
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
Given the function $f(x) = \sqrt{3}\sin^2x + \cos^2\left(\frac{\pi}{4} - x\right) - \frac{1 + \sqrt{3}}{2} \ (x \in \mathbb{R})$.  
(1) Determine the maximum value of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$.  
(2) In triangle $\triangle ABC$, if $A < B$, and $f(A) = f(B) = \frac{1}{2}$, find the value of $\frac{BC}{AB}$.

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