已知椭圆\(C\):\(\dfrac{{{x}^{2}}}{{{a}^{2}}}+\dfrac{{{y}^{2}}}{{{b}^{2}}}=1(\)\(a\)\( > \)\(b\)\( > 0)\)的离心率为\(\dfrac{\sqrt{2}}{2}\),长轴长等于圆\(R\):\(x\)\({\,\!}^{2}+(\)\(y\)\(-2)^{2}=4\)的直径,过点\(P\)\((0,1)\)的直线\(l\)与椭圆\(C\)交于两点\(A\),\(B\)\(.\)与圆\(R\)交于两点\(M\),\(N\).
\((1)\)求椭圆\(C\)的方程
\((2)\)求证:直线\(R\)\(A\),\(R\)\(B\)的斜率之和等于零;
\((3)\)求\(|\)\(AB\)\(|·|\)\(MN\)\(|\)的取值范围.