*a* median of the distribution to be *any* real number \(m\) such that

\[ \mathbb{P}_{X\sim p} (X\leq m) \geq \frac{1}{2}, \hspace{5pt} \mathbb{P}_{X\sim p} (X\geq m) \geq \frac{1}{2}.\]
Find the maximum possible value of \(m\).

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