number theory - Find all integer solutions to $a+b+c|ab+bc+ca|abc$

As you can see from the title, I am trying to find all integer solutions $(a,b,c)$ to $$(a+b+c) \ \lvert\ (ab+bc+ca) \ \lvert\ abc$$ (that is, $a+b+c$ divides $ab+bc+ca$, and $ab+bc+ca$ divides $abc$). Unfortunately, I could not find anything on this problem (although I find it hard to believe that nobody though of this before).

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What I've found so far

I have looked at the simpler case: $(a+b) \ \lvert\ ab$. I was able to solve this, and all solutions are $$(a,b)=(\alpha(\alpha+\beta)\gamma,\beta(\alpha+\beta)\gamma)$$ with $\alpha,\beta,\gamma\in\mathbb{Z}$.

I was also able to reduce the given problem to only one division. If we are able to solve $$(a_0b_0+b_0c_0+c_0a_0) \ \lvert\ (a_0+b_0+c_0)a_0b_0c_0$$ with $\gcd(a_0,b_0,c_0)=1$, then we know that
$$\begin{align} a&=a_0(a_0+b_0+c_0)\cdot k\\ b&=b_0(a_0+b_0+c_0)\cdot k\\ c&=c_0(a_0+b_0+c_0)\cdot k\\ \end{align}$$
For $k\in\mathbb{Z}$ are all solutions to the original problem. However, I was not able to solve this. I have computed a few solutions to the last (and the corresponding solutions for the original problem) but was not able to find a pattern. Any progress on the problem is welcome!