Etc. Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠆⠀⠊⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠊⠉⠉⠀⠀⠀⠵⠤⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠊⠉⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠠⠄⠐⠀⣈⠀⠤⠀⠀⠀⠐⠚⣢⠀⠀⠀⠀⠈⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠤⠓⡀⠠⠄⠒⢂⠀⣀⠠⠤⠒⠒⠛⠛⠋⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣄⣠⠀⠀⡄⠀⠀⠄
⠀⠀⠀⢠⠞⠀⠀⢀⡠⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣆⢸⡆⢧⠀⢹⣿⠀⠀⠹⠀⢠
⠀⠀⠀⠀⠉⠲⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠿⢿⡄⣧⢸⣄⣀⠏⠙⠂
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠀⠀⠃⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠘⣄⣀⣤⣴⠂⠀⠠⢀⡞⢹⠂⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⢈⡝⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡄⠀⠀⠘⡄⡜⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠒⠁⠀⠀⠀⠠⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣄⠐⠚⠚⠁⡠⠚⠉⠉⠀⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠢⠤⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡰⣵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠛⠛⠉⠉⠁⠈⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⢈⢋⡾⠓⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠎⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠉⠛⠁⠀⠀⠂⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⣤⣄⣠⠤⡤⡤⠤⠒⠊⠁⠀⠀⠀⡔⠆⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢦⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⠼⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠒⠂⠀⠀⠀⡐⠛⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⠈⠉⠛⠛⠻⠿⠶⠶⠶⢤⣤⣤⣤⣤⡤⠶⠶⢧
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠿⠿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠔⠁
⠀⠀⠀⠀⠀⠨⡄⠀⠀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀⠀⠀⠀⠀⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⠭⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⠇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣷⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠋⠀⣠⡏⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢠⠀⠀⡆⠀⣠⠟⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⢄⣼⣁⠴⠃
⠀⠀⠀⠀⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠐⠒⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⣀⣀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⢀⣾⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣄⠀⠀⠀⠀⠤⠟⠋⠀⠀⠀⠀⢀⠀⣠⠤⠴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣷⠿⠿⢿⣿⡀⠀⠀⠀⠀⠘⠉⡟⠒⠒⠂
⠀⣰⣶⣤⣀⠤⠀⣠⢮⣉⠙⣆⠀⠀⠀⠔⠿⢰⣦⠀⣾⡧⢿⣳⣄⠀⠀⠘⡄⢇
⠀⠈⠉⠉⠉⠂⠊⠀⠀⠈⡆⠤⠀⢠⢺⡾⢗⣀⣙⠁⠛⣠⣽⠛⠋⠳⡀⠀⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢳⣀⣇⣠⣤⣬⣭⣩⣤⣤⣿⣶⠿⠿⢶⣾⡶⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠋⠉⣩⡉⠹⣏⣩⣿⢠⡀⣶⣦⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢪⡄⢸⡇⢸⣷⡀⠙⣿⠁⣿⡇⢸⣿⡆⣤⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢺⡇⢸⣿⠈⣿⣇⠀⣿⠀⣿⡇⣼⣿⢼⣿⡇⢷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⡾⠀⣾⣷⢸⣿⡆⢻⣿⠈⣿⢸⣿⡇⣿⣿⢸⣿⡇⢸⡓⢮⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠰⢡⠁⡅⣀⢹⣿⠸⣿⣇⢸⣿⠀⣿⢸⣿⠃⣿⣿⢸⣿⢱⣾⠃⠀⢃⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢀⣿⣼⣿⠀⣿⣿⠈⣿⡇⣿⢸⣿⢸⣿⡏⣼⣿⣸⡷⠂⠀⠈⡌⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠇⠀⠈⢹⡇⣿⡆⢿⣿⡇⣿⡇⣿⢸⡏⣸⣿⠇⣿⡇⣿⡗⠇⠀⠀⠇⠈⠓⢆
⠀⠀⠀⠀⠀⡠⠃⠀⠇⠀⠀⡌⠁⢻⡇⢸⣿⣿⠙⠃⣿⠀⠀⣿⡿⢰⣿⠃⠙⡰⠀⠀⠀⠘⠢⠤⠪⢐⣤⣤⣤
⠀⠀⠀⠀⡰⣄⣠⠔⠀⠀⠀⠘⡄⠘⠛⠀⣿⣿⡆⠀⣿⠀⢨⣿⠇⠘⠋⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⣀⣠⣴⡮⠔⠁⠀⠀⠀⠀⠀⠀⢈⢦⡀⠀⢹⣿⡇⠀⣿⡄⣸⡿⠀⠀⣠⢼⡁
⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠋⠙⠢⡀⠻⠃⢠⣿⣇⠈⣁⡤⠚⠉⢱⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡀⠀⠀⠈⠙⠒⠚⠛⠛⠉⠁⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡀⠷⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠙⠦⡉⠓⣄⡀
⠀⠀⠀⠀⠀⣤⣶⡶⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⠗
⣤⣤⣤⣤⣤⣼⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠄⠀⢀⣠⡄⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠇⠀⠉⠀⠚⠛⢿⣧⣤⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠉⠉⠁⠀⢸⡧⠃
⠀⠀⠀⠀⠀⠰⠇⢧⣄⣀⠄⠀⠀⣀⣀⣴⣿⣿⢋⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣤⣴⣾⠄⠀⡠
⠀⠀⠀⠀⠀⣥⣤⣭⣭⣭⡠⣤⣤⣤⣤⣬⣅⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣠⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠃⠀⠀⠀⢀⣠⡿⠿⠿⠛⠛⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠁
⠀⠀⠀⢰⡆⣞⠅⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢠⠀⠀⠀⠠⠀⠂⠒
⠀⠀⠘⡄⢱⠐⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢣⠀⡀⠂⡀
⠀⠀⠀⠈⢪⡟⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⡂⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⡆⠀⠋
⠀⠀⠀⠀⠀⢡⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠓⠓⠲⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣅⣀⣀⣦⣡⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠒⠓⠒⠒⠂⢇
⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢠⠏⡽⠉⠉⠉⠙⡏⢆⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠃⠀⠀⠀⠀⢀⣋⣸⠀⠀⠀⠀⠀⢰⣊⣆⠀⠀⠀⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠥⡀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⣀⡇⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠠⣠⡤⠄⠀⠀⠀⠀⠀⣞⠟⠛⡄⠀⠀⠀⠀⠀⠀⠀⠘⠈⠉⠛⠛⠁⠀⠀⠹⢳⡀⠀⠀⠀⠀⢼⡓
⠀⠀⠀⠀⣀⣀⣀⡀⠐⠀⠀⠀⠀⢀⠜⢛⣠⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⠀⠁⢋⡇⠀⠀⠀⠀⠀⢼⠄
⠀⠀⢀⠞⠉⠀⠝⠹⣆⢄⣤⠀⠀⠁⣰⠛⣇⠀⠈⢱⡀⠀⠀⠀⠀⠉⡍⢁⣽⠁⠀⠀⠘⠀⠂⠋⠁⠀⠀⠀⢠
⠀⠀⢸⠀⠀⠿⠳⠠⠗⠸⡾⠁⠀⠀⡇⠀⠘⠿⠀⢀⡇⠀⠀⠀⠀⠀⠤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠻⠟
⠀⠀⠈⠦⣀⣀⣠⠜⠀⠰⠆⠀⠀⠀⠘⠦⣀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣒⣶⡾⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠦⢤⣄⡀
⠀⠐⠺⣿⡓⠲⠀⠀⠀⠀⠀⠀⠸⣿⣆
⠐⢶⣤⠘⡉⠁⠈⠸⣿⠶⠀⠀⢻⣼⣿⣦
⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣇
⠀⠀⠖⡀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣽⢣⣿⣿⡀
⠀⠀⢿⠁⠀⠀⠀⠀⠀⢠⠔⠀⠀⠉⠀⠉⠙⠃⠀⣀
⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠈⠡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠘⠾⢢⡀⠀⠀⠀⠘⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⣄⡀⠀⠀⠀⠂⠀⡀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣄⣙⡟
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠄⠘⠟⣷⣄⠀⠀⠀⠀⠫⢦⣌⠀⠢⢄⠀⠀⠀⠀⠀⠙⢦⡀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⢨⣞⣯⠝⢀⣦⣤⠓⠆⢤⢈⠙⠂⠀⠈⠐⠠⠀⠀⠀⠀⠙⡒⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠰⢈⡛⢺⣉⢯⢿⣦⣤⣈⡉⠛⠢⠤⢀⣀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⣀⠄⣀⡀⠈⠚⠿⢷⣶⣿⣼⣿⣿⣶⣶⣤⣄⣁⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⣿⡿⠉⢩⣿⠋⠉⠉⣡⣿⡾⠿⣟⠿⣿⣷⣶⣶⣶⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⣸⠇⠀⣠⣾⡿⠛⠁⠀⠩⢳⡍⠛⢿⣿⣿⡷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡞⠀⠀⢬⣟⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠉⢦⡀⠙⠗⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣢⠏⠀⣰⣧⣿⠿⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠪⠳⣄⠀⠙⢆⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣥⣘⢧⡴⣾⣿⣿⠿⠋⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⣹⣿⡿⣜⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠢⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣈⣁⣴⣼⣿⣧⠼⠋⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠁⢠⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⣶⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣰⠴⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠠⡄

⠀⠀⢀⠀⠤⠒⠐⣄⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⢸⠤⠄⠀⠂⠠⠄⠀⠈⠀⠈⠙⠓⠦⠾⠋⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⡌⠀⠁⠀⠀⠀⢁⠀⢘⡀
⠀⠀⡄⠀⠃⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⢰⠀⢾⡩⡭⡿⢯⠆
⠀⠀⠃⠀⠀⠀⠉⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣸⠀⢀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠘⠛⠛⠛⠛
⠀⠀⢰⠀⠐⠄⢀⠨⠘⠐⠠⢀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠧⠤⠦⠤⠤⠴⠯⠄⠄⠰⠤⠼⠀⢸⠀⢀⠀⠀⠀⢀⡀⠸⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠁⠈⠛⠛⠉⠛⠂⠀⠉
⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡤⠄⠀⠀⠀⠀⠠
⠀⠀⠀⡰⠉⠀⠀⠀⠀⢠⢁⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣀⣀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠁⠐
⠀⠀⡐⠀⠀⠀⠀⠀⠠⠄⠺⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠁⠀⠙⠊⠁⠁⢠
⠀⡰⠥⠤⠀⠀⠀⠒⠎⣸⠁⠀⡇⠀⠀⣀⢀⡀⡀⣀⢀⡀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠡⠄⠤⠤⠤⠄⠄⠠⠃⠀⠀⠙⠒⠓⠒⠒⠚⠛⠒⠒⠒⠒⠚⠁⠀⣧⣤⣀⣀⣀⣀⣀⡇⠀⢀⡄⠀⡀⢀⣀⣀⣀⠀⣤⣀⣤⠤⠤⠤
⠀⠀⠀⠀⡀⠀⠀⠤⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠀⠈⠉⠉⠉⠉⠀⠀⠁⠀⠀⠀⠁
⠀⠐⣞⡄⡀⠀⠀⠀⠀⢀⣨⡄⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀
⠀⠠⣏⠉⠒⠿⠔⠊⠀⡠⠐⠧⠀⠀⠀⠀⢠⣿⡏⠉⡟⢉⠻⢉⢹⠉⢻⠉⣽⣿⣦⠀⠀⠀⢸⡀⠀⠀⠀⠀⠸⠆⠀⡅⠀⠀⠀⠀⠁⠇
⠀⣰⠁⠉⠲⣤⠕⠂⠁⣀⠄⠺⡄⠀⠀⠀⣼⣿⡇⠀⡇⠸⣄⠸⣸⠀⠹⡆⢹⣿⣿⠀⠀⠀⢸⡄⠀⠀⠀⠀⠐⠆⠀⠁⠀⠀⠐⠒⠂
⠘⡇⠈⠐⠢⠄⣠⠄⠊⠁⢈⡤⠃⠀⠀⠀⢻⣿⣷⣾⡟⠞⠛⠛⠛⢺⢻⣷⣿⣿⣿⠀⠀⠀⢸⠄⠀⠀⠀⠀⠐⠂⠀⠘⠀⠀⠀⠀⠀⠀⣁
⠈⠓⠠⣀⠀⠀⠠⠀⣠⠔⠁⠀⠀⠀⠀⠀⠈⠻⢿⣿⣇⣤⣤⣡⣠⣸⣸⣿⣿⠿⠃⠀⠀⠀⠘⡆⠀⠀⠀⠀⣰⠃⠀⠀⠶⠬⠤⠖⠒⢚⡺
⠀⠀⠀⠀⠁⠑⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡀⠤⠄⠀⠀⠀⡀⠀⠀⠄⠄
⠀⢰⠀⢸⠤⡤⡇⠠⠠⠃⢘⡀⣠⠀⣠⠀⠴⠀⡀⡀⠀⠀⠔⠒⠒⠒⠒⠀⠠⠀⠀⠀⠀⢠⠒⠀⠀⠀⠀⠒⠆⠀⠠⠊⠉⠀⠀⠀⠀⠡⡄
⠀⠀⢠⡀⠀⠃⡄⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠈⠀⠀⠇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢜⠁
⠀⠠⠘⠀⠀⠀⡅⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠀⢺⡀
⠀⢀⠸⠀⠀⠀⡃⠀⠈⠀⠀⠀⠀⠀⢀⡀⣠⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠘⡀⢘⡀⣇⡇⠀⢐⡀⢈⠁⢁⠀⢠⠀⣁⢀⡀⢰⠀⠀⠀⠳⠭⠅⢛⠛⡖⢲⡞⠀⠀⠸⣀⣀⠀⠤⠤⠤⡜⠀⠀⣀⣀⣀⣀⣀⣀⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠄⠀⣀⠔⠀⠀⠁⠐⠀⠄⠀⠀⠀⠀⠀⠀⠰⠀⠈⠠⣀
⠀⠀⠣⡀⠀⠀⠊⠀⠀⠀⠀⠀⠰⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠈⢄⠂⠀⠀⠠⠀⠀⠈⠀⠀⠐⡄⠀⠀⠀⠀⠀⠅⠀⡈
⠀⠀⠀⠆⠈⠀⠃⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠰⡐⢀⠀⠸⠓⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡈⠠⠀⠀⢀⠀⠀⠀⠀⠂⣀⡂⠀⠐⠀⠀⠀⠀⠁⡀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠠⠀⠈⠁⠌⠁⠀⠐⠑⠄⠀⠀⠀⠀⢈⣽⢉⠳⠀⠀⠀⡀⠀⠁⠇⠀⠀⠀⠂
⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠁⠀⠤⠊⠀⠤⠀⠀⠀⠀⠣⠀⣆⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠊⠀⠀⠀⡀⠀⠀⢨⠀⠀⠐
⠀⠀⠀⠐⠁⠀⡂⠀⠀⠢⠄⠀⠀⠁⠀⢓⠀⠀⠀⠀⠀⠃⠌⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠑⢄⠀⠀⡄
⠀⠀⠀⠀⠈⠀⠀⠀⠠⣄⠀⠀⠀⡀⠍⠁⢀⠦⠤⠤⡀⠄⠀⠀⠁⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⡂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠣⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠶⠄⠀⠠⠎⠀⠀⠀⠀⠀⠀⠐⠀⠀⠔⠂
⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⢀⠔⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠐⡄⠀⠀⠌⠀⠒⠠⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠐⠈
⠀⠀⠀⠗⠂⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⢴⠄⠀⠀⠀⢀⠀⠁⠀⡠⠄⠀⠐⠐
⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⢀⠁⠀⠀⠁⠀⠀⠀⢀⠀⠀⠠⠀⠀⠠⠀⠁⠀⠁
⠀⠀⠀⠀⢁⠀⠀⠁⠒⠉⠀⠀⠀⢀⠂⠀⠀⢐⠃⠀⢐⠁⠀⠀⠀⠀⠀⢀⡠⠖⠈⠀⠀⠀⡀⠄⠀⠂⠀⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⢦⠀⠀⠀⠨⠀⠀⠀⠘⡀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠂⠀⠀⠀⢀⠈⠡⠀⠀⠀⠐⠒⠀⠒⠂
⠀⠀⢀⡀⠀⠀⠀⢀⠐⠂⠂⢀⠀⠰⠀⠀⠀⠀⠁⠀⠀⡀⠵⡀⠀⠀⡀⠀⠀⡁⠀⠀⣀⡤⠂⠀⠈⠀⠀⠂⠀⢀⣀
⠀⠀⠈⠀⠀⢀⠠⠘⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠐⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡧⠀⠀⢠⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⡀⠔⠀⠀⠀⠪⠀⠀⠀⠠⠄⠀⠉⠐⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠄⣀⠐⢄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⡀⠀⠠⠄
⠀⠈⢡⠀⠄⠀⠀⠀⠁⠀⠀⠢⢄⠀⠀⠀⡀⢴⠉⠑⠂⠐⠊⠀⠀⢀⠂⠀⠀⠐⠀⠀⠀⠀⠀⢡⠀⠌⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠐⠀⠀⠤⠄⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠈⢣⡀⠀⠸⡄⠀⠀⡇⠀⣼⠀⠀⣠⠋⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠈⠢⡀⠑⡄⢀⠹⡀⡀⡇⢃⢧⠂⡔⠁⢀⡴⠋⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠨⡀⠈⠀⠁⠁⠀⡇⠈⢀⠎⢔⠔⢉⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢄⣀⠅⠈⠀⠀⢀⠡⠀⠀⠄⠑⠀⠠⠒⠡⠀⡀⠤⠒⠋
⠀⠀⠀⠀⠀⠙⠛⠒⠒⠶⠤⠤⠀⢁⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⢤⣶⠤⠶⠖⠋⠑⠒⠚⠉⠑⠒
⢠⡤⠤⠤⠴⠒⠋⠉⠑⠒⠤⣀⡀⠀⠀⠈⢠⠀⠀⠀⠀⠀⣀⡄⠤⠤⠤⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢤⠒⠃⠈⠉⠀⠉⠁⠀⣀⣠⠔⠃⠴⠉⠀⠀⣀⡠⠔⠒⠀⠈
⠀⠀⡀⢀⡀⢀⠀⢉⠐⣂⡀⢀⣁⣀⣀⣐⣛⣯⡍⠉⠉⢉⣩⠋⠀⠐⡀⠉⠀⠈⡳⠨⡖⠢⡄⠄⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⢞⡤⠿⠼⠞⠲⢥⣀⣀⣀⡬⢤⣄⣈⣩⣄⣯⣄
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡠⠤⠤⠶⠪⠉⠑⣚⢋⠩⠅⠀⠒⠒⠒⠒⠒⠀⠒

⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⢀⣀⠀⣀⡀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠸⠄⢀⢀⣀⡀⠠⠂⠀⢸⠀⠁⠄⡀⠀⠀⠀⠀⠀⣠⠤⣤⡤⣤⡀⠈⠀⡇⠀⠀⠀⠠⠐⢈⠡⠐⠳⡀⠀⢠⠒⠀⠉⠉⠉⠉⠉⠉⠀⢢⠀⡇⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⣀⣀⣤⣤⡄⠀⢸⠀⠐⢚⣧⡁⠐⢁⢀⠤⣋⣉⣉⣉⢉⣁⠀⠀⠇⠀⠀⡠⠚⠅⠀⠰⣏⠀⠈⢄⠸⡁⠁⠒⠒⠒⠒⠒⠂⠈⠹⠀⡆⠀⠀⠄⠤⠠⠸
⠀⢠⡶⣿⢿⡟⣹⣹⡇⠀⢸⠀⠀⠂⠉⠉⠋⠛⠈⠂⠓⠉⠛⠛⠘⠙⠀⠀⡇⠀⠞⠀⠀⠀⣀⠀⠀⠀⠀⣨⠀⡀⠀⢔⣂⣉⠐⠒⣄⠀⡄⠀⠁⠀⠀⡤⣭⡅
⠀⢸⣇⣍⣼⡷⠷⠿⠃⠀⠈⠈⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠂⠠⠀⠀⠈⠀⢀⣤⡞⣿⠀⡇⠈⠻⠩⢭⣭⣿⠟⢀⠁⠀⣷⣟⡇⠐⠖⠂
⠀⠸⠛⠉⠁⠀⡀⠀⠀⠀⢠⠀⠀⠁⠉⠉⠀⠀⠐⡄⢰⡂⢐⡂⢘⡃⢰⠀⠀⠀⡂⠀⠀⠀⢠⣴⢿⣣⣿⠟⠀⢠⢸⠀⢰⣖⡆⠀⠰⠈⠀⠀⡿⣿⡏⠁⢁
⠀⠀⠀⠗⠂⠀⡀⠀⠀⠀⠈⠉⢀⣀⣒⣒⣒⠀⠉⡇⠘⠁⠈⢀⢀⠀⠀⡆⠀⠀⡁⠀⠀⠀⢸⣗⡷⠛⠁⠀⠀⠘⠀⠀⡃⠉⢙⣁⠂⠀⠀⠀⠁⢀⠀⠠⡀
⠀⢠⣄⣀⣀⣀⣀⣠⡀⠀⠀⢀⣿⣵⣼⣾⡿⣦⡀⡇⠇⠀⡀⠀⠀⢀⢤⡅⠀⠀⡇⠀⠀⠀⠸⠋⠀⡴⡆⢀⠀⠀⠑⢄⡀⠉⠀⣀⠔⠁⠀⠀⠀⡀⠀⠀⠀⠁⠇
⠀⠸⠷⠽⠯⠯⠭⠿⠃⠀⠀⢿⡧⣿⢭⣿⣸⣾⡗⠀⠂⠀⠑⢴⠞⠋⠈⢱⠀⠀⡇⠀⠀⠀⠀⠀⡜⠀⡘⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠑⠀⠀⠢⠀⠊⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⢀⡇⢀⠀⠄⠀⠀⠀⠀⢸⠀⠀⣇⠔⠉⠢⡀⠀⠓⣊⠤⠊⠀⠐⠁⠀⠀⠀⣏⠹⢄⠀⠀⢰⠃⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⣀⠠⢔⠂⢬⠏⠀⠁⠒⠠⠤⠤⠄⠒⠉⠀⠈⠐⠢⠤⠤⠀⠒⠃⠀⠀⠀⠀⠉⠐⠛⠂⠁⢀⡠⠔⠂⠡⠓⠒⠄⠀⠀⠘⢂⡄⠀⠸⠤⠤⠄⠠⠠⠀⠜
⠀⠀⠘⠉⠀⠁⣀⡀⢮⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠐⢠⡄⠀⠀⢀⠀⠀⠀⢹⠥⠖⠀⢸⢲⠈⠉⠳⢰
⠀⠀⡧⠴⢘⡁⠄⢂⠐⣀⣶⢹⠀⠀⠀⠀⣶⣶⠶⠶⡆⠐⠀⠀⠀⠀⠀⠀⣶⡶⣶⣶⡆⠀⠀⠀⠈⠠⠀⠀⢠⣶⣯⠿⠀⠀⠀⠀⠀⢀⣀⣈⣉⡒⠐⠙⠚
⠀⠀⠁⠀⠀⠀⠙⠀⣙⣩⡀⢼⠀⠀⠀⠀⣿⡇⠀⠠⣇⣠⣀⣀⠀⢀⣠⠀⢿⠀⠘⠻⡇⠀⠀⠀⠀⠠⠀⠀⠀⢀⣠⠀⢀⠀⣄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠀⠀⠀⠀⣿⠟⠖⠀⠀⠀⠀⠁⠀⠀⠀⠠⢤⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢎⡨⠃⠉⠀⢘⠉⠐⠒⠒⠲⠤⠒⠒⠂⠉⡻
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⢸⠀⠀⠀⠀⠿⠦⠤⠴⠷⠶⠴⠷⠴⠶⠶⢶⡾⡦⠴⠶⠆⠀⠀⠀⢀⠀⠀⠀⠆⠒⢚⡲⠅⠀⠀⢄⡀⠀⠁⠀⡄⠀⡀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⣘⠟⣰⡗⢸⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢⠴⠤⠒⠉⠀⠀⠀⠀⠀⠉⠢⠄⢀⡀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠸⣿⣟⠇⠀⢸⠀⠀⠀⡇⠈⠁⡀⢀⣨⣤⣴⣶⣿⣿⠭⣍⠉⠉⡇⠐⠐⠒⠶⠔⠂⠒⠒⠐⡒⠒⠀⠂⣀⠀⠀⡀⠀⢀⣶⡶⠂⠰⠒⠐⠲⠒⢲
⠀⠀⠂⠀⢀⣀⣐⣻⠁⠔⠊⠀⠀⠀⠠⡃⠀⡴⢛⡛⠟⠋⣍⠃⢶⢇⠻⣸⠆⠀⡂⠀⠀⠀⠀⠀⡇⠹⣿⠀⡇⢃⡇⠀⣿⢸⣧⢇⢀⣾⡟⠀⠐
⠀⠀⣀⡤⢀⢀⠀⢀⡀⣀⣀⣀⣀⡀⠀⡆⠘⣇⣛⣧⣦⣶⣤⣿⣶⡿⠟⠉⠀⠀⡇⠠⠤⠦⠤⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠀⠤⠴
⠀⠼⢀⠃⠈⠋⠈⠀⠡⡄⠀⠀⣴⡇⠀⠆⠀⠈⠙⠛⠛⠛⠉⠉⠀⢾⣿⣭⡷⠀⡇⢀⠤⠴⣀⢲⢒⣒⡐⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⠄⠠⣄⢤
⠀⠘⠚⠓⠒⠒⣒⠒⠛⠓⠒⠒⠛⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⡄⢳⣶⣧⣹⡭⡗⣱⢻⡔⡧⠀⠀⠀⠀⠀⠆⠀⠀⠘⡆⠬⡐⢊⠷⠸⠇
⠀⠸⠉⠉⠉⠭⠉⠉⠭⠉⠉⠉⠙⠇⠀⢀⣄⡄⣒⠲⡤⡄⠀⢠⢄⡤⠒⢢⡄⡄⠀⢣⠉⠐⠘⠱⠐⠛⠒⠂⢡⠃⠀⡀⠄⠂⠁⠒⠂⣄⠀⢀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠘⡆⠀⠈⠇⠦⣉⡡⠦⠇⠀⠹⠴⢄⣀⡠⠂⠛⠀⠀⠑⢤⡀⢀⠀⠀⣠⠔⠁⢸⠁⠐⢀⡄⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠉⣡
⠀⠀⠀⠀⠀⢈⡷⣹⠟⡄⠀⠀⠀⠇⠀⢠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⡄⠀⢀⠀⠈⠒⠒⠊⠁⡀⠀⠈⠠⠀⠀⠀⠀⠠⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠲⠂⠀⠲⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢸⠒⠒⠀⠀⡒⠒⡇⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⢸⢀⣶⢾⠿⢷⣦⢸
⠀⠀⠀⠀⣀⢂⡠⢄⠠⠀⡄⠀⠀⠀⠀⠘⠀⠀⢰⣿⣿⣹⣹⡏⣯⣩⣿⡆⠀⠀⠀⠀⢸⠀⠈⠄⠰⠁⠀⡇⠀⠀⠐⡀⠀⠀⠄⠀⠂⠀⢸⠈⠻⠶⠶⠾⠟⠸
⠀⠀⠀⠀⠈⠉⠳⣌⡰⠊⠀⠀⠀⡀⠀⠂⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡿⠥⠤⠀⠀⠀⢸⠀⠀⠶⠶⠆⠀⡇⠀⠀⠀⠐⠀⠇⠀⠠⠂⢑⢸⠀⠀⠀⠀⠀⠀⠰
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⠀⢸⠀⠀⠖⢴⠀⠀⡇⠀⢠⡀⠂⠀⠃⠀⡀⠔⠊⠈⢆⠀⠀⠀⠀⠠⠞
⠀⠘⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠁⠀⠀⠐⠒⠀⠀⠒⠊⠁⠀⠀⠉⠁⠙⠓⠁⠀⠀⠀⠀⠀⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠲⠶⠶⣤⡤⠶⠒⠒⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⣠⣶⣶⣶⣤⣤⣤⣴⣶⣦⣄⣀⣀⡈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠐⠂⠀⠀⡹⠃⠀⠀⠉⠉⠁⠀⠀⠈⠙⠛⠉⠁⠀⠈⠉⠉⠀⠀⠙⠆⠀⠀⠀⠀⠠⠀⣀
⠀⠀⠀⠀⠀⠀⠈⡔⠶⠶⠞⢻⣷⣦⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⢠⣶⣿⣿⣶⣶⠖⢨
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠴⠛⠿⢃⣼⠥⠀⠊⠉⠛⢢⣤⣀⣀⡼⣷⣄⣀⣤⠐⠝⢉⡑⠀⠀⠠⡌⢻⣿⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢲⡖⠀⠸⢁⢀⡀⠈⠙⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠉⠉⠉⠀⢀⠀⢠⠀⠈⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡬⠀⡴⢀⣿⡞⠉⡔⢂⠴⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠤⡔⠂⠉⢳⣼⡇⢸⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢠⣾⢃⢸⣿⢁⡾⠤⠄⠄⠀⠘⣄⣀⠀⠀⣠⢀⡅⠀⠀⠱⡤⠤⠬⣿⡇⡸⣷⡄⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡄⣿⣿⡟⠘⣿⢰⡇⠀⣦⡂⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⢀⣶⠃⢸⠀⣿⡇⢷⣿⣿⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠋⣿⡇⠀⠘⣸⣿⢀⠈⣻⡏⢩⣭⣦⣶⣶⣤⣼⡕⢠⣿⠉⣑⡿⢤⠃⠃⣾⣿⢻⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡶⠀⢹⣇⢰⠀⣇⡹⣿⠀⠛⢿⣼⣿⠿⣿⡟⠛⢿⣇⡿⢋⠀⣿⠇⣰⠀⢀⣿⡏⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⢀⡆⠻⡜⣇⢿⣧⡙⢆⢠⢸⣿⡇⠀⠘⠀⠀⢸⣿⡇⡼⢠⠟⣰⠏⠀⣼⠟⢠⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⢧⣨⠈⠘⣀⣙⣦⠈⠎⠛⠃⠀⠀⠀⠀⠸⠻⢣⠃⢨⣮⣍⡀⠚⢡⣆⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠋⢢⣸⠀⠈⡍⠀⢘⠀⢀⡏⠋⠑⢳⡇⠀⡃⠀⡞⠀⠀⢱⢦⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠇⠀⢸⠄⠀⠀⠀⠐⠄⠉⠤⡃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡁⡌⠳⠶⡿⠿⢶⣶⠶⠊⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⡰⢠⡀⠀⠀⢀⣄⣴⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠣⣴⡠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠆⠁⠀⠇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠄⠇
⠀⠀⠀⠐⠀⠀⢠⣶⡍⠁⡔⢳⠀⠀⠀⠀⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⢠⣶⠍⠁⡔⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀
⠀⠀⠂⠀⠀⠠⡀⠀⠸⠄⠀⠀⡀⠀⠤⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⡴⠆⠀⠀⢀⠐⠀⠀⠀⠠⡀⠀⠐⠂⠀⠀⡀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠆
⠀⠤⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⢠⣶⡁⠁⠀⠀⠀⠤⠀⠀⠤⣄⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠄⠀⢠⣶⠁⠁
⠀⠀⠀⠀⠐⠀⠄⢈⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠐⠀⢄⢨⠆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⠀⠈⠉⠙⠁⠀⠀⡄⠀⠀⠀⠈⠛⢁⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⢁⡀⠁
⡀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠓⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠁
⠁⠀⠠⠀⠀⢰⠀⠀⠀⢙⡈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠠⠀⠀⠰⠄⠀⠀⢑⡈⠀⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢂⣒⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤
⠀⠀⠀⠐⠀⢠⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀⡀⠈⠉⣰⠁⠀⠀⠀⠀⢄⠦⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⢀⡀⡀⠐⠉⣰⠁⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠠⣴⠖⠋⠉⠁⠀⢠⠡⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣤⡖⠋⠉⠁⠀⣰⠉⠓
⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠄⠈⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠄⠈
⠀⠈⠂⠀⠀⠐⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠈⠆⠀⠀⠐⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⢚⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠆⠀⡄⠀⠀⠀⠀⠐⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠆⠀⡄
⠈⠀⠀⠀⠀⠀⠀⣀⠻⠅⢀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⣀⠯⠅⢀⣄⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠁⠁
⠠⠀⠀⠀⠀⠀⠘⠛⠃⠀⠣⠘⠀⠀⠠⠀⠁⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠈⠁⠀⠀⠀⠈⠉⠀⠂⠀⠀⠀⠄⠀⠀⠀⣀⠍⠅⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠈⠁⠀⠀⠂⠀⠉⠀⠀⡀⠀⠀⠄⠀⠀⠀⣀⠯⠅
⠀⠈⠀⠀⠀⠓⠀⠰⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠟⠀⠀⠀⠀⠀⠉⠁⠀⠉⠒⠀⠠⡀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠐⠁⠀⠘⠏⠂
⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠠⠤⢤⠄⠀⠀⠀⠀⠀⠀⠰⣶⠄⠐⠗⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠰⣶⠄⠐⠗
⠀⠈⠀⠀⠘⠀⠀⠀⠀⡐⡆⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡀⣆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠆
⠆⠀⠀⠀⠀⢀⠀⠀⠀⢄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠄⠈⠀⠀⠘⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠈⠁⠀⠀⠰⠆⠀⠈⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠢⠤⢭⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠂⠤⢭⠄⠐
⠀⠀⠀⠀⠀⠈⠀⠀⠄⠃⠀⠀⠀⠀⠀⣀⡴⠤⠆⠀⢀⡆⡄⢀⠀⠀⠀⠑⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠤⠆⢈⢀⢆⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⢀⣼⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢀⣾⠀⠁⠀⠀⢠
⠀⠢⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠐⠢⡀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠
⢠⠀⠁⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠁⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣿⣷⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣇⣷⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⣀⡀
⠀⠀⠀⠀⣠⣶⣶⣿⣷⣶⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢢⡙⣦
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣠⣤⠤⢤⣀⡀⠀⠀⠀⠀⠀⢠⠁⢹
⠀⢀⣾⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠻⠧⠤⠒⠒⠊⠁⠀⠀⠀⠀⠀⠉⠻⢯⠉⠛⠲⠊⣠⡞
⠀⣼⣿⣿⠿⠋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⢤⣉⠃
⢀⡿⠋⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡟⠶⢼⡏
⠘⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⢠⡄⠀⠀⠀⣿⠐⠊
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠛⠛⠛⣻⠃⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢀⡇⠀⠀⠀⠀⠀⢀⣸⣁⡸⠀⠀⠀⡄⡿
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠟⠉⠀⠀⠐⣔⢤⡴⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⠐⢉⣈⠁⠀⠀⠀⢰⢻⡇
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠠⡧⠀⢀⠀⠀⠀⠂⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⣀⣠⣶⣿⡏⠱⡄⠀⠀⠠⣾
⠀⠀⠀⢀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣄⠀⣴⢁⠀⠈⠀⠀⣠⠶⣏⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠟⠚⠛⠛⡟⠋⢻⡁⠀⠙⠦⡴⠀⠹⡀
⠀⠀⠀⢸⣿⡇⣿⣿⠏⠀⠀⠀⠀⠀⠀⢫⡛⠋⢀⠇⠀⢠⠞⠁⠀⠈⢳⠀⠀⢦⣾⣾⠿⠀⠀⡄⡇⠀⠀⠀⠀⠀⠘⣄⠈⢇⠀⠀⠀⠱⣄⠀⢇
⠀⠀⠀⣸⣿⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⢿⠘⠦⢁⠀⢿⠀⠀⠀⠀⠸⡄⠘⢸⣿⣿⠀⠇⠀⡇⡇⠀⠀⠀⠀⠀⠀⣼⠀⢾⠁⠀⠀⠀⠙⡷⠌⡷
⠀⠀⢠⣿⣿⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⣞⠘⠇⢀⣤⡀⢸⠀⠀⠀⠀⠀⢳⠀⢸⣿⣿⠀⠘⡄⢱⡇⠀⠀⠀⠀⠀⠀⢸⢠⠏⠀⠀⠀⠀⠀⢇⢀⣾
⠀⠀⠀⢹⡉⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⡈⠀⠀⢰⠂⠀⠀⠀⠀⠨⠀⢸⣿⣿⠀⠀⡇⠘⡇⠀⠀⠀⠀⠀⠀⡼⢸⠀⠀⠀⠀⠀⠀⠀⡎⣇
⠀⠀⠀⠻⠤⣤⡿⢿⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠈⢰⠋⠀⠀⠀⠀⠀⠰⡄⡸⢛⣿⠷⣄⢡⠀⢡⠀⠀⠀⠀⠀⢠⠇⡏⠀⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠈⠛⠚⠋⠀⠀⠀⠀⠀⠀⠀⡌⠀⠂⠀⡏⠀⠀⠀⠀⠀⠀⠀⡇⡏⠈⠓⠒⠚⠈⠟⢛⠀⠀⠀⠀⠀⡌⢀⠏⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡘⢇⣬⠇⠀⠀⠀⠀⠀⠀⢠⠇⡇⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⡴⠁⢯⠀⠀⠀⠀⠀⠀⠀⠀⡗⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠇⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠘⠧⠖⠋⠀⠀⠀⠀⠀⠀⠀⣜⠁⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢀⡹⠀⠀⠀⠀⠀⠀⢸⠆⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢨⣀⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠛⠓⠲⣄⡴⠊⠉⠉⠉⠉⠓⢦⡀⢀⣤⢀⡔⠉⠀⢸⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠏⠘⠟⠀⠀⠀⠘⣦⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠛⣆⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠞⠀⠀⠈⣆⡶⡄⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢀⡄⠀⠘⣆⠀⢸⠀⠀⠀⠀⠉⠀⣹⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠞⠁⠀⢧⢀⣶⡀⠀⠀⠀⣀⠀⠀⠀⢸⡇⠀⠀⢠⢾⠃⠀⠀⠈⠢⣈⠣⣄⡀⢀⣀⡴⣣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢹⠏⢀⡀⠀⠈⠏⠀⢧⠀⠀⠀⢻⢢⠀⠀⡟⢇⠀⢰⠃⢸⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣠⠋⠹⡄⠀⠀⠀⠘⡆⠀⠀⢸⡀⢧⢀⡇⢸⢠⠇⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡼⠁⠀⠀⠹⡴⢳⡀⠀⣷⠀⠀⠀⡇⠘⣾⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣇⠀⠀⠀⠀⠁⢀⡇⢠⠇⠀⠀⠀⣧⠀⠛⢀⡀⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡘⠦⣄⣀⣀⡤⣊⢴⡃⠀⠀⠀⠀⢻⠀⠒⠋⠀⠀⣀⠀⢸⠀⢀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠛⠉⠁⠀⠑⠦⠤⡤⠴⢾⠀⠀⠺⠂⠀⠟⠀⠸⡤⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⡴⠦⣄⡤⠲⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠙⠋⠉⠙⠋⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠀⠀⠈⠁⠒⠒⠤⣄⣀⣴⣦⣀⡤⠔⠒⠂⠉⠀⠀⠠⠀⠀⠀⣀⡀
⠀⠀⠀⢀⣼⣿⣷⠀⠀⠀⠀⠀⠀⡠⠄⠰⠒⢽⣿⣿⣿⣿⠕⠒⠀⠤⡀⠀⠀⠀⠀⠀⢰⣿⣿⣄
⠀⠀⠀⠈⠻⣿⣿⣷⠀⠀⠀⣷⡆⠀⠀⠐⠀⡼⣿⣿⣿⡿⡆⠐⠀⠀⠀⢶⡇⠀⠀⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠻⢿⣧⡀⠀⠀⠀⢠⣶⢠⠎⠀⢿⣿⣿⡇⠘⢦⢰⣦⠀⠀⠀⠀⣠⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠈⠀⠈⣠⠋⠀⠀⢸⣿⣿⠐⠀⠈⢣⡈⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠶⡤⠊⣌⠔⠃⠈⢸⣿⣿⠘⠀⠓⢌⡌⠢⡴⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠁⠀⠀⠀⠈⢿⠏⠀⠀⠀⠀⠉⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠀⢀⠀⠀⠀⢀⠆⢆⠀⠀⠀⢀⠀⠈⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠁⠀⠀⠁⠒⠒⠈⠁

⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⣄⢠⣤⣤⣤⣤⣄⠀⠀⠘⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠞⣿⡎⣿⡆⣿⣿⣿⣿⣿⠀⠀⠀⢀⠎⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⡐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⠋⠈⠉⠉⠉⠉⠁⠀⠀⢀⠂⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⢠⣶⣶⣴⣶⣶⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠁⠀⠀⢀⣠⡤⣤⣀⠀⠀⠈⠀⠀⠀⠐⠀⠃⠈⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⣾⡿⠓⠻⠿⣧⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡉⡉⢙⠛⡛⢛⠛⠛⢛⠛⠛⠛⠉⠉⠀⠘⠃⠠⠀⠀⠀⠉
⠀⠀⠠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠳⡼⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣾⣿⠇
⠀⠀⠀⠀⠀⢐⣢⣀⠀⠀⠠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢋⠠⠀⡆⠠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋
⠀⠀⠀⡀⠀⡨⠻⢏⣿⢶⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢠⣀⡄⡤⣄⢠⣶⠄⣴⠆⠀⣴⡀⠀⠀⢀⠄⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣿⣿⣻⣻⣿⣿⡆⢸⠛⠆⠧⠞⠸⠛⠂⠻⠶⠘⠛⠃⠀⣀⡀⠁⠀⠀⠀⢲
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢰⣿⣷⣠⣤⣀⣤
⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡀⢠⠤⣄⠺⣏⡀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⣿⣟⡉⣿⡿
⠀⠀⠀⢰⡆⠐⣶⠅⠀⠀⠀⠀⠀⠀⠀⠻⣏⢙⡶⠚⠘⠛⠁⠀⠀⠈⠀⠀⠂⢠⡅⠀⠀⠋⠀⠀⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⢠⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠒⢐⣶⣶⡿⣷⣒⡆⠐⠂⠀⠀⠐⠀⠀⠐⠒⠂
⠀⠀⠑⢠⡄⠀⠼⡇⢸⡷⠀⠀⡄⠈⠀⠀⢀⠀⡼⠀⠀⠀⠀⠀⠐⠀⠘⠉⢿⠈⠉⠀⠂⠀⠀⠀⠀⠀⡸⠦⣤⠴⢇
⠀⠀⠀⠀⠐⠒⠓⠉⠋⠚⠃⠂⠀⠀⠀⠀⠘⣄⢣⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠈⠂⠀⠈⠀⠐⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢀⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡀⠀⢀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠐⣀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠁⢂⣰⠉⢧⣀⠈⠀⠀⠀⠀⠐⢄⡁⠶⢈⡠⠄⠀⠀⠠⠀⠀⠀⠀⢸⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⢠⠘⠧⡙⠂⢛⡠⠃⡠⠀⠀⠀⠀⡀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠰⣞⣆⡈⢿⠁⣰⣳⠆⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠙⠒⠀⠈⠀⠐⠊⠁⠀⠀⠀⠀⠀⠑⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠂⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⣀⡀⢀⢠⢠⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢠⠠⢤
⠀⠀⠀⠀⠈⠀⡿⠩⡇⠈⠀⠀⠀⠀⠀⠀⠐⢀⡠⠧⠀⠆⠀⠀⠀⠀⠈⡇⠀⠀⣀⣀⠀⠀⢸⠃⠀⠀⠀⠀⢸⡆⢸
⠀⠀⠀⠀⠀⢠⡚⠚⢢⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠬⠁⠀⠀⠀⠀⠀⢸⡄⣐⣿⣿⢂⢀⡟⠀⠀⠀⠀⠀⢸⡧⡞
⠀⠀⠀⠀⠀⠘⠄⠀⠜⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⢂⠀⠀⠀⠈⠸⣇⠈⠛⠟⠁⣰⠗⠃⠀⠀⠀⠀⢸⡓⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⡇⠀⠀⠀⠀⠈⠀⠘⠂⠀⠀⠐⠃⠀⠁⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂
⠀⠀⠀⠀⠢⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢰⠄⠄⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⢀⠯⠼⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠒⣋⠉⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⠊⠇⠀⠀⠀⢤⡀⠐⠒⠀⣠⠄
⠀⠀⠀⠀⠎⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠠⠀⢀⠘⠶⠃⡀⠀⡄⠀⠀⠀⠀⠀⠚⠑⠂⠱⠂⠀⠀⠐⠐⠀⠋⢈⠁⠀⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⣈⠁⢁⠈⢠⠇⠀⠀⠀⠀⠀⢠⠀⠀⠰⠈⠀⠀⠀⡀⢂⠀⠊⠀⠂
⠀⠀⠀⠀⠀⠦⢤⠤⢚⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠰⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡁
⠀⠀⠀⠀⠀⠈⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡀⠹⣬⣦⣴⣿⣿⣿⣶⣾⣥⠆⢀⣠⠞⠁⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢀⠠⠒⠚⠛⠛⠛⠿⣷⣶⣦⣄⣀⠀⠀⠹⠶⣼⣿⣿⣆⣤⣀⣿⣿⣷⡶⠟⢁⣀⣤⣴⣶⣿⣟⠛⠋⠉⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠒⠋⠉⣩⠛⠛⠛⢷⣶⡾⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡞⢯⣉⣀⣉⣲⣄⣀⠈⠉⠂⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡤⠒⠋⠙⣻⣶⡾⠛⠉⠀⠀⣸⣿⣿⣯⣽⣿⣧⠀⠀⠈⠂⠙⣟⠛⢥⡀⠈⠙⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣀⠜⢡⡧⠀⠀⠀⢀⣰⣾⣿⣿⣾⣿⣿⣿⣷⣄⠀⠀⠀⣀⣧⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣠⠴⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠓⢦⣤⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⢠⠏⢿⣿⣿⣿⣿⣿⣿⠋⢷⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠉⠛⠛⠛⠋⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣤⣠⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣷⣖⣲⣉⣿⣩⣽⣿⣷⣤⣾⣿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⢀⣄⠀⣄⣀⠀⠀⠀⣀⣠⠼⣻⡍⠉⠁⣤⡀⠀⠀⠀⣤⣬⡝⢻⣦⣄⣀⠀⠀⠀⠀⣠⣤⣄⣤⣤⣤⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡇⣴⡟⠹⠏⠀⢉⣀⠀⠀⠻⣿⡀⠀⢸⣿⠇⠀⠀⠀⠘⠛⠳⣆⠀⢸⣿⣿⣿⠿⠟⠛⠛⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⡿⣣⣴⣤⡂⠜⡉⢠⢀⣀⠈⡀⢀⠀⠙⠀⠀⠀⠙⠀⢀⣤⣌⠳⣌⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣹⠁⢴⡏⣁⢉⣼⣴⣶⢾⡄⠁⡰⡴⠚⢃⡠⣄⠂⢲⣦⣤⣤⡙⠋⠀⠹⣦⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⣏⣀⣎⣴⣤⣍⣭⣬⣉⣉⣡⣯⡴⢛⠶⣮⣿⢿⢿⣷⣽⣥⣶⣿⣄⡖⠀⢹⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⣿⣿⠛⠿⠻⠛⠛⢛⡟⠻⢿⡿⠿⠟⠿⣾⣿⣿⣿⢻⣿⢿⡟⢛⣿⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⣽⣻⣿⠏⢰⣿⡇⠃⠀⣠⠀⠸⢿⣷⣤⣤⣤⣿⡏⠀⣰⣯⡵⢿⣿⣯⢻⣯⡽⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡿⠇⣼⣿⠀⢸⣿⡇⠀⠀⣿⣇⠀⠈⡻⣿⣿⣟⣽⠃⣐⣿⠏⠐⠀⣿⣿⡎⢹⣷⡼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⣾⠃⢐⣿⡇⠀⣿⣿⡇⠀⠀⣿⣿⠀⠀⠁⢻⣿⣿⡌⠀⢿⡼⣇⠀⠀⣿⣿⠇⠈⣿⣳⢾⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢀⡟⠀⢼⣿⠀⢠⣿⣿⡃⠀⠀⠙⣿⡀⠀⠀⢸⣿⣿⠁⠀⢾⣷⡿⠀⠀⡞⣿⡆⠀⠰⣿⠈⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⢠⡃⠀⣿⡟⠀⠸⣿⣿⡇⠀⠀⠀⣿⣧⠀⠀⢸⣿⣿⠀⠀⡙⣿⡇⠀⠀⢿⣿⣷⠀⠀⣿⠀⢊⡞⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⡇⢰⡆⠀⣾⡁⠀⠀⣿⣹⠇⠀⠀⠀⢸⡋⠀⠀⠘⡏⠇⠀⠀⢈⡿⠀⠀⠀⢸⣽⡿⠀⠀⣿⠀⠐⡇⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢻⠀⠀⣷⡇⠀⠀⢻⡧⠀⠀⠀⠀⠸⢗⡀⠀⢴⡿⠀⠀⠀⣾⡇⠀⠀⠀⣽⣻⠇⠀⢀⣿⠁⠀⡇⢉⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢸⠀⠀⡑⠃⠀⠀⢸⣆⣠⠀⠀⠀⠐⣿⡄⠀⠸⣷⠀⠀⢀⣽⡇⠀⠀⠀⢿⠋⠆⠀⠀⣿⠀⠀⡇⢸⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡃⠀⣽⡃⠀⠀⢺⠟⠀⡂⠀⠀⠀⡾⠇⠀⠠⣿⠀⠀⠀⣇⠀⠀⠀⠀⠙⢷⠂⠀⢸⣿⠀⢸⡇⣸⣿⢺
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡇⢿⡇⠀⢯⠑⠀⠀⠀⠀⣈⠇⠀⠀⠀⠨⠂⠀⢀⣿⠀⠀⠠⣯⠀⠀⠀⠐⣿⣿⠁⠀⠀⣿⠀⠠⡇⣿⣿⡇
⠀⠀⣀⣤⣤⣶⣿⣿⣿⣷⣾⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⠀⠀⢀⣾⣷⠀⠀⢸⣿⠁⠘⢢⣿⣿⣇
⣴⡟⠋⠛⠛⠉⠉⢘⣏⢻⣿⣇⠈⣗⠀⢠⡇⠀⠀⠀⢠⠀⡃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢈⣿⡷⠀⠀⢸⡧⠀⢈⣿⣿⡟⢿⣷⣶⣄⣀
⠻⠀⠀⠀⠀⠀⠀⣿⣿⠛⠋⢹⠀⣿⠀⠀⣏⠀⠀⠀⠨⡾⣥⠀⠀⠀⠀⠐⠀⢀⣿⠀⠀⣾⠁⠀⠀⠀⢸⣿⡅⠀⠀⢸⡇⠀⠘⣿⣿⣟⣿⡯⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⠈⣇⣿⠀⠀⢸⣇⠀⠀⠀⣷⡿⠀⠀⠀⠀⠐⠀⠰⣿⠀⠀⡟⠀⠀⠀⠀⢿⣿⠀⠀⠀⣠⠄⠀⣴⡏⠻⠿⢿⣷⡄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠘⡎⢿⣆⠀⢹⣿⣦⠀⠘⣿⠃⠀⠀⠰⣅⠃⠀⠀⠀⢰⡆⢸⣿⠆⠀⣟⠀⠀⠀⢰⣿⡷⠀⠀⠰⡟⠀⢠⡟⠀⠀⠀⣾⣻⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⣿⣇⠀⢻⣼⡆⠀⠼⣣⠀⠀⠀⡏⠀⠀⠀⠀⢘⡇⢸⣿⡇⢰⡯⠀⠀⠀⣸⢿⠁⠀⢠⣿⠀⢠⡟⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡘⣿⣷⡀⢻⡟⣄⠀⣻⡆⠀⠀⢙⣛⠀⠀⠀⠘⡇⠸⣿⡇⢸⡇⠀⠀⠀⣻⡗⠀⠀⣸⠏⣰⠏⠀⠀⢀⣼⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⣿⣷⡀⠘⢿⣄⠘⢷⣄⠀⠈⠻⠂⠀⠀⠀⢿⢸⣿⡟⠸⠇⠀⠀⢰⣟⠁⠄⣼⣻⡾⠃⠀⠀⣰⣿⢧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⡇⠀⠀⠻⣧⣄⠙⢷⡄⣻⡄⠀⠀⠀⣮⢘⣿⣇⣾⡄⠀⠐⣼⣿⣼⣯⡾⠋⠀⠀⠀⣸⣿⠃⣾⠁
⠀⣀⣤⣤⣶⣤⣴⣾⡿⣿⡿⠋⠉⠉⠁⠀⠀⠀⠀⠙⠻⢾⣏⣿⣧⠀⠔⠰⡏⢸⣿⣿⣋⣶⡀⣞⣿⠿⠟⠋⠀⠀⠀⠀⠠⣿⣿⣰⣿⡀
⠀⡟⠋⠀⠛⠛⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢶⣦⣾⣧⣼⣿⣟⣿⣡⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢿⣧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣋
⣿⣿⣿⠟⠻⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣦⣶⣶⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠿⠿⠛⠛⠛⠉⢁⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠂⠀⠀⢀⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠊⡉⠀⡤⠀⠀⠀⣲⣦⣴⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠤⢁⠀⠈⢠⣬⠀⠈⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡶⠖⠄⣀⡐⠁⠀⠁⡖⢡⣎⢄⡰⠃⠀⠂⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣡⡶⣷⡞⠁⠉⠡⣮⢔⡫⣕⡠⣻⠟⠁⠀⠀⠈⠀⠂⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠙⢋⣀⣤⣴⣿⣷⣾⣭⡵⠋⠀⠀⠀⠀⠀⠀⢄⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡡⠶⠀⠳⠬⣿⣿⣯⣿⣿⣿⡏⠉⡀⠀⠀⠀⠀⠀⢐⣵⠛⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⢿⣶⣶⠿⠶⠾⡶⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⢀⠐⡶⠛⠀⠀⡠⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠁⠤⠞⠍⠿⠠⣷⠾⡄⡀⠀⠠⠀⠀⠀⠀⠀⠀⡠⠞⠛⠛⠛⣦⡴⢊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡀⠀⠀⠀⠀⠀⠀⣼⣯⣵⣿⠃
⠀⠀⠀⠀⠀⠀⣀⠔⠮⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢀⠠⣪⠾⠋⠉⠃
⠀⠀⠀⣀⢔⡪⢖⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣦⣶⠋⠁
⢀⣴⣿⢟⠟⣨⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣶⠿⣿⠿⠟⠉
⠈⠻⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⢰⡄⢀⢉⠛⠋⣉⡀⢰⠆⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⡄⠀⠀⠀⠀⠀⠑⢤⣆⠀⠀⣰⡤⠊⠀⠀⠀⠀⠀⢠⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⡸⡯⣴⣦⡽⢇⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⣠⢶⢟⣶⣾⡿⣿⣶⣾⡳⡦⣄⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⡁⢰⡿⣷⣿⡇⢸⣿⡾⢿⡅⢸⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡇⢸⠁⣿⣿⡇⢸⣿⡿⢈⡅⢸⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢉⣩⣿⣽⣇⠸⣿⡇⣾⣿⠇⣸⣷⣿⣉⡙⢷⣄
⠀⠀⠀⠀⠀⣀⣀⡠⠤⠴⠾⠟⠁⢠⣯⣤⡄⠉⠙⢿⡦⡤⢤⢴⠿⠋⠁⢠⣬⣽⠀⠙⠿⠷⠦⠤⠄⣄⡀
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⣷⢴⣾⡦⡿⣱⣮⢞⣰⠷⡖⢿⡿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢘⡯⣯⡽⣷⣌⢡⡆⢰⣾⢞⣳⣿⠇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢚⠛⣛⣿⣽⣯⢸⡆⣿⣗⣿⢹⠧⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣀⣃⣻⢤⣻⣹⡷⣏⡖⣟⣭⣤⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⣙⣶⡢⡿⢸⡇⢺⢳⡇⡅⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡉⡍⠫⠃⢸⡇⠙⠫⣵⢛⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠙⠛⠓⠒⠛⠓⠒⠚⠛⠋⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠰⣶⡄⠀⠂⠐⠂⠁⠤⠀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠈⣿⡀⢹⡿⠀⠺⠷⠷⠏⠀⢰⡋⠀⡼⠫
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢈⡁⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⣀⠂⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠃
⠀⠀⠀⠀⠐⢐⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠈⠀⠀⡂⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠂⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⠂⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡘⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢡⢀⡔⠙⣦⠀⠀⠀⣀⡀⠀⠀⢰⡶⠿⣦⠈⠁⠀⠀⠰⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢡⠶⡁⠀⡀⠈⠀⠁⢀⠀⠘⠋⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠋⠀⠀⠀⠀⠀⠉⠑⠈⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⢰⣿⡆⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⢶⣢⣶⣄⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣖⣶⠄⡀
⠀⠠⢀⡐⣒⠚⠍⠁⠀⠁⢳⣿⡞⠀⠀⠉⠹⠓⣀⢀⡀⠄⠀⠈⢧⣕⣿⣯⣃⣹⢁⣸⠗⠠⢶⠂⠚⣏⡀⣏⣘⣽⣿⣚⣵
⠀⠀⣀⣀⠈⠁⠐⠄⠀⠂⢾⢛⡇⠐⠀⠠⠂⠉⠁⣀⡀⠀⠀⠀⠀⠀⣿⣿⣖⡠⣶⣪⣿⣷⣈⣼⣿⣯⣲⢄⣰⣿⣿⠃
⠘⠓⠒⠂⠀⠀⠀⣀⣁⡤⠜⣍⠢⢤⣈⣀⠀⠀⠀⠐⠲⠚⠀⠀⠀⠀⠉⠻⢿⢷⠏⢡⡟⣯⣟⣯⡟⡄⠙⣶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡄⠑⠊⢀⡟⡄⠙⠒⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢶⢄⠃⠀⠀⠈⠻⡶⣶⠞⠁⠻⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠤⠄⢀⣑⣾⣭⣿⣊⡀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢹⢿⣤⡿⠏⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠇⢨⣿⡇⠘⣾⢃⠀⠀⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⢠⠀⡏⠀⠀⠀⢸⠀⣆⠀⠀⠀⠀⠀⠀⠁⠃⠃⢸⣿⣇⠈⠈⠈⠀⢸⠀⠀⢠⣄⠀⡇⠀⣰⠏⠙⠒⠂⢄
⠀⠀⠀⢀⣠⡤⠑⢮⡀⣶⣀⠥⠊⠠⣄⡀⠀⠀⠀⠀⠀⢀⡜⢻⣿⡟⢢⡀⠀⠀⠸⡗⠀⣼⣿⣿⣷⣶⣇⣠⠴⠓⠒⢤⡀
⠀⠀⠀⠀⠤⡤⠀⣀⣻⠭⣿⣁⠀⠤⠤⠀⠀⠀⠀⠈⠓⠋⠀⠀⠉⠀⠀⠙⠚⠁⠀⣈⡛⢻⣿⣟⢟⣿⣽⡷⠚⢧⡀⠀⠈
⠀⠀⠀⠀⢠⠘⠋⣩⣤⠛⣡⣍⠙⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⠈⢉⣻⡿⣿⠿⠇⠀⠀⢱
⠀⠀⠀⠀⠘⠰⡄⠙⢿⣶⠿⠋⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⣰⡏⠘⠁⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⣿⣛⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣻⣿⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠶⠶⠾⠇⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⢀⡽⠀⠀⠰⠶⠀⠀⠀⢰⣿⠍⠂⠀⢹⣇⡀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠤⠀⠉⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⣆⠀⠀⠀⠀⣤⡉⠃⠀⠀⠀⠀⠐⠠⠀⠉⠀⠁⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠁⠀⠀⠳⠬⣀⣼⠃⠀⠉⠁⠀⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣶⢚⣛⣿⣭⡄⠀⠀⠀⠀⠀⣴⣶⣖⣦⠀⠀⠀⠀⠀⢀⠀⢀⣀⣴⣾⣷⣾⣾⣶⡶⡖⠲⣤⣀⡀
⠀⠀⠀⠀⡀⣀⠀⢱⣿⣷⡄⠀⠀⠀⠚⣛⣛⠉⠁⠘⣛⣛⣃⡀⠀⠀⣿⣻⣛⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⣰⣿⣿⡄⣿⡏⠉⠁⠀⣀⣀⣔⣟⣉⢦⣤⠁⠈⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠊⠉⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣿⠀⢈⣷⠈⠁⠫⠄⠀⠉⠉⠉⠉⠉⠘⣿⣦⡀⠀⠀⠀⠀⠀⠰⠶⠶⠆⠀⣤⣀⣄⣤⣄⣀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠃
⠀⠀⠀⠀⠀⡄⠁⢀⢁⠀⠀⠀⠀⠀⢀⣤⣴⣶⣼⣩⣧⣤⡀⠀⠴⠶⠶⠄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠏⠀⠰⢶⡶⠂⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⠀⠂⠀⠀⠀⠀⠀⠿

