
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  $n$  be a fixed positive integer. For integer  $m$  satisfying  $|m| \le n$ , define  $S_m=\displaystyle\sum_{\substack{i-j=m 0 \le i,j \le n}} \dfrac{1}{2^{i+j}}$ . Then \[\lim_{n\to \infty} \left( S_{-n}^2 + S_{-n+1}^2 +... + S_n^2 \right)\] can be expressed in the form  $\dfrac{p}{q}$  for relatively prime positive integers  $p,q$ . Compute  $100p+q$ .

*Proposed by Vincent Huang*

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