
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
Given a sequence of positive terms \(\{a_n\}\) satisfying the following conditions:

\[ a_1 = 1, \]
\[ (n+1) a_n^2 - 2n a_{n+1}^2 + \sqrt{n^2 + n} a_n a_{n+1} = 0, \]
\[ z_n = \prod_{k=1}^n \left(1 - \frac{i}{a_k}\right) \quad \text{(i is the imaginary unit).} \]

Find the value of \(\left| z_{2019} - z_{2020} \right|\).

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