
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 $f$ and $g$ be quadratic functions where $g(x) = -f(120 - x)$. The graph of $g$ contains the vertex of the graph of $f$. Given that the four x-intercepts on the two graphs are $x_1$, $x_2$, $x_3$, and $x_4$, arranged in increasing order, and that $x_3 - x_2 = 180$, determine $x_4 - x_1$ and express it in the form $m + n\sqrt{p}$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime.
A) 800
B) 850
C) 900
D) 902
E) 950

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