
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, define the function $N(n)$ to represent the largest odd factor of $n$. For example, $N(3)=3$, $N(10)=5$, etc. Let $S(n)=N(1)+N(2)+N(3)+…+N(2^{n}).$ Then, $(1)S(4)=$ \_\_\_\_\_\_ $(2)S(n)=$ \_\_\_\_\_\_ .

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