
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 a non-zero common difference $d$, and $a_{k\_1}$, $a_{k\_2}$, ..., $a_{k\_n}$ form a geometric sequence. If $k\_1=1$, $k\_2=5$, $k\_3=17$, find the sum $k\_1+k\_2+...+k\_n$.

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