how can we use Sobolev embedding to get the following inequality?

Question

In $\mathbb{R}^3$, how can we use Sobolev embedding to bound the $L^\infty$ norm of $f$ by its $L^2$ norm and $L^4 \cap L^q$ norm of its derivative, where $\infty>q \gg 1$ is arbitrarily chosen. More precisely, how can we check that \begin{equation} \lVert f \rVert_{L^\infty} \lesssim \lVert \nabla f \rVert_{L^4+ L^q} + \lVert f \rVert_2, \end{equation} where \begin{equation} \lVert g \rVert_{L^4+ L^q} \triangleq \inf \left\lbrace \lVert g_1 \rVert_4 + \lVert g_2 \rVert_q: g=g_1+g_2, g_1 \in L^4, g_2 \in L^q \right\rbrace. \end{equation}