
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 $f(x)=\cos x(\cos x+ \sqrt{3}\sin x)$.

$(1)$ Find the minimum value of $f(x)$;

$(2)$ In $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $f(C)=1$ and the area of $\triangle ABC$ is $\frac{3 \sqrt{3}}{4}$ with $c= \sqrt{7}$, find the perimeter of $\triangle ABC$.

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