
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
Every quadratic equation can be reduced, by dividing it by the coefficient of the highest degree term, to the form

$$
x^{2}+c x+d=0
$$

The total number of distinct reduced quadratic equations in $p$-arithmetic equals $p^{2}$. Count how many of these equations have no roots, how many have one root, and how many have two distinct roots.

1) The expression $b^{2}-4 a c$ is called the discriminant of the equation $a x^{2}+b x+c=0$.

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