
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 \[ x_1^3 + x_2^3 + \cdots + x_{11}^3 \geq cM^3 \] whenever \( x_1, x_2, \ldots, x_{11} \) are real numbers such that \( x_1 + x_2 + \cdots + x_{11} = 0 \) and \( M \) is the maximum of \( x_1, x_2, \ldots, x_{11} \).

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