
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 \(\left\{a_{n}\right\}\) that satisfies
\[ a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad \text{for} \quad n \in \mathbf{Z}_{+}, \]
where \([x]\) denotes the greatest integer less than or equal to the real number \(x\). Let \(C\) be a real number such that for any positive integer \(n\),
\[ \sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C. \]
Find the minimum value of \(C\).

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