
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 that the sequence $\{a_{n}\}$ is an arithmetic sequence with a non-zero common difference, $a_{2}=5$, and $a_{4}$ is the geometric mean of $a_{1}$ and $a_{13}$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, find the sum of the first $n$ terms of the sequence $\{\frac{1}{{S}_{n}}\}$.

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