given by the following ansatz: $$\begin{split}
\Gamma_k[g,\bar g, A,\bar A,\xi,\bar \xi,\Upsilon,\bar \Upsilon]=\Gamma^{\textrm{EH}}_k[g]+\Gamma^{\textrm{YM}}_k[g,A]+\Gamma^{\textrm{gf}}_k[g-\bar g,A-\bar A;\bar g,\bar A]\\ +S_{\textrm{gh}}[g-\bar g,A-\bar A,\xi,\bar \xi,\Upsilon,\bar \Upsilon;\bar g,\bar A]\label{ansatz}
\end{split}$$ Here $$\Gamma^{\textrm{EH}}_k[g]=2 \kappa^2 Z_N(k)\int {\textrm{d}^d x\:}\sqrt{g}\,\left(-R(g)+2\bar \lambda(k)\right)$$ is a $k$-dependent form of the Einstein-Hilbert action. The corresponding dimensionful running parameters are the cosmological constant $\bar \lambda(k)$ and Newton’s constant $G(k)\equiv \hat G /Z_N(k)$ where $\hat G$ is a fixed reference value. Furthermore, $$\Gamma^{\textrm{YM}}_k[g,A]=\frac{Z_F(k)}{4\:\hat{g}^2_{\textrm{YM}}} \int {\textrm{d}^d x\:}\sqrt{g}\, g^{\mu\rho}g^{\nu\sigma} F^a_{\mu\nu}F^a_{\rho\sigma}$$ is the standard second-order Yang-Mills action, with a $k$-dependent prefactor $Z_F(k)$ though. Hence the (dimensionful, except in d=4) running gauge coupling is $\bar g_\textrm{YM}(k)=\hat{g}_\textrm{YM} Z_F(k)^{-1/2}$ with some constant $\hat{g}_\textrm{YM}$. Finally, $$\Gamma^{\textrm{gf}}_k[g-\bar g,A-\bar A;\bar g,\bar A]=\int {\textrm{d}^d x\:}\sqrt{\bar g}\, \left( \frac{Z_N(k)}{2 \alpha_{\textrm{D}}} \bar g^{\mu\nu} \textrm{F}_\mu\textrm{F}_\nu +\frac{Z_F(k)}{2\alpha_{\textrm{YM}}} \textrm{G}^a\textrm{G}^a\right)$$ implements the gauge fixing conditions for the diffeomorphisms, $\textrm{F}_\mu$, and the $SU(N)$ gauge transformations, $\textrm{G}^a$. As in [@mr] and [@nonabavact] we factored out the wave function renormalizations $Z_N$ and $Z_F$ from the gauge fixing parameters $\alpha_\textrm{D}$ and $\alpha_\textrm{YM}$, respectively. In principle the latter are still $k$-dependent but we shall neglect their running here. In fact, later on we set $\alpha_\textrm{D}=\alpha_\textrm{YM}=1$. Our choice for the gauge conditions complies with the requirements discussed in the previous section: $$\begin{aligned}
\textrm{F}_\mu(\bar h;\bar g)&=\sqrt{2}\kappa\left(\delta_\mu^\beta \bar g^{\alpha \gamma}\bar 