⠂⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⡀⠀⣤⠠⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⢿⣦⡀⠀⢀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠟⠛⠛⠛⠻⠿⣷⣿⡏⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⢙⣷⣾⡿⠛⠛⠉⠉⠉⠛⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣧⡀⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⣠⣿⠟⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠻⣿⡄⠈⠻⣷⣄⠀⠀⠀⠀⢠⣿⠃⠀⢰⣿⠃⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠟⠀⠀⠀⠀⢿⣷⠀⠀⠈⠻⣷⣄⢶⡦⣿⡟⠀⠀⣿⡏⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⢿⠟⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⠁⠀⣀⣿⣧⣤⣤⣶⣶⠾⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⠛⠛⢻⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠈⠙⡏⠀⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠤⠀⠀⠀⠀⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣶⣤⣴⣶⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣶⣶⣾⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠒⢺⣿⡷⢄⣼⣿⢫⡿⣿⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠒⣶⡄⠀⠀⠠⣾⣿⠀⠀⣿⡟⠁⠀⢠⣸⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠁⣿⡆⠀⠀⠀⠀⠀⠈⢿⣄⠈⠻⠻⠤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⢧⢻⡟⢇⠀⠀⣤⠀⠀⠠⠌⠻⣧⣔⡀⢠
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣬⠭⠥⠀⠬⣶⣷⡿⠆⠒⢰⣶⠈⠀⠉⠙⠲⢽⣆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⣵⡒⠛⡅⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠻⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣦⡀⠹⡄⠀⠀⠀⢿⡄⢐⠀⠀⠀⠀⠀⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⠄⢱⠀⠀⠀⠘⠿⣮⡀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⣥⣾⡟⣧⡀⠀⠀⠀⠀⠉⠙⠛⠿⣇⣘⡉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⣿⣿⣿⡆⠀⠀⠀⠀⣀⣴⠾⢚⣿⢿⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⡁⢸⣿⣿⡇⠀⣠⣴⠾⠋⠁⣰⡟⠁⠀⠙⣿⣦⢀⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠈⣿⢡⠇⣾⠛⣻⣶⠟⠋⠀⠀⢠⣾⠋⠀⠀⠀⣠⣾⣷⡹⠟⠛⠛⠛⠛⠿⣿⣷⣦⡀
⠀⠀⠀⢀⣤⣶⣿⣿⠿⠿⠿⣿⣿⣿⠏⣸⣿⣿⠉⠁⠀⠀⠀⣴⠟⠁⠀⠀⢠⣾⣿⠏⠹⣷⡠⡀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⢀⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⣸⡇⠀⣽⣾⣞⣆⠀⠀⢠⡾⠋⠀⠀⠀⠀⣾⣿⠃⠀⠀⠹⣷⡡⣀⠀⠀⠀⠀⠀⠀⢻⣿
⢀⣾⠻⠁⠀⠀⠀⠀⠀⠀⣤⣄⠘⠿⣿⣈⣿⣿⣼⣆⣸⣿⣥⠶⢂⣤⠀⢸⣿⣹⠀⠀⠀⠀⠹⣿⣿⣧⠀⠀⠀⠀⠀⠀⣿
⣼⣷⠃⠀⠀⠀⠀⣠⣾⣿⣷⡎⠻⢦⢈⡙⢿⣛⣽⣿⡿⣫⣴⡒⡧⠃⠀⠈⣿⡼⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢰⢿
⣿⣿⠀⠀⠀⠀⠀⢯⡻⢟⣿⠛⠉⠉⠙⢻⣦⣽⠉⠁⠚⠉⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⡿
⢹⣿⣧⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⣦⣄⣀⣀⣀⣀⣠⢴⣺⣷⠟⠁
⠀⠻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢾⣿⠶⠿⠛⠋
⠀⠀⠈⠛⢿⣿⣿⣶⣶⣶⣾⣿⡿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⠴⠂⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⢀⣸⠏⠳⣄⠈⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢢⡀⠀⠀⣀⣠⣤⣦⣤⣴⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠉⠉⠛⠛⢿⣿⣕⣤⡈⢷⣄⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⢀⡟⠀⠀⣹⣶⣿⡿⠟⠛⠋⠉⠛⠛⠻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⡞⠀⠙⢿⣿⣄⠙⢷⣄⠀⠀⠙⠳⣄⡀⠀⠀⡾⠀⢀⣼⣿⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣦
