What are the integers $k$ such that the following identity is always true for any real numbers $a, b$, and $c$?

$$
(a+b)(b+c)(c+a)=(a+b+c)(a b+b c+c a)+k a b c
$$
By developing:

$$
\begin{aligned}
(a+b)(b+c)(c+a) & =a b c+a b a+a c c+a c a+b b c+b b a+b c c+b c a \\
(a+b+c)(a b+b c+c a) & =a a b+a b c+a c a+b a b+b b c+b c a+c a b+c b c+c c a
\end{aligned}
$$

By substituting the two equalities, we get $2 a b c=3 a b c-k a b c$ which gives $k=-1$. The calculation ensures that this solution works.