
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 $f(x)=2\cos x\sin \left( x+ \frac{\pi}{3} \right)- \sqrt{3}\sin^2 x+\sin x\cos x$.

$(1)$ Find the smallest positive period of $f(x)$;

$(2)$ Find the intervals where $f(x)$ is increasing;

$(3)$ When $x\in\left[ 0, \frac{\pi}{4} \right]$, find the range of $f(x)$.

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