Quotations Emojis & Symbols

⠀⠀⠀⠀⠀⠀⡔⡀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣶⣒⣒⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⣠⣕⣒⣴⣤⡀⠀⠀⠀⠀⠀⠀⢸⠟⢻⣞⠷⠇⠀⠀⠀⠀⠀⠀⠀⣤⣁⢀⣠⡄
⠀⠀⠀⠀⠀⠉⢇⣉⢨⠉⠀⠀⠀⠀⠀⠀⠀⠼⢬⠎⢊⠆⡆⠀⠀⠀⠀⠀⠀⠀⠜⠃⢛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠃⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⣤⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠘⠀⠀⠀⠀⠀⠀⠀⠘⠞⠞⣶⡾⡗
⠀⠀⠀⠀⠀⠘⢻⠟⠛⠛⠃⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢐⡯⣝⣯⢱⣷
⠀⠀⠀⠀⠀⠀⠃⠉⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠴⠀⡀
⠀⠀⠀⠀⠠⡿⠋⠉⠉⠛⣦⠀⠀⠀⠀⠀⠀⠀⠑⠃⠚⠛⠇
⠀⠀⠀⠀⠀⢸⡠⠤⠡⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠖⠲⠶⠃⠀⠀⠀⠀⠀⠀⠀⠰⢶⣤⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠍

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠁⢸⡀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡠⠈⠀⠁⠈⠁⠉⠀⢠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠛⠘⠀⠈⡇⠃⠓⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡄⢀⡀⡀⠀⢠⠀⡄⠀⡄⢠⠀⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⣧⢸⡇⣧⠀⢸⠀⡇⠀⡧⢸⠀⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁⢀⠀⡌⡀⢁⡄⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡚⠛⠁⠘⡆⡇⡇⢸⡇⠀⠙⠛⡲⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⡠⠀⡄⠀⠀⢀⢠⠀⠀⠀⠀⠄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⠊⠀⠀⠘⠀⠃⠃⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠃⣿⢠⠃⢸⠀⣇⠀⢸⠀⡇⣄⢸⠀⡇⢶⠀⡇⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠛⠈⠂⠘⠀⠀⠀⠘⠀⠃⠘⠘⠀⠃⠛⠀⠃⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢒⣰⣢⡀⠀⠲⠀⠀⣀⢀⡀⣠⠇⠀⠠⠂⠀⢠⢀⣖⢒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠙⠿⣦⠀⠠⢠⡄⣠⣄⠀⡄⠀⠄⠀⣴⠟⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣛⡿⠂⠀⠸⡇⢸⢸⠀⡇⠀⠀⠸⣟⣻⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⢂⣃⡃⢠⠀⡀⠀⠀⠀⡀⠀⠀⢹⣉⡘⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠂⠓⠘⠀⠂⠀⠐⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⢀⡇⣧⠀⡇⣿⠀⢰⠁⢸⠀⢸⡅⣾⢸⡁⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠘⠁⠙⠀⠃⠋⠀⠈⠂⠘⠀⠈⠃⠋⠘⠃⠘⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡶⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⠀⣀⠤⢤⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠹
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣾⠁⠀⠘⠁⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣼⠟⠁⠔⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠻⢦⡤⠤⠒⠁⠀⠀⠀⠀⣀⣀⠀⠀⠘⠋⠀⢠⣿⢋⠔⠊⢀⣠⡴⢦
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀⠀⠀⣀⠀⢀⢤⠀⡀⠈⣰⣿⠃⣴⡗⠀⢠⣿⠋⠀⠀⣴⡿⢋⡤⠊⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠀⣼⡏⠔⢡⣿⠎⠀⣼⡿⠁⢰⡿⠀⢀⣿⡏⠀⠀⣸⡟⠉⠁⢀⠔⠁
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣹⡟⢠⣿⠎⢠⣿⠃⠀⣼⡟⠀⠠⢻⣇⠠⢹⣿⠁⠀⡰⠙⢷⡤⠔⠁⠀⣠⡀
⠀⠀⠀⠀⠀⠸⡀⠀⠀⣠⠟⠁⠈⠛⠀⠀⠁⠀⠀⢿⣀⠔⢁⡤⢲⠀⠈⠿⠶⠊⠠⠖⠒⠒⠲⠤⣴⡿⠁⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠈⢙⡉⣁⣀⠀⢤⠤⡄⢠⢀⡤⡄⠀⠀⢠⡟⡴⠁⠀⠀⠀⣀⣄⠀⣠⡶⠒⡂⣼⡿⠉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⡼⠁⢳⡄⢀⡞⣸⢓⣿⡟⢈⠁⠀⢠⡿⡫⠖⠲⣦⢠⣾⠁⠐⠘⣿⣄⠈⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⢀⣀⠀⠀⠀⢀⠉⠡⠤⠚⣿⡟⣡⠀⢠⠏⣿⡃⠀⡄⠀⠘⣿⠆⣾⡇⢀⠌
⠀⠀⠀⠀⠀⠀⠀⣼⢇⠔⢁⣿⠃⡠⠊⣽⡗⠀⠀⠀⠛⠀⢑⡶⡅⠀⠘⠓⠊⠀⠑⠚⠉⠀⠈⠛⣁⢀⠤⣄
⠀⠀⠀⠀⠀⠀⣼⣯⠋⢀⣾⣧⠎⢀⣼⡟⠐⣉⠉⠑⠒⣶⠛⣊⡁⠀⣠⡤⠄⠀⣴⠟⠀⣰⠏⣼⠋⠀⢀⣿
⠀⠀⠀⠀⠀⣸⣿⠃⢀⣾⡿⠁⢀⣾⢏⡴⠋⠀⣷⠆⣾⠗⠋⣩⢧⡾⢁⠄⠁⣼⠋⠀⣰⡏⣸⠃⢀⣠⡿⠁
⠀⠀⠀⠀⠀⣿⠇⠀⣼⡿⠁⢀⣾⠏⣾⠁⠀⣼⠃⣸⡿⢶⣍⠀⢻⣏⣀⠔⢰⡇⠀⡰⣿⢠⡏⠙⠛⠁
⠀⠀⠀⠀⢸⡟⠀⠀⣿⠃⠀⢸⣿⠘⢧⡠⠺⣏⠰⣿⠀⠀⠹⢷⣄⣀⣠⠔⠈⠓⣊⡀⠬⠺⡏⠉⢹⠄
⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠘⢿⣄⡀⠀⣄⠀⠀⠛⠀⠀⠀⠀⠉⣉⣠⠴⠒⠋⠁⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠒⠛⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⢛⣛⣛⡛⠛⠛⣛⣛⣛⠛⠛⠛⢛⣛⢛⣛⣛⣛⠛⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⡏⠈⢻⡆⣼⡏⠀⢹⣧⠀⠀⢸⣿⠀⢸⣿⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣧⣤⡾⠃⠹⢧⣤⡾⠃⠀⠀⠸⡿⠀⢸⡿⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠖⠶⣶⡶⢶⣶⡶⠖⠲⢶⣶⠲⣶⣶⠖⢶⣶⠖⣶⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡀⠹⡇⢸⡏⠀⣾⣶⠀⢻⡀⢸⡏⠀⠸⡏⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⡄⠀⢸⣇⠀⢿⡿⠀⣸⣇⠀⠀⣿⠀⠁⣼⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠀⠛⠛⠀⠘⠛⠂⠀⠀⠘⠛⠛⠀⠘⠛⠃⠀⠛⠃⠈⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⢼⣹⢾⣷⡾⡇⣿⡉⢹⢹⡇⣿⡾⡇⣿⡀⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡁⠀⠉⢠⣤⡄⢠⣭⣥⣤⢠⣤⣤⣤⠀⣬⣥⣬⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⠻⣷⠀⢸⣿⠀⢸⣿⣤⡄⠀⣿⣤⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⢴⠟⠛⢻⡇⠸⡿⠀⠸⣿⣤⣤⠀⣿⠀⠻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣲⡆⣶⡒⢠⡖⠦⣴⠒⣆⣶⣆⣴⡆⣶⣒⢲⣖⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠼⠇⠿⠶⠘⠧⠖⠹⠶⠏⠟⠹⠋⠇⠿⠦⠴⠮⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢸⡇⣾⣟⡛⠹⣧⢠⡟⢺⣟⡛⠁⣿⠛⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⡇⣿⣏⣁⠀⢻⣾⠁⢿⣏⣉⡀⣿⠙⣯⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠁⠀⠈⠉⠉⠀⠁⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣠⣤⣦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⣞⡳⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⢋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠖⠙⠫⠹
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠐⠁
⠀⠀⠀⠀⠀⢀⣴⠞⣿⡟⣀⣀⠀⠀⣠⣤⢄⡀⠀⣾⣆⡤⣤⡄⢰⣆⠤⣦⡄⢰⣦⠀⣾⡆⠀⣠
⠀⠀⠀⠀⢰⡿⠁⢰⣿⠋⣼⡿⠀⣰⡟⣡⣾⠇⣠⣿⢃⠄⣾⠏⣽⣇⢤⣿⠃⣿⣃⢼⣿⢁⣴⠏
⠀⠀⠀⠀⠈⠃⠀⠘⠃⠰⣿⣁⠔⠉⠉⠘⠯⠊⣾⠏⠘⠛⠉⢰⡿⠈⠋⠁⠀⢉⣡⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⣾⠃⠀⠀⢀⠔⢉⣾⠋
⠀⠀⠀⠀⠀⠀⠀⣆⠀⠠⣦⠤⢄⡀⠀⠀⠀⢿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠘⠤⠞⡁
⠀⠀⠀⠀⠀⠀⠀⣸⡄⣀⢧⠑⠦⡍⢆⠀⠀⠀⢀⠤⠤⡀⠀⠀⠀⢀⣠⠤⠒⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢬⣳⢄⣉⣾⠄⡜⢁⣹⡒⢢⡌⠳⡄⡀⢋⣀⠴⣋⠜⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⢉⣥⣲⠃⢸⡀⡁⢾⣫⡤⠓⡤⠁⢸⠛⢒⣈⣁⡀⠆⢀⠘⠛⠓
⠀⠀⠀⠀⠰⠾⠖⢀⠀⠀⠐⠛⠓⣫⡆⠀⠉⡍⠓⠚⠁⡠⠅⠀⢞⣶⡈⣿⢉⠉⠀⠘⠲
⠀⠀⠀⠀⠀⠀⠼⠟⠀⠀⢀⣴⠏⠺⠙⠦⡀⠉⠙⠂⠚⠁⡠⠊⠀⠑⠂⠻⠈⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢸⢀⣇⣀⠀⠀⣀⢀⡀⢀⣀⡀⢀⠀⡀⠀⠀⢀⠀⡀⢀⠀⢀⣄⢀⣀⣀⢾⡷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⠀⣿⡈⠀⠀⡟⢸⡇⣟⢹⣇⣼⢀⡇⢀⠀⢸⠀⡇⢸⢰⠋⣶⢸⡏⢹⢸⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⣷⠀⠀⠙⢻⡗⠙⠛⠉⠙⠋⠙⠁⠀⠘⠛⠙⠋⠈⠛⠛⠛⠁⠘⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡟⠀⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⢀⣀⠀⠀⢤⡄⠀⢠⡄⠀⢀⣀⣀⢀⣀⣀⣀⡀⣤⡄⠀⣀⡀⣀⡀⢀⣤⠀⢀⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠈⠀⣟⠈⢷⠀⢸⣧⠀⢸⣇⠀⡏⠉⠙⠈⠁⣿⠀⠁⢸⠀⠀⢽⠀⢸⠁⢸⣷⡀⢀⡇⢀⡞⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢰⡏⠳⢼⡤⢼⠹⣆⡏⢿⠀⡇⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⠀⢸⠀⢸⡏⣧⢸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢻⠘⡇⠀⣸⠀⢸⠀⠿⠁⢸⠀⡷⠆⠀⠀⠀⣿⠀⠀⢸⡶⠶⢿⠀⢸⡀⢸⡇⠸⣾⠀⢻⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣡⡼⠀⠙⠶⠋⠀⠸⠆⠀⠀⠘⠃⠷⠤⠗⠀⠀⠿⠄⠀⠸⠧⠀⠾⠆⢸⡇⠸⠇⠀⠻⠀⠈⠳⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠖⣗⣤⠀⣠⣄⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⣀⡟⢸⣇⣷⠏⣸⠁⣇⣠⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠁⠈⠉⠉⠉⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠄⠀⣼⣇⠠⣴⡤⠀⣠⡶⢦⡀⠀⠚⠛⡟⠙⠛⢶⣄⠀⠛⣿⠋⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⣿⠀⣿⠀⠀⣿⠀⠀⢻⡄⠀⠀⡇⠀⠀⠀⣿⠀⠀⣿⠀⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢻⡀⣿⠀⢰⡟⢧⣄⣸⡇⠀⣠⡧⠀⣠⡾⠃⠀⠀⣿⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣼⠃⢸⣷⡇⠀⢸⡇⠀⠉⠛⡿⠛⠉⣿⠛⣿⠀⠀⠀⠀⣿⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡿⠀⠘⣿⡇⠀⢸⡇⠀⠀⢸⡇⠀⠀⣿⠀⠘⣧⠀⠀⠀⣿⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⢿⣄⣠⡾⠁⠀⠀⣿⠀⠀⠘⠷⡤⠀⣿⠀⠀⠈⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠉⠀⠀⢀⠀⠉⠉⠀⠀⠀⠺⠿⠂⠀⢀⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡀⠀⠀⠀⠳⣼⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⢸⡟⢳⡿⣷⢸⢯⡇⠀⠀⡇⢸⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡌⠷⠟⠋⠘⠷⠃⠀⠳⠛⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⢀⠠⠂⣠⣠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⡀⠦⠶⠒⠀⠀⠀⠀⡄⢠⣂⠀⠀⠀⠀⣀⢀⣠⣀
⠀⢺⠙⣋⣉⣁⣈⠀⠀⠀⠐⠀⢥⠀⠀⠀⢘⠇⠦⠚⠚⠾⠇⠀⠀⠀⣷⡘⠇⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠁⢀
⠀⠀⡄⠠⠁⣉⣭⠀⠀⠀⣀⡀⡀⣠⡀⠀⠀⠶⠿⠰⠿⢼⠄⠀⠀⢀⡈⠉⠓⠀⠀⠀⠀⠀⠂⡐⠂⠀⠀⠀⠀⠀⡠⣤⣴⡆⣾⢰⣢
⠀⠀⠙⠈⠛⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠾⠰⠃⠀⠀⠀⠳⠴⠤⠄⠀⠀⠀⠀⠂⠉⠁⠉⠀⠀⠀⢰⣮⠇⠀⠂⠀⡠⢮⣁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⣇⢸⠃⠍⢷⠁⣏⠪
⠀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠖⠟⠛⠒⠀⠀⠀⠀⣀⠠⠀⠐⠀⠀⢠⣔⣀⣤⣷⠷⠄⠀⠀⠀⡇⣐⢸⣠⠀⠀⠘⠉⠉⠣⠤⠤⠞⠀⠂
⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⢌⠀⠀⠀⠈⠀⠀⠀⠰⠒⠚⠃⢀⠀⠸⡘⡅⣀⣤⣄⣄⠀⠀⡀⠁⣥⠼⣉⡠⠀⠀⠀⡀⠀⢠⠀⠘⢀⡠⢠
⠀⠀⠀⣀⢀⣀⠈⠁⠀⠘⠘⠒⠛⡓⠎⠀⠀⠑⣂⠐⣌⡹⠋⠀⠠⠡⠤⠼⠦⠬⠴⠀⠀⢁⠙⠉⠉⠉⠉
⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠁⠀⠀⠰⠤⣄⡶⠇⠀⠀⠀⠀⠓⠛⠓⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣄⠀⠀⠀⢠⣦⣤⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣿⣷⣄⣿⣿⠀⢴⣶⣾⣯⣽⣿⠀⢠⣿⡟⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠃⠀⠀⢸⣿⢿⡿⣿⡇⠀⢸⣿⠁⢹⣿⢀⣿⡏⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⠟⢿⣷⠸⣿⡄⠁⠘⠛⠀⠛⠋⠀⠛⠗⠸⣿⣤⡄⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠀⢿⣷⣿⠏⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠿⣷⣤⠀⢀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣬⡿⢃⠄⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⣠⡾⠃⠀⡀⠀⠀⠀⠀⠈⢡⡘⢯⡵⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣿⠁⠘⣀⡀⠐⠋⢉⣿⡟⠛⠒⠒⢛⠛⠉⠀⣠⣾⡟⣀⣀⠀⠀⠀⠈⣃⡈⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠃⢀⣴⣿⡿⠁⠀⣰⡾⢋⡿⠀⢠⣿⣿⠏⠁⣹⣧⠀⣀⣼⡏⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⢀⣹⣷⠊⠁⣼⣿⠃⢀⣼⡿⠗⠋⢀⣴⣿⡿⢁⣀⣴⣿⠟⠊⠁⢹⣷⡀⠀⠀⠀⣰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⣠⠖⠁⢸⣿⠅⢸⣿⣃⡴⠛⣿⡃⣀⡴⠋⣼⡿⣷⣉⣼⠟⠁⢀⡴⠒⠚⣿⣷⠶⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣇⠀⢀⣾⠟⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⣼⡿⠁⠈⠉⠀⠀⠀⢺⣀⢀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⢺⣿⣿⠀⢰⣿⣿⠇⠀⣿⣿⣿⣿⣷⠀⢰⣿⣿⠿⣿⣶⡄⠀⣿⣿⠂⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠙⠀⠀⣿⣿⠀⢸⣿⡟⠀⠀⣿⣿⣀⣄⠀⠀⢸⣿⡇⠀⣨⣿⡿⠀⣿⣿⡀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣤⣾⠀⠀⠀⢻⣿⡆⣼⣿⠃⠀⠀⣿⣿⠛⠁⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⢻⣿⣧⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠉⠀⠀⠀⢸⣿⣧⣿⣿⠀⠀⠀⣿⣿⠀⣀⣠⡄⢸⣿⡇⠀⢻⣿⡄⠀⠀⠉⠛⠛⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣴⡆⠀⠀⣿⣿⣿⣿⠀⠀⠀⠿⠿⠿⠿⠿⠇⢸⣿⡇⠀⢸⣿⡇⠀⠘⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣿⡇⠀⠀⠙⠉⠁⠀⠀⣠⣶⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠈⠛⠻⠀⠀⢿⣿⣦⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠁⠈⠉⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣶⠀⢀⡀⠀⠀⣀⠀⠀⠸⣿⣿⠁⠀⠀⠀⠀⠀⠘⣿⣷⠀⠀⣀⣤⡄⠀⢀⠀⢀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⣦⡿⣧⡈⠇⠀⣰⣿⣿⣷⣦⡄⠉⠀⢀⣀⣀⣀⣀⡀⢹⣿⡆⠀⢹⣿⣧⠀⠾⢁⣼⢿⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⠛⠀⣰⣿⣿⠉⠹⣿⣿⡄⠀⣿⣿⠿⣿⣿⡇⠘⣿⣿⡀⣸⣿⣿⡆⠀⠛⠳⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⣿⣿⠃⠀⢸⣿⣿⠃⣸⣿⣟⣀⣹⣿⣷⠀⠘⠿⣿⡿⢻⣿⣿⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣧⣤⣴⣿⣿⠏⢰⣿⣿⠿⠿⠿⣿⣿⣇⠀⠀⣤⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⠁⢠⣿⣿⡏⠀⠀⠀⢹⣿⣿⡄⠀⢻⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠘⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⣯⢸⡇⡟⠁⠀⠀⢠⡗⡲⢄⠀⠠⣿⠄⠀⠀⣀⣀⠀⠀⠀⠺⣄⡀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠸⡇⡇⠀⠀⠀⢸⡇⡆⠸⡟⠀⣿⠀⠈⠁⠉⠉⠙⠓⠒⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⠶⡇⡷⠆⠀⡠⢾⠀⡇⠀⠙⡆⣿⠀⡠⢴⡄⣇⣀⠀⢠⠤⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠙⠀⠃⠙⠒⠀⠓⠺⠄⠓⠘⠓⠃⠛⠃⠑⠚⠇⠋⠈⠳⠚⠒⠚⠣⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠔⠂⠤⠔⠂⠤⠔⠢⠤⠒⠠⠤⠒⠤⠤⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⡶⣄⢠⠖⢀⡖⠂⢲⠂⡆⠀⡆⣴⠂⢰⡀⢴⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⡀⠀⠀⠀⡇⢸⢸⠀⢸⡇⠀⢸⠀⡇⠀⡇⡯⠀⢸⢧⢹⠀⠀⠀⣀⣠⠴⠞⠋⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠖⠂⣈⠭⠅⠀⠀⣷⡋⢸⡤⠘⡧⠀⢸⡀⣧⣀⡇⣧⠄⢸⠈⣿⠀⠀⠀⠐⠦⣄⡙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠓⠋⠘⠓⠈⠓⠀⠘⠃⠙⠛⠃⠙⠆⠘⠀⠙⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠈⠒⠒⠉⠂⠒⠉⠒⠊⠑⠒⠊⠑⠒⠈⠐⠒⠉⠒⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣖⣾⠙⠆⠀⠀⡟⢦⢰⣟⡳⢸⠋⢰⢻⡄⡷⠼⡇⡞⠻⠄⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⣀⡄⠀⠀⠈⠻⠏⠀⣄⠀⢿⢹⡆⣯⠀⢸⠀⢸⠀⡇⡇⠀⡇⢳⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⢴⢳⠆⣆⣠⡆⡏⠃⢸⣺⢱⡟⡆⢸⠆⢸⣤⡇⡇⠀⡇⠀⢳⠀⠀⠀⢄⢠⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡇⠈⠉⠀⠀⠀⠁⠁⠀⣘⠁⡈⠁⠁⢈⣛⠈⣠⡁⠁⢀⡇⠈⠋⠀⠠⠄⢪⠖⠀⠒⢄⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡈⠛⠁⢠⢶⠠⡆⢢⢰⢦⠘⢇⣈⡼⠂⠀⣈⠀⠃⣿⠂⠐⠈⡏⠀⠀⠀⠐⣂⢇⡠⠄⠔⣠⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠤⣉⣈⡡⠌⠀⠀⢸⣸⠃⡉⣾⢸⢸⠀⢠⣁⠀⠀⠀⠀⢀⡀⣿⠁⡇⠀⡇⢰⣠⡆⠈⠀⠂⢨⠻⠉⢆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⡒⠂⢚⡓⠀⠀⠘⠛⠁⢃⠙⠘⠋⠀⢸⠉⢼⣛⣣⣞⡹⡇⣯⣠⣇⡀⣇⡄⠀⡇⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠦⠬⢭⠥⠬⠀⠀⠘⠩⠑⠒⠊⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠁⠈⠀⠀⠀⠈⢠⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⢠⣿⢀⣿⠀⠀⠀⠀⠀⡇⠀⢠⣄⠀⡤⣄⢤⡄⣭⢀⣤⡀⢠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢰⠀⢸⡁⠈⢳⡀⠀⠀⠀⠀⢸⡟⢸⡟⠀⠀⠀⠀⠀⡇⠀⣿⢸⢸⠃⠛⢸⡇⣿⢸⠖⣷⢸⣇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢳⠀⢸⣧⠶⢀⡤⣶⢸⡇⢸⠇⢠⣴⡄⣠⢾⡇⠀⢸⢸⢸⡄⠀⠸⡇⣿⢸⡋⣿⢸⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠘⠋⠙⠚⣀⡾⠙⠒⠚⠓⠙⠈⠓⠛⠓⠛⠒⠚⠙⠺⠇⠀⠼⠹⠀⠧⠇⠀⠇⠹⠘⠧⠏⠸⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠐

