
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 $3x^2 - 2xy + y^2 = 6$. Find the maximum possible value of $3x^2 + 2xy + y^2$. Express your answer in the form of $\frac{a + b \sqrt{c}}{d},$ where $a, b, c, d$ are positive integers, and find $a + b + c + d.$

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