
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
Choose one of the following three conditions to supplement the question below and complete the solution.<br/>①$a+c\sin B=b\cos C$；②$\sqrt{2}a=\sqrt{2}bcosC-c$；③$c+2b\cos B\sin C=0$.<br/>Let the sides opposite angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$, respectively. Given _____.<br/>$(1)$ Find $B$;<br/>$(2)$ If $b=\sqrt{5}$ and $a=1$, find the area of $\triangle ABC$. Note: If multiple conditions are chosen to solve separately, the first solution will be scored.

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