
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 lengths of the sides opposite to angles A, B, and C are a, b, and c, respectively. The area of triangle ABC is given by $\frac{\sqrt{3}}{6}b(b + c - a\cos C)$.

1. Find angle A.
2. If b = 1 and c = 3, find the value of $\cos(2C - \frac{\pi}{6})$.

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