
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 $F\_2(\sqrt{3},0)$, and the eccentricity $e=\frac{\sqrt{3}}{2}$.
(1) Find the equation of this ellipse;
(2) Let line $l$: $y=x+m$, if $l$ intersects with this ellipse at points $P$ and $Q$, and $|PQ|$ is equal to 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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