
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  $n \ge 3$  be an integer. Find the minimal value of the real number  $k_n$  such that for all positive numbers  $x_1, x_2, ..., x_n$  with product  $1$ , we have   $$ \frac{1}{\sqrt{1 + k_nx_1}}+\frac{1}{\sqrt{1 + k_nx_2}}+ ... + \frac{1}{\sqrt{1 + k_nx_n}} \le n - 1. $$  

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