WKB problem with 4 turning points?

Question

I was recently given a problem that asked to find the solvability conditions for $$\epsilon^2y''=(W(x)-E)y;\quad y\rightarrow0\text{ as }|x|\rightarrow0$$ where $W$ was some piecewise linear, $``W"$-shaped potential function (for simplicity, let the middle peak of the W be $(0,0))$. To summarize the setup, since each region is linear, an appropriate variable substitution $z_j$ as a linear function of $x$ for each linear region of $W$ recovers the Airy ODE in each region: $$y''=z_jy$$ with solutions $$y(x)=A_j\cdot \text{Ai}(z_j(x))+B_j\cdot\text{Bi}(z_j(x)),\quad j=1,2,3,4$$ in each of the four intervals where $W(x)$ is linear(say we label them from left to right, where $I_4=(-\infty,a]$, $I_4=(a,0]$, $I_4=(0,b]$, and $I_1=(b,\infty)$. We require continuity of both $y$, and $y'$, so we end up with a matrix for the undetermined coefficients: \begin{bmatrix} \text{Ai}(z_4(b)) & -\text{Ai}(z_3(a)) & -\text{Bi}(z_3(b)) & 0 & 0 & 0\\ z_4'(b)\cdot\text{Ai}'(z_4(b)) & -z_3'(b)\cdot\text{Ai}'(z_3(b)) & -z_3'(b)\cdot\text{Bi}'(z_3(b)) & 0 & 0 & 0\\ 0 & \text{Ai}(z_3(0)) & \text{Bi}(z_3(0)) & -\text{Ai}(z_2(0)) & -\text{Bi}(z_2(0)) & 0\\ 0 & z_3'(0)\cdot\text{Ai}'(z_3(0)) & z_3'(0)\cdot\text{Bi}'(z_3(0)) & -z_2'(0)\cdot\text{Ai}'(z_2(0)) & -z_2'(0)\cdot\text{Bi}'(z_2(0)) & 0\\ 0 & 0 & 0 & \text{Ai}(z_2(a)) & \text{Bi}(z_2(a)) & -\text{Ai}(z_1(a))\\ 0 & 0 & 0 & z_2'(a)\cdot\text{Ai}'(z_2(a)) & z_2'(a)\cdot\text{Bi}'(z_2(a)) & -z_1'(a)\cdot\text{Ai}'(z_1(a)) \end{bmatrix} Note that $B_1$ and $B_4$ must be $0$ to satisfy the $y$ bounded for large $|x|$. If the determinant of this matrix is $0$, then there are nontrivial solutions to the problem. Now, supposedly we can determine conditions on $E$ from the determinant, and we should be able to do it explicitly if we 1.) assume the case of a symmetric $W$ potential, and 2.) then use the first term asymptotic expansions of $\text{Ai}(z)$, $\text{Bi}(z)$, but I couldn't get it. Specifically for the this symmetric case with asymptotic approximations, does anyone know how to find the determinant?