
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  $S$  be the set of positive integers  $x$  for which there exist positive integers  $y$  and  $m$  such that  $y^2-2^m=x^2$ .

(a) Find all of the elements of  $S$ .
(b) Find all  $x$  such that both  $x$  and  $x+1$  are in  $S$ .

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