
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 $x_1,$ $x_2,$ $x_3,$ $\dots,$ $x_{50}$ be positive real numbers such that $k(x_1^2 + x_2^2 + x_3^2 + \dots + x_{50}^2) = 1$ where $k > 1$. Find the minimum value of
\[\frac{x_1}{1 - x_1^2} + \frac{x_2}{1 - x_2^2} + \frac{x_3}{1 - x_3^2} + \dots + \frac{x_{50}}{1 - x_{50}^2}.\]

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