
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 $\alpha, \beta, \gamma$ be three given quantities and $x_{1}, x_{2}$ be the two roots of the following equation:

$$
x^{2} + p x + q = 0
$$

What relationship must hold between $p$ and $q$ for the following equality to be true?

$$
\alpha x_{1}^{2} + \beta x_{1} + \gamma = \alpha x_{2}^{2} + \beta x_{2} + \gamma
$$

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