
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 $AC = 300$ and side $BC = 200$. Point $K$ is on $\overline{AC}$ such that $AK = KC$, and point $L$ is on $\overline{AB}$ such that $\overline{CL}$ is the angle bisector of $\angle C$. Let $P$ be the intersection point of $\overline{BK}$ and $\overline{CL}$. If $AM = 120$ where $M$ is a point on line $BK$ such that $K$ is the midpoint of $\overline{PM}$, determine the length of $LP$.

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