
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 $p$ and $q$ be fixed, and let $z_{1}, z_{2}$, and $z_{3}$ be the three roots of the polynomial $X^{3}+pX+q$. Calculate $\frac{1}{z_{1}^{2}}+\frac{1}{z_{2}^{2}}+\frac{1}{z_{3}^{2}}$ in terms of $p$ and $q$.

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