
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
Given functions f and g are quadratic, g(x) = -f(90 - x), and the graph of g contains the vertex of the graph of f. The four x-intercepts on the two graphs have x-coordinates x1, x2, x3, and x4, in increasing order, and x3 - x2 = 180. Find x4 - x1 = m + n√p, where m, n, and p are positive integers, and p is not divisible by the square of any prime. Calculate m + n + p.

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