
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
Let \( a \) be the proportion of teams that correctly answered problem 1 on the Guts round. Estimate \( A = \lfloor 10000a \rfloor \). The score is calculated by \(\max (0, \lfloor 20 - |A - E| / 20 \rfloor) \), where \( E \) is your estimate of \( A \).

For reference, problem 1 was: 
Two hexagons are attached to form a new polygon \( P \). What is the minimum number of sides that \( P \) can have?

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