11. Given the sequence $\left\{a_{n}\right\}$, the sum of the first $n$ terms is
$$
S_{n}=2 a_{n}-2\left(n \in \mathbf{Z}_{+}\right) \text {. }
$$
(1) Find the general term formula $a_{n}$;
(2) Let $b_{n}=\frac{1}{a_{n}}-\frac{1}{n(n+1)}, T_{n}$ be the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$, find the positive integer $k$ such that for any $n \in$ $\mathbf{Z}_{+}$, we have $T_{k} \geqslant T_{n}$;
(3) Let $c_{n}=\frac{a_{n+1}}{\left(1+a_{n}\right)\left(1+a_{n+1}\right)}, R_{n}$ be the sum of the first $n$ terms of the sequence $\left\{c_{n}\right\}$, if for any $n \in \mathbf{Z}_{+}$, we have $R_{n}<\lambda$, find the minimum value of $\lambda$.