
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 geometric sequence $\{a_n\}$ with a common ratio $q > 1$, the product of the third, fifth, and seventh terms is $512$, and these three terms, respectively subtracting $1, 3, 9$, form an arithmetic sequence.  
$(1)$ Find the first term and the common ratio of $\{a_n\}$.  
$(2)$ Let $S_n = a_1^2 + a_2^2 + \ldots + a_n^2$, find $S_n$.

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