Consider the polynomial $P(x) = x^3 + ax^2 + bx + c$. If the roots $\alpha, \beta, \gamma$ satisfy $\alpha^2 + \beta^2 + \gamma^2 = 7$ and $\alpha^3 + \beta^3 + \gamma^3 = 10$, and $\alpha + \beta + \gamma = 1$, find the value of the coefficient $b$.