⠀⠀⠀⠀⠀⣀⠤⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⣤⣤⢤⡴⡄⣬⣤⢭⣧⢤⢤⣤⣤⡤⣤
⠀⠀⠢⠖⠡⠀⠀⢠⠀⠀⠐⠈⠂⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⠉⠩⠉⠉⠨⠍⠛⠛⠋⠙⠉⠛⠙⠋⠙⠙⡟
⠀⡁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⣄⡀⠀⠀⠀⠀⡇⡇⠘⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂⠀⠘⠀⠀⡧
⡟⢀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢀⡇⢀⣈⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⣿
⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⡏⠀⠀⠀⣿⢓⣶⣦⣐⣖⣀⣶⣆⣶⣶⣾⣆⣶⣶⣳⣶⣶⣷
⠰⡔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠁
⠀⠙⢌⠢⠈⠀⠀⠀⠀⠀⠄⡀⢌⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠉⠒⠤⢭⣠⣤⠬⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⡍⠉⢡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠃⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⠀⢔⡢⠖⠊⢹⡄⠉⠛⠷⣠
⠀⠀⠀⠀⠀⠀⢀⡆⠈⠈⢰⠄⠈⠉⣏⣉⠁⠀⢰⡂⠁⠠⡀⠀⠀⠠⡞⠛⠓⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⢺⠋⠀⠀⠀⢾⡗⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠘⢲⡀⠖⣠⠔⠉⠀⠠⠀⠀⢉⠂⡝⢶⢀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢈⠀⠀⣀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⡼⠁⢠⣄⠀⠀⠀⢀⡀⠑⠘⡀⠁⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⠁⠀⠀⠀⠘⠁⠉⠋⠁⠀⠉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠀⠉⠳⣦⡞⠋⠀⠠⠀⠁⠀⠀⠀⠀⠀⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠫⢄⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⣐⡅⣀⡀⣀⡠⢦⣔⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠠⠀⠀⠄⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠿⡟⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠒⠒⠂⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠙⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⣻⠞⠁⠀⠀⠀⠀⢀⠀⠤⠀⠁⠼⠤⠥⢈⣀⣤⣀⡀
⠸⠧⠥⠴⠦⠤⠄⠄⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⠀⠀⠀⠀⠙⢦⣀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠰⠾⠿⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⡄⠀⠀⠀⠀⠀⢻⡇⢀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠘⣇⣾⡿⠁⠀⠀⠀⠀⠀⠈⠉⠏⢈⠁⠁⠀⡀⢀⠀⠀⠀⠙⢏⠁
⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣿⣶⠄⠀⠀⢠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡅⠀⠈⠂⠀⢀⢸
⠀⠀⠈⠐⠄⠂⠂⠈⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡧⠐⠾⢿⡗⢦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡀⠀⠀⢀⡀⠀⣀⠀⠠⢸⠈
⠀⠀⠀⡀⠀⠀⠀⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⣰⠀⠠⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⢸⠸
⠀⠠⢁⠄⠀⠀⠀⠐⠀⠑⡄⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⣔⠁⠀⠼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠠⠄⢀⠠⡀⠀⠀⡀
⠀⢃⠇⠀⠀⠀⠀⠀⠀⠱⢡⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⡏⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠂⠀⡞⢹⡏⠙⠂⠈⠀⠀⡀⡆
⠀⡀⠃⠀⠀⠀⠀⠀⠀⢈⢨⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠇⠉⣠⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠇⡈⠡⠀⠀⠀⠀⠀⠀⠇
⢠⣗⣐⣒⣆⣓⡒⠚⠒⠒⠚⠆⠀⠀⠀⠀⠈⠠⣈⣀⣠⣤⣼⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠚⠛⠋⠙⠛⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣀⠀⠀⠰⠆⠀⠀⢀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠈⠁⠀⢸⡇⠀⠀⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠀⢀⣴⠟⠁⠀⠀⠀⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣠⠈⠁⠀⠀⠀⠀⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠀⠀⢰⡆⠀⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣤⡴⠞⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠀⠀⠀⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⡄⠀⢤⣤⣠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⣁⣀⣸⣀⡙⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⣩⣷⡾⠟⠛⠛⠛⠛⠛⠿⢶⣿⣙⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡄⠀⠀⠈⠙⢢⣦⡄⠀⠤⠤⠄⢤⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⠷⣶⣶⣶⣶⡶⠿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠋⠉⠉⠉⠉⠉⠉⠑⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⣀⡤⠴⠒⢚⣉⣉⣉⣉⣉⣉⣉⡙⠒⠲⠤⣄⡈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⡠⠖⢋⡡⠔⠚⠉⠉⠀⠀⡄⠀⠤⣄⠀⠈⠉⠙⠒⠤⣉⠓⠦⣀⠉⠲⣄
⠀⠀⠀⠀⠀⢀⡴⠋⣀⠔⢉⡴⠊⡅⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢠⡄⠉⠢⣌⠑⢤⡈⠳⣄
⠀⠀⠀⠀⣠⠋⢀⠜⢁⠔⠁⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢄⠙⢆⠈⠣⡀
⠀⠀⠀⡼⠁⡰⠃⡴⠁⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠱⣄⠳⡀⠙⡄
⠀⠀⡜⠀⡼⢁⠞⢠⠀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⢆⠹⡄⠘⡄
⠀⡼⠀⡼⢁⠎⠀⢸⠀⣯⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢆⠹⡄⠸⡄
⢠⠃⢠⠃⡞⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⡆⢳⠀⢳
⡼⠀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢳⠘⡆⠸
⡇⠀⡇⢸⠀⢀⠄⢦⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⣀⣀⣤⠤⠤⠶⠒⠒⠻⠖⠉⠀⠐⠦⠀⢸⠀⡇
⡇⠀⡇⢸⠀⠈⢃⠟⠛⠧⠼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢄⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢘⠀⡇
⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⡇
⢫⠀⢧⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⢰
⠘⡄⠘⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⡼⠀⡜
⠀⢳⠀⢱⡈⢇⠀⠀⢦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠗⢀⠏⣰⠃⢰⠃
⠀⠀⢣⠀⠳⡈⢦⠀⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⠃⢠⠃
⠀⠀⠀⢳⡀⠱⣄⠳⡄⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠂⠀⠀⠀⡴⢃⡴⠁⣰⠃
⠀⠀⠀⠀⠙⣄⠈⠢⡈⠢⣀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⢀⡠⠊⡠⠊⢀⡜⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠈⠢⣌⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢤⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠑⠢⣌⡑⠲⠤⣄⣀⡀⠀⠀⠣⠘⠀⠀⣀⣀⡤⠴⠒⣉⡤⠖⠉⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡉⠑⠒⠤⠤⣍⣉⣉⣉⣉⣉⣉⡥⠤⠔⠒⠉⢁⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢤⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⣰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢠⣷⣶⣤⣤⠀⠀⠀⣀⣀⣠⡀⠀⠀⢠⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⡿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡤⠆⠘⢀⠃⠾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⣤⣆⡀⣀⣄⣠⡤⠀⠀⣀⣀⣀
⠀⠀⠀⢠⡇⠀⡋⠉⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠁⢄⠈⠉⠙⠉⠉⠀⠀⠀⢠⠿⠿⠿⠧⠃
⠀⠀⢠⣦⠀⠂⠀⠈⠉⠀⣸⡗⠀⠀⠀⠀⠀⠂⠄⣀⠔⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠶⠚⠁
⠀⠀⠈⠛⠳⢤⣤⣤⣤⣾⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣢⢰⠃⠀⠀⢀⠾⢿⡿⠿⡄⠀⠀⠀⠈⠀⠀⠀⠀⠞⠁⠓⢄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⣸⡂⠀⠰⣾⡿⠏⠀⠀⠈⠢⠤⠴⠖⠁⠀⠁⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠆⠀⠀⠀⠙⠷⠶⠶⠞⠛
⠀⠀⠘⡏⠃⠀⠌⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⡜
⠀⠀⠨⣇⣀⠀⠘⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢐⢂⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠂⣈⠸⠆⠀⠀⠀⠀⢀⢬⣧⠄
⠀⠀⠀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⢹⡏⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠈⠉⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠠⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠒⣶⠒⠦⣤⣀⠀⠀⢰⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⣀⡔⠾⣀⠀⠉⠀⠀⠀⠙⢷⣄⣈⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢤⣀⣬⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠹⣿⠂⠀⠀⡱⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⣸⣿⣿⣿⣿⡌⠉⠀⠀⠀⠀⠀⠀⠀⢡⢴⣿⣶⠐⠃⠱⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣠⣠⣶⣿⢏⣿⣿⣿⡆⢀⡀⠀⢀⠠⠤⠾⣞⠁⠀⠀⢿⡅⠆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡤⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣾⡓⠀⠀⠀⠀⠀⢫⠀⠀⠉⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⣀⣠⣿⣖⣻⣿⣾⣿⣟⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠘⠄⠀⢢⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠸⣶⠀⢸⣦⡾⣿⣤⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠛⠛⠋⠀⠈⠉⠉⠉⣉⣉⣭⣽⣿⣬⣽⠙⠓⠒⠒⠛⠛⠁⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⢿⣶⣶⣿⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠴⠖⠋⣰⣿
⠀⠀⠀⠀⠀⢠⣿⠛⠉⠀⣼⣿⠇⠀⢠⣿⠀⠀⣀⡀⠀⠀⢈⣉⣩⠉⢁⣤⠀⢀⣠⣄⠀⢠⣿⣟⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⣿⡏⠀⣠⣿⡇⣠⠊⣽⡇⢀⣾⡟⣿⣿⠀⣼⣿⢠⠎⣰⣿⠀⣾⣿⠏⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣰⠃⣿⣰⠃⣸⣿⠁⣾⡟⣸⣿⡇⢠⣿⣿⠏⣸⣿⡇⣰⣿⣿⠿⢿⣿⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣰⠃⠀⢻⡇⠀⣿⠇⣰⠛⠷⠟⣿⡇⡾⠹⠏⢠⣿⡿⢰⠇⢸⣿⠀⢻⣿⡀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣶⣶⠀⠀⠀⠻⠶⠃⣠⣄⠀⠻⡷⠃⠀⣤⠈⢿⣧⠏⣀⡀⠛⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢠⣿⡆⢀⣴⣾⡁⠈⡇⢠⡆⠀⢸⣿⠇⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⣠⠏⣿⣿⢻⣿⡇⠉⢛⡟⣿⡇⠀⣿⣿⠀⣠⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠞⢻⣿⡌⠻⢷⠶⠋⠀⠻⣧⠞⢿⣿⢰⠇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡾⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡾⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠉⠀⠀⠀⡀⠀⠠⠄⠀⢀⠀⠀⠈⠉⠙⠛⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⡀⠄⠒⠀⠀⠀⠀⠠⣤⠀⠠⠄⢦⡄⠀⠈⠀⠃⠠⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⠄⠈⠀⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⠀⢀⣀⣋⣀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⠀⢤⠀⠙⢿⣷⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠄⠀⠀⠀⠀⠀⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄⠀⠙⣿⣦
⠀⠀⠀⢠⣾⡟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢿⣷⡀
⠀⠀⢠⣿⠏⠀⠦⠀⢠⣤⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠆⠀⢻⣷⡀
⠀⢀⣿⡏⠀⡠⠀⠀⢸⡏⠀⣸⡃⠀⢸⡇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠤⠀⢻⣷
⠀⣼⡟⠀⢠⠀⠀⠀⢸⣇⠀⢹⡇⠀⢸⠇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⡀⠈⣿⣇
⢰⣿⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠚⠛⠀⠀⠀⢀⠀⢸⣿
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⣿⡇⠀⠀⠀⣠⠂⠘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠃⠀⢿
⣿⡇⠀⠆⠀⠙⠶⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣬⡀⠀⠀⠶⠀⢹
⣿⡇⠀⢀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣕⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⡄⠀⣿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠐⣿⣿⣿⡟⠁⢀⠀⠀⣿
⠸⣿⡄⠀⠀⠀⠀⠀⢀⡀⢀⡐⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣦⣤⣴⣶⣾⣿⣿⣿⣦⣈⠀⢸⣿
⠀⢻⣷⠀⠙⡀⠀⠀⠸⣆⣠⠃⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⡏
⠀⠈⢿⣇⠀⠐⠀⠀⣴⠉⠙⣷⣰⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⣦⠀⠖⠀⠈⠓⠚⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠈⢿⣧⡀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠲⠀⠀⠀⠈⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠘⣿⡧
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠃⢀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠠⣿⣿⣿⣿⣿⣿⣿⢠⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠁⢾⣿⣿⣿⣿⣷⡀⠀⠀⠀⢰⡇⠈⢻⡆⠀⠀⠀⠀⠀⠘⠂⠈⠁⢠⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠓⠀⠚⠁⠀⠀⠀⢀⠀⠤⠐⠈⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣧⠁⠉⠀⠘⠂⠀⠉⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋⠁⣼⣿⣿⣿⢿⣿⣿⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⣷⣶⣶⣶⣶⣾⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣏⣻⣿⣦⣈⣏⣙⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣑⣾⣿⣛⣿⣁⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡀⠀⢄⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠤⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠠⢤⠀⠀⢀⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⡐⠛⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠗⠀⠀⠀⠻⠂⠀⠀⠀⢰⠀⢀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⡄⠀⠀⠀⠀⠀⢂⡐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⡀⠄⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠐⠀⡀⢀⠀⠀⠀⠀⠊⠙⠐⠀⠐⡀⠀⠀⠘⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⡌⣀⡀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠠⠄⠠
⠀⠀⠀⠘⠂⠁⠀⠘⠀⠀⠀⢁⣉⣀⣀⣀⡀⠀⠀⠀⢀⡠⢀⢀⠀⠀⠀⠀⢀⠀⣂⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠠⠲⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈
⠀⠀⠀⠠⠒⠒⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠠⠀⠀⠀⠀⠠⠤⢂⠄⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠬⠡⠀⠀⠀⠀⠀⠠⠀⠀⡠⠀⠀⠀⠀⡁⠀⠠⢀⠀⠀⠀⠀⠰⠽⠠⠁⠂⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠴⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⡀⠠⠀⠀⠀⠀⠨⠀⡀⠁⠀⠀⠀⠘⠷⣤⢠⡾⠋⠀⠀⠀⠐⠀⡄⠀⠐⠀⠀⠀⠤⠄

⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢤⠤
⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠰⠀⠀⠀⠄⠀⠀⠀⠀⠎⡀⠀⠉⠁⠀⠀⠂⠰⢀⠀⠠⠈⠀⠀⢰⡃⠂⠀⠜⣳⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠠⡄⠀⠁⠀⣄⠀⠀⠂⠠⠈⠠⠄⠀⠃⠀⣠⣶⣿⣿⣷⣦⡄⠀⠀⠀⠠⠀⠠⠄⠀⠀⠀⠁⠀⡤⠄⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⡀⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⢂⣴⣾⣿⣿⣷⣼⣟⡹⣿⡿⣿⡿⢿⣿⢿⣿⠿⣿⡟⠻⣟⣙⣿⣶⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡋⠛⠻⢿⣿⣿⣿⣿⣷⣾⣶⣶⣿⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠠⣁⡌
⠀⠀⠀⠀⢀⢄⠀⠀⣰⣿⣿⣿⣿⣿⣶⣤⣄⡹⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠟⢻⣿⣿⣿⣟⢿⣿⣿⣿⣿⣦⠐⠈
⠀⠀⠀⠰⡂⣑⠂⣴⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣸⣿⣿⣯⣴⣾⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⣰⣿⣿⣿⢿⡿⠟⠛⠙⣿⣿⠓⢾⣿⣏⡼⠛⢿⣿⠛⢿⣿⡿⠙⣿⣿⣿⣭⠟⠛⠛⠛⠛⣿⣿⣆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⢰⣶⣿⣿⠇⠀⠘⣿⠟⠁⠀⣼⡿⠀⣸⣿⠇⢀⣿⠿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧⣄⠈⢿⣿⠀⢠⠀⠁⢠⠀⠀⣿⡇⢀⣿⡟⠀⣼⣿⣶⣿⡇⠀⣠⣀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠙⠿⠛⠀⣸⣏⠀⣾⣷⣶⣿⠀⣾⣿⠀⢸⣿⡇⠐⠟⠛⢛⣿⠀⠘⠛⠛⣛⣿⣿⣿⣿⣷⠰
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣖⣶⣿⣿⣿⣾⣿⣧⣿⣿⣶⣿⣿⣷⣿⣯⣷⣶⡶⡶⣿⣿⣷⣶⣶⣿⣭⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣟⡉⢿⣿⣿⣿⡿⢿⡿⣿⡿⢿⡿⠿⣿⢿⠿⣿⢿⡿⢿⡿⠿⠇⣷⣿⠿⣿⡿⢿⡿⣿⣿⡿⠉⣹⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⣐⣢⣷⠘⢡⡏⣐⣂⣽⢀⣾⡇⡼⠃⡼⢡⡶⢰⡟⣡⠆⣸⠁⡞⢀⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣭⣴⣿⣦⣾⣷⣬⣵⣷⣤⣿⣧⡴⢰⣷⣤⣦⣼⣷⣤⣦⡿⣦⠄⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⣀⣀⠸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠿⢫⣭⣵⣾⣯⣵⣿⠟⠋⠙⠋⠻⠿⢿⣷⢦⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠉⠛⠷⠙⣿⣿⣿⣿⣯⣶⣭⣶⣮⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡿⠟⠛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣶⣾⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡤⠠⠀⠀⢀⣤⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣠⡿⠀⣸⣶⠀⢸⢀⣾⡏⢠⣾⠃⠀⠀⠀⢀⣾⠟⣿⡟⠀⠀⣾⣷⠋⠙⠛⢠⣾⢛⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢚⣿⠟⠨⣿⠋⢉⡟⠀⢿⡴⣿⡏⠀⠀⡜⠀⠘⠿⢺⡿⠀⢀⠔⣿⡏⠀⠀⠀⢸⣏⠁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣹⡿⠀⠀⠙⠓⠋⠀⠀⠀⠀⣿⡇⢀⡜⠀⠀⠀⠀⠈⠳⠖⠋⠀⠙⠃⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⠀⠀⠀⠀⠀⠀⢀⣴⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⡜⠀⠀⠀⠀⠀⢀⣾⢃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠞⢹⡆⢀⣥⠀⡆⣰⡖⢠⡄⠀⢀⠀⣰⠖⠆⢠⣿⡞⠳⠆⣼⡧⠊⠀⣼⠃⣴⠂⠀⣾⠗⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣧⣾⠃⢼⡏⣹⠋⣿⣠⣿⠁⠀⠈⠉⢿⣉⡤⠺⣿⠀⠀⢀⣿⡄⠀⠈⠿⢾⡏⠀⢠⡿⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠈⠉⠁⠀⠈⠀⠻⡤⠖⠀⠀⠀⠀⠀⠀⠉⠀⠀⠸⣿⢸⠀⠀⠀⠸⣇⡤⠁⠙⠊
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢀⣤⣤⣀⣀⣀⣠⣿⠃⣀⣠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠀⠀⠀⠀⠀⠀⠀⣀⢤⣤
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⣀⠀⠈⣩⣿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⢇⣴⣿⠃⠀⠀⣴⣦⠶⣤⡄⣂⠁⢂⠀⣤⣠⠒⣶⠀⢀⣠⣄⠀⠀⠀⠸⣄⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠋⣼⡏⠀⠀⢠⣿⡏⠀⠀⣰⡿⠤⡼⠸⣿⠃⢸⡇⠀⣿⡵⢿⡦⠒⠀⣠⣿⣟⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⠀⣿⠃⠀⡰⠃⠻⠇⠀⠀⠹⠧⠞⠁⠀⠁⠀⠸⣧⡼⠁⢠⣿⠇⠀⢰⡿⠁⠈⠳⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣰⠟⠀⠹⠧⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣿⠀⠀⢺⡃⠀⢀⡜⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⡼⠃⠀⠀⠀⠙⠚⠉⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⣤⣿⣁⣀⣠⠤⢒⡶⢲⠀⠀⠀⠀⠀⠀⢀⣴⠋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢠⠀⢠⡿⢀⡞⠀⠀⠀⠀⠀⢀⣾⢃⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠔⠋⢀⣴⢲⠄⢀⡶⢳⡶⠀⢠⡿⢁⡶⠀⢀⣿⠃⠀⣠⡆⠀⣾⡧⠊⠀⣼⠇⣴⠆⠀⣸⡟⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢨⡗⠸⣯⣁⠤⠚⠓⣿⠁⡠⠺⠧⣾⡇⠀⣸⡏⠀⢀⣿⢀⣰⣿⡄⠐⠈⠿⢾⡟⠀⢀⣿⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠞⠁⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢻⣅⠔⠙⠷⠔⠃⠈⠉⢸⣿⢸⠀⠀⠀⠸⣇⡠⠂⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣤⡤⠤
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠹⠿⠛⡇⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢄⡀⡀
⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠘⢆
⠀⠀⠀⠀⠀⡈⢉⠉⠅⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⢠⣿⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠙⠚⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⢀⠀⠔⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠒⠄⠀⣼⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣭⣻⣿⡟
⠀⠀⠀⠀⠀⣴⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠰⠀⠀⢰⣿⠟⡑⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠈⠢⠙⠻⠿⠛⠁⠀⠀⠀⠀⣀⠀⠀⢹⠐⠓⠙⠘⠀⠈⢁⡔⠁⠀⠀⠀⠀⢀⢙⡋⠁⠀⠀⠀⠈⠉⠙⠛⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⠓⠠⠼⣇⣀⣀⣠⠤⠞⠉⠀⠀⠀⣀⣀⣀⣈⣀⣁⣁⡀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⠀⢸⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠉⢉⠀⠐⣛⠇⠀⠀⡶⢾⣶⣌⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡈⠀⢀⠰⢆⢿⣿⣶⣿⣷⠟⣠⡿
⠀⢠⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⠀⠀⠀⠁⠢⢄⡢⢀⡨⠭⠴⠂⠉⠁
⠀⢸⣿⣿⣿⡇⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠉
⠀⢸⣿⣿⣿⡇⠀⠀⠠
⠀⠀⠙⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⣈⠐⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠙⠻⢷⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⡟⠀⠀⠀⠀⠀⢤⣤⣤⣄⣀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡘⠀⠀⠀⡰⢻⣆⣌⢛⣦⠀⠘⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢰⣧⣶⣠⣦⣷⣿⣼⣿⣿⣿⠀⢈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⣦⠀⢸⡏⠸⡏⠀⢹⣿⣿⡇⠐⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡇⠀⡇⠀⢸⣿⣿⣷⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣅⣤⣡⣤⣬⣭⣭⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠋⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠷⠒⠒⠛⠛⠛⠛⠿⠦⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠴⠊⠀⠀⠀⠀⣠⠔⠚⡛⠥⠐⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⢐⠵⠁⠀⠀⠀⡀⠐⢉⠤⠂⠁⠐⠂⠀⠀⠀⠀⡶⠀⡷⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣀⡀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⣠⡖⠡⠀⠐⣄⢳⡀⠀⠀⠀⠀⢠⡄⣿⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠙⢷⣝⢦⡀⠀⠀⠈⢿⣆
⠀⠀⠀⡰⠁⠀⠀⠀⣴⢃⡴⠃⠈⠀⠀⠀⠈⠳⣳⡀⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡙⢶⣿⣦⡀⠄⠈⢿⣆
⠀⠀⡴⠀⠀⠀⢀⡾⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠔⠈⠀⠈⢻⣿⣿⣶⣆⠀⢿⣆
⠀⢠⠃⠀⠀⠀⠞⠠⠀⠀⠒⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣡⣴⠿⠛⠀⠀⢻⣿⣿⣿⣶⠘⣿⡄
⠀⡄⠀⠀⠀⡘⢀⠁⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⣠⠖⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⢻⣧
⡀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠸⣿
⠃⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⣿⡇⢿⡇⠐⣿
⠀⠀⠀⠐⣼⡇⠰⠀⠬⠍⠉⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠿⢀⡀⣿⣟⢸⣷⡀⣿
⠀⠀⠀⠘⡎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡿⣌⠧⠀⣿
⢠⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣸⣿⡇⢟⠁⢀⣿
⠈⡤⠀⠀⢹⡿⢣⠀⠀⢀⣄⢖⣫⢱⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣄⠀⠀⠀⠀⠀⠀⣾⣿⣶⣤⣦⡀⠀⢠⣿⡻⠼⠞⠀⣸⡏
⠀⠐⢀⠀⠘⣷⡉⣧⡀⡈⠻⣋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠉⠛⠶⠙⠀⢠⡿⡷⢡⠚⠀⣸⡟
⠀⠀⠀⢂⠀⢻⣷⡹⢿⡇⠀⠋⠀⠀⠀⠀⠀⣼⠷⠀⠀⠀⠀⢘⣧⡀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣏⡟⣡⠎⠀⣰⡟⠁
⠀⠀⠀⠀⡣⡈⣿⣿⣌⠢⣾⣦⠀⠀⠀⢀⣤⠏⠗⠀⠀⠘⠃⠈⠛⠉⠀⠀⠀⠈⠀⠻⣄⠀⠀⠠⠄⣠⣾⣯⢏⣴⠁⠀⣰⠟
⠀⠀⠀⠀⠘⢷⠈⢿⣿⣷⣝⠿⣷⣄⠀⠈⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠂⢀⣤⣾⣯⠟⣡⠟⠁⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢂⠉⢿⣿⣿⣮⡝⢷⣦⢄⡀⡀⠀⠀⠀⠘⠂⠰⡆⡀⠀⠠⠀⡀⣠⡴⣞⠿⠟⢋⣴⡟⠉⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⢻⣿⣿⣿⣷⣮⣘⣤⠉⡀⠐⠒⠂⠤⣤⣤⣶⣶⣶⣿⣿⠿⢞⣋⣴⣶⡿⠗⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣍⣉⣁⣀⣠⣤⣤⣶⣶⣿⢿⢛⣉⢤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣥⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢄
⠀⠀⠀⠀⠀⠀⠇⢠⣤⡾⡖⠀⡄⠀⠀⠀⠀⡠⢀⠀⢀⣠⢠⡀⠀⣄⠀⠀⠀⠰⡃⢴⣀⣶⣷⠿⡿⢶⠃
⠀⠀⠀⠀⠀⣰⢀⠁⡈⡄⣁⣠⠇⠀⠀⠰⠚⠛⣎⢷⡞⢹⠋⠛⠟⠸⠡⠤⠀⠀⠈⠦⢙⡁⡞⣆⣁⣆
⠀⠀⠀⠀⠀⢹⢯⠦⢱⡼⠏⠙⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⢱⠙⠛⠃⠸⠂
⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⡄⠀⠀⠀⠀⠀⢀⠆⠓⠰⠒⠀⠀⠀⠀⠀⠠⠀⠀⢠⠀⠀⠀⠀⢠⠀⢐
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠇⠘⠀⠀⠀⠀⠀⠀⠀⡂⡄⠀⣠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⢁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠆⢴⠰
⠀⠀⠀⠀⠀⠐⠀⠀⠒⠀⠓⠀⠀⠀⠀⠀⠀⠀⠄⠐⢁⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠘⠀⠓

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⢀⢰⡶⠀⡂⠀⠀⠀⠀⢠⠤⡄⠀⣀⢂⠀⣀⡠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠈⠲⠴⠴⢾⠠⢠⠀⠀⠀⠙⢾⣁⣤⣴⣦⣤⣶⣤⣴⢶⣤⡀⠀⠀⠀⣳⠼⡴⠏⢹
⠀⠀⠀⠀⠀⠖⣶⢀⢤⢶⢢⢤⡞⡀⠀⠀⠀⠀⠙⠁⠉⠤⠤⠤⠬⠀⠈⠋⠀⠀⠀⢠⣤⡄⣤⢠⣦
⠀⠀⠀⠀⠀⠀⠟⠺⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠗⠘⠛⠆

⠀⠀⠀⠀⢄⣄⠪⠦⠀⠀⠀⠀⢀⣀⡀⠀⠀⢠⠄⠀⡠⠀⠠⠈⠀⠂⠐⡄⢀⠀⠀⠀⠀⠀⣶⠀⢠⡄
⠀⠀⠀⠈⠤⠒⡁⠂⠒⠀⠀⠀⠈⠉⠁⠈⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⡨⣤⢎⠀⠀⠀⢰⡆⣿⢸⠸⡇
⠀⠀⠀⠀⢀⠇⠀⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⣴⠀⠀⠀⣀⠀⣀⠄⠀⡑⢠⠢⢁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⢰⡆⣿⠀⠀⠀⠁⠉⠁⠀⠃⠌⠠⢁⢌⠀⠁
⠀⠀⠀⢀⠀⠀⠃⠀⠀⠀⡀⢄⠀⠛⠀⠀⠁⠈⢠⠄⠀⡄⠀⠀⠀⠀⠀⠡⠈⠂⠀⡀⠁⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠸⡄⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⢠⡄⣄⣠⠀⣤⠀⠆⠀⠠
⠀⠀⠀⠀⠁⢀⠀⢀⠀⠈⠀⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣿⠻⠀⠻⠇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠊⠑⠉
⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠙⠞⠀⠉⠃⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡰⠖⣸⡖⠷⣆⠀⡀⠀⠀⠀⢠⠀⠈
⠀⠀⠀⠀⢀⠐⢂⡐⠂⠘⠃⠌⡈⠤⠔⠁⠀⠈⠀⠄⠀⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢰⡄⡤⡀
⠀⠀⠀⠀⡘⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿⣤⢰⡆⢴⣦⡆⠁⠀⣀⡀⠀⠠⠄⠀⠘⢎⠺⠁
⠀⠀⠀⠀⠁⡌⠀⠀⠂⠀⢰⡄⠀⠀⠀⠀⠈⠀⠀⠻⡀⠁⠀⠈⠁⠄⠁⠀⢤⡤⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠈⡄⠀⠐⣄⠀⠀⢠⠚⠙⠲⡀⠀⠘⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠀⠀⢠⠀⠀⢀⠇⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠄⠾⠰⠶⡿⠟⡾⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠣⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠏⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠸⠋⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠛⠀⠀⠛⠀⢀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠒⠶⣾⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⣷⣾⣿⣿⣾⣭⣍⠛⣦⠰⣧⣤⣼⠇⢾⣁⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⡄⢀⡞⣽⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣀⣿⣄⣀⣀⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣧⠀⢿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡿⠀⡾⠀⣿⣿⣿⣿⣿⡟⣉⣤⠶⣛⣫⣭⣭⣿⣿⣿⣿⣿⣷⣶⣿⡟⠻⣿⣿⣿⡇⢸⠄⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⢿⣿⡿⢋⡽⠛⣩⣶⣿⠿⠛⠛⢉⣉⠉⠉⠛⠻⠿⣿⣿⣿⣷⣄⠉⠛⠧⡿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠻⣤⠞⠁⣰⠋⣠⣾⡿⢋⡅⠀⠀⠀⢸⣿⠀⠀⠀⠀⠖⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣾⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⡿⢡⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠴⠶⠀⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢿⣸⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⢨⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡾⢸⣿⠁⢠⡏⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣇⠀⠹⣿⣿⣧⠀⡤⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⢡⣿⠏⢀⡾⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣦⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⠞⣱⣿⠋⢠⠞⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢳⡀⠈⢻⣿⣿⣷⣤⣘⠃⠀⠀⠀⣿⠄⠀⠀⣀⣤⠞⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢻⣿⣿⣿⣿⣶⣶⣶⣶⠶⣖⣛⣭⣴⡿⢿⣯⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠋⠛⠻⠿⢿⣿⣿⠿⠿⠟⠛⠉⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋

⢀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⣿⣿⡿⠻⣿⠛⡿⢿⠿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠆⠈⣿⠀⣀⣤⡀⠀⣄⢀⣄⠀⠘⠁⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣰⠏⣼⠋⢰⡇⢠⡿⢡⡟⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠛⠁⠀⠙⠶⠋⠀⠛⠀⠘⠗⠀⠀⠀⠻⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣴⡟⡇⣻⣇⣿⣿⣟⣷⣿⠀⣸⣿⢸⡷⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⢾⣏⡿⡿⣿⣿⣿⡟⠁⣿⣀⣿⣿⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣭⡄⠁⠈⠈⠁⠁⠀⠉⠉⠉⠈⠉⢠⣥⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⣶⡆⣶⣶⠀⣴⣶⣶⣦⡀⣜⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⡇⣿⣿⠘⣿⣧⣜⠛⠃⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢨⣿⡇⢸⣿⠀⣿⣿⠀⣈⡻⣿⣷⡄⢸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣴⣿⡇⢸⣿⣦⣿⣿⠸⣿⣧⣼⣿⠇⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⡀⠈⠉⠉⠉⠉⠀⠈⠉⢉⣁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡄⠉⢻⡆⠀⢀⡀⠀⠀⠀⠀⠈⣽⠁⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣾⢃⣾⠋⢹⡇⠀⠀⠀⢠⡟⠀⣽⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣁⠰⠟⠁⠸⣇⠠⠟⠀⠀⠀⢠⡟⠁⠀⣿⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⣤⡶⠶⠶⠶⠶⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⢋⡩⠴⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠉⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⢀⡠⠤⠤⠤⠤⠔⠋⠀⠀⠒⢲⠟⡟⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⢀⠀⢀⡀⠀⠀⠀⠀⢀⠀⣼⡼⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡾⢀⠷⣄⡀⠀⠀⠀⡎⢠⡟⠁⠀⡄⠀⠠⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣧⠏⣞⣽⣀⣀⡀⠐⣧⠛⡇⠀⡼⠁⠀⠘⢌⣀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠂⠉⠉⠉⢱⡆⠀⠀⢀⢤⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⠉⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠇⡞⠀⠀⠀⠀⠀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣜⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⡠⠂⠀⠀⠀⢰⡇⠀⢠⣆⡞⠂⠆⢀⠖⡿⢡⢞⡿⢀⠀⣶⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡿⠀⢸⣠⣧⢼⡇⣞⠜⡇⢸⡜⣸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⠖⡄⠀⣿⠀⠀⠈⠁⠀⠈⠁⠀⠀⠃⠈⠀⢷⡴⠛⢺⠓⠉⠉⠉⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠡⠀⠀⠀⢻⡀⠀⠀⠀⠀⡔⣻⠀⠀⠀⠀⣠⠋⠀⠀⡞⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⣺⢹⠏⠉⠙⠀⢰⡇⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄⠀⠀⠀⢀⠀⢀⣀⢀⣇⠏⠀⠀⠀⠀⠈⠣⠠⠚⠁⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠂⢰⢀⠟⣆⡿⠊⣸⠏⠀⢠⠄⠀⠀⡠⠒⢐⡄⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣮⢴⡟⠧⠜⢹⡄⢠⠏⠀⢀⠎⠀⢀⠞⢡⠊⠀⠀⠠⡀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢸⠁⠀⠀⠀⠑⠋⠀⠀⠘⠀⡴⠋⠀⠸⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣟⣀⠤⠤⠴⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠓⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢂⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁


⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⣊⣤⣶⣾⣿⣿⣿⣿⣥⠈⠉⠒⠚⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⣫⣴⣿⣿⠿⣻⡿⠛⠋⠉⠉⠉⠛⠢⠀⠀⠀⠈⠐⣶⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢥⣾⣿⡶⠛⠁⠞⠁⠄⣒⠒⠒⠂⣀⣀⢀⠀⠀⠀⠀⠀⠈⠃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢠⣾⣿⣟⠁⠀⠀⣠⡴⠟⣛⣻⣶⣜⢿⣿⢎⠳⡷⠀⠰⠶⣷⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⡀⢸⣿⣿⠃⠀⣠⣿⠋⣰⠟⠉⠉⠉⠉⠐⠀⠀⠀⠀⠀⠀⠀⠈⣿⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣇⠈⠐⣶⣆⣰⠃⢡⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢷⣿⣿⣿⠋⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠈⠻⠻⠷⣾⡷⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣷⣄⣷⣦⣾⣟⠁⠀⠀⠈⣀⣀⣀⣉⠙⠛⠃⠀⠀⠀⠴⣛⣉⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⣿⣿⡇⣙⢿⣷⣪⠂⠀⠀⠙⠻⣟⣿⠏⠳⠄⠀⠀⠀⡴⠻⡟⠉⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⡿⡟⠱⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣅⢻⣄⠁⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⣿⣿⣷⣦⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⡠⠞⠙⣿⣿⣿⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠿⠶⠾⡦⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠩⠷⠶⠖⠀⡰⠁
⠀⠀⠀⠀⠀⠀⣸⣿⡄⢀⡴⠞⠀⢰⣯⣾⣿⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠘⣿⣏⡀⣨⣿⠁⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢄⣻⣿⠃⢀⣨⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡿⢷⢾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⠟⡀⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠛⠋⡀⠀⠀⠁⠘⠢⠤⣀⠀⠈⠂⠀⠈⠻⠮⡳⣌⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠘⠻⠧⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠑⠀⠀⠈⠪⡓⡙⠛⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣶⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⢤⡯⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⠟⠛⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢿⣿⣿⣷⢟⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢋⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⣼⠛⢻⡏⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠈⠳⢤⣿⡹⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣀⣀⣀⣠⣿⣿⣆⣀⣀⣀⣰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⡿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢐⣟⣿⣻⣛⣛⣿⣟⣿⣻⣿⣻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠟⢿⠟⠁⠀⠀⠀⠀⠀⠙⢿⠇⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠠⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠠⠀⢄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⢀
⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠁⠀⠠⠌⠀⠀⢠⡴⠾⠟⠃⠸⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠈⠂⠀⢁⣀⠀⠀⠀⢀⣀
⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠀⠀⠀⠀⠐⠂⠀⢤⠄⢠⠴⠶⢄⠀⠀⠰⠸⡀⢀⠀⠀⠀⠀⠀⠀⢀⠉⣠⠀⡀⠀⢀⣀⡈⠈
⠀⠈⠤⢀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⡀⣀⠀⠀⠀⠙⠆⡅⠀⠄⠀⠀⠙⠮⠎⠁⣤⡄⠀⢷⠀⠀⠃⢁⠀⢢⠤⠴⠒⠂⠀⠀⠢⡀⠀⠀⡝⠢⡄⠁
⠰⣶⡆⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢔⠐⠂⡀⠀⠀⠀⠐⢁⠴⣂⠀⠀⠀⠀⠀⠀⠀⡄⠀⡾⠀⢀⠄⠀⠃⠈⠀⠀⢀⠄⢀⠀⠀⠈⠀⢠⠐⣍⠐⠀⠀⢀⠐
⠀⠈⠁⠀⠀⠀⠠⠌⠉⠁⠀⢀⣄⣟⠁⠈⠀⠀⠀⡀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⢈⡼⠁⠀⠈⠀⠀⠈⠀⠀⠀⢇⠀⠀⣴⣿⣿⣤⡀⢂⢠⠀⠀⢄⡆
⠀⡂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣥⣶⣶⣿⣿⣿⣧⠀⠈⠁⠀⡀⠄⠀⠢⠤⠖⠀⣰⠋⠐⠐⠀⠐⣒⠎⠉⠐⢠⠀⠀⠐⢈⣿⣿⣿⣿⠇⠀⠒⠀⠀⠠⣀⡀
⠀⠂⠐⠀⠃⠀⠀⠀⢀⣿⣄⠀⢿⡏⢾⣯⣽⣿⣿⡆⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠘⠒⢀⢀⣄⡀⢻⢃⠀⠈⠀⢁⠀⠀⠈⠉⠉⠉⣀⣀⣤⣀⠰⣀⣀⢀⠜
⠀⠀⠀⢐⠂⠑⢀⠀⠘⠿⠿⠀⠘⣿⡶⠿⠟⢛⠉⠠⢄⡀⠀⣶⣶⡖⢯⡾⠃⠠⢤⡘⢿⡎⠟⠁⠈⠊⠀⠤⠴⠮⠐⠀⠀⠉⠀⢰⡟⢸⣿⠘⣧
⠀⠁⠈⠁⠀⠀⠈⠀⠀⢤⣴⠆⠀⠀⡄⠂⠄⠘⠇⢸⡿⠃⢤⡄⢤⡀⠐⡦⣀⠀⢸⠧⣤⠀⢀⢠⣄⠠⣤⡠⣤⠀⢀⠄⠀⠀⠀⢸⡇⢸⣿⢀⣿⢠⣶⣶⣶
⠀⠀⠀⢀⡀⠀⠷⣶⣦⠀⠐⢢⠤⠀⠀⣀⠀⠁⢀⢸⡇⠀⢸⡇⢸⡇⠀⣶⠟⠁⠈⢸⡏⠀⢘⣠⣿⠰⠘⢱⡏⠀⠀⠀⠀⠀⠁⠈⢿⣸⣿⡾⠃⢸⣿⣿⣿
⠀⠀⠢⣭⡀⠀⠀⠙⣿⣷⠈⠀⠀⠈⠀⡀⠀⠀⢸⣸⡇⠀⣌⣁⣸⡇⣀⣿⠀⢀⣸⢧⣻⡄⣈⣳⡄⠀⠃⣿⠃⠀⠈⠄⠀⠀⠀⠀⠀⢻⠿⠁⠀⠀⢻⣿⠏
⠰⠶⠀⠿⠀⠀⢀⠀⠈⠉⠀⠚⠀⠀⣶⣦⡄⠀⠀⠉⠁⠀⠈⠉⠉⠁⠈⠉⠀⠀⢉⢀⠭⠀⠀⠉⠁⠰⣼⡟⠐⠢⠀⢈⢯⠀⢀⡀⠀⣜⣑⠀⠀⠀⠠⠓
⠀⠰⡀⠀⠀⣄⠀⠀⠤⠁⠀⠀⠀⠀⠙⠉⡀⠀⠊⠈⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⣤⣦⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁⡀⢂⠀⢀⣀⠀⠐⠰
⠀⠀⠀⠀⠀⠉⢺⡷⠂⢿⡗⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⠀⠠⢀⢤⠀⡀⢀⠈⢉⢠⠀⠇⠀⠀⠂⡠⡀⠀⠀⣀⡄⠀⡐⢠⡁⡄⣦⢃⡌⡄⡀⡆⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⢀⡀⠀⠀⠀⠀⣀⠾⠃⠀⠀⠀⠀⠀⠆⢘⣠⣀⡀⣈⠁⠐⠀⠀⠄⢒⡨⠴⠚⠀⠀⢠⢁⠑⣤⠁⢀⣤⠀⠘⡀⢁⠘⠃⢀
⠠⢂⠀⡀⠀⠀⠀⠻⣄⠀⢈⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢤⣀⡀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠈⠀⠀⠀⣄⠀⠀⠂⢈⡙⠈⢆⠈⠉⠀⠄⠱⠌⠀⣄
⠀⢤⠄⠀⠀⠀⠀⠀⠙⠛⠋⠀⡀⠀⠀⠀⠙⠷⠆⠀⠀⠀⢀⠈⠐⢦⡙⠛⠋⠀⠐⠶⠀⣀⠀⠁⠀⢀⠉⢀⡀⡀⠘⠃⠀⢀⢀⠀⠀⠀⠀⠀⠀⡡⡄
⠀⢬⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⡀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠳⠄⠀⠀⠐⠀⠉⠐⠃⠀⠛⠀⠠⠀⠀⠀⢠⠀⠂⠄⠀⠀⠀⠀⠀⢸⠁⠽⠔⣢
⠀⠈⣀⣀⠀⠀⠀⣀⡀⢠⡄⠀⣀⣀⠀⠃⠀⠀⢀⡤⠀⣄⠀⠀⠀⠀⠀⠀⠈⠙⠒⠀⠰⠇⠴⠆⠀⠀⠘⢷⣤⡾⠃⠘⢠⣶⡤⠠⣶⡤⠠⠀⣈⡀⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠄⠀⢢⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣄⣀⣀⡀⠀⣀
⠀⠄⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⣤⡀⠀⢀⡄⠀⠀⢒⠉⠀⠀⠴⠀⠀⣾⣿⣿⣿⠇⠀⠉⠀⡀⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣆⠀⠀⠀⠉⢁⠠⠉⠁⠀⠀⠈⠈⠃⠀⠀⠄⠀⡉⠉⠉⠁⢠⡄⡀⣈⠡⠿⠗
⠀⠀⠀⣤⠶⠀⠀⠀⠀⢸⣿⠇⠀⢈⣀⡀⠀⠀⠀⣴⣤⠀⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠁⠡
⠀⠀⠀⠀⠀⠀⠁⠐⢾⠀⣀⣤⢀⣾⡟⠀⣸⣷⠀⣫⠁⠀⠀⢀⠄⠃⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠐⠀⠀⠀⠚⠀⠀⠀⠉⠁⠈⠉⠃⠰⠿⠇⠀⠀⠀⠂⢀⡀⠁⠀⠀⠈⠁⠀⡀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⠀⢘⣧⢀⡀⣄⠀⡇⠐⡆⣤⠀⢀⣼⣇⠀⠀⠀⠀⢀⣤⡐
⠀⠄⠀⠀⠀⠀⠀⠀⢆⠀⠀⠂⠀⠀⠀⣿⠀⠸⣃⠀⠄⣿⠀⢇⠀⢠⡇⠀⠉⠙⠟⠀⣤⣄⡀⠸⠿⠇
⠀⢀⢤⠠⣤⠀⠀⠀⠈⠀⠁⠀⠀⠀⠘⢛⠀⠙⠛⠈⠛⠋⡈⡙⠡⣾⠁⠘⣷⣶⠂⢴⣷⣶⡇⢰⠆
⠀⠄⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠛⠐⠛⠀⢀⡀⠀⠈⠁⢀⣟⡶⣷⠏⠀⣤⣄
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠔⠀⠀⠀⠴⠀⠀⠀⢱⠆⠀⣰⡾⠯⣿⠂⠀⠙⠉
⠀⠀⠀⢀⢀⢉⠀⠀⠀⠀⠀⠀⠓⡄⠀⠀⠀⠀⠀⣶⣾⣿⣆⠀⠀⠐⠃⠉⠀⠀⢭⣿⣻⡗⠀⠀⢀⠺⠛
⢈⣉⠉⠍⠉⠈⠁⠀⠀⢀⣀⡂⠀⠀⠀⠐⠲⠆⡖⠉⠙⠉⠀⠀⢀⠀⠀⠚⠓⢐⣻⢶⢾⡇⠀⠀⠏⠥⠠
⠐⠦⠤⠤⠤⠠⠡⠀⠤⠤⠔⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠈⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣀⡀⠀⢉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡴⠋⠉⠙⢿⠂⡼⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⡀⢾⢳⢰⠇⠀⠀⠀⠀⠃⡀⠀⠐⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠧⠚⢁⠄⢸⠈⡄⠀⠀⠀⠀⠂⡁⠀⢈⣡⣤⡀⢤⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠃⠀⢀⠘⣮⡂⢸⠀⠐⠀⠀⠀⢀⠀⠀⢠⠏⠀⠀⠙⢮⠔⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⡤⠬⣄⠀⠁⠈⠻⢶⡆⠀⠀⢀⡴⠛⠀⡀⡾⠀⠀⠀⠀⠈⣼⠃⠀⠀⣴⢶
⠀⠀⠀⠀⠀⠀⠀⡔⢂⣰⠋⠀⠀⠈⢣⠐⠀⡔⠒⣦⠞⠉⠙⢦⠀⠀⠀⠸⡀⠀⠀⠀⠘⠁⠀⠀⢸⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠴⠛⢿⠀⠀⠀⠀⠘⣶⡄⠙⢺⡇⠀⠀⠀⠀⡧⠒⢲⡀⠓⢀⠀⠀⠀⠀⠀⢠⠏⠠⣣⠄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⡀⠀⠀⠀⠀⠈⢀⣑⠘⡇⠀⠀⠀⠀⠀⠐⠚⡅⠀⠸⠳⣀⣵⣄⡰⠷⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠘⢄⢠⠇⠈⣀⠀⠀⢀⣨⣴⡾⠛⠓⠄⠀⠀⣀⠀⡠⠀⠇⡄⠀⡼⠁⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠲⠟⠁⠀⢢⡈⠙⠛⣻⠿⠥⣄⠀⣠⠖⠲⡄⣿⡿⣦⣒⣦⢐⣈⡇⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠰⡀⢀⠐⠘⣿⣤⡼⠁⠀⠀⠈⢶⢡⠈⠀⠀⢹⣿⣖⠉⠡⠞⠉⢇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠓⣠⣉⡇⠀⠀⠀⠀⠸⢏⣨⠢⡀⢸⣿⣧⠀⠀⣰⣦⡌⣄⠀⣴⠊⠈⢀⢀⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠁⢝⡱⠀⠀⠀⢀⣾⣶⡯⣍⣀⠸⣿⣿⡀⠀⠀⠠⣄⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣁⠀⠀⠀⠘⠁⢫⠉⠀⠈⠛⣿⣿⣷⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⡠⠞⠀⠀⠀⠀⠸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠐⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀
⠀⠀⠀⠀⢰⣇⡔⣿⣴⣂⡉⢳⣄⠀⠀⠰⣀⣺⡾⢶⣰⢙⡶⡀⠀⠀⠀⣴⣶⡶⣶⣶⣶⣷⣶⡆⠀⠀⢠⣤⣬⣭⣥⣽⣤⣶
⠀⠀⠀⠀⠀⢁⣴⡏⢃⢉⡁⠻⠏⠀⠀⢘⣁⡃⠈⠈⠉⣀⣃⠁⠀⠀⠀⡻⡟⠛⠛⠿⠛⣛⢙⠉⠀⠀⢸⣟⣿⣿⠿⠿⠿⠿⠁
⠀⠀⠀⠀⢰⠏⡾⠀⠉⠄⠀⠀⠀⠀⠀⠫⠀⠛⠻⠘⠛⠂⠓⠀⠀⠀⠀⢳⠜⠺⠀⠞⢋⢎⠝⠀⠀⠀⠀⣤⠬⣐⣀⢓⠐⠂
⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⡴⠁⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡠⣀⣀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠰⠴⠶⢆⡀⠀⠀⠀⠀⠣⠤⠲⠁⠳⢿⣋⣠⠀⠀⠀⠀⢧⠤⠶⢼⠦⠶⠶⠀⠀⠀⠀⢣⣡⣤⢄⠮⠦⢤⠄
⠀⠀⠀⠀⢀⣤⢤⣤⣴⣤⣬⣤⣤⡄⠀⠀⡠⢔⡄⣀⣁⡠⠘⢳⠀⠀⠀⠀⣈⣀⣆⣀⡄⡰⡖⠀⠀⠀⢠⠄⠀⢀⣠⣄⣤⢠⡀
⠀⠀⠀⠀⢸⣯⣿⣼⣭⣽⣿⣽⣿⡇⠀⠈⢓⣿⣉⣦⠄⠈⠀⠁⠀⠀⠀⢠⠂⠀⠏⢉⠭⠀⠅⠀⠀⠀⠀⠐⡩⠯⠥⠅⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠢⡀⠀⡀⠀⠀⠀⠀⠀⠀⠔⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⡎⠨⢀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⡞⠉⠈⢹⣄⡀⠀⠀⠀⠀⡎⠀⢀⣠⣅⡀⣡⠀⠀⠀⠀⠀⡈⠢⡀⡤⠤⢴⡀⠀⠀⠀⠀⡐⠈⠶⠈⣉⢉⠇⣀
⠀⠀⠀⠀⠀⠀⢧⡀⢐⣋⣠⣴⠰⠀⠀⠀⠑⢄⠃⡼⣁⡤⡟⢁⠇⠀⠀⢀⠀⠄⠈⣠⠀⢪⠆⡀⠀⠀⠀⠰⡋⡀⠌⣀⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠔⡹⡇⠩⠥⠳⠂⠀⠀⠀⠀⠘⠁⠁⠙⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠃⠉⠈⠂⠀⠀⠀⠀⠀⢹⠃⢨⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣻⣻⣛⣝⣻⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣹⣿⣿⣟⣿⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⣤⡴⠶⠔⠿⠃⡀⢠⢠⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⡀⠀⢠⡄⠀⢦⣠⣆⠀⠀⠰⠁⠀⠀⠶⠋⢁⡀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠸⠟⡃⠄⢰⣶⠿⢈⣉⣭⣤⣤⣦⣴⣤⣿⣧⣀⡺⠻⣦⣤⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡈⠁⢀⣠⣴⣾⠛⠛⣻⣿⣿⣿⣿⠛⠛⠛⣻⣿⣿⣦⣄⡀⢠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠶⠀⠈⣁⣿⣿⣿⣿⣧⣄⡨⣽⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣠⡦⢀⣶⣿⣿⣿⣿⣿⣹⣿⡇⠋⠈⠛⠛⠋⠙⠛⠻⣿⣿⣼⣿⣿⣿⣿⣿⣷⣠⣬⠁⠀⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⢀⣾⣿⣿⣿⣿⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⣿⣿⣿⣿⣿⣿⡄⢘⡁⠀⠀⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⠀⠀⠺⢦⣾⣿⣿⣿⣿⣿⣿⣿⢸⣟⠠⣠⣤⣤⣤⣤⣤⣤⣤⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣞⣁⡀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠀⠙⢿⣿⣿⣿⣿⡿⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡻⠇⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⡄⠀⠙⢿⠿⠉⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡎⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡹⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣙⣿⣿⠀⠀⠈⠀⠀⢠⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⢠⠆⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣷⠀⠀⠀⠠⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠟⠁⠀⠀⠀⠀⠈⢻⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣟⣈⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣞⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢋⡈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠛⢶⠄⠘⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣴⣦⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⢋⡄⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠐⢤⡄⠈⢻⣿⣿⣿⣿⣿⢿⣿⣧⣀⣴⣿⣿⡿⢿⣦⣠⣿⣿⢿⣿⣿⣿⣿⣿⡿⠋⢘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡄⠙⠿⣿⣿⣿⣾⠟⠿⢧⣶⣿⣿⣷⠒⠿⠿⢿⣿⣾⣿⣿⣿⡿⠋⣄⡀⢊⡙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠟⢴⣌⣻⢿⣿⣤⣈⣿⣿⣿⣿⣿⣭⣥⣤⣿⣿⣿⡿⠟⠉⠠⣶⡈⡁⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⢨⡿⠻⢂⣤⠉⡙⢻⢻⠿⠿⡿⡿⠟⠛⣛⠉⢡⡄⢀⠛⠀⠂⢀⣄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⢰⣶⡄⠻⠆⠛⢃⣈⡁⠀⠀⣿⣿⠶⣿⠁⠈⠁⣿⠁⠛⠀⠙⠋
⠀⠀⠀⠀⠀⠐⠀⢱⡜⢂⣀⣀⣠⣄⠱⣆⠀⣄⠀⣰⠃⠀⡙⠁⠾⠂⠠⠚⠛⠋⢰⣿⠀⠁⢴⡷⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⠀⢀⣡⣼⣿⣿⣿⡏⣩⣦⣦⣶⣿⣿⡟⠀⠢⠉⠀⠀⠀⠀⠀⡀⠀⠀⠁⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢹⠀⠈⢸⡇⠁⣹⠀⣿⠀⢸⡇⡇⢸⠀⠈⡇⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠋⠘⠀⠀⠘⠃⠀⠚⠀⠛⠐⠘⠃⠓⠘⠀⠀⠃⠐⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⣀⣠⣾⣿⡿⠀⠘⣧⠀⣰⣷⠀⠀⠀⠀⣀⠀⠀⠀⠿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠁⠀⠀⢀⣰⣿⣿⡗⠀⠀⠀⠈⠀⠀⠀⠀⠉⠻⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠙⢃⣤⣴⣶⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣧⣤⣴⣄⠙⠛⠁⠀⠀⠶⢿⣄⠀⠁⠦⣄⣤⣿⣶⠀⠘⢋⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠿⠿⠿⣿⣿⣿⠀⢸⣶⠀⣀⣰⡄⡙⠳⠶⣦⣄⠙⡋⠀⠀⠀⠸⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⣌⠻⢿⠀⠀⠙⢷⠈⣿⡿⠛⠛⠛⠉⣥⣄⣁⡐⠀⠘⡲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠛⠁⠸⣿⣧⠈⠙⠛⢦⠸⣧⣿⠀⠀⢀⣴⣲⡿⠛⠉⣱⡆⠀⠁⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⢿⣿⣳⡄⠀⠀⠀⠛⣃⣠⣦⡌⢿⠏⠀⠀⣼⣿⠇⢀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣼⣿⣀⣀⣀⠛⣿⣿⣿⣿⣿⣿⠿⢛⣋⣤⣤⣄⣴⣾⡿⠿⢣⣼⣿⣩⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠉⠉⣼⠿⠛⠉⠉⠉⠁⠀⠈⠙⠛⠋⠿⠟⠻⢷⣄⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣄⣠⣶⡆⠀⠀⠐⠀⠒⡖⠒⠤⠔⢦⡤⠒⠲⠒⠂⠢⠀⠀⠐⣢⢄⣠⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢿⡿⣠⡀⠀⠀⠀⠀⢶⣤⠀⠃⢻⣦⣶⣶⠀⠀⠀⠀⣴⣄⢇⡸⠇⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡄⣠⣼⣛⡟⢿⡿⣶⣄⠈⠉⠹⠀⠈⣯⡟⠉⣠⣴⣿⠏⠿⣻⣆⡀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠏⠻⠀⢿⣿⠀⠻⣎⢻⣷⠄⠃⠀⠀⠻⣴⣿⣿⡿⠏⠀⠸⣿⣿⣷⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠠⠞⣿⣦⡀⢾⣺⡿⠀⡇⣤⡅⢸⠙⣿⣿⠆⠀⠀⣰⣿⣿⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⢀⠈⡟⣡⡌⠿⠃⠠⠀⠑⠋⠈⡄⢻⣿⠀⢠⡞⣿⣿⠿⠁⠜⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢼⣿⠆⣨⡟⢉⣙⠛⠀⠀⡀⢀⢿⣄⠀⠀⠀⠀⠀⣿⡟⠋⡀⢠⠂⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⢸⣿⡏⢿⣿⣿⣶⣄⠉⠈⠙⠋⠀⠀⣠⣶⣿⣿⡿⢺⣿⣆⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⣿⡇⠀⢻⣿⣿⣿⣷⣄⠀⠀⣰⣾⣿⣿⣿⡟⠁⠸⡯⢩⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠀⠐⣹⣿⣷⣦⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣀⣤⡴⣿⣿⣟⠋⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡇⣀⠐⠿⠋⠉⠙⠋⢻⣿⣿⣿⡿⣿⣿⣿⣿⠧⢾⣿⣿⣠⣿⣿⠅⣠⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠆⠀⠀⠀⠀⠀⠙⠛⠉⠐⠂⠉⠛⠋⠀⠀⠀⠀⠀⠛⠋⠘⠛⠂

⠀⠀⠀⠀⢀⠄⡤⣄⠀⢀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠒⡀
⠀⠀⠀⠀⠀⠸⠁⠸⡿⠌⠾⠀⡇⠀⠀⠀⠀⠀⠰⠉⠁⢀⡂⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢦⢸⠀⡞⡇⠎⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠆⠴⠀⠣⠄⠀⠀⠀⠀⠅⠀⠀⢸⠀⣶⠀⠟⠀⠰⠀⠀⠀⠀⠀⠀⠙⠄⠀⠐⡀⢠⠀⡅
⠀⠀⠀⣀⣴⡇⡄⢨⣤⢐⠀⡀⡦⢀⡔⢂⠀⠀⠀⠀⢀⠻⠀⠀⢂⠂⠀⠀⠇⠀⠀⠀⡞⠀⡀⢰⢰⣔⡇⣹
⠀⠀⠀⢿⣻⡇⠷⠞⢇⠉⢺⡟⠇⠘⠂⢸⠁⠀⠀⠀⡄⣀⠃⠀⢀⠮⠀⠞⠀⠀⠀⠀⢰⠀⢿⠈⠘⠼⡇⠘
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠜⠀⠀⠀⠀⠀⠃⡇⢠⠈⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⣀⢠⢄⡐⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡦⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠟⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⠂⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⢲⡰⣠⡆⠇⡇⠀⡔
⠀⠀⠀⠀⠀⠀⠀⣸⠀⡌⠋⡄⡀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢈⠹⠄⠀⠀⠀⡄⠀⠑⠁⠘⠉⠀⠀⠆⡁⣀
⠀⠀⠀⠀⠀⠀⡂⢸⠀⣇⠀⡇⠀⢷⠄⠀⠀⢸⠀⠀⢒⠀⡄⡞⠀⡗⠆⡏⠀⡇⠀⠀⣆⡀⢸⢆⣶⠰⠇⠀⢳
⠀⠀⠀⠀⠀⠸⢛⡀⠀⠈⠀⠈⢀⡇⠀⠀⠀⢹⡀⠆⢸⡀⠁⢇⠀⡇⠀⠧⠀⠁⠀⠈⡇⠀⠘⠌⠹⠄⡀⠀⠀⠇
⠀⠀⠀⠀⠀⢰⠁⠘⣤⠼⡄⢶⠹⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢄⡀⡄⢢⡰⡇
⠀⠀⠀⠀⠀⠈⠀⠀⡛⠂⠉⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠇⠇

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⣄
⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⡂⠀⠀⠀⣀⣠⣤⣤⣄⠀⠙⢦⣄⣀⡤⠞⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⣿⡆⠀⠀⠀⠀⠹⣄⠀⠀⠀⣀
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠇⠀⠀⠀⡿⠁⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠰⠋⢠⣄⣀⠀⠀⣄⡀⠀⣀⡀⣠⡀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⣼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣴⣾⠗⠁
⠀⠀⠀⠀⠈⣉⣉⢀⣄⡉⡁⣉⢉⡉⣩⡁⠀⠀⠀⠀⠀⠀⠛⠳⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠤⠄⠤⠠⠤⠤⠴
⠀⠀⠀⠀⠀⣧⣿⢸⣼⡇⡇⣿⣾⣿⡇⣍⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣄⡠⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⢴⡓⢲⡖⢲⡖⣞⠒⣞⡆
⠀⠀⠀⠀⠀⠧⠾⠸⠸⠇⠇⠧⠹⠟⢧⠟⠀⡇⠀⠀⠀⠀⠀⠀⣤⢴⣦⣤⣴⣤⡦⣴⢰⠀⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⢾⣅⠸⡇⠸⠇⣯⡠⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠒⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠀⠀⠘⠸⠟⠿⠛⠓⠞⠻⠸⠄⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠆
⠀⠀⠀⠀⠀⠀⣾⢳⣾⢳⣾⣾⣼⣾⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⠀⠀⠀⠀⠀⠀⢶⠶⠶⡟⠐⢞⣷⠀⠳⠶⠷⠶
⠀⠀⠀⠀⠀⠀⣿⡼⢿⡼⢸⡟⡏⢿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⡾⢁⣀⣀⣠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡱⠀⠀⠀⣿⡘⠋⠉⠙
⠀⠀⠀⠀⠀⠀⢠⠖⠒⠲⡄⠀⠀⠀⠀⠸⡁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠁⠀⠀⠰⣮⠛⠶⠤⠶⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠴⠷⣶⣆
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠛⠀⠀⠀⠀⠀⠛⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠓⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠛⠚⠀⠀⠀⠘⠓⠓⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢸⣿⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⣶⠿⠛⠛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⠀⣾⠟⠟⠟⠘⣿⡄⠀⢠⣿⠃⣾⠿⠛⠿⠀⣿⣀⣠⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⠀⣿⣤⣶⣦⠀⢿⣷⠀⣾⡏⠀⣿⣤⣶⣶⢸⣿⣿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣿⣿⠀⣿⠁⠀⠀⠀⠸⣿⣿⡟⠀⠀⣿⠇⠀⠀⠘⣿⡈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠻⠟⠀⠿⠷⠟⠛⠀⠀⠙⠟⠁⠀⠀⠻⠷⠟⠛⠀⠟⠃⠀⠙⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⢿⠀⠀⣿⡆⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠰⠆⢿⣧⠀⠀⢠⣿⢡⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢶⣶⡆⢸⡇⠸⣿⡄⢀⣿⠃⢸⣿⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⣿⠇⢸⡇⠀⢿⣷⣾⡏⠀⢸⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⡿⠀⠸⠇⠀⠘⢿⠿⠀⠀⠸⣿⣴⠶⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢠⣤⢠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡿⢸⡇⠀⠀⣹⡇⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⡇⢸⣿⣿⣿⡿⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢰⣿⡇⢸⡏⠉⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣴⣿⢿⣷⢸⡇⠀⠀⠀⠀⠈⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⢸⣿⣟⠀⣠⣀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠀⠈⡜⠀⠀⠸⣿⠋⠀⣸⣇⠀⠀⠀⠀⠀⠀⠢⠀⠑⢄⠀⠀⠀⠴⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡀⠈⡀
⠈⠀⠐⠀⠀⢤⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢰⣿⡀
⠀⠀⠀⠀⢠⣾⡏⠀⠀⠀⠀⠀⠀⠀⢀⡏⡀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡈⣿⣧
⠀⠀⠀⠀⢀⣽⠇⠀⠀⠀⠀⠀⠀⢀⡾⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡶⠀⣤⣀⠀⠀⠀⠀⠀⠀⢠⣤⡄⢸⣷⠹⣿⡆
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣾⡇⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢘⢻⠃⣸⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⠇⠸⣿⡇⢻⣿
⡄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠰⠛⠟⠁⢀⠀⠀⠀⠀⢠⣿⣩⢠⡄⢻⣿⠀⣿
⣷⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢠⠂⠀⡤⣝⢦⡀⠀⠈⠃⣟⡘⠁⠈⣿⣿⣿
⣿⣆⠀⢰⠀⠀⠀⠀⡂⠀⣰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⡸⠀⠀⠀⠀⠀⠑⠀⠀⡘⢠⣿⡇⠀⠀⠸⣿⣿
⢻⣿⡀⢾⠀⠀⠙⠜⠀⠀⠛⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢻⠃⠀⠀⠀⢿⣿
⠊⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⣾⣶⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣆⢀⡀⠀⠀⠀⠈
⣾⣿⡏⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡄⡌⠋⣼⣿⣆⠀⠀⠀⠀⢀
⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣼⣷⣶⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠘⣿⣿⡟⠀⠀⠀⢠⣿
⣿⡇⠀⠀⠖⠃⠀⠶⠿⠿⠿⠛⠉⠀⠀⢀⣿⣿⠇⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠖⡤⠀⠀⠸⣿⠟⠀⣸⣆⣽⡿⠀⠀⠀⠰⠿⣿
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠠⠿⠷⠁⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠋⠀⠠⣿⣿⣿⠃⠀⠀⠐⠃⠸⣿
⣿⡇⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢁⠂⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡿⠀⢀⣼⠃⠀⢴⡇
⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢇⡞⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣾⠂⠇⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⣿⣿⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠈
⣤⣠⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣇⣸⣿⣿⣀⣀⣠⣀⣀⣀⣀⣴⣿⣅⣀⣀⣦⣈⣤⣄⣀⣤⣀⣠⣄⣀⣀⣐⣿⣆⣀⣀⣠⣤
⣿⣿⣩⣽⣉⣉⣯⣋⣿⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⡀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠓⠁⠈⠛⠓⠚⠛⠓⠂


⠀⠀⠀⠀⠀⣠⣤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⡀⠀⢤⠄⢀
⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⣠⣿⣅⠀⠀⠀⠀⠀⢀⣴⣴⣄⠀⠀⠀⠀⣀⣶⣶⣶⣀⠀⠀⠀⠀⣴⢻⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⡃⠀⠀⠀⠀⠀⠉⠉⡁⠀⠀⠀⠀⠻⠟⠋⠿⠿⠃⠀⠀⠐⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠐⠒⠀⠈⠁⠀⠀⠚⠀⠚⠀⠓⠀⠀⠀⠀⠀⠛⠛

⠀⠀⠀⠀⠀⠀⣶⣶⣶⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣴⣿⣿⡟⠛⠀⠀⠀⠀⣀⣻⣇⡀⠀⠀⠀⠀⢀⣸⢿⣀⠀⠀⠀⠀⠀⣀⣿⣁⡀⠀⠀⢀⣄⣀⠛⢁⣀⣄
⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⢠⣿⣖⣼⣿⠀⠀⠀⠀⣿⣧⣥⣿⡇⠀⠀⠀⢸⣿⣄⣾⣧⠀⠀⠀⠹⡿⠂⢿⡿⠁
⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠉⠙⠛⠉⠁⠀⠀⠀⠈⠙⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣼⡇⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢀⣿⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠃⣀⠀⠀⠀⠀⠰⠋⠁⠀⡀⢸⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡘⠀⣬⠀⣤⠀⣤⡄⢀⡀⠀⣤⢸⣿⠀⢀⣤⢀⣤⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣿⠀⣿⣼⢹⣷⢛⣗⢸⡿⢸⣿⠀⣿⠌⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⠀⢰⠿⠌⠻⠇⠸⠟⢸⡇⣸⠷⠻⣿⠀⣿⣤⠞⠓⠺⣿⠀⡼
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠘⠧⠂⠀⠀⠀⠀⠀⠀⢸⣧⠃⣠⠀⠻⠞⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣿⠀⡀⠀⡀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢋⡀⠀⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢀⡤⢼⡇⢠⡆⠀⣿⠀⣴⡄⣶⠒⢦⣰⣷⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⠁⣿⡇⢸⡇⢸⡇⢰⣿⠀⣿⠀⣾⡉⢹⠸⣿⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡴⠻⠧⢿⡇⢸⣧⠞⣿⠀⣻⠞⠈⠑⠁⠀⠉⠀⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⢿⣆⠇⠀⠀⠀⠀⠀⠀⠘⣿⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠦⠀⢠⡆⠀⠈⠀⡀⠀⠀⠠⠀⡀⠖⠂⢠⠋⠀⢀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠑⠐⠒⠾⠄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⠀⢠⠀⠀⢀⡀⠂⠁⢀⠐⠉⢉⠛⢛⡀⠁⠠⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⡀
⠀⠀⣠⠄⠀⠀⠰⠯⠆⠀⠀⠀⠀⠀⠈⠉⠙⠈⠀⠀⠀⠉⢐⡒⠛⠀⠀⠁⠀⠀⢱⠀⠀⣀⢀⠀⠘⠀⠀⠀⠀⠀⢀
⠀⠀⢅⡀⠀⢀⠀⠀⠀⠀⠄⠠⢀⠀⡠⠀⡀⠀⠀⠀⠀⡆⠈⡃⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⣆⠈⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠚⠒⠤⢀⠀⠄⠀⠈⠓⠢⠤⠀⠙⠂⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠜⠂
⠀⠀⡀⠀⠀⠀⢀⠀⠐⠄⠀⠀⠀⠀⠀⠂⠐⠀⠉⠀⠀⠀⠁⡀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢛⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⢙⠂⠄⠀⢐⡆⠈
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⣀⠀⠄⠀⠀⢀⡄⠰⠀⠀⠀⢐⠀⣄⠀⠀⠀⠄⢀⡀⠀⠉
⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⠠⢤⡄⠀⠀⠀⠀⠀⠖⠋⠉⠠⠀⠀⠀⠶⠀⠀⣐⠀⠐⡀⠀⠰⠦⠀⠀⠲⠈⠉⠓⠖
⠀⠀⠀⣀⡠⠅⠠⠈⠊⠀⠀⠀⠀⠀⠀⠑⢤⣀⡀⠀⠠⣀⠀⠀⡠⠠⢥⡡⠠⡂⡀⡠⠀⠠⡀⠀⠀⠀⠁⠁⠀⠀⠀⢰⠠
⠀⠀⠈⢉⣉⠐⠀⢐⣩⣾⣿⠏⣍⣉⣉⣉⣉⡉⠉⠀⠀⠑⠑⢥⡐⡦⡀⢰⡖⢤⠸⠀⠀⠀⠸⢠⡀⣀⣀⣀⡈⢡⠔⠋⠁
⠀⠀⠢⠴⠦⠀⠐⠛⡟⠧⠃⡝⢸⠛⠃⠒⠀⠊⠗⠂⠀⠀⢠⠙⠁⠀⠌⠡⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⡀⠌⠁⠘⠁
⠀⠀⠀⡋⡏⡍⠄⡈⣁⢨⢀⡇⢈⠐⠀⢠⢀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠀⠀⠆⠀⠘⠀⠀⠀⣸⠀⠀⠦⠀⠠⠄⠀⠠⠄
⠀⠀⠆⠈⠉⠉⠉⠉⠌⠡⠼⠛⢷⠀⠀⠀⠀⠀⠘⡂⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠉⠀⠁⠀⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⣠⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠘⢶⣄⠀⣾⣷⡄⣠⡶⠃⠀⠀⠀⠀⠀⠀⠀⣰⠆⠿⡿⠃⠀⠀⠀⠀⠀⠿⡿⢋⡷⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⢛⣿⠋
⠀⠀⠀⣿⡇⠀⠀⠀⠙⢷⣭⣭⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢾⣿⣿⡷⣦⠀⠀⠀⠀⢰⣿⡏⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠛⠁
⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⡩⡿⠋⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣯⢻⣿
⠀⠀⠀⣿⣇⣶⣶⣆⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣃⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⠟⠘⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠸⣿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⢿⡿

