
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]


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Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, where $a_{1} \gt 0$. It is known that $\frac{{S}_{n+1}}{{a}_{n+1}}-\frac{{S}_{n}}{{a}_{n}}=\frac{1}{2}$.  
$(1)$ If $a_{1}=1$, find the general formula for the sequence $\{a_{n}\}$.  
$(2)$ If $\frac{1}{{S}_{1}}+\frac{1}{{S}_{2}}+\cdots+\frac{1}{{S}_{n}}＜1$ holds for any $n\in \mathbb{N}^{*}$, find the range of values for $a_{1}$.

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