
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
Draw the line $k x + y + k^{3} = 0$ for $k = 0, \pm 0.3, \pm 0.6, \pm 0.9$. How many different intersection points do these seven lines determine? How many intersection points would there be if we drew 21 lines by substituting $k$ with the values $0, \pm 0.1, \pm 0.2, \pm 0.3, \ldots, \pm 0.9, \pm 1.0$?

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