2023 Pascal Problem 24

Cube ABCDEFGH has edge length 100. Point P is on AB, point Q is on AD, and point R is on AF, as shown, so that AP = x, AQ = x+1 and AR = \dfrac{x+1}{2x} for some integer x. For how many integers x is the volume of triangular-based pyramid APQR between 0.04\% and 0.08\% of the volume of cube ABCDEFGH? (The volume of a pyramid is equal to one-third of the area of its base times its height.)

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Full credit to this problem is given to the CEMC, you may view all Pascal contests here.