
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

On the extensions of the sides of triangle \(ABC\), points \(A_1\), \(B_1\), and \(C_1\) are taken such that \(\overrightarrow{AB_1} = 2 \overrightarrow{AB}\), \(\overrightarrow{BC_1} = 2 \overrightarrow{BC}\), and \(\overrightarrow{CA_1} = 2 \overrightarrow{AC}\). Find the area of triangle \(A_1B_1C_1\), given that the area of triangle \(ABC\) is \(S\).

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