
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 $\alpha$ is an angle in the first quadrant, its terminal side intersects the unit circle at point P with a horizontal coordinate of $\frac{1}{3}$. By rotating the ray OP counterclockwise around the coordinate origin O by $\frac{\pi}{3}$, it intersects the unit circle at point Q. Determine the vertical coordinate of point Q.

A) $$\frac{2\sqrt{3} - 1}{6}$$
B) $$\frac{2\sqrt{2} - 3}{6}$$
C) $$\frac{2\sqrt{6} + 1}{6}$$
D) $$\frac{2\sqrt{2} + \sqrt{3}}{6}$$

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