
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 $f(x)= \sqrt{3}\cos^2x+\sin x\cos x$.

$(1)$ Find the value of $f\left( \frac{\pi}{6}\right)$;

$(2)$ Find the interval of monotonic increase for $f(x)$;

$(3)$ If $\alpha\in(0,\pi)$ and $f\left( \frac{\alpha}{2} \right)= \frac{1}{4} + \frac{ \sqrt{3}}{2}$, find the value of $\sin \left(\alpha+ \frac{7\pi}{12} \right)$.

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