
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 $C$ with foci $F_1(-1,0)$ and $F_2(1,0)$, and the endpoints of its minor axis are $B_1$ and $B_2$ respectively.  
(1) If $\triangle F_1B_1B_2$ is an equilateral triangle, find the equation of ellipse $C$;  
(2) If the length of the minor axis of ellipse $C$ is 2, and a line $l$ passing through point $F_2$ intersects ellipse $C$ at points $P$ and $Q$, and $\overrightarrow{F_1P} \perp \overrightarrow{F_1Q}$, find the equation of line $l$.

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