Limitation

$a$ ; $b$ ; $c$ are three numbers such that

$\large \displaystyle \left\{ \begin{aligned} a, b, c &\in [12; 18]\\ x = a + b + c &\in [42; 48] \end{aligned} \right.$

For every possible value of $x$ , the maximum value of $a^2 + b^2 + c^2$ is defined by the fuction $f(x) = x^2 - mx + n$ . Calculate the value of $m^2 - n$ .