Limit of a self-normalised uniform sum

Let $\{X_k\}_{k\ge 1}$ be a sequence of i.i.d. $\mathrm{U}(-1,1)$ random variables. Suppose $$Y_n=\sqrt{3n}\,\frac{\sum_{i=1}^n X_i}{\sum_{i=1}^n X_i^4}.$$ Then $\{Y_n\}_{n\ge 1}$ converges in distribution as $n\to\infty$ to a