
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
For a positive integer $p$, define the positive integer $n$ to be $p$-safe if $n$ differs in absolute value by more than $2$ from all multiples of $p$. The set of $5$-safe numbers, for instance, is $\{ 3, 4, 8, 9, 13, 14, 18, 19, \ldots\}$. Determine the number of positive integers less than or equal to $5000$ that are simultaneously $5$-safe, $7$-safe, and $9$-safe.

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