Bär Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠶⣄⣠⡶⠶⢦⡀⠀⠀⠀⠀⠀⠀⢠⡞⢉⡍⣿⠀⢠⡾⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⠞⢿⠂⡿⠃⡴⠷⠈⣷⣄⣀⣀⣴⠟⢷⡟⠀⡼⢡⡟⠷⠟⠀⠘⢋⡙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⡃⢠⠟⣠⠁⡄⠴⠂⣰⣤⣄⣀⣀⣀⣠⣾⠁⠈⣰⣿⣷⣶⠀⣰⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣷⣤⣶⠇⠰⠟⠷⠋⣍⢻⠋⡍⠙⠁⡤⠙⠀⡞⢁⡄⢀⡏⢠⡟⢠⡖⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢸⣿⠟⣁⠀⣰⠀⡆⢠⠃⡄⢰⡷⠀⠠⢁⡆⠀⡀⢈⡀⢘⣥⠸⣃⠨⠾⣡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⣤⣤⡶⠿⠶⠿⠶⠶⢶⡾⢿⣶⣾⣷⡶⠿⠟⠀⠉⣉⣁⣤⣤⡉⢿⡛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⣀⣀⣤⡴⠶⣄⣙⠛⠛⣋⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣠⣍⠉⠉⠉⠉⣠⠤⠤⠌⠉⠙⣋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⡟⠚⠛⠛⠋⠉⣻⠋⠙⠉⠉⢩⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣿⣿⡰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣘⠲⣶⠶⠖⣲⠷⠶⣶⣾⠿⢺⣿⢾⠾⠒⠶⢾⡿⣯⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢭⠉⠻⠷⠁⠀⠀⠀⠀⠉⠉⠐⠠⣉⡀⠀⠀⣀⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⠈⠁⠠⠄⠈⠁⠙⠃⠀⠀⢤⡤⠴⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣭⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣛⣩⣴⣶⣿⣿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣫⣼⡿⠿⠟⠛⠉⠀⠈⢻⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⠀⠀⠀⠀⣠⣴⠶⠄⠘⠏⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣶⡞⣿⣶⡆⠘⢿⣿⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣤⣤⠀⠈⡇⣼⢻⣿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⢠⣿⡽⠀⠀⠉⢙⠆⠳⣦⡀⠀⣠⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⢰⣿⡿⣷⣷⡀⣟⣿⡷⢛⠀⣀⣠⣆⠀⢻⣿⠉⠀⠀⢂⠀⣄⣀⡀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⡈⠣⠟⠁⠀⣿⣿⠿⢷⣾⣿⣿⣿⣿⠷⢼⣇⣀⠔⡡⠀⠀⣿⣿⣿⡏⠛⠛⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣿⣏⣡⣴⣶⠞⠉⠀⠀⢸⣿⣿⣿⣿⢧⣤⣄⣩⣿⣁⣴⣆⣼⣷⣿⣿⣿⠀⠀⠀⡈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣏⡽⠻⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⠀⠃⠀⡇⠈⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⡉⢳⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠛⠀⠉⠉⢳⣿⣿⣿⣿⣿⠀⢠⠀⠀⠀⢿⣅⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣿⡿⠶⠏⠀⣿⡟⠀⡆⠀⠀⢲⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⢠⡌⠀⠀⠀⠈⣿⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠻⣹⣼⣽⠓⣿⡇⠀⠈⠀⠐⣸⣿⣿⣿⣿⣿⣿⣿⡆⠀⠰⣰⣿⣿⣿⣿⣿⣿⣿⢸⢃⡄⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣿⡿⠖⣋⣿⠁⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡄⢰⣿⣿⣿⣿⣿⣿⣿⣿⢸⣾⣷⡀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⡀⠈⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⠀⠀⠀⠴⠊⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣭⣽⠁⠙⠛⣦⢈⣻⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⡟⠻⠟⠐⠟⡂⠠⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢉⣉⣥⢀⣝⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠟⠀⠀⠀⠈⢳⠀⣶⠚⠃⢰⠛⢦⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠁⡌⠁⠈⠻⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⡖⠉⡠⠊⣠⠀⢱⣿⣤⣽⡇⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢿⣆⣰⣥⠀⡀⠀⢀⣼⣿⣿⣏⣻⠿⠿⠿⣿⣿⣿⣿⣿⣧⠞⣫⣀⡔⡡⢎⣼⣿⣿⣿⡇⠀⣾⣿⣶⣶⣴⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠲⠿⣿⣿⣿⣿⠿⠿⢋⣉⣉⣉⣉⣀⣀⣀⣐⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣂⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⡄⠀⡇⠀⢸⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡄⠀⢸⠂⠀⡇⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢰⣿⡆⢀⣿⠀⢰⣧⢸⣿⠀⣾⣧⢠⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡆⣼⣧⡀⣾⣧⢰⣿⡄⢀⣇⠀⢸⣇
⠀⠀⠀⢸⣿⡇⣿⣿⣿⢸⣿⢸⣿⠀⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⡗⢸⣿
⠀⠀⢀⣸⣿⣇⣿⣿⣿⣸⣿⣸⣿⣄⣿⣿⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣇⣿⣿⣇⣿⣿⣸⣿⣇⣿⣿⣧⣼⣿⡀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠉⠁
⠀⠀⠀⠀⠀⣶⠀⠀⢀⣤⡀⠀⢠⣧⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⡀⢀⣿⡀⠀⣄⠀⢀⣆⡀⢠⣿⡄
⠀⠀⠀⣾⣿⣿⣿⣷⢸⣿⡇⢠⣿⣿⣧⢠⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⡇⢸⣿⡆⢿⣿⠇⢸⣿⡇
⠀⠀⢠⣼⣿⣿⣿⣯⣤⣿⣥⣬⣿⣿⣭⣤⣽⣿⣯⣄⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣿⣼⣿⣧⣼⣿⣧⣬⣿⣤⣬⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠙⠉⣿⣿⣿⡅⣠⣤⣤⣤⣤⣤⣄⢸⣿⣿⣿⣯⢀⣤⣤⣤⣤⣤⣤⡀⣽⣿⣿⣿⡅⣠⣤⣤⣠⣤⣤⣄⢨⣿⣿⣿⠉⠉
⠀⠀⠀⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⡏⣼⣿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿
⠀⠀⠀⣿⣿⣿⠛⠘⠿⠿⠿⠿⠿⠃⢛⣿⣿⣿⣟⠃⠻⠿⠿⠿⠿⠟⠘⣻⣿⣿⣿⡛⠘⠿⠿⠿⠿⠿⠃⢛⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣷⡄⣀⣀⣀⣀⡀⢠⣾⣿⣿⣿⣿⣦⢀⣀⣀⣀⣀⡀⣴⣿⣿⣿⣿⣷⡄⢀⣀⣀⣀⣀⢠⣾⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⢃⣿⣿⣿⣿⣿⡸⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣇⢻⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⡘⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⡟⢨⣭⣭⣭⣭⣭⡅⢿⣿⣿⣿⣿⠃⣭⣭⣭⣭⣭⣭⠘⣿⣿⣿⣿⡿⢨⣭⣭⣭⣭⣭⣅⢻⣿⣿⣿
⠀⠀⠀⣿⣿⣿⢃⣶⣶⣶⣶⣶⣶⣶⠸⣿⣿⣿⡿⣰⣶⣶⣶⣶⣶⣶⣆⢻⣿⣿⣿⠇⣶⣶⣶⣶⣶⣶⣶⡘⣿⣿⣿
⠀⠀⠀⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿
⠀⠀⠀⣿⣿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⣿⣿
⣶⣶⣶⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣶⣶⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⢸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠹⠀⠀⠀⢰⡓⡶⡳⢶⡲⠀⠀⠀⠀⠀⢠⠂⠘⡀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠶⠶⠤⡄⣤⣤⣤⣤⡄⠠⠄⠠⠠⠄⠀⠀⠀⠀⠀⠠⠤⣤⣤⠄⣤⠀⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠍⠉⠀⠀⠀⠀⠨⠄⠀⠀⠀⠠⠂⠛⠘⠀⠀⠐⠂⠀⣐⠀⠀⠾⠂⠘⠛⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⠀⠲⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⡀⣀⡀⣀⠀⠀⢀⠀⣤⡠⡀
⠈⠉⠙⡆⠀⠀⢰⠉⢙⠉⠁⢹⠀⠀⠀⠰⡏⢹⠉⡇⢹⠆⠀⢸⠁⠈⢸⢀⡇⠀⢸⠀⡃⠁⢸⠀⠀⢸⠀⡀⠁⡇
⠀⠀⢰⣃⡀⠀⠸⣆⣘⢀⣀⣸⠀⠀⠀⠀⣇⣘⣀⣇⣸⠀⠀⣼⣄⣠⣸⣼⠁⠐⢾⠤⣤⡴⣾⠀⡟⢺⢶⠖⢿⠁
⠀⠉⢹⠉⡇⠀⠀⣿⠉⠉⠉⣿⠀⠀⠀⠀⣿⠈⠁⠉⣿⠀⢰⢸⠀⠀⡇⢸⠀⢸⢸⠀⠀⠂⢰⢀⠁⢸⠘⠀⠘⡀
⠀⣀⣸⡀⡇⠀⢀⣿⣀⣀⣀⣻⠀⠀⠀⠀⣇⣀⣀⣀⣿⠀⢸⢸⠤⠄⣇⣸⠀⢸⢼⠖⠒⡧⢼⢸⡖⠙⠙⠃⠢⡇
⠀⠀⢸⣩⣷⠀⢸⣏⠀⠀⠀⣹⠆⠀⠀⠸⣇⣀⣀⣀⣿⠀⡟⢸⡤⠀⠧⢼⠀⠞⠺⡦⠤⠗⢺⠘⠈⠙⠓⠋⠉⢃
⠀⠉⠹⠀⠘⠀⢸⠉⠉⠉⠉⠹⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠘⠀⠀⠀⠈⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠄⢀⡀⣠⣠⡄⠀⠠⠶⡷⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠋⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⠿⢿⣷⣾⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⢀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣷⣿⠂⣾⣲⣾⣾⡷⠠⣼⣵⣾⡇⠰⣶⠴⢿⡆⢸⣷⣾⣥⠌⠩⡭⢯⠭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠋⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣯⣽⣾⣨⣥⠒⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣴⠶⣖⢀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠁⠉⠈⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢤⠤⠥⠄⠀⣀⡀⠐⠀⡠⠀⠀⠀⠀⣀⠂⢀⣡⡤⠤⠤⠤⢤⣤⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠂⠰⠶⠂⠦⠀⠈⠉⠁⠉⠤⠀⠀⠀⠀⠀⠤⠬⠤⠌⠙⠀⠁⠶⠴⠶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠀⠐⠒⠐⠒⠲⠂⠀⠒⠒⠐⠒⠒⠒⠒⠒⠒⠐⠒⠒⠒⠒⠒⠰⠶⠆


