
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
In rectangle \(ABCD\), \(AB=\sqrt{3}\), \(BC=a\), and \(PA \perp\) plane \(ABCD\), with \(PA=4\).

1. If there exists a point \(Q\) on side \(BC\) such that \(PQ \perp QD\), find the range of values for \(a\).
2. When there exists a unique point \(Q\) on \(BC\) such that \(PQ \perp QD\), find the angle between skew lines \(AQ\) and \(PD\).
3. If \(a=4\) and \(PQ \perp QD\), find the measure of the dihedral angle \(A-PD-Q\).

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