
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
Points \( A_1 \), \( B_1 \), and \( C_1 \) are taken on the extensions of the sides of triangle \( ABC \) 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}
user