The real numbers $x$ and $y$ satisfy the following system of equations:
\[
\begin{cases} 
\sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\
\sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt{2} 
\end{cases}
\]
Find the numerical value of the ratio $\frac{y}{x}$.

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