Example $\mathbf{3}$ Given the Fibonacci sequence $\left\{a_{n}\right\}$ satisfies
$$
\left\{\begin{array}{l}
a_{1}=a_{2}=1, \\
a_{n}=a_{n-1}+a_{n-2}(n \geqslant 3) .
\end{array}\right.
$$

Find the general term formula $a_{n}$ of the sequence $\left\{a_{n}\right\}$.