
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 in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ 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 $B$ as the right angle, and $|BC|= \sqrt {2}$, find the area of $\triangle ABC$.

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