The cubic equation \(2x^3 - 4x^2 + 3 = 0\) has roots \(\alpha, \beta, \gamma\). Let \(S_n = \alpha^n + \beta^n + \gamma^n\).

1. (a) State the value of \(S_1\) and find the value of \(S_2\).
2. (b)
   1. Express \(S_{n+3}\) in terms of \(S_{n+2}\) and \(S_n\).
   2. Hence, or otherwise, find the value of \(S_4\).
3. (c) Use the substitution \(y = S_1 - x\), where \(S_1\) is the numerical value found in part (a), to find and simplify an equation whose roots are \(\alpha + \beta, \beta + \gamma, \gamma + \alpha\).
4. (d) Find the value of \(\frac{1}{\alpha + \beta} + \frac{1}{\beta + \gamma} + \frac{1}{\gamma + \alpha}\).