Let $f:[1,\infty )\to R$ be a differentiable function such that $3f\left(1\right)=1$ and $3{\int }_1^{x}tf\left(t\right)dt=x^{2}f\left(x\right)-\frac{x^3}{3},x\in [1,\infty ).$ If e denotes the base of the natural logarithm, then $3f\left(e^2\right)$ equals.