
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 system of 729 switches, each switch can be in three positions: X, Y, and Z changing only from X to Y, Y to Z, or Z to X respectively. Initially, each switch is in position X. The switches are labeled with 729 different numbers of the form $2^x 7^y$ where $x, y$ range over $0, 1, 2, \ldots, 8$. At step $i$ of a 729-step process, the $i$-th switch advances one step, and all other switches whose numbers divide the number on the $i$-th switch also advance one step. After every 100 switches, additionally, five random switches revert back to X. How many switches will be in position X after the process is over?

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