
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The inequality $x^{2}\cos C + 2x\sin C + \frac{3}{2} \geqslant 0$ holds true for all real numbers $x$.  
$(1)$ Find the range of values for $\cos C$;  

$(2)$ When $\angle C$ is at its maximum value, and the perimeter of $\triangle ABC$ is $9$, find the maximum area of $\triangle ABC$.

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