
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 \( g(x) = 2f(200-x) \) and the graph of \( g \) contains the vertex of \( f \). Assume the four \( x \)-intercepts of the graphs of \( f \) and \( g \) are \( x_1, x_2, x_3, x_4 \) in increasing order, and \( x_3 - x_2 = 300 \). Determine \( x_4 - x_1 = m \sqrt{p} \), where \( m \) and \( p \) are integers, and \( p \) is not divisible by the square of any prime.
A) \( 150 \)
B) \( 300 \)
C) \( 450 \)
D) \( 600 \)

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