
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 six positions, labeled $A, B, C, D, E, F$. When the position of any switch changes, it cycles through these positions in the given order. Initially, each switch is in position $A$. The switches are labeled with the 1000 different integers in the form $(2^{x})(3^{y})(5^{z})$, where $x, y$, and $z$ range from $0$ to $11$. At each step $i$ of a 1000-step process, the $i$-th switch is advanced one position, and so are all 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}
user