
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 $f\left( x \right)=\sin x\cos x+\sqrt{3}{\cos ^{2}}x$, the graph is translated to the right by $\frac{\pi }{6}$ units and then downwards by $\frac{\sqrt{3}}{2}$ units to obtain the graph of $g\left( x \right)$.
(I) Find the analytical expression for $g\left( x \right)$;
(II) In an acute triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $g\left( \frac{A}{2} \right)=\frac{1}{2}$ and $a=1$, find the maximum area of $\triangle ABC$.

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