If $x, y, z, p, q, r$ are real numbers such that:
\[ \frac{1}{x+p} + \frac{1}{y+p} + \frac{1}{z+p} = \frac{1}{p} \]
\[ \frac{1}{x+q} + \frac{1}{y+q} + \frac{1}{z+q} = \frac{1}{q} \]
\[ \frac{1}{x+r} + \frac{1}{y+r} + \frac{1}{z+r} = \frac{1}{r} \]
Find the numerical value of $\frac{1}{p} + \frac{1}{q} + \frac{1}{r}$.

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