
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 \( k \) be a positive integer, and \( g(k) \) represent the greatest odd divisor of \( k \) (for example, \( g(3) = 3 \) and \( g(20) = 5 \)). Find the sum \( g(1) + g(2) + g(3) + \cdots + g(2^n) \) where \( n \) is a positive integer.

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