Biene Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠢⢄⣀⣀⣀⣀⡀⣀⡠⠔⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡀⠀⠀⢠⣬⣿⣟⣻⣿⣿⣄⡀⠀⠀⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⢺⣿⣿⣿⣿⣿⣷⡿⣡⣴⡟⠁
⠀⠀⠁⠀⠀⠤⠀⠒⡦⠤⣀⣀⠀⠀⠀⠈⠻⣿⣿⣟⣿⡛⢻⣿⣿⡟⠉⠀⠀⠀⡀⠀⠄⠠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠰⣁⠀⠠⢉⣑⣒⣦⣼⣿⣿⣿⣿⣧⣾⣭⣿⣷⡔⣒⢉⣁⠄⠈⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠀⡀⠈⠁⠈⢹⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⠍⠀⠈⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠉⠀⠀⢀⠀⣀⣌⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠻⢿⣿⣿⢿⠍⠉⠙⢿⣿⡿⠿⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠃⢀⣴⣾⣿⡛⠛⠛⠋⠛⠛⢻⣷⣶⣀⠈⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⡤⠒⠛⠋⠁⠀⣴⣿⠋⢹⣿⣯⣀⡀⠀⠀⣀⣾⡟⠛⣿⣦⡀⠉⠛⠓⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡏⠀⢸⣿⣿⣿⣿⠖⠺⣿⣿⡇⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠸⣿⣿⣭⣅⣀⢠⣯⣿⠁⠀⠈⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠀⠀⠀⢻⣿⣛⣛⠟⣽⣯⡏⠀⠀⠀⠘⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⠹⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣲⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣂⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠙⠓⢶⡖⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡷⠀⡔⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡇⠀⡱⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡼⠀⢠⠃⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⡠⠖⠛⠙⠓⠲⢄⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡰⠁⠀⡸⠀⠀⠀⠀⢀⡤⠚⠁⠁⠂
⠀⢸⡥⠴⠖⠒⠒⠦⠤⣍⣢⣧⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡏⢀⡜⠁⣀⣀⣧⠀⠀⠀⣠⠋
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣠⣤⣶⠚⠋⠉⠉⢩⣿⣿⣿⡟⠉⠉⠟⠁⠀⠀⠀⠉⢢⡔⠉⠀⠀⠈⠐⠄
⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠃⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⡇⢀⣄⠀⠀⠀⠈
⠘⢦⡀⠀⠀⠀⠀⢀⡾⠁⣾⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢀⡇⠈⠉⠀⠀⠀⡐
⠀⠀⠈⠓⠤⣄⣠⡟⠀⢸⣿⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠁⠈⠀⠀⠀⢤⠈
⠀⠀⠀⠀⠀⠀⣼⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⣀⠤⡇⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹
⠀⠀⠀⠈⠓⠤⣇⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⡑⠀⡄⠀⠀⠀⠤⠀⠂⠁⠀⡼
⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⢹⣿⣿⣿⡆⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⣿⣿⣿⣇⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠢⡖⢒⠤⠠⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠘⣿⣿⣿⡄⠀⠀⠀⠹⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣽⣿⣿⣿⡄⠀⠀⠀⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣦⣄⣀⣀⣹⣿⣿⣿⡦⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢰⡿⠀⠀⣿⡀⠹⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠁⠀⠀⠀⠀⢀⣠⠒⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡄⢀⠂⠀⠀⣠⡞⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠀⠀⣠⡾⠋⣠⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⠟⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣠⡾⠃⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡿⢁⠄⠀⠀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡴⠞⠉⢋⠙⠻⣿⠋⠀⠀⣠⡄⠀⠀⣀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣴⢶⣶⠾⣾⡛⢻⡿⣶⣤⣦⣀⡤⢀⣉⡆⠀⠀⣿⣿⡀⠀⢸⣿⡇⠀⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠙⣷⢹⣿⣿⣿⣼⣧⣰⣦⣭⣿⢹⣽⡟⣷⠀⠀⣻⣿⣧⣀⢿⢿⣇⢰⣿⣿⡏⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⡷⣼⣿⢿⣫⣷⡿⣿⣿⣿⣿⣿⢇⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠟⠀⠀⠈⠳⠹⣿⣿⣿⣯⠾⢡⣿⡟⢩⣴⡟⢻⣿⣍⣵⣿⣟⠟⣰⣿⣿⣿⡿⣫⣾⣿⡏⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⣯⢿⣆⡟⠉⢰⣿⡟⠀⠈⠛⣩⣾⢟⣡⣾⡿⠿⢿⣿⣿⣿⣿⣋⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢋⣠⡾⠃⠘⠛⠁⠀⠀⣴⡿⠟⣵⣿⡿⠋⠀⠀⠀⠈⠁⠘⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⡄⠠⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠗⠀⠈⠉⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡙⠄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⠈⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⢰⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠐⠒⠒⠬⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡆⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠀⠒⢄⣰⠁⠀⠀⠋⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢢⠘⡄⠀⠀⠀⠹⡀⢠⣠⠁⣰⠃⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠸⠤⠜⠀⢷⠀⠀⠀⠰⣤⣏⠈⢻⠁⠀⠀⠀⠀⠀⠀⠜⠀⠀⡔⠀⠀⡰⠀⠀⠀⣸⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣉⠀⠀⢀⡀⠀⠈⠱⣌⠀⠀⠀⠀⠀⢀⠂⢀⡴⠊⠀⠠⠊⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⠋⠈⠗⠋⢣⠹⠀⠀⠀⠹⡆⠀⠀⠀⠠⢃⡴⣋⡀⠀⠀⠤⠐⠉⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣰⣠⡰⠄⡸⠀⠀⢀⠀⢀⢧⠀⠀⡰⢡⢟⡨⠤⠤⠤⠴⣒⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢈⣀⠉⠉⠀⠀⢀⢼⠀⡜⠀⢣⠀⢠⠎⠁⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡁⠀⠀⠀⠐⣑⣮⣾⣄⣀⣤⣄⢀⣠⣴⣞⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣶⣾⣿⣿⣿⡿⠋⢻⣿⣿⡀⠙⢿⣿⣿⣿⡔⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⣿⣿⡿⠟⠋⠀⢢⢸⣿⣿⣧⠀⠀⢻⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠇⠈⠀⠀⠀⠀⠀⣸⠀⣿⣿⣿⠀⠀⠈⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠠⣄⢀⠀⠀⠂⠠⣼⣿⠀⣿⣿⡿⢤⡀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠓⠊⢀⡠⠇⠀⠀⠔⠐⠿⢲⢏⠉⢙⠢⣈⠒⣲⡳⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠠⡀⢑⡄⠀⠀⠙⡒⠀⠀⣌⡦⠤⢯⠿⠯⠎⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠰⣁⣀⡴⠗⢹⡇⠠⣑⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠛⠳⢶⡶⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⡇⠀⡠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠃⠘⣇⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡞⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤
⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡼⠁⠀⡎⠀⠀⠀⠀⠀⠀⣀⠤⠻⣄⡠
⠀⠀⡼⠉⠀⠀⠀⠈⠙⠲⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⡼⠁⠀⠀⡇⠀⠀⠀⠀⣠⠞⠁
⠀⣸⠗⠛⠉⠉⠉⠉⠙⠒⠦⢵⣿⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⢤⣤⣤⣯⣀⣜⣀⡠⠤⠤⠧⣀⠀⢀⣜⡡⠤⠤⠄⡀
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⣶⣿⣿⡿⠁⠀⠀⠀⠀⣼⣿⣿⣿⠋⡜⠁⠀⠀⠀⠀⠈⠳⡞⠁⠀⠀⠀⠀⠈⢢
⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣼⣿⣿⣿⠁⠀⠀⠀⠀⣸⣿⣿⣿⠇⠀⠀⠀⠀⢀⣀⠀⠀⠀⢣⠀⣤⠀⠀⠀⠀⠀⠄
⠈⠳⣀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣿⣿⡿⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⢆⠀⠉⠀⠀⠀⠀⡰
⠀⠀⠈⠑⠦⢄⣀⣰⠋⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⠈⠢⢀⠀⠀⣀⠠⠊⠀⠁⠠⠤⢤⡄⠊
⠀⠀⠀⠀⠀⠀⢨⡏⠀⠀⠀⢰⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⣠⠞⢹⠁⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢀⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠚⢧⡀⢸⡀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⠁⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⣀⡤⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢹⣿⣿⣿⣆⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⣻⣿⣿⣿⣆⠀⠀⠀⠀⠹⣿⣿⣿⣧⠀⠀⣀⣠⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⣄⣀⠀⣾⣿⣿⣿⣿⠆⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢓⣼⣿⠿⠓⢦⡴⠋⠀⠀⢀⡀⣀⠠⠤⠄⠀⠀⠤⠠⡀
⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⣿⣄⣀⣠⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⡿⢛⡁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⣤⣽⣿⣿⣿⣧⣬⣤⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⣼⠁⣾⣿⣿⣿⣭⣬⣽⡆⢸⡄⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⣿⣿⣿⣿⣿⣏⣾⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⣿⣿⣿⣧⣾⡇⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠙⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢀⣴⡞⠃⠉⢻⣦⡀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡄⢸⣿⡧⠀⡀⢸⣿⣟⠀⠀⡠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢼⣀⣉⣿⣶⢿⣿⣷⣤⣀⣐⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣼⠀⠀⠀⠀⠀⠀⠈⠷⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢸⣖⣀⠀⠀⠀⢀⠀⣀⣿⣄⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠤⣶⡶⠂⠶⠋⠀⡄⠿⠟⣿⣶⣼⠀⡴⣿⠿⡞⠯⠀⠀⠘⢶⠶⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⣰⣲⣞⠉⠀⠁⠐⠁⢀⣝⣷⣒⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⡍⣂⠉⠓⠒⠐⠐⠙⠉⠉⠀⠀⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠱⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠒⠄⠀⠀⠀⠀⠀⠐⠀⠃⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠜⢱⡀⡀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠉⠙⠀⠀⠀⠀⠐⠂⢈⠀⠀⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⡤⠀⢀⢀⣠⡤⠶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⣠⠔


⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⢳⡀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢸⡗⠉⠀⠀⠈⠳⣄
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠙⣆⢸⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢸⣥⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⣸⠁⠀⢠⣤⢀⡟⡟⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠈⠛⠀⠀⠀⠘⡆⠀⠀⠀⠀⢀⡏⠀⠀⠈⠛⠋⠀⡇⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⣼⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣠⠷⠒⠒⠒⠛⠲⣄⡀⠀⢀⣠⣤⣴⣶⣿⣟⠛⠛⠓⠛⢯⣅⡀
⠀⠀⠀⠀⠀⣠⢞⣁⠀⠀⠀⠀⠀⠀⠀⠙⢾⡉⠀⠈⢻⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⢦⡀
⠀⠀⠀⠀⣰⣿⢿⡿⣧⢠⢿⣿⢦⠀⠀⠀⠀⢳⡀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⢸⣿⣿⡇⠙⢦⠀⠀⢀⣼⠃
⠀⠀⠀⢠⡇⣇⠀⢀⡏⣿⠀⠁⣸⠀⠀⠀⠀⠀⣧⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣧⠀⢸⣦⣴⣿⡿
⠀⠀⠀⢸⡇⠈⠛⠉⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⢸⡿⠿⠛⠁
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⡟⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⡼⠁
⠀⠀⠀⠀⠘⢦⣀⣀⣀⣠⠴⠚⠁⠁⠀⠀⢠⡞⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⣼⣿⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠈⠛⢯⣄⣀⠀⠀⠀⣀⣠⠴⠋⠓⠲⠤⢤⣿⣿⣿⣿⣟⣀⣀⣴⡿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡏⠀⡼⠀⠀⠀⠀⠀⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⠀⠀⠀⠀⢧⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠹⡄⠀⠀⠀⠀⠈⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠁
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢈⠀⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠞⢀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠒⢶⡀⠀⠀⠀⠀⠀⠀⢸⢁⡀⢀⣤⠔⠉⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠇⠀⡷⠈⠁⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣴⣹⣿⣦⣴⣶⣿⣿⣷⣦⣄⣸⣧⢀⣤⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣿⡟⣿⡛⠢⣍⣿⣿⣿⣿⣿⣩⣶⣶⣅⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⢹⢟⣷⣿⣿⡀⢿⣿⣶⣿⣿⣿⣿⣿⢿⣿⣿⠿⣿⣿⢿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠤⠞⠉⠀⠀⠀⠘⠿⢿⣿⣿⣟⣈⡝⢿⣿⣿⡿⠟⠋⠸⣿⣷⣾⣿⡟⠀⠙⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⠤⣝⣿⣿⣷⣾⣶⣤⣶⣤⣥⣿⣿⢿⣿⠏⠀⠀⢀⣿⣿⡿⣿⣿⡷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⢠⠶⠒⠻⠷⠶⠛⠓⠾⣿⡟⢻⣿⢻⣿⣿⣿⣿⠟⣷⠊⠁⠀⠀⢠⣾⣿⡟⠀⣻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⡍⢀⣼⣓⢸⠟⠁⠻⣿⣄⠸⣆⣀⣠⣤⣴⡿⠏⠀⢠⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣉⡍⠀⣾⣿⠹⢤⠀⠀⠀⠉⠺⣇⠘⢿⣿⠿⠋⠀⢀⣴⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣁⠀⠞⢰⢧⣸⠓⠀⡄⠀⠀⠀⠀⠀⠀⢿⣦⠀⠙⢶⣤⡴⠛⣿⠿⢫⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠈⢸⣿⠀⠀⣸⡇⠀⠀⠀⠀⠀⠈⣜⢧⠀⠀⠈⠳⢶⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠇⠐⠋⠁⠄⠀⠄⠀⣮⡉⣠⠔⠀⢻⠀⠀⠀⠀⠀⠀⠘⢾⣧⠀⠀⠀⠀⠉⠛⠛⠿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠃⠄⢀⡤⠔⠊⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠤⠀⠒⠒⠂⡀⠀⠀⠀⣠⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡀⢧⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢙⢷⡄⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⢔
⠀⠀⠀⠀⠀⢀⠄⣄⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠈⡗⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣠⣤⡔⠂⠀⠀⠀⠀⠀⠀⠀⢾⣷⠢⢄
⠀⠀⣀⡴⠚⠉⠀⠈⣗⠀⡀⠀⠀⠐⣴⣿⣿⠀⢠⡟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⠀⠠⣧⡀
⠐⠚⠁⠀⠀⠀⠀⡰⠀⢙⡒⠀⢠⣿⣿⣿⠇⠀⣾⠇⢀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣿⣿⣥⠀⠀⠀⠀⠁⢀⠀⢈⣾⣿⠏⠀⣰⣿⠇⢀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠘⡇⢀⠈⠿⠿⠋⠀⠈⠀⢁⢶⣆⢀⡀⠀⠀⢠⣤⡜⢘⡳⠘⡃⠀⡐⢂⠰⠄⢈⡁⢿⣿⡏⣀⣤⣿⡟⢀⣿⡆
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣠⢞⡫⠄⠑⠃⠀⣠⣶⡶⣡⡾⠀⠉⠁⠀⠸⡷⢿⠂⡀⠀⠀⠀⣀⠤⢔⢟⣿⣿⠀⢻⣿⣿⣿⢏⣠⣾⠟⢱⡀
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠙⠞⠁⠀⠀⠀⣾⡾⠈⢠⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠚⠦⠊⡈⠙⣿⣇⠀⢹⣿⣿⣿⣟⣡⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⢈⣿⡿⠆⠀⠀⠀⢶⢷⣮⠂⠈⠛⠀⠀⢻⣿⣿⣿⣿⣵⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢀⣸⠃⠠⣾⣦⠀⠀⠀⠙⠿⢇⠀⠀⠆⠀⢸⣧⠉⠉⠉⠉
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠹⠿⠇⠀⠀⠀⠀⠈⢲⡠⠀⠀⠜⠋⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⡹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠒⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠶⣶⣦
⠀⠀⠀⠀⠀⣀⣀⣀⣤⡾⠃⠀⠀⢠⡄⠘⢿⣦⠤⠤⠤⠤⠴⣿⣤⣤⣀⡀⣌⢻⣧
⠀⠀⠀⠀⢺⣯⡉⠁⠈⠁⠀⠀⣠⠿⠆⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠺⣿⣧
⠀⠀⠀⠀⠈⠛⠿⣷⣶⣶⣶⣿⡁⠀⠀⠀⠀⠀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠤⣀⡀⠀⠙⠻⣶⡄
⠀⠀⠀⠀⢠⣶⣶⣶⣿⣿⠏⢽⠇⠀⠀⠀⠀⠀⠙⠟⢛⣛⠿⢷⣦⣀⠀⠀⠀⠙⠲⣽⡆⠘⠛⠷⣦⣄
⠀⠀⠀⠀⠀⢻⡄⠀⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⣭⣓⢦⣀⠀⠀⠙⣿⡀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⢠⣿⣿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠀⠀⠉⢻⣿⢿⣿⣶⣤⡼⠿⠶⢶⣶⣾⣿⠏
⠀⠀⠀⣰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⣀⡀⠀⠈⠑⠶⣯⣭⣭⠤⠶⣶⣶⣶⣽⣷⣄⡀
⠀⠀⣀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⢋⣵⡟⠀⣿⣿⣿⣿⣷⣦⡀⠀⠉⠃⠀⠀⠈⠙⠛⠿⣿⠈⠛⢷⣦⠀⣀
⢾⣿⣏⠁⠀⢹⣿⡄⠀⠀⠀⠀⠀⢀⣴⠟⠁⡠⠋⢸⠃⠀⣿⢿⡿⠛⠛⣿⡿⢦⡀⠀⢰⣦⡀⠀⠀⠀⠈⠀⠀⠀⠹⣿⡿⣷⡀
⠀⠙⢿⣷⣦⣾⣿⣿⣦⣸⠄⠀⣠⡾⠁⠀⠀⠀⠀⠈⠀⠀⣿⢠⣇⠀⢀⣼⡄⠀⠙⣶⣿⠃⢳⣄⣀⣠⣴⠿⣦⣤⣤⣿⠃⢸⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡆⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠙⣦⡞⠁⠈⠲⡞⢁⣏⣀⣠⣯⣿⣿⣀⣀⣻⡄⢸⡏⠀⠼⣷⡀
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢸⣧⡀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠲⢄⡀⠀⠘⢿⣦
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠟⠁⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠈⠃⠀⠀⠈⣿
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠛⠛⠛⠛⠿⢿⣷⣶⣤⣤⣤⣤⣾⣿
⠀⠀⣠⣶⡿⠋⠉⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠙⣿⣿⠿⠋
⠀⠀⠻⠿⣶⣤⣤⣴⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣼⠏⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣋⣠⣶⣋⣡⣤⣶⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⡿⠿⣷⣤⣀⡀⠀⠀⣀⣴⣧⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡇⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⢻⣿⣿⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⡿⠿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣼⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠷⣤⠶⣛⠋⠓⠅⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠉⠀⠁⠀⠀⠀⠀⠈⠢⢀⠘⠆⠈⠉⠀⠀⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢡⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠂⠀⠢⡀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⣀⣀⠜⠳⣄⠀⠀⠀⠀⠁⢀⠀⠀⠀⢀⠰⠀⠐⠀⠂⠀⠀⢨⠉⠒⢲⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⡅⠀⠀⠀⡇⠀⠀⠰⠈⠻⣅⠈⠁⠀⠀⢀⠈⢁⠀⠀⠀⠐⠀⠀⠀⠂⠚⠀⠀⢸⡀⠀⠀⠘⢀⠆
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⡜⠀⠀⠀⠀⢃⠁⠈⠑⢢⡀⠀⠀⠦⠀⠠⠀⠀⠀⠄⠐⠀⠀⢒⡀⠀⠀⢱⡀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠇⠀⠀⡌⠀⠀⠀⠀⠀⢈⢀⡤⠞⠁⢣⢀⠀⠐⠀⢀⠀⠀⠢⣗⠈⠙⠉⠀⠀⠀⠀⠐⢣⠐⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢨⠡
⠀⠀⠀⢀⠀⠔⠀⠈⠀⠈⠁⠈⠲⠀⠀⠀⠁⠀⠀⡠⠜⠀⠨⠉⣀⠀⠀⠀⠀⠈⠑⠢⡀⢀⠀⢀⠆⠀⠀⠀⠀⠀⢉⠀⠢⡄⠀⠀⠀⢀⠞⠀⡄
⠀⠰⠋⣀⠄⠀⠀⢀⠎⠀⠙⣄⡰⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⣰⣀⠄⠀⠀⠀⠀⠀⠀⠢⡉⢁⢀⠠⠒⠀⠀⡀⠀⡈⢄⠖⠀⣶⣂⡁⠀⠀⠃
⠀⠸⡞⡅⠀⣄⠀⠘⠲⣄⣀⠠⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⡠⠂⠈⠢⡀⠀⠀⠀⠀⠀⠀⢱⢬⠙⠀⠀⠰⡋⠀⠠⠃⠈⠀⠀⠀⠞⠀⠙⡀⡀
⠀⠀⠀⠰⣀⢈⠗⣀⠀⠚⠁⠀⠀⠀⠀⠘⠀⠀⠀⢠⡠⠔⠊⠀⠀⠀⠀⠈⠑⠂⢰⠀⠀⠀⠀⠀⢡⠀⠈⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⢋⢱⠃
⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠄⢠⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠰⢀⠀⢰⠄⠀⠚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡆⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠙⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢸⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⣀⡤⠄⢀⠒⠀⠉⠉⠐⠒⠤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠒⠉⠁⠀⠀⠈⠉⠈⠉⢲⠄⡀
⠀⠀⠀⠀⡮⠀⡁⠒⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠜⠛⢄⡀⢾
⠀⠀⠀⠀⠁⠈⠒⠀⠑⠒⠈⠀⠀⠈⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠁⠒⠀⠁⠒⠉⠁⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⣠⣤⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣴⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣶⡶⠶⣶⣶⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⢀⡰⠆⢉⣿⡶⠂⠀⠀⢠⣤⡀⠀⠈⠻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣤⠟⠳⣤⠀⣀⡀⠐⠒⢀⣀⠀⣠⣿⠿⠟⠛⠛⠛⠙⠛⠒⢿⡆⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⡀⠀⣤⡉⠀⢀⣀⣾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠚⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠿⠿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⡀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠙⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⢀⣀⡀⢀⣴⣆⣀⣠⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠙⠻⠿⠿⣿⣿⠻⢿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣀⣀⣠⣶⣶⣿⣿⣶⣶⣦⣄⠀⠛⣷⣄⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⠀⠰⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣹⡿⠟⠋⠁⠀⠀⠉⠻⣿⡻⢷⣦⡀⠹⣧⡈⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣀⢀⣠⡤⠙⣆⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠓⠛⠉⢀⣤⣤⣤⣤⣤⣄⣀⡜⠃⢀⠙⠿⣦⠈⢿⡄⢠⡿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠋⠛⠋⠁⠀⢻⣦⡀⠀⢀⣠⡟⣻⣿⡅⠛⠛⠓⣾⡿⠋⠙⠛⠛⠛⠿⠟⠛⠢⣾⣄⠀⠙⣷⡜⢃⡈⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠻⢇⡀⠀⢀⣀⣴⠶⠶⣿⡇⣠⡾⠋⢹⣿⠁⢀⣀⣀⣠⣿⣷⣴⣾⢃⣀⣀⣀⡄⠀⠀⠀⠙⢷⡄⠘⠇⠸⣷⡄⠀⣼⢦⡄
⠀⠀⠀⠀⠀⠀⠀⣶⠊⠻⠿⠟⠛⠉⠀⢀⣽⣿⣿⠁⢀⣄⠁⠶⠟⠻⠟⠛⣿⡏⠉⠛⠋⠉⠉⠻⡿⠾⠿⠶⣦⣀⣿⣄⠂⠀⢿⡇⠀⠟⠀⢳⡄
⠀⠀⠀⠀⠀⠀⢠⠙⢢⣤⣀⣀⣠⣤⠀⠘⢿⣿⣿⣿⠿⣿⣇⢀⣠⣶⣦⣼⣿⣿⣷⣾⡿⣷⣦⣴⣦⣀⣶⣦⡌⠁⠻⣿⠀⠀⠘⠃⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⢰⣶⠀⠉⠛⠛⠋⢀⣠⣴⠾⣿⡇⢀⠠⠿⠛⣛⣋⣀⣿⣇⣈⣉⣉⣀⣀⠉⡿⠛⠛⠛⠃⢀⠻⣦⣀⠘⣧⡄⢁⡀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⣴⡉⠁⢠⣶⠆⢀⣿⠁⢀⡠⢾⡷⣿⣷⣶⠿⠟⠛⠛⠛⠛⠿⠿⠿⠛⠿⠿⣷⣶⣶⣶⣄⣈⠀⠙⣿⢿⡟⢱⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⣿⣿⡀⠘⠁⠰⠟⣵⣾⢏⣡⣾⣿⠿⠛⠁⠀⠀⠀⠐⠲⣶⣶⡶⠖⠀⠀⠀⠀⠀⠈⠻⣿⣿⡖⢰⣷⣄⠙⣾⣿⣷⡀⡄⢠⣾⠃
⠀⠀⠀⠀⢼⠀⣿⢿⣷⠀⠀⢀⣀⣉⣠⣿⠋⠁⣀⣀⣀⣀⣐⣀⡛⢻⣶⣼⣿⡾⠛⠋⢁⣀⣈⠛⠳⠄⠈⠛⢿⣿⣿⡅⠀⣿⣿⡇⠀⡄⠈⠁
⠀⠀⠀⠀⢸⡇⠉⢰⣿⠀⣴⣿⣿⡿⠟⣁⡀⠀⠙⠻⢿⣿⣛⣻⣿⠛⢿⣿⣧⣤⠾⠛⢋⣉⣙⠻⢶⣤⣀⡀⠀⠙⠛⠿⣟⣿⣿⣧⠀⠀⣿
⠀⠀⠀⠀⣾⡇⠀⠘⣿⣶⣿⣉⣁⣤⣤⣉⠁⠀⠶⠶⠆⠈⣉⡁⠉⠓⢤⣿⣿⠳⠶⠂⠉⣉⡉⠱⠞⠛⠻⠿⣿⣿⣶⣦⣌⣻⣿⣿⠀⠀⠶
⠀⠀⠀⠰⠉⠛⠀⠀⠻⣿⣿⣿⣿⣯⣩⣭⠀⢠⣤⣤⡀⠀⠈⠉⠞⠒⠸⣿⣿⠆⠰⠖⠀⠉⢁⣀⣤⣠⡀⢰⠁⣴⣉⣭⣿⡟⠁⢻⣷⡀⢰⣧⡀⢀⣀
⠀⠀⠀⢰⡀⠀⠀⠀⠀⠘⠻⢿⣿⣿⠟⠉⠀⠘⢻⡿⣿⣷⣤⡀⠀⣶⠄⠀⠀⣈⠁⠀⣠⣿⡛⠿⢙⣿⡃⠀⠀⣹⣿⠛⠁⢸⡆⣸⣿⠟⠀⠈⠉⠁⠁
⠀⠀⠀⠈⢷⣿⠀⠀⠀⡀⠀⠀⣾⡟⠀⣄⢀⡀⠛⠁⢻⣷⣿⣷⠾⣷⣾⡛⠉⠛⠿⠟⠷⣶⣿⠀⠉⠛⠛⠀⠀⢻⣿⣦⣤⣼⣿⣿⡁⠀⠆⣀⣸⣤⡾
⠀⠀⠀⠀⠀⠙⣷⣀⣼⣿⣤⣴⣿⡇⠸⢿⠘⣷⡄⣴⡿⠇⠀⠁⠀⠈⠛⠀⠀⠀⠀⡀⠀⠈⠻⣷⣴⡶⢀⣀⣶⠀⢿⣿⣿⡟⠉⠉⣿⡿⠿⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⢨⣿⡏⠀⠀⠀⠸⣷⣾⣷⣀⣴⣶⡾⠃⠀⠀⠀⠘⣿⢡⣾⣿⡆⠀⣾⣿⠉⠁⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡎⠹⣿⣿⣧⠀⣿⢰⣄⣿⡇⠀⠀⠀⡀⠀⢿⣿⣿⣿⠇⠀⡀⠀⠀⠀⢠⣿⣿⡿⠋⠀⢠⣿⣿⣴⡄⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⢠⠾⠉⠀⠀⠹⣿⣿⠀⢻⡄⠹⣿⣷⣿⣶⣤⣀⣀⡀⣻⣟⣁⣀⣀⣁⣀⣠⣶⣿⡿⢋⣤⠄⠀⣰⣿⠿⠿⣿⣶⣄⠙
⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⢹⣿⡄⠘⣷⡄⠘⠿⣷⡈⢿⣏⠙⣧⣬⣭⣽⣿⡏⢸⣿⠟⢿⠋⠁⠞⠉⠀⠀⣽⡿⠀⠀⠀⢻⣿⣇
⠀⢀⣤⣄⣠⣶⣏⠀⠀⠙⠁⣙⣿⣿⣧⠀⠘⢧⣈⢠⣈⣿⡘⣿⣾⣿⠛⢿⠀⢹⣷⣿⠇⠀⡀⠀⣠⠄⠀⢀⣾⣿⣄⣠⠄⠀⠙⣿⣿⡄
⢠⡿⠉⠉⠁⠀⠉⠁⠀⠀⠈⠉⠉⢻⣿⣶⣄⠘⠻⣄⡹⣿⡇⢹⡏⠀⠀⠀⠀⠘⣿⠇⣼⡟⣠⡾⠁⠀⢀⣾⣿⡿⠿⠋⠀⠀⢰⠛⣿⡏
⣿⠀⣤⡾⠃⠀⠀⠀⡄⠀⠀⠀⠀⠘⠉⠉⠻⣷⣄⠉⠻⣿⡇⢸⡿⢆⡀⠀⠀⢀⣼⠇⣽⣿⠉⠀⢀⣴⡿⠟⠉⠀⠀⠀⢀⡴⠂⠀⣿⠇
⠙⢿⣿⣧⣼⠂⣠⠟⠀⢀⠀⠀⠀⠀⠀⢀⡀⠈⠿⠿⠿⣾⢿⡜⠣⣌⠛⠛⠛⣛⡁⣤⣿⣿⣿⣿⣿⠏⠀⠀⠀⣤⣤⡴⠋⢀⠀⠀⣿⡄
⠀⠀⠁⠀⢹⡿⠿⣷⣦⣻⣇⣀⣾⣆⣀⣟⢱⣶⠆⠀⠀⠁⠈⢻⣦⣈⡙⠛⠛⠛⣳⡿⢿⣿⣿⣿⣷⣶⠟⠁⠀⠀⣿⣡⣶⣿⣶⣴⡿
⠀⠀⠀⠀⠈⠀⠀⠀⠙⡿⠉⠉⠁⠉⠉⢹⡏⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⢾⡿⠏⢀⣾⣿⣿⣿⡿⠁⠀⠀⠀⣸⡿⠉⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁⠀⠀⠀⢀⡿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣾⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠉⢻⣿⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠗⠀⠀⠀⣸⠁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⡏⠀⠀⠀⢀⠀⠀⠀⠀⠀⣾⣄⡀⠀⠀⠀⠀⣤⣄⠙⠻⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⠟⠛⠉⣿⠇⠀⠀⠀⠉⢿⣦⠀⠀⠀⠈⠉⠻⣦⠀⠀⠀⠈⠙⠳⣄⣀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⢸⣿⡇⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⠀⠀⠀⣸⠛⠀⠀⠀⠀⠀⠸⣿⣿⠃⠰⡄⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠙⣷⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⢱⣶⣶⣾⡏⠀⠀⢀⡀⠀⠀⢀⣿⡇⠀⠀⠀⢀⠀⠈⢿⣷⡄⠀⠀⠀⣀⣠⣿⠋⠉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⢻⣦⣾⡇⢸⣷⣤⣾⣿⣧⡀⠀⢰⡟⢳⣄⠸⣿⣷⡀⠀⡀⠋⣽⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠛⢣⣼⠛⠉⠀⠘⠿⣧⣤⣼⡇⢸⡿⠃⠈⠻⢿⣿⣿⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠈⠛⠛⡷⠋⠀⠀⠀⠀⠀⠀⠘⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠿⠋⠻⠿⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⢀⣀⢀⣀⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠳⣤⠴⠶⠶⠦⢤⡔⠋⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠠⠖⠀⠀⠐⠦⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠶⣶⣦⠔⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠊⠉⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣤⣶⣶⣤⣤⣄⠀⠀⠀⠀⣀⠀⠀⠀⠠⢀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠈⣠⣀⣠⣶⣶⣶⣦⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣠⣄⣼⣦⣤⣀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⢀⡴⠛⠋⠙⠻⠿⠟⠋⠉⠙⠿⠿⠿⠛⠛⢻⣿⣿⡿⠛⠋⠉⠉⢻⡟⠛⠛⠛⢿⠉⠉⠙⠻⠿⣿⡿⠛⠉⠉⠉⠳⢄
⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠋⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈
⡇⠀⠀⢀⣀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠘⠇⠀⠀⠀⡴⠒⠲⠶⠃⠀⠰⠀⠀⠀⠁⠀⢠⣤⠀⠀⢠⣄⠀⠘⠳⡊⠉
⠘⣄⠀⠘⠛⡿⢆⠀⠀⠻⠏⢀⣄⠀⠀⠀⠀⠀⢀⠀⠻⠤⠄⠀⣀⢀⣤⠀⠀⠀⠀⠀⠈⠁⡤⠔⠻⡏⠙⠀⠀⡇
⠀⠈⠑⠒⠚⠁⠈⠑⠢⠤⠖⠁⠘⠦⣀⣀⣀⡤⠊⠑⠢⠤⠤⠤⠇⠉⣇⣀⣀⣀⣠⡤⠴⠦⠚⠀⠀⠙⠒⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣧⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢀⣴⡏⠍⠀⠈⢹⣷⣄⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⡄⢸⣿⣷⠀⠀⠀⣸⣿⣿⠀⠀⠀⡠⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣳⠀⠛⣿⣯⣾⣶⣿⣭⣍⡀⠀⠸⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠉⠋⠉⠁⠁⠉⠛⢿⣿⣿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⡄⠀⠀⠀⠀⠀⠀⠐⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣧⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⢀⣾⣏⣄⠄⠀⡀⠀⢀⢠⣤⣿⠷⣄⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣶⣖⣶⢊⠿⠏⠀⢢⠆⠯⠿⣿⣷⣾⡷⣿⣾⣿⡿⣿⣫⠅⠄⠀⠈⢳⣶⣶⠦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠃⡤⣶⣶⠛⠉⠍⠁⠛⠋⠁⣳⢶⣖⢤⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠘⢈⢿⠙⠛⡦⣤⠄⡤⠤⠴⠟⠛⠈⡍⠉⡀⠀⠀⠀⠀⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢀⠂⠀⠘⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠌⠀⠠⠀⢹⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢪⡟⠙⠃⠀⠄⠀⠠⠄⠀⠐⠁⣉⠀⠰⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣡⠋⢰⡀⣀⠀⠀⠀⠀⠀⠀⢠⠰⣿⠀⠀⠙⢮⠈
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⠛⠙⠂⠀⠀⠄⠠⠤⠖⠂⡉⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⣀⣤⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⠉⠋⠋⠉⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠶⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠍⠹⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡟⡠⡀⡀⡀⠋⠼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠋⣴⣣⠟⡻⢷⡿⢟⠻⣞⣎⢳⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠁⣠⡬⢽⣿⡛⠴⣯⣶⣶⣽⠦⣞⣿⡯⢥⣄⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠉⠀⢻⣿⠁⢹⠞⠢⠴⠛⡏⠈⣿⡟⠀⠉⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⢻⣆⠀⠀⠀⡈⡙⣿⡏⣵⠶⡶⣮⢽⣿⣟⡁⠀⢀⠀⢸⡟⣧
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠛⢇⠘⠀⠀⣿⣿⠻⣶⣿⣿⣷⠿⣻⣷⣁⠀⣆⣞⠞⠁⡘⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠄⠀⢈⣻⣤⣶⣿⣗⠀⠈⢻⣿⠋⠀⢚⣿⣿⣽⣿⡭⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠁⠀⠀⠉⠛⢿⣿⣿⣋⣼⣿⣌⣿⣿⣿⠏⠉⡀⠀⠀⢳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⣧⡆⡄⡀⠀⢶⢈⣿⣿⣿⣿⣿⣿⣿⣿⠧⡆⠈⠀⣤⣼⣾⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣁⠀⡄⠀⢿⣿⣿⣿⣿⣿⠟⠛⠀⠖⣮⣾⣫⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠛⠂⠉⠛⢿⣿⣿⣏⡀⠀⠀⢀⣹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣯⠀⠀⠀⣆⡀⣿⢿⣽⣳⣾⣧⣾⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣷⣿⣿⣭⡇⠉⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣂⣥⣤⣉⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣷⣤⣿⣶⣆⢠⡀⠻⣿⣶⣶⠆⢀⡄⣴⣿⣽⣿⣿⣟⠀⢀⢀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⢿⣿⣿⣿⡩⣅⠸⡿⢊⡴⣤⣦⡁⢾⠇⣡⠌⣿⣿⣿⣞⣿⣿⢯⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠚⠛⠿⠿⠷⠎⠳⠀⣸⣏⠻⣿⣇⠀⡞⠱⠞⠻⠻⠛⠛⠛⠉⠈⠀⢆⡀
⠀⠀⠀⡀⠀⣠⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡇⢀⣼⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠲⡦⠅⠂⢄
⠀⠀⠐⠀⠀⡘⣧⣕⡠⠀⠀⠀⢀⠀⠠⠄⠀⠀⠸⣿⣾⣿⣿⣿⡇⠀⠂⠠⠄⠀⣀⠀⠀⠀⠄⣊⣶⣄⣠⣰⡌
⠀⠀⡄⠀⢸⣿⣯⣿⣿⣶⣦⡄⠀⠠⡴⣓⢤⣦⠤⢿⠏⠿⣿⡿⠤⣰⡦⠘⢦⡄⡀⢀⣠⣶⣾⣿⣿⣿⣿⠟⠃⠆
⠀⠀⢄⠀⠼⣿⣛⣿⣿⡿⢟⣁⣀⠪⣵⢿⠇⡼⢶⣥⣴⣶⣾⣯⡴⢧⠘⣿⡦⠑⣀⣬⣙⠉⢿⣻⣿⣿⡤⠀⡀
⠀⠀⠀⠙⠂⠈⠛⢋⣥⣶⠿⠛⠉⠑⠢⠉⣐⠃⣼⣁⣠⣶⣶⣿⣅⠀⣃⠉⠀⠀⠉⠛⠿⣿⣦⡈⠉⠁⣠⠊
⠀⠀⠀⠀⠀⠀⠀⡌⠉⠁⠀⠀⠀⠀⠀⠀⡟⣴⢹⣥⣤⣴⣾⣿⡍⣤⢻⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡠⠏⡄⣿⡀⢧⣀⣬⣿⡦⠰⣷⠀⠹⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡃⢟⣮⡌⠻⢿⢟⣡⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡈⠛⠷⠞⠛⠋⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⡠⠚⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠦⣤⣤⣀⢀⣀⡀⡄⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣷⣦⣀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⡭⢽⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⢱⣆⠀⠀⢿⣷⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣶⣿⡏⠁⠀⠀⡀⠂⠀⣤⠞⠉⢠⠏⠀⠀⠀⣸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⣬⣿⡵⠋⠠⠀⠀⣼⢶⢦⣀⢷⣿⣿⣿⣯⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⡟⢉⠆⠀⠀⠀⣼⣷⢸⣿⣾⠏⢻⡌⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⡃⠀⠀⡀⠀⣼⡿⠉⠀⣿⡆⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⣶⣿⣿⣦⡶⠋⠀⠀⠀⠁⠀⠂⢀⣾⠉⠀⠀⠀⢸⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠈⠻⢿⣿⡿⠁⠀⢀⠀⠀⠀⢀⣤⢻⣿⠀⠀⠀⠀⠈⢿⡆⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⢀⣿⠇⠀⣿⠀⠀⠀⠀⠀⢀⠇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⡷⢀⡄⡀⠀⠀⠀⢾⠀⠀⠀⢀⣀⣴⣿⠇⠀⢀⣿⡆⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢷⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⠇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢏⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⢉⣼⣿⡏⠀⠀⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠿⣆⣿⣰⣶⣶⣵⣿⣿⣿⠿⠁⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠛⠛⠻⡿⣿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⣀⣠⣤⣠⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠉


