
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. A number $x$ is randomly selected from the interval $[0,4]$. What is the probability that the event $\{ -1 \leq \log_{\frac{1}{2}}(x+\frac{1}{2}) \leq 1 \}$ occurs?

2. $A$, $B$, $C$, and $D$ are four points on the surface of a sphere. In $\triangle ABC$, $\angle BAC = \frac{2\pi}{3}$, $AB=AC$, $AD \perp$ plane $ABC$, $AD=6$, and $AB=2\sqrt{3}$. What is the surface area of the sphere?

3. Given the function $f(x)=\frac{1}{x+1}$, $O$ is the origin, $A_{n}(n,f(n))(n \in \mathbb{N}^{*})$, $\overrightarrow{i} = (0,1)$, and $\theta_{n}$ is the angle between $\overrightarrow{OA_{n}}$ and $\overrightarrow{i}$. What is the value of $\frac{\cos\theta_{1}}{\sin\theta_{1}} + \frac{\cos\theta_{2}}{\sin\theta_{2}} + \dots + \frac{\cos\theta_{2017}}{\sin\theta_{2017}}$?

4. In quadrilateral $ABCD$, $AB=2$, $BC=2\sqrt{2}$, $AD=\sqrt{2}CD$, and $\overrightarrow{AC} \cdot \overrightarrow{CD} = 0$. What is the maximum value of $|\overrightarrow{BD}|$?

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