
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 the diagram, circle $P$ has a radius of 10 units. Chord $EF$ has a length of 12 units and is parallel to segment $LM$. Another chord, $GH$, also parallel to segment $LM$, has a length of 16 units. Points $L$, $M$, $P$, and $N$ are collinear with $LP = 24$ units. Calculate the area of triangle $LGH$. [asy]
draw(Circle((0,0),10));
dot((0,0));
label("$P$",(0,0),S);
label("$M$",(-10,0),SW);
label("$N$",(10,0),SE);
label("$L$",(-24,0),W);
draw((-24,0)--(10,0));
label("$E$",(-6,sqrt(64)),NW);
label("$F$",(6,sqrt(64)),NE);
draw((-24,0)--(-6,sqrt(64)));
draw((-24,0)--(6,sqrt(64)));
draw((-6,sqrt(64))--(6,sqrt(64)));
label("$G$",(-8,sqrt(36)),NW);
label("$H$",(8,sqrt(36)),NE);
draw((-24,0)--(-8,sqrt(36)));
draw((-24,0)--(8,sqrt(36)));
draw((-8,sqrt(36))--(8,sqrt(36)));
[/asy]

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