
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
Let an infinite geometric sequence \(\left\{a_{n}\right\}\) have a common ratio \(q\) such that \(|q|<1\) and the first term \(a_{1}=1\). If \(a_{n} = k\left(a_{n+1} + a_{n+2} + \cdots\right)\), find the range of the real number \(k\).

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