
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]


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In a positive geometric sequence $\{a_{n}\}$, it is known that $a_{2}$, $3a_{1}$, $a_{3}$ form an arithmetic sequence. The sum of the first $n$ terms is $S_{n}$, and if $S_{n}=ka_{n}-1$, then calculate the value of $a_{2022}$.

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