
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 $q$, define the positive integer $m$ to be $q$-safe if $m$ differs in absolute value by more than $3$ from all multiples of $q$. For example, the set of $9$-safe numbers is $\{ 4, 5, 6, 7, \ldots\}$. Find the number of positive integers less than or equal to $20,000$ which are simultaneously $9$-safe, $10$-safe, and $14$-safe.

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