Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $f$ is a root of $g,$ and the $x$-coordinate of the vertex of $g$ is a root of $f,$ and both $f$ and $g$ have the same minimum value.  If the graphs of the two polynomials intersect at the point $(100,-100),$ what is the value of $a + c$?
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