What Is a Permutation?

A permutation $a_1, a_2, \dots, a_{13}$ of the numbers from $1$ to $13$ is given such that $a_i > 5$ for $i=1,2,3,4,5$. Determine the maximum possible value of $$a_{a_1} + a_{a_2} + a_{a_3} + a_{a_4} + a_{a_5}.$$

Source: 2012-2013 Winter Online Math Open