Getting wrong number of Wick contractions

Consider this lagrangian:

$$mathcal{L} = dfrac{1}{2} (partial_{mu}phi_{1})^2 + dfrac{1}{2} (partial_{mu}phi_{2})^2 + dfrac{m^2}{2}(phi_{1}^2 + phi_{2}^2) + dfrac{g}{4!}(phi_{1}^4 + phi_{2}^4) + dfrac{h}{4}phi_{1}^2phi_{2}^2 $$

It has symmetries: $ phi_{1,2} longleftrightarrow -phi_{1,2} $ and $ phi_{1} longleftrightarrow phi_{2} $

I am trying to compute the symmetry factor of this $O(gh)$ diagram:

Here, the left vertex corresponds to $phi_{1}^2phi_{2}^2$ interaction (which is proportional to $h$), and the right one corresponds to $phi_{2}^4$ interaction (which is proportional to $g$).

I am trying to calculate the symmetry factor by counting the possible number of Wick contractions. (I am getting a symmetry factor of 1/4 but I have checked from many sources that the correct symmetry factor is 1/2 ). In order to make this question concise and easily answerable, let me frame my question like this:

How many Wick contractions are there for this integral? : $$left(frac{gh}{2!4!4}right)phi_{1}(w)phi_{1}(x)phi_{2}(y)phi_{2}(z)int{}d^4v,phi_{2}phi_{2}phi_{2}phi_{2}int{}d^4u,phi_{1}phi_{1}phi_{2}phi_{2}$$

To get a symmetry factor of 1/2, there should be 96 possible ways to contracts this integral. But I am getting only 48 possible contractions ( hence a symmetry factor of 1/4 ).

2 ways for contractions among $phi_{1}$ fields ( without forming loops )

4 ways to contract $phi_{2}(y)$ with 4 $phi_{2}$ fields at $v$.

3 ways to contract $phi_{2}(z)$ with remaining 3 $phi_{2}$ fields at $v$.

2 ways to for the remaining $phi_{2}$ fields (at $u$ and $v $) to contract (without forming loops).

That’s 48 ways. I don’t see any other contraction that would lead to the diagram I am interested in.

Can anyone please help me in figuring out what am I doing wrong???
Let me know if I need to add more details.

(FYI, this question is very similar to Peskin Schroeder 12.3.)