
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
The graph of the function \(y = ax^2 + bx + c\) is a downward-opening parabola. The coefficients \(a\), \(b\), and \(c\) are distinct and each takes values from the set \(A = \{n \mid n \leq 5, n \in \mathbb{Z}\}\). How many of these parabolas pass through the point \((0, -1)\)?

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