Contrapositive and Contradiction · Hard problem

A minimal-counterexample argument proves that every integer $n \ge N$ is a sum of $5$s and $7$s (any number of each, including none): check $N, N + 1, \ldots, N + 4$ by hand, and a smallest failure $m \ge N + 5$ would make $m - 5$ a smaller failure. What is the smallest possible $N$?