$h$ is a trigonometric function of the form $h(x)=a \cos (b x+c)+d$. Below is the graph of $h(x)$. The function has a maximum point at $(-2 \pi, 3)$ and a minimum point at $\left(-\frac{\pi}{2},-4\right)$. Find a formula for $h(x)$. Give an exact expression.

Please directly answer the question and provide the final value, e.g., 1, 2.5, 300.
Question: $h$ is a trigonometric function of the form $h(x)=a \cos (b x+c)+d$. Below is the graph of $h(x)$. The function has a maximum point at $(-2 \pi, 3)$ and a minimum point at $\left(-\frac{\pi}{2},-4\right)$. Find a formula for $h(x)$. Give an exact expression.

Please first conduct reasoning, and then answer the question and provide the final value, e.g., 1, 2.5, 300, at the end.
Question: $h$ is a trigonometric function of the form $h(x)=a \cos (b x+c)+d$. Below is the graph of $h(x)$. The function has a maximum point at $(-2 \pi, 3)$ and a minimum point at $\left(-\frac{\pi}{2},-4\right)$. Find a formula for $h(x)$. Give an exact expression.
