The parabola \(f(x)=x^2\) is tangent to the graph of \( g(x) = x^4 + ax^3 + \color{red}cx^2+ bx +1\) at two distinct points. Given that the **\({\color{purple}{minimum}}\)** area enclosed by these two curves is \(\frac{p}{q}\), where \(p\) and \(q\) are coprime positive integers, find the value of \(p+q\).

###### This problem is part of Curves... cut or touch?

**Remark**: try the case for \(c=1\) here

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