The sum of the real roots of \(f(x)=x^4-3x^3-4x^2-3x+1=0\) can be represented as \(\large{\frac{a+\sqrt{b}}{c}}\).

If \(a^{-1}+b^{-1}+c^{-1}=\large{\frac{p}{q}}\) such that \(p\) and \(q\) are relatively prime positive integers, find the sum of the digits of \((p+q)\).

**Details and Assumptions**:

- \(a^{-1}\) is the multiplicative inverse of \(a\).

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