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢻⣿⣆⠀⠙⢧⡀⠀⠀⠈⠙⢦⣼⠃⠀⣾⣿⠃⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⡼⠁⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠙⢦⡀⠀⣀⣰⡇⠀⣸⣿⡇⠀⠀⠀⠀⠀⠱⣄⣀⠀⠀⠀⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⠺⠿⠂⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠀⠀⠀⣽⢿⣉⡟⠳⡄⢛⣿⣁⣀⣈⡤⠥⠶⢾⠿⠿⠂⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠀⠀⠀⣧⣠⠿⢷⣉⡏⠉⢻⣇⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⣀⣠⠞⠛⠶⠞⠛⢦⡀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣤⣤⣤⣶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣶⣤⣤⣤⣶⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠒⠒⣀⣀⣀⣐⣒⣂⡠⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣯⣴⣂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣅⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣅⢄
⠀⠀⠀⠀⠀⠀⠀⠧⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣎⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⢂⠉⢳⡀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⢀⣣⣌⢰⣾⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠊⠀⠙⣎⠙⠫⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⡄⢄⠈⢀⣠⣴⣶⣤⡼⠦⢠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠊⠉⠀⢠⣿⣿⣿⣿⢏⣁⣐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠋⠉⠉⠙⠲⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣀⡀⠀⠀⠀⠹⡿⠿⠿⣿⣿⣦⣄⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢣⢀⣴⣤⠙⠿⣿⣿⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⢦⠀⠈⣿⣿⣿⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠂⠀⠠⢀⠈⣾⣍⣤⣀⣀⡀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠈⣡⡶⠶⠾⠷⠶⠤⣤⣌⡙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣁⣀⣀⣀⠀⠀⠀⠰⢈⠛⢦⡙⣧⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⣿⠋⠥⠀⠀⠈⠹⡶⣄⠀⠋⠀⠀⠙⣿⡃⡀⠂⠀⠀⡀
⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠷⠀⣦⠀⠀⠀⠀⠐⠦⠑⡄⠀⠀⠀⢸⣁⡆⠀⠀⡰⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠰⠶⠀⠀⠑⠀⠀⠀⣸⠈⠑⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⢄⣀⣀⡀⠀⠀⠀⠀⠀⣆⡀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠄⣠⠴⠁⠀⠀⠀⠀⡠⠿⠛⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣀⢀⣀⣀⣠⡤⠖⠲⢄⡐⠒⠛⢉⠆⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣙⠉⣉⡁⠀⠀⠀⠀⠀⠑⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠀⣀⡀⠈⠁⠢⠄⠀⣠⠀⠀⠀⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠈⠁⠀⡰⠁⠀⠀⠂⠚⠁⠒⠛⠂⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢈⡽⠋⠑⢺⠍⠁⠀⠀⠉⠁⠠⠀⠉⠙⠋⠉⠈⠱⠦⢄⣀⠀⠀⣀⣠⠎⡏
⠀⠀⢀⣀⣀⠤⠚⠁⠀⠀⠀⠀⢸⠃⠀⠂⢠⢄⡀⠀⢀⣀⣀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⢀⠃
⠉⠉⠉⠁⠀⠀⠀⣀⡀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠙⡴⠉⠀⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀
⠀⠀⠀⠀⠀⡠⠊⢁⠟⡇⠀⢀⠇⠀⠀⡄⠀⢀⣀⡀⠀⠀⢀⡎⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣥⠽⢟⡆
