
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 \(\left\{a_{n}\right\}\) is an increasing sequence of non-negative integers, with \(a_{1}<a_{2}<\cdots<a_{k}\), such that 
\[
\frac{2^{289}+1}{2^{17}+1}=2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{k}},
\]
calculate the value of \(k\).

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