Question 71
Let $ABCD$ be a parallelogram and let $\angle BAD$ be acute. Denote by $E$ and $F$ the feet of the perpendiculars from the vertex $C$ to the lines $AB$ and $AD$, respectively. A circle through $D$ and $F$ is tangent to the diagonal $AC$ at a point $Q$ and a circle through $B$ and $E$ is tangent to the segment $QC$ at its midpoint $P$. Find the length of diagonal $AC$ if $AQ=1$.