Geometric Mean of Random Multipliers via LLN and CLT

An investment grows by a random factor each year: in year $i$, the portfolio is multiplied by $X_i$, where $X_i$ are i.i.d.\ with $P(X_i = 2) = P(X_i = 4) = 1/2$. After $n$ years, the annualized growth factor is the geometric mean $$G_n = \left(\prod_{i=1}^n X_i\right)^{1/n}.$$

(a) Find $\lim_{n \to \infty} G_n$ almost surely.

(b) For $n = 100$, use the CLT to approximate $P(G_{100} > 3)$.

You may use $\ln 2 \approx 0.6931$, $\ln 3 \approx 1.0986$, and $\Phi(1.70) \approx 0.9554$.