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

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