
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

Given the following events and probabilities:

\[ A = \{\text{The scientist hits the first basket}\} \]
\[ B = \{\text{The scientist hits the second basket}\} \]

Find the probability of the event \(A \cap B\):
\[ \mathrm{P}(A \cap B) = \mathrm{P}(A) + \mathrm{P}(B) - \mathrm{P}(A \cup B) \]

Assume the probability that the scientist does not hit a particular basket in a single throw is given by \(1-p\). The probability that the scientist does not hit the basket in six throws is \((1-p)^6\). Therefore:
\[ \mathrm{P}(A) = \mathrm{P}(B) = 1 - (1-p)^6 \]

Given that hitting both baskets simultaneously is impossible, the probability that the scientist hits at least one basket is \(2p\). The probability of not hitting either basket in a single throw is \(1-2p\), and the probability of not hitting either basket in six throws is \((1-2p)^6\). Hence:
\[ \mathrm{P}(A \cup B) = 1 - (1-2p)^6 \]

As a result, we get:
\[ \mathrm{P}(A \cap B) = 2 - 2(1-p)^6 - 1 + (1-2p)^6 = 1 - 2(1-p)^6 + (1-2p)^6 \]

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