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

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