
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 the sequence \( \{a_{n}\} \) have the sum of its first \( n \) terms denoted by \( S_{n} \). It is given that \( S_{2} = 3 \) and \( 2S_{n} = n + n a_{n} \) for \( n \in \mathbf{N}^{*} \).

(1) Find the general term of the sequence \( \{a_{n}\} \).

(2) For the sequence \( c_{n} \) defined as
\[ c_{n} = 
\begin{cases} 
a_{n+1}, & n \text{ is odd}, \\
3 \times 2^{a_{n-1}} + 1, & n \text{ is even}
\end{cases}
\]
find the sum of the first \( 2n \) terms, denoted by \( T_{2n} \).

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