7.
如图,在长方体\(ABCD-A_{1}B_{1}C_{1}D_{1}\)中,\(AB=\sqrt{3}AD=\sqrt{3}A{{A}_{1}}=\sqrt{3}\),点\(P\)为线段\(A_{1}C\)上的动点\((\)包含线段端点\()\),给出下列结论:
\(①\)当\(\overrightarrow{{{A}_{1}}C}=3\overrightarrow{{{A}_{1}}P}\)时,\(D_{1}P/\!/\)平面\(BDC_{1}\);
\(②\)当\(\overrightarrow{{{A}_{1}}C}=5\overrightarrow{{{A}_{1}}P}\)时,\(A_{1}C⊥\)平面\(D_{1}AP\);
\(③∠APD_{1}\)的最大值为\(90^{\circ}\);
\(④AP+PD_{1}\)的最小值为\(\sqrt{5}\).
其中正确的结论为________\(.(\)请填写所有正确结论的序号\()\)