
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 an obtuse angle $\alpha$ with its vertex at the origin and its initial side along the non-negative half of the x-axis, the terminal side of $\alpha$ intersects the circle $x^2+y^2=2$ at point $A(x_1, y_1)$. When the ray OA is rotated clockwise by $30^\circ$, it intersects the circle $x^2+y^2=2$ at point $B(x_2, y_2)$. Determine the value of $\cos\alpha$ when $|x_1 - x_2|$ is maximum.

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