
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 500 switches, each of which has five positions, labeled $A, B, C, D, E$. The positions transition in a cyclic order: $A$ to $B$, $B$ to $C$, $C$ to $D$, $D$ to $E$ and then directly back to $A$. Initially, each switch is in position $A$. The switches are labeled with the 500 different integers $(2^{x})(3^{y})(7^{z})$, where $x, y$, and $z$ take on the values $0, 1, \ldots, 4$. In a 500-step process, at step i, the $i$-th switch is advanced by one step, and so are all other switches whose labels divide the label on the $i$-th switch. After the completion of step 500, how many switches will be in position $A$?

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