
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 each term is positive, and the sum of the first $n$ terms is $S_n$, it satisfies that $a_1 = 1$ and $a_{n+1} = 2\sqrt{S_n} + 1$ for $n \in \mathbb{N}^*$.  
(1) Find the value of $a_2$.  
(2) Find the general term formula for the sequence $\{a_n\}$.  
(3) Determine whether there exists a positive integer $k$ such that $a_k$, $S_{2k-1}$, and $a_{4k}$ form a geometric sequence. If it exists, find the value of $k$; if not, provide an explanation.

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