
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 denoted by $a$, $b$, $c$ respectively. It is given that $A = \frac{\pi}{4}$ and $b^2 - a^2 = \frac{1}{2}c^2$.  
(1) Find the value of $\tan C$;  
(2) If the area of $\triangle ABC$ is $3$, find the value of $b$ and the circumference of the circumcircle of $\triangle ABC$.

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