# The Iterative Degree

Algebra Level 5

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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