
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 a monotonically decreasing geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms being $S_{n}$, ${a_1}+{a_3}=\frac{5}{8}$, $S_{3}=7a_{3}$.  
$(1)$ Find the general term formula for the sequence $\{a_{n}\}$;  
$(2)$ Find all positive integers $n$ that satisfy ${S_n}≤\frac{{999}}{{1000}}$.

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