
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 a quadratic equation \(x^{2} + px + q = 0\), with roots \(x_{1}\) and \(x_{2}\). Find the equations for which the roots would be:

1. \(x_{1} + \frac{1}{x_{1}}\) and \(x_{2} + \frac{1}{x_{2}}\)
2. \(\frac{x_{1}}{x_{2}}\) and \(\frac{x_{2}}{x_{1}}\)
3. Roots that are \(\frac{p}{2}\) greater than the roots of the given equation.

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