
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 $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ with eccentricity $\frac{\sqrt{3}}{2}$, and it passes through the point $\left( 1, \frac{\sqrt{3}}{2} \right)$.

$(1)$ Find the equation of $E$;

$(2)$ If the line $l: y = kx + m \left( k > 0 \right)$ intersects $E$ at points $P$ and $Q$, and the sum of the slopes of $OP$ and $OQ$ ($O$ is the origin) is $2$, find the range of distances from $O$ to the line $l$.

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