
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 the sequence \( \left\{a_{i}\right\}_{i=0}^{\infty} \) be defined by \( a_{0}=\frac{1}{2} \) and \( a_{n}=1+\left(a_{n-1}-1\right)^{2} \). Find the product
\[ \prod_{i=0}^{\infty} a_{i} = a_{0} a_{1} a_{2} \ldots \]

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