
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 $\triangle{ABC}$, $AB = 12, \angle{A} = 45^\circ$, and $\angle{C} = 60^\circ$. Let $H, D,$ and $M$ be points on line $BC$ such that $AH \perp BC$, $\angle{BAD} = \angle{CAD}$, and $BM = CM$. Point $N$ is the midpoint of segment $HM$, and point $P$ is on ray $AD$ such that $PN \perp BC$. Calculate $AP^2$.

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