Given scalars $c_1, c_2 > 0$, how would one go about solving, for non-expansive (i.e 1-Lipschitz) $\phi: \mathbb R \rightarrow (-\infty,+\infty]$, the following equation

$$ \begin{split} \mathbb E_{z \sim \mathcal N(0, 1)}[\phi(z)^2] &= c_1\\ \mathbb E_{z \sim \mathcal N(0, 1)}[\phi'(z)] &= c_2. \end{split} $$

**Notation:**

- $\mathcal N(0, 1)$ is the standard Gaussian normal distribution so that $$\mathbb E_{z \sim \mathcal N(0, 1)} [g(z)] = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \exp(-z^2/2)g(z)dz $$