
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 \( p(x) = a x^3 + b x^2 + c x + d \) is a cubic polynomial, and it satisfies \( p\left( \frac{1}{2} \right) + p\left( -\frac{1}{2} \right) = 1000 p(0) \). Let \( x_1, x_2, x_3 \) be the three roots of \( p(x) = 0 \). Find the value of \( \frac{1}{x_1 x_2} + \frac{1}{x_2 x_3} + \frac{1}{x_1 x_3} \).

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