
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 the conditions $b^2+\sqrt{2}ac=a^2+c^2$, $a\cos B=b\sin A$, and $\sin B+\cos B=\sqrt{2}$, choose one of these conditions to fill in the blank below and solve the problem. In triangle $\triangle ABC$, _______, $A=\frac{π}{3}$, and $b=\sqrt{2}$. Find:

$(1)$ The measure of angle $B$.

$(2)$ The area of triangle $\triangle ABC$.

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