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣿⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣇⣶⣿⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⠉⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢻⣿⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⡁⠀⠰⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠈⠹⢳⢦⣀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⣱⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡄⠁⢄⠀⠀⠀⠈⠲⣀⢬⣷⡴⠃⠀⠀⠀⠀⢺⢹⢸⠀⠀⠀⠚⣲⢾⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠐⠛⠢⠀⡀⠤⠉⠒⠦⠀⠀⠀⠀⠀⠀⠸⠼⠞⠀⠀⠀⠀⠈⠫⢤⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⡀⠀⠐⠀⡀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠎⢀⠀⠠⠀⠰⢠⡆⠀⠜⠀⠁⠇⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⡀⢀⠀⢀⣔⣁⠠⠠⠌⡘⢰⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⢀⠀⠡⠄⠂⠀⠀⠀⢶⣦⡶⠀⡄⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⠛⠃⠀⠈⠐⠄⠀⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⣀
⠀⠀⠀⠀⠰⠀⠒⠢⢄⡀⣿⡟⠛⠇⠠⠀⡀⠀⠀⠀⡴⠃⠀⠳⡄⢀⠂⠀⣀⠤⠒⠓⠀⠃⠀⣀⠤⠐⠒⠶⠀⠀⠀⠀⠀⠀⠀⠳⡻⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠈⢢⡴⠋⠀⠀⠀⠀⠈⠳⣀⠞⠀⠀⠀⠀⠀⢸⠊⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢤⣶⡄
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡎⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⡱⣄⡀⠀⠀⠐⢄⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⠞⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⢀⡠⠖⠉⠀⠀⠉⣓⠦⠤⣄⣑⣠⣀⠀⠘⢄⠀⠀⠀⠀⠀⠀⣀⠔⢁⣀⡤⠞⢁⣀⠤⠴⢋⠁⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠋⠁
⠲⢎⣁⡀⠀⢀⠤⠂⠁⠀⠀⠀⣠⠈⠉⠉⠉⠙⡛⠛⠒⠒⠀⠚⠻⢟⡉⠉⠉⠉⠉⠠⠀⠀⠀⠀⠀⠢⣀⠂⣀⡈⣓⡤
⠀⠀⠀⢁⠜⠃⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠⠀⠈⣄⡀⠀⠀⠀⠈⠣⣌⡀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠉⠐⠒⠒⠒⠀⢉⠏⠀⠀⠀⠀⣀⠴⠳⣀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣀⠀⠀⠀⠀⠘⡀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠒⠉⠁⠀⠀⠀⠁⠢⡄⢀⠔⠊⠁⠀⠀⠀⠉⠓⠒⠖⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣮⣤⣤
⣿⣿⣏⣉⣛⣉⣙⣙⣏⣋⣽⣻⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢰⠀⢸⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣷⠀⠀⠀⠀⠈⠀⠂⠠⠄⡇⠀⠀⠀⢀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⢠⠀⠀⠰⣧⣇⠀⠀⠀⠋⡘⠀⠀⢸⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⢻⠀⠀⠀⠚⣿⡟⡀⠀⢀⣴⠄⠀⠀⢸⣟⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡿⡆⠀⢰⠐⣿⡇⠀⠀⠀⢸⡇⠀⠀⣿⡆⠀⠀⡌⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡱⠀⢻⠸⡿⡗⠀⠀⠀⢸⠁⡀⠀⡜⠇⠀⠀⠁⠀⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡀⡇⣧⠀⢸⡄⡇⠀⠀⠀⠀⣸⡇⠀⠼⣿⠀⠀⠀⠀⣄⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢣⣿⣿⣾⠃⡁⠀⠸⠀⣾⣿⣿⢸⣆⣾⠁⡄⢠⣧⣧⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢻⣿⣷⡼⣾⡇⠘⡄⢸⣿⣿⣿⣿⣿⣿⣀⡀⢸⣟⢩⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⣿⣿⣇⣼⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣾⣿⡇⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣽⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠁⠀⠁⠀⠀⠹⡿⣿⣿⣿⣿⣿⣿⣿⢸⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⢸⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡿⣿⣿⣿⣿⣠⣾⣿⣿⣿⣤⣿⣿⣿⠿⠟⣹⣿⣿⣟⣸⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠆⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢩⣿⣿⣿⣿⣏⠉⠁⠀⠀⢠⣤⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⢻⣿⣿⣿⣿⣿⣿⣷⠀⠀⣬⠉⣿⣿⣿⣿⣿⣿⠈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢸⣿⣿⣿⣿⣿⣟⠀⠀⠀⠿⠛⠿⠿⣿⣿⣿⣿⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡟⠃⠀⠀⠀⠀⣶⣶⣦⣤⣀⣴⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⢉⡍⠙⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣤⠤⠉⠀⠈⠹⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⣰⣿⢿⣿⣄⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⣿⠆⣿⡇⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠾⠛⠃⠘⢻⡻⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⡿⠛⣿⣧⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠴⣦⣛⠛⡿⠛⡁⡶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠓⢺⣿⣿
⠀⠀⠀⠀⠀⣰⠏⢀⣿⣿⣴⣷⡂⣇⣡⡀⠳⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡃⠀⢸⣿⡏⠑⡀
⠀⠀⠀⠀⠁⠀⢰⠏⢻⣿⣿⣿⣿⣿⠏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣰⣀⣾⣿⣷⡘⠃
⠀⠀⠀⠀⠀⢀⡿⠀⠀⠙⠛⠛⠛⠁⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⡄⠻⡿⠿⢁⠇
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⡇⢀⠀⣿⠀⠀⠘⡄⡷⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢷⣾⣶⣿⣿⣷⡾⢀⡗⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠶⢒⣒⣶⡶⠦⠤⣄⡀⠑⠐⠦⣀⠙⣿⣩⣬⣿⣿⣿⣿⣋⡠⠖⠂⠂⠀⢀⣀⢤⣤⣶⣶⠶⠶⠤⠤⢀⡀
⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈⡉⠙⠒⠘⢿⣷⣲⡶⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡿⠻⠲⠖⠊⠉⠁⠀⠀⠀⠈⠀⠈⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠁⠈⠈⠉⠀⢀⡜⠉⠉⢁⡟⠛⣿⣿⣿⢻⡿⠻⣿⡯⢿⡉⠉⠙⠛⠑⠛⠒⠄⡀
⠀⠀⠀⠀⠀⠈⠀⠀⡠⠊⠀⠀⠀⠀⠀⣠⠴⠛⣹⠀⡀⠀⠜⠙⣼⣿⣿⣿⣦⡛⠀⢦⠳⠰⣿⠓⢤⡀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠊⠀⠀⠀⣷⠈⠂⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄⠀⢈⣧⡿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠁⠹⣷⡶⢿⠻⣿⣯⡿⣿⡿⣿⣿⠟⣶⣼⣿⠏⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠀⠙⢿⣿⣿⣿⠟⠁⠀⡇⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠉⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⡀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⢀⠄⡟⢸⡿⢿⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⢛⣝⡛⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠙⠻⣷⣮⠟⠁⠀⠱
⠀⠀⠀⠀⠀⠀⢀⢴⡄⠀⠈⠉⠁⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡴⠟⠛⢻⡟⠛⠛⠆⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠿⠆⠀⠸⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡇⣰⣄⠘⡃⣠⣆⢰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠠⢤⡄⠀⠀⢀⣤⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠘⢙⠉⣿⠇⠙⢿⠉⡋⠃⠈
⠀⠀⠀⠀⠀⠀⢀⠇⠈⠑⠀⠄⠀⠉⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⡆⠹⣦⣴⠃⢰⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢀⣤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣴⣿⣵⣿⣥⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢛⡿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣼⣻⣄⠁⢀⡩⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠋⠁⠀⢈⣿⠁⠰⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢀⠬⠁⡙⣭⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡰⠷⠆⣰⣞⣷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠂⠀⠀⣀⣼⠳⣶⡻⢮⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⢠⠀⠸⣿⣿⢄⠛⣬⣙⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⡐⠃⠀⢸⣿⢠⡈⢻⣬⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⣀⣠⡞⠃⠀⢠⣿⣿⣷⣼⣻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⣥⣀⣀⡀⣀⡘⣯⡄⢉⣻⡗⠠⣾⣿⣿⣝⢷⡼⠋⠀⠀⠀⠀⠀⠈⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢿⣿⢉⠘⠛⠛⠻⣿⣿⣿⣿⣾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣷⣦⣠⣿⣿⡠⡀⡄⠀⠀⠈⠻⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠆⠛⠉⠉⠀⠉⠛⠻⠉⠻⣥⣡⡀⠀⠀⠀⠀⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⣀⡀⠤⠒⠉⣀⣤⣔⣾⣾⣷⣦⡌⠡⣄⢘⠅⠀⡄⣥⡁⠑⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠁⠈⠀⢰⣦⣴⡒⢘⡷⣺⣿⡿⠛⠉⠀⠀⠘⣿⣿⣿⣠⣼⡧⢀⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠘⡿⣇⢀⣀⣴⣤⡶⢿⢿⣿⣿⣧⣿⠞⠋⠀⠀⠀⠀⠀⠀⠨⣷⠙⠛⣳⠼⠁
⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣷⣠⣸⢾⡇⡛⡽⣿⣿⣾⣝⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣉⢿⣾⣾⣽⣾⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠋⠉⠁

⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢠⣴⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⡐⡐⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⢠⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⡉⢹⣏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆
⠀⢨⣿⣿⣿⣿⣿⣿⡀⠀⢀⣴⣞⡻⣿⣘⣙⣿⣦⣄⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⠃
⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⠉⢀⡀⠉⠙⠋⠉⠈⢙⠆⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⢸⠹⣿⣿⣿⣿⢹⡇⠀⠘⢦⡬⠧⢤⡀⡅⣠⣅⣤⢢⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⡀
⠀⠘⢀⣿⠈⠁⣿⡀⠁⠀⠈⠀⠋⠈⣈⠐⢀⠚⠐⠁⠀⠀⠀⠚⠛⠋⠉⠿⠿⠧⠄⠀⠲⡿⠿⠿⡿⣿⠿⠟
⠀⠀⠈⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⣶⣤⠀⠀⠀⠀⠀⢀⡀⣀⠀⡄⠀⠀⠀⠈⢿⣶⡤⠀⢠⢿⣿⠀⠀⠀⠀⠀⠀⣷⣄⣠⣄⣠⣾
⠀⠀⢠⣶⣾⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷⣴⣿⣦⡌⠀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢈⡿⠿⠿⡇⠀⠀⠀⠀⢀⣿⣿⣿⡛⢿⢿⣶⣿⣷⠆⠀⠀⢿⣭⣿⣿⣭⣿
⠀⠀⠽⢿⣿⣿⣿⣄⠀⠀⠀⣠⣼⣧⣤⣼⣧⣄⠀⠀⠀⠀⠀⠈⠻⣿⣦⣤⣾⡟⠀⠀⠀⠀⠘⡍⠉⠉⢉⠏
⠀⠀⠀⠘⠋⠁⠀⠈⠀⠀⠀⠀⠙⠛⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡠⣲⡌⠉⣤⡈⢡⣖⢄⠀⠀⠀⠀⡠⣲⣯⣥⠬⣬⣽⣶⢄⠀⠀⠀⢀⣴⣾⣿⣀⣠⣄⣦⣀⣸⣿⣦⣄
⠀⢰⢸⣿⣿⣾⣿⣷⣿⣿⡷⡇⠀⠀⠀⣼⠿⠿⣿⠿⣷⠾⠿⣧⡇⠀⠀⢸⣿⣿⠿⣯⡉⠿⢋⣹⡿⢿⣿⣿
⠀⠀⠑⠝⣀⠀⠛⠀⣀⠫⠃⠀⠀⠀⠀⠑⠢⠄⣀⣉⣁⠀⠄⠊⠀⠀⠀⠀⠙⠷⠀⠈⠈⠳⠃⠈⠀⠸⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠾⡷⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠲⠩⡷⠱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢀⣆⢴⣿⣠⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠃⠀⠀⢼⣤⣼⣿⢥⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡞⠢⠠⠀⠀⣼⠽⣋⡿⠿⣯⠙⢉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣸⢷⡫⣼⠁⠀⢹⡔⣋⡺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣰⣗⣲⣶⣟⣒⣒⣒⡷⡛⢗⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣟⣟⣿⣟⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⣋⣿⣿⠿⠿⠛⠿⢿⣿⣏⠹⡋⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠐⠁⣼⠟⠁⠀⠀⠀⠀⠀⠈⢻⣆⠙⠣⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠩⠔⠊⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠐⠃⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠠⣢⡴⣔⡂⠀⠀⠀⠀⠀⠀⢨⠄⠀⠀⠀⠀⠀⠀⠀⡄⢦⡴⠀⠀⠀⠀⢈⣋⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠰⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⢤⡤⠀⠀⠮⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⢀⠀⠀⠈⠇⠀⠀⠀⠈⠀⠀⠀⠀⠃⢲⠈⠀⠀⠀⠀⠃⡖⠘⠀⠀⠃
⠀⠀⠀⡀⠀⠀⢀⣀⡀⠀⠀⠀⠛⠛⠛⠛⢛⢂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⢀⠀⠀⠁⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⣔⠂⠀⠀⠀⠈⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠐⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠂⠀⠀⡋⡇⣿⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⠃⠋⠀⠀⠀⠀⠈⠉
⠀⠀⠠⠤⠤⠤⠤⠤⠠⠤⠤⣤⠤⠀⠀⠀⢤⡄
⠀⠀⠀⠈⠍⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⡤⡄⠀⠐⠀⠀⣤⢰
⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠒⠐⠂⠀⠀⠀⡀⠄⡀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠃⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠸⠇⠀⠸⠿⠄

⣀⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⣀⡀⠴
⢩⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣫⣁
⠈⠉⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢳⣿⠞⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡙⠈⠉⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⢀⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⣿⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡿⠇
⠀⢀⠠⠒⣀⡰⡖⠀⠀⠘⡀⠀⠰⢦⣀⠐⠢⠀⠀⠀⠀⠀⠊⣌⠋⣬⠁⠀⠀⠀⢀⠠⠒⣀⡐⡖⠀⠀⠈⠀⠀⠲⢦
⠀⠁⠀⠀⠀⠀⠀⠐⡇⠀⡰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⠠⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⣀⣤⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣄⡀⠀⢀⠀⡀⠀⠀⣀⣤⣤⣦⡄
⣩⡽⣻⣿⠟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡽⣦⣍⡓⢄⢤⡤⣒⣩⡽⣻⣿⠟⠀⠀⠀⠀⡀⠄⢀
⣦⣼⡿⠃⠀⠀⠀⠀⠀⣤⡀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣶⠛⣩⢤⡙⠳⣦⣼⡿⠃⠀⠀⠀⠀⠀⣤⡀⢠⣄
⣤⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡞⠹⡜⡌⣦⣮⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃⠸⣟
⠾⠋⠀⠀⠀⠀⠀⠀⠀⠠⠃⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠃⠀⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠩⠃⠘⠏

⣀⡀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣀⡀⠘
⢵⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣰⢵⣥
⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠿⠋⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡌⠀⠈⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠢⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂

⠀⢀⣀⣀⣀⡀⠀⠀⣠⣶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡆⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠿⠿⢿⡇⠀⣠⣮⣿⠟⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠤⠄⠀⠤
⠀⠀⠀⢸⠁⢠⣽⠟⠁⠀⠀⠀⠀⠀⠀⢉⣶⣶⣆⠀⢀⢺⣿⠟⠀⠀⠀⢀⠄⣀⠠⠄⠘⠛⡀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢰⣷⣷⢸⣶
⠀⠀⠀⢸⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡗⢸⠟⠁⠀⠀⠀⠀⢠⡀⠶⠀⣴⠢⠶⠄⣁⠀⠀⠀⠀⢀⡀⠀⠀⢐⣧⣕⢸⣨
⠀⠀⠀⠘⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠃⠃⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠶⠖⠊⠁⠀⠀⠀⠠⠟⠁⠀⠀⠘⠿⠟⠸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⢀⣀⢀⣀⣠⡤⡀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣼⣧⣶⠆⠀⠀⠀⠀⠀⠀⠀⡞⠛⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣾⣿⡆
⠏⠉⠉⠉⣀⠀⠃⢀⠉⠛⠒⠲⠀⠀⠀⠀⣰⠋⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟⣀⣠⣀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⣴⣶
⢸⠈⠁⠀⠀⠀⢑⢈⡀⠐⠀⠌⠀⠀⠀⠀⣏⠀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡫⣽⢿⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣿⠛⠁⠈⠉
⠈⠢⡐⠶⠞⡰⠣⡈⢉⣠⠎⠀⠀⠀⠀⠀⠙⢻⣿⣿⡛⠋⠀⠀⠀⠀⠀⠀⢸⣿⣷⣝⣻⣾⣿⡇⠀⠀⠀⠀⠀⠀⣠⠏⠙⣆
⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢰⣷⠄⠀⣀⠀⠀⠀⠀⣀⠀⢀⣀⣘⣿⣀⡀⣀⠀⠀⠀⠀⠀⢀⣙⣲⠤⣞⣋⡀
⠀⠀⡇⠀⠀⣽⣿⡀⠀⢘⠀⠀⠀⠀⠀⠉⠛⠿⣷⣇⢸⡟⠁⠀⠀⠀⠀⠈⠙⠶⡄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣶⠀⠉⠁
⠀⠀⡇⣤⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢾⣽⡶⠶⠶
⠀⠀⣃⣸⣿⣿⣿⣿⣟⣐⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧
⢰⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⡿⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⠇


