Unit square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. The point $P$ is the intersection of $EF$ and $GH$. It is given that $BF = \frac{1}{4}$ and $BF + DH = FH$. Calculate the nearest integer to the degree measure of $\angle FAH$.

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