
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 $\triangle ABC$, if $|\vec{AC}| = 2\sqrt{3}$ and $\vec{BC} \cdot \cos A + \vec{AB} \cdot \cos C = \vec{AC} \cdot \sin B$:  
(1) Find the size of angle $B$;  
(2) Calculate the area $S$ of $\triangle ABC$.

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