Let \( S = (-1, \infty) \) and \( f : S \to \mathbb{R} \) be defined as \( f(x) = \int_{-1}^{x} (e^{t} - 1)^{11}(2t - 1)^{5}(t - 3)^{12}(2t - 10)^{61} dt \).
Let \( p \) = Sum of squares of the values of \( x \), where \( f(x) \) attains local maxima on \( S \), and \( q \) = Sum of the values of \( x \), where \( f(x) \) attains local minima on \( S \). Then, the value of \( p^{2} + 2q \) is __________.
