
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 $\left\{a_{n}\right\}$ defined by
$$
a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad (n \in \mathbf{Z}_{+}),
$$
where $[x]$ denotes the greatest integer not exceeding the real number $x$, determine the minimum value of the constant $C$ such that for any positive integer $n$, the following inequality holds:
$$
\sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C.
$$

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