George throws two coins, \(A\) and \(B\), at the same time. Coin \(A\) is biased so that the probability of obtaining a head is \(a\). Coin \(B\) is biased so that the probability of obtaining a head is \(b\), where \(b\lt a\). The probability generating function of \(X\), the number of heads obtained by George, is \(\mathrm{G}_{X}(t)\). The coefficients of \(t\) and \(t^{2}\) in \(\mathrm{G}_{X}(t)\) are \(\frac{5}{12}\) and \(\frac{1}{12}\) respectively.
(a) Find the value of \(a\).

The random variable \(Y\) is the sum of two independent observations of \(X\).
(b) Find the probability generating function of \(Y\), giving your answer as a polynomial in \(t\).

(c) Find \(\operatorname{Var}(Y)\).