2003 AIME II Problem 15

Let

$$P(x)=24 x^{24}+\sum_{j=1}^{23}(24-j)\left(x^{24-j}+x^{24+j}\right)$$

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

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

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