
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 $ABC$, side $AC = 900$ and side $BC = 600$. Points $K$ and $L$ are located on segment $AC$ and segment $AB$ respectively, such that $AK = CK$, and $CL$ is the angle bisector of $\angle C$. Let point $P$ be the intersection of line segments $BK$ and $CL$. Point $M$ is located on line $BK$ such that $K$ is the midpoint of segment $PM$. If $AM = 360$, find $LP$.

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