
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 \(\left\{a_{i}\right\}_{i \geq 0}\) be a sequence of real numbers defined by
$$
a_{n+1}=a_{n}^{2}-\frac{1}{2^{2020 \cdot 2^{n}-1}}
$$
for \(n \geq 0\). Determine the largest value for \(a_{0}\) such that \(\left\{a_{i}\right\}_{i \geq 0}\) is bounded.

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