
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 that $a$ is a rational number that is not equal to $1$, we call $\frac{1}{1-a}$ the reciprocal difference of $a$. For example, the reciprocal difference of $2$ is $\frac{1}{1-2}=-1$. It is known that $a_1=\frac{1}{2}$, $a_{2}$ is the reciprocal difference of $a_{1}$, $a_{3}$ is the reciprocal difference of $a_{2}$, and $a_{4}$ is the reciprocal difference of $a_{1}$.

$(1)$ Fill in the blanks: $a_{2}=$______, $a_{3}=$______, $a_{4}=$______;
$(2)$ Based on the results from $(1)$, find the value of $a_{2021}a_{2022}a_{2023}$.

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