
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
In triangle $ABC$, let $A$, $B$, and $C$ denote the angles, and let $a$, $b$, and $c$ denote the lengths of the sides opposite to the respective angles. Given that $a\cos B-b\cos A= \frac {1}{2}c$.

1. Find the value of $\frac{\tan A}{\tan B}$;
2. Find the maximum value of $\tan (A-B)$.

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