
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 vertex of angle $\alpha$ is at the origin, its initial side coincides with the non-negative half-axis of the x-axis, and its terminal side intersects a unit circle centered at the origin at point A. When the terminal side of angle $\alpha$ is rotated counterclockwise by $\frac{\pi}{6}$, it intersects this unit circle at point B. Let A$(x_1, y_1)$ and B$(x_2, y_2)$. If A$(-\frac{1}{2}, \frac{\sqrt{3}}{2})$, determine the value of $x_2$.

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