
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 that the sequence $\{a_n\}$, whose terms are all real numbers, is an arithmetic sequence with a common difference $d$ and the sum of its first $n$ terms is $S_n$, it satisfies $S_4 = 2S_2 + 8$.
(1) Determine the value of the common difference $d$;
(2) If the square of the first term of the sequence $\{a_n\}$ plus the sum of the rest of the terms is at most 10, find the maximum number of terms that such a sequence can have;
(3) Directly write down a sequence that satisfies condition (2) with the maximum number of terms (no calculations steps needed).

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