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠶⢶⣶⠀⠀⠀⣗⠒⢒⡖⠀⠀⠐⣷⠖⠲⡇⠀⠀⠘⢲⣤⠴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⡏⠀⠀⠃⠀⠀⢀⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡤⡄⠀⢰⡇⠀⠠⣧⠀⠀⢠⡇⠀⢰⣵⠀⠀⣸⢠⠇⠄⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠒⠒⠇⢣⠀⣼⠓⠒⠒⢿⠀⠀⢸⠇⠀⠸⠿⠀⢀⠏⢺⠒⠒⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⢹⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠁⠀⠀⠀⡆⠀⠁⡆⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠈⢃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⡤⠚⠀⠀⠀⠀⠀⠀⠀⡔⢉⠙⢢
⠀⣺⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠁
⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟
⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠿⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⡀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢤⡄⢠⡄⢠⣄⣀⣤⢀⣤⡀⣤⣄⣠⡄⢠⣄⢠⡤⠀⠀⠀⠀⠘⠛⠛⠛⠛
⣴⣿⣿⡀⠀⠀⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢩⡭⢙⠋⣭⠙⢩⠭⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣤⣤⣤⣄
⠈⠻⢿⣿⣿⣿⣟⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣤⣭⣴⣴⣶⣤⣼⣦⣽⣿⣿⡿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⣿⠉⢩⣭⣍⠉⠉⠉⠉⠉⢹⣿⠬⡧⢼⠤⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⣼⣿⡿⡄⠀⠀⠀⠀⢸⣿⣶⣷⣾⣶⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⡏
⠀⠀⠻⢿⣿⣿⠟⠁⠀⠀⠀⠀⣿⠀⠛⠿⠏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢻⡟
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡀⡤⠶⠀⢸⡿⢿⣿⣿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠆⠒⠋⠁⠀⠀⠀⢸⣿⣿⣽⣿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣤⣶⣿⣷⣦⠄⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⢿⣿⣭⣭⣽⣿⣿⣯⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠓⠛⠛⠛⠛⠀⠀⠀⠀⠐⠚⠛⠛⠒⠂
⠀⢰⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄
⠀⣾⣿⣿⣿⣿⣿⣿⠟⠛⢿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⡿
⠀⢹⣿⣿⣿⣿⣿⣿⣶⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏
⠀⠘⠻⠿⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣷⣦⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡈⠀⠀⡇⠰⠀⣏⢹⣿⠈⠉⠉⠉⠉⠉⣹⠉⢹⡏⠉⠉⠉⠉⢁⡀⠀⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠼⠧⡶⢐⠀⡆⢲⠡⣇⢻⡜⢃⡲⢀⡾⢂⣁⡐⢿⣧⢆⠶⣴⢼⢾⠃⡌⢿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣄⣃⣸⣀⣧⣸⣤⣗⣘⣷⣾⣄⣸⣄⣀⣿⣤⣤⣻⣸⣀⣼⣼⣿⣤⣧⣾⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⡴⢤⡄⢀⠀⠂⢈⠀⠀⠀⠀⠠⠀⠀⡄⢠⣏⢠⠀⠤⡀⠀⣀⡀⠀⣸⠁⡀⠀⡄⠀⢀⢀⢀⣼⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠂⡇⢁⡸⡇⢶⢼⡇⠀⠀⠀⠀⣇⢀⠺⣰⡇⡾⢻⠀⠀⠀⢨⠴⡗⡷⠀⠀⠈⣿⢷⣳⡛⣄⡯⣎⣻⠀⢀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠈⣇⢻⣸⣇⣇⣿⣸⡃⠀⢠⢀⢧⡝⣞⢸⢋⠇⣿⢼⡀⠀⢀⡞⢳⢑⠇⠀⠀⠀⣿⣸⣿⣄⡍⣅⡿⠟⠀⣸⠀⠄⠀⠀⠘⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠠⠤⢤⠤⠤⠤⠤⠀⢰⡎⢋⡀⣆⢹⡜⠃⠁⠉⠉⠉⠓⠛⠣⣜⡸⢀⠄⠀⢸⡝⡛⠛⠛⠋⠉⠉⢉⢫⠃⢸⡀⠀⠀⠀⠙⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠈⠉⠉⠉⢁⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠢⠐⠷⠭⠦⠤⢄⣀⣀⣨⠻⠀⠀⣇⠀⠀⠀⠀⢸⣔⣀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠹⡀⠘⠁⠉⠉⠉⠉⠉⢁⠨⠼⠠⠤⠤⠀⠀⠀⠀⡄⡠⢴⠶⠦⠤⣤⣤⠤⠖⠒⠒⢦⠑⡀⠸⡀⢃⡞⠉⠉⠉⠉⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠓⠀⢰⠃⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠸⡀⠞⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡇⡀⡿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢢⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢈⣰⠀⢽⣷⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⡀⣿⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⢸⠹⠂⢤⣯⣀⣠⣤⣤⡤⠤⢤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⢴⡶⢖⠒⠒⠚⠛⢻⠛⠀⠈⠟⠒⠒⠒⠛⠛⡏⠉⠉⠀⠀⢸⠀⠀⠀⡇⡄⢀⠇⠘⡆⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣃⣀⠅⠀⠀⠀⢸⢸⡆⠀⡇⡄⠀⠀⠀⠀⣧⠀⡆⠀⠀⠸⢰⠀⢀⡧⡦⣮⡤⣻⡇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⢋⣉⡉⠁⠠⠤⢽⠈⣷⢰⡇⡇⠠⠤⠶⠶⣿⠀⡇⡀⠀⢰⣿⠛⠉⣷⡏⠉⠉⢹⣿⣦⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠠⠤⠤⢼⢠⣿⣺⡇⡇⠀⣉⣉⣉⣽⣵⣿⡄⠀⢸⣿⠀⠀⡇⡷⠚⠛⢻⡟⠉⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡗⠀⠀⠉⠉⠉⠁⢹⡸⣿⠘⡇⡏⠉⠉⠉⠁⢉⡀⢿⡇⠀⢸⡟⠀⠀⣇⡇⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⡇⡇⠀⠀⠀⠀⠸⡇⢸⡇⠀⠘⠃⠀⠀⡇⡇⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠓⠁⠀⠳⠃⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⣡⠔⠳⣌⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⢿⣄⠈⠀⠀⠀⠈⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠔⠋⠙⣆⠙⣦⠀⢀⠀⠀⠀⢀⣽⣧⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠻⣄⠀⠀⠀⠀⠘⠀⠘⣧⡈⢁⡤⠖⣩⣄⠻⣷⡀
⠀⠀⠀⠀⢠⡞⠁⠀⠊⠉⠣⠘⣦⠀⢀⠀⠀⠀⢀⡿⢿⣇⠀⠈⠀⠈⠁⠘⢷⡀
⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠈⢷⣌⣉⡴⢚⡥⠦⡄⠻⣦⠀⠀⠀⠀⠀⠀⠳⠂⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠹⣿⡄⠠⣄⡠⠀⣠⠼⣿⣇⠀⠀⠀⠀⠀⠀⠹⢧⠀⢀⠀⠀⠁⠀⠀⢀⡀⠈⠉⠉⣉⠖
⠀⠀⠀⠀⠀⠀⠘⣿⣦⡤⠖⠫⠖⠲⣄⠹⣧⠀⢀⡀⠀⠀⣀⠤⠎⠁⠀⠀⠓⢀⠀⣀⣻⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠙⢷⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠁⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠳⠀⠀⣀⢤⡞⠁⠴⣿⣶⡄⢀⣀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠔⢋⣡⡴⠿⠶⠾⠋⠋⠀⠀⠈⠒⢦⣤⠏⠀⠀⠀⠀⠀⠀⠁⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠄⠀⢀⡠⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣂⢹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠔⠁⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⠔⠋⡠⠊⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⠀⠀⠀⠀⡠⢊⡔⢈⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⠜⠀⠊⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⣠⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⢀⣠⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⡄
⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⣠⠏⠀⠀⠀⠀⠼⠃⠀⠀⠀⠈⠓⢶⣀⠀⠀⠇
⠀⠀⠀⢀⠐⢄⠀⠠⣤⠠⢥⣀⣀⣀⡼⣀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⠀⠀⠑⠶⣶⢁⠂⠀⠀⠀⠀⠀⠓⠨⠁⠀⡠
⠀⠀⠀⠉⠑⠀⡈⠑⠀⠉⠌⠀⠧⣄⡴⠀⠉⠉⢉⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢠⠏⠀⠀⠀⢰⣣⠃⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠲⢦⣉⠂⠀⠀⠀⠀⠀⠈⠋⠀⠀⠄⢸⣄⠀⠀⠀⠀⠄⣀⣤⣄⡰⡏⠀⠀⠀⢠⢒⠏⠀⠀⠬⡁⠀⢢⠐⡄⠀⠀⠀⠀⠀⢀⣀⡠⡆
⠀⠀⠀⠀⢾⠂⠀⠉⠃⠐⠠⠀⠀⠀⠉⠀⠐⠀⡀⠀⠀⠀⠠⠂⠀⠀⠈⡙⠁⠛⠶⢶⣤⡋⠎⠀⠀⠀⣄⠈⢦⠀⠇⡇
⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⣨⡋⠀⠀⠀⠀⠉⠓⠚⠂⠃⠀⠀⠀⠀⠙⠓⠒⠂
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠃⠀⠀⢶⣤⣀⣀⣴⠏⠀⠀⠀⠀⢠⠟⠀⠀⣠⣤⢶⣤⣄⠀⠀⠀⠀⠀⠀⢠⡄⢄⢄
⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠻⠿⣦⣄⣀⠀⣴⡎⠀⠀⢪⠞⡵⠚⠿⣯⣧⠀⠀⠀⠀⠰⠀⠀⠉
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠀⠀⣴⣄⢘⢀⠀⠈⠁⣠⠦⠀⠀⠀⠸⣤⣠⣀⡀⡰
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠈⠀⠀⠀⠀⣴⠄⠀⠀⠉
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠏⠀⠀⠀⠀⠄
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠂⠀⢦⣄⣠⣔⣺⡼⣷⣄⠀⠀⠀⠀⠀⠀⣠⠤⠂⢩⣿⡍⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠙⠒⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠇⠀⠀⣄⠀⠀⠄⢀⡿⢟⣯⠾⠛⣫⠼⢆⡀⠀⠀⡀⢺⣿⣦⡀⠈⠙⢯⣌⠙⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⢗⣋⣽⡔⢛⣯⣴⣿⣭⣉⣙⡿⠆⠁⠀⠀⠙⢯⡻⡯⠈⠀⠈⠛⠒⠤⣉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠰⠄⠀⠀⠀⠀⠉⠻⠻⠟⠊⠉⠛⠂⠓⠛⠋⠹⠁⠀⠀⠀⠀⠀⠀⠙⠂⠈
⠀⠀⠀⠀⠂⠀⠀⢐⡎⠀⠀⠀⡄⢀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⠀⠀⡟⠒⠂⠀⣸⠎⠀⠀⠈
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⠀⣠⠄⠉⠍⠉⣇⠀⠈⠲⠗⡉⠁⣀⣀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⢲⠁⠀⠀⠀⠠⡀⠀⠀⠠⠄⢨⠇⠀⠀⠀⠀⢠⠦⠀⠀⠀⠀⠀⠀⢀⡀⠀⠁⠓⠀⠀⠀⡀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠐⢄⣀⣤⣤⠤⠃⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠁⠀⠹⢦⡘⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠢⡀⠠⠄
⠀⠀⠀⢠⣀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠈⠀⡖⠀⠀⢰⣦⡤⢄⡀⠀⣠⢟⠦⡀⠀⢄⠀⢀⠀⢀⡴⠀⠀⢀⡀⠀⠠⠀⠐⡇⠀⠀⠠⠀⢀
⠀⠀⠀⠁⠀⠀⠀⠆⠀⠀⠀⢈⠀⠀⠀⢀⣷⠄⠀⠈⠙⠣⠀⠈⢀⡴⠁⠀⠈⢢⣄⣧⣤⠆⠈⠀⠀⠀⢋⠀⠀⠀⠐⠔⣀⡀⠴⠤⠖⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠉⠀⠋⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠔⠚⠩⢤⣄⣀⡐⠂⠨⠉⠑⢒⣢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣴⡉⠧⠤⣶⣶⠮⠟⠶⠄⠉⢈⣁⡠⠔⢚⣉⡭⠴⡓⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢙⢳⡿⢟⡲⠤⣌⣙⠒⠲⠚⣋⣡⠤⢖⡛⢩⢴⠚⠋⢁⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣲⣤⣆⣂⠙⠹⡗⠪⣭⣉⣉⢽⠂⠏⠠⠀⣈⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢎⡀⠀⠐⠻⠽⣒⡦⣄⣈⢘⣀⣉⣤⣴⢶⣛⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠹⢓⠺⣝⡲⠤⣄⣀⠩⢑⠺⣭⠭⡝⠛⠋⣉⣩⣤⣾⣡⠵⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠙⠛⢶⣅⣈⠉⠛⠓⡻⣿⣖⠤⠬⠤⣶⣞⣫⢷⠿⠩⠝⠛⣀⣸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠫⡧⡀⢪⣍⣛⠶⠮⣄⡉⠁⠉⡿⠉⠁⣘⣀⡤⡤⣖⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡁⣝⣄⠳⡤⠵⠒⠒⠋⠓⢶⣷⣶⡿⣭⣔⣒⣺⣸⣦⠀⠀⠀⠀⢀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠙⢦⡀⠀⢣⡙⢮⣒⣀⣖⣶⣬⡽⠿⠳⠒⠚⠋⠉⢉⠙⣁⣤⡼⠆⠀⠀⠠⠋⠀⣠⣽
⠀⠀⠀⠀⠀⠀⠀⣡⠘⢓⣿⠀⠀⠑⣬⣉⣁⣁⣬⣬⣤⠶⠶⠖⠻⠿⠛⠛⠋⠋⠉⠀⣀⠀⠀⠙⠻⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠦⣄⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⣂⣛⣦⠀⠀⢜⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠙⠒⠲⠤⢤⣀⣀⠀⠀⠀⣠⠴⠋⢧⣀
⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠷⠒⠃⠀⠀⠀⠈⠱⠤⡀⠀⠀⣀⡀
⠀⠀⠀⠀⣤⣞⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠒⠋⠁⠀⠀⠀⠀⠀⢠⣄⣤⠖⠚⢍⠁⠔⠁
⠀⠀⠀⣸⠁⠀⠉⠙⠒⠢⠤⣀⣀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢀⡀⣠⣢⠏⡉⠉⠀⠅⠀⢅⡀⠄⠈⡄⠀⠂⡠⡀
⠀⠀⠀⠛⠬⢄⣀⠀⠀⠀⠀⠀⠈⠙⠻⡍⠉⠉⠁⠀⠀⢀⣒⢵⡿⠯⠀⡁⠉⠐⠉⠄⠂⣀⠤⠁⡉⠀⠆⠁⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⣀⡀⠀⠓⠐⢲⡀⠀⠄⡸⠾⠋⡁⠀⠉⠀⠤⢐⢀⡰⠌⠀⠈⠑⠊⠀⠀⠀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⢤⠬⠃⠀⠳⣄⠀⠀⠉⠐⠀⠠⠈⡀⢀⠤⠂⠄⠀⠀⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠑⠘⠦⣄⣀⠀⠀⠀⠀⠀⣕⣀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠙⠦⣀⣀⣀⡀⠀⠈⢏⠓⠒⠦⢬⣿⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠐⠀⠐⣶⡈⠉⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣆⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠛⣻⡜⡂⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⢠⣤⣤⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⢶⠤⣉⣻⡐⢠⣤⣾⡿⠿⠿⠟⠛⢿⡛⠛⠉⠉⠉⠉⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠉⠁⠚⠨⣷⡸⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠩⡒⠦⣤⣘⣷⢹⡄⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀⣀⣀⠀⡄
⠀⠀⠀⠀⢺⣿⣳⣤⡀⠀⠀⢤⣄⠺⠿⠟⣣⢷⡀⠠⠤⠶⠒⣒⣦⣼⠤⠤⠔⣒⣒⣛⡿⢃⠀⢀
⠀⠀⠀⠀⠈⠻⢿⣷⣤⠠⣂⠉⠀⠀⠀⠀⠨⠿⠿⡯⠭⠭⠽⠖⠒⠶⠛⠋⢍⠀⠀⠀⣀⠀⠀⣺⣷
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣮⣟⠦⠀⠀⠀⠂⠈⡤⢳⠀⢰⣞⢦⣄⠄⠀⠀⡀⠙⣦⠐⠁⠀⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⢠⠄⢀⠄⠀⠈⠀⠈⠻⣷⣍⠂⠀⠀⡐⠚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⠘⠀⠀⠈⠻⣿⠂⢸⠟⢠⣿⣄⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠠⠤⠤⣤⡄⠒⠈⠉⠉⠁⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣇⠀⠀⠀⠀⠀⠀⡀⠁⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠩⢆⠀⠀⠀⠐⠐⠄⠀⠄⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣇⢨⠀⠀⠀⠀⠀⠀⢀⣀⣀⠈⢿⣄⠀⠀⢀⣒⣢⣠⣤⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠘⠻⣆⢣⢠⣤⣤⣭⣷⣿⡶⠿⠵⠟⣍⡴⠛⠛⠋⠉⠉⠉⠁⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⡷⣄⣀⠀⠘⡆⢂⣏⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡁⠃⠀⠏⣑⠻⡄⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⢩⠃⢲⠠⣍⣛⣻⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠤⠤⠔⠂⠄⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠉⠐⠂⠹⠟⠛⡘⣆⠀⠀⠰⠖⠢⠘⣛⣛⣂⡪⠤⠤⠆⠒⠚⣋⣉⣉⡭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠸⣸⠖⣒⣚⣋⣉⣡⠤⠤⠴⠒⠒⠒⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⢸⡟⠀⠉⠉⠉⠀⠀⢰⠢⠀⣾⡅⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠶⣦⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⡾⠟⠀⣠⣶⠶⣶⣄⠀⠉⠁⠀⠈⠉⠛⠻⢷⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢯⣭⡛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣫⡞⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢌⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣴⠟⣡⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦
⠀⠀⠀⠀⠀⣸⠋⣼⠋⠀⠀⠀⠀⢠⣶⣶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣷⣄
⠀⠀⠀⢀⡴⢁⡾⠋⠀⠀⠀⠀⠀⢺⣿⣿⣷⣌⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣷⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⢀⢹⣦
⠀⠀⣠⡞⡴⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠉⠛⠋⠀⠀⠀⣼⣿⣿⣿⠿⣦⠀⠀⠀⠘⡇⣿⡇
⠀⢰⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠛⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⡿⠀⠀⠀⢰⡇⣿⡇
⠀⣿⠃⠀⢤⣤⣶⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠟⠋⠀⠀⠀⠀⠹⣇⢻⣇
⠀⡿⠀⠀⠈⢿⣿⣧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢰⠃⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢳⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⣿⣇⡇⠀⠀⠀⣠⣾⣷⠸⡆⢸
⣿⣸⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⣿⠿⠋⠈⠀⠀⢀⣼⣻⣿⣿⢀⡇⣼
⣿⣾⡆⠀⠀⠀⠀⠀⢰⣿⣿⣿⣜⣿⠀⠀⠀⠀⠈⠻⢿⣿⠿⢟⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⣸⣷⡟
⠘⣧⢻⣦⡀⢀⣀⠀⠘⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢫⡟
⠀⠘⢷⣾⣿⢷⣾⣿⠿⣶⣿⣿⣍⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣰⣿⡇
⠀⠀⠈⠉⠲⣆⠙⠁⢠⡌⠉⠉⣿⠿⣷⣤⠀⢠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢶⡀⠀⠀⠀⢀⢞⣿⠇⣿⡏
⠀⠀⠀⠀⠻⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⢿⣿⣿⣿⡟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⡇⠀⠀⣠⣵⡿⢋⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⣝⡿⠿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣛⡫⠀⠀⣼⡿⣫⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠌⢻⡆⠀⠉⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣼⡟⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠢⣾⡃⠀⠀⠀⠀⢰⣿⣿⣥⣿⣧⠀⢀⣀⣠⣶⣴⠾⠷⠿⠟⣛⣵⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⣽⣭⣤⣬⣙⢿⣿⣿⣿⠏⡰⠟⡭⣉⣭⣷⣶⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⢿⣶⣶⣶⣶⣦⣭⣭⣤⣶⣶⠿⠿⠿⠛⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣬⣿⣿⣯⣭⣤⣿⣿⣯⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⡀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣷⠈⠁⢸⣧⡄⠀⡇⠀⠀⠀⠀⢀⣯⣄⠀⣾⣴⡀⠐⣇⡄⠀⡄
⠀⠀⠀⠀⠀⠀⢸⠧⣶⠶⣧⠴⠶⢾⡷⠴⠤⡇⠀⠀⠀⠀⢸⡧⢶⠦⢷⠮⠵⠤⡥⠤⠤⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠰⠆⢸⠟⠀⠀⡇⠀⠀⠀⠀⢸⡇⠉⠀⢸⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠖⠶⠚⠗⠺⠟⠺⠖⠚⠀⠁⠀⠀⠀⠀⢸⡷⠗⠒⣰⠶⠶⠶⡴⠶⠖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠲⢸⡇⠰⠀⡷⠋⢠⠸⡟⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠘⣷⣯⠉⣮⣛⡛⢲⠓⠓⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣰⣄⣀⡄⣀⣀⢠⠂⠀⠀⠀⠀⣟⣻⣀⢱⣛⣃⡸⣀⣀⠀⡇
⠀⢀⠠⠄⠀⠀⠀⣁⣤⣾⣿⣿⣿⡿⠟⠋⠁⠈⠀⠀⠀⠀⠀⣏⠉⠉⡏⠉⡉⢹⠉⠉⠉⡇⠀⠀⠀⣀⠤
⠀⣘⠷⠤⠶⠒⢻⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣤⣤⣡⣬⣤⣘⣤⣤⣠⡃⠀⠐⠻⢷⣄⣤⡶⡶⠷
⠀⠁⠀⠀⠀⠀⢸⣿⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠇⠀⠀⢸⡄⠀⠀⠃⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠁⠀⠀⠀⠀⠀⠃⠐⠒⠒⠀⠀⠚⠓⠒⠒⣠⣤⡀⠀⠰⣶⣀⣀⣀⣄
⠀⠀⠀⠀⠀⢀⠠⢰⣥⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣒⢀⠀⡀⠀⠀⠀⠸⣿⡛⠋⠉⠉⢿⡵⣀⣀⣠⡄
⠀⠀⣤⣶⣤⣼⣿⠟⠿⢽⣿⣿⣶⣤⠂⢀⣤⠀⢰⣖⠇⢀⡙⠦⣔⠖⠋⠛⣧⣀⢠⡤⠀⠀⢸⣷⣿⡿⠿⠩⠞⠙⠛⠂⠉
⠀⠀⠙⠿⣿⣿⣿⠿⣶⣿⣯⣤⡠⠼⢫⡭⠿⢷⡀⠙⣿⣤⢀⣤⡬⠓⢶⡊⠁⢈⡳⢤⣤⡶⠄
⠀⠀⠀⠀⠈⠙⠿⣆⡚⠛⠻⠿⠿⠛⠁⠀⠈⠉⠀⠀⠈⢿⣿⣶⣅⣤⣴⣯⣾⣿⣿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠒⠚⠉⠉⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠚⠛⢶⣏⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣄⣠⠎⠉⠁⠀⠀⠀⠀⠈⢿⡀⠀⠀⠄⠀⠀⠀⠀⢈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠚⠋⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⣠⣤⣶⡾⠿⢛⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠧⠐⠛⣩⠥⠒⠈⠁⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣆⢀⣾⠿⠟⠋⠩⠔⠂⠰⣶⡀⠀⢡⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢣⣠⣴⣶⣾⠿⠟⣉⣊⠀⠊⠁⠀⠀⠀⠀⠀⠘⣿⡄⠀⢡⠀⠀⠀⠀⠀⠀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⡿⠟⣋⠅⠀⠂⠉⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠳⣠⣤⣶⣾⠿⢛⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣦⠀⢛⣩⣤⢴⠺⣯⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⢀⣶⡾⠿⣛⣋⢭⡥⡤⣴⣿⣮⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢀⣀⣤⣴⣶⡿⠿⣛⣎⣥⣲⣽⢷⣔⣉⣛⡤⠗⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⡀⢸⣿⣛⣩⣥⣾⠯⡩⢿⣧⡽⠾⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣁⣞⢿⣿⣯⣿⠿⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠉⠁




