In $\triangle ABC$ with side lengths $AB = 13,$ $AC = 12,$ and $BC = 5,$ let $O$ and $I$ denote the circumcenter and incenter, respectively. A circle with center $M$ is tangent to the legs $AC$ and $BC$ and to the circumcircle of $\triangle ABC.$ What is the area of $\triangle MOI$ ?

$\dfrac{11}{4}$
$\dfrac{7}{2}$
3
$\dfrac{5}{2}$
$\dfrac{13}{4}$

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