Two towns, $P$ and $Q$, are connected by a straight road, $21$ miles long. Traveling from town $P$ to town $Q$, the speed limit changes every $7$ miles: from $30$ to $50$ to $20$ miles per hour (mph). Two cars, one at town $P$ and one at town $Q$, start moving toward each other at the same time. They drive at exactly the speed limit in each portion of the road. How far from town $P$, in miles, will the two cars meet?