general relativity - Derivation of $f(R)$ field equations, problem with integration by parts

I am following the derivation of the field equations on the the Wikipedia page for $f(R)$ gravity.

But I do not understand the following step:

$$\delta S = \int \frac{1}{2\kappa} \sqrt{-g} \left(\frac{\partial f}{\partial R} (R_{\mu\nu} \delta g^{\mu\nu}+g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) -\frac{1}{2} g_{\mu\nu} \delta g^{\mu\nu} f(R) \right)$$ the wiki article says, the next step is to integrate the second and third terms by parts to yield: $$\delta S = \int \frac{1}{2\kappa} \sqrt{-g}\delta g^{\mu\nu} \left(\frac{\partial f}{\partial R} R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} f(R)+[g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu] \frac{\partial f}{\partial R} \right)\, \mathrm{d}^4x$$ In other words, integrating by parts should yield: $$\int \sqrt{-g} \left(\frac{\partial f}{\partial R} (g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) \right)\, d^4x$$ $$= \int \sqrt{-g}\delta g^{\mu\nu} \left([g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu] \frac{\partial f}{\partial R} \right) \mathrm{d}^4x$$ From there getting the usual f(R) field equations is trivial. What I'm confused by is how to integrate by parts to get that.

I have tried many different ways the one I think is most correct is: assuming $g_{\mu \nu} \Box$ and $\nabla_\mu \nabla_\nu$ are differential operators then $u' = g_{\mu \nu} \Box \delta g^{\mu\nu}$ and $v = f'$, similarly with the $\nabla_\mu \nabla_\nu$ so using the formula for integration by parts:

$$\int u'v = uv -\int uv'$$ I get: $$\int \sqrt{-g} \left(f' (g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) \right)\, d^4x$$ $$= -\int \sqrt{-g}\delta g^{\mu\nu} \left([g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu] f' \right) \mathrm{d}^4x$$ because the $uv$ term will disappear.

So can any one explain to me why I have the minus sign and Wikipedia doesn't? Is it ok to use $g_{\mu \nu} \Box$ as a differential operator? I have tried other ways such as writing $\Box$ explicitly and using integration by parts twice but I also couldn't get the correct answer as i end up with terms such as $\nabla_\nu \nabla_\mu$ which cant be correct.

There is a similar post on physics forums on this step but it does not answer my question and is now closed.