
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
Suppose you have a set of 500 switches, each with four positions: $A, B, C$, and $D$. Changes to the position of any switch occur in a sequence from $A$ to $B$, $B$ to $C$, $C$ to $D$, and $D$ to $A$. Initially, each switch is in position $A$. The switches are labeled with integers of the form $(2^{x})(3^{y})(5^{z})(7^{w})$, where $x, y, z, w$ range from $0$ to $9$. For a process comprised of 500 steps, at each step $i$, the $i$-th switch advances one position, as do all other switches whose labels divide the label of the $i$-th switch. After the 500 steps are completed, how many switches are in position $A$?

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