Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers a, b, c, respectively. They then compute the polynomial f(x)=x^{3}+p x^{2}+q x+r with roots a, b, c. If the expected value of the sum of the squares of the coefficients of f(x) is \frac{m}{n} for relatively prime positive integers m, n, find the remainder when m+n is divided by 1000.