
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
Once, the beautiful Queen Guinevere, while in her parents' castle, asked King Arthur to send her 20 pearls. The roads are unsafe, and Arthur decided to send 40 pearls with different messengers, instructing them to take different routes. Along the way, the messengers may be ambushed by robbers. The probability that any given messenger will be robbed is \( p \), independently of the chosen route and the fate of the other messengers \((0 < p < 1)\).

King Arthur is contemplating: should he send two messengers, giving each 20 pearls; send three messengers, giving one 20 pearls and the other two 10 pearls each; or send four messengers, giving each 10 pearls? Which option should the king choose to maximize the probability that Queen Guinevere receives at least 20 pearls?

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