
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 $m \ge 7$ be an odd integer, and let $E(m)$ denote the number of quintuples $(a_1, a_2, a_3, a_4, a_5)$ of distinct integers with $1 \le a_i \le m$ for all $i$ such that $m$ divides $a_1+2a_2+3a_3+4a_4+5a_5$. There is a polynomial
\[ r(x) = d_4x^4 + d_3x^3 + d_2x^2 + d_1x + d_0 \]
such that $E(m) = r(m)$ for all odd integers $m \ge 7$. What is $d_2$?
A) 10
B) 12
C) 14
D) 16
E) 18

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