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⢀⣀⣀⣠⣠⣤⣲⡦⠤⠷⠒⠒⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⡀⠤⠤⠄⠒⠂⠹⠾⠖⠒⠚⢻⣽⡏⠉⠁⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⢀⡀⠀⢀⣀⠠⠤⠤⠄⠒⠒⠈⠉⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠸⣿⡅⠀⠀⠀⠀⠀⣿⣿⠀⢀⣀⣀⣀⣼⡇
⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣀⣀⣀⣀⣤⣿⠷⠤⡶⠶⠒⢺⣛⣟⣋⣩⣭⡵⠶⢤⣷
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣩⣭⣭⡵⠶⠤⢾⣷⠖⠚⠋⠉⠉⢹⣿⡅⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⣻⡄
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⣿⡀⠀⠀⠀⠀⣸⣿⣇⣀⣠⣤⠤⠤⢿⡿⠶⠲⣛⣛⣋⣭⡇
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣿⠗⠚⣛⣛⣫⣭⣽⣧⠶⠤⠵⠒⠒⢳⣿⡋⠉⠉⠁⠀⠘⣧
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⣿⡟⠉⠉⠁⠀⠀⢿⣿⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⣿⣿⠄⠀⢀⡀⠀⢸⣿⡇⠀⢀⣀⣀⣠⣿⣷⡤⠤⣴⡲⠶⠿⠇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠄⢀⣠⠞⠇⡉⠀⠀⠀⠀⠀⡈⠐⢾⡻⠛⠛⠉⠉⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣀⣀⠀⡴⠚⢍⠀⠀⠀⠀⠀⠀⠈⠂⣄⠈⢢⠄⠂⠈⠀⣀⡤⠴⠛⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠠⠒⠒⠂⠉⠁⠀⠀⠃⠀⢀⠀⠀⠁⢠⠀⠀⠀⠀⠈⠁⣀⣳⣬⣀⡴⠚⡉⠁⠀⠀⠀⠀⢀⠠⠐⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠠⡄⠀⡠⡶⠒⠛⠉⠉⠀⠀⠈⠑⠄⡀⠑⢠⠀⠀⠈⢀⣀⡠⠂⠉⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠂⡈⠀⠈⠂⢄⠀⠀⢀⡀⠄⠀⠐⠉⣦⣰⡀⣤⠖⠉⠁⠀⠀⠀⠀⢀⠠⠈⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⢀⠀⠑⠈⠀⢀⢠⡠⠔⠂⠁⠉⠑⠧⡀⠈⠂⢄⡀⠄⠂⠀⠠⠐⠀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢑⢔⡦⢲⠞⠊⡁⠀⠀⠀⠀⠀⠀⢀⠠⠐⣄⢀⢀⠀⠀⡁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢅⠀⡀⡀⠈⠠⢀⡀⠀⠐⠈⠀⠀⠀⠉⢺⠅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠊⠢⡀⠈⢀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠍⡀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⣀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣶⣿⠉⣷⣄⠀⠀⠀⠀⡧⣿⢯⠷⡾⡿⢭⠂⠀⠀⠀⢾⡟⠉⠉⠉⣯⡆⠀⠀⠀⢀⣂⣂⣉⠈
⠀⠀⣿⣤⣿⣻⣏⣸⡆⠀⠀⠀⢈⣿⣯⣤⣼⣿⣁⠀⠀⠀⠀⡱⠆⢸⡷⠀⠶⡇⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠹⢧⣼⣏⣨⠟⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠻⠀⠀⠀⠀⢾⣧⣀⣀⣠⣟⡇⠀⠀⠀⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⣰⠁⠀⢘⡄⠀⠀⠀⢸⠙⠋⠋⢳⠃⠋⠋⢻⡇⠀⠀⣼⣛⣛⣛⣻⡄⠀⠀⠀⡠⠴⠟⠙⠋⠹⢶⣄⡀
⠀⠀⡼⡌⠀⠀⠀⡿⣆⠀⠀⢸⢀⣀⡀⣰⡀⣀⣀⣸⡇⠀⠀⠀⠙⣤⡜⠁⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠘⠚⠒⠒⠒⠒⠚⠚⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠁⠀⠀⠠⠊⠀⠈⠢⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⣀⣀⢀⡀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⣀⡠⠤⣲⣷⠀⠀⠀⣴⠤⣤⢦⡤⢶⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠻⡇⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⣭⣶⣾⣿⣿⣿⠀⠀⠀⢹⠀⣿⢸⡇⣾⠀⠀⠀⠛⢷⣾⢫⡙⣶⠟⠃
⠀⣴⡇⠀⠀⠀⠀⠀⠀⣷⡄⠀⠀⠋⡟⠻⢿⣿⣿⣿⠀⠀⠀⢸⡆⣿⢸⡇⡿⠀⠀⠀⠀⢀⣿⡿⢿⣧
⠀⠉⠉⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠈⠉⠀⠀⠀⠘⠧⠿⠼⠶⠇⠀⠀⠀⠀⠚⠁⠀⠀⠙

⠀⠀⣀⣤⠤⣄⡀⠀⠀⠀⠀⢠⣺⣭⠷⣠⣚⣯⡷⡄⠀⠀⠀⢀⡤⡤⠤⠤⢤⢤⡀⠀⠀⠀⠀⡀⡀⠄⠄⠀⠂
⠀⣼⠛⣿⣤⡾⠻⡆⠀⠀⠀⠘⠯⣯⠝⠉⢯⣿⠽⠁⠀⠀⠀⠠⣛⠀⢠⣀⠀⣛⡇⠀⠀⠀⢰⣶⣶⣦⣦⣦⣤
⠀⣿⠟⢻⡾⠷⣶⡇⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣴⠀⠀⠀⠀⠐⣭⠀⠘⠉⢨⣭⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠈⠳⠾⠷⠴⠋⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠉⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠛⠛⠿⠿⠿⢿⡟

⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⣬⠀⠀⠀⣣⠀⠀⠀⠀⡇⠁⠁⠁⠑⠁⠁⠁⠁⡇⠀⠀⠠⣯⣭⣭⣭⣽⠄⠀⠀⠀⡤⣶⣿⣁⠍⠠⠿⠷
⢀⣞⡇⠀⠀⠀⢨⢳⡀⠀⠀⡇⡀⡀⡀⡠⡀⡀⡀⡠⡇⠀⠀⠀⠀⢑⣴⡋⠀⠀⠀⠀⠀⢺⣿⣿⣿
⠘⠚⠒⠒⠒⠒⠚⠓⠃⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠔⠁⠀⠉⠢⠀⠀⠀⠀⠓⠒⠒⠂⠉⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣶⠀⠀⠀⢠⡬⣥⡤⣤⡬⣦⠀⠀⠀⠀⠀⠀⢰⣧
⢿⣿⠀⠀⠀⠀⠀⠀⢹⡿⠀⠀⢰⣖⣫⣥⣶⣾⣿⣿⠀⠀⠀⠘⡇⢸⡇⣸⠀⣿⠀⠀⠀⠲⣶⣶⠟⡟⢶⣶⠶
⣠⣿⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠘⠻⡿⢿⣿⣿⣿⣿⠀⠀⠀⠀⣷⢸⡇⣿⠀⡏⠀⠀⠀⠀⠀⣽⣾⣿⣾⡁
⠛⠛⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠈⠓⠒⠋⠉⠙⠻⠀⠀⠀⠀⣿⣸⣇⣿⣰⡇⠀⠀⠀⠀⠰⠟⠁⠈⠙⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⢸⣇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠚⠿⠛⠛⢲⣄⣄⠀⠀⠀⠀⡀⠀⠉⣹⣿⠇⣀⠀⠀⠈⢳⣶⡄
⠀⠀⠀⠀⣿⡀⢀⠐⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡞⠀⠀⣆⠀⣿⣷⢇⠀⠀⠘⢷⣷⠶⠟⠁⠀⠙⠳⠶⠶⠾⠛⢁⣀
⠀⠀⠀⠀⣿⡄⠀⠠⠀⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⢐⠀⣾⠃⠀⠀⢹⠀⢸⣾⡘⠀⠀⠀⠀⡀⠀⠀⠐⣿⣿⡆⠀⠀⠀⠁⠈⣸⡇
⠀⠀⠀⠀⠘⢿⣶⡀⠃⢀⠀⠁⠆⢀⠀⠀⠀⠀⠀⢸⢠⡇⠀⠀⠀⠸⠀⢸⣧⡇⡇⠀⠀⠀⠷⣤⣤⡾⠿⠋⠁⠀⢠⡀⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢻⣧⣆⠁⠊⣀⠀⢊⡄⠀⠀⠀⠀⠈⠘⠁⠸⠀⠀⠀⠀⢸⡇⣇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠡⢆⡄⠠⢆⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⡇⢰⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⣤⡄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣾⣦⣈⠰⣦⠉⠔⠀⠀⠈⣆⣿⣤⠀⠀⢰⣇⣼⣿⡇⠀⠀⠀⠀⠀⠀⠐⢲⣦⢴⣾⣯⣁⣠⣿⡂⠁⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣬⣁⠸⠄⠀⠀⣿⡸⣿⣧⡀⢸⣿⣿⡿⠁⠀⠀⣀⡀⠐⠳⣶⡏⣀⣼⣿⣍⣤⣝⣿⣆⠉⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠑⠀⠀⠘⢷⣿⣿⣇⣾⣿⡟⠀⢀⣀⠈⢉⣿⣶⣞⣫⣤⣴⣿⣿⢙⣿⣿⣻⣷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠖⣐⠲⠶⠶⠶⣂⠴⠶⠶⠶⣌⠙⠛⠛⠉⠀⠀⠀⠈⠙⠻⢾⣷⣿⣯⣭⣽⣿⣿⠷⠾⠿⠿⠋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠤⠤⣿⠤⠤⠤⠤⣽⠄⡤⠤⢌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠈⠀⢀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⠀⠀⠀⠂⣿⠇⠀⠐⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⡴⣯⢤⣤⡠⢤⣿⢤⣤⡤⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⢰⣿⠀⠀⠐⢻⡯⡀⢀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠠⣇⣠⠶⠆⠛⠷⠖⠰⠖⠿⠶⠖⢲⣔⠏⢻⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡟⠀⠀⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠛⠀⢠⡴⠁⠀⠀⠀⠀⠀⠤⣴⣶⣤⣴⠒⢶⣖⡖⠚⠃⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⠠⣶⣦⣾⣤⣔⣋⣤⣽⣏⠀⡀⠀⠛⠗⢦⣄⠰⣶⣤⠤⣤⣤⣀⣁⡀⠘⡃⣾⣿⣷⠄⠀⠀⣀⣠
⠀⠀⠲⣦⡀⢀⠀⠀⢠⣄⣸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣙⠶⣽⣶⠤⣌⡉⠉⠛⠂⠀⣤⠨⠿⠿⠛⠉⠉⠉⢉⣻⣶⣿⣿⣿⠆
⠀⠀⠜⠋⠻⠮⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠓⢄⠈⠳⢌⣽⠿⠶⠶⣤⣤⣀⡈⠁⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄
⠀⢀⡴⠃⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠢⠀⠹⢦⣄⣠⣤⣴⣿⣻⣷⣶⣄⣸⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⣿⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠑⠢⣾⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊⠀⢀⡀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⡀⠀⠀⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣹⠇⠀⠀⠀⠀⠀⠀⠀⢦⣤⣾⢃⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠸⣿⠟⠋⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠛⢋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢱⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⣤⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⢀⡸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠚⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⢠⣿⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⣤⣴⣾⣿⡄
⠀⢀⣼⠏⡀⠁⡈⠁⡀⢀⡀⢀⡈⢓⡦⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣸⠋⡄⠁⡄⠁⡄⠁⡄⠁⡄⠁⡄⢁⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠃⠄⠀⠄⠀⠄⠠⠄⠠⠄⠠⠄⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⢸⠂⠀⠂⠀⠂⠀⠂⠐⠂⠐⠂⠐⠂⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠚⣷⡃⡀⠃⣀⣧⣄⠃⡐⠃⡀⠃⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠈⠙⠷⣶⠟⠁⠉⠿⣅⡀⢁⡀⢁⣈⣥⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠛⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠇⠈⠿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠁⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣴⣦⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢸⣧⣄⠀⡀⠀⢀⣶⣿
⠀⠀⠀⢠⣴⣾⣿⠿⠛⠿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⣾⣿⣯⠋⠉⡁⠈⠹⣿⣿⠃⠀⠀⢀⠔⠛⠻⣆⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠀⠀⣿
⠀⠀⠀⠘⢿⣿⡄⠐⠛⠆⢦⡿⠋⠀⠀⠀⠊⢠⡀⢠⣻⠀⠀⢠⣴⡞⠉⠙⠛⠙⠃⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣌⠟⠁
⠀⠀⠀⠀⠀⠉⣷⢉⢦⣼⣥⠀⠀⠀⠀⠘⠀⠀⢤⠿⣸⠀⠀⢺⣿⡇⠀⠀⠀⠀⢀⣿⣿⠆⠀⣀⡤⠤⠤⢦⣄⠀⠀⠀⠀⠠⠖⣨⡿⠉⠟⠃⠀⠈
⠀⠀⠀⠀⢠⣿⣿⣿⣆⠺⣴⣶⣶⣦⡀⠀⠀⠐⠒⠁⠇⠀⠀⠀⠻⠇⢀⡀⣀⠀⠸⠿⠋⠀⣼⠏⠀⠀⠀⠀⠹⣆⠀⠀⠀⢀⣬⣥⡤⠚⣞⣶
⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠝⣻⣿⣹⣵⢲⡀⠲⣶⡆⠀⢀⡀⠀⡃⠰⡀⣒⠀⡄⠀⠈⠀⡀⣿⠢⢤⠀⢠⠤⠀⠟⠁⠀⣿⡇⣿⣿⢷⡶⣿⣿
⠀⢸⣿⠟⢋⢓⣾⣿⣿⣧⣤⣿⠟⠋⠁⠸⣿⣦⣽⣏⡀⠘⣷⡀⠀⢀⠀⠭⠀⠀⠖⠀⡠⠻⣜⣷⣾⣤⡄⠀⠀⠀⠀⢰⢁⣼⣋⠿⣤⡽⠿⢾⡇
⠀⠈⠛⠳⣤⣿⣿⣿⣻⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⢷⡤⣀⡙⢸⣋⣡⣀⡔⠁⠀⠈⢿⣿⣟⡛⠀⠰⠀⠀⢸⣾⣿⣿⡒⢂⣠⣐⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⠁⠀⣄⣄⣰⢿⣿⣽⣿⡿⠿⠋⠉⠀⠀⠀⠙⠁⠁⠀⠀⠀⠀⠺⢿⣻⡿⠿⠀⠀⠐⠈⠁⠀⢹⠿⣿⣿⣿⣿⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⡽⠖⠚⡎⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠹⣿⣭⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢉⣩⣭⣤⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣉⣉⣉⣉⣉⣿⡟⢉⣉⣉⣉⣉⣽⡇⠈⣿⠸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠃⢀⡸⡆⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣉⢻⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⠀⠁⣧⢸⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠃⢸⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⠀⠐⢸⣾⣿⠿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢸⣷⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⡏⣥⣭⣭⣭⣭⣿⡏⢨⣭⣭⣬⣥⣿⣿⢉⣭⣭⣭⣭⣽⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⡟⠀⢸⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠁⢸⡇⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⡇⢀⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠛⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠙⠛⠷⣶⣶⣶⣶⣶⣿⡇⢰⣶⣶⣶⣶⣾⣿⢠⣶⣶⣶⣶⣾⠷⠖⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠠⢍⡙⠻⢿⣿⣷⣾⡿⠿⠿⠿⣿⡿⠾⠛⢋⣉⠥⠆⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣄⣀⠉⢛⠛⠾⠁⠀⠀⠤⣤⡤⣤⣀⣠⣭⠁⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠉⠉⠑⠉⠉⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⠤⢤⠤⠤⠤⠤⠤⠤⠤⣤⣤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠼⠃



