
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
Is it possible to find constants $a$, $b$, $c$ such that the equation $1(n^2-1^2) + 2(n^2-2^2) + \ldots + n(n^2-n^2) = an^4 + bn^2 + c$ holds true for all positive integers $n$? If such constants exist, determine the values of $a$, $b$, $c$; if not, explain why.

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