
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 $ABC$, angle $A$, $B$, $C$ correspond to sides $a$, $b$, $c$ respectively. It is known that $\cos 2C - 3\cos(A + B) = 1$.
1. Find the angle $C$.
2. If $c = \sqrt{7}$ and $b = 3a$, find the area of triangle $ABC$.

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