
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 largest real number \(d\) such that \[x_1^2 + x_2^2 + \dots + x_{51}^2 \geq d\left(\frac{x_1+x_2+\dots+x_{51}}{51}\right)^2\] whenever \(x_1, x_2, \ldots, x_{51}\) are real numbers such that \(x_1+x_2+\cdots+x_{51} = M\) and \(M\) is the median of \(x_1, x_2, \ldots, x_{51}\).

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