
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 are positive reals  $x_1, x_2,..., x_n$  such that  $\sum\frac {1}{1+x_i^2}=1$ . Find the minimal value of the expression  $\frac{\sum x_i}{\sum \frac{1}{x_i}}$  and find when it is achieved. 

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