
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 that $x$ and $y$ are positive real numbers such that $x^2 - 3xy + 5y^2 = 9$. Find the maximum possible value of $x^2 + 3xy + 5y^2$ and express it in the simplest form as $\frac{a + b \sqrt{c}}{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers. Find $a + b + c + d.$

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