Let \(A\) and \(B\) be the points of intersection between the line \(x+y=-1\) and the circle with radius \(5\) centered at \((0,0).\) Let \(f(x) = ax^2+bx+c,\) where \(a, b, c\) are real numbers with \(a > 0\) pass through \(A\) and \(B.\) Additionally, let the minimum value of \(f(x)\) be located at \(C\) and the \(x\)-coordinate of \(C\) be \(1.\)
Let \(k\) be the sum of roots of \(f(x)\) and let \(D\) and \(E\) be the intersection between the line \(x+y=k\) and the circle with radius \(5\) centered at \((0,0).\) The value \(CD^2 + CE^2\) can be expressed as \(\frac{m}{n}\) where \(\gcd(m,n)=1\). Find \(m+n.\)