
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
Given infinite real sequences \(a_1, a_2, \ldots\) and \(b_1, b_2, \ldots\), for \(n \geq 1\), the following conditions hold:

\[a_{n+1} + b_{n+1} = \frac{a_n + b_n}{2}\]
\[a_{n+1} b_{n+1} = \sqrt{a_n b_n}\]

Given \(b_{2016} = 1\) and \(a_1 > 0\), find all possible values of \(a_1\).

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