Dictionary Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣻⡶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣿⣹⢟⢿⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⢠⡿⠛⠿⠿⠛⠛⢻⣿⣹⠿⣇⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⡅⠀⠀⠀⠀⢀⣼⣿⢿⣿⣮⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⢀⡀⢻⡟⠘⣧⢸⡀⠀⠀⠀⠀⢨⣟⣼⣿⣿⣿⣮⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠷⣌⠙⣶⠅⠀⣿⡄⣧⣤⣤⣴⣶⣶⣾⣷⣶⣶⣶⣿⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⠈⠘⢶⣬⣿⣿⣿⡇⣾⡆⠀⣠⣬⣿⣿⣿⣿⣯⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⢉⣿⣿⣿⢀⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⡠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠉⠇⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⡿⣛⣿⣿⡿⠋⠀⢴⣾⣃⣤⣠⣾⣿⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⠑⢦⣤⡄⢠⣿⠟⠉⠁⠹⡿⢻⡇⠈⠉⠛⠛⠉⠀⠀⢰⣿⢟⣿⣿⣿⣿⣿⢻⣿⣿⣧
⠀⠀⠀⢠⣶⣞⢻⣿⡹⣿⣷⡀⠘⣣⣿⢃⣀⣼⣏⣿⡻⢿⣷⣄⠀⠀⢀⣴⡗⢀⣴⣶⣿⣿⣿⣿⢿⣽⣿⣿⣟⣿⡗
⠀⠀⠀⣾⣿⣽⣯⣼⣿⣻⣇⡿⢀⣿⣿⡿⠿⠿⠟⠛⠛⡿⠿⣿⡆⠀⣸⣫⣤⠿⣿⣿⣿⣿⣯⣿⣿⡿⣿⣿⣿⣿⡷
⠀⠀⠀⣿⠟⠋⠉⠉⠉⠉⠉⣛⣿⣿⣶⣤⣤⣀⠀⡙⠘⠇⠀⣀⣼⣿⣿⣤⣤⣬⣿⣶⣖⣯⣽⣿⣿⣿⣮⣿⣿⠟⠁
⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⣼⣿⣴⣿⣿⣿⣿⣿⣧⣇⣶⠶⠿⠛⠛⠿⠿⠿⢿⣿⣽⣿⣻⣟⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⣤⣦⣄⣠⣤⣤⣤⣄⠈⠙⠛⠛⠻⣍⠉⠙⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣾⣿⣿⣿⣿⣟⣿⣿⣿⣿⢻⣿⣿⣿⣿⣮⠙⠿⣿⣷⡄⠀⠀⠀⠹⢀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡏⠈⠻⢿⣿⣿⣿⣿⡇⠀⢻⠻⢿⣿⣿⡇⠂⠮⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⣿⠅⠀⠀⠀⠈⠙⢿⣿⠂⠀⠰⡆⠀⠉⠹⠇⠀⠀⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⢿⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠄
⣤⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄⠀⠀⡰⠁⠀⠀⠂⢀⠂⠀⠀⡀⠀⡠⠄⠀⠀⠀⣀
⢸⣿⡏⠁⠈⡓⣤⡂⡀⠀⠀⠀⠀⢇⠀⠰⡀⢠⠃⠀⠀⠀⠈⠅⠀⠀⠀⠐⡊⠀⠀⢩⠈⢻⣿⣆
⠀⣿⣿⡔⡀⢱⣿⣧⠙⢄⠀⠀⠀⢸⡀⠀⢡⡌⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠱⠀⠀⠀⢂⠀⡹⣿⣆
⠀⢸⣿⣿⠂⠀⢿⣿⡆⠀⠙⠦⡀⠀⡇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢂⠀⠀⠈⠄⠐⠹⣿⡆
⠀⠀⣿⣿⣏⠂⠈⣿⣿⡀⠀⠀⠈⣦⡇⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⠘⡀⠁⢹⣿⡄
⠀⠀⠸⣿⡷⡁⠀⠸⣿⣧⠀⠀⠀⠘⡆⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠡⢻⣿⡀
⠀⠀⠀⢿⣿⡐⡐⡀⢹⣿⡆⠀⠀⠀⢹⡄⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠸⣿⣇⢀⢈⠀⢸⣿⡀⠀⠀⠀⢳⡀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢡⠀⠀⠀⢀⠀⠀⢻⣅⡀
⠀⠀⠀⠀⢿⣿⡆⠊⣧⡹⣿⣧⠀⠀⠀⠈⢧⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢣⠀⠀⠀⠄⠀⠌⢿⣷⡀
⠀⠀⠀⠀⠸⣿⣿⠰⡸⣷⢿⣿⡇⠀⠀⠀⠘⣆⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠆⠀⠀⠘⡄⠘⡌⢿⣧
⠀⠀⠀⠀⠀⢿⣿⣇⢳⢻⣯⢿⣿⡄⠀⠀⠀⠸⡄⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠘⡆⠀⠀⠸⡀⠘⡜⣿⣧
⠀⠀⠀⠀⠀⠘⡼⣿⡌⢯⢿⡎⣿⣷⠀⠀⠀⠀⢳⡀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⡄⠀⠀⢱⠀⠸⡘⣿⣧
⠀⠀⠀⠀⠀⠀⢳⢹⣷⠘⡌⢿⡘⢿⣧⠀⠀⠀⠈⢇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠱⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠘⡎⣿⣇⠠⠼⠳⡏⠻⣆⠀⠀⠀⠈⡄⠀⠀⢀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠤⠊⠁⠀⠀⠀⢀⡠⠀⠀⢁⡹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢣⡀⠀⠘⠦⣈⠳⣄⡀⠀⠠⠀⠀⠈⡄⠀⠀⠀⠆⠀⠀⠄⠀⠀⠀⠀⠐⠂⠉⣀⡤⠔⠊⢀⣠⣽⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡈⠢⡀⠀⠈⠑⠪⣛⢷⣄⠡⠀⠀⠰⠀⠀⠈⠀⣠⠆⠂⠀⣀⠠⠤⠒⠊⠉⣀⣤⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠢⡀⠀⠐⠤⡑⢽⢷⡆⠀⢸⢀⠄⣠⢞⠡⠔⠊⠉⠀⢀⣀⣤⡶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠢⡀⠀⠈⠀⠑⣿⠀⡎⢈⠴⣫⠅⠀⢀⣠⣴⠾⢛⡫⠝⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠢⡀⠀⠀⠈⢬⠖⠑⣉⣤⣴⣾⣛⡭⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠢⣠⣤⣦⣶⢿⣿⡿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣦⣿⣿⣽⣚⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⡾⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢶⣤⡀⠀⠀⢀⣤⡶⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⢷⣄
⠀⠀⠀⠀⠀⣾⠏⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠹⣷
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠄⠀⠀⢀⣴⣾⠿⠛⠋⠉⠉⠙⠛⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⣴⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⠃⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡄⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⣿⣦⣤⣤⣼⣿⣿⣦⣤⣀⣀⣀⣀⣤⣴⣿⡿⢿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠻⣯⣸⣇⣼⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢶⣤⡀⠀⠀⠈⢿⡿⠁⠀⠙⢿⣿⣿⣿⣟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⣀⣈⣁⣀⣀⣴⠟⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠙⢿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠶⢶⣶⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠷⠶⢦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⢰⣦⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⣶⡄⠀⠀⠀⠀⠀⣰⠏⣠⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⠋⡜⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠠⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⢶⣦⣤⣄⠀⠀⠀⠀⣰⠏⣰⢃⡜⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠶⠶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢰⠏⡰⢃⡞⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠷⣶⡄⠀⠀⠀⠀⠀⠀⢠⠏⡼⢡⡞⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡼⢡⠏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡼⢡⠏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡜⣠⢏⡞
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⣱⢏⡾
⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡜⣰⢏⡾⠁
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡜⣰⢯⡿⠁
⠀⠀⠀⠀⠀⣾⣷⣶⣶⣾⣿⣿⣶⣶⣶⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡜⣰⢯⡿⠁
⠀⠀⠀⠀⠀⢻⣿⠉⢻⣿⣄⡀⠈⠉⠉⠙⠛⠛⠿⠿⠿⢿⣶⣶⣶⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠎⡜⣱⢯⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣆⠀⣿⣟⠛⠛⠲⠶⠶⠦⣤⣤⣤⣀⣀⣀⣀⠀⠈⠉⠉⠉⠛⠛⠛⠻⠿⠷⢶⣶⣏⡞⣰⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⠛⠛⠲⠶⠦⣤⣤⣀⣀⠉⠉⠉⠛⠛⠛⠿⠷⠶⣶⣶⣶⣤⣤⣤⣼⣿⡟⣰⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⢶⣦⣤⣤⣀⣉⡉⠙⠛⠓⠶⠶⠤⣤⣄⣀⣀⠀⠀⠉⠉⠙⠛⢻⣷⢏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠻⠷⠶⢶⣤⣤⣤⣉⣉⣙⠛⠛⠲⠶⠶⣾⣏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠒⠶⠾⢿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣵⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⣿⢷⣿⣯⣾⣿⣾⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣛⡉⢭⢐⠮⣝⠷⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣛⡩⢤⡐⠮⣧⢲⢮⡑⡣⣋⠺⠜⣑⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠿⠻⢛⣉⠥⡕⡲⢬⣛⠶⣍⡣⢬⣛⠶⢝⣃⡩⠍⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣠⢢⢝⣱⣬⣥⢢⢜⡵⠄⠃⣂⠭⠔⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠸⡜⡪⢻⣿⣿⣧⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠂⠂⠁⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣧⣤⣄⡄⣤⢠⠀⣤⣠⢠⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠉⠉⠉⠉⠉⠉⠋⠉⠙⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠃⠐⠂⠀⠀⠁⠐⠀⠆⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁


⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⢀⡄⠀⢠⡀⠀⠀⠀⠀⠀⣼⣻⠷⠄⢀⢀⣀⡀⠄⠂⣈⣠⠄
⠀⠀⠀⠓⣶⠀⠀⠀⠀⢠⣤⣶⣶⣦⣼⡿⠒⠛⣷⣶⣶⣶⣶⣀⣤⣶⡄⢠⡄⠀⢄⡀⠀⠀⠈⢠⣼⣟⠿⡿⠷⠾⢚⣝⢳⢶⡟⠉⠀⣠⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣆⣴⣿⣿⣿⣿⣁⣽⣿⣿⣷⢀⡇⢀⡜⢆⡐⠀⠀⡄⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣷⣶⣴⣿⣿
⠀⠀⠀⣰⣿⣆⣀⠀⠀⠀⣹⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡏⢹⡏⠉⣿⣿⣿⠀⣾⣅⠈⠀⢆⣨⢡⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⢿⡿⠶⣿⣿⠿⣿⣷⣾⣯⣠⣽⣿⡇⢸⣿⠇⣿⠀⣾⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿
⣻⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠛⢿⣧⣽⣄⢀⡛⡀⢿⣿⣿⣾⣿⣿⣿⣧⣾⣿⣦⣏⢰⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⡛⢻⣿⣧⣀⡄⠚⠻⣿⣶⣤⣙⠞⣞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣷⣿⣿⣷⣾⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣦⣀⣤⣿⣿⣿⣿⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⡍⠿⠛⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⡿⠿⠻⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣷⠻⠇⠘⣿⣿⣿⣿⡿⣿⣿⣿⠷⠶⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣤⣿⡿⠋⠉⠀⣶⠀⢉⣿⡇⢤⢀⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢯⣷⣷⡶⠚⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣯⡀⠢⣘⠛⢁⣴⣿⠟⠉⠀⠈⠙⢿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⠟⣻⣯⣧⣼⣿⣿⣿⠿⠿⣿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣾⡟⠃⠠⠃⠀⠀⣌⠠⣿
⢸⣿⡿⢟⣻⣟⣿⣿⣿⣿⡿⠿⣿⣿⣷⣾⣿⣿⢋⠉⢻⣿⣿⣧⢋⠀⠀⢀⣼⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⠀⣦⣄⠈⢀⣴⣿
⣹⣿⡷⠘⣿⡟⣿⣿⣿⠋⠀⠀⡀⢹⣿⣿⡿⠛⠁⠀⠈⢿⣿⡿⠃⠀⠀⠛⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿
⣿⣟⣛⢧⡘⣧⣿⣿⣿⣧⡀⠀⠀⣠⣿⣿⡄⢀⠀⠀⢀⣼⣿⠁⠀⣤⡾⠷⠆⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣇⢘⣿⡿⠃⠂⠉⠙⣿⣿⣿⣶⣿⣿⣿⡿⠷⢾⣿⣷⣶⣶⣾⣿⣿⡟⠉⠛⠛⠛⠉⢡⣆⣹⣿⣿⣽⣿⣿⣿⢷⣿⣿⣿
⢻⣿⣿⣿⡏⠟⠛⢻⣿⣿⣧⡀⠀⠘⠀⣸⣿⣿⣿⡿⣿⣿⣿⢦⣤⣬⣿⣿⣿⣿⣿⣿⡇⣀⡁⠀⢠⣼⡟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⠿⣿⠟⢠⡿⣿⣿⠟⠉⠻⣷⣶⣿⡟⠛⣿⣿⠁⠘⣿⣿⡿⠷⣶⣦⣍⡙⠻⠿⢿⣿⣿⡏⠀⠈⣽⠇⠀⠀⠛⠉⠉⢱⡍⣿⣿⠉⠿⠿
⠀⠀⠀⠀⠀⠀⠸⠙⠀⠀⠀⠀⠈⠛⠻⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠟⡿⠛⠀⠈⠛⠛⠁⠀⠀⠙⠳⠀⠀⠀⠀⠀⠨⠴⠟⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠶⠶⠶⠲⣶⠶⠶⠶⠲⡶⠶⠲⠖⠲⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠖⠿⠶⠳⠶⠶⠓⠶⠞⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⢀⡀⠀⡀⢄⡀⣀⢀⡀⢀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠆⠠⠄⠠⠄⠄⠰⠄⠄⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠊⠉⠉⠁⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢧⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡴⢦⣠⠏⠀⠀⠀⠀⢹⡈⠄
⠀⠀⠀⢀⠴⠦⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⢾⢷⡈⣿⣴⡁⠀⢻⡀⠀⠀⠀⠈⡉⡇
⠀⠀⠀⠘⡀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⠀⡶⣦⢸⣷⣼⡇⢸⡶⢧⡹⠀⠙⠂⠈⠀⠀⠀⠀⠀⠃⠀⢀
⠀⠀⠀⣠⡇⠘⠀⠀⠀⠀⠀⠀⠀⢀⣤⡔⢻⡏⠀⣿⠰⡅⢸⠀⡧⠙⠟⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠜⡌⠃
⠀⠀⠠⠖⣃⠀⠆⠀⢰⡶⢶⡄⣧⢸⡇⠀⢸⡇⠀⢼⠄⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⢖⣻⠀⠀⠀⠀⣇⢠⡇⣿⠘⠷⠇⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠰
⠀⠀⠀⣩⠼⡀⠐⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃
⠀⠀⠀⠠⢲⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢀⠀⢠⠀⢠⠀⠀⢨⣿⢿⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⢈⡡⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢿⡁⠠⠄⢀⠴⠒⠘⠲⢄⣁⠀⠈⣿⠟⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠆⡀
⠀⠀⠀⠀⠖⣻⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡿⠟⢠⡤⣇⣀⣀⡠⠒⠉⠉⡄⠀⠈⢳⣸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠂
⠀⠀⠀⠀⢉⠴⡆⠈⠀⠀⠀⠀⠀⠀⠀⢾⡿⣿⣶⠆⠀⡆⠀⠀⢹⣇⠀⠀⠀⢧⠀⠰⢿⣏⡥⠋⠀⠀⠀⠀⠀⠀⠀⠈⡸⠈
⠀⠀⠀⠀⠐⣚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣸⢁⡄⠀⠇⠀⠀⠈⣿⠀⠀⠀⠸⡀⣠⡈⣏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆
⠀⠀⠀⠀⠈⠗⣻⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠘⣿⠋⠀⢸⣄⠀⠀⢹⡆⠀⢀⠜⠃⠙⢿⢋⣫⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⢳⠈
⠀⠀⠀⠀⠀⢺⢽⡀⢸⠀⠀⠀⠀⠀⠀⠀⠉⢳⣻⣴⠻⠀⠉⠉⠒⢬⣿⡞⢁⡀⢻⢦⡸⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡄⠁
⠀⠀⠀⠀⠀⠐⣋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠭⢤⣻⣀⣿⠆⠀⣤⣈⠠⣟⣥⢀⣿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⠈
⠀⠀⠀⠀⠀⠀⢕⡧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣛⣉⡹⣦⣤⣭⠿⣆⣿⣥⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠃
⠀⠀⠀⠀⠀⠀⡥⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⢻⣿⢮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⠸
⠀⠀⠀⠀⠀⠀⢚⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⡴⠃
⠀⠀⠀⠀⠀⠀⢘⡩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⡀⢇
⠀⠀⠀⠀⠀⠀⠐⣊⣅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⢿⣿⡿⢿⣹⣿⡙⣧⠘⡄
⠀⠀⠀⠀⠀⠀⢈⡡⣼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⡿⣿⠿⣟⣋⣭⣔⠒⣉⠴⢞⡛⠭⢻⣧⢿⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠖⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⠿⢟⣻⣫⠭⠁⠀⠁⠂⠙⠉⠉⠀⠈⠁⠀⣈⣤⡤⠴⠾⠻⣦⣣⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣭⠴⠀⠀⠀⠀⢀⣠⣤⣶⣾⡿⢿⣿⣏⡿⠭⠑⠲⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠒⠊⠉⠁⠀⠀⣀⣤⣬⠍⠟⢃⡽
⠀⠀⠀⠀⠀⠀⠀⢒⣉⡷⣴⣿⣿⣿⣿⣿⣿⣼⠷⠿⠏⠉⠠⠀⠀⠀⠀⠀⢀⣠⠤⠐⠂⠉⠁⠀⠀⢀⣀⣤⡴⠲⠚⣛⠉⠥⠀⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠤⢺⣸⣿⣿⣿⣿⣟⣭⠝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⣠⣤⠶⠶⠛⣛⡁⠥⠀⣶⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠡⠹⣿⣿⣿⡍⠀⠉⠀⠀⢀⠀⠄⠀⠀⣀⣠⡴⠶⠛⣛⠉⠥⠐⠂⠈⠁⠀⠀⠀⠸⠷⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣈⣿⣟⠻⣤⠐⡂⣈⣠⣤⠶⠖⠛⣋⠩⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⢂⢹⣿⣷⣍⠛⠛⡉⠍⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠂⠿⠟⠛⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⡠⠴⠒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⠲⠦⣄⣠⠴⠖⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠓⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈


⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⡟⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣀⡀⢀⣀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⣀⡤⠶⠚⠋⠉⢙⡒⠨⠭⠭⠭⠭⠿⣿⣷⣶⣶⣾⣿⠿⠭⠭⠭⠭⠅⢒⡊⡉⠙⠓⠶⢤⣄⣿
⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



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

⠀⠀⣯⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣶⢸⣿⣷⡇⠀⠀⠀⠀⠀⣴⣤⣅⢀⣀⣩⣴⡄⠀⠀⠀⢀⡄⠀⣀⡀⠀⢠
⠀⠀⣿⣧⣤⣤⣼⡇⠀⠀⠀⠀⣿⣿⣸⡟⣩⠙⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⠃⠐⢅⣨⡀⠘⡆
⠀⠀⡿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣶⡞⢤⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣯⣤⣤⣄⣠⣭⣤⣷
⠀⠀⠋⠛⠛⠛⠛⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁

