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Math·Geometry and Trigonometry·Circles

hard

The circle shown has center $Q$, and points $B$, $C$, and $D$ lie on the circle. Line $m$ is tangent to the circle at point $C$, and line $m$ is parallel to line segment $BD$ (not shown). Line $DQ$ (not shown) intersects line $m$ at point $A$, and line $BQ$ (not shown) intersects line $m$ at point $P$. If the length of line segment $AQ$ is $90$ and the length of line segment $AC$ is $72$, what is the perimeter of triangle $CPQ$?

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