
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 Fibonacci sequence is defined by  $F_1 = F_2 = 1$  and  $F_{n+2} = F_{n+1}+F_n$  for every integer  $n$ . Let  $k$  be a fixed integer. A sequence  $(a_n)$  of integers is said to be  $\textit{phirme}$  if  $a_n + a_{n+1} = F_{n+k}$  for all  $n \geq 1$ . Find all  $\textit{phirme}$  sequences in terms of  $n$  and  $k$ .

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