
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 sequences  $(a_{n})$ ,  $(b_{n})$  are deﬁned by  $a_{1} = \alpha$ ,  $b_{1} = \beta$ ,  $a_{n+1} = \alpha a_{n} - \beta b_{n}$ ,   $b_{n+1} = \beta a_{n} + \alpha b_{n}$  for all  $n > 0.$  How many pairs  $(\alpha, \beta)$  of real numbers are there such that  $a_{1997} = b_{1}$  and  $b_{1997} = a_{1}$ ?

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