
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 343 switches, each with six positions: $A, B, C, D, E, F$. Changes in position occur sequentially from $A$ to $B$, $B$ to $C$, $C$ to $D$, $D$ to $E$, $E$ to $F$, and from $F$ back to $A$. All switches start in position $A$. The switches are numbered with the different integers $(2^x)(3^y)(7^z)$, where $x, y$, and $z$ take values from $0, 1, \ldots, 6$. Each step of a 343-step process involves advancing the $i$-th switch one step, as well as all switches whose labels divide the label of the $i$-th switch. After completing all steps, how many switches are in position $A$?

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