In the first example, the strings initially have the following configuration:

\begin{center}
    \includegraphics[scale=0.7]{test.png}
\end{center}

Since strings 3 and 4 already intersect, no operations are needed to achieve beauty at least $1$ and beauty at least $2$.

To achieve beauties $3$ and $4$, you can perform a \textit{restringing} operation on strings 2 and 5 to obtain strings $(4, 29)$ and $(5, 15)$.

\begin{center}
    \includegraphics[scale=0.7]{test_one.png}
\end{center}

To achieve beauty $5$, you can additionally perform an operation on strings 3 and 6 to obtain $(7, 16)$ and $(12, 23)$.

\begin{center}
    \includegraphics[scale=0.7]{test_two.png}
\end{center}

To make all 6 strings intersect, you can perform operations on strings 1 and 5, 2 and 3, 4 and 6 to obtain the following configuration:

\begin{center}
    \includegraphics[scale=0.7]{test_three.png}
\end{center}
