
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 as focus the point $(0,0)$ and the directrix lines have the form $y=ax+b$ with $a$ and $b$ integers such that $a\in \{-3,-2,-1,0,1,2,3\}$ 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 two of these parabolas?

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