
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two distinct quadratic polynomials with real coefficients. It is given 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 $f$. If both polynomials have the same minimum value and intersect at the point $(50, -200)$, find the value of $a + c$.

Present the answer in LaTex format: \boxed{Your answer}
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