⠀⠀⠀⠀⢸⠅⡠⣿⡼⠁⠠⡟⢦⣀⣤⠷⠛⠋⢉⠅⠀⠀⡾⠀⣸⠀⠀⠀⣀⣀⣀⣤⣤⡶⠾⣛⣫⢭⣖⣭⡁
⠀⠀⠀⠀⠘⣆⠀⢩⣧⣄⣀⣈⠓⣬⠗⠒⣲⡀⠸⡀⢀⣼⣥⡶⠟⠛⣛⣩⠽⢟⣫⠭⠖⢚⣉⠝⠉⠀⢁⣠⠏
⠀⠀⠀⠀⠀⠈⠑⠠⠍⠻⡷⢶⡘⠁⠠⢿⠦⣌⠀⠤⣉⣩⠤⢶⣚⣭⡵⠒⣋⡡⠴⢒⡫⢭⣶⡤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡆⢰⡀⠀⠉⠲⠷⠚⠋⡝⢈⠠⣟⠭⠓⣊⣭⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣸⣄⠙⠬⢭⣉⡉⠭⠤⢓⣁⣤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣡⣿⣿⠿⢿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⢉⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠁⠀⠀⠀⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢰⣾⣿⡛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⡒⠤⣄⣀⡀⣸⣿⣿⣷⡀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠈⠙⣦⡴⣮⣙⠻⣿⠟⣿⣷⡾⠋⠉⠉⠉⠛⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⢶⣶⣿⡀⠀⠀⠀⠘⢧⡀⠉⢻⣿⣦⣿⣿⣇⡀⣠⣄⡀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⠀⠀⠀⣸⠋⢻⣦⠀⠀⠀⠈⢻⣷⣾⣿⣿⣿⣿⣬⢽⣿⣿⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢠⢲⣶⣧⠀⠀⢻⡆⠀⠀⠀⠘⠛⠛⠿⠿⢻⣇⠀⠀⠰⠋⠉⠀⠀⣸⡯
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠐⠿⠿⠋⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣀⣠⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⢶⡶⠾⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠑⢄⠀⠙⢆⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡑⢌⠠⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣁⣈⣈⠀⠐⢁⠈⠀⢁⡤⠞⢉⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠈⢂⠔⢁⠔⠉⠀⠀⠀⠈⠐⢄⠠⡒⠁⠐⢊⠁⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠅⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠛⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⡀⠀⠀⠈⠀⢣⣀⣤⡤⣄⣀⡤⠴⠚⠉⠁⠉⠉⠀⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠒⠪⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠚⢉⠰⠲⠁⠀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⠻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢬⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢴⣐⣋⡂⠆⠀⠀⠐⢉⠀⠞⠁⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠶⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠭⠍⠭⠟⠉⠉⢰⡄⠒⠐⠒⠒⠊⠉⠐⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠺⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣾⡿⣿⣿⡿⠛⠛⠛⠛⣛⠛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣷⠒⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⣿⣿⣷⣶⣶⣾⣷⣶⣦⣤⢴⣶⢤⣄⣀⣠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠿⠿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠶⠶⠶⠶⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣿⣿⠀⠀⠀⠀⣀⣬⣾⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣴⡿⣵⣿⡿⠏⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣃⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⢀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣻⣿⡿⠿⠟⣿⡇⠀⠀⠸⣿⣿⣿⠉⠀⣘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣏⠀⠀⠀⢠⣿⠟⠛⠛⠋⠛⣿⣿⣏⢷⣿⣿⣿⡿⢻⣿⣄⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣳⣶⣆⢠⠿⣷⡄⠀⠀⠀⠘⢿⣿⣿⣿⣿⣏⠀⣸⠁⠹⣆⢀⣤⣤⣶⣶⣶⣶⣦⣭⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠿⠛⠛⠛⠛⢻⡾⣶⣬⡻⣦⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣯⠀⣠⣾⢿⠋⠉⠀⠀⠀⠀⠉⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⢠⡿⠀⠀⠙⢿⣮⠳⣄⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⠏⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢻⣷⠈⣳⣶⣾⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠹⣄⠀⢀⠀⠀⠀⠀⠀⠀⠹⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠈⣿⡆⠛⠿⠿⣿⣍⡤⣼⣿⣿⣿⣷⣆⣀⣀⣠⣽⣿⣿⣄⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡿⣷⣿⣾⣿⣿⣿⣿⡿⠉⠁⠈⠻⣿⣁⡀⠀⠀⠀⠀⢀⣿⣿⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠈⠳⣬⣿⣿⣿⣿⠋⠀⠀⠀⠀⠂⠀⠈⠉⠁⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠷⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣤⣤⣄⣀⣠⣤⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⣻⣿⣶⣦⣤⣤⣤⣤⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠻⠿⠿⠿⠿⢿⣿⣿⡶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣯⣭⣿⣯⣿⣯⣯⣭⣿⣶⣶⡤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⣲⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣲⢤⡀
⠀⠀⠀⠀⠀⣠⣾⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢤⡀
⠀⠀⠀⣠⣮⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠻⢿⣿⣿⣿⣿⣷⣟⢦⡀
⠀⠀⡴⣵⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⢦⡀
⠀⠀⢿⡿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⣵⡄
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣯⡽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢀⣴⠿⣷⣦⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣼⣭⣤⣍⣾⡇⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣟⣠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⢻⣿⡿⠛⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠈⠛⠻⠿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⣴⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⣼⣿⣿⢟⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀⣾⣿⣿⢏⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⣼⣿⣿⣯⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠵⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠄⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠁⡀⢀⣀⢠⣀⠠
⠀⠀⠀⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢀⡀⢘⠂
⠀⠈⠄⠠⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⢀⡀⣀⠠⠤⠐⠒⠒⠒⠦⠤⣀
⠀⠀⠀⠀⠀⠀⣀⡀⣠⣔⣀⡂⠐⢀⠰⠤⠀⠚⠛⠉⢁⡐⠢⡀⠀⠀⣀⠤⠶⠒⠙⠒⠚⠓⠒⠲⠤⠤⣀
⠀⠀⠄⠂⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠁⠀⠀⢁⣀⠥⠤⣁⣉⡐⠒⠲⠀⡀⣀⡠⠤⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⣀⡠⠤⠤⠑⠐⠒⠚⠃⠀⠈⠑⣄⣀⠤⠉⠁⠠⢍⡀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⢀⣀⡀⠠⠄⠀⣀⡀⡀⣈⡆⡴⠒⠉⠉⠀⢄⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠈⢦⡠⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⢊⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⡠⠔⠒⠒⢻⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣕⠢⠤⠤⠤⢽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡱⠋⠉⠉⠀⠀⠀⠀⢀⣀⠤⣒⠭⠓⢋⡵⡻⣸
