
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
The people in an infinitely long line are numbered  $1,2,3,\dots$ . Then, each person says either ``Karl" or ``Lark" independently and at random. Let  $S$  be the set of all positive integers  $i$  such that people  $i$ ,  $i+1$ , and  $i+2$  all say ``Karl," and define  $X = \sum_{i \in S} 2^{-i}$ . Then the expected value of  $X^2$  is  $\frac{m}{n}$ , where  $m$  and  $n$  are positive integers with  $\gcd(m,n)=1$ . Compute  $100m+n$ .

*Proposed by Ankit Bisain*

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