
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 $\triangle ABC$, $A$ is an acute angle, and $4\sin (5\pi-A)\cos ^{2}\left( \dfrac {A}{2}- \dfrac {\pi}{4}\right)= \sqrt {3}(\sin \dfrac {A}{2}+\cos \dfrac {A}{2})^{2}$.  
$(1)$ Find $A$;  
$(2)$ If $AC=1$ and the area of $\triangle ABC$ is $\sqrt {3}$, find the height from $BC$.

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