
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 a competition with 557 students, each student answers questions from a set of 30 problems. Estimate \( A \), the number of subsets \( R \subseteq P \) where some student answered only the questions in \( R \) correctly and no others. An accurate estimate \( E \) earns points based on the formula \(\max \left(0,\left\lfloor 20-\frac{2}{3}|A-E|\right\rfloor\right)\).

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