
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 the sequence $\{a_n\}$. If $\frac{S_{2n}}{S_n}$ is a non-zero constant, then the sequence is called a "sum geometric sequence".

(1) If the sequence $\{2^{b_n}\}$ is a geometric sequence with the first term as $2$ and the common ratio as $4$, determine whether the sequence $\{b_n\}$ is a "sum geometric sequence".

(2) If the sequence $\{c_n\}$ is an arithmetic sequence with the first term as $c_1$ and the common difference as $d (d \neq 0)$, and the sequence $\{c_n\}$ is a "sum geometric sequence", explore the relationship between $c_1$ and $d$.

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