
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 acute triangle \( \triangle ABC \), the sides opposite angles \( A \), \( B \), \( C \) are \( a \), \( b \), and \( c \), respectively, satisfying \( \frac{2b-c}{a} = \frac{\cos C}{\cos A} \).  
(1) Find the measure of angle \( A \).  
(2) If \( a = \sqrt{13} \) and the area of \( \triangle ABC \), \( S_{\triangle ABC} = 3\sqrt{3} \), find the value of \( b + c \).  
(3) Given the function \( f(x) = 2\sin x \cos \left(x+ \frac{\pi}{6}\right) \), find the range of \( f(B) \).

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