There are real numbers $x$, $y$, and $z$ such that the value of \(x + y + z - \left(\frac{x^2}{5} + \frac{y^2}{6} + \frac{z^2}{7}\right)\) reaches its maximum of \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. Find \(m + n + x + y + z\).

Let's think step by step and output the final answer within \boxed{}.
user