
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 right triangle \( ABC \) with right angle at \( C \), let \( P_1, P_2, P_3 \) be equilateral triangles constructed on sides \( AB, BC, \) and \( CA \). Determine the ratio of the area of \( P_1 \) to the sum of the areas of \( P_2 \) and \( P_3 \).

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