
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
(i) Find all integers \( m \geq 1 \) such that \( m(m+1) \) is a power of a prime number.

(ii) Find all integers \( m \geq 1 \) such that there exist two integers \( a \geq 1 \) and \( k \geq 2 \) such that \( m(m+1) = a^k \).

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