
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
In terms of  $n\ge2$ , find the largest constant  $c$  such that for all nonnegative  $a_1,a_2,\ldots,a_n$  satisfying  $a_1+a_2+\cdots+a_n=n$ , the following inequality holds:
\[\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.\]

*Calvin Deng.*

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