Algebra Level 5

$\large P(x)=4x^{4}+\sum_{j=1}^4 (5-j)(x^{4+j}+x^{4-j})$

Let $$z_{1},z_{2},z_{3}, \ldots,z_{k}$$ be the distinct roots of $$P(x)$$, and let $$z_{n}=a_{n}+b_{n}i$$ for $$n=1,2,3,\ldots,k$$, where $$i=\sqrt{-1}$$ and $$a_{n}$$ and $$b_{n}$$ are real numbers. Let

$\sum_{n=1}^k \left|b_{n}\right|=m+p\sqrt{q}\$

where $$m$$, $$p$$ and $$q$$ are positive integers and $$q$$ is not divisible by the square of any prime. Find $$m+p+q$$.

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