= Solution
Set $z=z_{1-\alpha}$. Under the <null hypothesis> rejection requires both $W_1\ge f$ and $W_2>z$. Since $W_2$ has the <standard normal distribution>,
$$
P_0(\text{reject})=\alpha-P_0(W_1<f,\ W_2>z).
$$
The <bivariate normal distribution> in the preceding calculation has a nonsingular <covariance matrix> and strictly positive <statistical probability density> everywhere. For every finite <futility boundary> $f$ and $0<\alpha<1$, the open rectangle $\{W_1<f,W_2>z\}$ has positive <probability>. Therefore
$$
\boxed{0<P_0(\text{reject})<\alpha.}
$$
For an explicit expression, conditional on $W_1=w$ the full-sample statistic is $N(w/\sqrt2,1/2)$ under the <null hypothesis>, yielding
$$
P_0(\text{reject})=\int_f^\infty\phi(w)\left[1-\Phi(\sqrt2z-w)\right]dw.
$$
The <Type I error> is reduced because some otherwise rejecting paths stop for futility. Equality is approached as $f\to-\infty$, but does not hold at a finite <futility boundary>.
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