
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with eccentricity $\frac{\sqrt{3}}{2}$ and passing through the point $(1,\frac{\sqrt{3}}{2})$.<br/>$(Ⅰ)$ Find the equation of the ellipse $C$;<br/>$(Ⅱ)$ The line $l$ passing through the point $M(1,0)$ intersects the ellipse $C$ at points $A$ and $B$, with point $N(\frac{1}{2},0)$. If the area of triangle $ABN$ is $\frac{3}{10}$, find the slope $k$ of the line $l$.

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