Let $ABCDE$ be a convex pentagon that satisfies all of the following: There is a circle $\Gamma$ tangent to each of the sides. The lengths of the sides are all positive integers. At least one of the sides of the pentagon has length $1$. The side $AB$ has length $2$. Let $P$ be the point of tangency of $\Gamma$ with $AB$. (a) Determine the lengths of the segments $AP$ and $BP$. (b) Give an example of a pentagon satisfying the given conditions.