
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 real number $k$ such that \[x_1^2 + x_2^2 + \dots + x_{51}^2 \geq kA^2\] whenever $x_1, x_2, \ldots, x_{51}$ are real numbers such that their mean is 0 and $A$ is the mean absolute deviation (i.e., $ A = \frac{1}{51} \sum_{i=1}^{51} |x_i - \mu|$ where $\mu = \frac{1}{51} \sum_{i=1}^{51} x_i = 0$).

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