
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
Find the number of functions \( g : \mathbb{R} \to \mathbb{R} \) such that
\[ g(x + y) g(x - y) = (g(x) + g(y))^2 - 4x^2 g(y) + 2y^2 g(x) \]
for all real numbers \(x\) and \(y\).

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