
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 729 switches, each with four positions, labeled $A, B, C$, and $D$. The position of any switch can change only in the sequence $A \to B \to C \to D \to A$. Initially, each switch is in position $A$. The switches are labeled with the 729 different integers $(2^{x})(3^{y})(7^{z})$, where $x, y$, and $z$ take on the values $0, 1, \ldots, 5$. At step i of a 729-step process, the $i$-th switch is advanced one step, and so are all other switches whose labels divide the label on the $i$-th switch. After step 729 has been completed, how many switches will be in position $A$?

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