⠀⢀⡠⢀⣀⠀⠀⣀⣀⢠⣀⠀⠀⠀⠀⠀⢀⡴⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠆⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿
⠀⣿⠀⠀⠀⠙⡏⢸⡇⠀⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢠⠲⡤⢾⣂⡛⢠⠋⠀⠀⠀⠀⠀⠀⠘⡿⣿⡿⠂
⠀⣿⠀⠀⠀⠀⡇⠈⠁⠀⣿⡇⠀⠀⠀⠀⠀⢣⢀⣀⡎⠀⠀⠀⠀⠀⠀⢺⠉⣠⣦⡈⣹⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⣿⣤⣤⣀⣀⣇⣀⣤⣤⣿⡇⠀⠀⠀⠀⠀⠈⣾⣿⠁⠀⠀⠀⠀⠀⠀⠺⢤⠈⡋⢠⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡄
⠀⠉⠉⠉⠉⠙⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠟⠁
⠀⠀⠀⠀⠀⣠⠾⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⡠⠊⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣶⣶⡄
⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠟⠉⠙⣿⣾⣿⣿⣿⣿⣿⣿⠿⣿⢿⣿⡿⠂
⠀⠀⠀⢠⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⡿⣿⠙⠙⠃
⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣷⣿⣆⣀⡀
⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠁⠨⠃
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣌⣉⠁
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⣤⠖
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⡉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢟⣿⣿⣿⣞⣫⣿⣿⣿⣿⡏⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣷⣤⣿⣟⣿⣿⣿⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⠿⣿⡉⠙⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣿⣿⣿⣿⣿⣿⢻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠘⢀⠀⣭⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢠⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⠉⢹⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢸⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠈⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡷⣾⠟⠉⢣⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⢾⡿⣯⣏⣀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⢀⠈⡏⠸⠟⠉⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⢀⣄⣴⡏⣾⣷⣴⢤⣤⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠀⠟⠀⠋⠈⢱⡈⣱⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣌⢻⡷⠞⠢⠄⢀⣸⠇⢹⢇⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣿⠛⣿⣿⡈⠃⠀⠀⠀⠀⠀⠀⠈⢈⡹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠛⣡⣞⣿⢿⣷⠀⢿⠀⠀⢀⣦⠀⢀⠋⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⡘⠷⢸⡟⠸⠟⢹⣿⠿⠀⠀⠀⠀⢸⡟⠘⠛⡀⢘⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠼⠛⢇⣉⣥⠆⠀⠀⠀⠀⠈⠀⢡⣾⡇⢸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡤⠀⣠⡴⢯⡼⠇⢀⣀⣙⠀⠀⠀⢀⡼⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⣤⠀⠿⠁⣼⢟⣾⣿⣿⠀⣜⠉⠁⢰⡆⠀⠀⣠⡿⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⣡⡟⠸⠇⣠⣾⣿⣿⣿⠀⠟⠋⠀⠠⡤⠄⠀⠉⠁⠘⢿⣿⡇⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⢠⣾⣴⣿⣿⣿⣿⠃⣸⠖⠂⠀⠀⠀⠀⠰⢀⡄⠀⢶⡟⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢸⡷⢠⣼⡿⣿⣿⣿⡏⣸⣟⡛⠁⣰⠀⠈⢹⡄⠄⠀⢆⣈⠳⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠘⠁⢈⠻⣷⣿⣿⣿⣿⡟⡟⠉⠀⠈⠀⠀⠀⣤⡄⢀⣸⠇⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⣦⠀⠈⢱⡞⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠈⠉⠀⣼⡁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣏⠘⡿⣾⣤⣿⣿⣭⣾⣿⡿⠖⠀⠂⠀⠀⠀⠀⠀⠀⣰⠛⠁⠀⡜
⠀⠀⠀⠀⠀⠀⢠⣿⣹⡶⠃⠙⣿⣿⣿⣿⣿⣿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠐⠣⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣾⡏⢩⣥⢀⡀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠟⠧⠘⣿⣼⡇⣿⣿⣿⣿⣿⠉⠋⠀⠀⠛⠠⠀⣠⠋⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⢠⠀⣀⡸⠃⣰⣿⣿⣿⣿⠃⠀⠀⠀⠀⠁⢀⡄⠁⠀⠀⣰⠏
⠀⠀⠀⠀⣸⣿⣍⠀⣹⠇⣼⣿⣿⣿⣿⣿⠀⠀⠀⢀⣤⣴⠋⠀⠀⠀⠰⠋
⠀⠀⠀⢰⣿⣿⠃⠀⣿⣾⣿⣿⣿⣿⣿⡿⢀⡀⢀⣶⣿⠟⠀⣠⣷⠘
⠀⠀⠀⠸⠛⠁⠀⡄⣿⣿⣿⣿⣿⣿⣏⡿⠟⣻⣿⠁⣿⣯⣾⠟⡿⠀⢀
⠀⠀⠀⠀⠀⠀⠰⢁⣿⣿⣿⣿⣿⣳⠏⡀⢠⣿⠷⢀⣈⠛⠓⣶⣷⣤⣀⣀⠁⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠇⢍⢤⡛⢹⡤⣶⡏⠉⢉⠉⢉⢙⡁⠈⡏⢉⠉⠍⠉⠛
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢀⡜⣀⠀⠉⢰⣿⡇⠀⠀⠁⢸⡎⠁⠁⠁⢹⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣹⣿⣼⣿⠀⠀⠀⢿⡇⢰⣀⡇⢸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⢀⣿⣿⣿⡿⠀⠀⠀⢸⡇⠈⢿⠃⠘⣿⢠⠀⠀⢸⡀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⣿⣿⢿⣿⠇⠀⠈⠀⠘⣧⠀⣸⠁⢰⣿⡇⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⣾⡏⠀⣼⡟⠡⣬⡟⠀⠀⣸⠇⢀⣿⠀⠁⠀⣸⢹⡇⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⢀⡿⠀⣸⣿⠃⣴⠃⠁⠀⣸⡏⠀⡼⢿⠀⠀⠀⠹⠀⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⢸⡇⠀⣿⡏⡸⣏⠀⠀⠀⢻⣿⣶⣧⣼⠀⠰⠀⠀⠀⠃
⠀⠀⠀⠀⠀⣼⠀⣰⡟⢠⠃⠈⠀⠀⠀⠈⣿⠁⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠘⠀⠾⣧⢆⠀⠀⠀⠀⠀⠐⣿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⡷⠀⠀⠀⠈⠺⡿⢿⣿⠿⣿⠀⠠⠠⢀⠀⠀⠀⠀⠀⢀⠠⢠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣻⣿⡄
⠀⠀⣊⠰⠆⠲⠂⣟⢂⠀⠀⠀⠀⣿⣿⣹⣿⡷⠀⠦⠘⠁⠀⠀⠀⠙⠀⣀⣤⢤⠉⣿⡟⡀⠀⠀⠀⢀⣴⣾⣷⠜⠛⠘
⠀⡆⣍⠁⠀⠀⠀⠿⢀⠀⠀⠀⠀⠘⠷⠶⠿⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠟⠀⠀⠀⢀⠴⠛⠛⠉
⠀⠀⢉⠲⢀⠠⠂⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠠⣶⣖⣲⣶⠀⡁⠀⠀⠀⠀⣀⣤⣵⣾⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⡯⣝⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⢹⣾⠀⠀⠀⠀⠀⠀⡀⢹⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⠠⢀
⠀⠀⠸⠤⠩⠄⠀⡀⠀⠀⠀⠀⣿⠸⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡏⠸⢄⡀⠀⡀⠀⠀⠀⠫⡒⢟⠛⠛⠀⡀⠜
⠀⢀⠐⠀⠀⠂⡀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠒⠛⠛⠀⠀⠀⠀⠀⢸⣾⠃⠀⠀⠙⢧⣼⣆⠀⠀⠀⠈⠐⠎⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⢠⢴⠀⢐⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⣀⡴⠐⠒⠂
⠀⢀⣶⠘⡂⢃⢈⡀⡀⠀⠀⠀⠰⠈⠀⠈⠒⡇⠀⠀⠀⠀⠀⠀⢀⡤⢢⠎⣦⢳⡐⢤⡀⠀⠀⠀⠀⢠⠏⡏⠉⢷⠃⠐
⠀⢰⣾⣾⣷⣿⣾⣷⡇⠀⠀⠀⠠⠀⠀⠶⣶⣶⣶⣶⠆⠀⠀⠀⢛⡤⠛⣤⠛⣴⠛⣴⠋⠀⠀⠀⠀⣾⡀⡇⠀⢸⡆
⠀⢸⣿⣏⣽⣟⣿⣿⡇⠀⠀⠀⠀⢁⠀⢠⠈⢹⡉⠀⠀⠀⠀⠀⢸⡇⠈⠁⠀⠉⠀⢹⡇⠀⠀⠀⠀⠘⠓⣿⣖⠛⠁
⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣻⣿⢿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⢃⣴⣿⡣⡤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⠓⠌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣐⣤⣈⠂⠨⢙⠻⢿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣮⣔⠬⣄⢺⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣿⣿⣿⣿⣮⣼⣿⡇⢯⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⢿⣏⠀⢾⣿⣿⣿⣿⡿⠃⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡍⠽⢳⡌⣿⣿⣿⣿⣿⡖⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠛⣿⣿⣿⠻⡿⠻⣧⡿⡻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⡟⠀⠙⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠈⠈⣿⣿⣿⡿⠟⡫⠂⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠐⠤⣔⡘⠻⠭⠥⠚⢉⡀⠀⡰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡷⠀⠀⠔⠀⠐⢒⠠⠠⠄⢐⣠⠀⣤⣾⣿⣷⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢉⡻⢿⣿⣎⣿⣯⢻⠏⠉⢡⠀⠀⠄⠀⠔⠓⠃⢀⠟⠉⣰⣿⣿⣿⡿⢾⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠛⣼⣿⣾⣿⣎⣿⣿⢿⣁⣰⢧⣦⠒⠀⠀⠀⢀⡌⠿⣵⣸⣿⣿⠟⢻⢴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣽⣿⣿⢿⣿⢸⣿⡟⠉⡿⠜⣻⣮⣴⣶⣦⢋⡂⠀⣾⣿⣟⡥⣿⣼⡿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠾⠛⣟⢛⡛⢫⣞⣋⣰⣶⣷⣾⣭⣵⣾⣿⡿⠈⣻⣿⣿⣦⣿⣿⣿⡿⢛⣽⣾
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠂⣨⣿⣿⣷⣿⣻⣿⢿⡗⣷⢹⡿⢿⣿⡧⣪⣾⣿⣿⣿⣿⡷⢙⢻⣿⡿⢚⣵
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡵⣿⣿⣿⣿⣿⣿⣿⣾⣻⣾⣿⠾⣟⣿⣿⣿⡟⣹⣿⣿⢛⣷⣿⣾⠟⣵⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣾⣿⡿⣾⡿⡿⢃⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣥⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠉⠙⠷⠾⠋⠉⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢣⠋⠀⠀⠙⡜⢀⠜⠁⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⡷⠀⠀⢾⣿⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠀⠀⠀⢀⡠⠻⢖⣾⣷⡶⠛⢤⡀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⣠⢶⢟⣿⡾⣾⡿⢿⣷⢷⣿⠻⡳⣄⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣾⠁⢈⣾⣿⣾⣿⡇⢸⣿⣷⢿⣷⡑⠈⣷⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠸⠏⢸⣿⣿⡇⢸⣿⣿⡇⠹⠁⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠿⣇⢾⡆⠘⣿⣿⡇⢸⣿⣿⠁⢰⡷⣸⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⢀⡤⠾⠿⣿⣷⣄⡹⠿⠟⠺⠿⢋⣠⣾⣿⠿⠷⢦⣌⠻⣷⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠴⠶⠿⠿⠋⠀⠀⡏⣶⣶⡄⠀⠈⠙⢿⡲⡖⢲⠖⡻⠛⠁⠀⠀⣦⣴⣽⠀⠈⠻⠿⠓⠲⠶⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠹⣿⣷⢰⣦⣷⢤⣷⢻⡜⡼⣂⣴⠧⣖⣿⣿⠏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⡗⣞⣙⣷⣌⠒⢩⠱⠱⣭⣿⡶⢧⣾⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡆⣾⣯⡿⣿⡔⣟⠲⢸⡇⢰⣽⣧⣾⢻⣶⢣⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡤⠘⡶⣻⡕⠯⣯⢼⡇⣻⠽⡳⣟⣸⠉⠙⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢠⣤⣥⠽⠓⣎⣿⢼⡏⢷⣼⠉⡯⣶⠶⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣈⢲⡞⢺⡯⢸⡇⢸⡟⢻⡆⠕⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡖⠛⢼⠙⢨⠀⢸⡇⠘⡛⣏⣧⠚⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠶⣿⣦⣀⣤⣼⣧⣄⣀⣶⣿⠿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡄⡀⢰⠂⠀⣿⠹⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⣼⠀⢠⣿⠀⡄⠀⢀⡄⠀⣿⣷⡀⠀⠀⠀⠀⠀⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⢸⣿⣇⠋⠀⢸⣿⠀⢿⣄⠌⢣⣴⠃⢻⣷⠆⠀⠀⠀⠀⠈⣿
⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣷⠀⠀⢸⡏⣿⡇⠀⢸⣿⠀⠘⣿⠀⠀⠁⠀⠒⣿⠀⠀⠀⠀⠃⠀⡽
⠀⠀⠀⢿⡀⠀⠰⡀⠀⠀⢸⡀⠀⠀⡇⠀⠀⣸⠁⡆⣤⠀⢸⣿⡄⠀⢿⣷⠇⠀⡴⠀⣿⡀⠀⢰⡆⠀⠀⡄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠹⣦⣀⡁⠀⠀⠀⢿⡄⠀⡇⠀⠀⠠⠀⡟⢸⡄⢸⣿⡇⠄⠀⣿⡇⢸⡇⠀⢹⡇⢀⣾⠃⠀⠸⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀⠀⠈⢿⣀⡇⠀⠀⠆⠀⣿⡰⣿⣼⢹⡧⣤⠀⠸⣧⡿⠀⠀⠘⣷⣼⠏⢀⡜⠀⠀⠀⠀⠀⠀⡄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢆⡀⢸⣿⠃⡆⠀⠀⠀⣿⣷⠈⣿⣸⣇⢰⢀⠆⢸⡇⠀⠀⠀⣹⡇⣄⠞⠁⠀⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⣿⠀⡇⠀⢠⢀⠈⡿⠀⠏⢹⡟⠰⡎⠀⣿⠃⠀⠀⠀⣿⠇⡇⡀⠀⠀⠀⢀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⠃⠀⠀⢸⣰⣇⠘⠀⢸⡟⠘⠇⠀⣿⠀⠀⠀⢀⣿⠀⣇⡇⠀⠔⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⢸⣿⠙⡃⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣸⡏⢠⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣇⢰⡆⠀⣿⠀⢿⠀⣾⠇⠀⢻⡀⠁⠀⠀⠀⣿⠀⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢾⠻⡇⠀⢿⡆⢺⣧⣿⠀⠀⠀⠁⠀⠀⠀⠸⠇⣸⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣠⣷⣀⣤⡄⠨⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⢿⣿⣿⣿⡧⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣸⢃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣏⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⣿⣿⣿⣷⣶⣶⣤⣄⡀⠀⢀⣤⣤⣴⣶⣾⣶⣿⠿⣿⣷⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠻⠯⢿⣿⣷⡄⠈⣿⣿⡿⠁⠘⣻⣿⠀⣛⣿⣿⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣼⣿⣟⠙⠿⠷⣶⣾⣿⣿⠇⣾⣿⠿⣷⣶⣾⠟⠛⠀⣿⣯⣵⠇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇⣀⣤⣴⣦⣿⣿⠀⢻⣿⠀⠀⠉⠀⠀⠀⢸⣿⣻⣿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋⠉⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣤⣀⣠⣦⣦⣀⠀⠀⠀⢨⢙⣿⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⢹⣦⠀⠀⠀⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠟⠂⠀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣿⣿⣿⣿⣍⣩⣽⣯⣿⣶⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄⠀⠉⠁⠉⣽⣿⣿⣿⠁⢀⣨⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⢶⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⡟⢿⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠃⢀⠿⣿⣧⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡍⣩⣿⠀⠉⠀⠀⠈⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢙⣁⠀⠀⠀⠀⠀⠀⠀⢻⣷⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⢀⢨⢿⠠⠶⣾⣁⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣼⣿⡎⠀⠀⠀⠐⣿⣿⡄⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢆⠀⣿⣿⣷⢀⢀⠀⣨⣿⣿⡇⠀⠀⢀⡴⣡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⡆⠈⢛⣾⣾⣾⣷⣿⣭⣤⣁⡀⢀⢾⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠟⠙⠛⠉⠁⠉⠉⠙⠹⣿⣻⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠈⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢊⣶⠔⢀⡀⠀⠀⠀⠀⠀⣀⢠⣿⣤⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⣤⡴⠊⡐⡂⣺⢿⣿⣾⣤⣴⣆⡄⣄⣮⣷⣿⣭⠭⢳⠀⠈⠳⣤⣄⣄⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠚⠓⢚⠍⠁⠀⡰⠃⠉⢉⣹⣿⠻⠿⡏⡿⣟⠿⢻⣙⣓⠚⠃⢣⠀⠀⠈⠛⠒⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⢁⣞⣯⣽⡃⠀⠀⠁⠀⠈⠁⣀⣠⡏⠿⣭⣛⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠈⢠⡙⣆⠋⠹⠓⠶⠠⠖⠒⠚⠛⠉⠁⠘⠄⢀⡀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢠⠁⠀⠈⣇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠇⠀⠰⠀⠘⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠞⣽⠟⠒⠀⠄⠄⠀⠀⠤⠠⠐⠊⠈⠙⠀⢲⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⢇⡾⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⣧⠀⠀⠻⣝⡂⠃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⠀⠀⠿⠾⠧⡀⠀⠀⠀⠀⣀⣀⡥⠘⠏⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⢀⡰⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⠶⠤⢄⡀⣄⠤⡤⠾⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⢀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠠⠀⠀⠂⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⡠⠲⠶⣀⠈⠁⡀⠘⠀⠀⠐⠀⠀⠡⠀⠯⠀⢠⡠⠀⢀⡀⠙⠀⡟⠲⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⣴⣦⣀⠎⠺⣏⠁⠄⠀⠀⠀⡀⡀⢈⡽⠁⣾⣿⣿⣷⡄⡅⠀⠀⠀⠀⣴⠀⠀⠀⠀⡴⠀⢀
⠀⠀⠀⠀⠀⠀⡐⠠⠄⠀⣡⡌⠨⡀⢀⣍⠀⠄⠈⢉⡦⠀⢠⣤⠀⣀⣤⠦⠀⠄⠀⢿⣿⣗⠁⠀⠀⠉⠀⠀⢀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠐⠉⠀⠑⠂⣀⡉⠀⡙⠃⢀⠀⣤⠐⠄⠀⠃⠀⠀⠰⢿⣷⣶⣿⠟⡀⡀⠀⠀⣀⣰⣿⣦⣀⠂⠀⠀⠉⣄
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡌⠀⠐⠋⡂⠄⡊⠀⠈⠐⠛⠉⣀⢠⡘⢀⠠⡈⠙⢹⣿⣿⣷⢾⡠⠀⣀
⠀⠀⠀⠀⠀⠈⡡⠒⢄⠠⠚⢢⡠⠄⠰⡌⢃⡀⠔⢶⡆⠀⠀⠆⡉⠁⡤⢑⡦⡀⠀⠀⢀⠇⠐⠀⠘⠛⢷⢴⣼⣿⣿⣿⣾⡄⣠
⠀⠀⠀⠀⠀⠀⠤⣠⡖⠒⡤⡔⠒⣤⡦⠬⣉⡉⣆⠀⡀⠰⠤⣋⠁⠀⣄⢀⡀⢀⣀⠦⠐⢀⠐⠦⠀⠀⠀⣠⡍⠙⠛⠛⠙⣢⠐
⠀⠀⠀⠀⠀⡘⠁⡅⠀⠀⣄⢠⢀⡀⠐⢀⠃⣀⡄⢀⢤⠂⠒⠉⠉⢰⠠⠄⠀⠀⠀⠀⠀⢞⢸⣸⠀⠀⠀⠉⠀⠀⠀⠀⠐⠊⠘⠚⠄
⠀⠀⠀⠀⠀⠈⡀⣷⠀⠀⠱⣶⢹⠃⢳⡏⡀⢑⡼⠃⠺⠑⠄⢀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠾⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡅⢹⣠⠀⣂⣿⣲⠀⡁⠈⠀⣿⡗⠈⠀⠐⠤⢌⠁⡀⢘⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⢐⠘⢿⢻⢟⡟⣷⢒⢵⣠⠄⡙⢷⣤⣄⡀⠘⠉⠈⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡖⠂⠀⠀⠀⡘⠉⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡌⣾⣿⣯⡟⠓⠄⠐⠋⠆⢈⠨⠑⡄⠀⡀⢁⡰⠀⠀⠀⠶⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠴⣀⠌⢽⠰⡖
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠈⠉⠉⠀⠓⠀⠀⢤⣄⣈⣷⡀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠸⠀⠀⢠⣠⠄⠀⠀⠂⠀⠀⠀⠀⡼⣆⣻⣎⠌⢧⠁
⠀⠀⠀⠀⠀⠀⠀⠤⡁⢫⠵⢦⡄⠐⢣⡄⠀⢀⡄⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠠⠁⠀⠀⣐⠄⠀⠒⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⢉⣋⣈⠃⢈⠣⠤⠤⠤⠤⢤⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⠀⠀⠀⢉⢷⢖⣸⠃⢈⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡿⠿⢍⠙⣿⠿⣿⣯⣟⠻⢿⠿⠂⣠⡄⠀⠀⠀⠀⠀⠉⠁⠨⠀⠀⠀⠀⠠⣴⢺⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢳⢧⣌⣉⠈⠈⠀⠀⠀⠀⠉⠳⡄⠔⢨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣿⣷⡄⠠⠄⠀⡀⠀⣄⠀⠀⢆⡀⠀⠀⢆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠏⢿⡁⠀⠀⠀⠙⠀⠈⠀⠀⠀⠳⠀⠀⠈⢠⡄⠀⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠄⠳⡀⠀⠀⢀⠀⢀⡀⠀⠀⠀⢇⠀⠀⠘⠇⢰⠇⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠸⣆⠀⠸⠀⣿⠁⠀⠀⠀⢸⡀⠀⠀⢸⡇⠀⢀⡏⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠻⠀⠀⠈⡄⠀⠃⢾⣀⡀⠈⡁⠈⢳⣄⢰⡾⠀⠀⢘⠃⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠂⠀⢀⡤⠀⠀⠲⢶⡆⠀⠀⢫⡼⡀⠀⠀⠉⢿⣿⠃⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠹⢦⡀⠘⣇⠈⢆⠀⠀⠈⣿⠀⠀⠀⢿⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠟⠀⠀⠀⠀⠀⢷⡀⠙⢷⣄⡇⠰⠆⣿⠀⠀⠈⣿⠁⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢘⣧⠐⠀⠀⠀⠀⠀⠀⢹⣦⣦⡿⣷⣶⡴⢿⣷⣶⣶⣿⣶⡶⣶⣶⣶⠶⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⠿⣿⡿⠛⣿⠃⣼⣿⢻⣿⠋⡟⢱⣿⠿⠿⡷⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢰⡟⢺⡇⠘⣷⡀⣻⣧⣸⣿⢸⡏⠀⣿⡟⣿⠀⡀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⡆⠀⠀⠀⠀⣼⣃⣚⣓⣻⣟⣻⣿⣿⣿⣿⣸⣿⣐⣛⣛⣛⣈⣋⣁⣘⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣘⣛⣃⣀⣀⣀⠀⣿⡉⠉⠉⠉⠉⠉⠉⢻⣿⡏⣹⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⠀⢀⢀⡀⠀⠀⠀⢿⠉⢉⠍⠉⠉⠉⢹⡿⠀⣿⠀⠀⠀⣀⣀⠀⠐⢿⣿⣿⣿⠃⠂⢠⣄⠠⡄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠖⢤⠘⢷⡀⠀⠀⠸⣷⣼⣿⠛⠓⢰⠟⠇⠀⡿⠀⠀⠂⣹⡏⠓⠀⢾⣿⣿⡿⠤⠀⠨⡿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠐⠀⠘⠳⡤⠃⠀⠀⠄⠙⣻⣷⣦⣐⠚⠂⢀⠀⠁⠤⠀⠀⢸⡇⠀⣠⣼⣿⣿⣗⣲⣾⣶⣿⣿⣷⣦⣦⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢀⣤⡀⠀⠀⠠⠘⢿⣄⠆⠀⣻⣿⠋⠉⠉⠙⢷⢏⡀⠀⡀⠀⠠⡈⠡⠀⠠⣿⣿⣿⣿⣤⣿⣭⣽⣿⣯⣽⣿⣧⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⢀⠀⠙⢢⡀⠁⠀⣙⣿⣿⠀⠀⠀⠀⢨⡾⢁⣾⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿⣿⠛⡙⠛⠙⠛⢙⠛⠛⢉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⠀⠀⠀⢻⣶⣿⣿⣿⠾⠂⠲⣶⢻⣿⣿⡟⠉⠀⠀⣶⣶⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠸⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⢠⡀⢈⡉⢳⣦⡀⢻⣿⣟⢻⣉⣎⣷⣿⣯⣿⣋⣽⣿⠐⠂⠀⣿⣇⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠰⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⢿⡿⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⣀⠀⢰⡄⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡟⠓⠀⠀⠂⠀⠋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠻⢿⠗⠀⠀⠉⠈⠉⢻⣿⣿⡿⠏⠙⠀⠉⠀⣙⠿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠛⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⢨⡄⣠⣤⣤⣼⣿⡟⢧⣄⡰⠀⠰⣶⣜⠀⢹⣿⠺⠻⠃⠀⢿⣷⡆⠀⢿⣿⣿⣿⠆⠀⠀⠀⠀⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣷⠉⠙⠋⢹⣿⣶⣿⣿⣿⣄⢰⣿⣿⣷⣿⣿⡀⠀⠀⠀⢸⣿⡇⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠘⠋⠀⠀⠀⠜⠛⠉⠁⠉⠉⠉⠁⠈⢩⡍⠙⠛⠄⠀⠀⠀⢸⣿⡅⠀⣿⣿⣿⣿⠽⠄⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠐⠒⠂⠀⠠⠤⠤⠤⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠏⠁⠀⣿⣿⣿⣷⠀⠀⠀⠲⠒⠤⠔⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⠀⠀⠀⠤⠀⠶⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⠘⠉⠀⠉⠛⠷⠄⠀⠘⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠁⠘⠛⠚⠶⠦⠤⢤⡤⠤⠤⢤⣤⣤⣤⣤⣶⣶⣶⣶⡶⠾⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠂⠀⠂⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠠⠀⠨⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄






⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣉⣿⣋⣙⣋⣉⣉⣀⣩⣉⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣥⣤⣥⣤⣤⣤⣿⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢰⣾⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣷⣦⣄⡀⠠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣿⣿⣶⣦⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⣀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣯⣉⡙⠿⠿⣿⣿⣿⡿⠿⠋⠙⠻⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠈⠓⠂⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣍⣭⣿⠁⠀⠀⣀⣀⣀⣉⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣄⣠⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠈⠻⣿⠻⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⢳⡤⣀⠀⠀⡞⢹⣷⣶⣄⣀⠀⠀⠀⠀⠈⢳⣄⠀⣩⣽⣿⣿⣿⣿⣿⣟⣋⣉⣤⣤⣴⣦⡀⠸⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡑⡼⠃⠀⠛⢿⣿⣿⣶⣄⢀⣰⣶⣶⣽⣿⣿⣟⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⢻⣷⣤⣬⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡆⠀⢀⣠⣬⠿⠿⠿⠻⠟⠉⠛⣿⣿⣿⣿⣿⣿⣿⡿⢻⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢷⣮⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⡟⠻⠛⠋⠀⠀⠀⢠⣤⣴⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣋⡁⣀⣠⣤⣶⣾⠶⠿⢿⡯⣁⣀⣀⡁⡀⢀⣠⣤⡄⣀⣀⣠⣆⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⣸⣿⡿⠟⠉⠁⠀⠀⠀⠀⠙⠋⠉⣹⢛⣟⠛⠣⡈⢑⣄⠀⠈⢮⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠾⠷⠁⢠⡿⣾⣨⣦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡇⠀⠀⠈⠃⠀⠘⠁⠀⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣰⠇⠀⠀⢀⠦⠀⢤⡀⠀⠀⠀⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠈⢙⢃⢀⣘⡃⠁⣀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠛⣱⢫⣿⠸⣿⣻⡍⠙⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡄⠉⣿⣿⠐⣿⣿⠁⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠛⠓⢤⣹⣿⠠⣿⢃⠴⠟⢛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣿⣿⣦⡀⠈⢦⠤⡖⠁⢰⣿⣿⣿⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠏⠀⢻⣿⣿⠃⠀⢸⣀⠁⠀⢰⣿⣿⡇⠀⠹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⣿⡿⠀⠀⠀⣿⠀⠀⠈⢿⣿⡀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢼⣿⣿⡆⠀⠀⠀⡿⠀⠀⠀⣿⡿⣿⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢻⣿⣻⠀⠀⠀⡇⠀⠀⢀⣿⡏⢻⡇⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⠘⣿⣿⣷⣄⠀⣇⠀⣠⣿⣿⡇⣸⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢻⣿⣿⣿⡆⡧⣼⣿⣿⣿⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠈⢿⣿⣿⣧⣿⢻⣿⣿⠏⠀⢿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠁⠀⠀⠈⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜

⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢸⡅⠀⠀⢀⡗⠀⠀⠀⣠⡤⢠⢀⢄⣠⣀⢄⢠⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⢢⠀⢻⢛⡛⢿⠃⡀⠀⠀⠙⢦⡀⢢⣾⡷⣿⣦⠂⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠣⢼⣯⣽⡯⠔⠁⠀⠀⠀⠀⠹⣦⣀⣹⣋⣠⡼⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢎⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⢀⣺⣿⣷⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠸⣶⣧⡶⠄⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡠⣼⠿⢿⢤⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⣯⡧⠀⠀⠀⠀⠀⠈⡜⢹⣄⣸⢱⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⠈⠁⠈⠻⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣽⣷⣇⠀⠀⠀⠀⠀⢐⣛⡂
⠀⠀⠸⣿⠇⠀⠀⠀⢀⣿⣷⣾⣧⠀⠀⢀⡜⠻⡿⠛⡀⠀⠀⠀⣿⣽⣿⣯⡟⠀⠀⠀⠀⠹⣿⣿⡐⠄
⠀⠀⠀⣉⠀⠀⠀⠀⢸⢀⢠⢇⣭⡇⠀⠀⣵⣦⣶⣶⡉⠀⠀⠪⡇⠈⠛⢀⣶⡅⠀⠀⠈⠁⢸⡏
⠀⠀⠀⠉⠀⠀⠀⠀⠰⠻⠾⠾⠣⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠰⡹⣴⣶⣿⢿⠃⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣟⣛⣛⣛⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣠⣸⠒⠒⠒⠒⠒⠚⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡅⠀⠀⠀⠀⠀⠠⠲⣶⠶⠶⣿⢿⠀⠀⠀⠀⡀⢸⠼⣿⠶⠶⣶⠖⠄⠀⠀⠀⠀⠀⢈⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠘⣭⠃⢚⣍⢻⠀⣇⣾⣦⡇⢸⢸⣩⡓⠘⣭⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠈⠸⣿⠠⠾⣿⢿⡖⣻⣿⣿⡛⣾⠸⣿⠷⠀⣿⠇⠁⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⡃⠀⠀⠀⠀⠀⠨⠈⣭⠃⢚⣭⢻⣳⡏⣭⣍⣿⣿⢸⣭⡓⠘⣭⠁⠁⠀⠀⠀⠀⠀⢚⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣿⣾⣶⣷⣿⣷⣶⣶⣶⣿⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣾⣷⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⣿⣷⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⢹⣿⣿⡿⠛⠟⠉
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣾⣷⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⢤⣤⣀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠘⣿⣳⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣿⣯⣦⣀⠀⠀⠀⠀⣀⣠⣾⣿⠏⠈⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡟⠀⢈⣿⠛⠉⠁⠉⢩⣉⠉⠉⠉⣶⣂⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣿⡀⠀⢀⠘⢻⠋⢀⣀⢀⣼⣿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⣿⣿⣿⣿⣤⣤⣤⣿⣿⣿⣿⡏⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⣟⠛⠛⠋⠉⠀⠀⠉⠙⠛⠛⢳⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣤⣠⡀⡀⢰⣾⣶⣦⣤⣤⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢁⣇⢸⠟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡘⢿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⠁⢀⣴⣮⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⣿⠂⠉⠉⠁⠀⠀⣤⣠⠈⠁⠁⠀⣀⣻⡟⠀⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⣦⣤⣤⣤⣀⣀⣨⣧⣄⣤⣴⣶⣿⣿⣧⠀⢸⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠟⠋⠃⠿⣿⡇⠸⠇⠶⠖⠗⠒⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠖⠛⠛⠳⠓⠚⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⡟⣏⡟⣏⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⣧⢹⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣿⠁⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢹⣿⣾⣿⣿⣿⣿⣿⣿⠀⠀⢠⣄⣀⣾⡟⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣶⣷⣼⣿⣿⣿⠃⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠷⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠿⠛⠛⠿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠃⠘⠳⠆⠀⠀⠀⠈⠟⢉⡀⠠⠀⠚⠋⠙⠏⡙⠉⠟⣙⣋⠝⣻⣭⣽⣿⣿⣏⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠋⠀⠈⠀⠀⠀⠀⣀⣀⣀⣀⠐⢀⣀⡀⠴⣦⣄⡙⢿⣿⠟⢋⣼⣿⣿⣿⣿⣿⣿⡄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠋⣭⣭⣤⣶⣶⣶⡞⠀⣀⡀⠀⢻⣿⣿⣿⢾⣿⣿⣷⣶⣶⣿⣿⣦⣄⡀⢈⣿⣿⢿⣿⣿⣿⣿⣿⡄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⠟⢰⣿⣿⡿⣿⠀⠻⣿⡟⠀⠿⢟⣿⣿⣿⣿⣿⣏⠁⠹⣟⣿⣅⣠⣾⣿⣿⣿⣿⣿⣄⠀⠙⢟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⠋⠀⠸⣿⠟⠁⠄⠀⠀⠀⠀⠀⠠⠟⠙⢿⣿⣿⣿⣿⣇⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⠜⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⢿⣿⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⢀⠎⠀⠀⠀⠈⢻⣿⣿⣿⣿⣆⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢏⠀⡅⠀⠀⢀⣼⣿⠁⠀⠀⠀⡔⠃⠀⠀⠀⠀⠀⠀⢿⣉⣿⡏⢿⣆⠀⠀⠀⠀⠐⡼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⡀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⢀⣽⣟⠀⠀⠀⠀⠀⠀⢘⣿⣼⡇⠈⣿⡀⠀⠀⠀⠀⠈⢻⣻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠁⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⣄⣀⣼⣿⠏⠀⠀⠀⠀⠈⠀⢻⡄⠀⠀⣀⣀⣠⣼⣿⠉⡇⠀⢸⣧⠀⠀⠀⠀⣷⠀⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⠀⠀⠀⠙⣻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⢿⣯⣽⣿⣿⡗⠀⡄⠀⢸⡿⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠈⠀⡉⣶⢻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠙⠿⠛⣿⠇⣰⡇⠀⢸⡇⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣆⠀⠀⢳⡀⣾⢻⣿⣿⣿
⠀⠀⠀⠀⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢏⠀⠿⠀⠇⠘⡇⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⢻⣿⣿⣿⣿⣆⠀⠈⢲⡙⣿⣿⣿⣿
⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠈⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡜⣿⣿⣿⣿⣿⣧⠀⠀⢹⠿⣿⣿⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠃⠘⣭⣿
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⢐⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣻⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠐⣸⣿
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣷⢾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠩⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣹⣿⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢿⣿⣿⣟⣿⣿⣿⣿⣷⡀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣤⣈⡟⠻⠛⠻⠀⣶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠹⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣷⣦⣄⠀⠀⢹⣧⣀⠀⠀⠀⠀⣼⣿⣷⡄⠀⢹⣿⣷⣼⣻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⣾⡿⢼⣷⣤⠀⢸⣿⣿⣿⣿⡀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠽⢿⣿⣿⣿⣿⣿⠀⢻⣦⡀⣿⣿⣷⡜⢿⣿⣿⣿⣇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡁⠉⣿⠀⡿⣿⣿⡽⣿⣿⣇⢸⣿⣿⡿⢿⣧⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⢻⠀⠗⠈⢿⣿⣽⣿⣿⢿⣿⣿⣷⣄⠈⡇⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⠀⢿⢿⣿⣷⡀⣮⣤⡙⢿⡇⢠⣶⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡞⠀⠀⠇⠀⠀⠀⠀⠈⠘⢿⣿⣷⡈⠻⢿⠆⠀⢰⣿⠆⠛⢣⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣍⠻⢿⣤⠀⠀⠀⣰⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⣻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣯⣽⣿⣽⣿⣿⣿⣿⣿⣿
⣶⣴⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢣⡀⠀⡀⢠⣠⠠⣤⠀⡄⠀⡰⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠑⢄⣠⣶⡟⠂⢾⣷⡡⠚⠀⠀⠀⠀⠀⢠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⢙⠀⠃⠀⠀⠸⢣⠀⠀⠀⠀⠀⠒⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⣠⡼⠄⣀⡀⠚⠃⠧⡀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢄⡀⢠⠏⠀⠀⠀⠀⠀⠀⢀⢸⡄⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⠀⡄⠀⠀⠑⠀⠐⠟⣼⡷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣃⠀⣀⣘⣦⣀⣈⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠉⠁⢸⠈⠉⠹⠇⣽⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⢻⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠻⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠁⠀⣿⣆⠀⠀⠀⠀⠀⡆⣿⣾⡇⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣿⠀⠀⣄⢀⠸⡷⣿⣿⣿⠀⠀⠀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⢴⣿⣿⣼⣆⠹⢸⣷⣷⣿⣿⣿⣷⡀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠏⠀⠀⢠⠜⠹⣏⣿⡏⣸⢸⣿⣿⣿⢻⡏⠀⢧⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⢿⣿⣷⡀⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠈⠻⢿⣧⣿⣿⣿⠟⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠐⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄

⠀⠀⠄⠀⠀⠀⣤⡤⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠤⣶⠄⠀⠀⢀⠊
⠀⠀⠘⡂⠀⠀⠈⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⣀⠀⠀⠀⠀⣠⡴⣦⡄⠀⠀⠀⣀⡤⠔⠂⢀⣯⡄⡀⠀⠀⠠⠦⠀⠀⠀⡠⠃
⠀⠀⠀⠈⢆⡀⠀⠀⠐⠀⠀⠻⠱⡂⣄⡀⠂⠨⢉⣓⣼⣏⣥⣼⣧⣉⣾⣴⡫⢛⠀⠀⠰⠬⠛⠊⠀⠀⠠⠀⠁⠀⠠⠔
⠀⠀⠀⠀⠀⠱⠄⢆⠀⠀⠀⠀⣠⣀⣀⣀⣀⣈⡠⠾⣤⣘⣿⣷⣿⣟⣃⣽⠷⣤⢤⢄⣠⣤⣄⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠙⢢⣿⣿⣿⡛⢻⣿⣿⣿⠂⠁⠉⠁⠀⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⡿⣿⣿⣟⢿⢿⡋⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢣⣿⠿⣿⡿⢿⡗⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⠛⠂⣽⡀⠙⡑⠏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⡿⠿⢿⣋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠑⣒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣤⣄⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⡂⠀⠀⢀⣴⡿⠿⠯⠥⠤⠤⠴⠿⣦⣄⡐⢤⡐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⣷⣭⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠚⠀⢉⣹⣦⣍⣿⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠈⠉⠻⣌⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢾⣆⣼⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠐⠲⠲⢶⡿⠃⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠠⠀⠀⠀⠀⠈⠀⠀⠉⠛⢦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢠⣾⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢹⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠀⣴⠏⣠⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸
⠀⠀⠀⠀⠀⠀⢰⡈⠀⠰⠀⠘⣧⠀⡰⠁⠀⠀⠀⠀⡀⣠⠒⠁⠈⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠆⢰⠀⢇⠀⢛⣼⠃⠀⠀⡀⠔⠂⠀⠀⠀⢄⡀⠀⣀⣄⣳⣄⣠⠀⠀⠀⠀⠉⠛⠷⠀⠀⠀⠀⣀⣄⣠⣀⣸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣌⢷⠀⣿⠀⠀⠊⠀⠀⠈⡠⡇⠀⠈⠻⠿⢿⣿⣿⣿⣭⣙⢢⡀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠉⠙⠻⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣷⣷⣿⠀⢀⣤⠊⠀⠤⠀⠁⠘⠃⣤⣤⡀⢹⠾⢻⠃⠈⠱⣷⠤⠤⣄⠀⠀⠀⢀⣀⣤⣄⠀⣠⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣁⠁⠀⠈⢿⡟⠀⣾⠃⣠⢢⡏⢠⠀⠀⠀⠀⠉⠻⣿⠀⠀⠀⠀⠀⠉⠀⠀⠘⠇⠀⠚⢿⣯⡷⢌⢻⣿⣷⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠱⡀⠟⢸⡇⡎⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠷⢶⣦⡞⠁⠀⠀⠽⠟⢧⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⡇⠀⠀⠀⠀⠷⢾⡇⠀⠆
⠀⠀⠀⠀⠀⠀⠘⣆⠱⡄⠀⠀⢤⡀⣘⠀⡿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⣀⡴⠟⠋⠒⠄⠀⠀⠰⠁⠀⠀⠀⠀⣠⠾⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠷⣌⣢⣀⠀⠳⣿⣦⣷⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣠⡞⠁⠀⡿
⠀⠀⠀⠀⠀⠀⢠⠃⡐⠉⠌⠙⠛⢷⣿⠋⡍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⠈⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠈⢻⣌⠀⢳⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⣤⠀⠠⢔⣀⣀⣀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣠⣾⠛⣦⠀⠀⠀⠀⡄⠀⠀⠀⠀⠛⠀⠀⠀⠉⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢸⡿⢣⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢠⢸⡟⠘⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠐⠄⢄⠀⠀⢀⡀⡀⢾⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡦⣤⣘⣾⠏⠀⠀⠀⠈⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀⠘⢀⡬⠇⡌⠁⠀⠀⠀⠀⠀⠀⠀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢹⡯⠀⢰⣂⣴⢇⢠⡀⠀⡎⠀⠀⠀⠐⠀⢤⡤⡄⠐⣿⡰⠏⣀⢰⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢋⢾⣧⣼⡟⠁⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠻⣤⡶⠋⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠉⠹⣦⠁⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣸⠿⣦⣄⣘⣧⠀⠀⣿⠃
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠆⢀⠀⢹⡆⠂⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⣠⠈⣮⠿⠋⢀⡶⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢀⠹⢿⡟⣀⠆⢸⡏⠀⠀⠀⠀⠀⠀⠨⣷⣾⡇⢀⣶⣾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡉⠒⣾⣿⡇⢰⣼⡃⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡎⠿⣿⡿⠛⡇⠀⠀⠀⠀⠀⠀⡴⠋⡰⢶⡦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡤⣤⡔⠁⢢⡀⠀⠀⠀⡘⢂⠀⣿⠀⢸⡏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⠁⢻⡷⠀⠀⢻⣄⠀⢰⣿⡎⢰⣿⠂⣸⠁⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡞⠀⠈⠀⢃⠀⠈⣿⡄⠸⣿⢠⣿⠋⢀⡿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠃⠀⠀⢸⠸⡆⠀⠹⠃⠀⢱⡟⡏⢀⡾⢱⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠀⢠⡐⣷⡁⠀⠀⡇⠀⠈⣹⡇⣼⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⢹⡞⣿⠀⠈⡇⠀⠀⠹⣷⠉
⣀⣀⣀⡀⢀⡀⢀⣀⣀⣀⣀⣀⣀⠀⣀⢀⠀⠀⢀⡀⠀⠀⢀⠀⣁⣀⠱⢦⣥⡀⣀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⣿⡿⠿⢿⡿⠿⢿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⡞⠀⣠⡄⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⡠⠄⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢹⡇⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠈⢳⣴⣿⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⠀⠀⠀⢹⣇⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣵⡀⠀⠀⡐⠂⠀⠤⠤⠴⠞⠻⣍⠛⠳⠶⢾⣿⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠙⠓⠶⠤⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠀⣒⠉⢀⠄⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢄⠀⠀⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡠⠤⣤
⣤⣤⣶⣶⣦⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣄⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣿⣿⠛⠛⡛⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠿⣿⣿⣿⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠿⠿⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣟⡾⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⣿⣿⡿⣿⣿⣿⣿⣿⣟⣿⣥⣴⣿⣛⡻⠏⠁⠝⢹⣿⣤⡀⠀⠀⠀⠀⠀⠀⠐⠛⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣧⣴⠢⠾⠟⠈⠀⠁⠀⠀⠀⢀⣀⣠⣶⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⣾⢿⣿⣿⣿⣷⣿⣿⡙⠙⠋⢉⣻⣿⣻⣥⣾⣶⣦⣄⠀⠀⠀⠀⠈⠀⠻⣿⣭⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠿⠿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣯⣭⣉⣩⣅⣐⡙⣶⣶⠄⠀⠀⢀⣀⣺⣍⣬⣿⣿⣿⣿⣿⣿⣤⣤⣤⣍⣩⣅
⣀⣀⣀⣸⢟⣻⣿⣿⣿⣿⡿⣧⢿⣯⣿⣿⣿⠉⠽⠿⠛⠃⠁⠀⠀⠀⠀⠈⠉⠿⠐⢠⣤⡀⠭⣿⡿⠿⠛⠛⠛⠋⠉
⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⢛⣛⣋⣀⣀⣈⣀⣀⣀⡉⠉⢀⣀⡐⠊⠀⠛⢿⣶⣶⣾⣿⣿⣷⣶⣧⣶⣿⡿⠿⢿⣿⣿
⣿⣿⣾⣟⣦⣤⣤⣤⣴⣾⣿⣿⠿⠿⠿⠿⠛⠿⢿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠉⠉⢉⣙⣓⣒⣉⠀⣰
⠛⠛⠉⠙⠛⠋⠉⣉⣡⣴⣶⣶⣦⣤⣤⣶⣤⣤⣴⢶⣶⣆⣴⣄⣀⣘⠓⠢⠔⠂⠐⠒⠾⠖⠛⠓⠉⠉⠉⠉⠉⠉⠁
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠙⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡻⣶⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣾⣿⣎⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣝⡛⣷⡙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⣯⣝⣿⣿⣿⣞⣿⡜⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠿⣯⡛⢻⣅
⠀⠀⠀⠀⠀⠀⢀⣴⣶⠿⣧⣙⣿⣯⣿⣿⣻⡿⣞⢷⣿⢷⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣶⣍⠻⢧⡈⢙⠎⢻⣆
⠀⠀⠀⠀⠀⣠⣾⣿⣝⣿⣿⣻⢷⣟⢿⣏⡹⣿⣿⣏⢿⡸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⡻⢦⣝⢳⡌⠳⠈⠆⢰⡿⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣾⣿⣷⣯⣿⢯⣟⡿⢿⣿⣾⣿⣷⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣯⣿⣦⣙⣆⠙⣦⣬⣵⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣮⣿⣿⣿⣿⣿⡿⢿⣿⣿⡯⣿⣆⡀⠀⠀⠀⠀⠀⠀⣰⣛⣛⣿⣿⣿⣟⣶⣼⣾⣿⣿⢿⣿⢿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣯⣽⣿⣿⣿⣿⣿⠟⣫⣷⡾⠯⢿⣶⣤⣤⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠉⠹⠉⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣯⡽⣿⡶⢾⣿⠚⣽⣿⣿⣿⣿⣿⣷⣾⣿⣛⡋⠛⠀⢹⣿⢿⣿⣿⣿⠦⠀⠲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣻⡷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⡷⠯⠆⠁⡠⠄⠲⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣟⢿⡾⣥⣽⠶⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⡶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣷⣶⣶⣤⣤⣶⣶⣤⣆⡟⡙⠚⣯⣿⠛⢭⡁⣬⣭⣡⡉⢙⠻⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⡷⡀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⣿⣿⣯⣿⡿⣇⣚⢿⣮⠾⢿⣾⠛⢿⣿⣦⠉⢮⡙⢿⣿⣿⡯⣽⡿⣿⣿⣿⠉⢉⠘⠻⣿⠶⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣮⣿⣷⢏⠀⢈⣿⣟⣦⣿⣤⣀⡆⡀⠘⣿⣿⣿⡿⠟⡊⠉⢂⠀⣀⢀⣀⣤⠀⣾⣷⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⡆⠛⠻⣻⣏⢡⣼⠛⠁⠀⠀⠈⡯⠀⠐⢦⣴⣶⠏⠁⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠎⣧⢛⢦⣟⠂⢀⣴⣿⣤⣶⣴⡷⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠁⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⣄⣀⡀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠩⠭⠩⣉⡉⠑⠒⠦⣄⣀⡀⠀⠀⢀⢀⢠⣈⡋⢗⠦⣄⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠈⠉⠑⠒⠤⣌⣉⠓⠶⣦⣿⣿⡪⠇⠀⠀⠈⠻⡶⣤⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣾⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣍⠋⠀⠀⠀⠀⠀⡀⢋⣤⡶⠶⢞⡁⠀⢀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣦⣤⣤⡴⠖⠛⢁⡧⢀⣾⣿⣾⣽⣿⡾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣟⠂⠄⣀⠘⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡌⢻⡿⠁⠀⠀⠙⠙⠧⣁⡹⣿⡻⣿⢿⣶⡍⢿⣷⡕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣶⣼⣕⣷⣭⡉⠳⠦⢄⡀⠀⢀⠀⣠⣿⠇⡾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⣷⣄⣲⣿⣆⠙⣿⣦⣆⢀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣿⣞⣔⢅⣌⢲⣤⣿⣿⢋⣾⢱⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⣿⣿⣿⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣾⣿⡿⢣⣾⡏⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠟⡵⢿⣻⣦⣤⣀⣀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣶⣦⣤⣤⣬⣭⣭⡯⠵⣿⣿⣧⠰⠄⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣏⡞⣠⣤⣿⣿⣧⡉⠙⠳⠤⠈⠃
⠀⠀⠀⠀⠀⣀⣠⣤⠶⠛⠁⠈⠻⠿⣿⣯⣽⣿⣿⣿⠿⠛⠀⠀⢈⣉⣽⣧⠀⠀⠀⠀⠡⣴⣾⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⡾⡿⢿⡿⠃⠀⠀⠀⠀⠤⠀⠈⣽⣿⣟⠛⠻⣿⣼⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣿⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣳⡄⠘⢿⣿⣿⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣻⡄⠀⠀⠈⢻⣿⡛⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢡⡀⠀⠀⠘⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⣀⡴⠞⠋⠁⠈⠉⠉⠙⠻⣿⣷⣼⡟⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⠀⠀⠹⣿⣿⢷⣾⣩⣾⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⢄⠀⠀⠀⠀⣿⣿⡟⣿⣿⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠠⣿⣿⢿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠄⡀⠀⠉⠛⠚⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣻⣿⡇⠀⠔⣨⣶⣾⣶⡀⠀⠪⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⠀⠀⢰⠀⠉⠻⣿⣿⣿⣿⣟⠿⠟⣸⡟⢀⠎⠀⣿⣿⣿⡟⠃⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠁⣿⠀⠀⢸⠀⠀⡀⣿⣿⣿⣿⣿⣿⣿⣿⡆⣾⠀⠀⠀⠀⠉⢷⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⠀⢿⡄⠀⠸⣆⠀⠳⣌⠻⣿⣿⣿⣿⡏⣿⢡⣿⢨⠀⠀⠀⢠⣾⣿⣷⣦⣄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⠁⢸⠃⠀⠘⣧⠀⠀⠙⢄⠀⠈⠳⢬⡻⢿⣿⢱⣏⣾⣿⣿⠀⠀⢀⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠀⠀⠀⠁⠀⠀⠀⠘⢧⡀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠛⠛⠉⣷
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠁⠘⡿⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠀⠄⠀⢀⣿
⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡄⠀⠀⠀⢸⠀⣼⠏
⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣞⣼⡏
⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠉⠉⠉⠉⠙⠳⣦⣀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣴⣿⣿
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠒⠶⣄⠀⠀⠙⢷⣄⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢀⣼⣿⢿⡿
⠀⠀⢀⣿⠃⠀⢀⣤⡶⠞⠃⠀⠀⠀⠈⠳⣄⠀⠈⢻⣿⣦⣄⡳⣄⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⠏⢸⡇
⠀⠀⣸⡏⠀⣴⠟⠉⠀⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢸⣿⣿⣿⣿⣎⢧⡀⠀⠀⠀⠀⠀⢸⣿⠋⠀⢸⡇
⠀⢀⣿⠀⠸⠁⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣸⣿⣿⣿⣿⣿⣇⢷⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇
⠀⣸⡏⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢠⠞⡁⠀⣸⣿⣿⣿⣿⣿⣿⣿⣼⡆⠀⠀⠀⠀⠈⣿⡄⠀⠈⣷⡀
⣸⣿⠁⠀⠀⠘⠀⢀⣤⣤⣄⣀⠀⠀⠀⠾⢷⣤⡹⣿⣿⣿⣿⣿⣿⠿⠇⢳⡀⠀⠀⠀⠀⠹⣆⠀⠀⠘⠷⣄⢀⡀
⣿⣿⡀⠀⠀⠀⣰⠋⠁⠀⠉⠛⠷⣦⡤⠤⠀⠈⠛⠶⣿⣿⣿⣍⠀⠛⠛⠦⠳⠀⠀⠀⠀⠀⠈⠢⢠⣤⡀⠀⢹⣿⣆
⠘⢿⣿⣦⣀⣀⣁⣀⣀⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⡀⠈⣷⣿⣿⣷⡶⠶⠶⠖⢠⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⡛⠿⠛
⠀⠀⠉⠛⠛⠛⠛⠻⠿⢯⣭⣤⣤⠶⠿⠿⣷⣶⣶⣷⣶⣿⣿⡿⠀⠀⠀⠀⠀⠈⠳⣄⣀⡤⣄⢰⡄⢸⣄⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢦⡙⠀⠀⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣻⣴⣆⣸⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⢀⡨⠛⠊⠀⢺⠅⠀⠠⣤⠀⠐⠂⠈⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡄⠀⠀⠀⠠⡧⢂⠀⠀⠀⡸⠶⡀⠀⠐⠮⠁⠂⠀⢄⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠄⠚⡄⠀⠀⣉⡀⠀⠉⢦⠋⢠⣈⣁⡀⠀⡠⢄⠠⢀⡴⠥⠹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣷⡀⠚⠛⠉⢁⣠⠾⣆⡨⠉⠙⠑⢴⠀⣴⠀⢸⠠⡀⠀⠀⡠⠄⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣄⠀⠀⠀⠀⢠⡌⠥⣹⠧⣴⠀⠀⠀⠤⢰⣃⣸⡷⠀⠀⠀⡔⠸⠃⢠⠎⠙⢡⠆⢈⡀⠁⠂⢰⡄⡁
⠀⠀⠀⠀⠀⠀⢀⡀⣠⢞⠟⣉⢙⠆⠀⢀⣶⡌⠁⠀⠙⡈⠗⣀⡄⠀⠴⠾⠁⠘⣆⠀⠀⢀⡇⠀⡃⠰⠀⠀⢀⠀⠀⠻⢮⢤⣊⠜
⠀⠀⠀⠀⠀⠀⢇⢀⡞⡏⡔⡽⠉⢢⡀⠁⠈⠓⡄⠀⠀⢼⣷⠴⠛⠚⠛⠉⠋⠉⠉⠉⠙⢛⠿⡦⣟⡓⣀⣆⠞⢳⣤⡂⠄⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢚⢱⠀⢭⠠⠀⣼⢇⠀⠀⣀⠃⢺⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣄⣈⡀⢀⡀⠒⠊⢹⢿⣉⣶⠄⠉⠀⠀⠘⢯⣔⣠⣤⡀⢔⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢘⡤⠀⢉⠖⠈⢱⡠⢀⠴⠜⢪⣹⡇⠀⠠⠂⠀⠀⠀⠀⠀⠀⢀⣠⠤⠍⣈⠐⣢⠈⠀⢹⡅⠀⢠⠀⠀⠀⢀⠜⢣⠎⠁⠘⣅⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠈⠀⠀⣨⣦⠅⠁⠀⠘⡏⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠉⠁⠀⠀⠀⣷⡀⢆⠀⠀⠀⢷⡀⠘⣦⡁⠀⢈⡒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠁⠀⠀⠀⢨⡅⠀⠀⠀⠀⠄⢄⣀⣀⠀⠀⢠⠀⣀⡀⠀⠀⠀⠀⠀⢨⣿⠶⠛⢆⠀⢀⠜⠉⠑⠅⡼⠛⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠀⣲⠰⠷⠖⢀⠅⢿⠃⠈⢳⠒⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⡇⠐⠀⠀⠉⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠠⠀⡽⠶⠒⠺⠠⠀⢿⡧⠀⠀⣄⡀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣿⢃⠉⢻⡁⡐⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠁⠀⢀⡀⠀⠀⠀⠈⣧⡀⠀⠀⠉⣄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢺⠟⠚⠳⢇⣠⠁⠀⠀⡰⠋⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡋⠀⣠⠇⠲⡙⢃⡀⠀⠀⠀⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣈⠁⠀⠀⠉⠐⠀⠀⣅⠁⢈⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡉⠃⠀⠑⡀⢸⠎⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡷⢄⡡⡠⢀⠀⠀⣠⣶⡂⠈⠓⢼⡤⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡊⠁⠉⠃⠀⡀⠐⠐⠁⠜⠳⢭⣿⠶⠶⢦⣄⣰⣦⣭⣤⣟⢻⣇⠀⠈⢦⠊⠀⠀⠀⢀⠄⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⢠⠏⠰⣦⡀⠀⠘⠀⠀⠀⡀⠀⠀⣀⡀⢠⠄⠀⠐⡋⡄⢁⠁⠀⠀⠡⠀⣻⠟⡉⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠠⠜⠀⠀⠙⠋⢀⢀⠀⠀⠀⠀⠀⠀⡥⠘⠀⡄⠀⠀⠰⠓⠈⡙⠄⣴⣆⠐⠩⡈⠀⢀⠜⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡁⢲⢤⠡⠲⣤⠄⢁⣲⢌⡀⠀⢶⠊⢦⠀⠛⠂⠉⠁⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡸⠃⠀⠀⠀⠀⡸⠁⠀⠐⠆⠀⠀⠘⠠⠠⠐⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠈



