
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 that the angles $A$, $B$, $C$ in $\triangle ABC$ have opposite sides $a$, $b$, $c$ respectively, and $\sin^2 B = 2 \sin A \sin C$.
1. If $\triangle ABC$ is an isosceles triangle, find the cosine value of the vertex angle $C$.
2. If $\triangle ABC$ is a right triangle with right angle at $B$ and $|BC| = \sqrt{2}$, find the area of $\triangle ABC$.

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