Inverse of Inverses

\[\begin{cases} \dfrac{1}{a} - \dfrac{1}{b + c} = \dfrac{1}{4}\\ \dfrac{1}{b} - \dfrac{1}{a + c} = \dfrac{1}{10}\\ \dfrac{1}{c} - \dfrac{1}{a + b} = \dfrac{1}{56}\\ \dfrac{1}{ab} + \dfrac{1}{bc} + \dfrac{1}{ca} = \dfrac{1}{x} \end{cases}\]

Given that \(a,b\) and \(c\) are integers satisfying the system of equations above, find \(x\).

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