
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 $p(x) = x^2 + px + q$ and $r(x) = x^2 + sx + t$ be two distinct polynomials with real coefficients such that the $x$-coordinate of the vertex of $p$ is a root of $r,$ and the $x$-coordinate of the vertex of $r$ is a root of $p,$ and both $p$ and $r$ have the same minimum value. If the graphs of the two polynomials intersect at the point $(50, -50),$ what is the value of $p + s$?

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