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

⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣯⣤⡶⠀⠂⠀⣄⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣨⡷⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣤⡀⠀⠀⢹⢡⠙⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣄⠄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⡙⢂⡆⠀⠄⠀⠀⠂⠈⠠⠀⠀⠀⠀⠀⠀⡤⠞⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣷⣾⠃⠄⠀⠀⠀⠂⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣳⣀⣹⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⣵⢧⠋⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣛⠛⠛⡂⠀⣀⣀⣤⣾⣿⣿⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⢀⣀⠛⠛⠉⠀⠀⠿⡿⣿⡏⢍⡊⠼⢽⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢝⠧⠻⣷⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣄⠰⡱⡑⣕⢮⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣶⠔⠔⣱⢲⢿⢸⢹⠈⢢⣤⢀⡆⠈⠁⠓⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠴⢪⢦⢃⣇⣎⣄⢤⠖⠉⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡈⠀⢄⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠑⠋⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⡿⠿⣿⣶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠟⠃⠀⠒⠒⢲⣤⣶⣿⣿⣧⣄⣀⠀⠀⠈⠀⠈⠉⠙⢛⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠞⠉⠀⣀⣴⣾⣿⡶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣿⣿⡟⠻⠿⠋⠛⠻⠿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢲⠋⣿⣿⡿⠃⣠⣴⡶⢖⣴⣾⣿⣿⣿⣿⠀⣾⠃⠀⣀⡀⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣟⠛⠢⠀⣄⡀⠉⠙⠿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡿⢁⡞⠉⠀⠀⢈⣿⣿⣿⣿⡿⠁⣰⣿⣷⣿⣿⣿⣿⣷⣦⣄⠀⠈⠙⢿⣿⣿⣿⣿⠀⠀⠀⣴⣭⣄⣤⡀⠈⢧⠉⠻⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠀⣼⣿⡇⢨⡅⠀⠀⠀⠈⠛⢿⣿⣏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠙⢿⣿⣿⣄⣀⣤⣿⣿⣿⣿⣿⣦⣈⠀⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⢾⣴⣿⣿⡇⢸⣿⣿⠀⣶⣶⣶⣿⣿⠏⢠⣿⣿⠋⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣷⣦⣄⣀⣀⣭⣽⣿⣿⣿⠇⠈⣻⣿⣿⣯⢧⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠷⠈⠻⣯⣁⣻⣿⣿⣿⣿⡄⢸⣿⣿⣷⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⣾⣿⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⣿⣧⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡳⠀⠻⣿⣿⣿⣿⣸⡿⠉⠻⢿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠉⠉⠙⠻⣿⣿⣿⣿⡃⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠒⠤⣈⠙⠛⢻⣿⣿⣦⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣾⣿⣿⣿⣇⠘⣿⣿⣿⣽⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠋⠀⠀⠀⠀⠉⠒⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠙⣿⣿⣏⣤⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢀⣶⣿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠙⠛⠻⠛⠿⠟⠻⢿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠟
⣀⣀⣠⡤⠶⠚⠉⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠿⠿⠓⠒⠶⠿⠿⠿⠿⠿⠛⠛⠉⠁
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣦⣤⣤⣤⣴⣯⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠙⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠉⠉⠛⠛⠳⠿⠖⠺⠟⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣤⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣯⣽⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠛⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠇⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⡿⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⣿⣿⡃⠀⠀⠀⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣷⣾⣶⣦⣿⣯⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣽⣿⣿⣿⣟⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠻⢿⣿⣿⣷⣀
⠀⠀⢀⡀⠀⢠⡄⠀⠀⠀⢀⣠⣄⣸⠇⢘⡀⠄⣦⣄⣀⣴⣯⠿⢠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡄⠀⠀⢺⡗⠇⠀⢠⠄
⠂⠀⠀⡃⠀⠈⠁⠀⠂⣧⣼⣅⣹⣇⣀⢮⣵⣇⣼⠮⡾⣿⣚⡷⣿⣿⠿⣶⣾⣧⣀⣀⣀⣀⡀⠀⠠⠸⢿⣿⢿⡿⣦⠀⠈⠷⠠⠄
⠐⠆⠀⠁⠈⠋⠉⠉⠁⠉⠁⠈⠀⠈⠈⠀⠀⣀⡤⢶⣝⣻⠿⣵⠟⠟⠛⢉⣭⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢤⣔⣂⣤⡷⣺⠿⠛⠉⠈⠀⢠⣶⣿⣦⣸⡿⠏⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⢀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠒⣈⡁⣴⣴⣿⡛⠭⠙⠀⠀⠀⠀⣀⡀⠀⠀⣿⣿⣿⠋⢹⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣸⣿⣦⠿⠞⠛⠁⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⠆⢙⡏⡁⠀⠀⠄
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣟⣿⣿⠛⡛⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣯⣤⣽⣋⣀⡿⠍⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣛⣛⠋⠉⠧⠶⠤⠀⠩⠨⠅⢀⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠄⠐⠚⠛⠁⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢺⣤⠀⠀⠠⠬⠁⠄⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⣶⠤⠀⢤⡤⠤
⣤⣤⣤⣤⣤⣤⣦⣴⣶⣦⣬⣯⣤⣤⣤⣭⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡥⢬⣟⡛⠛⢛⡛⣛⠛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⡷⣦⣄⣀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠈⣿⣮⣻⣿⣷⣄⣼⣿⢟⣵⣿⡟⠁
⠀⠀⠀⠀⠘⣿⣿⣾⢻⣿⢽⣯⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠙⠿⠏⢫⣼⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣠⣴⣦⢶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⣴⣾⣿⣿⠀⠀⠀⣠⣶⣿⣿⣿⣿⣮⡻⣿⠿⢿⣷⡀
⠀⠀⠀⠀⢀⣀⣀⣿⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣩⣟⡛⠛⣠⡀
⠀⠀⠀⠀⠀⠉⠛⢿⡆⢀⡤⠀⠈⠉⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⢩⣼⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣼⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢠⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣺⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣧⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣦⣤⡀⢀⣤
⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣿⡿⠛⠉⠉⠙⣿⣿⣿⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⢻⡆⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢟⣟⢿⣽⣟⣷⣿⣿
⠀⠀⠀⠀⢀⣾⡟⠉⠀⣾⠋⠀⠀⠀⠀⠀⢻⣿⡿⠀⣿⡄⠙⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⡈⠛⠛⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣯⡿⣿⡿⣾⣿⣿⣿
⠀⠀⠀⢠⣿⠏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣠⣼⣿⣧⠠⣌⠻⣿⣿⣿⣿⣿⢸⠀⣿⣿⣷⣶⣿⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣷⣿⣾⣧⠙⠿⠟⠃
⠀⠀⠀⣼⣿⣷⠀⠀⠀⢻⠀⠀⢀⣀⣀⡀⠀⠈⢻⣿⣿⣿⠀⣿⣦⠈⠻⣿⣿⣿⠘⣇⠘⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠟⠀⣦
⠀⠀⠀⣿⡏⢿⣇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⠹⣿⠿⠀⣼⣿⣧⠀⡘⢿⣿⠀⢻⣦⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⠄
⠀⠀⠀⢹⣷⠀⠙⢧⡀⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⠀⠀⢀⣴⣿⣿⣿⡇⢹⣆⠹⠀⠈⠻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⢠⢿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠁⣼⣿⡆⢠⠸⣦⣈⢉⣀⣶⣄⠀⠀⠀⣀⣀⣤⠖⠀⠀⠀⠀⠀⣠⡾⠃
⠀⢀⡴⠋⠀⠻⣦⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣄⣀⡀⠀⠀⠀⠀⠀⠠⢾⣿⣿⡗⢸⣧⡙⠿⠿⠉⠻⢿⣿⣾⣽⣟⣋⣁⣀⣀⣀⣤⣴⡾⠟⠁
⠀⠉⠀⠀⠀⠀⠘⠳⣄⡀⠀⣿⣿⣿⡿⣿⣯⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⢫⣥⣾⣿⣿⣶⣶⡄⠀⠀⠈⠉⠛⠛⠻⠿⠿⢿⣛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢘⡛⢿⣶⣿⣿⣮⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡽⣿⣻⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⡏⣿⣯⡻⣟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠾⠿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢹⣿⣷⡘⢯⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⠈⣿⣿⣿⡌⢳⡝⣷⣄⠀⠀⠀⠀⠀⠀⣼⣿⣿⡌⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢹⡆⠸⣿⣿⣷⡀⢻⣎⢿⣷⣄⠀⠀⠀⠀⢿⣯⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡘⣷⡀⠹⣿⣿⣧⠀⢿⣆⠻⣿⢷⣄⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡇⢻⣧⡀⠙⢿⣿⠀⠘⣿⡆⠹⣧⡙⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢻⡄⠹⣷⣄⠀⠁⠀⠀⣿⣿⠀⢹⣷⠈⢿⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠻⠀⠙⠶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠘⠃⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⢆⣤⣐⢒⠂⣠⣺⣟⢙⣷⣷⣶⢶⣤⠀⠀⠀⡀⣤⣴⣶⣞⡟⡟⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠲⠋⣻⡉⠁⠚⢀⠛⠀⠀⢦⣁⠛⢓⣼⣿⣯⣿⣷⡿⢿⣻⡍⠂⠀⣰⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠐⠛⢻⠶⣯⣯⡀⣼⡤⣶⣽⡿⣿⣷⡝⠻⣉⣵⣿⣟⢹⣧⡗⢡⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠃⠈⠓⠛⠛⠛⠀⢒⣾⠓⠖⠚⢿⠝⣪⠟⠛⠘⠷⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⠄⠂⠛⠀⠀⠀⠀⠀⠀⢀⡴⠋⣇⠀⠀⠠⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠈⠙⠆⠘⠉⢿⣿⠚⣖⣖
⠀⠀⠀⠀⠀⠀⠀⢉⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃
⠀⠀⠀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⣿⢷⣄⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣠⣶⣾⣭⢷⠆⢀⡀⠀⠀⢀⣠⣶⡿⠋⠛⠁⡶⢿⣯⠟⠻⢦⡀⣤⣾⣿⣿⣦⣀
⠀⠀⣠⠞⡱⢻⣿⡟⢿⡻⢮⣛⡲⣴⣿⡟⠉⠀⠀⠀⠠⠇⠀⠥⢂⣴⠫⠚⣡⡾⣻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠈⠰⢃⣿⠱⠈⣗⠀⠻⡿⢛⠉⠷⣦⣤⣂⠀⠀⢀⣤⠖⠛⣁⠴⢚⣡⣊⢔⠍⠙⣿⣿⣷⣽⣷⣄⡀
⠀⠀⠀⠈⠀⡜⠘⠀⠀⠙⢧⣁⡀⠂⠙⠢⠤⠉⠚⢗⠾⢿⣩⣀⠼⠁⠴⠊⡔⠁⡌⠀⠀⠈⠛⠿⣿⣿⣿⡛⢦⣄⡀
⠀⠀⠊⠄⡀⠀⠀⠀⠀⠀⠀⠏⠲⣌⠢⠀⠐⠀⠀⠠⠈⠲⠬⣿⣷⠾⣄⣾⣮⠀⡀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣵⣮⣛⠶
⠀⠀⠈⠐⠁⠀⣀⣠⣤⣤⣤⣬⣤⣜⣛⣦⡀⠀⠀⠁⠢⠄⠀⠀⠀⠀⠐⢽⣗⣚⠧⣤⡀⠀⠀⠀⠀⠀⠀⢹⡻⡿⢟⠻
⠀⢀⣤⣶⣾⣿⣿⣿⢯⣾⣿⠿⣿⣭⣉⠽⣿⣷⣶⣒⠒⠖⠲⠤⣅⣀⣠⣀⠠⠌⣉⠒⢭⣟⡢⣤⣬⣄⠀⠠⡁⠝⢦
⠀⠈⠻⢳⣉⣃⣘⡀⣺⣉⣀⣀⣀⣋⣁⣄⣉⣻⣟⣿⣽⣶⣶⣤⣀⣬⣍⣹⡿⢶⠦⠄⢀⣀⠬⠉⠓⠒⠯⢍⣳⣄⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⠚⠓⠄⠐⠒



