
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 $f(x) = x^2 + ax + 3$ and $g(x) = x^2 + cx + 6$ 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 $f$. The graphs of the two polynomials intersect at the point $(50,-50)$. If $f(x)$ and $g(x)$ have minimum values of $3$ and $6$, respectively, what is the value of $a + c$?

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