Another cubic, with interesting roots

Algebra Level 4

Let \(f(x) = x^3 + ax^2 + bx + a,\) for some nonzero integers \(a\) and \(b\). Let \(r_1, r_2,\) and \(r_3\) be the roots of \(f(x)\). It is given that

\[ (r_1 + r_2)(r_2 + r_3)(r_3 + r_1) = -1. \]

Find the value of \(b\).

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