The diameter $[AC]$ of a circle is divided into four equal segments by points $P$, $M$, and $Q$. Consider a segment $[BD]$ that passes through $P$ and intersects the circle at points $B$ and $D$, such that $PD = \frac{3}{2} AP$. Given that the area of triangle $[ABP]$ is $1$ cm$^2$, calculate the area of quadrilateral $[ABCD]$.

Let's think step by step and output the final answer within \boxed{}.
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