
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
If for a positive integer $k$, $g(k)$ represents the largest odd factor of $k$, for example, $g(3)=3$, $g(10)=5$. Let $S_{n}=g(1)+g(2)+g(3)+g(4)+\ldots +g(2^{n})$.

(I) Find the values of $g(6)$ and $g(20)$.

(II) Find the values of $S_{1}$, $S_{2}$, $S_{3}$.

(III) Find the general formula for the sequence $\{S_{n}\}$.

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