
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
An equation of the form \( x^{2} + px + q = 0 \) with nonzero integer coefficients \( p \) and \( q \) was written on the board. At times, different students came up to the board, erased the equation, then constructed and wrote another equation of the same form, whose roots were the coefficients of the erased equation. At some point, the newly constructed equation coincided with the one that was originally written on the board. What was the original equation written on the board?

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