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 $\tfrac{m}{n},$ 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{}.
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