For a complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy the equation:
\[
x^3 + cx + p(1 + qi) = 0
\]
Suppose $c$, $p$, and $q$ were chosen such that all three complex roots $x$ satisfy the inequality:
\[
\frac{5}{6} \leq \frac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \frac{6}{5}
\]
where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real parts of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction \( \frac{m}{n} \), where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

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