
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  $a_1, a_2, a_3,...$  and  $b_1, b_2, b_3,... $ suffices for all positive integers  $n$  of the following recursion: $a_{n+1} = a_n - b_n$  and  $b_{n+1} = 2b_n$ ,  if  $a_n \ge b_n$ , $a_{n+1} = 2a_n$  and  $b_{n+1} = b_n - a_n$ , if  $a_n < b_n$ .
For which pairs  $(a_1, b_1)$  of positive real initial terms is there an index  $k$  with  $a_k = 0$ ?

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