
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

Refer to triangle \( \triangle PQR \) and triangle \( \triangle DEF \):

1. If \( \triangle PQR \) is the inscribed triangle of \( \triangle ABC \), and \( \triangle DEF \) is a circumscribed triangle around \( \triangle ABC \), with \( AF : FB = \lambda_1 \), \( BD : DC = \lambda_2 \), and \( CE : EA = \lambda_3 \), then:

   (1) \( \frac{S_{\triangle PQR}}{S_{\triangle ABC}} = \frac{\left(1 - \lambda_1 \lambda_2 \lambda_3 \right)^2}{\left(1 + \lambda_3 + \lambda_1 \lambda_3 \right) \left(1 + \lambda_1 + \lambda_1 \lambda_2 \right) \left(1 + \lambda_2 + \lambda_2 \lambda_3 \right)} \).

   (2) \( \frac{S_{\triangle DEF}}{S_{\triangle ABC}} = \frac{1 + \lambda_1 \lambda_2 \lambda_3}{\left(1 + \lambda_1 \right) \left(1 + \lambda_2 \right) \left(1 + \lambda_3 \right)} \).

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