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Geometry Level 5

In triangle \(\triangle ABC\), let \(BC=7\), \(AC=6\) and \(AB=5\). Let \(D\), \(E\) and \(F\) be the centers of the excircles relatives to \(BC\), \(AC\) and \(AB\) respectively.

If \(EF=\dfrac{c \sqrt{a}}{g}\), \(DF=\dfrac{d\sqrt{ab}}{g}\) and \(DE=f\sqrt{b}\), where \(ab\) is square free, \(\gcd(c,g)=1\) and \(\gcd(d,g)=1\), find \(abc-dfg\).

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