
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 $a_{k_1}$, $a_{k_2}$, $a_{k_3}$, ... form a geometric sequence $\{a_{k_n}\}$, 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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