
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
Let the sequence \(\{a_n\}\) be defined by:
\[
\begin{array}{l}
a_1 = 3, \\
a_2 = 7, \\
a_{n-1}(a_{n-1} - 2) = a_{n-2}a_n - 4 \quad (n = 3, 4, \ldots).
\end{array}
\]
Let \([x]\) denote the greatest integer less than or equal to the real number \(x\), and \(\{x\} = x - [x]\). Find the value of \(\lim_{n \to \infty} \{\sqrt{a_n}\}\).

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