
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
In an arithmetic sequence $\{a_{n}\}$, all terms are positive, $a_{1}=1$, and the sum of the first $n$ terms is $S_{n}$. In a geometric sequence $\{b_{n}\}$, $b_{1}=1$, and $b_{2}S_{2}=6$, $b_{2}+S_{3}=8$. <br/>$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$; <br/>$(2)$ Find $\frac{1}{{{S_1}}}+\frac{1}{{{S_2}}}+…+\frac{1}{{{S_n}}$.

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