⣿⣿⠿⠿⡿⢻⠿⡿⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠤⠴⣄⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠁⠀⠀⠀⠺⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⣀⣴⠟⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣴⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠔⠀⠀⢀⣴⠟⡀⠊⠀⠀⠀⠀⠀⢀⠠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢀⣴⠿⠅⠈⠁⠀⠀⢀⡠⠔⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢀⣴⠿⠥⠄⠀⢀⠠⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠋⠉⢿⣦⠀⠀⢀⣤⣤⣤⡀⠀⢠⣿⠁⣠⣿⡏⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⣠⠴⣶⣄⡀⢻⡗⣺⢟⣻⠻⣿⣿⡄⢸⠃⢰⣿⣿⣟⠄
⠀⠀⠀⠀⠀⠈⠋⢀⣴⠟⠁⠀⠀⠙⠻⣶⣷⡏⣿⣿⡀⢹⣿⠧⠀⠀⠘⠁⠏⠔⠀⣀⣀⣠⣤⣤⡀⠲⣶⣦⣄⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⢿⣿⣇⣿⣿⣥⣼⣯⣄⠰⣀⠐⣂⣠⣤⣿⣿⣦⣾⣦⡼⠦⠄⠈⢻⣿⣧⠀⠈⢹⣿⣦⡀
⠀⠀⠀⠀⠀⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣡⣻⣿⢫⣟⣢⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠀⣀⣼⣿⣿⠀⠀⢬⣿⣿⣷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⠃⠀⢠⣾⣿⣿⡿⠀⡸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠏⠀⢻⣿⣿⣿⣿⣯⣻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣀⣴⣿⣿⣿⡿⠁⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠁⠀⠀⠸⠿⣿⣿⣿⠿⣿⣿⡏⣻⣿⣿⡟⣻⣿⣿⣀⣺⣾⣿⣿⣻⠟⢭⠀⣾⣿⣿⣿⠏⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠀⠀⠀⠀⠻⣾⣯⣝⠣⣿⣿⣇⠏⢛⡏⣹⣿⣿⡿⡿⣿⡿⠿⠟⣫⣶⡟⢰⣿⣿⣿⢏⣳⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⡟⠭⠃⠀⢸⡿⣿⣿⣿⣿⡔⠋⠑⣀⣼⣿⣿⣶⣷⣿⠋⣱⣾⣿⡿⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣾⡿⠞⡿⢋⣴⣾⣿⣿⣏⣩⣧⣸⣾⣿⣿⠿⢛⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣻⢗⣴⣶⣿⣿⣟⡉⡈⠛⠿⣿⣿⣯⣭⣿⣿⣿⠿⣛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣯⣇⢱⠀⠀⠀⠈⠐⠻⢿⣷⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠆⠻⣿⠸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⠀⠙⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠗⠃⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢲⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿
⠀⠙⢿⣭⣙⠛⠛⠛⠛⠿⠿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⠾⠿⠟⠛⠋⣰⡿⠁
⠀⠀⠀⢉⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠈⠙⠷⣦⡀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠈⠻⣷⣄⣀⣤⣤⣤⣤⣤⣄⣀⣠⣾⡿⠋⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⣾⣏⡀
⠀⠀⠀⠀⠀⠲⣿⣟⠛⠂⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠈⣿⠿⠛⠉⠉⠉⠙⠛⠛⠿⣯⣀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠚⢋⣿⠟⠁
⠀⠀⠀⠀⣀⣀⣠⣽⣷⣦⡀⠀⠀⠀⠀⠹⣿⣿⡦⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀⢰⣿⡇⠀⠀⠀⠀⣠⣶⣿⡷⠂
⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⣀⡀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⣁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣶⠾⠛⠋⠙⣷⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠠⠾⢿⣿⠟
⠀⠀⠀⠀⠀⠀⢠⣾⠏⢁⣠⠶⠚⠛⢻⡶⠀⠐⠲⠶⣦⣤⣀⠀⠀⠀⠘⡇⡀⠀⠀⠀⠀⣀⣤⣤⠤⠤⠙⣿⣆⣠⣶⡿⠃
⠀⠀⠀⠀⠀⠀⣼⠃⢠⡏⢻⣧⣀⣠⡿⠁⠀⠀⣀⣤⣤⣍⣿⣷⣄⣀⣀⠁⣷⡀⣠⣴⣿⣿⣉⣀⠀⠀⠀⠘⢿⣿⠋
⠀⠀⠀⠀⠀⠀⢻⡄⠸⣇⡀⠉⢻⣿⠃⠀⠀⠀⠈⠃⠘⠿⠭⠟⠛⢉⣡⡾⠿⠿⣿⣿⡻⠽⠃⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠈⠻⣄⡈⠙⠛⠛⢹⣿⠗⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠋⠀⠀⠀⠈⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⣀⡀⠀⠀⣦⣼⡟⠀⣰⣿⣷⡄⠀⠀⣰⣿⣿⡄⠀⢻⣧⠀⣰⣧⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⣰⣿⣉⠀⠀⠳⣹⣿⡇⠀⣿⡇⠀⠀⠀⠀⡿⠉⢿⣷⠀⠈⣿⣠⠏⣿⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠟⠀⢿⡷⣦⣄⣙⣿⣷⠀⠙⣷⡀⠀⠀⠀⠀⠀⢸⡏⠀⢠⡿⠋⢀⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⢷⡄⠉⠛⢿⣿⣧⡀⠈⠑⠀⠀⠀⠀⠀⠈⣀⣤⣿⠃⢀⣼⠇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠙⢿⣿⣶⣦⠶⠶⢛⣛⠿⣿⣿⣿⣯⣴⡾⠃⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⣤⣿⣟⠲⣶⢿⡛⠛⡏⠉⢻⠉⢹⠉⣹⣁⡟⣿⢻⣷⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⢸⡏⠻⣦⡘⢯⡳⢖⣛⣒⢛⣛⣛⣉⣩⣥⣰⡏⠀⣿⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⠀⠀⠀⣿⡀⠈⠻⣦⣁⣾⠋⠉⡏⠉⣏⠉⡇⢸⣿⣀⣴⡏⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣼⣧⣀⠀⠈⣻⣿⣦⣴⣷⡤⠿⠶⠟⠛⣻⡏⠉⠀⢀⣠⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣭⣉⠛⠛⠋⣀⣤⠀⠀⠀⠀⠀⠀⣾⣯⣤⡶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⡋⠁⠀⠀⠀⠀⣀⣼⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣶⣶⡿⠟⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡈⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣍⠁⠨⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡑⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠠⠔⠒⠉⠉⠉⠉⠁⠐⠠⢌⡀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠐⠤⣄⢀⠀⢀⣀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠑⠠⠤⣤⣄⣀⣀⡀⠒⠦⠁⠀⠈⠂⢀⠹⣧⣾⢷⣾⣷⣤⣤⣄⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣿⣷⣤⡶⢶⣄⣀⣘⠢⣴⣀⣀⣿⣿⣿⣿⣿⡍⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⣇⣤⣿⣿⣿⣿⣿⡛⡿⠿⠿⣰⣿⣿⣷⣸⣿⠟⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢿⣿⣆⠀⢻⣿⣿⣿⣿⣿⣿⣿⡘⢶⣶⣿⡿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⠀⠈⢿⣿⣷⣴⣿⣿⣿⠿⣷⣾⠛⠿⢻⡟⡟⢡⡈⢿⣯⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠺⣷⣦⣤⠀⠙⢿⣟⣵⣿⠏⠀⣾⣿⠀⠀⠈⠃⠁⠈⢷⡄⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠈⠻⣿⣿⡿⣷⣿⡿⠁⢀⣴⡟⠃⠀⠀⠀⠀⠀⠀⠀⠹⣦⠔⠆⠀⠀⢀⠔⠀⠀⣀⣴⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣦⣴⣩⣿⣿⡏⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁⠀⠀⠀⠉⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⣿⣿⠏⣿⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⡿⠊⣡⢂⠀⡀⠀⡠⠔⠊⠈⠁⠀⠀⠉⠐⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⠀⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠿⠇⠸⢧⣾⣪⠄⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⢠⠔⠀⠀⠀⠀⠀⢻⣾⡇⠔⠊⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣻⣿⡍⠃⠀⠀⢀⠈⠀⣰⣿⣿⠿⠿⠿⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠃⡠⠀⠀⣈⣴⣾⣿⣏⠙⠓⠂⠀⠠⠀⠀⠀⠉⠐⠒⠄⠀⠡⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣝⣿⣿⣿⠟⠿⢿⣮⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⡀⠈⠁⡾⢫⢯⠏⠸⣿⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠐⠀⠀⠀⠀⠯⠉⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⢄⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⣎⣷⠀⠀⠀⠀⠀⠒⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡧⠊⠀⠀⠀⠀⠀⠻⡅

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣁⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠈⠢⣤⣤⣤⣤⡔⠁⠀⠀⢀⣾⠟
⠀⠀⠀⠔⠐⠒⠐⠲⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⣴⣾⣿⣿⣿⣷⣷⣦⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠖⠂⠒⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠓⠢⠄⡀⠀⠀⠀⠀⠘⣿⣦⣀⣿⠿⠟⠛⠟⠿⠿⣿⣀⣴⣿⠃⠀⠀⠀⠀⢀⡠⠔⠚⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠈⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⢀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠷⢶⣿⣦⣤⣤⡆⣠⣄⢰⣤⣤⣴⣿⡶⠾⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⢠⣾⣿⡻⠿⠿⠿⠿⠟⣋⣙⠻⠿⠿⠿⠿⢙⣿⣷⡄⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢀⡿⠛⠁⠀⠰⠶⡶⣶⣾⡻⢿⣷⣶⢶⠶⠆⠀⠈⠛⢿⡀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠈⠀⠀⢸⣷⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⣿⡇⠀⠀⠑⠶⣤⣤⣤⣤⠶⠦⢤⣤⣤⣤⡶⠈⠀⠀⢸⣿⡆⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⢸⡟⠃⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⢻⡇⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣷⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣾⠃⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠈⠻⢿⣿⣿⢿⡿⣿⣿⡿⠟⠁⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⢸⡄⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠈⠓⣄⣠⣤⣴⡃⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⡄⠀⢠⣿⣾⣿⣿⣷⡄⠀⠀⡞⡸⠀⠀⠀⠀⠀⡀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢿⣠⣾⡿⠿⠟⠻⠿⠟⢆⣠⡗⠃⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⢄⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⣀⣤⠀⡒⠲⡶⣶⠤⣻⡄⠀⠀⠁⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠠⠼⣧⡾⡿⠄⠀⠀⠀⠀⢼⣷⡧⢔⣒⣀⡀⠀⠠⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⣿⣿⣫⡁⠀⠀⠀⠀⡨⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠁⠁⠀⠈⠉⠉⠉⣿⡺⣧⣮⣡⣤⣶⣷⣿⣿⣷⡶⢶⣶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣤⣾⠟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋⠁⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣚⣯⡿⡿⡾⣮⠙⡁⢉⡉⢹⣻⣷⣿⣿⣿⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣉⡽⠛⠋⠁⣼⡟⠟⠋⢤⣄⠀⡈⢬⣷⣾⠿⠟⠋⠁⢨⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠁⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣭⣧⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠹⣿⣯⠜⢘⡀⠀⢒⣲⣘⡿⣿⣶⣶⡾⠁⠀⠳⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠹⢟⠿⢾⣖⡘⠼⣧⣿⣷⡿⣿⠿⠁⠀⠀⠀⠀⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠀⠀⠀⠀⠑⠢⠄⡈⣀⡉⣉⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⡀⠘⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢦⠘⢦⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠓⢤⡀⠀⠀⢣⡀⢻⡆⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⡀⠉⠓⢄⠈⢷⠈⣷⠀⠀⠀⣧⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢢⠈⢧⢹⣇⠀⢠⣿⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⣆⡑⢬⡳⣿⣦⠘⢇⣤⣤⣶⣦⣄⡟⢀⡤⠤⡄⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠝⣶⣭⣿⣿⣿⣧⡈⠉⠻⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⢠⣶⣶⣶⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠹⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠟⠛⠛⠛⠛⠁⠙⠛⣿⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣸⣷⣶⣶⣶⢦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣴⠿⣿⡿⠋⠙⠉⢿⣿⣿⣿⣿⣿⣴⣻⣿⣿⣿⡠⣦⣌⡀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⠟⠋⠁⠀⣰⣿⡃⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡻⣧⢾⣝⠻⢷⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣼⢟⠟⠁⠀⠀⠀⣰⣿⣿⣿⣄⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣟⠛⠋⠈⠳⣮⠛⠦⣬⡉⠛⠒⠠⣄⡙⠛⠏⠩⠭⠙⢲⡀
⠀⠀⡀⣀⡴⠞⠉⠋⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⢻⣿⢿⣄⠀⡀⠀⠠⠀⠀⢀⡉⠉⠀⠂⠀⠉⠉⡁⠉⠐⠀⣸⠟
⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⢀⣸⣿⠈⢿⣷⣷⠀⠀⠀⠐⠠⠙⢶⣤⣄⣀⠀⠀⢀⢑⣤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢄⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠤⣀⠙⢿⣿⣿⣿⠿⠋⠀⣾⡏⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠛⠛⠉⠀⠀⣠⣾⡿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣾⣿⡟⠁⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠓⢀⠀⢀⠐⠋⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠀⡰⣦⣨⣤⣬⣠⣶⡄⠀⢀⡴⠞⠋
⠀⢀⡤⠐⠲⣒⣒⣒⡒⠢⠤⢀⣀⠀⠀⠀⠀⠀⠹⣷⣄⠻⣿⡿⠿⣿⣿⡿⢃⣴⡽⠁⠀⠀⠀⠀⢀⣀⡠⠤⠒⣒⣒⣒⡲⠒⠠⣄
⠀⠸⣄⡀⣀⣀⢴⡣⠤⠭⠴⣣⠤⠯⡑⠒⠤⢄⣀⡈⠻⣶⠇⠀⠀⠨⢿⣷⣾⠋⣀⣀⠤⠔⠒⡩⠯⠤⡻⠬⠥⠤⡳⢄⣀⣀⣀⡼⠁
⠀⠀⠀⠐⢄⣀⣀⡩⠖⠒⠂⠺⣍⣉⠉⠉⠛⢚⣿⣿⣿⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣟⠚⠋⠉⢉⣉⣹⠂⠒⠒⠮⣁⣀⣀⠔
⠀⠀⠀⠀⠀⠈⠢⠤⡤⠒⣊⡁⢶⣁⡬⠭⢽⠿⣛⣿⣿⣿⣧⡤⠀⣲⣿⣿⣿⣿⣟⡻⢿⠭⠭⣄⣑⠆⣉⡓⠢⡤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠤⣔⣁⣤⣚⡡⠤⣒⣭⣭⣬⣿⣿⣿⣿⣿⣯⣬⣭⣕⡢⠤⣉⣢⣄⣑⡦⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⢉⣭⢟⣿⣿⡏⠙⣿⣿⣿⣟⢿⣍⠙⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠙⠋⠉⠀⢀⣾⣵⢿⠟⠿⠿⢿⣿⣿⣿⢿⢷⣽⣆⠀⠈⠉⠛⠑⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⣾⣀⡀⠀⠀⠻⣿⣿⡏⡆⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⣿⠻⠿⠿⠿⢿⣿⣿⡇⡇⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⢹⠷⣶⣦⣤⣾⣿⣿⢷⠃⠀⠀⠻⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⠀⠀⠀⠀⠘⣷⣤⣤⣤⣽⣿⣟⡞⠀⠀⠀⠀⠈⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣾⣿⣿⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡢⠄⣲⣂⠤⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⣿⣿⣿⣇⡀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠤⠃⡀⠀⢀⡿⢋⣾⣿⣿⣯⠻⣇⠀⠀⡀⠣⠄⠠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡧⣾⣿⣿⣿⣿⡦⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⣿⡿⣿⡿⣿⣷⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢾⣳⣶⣿⣿⣶⣻⠇⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡼⢁⠄⠨⠿⡛⠛⠛⡻⠯⠀⢄⠹⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⠀⢀⣤⣼⣷⣿⣥⣄⠀⠈⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢠⡏⠁⢉⣿⣯⠁⠉⣷⡀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡔⠁⠊⠹⠛⠷⢾⣿⣿⠶⠟⠻⡉⠂⠜⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢀⠚⠀⠀⠀⠂⠸⠗⠸⣿⡿⠀⠿⠂⢃⠀⠀⠘⠢⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠐⠀⠀⠀⠀⠘⠛⠶⠶⢶⣶⣴⠶⠶⠞⠛⠀⠀⠀⠀⠐⠈⠄
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠇⠀⠀⠀⢀⡀⠘⠃⠈⢿⠟⠀⠛⠀⣀⠀⠀⠀⠀⠇⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠛⢛⠓⣶⡖⢚⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠶⠤⠤⣬⡥⠤⠴⠶⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⠤⣬⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠗

