
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 the sequence $\{a_{n}\}$ is a geometric sequence with a common ratio $q\neq 1$, $a_{1}=3$, $3a_{1}$, $2a_{2}$, $a_{3}$ form an arithmetic sequence, and the terms of the sequence $\{a_{n}\}$ are arranged in a certain order as $a_{1}$, $a_{1}$, $a_{2}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $\ldots$, determine the value of the sum $S_{23}$ of the first 23 terms of the new sequence $\{b_{n}\}$.

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