
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 $\triangle ABC$, the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given vectors $\overset{→}{m}=( \frac{a}{2}, \frac{c}{2})$, $\overset{→}{n}=(\cos C,\cos A)$, and $\overset{→}{n}· \overset{→}{m}=b\cos B$.
(1) Find the value of $B$;
(2) If $\cos \frac{A-C}{2}= \sqrt{3}\sin A$, and $|\overset{→}{m}|= \sqrt{5}$, find the area of $\triangle ABC$.

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