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠤⠶⢀⣴⢘⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠃⠀⢀⡈⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⢦⡀⠀⠀⣿⡇⠀⠀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡉⣿⡆⠒⣾⣿⡇⠀⣿⣿⡇⠻⢃⣠⠴⣿⡇⠀⢾⣏⠀⠁⠀⣍⡷⠸⠏⠻⠰⠭⠆
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣇⣸⠃⠀⣿⣿⣿⠀⣿⣿⣷⢠⣿⡏⠀⣿⡇⠀⠀⢹⣷⠀⠀⣰⡄⠰⡦⣤⢠⠶
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⠹⣷⡄⣿⣿⣿⠀⣽⣿⣿⠸⣿⣇⢀⣿⣇⢰⡄⣸⡿⠀⠀⠘⠶⠈⠁⠘⠈⠛⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡠⠟⢁⡼⠟⠛⠀⠋⠉⠉⠀⠈⠉⠉⠉⠉⠀⣁⣀⡀⠀⢀⣤⢤⡄⠀⣴⡶⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣾⡏⢙⡿⠀⠀⢠⣿⣟⣨⡿⠀⣿⣿⠚⠛⠀⠙⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠉⠙⣷⡄⠸⣿⣏⠀⣀⡄⠻⣿⣀⡠⠠⡄⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠀⠛⠉⠁⠀⠈⠉⠁⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠄⠀⠀⡠⠀⠀⠀⠠⢀⠀⢀⢀⡀⠀⡀⢰⠃⡴⡀
⠀⠀⠀⠀⠀⠀⢠⠒⠰⢢⡆⢶⠎⡁⢴⠦⣞⠀⠸⠂⣶⡈⠱⠖⠟⠂⠀⠉⠖⠛⠃⠀⠀⠀⠘⠁⠒⠙
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠁⠈⠀⠀⠀⠠⠙⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢀⢀⡀⢀⢠⡀⢀⡀⢠⡀⠄⠀⣄⡠⡄
⠀⠀⠀⠀⠀⢙⣠⠄⢦⡤⡤⣶⡣⡄⠀⠀⢼⣷⠶⠷⠀⠶⠆⠹⠂⠄⡇⠀⠂⠃⠉⠀⠋⠓⠈⠀⠄⠉⠉⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠛⠲⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠄⢀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠆⢳⡼⢇⠠⡀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢠⠃⠀⠈⢲⡃⠀⢸⠀⠀⠀⣸⠋⢹⡆⢀⠖⢳
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠻⢲⣄⠛⣠⠃⠀⠀⠀⠀⠹⣆⠀⠀⠀⠸⠀⠀⠀⠀⢷⠀⠈⠀⠀⠀⢺⡀⠺⠿⡎⢀⡆
⠀⠀⠀⠀⠀⠀⠀⣊⣳⠞⢉⣈⠉⠁⠀⠀⠀⠀⢠⠖⠉⠉⠉⠑⠢⢄⠀⠀⠀⠀⢇⠀⠇⠀⠀⠘⣧⣀⣀⡇⠘⠋
⠀⠀⠀⠀⣠⠖⠻⡉⠁⠀⠀⠀⠙⠂⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠘⣴⣀⠀⡴⠋⠄⢛⠳⣾⠖⢢
⠀⠀⠀⠘⠁⠀⠀⠘⣠⣀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠂⠀⡀⠀⠹⣿⡼⠀⠐⠦⣍⡵⢧⣭⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠴⠋⠳⠂⠀⠀⣸⠀⣴⣾⣷⣾⡛⠛⠀⠀⠔⠂⠈⡷⡼
⠀⠀⠀⠀⠀⠀⠀⢠⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⠏⢠⣿⠉⠈⠛⠋⠀⠀⠀⠀⠀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠠⠾⣷⣿⡟⠣⡀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⣠⡾⢻⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣧⠸⣧⠹⣦⠀⠀⠀⠀⣀⣠⡔⠻⠶⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡄⢻⣆⠈⠳⣤⣙⡻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⣤⠀⠠⢴⡆⢦⣰⢶⠛⢷⡀⠻⣦⡀⠈⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⢸⠹⣗⠡⢺⣿⠆⢸⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠀⠀⠀⢰⠈⡄⠘⠆⢸⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⣀⡄⢃⣀⡀⣠⡄⢡⣄⣤⢸⣿⣀⢸⣧⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢿⣿⣭⣕⣾⣿⣿⣿⣦⠾⠃⠀⠈⠉⠉⠁⠀⠀⠘⠛⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠋⢁⣠⣄⡈⠙⠻⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠟⠋⢁⣠⣴⣾⣿⠟⠿⣿⣷⣦⣄⡈⠙⠻⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠟⠋⢁⣠⣴⣾⣿⠿⠛⠉⠀⠀⠀⠀⠉⠛⠿⣿⣷⣦⣄⡈⠙⠻⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⠋⢁⣠⣴⣾⡿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣷⣦⣄⡈⠙⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣾⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣷⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠃⠀⠀⠀⠀⠀⣠⣤⡾⢋⣉⣉⡩⠒⠒⢍⣉⣉⡙⢷⣤⣄⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⣿⠀⠀⠀⠀⠀⢸⣇⣿⠃⠸⢛⡟⠛⠓⠚⠛⢻⡛⠇⠘⣿⣸⡇⠀⠀⠀⠀⠀⣿⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣾⡀⡴⣛⣛⢦⢀⣷⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⠀⣀⣤⣤⣶⣿⠀⠀⣇⣿⣷⣿⠻⠟⣿⣿⣿⣸⠀⠀⣿⣶⣤⣤⣀⠀⠀⢹⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣿⣧⣴⠟⠋⠵⣾⣿⡇⢀⢳⣿⣿⣿⠶⠻⠿⠶⣿⣿⣿⡞⡀⢸⣿⣷⠮⠙⠻⣦⣼⣿⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡿⠗⠂⣀⣽⣿⣿⣷⣈⣻⣿⠉⢿⡷⢶⡶⢾⡿⠉⣿⢛⣁⣾⣿⣿⣯⣀⠐⠺⢿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣼⠟⠲⣴⣾⣿⣿⣿⣿⢻⡿⣿⣿⣷⣈⢡⡀⢀⡌⣁⣾⣿⣿⢿⡟⣿⣿⣿⣿⣷⣦⠖⠻⣧⡀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡞⠛⣿⢶⣮⠙⢿⣿⣿⣿⠈⠀⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⠀⠃⣿⣿⣿⡿⠋⣵⡶⣿⠛⢳⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠒⡾⢁⣽⣿⣷⠀⣿⣿⣿⣦⣷⡈⠃⠘⠇⢻⣿⣿⡟⠸⠃⠘⢁⣾⣴⣿⣿⣿⠀⣾⣿⣯⡈⢳⠒⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣮⡉⣽⣁⣼⣿⡇⣹⣿⣯⠻⣿⣿⣄⣷⡀⠀⠙⠋⠀⢀⣾⣠⣿⣿⠟⣽⣿⣇⢸⣿⣧⣈⣯⢉⣵⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣽⣃⣤⡷⢿⣿⣿⣧⣦⣈⠙⠿⠿⣦⣀⣀⣴⠿⠿⠋⣁⣴⣼⣿⣿⡿⢾⣤⣘⣯⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⡀⠀⠉⠛⢿⣿⣿⣷⣦⣄⣀⠙⠋⣀⣠⣴⣾⣿⣿⡿⠟⠉⠀⢀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⣀⠀⠀⠉⠙⠻⠟⠋⠉⠀⠀⢀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⢀⣠⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠻⣦⡀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣤⠾⠛⠉⠉⠈⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠈⢿⡄⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠻⠟⠀⠙⢿⡄⠀⢸⡇⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠷⠿⠷⣤⣀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠉⢳⣤⣴⡶⠶⠺⣿⣷⣤⣾⣋⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⡀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢀⣿⣿⣿⡏⠉⢻⣿⣿⡟⠛⢿⣿⣛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠋⠀⠁⠀⠀⠀⠀⣸⣿⣿⠃⠀⢸⣿⣿⣿⠇⠀⣾⣿⣿⡇⠀⢸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⢠⣿⣿⣿⡟⠀⢀⣿⣿⣿⠇⠀⣾⣿⣿⠃⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠒⠛⠻⣿⠃⠀⣠⣿⣿⣿⡟⠀⠀⣼⣿⣿⡿⠀⢰⣿⣿⡟⠀⢸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⣿⣿⠟⠀⠀⣼⣿⣿⡿⠁⢀⣿⣿⣿⠃⢀⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡃⠀⣠⣾⣿⣿⡿⠁⢀⣾⣿⣿⠃⢠⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣟⣀⣠⣿⣿⣿⡷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠙⣆⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢀⡞⠉⠀⠉⠉⠛⠶⣤⣀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⡼⠁⠀⣈⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⣸⠴⢒⣾⣷⢴⡏⠁⠀⠘⠛
⠀⠀⠀⠀⠀⠀⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣠⠋⠀⡜⠁⢀⠈⢳⣈⣦⡀
⠀⠀⠀⠀⠀⠀⠙⢄⡈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠇⠀⠀⢄⠻⠿⠃⡜⣀⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠂⠉⠲⠄⠉⠓⠢⠤⣄⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠑⠒⠉⠐⠭⣁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠾⣿⣷⣍⠉⠉⠻⣿⣯⠱⣄⠀⠐⢦⣀⣀⣀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⠀⠀⠈⢿⣿⣦⠀⠀⠙⣿⣧⠈⢳⡦⠤⠭⠥⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⡄⠀⠀⢻⣿⣇⠀⠀⢹⣿⠀⡼⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠈⣿⣿⠀⠀⢸⣿⢟⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⢿⣿⣿⣷⠀⠀⣿⣿⣀⣤⠞⢁⣼⠇⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⠛⠛⢩⡿⠁⣾⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠟⢠⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠹⡄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠙⣄⠙⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠈⠆⡈⢿⡀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⡀⠍⠲⡄⠘⣇⠸⣧⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠳⡄⠀⠈⢄⠘⣆⢿⡄⠀⢰⡏⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢶⣄⠱⣌⢾⣿⣄⠸⣧⣄⣴⣶⣤⣰⠇⣠⠤⣤⢀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣦⣝⣿⢿⣿⣄⡉⠙⠻⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⢤⣶⣶⣦⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⡀⠈⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠿⠿⠟⠛⠛⠉⠛⠿⣷⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀⣹⣧⣶⣶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣼⠷⣾⡿⠋⠛⠛⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⢠⣭⣉⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⠞⠋⠁⠀⣼⣿⡀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣔⣯⡛⢿⣶⣶⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣟⠟⠁⠀⠀⠀⣸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡟⠛⠃⠙⢷⣝⠷⢦⣽⠙⠒⠂⢄⡈⠛⠻⠭⠭⢙⡒⣄
⠀⠀⠀⣀⡤⠞⠋⠋⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣆⡀⠀⠀⠀⠀⠈⢿⣿⣷⡄⠀⠀⠀⠇⠀⠀⡀⠉⠀⠒⠀⠉⠉⢉⠁⠐⠀⢈⡿⠃
⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡀⠀⣸⡇⠸⣿⣶⣇⠀⠀⠀⠂⠌⠓⣦⣤⣀⠀⠀⠀⠉⢢⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠀⢸⣿⠁⠀⠀⠀⢀⠀⠀⢀⡙⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢈⠀⢤⡈⠛⣿⣿⣿⣿⠿⠋⢀⣿⠇⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠉⠳⠛⠛⠋⠀⠀⣠⣾⡟⠀⠀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡾⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠤⡧⡧⠤⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢌⣄⡀⠾⠆⣀⣌⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠂⠂⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠄⠀⠀⠀⠀⣼⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⠞⠕⠂⠠⠀⠒⡝⢮⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢋⠸⠀⠀⠀⠀⠀⠸⢈⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⡇⠀⠠⠀⠠⠄⠀⣾⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⢸⢣⡀⠀⠀⠀⠀⣠⢻⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⡠⠂⠀⠢⡀⠀⠀⡠⠂⠀⠢⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠈⠂

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠞⠋⠉⠙⣟⠛⠚⢯⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡿⠟⠛⠉⣏⠅⡒⠦⠘⠁⢀⣦⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⡀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⡍⡀⢀⠀⢠⠒⠻⣭⣁⣀⠨⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣟⡛⠻⠿⣿⠿⠛⠛⠟⠛⢉⡩⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠂⠉⠉⢙⣻⣶⣦⣶⣶⣯⣥⣠⡀⡀
⠀⠀⠀⣠⡴⠾⠞⠛⠓⠛⠒⢾⣿⣿⣆⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡔⢲⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠐⠃⠀⠀⠀⠀⣠⠴⠖⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣗⡉⠙⣿⣿⢉⠻⣿⣿⣛⠒⠶⠦⠀⢀⡀
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⣽⡿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣿⣇⣀⢰⣿⡧⣀⢠⣾⠃⢸⣤⠀⠀⠀⢄⡈⠑⠠⢄⡀
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢠⣾⡟⠀⠈⣡⣼⣿⠉⣿⣏⠻⣿⣿⡹⣿⣄⠛⠻⣿⣿⣿⣿⣿⡟⢻⣿⣿⣧⠉⣿⣿⡧⢤⣿⣻⣄⣾⣻⣄⠀⠀⠰⡇⠬⠬⠟⠁
⠀⠀⠀⠀⠉⠀⠀⠀⣤⣴⠿⠉⠀⠀⣾⡿⠉⠀⠀⢿⣿⣤⣨⣙⢷⣮⣽⣵⣤⠄⠙⠿⣿⣿⣷⣿⣿⣿⡇⠀⣼⣿⢀⣼⡿⣩⣿⠿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠉⠀⠸⣿⡆⠙⠛⠿⠿⠛⠓⢿⣦⡠⣯⡙⠛⠻⠷⠾⢿⣿⣿⣿⣾⣟⣡⡴⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠘⣿⣿⢯⠛⠻⠿⣦⡀⠀⠀⠀⠉⠛⠻⣿⣥⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡷⡄⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⡀⠈⠛⠛⠳⠶⠶⠦⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣄⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⣄⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠙⠛⠛⠐⠂

⠀⠀⠀⠀⠀⠀⡔⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⣦
⠀⠀⠀⠀⠀⠀⠳⠤⠚⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠿⠇⠀⠀⠀⠀⠀⠾⢶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠀⠉⠑⢆⡠⠒⠀⠙⠢⡀
⣰⠚⠉⠉⠉⠑⠲⠄⡀⠀⠀⠀⡇⠀⠀⢀⣄⠀⠈⣆⡀⠀⠀⠀⢱⠀⠀⠀⠀⢀⡠⠤⠒⠚⠓⠒⢦
⡇⠀⠀⠀⠀⠀⠀⠀⠈⢑⡤⠞⢇⠀⠀⠈⠁⠀⣰⢅⠁⠀⠀⢀⠎⠛⠦⣤⠒⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡠⢤⡀
⠸⡀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠑⠂⠤⠔⠚⠁⠀⠁⠂⠈⠁⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠆⠀⡃
⠀⠱⡀⠀⠀⠀⢀⡞⡠⠴⠂⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⡀⠀⡣⠒⠀⠠
⠀⠀⠘⢆⠀⣠⡞⠀⠀⠘⡀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⡰⠃⠀⠸⡄⠀⠀⢠⠊⠀⠀⠀⠀⢼⠀⠀⢱⠤⠤⠚
⠀⠀⣠⡴⠛⡽⠀⠀⠀⠀⠘⡦⢀⠀⠑⠤⣀⡀⠀⠀⢀⡀⠴⠊⣀⠤⠊⠀⠀⠀⠀⢻⢲⠶⠧⣤⣤⣤⣤⣤⠀⠀⠀⡝⠢⠤⠴
⠀⣾⣁⠴⢲⡇⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢀⠖⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⣅⣀⠀⠀⠀⠀⠀⠀⠀⠔⠑⠓⠀⠐⠁
⠀⡜⣧⠀⢸⠁⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠑⢄
⢸⠀⠈⠷⢸⣄⣀⣀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠀⣀⣁⠠⠜⠻⠇⠀⠀⠀⠀⠀⣀⣀⣰⡇⠀⠀⠈⡆
⢀⣗⠞⠀⠸⡇⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⠉⠉⢸⠁⠀⣀⠜⠁
⢇⠸⡀⠀⠀⢳⡤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠛⠉⠉⠁
⠀⠑⠠⠤⠤⠚⢧⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⡬⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⢀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⢲⡖⠒⠒⣶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣶⠶⣒⡒⠲⠾⣇⠀⣠⠿⠒⢒⣒⢒⡶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡁⠀⠀⢙⣀⠀⢀⣠⠞⠈⠳⢄⡀⠀⣠⣋⠀⢀⡀⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣈⣩⡤⠴⠚⠉⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣤⡤⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⡄⣸⠇⢸⠏⢹⡏⠴⣷⣿⣿⣿⣷⣄⣄
⠀⠀⠀⠠⣼⡇⢻⡄⢹⡆⢸⡄⢲⣿⣿⣟⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠈⠻⠘⣿⢨⠷⠬⠷⠀⡟⠛⢿⡿⢿⣟⠈
⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠙⡆⠀⢁⠀⢋⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⢀⡀
⠀⠀⠀⠐⠒⠉⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠉⠀⠀⠀⠀⠀⡀⣸⣿⣿⣟⣿⣥⣤⠂⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣿⣿⣿⣿⣿⣿⣿⡏⢀⣼⡟⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠋⠻⡟⠋⢹⠙⠛⣿⣿⠿⠛⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣦⣄⠀⠀⠀⠀⠀⠈⠀⡝⠀⠀⠈⠀⠋⠀⢀⡴⢿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠏⠀⠰⣶⡶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⡿⠟⢻⣻⡇⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣖⣿⣿⡶⣾⣿⣧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣭⡛⠒⢛⣫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡴⢶⣌⠉⠛⢋⣡⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠞⠁⠳⣬⣙⣛⣩⠏⣟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⢬⡽⠆⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⣀⣀⣀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄⢠⣬⣭⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡷⠀⠀⢰⡿⢿⡟⣀⡀⢀⣀⢻⡿⠿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣢⡴⠓⣿⣿⣿⣿⢿⣿⣾⡻⠷⢶⣤⣣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠃⣀⣿⣿⣿⡏⠀⠙⣿⣿⡁⠘⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠀⠀⠀⠀⡖⢠⣿⣿⣿⣿⡟⠛⠀⠘⣿⣟⣿⣿⡄⢠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⡄⢄⠀⢀⡷⠁⠀⢼⣿⣿⣷⡤⠠⣾⣿⣿⡷⠀⠘⢿⡄⠀⡀⢆⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠨⣣⢎⡠⠀⢰⠸⡆⠻⢿⣷⡴⠿⠏⠰⠆⢀⡈⢄⡹⣖⡧⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠗⠉⠀⠘⠀⠀⢳⣤⣴⣄⣠⣆⣀⡄⠀⠈⠃⠀⠁⠈⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⢀⣀⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠁⠀⠀⠀⠀⠧⠀⡸⠛⠛⠛⠀⠉⠙⠃⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣥⣴⣤⣤⣄⣄⠀⢼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣭⣥⣤⣄⣀⣀⣀⠀⣀⣀⣀⣴⣿⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠇⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠃⠀⠻⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠀⢿⣶⣤⣤⣤⣤⣄⡀⢀⣀⣤⣴⣾⡇⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⠈⢻⣟⠛⠛⠛⠛⠃⠈⠉⠙⣻⡏⠀⠀⠀⠀⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⣴⣶⣴⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⠿⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⣦⣶⣶⣤⠤⠤⠤⠄


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

⠀⠀⣴⣶⣶⣤⢤⣤⣤⣤⣤⣤⣤⣤⣄⢀⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀
⠀⠀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠛⠉⠉⠁⠀⠈⠉⠿⠿⠟⠙⢋⠁⢉⠛⢻⢯⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⠤⠄⢀⣠⡠⡄⠠⠤⣤⣼⣷⣿⣿⣾⣾⣧⡐⢽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⣾⣿⣾⣿⡇⠈⣻⣷⠂⣀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⡷⣾⣿⣿⣿⣿⠇
⠀⠀⣿⣿⣿⣿⣿⣿⠛⠛⠂⢰⡿⢛⣿⣽⠃⣼⣿⣿⡁⠀⠀⠴⠾⢾⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⠇
⠀⠀⣿⣿⣿⣿⡿⠟⠂⠀⢀⣸⣿⣯⣿⣿⣾⢛⣷⡟⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⢹⣿⠿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⠄⠀⢸⣿⣿⣿⣿⣯⣼⣿⣿⣏⠿⠃⠀⠀⢀⣾⡏⣿⣿⣿⣿⣿⣯⣿⡟⢾⣿⣟⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣧⡀⠀⢹⣿⣿⣿⣿⣿⣿⢿⣿⠗⠀⢀⡄⢠⣾⣿⡇⣿⣿⣿⣿⣿⣿⡿⡁⢸⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣇⠀⢸⣏⣿⣿⣿⣿⣿⣹⠻⡄⢀⣼⠁⢸⣿⡏⢰⣿⢿⣿⣿⣿⡟⣳⡏⢸⣿⣿⡟⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡌⢈⡀⢰⣾⣿⠀⢸⡟⠁⣿⡏⣠⡷⢿⣾⣷⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣮⣉⡙⢿⣿⡇⠈⠁⠼⠟⢋⣁⣴⣶⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⡿⢹⠋⠙⠻⢿⣿⣧⣠⣴⣶⣿⡏⠀⠛⠀⠀⢿⣿⡧⢻⣿⣟⢻⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⡇⠀⠄⠤⠼⣿⣿⡿⠿⠿⠶⠀⠀⠀⠶⠸⠿⠿⠿⠿⠿⠾⠿⠃
⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⣴⠀⠠⠆⠠⠄⣀⣀⡀⠤⠔⠒⠊⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠁⠀⠀⠀⠀⠘⠀⠁⠀⠀⢀⠀⠀⢀⡀⠀⠀⡄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠠⠎⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡠⠄⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠤⣀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠐⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠚⠀⠈⠁⠂⠤⢀⣀⡄⠐⢁⣀⠠⠤⠒⠉⠀⠀⢀⣀⣀⣀⣀⣀⣀⣄⣀⠤⠤⠤⠤⠤⠤⠤⠤⠶⠄⠐⠒⣂⡸⠂
⠀⢀⠀⠀⠀⢀⣀⣀⣀⡀⢤⣤⠄⠁⠀⠀⠰⠒⠒⠒⠒⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⢀⣀⣀⠤⠤⠔⠒⠉⠁
⠀⠙⠪⠍⠩⠄⢀⣀⣀⣀⣈⣃⣄⣀⣠⣔⣁⣀⣀⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠒⠒⠒⠒⠚⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠦⠤⢤⣠⣶⡿⠷⢤⡷⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡋⣁⡀⠀⢀⠀⠀⢻⣦⣾⣧⡀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠶⣒⣒⠛⢛⣋⡉⠭⢨⣟⢲⣿⡟⣿⡞⠋⠙⣷⠼⣿⣿⠟⣿⣦⠟⠉⠛⠛⠒⠶⠶⠶⠴⠖⢤⣄
⠀⠀⠀⠀⢀⡤⠒⡡⠊⠉⢑⠀⠠⡔⠀⠈⠁⠀⠛⠉⢀⣤⡾⠉⠾⣿⡿⣿⣶⣾⣿⣿⣷⠈⠿⣿⣿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢀⣠⣴⠞⢉⡞⣰⣇⠈⣿⣧⣙⢿⡿⣿⣿⡋⠀⠀⠘⡇
⠀⠀⠀⠀⠁⠀⠄⠀⠀⣀⣀⣀⣀⣠⠴⠺⢿⡿⠃⢠⣾⠅⢿⣿⣶⡍⠛⠿⣿⡞⠲⡏⠀⠀⣾⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡹⢿⣿⣄⠈⠣⠴⠛⢿⣦⠘⣿⣿⣏⣀⣀⣼⣷⠚⠉⣀⠞⠙⠛⢭⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣶⣡⡀⠘⢿⣿⡀⠀⠀⢀⣼⡿⠷⢟⣿⢟⣟⠟⡼⣿⡷⠊⠁⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⢀⣠⠿⡭⠭⣯⡈⢿⣿⣷⡀⠙⢿⣦⠀⠘⣿⣀⡰⠋⢡⣾⣧⣾⣠⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠋⠁⠀⠀⣬⣿⣿⡌⠻⣿⣷⡀⠘⣿⣿⡶⠋⢘⢷⣴⠟⠁⠽⠁⣿⠻⣟⡒⢢⣄⡀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡿⣿⣄⠙⣿⣿⣦⣘⡿⠅⠀⠉⠛⢳⣄⡔⠰⣺⠇⠀⠈⠻⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣷⣌⢿⣷⣌⣿⣿⣿⠀⠀⠀⢀⡔⢣⠋⠀⢠⡏⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠃⠉⠉⠉⣠⡏⠀⠠⠖⠉⢀⠄⠁⠶⡞⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡇⠀⠀⠤⠐⠁⠊⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠈⣀⠖⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠦⣠⠾⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠑⠀⠉⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠃⠀⠀⠈⠐⠐⣠⠈⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠇⠀⠒⠰⠤⠒⠉⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠡⠆⠠⠌⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡺⣿⠕⠁⠀⠀⠀⠀⣀⣔⣤⣂⣠⢄⡤⢒⣈⡩⠤⣔⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⡿⣿⠿⣶⣄⠀⣀⣔⣮⣷⠿⢶⣀⡤⠝⠫⠤⠤⠜⠒⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣴⣿⢿⡿⣿⣿⣿⣿⣿⡿⣟⡿⠿⣿⣭⣶⣿⣿⣅⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠨⣼⣿⣿⣿⣿⡵⣫⣵⣶⣿⣿⣿⣿⣿⣿⣿⢻⣷⣶⢀
⠀⠀⠀⠀⠀⠀⢸⡟⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠛⠃⣼⣿⠟⢰⣷⡀
⠠⠄⠒⡨⠑⠓⢾⡇⣿⣾⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠐⡂⠀⠛⠃⢠⣿⣿⡿⣦
⠀⢠⠎⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣻⣿⣿⣿⣿⣿⣿⣾⣿⡶⢀⡀⠈⠙⠟⣹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠋⣁⣩⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⣀⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⣼⣿⠟⠛⠋⠙⠛⢻⣿⣿⣌⠛⠛⢻⣿⣿⣿⣿⣿⣿⣶⣿⡿⢛⣩⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣿⠀⠀⠀⠀⠀⣾⣿⠛⢿⣆⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⢻⣆⠀⠀⠀⣠⣿⠏⠀⠘⠛⠷⠄⣽⣿⡈⠻⣿⣿⣿⣿⣿⣷⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⠈⢿⡄⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠈⠙⠿⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠼⠋
⣿⣿⠛⢿⢻⠿⡿⢻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠠⠀⠄⣀⠲⣄⢠⣶⣧⡀⣠⢂⡀⠄⠠⠄⠈⠢⣰⣾⣶⣦⠎⠁⠠⠀⠀⠄⠰⢄⣰⢯⣷⣀⠴⠀⠄⠀⠠
⠀⠢⢈⠀⠀⠀⠘⣿⣭⣽⡟⠁⠀⠀⠀⡠⠂⠀⡠⢺⣿⣿⡇⡀⠀⠀⠠⠀⠀⠀⢐⣷⣿⣿⣗⠀⠀⠀⡠⠈
⠀⠀⠈⠂⠀⣠⢾⣿⠷⢾⣿⢦⡀⠀⠊⠀⠠⠖⠀⢸⡧⢾⠈⠀⢦⠀⠀⠀⠀⡰⢿⡿⠭⢽⣿⠧⡀⠈
⠀⠀⠀⠀⠀⠀⢸⢿⡛⣻⢿⠃⠀⠀⠀⠀⠀⠀⣀⣼⠓⢺⣦⡀⠀⠀⠀⠀⠈⠁⢣⢻⣑⣻⢳⠂⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠂⠟⠘⠏⠟⠁⢏⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡎⠀⠉⠁⠉
⠀⠀⠀⠀⢀⠀⠄⠒⠒⠢⠤⢤⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠱⡄⢠⣷⠦⡀
⠀⠀⢀⡼⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠟⠿⠋⠛⣛⡾⣶⣿⣿⣶⣶⣶⣦⣼⣶⡿⢿⣿⡷⡟
⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⢿⡛⠓⠶⠤⣄⡀⠀⠀⣀⣀⡀
⠀⠀⠀⡀⠴⠚⠛⠋⠉⠉⣉⣧⣷⣿⣟⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⠿⢭⠟⢀⠏⠀⠸⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉
⠀⢠⠈⠀⠀⠀⠀⠀⢠⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠋⣠⠏⠀⠀⠈
⠀⠈⠀⠀⠀⢀⡴⣾⣟⣻⣟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⠞⠁⠘⠃⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠋⠙⠙⠋⠛⠧⣶⣼⣿⣿⡿⠿⠟⢛⣿⠟⠑⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠎⣤⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⡀⣠⠞⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢄⡤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⣧⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⣬⣀⣀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠀⠀⠈⠑⠢⣍⢲⡀⠀⠀⠀⠀⠀⢀⡖⠀⣀⣤⠀⠀⢀⡔⣩⠴⠊⠁⠀⠀⠃⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣦⠀⣀⡄⢀⣿⣷⣾⣿⠀⠀⣴⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠗⢣⡈⠙⠿⢿⣿⣿⣷⣲⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡤⠶⠦⠁⠀⠀⠀⠉⠀⠴⠶⢤⠙⢦
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠈⠉⠒⠢⢄⡀⠀⠀⢰⡇⠠⢡⣶⡖⠁⡄⠀⠀⠀⠀⢠⣾⠖⠀⡁⡄⢸⡆⠀⠀⢀⡠⠔⠒⠉⠀⠀⠀⠀⠀⠉⠲⡄
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣸⡆⢃⠸⣿⣿⣾⡟⠀⠠⠀⠀⢻⣿⣷⣾⠇⡄⢠⣇⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠓⠬⠭⠓⠁⠀⠀⠀⠀⠈⠺⠭⠥⠚⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⢤⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠉⠓⠒⠒⠋⠁⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠐⠈⠉⠉⠉⠉⠒⠒⠀⠀⠤⢄⣈⣿⣷⣶⣶⣶⣶⣾⣿⣁⡠⠄⠀⠀⠐⠒⠈⠉⠉⠉⠁⠒⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⡸⢋⣟⠀⠉⠛⠛⠛⠛⠉⠁⢹⡝⢇⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⣀⣀⠠⠤⠐⠂⠁⠀⢀⡠⢞⣠⣿⣿⣷⣶⣶⣤⣤⣶⣶⣾⣿⣿⣤⡓⢄⡀⠀⠈⠐⠂⠠⠄⣀⣀⠀⢀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣿⣿⣾⣭⠔⢁⣾⠀⠉⠛⠛⠿⠿⠿⠿⠛⠛⠉⠀⣳⡈⠲⣬⣷⣿⣿⢽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠽⠛⣡⠊⢀⡎⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⠱⡀⠘⣌⠙⠯⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡜⠀⢯⠙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠋⣽⠀⢳⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣰⠇⠀⠸⣶⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠷⠀⠸⣇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠚⢹⠁⢠⠀⠙⣿⢿⣿⣷⣶⣶⣾⣿⡿⣿⠏⠀⠀⠀⡏⠑⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢠⠀⠀⠈⠳⢤⣉⣉⣉⣉⡤⠞⠁⠀⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⡂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠐⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠜⠋

⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠐⢚⠑⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠀⠹⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠦⣦⡄⡴⠀⠀⠐⠆⠀⡐⠀⠠⠴⢦⠉⠁
⠀⠀⠀⠀⠰⠡⠄⣤⡄⠀⠀⠀⢀⠀⠀⠀⠈⠓⠈⠑⠈⠒⠀⠀⠀⠀⢠⠂⠀⠀⠀⢫⣤⠑⠀⠈⠀⠀⠙
⠀⠀⣤⠀⠛⠋⠈⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⡀⢀⠀⠀⡄⠀⠀⠀⠀⠐⡂
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠂⠀⠐⠟⠀⢠⣤⠀⠀⠀⠘⠃⠘⠋⠀⠀⠀⠘⠃⠀⠙
⢀⠈⠉⠉⠀⢴⡄⠀⠀⠀⠀⠀⠄⠀⡜⠛⡀⣀⡀⡀⠀⠀⢀⢀⠀⣶⠈⠛⠃⠀⠴⠐⠂⠀⠀⠓
⠠⡿⣴⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠁⢁⠉⠉⠀⠁⠉⠈⠀⠀⠀⠀⠠⠆⠂⡇⣦⠀⠀⠀⢀
⠀⠀⠤⠀⠀⠁⠀⠈⠋⠃⠀⠀⠀⠁⠀⠐⠛⠚⠉⠛⠚⠘⠘⠚⠲⠀⠁⠀⠀⠐⠈⡇⠉⠀⠀⢀⡁
⠀⢱⣖⡖⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⣾⡗⠀⠀⠀⢠⠀⢠⠀⠀⢠⠀⠀⠰⠆
⠀⢸⣽⡗⠀⠀⠀⠀⠀⠀⠀⢀⣧⣄⠈⢰⣦⡈⠀⠀⠃⠀⠋⠀⠀⠸⠿⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖
⠀⠀⠉⠀⠈⢉⢉⡀⢀⠀⠀⣿⣿⣿⡇⠈⠛⠁⠀⠀⢀⠀⠀⠀⠈⠉⠈⠋⠀⢰⡀⠀⠀⠠⠐⠀⠀⢠⡄
⠀⡆⡀⠀⢠⠀⠀⣀⣀⠀⣤⡈⠛⠋⠀⠀⠀⠀⠐⠀⢿⠆⠀⠀⠀⠈⠁⠀⠀⠸⠃⠐⡐⠒⠀⠁⢂⡈⠉
⠀⠆⠃⠀⠊⠀⠀⠻⠟⠀⠈⠀⢿⠇⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠘⠟⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⢢⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⢡⠀⢀⠎⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣄⠀⠀⠂⠄⡀⡜⠀⠀⢀⣀⣀⣀⣀⣀⣛⠁⠀⠀⠀⠰⡤⠤⠒⠒⠒⠒⠋⠀⢰
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢹⠀⠀⠀⣀⡼⠗⠊⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠶⢬⡀⠀⠀⠀⠉⠑⠢⢄⣸⣀⠴⠋⠁⠀⢀⣠⣤⣤⣤⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠉⢉⠙⠋⠀⠀⠀⠀⠘⠛⠻⢧⠀⠀⠀⠀⠀⠙⢆⠀⠀⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠈⠢⡀⠀⡤⠒⠈⠐⠢⡀⠀⠀⠀⠀⠀⠀⠳⡌⠉⠑⠢⢤⣀⣠⠆
⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠸⡞⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⡀⠤⠛⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡇⣐⣦⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⡠⠂
⠀⠀⡰⠉⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠈⠐⠀⠀⠇⠙⠋⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⠰⢭⡀
⠀⠘⠉⠉⠳⡄⠀⠀⣾⠀⠀⠀⠀⡠⣶⠒⠀⠀⠀⠳⡀⠀⠀⠀⢀⠖⠘⠢⢄⣀⡠⠞⠁⠀⠀⠀⠀⢰⠃⠀⠀⠀⢸⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠈⠢⢀⣿⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⠁⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢸⡆⠀⠀⠀⠓⢤⣀⣀⣤
⠀⠀⠀⠀⠀⡠⠔⠊⣿⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡰⠁
⠀⠀⠀⡠⠊⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢛⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡀⠔⠋
⠀⠀⡼⠁⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠈⢆⡇⠈⠉⠓⡖⠒⠒⢶⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢷⡖⠈⠉⠁
⠀⢰⢁⠤⠤⢄⡀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⢣⣤⠀⠄⠒⢒⡎⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢻⡄
⠀⠘⠀⠀⠀⠀⠈⠑⠒⢢⠌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣖⣚⣁⡱⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠛⠦⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠡⠀⠀⠀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡴⠒⠂⢙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⡌⠀⠈⠙⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠖⠛⠁⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⡭⠻⡉⠉⠉⠉⠉⡩⠛⢦⡀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⣠⠖⠒⠉⠁⠀⠀⠱⡀⠀⠀⡼⠁⠀⠀⠉⠢⢤⣀⡀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣳⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⢤⡀⢰⠁⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⣀⡀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢂⣂⠴⠚⠁
⠀⠀⢧⠀⠈⢦⣧⢠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣹⣶⣿⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣯⠠⠈
⠀⠀⠀⠙⠲⢄⣳⡍⢇⣀⠀⠀⠀⠀⡤⣊⡵⢆⠜⣷⠶⣿⠦⡱⢮⣑⢤⠀⠀⠀⠀⢾⣿⢀⡍⡔⣇⣦⡀
⠀⠀⠀⢠⣾⣇⡮⠇⠘⠟⠀⠀⠀⠈⠉⠛⠚⠁⠀⢘⡿⠍⠀⠈⠓⠛⠉⠁⠀⠀⠀⠠⠂⠈⡀⠸⡙⢟⠇
⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠨⠁⠂⠄⠀⣰⣆⠀⠠⠐⠈⡅⠀⠀⠤⣄⣀⡀⣠⠶⣤⢀⣀⣀⠤⠤⡀⢀⠤⠤⠄⣀⢠⣶⣄⢀⠀⠤⠤⣄
⠀⠀⠹⠛⠓⢿⠷⠾⡿⠚⠛⠏⠀⠀⠀⠀⠀⠀⠀⣽⠒⣿⠀⠀⡀⠄⠀⠀⠈⠒⠶⡒⣨⢿⠿⢿⢭⡐⡲⠖⠊
⠀⠀⠀⠀⠀⠈⢽⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠢⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠉⠳⣄⢰⣄⡠⠞⠁⠀⠀⠀⠀⠀⢀⣀⣀⢠⢢⢀⣀⣀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⡱⡎⠀⣀⣀⣀
⠀⠀⠈⠓⠒⠂⣡⣈⠒⠒⠚⠁⠀⠀⠀⠀⣏⠀⢀⣨⠞⠉⠀⠀⡷⠀⠀⠀⠀⠘⠤⣄⢈⣁⡷⢞⣉⠀⣤⠔⠁
⠀⠀⠀⠀⠀⠘⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢫⡝⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣯⣵⠀⠉
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠐⠲⣕⣦⡔⣩⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠤⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⡔⠉
⠀⠀⠀⠓⢒⣾⣿⣿⣶⡒⠚⠀⠀⠀⠀⠀⠀⣀⠴⣞⣙⣳⠤⡀⠀⠀⠀⠀⠀⠀⠂⠉⣩⣤⡼⣦⣬⣉⠁⠂
⠀⠀⠀⠚⠁⠐⣟⡿⠀⠉⠂⠀⠀⠀⠀⢀⠜⠁⡠⢧⠀⡼⣄⠈⢢⡀⠀⠀⠀⠀⠲⠟⠋⠐⣛⡛⠈⠛⠷⠂
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⠀⠀⠉⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⣠⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣠⡀⠀⢀⣴⠿⡍⠈⠁⣽⢻⣦⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠸⣿⣿⣿⡀⢀⣿⣿⡿⢀⣠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⠿⠛⠛⠿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡞⠀⠀⠀⠀⠲⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⣾⣿⡁⠀⠀⠀⠀⣠⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣩⠟⠛⡛⣻⣿⣴⢤⣄⣶⢿⣟⠛⠛⢿⡛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⣠⠾⠃⠀⣶⢿⣟⣁⠀⠀⣈⣀⣼⣿⣿⣦⠈⠳⡀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠠⣿⣸⠿⢿⣿⣿⣿⣿⡿⠟⡏⢹⡆⠀⠙⢦⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠠⠀⠀⠀⣸⡟⢹⣶⣦⣤⣤⣤⣤⣶⣿⣷⠘⣿⡀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢀⡾⠛⠁⠸⣬⣉⣉⣉⣉⣩⣭⣭⡏⢀⠘⠿⣆⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⡼⠀⠀⠀⢠⡼⠋⠀⠀⠀⠀⢫⣽⣛⣛⣛⣫⣭⣽⠁⠀⠓⠀⠈⠳⣦⡀⠀⠁
⠀⠀⠀⠀⢰⣇⣀⣀⠦⠀⠀⠀⠀⠀⠀⠀⠈⢻⡛⠛⣛⣛⣻⠏⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⢔⡭⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠛⠈⠁⢀⠀⣴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠻⣽⣿⣿⣷⣦⣠⣄⣹⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠚⢹⣿⣶⣷⣶⣆⣙⢧⣻⢿⣿⡄⠀⠙⣿⡖⠂⠷⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⡨⢟⣛⢻⣿⣿⣿⣷⣿⣿⣿⣿⡄⠀⣾⡇⠀⠀⠈⠫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⢠⡟⢉⣷⢻⣿⣿⣿⣿⣿⣿⣿⣇⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣿⣷⣾⣿⣜⠿⢿⣿⣿⣿⡟⣻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠿⠛⠛⣻⣿⣿⢿⣇⠀⠙⣿⡟⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣠⣿⣽⣿⣿⣿⠘⠙⠆⠀⠈⣿⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣯⣝⣽⣿⣿⣟⠈⡄⠘⡄⠘⢿⠟⢷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠻⢿⣿⣿⣿⠀⢠⠀⠀⢀⠰⢅⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡾⣿⣿⣿⣧⠀⠐⡄⠈⠂⠂⠡⡀⢄⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠐⠰⣟⣠⣼⣿⣿⢿⣷⡀⠀⠀⠈⡄⠂⠈⢀⠁⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⢟⣋⣥⣶⣿⣿⣿⣄⠀⠀⠘⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠈⠀⢰⠂⠙⠈⠀⠀⠀⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⢀⠀⠀⠀⠈⠀⢀⠀⠂
⠀⠀⠀⠀⠀⡎⠁⠀⠀⢀⢀⢼⣟⣿⡿⠿⠿⠿⠋⣿⣿⣿⣿⣿⣿⣿⡆⢱⣄⠀⠀⠀⠀⠢⡀⠈⠀⣐⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠀⠀⣿⡘⣧⣠⣤⠶⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⢿⣇⠐⠄⠀⠀⠈⠀⡀⠀⠠
⠀⠀⠀⠀⠘⠀⠀⠀⠊⠁⠀⠀⢹⣧⠈⣿⣅⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⡆⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠈⣿⠂⠈⠻⣿⣿⣯⣙⣿⣿⣿⣿⣿⣿⠇⠀⠸⣷
⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠈⠙⠀⠙⠛⠋⠙⠋⠁⠁⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⡊⢉⣩⠥⠀⠒⠋⠉⠉⠉⠉⠉⡉⠉⠁⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡠⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡴⠊⠀⢀⠤⠐⠀⠀⠀⠀⠀⠀⠐⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠕⠁⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⡠⠴⣇⣄⣀⣀⣀⣀⡀⠀⠄⠀⠠⠤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣀⣴⡿⢻⠏⢠⣴⠏⠈⣻⠟⠛⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡏⠟⠛⣿⢿⣻⣿⣯⠌⢸⣏⠀⣰⣿⣆⣤⣾⢏⢀⣽⣯⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣄⣰⡟⠀⠉⢳⣾⡶⠒⠛⠛⢻⡟⠛⠿⣧⣷⣿⣏⠈⣿⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⡠⢫⠞⠀⠀⠀⠀⢸⢹⡀⠀⠘⢽⢿⠯⣼⣟⣿⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡃⠀⠀⠀⠀⠀⡆⠀⠃⠀⢐⠁⢀⠟⠒⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠚⢉⡳⣀⢠⣴⣉⣀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⡻⠿⠿⠛⠛⠛⠻⠿⠋⣀⣜⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⡿⠀⠀⠀⠀⠀⠈⢈⣻⡟⠛⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠦⣤⣿⣁⠀⠀⠀⠀⠀⠀⠀⣣⣿⣾⣯⡿⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢋⢝⣿⣻⢄⣀⠴⠐⠦⢤⢘⣿⣷⣷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣥⣿⣿⢱⠾⢾⣗⣲⣾⣽⢾⡌⢻⣮⠳⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠅⢾⠌⠘⢃⢶⢿⢿⣯⣭⣽⣿⣷⡾⡘⡷⣤⠀⠙⢟⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢈⡞⢠⣿⣯⢶⣬⣷⡄⣮⡡⢶⣧⠇⣧⠹⣜⢳⣄⠀⠱⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠗⣼⡰⡿⠁⡿⢿⣶⣥⣀⣀⣠⣬⣷⡿⢻⠀⠙⣧⡡⡑⢶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⢜⣿⡜⠁⠀⣿⣮⣭⠙⠛⠛⢛⣫⣭⣶⣿⠀⠀⠈⢷⣴⣄⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣴⠗⣩⠋⠀⠀⠀⠸⣿⣿⣿⣶⣶⣿⡿⢟⣯⠇⠀⠀⠀⠀⠙⢯⡂⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡠⠞⠁⠀⠀⠀⠀⠀⠙⢿⡷⣶⣤⣾⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂
⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⣀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⡀
⠀⠀⠀⠀⠔⢀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠀⠀⠀⠁⠁⠈⠋⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠁⠀⢀⠀⠊⠠⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⢀⠐⠁⠁⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠑⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠆⠀⠀⠀⠀⠈⠁⠀⠀⢀⠠⡒⠒⠄⡀⠀⠀⠄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣀⠀⠀⠀⠀⠀⠀⠀⠢⠉⠉⠐⡀⠀⠀⠀⠸⡀⠁⠀⠀⠔⠀⠀⠀⠀⠀⠀⡐⠄⠃⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠐⠠⠀⠌⠀⠀⠀⠀⠀⠀⠀⣈⠅⠀⠀⠂⠀⡀⠀⢀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⡠⠖⠂⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠄⠀⠀⠀⢀⠀⠁⠀⠀⠐⡇⠆⠲⠂⠀⠀⢀⠤⢠⢀⠀⠀⠐⢂⠕⡀⠡⠐⠉⠀⠄⠀⡀⠀⠈⠱
⠀⠀⠀⠀⠠⠠⠄⠂⠁⠀⠀⠠⠈⠂⠁⠀⠀⠀⡃⠈⢓⠀⠥⡄⠀⡆⠀⠂⢌⠀⠀⢀⣘⠕⠈⠈⠀⠀⠱⠀⠀⠀⣀⡈⠁⠀⡀⠈⡁⠐⠗⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⡄⠀⠀⠐⠀⠀⠌⠀⠸⠀⠀⣓⠉⠀⠀⠀⠀⠇⠈⠒⡆⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⡀⠂⠀⠀⠀⠠⠐⠈⠀⠠⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⡠⠀⠀⢀⠀⠈⠈⠀⣠⠀⠀⠪⠄⠀⡀⣀⠀⢀⡶⠠⢀⠀⠈⠁⠀⢀⠈⠀⠀⠀⠀⣄⣀⠀⠐⠀⠀⠄⡀⠇⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠈⠈⠈⠀⠠⠀⠀⠀⠀⠀⠂⠤⠈⠄⠹⠄⣄⠀⠉⠀⠀⡉⡚⠚⠋⠻⠶⠚⠏⣤⣿⡓⣩⣦⡀⠘⠓⢺⡟
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠈⠀⢀⡠⠃⠀⠀⠀⠁⠃⠀⠚⠀⠀⠁⠀⠃⢀⠀⠲⠆⠀⢤⣙⠰⠶⠤⠔⠂⠀⠘⠋⠀⠻⠿⠆⠀⣴⡟⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⣠⢤⠀⡀⠀⠀⠀⠀⠀⠰⡠⣀⢀⣄⢄⡀⠀⠀⠀⢠⣐⣶⣤⠀⠀⠒⠈⠀⠉⠀⠐⠛⠛⠛⠛⠓⠛⠠⠰⠶
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠛⠙⠛⠀⠀⡈⠁⠀⠀⠑⠻⠋⠒⠀⠀⠀⢀⠁⠉⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⢠⡀⠆⢰⠂⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠄⠀⠀⠀⠀⠀⠀⢄⣏⠀⠀⠀⠒⠲⠎⠜⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⢀⢧⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠈⢍⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢴⠆⠀⠄⠀⣬⡄⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠑⢤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⢋⡉⠀⣸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢉⣀⠀⠀⠈⠻⠦⠂⠐⠢⠀⠀⢀⢀⡎⣸⠃⣀⠤⠊⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⠟⠉⠙⠒⠤⠤⠤⠀⠠⠀⠀⠀⡚⠉⠀⠀⠉⠁⠀⠀⢢⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⢤⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢸⡤⣠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡔⢶⠁⠀⠀⠀⠀⠀⢀⣤⡶⢋⠀⠀⠶⣿⣿⣿⠀⢸⠀⠸⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⣀⠠⠔⠊⠀⠀⠋⠀⠀⠀⠱⠀⣾⣿⣿⠏⠀⣴⣾⡿⠟⠁⠀⣸⡀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⡉⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⢀⠠⠞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡠⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠔⢺⠃⠀⠀⢜⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠀⠀⠀⢀⠌⣽⠉⠉⠒⠒⠒⠒⠖⠒⠒⠒⠉⠉⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣀⠊⠀⣰⡀⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠞⠉⠀⢀⠞⠀⠀⠀⢀⡞⠀⠀⠈⢧⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⢢⡤⢤⡤⠎⠀⠀⠀⠀⠈⠦⢤⡤⠶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣇⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⣉⠉⠉⢉⠉⠉⠁⠈⠓⠒⠢⢤⣀⡴⠒⠒⠋⠀⠈⠉⠁⢀⠀⠉⣉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡴⠋⢀⠴⠊⣀⡤⠖⠚⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠒⠦⣄⠙⠢⣄⠙⢦⡈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠧⠲⢧⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠼⠓⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡼⠁
⠀⠀⢀⠴⠒⠛⠛⠒⠲⠤⣄⡀⠀⠀⠀⠀⢀⡼⠟⠉⠉⠻⢧⡀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠛⠒⠦⡀
⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠦⣄⣠⠏⠀⠀⠀⠀⠀⠀⠹⣄⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣤⣤⣤⣤⣤⣤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⠟⠛⠛⠻⣯⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠙⠒⠦⠤⠤⠤⣴⡶⠶⠛⠋⠉⢠⣿⣶⣶⣶⣶⣿⡄⠉⠙⠛⠶⢶⣤⠤⠤⠤⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⣠⠋⠉⠉⠉⠉⠉⠉⠛⣄⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢀⣴⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣦⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠖⠋⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠃⠙⠲⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⣰⠁⠀⢈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠈⠓⢴⠁⠀⠀⠋⠈⡇⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⢱⠀⠀⠀⠈⡃⣠⠶⣀⡜⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠸⠇⠀⠀⠀⠒⠧⣄⢸⠀⠀⠀⠀⠀⠀⠐⠁⢀⡴⠁⠀⡰⠁⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠴⢆⣀⣔⢦⠀⠀⠈⢦⡀⠀⠀⠀⠀⠌⢀⡴⠋⠀⠀⠀⠠⠄⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡌⡅⡀⠈⡀⠨⠈⠀⠀⠀⠈⡇⠀⠀⢀⢊⡼⠁⠐⠒⠀⣈⡡⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠑⠑⠣⠥⠜⠀⠀⢰⠆⢠⠙⡄⠠⠂⡾⠐⠈⠉⠉⡩⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠒⠂⠀⠀⢀⠔⡩⣠⡃⠀⠈⠀⠊⠀⣀⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⣤⣴⣶⣿⣿⠟⣿⣿⣯⠙⢿⣿⣿⣷⣆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣹⣿⣿⣿⠟⠁⠄⢸⣿⣿⡆⠀⠹⣿⣿⣿⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠉⠉⠀⠀⠀⢸⠀⣿⣿⣿⠀⠀⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⠀⠀⠀⢀⣠⣿⠀⣿⣿⣿⣄⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠑⠊⣀⡱⠀⠀⡠⠘⢿⡤⣟⠛⡓⢌⠑⠠⣖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⡄⠙⡄⠀⠀⠑⠲⠀⠀⣘⡠⠼⣦⠿⠭⡊⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠁⠀⠰⣉⣀⡠⠗⢹⡇⠀⢃⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠓⠒⠒⠊⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⡀⠀⠀⣼⠃
⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⠶⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠛⠉⠉⠛⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠶⠶⢦⣤⡀
⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣄⡀⠀⠀⠀⠀⣠⠟⠁⠉⠀⠀⠀⠀⠉⠈⠻⣄⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⣿⡟⠉⠉⠉⠉⢻⣿⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠾⠟⠛⠉⠁⣴⣯⣤⣤⣤⣤⣤⣤⣽⣦⠈⠉⠛⠻⠷⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣩⡿⠛⠉⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⠙⠛⢿⣍⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢀⣾⣧⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣼⣷⡀⠀⠀⠀⠀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⣀⣴⠛⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠛⢦⣀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠛⠋⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠙⠛⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣶⣶⣶⣿⣿⣿⣷⣷⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠛⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠈⢳⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⡀⢴⠋
⠀⠀⠀⠀⠀⠈⠙⢶⡀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⣠⣴⣿⣿⡷⢤⡀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣰⡗
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠠⣼⡅⠀⢰⣿⣿⡈⣿⣿⣁⣄⣩⣷⠀⢀⣷⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠈⢿⡀⠘⠿⠿⣷⣿⣿⣿⣿⡿⠟⢠⡾⠁⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠈⢻⣾⣿⣿⣿⣿⣿⣯⢸⡿⣿⠋⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⢹⣷⣿⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣀⢀⣠⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⣻⣿⣿⣿⣿⣻⣿⣿⣧⣿⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠇⠘⣿⣿⣿⣿⣿⣿⣿⣾⢿⢨⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡐⠌⠠⡔⠘⢿⣽⣯⣽⣯⡽⠃⢸⡜⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⠃⠀⠀⢸⣄⣘⣿⣿⣿⢿⡀⢀⡀⠀⢰⡈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠅⢟⠀⠀⢰⣿⣤⣎⠂⠀⡆⢨⡅⠀⢀⠀⠀⡞⡂⠝⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⠈⠄⠈⠛⠀⠘⣇⠛⢻⡴⠿⠠⡞⠘⢃⢠⠀⠀⠁⢀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⡄⠐⠀⠀⠀⠀⣻⣷⢿⡇⣤⠠⠀⠂⢀⡈⠀⠀⠀⠀⠂⠓⠈⣇
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⡀⣠⠀⠘⠀⠀⢼⣿⣷⣼⣧⣿⡁⠀⢀⠈⠁⠀⠀⠀⠀⠀⡄⠀⢻⡀
⠀⠀⠀⠀⠀⢀⣴⠃⠈⠈⠡⠁⠀⡀⠀⠀⡈⣿⡿⣹⣏⣿⣷⠣⠊⠲⠀⠀⠀⠀⠀⠀⠁⡘⡀⠙⢷⣄⡀
⠀⢀⣀⡴⠚⠃⠀⠀⠰⠁⠀⠀⠀⢈⡤⠔⢫⠹⣿⣵⣿⣿⣿⣷⣄⡄⠀⠤⠀⠀⠀⠀⠀⡁⠁⠀⠀⠙⠛⣶⣄⣀
⠿⠋⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠴⢿⣿⣿⣿⣟⣛⣿⠃⠀⠀⠀⢀⠀⠀⢠⠁⣷⠀⠀⠀⠀⠈⠈⠙⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⠀⠀⢀⠨⠀⠠⠃⠀⠈⠻⣿⣿⣿⡿⠃⠀⠀⢄⠀⠈⠀⠀⠈⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⢀⡁⠌⡆⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⠂⡈⠂⠀⢀⢈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣈⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣿⣿⡟⠋⠁⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⠉⠉⠉⠀⠀⠠⡔⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⡶⢦⡄⠀⡆⠁⠀⠈⣲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⢄⠀⢬⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠁⠀⠀⢀⣤⣤⠀⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡅⠀⠀⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⠉⠛⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⡤⢤⣿⣿⣿⣷⣴⡀⠀⠀⠀⢠⣿⣿⣿⣿⡟⣥⠂⠀⣿⡟⠠⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⡤⢶⠆⠉⢹⣿⣿⣿⣿⣿⡅⠽⠿⣿⣿⣟⣿⠋⢠⠀⣰⣿⣿⣿⣿⣿⣇⣁⠀⠀⠈⠉⢦⠀⢣
⠀⠀⠀⠀⠀⣴⡋⠈⠉⠈⠉⠛⠋⢛⡁⠄⢼⡆⠀⠉⠉⢻⣿⣿⣿⣷⣾⣿⣿⡞⠉⠙⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣄⠀⠀⠈⢃⢸
⠀⠀⠀⣠⣾⡦⠀⠈⡝⠁⠀⡤⠀⣀⣀⠠⣿⣿⣤⠶⠺⣻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠩⡿⠋⠻⣏⢻⠟⠙⠉⢿⣿⡀⠀⠀⠸⣿⡇
⠀⠀⣴⣿⣿⣤⡴⠚⠓⠈⣉⡭⣽⣥⠴⢾⣿⣿⣿⣦⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣤⣾⣧⠐⢳⠘⢯⠇⠀⠀⠈⠻⠇⠀⠀⠀⣿⠇
⠀⣼⣿⣿⣟⣿⠷⣶⡲⠽⠞⠋⠁⠀⠀⢸⣿⣿⣿⡋⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⡸⠟⠉⠁⠀⠱⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡄
⠀⠉⠙⠛⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣿⣿⠃⠀⣿⣿⡇⠀⠀⠀⠀⠀⢃⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⡄⠀⠘⠻⣧⠀⠀⠀⠀⠀⠈⠳⡤⠄⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣄⠀⠈⣿⣿⣷⠀⠀⠂⠈⠃⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣆⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆⠀⡈⢿⣷⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠁⠀⠙⠛⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠘⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠖⠒⠈⠉⠉⠉⢉⠉⠁⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠤⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⠀⠀⠀⠀⠀⠀⣀⢀⣅⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠒⠚⠉⠉⠉⠀⠀⠀⠀⢀⣀⡀⠀⡤⠤⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢈⡟⠉⠉⠁⠀⠀⠀⡄⠁⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⡀⢠⠏⢀⡌⠄⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠠⠀⠠⡧⠀⡄⢾⠀⠀⠀⠀⠀⣇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠙⠃⠀⠉⣤⣄⣀⠈
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⡊⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡴⠤⠀⡸
⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡜⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⠱⠖⠉
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢹⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠆⢀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠄⠢⠄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡿⠛⠉
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠀⠈⠡⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠸⠀⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣠⡾⢻⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸
⠀⣴⠟⠉⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⢿⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠈⢻⡄⢀⣸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸
⠀⣠⡴⠀⠈⢹⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸
⡎⠱⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡆
⠈⠀⢁⣀⠤⢼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⠠⠔⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠳⠄⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠷⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣶⠒⠂⢶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣿⡀⢀⣸⡧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠒⠀⠀⡇⢘⠀⠀⠲⡶⠶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⢋⡀⠀⢀⡬⠃⠸⢤⡀⠀⢀⡉⠀⠀⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠈⠑⠲⠤⢤⣤⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠳⢄⣀⡤⣤⣄⣠⠊⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠾⡤⠤⠤⠤⡼⠇⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣦⣤⣄⣀⣤⣤⣶⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢤⣄⣀⣀⣀⣀⣠⠤⠖⠚⡻⣿⣯⡟⠻⣿⠿⢻⢿⡿⠟⣿⣿⢿⠓⠒⠤⣄⣀⣀⣀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣴⣿⣿⢿⠻⡿⣿⣿⣾⣿⡿⠿⡹⣿⣷⣧⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡩⢱⢿⠈⣷⣄⢧⣼⣿⣤⢇⣴⠃⡟⡏⢏⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢠⡁⡎⡆⢇⠘⣿⡆⠻⠹⠅⣿⠇⠰⠠⢸⡈⡆⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⣼⡠⠃⡟⠁⢁⠘⢆⢳⣶⣶⣶⣶⡞⣠⠃⠰⠀⢻⠘⢆⣻⡙⢷⣤⣀
⠀⠀⣠⡤⠶⠚⠋⠉⠀⠀⡜⡋⠀⣰⣧⢼⠸⡀⠀⠙⣦⣤⣤⣤⠟⠁⠀⡆⡄⢸⣧⠀⢩⢣⠀⠀⠉⠙⠓⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣸⠁⡔⢹⢻⢸⠀⠱⡀⠀⣿⠛⠛⠿⠀⢀⡜⠀⠇⡌⣏⢢⠀⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣿⠜⠀⣼⠘⠈⡇⠀⠙⢦⣿⣿⣿⣿⡴⠋⠀⢰⠃⡇⡹⠀⠱⣿⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⡉⠀⠰⢸⠀⡇⠘⡄⠀⡰⢸⣭⣭⡇⢣⠀⢀⠇⢠⠁⡇⢇⠀⢹⢇⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠆⠀⡇⠈⡄⠸⠀⠘⣴⠁⢸⡟⢻⡇⠀⣧⠋⠀⡞⠀⠇⠘⡀⢸⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢀⢠⠀⠀⢣⠀⠣⡴⠉⢢⢠⣿⣿⡇⡰⠉⢧⡜⠀⡸⠀⠀⡇⠸⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠅⢸⠸⠀⠀⠈⣦⠾⡀⠀⠀⢹⣶⣶⣏⠀⠀⢀⠷⣤⠃⠀⠀⡇⡆⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠙⢦⠤⠒⢇⠀⠘⣄⢠⠛⠿⠿⠟⡄⢠⠊⠀⣨⠓⠠⠴⠋⡅⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢸⠀⠸⡄⠀⠈⠣⡀⣨⠃⠀⠀⠀⠀⠈⢇⢀⠔⠁⠀⢀⠇⠀⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡀⠀⠱⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢀⠎⠀⠀⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢣⠀⠀⠘⢤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⠊⠀⠀⡸⠁⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠀⠈⢢⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡰⠁⠀⡎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣃⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣸⣠⠎


