
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
Let $F_1$ and $F_2$ respectively be the left and right focal points of the ellipse $C: \frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$ ($a>b>0$), point $A$ be the left vertex of ellipse $C$, and point $B$ be the top vertex of ellipse $C$, with distance $|AB| = \sqrt{3}$. Triangle $\triangle BF_1F_2$ is a right triangle.  
(1) Find the equation of the ellipse $C$.  
(2) Let the line $y=kx+2$ intersect the ellipse at points $P$ and $Q$, and let $OP \perpendicular OQ$, find the value of the real number $k$.

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