
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 $A$, $B$, $C$ are $a$, $b$, $c$ respectively, and the area is $S$.
(1) If $\overrightarrow{AB} \cdot \overrightarrow{AC} = 2\sqrt{3}S$, find the value of $A$;
(2) If $\tan A : \tan B : \tan C = 1 : 2 : 3$, and $c=1$, find the value of $b$.

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