
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 that \( a_{1} = a > 0 \), for \( n \geq 2 \), \( a_{n} = \prod_{i=1}^{n-1}(1 + a_{i}) \). Find the real number \( a \) such that for all positive integers \( n \), 

\[
\sum_{k=1}^{n} \frac{1}{1 + a_{k}} < \frac{1}{2021}
\]

and for any \( c < \frac{1}{2021} \), there exists a positive integer \( n \) such that 

\[
\sum_{k=1}^{n} \frac{1}{1 + a_{k}} > c.
\]

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