
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
Let $n$ be an even integer greater than or equal to 4, and let $E(n)$ denote the number of quintuples $(b_1, b_2, b_3, b_4, b_5)$ of distinct integers with $1 \leq b_i \leq n$ for all $i$, such that $n$ divides $b_1 + b_2 + b_3 + b_4 + b_5$. Find the coefficient of $x$ in the polynomial $p(x) = d_4x^4 + d_3x^3 + d_2x^2 + d_1x + d_0$ such that $E(n) = p(n)$ for all even integers $n \geq 4$.

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