
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 \(f(x)\) be a function such that \(f(0) = 2\) and 
\[f(xy) = f \left(\frac{x^2+y^2}{2}\right) + (x+y)^2\] for all real numbers \(x\) and \(y\). Find \(f(x)\).

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