
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 $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\log a - \log b = \log \cos B - \log \cos A$,
1. Determine the shape of $\triangle ABC$;
2. If $a$ and $b$ satisfy the condition: the graph of the function $y = ax + 3$ is symmetric to the graph of the function $y = \frac{1}{3}x - b$ with respect to the line $y = x$, find the length of side $c$.

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