
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
From the eight numbers $-3, -2, -1, 0, 1, 2, 3, 4$, randomly select three different numbers as the coefficients of the quadratic function $f(x) = ax^2 + bx + c \ (a \neq 0)$. Given that the graph of the quadratic function passes through the origin, and its vertex is in either the first or third quadrant, how many such quadratic functions are there?

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