
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 ellipse $C$ are $F\_1(-\sqrt{3},0)$ and $F\_2(\sqrt{3},0)$, and the two endpoints of the minor axis are $B\_1$ and $B\_2$,

(1) If $\triangle F\_1B\_1B\_2$ is an equilateral triangle, find the equation of ellipse $C$;

(2) Under the condition of (1), a line $l$ passing through point $F\_2$ intersects with ellipse $C$ at points $P$ and $Q$, and the slope of $l$ is $1$, find the length of $|PQ|$.

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