
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 how many natural numbers \( n \) not exceeding 600 are the triples of numbers

\[ \left\lfloor \frac{n}{2} \right\rfloor, \left\lfloor \frac{n}{3} \right\rfloor, \left\lfloor \frac{n}{5} \right\rfloor \quad\text{and}\quad \left\lfloor \frac{n+1}{2} \right\rfloor, \left\lfloor \frac{n+1}{3} \right\rfloor, \left\lfloor \frac{n+1}{5} \right\rfloor \]

distinct? As always, \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \( x \).

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