four variable ratio of product over sum, distinct variables In this question the form $xyz/(x+y+z)$ is discussed, and positive integer values of $x,y,z$ which make the form integral. My question relates to the four variable version $wxyz/(w+x+y+z).$ I wanted to find distinct positive integer values $w,x,y,z$ to produce a given positive integer result.
I found it was relatively easy to come up with formulas to do this, as long as the goal integer was at least $4.$ However I found no way to do it for the goal integers $1,2,3.$ So I'm asking for either examples or impossibility proofs for these cases.
In the case of goal $1$ there is $(1,1,2,4).$ But I could neither find nor disprove the existence of distinct positive integers $a,b,c,d$ for which $abcd=a+b+c+d.$ Any help on these questions appreciated.

To show that this quotient cannot be $1$, assume WLOG that the numbers are in increasing order. Then $abcd < 4d \implies abc < 4$. But this has no solution in the positive integers for $a,b,c$ distinct (the smallest possible product of three distinct positive integers is $1\cdot 2 \cdot 3 =6$).
To show that it cannot be 2, the same argument as above leads us to $abc < 8$, which means the only possibility is $a=1, b=2, c=3$; this gives the equation $6d = 2(6 + d)$, which gives $d=3$, so there is no such solution in distinct positive integers for 2. However, replacing the coefficient in fact provides a solution $(1,2,3,6)$ for the quotient to equal 3; their product is $36$ and their sum is $12$.
