
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
(1) In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. If $2b\cos B = a\cos C + c\cos A$, find $B$.

(2) In an arithmetic sequence $\{a_n\}$, if $a_1 + a_7 + a_{13} = 6$, then find $S_{13}$, the sum of the first 13 terms.

(3) In triangle $ABC$, the sides $a$, $b$, $c$ are opposite angles $A$, $B$, $C$ respectively. Given that $b=2$, $c=2\sqrt{2}$ and $C=\frac{\pi}{4}$, find the area of triangle $ABC$.

(4) Let $a > 0$ and $b > 0$, and $a + b = 4$. Find the minimum value of $\frac{1}{a} + \frac{4}{b}$.

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