
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
If every multiple of a positive integer $N$ denoted as $\overline{\mathrm{abc}}$ satisfies that $\overline{\mathrm{bca}}$ and $\overline{\mathrm{cab}}$ are also multiples of $N$ (where $a$, $b$, and $c$ are integers from 0 to 9, and by convention 123 represents 123, 028 represents 28, 007 represents 7), then $N$ is called a "perfect divisor" (for example, 9 is a "perfect divisor"). How many such "perfect divisors" are there?

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