
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
Find the smallest non-negative value of $k$ such that \[x_1^2 + x_2^2 + \dots + x_{50}^2 \geq kx_{25}^2\] whenever $x_1, x_2, \ldots, x_{50}$ are real numbers such that $x_1 + x_2 + \cdots + x_{50} = 0$ and $x_1 \le x_2 \le \ldots \le x_{50}$.

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