
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
Let \( a_{i} \) (where \( i \in \mathbf{Z}_{+} \) and \( i \leq 2020 \)) be non-negative real numbers such that \( \sum_{i=1}^{2020} a_{i}=1 \). Find the maximum value of \( \sum_{\substack{i \neq j \\ i, j}} a_{i} a_{j} \).

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