
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
Consider a set of 24 parabolas defined as follows: all parabolas have the focus at the point $(0,0)$, and their directrix lines are of the form $y = ax + b$ with $a$ and $b$ integers such that $a \in \{-3, -2, -1, 0, 1, 2, 3\}$ and $b \in \{-2, -1, 0, 1, 2\}$. No three of these parabolas intersect at a common point. How many points in the plane are on exactly two of these parabolas?

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