Let $ABCDEF$ be a regular hexagon with a side length of $3$. Points $X$, $Y$, and $Z$ are located on segments $AB$, $CD$, and $EF$ respectively, such that $AX = CY = EZ = 1$. The area of triangle $XYZ$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $\gcd(a, c) = 1$. Find $100a + 10b + c$.

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