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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