
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
In a terrifying thunderstorm, \(n\) gnomes are climbing up a rope ladder in a chain. If there is a lightning strike, then out of fear, each gnome can fall with a probability \(p\) (where \(0 < p < 1\)). If a gnome falls, it also knocks down all the gnomes below it. Determine:

a) The probability that exactly \(k\) gnomes will fall.

b) The expected number of fallen gnomes.

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