
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 were unsafe, so Arthur decided to send 40 pearls via different couriers, instructing them to take different routes. Couriers might encounter robbers on their way. The probability that each individual courier will be robbed is $p$, independent of the route taken and the fate of other couriers $(0<p<1)$.

The king is considering: should he send two couriers, giving each 20 pearls; send three couriers, giving one 20 pearls, and two others 10 pearls each; or send four couriers, 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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