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣭⣥⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣽⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠆⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣶⣶⣶⣶⣶⣶⣦⣶⣶
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣧⣿⣿⣅⠹⠏⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠉⢿⣦⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠀⠘⠛⢿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⡀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣿⣏⡶⣿⣿⠍⣳⣷⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⠀⠈⠛⢿⣿⠟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣯⣙⠿⠃⡨⠛⢦⠹⢿⣾⡘⠻⡧⠀⠀⠀⠀⠀⠀⢀⢈⡛⢿⡟⠀⢀⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠙⠧⠈⢰⣆⣀⣠⣿⣯⣿⣿⠶⠀⠀⠀⣴⡆⢀⣿⣿⠁⡿⠛⠀⣨⣿⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠑⠪⣀⠀⠀⠚⣷⣞⣿⣥⡘⠟⠻⣿⣧⠀⠀⠀⣉⣁⣾⣟⠃⠨⠀⠢⠈⠉⣭⢾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⠀⠹⠿⣿⣟⠁⠀⠀⠈⠉⠀⠀⢘⣿⣿⣶⡿⠧⣤⣼⢷⣔⠐⠃⠱⠄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣈⠛⢁⣴⣞⢻⠇⠀⠀⠀⠀⠀⠀⠀⢼⡿⠋⠉⢡⣶⣿⣽⣿⠍⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣙⢿⡿⠿⢿⣫⣿⡿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠃⢠⣤⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠙⡙⠐⠀⢘⡏⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣯⣼⣷⣦⣿⣿⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⣿⢷⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠛⠿⠻⣿⣧⣿⠛⠛⣋⣥
⠀⠀⠀⠀⠀⠀⠀⠲⠔⠱⠟⠿⣿⡟⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣿⣦⡀⢠⣍⣈⣧⡀⣁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠦⠀⣄⣰⣦⣈⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣾⣿⣿⣻⣿⣟⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⡟⠙⠃⠈⠋
⠀⠀⠀⠀⠀⠀⠀⢀⠙⣛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣭⣿⣷⣴⣄⣘
⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣯⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⣿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⡿⠿⠿⠿⠿⢿⠿⡧⠁⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠁⢠⡀⢦⡠⠐⠠⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠀⠀⠀⠀⠸⠃⢉⢁⡡⠄⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢶⣾⠀⢹⣿⣧⡀⠀⠐⡆⠀⠀⠸⡀⠀⣰⠄⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠑⢂⡍⠈⢿⣦⡀⠻⠿⠿⠦⣠⠟⡀⠀⢰⡇⢺⠃⠘⣿⣷⣈⢳⢄⠄⠠⠤⠄
⠀⠀⠀⠀⠀⠀⡠⠊⠈⠀⠀⠘⣷⣄⣻⠿⢮⠻⣖⣰⠂⢁⣈⢉⡰⢁⣨⠟⠀⠸⣿⡼⠁⢈⠒⢣⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠸⣦⠀⠙⠻⢷⡶⣟⢭⡶⠈⠋⡠⠊⠀⠀⠐⣾⣧⡤⠘⠡⡤⠄⠂⠺⡇
⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢲⡈⠛⠿⣶⠞⠙⠁⢡⡀⠀⠀⡷⠀⠀⠀⠀⠀⢿⣷⣦⡠⠃⠀⠀⠐⢳
⠀⠀⠀⡌⠀⠀⠀⠀⣀⣰⣶⡉⡓⠂⠀⠀⠀⠀⠀⠣⠀⣰⣿⡀⠀⠀⠀⢀⠤⠘⢻⣿⣀⡀⠀⠀⠸
⠀⠀⠠⠁⠀⠀⣤⠴⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⣀⡶⠞⠁⠘⣇⢀⢀⠀⠀⠀⠀⠈⠘⠿⣄⡀⠀⠰
⠀⠀⠂⣀⡶⠛⠃⠀⠀⠀⠀⣠⣀⠀⠆⠀⢚⠍⠁⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠐⡄
⢀⣾⠞⠑⠂⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠈⠉⠚⠒⠤⠤⣠⣀⣀⠙⠦⣰⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠻⢷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡠⠴⠶⠮⣁⣤⠽⠦⣄
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢦⣤⡴⠒⢻⡄⠀⠀⠀⠳⡀
⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠁⠒⠒⠂⠄⠠⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣾⣿⡇⠀⢸⠇⠀⠀⠀⠈⡇
⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠙⠛⠒⠦⡀⠐⠀⠀⢀⡹⣟⠉⠀⣠⡞⠀⠀⠀⠀⠀⡷
⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡀⠀⠈⣢⠶⠚⠁⠀⠸⡉⠋⠕⠀⠀⠀⠈⠐⢲⠃⠀⠠⡀⢠
⠻⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣡⠞⠁⠀⠄⠀⠀⢀⣹⣆⡐⠆⡀⠀⣀⡴⠋⠀⠀⠠⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠈⠉⠉⠉⠁⠈⠁⠉⠉⠉⠉⠉⠉⠑⠒⠠⠖⠊⣡⠏⠀⠀⠀⠀⠀⣠⡞⠁⠘⠟⠳⡚⠛⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠒⠒⠈⣡⠃⠀⠀⠀⢐⡤⠞⠁⠁⠀⠀⠀⠀⠈⠀⢄⠀⠀⠐⠆⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠐⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⢀⠞⢧⡀⠀⣠⠶⠏⠝⠃⠀⠀⠀⠀⠀⠠⠆⠈⠁⠒⢁⠀⠀⠀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣈⠔⠁⠀⢀⡟⠉⠁⠀⠀⡇⠇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠁⡠⠚⢇⠀⠀⡰⠋⠀⠀⠀⠀⠀⠁⠀⠈⠄⠀⠀⠀⠠⡀⡀⢀⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠄⠄⠐⠊⠁⠀⢀⡴⠊⠀⠀⢀⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠣⠑⠔⠆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠘⢦⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠛⠉⠁⠀⠀⢤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡇⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢂⠤⢀⣀⡀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠙⣦⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢆⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢙⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⢄⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠢⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢀⣠⠤⠤⡄
⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⢤⣀⣠⠟
⣞⠉⠉⠓⣶⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠙⠒⠒⠋⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠉⠀⠀⠈⠉⠙⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⣟⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠶⠛⠋⠉⠉⠀⠀⠀⠀⠉⠉⠙⠛⠶⣦⣄⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠉⠙⠶⣄⡀
⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠁⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠛⠛⠛⣿⣶⣶⣦⣤⣀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡏⠉⠛⠷⣴⣞⠉
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣷⣦⡀
⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠉⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠶⠶⠶⣶⠟⠛⠋⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⡇⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⡇⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠃⠀⠀⠀⢀⣿⣿⣿⣿⣿⠁⠀⠀⠀⣸⣿⣿⣿⠃⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⣼⣿⣿⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⢀⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⣼⣿⣿⣿⠟⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠁⠀⠀⠀⣸⣿⣿⣿⠃⠀⠀⢀⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣿⣿⣿⠏⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠃⠀⠀⠀⢰⣿⣿⣿⠏⠀⠀⣠⡾⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣧⣄⡀⠀⠀⢠⣿⣿⣿⣿⡿⠃⠀⠀⠀⢠⣿⣿⣿⠏⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣿⣿⣿⣿⣟⣁⣀⣀⣀⣰⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⠀⠀⠠⠠⠄⠠⠀⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢡⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠋⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡀⠄⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠁⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⡀⠄⠂⣠⠖⠉⠀⠒⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠄⠂⠁⢀⡴⠋⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠲⠶⠶⠶⠖⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⡴⠋⠀⠀⠀⠀⣀⡠⠤⠒⠊⠉⠉⠉⠑⠲⢄⡀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢤⡀⢹⡀⡀⠐⣀⣀⠐⠦⡀⠀⠀⠀⠀⠀⢀⠎⠀⣠⠖⠁⠀⣀⠤⠖⠊⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡁
⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⠀⠉⢳⢟⡴⢛⣿⣿⢹⡄⠈⢣⣀⡤⢠⡴⢅⡴⢊⡡⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⣤⡠⠐⠈⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢢⣾⣵⣿⡿⢃⠞⠀⡠⠊⠋⠀⢂⡴⠟⠈⠁⠀⠐⠂⠠⣀⠀⠀⠤⠤⠒⠖⠀⠀⠀⠀⠀⣀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⣿⣿⣏⡴⠋⠀⡔⠠⡘⠀⢀⠯⣀⣀⣀⣀⢤⠤⢤⢤⠤⠤⠤⠴⠶⠶⠒⠒⠚⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠭⠚⠁⠀⣀⣴⢳⡙⢀⠠⠂⠀⠀⠐⠌⠀⠠⡀⠀⠑⡗⠐⢒⣖⡀⠤⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣠⠴⠊⠁⢸⠀⢀⠢⠑⢀⠨⢐⠀⢀⠂⠌⠂⢀⡤⣇⠀⢸⣿⣿⠃⠀⠀⣬⣐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⠁⠀⠀⠀⠀⢦⠄⡟⡆⣐⠀⠀⢀⡐⠈⣡⣶⠽⡇⠈⢧⢻⣿⠃⠀⠀⢠⣿⣿⣷⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⢷⠃⣇⠀⠀⠈⠘⡸⡄⠉⣡⠖⢳⠀⠸⡟⠁⠀⠀⢠⣿⣿⣿⠏⠀⢀⣬⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡖⠚⡏⠉⠉⠒⡲⢂⣧⣹⠘⣄⣴⣾⡄⠀⣇⠀⠀⣠⣿⣿⣿⠋⠀⠀⣼⣿⡗⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢹⡱⣇⠀⠀⣼⠕⠋⣠⠏⠀⢻⠋⠁⢧⡀⣸⢀⣾⣿⣿⠟⠁⠀⠀⣰⣿⡟⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣣⠙⣆⢸⣥⣤⡞⠣⡀⠀⠈⢧⠀⢠⠋⢇⣿⣿⠟⠁⠀⠀⣠⣾⡿⠋⠀⢀⣾⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⡎⢆⠐⣜⢦⠙⡄⠀⠸⠻⣿⡆⢸⠉⠀⠀⢀⣴⣾⡿⠋⠁⠀⣠⣿⡿⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⢧⡘⣆⠙⢃⠀⠀⡇⣸⢣⠀⣇⢤⣾⡿⠛⠉⠀⢀⣠⣾⠿⠋⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⢹⡄⠀⠘⣿⡈⢰⡳⡐⠤⢄⣀⢤⡶⠿⠋⠁⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢿⡄⠀⠈⢷⣄⠙⢜⢆⠀⠈⠑⠒⠢⠤⠤⠼⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠚⠁⠀⠀⠈⠻⢦⡌⠳⣕⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡠⠤⡀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠄⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡀⠐⠚⠁
⠀⠀⢧⡀⠘⢾⢆⢸⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⡼⠯⣿⠤⡤⣀⠀⠀⠀⠀⠀⠀⣠⡠⢧⣮
⠀⠀⠀⠉⢒⡨⣷⠜⢤⡄⠀⠀⠀⣴⣪⠴⡣⠞⢽⣛⣿⠳⢜⠦⣕⣦⠀⠀⠀⠘⢿⠇⡴⢴⣩⡴⡄
⠀⠀⠀⡰⣷⠮⠊⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠨⡟⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠆⠀⢇⠉⠁
⠀⠀⠀⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠐⢀⣀⡀⣤⡟⣆⣄⣈⣀⠄⠀⠀⠀⠒⠀⠀⢸⣋⡷⠠⠀⠀⠀⠀⠀⢐⠀⢒⠀⢠⣾⣷⡤⠀⡐⠀⡢
⠀⠀⠀⠀⠀⠉⣭⡝⠅⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣒⣿⠀⠀⠀⠀⠀⠀⠀⠈⠑⠐⠊⡯⢽⢋⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠐⠲⣄⠀⣀⠀⣠⠔⠒⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠂
⠀⠀⠢⠤⠬⠑⣉⠛⠥⠤⠔⠀⠀⠀⠀⣴⠋⠉⠘⡨⠖⠋⠑⣆⠀⠀⠀⠀⢎⡉⠑⠂⢼⣸⠄⠂⠉⣩⠂
⠀⠀⠀⠀⠀⢸⣡⡁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⢛⣲⠖⠐⠒⠁⠀⠀⠀⠀⠀⠈⠒⠈⢵⣶⠍⠒⠋
⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟

