
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
Define a sequence  $(a_n)$  by  $a_1 =1, a_2 =2,$  and  $a_{k+2}=2a_{k+1}+a_k$  for all positive integers  $k$ . Determine all real numbers  $\beta >0$  which satisfy the following conditions:

(A) There are infinitely pairs of positive integers  $(p,q)$  such that  $\left| \frac{p}{q}- \sqrt{2} \, \right| < \frac{\beta}{q^2 }.$ (B) There are only finitely many pairs of positive integers  $(p,q)$  with  $\left| \frac{p}{q}- \sqrt{2} \,\right| < \frac{\beta}{q^2 }$  for which there is no index  $k$  with  $q=a_k.$ 

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