⠀⠀⠀⠀⠀⠀⣀⠤⢐⣒⣒⣒⣒⠰⢵⡁⠀⠀⠀⣀⡠⣔⣪⢽⠊⠉⠀⢀⢔⡯⠊⣀⠧⠧⠤⠤⠤⢀⡀
⠀⠀⠀⠀⡴⣪⠔⠉⠅⠀⢠⠀⢠⣿⠓⢬⡲⣖⠭⠒⠉⠁⢸⢸⠀⢀⢔⡵⣫⢔⡩⠒⢺⢹⠉⠉⠉⠒⠪⣑⢤⡀
⠀⠀⢀⢎⠜⠁⠠⡀⠈⠀⠠⢠⣿⠃⡠⠊⠙⢌⢆⠀⣀⠤⠼⠾⣔⠕⠋⡔⡱⠛⢄⠀⠘⡏⣆⠀⠠⠁⢀⠌⠳⡑⡄
⠀⠀⡎⡜⠐⠠⠀⠈⠢⠐⢰⣿⡧⠊⠀⠀⠄⠚⡎⡾⠁⢀⣀⠀⠈⡆⡜⡼⠠⢀⠀⠑⢄⢹⢹⠀⢀⠔⠁⠀⠀⠸⡘⡄
⠀⠀⡇⡇⠄⠀⠀⠀⠁⢰⡛⣻⣶⣿⡶⠭⠭⠭⡧⡟⠲⢞⠖⠁⠀⣿⡇⡇⠀⠀⠀⠀⠂⢱⠿⢧⠁⠐⠀⠀⠀⠀⡇⡇
⠀⠀⡇⡇⠀⠀⠄⠂⡨⢊⠙⢉⠀⡐⠀⠄⡀⠀⡇⡏⠓⢄⡱⠔⠊⠀⡇⡇⠀⠀⠀⠄⢀⠘⠓⠛⢄⠀⠀⡀⠀⠀⡇⡇
⠀⠀⠸⡘⡄⠀⡠⠊⠀⠂⢀⠀⠄⠀⠠⡀⠀⡜⡸⠀⠀⠀⠀⠀⠀⠀⢱⠱⡈⠀⢀⠔⠁⠌⠀⠈⠀⠑⢄⠀⠀⡰⢱⠁
⠀⠀⠀⠘⢌⠪⣀⠀⠄⠀⢈⠀⠘⡀⢀⡠⢊⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢄⠁⠀⡐⠀⠀⠀⠐⠀⠀⢁⠔⡱⠁
⠀⠀⠀⠀⠀⠙⠪⢝⣒⠢⠴⠤⢒⣚⠭⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢝⡒⠤⢄⣠⣀⠤⠗⣚⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠳⠈⠈⡙⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠄⢀⡀⢴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⢀⠔⠯⢄⢐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⢀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠁⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢀⣀⠀⠠⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡋⣉⠀⠉⠙⠋⠒⢄⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠈⢀⠀⠙⢷⢴⠴⣓⣤⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣀⣨⡗⠀⠀⣠⠴⠋⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠈⡿⠀⠀⠀⠃⡄⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⣆⣠⠆⠀⠄⣄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠘⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢀⠴⢋⡽
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡌⡖⠀⠀⠀⠀⠀⠛⠉⠀⠀⠇⠀⠀⢀⡠⠔⡠⠔⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚
⠀⠀⢠⠋⠙⣦⡀⢠⣴⡦⡟⠉⠀⠈⠙⢄⠀⠀⠀⠀⣄⣀⣀⠐⢆⢀⠠⠂⠡⠐⠉⠀⠀⣠⠎⠀⠀⠐⡀⠀⠀⠀⢀⠔⠋
⠠⡎⠉⠀⠀⢀⡜⢧⠊⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡮⠉⠉⢳⡋⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠙⢮⣓⡀⠀⠉⠈⠀⠀⠀⣸⠿⠀⠀⠀⠀⠀⢋⡨⠇⢠⡶⡼⠁⠀⠀⠀⣀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠈⠛⣄⢀⣇⠀⡴⠊⠉⢦⡱⢄⠀⠠⠀⡁⠐⠙⠤⠴⠁⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠘⢇⣫⡶⠁⠀⠀⣀⠝⠊⡡⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢋⠡⠒⠁⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢀
⠀⠀⠀⠀⠀⢀⡠⠒⣉⠄⠊⠁⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠚⢉⣤⣶⣿⠿
⠀⠀⠀⠄⠊⣁⣴⠞⠁⠀⠀⠀⠀⢀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⢀⣠⡼⠯⠵⠒⠛
⢴⣊⣠⣴⣿⡒⠁⠀⠀⣀⡠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⢀⣴⣶⣿⠏
⠀⠉⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⢠⣤⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⣠⣴⣾⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢤⠤⠖⠉⠀⢀⣤⣾⠿⠉
⠀⠀⠀⠀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⠿⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⢐⠆⣉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠉⠁⠀⠈⠉⢐⡄⠀⠀⠀⠀⠀⢠⢣⠎⠀⠑⡌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢤⣤⡤⠤⠒⠉⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⣄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠆⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⣀⡎⡴⡘⢍⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⣨⠤⠤⠤⠤⢄⡀
