
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 triangle $ABC$ with the sides opposite to angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, let vectors $\overrightarrow{m}=(1-\cos(A+B), \cos \frac{A-B}{2})$ and $\overrightarrow{n}=(\frac{5}{8}, \cos \frac{A-B}{2})$, and it's known that $\overrightarrow{m} \cdot \overrightarrow{n} = \frac{9}{8}$.
1. Find the value of $\tan A \cdot \tan B$.
2. Find the maximum value of $\frac{a b \sin C}{a^2 + b^2 - c^2}$.

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