
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 a point A is on plane α, and a point P is not on plane α. The vector $\overrightarrow {PA} = \left(- \frac{\sqrt {3}}{2}, \frac{1}{2}, x\right)$ where $x > 0$, and the magnitude of vector $\overrightarrow{PA}$ is $\sqrt {3}$. A normal vector to the plane $\alpha$ is given by $\overrightarrow {n} = (0, - \frac{1}{2}, - \sqrt{2})$.
1. Find the value of $x$.
2. Determine the angle between line PA and plane α.

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