
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
A set  $S \subseteq \mathbb{N}$  satisfies the following conditions:
	
(a) If  $x, y \in S$  (not necessarily distinct), then  $x + y \in S$ .
(b) If  $x$  is an integer and  $2x \in S$ , then  $x \in S$ .

Find the number of pairs of integers  $(a, b)$  with  $1 \le a, b\le 50$  such that if  $a, b \in S$  then  $S = \mathbb{N}.$ *Proposed by Yang Liu*

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