Example 5 Express $\frac{107}{95}$ as a continued fraction.
$$\begin{aligned}
\frac{107}{95} & =1+\frac{12}{95}=1+\frac{1}{\frac{95}{12}}=1+\frac{1}{7+\frac{11}{12}} \\
& =1+\frac{1}{7+\frac{1}{1+\frac{1}{11}}}=[1,7,1,11]
\end{aligned}$$

We have
$$\begin{aligned}
\frac{107}{95} & =1+\frac{12}{95}=1+\frac{1}{\frac{95}{12}}=1+\frac{1}{7+\frac{11}{12}} \\
& =1+\frac{1}{7+\frac{1}{1+\frac{1}{11}}}=[1,7,1,11]
\end{aligned}$$