
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 a set of real numbers with mean  $M$ . If the means of the sets  $S\cup \{15\}$  and  $S\cup \{15,1\}$  are  $M + 2$  and  $M + 1$ , respectively, then how many elements does  $S$  have?

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