
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 angles $\angle A, \angle B, \angle C$ are the interior angles of triangle $ABC$, and vector $\alpha=\left(\cos \frac{A-B}{2}, \sqrt{3} \sin \frac{A+B}{2}\right)$ with $|\alpha|=\sqrt{2}$. If when $\angle C$ is maximized, there exists a moving point $M$ such that $|MA|, |AB|, |MB|$ form an arithmetic sequence, the maximum value of $|AB|$ is ____.

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