
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
Let  $X_1X_2X_3$  be a triangle with  $X_1X_2 = 4, X_2X_3 = 5, X_3X_1 = 7,$  and centroid  $G$ . For all integers  $n \ge 3$ , define the set  $S_n$  to be the set of  $n^2$  ordered pairs  $(i,j)$  such that  $1\le i\le n$  and  $1\le j\le n$ . Then, for each integer  $n\ge 3$ , when given the points  $X_1, X_2, \ldots , X_{n}$ , randomly choose an element  $(i,j)\in S_n$  and define  $X_{n+1}$  to be the midpoint of  $X_i$  and  $X_j$ . The value of

\[ \sum_{i=0}^\infty \left(\mathbb{E}\left[X_{i+4}G^2\right]\left(\dfrac{3}{4}\right)^i\right) \]

can be expressed in the form  $p + q \ln 2 + r \ln 3$  for rational numbers  $p, q, r$ . Let  $|p| + |q| + |r| = \dfrac mn$  for relatively prime positive integers  $m$  and  $n$ . Compute  $100m+n$ .

Note:  $\mathbb{E}(x)$  denotes the expected value of  $x$ .

*Proposed by Yang Liu*

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