Two Symmetric Equations

Algebra Level 3

If $a, b$ and $c$ are non-zero real numbers, that satisfy the equations $a^2 + b^2 + c^2 = 1$ and $a\left(\frac {1}{b} + \frac {1}{c}\right) + b\left(\frac {1}{c} + \frac {1}{a}\right) + c\left( \frac {1}{a} + \frac {1}{b}\right) = -3$ , how many possible values are there for $a + b + c$ ?