
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 20 parabolas with focus at $(2,-1)$, where the directrix lines are given by the equations $x=cy+d$ with $c \in \{-1, 0, 1\}$ and $d \in \{-5, -3, 0, 3, 5\}$. Every pair of directrix lines with non-equal slopes intersects, and no three parabolas share a common point. Determine how many points in the plane are on exactly two of these parabolas.

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