
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
How many distinct lines pass through the point  $(0, 2016)$  and intersect the parabola  $y = x^2$  at two lattice points?  (A lattice point is a point whose coordinates are integers.)

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