Given a triangle $ABC$ with an area of $1$. Let $A'$, $B'$, and $C'$ be the midpoints of the sides $[BC]$, $[CA]$, and $[AB]$ respectively. What is the minimal possible area of the common part of the two triangles $A'B'C'$ and $KLM$, if the points $K$, $L$, and $M$ are lying on the segments $[AB']$, $[CA']$, and $[BC']$ respectively?

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