
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 35 parabolas each having the focus at the origin \((0, 0)\) and their respective directrix of the form \(y = ax + b\) with \(a \in \{-3, -2, -1, 0, 1, 2, 3\}\) and \(b \in \{-4, -3, -2, -1, 1, 2, 3, 4\}\). Assuming that no three parabolas intersect at a common point, how many points in the plane lie on exactly two of these parabolas?

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