
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 there a constant $a$, $b$, $c$ such that the equation $1^2+2^2+3^2+…+n^2+(n-1)^2+…+2^2+1^2 = an(bn^2+c)$ holds for all $n\in \mathbb{N}^*$? If there is, find the values of $a$, $b$, $c$ and provide proof; if not, explain why not.

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