
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 that all terms of the geometric sequence $\{a_n\}$ are positive, with the first term $a_1 = 1$, and $a_4$, $3a_3$, $a_5$ form an arithmetic sequence.

(1) Find the general formula for the sequence $\{a_n\}$.
(2) The sum of the first $n$ terms of the sequence $\{a_{n+1} - \lambda a_n\}$ is $S_n$. If $S_n = 2^n - 1$ ($n \in \mathbb{N}^*$), find the value of the real number $\lambda$.

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