
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 $f\left(x\right)=\sqrt{3}\sin x\cos x-\cos ^{2}x+\frac{1}{2}$.<br/>$(1)$ If $x\in [0$,$\frac{π}{2}]$, find the range of $f\left(x\right)$;<br/>$(2)$ Find the interval where $f\left(x\right)$ is monotonically increasing on $\left[0,\pi \right]$;<br/>$(3)$ Let $\triangle ABC$ have sides $a$,$b$,$c$ with perimeter $1$, if $f\left(B\right)=-\frac{1}{2}$, find the maximum area of $\triangle ABC$.

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