⣿⣿⣿⢿⣿⠿⢿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣶⣶⣷⣼⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠆⠰⠀⠐⠀⠆⠄⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠁⠀⠀⠄⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠘⠁
⠀⠀⡈⠀⠀⠀⠀⠀⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⡠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⠠⠂
⠀⠀⠐⠂⡀⣄⣤⡄⢠⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⢴⡄⠀⠀⠀⣲⠂
⠀⠀⠀⠀⡀⠈⠐⠀⠀⠈⠉⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠉⡔⡁
⠀⠀⠀⠀⠁⠖⣀⠶⢄⡀⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠆⢸⡆⠀⡼⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⢀⠉⠐⠒
⠀⠀⠀⠀⠀⠂⠀⠙⠉⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⡆⢰⠁⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠐⠒⠠⢄⠄⠀⠈
⠀⢰⠋⠃⡇⣔⠀⡀⡀⢤⢠⠀⢆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠘⠆⠀⠙⠂⠀⠉⠁⠀⠻⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⠀⡀⠀⠛⠛⣤⣀⠀⣀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠠⠀⢀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⢀⡀⢆⠀⢀⠈⠉⡀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣼⣦⡀⠀⠀⢀
⠀⠀⠈⠀⠀⠀⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢿⠀⡀⡞⠀⠉⢠⠸⢃⠗⢠⣀⡴⠀⠀⠀⠀⠀⢸⣷⠒⠋⢸⣧⠈⠒⣲⣿⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠘⢀⡀⠈⠀⠁⠀⠠⠄⠀⠀⠀⠀⠀⢐⣞⠠⠯⠬⠅⠡⠌⠋⢅⣲⠚⣄⡠⡴⠃⠀⠀⠀⠀⠀⢻⠙⠀⠀⠀⠀⠀⠀⠈⠀⢣⡀
⣧⣄⣀⠀⡀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠠⠠⠤⠀⠀⠀⠀⢀⠈⠉⠚⠁⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠄⠠⠠⠀⠀⠄⠀⠱
⠙⢿⣿⣤⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀⠁⢂⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣌⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⣄⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠁⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠁⠀⠀⠀⠀⠉⢻⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⣾⣧⣄⠀⠀⠀⠀⢀⡴⡫⣪⠾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠉⠙⢿⣿⣶⡄⢀⡴⡫⣪⠞⠁⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠁⠀⠀⠀⠀⠙⢿⣿⡫⣪⠞⠁⠀⠀⠀⠀⠀⢉⡿⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣺⣾⣧⡀⠀⠀⠀⠀⢀⡴⡫⣪⢿⣿⣶⣀⠀⠀⠀⢀⡴⡫⣪⠞⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠉⠻⣿⣿⣦⡀⢀⡴⡫⣪⠞⠁⠀⠙⢿⣿⣦⣀⡴⡫⣪⠞⠁⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠞⠁⠀⠀⠀⠈⠻⣿⣿⡫⣪⠞⠁⠀⠀⠀⠀⠀⢙⡿⣯⣪⡞⠁⠀⠀⠀⠀⠀⢀⡼⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠳⢨⡀⠀⠀⠀⠀⢀⡴⡫⣪⣿⣿⣦⡀⠀⠀⠀⢀⡴⡫⣪⠾⢿⣿⣦⡀⠀⠀⢀⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⠀⠑⡙⣦⠀⢀⡴⡫⣪⠞⠁⠈⠻⣿⣿⣦⣀⡴⡫⣪⠞⠁⠀⠀⠙⠿⢿⢦⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⠀⠀⠀⠀⠐⣸⣿⡫⣪⠞⠁⠀⠀⠀⠀⠈⢻⡿⣯⢪⡞⠁⠀⠀⠀⠀⠀⢀⡴⡃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⢿⣿⣄⠀⠀⠀⠀⢀⡴⡫⣪⠾⣿⣿⣦⡀⠀⠀⢀⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⢻⠛⠳⣄⣀⡴⡫⣪⠞⠁⠀⠈⠻⣿⣿⣦⡴⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠪⣣⡿⣯⣪⡞⠁⠀⠀⠀⠀⠀⢈⡿⡻⠊
⠀⠀⠀⠀⠀⠀⠀⡠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣪⡾⣿⣷⣄⠀⠀⠀⢀⡴⡫⠊
⠀⠀⠀⠀⠀⡠⡪⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡙⢄⠈⠻⣻⣷⣤⡴⡫⠊
⠀⠀⠀⡠⡪⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢈⡿⡫⠊
⠀⡠⡪⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣴⡫⠊
⠲⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡸⠊
⠀⠈⠮⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡨⠊
⠀⠀⠀⠉⢪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠊
⠀⠀⠀⠀⠀⠈⠪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⡢⡀⠀⠀⠀⠀⠠⡨⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⡢⡀⡠⡢⠌
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣪⣚⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠒⠋⠉⠉⠀⠀⠀⠈⠉⠙⠒⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠒⠶⠶⣯⣬⣟⣳⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠽⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⠋⠙⢻⢿⡏⠛⠲⠦⣤⠴⠒⠊⠉⠀⠀⢀⠀⢸⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠐⣵⠋⡻⣦⣀⠀⠏⢀⣀⠤⢒⣊⣥⣤⠞⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢀⢠⣿⠁⠀⣇⣾⠷⢦⣤⠌⠒⠋⠁⠀⢸⢃⣠⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⡠⢃⡟⢸⠀⢀⣿⣇⠀⠀⠀⠀⠀⣀⣠⣴⡾⠟⠉⠱⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡜⣡⢿⡀⣸⡶⠛⠛⠿⠿⠶⣶⣖⡛⠹⠛⠉⠀⢀⠀⢀⡔⡿
⠀⠀⠀⠀⠀⢠⡟⠉⠓⠲⠦⢤⣀⣀⠀⠀⠀⠀⠀⣠⢎⡼⡇⠀⣡⣿⣧⣄⣀⣀⠀⠀⠀⠀⠉⡆⢀⡠⠚⣡⡖⡽⠀⢣
⠀⠀⠀⠀⠰⡿⣭⣓⡂⠤⠀⡀⠀⠈⠉⢳⡄⢀⡴⢃⣾⠀⣿⣴⡟⠛⠛⠓⠢⠤⢭⣭⣁⡀⠀⠀⢈⣴⠞⡇⢼⣡⡶⠋
⠀⠀⠀⠀⠀⠹⡀⢈⠉⢛⠳⣤⣀⡀⠀⠠⠿⠟⣡⡞⣿⢀⣾⣿⣤⣄⣀⡀⠀⠖⠉⠋⠉⠉⠉⠙⢻⡅⢀⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠑⠚⠦⠬⠷⢄⡀⠙⢿⡒⠒⣾⣏⠀⢠⣿⠏⠉⠉⠉⠉⠉⠉⠛⠒⠶⠤⣴⣂⣀⢸⣧⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠴⣤⣘⣏⣢⣾⠟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⡀⠀⣀⠀⠀⠉⠉⠁⠀⡠⠊⢉⢻⣦⡀
⠀⠀⠀⠀⢀⡄⠈⣈⠹⣾⡿⢂⣡⠄⢴⡏⠷⢄⡀⠀⣰⠋⠀⠀⣸⡤⢿⣿⣀
⠀⠀⠀⠀⠀⠙⠚⠲⠾⠿⢄⣈⢀⣀⡃⣙⣤⡾⠁⠀⠣⢄⣀⣈⣉⣠⡼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠂⠀⠀⠉⠉⠉⠁⠈⠉⠁⠀⠀⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣴⣀⡁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣽⣿⣿⠿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢟⣿⣟⣹⣿⣿⣻⣿⡤⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⣿⣯⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣳⣿⣿⣿⣿⣯⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠔⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣏⡛⣛⣽⢿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠈⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠤⠀⠄⠀⢀⠀⢠⡄⠤⠄⠒⠉
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⢠⣄⣀⣠⣴⣶⣷⣞⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠋⠁⠀⠀⠀⠁⠀⠀⣠⠊
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠛⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡄⠤⠛⠉⠀⠀⠀⠀⠀⠈⢧⣄⣀⡀⠀⢀⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠶⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⠀⠀⠀⠀⠀⠠⠄⠤⠤⠤⠭⠀⠀⠀⠶⠶⠶⠶⠴⠦⠀⠀⠀⠀⠀⣤⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣦⣞⣠
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠐⠒⠒⠒⠒⠒⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣀⣀⣰⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣖⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠦⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣦⠀⠀⢀⣀⠀⠀⠈⠀⠀⠶⠄⠀⣖⠀⠀⠀⠀⠀⢃⠄⠀⠀⠀⠀⠉⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⣧⡠⢺⣿⠀⢸⣿⠀⠀⣶⠀⣤⠄⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠛⠁⣸⡇⠀⢸⣏⠀⠀⡿⠀⠙⢷⡀⠀⣿⠁⣰⡞⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠉⠉⠁⠐⠂⠼⠃⠠⣿⠀⠹⣇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠂⠐⠀⠠⠄⣀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠀⠸⠀⠠⠄⠀⡄⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠐⠂⠀⠤⠀⢉⠙⠀⠀⠀⠀⡂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⠀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠁⠘⠂⠐⠒⠠⠄⣤⡄⢀⠰⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⢛⠀⠀⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣀⡀⠠⣤⠀⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣼⠳⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⢿⠃⠀⠘⡇⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠟⢀⢀⣤⣴⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣀⠰⡀⢣⠀⠀⠒⢒⢻⣭⣈⣘⣻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣶⣶⣾⣷⣶⣄
⠀⠀⠀⠀⠁⣦⠤⢠⣤⣴⡖⠛⠓⠀⠀⠀⠐⠂⢳⠛⠛⠫⠾⠡⣄⠀⠀⠀⡄⠀⠀⠒⠠⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⠜⢻⠻⠷⢶⣶⡿
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⢠⠀⠈⠀⠢⢸⠘⠒⠒⠒⠂⠒⠃⠀⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠒⠀⠒⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉
⠀⠀⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⣤⠆
⠀⠀⠀⣞⢫⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠜⠁
⠀⠁⠀⠈⠀⠀⡀⠀⠈⠀⠀⢀⠀⠀⠀⠀⡠⠀⠰⣉⠉⣥⣘⢦⡀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⢀⠈⠀⡄
⠀⠀⢀⡇⠀⠀⠀⡆⠀⢀⡴⣿⠆⠀⠀⠀⡇⠀⠀⢏⢢⡘⣿⣷⠙⠳⣄⠀⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠶⠒⡤⠤⠀⣠
⠀⠁⢸⣿⣦⡀⠀⡇⣼⡿⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢹⣬⠅⠀⢱⡈⡇⠓⠒⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠒⢨⢁⣠
⠀⠀⡄⠙⠻⡁⠀⡿⠋⠀⠀⠀⠀⠀⠀⢠⡂⠀⠀⠀⠀⠀⠀⠀⠰⠾⠃⠷⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⠂⠷
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠄⠃⠀⠀⠢⡀⠀⠀⠀⣀⣶⡄⠤⠤⠤⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢸⢀⠀⠀⢁⣄⣀
⠀⠁⠀⠀⠀⠐⠂⠀⠀⠠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠀⠀⠐⠋⠆⠀⠆⠖⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠶⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣦⡤⣀⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠋⠀⢀⠀⠀⢤⡤⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠉⠳⣤⣀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡃⢽⡶⣄⠈⠑⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⡀⠀⠀⠀⠀⠰⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⢯⡛⠶⣿⣯⣌⣲⣤⣄⠲⣴⡆⣀⡀⢤⣤⠤⠀⣀⡀⠀⡠⠀⠀⠈⣊⣤⠎⠳⠠⠄⠀⠙⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠓⠊⢤⣴⡒⠦⣛⣴⣤⣤⣄⡀⠀⠈⠓⣤⣤⣀⠞⠀⠀⡀⠚⢩⡼⠂⠀⠁⠀⣀⠴⢚⠤⣴⠶⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠁⠀⠀⡐⠦⣤⣀⡀⠀⠈⠛⠷⠾⠇⣻⢿⣤⣿⣿⣟⡷⣮⣞⠉⠉⠒⠊⠀⠀⣀⣀⠠⠀⠂⠩⠐⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠆⠠⠄⣸⠍⠙⠻⠻⢏⣹⠷⢋⣙⠛⠛⠁⠀⠀⠠⠖⠋⠉⠀⠀⠀⠒⠀⠁⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠤⡀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠀⠈⠉⠉⠴⠶⠾⠀⠀⠉⠀⠀⡀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠙⠛⠓⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠊⠳⡈⠳⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠤⠤⠤⠤⠀⠀⠀⠒⠂⠈⠉⠉⠉⠉⠐⣀⠤⠷⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⡀⠸⡄⠀⡀⠐⠠⣄⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⣖⠴⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠁⠀⣧⣄⣈⣓⠒⠀⠉⠉⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠂⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠀⠤⠄⠉⠉⠉⠈⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠼⠴⢦
⠀⠀⠀⠀⠀⠀⠀⡴⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⣴⢉⡁⠁⣦⣄⠶⡄
⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡂⠀⠀⠠⣀⠋
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠁⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣝⡛⣋⡙⢻⡟⣫⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⢀⡀⢀⣀⠀⢀⣀⠀⢀⡀⠀⡀⠀⠀⣀⠀⣀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⠇⣿⢸⡇⣿⢹⡇⣿⠹⠇⡏⢹⡆⡇⠀⢰⣿⡆⢹⡏⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡄⣿⢹⡇⣿⢸⡇⣿⢠⡄⡇⢸⡇⡇⠀⣸⣭⣧⢸⡇⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠛⠘⠃⠙⠛⠁⠙⠛⠁⠙⠛⠀⠛⠛⠛⠀⠛⠘⠃⠈⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠲⠤⠤⠤⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢹⡿⡀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡟⠀⠩⠀⠀⠀⠉⠉⠙⠒⠒⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⡞⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠂⠀⠀⠀⠈⠉⠉⠀⠈⠀⠀⠄⠜⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠥⢐⡒⠀⠀⣀⡠⠄⣀⣀⣁⢈⠞⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠫⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢃⠎⠀⢰⠫⢄⣀⡀⠉⠉⣒⠶⠠⢤⣦⠾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠑⠁⠀⠀⠀⠀⠀⠀⢀⠊⠀⠠⢠⠎⠀⢠⠇⠀⠀⠀⠈⠁⠈⠀⠀⠠⠂⠵⠊⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠃⠤⠀⠀⠀⢰⠂⠀⠀⠙⠒⠢⠖⠻⠤⠠⠿⠁⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⣠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠃⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡦⠤⣤⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢤⡼⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡏⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠉⠀⣀⠀⠂⢂⡀⠀⡎⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠈⡡⢸⢁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠲⠦⠤⣤⣤⣀⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠯⣅⡀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠦⢄⡀⠈⠁⠀⢀⠤⠒⠊⠁⠀⠀⠀⠤⢐⣂⡭⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⡒⠢⢅⡀⠀⢀⣄⡤⠄⢂⣀⡠⠒⠛⠙⠙⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⣶⣤⡈⠀⠀⢾⠀⡴⠒⠋⢉⠜⢀⣰⣦⡤⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢟⠻⣶⣤⣀⢰⣆⣀⡤⠟⠒⠉⠀⠀⠀⢀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠈⢲⡸⡧⡈⠂⢜⠉⠉⠁⠀⠀⣀⣀⣠⠬⠭⣽⠓⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠈⠻⡁⠈⠶⡔⠒⠋⠙⠋⠉⢹⠞⣠⠆⠰⠁⠀⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠤⠾⡗⠉⣹⠠⡑⢄⠃⡇⠠⠀⠀⣀⣤⠯⠞⠃⠒⠓⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢉⣟⡶⠀⡐⠀⠈⠱⣄⢻⠆⠉⠉⠉⠉⠀⠠⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠀⠘⠋⠀⠀⠀⠀⠀⠚⠛⠋⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡄⠀⠀⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣿⣿⠀⠀⠀⡇⠀⠀⠈⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⡇⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⢀⠀⠀⠀⡀⣀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢈⡀⣀⠀⠀⠘⠁⠀⠈⠀⠀⣀⣀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⢎⠀⠀⠁⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠃⠀⠀⠀⠁⠘⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣄⣼⣿⡇⠀⢀⣶⡄⠀⢸⡆⠐⠉⢠⡞⠀⠀⠀⠀⣴⠉⠉⢢⡀⠊⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣳⣾⢷⣶⣾⣿⠀⠀⢸⡇⠀⠀⡞⠀⠀⠀⠀⠀⡇⠀⠀⠈⡇⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣏⣛⣫⣾⣿⣿⠃⠀⠀⢸⡇⠀⠸⡇⠀⠀⠀⠀⠀⣷⠀⠀⢀⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣻⠿⣿⣿⡀⠀⠀⢸⡇⠀⠀⠣⠀⠀⠠⢀⣄⠘⠷⠤⠊⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣫⣭⣶⣿⣿⣧⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢸⢸⠀⠤
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣻⣿⣛⣛⣽⣿⣿⣿⡆⠀⢠⠆⠀⣤⣀⡀⢀⠄⣸⡈⠀⣤⠀⠀⡖⢐⡄
⠀⠀⠀⠀⠀⠀⠘⢿⣏⣙⣛⡛⢛⣛⣹⣿⡿⠃⠀⢼⠀⡠⠏⠀⢱⢸⡀⠈⣇⠀⣿⡀⢀⣧⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠉⠁⠒⠀⠘⠊⠙⠛⠈⠉⠁⠉⠁⠀⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣲⠄⠀⠀⠴⠦⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠉⠀⣰⠆⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠋⠉⠉⠒⠂⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠠⠤⠷⠧⠤⡤⠤⠀⠳⣀⠆⠀⠀⠰⣏⡂⣖⠀⠀⠀⣀⠈⡍⠁
⠀⠀⠀⢀⡀⠀⠠⠔⠂⠀⠀⢀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠒⠀⠦⠤⠄⠒⠂⠀⠁⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠈⠛⠋⠉
⠀⠀⠀⠨⡀⠲⢄⣀⢠⡴⠚⠋⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠉⠒⠈⠿⣠⣒⣦⣅⡵⡀⠨⠐⠀⠀⠀⠀⠀⠒⠁⢀⡁⣄⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣂⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢔⠀⢀⠀⠀⠀⢠⡆⠄⣤⡀⠀⠀⠀⠀⠐⡦⠈⣶⡬⠔⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠰⣌⠀⠀⢐⣀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠀⠀⠀⢰⡅⠠⡔⡄⠤⠁⠔⠉⠔⡀⠀⠀⠀⠘⢆⢛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠀⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⢦⣱⡀⠀⠴⢂⡄
⠀⠀⠀⢀⣀⣀⣈⠀⢠⠥⠒⠂⠀⠀⢬⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣡⣴⢺⠿
⠀⠀⢀⠂⡴⢀⠌⠾⠃⠀⠀⢬⣴⣤⣈⠼⡦⠃⠀⠀⠀⢀⡠⠤⠢⢂⠠⠴⢊⠍⠁⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠠⠠⡀⡀⠀⠀⠀⠙⠉
⠀⠀⠘⠶⠶⠾⣇⡀⣀⣵⠠⡶⣜⣩⠜⠀⠀⠀⠀⠀⠀⠀⡴⢘⣉⣠⠤⠚⣁⣀⣄⣀⠀⠀⠀⠀⠀⢸⡀⠉⠀⠒⠳⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠽⠛⠊⢁⢉⢀⠨⢄⢈⡇⠀⠀⠀⠀⠀⠷⢃⣀⣀⣀⣄⡀⢷⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠤⠴⠾⠶⠒⠋⠉
⠀⠀⢠⠀⠒⡛⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠈⡀⠠⠄⠈⠾⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢯⠀⠀⠀⠀⢠⡋⠀⠄⠀⠀⠂⠈⠀⡀
⠀⠀⠀⢙⠂⠀⠀⠀⠰⠀⠀⠀⠀⠀⠓⠂⠈⠀⣀⠒⠀⠀⠀⠀⠐⠒⠆⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⣸⠏⠀⠀⠀⠀⠘⡴⠀⠂⠄⠲⡑⠚⣠⠖
⠀⠀⠀⠈⣆⡀⠀⣀⠴⠣⠀⠀⠀⠀⢠⢴⢤⣀⡠⠜⠃⣀⡈⠀⣠⡶⠀⠀⠀⠀⠀⠠⣉⣀⣤⠔⢢⠎
⠀⠀⠀⠀⢙⠋⠀⠀⠀⠀⠁⠀⠀⠀⠘⡀⡀⢠⢍⠠⠶⠒⢦⠚⠘⠓⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢧⡀⠐⠂⣀⠽⠀⠀⠀⠀⢇⠀⣠⣀⣀⡀⠤⠴⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠠⡀
⠀⠀⠀⠀⠀⠈⠓⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠀⢤⡒⠚⠉⠁⠙⢧⠀⠀⣀⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢴⡒⢞⠉⠁⠙⣄⠀⠀⣀⣄⣱⣀⣤⠾⠷⠾⢿⡉⠉⡉⠈⠢⡀
⠀⠄⠀⠀⠒⠂⢀⠀⠀⠀⠀⠀⢀⡄⢀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠸⢷⡀⠀⠀⣈⠳⣤⣤⣤⠶⠾⢶⠛⠛⡉⠁⢻⡀⠀⠀⢀⣀⣻⣠⠤⠴⠷⠓
⣄⠤⢄⡠⠀⠠⠘⢆⠀⠀⠀⠀⠋⠀⠀⠀⠉⠑⠁⠁⠀⠀⠀⠀⠀⡈⣷⠚⠛⠋⠉⢩⠀⠠⠀⠀⠀⠄⠀⣁⣤⡶⠷⠔⠋⢉⠁⠀⢀⡄⠠⠤⠔
⠰⠀⠤⠤⠔⠇⠐⠀⠀⠀⠀⠀⠠⣦⡅⣡⣀⣮⡀⠤⠔⠁⠀⠀⠀⠘⡇⠣⢀⣀⣠⣤⠧⠔⠚⢛⠉⠉⠁⠀⢀⣠⠾⠤⠾⠃⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠑⠄⠂⠘⢛⠁⡄⠀⠀⠀⠀⠀⠀⠹⣾⠉⠁⣀⣀⡴⠶⠄⠒⠛⠓⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣄⡀⣉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⠀⡄⣠⣄⢀⣤⡀⣤⣄⢠⠀⠀⣤⠠⣤⡄⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⢸⣤⡇⡇⢸⢸⠀⠁⡇⣿⢸⠀⢰⢿⡆⢸⠀⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠟⠸⠐⠇⠷⠾⠸⠶⠇⠧⠟⠸⠦⠼⠛⠇⠸⠀⠿⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠒⠒⠂⠀⠒⢒⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠳⠒⠒⠒⠦⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠉⠀⠀⠀⠀⢂⠀⢀⠎⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠤⣀⣀⣠⢀⠎⠀⣼⠇⠀⠀⠀⠀⠀⠀⡇⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡃⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⠊⠀⡜⠓⠀⠄⡈⠁⠝⠃⠒⠷⢛⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⢀⠀⠀⢤⣀⢀⣠⠇⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⢐⠇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⢤⣀⣀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⢤⡼⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢰⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡼⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠤⠤⠤⢄⣀⣀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⣄⡀⢠⡀⠀⣠⡄⠀⠀⠀⠀⠀⣶⡦⢀⠀⠀⠀⠀⠀⢀⣤⣾⣦⣄⠀⠀⠀⠀⠠⠐⠒⣄⡒⠀⠄
⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⣼⣿⡗⣲⡄⠀⠀⠀⠀⠛⠛⢻⠛⠛⠃⠀⠀⠀⢃⢠⡦⢐⣴⡆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⠀⠀⠀⠀⠀⣐⣀⣸⣀⣐⡀⠀⠀⠀⠨⢑⡨⠤⢂⡰⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⣾⡟⠛⠛⠛⣿⡆⠀⠀⠀⢸⡏⠉⠉⢹⡇⠀⠀⠀⠀⣴⡺⡽⣖⢄⠠⠀⠀⠀⢀⣤⣤⡄⣤⡤⣄
⠀⠀⡟⠀⠰⡇⠀⠙⡇⠀⠀⠀⢸⡷⠤⠤⢼⡇⠀⠀⠀⢸⢞⢎⢏⣎⣾⠀⠀⠀⠀⢸⣿⣿⣷⣿⣤⣿
⠀⠀⣿⣲⣴⢳⣴⣾⡇⠀⠀⠀⢸⣇⠀⠀⢸⡇⠀⠀⠀⠈⣯⢗⣷⣮⣎⠀⠀⠀⠀⠻⠟⠃⠀⠈⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉

