
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
Given that $A \in \alpha$, $P \notin \alpha$, $\overrightarrow{PA} = (-\frac{\sqrt{3}}{2}, \frac{1}{2}, x)$ where $x > 0$, and $| \overrightarrow{PA} | = \sqrt{3}$, and a normal vector of plane $\alpha$ is $\overrightarrow{n} = (0, -\frac{1}{2}, -\sqrt{2})$.

1. Find the value of $x$.
2. Find the angle between line $PA$ and plane $\alpha$.

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