Limiting probabilities under WLLN

Let $X_1, X_2, \dots$ be a sequence of i.i.d. random variables such that $$P(X_1=0)=P(X_1=1)=P(X_1=2)=\frac{1}{3}.$$ Let $S_n = \frac{1}{n}\sum_{i=1}^n X_i$ and $T_n = \frac{1}{n}\sum_{i=1}^n X_i^2, n=1, 2, \dots$. Suppose that $$\alpha_1 = \lim_{n\to\infty} P\left(|S_n - 1/2| < 3/4\right),\quad \alpha_2 = \lim_{n\to\infty} P\left(|S_n - 1/3| < 1\right),$$ $$\alpha_3 = \lim_{n\to\infty} P\left(|T_n - 1/3| < 3/2\right)\quad\text{and}\quad \alpha_4 = \lim_{n\to\infty} P\left(|T_n - 2/3| < 1/2\right).$$ Then, the value of $\alpha_1+2\alpha_2+3\alpha_3+4\alpha_4$ equals