
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
Let $\{a_{n}\}$ be an arithmetic sequence with a common difference not equal to $0$, and let $S_{n}$ denote the sum of the first $n$ terms. If $S_{3}=9$ and $a_{2}$, $a_{5}$, $a_{14}$ form a geometric sequence, find:<br/>
$(1)$ the general formula for the sequence $\{a_{n}\}$;<br/>
$(2)$ the maximum value of the positive integer $n$ that satisfies $(1-\frac{1}{{S}_{2}})(1-\frac{1}{{S}_{3}})\ldots (1-\frac{1}{{S}_{n}})\geqslant \frac{1013}{2022}$, where $n\in \mathbb{N}^{*}$ and $n\geqslant 2$.

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