
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 the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \frac{\sqrt{3}}{2})$, and the eccentricity of ellipse $C$ is $e= \frac{\sqrt{3}}{2}$.

(Ⅰ) Find the equation of ellipse $C$;

(Ⅱ) If points $M$, $N$ are two moving points on ellipse $C$, and $k_1$, $k_2$ are the slopes of lines $OM$, $ON$ respectively, and $k_1k_2= -\frac{1}{4}$, investigate whether the area of $\triangle OMN$ is constant, and explain the reason.

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