
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, denoted as $A, B, C$, and $D$. The switches change position sequentially from $A$ to $B$, $B$ to $C$, $C$ to $D$, and $D$ to $A$. Initially, each switch is in position $A$. These switches are uniquely labeled with the numbers $(2^{x})(3^{y})(7^{z})$, where $x, y$, and $z$ take on the values $0, 1, \ldots, 9$. In a 1000-step process, at each step $i$, the $i$-th switch and all switches whose labels divide the label of the $i$-th switch are toggled one position forward. After completing 1000 steps, how many switches will be in position $A$?

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