Two points $A$ and $B$ are randomly chosen on a circle with a radius of $100$. For a positive integer $x$, denote $P(x)$ as the probability that the length of $AB$ is less than $x$. Find the minimum possible integer value of $x$ such that $P(x) > \frac{2}{3}$.

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