
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
With two cannons that we always fire simultaneously, we shoot at a stationary target. The probabilities of a short shot, a hit, and a long shot for one cannon are $0.3, 0.2, 0.5$ respectively; for the other cannon, these probabilities are $0.6, 0.1, 0.3$. What is the probability that:

a) Out of 20 pairs of shots, we achieve at least one pair of hits?

b) Firing 3 pairs of shots, each time achieving at least one hit with either cannon?

c) Out of 2 pairs of shots, there are at most 3 short shots?

d) How many pairs of shots must be fired so that the probability of achieving at least one "simple hit" is at least 0.9?

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