
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 $\{a_n\}$ that satisfies $a_1=1$ and $a_{n+1}=pS_n+r$ (where $n\in\mathbb{N}^*$, $p$, $r\in\mathbb{R}$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$):
1. When $p=2$ and $r=0$, find the values of $a_2$, $a_3$, and $a_4$.
2. Determine whether there exist real numbers $p$ and $r$ such that the sequence $\{a_n\}$ forms a geometric sequence. If such $p$ and $r$ exist, find the conditions they must satisfy; if not, explain why.

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