
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
Consider a sequence defined as follows:
\[ a_0 = 1, \quad a_{k+1} = \frac{a_k}{4p} \text{ for } k \geq 0 \]
Calculate the sum of the first 50 terms of this sequence when $p=1004$ and express your result in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Provide the remainder when $m+n$ is divided by 1004.

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