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)