
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 having four positions, labeled $A, B, C$, and $D$. The switches advance positions similarly from $A$ to $B$, $B$ to $C$, $C$ to $D$, and $D$ back to $A$. Initially, each switch is in position $A$. These switches are numbered with the integers $(2^x)(3^y)(7^z)$, where $x, y, z$ range from $0$ to $5$. In a 729-step process, each step $i$ advances the $i$-th switch and all others whose labels divide the $i$-th switch's label. Determine how many switches are in position $A$ after all steps.

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