
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
A $\frac{1}{2011}$-array is an infinite structured collection of numbers, constructed as follows:
\begin{align*}
1 \qquad \frac{1}{2011}\,\ \qquad \frac{1}{2011^2}\,\ \qquad \frac{1}{2011^3} \qquad &\cdots\\
\frac{1}{4022} \qquad \frac{1}{8044}\,\ \qquad \frac{1}{16088} \qquad &\cdots\\
\frac{1}{16088} \qquad \frac{1}{32176} \qquad &\cdots\\
\frac{1}{64352} \qquad &\cdots\\
&\ddots
\end{align*}

Where the first entry of each row is $\frac{1}{2} \times (\text{the first entry of the previous row})$, and each subsequent term in a row is $\frac{1}{2011}$ times the previous term in the same row. Determine the sum of all terms in the array, expressed as a simplified fraction $\frac{m}{n}$, and find the remainder when $m+n$ is divided by $2011$.

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