In $\triangle PAT,$ $\angle P=36^{\circ},$ $\angle A=56^{\circ},$ and $PA=10.$ Points $U$ and $G$ lie on sides $\overline{TP}$ and $\overline{TA},$ respectively, so that $PU=AG=1.$ Let $M$ and $N$ be the midpoints of segments $\overline{PA}$ and $\overline{UG},$ respectively. If the answer is $x^\circ$, what is the value of $x$ for the degree measure of the acute angle formed by lines $MN$ and $PA?$

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