⣿⣿⢿⠿⢛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣿⣾⡿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣠⣷⡚⢛⡾⠛⠛⠁⠀⠈⠙⠲⠦⢤⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣤⣤⣤⣠⡴⠒⠛⠏⡀⠱⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡑⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢹⠃⠛⡛⠁⠉⣤⡴⠚⠋⠀⠀⡼⠀⠀⠀⣀⠀⠀⣀⣀⠀⠀⠸⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢸⡄⣤⣰⡄⢠⢿⣷⠀⠀⢰⠞⠁⠀⢾⣿⣿⣷⣾⠛⠉⠁⠀⣤⣌⠻⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢿⣿⣿⣷⣀⣾⣿⣶⣶⣿⡟⣠⣴⣿⠍⠛⣉⠉⢁⣀⡤⠠⠄⠉⣁⢿⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⣿⠿⠿⠟⣻⡿⢻⡇⠀⠘⢿⣿⣿⣷⣦⣴⣾⣿⢃⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢿⣿⣟⠸⣆⠸⡀⢠⢀⣯⣴⠿⣿⣶⣿⠘⣿⣿⣿⣿⣿⣿⡟⠆⠰
⠀⠀⠀⠀⠀⣤⠤⣄⣨⠀⣧⠀⠀⢠⣆⣼⠀⠈⠀⠋⠀⣿⠘⣿⡟⣿⠏⢀⣾⡏⠿⠛⢷⡄⢹⡿⣿⣿⣿⣿⡀⠞
⠀⠀⠀⠀⠀⠀⠲⢄⠉⠀⣿⡆⠀⠸⠋⠇⠀⠀⠀⠄⠈⣷⡀⢻⣾⢸⣿⣾⣿⡆⠀⠀⢸⡇⢹⣿⣿⠿⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢠⣿⡄⠀⠀⠀⠀⠀⠈⠁⠈⠻⣖⠛⢿⣿⣷⠠⣴⣯⣽⣾⣿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠉⠙⠛⠃⠀⠀⠀⠀⠀⠈⠃⠀⠹⡿⠶⠭⢿⣧⡿⠾⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⣠⠶⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡯⠁⠈⠐⠟⠀⠀⠀⠀⠀⠀⠀⢀⣻⣴⣿⣻⣿⣿⢿⣍⣉⣉⢡⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣿⠷⠖⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠈⠛⣻⣿⠋⢻⣿⣾⣿⣿⠯⠘⠁⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡌⠀⣀⣿⣦⣴⡦⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠻⣧⡃⠟⢩⣿⣛⣠⡜⠒⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⣴⠛⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⡀⠸⠿⡿⣿⠿⣏⡉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠋⠀⠀⠀⠀⠀⠉⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣘⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⢹⣏⠉⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⡀⠀⠀⠒⠈⢿⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡀⠐⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⡤⢁⡀
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠁⡘⠁⠀⣀⡙⠃⠃⠀⠠⢔⠀⠀⠀⠈⠂⣀⠀⠀⠘⣆⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠇⢀⡊⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢤⡘⠦⠰⣦⠘⠁⠀⠢⠀⠀⠀⠀⠉⠀⠄⠀⠀⠈⠙⠦⣄⠹⣆⠀⠸⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⣴⡟
⠀⠀⠀⠀⠘⠂⠀⠀⣀⢀⡀⠴⠀⡏⠀⠀⢀⠀⠀⠀⠒⠒⠠⠤⠤⠔⠂⠀⠀⠀⠈⠻⣿⣤⡷⢀⡤⠖⠉⢛⠋⠛⠓⠢⠄⠀⠀⢠⡿⡘⠋⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⡇⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣧⡞⠁⠀⢶⠈⠀⠀⠀⠀⠀⠳⠶⢿⣷⣤⣀⣀⣀⡀⡀⣠⠛⢉
⠀⠀⠀⠐⠀⠁⠀⣠⣴⠶⡆⠀⠀⠰⡀⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣍⣩⣍⡙⠟⠀⠉
⠀⠀⠀⠀⠀⠀⠄⠙⠆⠔⢻⡄⠀⠀⠻⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠹⣿⣿⣛⠛⡁
⠀⠀⠀⠀⠀⠀⠐⠘⠂⠀⠈⠆⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢹⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠷⣀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣖⣶⣠⣤⣀
⠀⠀⠀⠀⢀⣤⣾⣩⣛⣶⠟⣡⣾⡿⣿⣿⡿⣶⣿⣶⠶⠶⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠶⠦⢄⣀⡀⠀⠀⠀⣀⣀⣠⣤⣤⡶⠶⠷⣶⣦⣤⣀
⠀⠠⡴⢿⠟⣿⣿⣿⣿⣷⡿⣛⣿⣆⣹⣿⣄⢨⠛⢿⡛⠻⣷⣤⣉⣋⣑⡲⠶⣶⣾⣯⣉⣀⠀⠀⠀⠀⠠⠶⠾⢻⠿⠟⠋⠿⠛⠋⠉⠁
⠀⠀⠐⠹⠟⠿⡛⠛⣟⣛⣁⣩⠋⡏⠙⠿⠙⢆⠑⢿⡻⠶⣌⡙⠻⠟⡙⠛⠿⠿⣗⠲⠮⠭⠝⢛⣲⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤
⠀⠀⢾⢶⣿⣿⣿⡟⣛⣭⣭⣿⣯⣍⡿⢿⣿⣿⣷⡒⣻⣶⣶⠶⠶⢶⣦⣶⣦⣶⣶⣶⣶⣾⣿⣿⣹⣋⣉⣉⣉⣹⣿⣷⣾⡿⣛⠋⣽⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠉⢩⣯⣤⣾⣿⡟⠛⠿⠛⠲⠀⠀⠀⠀⠛⠿⠟⠓⠚⠛⠃⠀⠀⠰⣿⣿⡭⠉⢭⣽⢿⣿⣿⣯⣭⡽⠇⠉⠉⣤⣭⣭⣽⡿⣿⣿⣿⣿⣶⣶⡶
⠀⠀⠀⠤⠤⠶⣶⣶⣾⣛⡛⠛⠓⠲⠶⠶⠖⠲⠾⠿⠿⠿⠛⠛⠛⠋⠓⠒⠶⠤⠤⠤⣤⣽⣧⣄⣀⣀⣙⣻⣀⣤⣤⣤⣭⣭⡥⢤⢤⡤⠼⠿⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣆⣰⣤⣤⡄⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠓⠾⠤⢤⣤⣴⣶
⠀⠀⠀⠀⢤⣄⢀⣦⢧⢦⣿⢿⣷⠀⢾⠠⠘⠴⣟⣀⣯⣯⢿⣿⠿⣹⣟⢰⣧⢿⣲⢧⣿⣿⠖⠀⣶⣄⡀⠾⠗⠶⢰⣼⡆⣴⣠⠀⢀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠁⠉⠈⠈⠉⠈⠁⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠁⠉⠁⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡀⠀⠀⠀⠀⠀⠀⡄⠀⢀⣀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣸⣟⡣⠀⠀⠀⠀⠀⢀⣀⣀⠀⠤⠤⠤⠠⠷⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠈⠀⠀⢀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢢⣤⢶⣦⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠀⠀⠀⠈
⠀⠀⠀⠀⠠⣀⠀⠀⠀⠒⢌⣐⠠⠀⢤⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠀⠿⠃⠀⠀⡅⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠉⠒⠄⣀⡈⠑⠲⢶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢘⡂⠀⠀⠀⢀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠃⠐⠀⠀⡆⠀⠑⠂⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠚⠇⠐⠟⠂
⠀⠀⠀⠀⠈⠀⠂⢀⡄⠀⠃⠀⠀⠀⠀⠀⠀⠐⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⣔⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠄⡀⠀⠀⠀⠀⣉⣅⡐⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⠄⣾⠃⠰⣸⠴⠀⢀⣀
⠀⠀⠀⠀⠀⡀⡄⡀⣀⣰⣛⣇⠀⢀⠀⠨⠬⣭⠅⠈⠉⠉⠁⠀⠐⣏⣻⠀⠹⠀⡁⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠆⠀⢀
⠀⠀⠀⠀⠀⡁⠈⠑⢋⢙⢲⠙⠛⢛⡟⠀⣀⣁⡀⠀⢀⠀⠀⣶⠀⢿⣿⠀⠲⣯⠩⠄⠀⠀⠹⢶⠤⠤⠄⠀⠀⠄⣂⠀⠋⠁
⠀⠀⠀⠀⠀⠃⠀⠈⢀⣈⠚⠀⠀⢰⡁⠰⡇⣀⠤⠠⢸⠀⠀⡉⢁⠘⠋⣤⣤⣤⣀⣀⢠⣤⣤⣏⡅⢀⠉⠉
⠀⠀⠀⠀⠀⠀⠠⠤⠟⠑⣿⡄⠁⠀⣨⡀⠴⠀⠀⠀⡠⠆⠀⠀⢘⠆⠀⡆⢀⡄⣀⡀⠀⡠⣄⠛⡋⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⣹⣋⠀⠀⠉⠃⠀⠏⠙⢳⢀⣁⢂⡘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⡆⠀⠀⠀⠀⠀⠀⢰⣶⣷⣲⠀⠀⠀⠀⠀⠸⠃⠈⠃⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠀⠠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⣀⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡄⠀⡀
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣹⡀⠀⠀⠀⠁⠈⢷
⠀⠀⠀⠀⠡⣤⢆⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡀⠈⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢄⣀⣤⣾⣶⣾⠟⠁⠀⢸⣿⣿⣤⡄⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⠤⣤⣴⡿⠻⡿⢛⣿⣶⣆⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⠀⠀⢨⠹⢶⣶⣿⡤⢴⠓⡞⣻⣿⣁⣴⣤⣤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢔⣯⣀⠀⠀⠘⢿⣯⡷⡆⣠⡾⢋⠜⣡⡞⠀⣐⠀⠀⠉⠑⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⣹⣿⣿⡿⢚⣵⣿⣶⣷⡤⠃⠀⠠⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠛⠁⣻⣶⡿⣈⣸⣿⣿⣦⡄⠚⠀⠈⠛⠙⡄⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠹⣿⣯⣃⣠⠿⣿⣧⣄⢴⠳⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠻⣿⣯⣠⣬⠿⣿⣦⣤⠒⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢰⡋⠙⢿⣶⣤⣾⡻⣿⡀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠙⢦⡀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⣀⠤⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤
⠀⢣⠀⠀⠀⠀⠈⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⡀
⠀⠀⢑⡆⠀⠀⠀⠠⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⢐⡊
⠀⠀⠈⢷⡀⠀⠉⠀⠈⠦⣀⣙⢦⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣶⠆⠀⠀⠀⠀⠰⣾⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⡴⣋⢀⠴⠁⠀⠉⠀⢀⡾⠁
⠀⠀⠀⠈⢿⠤⠀⠲⠦⣄⣘⠛⢣⡷⣄⠀⠀⠀⢠⠞⠀⠈⠉⠑⣤⣶⣶⣶⣶⡴⠊⠉⠁⠀⠳⡄⠀⠀⠀⣠⢾⡜⠻⣓⣠⠴⠖⠀⠠⡿⠁
⠀⠀⠀⠀⠘⣀⣀⠀⠀⠀⠀⠤⡆⠐⢌⢣⡀⠀⠸⣄⡀⠀⠀⣠⡿⣿⣿⣿⣿⢿⣄⠀⠀⢀⣠⠇⠀⢀⠜⡩⠂⢰⠤⠀⠀⠀⠀⣀⣄⠃
⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠋⠃⠀⠀⡀⠙⢦⡀⢸⣿⢀⣾⢽⣾⣿⣿⣾⣿⣷⡯⣷⡀⣿⡇⢀⡴⠋⢀⠀⠀⠈⠙⠂⠀⠀⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠙⣦⡹⢿⣿⣤⣿⣿⣿⣿⣿⣿⣤⣿⡿⣏⣴⠋⣠⠖⠁⠀⠁⠀⠀⠀⠀⡀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠀⠴⠶⠿⢿⣷⣾⣿⡛⡿⠟⢛⣿⣿⣿⣿⡛⠻⠿⢛⣿⣷⣾⡿⠟⠶⠦⠀⠀⠀⠰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠀⠔⠉⠛⣿⣿⡿⡀⠀⢘⣷⣿⣿⣾⡃⠀⢀⢿⣿⣿⠛⠉⠢⠄⠀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⢀⠈⣸⣿⡇⡀⠀⡚⣿⢿⡷⣿⢃⠀⢈⢸⣿⣎⠁⡀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀⠈⣸⢻⡇⠀⠠⠋⢻⣼⣇⡗⠙⠄⠀⢸⡟⣇⠑⢀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⣼⣿⣿⡇⠀⠀⠼⢻⣻⣟⡿⠇⠀⠀⢸⣿⣿⣧⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣷⡇⣿⣇⠂⢀⠀⡐⣿⣿⢂⠀⠀⠈⣸⣿⢸⣾⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢻⣿⣾⣄⠓⢿⣿⣿⡿⠚⣠⣷⣿⡟⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠉⠻⠶⠶⠟⠉⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠠⠀⠀⢀⠀⠠⠀⢀⠀⠀⠀⢀⠄⠐⠠⠀⠤⢂⠀⠠⠄⡀⠂⠀⠁⢀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠠⠀⣄⠀⠀⠀⠀⠀⣠⠤⡐⠃⡀⢀⠀⢠⣠⢃⣴⣶⣦⡌⢂⡀⠀⠈⡀⠄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠊⢀⠏⣱⠓⠀⣠⡄⢧⣄⣥⠀⢳⣤⣤⡾⠻⠸⣿⣏⠛⠁⣸⣿⣷⣶⠟⣩⣤⣤⣤⣄⠠⠰⣄⠂⡑
⠀⠀⠀⢱⠄⢠⠖⢂⡞⣡⣤⣤⣄⢀⡁⠂⠉⠻⢷⡊⠡⡙⠻⢷⣾⠃⠄⡻⠿⠘⢧⣹⣿⡿⠋⢄⠀⠊⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠀⡱⢭⣿⣯⡿⠇⠀⢠⡀⠂⠀⠳⠄⢠⠐⠀⠛⢀⠐⠄⣠⢀⡜⠈⠋⠠⠀⡄⠜⢠⠀⠈⠍⢂⠁⢀⠰⡁
⠀⠀⠢⡀⠀⠀⠘⡁⠀⠈⢻⡏⣴⠛⠲⡎⠇⠈⠀⢀⣧⠅⠀⠀⠂⠀⢠⠄⠈⠎⠀⠀⠈⠆⠀⠠⠀⠁⡈⢁⠄⠃⠀⠢⠑⠈
⠀⠀⠀⠀⣠⣴⣶⡆⠀⠀⠀⠠⣙⣀⠠⠐⠀⠀⢀⡌⠩⡔⠀⣀⠀⠁⡄⠀⠃⠀⠈⠀⠀⣼⠦⠠⠄⢀⡠⢺
⠀⠀⠊⠠⠛⣿⣿⡇⠀⢀⣠⠴⠚⠁⠀⠀⢀⢠⣚⠛⣂⠘⠒⠂⢴⣾⣧⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠠
⠀⠠⠀⠀⠸⣿⡿⠗⠊⢉⢁⠀⣀⠤⠖⢿⡿⢐⠀⠀⠀⣿⠿⠴⢿⠛⠛⠓⠚⠦⣀⠀⠀⠀⠀⣀⢐⠈⠀⠄⢀⢀⡄⠊⠔⠁
⠀⠀⠁⠀⠰⠏⠀⣴⢛⠻⢄⠀⠛⠛⠒⢃⢰⣿⠷⠶⠿⠣⠿⠏⢈⣴⡿⢟⠁⠐⢦⠌⣈⡉⠁⠉⢁⡤⠊⢉⣉⣀⣬⣦⠀⠐
⠀⠀⠔⠁⢘⠀⠀⠰⣶⡌⢂⢹⣶⣾⠟⢭⠀⠀⠀⠠⠰⠀⠀⢰⠐⠐⠒⠸⢳⣤⣀⠁⠈⡿⠌⠀⠁⠀⠀⠀⠌⠻⣿⣿⣷⠄
⠀⠀⠀⢰⡏⠠⣀⠐⠀⠀⠀⠜⢛⠿⢇⠀⠀⠀⠀⠐⠀⠀⠀⠌⠀⠀⢄⡄⠉⢉⡽⠻⠇⠘⠚⢧⠄⡈⠀⠐⣶⡀⠐
⠀⠁⠑⣬⠻⠄⠈⠑⠲⡾⠀⠐⠁⠀⠀⠑⠀⠀⣀⣀⣀⠄⠊⠀⠀⠀⠀⠑⢂⠉⠁⡀⠀⠀⠀⠣⠄⢈⣤⣁⡘⣿⣷⡀⠁
⠀⠀⠀⡤⡀⠀⢱⣤⡜⠁⠀⣄⠀⠀⠀⠀⠀⠀⢸⣟⣿⠘⠀⠀⠀⠀⢤⠚⢤⣀⣤⣈⣠⣀⣀⣴⣶⠞⠛⣫⣿⣿⣿⣏⢈⠂⡆
⠀⡄⢄⠳⢥⣴⡿⠟⠁⠀⠀⠈⡦⣄⣀⠀⢀⣠⠊⠀⠉⠓⢌⡂⠄⢀⠀⢀⠈⠟⠁⠉⠀⠈⠋⠁⢡⡴⠊⠀⠈⠛⣟⠋⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠈⠈⠇⠘⢏⠂⠁⢄⠈⢻⡦⢀⡿⢮⠈⠁⡀⠀⠀⠀⠀⠐⠋⠀⠈⠁⠠⠀⠀⣆⢠
⠀⠀⠀⡂⠀⠈⠀⠈⢛⠀⠀⢀⠀⠀⢠⢠⠀⡀⢸⠀⡀⡀⢂⠀⠰⡏⠁⠄⢳⡈⢀⡀⠀⠀⠤⠈⠢⡀⢀⠀⠀⡁⠀⠘
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠂⠀⠀⢚⣃⡌⠴⠁⢀⠀⠘⠀⢁⠐⢤⡀⠊⡄⠙⠄⠠⣠⣴⠟⠀⠀⡇⠠⣀⢤⠯⢢⡌⢀⠄
⠀⠀⠁⠠⡀⠨⠀⠀⠀⠀⢲⣶⣶⠀⠱⢠⢹⣦⡀⠄⠀⠀⠁⠀⠀⠀⡠⢄⠀⢄⠀⠂⠉⠁⠀⠀⠄⠁⡀⠀⠠⣀⠌⠀⢀⠄
⠀⠀⠀⠀⠉⠂⠀⢠⠀⢀⠀⠉⠉⠀⠀⠀⣾⣿⠿⢷⣶⣶⡶⠦⡂⠣⠈⠐⠠⢘⡀⠀⢀⢀⠀⠀⢀⠁⣄⡑⢀⣿⠄
⠀⠀⠀⡀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠈⠀⢀⠈⠉⠈⠐⠤⠙⢆⡀⠈⠀⠄⡀⠀⠢⢈⡐⠒⠀⠀⣠⣼⣦⠄⠀⣂⠄⢀
⠀⠀⠀⠀⠀⠀⠄⠂⠄⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠈⠁⢐⠉⠀⠀⠈⢂⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠁⠐⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣭⣿⣿⣟⠻⠂⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⢉⣴⣄⠀⠀⠀⠙⣟⠑⣰⣿⣿⣿⣿⣿⣷⢦⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⡇⠀⠀⠀⠀⠀⢀⣴⢋⣿⣿⣿⣾⣿⣿⣿⠅⡄⠀⠀⠀⠾⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⣿⣿⣾⣮⣵⠂⠀⣠⡿⢃⣼⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⢀⣿⣿⡟⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⠀⡼⠋⢀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣷⣿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡿⠃⠀⠞⠀⠀⣾⣿⣿⣿⡟⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡗⠓⠿⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠏⠉⠉⠘⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠥
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣾⡄⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠲⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⣀⠀⠀⠀⠀⠀⢿⢿⡿⢿⠟⣿⡾⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠟⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠻⠺⠿⠛⠃



⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣼⣦⣴⣶⣴⣷⣶⣯⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣿⠿⣿⣦⠉⣿⣿⣷⡌⣻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠙⠒⠏⢰⣿⣿⠀⣿⣿⣟⣿⡌⠹⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢳⣢⣤⢶⣾⠟⠁⢈⣿⣿⣷⣿⣿⡀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠉⠉⠁⠀⠀⢠⣼⣿⣿⣿⣿⡟⣈⣼⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠀⠀⠀⠀⣤⣼⣿⣿⠟⠩⡇⢳⣿⣿⣿⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⠻⣤⣀⣤⣿⠗⠋⠀⠀⡸⢂⡞⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⢇⠀⠀⠀⠀⠀⠀⡀⠒⠧⢌⢿⣿⣿⣿⡟⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⠓⠀⠀⠤⠀⠂⡠⢂⣦⣾⣿⣿⠷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⣟⢰⡦⣰⣾⣗⣼⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⡿⠿⠟⠛⠉