
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
Let's define the distance between numbers as the absolute value of their difference. It is known that the sum of the distances from sixteen consecutive natural numbers to a certain number \(a\) is 636, and the sum of the distances from these same sixteen numbers to the number \(a^2\) is 591. Find all possible values of \(a\).

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