
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 $a_n$ denote the largest odd factor of the natural number $n$. For example, the factors of $9$ are $1$, $3$, and $9$, so $a_9 = 9$; the factors of $10$ are $1$, $2$, $5$, and $10$, so $a_{10} = 5$. Let $S_n$ denote the sum of the first $n$ terms of the sequence ${a_n}$. Find $S_{2^{2016} - 1}$.

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