
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 $x_1, x_2, \dots, x_{50}$ be real numbers such that $x_1 + x_2 + \dots + x_{50} = 1$ and
\[
\frac{x_1}{1 - x_1} + \frac{x_2}{1 - x_2} + \dots + \frac{x_{50}}{1 - x_{50}} = 1.
\]
Also, $x_1^2 + x_2^2 + \dots + x_{50}^2 = \frac{1}{2}$. Find
\[
\frac{x_1^2}{1 - x_1} + \frac{x_2^2}{1 - x_2} + \dots + \frac{x_{50}^2}{1 - x_{50}}.
\]

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