The sequence $\left\{ a _ { n } \right\}$ satisfies $a _ { 1 } = 3 , a _ { n + 1 } = 3 a _ { n } - 4 n$ .
(1) Calculate $a _ { 2 } , a _ { 3 }$ , conjecture the general term formula of $\left\{ a _ { n } \right\}$ and prove it;
(2) Find the sum $S _ { n }$ of the first $n$ terms of the sequence $\left\{ 2 ^ { n } a _ { n } \right\}$ .