The exact value of $\cot\left(\frac{7\pi}{3}\right)$ is $\frac{\sqrt{3}}{3}$.

Thus, the expression $g \cdot \cot\left(\frac{7\pi}{3}\right)$ becomes:

$g \cdot \frac{\sqrt{3}}{3}$

Would you like further clarification or more details on the steps? Here are some related questions you might explore:

1. How is the cotangent function defined in terms of sine and cosine?
2. What is the period of the cotangent function, and why does it affect the angle $\frac{7\pi}{3}$?
3. Can we simplify this result further depending on the value of $g$?
4. What are the key differences between cotangent, tangent, and their reciprocals?
5. How does this solution change if we use degrees instead of radians?

Tip: When dealing with trigonometric functions like cotangent, it's useful to remember their periodic nature to simplify large angles.