
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
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Find the number of ordered 11-tuples $(a_1, a_2, a_3, ..., a_{11})$ of integers, each such that the square of any number in the 11-tuple is equal to the sum of the other ten numbers.

**

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