在直角坐标系\(xOy\)中, 动圆\(M\)与圆\({{O}_{1}}:{{x}^{2}}+2x+{{y}^{2}}=0\)外切,同时与圆\({{O}_{2}}:{{x}^{2}}+{{y}^{2}}-2x-24=0\)内切.
\((1)\)求动圆圆心\(M\)的轨迹方程;
\((2)\)设动圆圆心\(M\)的轨迹为曲线\(C\),设\(A,P\)是曲线\(C\)上两点,点\(A\)关于\(x\)轴的对称点为\(B(\)异于点\(P)\),若直线\(AP,BP\)分别交\(x\)轴于点\(S,T\),证明:\(\left| OS \right|\cdot \left| OT \right|\) 为定值.