
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
The sequence of positive integers $\{f(n)\}$ satisfies: $f(2)=15$ and
$$
\begin{aligned}
2 +\frac{4}{f(n)+1}<\frac{f(n)}{f(n)-4 n+2}+\frac{f(n)}{f(n+1)-4 n-2}<2+\frac{4}{f(n)-1},
\end{aligned}
$$
Calculate $S(n)= \frac{1}{f(1)}+\frac{1}{f(2)}+\cdots+\frac{1}{f(n)}.$

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