
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  $ n $   $(n\geq2)$  be an integer. Find the greatest possible value of the expression   $$ E=\frac{a_1}{1+a_1^2}+\frac{a_2}{1+a_2^2}+\ldots+\frac{a_n}{1+a_n^2} $$   if the positive real numbers  $a_1,a_2,\ldots,a_n$  satisfy  $a_1+a_2+\ldots+a_n=\frac{n}{2}.$  What are the values of  $a_1,a_2,\ldots,a_n$  when the greatest value is achieved?

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