
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 a triangle $\triangle ABC$ with side lengths $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively. The perimeter of $\triangle ABC$ is $\sqrt{2} + 1$, and it is known that $\sin A + \sin B = \sqrt{2} \sin C$.

(1) Find the value of $c$.
(2) If the area of $\triangle ABC$ is $\frac{1}{6} \sin C$, find the measure of angle $C$ in degrees.

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