
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, staying at her parents' castle, asked King Arthur to send her 20 pearls. The roads are unsafe, so Arthur decided to send 40 pearls with different messengers, instructing them to take different routes. Each messenger might encounter robbers along the way. The probability that any given messenger will be robbed is $p$, independently of the chosen road and the fate of other messengers $(0 < p < 1)$.

King Arthur is contemplating the following options: 

- Send two messengers, giving each 20 pearls.
- Send three messengers, giving one 20 pearls and two 10 pearls each.
- Send four messengers, giving each 10 pearls.

Which option should the king choose to maximize the probability that the queen receives at least 20 pearls?

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