Let
$$P(x) = 24x^{24} + \sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).$$
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 \(k = 1,2,\ldots,r,\) where \(i = \sqrt { - 1},\) and \(a_{k}\) and \(b_{k}\) are real numbers. Let
$$\sum_{k = 1}^{r}|b_{k}| = m + n\sqrt {p},$$
where \(m,\) \(n,\) and \(p\) are integers and \(p\) is not divisible by the square of any prime. Find \(m + n + p.\)
(第二十一届AIME2 2003 第15题)