设\(\{a _{n} \}\)是等差数列，\(\{b _{n} \}\)是等比数列．已知\(a _{1} =1\)，\(b _{1} =2\)，\(b _{2} =2a _{2}\)，\(b _{3} =2a _{3} +2\)．
\((1)\)求\(\{a _{n} \}\)和\(\{b _{n} \}\)的通项公式；
\((2)\)数列\(\{c _{n} \}\)满足\(c _{n} = \begin{cases} {1,n=2^{k}} \\ {a_{n},n\neq 2^{k}}\end{cases} (k∈N)\)，设数列\(\{c _{n} \}\)的前\(n\)项和为\(S _{n}\)，求\(S _{2^{n}}\)．