解析这个题目**Question Stem:**
如图，在平行四边形 ABCD中，AB = 2, AD = 3, 点 E 是 AB 的中点，F 是线段 AD 上靠近点 A 的三等分点，$\vec{BG} = \lambda\vec{BC}$，设 $\vec{AB} = \vec{a}, \vec{AD} = \vec{b}$.
**Diagram Description:**
* Type: Geometric figure.
* Main Elements:
* A parallelogram ABCD with vertices labeled A, B, C, D in counter-clockwise order.
* Point E is on side AB.
* Point F is on side AD.
* Point G is inside or on the boundary of the parallelogram. A line segment connects B to G. A line segment connects E to F, E to G, and F to G, forming a triangle FEG.
* Vertices A, B, C, D, E, F, G are labeled.
* Lines are straight segments connecting the points as described (sides AB, BC, CD, DA, internal segments EF, EG, FG, BG).
**Questions:**
**(1)** 若 $\lambda = \frac{1}{3}$, 求 $\angle FEG$ 的大小;
**(2)** 若 $\lambda = \frac{1}{2}$, $\vec{EF} \cdot \vec{EG} = \frac{1}{3}$, 求 $\cos\langle\vec{a}, \vec{b}\rangle$.