
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 $\frac 15$ -array is a structured, infinite, collection of numbers. For example, a $\frac 15$ -array is constructed as follows:
\[
\begin{align*} 
1 \quad \frac 15\, \quad \frac 1{25}\, \quad \frac 1{125} \quad &\cdots\\ 
\frac 1{10} \quad \frac 1{50}\, \quad \frac{1}{250} \quad &\cdots\\ 
\frac 1{100} \quad \frac 1{500} \quad &\cdots\\ 
\frac 1{1000} \quad &\cdots\\ 
&\ddots 
\end{align*}
\]
In general, the first entry of each row is $\frac{1}{5}$ times the first entry of the previous row, and each succeeding term in a row is $\frac{1}{5}$ times the previous term in the same row. If the sum of all the terms in a $\frac{1}{1001}$ -array can be written in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers, find the remainder when $m + n$ is divided by $1001$.

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