
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 a sequence $\left\{a_{n}\right\}$ which satisfies:
$$
\begin{array}{l}
a_{1}=a_{2}=1, \\
a_{n+1}=\frac{n^{2} a_{n}^{2}+5}{\left(n^{2}-1\right) a_{n-1}} \text{ for } n > 1,
\end{array}
$$
is there a real number $x, y$ (with $x \neq 0$) such that the sequence $\left\{\frac{b_{n+2}+b_{n}}{b_{n+1}}\right\}$ (where $b_{n}=(n x+y) a_{n}, n \in \mathbf{Z}_{+}$) is a constant sequence?

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