
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 $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively. The area of $\triangle ABC$ is $30$, and $\cos A= \frac{12}{13}$.
(1) Find $\overrightarrow{AB} \cdot \overrightarrow{AC}$;
(2) If $c-b=1$, find the value of $a$.

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