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

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