
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
On a windy day, Jill, struggling to ensure her notes don't fly away, accidentally mixes up several pages containing polynomial equations of a similar kind. She recalls that one of the equations is a monic polynomial of degree \(n\) with real coefficients where the first two terms after \(x^n\) are \(a_{n-1}x^{n-1}\) and \(a_{n-2}x^{n-2}\). She can barely make out that \(a_{n-1} = a_{n-2}\) for this particular polynomial. Despite this mix-up, Jill aims to find the sum of the squares of the roots of the polynomial. Although she doesn’t know the actual values of \(n\) or \(a_{n-1}\), she attempts to determine the greatest lower bound of this sum.

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