Entropy lower bound

Question

I have the following observation: $$Z=\sqrt{P}H+W,$$ where $$H \sim \mathcal{N} (0,\sigma^2). $$ $P$ is given and fixed. I am trying to find $W$ that minimizes $h(Z)$. Given that $E[|W|^2]=\sqrt{\Gamma}$, one case is when $\Gamma\geq P$ we can use $W=kH$ and $k=-\sqrt{P}$. What I am struggling with is how to find the optimal $W$ that minizes $Z$ for the case when $P>\Gamma$.