\[
f(x)=2 \cos (3 x+2)
\]
approximate (to the nearest thousandth) the solution of
\[
f(x)=-0.8
\]
on the interval $\left[-\frac{2}{3}, \frac{\pi-2}{3}\right]$.

Step 1 ：We are given the function \(f(x)=2 \cos (3 x+2)\) and we are asked to approximate the solution of \(f(x)=-0.8\) on the interval \(\left[-\frac{2}{3}, \frac{\pi-2}{3}\right]\).

Step 2 ：To solve this problem, we need to set the function equal to -0.8 and solve for x. Since the function involves cosine, we will likely need to use a numerical method to find the solution.

Step 3 ：We can use the bisection method, which is a root-finding method that applies to any continuous function for which one knows two values with opposite signs. The method consists of repeatedly bisecting the interval and then selecting the subinterval in which the function changes sign, and therefore must contain a root.

Step 4 ：We will use the interval given in the question as the initial interval for the bisection method. The initial interval is \(\left[-\frac{2}{3}, \frac{\pi-2}{3}\right]\).

Step 5 ：By applying the bisection method, we find that the root of the function on the given interval is approximately -0.006.

Step 6 ：Final Answer: The solution to the equation \(f(x) = -0.8\) on the interval \(\left[-\frac{2}{3}, \frac{\pi-2}{3}\right]\) is approximately \(\boxed{-0.006}\).