Assume that following inequality holds $$\mathsf{w,x,y,z<AB,AC,AD,BC,BD,CD<ABC,ABD,ACD,BCD<wx,wy,wz,xy,xz,yz}$$ with $$\mathsf{gcd(A,B)=gcd(A,C)=gcd(A,D)=gcd(B,C)=gcd(B,D)=gcd(C,D)=1}$$

where we know $\mathsf{A,B,C,D}$ and we have following residue values: $$\mathsf{xz\bmod A,\quad wz\bmod C}$$ $$\mathsf{xy\bmod B,\quad wy\bmod D}$$

Is there a standard procedure to find $\mathsf{w,x,y,z}$?

Only standard procedure I can think of is exhaustive search which needs $\mathsf{(ABCD)^2}$ arithmetic operations. Is there a way to do this is say at most $\mathsf{(ABCD)^{\frac{1}\beta-\epsilon}}$ arithmetic operations where $\epsilon\in(0,\frac{1}\beta)$ with $\beta>2$ holds?