
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 are 1250 switches each with five positions, named $A, B, C, D, E$, with transitions as follows: $A \to B$, $B \to C$, $C \to D$, $D \to E$, $E \to A$. Initially, each switch is in position $A$. The switches are labeled with a unique integer of the form $(2^{x})(3^{y})$, where $x, y$ can be any integer from 0 to 9. In a sequence of 1250 steps, during step $i$ the switch and its divisors will advance one position. Determine the number of switches in position $A$ after all steps are completed.

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