
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
 $n>1$  is an integer. The first  $n$  primes are  $p_1=2,p_2=3,\dotsc, p_n$ . Set  $A=p_1^{p_1}p_2^{p_2}...p_n^{p_n}$ .
Find all positive integers  $x$ , such that  $\dfrac Ax$  is even, and  $\dfrac Ax$  has exactly  $x$   divisors

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