Working with Wolfram Alpha again, requiring an upgrade to Pro Premium this time, leads to the set of values for x, y and z (using x, y and z instead of a, b and c) (obviously order doesn't matter) being 1, 1-sqrt(2) and 1+sqrt(2). Wolfram doesn't specifically tie the solutions for x, y and z together, but it seems the first solution given for x goes with the first solution for y, etc., even though all the x solutions are listed first, then all the y solutions, then all the z solutions.

In any event m = 83. And BTW the total of squares is 7, which is indeed under 10.