⠀⠀⠀⠀⠀⢀⠴⢊⠤⠒⠊⢹⠉⢐⠦⣌⡓⢌⢎⢢⡀⠀⠀⠀⠀⠀⡠⢒⡡⢴⢲⠉⡏⠑⠒⠤⡑⠦⡀
⠀⠀⠀⠀⢠⢋⠔⠣⡀⠀⠀⢸⠀⡜⡜⡠⠛⢆⠳⡣⡑⡄⠀⠀⢀⠜⡰⠛⢄⠀⠆⢃⡇⠀⠀⢀⠜⠢⡙⡄
⠀⠀⠀⢀⠃⡎⠀⠀⠈⠢⡀⢸⡜⡰⠋⠀⠀⠈⠆⢣⠑⡌⢆⠀⡜⠰⠀⠀⠀⠑⢼⡘⡇⢀⠔⠁⠀⠀⢱⠘⡀
⠀⠀⠀⢸⠀⡧⠤⠤⠤⠤⠬⣾⣿⣷⣶⣶⣶⣶⣾⣺⣒⣚⠖⠉⢨⡧⠤⠤⠤⠤⠤⢵⣷⠥⠤⠤⠤⠤⢼⠀⡇
⠀⠀⠀⠘⡀⡇⠀⠀⠀⡠⠊⢸⠈⠢⡀⠀⠀⠀⡌⡸⠀⠘⠆⠚⢇⢡⠀⠀⠀⢀⠔⠁⡇⠑⢄⠀⠀⠀⢰⢀⠃
⠀⠀⠀⠀⠱⡘⢆⡠⠊⠀⠀⢸⠀⠀⠈⠢⣀⠜⡰⠁⠀⠀⠀⠀⠈⢆⠣⣀⠔⠁⠀⠀⡇⠀⠀⠑⢄⡰⢃⠎
⠀⠀⠀⠀⠀⠈⠢⡑⠢⣄⣀⣸⣀⣀⡠⠔⣡⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⢌⠢⢄⣀⣀⣇⣀⡠⠔⢊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠤⠤⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠄⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⠋⠉⠁⠀⠀⢀⡄⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⡁⠠⠀⣀⢀⢬⡐⡘⠂⠀⣿⢸⠐⡄⠀⡄⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠠⡄⡄⡁⡻⣩⢳⡜⢶⣣⣾⣴⣦⣇⣿⠠⣧⠌⣆⠀⠀⠈⡀⢠⠀⠀⠀⠀⠀⠂⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⠀⢀⣔⣦⣃⢠⠈⢂⡘⢷⡄⣷⡠⣷⣿⣿⢻⣟⠹⣆⢻⡜⣯⢣⡂⠀⠅⠀⡀⣀⠀⠀⠀⠀⠘⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠐⡀⠈⠀⠺⣿⠞⣇⢱⡄⠹⣧⠘⢧⠘⣷⡺⣟⢽⡏⢻⡇⠳⠠⢇⠀⡋⢻⡒⣒⣋⣇⣺⣐⣀⣀⡀⢀⠀⠂⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠁⡀⠰⠀⣪⣄⢿⣗⠘⣆⢫⣢⠩⢷⠈⠳⠘⢆⠘⣂⠩⠤⠓⠒⠓⠋⠉⠉⠉⠉⠀⠈⠁⠀⠉⠉⠉⠉⠛⠛⠲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠀⢁⠀⠀⡀⣽⡫⡻⣎⢿⣿⡜⢮⠛⣃⡠⠥⠖⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠐⢀⣸⣌⠙⢜⣛⣯⠭⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠌⠉⠠⡚⢫⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠻⣤⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠞⠀⢀⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠉⠉⢻⣴⣾⡧⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣾⡿⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣶⣶⣴⡤⠔⠂⠀⠠⠶⠶⠀⠀⠀⠠⠴⢿⣥⣿⣥⣤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣦⣭⣶⣮⣬⣾⣶⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣶⣤⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠉⠿⠿⢿⡿⠟⠛⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣀⠀⠀⠀⠀⠀⠈⣿⡿⢟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡟⢶⣶⣶⣶⠶⠋⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠚⠛⠂⠘⢿⡀⠀⠙⠛⠋⠀⢠⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠉⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⣠⣶⣿⣿⣿⡟⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⣿⣿⣿⣿⡿⠋⠴⠁⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣤⠀⢀⣀⣤⠞⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠋⠀⢉⣿⠟⠁⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠿⠃⠀⣠⠞⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⡄⣴⡞⠁⠀⠀⠀⢀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢻⡏⠁⠀⠀⠀⣤⡌⠹⠟⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠛⠻⣧⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⣠⠏⠀⠀⠀⠀⣸⠛⠁⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⠏⠀⠀⠛⣄⢀⣀⣠⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⣠⡾⠁⠀⠀⠀⠀⡼⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠾⠛⣉⣴⣿⠏⠀⠀⠀⠀⠈⢿⣿⠟⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⡾⠋⠀⠀⠀⠀⠀⣶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⠶⢛⣉⣿⠏⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤⡄⠒⠋⠁⠀⣾⠀⠀⠀⠀⠀⠀⢀⣿⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣪⣿⡇⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠹⠀⠘⢀⣀⣠⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⣈⠳⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣀⣾⣿⣿⣿⣶⣦⣄⠀⠀⣼⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⠀⠀⠀⠀⠀⠠⠟⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠳⣾⣿⣿⣿⣿⡿⢷⣾⣿⣿⣿⣿⣿⣿⠁⠀⠉⠛⠳⠶⠶⢦⣤⣄⣉⡉⠛⢻⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠻⠿⠿⢷⣬⣿⣿⣿⡟⠛⠻⢿⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠈⠙⠃⠾⣿⢄⡤⢀⣴⣀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣶⡿⠿⠿⠛⠛⢻⣿⣿⣿⣶⣶⠂⠀⠁⠈⠉⠙⢃⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⣿⣿⣿⣿⣿⣿⠻⣿⣿⡇⠉⠀⠠⠴⠾⠟⠛⢿⣿⣇⣀⠙⢧⣀⡀⣠⡀⢶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠏⣠⣾⣿⣷⡖⠀⠀⠀⠀⠀⠀⠈⠛⠉⢹⣧⠈⠉⠉⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⢰⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⢠⣿⣿⣿⣿⡿⠁⣿⠓⡄⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣛⣛⣋⣴⣦⣸⣿⣷⠈⣿⠃⠀⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⠟⣿⣿⡿⢿⣿⣿⡟⣷⠘⣇⠀⢹⣯⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠇⠀⣿⣿⣿⣿⣿⠛⣷⣿⣧⠸⠇⢈⡟⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣷⣶⣤⣿⣿⣿⣿⠃⠾⢿⣯⢿⣧⠘⢋⣷⣤⣤⣀⡀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢠⣿⣃⣶⡀⣾⣿⣾⣿⣇⠘⣭⡄⠀⠈⠉⠛⠷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡯⠛⠿⢿⣿⣿⠸⣧⠹⣿⡇⢿⣿⣿⣿⣿⣧⢻⣷⡄⠀⠀⠀⠀⣿⣋⡁
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⢁⣾⣿⣿⠁⢀⣄⣠⣀⣤⡄⢀⡀⢹⣇⢸⣿⣿⣿⣿⣿⣇⢹⣷⡀⢀⣀⠀⠘⢿⣿⣶⣀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠋⢀⣾⡟⣾⠋⠀⠈⠛⠛⠙⠛⠑⠏⠀⠘⠏⠘⣿⠟⠻⣿⡟⢻⡆⠛⠇⠸⠧⠀⠀⠈⠻⠿⢿⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠁⠀⣼⡟⣿⠋⠀⠀⠤⣶⣿⡿⢠⣷⣄⡀⢰⣷⢰⣿⣿⣿⣿⡷⠻⣷⣶⣿⣆⠀⠀⠀⠀⢼⣿⣿⣷
⠀⠀⠀⢀⣿⣿⣿⣿⡟⠀⠀⣼⣿⣿⠃⠀⠀⠀⢠⣿⡿⠁⣸⣿⣿⣄⢸⣿⠀⣿⠛⢿⡿⠀⣼⣿⡏⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⢸⣿⠃⣰⣿⡿⢻⣿⠀⣿⠀⣿⠀⠸⠇⢰⣿⣿⣧⠻⣿⡆⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⣸⣿⣿⣿⣿⠁⠀⢰⣿⣿⠿⠀⢠⣴⣆⣀⣿⠃⡀⢿⣿⡇⢸⣿⠀⣿⠀⣿⣤⠇⢰⣿⣿⣿⣿⡄⣿⡇⠀⠀⢰⣿⣿⣿⣿⡇
⠀⢠⣿⣿⣿⣿⣧⣤⣴⣿⣷⡏⠀⣴⣟⡁⢈⣿⠃⢰⡇⣼⣿⠃⢸⣿⣤⣽⠀⣿⡏⢰⣿⣿⣿⠟⢿⡿⢸⣿⠀⠀⡈⢿⣿⣿⣿⡇
⠀⣼⣿⣿⣿⣿⠿⠿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡃⣠⣿⠃⣿⣿⠀⢸⣿⣿⣿⡀⢹⣷⣿⣿⣿⡿⠀⣤⣀⣸⡟⠀⠀⡇⣼⣿⢿⣿⠁
⢠⣿⣿⣿⣿⠇⠀⠀⠈⠻⠟⢹⣿⣿⣿⠹⣿⣿⣿⣿⣾⣿⡇⠀⣼⣿⣿⣿⡇⠘⢿⣿⣿⡿⠀⣰⣿⣿⣿⣇⣀⣤⣿⣿⣿⢸⡿
⢸⣿⣿⣿⣿⠀⠀⠀⠀⠈⠀⢸⣿⠻⢟⣀⠉⢹⣿⢿⣿⣿⡷⠀⣿⣿⣿⣿⣷⣶⣾⣿⣿⢱⠿⠿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣼⡇
⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢿⣴⡞⠁⢀⣿⠏⣼⣿⠇⠀⠀⢿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣶⣶⡦⠙⠛⢫⣽⣿⠿⢿⣿⣧
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣰⣿⡏⠀⠀⠀⠀⠀⠉⠀⣾⣿⣿⣿⣿⣿⣿⡿⢻⠀⠀⠀⣼⣿⣿⠀⣾⡏
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢿⣞⡀⠀⣼⣿⣿⠉⣶⡿
⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢁⣾⣿⣿⣿⣿⣿⠟⢰⡿⠁
⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡿⢋⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⠃
⠘⣿⣿⣿⣿⣇⠀⠀⠀⢀⣠⡾⢋⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠾⣿⣿⣿⣿⣩⡤⣴⣿⠃
⠀⢹⣿⣿⣿⣿⣷⣶⣿⣿⣭⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⣈⣽⣿⣿⣿⣿⡿⠁
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠉⠛⠿⠾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠴⡔⣄⠄⠀⡀⠀⠆⠆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠐⠅⢤⡈⠊⠂⠈⢌⢆⢀⠱⡘⢀⠀⠀⠀⠇⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠬⢀⠑⠢⣀⠀⠑⣤⠠⠤⠤⠦⠒⡲⣿⡄⣤⠀⠀⠀⠀⡄⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠭⣑⡒⠤⢄⣭⣿⣈⣽⣍⣡⣤⢶⣖⣺⣿⣻⣿⣿⢹⡾⣶⡇⣰⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠄⣀⡀⠩⠁⠐⠀⠍⠛⠫⠛⠉⢋⠙⠉⠛⠙⠍⠉⢣⣿⣿⣿⢠⡏⠀⠀⠀⡠⣪⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢀⠉⠐⠒⠤⠄⢀⡀⠐⢂⠤⢄⣀⠀⠠⣤⣤⣜⣿⣿⣿⣾⡄⢀⣠⡿⣛⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⠉⢁⡀⠀⠒⠂⠠⠄⠀⠀⠀⢈⡙⠒⢞⠴⣶⣿⣗⢊⡿⡋⠊⠁⠈⠻⣿⣿⣷⡞⠅⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠬⠄⠀⢀⣴⣰⣶⡒⠒⠒⠚⠛⠻⢿⣿⡟⠀⠀⠀⠀⠐⢸⣿⣿⣯⡉⠁⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⢀⡀⢀⠀⢀⣠⣤⠬⣑⣣⣶⣤⣭⣒⡁⠑⣮⢍⣭⣿⣷⡣⡀⠀⢀⣠⣷⢛⣯⣿⠟⣁⣄⣮⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠊⣉⠰⣚⡶⠛⠺⣯⠟⠻⠿⠟⢩⡿⢶⣭⠿⣾⣿⠟⣿⣶⣿⣷⣿⣯⣿⡶⣾⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⣡⣾⣿⣿⣝⣥⣦⣲⣤⣶⣴⣆⣦⣽⣶⣶⣤⣄⡻⢏⣻⣴⠕⣿⣿⣿⣿⣿⣿⣍⢢⡪⡪⣑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡹⠿⣛⠽⠛⠟⡻⠟⠛⠛⡩⢟⠻⣻⣿⣻⣿⣿⠿⢛⡟⡱⠋⠀⠈⣟⡿⡹⡟⢮⡿⢧⡘⠊⠪⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠩⠂⠬⡒⢕⠪⣐⠌⠀⢂⢵⣿⠿⣻⢵⠟⢁⢤⢎⠂⠡⠀⠀⠀⠙⢻⠀⠈⠈⠵⣜⢵⡔⢕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⡒⠅⢢⡐⢝⣡⡤⣨⣾⣿⡧⡊⡤⡡⣢⣪⢫⢆⢠⢂⢃⢂⢀⡄⡄⠠⢂⠀⠀⢌⢦⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⣺⣵⣾⣿⣿⣾⣿⣿⣿⣿⡿⢾⡷⢷⣧⣯⣎⣞⣾⣼⣾⣾⣿⣿⣿⣿⣿⣿⣿⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⠾⠦⠄⠀⠛⢉⠉⣙⢛⠿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣟⣉⣉⡤⠀⠀⠒⡨⠑⣩⠐⡻⠻⢷⣮⣟⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢻⣿⣧⣬⣉⣛⠷⠔⣂⠍⢐⠩⠒⠍⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠈⠑⠂⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁⠉⠒⠤⣄⡀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠉⠒⠤⣀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠉⠑⠢⠤⣀
⠀⠀⠀⢀⣴⠊⠁⠀⠉⠒⢄⡀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠉⠒⢢⡄
⠀⠀⠀⠈⠙⢷⣦⣄⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠀⠉⠒⣤⡤⡪⠕⠁⠀⠀⣀⣀⣀⣀
⠀⠀⢠⣦⣄⠰⢸⠙⡿⢶⣤⡀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠉⠲⢄⣀⢴⡪⠟⠉⡇⠀⠀⣰⠯⡅⠀⣈⠙⠻⣦
⠀⠀⠘⠤⡙⠟⣦⣴⠃⠀⠉⠛⢷⡦⣀⡀⠀⠀⠈⠓⢤⡀⢀⣤⡾⠟⠉⠓⠲⣞⢳⡦⢒⡡⠾⠵⠊⠉⠑⡄⠹⡇
⠀⠀⠀⠀⠈⠑⢬⡙⢷⣦⡀⠀⠀⠈⠑⢯⣢⢄⡀⢀⡴⡺⠗⠉⠀⠀⠀⢀⡠⢞⡩⠶⣉⠳⢦⣄⡀⠀⢀⠇⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⡻⢷⣤⡀⠀⠀⡏⡗⠺⢷⡏⠀⣀⣀⣀⣠⢖⡩⠖⠁⠀⠀⠀⠙⠲⠬⣉⣉⡡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢍⡻⢶⣤⡇⡇⠀⡼⢠⠊⠁⠀⠈⠙⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢌⠃⢷⢎⡀⠈⠦⠤⠴⠒⠲⢄⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣕⠭⡢⣄⠀⠀⠀⠀⠈⠆⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡓⠷⣦⣄⣀⣀⠆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣉⣉⣁⠴⠋