
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 $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it is given that $2\sqrt{3}\cos^2\frac{C}{2} = \sin C + \sqrt{3} + 1$.
1. Find the size of angle $C$.
2. If $a=2\sqrt{3}$ and $c=2$, find $b$.

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