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

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