
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 real number $c$ such that \[x_1^2 + x_2^2 + \dots + x_{51}^2 \geq c\bar{X}^2\] wherever $x_1, x_2, \ldots, x_{51}$ are real numbers satisfying $x_1 + x_2 + \dots + x_{51} = 0$, and $\bar{X}$ is the average of the absolute values of $x_1, x_2, \ldots, x_{51}$.

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