
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 with g(x) = -f(120-x). The graph of g contains the vertex of the graph of f. The four x-coordinates of their intercepts are x_1, x_2, x_3, and x_4 (in increasing order), and it's given that x_3 - x_2 = 180. Find x_4 - x_1 = m + n√p where m, n, and p are positive integers, and p is not divisible by the square of any prime. What is m + n + p?

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