
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
Given an integer  $n \geq 3$ , determine the maximum value of product of  $n$  non-negative real numbers  $x_1,x_2, \ldots , x_n$  when subjected to the condition
\begin{align*} \sum_{k=1}^n \frac{x_k}{1+x_k}  =1 \end{align*}

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