Funny Polynomial!

Algebra Level 4

\[P(x) = 24x^{24} +\displaystyle \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).\] Let \(z_{1},z_{2},\ldots,z_{r}\) be the distinct zeros of \(P(x)\) and let \(z_{k}^{2} = a_{k} + b_{k}i\) for real \(a_k ,b_k \), \(k = 1,2,\ldots,r,\) Let

\(\displaystyle\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},\) where \(m,n,\) and \(p\) are integers and \(p\) is not divisible by the square of any prime. Find \(m + n + p.\)

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