
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  $n$ , two payers  $A$  and  $B$  play the following game: Given a pile of  $s$  stones, the players take turn alternatively with  $A$  going first. On each turn the player is allowed to take either one stone, or a prime number of stones, or a positive multiple of  $n$  stones. The winner is the one who takes the last stone. Assuming both  $A$  and  $B$  play perfectly, for how many values of  $s$  the player  $A$  cannot win?

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