
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
In the sequence $\left\{a_n\right\}$, each term $a_n$ represents the largest odd divisor of the positive integer $n$. For example, the divisors of 5 are $1$ and $5$, so $a_5 = 5$. The divisors of 6 are $1, 2, 3, 6$, so $a_6 = 3$. 

Let $S_n$ be the sum of the first $n$ terms of the sequence $\left\{a_n\right\}$. What is $S_{2^{2000}-1}$?

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