Parameter values from degenerate limit CDF

Let $X_1, X_2, \dots$ be a sequence of i.i.d. random variables, each having the probability density function $$f(x)=\begin{cases} \frac{x^2 e^{-x}}{2}, & \text{if }x\ge 0, \ 0, & \text{otherwise.} \end{cases}$$ For some real constants $\beta, \gamma$ and $k$, suppose that $$\lim_{n\to\infty} P\left(\frac{1}{n}\sum_{i=1}^n X_i \le x\right) = \begin{cases} 0, & \text{if }x < \beta, \ kx, & \text{if }\beta \le x \le \gamma, \ k\gamma, & \text{if }x > \gamma. \end{cases}$$ Then, the value of $2\beta+3\gamma+6k$ equals __________