
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
Denote by \( P(n) \) the greatest prime divisor of \( n \). Find all integers \( n \geq 2 \) for which

\[ P(n) + \lfloor\sqrt{n}\rfloor = P(n+1) + \lfloor\sqrt{n+1}\rfloor . \]

(Note: \( \lfloor x\rfloor \) denotes the greatest integer less than or equal to \( x \).)

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