
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 $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The areas of the equilateral triangles with side lengths $a$, $b$, and $c$ are denoted as $S_{1}$, $S_{2}$, and $S_{3}$, respectively. Given that $S_1+S_2-S_3=\sqrt{3}$ and $ab=\frac{3\sqrt{2}}{2}$. Find:<br/>
$(1)$ The value of $\cos C$;<br/>
$(2)$ If $\sin A \sin B = \frac{\sqrt{2}}{3}$, find the length of side $c$.

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