
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  $c$  such that \[\sum_{i=1}^{101}x_i^2\geq cM^2\] whenever  $x_1,\ldots,x_{101}$  are real numbers such that  $x_1+\cdots+x_{101}=0$  and  $M$  is the median of  $x_1,\ldots,x_{101}$ .

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