
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 $ABC$, $AC = 540$ and $BC = 360$. Points $M$ and $N$ are located on $\overline{AC}$ and $\overline{AB}$ respectively, such that $AM = MC$, and $\overline{BN}$ is the angle bisector of angle $B$. Let $P$ be the point of intersection of $\overline{CM}$ and $\overline{BN}$, and let $K$ be the point on line $CM$ for which $M$ is the midpoint of $\overline{PK}$. If $AK = 240$, find $NP$.

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