
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 an ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\frac{1}{2}$, and it passes through point $A(1, \frac{3}{2})$.
(1) Find the equation of the ellipse $C$;
(2) If point $B$ is on the ellipse, point $D$ is on the $y$-axis, and $\overrightarrow{BD} = 2\overrightarrow{DA}$, find the equation of line $AB$.

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