In cube ABCD_ In a1b1c1d1, if the edge length is a, P is the midpoint of edge Aa1, and Q is any point on edge BB1, then the minimum value of PQ + QC is?

In cube ABCD_ In a1b1c1d1, if the edge length is a, P is the midpoint of edge Aa1, and Q is any point on edge BB1, then the minimum value of PQ + QC is?

Expand aa'b'b and bb'c'c into a plane
When p, Q and C are collinear, PQ + QC = PC is the minimum
PC=√(AP²+AC²)=√17·a/2

In the cuboid abcd-a1b1c1d1, e and F are the midpoint of B1B and A1B1 respectively. Verification: the lines EF and b1d1 are out of plane lines
I'm not good at writing the steps of solving this problem. What's the train of thought!

Proof 1: line b1d1 intersects plane abb1a1 at point B1, EF is in plane abb1a1, and B1 is not on EF,
The b1d1 and EF are straight lines out of plane
Proof 2: the straight line EF and plane bdd1b1 intersect at point E, but e is not on b1d1, EF and b1d1 are non planar straight lines
Judgment basis: a straight line intersects with a plane, and the straight line intersecting with the plane is a non planar straight line