
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
A sequence \( a_{1}, a_{2}, a_{3}, \ldots \) of positive reals satisfies \( a_{n+1} = \sqrt{\frac{1 + a_{n}}{2}} \). Determine all \( a_{1} \) such that \( a_{i} = \frac{\sqrt{6} + \sqrt{2}}{4} \) for some positive integer \( i \).

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