
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 the set of 50 parabolas defined as follows: all parabolas have a common focus at the point $(0,0)$, and the directrix lines have the form $y = ax + b$, where $a \in \{-3, -2, -1, 0, 1, 2, 3\}$ and $b \in \{-4, -3, -2, -1, 0, 1, 2, 3, 4\}$. No three of these parabolas have 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}
user