LRT after an exponential transformation

Let $X_1,X_2,…,X_n$ be a random sample from a $N(\theta,1)$ distribution. Instead of observing $X_1,X_2,…,X_n$, we observe $Y_1,Y_2,…,Y_n$, where $Y_i=e^{X_i}$, $i=1,2,…,n$. To test the hypothesis

$$H_0:\theta=1 \text{ against } H_1:\theta\ne1$$

based on the random sample $Y_1,Y_2,…,Y_n$, the rejection region of the likelihood ratio test is of the form, for some $c_1<c_2$,