Let $\triangle ABC$ be a triangle with points $D$, $M$, and $N$ on sides $BC$, $AB$, and $AC$, respectively. Suppose that $MD$ is parallel to $AC$ and $ND$ is parallel to $AB$. Given that the area of $\triangle BMD$ is $9\, \text{cm}^2$ and the area of $\triangle DNC$ is $25\, \text{cm}^2$, compute the area of $\triangle AMN$.

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