
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 triangle \( \triangle ABC \), the ratio \( AC : AB = 1 : 2 \). The internal and external angle bisectors of \( \angle A \) are \( AE \) and \( AF \), respectively. Determine the ratio of the areas \( S_{\triangle ABC} : S_{\triangle ABE} : S_{\triangle ABF} \).

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