
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 $k$ such that \[x_1^2 + x_2^2 + \dots + x_{101}^2 \geq kN^2\] whenever $x_1,x_2,\ldots,x_{101}$ are real numbers such that $x_1+x_2+\cdots+x_{101}=0$ and $N$ is the mean of the absolute values of $x_1,x_2,\ldots,x_{101}.$

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