
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 $\{a_n\}$ satisfying: $a_1=a_2=1$, when $n \geqslant 2$, $a_n^2$ is defined as follows:
$$
a_n^2=
\begin{cases} 
a_{n-1}a_{n+1}-1, & \text{if } n=2k, k\in \mathbb{Z},\\
a_{n-1}a_{n+1}+1, & \text{if } n=2k+1, k\in \mathbb{Z}.
\end{cases}
$$
Find the value of $\frac{a_5}{a_6}$.

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