
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
Given a triangle \(ABC\) with points \(A_1\), \(B_1\), \(C_1\) on the sides \(a\), \(b\), \(c\) of the triangle such that:

\[
\frac{BA_1}{A_1C} = k_1, \quad \frac{CB_1}{B_1A} = k_2, \quad \frac{AC_1}{C_1B} = k_3 
\]

with \(k_1 > 1\), \(k_2 > 1\), and \(k_3 > 1\). Given the area \(S\) of triangle \(ABC\), find the area of triangle \(PQR\) formed by the lines \(AA_1\), \(BB_1\), and \(CC_1\).

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