
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
Eighteen scientists attended a conference, and exactly 10 of them know an astonishing piece of news. During the coffee break, all the scientists split into random pairs, and in each pair, anyone who knows the news will share it with the other person if they do not already know it.

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

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

c) Let $X$ denote 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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