
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 sequence $\{a_n\}$ that satisfies $a_1=a$, and $$a_{n+1}= \frac{(4n+6)a_{n}+4n+10}{2n+1},$$ for $n \in \mathbb{N}^*$.
(1) Determine if the sequence $\left\{ \frac{a_{n}+2}{2n+1} \right\}$ is a geometric sequence. If not, please explain why; if it is, find the general term $a_n$.
(2) If $a=1$, and the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n$, find $S_n$.

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