
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 $\frac{1}{1013}$-array is constructed in a similar manner to a $\frac{1}{p}$-array where each entry's value in the $r$th row and the $c$th column is given by $\left(\frac{1}{(2 \times 1013)^r}\right)\left(\frac{1}{1013^c}\right)$. Determine the sum of all terms in this array in the form of $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers, and find the remainder when $m+n$ is divided by $1013$.

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