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀
⠀⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⣠⣶⣳⣶⣤⠀⠀⠀⠀⠀⣴⠶⠶⠾⠻⠷⣶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠸⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃
⠀⠘⠿⣿⠉⣿⢸⡇⠀⠀⠀⣸⡏⠀⠀⠀⠀⠸⣿⠇⢹⣆⠀⠀⠀⠀⠀⣴⣿⡏⣿⣦⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣶⣾⣿⡄⣠⣶
⠀⠀⠀⣿⣿⣿⢘⣇⠀⠀⢰⡿⣶⣿⣿⣿⣿⣿⣿⣿⣶⢿⡆⠀⠀⢀⣤⣬⣿⣧⣿⣤⣽⣿⣿⣄⠀⠀⠐⠒⢶⣶⡿⠛⠛⢉⣹⣿⡇⣿⡏
⠀⠀⠀⣿⣿⣿⠸⠿⠀⠀⢸⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⡇⠀⠀⠀⢹⢹⣿⢿⢿⣿⠉⠉⢹⡇⠀⠀⠀⠀⠀⣿⠀⢠⣾⠟⣛⣿⣥⣿⡇
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢘⣿⡟⠉⠉⠉⠉⠉⠉⠉⢻⣿⠃⠀⠀⠀⣾⠈⣿⣾⣼⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣼⡇⣾⣿⣿⣿⣿
⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠉⠈⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣐⠯
⠀⢰⣾⣿⡄⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣀⠠⠊⠃
⠀⠀⢻⣿⣿⡄⣾⣶⡶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣷⡿⢶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣏⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢰⣶⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣶⡄⢀⠀⠀⠀⢸⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠘⠿⠿⠛⠛⠻⠿⠟⠸⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠿⠃⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠸⣿⡇
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠉⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⣀⣀⣀⣤⣴⣶⣶⣿⡿⠋⢻⡇⠀⢀⣤
⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣿⡇⠀⢾⣿
⠉⠛⠿⠿⣿⣿⣿⣿⣿⣷⣽⣟⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⢸⣟⣀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠻⠿⠿⢿⣿⣏⡁⠀⠈⢻⣷⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣦⣾⣃⣔⣀⣼⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣶⣿⠆⣿⣿⣿⣿⡏⠉⠁⠉⠁⠈⠀⠉⠉⠉⠁⠉⠁⠉⢹⣷⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⢿⢧⣿⣟⣛⡿⣧⣤⡤⠤⠤⠤⠤⠤⠴⠤⣤⣄⣠⣤⣼⣿⣿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⢸⠷⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠿⠙⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⡈⠋⠀⠀⠀⠀⠀⢶⣾⠡⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢄⠀⠀⠀⠀⢀⡾⠁⠁⠀⢀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣾⣿⠀⠀⠀⠀⠀⢄⠀⢀⡮⠊⠀⠀⠀⠈⠷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⢿⣷⠀⠀⠀⠀⡀⠈⢸⠀⠑⠀⠀⠀⠀⠀⡾⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠈⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠈⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⡃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠀⠀⠑⢾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⡇⣸⣦⢤⣤⡤⠀⠀⠀⠀⠀⠠⠧⣴⣃⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣾⣾⡞⣿⣯⣽⣷⡟⠛⠳⠒⠒⠒⠒⠲⠖⠒⠛⠋⠑⠛⢻⣷⣶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⣿⢆⣿⣿⣿⣿⣇⡀⠀⠀⡀⢀⠀⠀⢀⡀⣀⣀⣀⣀⣸⣿⠿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀
⠀⠀⠀⠀⢀⡶⠀⠀⠀⠀⠠⠋⠀⠀⠀⠙⢆⠀⠀⠀⠀⢀⣀⣀⡀⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣿⣿⣿⣿⣧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⡄⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿⣿⢠
⠀⠀⠀⡾⠀⣀⠤⠀⠀⠀⠘⠦⣀⣤⣀⣀⡘⡄⠀⠀⠈⣻⣿⣿⣿⡟⠀⠀⠂⠀⠀⠀⢐⡀⠀⠀⡀⠀⢉⡿⢿⡗⠂⠀⠀⠀⠀⠙⠿⣿⠿⠃
⠀⠀⠀⠀⡀⢀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⣾⣿⣿⠟⠋⠀⠘⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢠⣤⣿⣤⡄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⢠⣶⣦⣀⠀⠀⠀⠀⣿⡟⡃⠀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣤⣶⣶⣶⡄⠀⠂
⠀⠀⢠⣦⣤⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣤⣄⠈⠛⠛⠀⠀⠀⢰⣾⣿⣆⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣶⣻⡀
⠀⠀⣿⣿⣿⣿⠀⠠⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⠃⠠⠀⠀⠀⠀⠸⣿⣿⣿⠀⠠⠀⠐⠀⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⠀⠄⠠⠀⠀⣿⣿⣿⡇
⠀⠀⣻⣿⣿⣿⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣠⣤⡀⠀⠀⠀⠀⢸⣟⡇⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠙⠛⠻⠟⠀⠀⣰⠏⠀⠈⠀⢀⣤⣄⠀⠀⣛⣹⣿⢿⣿⣦⣠⣀⠀⢨⣿⣄⠀⠀⠀⡤⠖⠒⢲⡾⠙⡟⠉⣿⣄⢀⡴⢦⣴⣿⣦⣀⠀⢻⠛⠉
⠀⠀⠀⠀⠄⢀⡴⣋⣿⢷⣦⣤⠞⠋⣀⣼⠿⣿⣿⣟⠁⠈⠉⢻⡇⠸⣷⣾⣶⣿⠷⠤⢴⡇⢰⣾⣿⡇⠀⣧⣾⣿⠛⣿⣠⣾⡿⠋⣿⣿⠟⢉⣿⣦⣄
⠀⠀⠀⠀⠀⣿⡾⣿⣷⡞⠋⠀⠀⣾⣿⠃⣴⡿⠛⣿⠀⣼⠃⢠⣷⢀⣿⠟⢸⣧⡄⠀⣴⣷⣾⣿⣿⣇⣼⡟⢻⣿⢀⣿⡿⠋⡇⠀⣿⠃⢠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠻⣄⠈⠻⣦⣾⣧⣾⣿⡇⣼⣿⣿⠀⣿⠀⢀⣴⣿⣿⡟⠁⢠⣿⣿⣧⣾⣿⠁⠈⣻⣿⡿⠛⠀⣸⣿⣾⣿⠀⠀⠀⠀⣿⣾⣿⡿⣯⡉
⠀⠀⠀⢰⣤⡶⢮⣳⡄⠸⣿⡟⠀⣿⣿⡿⠟⠁⣰⡧⠀⠈⣿⣿⠉⡿⠀⣼⣿⣿⠏⢸⡿⠀⣿⣿⣿⡇⠀⠀⣿⣿⠁⣿⠀⠀⠀⣰⣿⠃⢸⣤⠀⣹⣷⡄
⠀⠀⠠⠞⢿⢧⣄⣀⣠⣼⣿⣧⣠⣿⣿⢶⣶⣾⣿⡇⢀⣶⡿⠻⣷⡇⢀⣿⣿⢸⣀⣸⣇⠀⠛⠛⠿⡇⢠⡀⠈⣉⠀⣿⠀⢿⣴⣿⣿⣧⡈⣁⣠⣿⣿⠁
⠀⠀⠀⠀⠘⠀⠈⠛⠿⢿⣿⣿⣿⢻⠋⠀⠈⠉⠙⢷⣾⣿⣷⣄⣹⣿⣿⣿⠟⠀⠉⠻⠿⠷⣶⣶⣶⣇⣀⣉⣿⣿⠀⣻⣶⣿⣿⣿⠿⠋⠙⠻⣿⣿⡿⠧
⠀⠀⠀⠀⠂⣀⠀⠀⠀⣾⣿⣿⣿⣆⠣⠀⠀⠀⠀⣀⠉⠛⠋⠉⠛⠿⠃⠀⢀⡀⣀⢀⠀⠀⠀⠈⠉⠉⠛⠿⠿⠟⠛⢿⣿⣿⣿⡈⠄⠀⠀⠀⠈⠀⡀⢸
⠀⠀⠀⢀⣠⣤⣤⡀⠀⢻⣿⣿⣿⣿⣦⡱⡀⠀⣠⣿⣶⠖⠀⠀⠀⠀⠠⠄⠸⠃⠿⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⢠⡀⠀⠄⠰⡀⣓⣰⠄
⠀⠀⠀⣾⣿⣿⣿⣿⠀⠀⠹⣿⣿⣿⣿⣿⡗⡌⢩⡿⢷⠀⠀⡀⠀⠀⠀⠀⠴⠤⠴⣄⠀⠀⠀⠄⠀⠀⠒⠆⠁⠀⣿⢿⣿⣿⣿⣿⣧⡟⠤⠀⠀⠉
⠀⠀⠀⠻⣿⣿⣿⡟⠀⠀⠀⠈⢿⣿⡿⢏⠔⠁⠈⠀⠀⠀⢊⠀⠐⠁⠀⠀⠀⠀⢀⡋⣈⠂⢀⡀⠀⠀⠀⠀⠀⠀⠹⠟⣿⣿⣿⣿⣟⣿⠗⠀⠀⡀
⠀⠀⠀⠀⠘⢿⣿⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠁⠀⠀⠀⢀⠄⠀⠁⠀⠃⠀⠳⠤⠞⠁⠀⠀⢞⢢⣗⡂⠀⠀⢀⠀⢠⣆⠸⣿⣿⣿⠟⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⣤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠈⢙⡂⣚⠟⠁⣠⣤⡈⠀⠀⠻⣆⠹⣿⢋⠄⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢲⠁⢩⡡⣧⠀⠀⠀⣤⠀⠀⠀⠀⣴⣿⣿⣿⣷⣄⡀⠀⠰⠀⠀⠀⠀⠐⣿⣿⣿⠀⠀⠀⠛⠀⠈⠁⠀⠀⠀⠈⠁⠈⠗
⠀⠀⠀⠀⠁⠀⠀⠀⣠⡄⠀⠈⠣⠤⠥⠂⠠⠤⣴⣿⣶⠦⠄⠸⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠠⡆⢰⠀⡇⣶
⠀⠀⠀⢀⣠⣤⣴⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⡞⠙⢧⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⢟⠟⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⡄⠀⠀⠀⡀⠃⠘⢀⠃⡋⡀
⠀⠀⠀⠸⡿⣿⣿⣿⣿⠟⡣⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⡿⢃⠂⠀⠀⢀⡀⠀⠀⠀⠀⠈⢛⡿⣿⠶⠀⠀⠰⠳⠀⠃⠐⠀⠇⠐
⠀⠀⠀⠀⠙⣿⠟⡻⠓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤
⠀⠀⠀⢸⡆⠀⢘⠂⢀⣤⣤⠀⠸⢮⠛⠄⠀⠀⠀⠀⣶⣰⢦⡄⣴⠀⡦⢰⠒⡆⠀⣰⣷⣶⠀⢸⣿⣿⣿⢿⡿⣿⣿⡏⠉⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⢸⣀⣈⡛⢻⡇⠀⠀⠀⠐⠄⢄⠀⠀⡿⠹⠾⠇⠿⠤⠷⠜⠿⠃⠀⠉⠻⠋⠁⢸⣇⡺⣿⢠⡀⣿⣿⡇⠸⢿⡇⣾⡿
⠀⠀⡄⠀⢩⡄⢠⠭⣄⠙⠋⠀⠀⠀⠀⠀⠑⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣷⣶⣿⣼⣧⣿⣶⣿⣤⣤⣷⣾⡇
⠀⠀⣷⣦⠘⠃⠹⣶⡾⠁⢠⣀⡤⠀⠀⠀⣀⣤⣶⣶⣶⣶⣤⣀⠀⠰⣧⡀⠀⠀⠀⠀⢀⣽⡇⠀⠉⠉⠉⣉⠉⠉⠙⠛⠛⠛⠛⠛⠃
⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠾⣿⠿⠆⢀⣾⡿⢿⢿⢿⠟⢿⡿⠻⣷⡄⠉⠛⠷⣦⣤⡶⠀⣮⡍⠀⠀⢀⠘⠿⠇⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⣼⣿⣐⣻⣘⣸⣇⠛⣇⢐⣿⣿⠀⠀⠀⠀⠁⠀⠀⠴⠗⠀⠀⢿⡀⠀⢰⠆⢿⡀⣿⠀⡆
⠀⠸⠀⠙⠂⠀⡂⠀⠉⠓⠚⠀⠀⠀⠸⣿⣉⣛⣙⡋⣏⢿⠉⣉⣿⠏⠀⠀⠀⡀⠀⠀⠀⠤⡄⠀⠀⣈⣉⢈⢈⣉⢈⣁⣉⢁⡁
⠀⠐⢇⢰⡷⢢⣿⠰⡟⠋⢰⠆⠀⠈⠀⠈⠻⢿⣿⣿⣿⣿⣿⠟⠋⢀⠉⠠⠀⠁⡀⠀⠀⠚⠋⠀⠀⠏⠹⠷⠛⠻⠱⠘⠘⠸⠇
⠀⠈⠉⠈⠁⠘⠋⠂⠛⠛⠈⠃⠀⡠⠂⠀⠶⠀⠀⠉⠉⠁⠀⠀⠀⠛⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⢰⡦⠀⢠⣤⢀⠛⢫⠙⠋⠉⢤⠀⢤⡀⠀⣷⠖⢠⠀⠀⣶⣶⢠⣥⡍⣆⢰⠀⠄⠀⠀⠀⠀⢠⠀⣤⡄⣰⡀⣴⠶⢀⡄
⠀⠀⢀⠀⠀⠀⠈⠋⠨⠶⢮⣴⠦⡤⠚⠒⠋⠁⠀⠏⠀⠸⠼⠀⠟⠿⠸⠻⠃⠸⠏⢠⡄⣄⣤⢠⡄⠒⠃⠟⠁⠟⠇⠻⠶⠘⠇
⠀⠀⢸⣆⡀⢀⡤⢤⠀⣠⠖⢦⠀⠙⠄⢠⡀⣤⢀⣤⡀⣀⣀⡀⣀⣀⣤⣤⢠⡄⣀⣀⢀⣥⠈⠁⠀⡶⠂⡆⠀⢠⣤⢠⡄⣶
⠀⠀⠸⠿⠟⠈⠻⠛⠀⠙⠷⠟⠁⠀⠀⢸⠛⢿⢾⣄⣿⢻⣿⡇⡻⣤⢸⡇⢸⣃⣿⡞⢩⣓⡆⠀⠀⠁⠀⠛⠀⠐⠿⠉⠀⠈⠁
⠀⠀⠀⠀⢠⡀⠀⠠⠄⠀⠀⢀⠀⠀⠀⠘⠀⠘⠈⠛⠃⠈⠈⠁⠉⠁⠀⠁⠀⠀⠙⠈⠁⠉⠀⣀⣤⣴⣶⣶⣶⣶⣦⣤⣀⠀⠿⠂
⠀⠀⠀⠀⠈⠀⣀⠀⢀⣀⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡒⠀⢤⠤⠀⠀⣴⣿⠉⣿⣿⣿⢻⡿⠻⣿⠉⣿⣿⣦
⠀⢘⢯⢸⡿⢢⡿⡀⣿⠉⠀⡏⠀⢸⣿⢿⡿⡟⣿⢟⡿⡶⡶⢶⢶⡄⠀⢥⠀⠒⠶⠀⣼⣿⣿⠈⠻⣟⣷⠸⢻⡘⢻⡆⠛⣇⣿⣧
⠀⠈⠉⠈⠁⢈⠃⠃⠙⠛⠁⠉⠀⢸⣿⣶⣷⣧⣽⣮⣷⣴⣷⣾⣾⡇⠀⠴⠀⠍⡓⠃⢻⣿⣿⢿⠿⣿⠻⠉⡟⡽⢻⢛⣟⡛⣿⡿
⠀⠸⠸⠧⠟⢻⡜⠶⠃⠇⠧⠟⠀⠀⣿⣿⣛⣿⣏⣏⣍⣸⡱⣏⣿⡇⠀⠈⠀⠏⠹⠂⠈⠻⣿⣼⣤⣧⣾⣄⣱⣶⣿⣾⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠿⠛⠛⠛⠛⠛⠛⠋⢉⡉⠉⠀⠀⠚⠀⠋⠉⠀⠀⢀⣈⠙⠻⠿⠿⢿⠿⠿⠿⠟⠋⠁⢤⡄
⠀⣰⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⢀⠀⡀⠀⡀⢠⡀⠀⢀⣀
⠀⣿⣇⣿⣻⣁⣇⣿⣧⣧⣽⣿⡇⠀⠀⠀⠀⠸⠄⠆⠀⠣⠒⠂⠸⠹⠺⣼⠖⢧⡷⠹⠇⠀⠀⠀⢰⡏⠁⣿⢸⡏⢻⡃⣀⣿⢿⡃
⠀⣿⢛⣿⠛⣿⠙⡿⢋⣻⡿⣿⡇⠀⡀⢀⠀⠀⠀⢠⢀⣀⢀⡄⠀⡀⠀⣄⣠⠄⣶⠂⠀⠀⠛⠀⢈⡉⠁⠉⠋⣀⢈⡙⢃⡉⣛⠁
⠀⣿⣥⣿⣰⣏⣠⣿⣈⣛⣇⣿⡇⠸⢿⡿⠀⠀⠀⠛⠂⠙⠈⠁⠀⠙⠓⠋⠛⠆⠿⠆⠀⣀⠀⠀⠘⠙⠻⠛⠣⡿⠸⣥⠾⡯⠿⠅
⠀⠀⠀⠉⠉⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠴⠒⠛⠢⣄
⠀⠀⠀⠀⢀⣠⡤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⢀⡞⠉⢀⢰⣶⣠⡀⠀⠀⡰⢅⠀⠀⢀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⣼⠀⠀⠀⠀⠀⢙⣿⣀⣤⣶⣀⠲⡠⠁⣠⣢⠤⠀⠀⠀⠀⠘⠳⢦⣄
⠀⠀⠘⣄⠀⠀⠐⠋⠋⠉⠉⣙⣻⣿⡷⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⠶⣄
⠀⠀⢀⡞⠱⠦⠤⠒⠊⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⠝⠳⡄
⠀⡰⡯⠐⠲⣆⠀⠀⠀⠾⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢰⡄⠀⢀⡃⠒⠀⠀⠀⣈⣴⡶⠶⠶⠾⢤⡄⣠⡤⣴⢂⠠⠲⠆⢀⠀⠀⠀⠀⠀⠀⠃⠠⡌⢳⡶⠾⠟⠒⠦⠴⢿⡓⣦⣄⣀
⠀⠙⣺⠛⠳⠶⢒⣋⣉⣉⣀⣀⢀⣤⣴⠿⣿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠘⣷⣆⢸⠀⠀⠀⠬⢷⠁⠍⠏⠉⠓⠲⠦⢤⣤⣤
⠀⢀⣇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⡔⠀⠘⣉⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣋⢸⠀⠀⠀⢀⣸⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈
⠀⠀⠻⣧⣸⣠⣄⣀⢀⣐⠀⡉⠠⣩⣴⡼⠟⢫⣤⡙⢿⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡧
⠀⠀⠀⠈⢹⡋⠛⠛⠛⠛⠒⠛⠛⠛⢩⠀⠀⣼⡯⠔⠉⠉⠓⠆⣤⣶⣤⣀⡀⠀⠄⢄⡀⢀⠀⠀⠀⡞⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⢷⣦⣀⡀⡀⠀⠀⣄⣀⣷⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠰⠀⠀⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠁⠈⠙⠿⢼⣐⡜⣁⢀⠀⠀⠀⠀⠀⠀⠉⠀⠁⣠⣀⠀⠀⢠⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠚⠿⢧⣤⣥⣶⠾⠶⠛⠛⢶⣙⡀⡏⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣿⠀⠀⠀⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣄⣀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣻⣿⡟⡇⡀⡄⠀⠂⠀⠂⠀⡐⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣱⣹⡡⡐⣐⡀⡀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣵⣿⣿⣿⣟⡙⠐⢡⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠾⠮⢼⣔⣐⣃⣩⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠶⡃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡭⠉⠉⠁⠀⠐⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⣩⡿⢷⠄⣀⣴⡇⠀⠉⠀⢀⣀⠀⠀⠀⠚⣷⣾⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠈⠀⠀⠀⢠⡝⢿⡉⠀⠀⠀⠉⠉⠉⠁⠀⠸⣿⣛⠛⠶⠟⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡀⠀⢀⠐⡄⠀⠀⠀⠀⠀⠈⡛⣚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣰⣿⡿⣭⡀⣀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠛⠛⠛⠛⠛⠪⣽⣟⣿⣛⣛⣭⣿⠟⠀⠀⠈⠀⠉⠙⠒⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⡶⢶⣶⣶⠠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠿⠿⠿⠿⠿⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠿⠛⠉⠀⠀⠀⠀⣿⢸⡀⠀⠀⠈⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠞⠁⠀⢦⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⡴⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡞⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠤⠴⠄⠀⠀⢀⣀⣤⣤⣤⣤⣴⡛⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⡼⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⡜⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠋⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠁⣀⡴⣫⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣀⣀⠀⠀⠘⠃⠀⢀⣀⣠⣴⣾⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡄⠀⠙⢿⣷⣦⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠙⠻⣾⠟⠻⡿⢿⣷⡈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣄⡀⢸⠗⠂⠀⣜⠛⢇⣀⠘⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢶⠀⣠⢟⣴⣿⠟⠁⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣵⣿⠋⠁⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠉⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠀⠀⠈⢿⠀⢠⡄⠀⠀⠀⠐⣆⠀⠀⠀⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡃⠀⠀⠀⢀⠁⣿⣿⡄⠀⠀⠀⣿⣇⠀⠀⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠘⣷⡜⢿⣿⡀⠀⠀⣿⣿⠀⠀⠙⠉⠉⠉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣭⣝⣈⢿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⣿⣿⣿⡜⠇⠀⢰⣿⣿⠐⣿⣿⡇⣶⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠛⠛⠛⢸⣿⣿⣿⡇⣦⠀⠈⠁⠀⡄⣿⣿⣇⣿⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢇⣿⡇⠀⠀⢸⡇⣿⣿⢸⡇⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⢀⣀⠀⠀⢸⣿⣿⣿⢸⡟⠀⠀⢀⣿⡇⣿⣿⣸⡇⠀⠀⠀⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⢸⣿⣷⡦⢸⣿⣿⡇⡾⠁⠀⢀⣾⣿⢧⣿⡇⣿⢁⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⠃⣾⣿⣿⡇⠁⠀⣠⣾⣿⡏⣾⣿⣧⣿⠈⢻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠏⠀⠀⠀⢀⣼⣿⣿⠃⠀⢿⣿⣿⣴⣶⠆⣨⣭⣭⣼⣿⣿⢸⡟⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⡿⠁⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠸⣿⣿⡈⠛⠛⠿⠿⣿⣿⠼⠇⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠻⢿⣷⣶⣤⣝⡋⠀⠀⣀⣴⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡄⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⣀⣤⣦⣤⣤⡀
⢰⣶⣶⣶⣶⣶⣶⣶⣿⣋⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠚⠿⠿⣿⣿⣿⣿⣆
⣾⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠛⠙⠛⠛⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⡀
⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠙⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠀⠀⠙⢿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠙⠄
⣿⣿⣿⣿⣿⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠙⢿⣿⣿⣿⣦⣀⠀⠀⠀⣠⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠀⠙⢿⣿⣿⣿⣷⣦⡀⠹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⠻⣿⣿⣿⡿⠃⢹⣿⣿⣿
⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠛⠁⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣶⣾⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⣄⣤⠀⠀⠀⠀⢠⠀⣄⢠⡀⡄⠠⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⡇⠀⡇⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠘⠀⠉⠘⠀⠃⠀⠃⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢤⣄⣠⣶⡇⠰⡄⣰⡄⠀⠀⡀⠀⢰⣶⣶⡦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠈⢿⡿⠿⣁⣀⡴⠿⠋⠀⠀⠀⠀⠀⠈⣿⣿⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⣰⣾⣦⣭⡙⠋⠀⠰⢶⣀⠐⠤⣶⠄⠘⣵⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠘⡉⠉⣙⠿⡆⠙⣦⢺⡶⠮⠟⣃⣀⠀⠠⣉⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠘⢛⠁⢹⣧⡈⠙⠸⡾⢀⠰⡶⠋⢉⡷⠀⠙⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠠⣿⣆⣀⢻⣷⣶⣶⡿⢿⣥⣅⣴⡾⢇⣼⣷⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⣄⡈⣡⠀⠈⠁⠠⢀⡠⣄⡀⡍⠉⠙⠂⣌⢉⣡⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⠛⢿⣄⣀⡀⠰⢦⠀⢻⣶⠆⢀⣀⣦⡁⠋⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⡟⠗⠛⣿⠘⢟⣷⠄⠀⠘⣵⣾⡿⠁⢿⣷⡜⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⣦⠀⠹⣷⡘⡿⠀⠲⠄⠈⣿⠀⣠⣾⡿⢀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢾⣻⢆⣜⣉⣙⠀⠀⢰⡄⠀⠀⣰⣟⣉⠰⠐⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠈⠰⢾⣟⠙⣿⣷⣦⡀⢀⣴⣿⣿⠋⢿⣳⣤⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢼⡃⢀⡿⠿⠾⣿⣿⣿⣿⣿⣟⣡⣶⣺⣿⢉⡶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠘⠛⠂⠀⠀⠀⠈⠛⠉⠉⠛⠁⠉⠉⠙⠋⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠂⠈⠉⠉⠉⠉⠉⠉⠉⠓⠒⠢⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢸⠀⠉⠓⣄⠀⠀⠀⣿⠀⠀⠀⡇⠉⠙⠲⡄⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣆⣀⡠⠏⠀⠀⠀⣿⠀⠀⠀⣇⣀⡤⠔⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠿⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠈⠳⣤⠀⡄⠀⡏⠠⠄⠰⠇⠀⠤⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢆⠀⢀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠁⠈⠄⣀⣀⣀⣁⣀⣀⣀⣀⣒⣀⠀⣀⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠥⠤⣄⠀⠀⠀⠊
⠀⠀⠐⠲⡄⠀⢀⠟⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠒⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠂⢀⡤
⠀⠈⢷⡄⢠⡆⡸⠀⡀⠀⢠⡀⠀⠀⢀⣀⠢⢀⠐⠀⢇⢈⡀⠀⠀⠀⣄⠀⢀⠄⠀⠀⡠⠒⡀⢠⠀⠀⠀⢇⢔⠜⠁
⠁⠀⠘⡘⠎⠂⠀⠨⠴⠀⠠⠱⠰⠄⡈⠀⠙⠀⠹⠇⠂⠀⠛⠒⠀⠀⠈⠀⠑⠈⠠⠊⠄⠀⠣⠀⠃⠀⢀⠀⠀⠒⠃⠀⢀
⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠐⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⠀⠀⡠⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡁⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⣶⠆⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠈⣿⡇⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠛⠁⣀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡠⢀⠐⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠟⠂⠀⠰⠟⠶⠲⠾⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠠⡀⢄
⠀⠀⠀⠀⠀⠜⡠⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠣
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠
⠀⠀⠀⠄⡐⠀⠀⢀⡀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢂
⠀⠀⠀⠐⠀⠀⠀⢸⡇⠀⣠⣟⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⠘⣿⠀⠀⠀⠂⠀⡀
⠀⡀⠀⠁⠀⠀⠀⢸⡇⠀⢸⡿⢿⣿⣎⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠞⠁
⢀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠈⠿⠄⠙⠿⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢦⣤⣄⡀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⠀⢠⡄⠐⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣤⣀⠀⢈⣿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀
⡇⠀⠀⠀⣿⡀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣿⣷⣦⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣠⠇
⡇⠀⠀⠀⢈⡝⠂⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠻⣿⣿⠟⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⣶
⠇⠀⠀⠀⠈⠃⠀⠚⠁⠀⠀⠀⠀⠀⠀⠠⠄⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠤⠼⠋
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠈⠀⠀⠀⠀⠀⠀⣠⠄⠀⢤⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠃⠀⡀⠀⠀⠀⢿⣦⣀⡘⠀⠀⠁⠂⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠠⠀⠀⢰⠃⠀⠈⢻⡆⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⣿⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠃⠀⠈⠓⠀⠀⠛⠁⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⡈
⠀⠀⠀⠀⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰
⠀⠀⠀⠀⠀⢢⠑⠀⠀⠀⠀⠀⠀⠀⣼⡟⣴⠶⠶⠶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡔
⠀⠀⠀⠀⠀⠀⠑⠀⠠⡀⠀⠀⠀⢸⣿⠃⢀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⢦⣄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠢⡀⠀⢼⡿⠀⢈⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣰⣏⡀⢈⡉⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⢀⣽⠀⠀⠀⠀⠀⠐⢁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢄⠀⠀⢠⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⠛⠐⠋⠁⠀⠀⡠⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢂⡁⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⠀⠀⠀⠐⠂⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢂⠁⢄
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⢃
⠀⠀⠀⠀⠠⠀⠀⢀⣀⣀⡀⠀⠀⠀⣈⢀⡀⠀⢀⡠⣄⣀⠠⢁⢀
⠀⠀⠀⠀⠁⠀⠀⠀⡇⠁⡉⠄⠠⠀⢸⠀⠀⠀⢨⠂⠀⡀⠢⠀⠠⠁
⠀⠀⠀⠀⠃⠆⠀⠀⡇⠟⢕⠀⠀⠀⢸⠀⠀⠀⢸⠠⠛⠓⠀⠀⠰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠆⠸⠠⠠⠄⠼⠰⠄⠄⠼⠰⠄⠤⠈⠀⡠⠪
⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡂
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⢀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡩
⠀⠀⠀⠀⠇⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠠⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⡄⠚⠚⠐⠐⠉⠉⠉⠀⣏⠓⠒⣐⣤⡤⣠⣀⠀⠀⢀⡇
⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠓⠀⠀⠀⠂⠀⠀⠧⢀⣋⡀⠲⢙⠽⣆⣀⡆
⠐⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠒⡠⣰⠀⠴⡈⠉⢏⣿⡟⢶⣤⡀
⠀⠀⠹⡀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⡀⠈⠂⠀⠁⠈⠀⠿⢓⣘⠂⠟⠲⠖⠙⠛⣦⡀⠄⠊⠡⠐⠀⢀
⠀⠀⠀⠈⠓⠤⢄⢀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠋⠲⣖⢐⣠⠔⢺⠞⠁⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠖⠲⠖⠲⠒⠲⠶⠦⠦⠖⠛⠒⠖⠓⠉⠀⠈⠁⠈

