
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 that the two foci of an ellipse are $F_1(-\sqrt{3},0)$ and $F_2(\sqrt{3},0)$, with eccentricity $e= \frac{\sqrt{3}}{2}$.

$(1)$ Find the equation of this ellipse.

$(2)$ Suppose the line $y= \frac{x}{2}+m$ intersects the ellipse at points $P$ and $Q$, and $|PQ|$ equals the length of the minor axis of the ellipse. Find the value of $m$.

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