How to use Ricci identity to prove exchange of laplacian beltrami operator and derivative with Levi-Civita tensor involved in 2D

Question

I am trying to prove the following equation using Ricci identity for the 2D manifold: $$\nabla^{2}(\mathbf{\epsilon}\cdot\nabla \Phi)-\mathbf{\epsilon}\cdot\nabla (\nabla^{2}\Phi) = K\mathbf{\epsilon}\cdot\nabla\Phi,$$ where $\Phi$ is a scalar function, $\mathbf{\epsilon}$ is 2D Levi-Civita tensor, $K$ is the Gaussian curvature of the 2D manifold.
My attempt: I figured out how to prove this one: $$\nabla^{2}(\nabla\Phi)-\nabla(\nabla^{2}\Phi)=K\nabla\Phi.$$ But with a $\mathbf{\epsilon}$ involved, I do not know how to prove it. Can someone help, thank you.