Joint probability density function of two dependent gaussian variables

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I have faced the following dilemma while doing some research, so your kind support will be highly appreciated.

I have the following two random variables: $$X_1 \sim N(\mu_1,\sigma^2_1)$$ $$X_2\sim N(\mu_2,\sigma^2_2)$$

and giving $$Z=\frac {X_1 + X_2}{2} $$
I am looking for the joint probability density function $ f(Z,X_1) $

suppose that $\mu_1= \mu_2=a $ and $\sigma^2_1=\sigma^2_2=2a$.
which would give us $Z\sim N(a,a)$

Anyone could help
Regards.