
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 positive real number $d$ such that \[x_1^2 + x_2^2 + \dots + x_{51}^2 \geq d\bar{x}^2\] where $\bar{x}$ is the average (mean) of $x_1, x_2, ..., x_{51}$ and the $x_i$ numbers satisfy $x_1 + x_2 + \dots + x_{51} = 100$.

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