⠀⠀⠀⣠⠤⠤⣄⠀⠀⠀⠀⠀⠀⢀⢤⠴⠦⡤⡀
⠀⠀⠈⣴⡀⠀⡖⠑⠀⠀⠀⠀⢠⠓⢃⠚⠂⢸⠚⡄⠀⠀⠀⢸⠀⣀⠀⠀⢤⠈⡇⠀⠀⠀⢰⢃⣦⢀⡀⡲⢸⡀
⠀⠀⢤⣄⠁⠈⣩⣤⠀⠀⠀⠀⠸⡁⡊⢃⡁⢃⢈⠇⠀⠀⠀⢸⠸⠛⠆⠁⠮⠀⡇⠀⠀⠀⡼⠈⠉⠁⠀⠃⠀⣇
⠀⠀⠘⠿⠆⠰⠿⠃⠀⠀⠀⠀⠀⠙⠥⣌⣡⠯⠋⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠛⠛⠛⠓⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⣷⣿⢿⣿⢿⡇⠀⠀⠀⠀⠠⣏⣉⣉⣉⣉⣹⠀⠀⠀⠀⣰⢋⡁⠈⡹⡆⠀⠀⠀⠀⠀⠀⡄⣶⡀⠀⠀⢢
⠀⠀⣿⣿⣾⣿⣽⡇⠀⠀⠀⠀⢾⣉⣉⣉⣉⣙⡋⠀⠀⠀⠀⠹⣌⠁⠀⣠⡏⠀⠀⠀⠀⠀⢸⠁⠀⣠⡄⠧⠈⡇
⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣛⡁⠀⠀⠀⠀⠉⠛⠛⠉⢡⣄⠀⠀⠀⠀⣿⣷⣶⣿⣧⣶⣾⣿
⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⣖⢴⢒⠒⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⣠⠶⠲⠖⠶⠖⠶⡄
⠀⠀⠀⠏⢾⠉⡟⢱⠀⠀⠀⠀⠀⣤⣔⣀⣁⣢⣤⠀⠀⠀⠀⢠⡞⠉⢩⡁⠀⢠⠄⠀⠀⠀⡇⢀⣾⡀⠄⡹⠃⢸
⠀⠀⢸⣸⢸⣃⣇⡸⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠸⣇⠀⠋⠃⣩⠚⠀⠀⠀⠀⠳⣈⢀⣁⣀⣉⣡⠞
⠀⠀⠋⠇⠐⠿⠾⠗⠀⠀⠀⠀⠀⠿⠿⢿⡿⠿⠿⠀⠀⠀⠀⠀⠙⠓⠶⠚⠉⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠛⠋⠉⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⢠⡟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⣼⡇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡆⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⢿⣷⠏⢿⣇⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⡄⠀⠀⢸⡟⣷⡄⠀⠀⠰⠚⠛⠃⠀⣿⣷⠿⠟⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⣿⡄⠀⢹⣧⠀⠀⢸⡿⠛⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⣿⡆⠀⠘⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣸⣿⣇⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⣛⣩⣥⠶⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠘⣿⣤⣤⣶⠾⠟⣛⣉⣥⡴⠶⢟⣛⣩⣥⠶⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⣰⣿⣩⡤⠶⢞⣛⣫⣭⠶⠶⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣭⠶⠶⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄
⡏⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀
⣿⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⣿⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣀⣀⡀⣀⡀⣀⣀⣹⣄⣀⣀⣀⡀⢀⣀⡀⢀⣈⢳⣄⡀⣀⡀⢀⣀⣀⣀⣀⢀⣀⠙⢦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠙⢿⣝⣾⣿⣿⣿⣍⢉⣻⣿⣍⠛⢾⡿⣍⡳⣿⣿⣿⣿⣿⣿⣶⣝⣿⣿⣮⡛⠿⣆⠀⠙⢦⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠈⠛⠋⠈⠙⠈⠉⠛⠁⠙⢿⣀⠙⠀⠉⠛⠁⠙⠃⠉⠛⢯⠉⠙⠙⠃⠙⠃⠈⠃⠀⠀⠙⠶⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦
⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚
⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⢿⠿⠿⣿⢻⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣟⣻⣿⣛⣛⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠶⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠁⠀⣀⣀⣠⣤⣄⣀⡀⠉⠛⠷⣤⣀⣀⣠⣤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠶⠶⢦⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡏⣿⠀⠛⢉⣉⣀⣀⣈⡉⠛⠳⢶⣄⡀⢹⡏⠁⠀⠀⢠⣤⣤⠶⠶⢤⣤⡄⠀⠀⠀⢸⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠛⠛⠉⠉⠉⠉⠙⠛⠶⣤⣈⠁⢸⡇⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡇⣿⠀⠶⠞⠛⠛⠛⠛⠛⠶⣦⣄⡉⠁⢸⡇⠀⠶⠟⠛⠋⠉⣉⣉⣉⡉⠉⠛⠛⠀⢸⡇⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠴⠶⠟⠛⠛⠛⠷⢶⣤⣈⠛⠂⢸⡇⠀⠴⠶⠛⠛⠛⠋⠉⠉⠛⠛⠛⠓⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⣤⣴⠶⠶⠶⠶⢦⣤⣈⡉⠻⠆⢸⡇⠀⣤⠶⠶⠛⠛⠛⠛⠛⠛⠛⠳⠶⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⣀⣤⣤⡶⠶⢶⣤⣄⣉⠛⠷⠄⢸⡇⠀⣠⣤⠶⠶⠾⠛⠛⠛⠛⠶⠶⠦⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⢉⣀⣤⣤⣤⣤⣀⡉⠙⠻⢦⡄⢸⡇⠀⣀⣤⣤⠶⠶⠶⠶⠶⠶⠶⢶⣤⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠉⣉⣀⣀⣀⣀⡉⠙⠻⢶⣤⡀⢸⡇⠀⣉⣠⣤⣤⡴⠶⠶⠶⠶⣤⣤⣤⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠛⠋⠉⠉⠉⠉⠙⠛⠶⢦⣌⡁⢸⡇⠀⢉⣁⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⠀⠲⠛⠛⠛⠛⠛⠻⠶⢦⣤⡉⠁⢸⡇⠀⠛⠉⢁⣀⣀⣀⣀⣀⣀⣀⣀⠉⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⣿⣀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠈⠙⠃⢸⡇⠀⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠋⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣧⣭⣭⡥⠴⠶⠶⠶⠦⢭⣭⣿⣷⣦⣤⣸⣇⣀⣠⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠦⣤⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠿⠶⠶⠶⠶⠖⠒⠒⠚⠛⠛⠛⠛⠛⠻⠿⠿⠟⠛⠛⠛⠛⠛⠛⠒⠒⠶⠶⠶⠶⠶⠾⠷⠟





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⡉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁⠀⠂⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠾⠿⠿⠟⠉⠩⠏⠁⠀⠀⠐⡟⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡶⢶⣶⣶⠶⢲⣶⢢⡖⠒⠒⠒⠒⠒⢶⠶⡶⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣥⣿⣤⣴⣦⣼⣄⢠⡷⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠈⠛⠛⠿⠿⢿⣿⣿⣿⢿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⢀⣠⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⠟⠛⠉⠉⠉⠉⡥⠖⠒⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠄⠠⣀⣀⣀⣀⡀⠀⢀⡤⠖⢒⠉⠀⠀⠀⠀⠀⣼⣷⡤
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠀⡏⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠑⠒⠒⠢⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠢⠤⠄⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠋⠉⠉⠀⠀⠀⠀⠉⠉⠛⠛⠶⣤⣀
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠛⠛⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣰⣿⣄⣀⣀
⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠈⠉⠉⠛⠷⢦⣄⡀
⠀⠀⠀⠀⢀⡾⠁⠀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⡴⠾⠛⠁⠀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⢀⣿⣥⡶⠛⠉⠀⠀⠈⠉⠉⣿⠛⠉⠉⠉⠉⠀⠘⠛⢛⡟⠛⠛⠛⢛⣿⠛⠛⠂⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠉⠛⢶⣤⣀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣄⡀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠛⢶⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠄⠀⠐⠒⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠶⠶⠶⠞⠻⠛⠛⠛⡏
⠀⠀⠀⠀⠀⠈⠓⠲⣤⣀⠀⠀⠀⠀⠀⠀⠘⠲⣶⣿⠿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠙⠲⠦⣄⡀⠀⠀⠘⣿⠄⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⣶⣶⣒⣺⣿⣯⣽⠇
⠀⠀⠀⠀⠀⠈⠁⠢⠑⠢⢄⡀⠀⠀⠉⠙⠲⠤⣿⣿⣶⣶⣖⣺⣭⡭⠷⠶⠚⠛⠋⠉⠉⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⣑⠢⢄⡀⠀⠀⠀⠈⣿⡯⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⡴⣖⣿⠂
⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠈⠑⠢⣌⠑⠢⢀⠀⣿⣄⠀⢀⣀⣀⣤⣴⣶⣺⣿⣽⠷⠾⠛⠛⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠉⠒⢤⠈⣙⣻⣭⡽⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠐⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⢠⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⠀⢸⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢸⣷⣆⣀⣠⣤⡤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡿⢛⣻⣿⣶⣶⣤⣄⡀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⢛⣩⣄⢰⣟⣿⠍⠉⠉⠉⠉⠻⢦⣴⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠺⣿⡿⠛⠉⠉⠀⢀⣀⡀⠀⡀⠀⠙⣧⡈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢠⡟⣻⡶⣞⢷⣷⣤⣼⠿⢿⡞⢿⡅⠀⠈⢷⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣼⠟⠛⠋⠙⢿⣧⣤⣽⣿⡷⠀⠀⠀⠀⠀⠘⣧⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠰⢿⣶⡴⣄⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⣴⠟⠀⢻⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠈⢷⣄⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠛⢻⡿⠀⠀⢸⡏⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣆⠀⠀⠈⠉⠀⠈⣿⡀⠀⠀⠀⠀⢠⡶⠋⠀⠀⢀⡿⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢀⣿⣰⡆⠀⣰⠟⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠻⢦⣄⡀⠀⠀⠀⢿⣄⠀⣠⡟⢹⣋⣴⠞⢉⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠙⢶⣄⡉⠛⠳⠶⢦⣤⣿⣿⠿⠶⠛⢋⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠙⠻⠶⠦⣤⣤⣤⣤⡴⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣄




⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣀⣠⣤⠴⠶⠶⠶⠒⠲⠶⢶⡶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢿⢻⣤⡀⠂⠉⠀⠀⠀⡀⠐⠂⠄⠀⠀⠍⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢛⠁⠅⣿⣷⠚⠃⢀⡀⢰⣟⡷⣀⡀⠀⠀⠀⢀⣤⣌⣡⣈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠅⠉⠀⠀⠀⢸⠟⣷⠦⣼⢳⣞⠙⢚⣉⡽⠂⢀⣁⠈⠉⠙⡆⠙⠶⣤⡙⢦
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⣀⢠⠅⠀⢉⠸⣾⠽⣶⠲⠶⠶⠚⡉⠀⠀⠀⢄⠙⠻⢿⣾⣡⣤⠢⠀⠙⠛⢳⡀⠀⠀⠀⠐⣧
⠀⠀⠀⠀⠀⠀⣼⠁⠜⠀⠁⠀⠀⠀⠀⠀⣼⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠑⢼⠀⠀⠀⠈⢀⣹⡤⠖⠋⠀⠈
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⠀⠀⠀⢣⠇⠀⠀⠘⣆⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠴⠞⠉⠀⢳
⠀⠀⠀⠀⢠⡇⠀⡄⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠄⠀⠀⢀⣠⡴⠖⢻⡉⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣀⠽⡞⠓⢦⣀⡤⡶⠚⠉⠀⠐⠀⠀⢧⠀⣀⠀⡀⠀⠂⣇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠈⠉⠛⠋⢁⣠⠽⡖⠋⠛⢦⡸⡀⠀⠀⠀⢀⠀⢸⠸⡏⡇⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⢀⣀⡤⠶⠛⠉⠀⠀⢻⡄⠀⠀⠹⣧⠀⠀⠀⠈⢀⡏⠈⠷⠃⠀⠀⡄⡏
⠀⠀⠀⠀⠸⡶⠒⠛⠙⣆⢀⣀⣠⠴⠚⠉⠁⠀⠀⠂⠀⠀⠀⠀⢳⡄⠀⠀⠻⣄⡀⠀⠀⡟⠀⠀⠄⠀⠂⠀⢰⠇
⠀⠀⠀⠀⠀⢷⣀⣀⡵⠞⠛⠷⠶⠲⣄⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⢳⡀⠀⠀⠈⢳⡀⢰⠏⠀⠀⠀⠀⠀⢀⡟
⠀⣠⡀⣤⠴⠚⣯⠁⠀⠀⠀⠀⠡⡀⠸⣄⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠗⠋⢘⠀⢀⣀⠄⢀⡿⠁
⠙⣿⠷⠀⠀⠀⠈⢧⡀⠀⠀⣀⠀⠘⠄⢹⠀⠈⠀⠀⠀⠀⠁⠄⠀⠀⠈⢧⠀⠀⠈⡀⠀⠔⠀⠈⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠳⣄⠠⠀⠀⢹⡄⠀⠀⠑⠀⠀⠀⠈⠀⠀⠀⠈⢷⠀⠀⠁⠀⡀⡰⢒⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠳⣈⠠⠄⢻⡅⠀⠀⠀⠀⣀⠀⠀⠃⠀⠀⢈⡧⠤⠶⠾⢶⡵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣟⡒⢮⣳⣤⣄⠉⠀⠀⢀⣀⡤⢀⡽⠋⢀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⢧⣤⣬⣥⣬⠤⠾⠗⠚⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠉⠀⠒⠤⣀
⠀⠀⠀⠀⣠⠖⠒⠁⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠈⠁⡀⠀⠀⢀⡤⠤⠤⣄⣀⠤⠒⠀⣀
⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡔⠁⠀⠀⠀⠀⡇⠀⠀⠀⢈
⠀⠀⠀⠸⠀⠉⠗⠤⢠⣀⣀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⡸⠀⣧⠀⠀⠀⠀⠀⡘⠦⣄⢠⡎
⠀⠀⠀⠀⢀⢼⣀⠔⠁⠀⠴⢠⠃⠀⠀⠀⠀⢀⠤⠂⠈⠁⠀⠀⠑⢄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢰⠁⠀⢇⠙⠚⠓⠤⣀⡷⡀⠀⡟⠉
⠀⠀⠀⠘⡐⢼⠁⠢⡀⠀⠉⢸⠀⠀⠀⠀⡰⠁⠀⠠⠤⠤⠤⠤⠤⠄⠝⠶⠠⠤⢠⡀⠠⠤⠒⠢⣀⣀⡣⠀⠸⠈⠀⠀⠀⠈⢱⠆⡰
⠀⠀⠀⠀⠈⣦⡀⠀⠈⠻⣹⠁⠀⠀⠀⠀⡁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡸⠀⠀⢀⢾⠊⠁
⠀⠀⠀⠀⢠⠋⠛⠲⠦⡴⠊⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠖⠋⠁⣸⣀
⠀⠀⢀⠔⠁⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠒⠉⠀⠈⠳⡀
⠀⠔⡕⠒⠢⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠦⠤⠤⠒⠒⠒⠀⠄⠀⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠠⢀⠀⠸⡀
⠘⠘⠀⠀⠀⠘⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⡠⠁⠀⠀⠃⠀⢃
⠀⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠙⠞⠀⠀⢟⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢠⠁⠀⠀⠀⠀⠀⢸
⢠⢃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⢀⡠⢲⣗⠥⠂⠀⠀⠀⠀⢱⠴⣗⠠⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸
⢸⠸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⣠⠖⠁⠐⢋⡴⠃⠀⠀⠀⠀⠀⠀⠑⢌⡁⠀⠑⢄⠀⠀⠀⠀⠸⠀⠰⠀⠀⠀⠀⠀⠆⠀⢸
⠸⠀⢇⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⣀⠠⠔⠊⢁⡀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢄⡉⠀⠐⠠⢠⣄⠆⠀⠀⠀⠀⡌⠀⠀⢸
⠈⠀⠈⡄⠀⠀⠀⠀⢀⠧⠒⠉⠁⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠈⠢⠀⠀⠀⡰
⠀⠀⠀⠘⡄⠀⠀⠀⠎⠀⠀⢀⣠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠢⠄⣄⠀⡐⠁
⡄⠀⠀⠀⠘⢄⣀⣼⠠⠔⠊⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⠤⣀⣀⠤⠒⠒⠲⠤⢤
⠸⡆⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡄
⠀⢰⠁⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢡
⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⠛⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢸⡃⠈⡷⠴⡏⠀⢹⠀⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⠀⠙⠛⠁⠀⠉⠛⠉⢀⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⡀⠀⠀⠀⠀⠀⢀⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠈⡟⠒⠶⠶⠶⠚⡏⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⣇⠀⠀⠀⠀⢀⡇⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⣻⠇⠀⢻⠉⣻⠿⡏⢹⠃⠀⢿⡳⠤⣄⣀⡤⠖⠚⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢴⡃⠀⠀⠈⣇⢳⣶⢋⡏⠀⠀⠀⣹⠄⣰⠋⠳⡄⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠙⢦⣀⠀⢹⡞⠘⣿⠀⠀⣠⠞⠁⢠⡇⣾⠁⢿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢷⢰⢇⡴⠋⠁⠀⠀⠘⡇⠉⠀⢸⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⠀⠀⠀⠀⠀⠀⠹⣄⠀⠈⠇⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣤⠴⠞⠙⡶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⠢⢭⣙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣈⠒⠦⣌⡉⠒⠦⣍⡛⠿⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡐⠤⢍⣒⠦⢌⣙⠲⠤⣀⡀⠀⠈⠙⠻⣦⣄⠠⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠒⢬⣝⡲⠬⣉⡒⠤⢌⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⣙⣳⣦⣍⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣛⢻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⠉⠉⠉⠉⠛⠷⢶⣤⣄⣀⠀⠀⢀⣀⣼⠇⢀⣀⣠⣤⣶⠶⠞⠛⠉⠉⠉⠉⠉⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⢶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡟⢧⡛⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠾⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠸⠀⠈⢷⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⠀⠀⢀⣀⡈⣻⣿⣤⣴⣶⣶⣶⠶⠶⠶⠾⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣀⣠⣴⠶⠞⠛⠋⢹⣿⣴⠾⠟⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⠶⠶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣆
⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⣛⣒⣒⣒⣒⣒⡒⠲⠶⠶⠤⠤⠤⠭⠭⠭⠭⠉⠉⠉⠉⢉⣿⡉⠉
⢠⡿⠁⠀⠀⠤⣄⡀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣂⣀⣀⡀⠀⠀⠀⠀⠈⠭⠭⠭⣭⣭⣭⣭⣅⣒⠒⠀⠀⠀⠀⣾⠀⠉⠓⠦⣄⡀
⣿⠁⠀⠀⠀⠀⠀⠈⠉⠂⠀⠘⣷⡀⠀⠀⠀⠐⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⢭⣍⣉⣉⣉⣉⣒⡒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠙⠳⣦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣐⡒⠲⠶⠶⠶⠶⣤⣤⣤⣤⣄⣀⣀⣀⣒⡒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢷⣀⣀⣤⣤⡴⠶⣶⠟⠛
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠷⠶⠶⠶⠾⠿⠿⠿⠿⣿⣿⣽⣿⣭⣭⣤⣀⣀⣀⣀⣀⣀⣐⣒⣀⣀⣀⣙⣏⠁⠀⠀⢰⠇
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠠⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠐⠂⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡴⠶⠶⠶⠚⣛⣋⣉⣥⣤⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣌⣛⣬⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡴⠶⠶⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠀⠠⠤⢀⠀⠀⠀⢀⠤⠒⠀⠈⠉⠉⠁⠐⠢⢄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⡀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠐⠀⠀⠐⠆⠁⠉⢂
⠀⠀⠀⠀⠀⠈⡆⠠⡄⠄⠤⠞⠀⡄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⣤⠀⠀⠀⠀⡦⢀⢠⡎
⠀⠀⠀⠀⠀⠐⠇⢰⠀⠈⠄⡆⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢀⠔⠉⠑⠄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠈⡀⠀⠑⠚⠂⢄⡧⠀⡛⠁
⠀⠀⠀⠀⠈⢬⠂⠀⠐⢶⢲⠂⠀⠎⠄⠠⠒⠂⠤⠤⠤⠄⠐⠊⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈⠀⠀⠉⠠⡀⡄⠀⠀⣟⠴⠁
⠀⠀⠀⠀⠀⠀⡑⠠⠄⢠⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢥⠄⠊⣃
⠀⠀⠀⠀⣠⣊⡉⠂⠴⠊⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠊⠀⠑⡄
⠀⠀⠀⡼⠁⠀⠈⡄⠀⠀⠀⠘⡡⡀⠀⠀⠀⠀⢀⡴⣟⣦⠞⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠑⠘⡄
⠀⠀⡜⠀⠀⠀⠀⠃⠀⠀⠀⠀⠇⠈⢆⠀⡠⠂⠁⠈⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠆⠀⠀⠀⡇⡆
⠀⡘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⢯⠂⢀⠀⠀⠀⠀⢰⠀⠐⠀⠀⠀⠀⠁
⠀⠂⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢀⠑⠀⠀⡀⠘⠀⠁⠀⠀⠀⡐
⢠⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⡀⠀⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠐⡄⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠤⢰⡀⡠⠁⠀⠀⠈
⣆⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣰⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⢠⡾⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⢷⡄
⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣠⣶⠿⠛⠛⠉⠙⠛⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⠀⢠⣾⠟⠁⠀⣠⣤⣤⣤⣤⡀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⠀⣿⡏⠀⠀⣾⡿⠋⠉⠉⢻⣿⡆⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⠈⣿⡆⠀⠀⠀⠀⠰⠿⠿⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⠀⠹⣿⣄⠀⠀⠀⠰⣿⡷⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⣿⣤⣤⣬⣿⣷⣤⣀⣀⣀⣀⣀⣤⣶⡿⣿⣦⣄⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⢸⡇
⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⡀⠉⠛⠛⠛⠛⠛⢯⣹⣇⡿⢿⣿⣿⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⣤⡀⠀⠀⢻⡟⠀⠀⢙⣿⣿⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⠤⠤⠤⠤⠴⠟⠁⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⠶⠶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⣿⡇⠀⠀⠀⢀⣴⣶⣶⣶⣶⣶⣶⣾⡧⠀⠀⠀⢸⣿⣿⣉⣉⣉⣀⣉⡉⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⣾⡎⢿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠿⠿⠿⠿⣿⣿⡿⢃⠀⠀⠀⢸⣿⣿⣿⡙⣿⣿⣿⢃⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠘⠛⠛⠘⣿⣿⣿⡇⠀⠀⠀⡟⣴⣿⣿⣿⣷⣶⣶⣶⡜⠃⠀⠀⢸⣿⣿⣿⣷⡙⣿⢃⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣰⣿⣿⣿⣷⠹⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡿⢋⣤⡛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⣀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣴⣿⣿⣿⣤⣾⣿⣿⣿⣷⣬⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣦⣝⣛⣡⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠚⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⣀⣄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⢀⣠⡶⠛⠉⣙⣼⡦⠔⠂⠀⠉⠀⠀⠠⢀⡘⢋⡉⠬⠀⠒⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⡴⠛⠁⣠⡴⠛⠡⠞⠁⠀⠀⠂⠀⠀⠀⡆⣀⢡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⡾⠉⢀⡴⠛⠋⣁⠀⠁⠀⠀⣀⠄⠤⢀⢠⠠⠱⠛⢻⠗⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠉⢢⡄
⠀⠀⠀⠀⢠⡞⠁⠀⢀⡴⠋⢀⠔⠁⠀⠀⠈⠁⠀⣀⡤⠊⢡⣂⣸⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠷⠀⠀⠀⠀⠙⣃
⠀⠀⠀⢠⠏⠀⠀⢠⠎⢀⡴⠃⠀⠀⠀⠀⣠⠆⢰⣣⡶⢁⡨⠜⠋⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠀⠀⠀⠀⠀⠫⡆⠰⡀
⠀⠀⢠⠏⠀⠀⣠⠋⢠⠞⠀⠀⠀⠀⠘⣷⢃⠀⠿⠩⠐⠉⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠒⡜⢄
⠀⢠⡏⠀⠀⢰⠃⢠⠏⠀⠀⠀⠀⠀⡬⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⠤⡀⢀⡠⠀⠀⠈⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠐⠈⠆
⠀⡼⠀⠀⢠⠏⢠⡏⡀⠀⠀⢠⡶⠖⣣⢦⢿⣠⣤⡀⠀⠀⡠⠲⡿⠾⣅⠀⠐⡁⠲⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠘⡄
⢠⠇⠀⠀⡾⠀⣼⠀⠀⠀⠀⣷⡵⠋⠁⠀⠀⢻⣧⢽⣦⣤⠷⠔⠒⠄⠖⠁⠀⢸⡂⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⡄⢱
⣸⠀⠀⠀⡇⢠⡇⠀⠀⠀⡠⠗⠓⠒⠈⣳⠀⠀⠃⠘⠏⠘⢦⡶⠶⡖⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠹⠁⠈
⡏⠀⠀⢠⠀⢸⠁⠀⢀⠼⠆⠀⠀⠀⠀⠉⠢⡀⢠⡤⣄⠀⠀⢀⣀⠇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠳⣤⠀⢸
⡇⠀⠀⢸⠀⢸⠀⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠈⠠⣤⣆⠀⠀⠀⠀⠩⡈⠙⠶⢳⠤⠒⠋⠉⠣⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⢾⠙⠰
⡇⠀⠀⢸⠀⢸⠀⣿⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢶⣀⠀⠀⠀⠘⠉⠚⠀⢁⡆⠀⠀⠀⠀⠙⢦⡦⠐⡇⠀⠀⠀⠀⠐⢠⢸
⢳⠀⠀⠀⡆⠸⡇⣼⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠻⣦⡀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠹⣀⠀⡀⠀⠀⠀⢐⣠⡀
⢸⡄⠀⠀⣣⠀⣧⠘⣿⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⢿⣆⡀⣀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⠈⢽
⠀⢧⠀⠀⠸⡄⠸⡆⠘⢿⢇⣐⠀⡀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⠷⢙⣉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠸⡆⠀⠀⠱⡀⢻⡄⠈⠻⠞⠻⣭⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠹⡄⠀⠀⢳⡀⠳⡄⠀⣀⠀⠀⢉⡗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠹⡄⠀⠀⠱⡄⠹⣆⠛⣤⠀⠀⠻⣿⡄⠀⠀⠀⠀⠀⠀⠀⡤⡌⠀⠀⣀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠹⣆⠀⠀⠘⢦⠈⢳⣌⠳⣄⠀⠘⢧⣷⣀⠀⠀⠀⠀⣤⡙⠁⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠑⢄⠙⠳⣌⠃⠀⢈⣿⣩⠃⠀⠀⠀⠀⠃⠰⣄⠈⢣⡀⠀⠀⠶⡀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠑⢦⡈⠻⢦⣄⡙⠷⣤⠀⡂⠀⠀⠀⠀⠉⣠⠔⠁⠀⡄⠀⠀⠀⠀⠀⠀⠊⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠓⠤⣈⠛⠶⢮⣿⣧⣄⡀⣀⡀⢰⡇⠠⠄⠈⠟⠀⠀⠀⠀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠉⠒⠠⠌⣉⡙⠛⠻⠿⠿⠤⠤⠤⠤⠤⠒⠒⠚⣉⡡⠼⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠒⠒⠒⠒⠈⠉⠁⠀⠀⠀⣀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠤⠤⠠⣿⡇⠀⠀⠸⠄⠠⠤⠤⠄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠉⠁⠀⠀⠀⠠⠤⢤⣴⡿⡇⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠂⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣈⠉⠑⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⠒⠉⢁⣀⡤⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⣀⣀⢀⠀⡀⠀⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠰⡄⠇⡰⢄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⡿⠁⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⠊⠀⠀⠀⠀⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠠⡀⡆⢠⠄⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠈⠀⠘⡶⠁⠀⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠇⡠⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢠⡀⡇⢠⡀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠈⢰⠂⠐⠂⠁⠬⠐⠀⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⠂⠀⠱⣶⠃⠈⠂⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠂⠀⠀⡨⠠⠞⠀⠆⠇⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠁⠦⠀⠅⠀⠔⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠈⡌⡔⠀⡤⠤⠤⢤⣀⠀⠀⠈⠀⢳⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣃⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡫⠭⠭⣙⣒⠀⠸⣿⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡪⠭⢝⣒⣂⠀⠰⠿⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣊⣩⣍⣙⣓⣒⠦⠥⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣒⣛⣛⣒⣒⣒⠭⠭⣓⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣒⣒⣒⣒⣒⡦⠭⢭⣒⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠈⠉⠙⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣁⣉⣀⣉⣉⣉⣉⣓⣒⣤⣼⣤⣤⣐⣀⣀⣀⣀⣀⣀⣀⣈⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠠⢷⡄⠀⠀⢠⣴⡿⠋⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠉⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⣿⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⢷⣾⣿⣿⣶⣄⡀⠀⠈⣿⡇
⠀⠀⠀⠸⣧⠄⠀⠀⠉⣼⡛⣿⣶⣿⣿⣿⣿⣿⣿⡿⠟⢛⣙⡛⠛⠛⠛⠛⢛⣿⣿⣷⣾⣿⣧⣤
⠀⠀⣷⣀⣀⡀⠀⠀⠐⣽⣧⠘⣿⣿⣿⣿⣿⣿⣏⣤⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣽⣿⣿⣿⣿⣿⣧
⠀⠘⠟⢿⠟⢁⡄⠀⣼⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠠⣤⣾⠿⠲⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⡛⠛⠀⠈⠁⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠉⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠘⠲⠰⠋⠈⢿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠈⠁⠐⠃⢠⣆⣉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡯
⠀⠀⠀⠶⠆⠀⠀⠙⠁⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣀⠈⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣤⣤⣬⣭⣽⣾⠆⣿⣿⣿⣿⣿⣦⣶⣤⡀
⠀⠀⠀⠀⠀⢀⣾⡷⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠛⠛⠿⣿⢯⢸⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠖⣶
⠀⠀⠀⡴⠚⠉⠉⣡⣶⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣤⣄⣼⣧⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠾⠃⠀⢾⡋⣼⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⢟⣿⣿⣿⣿⣿⣷⣾⣿⣷⢋⣹⡏⠆
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠛⠠⣬⣾⣿⣹⣿⣿⠇⣼⣿⣿⣿⣷⣶⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢹⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠉⠁⠂⠀⣿⣿⣿⣿⣿⡟⣿⡾⢿⡜⢿⣿⣿⣿⣟⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⣿⣿⣿⡏⡡⠉⠙⠊⣥⠩⣿⠙⠛⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠛⠛⡿⠀⠀⠀⠀⠀⢠⠁⣿⠀⠀⠐⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣾⣷⣄⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣠⣄⠀⠈⢻⣿⣧⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⢹⣿⣧⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⢿⣿⡆⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⣿⡇⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢸⣿⡇⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠀⠀⣿⣿⠃⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁⠀⣼⣿⡏⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣠⣾⣿⠏⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⡿⠋⢄⡈⠑⠤⣀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⡫⢈⠔⠁⠀⠀⠀⠀⠉⠒⠄⡀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡫⠊⠋⠀⠤⡁⠀⢰⠀⠀⠀⠀⡀⠀⠀⢈⠑⠢⣄⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⣠⡶⠶⣬⡑⠢⢄⠀⠠⠀⠀⠀⠄⠀⠈⡀⠀⠁⠀⢄⡀⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠋⠀⡒⣾⠛⠀⠀⠀⠍⠓⢦⣉⠢⢄⠐⠂⠀⠀⠊⠀⢀⠔⠠⠀⠈⠒⠤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⣠⡾⠋⠀⠀⠈⠑⢢⠀⠐⠤⠤⠌⠛⠦⣍⠀⢀⡐⠠⡴⡡⠊⠤⠀⠀⠐⠀⠈⣁⠀⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠋⠉⢀⣾⠏⠀⠀⠀⠰⣶⣤⠀⠀⠀⠠⠈⠁⠀⠀⠈⠑⠦⣉⠒⢄⡀⠀⠀⠀⠁⠠⢀⠞⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⣪⠆⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⠀⠀⠀⠈⠓⠦⡈⠑⠤⡀⢂⠔⢁⣼⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⠞⠁⣰⠟⡅⡆⠀⠠⠂⢀⡆⠐⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢈⣴⡿⣫⢞⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⣠⠾⣱⠐⠿⢀⠸⣄⠒⠺⣵⣮⠀⠾⠂⡨⠀⢄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣴⡿⢋⢜⣵⣿⠿⣷⡿⠂
⠀⠀⠀⠀⠀⢀⠞⣹⠞⠁⣠⠞⠁⠀⠛⠃⢠⣦⠠⡌⠁⡠⢄⢀⠅⣠⠘⡀⡠⠶⢆⣀⡘⠀⢠⠀⣀⢀⣤⠞⢁⣴⡯⢋⢔⣵⣿⣯⣜⣷⣾⠁
⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠈⠻⢦⣀⠠⢶⡂⠷⠖⢘⡏⢁⡤⠠⢤⢠⡶⡎⠀⠀⠘⠁⠈⠁⢻⢔⣵⠟⢁⡴⡯⢋⢔⣵⣿⡻⢝⢷⣽⡿⣻⡀
⠀⠀⠀⠀⠀⠈⣿⢨⡙⠳⣦⣄⠢⢝⠳⣤⡀⠘⠉⢻⡁⢞⣻⠁⠄⢿⡉⣳⠈⠑⠠⡄⢌⡴⣵⠟⢁⣴⠏⢀⢔⣵⣿⠻⢷⣌⣳⣿⠏⢈⣼⣿⠆
⠀⠀⠀⠀⠀⠀⣿⡀⢌⡑⢤⡙⠻⣦⣍⠪⢕⠒⠘⠟⠙⣼⡧⢂⠄⡼⢄⠢⠁⠐⠈⠀⣡⠞⢁⣴⢟⡵⠀⣰⣿⠿⣗⢍⣢⣽⣿⢇⢔⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣶⣌⡓⢌⡒⢄⡙⠳⣦⣉⠲⢄⡐⠋⡣⠉⠋⠃⢌⡗⠀⠀⢠⡞⠁⣰⣿⡿⠋⣠⣾⣿⣭⣿⢾⣿⣿⠟⠕⣱⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣌⡓⢬⡂⢌⠉⠙⠲⣍⡢⠀⠺⢦⣀⠀⠀⠀⡴⠋⣠⡾⠿⠉⣰⡾⠟⡧⣊⢝⣷⣿⡟⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣥⣬⣙⠻⣿⣿⣿⣿⣦⠈⠃⠉⣀⠈⠒⠝⣢⣌⠓⢍⡳⠆⠈⣠⡾⣫⠄⣠⣾⣟⡒⢕⡪⣠⣾⡻⢋⢤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⢸⣏⣉⠻⢶⣽⣢⣭⣻⢿⣦⣄⡈⡑⠢⣐⠬⡙⠻⣦⣌⣠⡾⣫⠞⣡⣾⠟⢓⠬⡻⣵⣾⢿⢟⢔⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠸⣏⠠⣉⠢⢌⡙⠿⣯⣻⢮⣝⠻⣿⣗⠤⡉⠢⢕⠀⡽⢋⠜⣡⣾⡟⢁⠪⣑⢉⣾⠟⠕⡱⣫⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣍⠢⢍⡢⢌⠙⠳⣮⣟⡒⠿⢿⣮⣕⠢⠀⡷⣡⣾⠟⡁⢌⡒⠙⣢⣿⠋⠀⣪⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣍⠒⢭⡒⢄⡁⠉⠻⢷⣮⢝⠿⣶⣾⠟⢛⣶⣮⣕⣭⣾⡿⠋⢜⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⡈⠑⠉⢀⠀⠑⠪⢿⣿⣦⢤⡲⣿⣿⣿⣿⣿⠏⡰⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⢉⠒⢤⡒⢬⡙⠿⣯⣄⣠⣿⠟⡡⣪⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⡢⣌⡑⠪⣕⠄⣽⠿⠃⢔⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣭⡓⠤⡁⡇⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⢀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡄⢠⡍⡄⠀⠀⠄⠀⢀⠀⠀⠀⠀⡀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡄⢀⠀⠀⠀⠣⡔⠈⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠈⠉⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠐⣤⣁⣀⠔⠄⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣄⡄⠀⣠⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠀⠸⠙⠞⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠟⠙⠻⠸⡊⠠⡀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠜⣔⠀⡀⠀⣰⣥⣠⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠈⠀⠂⠀⠀⠀⠡⠐⠒⢫⡆⠀⠀⠀⠀⠀⠀⠀⠉⠱⠶⠚⢉⡉⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠠⠀⢴⠘⢇⠀⠀⠀⠤⠀⠴⠾⠷⠄⠀⠠⠾⠶⠱⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠠⢀⠣⣀⠘⠀⠠⠀⠄⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⢉⣠⡆⠀⠀⠀⠀⠀⠁⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣃⣈⣀⣀⣀⣓⣤⡀⠇⠀⠀⠀⠀⠀⠈⠡⠤⠤⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠄⠀⠀⠃⠠⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠆⠀⠤⠸⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢃⠀⠤⠀⠀⠀⠘⠀⣀⠐⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠊⡀⠠⡀⠀⡠⢘⠇⠀⠄⠂⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣢⡀⠈⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⢠⠀⠀⠶⠥⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⣀⠁⠂⠍⢈⡉⡐⢄⠀⠁
⠀⠀⠀⠀⠀⢀⠐⢎⣪⢗⡋⢉⡱⠂⢄⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠉⠉⠈⠉⠈⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⣼⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠂⠀⠀⠀⠀⠸⢿⣿⣿⣦⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠦⣤⣀⣀⣤⠤⠦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡏⠉⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠈⠉⢹⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣀⣀⣤⣤⣀⢸⣇⣀⣤⣤⣀⣀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠛⠉⠁⠀⠀⠉⠛⠛⠉⠀⠀⠈⠉⠛⠛⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⠉⠉⠉⠉⠉⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇
⠀⠸⣿⡿⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠀⠀⠀⠀⠀⢉⡉⠀⠙⠻⠃⠀⠀⠀⠀⢸⣿⡇
⠀⢰⣿⣷⠀⠀⠀⠀⠀⣿⣿⠀⡿⠋⢀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⢀⣼⣿⡿⠋⠉⠀⣤⣤⠀⠉⠙⢿⣿⣦⡀⠀⢸⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⡏⢠⣿⣿⠋⠀⠀⠀⠀⣿⣿⠀⣿⣷⡄⠙⣿⣿⡄⢸⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⠀⣾⣿⠇⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⢸⣿⣧⠀⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⢨⣿⣿⠀⣿⡇
⠀⢸⣿⣿⢀⣀⣀⣀⣀⣿⡀⢻⣿⣇⠀⠀⢀⣀⣀⣿⣿⠀⣿⣿⡇⠀⣸⣿⡏⢀⣿⡇
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⢿⣿⣦⡈⠻⢿⣿⣿⣿⠀⠿⠟⢁⣼⣿⣿⣀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣀⣀⣀⣀⣤⣶⣿⡿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠟⠛⠉⠀⠈⠛⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠀⠠⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶⠦⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠐⠒⠦⠤⢍⣁⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠬⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠋⠉⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠲⠤⢤⣤⣠⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀
⠀⠀⠀⠀⢀⡀⠨⠶⠒⠶⠶⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠉⠀⠀⠠⠤⢀
⠀⠀⠀⠀⠀⢻⠓⠒⠤⢤⣀⡀⠀⠀⠈⠉⠙⠓⠲⠶⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⠀⠈⠉⠻⣿⣆⣠⣤⣶⡾⠿⠿⠓⠒⠊⠉⠁⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠻⠶⢤⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⣤⣤⣀⣀⡀⠹⡆⣀⣤⣴⢾⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢦⣤⣤⣤⠴⠶⠒⠚⠋⠉⠉⠀⠀⠉⠉⠛⠛⠛⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⢶⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⣮⣄⣀
⠀⢠⣶⣖⣋⣭⣉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠐⠒⠊⢉⣁⣠⣬⡵⣿⡞⠉
⢀⣿⠏⠀⠉⠉⠛⠻⠶⣤⣍⣛⡳⢦⣤⣀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠖⠒⠚⣉⣉⣡⣤⠤⠴⠖⠒⠋⠉⠉⠉⠁⠀⠀⣾⠁
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣶⣤⣀⡜⢁⣆⣠⣤⣤⣶⠶⠾⠛⢛⣋⣉⣭⣥⣶⡶⠶⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣯⣥⣶⣶⠾⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣐⣈⣧⣤
⠀⠀⠙⠛⠿⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⢷⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡾⠶⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⢶⣤⣀⣀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠶⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣮⣤⡶⠶⠟⠛⠋⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠀⠄⠀⠀⠀⠠⠀⠄⠄⠀⠀⠀⠀⠀⠐⠠⠂⠀⠆⠄⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠿⠟⠛⠛⠛⠻⠿⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣤⠈⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⡿⣦⠿⠀⠀⠈⢿⣿⣿⡄
⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠛⣿⠉⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⣴⡟⢻⡆⠀⠀⠀⣠⣾⠛⠛⠁⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠈⠛⠛⠛⢷⡟⢻⡟⢀⣤⣤⣤⠀⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣠⣤⣤⡀⠈⠛⠛⠁⢸⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⣿⣿⣿⡇⢀⣀⣀⡀⢸⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣄⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⣴⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⢿⣿⣭⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠙⢿⣶⣿⣶⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣌⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢣⣭⡸⣿⣶⣿⣣⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣧⣭⣭⣿⢟⣫⣽⣶⣤⣄⡀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⢳⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣩⣿⣽⣿⣿⣿⠀⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⡹⣿⣿⡿⣿⢹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⡇⣿⢸⣿⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣷⣿⣾⣿⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠼⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⠓⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠐⣖⠔⠶⢶⣖⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠖⠀⠀⠀⢀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠘⠆⠀⠀⢀⡀⠀⣣⡀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢸⡈⠋⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⠿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠲⢄⣠⠖⠤⣰⠃⠀⠀⠀⠀⠀⠀⢀⣀⡠⠃⠈⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠑⣄⣓⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢀⣀⠀⠀⠈⠛⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⡏⠁⠀⠀⠀⣀⣀⣀⠴⠂⠀⠀⢈⡥⠎⠀⠀⠐⠔⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢳⠀⠀⠀⡜⢀⢉⢪⠃⠀⠀⠀⠸⣀⣀⡀⠀⠀⢠⠟⠋⠁
⠀⠀⠀⠀⠀⢀⡏⠀⠀⢰⠀⠀⠀⢸⠀⠀⢠⠃⣸⠷⠋⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠁⠀⠀⢸⡀⠀⠀⢸⠀⢠⠯⣲⠃⠀⠀⠀⠀⠀⢀⣴⣶⢿⡻⣿⡟⣻⣷⡶⣦
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⡇⠀⠀⢸⢀⡏⠀⢸⠇⠀⠀⠀⠀⢀⠞⣗⡀⢉⣼⡿⣿⡤⡨⣼⠋⠲⣄
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⡇⠀⠀⢸⡞⠀⢀⡟⠀⠀⠀⠀⠀⡞⠀⡌⠳⠋⢿⠃⠽⠌⠛⠃⠀⠀⠸⡆
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢷⠀⠀⠸⠁⠀⡾⠀⠀⠀⠀⠀⢸⠁⢸⡇⠀⣴⠂⢀⣠⡄⠀⠀⢸⠀⠀⣷
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣸⠃⠀⠀⠀⠀⢀⡏⠀⠀⢣⠎⠛⠙⣾⣿⣾⠅⠀⡎⠀⠀⣿
⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢳⡀⠀⢠⡏⠀⠀⠀⠀⢠⠞⠀⢀⡴⢁⡠⠔⠒⠋⠉⢉⠆⡼⡇⠀⠀⣿
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡷⣤⠞⠀⠀⠀⠀⢠⡏⠀⣠⠊⡴⠉⢢⠘⡃⠀⠈⡟⡼⠁⢇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢈⠇⡖⢁⡜⠁⠀⠈⡇⠋⠀⢲⠂⡇⠀⢹⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠣⠜⠀⠀⠀⡼⢙⠋⠀⠙⡅⢳⠀⢻⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⢀⡜⠁⡈⠁⠀⠛⡇⠈⢧⠸⣃⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉⠀⠀⠈⠸⡄⠀⠃⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣽⣿⣿⡿⠯⢉⣙⣶⠶⠆⠒⠂
⠀⠀⠀⠀⠀⠀⢠⣶⣛⢛⣙⣈⣱⣛⢛⡛⠛⠋⠈⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣯⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠈⠳⣄⠀⠀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠳⣬⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⢤⡀⠀⠀⠸⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣤⣀⠀⠀⢁⣤⠶⠶⠶⢤⣄⠁⠀⢀⣠⡄⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠉⠁⣴⠋⠀⠀⠀⠀⠀⠈⢳⡄⠉⠁⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠻⠶⠶⢶⡀⠀⠀⠠⠤⠄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠤⠄⠀⠀⠀⢀⡶⠶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣇⠀⠰⣆⠀⢠⡶⠀⢀⡿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠹⣦⠀⢹⡀⣼⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⡇⡿⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠿⠷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⠀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠛⠋⠉⠀⠉⠙⠛⠻⢿⣿⣿⣿⣷⡄⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⢉⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠛⠿⠏⠀⠀⠀⠀⠀⡀⠿⠇⡀⠀⠀⠀⠀⢀⣴⠟⠁⠸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠂⠀⠀⢀⡀⠾⠋⠀⠀⠻⣦⠀⠀⣴⠟⠁⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠶⠶⣤⣄⡀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⢾⡷⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣦⣌⠻⣦⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠹⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠛⠛⠛⠛⢛⣛⣛⣛⣛⡛⢿⠀⠀⠀⠀⠙⣿⣿⣿⣷⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⠀⠀⠀⠀⠀⠈⠻⠟⣡⣾⣿⣶⣤⣤⣀⣀⣀⡄⢠⣶⣿⣿⣷⣌⠛⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣍⣉⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣶⣌⡙⠿⢋⣴⣶⠀⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇
⠀⠀⠀⠀⢀⣠⣶⣿⣿⣦⠹⡿⢃⣤⠙⡿⢃⣾⣿⣷⣦⣄⠀⠀⠀⠀⢹⣿⣿⣷⣄⡀
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠙⢿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⡆⢸⣯⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿
⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣈⣉⣉⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⢸⣿⣟⣿⡟⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⣤⠶⢆⡴⠶⣄⢴⣄⢰⢀⡴⠆⢰⠀⢰⠀⣶⡀⣴⡄⣶⠶⢠⡶⣦⠀⠀⣴⣆⠀⣦⡀⡆⢀⣶⡄⢰⡄⠰⣄⣴⢠⡶⠆⣴⢠⡶⠆
⠀⢯⣀⡸⣇⣠⡟⣼⠙⣿⢀⣙⡷⢸⣅⣼⢀⡟⣷⢻⡇⣿⣋⢸⡟⣿⠀⢰⡷⢿⡀⡏⢻⡇⣸⠷⣧⢸⣇⡀⢹⠃⢈⣹⡆⣿⢈⣙⡦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣰⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣦
⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠠⣶⠶⣶⠟⠃⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⢠⣤⣤⣿⣤⣤⡀⣿⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣸⣿⣀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⠟⣿⠙⠿⠀⣿⠀⠀⢸⣯⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⡆⠀⠀⠙⠋⠀⣿⠀⠀⠐⣿⣤⣤⣼⣯⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠛⠛⠛⠛⠛⣿⡄
⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠿⠀⠀⠈⠛⠀⠀⠈⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠙⢿⣦⣤⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢹⣿⡆⠀⠀⠀⠀⣠⡴⠒⠲⣶⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢠⣿⢃⣠⣾⠟⠉⠉⠉⠉⠉⠉⠉⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢻⣿⡀⠀⠀⠀⠻⠋⠀⠀⣸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠰⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢠⡟⠛⠛⠛⠛⢿⣷⠀⠀⠀⣠⡶⠚⠉⢹⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠘⣿⣧⡀⢸⣿⡇⠀⢀⣼⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠀⠙⠛⠛⠉⠈⠛⠛⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠠⣤⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠄





