
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
For an arithmetic sequence $\{a_n\}$ with a non-zero common difference, some terms $a_{k_{1}}, a_{k_{2}}, a_{k_{3}}, \ldots$ form a geometric sequence, and it is given that $k_{1} \neq 1$, $k_{2} \neq 2$, $k_{3} \neq 6$. Find the value of $k_{4}$.

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