
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 36 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 \(b \in \{-4, -3, -2, -1, 1, 2, 3, 4\}\). 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}
user