
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  $\{a_i\}^\infty_{i=0}$  be defined by  $a_0 =\frac12$  and  $a_n = 1 + (a_{n-1} - 1)^2$ . Find the product   $$ \prod_{i=0}^\infty a_i=a_0a_1a_2\ldots $$  

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