Example 6 (Question from the 10th "Hope Cup" Invitational Competition) The graph of the function $f(x)=\sin 2x$ is shifted to the left by $\frac{\pi}{3}$ units, and then the horizontal coordinates of each point on the resulting graph are compressed to $\frac{1}{2}$ of their original values. The function expression of the resulting graph is $\qquad$.
Solve: Fill in $\sin \left(4 x+\frac{2 \pi}{3}\right)$. Reason: The graph of the function $f(x)=\sin 2 x$ is shifted to the left by $\frac{\pi}{3}$ units, and the function expression of the resulting graph is $\sin 2\left(x+\frac{\pi}{3}\right)=\sin \left(2 x+\frac{2}{3} \pi\right)$. After compressing the horizontal coordinates by half, the function expression of the resulting graph is $\sin \left(4 x+\frac{2 \pi}{3}\right)$.