
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
For a sequence $\{a_n\}$ where each term is positive and satisfies $a_n^2 = 4S_n - 2a_n - 1$ ($n \in \mathbb{N}^*$), where $S_n$ is the sum of the first $n$ terms of $\{a_n\}$.

(1) Find the values of $a_1$ and $a_2$;  
(2) Find the general formula for the sequence $\{a_n\}$;  
(3) Determine whether there exist positive integers $m$ and $n$ such that the vector $a = (2a_{n+2}, m)$ is perpendicular to the vector $b = (-a_{n+5}, 3+a_n)$. Provide a reason.

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