CLT limit of two binomial probabilities

For $n\in\mathbb{N}$, let $X_n$ be a random variable having the $\mathrm{Bin}\bigl(n,\frac14\bigr)$ distribution. Then $$\lim_{n\to\infty}\Bigl[P\Bigl(X_n\le\frac{2n-\sqrt{3n}}{8}\Bigr)+P\Bigl(\frac n6\le X_n\le\frac n3\Bigr)\Bigr]$$ is equal to (You may use $\Phi(0.5)=0.6915$, $\Phi(1)=0.8413$, $\Phi(1.5)=0.9332$, $\Phi(2)=0.9772$)