
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 $f(x)=\cos(2x+\frac{\pi}{3})+1$, in triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively.

$(I)$ If angles $A$, $B$, and $C$ form an arithmetic sequence, find the value of $f(B)$;

$(II)$ If $f\left(\frac{B}{2}-\frac{\pi}{6}\right)=\frac{7}{4}$, and sides $a$, $b$, and $c$ form a geometric sequence, with the area of $\triangle ABC$ being $S=\frac{\sqrt{7}}{4}$, find the perimeter of $\triangle ABC$.

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