
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 $S_{n}$ be the sum of the first $n$ terms of a geometric sequence $\{a_{n}\}$ with a common ratio $q \gt 1$, and $a_{2}+a_{3}+a_{4}=39$, $a_{3}+6$ is the arithmetic mean of $a_{2}$ and $a_{4}$.  
$(1)$ Find $S_{n}$;  
$(2)$ Determine if there exists a constant $\lambda$ such that the sequence $\{S_{n}+\lambda\}$ forms a geometric sequence. If it exists, find the value of $\lambda$; if not, explain the reason.

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