
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
Real numbers \(x, y\) satisfy \(4x^2 - 5xy + 4y^2 = 5\). If \(S = x^2 + y^2\), let the maximum value of \(S\) be \(p\) and the minimum value of \(S\) be \(q\). Find the value of \(\frac{1}{p} + \frac{1}{q}\).

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