
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 $\triangle{XYZ}, XY=12, \angle{X}=45^\circ$, and $\angle{Z}=60^\circ$. Let $H, D,$ and $M$ be points on the line $YZ$ such that $XH\perp{YZ}$, $\angle{YXD}=\angle{DXZ}$, and $YM=MZ$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $XD$ such that $PN\perp{YZ}$. Then $XP^2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

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