
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
Let the function $f(x)=1-2\sin^2x-\cos(2x+ \frac{\pi}{3})$.

$(1)$ Find the smallest positive period of the function $f(x)$;  
$(2)$ For triangle $ABC$, the sides $a$, $b$, $c$ are opposite to the angles $A$, $B$, $C$ respectively. Given $b=5$, and $f\left( \frac{B}{2}\right)=1$, find the maximum area of $\triangle ABC$.

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