
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
There is a set of 1000 switches, each of which has four positions, called  $A, B, C,$  and  $D.$   When the position of any switch changes, it is only from  $A$  to  $B,$  from  $B$  to  $C,$  from  $C$  to  $D,$  or from  $D$  to  $A.$   Initially each switch is in position  $A.$   The switches are labeled with the 1000 different integers  $2^x3^y5^z,$  where  $x, y,$  and  $z$  take on the values  $0, 1, \ldots, 9.$   At step  $i$  of a 1000-step process, the  $i$ th switch is advanced one step, and so are all the other switches whose labels divide the label on the  $i$ th switch.  After step 1000 has been completed, how many switches will be in position  $A$ ?

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