
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 $x_{1}=a$ and $x_{2}=1$, with the recurrence relation
$$
x_{n+2}=\frac{x_{n-1}^{2}-5}{x_{n}} \quad (n \in \mathbf{N}^{*}),
$$

determine the value of $a$ such that the sequence $\left\{x_{n}\right\}$ is a periodic sequence with a period of 4.

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