
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 $ABC$, with angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$ respectively, it is given that $a\cos C + (c-3b)\cos A = 0$.
1. Find the value of $\tan A$.
2. If the area of $\triangle ABC$ is $\sqrt{2}$ and $b-c=2$, find the value of $a$.

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