
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
Let $g(x)$ be a function such that $g(0) = 2$ and
\[ g(xy) = g \left( \frac{x^2 + y^2}{2} \right) + c(x - y)^2 \]
for all real numbers $x$ and $y.$ Find $g(x)$ if $c = 3$.

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