
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
Xiao Ming selects an arithmetic sequence \(\{ a_n \}\) in which all terms are non-zero integers, and then lists 2020 quadratic equations in \(x\): \( E_i : a_i x^2 + a_{i+1} x + a_{i+2} = 0 \) for \(i = 1, 2, \cdots, 2020\). He then writes all the roots of these equations on the blackboard. What is the maximum possible number of distinct real roots written on the blackboard?

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