
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 50 parabolas, each having the focus at the point \( (0,0) \) and directrix lines of the form \( y = ax + b \), where \( a \) and \( b \) are integers such that \( a \in \{-3,-2,-1,0,1,2,3\} \) and \( b \in \{-4,-3,-2,-1,0,1,2,3,4\} \). Determine how many points in the plane are shared by exactly two of these parabolas, assuming no three parabolas have a common intersection point.

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