
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
Given the function $f(x)=-2x^{2}+22x$, the sum of the first $n$ terms of the sequence $\{a_n\}$ is represented as $S_{n}$, and the points $p_{n}$ $(n, S_{n})$ $(n∈N_{+})$ all lie on the curve of the function $y=f(x)$.

(1) Find the general term formula $a_n$ for the sequence $\{a_n\}$.

(2) There exists $k∈N^{*}$, such that $\frac{S_{1}}{1} + \frac{S_{2}}{2} + … + \frac{S_{n}}{n} < k$ holds true for any $n∈N^{*}$, find the minimum value of $k$.

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