
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
Suppose \( x \) and \( y \) are positive real numbers satisfying \( 2x^2 - xy + y^2 = 15 \). Find the maximum possible value of \( 2x^2 + xy + y^2 \), which can be expressed as \( \frac{p + q \sqrt{r}}{s} \), where \( p, q, r, s \) are positive integers. Compute \( p + q + r + s \).

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