
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
Triangle $ABC$ has $AC = 360$ and $BC = 240$. Points $K$ and $L$ are located on $\overline{AC}$ and $\overline{AB}$ respectively such that $AK = CK$. The line $\overline{CL}$ is the angle bisector of angle $C$. Let $P$ be the point of intersection of $\overline{BK}$ and $\overline{CL}$. Point $M$ is on line $BK$, such that $PM = 3 \times MK$ and $AM = 144$. Find the length of segment $LP$.

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