
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 a positive integer \( a \), define a sequence of integers \( x_1, x_2, x_3, \ldots \) such that \( x_1 = a \) and \( x_{n+1} = 2x_n + 1 \). Let \( y_n = 2^{x_n} - 1 \). Determine the largest possible integer \( k \) such that for some positive integer \( a \), the numbers \( y_1, y_2, \ldots, y_k \) are all prime numbers.

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