Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers $1$ through $12$. Each throw hits the target in a region with a different value, and all regions hit across all players have different values. The scores are: Alex $16$ points, Briana $6$ points, Chris $9$ points, David $11$ points, and Emily $13$ points. Who hits the region worth $4$ points?