⠀⠀⠀⠒⢬⡒⣄⠔⡠⠒⠂⠀⠀⠀⠀⠀⠀⠈⢑⠤⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢂⣠⠒⠁
⠀⠀⠐⠒⢢⢽⣿⣯⣔⠒⠂⠀⠀⠀⠀⠀⢀⠤⣞⣛⣳⢤⡀⠀⠀⠀⠀⠀⠀⠀⢉⣥⣬⣥⡤⣉⠑
⠀⠀⠀⠈⠁⠐⣛⠆⠈⠁⠀⠀⠀⠀⠀⡔⠁⡠⢧⠀⡼⢄⠈⢢⠀⠀⠀⠀⠀⠺⠛⠁⢛⡛⠊⠙⠻⠂
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠉⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣀⢀⠀⠀⢀⣀⠀⠀⣀⠀⣀⠀⣀⠀⠀⣀⠀⡀⠀⠀⢀⡀⠀⢀⡀⠀⡀⠀⢀⣀⠀⠀⠀⣀⠀⢀⡀⠀⢀⡀⠀⣀⠀⢀⡀
⠀⢘⣯⠘⣷⠇⣟⣭⠀⢸⣟⢸⠿⢛⣿⡄⢸⡿⠃⡟⣾⡷⣏⣥⠀⢸⣹⡇⣹⣏⢸⠯⠀⣿⢸⡿⠳⣏⣥⠀⣿⠿⢸⣉⡇⣿⠅
⠀⠈⠁⠀⠈⠀⠈⠉⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠁⠀⠁⠈⠈⠀⠀⠛⠈⠀⠀⠈⠁⠀⠈⠀⠈⠉⠀⠁


⠀⠀⢀⣤⠶⠦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠒⠛⠓⢦
⠀⠀⠰⠐⠂⠈⠈⠉⠉⠙⠳⢦⣄⡀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⢀⡤⠒⠁⠀⠀⠀⠀⣀⣤⣶⢟⣋⡁⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠁⣀⠀⠀⠀⠠⠤⠤⠤⠭⡝⠳⢦⣄⡀⠀⠀⠀⠉⠲⡤⢤⡞⠁⠀⠀⠀⢀⣠⣴⠾⢋⠏⠤⠄⠠⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠒⠒⠚⠢⠤⠈⣙⠛⠶⣤⡀⢻⠛⠛⢿⢀⣠⡴⠾⣛⡡⠤⠔⠊⠀⠀⠀⠐⠒⠀⠠⡀⠂
⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠈⠀⠠⠄⢀⠀⠀⠀⠈⣉⣉⡛⠏⡚⢹⠛⣿⠛⣊⣩⠍⠁⠀⠀⠀⠀⠀⠀⠀⢈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣲⠆⠉⠉⠒⢦⡤⢤⡤⠤⠀⢰⡖⣋⢉⣦⡿⠈⠠⠤⢤⠴⠖⠂⠒⠈⢙⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢐⠨⢀⠀⠋⠋⠉⠡⢊⣿⣉⠈⢿⣎⠀⢌⠉⡋⠑⠄⡀⠁⠐⠠⠈⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠤⢀⡠⠊⠀⠀⠀⠀⠾⠀⠀⠀⠀⠇⢢⠀⠑⢌⠂⠀⡄⠠⠄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⡔⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡄⠀⠀⠀⠀⠀⢀⣠⣴⣴⣶⣶⣤⣿⣿⣿⣿⣿⡆⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢲⣿⣿⡿⠿⢝⣧⣹⣦⣿⣿⣯⡿⣿⣷⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣾⠃⠈⠐⠐⠊⢻⠿⣻⠿⢷⣷⣿⣿⣯⡍⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⣴⡞⠹⠓⢿⣦⡄⠀⠀⡰⢃⠞⠀⢐⣿⠿⡟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠙⣷⣿⣻⡄⠀⢹⡇⠐⡼⠁⣌⣰⡧⣌⣪⣿⣟⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⣳⡶⠶⠶⠶⠚⠉⣹⣿⣧⠀⠀⠀⠘⢿⣏⡾⣄⡀⣳⣄⣅⣠⣯⣿⣿⢿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠊⠻⠀⠁⠀⠀⢀⠐⠀⣀⣙⣟⣩⡄⠀⠀⠀⠀⢹⣼⣏⡼⢿⣿⣯⣭⣯⣽⣞⣯⠿⢿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡏⠀⠚⡆⠀⠀⠈⢀⣀⠤⠀⡀⢸⣏⣭⢦⡀⠀⠀⠼⢖⡡⠀⠀⠀⡿⣿⡿⠟⠉⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⢃⠀⠸⢷⡀⠀⡈⠁⢀⡠⠂⠁⠀⠩⢷⡿⠋⠁⠄⠀⠈⠀⠀⢀⣿⡉⠀⠀⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣻⣿⡹⡀⠘⡆⠀⠘⢳⡀⠁⠀⠁⠀⠀⠀⢫⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⢿⣿⠀⠀⠀⠠⡞⠁
⠀⠀⠀⠀⠀⠀⣰⣷⢺⡿⣧⠽⡀⠘⡄⠀⢈⡑⠄⠀⠀⢀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⢈⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣇⢃⡚⢯⣷⡹⢆⠀⢆⠈⠁⠀⠙⠲⠌⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⢸⡕⠀⠊⠳⡜⢫⣿⢦⡳⠀⠑⠄⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯
⠀⠀⠀⠀⠀⢸⣿⣄⢀⠐⢈⠆⡘⢿⣿⣿⣗⣤⣜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠸⡿⢿⢷⣴⢾⣿⣿⠦⠿⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠠
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⡖⠒⠒⣢
⢰⠁⠀⠸⡀⠙⡑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⢿⣤⣶⣶⣶⣶⣾⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡠⠄⢂⡉⠀⠄⢴⠃⢀⠔⢸
⠈⢆⠀⠀⠳⡄⠈⠓⠀⠀⠉⠶⠀⢄⡀⠀⠀⠸⣿⣦⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣾⣿⠃⠀⣀⠴⠒⣹⠆⠁⡀⠀⢠⠔⡡⠖⠁⣰⠃
⠀⠈⡇⠀⠀⠈⠓⢤⠄⣐⢀⡘⠦⣄⡈⠑⠢⢄⡙⢿⣧⣄⣿⠟⡋⠉⠉⠉⠉⠛⠛⠿⣄⣠⣿⣃⠔⠊⠔⢈⡤⣚⣩⡤⠖⠉⣽⠏⠀⠀⢸⠃
⠀⠀⠈⠲⣤⣄⣀⣀⠐⢤⣈⠉⠁⠘⡙⠢⢄⣀⠈⠛⠿⣿⣱⣈⣷⣾⣦⣾⣾⣷⣤⣀⣈⡿⠋⣁⡨⠴⢊⡡⣃⣕⡥⢶⠖⢩⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠑⢤⡀⠀⠀⠈⠢⢌⣑⡒⠬⠐⡦⠤⠭⣥⣖⣦⣽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣖⡭⢴⡾⠋⣑⡨⢿⡷⠖⢃⡴⠟⠂⣨⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠣⣅⣀⡈⠁⠂⣬⣍⣉⣀⣉⣀⠽⠾⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡾⠯⠭⠤⠭⣤⠦⠥⣤⣴⣭⠶⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⠉⠉⢩⠥⠀⠤⠤⣂⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡫⠥⡒⠲⠒⠛⠢⠀⠀⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠧⠤⠔⢒⡩⠔⢊⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⠦⣌⡐⠚⠚⠃⠀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠖⠊⠁⣠⣾⡟⠀⠀⠈⠀⠋⢹⠟⠩⠙⠉⠈⠹⢽⣿⣷⣄⠉⠑⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⣿⣃⡀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⢰⣿⣿⣿⣷⣷⣶⣶⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⠀⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢻⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣻⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠘⡟⠉⢻⣦⡹⣇⠀⠀⠁⠀⠑⠀⢀⡼⠋⠀⢻⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠻⠯⣤⣠⣃⣠⡴⠞⠋⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠘⠀⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣟⣀⣤⣤⣤⣀⣀⣀⣀⣠⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠔⠋⠉⠉⠉⠉⠉⣉⣙⡛⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⡠⠀⠀⣀⣠⣄⣀⠀⠀⠈⠉⠛⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⣿⣿⣶⣶⣟⣩⠟⠙⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣷⣶⠖
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠠⠶⠈⠉⣉⣙⡃⠁⠀⠀⠀⠈⠙⠳⣦⣄⠀⠤⠴⠾⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣠⡤⣶⣶⣬⣝⠻⣿⣦⠀⠀⠀⠐⢶⣾⣿⣷⡄⢠⡀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣿⣷⣾⣿⣷⡌⢳⡄⠀⠀⠀⠀⠹⣿⡟⠻⠈⣿⣶⣦⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠛⢿⣿⣿⣿⡂⣿⡀⠀⠀⢀⠀⢻⣿⠀⠀⢸⣧⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡟⣿⠻⣾⡇⠀⠀⢸⣷⣾⣿⣦⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣁⣡⣴⡎⠃⠀⢀⣼⡟⠙⠃⠘⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⣹⣟⣀⣠⣴⡿⠋⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢦⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢄⠀⠀⠈⠇⢰⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠑⠀⠀⠠⣟⣀⢾⣿⣶⡀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣑⣦⣀⠀⠀⠀⢸⣿⣿⣷⡝⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣨⣅⠘⢷⡼⢿⣿⣿⠿⠶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣉⠘⢿⣿⣦⡈⠛⢷⠀⠀⠀⠀⠈⠉⠉⠓⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠈⠻⣿⣿⣶⠤⣆⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣷⣤⡀⠉⣽⠀⢸⢆⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⠋⠁⡼⠁⠀⠨⠈⠳⠤⣄⣀⣠⡽⠲⠤⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘





⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣶⣶⣶⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠖⠛⠉⢹⣿⣿⣿⠋⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⢀⣤⣤⣀⠀⢰⣏⡀⠀⢀⣰⣾⣿⣿⣿⣤⠀⠀⠀⠹⣆⠀⠀⣀⣤⣀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠟⠋⠀⠻⣷⣎⠻⠿⠷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠛⢁⣴⣻⠟⠉⠛⠻⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣼⣿⠋⠁⠀⠈⠉⠙⠛⢿⡶⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣦⣤⣤⣤⣤⣤⣶⡟⢸⠆
⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⠋⠉⠉⠉⠉⠉⠛⢿⣿⣿⣦⣀
⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠙⢿⣦⣄⡀⠀⣠⣤⣾⢿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠠⡿⡎⠻⣿⣷⣦⣄⠀⠀⠀⢀⣠⣴⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠞⠛⠉⣡⣿⠋⢸⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⣴⡇⣷⠐⡌⢿⢿⡛⢷⣶⠞⠛⠉⠀⠀⠀⠈⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⣼⣼⣿⣿⣿⣿⣦⣤⣀⢀⣠⣤⣯⢸⣏⣧⣧⡈⠆⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⡟⢀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⣽⣿⡟⢿⡙⠻⣿⣦⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣡⠎⠀⡈⢠⣿⣿⣿⣿⣿⣿⠁⠀⠈⠙⣿⣿⣿⣷⠈⡇⢣⠙⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢣⣿⡴⠁⠀⢀⠁⢼⣻⣿⣿⣿⣿⣟⣷⢦⣾⣿⣿⡿⠃⠻⢀⣧⠀⢆⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠿⠟⢡⡟⡘⠀⢰⠀⡸⠀⢸⣿⣿⣦⣭⣭⣿⣿⣿⣿⣿⣧⣤⣶⣿⣿⡟⡄⠈⢧⡀⠀⢸⠙⣷
⠀⠀⠿⠦⠶⠾⠿⠟⠉⠀⠀⢀⡾⢠⠇⠀⡆⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠿⣿⢿⠃⠐⡀⠀⠙⢆⠸⡀⠹⣷⢶⣶⣤⣤⣀⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡊⠀⡘⠀⢸⠀⠀⠸⣿⢽⣿⣿⣿⠿⢹⠏⠁⠀⠘⡅⠶⢸⡜⠀⠀⠐⠀⠀⠀⠙⢧⠀⢹⡆⠀⠉⠙⠛⠛⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣋⠎⠠⠰⠁⠀⠈⠀⠀⠀⢻⣼⡿⣿⠿⠀⠀⠀⠀⠀⠀⢳⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠠⠁⠀⠀⢠⠀⠀⠀⠈⢿⣿⣷⣦⣤⣀⣀⣠⣤⣴⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡰⠀⠀⠀⠀⠘⠀⠀⠀⢠⠈⢻⣿⣿⣿⢻⡏⢿⢻⠿⣿⠟⢢⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠙⠿⣉⠉⠀⠀⢋⡤⠋⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡆⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠙⠂⠖⠋⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⣆⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⡀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠘⠁



