The complex number $z = re^{i \theta}$ satisfies
\[\left| 2z + \frac{1}{z} \right| = 1.\]Find the minimum value of $\sin^2 \theta.$The answer is in the form rac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

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