已知数列\(\left\{ {{a}_{n}} \right\}\)满足：\({a}_{1}=1,{a}_{n+1}=2{a}_{n}+1 \)．
\((1)\)求证：数列\(\left\{{a}_{n}+1\right\} \)是等比数列；并求数列\(\left\{ {{a}_{n}} \right\}\)的通项公式；
\((2)\)设\({c}_{n}= \dfrac{{a}_{n}+1}{n\left(n+1\right){2}^{n}} \)，求数列\(\left\{{c}_{n}\right\} \)的前\(n\)项和\({T}_{n} \)的取值范围．