
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 \[x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 \geq cM^2\] whenever $x_1,x_2,x_3,x_4,x_5$ are real numbers such that $x_1+x_2+x_3+x_4+x_5=0$ and $M$ is the median of $x_1,x_2,x_3,x_4,x_5.$

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