
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 \( a, b, c \) are all distinct and take values from the set \( A = \{n \mid -5 \leq n \leq 5, n \in \mathbf{Z}\} \). Determine the number of these parabolas that pass through the point \( (0, -1) \).

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