In an equilateral triangle $ABC$ with side length $6$, a point $P$ is chosen randomly inside the triangle. Lines are drawn from $P$ to each vertex of the triangle, forming three smaller triangles $\triangle ABP$, $\triangle BCP$, and $\triangle CAP$. What is the probability that the area of $\triangle ABP$ is greater than the area of $\triangle BCP$ and $\triangle CAP$? Let's think step by step and output the final answer within \boxed{}.
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