
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 function \( f(x) \) is defined on the positive half-line and takes only positive values. It is known that \( f(1) + f(2) = 10 \) and \( f(a+b) = f(a) + f(b) + 2 \sqrt{f(a) f(b)} \) for any \( a \) and \( b \). Find \( f\left(2^{2011}\right) \).

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