设数列\(\{a_{n}\}\)的前\(n\)项和为\(S_{n}\)，已知\(a_{1}=1\)，\(S_{n}-a_{n+1}=-1.\)
\((1)\)求\(\{a_{n}\}\)通项公式；
\((2)\)对任意的正整数\(n\)，设\(c_{n}=\left\{\begin{array}{ll}\dfrac{2}{lo{g}_{2}{a}_{n+1}⋅lo{g}_{2}{a}_{n+3}},n为奇数\\ \dfrac{lo{g}_{2}{a}_{n}}{{a}_{n+1}},n为偶数\end{array}\right.\)，求数列\(\{c_{n}\}\)的前\(2n\)项和．