
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 \( f : \mathbb{R} \to \mathbb{R} \) such that
\[ f(x + y) f(x - y) = c(f(x) + y^2)^2 - cx^2 y^2 \]
for all real numbers \( x \) and \( y \), where \( c \) is a nonzero real number.

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