
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 an arithmetic sequence with the first term $a_1=1$ and common difference $d\neq 0$. If $a_{k_1}$, $a_{k_2}$, $a_{k_3}$, ..., $a_{k_n}$, ... form a geometric sequence, and $k_1=1$, $k_2=2$, $k_3=5$, then the general formula for the sequence $\{k_n\}$ is $k_n=$______.

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