Consider a non-empty set of segments of length 1 in the plane which do not intersect except at their endpoints. (In other words, if point $P$ lies on distinct segments $a$ and $b$, then $P$ is an endpoint of both $a$ and $b$.) This set is called $3-amazing$ if each endpoint of a segment is the endpoint of exactly three segments in the set. Find the smallest possible size of a 3-amazing set of segments.