
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 $\frac{1}{1004}$-array is constructed similarly to a $\frac{1}{2008}$-array, where each succeeding term in a row is $\frac{1}{1004}$ times the previous term in the same row, and the first term of each row is $\frac{1}{2 \times 1004}$ times the first term of the previous row. If the sum of all the terms in this array is expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find the remainder when $a+b$ is divided by $1004$.

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