⠀⡉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⡈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⢀⣴⠀⣠⠖⠃⣠⡙⢶⣆⡀⢈⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠨⠉⣤⡭⢾⠟⠛⣿⣿⣯⢻⣾⢟⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⡁⠨⡝⠒⠐⠶⣿⣧⣾⣛⣵⣺⡿⠄⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣷⣦⣤⣾⣿⠀⠀⠀⠰⣤⣨⣝⠻⠦⢴⣶⣶⣦⣼⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣝⠻⢿⣿⣶⡄⠀⠀⠿⣿⣷⣶⣶⣶⣯⣽⣿⣿⠿⣁⣤⣤⣴⡞⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠗⢲⣬⣿⣷⣦⣴⣶⠶⣽⣿⣿⣷⣶⣿⣿⡿⠿⣿⡳⠘⠿⠦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠙⠈⠋⠺⠟⠁⠀⠉⠿⠟⠋⢻⣿⣿⣷⡀⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣍⣁⠂⠀⠀⠀⠀⢀⡰⣾⣿⣶⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⣿⣷⣀⡀⠀⠀⡉⣾⣿⣽⣿⣿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣀⡫⣥⣀⢸⣷⣴⡆⠀⣿⡧⣿⣯⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠿⠻⣿⢛⠟⠢⣝⡇⠋⠉⠀⣀⠀⠀⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⡟⠁⠀⣴⣾⣿⡄⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠌⣠⣶⣾⣿⠋⡆⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢰⣿⣿⣿⢈⣀⣿⡿⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⢛⠿⠻⢿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⢺⣿⣟⣿⡗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⢈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠛⢄⠀⠀⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠄⢴⡁⠀⠀⠀⠈⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠑⠢⡀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠤⠄⠠⠄⡀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠰⠁⡠⠐⠢⡀⠙⡄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢃⠀⢠⡀⣀⠄⠀⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠢⢀⣈⣁⡠⠊⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠈⠐⠠⣤⠀⠀⠤⠔⠚⠣⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡃⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣙⠳⣤⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⠤⠄⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣭⠆⠀⠈⢩⠋⠝⣋⡛⢍⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢁⢠⠸⡘⢅⠸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢀⠔⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢠⣤⡄⠊⠀⡈⢆⠢⢍⣒⣒⡡⠊⠳⡀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⢄⠁
⠀⠀⠀⠀⠀⠀⡜⠠⠁⠀⠀⢀⠀⠘⡌⢆⠀⠀⠉⠢⡀⠀⠈⠢⡀⠀⢀⠊⠀⠀⠀⢠⠊⠉⠑⡄⠀⠀⠈⢢
⠀⠀⠀⠀⠀⡼⠠⠁⠀⠀⠀⠑⠃⠀⠘⡌⢆⠀⠀⠀⠈⠦⡀⠀⠈⢦⡌⠀⠀⠀⠀⠐⢄⣀⡠⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡰⢡⠃⠀⢠⡆⠀⠀⠀⠀⠀⠘⡜⡆⠀⠀⠀⠀⠈⠢⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡰⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣦⠀⠀⠀⠀⠀⠈⢢⡇⠀⠀⠀⠀⠀⣀⡀⠀⢤⣄⣀⡠⠤⠿⠐⠒⠺⠄
⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠀⡔⠉⡆⠀⠀⠀⠱⡘⣄⠀⣀⣀⡀⠤⢤⣗⠒⠦⠭⠙⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢰⢃⠂⠀⢠⢢⠀⠀⠀⠀⠈⠛⠁⠀⡠⠠⡀⠱⡘⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢠⢃⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⠁⠀⠡⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣣⣤⣭⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠋⠉⠉⠉⣽⠉⢉⡽⠋⠙⠒⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠈⣠⠖⠶⠶⣿⠊⠉⠀⠈⠀⣾⠁⠀⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⢰⠋⡳⣠⠞⠁⠀⠀⠀⣿⠀⡴⠦⣤⡖⠚⠀⢀⠐⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢠⡞⠃⠀⠀⡴⠒⢫⢴⡿⠁⠀⠀⠀⠀⠀⣿⡴⢃⣶⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⣰⠏⢀⣀⣠⣼⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⢓⣛⣃⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙⣆
⠀⠀⠀⠀⣰⣧⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠋⠈⠉⠀⠀⠀⣴⠛⣦⠀⠀⠀⠀⠀⠀⢀⠘⣆
⣄⣀⡀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣿⠒⠦⣜⣿⡄⣠⢳⠖⣋⠀⠀⠀⠀⠀⢀⣀⠀⠘⡆
⠿⠿⠃⣟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠯⣿⠤⣄⠈⠛⠙⠃⠈⢉⣩⠇⠀⠺⠯⣟⠉⠈⠱⣄⣿
⠀⠀⢰⣋⡻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠏⠀⠀⣿⠀⠈⠓⠚⠋⠉⠉⠉⠱⡖⢤⡀⢀⡜⠀⠀⡠⠻⣼⡇
⠀⠀⢸⡏⠉⠺⣗⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣵⣞⣤⡔⠋⠀⠀⠀⡇
⠀⠀⢸⡇⠀⠀⠈⠑⡆⠉⠓⠲⠤⣄⣀⣀⣰⠃⠀⠀⠀⠀⢿⠀⠀⠀⠀⣀⣀⣀⡤⠤⠖⠒⠋⠁⢠⡇⠀⠀⠀⢰⡇
⠀⠀⠀⣇⠀⠠⠀⣠⠇⠀⠀⠀⠀⠀⠀⠈⠹⣉⠉⠉⠉⠋⣿⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢻⡀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠖⣿⠲⢶⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠞⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢷⠀⡀⠀⠈⠑⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⣰⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢶⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢀⡴⠋⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢷⣀⡴⠟⠁⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣄⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠒⣿⠒⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⢀⡄⡆⠆⠀⠠⡄⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⣀⣮⠄⠀⠀⠀⠀⠀⣀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄
⢸⡇⠇⢰⣷⠀⡇⠀⠀⠀⠀⠀⣿⣿⣶⣭⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⠐⣾⡇⣢⡄⠀⠀⠀⠀⠀⠀⡝⠛⠛⠓⠒⡄
⢸⡇⣄⢸⣿⢀⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢈⣿⢀⣿⡇⣻⡇⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⡦



⢀⣴⡿⠻⢿⣦⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣴⣺⡭⣭⣶⣄⠀⠀⠀⠀⠀⠀⣠⠖⣶⢐⡦⣄
⣿⡏⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢸⣿⣿⠟⣿⣿⣿⠀⠀⠀⠀⠀⢸⢹⠋⠀⠀⠁⠸⠀⠀⠀⠀⠀⢰⠧⣉⠿⢠⣇⢸
⠻⣷⣄⣀⣠⣬⠋⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⣷⠗⡄⠀⠀⠰⠀⠀⠀⠀⠀⠈⢋⢵⡫⠝⣾⡝
⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉


⠀⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣾⣾⡆⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⠄⢸⡇⡜⡆⠀⠀⠀⠀⢲⢂⠒⡰⡾⢛⢟⣌⠀⠀⠀⠀⢠⢄⣀⣤⣎⢀⣱
⣿⣿⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠈⠛⠀⡀⠘⠃⠣⠂⠀⠀⠀⠀⠐⠘⠻⠓⡐⠠⠂⠙⠃⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿
⠉⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉


⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢰⣶⣶⠶⢶⣶⣶⠀⠀⠀⠀⠀⢀⣀⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣦
⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⡿⢠⡜⠛⣿⠀⠀⠀⠀⠀⠠⠤⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⡇⣿⠘⣿
⢻⡿⡫⢸⣿⣿⠇⠀⠀⠀⠀⠀⢸⠏⠹⣿⣷⣶⣿⠀⠀⠀⠀⠀⠐⠒⠐⠒⠐⠒⠒⠒⠂⠀⠀⠀⠀⣿⢸⡇⣿⢸⣿
⠀⠎⣀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠘⠓⠚⠛⠛⠛⠛⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉⠀⠉
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠰⣶⡆⠀⡀⠈⠀⠀⡗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⡄⠀⠀⡀⣀⠁⢳⢰⡇⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠂⠈⠀⡇⢻⡴⡾⢨⠋⣸⠶⣁⡠⢤⠀⠀⠐⣒⣢⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠐⠤⢢⣼⣾⡄⠷⡀⠅⠀⠋⠀⠈⢻⣭⡤⠟⠹⣄⢳⣀⠊⠉⡀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠋⠀⠚⠾⠏⠁⠀⠀⠙⠀⠁⠀⠀⠀⠀⠙⠂⠁⠒⠟⢿⣗⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠤⠀⣐⠠⠀⠀⢀⡀⠐⠦⠀⠀⠈⠁⣁⠀⡴⠀⠀⠀⠀⠃⠀⠌⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠰⠽⠙⣷⡟⠋⠐⣲⣟⣓⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⢸⣦⣾⣿⣿⣿⣿⡧⠐⠌⢡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠔⠁⠀⠀⠀⢈⣿⣿⣇⣼⡿⠛⠁⠾⡀⠀⠻⠖⠀⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠠⠴⢿⣘⠿⠿⣿⣿⠄⠀⡀⠛⠁⢐⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣍⣁⣄⣮⣤⠤⡤⠀⠄⣘⡛⠯⢅⡄⠀⠴⠟⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢩⡿⣧⠇⡐⠃⠀⠀⠀⡀⠀⢠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡟⠥⢀⣠⡴⠞⠁⠀⠀⠀⠀⢀⢐⢙⡀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠐⠋⠁⣀⡠⠤⠆⠐⠦⠂⠀⢻⣾⠟⠓⣠⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢨⣭⣄⠹⣷⣶⡆⢠⣦⠀⡈⠁⠀⠐⠉⠀⠀⡠⠐⠉⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠲⢤⡛⠈⢳⢠⣉⡋⠂⠉⢸⠃⠰⢤⠁⠠⢀⠞⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣂⢯⣀⣎⡸⣅⡑⠀⣸⠃⠀⣥⣼⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠁⠉⠉⠉⠈⠩⠞⠛⢴⠏⢅⣊⡈⢽⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠠⢽⡥⣀⠄⠙⢭⣋⡀⠐⠂⠀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠿⢟⠂⠒⠤⠀⡈⠁⠀⠈⢁⣀⡀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢐⡷⠃⠀⠀⠀⠀⠁⠉⠀⠉⠐⠐⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⣤⣀⣀⣠⣄
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⣼⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠟⠛⣿⣿⣿⣿⠟⠛⠒⠀⡀
⣿⣿⣿⣿⣿⡇⠀⢀⣀⠀⠀⢀⡀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣶⣴⣶⣾⣄⢻⡿⢋⣉⣴⣿⣿⣷⣜⣆
⢿⡟⠙⢿⠏⠀⠀⠀⣤⠀⠀⢠⡄⣼⠃⠀⠀⠀⢀⣾⣷⣶⣤⡍⢻⣿⣇⣘⡁⢨⣉⣤⡄⢹⣿⣿⣿⣷⡀
⠘⣿⣄⡀⠀⠀⠀⠀⠉⠀⠈⠈⠁⠁⠀⠀⠀⠀⣼⡿⠟⣡⣤⣴⣿⣿⣿⣿⣿⠰⣿⣿⣷⣆⣈⣥⣌⣿⣧
⠀⠈⠻⠷⠀⠀⠀⠀⠠⣄⣠⠆⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠿⢃⣤⣌⠛⢿⣿⣷⠄⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⣤⣾⣿⣿⣿⠟⣠⣿⣿⣤⣍⠹⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⢋⣡⣤⣼⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡋⠰⣿⣿⣿⣿⣤⣥⡆⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣶⣌⢻⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣦⡀⠀⠀⢀⡔⠻⠉⠙⠿⣿⣿⣿⣧⣾⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⠿⠟⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣦⣤⣀⣀⣤⣀⡀⠀⠀⠚⠁⠀⠀⠉⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⠦⠤⠤⠤⠴⠶⠾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⠠⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣴⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⣿⣿⣿⡇⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠸⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢻⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⣾⣿⣿⢸⣿⣿⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣭⣷⣶⣦⡝⠛⠘⠛⠛⠛⠃⠛⠛⢉⡤⠖⠒⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢀⣠⣄⣀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⢿⣿⠟⣁⠀⠀⠘⠲⣄⡀⠀⠀⠀⣀⡤⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠉⠉⠉⠉⠁⠀⠀⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡗





