
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 $n\ge 7$ be an odd integer, and let $E(n)$ denote the number of quadruples $(b_1, b_2, b_3, b_4)$ of distinct integers with $1\le b_i \le n$ for all $i$ such that $n-2$ divides $b_1+b_2+b_3+b_4$. There is a polynomial \[r(x) = d_3x^3+d_2x^2+d_1x+d_0\]such that $E(n) = r(n)$ for all odd integers $n\ge 7$. Find the value of $d_1$.

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