Grouping, Partitions, and Coloring

Let $S$ be the set of all lattice points $(x, y)$ in the plane satisfying $|x|+|y|\le 10$. Let $P_1,P_2,\ldots,P_{2013}$ be a sequence of 2013 (not necessarily distinct) points such that for every point $Q$ in $S$, there exists at least one index $i$ such that $1\le i\le 2013$ and $P_i = Q$. Suppose that the minimum possible value of $|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|$ can be expressed in the form $a+b\sqrt{c}$, where $a,b,c$ are positive integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. (A lattice point is a point with all integer coordinates.)

Source: 2012-2013 Winter Online Math Open