
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
Define the sequence \(x_{i}\) by \(x_{1}=a\) and \(x_{i+1}=2x_{i}+1\). Define the sequence \(y_{i}=2^{x_{i}}-1\). Determine the largest integer \(k\) such that \(y_{1}, \ldots, y_{k}\) are all prime numbers.

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