
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
Queen Guinevere once asked King Arthur to send her 20 pearls. The roads are unsafe, so Arthur decided to send 40 pearls using different couriers and instructed them to take different routes. On the way, the couriers might encounter robbers. The probability that any given courier will be robbed is $p$, independently of the chosen route and the fate of the other couriers $(0 < p < 1)$.

King Arthur is contemplating whether to send:
1. Two couriers, giving each 20 pearls,
2. Three couriers, giving one 20 pearls and each of the other two 10 pearls, or
3. Four couriers, giving each 10 pearls.

Which option should King Arthur choose to maximize the probability that Queen Guinevere will receive at least 20 pearls?

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