
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 be an odd integer greater than or equal to 5, D'(m) denote the number of quadruples (a1, a2, a3, a4) of distinct integers with 1 ≤ ai ≤ 2m for all i, such that m divides a1 + a2 + a3 + a4 and ai + aj ≤ 2m for all i ≠ j, and q(m) = c3m^3 + c2m^2 + c1m + c0 is a polynomial such that D(m) = q(m) for all odd integers m greater than or equal to 5. Find the coefficient c2.

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