Let \(f:[1, \infty) \rightarrow \mathbb{R}\) be a differentiable function such that \(f(1)=\frac{1}{3}\) and \(3 \int_1^…

Let \(f:[1, \infty) \rightarrow \mathbb{R}\) be a differentiable function such that \(f(1)=\frac{1}{3}\) and \(3 \int_1^x f(t) d t=x f(x)-\frac{x^3}{3}, x \in[1, \infty)\). Let \(e\) denote the base of the natural logarithm. Then the value of \(f(e)\) is

[JEE Advanced 2023]