
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given points $A(0,2)$ and $B(3,1)$ on the ellipse $G$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$.
(1) Find the eccentricity of ellipse $G$;
(2) Given that line $l$ passes through point $B$ and intersects ellipse $G$ at another point $C$ (different from point $A$), if a circle with $BC$ as the diameter passes through point $A$, find the equation of line $l$.

Present the answer in LaTex format: \boxed{Your answer}
user