Let $a, b, c, d$ be the four roots of the polynomial:
\[
x^4 + 3x^3 - x^2 + x - 2.
\]
Given that:
\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} = \frac{1}{2}
\]
and
\[
\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} + \frac{1}{d^2} = -\frac{3}{4},
\]
the value of:
\[
\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} + \frac{1}{d^3}
\]
can be expressed as \(\frac{m}{n}\) for relatively prime positive integers \(m\) and \(n\). Compute \(m+n\).

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