Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(3,4)$. The equation of $\mathcal{C}_{1}$ is $x^2 + y^2 - 6x + 8y - 11 = 0$ and the radius of $\mathcal{C}_{2}$ is $5$. Find the distance between the centers of $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$. Let's think step by step and output the final answer within \boxed{}.
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