如图，四棱柱\(ABCD-A _{1} B _{1} C _{1} D _{1}\)的底面\(ABCD\)是正方形，\(O\)为底面中心，\(A _{1} O⊥\)平面\(ABCD\)，\(AB=AA _{1} = \sqrt {2}\)．
\((\)Ⅰ\()\) 证明：平面\(A _{1} BD/\!/\)平面\(CD _{1} B _{1}\)；
\((\)Ⅱ\()\) 求三棱柱\(ABD-A _{1} B _{1} D _{1}\)的体积．