Consider an ideal I ⊆ K [x, y, z] corresponding to a point configuration in P 2 where all but one of the points lies on a single line. In this paper we study the symbolic generic initial system {gin(I (m) )}m obtained by taking the reverse lexicographic generic initial ideals of the uniform fat point ideals I (m) . We describe the limiting shape of {gin(I (m) )}m and, in proving this result, demonstrate that infinitely many of the ideals I (m) are componentwise linear. arXiv:1304.7541v1 [math.AC]