
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 a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their corresponding opposite sides $a$, $b$, $c$. It is known that $2a\sin (C+ \frac{\pi}{6})=b+c$.

1. Find the value of angle $A$.
2. If $B= \frac{\pi}{4}$ and $b-a= \sqrt{2}- \sqrt{3}$, find the area of $\triangle ABC$.

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