
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 an arithmetic sequence $\{a_{n}\}$ with a non-zero common difference, it is known that $a_{1}=1$ and $a_{2}$ is the geometric mean of $a_{1}$ and $a_{5}$.
(I) Find the general term formula for the sequence $\{a_{n}\}$;
(II) Let $b_{n}=2^{a_{n}}$. Determine if the sequence $\{b_{n}\}$ is a geometric sequence. If it is, find the sum of the first $n$ terms, $S_{n}$. If not, provide the reason.

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