
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 20 parabolas defined with the focus at the point $(0,0)$. The directrices of these parabolas are of the form $y = ax + b$ with $a$ and $b$ being integers. Here, $a \in \{-1,0,1\}$ and $b \in \{-2,-1,0,1,2\}$. Determine how many points in the plane are on exactly two of these parabolas, assuming no three parabolas share a common point.

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