
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
In the Cartesian coordinate system $xOy$, it is known that ellipse $C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a > b > 0$) passes through the point $(1, \frac{3}{2})$, and its eccentricity is $\frac{1}{2}$.
(1) Find the equation of ellipse $C$.
(2) The line $l$ passing through the point $(1,0)$ intersects ellipse $C$ at two points $A$ and $B$. If $\overrightarrow{OA} \cdot \overrightarrow{OB} = -2$, find the equation of line $l$.

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