
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
In the Cartesian coordinate system xOy, if the line $ax + y - 2 = 0$ intersects a circle centered at C, given by $(x - 1)^2 + (y - a)^2 = \frac{16}{3}$, at two points A and B, and triangle $\triangle ABC$ is equilateral, find the value of the real number $a$.

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