
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
(1) Does there exist positive integers $m, n$ such that $m(m+2) = n(n+1)$?  
(2) Let $k (k \geq 3)$ be a given positive integer, does there exist positive integers $m, n$ such that $m(m+k) = n(n+1)$?

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