
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 sequences  $a_1,a_2,a_3,\cdots$  and  $b_1,b_2,b_3,\cdots$  of real numbers satisfying  $\displaystyle a_{n+1}+b_{n+1}=\frac{a_n+b_n}{2}$  and  $\displaystyle a_{n+1}b_{n+1}=\sqrt{a_nb_n}$  for all  $n\geq1$ . Suppose  $b_{2016}=1$  and  $a_1>0$ . Find all possible values of  $a_1$ 

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