⠀⠀⡀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠌⠀⠀⠀⠈⠁⠀⠅⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⢠⢦⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠐⠂⠚⠓⠂
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⣰⠋⠱⡄⠀⣇⣸⢰⠋⠙⢦⠀⣿⠘⢀⡎⣿⠀⠀⠀⠀⢀⠁⢀⠀⠀⡀⡠
⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠰⠄⠂⠀⠀⠀⠀⢹⡐⠆⠻⣄⣿⠁⣘⡷⠤⢼⣿⣿⠀⢸⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈
⠀⠀⡄⠀⠀⠀⠀⢀⡀⡄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠤⠤⣿⢴⡇⠁⠀⢸⠏⣿⡄⠘⣧⠐⠯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢀⠀⡀⡄
⠀⠀⡇⡄⠀⠀⠀⡈⠁⠈⢁⠀⠀⠀⣦⠀⠀⠀⠙⠄⠀⢀⡤⠏⠈⠳⢄⠤⠋⠀⠘⠿⠖⠙⠂⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠄⠁⠁
⠀⠰⠁⠇⠠⠀⢸⠁⡇⢸⠾⠀⠗⢤⠿⠀⠀⠀⠀⠀⠀⠀⣔⠀⠀⠀⠂⠰⠁⡎⠐⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠐⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠘⢂⠈⠀⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⡀⠄⠀⠀⣆⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢠⡇⢠⠋⠀⠀⠀⠀⠀⠀⠀⢀⠐⣆⡀⡄⢀⡄⠀⠠⠀⡀⠠⡠⠀⠀⠧⠆⠀⠄
⠀⢣⠀⠀⠣⠙⠈⠀⢀⠠⠸⠓⠀⠀⢾⡀⠀⠀⢚⢀⠀⢀⣀⡀⠁⠁⠁⠈⢁⡁⢀⡀⠀⠀⠀⠀⡀⢉⢠⠹⣇⠀⠉
⠀⠂⠁⠀⡀⠀⡀⠂⠈⠀⠀⠀⠀⠀⠀⠉⢠⡖⢉⡄⡶⠥⠴⢳⣀⣀⠀⣴⢹⠂⢸⠐⠀⢱⠀⣇⡹⡄⠉⣇⡸⠀⠀⠀⠌⠀⠀⠑
⠀⠀⠀⠀⠚⠀⠀⡀⠰⠄⠀⠀⢰⢠⠂⠀⠈⠛⠉⠈⠳⠶⢊⡞⠉⣽⣀⣿⢸⡄⢸⡂⠻⠈⠷⠉⠀⠈⠁⠈⠀⠀⡤⠀⠁⠒⠆⠰⠀⠘⠈⢂⠀⡀
⠀⠀⠀⠀⣰⠂⠀⠐⠀⠀⢠⠰⡄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⢡⣖⢋⡿⠈⠳⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⢀⠀⡀⢀⠀⠐⡀⡈⠃
⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣁⣀⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣈⣀⣃⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡗⠁⢠⡿⠀⠀⢀⣀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⢴⣯⡀⠀⢠⣿⣏⣠⠇⣾⡿⠥⠟⣠⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⣀⣘⣿⡀⣸⡟⠀⢀⡼⣿⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢀⣥⡴⠃
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣄⠀⠀⠀⢿⡟⠰⣯⣤⣿⠟⠁⠈⠛⠒⠉⠀⠉⠋⠉⠀⠀⠀⠀⠀⢀⢤⣐⣾⠿⣽⣇
⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⣿⡇⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⣺⢵⣛⣛⢻⡿⣧
⠀⠀⠀⠀⠀⢸⣿⠇⣠⣾⠟⢡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠈⢫⡭⠟⠳⠼
⠀⠀⠀⠀⠀⣿⡿⣿⣍⠀⠀⣶⡅⠀⣴⡄⢀⣤⣤⠀⢀⣴⣦⢸⣿⠇⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⢸⣿⠃⢸⣿⡆⢀⣿⠃⣠⣿⡷⠋⣼⡟⠀⣿⡏⠀⣿⡿⠀⡀⠀⠐⠐⠀⢀⠀⠀⠂⠈
⠀⠀⠀⠀⢸⣿⠀⠸⣿⡇⠀⠛⠚⠉⠿⠁⠀⢿⣅⠴⠙⠳⠚⢿⣇⣠⠟
⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣿⣿⡿⠭⠉⢽⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣱⡦⠤⣊⣽⣿⠟⠉⠀⡐⠂⣉⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠍⢻⣿⣿⣿⣷⣆⣀⣠⣶⣿⡿⠟⠉⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢴⡿⣿⣿⣿⣿⢿⣯⣭⣾⡏⠀⣀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣿⡿⣿⣿⣿⣿⡟⠏⠁⠀⠀⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣡⣾⣿⣿⣿⣿⠿⣧⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣡⣴⣿⣿⣿⡏⠀⠈⢷⣤⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⡿⠁⠀⠀⠸⠁⠀⠀⠀⢀⣀⣤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⠿⣾⢿⠟⠁⣀⣠⠤⠴⣶⡶⠾⠛⣋⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠒⠋⠉⠩⠐⠂⠉⠭⠐⠂⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢒⣿⠷⢾⣦⣴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣭⡄⢸⣿⡿⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣖⢾⠟⠀⠀⠀⠘⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⢿⣿⣿⣿⡷⠋⣀⣴⠦⢀⣤⣿⣿⣿⣔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣬⣿⡜⠟⣿⡀⠈⠁⠀⠈⠙⣿⣿⡻⣿⣆⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣽⣿⡗⣄⠑⠀⠀⣀⣤⣰⣿⡷⡝⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠈⢷⣤⣀⠀⢉⣿⣿⡇⣷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡵⢻⡿⠀⠈⠀⠈⣻⣿⣿⣿⣿⣿⣿⢟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠁⠀⠐⠊⠉⠉⠁⠈⠉⢻⣿⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣽⣿⣷⣤⡎⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣏⢹⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣴⣿⣿⣿⣿⠟⠛⣸⡿⠛⠙⠻⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠒⠶⠶⢤⣀⠀⠀⠀⢠⣾⣷⣿⣿⣿⣿⡿⠛⠁⣤⣶⡉⣶⣄⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡡⠀⠀⡰⠋⠀⠀⠀⣈⣭⠓⢒⣾⠏⢉⡿⠛⡋⠁⠀⢠⣾⣿⡇⠉⣦⣀⣱⣀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⡈⠀⠀⣰⡔⠀⠀⠀⠀⠀⠁⠐⠋⠁⠀⠜⠀⡞⠀⠀⠔⢋⡀⢸⣇⣴⣯⡿⣏⠃⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⢸⡄⢀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣴⡟⠀⣿⡟⠛⢦⣏⠈⢀
⠀⠀⠀⠀⠀⠒⠶⣾⣿⣿⣿⣿⠀⣇⢸⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡀⠀⠀⠁⠀⣼⡟⢀⣼⠏⠀⠀⠀⠙⣦⣛⣷⡞
⠀⠀⠀⠀⠀⠀⠞⣿⣿⣿⣿⣿⣧⡈⠀⣿⣧⠈⠻⢦⡐⢤⣄⣠⣾⠃⠀⣧⠀⠀⠀⣼⡿⢡⡟⢁⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠙⠛⠋⠉⠀⠁⠁⠘⣧⢠⠾⠿⢷⣄⠀⠈⠑⠈⠉⣿⠀⣠⡞⠳⣤⡀⠹⠵⠋⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣱⣶⠇⠈⢳⣶⣶⣤⣤⣿⣾⢿⡇⣰⡿⠧⢀⣐⡺⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⣿⠋⠀⠀⡼⡿⢻⠋⠙⢻⠁⠀⠹⡿⠷⠻⢿⣟⠓⢤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣻⡇⠀⠀⠀⠹⣼⢿⡀⠀⠘⣧⡄⠐⡇⠀⠀⠀⠉⠻⣆⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠈⠻⡹⣄⡀⠘⣿⡄⢃⠀⠀⠀⠀⠀⠘⢿⡦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡽⣆⠀⠀⠀⠀⠀⠙⣿⣷⡀⢸⡅⠈⡆⠀⠀⠀⠀⠀⠀⠹⡜⣦
⠀⠀⠀⠀⠀⠀⠉⠁⢀⡀⠀⠀⠀⠀⠀⢷⣿⣇⠀⠀⠀⠀⠀⢹⣿⠛⣸⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⢻⡈⣷
⠀⠀⠀⠀⠈⠙⠓⠀⣈⣛⠛⠛⣓⠒⠒⣾⡟⠛⡆⠀⠀⠀⠀⢠⡾⣏⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠙⠊⠉⠛⠁⠀⠀⠀⠀⢸⣿⠟⠉⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠘⠶⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠙⠛⠛⠓⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠰⣳⢟⣼⣟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠆⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡓⢮⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠁⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣴⠞⢁⣿⣏⠤⣤⡀⠀⣠⣶⠖⣠⡀⠀⣿⣧⠔⢲⣦⣠⣿⡔⠚⣷⡄⣸⣿⠀⣼⡇⠀⢠⡆
⠀⠀⠀⠀⠀⣿⠏⠀⢸⡿⠁⣼⡿⠁⢠⣿⣁⣴⡿⢁⣴⣿⢣⠒⣽⡟⢰⡿⡠⣲⡿⠉⣿⣡⢾⣿⣃⣴⡿⠁
⠀⠀⠀⠀⠀⠙⠀⠀⠘⠃⠰⣿⣁⠔⠁⠉⠁⠻⠗⢃⣿⠇⠈⠛⠉⢀⣿⠇⠈⠉⠀⠀⢀⣠⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⢸⡟⠀⠀⠀⢀⠞⢉⣾⠏
⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢠⣤⢤⣀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣯⢳⣄⡛⢢⠀⠀⠀⠉⠀⣀⣀⠀⠀⠈⠁⠀⠀⣀⡠⡴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⢿⡘⢦⠠⠽⢌⣇⣀⡴⠒⢪⡠⣄⠙⢢⡀⠀⣠⢊⠁⡠⠗⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢨⡷⠋⢍⡹⠋⡜⢡⠞⣿⣲⠌⡧⡀⡏⠑⣦⣞⣈⠵⠚⠀⡔⠑⡦⣤⣖⡤
⠀⠀⠀⠀⠀⣀⣤⣀⣏⠀⡂⠄⣴⣿⡻⢇⠀⠳⢼⠦⣑⣺⠏⠀⡯⠀⢀⣇⡐⠲⡾⠷⠦⠑⢰⣥
⠀⠀⠀⠀⠈⠉⠋⣥⣤⠀⠀⠀⠈⣩⢴⡿⡄⠀⠘⢤⣌⠀⢀⡜⠁⢀⠜⢽⣿⢸⣯⢢⣄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⢠⡾⠏⠘⠃⠈⠢⢄⣀⡀⢉⣕⣁⠔⠁⠀⠀⠉⠁⠋⠀⠻⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣒⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠛⠻⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡏⠀⠀⠀⠀⠉⠛⣳⣦⣤⣄⠀⠀⠀⣀⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠇⠀⠀⠐⠿⠿⣿⣿⣿⣿⠿⠟⣰⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⠿⢿⣷⡄⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⢿⠉⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣌⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣄⣠⣿⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣬⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠱⢶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣇⠀⠀⠈⢿⣿⣶⣤⣀⡀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠘⣿⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢻⡄⠀⠀⣰⠟⠈⠻⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠃⠀⢠⡟⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠉⠉⠛⠶⣤⣀
⠀⠀⠀⢀⣿⣿⠿⠿⠛⠻⠿⠁⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠀⠤⠄⠀⠀⠐⠒⠒⠛⠛⠛⠂⠉⠉
⠀⠀⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠚⠓⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠤⠄⡀⠀⠀⡀⣀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢦⣄⠀⠸⣿⣿⣿⣄⡀
⠀⣰⠏⠀⠀⠈⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠂⠀⠈⢻⣷⣄⠉⠿⣿⣿⣿⣆
⢠⡏⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⣿⣿⡀
⣾⠀⠠⢄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠈⠞⢑⠀⠸⡇⠙⠿⠟⣿⣿⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣲⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣿⣿⣿⡀
⣿⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⠀⠀⢀⠠⠄⠀⢠⡇⠀⠀⣠⣾⣿⣿⣿⡇
⢸⡆⠀⠓⠒⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⣸⠁⣰⣿⣿⣿⠿⣿⣿⣷
⠀⢻⡄⠀⠀⠀⠠⣄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⣰⠇⠸⣿⣿⠋⠀⠀⠹⣿⣿⣆
⠀⠀⠹⣄⠀⠘⠓⠲⠀⠀⣀⠀⠀⠀⢀⡀⡀⠀⠀⠁⠈⠡⢀⡼⠃⠀⠀⢻⣿⣇⠀⠀⠀⠙⢿⣿⣷⣦⡀
⠀⠀⠀⠈⠳⣤⡀⠀⠀⠨⠬⠑⠀⠀⠐⠪⠍⠀⠀⠀⣀⡴⠛⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠈⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⡀⠀⢀⣀⣀⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⢀⣹⣿⣧⠀⠀⠀⠀⠀⠀⠀⢉⣿⠿
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣦⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣾⣿⣶⣿⡿⣿⣿⠟⢋⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣴⣿⡟⢹⣷⣾⠃⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣿⡿⣿⣤⣾⣿⣿⣿⢿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢉⣴⣿⣿⣿⣿⣿⡇⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡿⠏⣠⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣤⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣫⣼⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣏⣴⣿⣿⣿⣿⠟⠁⢸⣿⣿⣿⣿⣿⡇⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣽⣾⣿⣿⣿⡟⠁⠀⠀⢨⣿⣿⣿⣿⣿⡗⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡷⣹⡇
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⣫⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⣿⣿⡟⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣧⣤⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣸⣷⣿
⠀⠀⠀⠀⠀⠹⣿⣷⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣼⣿⣽⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣯⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣿⡇⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢹⡋⣿⣿⢾⡇⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣷⣿⣿⣻⡇⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⣿⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠘⣿⣿⣿⡇⢺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠙⣿⠀⠀⢸⣿⣿⠁⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⢿⡀⠀⢈⣿⣿⣧⡀⣿⡛⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠯⣴⣶⣶⣿⣟⣹⡿⣿⣟⣹⣷⣽⡇⠈⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣴⣶⣿⣯⣶⣾⣿⣿⣤⣾⣿⣽⣿⣿⣷⣤⣤⣬⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣉⣹⣟⣿⣿⣿⣿⣿⣶⣶⡿⠿⠷⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣿⣷⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠉⠙⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠙⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡄⠀⠀⠀⢀⣿⣿⣧⠀⠀⢘⣦⣄⠀⠀⠀⠠⣦⣀⠀⢲
⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢠⣿⣿⡌⣿⣦⠀⣼⡏⠈⢳⡀⠀⠀⣿⡗⠀⣿⠄⣘⠀⣀
⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣺⠿⢀⣀⣰⣧⡀⠀⣿⡇⠀⣿⣬⢙⡉⣲⣤⣠⣀
⠀⠀⠀⠀⠀⣫⣾⡉⠨⠉⠉⠉⠁⢸⣿⣿⠿⢋⣹⡿⠋⠀⢹⡏⢁⠀⣠⠼⢧⣼⣿⣏⣠⡊⠛⠉⠻⣿⣿⡆
⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⠀⠀⠀⣨⣿⣿⣴⡿⣟⢉⠙⣦⣾⣟⡛⠉⠻⣄⢈⠿⣿⣿⡇⠀⠀⠀⣀⠿⣿⣿⡀
⠀⠀⡸⣷⣦⣾⣿⣧⡀⠀⠀⠀⠀⢩⣯⣝⣛⡷⠤⢤⠴⡧⠿⠿⠷⠦⡄⣻⣩⣿⣿⣿⣗⣦⡭⠊⠁⠀⢹⣿⠂
⠀⠀⢽⣿⣿⡟⠛⠋⠉⠀⠀⠀⣴⢸⡿⠻⣿⡿⣿⣿⡿⣛⡊⠑⠉⠉⢉⠕⣿⣿⣻⣿⣿⣧⣤⣤⣆⣶⣿⣿⠆
⠀⠀⣿⣿⣿⡷⢶⣠⢴⣦⡀⣤⣿⣼⠿⣷⣿⣶⣤⣧⣤⣼⣤⣤⣧⣤⣦⣴⣿⡿⣿⣿⣧⡉⠛⣛⢉⣽⣿⣿
⠀⠀⠸⣿⣿⣷⣿⣿⣿⣿⣟⣿⣻⣿⡏⠉⠙⣿⠿⡿⣿⣿⣍⣛⡛⢿⣟⣿⣿⣽⠿⣾⡿⡻⣿⠿⢿⣿⣿⠃
⠀⠀⠐⢟⣿⣻⣷⢿⣿⣯⣿⣿⣿⣿⣷⣤⣴⠍⠑⠒⠚⠉⠉⣅⣾⣿⣿⣽⣻⡟⣿⣚⠷⢗⡥⣀⠒⢻⡟
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠯⢿⣞⣿⡽⠉⠛⠂⠠⠤⣄⣠⡸⠟⠛⠒⠙⠓⠛⠻⠿⠽⠻⠋⠠⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣲⡖⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⠀⠀⠀⠀⠀⣿⣂⠘⠁⡰⠷⣤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠟⠊⠀⠀⠀⠀⠀⠀⠻⣐⠒⢐⣗⠠⠿⠅⢰⡇⠀⠀⠀⠀⠀⠀⡔⠉⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⣡⣒⣁⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠅⠒⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠥⢬⠁⠀⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠈⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⣀⠀⢀⠀⠀⠀⠀⣀⠦⢒⣂⣠⣤⣤⣀⣒⠠⢇⡀⠀⠀⠀⠀⠀⠀⡯⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⠖⠋⠉⠀⠀⢰⠀⠀⠈⠉⠓⢮⡑⢄⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣡⠞⠁⠀⠙⠀⠀⠀⠘⠀⠀⠀⠘⠁⠀⠈⢢⡱⡀
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠐⠂⡜⣴⠁⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⠄⢱⡜⡄⠒⠈⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⢰⠃⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢳⢠⠀⠀⠀⠀⠀⢀⠔⡣⠎⣴⣉⠙⡦⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢸⠀⣀⡀⠀⠀⠀⠀⠈⠓⢤⠖⠁⠀⠀⠀⠀⠀⣀⡀⢸⠌⠀⠀⠀⠀⠀⡒⢰⠥⣰⡀⢸⠉⠓⢲
⠀⠀⠀⣸⣷⣶⣶⣶⣶⣶⣶⣶⣷⠀⠀⠀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠦⢼⢀⡇⢀⠋⢐⠗⡼
⠀⠀⢠⣉⣀⣀⣶⣶⣶⣶⣎⣁⣉⡆⠀⠀⠀⡈⣆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡎⡈⠀⠀⠀⠀⠀⠈⠲⣤⣁⣍⡱⣓⠔⠁
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠒⠉⠑⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡐⠉⠑⠂⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⡳⣄⠀⠖⠀⠀⠀⢰⠀⠀⠀⠐⠆⢀⡴⡫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠢⢝⠲⠤⣤⣀⣈⣀⣀⡤⠴⣚⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠓⡽⡂⠀⠀⠀⠀⢀⠋⠁⠒⠂⠀⠀⠒⠂⠉⢃⠀⠀⠀⠀⠀⠀⠀⡠⡺⡉⢱⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣎⠵⣳⣚⠳⡲⠽⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⠅⠸⣗⠀⠇⢣⣬⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣄⠀⡈⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⣖⠒⢒⣶⣒⠀⠀⠀⠀⠀⠀⠀⢀⢖⡭⠚⠂⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠣⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡆⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠃⠐⣶⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠶⠶⠴⠭⠯⠯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⡶⠂⣷⣥⠀⠠⠴⢒⣠⢄⣀⣀⡀⡀⠀⢀⠠⣴⣶⣢⠄⠀⡢⠀⠛⡹⠎⣁⣀⣀⢀⡀⠀⠀⠀⠀⠈
⠀⠐⠀⠀⠀⡀⠀⢀⣀⠀⠀⢀⠁⢀⣻⣤⣿⣿⣯⣷⣿⣿⣿⣱⣿⣿⣮⣹⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣖⣺⣶⣀⡀
⠀⢸⢆⡘⢀⣏⣸⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣧⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⣿⣿⣯⣧⣾⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⠿⠏⠘⢾⣿⣿⣿⠁⠄⢺⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⣾⣿⣿⣿⣾⣶⣟⠻⡟⢹⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠅⠤⠀⠩⠍⠉⠽⣿⣧⠴⡿⢻⣿⣿⣿⣿⣿⣿⣿⠀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⡿⢰⣃⢸⡇⢸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣷⣅⣀⢀⠀⠀⣀⣀⣀⣬⣿⣴⣦⠈⣿⣿⣿⣿⣿⣿⢿⠁⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⢿⣻⡟⢛⡧⠾⠛⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣷⠟⢦⣙⣾⣦⡭⣻⣿⣿⣷⣶⣿⣿⣿⣿⣿⢏⠉⠀⢁⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⠟⣰⣿⣶⠟⠛⠀⠄⠐⠤⠈⠿⣷⣿⣟⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡟⢳⡶⢻⣝⣿⣕⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⠻⣿⢻⠏⠙⣶⠻⠖⠶⠚⢒⡶⡿⣿⣹⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢴⣿⣻⣿⣿⣾⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⠀⡷⢺⡛⢦⣿⠀⠀⢷⣲⣇⡾⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣶⡗⣿⣿⡿⠉⢷⣄⣠⠿⢿⣤⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠄⡀⡉⢉⡩⣁⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⢿⣿⣶⢧⡾⢸⣯⣶⡦⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⠛⠟⠛⣿⡟⠻⠿⠟⠛⠻⠋⠐⣦⣴⣶⣤⣦⣹⣹⣿⣴⣾⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣽⡋⡸⢅⡈⢧⣿⣵⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⠆⡆⠠⠷⣶⣿⡼⢿⣿⠏⡼⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣷⡜⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡆⠲⣔⣲⣽⡅⠺⣿⡇⢸⣧⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣽⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⣰⣿⠹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣜⠒⠉⠥⠈⣱⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠸⡿⠿⠿⠶⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋⠛⠿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⠟⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟
⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠂⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣥⢲⠴⠶⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠾⢻⢾⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠦⠤⠴⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠲⠾⢶⠲⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠶⠾⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣦⢰⢢⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠠⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣒⠆⠀⠀⠀⢑⢒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠄⠀⠀⠀⠀⠀⠢⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢈⢡⢸⠖⣓⣶⡈⢁⡀⡀⢀⡤⡞⠻⡦⠲⠀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠠⠸⡿⢿⠹⢿⠻⠹⠿⠿⠿⠿⡟⢿⠇⠠⢒⡔⠀⡰⠹⠷⡿⠿⠾⠾⢾⡷⠇⠠⠤⠄
⠀⠀⠀⠀⠀⠀⢠⣶⣺⣶⣶⢲⢶⣶⣷⣶⣶⣿⢸⢼⡷⣇⡇⣟⢻⢼⠼⢿⡿⢧⢤⠤⣦⢤⣦⡄⣬⡥⠐⠒⠒⠂
⠀⠀⠀⠀⠒⠒⠒⠶⠀⠂⠶⠆⠲⠒⠶⠶⠶⠠⠶⠖⠶⠶⠦⠆⠶⠶⠲⠶⠒⠈⠰⠶⠖⠰⠶⠠⠶⠖⠶⠆

⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⠉⠉⠀⠀⠀⠉⠉⠙⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣠⣶⣶⣤⡀⠀⠀⣤⡀⠀⢠⣶⣶⣦⡀⠀⠀⠀⣧⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣿⣿⣬⣿⠃⠀⠀⣿⡇⠀⢸⣿⣀⣽⡿⠀⠀⠀⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿⠻⣷⡄⠀⠀⣿⡇⠀⢸⣿⠛⠛⠁⠀⠀⠀⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠸⠟⠀⠻⠇⠀⠀⠻⠃⠀⠘⠿⠀⠀⠀⠀⠀⠀⣿⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠀⠀⣟⠳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣞⡿⠞⠃⠀⢰⣯⣙⡏
⠀⠀⠀⠀⠀⠀⣄⠀⣿⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⢸⣧⣹⡇
⠀⠀⠀⠀⠀⢤⣿⢷⣿⢀⣾⣇⣠⡶⢀⡤⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⢠⣄⠀⢸⠷⣽⡇⣀
⠀⠀⠀⠀⠀⠈⢿⡄⠙⠚⠙⠛⢻⣷⣯⣤⣤⣤⣤⣿⣦⣸⣧⣤⣤⣿⡿⢿⠳⣾⠞⠿⣿⡿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠁

⠀⠀⠀⠁⠀⠀⠀⡀⠀⢀⠀⠀⠀⠒⠀⠀⠀⠀⢀⠐⣀⠀⠀⢠⠀⢀⢠⠀⠀⠀⢉⠁
⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠴⠾⠿⠶⠄⠀⠀⠀⠀⠀⠀⠈⠀⠤⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⢀⠀⡀⠀⢀⡑⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠄⠀⠀⠀⠤⠄⠀⠀⢀⡀⠀⢀
⠀⠀⠀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⡀⠀⠀⠀⢸⠓⠒⠲⠀⠀⢾⣿⣿⣷⠀⢀⣆⣤⣴⠆
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠁⠀⠈⠁⠈
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠰⠂⠀⠰⠂⠀⠀⡁⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⣷⣦⠀⠀⠀⠀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠾⠶⢘⣿⣀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⡠⣦⢀⣸⣿⣧⡀⠀⠀⢀⡴⠾⠗⠒⢳⣦
⠀⠀⠀⠐⢶⡄⠀⢸⣿⣿⣧⡀⠀⢠⣤⣤⣿⣿⣿⠶⣤⠀⠀⢠⡞⢋⣿⠉⢷⡄⠀⠀⢸⣿⣿⣾⣿⠛⢿⡶⠄⠀⢈⡿⣰⣺⡆⢸⡇
⠀⠀⠀⠀⣾⣵⠃⣿⢯⣅⡀⠀⠀⠀⠀⢘⢿⣿⣿⣀⡀⠀⠀⠈⣷⠟⠁⢀⣾⠇⠀⠀⣼⠿⠿⢸⣿⣿⣿⣧⠀⠀⢸⡇⠻⠿⠇⢸⣿
⠀⠀⠀⠀⠻⠛⠛⠁⠀⠙⠿⠆⠀⠀⠀⠿⢷⣭⣽⡯⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠛⠁⠀⣿⠀⠀⠀⠸⠇⠀⠀⠰⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠴⠒⠒⠒⠛⠃⠈⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠈⠙⡇⢀⣠⣤⣤⣤⣤⣴⡶⠶⢶⣦⠄⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣶⣶⠾⠿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠋⠁⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠘⢆⣶⡄⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣠⠴⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠀⠀⠀⠀⠀⠀⠘⠃⢸⡆⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡏⠀⣤⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠛⠉⠳⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⣧⣀⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡶⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⣀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⣇⣼⠏⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠘⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠓⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⢦⣄⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣧⡀⢀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠙⠻⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⣀⣀⣀⣤⣴⣾⣿⣿⡿⠋⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣀⣠⣀⠀⢿⡛⢿⣿⣿⡿⣿⣿⣿⢿⣿⣿⠿⠿⠛⠛⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡞⠉⠀⠼⡇⠀⠙⠿⣧⣴⣟⡩⠴⠶⠚⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⣈⣷⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡀⠈⠳⣤⣤⣤⣿⡇⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡄⠀⠀⢸⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⢹⣿⠟⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣆⠰⠚⢧⡍⠁⢙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣀⠀⠀⣙⡋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣩⣆⣤⣾⡡⠖⣩⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠙⠛⠿⣿⣿⣶⣶⣶⣶⣶⠶⠿⠛⢻⣿⣿⣿⡿⠃⠈⠋⣠⠴⠋⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧⣤⣾⣿⣿⠿⠿⠿⣿⣿⣷⣄⡤⠿⣿⣿⣿⣧⣀⡴⠚⠁⠀⢀⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠴⠿⠛⠁⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⣿⡿⠟⠁⠙⢦⣄⣀⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢬⣿⡛⠳⢿⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠛⠀⠈⠁⠀⠀⠀⠀⠻⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⣀⠤⠔⠒⠒⠀⠐⠒⠒⠠⢄⣀⠀⠀⠀⣴⡏⠤⠄⠒⠂⠀⠂⠤⠄⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠘⢁⣤⠴⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠙⠶⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣀⠈⠳⡀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣤⣼⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣼⣿⣿⣄⡄⢳⠀⠀⠀⠀⢰⡇⢰⣾⣿⣇⣀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢻⣄⡀
⠀⠀⣼⠁⢸⡇⠀⠀⠘⠉⠉⠛⠻⢤⡀⠀⠀⠻⣿⣿⡿⠃⣸⠀⠀⠀⠀⢸⡇⠹⣿⣿⣿⠏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣺⠉⣷
⠀⠀⢹⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⢄⣀⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠻⣄⣀⣉⣀⣠⣤⠖⠋⠉⠉⠛⠒⠀⠀⠀⣿⢀⣿
⠀⠀⠀⠙⠶⣿⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣄⣰⠟⠉⠱⡄⠀⠀⠉⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠾⠁
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣴⣟⣻⣄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⣼⠐⢻⣶⣈⢉⢩⡏⢹⡆⢀⣠⡤⠖⢳⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣷⣶⣿⣼⣼⣼⣿⣾⣼⣗⣺⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⢀⣤⠶⠚⠋⣉⣀⠤⠤⠤⠤⠤⢄⣈⡉⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠾⠋⠁⠀⠊⠉⠀⠀⠀⠂⠀⠀⠀⢀⠆⠈⠉⠒⠄⠈⠛⢦⡀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡤⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡀⠀⠀⠀⠀⠲⣄⠙⢶⡾⢧⣄⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠋⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠚⠈⢢⡈⠻⣤⠿⠻⣷⣿⡿⣆
⠀⠀⠀⠀⣠⡾⢋⣾⠁⡰⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢹⣆⠀⢹⣿⢿⣿⡄
⠀⠀⣠⠞⠁⠀⣼⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡱⠀⢿⣄⣠⡿⠈⠉⠙⢦⡀
⢀⡾⠃⣴⠛⠛⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⣿⠷⢶⣄⠀⠀⠀⠻
⣾⠁⠘⣃⣀⣴⡋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⢹
⠹⢦⣀⢠⡟⠁⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣴⠟⠛⣿⠋⠀⠀⣠⡴
⠀⠀⠉⠛⢻⣤⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⣀⣀⣼⠷⠞⠛⠉
⠀⠀⠀⠀⢿⡀⠀⣿⠰⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣼⠋⠙⣧
⠀⠀⠀⠀⠈⠛⠛⢿⡀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠠⢄⢠⠃⣰⠿⠦⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠱⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡰⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡠⠊⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠑⠢⣄⣀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⡤⠒⠁⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⣄⣀⡙⠀⠒⠂⠶⠆⠳⠀⠚⢉⣀⣤⣶⠛⠁⠸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠉⢻⡟⠓⠶⠶⠶⠶⠛⠛⠋⠉⠀⣿⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠘⠳⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⣠⡄⠀⠀⠀⣠⠶⡆⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡤⢦⠀⠀⠀⢀⣼⠃⠀⣰⠋⡿⠀⡟⠀⢀⡼⠁⢸⠇⠀⢀⣾⠃⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣶⠀⠀⢀⡴⠋⣍⡴⠋⠀⡼⠀⠀⢠⢾⠋⠀⣰⠁⠸⡇⢠⠃⣠⠟⠀⢀⡟⠀⣰⢋⠟⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣁⡴⠋⠀⣠⠞⠁⢰⡟⠀⠀⢰⠃⢀⡴⠃⡞⣠⠾⡇⠀⢀⡽⠛⠛⠁⠀⠀⠸⠦⠞⢁⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⢀⡴⠃⠀⢠⡟⠀⠀⠀⣇⣠⠞⠀⣸⠖⠁⠀⠙⠒⠋⠀⠀⠀⠀⠀⠀⠀⣀⡴⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣠⠴⠋⠀⠀⠀⠛⠀⠀⠀⠀⠉⠀⣠⣾⠃⠀⠀⣠⠋⡹⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⡏⠀⠀⢰⠏⡰⠃⠀⠀⠀⠀⠀⠀⡞⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⢠⠟⠀⠀⢀⢏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠙⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠎⠀⠀⠀⣼⠏⢀⠀⠀⠀⠀⢀⣤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠤⠦⠀⠀⣼⣡⠞⠉⡳⠀⠀⣠⠏⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢠⣾⡿⠁⠀⣠⠃⠀⣠⡿⠊⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⡰⢃⡞⠀⠀⢠⡏⢀⡜⢫⡀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣠⠞⠁⣼⠁⠀⠀⠘⠗⠋⠀⠀⠉⠉⠀⠀⠀⢠⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⡏⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣠⠇⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⡜⠀⠀⠀⠀⣾⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣡⠃⠀⠀⠐⠋⠀⠀⡾⠀⠀⠀⣀⣰⣁⣠⠀⠀⣸⠟⠀⠀⢠⠖⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃⠀⠀⢠⠀⠀⠴⣾⠟⠛⠃⠀⣹⠏⠉⠀⠀⣰⠏⠀⠀⢠⣃⡴⠋⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣴⠏⠀⠀⣴⠏⠀⠀⢀⡴⠻⠀⠀⢀⡼⡟⠀⢀⣰⡏⠀⠀⢀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢠⠞⡏⠀⢠⠞⢸⠀⠀⣠⠎⢠⡇⠀⣰⠏⠀⣇⣠⠞⠁⠷⠤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡴⠋⢸⢀⡴⠋⠀⠸⣤⠞⠁⠀⠀⠓⠊⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠚⠀⠀⠘⣉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡞⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⢰⣧⠎⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⣀⣤⣤⠀⠀⠀⢀⡴⠊⢉⡟⠀⠀⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⡿⢃⣠⡄⠀⠀⠀⢀⡶⠀⠀⢀⡾⠉⣇⡴⠋⠀⡞⠀⠀⣰⠋⠀⢀⡾⠁⢀⡾⠁⢸⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠷⠒⠃⠀⢠⣞⡴⠋⠀⡇⠀⠀⣠⡿⠁⠀⣠⠋⠀⣼⠏⠀⠀⡼⠁⢀⣾⠃⠀⢀⣾⢃⣴⠋⠀⢀⣼⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⡰⢻⠟⠀⠀⣸⠁⢀⡴⢹⠁⢀⡼⠁⠀⣰⠏⠀⠀⢸⣇⡴⠋⠳⣤⠴⣫⠟⠁⠈⠛⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⢠⠞⢡⠏⠀⠀⢰⡇⡠⠏⠀⢯⡴⠋⠀⠀⠀⠋⠀⠀⠀⠀⠉⠀⠀⢀⡴⢊⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡴⠋⠀⠟⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⢢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢬⣄⣀⣀⡀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠊⠋⠁⠀⠀⠉⠀⠈⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠖⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠎⠀⠀⠰⡀⠀⠀⢠⠀⠀⠀⢦⢸⢠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠄⢂⣠⣜⢀⣨⣥⠐⠊⣵⣤⣌⣤⣃⡌⠠⠀⠙⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠦⡀⢆⠌⠒⠀⠙⠎⠉⠉⠀⠀⠉⠉⠱⠋⠀⠂⠭⡰⣁⠴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠠⠄⠀⠈⠂⠄⠈⠀⠠⠐⠁⠀⠠⠄⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢂⠀⠀⠀⠀⠀⠀⢀⠀⠡⣣⡀⢀⣜⠌⠀⡀⠀⠀⠀⠀⠀⠀⡐⠄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠈⠄⠀⠀⠀⠀⠀⢈⠶⠀⠀⠑⠛⠀⠀⠶⡉⠀⠀⠀⠀⠠⠠⠁⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠘⠄⠀⠀⡠⠆⠀⠀⠔⠃⠀⠀⠈⢦⣀⠀⡰⢄⠀⠀⠠⠁⠀⠄⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⡠⠇⣒⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠐⠌⠒⠼⢄⡀⠎
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⡀⢀⠀⠀⢀⠀⠀⠀⠀⢀⣀⠂⠀⠀⠈⢄⠀⠀⠀⠀⠀⢀⠈⠐⡄
⠀⠀⠀⠀⠀⢀⠠⣋⠀⠖⠀⢰⠀⠀⠀⢀⠃⢠⠤⠒⠊⠁⠂⠑⣄⠀⠀⠈⠄⠀⠠⢈⣁⠀⠘⠓⠛⠦⠄
⠀⠀⠀⠀⠢⣕⠊⠀⠀⠠⠊⠉⢢⣀⢀⡍⠀⠈⠁⡘⠂⢀⠠⢺⠏⠀⢀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠈⠡⡄
⠀⠀⠀⠀⢀⡖⠀⠀⢰⠃⠈⢁⣠⠜⡸⠁⠀⠀⠈⠗⠊⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠁⠀⠐⡀
⠀⠀⠀⢀⠎⠀⠀⠐⠉⠀⠈⠉⠀⠁⣨⡠⠀⠲⠀⣀⠀⠀⡄⠀⠀⠀⠀⠀⠄⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⢀⠂⠀⠀⠀⠀⢄⠡⡀⠀⠀⠈⠈⢁⠴⠁⠐⠁⢩⠀⠊⠉⠢⣀⡀⠀⠀⠀⠀⠀⠳⠤⠀⡂⢭⠀⠁⠀⠰⡀
⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠈⢁⠄⢂⡆⡀⣠⠗⠠⡀⢀⠔⣀⠑⠢⢀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠠
⠀⠀⠢⠀⣁⣀⣀⣠⣤⣀⣀⣂⣀⠀⠌⠐⠈⠀⠑⠂⠁⠀⠀⠈⠁⠉⠀⠉⠂⠠⠌⣀⣀⣀⣀⣤⣄⣀⣀⡀⠊⠔

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

⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⢤⣤⡴⠗⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠛⢦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⠀⠀⠀⢰⠃⠀⠀⠱⠦⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣯⣥⣴⣶⣶⡿⠿⠛⠉⠁⣀⡀⠀⢸⡧⠀⠀⠞⠂⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠙⠿⣿⠀⢀⣀⠀⠉⠛⠀⠤⣤⣭⣙⣛⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⠁⣀⣠⣴⣾⣿⣿⡇⠀⠘⠁⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⣤⣦⣄⠀⠀⠀⠺⣿⣿⣷⣦⣤⣀⠀⠉⠛⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣴⣿⡄⠀⠀⠀⠀⡇⠀⠀⠀⠈⢿⡐⣿⣿⣿⣿⣦⣀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡿⣻⣯⣅⢢⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣉⣩⣈⡅⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠴⠰
⠀⠀⠀⠀⠈⠳⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡉⠉⠉⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⠀⠀⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣸⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠒⠔⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠠⣀⢀⡀⠀⠀⠴⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⢄⡀⠀⢀⣿⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣛⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⠤⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈⢀⡀⠀⠀⢀⣽⣿⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢄⠰⢿⣻⣿⣿⡿⡄⠀⠀⠀⢀⠀⠄⠀⠈⣄⠀⣠⡠⡁⡄⠀⠸⠟⠁⡰⠀⠀⠀⠀⠀⠀⠸⣆⡄⠤⠀⠀⠀⠚⠿⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠙⠀⠉⠉⠉⠀⠁⠀⠀⠀⠀⠱⣄⠀⠀⢸⡄⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠠⡄⠥⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⠉⠒⠢⣧⣤⣤⡤⠴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠊⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠂⡀⢀⡀
⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠒⢤⡀⠨⡈⠀⢒⣛⣠⡤⠔⠒⠂
⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡐⠁⠠⠀⠀⠀⠀⠈⠑⠤⠚⠉
⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠈
⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠂
⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠢⠀⠀⠀⠀⣠⡴⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⢸⡇⠀⠀⠀⠠⠀⠀⠀⢠⡟⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢺⡇⠀⠠⠐⠁⠈⠑⠤⣸⣇⣀⡠⢪⣄⠀⠀⠀⠀⠀⠀⢰⡟⠳⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⢄⠀⣠⡶⠓⢦⢸⡇⠀⠀⢨⡁⢀⣴⠒⡆⠀⣦⡘⢧⣀⣇⣠⣿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⠢⠦⣿⡇⠀⠂⢸⣷⠀⠀⣾⡇⢸⣿⠀⠀⠀⢸⣇⠀⢸⠁⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢀⠆⠙⢷⣄⣀⠸⣿⡀⠀⢿⣇⠈⢿⣆⣀⠔⠈⢿⣤⠇⠀⠈⠻⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠈⠉⣁⠀⠀⡀⠈⠁⠀⠈⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣟⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢸⡧⠊⣼⠂⠀⠀⡊⢸⡆⢠⡀⢠⡾⢰⣓⣠⣦⠀⣷⠀⠙⠻⣻⡟
⠀⠀⠀⠀⠀⠀⡴⠒⠂⠀⣿⠀⠀⢼⡇⣸⡏⠀⠀⠀⠉⣽⠇⢸⡇⢻⡇⠀⡰⠸⣟⠀⣿⣀⡠⢣⣿⠁
⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠛⠀⠀⠘⢀⠘⠃⠀⠀⠀⠀⠙⠦⢺⡇⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠈⢿⣦⣤⠞⠁
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢠⡾⠉⠉⠂⠀⠀⠀⢀⡴⠋⢹⡗⠦⣄⠀⠀⠾⠉⢱⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢄⣀⣼⣇⠠⠔⠁⠀⠀⠀⢸⡀⠀⣸⠃⠀⠈⠓⠦⣤⠤⠊⠀⠀⠀⣿⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⢿⡇⢶⠓⡄⣠⠄⢠⡤⢍⠉⣡⠖⢦⠀⢰⡤⠒⣦⠔⣦⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣸⡇⠈⢉⢫⡟⠀⣿⠒⠉⢸⡏⠀⣷⠀⢸⡇⢸⡟⠰⡟⠀⢀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠚⠙⢧⡠⠊⠘⠧⠀⠻⠦⠤⠘⠷⠄⠻⡧⠘⠇⠈⠃⠀⠑⠒⠁⠀⡠⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⣠⡶⣾⣟⢿⣷⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⢰⣿⡇⢿⣿⡈⠛⢿⣿⣿⣿⣿⣭⣉⣙⣉⣉⣉⣉⣤⣴⣶⣾⣿⣿⠿⠟⢛⣛⣷⣄⡀
⣰⡘⣿⣷⡜⣿⣿⣤⡀⠈⠙⠛⠿⠿⠿⣿⣿⣿⠿⠿⠿⠛⠛⢉⣡⣤⣶⣿⣿⣿⠿⠿⠿⢆⡀
⣿⣷⡙⢿⣿⣮⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⡿⢟⣛⣩⣥⣴⣶⣶⣾⣿⣿⣷⡄
⢻⣿⣿⣶⣽⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⣉⣉⣉⣉⠛⠛⠻⠿⣿⣿⡆
⠀⠹⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⣿⡄
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢀⣠⣴⣿⡿⠟⠋⠉⠉⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠁⠀⠴⠶⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠈⠛⣛⣩⣶⡶⠒⠓⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⣴⣶⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠟⠋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀⠻⠇⠀⠀⢠⣤⣤⣶⠿⢋⣿⢻⣿⣿⣿⣿⣿⣿⠃
⠀⠘⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡏⠉⠀⠀⠸⠃⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣿⣿
⠀⠀⠀⠀⠀⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠸⣇⡀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠛⠛⠛⢿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⢀⣠⣶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣰⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣰⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡿⠟⠉⠀⣠⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠉⠀⣠⣶⣿⣿⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠃⠀⠀⣼⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠃⠋⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢄⡀⢸⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⣀⠠⢚⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠉
⠀⠀⠀⠀⠀⡀⢠⡂⠀⠈⢁⢀⠐⠉⠀⣀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠠⠄⠀⢁⡨⠂⠁
⠠⠄⠂⠀⠁⠀⠀⠙⠂⠊⣀⠤⠐⠈⠁⠀⠀⠀⢀⠀⠀⠤⠠⠄⠒⠊⠀⠀⠀⠀⠀⠂⠂⠉⠘⢠
⠘⡤⡀⠀⠀⠀⢀⠄⠈⣁⠀⢀⠠⠄⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠂⠀⠀⠐⢄⠰⣅⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠢⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠂⠀⠐⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠑⡴⠂