Equilateral triangle $ABC$ has an area of $1$. Points $A'$, $B'$, and $C'$ are the midpoints of sides $BC$, $CA$, and $AB$, respectively. Points $A''$, $B''$, and $C''$ are the midpoints of segments $B'C'$, $C'A'$, and $A'B'$, respectively. The area of trapezoid $BB''C''C$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Determine the value of $m+n$.

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