如图,在四棱柱\(ABCD-A_{1}B_{1}C_{1}D_{1}\)中,侧棱\(AA_{1}⊥\)底面\(ABCD\),地面\(ABCD\)为梯形,\(AD/\!/BC\),\(AB=DC= \sqrt {2}\),\(AD=AA_{1}= \dfrac {1}{2}BC=2\),点\(P\),\(Q\)分别为\(A_{1}D_{1}\),\(AD\)的中点.
\((\)Ⅰ\()\)求证:\(CQ/\!/\)平面\(PAC_{1}\);
\((\)Ⅱ\()\)求二面角\(C_{1}-AP-D\)的余弦值;
\((\)Ⅲ\()\)在线段\(BC\)上是否存在点\(E\),使\(PE\)与平面\(PAC_{1}\)所成角的正弦值是\( \dfrac {2 \sqrt {14}}{21}\),若存在,求\(BE\)的长;若不存在,请说明理由.