
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
Determine the largest constant \(c\) such that \[ x_1^2 + x_2^2 + \dots + x_{100}^2 \geq cM^2 \] whenever \(x_1, x_2, \ldots, x_{100}\) are real numbers satisfying \(x_1 + x_2 + \cdots + x_{100} = 0\) and \(M\) is the median of \(x_1, x_2, \ldots, x_{100}\).

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