If \(P\) is a polynomial of degree \(2\), and if \(f(x)\) is a function of degree \(4\), the degree of \( f^{(19)}(P \cdot f^{(19)}(P\))) can be expressed in the form \(a^{b}\) \(+\) \(a^{c}\) , where \(a\), \(b\), and \(c\) are coprime, positive integers. Find \(a+b+c\).

**Note**:
The notation \(f^{(n)}(x)\) means that \(f(x)\) is **iterated**, or repeatedly carried out, \(n\) times. For example, \(f^{(3)}(x)\) means \(f(f(f(x)))\) .

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