
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 any nonnegative integer  $ n$ , let  $ f(n)$  be the greatest integer such that  $ 2^{f(n)} | n \plus{} 1$ . A pair  $ (n, p)$  of nonnegative integers is called nice if  $ 2^{f(n)} > p$ . Find all triples  $ (n, p, q)$  of nonnegative integers such that the pairs  $ (n, p)$ ,  $ (p, q)$  and  $ (n \plus{} p \plus{} q, n)$  are all nice.

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