Let \( f : [1, \infty) \rightarrow \mathbb{R} \) be a differentiable function such that \( f(1) = \frac{1}{3} \) and\( 3 \int_{1}^{x} f(t)\, dt = x f(x) - \frac{x^{3}}{3} \), \( x \in [1, \infty) \).
Let \( e \) denote the base of the natural logarithm. Then the value of \( f(e) \) is:
