
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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