Let \(a\), \(b\), and \(c\) denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked 1, 2, 3, 4. If the probability that \(ax^2 + bx + c = 0\) has all real roots is \(\frac{m}{n}\), \(\text{gcd}(m, n) = 1\), then \(m + n\) is equal to ______.