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣐⡦⠾⠖⣷⠈⠉⠉⠃⡾⠻⡿⣻⠿⠡⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣵⣤⣀⡉⠛⠶⢿⠸⠠⡀⢠⣿⢽⠯⣒⠕⠋⣠⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢛⡉⠉⠡⠤⠤⠴⠈⠀⠰⠄⠭⠈⠤⠤⠬⠁⠐⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠡⠐⠂⣀⢊⣈⣀⠀⠀⠄⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣨⡤⣥⡄⠀⠀⡀⣁⠙⠀⠀⠈⢡⣀⠐⢖⣢⣶⣷⣾⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠋⠁⡠⣒⣥⣶⡛⡇⠚⡁⠘⠸⡏⠑⢤⡈⠑⠟⠫⠂
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡭⠶⠿⠿⠿⢶⣶⣢⣕⠿⠍⠿⠳⡿⠶⠇⢦⡥⠤⠧⠷⠷⠿⠛⠂⠁
⠀⠀⠀⠀⠀⠀⢶⠋⡔⠁⠀⢀⡤⣠⣀⠀⠈⠙⢝⢽⢦⠀⣄⠀⠰⠎⢀⡀
⠀⠀⠀⠀⠀⣼⠏⠘⢀⡼⢟⡩⠚⠁⠙⢷⣦⣇⠈⠆⢲⡇⠀⠈⡩⢂⠤⠊
⠀⠀⠀⠀⢠⣿⠠⠧⠔⠚⠁⠀⠀⠀⠀⠀⠉⠛⢤⢼⣀⡿⠀⠘⢢⠌⣒⠄
⠀⠀⠀⠀⣸⠃⠀⣆⠀⠠⣀⡀⠀⠀⢀⡀⣀⠀⢸⡄⠀⢱⡄⠖⠓⠊⠁
⠀⠀⠀⠀⠙⢔⠰⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢸⡇⠔⡜⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡘⠛⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠄⡀⠠⠔⠁⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⢲⢄⣀⣀⣀⡴⡿⢦⢄⡀
⠀⠀⠀⠀⠀⢀⠔⠡⠤⠀⢸⣄⣀⠑⠁⣰⡏⠐⠁⡈⠲⢄
⠀⠀⠀⠀⡰⠁⡀⢁⠀⠐⢽⡝⠂⠀⠐⢚⡯⢐⠠⡂⠄⠀⠣⡀
⠀⠀⠀⠐⠀⠄⡠⢀⠀⡀⣸⡇⠀⠀⠀⢰⡪⢐⠨⢂⠅⡂⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠐⠈⠈⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⠛⠛⠋⠉⠉⠉⠉⠙⠛⠛⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄
⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣤⣤⣤⣤⣼⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⣤⣄⡀⠀⢿⣿⣄⣀⣀⣤⣴⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠈⠙⠛⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⢷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣙⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣲⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡦⡤⢄
⠀⠀⠀⠀⢠⣶⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣸⠆⠀⠀⠀⠀⠀⠀⣿⢠⡝⠛⠉⠈⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇⠃⠀⠱⡀
⠀⠀⠀⠀⠙⠿⠟⠀⠅⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠦⠾⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠙⠗⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⢟⣭⣲⡤⢄⡇
⠀⠀⢀⣼⣿⣿⣟⣛⣿⣿⣿⣅⠀⠀⠀⠀⠈⠛⠛⢛⣿⣿⡛⠙⠛⠦⠀⠀⠀⠀⠀⠀⣿⢀⣤⣠⣴⠆⡇⠀⠀⠀⠀⠀⠀⠈⢿⣵⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁


⠀⠀⠀⣀⣀⣀⣠⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⡴⢶⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⢤
⠀⠀⢸⡏⠉⠩⠉⠉⠍⡉⠉⡇⠀⠀⠀⠀⠀⣸⣇⣠⡿⣤⢽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⢻⡇
⠀⠀⢸⡇⢐⢸⣦⣥⣿⣶⠀⡇⠀⠀⠀⠀⠀⠸⣏⡀⣹⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⣤⣾⡇
⠀⠀⢸⣇⣘⣛⣛⣛⣛⣛⣀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠉⠋⠛⠙⠃
⠀⠀⠀⠈⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠚⠛⠛


⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⢰⣤⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⢀⡞⠉⢠⣭⣷⣴⣦⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠈⠟⢿⡄⠀⠀⠀⠀⠀⠀⠀⡆⡀⢈⠁⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⠶⢤⡀⠄
⠀⠀⠐⠻⠟⢿⡻⠃⢀⡾⠈⠁⠀⠀⠀⠀⠀⢸⡇⠀⣀⡀⣤⣾⢈⣿⠀⠀⠀⠀⠀⠀⠀⠇⡅⢨⡁⢨⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⣴⣿⢹⠂
⠀⠀⠀⠀⠀⠀⠉⠛⠉⠙⢦⡀⠀⠀⠀⠀⠀⠈⢿⣜⠙⠙⠙⢛⣾⠏⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠞⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠛⠃


⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣤⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡤⣤⡀
⠀⠀⢸⠿⠿⠿⠿⠿⠿⠻⠿⣇⠀⠀⠀⠀⠀⣼⠋⠀⣠⣾⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠎⠾⠀⡧⠀⠦⠢⡀
⠀⠀⢸⢀⣄⢀⣄⡔⢉⣉⢦⣿⠀⠀⠀⠀⠀⢻⣜⠿⠿⠟⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠀⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⡦⢰⡆⠀⡧⠀⢰⠀⠆
⠀⠀⢸⠈⠉⠉⠉⠙⠯⠽⢇⡟⠀⠀⠀⠀⠀⠀⠙⠛⠒⠛⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠰⡔⠦⠀⣗⠉⢁⡴
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠷⠟⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠂⠐⠒⠒⠂⠤⠤⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢔⣾⠯⠷⠦⢰⠋⣤⣔⣒⣀⠀⠀⡀⠄⢀⣀⣀⣀⣀⣠⣤⣤⣔⣲⣬⣷⡿⠟⠃
⠀⠀⠀⠀⠀⢸⣼⠂⠀⠀⠀⠀⠀⠀⠤⠄⣉⡉⠉⠙⠓⠛⢛⡞⠛⢙⠩⠩⡫⡀⠀⠐⡨⡇
⠀⠀⠀⠀⠀⠀⡾⠶⢦⢤⣤⣄⣀⣀⣀⠉⠉⠉⠉⠀⠀⠀⢸⡈⢈⠀⣨⣪⣠⡦⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠁⠉⠉⠙⠙⠒⠒⠛⠲⠲⠶⣿⣿⣿⣛⣛⣛⣭⣉⣓⣦⡀
⠀⠀⠀⠀⠀⠀⠴⠁⢑⣦⣁⣀⣀⣤⣤⣤⣤⣶⡶⡶⠲⡻⡻⣻⠞⠝⠝⡝⠝⠝⢝⢍⠻⣧
⠀⠀⠀⠀⠀⠀⠈⢔⠄⢫⠙⡯⡉⠁⠑⠑⠁⢔⠕⢔⢌⢔⠩⠪⠊⠂⠪⠂⠢⠢⡢⡀⣩⣽⡇⢀⡀
⠀⠀⠀⠀⢀⠠⠔⠚⢮⡄⠀⡏⡨⣠⣢⣢⣢⣠⣠⣡⣵⣰⣵⣵⣵⣵⣵⣴⢷⣶⣾⣾⣯⣿⣴⡿⠃
⠀⠀⣠⣞⣑⣧⣤⣤⢀⡙⠛⡻⠿⠿⠟⠛⠛⢙⣛⠹⠭⣍⡉⠈⠉⠅⠀⠉⠠⠀⢂⣴⡾⠛⢻
⠀⢰⣿⠉⠁⢨⢉⡉⠉⠒⠒⠓⠾⠭⠶⣶⣒⣒⡠⠤⠤⢀⡀⠀⠀⠀⠀⠀⣠⣴⠻⠏⠂⡄⣬⣶⣄⡀
⠀⠘⢿⣄⣀⡳⣾⠀⠠⠤⠀⣈⠀⠰⠂⡀⠀⠀⠉⠉⠛⠒⠒⠿⠽⣦⣶⢿⡏⠀⡁⣀⣶⣾⡛⠁⠀⠀⢉⣴
⠀⠀⠀⠀⢉⣩⣿⣿⠷⠶⢶⣦⣤⣀⣉⢀⡒⠀⠉⠁⠀⠀⠀⠀⠀⡇⠁⠨⠁⣤⣷⠟⢉⡀⢉⣠⣴⠞⡿⠁
⠀⠀⠀⢠⣻⠛⠛⠛⠯⣔⡂⠠⢀⡈⠉⠛⠛⠛⠲⠲⠶⣤⣄⣀⣀⣷⣠⣶⠟⠛⢂⣠⣶⢾⡿⠫⠁⠁⣧
⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠌⡛⠓⠦⢅⣒⠠⠀⡀⠀⠀⠀⠈⠉⠉⠛⢋⡹⣖⡿⠟⢙⠍⡠⠁⠐⢐⣠⣿⠦⠍
⠀⠀⠀⢀⠻⣦⡀⠀⠀⠄⢀⠀⠑⠢⢀⠈⠉⠓⠤⢭⣐⡀⢐⣤⣴⢾⠻⠫⡁⠎⠀⠀⣂⣱⣴⣾⠟⠉⠁⠀⠭⠁
⠀⠀⠀⠀⠀⠐⠙⠶⣦⣀⠈⠀⠒⢤⣀⠀⠁⠀⠀⠀⠀⡝⠉⠩⡢⡁⡁⠀⢌⣂⣶⣻⡽⠚⠓⠂⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠒⠶⢢⣄⡉⠙⠀⠀⠀⠀⡟⡑⠐⠔⣔⣴⣾⠿⠟⠉⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠪⢤⣒⠤⢹⣤⣾⣾⠿⠋⠉⠁⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠁⢀⣀⠀⠈

⠠⠄⢀⠀⠀⠀⠀⠠⠀⢠⡄⠀⣠⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⠠⡤⠀⢀⣀⡀
⠀⠄⠀⢀⣤⡞⣦⢄⡀⠀⢠⣾⣻⣽⣾⠃⠀⠀⣦⣶⠀⠀⠀⠀⠀⢀⣾⣿⣷⠀⠀⠀⠧⠞⠀⠀⠠⠄⠀⠀⠘⣿⢿⡦
⠀⣠⣾⣿⢿⣯⣥⡿⣿⣷⣿⣿⣷⠟⠁⠀⠀⠀⠿⠛⠀⢠⢀⠀⠀⢀⢀⠛⠋⠀⢠⣶⠀⠀⠀⠂⠀⣰⣶⣦⠀⠙⣧
⠀⠈⠛⢿⣿⣿⣿⣿⠟⠁⠈⠉⠁⣠⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠈⠉⠁⠁⢀⣦⡀
⠀⠀⠀⠀⠙⠿⠟⠁⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⣀⠀⠀⠀⢸⡏⠀⠖⠂⠖⠀⠀⠀⠀⠀⠀⠀⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢳⣄⡀⠀⠀⠸⡇⠀⢀⡀⠀⠀⠈⠀⠀⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠋⢁⠀⠀⢰⡇⠀⣾⠟⣀⠀⠀⠀⠀⠀⢰⡄⠀⣠⢖⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⢀⣠⠟⣠⣆⡩⠀⠀⣚⣵⠟⠀⠀⠀⢻⠀⢿⣍⡽⠁⠀⢀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠖⠛⠋⠁⢈⠙⠛⠒⠺⠿⠟⠁⠀⠁⠀⠀⠈⣧⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠂⠀⠀⠀⠈⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠉⠛⣯⠀⠀⣄⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠠⣿⡤⠀⠀⣰⠀⠀⢀⠇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⣽⢿⠂⠀⣿⠆
⠀⠀⠀⠀⢢⠀⠀⠀⡿⠀⠀⠘⣆⠀⠉⣹⡐⡟⡁⢎⡱⢜⣱⢸⢰⣹⠦⢯⡱⠋⠇⢆⡇⠀⠀⠻⠇⠉⠀⠀⠀⠋
⢀⣀⠀⠀⠈⣧⡀⢰⡧⢀⣤⠆⠈⠉⠉⠀⠁⠈⠀⠀⠀⢈⣽⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇
⠀⠈⠉⠒⢶⠤⣷⣾⣖⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⠀⣀⣀⣤⡀⠀⠈
⠀⠀⢀⣠⣴⡜⢩⣿⡌⠛⠛⠓⠓⠂⠀⠀⠀⢠⡇⢱⠀⠀⠀⢀⣴⡋⠹⣿⣿⡷⣷⣔⠞⠃⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⡼⠁⠙⢦⠀⠀⣤⠀⠀⠀⠀⠈⢻⡬⠀⠀⢀⣾⠿⠇⠀⠙⠛⠁⠙⠟⠀⠀⠀⠠⠴⠀⡀⠄⠠⠈⢷⡀
⠀⣾⠿⣿⡆⠸⠁⠀⢰⣆⠀⣀⣿⡆⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣤⡰⣶⣦⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀
⠀⠘⣷⣿⠃⢀⣴⣤⣼⣿⠂⠀⠈⠁⠀⠀⠀⢠⣾⠃⠀⠀⣠⡄⠀⣠⣾⣿⣿⢻⣿⣿⠟⠀⠠⠀⠀⡠⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠈⠋⠀⠈⠉⠈⠉⠁⠀⠀⠀⠐⠒⠀⠀⠼⣿⣧⣤⠄⠁⠀⠀⠛⠩⠁⠀⠸⠟⠉⠀⠀⢀⠐⠂⠀⠀⠄⠀⠰⠿⠿⠃
⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠐⢶⡄⠀⠀⠀⠀⠀⠈⠉⠉⢠⣦⣄⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠸⣆⠀⠀⠀⠈⠛⠻⢿⣿⣦⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠦⠠⠀⠀⢻⡂⠀⢰⣴⣾⡇⠀⠀⠉⠁⠀⠴⠾⢿
⠀⠀⠀⠀⠰⠚⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠤⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠁⠀⠉⢙⡆⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡤⠒⠒⠒⠲⢄⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⡠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⠇⠈⡷⢤⡀⠀⠀⠀⠀⠘⣆⠀⠈⢦⠞⠁⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠭⠤⣼⠤⠤⢷⠤⠬⠳⠆⠀⠀⠀⠸⡄⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⣁⣀⣈⣀⣀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢹⠉⣢⣴⠋⢿⠳⢄⡀⠀⠀⠀⠐⣗⣒⣒⣒⣒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣎⣀⣀⣷⣎⣁⣀⣙⣾⣄⣀⣙⣦⡀⠀⠀⠙⢀⠀⠀⡌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢉⡶⢶⣋⠁⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠚⠁⠀⡄⠈⠑⢺⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⡇⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠻⢷⣶⣤⣴⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⣤⣦⣠⣤⣤⣤⣬⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠑⠂⠀⠀⣸
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠒⠒⠲⠶⣤⣤⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⢻⣿⣿⣿⣿⡟⠩⠐⠆⢻⣿⣿⠇⠀⠀⠀⠐⠛⠻⢷⣤⣄⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⣆⣴⠀⠸⣿⣿⠀⠀⠀⠀⠀⠲⠆⠀⢸⠛⠛⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⢸⣿⣿⣿⣿⡆⠸⠇⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠠⠋⠀⠈⠙⠻⢿⣿⣿⣿⡿⠠⡄⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⠆⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⣠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡟⠲⠄⠀⠀⠀⠀⠀⠀⣈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣄⠀⠉⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠓

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣸⡆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢠⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠉⠉⠉⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣾⣿⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣆⣨⣿⣿⣿⣿⣿⣿⠟⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢁⣠⣾⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠛⠋⠛⠛⠿⠿⠿⣿⣿⣿⠿⠿⠿⠛⠛⠉⣀⣤⣶⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣴⣶⣾⣉⣉⣉⠉⣉⣁⣀⣉⣉⠉⢉⣉⣉⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠹⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⡿⠋⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠟⠋⢁⣠⣶⣿⣶⣄⡈⠙⠻⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⡄



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

⠀⢀⣴⢾⣿⡷⣦⠀⠀⠀⠀⣴⠋⠝⣦⠀⠀⠀⠀⢸⣶⣶⣶⣶⠀⠀⠀⠀⡠⣴⣾⡶⠆⠀⠀⠀⠠⠒⠒⠖⠒⠆
⠀⠘⢧⣦⣿⣼⠟⠀⠀⠀⠀⠘⠦⢴⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠇⠄⠀⢄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⢀⣴⣶⣤⡀⠀⠀⠀⢠⠒⠒⠒⠒⡄⠀⠀⠀⢠⣄⣤⣄⣤⠀⠀⠀⠀⠀⣀⠴⢤⡀⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⡧⡧⡧⠯⠵⠀⠀⠀⠸⣄⣶⣶⣤⠇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠸⣧⣤⣾⠅⠀⠀⠀⠠⣶⣞⡉⠈
⠀⠀⠙⠿⠿⠟⠁⠀⠀⠀⠸⠶⠄⠠⠶⠧⠀⠀⠀⠐⠚⠓⠛⠒⠀⠀⠀⠀⠈⠻⠽⠋⠀⠀⠀⠀⠀⠀⠀⠉⠿⠇

⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⣶⣶⡄
⠀⠸⣿⣿⣟⣿⡦⠀⠀⠀⢸⣿⣿⣧⣿⡆⠀⠀⠀⣘⣉⠋⢋⣙⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⢐⠠⣠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠲⡄⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⣶⣶⣖⣶
⠀⠀⢈⠀⠈⢀⠀⠀⠀⠀⢠⣶⠀⠀⢰⡆⠀⠀⠀⠸⡀⠫⠃⡸⠀⠀⠀⠀⡿⠼⠿⠿⢿⠀⠀⠀⠀⡯⠀⠉⢹
⠀⠀⠈⠛⠛⠛⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡋⢹⣿⡋⣻⣿⠉⣿⣿⣿⣿⠄⠀⠀⠈⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄⠀⠀⠀⠀⠚⠋⠉⠀⠀⠀⠈⠛⠶⢤⣤⣤⣤⣤⣤⣄⣺⡏⠁⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣛⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⢸⣿⣿⣿⣿⡟⠿⢿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⡿⠿⢻⣿⣿⣿⣿⡇⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠙⠿⠿⠷⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠾⠿⠿⠋⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⣿⣿⣿⣿⣧⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⣀⣼⣿⣿⣿⣿⡇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⣿⣿⣿⣿⣿⡏⠉⠙⢿⣿⣷⡀⠀⠀⢸⡇⠀⠀⢀⣾⣿⡿⠋⠉⢹⣿⣿⣿⣿⣿⣿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣿⣿⣷⠀⠀⢸⡇⠀⠀⣾⣿⣿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⣿⣿⣧⠀⢸⡇⠀⣼⣿⣿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⣿⡇⢸⡇⢸⣿⣿⠇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠀⠀⠈⠉⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⣿⣿⠇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡿⠋⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⠻⢿⣿⣿⣿⠀⠀⠀⠈⣁⣴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣀⣁⣤⣤⣴⠾⠛⠁⠙⣷⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣠⣀⣉⣉⣉⡉⠁⠀⢰⣶⣄⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣷⣄⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⢤⣤⣔⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⡖⢠⣶⣿⣿⣷⣶⣿⣶⣦⡐⢦⠘⣿⡖⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢸⡋⠱⣾⠋⣴⣾⡏⠴⠶⠶⣤⠀⣨⢀⣿⠇⠀⠀⢡⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⣿⡟⡆⠈⠻⣦⠙⠛⠒⢒⣋⣡⣾⣿⡿⠗⠀⠀⢠⣿⠟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠸⣇⠘⣦⣀⣈⠓⠲⣾⣛⠛⠛⠉⠉⠀⠀⣀⣴⣿⣿⠇⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣌⠳⢮⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⠿⢿⣿⣿⠟⠀⣼⡶
⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢹⣿⣷⣶⣤⣶⢿⣿⣿⣽⡻⣿⠙⠛⠉⠀⣼⣿⠋⣠⣾⣿⠛⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠸⢿⣿⣿⣦⣴⣬⣿⣿⣿⣎⠳⣀⣠⣾⣯⣵⣾⣿⣟⠁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠟⠉⠋⠉⠋⣹⢇⣿⡿⠃
⠀⠀⠀⠀⠀⢀⡄⠀⠻⣿⣷⣴⣤⣤⣶⣶⣶⣶⣿⣿⣇⢦⡀⠀⠀⠀⢁⣼⠟⠁⠀⠀⡔
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠉⠹⣿⣿⣿⣿⣦⣶⣽⣿⣿⣿⣆⢰⣮⡤⠔⠋⠁⠀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠈⠙⢻⠿⠿⠛⠛⠻⠿⠛⠛⢋⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣴⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠉⢹⠏⠀⣿⣿⣿⣿⣿⡟⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢁⣬⣿⡇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠘⣿⣿⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣠⣤⣶⣶⣶⣄⣀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣾⣷⣧⣷⣹⡿⢷⡀⣷⠀⠀⢀⣤⣶⣾⣿⣿⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡿⣣⣛⣯⢀⣴⣿⠿⢿⣛⣯⣤⣛⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⣿⠎⣩⣶⣿⣿⣿⣮⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠛⠻⠟⠛⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⣄⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⢤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠧⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣀⣀⣠⡶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡆⠐⠀⠘⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⢿⡟⠡⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢀⣴⣂⠀⠀⠀⠀⢿⣿⣇⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣧⣄⢸⡇⢀⡀⠀⠀⠁⢀⢽⠇⠀⠀⠀⢘⣿⣦⣄⠀⢀⣠⣀⠀⠀⠥⠀⣴⣿⣿⠟⠁⣀⠀⠀⣿⣿⡇⠀⠀⢠⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⡇⠀⣀⣴⠀⠀⢠⣀⡆⠀⠀⠀⢰⣮⣟⡯⡱⠟⠉⣰⡀⠀⠀⢰⣿⣿⠃⢠⣶⠤⠀⠀⢿⣿⡇⠀⠀⠈⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡷⣿⡇⣿⡿⣿⠀⠀⢸⣿⡇⠀⠀⠀⣼⡇⠨⣿⢠⠀⢐⣿⣿⡇⠀⠈⠻⢿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⠻⢿⡇⠉⠀⣿⠀⠀⢸⣿⡇⠀⠀⠀⡝⠋⠐⢿⠈⡀⢐⣿⣿⡇⠀⠀⠀⠀⢤⣾⣷⣤⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⣿⣸⣷⣾⡇⣾⣿⣿⠀⠀⠀⠉⡀⠀⠀⠀⣿⣷⠀⠀⠀⡂⠠⣿⣿⡇⠀⠸⢦⡄⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠈⠠⠃⣿⢙⠻⢿⡇⠁⠀⣿⠀⣠⠰⠃⠑⠀⠀⠒⣿⣿⠀⠀⠀⣤⡾⣿⣿⡇⠀⠀⠀⡀⠔⠚⡃⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣷⣾⡇⣶⣿⣿⣾⣿⡷⠉⠀⠀⠀⠈⣿⣿⠀⠀⠊⠀⠀⠙⡿⡇⠀⠀⠀⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⠙⢻⣋⠉⠉⢤⣿⣿⡟⠀⣄⡀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢹⣿⣿⠀⢸⣿⣿⠀⠘⠻⡇⠠⣦⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡅⠿⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣦⡤⠈⠀⢸⣿⣿⠀⢸⣿⡇⠀⠰⣿⣶⠖⠀⠀⠀⠈⠁⠀⠀⠀⢰⠈⣿⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⡇⠀⠀⠀⠙⠿⠀⠈⠙⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢠⣤⡞
⠀⢠⣄⠀⠀⠀⠀⢀⣀⢰⣿⡇⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢸⣿⡇⠀⢴⣾⡿⠛⢹⣿⡇⠀⠀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⣿⡇⠄⠂
⠀⢸⣿⡇⠀⠉⠁⣀⣤⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠐⠋⠁
⠀⢸⣿⡇⠀⣴⣿⠟⠋⠉⠉⠓
⠀⢸⣿⡇⠀⠈
⠀⠈⠻⠇

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢦⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠻⢰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡙⢄⠀⠀⠀⠡⡀⢆⠀⠈⠄⢁⠀⠀⡀⠀⠈⡄⠀⡄⠀⠂⠀⠀⡂
⠀⠀⠀⠀⠢⡀⠀⢄⠡⡀⢄⠀⠑⠈⢂⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠁⠀⡀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠄⠀⠁⢀⠎⢠⠎⠔⢀⠔
⠀⠀⠀⠀⠀⠈⠢⠀⠠⡀⢄⠀⠀⠀⠄⢂⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠃⠀⠀⢈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠂⠀⡡⠀⠀⠉⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠐⡄⠀⠄⢠⠀⠀⢠⠀⠀⢰⠀⠀⡇⢸⠀⠀⠀⡞⢀⠀⠁⢀⠆⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡐⢌⠆⠈⢆⢣⡀⠈⣶⡀⠀⡔⠀⣧⢸⡁⡀⠁⡇⡌⠀⢄⠎⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢢⡢⠠⡨⣦⢫⢤⡸⣇⢃⢁⠀⡇⢸⠇⠅⢸⣷⢀⣈⡎⠀⠀⠀⠐⠀⡔⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢄⠀⠢⡀⠐⠄⠄⠘⣿⡏⢧⢻⡸⣼⡇⣇⣸⣴⣰⢿⣇⢂⢟⠀⣤⠂⠀⢀⢌⢔⠄⢀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠑⢦⡘⠦⠈⠪⡈⣾⣿⣾⣷⣷⣿⡧⣹⣿⣾⣿⣿⣧⣾⢇⡼⠁⠀⣴⣷⠕⣀⠐⠁⠀⠀⠀⣀⠠⠀⠀⠀⠀⢀⡀⠄
⠀⠀⣀⠀⠀⠀⠀⠀⠉⠂⠄⡀⠀⠙⠷⣜⣢⣤⣿⣿⣿⣿⣿⣿⠿⡿⠾⢿⠿⣿⣿⣿⣿⡵⣡⣾⠟⣡⣞⠁⠀⢁⠠⠔⠊⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠐⠤⢀⡀⠀⠀⠈⣑⡂⢀⠈⢻⣿⣿⣿⣿⡟⠏⠈⠀⠁⠀⠈⠀⠈⣜⡽⢿⣿⢟⡵⣪⣽⣯⣴⡦⠞⠀⠀⠀⠀⣀⠤⠔⠂⠈⠠⠄
⠀⠀⠀⠀⠀⠤⠀⠀⠈⠉⠒⠲⢮⣝⣳⣮⣶⣿⣿⣿⠫⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⣺⣿⣷⣾⣿⣿⣋⣥⣶⣬⡤⠔⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⢙⡲⠶⢭⣽⣿⣿⣿⠃⠁⠀⣀⡀⠐⠒⠒⢒⣀⠀⠀⠁⠈⠙⣽⣿⣿⣿⣿⣏⡩⠄⠒⠒⣉⡉⠀⠀⠀⠀⣐⠂⠉⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⢰⣿⣿⡏⢁⣤⣀⣀⣤⣌⡀⠀⠀⣀⣤⣤⣀⣀⣬⡚⣿⣭⣿⣷⣾⣯⠭⠋⠉⠉⠀⠀⠐⣂⣉⣉⠁⣀⠀⢤⠤
⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⣒⡒⣒⠚⣻⣿⣿⠀⢤⣤⣽⣟⣿⣧⠀⣼⣿⣿⣿⣿⣶⠬⠷⣿⣏⡉⠍⠉⠉⠀⠩⠭⠥⠤⠤⠤⠤⢤⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠈⠉⢁⡀⢁⠤⠠⠀⠀⢀⡀⠤⠻⣿⡿⠀⠀⠈⠙⠛⣿⢻⠀⣿⣿⣿⠛⠋⠀⢁⡸⣿⣿⠟⢀⡐⠀⠈⠥⠄⠀⣀⣀⠀⠀⠀⠀⠀⠶⠒⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠠⠒⠀⡀⠅⠀⠀⠠⠀⣿⡇⠀⠀⠀⠀⠐⠁⢸⠂⢿⣿⡇⠀⠀⠀⠘⡇⣿⡿⠤⢀⠀⠉⠁⠀⠒⠠⢄⣀⡉⠉⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⢿⠀⢤⣤⡄⠀⠀⠀⠀⢸⣿⣿⣦⣤⣄⣫⣽⡿⠗⠈⠀⠈⠙⠒⠤⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣧⡀⣙⠃⢀⠴⠛⢶⣾⠿⢿⣿⣿⣿⢷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣏⠉⣧⣤⣤⣤⣇⣧⣤⣬⣽⣿⣿⡿⠥⣀⠐⠢⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⠀⠙⠻⣩⡿⠟⣧⣿⣿⡟⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠒⠛⠿⠛⢓⠾⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⣤⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠷⠶⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣯⣭⣀⣀⣀⣀⣛⣹⣿⣿⣧⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⡙⠻⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣀⠈⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⣿⣿⣿⣿⣆⠂⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣧⣿⣿⣿⣿⣿⣿⣴⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢿⣿⣿⣿⣿⣧⠈⠛⢷⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣷⣦⣙⢿⣟⢿⢿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠿⠿⠿⢿⣿⣷⣿⣿⣿⣿⣿⣿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠃⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⢹⠀⠀⠀⠀⠈⠉⠛⠶⣞⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢠⡞⠀⠀⠀⠀⢀⣀⡀⠀⠈⠻⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣄⡀⠀⠀⠀⡾⠀⠀⠀⠀⢀⡟⠉⠉⢳⣄⡀⠈⢳⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⠲⠶⠚⠁⠀⠀⠀⠀⠈⣇⠀⠀⠀⠉⠙⡇⠀⢷⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⢀⡇⠀⠸⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⡞⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠀⠀⠀⢰⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠸⠦⠶⠶⠦⣄⠀⠀⣠⠶⢦⣄⠀⠀⠀⠀⢀⡾⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠈⠙⢳⣀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠙⠦⣤⣀⡀⠀⠀⠀⠀⣀⣀⡤⠞⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣈⣉⡉⠛⠋⣉⣉⣡⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠀⢸⠀⠀⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣤⣄⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣠⡬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⠐⠂⠈⠉⠉⠁⠐⠒⠒⠤⠤⢤⣀⣀⣀⡀
⠀⠀⠀⢀⡤⠴⣒⣀⡉⠁⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠓⠒⠲⠦⠤⣤⣀⡀
⠀⠀⠀⡘⠀⠀⠀⠀⠈⠉⠑⠒⠒⠦⠤⢤⣀⣐⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⢙⡏
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠦⠤⣤⣀⣀⣀⣀⠤⠴⠚⠋⠉⠀⠀⠀⠀⠸⡄
⠀⠀⠀⣄⣭⠷⠶⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⡀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⢉⣒⡢⠤⡄
⠀⠀⠀⢸⡏⠙⠲⠤⣄⣀⠀⠀⠉⠉⠓⠲⠶⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠉⠙⣷⣦⣤⠶⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⢹
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠉⠙⠒⠦⣤⣀⡀⠀⠀⣀⣀⣉⣭⠽⠷⠶⠶⢤⣼⣤⡶⠟⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎
⢀⣴⣶⣏⡙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⢒⣒⣫⣭⣽⣿⠟
⣾⠇⠀⠉⠙⠛⠷⢮⣿⣿⣶⣤⣀⡀⠀⠀⡾⠐⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠶⠒⣚⣋⣩⡥⠤⠶⠖⠛⠛⠉⠉⠁⠀⠀⡏
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣶⣥⣧⡴⠶⢒⣛⣛⣭⣭⣴⠶⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠘⠿⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠛
⠀⠀⠀⠀⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣀⣀⣧⠀⠀⣀⣀⣠⣤⣴⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠾⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⢋⣭⡀⠁⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢙⡛⢂⣶⠟⠋⣰⡀⠀⠀⠁⢠⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⣰⣷⢮⣄⠐⠒⠁⢰⠄⢸⠀⠀⡈⠃⢀⠀⠀⠙⠳⣤⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣹⣦⣴⣯⣿⠙⠋⠛⠋⢂⠲⠀⠀⣾⡆⠀⠰⡄⢈⠀⢀⠀⢰⡧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣿⣿⣿⡇⠀⠀⠀⣂⠒⠀⠀⠀⠹⣷⠀⠀⢱⣼⣴⡞⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⡀⠈⠻⣿⠁⠀⠀⠀⠈⠀⠀⢰⠀⠀⢻⣇⠀⠈⣿⡿⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢴⣶⣷⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠘⣿⣆⠀⠻⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⣤⣖⡐⢎⡉⠓⠈⠛⡿⣟⡘⢷⣄⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⣸⣿⣶⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠿⣟⢂⡀⠀⠈⢻⣿⣿⡇⠀⠀⠸⣧⠈⢀⠀⠀⠀⠀⠀⠰⣾⡿⠁⡜⢫⡿⢿⡿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡀⠀⠀⠸⣿⣿⠇⠀⠀⠀⠻⠦⣀⣿⣀⣀⠤⣡⣹⡿⠁⣼⣷⣼⣹⡾⠛⠁⠀⣀⠀⣠⣀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠻⠟⠀⣜⣽⠿⡿⡜⢡⣦⣾⠞⠛⠉⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⣦⣵⡹⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡃⢠⣄⡴⢴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣸⠿⠏⠙⠚⠻⢟⣩⣗⣶⣤⣤⣀⣀⣀⠀⠀⠐⣴⠋⡾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢂⠽⣿⠟⠉⠀⢻⡏⣿⠿⠀⠀⠀⠇⠈⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠐⡇⠀⠀⠐⠊⣁⡷⠀⠀⠀⠀⠀⠂⠢⠌⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣀⡀⢀⣠⠄⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡤⠃⣀⣀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠁⠀⣄⣰⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠊⠀⠂⠂⢀⠀⠀⠀⡇⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠙⠀⠐⠦⣄⠀⢈⠁⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠄⠻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⡆⠀⠀⣵⡠⠀⢀⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⣀⢀⡷⠫⢠⠌⠻⠀⠀⠋⣤⣀⠜⠡⣀⡨⠿⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠹⣾⡀⠀⠀⠀⠀⠀⠀⠀⠃⡜⠀⣹⡅⠀⠀⠀⠀⢈⡶⠹⢦⡀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⡜⠁⠀⠀⠀⡶⠉⠀⠈⠙⠳⠶⣦⣠⡄⠁⡄⣰⠂⢰⣍⠀⣤⠶⠾⠛⠋⠁⠀⠀⠳⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠱⠀⡄⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡈⡁⢈⡿⠁⣉⠠⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣰⠀⠈
⠀⠀⠞⠀⠀⠀⢹⡮⢠⣦⣄⡀⢀⠀⣤⡄⠀⠀⠀⢸⡗⠈⢺⠺⡗⠢⢯⡀⢀⣀⣠⣤⣤⣄⡠⠼⣦⡟⠀⠀⠀⠘⠄
⣰⠀⢦⡀⢀⡀⠈⣇⣨⠁⠉⠋⠉⠙⠉⠀⠳⠀⢀⠈⠑⡅⠼⣆⡃⡤⢾⠁⠀⠉⠀⡁⠀⠄⠀⠚⣛⡇⠀⠠
⠡⠄⠀⠁⠀⠃⢀⠻⣄⠀⠀⠀⠀⠀⠀⠐⠀⠑⠚⠃⠀⠀⢾⣀⢁⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠞⢀⠀⠀⠀⠆
⠀⠐⠀⠀⠀⠀⢾⡗⠌⠑⢂⠀⡀⠀⠀⠀⠠⠒⠀⠀⡀⠀⡇⢸⣾⢃⠰⣦⣄⣔⡁⢀⠠⠞⡋⠁⠆⡿⠆⠀⠘
⠀⠀⢠⡀⠀⠀⠀⠃⠀⠀⠀⠑⠀⠂⠀⠀⠆⠀⢀⡇⠀⠀⢾⣇⡘⡆⣲⡝⣿⡿⣷⠃⠀⠀⠀⠠⣸⠀⢀⡀⠀⠀⢂
⠀⠃⠀⠐⠀⠀⢱⣼⠃⠀⠀⠀⠀⠀⠀⠨⠀⠀⢸⠀⠀⠁⣎⠑⠇⠃⣷⡆⢸⣿⣿⠀⠀⠀⠀⠀⣟⡶⠊⠀⠀⠀⠐
⠀⠀⠃⠀⠊⠐⢀⡀⠷⠀⠀⠀⠀⠀⡀⠐⠀⠀⢰⠠⣶⡾⠏⢹⣶⣄⡉⠁⢸⣿⣿⡇⠀⠀⢠⢸⠏⣠⠦⠊⢀⠔
⠀⠀⠀⠀⠡⠄⠀⠛⢛⡄⠀⠀⠀⠈⠷⡤⠀⢀⢀⠓⡈⠑⣣⢸⣶⣾⣿⣷⡾⠿⡉⠃⠀⣐⣳⡿⠂⠁⢀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⣴⣤⡽⠆⠀⠀⢀⣤⣤⣠⠤⠖⠚⠚⠉⠉⠋⠙⢟⢿⣿⣷⡶⣤⣷⡇⣱⡿⠿⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢧⣀⣼⢣⢮⣧⡶⠞⠋⠀⡀⣈⣀⣉⣻⣿⣿⡷⣧⣻⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡻⡥⣤⣭⢹⠀⠀⠀⡅⣿⢦⣉⠉⢩⣻⣷⣶⣿⣤⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠓⢋⡹⠾⠆⠉⠉⠁⣭⢩⣭⡕⣺⣿⢋⡟⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣬⢏⢸⠑⠀⣀⡀⠿⠈⣚⣓⣿⣿⣿⣿⣽⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣅⡘⠛⠈⠀⠀⢠⠐⠚⠛⠩⣶⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣻⣯⣼⠗⠀⣴⣿⠟⠞⠀⢶⣴⣶⣿⣾⢿⡿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⡶⠀⠀⠀⠈⢹⠀⠀⣩⣿⣿⣿⡖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠖⢶⡬⠗⠀⠀⠻⠀⠈⠹⡽⣿⢿⠟⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⡇⣠⡅⢀⣉⣉⣀⣉⣉⡉⠠⡼⠸⠇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣕⠉⠀⠁⢿⠅⠀⠉⢻⠀⠀⠈⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⡃⠀⠀⢡⠀⠀⠀⣘⠂⠀⠀⣜⠀⠀⠀⠇⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢀⠀⢹⣄⠀⠀⠲⣀⠀⠀⠀⢸⡴⡃⠀⢠⣼⡀⢀⣀⠜
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣤⣥⣿⣶⣯⣥⣭⣯⣭⣼⣥⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣰⣯⣍⣩⣭⣭⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣀⣀⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠛⠛⠛⠛⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠿⠿⠷⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢀⣠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠟⠉⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⠏⢀⣤⣾⠿⠋⠉⠉⠛⠛⠻⠿⢿⣿⡶⠶⠿⠿⠿⠛⠛⠋⢩⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⢾⣿⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠉⠀⠀⠙⠻⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠃⠀⠀⠀⠀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⡶⠿⠟⠛⠿⣶⣤⡀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠀⠀⠀⠀⠈⠑⢦⣠⣴⣶⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⡞⣠⠖⠒⠖⠒⠊⠁⠈⠻⢿⣿⣿⣿⣿⣿⢿⣿⣦⣀⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠘⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⡿⠃⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠔⠋
⠀⠀⠀⠀⢠⡞⠛⠛⠛⠛⠛⠛⠛⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠉⠉⠉⠉⠉⢹⡄
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣾⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⢿⠀⠀⡄⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠈⠧⢼⡃⠀⢸⠀⠀⢹⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣿⠀⠀⡇⠀⠀⡷⠞⠁
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡅⠀⢸⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⡇⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠉⠙⠉⠉⠠⠦⠄⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠈⠉⠙⠉⠛⡀
⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠰⠧⠤⠄⣀
⠀⠀⢀⣠⡄⠒⠛⠓⠒⠒⠒⠒⠒⠂⠀⠀⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠑⢢
⣀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠛⠦⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣌⠳⠆⠈⠉⠉⠉⠉⠠⠾⢋⡴⢆⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠴⠖⠉⠁
⠀⠀⠀⠀⣿⠛⡟⠀⠠⠤⠤⠤⠤⢀⡴⣛⡾⠁⣀⡀⣀⠀⢀⠀⠀⠀⠀⠀⠠⠴⠶⢌⡷⡄⠚⠛⠋⠁⠈⠻⢄⣘⡷
⠀⠀⠀⡌⢿⠏⠀⠀⠀⠀⠀⠀⢀⢮⡙⠁⠀⠀⠀⠀⠀⢰⠄⠀⢴⠄⠀⠀⠀⠀⠀⠈⠳⣡⡄⠀⠀⠀⠀⠀⠀⠹⠇⣴
⠀⠀⣴⠃⠘⠀⠀⠀⠀⠀⠀⢠⡙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠋⡸⠂⠀⠀⠀⠀⠀⠀⢸⠉⡆
⠀⠀⣅⢠⡆⠀⠀⠀⠀⠀⠀⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣴⠀⠀⠀⠀⠀⠀⢸⠀⡘
⠀⠀⠈⠲⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠟⠁
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠻⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⢰⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⢴⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠃⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠙⢾⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢾⠓⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣀⠀⠀⠀⠀⠙⠿⣦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⣴⠷⠋⠁⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⡀⠀⠀⠀⠈⠛⠛⢷⡾⠷⠾⠟⠛⠉⠀⠀⠀⠀⢀⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠂⢤⣤⡄⠀⢸⡂⠀⠀⣿⠂⢠⣤⡤⠰⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢹⠃⠀⠀⣟⠀⠀⠀⠀⢀⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣼⡁⠀⠀⠉⠉⠉⢹⡇⠀⠀⣿⣉⡉⠉⠉⠀⠙⣧⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢸⠉⠳⣄⠀⠀⠀⠘⣇⠀⠀⠟⠈⢡⠀⢀⡴⠛⠋⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢀⡌⠀⠀⠈⢳⠄⢀⡜⠛⠳⠤⠀⠀⠈⠾⠯⠤⢀⣸⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⣼⣧⣤⡤⠰⠛⠲⠞⠙⠲⠀⠀⠀⠀⠸⠀⠀⠀⠈⠉⠑⠒⢠⣄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠷⠶⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠶⠆⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡦⠤⠤⠀⣾⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠈⢤⠤⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⡀⠀⡀⠀⠀⣇⣀⣀⣀⡈⠉⠿⢂⠒⠒⠀⢤⣤⣄⣀⡀⡀⢀⡀⢀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠉⣟⠉⠉⢻⠋⠹⠟⠛⢩⣽⠛⠏⣭⡅⠴⡉⢹⠖⠒⠐⡟⠈⢿⡁⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠒⠋⢉⠉⠹⢇⠿⠲⠶⢶⡄⠀⠀⣰⠴⠶⢧⣸⠇⠀⠈⠙⠆⠛⢃⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⣄⣠⣾⡦⢾⣿⣦⡀⠀⠀⠀⣠⡟⢹⡟⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠠⣿⡟⠛⠿⠀⠙⠁⣠⡛⠀⠱⠀⠀⠀⡛⠁⡾⢷⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢿⣥⠄⠒⠒⠈⢀⣍⠉⠀⢿⠀⠀⣠⠃⢸⠀⠀⠳⡀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠲⠂⠈⠉⣹⠏⠉⢹⣿⠀⢠⡋⠀⠿⢢⡄⠀⢣⣀⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠶⠛⠉⠠⠷⠤⠰⠾⠛⠳⡄⠙⠾⠆⠾⠇⠀⠈⠁⠖⠁⠶⠶⠄⠀⠀⠀⠀⠠⠶⣷
⠀⠀⠀⠀⠀⠀⢰⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠
⠀⠀⠀⠀⠀⢀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄
⠀⠀⠀⠀⠀⡌⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⡇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢡⡀
⠀⠀⠀⠀⠀⠏⠀⠀⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠈⢃
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠧⠤⡤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⡀⠐⠀⣠⣄⣠⣀⣀⣀⣀⣀⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⢞⣛⡿⢿⠿⣛⣛⣫⣿⣬⡁⣿⡏⣨⣿⣯⣛⣛⣛⠿⠿⢿⣛⠢⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡷⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢥⠶⢂⣀⣀⣀⣀⡰⠶⠬⠓⠉⠘⠫⡴⠖⣀⣀⣀⣀⣀⠶⢦⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠂⠀⠀⠀⣀⡀⢀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠤⠤⠤⠀⠀⢀⠀⠀⢹⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠩⠉⢉⣁⣀⠀⠠⢼⡯⡀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠉⠐⢄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⣒⡒⠒⠆⠀⠀⡜⠧⡀⠰⠀⠀⠀⠀⠀⠌⠀⠀⠁⠀⠀⠄⠀⠠⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠤⠀⠤⠭⠭⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⡀⠁⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠉⠉⢁⣐⡒⠢⠠⠭⣁⣺⢀⣀⠀⠀⠀⡂⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣒⡒⠒⠢⠬⠍⣉⣒⠒⠼⣿⠉⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⡠⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠤⠤⠤⠭⠍⣁⣒⠂⠠⠬⢹⠿⠀⠀⠀⠀⠀⠂⠁⠂⠀⠀⠀⠐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠉⠉⠃⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠙⠒⠂⠤⢤⡨⣄⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⣴⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⢿⣿⢏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣛⣛⣃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠿⠦⠉⠾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⡇⠀⠀⠀⣯⡹⡿⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣩⣴⣶⣶⣶⣶⣭⣙⠿⡇⠀⠀⠀⠛⠛⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⡞⣵⣿⣶⣶⣶⣶⣶⣶⣶⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⠿⠿⢿⡟⣼⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⡄⠀⠀⣠⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⡟⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⣇⢿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠇⠀⠀⣿⣌⠛⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⡻⠿⣿⣿⣿⡿⠿⣛⣱⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣷⡎⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠈⠻⠿⠇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠠





