
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 $f(x)= \sqrt{3}\sin^2x+\cos^2\left( \frac{\pi}{4}-x\right)- \frac{1+ \sqrt{3}}{2}$ $(x\in\mathbb{R})$.

$(1)$ Find the maximum value of the function $f(x)$ in the interval $\left[ 0,\frac{\pi}{2} \right]$;

$(2)$ In $\triangle ABC$, 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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