
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 \( f(x) = x^2 + 2x + 1 \). Let \( g(x) = f(f(\cdots f(x))) \), where there are 2009 f's in the expression for \( g(x) \). Then \( g(x) \) can be written as
\[ 
g(x) = x^{2^{2009}} + a_{2^{2009}-1} x^{2^{2009}-1} + \cdots + a_1 x + a_0 
\]
where the \( a_i \) are constants. Compute \( a_{2^{2009}-1} \).

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