⠀⠀⣦⣬⣀⣀⣥⣴⡄⠀⠀⠀⢠⣶⣾⣷⠂⠀⠀⠀⠀⠀⠀⢀⣒⣒⡀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠱⢆⣾⣿⠀⠀⠀⠀⢀⣿⡿⣿⣷⠄⠀⠀⠀⢸⣿⡏⣄⡌⣿⣿
⠀⠀⠿⠿⢿⡿⡿⠿⠃⠀⠀⠀⠈⠀⣾⣿⠿⠃⠀⠀⠀⠀⣼⣿⣷⣶⣶⣆⠀⠀⠀⠀⠉⢡⣟⣣⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⠉⠃⠀⠀⠀⠀⢠⠖⢻⠛⠲⡄⠀⠀⠀⠀⠾⠮⠴⠧⠷⠇
⠀⠀⠀⢸⡁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣹⠶⠀⠀⠀⠀⠀⠀⡟⠑⣾⣷⠊⣻⠀⠀⠀⢠⠀⢴⠄⢴⠄⢠
⠀⠀⣀⣀⣙⣲⣒⣋⡀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠘⠾⣅⣨⠶⠃⠀⠀⠀⠈⢶⣾⣶⣾⣶⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠙⠲⠶⠖⠛⠻⢦⡼⠛⠚⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠳⢤⡀⠀⠀⣠⡜⠋⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡙⣦⣾⠋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡏⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠳⣦⣀⠀⠀⡀⢀⣶⠟⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢤⣠⠴⠓⠾⠛⠛⢶⣄⡀⣠⡾⠋⠀⠀⠀⠀⠈⠻⢷⣤⣵⠟⠉⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣠⣤⡤⣤⣄⡒⠒⠀⠰⡌⢻⣿⣀⡀⠀⠀⠀⠀⠀⢀⣴⠟⠻⢷⡀⡀⠀⠈⢠⣾⣏⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢟⠁⠀⠀⠀⠀⣿⡉⠉⣸⣥⣄⣿⠉⠻⣤⣀⡀⠀⣸⡟⠃⠀⠀⠀⠙⢿⣄⣴⣿⣳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢸⠀⠀⠀⠀⠀⡟⠙⠛⠁⠀⢩⠿⣶⠂⠉⠩⡍⠙⠙⢧⣤⣄⡀⠀⠀⣠⡞⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣠⠞⢡⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡸⠀⠈⠛⠛⣿⠁⠒⣠⢤⠀⠙⢧⣀⣼⣿⡽⠋
⠀⠀⠀⠀⣠⡞⠁⠀⡼⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⣰⠏⣴⢶⡀⢠⢤⠀⠈⠻⣿⡿⣧
⠀⠀⣠⡾⠋⠀⠀⠀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣽⠟⠁⣈⣳⢄⠀⣠⣄⠀⣤⠀⠙⣧
⢠⡾⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⢳⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠙⠛⢛⡿⢻⣿⠃⢀⣠⡄⣸⣷⡾⠃
⢿⣇⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⢀⡶⠋⠀⣸⣿⠀⣀⣿⣄⣾⠋
⠀⠙⢦⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀⡴⠋⠀⣠⠞⠁⠘⠳⠏⠀⠉⠁
⠀⠀⠀⠙⢦⣄⠀⠀⠈⡇⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠙⢷⣄⠀⢹⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⡴⠋⠀⢠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⢧⠀⠀⠀⠀⠀⠱⡀⠀⣰⠏⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣠⠞⠁⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣠⠞⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣀⣼⠏

⠀⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⢘⣿⠀⠀⠀⠀⠺⢷⡀⠀⢰⣷⠀⢰⡗⠀⣶⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣦⣀⢸⣿⣤⣉⡀⢀⣴⣿⣅⠀⠀⢹⣧⣿⣥⣰⣿⣿⣟⡧
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡿⠻⣿⠁⠈⣛⠉⣿⣠⣄⣈⠛⢛⣿⡃⠘⠉⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡟⣽⣧⠀⣿⡆⢠⣿⠀⣿⣿⣭⡻⢀⠙⢹⡇⠀⠀⣼⠋⠉⠉
⠀⠀⠀⠀⣤⣤⣤⣤⣴⣿⣧⣾⣯⣼⣿⣿⣿⣿⣤⣿⣷⣾⣯⣬⣴⣾⣿⣤⣤⣽⣷⣶⣦⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠉⠉⠙⠋⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠿⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⠏⢿⣿⣷⣶⣾⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⠿⠿⠟⡈⠛⢿⣟⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣰⣿⣿⣿⣿⣿⣿⡿⣯⡿⠛⢠⠘⢿⣿⣿⠛⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣻⣿⣿⣿⣿⣿⣿⡋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣿⣿⣿⣿⣿⡛⠯⠁⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠛⠉⠊⠁

