
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
In a modified scenario, there is a set of 729 switches, each with four positions labeled $A, B, C, D$. Changes occur strictly in the sequence $A \rightarrow B \rightarrow C \rightarrow D \rightarrow A$. Initially, each switch is in position $A$. These switches are labeled with the 729 different integers $(2^{x})(3^{y})(7^{z})$, where $x, y, z$ range from $0$ to $8$. In a 729-step process, the $i$-th switch advances one step, as do all other switches whose labels divide the label on the $i$-th switch. After completing all steps, how many switches are in position $A$?

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