
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
For every positive integer  $ n$  determine the highest value  $ C(n)$ , such that for every  $ n$ -tuple  $ (a_1,a_2,\ldots,a_n)$  of  pairwise distinct integers

 $ (n \plus{} 1)\sum_{j \equal{} 1}^n a_j^2 \minus{} \left(\sum_{j \equal{} 1}^n a_j\right)^2\geq C(n)$ 

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