⠠⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄

⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⠤⠤⠤⠤⡤⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣛⣛⣀⡀
⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣶⣶⣶⣾⣶⣶⣶⣶⣷⣶⣶⣶⣶⣿⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡀
⢸⣿⣿⣿⣿⣿⣿⣠⠴⣶⣶⣄⠀⠀⠀⠺⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠟⠃⠀⠀⠀⠐⣖⠘⠛⣿⣿⣿⣟⠛⠃
⢸⣿⣿⣿⣿⣿⣿⡁⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣤⣴⣶⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡏⠑⠲⣿⣿⣿⡷⠖⠉
⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡀⠀⣿⣿⣿⡇⠀⣠
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⢀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⣾⣿⣿⣿⣿⣿⣿⡧⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⠲⢿⣷⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⣤⣴⣶⣾⣿⣇⣶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⠈⠙⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⢶⡶⠻⢻⣦⣤⣾⣿⡆⠀⣠⣴⣶⣶⡄⠀⠀⠀⢻⣿⣛⣽⣿⣮⣽⢸⣿⡛⢿⣿
⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣴⡾⠛⠓⢿⣯⣄⣠⡿⣣⣾⡿⠿⠛⠿⠀⠀⠀⠀⠈⢿⣿⣿⠿⠿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⣭⣟⣒⣒⣛⣻⣯⣵⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣾⣿⡿⠸⣿⣝⠛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣸
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢸⣿⣾⡇⣾⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣿⡇
⣿⣿⣿⠀⠀⠈⠻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢹⡏⠁⠹⣿⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣷⣶⣶⣶
⣿⣿⠏⠀⠀⠀⠀⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⡟⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⢦⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣼⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠷⠄⠀⠀⠀⠀⠀⢠⢧⠄⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠿⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⠿⠟⠃⠀⠀⢠⣏⡿⣆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⣑⠶⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣦⡀⠀⠀⣠⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠛⠊⠀⠀⣼⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣷⣶⣿⣿⣿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠈⠙⠗⠀⠀⠀⡰⠳⡝⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣀⣤⣶⣶⣶⡄⠀⣀⠀⠀⠀⢠⡐⠟⢩⣄⡌⠉⣄⣤⠀⠈⣩⠄⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠌⠀⠐⠾⠿⠁⠐⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⣀⣹⣿⣿⣿⣿⢿⡿⠋⠀⠀⠀⠈⢳⣾⣧⡿⣶⣿⡼⣧⣿⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤
⠀⣿⣯⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢹⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡄⣤⢠⢶⡀⡄⡄⢴⢠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠌⠀⢸⠿⢻⢸⢸⡇⣿⠇⢸⢸⡇⠸⠓⡀⠰⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠢⠔⠀⢁⠀⠀⠆⠈⠠⠀⠠⠉⠁⠒⢠
⠀⠠⣐⣦⣤⣄⡢⡌⠁⡀⠀⠘⠀⠘⠈⠛⠁⠘⠀⠚⠈⠓⠀⠀⠀⠁⠀⠀⠈⠀⠀⠃⠐⢤⠀⠈⠂⠀⠀⡠⠃⠈⠇⠀⠀⢐⡀⣥⠀⣀⠴
⠀⠌⠛⠛⠋⠉⠀⠃⠀⠈⡆⠀⠈⠄⠀⢀⠄⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠆⡠⠃⠀⣬⠆⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⢀⢐⡀⣀
⠀⠄⠀⢿⣿⡗⠀⡬⠀⢂⣀⠀⠀⡼⠰⠄⠀⡀⣀⠀⠀⠃⠀⠀⠠⠞⠀⠀⠀⢀⠑⠐⡂⠡⠄⣀⠀⠀⠣⠀⡔⠉⠢⡀⠀⡰⠊⠁⠀⠀⣄⠑⡄
⠀⠈⠂⠠⠭⠀⠊⠀⡈⣤⣴⣶⣶⣶⣶⣦⣤⠀⠀⠈⠀⠃⠀⠀⢀⢀⡀⠀⠀⠀⡄⠙⠤⢄⠀⣻⠈⢈⠁⠸⠀⠀⠀⠓⡀⣄⠀⢁⠀⠌⠠⢀⠇
⠀⢀⠄⠠⠠⡀⠀⠀⡇⣿⣿⣷⣿⣟⣿⣿⣿⠀⠀⠀⡆⢰⠀⠀⠙⠺⠹⠈⠠⠀⠰⠀⠀⢛⠒⠚⠀⠂⡃⠘⡐⠀⠀⣈⠃⠈⠐⠀⠀⠀⠀⠁
⠀⠊⠀⠀⠀⠈⢂⠀⡇⣿⣿⡟⡿⣿⡿⢿⣿⠀⠀⠀⠆⠀⣤⠆⠂⠐⠒⠒⠒⠒⡖⠀⠀⠸⠀⠀⠀⠨⠁⠀⢱⠀⠀⡄⠀⠀⢠⢴⣯⡀⠀⠈⠢
⠰⣅⣀⣀⢀⣀⠃⠀⠇⠛⠿⠿⠿⠿⠿⠟⠣⠀⠀⢀⠀⠀⠉⠀⢀⡠⠄⠐⠀⠤⡀⠀⠀⠀⠆⠀⠀⡠⠀⠀⠈⠒⠒⠁⠀⠀⠁⠀⠈⢿⣄⠐⡰
⠀⢽⠔⠛⢸⡇⠀⠀⠸⠋⠉⠉⢉⣉⡉⠉⡈⠉⠻⠅⡠⠐⡠⢤⣌⡰⢄⠀⠀⡠⠄⢹⠒⠄⡀⠀⠀⡴⠒⠒⢠⡀⠀⠐⠄⠀⠐⢀⠀⠸⠿⠋
⠀⢺⢤⡄⢾⠀⠀⠀⠀⢸⠉⠀⠀⣠⡤⠀⠃⠀⢀⡐⠀⠠⡀⠀⠀⠀⢀⠀⢸⠀⠀⣘⠢⠱⠈⢄⠀⠀⠀⠒⡆⣶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠙⠲⠧⠟⠂⠀⠀⠀⠈⡄⠀⠠⡤⠛⠀⠸⠀⠑⢄⠀⠖⡱⠆⢩⠀⢊⠐⢚⡁⢎⠠⢊⠜⢀⠒⠀⠢⠄⠤⠬⠤⠤⠤⠊
⠀⠀⠀⢀⣤⣶⣶⣦⣀⠀⢃⠀⠀⠶⠐⠂⠀⡆⠀⠀⠑⠠⢀⠀⠀⠉⠉⠀⠀⠀⠀⢁⠠⠐⠁⠀⠀⠀⠀⢀⠠⢀⠀⠀⣿⣻⠏⣿⡟⣿⡏⡉⣿
⠀⠀⣴⢟⣭⡶⣶⣭⡻⣆⠘⠒⠒⠈⠉⠉⠀⠀⠀⠀⢀⣀⡤⡀⠀⠀⠀⠀⠀⢀⢠⠶⠤⠤⠀⢄⠀⠀⢠⠁⠀⠀⢑⠀⠟⠛⠛⠃⡀⠈⠛⠛⠛
⠀⠰⠿⠟⠁⠀⠈⠙⢳⢿⠀⢠⣴⣶⣶⣶⣶⣶⣶⡄⠈⢥⠜⠁⣀⡤⡄⠀⠼⢁⣄⡀⠀⠘⠃⠃⠀⠀⢸⠀⠀⠐⠘⡄⠀⠀⢘⠀⠉⠈⠈
⠀⢃⣀⡀⠀⠀⠀⠀⣄⣠⠀⢸⡏⠁⠸⠀⡏⠠⢸⡇⢠⣀⢶⡄⢈⣀⠀⠀⠄⠈⠛⢀⠔⠒⠤⣀⡸⠀⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄
⠀⠀⢹⣷⣄⠀⢀⣼⣿⠀⠀⢸⣇⠀⡀⡄⠃⠰⢺⡇⠈⠁⠀⢠⣼⣙⠃⠀⠈⠒⠒⠁⠀⣀⠀⠀⢀⡀⠀⠑⢰⡶⠒⠁⢐⠀⠀⠀⠐⠈⠀⠐
⠐⠂⠘⠋⡻⠀⠘⢋⡙⠀⠀⢸⣿⢾⣷⣿⣾⣤⣾⣿⠀⠀⢀⠀⠈⡉⠀⠀⠀⠀⠀⠀⢺⠁⠋⠙⠛⠛⣿⣆⠘⠷⠇⠀⠺⣶⣿⣛⣓⣚⣻⣷⣾
⠀⠈⣠⣉⠤⡀⠀⠀⣈⡄⠀⣰⣿⠀⡇⡆⢀⠘⢸⣿⠀⢰⠓⠲⠾⠶⢾⠄⢄⠀⠀⠀⠇⠀⣘⢀⡀⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⢀⠿⠏⢀⡠⣿⣿⠿⠷⠀⠉⢿⣶⡷⠶⠿⠶⠾⠟⠀⢸⠀⠀⠄⠠⠀⠀⠀⠀⢀⡞⠋⡧⠰⠎⠀⡤⠢⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢈⠅⡇
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⡀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠀⢠⠆⠌⠀⠸⡄⠀⡀⠀⠀⠀⠃⠀⢸⠀⠂⠀⠞⣀⣀⡀⠙⠀⠈⠀⠁
⠀⢀⡀⠤⠤⠤⠀⠀⠀⠀⠀⡘⠀⠀⠀⠘⠀⠀⢀⣀⡀⠀⠑⠀⠀⠀⠊⠀⠀⠰⠀⠘⠒⠁⠀⠀⠀⠧⠴⠃⠐⠀⠀⢼⣿⣿⣿⡗⠀⡠⠀⠐⠄
⠀⠂⠀⠀⠀⠀⠀⣳⠀⠀⡖⠃⠀⣀⠀⠀⠑⡄⢸⠀⠀⠉⠐⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠄⠀⠀⡄⠀⠀⠙⡍⠉⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠇⠀⠈⠿⢿⠀⠀⠃⠈⡀⠀⠀⠀⠀⠀⡥⠐⠭⠂⠀⠀⠀⢀⠄⠠⣀⠀⠀⣀⠀⢀⠇⠀⠀⢠⠇
⠀⠀⠀⠀⢶⠂⠀⢸⡀⠀⠀⠀⣀⠀⢀⣀⡀⠀⠀⠇⠀⠀⢀⠔⡨⢔⠁⠀⢀⠀⠀⠐⣵⡿⢷⡌⡆⠀⠜⠀⠣⡀⠀⢔⠺⠂⡄⠀⢸⠀⠘⠃
⠀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠟⠣⠞⠃⠈⠀⢀⠀⠀⠈⠁⠀⠸⠀⠀⠀⠠⡊⠀⠙⠾⠋⠈⠀⢸⡀⠀⠀⡄⠀⠀⠀⣤⡆⠀⠈⡄
⠀⠑⠚⠚⠛⠛⠛⠋⠀⠀⠑⠆⠀⠀⠀⠀⠀⠐⠃⠀⠉⠑⠒⠒⠢⠤⠆⠀⠀⠀⠈⠢⠠⠊⠀⠀⠀⠈⠑⠀⠊⠁⠀⠀⠀⠈⠁⠀⠀⠐⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢹⠶⠂⠦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠒⠷⣦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠖⣤⡜⠛⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠈⣟⢦⣽⣎⣒⢠⣈⡉⢄⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠫⠆⡠⢄⣀⠀⠈⠶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠓⠪⠋⣙⡯⣭⡹⣠⣉⣀⡉⠉⠉⠣⣀⣁⠀⠔⠁⠄⠠⠞⣡⠅⠀⠀⠀⠀⡮⢄⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠑⠀⠀⠐⢤⣄⡀⠀⠈⠛⠷⠼⡏⣻⣯⣽⣿⣷⣶⣼⡭⠙⠂⠒⠀⠀⠀⢁⡀⠀⠐⠃⠅⠂⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠳⣄⠀⠀⠀⠀⠀⡀⠈⣙⠳⢂⣦⣜⣎⡙⠛⠻⣛⣿⢾⣍⢛⡛⠁⠀⠀⠠⠚⠉⠁⠀⠀⠂⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⢤⣀⠀⠀⠀⠀⠀⠙⠒⠀⠈⠀⠀⢈⠐⡓⠺⠛⠃⠙⠁⠠⠤⠀⣀⣤⣅⡀⠀⠈⠀⡘⠋⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⢧⠈⢷⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠓⠒⠒⠂⣀⠈⠉⠉⠀⠀⠈⠀⠀⣤⡐⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠙⢧⠀⡇⠀⢠⣈⠒⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠸⡗⠃⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠟⠓⠲⠤⠭⠀⠀⠀⠈⠶⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⢄⡀⣀⡈⠁⠉
⠀⠀⠀⠀⠀⠀⡉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠍⠉⠀⡈⠝⠀⠁
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⠈⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⡿⠦⣤⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠢⡀⠀⠀⠁⠠⡀⠀⠀⠀⠀⣠⠖⡥⠊⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⡤⢈⠢⡀⠀⠀⠈⠁⠂⠒⢋⡹⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡻⠀⢠⠑⠠⡀⠀⠀⠠⠊⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠊⡛⢠⡅⢉⡑⠎⠀⠀⢀⠼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠈⠢⡃⢨⠐⠀⠦⢒⡁⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⢠⠆⠁⢀⣸⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠀⠀⠀⠀⠉⠑⢎⠒⢀⠾⢉⠢⣀⠀⠀⠈⠀⠄⠀⠀⢀⠔⠁⠀⡤⣋⠐⠚⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠋⠁⡴⠀⠀⠀⠀⠀⠀⠀⠀⢂⠒⣮⡀⢬⠠⠍⠑⢠⡀⠀⠀⠀⠉⠀⠀⣰⡼⠄⡄⣻⡴⠛
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⠔⡡⠞⠉⢛⣿⢗⠘⠛⠤⠈⣓⡄⡀⢠⠔⣈⠃⠶⣡⠟⠉
⠀⠀⠀⠀⠀⠀⡼⠃⠁⠑⠒⠠⠤⠀⣀⣀⡀⢀⡠⠐⠉⠀⢀⣶⢚⣅⠶⢁⣤⡀⢹⠇⡴⠀⣙⠈⠉⡤⠚⠁
⠀⠀⠀⠀⠀⡼⠓⠀⢀⡀⠀⠀⠀⠀⠀⠐⠀⡄⠀⢀⠤⠊⢡⡄⣀⡵⠚⠁⠀⠉⠳⣌⡋⠰⢂⡰⠋
⠀⠀⠀⠀⠀⡇⠰⠦⠎⢡⣄⢀⡇⠠⠐⣂⠤⠟⢊⠃⠤⠴⣢⠞⠉⠀⠀⠀⠀⠀⠀⠈⠛⠷⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠒⠢⠭⢤⣐⣒⡀⠩⠄⡄⠚⣄⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣴⣶⡆⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⣿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢠⣿⣿⣿⣶⠀⠀⠀⠈⢻⣿⠿⣿⣿⡟⣱⣷⣶⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡿⣿⣿⣿⡄⠀⠀⠀⠀⢁⣤⣻⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⢠⣶⣿⡇⠋⠀⢸⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠀⣨⣍⡛⢧⡹⣿⣿⣿⣿⣇⡇⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠐⠻⢿⣿⢿⡏⢸⣿⣿⣿⣷⣄⣸⣿⣿⣿⢹⡇⠀⣸⣦⣽⣿⣿⣿⣿⡟⠿⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢘⣯⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⣿⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣷⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣯⣥⣷⣾⣧⣀⠀⣠⢽⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣫⡽⠖⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣰⣿⣍⠳⣜⠿⣿⠿⣟⣛⣛⡿⣿⣿⡿⣫⡵⢋⣥⣶⣿⡧⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣍⡛⢛⣋⣭⣭⣭⣙⠲⠶⠚⣫⣴⣿⡿⠋⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣠⣤⣴⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⡄
⠀⢺⣿⣿⠟⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⠏⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⢠⣾⡿⠋⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢰⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⠘⣿⠇⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⡇⠈⠟⠛⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠟⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢀⣀⡀⣠⠆⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢠⠄⣿⠁⠀⣤⣤⣶⣳⣎⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠙⣆⠀⠐⣤⣷⣾⣿⣆⠀⠈⢹⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠲⣀⣄⠘⠾⡁⠀⣀⠤⢉⢻⣷⣠⣾⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢆⢀⡼⢿⡄⠛⣉⣑⠛⠿⣋⢸⣿⢏⣽⠛⠋⣻⣯⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⢀⡿⠒⠋⢻⠀⣻⠿⢤⢈⡿⢋⣴⡁⣿⣷⣤⣄⢽⣎⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣥⢀⣞⣤⣤⠀⣸⠀⡍⣩⣼⣿⣇⣘⠇⣽⡌⡏⢀⣬⡞⢩⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢠⣆⠳⣤⣍⡳⣶⣾⣇⡩⠝⠿⣯⣷⡾⣿⣣⣏⡘⣛⣴⢫⣾⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠐⠶⣶⡟⠉⣁⡿⢿⡿⡇⠀⢠⡾⠋⢸⣯⢀⣴⣾⠟⣻⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣬⣑⠺⠿⡚⢻⣿⡿⠴⠶⠂⠲⠾⣦⣲⣾⣾⣿⣷⡶⠞⢛⣩⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⠐⠒⡤⣭⣭⣭⣁⣀⣀⣈⣉⣉⣉⣭⣥⣶⣶⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠿⡄⠀⣇⠀⠀⠈⠙⠟⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢿⠁⢸⡄⢐⢰⣷⠆⠾⣿⣿⣿⣿⣟⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠸⡏⢸⣼⣿⣧⢽⣿⣿⣹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡄⣿⡄⣾⣿⣿⣸⣿⣿⣿⣿⢸⣽⣿⣿⣿⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠧⠧⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠿⠿⠿⠿⠙⡂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠗⠀⠀⣼⣿⡇⢠⡟⣀⡀⠀⠀⠀⠀⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⢸⣿⣿⣤⠟⠹⡿⠟⠀⣀⢀⣀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⣾⣿⣿⠀⣾⣿⡇⠀⣾⣿⠁⡿⢠⣾⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣿⡟⢀⡜⠻⢿⠼⠃⠸⢿⣏⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠈⣉⣭⡤⠶⠶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠆⠀⣼⣿⣿⣤⠶⠛⠉⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣴⣿⠿⠋⠁⢀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⣠⣾⣿⠛⠁⢸⣿⡇⣴⣶⣦⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⡀⠀⢀⡾⣿⣿⠄⢀⣤⣤⠀⢀⣠⡄⢰⣿⣿⡇⠀⢠⡿⠋⣠⣬⠉⢁⣴⣿⡿⠋⢻⣷
⠀⠀⠀⠀⠀⠀⢀⣿⣷⢀⡞⢹⣿⣦⡟⣸⣿⡿⢠⣿⣿⠇⢠⣾⣿⡇⠈⢿⣿⣷⡀⠉⢀⣾⣿⡟⠀⣿⣿⡟⠿⣦⡿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡾⠀⣿⣿⡟⢀⣿⣿⠃⢸⣿⣿⠀⢸⣿⣿⠀⢠⡞⢻⣿⣷⡀⢸⣿⣿⢁⡴⢿⣿⣇⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁⠀⠛⠛⠁⢸⣿⡏⣠⡟⠻⠷⠴⠋⠙⠛⠒⢋⣠⠤⣿⣿⣧⡀⠉⠛⠋⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣼⣿⠏⠉⠻⣷⣤⣀⠀⠀⠀⠀⢀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⠿⣷⣦⣤⣤⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠠⣄⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡄⢠⠀⡂⠀⠀⠀⣀⣤⣠⡈⡆⠀⠀⠀⠀⢰⠀⡄⠀⠀⢀⣴⠿⢿⣦⠀⠀⠀⠀⠀⡜⢃⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣿⠻⠏⣿⡇⠀⠀⠀⣈⠀⠶⠀⠀⠀⣿⣿⠛⣿⡆⠀⠀⠐⣻⡏⠁⠁⠀⠀⢰⡇⢸⢀⡇⠀⠀⠘⣿⣤⣴⡿⠀⠀⠀⠀⠋⠠⠋⠀⠀⠀⣿⣫⣿⣻⡇
⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠁⠈⠀⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⣒⢴⣲⠀⠀⠀⣠⣴⡞⢿⡆⠀⠀⠀⠉⠸⢃⡀⠀⠀⣾⢷⣶⡶⣶⠀⠀⢠⡤⡤⢠⢤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣠⣄⣄⠀⠀⠀⠀⠴⣷⡄
⠀⠀⢶⣿⣾⣿⠀⠀⠀⣷⣴⠿⠟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⠀⠀⢸⡆⡇⢠⣾⠀⠀⠰⣷⠀⠀⣾⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⢰⣿⡇



⠀⠰⠶⠆⠰⠆⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⣀⣠⣞⣟⡷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⢀⠀⠀⡀⠀⠀⢠⠤⠤⠤⠤⢤
⠀⢀⡀⠰⢆⡰⠆⠀⠀⢠⣄⢠⣤⠀⠀⠈⢿⡾⠾⠁⠁⠀⠀⣶⣦⣶⣤⣤⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢺⣧⣼⠇⠀⠀⠀⢸⢡⡆⡟⠀⠀⢸⠹⠀⠇⠳⢸
⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠁

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⣐⡿⢣⣤⡄⠀⠀⠀⠄⠛⣿⡆⠀⠀⣀⣀⣤⣄⡀⠀⠀⠠⠀⠀⢂⠀⠀⠀⠀⠀⢠⣮⠛⠀⠀⢠⠊⡉⣳⣀⡀⠀⠀⠀⢰⣦⡤⠀⠀⠀⣄⣰⡀⡄
⠀⠘⠿⢣⣾⠍⠀⠀⠘⣿⣦⡄⠈⠀⠀⠀⣿⡏⣶⢎⣿⠀⠀⠀⠱⣖⣻⡆⠀⠀⠀⡴⠟⠁⠀⠀⠀⠘⢄⣀⠜⠉⠁⠀⠀⠛⢻⣿⠟⠀⠀⠘⡘⢤⢼⡘
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠈


⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐
⠀⠀⣼⣿⣿⣧⠀⠀⢸⢾⡯⡭⡿⠇⠀⢀⣾⣶⣶⣾⣷⠀⠀⠠⠤⢤⠤⡄⠀⠀⢠⣼⣟⡷⣤⠀⠀⠀⠀⡖⠿⠀⠀⠀⠀⠀⡠⠚⠁⡀⠀⠀⣤⣶⠀⠀⡀
⠀⠀⠘⠾⠷⠃⠀⠀⠈⠀⠀⠁⠁⠀⠀⠈⠻⠟⠉⠻⠟⠀⠀⠠⠤⠦⠹⠄⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠶⠀⠚⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠛⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠐⠠⠢⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⠀⠘⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄
⠀⢸⣿⡇⠈⠁⠀⠀⢰⣋⣟⣹⣏⡇⠀⠀⠘⠁⠀⠁⠀⠀⠀⠀⡏⠀⢹⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠦⢀⢀⠂
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠁


⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣼⣿⣿⣼⣷⣶⣧⣼⣧⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣷⣶⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⢿⠀⠉⠙⢿⡆
⠀⠀⠀⠸⣿⡇⣿⣿⡟⠘⠋⠋⠅⠀⠀⠀⠉⠀⠈⡇⠀⠀⢀
⠀⡀⠀⢀⣿⣧⣸⣿⡇⠀⠀⠀⡆⠀⠀⠀⠂⠁⠀⡇⠀⠀⡄⠂⡄
⢆⠀⣠⣾⣿⣿⠏⢻⡇⠀⠀⠀⢀⠀⠀⠀⠀⡠⠀⢧
⠈⠛⠛⠛⠛⠁⠀⠀⢹⣄⢀⠀⠀⢸⣶⣶⡇⠀⠀⡸⠀⢀⠁⠇
⠀⠀⠀⠨⠐⠄⠀⠀⠘⠮⣅⡀⠀⠈⠁⠁⠀⣀⡜⠓⠆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠄
⠀⠀⠀⠐⠂⠨⡢⠀⠀⡀⡔⠉⠑⠒⠒⠒⠋⠁⠘⡆⠁⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠖
⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠉⢲⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠶⠎⠀⠀⠁⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣤⠤⠤⠶⢲⠾⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠒⠲⣶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠀⠀⢀⣀⣴⣏⣀⣾⠄⡼⠀⠀⠀⠀⢸⣿⣭⣭⣍⣉⣹⣿⣿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢑⢦⠞⠉⠉⠁⠀⠀⣠⢣⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠱⡀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣵
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢠⡌⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠇⠀⠈⠀⠀⠀⠀⠀⢀⠐⠁⠠⠛⠉⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⠀⠘⠶⣤⡈⠐⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⡠⠀⠀⠀⠈⠙⠳⣄⡀⠀⡠⠃⡀⠊⠀⠀⠀⠀⠀⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡟⢶⡦⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣬⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠉⠀⣠⠟⠛⣀⠢⣀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠍⠓⢤⡈⠐⢄⠰⠒⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠚⠁⠀⠘⢿⡄⠁⠈⠳⣄⠙⠢⠄⠀⠀⢀⠞⠁⠖⠁⠀⠀⠀⠉⠳⣄⣁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠀⢀⠀⠀⠙⢦⡀⠀⠀⠉⠢⣀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠹⠛⢷⠀⠙⠦⣀⠀⠀⠈⢿⣥⣄⠀⠀⠀⠉⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢠⠔⠠⠀⠀⠸⠖⢳⠀⣨⣷⣄⡀⠂⠀⠈⠳⠤⠀⠀⠀⠀⢂⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠰⠁⠈⠁⠓⢂⡀⠀⠠⠾⠁⣀⣸⡙⠲⣄⡀⠀⠀⠀⢤⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠑⠊⡅⣰⠋⣼⢤⡀⠀⠻⠽⠋⢹⠂⠀⠉⠓⠦⣤⣠⠏⣹⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠉⠡⠞⡁⢖⣮⡤⠂⠀⢀⢎⣀⣠⠶⣤⠴⢤⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⡏⠺⣈⠅⠽⡏⠈⠋⠀⢀⡴⠋⠛⠒⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢁⠨⢄⠇⠠⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⠂⢠⣤⣤⣄⣀⠀⠀⣰⣧⣬⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣽⣿⣶⣿⣿⣿⣿⣿⣆⡀⣿⣯⣄⢠⣀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠹⠇⠹⢿⣿⣿⣿⣷⠸⡅⣩⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡉⠀⣘⠃⠸⣿⣿⣿⣿⠤⣧⡙⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣦⣁⡴⠀⠻⣿⣿⣿⣶⠟⣿⠟⢀⣾⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⢻⠯⣤⡟⠀⠀⣀⣈⣋⠀⠀⠀⠘⣇⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠄⣌⢠⢠⠀⢈⠄⠀⠐⠙⠿⠈⠳⡆⠀⠀⢻⣿⣿⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⡾⠃⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢇⠀⠀⢸⣿⣿⡆⠀⢌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣰⣤⣞⠰⣧⠚⣷⣄⣄⠀⠀⠃⠀⢀⣲⡏⢳⣿⠀⢸⢸⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠈⢿⡟⠁⢠⣿⣿⡿⠀⢰⠄⠀⣨⢷⡅⢈⣻⠀⢸⣾⣿⣿⡇⢠⣟⠃⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⣠⠟⠈⢻⡀⣰⠃⠀⣴⠋⣾⣯⣨⣿⠀⢘⣿⣿⣿⡇⠀⠈⢁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⠶⢀⠈⠃⠋⠐⠈⣧⠚⠉⠛⢿⡿⢀⣾⠛⠉⢹⡇⠐⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠓⠧⣀⣠⡀⡇⠀⢀⠄⠈⡠⠊⠃⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡛⣡⣤⣴⡾⠁⠀⠀⢀⣴⡟⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⢠⣤⣴⠋⠀⠀⠀⠀⠠⠽⠋⠙⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢀⠀⣿⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣰⡏⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠀⠀⠀⢠⠀⠀⠀⠀⠱⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⣀⢀⡀⠉⠀⣀⠴⠦⠀⠀⠀⠀⠀⠀⠕⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠠⣻⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⡟⠃⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣭⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠀⣗⡆⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣿⣧⡀⢫⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡛⢃⣺⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⠉⠀⢠⡤⢠⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠛⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠋⠉⠉⠉⠛⠒⠶⠤⣄⣀⡀⠀⠀⠀⡤⠾⠃⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢉⡿⠷⠞⠁⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡤⠽⡒⠊⢉⠽⠁
⠀⠀⠀⠀⣼⠋⠙⠓⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡰⠖⠒⠫⠁⠂⡀⠌⢁⣀⠀⠒⢀⠠⢀⡀
⠀⠀⠀⢠⡇⠤⡀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣄⣤⠤⠤⠤⠼⠀⠀⠀⠀⢀⠤⣪⣴⠾⠚⠥⢰⣀⡠⠦⢁⣀⠁⠃⢨⠀⠤⢃⠀⢠⠐⠙⡄
⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠀⠀⠀⠈⢉⡇⠀⠀⠀⠀⠀⠀⣠⠵⣚⣿⠍⠂⣀⠤⠀⡀⣈⠙⠂⢠⠤⠔⣃⠠⢄⠀⠋⠀⠀⠀⢘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣄⣀⠀⠉⠊⠙⣄⡀⠀⠆⡽⠺⠋⢠⣀⠀⠀⠀⠑⠂⣁⠠⣚⠢⠤⣀⠤⠋⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⡤⠜⠀⠣⠳⣄⠀⠀⠀⠉⠉⠀⠠⠎⣀⠀⢀⠭⠒⠠⠀⠀⠀⡀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠓⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠔⠒⢦⡀⠀⠀⠀⠀⠈⠉⠛⡶⠦⢤⣀⡀⠰⡟⢋⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠚⠙⠲⠤⠤⠤⠵⠀⠀⠀⠉⠙⠋⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧⢤⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⡇⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢹⠀⣿⣿⣿⣿⣿⣿⡆⢉⢻⣿⣿⣿⣿⠀⣿⣩⠟⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⣿⣿⣾⣿⣿⣟⣟⡏⣹
⣿⣾⣿⣷⣦⣿⣿⣤⣤⣤⣤⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡍⣿⠀⣿⣿⣿⣿⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠀⣿⣻⣿⣿⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡛⢻⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡟⢛⡟⠛⢛⡻⠟⠛⠛⠛⠋⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⣿⠘⢿⣿⢿⣿⠿⣿⠏⠁⣸⣿⣿⠀⣷⣮⣿⡾⠋⣠⣴⢶⣶⣦⣄⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⠀⣿⠀⠈⠋⣡⠵⠯⠉⠁⠈⣱⣼⣿⠀⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⣿⣿⣿⣹⣾⣽⣿⣿⣿⠀⣿⢀⣾⣤⣁⣗⣀⡀⣠⣧⣉⣾⣿⠀⣿⣿⠃⣄⣴⢟⡛⠛⠲⡄⠀⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡘⣿⣿⡿⠈⢭⠰⣿⣿⣿⣿⣿⠀⣿⣻⣇⣿⣷⣿⡏⠑⣄⣿⣾⣿
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣥⣿⣿⣷⢦⣌⣤⣹⣿⣿⣿⣿⠀⣿⣷⣿⢾⣿⣿⣿⣾⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⡟⡟⠈⣿⡟⣿⣿⣿⣿⠀⣿⠏⠉⣿⣿⠿⢿⣿⠿⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢱⣿⡇⣿⣾⣿⣇⣿⣿⣿⣿⠀⠛⠒⠛⠛⠛⠛⠛⠓⠒⠒⠒⠋

⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣀⡀⠀⢀⠤⠒⡤⡀⠀⢀⡠⠤⣀⠀⢀⡀
⠀⠀⢸⠏⢳⣝⣿⣦⣙⣷⣌⠻⣿⣕⡄⠈⣁⠤⠐⠒⠉⠁⢡⡁⡇⠀⠘⠻⠅⠀⡓⢓⣈⣅⢡⠔⠡⡄⢖⡄⠱⡈⠉
⠀⠀⡾⢄⣀⣦⣤⢀⣀⠀⠈⠉⠉⠉⠁⢂⢀⠀⡀⠀⠀⠀⠘⠞⡇⢸⠲⠤⣔⣒⠽⣼⣧⠔⠉⡇⠉⡥⡠⣒⢂⠇
⠀⢠⠁⠉⠙⠃⠶⠾⣛⣩⣵⡶⠦⣤⡄⠀⠉⡏⡝⠉⣀⡒⠒⠓⠁⠘⠚⡲⠢⢌⡉⠑⠺⣿⠒⠸⢦⣭⡡⠴⠊⢀⡀
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠛⠫⠃⠀⢀⠑⠃⠀⠉⣁⢀⡀⠀⠀⢠⠃⠀⠀⠀⠉⠡⠀⠀⠀⠀⠀⠈⢳⠀⠛⠃
⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠽⢿⠃⡤⠂⠉⠀⢀⠈⠉⠢⣌⠒⡦⠤⣀⡀⠀⠀⢀⠀⠀⠀⢀⠎⠉⠁⡹
⠀⢀⣍⠁⠒⠠⠤⢁⣀⠀⠀⠀⠀⡇⠀⠀⢀⠎⠀⠀⢆⠀⠸⠁⢀⡠⠈⢣⠈⠁⡒⢩⡝⡒⠤⢌⡀⠀⣜⡀⠀⢰⠁
⠀⠘⠉⣀⡤⠔⠊⠠⣀⣈⠉⠒⠒⠀⠀⣠⡿⠀⣀⣀⠠⠆⣘⠂⠁⢀⡀⠈⡆⠹⣛⣀⣉⠐⠚⠓⢡⣌⠀⠈⠲⠃⣤
⠀⢠⠊⡀⠀⠐⠑⠂⠀⠌⢳⠀⠲⠖⠛⠁⢠⡀⠀⢀⣠⢴⡀⠺⡁⡡⠁⢀⠇⡴⠇⠤⢈⠓⡄⠀⡠⡔⠂⠀⢲⣄⠉
⠀⢹⣦⠤⣀⣁⣀⣀⣀⣀⣼⠀⢠⠿⠴⣄⢄⢢⡀⠉⠒⠰⠇⠀⠑⠁⢀⠜⠀⣇⢐⡒⢡⡅⡇⢔⠑⠒⠒⣂⡭⡊
⠀⠸⣿⠀⣿⠀⣿⠇⣿⢸⠇⢀⡖⢔⣒⠨⡉⠃⠙⠢⢄⣀⣀⣀⡠⠔⠃⠰⠀⣿⢨⠬⢸⡇⢱⠀⢷⢋⣴⢟⣴⢷
⠀⠀⣿⡄⣿⠀⣿⢰⡏⡞⠀⡾⢁⡨⠙⠃⡉⠀⡀⠖⠠⢀⠶⠰⣖⠀⠉⣿⠀⢫⠃⠫⠄⡧⢸⠀⠘⣟⣵⠟⣡⠾
⠀⣤⠙⠓⠻⠶⠿⠾⠓⠁⣸⠐⢌⠛⣣⡐⠀⡾⠂⡖⠄⡘⠄⠀⣿⠆⠀⢻⡆⠘⢄⡀⣓⣠⠚⢠⡄⠘⣥⢞⣥⡾
⠀⣉⣤⣀⣠⣄⣠⣒⣂⣄⡈⠐⠀⠀⡰⡠⠊⠀⠠⠛⠂⠌⣃⣄⢩⣇⣀⣸⣇⡀⠀⣍⠉⠀⠀⢠⡀⠀⠀⠉⠀⢤
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣠⠀⠘⢠⣤⢄⠀⠀⠀⠀⠉⢉⠾⣿⣴⣟⣻⢆⢀⣸⣧⠰⠆⢠⣀⣀⡀⠀⠀⣠
⠀⠉⠢⠠⠠⠂⠒⠤⠤⠔⠊⠘⠛⠁⠘⠠⠀⠀⠈⠑⡇⢀⠔⠁⢦⢆⠀⣤⠈⣿⠀⠀⢿⡄⢰⣽⡃⠀⠀⠀⡀⣹⠢
⠀⠐⠂⠒⠂⡆⠀⠀⠀⢀⠀⠀⠀⠐⠈⠐⠐⠉⠁⠀⠉⠁⡀⢴⠟⢢⠠⠁⠀⠶⡇⠀⢹⡇⢰⣏⢁⣀⠀⢤⣠⠀⣽
⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠐⠦⡀⠀⠀⠀⠀⠐⢿⢶⠶⠥⢐⠡⠀⠠⠱⠑⡀⢸⣥⡀⠸⠗⣀⡙⢦⠑⣷⡺⢃⡴⠁
⠀⢰⠀⠀⠐⠀⣸⣿⣷⣿⣬⣀⣵⠄⠃⠀⠀⡄⠈⢢⠣⠀⠀⠰⢐⡂⡂⠱⠝⠀⠀⣠⠚⠁⠐⡵⡌⢻⣟⢻⣟⠐⠗
⠀⢸⣄⡠⠂⠀⠸⣄⡜⢛⣻⣿⠏⠀⠈⢀⣀⡇⠀⠀⢣⠑⡈⠚⣉⠡⠌⣁⠂⠀⢸⡠⠉⢀⢀⡤⣧⠋⡹⢧⡑⢷⡀
⠀⢸⠻⣄⠀⠀⠀⠈⠉⠸⠋⠁⠀⠀⠀⣠⠟⡇⠰⣿⡇⠛⠊⠀⢈⢦⣤⣤⠚⢥⠺⠓⠭⠔⡉⠀⢡⡜⠁⠀⠳⣾
⠀⠈⠓⠚⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠒⠂⠁⠀⠈⠙⠂⠀⠘⠋⠀⠊⠉⠉⠉⠁⠙⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⢒⡲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠁⠀⠀⠀⠀⠌⡂⠂⣉⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⢱⡄⢀⠀⠀⢀⣴⢟⣩⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠠⣀⢀⣬⣷⣀⣸⡄⠈⣀⣠⣴⣿⣿⠿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣄⠀⠈⠀⠀⠀⠮⣙⠷⣮⡙⢿⡇⠀⢻⣿⡿⠯⠔⠒⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣝⠷⢄⠀⠀⣀⠠⢄⠤⣜⣿⣷⣌⡑⠛⡁⠀⡄⠀⢉⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡂⠐⠪⠿⣿⢳⣦⠄⡥⠐⠊⠉⠉⠀⠀⠈⠛⣿⡈⡏⠀⠜⢒⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⡹⠀⠁⢀⣌⠻⣿⣷⣜⠷⣤⡀⠀⠀⠀⠀⠀⠯⠟⠧⠀⡒⠵⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⠃⠀⠀⠀⢹⠆⣈⠻⣿⣷⣜⠓⠄⠀⠠⠀⠒⠂⢈⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠰⠯⠆⠈⠻⣿⡷⣤⢠⡤⠀⢆⠀⢩⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠈⠧⠀⠘⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣼⠣⡘⠿⠀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⡉⢔⡲⠈⢢⠘⠒⡩⠔⣉⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢤⣀⠴⢚⢁⠈⢄⠑⣤⠴⢊⠡⡒⢌⠣⡘⢛⡠⢸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢣⡁⠈⠂⠉⢄⠈⠉⣀⠔⣣⡄⢊⠀⠻⠇⢈⠆⠉⣵⠊⣵⡆⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⢡⠀⠀⠄⠀⠀⠳⡭⠒⡭⠄⠈⢢⠱⢄⡪⢂⣉⠳⡈⢦⡨⣔⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡂⡁⠀⠀⠀⠠⡀⠀⡀⠀⠀⠀⠙⢦⣈⡡⠔⠪⠦⡧⡀⠀⢊⣠⡷⠦⡘⣄⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠋⢀⠃⣀⠀⠀⠀⠈⠢⡀⠁⠢⡀⠀⠀⠱⣎⠐⠯⠂⣸⠎⠊⡟⠡⢦⢀⠽⡸⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠀⠀⠀⠀⢸⠀⣬⠾⢒⣆⠘⠄⠀⠀⢱⠀⠀⢀⠀⠀⠀⠈⠢⣔⡮⢚⡉⢦⠈⢦⡪⠔⠟⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣆⠀⠀⠀⠀⣘⢀⣠⠖⣏⡜⢠⣠⢤⠀⠀⡆⠀⠀⠉⠐⡀⠀⠀⠈⢦⡉⢊⣴⠗⢎⠣⡜⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢏⢧⡀⠀⠀⠈⠁⠈⢣⡄⣤⢡⠇⡸⠀⠀⠘⠦⢄⠀⠀⢁⠀⠀⠀⠀⠹⡇⠐⠯⢀⠷⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢳⡀⠀⠀⠀⠀⠀⡄⠘⠋⣠⠗⡄⠀⠀⠀⠀⠙⣆⠘⠒⠒⠲⡀⢠⠘⡦⣂⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡸⣄⠀⠀⠀⠀⠓⢤⠐⢡⣊⡡⣀⠀⠀⠀⠀⢘⠀⠀⠀⠾⠁⢀⡰⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡜⣦⠀⠀⠀⠀⠘⡄⠀⢡⢔⡡⡀⠀⠀⠀⠈⣉⣵⣀⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡳⡄⠀⠀⠀⠘⡀⠈⢉⠔⠃⢀⡠⢐⣫⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡘⢦⡀⠀⠀⠙⢊⣡⡴⣺⡥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⠙⣄⡤⢖⣫⠵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡾⠒⠉





