
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
Let  $ P,\ Q,\ R$  be the points such that  $ BP: PC=CQ: QA=AR: RB=t: 1-t\ (0<t<1)$  for a triangle  $ ABC$ .
Denote  $ K$  by the area of the triangle with segments  $ AP,\ BQ,\ CR$  as side lengths and  $ L$  by triangle  $ ABC$ , find  $ \frac{K}{L}$  in terms of  $ t$ .

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