
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
The infinite sequence  $a_0,a_1,\ldots$  is given by  $a_1=\frac12$ ,  $a_{n+1} = \sqrt{\frac{1+a_n}{2}}$ .  Determine the infinite product  $a_1a_2a_3\cdots$ . 

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