
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 $\triangle ABC$, with angles A, B, and C, the sides opposite these angles are labeled as $a, b,$ and $c$ respectively. Given the vectors $\overrightarrow{m}=\left( \frac{a}{2}, \frac{c}{2} \right)$ and $\overrightarrow{n}=(\cos C, \cos A)$, it is also known that $\overrightarrow{n} \cdot \overrightarrow{m} = b\cos B$.
1. Determine the value of angle B;
2. If $\cos \left( \frac{A-C}{2} \right) = \sqrt{3}\sin A$ and $|\overrightarrow{m}|=\sqrt{5}$, find the area of $\triangle ABC$.

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