
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 $\triangle ABC$, let $a$, $b$, and $c$ be the sides opposite to angles $A$, $B$, and $C$, respectively. Given that $2\cos^2\left(\frac{A-B}{2}\right) - 2\sin A \sin B = 1 - \frac{\sqrt{2}}{2}$, find:<br/>
$(1)$ The measure of angle $C$;<br/>
$(2)$ If $c=1$, find the maximum value of the area $S_{\triangle ABC}$.

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