
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
18 scientists attended a conference, 10 of whom know an astounding piece of news. During a coffee break, all scientists pair up randomly, and in each pair, anyone who knows the news shares it with the other if the other does not already know it.

a) Find the probability that after the coffee break, the number of scientists who know the news is 13.

b) Find the probability that after the coffee break, the number of scientists who know the news is 14.

c) Let \( X \) represent the number of scientists who know the news after the coffee break. Find the expected value of \( X \).

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