
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
Define a $\frac{1}{p}$-array with a slightly altered structure, as follows: each first entry of a row is $\frac{1}{3p}$ times the first entry of the previous row, and each succeeding term in a row is still $\frac{1}{p}$ times the previous term in the same row. Using this structure, calculate the sum of all terms in a $\frac{1}{2008}$-array. If the sum can be expressed as $\frac{m}{n}$ where $m$ and $n$ are coprime, find the remainder when $m+n$ is divided by $2009$.

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