11. (14 points) Let the sequence $\left\{a_{n}\right\}$ have the sum of the first $n$ terms as $S_{n}$, and satisfy
$$
S_{2}=3,2 S_{n}=n+n a_{n}\left(n \in \mathbf{N}_{+}\right) \text {. }
$$

Find: (1) The general term formula of the sequence $\left\{a_{n}\right\}$;
(2) The sum of the first $2 n$ terms $T_{2 n}$ of the sequence
$$
c_{n}=\left\{\begin{array}{ll}
a_{n+1}, & n \text { is odd; } \\
3 \times 2^{a_{n-1}}+1, & n \text { is even }
\end{array}\right.
$$
11. (1) From the problem, we know that when $n=1$, $a_{1}=1$.

When $n \geqslant 2$,
$$
\begin{array}{l}
2 S_{n}=n+n a_{n}, \\
2 S_{n-1}=n-1+(n-1) a_{n-1} .
\end{array}
$$

Subtracting the above two equations gives
$$
2 a_{n}=1+n a_{n}-(n-1) a_{n-1} \text {. }
$$

Then $2 a_{n+1}=1+(n+1) a_{n+1}-n a_{n}$.
Subtracting (1) from (2) gives
$$
\begin{array}{l}
(n-1) a_{n+1}+(n-1) a_{n-1} \\
=2(n-1) a_{n} .
\end{array}
$$

That is, when $n \geqslant 2$,
$$
a_{n+1}+a_{n-1}=2 a_{n} \text {. }
$$

Also, since $S_{2}=3, a_{1}=1$, we know $a_{2}=2$.
Therefore, $\left\{a_{n}\right\}$ is an arithmetic sequence with the first term 1 and common difference 1.
Hence $a_{n}=n\left(n \in \mathbf{N}_{+}\right)$.
$$
\begin{aligned}
\text { (2) } T_{2 n}= & \left(a_{2}+a_{4}+\cdots+a_{2 n}\right)+ \\
& 3\left(2^{1}+2^{3}+\cdots+2^{2 n-1}\right)+n \\
= & n(n+1)+2^{2 n+1}-2+n \\
= & 2^{2 n+1}+n^{2}+2 n-2
\end{aligned}
$$