
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 \( x \) and \( y \) satisfy the equation \( 4x^{2}-5xy+4y^{2}=5 \). Let \( S = x^{2} + y^{2} \). Find the value of \( \frac{1}{S_{\text{max}}} + \frac{1}{S_{\text{min}}} \).

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