
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  $\{fn\}$  be the Fibonacci sequence  $\{1, 1, 2, 3, 5, \dots.\}. $ (a) Find all pairs  $(a, b)$  of real numbers such that for each  $n$ ,  $af_n +bf_{n+1}$  is a member of the sequence.

(b) Find all pairs  $(u, v)$  of positive real numbers such that for each  $n$ ,  $uf_n^2 +vf_{n+1}^2$  is a member of the sequence.

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