⣿⣿⡿⠛⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠓⢀⠀⢀⠐⠃⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣀⠀⠀⡰⣦⣨⣤⣬⣠⣶⡀⠀⢀⡰⠞⠉
⠀⢀⡤⠐⠲⣂⣀⣒⠒⠂⠤⢀⣀⠀⠀⠀⠀⠀⠹⣵⣄⠻⣿⡿⠿⣿⣿⡿⢃⣴⡽⠁⠀⠀⠀⠀⢀⣀⠠⠄⠒⢒⣂⣀⡲⠒⠠⣄
⠀⠸⣄⡀⢀⣀⠴⡣⠤⠭⠔⡣⠄⠯⡑⠒⠤⢄⣀⠈⠻⣶⠃⠀⠀⠠⢿⡷⡾⠋⢀⣀⠤⠔⠒⡩⠇⠤⡻⠬⠤⠤⡳⢄⣀⣀⣀⡼
⠀⠀⠀⠐⢄⣀⣀⡨⠒⠒⠂⠲⣉⣉⠉⠉⠐⠚⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⠒⠈⠉⢈⣉⡱⠂⠒⠒⠪⣀⣀⣀⠔
⠀⠀⠀⠀⠀⠀⠢⠤⡤⠒⣈⠁⢖⣁⠬⠭⢽⠾⣛⣹⣿⣿⣧⡤⠀⢲⣿⣯⣿⣿⣝⡻⢾⠭⠥⢄⣑⠆⢉⡐⠢⡤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⠤⣐⣁⡤⣊⠡⠄⣒⣭⣥⣬⣿⣿⣿⣿⣿⣯⣤⣭⣕⡂⠤⣉⡢⣄⣑⡠⠤⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠟⢉⣭⢟⣿⣿⣋⠙⣻⣿⣿⣟⢯⣍⠙⢿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠉⠉⠉⠀⢀⣾⣵⢿⠟⠿⠿⠿⣿⣿⣿⢻⢷⣽⣆⠀⠈⠉⠉⠑⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⣼⣀⡀⠀⠀⠻⣿⣿⡏⡆⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⣿⠛⠿⠿⠿⠿⣿⣿⡇⡇⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠁⠀⠀⢹⠷⣶⣦⣤⣼⣿⣿⢶⠁⠀⠀⠹⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠘⣷⢤⣤⣤⣭⣿⣟⡞⠀⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣴⣶⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⣧⠀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠣⡀⠀⠀⠀⠀⢸⠀⠀⣰⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠢⡀⠀⠀⠸⠇⢠⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠀⠀⠀⠠⣟⣀⠰⣿⣿⡦⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⢰⣿⣿⣿⣮⢻⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⠀⢤⣤⢹⣿⣿⣿⣿⣇⣠⡴⠖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠻⣿⣿⣆⠀⠹⣦⣍⠙⠛⠉⠈⠡⠤⠤⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠉⢀⣦⠀⠹⣿⣿⣷⣄⡈⠙⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣷⡀⠈⠻⢿⣿⣿⠂⢷⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⢿⣿⣶⣄⡀⢉⡾⠁⢸⡷⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠉⠛⠋⢀⡾⠁⠀⠘⠁⠙⠦⣄⣀⣀⣀⣈⡳⢦⣤⣀⣤⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣋⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠳⡄⠀⢀⡔⠉⠙⠁
⠀⠀⠀⠀⠀⢰⣿⡿⠿⢶⣤⣀⠀⠀⠀⠀⠐⢷⡄⠀⠀⠀⣹⣤⣼⡀⠀⠀⠀⣴⠗⠀⠀⠀⠀⢀⣠⣴⠾⠿⣿⣷
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠙⠿⣦⣄⠀⠀⠈⣿⠀⣾⣿⣿⣿⣿⣿⣷⡄⢸⡇⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠙⠷⣦⡀⠻⣆⠻⠿⠿⣿⡿⠿⠟⣡⡾⠁⣠⡾⠛⠁⠀⠀⠀⠀⠀⢀⣿⠏
⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣍⣻⣾⣿⣿⣿⣿⣾⢋⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣹⠿⠷⠶⠒⠒⢀⡈⢛⣛⣭⣝⣛⠋⣀⠒⠒⠲⠶⠿⣯⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⣠⣴⠛⣵⣿⣿⣿⣿⣿⣷⡙⢷⣄⡀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⣠⠴⠊⣿⠁⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⣿⡝⠲⣤⣀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠴⠖⠛⠉⠀⠀⣠⡟⠀⢠⣴⣶⣶⣶⣶⣶⣦⡤⠀⠸⣧⡀⠀⠉⠙⠓⠶⠤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠚⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⣠⠏⠀⠀⢠⠖⠒⠒⠢⣄⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡏⠁⠀⠀⠀⢿⡀⠀⠀⠀⢸⠆⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⠿⠳⠶⠖⠋⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⢀⡠⢔⣒⡒⠿⠙⠛⠒⠒⠤⣴⡃⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⣨⣴⣥⠤⠤⠤⢄⣀⡀
⠀⠀⣠⠚⢁⠎⣡⣤⣍⡆⠀⣀⣀⠈⢷⠀⠙⢆⠀⠀⠀⢀⡟⠀⠀⢀⣤⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⠍⠲
⠀⡼⠁⠀⠘⢆⣿⣷⠼⡗⢉⣠⣌⠱⡈⠀⠀⠈⣇⠀⢀⡞⢀⡤⠚⠉
⢸⠃⠀⠀⠀⠀⠀⠀⠀⣇⢿⣿⣥⣥⠇⠀⠀⠀⢹⢠⣾⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⢸⡄⠀⠀⢰⠢⢄⣀⡀⠈⣓⠒⠚⠁⠀⠀⠀⠀⡿⠛⠷⢶⣤⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠖⠋⠁
⠀⢳⡀⠀⠸⣶⣦⡤⠍⠤⡼⠁⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠙⣧⣠⠤⠤⣤⣬⣭⣉⡉⠉⠉⠁
⠀⠀⠙⠦⣄⡉⠓⠤⠔⠚⠁⠀⠀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⣿⣿⣿⠉⠓⢦⣄⡀
⠀⠀⠀⠀⢠⠏⣹⠛⠒⠒⢲⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢸⣿⣿⣿⠀⠀⠘⣿⣿⣦⣄
⣎⠉⠒⠚⣁⠞⠁⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠂⢀⡾⠁⠀⠀⢀⣾⣿⣿⡏⠀⠀⢸⣿⣿⣿⠈⠳⣄
⠘⠳⠶⠚⠁⠀⠀⠀⠀⡠⠞⣠⠟⠷⢤⣄⣀⣀⣤⣤⠞⠋⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⢀⣿⣿⣿⡏⠀⠀⢹⣷
⠀⠀⠀⣠⠤⠤⠤⠐⠋⣠⡾⠁⠀⠀⢀⡠⠝⣋⡭⣧⠀⠀⣀⣤⣾⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⣿⠁⠀⠀⣾⣿
⠀⠀⠀⠹⣦⣤⣤⡴⠞⠋⠀⠀⢠⠞⠁⣴⠚⠁⣀⠜⣷⣿⣿⣿⡿⠟⠁⠀⠀⢀⣴⣿⣿⣿⡿⠃⠀⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣰⠇⢠⠊⢠⠞⠁⠉⠻⢧⣤⣀⣀⣤⣶⣿⣿⣿⡿⠟⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣇⠀⠋⠛⢦⠀⠀⠀⠈⠉⠛⡿⢿⡿⣿⣯⣤⣤⣤⣴⠾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠛⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⢸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⡇⠀⠀⠀⡇⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⡇⠀⠀⠀⣇⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡼⠃⠀⠀⠀⠸⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣴⠃
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣤⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢀⣤⣄⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⠀⣴⣿⣿⡿⠖⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⡀⠀⠀⠀⠈⠈⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣬⣥⣬⣵⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣠⣿⠋⢁⣤⠀⠀⠀⠀⠈⠛⣻⡀⣧⡄⠀⢰⢣⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣧⣿⣿⣏⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢼⣯⠍⠙⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢠⡴⠋⣽⣿⣿⣿⣿⣍⠛⢦⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⢇⣘⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢸⠀⠀⠀⠀⣤⠶⢦⣄
⠀⠀⠀⠀⠀⢀⠉⢀⡾⠻⠛⠿⠟⠿⠟⣧⠀⠁⠀⢠⡟⠙⢿⡟⢟⣿⣷⣿⣿⢟⠛⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣥⣤⣤⣤⣤⣼⡿⢶⡛⣟
⠀⠀⠀⠀⠀⠀⠀⡾⠁⢠⣤⣀⣀⣤⡄⠘⠇⠀⠀⢿⣿⣧⠰⠇⣼⠿⠋⠀⠟⠀⣶⣎⠀⠀⠀⠀⠀⠀⠉⠉⣻⣿⢿⣿⣿⠿⣿⣿⣿⣿⣿⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠹⠶⢷⡶⠀⠈⠻⣿⡅⠀⠀⠀⠀⢀⣠⠚⠫⣟⣫⠿⣯⡛⠷⠦⠤⣨⣿⣿⠃
⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡰⢊⢳⡾⠇⠀⣤⣴⣾⣟⠁⠀⢠⣼⣿⠆⠈⣹⡿⠿⣿⠿⠟⠁
⠀⢾⣿⠿⠋⠀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⣷⡄⠀⠀⡿⠛⠁⢀⣴⠿⠛⠁⠐⢿⣿⣎⣁
⠀⠈⣿⢶⣤⣤⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡋⠃
⠀⠀⠀⠀⡟⢿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠤⢤⣤⣶⣯⡙⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⣟⣦⡀⠀⠀⠀⠀⡇
⠀⠀⢶⠀⠙⢀⣍⣛⣿⣿⣿⣿⣆⠀⠀⢀⠤⣶⢶⣿⣿⣿⣻⡞⠛⠉⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠈⣷⣤⣿⣿⡇⠀⠀⠀⠰⠀⠀⠀⣀⢠⡀
⠀⠀⠘⢿⡗⠂⠛⢻⣿⣿⣿⣿⣿⣆⠀⠀⢰⠎⠣⣻⢿⡟⢿⠈⠀⠀⠀⠀⠀⠀⠘⢿⡿⣿⡆⠀⠀⢠⠿⢛⣿⣿⣧⡀⠀⠀⠀⠀⠠⣌⢿⣷⣿⠄
⠀⠀⠀⠘⣿⡀⠀⠀⢿⣿⣿⣿⡏⢻⡄⠀⠘⠦⠴⠜⣻⠃⠸⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠈⡇⠀⠀⢤⣀⠀⢸⣿⣿⣿⣄⠀⠀⠀⠀⠈⠲⠟⠻⠆
⠀⠀⠀⠀⠈⠳⣆⣤⠜⠿⣿⣿⣷⡀⢻⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢃⠀⠀⢠⡀⠀⠸⠿⣿⢟⣫⣿⣿⣽⡆
⠀⠀⠀⠀⣴⠾⣿⣿⡄⠀⠈⠻⣿⣿⣤⠓⠀⠀⠀⢀⡀⠲⠶⣦⡀⠀⠀⣀⣾⡿⣟⣾⣧⣴⠞⠀⠀⠀⠻⠏⠉⠉⠉⢻⣿⣷⣄⣀⣀⣤⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠁⠀⣿⡟⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠚⠿⠽⠒⣠⠀⠉⣰⡾⣿⣿⣾⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠻⣿⣿⣟⡿⠗⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣸⡿⠃⢀⡀⠀⠀⠀⢹⣿⣿⣆⠀⠀⠀⠀⠚⠁⠠⠞⠉⣸⢟⣿⡟⠁⠀⠀⠑⠤⣄⣀⣠⣴⠀⢠⠟⠉⠉⠟⠋⠙⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠹⣶⣄⠀⠀⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢰⠃⣼⡟⠀⠀⠀⣀⣚⠏⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠘⠃⠀⢀⣷⠀⠄
⠀⢀⡀⠀⠀⠘⢶⣤⣀⠀⠙⢿⡧⠀⠘⢿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢉⠏⢐⡣⠴⡶⠶⠀⠀⠀⠈⣽⣟⣭⡽⠂
⠀⠸⣿⣿⣦⣄⠀⠙⢿⣷⡴⢾⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⠇⠀⢶⣦⣄⣦⠶⡦⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠒⠁⠙⠇
⠀⠀⠀⠈⠙⠛⢿⣶⣶⣿⣿⣦⢫⣁⣤⣄⠀⠀⠀⢠⡄⣠⡏⠀⠀⣼⡟⠀⠀⠀⠙⣿⣿⣶⣿⣷⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⠛⢿⣿⠞⢻⠗⠀⠀⠈⠀⠈⠀⠀⣸⣿⠃⠀⠀⠀⠘⠡⣷⡎⢎⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠞⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠐⠶⠞⠛⠉⠀⠀⠀⠀⢀⠀⠆⠀⠀⢿⠇⠀⣴⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⠟⠉⠀⠀⢀⣤⣠
⠀⠙⢿⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠀⣀⣀⠠⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⢿⣿⣶⣶⣦⣤⣤⣶⣿⣿⡿⠁⣠⣤⣤⡀⠀⠸⠟⠿⠃
⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⣿⣿⣿⠟⢊⣁⠀⠀⠀⠀⠹⠿⠂⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠉⠉⠉⠉⠛⠛⠃⠘⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⣀⠀⠭⠭⠭⠅⢁⡒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠒⠁⠀⠀⠀⠀⠀⠀⡀⠀⠑⠢⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠠⠊⠀⠀⠀⠈⣀⣀⡀⣀⣀⡀⠀⠀⠀⠈⠢⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡐⠁⠀⠀⠀⠀⠀⣌⡛⠷⠛⣉⠄⠀⠀⠀⠀⠀⠑⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠈⢄⠢⡀
⠀⠀⠀⣀⠀⠄⠒⡁⠔⠁⠀⠀⠀⠀⣀⠤⠖⠒⠈⠉⠉⠉⠉⠑⠒⠢⢄⡀⠀⠀⠀⠀⠑⠄⡑⠂⠄⣀⡀
⡔⢈⠡⠀⠀⠂⠁⠀⠀⠀⠀⠀⢠⠎⠀⠀⣀⣀⠤⠤⠄⠀⠤⢄⣀⡀⠀⠈⢣⡀⠀⠀⠀⠀⠈⠁⠂⠀⠤⢉⠐
⠑⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠔⠉⠉⠉⠑⠀⠀⠀⠐⠉⠀⠉⠑⢤⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗
⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡏⠶⣤⡀⠀⣠⡄⣦⡆⢀⡀⠀⣠⣴⠎⡷⡌⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⢀⠈⡁⠓⢀⡌⢻⣾⢿⡟⠁⠁⠛⠉⠉⠁⠀⢸⡏⡁⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠀⢹⠀⠀⠉⠙⠃⣈⣷⠀⢰⣇⡤⢀⣀⠠⠂⢸⡁⠘⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢦⣾⡀⠀⢀⠔⡾⠊⣻⠀⢸⡋⠚⡖⢄⠀⠀⣸⣦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢘⣷⣿⣂⣠⣷⣴⣯⣿⠀⢸⣯⣷⣴⣧⣀⣺⣷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⠿⣿⣿⡉⠁⠋⢿⡿⠿⠤⠼⠿⣿⠏⠃⠉⣹⣿⠟⢶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠿⠛⠳⡀⠀⠱⡳⢄⣀⣀⡴⡳⠁⠀⡰⠛⠻⠷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⠀⠈⠓⠒⠒⠋⠀⢈⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣖⣿⣗⡆⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏
⣿⡿⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡅⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠄⠀⠀⠀⣖⡆⠖⢲⢄⡱⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⢆⠄⠀⢀⣡⣴⡞⢵⡎⢣⢋⣧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢁⣋⣀⣾⣿⣻⣷⣌⣡⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣺⣿⣽⡿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠨⠿⠻⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠩⣶⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠐⠀⠀⠊
⠀⠀⡀⠐⠀⠀⢈⠀⠀⠀⠀⠀⣤⣤⣤⣤⣀⡠⡪⢖⠝⢊⠙⢂⠜⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠀⢀⣀⠠⣀⠀⠀⠁⠀⠀⠀⠀⡠
⠐⠂⢠⠊⠉⠀⢤⠤⠬⠍⠷⠒⠟⠛⣿⣽⣺⠀⠓⡤⠛⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠁⠀⣀
⠀⠀⠈⠢⢄⣁⠈⡅⠙⢆⠲⠀⠀⢰⣩⢭⣥⡕⣠⡔⠈⡐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠊⡱⠋⡲⠂⠀⢰⢀⣀⠤⠤⠒⠒⠤⡀
⠀⠀⠁⠠⠀⡀⠀⡉⠉⠉⠹⣿⡷⢩⣉⠲⣼⠁⣭⣽⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡶⢡⠴⠠⢊⠤⠛⠁⣠⠔⠂⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣼⣛⣤⠟⣶⣯⠭⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢰⠁⣜⣠⠎⢀⠜⠁⣀⢖⠉⠀⡤⠄⠘⠀⠀⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇⠬⠥⢀⡔⠁⡰⢎⠀⡠⠷⠁⣠⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⢠⠎⡠⠪⣀⠔⢫⡠⠐⣣⠐⠂⡠⠊⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠱⣿⣿⣷⣀⡎⠐⢠⡔⠉⠔⠉⠐⠈⣀⡤⠊⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⢀⣯⠿⢿⣜⣶⣀⠀⠈⢀⣤⣶⣾⣿⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣡⢎⡠⣤⡐⠈⣿⣾⣶⣶⣟⣟⣿⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡤⠛⢹⣀⣉⣁⣤⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⡷⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣞⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⠛⡝⢌⠻⠳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⡁⠀⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠈⠀⠀⠀⠈⣉⣛⠶⢦⣤⣀⠀⠀⢀⣀⡤⠤⠶⠶⠆⢤⣗⡂⠓⠳⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠦⠈⠀⣀⣀⣀⡉⠲⠶⢿⣿⣶⣬⣉⠁⢀⢀⣀⣠⠎⠀⠀⢁⠀⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠈⠁⠀⢠⠯⢦⣁⡄⢠⣿⣿⣿⠀⢀⣦⡔⡦⠗⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⠉⠁⠀⠀⠀⠀⠤⠤⠤⣤⣭⣄⣠⣬⣽⣮⢉⣠⣿⣿⣿⣿⣷⣿⣿⣿⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠠⠤⠤⠠⠁⠒⡀⠀⠀⣀⣐⣻⣿⣿⠿⠟⠛⢉⣨⣿⣿⣿⣿⣿⣿⠟⠿⣿⣛⣴⡏⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⣒⠂⠠⠴⠞⣻⠟⠉⠉⢱⣗⡔⢶⣿⣝⣸⣿⠿⣿⣿⣟⣡⣴⣶⠾⢿⣏⠀⠀⠉⠛⠦⠶⠖⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠂⠀⠉⠁⠀⠀⠀⢀⡜⢱⡀⠀⣿⣿⣿⡏⣭⣭⣽⣿⣿⢿⢶⣌⠿⠟⠋⠁⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣸⣇⠰⣺⣿⣿⣷⣿⣿⡿⢿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣧⣤⣤⣄⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢡⣦⣙⣿⣷⣾⣿⣿⣿⣿⣿⡷⠾⠀⠀⠀⠈⠓⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⣿⣿⣿⡿⣛⣩⣭⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠈⠀⡞⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⢀⡴⠃⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠴⠋⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁



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

⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⣁⠈⠀⠀⠀⡄
⠀⠀⠀⠀⢀⠣⠄⠀⠀⡄⠒⠀⠀⠀⠀⠀⠛⠁⠃⠈⠀⠀⢰⣧⡄
⠀⠀⠀⠀⠘⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠆
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋⠀⠀⢀⣀⣤⡄⠀⣄⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣡⣠⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⢀⡀⢶⣦⢰⡄⠀⣆⣶⠀⠀⠈⢻⡟⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠈⠀⠀⣴⣶⣶⡤⣤⣀⣀⡀
⠀⠀⠈⠁⠘⠆⠀⠀⠀⡈⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠟⠛⠃
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢾⣇⡄⠀⠀⢀⠨⣷⡶⡄⠀⣠⠀⠀⠀⠀⢠⣼⣿⡿⠟⠛⠋⠁
⠀⠀⠀⠈⠀⠀⢀⠀⠈⠀⠀⠁⢀⠤⠚⠉⠙⡄⠀⢱⡿⡟⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠐⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠆⢀⣤⣸⣅⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠛⠋⠀⠸⣿⣿⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⠿⠃
⠀⠀⠀⠒⢴⡟⠸⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣤⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢱⣦⡄⠀⠀⠀⠀⠀⢀⠙⠛⠉⠀⠀⠀⠀⢴⣴⣶⠦⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢄⡀⠀⠀⠈⠻⠶⠟⠁⠀⠀⠀⠀⠀⠨⠁⠃⠀⠀⠀⠀⠀⣀⡉⠻⠟⠁⠀⠀⣴⣮
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠋⠀⠀⠀⠀⠐⠮⠍

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⣎⠀⢉⣓⠐⠲⠤⣀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣀⣤⠔⠒⣋⠉⠉⢱
⠀⠀⠀⠘⠄⠀⠀⠓⠢⡀⠀⠉⠒⠤⣀⠀⠈⢦⠀⢳⣠⣀⣄⡞⠀⡴⠁⠀⣠⠔⠊⠁⣠⡞⠚⠃⠄⢠⠃
⠀⠀⠀⠀⠈⠀⣉⠄⠀⠌⠳⣔⠱⣄⠀⠑⠦⡄⠑⣤⡟⠉⣻⣤⠞⣀⠴⠋⠀⠀⡤⢊⠥⠈⠉⠉⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠐⠈⢉⠲⡒⠠⣄⠀⢐⣮⠷⠤⢿⣧⠊⢀⠆⠴⢒⢋⡁⠀⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠠⢠⡈⠒⠤⣤⡞⣴⠟⠛⠳⣜⢦⡤⠂⠀⣀⠠⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⢤⣬⠽⣿⣷⣟⠀⠀⠀⢹⢸⡿⢧⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣶⣦⠻⣶⣤⣴⠟⣼⣴⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⣴⡎⣥⣶⣶⣶⣤⡝⣦⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⢀⣾⡔⢷⣤⣭⣭⣭⣥⣴⣯⣿⣆⠀⠓⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢸⣇⢣⣭⣉⣉⣉⣉⣵⡏⣿⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠈⣟⠈⢨⣭⣉⣫⣭⡟⠁⣻⠁⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠙⠻⠟⠋⢀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠈⠙⠶⢤⡀⠀⢀⡴⠞⠋⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⢀⣿⣤⢿⣀⠀⠀⠀⣠⡶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⣿⣿⣮⣿⣿⡇⠀⣰⡻⠁
⠀⠀⠀⠀⣠⡤⠤⠤⠠⠤⣤⠤⣤⣄⣀⣀⣀⡹⣦⣼⣿⣯⢋⡀⢸⣽⣠⣿⣁⣀⣀⠤⣤⠄⢲⡀⠀⢀⠀⣲⠂⡄
⠀⠀⠀⠈⡆⠀⢀⠂⡰⠀⠤⠃⣀⣑⢦⠤⠭⠭⢿⣿⣿⣧⣾⠧⣤⣿⣟⠓⠂⣀⣴⠆⠀⠀⡨⠆⠨⡌⠄⠀⣠⠇
⠀⠀⠀⠀⠈⠢⣈⠁⠑⠤⠄⠛⣤⠄⠐⣋⠩⢕⣻⢿⣿⣿⣿⣼⣿⣿⢌⠉⠒⠀⠀⢁⡀⡔⠃⠠⠔⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⡞⠩⠏⡁⠀⣒⡷⠖⢹⣿⣿⣟⣙⣛⣳⡌⢲⣆⢉⠀⠐⠂⠀⣈⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⢂⣞⠟⢀⣴⣿⣿⣿⡯⠀⠀⢸⡟⢆⢳⢀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⣰⡿⠟⣿⣿⣿⡶⠚⠋⠹⡟⢶⣆⢳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠀⠀⢰⣿⠀⠀⢹⣿⣯⣵⡤⠤⣴⠇⠈⣉⡀⠈⠉⠻⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢀⣮⡿⠀⠀⠘⣿⣿⣇⣀⣠⡿⠀⠀⢹⣥⡄⠀⠀⠈⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣾⠃⠀⠀⠀⢻⣿⣿⣀⣼⠃⠀⠀⠀⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠙⠻⠟⠃⠀⠀⠀⠀⠘⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