
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 \( x_{1}, x_{2} \) be the roots of the equation \( x^{2} + px + q = 0 \). Express the following expressions in terms of \( p \) and \( q \):

a) \( \frac{1}{x_{1}} + \frac{1}{x_{2}} \)

b) \( \frac{1}{x_{1}^{2}} + \frac{1}{x_{2}^{2}} \)

c) \( x_{1}^{3} + x_{2}^{3} \)

d) \( \frac{1}{\left( x_{1} + p \right)^{2}} + \frac{1}{\left( x_{2} + p \right)^{2}} \)

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