
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 A and B are two interior angles of $\triangle ABC$, and $\overrightarrow{a} = \sqrt{2}\cos\frac{A+B}{2}\overrightarrow{i} + \sin\frac{A-B}{2}\overrightarrow{j}$ (where $\overrightarrow{i}$ and $\overrightarrow{j}$ are mutually perpendicular unit vectors), if $|\overrightarrow{a}| = \frac{\sqrt{6}}{2}$.

(1) Is $\tan A \cdot \tan B$ a constant value? If it is, please find it; if not, explain why.

(2) Find the maximum value of $\tan C$ and determine the shape of the triangle at this time.

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