
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 an arithmetic sequence $\{ a_n \}$, given that $a_1 + a_7 + a_{13} = 6$, find the value of $S_{13}$.

(2) Determine the interval of decrease for the function $y = \log_{\frac{1}{2}}(x^2 + 2x - 3)$.

(3) In triangle $ABC$, given $a = \sqrt{3}$, $b = 1$, and $\angle A = \frac{\pi}{3}$, find the value of $\cos B$.

(4) Point $A$ lies on circle $O$ with diameter $BC$ and is distinct from points $B$ and $C$. Point $P$ is outside the plane $ABC$ and plane $PBC$ is perpendicular to plane $ABC$. Given $BC = 3$, $PB = 2\sqrt{2}$, and $PC = \sqrt{5}$, find the surface area of the circumscribed sphere of the tetrahedron $P-ABC$.

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