
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 the matrix $$\begin{pmatrix} 1 & a_{12} & … & a_{1i} & … & a_{1n} \\ 2 & a_{22} & … & a_{2i} & … & a_{2n} \\ 3 & a_{32} & … & a_{3i} & … & a_{3n} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ n & a_{n2} & … & a_{ni} & … & a_{nn}\end{pmatrix}$$ each row forms a geometric sequence with a common ratio of 2. The sum of the elements in the $i$-th column is $S_i$. Then, the value of $$\lim_{n\rightarrow \infty} \frac {S_{n}}{n^{2}\cdot 2^{n}}$$ is \_\_\_\_\_\_.

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