如图，在底面为平行四边形的四棱柱\(ABCD-A_{1}B_{1}C_{1}D_{1}\)中，\(D_{1}D⊥\)底面\(ABCD\)，\(AD=1\)，\(CD=2\)，\(∠DCB=60^{\circ}\)．
\((\)Ⅰ\()\) 求证：平面\(A_{1}BCD_{1}⊥\)平面\(BDD_{1}B_{1}\)；
\((\)Ⅱ\()\)若\(D_{1}D=BD\)，求四棱锥\(D-A_{1}BCD_{1}\)的体积．