⣿⠿⠿⡿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣿⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠛⠛⠛⠻⠗⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣤⣾⣿⣷⡀
⠀⠀⠀⠀⠀⣤⣀⣀⣰⣿⠷⠶⠿⠿⢿⡏⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣀⣤⣴⡆
⠀⠀⠀⠀⠀⣿⠿⠿⢿⣿⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠈⠙⣻⣿⣿⣿⣿⣿⢶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⢿⣿⠏⠉⠈⠁⠉⠉⠛⠿⠇
⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠖⠀⣤⣽⣿⣧⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠹⣯
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢴⠆⢶⠆⢠⣾⣷⣶⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⡀
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⣾⠇⠀⢹⡇
⠀⠀⠀⠀⠀⠈⠉⢻⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠰⣦⠈⠻⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡯⢀⣤⠄⣠⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢤⣶⣾⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⢀⠔⢹⠑⢄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠻⣘⡿⠿⠿⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠂
⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀⣀⣀⣀⡀
⠀⠀⠀⠀⢹⡿⣿⣿⢿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠯⠿⠿⠽⠽⠽⠿⠿⠯⠯⠯⠿⠿⠽⠽⠽⠽⠿⠇
⠀⠀⠀⠀⢸⡁⢀⣀⢈⣿⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣼⣿⣼⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣶⣶⣶⢲⢲⢲⣶⣶⡖⡖⣶⣶⣶⣶⢲⣲⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⢠⢀⡄⣠⣄⢠⣤⡀⣤⣄⢠⠀⠀⣤⠠⣤⡄⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⢿⢼⡇⡇⢸⢸⠈⠉⡇⣿⢸⠀⢰⣿⡆⣿⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠟⠹⠘⠇⠷⠾⠸⠶⠋⠷⠟⠸⠦⠼⠛⠇⠻⠀⠷⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠁⠛⠓⠒⠒⠒⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠳⠊⠑⠒⠲⠤⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠐⠀⠀⠀⠠⡤⠤⡠⢠⠂⢀⡾⠃⠀⠀⠀⠀⠀⠈⡃⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⡀⠁⠀⢠⠂⢀⡎⠑⠂⠄⠀⠨⠝⠉⠒⠗⡉⣷
⠀⠀⠀⠀⠀⠀⠀⢀⢞⠁⠀⠀⠀⣠⠄⠀⠤⢄⣀⣠⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠉⠍⠀⢀⠀⠀⠀⠐⠀⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡦⠤⣄⣀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⢤⡞⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣸⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡁⠁⠀⠀⠀⠀⠀⠀⠀⠨⡄⢠⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠤⠤⠤⢤⣀⣀⣀⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⣀⡀⡀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠰⠒⠒⠒⠂⠂⠲⠀⠒⠒⠒⠒⢦⠀⡠⠠⠀⠀⢀⡀⠀⠀⢀⠤⠄⣄⠀⠐⠒⠒⠒⠀⡀⠀⡀⢀⢀⣀⠒⡀⢀⢀⠂⠀⠀⠒⠒⠒⡂⠂⢀⡀⠀⠀⠀⠀⢠
⢰⠀⠀⠠⡂⠀⠢⠀⠀⢄⠀⠀⣼⢸⠇⠂⠀⠀⠀⠈⢱⠄⢉⠀⠀⡸⠀⠀⠀⠀⠀⠘⡀⠻⠧⠇⠸⠸⠀⠟⠀⠸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠉⠀⠈⠉
⢠⡤⠀⢀⡄⠀⣠⢀⠀⣰⠀⠠⡌⠈⢀⣀⠀⠀⠀⠉⠉⠀⢀⣈⣩⣤⣤⣄⣒⡁⠀⠀⠁⢀⣀⠀⠀⠀⠀⠀⣀⡍⡁⠈⠉⠁⢀⠀⠁⠃⠀⢰⣮⣼⣿⣷⣦
⠀⠀⠀⢀⣠⣤⣴⣤⣤⣀⠀⠀⠀⢐⣿⣿⣿⣿⣷⣄⠂⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⠖⠠⡀⢜⠷⠀⢄⡓⠒⠓⠘⢀⠊⠙⣀⠺⠐⠄⠀⣿⣿⣿⣿⣿⡇
⢐⠡⢉⣿⣿⣟⣿⣟⣿⣿⡗⠀⠀⢸⣿⣿⣿⣿⣿⡋⠀⢸⠏⣿⢻⡿⢹⣟⢻⡏⣿⠘⠠⢀⠛⠉⠐⣶⠤⡫⠜⢧⠀⠈⠧⡀⣀⣀⣠⠆⠀⢻⣿⣿⣿⣿⠃
⠀⢀⠃⢇⠙⠿⣿⡿⠿⠋⠴⡀⠀⢟⣿⣿⣿⣿⣿⠁⢀⠣⡀⠉⠀⠀⠘⠋⠀⠀⡸⠂⠀⠀⠈⠑⠂⠤⣀⣃⠤⠊⠀⠀⠀⠀⠉⠉⠀⠀⠀⣘⠿⢿⡿⢟⡄
⠀⢀⠈⠉⠓⢀⠀⠀⠠⠞⠀⡇⠀⠀⠈⠈⠉⠃⠁⣄⡄⠑⠨⠠⠄⢀⡀⠀⠀⠁⠂⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡠⠄⢔⢶⡲⣤⠘⠀⠤⠰⠐⠁
⠀⠀⠑⠤⣀⠀⠀⠀⠀⣀⠔⢀⡠⠔⠒⠒⣢⡀⠜⡜⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣨⠖⢗⠂⡄⢸⡍⠁⠉⠉⣍⠀⣰⠈⢉⣀⠄⠓⠈⠁⠀⠅⢀⠤⠄⣤⡀
⠀⡤⠀⠄⠤⡄⠈⠀⠀⢠⠆⠁⠀⠐⠀⢀⠔⣄⠋⠰⠒⠢⡀⠀⣈⠁⣘⢈⠆⠀⠈⠉⠀⠀⠀⠎⠀⠠⠄⠀⠈⢀⠡⠀⣺⣀⡀⠄⠒⠁⠀⠀⣏⡘⣀⠠⡾
⠀⠁⢉⠉⠉⠃⠀⠀⢰⠁⠂⠈⣀⠤⢊⣱⠋⢸⠀⡇⠀⠂⠘⠆⠀⢀⣤⣶⣿⣿⣿⣿⣶⡀⠀⠃⠀⣀⠀⣄⡀⢸⠀⠀⢉⣀⡀⠀⢀⠔⠠⡦⠌⠒⠒⠒⠁
⠀⠀⢙⠀⠀⠀⠀⢀⣭⣈⠉⠀⠀⡔⠁⠈⢃⠼⢀⢸⠀⠈⠀⠰⠀⢸⣿⣿⣭⣭⣾⣿⡿⡃⠀⠃⡐⠦⠖⠢⠄⢸⠀⡾⠁⠄⠉⡄⠸⠈⡑⠦⠘⠦⠰⠆⣒
⠀⠉⢻⠀⠠⠀⢠⡾⠿⡟⠧⠀⡜⠀⠀⠀⢲⡶⡦⠰⢁⠀⢁⢠⠀⠈⠛⢛⣙⣋⠁⠄⠊⠀⠀⢓⠛⠱⠌⠁⢀⡼⠀⠡⢌⣐⣢⠇⠀⠱⣄⣺⡝⠂⠡⢄⣫
⠀⠀⢠⡀⠀⠀⠘⠢⠀⠄⠊⢰⠀⠀⠀⠀⠈⢆⢸⠀⠀⠑⣄⡜⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣍⣉⠉⠀⠀⠰⠈⠁⠆⠀⠀⠀⠀⠀⣀⠈⢁⠈⠉
⠀⢠⠶⠁⠀⠀⠀⣠⣶⣷⣄⠀⢣⡀⠀⣰⡀⢝⡈⣧⠰⠀⢈⡴⣛⡟⡿⠿⣛⠿⢛⣻⣛⡻⢻⣿⠻⡟⣛⠛⣶⡀⠣⢀⡀⠄⠀⠀⠀⠠⡎⠋⡂⠄⣈⠑⣄
⠀⠀⠃⠁⠀⡅⠸⡑⠟⠟⠻⠀⠀⠑⠋⠀⠐⠄⠣⢚⠆⠀⢸⣷⣭⣧⣶⣬⣯⣶⣭⣵⣝⣵⣬⣧⣶⣧⣿⣬⣿⣷⠀⠀⠀⢀⡛⠁⠀⠀⠉⠒⠨⢐⠊⠥⡹
⠀⠇⠀⠈⠸⠇⣀⠀⠤⠤⠁⢀⣀⠀⠤⢀⡀⠀⠈⠉⠀⠀⠘⢿⣿⣿⣿⣿⣟⣻⢟⣿⢛⣹⡛⢻⣿⣿⣿⣿⣿⠇⢀⡀⢀⡶⠣⡬⠃⠐⠠⡀⠀⠀⠀⡸⠁
⠀⠀⢠⣀⠐⣸⣞⣋⣋⡋⣉⣈⣈⣈⣙⡃⡌⢐⡄⣀⠴⠂⠶⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⢁⠆⠙⠈⠀⠸⠻⠀⠀⡜⠠⠈⡝⠀⡰⠁
⠀⠀⠀⠀⣻⡊⠀⠁⠀⠀⠂⠀⠀⠀⠀⠨⡰⢛⡈⠠⠈⣀⠃⣀⡀⢤⣐⣄⣠⣄⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠰⠶⡀⡀⡆⠀⢴⠂⠀⣄⠀⠀⠀⢀⠅
⠀⢀⠐⠉⠀⠚⠂⠤⣤⡄⠀⠀⢠⣠⡤⢚⡃⢀⣀⠠⠖⠁⢘⠻⠏⠶⠦⠿⠽⠚⠁⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⣤⢠⣦⣴⣶⡎⠀⠀⠀⠑⠒⠐⠋
⠀⠛⠀⠒⠒⠒⠈⠁⠀⠈⠂⠤⠀⡀⢐⠀⣡⡀⡠⣄⠀⠀⠘⠒⠐⠖⠒⠂⠄⠒⠁⠀⠀⢛⣿⢹⣿⣿⠃⠀⢰⣿⣿⡿⠟⠉⣧⠀⢰⡆⠉⢋⠉⢉⡉⢹
⠀⠴⠒⠲⠖⠐⠲⠒⠲⡖⠒⠲⠀⠴⠄⠤⣈⡝⠠⢹⠀⠀⢘⠀⠀⢠⠀⠠⡄⠰⠀⠀⠰⠀⠛⠀⠸⠟⠀⠀⠀⣯⣀⣤⣶⣾⡍⠀⠀⡏⠀⠉⠀⠈⠀⠈⡀
⠀⢀⠀⠀⠀⠀⢈⠀⢀⡇⠀⣀⠃⠑⠄⣁⣠⡱⢢⣮⠠⠀⡈⠁⠀⠀⠀⠀⠀⠀⢢⠀⠈⡀⠀⠀⠀⢀⠀⠀⠀⠘⣿⡿⠿⡟⠀⠀⠸⠀⠀⠀⠀⣀⣤⣶⣧
⠀⠈⠀⠈⠁⠀⠉⠰⢎⠁⠀⠈⠀⠀⠀⠑⠮⠽⠭⠮⢉⡧⠈⠁⢐⡀⠀⢀⡈⠁⠀⠀⠀⠈⢩⠉⠉⠀⠀⠀⠀⢠⠒⣀⡘⢤⠀⠀⠀⠀⢳⣾⡿⣻⣭⣴⣿
⠀⠉⡏⢉⣉⠉⠙⣄⠩⠉⠉⠉⠋⠉⠉⠙⠀⠀⠾⠭⢱⠀⠀⠰⢥⣴⣶⣦⣤⣵⡀⠀⠀⠀⠘⠔⡄⠀⠀⠀⠀⣀⠈⠩⢉⠀⠀⠀⢠⠀⢸⣿⠿⠛⠋⠁
⠀⠀⡇⢀⣄⡀⠀⠘⢆⠘⢂⠀⠄⠀⠀⠰⠀⡐⠁⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠆⣀⣤⣶⣼⣷⣦⡀⢎⠀⢱⣐⡈⠇⠠⡀⠸⡀⡌
⠀⡀⣇⠁⠘⠑⠈⠃⠊⢣⠒⡀⣀⠀⠀⣠⠀⠴⠶⠿⠶⠶⡅⢀⠀⠂⠒⠀⠋⢹⢀⠔⢸⣿⢿⢿⡟⣿⠙⢣⠀⠑⠄⡁⠈⡖⠀⢹⠀⠈⠁⠒⠒⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠜⠁⠀⠀⠀⠀⠰⢶⣆⢰⡆⠀⠀⢣⣀⣀⣀⡤⠁⠀⢰⠈⠥⠤⠬⠥⠐⠒⢁⠠⠁⠀⠈⠑⠓⢂⡁⠀⣠⡄⠐⠴⠪⡁⠑⢄
⠀⢰⠰⠊⣭⢱⢲⠀⢠⠖⠖⠀⠑⡖⢰⠀⠀⢦⣤⣤⣤⣤⠆⠀⣤⣤⢡⣅⢀⣀⢀⣀⡁⣀⡀⣀⠀⠀⠁⠀⣀⢠⡒⡆⠀⠀⡊⠑⡤⠌⢪⡖⠀⠙⠞⠀⢀
⠀⠈⠋⠑⠐⠋⠘⠀⠀⠓⠑⠂⠐⠉⠈⠀⠀⠯⠿⠿⠿⠿⠂⠈⠮⠿⠦⠿⠦⠿⠧⠽⠧⠼⠷⠼⠃⠀⠀⠘⠒⠚⠠⠕⠀⠀⠐⠠⠧⠄⠀⠀⠀⠐⠀⠉

⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⣰⠀⠀⠀⢀⣀⠀⠀⠀⢠
⠀⠀⠀⠢⡀⠀⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠤⠴⠃⣀⣀⣀⠄⠃⠀⠀⠀⠀⣈⠷⠀⠀⠈
⠀⠀⠀⠤⡈⠀⠌⡉⠆⠓⢒⡒⢚⠡⠤⠤⢼⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠂⠀⠀⢈⣏⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⣄⠒⢄⠀⠈⠉⠈⠉⠺⠄⠀⡀⢠⠀⠀⠀⠀⣐⢀⣀⠀⣠⠏⠀⠀⠀⡨⡎⠀⢀⠔⠐⢀⢢⡀⠀⠀⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠆⠉⠐⠆⠠⣀⠀⠀⠒⠀⠀⠀⠈⠁⠀⠀⡄⠀⠉⠙⠻⠐⠦⣤⣤⡜⠜⠀⠀⢀⠈⢢⠀⠇⡇⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⢈⡏⠀⠀⠀⠈⠓⠒⠂⠂⠁⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰⢦⣄⣀⣴⠇⠀⠀⠀⢀⡟⠀⢀⣤⠴⢶⣤⡀⠀⠀⠀⠀⠀⣤⢰⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⣠⡾⠀⠀⠠⢋⠖⠛⣻⠿⡄⠀⠀⠀⣂
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠲⢦⡤⡤⠀⠀⡤⠗⠀⠀⠀⠙⠶⠦⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣦⡀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣰⠟⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠧⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣣⣼⢗⣫⠼⢙⡄⠀⠀⠀⠀⣴⣆⠀⠐⠿⣊⠓⢦⣄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢣⣴⠖⣺⣍⢶⣏⣴⣞⣛⠛⣷⡄⠀⠀⠙⢿⡻⣦⠐⠉⠹⠆⢌⡙⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠄⠀⠀⠀⠈⠻⡿⠯⠒⠛⠓⠉⠍⠛⠋⠇⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⢀⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡆
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢌⠀⠂⢀⠵⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⡰⠁⠉⠀⠈⠄⠀⠙⠋⠅⠀⠠⠄⠀⠠⠔⢀
⠀⠀⠀⠀⢨⠃⢀⠀⠀⠰⣄⠀⠀⠂⠰⠃⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⢤⠀⠀⠉⠀⠀⠂⢤
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢧⡈⢠⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⡁⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢨⡆⠀⠀⡀
⠀⠀⠰⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⠛⠿⠈⠑⠀⣊⠀⠉⠢⣈⢠⢀⡀⠐⠁⠀⠀⠃⠈⠀⠠⡀⠁⢀⣄⣠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠑⠂⠒⠀⠘⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠏⣀⠁⢄⠀⠀⢀⡆⡐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡄⠈⠉⠀⠀⠀⠀⠀⠀⠁⠒⠀⠐⠂⠀⠐⠈⠀⠀⠀⠈⠉⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠾⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠒⠂⠀⠀⠘⠆⠓⠈⠀⠀⠀⠂⠀⠀⡠⢀⠀⠀⠰⡤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠘⡤⠀⢀⡀
⠀⠀⠀⠀⠀⠄⢀⠢⠎⠀⢠⣀⠳⡤⠀⠀⢀⠀⠀⠀⠁⠒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠛⠉⠁
⠀⠀⠀⠀⠀⠓⠛⠦⠐⠓⠒⠑⠁⠀⠀⠀⠠⢢⡤⠤⠅⠠⠀⡀⠀⠀⠨⡀⠀⠑⠂⣂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠶⠒⠊⠁⠀⠀⠀⠀⠈⠉⠁⠈
⠀⠀⠀⠀⠐⠈⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡆⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢛⠀⠀⢀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠐⡂⠀⠀⠀⡄⠀⢀⡼⠁⠀⠀⠐⠀⠂⠂⠀⠒
⠀⠀⠀⠀⠀⠘⡆⠀⠁⠀⠀⠀⢋⠁⠀⠠⠤⠄⠺⠀⠀⠀⠀⠒⠛⠚
⠀⠀⠀⠀⠀⠀⠱⠄⠄⠘⠀⠀⠐⠀⠂⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠠⡔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠐⠀⠘⣄⣀⣬⣧⠤⠶⠒⣎⠉⠁⡀
⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠠⠂⠀⠀⠄⠀⠀⠀⠈⣧⠤⠖⠱⠛⠉⠙⠁⢀⣀⠧⠀⠐⠚⢂⠉⠉
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠶⠦⢴⡄⠂⠀⠀⠀⢣⡀⠤⠔⢃⠈⣉⡉⠠⠀⠒⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⣼⢃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣶⡄⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⢱⣾⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⢠⣤⣶⠟⠋⠁⣿⣿⣿⠟
⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠈⢿⠀⠀⣠⣾⣿⣿⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⣿⣿⠛⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⠁⣸⣿
⠀⠀⠀⠀⠀⢿⣏⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠈⠙⢶⣤⠀⠀⠀⠀⠀⠀⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢴⡟⠿⠿⣷⡄⢀⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠲⡄⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣷⡤⠜⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⢠⡇⠀⠀⠀⠃⠀⠀⠄
⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣀⠤⠶⠒⠒⢤⣄⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠠⣤⣀⠀⠀⠀⣰⠏⠀⠀⢀⠊⠀⠀⠀⠀⠀⡴⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠛⠢⠚⠋⠀⠀⠀⠀⠂⠁⠀⠀⠀⣰⣃⣄⣠⠖⠛⢇
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠂⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠿⠿⣆⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠤⠤⠄⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣽⡧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢻⣿⡏⠉⠉⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣏⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣟⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠒⠶⠶⠾⠿⠛⠻⠿⠿⠿⠿⠷⠶⠶⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⣀⣀⡀⣀⢀⣤⡀⡤⣄⡴⣤⢴⠀⢰⣦⠖⣖⢲⡞⠶⣿⣶⣶⣾⡀
⠀⠀⠀⠀⠀⢸⣾⣿⣿⣅⣧⠃⣷⣹⣸⡇⣧⣷⠈⣷⢹⡼⡇⣸⣿⡆⢻⡀⢿⠒⢾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣏⣿⣀⣿⢸⡇⣧⣿⠸⣤⠾⣼⠇⠿⠿⠇⠿⠘⠇⠘⠓⠓⣿⣿⣿⣾⣇
⠀⠀⠀⠀⠀⠀⣟⣿⣿⣿⣨⣉⣈⢠⣤⡀⢀⡒⣒⣦⢤⣄⢤⡶⡄⢠⢠⣼⡗⣶⣄⣾⠏⠝⠋⠙
⠀⠀⠀⠀⠀⠀⠉⠉⠁⠹⠟⠃⠛⠚⠥⠈⠀⠀⠉⠋⠈⠙⠋⠀⠁⠀⠁⠈⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⢀⣠⣴⣶⠿⠛⠒⠚⠛⠙⠓⠒⠚⠿⣶⣦⣄⡀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⡄⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⢠⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣊⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣑⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⢿⡀⣼⠁⣿⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠸⠇⠈⠿⠏⠀⠿⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⡀⠀⠀⠀⣀⠀⢀⡀⢀⡀⢀⡀⢀⣀⣀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⡀⠀⢹⣆⠀⠀⣿⢷⠿⡇⢸⡇⢸⡇⠙⠷⣬⠸⡇⢼⠁⠀⠀⠀⣰⡏⠀⢀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠰⠀⠀⢿⣆⠀⠛⠀⠀⠛⠈⠛⠛⠁⠘⠛⠛⠈⠃⠈⠛⠛⠀⣰⡿⠀⠀⠄⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠆⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠐⠀⠀⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠘⡄⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⢠⠃⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⢱⡄⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⣀⠎⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡀⡄⢻⣿⣀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢡⣿⣿⢠⢀⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠿⡿⣿⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⣿⡿⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠠⢀⠄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⡠⠖⠁⠀⠀⠀⠀⠀⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠖⠚⣛⣉⣉⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⢶⣶⠶⠶⠶⠶⠶⠶⣶⡄⠀⣿⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠇⠀⢹⡆⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⣼⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡀⠐⠆⣧⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⡖⢸⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇⠀⢀⢹⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠃⢸⡇⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⡇⠀⠈⠸⠿⠟⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡇⣶⣶⣶⣶⣶⣾⣿⠀⣶⣶⣶⣶⣶⣿⣿⠠⣶⣶⣶⣶⣶⣿⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣿⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠞⠗⡀⢸⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠐⠁⢸⡇⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⠀⠈⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠀⠙⠆⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠳⠾⣧⣤⣤⣤⣬⣥⣽⣿⠁⣤⣤⣤⣤⣤⣿⣿⢀⣤⣤⣤⣤⣬⣿⣤⢴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢉⡛⠿⢿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣴⡿⠿⠟⢛⣩⠁⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠉⠳⢦⡬⣿⡿⠋⠉⠉⣀⠀⣀⡉⠉⠠⠒⠚⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠒⠦⠴⠦⠤⠤⠤⠤⠈⠉⠛⠉⠙⠚⠛⠉⠀⠀⠀⣶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