
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 with its focus on the x-axis: $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{m}=1$, and eccentricity $\frac{1}{2}$.<br/>$(1)$ Find the value of the real number $m$;<br/>$(2)$ A line $l$ passing through the point $P(0,2)$ intersects the ellipse at points $A$ and $B$, with the midpoint of line segment $AB$ being $M$. If the slope of line $OM$ is $\frac{3}{4}$ (where $O$ is the origin), find the equation of line $l$.

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