
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
Functions \( f \) and \( g \) are quadratic, with \( g(x) = -f(120 - x) \), and the graph of \( g \) contains the vertex of the graph of \( f \). The four \( x \)-intercepts on the two graphs have \( x \)-coordinates \( x_1, x_2, x_3, \) and \( x_4 \), in increasing order, and \( x_3 - x_2 = 120 \). Compute \( x_4 - x_1 \), where the resulting expression involves square roots and integers.

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