Snake lemma for derived functors

Assume we are in some abelian category with enough injectives, and we are given a short exact sequence: $$0\longrightarrow A'\longrightarrow A\longrightarrow A''\longrightarrow 0$$

And a left exact functor $$F$$. I would like to prove that it induces a long exact sequence in derived functors, i.e.: $$0\longrightarrow R^{0}F(A') \longrightarrow R^{0}F(A)\longrightarrow R^{0}F(A'')\longrightarrow R^{1}F(A')\longrightarrow ...$$

In order to do that, I observed injective resolutions of the 3 objects $$A'\longrightarrow I^*, A\longrightarrow J^*, A''\longrightarrow K^*$$

And say I apply a Lemma such as https://stacks.math.columbia.edu/tag/013G, which tells me that one can construct a corresponding exact chain of complexes: $$0\longrightarrow I^*\longrightarrow J^*\longrightarrow K^*\longrightarrow 0$$ With exact rows and everything. The snake Lemma tells us that this translates to an exact sequence in cohomology, however, working with derived functors, one must first apply the functor $$F$$ before taking cohomologies.

Then naively, I obtain a short left exact sequence of complexes: $$0\longrightarrow I^*\longrightarrow J^*\longrightarrow K^*$$

For which, the snake Lemma does not apply. Any answer or reference to text on the subject would be appreciated.