
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
We have a set of 512 switches, each with four positions $A, B, C$, and $D$. They can only move sequentially from $A$ to $B$, $B$ to $C$, $C$ to $D$, and $D$ back to $A$. Initially, each switch is at position $A$. The switches are labeled with the integers $(2^x)(3^y)(7^z)$, where $x, y, and z$ each range from $0$ to $7$. During a 512-step process, during step $i$, the $i$-th switch and all switches whose labels are divisible by the $i$-th switch's label are advanced one position. After all steps, determine the number of switches in position $A$.

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