
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) that passes through point $M(2, 0)$, with an eccentricity of $\frac{1}{2}$. Points $A$ and $B$ are on ellipse $C$, and the product of the slopes of lines $OA$ and $OB$ is $-\frac{3}{4}$, where $O$ is the origin of the coordinate system.  
(Ⅰ) Find the equation of ellipse $C$;  
(Ⅱ) If point $P$ on ray $OA$ satisfies $|PO| = 3|OA|$, and line $PB$ intersects the ellipse at point $Q$, find the value of $\frac{|BP|}{|BQ|}$.

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