
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 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 $f$. Both $f$ and $g$ intersect at the point $(50, -50)$, and the minimum value of $f$ is the same as $g$. Determine the value of $a + c$.

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