
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  $ n>1$  and for  $ 1 \leq k \leq n$  let  $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$  be the sum of the products of all possible combinations of k of the numbers  $ a_1,a_2,...,a_n$ . Furthermore let  $ P \equal{} P(a_1, a_2, . . . , a_n)$  be the sum of all  $ p_k$  with odd values of  $ k$  less than or equal to  $ n$ .

How many different values are taken by  $ a_j$  if all the numbers  $ a_j (1 \leq j \leq n)$  and  $ P$  are prime?

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