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠈⠉⠉⠀⠀⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠠⠲⠠⠒⡂⠀⠄⡀⠀⠀⠉⠢⣄⡄⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠠⠄⠐⠂⠠⠀⡀⠀⠀⠀⠑⠀⠀⡐⠋⠀⢀⣠⢤⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⣀⠀⠀⠂⠈⠴⣂⠥⣂⠍⡢⠄⠀⠀⠜⠀⠀⠘⠁⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡼⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠠⠁⠀⠂⠤⢐⡂⠄⡀⠀⠁⠀⠀⢀⠎⠀⠀⠈⠀⠈⠉⠈⠁⠠⠄⠀⠈⠁⡐⠂⠀⠀⠀⠀⡜⣡⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⡀⢀⣀⠀⠤⢈⡑⠂⠅⠂⠤⡑⠪⡑⠀⠀⢀⠊⠀⣀⠨⠀⠀⠀⠒⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡜⣰⢿⣿⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠄⢐⡂⠤⢁⠁⠂⠍⡒⠤⡀⠈⠑⠀⠀⠠⠁⠀⠠⠐⠒⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⢀⢞⣼⣯⣿⡏
⠀⠀⠀⠀⠠⢎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⡂⠀⠈⠀⠈⠂⠀⠀⡐⠁⠀⠀⢁⡀⠠⠤⠄⠉⠉⠁⠰⠆⢀⡂⠀⠀⠀⠀⢠⢮⢮⢯⣾⣿⢫⡇
⠀⠀⠀⠀⠠⣮⣌⣉⣉⠒⠢⠤⣀⠀⠤⠀⠀⠀⠈⠑⠄⠀⠀⠀⢀⠜⠀⢀⡄⠰⠀⠤⠤⠤⠤⠀⠀⠀⠉⠁⠒⠀⠀⠀⠀⡰⢣⢣⣣⣿⡿⣣⡻⠁
⠀⠀⠀⠀⢴⢥⡤⠤⢭⢭⣍⣒⡢⠝⡒⣤⡀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⡱⢳⣿⡿⣱⡗⠁
⠀⠀⠀⠀⢹⠥⡆⢀⡀⠀⠉⠁⠛⠿⠷⢿⣾⣷⣤⡀⢀⠀⡠⠗⠀⠀⣀⣀⣤⣤⡤⡤⡤⠤⣤⣤⣀⣀⠀⠀⠀⠀⠀⡰⡱⡽⣱⣿⡟⣴⠟
⠀⠀⠀⠀⠀⢱⢘⣕⠯⠕⡖⠚⠓⡒⠷⣶⣟⢛⣻⣿⡷⣴⣶⢶⡿⢟⣳⣾⣭⣭⣥⣤⡴⠍⡠⡔⠂⠌⣉⠓⠶⣬⡴⣼⡝⣵⢿⠏⣾⠿
⠀⠀⠀⠀⠀⢯⣙⣨⣭⢤⡴⣶⣶⣮⣷⣾⡿⣿⣛⣻⣛⣛⡛⡻⣿⣿⣟⢛⣿⡿⡭⣭⡴⣒⣠⠀⡉⠛⠓⢦⠴⣠⢜⣟⣾⣿⢋⣾⡟⠄
⠀⠀⠀⠀⠀⡎⠟⢹⣧⣼⣏⠁⣑⡁⠀⢉⠛⠒⠒⠲⠲⠭⢭⠭⠿⠿⠻⠿⢿⠟⣫⣭⣍⣑⣨⣍⣛⣲⣤⣄⣈⣦⣭⣿⣼⣡⣿⣓⣀⣅⣂⣬⣐⡠⢀⡀
⠀⠀⠀⠀⠀⣤⠱⠂⢻⡗⡇⢒⣚⡿⠯⠭⠽⠿⠿⠟⠛⠋⠉⠉⠙⠓⠒⢲⣖⠿⣿⣿⢟⡵⢿⣃⣠⣌⣁⠨⣍⢻⣭⣿⣷⣿⣿⣿⡽⣛⣯⣽⣷⣾⣿⠇
⠀⠀⠀⠀⠀⢿⠀⡟⠸⣏⢏⣥⡤⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⣸⣾⡟⣡⣟⠀⢽⡤⣤⡥⣼⣵⠤⣅⣬⣤⣭⣭⣤⣼⠿⠿⣿⢿⠿⠏
⠀⠀⠀⠀⠀⠹⡄⠋⠀⢺⣿⠻⠛⠿⠿⠶⡶⠾⢶⢴⣤⣀⣀⡤⢤⣒⡃⣽⡿⣣⠾⠷⡾⢀⠡⠂⢁⠩⠁⡀⠌⠀⣊⠍⠑⡀⠅⡀⠠⢁⠐
⠀⠀⠀⠀⠀⠀⣣⡀⠀⢀⠽⡬⢴⣖⣂⡤⠾⠾⠛⠻⠿⠋⠉⠉⠛⠛⡱⠶⠚⠉⠉⣼⢁⠀⠀⢄⠂⢉⡤⠶⠖⢃⡠⢤⣂⣤⣂⠜⠊
⠀⠀⠀⠀⠀⠀⠘⣯⢲⢪⢿⠉⠁⠤⠔⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⢹⣧⠆⠁⠐⢊⠰⣒⠧⠖⠋⢩⢁⡠⠍
⠀⠀⠀⠀⠀⠀⠀⠙⣭⢿⣽⠀⢒⣒⡊⠉⠀⠀⠀⠀⢀⣀⠀⠀⢀⢀⣀⡀⢒⠐⠉⠡⣟⠷⠀⢊⡉⠠⡐⣪⡤⠆⡅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣛⡼⠶⠶⣒⢒⢂⣛⣙⣉⠭⠬⠽⠫⠖⠔⠃⢘⠗⡫⢑⠩⠑⡂⠌⢀⠌⢁⠭⢀⠠⢁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⣤⣀⣀⡀⠀⠀⣠⣾⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣾⠏⠻⠉⣿⣿⠿⠿⢿⣿⡙⠻⣿⣾⡿⣿⠻⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢻⣏⣩⡒⣏⣰⣷⣶⣿⣶⡶⣿⣿⡀⠀⠻⠯⣤⢾⣽⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣿⡞⠋⣿⣿⠟⠟⠋⠉⢩⠍⠁⠉⠐⠀⠈⠀⠀⠀⠀⠀⠈⠈⠟⣿⣧⡝⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣼⢿⣿⠃⠈⠁⠀⠀⠀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⣶⣿⣿⢿⣎⢻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢹⣿⡟⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣯⣻⣿⣆⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣤⣿⣿⣶⣦⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣾⣿⡟⣻⣃⠀⠹⠆⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢰⣿⣿⣼⠏⠙⢳⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠘⣿⠷⠾⢻⣿⣶⣧⣙⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡷⠀⢀⣸⣿⣙⠢⠌⠐⢧⣜⠳⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣥⣶⣿⣿⠫⣝⠙⢿⡆⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⢻⣿⣅⡁⠀⠀⠀⢄⠘⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⣾⣷⠿⠟⠠⠇⠙⡛⢻⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣶⠀⠈⠘⣿⣿⠋⢰⡀⠀⢻⡦⢫⡻⣿⣆⠀⠀⠀⡰⠃⠀⠀⠠⣿⢿⣯⠘⠃⠲⠆⠀⡀⢰⣻⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣧⠀⢘⠀⣿⣿⡇⢺⠠⠀⠐⠀⣀⣀⣿⣿⡄⠀⠀⠀⠐⠀⠀⡀⠉⠻⢿⣷⣦⠀⠀⠐⠀⣾⣿⠁⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣀⠈⣰⣿⡿⢷⣿⣠⣸⣿⣾⣿⠟⠚⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠤⠸⣿⣿⣀⠀⣄⣿⡿⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠲⣿⣿⣷⣼⠷⠟⠛⠡⡍⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣹⣿⠃⢾⣟⡞⢡⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡝⢿⣿⠀⠂⠀⢀⠀⠀⠀⠀⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⢿⠟⠀⢾⠎⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣻⣦⣴⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⣰⣿⣿⣆⡔⢁⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣷⣤⣁⡀⠂⠀⠀⠀⠀⠚⠀⡀⠀⠀⠀⠀⣤⣶⣯⣽⠝⢉⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣶⣤⣤⠴⠆⠀⠀⣀⣠⣠⣤⣬⠟⢋⣡⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣿⣤⣍⣙⡛⠛⠛⠚⠛⠛⠛⣋⣋⣥⣴⣾⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣭⣿⣿⣿⣿⣡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣤⣿⠛⠛⢻⣟⣿⣭⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣯⣽⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠒⣋⣿⣼⣿⣏⣘⠲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣛⣭⣿⣿⣷⣶⣤⣬⣿⣿⡿⢟⠉⢙⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠤⠿⠷⣶⣿⠋⢀⣀⡀⢀⣙⣻⣿⣭⣶⢿⣏⠀⢙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣦⠁⠀⠀⠉⠉⠉⠙⠋⠛⠉⠛⢄⢀⣁⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠳⠶⠶⠶⠦⠤⠤⠔⠟⠛⠛⠉

⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣶⣶⣴⣶⣾⣶⣶⣶⣷⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣾⣿⣷⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣾⣷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⠇⠀⠀⠀⠀⠀⣾⣅⣴⢏⣀⣿⣧⠈⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡆⢻⡋⣟⢲⣄⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⣠⠀⠀⠀⣸⣏⣦⣻⠛⢻⣿⢶⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣶⠿⣿⣿⣦⠀⠀⢡⡮⣿⠋⠂⢀⣼⡇⠀⠀⣠⣿⠏⠀⠀⠀⠀⣿⣳⣷⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⣩⣌⠉⢰⣷⣼⣣⣿⠈⢠⣿⣿⡇⠀⠀⢹⣿⡄⠀⠀⠀⢰⣵⣿⣽⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣀⢰⣟⡛⢿⣧⠈⣿⣿⡿⡏⠀⠸⣿⠟⠀⠀⠀⠸⣿⣧⣤⣀⣼⣿⣿⡿⠥
⠀⠀⠀⠀⠀⠀⠀⠘⠁⣠⣶⢶⡆⢿⣿⣄⡈⢿⣆⠉⢁⣀⡇⠀⢸⡏⠀⠀⠀⠀⠀⣹⠿⣯⣽⣿⠿⠋⠀⣠
⠀⠀⠀⠀⠀⠀⢀⣄⣚⠛⠻⠸⡇⣸⣿⣿⣿⣷⣿⡆⢿⣯⣿⣰⣾⠀⠀⠀⠀⠀⡾⠚⣿⠿⠋⣠⣶⣶⣿⣧⣴⣶⡞
⠀⠀⠀⠀⠀⠀⠈⣿⠉⠁⠀⢺⣧⠉⣿⣿⡽⢿⣿⣿⠚⠛⢿⠿⡟⠃⠀⠀⠀⢸⡇⣼⠃⠀⢠⣿⣿⣿⣟⣿⣿⣿⡿
⠀⠀⠀⢀⣀⣀⠀⢸⠄⠀⠀⠀⡹⣟⠷⣿⣳⣾⣿⡇⢀⣷⢸⡆⡇⠀⠀⠀⠀⠀⣰⠃⠀⣠⣿⡟⣿⣿⣿⢿⣿⣿⠁
⠀⠀⠀⠋⢻⣁⠀⠀⠀⠒⢢⣴⡾⣿⣦⣻⡿⠟⠋⣀⠀⢤⣾⣿⣷⣶⣶⣄⡀⢠⡏⠀⣼⣿⣿⣿⣿⣿⡿⣈⡽⠂
⠀⠀⠀⠀⠀⠛⠿⠿⢿⣖⣀⣈⠙⠻⡟⢋⣀⣾⣿⠛⢆⠘⣿⣿⣿⣿⣿⣿⣿⣯⢀⠞⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⡿⠙⡟⠻⣆⠈⠆⠸⣿⣿⣿⣿⣿⣿⣿⡇⢀⣬⣿⣿⣯⣉⣥⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠤⢅⡀⠲⣎⡗⠀⠁⠀⠳⠀⠹⡆⠀⠀⢸⡯⠙⠛⠻⠿⠿⠴⠿⠿⠟⠓⢿⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⡔⢋⣵⡆⣀⡜⠂⣨⣿⣿⡆⢠⡄⢀⣄⣿⣶⠟⡙⢷⡶⠆⢀⣀⢲⡶⢶⣦⡰⡞⢹⣿⣿⣿⣿⣿⣧
⠀⠜⠀⠀⠠⡙⢾⣭⡗⠩⠐⠿⠃⣄⣿⣧⠀⠿⠄⠹⠟⢡⣾⡇⠀⠀⠀⢉⣉⣈⠁⣠⠑⠚⠁⠺⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠔⠆⢀⡀⠂⠀⠀⠀⠠⢫⣿⣿⡀⢀⠐⠀⢰⣿⣿⠃⠞⢰⡿⢿⣋⣥⠤⠷⢤⡗⠲⢶⡦⠠⠴⠖⢤
⠀⠀⠀⠀⢀⠻⣿⡀⠀⠀⢰⣮⠴⠛⣛⠻⣧⠸⡇⠠⡆⣼⣿⠆⣰⣧⣷⡾⡍⠁⠀⠀⠀⠻⣦⡀⠀⠲⣦⠄
⠀⠀⠀⠀⠙⠓⠛⠛⠒⡀⢋⣡⠖⠋⠁⢹⣿⡿⠁⠀⠠⠟⢁⣴⣯⡼⠏⠀⠀⠀⠠⠦⢄⡃⢉⣻⡄⠀⠀⣀⠀⢰⣶⣶⣶⡄
⠀⠀⠀⠀⠁⠀⠈⠛⠛⣰⣿⣏⣠⣤⡴⠟⠉⠀⡀⡀⠷⡄⢿⣯⣤⣠⡄⠀⠀⠰⣾⣿⢦⠄⠚⣉⡻⣅⠀⣿⠀⠈⣿⣿⡏
⠀⠠⠤⠀⡀⠀⠀⢠⠾⢛⠿⠉⠁⠀⣤⡆⠀⢾⣦⡇⠀⠀⢸⣿⡧⢻⣤⠀⢀⠳⠨⠱⠟⣙⡔⢉⠕⠉⣠⠾⠶⣠⡿⠛
⠀⠀⠀⠀⠀⠐⠀⠀⢁⠉⠀⣴⣿⣆⡹⠀⣷⣤⣉⣁⠀⠀⠀⠻⢿⣦⠛⡷⠸⢷⣈⣥⣤⠟⠠⠞⢀⣾⡟⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⣾⣿⣿⣿⣦⠀⢿⣿⣿⡏⠀⢀⣤⡶⠛⡍⠙⠿⣿⣿⣷⣶⣶⠶⠚⣠⣎⡁⢄⡐⣉⠉
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡿⣿⣿⣿⣤⣦⣉⡉⠁⠀⠀⠈⢀⢰⡇⠀⠀⠀⠈⠉⠡⣶⠶⠋⠉⣀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣀⢰⣿⡟⣿⣿⣿⣿⣦⢰⣄⠀⠀⠀⠘⠛⠛⠂⠀⠀⠀⠀⢀⣴⣾⣿⠋⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⣤⣶⡾⣻⣿⣿⠘⣏⣹⣿⣿⣿⣿⣿⣿⣿⡇⣄⢀⡤⠤⠀⢠⠶⠟⠐⠺⢻⣿⠿⠃⠀⣠⣤⣾⣿⣶⣄
⠀⠀⠀⠀⠀⣼⣟⢫⣿⡿⣿⡽⠀⠸⣿⡿⢿⣿⣿⣿⢹⡁⣿⠃⣀⢤⣦⣤⡐⣶⣶⣄⡀⠀⠘⢀⣤⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣼⣿⢛⣿⣿⣿⣿⠇⠀⠀⢹⣷⣿⣿⣿⡟⣿⣄⣿⣿⣿⣷⣬⣿⡇⠸⣿⣿⣿⣦⣴⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠙⠛⠿⠇⣿⢾⣿⡜⠳⠾⢿⣿⡇⠀⠘⠿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠙⠛⠉
⠀⠀⠀⠘⣿⣿⢿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠹⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢸⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⣷
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⢸⡆⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢀⠠⠊⢡
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠯⣕⣦⡀⢀⡄⣊⣀⣀⡀⡠⠂⠁⠀⢀⡾⠃
⠀⠀⢸⡷⠦⠤⢤⣀⠀⠀⠃⢀⣠⣤⣴⣒⡈⣿⠀⠀⠀⠀⠀⡩⠛⠿⠿⣟⡒⠤⣁⠀⠀⢀⠔⠉
⠀⠀⠺⠿⠿⠷⠶⠶⠿⠷⡾⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠓⠾⡦⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣬⡓⢤⡀⠀⠀⠀⠀⠀⢑⠥⠆
⠀⠀⠀⣶⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⣞⡂⠀⠙⠲⣌⡒⣤⡦⠐⣈⠤⠒⡀⠒⠂⠀⠠⠤
⠀⠀⢀⠱⡈⠑⢄⠀⠀⠙⠢⣄⠀⠀⠀⣴⣿⠽⣮⡄⠀⠀⠈⠙⠷⠛⡍⠀⣠⠞⠁⠀⠀⠀⠀⢀⡰⡯
⠀⠀⠙⢇⠈⠢⡀⠀⠤⠄⠀⢈⣻⣆⣼⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⡠⢊⠞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⠰⣾⣻⠿⠟⠛⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠦⢉⣁⠒⠂⠤⢎⠔⠁
⠀⠀⠀⠀⠀⠈⠳⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣀⣠⡤⠴⠶⢖⠖⠒⠒⠶⠶⡦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⢯⢻⣤⡀⠂⠉⠀⠀⠀⡀⠐⠂⠄⠀⠀⠍⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⠁⠄⣾⣷⠚⠁⢀⠀⢰⣞⡷⢀⡀⠀⠀⠀⢀⣤⣌⣀⡈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠅⠈⠀⠀⠀⢸⠍⣶⠦⣼⢳⡎⠙⢚⣉⡼⠂⢀⣁⠈⠉⠙⡆⠙⠶⣄⡙⢦
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⢀⠄⠀⠉⠸⢾⠽⣶⠢⠶⠶⠚⠉⠀⠀⠀⠀⠙⠻⢿⣾⣡⣤⠢⠀⠙⠛⢳⡀⠀⠀⠀⠐⣧
⠀⠀⠀⠀⠀⠀⡼⠁⠐⠀⠁⠀⠀⠀⠀⠀⣼⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢸⠀⠀⠀⠈⢀⣱⡤⠖⠋⠀⠈
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⠀⠀⠀⢠⠇⠀⠀⠘⣆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠴⠞⠉⠀⢳
⠀⠀⠀⠀⢠⡇⠀⡀⠀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠄⠀⠀⢀⣠⡤⠖⢻⡉⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣀⠽⡞⠓⢦⣀⡤⡶⠚⠉⠀⠀⠀⠀⢧⠀⣀⠀⡀⠀⠂⣧
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠈⠉⠛⠋⢁⣠⠽⡖⠋⠛⢦⡐⡀⠀⠀⠀⢀⠀⢸⠘⡏⡇⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⢹⡀⠀⠀⠹⣇⠀⠀⠀⠈⢀⡏⠀⠷⠃⠀⠀⡀⡏
⠀⠀⠀⠀⠸⡶⠒⠋⠙⣆⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠻⣄⡀⠀⠀⡟⠀⠀⠄⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢷⢀⣀⡥⠞⠛⠻⠶⠲⣄⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢳⡀⠀⠀⠈⢳⡀⢠⠏⠀⠀⠀⠀⠀⢀⡞
⠀⣀⡀⣠⠴⠚⣯⠁⠀⠁⠀⠀⠁⡀⠸⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠈⠓⠋⢀⠀⢀⠀⠄⢀⡿⠁
⠙⣿⠷⠀⠀⠀⠈⢧⡀⠀⠀⡀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢧⠀⠀⠀⡀⠀⠐⠀⠈⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠳⣄⠠⠀⠀⢹⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡀⡠⢂⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠣⣀⠠⠄⢻⡅⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⢈⡧⠤⠶⠾⢶⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣟⡒⢮⣳⣤⡄⢁⠀⠀⢀⣀⠤⢀⡴⠋⢀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢧⣤⣬⣥⣬⠤⠼⠗⠚⠉⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦

⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡗⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⢷
⠀⠀⣰⣿⣿⡿⠟⠛⠛⠋⠐⠀⣀
⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣀⣀⣀⡀
⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⣤⠒⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⢿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠁⠐⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠙⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⣿⣿⣿⣿⠏⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠋⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⢶⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣶⣦⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣇⣀⣸⣿⣿⣿⣿⣿⣿⠿⠛⣫⣄⠰⢿⣶⡿⠟⣡⣾⣷⣤⣉⣩⣥⣶⣌⠻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡋⠱⣀⣦⣡⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠿⢮⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡁⣂⣴⣾⣿⣿⣛⣩⣤⣌⣙⠛⠉⢁⣈⣉⡙⠉⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣟⡈⣬⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠛⠛⠛⠋⠘⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⡉⠭⠍⣁⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠊⠁⠀⠀⠀⠀⠉⢢⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡄⣂⠉⠉⠉⠉⠉⠉⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡑⢌⠢⢄⣀⣀⠤⢊⠜⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣁⠂⠀⠀⢂⡡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠉⠉⠉⠢⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣹⠉⡉⠉⢹⡁⠀⣀⠀⠀⠀⠀⢸⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠉⠀⠘⠋⠀⠀⠈⠁⠢⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠂⢀⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠱⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠢⣀⠌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⡎⢐⠄⢀⠠⡀⠀⠀⠀⠘⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠁⢀⢀⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃⠠⡄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠉⠰⠒⠉⠑⠂⠀⠁⢠⡍⠉⠒⠊⠉⠃⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⢎⠈⡳⠸⠏⠒⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠋⡀⢸⠀⠀⠀⠀⠀⠀⠀⡠⠋⠑⢌⣉⡥⠚⠀⠤⢄⡀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠘⠤⢂⢌⠔⠀⢸⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢀⠌⠀⠀⠀⠀⠀⠈⠑⢠⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠈⠢⣑⠤⣀⠀⠀⢎⡰⠀⢀⡠⢐⠥⠊⠀⠀⢸⠀⠀⠀⠀⢠⣎⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⡀
⠀⠀⠀⠀⠀⠘⡤⠤⠤⠤⣬⠃⠀⠀⠀⠁⠒⠨⠭⠬⠭⠭⠐⠊⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠪⣓⢦⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠱
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠇⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠈⢁⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀
⠀⠀⠀⠀⠀⡰⠡⢤⠬⠥⡌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠂⡶⡰⣒⠂⠲⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⡜
⠀⠀⠀⠀⡰⠱⠁⠀⠠⡄⠀⠌⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⠃⡣⡘⣢⡀⠀⠀⠀⠀⢀⠤⠐⠒⠂⠤⣀⠀⢀⠔⠁⢀⠜
⠀⠀⠀⠰⠡⠃⢀⡀⠀⠀⠀⠐⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛⢊⡁⠈⠓⠤⣀⠜⠁⠀⠀⠀⠀⠀⠀⠑⡁⠀⡠⠊
⠀⠀⢠⢁⠂⠀⠀⠀⠀⠀⠀⠀⠐⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠲⢄⡀⠸⠀⠀⠀⠔⠁⢑⠀⠀⠀⢰⠈
⠀⢀⢂⠂⠀⠀⠀⠀⠰⣉⡁⠀⠀⠑⡐⡤⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠸⠤⠤⠀⠤⠄⢤⠀⠀⠀⢙⠀⠀⠀⠈⠐⠈⠀⠀⠀⢸
⢀⢂⠂⠀⠘⠃⠀⠀⠀⠀⠀⠰⠉⠆⠱⡑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣆⣘⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣁⣵⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡆⡄
⠀⠀⠀⠀⠀⠀⠀⢧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⠙⣿⡾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⣿⣷⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⣿⣇⣼⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣎⣿⣿⣨⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣋⠘⣻⠇⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⡏⠀⠀⠀⠀⠀⠀⠀⠰⡀⢀⠄⠈⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⢀⣀⣀⣠⣹⡿⡆
⠀⣀⣀⣤⣤⣀⡀⠡⣀⠁⠀⠲⢽⡀⠀⠀⠀⠀⢀⣶⠞⠀⠀⢠⡟⠙⠛⢿⣡⣴⣷⣶⣤⣤⡄
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⢷⣄⢀⣵⣖⠀⠀⠀⣾⠋⠀⢀⣴⠟⠀⣤⣶⣼⠟⢱⣧⣘⣿⣯⠛⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣿⡏⠀⠀⠸⠹⠀⠀⠘⡇⠀⢠⡿⠉⠀⠀⠀⢹⡿⠸⣿⣦⣷
⠀⠀⠀⠘⠒⢄⠀⠀⠀⠀⠀⠀⠀⢾⣷⠀⠀⢸⡟⠀⣾⣴⠿⡀⠘⣧⠀⠀⠀⠠⠚⡶⣀⣹⣿⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣦⣄⠀⠀⣴⢷⣮⣤⡆⠀⠈⣇⣸⠟⠁⠀⣿⠀⠸⣷⠀⢀⡠⠞⠛⠛⠙⢿⣏⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⢀⣀⣹⠏⠐⠀⣶⣷⡏⠀⠀⢀⣿⣷⢀⣧⠞⠋⠀⡀⠀⠀⢀⣤⣹⣶⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠁⠀⠀⠀⣿⣿⠂⠀⢀⣾⣉⠉⠉⠀⣠⡴⠞⠿⠲⠛⣿⣋⣴⡙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠈⠀⠀⠀⣀⡀⠀⣧⣿⡀⢀⠌⠉⠘⠛⠟⠋⠉⠀⠀⠘⠓⢀⣿⡟⢉⠖⠁
⠀⠀⠀⠀⢠⠄⢰⠏⠀⠀⠀⠀⣔⣨⠉⠻⠻⠿⢿⣧⣿⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠈⠐⠊
⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⢸⣿⣡⣶⠒⠂⠀⢲⡏⠹⣾⣦⣠⠄⣀⣀⣠⣤⠴⠟⠉
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠘⠛⠟⠛⠀⢠⡟⠁⠀⠹⣿⣿⢿⡟⠿⠟⢻⡆
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠠⢀⣿⣇⠀⠀⠀⣱⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣠⠾⠛⠃⠉⠙⠁⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠤⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢀⣔⣁⣀⣀⣀⣀⣀⣀⣉⣢⡀⠀⠀⢶⠆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠟⠛⠛⠛⠛⠛⠛⣿⣿⠁
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⣠⣤⣴⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠔⠊⠀⢠⣶⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠈⢉⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠐⠢⣀⠎⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠻⡟⠿⠻⠿⠻⠛⠿⠿⠿⢿⠋⠀⠂⡈⣩⣦⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⣀⣀⡀⠤⠊⠁⠀⢆⣴⣾⡟⠋⠛⠁⠀⠀⢸⣷⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⡿⣿⠉⠀⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢊⠁⠂⠀⠀⢤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠙⠛⠁⠀⠈⠀⠀⠀⢀⡠⠐⠉⠀⠀⠐⡋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⣠⣤⡀⠀⠂⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⢠⣿⣦⣠⣶⣦⠀⠀⠀⠀⠀⠀⢸⣿⣿⣹⣿⣿⣿⡇⠀⠀⠆⠀⠿⠿⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⢸⢀⣿⣿⡏⣿⡇⠀⠀⠀⠀⠀⢸⣿⡿⠋⠉⠙⢿⡇⠀⠀⠀⠂⠀⠀⠀⠀⣷⣿⣷⣿⡿
⠀⠀⠀⠀⡐⠀⠰⠆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣼⣿
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⡆⢀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠰⡿⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣂⢸⣿⣛⣛⣛⣛⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⢺⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢾⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⣠⠏⣬⣭⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠈⠋⠀⢤⣿⣿⣹⣿⡇⢠⣤⣤⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⣠⣾⣿⣿⡿
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢰⣀⣨⣉⣉⣉⣉⡇⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢀⣼⣿⣿⡿⠋
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⡇⢸⣿⠿⠄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⠿⠿⢿⡇⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⢀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣧⣀⢼⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣿⣿⣀⣀⣀⣀⣀⣻⣿⣿⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⣿⣿⣿⣿⠶⢾⣿⣿⣿⣿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡏⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⢿⠇⢸⡿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⣀⠀⠀⣿⣿⣿⣧⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⢠⣶⡄⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣼⣿⣧⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠛⢹⣿⡟⠛⢻⣿⠋⠙⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⠀⢸⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣿⡇⠀⢸⣿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢰⣿⣦⡀⢸⣷⠀⠀⣿⣷⣄⣠⣾⣿⠆⠀⠀⢸⣿⡇⠀⢸⣿⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⡇⠀⢀⣤⣾⡿⢿⣷⣤⡀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣧⣾⡿⠟⠉⠀⠀⠉⠻⢿⣷⣤⡀⠀⠈⢻⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣶⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠘⠿⠟⠉⠀⢀⣤⣶⣶⣤⡀⠀⠉⠻⢿⣷⣄⣸⣿⠈⠙⠛⠻⠿⠿⠿⣿⡿⠿⠿⢿⣿⡟⠉⢹⣿
⠀⠀⠀⠀⠀⢀⣤⣾⡿⠏⠀⠀⠀⠀⠀⢼⣿⠉⠉⣿⡇⠀⠀⠀⠀⠉⠻⢿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⢀⣤⣾⣿⣟⡉⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢉⣻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠘⠛⠛⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⡆⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⣿⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⣿⠀⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⢸⣿⣶⣶⣿⣿⣶⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⣸⣿⠀⠀⠀⢀⣿⡇⠀⠀⠈⠈⠉⠈⠉⠁⠁⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