
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 the sequence \( \{a_{n}\} \) defined as follows:
\[
\begin{align*}
a_{1} = 6, \quad a_{2} = 20, \\
a_{n}(a_{n} - 8) = a_{n-1} a_{n+1} - 12,
\end{align*}
\]
where \(\{x\}=x-\lfloor x \rfloor\) and \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\). Find the value of
\[
\lim_{n \rightarrow +\infty} \left\{ \sqrt{a_{n}} \right\}.

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