For how many integers $k$ does the following system of equations have a solution other than $a=b=c=0$ in the set of real numbers?
\[
\begin{cases}
  a^2 + b^2 = kc(a+b), \\
  b^2 + c^2 = ka(b+c), \\
  c^2 + a^2 = kb(c+a).
\end{cases}
\]

Let's think step by step and output the final answer within \boxed{}.
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