
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 \(\triangle{ABC}\), \( AB = 8 \), \(\angle{A} = 45^\circ\), and \(\angle{C} = 30^\circ\). Let \( H, D, \) and \( M \) be points on the line \( BC \) such that \( AH \perp BC \), \(\angle{BAD} = \angle{CAD}\), and \( BM = CM \). Point \( N \) is the midpoint of the segment \( HM \), and point \( P \) is on ray \( AD \) such that \( PN \perp BC \). Determine \( AP^2 \).

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