
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  $a_1, a_2, \ldots$  and  $b_1, b_2, \ldots$  be sequences such that  $a_ib_i - a_i - b_i = 0$  and  $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$  for all  $i \ge 1$ .  If  $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$ , then what is  $b_{6}$ ?

*Proposed by Andrew Wu*

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