
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 an acute triangle $\triangle ABC$ with sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively. If $c=\sqrt{3}$, find the following:<br/>
$(1)$ Determine angle $C$;<br/>
$(2)$ Find the range of possible values for the area of $\triangle ABC$.<br/>
Choose one of the conditions from $a^{2}+b^{2}-ab=3$ and $2c\cos C=a\cos B+b\cos A$, and fill in the blank line to complete the solution.<br/>
Note: If multiple conditions are chosen and answered separately, only the first answer will be scored.

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