
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
How many integers between $500$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $7$ between $500$ and $999?$ For example, $539$ and $935$ are counted if any permutation of its digits (such as $593$, $935$, etc.) is a multiple of 7.

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