
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
Let the function $f(x)= \sqrt{3}\sin x\cos x+\cos^2x+a$  
$(1)$ Write out the smallest positive period and the interval of monotonic decrease for the function $f(x)$;  
$(2)$ When $x\in\left[- \dfrac{\pi}{6}, \dfrac{\pi}{3}\right]$, the sum of the maximum and minimum values of the function $f(x)$ is $\dfrac{3}{2}$. Find the solution set of the inequality $f(x) > 1$.

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