
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 functions $f$ and $g$ be defined such that $g(x) = -f(200 - x)$. It is given that the vertex of the graph of $g$ is on the graph of $f$. If the four $x$-intercepts of the combined graphs of $f$ and $g$ are at $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 200$, determine $x_4 - x_1$ which expresses as $m + n\sqrt{p}$, where $m$, $n$, and $p$ are integers, and $p$ is not divisible by the square of any prime. Find $m + n + p$.
A) 1720
B) 1750
C) 1802
D) 1803
E) 1823

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