
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
Given a sequence $\{a_n\}$ where all terms are positive, and for any $n \in \mathbb{N}^*$, $a_{n+1}^2 = a_n a_{n+2} + k$ ($k$ is a constant),

(1) If $k=0$ and $8a_2, 3a_3, a_4$ form an arithmetic sequence, find the value of $\frac{a_2}{a_1}$.

(2) Given $a_1 = a, a_2 = b$ ($a, b$ are constants), is there a constant $\lambda$ such that $a_n + a_{n+2} = \lambda a_{n+1}$ holds for any $n \in \mathbb{N}^*$? If it exists, find the value of $\lambda$; if not, explain why.

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