
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 $\left\{ f_{n} \right\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \ldots\}$.
(a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $a f_{n} + b f_{n+1}$ is a member of the sequence.
(b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $u f_{n}^{2} + v f_{n+1}^{2}$ is a member of the sequence.

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