
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
When $n$ is a positive integer, the function $N(n)$ represents the maximum odd factor of $n$. For example, $N(3)=3$, $N(10)=5$, etc. Let $S_{n}=N(1)+N(2)+N(3)+N(4)+…+N(2^{n}-1)+N(2^{n})$, find the value of $S_{n}$.

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