
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 $\triangle ABC$, it is known that $a^2+c^2=b^2+\sqrt{2}ac$. Find the value of angle $B$. If $c=\sqrt{3}+1$, and one of the following conditions is chosen to ensure the existence and uniqueness of triangle $\triangle ABC$, find the area of $\triangle ABC$. The conditions are as follows: Condition ①: $\sin A=\frac{1}{2}$; Condition ②: $b=2$; Condition ③: $\cos A=\frac{\sqrt{3}}{2}$. Note: If the chosen condition does not meet the requirements, the second part will score 0 points. If multiple conditions are met, the first one will be scored.

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