
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 $\angle B = 60^\circ$. Let $H$, $D$, and $M$ be points on line $BC$ with $AH \perp BC$, $\angle BAD = \angle CAD$, and $BM = MC$. Point $N$ is the midpoint of segment $HM$, and point $P$ is on ray $AD$ such that $PN \perp BC$. Compute the length of $AP$.

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