
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 that the eccentricity of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ is $\frac{2}{3}$, and the left vertex is $A(-3, 0)$.  
$(Ⅰ)$ Find the equation of the ellipse;  
$(Ⅱ)$ Let the line $l: y = kx (k > 0)$ intersect the ellipse in the first quadrant at point $P$, and the line passing through point $A$ denoted as $l'$ intersects the ellipse at point $Q$. If $l'$ is parallel to $l$ and $|AQ| = \frac{1}{2}|OP|$ (where $O$ is the origin), find the value of $k$.

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