
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
Consider a set of 500 switches, each having five positions labeled $A, B, C, D$, and $E$. The transition between positions for each switch is sequential from $A$ to $B$, $B$ to $C$, $C$ to $D$, $D$ to $E$, and from $E$ back to $A$. Initially, each switch is in position $A$. These switches are uniquely labeled with integers of the form $(2^{x})(3^{y})(7^{z})$, where $x, y$, and $z$ can have values from 0 to 6. In a sequence of 500 steps, for step $i$ (1 through 500), the $i$-th switch is moved forward by one position, and so are all other switches whose labels divide the label of the $i$-th switch. How many switches will be in position $A$ after step 500?

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