Let $\omega$ be a circle of radius $6$ with center $O$. Let $AB$ be a chord of $\omega$ having length $5$. For any real constant $c$, consider the locus $\mathcal{L}(c) $ of all points $P$ such that $PA^2 - PB^2 = c$. Find the largest value of $c$ for which the intersection of $\mathcal{L}(c)$ and $\omega$ consists of just one point.

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