
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 the pyramid \( P-ABCD \), the base \( ABCD \) is a rectangle. The lateral edge \( PA \) is perpendicular to the base \( ABCD \), with \( AB = \sqrt{3} \), \( BC = 1 \), and \( PA = 2 \). Let \( E \) be the midpoint of \( PD \).

1. Find the cosine of the angle between the lines \( AC \) and \( PB \).
2. In the face \( PAB \), find a point \( N \) such that \( NE \perp \) plane \( PAC \), and calculate the distances from point \( N \) to \( AB \) and \( AP \).

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