If $\frac{1}{a} + \frac{1}{b} + \frac{1}{a+x} = 0$ ; $\frac{1}{a} + \frac{1}{c} + \frac{1}{a+y}$ ; $\frac{1}{a} + \frac{1}{x} + \frac{1}{y} = 0$. 
If $a \neq 0$ , $b \neq 0$ , $c \neq 0$ and if :- $\frac{1}{a} + \frac{1}{b} + \frac{1}{a+x} = 0$ ; $\frac{1}{a} + \frac{1}{c} + \frac{1}{a+y}=0$ ; $\frac{1}{a} + \frac{1}{x} + \frac{1}{y}  = 0$ , find $(a+b+c)$ .

What I Tried :- No information is given about $x$ and $y$ . So I thought of putting $x = y = 1$ , and this silly thing came out in the end .
Now, as $x = y = 1$ , I have $a = \frac{1}{-2}$ from the $3$rd equation .
So from the $1$st equation I get :- $$\frac{1}{a} + \frac{1}{b} + \frac{1}{a+x} = 0$$
$$ \rightarrow -2 + \frac{1}{b} + 2 = 0$$
$$ \rightarrow \frac{1}{b} = 0$$
This definitely looks absurd (also it's given that $b \neq 0$), so I guess putting $x = y = 1$ was a big mistake .
I don't have any other cool ideas for now as I see that doing it algebraically is going to include a lot of simplification and stuffs, and since there are $5$ variable there must be some shortcut of this .
Can anyone help?

Ok so by @player3236's hint , he said that taking $(x = y)$ will create a problem as then it would imply that $\frac{1}{a} + \frac{1}{a+x} = 0$ which is giving $\frac{1}{b} = 0$ and stopping me to move on.
For now, I am taking say $x = 1$ , $y = 2$ .
This gives :- $a = -\frac{2}{3}$ in the $3$rd equation .
Now in the $1$st equation, we have :-
$$\frac{1}{a} + \frac{1}{b} + \frac{1}{a+x} = 0$$
$$\rightarrow \frac{-3}{2} + 3 + \frac{1}{b} = 0$$
$$\rightarrow \frac{1}{b} = \frac{-3}{2}$$
$$\rightarrow b = -\frac{2}{3}$$
Similarly solving for $c$ in the $2$nd equation we get :- $c =  \frac{4}{3}$ .
This gives $(a + b + c) = 0$.
Fortunately I got my answer to this question, but can I get some generalization of why $(a + b + c) = 0$ every time, i.e. , instead of putting $x = 1$ , $y = 2$ can we find $(a + b + c)$ .
