Hypothesis Tests · Superchallenge problem

A researcher collects $n$ observations from a normal population with known $\sigma$ and runs a two-sided $z$ test of $H_0\!: \mu = \mu_0$ at the 5% level ($|z| > 1.96$). If it is not significant, she collects $n$ MORE observations and tests again on all $2n$, with the same rule, and stops there. If $H_0$ is true, what is the probability that she ends up rejecting $H_0$? Round to three decimal places.