Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\frac{1}{\sqrt[4]{3}}$. What is the value of $900p$, rounded to the nearest integer?

Let's think step by step and output the final answer within \boxed{}.
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