
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
In a set of numbers $a_{1}$, $a_{2}$, $a_{3}$, $\ldots a_{n}$, $a_{n+1}$, $\ldots $, where $a_{1}=x$, $a_{n+1}=1-\frac{1}{a_n}$ (where $n$ is a positive integer),<br/>$(1)$ Express $a_{2}$, $a_{3}$, $a_{4}$ in terms of $x$ using algebraic expressions and state the range of values for $x$. <br/>$(2)$ When $x=2023$, find the value of $a_{2024}$.

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