
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 $f(x)=a\sin x\cdot\cos x- \sqrt {3}a\cos ^{2}x+ \dfrac { \sqrt {3}}{2}a+b$ ($a > 0$)  
$(1)$ Write down the interval of monotonic decrease for the function;  
$(2)$ Let $x\in\left[0, \dfrac {\pi}{2}\right]$, the minimum value of $f(x)$ is $-2$, and the maximum value is $\sqrt {3}$. Find the values of the real numbers $a$, $b$.

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