
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 $f(x)=\cos x( \sqrt {3}\sin x+\cos x)$, where $x\in\mathbb{R}$.  
$(1)$ Find the maximum value of the function $f(x)$.  
$(2)$ If $f\left( \frac {\theta}{2}\right)= \frac {3}{4}$, where $\theta\in\mathbb{R}$, find the value of $f\left(\theta+ \frac {\pi}{3}\right)$.

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