设数列\(\left\{\begin{array}{l}a_{n}\end{array}\right\}\)的前\(n\)项和为\(S_{n}\)，已知\(a_{1}=4\)，\(S_{n}=a_{n+1}+2n-4\)，\(n\in N^{*}.\)

\((1)\)求数列\(\left\{\begin{array}{l}a_{n}\end{array}\right\}\)的通项公式；

\((2)\)设\(b_{n}=\dfrac{a_{n}-2}{\left({2^{n}+1}\right)\left({2^{n+1}+1}\right)}\)，数列\(\left\{\begin{array}{l}b_{n}\end{array}\right\}\)的前\(n\)项和为\(T_{n}\)，求满足\(T_{n}>\dfrac{13}{40}\)的正整数\(n\)的最小值．