
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ with $a > b > 0$ and eccentricity $\dfrac{1}{2}$, and an ellipse $C$ shares a chord of length $4$ with the circle $M$: $x^2 + (y-3)^2 = 4$,  
(1) Find the equation of ellipse $C$;  
(2) Let $O$ be the origin, and through the right vertex $A$ of ellipse $C$ a line $l$ is drawn that's tangent to the circle $x^2 + y^2 = \dfrac{8}{5}$ at another point $B$, find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$.

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