
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 point $A(0,1)$, line $l$: $y=kx-m$ intersects circle $O$: $x^{2}+y^{2}=1$ at points $B$ and $C$, the areas of $\triangle ABC$ and $\triangle OBC$ are $S_{1}$ and $S_{2}$ respectively. If $\angle BAC=60^{\circ}$, and $S_{1}=2S_{2}$, then the value of the real number $k$ is \_\_\_\_\_\_.

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