Probability that a truncated Cauchy average is large

For $n\in\mathbb{N}$, let $X_1,X_2,…,X_n$ be a random sample from the Cauchy distribution having probability density function $$f(x)=\frac1{\pi(1+x^2)},\quad -\infty<x<\infty.$$ Let $g:\mathbb{R}\to\mathbb{R}$ be defined by $$g(x)=\begin{cases}x,&\text{if }-1000\le x\le 1000,\ 0,&\text{otherwise.}\end{cases}$$ Let $$\alpha=\lim_{n\to\infty}P\Bigl(\frac1{n^{3/4}}\sum_{i=1}^n g(X_i)>\frac12\Bigr).$$ Then $100\alpha$ is equal to __________ (answer in integer)