
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
The real function \(f\) has the property that, whenever \(a,\) \(b,\) \(n\) are positive integers such that \(a + b = 2^n,\) the equation
\[f(a) + f(b) = n^2\] holds. What is \(f(1002)\)?

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