
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
Grogg’s favorite positive integer is  $n\ge2$ , and Grogg has a lucky coin that comes up heads with some fixed probability  $p$ , where  $0<p<1$ . Once each day, Grogg flips his coin, and if it comes up heads, he does two things:
[list=1]
[*] He eats a cookie. [/*]
[*] He then flips the coin  $n$  more times. If the result of these  $n$  flips is  $n-1$  heads and  $1$  tail (in any order), he eats another cookie. [/*]
[/list]
He never eats a cookie except as a result of his coin flips. Find all possible values of  $n$  and  $p$  such that the expected value of the number of cookies that Grogg eats each day is exactly  $1$ .

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