
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), and a line $l: y = kx - m$ intersects with the 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}
user