The complex number $z = re^{i \theta}$ satisfies
\[\left| 2z + \frac{1}{z} \right| = 1.\]Find the minimum value of $\sin^2 \theta.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
The complex number $z = re^{i \theta}$ satisfies
\[\left| 2z + \frac{1}{z} \right| = 1.\]Find the minimum value of $\sin^2 \theta.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{7}{8}
The complex number $z = re^{i \theta}$ satisfies
\[\left| 2z + \frac{1}{z} \right| = 1.\]Find the minimum value of $\sin^2 \theta.$
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