
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 the arithmetic sequence $\{a_n\}$, $a_n \neq 0$, for $n \geq 2$, $a_{n+1} - a_n^2 + a_{n-1} = 0$. If $S_{2k-1} = 46$, then the value of $k$ is.

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