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

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